Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information: Algorithms and Applications (Studies in Fuzziness and Soft Computing, 430) 3031436350, 9783031436352

Table of contents :
Foreword
Preface
Contents
About the Authors
List of Figures
List of Tables
1 Hybrid Multi-polar Fuzzy Models
1.1 Introduction
1.2 m-Polar Fuzzy Sets
1.3 Rough m-Polar Fuzzy Sets
1.3.1 Selection of Flats ch1DAk2
1.3.2 Selection of Employees for Promotion and Bonus
1.4 m–Polar Fuzzy Soft Sets
1.4.1 Selection of an Employee in an Organization
1.4.2 Selection of Suitable Site for a Resort
1.5 Similarity Measure for m-Polar Fuzzy Sets
1.5.1 Pattern Recognition Problem
1.5.2 Medical Diagnosis of Anemia
1.5.3 Medical Diagnosis of Dengue Fever
1.6 m-Polar Fuzzy Rough Sets
1.6.1 Selection of Prints and Shades for Variety of Fabrics
1.6.2 Selection of Features for Different Models of Mobiles
1.7 m-Polar Fuzzy Soft Rough Sets
1.7.1 Selection of a Hotel
1.7.2 Selection of a Place
1.7.3 Selection of a House
1.8 Soft m-Polar Fuzzy Rough Sets
1.8.1 Comparison of Popular Mobile Phones for Selection
1.8.2 Selection of a Site for Construction of a Grid Station
1.8.3 Comparison of Patients for Recovery of Heart Disease
1.9 Conclusion
References
2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets
2.1 Introduction
2.2 Multi-criteria Decision Making Methods
2.3 m–Polar Fuzzy ELECTRE-I Method
2.3.1 Selection of a Suitable Location for a Diesel Power Plant
2.3.2 Selection of a Site for the Airport
2.3.3 Performance Evaluation of Physical Sciences Instructor
2.4 Contribution, Sensitivity and Comparison Analysis
2.5 The Concept of m-Polar Fuzzy Linguistic Variable
2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM
2.6.1 Salary Analysis of Companies
2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM
2.7.1 Selection of Most Corrupted Country
2.8 Discussion of the Proposed Approach
2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM
2.9.1 Models Ranking According to their Appearance
2.9.2 Ranking of High Speed Racing Cars
2.10 Comparison Analysis of Proposed Approach
2.11 Conclusion
References
3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models
3.1 Introduction
3.2 m-Polar Hesitant Fuzzy Set
3.2.1 Basic Operations of m–Polar Hesitant Fuzzy Set
3.2.2 Comparison Laws of m–Polar Hesitant Fuzzy Elements
3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach
3.3.1 Selection of a Perfect Brand Name
3.3.2 Selection of Suitable Product Design for a Company
3.4 An m-Polar Hesitant Fuzzy ELECTRE-I Approach
3.4.1 Selection of Bricks for Construction
3.5 Hesitant m–Polar Fuzzy Set
3.5.1 Basic Operations for Hesitant m–Polar Fuzzy Set
3.5.2 Comparison Laws of Hesitant m–Polar Fuzzy Set
3.6 Hesitant m-Polar Fuzzy TOPSIS Approach
3.6.1 Comparison of Top Five Populous Countries
3.6.2 Comparison of Different Types of Textiles or Clothing
3.7 Hesitant m-Polar Fuzzy ELECTRE-I Approach
3.7.1 Site Selection for Farming Purposes
3.8 Conclusion
References
4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets
4.1 Introduction
4.2 An m–Polar Fuzzy ELECTRE I Method
4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall
4.4 The m–Polar Fuzzy ELECTRE II Method
4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant
4.5.1 Available Alternatives
4.5.2 Selection of m–Polar Criteria
4.5.3 Stepwise Procedure
4.6 Comparison Analysis
4.7 Conclusion
References
5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets
5.1 Introduction
5.2 An m–Polar Fuzzy ELECTRE III Method
5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm
5.4 Comparative Study
5.4.1 Comparison with m–Polar Fuzzy ELECTRE I Method
5.4.2 Comparison with m–Polar Fuzzy ELECTRE II Method
5.4.3 Discussion
5.5 Insights of m–Polar Fuzzy ELECTRE III Method
5.6 Conclusion
References
6 Extended ELECTRE IV Method with Multi-polar Fuzzy Sets
6.1 Introduction
6.2 An m–Polar Fuzzy ELECTRE IV Method
6.3 Case Study: Islamic Azad University Qazvin Branch Innovation Park Project, Iran
6.4 Comparison Analysis
6.4.1 Discussion
6.5 Conclusion
References
7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets
7.1 Introduction
7.2 Basic Concept
7.3 Analytical Hierarchy Process
7.4 m–Polar Fuzzy PROMETHEE Method
7.4.1 Ranking the Sites of Hydroelectric Power Stations
7.4.2 Criteria Weights by AHP
7.4.3 Ranking Through m–Polar Fuzzy PROMETHEE
7.5 Comparative Analysis
7.5.1 With Usual Criterion Preference Function
7.5.2 m–Polar Fuzzy ELECTRE I
7.6 Conclusion
References
8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets
8.1 Introduction
8.2 m–Polar Fuzzy Dombi Aggregation Operators
8.2.1 m–Polar Fuzzy Dombi Arithmetic Aggregation Operators
8.2.2 m–Polar Fuzzy Dombi Geometric Aggregation Operators
8.3 m–Polar Fuzzy Hamacher Aggregation Operators
8.3.1 m–Polar Fuzzy Hamacher Arithmetic Aggregation Operators
8.3.2 m–Polar Fuzzy Hamacher Geometric Aggregation Operators
8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators
8.4.1 Agriculture Land Selection
8.4.2 Performance Evaluation of Commercial Banks
8.4.3 Assessment of Health Care Waste Treatments Alternatives
8.4.4 Selection of a Best Company for Investment
8.4.5 Selection of Most Affected Country by Human Trafficking
8.5 Comparison Analysis and Discussion
8.5.1 Effectiveness Test
8.6 Conclusion
References
9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators
9.1 Introduction
9.2 Preliminaries
9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators
9.4 2-Tuple Linguistic m–Polar Fuzzy Hamacher Geometric Aggregation Operators
9.5 Mathematical Approach for MADM Using 2-Tuple Linguistic m–Polar Fuzzy Information
9.6 Best Location for the Thermal Power Station: Case Study ch9no1
9.6.1 Influence of the Parameter λ on Decision Making Results
9.7 Comparative Analysis
9.7.1 Comparison with Existing Techniques
9.7.2 Discussion
9.8 Conclusions
References
10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets
10.1 Introduction
10.2 m–Polar Fuzzy Soft Expert Sets
10.3 Mathematical Approach for MCGDM with m–Polar Fuzzy Information
10.4 Applications
10.4.1 Selection of a Suitable Site for a Dam
10.4.2 Country Most Affected by Human Trafficking
10.5 Comparative Analysis
10.6 m–Polar Fuzzy N–Soft Sets
10.7 m–Polar Fuzzy N–Soft Rough Sets
10.8 Applications
10.8.1 Selection of a Restaurant
10.8.2 Selection of a Hotel
10.8.3 Selection of a Resort
10.8.4 Selection of a Laptop
10.9 Discussion
10.10 Conclusion
References
Index

Citation preview

Studies in Fuzziness and Soft Computing

Muhammad Akram Arooj Adeel

Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information Algorithms and Applications

Studies in Fuzziness and Soft Computing Volume 430

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Muhammad Akram · Arooj Adeel

Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information Algorithms and Applications

Muhammad Akram Department of Mathematics University of the Punjab, Quaid-e-Azam Campus (New Campus) Lahore, Pakistan

Arooj Adeel Department of Mathematics, Division of Science and Technology University of Education (Bank Road Campus) Lahore, Pakistan

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-031-43635-2 ISBN 978-3-031-43636-9 (eBook) https://doi.org/10.1007/978-3-031-43636-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

We dedicate this book to the memory of Professor Lotfi Zadeh!

Foreword

Multi-polar fuzzy sets (m-polar fuzzy sets) are an extension of fuzzy set theory that allow multi–polar information to have practical applications in both theoretical and real-world problems. Multi-criteria decision making with m-polar fuzzy sets has been an attractive research area in recent years because of its outstanding capacity as a tool for representing fuzziness and uncertainty under multi-polar information in MCDM problems. In the past few years, the average number of papers on the application of m–polar fuzzy sets published in the leading journals has become about 25. Professor Dr. Muhammad Akram from University of the Punjab is the leading researcher in this area in the whole world. The subject areas of the published m–polar fuzzy sets are mathematics, computer science, engineering, and chemistry. This book is an excellent source for researchers studying hybrid m–polar fuzzy models such as m–polar fuzzy soft sets and m–polar fuzzy soft rough sets. TOPSIS, ELECTRE I, ELECTRE II, ELECTRE III, ELECTRE IV, and PROMETHEE methods with m–polar fuzzy linguistic sets including the hesitancy concept together with several illustrative examples make this book a unique book in the literature so far. MCDM applications using 2–tuple linguistic m–polar fuzzy Hamacher aggregation operators with comparative analysis are another superiority of this book among the rare competitive sources in the literature. Hybrid models based on m–polar fuzzy soft sets based on m–polar fuzzy soft expert sets present several useful and didactic multiple criteria decision making examples such as selection of a restaurant, selection of a hotel, selection of a resort, and selection of a laptop. The book includes 72 figures and 348 tables that are indicators of how meticulously and devotedly the book was written. Researchers reading this wonderful book will find doors open to new research opportunities such as Pythagorean m–polar fuzzy sets, picture m–polar fuzzy sets, and spherical m–polar fuzzy sets. Each of these extensions can be integrated with different MCDM methods such as VIKOR, EDAS, CODAS, COPRAS, WASPAS, and MOORA. I congratulate the authors of the book, Prof. Dr. Muhammad Akram

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and Dr. Arooj Adeel, for successfully completing this project, which requires patience and perseverance. Prof. Cengiz Kahraman Department of Industrial Engineering Istanbul Technical University ˙Istanbul, Türkiye

Preface

This monograph is at the crossroads of two disciplines, namely an extension of fuzzy set theory allowing for multi-polar information, and a branch of operational research called Multi-criteria Decision Making (MCDM). A typical MCDM problem deals with the evaluation of a set of alternatives according to a set of decision criteria. Fuzzy set theory owes its origins to the groundbreaking work of Lotfi A. Zadeh in 1965. It was conceived as a tool for representing uncertainty and vagueness in real-world systems. Extensions of fuzzy set theory quickly proliferated. They are still one of the most important approaches with a non-probabilistic nature for the representation of uncertain, incomplete, imprecise, or vague information. Specifically, real-world models often contain multi-attribute, multi-index, multi-object, and multi-information data. With this practical motivation, Chen et al. introduced multi-polar fuzzy sets (m–polar fuzzy sets) in 2014. Another inspiration for the idea behind this extension of fuzzy sets is that “multi-polar information” may come from various sources. In particular, fuzziness has proved exceptionally adept in optimization issues. This book focuses on its impact on the theoretical and practical development of MCDM. The goal of MCDM is to produce the best results when several conflicting targets and criteria are explicitly considered. But many different principles intervene in the debate about which procedures have advantage over other approaches. As a result, MCDM attempts to devise systematic strategies to “optimize” feasible options and explain why certain alternatives can be declared “best.” This is a difficult and controversial task when goals and choices are objective and precisely stated. However, this is not always the case. All too often, we must make decisions in an uncertain environment and such inconvenience gives rise to a much more elaborated scenario. In this monograph, we consider some models that can be flexibly adapted to the lack of certainty, and then we explore some valuable strategies for making decisions under a variety of criteria. This book is based on some of the authors’ papers, which have been published in various scientific journals. It may be useful to mathematical researchers, computer scientists, and social scientists, among others. Let us briefly describe its contents. In Chap. 1, hybrid models are presented that combine m–polar fuzzy sets with rough and soft sets. These combinations produce rough m–polar ix

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fuzzy sets and m–polar fuzzy soft sets. Similarity measures of m–polar fuzzy sets are considered, and a variety of applications are presented. Furthermore, we set forth the notions of m–polar fuzzy rough sets and m–polar fuzzy soft rough sets. Related concepts and some of their basic properties and fundamental operations are explored. Moreover, we discuss the potential applications of these models in MCDM. In Chap. 2, a purely m–polar fuzzy context is considered. We elaborate on the corresponding m-polar fuzzy ELECTRE I solution. Then the linguistic features for the formulation of an m–polar fuzzy linguistic ELECTRE I methodology are incorporated. Relatedly, an alternative m-polar fuzzy linguistic TOPSIS method is considered where the evaluations of the alternatives are expressed in terms of suitable linguistic values. Furthermore, the efficiency of these techniques is validated by respective applications to real-life examples. Finally, we present algorithms of these approaches, and we also give their computer programming codes. Chapter 3 concerns the hybrid models known as m–polar hesitant fuzzy sets and hesitant m–polar fuzzy sets. We establish some of their fundamental properties and formulate their basic operations. Moreover, this chapter develops m–polar hesitant fuzzy TOPSIS and m–polar hesitant fuzzy ELECTRE I approaches for MCGDM. Furthermore, it develops hesitant m–polar fuzzy TOPSIS and hesitant m–polar fuzzy ELECTRE I approaches for MCGDM. The efficiency of these decision making procedures is discussed by their application to real situations in industrial fields. Finally, algorithms and computer programming codes of these methodologies are presented. Chapter 4 unfolds some decision making approaches based on the ELECTRE spirit for group decision making. A major contribution of this chapter is the redesigning of the m–polar fuzzy ELECTRE I method for collective scenarios. Another goal of this chapter is to employ the ELECTRE II method to deliver a procedure of m-polar fuzzy ELECTRE II that efficiently deals with the multi-polar information in group decision making environments. We also provide a comprehensive comparison of both approaches to highlight the insights and limitations of the techniques presented in this chapter. Chapter 5 is designed to present the significant procedure of the m–polar fuzzy ELECTRE III method by exploiting the tremendous theory of the ELECTRE approach. Further, the dominant features and captivating structure of the m–polar fuzzy sets empower the presented approach to address the multi-polar ambiguous information of real-world problems competently. To exhibit the applicability of m– polar fuzzy ELECTRE III method, a case study for the selection of the most competent hazardous waste carrier company is presented. Moreover, a comparison analysis with existing outranking approaches is conducted to highlight the salient features and superiority of the presented methodology. Chapter 6 is devoted to studying an extended version of the ELECTRE IV method. The presented procedure takes the advantage of outranking principles of the ELECTRE family to specify the optimal alternative but the noticeable edge of the m–polar fuzzy ELECTRE IV method is its unique procedure that does not need the criteria weight for operation. The presented technique competently captures the multi-polarity and fuzziness of the real-world problems, simultaneously. Further, the

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presented variant is competent enough to filter out five different classes of dominance to rank the alternatives. The ranking is achieved by two different distillation procedures to examine the ranking order from both perspectives including ascending and descending distillations. The application of the presented technique is manifested by a real-life case study for the selection of a qualified and efficient contractor for the Islamic Azad University-Qazvin Branch Innovation Park project. Chapter 7 is devoted to studying an extended version of the PROMETHEE strategy that benefits from the AHP (for Analytic Hierarchy Process) technique in an m–polar fuzzy setting. It produces the AHP-based m–polar fuzzy PROMETHEE method that consists of two parts. First, thanks to the AHP technique, we quantify the normalized weights of the attributes by pairwise comparisons of the criteria. In the second part, the m–polar fuzzy PROMETHEE approach is used to rank the alternatives on the basis of conflicting criteria. A comparative study considers the usual criterion preference function for all the criteria in order to check the influence of different types of preference functions on the output. In Chap. 8, we focus on aggregation to handle uncertainty in m–polar fuzzy information. We present arithmetic and geometric aggregation operators using Dombi and Hamacher t-norms and t-conorms. Then we study some of their properties and elaborate on corresponding algorithms to solve MCDM issues. To prove the validity and feasibility of these models, we produce numerical solutions for some examples. Then we perform a comparison with the m–polar fuzzy ELECTRE I approach. The effectiveness of the m–polar fuzzy Dombi aggregation operators is checked by validity tests. Chapter 9 is devoted to present the concept of a two-tuple linguistic m–polar fuzzy set. It introduces this model and establishes some of its fundamental operations. The following aggregation operators are presented: two-tuple linguistic m– polar fuzzy Hamacher weighted average operator, two-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator, two-tuple linguistic m–polar fuzzy Hamacher hybrid average operator, two-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator, two-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator, and two-tuple linguistic m–polar fuzzy Hamacher hybrid geometric operator. An algorithm is formulated to solve multi-criteria decision making problems. Moreover, a comparative study with existing methods is performed in order to show the applicability of the presented model. In Chap. 10, MCGDM models are presented which cover criteria evaluation by different experts. Hybrid models for soft computing, namely m–polar fuzzy soft expert sets, m-polar fuzzy N-soft sets, and m–polar fuzzy N–soft rough sets are discussed. Their characteristics are explored with the aid of numerical examples. Further, their basic properties and operations are investigated. The problems are solved with the help of these hybrid models. The algorithms provide evidence of their efficiency and cogency. Lahore, Pakistan

Muhammad Akram Arooj Adeel

Contents

1

Hybrid Multi-polar Fuzzy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 m–Polar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Rough m–Polar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Selection of Flats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Selection of Employees for Promotion and Bonus . . . . 1.4 m–Polar Fuzzy Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Selection of an Employee in an Organization . . . . . . . . 1.4.2 Selection of Suitable Site for a Resort . . . . . . . . . . . . . . . 1.5 Similarity Measure for m–Polar Fuzzy Sets . . . . . . . . . . . . . . . . . 1.5.1 Pattern Recognition Problem . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Medical Diagnosis of Anemia . . . . . . . . . . . . . . . . . . . . . 1.5.3 Medical Diagnosis of Dengue Fever . . . . . . . . . . . . . . . . 1.6 m–Polar Fuzzy Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Selection of Prints and Shades for Variety of Fabrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Selection of Features for Different Models of Mobiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 m–Polar Fuzzy Soft Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Selection of a Hotel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Selection of a Place . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Selection of a House . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Soft m–Polar Fuzzy Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Comparison of Popular Mobile Phones for Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Selection of a Site for Construction of a Grid Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Comparison of Patients for Recovery of Heart Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multi-criteria Decision Making Methods . . . . . . . . . . . . . . . . . . . 2.3 m–Polar Fuzzy ELECTRE-I Method . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Selection of a Suitable Location for a Diesel Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Selection of a Site for the Airport . . . . . . . . . . . . . . . . . . 2.3.3 Performance Evaluation of Physical Sciences Instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Contribution, Sensitivity and Comparison Analysis . . . . . . . . . . . 2.5 The Concept of m–Polar Fuzzy Linguistic Variable . . . . . . . . . . 2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Salary Analysis of Companies . . . . . . . . . . . . . . . . . . . . . 2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Selection of Most Corrupted Country . . . . . . . . . . . . . . . 2.8 Discussion of the Proposed Approach . . . . . . . . . . . . . . . . . . . . . . 2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM . . . . . 2.9.1 Models Ranking According to their Appearance . . . . . . 2.9.2 Ranking of High Speed Racing Cars . . . . . . . . . . . . . . . . 2.10 Comparison Analysis of Proposed Approach . . . . . . . . . . . . . . . . 2.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introducing Hesitancy: TOPSIS and ELECTRE-I Models . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 m–Polar Hesitant Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Basic Operations of m–Polar Hesitant Fuzzy Set . . . . . 3.2.2 Comparison Laws of m–Polar Hesitant Fuzzy Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach . . . . . . . . . . . . . . 3.3.1 Selection of a Perfect Brand Name . . . . . . . . . . . . . . . . . 3.3.2 Selection of Suitable Product Design for a Company . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 An m–Polar Hesitant Fuzzy ELECTRE-I Approach . . . . . . . . . . 3.4.1 Selection of Bricks for Construction . . . . . . . . . . . . . . . . 3.5 Hesitant m–Polar Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Basic Operations for Hesitant m–Polar Fuzzy Set . . . . . 3.5.2 Comparison Laws of Hesitant m–Polar Fuzzy Set . . . . 3.6 Hesitant m–Polar Fuzzy TOPSIS Approach . . . . . . . . . . . . . . . . . 3.6.1 Comparison of Top Five Populous Countries . . . . . . . . . 3.6.2 Comparison of Different Types of Textiles or Clothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 86 89 92 96 100 106 106 110 114 118 121 129 137 141 145 148 151 152 157 158 160 162 168 170 173 178 183 185 192 194 199 202 205 209

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Hesitant m–Polar Fuzzy ELECTRE-I Approach . . . . . . . . . . . . . 3.7.1 Site Selection for Farming Purposes . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 An m–Polar Fuzzy ELECTRE I Method . . . . . . . . . . . . . . . . . . . . 4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The m–Polar Fuzzy ELECTRE II Method . . . . . . . . . . . . . . . . . . 4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Available Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Selection of m–Polar Criteria . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Stepwise Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 An m–Polar Fuzzy ELECTRE III Method . . . . . . . . . . . . . . . . . . 5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Comparison with m–Polar Fuzzy ELECTRE I Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Comparison with m–Polar Fuzzy ELECTRE II Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Insights of m–Polar Fuzzy ELECTRE III Method . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended ELECTRE IV Method with Multi-polar Fuzzy Sets . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 An m–Polar Fuzzy ELECTRE IV Method . . . . . . . . . . . . . . . . . . 6.3 Case Study: Islamic Azad University Qazvin Branch Innovation Park Project, Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 238 241 248 259 267 271 271 273 278 279 279 283 283 286 292 302 302 307 308 310 311 311 315 315 317 324 330 335 337 339

xvi

7

8

9

Contents

Extended PROMETHEE Method Under Multi-polar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Analytical Hierarchy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 m–Polar Fuzzy PROMETHEE Method . . . . . . . . . . . . . . . . . . . . . 7.4.1 Ranking the Sites of Hydroelectric Power Stations . . . . 7.4.2 Criteria Weights by AHP . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Ranking Through m–Polar Fuzzy PROMETHEE . . . . . 7.5 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 With Usual Criterion Preference Function . . . . . . . . . . . 7.5.2 m–Polar Fuzzy ELECTRE I . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 m–Polar Fuzzy Dombi Aggregation Operators . . . . . . . . . . . . . . . 8.2.1 m–Polar Fuzzy Dombi Arithmetic Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 m–Polar Fuzzy Dombi Geometric Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 m–Polar Fuzzy Hamacher Aggregation Operators . . . . . . . . . . . . 8.3.1 m–Polar Fuzzy Hamacher Arithmetic Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 m–Polar Fuzzy Hamacher Geometric Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Agriculture Land Selection . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Performance Evaluation of Commercial Banks . . . . . . . 8.4.3 Assessment of Health Care Waste Treatments Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Selection of a Best Company for Investment . . . . . . . . . 8.4.5 Selection of Most Affected Country by Human Trafficking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Comparison Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Effectiveness Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343 344 346 348 350 356 357 359 365 365 369 371 371 375 375 377 378 385 391 392 402 410 412 413 418 422 426 429 433 433 434

2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Contents

2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 2-Tuple Linguistic m–Polar Fuzzy Hamacher Geometric Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Mathematical Approach for MADM Using 2-Tuple Linguistic m–Polar Fuzzy Information . . . . . . . . . . . . . . . . . . . . . 9.6 Best Location for the Thermal Power Station: Case Study . . . . . 9.6.1 Influence of the Parameter λ on Decision Making Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Comparison with Existing Techniques . . . . . . . . . . . . . . 9.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

9.3

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 m–Polar Fuzzy Soft Expert Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Mathematical Approach for MCGDM with m–Polar Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Selection of a Suitable Site for a Dam . . . . . . . . . . . . . . 10.4.2 Country Most Affected by Human Trafficking . . . . . . . 10.5 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 m–Polar Fuzzy N –Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 m–Polar Fuzzy N –Soft Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 Selection of a Restaurant . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.2 Selection of a Hotel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.3 Selection of a Resort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.4 Selection of a Laptop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

440 454 460 462 467 469 469 471 472 473 475 475 478 490 491 491 494 498 502 512 518 518 523 527 529 531 532 532

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

About the Authors

Muhammad Akram has received M.Sc. degrees in Mathematics and Computer Science, M.Phil. in (Computational) Mathematics and Ph.D. in (Fuzzy) Mathematics. Dr. Muhammad Akram presently holds the esteemed position of Chairman/Professor within the Department of Mathematics at the University of the Punjab, Lahore. His scholarly contributions extend to his service at the Punjab University College of Information Technology. Dr. Akram’s academic pursuits center around fuzzy numerical methods, fuzzy graphs, fuzzy algebras, and fuzzy decision support systems. With an illustrious record, he has authored 11 influential books in the realm of Fuzzy mathematics, published by reputable sources, alongside an impressive portfolio of over 500 research articles in esteemed international scientific journals. Evidencing his impact, Dr. Akram boasts an H-index of 60 on Google Scholar. Recognized for his exceptional expertise, Stanford University reports consistently position Dr. Akram is within the top 2% of scientists globally across the years 2020, 2021, 2022, and 2023 in the fields of Artificial Intelligence and Image Processing. His involvement in academia is multifaceted, having served as an Editorial Member for 23 distinguished international academic journals. A testament to his mentorship, Dr. Akram has successfully guided twenty students through their Ph.D. research endeavors, and presently oversees the scholarly pursuits of five Ph.D. candidates under his tutelage. His unwavering commitment to academia and prolific contributions stand as a testament to his invaluable presence within the academic community. Arooj Adeel is an Assistant Professor at the Department of Mathematics, University of Education, Lahore, Pakistan. She completed her Ph.D. degree in Mathematics from the University of the Punjab, Lahore, Pakistan. She has published over 22 research articles in international peer-reviewed journals. Her research interests include fuzzy sets and graphs, hybrid models, and decision making techniques. She has been an Editorial Member of international academic journals, Journal of New Theory and Mathematical Problems in Engineering, and reviewer/referee for 15 international journals.

xix

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16 Fig. 2.17 Fig. 2.18 Fig. 2.19 Fig. 2.20 Fig. 2.21 Fig. 2.22 Fig. 2.23 Fig. 3.1 Fig. 3.2

Order relation when m = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Order relation when m = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hierarchical structure of MCDM methods . . . . . . . . . . . . . . . . . . Hierarchical system for MADM . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of climatic and atmospheric conditions . . . . . . . . . . . . Attributes of social infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of government policies . . . . . . . . . . . . . . . . . . . . . . . . . Directed graph of outranking relation of locations . . . . . . . . . . . . Attributes of operational considerations . . . . . . . . . . . . . . . . . . . . Attributes of socioeconomic impacts . . . . . . . . . . . . . . . . . . . . . . . Attributes of ecological impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of expenditures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of risk and deliverability . . . . . . . . . . . . . . . . . . . . . . . . Directed graph of outranking relation of sites . . . . . . . . . . . . . . . . Attributes of teaching style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of social practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes of knowledge and expertise . . . . . . . . . . . . . . . . . . . . . . Attributes of facilitation skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directed graph of outranking relation of instructors . . . . . . . . . . . Flow chart of m–polar fuzzy ELECTRE-I Approach . . . . . . . . . . Graphical representation of outranking relation of companies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of outranking relation of countries . . . . m–polar fuzzy linguistic ELECTRE-I approach for MCDM and MCGDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow chart of m–polar fuzzy linguistic TOPSIS for MCGDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation of bricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation of sites for farming . . . . . . . . . . . . . . . . . . . .

5 5 76 87 88 93 93 93 93 96 97 97 97 97 97 99 100 101 101 101 104 105 117 126 130 149 191 225 xxi

xxii

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 4.13 Fig. 4.14 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 9.1 Fig. 9.2 Fig. 9.3 Fig. 9.4 Fig. 9.5 Fig. 9.6 Fig. 10.1

List of Figures

Graphical representation of relations in outranking decision graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of m–polar fuzzy ELECTRE I method . . . . . . . . . . . . Characteristics of original insulating wall . . . . . . . . . . . . . . . . . . . Structure of insulating wall for scheme x1 . . . . . . . . . . . . . . . . . . Structure of insulating wall for scheme x2 . . . . . . . . . . . . . . . . . . Structure of insulating wall for scheme x3 . . . . . . . . . . . . . . . . . . Structure of insulating wall for scheme x4 . . . . . . . . . . . . . . . . . . Multi-polar criteria and their poles . . . . . . . . . . . . . . . . . . . . . . . . Outranking graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iterative procedure for the forward ranking . . . . . . . . . . . . . . . . . . Iterative procedure for the reverse ranking . . . . . . . . . . . . . . . . . . Flow chart diagram of m–polar fuzzy ELECTRE II method . . . . Location of Fujian province . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decision criteria and their poles . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of m–polar fuzzy ELECTRE III method . . . . . . . . . . . Decision criteria and their poles . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong (a) and weak (b) outranking graphs . . . . . . . . . . . . . . . . . . Reverse strong (a) and weak (b) outranking graphs . . . . . . . . . . . Flowchart of m–polar fuzzy ELECTRE IV method . . . . . . . . . . . m–polar criteria with their poles . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preference function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outgoing flow of Rφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incoming flow of Rφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of m–polar fuzzy PROMETHEE method . . . . . . . . . . Partial relations of PROMETHEE I . . . . . . . . . . . . . . . . . . . . . . . . Partial relations of PROMETHEE I . . . . . . . . . . . . . . . . . . . . . . . . Graph representing the outranking relation of alternatives . . . . . . Flowchart of selecting the best option . . . . . . . . . . . . . . . . . . . . . . Outranking relation of banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation of treatments alternatives . . . . . . . . . . . . . . . Outranking relation of companies . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of selecting the most worst country affecting by human trafficking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of first application in Sect. 8.4 . . . . . . . . . . . . . . . . . Comparison of second application in Sect. 8.4 . . . . . . . . . . . . . . . Flowchart for decision making . . . . . . . . . . . . . . . . . . . . . . . . . . . . Criteria representation in selected 3–polar environment . . . . . . . Score values based on average operator . . . . . . . . . . . . . . . . . . . . Score values based on geometric operator . . . . . . . . . . . . . . . . . . . Comparison chart 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison chart 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of the strategy of solution in Sect. 10.3 . . . . . . . . . . . .

248 249 250 251 252 252 253 254 259 267 268 269 270 273 293 296 306 309 309 325 333 336 352 354 354 356 361 366 371 416 418 422 427 429 431 432 463 464 467 468 471 471 491

List of Figures

Fig. 10.2 Fig. 10.3

Comparison under different values of m in the application described in Sect. 10.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison under different values of m in the application described in Sect. 10.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii

501 501

List of Tables

Table 1.1 Table 1.2 Table 1.3 Table 1.4 Table 1.5 Table 1.6 Table 1.7 Table 1.8 Table 1.9 Table 1.10 Table 1.11 Table 1.12 Table 1.13 Table 1.14 Table 1.15 Table 1.16 Table 1.17 Table 1.18 Table 1.19 Table 1.20

3–polar fuzzy set C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy evaluation of each car based on price . . . . . . . . Approximations of the 3–polar fuzzy set d about (U, M) . . . . Mediation mass assignment and its focal elements approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy objective decision information system . . . . . . . Approximations of the 3–polar fuzzy partition . . . . . . . . . . . . . Mediation mass assignment of D1 and its approximations . . . Mediation mass assignment of D2 and its approximations . . . Approximations of the 3–polar fuzzy partition by D in (U, ind(c1 )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations of the 3–polar fuzzy partition generated by D in (U, ind(c2 )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D1 and its approximations in (U, ind(c1 )) . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D2 and its approximations in (U, ind(c1 )) . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D1 and its approximations in (U, ind(c2 )) . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D2 and its approximations in (U, ind(c2 )) . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy objective decision information system . . . . . . . Approximations of the 3–polar fuzzy partition generated by D in (U, ind(r1 , r2 )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D1 and its approximations . . . Mediation mass assignment of D2 and its approximations . . . Approximations of the 3–polar fuzzy partition generated by D in (U, ind(r1 )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations of the 3–polar fuzzy partition generated by D in (U, ind(r2 )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 14 14 15 16 16 17 17 17 18 18 18 18 18 19 20 20 20 20 20

xxv

xxvi

Table 1.21 Table 1.22 Table 1.23 Table 1.24 Table 1.25 Table 1.26 Table 1.27 Table 1.28 Table 1.29 Table 1.30 Table 1.31 Table 1.32 Table 1.33 Table 1.34 Table 1.35 Table 1.36 Table 1.37 Table 1.38 Table 1.39 Table 1.40 Table 1.41 Table 1.42 Table 1.43 Table 1.44 Table 1.45 Table 1.46 Table 1.47 Table 1.48 Table 1.49 Table 1.50 Table 1.51 Table 1.52 Table 1.53 Table 1.54 Table 1.55 Table 1.56 Table 1.57 Table 1.58 Table 1.59 Table 1.60 Table 1.61

List of Tables

Mediation mass assignment of D1 and its approximations in (U, ind(r1 )) . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D2 and its approximations in (U, ind(r1 )) . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D1 and it approximations in (U, ind(r2 )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mediation mass assignment of D2 and its approximations in (U, ind(r2 )) . . . . . . . . . . . . . . . . . . . . . . . . . Tabular representation of 1st pole . . . . . . . . . . . . . . . . . . . . . . . Comparison table for the 1st pole . . . . . . . . . . . . . . . . . . . . . . . Membership score table of 1st pole . . . . . . . . . . . . . . . . . . . . . . Tabular representation of 2nd pole . . . . . . . . . . . . . . . . . . . . . . Comparison table for the 2nd pole . . . . . . . . . . . . . . . . . . . . . . Membership score table of 2nd pole . . . . . . . . . . . . . . . . . . . . . Tabular representation of 3rd pole . . . . . . . . . . . . . . . . . . . . . . . Comparison table for the 3rd pole . . . . . . . . . . . . . . . . . . . . . . . Membership score table of 3rd pole . . . . . . . . . . . . . . . . . . . . . Tabular representation of 4th pole . . . . . . . . . . . . . . . . . . . . . . . Comparison table for the 4th pole . . . . . . . . . . . . . . . . . . . . . . . Membership score table of 4th pole . . . . . . . . . . . . . . . . . . . . . Final score table of candidates . . . . . . . . . . . . . . . . . . . . . . . . . . 1st pole’s tabular representation . . . . . . . . . . . . . . . . . . . . . . . . 1st pole’s comparison table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1st pole’s membership score table . . . . . . . . . . . . . . . . . . . . . . . 2nd pole’s tabular representation . . . . . . . . . . . . . . . . . . . . . . . . 2nd pole’s comparison table . . . . . . . . . . . . . . . . . . . . . . . . . . . 2nd pole’s membership score table . . . . . . . . . . . . . . . . . . . . . . 3rd pole’s tabular representation . . . . . . . . . . . . . . . . . . . . . . . . 3rd pole’s comparison table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3rd pole’s membership score table . . . . . . . . . . . . . . . . . . . . . . Final score table of resorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy set for Rock Fields . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft set (, T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft set for patient P1 . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft set for patient P2 . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft set for patient P3 . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft set for patient P4 . . . . . . . . . . . . . . . . . . . . . Distance between 3–polar fuzzy soft sets . . . . . . . . . . . . . . . . . Similarity measure of 3–polar fuzzy soft sets . . . . . . . . . . . . . . 3–polar fuzzy relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intersection ξ = ξ1 ∩ ξ2 of two 3–polar fuzzy relations . . . . . 3–polar fuzzy relation ξ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy relation ξ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 22 22 22 28 28 29 29 29 29 30 30 30 30 31 31 31 32 32 33 33 33 33 34 34 34 34 41 44 44 44 45 45 45 45 46 46 52 52 52 54

List of Tables

Table 1.62 Table 1.63 Table 1.64 Table 1.65 Table 1.66 Table 1.67 Table 1.68 Table 1.69 Table 1.70 Table 1.71 Table 1.72 Table 1.73 Table 1.74 Table 1.75 Table 1.76 Table 1.77 Table 1.78 Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table 2.6 Table 2.7 Table 2.8 Table 2.9 Table 2.10 Table 2.11 Table 2.12 Table 2.13 Table 2.14 Table 2.15 Table 2.16 Table 2.17 Table 2.18 Table 2.19 Table 2.20 Table 2.21 Table 2.22 Table 2.23 Table 2.24 Table 2.25 Table 2.26

xxvii

Choice value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–polar fuzzy relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . An 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . An 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . An 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy soft relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k—matrix format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy weighted decision matrix . . . . . . . . . . . . . . . . A 3–polar fuzzy concordance set . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy discordance set . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy weighted decision matrix . . . . . . . . . . . . . . . . . . A 4–polar fuzzy concordance set . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy discordance set . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy weighted decision matrix . . . . . . . . . . . . . . . . . . A 4–polar fuzzy concordance set . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy discordance set . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy linguistic variable . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy linguistic decision matrix . . . . . . . . . . . . . . A 4–polar fuzzy linguistic decision matrix . . . . . . . . . . . . . . . . A weighted 4–polar fuzzy linguistic decision matrix . . . . . . . . A 4–polar fuzzy linguistic concordance set . . . . . . . . . . . . . . . A 4–polar fuzzy linguistic discordance set . . . . . . . . . . . . . . . . Tabular representation of comparison of companies . . . . . . . . An of m–polar fuzzy linguistic decision matrix . . . . . . . . . . . . An aggregate m–polar fuzzy linguistic decision matrix . . . . . . A 4–polar fuzzy linguistic group decision matrix . . . . . . . . . . An aggregated 4–polar fuzzy linguistic group decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weights assigned by decision makers . . . . . . . . . . . . . . . . . . . . Weighted aggregated 4–polar fuzzy linguistic group decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 57 58 59 59 60 62 66 68 69 72 75 75 77 77 78 78 89 94 94 94 94 98 98 98 98 100 102 102 102 109 111 115 116 116 116 118 119 120 123 123 124 124

xxviii

Table 2.27 Table 2.28 Table 2.29 Table 2.30 Table 2.31 Table 2.32 Table 2.33 Table 2.34 Table 2.35 Table 2.36 Table 2.37 Table 2.38 Table 2.39 Table 2.40 Table 2.41 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 3.6 Table 3.7 Table 3.8 Table 3.9 Table 3.10 Table 3.11 Table 3.12 Table 3.13 Table 3.14 Table 3.15 Table 3.16 Table 3.17 Table 3.18 Table 3.19

List of Tables

The 4–polar fuzzy linguistic concordance sets . . . . . . . . . . . . . The 4–polar fuzzy linguistic discordance sets . . . . . . . . . . . . . Comparison of countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MATLAB computer programming code of the proposed approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy linguistic decision matrix . . . . . . . . . . . . . . An aggregated m–polar fuzzy linguistic decision matrix . . . . . A 3–polar fuzzy linguistic group decision matrix . . . . . . . . . . An aggregate 3–polar fuzzy linguistic decision matrix . . . . . . Tabular representation of weights assigned by decision makers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted aggregate 3–polar fuzzy linguistic decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tabular representation comparison of appearance of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy linguistic decision matrix . . . . . . . . . . . . . . . . Weighted 4–polar fuzzy linguistic decision matrix . . . . . . . . . Comparison of cars according to their speed . . . . . . . . . . . . . . Computer programming code of proposed approach for MCGDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A generic m–polar hesitant fuzzy decision matrix . . . . . . . . . . A generic weighted m–polar hesitant fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 4–polar hesitant fuzzy decision matrix . . . . . . . . . . . . . . . . . Optimistic 4–polar hesitant fuzzy decision matrix . . . . . . . . . . Weighted optimistic 4–polar hesitant fuzzy decision matrix of Table 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar hesitant fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Pessimistic 3–polar hesitant fuzzy decision matrix . . . . . . . . . Weighted pessimistic 3–polar hesitant fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar hesitant fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Optimistic 3–polar hesitant fuzzy decision matrix . . . . . . . . . . Weighted optimistic 3–polar hesitant fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar hesitant fuzzy concordance set . . . . . . . . . . . . . . . . . . . 3–polar hesitant fuzzy discordance set . . . . . . . . . . . . . . . . . . . Comparison of bricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A generic hesitant m–polar fuzzy decision matrix . . . . . . . . . . A generic weighted hesitant m–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hesitant 3–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Optimistic hesitant 3–polar fuzzy decision matrix . . . . . . . . . . Weighted optimistic hesitant 3–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 124 127 131 139 139 142 142 143 143 144 146 146 147 150 170 171 175 176 177 179 180 181 188 189 190 190 190 192 203 204 207 208 209

List of Tables

Table 3.20 Table 3.21 Table 3.22 Table 3.23 Table 3.24 Table 3.25 Table 3.26 Table 3.27 Table 3.28 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9 Table 4.10 Table 4.11 Table 4.12 Table 4.13 Table 4.14 Table 4.15 Table 4.16 Table 4.17 Table 4.18 Table 4.19 Table 4.20 Table 4.21 Table 4.22 Table 4.23 Table 4.24 Table 4.25 Table 4.26 Table 4.27 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 5.8

xxix

Hesitant 4–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Pessimistic hesitant 4–polar fuzzy decision matrix . . . . . . . . . Weighted pessimistic hesitant 4–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hesitant 4–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Pessimistic hesitant 4–polar fuzzy decision matrix . . . . . . . . . Weighted pessimistic hesitant 4–polar fuzzy decision . . . . . . . Hesitant 4–polar fuzzy concordance set . . . . . . . . . . . . . . . . . . Hesitant 4–polar fuzzy discordance set . . . . . . . . . . . . . . . . . . . Comparison of sites for farming . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e1 . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e2 . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e3 . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e4 . . . . . . . . . . . . Aggregated m–polar fuzzy decision matrix . . . . . . . . . . . . . . . Weights of criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated weighted m–polar fuzzy decision matrix . . . . . . . Score degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The m–polar fuzzy concordance sets . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy discordance sets . . . . . . . . . . . . . . . . . . . . . . Euclidean distance measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of outranking graph . . . . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e1 . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e2 . . . . . . . . . . . . An m–polar fuzzy decision matrix of expert e3 . . . . . . . . . . . . Aggregated m–polar fuzzy decision matrix . . . . . . . . . . . . . . . Weights of decision criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated weighted m–polar fuzzy decision matrix . . . . . . . Score degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy concordance sets . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy indifferent sets . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy discordance sets . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy concordance matrix . . . . . . . . . . . . . . . . . . . Euclidean distance measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy discordance matrix . . . . . . . . . . . . . . . . . . . Outranking relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m–Polar fuzzy decision matrix of expert e1 . . . . . . . . . . . . . . . m–Polar fuzzy decision matrix of expert e2 . . . . . . . . . . . . . . . m–Polar fuzzy decision matrix of expert e3 . . . . . . . . . . . . . . . Score degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projection values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized weights of criteria . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated m–polar fuzzy decision matrix . . . . . . . . . . . . . . . Aggregated weighted m–polar fuzzy decision matrix . . . . . . .

212 213 214 221 222 223 223 224 225 255 255 255 255 255 256 256 256 256 257 257 259 273 274 274 274 275 275 275 275 276 276 276 276 277 277 277 297 297 297 298 298 298 298 299

xxx

Table 5.9 Table 5.10 Table 5.11 Table 5.12 Table 5.13 Table 5.14 Table 5.15 Table 5.16 Table 5.17 Table 5.18 Table 5.19 Table 5.20 Table 5.21 Table 5.22 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 6.9 Table 6.10 Table 6.11 Table 6.12 Table 6.13 Table 6.14 Table 6.15 Table 6.16 Table 6.17 Table 6.18 Table 6.19 Table 6.20 Table 6.21 Table 6.22 Table 6.23 Table 6.24 Table 6.25 Table 6.26 Table 7.1 Table 7.2 Table 7.3 Table 7.4 Table 7.5

List of Tables

Threshold values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairwise comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weights of decision criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated weighted m–polar fuzzy decision matrix . . . . . . . Score degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m–Polar fuzzy concordance sets . . . . . . . . . . . . . . . . . . . . . . . . m–Polar fuzzy discordance sets . . . . . . . . . . . . . . . . . . . . . . . . . Euclidean distance measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of outranking graph . . . . . . . . . . . . . . . . . . . . . . . . . . . m–Polar fuzzy indifferent sets . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The m–polar fuzzy decision matrix Z (1) . . . . . . . . . . . . . . . . . . The m–polar fuzzy decision matrix Z (2) . . . . . . . . . . . . . . . . . . The m–polar fuzzy decision matrix Z (3) . . . . . . . . . . . . . . . . . . Aggregated m–polar fuzzy decision matrix Z . . . . . . . . . . . . . Score degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference of score degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . Threshold values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preference relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dominance relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Credibility indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ascending distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Descending distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ranking of alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projection values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized weights of criteria . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated weighted m–polar fuzzy decision matrix . . . . . . . Threshold values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Score degrees of aggregated weighted decision matrix . . . . . . Pairwise comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial concordance indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comprehensive concordance index . . . . . . . . . . . . . . . . . . . . . . Discordance indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Credibility indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ranking of alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saaty (1–9) preference scale . . . . . . . . . . . . . . . . . . . . . . . . . . . Random index for different values of n . . . . . . . . . . . . . . . . . . The pairwise comparison of criteria . . . . . . . . . . . . . . . . . . . . . Types of criteria and corresponding parameters . . . . . . . . . . . . Decision values of alternatives by decision maker D1 . . . . . . .

299 299 302 303 303 303 303 304 304 306 308 308 309 310 327 327 328 328 328 329 329 330 331 331 331 332 332 332 332 333 334 334 334 335 337 337 338 338 338 339 348 350 358 359 359

List of Tables

Table 7.6 Table 7.7 Table 7.8 Table 7.9 Table 7.10 Table 7.11 Table 7.12 Table 7.13 Table 7.14 Table 7.15 Table 7.16 Table 7.17 Table 7.18 Table 7.19 Table 8.1 Table 8.2 Table 8.3 Table 8.4 Table 8.5 Table 8.6 Table 8.7 Table 8.8 Table 8.9 Table 8.10 Table 8.11 Table 8.12 Table 8.13 Table 8.14 Table 8.15 Table 8.16 Table 8.17 Table 8.18 Table 9.1 Table 9.2 Table 9.3 Table 9.4 Table 9.5

xxxi

Decision values of alternatives by decision maker D2 . . . . . . . Aggregated decision values of alternatives . . . . . . . . . . . . . . . . Deviation of alternatives with respect to criteria . . . . . . . . . . . Generalized criteria preference function . . . . . . . . . . . . . . . . . . Multi-criteria preference index . . . . . . . . . . . . . . . . . . . . . . . . . Positive and negative outranking flows . . . . . . . . . . . . . . . . . . . Net flow of alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Usual criterion preference function . . . . . . . . . . . . . . . . . . . . . . Multi-criteria preference index . . . . . . . . . . . . . . . . . . . . . . . . . Positive and negative outranking flows . . . . . . . . . . . . . . . . . . . Net flow of alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final ranking of hydroelectric power plants . . . . . . . . . . . . . . . Weighted aggregated decision matrix . . . . . . . . . . . . . . . . . . . . m–polar fuzzy ELECTRE I results for selection of hydroelectric power plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy weighted decision matrix . . . . . . . . . . . . . . . . . . 4–polar fuzzy concordance sets . . . . . . . . . . . . . . . . . . . . . . . . . 4–polar fuzzy discordance sets . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy decision matrix of treatments . . . . . . . . . . . . . . . 3–polar fuzzy weighted decision matrix of treatments . . . . . . 3–polar fuzzy concordance sets of treatments . . . . . . . . . . . . . 3–polar fuzzy discordance sets of treatments . . . . . . . . . . . . . . 4–polar fuzzy decision matrix of companies . . . . . . . . . . . . . . 4–polar fuzzy weighted decision matrix of companies . . . . . . 4–polar fuzzy concordance set of companies . . . . . . . . . . . . . . 4–polar fuzzy discordance set of companies . . . . . . . . . . . . . . 3–polar fuzzy decision matrix of countries . . . . . . . . . . . . . . . . Comparison of m–polar fuzzy Dombi and Hamacher aggregation operators in land selection . . . . . . . . . . . . . . . . . . . Comparison of m–polar fuzzy Dombi and Hamacher aggregation operators in bank selection . . . . . . . . . . . . . . . . . . 3–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 3–polar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . Decision matrix for 2–tuple linguistic 3–polar fuzzy information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assembled assessment by using Hamacher weighted average operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scores values for all the 2–tuple linguistic 3–polar fuzzy numbers pˆi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assembled assessment by using Hamacher weighted geometric operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scores values for all the 2–tuple linguistic 3–polar fuzzy numbers pˆi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

360 360 362 363 364 364 365 367 368 368 368 368 369 370 412 414 416 417 417 419 421 421 421 423 425 425 425 428 430 431 434 434 464 465 465 466 466

xxxii

Table 9.6 Table 9.7 Table 9.8 Table 9.9 Table 9.10 Table 10.1 Table 10.2 Table 10.3 Table 10.4 Table 10.5 Table 10.6 Table 10.7 Table 10.8 Table 10.9 Table 10.10 Table 10.11 Table 10.12 Table 10.13 Table 10.14 Table 10.15 Table 10.16 Table 10.17 Table 10.18 Table 10.19 Table 10.20 Table 10.21 Table 10.22 Table 10.23 Table 10.24 Table 10.25 Table 10.26 Table 10.27 Table 10.28

List of Tables

Alternative ranking order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scores values based on Hamacher weighted average operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scores values based on Hamacher weighted geometric operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative analysis of presented operators with existing ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics comparison of presented operator with existing structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . An agree-3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . Disagree-3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . Complement of 3–polar fuzzy soft expert set . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . Union of the 3–polar fuzzy soft expert sets . . . . . . . . . . . . . . . Intersection of the 3–polar fuzzy soft expert sets . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . AND operation between the 3–polar fuzzy soft expert sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OR operation between 3–polar fuzzy soft expert sets . . . . . . . A 4–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . An agree-4–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . Score values for agree-4–polar fuzzy soft expert set . . . . . . . . Disagree-4–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . Score values for disagree–4–polar fuzzy soft expert set . . . . . Final score table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . . . . . . . An agree-3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . Score values for agree-3–polar fuzzy soft expert set . . . . . . . . Disagree-3–polar fuzzy soft expert set . . . . . . . . . . . . . . . . . . . Score values for disagree-3–polar fuzzy soft expert set . . . . . . Final score table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between different values of ‘m’ in Application 10.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between different values ‘m’ in Application 10.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 468 469 470 470 480 481 481 482 482 483 484 484 485 486 487 488 488 489 493 493 494 494 495 495 497 497 498 498 499 499 500 500

List of Tables

Table 10.29 Table 10.30 Table 10.31 Table 10.32 Table 10.33 Table 10.34 Table 10.35 Table 10.36 Table 10.37 Table 10.38 Table 10.39 Table 10.40 Table 10.41 Table 10.42 Table 10.43 Table 10.44 Table 10.45 Table 10.46 Table 10.47 Table 10.48 Table 10.49 Table 10.50 Table 10.51 Table 10.52 Table 10.53 Table 10.54 Table 10.55 Table 10.56 Table 10.57 Table 10.58 Table 10.59 Table 10.60 Table 10.61 Table 10.62 Table 10.63 Table 10.64 Table 10.65 Table 10.66 Table 10.67 Table 10.68 Table 10.69 Table 10.70 Table 10.71 Table 10.72 Table 10.73

xxxiii

Comparison with existing hybrid models . . . . . . . . . . . . . . . . . Information extracted from the related data . . . . . . . . . . . . . . . Tabular representation of 6–soft set . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 6–soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The weak complement of 3–polar fuzzy 6–soft set . . . . . . . . . The top weak complement of 6–polar fuzzy N –soft set . . . . . The bottom weak complement of 3–polar fuzzy 6–soft set . . . A 3–polar fuzzy 5–soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 4–soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restricted intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restricted union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An m–polar fuzzy N –soft relation . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 5–soft relation . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 5–soft relation . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 5–soft relation . . . . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 5–soft relation . . . . . . . . . . . . . . . . . . . . . . . . . Information obtained from the related data . . . . . . . . . . . . . . . . Tabular representation of 6–soft set . . . . . . . . . . . . . . . . . . . . . . A 4–polar fuzzy 6–soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tabular representation of 1st pole . . . . . . . . . . . . . . . . . . . . . . . Comparison table for the 1st pole . . . . . . . . . . . . . . . . . . . . . . . Membership score table of the 1st pole . . . . . . . . . . . . . . . . . . . Tabular representation of the 2nd pole . . . . . . . . . . . . . . . . . . . Comparison table for the 2nd pole . . . . . . . . . . . . . . . . . . . . . . Membership score table of the 2nd pole . . . . . . . . . . . . . . . . . . Tabular representation of the 3rd pole . . . . . . . . . . . . . . . . . . . . Comparison table for the 3rd pole . . . . . . . . . . . . . . . . . . . . . . . Membership score table of the 3rd pole . . . . . . . . . . . . . . . . . . Tabular representation of the 4th pole . . . . . . . . . . . . . . . . . . . . Comparison table for the 4th pole . . . . . . . . . . . . . . . . . . . . . . . Membership score table of the 4th pole . . . . . . . . . . . . . . . . . . Final score table with grades . . . . . . . . . . . . . . . . . . . . . . . . . . . Information obtained from the related data . . . . . . . . . . . . . . . . Tabular representation of 6–soft set . . . . . . . . . . . . . . . . . . . . . . A 3–polar fuzzy 6–soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tabular representation of the 1st pole . . . . . . . . . . . . . . . . . . . . Comparison Table for the 1st pole . . . . . . . . . . . . . . . . . . . . . . . Membership score Table of the 1st pole . . . . . . . . . . . . . . . . . . Tabular representation of 2nd pole . . . . . . . . . . . . . . . . . . . . . . Comparison table for the 2nd pole . . . . . . . . . . . . . . . . . . . . . . Membership score table of the 2nd pole . . . . . . . . . . . . . . . . . . Tabular representation of the 3rd pole . . . . . . . . . . . . . . . . . . . . Comparison Table for the 3rd pole . . . . . . . . . . . . . . . . . . . . . .

502 503 504 504 506 507 508 508 508 509 509 510 511 513 513 514 515 517 519 519 519 519 520 520 520 520 521 521 521 521 521 522 522 522 524 524 524 524 525 525 525 525 525 526 526

xxxiv

Table 10.74 Table 10.75 Table 10.76 Table 10.77 Table 10.78 Table 10.79

List of Tables

Membership score table of the 3rd pole . . . . . . . . . . . . . . . . . . Final score table with grades . . . . . . . . . . . . . . . . . . . . . . . . . . . Information obtained from the related data . . . . . . . . . . . . . . . . A 3–polar fuzzy 6–soft relation . . . . . . . . . . . . . . . . . . . . . . . . . Information extracted from the related data . . . . . . . . . . . . . . . Tabular representation of 3–polar fuzzy 6–soft relation . . . . . .

526 526 527 528 530 530

Chapter 1

Hybrid Multi-polar Fuzzy Models

In this chapter, the advantages of multi-polar (m–polar) fuzzy information are explored. In fact hybrid models are presented by combining m−polar fuzzy sets with rough and soft sets, which produce rough m−polar fuzzy sets and m−polar fuzzy soft sets. Some of their fundamental properties are investigated. Their importance is studied by way of some real-life applications to MCDM. Moreover, similarity measures of m−polar fuzzy sets are considered and a variety of applications are presented, including medical diagnosis, pattern recognition, coding theory, game theory and region extraction. Furthermore, the notions of m−polar fuzzy rough sets, and m−polar fuzzy soft rough sets are presented for soft computing and modeling of MCDM problems. Related concepts and some of their basic properties and fundamental operations (includsive of union, intersection, and composition) are explored. The relationship between m−polar fuzzy soft rough approximation operators and crisp soft rough approximation operators is examined. Moreover, potential applications of presented models are explored in MCDM. Efficient algorithms are developed to solve MCDM problems based on the hybrid models presented in this chapter. This chapter owes to [1–5, 21].

1.1 Introduction Fuzzy set theory offered an escape to the rigidity of crisp models. Many other models followed the path initiated by Zadeh’s [58] pioneering work, and examples have appeared so far in this monograph. But these tools are unable to approach ambiguities of all types. Two other totally different expressions of vague knowledge are captured by soft set theory and rough set theory. The idea of soft set theory [43, 44] dates back to 1999. It states that we must often describe or characterize objects as having © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_1

1

2

1 Hybrid Multi-polar Fuzzy Models

(or failing to have) a collection of features. Therefore, formally speaking a soft set is determined by a set-valued mapping, that associates exactly one subset of the universe of alternatives with each parameter under consideration. Applications to various fields, including decision making [27], forecasting [54], and data analysis [63], soon justified its interest for both pure and applied researchers. Some basic algebraic operations on soft set theory were presented by Maji et al. [38] and Ali et al. [17]. The later work introduced some new soft set theoretic operations for soft sets and checked that in terms of them, certain De Morgan’s laws hold true. Maji et al. [36] extended the idea of soft sets and proposed a hybrid model called fuzzy soft sets which allow for partial satisfaction of the attributes. Furthermore, Maji et al. [39] presented the concept of intuitionistic fuzzy soft sets. The uncertainty measures of soft sets, fuzzy soft sets and intuitionistic fuzzy soft sets were introduced by Majumdar and Samanta [40–42]. Some set theoretic operations based on similarity measures of soft sets were presented by Kharal [32]. Jiang et al. [31] initiated the concept of entropy in both the intuitionistic fuzzy soft and interval-valued fuzzy soft environment. Feng et al. [27] and Alcantud et al. [14] provided deeper insights into the decision making processes based on fuzzy soft sets. The model has been linked to topological ideas too [10]. Rough set theory, proposed by Pawlak [45–47] in 1982, is another totally redesigned tool for the mathematical treatment of incomplete, imprecise and uncertain information. In rough set theory, two precise boundary lines are established in order to describe imprecisely defined objects. Therefore, rough set theory is a precise mathematical tool for the solution of uncertain problems. Yao [57] introduced constructive and algebraic methods for theory of fuzzy sets. In recent years, for both theoretical and practical needs, many authors have produced generalized rough set models [15, 16, 29, 59]. The study of hybrid models combining the spirit of rough sets with other mathematical models soon emerged as an active research topic within rough set theory. Dubois and Prade [24] first introduced the concepts of rough fuzzy set and fuzzy rough set, which are the combination of fuzzy sets and rough sets. A rough fuzzy set is the approximation of a fuzzy set in a crisp approximation space. Therefore, rough sets not only have ample applications into mathematical theories but also in real-world problems. The combination of Pawlak’s rough sets with soft set theory led Feng et al. [25] to propose the novel idea of rough soft sets. They can be regarded as a collection of rough sets that share a common Pawlak approximation space. In 2011, Feng et al. [28] introduced the concept of soft rough sets. Zhang and Zhou [61] proposed a general framework for intuitionistic fuzzy rough sets. Recently, novel relationships between N -soft set theory and rough sets have been established in Alcantud et al. [13]. The decision making based on fuzzy sets and its extensions, soft sets, N -soft set, fuzzy soft sets, intuitionistic fuzzy soft sets and rough sets has been considered by many researchers, see [6, 11, 20, 35, 37, 52, 60] as a short sample. Let us return to the topic of extended fuzzy sets theories after that brief digression. Atanassov [18] introduced the concept of intuitionistic fuzzy set, an extension of fuzzy set, that together with a membership grade μ, make use of a non-membership grade ν, and they are jointly subject to the condition μ + ν < 1. Although the mathematical performance of intuitionistic fuzzy sets is quite good, still they fail to capture

1.2 m−Polar Fuzzy Sets

3

many situations where the sum of membership and non-membership degrees exceeds 1 (at some option). Yager [55, 56] presented the less stringent concept of Pythagorean fuzzy sets. Nevertheless real world models often contain multi-attribute, multi-index, multi-object and multi-information data. For example, multi-polar technology, which can be used for the management of large scale applications of information technology. With this practical motivation, Chen et al. [21] introduced the more general concept of m−polar fuzzy sets in 2014. In fact these authors showed that 2-polar fuzzy sets and bipolar fuzzy sets are cryptomorphic mathematical notions. The idea behind this extension is that “multi-polar information” exists because data from real-world experiences sometimes come from n agents (n ≥ 2). There are many other motivations, such as truth degrees of logic formulae which are based on n logic implication operators (n ≥ 2). So also under this motivation, the m–polar case goes beyond the idea that bipolar information may correspond to two-valued logic. Practical examples of this sort of information include the exact degree of telecommunication safety of mankind, which is a point in [0, 1]n (n ≈ 7 × 109 ) because different persons have been monitored different times; the ordering results of magazines or movies; the rankings of universities; and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures and decision preformation evaluations) of rough sets. Akram and Waseem [3] introduced the idea of similarity measures for m–polar fuzzy sets and m–polar fuzzy soft sets. Akram [1] extended his work on graphs under m–polar fuzzy environment. Furthermore, Akram et al. [2, 4, 5, 7, 8] proposed some novel hybrid models by the combination of m–polar fuzzy sets with rough sets and soft sets.

1.2 m−Polar Fuzzy Sets Set theory and logical systems are coupled in the development of modern logic. Classical logic corresponds to classical set theory, while fuzzy logic corresponds to fuzzy set theory proposed by Zadeh in his seminal work [58]. Chen et al. [21] first considered the notion of m−polar fuzzy sets. The grade of membership of m−polar fuzzy sets ranges over the interval [0, 1]m , and it represents m different properties of an object. In order to study their model, some formal concepts are in order. Definition 1.1 ([49]) A lattice L is a triple (U, ∨, ∧) where U is a non-empty set, ∨ and ∧ are binary operations on U such that 1. 2. 3. 4.

x x x x

∨ y = y ∨ x, x ∧ y = y ∧ x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z ∨ (x ∧ y) = x, x ∧ (x ∨ y) = x ∨ x = x, x ∧ x = x for all x, y, z ∈ U.

Definition 1.2 ([49]) A lattice L is said to be complete if L contains two distinguished elements denoted by 1 and 0, which are distinct and satisfy

4

1 Hybrid Multi-polar Fuzzy Models

5. x ∨ 1 = 1, 6. x ∨ 0 = x,

x ∧ 1 = x, x ∧ 0 = 0,

for all x in L. Example 1.1 1. Let I = [0, 1] be a lattice with definitions: x ∨ y = max(x, y) and x ∧ y = min(x, y). It is a complete lattice. 2. Let L = I m = I × I × · · · × I (m times) be lattice with x ∨ y = max(x, y) and x ∧ y = min(x, y). Let L be a complete lattice with a smallest element 0 and a largest element 1. Then L U , the set of all mappings A : U → L is also a complete lattice with point-wise order. Definition 1.3 ([49]) For any two n-tuples (a1 , a2 , . . . , an ) and (b1 , b2 , . . . , bn ), where ai , bi , for each 1 ≤ i ≤ n, are real numbers then, • (a1 , a2 , . . . , an ) = (b1 , b2 , . . . , bn ) ⇔ ai = bi , for each 1 ≤ i ≤ n, • (a1 , a2 , . . . , an ) ≤ (b1 , b2 , . . . , bn ) ⇔ ai ≤ bi , for each 1 ≤ i ≤ n, • (a1 , a2 , . . . , an ) ≥ (b1 , b2 , . . . , bn ) ⇔ ai ≥ bi , for each 1 ≤ i ≤ n, • (a1 , a2 , . . . , an ) × (b1 , b2 , . . . , bn ) = (a1 b1 , a2 b2 , . . . , an bn ). The main concept in Chen et al. [21] is captured by the next definition: Definition 1.4 ([21]) An m− polar fuzzy set C on a non-empty set U is characterized by a mapping C : U → [0, 1]m . The membership value of every element x ∈ U is denoted by C(x) = ( p1 ◦ C(x), p2 ◦ C(x), . . . , pm ◦ C(x)), where pi ◦ C : [0, 1]m → [0, 1] is defined as the i−th projection mapping. Note that [0, 1]m (the m–th power of [0, 1]) is a partially ordered set with the point-wise order ≤, where m is an arbitrary ordinal number (we make the convention that m = {n|n < m} when m > 0), ≤ is defined by x ≤ y ⇔ pi (x) ≤ pi (y) for each i ∈ m ( x, y ∈ [0, 1]m ), and pi : [0, 1]m → [0, 1] is the i−th projection mapping (i ∈ m). Henceforth 1 = (1, 1, . . . , 1) denotes the greatest value and 0 = (0, 0, . . . , 0) denotes the smallest value in [0, 1]m . mF U represents the power set of all m−polar fuzzy subsets on U . (i) When m = 2, [0, 1]2 is the ordinary closed unit square in R2 , the Euclidean plane. The righter (resp., the upper), the point in this square, the larger it is. Let x = (0, 0) = 0 (the smallest element of [0, 1]2 ), a = (0.35, 0.85), b = (0.85, 0.35) and y = (1, 1) = 1 (the largest element of [0, 1]2 ). Then x ≤ c ≤ y, ∀ c ∈ [0, 1]2 , (especially, x ≤ a ≤ y and x ≤ b ≤ y hold). It is easy to note that a  b  a because p0 (a) = 0.35 < 0.85 = p0 (b) and p1 (a) = 0.85 > 0.35 = p1 (b) hold. The “order relation ≤” on [0, 1]2 can be described in at least two ways. It can be seen in Fig. 1.1.

1.2 m−Polar Fuzzy Sets

5

Fig. 1.1 Order relation when m = 2

y

y a

b

x y

Fig. 1.2 Order relation when m = 4

y

b

a

a

b x

x y

b

a

a

b x

x

(ii) When m = 4, the order relation can be seen in Fig. 1.2. (iii) A 2-pole fuzzy set C on a non-empty set U is characterized by the map C = (μ1 , μ2 ) : U → [0, 1]2 . The 2-pole fuzzy set C on the non-empty set U can be expressed as C = {(x, μ1 (x), μ2 (x)) : x ∈ U } , where μ1 (x), μ2 (x) are independent, and μ1 (x), μ2 (x) ∈ [0, 1]. Notice 1=(1,1) represents the maximum value in [0, 1]2 and 0=(0,0) represents the minimum value in [0, 1]2 . This 2-pole fuzzy set is different from Zhang’s bipolar fuzzy set [62]. A bipolar fuzzy set is a pair of fuzzy sets that represent the positive and negative aspects of a given information. The name “bipolar fuzzy set” is used for objects introduced by Zhang, originally called (yin)(yang) bipolar fuzzy sets. The main difference between these models is the independence of poles in 2-polar fuzzy sets whereas in bipolar fuzzy sets both necessarily represent the counter property of each other. Thus, one pole is assigned a membership grade from the interval [0,1], while its counter property is assigned a membership grade from [-1,0]. On the other hand, in a 2-pole fuzzy set, both poles are assigned membership grades starting from [0,1] since they correspond to independent features. For example, let U = {u 1 , u 2 , u 3 } be a set of houses. In a 2-polar fuzzy set, we can monitor two independent aspects of a house, i.e., its beauty and cost. Whereas in bipolar fuzzy sets, we can model two counter properties at a time that can be beauty and ugliness. Example 1.2 ([21]) Suppose that a democratic country wants to elect its leader. Let C = {Irtiza, Moeed, Ramish, Ahad} be the set of four candidates and Y = {a, b, c, . . . , s, t} be the set of voters, and let the voting be assumed weighted. Let A(a) = (0.8, 0.6, 0.5, 0.1) (which shows that the preference degrees of a corresponding to Irtiza, Moeed, Ramish and Ahad are 0.8, 0.6, 0.5 and 0.1, respectively),

6

1 Hybrid Multi-polar Fuzzy Models

Table 1.1 3−polar fuzzy set C Company Profit x p1 ◦ C(x) T1 T2 T3 T4 T5

0.7 0.5 0.9 0.8 0.6

Market power p2 ◦ C(x)

Price control p3 ◦ C(x)

0.4 0.3 0.7 0.7 0.3

0.9 0.6 0.5 0.6 0.6

A(b) = (0.9, 0.7, 0.5, 0.8), A(c) = (0.9, 0.9, 0.8, 0.4), . . . , A(s) = (0.6, 0.7, 0.5, 0.3) and A(t) = (0.5, 0.7, 0.2, 0.5). Thus, a 4−polar fuzzy set A : Y → [0, 1]4 is obtained, which can also be written as A = {(a, (0.8, 0.6, 0.5, 0.1)), (b, (0.9, 0.7, 0.5, 0.8)), (c, (0.9, 0.9, 0.8, 0.4)), …, (s, (0.6, 0.7, 0.5, 0.3)), (t, (0.5, 0.7, 0.2, 0.5))}. Example 1.3 ([21]) Let U = {T1 , T2 , T3 , T4 , T5 } be a set of companies which may have different reputations in the market due to their annual profits, market power and price control of their products. This data produces multi-polar information which are fuzzy in nature. Let C be a 3−polar fuzzy set on U . The degree of membership of each company is shown in Table 1.1. The membership value of T1 is (0.7, 0.4, 0.9) which shows that T1 has 70% annual profit, 40% power in business market and 90% price control of its product. The fuzzy strategies in Table 1.1 can be represented by a 3−polar fuzzy set as follows: C = {(T1 , 0.7, 0.4, 0.9), (T2 , 0.5, 0.3, 0.6), (T3 , 0.9, 0.7, 0.5), (T4 , 0.8, 0.7, 0.6), (T5 , 0.6, 0.3, 0.6)}. Definition 1.5 ([1]) Let C and D be two m−polar fuzzy sets on U . Then the operations C ∪ D, C ∩ D, C ⊆ D and C = D are defined as 1. 2. 3. 4.

pi ◦ (C ∪ D)(x) = sup{ pi ◦ C(x), pi ◦ D(x)} = pi ◦ C(x) ∨ pi ◦ D(x). pi ◦ (C ∩ D)(x) = inf{ pi ◦ C(x), pi ◦ D(x)} = pi ◦ C(x) ∧ pi ◦ D(x). C ⊆ Dif and only if pi ◦ C(x) ≤ pi ◦ D(x). C = Dif and only if pi ◦ C(x) = pi ◦ D(x),

for all x ∈ U , for each 1 ≤ i ≤ m.

1.3 Rough m−Polar Fuzzy Sets Rough set theory was proposed by Pawlak [45–47] in 1982. It is based on the assumption that every object in the universe is associated with some information. Objects

1.3 Rough m−Polar Fuzzy Sets

7

described by the same data are indistinguishable. Therefore, the indiscernibility relation induced in this way forms the mathematical basis of rough set theory. In Pawlak’s model, the indiscernibility relation is conceived as an equivalence relation on the universe of alternatives. Then any subset of this universe can be characterized by two definable, or observable subsets called its upper and lower approximations. Definition 1.6 ([45]) Let U be a universe of discourse and let Q be an equivalence relation on U . A pair (U, Q) is called a Pawlak approximation space. For any X ⊆ U , the lower approximation Q ∗ (X ) and upper approximation Q ∗ (X ) of X are defined by Q ∗ (X ) = {x ∈ U | [x] Q ⊆ X }, Q ∗ (X ) = {x ∈ U | [x] Q ∩ X = ∅}. If Q ∗ (X ) = Q ∗ (X ), X is said to be definable in (U, Q), otherwise X is referred to as a rough set. Akram et al. [4] proposed the concept of rough m−polar fuzzy set in 2018. Definition 1.7 ([4]) Let U be a universe and M a crisp equivalence relation on U . A pair (U, M) is said to be a crisp approximation space. For each X ∈ mF U , the lower and upper approximations of X , denoted by M∗ (X )(u) and M ∗ (X )(u), respectively, which are m−polar fuzzy sets in U defined by M∗ (X )(u) =

 

 pi ◦ X (v) ,

v∈[u] M

M ∗ (X )(u) =

 

 pi ◦ X (v) ,

v∈[u] M

for all u ∈ U , where [u] M = {v | (u, v) ∈ M} is an equivalence class of u ∈ U and i = 1, 2, . . . , m. The operators M∗ (X ) and M ∗ (X ) are called the lower and upper approximation operators of X , respectively. If M∗ (X ) = M ∗ (X ), then X is said to be definable, otherwise X is called a rough m−polar fuzzy set. Example 1.4 Let U = {1, 2, 3, 4, 5, 6} be a universe, M = {(1, 3), (3, 1), (2, 6), (6, 2), (4, 5), (5, 4)} be an equivalence relation on U . Then [1] M = {1, 3} = [3] M , [2] M = {2, 6} = [6] M , [4] M = {4, 5} = [5] M . Consider a 3−polar fuzzy set X as follows: X=

 (0.8, 0.8, 0.8) (0.3, 0.3, 0.3) (0.5, 0.5, 0.5) , , , 1 2 3 (0.1, 0.1, 0.1) (0.4, 0.4, 0.4) (0.3, 0.3, 0.3)  , , . 4 5 6

By Definition 1.7, lower and upper approximations are as follows:

8

1 Hybrid Multi-polar Fuzzy Models

M∗ (X )(1) = (0.5, 0.5, 0.5) = M∗ (X )(3), M ∗ (X )(1) = (0.8, 0.8, 0.8) = M ∗ (X )(3), M∗ (X )(2) = (0.3, 0.3, 0.3) = M∗ (X )(6), M ∗ (X )(2) = (0.3, 0.3, 0.3) = M ∗ (X )(6), M∗ (X )(4) = (0.1, 0.1, 0.1) = M∗ (X )(5), M ∗ (X )(4) = (0.4, 0.4, 0.4) = M ∗ (X )(5). Thus,  (0.5, 0.5, 0.5) (0.3, 0.3, 0.3) (0.5, 0.5, 0.5) , , , 1 2 3 (0.1, 0.1, 0.1) (0.1, 0.1, 0.1) (0.3, 0.3, 0.3)  , , , 4 5 6  (0.8, 0.8, 0.8) (0.3, 0.3, 0.3) (0.8, 0.8, 0.8) M ∗ (X ) = , , , 1 2 3 (0.4, 0.4, 0.4) (0.4, 0.4, 0.4) (0.3, 0.3, 0.3)  , , . 4 5 6 M∗ (X ) =

Clearly, M∗ (X ) = M ∗ (X ). Hence, X is a rough 3−polar fuzzy set. Theorem 1.1 ([4]) Let (U, M) be a crisp approximation space and X, Y ∈ mF U . Then the lower and upper approximations of X and Y have the following properties: 1. 2. 3. 4. 5. 6. 7.

M ∗ (X ∪ Y ) = M ∗ (X ) ∪ M ∗ (Y ), M∗ (X ∪ Y ) ⊇ M∗ (X ) ∪ M∗ (Y ), X ⊆ Y ⇒ M∗ (X ) ⊆ M∗ (Y ) and M ∗ (X ) ⊆ M ∗ (Y ), M∗ (X ∩ Y ) = M∗ (X ) ∩ M∗ (Y ), M ∗ (X ∩ Y ) ⊆ M ∗ (X ) ∩ M ∗ (Y ), M∗ (∼ X ) =∼ M ∗ (X ) and M ∗ (∼ X ) =∼ M∗ (X ),

M∗ (M∗ (X )) = M ∗ (M∗ (X )) = M∗ (X ) and M ∗ (M ∗ (X )) = M∗ (M ∗ (X )) = M ∗ (X ),

where ∼ X denotes the complement of X . Proof 1. By Definition 1.7, ∀ u ∈ U , 

M ∗ (X ∪ Y )(u) = =

 

pi ◦ X (v) ∨ pi ◦ Y (v)

v∈[u] M

=



pi ◦ (X ∪ Y )(v)

v∈[u] M

pi ◦ X (v) ∨

v∈[u] M

 v∈[u] M

  = M ∗ (X ) ∪ M ∗ (Y ) (u). Thus, M ∗ (X ∪ Y ) = M ∗ (X ) ∪ M ∗ (Y ).



pi ◦ Y (v)

1.3 Rough m−Polar Fuzzy Sets

9

2. By Definition 1.7, ∀ u ∈ U , M∗ (X ∪ Y )(u) = ≥



 

pi ◦ X (v) ∨ pi ◦ Y (v)



v∈[u] M

pi ◦ X (v) ∨

v∈[u] M



pi ◦ Y (v)

v∈[u] M

  = M∗ (X ) ∪ M∗ (Y ) (u).

Hence, M∗ (X ∪ Y ) ⊇ M∗ (X ) ∪ M∗ (Y ). 3. It follows promptly from Definition 1.7. The remaining parts can be proved similarly.  Remark 1.1 For any X ∈ mF U , 1. M∗ (X ) ⊆ X ⊆ M ∗ (X ), 2. M∗ (∅) = ∅ = M ∗ (∅), M∗ (U ) = U = M ∗ (U ), 3. In the Theorem 1.1, the equality in part (4) and (7) holds only if X ∪ Y ⊆ Y or X ∪ Y ⊆ X and X ⊆ Y or Y ⊆ X , respectively. Definition 1.8 Let H = {σ = (a1 , a2 , . . . , am ) | ai ∈ [0, 1], i = 1, 2, . . . , m}. Then, σ is called an m−polar fuzzy number. For all σ = (a1 , a2 , . . . , am ) ∈ H , the mediation value of σ is defined as σA =

a1 + a2 + · · · + am . m

Definition 1.9 Let (U, M) be a crisp approximation. For X ∈ mF U , σ ∈ H and 0 < σ < 1, the σ -lower level boundary of X is defined as M L σ∗ (X ) = {u ∈ U | M∗ (X )(u) < σ }. Definition 1.10 Let (U, M) be a crisp approximation. For X ∈ mF U , σ ∈ H and 0 < σ < 1, the σ -upper level boundary of X is defined as M L ∗σ (X ) = {u ∈ U | M ∗ (X )(u) > σ }. Definition 1.11 Let (U, M) be a crisp approximation. The following expressions for X, Y ∈ mF U , σ ∈ H, 0 < σ < 1, defined as 1. W (X, Y )σ = {u ∈ U | M∗ (X ∪ Y )(u) = σ, u ∈ M L σ∗ (X ) ∩ M L σ∗ (Y )}. 2. W (X, Y )σ = {u ∈ U |M ∗ (X ∩ Y )(u) = σ, u ∈ M L ∗σ (X ) ∩ M L ∗σ (Y )}. σ u ∈ W (X, Y )σ , 3. M∗ (W (X, Y )σ )(u) = 0 u∈ / W (X, Y )σ .  σ u ∈ W (X, Y )σ , 4. M ∗ (W (X, Y )σ )(u) = 1 u∈ / W (X, Y )σ . 5. M∗ (W )(X, Y ) = σ M∗ (W (X, Y )σ ).

10

1 Hybrid Multi-polar Fuzzy Models

6. M ∗ (W )(X, Y ) =

σ

M ∗ (W (X, Y )σ ).

Theorem 1.2 Let (U, M) be a crisp approximation space, then lower and upper approximations of m−polar fuzzy sets X and Y have the following properties: 1. M ∗ (X ∩ Y ) = M ∗ (X ) ∩ M ∗ (Y ) ∩ M ∗ (W )(X, Y ). 2. M∗ (X ∪ Y ) = M∗ (X ) ∪ M∗ (Y ) ∪ M∗ (W )(X, Y ). Proof 1. From Theorem 1.1, M ∗ (X ∩ Y ) ⊆ M ∗ (X ) ∩ M ∗ (Y ) ∀ u ∈ U, set M ∗ (X ∩ Y )(u) = σ. If there exists u ∈ W (X, Y )σ , then M ∗ (W )(X, Y ) = σ. Otherwise Thus, ∀ u ∈ U ,

That is,

M ∗ (W )(X, Y )(u) = 1. M ∗ (X ∩ Y )(u) ≤ M ∗ (W )(X, Y )(u). M ∗ (X ∩ Y ) ⊆ M ∗ (W )(X, Y ).

If M ∗ (W )(X, Y )(u) = 1, there exists σ such that M ∗ (X ∩ Y )(u) = σ / M L ∗σ (Y ), this implies that and u ∈ / M L ∗σ (X ) or u ∈ M ∗ (X )(u) = σ or M ∗ (X )(u) = σ. That is, M ∗ (X ∩ Y )(u) =M ∗ (X )(u) ∩ M ∗ (Y )(u) ∩ M ∗ (W )(X, Y )(u). If

M ∗ (W )(X, Y )(u) = σ = 1,

there exists σ satisfying M ∗ (X ∩ Y )(u) = σ and u ∈ M L ∗σ (X ), u ∈ M L ∗σ (Y ), followed by M ∗ (X )(u) > σ and M ∗ (X )(u) > σ. Therefore, it is proved that

1.3 Rough m−Polar Fuzzy Sets

11

M ∗ (X ∩ Y )(u) = M ∗ (X )(u) ∩ M ∗ (Y )(u) ∩ M ∗ (W )(X, Y )(u), = M ∗ (W )(X, Y )(u) = σ. Hence, M ∗ (X ∩ Y ) = M ∗ (X ) ∩ M ∗ (Y ) ∩ M ∗ (W )(X, Y ). 2. It can be proved similarly.



Proposition 1.1 ([4]) Let (U, M) be a crisp approximation. Then the lower and upper approximations of the m−polar fuzzy sets X and Y satisfy the following laws:   1. ∼ M∗ (X ) ∪ M∗ (Y ) = M ∗ (∼ X ) ∩ M ∗ (∼ Y ).   2. ∼ M∗ (X ) ∪ M ∗ (Y ) = M ∗ (∼ X ) ∩ M∗ (∼ Y ).   3. ∼ M ∗ (X ) ∪ M∗ (Y ) = M∗ (∼ X ) ∩ M ∗ (∼ Y ).   4. ∼ M ∗ (X ) ∪ M ∗ (Y ) = M∗ (∼ X ) ∩ M∗ (∼ Y ).   5. ∼ M∗ (X ) ∩ M∗ (Y ) = M ∗ (∼ X ) ∪ M ∗ (∼ Y ).   6. ∼ M∗ (X ) ∩ M ∗ (Y ) = M ∗ (∼ X ) ∪ M∗ (∼ Y ).   7. ∼ M ∗ (X ) ∩ M∗ (Y ) = M∗ (∼ X ) ∪ M ∗ (∼ Y ).   8. ∼ M ∗ (X ) ∩ M ∗ (Y ) = M∗ (∼ X ) ∪ M∗ (∼ Y ). 

Proof Proof is obvious.

Definition 1.12 Let (U, M) be a crisp approximation. The following expressions for X ∈ mF U , σ ∈ H , are defined as M∗ (X )σ = {u ∈ U | M∗ (X )(u) ≥ σ }, M ∗ (X )σ = {u ∈ U | M ∗ (X )(u) ≥ σ }. Theorem 1.3 Let (U, M) be a crisp approximation space and X, Y ∈ mF U . Then, 1. 2. 3. 4.

M∗ (X ∩ Y )σ M ∗ (X ∪ Y )σ M ∗ (X ∩ Y )σ M∗ (X ∪ Y )σ

= M∗ (X )σ ∩ M∗ (Y )σ , = M ∗ (X )σ ∪ M ∗ (Y )σ , = M ∗ (X )σ ∩ M ∗ (Y )σ ∩ M ∗ (W )(X, Y )σ , = M∗ (X )σ ∪ M∗ (Y )σ ∪ M∗ (W )(X, Y )σ .

Proof Its proof follows immediately from Theorems 1.1 and 1.2.



Definition 1.13 Let (U, M) be a crisp approximation space. For 0 < τ ≤ σ ≤ 1, the (σ, τ )-related accuracy degree of the m−polar fuzzy set X is defined as γ σ,τ =

|M∗ (X )σ | , |M ∗ (X )τ |

and the corresponding (σ, τ )-related rough degree of the m−polar fuzzy set X is given by

12

1 Hybrid Multi-polar Fuzzy Models

ρ σ,τ = 1 − γ σ,τ = 1 −

|M∗ (X )σ | . |M ∗ (X )τ |

Definition 1.14 Let X ∈ mF U , if there exists v ∈ U such that X (v) = 1, then X is said to be a normal m− polar fuzzy set. Suppose that the range of the mediations of X is Range(X A ) = {a1 , a2 , . . . , am }, where ai > ai+1 > 0, for i = 1, 2, . . . , m − 1. Definition 1.15 Let X ∈ mF U denotes the mediation mass assignment of X , denoted by m and defined as 1 − a1 , m m(X i ) = ai − ai+1 , m(∅) =

i = 1, 2, . . . , m,

where X i is a Pawlak set, and X i = {u ∈ S | X A (u) ≥ ai }, i = 1, 2, . . . , m. m are the focal elements of m. {X i }i=1

Next, the notion of mediation mass assignment of m−polar fuzzy sets is used to propose the parameter-free rough degree of the normal m−polar fuzzy sets. Definition 1.16 Let (U, M) be a crisp approximation space. For a normal m−polar fuzzy set X , the parameter-free rough degree of X about (U, M) is defined as ρ(X ) = =

 |M∗ (X i )|  , m(X i ) 1 − |M ∗ (X i )| i=1

m 

m 

m(X i )ρ(X i ).

i=1

Remark 1.2 1. X is called rough m− polar fuzzy set if and only if 0 ≤ ρ(X ) ≤ 1. 2. X is definable if and only if ρ(X ) = 0. Proposition 1.2 Let (U, M) be a crisp approximation space and X ∈ mF U , then ∀ ai ∈ Range(X A ), the following properties hold. 1. M ∗ (X i ) ⊆ M ∗ (X )i , 2. M∗ (X )i ⊆ M∗ (X i ). Proof Proof is obvious.



1.3 Rough m−Polar Fuzzy Sets

13

Definition 1.17 Let (U, M) be a crisp approximation. For X ∈ mF U , the parameterfree rough degree of X about (U, M) is defined as ρ(X ) =

m 

m(X i )ρ(X i ) + m(∅)ρ(∅).

i=1

If A is normal and ρ(∅) = 0, then m(∅)ρ(∅) = 0 , but γ (∅) = 1, by convention. So, the parameter-free rough accuracy degree of an m−polar fuzzy set in a crisp approximation space is defined. Definition 1.18 Let (U, M) be a crisp approximation space. For X ∈ mF U , the parameter-free rough accuracy degree of X about (U, M) is defined as γ (X ) =

m 

m(X i )γ (X i ),

i=1

where X is a normal m−polar fuzzy set. If X is not a normal m−polar fuzzy set, then m  m(X i )γ (X i ) + m(∅)γ (∅). γ (X ) = i=1

Remark 1.3 For any normal m−polar fuzzy set X ∈ mF U in (U, M), γ (X ) = 1 − ρ(X ). The parameter related rough degree and parameter-free rough degree are different. The difference between them is investigated through the following example. Example 1.5 Let U = {c1 , c2 , c3 , c4 , c5 , c6 } be a set of six cars under consideration, and let “q1 =Price of the car” be the attribute related to the universe U . Three further characteristics of the attribute are described as follows: • The “Price of the car” may be cheap, costly, or very costly for the buyers. Consider a 3−polar fuzzy evaluation for each car is defined in Table 1.2. Let (U, M) be a crisp approximation space where U = {c1 , c2 , c3 , c4 , c5 , c6 } and M an equivalence relation such that   U/M = {c1 , c3 }, {c2 , c4 , c5 }, {c6 } . The lower and upper approximations of d are determined in Table 1.3. From Definition 1.13, (σ, τ )-related rough degree of d is given by ρ

σ,τ

 (d) =

1 0.67

f or σ ≥ (0.6, 0.6, 0.6) , f or σ = (0.43, 0.43, 0.43)

14

1 Hybrid Multi-polar Fuzzy Models

Table 1.2 3−polar fuzzy evaluation of each car based on price U c1 c2 d U d

(1.0,0.3,0.0) c4 (0.4,0.7,0.7)

(0.8,0.2,0.3) c5 (0.0,1.0,1.0)

c3 (0.4,0.7,0.2) c6 (1.0,1.0,1.0)

where it is always assumed that σ ≥ τ > (0, 0, . . . , 0). From Definition 1.15, mediation mass assignment for d, and the approximations of its focal elements are computed in Table 1.4. Using Definition 1.18, parameter-free accuracy degree of d is given by: ρ(d) =

 |M∗ (di )|  , m(di ) 1 − |M ∗ (di )| i=1

m 

= 0.33 × 1 + 0.07 × 1 + 0.17 × 1 + 0.06 × 0.67, = 0.6102. Now, it is clear from Example 1.5 that (σ, τ )-related rough degree in Definition 1.13 depends on σ and τ , and the rough degree in Definition 1.16 does not depend on parameters.   Definition 1.19 Let U, C, {Ci }, D, {D j } be an m−polar fuzzy objective decision   information system, where U, C, {Ci } is an information system, and D j is the set of m−polar decision values for attributes d j , where d j ∈ D, j =  1, 2, . . . , n. If the con- ditional attribute values {Ci } are m−polar fuzzy numbers, then U, C, {Ci }, D, {D j } is called an m−polar fuzzy objective decision information system.   Definition 1.20 Let U, C, {Ci }, D, {D j } be an m−polar fuzzy objective decision information system and B ⊂ C, then B generates a crisp approximation space (U, M B ). Decision attributes set {d1 , . . . , dn } on U can be seen as an m−polar fuzzy partition P B = {D1 , . . . , Dn } of S. Approximation of P B about M B is denoted by η(P B) and is defined as Table 1.3 Approximations of the 3−polar fuzzy set d about (U, M) U/M {c1 , c5 } {c2 , c4 } M∗ (d) M ∗ (d)

(0.0,0.0,0.0) (1.0,1.0,1.0)

(0.4,0.2,0.2) (0.7,0.7,0.7)

{c3 , c6 } (0.4,0.7,0.2) (1.0,1.0,1.0)

1.3 Rough m−Polar Fuzzy Sets

15

Table 1.4 Mediation mass assignment and its focal elements approximations dA 1 0.67 0.6 0.43 di m(di ) M∗ (di ) M ∗ (di )

{c6 } 0.33 ∅ {c3 , c6 }

{c5 , c6 } 0.07 ∅ {c1 , c3 , c5 , c6 }

1  m(D kj )|(M∗ ) B (D kj )|, |U | j=1 K =1 n

η(P B) =

{c4 , c5 , c6 } 0.17 ∅ {c1 , . . . , c6 }

{c3 , c4 , c5 , c6 } 0.06 {c3 , c6 } {c1 , . . . , c6 }

l

(1.1)

where m(D kj ) and D kj represents the mediation mass assignment of D j and its focal elements, respectively.   Definition 1.21 Let U, C, {Ci }, D, {D j } be an m−polar fuzzy objective decision information system, if U/MC ≤ U/M D , i.e., for every [v]C , there exists [v] D , such that [v]C ⊆ [v] D , then the m−polar fuzzy objective decision information system is called consistent . If U/MC ≤ U/M D , i.e., for every [v]C , there exists [v] D , such that [v]C  [v] D , then the m−polar fuzzy objective decision information system is called inconsistent. In the next subsections, some applications of the rough m−polar fuzzy model to MCDM are described.

1.3.1 Selection of Flats [4] Nowadays, the selection of suitable flats for investment and personal use is a MCDM problem that has great importance for the buyers. It is a very complicated decision due to high cost of reconfiguration and relocation. There are a number of factors that need to be taken into account when purchasing a flat such as location of the flat and size of the flat. Such factors among many others influence house buyers before they even get to start thinking about purchasing a new flat, because the location and size are two things about a property which cannot really be altered. Suppose a buyer (Mr. Tabish) wants to find two suitable flats to buy simultaneously. One is for living in and the other as an investment in the city Z . Mr. Tabish has 10 alternatives in the city Z . The alternatives are f 1 , f 2 , . . . , f 10 . Let U = { f 1 , f 2 , . . . , f 10 } be the set of flats, and let C = {c1 , c2 } be the set of attributes related to the flats in U , where, • c1 stands for the location of the flat. • c2 stands for the size of the flat.

16

1 Hybrid Multi-polar Fuzzy Models

Table 1.5 3−polar fuzzy objective decision information system U c1 c2 d1 f1 f2 f3 f4 f5 f6 f7 f8 f9 f 10

3 3 2 1 1 3 1 2 2 2

3 1 3 1 2 3 2 3 3 1

(0.2,0.6,0.7) (0.3,0.6,0.0) (0.1,0.9,0.7) (0.5,0.0,0.3) (0.4,0.1,0.9) (1.0,0.2,0.6) (0.1,0.1,0.4) (0.3,0.2,1.0) (0.9,0.1,0.5) (0.7,0.3,0.2)

Table 1.6 Approximations of the 3−polar fuzzy partition U/M { f1 , f6 } { f2 } M∗ (D1 ) M∗ (D2 ) U/M M∗ (D1 ) M∗ (D2 )

(0.2,0.2,0.6) (0.8,0.0,0.0) { f4 } (0.5,0.0,0.3) (0.4,1.0,0.4)

(0.3,0.6,0.0) (0.6,0.2,0.7) { f5 , f7 } (0.1,0.1,0.4) (0.4,0.9,0.5)

d2 (1.0,0.2,0.4) (0.6,0.2,0.7) (0.0,0.0,0.0) (0.4,1.0,0.4) (0.7,1.0,0.5) (0.8,0.0,0.0) (0.4,0.9,0.6) (0.9,0.5,0.1) (0.4,0.3,0.9) (0.8,0.4,0.9)

{ f3 , f8 , f9 } (0.1,0.1,0.5) (0.0,0.0,0.0) { f 10 } (0.7,0.3,0.2) (0.8,0.4,0.9)

These attributes are characterized into further more characteristics, which are represented as • The “Location” of the flat include, close to workplace, close to the city center and near to main road. • The “Size” of the flat may be, small, large, very large. Let D = {d1 , d2 } be the set of decision attributes, where, • d1 represents to the flat to live in. • d2 represents to the flat to invest. Now all given information on these flats under consideration can be described as a consistent 3−polar fuzzy objective decision information system, which is given in Table 1.5. Thus, Table 1.5 describes the rough 3−polar fuzzy sets in which location and size of the flats are considered. For example, if the “Location of the flat”, is considered means that the flat f 1 is 20% near to workplace, 60% near to city then (0.2,0.6,0.7) f1 center and 70% near to main road. Approximations of the 3−polar fuzzy partition generated by D in (U, ind(c1 , c2 )) are given in Table 1.6. Mediation mass assignment of D j ( j = 1, 2) and corresponding approximations of focal elements are computed in Tables 1.7 and 1.8, respectively.

1.3 Rough m−Polar Fuzzy Sets

17

Table 1.7 Mediation mass assignment of D1 and its approximations D1A

0.4

0.33

0.3

0.27

0.23

m(D1l )

0.07

0.03

0.03

0.04

0.03

M∗ (D1l )

{ f 10 }

{ f 1 , f 6 , f 10 }

{ f 1 , f 2 , f 6 , f 10 }

{ f 1 , f 2 , f 4 , f 6 , f 10 }

{ f 1 , f 2 , f 3 , f 4 , f 6 , f 8 , f 9 , f 10 }

Table 1.8 Mediation mass assignment of D2 and its approximations D2A

0.7

0.6

0.5

0.47

0.26

m(D2l )

0.1

0.1

0.03

0.21

0.26

M∗ (D2l )

{ f 10 }

{ f 5 , f 7 , f 10 }

{ f 2 , f 5 , f 7 , f 10 }

{ f 2 , f 4 , f 5 , f 7 , f 10 }

{ f 1 , f 2 , f 4 , f 5 , f 6 , f 7 , f 10 }

Table 1.9 Approximations of the 3−polar fuzzy partition by D in (U, ind(c1 )) U/M { f1 , f2 , f6 } { f4 , f5 , f7 } { f 3 , f 8 , f 9 , f 10 } M∗ (D1 ) M∗ (D2 )

(0.2,0.2,0.0) (0.6,0.0,0.0)

(0.1,0.0,0.3) (0.4,0.9,0.4)

(0.1,0.1,0.2) (0.0,0.0,0.0)

η(PC) = ηC={c1 ,c2 } (D) 1 0.07 × 1 + 0.03 × 3 + 0.03 × 4 + 0.04 × 5 + 0.03 × 8 + 0.1 × 1+ = 10  0.1 × 3 + 0.03 × 4 + 0.21 × 5 + 0.26 × 7 ,

= 0.411. From Tables 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, 1.13 and 1.14, compute η(P Bi )(i = 1, 2), η(P B1 ) = η B1 ={c2 } (D) = 0.304, η(P B2 ) = η B2 ={c1 } (D) = 0.24. Set β = 0.6. From ξi =

η(P Bi ) , η(PC)

(i = 1, 2),

ξ1 = 0.7397 ≥ β, ξ2 = 0.58 ≤ β. This implies reduced condition attribute set is B = {c2 }.

18

1 Hybrid Multi-polar Fuzzy Models

Table 1.10 Approximations of the 3−polar fuzzy partition generated by D in (U, ind(c2 )) U/M { f1 , f3 , f6 , f8 , f9 } { f 2 , f 4 , f 10 } { f5 , f7 } M∗ (D1 ) M∗ (D2 )

(0.1,0.1,0.5) (0.0,0.0,0.0)

(0.3,0.0,0.0) (0.4,0.2,0.4)

(0.1,0.1,0.4) (0.4,0.9,0.5)

Table 1.11 Mediation mass assignment of D1 and its approximations in (U, ind(c1 )) D1A 0.17 0.13 m(D1l ) M∗ (D1l )

0.04 { f 3 , f 8 , f 9 , f 10 }

0 { f 1 , f 2 , f 3 , f 6 , f 8 , f 9 , f 10 }

Table 1.12 Mediation mass assignment of D2 and its approximations in (U, ind(c1 )) D2A 0.57 0.2 m(D2l ) M∗ (D2l )

0.37 { f4 , f5 , f7 }

0.2 { f1 , f2 , f4 , f5 , f6 , f7 }

Table 1.13 Mediation mass assignment of D1 and its approximations in (U, ind(c2 )) D1A 0.23 0.2 m(D1l ) M∗ (D1l )

0.03 { f1 , f3 , f6 , f8 , f9 }

0.1 { f1 , f3 , f5 , f6 , f7 , f8 , f9 }

Using Table 1.6, the indiscernible classes of U are determined for each attribute represented by the Tables 1.9 and 1.10, respectively. The decision rules based on B = {c2 } are described from Table 1.10 as follows: 1. 2. 3. 4. 5. 6.

If c1 If c1 If c1 If c1 If c1 If c1

= 3, c2 = 1, c2 = 2, c2 = 2, c2 = 2, c2 = 3, c2

= 1, then buy the flat to invest. = 1, then buy the flat to invest. = 1, then buy the flat to invest. = 2, then buy the flat to invest. = 3, then buy the flat to live in. = 3, then buy the flat to live in.

Mr. Tabish will buy the flats f 1 , f 3 , f 6 , f 8 , f 9 to live in and will buy the flats f 2 , f 4 , f 5 , f 7 , f 10 for investment. Table 1.14 Mediation mass assignment of D2 and its approximations in (U, ind(c2 )) D2A 0.57 0.33 m(D2l ) M∗ (D2l )

0.24 { f5 , f7 }

0.33 { f 2 , f 4 , f 5 , f 7 , f 10 }

1.3 Rough m−Polar Fuzzy Sets

19

Table 1.15 3−polar fuzzy objective decision information system U r1 r2 d1 e1 e2 e3 e4 e5 e6

2 3 2 1 2 3

3 3 2 1 3 2

(0.7,0.5,0.2) (0.3,0.3,0.2) (0.5,0.1,0.9) (0.2,0.6,0.3) (1.0,0.1,0.9) (0.9,0.9,0.2)

d2 (0.2,0.0,0.7) (0.4,0.6,0.7) (0.8,0.2,1.0) (0.1,0.0,0.2) (0.6,1.0,0.1) (1.0,1.0,1.0)

1.3.2 Selection of Employees for Promotion and Bonus The selection of employees for promotion and bonus is a MCDM problem. Assume that a state department holds a meeting with higher officials of the concerned ministry for the selection of employees to promote and bonus, technically called departmental promotion committee. As per rules of promotion, 5E, i.e., There are two selectors, one is representative of the director general of the department and the other represents the ministry under which department is functioning. The selection is made on the basis of evaluation reports submitted by the immediate bosses of the employees under consideration. Suppose that the set of employees under consideration is U = {e1 , e2 , e3 , e4 , e5 , e6 } and the set of attributes related to employees characteristics is Q = {r1 , r2 }, where • r1 stands for the Personal characteristics. • r2 stands for the Business characteristics. These attributes are characterized into further more characteristics, which are represented as • The “personal characteristics” of the employee include, self motivation, good personality, confidence. • The “business characteristics” of the employee include, good knowledge of financial matters, leadership qualities, fluency in English language. • d1 represents to the employee selected for bonus. • d2 represents to the employee to promote. Now all given information can be described as a consistent 3−polar fuzzy objective decision information system is given in Table 1.15. Thus, Table 1.15 describes the rough 3−polar fuzzy sets in which personal and business characteristics of the employees are considered. For example, if the “Permeans that the employee e1 has, sonal Characteristics” is considered then, (0.7,0.5,0.2) e1 50% self motivation, 20% good personality and 10% confidence. Approximations of the 3−polar fuzzy partition generated by D in (U, ind(r1 , r2 )) are given in Table 1.16.

20

1 Hybrid Multi-polar Fuzzy Models

Table 1.16 Approximations of the 3−polar fuzzy partition generated by D in (U, ind(r1 , r2 )) U/M {e1 , e5 } {e2 } {e3 } {e4 } {e6 } M∗ (D1 ) M∗ (D2 )

(0.7,0.5,0.2) (0.2,0.0,0.1)

(0.3,0.3,0.2) (0.4,0.6,0.7)

(0.5,0.1,0.9) (0.8,0.2,1.0)

(0.2,0.6,0.3) (0.1,0.0,0.2)

Table 1.17 Mediation mass assignment of D1 and its approximations D1A 0.67 0.5 0.47 m(D1l ) M∗ (D1l )

0.17 {e6 }

0.03 {e3 , e6 }

0.1 {e1 , e3 , e5 , e6 }

Table 1.18 Mediation mass assignment of D2 and its approximations D2A 1 0.67 0.57 m(D2l ) M∗ (D2l )

0.33 {e6 }

0.1 {e3 , e6 }

0.47 {e2 , e3 , e6 }

(0.9,0.9,0.2) (1.0,1.0,1.0)

0.37 0.2 {e1 , e3 , e4 , e5 , e6 }

0.1 0 {e1 , e2 , e3 , e5 , e6 }

Table 1.19 Approximations of the 3−polar fuzzy partition generated by D in (U, ind(r1 )) U/M {e1 , e3 , e5 } {e4 } {e2 , e6 } M∗ (D1 ) M∗ (D2 )

(0.5,0.1,0.2) (0.2,0.0,0.1)

(0.2,0.6,0.3) (0.3,0.0,0.2)

(0.3,0.3,0.2) (0.4,0.6,0.7)

Mediation mass assignment of D j ( j = 1, 2) and corresponding approximations of focal elements are computed in Tables 1.17 and 1.18, respectively.

η(P Q) = η Q={r1 ,r2 } (D), 1 = 0.17 × 1 + 0.033 × 2 + 0.1 × 4 + 0.2 × 5 + 0.33 × 1 8  + 0.1 × 2 + 0.47 × 3 , = 0.446.

Table 1.20 Approximations of the 3−polar fuzzy partition generated by D in (U, ind(r2 )) U/M {e1 , e2 , e5 } {e3 , e6 } {e4 } M∗ (D1 ) M∗ (D2 )

(0.3,0.3,0.2) (0.2,0.0,0.1)

(0.5,0.1,0.2) (0.8,0.2,1.0)

(0.2,0.6,0.3) (0.1,0.0,0.2)

1.3 Rough m−Polar Fuzzy Sets

21

From Tables 1.21, 1.22, 1.23 and 1.24, we compute η(P Bi )(i = 1, 2), η(P B1 ) = η B1 ={r2 } (D) = 0.155, η(P B2 ) = η B2 ={r1 } (D) = 0.139. Set β = 0.3. From ξi =

η(P Bi ) , η(P Q)

(i = 1, 2),

ξ1 = 0.347 ≥ β, ξ2 = 0.3 ≤ β. Thus, the reduced attribute set is B = {r2 }. Using Table 1.16, the indiscernible classes of U are determined for each attribute represented by the Tables 1.19 and 1.20, respectively. The decision rules based on B = {r2 } are described from Table 1.20 as follows: 1. 2. 3. 4. 5.

If r1 If r1 If r1 If r1 If r1

= 1, r2 = 2, r2 = 3, r2 = 3, r2 = 2, r2

= 1, then employee is selected for bonus. = 2, then employee is promoted to next grade. = 2, then employee is promoted to next grade. = 3, then employee is selected for bonus. = 3, then employee is selected for bonus.

Thus, the departmental promotion committee will promote employees e3 , e6 to next grade and will select employees e1 , e2 , e4 , e5 for bonus. The method for selection of best objects under rough m–polar fuzzy model is described in Algorithm 1.1. Algorithm 1.1 Rough m–polar fuzzy sets 1. Input U as universe of discourse. 2. Input different features of universe U . 3. Consider a consistent 3−polar fuzzy objective decision information system in tabular form. 4. Compute approximations of the 3−polar fuzzy partition generated by D in (U, ind(c1 , c2 )) where ind(c1 , c2 ) represents the equivalence relation based on c1 and c2 . 5. Compute the mediation mass assignment of D1 , D2 and approximations of corresponding focal elements in (U, ind(c1 , c2 )). 6. Determine the quality of approximation η(P Q) of P Q by M Q . 7. Let Bi = Q − {ri }, determine η(P Bi ) of P Bi by M Bi (i = 1, 2). 8. For β ∈ [0, 1], determine η(P Bi ) , i = 1, 2. ξi = η(P Q) If ξi ≥ β, then remove ri from Q. Remove the condition attributes which are not necessary and obtain a new reduced attribute set B. 9. Compute the equivalence classes of U from the reduced attribute set B.

22

1 Hybrid Multi-polar Fuzzy Models

Table 1.21 Mediation mass assignment of D1 and its approximations in (U, ind(r1 )) D1A 0.37 0.27 m(D1l ) M∗ (D1l )

0.1 {e4 }

0 {e1 , e3 , e4 , e5 }

Table 1.22 Mediation mass assignment of D2 and its approximations in (U, ind(r1 )) D2A 0.57 0.17 m(D2l ) M∗ (D2l )

0.4 {e2 , e6 }

0.07 {e2 , e4 , e6 }

Table 1.23 Mediation mass assignment of D1 and it approximations in (U, ind(r2 )) D1A 0.37 0.27 m(D1l ) M∗ (D1l )

0.1 {e4 }

0 {e1 , e2 , e4 , e5 }

Table 1.24 Mediation mass assignment of D2 and its approximations in (U, ind(r2 )) D1A 0.67 0.1 m(D1l ) M∗ (D1l )

0.57 {e3 , e6 }

0 {e1 , e2 , e3 , e5 , e6 }

10. For every u ∈ U , there exists l ≤ n satisfying for every j = l, (M∗ ) B (Dl )(u) > (M∗ ) B (D j )(u). Then, the decision can be determined as ∀ u ∈ U if v ∈ [u] B , then the decision of v is dl .

1.4 m–Polar Fuzzy Soft Sets Molodtsov [43] has developed the concept of soft sets. Soft sets play a substantial role in handling uncertain data and information. Let U be a universe of discourse and T be the set of parameters. The parameters are typically the attributes or characteristics of elements in U . The pair (U, T ) is called a soft universe and P(U ) denotes the power set of U . Definition 1.22 A pair (F, A) is called a soft set over U , where A ⊆ T and F is a mapping given by F : A → P(U ).

1.4 m–Polar Fuzzy Soft Sets

23

In other words, a soft set over U is a parameterized family of subsets of the universe U . For ∈ A, F( ) may be considered as the set of -approximate elements of the soft set (F, A). Definition 1.23 ([36]) A pair (F, A) is called a fuzzy soft set over U , where F is a mapping given by F : A → P(U ) from A into P(U ), P(U ) = I U = ([0, 1])U = The collection of all fuzzy soft sets on U. In general, for every ∈ A, F( ) is a fuzzy set of U and it is called fuzzy value set of parameter . The set of all fuzzy soft sets over U with parameters from T is called a fuzzy soft class. Akram and Waseem [3] introduced the concept of m−polar fuzzy soft sets. Definition 1.24 ([3]) Let U be a universe, T a set of parameters and N ⊆ T . Define ζ : N → mF U . Then (ζ, N ) is called an m−polar fuzzy soft set over a universe U , which is defined by, (ζ, N ) =



x, pi ◦ N (x) : x ∈ U and ∈ N . 

Example 1.6 Let U = {b1 , b2 , b3 , b4 , b5 } be the set of five bungalows, and let T = {t1 , t2 , t3 , t4 } be the set of parameters, where the parameter, ‘t1 ’ stands for the Location of Bungalow, ‘t2 ’ stands for the Price of Bungalow, ‘t3 ’ stands for the Architecture of Bungalow, ‘t4 ’ stands for the Beauty of Bungalow. The additional attributes of these parameters are described as follows: 1. The parameter “Location of Bungalow” relies upon centrality, neighborhood and commercial development. 2. The “Price of Bungalow” might be expensive, exorbitant and cheap for the buyer. 3. The “Architecture of Bungalow” might be modern, contemporary and colonial. 4. The “Beauty of Bungalow” is dictated by the landscape, furniture and wall decorations. Suppose that a family wants to buy a bungalow of U . They consider three parameters t2 , t3 , t4 for the selection of a bungalow. Let N = {t2 , t3 , t4 } be subset of T . At that point all accessible data on these bungalows viable can be detailed as a 3−polar fuzzy soft set (ζ, N ). (ζ, N ) = ⎧  (0.7, 0.5, 0.3) (0.6, 0.8, 0.5) (0.6, 0.4, 0.7) (0.5, 0.6, 0.3) (0.7, 0.6, 0.4)  ⎪ ζ (t2 ) = , , , , , ⎪ ⎪ b1 b2 b3 b4 b5 ⎨  

⎫ ⎪ ⎪ ⎪ ⎬

b1 b2 b3 b4 b5 ⎪   ⎪ ⎪ ⎩ζ (t4 ) = (0.8, 0.6, 0.4) , (0.6, 0.5, 0.5) , (0.4, 0.3, 0.5) , (0.6, 0.6, 0.5) , (0.7, 0.6, 0.7) .

⎪ ⎪ ⎪ ⎭

ζ (t3 ) =

(0.9, 0.6, 0.7) (0.6, 0.5, 0.6) (0.5, 0.6, 0.7) (0.7, 0.8, 0.6) (0.4, 0.3, 0.6) , , , , ,

b1

b2

b3

b4

b5

24

1 Hybrid Multi-polar Fuzzy Models

Thus, (ζ, N ) is a 3−polar fuzzy soft set in which the Price, Architecture and Beauty of the Bungalow are chosen as desired parameters for the selection of Bungalow. For example, if the parameter “Beauty of Bungalow” is considered, then (0.6,0.5,0.5) b2 shows that according to the family bungalow b2 has 60% beautiful landscape, 50% stylish furniture and 50% good wall decorations. Definition 1.25 An m−polar fuzzy soft class is the collection of all m−polar fuzzy soft sets on U with attributes from T. An m−polar fuzzy soft set (ζ, N ) over U is said to be a null m−polar fuzzy soft set, denoted by ∅, if ∀ t ∈ N , ζ (t) is the null m−polar fuzzy set 0 of U , where 0(u) = (0, 0, . . . , 0), ∀ u ∈ U. An m−polar fuzzy soft set (ζ, N ) over U is said to be an absolute m−polar fuzzy soft set, denoted by ω, if ∀ t ∈ N , ζ (t) is the absolute m−polar fuzzy set 1 of U , where 1(u) = (1, 1, . . . , 1), ∀ u ∈ U. Definition 1.26 Let (ζ, N ) and ( , O) be two m−polar fuzzy soft sets over the same universe U . Then (ζ, N ) is said to be an m−polar fuzzy soft subset of ( , O), denoted by (ζ, N )  ( , O), if 1. N ⊆ S, 2. For any t ∈ N , ζ (t) ⊆ (t). Obviously, (ζ, N ) = ( , O) if (ζ, N )  ( , O) and ( , O)  (ζ, N ). Definition 1.27 The union of two m−polar fuzzy soft sets (ζ, N ) and ( , O) over the universe U is an m−polar fuzzy soft set (υ, P) = (ζ, N )  ( , O), where P = N ∪ O and ∀ t ∈ P, ⎧ if t ∈ N \ O, ⎨ ζ (t) if t ∈ O \ N , υ(t) = (t) ⎩ ζ (t) ∪ (t) if t ∈ N ∩ O. Definition 1.28 The intersection of two m−polar fuzzy soft sets (ζ, N ) and ( , O) over the universe U is an m−polar fuzzy soft set (υ, P) = (ζ, N )  ( , O), where P = N ∩ O = ∅ and υ : P → mF U is defined by υ(t) = ζ (t) ∩ (t), ∀ t ∈ P. Proposition 1.3 Let (ζ, N ) be an m−polar fuzzy soft set over a universe U. Then 1. 2. 3. 4.

(ζ, N )  (ζ, N ) = (ζ, N ). (ζ, N )  (ζ, N ) = (ζ, N ). (ζ, N )  ∅ = (ζ, N ), where ∅ is a null m−polar fuzzy soft set. (ζ, N )  ∅ = ∅, where ∅ is a null m−polar fuzzy soft set.

Proof Proof is obvious. Lemma 1.1 Absorption property of m−polar fuzzy soft sets (ζ, N ) and ( , O). 1. (ζ, N )  ((ζ, N )  ( , O)) = (ζ, N ). 2. (ζ, N )  ((ζ, N )  ( , O)) = (ζ, N ).



1.4 m–Polar Fuzzy Soft Sets

25

1. An intersection of two m−polar fuzzy soft sets (ζ, N ) and ( , O) is (υ, P), where P = N ∩ O. (υ, P) = (ζ, N )  ( , O), where P = N ∩ O.

(1.2)

υ(t) = ζ (t) ∩ (t), if t ∈ P = N ∩ O. Let m−polar fuzzy soft set (ψ, V ) be the union of two m−polar fuzzy soft sets (ζ, N ) and (υ, P) which is (ψ, V ) = (ζ, N )  (υ, P), where V = N ∪ P. It is defined

(1.3)

⎧ if t ∈ N \ P, ⎨ ζ (t) if t ∈ P \ N , ψ(t) = υ(t) ⎩ ζ (t) ∪ υ(t) if t ∈ N ∩ P.

Thus, there are the following three cases: Case(i): If t ∈ N \ P, then ψ(t) = ζ (t), if t ∈ N \ P, = ζ (t), if t ∈ N , ψ(t) = ζ (t), (ψ, V ) = (ζ, N ), from (1.3). Case(ii): If t ∈ P \ N = N ∩ O − N = ∅, then ψ(t) = υ(t), if t ∈ P \ N = ∅, = ∅, if t ∈ ∅, ψ(t) = ∅, if t ∈ ∅, (ψ, V ) = ∅, from (1.3). Case(iii): If t ∈ N ∩ P, then ψ(t) = ζ (t) ∪ υ(t), if t ∈ N ∩ P and P = N ∩ O, = ζ (t) ∪ (ζ (t) ∩ (t), from (1.2), = ζ (t), since (ζ (t) ∩ (t)) ⊂ ζ (t), ψ(t) = ζ (t), (ψ, V ) = (ζ, N ), from (1.3). Proof All cases are satisfied. Hence, (ζ, N )  ((ζ, N )  ( , O)) = (ζ, N ).



26

1 Hybrid Multi-polar Fuzzy Models

Theorem 1.4 Commutative property of m−polar fuzzy soft sets (ζ, N ) and ( , O). 1. (ζ, N )  ( , O) = ( , O)  (ζ, N ). 2. (ζ, N )  ( , O) = ( , O)  (ζ, N ). Proof Proof is obvious.



Theorem 1.5 Associative law of m−polar fuzzy soft sets (ζ, N ), ( , O) and (υ, P). 1. (ζ, N )  (( , O)  (υ, P)) = ((ζ, N )  ( , O))  (υ, P). 2. (ζ, N )  (( , O)  (υ, P)) = ((ζ, N )  ( , O))  (υ, P). Proof Proof is obvious.



Theorem 1.6 Distributive law over m−polar fuzzy soft sets (ζ, N ), ( , O) and (υ, P). 1. (ζ, N )  (( , O)  (υ, P)) = ((ζ, N )  ( , O))  ((ζ, N )  (υ, P)). 2. (ζ, N )  (( , O)  (υ, P)) = ((ζ, N )  ( , O))  ((ζ, N )  (υ, P)). Proof Its proof is similar to Lemma 1.1.



Lemma 1.2 Let (ζ, N ) and ( , O) be two m−polar fuzzy soft sets. 1. (ζ, N ) ⊂ ( , O) ⇒ (ζ, N )  ( , O) = (ζ, N ). 2. (ζ, N ) ⊂ ( , O) ⇒ (ζ, N )  ( , O) = ( , O). Proof Proof is obvious.



Definition 1.29 The AND operation of two m−polar fuzzy soft sets (ζ, N ) and ( , O) over a common universe U is denoted by (ζ, N )  ( , O) and is defined by (ζ, N )  ( , O) = (υ, N × O), where υ(t j , tk ) = ζ (t j ) ∩ (tk ) for all (t j , tk ) ∈ P = N × O, where ∩ is the intersection operation of m−polar fuzzy sets. Definition 1.30 The OR operation of two m−polar fuzzy soft sets (ζ, N ) and ( , O) over a common universe U is denoted by (ζ, N )  ( , O) and is defined by (ζ, N )  ( , O) = (υ, N × O), where υ(t j , tk ) = ζ (t j ) ∪ (tk ) for all (t j , tk ) ∈ P = N × O, where ∪ is the union operation of m−polar fuzzy sets. Proposition 1.4 Idempotent property of two m−polar fuzzy soft sets (ζ, N ) and ( , O). 1. (ζ, N )  (ζ, N ) = (ζ, N ), 2. (ζ, N )  (ζ, N ) = (ζ, N ). Proof Proof is obvious.



In next subsection, the applications of m−polar fuzzy soft model to MCDM problems are described.

1.4 m–Polar Fuzzy Soft Sets

27

1.4.1 Selection of an Employee in an Organization The selection of an employee in an organization is the job of human resource department. Depending on the nature of job, suitable candidates are selected out of many aspirants. It is very important to make sure that the candidate will fit in with your organization’s culture and the team. Suppose a multi-national company XYZ wants to hire a manager. The human resource department wants to select the most capable, effective, suited, experienced and qualified, person for the post of manager. Suppose U = {c1 , c2 , c3 , c4 , c5 } is the set of candidates who appear for the interview and T = {t1 , t2 , t3 , t4 } is the set of parameters related to the candidates of U, where, ‘t1 ’ denotes the Personal Characteristics, ‘t2 ’ denotes the Business Characteristics, ‘t3 ’ denotes the Communication Qualities, ‘t4 ’ denotes the Relationship Qualities. The further characteristics of these parameters are described as follows: • The “Personal Characteristics” of a candidate include Self Motivation, Reliability, Confidence and Flexibility. • The “Business Characteristics” of a candidate include Industry Knowledge, Legal Implications, Basic Money Management and Business Hierarchy. • The “Communication Qualities” of a candidate include Written Communication, Public Speaking, Active Listening and Presentation Skills. • The “Relationship Qualities” of a candidate include Mediator, Team Player, Collaborations and Customer Service. Suppose that the company XYZ wants to hire a manger on the basis of the parameter set N = {t1 , t2 , t3 }. Main aim is to find out the good manager for the company. Consider the 4−polar fuzzy soft set as (ζ, N ) = ⎧  (0.5, 0.6, 0.5, 0.4) (0.6, 0.7, 0.8, 0.7) (0.7, 0.7, 0.6, 0.7) ⎪ ⎪ ζ (t1 ) = , , , ⎪ ⎪ c c2 c3 ⎪ 1 ⎪  ⎪ (0.6, 0.5, 0.5, 0.4) (0.6, 0.6, 0.4, 0.5) ⎪ ⎪ , , ⎪ ⎪ ⎪ c4 c5 ⎪ ⎪  (0.7, 0.6, 0.6, 0.7) (0.8, 0.9, 0.7, 0.8) (0.6, 0.5, 0.8, 0.7) ⎪ ⎪ ⎨ζ (t2 ) = , , , c1 c2 c3  (0.5, 0.7, 0.6, 0.6) (0.6, 0.6, 0.7, 0.5) ⎪ ⎪ , , ⎪ ⎪ c4 c5 ⎪ ⎪ ⎪  (0.4, 0.6, 0.5, 0.5) (0.7, 0.8, 0.7, 0.6) (0.5, 0.7, 0.8, 0.7) ⎪ ⎪ ⎪ , , , ⎪ζ (t3 ) = ⎪ c1 c2 c3 ⎪ ⎪ ⎪ (0.6, 0.5, 0.6, 0.4) (0.6, 0.5, 0.7, 0.8)  ⎪ ⎩ . , c4 c5

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Thus (ζ, N ) is a 4−polar fuzzy soft set in which Personal Characteristics, Business Characteristics and Communication Qualities of the candidates are considered. For

28

1 Hybrid Multi-polar Fuzzy Models

example, if “Communication Qualities” is considered then, (0.4,0.6,0.5,0.5) means that c1 the candidate c1 has 40% written communication skills, 60% public speaking skills, 50% active listening skills and 50% presentation skills. The tabular representation of the 1st pole is given in Table 1.25. Now, the comparison table for the 1st pole is presented in Table 1.26 . The membership score for each candidate is given in Table 1.27, which is obtained by subtracting the column sum from the row sum of Table 1.26. Similarly, the remaining three poles are represented in tabular form and calculated the membership scores with the help of comparison tables for each pole, respectively (see Tables 1.28, 1.29, 1.30, 1.31, 1.32, 1.33, 1.34, 1.35 and 1.36). The final score for each candidate is displayed in Table 1.37 and is obtained by adding the membership scores of all four poles. It is evident from the calculations that the maximum score is 32 scored by c2 . Hence, the company X Y Z will hire c2 for the post of manager as he is the most suitable candidate for this post.

1.4.2 Selection of Suitable Site for a Resort Selecting a resort location is a MCDM problem and is of great importance for the resort management. It is a very critical decision due to the high cost of reconfiguration and relocation. A tourist company has decided to build a new tourist resort in the city Y . The company has four resort site alternatives around the city Y . The alternatives are s1 , s2 , s3 and s4 . The company wants to select the best site for new resort. The

Table 1.25 Tabular representation of 1st pole p1 t1

c1 c2 c3 c4 c5

0.5 0.6 0.7 0.6 0.6

Table 1.26 Comparison table for the 1st pole . c1 c2

c1 c2 c3 c4 c5

3 3 2 2 2

0 3 1 1 1

t2

t3

0.7 0.8 0.6 0.5 0.6

0.4 0.7 0.5 0.6 0.6

c3

c4

c5

1 2 3 1 2

1 3 2 3 3

1 3 2 2 3

1.4 m–Polar Fuzzy Soft Sets

29

Table 1.27 Membership score table of 1st pole . Row sum (a) Column sum (b)

c1 c2 c3 c4 c5

6 14 10 9 11

c1 c2 c3 c4 c5

0.6 0.7 0.7 0.5 0.5

Table 1.29 Comparison table for the 2nd pole c1 c2

c1 c2 c3 c4 c5

3 3 2 1 1

−6 8 1 −3 0

12 6 9 12 11

Table 1.28 Tabular representation of 2nd pole p2 t1

0 3 1 0 0

t2

t3

0.6 0.9 0.5 0.7 0.6

0.6 0.8 0.7 0.5 0.5

c3

c4

c5

1 3 3 1 1

2 3 2 3 2

3 3 2 3 3

Table 1.30 Membership score table of 2nd pole . Row sum (c) Column sum (d)

c1 c2 c3 c4 c5

9 15 10 8 7

O1 = a − b

10 4 9 12 14

S2 = c − d −1 11 1 −4 −7

30

1 Hybrid Multi-polar Fuzzy Models

Table 1.31 Tabular representation of 3rd pole p3 t1

c1 c2 c3 c4 c5

0.5 0.8 0.6 0.5 0.6

Table 1.32 Comparison table for the 3rd pole . c1 c2

c1 c2 c3 c4 c5

3 3 3 3 3

0 3 2 0 2

t2

t3

0.6 0.7 0.8 0.6 0.7

0.5 0.7 0.8 0.6 0.7

c3

c4

c5

0 1 3 0 1

2 3 3 3 3

0 3 3 0 3

Table 1.33 Membership score table of 3rd pole . Row sum (e) Column sum ( f )

c1 c2 c3 c4 c5

5 13 14 6 12

Table 1.34 Tabular representation of 4th pole p4 t1

c1 c2 c3 c4 c5

0.4 0.7 0.7 0.4 0.5

15 7 5 14 9

S3 = e − f −10 6 9 −8 3

t2

t3

0.7 0.8 0.7 0.6 0.5

0.5 0.6 0.7 0.4 0.8

1.4 m–Polar Fuzzy Soft Sets

31

Table 1.35 Comparison table for the 4th pole . c1 c2

c1 c2 c3 c4 c5

3 3 3 1 2

0 3 2 0 1

c3

c4

c5

1 2 3 0 1

3 3 3 3 2

1 2 2 1 3

Table 1.36 Membership score table of 4th pole . Row sum (g) Column sum (h)

c1 c2 c3 c4 c5

8 13 13 5 9

Table 1.37 Final score table of candidates . S1 S2

c1 c2 c3 c4 c5

−6 8 1 −3 0

−1 11 1 −4 −7

S4 = g − h −44 7 6 −49 0

12 6 7 14 9

S3

S4

Final 4 score ( i=1 Si )

−10 6 9 −8 3

−4 7 6 −49 0

−21 32 17 −24 −4

geographical conditions, transportation facilities and operation management are the main parameters for the site selection of a resort. In geographical conditions of the site, the company wants to check whether the site has availability of resources, accessibility to other facilities such as hospitals, parks, etc., and is expandable for constructing additional buildings. It is very important that the location has good transportation facilities so that the tourists have accessibility to the airports, bus stands and other tourists attractions in the city. Lastly, the operation management is an important criterion for the site selection. It includes the land cost, labor cost and sufficient human resources. Let U = {s1 , s2 , s3 , s4 } be the set of resort site alternatives around the city Y and let T = {t1 , t2 , t3 } be the set of parameters related to the sites in U , where, ‘t1 ’ stands for the Geographical Conditions, ‘t2 ’ stands for the Transportation Facilities, ‘t3 ’ stands for the Operation Management. The further characteristics of these parameters are described as follows:

32

1 Hybrid Multi-polar Fuzzy Models

Table 1.38 1st pole’s tabular representation p1 t1

s1 s2 s3 s4

0.6 0.6 0.8 0.5

Table 1.39 1st pole’s comparison table . s1 s2

s1 s2 s3 s4

3 1 3 0

3 3 3 2

t2

t3

0.7 0.4 0.7 0.6

0.7 0.6 0.7 0.6

s3

s4

2 0 3 0

3 2 3 3

• The “Geographical Conditions” of the site include, proximity to public facilities, availability of natural resources and easily expandable. • The “Transportation Facilities” of the site include, accessibility to airport, accessibility to bus stands and accessibility to tourists attractions. • The “Operation Management” of the site include, land cost, labor cost and sufficient human resources. All information available on these sites under consideration can be formulated as a 3−polar fuzzy soft set (ζ, N ). (ζ, N ) = ⎧  (0.6, 0.5, 0.7) (0.6, 0.7, 0.5) (0.8, 0.7, 0.8) (0.5, 0.6, 0.7)  ⎪ ⎪ ζ (t1 ) = , , , , ⎪ ⎪ s1 s2 s3 s4 ⎪ ⎨   (0.7, 0.6, 0.6) (0.4, 0.5, 0.7) (0.7, 0.8, 0.7) (0.6, 0.7, 0.5) ζ (t2 ) = , , , , ⎪ s1 s2 s3 s4 ⎪   ⎪ ⎪ (0.7, 0.6, 0.5) (0.6, 0.7, 0.7) (0.7, 0.7, 0.9) (0.6, 0.7, 0.7) ⎪ ⎩ζ (t3 ) = . , , , s1 s2 s3 s4

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Thus, (ζ, N ) is a 3−polar fuzzy soft set in which geographical conditions, transportation facilities and operation management of the sites are considered. For example, if “Transportation Facilities” is considered then, (0.4,0.5,0.7) means that the site s2 has s2 40% accessibility to airport, 50% accessibility to bus stands and 70% accessibility to tourists attractions. The tabular representation of the 1st pole is given in Table 1.38. Now, the comparison table for the 1st pole, is displayed in Table 1.39. The membership score for each site is obtained by subtracting the column sum from the row sum of Table 1.39, which is given by Table 1.40.

1.4 m–Polar Fuzzy Soft Sets

33

Table 1.40 1st pole’s membership score table . Row sum (a)

s1 s2 s3 s4

11 6 12 5

Table 1.41 2nd pole’s tabular representation p2 t1

s1 s2 s3 s4

0.5 0.7 0.7 0.6

Column sum (b)

S1 = a − b

7 11 5 11

4 −5 7 −6

t2

t3

0.6 0.5 0.8 0.7

0.6 0.7 0.7 0.7

Table 1.42 2nd pole’s comparison table . s1 s2

s1 s2 s3 s4

3 2 3 3

1 3 3 2

Table 1.43 2nd pole’s membership score table . Row sum (c)

s1 s2 s3 s4

4 9 12 9

s3

s4

0 2 3 1

0 2 3 3

Column sum (d)

S2 = c − d

11 9 6 8

−7 0 6 1

Similarly, the remaining two poles are represented in tabular form and calculated the membership scores with the help of comparison tables for each pole, respectively (see Tables 1.41, 1.42, 1.43, 1.44, 1.45 and 1.46). The final score for each candidate is given in Table 1.47, which is obtained by adding the membership scores of all four poles. It is clear from the calculations that the maximum score is 21 scored by s3 . Thus, the resort management company will select the site s3 for the construction of a new resort. The method of selecting the best object under m–polar fuzzy soft information is described in the following Algorithm 1.2.

34

1 Hybrid Multi-polar Fuzzy Models

Table 1.44 3rd pole’s tabular representation p3 t1

s1 s2 s3 s4

0.7 0.5 0.8 0.7

t2

t3

0.6 0.7 0.7 0.5

0.5 0.7 0.9 0.7

Table 1.45 3rd pole’s comparison table . s1 s2

s1 s2 s3 s4

3 2 3 2

1 3 3 2

Table 1.46 3rd pole’s membership score table . Row sum (e)

s1 s2 s3 s4

6 8 12 7

Table 1.47 Final score table of resorts . S1 S2

s1 s2 s3 s4

4 −5 7 −6

−7 0 6 1

s3

s4

0 1 3 0

2 2 3 3

Column sum ( f )

S3 = e − f

10 9 4 10

−4 −1 8 −3

S3

Final 4 score ( i=1 Si )

−4 −1 8 −3

−7 −6 21 −8

Algorithm 1.2 m–polar fuzzy soft sets 1. 2. 3. 4. 5. 6.

Input the set N ⊆ T of parameters of the resort management company. Consider the m−polar fuzzy soft set in tabular form. Compute the comparison table of information function for all m−poles. Compute the information score for all m−poles. Compute the final score by adding the scores of all m−poles. Find the maximum score, if it occurs in jth row, then company will select the site s j , 1 ≤ j ≤ 4.

1.5 Similarity Measure for m−Polar Fuzzy Sets

35

1.5 Similarity Measure for m−Polar Fuzzy Sets Akram and Waseem [3] introduced distances between two m−polar fuzzy sets and similarity measure for m−polar fuzzy sets. Definition 1.31 ([3]) Let N and S be two m−polar fuzzy sets on U = {u1 , u2 , u3 , . . . , un }. Then the distance between N and S is defined as: 1. Hamming distance: 1 d H (N , S) = m

 m n       pi ◦ N (u j ) − pi ◦ S(u j ) . i=1

j=1

2. Normalized Hamming distance:  m n    1   d N H (N , S) =  pi ◦ N (u j ) − pi ◦ S(u j ) . mn i=1 j=1 3. Euclidean distance:

   m n  2  1 d E (N , S) =  . pi ◦ N (u j ) − pi ◦ S(u j ) m i=1 j=1

4. Normalized Euclidean distance:    n  m 2   1  d N E (N , S) = . pi ◦ N (u j ) − pi ◦ S(u j ) mn i=1 j=1 Theorem 1.7 The distances between N and S satisfy the following inequalities. 1. 2. 3. 4.

d H (N , S) ≤ n. d N H (N , S) ≤√1. d E (N , S) ≤ n. d N E (N , S) ≤ 1.

Theorem 1.8 The distance functions d H , d N H , d E , and d N E , defined from mF U → R + , are metric. Proof Let N , S and R be three m−polar fuzzy sets over U , then 1. d H (N , S) ≥ 0. 2. Suppose d H (N , S) = 0.



36

1 Hybrid Multi-polar Fuzzy Models

 m n  1     ⇔  pi ◦ N (u j ) − pi ◦ S(u j ) = 0, m i=1 j=1   ⇔  pi ◦ N (u j ) − pi ◦ S(u j ) = 0, ⇔ pi ◦ N (u j ) = pi ◦ S(u j ), for all 1 ≤ i ≤ m, 1 ≤ j ≤ n, ⇔ N = S. 3. d H (N , S) = d H (S, N ). 4. For any three m−polar fuzzy sets N , S and R,    pi ◦ N (u j ) − pi ◦ S (u j )   =  pi ◦ N (u j ) − pi ◦ R(u j ) + pi ◦ R(u j ) − pi ◦ S (u j ), for all i, j.     ≤  pi ◦ N (u j ) − pi ◦ R(u j ) +  pi ◦ R(u j ) − pi ◦ S (u j ), for all i, j.

Thus d H (N , S) ≤ d H (N , R) + d H (R, S). Definition 1.32 ([3]) The similarity measure of two m−polar fuzzy sets N and S is defined as 1 , S(N , S) = 1 + d(N , S) where d(N , S) is any of the above distances defined in Definition 1.31. Definition 1.33 The similarity measure of two m−polar fuzzy sets N and S is defined as S (N , S) = exp−βd(N ,S) , where β > 0 is called the steepness measure. Definition 1.34 ([3]) The two m−polar fuzzy sets N and S are β similar if and only if S(N , S) ≥ β, i.e., N ≈β S ⇔ S(N , S) ≥ β, β ∈ (0, 1). N and S are significantly similar if S(N , S) ≥ 21 . Theorem 1.9 The similarity measure of two m−polar fuzzy sets N and S satisfies the following. 1. 0 ≤ S(N , S) ≤ 1, 2. S(N , S) = S(S, N ), 3. S(N , S) = 1 ⇔ N = S. Definition 1.35 Let U be a universe, T a set of parameters and N ⊆ T . Define ψ : N → m F U , where m F U is the collection of all m−polar fuzzy subsets of U . Then (, N ) is called an m−polar fuzzy soft set over a universe U , which is defined by

1.5 Similarity Measure for m−Polar Fuzzy Sets

N = (, N ) =

37

  t, ψ N (t) : t ∈ T, ψN (t) ∈ m F U ,

and ψN (t) is an m−polar fuzzy set, denoted by ψN (t) =

  u, pi ◦ N (u) : u ∈ U .

Example 1.7 Let U = {o1 , o2 , o3 , o4 , o5 } be the set of five couches, and let T = {t1 , t2 , t3 , t4 } be the set of parameters, where the parameter ‘t1 ’ denotes the Fabric of Couch, ‘t2 ’ denotes the Style of Couch, ‘t3 ’ denotes the Frame of Couch, ‘t4 ’ denotes the Price of Couch. The further characteristics of these parameters are described as follows: • • • •

The “Fabric of Couch” may be leather, polyester and velvet. The “Style of Couch” may be modern, contemporary and sectional. The “Frame of Couch” may be of hardwood, particle board and metal. The “Price of Couch” may be very costly, costly and cheap for the buyer.

Suppose that a family wants to purchase a couch from U . They consider three parameters t1 , t2 , t4 for the selection of a couch. Let N = {t1 , t2 , t4 } be subset of T . Then we can formulate all possible information on these couches as a 3−polar fuzzy soft set (, N ). (, N ) = ⎧ ⎫  ⎪ ⎪ ψ( t ) = ( o , 7/10, 1/2, 3/10), ( o , 1/5, 1/2, 3/10), ( o , 3/5, 2/5, 7/10), ⎪ ⎪ 1 1 2 3 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( o , 1/2, 3/5, 3/10), ( o , 7/10, 3/5, 2/5) , ⎪ ⎪ 4 5 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ψ(t2 ) = (o1 , 9/10, 3/5, 7/10), (o2 , 3/5, 1/2, 2/5), (o3 , 1/2, 3/5, 7/10), ⎪ ⎬  . ⎪ ⎪ (o4 , 7/10, 7/10, 3/5), (c5 , 2/5, 3/10, 3/5) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ψ( t ) = ( o , 4/5, 3/5, 2/5), ( o , 3/5, 1/2, 3/5), ( o , 2/5, 3/10, 1/2), ⎪ ⎪ 4 1 2 3 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (o4 , 3/5, 2/5, 1/2), (o5 , 7/10, 3/5, 7/10) . Thus, (, N ) is a 3−polar fuzzy soft set in which we have chosen the Fabric, Style and Price of the Couch as desired parameters for the selection. For example, if the parameter “Fabric of Couch” is chosen then, (o2 , 0.2, 0.5, 0.3) shows that according to the family couch c2 has 20% leather, 50% polyester and 30% velvet fabric. Definition 1.36 Let U = {u1 , u2 , . . . , un } be a universe, T = {t1 , t2 , . . . , tq } a set of parameters, N , S ⊆ T and N , S two m−polar fuzzy soft sets on U with their m−polar fuzzy approximate functions ψN (t j ) =



  u, pi ◦ N (u) : u ∈ U ,

38

1 Hybrid Multi-polar Fuzzy Models

ωS (t j ) =



  u, pi ◦ S(u) : u ∈ U ,

respectively. Then the distance between N and S are defined as: 1. Hamming distance: d H (N , S ) =

1 mq

 q m n       p ◦ N ( t )( u ) − p ◦ S( t )( u )  i j k i j k  . i=1

j=1

k=1

2. Normalized Hamming distance: 1 d N H (N , S ) = mqn

 q n  m       pi ◦ N (t j )(uk ) − pi ◦ S(t j )(uk ) . i=1

j=1

k=1

3. Euclidean distance:    q n  m 2    1  d E (N , S ) = . pi ◦ N (t j )(uk ) − pi ◦ S(t j )(uk ) mq i=1 j=1 k=1 4. Normalized Euclidean distance:    q n  m 2    1 d N E (N , S ) =  . pi ◦ N (t j )(uk ) − pi ◦ S(t j )(uk ) mqn i=1 j=1 k=1 Example 1.8 Let U = {u1 , u2 , u3 , u4 , u5 } be the feature space, T = {t1 , t2 , t3 , t4 , t5 } a set of parameters and N = S = {t2 , t3 , t4 } ⊆ T . The 3−polar fuzzy soft sets (, N ) and (, S) over U are defined as follows: (, N ) = ⎧ ⎫  ⎪ ⎪ ψ( t ) = ( u , 7/10, 1/2, 3/10), ( u , 3/5, 4/5, 1/2), ( u , 3/5, 2/5, 7/10), ⎪ ⎪ 2 1 2 3 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( u , 1/2, 3/5, 3/10), ( u , 7/10, 3/5, 2/5) , ⎪ ⎪ 4 5 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ψ(t3 ) = (u1 , 9/10, 3/5, 7/10), (u2 , 3/5, 1/2, 3/5), (u3 , 1/2, 3/5, 7/10), ⎪ ⎬  , ⎪ ⎪ (u4 , 7/10, 4/5, 3/5), (u5 , 2/5, 3/10, 3/5) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ψ(t4 ) = (u1 , 4/5, 3/5, 2/5), (u2 , 3/5, 1/2, 1/2), (u3 , 2/5, 3/10, 1/2), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (u4 , 3/5, 3/5, 1/2), (u5 , 7/10, 3/5, 7/10) .

1.5 Similarity Measure for m−Polar Fuzzy Sets

39

(, S) = ⎧ ⎫  ⎪ ⎪ ω( t ) = ( u , 4/5, 4/5, 1/2), ( u , 3/5, 7/10, 7/10), ( u , 9/10, 3/10, 1/2), ⎪ ⎪ 2 1 2 3 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( u , 3/5, 3/5, 1/2), ( u , 3/5, 4/5, 2/5) , ⎪ ⎪ 4 5 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ω(t3 ) = (u1 , 4/5, 3/5, 1/2), (u2 , 9/10, 2/5, 1/2), (u3 , 1/2, 7/10, 4/5), ⎪ ⎬  . ⎪ ⎪ (u4 , 4/5, 9/10, 3/5), (u5 , 7/10, 1/5, 3/5) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ω( t ) = ( u , 9/10, 7/10, 1/2), ( u , 3/5, 1/2, 1/2), ( u , 1/2, 3/5, 1/2), ⎪ ⎪ 4 1 2 3 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (u4 , 3/5, 1/2, 0.7), (u5 , 1/2, 7/10, 7/10) . Then, by using Definition 1.36, the distance between N = (, N ) and S = (, S) can be calculated as 1. 2. 3. 4.

d H (N , S ) = 0.555, d N H (N , S ) = 0.1111, d E (N , S ) = 0.3266, d N E (N , S ) = 0.1461.

Theorem 1.10 The distances between (, N ) and (, S) satisfy the following inequalities. 1. 2. 3. 4.

d H (N , S ) ≤ n, d N H (N , S ) ≤√1, d E (N , S ) ≤ n, d N E (N , S ) ≤ 1.

Theorem 1.11 The distance functions d H , d N H , d E , and d N E , defined from mF U → R + , are metric. Proof Let N = (, N ), S = (, S) and R = (, R) be three m−polar fuzzy soft sets over U , then  1. d H (N , S ) ≥ 0. 2. Suppose d H (N , S ) = 0.  q n  m    1    pi ◦ N (t j )(uk ) − pi ◦ S(t j )(uk ) = 0, for all i, j, k, qm i=1 j=1 k=1   ⇔  pi ◦ N (t j )(uk ) − pi ◦ S(t j )(uk ) = 0, ⇔ pi ◦ N (t j )(uk ) = pi ◦ S(t j )(uk ),

⇔

⇔ N = S . 3. d H (N , S ) = d H (S , N ).

40

1 Hybrid Multi-polar Fuzzy Models

4. For any three m−polar fuzzy soft sets  N ,  S and  R ,    pi ◦ N (t j )(uk ) − pi ◦ S(t j )(uk )   =  pi ◦ N (t j )(uk ) − pi ◦ R(t j )(uk ) + pi ◦ R(t j )(uk ) − pi ◦ S(t j )(uk ), for all i, j, k.     ≤  pi ◦ N (t j )(uk ) − pi ◦ R(t j )(uk ) +  pi ◦ R(t j )(uk ) − pi ◦ S(t j )(uk ), for all i, j, k.

Thus, d H (N , R ) ≤ d H (N , S ) + d H (S , R ). Definition 1.37 The similarity measure of N and S is defined as S(N , S ) =

1 , 1 + d(N , S )

where d(N , S ) is any of the above distances defined in Definition 1.36. Definition 1.38 The similarity measure of N and S is also defined as S (N , S ) = exp−βd(N ,S ) , where β > 0 is called the steepness measure. Definition 1.39 The two m−polar fuzzy soft sets N and S are β similar if and only if S(N , S ) ≥ β, i.e., N ≈β S ⇔ S(N , S ) ≥ β, β ∈ (0, 1). N and S are significantly similar if S(N , S ) ≥ 21 . Example 1.9 Consider the two 3−polar fuzzy soft sets N and S as in Example 1.8. The similarity measure of N and S by using Euclidean distance is calculated as, 1 S(N , S ) = 0.7538 ≥ . 2 Hence N and S are significantly similar. Theorem 1.12 The similarity measure of N and S over U satisfies the following. 1. 0 ≤ S(N , S ) ≤ 1. 2. S(N , S ) = S(S , N ). 3. S(N , S ) = 1 ⇔ N = S . In next subsections, the applications of similarity measure for m−polar fuzzy sets are presented in order to classify hybrid rocks.

1.5 Similarity Measure for m−Polar Fuzzy Sets

41

1.5.1 Pattern Recognition Problem Suppose that there are four types of rock fields denoted by R1 , R2 , R3 and R4 . Let U = {u1 = Texture, u2 = Fracture, u3 = Grain Size, u4 = Crystalline Structure} be the feature space of rock fields. The feature “Texture” of the rock refers to the arrangement, shape and distribution of minerals in the rock. The minerals within the rock may have uneven, conchoidal or hackly “Fracture.” Rocks may have no visible grain, medium or very coarse grain size. “Crystalline Structure” is another feature to classify what type of rock it is. The rock may have angular, medium or rounded crystalline structure. Table 1.48 represents the four types of rock fields by 3−polar fuzzy sets in the feature space U . Let N be an unknown hybrid rock, which is to be recognized.  N = (u1 , 0.10, 0.92, 0.22), (u2 , 0.85, 0.16, 0.35),  (u3 , 0.22, 0.81, 0.20), (u4 , 0.90, 0.06, 0.17) . The Euclidean distance between R j and N is calculated as d E (R1 , N ) = 0.8178, d E (R2 , N ) = 0.7563, d E (R3 , N ) = 0.0294, d E (R4 , N ) = 1.0463. The similarity measure of R j and N is calculated as S(R1 , N ) = 0.5501, S(R2 , N ) = 0.5694, S(R3 , N ) = 0.9714, S(R4 , N ) = 0.4887. Since S(R3 , N ) is highest, so R3 and N have same pattern. Thus, hybrid rock N belongs to the rock field R3 .

Table 1.48 3−polar fuzzy set for Rock Fields . u1 u2 R1 R2 R3 R4

(0.55, 0.27, 0.73) (0.76, 0.54, 0.34) (0.11, 0.91, 0.25) (0.78, 0.36, 0.45)

(0.33, 0.83, 0.24) (0.56, 0.44, 0.21) (0.85, 0.15, 0.35) (0.34, 0.26, 0.83)

u3

u4

(0.13, 0.76, 0.65) (0.79, 0.10, 0.33) (0.20, 0.80, 0.22) (0.93, 0.20, 0.10)

(0.78, 0.46, 0.22) (0.89, 0.36, 0.11) (0.90, 0.05, 0.15) (0.10, 0.71, 0.09)

42

1 Hybrid Multi-polar Fuzzy Models

1.5.2 Medical Diagnosis of Anemia The concept of similarity measure is applied for soft sets in medical diagnosis. An application is presented to show that the distance based similarity of two m−polar fuzzy soft sets can be used to decide whether a patient has anemia or not. Consider the universal set U = {u 1 = anemia, u 2 = not anemia} consisting of only two elements, and let T = {t1 , t2 , t3 } be the set of parameters where the parameter, ‘t1 ’ stands for the General Fatigue, ‘t2 ’ stands for Heart Symptoms, ‘t3 ’ stands for Strange Cravings, The further characteristics of these parameters are described as follows: • The symptom “General Fatigue” can cause headache, dizziness, poor concentration and irritability. • The “Heart Symptoms” of the patient may include dyspnea, chest pain, hypotension and arrhythmia. • The patient may have “Strange Cravings” to eat items that are not food such as clay, dirt, ice and starch. Then, all attainable information can be formulated on these symptoms under discussion as a 4−polar fuzzy soft set (ω, N ) and this 4−polar fuzzy soft set can be constructed with the help of Anemia Specialist.  = (ω, N ) = ⎧   ⎫ ⎪ ⎨ω(t1 ) = (u1 , 3/5, 7/10, 3/5, 7/10), (u2 , 1/2, 3/5, 3/5, 2/5), ⎪ ⎬ ω(t2 ) = (u1 , 7/10, 4/5, 3/5, 4/5), (u2 , 3/5, 1/2, 7/10, 3/5) , . ⎪   ⎪ ⎩ ⎭ ω(t3 ) = (u1 , 3/5, 7/10, 7/10, 3/5), (u2 , 1/2, 2/5, 1/2, 3/5) . Now, a 4−polar fuzzy soft set (ψ, S) is constructed, based on the medical reports of the patient.  = (ψ, S) = ⎧   ⎫ ⎪ ψ( t ) = ( u , 3/5, 3/5, 3/5, 1/2), (u2 , 1/2, 1/2, 3/5, 3/10) , ⎪ 1 1 ⎨   ⎬ . ψ(t2 ) = (u1 , 4/5, 9/10, 7/10, 4/5), (u2 , 1/2, 2/5, 3/5, 1/2) , ⎪   ⎪ ⎩ ⎭ ψ(t3 ) = (u1 , 7/10, 3/5, 3/5, 7/10), (u2 , 1/2, 2/5, 3/10, 1/2) . The Hamming distance between (ω, N ) and (ψ, S), is calculated as d H (N , S ) = 0.1583. The similarity measure between (ζ, N ) and (ψ, S) is

1.5 Similarity Measure for m−Polar Fuzzy Sets

S(N , S ) =

43

1 1 = 0.8633 > 1.1583 2

Since S(N , S ) > 21 , so the two 4−polar fuzzy soft sets are significantly similar. Thus, it is concluded that the patient has the disease anemia. The method for calculating the similarity measure of 4−polar fuzzy soft sets is presented in Algorithm 1.3. Algorithm 1.3 Similarity measure of 4−polar fuzzy soft sets 1. Construct a 4−polar fuzzy soft set N with the help of Anemia Specialist. 2. Construct a 4−polar fuzzy soft set S based on the medical reports of ill person. 3. Calculate the Hamming distance between N and S , using the formula, d H (N , S ) =

q m n  1       pi ◦ N (tk )(u j ) − pi ◦ S(tk )(u j ). mq i=1 j=1 k=1

4. Calculate the similarity measure of N and S . 5. Evaluate result by using similarity.

1.5.3 Medical Diagnosis of Dengue Fever Now, another application of similarity measure is explained in medical diagnosis to determine which patient is running a dengue fever. Suppose that there are four patients in a hospital with symptoms, High Fever, Severe Pain and Bleeding. Let U = {s = severe, m = mild, n = normal} be the universal set and T = {t1 , t2 , t3 } be the set of parameters, which are symptoms of dengue fever. The parameter ‘t1 ’ stands for the High Fever, ‘t2 ’ stands for the Severe Pain, ‘t3 ’ stands for the Bleeding. The further characteristics of these parameters are described as follows: • The patient suffering from “High Fever” may also have headache, irritability and loss of appetite. • The patient may have “Severe Pain” in muscles, joints or pain behind the eyes. • The symptom “Bleeding” may include the bleeding of nose, gums and bleeding under the skin. The 3−polar fuzzy soft set is constructed for dengue fever with the help of doctor, which is given in the Table 1.49. Now, the 3−polar fuzzy soft set is constructed, based on the medical reports of these four patients (see Tables 1.50, 1.51, 1.52 and 1.53). The distances between (, T ) and ( j , T ) obtained from the medical reports of four patients is calculated in Table 1.54.

44

1 Hybrid Multi-polar Fuzzy Models

From Table 1.55, it is cleared that the patient P3 suffers dengue fever. The procedure for calculating the similarity measure of 3-polar fuzzy soft sets is presented in Algorithm. 1.4. Algorithm 1.4 Similarity measure of m-polar fuzzy soft sets 1. Construct an m−polar fuzzy soft set T = (ψ, T ) for dengue fever with the help of doctor. 2. Construct m−polar fuzzy soft sets rN = (ωr , N ) based on the medical reports of patients Pr . 3. Calculate the distance between (ψ, T ) and (ωr , N ) using the formula   1 2 m (T , rN ), d∞ (T , rN ), . . . , d∞ (T , rN ) , d∞ (T , rN ) = d∞ where,

    i (T , rT ) = sup  pi ◦ T (t j )(u k ) − pi ◦ N (t j )(u k ). d∞

 j j j  4. Calculate the similarity measure S j = S1 , S2 , . . . , Sm of T and rN . 5. Put S = inf S j . 6. The patient Pr is suffering from dengue fever if S(T , rN ) is maximum for each i ∈ m. Table 1.49 3−polar fuzzy soft set (, T ) (, T ) t1 Severe Mild Normal

(9/10, 4/5, 4/5) (3/5, 2/5, 1/2) (2/5, 2/5, 1/5)

t2

t3

(7/10, 4/5, 7/10) (1/2, 3/5, 1/2) (2/5, 3/10, 1/5)

(4/5, 7/10, 4/5) (1/2, 2/5, 2/5) (1/5, 3/10, 0)

Table 1.50 3−polar fuzzy soft set for patient P1 (1 , N ) t1 t2 Severe Mild Normal

(1/2, 2/5, 1/2) (2/5, 1/5, 1/5) (2/5, 3/10, 1/5)

(2/5, 3/10, 3/5) (2/5, 1/2, 1/5) (3/10, 1/5, 1/10)

Table 1.51 3−polar fuzzy soft set for patient P2 (2 , N ) t1 t2 Severe Mild Normal

(7/10, 3/5, 7/10) (1/2, 3/5, 3/5) (3/10, 1/5, 1/5)

(3/5, 1/2, 7/10) (2/5, 1/2, 1/2) (1/5, 1/10, 1/10)

t3 (3/10, 2/5, 3/10) (2/5, 3/10, 2/5) (1/5, 3/10, 0)

t3 (7/10, 3/5, 1/2) (2/5, 3/10, 2/5) (1/10, 1/5, 0)

1.6 m−Polar Fuzzy Rough Sets

45

Table 1.52 3−polar fuzzy soft set for patient P3 (3 , N ) t1 t2 Severe Mild Normal

(8/10, 7/10, 4/5) (7/10, 3/5, 3/5) (1/2, 2/5, 1/2)

t3

(4/5, 9/10, 7/10) (3/5, 3/5, 3/5) (3/10, 1/2, 3/10)

Table 1.53 3−polar fuzzy soft set for patient P4 (4 , N ) t1 t2 Severe Mild Normal

(3/5, 1/2, 3/5) (1/2, 3/10, 2/5) (1/5, 1/5, 1/5)

t3

(7/10, 1/2, 2/5) (3/10, 1/2, 3/10) (1/10, 1/5, 1/5)

Table 1.54 Distance between 3−polar fuzzy soft sets 1 2 . d∞ d∞ P1 P2 P3 P4

(2/5, 2/5, 3/10) (1/5, 1/5, 1/10) (1/10, 1/5, 3/10) (3/10, 3/10, 1/5)

(7/10, 7/10, 7/10) (1/2, 3/10, 2/5) (3/10, 3/10, 1/10)

(3/5, 1/2, 1/2) (3/10, 1/5, 3/10) (1/10, 1/5, 1/10)

3 d∞

(3/10, 1/2, 3/10) (1/5, 3/10, 1/10) (1/10, 1/5, 1/10) (3/10, 3/10, 3/10)

(1/2, 3/10, 1/2) (1/10, 1/10, 3/10) (1/10, 1/10, 1/10) (1/5, 1/5, 3/10)

Table 1.55 Similarity measure of 3−polar fuzzy soft sets . S1 S2 S3 P1 P2 P3 P4

(0.71, 0.71, 0.7) (0.83, 0.83, 0.91) (0.91, 0.83, 0.77) (0.77, 0.77, 0.83)

(0.77, 0.67, 0.77) (0.83, 0.77, 0.91) (0.91, 0.83, 0.91) (0.77, 0.77, 0.77)

(0.67, 0.77, 0.67) (0.91, 0.91, 0.77) (0.91, 0.91, 0.91) (0.83, 0.83, 0.77)

S= inf{S1 , S2 , S3 } (0.67, 0.67, 0.67) (0.83, 0.77, 0.77) (0.91, 0.83, 0.77) (0.77, 0.77, 0.77)

1.6 m−Polar Fuzzy Rough Sets In this section, hybrid model called m–polar fuzzy rough sets is presented and its properties are discussed. The proposed model emerges from the hybridization of m–polar fuzzy set theory with rough sets. The fundamental and essential concept behind proposed model is, the approximation of lower and upper spaces of a set with multi-polar information under m–polar fuzzy relation. Definition 1.40 ([2]) Let S and T be two nonempty universes, an m–polar fuzzy set ξ ∈ m F(S × T ) of the universe S × T is called an m–polar fuzzy relation from S to T. In general, for any s ∈ S, t ∈ T, the degree of the membership ξ(s, t) = ( p1 ◦ ξ(s, t), p2 ◦ ξ(s, t), · · · , pm ◦ ξ(s, t)) denotes the degree of the relations of

46

1 Hybrid Multi-polar Fuzzy Models

s and t. If S = T, then the m–polar fuzzy relation ξ ∈ m F(S × T ) is called an m–polar fuzzy relation on S. Example 1.10 If S = {s1 , s2 , s3 } and T = {t1 , t2 , t3 } are two universes then a 3−polar fuzzy relation ξ : S → T of the universe S × T is given in Table 1.56. Definition 1.41 Let S and T be two finite universes of discourses and ξ be an m– polar fuzzy relation from S to T, the triple (S, T, ξ ) be called m–polar fuzzy approximation space. For any set X ∈ m F(T ), the lower and upper approximations ξ (X ) and ξ (X ) w.r.t. approximation space (S, T, ξ ) are m–polar fuzzy sets of S, whose membership functions for each s ∈ S are defined as  (1 − ξ(s, t)) ∨ X (t) , ξ (X )(s) = t∈T

ξ (X )(s) =

 ξ(s, t) ∧ X (t) . t∈T

The pair (ξ (X ), ξ (X )) is called an m-polar fuzzy rough set of X w.r.t. (S, T, ξ ) and ξ , ξ : m F(T ) → m F(S) are called lower and upper m–polar fuzzy rough approximation operators, respectively. Furthermore, if ξ (X ) = ξ (X ), then X is said to be definable. Remark 1.4 If S = T, then the pair (ξ (X ), ξ (X )) is called an m–polar fuzzy rough set of X w.r.t. (S, ξ ) and ξ , ξ : m F(S) → m F(S) are called lower and upper m–polar fuzzy rough approximation operators, respectively.

Table 1.56 3–polar fuzzy relation ξ t1 s1 s2 s3

(0.6,0.3,0.1) (0.5,0.3,0.2) (0.3,0.2,0.1)

Table 1.57 4–polar fuzzy relation ξ t1 s1 s2 s3 s4

(0.3,0.4,0.2,0.7) (0.2,0.9,0.1,0.5) (0.3,0.6,0.8,0.2) (0.5,0.7,0.7,0.2)

t2

t3

(0.4,0.7,0.6) (0.5,0.2,0.8) (0.3,0.4,0.8)

(0.4,0.6,0.2) (0.6,0.9,0.6) (0.7,0.3,0.5)

t2

t3

(0.8,0.2,0.6,0.5) (0.1,0.3,0.8,0.9) (0.2,0.4,0.6,0.8) (0.8,0.9,0.5,0.7)

(0.1,0.3,0.2,0.5) (0.2,0.9,0.1,0.6) (0.8,0.9,0.5,0.7) (0,0.5,1,0.9)

1.6 m−Polar Fuzzy Rough Sets

47

Example 1.11 Let S = {s1 , s2 , s3 , s t2 , t3 } be two universes of dis4 } and T = {t1 , 0.5,0.6,0.7,0.2 , 0.3,0.2,0.1,0.8 , 0.5,0.4,0.6,0.1 be a 4−polar courses and X = t1 t2 t3 fuzzy set, for these two universes a 4−polar fuzzy relation ξ : S → T is given in Table 1.57. By Definition 1.41, ξ (X )(s1 ) = (0.3, 0.6, 0.4, 0.3),

ξ (X )(s1 ) = (0.3, 0.4, 0.2, 0.5),

ξ (X )(s2 ) = (0.8, 0.4, 0.2, 0.4),

ξ (X )(s2 ) = (0.2, 0.6, 0.1, 0.8),

ξ (X )(s3 ) = (0.5, 0.4, 0.4, 0.3),

ξ (X )(s3 ) = (0.5, 0.6, 0.7, 0.8),

ξ (X )(s4 ) = (0.3, 0.5, 0.6, 0.1),

ξ (X )(s4 ) = (0.5, 0.6, 0.7, 0.2).

Thus, ξ (X ) = ξ (X ) =



 

0.3,0.6,0.4,0.3 s1

,

0.3,0.4,0.2,0.5 s1

,



 0.8,0.4,0.2,0.4 s2

,

0.2,0.6,0.1,0.8 s2

,



 0.5,0.4,0.4,0.3 s3

,

0.5,0.6,0.7,0.8 s3

,



0.3,0.5,0.6,0.1 s4 0.5,0.6,0.7,0.2 s4

, .

Hence, the pair (ξ (X ), ξ (X )) is referred as a 4−polar fuzzy rough set. Theorem 1.13 Let (S, T, ξ ) be an m–polar fuzzy approximation space, the lower and upper approximations ξ (X ) and ξ (X ) satisfy the following properties for any X, Y ∈ m F(T ), 1. 2. 3. 4. 5. 6. 7. 8.

ξ (X ) =∼ ξ (∼ X ), X ⊆ Y ⇒ ξ (X ) ⊆ ξ (Y ), ξ (X ∪ Y ) ⊇ ξ (X ) ∪ ξ (Y ), ξ (X ∩ Y ) = ξ (X ) ∩ ξ (Y ), ξ (X ) =∼ ξ (∼ X ), X ⊆ Y ⇒ ξ (X ) ⊆ ξ (Y ), ξ (X ∪ Y ) = ξ (X ) ∪ ξ (Y ), ξ (X ∩ Y ) ⊆ ξ (X ) ∩ ξ (Y ).

Proof The properties of the lower approximation operator ξ , for any X, Y ∈ m F(T ) are proved as follows: 1. For all s ∈ S,

48

1 Hybrid Multi-polar Fuzzy Models

ξ (X ) =

"  ! 1 − ξ(s, t) ∨ X (t) t∈T

"  ! = 1 − ξ(s, t) ∨ ∼ X (t) t∈T

=∼

"  ! 1 − ξ(s, t) ∨ ∼ X (t) t∈T

" ! ξ(s, t) ∧ X (t) . = t∈T

Thus, ξ (X ) =∼ ξ (∼ X ). 2. It can be proved directly by using Definition 1.41. 3. For all s ∈ S,  "  ! 1 − ξ(s, t) ∨ X ∪ Y (t) ξ (X ∪ Y ) = t∈T

 "  ! 1 − ξ(s, t) ∨ X (t) ∨ Y (t) ⊇ t∈T

 "  ! (1 − ξ(s, t)) ∨ X (t) ∨ (1 − ξ(s, t)) ∨ Y (t) = t∈T

! " ! " = (1 − ξ(s, t)) ∨ X (t) ∨ (1 − ξ(s, t)) ∨ Y (t) t∈T

t∈T

= ξ (X ) ∪ ξ (Y ). Thus, ξ (X ∪ Y ) ⊇ ξ (X ) ∪ ξ (Y ). 4. For all s ∈ S,  "  ! 1 − ξ(s, t) ∨ X ∩ Y (t) ξ (X ∩ Y ) = t∈T

 "  ! 1 − ξ(s, t) ∨ X (t) ∧ Y (t) = t∈T

 "  ! (1 − ξ(s, t)) ∨ X (t) ∧ (1 − ξ(s, t)) ∨ Y (t) = t∈T

" ! " ! (1 − ξ(s, t)) ∨ X (t) ∧ (1 − ξ(s, t)) ∨ Y (t) = t∈T

= ξ (X ) ∩ ξ (Y ). Thus, ξ (X ∩ Y ) = ξ (X ) ∩ ξ (Y ).

t∈T

1.6 m−Polar Fuzzy Rough Sets

49

Similarly, the properties (5 − 8) for the upper approximation operator ξ , for any X, Y ∈ m F(T ), can also be proved by using above arguments.  Definition 1.42 Let (S, T, ξ1 ) and (S, T, ξ2 ) be two m–polar fuzzy approximation spaces. • The m–polar fuzzy approximation space (S, T, ξ1 ∪ ξ2 ) is called the union of (S, T, ξ1 ) and (S, T, ξ2 ). • The m–polar fuzzy approximation space (S, T, ξ1 ∩ ξ2 ) is called the intersection of (S, T, ξ1 ) and (S, T, ξ2 ). Theorem 1.14 Let (S, T, ξ1 ) and (S, T, ξ2 ) be two m–polar fuzzy approximation spaces and ξ = ξ1 ∪ ξ2 , for any s ∈ S and X ∈ m F(T ), 1. ξ = ξ1 ∪ ξ2 , 2. ξ (X ) = ξ 1 (X ) ∪ ξ 2 (X ), 3. ξ (X ) = ξ 1 (X ) ∩ ξ 2 (X ). Proof 1. For all s ∈ S and t ∈ T , ξ(s) = ξ(s, t) = (ξ1 ∪ ξ2 )(s, t)  = (ξ1 )(s, t) ∨ (ξ2 )(s, t)  = (ξ1 )(s) ∨ (ξ2 )(s)  = ξ1 ∪ ξ2 (s). Thus, ξ = ξ1 ∪ ξ2 . 2. For all s ∈ S,  ξ(s, t) ∧ X (t) (ξ (X ))(s) = t∈T

"  ! (ξ1 )(s, t) ∨ (ξ2 )(s, t) ∧ X (t) = t∈T

 "  ! (ξ1 )(s, t) ∧ X (t) ∨ (ξ2 )(s, t) ∧ X (t) = t∈T

" ! " ! (ξ1 )(s, t) ∧ X (t) ∨ (ξ2 )(s, t) ∧ X (t) = t∈T

 = (ξ 1 (X ))(s) ∨ (ξ 2 (X ))(s)  = ξ 1 (X ) ∪ ξ 2 (X ) (s).

t∈T

50

1 Hybrid Multi-polar Fuzzy Models

Thus, ξ (X ) = ξ 1 (X ) ∪ ξ 2 (X ). 3. By using the property ξ (X ) =∼ ξ (∼ X ), we have ξ (X ) =∼ ξ (∼ X )  =∼ ξ1 (∼ X ) ∪ ξ2 (∼ X )   = ∼ ξ1 (∼ X ) ∩ ∼ ξ2 (∼ X ) = ξ1 (X ) ∩ ξ2 (X ). Thus, ξ (X ) = ξ1 (X ) ∩ ξ2 (X ).



Corollary 1.1 Let (S, T, ξ1 ) and (S, T, ξ2 ) be two m–polar fuzzy approximation spaces. If ξ1 ⊆ ξ2 , then for any X ∈ m F(T ), the following property hold. ξ 1 (X ) ⊆ ξ 2 (X ),

ξ 1 (X ) ⊇ ξ 2 (X ).

Proof It can be proved directly by using Definition 1.41.



For n different m–polar fuzzy relations Theorem 1.14 can be generalized as, Theorem 1.15 Let (S, T, ξ j ) be m–polar fuzzy approximation spaces and ξ = ∪nj=1 ξ j . Then for any s ∈ S and X ∈ m F(T ), 1. ξ = ∪nj=1 ξ j , 2. ξ (X ) = ∪nj=1 ξ j (X ), 3. ξ (X ) = ∩nj=1 ξ j (X ). Proof It is easy to prove by using similar arguments, as used in Theorem 1.14.  Theorem 1.16 Let (S, T, ξ1 ) and (S, T, ξ2 ) be two m–polar fuzzy approximation spaces and ξ = ξ1 ∩ ξ2 , for any s ∈ S and X ∈ m F(T ), 1. ξ = ξ1 ∩ ξ2 , 2. ξ (X ) ⊆ ξ 1 (X ) ∩ ξ 2 (X ), 3. ξ (X ) ⊇ ξ 1 (X ) ∪ ξ 2 (X ). Proof 1. For all s ∈ S and t ∈ T, ξ(s) = ξ(s, t) = (ξ1 ∩ ξ2 )(s, t)  = (ξ1 )(s, t) ∧ (ξ2 )(s, t)  = (ξ1 )(s) ∧ (ξ2 )(s)  = ξ1 ∩ ξ2 (s).

1.6 m−Polar Fuzzy Rough Sets

51

Thus, ξ = ξ1 ∩ ξ2 . 2. For all s ∈ S, (ξ (X ))(s) =

 ξ(s, t) ∧ X (t) t∈T

=

 ! t∈T

=

 !

" (ξ1 )(s, t) ∧ (ξ2 )(s, t) ∧ X (t)  " (ξ1 )(s, t) ∧ X (t) ∧ (ξ2 )(s, t) ∧ X (t)

t∈T

" ! " ! (ξ1 )(s, t) ∧ X (t) ∧ (ξ2 )(s, t) ∧ X (t) ≤ t∈T

 ≤ (ξ 1 (X ))(s) ∧ (ξ 2 (X ))(s)  = ξ 1 (X ) ∩ ξ 2 (X ) (s).

t∈T

Thus, ξ (X ) ⊆ ξ 1 (X ) ∩ ξ 2 (X ). 3. By using the property ξ (X ) =∼ ξ (∼ X ), we have ξ (X ) =∼ ξ (∼ X )  ⊇∼ ξ1 (∼ X ) ∩ ξ2 (∼ X )   = ∼ ξ1 (∼ X ) ∪ ∼ ξ2 (∼ X ) = ξ1 (X ) ∪ ξ2 (X ). Thus, ξ (X ) ⊇ ξ1 (X ) ∪ ξ2 (X ).



Example 1.12 Consider (S, T, ξ ) is an m–polar fuzzy approximation space, where S = {s1 , s2 , s3 } and T = {t1 , t2 , t3 } are two universes of discourses and ξ1 , ξ2 are two 3–polar fuzzy relations given in Tables 1.59 and 1.60. Intersection ξ = ξ1 ∩ ξ2 of two 3–polar  ξ1 and ξ2 isgiven in Table 1.58. fuzzy relations If X =

,

0.2,0.5,0.4 t1

,

0.3,0.4,0.1 t2

using Definition 1.41, we have  ξ 1 (X ) =



0.5,0.5,0.4 s1

,



0.4,0.5,0.2 s2

0.7,0.5,0.2 t3

 ,



0.4,0.4,0.5 s3

ξ 2 (X ) = , , 0.8,0.5,0.3 s3    0.7,0.7,0.5 ξ (X ) = , 0.7,0.6,0.7 , 0.8,0.5,0.8 s1 s2 s3 0.2,0.4,0.4 s1

0.3,0.5,0.2 s2

, , ,

is a 3−polar fuzzy set, then by

52

1 Hybrid Multi-polar Fuzzy Models

Table 1.58 Intersection ξ = ξ1 ∩ ξ2 of two 3–polar fuzzy relations ξ = ξ 1 ∩ ξ2 t1 t2 s1 s2 s3

(0.1,0.2,0.5) (0.3,0.4,0.1) (0.2,0.8,0.1)

(0.1,0.2,0.2) (0.4,0.3,0.3) (0.1,0.2,0.2)

Table 1.59 3–polar fuzzy relation ξ1 ξ1 t1 s1 s2 s3

(0.1,0.5,0.6) (0.3,0.4,0.2) (0.5,0.8,0.1)

Table 1.60 3–polar fuzzy relation ξ2 ξ2 t1 s1 s2 s3

(0.8,0.2,0.5) (0.7,0.6,0.1) (0.2,0.9,0.5)



 ξ 1 (X ) =



0.5,0.5,0.4 s1

,



,

t2

t3 (0.5,0.3,0.1) (0.7,0.2,0.8) (0.5,0.4,0.2)

t2

t3

(0.1,0.6,0.2) (0.4,0.3,0.8) (0.1,0.2,0.7)

(0.3,0.5,0.6) (0.9,0.7,0.2) (0.1,0.8,0.2)



0.5,0.5,0.2 s3

, , 0.2,0.5,0.4 ξ 2 (X ) = s3    0.3,0.3,0.4 , 0.7,0.4,0.2 , 0.2,0.5,0.2 ξ (X ) = s1 s2 s3 0.3,0.5,0.4 s1

0.7,0.5,0.2 s2

(0.3,0.3,0.1) (0.7,0.2,0.2) (0.1,0.4,0.2)

(0.5,0.2,0.6) (0.6,0.5,0.3) (0.6,0.7,0.2)

 0.7,0.4,0.2 s2

t3

, , .

It is easy to see that ξ (X ) = ξ 1 (X ) ∪ ξ 2 (X ),

ξ (X ) = ξ 1 (X ) ∩ ξ 2 (X ).

For n different m–polar fuzzy relations Theorem 1.16 can be generalized as, Theorem 1.17 Let (S, T, ξ j ) be m–polar fuzzy approximation spaces and ξ = ∩nj=1 ξ j , for any s and X ∈ m F(T ), 1. ξ = ∩nj ξ j , 2. ξ (X ) ⊆ ∩nj=1 ξ j (X ), 3. ξ (X ) ⊇ ∪nj=1 ξ j (X ). Proof It is easy to prove by using similar arguments as used in Theorem 1.16.



Definition 1.43 Let (S, ξ1 ) and (S, ξ2 ) be two m–polar fuzzy approximation spaces. The approximation space (S, ξ1 ◦ ξ2 ) is called the composition of (S, ξ1 ) and (S, ξ2 ).

1.6 m−Polar Fuzzy Rough Sets

53

Theorem 1.18 Let (S, ξ1 ) and (S, ξ2 ) be two m–polar fuzzy approximation spaces and ξ = ξ1 ◦ ξ2 , for any X ∈ m F(S), 1. ξ (X ) = [ξ 1 ◦ ξ 2 ](X ) = ξ 1 (ξ 2 (X )), 2. ξ (X ) = [ξ 1 ◦ ξ 2 ](X ) = ξ 1 (ξ 2 (X )). Proof 1. For all s ∈ S, (ξ 1 (ξ 2 (X )))(s) =

!

" (ξ1 )(s, t) ∧ ξ 2 (X )(t)

t∈S

=

!

(ξ1 )(s, t) ∧

t∈S

=

!



" (ξ2 )(t, r ) ∧ X (r )

r ∈S

 " (ξ1 )(s, t) ∧ (ξ2 )(t, r ) ∧ X (r )

t∈S r ∈S

=

" ! (ξ1 )(s, t) ∧ (ξ2 )(t, r ) ∧ X (r )

r ∈S

=

!

t∈S

" (ξ )(s, r ) ∧ X (r ) = (ξ (X ))(s).

r ∈S

Thus, ξ (X ) = [ξ 1 ◦ ξ 2 ](X ) = ξ 1 (ξ 2 (X )). 2. For all s ∈ S, (ξ 1 (ξ 2 (X )))(s) =

" ! (1 − ξ1 (s, t)) ∨ ξ 2 (X )(t) t∈S

 " ! = (1 − ξ2 (t, r )) ∨ X (r ) (1 − ξ1 (s, t)) ∨ t∈S

=

!

r ∈S

 " (1 − ξ1 (s, t)) ∨ (1 − ξ2 (t, r )) ∨ X (r )

t∈S r ∈S

=

!

r ∈S

" (1 − ξ1 (s, t)) ∨ (1 − ξ2 (t, r )) ∨ X (r )

t∈S

" ! (1 − ξ(s, r )) ∨ X (r ) = (ξ (X ))(s). = r ∈S

Thus, ξ (X ) = [ξ 1 ◦ ξ 2 ](X ) = ξ 1 (ξ 2 (X )).



Theorem 1.19 Let (Q, S, ξ1 ) and (S, T, ξ2 ) be two m–polar fuzzy approximation spaces and ξ = ξ1 ◦ ξ2 , for any X ∈ m F(T ), 1. ξ (X ) = [ξ 1 ◦ ξ 2 ](X ) = ξ 1 (ξ 2 (X )), 2. ξ (X ) = [ξ 1 ◦ ξ 2 ](X ) = ξ 1 (ξ 2 (X )). Proof It can be proved easily by using similar arguments as used in Theorem 1.18.  In next subsections, the applications of m–polar fuzzy rough sets are presented.

54

1 Hybrid Multi-polar Fuzzy Models

1.6.1 Selection of Prints and Shades for Variety of Fabrics Nowadays, the selection of suitable patterns, colors and shades for fabrics is difficult task for designers. In order to handle this difficult situation, we present the concept of m–polar fuzzy rough set model, which provides us information about the selection of stuffs and variety of colors combination with different patterns and shades. It also provides us information about the variety of fabrics. Suppose a textile designing company wants to manufacture different types of fabrics with suitable patterns, colors and shades. Company also wants to prepare such a kind of fabrics with same design and different variety of materials. So, company handovers this task to a designer for such types of fabrics. Let P and T be the two universes of discourses, with P = {pattern, colour, shade} the set of prints and shades of fabrics, T = {cotten, wool, silk, linen} the set of types of materials, used in manufacturing of fabrics and ξ : P → T be a 4−polar fuzzy relation. The universe P is further classified as • The “Patterns” of fabrics include tartan, cross tee, polka dotted and chevron. • The “Colors” of fabrics include pink, yellow, grey and peach. • The “Shades” of fabrics include light, dark, dull and bright. A 4−polar fuzzy relation is given in Table 1.61 as follows: A 4−polar fuzzy relation ξ ∈ (P × T ) provides us information about the patterns, colors and shades of different fabrics. For example, if we consider • “Patterns in cotton” then cotton has 60% tartan, 80% cross tee, 20% polka dotted and 10% chevron pattern. • “Colors in cotton” are classified as 50% pink, 30% yellow, 10% grey and 50% peach. • “Shades in cotton” are 40% light, 80% dark, 20% dull and 30% bright. Similarly for all other fabrics, patterns, colors and shades can be selected. For different variety of  fabrics a set M is  taken as   M=

0.7,0.8,0.1,0.5 cotton

,

0.6,0.2,0.3,0.7 wool

,

0.6,0.4,0.3,0.1 silk

,

0.3,0.9,0.1,0.7 linen

,

which

describes the further types of fabrics, classified as • The “types of cotton” include drill cotton, duch cotton, gauze cotton and flannel cotten.

Table 1.61 4−polar fuzzy relation ξ Cotton Pattern Color Shade

(0.6,0.8,0.2,0.1) (0.5,0.3,0.1,0.5) (0.4,0.8,0.2,0.3)

Wool

Silk

Linen

(0.4,0.2,0.9,0.5) (0.6,0.6,0.2,0.1) (0.4,0.9,0.2,0.6)

(0.6,0.5,0.2,0.2) (0.8,0.1,0.7,0.6) (0.1,0.1,0.3,0.4)

(0.9,0.7,0.3,0.6) (0.8,0.2,0.6,0.1) (0.6,0.7,0.3,0.2)

1.6 m−Polar Fuzzy Rough Sets Table 1.62 Choice value ξ (M) Pattern Color Shade

(0.3,0.5,0.3,0.7) (0.3,0.4,0.3,0.4) (0.4,0.2,0.7,0.6)

55

ξ (M)

Choice value ρs

(0.7,0.8,0.3,0.6) (0.6,0.3,0.3,0.5) (0.4,0.8,0.3,0.6)

ρ p =(0.7,0.8,0.3,0.7) ρc =(0.6,0.4,0.3,0.5) ρs =(0.4,0.8,0.7,0.6)

• The “types of wool” include merino wool, alpaca wool, mohair wool and lama wool. • The “types of silk” include charmeuse silk, filament silk, georgette silk and habutai silk. • The “type of linen” include damask linen, blended linen, bird’s eye linen, cambric linen. Now to decide the fabrics of different variety with same patterns, colors and shades the lower and upper m–polar fuzzy rough approximation operators are applied on a 4−polar fuzzy set M by using Definition 1.31. Further, for final decision the choice value is defined as ρs = max( pi ◦ ξ (M), pi ◦ ξ (M)), i ∈ m. Thus, from choice value ρ p =(0.7, 0.8, 0.3, 0.7) calculated in Table 1.62, it is concluded • 70% “Tartan Pattern” is suitable for drill cotton, merino wool, charmeuse silk and damask linen. • 80% “Cross Tee Pattern” is suitable for duch cotton, alpaca wool, filament silk and blended linen. • 30% “Polka Dotted Pattern” is suitable for gauze cotton, mohair wool, georgette silk and birds eye’s linen. • 70% “Chevron Pattern” is suitable for flament cotten, lama wool, habutai silk and cambric linen. Similarly, from other choice values as calculated in Table 1.62, one can easily find the suitable colors and shades for different types of fabrics.

1.6.2 Selection of Features for Different Models of Mobiles With the advent of new technology, the way of communication is also changed. Today is the era of wireless communication which gives rise to mobile phones. Mobiles are the latest invention and common way to communicate now-a-days. Mobile phones are now inexpensive, easy to use, comfortable and equipped with almost every latest features we desire. Feature specifications of different types of mobiles is a complicated

56

1 Hybrid Multi-polar Fuzzy Models

task for a company. For this MCDM the concept of m–polar fuzzy rough sets is used. Suppose a mobile company wants to launch a mobile phone with different features and specifications. Let (F, T, ξ ) be an m–polar fuzzy approximation space, where F and T are two universes of discourses and ξ : F → T be a 5−polar fuzzy relation . Let F = {os, size, battery, processor, memory, network, displays, sensors, camera} be the set of features of mobiles and T = {classic, flip, slider, qwerty, touch} be the set of types of mobiles. The universe F is further classified into five different features as • The “Os” includes android, blackberry, java, symbion and window. • The “Size” includes 3.5 inch, 4 inch, 4.5 inch, 5 inch and 5.5 inch. • The “Battery” includes lithium polymer, nickel cadmium, nickel metal hydride, lithium ion and new lithium technology. • The “Processor” includes dual core, quad core, octa core, intel and any other. • The “Memory” includes drum, floating body, MRAM, NAND and ReRAM. • The “Network” includes wifi, G, 2G, 3G and 4G. • The “Displays” include LCD, amoled, OLED, IPS LCD and retina. • The “Sensor” includes vibrations, motions, contact switch, ambient light and sound. • The “Camera” includes ultrawide angle, wide angle, normal, telephoto and super telephoto. A 5−polar fuzzy relation ξ ∈ (F × T ) is given in Table 1.63 as follows: For different models of mobiles, 5−polar fuzzy set M is taken as  M=

  0.8, 0.2, 0.6, 0.5, 0.7 0.5, 0.9, 0.2, 0.6, 0.7 0.3, 0.4, 0.5, 0.8, 0.6 , , , classic f li p slider   0.6, 0.8, 0.6, 0.2, 0.3 0.6, 0.2, 0.3, 0.5, 0.9 , . qwer t y touch

The models of mobiles are classified as • The “models of classic mobile” include samsung S, motorola raza V 3, nokia 3310, motorola 8000 dyna TAC and nokia 1110. • The “models of flip mobile” include samsung convoy, blackberry style, LG 450, samsung gusto and nokia 6350. • The “models of slider mobile” include samsung G600, C205, LG cosmos slide, motarola milestone and nokia N 95. • The “models of qwerty mobile” include black berry classic, black berry bold 9790, nokia asha 210, blackberry Q10 and nokia C3. • The “models of touch mobile” include samsung glaxy E7, black berry DTE K 50, LGK 10, lenovo A6000 and HTC tough HD. Now to decide the mobiles of different models with same set of features, the lower and upper m–polar fuzzy rough approximation operators are applied on a 5−polar

1.6 m−Polar Fuzzy Rough Sets Table 1.63 5−polar fuzzy relation ξ Classic OS Size Battery Processor Memory Network Displays Sensors Camera OS Size Battery Processor Memory Network Displays Sensors Camera

(0.1, 0.3, 0.6, 0.7, 0.1) (0.8, 0.7, 0.5, 0.5, 0.2) (0.3, 0.4, 0.6, 0.7, 0.5) (0.3, 0.1, 0.1, 0.2, 0.6) (0.3, 0.5, 0.3, 0.2, 0.6) (0.1, 0.2, 0.1, 0.3, 0.1) (0.5, 0.6, 0.5, 0.2, 0.1) (0.3, 0.5, 0.2, 0.8, 0.7) (0.2, 0.3, 0.5, 0.4, 0.9) (0.5, 0.7, 0.5, 0.3, 0.2) (0.1, 0.3, 0.8, 0.7, 0.5) (0.2, 0.5, 0.9, 0.3, 0.5) (0.5, 0.9, 0.7, 0.2, 0.1) (0.7, 0.2, 0.1, 0.6, 0.2) (0.8, 0.3, 0.9, 0.7, 0.6) (0.6, 0.7, 0.8, 0.9, 0.2) (0.8, 0.3, 0.5, 0.6, 0.7) (0.8, 0.9, 0.5, 0.2, 0.3)

57

Flip

Slider

(0.3, 0.1, 0.5, 0.8, 0.2) (0.7, 0.8, 0.6, 0.3, 0.6) (0.6, 0.2, 0.3, 0.5, 0.8) (0.5, 0.3, 0.4, 0.6, 0.1) (0.2, 0.5, 0.6, 0.1, 0.3) (0.3, 0.4, 0.5, 0.2, 0.1) (0.8, 0.9, 0.7, 0.3, 0.1) (0.6, 0.7, 0.8, 0.9, 0.2) (0.6, 0.8, 0.9, 0.5, 0.4) (0.9, 0.7, 0.6, 0.5, 0.8) (0.2, 0.3, 0.5, 0.7, 0.9) (0.2, 0.6, 0.7, 0.2, 0.7) (0.5, 0.4, 0.7, 0.3, 0.1) (0.3, 0.4, 0.8, 0.1, 0.2) (0.5, 0.6, 0.8, 0.3, 0.1) (0.3, 0.5, 0.8, 0.9, 0.7) (0.3, 0.2, 0.7, 0.8, 0.7) (0.7, 0.6, 0.5, 0.2, 0.3)

(0.4, 0.2, 0.3, 0.3, 0.4) (0.1, 0.3, 0.4, 0.5, 0.2) (0.6, 0.2, 0.5, 0.9, 0.1) (0.4, 0.8, 0.2, 0.1, 0.3) (0.8, 0.2, 0.3, 0.4, 0.6) (0.5, 0.2, 0.6, 0.3, 0.2) (0.8, 0.5, 0.2, 0.9, 0.7) (0.8, 0.2, 0.3, 0.7, 0.5) (0.3, 0.5, 0.4, 0.3, 0.2)

fuzzy set M, by using Definition 1.41. Further, for final decision the choice value is defined as ρs = max( pi ◦ ξ(M), pi ◦ ξ (M)), i ∈ m. Thus, from choice value ρos = (0.6, 0.7, 0.5, 0.7, 0.8) calculated in Table 1.64, it is concluded: • 60% “OS Android” is suitable for samsung s, samsung convoy, samsung G600, black berry classic and samsung glaxy E7. • 70% “OS Blackberry” is suitable for motorralla raza V 3, black berry style, C205, black berry bold 9790 and black berry DT E K 50. • 50% “OS Java” is suitable for nokia 3310,LG450, LG cosmos slide, nokia asha 210 and LGK10. • 70% “OS Symbion” is suitable for motorola 8000 dyna TAC, samsung gusto, motorola milessstone, black berry Q10 and lenovo A6000. • 80% “OS Window” is suitable for nokia 1110, nokia 6350, nokia N 95, nokia C3 and H T C tough H D.

58

1 Hybrid Multi-polar Fuzzy Models

Table 1.64 Choice value ξ(M) OS Size Battery Processor Memory Network Displays Sensors Camera

(0.6, 0.3, 0.4, 0.5, 0.7) (0.3, 0.2, 0.5, 0.3, 0.5) (0.5, 0.4, 0.3, 0.5, 0.5) (0.6, 0.6, 0.3, 0.5, 0.6) (0.5, 0.5, 0.3, 0.6, 0.6) (0.5, 0.4, 0.3, 0.3, 0.4) (0.5, 0.2, 0.3, 0.2, 0.7) (0.5, 0.3, 0.3, 0.4, 0.3) (0.6, 0.2, 0.2, 0.5, 0.6)

ξ (M)

Choice value ρs

(0.6, 0.7, 0.5, 0.7, 0.8) (0.7, 0.7, 0.6, 0.5, 0.8) (0.7, 0.4, 0.6, 0.7, 0.7) (0.5, 0.8, 0.6, 0.5, 0.6) (0.6, 0.4, 0.6, 0.4, 0.6) (0.6, 0.3, 0.6, 0.3, 0.3) (0.8, 0.7, 0.6, 0.6, 0.7) (0.6, 0.4, 0.6, 0.8, 0.7) (0.6, 0.8, 0.6, 0.5, 0.7)

(0.6, 0.7, 0.5, 0.7, 0.8) (0.7, 0.7, 0.6, 0.5, 0.8) (0.7, 0.4, 0.6, 0.7, 0.7) (0.6, 0.8, 0.6, 0.5, 0.6) (0.6, 0.5, 0.6, 0.6, 0.6) (0.6, 0.4, 0.6, 0.3, 0.4) (0.8, 0.7, 0.6, 0.6, 0.7) (0.6, 0.4, 0.6, 0.8, 0.7) (0.6, 0.8, 0.6, 0.5, 0.7)

Similarly, from other choice values as calculated in Table 1.64, one can easily find the other suitable features for different models of mobiles. The method of selecting the best object under m–polar fuzzy rough information is described in the following Algorithm 1.5 Algorithm 1.5 m–Polar fuzzy rough sets # as different features of universe S. 1. Input S and T , as universes of discourses, and F 2. Input X , as an m–polar fuzzy set such that X ∈ m F(T ) and # X as different types of set X . 3. Compute m–polar fuzzy relation ξ : S → T. 4. Compute lower and upper approximations ξ (X ) and ξ (X ), for any set X ∈ m F(T ) w.r.t. approximation space (S, T, ξ ) as  (1 − ξ(s, t)) ∨ X (t) , s ∈ S, ξ (S)(s) = t∈T

ξ (S)(s) =

 ξ(s, t) ∧ X (t) , s ∈ S. t∈T

5. Compute choice value ρs as ρs = max( pi ◦ ξ(X ), pi ◦ ξ (X )), i ∈ m. 6. Repeat this process for different varieties and features. 7. Evaluate Sm , the alternative for which ρs is maximum.

1.7 m−Polar Fuzzy Soft Rough Sets

59

1.7 m−Polar Fuzzy Soft Rough Sets In this section, the concept of m−polar fuzzy soft relation and m−polar fuzzy soft rough sets is presented. Further, some fundamental properties are investigated. Definition 1.44 ([1]) Let (τ, U ) be an m−polar fuzzy soft set over U . Then, an m−polar fuzzy subset ζ of U × T is referred to as an m−polar fuzzy soft relation from U to T is given by ζ =

%  $ (x, t), pi ◦ ζ (x, t) | (x, t) ∈ U × T ,

where ζ : U × T → [0, 1]m . If U = {x1 , x2 , . . . , xn }, T = {t1 , t2 , . . . , tn }, then an m−polar fuzzy soft relation ζ over U × T can be presented in Table 1.65 as follows: Example 1.13 Let U = {x1 , x2 , x3 } be a universe, T = {t1 , t2 , t3 } a set of parameters. A 3−polar fuzzy soft relation ζ : U → T of the universe U × T is given by Table 1.66 as follows: We now define m−polar fuzzy soft rough sets. Definition 1.45 Let U be a nonempty set called universe, T a universe of parameters. For any m−polar fuzzy soft relation ζ on U × T , the pair (U, T, ζ ) is referred to as an m−polar fuzzy soft approximation space. For an arbitrary Q ∈ mF T , the lower and upper soft approximations of Q about (U, T, ζ ), denoted by ζ (Q) and ζ (Q), respectively, are defined as follows: $ %  v, Q ζ (v) | v ∈ U , $ %  ζ (Q) = v, Q ζ (v) | v ∈ U ,

ζ (Q) =

Table 1.65 An m−polar fuzzy soft relation ζ t1 t2 x1 x2 .. . xn

pi ◦ ζ (x1 , t1 ) pi ◦ ζ (x2 , t1 ) .. . pi ◦ ζ (xn , t1 )

pi ◦ ζ (x1 , t2 ) pi ◦ ζ (x2 , t2 ) .. . pi ◦ ζ (xn , t2 )

Table 1.66 An 3−polar fuzzy soft relation ζa t1 x1 x2 x3

(0.6, 0.3, 0.1) (0.5, 0.3, 0.2) (0.3, 0.2, 0.1)

···

tn

··· ··· .. .

pi ◦ ζ (x1 , tn ) pi ◦ ζ (x2 , tn ) .. . pi ◦ ζ (xn , tn )

···

t2

t3

(0.4, 0.7, 0.6) (0.5, 0.2, 0.8) (0.3, 0.4, 0.8)

(0.4, 0.6, 0.2) (0.6, 0.9, 0.6) (0.7, 0.3, 0.5).

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Table 1.67 An 3−polar fuzzy soft relation ζ t1 t2

t3

t4

(0.4, 0.6, 0.2) (0.6, 0.9, 0.6) (0.7, 0.3, 0.5) (0.3, 0.1, 0.0) (0.4, 0.0, 0.7)

(0.4, 0.6, 0.2) (0.7, 0.3, 0.6) (0.2, 0.9, 0.9) (0.6, 0.4, 0.4) (0.8, 0.9, 0.0)

(0.6, 0.3, 0.1) (0.5, 0.3, 0.2) (0.3, 0.2, 0.1) (0.4, 0.3, 0.6) (0.2, 0.7, 0.3)

x1 x2 x3 x4 x5

(0.4, 0.7, 0.6) (0.5, 0.2, 0.8) (0.3, 0.4, 0.8) (0.5, 0.1, 0.4) (0.4, 0.8, 0.1)

where Q ζ (v) =

 &  ' 1 − pi ◦ Q ζ (v, w) ∨ pi ◦ Q(w) , w∈T

Q ζ (v) =



 pi ◦ Q ζ (v, w) ∧ pi ◦ Q(w) .

w∈T

The pair (ζ (Q), ζ (Q)) is called m−polar fuzzy soft rough set of Q about (U, T, ζ ), and ζ , ζ : mF T → mF U are, respectively, said to be lower and upper m−polar fuzzy soft rough approximation operators. Moreover, if ζ (Q) = ζ (Q), then Q is said to be definable. Example 1.14 Let U = {x1 , x2 , x3 , x4 , x5 } be the set of five laptops and let T = t1 = si ze, t2 = beauti f ul, t3 = technology, t4 = price be the set of parameters. Consider a 3−polar fuzzy soft relation ζ : U → T is given by Table 1.67 as follows: Consider a 3−polar fuzzy subset Q of T as follows:   Q = (t1 , 0.3, 0.1, 0.7), (t2 , 0.3, 0.6, 0.4), (t3 , 0.5, 0.6, 0.1), (t4 , 0.9, 0.1, 0.4) . From Definition 1.45, the lower and upper soft approximations are given by Q ζ (x1 ) = (0.4, 0.6, 0.4), Q ζ (x1 ) = (0.4, 0.4, 0.4), Q ζ (x2 ) = (0.7, 0.6, 0.4), Q ζ (x2 ) = (0.5, 0.6, 0.4), Q ζ (x3 ) = (0.5, 0.4, 0.4), Q ζ (x3 ) = (0.5, 0.1, 0.4), Q ζ (x4 ) = (0.6, 0.1, 0.6), Q ζ (x4 ) = (0.5, 0.6, 0.6), Q ζ (x5 ) = (0.8, 0.6, 0.3). Q ζ (x5 ) = (0.6, 0.1, 0.3), Now,  ζ (Q) = (x1 , 0.4, 0.4, 0.4), (x2 , 0.5, 0.6, 0.4), (x3 , 0.5, 0.1, 0.4), (x4 , 0.5, 0.6, 0.6),  (x5 , 0.6, 0.1, 0.3) ,  ζ (Q) = (x1 , 0.4, 0.6, 0.4), (x2 , 0.7, 0.6, 0.4), (x3 , 0.5, 0.4, 0.4), (x4 , 0.6, 0.1, 0.6),  (x5 , 0.8, 0.6, 0.3) .

1.7 m−Polar Fuzzy Soft Rough Sets

61

Hence, the pair (ζ (Q), ζ (Q)) is called a 3−polar fuzzy soft rough set. Theorem 1.20 Let (U, T, ζ ) be an m−polar fuzzy soft approximation space. Then, the lower and upper soft rough m−polar fuzzy approximation operators ζ (Q) and ζ (Q), respectively, satisfy the following properties, for any Q, R ∈ mF T , 1. 2. 3. 4. 5. 6. 7. 8.

ζ (Q) =∼ ζ (∼ Q), Q ⊆ R ⇒ ζ (Q) ⊆ ζ (R), ζ (Q ∩ R) = ζ (Q) ∩ ζ (R), ζ (Q ∪ R) ⊇ ζ (Q) ∪ ζ (R), ζ (Q) =∼ ζ (∼ Q), Q ⊆ R ⇒ ζ (Q) ⊆ ζ (R), ζ (Q ∪ R) = ζ (Q) ∪ ζ (R), ζ (Q ∩ R) ⊆ ζ (Q) ∩ ζ (R),

where ∼ Q denotes the complement of Q. 1. From Definition 1.45, (   ) ∼ ζ (∼ Q) = v, 1 − (∼ Q)ζ (v) | v ∈ U , (   ) = v, 1 − pi ◦ (∼ Q)ζ (v, w) ∧ pi ◦ (∼ Q)(w) | v ∈ U , = = =

w∈T

(

v, 1 ∧

(

)    1 − pi ◦ Q ζ (v, w) ∨ pi ◦ Q(w) | v ∈ U ,

w∈T

v,

(



)   1 − pi ◦ Q ζ (v, w) ∨ pi ◦ Q(w) | v ∈ U ,

w∈T

)  v, Q ζ (v) | v ∈ U ,

= ζ (Q). Thus, ζ (Q) =∼ ζ (∼ Q). 2. It can be proved directly by Definition 1.45. 3. By Definition 1.45, ( )  v, (Q ∩ R)ζ (v) | v ∈ U , (   )   1 − pi ◦ (Q ∩ R)(v, w) ∨ pi ◦ (Q ∩ R)(w) | v ∈ U , = v,

ζ (Q ∩ R) =

w∈T

 (      ) 1 − pi ◦ Q(v, w) ∧ R(v, w) ∨ pi ◦ Q(w) ∧ R(w) | v ∈ U , = v, w∈T

=

( )  v, Q ζ (v) ∧ Rζ (v) | v ∈ U ,

= ζ (Q) ∩ ζ (R).

Hence, ζ (Q ∩ R) = ζ (Q) ∩ ζ (R).

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Table 1.68 An 3−polar fuzzy soft relation ζ v1 w1 w2 w3 w4

(0.6, 0.3, 0.1) (0.5, 0.3, 0.2) (0.3, 0.2, 0.1) (0.4, 0.3, 0.6)

v2

v3

(0.4, 0.7, 0.6) (0.5, 0.2, 0.8) (0.3, 0.4, 0.8) (0.5, 0.1, 0.4)

(0.4, 0.6, 0.2) (0.6, 0.9, 0.6) (0.7, 0.3, 0.5) (0.3, 0.1, 0.0)

4. Using Definition 1.45, ( )  v, (Q ∪ R)ζ (v) | v ∈ U , )  (    1 − pi ◦ (Q ∪ R)(v, w) ∨ pi ◦ (Q ∪ R)(w) | v ∈ U , = v,

ζ (Q ∪ R) =

w∈T

⊇

(       ) v, 1 − pi ◦ Q(v, w) ∨ R(v, w) ∨ pi ◦ Q(w) ∨ R(w) | v ∈ U , w∈T

( )  = v, Q ζ (v) ∨ Rζ (v) | v ∈ U , = ζ (Q) ∪ ζ (R).

Thus, ζ (Q ∪ R) ⊇ ζ (Q) ∪ ζ (R). Proof The properties (5–8) can be proved by using similar arguments.



Example 1.15 Let U = {w1 , w2 , w3 , w4 } be the set of four cars and let T =   v1 , v2 , v3 be the set of parameters, where • v1 denotes the Fuel efficiency, • v2 denotes the Price, • v3 denotes the Technology. Consider a 3−polar fuzzy soft relation ζ : U → T given by Table 1.68 Consider 3−polar fuzzy subsets Q, R of T as follows:   Q = (v1 , 0.2, 0.1, 0.9), (v2 , 0.7, 0.5, 0.3), (v3 , 0.5, 0.6, 0.1) ,   R = (v1 , 0.4, 0.2, 0.5), (v2 , 0.6, 0.7, 0.3), (v3 , 0.4, 0.7, 0.8) . Then,   ∼ Q = (v1 , 0.8, 0.9, 0.1), (v2 , 0.3, 0.5, 0.7), (v3 , 0.5, 0.4, 0.9) ,   Q ∪ R = (v1 , 0.4, 0.2, 0.9), (v2 , 0.7, 0.7, 0.3), (v3 , 0.5, 0.7, 0.8) ,   Q ∩ R = (v1 , 0.2, 0.1, 0.5), (v2 , 0.6, 0.5, 0.3), (v3 , 0.4, 0.6, 0.1) .

1.7 m−Polar Fuzzy Soft Rough Sets

63

By Definition 1.45,   ζ (Q) = (w1 , 0.4, 0.5, 0.4), (w2 , 0.5, 0.6, 0.3), (w3 , 0.5, 0.6, 0.3), (w4 , 0.6, 0.7, 0.6) ,   ζ (Q) = (w1 , 0.4, 0.6, 0.3), (w2 , 0.5, 0.6, 0.3), (w3 , 0.5, 0.4, 0.3), (w4 , 0.5, 0.1, 0.6) ,   ζ (R) = (w1 , 0.4, 0.7, 0.4), (w2 , 0.4, 0.7, 0.3), (w3 , 0.4, 0.7, 0.3), (w4 , 0.6, 0.7, 0.5) ,   ζ (R) = (w1 , 0.4, 0.7, 0.3), (w2 , 0.5, 0.7, 0.6), (w3 , 0.4, 0.4, 0.5), (w4 , 0.5, 0.2, 0.5) ,   ∼ ζ (∼ Q) = (w1 , 0.4, 0.6, 0.3), (w2 , 0.5, 0.6, 0.3), (w3 , 0.5, 0.4, 0.3), (w4 , 0.5, 0.1, 0.6) ,   ∼ ζ (∼ Q) = (w1 , 0.4, 0.5, 0.4), (w2 , 0.5, 0.6, 0.3), (w3 , 0.5, 0.6, 0.3), (w4 , 0.6, 0.7, 0.6) ,   ζ (Q ∪ R) = (w1 , 0.4, 0.7, 0.4), (w2 , 0.5, 0.7, 0.3), (w3 , 0.5, 0.7, 0.3), (w4 , 0.6, 0.7, 0.6) ,   ζ (Q ∪ R) = (w1 , 0.4, 0.7, 0.3), (w2 , 0.5, 0.7, 0.6), (w3 , 0.5, 0.4, 0.5), (w4 , 0.5, 0.2, 0.6) ,   ζ (Q ∩ R) = (w1 , 0.4, 0.5, 0.4), (w2 , 0.4, 0.6, 0.3), (w3 , 0.4, 0.6, 0.3), (w4 , 0.6, 0.7, 0.5) ,   ζ (Q ∩ R) = (w1 , 0.4, 0.6, 0.3), (w2 , 0.5, 0.6, 0.3), (w3 , 0.4, 0.4, 0.3), (w4 , 0.5, 0.1, 0.5) .

Now,   ζ (Q) ∪ ζ (R) = (w1 , 0.4, 0.7, 0.4), (w2 , 0.5, 0.7, 0.3), (w3 , 0.5, 0.7, 0.3), (w4 , 0.6, 0.7, 0.6) ,   ζ (Q) ∪ ζ (R) = (w1 , 0.4, 0.7, 0.3), (w2 , 0.5, 0.7, 0.6), (w3 , 0.5, 0.4, 0.5), (w4 , 0.5, 0.2, 0.6) ,   ζ (Q) ∩ ζ (R) = (w1 , 0.4, 0.5, 0.4), (w2 , 0.4, 0.6, 0.3), (w3 , 0.4, 0.6, 0.3), (w4 , 0.6, 0.7, 0.5) ,   ζ (Q) ∩ ζ (R) = (w1 , 0.4, 0.6, 0.3), (w2 , 0.5, 0.6, 0.3), (w3 , 0.4, 0.4, 0.3), (w4 , 0.5, 0.1, 0.5) .

From the above calculations, ∼ ζ (∼ Q) = ζ (Q), ζ (Q ∩ R) = ζ (Q) ∩ ζ (R), ζ (Q ∪ R) = ζ (Q) ∪ ζ (R),

∼ ζ (∼ Q) = ζ (Q), ζ (Q ∪ R) ⊇ ζ (Q) ∪ ζ (R), ζ (Q ∩ R) ⊆ ζ (Q) ∩ ζ (R).

Remark 1.5 1. In Theorem 6.1, properties (1) and (5) show that the lower and upper m−polar fuzzy soft rough approximations operators ζ and ζ , respectively, are dual to one another. Proposition 1.5 Let (U, T, ζ ) be an m−polar fuzzy soft approximation space. Then, the lower and upper soft rough approximations of m−polar fuzzy sets Q and R satisfy the following laws:   1. ∼ ζ (Q) ∪ ζ (R) = ζ (∼ Q) ∩ ζ (∼ R),   2. ∼ ζ (Q) ∪ ζ (R) = ζ (∼ Q) ∩ ζ (∼ R),   3. ∼ ζ (Q) ∪ ζ (R) = ζ (∼ Q) ∩ ζ (∼ R),   4. ∼ ζ (Q) ∪ ζ (R) = ζ (∼ Q) ∩ ζ (∼ R),   5. ∼ ζ (Q) ∩ ζ (R) = ζ (∼ Q) ∪ ζ (∼ R),   6. ∼ ζ (Q) ∩ ζ (R) = ζ (∼ Q) ∪ ζ (∼ R),

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1 Hybrid Multi-polar Fuzzy Models

  7. ∼ ζ (Q) ∩ ζ (R) = ζ (∼ Q) ∪ ζ (∼ R),   8. ∼ ζ (Q) ∩ ζ (R) = ζ (∼ Q) ∪ ζ (∼ R). Proof Its proof follows immediately from Definition 1.45.



Definition 1.46 Let U be a universe, Q = {(v, pi ◦ Q(v)) | v ∈ U } ∈ mF U , and σ ∈ [0, 1]m . The σ -level cut set of Q and the strong σ -level cut set of Q, denoted by Q σ and Q σ + , respectively, are defined as follows: Q σ = {v ∈ U | pi ◦ Q(v) ≥ σ }, Q σ + = {v ∈ U | pi ◦ Q(v) > σ }. Definition 1.47 Let ζ be an m−polar fuzzy soft relation on U × T , it is defined ζσ = {(v, w) ∈ U × T | pi ◦ ζ (v, w) ≥ σ }, ζσ (v) = {w ∈ T | pi ◦ ζ (v, w) ≥ σ }, ζσ + = {(v, w) ∈ U × T | pi ◦ ζ (v, w) > σ }, ζσ + (v) = {w ∈ T | pi ◦ ζ (v, w) > σ }. Then, ζσ and ζσ + are two crisp soft relations on U × T . It is proved that the m−polar fuzzy soft rough approximation operators can be described by crisp soft rough approximation operators. Theorem 1.21 Let (U, T, ζ ) be an m−polar fuzzy soft approximation space and Q ∈ mF T . Then, the upper m−polar fuzzy soft rough approximation operator can be described as follows, ∀ v ∈ U : 1. Q ζ (v) =

   σ ∧ ζ σ (Q σ )(v) = σ ∈[0,1]m

=

   σ ∧ ζ σ (Q σ + )(v) , σ ∈[0,1]m

   σ ∧ ζ σ + (Q σ )(v) =

σ ∈[0,1]m

 

 σ ∧ ζ σ + (Q σ + )(v) .

σ ∈[0,1]m

2. [ζ (Q)]σ + ⊆ ζ σ + (Q σ + ) ⊆ ζ σ + (Q σ ) ⊆ ζ σ (Q σ ) ⊆ [ζ (Q)]σ . 1. For all v ∈ U ,

1.7 m−Polar Fuzzy Soft Rough Sets 



σ ∈[0,1]m

65

 σ ∧ ζ σ (Q σ )(v) = sup{σ ∈ [0, 1]m | v ∈ ζ σ (Q σ )}, = sup{σ ∈ [0, 1]m | ζσ (v) ∩ Q σ }, = sup{σ ∈ [0, 1]m | ∃ w ∈ T [w ∈ ζσ (v), w ∈ Q σ ]}, = sup{σ ∈ [0, 1]m | ∃ w ∈ T [ pi ◦ Q ζ (v, w) ≥ σ, pi ◦ Q(w) ≥ σ ]},    pi ◦ Q ζ (v, w) ∧ pi ◦ Q(w) , = w∈T

= Q ζ (v).

By similar arguments, it is computed Q ζ (v) =

 σ ∈[0,1]m

  σ ∧ ζ σ (Q σ + )(v) =

 σ ∈[0,1]m

  σ ∧ ζ σ + (Q σ )(v) =

 σ ∈[0,1]m

  σ ∧ ζ σ + (Q σ + )(v) .

2. By Definitions 10.20 and 1.47, it is directly verified that ζ σ + (Q σ + ) ⊆ ζ σ + (Q σ ) ⊆ ζ σ (Q σ ). Now, it is sufficient to show that [ζ (Q)]σ + ⊆ ζ σ + (Q σ + ) and ζ σ (Q σ ) ⊆ [ζ (Q)]σ . *  pi ◦ Q ζ For all v ∈ [ζ (Q)]σ + , we have Q ζ (v) > σ . By Definition 1.45, w∈T  (v, w) ∧ pi ◦ Q(w) > σ . Then, there exists w0 ∈ T , such that pi ◦ Q ζ (v, w0 ) ∧ pi ◦ Q(w0 ) > σ , that is, pi ◦ Q ζ (v, w0 ) > σ and pi ◦ Q(w0 ) > σ . Thus, w0 ∈ ζσ + (v) and w0 ∈ Q σ . It follows that ζσ + (v) ∩ Q σ = ∅. It is given v ∈ ζ σ + (Q σ + ). Hence, [ζ (Q)]σ + ⊆ ζ σ + (Q σ + ). To prove ζ σ (Q σ ) ⊆ [ζ *(Q)]σ , let an arbitrary v ∈ ζ σ (Q σ ), it is given ζ σ (Q σ )(v) = [ζ σ (Q σ )(v)] ≥ σ ∧ ζ σ (Q σ )(v) = σ , it is obtained v ∈ 1. Since Q ζ (v) = σ ∈[0,1]m

[ζ (Q)]σ . Hence, ζ σ (Q σ ) ⊆ [ζ (Q)]σ . Theorem 1.22 Let (U, T, ζ ) be an m−polar fuzzy soft approximation. If ζ is serial, then the lower and upper m−polar fuzzy soft rough approximation operators ζ (Q) and ζ (Q), respectively, satisfy the following: 1. ζ (∅) = ∅. ζ (T ) = U, 2. ζ (Q) ⊆ ζ (Q), for all Q ∈ mF T . Proof Its proof follows directly by Definition 1.45.



Definition 1.48 Let Q be an m−polar fuzzy set of the universe set U and let ζ (Q), ζ (Q) be the lower and upper soft rough approximation operators. Then, ring sum operation about m−polar fuzzy sets ζ (Q) and ζ (Q) is defined by ζ (Q) ⊕ ζ (Q) =

   v, pi ◦ Q ζ (v) + pi ◦ Q ζ (v) − pi ◦ Q ζ (v) × pi ◦ Q ζ (v) | v ∈ U .

In next subsections, the applications of m−polar fuzzy soft rough model to MCDM problems are presented.

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1.7.1 Selection of a Hotel The selection of the right hotel to stay is always a difficult task. Since every person has different needs when searching for a hotel. The location of the hotel is something that is very important for an enjoyable stay. There are a number of factors to take into consideration for selecting the right hotel, whether we are looking for a great location, a great meal option or a great service. Suppose a person (Mr. Adeel) wants to stay in a hotel for a long period. There are four alternatives in his mind. The alternatives are y1 , y2 , y3 , y4 . He wants to select the most suitable hotel. The location, meal options and services are the main parameters for the selection of a hotel. Let U = {y1 , y2 , y3 , y4 } be the set of four hotels under consideration and let T = {z 1 , z 2 , z 3 } be the set of parameters related to the hotels in U , where, ‘z 1 ’ represents the Location, ‘z 2 ’ represents the Meal Options, ‘z 3 ’ represents the Services. The more features of these parameters are described as follows: • The “Location” of the hotel includes close to main road, in the green surroundings, in the city center. • The “Meal options” of the hotel include fast food, fast casual, casual dining. • The “Services” of the hotel include Wi-Fi connectivity, fitness center, room service. Suppose that Adeel explains the “attractiveness of the hotel” by forming a 3−polar fuzzy soft relation ζ : U → T , which is given by Table 1.69. Thus, ζ over U × T is the 3−polar fuzzy soft relation in which location, meal option and price of the hotels are considered. For example, if we consider “Location” of the hotel, ((y1 , z 1 ), 0.2, 0.6, 0.1) means that the hotel y1 is 20% close to the main road, 60% in the green surroundings and 10% in the city center. It is assumed that Adeel gives the optimal normal decision object Q, which is a 3−polar fuzzy subset of T as follows:   Q = (z 1 , 0.5, 0.6, 0.7), (z 2 , 0.7, 0.6, 0.9), (z 3 , 0.9, 0.6, 0.8) .

Table 1.69 3−polar fuzzy soft relation z1 ζ y1 y2 y3 y4

(0.2, 0.6, 0.1) (0.4, 0.5, 0.7) (0.7, 0.8, 0.3) (0.5, 0.6, 0.4)

z2

z3

(0.3, 0.4, 0.7) (0.4, 0.5, 0.5) (0.8, 0.9, 0.4) (0.6, 0.7, 0.1)

(0.7, 0.3, 0.2) (0.7, 0.4, 0.1) (0.6, 0.2, 0.6) (0.8, 0.5, 0.3)

1.7 m−Polar Fuzzy Soft Rough Sets

By Definition 1.45, Q ζ (y1 ) = (0.7, 0.6, 0.8), Q ζ (y2 ) = (0.6, 0.6, 0.7), Q ζ (y3 ) = (0.5, 0.6, 0.7), Q ζ (y4 ) = (0.5, 0.6, 0.7),

67

Q ζ (y1 ) = (0.7, 0.6, 0.7), Q ζ (y2 ) = (0.7, 0.5, 0.7), Q ζ (y3 ) = (0.7, 0.6, 0.6), Q ζ (y4 ) = (0.8, 0.6, 0.4).

Now, 3−polar fuzzy soft rough approximation operators ζ (Q), ζ (Q), respectively, are given by   ζ (Q) = (y1 , 0.7, 0.6, 0.8), (y2 , 0.6, 0.6, 0.7), (y3 , 0.5, 0.6, 0.7), (y4 , 0.5, 0.6, 0.7) ,   ζ (Q) = (y1 , 0.7, 0.6, 0.7), (y2 , 0.7, 0.5, 0.7), (y3 , 0.7, 0.6, 0.6), (y4 , 0.8, 0.6, 0.4) .

These operators are very close to the decision alternatives yn , n = 1, 2, 3, 4. By Definition 1.48, the choice set is described as follows:  ζ (Q) ⊕ ζ (Q) = (y1 , 0.91, 0.84, 0.94), (y2 , 0.88, 0.8, 0.91),  (y3 , 0.85, 0.84, 0.88), (y4 , 0.9, 0.84, 0.82) . Thus, Mr. Adeel will select the hotel y1 to stay because the optimal decision in the choice set ζ (Q) ⊕ ζ (Q) is y1 .

1.7.2 Selection of a Place Choosing a place to go when some people have the opportunity to travel can sometimes be very difficult task. Suppose that a group of ten people plan a tour to a suitable place in a country Z . There are four alternatives in their mind. The alternatives are q1 , q2 , q3 , q4 . They want to select the best place for the tour. It is a challenge to find advice in one place. The environment and cost are the main parameters for the selection of a suitable place. In the environment of the place, they want to check whether the place has availability of built environment, natural environment and social environment. The term built environment refers to the man-made surroundings. Built environment of the place includes buildings, parks and every other things that are made by human beings. Natural environment of the place includes forests, oceans, rivers, lakes, atmosphere, climate, weather, etc. The social environment includes the culture and lifestyle of the human beings. Lastly, the tour cost is an important criteria for the place selection. It includes low, medium and high. Let U = {q1 , q2 , q3 , q4 } be the set of four places and T = {a1 , a2 } be the set of parameters, where ‘a1 ’ represents the Environment, ‘a2 ’ represents the Tour Cost.

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Table 1.70 3−polar fuzzy soft relation ζ a1 q1 q2 q3 q4

(0.8, 0.8, 0.9) (0.5, 0.7, 0.6) (0.8, 0.6, 0.7) (0.7, 0.9, 0.6)

a2 (0.4, 0.7, 0.6) (0.5, 0.7, 0.8) (0.8, 0.9, 0.4) (0.6, 0.7, 0.8)

The more characteristics of these parameters are described as follows: • The “Environment” of the place includes built environment, natural environment, and social environment. • The “Tour Cost” of the place may be low, medium, or high. Suppose that they describe the “attractiveness of the place” by constructing a 3−polar fuzzy soft relation ζ over U × T , which is given in Table 1.70. Thus, ζ : U → T is the 3−polar fuzzy soft relation in which environment and tour cost of the places are considered. For example, if we consider “Environment” of the place, ((q1 , a1 ), 0.8, 0.8, 0.9) means that the place q1 includes 80% built environment, 80% natural environment and 90% social environment. It is assumed that they give the optimal normal decision object Q, which is a 3−polar fuzzy subset of T as follows:   Q = (a1 , 0.8, 0.7, 0.9), (a2 , 0.7, 0.6, 0.8) . From Definition 1.45, Q ζ (q1 ) = (0.7, 0.6, 0.8), Q ζ (q1 ) = (0.8, 0.7, 0.9), Q ζ (q2 ) = (0.7, 0.6, 0.8), Q ζ (q2 ) = (0.5, 0.7, 0.8), Q ζ (q3 ) = (0.8, 0.6, 0.7), Q ζ (q3 ) = (0.7, 0.6, 0.8), Q ζ (q4 ) = (0.7, 0.7, 0.8). Q ζ (q4 ) = (0.7, 0.6, 0.8), The 3-polar fuzzy soft rough approximation operators ζ (Q), ζ (Q) are described as follows:   ζ (Q) = (q1 , 0.7, 0.6, 0.8), (q2 , 0.7, 0.6, 0.8), (q3 , 0.7, 0.6, 0.8), (q4 , 0.7, 0.6, 0.8) ,   ζ (Q) = (q1 , 0.8, 0.7, 0.9), (q2 , 0.5, 0.7, 0.8), (q3 , 0.8, 0.6, 0.7), (q4 , 0.7, 0.7, 0.8) .

These operators are very close to the decision alternatives qn , n = 1, 2, 3, 4. By Definition 1.48,  ζ (Q) ⊕ ζ (Q) = (q1 , 0.94, 0.88, 0.98), (q2 , 0.85, 0.88, 0.96),  (q3 , 0.94, 0.84, 0.94), (q4 , 0.91, 0.88, 0.96) .

1.7 m−Polar Fuzzy Soft Rough Sets

69

Thus, the optimal decision in the choice set ζ (Q) ⊕ ζ (Q) is q1 . Therefore, they will select the place q1 for the tour.

1.7.3 Selection of a House Buying a house is an exhilarating time in many people lives, but it is also a very difficult task to those who are not particularly real estate savvy. There are a number of factors to take into consideration for buying the house such as location of the house, size of the house and price of the house. These factors among many others influence house buyers before they even get to start thinking about buying a new house. Suppose a person (Mr. Ali) wants to buy a house. The alternatives in his mind are u 1 , u 2 , u 3 . The size, location and price are the main parameters for the selection of a suitable house. Let U = {u 1 , u 2 , u 3 } be the set of three houses and let T = {t1 , t2 , t3 } be the set of parameters related to the houses in U , where ‘t1 ’ represents the Size, ‘t2 ’ represents the Location, ‘t3 ’ represents the Price. We give further characteristics of these parameters. • The “Size” of the house includes small , large, and very large. • The “Location” of the house includes close to the main road, in the green surroundings, and in the city center. • The “Price” of the house includes low, medium, and high. Suppose that Ali describes the “attractiveness of the house” by forming a 3−polar fuzzy soft relation ζ : U → T , which is given by Table 1.71. Thus, ζ over U × T is the 3−polar fuzzy soft relation in which size, location and price of the houses are considered. For example, if we consider “Location” of the house, ((u 2 , t1 ), 0.8, 0.9, 0.1) means that the house u 1 is, 80% close to the main road, 90% in the green surroundings and 10% in the city center. It is assumed that Ali gives the optimal normal decision object Q, which is a 3−polar fuzzy subset of T as follows:

Table 1.71 3−polar fuzzy soft relation ζ t1 u1 u2 u3

(0.5, 0.7, 0.9) (0.8, 0.9, 0.1) (0.9, 0.7, 0.6)

t2

t3

(0.7, 0.6, 0.8) (0.6, 0.8, 0.9) (0.9, 0.8, 0.9)

(0.5, 0.6, 0.9) (0.8, 0.4, 0.2) (0.4, 0.6, 0.3)

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1 Hybrid Multi-polar Fuzzy Models

  Q = (t1 , 0.6, 0.8, 0.7), (t2 , 0.5, 0.8, 0.8), (t3 , 0.9, 0.8, 0.7) . By Definition 1.45, Q ζ (u 1 ) = (0.5, 0.8, 0.7), Q ζ (u 2 ) = (0.5, 0.8, 0.8), Q ζ (u 3 ) = (0.5, 0.8, 0.7),

Q ζ (u 1 ) = (0.5, 0.7, 0.8), Q ζ (u 2 ) = (0.8, 0.8, 0.8), Q ζ (u 3 ) = (0.6, 0.8, 0.8).

Now, 3-polar fuzzy soft rough approximation operators ζ (Q), ζ (Q), respectively, are given by   ζ (Q) = (u 1 , 0.5, 0.8, 0.7), (u 2 , 0.5, 0.8, 0.8), (u 3 , 0.5, 0.8, 0.7) ,   ζ (Q) = (u 1 , 0.5, 0.7, 0.8), (u 2 , 0.8, 0.8, 0.8), (u 3 , 0.6, 0.8, 0.8) . These operators are very close to the decision alternatives u n , n = 1, 2, 3. Using Definition 1.48,   ζ (Q) ⊕ ζ (Q) = (u 1 , 0.75, 0.94, 0.94), (u 2 , 0.9, 0.96, 0.96), (u 3 , 0.8, 0.96, 0.94) .

Hence, Ali will buy the house u 2 because the optimal decision in the choice set ζ (Q) ⊕ ζ (Q) is u 2 . The method of selecting a suitable object is explained in the following Algorithm 1.6. Algorithm 1.6 m–polar fuzzy soft rough sets 1. Input U as universe of discourse. 2. Input T as a set of parameters. 3. Construct an m−polar fuzzy soft relation ζ : U → T according to the different needs of the decision maker. 4. Give an m−polar fuzzy subset Q over T , which is an optimal normal decision object according to the various requirements of the decision maker. 5. Compute the m−polar fuzzy soft rough approximation operators ζ (Q) and ζ (Q) by Definition 1.45. 6. Find the choice set S = ζ (Q) ⊕ ζ (Q) by Definition 1.48. * 7. Select the optimal decision u k . If pi ◦ S(u k ) ≥ M, where M = pi ◦ S(u k ), 1≤k≤n

n is equal to the number of objects in U , and then the optimal decision will be u k . If there exist too many optimal choices in step 7 of Algorithm 1.6, that is, u ki = u k j , where 1 ≤ ki = k j ≤ n, change the optimal normal decision object Q and repeat the Algorithm 1.6 so that the final decision is only one.

1.8 Soft m−Polar Fuzzy Rough Sets

71

1.8 Soft m−Polar Fuzzy Rough Sets In this section, the concept of pseudo m–polar fuzzy soft sets is presented, which provide the information about the features of alternatives with multi-polar information. Further, soft m–polar fuzzy rough sets are presented which are the combination of soft sets and m–polar fuzzy rough sets and the generalization of previously defined models. Definition 1.49 ([5]) Let S be a universe and A be a set of parameters. A pair (ζ˜ −1 , A) is called a pseudo m–polar fuzzy soft set over the universe S if and only if ζ˜ −1 : S → m F(A) is a mapping of S into all m–polar fuzzy subsets of the set A, where m F(A) expresses all m–polar fuzzy subsets of parameter set A. i.e., ζ˜ −1 (s, a) ∈ [0, 1]m , ∀s ∈ S, a ∈ A. Remark 1.6 From Definition of pseudo m–polar fuzzy soft set, it is known that the pseudo m–polar fuzzy mapping ζ˜ −1 : S → m F(A) is a binary m–polar fuzzy relation defined between the universe S and parameter set A. i.e., for any s j ∈ S, ak ∈ A, ζ˜ −1 (s j , ak ) ∈ m F(S × A). In general, reflexive, symmetric and transitive properties do not hold in ζ˜ −1 (s j , ak ). Therefore, ζ˜ −1 (s j , ak ) is an arbitrary m–polar fuzzy binary relation. Example 1.16 Let C = {c1 , c2 , c3 , c4 , c5 } be a universe of five cars under observation, and A = {a1 , a2 , a3 , a4 } be the set of parameters, where the parameter, “a1 ” represents the price of car, “a2 ” represents the color of car, “a3 ” represents the body types of car, “a4 ” represents the attractiveness of car. We give further characteristics of these parameters. • • • •

The “price of car” may be cheap, costly, very costly. The “color of car” may be the combination of white, grey, silver. The “body type of car” may be sedan, coupe, hatchback. The “attractiveness of car” may include flexibility, comfort, speed.

In order to define m–polar fuzzy soft set, it is meant to specify the characteristics of price, color, body type and attractiveness of a car. The m–polar fuzzy soft set (ζ, N ) expresses the characteristics of car that Mr. Z(say) wants to buy. It is shown in Table 1.72 as follows: The pseudo m–polar fuzzy soft set specifies the features of each car including price, color, body type and attractiveness with their different characteristics. By Definition 1.49, the results are as follows:  0.2, 0.3, 0.7 0.3, 0.8, 0.7 0.8, 0.3, 0.7 0.4, 0.3, 0.2 ), ( ), ( ), ( ) , ζ˜ −1 (c1 ) = ( a1 a2 a3 a4

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1 Hybrid Multi-polar Fuzzy Models

 0.8, 0.3, 0.5 0.6, 0.4, 0.3 0.2, 0.1, 0.3 0.5, 0.4, 0.6 ζ˜ −1 (c2 ) = ( ), ( ), ( ), ( ) , a1 a2 a3 a4  0.6, 0.7, 0.8 0.8, 0.7, 0.5 0.5, 0.3, 0.2 0.6, 0.8, 0.9 ˜ζ −1 (c3 ) = ( ), ( ), ( ), ( ) , a1 a2 a3 a4  0.9, 0.3, 0.5 0.9, 0.2, 0.3 0.8, 0.9, 0.1 0.2, 0.5, 0.4 ˜ζ −1 (c4 ) = ( ), ( ), ( ), ( ) , a1 a2 a3 a4  0.7, 0.5, 0.4 0.5, 0.4, 0.2 0.7, 0.8, 0.4 0.4, 0.2, 0.3 ˜ζ −1 (c5 ) = ( ), ( ), ( ), ( ) . a1 a2 a3 a4 This means that the car “c1 ” has different features as follows: • The “price of car” shows, it is 20% cheap, 30% costly and 70% very costly for most of the customers. • The “color of car” shows, it may have combination of colors as 30% white, 80% grey and 70% silver. • The “body type of car” shows, its shape is 80% sedan, 30% coupe and 70% hatchback. • The “attractiveness of car” shows, it has 40% flexibility , 30% comfort and 20% speed. Definition 1.50 Let S be a universe and (ζ˜ −1 , A) be a pseudo m–polar fuzzy soft set over universe S, where ζ˜ −1 be a mapping defined as, ζ˜ −1 : S → m F(A). The triple (S, A, ζ˜ −1 ) is called soft m–polar fuzzy approximation space . For any set X ∈ m F(A), the lower and upper approximations of X, ζ (X ) and ζ (X ) w.r.t. soft m–polar fuzzy approximation space (S, A, ζ˜ −1 ) are the m–polar fuzzy sets of S whose membership functions for each s ∈ S are defined respectively, as follows: ζ (X )(s) =

 (1 − ζ˜ −1 (s, a)) ∨ X (a) , a∈A

Table 1.72 3-polar fuzzy soft relation C/A a1 a2 c1 c2 c3 c4 c5

(0.2, 0.3, 0.7) (0.8, 0.3, 0.5) (0.6, 0.7, 0.8) (0.9, 0.3, 0.5) (0.7, 0.5, 0.4)

(0.3, 0.8, 0.9) (0.6, 0.4, 0.3) (0.8, 0.7, 0.5) (0.9, 0.2, 0.3) (0.5, 0.4, 0.2)

a3

a4

(0.8, 0.3, 0.7) (0.2, 0.1, 0.3) (0.5, 0.3, 0.2) (0.8, 0.9, 0.1) (0.7, 0.8, 0.4)

(0.4, 0.3, 0.2) (0.5, 0.4, 0.6) (0.6, 0.8, 0.9) (0.2, 0.5, 0.4) (0.8, 0.2, 0.3)

1.8 Soft m−Polar Fuzzy Rough Sets

ζ (X )(s) =

73



ζ˜ −1 (s, a) ∧ X (a) .

a∈A

The pair (ζ (X ), ζ (X )) is called soft m-polar fuzzy rough set of X w.r.t. (S, A, ζ˜ −1 ) and ζ , ζ : m F(A) → m F(S) are called lower and upper soft m–polar fuzzy rough approximation operators respectively. Furthermore, if ζ (X ) = ζ (X ), then X is said to be definable. Example 1.17 Re-consider the Example 1.16 and define a 3−polar fuzzy set of attributes, X as     0.6, 0.5, 0.8 0.9, 0.7, 0.2 0.8, 0.2, 0.3 0.3, 0.4, 0.5 , , , . X= a1 a2 a3 a4 From Definition 1.50, lower and upper approximations of X can be calculated respectively, as follows:

Thus, ζ (X ) = ζ (X ) =

ζ (X )(c1 ) = (0.7, 0.5, 0.3),

ζ (X )(c1 ) = (0.8, 0.5, 0.8),

ζ (X )(c2 ) = (0.3, 0.6, 0.4),

ζ (X )(c2 ) = (0.6, 0.4, 0.5),

ζ (X )(c3 ) = (0.4, 0.2, 0.3),

ζ (X )(c3 ) = (0.6, 0.5, 0.5),

ζ (X )(c4 ) = (0.3, 0.5, 0.5),

ζ (X )(c4 ) = (0.8, 0.7, 0.5),

ζ (X )(c5 ) = (0.3, 0.5, 0.6),

ζ (X )(c5 ) = (0.8, 0.7, 0.4).



 

0.7,0.5,0.3 c1

,

0.8,0.5,0.8 c1

,



 0.3,0.6,0.4 c2

,

0.6,0.4,0.5 c2

,



 0.4,0.2,0.3 c3

,

0.6,0.5,0.5 c3

,



 0.3,0.5,0.5 c4

,

0.8,0.7,0.5 c4

,



0.3,0.5,0.6 c5 0.8,0.7,0.4 c5

, .

Hence, ζ (X ) and ζ (X ) are the lower and upper approximations of 3−polar fuzzy subset X in parameter set A, and the pair (ζ (X ), ζ (X )) specifies the soft 4−polar fuzzy rough set. Theorem 1.23 Let (S, A, ζ˜ −1 ) be the soft m–polar fuzzy approximation space. The lower and upper approximations ζ (X ) and ζ (X ) satisfy the following properties for any X, Y ∈ m F(A), Proof It can easily be proved by using Definition 1.50.



In next subsection, the concept of soft m–polar fuzzy rough sets is applied to the real life examples

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1 Hybrid Multi-polar Fuzzy Models

1.8.1 Comparison of Popular Mobile Phones for Selection Mobile phones are essential part of our daily communications. All mobile phones have range for voice and simple text messaging services. Recently, mobiles with many more features and functions have become available. So, in this age of competition it becomes difficult to compare the features of mobiles for selection. An Apple’s iPhone company launches a new mobile phone with different features and specifications. Company wants to compare its new launched mobile phone with latest mobile phones of other companies. For this purpose, the idea of soft m–polar fuzzy rough sets is presented. Let (M, A, ζ˜ −1 ) be a soft m–polar fuzzy approximation space, where M = {m 1 , m 2 , m 3 , m 4 , m 5 } be a universe of five mobile phones specified as • • • • •

m1 m2 m3 m4 m5

= Apple’s iPhone 6, = Amazons Fir Phone, = Samsung Galaxy S6, = Motorola Moto X (2nd gen.), = HTC One (MB).

Let A = {a1 , a2 , a3 , a4 } be the set of parameters related to the mobile phones in M, where “a1 ” represents the Measurements, “a2 ” represents the Key Facts, “a3 ” represents the Visual Effects, “a4 ” represents the Price. The further characteristics of these parameters are described as follows: • • • •

The “Measurements” include dimensions, weights, slimness. The “Key Facts” include operating system, processor, memory. The “Visual Effects” include camera, display, sensor. The “Price” includes cheap, costly, very costly.

However, for such a multi-criteria decision making problem, one wishes to determine the decision substitute in universe with the estimation value as greater as possible on the whole estimated index. Thus, an ideally conventional decision object X is constructed on the m–polar fuzzy set of parameters A as follows: X = max{ pi ◦ ζ˜ −1 (m j , ak )|m j ∈ M}, i ∈ m. Now, the soft m–polar fuzzy rough lower approximation ζ (X ) and upper approximation ζ (X ) of the ideally conventional decision object X are calculated in Table 1.73, by using the Definition 1.50. Moreover, the rough lower and upper approximations are relatively close values to the approximated set of universe of mobiles. Thus, the relatively close values ζ (X )(m j ) and ζ (X )(m j ) are attained to the decision substitute m j ∈ M, by the soft m–polar fuzzy rough lower and upper approximations

1.8 Soft m−Polar Fuzzy Rough Sets

75

Table 1.73 3−polar fuzzy soft relation Measurements

Key Facts

Visual Effects

Price

m1

(0.2, 0.6, 0.8)

(0.8, 0.5, 0.6)

(0.9, 0.6, 0.2)

(0, 0.9, 0.1)

m2

(0.1, 0.5, 0.7)

(0.2, 0.9, 0.1)

(0, 0.2, 0.6)

(0.1, 0.1, 0.2)

m3

(0.6, 0.2, 0.1)

(0.4, 0.3, 0.6)

(0.2, 0.3, 0.3)

(0.2, 0.8, 0.1)

m4

(0.8, 0.1, 0.7)

(0.3, 0.5, 0.4)

(0.5, 0.2, 0.5)

(0.6, 0.4, 0.4)

m5

(0.2, 0.3, 0.2)

(0.8, 0.7, 0.6)

(0.6, 0.7, 0.1)

(0.2, 0.3, 0.4)

X

(0.8, 0.6, 0.8)

(0.8, 0.9, 0.6)

(0.9, 0.7, 0.6)

(0.6, 0.9, 0.4)

ζ˜ −1

Table 1.74 Choice values ζ (X )

ζ (X )

Choice value (ρm j )

m1

(0.8, 0.6, 0.6)

(0.9, 0.9, 0.8)

ρm 1 =(1.7, 1.6, 1.4)

m2

(0.8, 0.6, 0.6)

(0.2, 0.9, 0.7)

ρm 2 =(1.0, 1.5, 1.3)

m3

(0.8, 0.7, 0.6)

(0.6, 0.8, 0.6)

ρm 3 =(1.4, 1.5, 1.2)

m4

(0.6, 0.8, 0.6)

(0.8, 0.5, 0.7)

ρm 4 =(1.4, 1.3, 1.3)

m5

(0.8, 0.7, 0.6)

(0.8, 0.7, 0.6)

ρm 5 =(1.6, 1.4, 1.2)

of the m–polar fuzzy subset X . Thus, the choice value ρm j is enumerated for the decision substitute m j on the universe of sites M as follows: ρm j = pi ◦ ζ (X )(m j ) + pi ◦ ζ (X )(m j ), i ∈ m, m j ∈ M. Finally choosing the mobile phone m j ∈ M, which has the maximum choice value ρm j as the most favorable decision for the given MCDM problem. From Table 1.74. it is easy to compare the choice values of all the mobile phones. m 1 =Apple’s iPhone 6, has the maximum choice value as compared to all other mobile phones. So, Apple’s iPhone 6 is best for selection as compared to all other mobile phones. Generally, if there occurs two or more items m j ∈ M with the same maximum choice value ρm j , then take one of them according to your choice as the ideal decision for the given MCDM problems.

1.8.2 Selection of a Site for Construction of a Grid Station An electricity grid is an interdependent chain for providing the electricity from source to user. It is an amenity project aiming to provide relief to citizens, rather than a commercial activity. Selection of site for construction of grid station is the early and significant process. This requires accurate planning, skillful investigation and admin-

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1 Hybrid Multi-polar Fuzzy Models

istration so that the selected site is mechanically, economically, environmentally and socially perfect for requirements. For this purpose, the concept of soft m–polar fuzzy rough set theory is used. Let (S, A, ζ˜ −1 ) be a soft m–polar fuzzy approximation space, and S = {Ss , Sn , Se , Sw , Sc } be a set of sites for grid station specified as • • • • •

Ss = Site in “south” of city, Sn = Site in “north” of city, Se = Site in “east” of city, Sw = Site in “west” of city, Sc = Site in “center” of city.

Let A = {a1 , a2 , a3 , a4 } be the set of parameters related to the sites of grid station in S, where, “a1 ” represents the Energy Sources, “a2 ” represents the Transportation, “a3 ” represents the Area Attributes, “a4 ” represents the Energy Storage Stations. Further characteristics of parameters are given in Fig. 1.3, that explains four different parameters with the deep classification of characteristics, each parameter is further classified in three different characteristics. An ideally conventional decision object X on the m–polar fuzzy set of parameters A is calculated in Table 1.75. The choice value ρs j , for the decision substitute s j on the universe of sites S is calculated in Table 1.76. From Table 1.76, it is easy to compare the choice values of all the required sites. Site in “west” of city, has the maximum choice value as compared to all other sites. So, it is best site for construction as compared to all other sites.

Parameters

Energy Storage Stations Energy Sources

Area Attributes Transportation Substations

Construction availability

Neighborhood battery storage

Fossil-fired power plant Wind+battery storage Harmonic pollution Accessibility to energy sources Pumped hydro-storage Commercial campus with thermal storage Wire connections through poles

Accessibility to substations

Accessibility to residential + commercial areas

Fig. 1.3 Characteristics of parameters

1.8 Soft m−Polar Fuzzy Rough Sets

77

Table 1.75 3−polar fuzzy soft relation ζ˜ −1 Energy sources Transportation Ss Sn Se Sw Sc X

(0.45, 0.32, 0.81) (0.51, 0.68, 0.77) (0.23, 0.82, 0.55) (0.53, 0.82, 0.29) (0.88, 0.21, 0.66) (0.88, 0.82, 0.81)

(0.52, 0.58, 0.23) (0.12, 0.76, 0.77) (0.52, 0.80, 0.63) (0.92, 0.36, 0.27) (0.89, 0.76, 0.54) (0.92, 0.80, 0.77)

Table 1.76 Choice value ζ (X )

Area attributes

Energy storage stations

(0.35, 0.76, 0.89) (0.22, 0.78, 0.70) (0.57, 0.65, 0.27) (0.20, 0.72, 0.88) (0.70, 0.20, 0.73) (0.70, 0.78, 0.89)

(0.35, 0.81, 0.48) (0.72, 0.76, 0.59) (0.67, 0.39, 0.23) (0.73, 0.58, 0.55) (0.55, 0.46, 0.60) (0.73, 0.81, 0.60)

ζ (X )

Choice value (ρs j )

Ss

(0.70, 0.78, 0.60)

(0.52, 0.81, 0.81)

Sn

(0.73, 0.78, 0.60)

(0.72, 0.78, 0.77)

Se

(0.70, 0.78, 0.77)

(0.67, 0.82, 0.63)

Sw

(0.73, 0.78, 0.60)

(0.92, 0.82, 0.88)

Sc

(0.70, 0.78, 0.60)

(0.89, 0.76, 0.73)

ρss = (1.22, 1.59, 1.41) ρsn = (1.45, 1.56, 1.37) ρse = (1.37, 1.60, 1.40) ρsw = (1.65, 1.60, 1.48) ρsc = (1.59, 1.54, 1.33)

1.8.3 Comparison of Patients for Recovery of Heart Disease Traditionally, health plans, medicare and medicaid pay providers for whatever services they deliver, regardless of whether the services truly benefits the patient. How long some one takes to recover after an episode in intensive care depends on many things, including their age, prevention, health care and medication etc. For such a comparison in patients that which patient will recover soon with prevention and medication, the approach of soft m–polar fuzzy rough sets is used. Let (P, A p , ζ˜ −1 ) be a soft m–polar fuzzy approximation space and P = { p1 , p2 , p3 , p4 , p5 , p6 } be a set of six patients. Let A p = {a1 , a2 , a3 , a4 } be the set of parameters of prevention and treatment of heart disease related to the patients in P, where “a1 ” represents the Vaccination, “a2 ” represents the Health care, “a3 ” represents the Medication, “a4 ” represents the Surgery. The further characteristics of these parameters are described as follows:

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1 Hybrid Multi-polar Fuzzy Models

Table 1.77 4−polar fuzzy soft relation ζ˜ −1 Vaccination Healthcare p1 p2 p3 p4 p5 p6 X

(0.3, 0, 0.8, 0.4) (0.4, 0.7, 0.1, 0.3) (0.4, 0.4, 0.1, 0.6) (0.7, 0.5, 0.2, 0.4) (0.1, 0.6, 0.1, 0.5) (0.5, 0.6, 0.1, 0.3) (0.7, 0.7, 0.8, 0.6)

Table 1.78 Choice value ζ (X ) p1 p2 p3 p4 p5 p6

(0.7, 0.8, 0.7, 0.6) (0.7, 0.7, 0.7, 0.6) (0.7, 0.7, 0.6, 0.6) (0.7, 0.7, 0.6, 0.6) (0.7, 0.7, 0.7, 0.6) (0.7, 0.7, 0.6, 0.7)

(0.7, 0.2, 0.6, 0.8) (0.1, 0.1, 0.5, 0.7) (0.7, 0.2, 0.5, 0.8) (0.6, 0.8, 0.5, 0.5) (0.2, 0.7, 0.5, 0.7) (0.5, 0.6, 0.7, 0.8) (0.7, 0.8, 0.7, 0.8)

Medication

Surgery

(0.5, 0.3, 0.2, 0.7) (0.4, 0.6, 0.3, 0.2) (0.7, 0.5, 0.6, 0.2) (0.5, 0.8, 0.5, 0.3) (0.2, 0.8, 0.3, 0.2) (0.5, 0.6, 0.4, 0.2) (0.7, 0.8, 0.6, 0.7)

(0.6, 0.2, 0.8, 0.7) (0.5, 0.4, 0.6, 0.7) (0.6, 0.7, 0.5, 0.6) (0.7, 0.3, 0.8, 0.8) (0.6, 0.8, 0.7, 0.5) (0.5, 0.8, 0.6, 0.6) (0.7, 0.8, 0.8, 0.8)

ζ (X )

Choice value ρ p j

(0.7, 0.3, 0.8, 0.7) (0.5, 0.7, 0.6, 0.7) (0.7, 0.6, 0.6, 0.8) (0.7, 0.8, 0.8, 0.8) (0.6, 0.8, 0.7, 0.7) (0.5, 0.8, 0.7, 0.8)

ρ p1 = (1.4, 1.1, 1.5, 1.3) ρ p2 = (1.2, 1.4, 1.3, 1.3) ρ p3 = (1.4, 1.3, 1.2, 1.4) ρ p4 = (1.4, 1.5, 1.4, 1.4) ρ p5 = (1.3, 1.5, 1.4, 1.3) ρ p6 = (1.2, 1.5, 1.3, 1.5)

4 i=1

ρ p1 ρ p2 ρ p3 ρ p4 ρ p5 ρ p6

pi ◦ ρ pj

= 5.2 = 5.2 = 5.3 = 5.7 = 5.5 = 5.5

• The “Vaccination” includes live-attended vaccines, inactivated vaccines, live nonpathogenic vaccines and live active vaccines. • The “Health Care” includes exercise, balanced diet, rest and plenty of liquids. • The “Medication” includes statins, beta blockers, anti-platelet and ACE inhibitors. • The “Surgery” includes coronary artery bypass grafting, heart valve repair or replacement, aneurysm repair and heart transplant. An ideally conventional decision object X on the m–polar fuzzy set of parameters A p is calculated in Table 1.77. The choice value ρ p j , for the decision substitute p j on the universe of patients P is calculated in Table 1.78. From Table 1.78, it is easy to see that no choice value is maximum, so it is difficult for some one to take a decision that whose patient will recover soon. For taking such a decision it is calculated 4 

pi ◦ ρ pj ,

j = 1, 2, . . . , 6.

i=1

Finally, taking the patients p j ∈ P with the maximum sum of poles of choice value ρ p j as the ideal decision for the given multi-criteria decision making problem. From Table 1.78, it is easy to compare the choice values of all the required patients. 4th patient has the maximum sum of poles of choice value as compared to all other

1.9 Conclusion

79

patients. So, he will recover soon from heart disease with prevention and treatment as compared to others. The method of selecting the best object under soft m–polar fuzzy rough information is described in the following Algorithm 1.7. Algorithm 1.7 Soft m–polar fuzzy rough set # different characteristics 1. Input S, as a universe, A, as a set of parameters, and C, of parameters set A. 2. Compute the ideally conventional decision object X X = max{ pi ◦ ζ˜ −1 (s j , ak )|s j ∈ S}, where i ∈ m, j = 1, 2, 3, . . . , n, k = 1, 2, 3, . . . , l. 3. Compute the lower and upper approximations ζ (X ) and ζ (X ) for any X ∈ m F(A) w.r.t. approximation space (S, A, ζ˜ −1 ) as ζ (X )(s) =

 (1 − ζ˜ −1 (s, a)) ∨ X (a) , s ∈ S, a∈A

ζ (X )(s) =

 ζ˜ −1 (s, a) ∧ X (a) , s ∈ S. a∈A

4. Compute the choice value ρs j as ρs j = pi ◦ ζ (X )(s j ) + pi ◦ ζ (X )(s j ), s j ∈ S. 5. Compute the maximum choice value ρsk as ρsk = max j ρs j , j = 1, 2, . . . , |S|. 6. Output S M , the alternative for which ρsk is maximum. 7. If no choice value is maximum, compute m  pi ◦ ρs j , where j = 1, 2, 3, . . . , n. Psum = i=1

8. Evaluate S+ M , the alternative for which Psum is maximum.

1.9 Conclusion This chapter is motivated by the existence of m−polar fuzzy information which is sometimes given in combination with other forms of vague knowledge. For this reason the theory of hybrid m−polar fuzzy models has a growing number of applications in several fields, including engineering, medicine, data analysis and artificial intelligence. The combination of said information is focused with features from the rough and soft set theories. In this way, firstly the concepts of rough m−polar fuzzy sets and m−polar fuzzy soft sets have been presented, and some of their properties have been investigated. In particular, the rough degrees of the m−polar fuzzy sets have deserved special attention. Further, the similarity measures of m−polar fuzzy

80

1 Hybrid Multi-polar Fuzzy Models

sets have been investigated. A variety of applications, including medical diagnosis, pattern recognition, coding theory, game theory and region extraction have been presented. In addition, the justification of hybrid models called m−polar fuzzy rough sets, m−polar fuzzy soft rough sets and soft m−polar fuzzy rough sets have been presented. Some fundamental properties of these models have bee discussed. The relationship between m−polar fuzzy soft rough approximation operators and crisp soft rough approximation operators has been examined. In addition, the justification of hybrid models called m−polar fuzzy rough sets, m−polar fuzzy soft rough sets and soft m−polar fuzzy rough sets have been presented. Some fundamental properties of these models have been discussed. The relationship between m−polar fuzzy soft rough approximation operators and crisp soft rough approximation operators has been examined. To extend the range of the number of parameters with multi-polar information in practical applications, respective novel approaches have been presented to MCDM based on these models. The applications of these models have also been discussed to decision information systems. The computational processes have been presented by means of some practical examples. The complexity of approximating the data under multi-polar information has been overcome with the presented approaches. The m−polar fuzzy models developed in this chapter provide more exactness, flexibility, and compatibility with a system when compared with the other mathematical models. All in all, the ground for deeper analysis of these general models has been settled down in the future.

References 1. Akram, M.: m–Polar fuzzy graphs. Stud. Fuzziness Soft Comput. 371 (2019). Springer 2. Akram, M., Adeel, A.: Novel hybrid decision making methods based on m F rough information. Granular Comput. 5(2), 185–201 (2020) 3. Akram, M., Waseem, N.: Similarity measures for new hybrid models: m–polar fuzzy sets and m–polar fuzzy soft sets. Punjab Univ. J. Math. 51(6), 115–130 (2019) 4. Akram, M., Ali, G., Waseem, N., Davvaz, B.: Decision making methods based on hybrid m–polar fuzzy models. J. Intell. Fuzzy Syst. 35(3), 3387–3403 (2018) 5. Akram, M., Ali, G., Alshehri, N.O.: A new multi-attribute decision making method based on m−polar fuzzy soft rough sets. Symmetry 9(11), 271 (2017) 6. Akram, M., Ali, G., Alcantud, J.C.R.: New decision making hybrid model: intuitionistic fuzzy N –soft rough sets. Soft Comput. 23(20), 9853–9868 (2019) 7. Akram, M., Ali, G., Alcantud, J.C.R.: Hybrid multi-attribute decision making model based on (m, N )-soft rough sets. J. Intell. Fuzzy Syst. 36(6), 6325–6342 (2019) 8. Akram, M., Ali, G., Shabir, M.: A hybrid decision making framework using rough m–polar fuzzy bipolar soft environment. Granular Comput. 6(3), 539–555 (2021) 9. Akram, M., Shahzadi, S.: Novel intuitionistic fuzzy soft multiple-attribute decision making methods. Neural Comput. Appl. 29, 435–447 (2018) 10. Alcantud, J.C.R.: Soft open bases and a novel construction of soft topologies from bases for topologies. Mathematics 8(5), 672 (2020) 11. Alcantud, J.C.R., de Andres, R.: The problem of collective identity in a fuzzy environment. Fuzzy Sets Syst. 315, 57–75 (2017) 12. Alcantud, J.C.R., de Andres, R., Cascon, J.M.: On measures of cohesiveness under dichotomous opinions: some characterizations of approval consensus measures. Inf. Sci. 240, 45–55 (2013)

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41. Majumdar, P., Samanta, S.K.: On distance based similarity measure between intuitionistic fuzzy soft sets. Anusandhan 12(22), 41–50 (2010) 42. Majumdar, P., Samanta, S.K.: Generalised fuzzy soft sets. Comput. Math. Appl. 59(4), 1425– 1432 (2010) 43. Molodtsov, D.A.: Soft set theory-first results. Comput. Math. Appl. 37(4–5), 19–31 (1999) 44. Molodtsov, D.A.: The theory of soft sets. URRS Publishers, Moscow, Russia (2004). (Russian) 45. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982) 46. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Boston (1991) 47. Pawlak, Z.: Rough set theory and its applications. J. Telecommun. Inf. Technol. 3, 7–10 (2002) 48. Rajarajeswari, P., Dhanalakshmi, P.: An application of similarity measure of fuzzy soft set based on distance. J. Math. 4(4), 27–30 (2012) 49. Rosen, K.H.: Discrete Mathematics and Its Applications, 7th edn. McGraw Hill (2011) 50. Sarwar, M., Akram, M., Shahzadi, S.: Distance measures and δ-approximations with rough complex fuzzy models. Granular Comput. 8, 893–916 (2023) 51. Shahzadi, S., Akram, M.: Intuitionistic fuzzy soft graphs with applications. J. Appl. Math. Comput. 55, 369–392 (2017) 52. Szmidt, E., Kacprzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst. 114(3), 505–518 (2000) 53. Wang, W., Xin, X.: Distance measure between intuitionistic fuzzy sets. Pattern Recognit. Lett. 26(13), 2063–2069 (2005) 54. Xiao, Z., Gong, K., Zou, Y.: A combined forecasting approach based on fuzzy soft sets. J. Comput. Appl. Math. 228(1), 326–333 (2009) 55. Yager, R.R.: Pythagorean fuzzy subsets. In: Proceedings of 2013 Joint IFSA World Congress and NAFIPS Annual Meeting, pp. 57–61 (2013). https://doi.org/10.1109/IFSA-NAFIPS.2013. 6608375 56. Yager, R.R.: Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 22(4), 958–965 (2014). https://doi.org/10.1109/TFUZZ.2013.2278989 57. Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Inf. Sci. 109(1–4), 21–47 (1998) 58. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 59. Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46(1), 39–59 (1993) 60. Zhang, H., Shu, L., Liao, S.: Intuitionistic fuzzy soft rough set and its application in decision making. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/287314 61. Zhang, X.H., Zhou, B., Li, P.: A general frame for intuitionistic fuzzy rough sets. Inf. Sci. 216, 34–49 (2012) 62. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. IEEE Int. Conf. Fuzzy Syst. 835–840 (1998) 63. Zou, Y., Xiao, Z.: Data analysis approaches of soft sets under incomplete information. Knowl.Based Syst. 21(8), 941–945 (2008)

Chapter 2

TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

This chapter contributes to multi-criteria decision analysis (MCDA) under the multipolar (m–polar) fuzzy perspective, it also works as an enhancement to the expression of reliable linguistic assessments. Different decision making methods for evaluating the performance of the alternatives are presented. Firstly, a purely m–polar fuzzy context is considered, with the corresponding m−polar fuzzy ELECTRE-I solution. Then linguistic features for the formulation of an m−polar fuzzy linguistic ELECTRE-I methodology are incorporated and relatedly, an alternative m−polar fuzzy linguistic TOPSIS method is considered. Under these approaches, pairwise comparisons of the alternatives are made by using outranking relations. Further, it is shown how these strategies can also be followed in numerical examples. The rankings of the alternatives are presented through directed graphs, showing which alternative is preferable. The evaluation of the alternatives is made by the decision makers in terms of suitable linguistic values. As an extension of the standard TOPSIS method, this chapter develops an m−polar fuzzy linguistic TOPSIS approach for MCGDM. Thus also in this approach, the decision makers are allowed to submit their estimations in the form of linguistic term sets. Furthermore, the efficiency of the presented techniques is validated by their respective applications on real life examples, and by appropriate comparisons with previously existing approaches. Finally, algorithms of these approaches are set forth, and their computer programming codes are also given. This chapter is based on [1, 3, 12].

2.1 Introduction Decision making [41] can be regarded as the concluding step of an intellectual and psychological process that leads to the selection of one among several different alternatives with the help of information. A number of theories have been developed to approach problems, when they are exactly and inexactly posed. Although probabilis© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_2

83

84

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tic ideas may be definitely useful, uncertainty is not always of probabilistic nature. Instead it is often due to relatively vague or imprecise descriptions. Thus other theories such as fuzzy set theory and fuzzy logic [40, 49, 74, 87] have been successfully applied in cases of imprecise and vague information from many contexts [15]. As a short sample from recent years, many researchers have proposed very interesting mixed methodologies [83, 84] applied to multi-criteria decision making (MCDM) evaluations. Antucheviciene et al. [22] solved a civil engineering problems by means of fuzzy and stochastic MCDM methods. First, TOPSIS is a well known MCDM technique that was introduced by Hwang and Yoon [48]. It determines the performance of the alternatives through similarity with the help of ideal solutions. Both positive and negative ideal solutions are determined. The positive ideal solution contains the best values, and the negative ideal solution consists of the worst values among all the alternatives. Then the main concept of the technique is that the selected object not only should be at a shortest distance from the positive ideal solution, but also at a farthest distance from the negative ideal solution. TOPSIS is arguably the most implemented technique for decision making problems, especially in the medical sciences. However its design did not mean to incorporate bipolar uncertainty, which is fairly natural in the perception of the decision makers. In the presence of this class of information, TOPSIS does not yield sufficiently accurate results. TOPSIS is a very appropriate and rational approach which combines computational efficiency and a simple mathematical form. In existing specifications of the standard TOPSIS, the results of the decision making process are determined by numerical values. However the perception about many problems in the real world is completely uncertain. In relation with this issue, the application of fuzzy set theory in decision making had been first demonstrated by Bellman and Zadeh [27] in 1970. In 2014, m−polar fuzzy set theory was introduced by Chen et al. [31]. An m−polar fuzzy set is a mapping μ : X → [0, 1]m whose motivation is that multi-polar information occurs because data about actual problems sometimes come from multiple sources or agents. The membership value in m−polar fuzzy sets yields better understanding of the uncertainty of data. Akram [4] introduced several notions based on m−polar fuzzy sets and m−polar fuzzy graphs, including metrics in m−polar fuzzy graphs, and certain types of irregular m−polar fuzzy graphs and m−polar fuzzy hypergraphs. Furthermore, Akram and Adeel [7] introduced novel hybrid decision making methods based on m−polar fuzzy rough information. Akram et al. [11] introduced a new multi-attribute decision making method based on m–polar fuzzy soft rough sets. For the study of decision making methods, the readers are refereed to [65, 82]. The preceding fuzzy tools are suitable for problems that are defined in purely quantitative situations. But because uncertainty is generally due to vagueness, in a sense it is presumed that decision makers are also interested in problems whose description is relatively qualitative. As a reaction to this situation, Zadeh [88–90] introduced the fuzzy linguistic approach which soon produced favorable outputs in many areas and applications [64]. As the linguistic classification and characterizations are generally less precise than their numerical counterparts, there is no denying that they have interesting uses in group decision analysis [46, 47, 60, 62, 82]. The linguis-

2.1 Introduction

85

tic approach is worthy of attention in contexts of personalized individual semantics in computing with words (CWW) [45, 59]. Selvachandran and Salleh [72] went beyond this framework and proposed the concept of intuitionistic fuzzy linguistic variables and intuitionistic fuzzy hedges (cf., [16, 56] for updated information about the intuitionistic framework). In recent years, a number of researchers including Liao et al. [54] discussed the distance and similarity measures for hesitant fuzzy linguistic term sets and their application in MCDM. Later, Riera et al. [63] introduced some interesting properties of the fuzzy linguistic model based on discrete fuzzy numbers in order to manage hesitant fuzzy linguistic information. Elimination and choice translating reality (ELECTRE) is one of the top MCDM methods. The ELECTRE approach was inaugurated by Benayoun et al. [28]. To quickly grasp its impact in the literature, a comprehensive review on methodologies and applications of ELECTRE and ELECTRE-based solutions is Govindan et al. [42]. Some improvements of the original methodology soon followed. A modified concept known as ELECTRE-I was introduced by Roy [69]. We refer the readers to [78] for a summary of multiple attribute decision making methods, applications and most prominent versions of ELECTRE-I. Further, this approach was also expanded into sundry alternative variants. It is claimed that the majority of these methods have been combined with fuzzy set theory and its extensions. Hatami-Marbini and Tavana [43] expanded ELECTRE-I and introduced the method of fuzzy ELECTRE-I with numerical examples to illustrate its effectiveness. Sevkli [73] compared crisp and fuzzy ELECTRE methods for supplier selection problem. For the selection of academic staff, Rouyendegh and Erkan [68] used fuzzy ELECTRE too. Devi and Yadav [39] proposed intuitionistic fuzzy ELECTRE to choose the proper location of a plant in a group decision making environment. Vahdani et al. [80] presented a comparison of the fuzzy and intuitionistic fuzzy ELECTRE methods. Hatami-Marbini et al. [44] applied the fuzzy group ELECTRE method. Zandi and Roghanian [91] introduced a novel fuzzy ELECTRE method that made use of the VIKOR method [76]. Kheirkhah and Dehghani [52] applied the fuzzy group ELECTRE method for the evaluation of quality of public transportation facilities. Vahdani and Hadipour [79] presented the technique of interval-valued fuzzy ELECTRE. Asghari et al. [23] used the fuzzy ELECTRE-I method for the analysis of mobile payment models. Fuzzy ELECTRE-I techniques were also applied in the evaluation of catering firms (Aytac et al. [25]) and environmental effect evaluation (Kaya and Kahraman [50]). Adeel et al. [2] proposed the m−polar hesitant fuzzy ELECTRE-I and hesitant m−polar fuzzy ELECTRE-I methods for MCDM. For handling MCDM problems, Wu and Chen [85] developed the concept of intuitionistic fuzzy ELECTRE-I method. Chen and Xu [35] proposed a novel MCDM technique by combining hesitant fuzzy sets with ELECTRE-II method. Lupo [58] calculated the service quality of three international airports using ELECTRE-III approach. In conclusion, ELECTRE methods have played a very significant role in the class of outranking methods. These methodolgies enable us to take advantage of incomplete knowledge. For other notations, terminologies and applications of this area of research, the readers are referred to [9, 20]. In addition to ELECTRE, TOPSIS gives another effective, renowned and widely used approach to MCGDM. It was proposed

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in [48] and it posits that selected alternatives should have the most suitable distances from the positive and negative ideal solutions. Since then, several extended TOPSIS methods including [13, 75] have been applied to different MCDM problems. And combinations with fuzzy-inspired sources of information have proliferated. Chu and Lin [38] used the fuzzy TOPSIS method for the selection of a robot. Nadaban et al. [61] discussed a general view on fuzzy TOPSIS. Yue [86] proposed a method for group decision making based on determining weights of decision makers using TOPSIS. By considering the triangular fuzzy numbers and defining the crisp Euclidean distance between two fuzzy numbers, Chen [30] introduced an extended TOPSIS method for MCDM. Roszkowska [66] studied MCDM models by applying the TOPSIS method to crisp and interval data. Further, Roszkowska and Wachowicz [67] applied the fuzzy TOPSIS method to score the negotiation offers in ill-structured negotiation problems. Ren et al. [62] developed a new hesitant fuzzy linguistic TOPSIS method for group multi-criteria linguistic decision making. Ashtiani et al. [24] extended the fuzzy TOPSIS method based on interval-valued fuzzy sets. Beg and Rashid [26] introduced TOPSIS for hesitant fuzzy linguistic term sets. Boran et al. [29] proposed the multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Chu [37] discussed the selection of facility location using fuzzy TOPSIS under group decisions. Akram et al. [6, 8] introduced TOPSIS method based on m−polar hesitant fuzzy and hesitant m−polar fuzzy models. Wang [81] extended hesitant fuzzy linguistic term sets and their aggregation in group decision making. Liu and Su [57] introduced an extended TOPSIS based on trapezoidal fuzzy linguistic variables.

2.2 Multi-criteria Decision Making Methods Decision making is the process of finding the best option(s) among a set of feasible alternatives. A particular instance is MCDM, which is concerned with structuring and formulating optimization problems that take account of multiple criteria. Many MCDM models have been developed and implemented in various fields such as engineering, economics, management, business and information technology, also under a fuzzy formulation [87]. A reason for the variety of approaches in this field is that the extent to which a good performance in terms of some criteria can offset bad performances for others, is open to debate. In addition, the structure of the data affects the discussion about the elements that determine the conclusion. The advancement in MCDM methods is closely associated to the development of computer technology, that has made it simple and easy to deal with complicated and large sources of information or data. Decision making situations can be categorized in different groups according to certain characteristics, sources of information and preference representations. 1. Single criteria decision making is concerned with a situation where we have only one source of information (or criteria) to define the decision problem. In these

2.2 Multi-criteria Decision Making Methods

87

MCDM Methods

AHP

ELECTRE-I

ELECTRE

ELECTRE-II

TOPSIS

ELECTRE-III

ELECTRE-IV

VIKOR

PROMETHEE

PROMETHEE I

PROMETHEE II

Fig. 2.1 Hierarchical structure of MCDM methods

situations, the solution of the problem comes directly and exclusively from the information provided. 2. In decision making processes, an expert typically needs to compare a finite set of alternatives xi (i = 1, 2, . . . , n) and construct a preference relation that is subsequently optimized [18]. Sometimes utilities can be associated with preferences, and if these have a poor structure, other type of tools may be used (e.g., weak utilities [17] or multi-utilities [19]). 3. However, decision making is not only concerned with the case of a single expert, since many problems arise from a group of experts who jointly attempt to agree on the best alternative(s) from a set of feasible alternatives under certain characteristics. Decision making with multiple experts is called group decision making (GDM) which is also known as multi-person decision making. 4. Real-world decision making problems are usually too complex and ill-structured to believe that the examination of a single criterion, attribute, or point of view will lead to the optimum decision. In fact, such uni-dimensional approach is merely an oversimplification that can lead to unrealistic decisions. A more appealing approach would be the simultaneous consideration of all the factors that are related to the problem. MCDM is a discipline in its own right, which deals with decisions involving the choice of a best alternative from several potential candidates in a decision, subject to several criteria or attributes that may be either concrete or vaguely defined. MCDM provides strong decision making in domains where the selection of a best alternative is highly complex. This section reviews the main streams of investigation in multi-criteria decision making theory and practice in detail. MCDM methods have been used in different applications. The most widely used MCDM methods are described in Fig. 2.1.

88

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets Overall objective

Goal

Aspect

Dimension 1

Criteria

C11

Alternatives

A1

···

Dimension j

···

C1r

···

Cj1

···

Ai

Dimension k

···

Cjs

Ck1

···

···

Ckt

An

Fig. 2.2 Hierarchical system for MADM

Dubois and Prade [40] summarized the procedures of MCDM in the following five main steps: Step 1: Step 2: Step 3: Step 4:

Define the nature of the problem. Construct a hierarchy system for its evaluation, as shown in Fig. 2.2. Select the appropriate evaluation model. Obtain the relative weights and performance score of each attribute with respect to each alternative. Step 5: Determine the best alternative according to the synthetic utility values, which are the aggregation value of relative weights, and performance scores corresponding to alternatives. If the overall scores of the alternatives are fuzzy, we can add Step 6 to rank the alternatives for choosing the best one. Step 6: Outrank the alternatives referring to their synthetic fuzzy utility values from Step 5.

Keeney and Raiffa [51] suggested five principles that must be followed when criteria are being formulated: 1. 2. 3. 4. 5.

completeness operationality decomposability non-redundancy minimum size.

The Matrix Representation of the MCDM Problem: The MCDM problems can be divided into two kinds. One is the classical MCDM set of problems among which the ratings and the weights of criteria are measured in crisp numbers. Another one is the multiple criteria decision making set of problems where the ratings and the weights of criteria evaluated on incomplete information, imprecision, subjective judgment

2.3 m–Polar Fuzzy ELECTRE-I Method Table 2.1 k—matrix format C1 A1 A2 .. . Am

k x11 k x21

.. . k xm1

89

C2

...

Cn

k x12 k x22

... ... .. .

k x1n k x2n .. . k xmn

.. . k xm2

...

and vagueness are usually expressed by interval numbers, linguistic terms, fuzzy numbers or intuitive fuzzy numbers. The solution of each multi-criteria problem (individual or group decision) begins with the construction of a decision making matrix (or matrices). In such matrices, the values of the criteria for alternatives may be real, intervals numbers, fuzzy numbers or qualitative labels. Let us denote by D = {1, 2, . . . , k} a set of decision makers or experts. The multicriteria problem can be expressed in k-matrix format as given in Table 2.1. where, • A1 , A2 , . . . , Am are the possible alternatives that decision makers have to choose. • C1 , C2 , . . . , Cn are the criteria for which the alternative performance is measured. • xikj is the k-decision maker rating of alternative Ai with respect to the criterion C j ( xikj is numerical, interval or fuzzy number). In this way for m alternatives and n criteria, we have matrix X k = (xikj ) where xikj is the value of i-alternative with respect to j-criterion for k-decision maker, j = 1, 2, . . . , n, k = 1, 2, . . . , K . The relative importance of each criterion is given by a set of weights which are normalized to sum to one. Let W k = [w1k , w1k , · · · , wnk ] denotes a weight vector for k-decision maker, where wkj ∈ R is the k-decision maker weight of criterion C j and w1k + w1k + · · · + wnk = 1. In the case of one decision maker, we write xi j , w j , X, respectively. Multi-criteria analysis focuses mainly on three types of decision problems: 1. choice—select the most appropriate (best) alternative. 2. ranking—draw a complete order of the alternatives from the best to the worst. 3. sor ting—select the best k alternatives from the list.

2.3 m–Polar Fuzzy ELECTRE-I Method This section establishes the steps of the m–polar fuzzy ELECTRE-I method. Let A = {x1 , x2 , . . . , xr } be the set of alternatives and T = {t1 , t2 , . . . , ts } be the set of criteria. The following computations produce the desired output of the m–polar fuzzy ELECTRE-I methodology:

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

1. The rating values of the alternatives with respect to each criterion are represented by the decision matrix Z = (z i j ), where, Z = (z i j ) = (z i1j , z i2j , z i3j , . . . , z imj ). 2. The weights are given by the experts and they must obey the standard condition of normality i.e., s  w j = 1. j=1

3. The weighted m–polar fuzzy decision matrix Y = (yi j ) is constructed as Y = (yi j ) = (yi1j , yi2j , yi3j , . . . , yimj ), where, yi j = w j z i j . 4. The m–polar fuzzy concordance sets are defined as F pq = {1 ≤ j ≤ s : v pj ≥ vq j , p = q; p, q = 1, 2, 3, . . . , r }, where vi j = yi1j + yi2j + yi3j + · · · + yimj . 5. The m–polar fuzzy discordance sets are defined as G pq = {1 ≤ j ≤ s : v pj ≤ vq j , p = q; p, q = 1, 2, 3, . . . , r }, where vi j = yi1j + yi2j + yi3j + . . . + yimj . 6. The m–polar fuzzy concordance indices (denoted by f pq ) are obtained as follows:  w j , for all p, q. f pq = j∈F pq

7. The m–polar fuzzy concordance matrix F can be constructed in the following manner: ⎛ ⎞ − f 12 f 13 ... f 1r ⎜ f 21 − f 23 ... f 2r ⎟ ⎜ ⎟ ⎜ ⎟ F = ⎜ f 31 f 32 − ... f 3r ⎟ . ⎜ .. .. .. .. ⎟ ⎝ . . . ... . ⎠ fr 1 fr 2 fr 3 ... −

8. The m–polar fuzzy discordance indices g pq  s are computed as:

2.3 m–Polar Fuzzy ELECTRE-I Method

91



 1 (y 1 − y 1 )2 + (y 2 − y 2 )2 + . . . + (y m − y m )2 m pj q j pj q j pj q j j∈G pq g pq =

 , for all p, q. m )2 max m1 (y 1pj − yq1 j )2 + (y 2pj − yq2 j )2 + . . . + (y m − y pj qj max

j

9. The m–polar fuzzy discordance matrix G can be constructed in the following manner: ⎛ ⎞ − g12 g13 ... g1r ⎜ g21 − g23 ... g2r ⎟ ⎜ ⎟ ⎜ ⎟ G = ⎜ g31 g32 − ... g3r ⎟ . ⎜ .. .. .. .. ⎟ ⎝ . . . ... . ⎠ gr 1 gr 2 gr 3 ... −

10. The concordance level and discordance levels are computed. The m–polar fuzzy concordance level f and the m–polar fuzzy discordance level g are respectively defined as the averages of the m–polar fuzzy concordance and m–polar fuzzy discordance indices. r  r  1 f = f pq , r (r − 1) p=1 q=1 p=q q= p

 1 g pq . r (r − 1) p=1 q=1 r

g=

r

p=q q= p

11. The m–polar fuzzy concordance dominance matrix and the m–polar fuzzy discordance dominance matrix according to concordance and discordance levels are constructed as: ⎞ ⎛ − h 12 h 13 ... h 1r ⎜ h 21 − h 23 ... h 2r ⎟ ⎟ ⎜ ⎟ ⎜ H = ⎜ h 31 h 32 − ... h 3r ⎟ , ⎜ .. .. .. .. ⎟ ⎝ . . . ... . ⎠ h r 1 h r 2 h r 3 ... −

where h pq =

1, f pq ≥ f , , and 0, f pq < f ⎛

− ⎜ l21 ⎜ ⎜ L = ⎜ l31 ⎜ .. ⎝ .

l12 − l32 .. .

l13 l23 − .. .

lr 1 lr 2 lr 3

⎞ ... l1r ... l2r ⎟ ⎟ ... l3r ⎟ ⎟, .. ⎟ ... . ⎠ ... −

92

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets



1, g pq < g, 0, g pq ≥ g. 12. Now peer to peer multiplication of the entries of H and L is performed to construct the m–polar fuzzy aggregate dominance matrix M, namely, where l pq =

⎛

− ⎜ m 21 ⎜ ⎜ M = ⎜ m 31 ⎜ .. ⎝ .

m 12 − m 32 .. .

m 13 m 23 − .. .

mr 1 mr 2 mr 3

⎞ ... m 1r ... m 2r ⎟ ⎟ ... m 3r ⎟ ⎟. .. ⎟ ... . ⎠ ... −

13. The alternatives are ranked according to the outranking values of matrix M. There is a directed edge from the entry x p to xq if and only if m pq = 1. Thus, the following three cases are as follows: (i). There exists a unique directed edge from x p to xq . (ii). There exists a directed edge from x p to xq and xq to x p . (iii). There does not exist any edge between x p and xq . In the first case, we say that x p is preferred over xq . For the second case, we say that x p and xq are indifferent. And we say that x p and xq are incomparable in the third case. In next subsections, the m−polar fuzzy ELECTRE-I approach is applied to real life examples.

2.3.1 Selection of a Suitable Location for a Diesel Power Plant Diesel power plants are used to generate electricity by converting the chemical energy of fuel into mechanical energy. Assume that a company wants to decide a location for the plant. Suppose that we have four alternatives A1 , A2 , A3 and A4 for the location of its diesel power plant. A team of skilled engineers is formed to select the best alternative. They considered the following four main criteria for the selection of the best alternative: T1 = Infrastructure, T2 = Climatic and Atmospheric Conditions, T3 = Social Infrastructure, T4 = Government Policies. Further each criteria is subdivided in three sub-criteria so that a 4-polar fuzzy set is formed. The infrastructure depends on the availability of fuel, availability of water and availability of transportation facilities. The Climatic and Atmospheric Conditions depends on ambient temperature, humidity and wind velocities. The social infrastructure includes educational institutions, hospitals, health care and recreation facilities. The government policies include the licensing policies, institutional finance and the government subsidies. The four criteria, and their attributes are described in Figs. 2.3, 2.4, 2.5 and 2.6.

2.3 m–Polar Fuzzy ELECTRE-I Method

93

Fig. 2.3 Attributes of infrastructure

T1 Infrastructure

Availability of Fuel

Transportation Facilities

Availability of Water

Fig. 2.4 Attributes of climatic and atmospheric conditions

T2 Climatic and Atmospheric Conditions

Ambient Temperature

Wind Velocity Humidity

Fig. 2.5 Attributes of social infrastructure

T3 Social Infrastructure

Health Care and Recreation Facilities

Educational Institutions Hospitals

Fig. 2.6 Attributes of government policies

T4 Government Policies

Government Subsidies

Licensing Policies Institutional Finance

94

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.2 A 3–polar fuzzy decision matrix . T1 T2 A1 A2 A3 A4 A5

(0.5, 0.4, 0.55) (0.7, 0.85, 0.75) (0.35, 0.5, 0.6) (0.6, 0.4, 0.5) (0.5, 0.65, 0.4)

(0.4, 0.5, 0.5) (0.8, 0.95, 0.9) (0.75, 0.7, 0.65) (0.35, 0.6, 0.4) (0.64, 0.32, 0.6)

Table 2.3 A 3–polar fuzzy weighted decision matrix . T1 T2 A1 A2 A3 A4 A5

(0.225, 0.18, 0.2475) (0.315, 0.3825, 0.3375) (0.1575, 0.225, 0.27) (0.27, 0.18, 0.225) (0.225, 0.2925, 0.18)

(0.06, 0.075, 0.075) (0.12, 0.1425, 0.135) (0.1125, 0.105, 0.0975) (0.0525, 0.09, 0.06) (0.096, 0.048, 0.09)

Table 2.4 A 3–polar fuzzy concordance set j 1 2 F1 j F2 j F3 j F4 j F5 j

– {1, 2, 3, 4} {1, 2, 4} {1, 4} {1, 2, 4}

{} – {} {} {}

Table 2.5 A 3–polar fuzzy discordance set j 1 2 G1 j G2 j G3 j G4 j G5 j

– {} {3} {2, 3} {3}

{1, 2, 3, 4} – {1, 2, 3, 4} {1, 2, 3, 4} {1, 2, 3, 4}

T3

T4

(0.6, 0.65, 0.7) (0.5, 0.8, 0.9) (0.5, 0.7, 0.4) (0.5, 0.6, 0.5) (0.5, 0.7, 0.6)

(0.35, 0.5, 0.44) (0.9, 0.95, 0.8) (0.64, 0.5, 0.6) (0.7, 0.6, 0.4) (0.5, 0.7, 0.5)

T3

T4

(0.15, 0.1625, 0.175) (0.125, 0.2, 0.225) (0.125, 0.175, 0.1) (0.125, 0.15, 0.125) (0.125, 0.175, 0.15)

(0.0525, 0.075, 0.066) (0.135, 0.1425, 0.12) (0.096, 0.075, 0.09) (0.105, 0.09, 0.06) (0.075, 0.105, 0.075)

3

4

5

{1, 3} {1, 2, 3, 4} – {1, 3} {1, 3}

{2, 3} {1, 2, 3, 4} {2, 3, 4} – {1, 2, 3, 4}

{3} {1, 2, 3, 4} {2, 4} {4} –

3

4

5

{2, 4} {} – {2, 4} {2, 4}

{1, 4} {} {1} – {}

{1, 2, 4} {} {1, 3} {1, 2, 3} –

2.3 m–Polar Fuzzy ELECTRE-I Method

95

1. The 3–polar fuzzy decision matrix that the experts submit is given in Table 2.2. 2. The normalized weights of the criteria are given below, and a 3-polar fuzzy weighted decision matrix is calculated in Table 2.3: w1 = 0.45, w2 = 0.15, w3 = 0.25, w4 = 0.15, which satisfy

4 

w j = 1.

j=1

3. Tables 2.4 and 2.5 represent the 3–polar fuzzy concordance and 3–polar fuzzy discordance sets, respectively. 4. The 3–polar fuzzy concordance matrix is evaluated as follows: ⎛

⎞ 0 0.7 0.4 0.25 − 1 1 1 ⎟ ⎟ 0 − 0.55 0.3 ⎟ ⎟. 0 0.7 − 0.15 ⎠ 0 0.7 1 −

− ⎜ 1 ⎜ F =⎜ ⎜ 0.75 ⎝ 0.6 0.75

5. The 3–polar fuzzy concordance level is f = 0.5425. 6. The 3–polar fuzzy discordance matrix is evaluated as follows: ⎛

− ⎜ 0 ⎜ G=⎜ ⎜ 0.9505 ⎝ 1 0.2858

⎞ 1 0.76636 0.3740 1 − 0 0 0⎟ ⎟ 1 − 1 1⎟ ⎟. 1 0.5596 − 1⎠ 1 0.4558 0 −

7. The 3–polar fuzzy discordance level is g = 0.6196. 8. According to the concordance level and discordance level, the 3–polar fuzzy concordance dominance and 3–polar fuzzy discordance dominance matrices are computed as follows: ⎛ ⎞ − 0 1 0 0 ⎜1 − 1 1 1⎟ ⎜ ⎟ ⎟ H =⎜ ⎜ 1 0 − 1 0 ⎟, ⎝1 0 1 − 0⎠ 1 0 1 1 − ⎛

− ⎜1 ⎜ L=⎜ ⎜0 ⎝0 1

0 − 0 0 0

0 1 − 1 1

1 1 0 − 1

⎞ 0 1⎟ ⎟ 0⎟ ⎟. 0⎠ −

96

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets A2

Fig. 2.7 Directed graph of outranking relation of locations

A5 A1

A3

A4

9. The 3–polar fuzzy aggregated dominance matrix is computed as: ⎛

− ⎜1 ⎜ M =⎜ ⎜0 ⎝0 1

0 − 0 0 0

0 1 − 1 1

0 1 0 − 1

⎞ 0 1⎟ ⎟ 0⎟ ⎟. 0⎠ −

10. The outranking relation of the alternatives is shown in the Fig. 2.7. Therefore, it is concluded that the most favorable location for the diesel power plant is A2 .

2.3.2 Selection of a Site for the Airport Suppose that a government wants to construct a new airport in a city. The government has four alternatives for the site of this facility. A group of decision makers evaluated each site by the following criteria: T1 = Operational Considerations, T2 = SocioEconomic Impacts, T3 = Ecological Impacts, T4 =Expenditures and T5 = Risk and Deliverability. All criteria depend on four further attributes. The operational considerations include the airspace availability, hazards and obstacles, meteorology and holding capacity. The socioeconomic impacts of the airport include benefits to the local economy, noise or air pollution, population relocation and loss of heritage sites. The ecological impacts of the airport are water quality, landscape, CO2 emissions and habitat loss. The expenditure of the airport includes, land acquisition and compensation, construction cost, access and utilities connections and operations. The risk and deliverability factors include construction risks and technical challenges, potential for legal challenge, potential for political hurdles and phasing potential. The five criteria and their constituent attributes are described in Figs. 2.8, 2.9, 2.10, 2.11 and 2.12. 1. A 4–polar fuzzy decision matrix is given in Table 2.6 and suppose that the group of experts assigned the following weights to each criteria: w1 = 0.25, w2 = 0.25, w3 = 0.1, w4 = 0.2, w5 = 0.2. The weights satisfy the normalization condition.

2.3 m–Polar Fuzzy ELECTRE-I Method

97

Fig. 2.8 Attributes of operational considerations

T1 Operational Considerations

Availability of Airspace

Holding Capacity

Hazards and Obstacles

Fig. 2.9 Attributes of socioeconomic impacts

Meteorology

T2 Socioeconomic Impacts

Benefits to Local Economy

Loss of Heritage Sites

Noise or Air Pollution

Fig. 2.10 Attributes of ecological impacts

Population Relocation

T3 Ecological Impacts

Water Quality

Habitat Loss

Landscape

Fig. 2.11 Attributes of expenditures

Co2 Emissions

T4 Expenditure

Land Acquisition and Compensation Construction Cost

Operations

Access and Utilities Connections

T5 Risk and Deliverability Factors

Fig. 2.12 Attributes of risk and deliverability

Construction Risks and Technical Challenges

Potential for Legal Challenge

Phasing Potential Potential for Political Hurdles

98

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.6 A 4–polar fuzzy decision matrix . T1 T2 S1 S2 S3 S4

(0.9, 0.3, 0.86, 0.84) (0.6, 0.6, 0.4, 0.5) (0.85, 0.5, 0.7, 0.8) (0.7, 0.2, 0.5, 0.5)

(0.85, 0.4, 0.5, 0.3) (0.5, 0.62, 0.65, 0.7) (0.79, 0.5, 0.6, 0.4) (0.68, 0.6, 0.7, 0.6)

T3

T4

T5

(0.94, 0.85, 0.4, 0.4) (0.7, 0.65, 0.7, 0.6) (0.9, 0.8, 0.5, 0.61) (0.6, 0.7, 0.6, 0.5)

(0.7, 0.6, 0.85, 0.9) (0.4, 0.35, 0.5, 0.62) (0.5, 0.2, 0.8, 0.7) (0.37, 0.6, 0.7, 0.5)

(0.3, 0.8, 0.85, 0.9) (0.5, 0.6, 0.65, 0.4) (0.4, 0.75, 0.8, 0.8) (0.52, 0.65, 0.7, 0.6)

Table 2.7 4–polar fuzzy weighted decision matrix . T1 T2 T3 S1 S2 S3

S4

(0.225, 0.075, 0.25, 0.21) (0.15, 0.15, 0.1, 0.125) (0.2125, 0.125, 0.175, 0.2) (0.175, 0.05, 0.125, 0.125)

(0.2125, 0.1, 0.125, 0.075) (0.125, 0.155, 0.1625, 0.175) (0.1975, 0.125, 0.15, 0.1) (0.17, 0.15, 0.175, 0.15)

T4

T5

(0.94, 0.85, 0.4, 0.4) (0.7, 0.65, 0.7, 0.6) (0.9, 0.8, 0.5, 0.61)

(0.14, 0.12, 0.17, 0.18) (0.08, 0.07, 0.1, 0.124) (0.1, 0.04, 0.16, 0.14)

(0.6, 0.16, 0.17, 0.18) (0.1, 0.12, 0.13, 0.08) (0.08, 0.15, 0.16, 0.16)

(0.6, 0.7, 0.6, 0.5)

(0.074, 0.12, 0.14, 0.1)

(0.104, 0.13, 0.14, 0.12)

Table 2.8 A 4–polar fuzzy concordance set j 1 2 F1 j F2 j F3 j F4 j

– {2, 3} {2, 3} {2}

{1, 4, 5} – {1, 3, 4, 5} {2, 4, 5}

Table 2.9 A 4–polar fuzzy discordance set j 1 2 G1 j G2 j G3 j G4 j

– {1, 4, 5} {1, 4, 5} {1, 3, 4, 5}

{2, 3} – {2} {1, 3}

3

4

{1, 4, 5} {2} – {2}

{1, 3, 4, 5} {1, 3} {1, 3, 4, 5} –

3

4

{2, 3} {1, 3, 4, 5} – {1, 3, 4, 5}

{2} {2, 4, 5} {2} –

2. The 4–polar fuzzy weighted decision matrix is given in Table 2.7. 3. Tables 2.8 and 2.9 represent the 4–polar fuzzy concordance and 4–polar fuzzy discordance sets, respectively. 4. The 4–polar fuzzy concordance matrix is evaluated as follows:

2.3 m–Polar Fuzzy ELECTRE-I Method

99

⎛

⎞ − 0.65 0.65 0.75 ⎜ 0.35 − 0.25 0.35 ⎟ ⎟ F =⎜ ⎝ 0.35 0.75 − 0.75 ⎠ . 0.25 0.65 0.25 − 5. The 4–polar fuzzy concordance level is f = 0.5. 6. The 4–polar fuzzy discordance matrix is evaluated as follows: ⎛

⎞ − 0.9298 0.4633 0.2224 ⎜1 − 1 0.3807 ⎟ ⎟. G=⎜ ⎝ 1 0.3410 − 0.1921 ⎠ 1 1 1 − 7. The 4–polar fuzzy discordance level is g = 0.7108. 8. According to the concordance level and discordance level, the 4–polar fuzzy concordance dominance and 4–polar fuzzy discordance dominance matrices are computed as follows: ⎛

− ⎜0 H =⎜ ⎝0 0 ⎛

− ⎜0 L=⎜ ⎝0 0

1 − 1 1 0 − 1 0

1 0 − 0 1 0 − 0

⎞ 1 0⎟ ⎟, 1⎠ − ⎞ 1 1 ⎟ ⎟. 1 ⎠ −

9. The 4–polar fuzzy aggregated dominance matrix is computed as: ⎛

− ⎜0 M =⎜ ⎝0 0

0 − 1 0

1 0 − 0

⎞ 1 0⎟ ⎟. 1⎠ −

10. The outranking relation of the alternatives is shown in the Fig. 2.13. It is clear from the graph that the most favorable site for the airport is S1 . Fig. 2.13 Directed graph of outranking relation of sites

S1

S2

S3

S4

100

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.10 A 4–polar fuzzy decision matrix .

T1

T2

T3

T4

I1

(0.47, 0.58, 0.62, 0.45)

(0.37, 0.50, 0.63, 0.70)

(0.50, 0.70, 0.67, 0.50)

(0.44, 0.56, 0.35, 0.66)

I2

(0.35, 0.68, 0.74, 0.54)

(0.69, 0.59, 0.70, 0.58)

(0.64, 0.72, 0.80, 0.59)

(0.70, 0.84, 0.56, 0.43)

I3

(0.80, 0.90, 0.80, 0.80)

(0.70, 0.82, 0.91, 0.64)

(0.64, 0.66, 0.71, 0.66)

(0.94, 0.86, 0.90, 0.82)

I4

(0.63, 0.77, 0.80, 0.50)

(0.38, 0.69, 0.59, 0.57)

(0.74, 0.29, 0.65, 0.71)

(0.48, 0.61, 0.58, 0.64)

I5

(0.59, 0.49, 0.39, 0.70)

(0.54, 0.41, 0.62, 0.43)

(0.49, 0.57, 0.39, 0.59)

(0.65, 0.49, 0.60, 0.55)

I6

(0.7, 0.85, 0.92, 0.76) (0.46, 0.54, 0.80, 0.57)

(0.83, 0.65, 0.72, 0.77)

(0.91, 0.75, 0.8, 0.74)

2.3.3 Performance Evaluation of Physical Sciences Instructor Performance evaluation of instructors is very important for university administration. It can help the university administration to improve the overall performance of the instructors. Since selecting the best instructor is a complex problem, we proceed to show how one can use the m–polar fuzzy ELECTRE-I approach in order to evaluate the performance of instructors in a physical science department. Therefore suppose that we have six instructors I1 , I2 , I3 , I4 , I5 and I6 . We intend to select the best instructor based on the following four qualities: T1 = Teaching Style, T2 = Social Practices, T3 = Knowledge and Expertise and T4 = Facilitation Skills. Each quality has four constituent attributes which are described in Figs. 2.14, 2.15, 2.16 and 2.17. The methodology suggests to proceed as follows:

T1 Teaching Style

Ability to explain concepts

Maintaining a lesson plan

Fig. 2.14 Attributes of teaching style

Ability to motivate students in learning and research

Allow student participation in class discussion

2.3 m–Polar Fuzzy ELECTRE-I Method

101 T2 Social Practices

Consistency in thoughts and actions

Availability at non-class time

Polite dealing with trainees

Empathetic to the problems of students

Use of study trips for teaching

Presentation of new topic relevant to field

Time Management

Handling diversity

Fig. 2.15 Attributes of social practices T3 Knowledge and Expertise

Mastery over the lesson

Usage of audio-visual gadgets

Fig. 2.16 Attributes of knowledge and expertise T4 Facilitation Skills

Handling questions in training

Incorporate adult learning styles

Fig. 2.17 Attributes of facilitation skills

102

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.11 4–polar fuzzy weighted decision matrix .

T1

T2

T3

T4

I1

(0.11609, 0.14326, 0.15314, 0.1111)

(0.06734, 0.091, 0.11466, 0.1274)

(0.173, 0.2422, 0.23182, 0.173)

(0.099, 0.126, 0.07875, 0.1485)

I2

(0.08645, 0.16796, 0.18278, 0.13338)

(0.12558, 0.10738, 0.1274, 0.10556)

(0.22144, 0.24912, 0.2768, 0.20414)

(0.1575, 0.189, 0.126, 0.09675)

I3

(0.1976, 0.2223, 0.1976, 0.1976)

(0.1274, 0.14924, 0.16562, 0.116481)

(0.22144, 0.22836, 0.24566, 0.22836)

(0.2115, 0.1935, 0.2025, 0.1845)

I4

(0.15561, 0.19019, 0.1976, 0.1235)

(0.06916, 0.12558, 0.10738, 0.10374)

(0.25604, 0.10034, 0.2249, 0.24566)

(0.108, 0.13725, 0.1305, 0.144)

I5

(0.14573, 0.12103, 0.09633, 0.1729)

(0.09828, 0.07462, 0.11284, 0.07826)

(0.16954, 0.19722, 0.13494, 0.20414)

(0.14625, 0.11025, 0.135, 0.12375)

I6

(0.1729, 0.20995, 0.22724, 0.18772)

(0.08372, 0.09828, 0.1456, 0.10374)

(0.28718, 0.2249, 0.24912, 0.26642)

(0.20475, 0.16875, 0.18, 0.1665)

Table 2.12 A 4–polar fuzzy concordance set j

1

2

3

4

5

6

F1 j

–

{}

{}

{}

{2, 3}

{}

F2 j

{1, 2, 3, 4}

–

{3}

{2, 3, 4}

{1, 2, 3, 4}

{2}

F3 j

{1, 2, 3, 4}

{1, 2, 4}

–

{1, 2, 3, 4}

{1, 2, 3, 4}

{1, 2, 4}

F4 j

{1, 2, 3, 4}

{1}

{}

–

{1, 2, 3, 4}

{}

F5 j

{1, 4}

{}

{}

{}

–

{}

F6 j

{1, 2, 3, 4}

{1, 3, 4}

{3}

{1, 2, 3, 4}

{1, 2, 3, 4}

–

Table 2.13 A 4–polar fuzzy discordance set j

1

2

3

G1 j

–

{1, 2, 3, 4}

{1, 2, 3, 4}

4

{1, 2, 3, 4}

{1, 4}

{1, 2, 3, 4}

G2 j

{}

–

{1, 2, 4}

{1}

{}

{1, 3, 4}

G3 j

{}

{3}

–

{}

{}

{3}

G4 j

{}

{2, 3, 4}

{1, 2, 3, 4}

–

{}

{1, 2, 3, 4}

G5 j

{2, 3}

{1, 2, 3, 4}

{1, 2, 3, 4}

{1, 2, 3, 4}

–

{1, 2, 3, 4}

G6 j

{}

{2}

{1, 2, 4}

{}

{}

–

1. The 4–polar fuzzy decision matrix is an input given in Table 2.10. 2. Suppose that the group of experts assigned the following weights to each quality. w1 = 0.247, w2 = 0.182, w3 = 0.346, w4 = 0.225, where,

4 

w j = 1.

j=1

3. The 4–polar fuzzy weighted decision matrix is given in Table 2.11. 4. Tables 2.12 and 2.13 represent the 4–polar fuzzy concordance and 4–polar fuzzy discordance sets, respectively.

2.3 m–Polar Fuzzy ELECTRE-I Method

103

5. The 4–polar fuzzy concordance matrix is evaluated as follows: ⎞ − 0 0 0 0.528 0 ⎜ 1 − 0.346 0.753 1 0.182 ⎟ ⎟ ⎜ ⎜ 1 0.654 − 1 1 0.654 ⎟ ⎟. ⎜ F =⎜ ⎟ 0.247 0 − 1 0 ⎟ ⎜ 1 ⎠ ⎝ 0.472 0 0 0 − 0 1 0.818 0.346 1 1 − ⎛

6. The 4–polar fuzzy concordance level is f = 0.5. 7. The 4–polar fuzzy discordance matrix is evaluated as follows: ⎞ − 1 1 1 1 1 ⎜0 − 1 0.4550 0 1 ⎟ ⎟ ⎜ ⎜ 0 0.3180 − 0 0 1⎟ ⎟. ⎜ G=⎜ 1 1 − 0 1⎟ ⎟ ⎜0 ⎝1 1 1 1 − 1⎠ 0 0.3914 0.935 0 0 − ⎛

8. The 4–polar fuzzy discordance level is g = 0.6033. 9. According to the concordance and discordance levels, the 4–polar fuzzy concordance and discordance dominance matrices are computed as follows: ⎛

− ⎜1 ⎜ ⎜1 H =⎜ ⎜1 ⎜ ⎝0 1 ⎛

− ⎜1 ⎜ ⎜1 L=⎜ ⎜1 ⎜ ⎝0 1

0 − 1 0 0 1

0 0 − 0 0 0

0 1 1 − 0 1

1 1 1 1 − 1

⎞ 0 0⎟ ⎟ 1⎟ ⎟, 0⎟ ⎟ 0⎠ −

0 − 1 0 0 1

0 0 − 0 0 0

0 1 1 − 0 1

0 1 1 1 − 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 0⎠ −

10. The 4–polar fuzzy aggregated dominance matrix is computed as: ⎛

− ⎜1 ⎜ ⎜1 M =⎜ ⎜1 ⎜ ⎝0 1

0 − 1 0 0 1

0 0 − 0 0 0

0 1 1 − 0 1

0 1 1 1 − 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 0⎠ −

104

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Fig. 2.18 Directed graph of outranking relation of instructors

I1

I2

I3

I5

I4

I6

11. The outranking relation of the alternatives is shown in the (Fig. 2.18). Therefore, both I3 and I6 are a best choice. To facilitate the computations, the Algorithm 2.3.1 of an m−polar fuzzy ELECTRE-I method is described as follows: Algorithm 2.3.1 The algorithm of an m−polar fuzzy ELECTRE-I approach 1. Input A as the set of alternatives, T as the set of criteria, Z as the decision matrix and W as the normalized weights. 2. Compute the weighted m–polar fuzzy decision matrix Y . 3. Compute the m–polar fuzzy concordance sets F pq . 4. Compute the m–polar fuzzy discordance sets G pq . 5. Compute the m–polar fuzzy concordance indices f pq  s . 6. Compute the m–polar fuzzy concordance matrix F. 7. Compute the m–polar fuzzy discordance indices g pq  s . 8. Compute the m–polar fuzzy discordance matrix G. 9. Compute the m–polar fuzzy concordance level f and discordance level g. 10. Compute the m–polar fuzzy concordance dominance matrix H and discordance dominance matrix L. 11. Compute the m–polar fuzzy aggregated dominance matrix M. 12. Draw the directed decision graph. 13. The recommendation is A, an alternative with maximum value. The flowchart of an m−polar fuzzy ELECTRE-I approach is shown in Fig. 2.19.

2.3 m–Polar Fuzzy ELECTRE-I Method Fig. 2.19 Flow chart of m–polar fuzzy ELECTRE-I Approach

105

Identifications of the alternatives and criteria

Construct the m-polar fuzzy decision matrix

Assess the weights of the criteria

Construct the weighted m-polar fuzzy decision matrix

Calculate the Euclidean distance between alternatives

Evaluate the m-polar fuzzy concordance matrix

Evaluate the m-polar fuzzy discordance matrix

Compute the concordance and discordance levels

Construct the m-polar fuzzy concordance dominance matrix

Construct the m-polar fuzzy discordance dominance matrix

Construct the m-polar fuzzy aggregate dominance matrix

Rank the alternatives according to the decision graph

106

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

2.4 Contribution, Sensitivity and Comparison Analysis The contribution, sensitivity and comparison analysis of the presented approach are summarized as follows: 1. m–polar fuzzy sets are used to represent imprecise, inexact, uncertain information involving multiple attributes. 2. The m–polar fuzzy ELECTRE-I approach proposed here is able to handle MCDM problems characterized by vague and uncertain data under multi-polar format. 3. Three applications are studied, including selection of suitable location of a diesel power plant, site selection of an airport, and performance evaluation of physical science’s instructors. They demonstrate the efficacy of the m–polar fuzzy ELECTRE-I approach. 4. The bipolar fuzzy ELECTRE-I approach is unable to handle the complexity of ambiguous data under multi-polar information. The approach in Sect. 2.3 overcomes such a drawback easily because it is based on m–polar fuzzy set theory which is designed to handle multi-polar information. 5. This procedure consists of a binary outranking approach in which the alternatives can be compared regardless of their clear preference. It is more reliable as it does not depend on the expert’s personal opinions. 6. This method does not always lead to a unique selection, but rather it leads to a small subset of preferable alternatives.

2.5 The Concept of m−Polar Fuzzy Linguistic Variable This section considers the case of multi-polar information that adheres to a linguistic pattern. Technically, the foundation depends upon the next known idea: Definition 1 Linguistic variables are variables, whose values are words or sentences in a natural or artificial language. If these words are described by fuzzy sets that are defined over a universal set, then the variables are called fuzzy linguistic variables. Definition 2 A linguistic term set is defined by means of an ordered structure providing the term set that is distributed on a scale at which a total order has been defined. For example, a set S of seven terms, could be written as follows: S = {so = nothing, s1 = ver y low, s2 = low, s3 = medium, s4 = high, s5 = ver y high, s6 = per f ect}. An m−polar fuzzy linguistic variable is a variable that considers words in natural language(s) as its values. Now the values of such a variable are characterized by

2.5 The Concept of m−Polar Fuzzy Linguistic Variable

107

m–polar fuzzy sets that are defined in a universe that contains the variable. Here is the formal definition: Definition 3 An m−polar fuzzy linguistic variable is characterized by a 4−tuple (L v , V, Pd , M) such that • L v is the name of an m–polar fuzzy linguistic variable, • V is the set of linguistic values of L v , • Pd = [0, ∞) is the physical domain in which an m–polar fuzzy linguistic variable takes its crisp values, • M is the semantic rule that relates every linguistic value in V with m–polar fuzzy set. The linguistic variable is called an m–polar fuzzy linguistic variable, because its linguistic values are further classified by m different characteristics. Whereas Pd is the physical domain in which an m–polar fuzzy linguistic variable takes its crisp values. This domain can be arranged in sections for linguistic values according to particular requirements. Finally, a semantic rule M is described. This is actually a rule, that differentiates the m–polar fuzzy linguistic variable from previously defined linguistic variables. This rule relates the linguistic values (Vl s|l = 1, 2, . . . , k) of an m–polar fuzzy linguistic variable with an m–polar fuzzy set, which shows that each linguistic value is further classified by m different characteristics. The degree of linguistic values is defined by dli = pi ◦ dl (Vl ) ∈ [0, 1], where i = 1, 2, · · · , m. It clearly shows m different characterizations of each linguistic value. The existence of m–polar fuzzy linguistic variables is discussed in real life by the following example: Example 2.1 Consider a linguistic variable “age” and describe its different stages by a set V = {EY, Y, O, E O}, which is the set of linguistic values of “age”, where • • • •

“EY” stands for extremely young, “Y” stands for young, “O” stands for old, “EO” stands for extremely old.

The physical domain for our linguistic variable is Pd = [1, 100], which is a subset of the real non-negative numbers, and each linguistic value has different range of physical domain given as follows: • • • •

For extremely young age, the physical domain is 1–20, For young age, the physical domain is 20–50, For old age, the physical domain is 50–80, For extremely old age, the physical domain is 80–100.

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

This linguistic variable is called a 4–polar fuzzy linguistic variable, because a semantic rule M is presented which describes each linguistic value in the set V by a 4−polar fuzzy set. According to the 4–polar fuzzy set, each linguistic value is characterized as: • p1 ◦ dk (Vk ) serves as “Physically energetic”, it describes the ability of someone that is active enough, with enthusiasm and eagerness to do any task. • p2 ◦ dk (Vk ) serves as “Mental health”, it means being generally able to think, feel and react in the ways that someone needs and wants to conduct his or herlife. • p3 ◦ dk (Vk ) serves as “Ability”, it is an acquired or natural capacity or talent that enables an individual to perform a particular job or task successfully. • p4 ◦ dk (Vk ) serves as “Effort”, it is an earnest or strenuous attempt to achieve a specified purpose. where k = 1, 2, 3, 4. Thus, L v = { EY, (0.89, 0.90, 0.95, 0.82), Y, (0.69, 0.58, 0.61, 0.55),

O, (0.33, 0.29, 0.32, 0.30), E O, (0.10, 0.13, 0.11, 0.10)}. In terms of the linguistic variable (age), a linguistic value such as extremely young is characterized by four different specifications including physical energy, mental difficulty, ability and effort which show the 4–polar fuzzy restrictions on the linguistic values of the base variable. These 4-polar fuzzy restrictions show different states considered for the linguistic values in order to evaluate each base variable in terms of 4–polar fuzzy set. Thus, age is called a 4–polar fuzzy linguistic variable by Definition 3. Its tabular representation is described by Table 2.14, which shows the range of each linguistic value in terms of a 4–polar fuzzy set. The tabular representation clearly shows the range of each linguistic value in terms of a 4–polar fuzzy set and shows how each 4-polar fuzzy set is related with the physical domain Pd . It simply defines the meaning of a 4–polar fuzzy linguistic variable, which shows that the 4–polar fuzzy linguistic variable (age) has four linguistic values, which have specified physical domains according to their limits. Each linguistic value is further classified by four different specifications according to states of the linguistic values. From Table 2.14, in extremely young age the range of physical domain is from 1 to 20 and each physical domain is associated with specific values of a 4–polar fuzzy set. For example, the physical domain value “10” is associated with the tuple (0.39, 0.60, 0.56, 0.60) which describes that a person (as compared to others) is considered extremely young at the age of 10. It thus has the following specifications: • The membership value 0.39 shows that an extremely young person at the age of 10 is 39% physically energetic. • The membership value 0.60 shows that the mental health of an extremely young person at the age of 10 is 60%. • The membership value 0.56 shows that the ability of an extremely young person to do a task successfully at the age of 10 is 56%.

2.5 The Concept of m−Polar Fuzzy Linguistic Variable Table 2.14 A 4–polar fuzzy linguistic variable 4−polar Linguistic Physical 4−polar fuzzy values of fuzzy set domain 4–polar Physically linguistic fuzzy energetic variable linguistic (L v ) variable Age

Extremely young

Young

Old

Extremely old

109

Mental health

Ability

Effect

1

0.05

0.09

0.08

0.07

.. .

.. .

.. .

.. .

.. .

10 .. .

0.39 .. .

0.60 .. .

0.56 .. .

0.60 .. .

20 .. .

0.89 .. .

0.90 .. .

0.95 .. .

0.82 .. .

30 .. .

0.80 .. .

0.81 .. .

0.72 .. .

0.75 .. .

40 .. .

0.69 .. .

0.58 .. .

0.61 .. .

0.55 .. .

50 .. .

0.40 .. .

0.49 .. .

0.45 .. .

0.50 .. .

60 .. .

0.33 .. .

0.29 .. .

0.32 .. .

0.30 .. .

70 .. .

0.25 .. .

0.23 .. .

0.31 .. .

0.29 .. .

80 .. .

0.20 .. .

0.19 .. .

0.21 .. .

0.19 .. .

90 .. .

0.10 .. .

0.13 .. .

0.11 .. .

0.10 .. .

100

0.03

0.02

0.01

0.02

110

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

• The membership value 0.60 shows that the effort of an extremely young person to achieve a specified purpose at the age of 10 is 60%. Example 2.2 Let “temperature” be a linguistic variable and V = {H ot, W ar m, Cool, Cold} be the set of its linguistic values. The physical domain for the linguistic variable is Pd = [0 ◦ C, 50 ◦ C], which is the set of real non-negative numbers, and each linguistic value has a different range of physical domain given as follows: • • • •

For cold temperature, physical domain is 0 ◦ C to 10 ◦ C, For cool temperature, physical domain is 10 ◦ C to 20 ◦ C, For warm temperature, physical domain is 20 ◦ C to 30 ◦ C, For hot temperature, physical domain is 30 ◦ C to 50 ◦ C.

We call this linguistic variable a 3–polar fuzzy linguistic variable (3–polar fuzzy linguistic variable), because we describe a semantic rule M that relates each linguistic value in set V with a 3−polar fuzzy set (3–polar fuzzy set). According to the 3–polar fuzzy set, each linguistic value is characterized as • p1 ◦ dl (Vl ) serves as “heat energy”, • p2 ◦ dl (Vl ) serves as “air pressure”, • p3 ◦ dl (Vl ) serves as “water vapors”, where l = 1, 2, 3. Thus, we have L v = { H ot, (0.90, 0.71, 0.20), W ar m, (0.79, 0.61, 0.29),

Cool, (0.45, 0.39, 0.69), Cold, (0.12, 0.19, 0.89)}. In terms of the variable (temperature), four different linguistic values are discussed in Example 2.1 and each linguistic value is further classified by three different criteria or properties on which the linguistic value shows its dependence. It shows the 3–polar fuzzy restrictions on the values of the base variable. These 3-polar fuzzy restrictions clearly show that each linguistic value totally depends on heat energy, air pressure and water vapors. Thus, it can be claimed that temperature behaves as a 3–polar fuzzy linguistic variable.

2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM We now present an m−polar fuzzy linguistic ELECTRE-I approach for MCDM problems, which is based on the concept of m–polar fuzzy linguistic variable. This approach is applied on real life examples, to show its importance and feasibility. In this approach, L v is chosen as an m–polar fuzzy linguistic variable, and A = {a1 , a2 , . . . , an }, the set of m–polar fuzzy linguistic variable of different alternatives. According to this m–polar fuzzy linguistic variable, {Vl |l = 1, 2, . . . , k} is taken as the set of linguistic values. These linguistic values are classified by m different

2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM

111

Table 2.15 An m–polar fuzzy linguistic decision matrix m−polar fuzzy linguistic variable Lv

Physical domain

Pd1

Pd2

···

Pdk

m−polar fuzzy linguistic values a1 a2 . . .

V1

V2

···

Vk

1 , d2 , · · · , dm ) (d11 11 11 1 , d2 , · · · , dm ) (d21 21 21

1 , d2 , . . . , dm ) (d12 12 12

···

1 , d2 , . . . , dm ) (d1k 1k 1k

1 , d2 , . . . , dm ) (d22 22 22

···

. . .

. . .

1 , d2 , . . . , dm ) (d2k 2k 2k

1 , d2 , · · · , dm ) (dn1 n1 n1

1 , d2 , · · · , dm ) (dn2 n2 n2

···

. . .

an

. . .

1 , d2 , · · · , dm ) (dnk nk nk

characteristics. The degree of each alternative (a p ∈ A, p = 1, 2, . . . , n) over all the linguistic values Vl s is given by m–polar fuzzy set ℘ p = {(a p , d ipl )|i = 1, 2, . . . m}, where d ipl = pi ◦ d pl (a p , Vl ) ∈ [0, 1] and d ipl classify the different characteristics or properties of linguistic values. Pd is the actual physical domain in which the m–polar fuzzy linguistic variable takes its quantitative (crisp) values, i.e., Pd = [0, +∞). In this case, the most suitable m values are chosen from the physical domain of each linguistic value. A decision maker is responsible for evaluating the m–polar fuzzy linguistic variable of n different alternatives under k different linguistic values. (i) The suitable ratings of alternatives are assessed in term of m different characteristics under the common physical domain Pd . The tabular representation of the m–polar fuzzy linguistic decision matrix is given by Table 2.15. (ii) The decision maker is allowed to assign weights to each linguistic value of m– polar fuzzy linguistic variable of alternatives according to his or her opinion about the importance of each linguistic value. The case of an m–polar fuzzy linguistic variable is discussed, so the decision maker has to assign the weights in terms of the linguistic term set L = {L 1 = ver y low, L 2 = low, · · · , L k = extr emly high}. It is supposed that the weights (wl ∈ (0, 1]) assigned by the k

decision maker satisfy the normalized condition. i.e., wl = 1. l=1

(iii) The weighted m–polar fuzzy linguistic decision matrix is calculated as W = [(e1pl , e2pl , · · · , empl )]n×k , where e1pl = wl d 1pl , e2pl = wl d 2pl , . . . , empl = wl d m pl . (iv) The m–polar fuzzy linguistic concordance set is defined as Yuv = {1 ≤ l ≤ k|eul ≥ evl , u = v; u, v = 1, 2, . . . , n},

112

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

where e pl = e1pl + e2pl + · · · + empl . (v) The m–polar fuzzy linguistic discordance set is defined as Z uv = {1 ≤ l ≤ k|eul ≤ evl , u = v; u, v = 1, 2, . . . , n}, where e pl = e1pl + e2pl + · · · + empl . (vi) The m–polar fuzzy linguistic concordance indices are determined as yuv =



wl ,

l∈Yuv

therefore, the m–polar fuzzy linguistic concordance matrix is computed as ⎛

− y12 y13 ⎜ y21 − y23 ⎜ ⎜ Y = ⎜ y31 y32 − ⎜ .. .. .. ⎝ . . . yn1 yn2 yn3

⎞ · · · y1n · · · y2n ⎟ ⎟ · · · y3n ⎟ ⎟. . ⎟ · · · .. ⎠ ··· −

(vii) The m–polar fuzzy linguistic discordance indices are determined as  max

z uv =

l∈Z uv

max l



1 1 [(eul m

m m 2 1 2 2 2 2 − evl ) + (eul − evl ) + · · · + (eul − evl ) ]

1 1 [(eul m

m m 2 1 2 2 2 2 − evl ) + (eul − evl ) + · · · + (eul − evl ) ]

,

therefore, the m–polar fuzzy linguistic discordance matrix can be computed as ⎞ ⎛ − z 12 z 13 · · · z 1n ⎜ z 21 − z 23 · · · z 2n ⎟ ⎟ ⎜ ⎟ ⎜ Z = ⎜ z 31 z 32 − · · · z 3n ⎟ . ⎜ .. .. .. .. ⎟ ⎝ . . . ··· . ⎠ z n1 z n2 z n3 · · · − (viii) For the rankings of alternatives, we compute threshold values known as m– polar fuzzy linguistic concordance and discordance levels. The m–polar fuzzy linguistic concordance and discordance levels are the average of the m–polar fuzzy linguistic concordance and discordance indices:  1 y¯ = yuv , n(n − 1) u=1 v=1 n

n

u=v u=v

2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM

 1 z uv . n(n − 1) u=1 v=1 n

z¯ =

113

n

u=v u=v

(ix) The m–polar fuzzy linguistic concordance dominance matrix according to its m–polar fuzzy linguistic concordance level is computed as ⎛

− r12 r13 ⎜ r21 − r23 ⎜ ⎜ R = ⎜ r31 r32 − ⎜ .. .. .. ⎝ . . . rn1 rn2 rn3

where ruv =

⎞ · · · r1n · · · r2n ⎟ ⎟ · · · r3n ⎟ ⎟, . ⎟ · · · .. ⎠ ··· −

1, yuv ≥ y¯ ; 0, yuv < y¯ .

(x) The m–polar fuzzy linguistic discordance dominance matrix according to its m–polar fuzzy linguistic discordance level is computed as ⎛

− s12 s13 ⎜ s21 − s23 ⎜ ⎜ S = ⎜ s31 s32 − ⎜ .. .. .. ⎝ . . . sn1 sn2 sn3

where suv =

⎞ · · · s1n · · · s2n ⎟ ⎟ · · · s3n ⎟ ⎟, .. ⎟ ··· . ⎠ ··· −

1, z uv < z¯ ; 0, z uv ≥ z¯ .

(xi) The aggregated m–polar fuzzy linguistic dominance matrix is computed as ⎛

− t12 ⎜ t21 − ⎜ ⎜ T = ⎜ t31 t32 ⎜ .. .. ⎝ . . tn1 tn2 where tuv is defined as

t13 t23 − .. . tn3

⎞ · · · t1n · · · t2n ⎟ ⎟ · · · t3n ⎟ ⎟, .. ⎟ ··· . ⎠ ··· −

tuv = ruv suv .

(xii) Finally, rank the alternatives according to the outranking values of matrix T . For each pair of alternatives there exist a directed edge from alternative au to av if and only if tuv = 1. Thus the following three cases arises.

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

1. There exists a unique directed edge from au to av , which shows au is preferred over av . 2. There exists directed edges from au to av and av to au , which shows au and av are indifferent. 3. There does not exist any edge between au and av , which shows au and av are not comparable. In the next subsection, the m−polar fuzzy linguistic ELECTRE-I method is applied for MCDM to a real life example.

2.6.1 Salary Analysis of Companies Salary analysis of companies is considered as one of the proxies to compare the economic situation of companies and its evaluation is far from being easy. The m– polar fuzzy linguistic ELECTRE-I method is presented for MCDM, in which salary is a linguistic variable and S = {Sc1 , Sc2 , Sc3 , Sc4 , Sc5 } is the set of salary package of five different well known companies. V = {Low, Moderate, Good, V er y good} is taken as the set of linguistic values of salary. The decision maker has to evaluate the companies on the basis of the linguistic values of their salary package and he has to design a physical domain in which salary package takes its quantitative values, i.e., Pd = [10k, 100k]. The physical domain for the linguistic values of salary package is given as follows: • • • •

For low salary, the physical domain is below 30k, For moderate salary, the physical domain is 30k–50k, For good salary, the physical domain is 50k–70k, For very good salary, the physical domain is above 70k.

The physical domain of each linguistic value shows the range of salary given by decision maker. The degree of salary of each company, over all the linguistic values is given by the 4–polar fuzzy set ℘ p = {(Sc p , d ipl )|i = 1, 2, 3, 4}, where • • • •

d 1pl d 2pl d 3pl d 4pl

= = = =

p1 ◦ d pl (a p , Vl ) serves for career, p2 ◦ d pl (a p , Vl ) serves for labor market, p3 ◦ d pl (a p , Vl ) serves for experience, p4 ◦ d pl (a p , Vl ) serves for credential,

where p = 1, 2, . . . , 5, and l = 1, 2, 3, 4. The 4–polar fuzzy set shows the further classifications or properties on which the linguistic values depend. (i) The tabular representation of the 4–polar fuzzy linguistic decision matrix is given by Table 2.16. It shows the different ratings of linguistic values assigned by a decision maker, in which he assigns ratings according to his expertise.

2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM Table 2.16 A 4–polar fuzzy linguistic decision matrix 4−polar fuzzy Physical domain linguistic variable (Salary) Below 30 k 30–50 k 4−polar fuzzy linguistic values Low Moderate Sc1 Sc2 Sc3 Sc4 Sc5

(0.3,0.4,0.5,0.4) (0.2,0.5,0.4,0.3) (0.4,0.3,0.5,0.5) (0.3,0.3,0.2,0.4) (0.2,0.5,0.4,0.4)

(0.4,0.6,0.6,0.3) (0.5,0.5,0.6,0.4) (0.6,0.5,0.4,0.5) (0.7,0.4,0.6,0.5) (0.6,0.5,0.4,0.7)

50–70 k

115

Above 70 k

Good

Very good

(0.6,0.7,0.8,0.7) (0.4,0.6,0.7,0.5) (0.6,0.6,0.7,0.7) (0.7,0.4,0.5,0.6) (0.4,0.5,0.5,0.6)

(0.8,0.9,0.8,1.0) (0.7,0.8,1.0,0.9) (0.6,0.8,0.9,0.9) (0.7,0.7,0.8,0.8) (0.6,0.7,0.8,1.0)

(ii) The normalized weights assigned to each linguistic value of 4–polar fuzzy linguistic variable by the decision maker are given as follows: wl = (0.15, 0.19, 0.27, 0.39). (iii) The weighted 4–polar fuzzy linguistic decision matrix is calculated in Table 2.17. (iv) The 4-polar fuzzy concordance sets are calculated in Table 2.18. (v) The 4-polar fuzzy discordance sets are calculated in Table 2.19. (vi) A 4–polar fuzzy linguistic concordance matrix is calculated as follows: ⎛ ⎞ − 0.81 0.66 0.81 0.81 ⎜ 0.19 − 0.58 0.81 0.66 ⎟ ⎜ ⎟ ⎟ Y =⎜ ⎜ 0.34 0.61 − 0.81 0.81 ⎟ . ⎝ 0.19 0.46 0.19 − 0.27 ⎠ 0.19 0.34 0.19 0.73 − (vii) A 4–polar fuzzy linguistic discordance matrix is calculated as follows: ⎛ ⎞ − 0.3189 0.6639 0.6488 0.8293 ⎜1 − 1 1 0.6890 ⎟ ⎜ ⎟ ⎜ − 0.5451 0.4451 ⎟ Z = ⎜ 1 0.7222 ⎟. ⎝1 1 1 − 1 ⎠ 1 1 1 0.9791 − (viii) A 4–polar fuzzy linguistic concordance level y = 0.5230, and a 4–polar fuzzy linguistic discordance level z = 0.8421, are calculated.

116

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.17 A weighted 4–polar fuzzy linguistic decision matrix 4−polar fuzzy Physical domain linguistic variable (Salary) Below 30 k 30–50 k 4−polar fuzzy linguistic values Low (0.15) Moderate (0.19) Sc1 Sc2 Sc3 Sc4 Sc5 4−polar fuzzy linguistic variable (Salary)

(0.045,0.060,0.075,0.060) (0.030,0.075,0.060,0.045) (0.060,0.045,0.075,0.075) (0.045,0.045,0.030,0.060) (0.030,0.075,0.060,0.060) Physical domain

50–70 k Above 70 k 4−polar fuzzy linguistic values Good (0.27) Very good (0.39) (0.162,0.189,0.216,0.189) (0.312,0.351,0.312,0.390) (0.108,0.162,0.189,0.135) (0.273,0.312,0.390,0.351) (0.162,0.162,0.189,0.189) (0.234,0.312,0.351,0.351) (0.189,0.108,0.135,0.162) (0.273,0.273,0.312,0.312) (0.108,0.135,0.135,0.162) (0.234,0.273,0.312,0.390)

Sc1 Sc2 Sc3 Sc4 Sc5

Table 2.18 A 4–polar fuzzy linguistic concordance set v 1 2 3 Y1v Y2v Y3v Y4v Y5v

− {2} {1, 2} {2} {2}

{1, 3, 4} − {1, 2, 3} {2, 3} {1, 2}

{3, 4} {2, 4} − {2} {2}

Table 2.19 A 4–polar fuzzy linguistic discordance set v 1 2 3 Z 1v Z 2v Z 3v Z 4v Z 5v

(0.076,0.114,0.114,0.057) (0.095,0.095,0.114,0.076) (0.114,0.095,0.076,0.095) (0.133,0.076,0.114,0.095) (0.114,0.095,0.076,0.133)

− {1, 3, 4} {3, 4} {1, 3, 4} {1, 3, 4}

{2} − {2, 4} {1, 3, 4} {3, 4}

{1, 2} {1, 2, 3} − {1, 3, 4} {1, 3, 4}

4

5

{1, 3, 4} {1, 3, 4} {1, 3, 4} − {1, 2, 4}

{1, 3, 4} {3, 4} {1, 3, 4} {3} −

4

5

{2} {2, 3} {2} − {3}

{2} {1, 2} {2} {1, 2, 4} −

2.6 m–Polar Fuzzy Linguistic ELECTRE-I Approach for MCDM

117 Sc1

Fig. 2.20 Graphical representation of outranking relation of companies

Sc5

Sc2

Sc3

Sc4

(ix) A 4–polar fuzzy linguistic concordance follows: ⎛ − 1 1 ⎜0 − 1 ⎜ R=⎜ ⎜0 1 − ⎝0 0 0 0 0 0

dominance matrix is calculated as

(x) A 4–polar fuzzy linguistic discordance follows: ⎛ − 1 1 ⎜0 − 0 ⎜ S=⎜ ⎜0 1 − ⎝0 0 0 0 0 0

dominance matrix is calculated as

(xi) An aggregated 4–polar fuzzy linguistic follows: ⎛ − 1 1 ⎜0 − 0 ⎜ T =⎜ ⎜0 1 − ⎝0 0 0 0 0 0

dominance matrix is calculated as

1 1 1 − 1

1 0 1 − 0

1 0 1 − 0

⎞ 1 1⎟ ⎟ 1⎟ ⎟. 0⎠ −

⎞ 1 1⎟ ⎟ 1⎟ ⎟. 0⎠ −

⎞ 1 1⎟ ⎟ 1⎟ ⎟. 0⎠ −

(xii) Finally, the companies can be ranked according to the outranking values of aggregated 4–polar fuzzy linguistic dominance matrix T . A directed graph for each pair of companies is drawn as shown in Fig. 2.20. From the directed graph of companies shown in Fig. 2.20, the following cases arise:

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.20 Tabular representation of comparison of companies Comparison of companies

Yuv

Z uv

yuv

z uv

ruv

suv

tuv

Ranking

(Sc1 , Sc2 )

{1, 3, 4}

{2}

0.81

0.3189

1

1

1

Sc1 → Sc2

(Sc1 , Sc3 )

{3, 4}

{1, 2}

0.66

0.6639

1

1

1

Sc1 → Sc3

(Sc1 , Sc4 )

{1, 3, 4}

{2}

0.81

0.6488

1

1

1

Sc1 → Sc4

(Sc1 , Sc5 )

{1, 3, 4}

{2}

0.81

0.8293

1

1

1

Sc1 → Sc5

(Sc2 , Sc1 )

{2}

{1, 3, 4}

0.19

1

0

0

0

Incomparable

(Sc2 , Sc3 )

{2, 4}

{1, 3, 4}

0.58

1

1

0

0

Incomparable

(Sc2 , Sc4 )

{1, 3, 4}

{1, 3, 4}

0.81

1

1

0

0

Incomparable

(Sc2 , Sc5 )

{3, 4}

{1, 2, 4}

0.66

0.6890

1

1

1

Sc2 → Sc5

(Sc3 , Sc1 )

{1, 2}

{3, 4}

0.34

1

0

0

0

Incomparable

(Sc3 , Sc2 )

{1, 2, 3}

{2, 4}

0.61

0.7222

1

1

1

Sc3 → Sc2

(Sc3 , Sc4 )

{1, 3, 4}

{2}

0.81

0.5451

1

1

1

(Sc3 , Sc5 )

{1, 3, 4}

{2}

0.81

0.4451

1

1

1

Sc3 → Sc4 Sc3 → Sc5

(Sc4 , Sc1 )

{2}

{1, 3, 4}

0.19

1

0

0

0

Incomparable

(Sc4 , Sc2 )

{2, 3}

{1, 3, 4}

0.46

1

0

0

0

Incomparable

(Sc4 , Sc3 )

{2}

{1, 3, 4}

0.19

1

0

0

0

Incomparable

(Sc4 , Sc5 )

{3}

{1, 2, 4}

0.27

1

0

0

0

Incomparable

(Sc5 , Sc1 )

{2}

{1, 3, 4}

0.19

1

0

0

0

Incomparable

(Sc5 , Sc2 )

{1, 2}

{3, 4}

0.34

1

0

0

0

Incomparable

(Sc5 , Sc3 )

{2}

{1, 3, 4}

0.19

1

0

0

0

Incomparable

(Sc5 , Sc4 )

{1, 2, 4}

{3}

0.73

0.9791

1

0

0

Incomparable

1. There exist directed edges from Sc1 to Sc2 , Sc3 , Sc4 and Sc5 which show Sc1 is preferred over all other companies according to its salary package. 2. Similarly, Sc1 is preferred over Sc5 . 3. Similarly, Sc3 is preferred over Sc2 , Sc4 and Sc5 . 4. There does not exist any edge from Sc4 to any other company, which shows Sc4 is incomparable to others. 5. Similarly, Sc5 is incomparable to others. Hence, Sc1 is the most dominated company as compared to the others and it has highest ranking according to its salary package. The comparison of companies is presented and the whole procedure is summarized in Table 2.20.

2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM In this section, an m–polar fuzzy linguistic ELECTRE-I approach is described for MCGDM problems, which is based on the concept of m–polar fuzzy linguistic variable. This approach is applied on real life examples, to show its importance and feasi-

2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM

119

Table 2.21 An of m–polar fuzzy linguistic decision matrix m−polar fuzzy linguistic variable Lv

Decision makers

Physical domain

Pd1

···

Pd2

Pdk

m−polar fuzzy linguistic values V1 Dg

···

Vk

g1

g2

gm

···

g1

g2

gm

(d1k , d1k , . . . , d1k )

···

(d2k , d2k , . . . , d2k )

V2

g1

g2

gm

g1

g2

gm

a1

(d11 , d11 , . . . , d11 )

(d12 , d12 , . . . , d12 )

a2

(d21 , d21 , . . . , d21 )

(d22 , d22 , . . . , d22 )

. . . an

. . .

g1

g2

gm

(dn1 , dn1 , . . . , dn1 )

. . .

g1

g2

gm

(dn2 , dn2 , . . . , dn2 )

. . . ···

g1

g2

gm

g1

g2

gm

g1

g2

gm

. . .

(dnk , dnk , . . . , dnk )

bility. In this approach, a group of r decision makers (Dg , g = 1, 2, · · · , r ) is responsible for evaluating m–polar fuzzy linguistic variable of n different alternatives (a p ∈ A, p = 1, 2, . . . , n) under k different linguistic values of L v (Vl , l = 1, 2, . . . , k). In the same sense, as we used in MCDM the degree of each alternative over all the lingi guistic values Vl s is given by an m–polar fuzzy set ℘ p = {(a p , d pl )|i = 1, 2, . . . m}, gi

g

where d pl = pi ◦ d pl (a p , Vl ) ∈ [0, 1] and d ipl classify the different properties or criteria’s of linguistic values according to each decision maker. (i) In this case, a group of r decision makers is responsible for evaluating m– polar fuzzy linguistic variable of n different alternatives. They are assessed in term of m different characteristics under a physical domain Pd . The tabular representation of the m–polar fuzzy linguistic decision matrix under a group of r decision makers is given by Table 2.21, which describes the ratings given by each decision maker. Table 2.21 shows the different ratings of linguistic values assigned by a group of r decision makers according to their respective expertise. The final m–polar fuzzy linguistic decision matrix under the group of decision makers is the aggregated m–polar fuzzy linguistic decision matrix, that is the average of the ratings of all decision makers. These aggregated ratings are calculated as follows: 

d 1pl =

1  g1 2 1  g2 1  gm  d pl , d pl = d pl , . . . , d mpl = d . r g=1 r g=1 r g=1 pl r

r

r

p = 1, 2, . . . , n and l = 1, 2, . . . , k. For the final decision and ratings, the aggregated m–polar fuzzy linguistic decision matrix is calculated in Table 2.22 by using the above averaging procedure.

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.22 An aggregate m–polar fuzzy linguistic decision matrix m−polar fuzzy linguistic variable Lv

Physical domain

Pd1

···

Pd2

Pdk

m−polar fuzzy linguistic values and weights

a1 a2 . . . an

V1

V2

···

Vk

w1

w2

···

wk

···

1 , d2 , . . . , dm ) (d1k 1k 1k













1 , d2 , . . . , dm ) (d11 11 11

1 , d2 , . . . , dm ) (d12 12 12

. . .

. . .

1 , d 2 , . . . , d m  ) (d21 21 21

1 , d 2 , . . . , d m  ) (d22 22 22

1 , d 2 , . . . , d m  ) (dn1 n1 n1

1 , d 2 , . . . , d m  ) (dn2 n2 n2







1 , d 2 , . . . , d m  ) (d2k 2k 2k

··· . . .

. . .







1 , d2 , . . . , dm ) (dnk nk nk

···

(ii) Decision makers have an authority to assign weights to each linguistic value of alternatives according to their choice and importance of each linguistic value, but the case of m–polar fuzzy linguistic variable is discussed so decision makers have to assign the weights in terms of linguistic term set L = {L 1 = extr emely low, L 2 = medium, . . . , L k = ver y high}. It is supposed that the weights assigned by the decision makers are g

g

g

W g = (w1 , w2 , . . . , wk ) ∈ (0, 1], g = 1, 2, . . . , r. The weights assigned by the decision makers satisfy the normalized condition, i.e., k  g wl = 1, g = 1, 2, . . . , r. l=1

The aggregated weights according to the decision makers are W  where, r 1 g  wl = w , l = 1, 2, . . . , k. r g=1 l







= (w1 , w2 , . . . , wk ),



(iii) The weighted aggregated m–polar fuzzy linguistic decision matrix W = [(e1pl ,   e2pl , . . . , empl )]n×k under the group decision making is calculated as 

















e1pl = wl d 1pl , e2pl = wl d 2pl , . . . , empl = wl d mpl . Steps (iv)–(xii) are the same as those described in Sect. 2.6.

2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM

121

In the next subsection, the m−polar fuzzy linguistic ELECTRE-I method is applied for MCGDM to a real life example.

2.7.1 Selection of Most Corrupted Country Usually, corruption is considered as a criminal activity or offence initiated by a person or organization endowed with an official position of authority, often to attain unauthorized benefits. Corruption may carry several activities including misappropriation, extortion and bribery, though it may also involve practices that are enforced in several countries. Corruption can appear on multiple scales. It ranges from poor level consideration between a small number of people (“petty corruption”), to large scale corruption that influences the government (“grand corruption”), and corruption that has become a part of the society, carrying corruption as one of the evidences of coordinated crime. Crime and corruption are regional sociological junctures which consistently occur in all countries on a global scale albeit in varying proportion and degree. Increasingly, a number of tools and indices have been developed which can rate several forms of corruption with growing accuracy. • Petty corruption Petty corruption appears at a lower scale and occurs at the practice end of civil services when civil authoritative person accommodates the civil people. For example, in several small places such as police stations, registration offices, state licensing boards, and a number of other government and private sectors, it indicates the daily fault of commended power by low and mid level public officials in their interactions with frequent civilians, who are trying to approach basic services or goods in public places like schools, police departments, hospitals and other agencies. • Grand corruption Grand corruption occurs at the highest scale of government in a way that depends upon expressive overthrow of the legal, political and economic systems. Such a type of corruption is usually found in countries with dictatorial or authoritarian governments but also in those without sufficient policing of corruption. • Systemic corruption Systemic corruption or endemic corruption is primarily due to the weaknesses of an institution, organization or management. It can be differentiated with agents or individual officials, who perform corruptly with in the system. Factors which encourage systemic corruption include elective powers, lack of transparency, conflicting stimulus, monopolistic powers, low pay and a culture of immunity. Measuring the corruption at country level is a very difficult phenomenon, for anticorruption agencies because it is willfully hidden, it is far impossible to evaluate it directly. Corruption inside a rustic government undermines the solidity of its establishments and has a tendency to cause popular unrest. To overcome this difficulty

122

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

and to measure the corruption on country level, we use m–polar fuzzy linguistic ELECTRE-I approach for MCGDM, in which corruption is the linguistic variable and C = {C1 , C2 , C3 , C4 , C5 , C6 , C7 } is the set of seven countries from which corruption have to be measured. Let V = { pett y corr uption, grand corr uption, systemic corr uption} be the set of linguistic values of corruption. Anti-corruption agencies and media sources work as decision makers, they have to evaluate the countries on basis of the linguistic values of corruption and design a physical domain in which corruption takes its quantitative values, i.e., Pd = [10%, 100%]. The physical domain for linguistic values of corruption is given as follows: • For petty corruption, physical domain is 40 − 60%, • For grand corruption, physical domain is 50 − 80%, • For systemic corruption, physical domain is 30 − 70%. Physical domain of each linguistic value shows the scale of corruption given by group of decision makers. The degree of corruption of each country over all the linguistic values are given by 4–polar fuzzy set ℘ p = {(C p , d ipl )|i = 1, 2, 3, 4}, where • • • •

d 1pl d 2pl d 3pl d 4pl

= = = =

p1 ◦ d pl (a p , Vl ) serves for personal greed, p2 ◦ d pl (a p , Vl ) serves for cultural environment, p3 ◦ d pl (a p , Vl ) serves for Institutional scale, p4 ◦ d pl (a p , Vl ) serves for organizational level,

where p = 1, 2, . . . , 7, and l = 1, 2, 3. The 4–polar fuzzy set shows the further criteria’s or properties on which linguistic values depend. (i) Tabular representation of 4–polar fuzzy linguistic group decision matrix is given by Table 2.23. It shows the different ratings of linguistic values assigned by a group of two decision makers, in which each decision maker assigns ratings according to his choice. For final decision and ratings, aggregated 4–polar fuzzy linguistic decision matrix is calculated in Table 2.24. (ii) To measure the corruption at country level anti-corruption agencies and media sources are considered as decision makers and the weights assigned by decision makers are given by Table 2.25. (iii) The weighted aggregated 4–polar fuzzy linguistic group decision matrix is calculated in Table 2.26. (iv) The 4−polar fuzzy concordance sets are calculated in Table 2.27. (v) The 4−polar fuzzy discordance sets are calculated in Table 2.28. (vi) A 4–polar fuzzy linguistic concordance matrix is calculated as follows:

2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM

123

Table 2.23 A 4–polar fuzzy linguistic group decision matrix Decision 4−polar fuzzy Physical domain makers linguistic variable (Corruption) 40–60% 50–80% 30–70% 4−polar fuzzy linguistic values Petty corruption Grand corruption Systemic corruption Ratings according to anti-corruption agencies C1 (0.6,0.5,0.3,0.4) C2 (0.5,0.7,0.5,0.6) C3 (0.3,0.4,0.4,0.7) D1 C4 (0.3,0.6,0.5,0.4) C5 (0.4,0.4,0.5,0.7) C6 (0.6,0.4,0.5,0.7) C7 (0.7,0.3,0.2,0.5) Ratings according to media sources C1 (0.5,0.4,0.3,0.5) C2 (0.6,0.4,0.5,0.7) C3 (0.3,0.5,0.7,0.6) D2 C4 (0.5,0.6,0.5,0.3) C5 (0.5,0.6,0.5,0.7) C6 (0.7,0.4,0.4,0.5) C7 (0.6,0.5,0.3,0.5)

(0.5,0.6,0.7,0.8) (0.4,0.6,0.8,0.9) (0.6,0.5,0.7,0.8) (0.7,0.6,0.8,0.7) (0.4,0.5,0.8,0.8) (0.6,0.7,0.5,0.7) (0.4,0.5,0.6,0.8)

(0.2,0.1,0.5,0.6) (0.3,0.4,0.7,0.8) (0.3,0.3,0.5,0.7) (0.2,0.4,0.4,0.6) (0.1,0.4,0.3,0.5) (0.5,0.5,0.4,0.7) (0.4,0.3,0.5,0.8)

(0.6,0.6,0.8,0.6) (0.8,0.7,0.7,0.8) (0.4,0.5,0.6,0.8) (0.5,0.7,0.6,0.8) (0.7,0.4,0.7,0.7) (0.6,0.6,0.6,0.6) (0.5,0.7,0.7,0.6)

(0.3,0.2,0.6,0.7) (0.4,0.3,0.7,0.6) (0.5,0.4,0.6,0.6) (0.3,0.5,0.6,0.5) (0.3,0.4,0.5,0.4) (0.4,0.3,0.6,0.5) (0.4,0.4,0.7,0.7)

Table 2.24 An aggregated 4–polar fuzzy linguistic group decision matrix 4−polar fuzzy Physical Domain linguistic variable (Corruption) 40–60% 50–80% 30–70% 4−polar fuzzy linguistic values Petty corruption Grand corruption Systemic corruption C1 C2 C3 C4 C5 C6 C7

(0.55,0.45,0.3,0.45) (0.55,0.55,0.5,0.65) (0.3,0.45,0.55,0.65) (0.4,0.6,0.5,0.35) (0.45,0.5,0.5,0.7) (0.65,0.4,0.45,0.6) (0.65,0.4,0.25,0.5)

(0.55,0.6,0.75,0.7) (0.6,0.65,0.75,0.85) (0.5,0.5,0.65,0.8) (0.6,0.65,0.7,0.75) (0.55,0.45,0.75,0.75) (0.6,0.65,0.55,0.65) (0.45,0.6,0.65,0.7)

(0.25,0.15,0.55,0.65) (0.35,0.35,0.7,0.7) (0.4,0.35,0.55,0.65) (0.25,0.45,0.5,0.55) (0.2,0.4,0.4,0.45) (0.45,0.4,0.5,0.6) (0.4,0.35,0.6,0.75)

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.25 Weights assigned by decision makers Decision 4−polar fuzzy linguistic values makers Petty corruption Grand corruption D1 D2  D

0.3251 0.2915 0.3083

0.3453 0.3801 0.3627

Systemic corruption 0.3296 0.3284 0.3290

Table 2.26 Weighted aggregated 4–polar fuzzy linguistic group decision matrix 4−polar fuzzy linguistic variable (Corruption)

Physical domain

40–60%

50–80%

30–70%

4−polar fuzzy linguistic values Petty corruption

Grand corruption

Systemic corruption

C1

(0.1696,0.1387,0.0925,0.1387)

(0.1995,0.2176,0.2720,0.2539)

(0.0823,0.0494,0.1810,0.2138)

C2

(0.1696,0.1696,0.1542,0.2004)

(0.2176,0.2358,0.2720,0.3083)

(0.1152,0.1152,0.2303,0.2303)

C3

(0.0925,0.1387,0.1696,0.2004)

(0.1814,0.1814,0.2358,0.2902)

(0.1316,0.1152,0.1810,0.2138)

C4

(0.1233,0.1850,0.1542,0.1079)

(0.2176,0.2358,0.2539,0.2720)

(0.0823,0.1481,0.1645,0.1810)

C5

(0.1387,0.1542,0.1542,0.2158)

(0.1995,0.1632,0.2720,0.2720)

(0.0658,0.1316,0.1316,0.1481)

C6

(0.2004,0.1233,0.1387,0.1850)

(0.2176,0.2358,0.1995,0.2358)

(0.1481,0.1316,0.1645,0.1974)

C7

(0.2004,0.1233,0.0771,0.1542)

(0.1632,0.2176,0.2358,0.2539)

(0.1316,0.1152,0.2139,0.2468)

Table 2.27 The 4–polar fuzzy linguistic concordance sets v 1 2 3 4 Y1v Y2v Y3v Y4v Y5v Y6v Y7v

− {1, 2, 3} {1, 3} {1, 2, 3} {1} {1, 3} {1, 3}

{} − {} {} {} {} {3}

{2} {1, 2, 3} − {2} {1, 2} {1, 2, 3} {3}

{} {1, 2, 3} {1, 3} − {1} {1, 3} {3}

Table 2.28 The 4–polar fuzzy linguistic discordance sets v 1 2 3 4 Z 1v Z 2v Z 3v Z 4v Z 5v Z 6v Z 7v

− {} {2} {} {2, 3} {2} {2}

{1, 2, 3} − {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 2, 3}

{1, 3} {} − {1, 3} {3} {2, 3} {1, 2}

{1, 2, 3} {} {2} − {2, 3} {2} {1, 2}

5

6

7

{2, 3} {1, 2, 3} {3} {2, 3} − {3} {3}

{2} {1, 2, 3} {2, 3} {2} {1, 2} − {3}

{2} {1, 2, 3} {1, 2} {1, 2} {1, 2} {1, 2} −

5

6

7

{1} {} {1, 2} {1} − {1, 2} {1, 2}

{1, 3} {} {1, 2, 3} {1, 3} {3} − {1, 2}

{1, 3} {3} {3} {3} {3} {3} −

2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM

⎛

− 0 ⎜ 1 − ⎜ ⎜ 0.6373 0 ⎜ 1 0 Y =⎜ ⎜ ⎜ 0.3083 0 ⎜ ⎝ 0.6373 0 0.6373 0.3290

0.3627 1 − 0.3627 0.6710 1 0.3290

0 1 0.6373 − 0.3083 0.6373 0.3290

0.6917 1 0.3290 0.6917 − 0.3290 0.3290

125

0.3627 1 0.6917 0.3627 0.6710 − 0.3290

⎞ 0.3627 1 ⎟ ⎟ 0.6710 ⎟ ⎟ 0.6710 ⎟ ⎟. 0.6710 ⎟ ⎟ 0.6710 ⎠ −

(vii) A 4–polar fuzzy linguistic discordance matrix is calculated as follows: ⎛ ⎞ − 1 1 1 0.8900 1 1 ⎜ 0 − 0 0 0 0 0.3813 ⎟ ⎜ ⎟ ⎜ 0.5221 1 − 0.644 0.5009 1 0.2448 ⎟ ⎜ ⎟ 1 1 − 1 1 0.7079 ⎟ Z =⎜ ⎜ 0 ⎟. ⎜ 1 ⎟ 1 1 0.6792 − 0.9193 1 ⎜ ⎟ ⎝ 0.7328 1 0.8089 0.5182 1 − 0.9071 ⎠ 0.5693 1 1 1 0.8811 1 − (viii) A 4–polar fuzzy linguistic concordance level y = 0.5243, and 4–polar fuzzy linguistic discordance level z = 0.7359 are calculated. (ix) A 4–polar fuzzy linguistic concordance dominance matrix is calculated as follows: ⎞ ⎛ − 0 0 0 1 0 0 ⎜1 − 1 1 1 1 1⎟ ⎟ ⎜ ⎜1 0 − 1 0 1 1⎟ ⎟ ⎜ ⎟ R=⎜ ⎜ 1 0 0 − 1 0 1 ⎟. ⎜0 0 1 0 − 1 1⎟ ⎟ ⎜ ⎝1 0 1 1 0 − 1⎠ 1 0 0 0 0 0 − (x) A 4–polar fuzzy linguistic discordance dominance matrix is calculated as follows: ⎞ ⎛ − 0 0 0 0 0 0 ⎜1 − 1 1 1 1 1⎟ ⎟ ⎜ ⎜1 0 − 1 1 0 1⎟ ⎟ ⎜ ⎟ S=⎜ ⎜ 1 0 0 − 0 0 1 ⎟. ⎜0 0 0 1 − 0 0⎟ ⎟ ⎜ ⎝1 0 0 1 0 − 0⎠ 1 0 0 0 0 0 − (xi) An aggregated 4–polar fuzzy linguistic dominance matrix is calculated as follows:

126

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

⎛

− ⎜1 ⎜ ⎜1 ⎜ T =⎜ ⎜1 ⎜0 ⎜ ⎝1 1

0 − 0 0 0 0 0

0 1 − 0 0 0 0

0 1 1 − 0 1 0

0 1 0 0 − 0 0

0 1 0 0 0 − 0

⎞ 0 1⎟ ⎟ 1⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 0⎠ −

(xii) Finally, to rank the countries according to the outranking values of aggregated 4–polar fuzzy linguistic dominance matrix T . A directed graph is drawn for each pair of countries as shown in Fig. 2.21. From the directed graph of countries as shown in Fig. 2.21, the following cases arise. 1. There does not exist any edge from C1 to any other country, which shows C1 is incomparable to others. 2. There exist directed edges from C2 to C1 , C3 , C4 , C5 , C6 and C7 which show that C2 is preferred over all other countries. 3. Similarly, C3 is preferred over C1 , C4 and C7 . 4. Similarly, C4 is preferred over C1 and C7 . 5. There does not exist any edge from C5 to any other country, which shows SC5 is incomparable to others. 6. C6 is preferred over C1 and C4 . 7. Similarly, C7 is preferred over C1 . Hence, the country C2 is most dominated as compared to others. Hence C2 is the most corrupted country. The comparison of countries is shown and the whole procedure is summarized in Table 2.29. The next Algorithm 2.7.1 corresponds to the m−polar fuzzy linguistic ELECTREI method for MCDM and MCGDM:

C1

Fig. 2.21 Graphical representation of outranking relation of countries

C7

C6

C2

C3

C4

C5

Yuv

{} {2} {} {2, 3} {2} {2} {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 3} {} {1, 3} {3} {2, 3} {1, 2} {1, 2, 3}

Comparison of countries

(C1 , C2 ) (C1 , C3 ) (C1 , C4 ) (C1 , C5 ) (C1 , C6 ) (C1 , C7 ) (C2 , C1 ) (C2 , C3 ) (C2 , C4 ) (C2 , C5 ) (C2 , C6 ) (C2 , C7 ) (C3 , C1 ) (C3 , C2 ) (C3 , C4 ) (C3 , C5 ) (C3 , C6 ) (C3 , C7 ) (C4 , C1 )

{1, 2, 3} {1, 3} {1, 2, 3} {1} {1, 3} {1, 3} {} {} {} {} {} {3} {2} {1, 2, 3} {2} {1, 2} {1, 2, 3} {3} {}

Z uv

Table 2.29 Comparison of countries

0 0.3627 0 0.6917 0.3627 0.3627 1 1 1 1 1 1 0.6373 0 0.6373 0.3290 0.6917 0.6710 1

yuv 1 1 1 1 1 1 0 0 0 0 0 0.3813 0.5221 1 0.6444 0.5009 1 0.2448 0

z uv 0 0 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 1

ruv 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1

suv 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1

tuv

(continued)

Incomparable Incomparable Incomparable Incomparable Incomparable Incomparable C2 → C1 C2 → C3 C2 → C4 C2 → C5 C2 → C6 C2 → C7 C3 → C1 Incomparable C3 → C4 Incomparable Incomparable C3 → C7 C4 → C1

Ranking

2.7 m–Polar Fuzzy Linguistic ELECTRE-I Method for MCGDM 127

Yuv

{} {2} {2, 3} {2} {1, 2} {1} {} {1, 2} {1} {1, 2} {1, 2} {1, 3} {} {1, 2, 3} {1, 3} {3} {1, 2} {1, 3} {3} {3} {3} {3} {3}

Comparison of countries

(C4 , C2 ) (C4 , C3 ) (C4 , C5 ) (C4 , C6 ) (C4 , C7 ) (C5 , C1 ) (C5 , C2 ) (C5 , C3 ) (C5 , C4 ) (C5 , C6 ) (C5 , C7 ) (C6 , C1 ) (C6 , C2 ) (C6 , C3 ) (C6 , C4 ) (C6 , C5 ) (C6 , C7 ) (C7 , C1 ) (C7 , C2 ) (C7 , C3 ) (C7 , C4 ) (C7 , C5 ) (C7 , C6 )

Table 2.29 (continued)

{1, 2, 3} {1, 3} {1} {1, 3} {3} {2, 3} {1, 2, 3} {3} {2, 3} {3} {3} {2} {1, 2, 3} {2, 3} {2} {1, 2} {3} {2} {1, 2, 3} {1, 2} {1, 2} {1, 2} {1, 2}

Z uv 0 0.3627 0.6917 0.3627 0.6710 0.3083 0 0.6710 0.3083 0.6710 0.6710 0.6373 0 1 0.6373 0.3290 0.6710 0.6373 0.3290 0.3290 0.3290 0.3290 0.3290

yuv 1 1 1 1 0.7079 1 1 1 0.6792 0.9193 1 0.7328 1 0.8089 0.5182 1 0.9071 0.5693 1 1 1 0.8811 1

z uv 0 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0

ruv 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0

suv 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0

tuv Incomparable Incomparable Incomparable Incomparable C4 → C7 Incomparable Incomparable Incomparable Incomparable Incomparable Incomparable C6 → C1 Incomparable Incomparable C6 → C4 Incomparable Incomparable C7 → C1 Incomparable Incomparable Incomparable Incomparable Incomparable

Ranking

128 2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

2.8 Discussion of the Proposed Approach

129

Algorithm 2.7.1 The algorithm of an m−polar fuzzy linguistic ELECTRE-I method for MCDM and MCGDM 1. Input n, the number of alternatives against the linguistic variable; k, the number of linguistic values; m, the number of membership values; and g, the number of decision makers. 2. Input Dg , m–polar fuzzy linguistic decision matrices according to decision makg ers, and wl , the weights according to decision makers. 3. Compute an aggregated m–polar fuzzy linguistic decision matrix D.  4. Compute aggregated weights W . 5. Compute the weighted aggregated m–polar fuzzy linguistic decision matrix W . 6. Compute m–polar fuzzy linguistic concordance set Yuv . 7. Compute m–polar fuzzy linguistic discordance set Z uv . 8. Compute m–polar fuzzy linguistic concordance indices yuv and concordance matrix Y . 9. Compute m–polar fuzzy linguistic discordance indices z uv and discordance matrix Z . 10. Compute m–polar fuzzy linguistic concordance and discordance levels y¯ and z¯ . 11. Compute m–polar fuzzy linguistic concordance dominance matrix R. 12. Compute m–polar fuzzy linguistic discordance dominance matrix S. 13. Compute aggregated m–polar fuzzy linguistic dominance matrix T . 14. Select the most dominating alternative having maximum value of T . The flowchart of m−polar fuzzy linguistic ELECTRE-I method is described in Fig. 2.22.

2.8 Discussion of the Proposed Approach In this section, the novelty of the presented concept and its decision making methods is presented. 1. Linguistic variables are considered as a valid generalization of numerical variables and take advantage of the concept of words in natural languages as its values. They associate human knowledge into various systems in an organized, efficient and productive manner. In many applications of decision making, where the situations totally depend upon uncertainty and imprecision the fuzzy linguistic variables are used extensively. All the previously defined concepts related to linguistic variables are insufficient to explain the situation, when given data is in form of sentences and words with m different numeric and fuzzy values with its crisp domain. It is actually a formal generalization of fuzzy linguistic variables, because in the proposed concept the linguistic values are further characterized by m different numerical and fuzzy values. 2. The ELECTRE-I method is preferred over all other MCDM methods, because it is a binary outranking method in which the alternatives can be compared without consideration of their preference. It is more reliable as it does not depend on

130

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets m−polar fuzzy linguistic

Rank the alternatives for final decision

ELECTRE-I method

aggregated mF linguistic For MCDM

For MCGDM dominance matrix

Identification of mFlinguistic variable

m-polar fuzzy linguistic discordance dominance matrix

Choose alternative for mFlinguistic variable m-polar fuzzy linguistic concordance dominance matrix Define linguistic values of

Set physical domain

mFlinguistic variable and define cretia’s

D = [0, = ∞) m-polar fuzzy linguistic concordance and discordance levels

Construction of m-polar fuzzy

Construction of aggregated m-polar fuzzy

linguistic decision matrix

linguistic decision matrix

Compute m-polar fuzzy linguistic discordance indices and matrix

Fix weights to linguistic values by decision makers

Compute m-polar fuzzy linguistic Compute weighted m-polar fuzzy

Compute weighted aggregated

linguistic decision matrix

m-polar fuzzy linguistic decision matrix

concordance indices and matrix

Compute m-polar fuzzy linguistic concordance set

Compute m-polar fuzzy linguistic discordance set

Fig. 2.22 m–polar fuzzy linguistic ELECTRE-I approach for MCDM and MCGDM

the expert personal opinions, and alternatives can be eliminated when they are dominated by other alternatives to a specified degree. 3. The comparison of m–polar fuzzy linguistic variable and m–polar fuzzy linguistic ELECTRE-I method with fuzzy linguistic approaches and decision making methods given in the literature is described by an example of salary analysis of companies. In this example, we consider salary as a linguistic variable and define its linguistic values such as low, moderate, good and very good, we call it 4F linguistic variable because we categorize these linguistic values in further four different fuzzy values such as career, labor market, experience and credential. Previous knowledge tells us about the linguistic values of a linguistic variable, but it is unable to deal the criteria on which the linguistic values depend. An m–polar fuzzy linguistic variable covers all the aspects on which salary based upon and an m–polar fuzzy linguistic ELECTRE-I approach is able to deal such a complex situation and provides flexible decision results.

2.8 Discussion of the Proposed Approach

131

Appendix A The computer programming code of Algorithm 2.7.1 is shown in Table 2.30 by using MATLAB R2014a. Table 2.30 MATLAB computer programming code of the proposed approach 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

clc n=input(‘no. of alternatives against linguistic variable’); k=input(‘no.of linguistic values’); m=input(‘no. of membership values’); g=input(‘no. of decision maker’); Rr=(1:n);Cr=1:m*k;Cw=1:k;A_g=zeros(n,m*k);w_g=zeros(1,k); for i=1:g D= input(‘enter the m-polar fuzzy linguistic decision matrix nxkxm’); A_g(Rr,Cr)=A_g(Rr,Cr)+D;w=input(‘enter the weights’)w_g(1,Cw)=w_g(1,Cw)+w; end A_g=A_g/g w=w_g/g W=zeros(n,m∗k);Sm=zeros(n,k);Y_uv=zeros(n,n∗k); Z_uv=zeros(n,n∗k); Y=zeros(n,n); Z=zeros(n ˆ 2,m∗k); for p=1:n l=1; for q=1:m∗k W(p,q)=A_g(p,q).∗w(1,l); if mod(q,m)==0 l=l+1; end end end W for p=1:n l=1; for j=1:m∗k Sm(p,l)=Sm(p,l)+W(p,j); if mod(j,m)==0 l=l+1; end end (continued)

132

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.30 (continued) 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73.

end Q=Sm’ Q=Q(:)’; for p=1:n for j=1:k∗n l=mod(j,k); if l==0 l=k; end if Sm(p,l)≥ Q(1,j) Y_uv(p,j)=1; end if Sm(p,l)≤ Q(1,j) Z_uv(p,j)=1; end end end fprintf(‘\n concordance Set Y_uv =\n’) for u=1:n v=0; for j=1:k∗n if mod(j,k)==1 v=v+1; end l=mod(j,k); if l==0 l=k; end if u==v if l==1 fprintf(‘ end elseif u =v if l==1 fprintf(‘ { ’) c=0; end if Y_uv(u,j)==1; c=c+1; fprintf(‘%d,’,l) end

’)

(continued)

2.8 Discussion of the Proposed Approach

133

Table 2.30 (continued) 74. if l==k & c==0 75. fprintf(‘ ,’,l) 76. end 77. if l==k 78. fprintf(‘\b} ’) 79. end 80. end 81. end 82. fprintf(‘\n’) 83. end 84. fprintf(‘\n discordance Set Z_uv =\n’) 85. for u=1:n 86. v=0; 87. for j=1:k∗n 88. if mod(j,k)==1 89. v=v+1; 90. end 91. l=mod(j,k); 92. if l==0 93. l=k; 94. end 95. if u==v 96. if l==1 97. fprintf(‘ ’) 98. end 99. elseif u∼=v 100. if l==1 101. fprintf(‘ { ’) 102. c=0; 103. end 104. if Z_uv(u,j)==1; 105. c=c+1; 106. fprintf(‘ %d,’ ,l ) 107. end 108. if l==k & c==0 109. fprintf(‘ ,’ ,l ) 110. end 111. if l==k 112. fprintf(‘\b} ’) (continued)

134

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.30 (continued) 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152.

end end end fprintf(‘\n ’) end for u=1:n v=0; for j=1:k∗n if mod(j,k)==1 v=v+1; end l=mod(j,k); if l==0 l=k; end if u =v if Y_uv(u,j)==1 Y(u,v)=Y(u,v)+w(1,l); end end end end fprintf(‘\nY=\n’) for u=1:n for v=1:n if u==v fprintf(‘ ’) else fprintf(‘%.4f ’,Y(u,v)) end end fprintf(‘\n ’) end v=0;r=0; l=1:m∗k; B=zeros(n,n∗k);r=0;D=zeros(n,n);Z=zeros(n,n); for u=1:n for q=1:n v=v+1; z(v,l)=(W(u,l)-W(q,l)). ˆ 2; end end (continued)

2.8 Discussion of the Proposed Approach

135

Table 2.30 (continued) 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192.

A=zeros(n ˆ 2,k);r=0; s=0; C=zeros(n ˆ 2,1);D=zeros(n,n);Z1=zeros(n,n); for i=1:n ˆ 2 q=0; for j=1:m∗k if mod(j,m)==1 q=q+1; end A(i,q)=A(i,q)+z(i,j); end A(i,:)=sqrt(A(i,:)/m); C(i,1)=max(A(i,:)); if mod(i,n)==1 r=r+1; end for j=1:k s=s+1; B(r,s)=A(i,j); end t=mod(i,n); if t==0 t=n; end Z1(r,t)=C(i,1); if mod(i,n)==0 s=0; end end for i=1:n q=0; for j=1:k∗n if mod(j,k)==1 q=q+1; end l=mod(j,k); if l==0 l=k; end if Z_uv(i,j)==1 D(i,q)=max(D(i,q),B(i,j)); (continued)

136

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.30 (continued) 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232.

end end end for u=1:n for v=1:n if u =v Z(u,v)=D(u,v)/Z1(u,v); end end end fprintf(‘\nZ=\n’) for u=1:n for v=1:n if u==v fprintf(‘ ’) else fprintf(‘%.4f ’,Z(u,v)) end end fprintf(‘ \n ’) end a=sum(Y); b=sum(a); a1=sum(Z); b1=sum(a1); R=zeros(n,n);S=zeros(n,n); y_bar=b/(n∗(n-1)) z_bar=b1/(n∗(n-1)) for u=1:n for v=1:n if u =v if Y(u,v)≥ y_bar R(u,v)=1; end if Z(u,v)< z_bar S(u,v)=1; end end end end fprintf(‘\nR=\n’) for u=1:n for v=1:n if u==v (continued)

2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM

137

Table 2.30 (continued) 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258. 259. 260. 261. 262. 263.

fprintf(‘-

’)

else fprintf(‘%d

’,R(u,v))

end end fprintf(‘ \n

’)

end fprintf(‘\nS=\n’) for u=1:n for v=1:n if u==v fprintf(‘else fprintf(‘%d end end fprintf(‘\n ’) end T=R.∗S; fprintf(‘\nT=\n’) for u=1:n for v=1:n if u==v fprintf(‘else fprintf(‘%d end end fprintf(‘ \n ’) end G=digraphs(T) plot(G)

’) ’,S(u,v))

’) ’,T(u,v))

2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM The MCGDM problem is analyzed and described that arises from the theory that have already been expressed. MCGDM is a procedure by which a number of decision makers have to choose the best alternative from a set of possible alternatives, and these are described in terms of some criteria related to the situation. MCDM is a

138

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

specific case of MCGDM in which a single decision maker is fully responsible for the choice of the best alternative. The corresponding algorithm is proceeded for this case. A computer programming code is also set forth for the required problems that is described by m–polar fuzzy linguistic variable. To prove their signification and efficiency, the TOPSIS method is applied to actual situations which are fully developed. TOPSIS (“Technique for Order Preference by Similarity to an Ideal Solution”) is a technique that recommends to select the alternative that has the shortest distance from the positive ideal solution (PIS) and the largest distance from the negative ideal solution (NIS). This technique is actually used for ranking objects and achieving the best performance in MCGDM. This method is expanded so that it can also take the information relating to m–polar fuzzy linguistic variables. Put shortly, an m–polar fuzzy linguistic TOPSIS approach deals with MCGDM problems based on m–polar fuzzy linguistic variable. Our proposed TOPSIS approach based on m–polar fuzzy linguistic variable deals with MCGDM problems, in which L v is chosen as an m–polar fuzzy linguistic variable and A = {a1 , a2 , . . . , a p } is the set of m–polar fuzzy linguistic variable of different alternatives. According to an m–polar fuzzy linguistic variable, {Vk |k = 1, 2, . . . , q} is taken as the set of linguistic values of L v , and these linguistic values are classified by m different characteristics. In such a case, a group of r decision makers (Dl , l = 1, 2, . . . , r ) is responsible for evaluating m–polar fuzzy linguistic variable of p different alternatives under q linguistic values and the suitable ratings of alternatives are according to all decision makers, assessed in term of m different characteristics under physical domain Pd . The degrees of each alternative (a j ∈ A, j = 1, 2, . . . , p) over all linguistic values Vk s are given by m–polar fuzzy set ℘ j = {(a j , d lijk )|i = 1, 2, . . . m}, where d lijk = pi ◦ d ljk (a j , Vk ) ∈ [0, 1] and d lijk classify the different characteristics of each criterion by decision makers. Pd is the actual physical domain in which the m–polar fuzzy linguistic variable takes its quantitative (crisp) values, i.e., Pd = [0, +∞). In this case, the most suitable m values are taken from physical domain of each linguistic value. Tabular representation of m–polar fuzzy linguistic decision matrix under group of decision makers is given by Table 2.31, which describes the ratings given by each decision maker. Table 2.31 shows the different ratings of linguistic values assigned by a group of r decision makers, in which each decision maker assigns ratings according to his choice. The final m–polar fuzzy linguistic decision matrix under group of decision makers is the aggregated m–polar fuzzy linguistic decision matrix, that is the average ratings of all decision makers. These Aggregated ratings are calculated as follows: 

d 1jk =

1  l1 2 1  l2 1  lm  d jk , d jk = d jk , . . . , d mjk = d . r l=1 r l=1 r l=1 jk r

r

j = 1, 2, · · · , p and k = 1, 2, · · · , q.

r

2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM

139

Table 2.31 An m–polar fuzzy linguistic decision matrix m−polar fuzzy linguistic variable (L v )

Physical domain

Decision makers

Pd1

···

Pd2

Pdq

m−polar fuzzy linguistic values V1 Dl

···

V2

Vq

a1

l1 , d l2 , . . . , d lm ) (d l1 , d l2 , . . . , d lm ) · · · (d11 11 11 12 12 12

l1 , d l2 , . . . , d lm ) (d1q 1q 1q

a2

l1 , d l2 , · · · , d lm ) (d l1 , d l2 , . . . , d lm ) · · · (d21 21 21 22 22 22

l1 , d l2 , . . . , d lm ) (d2q 2q 2q

. . .

. . .

. . .

ap

l2 lm l1 l2 lm (d l1 p1 , d p1 , . . . , d p1 ) (d p2 , d p2 , . . . , d p2 ) · · ·

. . .

. . .

l2 lm (d l1 pq , d pq , . . . , d pq )

Table 2.32 An aggregated m–polar fuzzy linguistic decision matrix m−polar fuzzy linguistic variable (L v )

Physical domain

Pd1

···

Pd2

Pdq

m−polar fuzzy linguistic values and weights V1

V2

···

Vq

w1

w2

···

wq







1

2

m











1

2

m











1

2







a1

1 , d2 , . . . , dm ) (d11 11 11

1 , d2 , . . . , dm ) (d12 12 12

···

1 , d2 , . . . , dm ) (d1q 1q 1q

a2 . ..

(d21 , d21 , . . . , d21 ) . ..

(d22 , d22 , . . . , d22 ) . ..

··· . ..

m ) (d2q , d2q , . . . , d2q . ..

ap

(d 1p1 , d 2p1 , . . . , d m p1 )

(d 1p2 , d 2p2 , . . . , d m p2 )

···

(d 1pq , d 2pq , . . . , d m pq )







For final decision and ratings, aggregated m–polar fuzzy linguistic decision matrix is calculated in Table 2.32 by using above average ratings. Decision makers are free to assign weights to each linguistic value of alternatives according to their expertise and the importance attached to each linguistic value. As the case of m–polar fuzzy linguistic variables is discussed that the decision makers have to assign the weights in terms of linguistic term set L = {l1 = extr emely low, l2 = medium, · · · , lq = ver y high}. The weights assigned by the decision makers are supposed W l = (w1l , w2l , . . . , wql ) ∈ (0, 1], l = 1, 2, . . . , r. These weights satisfy the usual normalization condition. i.e.,

140

2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets q 

wkl = 1, l = 1, 2, . . . , r.

k=1 







Aggregated weights according to the decision makers are W = (w1 , w2 , . . . , wq ), where, r 1 l  wk = w , k = 1, 2, . . . , q. r l=1 k The weighted aggregated m–polar fuzzy linguistic decision matrix under group deci   sion making E = [(e1jk , e2jk , . . . , emjk )] p×q is calculated as 

















e1jk = wk d 1jk , e2jk = wk d 2jk , . . . , emjk = wk d mjk . The m–polar fuzzy linguistic Positive ideal solution (m PI S ) and m–polar fuzzy linguistic negative ideal solution (m N I S ) of alternatives under m–polar fuzzy environment can be calculated as given in Eqs. 2.1 and 2.2. 





























m PI S = {((d11 )+ , (d12 )+ , . . . , (d1m )+ ), ((d21 )+ , (d22 )+ , . . . , (d2m )+ ), . . . , ((dq1 )+ , (dq2 )+ , . . . , (dqm )+ )},

(2.1) 





m N I S = {((d11 )− , (d12 )− , . . . , (d1m )− ), ((d21 )− , (d22 )− , . . . , (d2m )− ), . . . , ((dq1 )− , (dq2 )− , . . . , (dqm )− )}.

(2.2)

Here it is supposed 











(dk1 )+ = max{e1jk }, (dk2 )+ = max{e2jk }, . . . , (dkm )+ = max{emjk }, 











(dk1 )− = min{e1jk }, (dk2 )− = min{e2jk }, . . . , (dkm )− = min{emjk }. An m–polar fuzzy linguistic Euclidean distance is defined for each alternative a j from m–polar fuzzy linguistic positive ideal solution and m–polar fuzzy linguistic negative ideal solution by Eqs. 2.3 and 2.4.    q  1        De (a j , m PI S ) =  (e1jk − (d 1j )+ )2 + (e2jk − (d 2j )+ )2 + · · · + (emjk − (d mj )+ )2 , m 

k=1

(2.3)    q  1        (e1jk − (d 1j )− )2 + (e2jk − (d 2j )− )2 + · · · + (emjk − (d mj )− )2 . De (a j , m N I S ) =  m 

k=1

(2.4)

2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM

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The relative m–polar fuzzy linguistic closeness coefficient of each alternative a j can be computed by using following formula as described in Eq. 2.5. 



Ej =

De (a j , m N I S ) , De (a j , m PI S ) + De (a j , m N I S ) 

j = 1, 2, . . . , p.

(2.5)

The alternative with highest m–polar fuzzy linguistic closeness coefficient is best one and the ranking order of each alternative can be determined. In next subsections, the applications of presented model are discussed.

2.9.1 Models Ranking According to their Appearance Evaluating different personalities based on their physical appearance is a challenging and controversial task, especially in a multi-expert setting. We propose the method of m–polar fuzzy linguistic TOPSIS for MCGDM , in which L v is the linguistic variable i.e., appearance and M = {Am 1 , Am 2 , Am 3 , Am 4 , Am 5 } is the set of appearance of five different models, whereas V = {Less attractive, Fairly cute, Quite pretty, Very beautiful} is the set of linguistic values of appearance. decision makers have to evaluate the models on basis of the linguistic values of their appearance and they have to design a physical domain in which appearance takes its quantitative values, i.e., Pd = [10, 100]. The physical domain for linguistic values of appearance is given as follows: • • • •

For less attractive appearance, physical domain is 10–40/100, For Fairly cute appearance, physical domain is 40–70/100, For Quite pretty appearance, physical domain is 70–90/100, For Very beautiful appearance, physical domain is 90–100/100.

Physical domain of each linguistic value shows the range of marks given by group of decision makers out of 100. The degree of appearance of each model, over all the linguistic values are given by 3–polar fuzzy set ℘ j = {(Am j , d lijk )|i = 1, 2, 3}, where l • d l1 jk = p1 ◦ d jk (a j , Vk ) serves for “face features”, the distinguishing elements of a face having attraction, such as beautiful eyes, sharp nose and shaped lips. l • d l2 jk = p2 ◦ d jk (a j , Vk ) serves for “postures”, a way according to which someone arranges his limbs and positions his body. l • d l3 jk = p3 ◦ d jk (a j , Vk ) serves for “body figure”, it is the cumulative product of someone skeletal structure, the distribution and quantity of muscle and fat on his body.

where j = 1, 2, . . . , 5, k = 1, 2, 3, 4 and l = 1, 2, 3, 4. Tabular representation of 3–polar fuzzy linguistic group decision matrix is given by Table 2.33. It shows the different ratings of linguistic values assigned by a group of four decision makers, in which each decision maker assigns ratings according to his choice.

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.33 A 3–polar fuzzy linguistic group decision matrix Decision makers

3−polar fuzzy linguistic variable (Appearance)

Physical domain

10–40/100

40–70/100

70–90/100

90–100/100

3−polar fuzzy linguistic values D1

D2

D3

D4

Less attractive

Fairly cute

Quite pretty

Very beautiful

Am 1

(0.25,0.31,0.29)

(0.47,0.52,0.62)

(0.71,0.69,0.78)

(0.92,0.93,0.98)

Am 2

(0.30,0.23,0.33)

(0.39,0.51,0.70)

(0.69,0.73,0.75)

(0.86,0.90,0.89)

Am 3

(0.28,0.20,0.17)

(0.46,0.43,0.69)

(0.59,0.65,0.72)

(0.87,0.79,0.80)

Am 4

(0.18,0.24,0.29)

(0.27,0.60,0.65)

(0.60,0.73,0.79)

(0.93,0.85,0.95)

Am 5

(0.23,0.30,0.28)

(0.43,0.47,0.71)

(0.62,0.56,0.80)

(0.90,0.79,0.90)

Am 1

(0.20,0.34,0.19)

(0.46,0.57,0.70)

(0.75,0.65,0.72)

(0.89,0.95,0.97)

Am 2

(0.31,0.23,0.12)

(0.40,0.48,0.68)

(0.68,0.75,0.76)

(0.88,0.89,0.92)

Am 3

(0.25,0.29,0.15)

(0.51,0.47,0.58)

(0.63,0.59,0.70)

(0.82,0.85,0.92)

Am 4

(0.30,0.16,0.21)

(0.35,0.59,0.73)

(0.55,0.68,0.72)

(0.90,0.89,0.90)

Am 5

(0.15,0.21,0.32)

(0.29,0.53,0.72)

(0.44,0.62,0.81)

(0.89,0.90,0.92)

Am 1

(0.19,0.27,0.23)

(0.37,0.49,0.58)

(0.69,0.75,0.73)

(0.91,0.90,0.89)

Am 2

(0.24,0.24,0.20)

(0.43,0.48,0.71)

(0.68,0.72,0.76)

(0.89,0.89,0.93)

Am 3

(0.31,0.28,0.16)

(0.49,0.48,0.70)

(0.69,0.66,0.70)

(0.80,0.82,0.95)

Am 4

(0.18,0.20,0.21)

(0.36,0.58,0.75)

(0.64,0.70,0.79)

(0.90,0.87,0.90)

Am 5

(0.21,0.22,0.29)

(0.30,0.50,0.72)

(0.56,0.62,0.80)

(0.92,0.89,0.93)

Am 1

(0.22,0.31,0.23)

(0.43,0.57,0.59)

(0.73,0.68,0.70)

(0.87,0.90,0.95)

Am 2

(0.31,0.24,0.25)

(0.44,0.49,0.70)

(0.68,0.75,0.74)

(0.88,0.90,0.87)

Am 3

(0.30,0.28,0.17)

(0.49,0.47,0.59)

(0.62,0.60,0.70)

(0.85,0.80,0.90)

Am 4

(0.20,0.25,0.23)

(0.40,0.55,0.71)

(0.61,0.63,0.73)

(0.90,0.87,0.92)

Am 5

(0.21,0.30,0.28)

(0.40,0.50,0.69)

(0.55,0.58,0.79)

(0.90,0.87,0.90)

For the final decision and ratings, an aggregated 3–polar fuzzy linguistic decision matrix is calculated in Table 2.34.

Table 2.34 An aggregate 3–polar fuzzy linguistic decision matrix 3−polar fuzzy linguistic variable (Appearance)

Physical domain

10–40/100

40–70/100

70–90/100

90–100/100

3−polar fuzzy linguistic values Less attractive

Fairly cute

Quite pretty

Very beautiful

Am 1

(0.2150,0.3075,0.2350)

(0.4325,0.5375,0.6225)

(0.7200,0.6925,0.7325)

(0.8975,0.9200,0.9475)

Am 2

(0.2900,0.2350,0.2250)

(0.4150,0.4900,0.6975)

(0.6825,0.7375,0.7525)

(0.8775,0.8950,0.9025)

Am 3

(0.2850,0.2625,0.1625)

(0.4875,0.4625,0.6400)

(0.6325,0.6250,0.7050)

(0.8350,0.8150,0.8925)

Am 4

(0.2150,0.2125,0.2350)

(0.3450,0.5800,0.7100)

(0.6000,0.6850,0.7575)

(0.9075,0.8700,0.9175)

Am 5

(0.2000,0.2575,0.2925)

(0.3550,0.5000,0.7100)

(0.5425,0.5950,0.8000)

(0.9025,0.8900,0.9125)

2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM

143

The weights of linguistic values completely depend upon the decision makers and weights of each decision maker has its own importance. Weights assigned by decision makers are given in terms of linguistic term set as L = {l1 = low, l2 = medium, l3 = high, l4 = extr emely high}, whereas the numerical values assigned to this linguistic term set by each decision maker is given in Table 2.35 and these  values satisfy the normalized condition. In same Table, aggregated weights D are also calculated. The weighted aggregated 3–polar fuzzy linguistic decision matrix is calculated in Table 2.36. Use Eqs. 2.1 and 2.2 to determine the 3–polar fuzzy linguistic positive ideal solution and 3–polar fuzzy linguistic negative ideal solution respectively, for linguistic variable appearance. 3PI S ={(0.0558, 0.0592, 0.0563), (0.1121, 0.1334, 0.1633), (0.1869, 0.1914, 0.2077), (0.2886, 0.2925, 0.3013)}, 3N I S ={(0.0385, 0.0409, 0.0313), (0.0793, 0.1064, 0.1432), (0.1408, 0.1544, 0.1830), (0.2655, 0.2591, 0.2838)}.

Table 2.35 Tabular representation of weights assigned by decision makers Decision 3−polar fuzzy linguistic values makers Less attractive Fairly cute Quite pretty D1 D2 D3 D4  D

0.1957 0.2012 0.1830 0.1900 0.1925

0.2321 0.2259 0.2339 0.2280 0.2300

0.2512 0.2631 0.2703 0.2537 0.2596

Very beautiful 0.3210 0.3098 0.3128 0.3283 0.3180

Table 2.36 Weighted aggregate 3–polar fuzzy linguistic decision matrix 3−polar fuzzy linguistic variable (Appearance)

Physical Domain

10–40/100

40–70/100

70–90/100

90–100/100 Very beautiful

3−polar fuzzy linguistic values and weights Less attractive

Fairly cute

Quite pretty

0.1925

0.2300

0.2596

0.3180

Am 1

(0.0414,0.0592,0.0452)

(0.0995,0.1236,0.1432)

(0.1869,0.1798,0.1901)

(0.2854,0.2925,0.3013)

Am 2

(0.0558,0.0452,0.0433)

(0.0954,0.1127,0.1604)

(0.1772,0.1914,0.1953)

(0.2790,0.2846,0.2870)

Am 3

(0.0549,0.0505,0.0313)

(0.1121,0.1064,0.1472)

(0.1642,0.1622,0.1830)

(0.2655,0.2591,0.2838)

Am 4

(0.0414,0.0409,0.0452)

(0.0793,0.1334,0.1633)

(0.1557,0.1778,0.1966)

(0.2886,0.2766,0.2917)

Am 5

(0.0385,0.0496,0.0563)

(0.0816,0.1150,0.1633)

(0.1408,0.1544,0.2077)

(0.2870,0.2743,0.2902)

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Use Eqs. 2.3 and 2.4 to calculate, the 3–polar fuzzy linguistic Euclidean distance of appearance of each model from its 3–polar fuzzy linguistic positive ideal solution and 3–polar fuzzy linguistic negative ideal solution, that are calculated as follows: De (Am 1 , 3PI S ) = 0.0219,

De (Am 1 , 3N I S ) = 0.0442,

De (Am 2 , 3PI S ) = 0.0237,

De (Am 2 , 3N I S ) = 0.0398,

De (Am 3 , 3PI S ) = 0.0433,

De (Am 3 , 3N I S ) = 0.0262,

De (Am 4 , 3PI S ) = 0.0335,

De (Am 4 , 3N I S ) = 0.0326,

De (Am 5 , 3PI S ) = 0.0432,

De (Am 5 , 3N I S ) = 0.0290.

Use Eq. 2.5 to calculate the relative 3–polar fuzzy linguistic closeness coefficients E j of appearance of each model. E 1 = 0.6684,

E 2 = 0.6266,

E 3 = 0.3769,

E 4 = 0.4936,

E 5 = 0.4014. The comparison of appearance of models is shown and the whole procedure is summarized in Table 2.37. For the comparison, arrange the 3–polar fuzzy linguistic closeness coefficients of appearance of models {Am j | j = 1, 2, . . . , 5)} as shown in Table 2.37. Hence, model m 1 has the highest ranking according to her appearance and ranking of appearance of models is as follows: Am 1 > Am 2 > Am 4 > Am 5 > Am 3 .

Table 2.37 Tabular representation comparison of appearance of models Appearance De (Am j , 3PI S ) De (Am j , 3N I S ) Ej of models Am 1 Am 2 Am 3 Am 4 Am 5

0.0219 0.0237 0.0433 0.0335 0.0432

0.0442 0.0398 0.0262 0.0326 0.0290

0.6684 0.6266 0.3769 0.4936 0.4014

Ranking 1 2 5 3 4

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2.9.2 Ranking of High Speed Racing Cars In this subsection, the specific case of MCGDM is discussed, in which only one decision maker is responsible to choose the ratings of alternatives according to linguistic values. Choosing and comparing the high speed racing cars for different racing competitions is difficult task for experts. So, in this case “speed” is chosen as a linguistic variable and S = {Sc1 , Sc2 , Sc3 , Sc4 , Sc5 , Sc6 } is taken as the set of six different high speed racing cars, whereas V = {Slightly fast,Very fast,Completely fast} is the set of linguistic values of speed. Pd is the actual physical domain in which speed takes its quantitative values, i.e., Pd = [40, 100]. The physical domain for linguistic values of speed is given as follows: • For slightly fast speed, physical domain is 40–60 km/h, • For very fast speed, physical domain is 60–80km/h, • For completely fast speed, physical domain is 80–100 km/h. The degree of speed of each car, over all the linguistic values is given by 4–polar fuzzy set ℘ j = {(Sc j , d lijk )|i = 1, 2, . . . 4}, where l • d l1 jk = p1 ◦ d jk (a j , Vk ) serves for “power of engine”, it is the producer of the power to move the vehicle, basically it is the rate at which work is done and car moves. l • d l2 jk = p2 ◦ d jk (a j , Vk ) serves for “quality of tire”, it totally depends upon the speed ratings, tread wear warranty and typical wheel size. l • d l3 jk = p3 ◦ d jk (a j , Vk ) serves for “condition of car”, it describes the model specifications, history, outside look of a car. l • d l4 jk = p4 ◦ d jk (a j , Vk ) serves for “shape of car”, it describes the cars having straight line speed including the front splitter, rear wing, diffuser and underbody of the car.

where j = 1, 2, . . . , 6, k = 1, 2, 3 and l = 1. The tabular representation of the 4– polar fuzzy linguistic decision matrix is given by Table 2.38. The weights of linguistic values completely depend upon the decision maker. Weights assigned by a decision maker are given in terms of the linguistic term set as L = {l1 = low, l2 = medium, l3 = high}, where as the numerical values assigned to this linguistic term set is given in Eq. 2.6 , and these values satisfy the normalized condition. w = (0.23, 0.34, 0.43).

(2.6)

The weighted 4–polar fuzzy linguistic decision matrix is calculated in Table 2.39. Use Eqs. (2.1) and (2.2) to determine the 4–polar fuzzy linguistic positive ideal solution and 4–polar fuzzy linguistic negative ideal solution respectively, for the 4–polar fuzzy linguistic variable speed.

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Table 2.38 A 4–polar fuzzy linguistic decision matrix 4−polar fuzzy Physical domain linguistic variable (Speed) 40–60 km/h 60–80 km/h 4−polar fuzzy linguistic values Slightly fast Very fast Sc1 Sc2 Sc3 Sc4 Sc5 Sc6

(0.5,0.4,0.3,0.5) (0.4,0.5,0.4,0.5) (0.6,0.4,0.6,0.5) (0.4,0.3,0.7,0.6) (0.5,0.3,0.4,0.5) (0.5,0.5,0.2,0.6)

(0.6,0.7,0.6,0.6) (0.6,0.8,0.7,0.6) (0.5,0.6,0.7,0.5) (0.5,0.6,0.6,0.8) (0.6,0.6,0.7,0.6) (0.6,0.8,0.7,0.7)

80–100 km/h Completely fast (0.7,0.8,0.9,1.0) (0.8,0.8,0.7,1.0) (0.6,0.7,0.7,0.9) (0.7,0.6,0.9,0.8) (0.6,0.7,0.8,0.9) (0.8,0.8,0.9,0.8)

Table 2.39 Weighted 4–polar fuzzy linguistic decision matrix 4−polar fuzzy linguistic variable (Speed)

Physical domain

40–60 km/h

60–80 km/h

80–100 km/h

4−polar fuzzy linguistic values and weights Slightly fast

Very fast

0.23

0.34

Completely fast 0.43

Sc1

(0.1150,0.0920,0.0690,0.1150)

(0.2040,0.2380,0.2040,0.2040)

(0.3010,0.3440,0.3870,0.4300)

Sc2

(0.0920,0.1150,0.0920,0.1150)

(0.2040,0.2720,0.2380,0.2040)

(0.3440,0.3440,0.3010,0.4300)

Sc3

(0.1380,0.0920,0.1380,0.1150)

(0.1700,0.2040,0.2380,0.1700)

(0.2580,0.3010,0.3010,0.3870)

Sc4

(0.0920,0.0690,0.1610,0.1380)

(0.1700,0.2040,0.2040,0.2720)

(0.3010,0.2580,0.3870,0.3440)

Sc5

(0.1150,0.0690,0.0920,0.1150)

(0.2040,0.2040,0.2380,0.2040)

(0.2580,0.3010,0.3440,0.3870)

Sc6

(0.1150,0.1150,0.0460,0.1380)

(0.2040,0.2720,0.2380,0.2380)

(0.3440,0.3440,0.3870,0.3440)

4PI S = {(0.1380, 0.1150, 0.1610, 0.1380), (0.2040, 0.2720, 0.2380, 0.2720), (0.3440, 0.3440, 0.3870, 0.4300)}, 4N I S = {(0.0920, 0.0690, 0.0460, 0.1150), (0.1700, 0.2040, 0.2040, 0.1700), (0.2580, 0.2580, 0.3010, 0.3440)}. Use Eqs. 2.3 and 2.4 to calculate, the 4–polar fuzzy linguistic Euclidean distance of speed of each car from its 4–polar fuzzy linguistic positive ideal solution and 4–polar fuzzy linguistic negative ideal solution, that are calculated as follows: De (Sc1 , 4PI S ) = 0.0686,

De (Sc1 , 4N I S ) = 0.0853,

De (Sc2 , 4PI S ) = 0.0697,

De (Sc2 , 4N I S ) = 0.0929,

De (Sc3 , 4PI S ) = 0.0952,

De (Sc3 , 4N I S ) = 0.0632,

2.9 m–Polar Fuzzy Linguistic TOPSIS Method for MCGDM

147

De (Sc4 , 4PI S ) = 0.0834,

De (Sc4 , 4N I S ) = 0.0914,

De (Sc5 , 4PI S ) = 0.0868,

De (Sc5 , 4N I S ) = 0.0540,

De (Sc6 , 4PI S ) = 0.0747,

De (Sc6 , 4N I S ) = 0.0961.

Use Eq. 2.5 to calculate the relative 4–polar fuzzy linguistic closeness coefficients E j of speed of each car. E 1 = 0.5541,

E 2 = 0.5713,

E 3 = 0.3989,

E 4 = 0.5229,

E 5 = 0.3835,

E 6 = 0.5627.

The comparison of cars according to their speed is shown and the whole procedure is summarized in Table 2.40. For the comparison, arrange the speed of cars {Sc j | j = 1, 2, . . . , 6)} according to their values obtained from relative closeness coefficients as shown in Table 2.40. Hence, car c2 has the highest ranking according to its speed and ranking of cars according to their speed is as follows: Sc2 > Sc6 > Sc1 > Sc4 > Sc3 > Sc5 . Note the followings points: 1. The only condition imposed on the weights assigned by the decision makers is that they should be normalized. Experts are free to take the weights according to their choice. Four values are taken after decimal point for our convenience only. 2. Linguistic variables (appearance and speed) considered in this chapter can also depend on other characteristics, which may be incorporated to the set of criteria or conditions adopted by the users. The Algorithm 2.9.1 of an m−polar fuzzy linguistic TOPSIS Method for MCGDM is given as follows:

Table 2.40 Comparison of cars according to their speed Cars according De (Sc j , 4PI S ) De (Sc j , 4N I S ) to their speed Sc1 Sc2 Sc3 Sc4 Sc5 Sc6

0.0686 0.0697 0.0952 0.0834 0.0868 0.0747

0.0853 0.0929 0.0632 0.0914 0.0540 0.0961

Ej

Ranking

0.5541 0.5713 0.3989 0.5229 0.3835 0.5627

3 1 5 4 6 2

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

Algorithm 2.9.1 The algorithm of an m−polar fuzzy linguistic TOPSIS Method for MCGDM 1. Input p, the number of alternatives against linguistic variable; q, the number of linguistic values; m, the number of membership values; and r , the number of decision makers. Dl , m–polar fuzzy linguistic decision matrices according to decision makers. W l , weights according to decision makers.  2. Compute an aggregated m–polar fuzzy linguistic decision matrix Dl .  3. Compute aggregated weights W . 4. Compute the weighted aggregated m–polar fuzzy linguistic decision matrix E. 5. Compute m–polar fuzzy linguistic positive ideal solution m PI S . 6. Compute m–polar fuzzy linguistic negative ideal solution m N I S . 7. Compute m–polar fuzzy linguistic distance of alternatives from m PI S and m N I S . 8. Compute the relative m–polar fuzzy linguistic closeness coefficients. 9. Rank the alternatives for final decision and select the best one. The flow chart of the m−polar fuzzy linguistic TOPSIS under group decision making is described in Fig. 2.23.

2.10 Comparison Analysis of Proposed Approach In this section, the comparative analysis of decision making approach is discussed by m−polar fuzzy linguistic ELECTRE-I method. 1. All the TOPSIS methods for decision making are not suitable for such a situations, where the alternatives are assessed by the decision makers depending on the linguistic values of variable, which are further classified by m different characteristics. To handle such complex situations, we extend TOPSIS approach to m–polar fuzzy linguistic TOPSIS approach, which provides more flexible and precise results to choose the best alternative under complex fuzzy data. The proposed m–polar fuzzy linguistic TOPSIS approach is constructed according to observation and recognition of expertise idea about the linguistic variable and values of alternatives in form of words and sentences having multi-polar information. Its calculations are very simple and easy, it is also independent of the number of criteria associated in the problem. It provides us with the best alternative as well as the final ranking of all alternatives. 2. An m–polar fuzzy linguistic ELECTRE-I method is also considered as a flexible approach as compared to various other extensions of ELECTRE-I, because in this method a variable and its linguistic values are considered as a fixed criteria for the ranking and evaluation of alternatives. This method only provides us with the best alternative having maximum ranking value but it does not provide us any authentic knowledge about the ranking of alternative. This approach is more difficult to understand and has longer and more complex calculations as compared

2.10 Comparison Analysis of Proposed Approach

149

Fig. 2.23 Flow chart of m–polar fuzzy linguistic TOPSIS for MCGDM

m-polar fuzzy Linguistic TOPSIS approach for MCGDM

Identification of m-polar fuzzy linguistic variable

Choose alternatives for m-polar fuzzy linguistic variable

Define linguistic values

Set physical domain

of m-polar fuzzy linguistic variable

D = [0, ∞)

Assign ratings by r different decision makers

Compute aggregated m-polar fuzzy linguistic decision matrix

Fix weights to criteria’s by r different decision makers

Compute aggregated weighted m-polar fuzzy linguistic decision matrix

Compute mNIS

Compute mPIS

Calculate distance of alternatives from mPIS and mNIS

Calculate relative m-polar fuzzy linguistic closeness coefficient

Rank the alternatives for final decision

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2 TOPSIS and ELECTRE-I Methods Under Multi-polar Fuzzy Linguistic Sets

to m–polar fuzzy linguistic TOPSIS approach. This approach does not point out a unique alternative appearing among the others, but rather it points out the small subset of preferable alternatives, due to it decision makers face difficulty to choose the best one and rank the alternatives.

Appendix B The computer programming code of the presented approach (Algorithm 2.9.1) is shown in Table 2.41, which has been implemented with MATLAB R2014a. Table 2.41 Computer programming code of proposed approach for MCGDM MATLAB Computer Programming Code 1. clc 2. p=input(‘no. of alternatives against linguistic variable’); 3. q=input(‘no.of linguistic values’); 4. m=input(‘no. of membership values’); 5. r=input(‘no. of decision maker’); 6. Rr=(1:p);Cr=1:m*q;Cw=1:q;A_g=zeros(p,m*q);w_g=zeros(1,q); 7. W=zeros(p,m*q);mP_IS=zeros(1,m*q);mN_IS=ones(1,m*q); 8. Y1=zeros(p,m*q);Y2=zeros(p,m*q);Z1=zeros(p,q);Z2=zeros(p,q); 9. for i=1:r 10. D= input(‘enter the m-polar fuzzy linguistic decision matrix pxqxm’); 11. A_g(Rr,Cr)=A_g(Rr,Cr)+D;w=input(‘enter the weights’);w_g(1,Cw)=w_g(1,Cw)+w; 12. end 13. A_g=A_g/r 14. w=w_g/r 15. for j=1:p 16. k=1; 17. for s=1:m*q 18. W(j,s)=A_g(j,s).*w(1,k); 19. if mod(s,m)==0 20. k=k+1; 21. end 22. end (continued)

2.11 Conclusion

151

Table 2.41 (continued) 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

end W for j=1:p for s=1:(m*q) mP_IS(1,s)=max(mP_IS(1,s),W(j,s)); mN_IS(1,s)=min(mN_IS(1,s),W(j,s)); end end mP_IS mN_IS for j=1:p k=0; for s=1:m*q Y1(j,s)=(W(j,s)-mP_IS(1,s)). ˆ 2; Y2(j,s)=(W(j,s)-mN_IS(1,s)). ˆ 2; if mod(s,m)==1 k=k+1; end Z1(j,k)=Z1(j,k)+Y1(j,s);Z2(j,k)=Z2(j,k)+Y2(j,s); end end D_P=sqrt(sum(Z1,2)./m) D_N=sqrt(sum(Z2,2)./m) E=D_N./(D_P+D_N)

2.11 Conclusion ELECTRE has long been considered as one of the foremost MCDM technique. It is based on outranking relations, which induce a new preference relation called incomparability used to handle the situations in which the decision makers are unable to compare two alternatives. Pairwise comparisons of the alternatives are made by using outranking relations in numerical examples by introducing an m−polar fuzzy ELECTRE-I approach. Traditional methods are ineffective to study the imprecise behavior of linguistic computations and assessments, because it has become difficult to collect data about linguistic assessments in terms of numerical and fuzzy values. To deal with such a complexity, the concept of m–polar fuzzy linguistic variable has been presented, which is able to deal the situation when one has data in the for of linguistic assessments and m different numerical or fuzzy values as well. An m−polar fuzzy ELECTRE-I approach has been developed to deal the MCDM and MCGDM problems, which is used to handle the complex situation of m–polar fuzzy linguistic variable, and to compile the outranking relations of alternatives to choose and compare best alternative among others. The m−polar fuzzy ELECTRE-I method

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is used to produce more reliable and consistent results, when one has to eliminate the choices and to deal with systems with more than one arrangement. Moreover, the proposed method is arguably more efficient than the existing methods, when the alternatives that are dominated by other alternatives to a specified degree with linguistic values are eliminated. Further, in this contribution an m−polar fuzzy linguistic TOPSIS approach has also been presented for MCGDM problems, as an extension of TOPSIS method. Finally, the presented techniques have been applied on real life problems, operational algorithms have been developed, illustrative flowcharts have been presented and computer programming codes have been generated to facilitate implementation.

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Chapter 3

Introducing Hesitancy: TOPSIS and ELECTRE-I Models

In this Chapter, the hybrid models known as multi-polar (m–polar) hesitant fuzzy sets, and hesitant m–polar fuzzy sets are presented. Both the presented models are the hybridization of hesitancy with m–polar fuzzy sets, and the natural generalization of hesitant fuzzy sets. They facilitate the management of hesitation, uncertainty and vagueness motivated by multipolar information. Both have the fascinating advantages and characteristic, because hesitant framework is often preferred as compared to precise situations, and enable to tackle multi-polar information with hesitancy. Hesitancy incorporates symmetry into the treatment of the data, whereas the m– polar fuzzy format allows for differentiated or asymmetric sources of information. Basic key properties of m–polar hesitant fuzzy sets are highlighted and intrinsic operations are also formulated. Moreover, an m–polar hesitant fuzzy TOPSIS and an m–polar hesitant fuzzy ELECTRE-I approaches for multi-criteria group decision making (MCGDM) are developed, which are the natural extensions of the TOPSIS and ELECTRE-I methods to this framework. In hesitant m–polar fuzzy set model, the membership degrees of an element of given set deal with m different numeric and fuzzy values that enable it deal the hesitancy of multipolar information. Some of its useful properties are explored, fundamental operations are constructed and comparison laws are investigated. Moreover, hesitant m–polar fuzzy TOPSIS and hesitant m–polar fuzzy ELECTRE-I approaches for MCGDM are developed. The efficiency of proposed decision making approaches is discussed by applying them to the real situations and industrial fields, for selection of a perfect brand name, a suitable product design for a company and best bricks for construction in case of m–polar hesitant fuzzy sets, and for comparison of populous countries and different types of textiles or clothing, and for the selection of site for farming purposes in case of hesitant m–polar fuzzy sets. Finally, algorithms and computer programming codes, of presented approaches are described. This Chapter is based on [3, 7, 8].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_3

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3.1 Introduction Decision making is an action to choose and identify the alternatives on the evaluations and interpretation of decision maker or group of decision makers. Fuzzy set theory introduced by Zadeh [84] in 1965 to compose decision models with uncertain and vague data, it is also able to represent real-world problems in a more flexible and realistic way than using crisp decision making models. However, the modeling and representative tools of fuzzy sets are defined and limited, on the other hand two or more origins of vagueness can appear together. Thus certain well known extensions and generalizations have been developed based on, hesitant fuzzy sets and m−polar fuzzy sets. The concept of an m–polar fuzzy set theory is introduced by Chen et al. [30]. The basic concept regarding to this approach is, the multipolar knowledge occurs, because facts and knowledge for real world problems are sometimes from n agents (n ≥ 2), referred to [39]. There are several other examples, such as accurate degrees of a logic formula which are based on n logic implication operators (n ≥ 2). For example, the correct degree of telecommunication, assurance of human species are points in [0, 1]n (n ≈ 7 × 109 ), because distinct characters have been supervised by distinct times [47]. There are many other examples such as truth degree of two logic formula which are based on n logic indication operators (n ≥ 2) or many valued logics [25], regulating results of university, regulating results of magazines and inclusion degrees (rough measures, accuracy measures, fuzziness measures, approximation qualities and decision preformation evaluations) of rough sets. Akram [4] introduced many new concepts including m−polar fuzzy graphs, m−polar fuzzy line graphs, m−polar fuzzy labeling graphs and certain metrics in m−polar fuzzy graphs. Akram et al. [11, 12] proposed multi-attribute decision making (MADM) methods based on m–polar fuzzy rough and m–polar fuzzy soft rough information. Adeel et al. [1] introduced the novel concept of m–polar fuzzy linguistic ELECTRE-I method for group decision making. Further, Akram and Adeel [5] introduced the novel hybrid decision making methods based on m–polar fuzzy rough information. Particularly from another position, Torra and Narukawa [72, 73] introduced the notion of hesitant fuzzy sets to compose the generalization of fuzzy sets that is advancing expeditiously with its expansions, functions and utilizations to several fields [86]. This notion is reasonable for the conception of situations where decision makers have hesitancy in contributing their estimations and judgments over objects, or also when we combine the assumptions of distinct experts into an individual input. Certainly in most of the decision making cases, experts are generally hesitant or doubtful which forbid them from producing exclusive assessments [79]. The concept is preferable to acknowledge the hybridization with other theories to uncertainty and vagueness [16, 76]. The ideas of uncertainty or hesitancy in MCGDM had also been dealt within the evidential reasoning framework [82]. Alcantud and Torra [17] introduced the decomposition theorems and extension principles for hesitant fuzzy sets. Xia and Xu [79] developed some aggregation operators and presented applications to handle multi-criteria decision making (MCDM) problems under hesitant fuzzy environment. Xia et al. [80] also introduced some other hesitant fuzzy

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aggregation approaches and presented its importance in group decision making. Chen et al. [34] generalized the concept of hesitant fuzzy sets and induced the idea of interval-valued hesitant fuzzy sets. Mandal and Ranadive [58] proposed the concept of hesitant bipolar-valued fuzzy sets and bipolar-valued hesitant fuzzy sets and their applications in multi-attribute group decision making. Akram et al. [9] worked on hesitant fuzzy N -soft sets: a new model with applications in decision making. Garg and Arora [37, 38] introduced the aggregation operators, distance and similarity measures, and a robust correlation coefficient measure for dual hesitant fuzzy soft sets and their applications in MCDM problems. For the exploration of other hybrid models related to hesitant fuzzy sets, readers are referred to [7, 10, 36]. In real-world systems, usually we obverse activities and tasks in which its compulsory to adopt decision making techniques [50, 52–54]. Commonly, decision making is an intellectual process based on distinct reasoning and rational actions that leads to choose the reasonable alternative from a set of feasible alternatives in a decision situation [55, 57, 71]. In comparison to the existing decision making methods composed of distinct aspects, the method of TOPSIS proposed in [42] is an effective, favorable and widely used MCGDM method. Precisely, the method of TOPSIS derives from the concept that the preferred alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. Since then, several extended TOPSIS methods have been applied to different MCDM problems [14, 18, 26, 70]. Xu and Zhang [83] established a new approach based on TOPSIS and maximizing deviation method for the interpretation of MCDM problems. Ashtiani [21] developed the extensions of fuzzy TOPSIS method based on interval-valued fuzzy sets. Wang and Lee [77] introduced the generalized TOPSIS method for fuzzy MCGDM. Chen [26] introduced an extended TOPSIS method for MCDM by considering triangular fuzzy numbers and defining the crisp Euclidean distance between two fuzzy numbers. Roszkowska [63] proposed the MCDM models by applying the TOPSIS method to crisp and interval data. Further, Roszkowska and Wachowicz [64] applied the fuzzy TOPSIS method to rate the negotiation actions in poor formatted negotiation problems. Ren et al. [61] developed a novel hesitant fuzzy linguistic TOPSIS method for group multi-criteria linguistic decision making. Adeel et al. [2] developed the group decision making based on m−polar fuzzy linguistic TOPSIS method. Rodrí guez et al. [62] proposed the concept of a position and perspective analysis of hesitant fuzzy sets on information fusion in decision making, towards high quality progress. Zhang et al. [85] introduced the operations and integrations of probabilistic hesitant fuzzy information in decision making. Elimination and choice translating reality (ELECTRE) is one of the MCDM method, in which the decision maker desires to hold different criterions and there may be a robust collection associated with the nature of evaluation surrounded by a number of the standards. The ELECTRE approach was first introduced by Benayoun et al. [24]. After that, the modified concept of ELECTRE known as ELECTRE-I was introduced by Roy [65]. Further, this approach was expended into variety of alternative variants. Nowadays, the foremost widely used versions are referred as ELECTRE-II, ELECTRE-III and ELECTRE-IV. In literature, most of these methods have been combined with fuzzy sets by several researchers. For supplier selection

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problem, Sevkli [69] analyzed the classical and fuzzy ELECTRE methods. For the choice and evaluation of academic staff, Rouyendegh and Erkan [66] used the concept of fuzzy ELECTRE. Vahdani et al. [75] associated the fuzzy ELECTRE method with intuitionistic fuzzy ELECTRE method and compared them. To choose the proper location of plant under group decision making environment, Devi and Yadav [35] proposed the concept of intuitionistic fuzzy ELECTRE method. Vahdani and Hadipour [74] presented the technique of interval-valued fuzzy ELECTRE method. To deal the MCDM problems, Wu and Chen [78] developed the concept of intuitionistic fuzzy ELECTRE-I method. Chen and Xu [31] proposed a novel MCDM technique by combining hesitant fuzzy sets with ELECTRE-II method. Hatami-Marbini et al. [41] applied the method of fuzzy group ELECTRE for the interpretation of haphazard waste reprocessing of plants. Kheirkhah and Dehghani [44] applied the fuzzy group ELECTRE method for the assessment of quality of public transportation facilities. Hatami-Marbini and Tavana [40] expended the method of ELECTRE-I and introduced the method of fuzzy ELECTRE-I with numerical examples to illustrate the effectiveness of their proposed method. Asghari et al. [20] used fuzzy ELECTRE-I method for the analysis of mobile payment models. Further, fuzzy ELECTRE-I technique was applied in evaluating catering firm alternatives by Aytac et al. [22] and an environmental effect evaluation approach based on fuzzy ELECTRE-I was composed by Kaya and Kahraman [43]. Lupo [56] used the ELECTRE-III approach to calculate the service quality of three international airports. Akram et al. [13] introduced novel approach in decision making with m–polar fuzzy ELECTRE-I.

3.2 m−Polar Hesitant Fuzzy Set Definition 3.1 Let Z be a reference set, an m−polar hesitant fuzzy set on Z is a function m that returns a subset of values in [0, 1]m: m : Z → (P{[0, 1]m }). Mathematically, an m–polar hesitant fuzzy set is represented as follows: H = {z, m (z)|∀z ∈ Z }, 



where m (z) = {ζh |ζh ∈ p1 ◦ m (z)}, {ζh |ζh ∈ p2 ◦ m (z)}, . . . , {ζh |ζh ∈ pm ◦ m (z)} . This notation shows that m (z) is an m–tuple of sets, having possible membership degrees of each element z ∈ Z in set H , where m = m (z) is called an m–polar hesitant fuzzy element.

3.2 m−Polar Hesitant Fuzzy Set

161

It is apparent that, when m = 1, m–polar hesitant fuzzy elements are hesitant fuzzy elements, and considered as the m−polar hesitant fuzzy sets are standard hesitant fuzzy sets. The following Example 3.1 illustrates the concepts as follows: Example 3.1  Let Z = {z 1 , z 2 , z 3 } be a referenceset and m (z 1 ) = {0.3, 0.4}, {0.3, 0.5}, {0.4, 0.5, 0.65} ,   m (z 2 ) = {0.1, 0.3, 0.5}, {0.2, 0.3, 0.7}, {0.1, 0.4} ,   m (z 3 ) = {0.4, 0.55}, {0.5, 0.6}, {0.3, 0.4, 0.7, 0.75} , be respective 3-polar hesitant fuzzy elements. Then, a 3-polar hesitant fuzzy set H is given as  H=

  z 1 , {0.3, 0.4}, {0.3, 0.5}, {0.4, 0.5, 0.65} ,    z 2 , {0.1, 0.3, 0.5}, {0.2, 0.3, 0.7}, {0.1, 0.4} ,    z 3 , {0.4, 0.55}, {0.5, 0.6}, {0.3, 0.4, 0.7, 0.75} .

The following real life example illustrates the concept and shows its usefulness. Example 3.2 Let Z = {z 1 , z 2 , z 3 } be a reference set of candidates appearing for selection of job and m (z) represents the 3-polar hesitant fuzzy characterization of its evaluating criteria as • C.V evaluation • Interview evaluation • Knowledge evaluation These are three main evaluation features or criteria required for the selection of a candidate for job. Each candidate z ∈ Z has the following ratings classified by 3-polar hesitant fuzzy set according to the evaluating criteria and has the respective 3-polar hesitant fuzzy elements, represented as   m (z 1 ) = {0.34, 0.54}, {0.43, 0.45, 0.51}, {0.40, 0.58, 0.65} ,   m (z 2 ) = {0.29, 0.38, 0.57}, {0.21, 0.30, 0.70}, {0.18, 0.42} ,   m (z 3 ) = {0.43, 0.55}, {0.47, 0.60}, {0.34, 0.46, 0.70, 0.75} .  The 3-polar hesitant fuzzy element m (z 1 ) = {0.34, 0.54}, {0.43, 0.45, 0.51},  {0.40, 0.58, 0.65} shows, the candidate z 1 has the following ratings according to his evaluation criteria as

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

( C.V{0.34,0.54} , {0.43,0.45,0.51} , {0.40,0.58,0.65} ) and C.V{0.34,0.54} show evaluation I nter view evaluation K nowledge evaluation evaluation candidate z 1 has two hesitant values {0.34,0.54} according to his C.V evaluation, similarly he has the hesitant ratings according to other evaluation criteria. Remaining candidates are evaluated in the same sense and the 3-polar hesitant fuzzy set H is given as  H = z 1 , {0.34, 0.54}, {0.43, 0.45, 0.51}, {0.40, 0.58, 0.65} ,    z 2 , {0.29, 0.38, 0.57}, {0.21, 0.30, 0.70}, {0.18, 0.42} ,    z 3 , {0.43, 0.55}, {0.47, 0.60}, {0.34, 0.46, 0.70, 0.75} . 



The 3-polar hesitant fuzzy set H shows the complete information about the evaluation of candidates for a job. From Example 3.2, it is easy to understand the concept of the approach described in Definition 3.1, in which the multi-polar information under hesitant situation of each degree of membership of 3-polar fuzzy set is discussed separately. Some special m–polar hesitant fuzzy elements for z ∈ Z are given as follows: 1. Empty set: em = ({0}m ). f 2. Full set: m = ({1}m ). 3. Complete ignorance: (All values are possible) m = [0, 1], where 0 = (0, 0, . . . , 0) and 1 = (1, 1, . . . , 1). 4. Nonsense set: .

3.2.1 Basic Operations of m–Polar Hesitant Fuzzy Set In this subsection, the basic operations of m–polar hesitant fuzzy sets are constructed with example in the framework that has been defined in previous section. 1. Lower bound: − m (z) =



 inf{ζh |ζh ∈ p1 ◦ m (z)}, inf{ζh |ζh ∈ p2 ◦ m (z)}, . . . , inf{ζh |ζh ∈ pm ◦ m (z)} ,

for all z ∈ Z . 2. Upper bound: + m (z) =



 sup{ζh |ζh ∈ p1 ◦ m (z)}, sup{ζh |ζh ∈ p2 ◦ m (z)}, . . . , sup{ζh |ζh ∈ pm ◦ m (z)} ,

for all z ∈ Z .

3.2 m−Polar Hesitant Fuzzy Set

163

3. Complement:

  cm (z) = {1 − ζh |ζh ∈ p1 ◦ m (z)}, {1 − ζh |ζh ∈ p2 ◦ m (z)}, . . . , {1 − ζh |ζh ∈ pm ◦ m (z)} ,

for all z ∈ Z . 4. Union:   (H1 ) (H2 ) (H1 )− (H2 )− 1 ) ∪ (H2 ) )(z) = ((H {ζ ∈ p ◦  (z) ∪ p ◦  (z)|ζ ≥ sup{ (z),  (z)}} , h i i h m m m m m m (H1 ) (H2 ) (H2 ) 1) where pi ◦ (H m (z) ∈ m (z) and pi ◦ m (z) ∈ m (z), for all z ∈ Z and i ∈ m.

5. Intersection:

  (H1 ) (H2 ) 1 ) ∩ (H2 ) )(z) = 1 )+ (z), (H2 )+ (z)}} , ((H {ζh ∈ pi ◦ m (z) ∩ pi ◦ m (z)|ζh ≤ inf{(H m m m m

(H1 ) (H2 ) (H2 ) 1) where pi ◦ (H m (z) ∈ m (z) and pi ◦ m (z) ∈ m (z), for all z ∈ Z and i ∈ m.

6. Direct sum: (H1 )

(m

(H2 )

⊕ m

  (H ) (H ) )(z) = {ζh 1 + ζh 2 − ζh 1 ζh 2 |ζh 1 ∈ pi ◦ m 1 (z), ζh 2 ∈ pi ◦ m 2 (z)} ,

(H1 ) (H2 ) (H2 ) 1) where pi ◦ (H m (z) ∈ m (z) and pi ◦ m (z) ∈ m (z), for all z ∈ Z and i ∈ m.

7. Direct product:

  (H2 ) (H1 ) (H2 ) 1) ((H ⊗  )(z) = {ζ ζ |ζ ∈ p ◦  (z), ζ ∈ p ◦  (z)} , h1 h2 h1 i h2 i m m m m

(H1 ) (H2 ) (H2 ) 1) where pi ◦ (H m (z) ∈ m (z) and pi ◦ m (z) ∈ m (z), i ∈ m.

for all z ∈ Z and

The following example illustrates the operations defined. Example 3.3 Let Z = {z 1 , z 2 , z 3 } be the reference set. Then, two 3-polar hesitant fuzzy sets H1 and H2 on Z are, respectively, given as  H1 =





z 1 , {0.2, 0.3}, {0.4, 0.5, 0.6}, {0.4, 0.6} ,   z 2 , {0.3, 0.5}, {0.4, 0.6}, {0.7, 0.8} ,    z 3 , {0.1, 0.2}, {0.5, 0.6, 0.7}, {0.7, 0.8} .



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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

 H2 =

  z 1 , {0.4, 0.6, 0.7}, {0.6, 0.7}, {0.7, 0.8} ,    z 2 , {0.5, 0.6}, {0.2, 0.3, 0.4}, {0.3, 0.5, 0.8, 0.9} ,    z 3 , {0.3, 0.4}, {0.2, 0.4, 0.6}, {0.5, 0.7} .

The aforementioned operations on these two 3-polar hesitant fuzzy sets H1 and H2 are calculated as follows: 1. Lower bound: 1 )− (z 1 ) = (H m



 inf{0.2, 0.3}, inf{0.4, 0.5, 0.6}, inf{0.4, 0.6}

= (0.2, 0.4, 0.4),   2 )− (z ) = inf{0.3, 0.4}, inf{0.2, 0.4, 0.6}, inf{0.5, 0.7} (H 3 m = (0.3, 0.2, 0.5). 2. Upper bound: 1 )+ (H (z 2 ) = m



 sup{0.3, 0.5}, sup{0.4, 0.6}, sup{0.7, 0.8}

= (0.5, 0.6, 0.8),   2 )+ (z ) = sup{0.3, 0.4}, sup{0.2, 0.4, 0.6}, sup{0.5, 0.7} (H 3 m = (0.4, 0.6, 0.7). 3. Complement:   (H1 )c m (z 1 ) = {1 − 0.2, 1 − 0.3}, {1 − 0.4, 1 − 0.5, 1 − 0.6}, {1 − 0.4, 1 − 0.6}   = {0.8, 0.7}, {0.5, 0.6, 0.4}, {0.6, 0.4} ,

2 )c (H (z 2 ) m

 = {1 − 0.5, 1 − 0.6}, {1 − 0.2, 1 − 0.3, 1 − 0.4}, {1 − 0.3, 1 − 0.5,  1 − 0.8, 1 − 0.9}   = {0.5, 0.4}, {0.8, 0.7, 0.6}, {0.7, 0.5, 0.2, 0.1} .

3.2 m−Polar Hesitant Fuzzy Set

165

4. Union:   (H2 ) 1) ∪  )(z ) = sup (0.2, 0.4, 0.4), (0.4, 0.6, 0.7) = (0.4, 0.6, 0.7) ((H 1 m m   = {0.4, 0.6, 0.7}, {0.6, 0.7}, {0.7, 0.8} ,   (H1 ) (H2 ) (m ∪ m )(z 3 ) = sup (0.1, 0.5, 0.7), (0.3, 0.2, 0.5) = (0.3, 0.5, 0.7)   = {0.3, 0.4}, {0.5, 0.6, 0.7}, {0.7, 0.8} . 5. Intersection:   (H2 ) 1) ∩  )(z ) = inf (0.5, 0.6, 0.8), (0.6, 0.4, 0.9) = (0.5, 0.4, 0.8) ((H 2 m m   = {0.3, 0.5}, {0.2, 0.3, 0.4}, {0.3, 0.5, 0.7, 0.8} ,   (H2 ) 1) ∩  )(z ) = inf (0.2, 0.7, 0.8), (0.4, 0.6, 0.7) = (0.2, 0.6, 0.7) ((H 3 m m   = {0.1, 0.2}, {0.2, 0.4, 0.5, 0.6}, {0.5, 0.7} . 6. Direct sum:  (H2 ) 1) ⊕  )(z ) = {0.52, 0.68, 0.76, 0.79, 0.58, 0.72}, {0.76, 0.82, 0.8, ((H 1 m m  0.85, 0.84, 0.88}, {0.88, 0.82, 0.92} . 7. Direct product: 1) ((H m

⊗

2) (H m )(z 3 )

 = {0.03, 0.04, 0.06, 0.08}, {0.1, 0.2, 0.3, 0.12, 0.24,  0.36, 0.14, 0.28, 0.42}, {0.35, 0.49, 0.40, 0.56} .

The following propositions show that the union and intersection of m–polar hesitant fuzzy sets satisfy the commutativity, associativity, and idempotency properties, under limited conditions. (2) (3) Proposition 3.1 For any m–polar hesitant fuzzy elements (1) m , m , and m in H (Z ) and z ∈ Z , the following properties are defined as

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

1. Commutativity: (2) (2) (1) (i) ((1) m ∪ m )(z) = (m ∪ m )(z), (1) (2) (2) (ii) (m ∩ m )(z) = (m ∩ (1) m )(z).

2. Associativity: (2) (3) (1) (2) (3) (i) (((1) m ∪ m ) ∪ m )(z) = (m ∪ (m ∪ m ))(z), (1) (2) (3) (1) (2) (ii) ((m ∩ m ) ∩ m )(z) = (m ∩ (m ∩ (3) m ))(z).

3. Idempotency: (1) (1) (i) ((1) m ∪ m )(z) = m (z), (1) (1) (ii) (m ∩ m )(z) = (1) m (z).

Proof All three described properties are trivial to prove.



Some of the operational rules are stated, in the form of Propositions: Proposition 3.2 For any m ∈ H (Z ) and z ∈ Z , the following operational rules are defined as 1. (cm )− (z) = 1 − + m (z), 2. (cm )+ (z) = 1 − − m (z). Proof 1. (cm )− (z) = inf cm (z)  = inf{1 − ζh |ζh ∈ p1 ◦ m (z)}, inf{1 − ζh |ζh ∈ p2 ◦ m (z)},  . . . , inf{ζh |1 − ζh ∈ pm ◦ m (z)} , ∀z ∈ Z  sup{ζh |ζh ∈ p1 ◦ m (z)}, sup{ζh |ζh ∈ p2 ◦ m (z)},

=1 −

 . . . , sup{ζh |ζh ∈ pm ◦ m (z)} , ∀z ∈ Z =1 − + m (z).

2. (cm )+ (z) = sup cm (z)  = sup{1 − ζh |ζh ∈ p1 ◦ m (z)}, sup{1 − ζh |ζh ∈ p2 ◦ m (z)},  . . . , sup{ζh |1 − ζh ∈ pm ◦ m (z)} , ∀z ∈ Z  =1 −

inf{ζh |ζh ∈ p1 ◦ m (z)}, inf{ζh |ζh ∈ p2 ◦ m (z)},

 . . . , inf{ζh |ζh ∈ pm ◦ m (z)} , ∀z ∈ Z =1 − − m (z).



3.2 m−Polar Hesitant Fuzzy Set

167

Proposition 3.3 For any m ∈ H (Z ) and z ∈ Z , the following operational rules are defined as f

f

f

1. (m ∪ m )(z) = m (z) and (m ∩ m )(z) = m (z), 2. (m ∪ em )(z) = m (z) and (m ∩ em )(z) = em (z). Proof   1. (m ∪ mf )(z) = {ζh ∈ pi ◦ m (z) ∪ {1}i∈m |ζh ≥ sup{− (z), ({1} )}} , m m ∀ z ∈ Z and i ∈ m   = {ζh ∈ pi ◦ m (z) ∪ {1}i∈m |ζh ≥ ({1}m )} , ∀ z ∈ Z and i ∈ m =({1}m ) = mf (z),   (z), ({1} )}} , (m ∩ mf )(z) = {ζh ∈ pi ◦ m (z) ∩ {1}i∈m |ζh ≤ inf{+ m m ∀ z ∈ Z and i ∈ m   (z)} , ∀ z ∈ Z and i ∈ m = {ζh ∈ pi ◦ m (z) ∩ {1}i∈m |ζh ≤ + m =m (z).   2. (m ∪ em )(z) = {ζh ∈ pi ◦ m (z) ∪ {0}i∈m |ζh ≥ sup{− (z), ({0} )}} , m m ∀ z ∈ Z and i ∈ m   (z)} , ∀ z ∈ Z and i ∈ m = {ζh ∈ pi ◦ m (z) ∪ {0}i∈m |ζh ≥ − m =m (z),   (z), ({0} )}} , (m ∩ em )(z) = {ζh ∈ pi ◦ m (z) ∩ {0}i∈m |ζh ≤ inf{+ m m ∀ z ∈ Z and i ∈ m   = {ζh ∈ pi ◦ m (z) ∩ {0}i∈m |ζh ≤ ({0}m )} , ∀ z ∈ Z and i ∈ m =({0}m ) = em (z). 

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

3.2.2 Comparison Laws of m–Polar Hesitant Fuzzy Elements Definition 3.2 The score function s(m ) of the m–polar hesitant fuzzy elements of an m–polar hesitant fuzzy set is defined as  s(m ) =

1 γ pi ◦m (z)



 ζh , i ∈ m,

ζh ∈ pi ◦m (z)

where γ pi ◦m (z) is the number of elements in pi ◦ m (z). The score function helps us to compare m–polar hesitant fuzzy elements, according to the following rules: (2) Remark 3.1 For any two m–polar hesitant fuzzy elements (1) m and m of an m– polar hesitant fuzzy set:

• • • •

(2) (1) (2) If s((1) m ) > s(m ), then m is superior to (or finer than) m . (1) (2) (1) If s(m ) < s(m ), then m is inferior to (or weaker than) (2) m . (2) (1) (2) ) = s( ), then  is indifferent to  . If s((1) m m m m (2) If none of the above are true, then (1) m is totally different from m .

(1) m

 = {0.4, 0.5},

Example 3.4 Consider the two 3-polar hesitant fuzzy elements  {0.6, 0.8, 0.85}, {0.5, 0.7, 0.9}   (2) = {0.5, 0.6}, {0.7, 0.8, 0.9}, {0.5, 0.65, 0.7, 0.75} . Then, by Definition 3.2, m (2) the score functions of (1) m and m are calculated as follows:

 0.4 + 0.5 0.6 + 0.8 + 0.85 0.5 + 0.7 + 0.9 , , 2 3 3 = (0.45, 0.75, 0.7),   0.5 + 0.6 0.7 + 0.8 + 0.9 0.5 + 0.65 + 0.7 + 0.75 , , ) = s((2) m 2 3 4 = (0.55, 0.8, 0.65). s((1) m )=



(2) From these calculations, one readily concludes that (1) m is totally different to m .

However, the score function is not fully discriminative for a formal discussion of the cause, and this feature is exemplified as follows: Example  3.5 Consider two 4–polar hesitant fuzzy elements  (1) m = {0.2, 0.4, 0.6}, {0.1, 0.2}, {0.2, 0.3, 0.5, 0.6}, {0.7, 0.8, 0.9}

3.2 m−Polar Hesitant Fuzzy Set

169

  (2) = {0.1, 0.7}, {0.1, 0.15, 0.2}, {0.1, 0.5, 0.6}, {0.7, 0.8, 0.8, 0.9} . m (2) Then, by Definition 3.2, the score functions of (1) m and m are calculated as follows:

s((1) m ) = (0.4, 0.15, 0.4, 0.8), s((2) m ) = (0.4, 0.15, 0.4, 0.8). (2) From these calculations, one observes that (1) m is deemed indifferent to m , and (1) (2) unable to give formal support to the difference between m and m by the application of the score function alone.

In other words, the example shows that, sometimes it is not possible to perform a comparison when two m–polar hesitant fuzzy elements have coincident score functions, as computed by Definition 3.2. In order to break ties in such a situation, the deviation degree of an m–polar hesitant fuzzy element is defined. In case of indifference between two m–polar hesitant fuzzy elements, this figure may tell, which one is superior. Definition 3.3 The deviation degree (m ) of the m–polar hesitant fuzzy elements of an m–polar hesitant fuzzy set is defined as  (m ) =

1 γ pi ◦m (z)



(ζh − s(m ))2

1  2 , i ∈ m,

ζh ∈ pi ◦m (z)

where γ pi ◦m (z) is the number of elements in pi ◦ m (z). The criterion in Remark 3.1 is refined, using the following terms: (2) Remark 3.2 For any two indifferent m–polar hesitant fuzzy elements (1) m and m of an m–polar hesitant fuzzy set:

• • • •

(2) (1) (2) If ((1) m ) > (m ), then m is superior to (or finer than) m . (1) (2) (1) If (m ) < (m ), then m is inferior to (or weaker than) (2) m . (2) (1) (2) If ((1) m ) = (m ), then m is indifferent to m . (2) If none of the above are true, then (1) m is completely different from m .

(2) Example 3.6 By reconsidering Example 3.5, where s((1) m ) = s(m ), the deviation degrees of these 4–polar hesitant fuzzy elements are calculated as follows:

((1) m ) = (0.163, 0.05, 0.158, 0.082), ((2) m ) = (0.3, 0.041, 0.216, 0.071). (2) (1) From these calculations, it is observed that none of ((1) m ) > (m ), (m ) < (2) (1) (2) (1) (m ), or (m ) = (m ) are true. Thus, m is completely different from (2) m .

In this section, two different MCGDM approaches, known as m–polar hesitant fuzzy TOPSIS and m–polar hesitant fuzzy ELECTRE-I are developed, which are flexible and compatible with multi-polar data under hesitancy.

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach An m–polar hesitant fuzzy TOPSIS approach is based on m–polar hesitant fuzzy sets, that deals with MCGDM problems, in which Z = {z 1 , z 2 , . . . , z p } is considered as the set of different alternatives and C = {c1 , c2 , . . . , cq } as the set of criteria, which are distinguished by m different characteristics under a hesitant situation. The framework of the problem is as follows: The decision makers are subjected to evaluate the p different alternatives having q criteria, and the convenient valuations of the alternatives are determined having m different characteristics under r different membership values, due to hesitancy. The steps for the proposed approach are described as follows : Step 1: The degree of each alternative (z j ∈ Z , j = 1, 2, · · · , p) over all criteria (ck ∈ C, k = 1, 2, . . . , q) is given by m–polar hesitant fuzzy elements as jk jk m (z) = m =





jk jk jk {ζh |ζh ∈ p1 ◦ m (z)}, {ζh |ζh ∈ p2 ◦ m (z)}, . . . , {ζh |ζh ∈ pm ◦ m (z)} ,

jk

where ( pi ◦ m (z)|i = 1, 2, . . . , m) classify the several other characteristics of each criterion. The tabular representation of the m–polar hesitant fuzzy decision matrix H is given by Table 3.1, which describes the ratings of alternatives. For each possible j, k in Table 3.1,   jk jk jk jk jk m (z) = m = {ζh |ζh ∈ p1 ◦ m (z)}, {ζh |ζh ∈ p2 ◦ m (z)}, · · · , {ζh |ζh ∈ pm ◦ m (z)} .

Note that, in general, the number of m–polar hesitant fuzzy elements is not comparable in all m–polar hesitant fuzzy sets. In order to increase efficiency, the largest or smallest membership values are prolonged, until the lengths of all m–polar hesitant fuzzy elements become equal, as the decision makers want to choose the best alternative in an optimistic or pessimistic spirit. For this reason, the information fusion shows an optimistic or pessimistic response and improves the m–polar hesitant fuzzy

Table 3.1 A generic m–polar hesitant fuzzy decision matrix Alternatives Criteria c1 c2 ···

cq 1q

z1

11 m

12 m

···

m

z2 .. .

21 m .. .

22 m .. .

··· .. .

m .. .

zp

m

p1

m

p2

···

m

2q

pq

3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach

171

Table 3.2 A generic weighted m–polar hesitant fuzzy decision matrix Alternatives Criteria c1 c2 ··· Weights w1 w2 ···  11 m  21 m

z1 z2 .. .

.. .

p1

m

zp

 12 m  22 m

.. .

p2

m

cq wq

1q 

···

m

··· .. .

m .. .

···

m

2q 

pq 

data by adding the maximal or minimal values. Step 2: The decision makers have the ability to attach weights to each criteria of alternatives, according to their experience and the priority of each criteria. The desired weights assigned by the decision makers are W = (w1 , w2 , . . . , wq ) ∈ (0, 1]. Weights assigned by the decision makers satisfy a normalization condition. i.e., q 

wk = 1.

k=1

Note that the only condition for the weights assigned by decision makers is that the weights should be normalized. Readers are free to take the weights according to their own method and choice. It is not necessary to take two, three, or four digits after the decimal point, as taken in previous examples. Two or four values are chosen after the decimal point, for readers convenience, which satisfy the normalized condition. These weights totally depend upon the choice of the decision maker and the importance of the required criteria. In the case of a lack of information about these figures, the weights are equally divided. 

Step 3: The weighted m–polar hesitant fuzzy decision matrix H is calculated in Table 3.2: For each possible j, k in Table 3.2, jk 

jk

m =wk m   jk jk jk = wk {ζh |ζh ∈ p1 ◦ m (z)}, wk {ζh |ζh ∈ p2 ◦ m (z)}, . . . , wk {ζh |ζh ∈ pm ◦ m (z)}         jk jk jk = {ζh |ζh ∈ p1 ◦ m (z)}, {ζh |ζh ∈ p2 ◦ m (z)}, . . . , {ζh |ζh ∈ pm ◦ m (z)} .

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Step 4: The m–polar hesitant fuzzy positive ideal solution and m–polar hesitant fuzzy negative ideal solution of the alternatives, are calculated by the following Eqs. (3.1) and (3.2). 











m H PI S ={(1m )+ , (2m )+ , . . . , (qm )+ }, m H N I S ={(1m )− , (2m )− , . . . , (qm )− },

(3.1)

(3.2)

where 



(km )+ = max(km ) j      = max{ζh |ζh ∈ p1 ◦ mjk (z)}, max{ζh |ζh ∈ p2 ◦ mjk (z)}, j j    . . . , max{ζh |ζh ∈ pm ◦ mjk (z)} j      = {(ζh )+ |(ζh )+ ∈ p1 ◦ mjk (z)}, {(ζh )+ |(ζh )+ ∈ p2 ◦ mjk (z)},    . . . , {(ζh )+ |(ζh )+ ∈ pm ◦ mjk (z)} ,





(km )− = min(km ) j      = min{ζh |ζh ∈ p1 ◦ mjk (z)}, min{ζh |ζh ∈ p2 ◦ mjk (z)}, j j    . . . , min{ζh |ζh ∈ pm ◦ mjk (z)} j      = {(ζh )− |(ζh )− ∈ p1 ◦ mjk (z)}, {(ζh )− |(ζh )− ∈ p2 ◦ mjk (z)},    . . . , {(ζh )− |(ζh )− ∈ pm ◦ mjk (z)} . Step 5: The m–polar hesitant fuzzy Euclidean distances of each alternative a j from m–polar hesitant fuzzy positive ideal solution and m–polar hesitant fuzzy negative ideal solution respectively, are calculated by Eqs. (3.3) and (3.4).

3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach

173



De (z j , m H PI S )  

q  m  1  jk  k  )+ )2 + (ζ jk  − (ζ k  )+ )2 + · · · + (ζ jk  − (ζ k  )+ )2 , = (ζh1 − (ζh1 h2 h2 hr hr rm k=1

i=1

(3.3) jk 



where ζhl and (ζhlk )+ ∈ pi ◦ m (z). jk



De (z j , m H N I S )  

q  m  1  jk  k  )− )2 + (ζ jk  − (ζ k  )− )2 + · · · + (ζ jk  − (ζ k  )− )2 , (ζh1 − (ζh1 = h2 h2 hr hr rm k=1

i=1

(3.4) jk 



where, ζhl and (ζhlk )− ∈ pi ◦ m (z). jk

Step 6: The relative m–polar hesitant fuzzy closeness coefficient of each alternative a j is computed by using following formula, as described in Eq. (3.5). 



Ej =

De (z j , m H N I S ) ,  De (z j , m H PI S ) + De (z j , m H N I S )

j = 1, 2, . . . , p.

(3.5)

The alternative with highest m–polar hesitant fuzzy closeness coefficient is best one, and the ranking order of each alternative can be determined. In next subsections, the practical use of proposed model is discussed. In particular, it is shown how an m–polar hesitant fuzzy TOPSIS method is useful in the selection of a brand name and a product design for a company, respectively.

3.3.1 Selection of a Perfect Brand Name In this subsection, presented decision model is applied to a problem in strategic marketing; namely, the choice of a perfect brand name. This is one of the fundamental decisions when launching a new product into the market (especially if it is introduced under the umbrella of a new brand). A perfect brand name is not something that looks fine on a business card or a web banner, or is cool to say, or somebody likes it. It is perfect, when it conveys the right feelings to customers, from whom the demand for good brand names emanates. A good name can be the most prized property of a company. There exist many theories and have been many studies about what makes a good brand name, and common principles, which make a brand name simpler for the owner to use and easier for customers to remember, have also been established. Ideally, one must takes advantage of a fusion of information that derives from multipolar advice under hesitant directions. An m–polar hesitant fuzzy set handles all the

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

characteristics and tools to select a brand name, in terms of m different numeric values under hesitant situations, due to the guidelines of different decision makers or experts. For this purpose, B N = {Bn1 , Bn2 , Bn3 , Bn4 , Bn5 } is considered as the set of five different brand names and C = {c1 , c2 , c3 } as the set of three different evaluating criteria (or characteristics) of a brand name. Finding a good brand name can be exhausting, infuriating, and thrilling, but here are some criteria’s such as its articulate core identity, brainstorm and test which are further classified as: 1. “Articulate core identity”, which may include the following features: • • • •

The “Vision”, or why your company exists, the “Mission”, or what your company does, the “Value”, or how you do what you do, the “Direction”, or where it goes on.

2. “Brainstorm”, which may include the following features: • The “Founder”, a name based on a real or fictional person, • the “Description”, a name that describes what you do or make, • the “Magic spell”, a name that is a portmanteau (two words together) or a real word with a made-up spelling, • the “Fabricated”, a totally made-up name or word. 3. “Test”, which may include the following features: • • • •

“Sounds good”, it is good to hear, “Not confusing”, it is not linked with other brand names, “Not mispronounced”, it is easy to pronounce, “Related publicity”, it focuses on a targeted group of customers.

All these characteristics are assessed by a group of three different decision makers, who are responsible for evaluating the brand names. Due to their mutual decision, each criteria has a specified condition for further classification of described values into three hesitant values for any given candidate brand name. The decision makers have the authority to choose further membership values from the interval [0, 1]; their assigned values are described in Table 3.3. The desired computations of the 4–polar hesitant fuzzy elements are not proportionate with the 4–polar hesitant fuzzy-sets. In order to attain efficiency, the largest membership values are prolonged until the lengths of all 4–polar hesitant fuzzy elements become equal, as the company wants to base the perfect brand name on an optimistic spirit. For this reason, the information fusion shows an optimistic response and improves the 4–polar hesitant fuzzy data by adding the maximal values, as mentioned in Table 3.4. The weights that satisfy the normalized condition are given as w = (0.23, 0.34, 0.43). The weighted optimistic 4–polar hesitant fuzzy decision matrix is calculated in Table 3.5.

3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach Table 3.3 A 4–polar hesitant fuzzy decision matrix Brand names Articulate core identity Vision Mission Bn1 Bn2 Bn3 Bn4 Bn5 Brand names Bn1 Bn2 Bn3 Bn4 Bn5 Brand names

Bn1 Bn2 Bn3 Bn4 Bn5

{0.40,0.50} {0.60,0.70} {0.40} {0.70,0.80} {0.40,0.60} Brainstorm Founder {0.30,0.70} {0.10,0.20,0.30} {0.10,0.15} {0.40,0.50} {0.45,0.50} Test Sounds good {0.40,0.60} {0.30,0.40,0.60} {0.30,0.50,0.70} {0.20,0.50} {0.60,0.80,0.90}

175

Value

Direction

{0.30,0.60,0.70} {0.30,0.70} {0.40,0.50,0.80} {0.60,0.80} {0.55,0.70}

{0.20,0.70} {0.50,0.60} {0.20,0.30,0.50} {0.50} {0.40,0.50,0.70}

{0.30,0.70,0.80} {0.40,0.60,0.80} {0.60,0.80} {0.70,0.80,0.90} {0.75,0.80}

Descriptive {0.40,0.50,0.80} {0.50,0.60} {0.20,0.50} {0.65,0.70} {0.50,0.70}

Magic spell {0.60,0.80} {0.10,0.50} {0.40} {0.40,0.70} {0.10,0.20}

Fabricated {0.70,0.80} {0.60,0.80} {0.70,0.80,0.90} {0.70,0.80} {0.50,0.60,0.70}

Not confusing

Not mispronounced {0.30,0.50} {0.40,0.70} {0.60,0.90} {0.60,0.80} {0.70}

Related publicity

{0.70,0.80} {0.20} {0.50,0.80} {0.10,0.25} {0.50,0.60}

{0.60,0.70,0.90} {0.20,0.30,0.50} {0.50,0.70} {0.50,0.60,0.80} {0.20,0.30,0.35}

Use Eqs. (3.1) and (3.2) to determine the 4–polar hesitant fuzzy positive ideal solution and 4–polar hesitant fuzzy negative ideal solution, respectively:  {0.1610, 0.1840, 0.1840}, {0.1380, 0.1840, 0.1840}, {0.1150, 0.1610, 0.1610},  {0.1725, 0.1840, 0.2070} ,  {0.1530, 0.2380, 0.2380}, {0.2210, 0.2380, 0.2720}, {0.2040, 0.2720, 0.2720},  {0.2380, 0.2720, 0.3060} ,  {0.2580, 0.3440, 0.3870}, {0.3010, 0.3440, 0.3440}, {0.3010, 0.3870, 0.3870},  {0.2580, 0.3010, 0.3870} .  = {0.0920, 0.0920, 0.0920}, {0.0690, 0.1150, 0.1610}, {0.0460, 0.0690, 0.1150},

4H PI S =

4H N I S

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.4 Optimistic 4–polar hesitant fuzzy decision matrix Brand names Articulate core identity Vision Mission Value Bn1 Bn2 Bn3 Bn4 Bn5 Brand names Bn1 Bn2 Bn3 Bn4 Bn5 Brand names

Bn1 Bn2 Bn3 Bn4 Bn5

Direction

{0.40,0.50,0.50} {0.60,0.70,0.70} {0.40,0.40,0.40} {0.70,0.80,0.80} {0.40,0.60,0.60} Brainstorm Founder {0.30,0.70,0.70} {0.10,0.20,0.30} {0.10,0.15,0.15} {0.40,0.500.50} {0.45,0.50,0.50} Test Sounds good

{0.30,0.60,0.70} {0.30,0.70,0.70} {0.40,0.50,0.80} {0.60,0.80,0.80} {0.55,0.70,0.70}

{0.20,0.70,0.70} {0.50,0.60,0.60} {0.20,0.30,0.50} {0.50,0.50,0.50} {0.40,0.50,0.70}

{0.30,0.70,0.80} {0.40,0.60,0.80} {0.60,0.80,0.80} {0.70,0.80,0.90} {0.75,0.80,0.80}

Descriptive {0.40,0.50,0.80} {0.50,0.60,0.60} {0.20,0.50,0.50} {0.65,0.70,0.70} {0.50,0.70,0.70}

Magic spell {0.60,0.80,0.80} {0.10,0.50,0.50} {0.40,0.40,0.40} {0.40,0.70,0.70} {0.10,0.20,0.20}

Fabricated {0.70,0.80,0.80} {0.60,0.80,0.80} {0.70,0.80,0.90} {0.70,0.80,0.80} {0.50,0.60,0.70}

Not confusing

Related publicity

{0.40,0.60,0.60} {0.30,0.40,0.60} {0.30,0.50,0.70} {0.20,0.50,0.50} {0.60,0.80,0.90}

{0.70,0.80,0.80} {0.20,0.20,0.20} {0.50,0.80,0.80} {0.10,0.25,0.25} {0.50,0.60,0.60}

Not mispronounced {0.30,0.50,0.50} {0.40,0.70,0.70} {0.60,0.90,0.90} {0.60,0.80,0.80} {0.70,0.70,0.70}

{0.60,0.70,0.90} {0.20,0.30,0.50} {0.50,0.70,0.70} {0.50,0.60,0.80} {0.20,0.30,0.35}

 {0.0690, 0.1380, 0.1840} ,  {0.0340, 0.0510, 0.0510}, {0.0680, 0.1700, 0.1700}, {0.0340, 0.0680, 0.0680},  {0.1700, 0.2040, 0.2380} ,  {0.0860, 0.1720, 0.2150}, {0.0430, 0.0860, 0.0860}, {, 0.1290, 0.2150, 0.2150},  {0.0860, 0.1290, 0.1505} .

Use Eqs. (3.3) and (3.4) to calculate the 4–polar hesitant fuzzy Euclidean distances of the brand names from 4–polar hesitant fuzzy positive ideal solution and 4–polar hesitant fuzzy negative ideal solution , producing the following figures: 

De (Bn1 , 4H N I S ) = 0.2153,



De (Bn2 , 4H N I S ) = 0.0885,

De (Bn1 , 4H PI S ) = 0.1204, De (Bn2 , 4H PI S ) = 0.2045,





3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach

177

Table 3.5 Weighted optimistic 4–polar hesitant fuzzy decision matrix of Table 3.4 Brand names

Articulate core identity with weight 0.23 Vision

Mission

Value

Direction

Bn1

{0.0920,0.1150,0.1150}

{0.0690,0.1380,0.1610}

{0.0460,0.1610,0.1610}

{0.0690,0.1610,0.1840}

Bn2

{0.1380,0.1610,0.1610}

{0.0690,0.1610,0.1610}

{0.1150,0.1380,0.1380}

{0.0920,0.1380,0.1840}

Bn3

{0.0920,0.0920,0.0920}

{0.0920,0.1150,0.1840}

{0.0460,0.0690,0.1150}

{0.1380,0.1840,0.1840}

Bn4

{0.1610,0.1840,0.1840}

{0.1380,0.1840,0.1840}

{0.1150,0.1150,0.1150}

{0.1610,0.1840,0.2070}

Bn5

{0.0920,0.1380,0.1380}

{0.1265,0.1610,0.1610}

{0.0920,0.1150,0.1610}

{0.1725,0.1840,0.1840}

Brand names

Brainstorm with weight 0.34 Founder

Descriptive

Magic spell

Fabricated

Bn1

{0.1020,0.2380,0.2380}

{0.1360,0.1700,0.2720}

{0.2040,0.2720,0.2720}

{0.2380,0.2720,0.2720}

Bn2

{0.0340,0.0680,0.1020}

{0.1700,0.2040,0.2040}

{0.0340,0.1700,0.1700}

{0.2040,0.2720,0.2720}

Bn3

{0.0340,0.0510,0.0510}

{0.0680,0.1700,0.1700}

{0.1360,0.1360,0.1360}

{0.2380,0.2720,0.3060}

Bn4

{0.1360,0.1700,0.1700}

{0.2210,0.2380,0.2380}

{0.1360,0.2380,0.2380}

{0.2380,0.2720,0.2720}

Bn5

{0.1530,0.1700,0.1700}

{0.1700,0.2380,0.2380}

{0.0340,0.0680,0.0680}

{0.1700,0.2040,0.2380}

Brand names

Test with weight 0.43 Sounds good

Not confusing

Not mispronounced

Related publicity

Bn1

{0.1720,0.2580,0.2580}

{0.3010,0.3440,0.3440}

{0.1290,0.2150,0.2150}

{0.2580,0.3010,0.3870}

Bn2

{0.1290,0.1720,0.2580}

{0.0860,0.0860,0.0860}

{0.1720,0.3010,0.3010}

{0.0860,0.1290,0.2150}

Bn3

{0.1290,0.2150,0.3010}

{0.2150,0.3440,0.3440}

{0.2580,0.3870,0.3870}

{0.2150,0.3010,0.301}

Bn4

{0.0860,0.2150,0.2150}

{0.0430,0.1075,0.1075}

{0.2580,0.3440,0.3440}

{0.2150,0.2580,0.3440}

Bn5

{0.2580,0.3440,0.3870}

{0.2150,0.2580,0.2580}

{0.3010,0.3010,0.3010}

{0.0860,0.1290,0.1505}



De (Bn3 , 4H N I S ) = 0.1732,



De (Bn4 , 4H N I S ) = 0.1641,



De (Bn5 , 4H N I S ) = 0.1615.

De (Bn3 , 4H PI S ) = 0.1497, De (Bn4 , 4H PI S ) = 0.1550, De (Bn5 , 4H PI S ) = 0.1593,







Use Eq. (3.5), to calculate the relative 4–polar hesitant fuzzy closeness coefficients E j of the brand names:   E 2 = 0.3021, E 1 = 0.6413, 

E 3 = 0.5364,



E 4 = 0.5142,



E 5 = 0.5034. For the comparison, arrange the brand names {Bn j | j = 1, 2, . . . , 5} according to the ranking in the 4–polar hesitant fuzzy closeness coefficients; that is, Bn1 > Bn3 > Bn4 > Bn5 > Bn2 . Hence, Bn1 is the perfect brand name, according to this ranking.

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

3.3.2 Selection of Suitable Product Design for a Company In this subsection, a product design is focused, which is an action to determine the unique aspect and attributes of a product. The process of selection is also discussed, which is the development of techniques to manufacture the designed product, because these two processes are usually designed together. Product design and its selection process are induced by the quality of the product, its cost, and customer satisfaction. If the product design is not suitable and its manufacturing process is not appropriate, then the quality of the product may suffer. Further, products are composed and synthesized by using materials, machinery, and labor expertise, which should be valuable, productive, and profitable. It is called the product composition, according to which the product can be manufactured. Finally, if the product accomplishes customer satisfaction, it should have the associated features of good design, the capacity to fulfill the needs of a market, and competitive prices. This is actually true, whether the product is a car or pizza. An m–polar hesitant fuzzy set deals with all of the characteristics of a product, in terms of m different numeric or fuzzy values under hesitant situations by different decision makers or experts. For this purpose, PD = {Pd1 , Pd2 , Pd3 , Pd4 } is considered as a set of four different product designs and C = {c1 , c2 , c3 , c4 } as a set of four different evaluation criteria or characteristics of product design. Product design defines the aspects of a product, according to the specialized literature described in (www.google. com/search?q=selection+of+product+design) having characteristics such as appearance, the materials it is made of, its dimensions and tolerances, and its performance standards. These characteristics or criteria are further classified, as follows: 1. The “Appearance” of a product design may include the following features: • “Contrast and symmetry”, • “Color and shade”, • “Body texture and surface”. 2. The “Material” of a product design may include the following features: • “Fine quality”, • “Low cost”, • “Reversibility”. 3. The “Dimensions and Tolerances” of a product design may include the following features: • “Size and functions”, • “Flexibility”, • “Nominal geometry”. 4. The “Performance Standards” of a product design may include the following features: • “Market value”, • “Customer satisfaction”, • “Availability and evaluating report”.

3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach Table 3.6 3-polar hesitant fuzzy decision matrix Products design Appearance Contrast and Color and shade symmetry Pd1 Pd2 Pd3 Pd4 Products design Pd1 Pd2 Pd3 Pd4 Products design Pd1 Pd2 Pd3 Pd4 Products design

Pd1 Pd2 Pd3 Pd4

{0.25,0.45,0.47} {0.30,0.31,0.36} { 0.46,0.48,0.49} {0.47,0.49} { 0.51,0.53,0.57,0.60} {0.46,0.52,0.70} { 0.39,0.41,0.43} {0.60,0.68,0.71,0.73} Material Fine quality Low cost {0.45,0.49,0.51,0.59} {0.67,0.68,0.71} { 0.49,0.50} {0.71,0.74,0.79} { 0.71,0.73,0.77} {0.46,0.52,0.70} { 0.53,0.54,0.56,0.58} {0.60,0.63,0.73,0.79} Dimension and tolerance Size and functions Flexibility {0.85,0.86,0.87} {0.53,0.59,0.66} { 0.66,0.68,0.69} {0.47,0.50,51,0.64} { 0.51,0.55} {0.66,0.68,0.75,0.76} { 0.59,0.61,0.73,0.74} {0.26,0.38,0.41,0.43} Performance standards Market value Customer satisfaction {0.55,0.65} { 0.54,0.58,0.59,0.61} { 0.81,0.83,0.87} { 0.37,0.48,0.49,0.59}

{0.40,0.48,0.60,0.61} {0.77,0.79,0,84} {0.56,0.62,0.70} {0.26,0.38,0.41,0.43}

179

Body texture and surface {0.20,0.25,0.26} {0.55,0.60,0.61,0.63} {0.29,0.30,0.51,0.52} {0.50,0.67,0.69} Reversibility {0.50,0.56,0.63,0.64} {0.35,0.59,0.61,0.65} {0.29,0.30,0.51,0.52} {0.40,0.47,0.49} Nominal geometry {0.72,0.75,0.76,0.78} {0.65,0.66,0.81} {0.39,0.40,0.58,0.62} {0.51,0.77} Availability and evaluating report {0.80,0.85,0.86} {0.55,0.60,0.68} {0.69,0.70,0.76,0.82} {0.60,0.67}

In example, all these characteristics are assessed by a group of four different experts or decision makers, who are responsible for evaluating the product designs. Due to their mutual decision, each criteria has a specified condition for further classification of described values into four hesitant values for any single product design. The values assigned by the decision makers are given in Table 3.6. The desired computations of the 3-polar hesitant fuzzy elements are not proportionate with the 3-polar hesitant fuzzy sets. In order to attain efficiency, the smallest membership values are prolonged until the lengths of all 3-polar hesitant fuzzy elements become equal as, in this case, the company wants to base the best product design on a pessimistic decision. Now, the information fusion is pessimistically responsive and, the 3-polar hesitant fuzzy data is reformed by adding the minimal values, as given in Table 3.7. The weights that satisfy the normalized condition are given as w = (0.2012, 0.2259, 0.2631, 0.3098).

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.7 Pessimistic 3-polar hesitant fuzzy decision matrix Products design Appearance Contrast and Color and shade symmetry Pd1 Pd2 Pd3 Pd4 Products design Pd1 Pd2 Pd3 Pd4 Products design Pd1 Pd2 Pd3 Pd4 Products design

Pd1 Pd2 Pd3 Pd4

{0.25,0.25,0.45,0.47} {0.30,0.30,0.31,0.36} { 0.46,0.46,0.48,0.49} {0.47,0.47,0.47,0.49} { 0.51,0.53,0.57,0.60} {0.46,0.46,0.52,0.70} { 0.39,0.39,0.41,0.43} {0.60,0.68,0.71,0.73} Material Fine quality Low cost {0.45,0.49,0.51,0.59} {0.67,0.67,0.68,0.71} { 0.49,0.49,0.49,0.50} {0.71,0.71,0.74,0.79} { 0.71,0.71,0.73,0.77} {0.46,0.46,0.52,0.70} { 0.53,0.54,0.56,0.58} {0.60,0.63,0.73,0.79} Dimension and tolerance Size and functions Flexibility {0.85,0.85,0.86,0.87} {0.53,0.53,0.59,0.66} { 0.66,0.66,0.68,0.69} {0.47,0.50,0.51,0.64} { 0.51,0.51,0.51,0.55} {0.66,0.68,0.75,0.76} { 0.59,0.61,0.73,0.74} {0.26,0.38,0.41,0.43} Performance standards Market value Customer satisfaction {0.55,0.55,0.55,0.65} { 0.54,0.58,0.59,0.61} { 0.81,0.81,0.83,0.87} { 0.37,0.48,0.49,0.59}

{0.40,0.48,0.60,0.61} {0.77,0.77,0.79,0,84} {0.56,0.56,0.62,0.70} {0.26,0.38,0.41,0.43}

Body texture and Surface {0.20,0.20,0.25,0.26} {0.55,0.60,0.61,0.63} {0.29,0.30,0.51,0.52} {0.50,0.50,0.67,0.69} Reversibility {0.50,0.56,0.63,0.64} {0.35,0.59,0.61,0.65} {0.29,0.30,0.51,0.52} {0.40,0.40,0.47,0.49} Nominal geometry {0.72,0.75,0.76,0.78} {0.65,0.65,0.66,0.81} {0.39,0.40,0.58,0.62} {0.51,0.51,0.51,0.77} Availability and evaluating report {0.80,0.80,0.85,0.86} {0.55,0.55,0.60,0.68} {0.69,0.70,0.76,0.82} {0.60,0.60,0.60,0.67}

The weighted pessimistic 3-polar hesitant fuzzy decision matrix is calculated in Table 3.8. Use Eqs. (3.1) and (3.2) to determine the positive ideal solution and negative ideal solution, respectively. 3H PI S =

 {0.1026, 0.1066, 0.1147, 0.1207}, {0.1207, 0.1368, 0.1429, 0.1469},  {0.1107, 0.1207, 0.1348, 0.1388} ,  {0.1604, 0.1604, 0.1649, 0.1739}, {0.1604, 0.1604, 0.1672, 0.1785},  {0.1129, 0.1333, 0.1423, 0.1468} ,  {0.2236, 0.2236, 0.2263, 0.2289}, {0.1736, 0.1789, 0.1973, 0.2000},

3.3 An m–Polar Hesitant Fuzzy TOPSIS Approach

181

Table 3.8 Weighted pessimistic 3-polar hesitant fuzzy decision matrix Products design

Appearance with weight 0.2012 Contrast and symmetry

Color and shade

Body texture and surface

Pd1

{0.0503,0.0503,0.0905,0.0946}

{0.0604,0.0604,0.0624,0.0724}

{0.0402,0.0402,0.0503,0.0523}

Pd2

{0.0926,0.0926,0.0966,0.0986}

{0.0946,0.0946,0.0946,0.0986}

{0.1107,0.1207,0.1227,0.1268}

Pd3

{0.1026,0.1066,0.1147,0.1207}

{0.0926,0.0926,0.1046,0.1408}

{0.0583,0.0604,0.1026,0.1046}

Pd4

{0.0785,0.0785,0.0825,0.0865}

{0.1207,0.1368,0.1429,0.1469}

{0.1006,0.1006,0.1348,0.1388}

Products design

Material with weight 0.2259 Fine quality

Low cost

Reversibility

Pd1

{0.1017,0.1107,0.1152,0.1333}

{0.1514,0.1514,0.1536,0.1604}

{0.1129,0.1265,0.1423,0.1446}

Pd2

{0.1107,0.1107,0.1107,0.1129}

{0.1604,0.1604,0.1672,0.1785}

{0.0791,0.1333,0.1378,0.1468}

Pd3

{0.1604,0.1604,0.1649,0.1739 }

{0.1039,0.1039,0.1175,0.1581}

{0.0655,0.0678,0.1152,0.1175}

Pd4

{0.1197,0.1220,0.1265,0.1310}

{0.1355,0.1423,0.1649,0.1785}

{0.0904,0.0904,0.1062,0.1107}

Products design

Dimension and tolerance with weight 0.2631 Size and functions

Flexibility

Nominal geometry

Pd1

{0.2236,0.2236,0.2263,0.2289}

{0.1394,0.1394,0.1552,0.1736}

{0.1894,0.1973,0.2000, 0.2052}

Pd2

{0.1736,0.1736,0.1789,0.1815}

{0.1237,0.1316,0.1342,0.1684}

{0.1710,0.1710,0.1736, 0.2131}

Pd3

{0.1342,0.1342,0.1342,0.1447}

{0.1736,0.1789,0.1973,0.2000}

{0.1026,0.1052,0.1526, 0.1631}

Pd4

{0.1552,0.1605,0.1921,0.1947}

{0.0684,0.1000,0.1079,0.1131}

{0.1342,0.1342,0.1342, 0.2026}

Products design

Performance standards with weight 0.3098 Market value

Customer satisfaction

Availability and evaluating report

Pd1

{0.1704,0.1704,0.1704,0.2014}

{0.1239,0.1487,0.1859,0.1890}

{0.2478,0.2478,0.2633, 0.2664}

Pd2

{0.1673,0.1797,0.1828,0.1890}

{0.2385,0.2385,0.2447,0.2602}

{0.1704,0.1704,0.1859, 0.2107}

Pd3

{0.2509,0.2509,0.2571,0.2695}

{0.1735,0.1735,0.1921,0.2169}

{0.2138,0.2169,0.2354, 0.2540}

Pd4

{0.1146,0.1487,0.1518,0.1828}

{0.0805,0.1177,0.1270,0.1332}

{0.1859,0.1859,0.1859, 0.2076}

3H N I S

 {0.1894, 0.1973, 0.2000, 0.2131} ,  {0.2509, 0.2509, 0.2571, 0.2695}, {0.2385, 0.2385, 0.2447, 0.2602},  {0.2478, 0.2478, 0.2633, 0.2664} .  = {0.0503, 0.0503, 0.0825, 0.0865}, {0.0604, 0.0604, 0.0624, 0.0724},  {0.0402, 0.0402, 0.0503, 0.0523} ,  {0.1017, 0.1107, 0.1107, 0.1129}, {0.1039, 0.1039, 0.1175, 0.1581},  {0.0655, 0.0678, 0.1062, 0.1107} ,  {0.1342, 0.1342, 0.1342, 0.1447}, {0.0684, 0.1000, 0.1079, 0.1131},  {0.1026, 0.1052, 0.1342, 0.1631} ,

182

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models  {0.1146, 0.1487, 0.1518, 0.1828}, {0.0805, 0.1177, 0.1270, 0.1332},  {0.1704, 0.1704, 0.1859, 0.2076} .

Use Eqs. (3.3) and (3.4) to calculate the 3-polar hesitant fuzzy Euclidean distances of the product designs from 3-polar hesitant fuzzy positive ideal solution and 3-polar hesitant fuzzy negative ideal solution. They produce the following figures: 

De (Pd1 , 3H N I S ) = 0.0983,



De (Pd2 , 3H N I S ) = 0.1110,



De (Pd3 , 3H N I S ) = 0.1111,



De (Pd4 , 3H N I S ) = 0.0725.

De (Pd1 , 3H PI S ) = 0.1023, De (Pd2 , 3H PI S ) = 0.0854, De (Pd3 , 3H PI S ) = 0.0907, De (Pd4 , 3H PI S ) = 0.1311,









Use Eq. (3.5), to calculate the relative 3-polar hesitant fuzzy closeness coefficients E j of the product designs: 

E 2 = 0.5654,



E 4 = 0.3563.

E 1 = 0.4899, E 3 = 0.5504,





For the comparison, arrange the product designs {Pd j | j = 1, 2, . . . , 4} according to the ranking of their 3-polar hesitant fuzzy closeness coefficients; that is, Pd2 > Pd3 > Pd1 > Pd4 . Hence, the product design Pd2 is selected for manufacture. An Algorithm 3.3.1 of m–polar hesitant fuzzy TOPSIS approach is presented as follows: Algorithm 3.3.1 An m–polar hesitant fuzzy TOPSIS approach for MCGDM 1. Input p as a number of alternatives against m–polar hesitant fuzzy information, q as a number of criteria, m as a number of poles, according to characteristics, and r as a number of membership values due to hesitancy. 2. Input Dl as m–polar fuzzy linguistic decision matrices according to decision makers, and w as weight vector, according to decision makers. 3. Compute an m–polar hesitant fuzzy decision matrix H .  4. Compute the weighted m–polar hesitant fuzzy decision matrix H . 5. Compute the solution and m–polar hesitant fuzzy positive ideal solution. 6. Compute the m–polar hesitant fuzzy negative ideal solution.

3.4 An m−Polar Hesitant Fuzzy ELECTRE-I Approach

183

7. Compute the m–polar hesitant fuzzy distances of alternatives from m–polar hesitant fuzzy positive ideal solution and m–polar hesitant fuzzy negative ideal solution. 8. Compute the relative m–polar hesitant fuzzy closeness coefficients. 9. Rank the alternatives for final decision and select the best one.

3.4 An m−Polar Hesitant Fuzzy ELECTRE-I Approach In this approach, Z = {z 1 , z 2 , . . . , z p } is chosen as the set of alternatives and C = {Ck |k = 1, 2, . . . , q} as the set of criteria, which are further classified by the m–polar fuzzy information in terms of hesitancy. The structure of the m–polar hesitant fuzzy ELECTRE-I method, and steps from (i) to (iii) are same as described in Sect. 3.3. (iv) The m–polar hesitant fuzzy concordance set is defined as uk (z) ≥ evk (z), u  = v; u, v = 1, 2, . . . , p}, Yuv = {1 ≤ k ≤ q|em m jk

where em (z) =

 h



jk

ζh ∈ p1 ◦ em (z) +

 h



jk

ζh ∈ p2 ◦ em (z) + · · · +

 h



ζh ∈

jk

pm ◦ em (z). (v) The m–polar hesitant fuzzy concordance indices are determined as yuv =



wk ,

k∈Yuv

therefore, the m–polar hesitant fuzzy concordance matrix is computed as ⎛

⎞ · · · y1 p · · · y2 p ⎟ ⎟ · · · y3 p ⎟ ⎟. . ⎟ · · · .. ⎠ ··· −

− y12 y13 ⎜ y21 − y23 ⎜ ⎜ Y = ⎜ y31 y32 − ⎜ .. .. .. ⎝ . . . y p1 y p2 y p3

(vi) The m–polar hesitant fuzzy discordance set is defined as uk (z) ≤ evk (z), u  = v; u, v = 1, 2, . . . , p}, Z uv = {1 ≤ k ≤ q|em m jk

where em (z) =

 h

jk

pm ◦ em (z).



jk

ζh ∈ p1 ◦ em (z) +

 h



jk

ζh ∈ p2 ◦ em (z) + · · · +

 h



ζh ∈

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

(vii) The m–polar hesitant fuzzy discordance indices are determined as  max

z uv =

k∈Z uv



max k



1 rm

m  

 uk  − ζ vk  )2 + (ζ uk  − ζ vk  )2 + · · · + (ζ uk  − ζ vk  )2 (ζh1 h1 h2 h2 hr hr i=1

m   ,  1 uk  − ζ vk  )2 + (ζ uk  − ζ vk  )2 + · · · + (ζ uk  − ζ vk  )2 (ζ rm h1 h1 h2 h2 hr hr i=1

jk

where ζhluk ∈ pi ◦ em (z), ∀i ∈ m and l = {1, 2, . . . , r }. Therefore, the m–polar hesitant fuzzy discordance matrix can be computed as ⎛ ⎞ − z 12 z 13 · · · z 1 p ⎜ z 21 − z 23 · · · z 2 p ⎟ ⎜ ⎟ ⎜ ⎟ Z = ⎜ z 31 z 32 − · · · z 3 p ⎟ . ⎜ .. .. .. .. ⎟ ⎝ . . . ··· . ⎠ z p1 z p2 z p3 · · · − (viii) For the rankings of alternatives the threshold values are computed, known as m–polar hesitant fuzzy concordance and discordance levels. The m–polar hesitant fuzzy concordance and discordance levels are the average of m–polar hesitant fuzzy concordance and discordance indices.  1 yuv , p( p − 1) u=1 v=1 p

y¯ =

p

u=v u=v

 1 z uv . p( p − 1) u=1 v=1 p

z¯ =

p

u=v u=v

(ix) The m–polar hesitant fuzzy concordance dominance matrix according to its m–polar hesitant fuzzy concordance level is computed as ⎛

− r12 r13 ⎜ r21 − r23 ⎜ ⎜ R = ⎜ r31 r32 − ⎜ .. .. .. ⎝ . . . r p1 r p2 r p3 where,

 ruv =

⎞ · · · r1 p · · · r2 p ⎟ ⎟ · · · r3 p ⎟ ⎟, .. ⎟ ··· . ⎠ ···

1, yuv ≥ y¯ ; 0, yuv < y¯ .

−

3.4 An m−Polar Hesitant Fuzzy ELECTRE-I Approach

185

(x) The m–polar hesitant fuzzy discordance dominance matrix according to its m–polar hesitant fuzzy discordance level is computed as ⎛ ⎞ − s12 s13 · · · s1 p ⎜ s21 − s23 · · · s2 p ⎟ ⎜ ⎟ ⎜ ⎟ S = ⎜ s31 s32 − · · · s3 p ⎟ , ⎜ .. .. .. .. ⎟ ⎝ . . . ··· . ⎠ s p1 s p2 s p3 · · · − where,

 suv =

1, z uv < z¯ ; 0, z uv ≥ z¯ .

(xi) The aggregated m–polar hesitant fuzzy dominance matrix is computed as ⎛

− t12 t13 ⎜ t21 − t23 ⎜ ⎜ T = ⎜ t31 t32 − ⎜ .. .. .. ⎝ . . . t p1 t p2 t p3 where, tuv is defined as

⎞ · · · t1 p · · · t2 p ⎟ ⎟ · · · t3 p ⎟ ⎟, .. ⎟ ··· . ⎠ ··· −

tuv = ruv suv .

(xii) Finally, rank the alternatives according to the outranking values of matrix T . For each pair of alternatives there exist a directed edge from alternative z u to z v if and only if tuv = 1. Thus, the following three cases arises. a. There exists a unique directed edge from z u to z v , which shows z u is preferred over z v . b. There exists directed edges from z u to z v and z v to z u , which shows z u and z v are indifferent. c. There does not exist any edge between z u and z v , which shows z u and z v are not comparable.

3.4.1 Selection of Bricks for Construction The bricks selection is significant in the sense, that it regulates a project’s constancy and presentation, and crops in a durable impact. It is crucial to analyze and classify which criteria or properties of bricks are convenient to acknowledge in choosing the best bricks. Bricks having vast variety of size, color, strength, texture and shape are accessible. The designer, owner and engineers have to decide which aspects and

186

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

attributes of bricks are most demanding. This procedure of selection can precept the eminence and accomplishment of any project. An m–polar hesitant fuzzy set discusses the criteria or properties which are acknowledged in the selection of the convenient bricks for a project under the hesitant decision of project designers or engineers. Selection of bricks is based on several factors and criteria. It does not only depend upon durability importance, but absorption, strength, cost and availability are important to the designers, owner and contractors. The selection process may be challenging and tough since each group is trying to entertain several requirements. Generally, the ultimate selection depends upon the adjustment of all the including parties. To apply the concept of m–polar hesitant fuzzy model in real life situation, Br = {Br1 , Br2 , Br3 , Br4 , Br5 , Br6 } is considered as the set of six different types of bricks which have to be analyzed and C = {c1 , c2 , c3 } as the set of three main criteria or properties to select the bricks for the construction. For the evaluation the project dealers including owner, designer, constructor and engineer focus on three main criteria or properties of bricks such as physical properties, mechanical properties and durability, which are further classified into three different sub-criteria as 1. The “Physical Properties” may include • “Shape”, normally an ideal brick has absolutely rectangular shape. Its edges are sharp, well defined and having even and regular surface. • “Size and Color”, in construction the practiced size of brick differs from place to place and from country to country, where as the color of bricks may vary from dull red to light red and from buff to purple. • “Density”, the weight per unit volume or the density of bricks mostly depends on the process of brick molding and type of clay used to prepare it. 2. The “Mechanical Properties” may include • “Compressive Strength”, it is the highest considerable and crucial estate of bricks specifically when they are utilized in load-bearing walls, it depends on the degree of burning and formation of the clay. • “Flexure Strength”, usually bricks are utilized in directions and stages where tilting and twisting loads are feasible in a building. In essence, they maintain satisfactory strength across transverse loads. • “Slenderness Ratio”, in turn it depends upon the effective height, length and thickness of the wall or column. 3. The “Durability” may include • “Absorption Value”, this estate is depicted to the brick porosity. True Porosity is described as the rate of the volume of pores to the gross volume of the sample of the substance. • “Frost Resistance”, when bricks are utilized in cold climates, their decomposition due to this phenomenon of frost action may be a common process. therefore, it is significant that bricks in such areas should be accordingly protected from rain to decrease absorption.

3.4 An m−Polar Hesitant Fuzzy ELECTRE-I Approach

187

• “Efflorescence”, it is a natural distorting and depreciating process of bricks in humid and hot climates. All these criteria or properties are assessed by a group of project dealers, who are responsible for evaluating the best bricks for construction. Due to their collective decision each criterion is further classified by three sub criteria, which are evaluated by four different hesitant values assigned by project dealers. Project dealers assign hesitant values as described in Table 3.9. Obviously, the count of 3-polar hesitant fuzzy elements in general is not comparable in all 3-polar hesitant fuzzy sets. In order to gain efficiency and accuracy, the largest membership values are extended as far as the lengths of all 3-polar hesitant fuzzy elements become equal, because the required company wants to take bricks of class one on an optimistic spirit. For this reason, an optimistic response is shown, which improves the 3-polar hesitant fuzzy data by adding the maximal values as mentioned in Table 3.10. (i) Tabular representation of 3-polar hesitant fuzzy decision matrix is given by Table 3.9. (ii) The normalized weights assigned to each criteria are given as follows: wl = (0.234, 0.395, 0.371). (iii) The weighted optimistic 3-polar hesitant fuzzy decision matrix is calculated in Table 3.11. (iv) The 3-polar hesitant fuzzy concordance set is calculated in Table 3.12. (v) The 3-polar hesitant fuzzy concordance matrix is calculated as follows: ⎛

− ⎜ 0.2340 ⎜ ⎜ 1.0000 Y =⎜ ⎜ 0.6050 ⎜ ⎝ 1.0000 1.0000

0.7660 − 0.7660 0.7660 0.7660 0.7660

0.0000 0.2340 − 0.0000 0.3710 0.0000

0.3950 0.2340 1.0000 − 0.7660 1.0000

0.0000 0.2340 0.6290 0.2340 − 0.6290

⎞ 0.0000 0.2340 ⎟ ⎟ 1.0000 ⎟ ⎟. 0.0000 ⎟ ⎟ 0.3710 ⎠ −

(vi) The 3-polar hesitant fuzzy discordance set is calculated in Table 3.13. (vii) The 3-polar hesitant fuzzy discordance matrix is calculated as follows: ⎛

− ⎜ 1.0000 ⎜ ⎜ 0.0000 Z =⎜ ⎜ 1.0000 ⎜ ⎝ 0.0000 0.0000

0.6832 − 0.6075 0.4833 0.6978 0.4596

1.0000 1.0000 − 1.0000 0.7181 1.0000

0.7801 1.0000 0.0000 − 0.3702 0.0000

1.0000 1.0000 1.0000 1.0000 − 1.0000

⎞ 1.0000 1.0000 ⎟ ⎟ 0.0000 ⎟ ⎟. 1.0000 ⎟ ⎟ 0.6990 ⎠ −

(viii) The 3-polar hesitant fuzzy concordance level y = 0.5000, and 3-polar hesitant fuzzy discordance level z = 0.6833 are calculated.

188

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.9 3-polar hesitant fuzzy decision matrix Bricks Physical properties Shape Size and color Br1 Br2 Br3 Br4 Br5 Br6 Bricks Br1 Br2 Br3 Br4 Br5 Br6 Bricks Br1 Br2 Br3 Br4 Br5 Br6

{0.55,0.57,0.67,0.69} {0.46,0.58,0.59} {0.51,0.63,0.77,0.80} {0.39,0.41,0.53} {0.49,0.56} {0.50,0.61,0.63} Mechanical properties Compressive strength {0.65,0.66,0.69} {0.49,0.53,0.58,0.60} {0.61,0.73} {0.35,0.47,0.53,0.67} {0.59,0.61,0.63} {0.49,0.60,0.63,0.70} Durability Absorption value {0.25,0.36,0.37,0.40} {0.46,0.48,0.49} {0.31,0.33,0.45,0.46} {0.29,0.31} {0.49,0.51,0.53,0.56} {0.39,0.41,0.43}

Density

{0.39,0.46,0.66} {0.77,0.79,0.80,0.91} {0.66,0.72} {0.61,0.65,0.81,0.83} {0.60,0.65,0.71} {0.70,0.78}

{0.57,0.65,0.66} {0.70,0.75} {0.59,0.60,0.71,0.82} {0.54,0.65,0.69} {0.48,0.57,0.68,0.71} {0.47,0.67}

Flexure strength {0.40,0.61,0.66,0.70} {0.47,0.59} {0.56,0.58,0.70} {0.50,0.58,0.61,0.71} {0.61,0.68,0.73} {0.62,0.65,0.71}

Slenderness ratio {0.66,0.68,0.70} {0.55,0.62,0.67,0.69} {0.61,0.72} {0.54,0.64,0.69} {0.60,0.69} {0.60,0.77,0.79,0.80}

Frost resistance {0.73,0.74,0.76} {0.47,0.49,0.51,0.56} {0.66,0.68,0.70} {0.60,0.68} {0.60,0.68,0.71,0.73} {0.56,0.68,0.73,0.83}

Efflorescence {0.45,0.55,0.56,0.58} {0.55,0.61,0.66} {0.51,0.72} {0.60,0.67,0.69,0.73} {0.67,0.69} {0.50,0.56,0.67,0.69}

(ix) The 3-polar hesitant fuzzy concordance dominance matrix is calculated as follows: ⎞ ⎛ − 1 0 0 0 0 ⎜0 − 0 0 0 0⎟ ⎟ ⎜ ⎜1 1 − 1 1 1⎟ ⎟ R=⎜ ⎜ 1 1 0 − 0 0 ⎟. ⎟ ⎜ ⎝1 1 0 1 − 0⎠ 1 1 0 1 1 − (x) he 3-polar hesitant fuzzy discordance dominance matrix is calculated as follows:

3.4 An m−Polar Hesitant Fuzzy ELECTRE-I Approach

189

Table 3.10 Optimistic 3-polar hesitant fuzzy decision matrix Bricks Physical properties Shape Size and color Br1 Br2 Br3 Br4 Br5 Br6 Bricks Br1 Br2 Br3 Br4 Br5 Br6 Bricks Br1 Br2 Br3 Br4 Br5 Br6

{0.55,0.57,0.67,0.69} {0.46,0.58,0.59,0.59} {0.51,0.63,0.77,0.80} {0.39,0.41,0.53,0.53} {0.49,0.56,0.56,0.56} {0.50,0.61,0.63,0.63} Mechanical properties Compressive strength {0.65,0.66,0.69,0.69} {0.49,0.53,0.58,0.60} {0.61,0.73,0.73,0.73} {0.35,0.47,0.53,0.67} {0.59,0.61,0.63,0.63} {0.49,0.60,0.63,0.70} Durability Absorption value {0.25,0.36,0.37,0.40} {0.46,0.48,0.49,0.49} {0.31,0.33,0.45,0.46} {0.29,0.31,0.31,0.31} {0.49,0.51,0.53,0.56} {0.39,0.41,0.43,0.43}

⎛

− ⎜0 ⎜ ⎜1 S=⎜ ⎜0 ⎜ ⎝1 1 (xi) An aggregated 3-polar hesitant lows: ⎛ − ⎜0 ⎜ ⎜1 T =⎜ ⎜0 ⎜ ⎝1 1

1 − 1 1 0 1

Density

{0.39,0.46,0.66,0.66} {0.77,0.79,0.80,0.91} {0.66,0.72,0.72,0.72} {0.61,0.65,0.81,0.83} {0.60,0.65,0.71,0.71} {0.70,0.78,0.78,0.78}

{0.57,0.65,0.66,0.66} {0.70,0.75,0.75,0.75} {0.59,0.60,0.71,0.82} {0.54,0.65,0.69,0.69} {0.48,0.57,0.68,0.71} {0.47,0.67,0.67,0.67}

Flexure strength {0.40,0.61,0.66,0.70} {0.47,0.59,0.59,0.59} {0.56,0.58,0.70,0.70} {0.50,0.58,0.61,0.71} {0.61,0.68,0.73,0.73} {0.62,0.65,0.71,0.71}

Slenderness ratio {0.66,0.68,0.70,0.70} {0.55,0.62,0.67,0.69} {0.61,0.72,0.72,0.72} {0.54,0.64,0.69,0.69} {0.60,0.69,0.69,0.69} {0.60,0.77,0.79,0.80}

Frost resistance {0.73,0.74,0.76,0.76} {0.47,0.49,0.51,0.56} {0.66,0.68,0.70,0.70} {0.60,0.68,0.68,0.68} {0.60,0.68,0.71,0.73} {0.56,0.68,0.73,0.83}

Efflorescence {0.45,0.55,0.56,0.58} {0.55,0.61,0.66,0.66} {0.51,0.72,0.72,0.72} {0.60,0.67,0.69,0.73} {0.67,0.69,0.69,0.69} {0.50,0.56,0.67,0.69}

0 0 − 0 0 0

0 0 1 − 1 1

0 0 0 0 − 0

⎞ 0 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 0⎠ −

fuzzy dominance matrix is calculated as fol⎞ 1 0 0 0 0 − 0 0 0 0⎟ ⎟ 1 − 1 0 1⎟ ⎟. 1 0 − 0 0⎟ ⎟ 0 0 1 − 0⎠ 1 0 1 0 −

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.11 Weighted optimistic 3-polar hesitant fuzzy decision matrix Bricks

Physical properties with weight 0.234 Shape

Size and color

Density

Br1

{0.1287,0.1334,0.1568,0.1615}

{0.0913,0.1076,0.1544,0.1544}

{0.1334,0.1521,0.1544,0.1544}

Br2

{0.1076,0.1357,0.1381,0.1381}

{0.1802,0.1849,0.1872,0.2129}

{0.1638,0.1755,0.1755,0.1755}

Br3

{0.1193,0.1474,0.1802,0.1872}

{0.1544,0.1685,0.1685,0.1685}

{0.1381,0.1404,0.1661,0.1919}

Br4

{0.0913,0.0959,0.1240,0.1240}

{0.1427,0.1521,0.1895,0.1942}

{0.1264,0.1521,0.1615,0.1615}

Br5

{0.1147,0.1310,0.1310,0.1310}

{0.1404,0.1521,0.1661,0.1661}

{0.1123,0.1334,0.1591,0.1661}

Br6

{0.1170,0.1427,0.1474,0.1474}

{0.1638,0.1825,0.1825,0.1825}

{0.1100,0.1568,0.1568,0.1568}

Bricks

Mechanical properties with weight 0.395 Compressive strength

Flexure strength

Slenderness ratio

Br1

{0.2568,0.2607,0.2726,0.2726}

{0.1580,0.2410,0.2607,0.2765}

{0.2607,0.2686,0.2765,0.2765}

Br2

{0.1936,0.2094,0.2291,0.2370}

{0.1857,0.2331,0.2331,0.2331}

{0.2173,0.2449,0.2647,0.2726}

Br3

{0.2410,0.2884,0.2884,0.2884}

{0.2212,0.2291,0.2765,0.2765}

{0.2410,0.2844,0.2844,0.2844}

Br4

{0.1382,0.1857,0.2094,0.2647}

{0.1975,0.2291,0.2410,0.2804}

{0.2133,0.2528,0.2726,0.2726}

Br5

{0.2331,0.2410,0.2489,0.2489}

{0.2410,0.2686,0.2884,0.2884}

{0.2370,0.2726,0.2726,0.2726}

Br6

{0.1936,0.2370,0.2489,0.2765}

{0.2449,0.2568,0.2804,0.2804}

{0.2370,0.3042,0.3121,0.3160}

Bricks

Durability with weight 0.371 Absorption value

Frost resistance

Efflorescence

Br1

{0.0927,0.1336,0.1373,0.1484}

{0.2708,0.2745,0.2820,0.2820}

{0.1670,0.2041,0.2078,0.2152}

Br2

{0.1707,0.1781,0.1818,0.1818}

{0.1744,0.1818,0.1892,0.2078}

{0.2041,0.2263,0.2449,0.2449}

Br3

{0.1150,0.1224,0.1670,0.1707}

{0.2449,0.2523,0.2597,0.2597}

{0.1892,0.2671,0.2671,0.2671}

Br4

{0.1076,0.1150,0.1150,0.1150}

{0.2226,0.2523,0.2523,0.2523}

{0.2226,0.2486,0.2560,0.2708}

Br5

{0.1818,0.1892,0.1966,0.2078}

{0.2226,0.2523,0.2634,0.2708}

{0.2486,0.2560,0.2560,0.2560}

Br6

{0.1447,0.1521,0.1595,0.1595}

{0.2078,0.2523,0.2708,0.3079}

{0.1855,0.2078,0.2486,0.2560}

Table 3.12 3-polar hesitant fuzzy concordance set v 1 2 3 Y1v Y2v Y3v Y4v Y5v Y6v

− {1} {1, 2, 3} {1, 3} {1, 2, 3} {1, 2, 3}

{2, 3} − {2, 3} {2, 3} {2, 3} {2, 3}

{} {1} − {} {3} {}

Table 3.13 3-polar hesitant fuzzy discordance set v 1 2 3 Z 1v Z 2v Z 3v Z 4v Z 5v Z 6v

− {2, 3} {} {2} {} {}

{1} − {1} {1} {1} {1}

{1, 2, 3} {2, 3} − {1, 2, 3} {1, 2} {1, 2, 3}

4

5

6

{2} {1} {1, 2, 3} − {2, 3} {1, 2, 3}

{} {1} {1, 2} {1} − {1, 2}

{} {1} {1, 2, 3} {} {3} −

4

5

6

{1, 3} {2, 3} {} − {1} {}

{1, 2, 3} {2, 3} {3} {2, 3} − {3}

{1, 2, 3} {2, 3} {} {1, 2, 3} {1, 2} −

3.4 An m−Polar Hesitant Fuzzy ELECTRE-I Approach

191

Fig. 3.1 Outranking relation of bricks

Br1

Br2

Br6 Br3

Br4

Br5

(xii) According to outranking values of aggregated 3-polar hesitant fuzzy dominance matrix the bricks have the following relations, as shown in Fig. 3.1. Hence, Br3 category of bricks have the most outranking value as compared to others, and selected for construction. The whole procedure is summarized, and its comparison is shown in Table 3.14. An Algorithm 3.4.1 of m–polar hesitant fuzzy ELECTRE-I approach is presented as follows: Algorithm 3.4.1 An m–polar hesitant fuzzy ELECTRE-I approach for MCGDM 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Input the same data as described in Algorithm 3.3.1. Compute an aggregated m–polar hesitant fuzzy decision matrix D.  Compute aggregated weights W . Compute the weighted aggregated m–polar hesitant fuzzy decision matrix W . Compute m–polar hesitant fuzzy concordance sets Yuv . Compute m–polar hesitant fuzzy concordance indices yuv and concordance matrix Y . Compute m–polar hesitant fuzzy discordance sets Z uv . Compute m–polar hesitant fuzzy discordance indices z uv and discordance matrix Z. Compute m–polar hesitant fuzzy concordance and discordance levels y¯ and z¯ . Compute m–polar hesitant fuzzy concordance dominance matrix R. Compute m–polar hesitant fuzzy discordance dominance matrix S. Compute aggregated m–polar hesitant fuzzy dominance matrix T . Identify the most dominating alternative.

192

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.14 Comparison of bricks Comparison of CS Yuv of bricks

Z uv

yuv

z uv

ruv

suv

tuv

Ranking

(Br1 , Br2 )

{2, 3}

{1}

0.7660

0.6832

1

1

1

Br1 → Br2

(Br1 , Br3 )

{}

{1, 2, 3}

0

1

0

0

0

Incomparable

(Br1 , Br4 )

{2}

{1, 3}

0.3950

0.7801

0

1

0

Incomparable

(Br1 , Br5 )

{}

{1, 2, 3}

0

1

0

0

0

Incomparable

(Br1 , Br6 )

{}

{1, 2, 3}

0

1

0

0

0

Incomparable

(Br2 , Br1 )

{1}

{2, 3}

0.2340

1

0

0

0

Incomparable

(Br2 , Br3 )

{1}

{2, 3}

0.2340

1

0

0

0

Incomparable

(Br2 , Br4 )

{1}

{2, 3}

0.2340

1

0

0

0

Incomparable

(Br2 , Br5 )

{1}

{2, 3}

0.2340

1

0

0

0

Incomparable

(Br2 , Br6 )

{1}

{2, 3}

0.2340

1

0

0

0

Incomparable

(Br3 , Br1 )

{1, 2, 3}

{}

1

0

1

1

1

Br3 → Br1

(Br3 , Br2 )

{2, 3}

{1}

0.7660

0.6075

1

1

1

Br3 → Br2

(Br3 , Br4 )

{1, 2, 3}

{}

1

0

1

1

1

Br3 → Br4

(Br3 , Br5 )

{1, 2}

{3}

0.6290

1

1

0

0

Incomparable

(Br3 , Br6 )

{1, 2, 3}

{}

1

0

1

1

1

Br3 → Br6

(Br4 , Br1 )

{1, 3}

{2}

0.6050

1

1

0

0

Incomparable

(Br4 , Br2 )

{2, 3}

{1}

0.7660

0.4833

1

1

1

Br4 → Br2

(Br4 , Br3 )

{}

{1, 2, 3}

0

1

0

0

0

Incomparable

(Br4 , Br5 )

{1}

{2, 3}

0.2340

1

0

0

0

Incomparable

(Br4 , Br6 )

{}

{1, 2, 3}

0

1

0

0

0

Incomparable

(Br5 , Br1 )

{1, 2, 3}

{}

1

0

1

1

1

Br5 → Br1

(Br5 , Br2 )

{2, 3}

{1}

0.7660

0.6978

1

0

0

Incomparable

(Br5 , Br3 )

{3}

{1, 2}

0.3710

0.7181

0

0

0

Incomparable

(Br5 , Br4 )

{2, 3}

{1}

0.7660

0.3702

1

1

1

Br5 → Br4

(Br5 , Br6 )

{3}

{1, 2}

0.3710

0.6990

0

0

0

Incomparable

(Br6 , Br1 )

{1, 2, 3}

{}

1

0

1

1

1

Br6 → Br1

(Br6 , Br2 )

{2, 3}

{1}

0.7660

0.4596

1

1

1

Br6 → Br2

(Br6 , Br3 )

{}

{1, 2, 3}

0

1

0

0

0

Incomparable

(Br6 , Br4 )

{1, 2, 3}

{}

1

0

1

1

1

Br6 → Br4

(Br6 , Br5 )

{1, 2}

{3}

0.6290

1

1

0

0

Incomparable

3.5 Hesitant m–Polar Fuzzy Set Definition 3.4 Let Z be a reference set, a hesitant m−polar fuzzy set on Z is a function ℘h that returns a subset of values in [0, 1]m : ℘h : Z → P([0, 1]m ). Mathematical representation of a hesitant m–polar fuzzy set is as follows: M = {z, ℘h (z)|∀z ∈ Z },

3.5 Hesitant m–Polar Fuzzy Set

193

where ℘h (z) is a set of some different values in [0, 1]m representing the possible m membership degrees of the element z ∈ Z to set M, where ℘h (z) is called a hesitant m−polar fuzzy element. Note that ℘h (z) is a set of some different values in [0, 1]m and written as   ℘h (z) = ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) , for all z ∈ Z , where, m h (z) = ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)). The following example illustrates the above concepts: Example 3.7 Let Z = {z 1 , z 2 , z 3 , z 4 } be a reference set and ℘h (z 1 ) = {(0.2, 0.4, 0.5), (0.3, 0.4, 0.6)}, ℘h (z 2 ) = {(0.4, 0.6, 0.3), (0.5, 0.5, 0.4), (0.4, 0.6, 0.4)}, ℘h (z 3 ) = {(0.5, 0.2, 0.4), (0.6, 0.4, 0.5), (0.7, 0.5, 0.3)}, ℘h (z 4 ) = {(0.4, 0.8, 0.7), (0.5, 0.7, 0.9)}, be respective hesitant 3-polar fuzzy elements, then the hesitant 3-polar fuzzy set M is given as  M=

 z 1 , {(0.2, 0.4, 0.5), (0.3, 0.4, 0.6)} ,



 z 2 , {(0.4, 0.6, 0.3), (0.5, 0.5, 0.4), (0.4, 0.6, 0.4)} ,   z 3 , {(0.5, 0.2, 0.4), (0.6, 0.4, 0.5), (0.7, 0.5, 0.3)} ,   z 4 , {(0.4, 0.8, 0.7), (0.5, 0.7, 0.9)} ,

The following real life example illustrates the concept and shows its usefulness. Example 3.8 Let Z = {z 1 , z 2 , z 3 , z 4 } be a set of image blocks considered as a reference set and m h (z) represents the 3-polar fuzzy classification of its physical properties as • Color • Shape and size • Texture These are different features for the formation of an image block necessary to compose an image. Each block z ∈ Z is classified by 3-polar fuzzy set according to its physical properties and represented in respective hesitant 3-polar fuzzy elements as ℘h (z 1 ) = {(0.24, 0.44, 0.50), (0.34, 0.42, 0.61)}, ℘h (z 2 ) = {(0.54, 0.26, 0.33), (0.65, 0.75, 0.24), (0.34, 0.46, 0.64)}, ℘h (z 3 ) = {(0.51, 0.22, 0.24), (0.16, 0.34, 0.45), (0.78, 0.57, 0.39)}, ℘h (z 4 ) = {(0.41, 0.38, 0.57), (0.45, 0.27, 0.79)}

194

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

The hesitant 3-polar fuzzy element ℘h (z 1 ) = {(0.24, 0.44, 0.50), (0.34, 0.42, 0.61)} shows, the z 1 block of an image has the following characteristics as 0.24 , Shape0.44 , 0.50 ) and ( Color and si ze T extur e 0.34 , Shape0.42 , 0.61 ) is the hesitation part of hesitant 3-polar fuzzy element ( Color and si ze T extur e ℘h (z 1 ), similarly other image blocks are characterized in remaining hesitant 3-polar fuzzy elements and the hesitant 3-polar fuzzy set M is given as  M=

 z 1 , {(0.24, 0.44, 0.50), (0.34, 0.42, 0.61)} ,



 z 2 , {(0.54, 0.26, 0.33), (0.65, 0.75, 0.24), (0.34, 0.46, 0.64)} ,   z 3 , {(0.51, 0.22, 0.24), (0.16, 0.34, 0.45), (0.78, 0.57, 0.39)} ,   z 4 , {(0.41, 0.38, 0.57), (0.45, 0.27, 0.79)} . The hesitant 3-polar fuzzy set M shows the complete formation of an image by the characterization and classification of its blocks. From Example 3.8, it is easy to understand that the approach described in Definition 3.4, in which the multi-polar information under hesitant situation is discussed in form of m tuple degrees of membership of m–polar fuzzy sets. This approach is bound by the condition of an m tuple, its each degree of membership can not be handled individually or separately. Some special hesitant m–polar fuzzy sets for z ∈ Z are given as follows: 1. 2. 3. 4.

Empty set: ℘he = {0}, where 0 = (0, 0, . . . , 0). f Full set: ℘h = {1}, where 1 = (1, 1, . . . , 1). Complete ignorance: (all values are possible) ℘h = [0, 1]. Nonsense set: .

3.5.1 Basic Operations for Hesitant m–Polar Fuzzy Set In this subsection, the basic operations of hesitant m–polar fuzzy set are constructed and described by example. 1. Lower bound: ℘h− (z)

  = inf ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) , for all z ∈ Z .

3.5 Hesitant m–Polar Fuzzy Set

195

2. Upper bound: ℘h+ (z)

  = sup ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) , for all z ∈ Z .

3. Complement:   ℘hc (z) = (1 − p1 ◦ m h (z), 1 − p2 ◦ m h (z), . . . , 1 − pm ◦ m h (z)) , for all z ∈ Z .

4. Union: (M1 )

(M2 )

(℘h

∪ ℘h

  (M ) (M ) (M )− (M )− )(z) = m h (z) ∈ ℘h 1 (z) ∪ ℘h 2 (z)|m h (z) ≥ sup{℘h 1 (z), ℘h 2 (z)} ,

where, m h (z) =( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)), for all z ∈ Z .

5. Intersection: (M1 )

(℘h

(M2 )

∩ ℘h

  (M ) (M ) (M )+ (M )+ )(z) = m h (z) ∈ ℘h 1 (z) ∩ ℘h 2 (z)|m h (z) ≤ inf{℘h 1 (z), ℘h 2 (z)} ,

where, m h (z) =( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)), for all z ∈ Z .

6. Direct sum: (M1 )

(℘h

(M2 )

⊕ ℘h

 (M ) )(z) = m h (z)(M1 ) + m h (z)(M2 ) − m h (z)(M1 ) m h (z)(M2 ) |m h (z)(M1 ) ∈ ℘h 1 (z),  (M ) m h (z)(M2 ) ∈ ℘h 2 (z) ,

where, m h (z) =( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)), for all z ∈ Z .

7. Direct product: (M1 )

(℘h

  (M ) (M ) ⊗ ℘ (M2 ) )(z) = m h (z)(M1 ) m h (z)(M2 ) |m h (z)(M1 ) ∈ ℘h 1 (z), m h (z)(M2 ) ∈ ℘h 2 (z) , where, m h (z) =( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)), for all z ∈ Z .

Example 3.9 Let Z = {z 1 , z 2 , z 3 } be the reference set, two hesitant 3-polar fuzzy sets M1 and M2 on Z are respectively given as  M1 =

 z 1 , {(0.2, 0.4, 0.4), (0.3, 0.5, 0.6), (0.3, 0.6, 0.7)} ,   z 2 , {(0.3, 0.4, 0.7), (0.5, 0.6, 0.8)} ,   z 3 , {(0.1, 0.5, 0.7), (0.2, 0.6, 0.8)} .

196

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

 M2 =

 z 1 , {(0.4, 0.6, 0.8), (0.6, 0.7, 0.8)} ,



 z 2 , {(0.5, 0.2, 0.3), (0.6, 0.3, 0.5), (0.6, 0.4, 0.8)} ,   z 3 , {(0.3, 0.2, 0.5), (0.4, 0.4, 0.7)} .

The aforementioned operations on these two hesitant 3-polar fuzzy sets M1 and M2 are calculated as follows: 1. Lower bound: ℘h(M1 )− (z 1 ) = inf{(0.2, 0.4, 0.4), (0.3, 0.5, 0.6), (0.3, 0.6, 0.7)}, ℘h(M1 )− (z 1 ) = (0.2, 0.4, 0.4), ℘h(M2 )− (z 3 ) = inf{(0.3, 0.2, 0.5), (0.4, 0.4, 0.7)}, ℘h(M2 )− (z 3 ) = (0.3, 0.2, 0.5). 2. Upper bound: ℘h(M1 )+ (z 2 ) = sup{(0.3, 0.4, 0.7), (0.5, 0.6, 0.8)}, ℘h(M1 )+ (z 2 ) = (0.5, 0.6, 0.8), ℘h(M2 )+ (z 3 ) = sup{(0.3, 0.2, 0.5), (0.4, 0.4, 0.7)}, ℘h(M2 )+ (z 3 ) = (0.4, 0.4, 0.7). 3. Complement:  ℘h(M1 )c (z 1 ) = (1 − 0.2, 1 − 0.4, 1 − 0.4), (1 − 0.3, 1 − 0.5, 1 − 0.6),  (1 − 0.3, 1 − 0.6, 1 − 0.7)   = (0.8, 0.6, 0.6), (0.7, 0.5, 0.4), (0.7, 0.4, 0.3) ,   ℘h(M2 )c (z 3 ) = (1 − 0.3, 1 − 0.2, 1 − 0.5), (1 − 0.4, 1 − 0.4, 1 − 0.7)   = (0.7, 0.8, 0.5), (0.6, 0.6, 0.3) . 4. Union:   (℘h(M1 ) ∪ ℘h(M2 ) )(z 1 ) = sup (0.2, 0.4, 0.4), (0.4, 0.6, 0.8) = (0.4, 0.6, 0.8)

3.5 Hesitant m–Polar Fuzzy Set

197

  = (0.4, 0.6, 0.8), (0.6, 0.7, 0.8) . Remark 3.3 Note that the union of two hesitant m–polar fuzzy sets can be a nonsense set computed as follows: (℘h(M1 )

∪

℘h(M2 ) )(z 3 )

  = sup (0.1, 0.5, 0.7), (0.3, 0.2, 0.5) = (0.3, 0.5, 0.7) = .

Above calculations show that, none of the 3−tuple is greater and equal to (0.3, 0.5, 0.7), thus it is observed the union of hesitant 3-polar fuzzy elements ℘h(M1 ) (z 3 ) and ℘h(M2 ) (z 3 ) of two hesitant 3-polar fuzzy sets M1 and M2 is a nonsense set . 5. Intersection: (℘h(M1 )

∩

℘h(M2 ) )(z 2 )

  = inf (0.5, 0.6, 0.8), (0.6, 0.4, 0.8) = (0.5, 0.4, 0.8)   = (0.3, 0.4, 0.7), (0.5, 0.2, 0.3) ,

  (℘h(M1 ) ∩ ℘h(M2 ) )(z 3 ) = inf (0.2, 0.6, 0.8), (0.4, 0.4, 0.7) = (0.2, 0.4, 0.7) = . 6. Direct sum: (M1 )

(℘h

(M2 )

⊕ ℘h

 )(z 1 ) = (0.6, 1.0, 1.2) − (0.08, 0.24, 0.32) = (0.52, 0.76, 0.88), (0.8, 1.1, 1.2) − (0.12, 0.28, 0.32) = (0.68, 0.82, 0.88), (0.7, 1.1, 1.4) − (0.12, 0.3, 0.48) = (0.58, 0.8, 0.92), (0.9, 1.2, 1.4) − (0.18, 0.35, 0.48) = (0.72, 0.85, 0.92), (0.7, 1.2, 1.5) − (0.12, 0.36, 0.56) = (0.58, 0.84, 0.94),  (0.9, 1.3, 1.5) − (0.18, 0.42, 0.56) = (0.72, 0.88, 0.94)  = (0.52, 0.76, 0.88), (0.68, 0.82, 0.88), (0.58, 0.8, 0.92),  (0.72, 0.85, 0.92), (0.58, 0.84, 0.94), (0.72, 0.88, 0.94) .

198

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

7. Direct product: (℘h(M1 ) ⊗ ℘h(M2 ) )(z 3 )   = (0.03, 0.1, 0.35), (0.04, 0.2, 0.49), (0.06, 0.12, 0.40), (0.08, 0.24, 0.56) . The following proposition shows, that the union and intersection of hesitant m– polar fuzzy-sets satisfy the commutative, associative and idempotent properties under limited conditions. Proposition 3.4 For any hesitant m–polar fuzzy sets ℘h(1) (z), ℘h(2) (z) and ℘h(3) (z) in M(Z ) and z ∈ Z , the following properties are defined as 1. Commutativity: (i) (℘h(1) ∪ ℘h(2) )(z) = (℘h(2) ∪ ℘h(1) )(z), (ii) (℘h(1) ∩ ℘h(2) )(z) = (℘h(2) ∩ ℘h(1) )(z). 2. Associativity: (i) ((℘h(1) ∪ ℘h(2) ) ∪ ℘h(3) )(z) = (℘h(1) ∪ (℘h(2) ∪ ℘h(3) ))(z), (ii) ((℘h(1) ∩ ℘h(2) ) ∩ ℘h(3) )(z) = (℘h(1) ∩ (℘h(2) ∩ ℘h(3) ))(z). 3. Idempotency: (i) (℘h(1) ∪ ℘h(1) )(z) = ℘h(1) (z), (ii) (℘h(1) ∩ ℘h(1) )(z) = ℘h(1) (z). Proof All the three described properties are trivial to prove.



Some of the operational rules are stated, in the form of Propositions: Proposition 3.5 For any ℘h (z) ∈ M(Z ) and z ∈ Z , the following operational rules are defined as 1. (℘hc )− (z) = 1 − ℘h+ (z), 2. (℘hc )+ (z) = 1 − ℘h− (z). Proof 1. (℘hc )− (z) = inf ℘hc (z)   = inf (1 − p1 ◦ m h (z), 1 − p2 ◦ m h (z), . . . , 1 − pm ◦ m h (z)) , ∀ z ∈ Z   =1 − sup ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) , ∀ z ∈ Z =1 − ℘h+ (z),

2. (℘hc )+ (z) = sup ℘hc (z)   = sup (1 − p1 ◦ m h (z), 1 − p2 ◦ m h (z), . . . , 1 − pm ◦ m h (z)) , ∀ z ∈ Z

3.5 Hesitant m–Polar Fuzzy Set

199

  =1 − inf ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) , ∀ z ∈ Z =1 − ℘h− (z).



Proposition 3.6 For any ℘h (z) ∈ M(Z ) and z ∈ Z , the following operational rules are defined as f

f

f

1. (℘h ∪ ℘h )(z) = ℘h (z) and (℘h ∩ ℘h )(z) = ℘h (z), 2. (℘h ∪ ℘he )(z) = ℘h (z) and (℘h ∩ ℘he )(z) = ℘he (z). Proof 1. (℘h ∪

f ℘h )(z)

  − = m h (z) ∈ ℘h (z) ∪ 1|m h (z) ≥ sup{℘h (z), 1} , ∀ z ∈ Z   = m h (z) ∈ ℘h (z) ∪ 1|m h (z) ≥ 1 , ∀ z ∈ Z f

={1} = ℘h (z),   f (℘h ∩ ℘h )(z) = m h (z) ∈ ℘h (z) ∩ 1|m h (z) ≤ inf{℘h+ (z), 1} , ∀ z ∈ Z   = m h (z) ∈ ℘h (z) ∩ 1|m h (z) ≤ ℘h+ (z) , ∀ z ∈ Z =℘h (z).   2. (℘h ∪ ℘he )(z) = m h (z) ∈ ℘h (z) ∪ 0|m h (z) ≥ sup{℘h− (z), 0} , ∀ z ∈ Z   = m h (z) ∈ ℘h (z) ∪ 0|m h (z) ≥ ℘h− (z) , ∀ z ∈ Z =℘h (z),   (℘h ∩ ℘he )(z) = m h (z) ∈ ℘h (z) ∩ 0|m h (z) ≤ inf{℘h+ (z), 0} , ∀ z ∈ Z   = m h (z) ∈ ℘h (z) ∩ 0|m h (z) ≤ 0 , ∀ z ∈ Z ={0} = ℘he (z).



3.5.2 Comparison Laws of Hesitant m–Polar Fuzzy Set Scores are standard tools for comparing hesitant fuzzy elements, and in this section the concept is generalized to hesitant m-polar fuzzy set. These ideas can be exported to proposed case by the recourse to the following concepts:

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Definition 3.5 The score function s(℘h ) of a hesitant m–polar fuzzy set is defined as    1 1 p1 ◦ m h (z), ℘ (z) p2 ◦ m h (z), . . . , s(℘h ) = ℘ (z) h h λ λ p ◦m (z) p ◦m (z) 1

h

∈℘h (z)

1 λ℘h (z)



2

 pm ◦ m h (z) ,

h

∈℘h (z)

pm ◦m h (z) ∈℘h (z)

where λ℘h (z) is the total number of pi ◦ m h (z) ∈ ℘h (z) for each i ∈ m. The score function can help us to compare hesitant m-polar fuzzy set according to the following rules: Remark 3.4 For any two hesitant m-polar fuzzy sets ℘h(1) (z) and ℘h(2) (z) of a hesitant m–polar fuzzy set. • • • •

If s(℘h(1) ) > s(℘h(2) ) then ℘h(1) (z) is superior or finer to ℘h(2) (z). If s(℘h(1) ) < s(℘h(2) ) then ℘h(1) (z) is inferior or weaker to ℘h(2) (z). If s(℘h(1) ) = s(℘h(2) ) then ℘h(1) (z) is indifferent to ℘h(2) (z). If neither of the above is true then ℘h(1) (z) is totally different to ℘h(2) (z). 



Example 3.10 Consider ℘h(1) (z) = (0.5, 0.8, 0.7, 0.6), (0.8, 0.6, 0.6, 0.3), (0.6, 0.2, 0.3, 0.1)   (2) and ℘h (z) = (0.4, 0.6, 0.7, 0.9), (0.5, 0.5, 0.8, 0.3) are two hesitant 4–polar

fuzzy elements, then by Definition 3.5 the score functions of ℘h(1) (z) and ℘h(2) (z) are calculated as follows:   0.5 + 0.8 + 0.6 0.8 + 0.6 + 0.2 0.7 + 0.6 + 0.3 0.6 + 0.3 + 0.1 , , , s(℘h(1) ) = 3 3 3 3 = (0.63, 0.53, 0.53, 0.33),   0.4 + 0.5 0.6 + 0.5 0.8 + 0.7 0.9 + 0.3 , , , s(℘h(2) ) = 2 2 2 2 = (0.45, 0.55, 0.75, 0.6). From these calculations, one readily concludes that ℘h(1) (z) is totally different to ℘h(2) (z). However, the score function is not fully discriminative for a formal discussion of the cause, and this feature is presented in the next instance: ℘h(1) (z)

  = (0.7, 0.85, 0.9), (0.2, 0.4, 0.75), (0.3, 0.4, 0.6)

Example 3.11 Consider   (2) and ℘h (z) = (0.2, 0.5, 0.45), (0.5, 0.7, 0.8), (0.6, 0.3, 0.9), (0.3, 0.7, 0.85) are

3.5 Hesitant m–Polar Fuzzy Set

201

two hesitant 3-polar fuzzy elements, then by Definition 3.5 the score functions of ℘h(1) (z) and ℘h(2) (z) are calculated as follows: s(℘h(1) ) = (0.4, 0.55, 0.75), s(℘h(2) ) = (0.4, 0.55, 0.75). From these calculations, one observes that ℘h(1) (z) is deemed indifferent to ℘h(2) (z) and unable to give formal support to the difference between ℘h(1) (z) and ℘h(2) (z) by the application of the score function alone. In other words, the Example 3.11 shows that sometimes it is not possible to perform a comparison when two hesitant m-polar fuzzy sets have coincident score functions as computed by Definition 3.5. In order to break ties in such situation, the deviation degree of hesitant m-polar fuzzy sets is defined. In case of indifference between two hesitant m-polar fuzzy sets this figure may tell, which one is superior. Definition 3.6 The deviation degree (℘h ) of a hesitant m–polar fuzzy set is defined as  (℘h ) =

λ℘h (z) ...,





1

2 1 2 p1 ◦ m h (z) − s1 (℘h ) ,

p1 ◦m h (z) ∈℘h (z)

1 λ℘h (z)





1 λ℘h (z)





2 1 2 p2 ◦ m h (z) − s2 (℘h ) ,

p2 ◦m h (z) ∈℘h (z)

2 1  2 pm ◦ m h (z) − sm (℘h ) ,

pm ◦m h (z) ∈℘h (z)

where λ℘h (z) is the total number of pi ◦ m h (z) ∈ ℘h (z) for each i ∈ m. The criterion in Remark 3.4 is refined in the following terms: Remark 3.5 For any two hesitant m–polar fuzzy ℘h(1) (z) and ℘h(2) (z) of a hesitant m–polar fuzzy set. • • • •

If (℘h(1) ) > (℘h(2) ) then ℘h(1) (z) is superior or finer to ℘h(2) (z). If (℘h(1) ) < (℘h(2) ) then ℘h(1) (z) is inferior or weaker to ℘h(2) (z). If (℘h(1) ) = (℘h(2) ) then ℘h(1) (z) is indifferent to ℘h(2) (z). If neither of the above is true, then ℘h(1) (z) is completely different from ℘h(2) (z).

Example 3.12 By reconsidering the Example 3.11, where s(℘h(1) ) = s(℘h(2) ) the deviation degrees of these hesitant 3-polar fuzzy elements are calculated as follows: (℘h(1) ) = (0.216, 0.045, 0.015), (℘h(2) ) = (0.025, 0.027, 0.125). From these calculations, it is observed that neither (℘h(1) ) < (℘h(2) ) nor (℘h(1) ) > (℘h(2) ) nor (℘h(1) ) = (℘h(2) ) are true. Thus, ℘h(1) (z) is completely different from, ℘h(2) (z).

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Definition 3.7 The average value ν(℘h ) of a hesitant m–polar fuzzy set is defined as    m 1 ( pi ◦ m h (z)) . ν(℘h ) = m i=1 The average value can help us to compare hesitant value of each hesitant m–polar fuzzy. Example 3.13 By reconsidering the Example 3.10, the average values of hesitant 3-polar fuzzy elements are calculated as follows: 

0.5 + 0.8 + 0.7 + 0.6 0.8 + 0.6 + 0.6 + 0.3 0.6 + 0.2 + 0.3 + 0.1 , , = 4 4 4 = {0.65, 0.58, 0.3},   0.4 + 0.6 + 0.7 + 0.9 0.5 + 0.5 + 0.8 + 0.3 (2) , ν(℘h ) = 4 4 = {0.65, 0.53}. ν(℘h(1) )



In the next sections, the hesitant m–polar fuzzy TOPSIS and hesitant m–polar fuzzy ELECTRE-I approaches for MCGDM are proposed. These approaches are flexible and compatible to deal the hesitant situations motivated by the multi-polar information.

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach The hesitant m–polar fuzzy TOPSIS approach is based on hesitant m–polar fuzzysets that deals with MCGDM problems, in which Z = {z 1 , z 2 , . . . , z p } is chosen as the set of different alternatives and C = {c1 , c2 , . . . , cq } as the set of hesitant m–polar fuzzy criteria’s, which are classified by m different characteristics under hesitant situation. In such a case, decision makers are responsible for evaluating the p different alternatives under q hesitant m–polar fuzzy criteria’s, the suitable ratings of alternatives are according to decision makers, assessed in term of m different characteristics under r different membership values of hesitancy, where (l = 1, 2, . . . , r ). The following steps for proposed approach are described as follows: Step 1: The degree of each alternative (z j ∈ Z , j = 1, 2, . . . , p) over all hesitant m– polar fuzzy criteria’s (ck ∈ C, k = 1, 2, . . . , q) is given by hesitant m–polar fuzzy as   ℘h (z) = ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) , for all z ∈ Z ,

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach

203

Table 3.15 A generic hesitant m–polar fuzzy decision matrix Alternatives Hesitant m−polar fuzzy criteria’s c1 c2 ··· ℘h11 (z 1 ) ℘h21 (z 2 )

z1 z2 .. .

℘h12 (z 1 ) ℘h22 (z 2 )

.. .

.. .

p1

p2

℘h (z p )

zp

℘h (z p )

cq 1q

···

℘h (z 1 )

··· .. .

℘h (z 2 ) .. .

···

℘h (z p )

2q

pq

and m h (z) = ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) classify the different characteristics of each criteria. Tabular representation of hesitant m–polar fuzzy decision matrix is given by Table 3.15, which describes the ratings of alternatives. For each possible j and k, jk ℘h (z)

  jk jk jk = ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) .

Construct the optimistic or pessimistic hesitant m–polar fuzzy decision matrix by adding the maximal or minimal values, so the length of all the hesitant m–polar fuzzy elements become equal. Step 2: Decision makers have an authority to assign the weights to each hesitant m–polar fuzzy criterion of alternatives according to their choice and importance of each criterion. The weights assigned by the decision makers are W = (w1 , w2 , . . . , wq ) ∈ (0, 1]. Weights assigned by the decision makers satisfy the normalized condition. i.e., q 

wk = 1.

k=1

Step 3: The weighted hesitant m–polar fuzzy decision matrix is calculated in Table 3.16. For each possible j and k, jk 

jk

℘h (z) =wk ℘h (z)   jk jk jk =wk ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z))   jk  jk  jk  = ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) .

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.16 A generic weighted hesitant m–polar fuzzy decision matrix Alternatives Hesitant m−polar fuzzy criteria’s c1 c2 ··· Weights w1 w2 ···  )11

 )12

℘h (z 1  ℘h (z 2 )21 .. .  ℘h (z p ) p1

z1 z2 .. . zp

℘h (z 1  ℘h (z 2 )22 .. .  ℘h (z p ) p2

··· ··· .. . ···

cq wq



℘h (z 1 )1q  ℘h (2 )2q .. .  ℘h (z p ) pq

Step 4: The hesitant m–polar fuzzy positive ideal solution and hesitant m–polar fuzzy negative ideal solution of alternatives are calculated by Eqs. (3.6) and (3.7). 



q





q

H m PI S ={(℘h1 (z))+ , (℘h2 (z))+ , . . . , (℘h (z))+ }, H m N I S ={(℘h1 (z))− , (℘h2 (z))− , . . . , (℘h (z))− }.

(3.6)

(3.7)

where, 



(℘hk (z))+ = sup(℘hk (z)) j

  jk  jk  jk  = sup ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) j

     = (( p1 ◦ m kh (z))+ , ( p2 ◦ m kh (z))+ , . . . , ( pm ◦ m kh (z))+ ) , 



(℘hk (z))− = inf (℘hk (z)) j   jk  jk  jk  = inf ( p1 ◦ m h (z), p2 ◦ m h (z), . . . , pm ◦ m h (z)) j   k − k − k − = (( p1 ◦ m h (z)) , ( p2 ◦ m h (z)) , . . . , ( pm ◦ m h (z)) ) . Step 5: The hesitant m–polar fuzzy Euclidean distances of each alternative z j from hesitant m–polar fuzzy positive ideal solution and hesitant m–polar fuzzy negative ideal solution are calculated by Eqs. (3.8) and (3.9).

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach

205

  

q r  m    1   jk  ( pi ◦ m h (z) − pi ◦ m kh (z)+ )2 , D E (z j , H m PI S ) = r m k=1 l=1 i=1 l 

(3.8)   

q r  m    1   jk  k

− 2 ( pi ◦ m h (z) − pi ◦ m h (z) ) . D E (z j , H m N I S ) = r m k=1 l=1 i=1 l (3.9) Step 6: The relative hesitant m–polar fuzzy closeness coefficient of each alternative z j is computed by using following formula as described in Eq. (3.10). 



Cj =

D E (z j , H m N I S ) ,   D E (z j , H m PI S ) + D E (z j , H m N I S )

j = 1, 2, . . . , p.

(3.10)

The alternative with highest hesitant m–polar fuzzy closeness coefficient is best one and the ranking order of each alternative can be determined. In next subsection, the practical use of proposed model is discussed, and shown how hesitant m–polar fuzzy TOPSIS is useful in comparison of populous countries and different types of textiles or clothing respectively.

3.6.1 Comparison of Top Five Populous Countries The population is the count of all the organisms of the same species or group, which animate in a particular geographical area, and it refers to a collection of humans. Demography is a social science, which involves the analytical review of human populations. In simple terms, population is the count of people in a town, city, region, country or world. In a country, population is generally regulated by an action called census; a process of analyzing, collecting, compiling and publishing data, but at world level the comparison and ranking analysis of population of countries is a challenging task for demographers. For the comparison and ranking analysis of population of countries multi-criteria and multi-polar information is required under hesitant situations, for this purpose the concept of hesitant m–polar fuzzy sets is introduced, which deal the multipolar information under hesitant situations according to demographers. Further, hesitant m–polar fuzzy TOPSIS approach is used for comparison and data analysis. For this purpose, U = {Pakistan, U S, I ndia, I ndonesia, China} is considered as the set of five different populous countries and C = {c1 , c2 } as the set of two main criteria’s or factors, to calculate the population rate of a country. For the evaluation, comparison and attempt to forecast changes in the size of a population, the demographers typically focus on two main criteria’s or factors, such as natu-

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

ral criteria and planned criteria, which they further classified into three different sub-criteria’s as 1. The “Natural Criteria” includes • The “Fertility rate”, the factor which affects the growth of the population in the biggest way and is frequently measured by the number of children per one woman of child-bearing age. • The “Mortality rate”, it is the frequency of the number of deaths (in general, or due to a specific cause) in a peculiar population, scoped to the intensity of that population, per unit of time. • The “Life expectancy”, it is an assessment for the predicted life interval of an average new born child. 2. The “Planned Criteria” includes • The “Immigration”, when someone moves to a country from another place, it is known as immigration. • The “Emigration”, it is the action of leaving a resident country or place of residence with the intent to settle elsewhere. • The “Government Restrictions”, the restrictions imposed by government that would cause fewer resources to be used and prevent overpopulation. All these criteria’s and factors are assessed by the demographers, who are responsible to evaluate and compare the population of countries. Due to the collective decision of demographers each sub criteria of single country is further classified by three different hesitant values. Demographers are free to choose any membership value from interval [0, 1] according to the census result of each country, but in this case demographers want to compare the countries on optimistic decision. Thus, demographers assign hesitant values as described in Table 3.17. Obviously, the count of hesitant 3-polar fuzzy elements is not comparable in hesitant 3-polar fuzzy set. In order to take more efficiency, they prolong those 3-polar fuzzy membership values, whose average values are largest, so the length of all hesitant 3-polar fuzzy elements become equal for comparison. According to the moderation as described above, it is considered that the demographers show optimistic response in this case, and improve the hesitant 3-polar fuzzy data by adding the maximal 3-polar fuzzy membership values as mentioned in Table 3.18. Demographers assign the weights to each criterion according to their worth that satisfy the normalized condition, given as w = (0.59, 0.41). Weighted optimistic hesitant 3-polar fuzzy decision matrix is calculated in Table 3.19. Use Eqs. (3.6) and (3.7) to determine the hesitant 3-polar fuzzy positive ideal solution and hesitant 3-polar fuzzy negative ideal solution respectively.

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach

207

Table 3.17 Hesitant 3-polar fuzzy decision matrix Countries Hesitant 3–polar fuzzy criteria Natural criteria   (0.38, 0.45, 0.67), (0.47, 0.60, 0.52), (0.49, 0.65, 0.66) Pakistan   (0.45, 0.28, 0.70), (0.58, 0.30, 0.68), (0.63, 0.40, 0.57) US   (0.57, 0.38, 0.59), (0.44, 0.50, 0.71) India   (0.13, 0.54, 0.63), (0.33, 0.43, 0.75), (0.35, 0.62, 0.89) Indonesia   (0.74, 0.63, 0.69), (0.61, 0.66, 0.71) China Countries

Hesitant 3–polar fuzzy criteria Planned criteria   (0.76, 0.58, 0.25), (0.81, 0.63, 0.19), (0.69, 0.71, 0.36)   (0.43, 0.82, 0.61), (0.58, 0.76, 0.76)   (0.37, 0.58, 0.93), (0.43, 0.77, 0.89), (0.61, 0.82, 0.85)   (0.49, 0.56, 0.70), (0.53, 0.43, 0.61)   (0.72, 0.47, 0.58), (0.81, 0.53, 0.68), (0.65, 0.33, 0.70)

Pakistan US India Indonesia China



 (0.4366, 0.3717, 0.4130), (0.4366, 0.3717, 0.4425), (0.3717, 0.3894, 0.5251) ,   (0.3116, 0.3362, 0.3813), (0.3321, 0.3157, 0.3649), (0.2829, 0.3362, 0.3485) .   = (0.0767, 0.1652, 0.3481), (0.1947, 0.1770, 0.3068), (0.2065, 0.2360, 0.3363) ,   (0.1517, 0.1927, 0.1025), (0.1763, 0.2173, 0.0779), (0.2173, 0.1353, 0.147) .

H 3H PI S =

H 3H N I S

Use Eqs. (3.8) and (3.9) to calculate the hesitant 3-polar fuzzy Euclidean distance of countries from its hesitant 3-polar fuzzy positive ideal solution and hesitant 3-polar fuzzy negative ideal solution, that are calculated as follows: 

D E (Bn1 , H 3N I S ) = 0.1437,



D E (Bn2 , H 3N I S ) = 0.1721,



D E (Bn3 , H 3N I S ) = 0.2023,

D E (Bn1 , H 3PI S ) = 0.1948, D E (Bn2 , H 3PI S ) = 0.1602, D E (Bn3 , H 3PI S ) = 0.1363,







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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.18 Optimistic hesitant 3-polar fuzzy decision matrix Countries Hesitant 3-polar fuzzy criteria Natural criteria   (0.38, 0.45, 0.67), (0.47, 0.60, 0.52), (0.49, 0.65, 0.66) Pakistan   (0.45, 0.28, 0.70), (0.58, 0.30, 0.68), (0.63, 0.40, 0.57) US   (0.57, 0.38, 0.59), (0.44, 0.50, 0.71), (0.44, 0.50, 0.71) India   (0.13, 0.54, 0.63), (0.33, 0.43, 0.75), (0.35, 0.62, 0.89) Indonesia   (0.74, 0.63, 0.69), (0.74, 0.63, 0.69), (0.61, 0.66, 0.71) China Countries

Hesitant 3–polar fuzzy criteria Planned criteria   (0.76, 0.58, 0.25), (0.81, 0.63, 0.19), (0.69, 0.71, 0.36)   (0.43, 0.82, 0.61), (0.58, 0.76, 0.76), (0.58, 0.76, 0.76)   (0.37, 0.58, 0.93), (0.43, 0.77, 0.89), (0.61, 0.82, 0.85)   (0.49, 0.56, 0.70), (0.49, 0.56, 0.70), (0.53, 0.43, 0.61)   (0.72, 0.47, 0.58), (0.81, 0.53, 0.68), (0.65, 0.33, 0.70)

Pakistan US India Indonesia China



D E (Bn4 , H 3N I S ) = 0.1473,



D E (Bn5 , H 3N I S ) = 0.2264.

D E (Bn4 , H 3PI S ) = 0.1935, D E (Bn5 , H 3PI S ) = 0.1134,





Use Eq. (3.10) to calculate the relative hesitant 3-polar fuzzy closeness coefficients E j of countries.   E 1 = 0.4244, E 2 = 0.5179, 

E 3 = 0.5975,



E 4 = 0.4322,



E 5 = 0.6664. For the comparison, arrange the countries according to the ranking of relative hesitant 3-polar fuzzy closeness coefficients, i.e., China > I ndia > U S > I ndonesia > Pakistan. From comparison, it is easy to see • China is the most populous country, • India is at second number,

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach

209

Table 3.19 Weighted optimistic hesitant 3-polar fuzzy decision matrix Countries

Pakistan US India Indonesia China Countries

Pakistan US India Indonesia China

Hesitant 3–polar fuzzy criteria with weight 0.59 Natural criteria   (0.2242, 0.2655, 0.3953), (0.2773, 0.3540, 0.3068), (0.2891, 0.3835, 0.3894)   (0.2655, 0.1652, 0.4130), (0.3422, 0.1770, 0.4012), (0.3717, 0.2360, 0.3363)   (0.3363, 0.2242, 0.3481), (0.2596, 0.2950, 0.4189), (0.2596, 0.2950, 0.4189)   (0.0767, 0.3186, 0.3717), (0.1947, 0.2537, 0.4425), (0.2065, 0.3658, 0.5251)   (0.4366, 0.3717, 0.4071), (0.4366, 0.3717, 0.4071), (0.3599, 0.3894, 0.4198) Hesitant 3–polar fuzzy criteria with weight 0.41 Planned criteria   (0.3116, 0.2378, 0.1025), (0.3321, 0.2583, 0.0779), (0.2829, 0.2911, 0.1476)   (0.1763, 0.3362, 0.2501), (0.2378, 0.3116, 0.3116), (0.2378, 0.3116, 0.3116)   (0.1517, 0.2378, 0.3813), (0.1763, 0.3157, 0.3649), (0.2501, 0.3362, 0.3485)   (0.2009, 0.2296, 0.2870), (0.2009, 0.2296, 0.2870), (0.2173, 0.1763, 0.2501)   (0.2952, 0.1927, 0.2378), (0.3321, 0.2173, 0.2788), (0.2665, 0.1353, 0.2870)

• US is at third number, • Indonesia is at fourth number, • Pakistan is at fifth number.

3.6.2 Comparison of Different Types of Textiles or Clothing A textile is an extensible material subsisted by a network of natural or artificial fibres such as thread or yarn, whereas the narrated words “fabric” and “cloth” are usually practiced in textile assembly trades such as dressmaking and tailoring, as synonyms for textile. Suppose that an industry wants to compare the production of its different types of textiles or clothing and the whole decision depends upon a team of three competent textile engineers. They make a criteria to evaluate the different types of textiles or clothing, but in whole procedure they face the multipolar information related to textiles and they also suffer some hesitant situations about their ratings and industry decisions. The hesitant m–polar fuzzy set deals the multi-polar information and hesitant situations related to experts. For this purpose, Tc = {Tc1 , Tc2 , Tc3 , Tc4 , Tc5 } is considered as the set of five different types of textiles or clothing and C = {c1 , c2 , c3 , c4 } as the set of four different evaluating criteria’s or

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

characteristics of textiles assigned by team of three competent textile engineers. For the evaluation and comparison, the textile engineers choose the following criteria’s such as fibre properties, influencing factors, construction techniques and finishing processes, which they further classified in four different sub-criteria’s as 1. The “Fibre Properties” of textile and clothing may include • The “Length and appearance of the fibre”, this property of fibre tells us about the material used to make the fabrics, either it is made of staple fibres that have a rough look or made with filaments that have a smooth and shiny look. • The “Moister absorption”, this property tells us about the areas where the maximum comfort is needed. • The “Heat conductivity”, this property specifies the capacity of the fibre to conduct heat away from the body. • The “Strength”, the property of strength is committed in washing of the fabric. Comfort of washing a fabric depends upon the strength of the fibre when it is wet. 2. The “Influencing Factors” of textile and clothing may include • The “Climatic factors”, it tells to choose and maintain the clothes according to the atmosphere and climate of a region either it is hot or cold. • The “Occasion”, this factor deals the situations, which kind of clothes or dresses should be chosen according to an occasion such as a marriage ceremony and an interview. • The “Age”, it is noticed that the type of clothes worn changes with age and may vary with age difference such as infant wear, school going children, the later teen years, adults and old ages. • The “Profession/Occupation”, this factor tells us about, what kind of clothes should we wear according to our profession or occupation. Many professions have a specific dress code which gives them a special identity. 3. The “Construction Techniques” of textile and clothing may include • The “Knitting”, this technique provides elasticity and stretchability to the fabrics so it is suitable for socks, woollen sweaters and under-garments. • The “Weaving”, it provides smoothness, thickness and toughness of fabrics so it can be selected for making dresses accordingly. • The “Knotting”, it is the tying of two threads to each other or to tie thread to any other object and used to cord and yarn an object and to attach the rope. • The “Crocheting”, it is the modern sense and proceed by applying a loop on the hook, stretching another loop through the first loop, and so on to create a chain. 4. The “Finishing Processes” of textile and clothing may include • It is “Shrinkage control”, this process control the fabric shrinkage and finish it, that you do not have to worry about shrinkage.

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• It is “Mercerisation”, it is a process applied to cotton and sometimes to cotton blends to improve strength, to enhance the luster and to improve their compatibility for dyes. • It is “Water proofing”, the finishes are generally applied to improve look of a fabric and this process prevents and controls the water leakage in special king of clothes such as umbrellas and rain coats. • Its “Drying”, this process is used in various phases of textile processing, because water is considered an essential part to accumulates in fabrics, and the excess moisture must eventually be removed. All these criteria and characteristics are assessed by the team of textile engineers, who are responsible to evaluate and compare the textiles. Due to the collective decision of textile engineers each sub-criteria of single textile is further classified by three different hesitant values. Textile engineers are free to choose any membership value from interval [0, 1], but in this case industry wants to compare the textiles on pessimistic decision. Thus, textile engineers assign hesitant values as described in Table 3.20. Obviously, the count of hesitant 4–polar fuzzy elements is not comparable in hesitant 4–polar fuzzyset. In order to take more efficiency, they prolong those 4–polar fuzzy membership values whose average values are smallest, so the length of all hesitant 4–polar fuzzy elements become equal for comparison. According to the moderation as described above, the textile engineers show pessimistic response in this case according to industry decision, and improve the hesitant 4–polar fuzzy data by adding the minimal 4–polar fuzzy membership values as mentioned in Table 3.21. Textile engineers assign the weights to each criteria according to its worth that satisfy the normalized condition, given as w = (0.2156, 0.2451, 0.2627, 0.2766). Weighted pessimistic hesitant 4–polar fuzzy decision matrix is calculated in Table 3.22. Use Eqs. (3.6) and (3.7) to determine the hesitant 4–polar fuzzy positive ideal solution and hesitant 4–polar fuzzy negative ideal solution, respectively.  H 4H PI S =

(0.1876, 0.1682, 0.1250, 0.1703), (0.1595, 0.1682, 0.1595, 0.1725),  (0.1250, 0.2005, 0.1337, 0.1919) ,  (0.1887, 0.1765, 0.2034, 0.2157), (0.1985, 0.1936, 0.2157, 0.1863),  (0.1985, 0.2010, 0.2083, 0.1887) ,  (0.2207, 0.2049, 0.1813, 0.2364), (0.2154, 0.2128, 0.1865, 0.2312),

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Table 3.20 Hesitant 4–polar fuzzy decision matrix Alternatives Hesitant 4–polar fuzzy criteria C1 Fibre properties   (0.20, 0.75, 0.45, 0.67), (0.34, 0.66, 0.51, 0.72), (0.40, 0.68, 0.56, 0.66) Tc1   (0.48, 0.78, 0.47, 0.29), (0.58, 0.93, 0.38, 0.50) Tc2   (0.87, 0.39, 0.47, 0.59), (0.74, 0.50, 0.74, 0.60), (0.53, 0.66, 0.57, 0.34) Tc3   (0.23, 0.54, 0.36, 0.79), (0.43, 0.63, 0.45, 0.80), (0.44, 0.68, 0.41, 0.89) Tc4   (0.13, 0.35, 0.58, 0.63), (0.33, 0.43, 0.62, 0.75) Tc5 Alternatives Tc1 Tc2 Tc3 Tc4 Tc5 Alternatives Tc1 Tc2 Tc3 Tc4 Tc5 Alternatives Tc1 Tc2 Tc3 Tc4 Tc5

Hesitant 4–polar fuzzy criteria C2 Influencing factors   (0.77, 0.47, 0.69, 0.71), (0.81, 0.39, 0.59, 0.71)   (0.47, 0.72, 0.51, 0.88), (0.58, 0.74, 0.68, 0.76), (0.49, 0.64, 0.60, 0.74)   (0.48, 0.68, 0.73, 0.19), (0.43, 0.79, 0.80, 0.21), (0.50, 0.82, 0.85, 0.17)   (0.59, 0.66, 0.40, 0.59), (0.53, 0.58, 0.61, 0.49)   (0.26, 0.14, 0.83, 0.79), (0.33, 0.27, 0.88, 0.75), (0.25, 0.33, 0.82, 0.77) Hesitant 4–polar fuzzy criteria C3 Construction techniques   (0.84, 0.53, 0.69, 0.72), (0.82, 0.66, 0.71, 0.82), (0.72, 0.61, 0.70, 0.80)   (0.72, 0.58, 0.47, 0.31), (0.75, 0.53, 0.39, 0.45), (0.85, 0.65, 0.67, 0.56)   (0.76, 0.48, 0.63, 0.82), (0.83, 0.54, 0.61, 0.93)   (0.68, 0.57, 0.48, 0.90), (0.63, 0.60, 0.43, 0.88), (0.73, 0.63, 0.50, 0.95)   (0.13, 0.78, 0.44, 0.63), (0.33, 0.81, 0.48, 0.74), (0.25, 0.82, 0.59, 0.86) Hesitant 4–polar fuzzy criteria C4 Finishing processes   (0.74, 0.93, 0.69, 0.77), (0.71, 0.86, 0.71, 0.89), (0.75, 0.89, 0.62, 0.83)   (0.62, 0.46, 0.78, 0.82), (0.58, 0.53, 0.77, 0.79), (0.65, 0.43, 0.70, 0.87)   (0.76, 0.58, 0.49, 0.82), (0.83, 0.58, 0.55, 0.93), (0.63, 0.62, 0.67, 0.91)   (0.45, 0.68, 0.79, 0.43), (0.53, 0.70, 0.78, 0.38), (0.55, 0.62, 0.89, 0.29)   (0.23, 0.67, 0.54, 0.63), (0.29, 0.69, 0.43, 0.75)

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach Table 3.21 Pessimistic hesitant 4–polar fuzzy decision matrix Alternatives Hesitant 4–polar fuzzy criteria C1 Fibre properties   (0.20, 0.75, 0.45, 0.67), (0.34, 0.66, 0.51, 0.72), (0.40, 0.68, 0.56, 0.66) Tc1   (0.48, 0.78, 0.47, 0.29), (0.48, 0.78, 0.47, 0.29), (0.58, 0.93, 0.38, 0.50) Tc2   (0.87, 0.39, 0.47, 0.59), (0.74, 0.50, 0.74, 0.60), (0.53, 0.66, 0.57, 0.34) Tc3   (0.23, 0.54, 0.36, 0.79), (0.43, 0.63, 0.45, 0.80), (0.44, 0.68, 0.41, 0.89) Tc4   (0.13, 0.35, 0.58, 0.63), (0.13, 0.35, 0.58, 0.63), (0.33, 0.43, 0.62, 0.75) Tc5 Alternatives Tc1 Tc2 Tc3 Tc4 Tc5 Alternatives Tc1 Tc2 Tc3 Tc4 Tc5 Alternatives Tc1 Tc2 Tc3 Tc4 Tc5

Hesitant 4–polar fuzzy criteria C2 Influencing factors   (0.77, 0.47, 0.69, 0.71), (0.81, 0.39, 0.59, 0.71), (0.81, 0.39, 0.59, 0.71)   (0.47, 0.72, 0.51, 0.88), (0.58, 0.74, 0.68, 0.76), (0.49, 0.64, 0.60, 0.74)   (0.48, 0.68, 0.73, 0.19), (0.43, 0.79, 0.80, 0.21), (0.50, 0.82, 0.85, 0.17)   (0.59, 0.66, 0.40, 0.59), (0.53, 0.58, 0.61, 0.49), (0.53, 0.58, 0.61, 0.49)   (0.26, 0.14, 0.83, 0.79), (0.33, 0.27, 0.88, 0.75), (0.25, 0.33, 0.82, 0.77) Hesitant 4–polar fuzzy criteria C3 Construction techniques   (0.84, 0.53, 0.69, 0.72), (0.82, 0.66, 0.71, 0.82), (0.72, 0.61, 0.70, 0.80)   (0.72, 0.58, 0.47, 0.31), (0.75, 0.53, 0.39, 0.45), (0.85, 0.65, 0.67, 0.56)   (0.76, 0.48, 0.63, 0.82), (0.76, 0.48, 0.63, 0.82), (0.83, 0.54, 0.61, 0.93)   (0.68, 0.57, 0.48, 0.90), (0.63, 0.60, 0.43, 0.88), (0.73, 0.63, 0.50, 0.95)   (0.13, 0.78, 0.44, 0.63), (0.33, 0.81, 0.48, 0.74), (0.25, 0.82, 0.59, 0.86) Hesitant 4–polar fuzzy criteria C4 Finishing processes   (0.74, 0.93, 0.69, 0.77), (0.71, 0.86, 0.71, 0.89), (0.75, 0.89, 0.62, 0.83)   (0.62, 0.46, 0.78, 0.82), (0.58, 0.53, 0.77, 0.79), (0.65, 0.43, 0.70, 0.87)   (0.76, 0.58, 0.49, 0.82), (0.83, 0.58, 0.55, 0.93), (0.63, 0.62, 0.67, 0.91)   (0.45, 0.68, 0.79, 0.43), (0.53, 0.70, 0.78, 0.38), (0.55, 0.62, 0.89, 0.29)   (0.23, 0.67, 0.54, 0.63), (0.23, 0.67, 0.54, 0.63), (0.29, 0.69, 0.43, 0.75)

213

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Table 3.22 Weighted pessimistic hesitant 4–polar fuzzy decision matrix Alternatives

Tc1 Tc2 Tc3 Tc4 Tc5 Alternatives

Tc2 Tc3 Tc4 Tc5 Alternatives

Tc2 Tc3 Tc4 Tc5 Alternatives

Tc3 Tc4 Tc5

Hesitant 4–polar fuzzy criteria C3 with weight 0.2627 Construction techniques   (0.2207, 0.1392, 0.1813, 0.1891), (0.2154, 0.1734, 0.1865, 0.2154), (0.1891, 0.1602, 0.1839, 0.2102)   (0.1891, 0.1524, 0.1235, 0.0814), (0.1970, 0.1392, 0.1025, 0.1182), (0.2233, 0.1708, 0.1760, 0.1471)   (0.1997, 0.1261, 0.1655, 0.2154), (0.1997, 0.1261, 0.1655, 0.2154), (0.2180, 0.1419, 0.1602, 0.2443)   (0.1786, 0.1497, 0.1261, 0.2364), (0.1655, 0.1576, 0.1130, 0.2312), (0.1918, 0.1655, 0.1313, 0.2496)   (0.0342, 0.2049, 0.1156, 0.1655), (0.0867, 0.2128, 0.1261, 0.1944), (0.0657, 0.2154, 0.1550, 0.2259)

Tc1

Tc2

Hesitant 4–polar fuzzy criteria C2 with weight 0.2451 Influencing factors   (0.1887, 0.1152, 0.1691, 0.1740), (0.1985, 0.0956, 0.1446, 0.1740), (0.1985, 0.0956, 0.1446, 0.1740)   (0.1152, 0.1765, 0.1250, 0.2157), (0.1422, 0.1814, 0.1667, 0.1863), (0.1201, 0.1569, 0.1471, 0.1814)   (0.1176, 0.1667, 0.1789, 0.0466), (0.1054, 0.1936, 0.1961, 0.0515), (0.1226, 0.2010, 0.2083, 0.0417)   (0.1446, 0.1618, 0.0980, 0.1446), (0.1299, 0.1422, 0.1495, 0.1201), (0.1299, 0.1422, 0.1495, 0.1201)   (0.0637, 0.0343, 0.2034, 0.1936), (0.0809, 0.0662, 0.2157, 0.1838), (0.0613, 0.0809, 0.2010, 0.1887)

Tc1

Tc1

Hesitant 4–polar fuzzy criteria C1 with weight 0.2156 Fibre properties   (0.0431, 0.1617, 0.0970, 0.1445), (0.0733, 0.1423, 0.1100, 0.1552), (0.0862, 0.1466, 0.1207, 0.1423)   (0.1035, 0.1682, 0.1013, 0.0625), (0.1035, 0.1682, 0.1013, 0.0625), (0.1250, 0.2005, 0.0819, 0.1078)   (0.1876, 0.0841, 0.1013, 0.1272), (0.1595, 0.1078, 0.1595, 0.1294), (0.1143, 0.1423, 0.1229, 0.0733)   (0.0496, 0.1164, 0.0776, 0.1703), (0.0927, 0.1358, 0.0970, 0.1725), (0.0949, 0.1466, 0.0884, 0.1919)   (0.0280, 0.0755, 0.1250, 0.1358), (0.0280, 0.0755, 0.1250, 0.1358), (0.0711, 0.0927, 0.1337, 0.1617)

Hesitant 4–polar fuzzy criteria C4 with weight 0.2766 Finishing processes   (0.2047, 0.2572, 0.1909, 0.2130), (0.1964, 0.2379, 0.1964, 0.2462), (0.2075, 0.2462, 0.1715, 0.2296)   (0.1715, 0.1272, 0.2157, 0.2268), (0.1604, 0.1466, 0.2130, 0.2185), (0.1798, 0.1189, 0.1936, 0.2406)   (0.2102, 0.1604, 0.1355, 0.2268), (0.2296, 0.1604, 0.1521, 0.2572), (0.1743, 0.1715, 0.1853, 0.2517)   (0.1245, 0.1881, 0.2185, 0.1189), (0.1466, 0.1936, 0.2157, 0.1051), (0.1521, 0.1715, 0.2462, 0.0802)   (0.0636, 0.1853, 0.1494, 0.1743), (0.0636, 0.1853, 0.1494, 0.1743), (0.0802, 0.1909, 0.1189, 0.2075)

 (0.2233, 0.2154, 0.1839, 0.2496) ,  (0.2102, 0.2572, 0.2185, 0.2268), (0.2296, 0.2379, 0.2157, 0.2572),  (0.2075, 0.2462, 0.2462, 0.2517) .  H 4H N I S = (0.0280, 0.0755, 0.0776, 0.0625), (0.0280, 0.0755, 0.0970, 0.0625),

3.6 Hesitant m−Polar Fuzzy TOPSIS Approach

215

 (0.0711, 0.0927, 0.0819, 0.0733) ,  (0.0637, 0.0343, 0.0980, 0.0466), (0.0809, 0.0662, 0.1446, 0.0515),  (0.0613, 0.0809, 0.1446, 0.0417) ,  (0.0342, 0.1261, 0.1156, 0.0814), (0.0867, 0.1261, 0.1025, 0.1182),  (0.0657, 0.1419, 0.1313, 0.1471) ,  (0.0636, 0.1272, 0.1355, 0.1189), (0.0636, 0.1466, 0.1494, 0.1051),  (0.0802, 0.1189, 0.1189, 0.0802) .

Use Eqs. (3.8) and (3.9) to calculate the hesitant 4–polar fuzzy Euclidean distances of textiles or clothing from its hesitant 4–polar fuzzy positive ideal solution and hesitant 4–polar fuzzy negative ideal solution, that are calculated as follows: 

D E (Tc1 , H 4N I S ) = 0.1853,



D E (Tc2 , H 4N I S ) = 0.1642,



D E (Tc3 , H 4N I S ) = 0.1755,



D E (Tc4 , H 4N I S ) = 0.1460,



D E (Tc5 , H 4N I S ) = 0.1218.

D E (Tc1 , H 4PI S ) = 0.0912, D E (Tc2 , H 4PI S ) = 0.1288, D E (Tc3 , H 4PI S ) = 0.1225, D E (Tc4 , H 4PI S ) = 0.1319, D E (Tc5 , H 4PI S ) = 0.1789,











Use Eq. (3.10) to calculate the relative hesitant 4–polar fuzzy closeness coefficients E j of textiles or clothing. 

E 2 = 0.5605,



E 4 = 0.5253,

E 1 = 0.6700, E 3 = 0.5889,







E 5 = 0.4051. For the comparison, arrange the different types of textiles {Tcj | j = 1, 2, . . . , 5} according to the ranking of hesitant 4–polar fuzzy closeness coefficients. Hence, the ranking of types of textiles or clothing is as follows: Tc1 > Tc3 > Tc2 > Tc4 > Tc5 .

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The computer programming code for hesitant m–polar fuzzy TOPSIS approach is presented as follows: Computer programming code 1. clc 2. m=input(‘enter the total no of poles’); 3. H=input(‘enter the decision matrix in each entry as r×m’); 4. w=input(‘enter the weights as dimension 1×q’); 5. [u,q]=size(w); 6. [p,v]=size(H); 7. r = v/(q ∗ m); 8. if sum(w,2)==1 9. W=zeros(p,v); 10. for j=1:p 11. for k=1:q 12. for v1=k*r*m-(r*m-1):k*r*m 13. W(j,v1)=w(1,k).*H(j,v1); 14. end 15. end 16. end 17. W 18. mHPIS=zeros(1,v); mHNIS=ones(1,v); 19. for j=1:p 20. for v1=1:v 21. mHPIS(1,v1)=max(mHPIS(1,v1),W(j,v1)); 22. mHNIS(1,v1)=min(mHNIS(1,v1),W(j,v1)); 23. end 24. end 25. mHPIS 26. mHNIS 27. Y=zeros(p,v); Z=zeros(p,v); 28. for j=1:p 29. for v1=1:v 30. Y(j,v1)=(W(j,v1)-mHPIS(1,v1)). ˆ 2; 31. Z(j,v1)=(W(j,v1)-mHNIS(1,v1)). ˆ 2; 32. end 33. end 34. D_p=zeros(p,q);D_n=zeros(p,q); 35. for j=1:p 36. for k=1:q 37. for v1=k*r*m-(r*m-1):k*r*m 38. D_p(j,k)=D_p(j,k)+Y(j,v1); 39. D_n(j,k)=D_n(j,k)+Z(j,v1); 40. end 41. end 42. end 43. D=[sqrt(sum(D_p,2)./(r*m)) sqrt(sum(D_n,2)./(r*m))] 44. E=D(:,2)./sum(D,2) 45. end

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach

217

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach The hesitant m–polar fuzzy ELECTRE-I approach based on hesitant m–polar fuzzy set deals with MCGDM problems, in which A = {a1 , a2 , . . . , a p } is chosen as the set of different alternatives and {Ck |k = 1, 2, . . . , q} as the set of hesitant m–polar fuzzy criteria, which facilitate the management of hesitation, uncertainty and vagueness motivated by multipolar information. The structure of the hesitant m–polar fuzzy ELECTRE-I method, and steps from (i) to (iii) are same as described in Sect. 3.6. (iv) The hesitant m–polar fuzzy concordance set is defined as Yuv = {1 ≤ k ≤ q|ehuk (z) ≥ ehvk (z), u = v; u, v = 1, 2, . . . , p}, jk

where eh (z) =

r  l=1

jk

jk

jk

{( p1 ◦ eh (z) + p2 ◦ eh (z) + · · · + pm ◦ eh (z))l }.

(v) The hesitant m–polar fuzzy concordance indices are determined as yuv =



wk ,

k∈Yuv

therefore, the hesitant m–polar fuzzy concordance matrix is computed as ⎛

− y12 y13 ⎜ y21 − y23 ⎜ ⎜ Y = ⎜ y31 y32 − ⎜ .. .. .. ⎝ . . . y p1 y p2 y p3

⎞ · · · y1 p · · · y2 p ⎟ ⎟ · · · y3 p ⎟ ⎟. .. ⎟ ··· . ⎠ ··· −

(vi) The hesitant m–polar fuzzy discordance set is defined as Yuv = {1 ≤ k ≤ q|ehuk (z) ≤ ehvk (z), u = v; u, v = 1, 2, . . . , p}, jk

where eh (z) =

r  l=1

jk

jk

jk

{( p1 ◦ eh (z) + p2 ◦ eh (z) + · · · + pm ◦ eh (z))l }.

(vii) The hesitant m–polar fuzzy discordance indices are determined as  z uv =

k∈Z uv

max k

1 rm

max



1 rm

r  l=1 r  l=1

 

m  i=1 m  i=1





( pi ◦ ehuk (z) − pi ◦ ehvk (z))2 l

  , ( pi ◦ ehuk (z) − pi ◦ ehvk (z))2 l

therefore, the hesitant m–polar fuzzy discordance matrix ca be computed as

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

⎛

− z 12 z 13 ⎜ z 21 − z 23 ⎜ ⎜ Z = ⎜ z 31 z 32 − ⎜ .. .. .. ⎝ . . . z p1 z p2 z p3

⎞ · · · z1 p · · · z2 p ⎟ ⎟ · · · z3 p ⎟ ⎟. . ⎟ · · · .. ⎠ ···

−

Steps (viii)–(xii) are same as described in Sect. 3.4.

3.7.1 Site Selection for Farming Purposes Due to an increasing relevance of farming, land is considered to be a very significant element. Therefore, the selection of a site for farming purposes is a basic and fundamental analysis for the farmers. It is the fundamental step to initiate a farm when it has already been decided which crop should be grown. It also incorporates the selection of the suitable geographical location. This is the case with associate enterprising individuals and investors with acceptable dominates. The hesitant m–polar fuzzy set discusses the factors or criteria, which must be considered in the selection of the appropriate land site for farming purposes under the hesitant decision of farmers, investors and enterprising individuals. Site selection for farming purposes is based on a number of factors and criteria. To apply the concept of hesitant m–polar fuzzy model in a real life situation, consider S = {S f 1 , S f 2 , S f 3 , S f 4 , S f 5 } as the set of five different sites for farming which have to be analyzed and C = {c1 , c2 , c3 , c4 } as the set of four main factors or criteria to choose the site. For the evaluation the decision makers including farmers, investors and enterprising individuals focus on four main criteria or factors of sites such as climatic factor, socioeconomic factor, edaphic factor and other essential factors, which facilitate the hesitation and uncertainty motivated by multipolar information. 1. The “Climatic Factor” may include • “Rainfall”, which is the most frequent and familiar form of precipitation. The extent, measure and consistency of rainfall differ with area, climate and location types. It induces the influence of certain types of vegetation, growth of crop and its yield. • “Humidity”, which is the actual measure of water vapor in the air, considered as the percentage of the maximal capacity of water vapor it can dominate at usual temperature. It has different effects on the closing and opening of the stomata, which coordinates deficiency of water from the plant through photosynthesis and transpiration. • “Wind Pressure”, which is caused by differences in heating and due to the presence of pressure gradient on local and global scale. It compacts and the pressure raises, when the air close to the ground cools and it expands and drops pressure, when it warms.

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach

219

• “Temperature”, which has a great ascendancy on all growth processes of a plant such as respiration, photosynthesis, etc. At huge temperatures the alteration of photosyntheses is much more rapid and active so that plants tend to develop earlier. 2. The “Socioeconomic Factor” may include • “Infrastructure”, which is the requirement of large scale farming infrastructure to assure the highest yields per acre. Water movement towards the crops as well as away from the crops, is an analytical process to production. • “Land Tenancy”, which includes all models and plans of tenancy and ownership in any form. Land tenancy and land tenure affect the agricultural actions, activities and cropping patterns in many ways. The cultivators proceed the agricultural activities and farm management, by keeping in mind their benefits and occupancy duration on the land. • “Labor”, the availability and possibility of labor are major constraints in the use of agricultural land and cropping impressions of a region. It serves as all human maintenance except decision making and fundamentals. In decision making process of the farmer, the availability of labor, its quality and quantity at the periods of peak labor demand have a great significance. • “Marketing facility”, the accessibility and approach to the market is a major discussion. The concentration of agriculture and the production of crops descent as the location of cultivation takes away from the marketing centers. 3. The “Edaphic Factor” may include • “Structure”, to execute effectually as a growing medium, soils demand an open structure through the soil profile. For healthy plant growth, an effective soil structure allows water and air into the soil which are crucial. It improves drainage and lower the soil destruction due to excess surface run-off. • “Fertility”, which is the capacity of a soil to assist agricultural growth of plant, to maintain the plant surroundings and result in defend and homogeneous yields of immense quality. It supplies fundamental plant nutrients and water in sufficient amount and proportion for growth and reproduction of plant. • “Texture”,which is an essential soil exclusive that consequences storm water in filtration estimates. The texturing class of a soil is resolved by the ratio of clay, sand and slit. • “Porosity and Consistency”, soil porosity indicates the amount of pores and open spaces between soil particles. The soil compactness is the durability with which soil materials are held together or the resistance of soils to deformation. 4. The “Essential Factor” may include • “Environment”, the different operations of farming should not have a negative impact on the environment. The environment is not suitable or sometimes even harmful when the farming sites are close to an urban area. • “Government Policies”, it is in the interests of distinct governments to make policies that are convenient to attain growth in agriculture. It is possible to use

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

this influence and set up the farm in an area likely to gain from the performance of the policy. • “Biotic Interactions”, which reveal the existence or absence of some beneficial or harmful organisms. The natural population of certain organisms like bees and other pollinators have a great importance in site selection for farming purposes. • “Economic Agents”, this factor is considered as the most important to develop the agricultural business. It includes the benefits, terms of lease or acquisition and cost. All these criteria or factors are assessed by decision makers, who are responsible for the selection of site. Due to their collective decision, each factor is further classified by multi-polar information and evaluated by three different hesitant values assigned by decision makers, who are free to choose any membership value from the interval [0, 1]. Thus decision makers assign hesitant values as described in Table 3.23. Obviously, the count of hesitant 4–polar fuzzy elements in general is not comparable in all hesitant 4–polar fuzzy-sets. In order to gain efficiency and accuracy, they extend the smallest membership value such that the lengths of all hesitant 4–polar fuzzy elements become equal, because the required policy wants to select the site with the pessimistic prediction. For this reason they show pessimistic response and improve the hesitant 4–polar fuzzy data by adding the minimal values as mentioned in Table 3.24. (i) Tabular representation of hesitant 4–polar fuzzy decision matrix is given by Table 3.23. (ii) The normalized weights assigned to each criteria are given as follows: wk = (0.2501, 0.2458, 0.2633, 0.2408). (iii) The weighted pessimistic hesitant 4–polar fuzzy decision matrix is calculated in Table 3.25. (iv) The hesitant 4–polar fuzzy concordance set is calculated in Table 3.26. (v) The hesitant 4–polar fuzzy concordance matrix is calculated as follows: ⎛

− ⎜ 0.5091 ⎜ Y =⎜ ⎜ 0.5134 ⎝ 0.5134 0.7592

0.4909 − 0.7542 0.5134 0.5134

0.4866 0.2458 − 0.2458 0.2458

0.4866 0.4866 0.7542 − 0.7499

⎞ 0.2408 0.4866 ⎟ ⎟ 0.7542 ⎟ ⎟. 0.2501 ⎠ −

(vi) The hesitant 4–polar fuzzy discordance set is calculated in Table 3.27.

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach Table 3.23 Hesitant 4–polar fuzzy decision matrix Sites Hesitant 4-polar fuzzy factor as criteria C1 Climatic factors   (0.81, 0.65, 0.45, 0.69), (0.78, 0.66, 0.51, 0.74), (0.77, 0.62, 0.52, 0.67) Sf1   (0.68, 0.59, 0.67, 0.89), (0.61, 0.54, 0.63, 0.70) Sf2   (0.85, 0.79, 0.57, 0.87), (0.74, 0.60, 0.60, 0.80) Sf3   (0.43, 0.84, 0.66, 0.79), (0.49, 0.83, 0.75, 0.82), (0.54, 0.78, 0.71, 0.85) Sf4   (0.67, 0.75, 0.58, 0.75), (0.60, 0.63, 0.62, 0.75) Sf5 Sites Sf1 Sf2 Sf3 Sf4 Sf5 Sites Sf1 Sf2 Sf3 Sf4 Sf5 Sites Sf1 Sf2 Sf3 Sf4 Sf5

Hesitant 4-polar fuzzy factor as criteria C2 Socio-economic factors   (0.78, 0.57, 0.69, 0.46), (0.81, 0.69, 0.65, 0.49)   (0.67, 0.72, 0.51, 0.77), (0.68, 0.74, 0.64, 0.76), (0.66, 0.70, 0.60, 0.74)   (0.48, 0.68, 0.73, 0.19), (0.46, 0.69, 0.78, 0.21), (0.50, 0.82, 0.85, 0.27)   (0.49, 0.76, 0.39, 0.79), (0.53, 0.68, 0.41, 0.78)   (0.46, 0.44, 0.73, 0.79), (0.43, 0.57, 0.88, 0.75), (0.55, 0.53, 0.82, 0.77) Hesitant 4-polar fuzzy factor as criteria C3 Edaphic factors   (0.34, 0.83, 0.69, 0.38), (0.45, 0.86, 0.71, 0.40)   (0.72, 0.58, 0.77, 0.51), (0.75, 0.53, 0.79, 0.45), (0.80, 0.65, 0.67, 0.50)   (0.83, 0.54, 0.61, 0.93), (0.76, 0.48, 0.63, 0.82)   (0.78, 0.47, 0.49, 0.70), (0.73, 0.50, 0.49, 0.88), (0.73, 0.63, 0.50, 0.85)   (0.63, 0.78, 0.54, 0.66), (0.71, 0.81, 0.49, 0.64), (0.72, 0.82, 0.59, 0.76) Hesitant 4-polar fuzzy factor as criteria C4 Essential factors   (0.54, 0.73, 0.65, 0.77), (0.71, 0.86, 0.71, 0.89), (0.75, 0.89, 0.62, 0.83)   (0.62, 0.46, 0.71, 0.82), (0.58, 0.53, 0.77, 0.79)   (0.66, 0.78, 0.89, 0.82), (0.83, 0.58, 0.55, 0.93), (0.63, 0.62, 0.67, 0.91)   (0.58, 0.58, 0.49, 0.43), (0.55, 0.62, 0.89, 0.29)   (0.63, 0.67, 0.54, 0.63), (0.29, 0.69, 0.43, 0.75)

221

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.24 Pessimistic hesitant 4–polar fuzzy decision matrix Sites Hesitant 4-polar fuzzy factor as criteria C1 Climatic factors   (0.81, 0.65, 0.45, 0.69), (0.78, 0.66, 0.51, 0.74), (0.77, 0.62, 0.52, 0.67) Sf1   (0.68, 0.59, 0.67, 0.89), (0.61, 0.54, 0.63, 0.70), (0.61, 0.54, 0.63, 0.70) Sf2   (0.85, 0.79, 0.57, 0.87), (0.74, 0.60, 0.60, 0.80), (0.74, 0.60, 0.60, 0.80) Sf3   (0.43, 0.84, 0.66, 0.79), (0.49, 0.83, 0.75, 0.82), (0.54, 0.78, 0.71, 0.85) Sf4   (0.67, 0.75, 0.58, 0.75), (0.60, 0.63, 0.62, 0.75), (0.60, 0.63, 0.62, 0.75) Sf5 Sites Sf1 Sf2 Sf3 Sf4 Sf5 Sites Sf1 Sf2 Sf3 Sf4 Sf5 Sites Sf1 Sf2 Sf3 Sf4 Sf5

Hesitant 4-polar fuzzy factor as criteria C2 Socio-economic factors   (0.78, 0.57, 0.69, 0.46), (0.78, 0.57, 0.69, 0.46), (0.81, 0.69, 0.65, 0.49)   (0.67, 0.72, 0.51, 0.77), (0.68, 0.74, 0.64, 0.76), (0.66, 0.70, 0.60, 0.74)   (0.48, 0.68, 0.73, 0.19), (0.46, 0.69, 0.78, 0.21), (0.50, 0.82, 0.85, 0.27)   (0.49, 0.76, 0.39, 0.79), (0.53, 0.68, 0.41, 0.78), (0.53, 0.68, 0.41, 0.78)   (0.46, 0.44, 0.73, 0.79), (0.43, 0.57, 0.88, 0.75), (0.55, 0.53, 0.82, 0.77) Hesitant 4-polar fuzzy factor as criteria C3 Edaphic factors   (0.34, 0.83, 0.69, 0.38), (0.34, 0.83, 0.69, 0.38), (0.45, 0.86, 0.71, 0.40)   (0.72, 0.58, 0.77, 0.51), (0.75, 0.53, 0.79, 0.45), (0.80, 0.65, 0.67, 0.50)   (0.83, 0.54, 0.61, 0.93), (0.76, 0.48, 0.63, 0.82), (0.76, 0.48, 0.63, 0.82)   (0.78, 0.47, 0.49, 0.70), (0.73, 0.50, 0.49, 0.88), (0.73, 0.63, 0.50, 0.85)   (0.63, 0.78, 0.54, 0.66), (0.71, 0.81, 0.49, 0.64), (0.72, 0.82, 0.59, 0.76) Hesitant 4-polar fuzzy factor as criteria C4 Essential factors   (0.54, 0.73, 0.65, 0.77), (0.71, 0.86, 0.71, 0.89), (0.75, 0.89, 0.62, 0.83)   (0.62, 0.46, 0.71, 0.82), (0.62, 0.46, 0.71, 0.82), (0.58, 0.53, 0.77, 0.79)   (0.66, 0.78, 0.89, 0.82), (0.83, 0.58, 0.55, 0.93), (0.63, 0.62, 0.67, 0.91)   (0.58, 0.58, 0.49, 0.43), (0.58, 0.58, 0.49, 0.43), (0.55, 0.62, 0.89, 0.29)   (0.63, 0.67, 0.54, 0.63), (0.29, 0.69, 0.43, 0.75), (0.29, 0.69, 0.43, 0.75)

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach

223

Table 3.25 Weighted pessimistic hesitant 4–polar fuzzy decision Sites

Sf1 Sf2 Sf3 Sf4 Sf5 Sites

Sf1 Sf2 Sf3 Sf4 Sf5 Sites

Sf1 Sf2 Sf3 Sf4 Sf5 Sites

Sf1 Sf2 Sf3 Sf4 Sf5

Hesitant 4-polar fuzzy factor as criteria C1 with weight 0.2501 Climatic factors   (0.2026, 0.1626, 0.1125, 0.1726), (0.1951, 0.1651, 0.1276, 0.1851), (0.1926, 0.1551, 0.1301, 0.1676)   (0.1701, 0.1476, 0.1676, 0.2226), (0.1526, 0.1351, 0.1576, 0.1751), (0.1526, 0.1351, 0.1576, 0.1751)   (0.2126, 0.1976, 0.1426, 0.2176), (0.1851, 0.1501, 0.1501, 0.2001), (0.1851, 0.1501, 0.1501, 0.2001)   (0.1075, 0.2101, 0.1651, 0.1976), (0.1225, 0.2076, 0.1876, 0.2051), (0.1351, 0.1951, 0.1776, 0.2126)   (0.1676, 0.1876, 0.1451, 0.1876), (0.1501, 0.1576, 0.1551, 0.1876), (0.1501, 0.1576, 0.1551, 0.1876) Hesitant 4-polar fuzzy factor as criteria C2 with weight 0.2458 Socio-economic factors   (0.1917, 0.1401, 0.1696, 0.1131), (0.1917, 0.1401, 0.1696, 0.1131), (0.1991, 0.1696, 0.1598, 0.1204)   (0.1647, 0.1770, 0.1254, 0.1893), (0.1671, 0.1819, 0.1573, 0.1868), (0.1622, 0.1721, 0.1475, 0.1819)   (0.1180, 0.1671, 0.1794, 0.0467), (0.1131, 0.1696, 0.1917, 0.0516), (0.1229, 0.2016, 0.2089, 0.0664)   (0.1204, 0.1868, 0.0959, 0.1942), (0.1303, 0.1671, 0.1008, 0.1917), (0.1303, 0.1671, 0.1008, 0.1917)   (0.1131, 0.1082, 0.1794, 0.1942), (0.1057, 0.1401, 0.2163, 0.1843), (0.1352, 0.1303, 0.2016, 0.1893) Hesitant 4-polar fuzzy factor as criteria C3 with weight 0.2633 Edaphic Factors   (0.0895, 0.2185, 0.1817, 0.1001), (0.0895, 0.2185, 0.1817, 0.1001), (0.1185, 0.2264, 0.1869, 0.1053)   (0.1896, 0.1527, 0.2027, 0.1343), (0.1975, 0.1395, 0.2080, 0.1185), (0.2106, 0.1711, 0.1764, 0.1316)   (0.2185, 0.1422, 0.1606, 0.2449), (0.2001, 0.1264, 0.1659, 0.2159), (0.2001, 0.1264, 0.1659, 0.2159)   (0.2054, 0.1238, 0.1290, 0.1843), (0.1922, 0.1316, 0.1290, 0.2317), (0.1922, 0.1659, 0.1316, 0.2238)   (0.1659, 0.2054, 0.1422, 0.1738), (0.1869, 0.2133, 0.1290, 0.1685), (0.1896, 0.2159, 0.1553, 0.2001) Hesitant 4-polar fuzzy factor as criteria C4 with weight 0.2408 Essential factors   (0.1300, 0.1758, 0.1565, 0.1854), (0.1710, 0.2071, 0.1710, 0.2143), (0.1806, 0.2143, 0.1493, 0.1999)   (0.1493, 0.1108, 0.1710, 0.1975), (0.1493, 0.1108, 0.1710, 0.1975), (0.1397, 0.1276, 0.1854, 0.1902)   (0.1589, 0.1878, 0.2143, 0.1975), (0.1999, 0.1397, 0.1324, 0.2239), (0.1517, 0.1493, 0.1613, 0.2191)   (0.1397, 0.1397, 0.1180, 0.1035), (0.1397, 0.1397, 0.1180, 0.1035), (0.1324, 0.1493, 0.2143, 0.0698)   (0.1517, 0.1613, 0.1300, 0.1517), (0.0698, 0.1662, 0.1035, 0.1806), (0.0698, 0.1662, 0.1035, 0.1806)

Table 3.26 Hesitant 4–polar fuzzy concordance set v 1 2 3 Y1v Y2v Y3v Y4v Y5v

− {2, 3} {1, 3} {1, 3} {1, 2, 3}

{1, 4} − {1, 3, 4} {1, 3} {1, 3}

{2, 4} {2} − {2} {2}

4

5

{2, 4} {2, 4} {1, 3, 4} − {2, 3, 4}

{4} {2, 4} {1, 3, 4} {1} −

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

Table 3.27 Hesitant 4–polar fuzzy discordance set v 1 2 3 Z 1v Z 2v Z 3v Z 4v Z 5v

− {1, 4} {2, 4} {2, 4} {4}

{2, 3} − {2} {2, 4} {2, 4}

{1, 3} {1, 3, 4} − {1, 3, 4} {1, 3, 4}

4

5

{1, 3} {1, 3} {2} − {1}

{1, 2, 3} {1, 3} {2} {2, 3, 4} −

(vii) The hesitant 4–polar fuzzy discordance matrix can calculated as follows: ⎛

− ⎜ 0.7357 ⎜ Z =⎜ ⎜ 0.5634 ⎝ 0.7737 0.9074

1.0000 − 1.0000 1.0000 1.0000

1.0000 0.7269 − 0.8667 0.8059

1.0000 0.9763 1.0000 − 0.5249

⎞ 1.0000 0.9502 ⎟ ⎟ 1.0000 ⎟ ⎟. 1.0000 ⎠ −

(viii) The hesitant 4–polar fuzzy concordance level y = 0.5000, and hesitant 4–polar fuzzy discordance level z = 1.1995 are calculated. (ix) The hesitant 4–polar fuzzy concordance dominance matrix is calculated as follows: ⎛ ⎞ − 0 0 0 0 ⎜1 − 0 0 0⎟ ⎜ ⎟ ⎟ R=⎜ ⎜ 1 1 − 1 1 ⎟. ⎝1 1 0 − 0⎠ 1 1 0 1 − (x) The hesitant 4–polar fuzzy discordance dominance matrix is calculated as follows: ⎛ ⎞ − 1 1 1 1 ⎜1 − 1 1 1⎟ ⎜ ⎟ ⎟ S=⎜ ⎜ 1 1 − 1 1 ⎟. ⎝1 1 1 − 1⎠ 1 1 1 1 − (xi) An aggregated hesitant 4–polar fuzzy lows: ⎛ − 0 ⎜1 − ⎜ T =⎜ ⎜1 1 ⎝1 1 1 1

dominance matrix is calculated as fol⎞ 0 0 0 0 0 0⎟ ⎟ − 1 1⎟ ⎟. 0 − 0⎠ 0 1 −

(xii) According to outranking values of aggregated hesitant 4–polar fuzzy dominance matrix the sites for farming have the following relation as shown in Fig. 3.2.

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach

225

Fig. 3.2 Outranking relation of sites for farming

Sf 1

Sf 2

Sf 3

Sf 5

Sf 4

Table 3.28 Comparison of sites for farming Comparison of Yuv CS of bricks

Z uv

yuv

z uv

ruv

suv

tuv

Ranking

(S f 1 , S f 2 )

{1, 4}

{2, 3}

0.4909

1

0

1

0

Incomparable

(S f 1 , S f 3 )

{2, 4}

{1, 3}

0.4866

1

0

1

0

Incomparable

(S f 1 , S f 4 )

{2, 4}

{1, 3}

0.4866

1

0

1

0

Incomparable

(S f 1 , S f 5 )

{4}

{1, 2, 3} 0.2408

1

0

1

0

Incomparable

(S f 2 , S f 1 )

{2, 3}

{1, 4}

0.5091

0.7357

1

1

1

Sf2 → Sf1

(S f 2 , S f 3 )

{2}

{1, 3, 4} 0.2458

0.7269

0

1

0

Incomparable

(S f 2 , S f 4 )

{2, 4}

{1, 3}

0.4866

0.9763

0

1

0

Incomparable

(S f 2 , S f 5 )

{2, 4}

{1, 3}

0.4866

0.9502

0

1

0

Incomparable

(S f 3 , S f 1 )

{1, 3}

{2, 4}

0.5134

0.5634

1

1

1

(S f 3 , S f 2 )

{1, 3, 4} {2}

0.7542

1

1

1

1

Sf3 → Sf1 Sf3 → Sf2

(S f 3 , S f 4 )

{1, 3, 4} {2}

0.7542

1

1

1

1

Sf3 → Sf4

(S f 3 , S f 5 )

{1, 3, 4} {2}

0.7542

1

1

1

1

Sf3 → Sf5

(S f 4 , S f 1 )

{1, 3}

{2, 4}

0.5134

0.7737

1

1

1

Sf4 → Sf1

(S f 4 , S f 2 )

{1, 3}

{2, 4}

0.5134

1

1

1

1

Sf4 → Sf2

(S f 4 , S f 3 )

{2}

{1, 3, 4} 0.2458

0.8667

0

1

0

Incomparable

(S f 4 , S f 5 )

{1}

{2, 3, 4} 0.2501

1

0

1

0

Incomparable

(S f 5 , S f 1 )

{1, 2, 3} {4}

0.7592

0.9074

1

1

1

Sf5 → Sf1

(S f 5 , S f 2 )

{1, 3}

{2, 4}

0.5134

1

1

1

1

Sf5 → Sf2

(S f 5 , S f 3 )

{2}

{1, 3, 4} 0.2458

0.8059

0

1

0

Incomparable

(S f 5 , S f 4 )

{2, 3, 4} {1}

0.5249

1

1

1

Sf5 → Sf4

0.7499

Hence, the site S f 3 is best for farming purposes as compared to others. The whole procedure is summarized, and its comparison is shown in Table 3.28.

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3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models

The computer programming code of hesitant m–polar fuzzy ELECTRE-I approach is presented as follows: Computer programming code 1. clc 2. n=input(‘no. of alternatives against m–polar hesitant fuzzy sets or hesitant m– polar fuzzy sets’); 3. k=input(‘no.of criteria’s’); 4. m=input(‘no. of poles’); 5. r=input(‘no. of hesitation values’); 6. D= input(‘enter the m–polar hesitant fuzzy or hesitant m– polar fuzzy decision matrix nxkxm’); 7. w=input(‘enter the weights’); 8. Rr=(1:n);Cr=1:m∗r∗k;Cw=1:k;w_g=zeros(1,k); 9. W=zeros(n,m∗k);Sm=zeros(n,k);Y_uv=zeros(n,n∗k); Z_uv=zeros(n,n∗k); 10. for p=1:n 11. l=1; 12. for Cr=1:m∗r∗k 13. W(p,Cr)=D(p,Cr.*)w(l,1); 14. if mod(Cr,m∗r)==0 15. l=l+1; 16. end 17. end 18. end 19. W 20. for p=1:n 21. l=1; 22. for Cr=1:m∗r∗k 23. Sm(p,l)=Sm(p,l)+W(p,Cr); 24. if mod(Cr,m∗r)==0 25. l=l+1; 26. end 27. end 28. end 29. Q=Sm’ 30. Q=Q(:)’; 31. for p=1:n 32. for j=1:k∗n 33. l=mod(j,k); 34. if l==0 35. l=k; 36. end 37. if Sm(p,l)≥ Q(1,j) 38. Y_uv(p,j)=1; 39. end 40. if Sm(p,l)≤ Q(1,j) 41. Z_uv(p,j)=1; 42. end 43. end 44. end 45. Y=zeros(n,n);fprintf(‘\n concordance Set Y_uv =\n’)

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach Computer programming code 46. for p=1:n 47. v=0; 48. for j=1:k∗n 49. if mod(j,k)==1 50. v=v+1; 51. end 52. l=mod(j,k); 53. if l==0 54. l=k; 55. end 56. if u==v 57. if l==1 58. fprintf(‘ ’) 59. end 60. elseif p∼=v 61. if l==1 62. fprintf(‘ { ’) 63. c=0; 64. end 65. if Y_uv(p,j)==1; 66. c=c+1; 67. fprintf(‘%d,’,l) 68. end 69. if l==k & c==0 70. fprintf(‘ ,’,l) 71. end 72. if l==k 73. fprintf(‘\b} ’) 74. end 75. end 76. end 77. fprintf(‘\n’) 78. end 79. fprintf(‘\n discordance Set Z_uv =\n’) 80. for u=1:n 81. v=0; 82. for j=1:k∗n 83. if mod(j,k)==1 84. v=v+1; 85. end 86. l=mod(j,k); 87. if l==0 88. l=k; 89. end 90. fprintf(‘\nY=\n’) 91. for u=1:n 92. for v=1:n 93. if u==v 94. fprintf(‘ ’) 95. else

227

228

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models Computer programming code 96. fprintf‘%.4f ’,Y(u,v)) 97. end 98. end 99. fprintf(‘ \n ’) 100. end 101. fprintf(‘ \n Discordance Set Z uv =\n’) 102. for u=1:n 103. v=0; 104. for j=1:k∗n 105. if mod(j,k)==1 106. v=v+1; 107. end 108. l=mod(j,k); 109. if l==0 110. l=k; 111. end 112. if u==v 113. if l==1 114. fprintf(‘ ’) 115. end 116. else if u =v 117. if l==1 118. fprintf( ‘ ’) 119. c=0; 120. end 121. if Z uv (u,j)==1; 122. c=c+1; 123. fprintf(‘ %d, ’,l) 124. end 124. if l==k & c==0 126. fprintf(‘ , ’,l) 127. end 128. if l==k 129. fprintf(‘ ’) ¯ 130. end 131. end 132. end 133. fprintf(‘%.4f ’,Y(u,v)) 134. end 135. end 136. end 137. fprintf(‘\n ’) 138. end 139. z=zeros(n,ˆ 2,m∗r∗k); Cr=1:m∗r∗k; v=0; 140. for u=1:n 141. for q=1:n 142. v=v+1; 143. z(v,Cr)=(W(u,Cr)-W(j,Cr)). ˆ 2;

3.7 Hesitant m−Polar Fuzzy ELECTRE-I Approach Computer programming code 144. end 145. end 146. A=zeros(n ˆ 2,k);g=0; s=0; C=zeros(n ˆ 2,1);B=zeros(n,k);Z1=zeros(n,n); 147. for p=1:n ˆ 2 148. x=1; 149. for Cr=1:m∗r∗k 150. A(p,x)=A(p,x)+z(p,Cr); 151. if mod(Cr,m∗)==0 152. x=x+1; 153. end 154. A(p,:)=sqrt(A(p,:)/m∗r); 155. C(p,1)=max(A(p,:)); 156. if mod(p,n)==1 157. g=g+1; 158. end 159. for f=1:k 160. s=s+1; 161. B(g,s)=A(p,f); 162. end 163. t=mod(p,n); 164. if t==0 165. t=n; 166. end 167. Z1(g,t)=C(p,1); 168. if mod(p,n)==0 169. s=0; 170. end 171. end 172. D=zeros(n,n); 173. for p=1:n 174. q=0; 175. for j=1:k∗n 176. if mod(j,k)==1 177. q=q+1; 178. end 179. l=mod(j,k); 180. if l==0 181. l=k; 182. end 183. if Z_uv(p,j)==1 184. D(p,q)=max(D(p,q),B(p,j)); 185. end 186. end 187. end 188. for u=1:n 189. for v=1:n 190. if u∼=v 191. Z(u,v)=D(u,v)/Z1(u,v);

229

230

3 Introducing Hesitancy: TOPSIS and ELECTRE-I Models Computer programming code 192. end 193. end 194. end 195. fprintf(‘\nZ=\n’) 196. for u=1:n 197. for v=1:n 198. if u==v 199. fprintf(‘ ’) 200. else 201. fprintf(‘%.4f ’,Z(u,v)) 202. end 203. end 204. fprintf(‘ \n ’) 205. end 206. a=sum(Y); b=sum(a); a1=sum(Z); b1=sum(a1); R=zeros(n,n);S=zeros(n,n); 207. y_bar=b/(n∗(n-1)) 208. z_bar=b1/(n∗(n-1)) 209. for u=1:n 210. for v=1:n 211. if u∼=v 212. if Y(u,v)≥ y_bar 213. R(u,v)=1; 214. end 215. if Z(u,v)< z_bar 216. S(u,v)=1; 217. end 218. end 219. end 220. end 221. fprintf(‘\nR=\n’) 222. for u=1:n 223. for v=1:n 224. if u==v 225. fprintf(‘’) 226. else 227. fprintf(‘%d ’,R(u,v)) 228. end 229. end 230. fprintf(‘ \n ’) 231. end 232. fprintf(‘\nS=\n’) 233. for u=1:n 234. for v=1:n 235. if u==v 236. fprintf(‘’) 237. else 238. fprintf(‘%d ’,S(u,v))

3.8 Conclusion Computer programming code 239. end 240. end 241. fprintf(‘\n ’) 242. end 243. T=R.∗S; fprintf(‘\nT=\n’) 243. for u=1:n 244. for v=1:n 245. if u==v 246. fprintf(‘247. else 248. fprintf(‘%d 249. end 250. end 251. fprintf(‘ \n ’) 252. end 253. G=digraphs(T) 254. plot(G)

231

’) ’,T(u,v))

3.8 Conclusion In a short period of time, the powerful simplicity of hesitant fuzzy sets has fascinated many researchers, as a number of cases in different real-world problems depend upon a hesitant framework. Hesitant structures are generally preferred as compared to clear-cut situations. The hesitation on membership degrees can be manipulated using different types of information. In order to enable the practitioners to avail themselves of multi-polar information under hesitancy and to facilitate the management of hesitation, uncertainty and vagueness motivated by multi-polar information, the concepts of m–polar hesitant fuzzy sets and hesitant m–polar fuzzy sets have been presented. These models are capable of incorporating knowledge with m different numerical or fuzzy values in a hesitant environment. However in the present case of interest, these approaches are incapable of dealing with problems having multi-polar data. Basic operations and some of their properties are investigated. From a practical perspective and to handle MCGDM problems, m–polar hesitant fuzzy TOPSIS and m–polar hesitant fuzzy ELECTRE-I approaches have been presented, which are able to assess the alternatives depending on the hesitant situations of decision makers, under the conditions of huge data with multi-polar information. Further, hesitant m– polar fuzzy TOPSIS and hesitant m–polar fuzzy ELECTRE-I approaches have also been developed. Proposed methods have the fascinating advantages and characteristic, such as they are exposed as more flexible methods to be evaluated in multi-fold ways according to the practical interests and requirements than the existing generalizations of hesitant fuzzy sets. Proposed methods are used to handle the data having multi-polar information suggested by decision makers into account. The final decision of presented methods depends upon the optimistic or pessimistic opinion of decision maker, so it is limited when one has to take the decision independently.

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The differences and comparative analysis of discussed approaches have also been presented. Finally, the proposed techniques are applied to real life problems, algorithms are developed and their computer programming codes are presented by using MATLAB.

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Chapter 4

Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

In this chapter, the theory of ELECTRE methods is borrowed to construct two eminent and highly practical decision making approaches for multi-polar ambiguous information. The structure and potential of multi-polar (m–polar) fuzzy sets suit well to delineate the multi-polar ambiguous information. Therefore, this chapter keenly delivers the procedures of the m–polar fuzzy ELECTRE I method and m– polar fuzzy ELECTRE II method by deeply implementing the outranking principle. Although, the previous chapter thoroughly elaborates the methodology of m–polar fuzzy ELECTRE I approach that competently incorporates the multi-polar inexact data for all multi-criteria decision making scenarios where the decision has to be made by a single expert. There exist numerous real life situations where decision, depending on multiple attributes, has to be made by a panel of experts and the decision making in such scenarios is referred as multi-criteria group decision making (MCGDM). The multi-criteria decision making (MCDM) ELECTRE I strategy, elaborated in previous chapter, is incompetent to address the group decision making problems. Therefore, this chapter is designed to unfold some decision making approaches based on ELECTRE theory for group decision making scenarios. A major contribution of this chapter is the redesigning of m–polar fuzzy ELECTRE I method for group decision making scenarios. The presented m–polar fuzzy ELECTRE I method is demonstrated by a practical case study for the selection of the insulation scheme for the exterior walls of a building. Another goal of this chapter is to employ the striking theory of ELECTRE II method to deliver the procedure of m–polar fuzzy ELECTRE II method that efficiently deals with the multi-polar information in group decision making environments. The presented method is demonstrated by a case study to select the best location for a nuclear power plant. This chapter also provides a comprehensive comparison of both presented approaches to highlight the insights and limitation of proposed techniques.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_4

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4.1 Introduction Decision making, being an essential part of our daily life chores, contributes to the timely and smooth progression of any task. Decision making can be referred as a procedure of choosing the optimal alternative from the available choices. If the selection of the optimal alternative is governed by multiple decisive factors then such type of decision making is referred as MCDM. If a group of decision making experts is evaluating the potential of available alternatives regarding multiple attributes to figure out the best alternative such case is referred as MCGDM. A number of decision making methodologies, including TOPSIS [34], ELECTRE [22], PROMETHEE [14], AHP [34], have been developed to process the exact information to meet an authentic decision. ELECTRE family [22, 33] is a collection of outranking methodologies that deeply follow the outranking principle to find out the most significant alternative using the pairwise comparison of alternatives. The initiative for the establishment of ELECTRE methods was taken by Benayoun et al.[13] who presented the procedure of ELECTRE I approach. Later on, Grolleau and Tergny [24] proposed a modified variant of ELECTRE approach, namely, ELECTRE II method, after observing the shortcoming of ELECTRE I method. Duckstein and Gershon [21] solved the vegetation management problem using the ELECTRE II approach. Hokkanen et al. [26] employed the ELECTRE II method to select the most suitable solid waste treatment system for the Uusimaa region, Finland. Jun et al. [28] applied the ELECTRE II method to opt the appropriate site for solar/wind hybrid power station. Wen et al. [41] and Liu et al. [30] deployed the ELECTRE II procedure to assess the coal gasification techniques and to solve rank reversal problem, respectively. Later on, the researchers realized that human decisions are not exact always rather there exist ambiguity in human opinions. The revelation of the imprecision of human decisions highlighted the failure of crisp set theory and its decision making approaches to deal with ambiguous information. This created the need of a model that can present the ambiguous human decisions competently to solve the practical decision making problems. The credit for the establishment of the fuzzy set theory, a modern extension of classical set theory, goes to Zadeh [44] who laid the foundation of the fuzzy set owing to characteristic function whose range was restricted to the unit interval [0, 1]. Sevkli [36] employed the ELECTRE I procedure to opt the best supplier. Hatami-Marbini and Tavana [25] proposed a MCGDM approach employing the theory of ELECTRE I method under the fuzzy environment. Rouyendegh and Erkan [32] employed the fuzzy ELECTRE approach for staff selection. Shojaie et al. [38] opted the best supplier for a pharmaceutical company of Iran using the fuzzy ELECTRE method. Kaya and Kahraman [29] integrated the fuzzy AHP and ELECTRE methods to assess the environmental impacts. Govindan et al. [23] utilized the ELECTRE II approach to rank the third party logistics service provider. Dascal ˘ [20] redesigned the ELECTRE II method for fuzzy numbers. Mir et al. [31] integrated the fuzzy ELECTRE II method and fuzzy TOPSIS method to figure out the best lithium extraction process from brine and seawater. Akram et al. [8] and Alghamdi et al. [10] extended several multi-criteria decision making techniques, including TOPSIS

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and ELECTRE methods, under bipolar fuzzy environment. Shumaiza et al. [39] presented a modified version of ELECTRE method, namely, bipolar fuzzy ELECTRE II method, to capture the inexact bipolar information. Atanassov [11] introduced the concept of intuitionistic fuzzy set, an extension of fuzzy set, that together with a membership grade μ, make use of a non-membership grade ν, and they are jointly subjected to the condition μ + ν < 1. Although the mathematical performance of intuitionistic fuzzy sets is quite good, still they fail to capture many situations where the sum of membership and non-membership degrees exceeds 1 (at some option). Yager [42, 43] presented the less stringent concept of Pythagorean fuzzy sets for which the condition μ + ν ≤ 1 imposed by intuitionistic fuzzy sets is relaxed to μ2 + ν 2 ≤ 1. The aptitude of intuitionistic fuzzy set seemed to be inept after observing the multi-polarity of real life phenomena. Chen et al. [15] made the remarkable contribution in this regard by designing the structure of a novel model, namely, m–polar fuzzy set. The considerable edge of m–polar fuzzy set is the extension of range of membership function to [0, 1]m which enables it to capture the multi-polar ambiguous information covering m distinct aspects, simultaneously. Later on, Akram et al. [9] presented the m–polar fuzzy ELECTRE I technique to address the MCDM problems and unfolded the application of the proposed strategy in location selection and performance evaluation. Adeel et al. [1, 2] proposed the m–polar fuzzy linguistic ELECTRE I method and m–polar fuzzy linguistic TOPSIS method to solve MCGDM problems with verbal assessments. Akram et al. [7] extended the PROMETHEE method under the eminent structure of m–polar fuzzy set and employed the AHP method to assign weights to the decision criteria. Waseem et al. [40] proposed several aggregation operators on the basis of Hamacher operations to aggregate the m–polar fuzzy data. The ELECTRE methods seek the optimal alternative by outranking the inferior alternatives on the basis of crisp outranking relations and outranking graphs. The extra-ordinary performance and excellence of ELECTRE methods in decision making environments captivate to develop some advance and novel strategies on the grounds of ELECTRE theory. To enhance the authenticity, the daily life decisions are usually made on the preferences of a group of efficient and skilled decision making experts. In practical scenarios, it is difficult to find a person having expertise in all relevant fields. Therefore, the group decision making is preferred as it accounts the assessment of different experts efficient in their relevant fields. An m– polar fuzzy ELECTRE I technique [9] competently cumulates the multi-polar inexact evaluation but its calibre is limited to process the evaluations of a single expert. The broader structure and multi-polar characteristic of the m–polar fuzzy set motivated to extend the m–polar fuzzy ELECTRE I method for MCGDM problem. Thus, the presented approach integrates the advantageous theory of ELECTRE I approach with the authenticity of group decision making within the context of m–polar fuzzy sets. An m–polar fuzzy ELECTRE I method may not provide the ranking of the alternative due to the incomparable pairs of alternatives. Furthermore, the ELECTRE II approach (employing two outranking relations and five threshold values) is acknowledged to be a modified and improved variant of ELECTRE family that subsides the

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4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

limitations of ELECTRE I strategy by providing the complete ranking list in all situations. Here, we review some basic notions to understand the theory of m–polar fuzzy sets. Definition 4.1 ([15]) An m–polar fuzzy set ξ on a non-empty set X is a mapping ξ : X → [0, 1]m . The membership value of every element x ∈ X is denoted by ξ(x) = ( p1 ◦ ξ(x), p2 ◦ ξ(x), . . . , pm ◦ ξ(x)), where pi ◦ ξ : [0, 1]m → [0, 1] is defined as the i-th projection mapping. Definition 4.2 ([40]) The score degree of an m–polar fuzzy number ξ = ( p1 ◦ ξ, p2 ◦ ξ, . . . , pm ◦ ξ) is defined as follows: s(ξ) =

 m  1  ph ◦ ξ m h=1

(4.1)

Definition 4.3 ([40]) For any three m–polar fuzzy numbers ξ = ( p1 ◦ ξ, p2 ◦ ξ, . . . , pm ◦ ξ), ξ1 = ( p1 ◦ ξ1 , p2 ◦ ξ1 , . . . , pm ◦ ξ1 ) and ξ2 = ( p1 ◦ ξ2 , p2 ◦ ξ2 , . . . , pm ◦ ξ2 ), the elementary operations are defined as follows: 1. ξ1 ⊕ ξ2 = ( p1 ◦ ξ1 + p1 ◦ ξ2 − ( p1 ◦ ξ1 )( p1 ◦ ξ2 ), . . . , pm ◦ ξ1 + pm ◦ ξ2 − ( pm ◦ ξ1 )( pm ◦ ξ2 )), 2. ξ1 ⊗ ξ2 = ( p1 ◦ ξ1 . p1 ◦ ξ2 , p2 ◦ ξ1 . p2 ◦ ξ2 , . . . , pm ◦ ξ1 . pm ◦ ξ2 ), 3. μ(ξ) = (1 − (1 − p1 ◦ ξ)μ , 1 − (1 − p2 ◦ ξ)μ , . . . , 1 − (1 − pm ◦ ξ)μ ), μ > 0 4. ξ μ = ((( p1 ◦ ξ)μ , ( p2 ◦ ξ)μ , . . . , ( pm ◦ ξ)μ ), μ > 0. Definition 4.4 ([40]) For a finite collection ξi = ( p1 ◦ ξi , p2 ◦ ξi , . . . , pm ◦ ξi ), i = 1, 2, . . . , v, of m–polar fuzzy numbers, the m–polar fuzzy weighted averaging operator is defined as follows: m F W AOα (ξ1 , ξ2 , . . . , ξv ) = α1 ξ1 ⊕ α2 ξ2 ⊕ · · · ⊕ αv ξv   v v v    = 1− (1 − p1 ◦ ξi )αi , 1 − (1 − p2 ◦ ξi )αi , . . . , 1 − (1 − pm ◦ ξi )αi , i=1

i=1

i=1

(4.2) where the weight vector α = (α1 , α2 , . . . , αv ) represents the normalized weights of these m–polar fuzzy numbers.

4.2 An m–Polar Fuzzy ELECTRE I Method

241

4.2 An m–Polar Fuzzy ELECTRE I Method This section unfolds the detailed method of m–polar fuzzy ELECTRE I technique to deal with MCGDM problems in which the opinion of each expert in the decision making panel is expressed in terms of m–polar fuzzy numbers. Further, pairwise comparison of the available alternatives is performed in reference to concordance and discordance sets to check out their relative superiority or inferiority. Further, a flowchart diagram (Fig. 4.2) has been included to present a pictorial and casual view of the proposed methodology. Consider a panel of v experts E = {e1 , e2 , . . . , ev } which are appointed to assess the aptitude of r alternatives enclosed in the set A = {x1 , x2 , . . . , xr }. The normalized weights of the decision making experts are given by the weight vector α = (α1 , α2 , . . . , αv ). The decision making panel has considered the set of s multipolar decision criteria T = {t1 , t2 , . . . , ts } that influence the performance of the feasible alternatives. Furthermore, these selected criteria depend on m distinct attributes. To meet the accurate decision, the complete procedure is elaborated in the following steps: 1. Construction of independent decision matrices. Each expert in the decision making panel observes the performance and aptitude of the available alternatives in accordance to s multi-polar criteria and expresses the imprecise decision with the help of m–polar fuzzy number. Let (u1) (u2) (um) z i(u) j = (z i j , z i j , . . . , z i j ) be the m–polar fuzzy number assigned to the alternative xi by the expert eu with respect to the criteria t j . The kth pole z i(uk) of j (u) the m–polar fuzzy number z i j denotes the efficiency of the alternative xi with respect to kth attribute of the criteria t j according to expert eu . These m–polar fuzzy numbers, assigned by the expert eu , are organized in the matrix Z (u) as follows: ⎛

t1

t2

⎜(z (u1) , z (u2) , · · · , z (um) ) (z (u1) , z (u2) , · · · , z (um) ) x1 ⎜ 11 11 11 12 12 12 ⎜ ⎜ ⎜ (u1) (u2) (um) (u1) (u2) (um) ⎜(z x , z , · · · , z ) (z , z , · · · , z 21 21 21 22 22 22 ) Z (u) = 2 ⎜ ⎜ . ⎜ . . . ⎜ . . ⎜ . ⎜ . . (um) (u1) (u2) (um) ⎝ (u1) (u2) xr (zr 1 , zr 1 , · · · , zr 1 ) (zr 2 , zr 2 , · · · , zr 2 )

···

ts

⎞ (u1) (u2) (um) · · · (z 1s , z 1s , · · · , z 1s )⎟ ⎟ ⎟ ⎟ (u1) (u2) (um) ⎟ · · · (z 2s , z 2s , · · · , z 2s )⎟ ⎟ . ⎟ ⎟ . .. ⎟ . . ⎟ . ⎟ (u1) (u2) (um) ⎠ · · · (zr s , zr s , · · · , zr s )

Similarly, v decision matrices Z (1) , Z (2) , . . . , Z (v) are obtained that represent the decisions of the expert e1 , e2 , . . . , ev , respectively. 2. Construction of aggregated m–polar fuzzy decision matrix. The individual opinions of the decision making experts need to be merged in order to get a perfect and group satisfactory decision which is admissible for all experts. For the sake of aggregation, the m–polar fuzzy weighted averaging operator is used to evaluate z i j = (z i1j , z i2j , . . . , z imj ) as follows:

242

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets (1)

(2)

(v)

z i j = α1 z i j ⊕ α2 z i j ⊕ · · · ⊕ αv z i j   v  v  v  αu      (u2) αu (um) αu 1 − z i(u1) 1 − z 1 − z = 1− , 1 − , . . . , 1 − . j ij ij u=1

u=1

(4.3)

u=1

The cumulative decisions, obtained by aggregating the independent decisions, are arranged to form the aggregated m–polar fuzzy decision matrix as follows: t1

⎛

···

t2

ts

⎞

1 , z 2 , . . . , z m ) (z 1 , z 2 , · · · , z m ) · · · (z 1 , z 2 , . . . , z m ) x1 ⎜(z 11 11 12 1s ⎟ 11 12 12 1s 1s ⎟ ⎜ ⎟ ⎜ ⎜ 1 2 m ) (z 1 , z 2 , · · · , z m ) · · · (z 1 , z 2 , . . . , z m )⎟ ⎟ ⎜ (z , z , . . . , z x Z = 2 ⎜ 21 21 . 21 22 2s ⎟ 22 22 2s 2s ⎟ . . . ⎜ . . ⎟ . .. . . . ⎜ . . . . ⎜ ⎟ ⎠ ⎝ xr (zr11 , zr21 , . . . , zrm1 ) (zr12 , zr22 , · · · , zrm2 ) · · · (zr1s , zr2s , . . . , zrms )

3. Assignment of criteria weights. All the decision criteria may not be equally important. The panel of decision making experts assigns m–polar fuzzy weights to the decision criteria according to their significance and impact on the performance of alternatives. An m–polar fuzzy weight of a particular criterion t j assigned by the expert eu is represented by (u1) (u2) (um) w (u) ). Finally, the cumulative m–polar fuzzy weight j = (w j , w j , . . . , w j m 1 2 w j = (w j , w j , . . . , w j ) of criterion t j can be obtained by aggregating the individual weights assigned by all decision makers in the light of m–polar fuzzy weighted averaging operator as follows: (1)

w j = α1 w j ⎛

(2)

⊕ α2 w j

(v)

⊕ · · · ⊕ αdv w j

⎞

v  

v  v       (u1) αu (u2) αu (um) αu ⎠ = ⎝1 − ,1 − ,...,1 − . 1 − wj 1 − wj 1 − wj u=1 u=1 u=1

(4.4)

The normalized weight γ j of the criterion t j can be obtained from the aggregated m–polar fuzzy weight w j of that criterion by the following formula: γj =

s(w j ) , s  s(w j )

(4.5)

j=1

where s(w j ) presents the score of aggregated m–polar fuzzy weight w j of the criterion t j which can be determined as follows: m 

s(w j ) =

h=1

w hj

m

.

(4.6)

4.2 An m–Polar Fuzzy ELECTRE I Method

243

4. Construction of aggregated weighted m–polar fuzzy decision matrix. To obtain aggregated weighted m–polar fuzzy decision matrix , the aggregated weights of the criteria are multiplied with the corresponding entries of the aggregated m–polar fuzzy decision matrix. An entry yi j = (yi1j , yi2j , . . . , yimj ) of the aggregated weighted m–polar fuzzy decision matrix Y is computed as follows: yi j = (z i1j .w 1j , z i2j .w 2j , . . . , z imj .w mj ).

(4.7)

The aggregated weighted m–polar fuzzy decision matrix Y = (yi j )r ×s can be represented as follows: ⎛

t1

···

t2

m 1 2 x1⎜(y11 , y11 , . . . , y11 )

m 1 2 (y12 , y12 , . . . , y12 )

ts

⎞

m 1 2 (y1s , y1s , . . . , y1s )⎟

··· ⎟ ⎜ ⎟ ⎜ ⎜ 1 2 m m m ⎟ 1 2 1 2 Y = x2⎜(y21 , y21 , . . . , y21 ) (y22 , y22 , . . . , y22 ) · · · (y2s , y2s , . . . , y2s )⎟ . ⎟ ⎜ .⎜ . . . . ⎟ . .. .. .. .. .⎜ ⎟ ⎠ ⎝ 1 2 m m 1 2 1 2 xr (yr 1 , yr 1 , . . . , yr 1 ) (yr 2 , yr 2 , . . . , yr 2 ) · · · (yr s , yr s , . . . , yrms )

5. Construction of m–polar fuzzy concordance and discordance sets. To identify the optimal alternative, the theory of ELECTRE method is equipped with concordance and discordance sets to identify the superiority or inferiority between any pair of available alternatives. The alternatives are compared in pairs on the basis of their score degrees defined as follows: 1 s(yi j ) = m

 m 

 yihj

.

(4.8)

h=1

A m–polar fuzzy number with relatively higher score degree is considered to be superior and dominant than the others. The conditions for the determination of concordance and discordance sets are described as follows: (i) An m–polar fuzzy concordance set F pq represents the collection of indices of all those criteria that suggest the dominance of the alternative x p over the alternative xq . An m–polar fuzzy concordance set F pq , defined on the vital notion of score degree, can be evaluated as follows: F pq = { j : s(y pj ) ≥ s(yq j ), p = q; p, q = 1, 2, . . . , r ; j = 1, 2, . . . , s}. (4.9) (ii) On the other hand, the m–polar fuzzy discordance set G pq is utilized to unfold the inferiority of the alternative x p over the alternative xq with respect to decision criteria. An m–polar fuzzy discordance set G pq represents the

244

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

collection of indices of all those criteria that suggest the inferiority of the alternative x p over the other alternative xq . An m–polar fuzzy discordance set G pq , defined on the vital notion of score degree, can be evaluated as follows: G pq = { j : s(y pj ) < s(yq j ), p = q; p, q = 1, 2, . . . , r ; j = 1, 2, . . . , s}. (4.10) 6. Construction of m–polar fuzzy concordance matrix. An m–polar fuzzy concordance index f pq is a measure to exhibit the amount of dominance of the alternative x p over the alternative xq . The information embedded in m–polar fuzzy concordance sets is processed to evaluate the m–polar fuzzy concordance indices to highlight the level of superiority of the alternatives over the other. An m–polar fuzzy concordance index f pq is determined by using the following formula: f pq =



γj.

(4.11)

j∈F pq

All these m–polar fuzzy concordance indices are arranged in a matrix which is called m–polar fuzzy concordance matrix . An m–polar fuzzy concordance matrix F = ( f pq )r ×r can be represented as follows: ⎛

x1

x2

x1 ⎜ − f 12 ⎜ ⎜ ⎜ x2 ⎜ f 21 − ⎜ .. F = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜ f (r −1)1 f (r −1)2 ⎜ ⎜ ⎝ xr fr 1 fr 2

· · · xr −1 · · · f 1(r −1) · · · f 2(r −1) .. .. . . ··· − · · · fr (r −1)

xr

⎞

f 1r ⎟ ⎟ ⎟ ⎟ f 2r ⎟ .. ⎟ ⎟ . ⎟ ⎟ f (r −1)r ⎟ ⎟ ⎟ ⎠ −

7. Construction of m–polar fuzzy discordance matrix. An m–polar fuzzy discordance index g pq is a measure to reveal the inferiority of an alternative x p over the other the alternative xq . The information, compiled in m–polar fuzzy discordance sets , is processed to find out the discordance indices to measure the amount of subordinance of an alternative over the other. The discordance index g pq can be determined as follows: max d(y pj , yq j )

g pq =

j∈G pq

max d(y pj , yq j ) j

,

p = q.

(4.12)

4.2 An m–Polar Fuzzy ELECTRE I Method

245

Here, d(y pj , yq j ) represents the Euclidean distance between the m–polar fuzzy numbers which can be computed by using the following formula:  d(y pj , yq j ) =

(y 1pj − yq1 j )2 + (y 2pj − yq2 j )2 + · · · + (y mpj − yqmj )2 m

.

(4.13)

An m–polar fuzzy discordance matrix, comprising all the m–polar fuzzy discordance indices, can be represented as follows: ⎛

x1

x2

x1 ⎜ − g12 ⎜ ⎜ ⎜ x2 ⎜ g21 − ⎜ .. G = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜g(r −1)1 g(r −1)2 ⎜ ⎜ ⎝ xr gr 1 gr 2

· · · xr −1

xr

· · · g1(r −1) · · · g2(r −1) .. .. . . ··· − · · · gr (r −1)

⎞

g1r ⎟ ⎟ ⎟ ⎟ g2r ⎟ .. ⎟ ⎟ . . ⎟ ⎟ g(r −1)r ⎟ ⎟ ⎟ ⎠ −

8. Evaluation of m–polar fuzzy concordance dominance matrix. The m–polar fuzzy concordance indices are compared with a threshold value, called m–polar fuzzy concordance level, to check the effectiveness of the m– polar fuzzy concordance indices and outrank the inept alternatives. An m–polar fuzzy concordance level is evaluated by taking the average of all m–polar fuzzy concordance indices as follows: f =

r  1 r (r − 1) p=1, p =q

r 

f pq .

(4.14)

q=1, q = p

This threshold value is treated as a standard to check the outranking calibre of the feasible alternatives. The comparison of m–polar fuzzy concordance indices and m–polar fuzzy concordance level gives the m–polar fuzzy concordance dominance matrix H = (h pq )r ×r . The entries of m–polar fuzzy concordance dominance matrix can be evaluated by the following formula in accordance with m–polar fuzzy concordance level:  h pq =

1, f pq ≥ f ; 0, f pq < f .

(4.15)

The alternatives with higher m–polar concordance index than m–polar concordance level are considered to be preferable over the other. An m–polar fuzzy concordance dominance matrix H can be represented as follows:

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4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

⎛

x1

x2

x1 ⎜ − h 12 ⎜ ⎜ ⎜ x2 ⎜ h 21 − ⎜ .. H = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜h (r −1)1 h (r −1)2 ⎜ ⎜ ⎝ xr hr 1 hr 2

· · · xr −1

xr

· · · h 1(r −1) · · · h 2(r −1) .. .. . . ··· − · · · h r (r −1)

⎞

h 1r ⎟ ⎟ ⎟ ⎟ h 2r ⎟ .. ⎟ ⎟ . ⎟ ⎟ h (r −1)r ⎟ ⎟ ⎟ ⎠ −

9. Evaluation of m–polar fuzzy discordance dominance matrix. To determine the worth of discordance indices, the m–polar fuzzy discordance indices are checked on the basis of a threshold value, namely, m–polar fuzzy discordance level. An m–polar fuzzy discordance level g is computed by averaging all the m–polar fuzzy discordance indices as follows: g=

r  1 r (r − 1) p=1, p =q

r 

g pq .

(4.16)

q=1, q = p

An m–polar fuzzy discordance level establishes a standard value that is sufficient to suggest that an alternative is enough inferior than the other alternative to be outranked. The alternatives, possessing the m–polar fuzzy discordance indices less than the discordance level, are considered to be significant to outrank the compared alternative. An m–polar fuzzy discordance dominance matrix L = (l pq )r ×r can be obtained by employing the following formula:  l pq =

1, g pq < g; 0, g pq ≥ g.

(4.17)

Further, the m–polar fuzzy discordance dominance matrix L can be represented as follows: x1 x2 · · · xr −1 xr ⎞ ⎛ x1 ⎜ − l12 ⎜ ⎜ ⎜ x2 ⎜ l21 − ⎜ .. L = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜l(r −1)1 l(r −1)2 ⎜ ⎜ ⎝ xr lr 1 lr 2

· · · l1(r −1) · · · l2(r −1) . .. . .. ··· − · · · lr (r −1)

l1r ⎟ ⎟ ⎟ ⎟ l2r ⎟ .. ⎟ ⎟ . ⎟ ⎟ l(r −1)r ⎟ ⎟ ⎟ ⎠ −

4.2 An m–Polar Fuzzy ELECTRE I Method

247

10. Construction of m–polar fuzzy aggregated dominance matrix. The corresponding entries of the m–polar fuzzy concordance dominance matrix and m–polar fuzzy discordance dominance matrix are multiplied to obtain the m–polar fuzzy aggregated dominance matrix that exhibits the combined effect of the m–polar fuzzy concordance and discordance dominance matrices to highlight the outranking relation among any pair of alternatives. An m–polar fuzzy aggregated dominance matrix M = (m pq )r ×r can be evaluated by using the following formula: (4.18) m pq = h pq · l pq . An m–polar fuzzy aggregated dominance matrix can be represented as follows: ⎛

x1

x2

x1 ⎜ − m 12 ⎜ ⎜ ⎜ x2 ⎜ m 21 − ⎜ .. M = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜m (r −1)1 m (r −1)2 ⎜ ⎜ ⎝ xr mr 1 mr 2

· · · xr −1 · · · m 1(r −1) · · · m 2(r −1) .. .. . . ··· − · · · m r (r −1)

xr

⎞

m 1r ⎟ ⎟ ⎟ ⎟ m 2r ⎟ .. ⎟ ⎟ . ⎟ ⎟ m (r −1)r ⎟ ⎟ ⎟ ⎠ −

A non-zero entry m pq of m–polar aggregated dominance matrix indicates x p is strictly preferable over xq , or x p is outranking xq . 11. Sketching and exploitation of outranking graph. The outranking relations , embedded in m–polar aggregated dominance matrix, are portrayed by a directed graph, called outranking graph. In the outranking graph, the alternatives are represented by vertices and the arc between these vertices are drawn using m–polar aggregated dominance matrix in the light of following rules: • If m pq = 1 and m q p = 0 then a directed arc, pointing toward xq , is drawn from x p . This arc indicates that the alternative x p , being superior than xq , is outranking the alternative xq (i.e., x p xq ). • If m pq = 1 and m q p = 1 then a two way directed arc is drawn between alternatives x p and xq . This two way arc indicates that both of these alternatives exhibit same performance and thus they are indifferent to each other (i.e., x p ≈ xq ). • If m pq = 0 and m q p = 0 then no edge is drawn between the alternatives x p and xq . The absence of arc between any pair of alternative indicates that the corresponding alternatives are incomparable.

248

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Fig. 4.1 Graphical representation of relations in outranking decision graph

The visual representation of these relations is presented by the Fig. 4.1. Finally, identify the best alternative by examining the outranking graph. The flowchart diagram of the m–polar fuzzy ELECTRE I method is portrayed in the Fig. 4.2.

4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall This study explores the case for renovation of the external wall of a three-floor office building, situated in south of Beijing occupying the area of 1151 m2 . This building has been in practical use for office work since July 2004. Recently, Zhang et al. [45] determined the thermal indices and construction scheme of external walls through building information modeling (BIM). The analysis revealed that the heat transfer coefficient of the external wall was 0.87. Further, the existing insulation wall structure consists of the extruded polystyrene board, cement mortar, reinforced concrete and lime cement mortar. These thickness of these layers along with their thermal coefficients are represented in Fig. 4.3. According to Beijing public building energy efficiency design standard, the standard value for the heat transfer coefficient of external walls is at most 0.50. Thus, the original design was inefficient to meet the local energy-saving standards. This study is inclined to opt the most appropriate insulation scheme for the external walls to present an energy efficient design for the building. For this task, a panel of four decision-makers has been designated to keenly observe the performance and limitations of available insulation schemes. The details of each decision expert in the decision making panel is as follows:

4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall

Fig. 4.2 Flowchart of m–polar fuzzy ELECTRE I method

249

250

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Fig. 4.3 Characteristics of original insulating wall

• The expert e1 is a skilled and proficient technician who has been participating in many thermal insulation construction projects for 11 years. • The expert e2 is a constructor who possesses more than 10 years experience in many on-site construction for civil buildings. • The expert e3 is an experienced supervisor whose task is to check and ensure the quality of civil construction. • The expert e4 is chosen from academia who has worthy knowledge about the qualities and limitations of insulation materials. The weight vector of the experts is given by the weight vector α = (0.25, 0.25, 0.25, 0.25). The panel of decision makers considered the following insulating schemes as potential alternatives to replace the original insulating wall. Note that all of these insulation schemes possess the same thickness. • Scheme x1 : Expanded polystyrene (EPS) board as external insulation layer EPS insulation board, possessing organic closed cell structure, serves as an excellent thermal insulation structure for low-rise buildings. The significant features of this scheme are low technical requirements of workers and low construction cost. However, the considerable disadvantages of EPS board are being combustible, poor bonding with cement materials, severe heat expansion and cold shrinkage and production of poisonous gas after being burned. Further, heat transfer coefficient for this insulation scheme is 0.40. The exterior to interior structure of this insulation scheme is demonstrated by the help of Fig. 4.4. • Scheme x2 : Extruded polystyrene (XPS) board as external insulation layer XPS board of high thickness, possessing organic closed cell, is an excellent choice

4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall

251

Fig. 4.4 Structure of insulating wall for scheme x1

to be used as heat insulation material. Some major drawbacks of this scheme are being easily crackable , poor air permeability, complex construction procedure and poor bonding with hydrophobic surface. On the other hand, this scheme dominates owing to its relatively low water absorption rate, being crisp and hard to bend. Further, heat transfer coefficient for this insulation scheme is 0.35. The exterior to interior structure of this insulation scheme is given by Fig. 4.5. • Scheme x3 : Rigid polyurethane (PU) board as external insulation layer PU board, possessing organic interlinked closed-pore structure, is known for its tremendous properties of load bearing, heat preservation, fire prevention, waterproof, bonding ability, durability and ease of construction in short period. On the other hand, the uneven surface of PU board and its high cost can limit the practical use of this insulation technique. Further, heat transfer coefficient for this insulation scheme is 0.27. The exterior to interior structure of this insulation scheme is elaborated through Fig. 4.6. • Scheme x4 : Rock wool board as external insulation layer Rock wool board, possessing inorganic porous fibrous structure, is preferred due to small thermal conductivity coefficient, good bonding ability and good performance of heat absorption. A noticeable disadvantage of this insulation scheme is long, complex, highly dusty and health risky procedure of renovation of rock wool board. Further, high water absorption rate can reduce the service life of this insulation layer. Further, heat transfer coefficient for this insulation scheme is 0.43. The exterior to interior structure of this insulation scheme is demonstrated by Fig. 4.7. From the number of factors affecting the performance of insulation schemes, the decision-makers have short listed the following four significant m–polar criteria.

252 Fig. 4.5 Structure of insulating wall for scheme x2

Fig. 4.6 Structure of insulating wall for scheme x3

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall

253

Fig. 4.7 Structure of insulating wall for scheme x4

These m–polar fuzzy criteria along with their attributes are represented by the Fig. 4.8. The details of these m–polar criteria are as follows: t1 : Thermal insulation property The thermal insulation property of any insulation scheme depends on further three attributes, namely, heat transfer coefficient, thermal resistance and thermal conductivity. Heat transfer coefficient indicates the amount of heat transferred per unit area per unit time when the temperature difference of air on two sides of wall is 1 ◦ C under the condition of stable heat transfer. According to Beijing public building energy efficiency design standard, the value of heat transfer coefficient should not exceed 0.50. Thermal conductivity refers to the intrinsic ability of a material to transfer or conduct heat due to random molecular motion across the temperature gradient. Furthermore, thermal resistance is defined as the ratio of the temperature difference between two faces of a material to the rate of heat flow per unit area. t2 : Technique This criteria depends on primary attributes of on-site quality controllability, level of technical convenience and quality stability of service life. The attribute on-site quality controllability exhibits the level of convenience to ensure on-site quality performance. Level of technical convenience refers to the level of conveniences for on-site construction for renovation technique and convenience of employing technicians with relevant skills. The quality stability of service life looks for the quality stability of each insulation scheme during its service life. t3 : Durability The durability of insulation wall depends on waterproof ability, fireproof ability

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4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Fig. 4.8 Multi-polar criteria and their poles

and windproof ability. Waterproof ability refers to the ability to prevent the damage caused by rain, hail, frost, moisture, etc. Fireproof ability indicates the significant resistance ability to fire. Windproof ability is characterized by the resistance ability to wind pressure. t4 : Economic efficiency The attributes affecting the economic efficiency of the insulation schemes are unit cost, maintenance cost and Repairing/renovation cost. Unit cost is characterized by the average cost of construction, laborers wages and raw material of the insulation layer. Maintenance cost refers to the average cost spent on the maintenance of the insulation wall to increase its service life. Repairing/renovation cost is the average cost required for the repairing, refitting and renovation of the insulation wall. The solution of the elaborated multi-criteria group decision making problem is obtained in following steps: 1. The independent decision matrices Z (1) , Z (2) , Z (3) and Z (4) of the decision makers in the panel are represented by Tables 4.1, 4.2, 4.3 and 4.4, respectively. 2. Now, the independent opinions need to be aggregated to find a group satisfactory solution which is admissible of all experts. The task of the aggregation of m–polar fuzzy numbers is performed in the light of m–polar fuzzy weighted averaging operator. The aggregated m–polar fuzzy decision matrix Z , evaluated by Eq. 4.3, is shown in Table 4.5. 3. All the criteria may not equally influence the performance of alternatives. Therefore, the honorable experts manage to assign m–polar fuzzy weights to all deci-

4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall

255

Table 4.1 An m–polar fuzzy decision matrix of expert e1 Z (1) t1 t2 t3 (0.65, 0.70, 0.73) (0.71, 0.92, 0.88) (1.00, 0.72, 0.68) (0.64, 0.65, 0.69)

x1 x2 x3 x4

(0.50, 0.500.50) (0.30, 0.30, 0) (0.70, 1, 0.70) (0, 0, 0.70)

t4

(0.30, 0.30, 0.30) (0.50, 0.50, 0.50) (0.50, 0.50, 0.50) (0.50, 0.70, 0.30)

Table 4.2 An m–polar fuzzy decision matrix of expert e2 Z (2) t1 t2 t3 (0.65, 0.73, 0.69) (0.71, 0.89, 0.93) (1.00, 0.77, 0.64) (0.64, 0.71, 0.58)

x1 x2 x3 x4

(0.70, 0.50, 0.50) (0.30, 0.50, 0) (1.00, 0.70, 0.70) (0.30, 0.50, 0.30)

t4

(0.30, 0, 0) (0.50, 0.30, 0.70) (0.70, 0.70, 1.00) (0.30, 1.00, 0)

Table 4.3 An m–polar fuzzy decision matrix of expert e3 Z (3) t1 t2 t3 (0.65, 0.63, 0.59) (0.71, 0.9, 0.88) (1.00, 0.79, 0.62) (0.64, 0.59, 0.67)

x1 x2 x3 x4

(0.50, 0.50, 0.30) (0.30, 0.30, 0.50) (0.70, 1.00, 0.70) (0.30, 0, 0.50)

(0.65, 0.72, 0.63) (0.71, 0.86, 0.93) (1.00, 0.73, 0.65) (0.64, 0.61, 0.57)

(0.50, 0.70, 0.50) (0.50, 0.30, 0.30) (1.00, 0.70, 0.50) (0.50, 0, 0.30)

(1.00, 0.95, 0.87) (0.59, 0.61, 0.70) (0.31, 0.19, 0.24) (0.45, 0.60, 0.54)

t4

(0.50, 0, 0.30) (0.70, 0.30, 0.50) (1.00, 0.50, 0.70) (0.50, 0.70, 0.30)

Table 4.4 An m–polar fuzzy decision matrix of expert e4 Z (4) t1 t2 t3 x1 x2 x3 x4

(1.00, 0.91, 0.88) (0.59, 0.71, 0.65) (0.31, 0.17, 0.20) (0.45, 0.57, 0.41)

(1.00, 0.89, 0.82) (0.59, 0.52, 0.67) (0.31, 0.22, 0.17) (0.45, 0.51, 0.57)

t4

(0.30, 0.30, 0.30) (0.70, 0.50, 0.50) (0.70, 0.70, 0.70) (0.50, 0.70, 0.30)

(1.00, 0.90, 0.86) (0.59, 0.55, 0.69) (0.31, 0.26, 0.20) (0.45, 0.49, 0.44)

Table 4.5 Aggregated m–polar fuzzy decision matrix Z

t1

t2

t3

t4

x1

(0.6500, 0.6973, 0.6643)

(0.5599, 0.5599, 0.4561)

(0.3565, 0.1633, 0.2347)

(1.0000, 0.9161, 0.8592)

x2

(0.7100, 0.8946, 0.9083)

(0.3565, 0.3565, 0.2308)

(0.6127, 0.4084, 0.5599)

(0.5900, 0.6047, 0.6781)

x3

(1.0000, 0.7542, 0.6482)

(1.0000, 1.0000, 0.6591)

(1.0000, 0.6127, 1.0000)

(0.3100, 0.2107, 0.2029)

x4

(0.6400, 0.6431, 0.6313)

(0.2965, 0.1591, 0.4793)

(0.4561, 1.0000, 0.2347)

(0.4500, 0.5447, 0.4944)

sion criteria in accordance to their significance in multi-criteria group decision making problem. The independent opinions are merged by the help of m–polar fuzzy weighted averaging operator, defined in Eq. 4.4, to find m–polar fuzzy

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4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Table 4.6 Weights of criteria Criteria

Expert opinions e1

Aggregated m–polar

Normalized weights

t1

(0.45, 0.59, 0.49)

e2 (0.47, 0.43, 0.51)

e3 (0.38, 0.46, 0.43)

e4 (0.52, 0.35, 0.49)

(0.4573, 0.4648, 0.4808) 0.2258

Fuzzy weights

t2

(0.28, 0.36, 0.18)

(0.21, 0.32, 0.35)

(0.25, 0.36, 0.23)

(0.24, 0.21, 0.19)

(0.2454, 0.3151, 0.2407) 0.1290

t3

(0.54, 0.29, 0.41)

(0.42, 0.36, 0.38)

(0.43, 0.51, 0.48)

(0.38, 0.49, 0.54)

(0.4459, 0.4195, 0.4561) 0.2128

t4

(0.94, 0.83, 0.90)

(0.93, 0.82, 0.79)

(0.94, 0.87, 0.96)

(0.85, 0.91, 0.89)

(0.9216, 0.8624, 0.9020) 0.4324

Table 4.7 Aggregated weighted m–polar fuzzy decision matrix Y

t1

t2

t3

t4

x1

(0.2972, 0.3241, 0.3194)

(0.1374, 0.1764, 0.1098)

(0.1590, 0.0685, 0.1070)

(0.9216, 0.7900, 0.7750)

x2

(0.3247, 0.4158, 0.4367)

(0.0875, 0.1123, 0.0556)

(0.2732, 0.1713, 0.2554)

(0.5437, 0.5215, 0.6116)

x3

(0.4573, 0.3506, 0.3117)

(0.2454, 0.3151, 0.1586)

(0.4459, 0.2570, 0.4561)

(0.2857, 0.1817, 0.1830)

x4

(0.2927, 0.2989, 0.3035)

(0.0728, 0.0501, 0.1154)

(0.2034, 0.4195, 0.1070)

(0.4147, 0.4697, 0.4459)

Table 4.8 Score degrees Alternatives t1 x1 x2 x3 x4

0.3136 0.3924 0.3732 0.2984

t2

t3

t4

0.1412 0.0851 0.2397 0.0794

0.1115 0.2333 0.3863 0.2433

0.8289 0.5589 0.2168 0.4434

Table 4.9 The m–polar fuzzy concordance sets x1 x2 x1 x2 x3 x4

– {1, 3} {1, 2, 3} {3}

{2, 4} – {2, 3} {3}

x3

x4

{4} {1, 4} – {4}

{1, 2, 4} {1, 2, 4} {1, 2, 3} –

weights and normalized weights of the decision criteria. The individual m–polar fuzzy weight, aggregated m–polar weights and normalized weights of the criteria are organized in Table 4.6. 4. The aggregated weighted m–polar decision matrix, evaluated in the light of Eq. 5.9, is represented by Table 4.7. 5. The score degrees of the m–polar numbers in the aggregated weighted m–polar fuzzy decision matrix, evaluated with the help of Eq. 4.8, are given by Table 4.8. These score degrees are compared in accordance to Eqs. 4.9 and 4.10 to construct the m–polar fuzzy concordance and discordance sets, respectively. The m–polar fuzzy concordance sets are tabulated in Table 4.9. The m–polar fuzzy discordance sets are arranged in Table 4.10.

4.3 Case Study: Selection of Best Insulating Scheme for Exterior Wall Table 4.10 An m–polar fuzzy discordance sets x1 x2 x1 x2 x3 x4

– {2, 4} {4} {1, 2, 4}

{1, 3} – {1, 4} {1, 2, 4}

Table 4.11 Euclidean distance measure y11 y21 y31 y41 y11 y21 y31 y41 y13 y23 y33 y43

– 0.0874 0.0938 0.0174 y13 – 0.1233 0.2827 0.2043

– – 0.1117 0.1040 y23 – – 0.1607 0.1718

– – – 0.0997 y33 – – – 0.2627

– – – – y43 – – – –

y12 – y22 y32 y42 y14 y24 y34 y44

257

x3

x4

{1, 2, 3} {2, 3} – {1, 2, 3}

{3} {3} {4} –

y12

y22

y32

y42

– 0.0564 0.1053 0.082 y14 – 0.2838 0.6123 0.3949

– – 0.1599 0.0505 y24 – – 0.3492 0.1249

– – – 0.1843 y34 – – – 0.2371

– – – y44 – – – –

6. An m–polar fuzzy concordance index can be computed by the dint of Equation 4.11, employing the normalized weights of the decision criteria. The m–polar fuzzy concordance indices are organized to construct the m–polar fuzzy concordance matrix as follows: ⎛ ⎞ − 0.5614 0.4324 0.7872 ⎜ 0.4386 − 0.6582 0.7872 ⎟ ⎟. F =⎜ ⎝ 0.5676 0.3418 − 0.5676 ⎠ 0.2128 0.2128 0.4324 − 7. An m–polar discordance index is evaluated in the light of the corresponding discordance set using Eq. 4.12 and the Euclidean distance between the alternatives. The Euclidean distances between the entries of aggregated weighted m–polar decision matrix , evaluated by Eq. 4.13, are given by Table 4.11. Further, these discordance indices are arranged to construct the m–polar discordance matrix as follows: ⎞ ⎛ − 0.4345 0.4617 0.5173 ⎜1 − 0.4602 1 ⎟ ⎟. G=⎜ ⎝1 1 − 0.9026 ⎠ − 1 0.7270 1

258

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

8. An m–polar concordance level to check the significance of the concordance indices can be evaluated by Eq. 4.14 as follows: 1  0.5614 + 0.4324 + 0.7872 + 0.4386 + 0.6582 + 0.7872 + 0.5676 4(3)  + 0.3418 + 0.5676 + 0.2128 + 0.2128 + 0.4324

f =

(4.19) (4.20)

= 0.5000.

An m–polar concordance dominance matrix H , obtained by comparing the concordance indices with m–polar concordance level in the light of Eq. 4.15, is represented as follows: ⎛

− ⎜0 H =⎜ ⎝1 0

1 − 0 0

0 1 − 0

⎞ 1 1⎟ ⎟. 1⎠ −

9. An m–polar discordance level g can be determined by employing Eq. 4.16 as follows: 1  0.4345 + 0.4617 + 0.5173 + 1 + 0.4602 4(3)  + 1 + 1 + 1 + 0.9026 + 1 + 0.7270 + 1 = 0.7919.

g=

(4.21)

An m–polar fuzzy discordance dominance matrix L, constructed by the help of Eq. 4.17, is given by: ⎛

− ⎜0 L=⎜ ⎝0 0

1 − 0 1

1 1 − 0

⎞ 1 0⎟ ⎟. 1⎠ −

10. An m–polar fuzzy aggregated dominance matrix, obtained by merging the m–polar concordance dominance matrix and m–polar discordance dominance matrix according to Eq. 4.18, is given by: ⎛

− ⎜0 M =⎜ ⎝0 0

1 − 0 0

0 1 − 0

⎞ 1 0⎟ ⎟. 1⎠ −

11. The outranking graph plotted among all the alternatives in accordance to their mutual relations is represented by Fig. 4.9. The information, inferred from the outranking graph, is organized in Table 4.12. According to outranking relations, scheme x1 is the best insulation technique for the external wall.

4.4 The m–Polar Fuzzy ELECTRE II Method

259

Fig. 4.9 Outranking graph Table 4.12 Analysis of outranking graph Alternatives Submissive alternatives x1 x2 x3 x4

x2 , x4 x3 x4 –

Incomparable alternatives x3 x4 x1 x2

4.4 The m–Polar Fuzzy ELECTRE II Method This section presents the complete procedure of m–polar fuzzy ELECTRE II method to address the complex multi-criteria group decision making problems comprising the uncertain multi-polar information. The proposed methodology preserves the outranking principles of the ELECTRE theory and evaluates the relative performance of alternatives on the basis of three non-intersecting distinct sets, namely, m-polar fuzzy concordance, indifferent and discordance sets. Further, the procedure is summarized in a comprehensive flowchart diagram. The mathematical description of the a multi-criteria group decision making problem can be elaborated as follows: The core aim of a multi-criteria group decision making problem is to opt the best or most favorable alternative as the solution from a set of r available alternatives A = {x1 , x2 , . . . , xr }. For the sake of authentic decision making, a panel of v intelligent and experienced decision making experts e1 , e2 , . . . , ev has been selected to check the calibre of alternative and relative significance of the criteria. The normalized weights of the decision making experts are given by the weight vector α = (α1 , α2 , . . . , αv ). The performance and efficiency of the feasible

260

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

alternatives depend on the s m–polar criteria t1 , t2 , . . . , ts , which are selected by decision making panel. The methodology of the m–polar fuzzy ELECTRE II method is delineated in following steps: 1. Construction of independent m–polar fuzzy decision matrices. The appointed experts judge the performance of the available alternatives with respect to s m–polar criteria and express their decisions in the form of m–polar fuzzy numbers. Thus, the decision of the expert eu about the efficiency of the alternative xi corresponding to the criterion t j is given by the m–polar fuzzy (u1) (u2) (um) number z i(u) j = (z i j , z i j , . . . , z i j ). The individual decisions of the expert eu are compiled in matrix form to construct the m–polar decision matrix Z (u) as follows: t1

⎛ (u1)

Z (u)

(u2)

···

t2 (um)

(u1)

(u2)

(um)

x1 ⎜ ⎜(z 11 , z 11 , . . . , z 11 ) (z 12 , z 12 , . . . , z 12 ) ⎜ ⎜ (u1) (u2) ⎜(z , z , . . . , z (um) ) (z (u1) , z (u2) , . . . , z (um) ) x 2 ⎜ 21 21 21 22 22 22 = ⎜ . ⎜ . . . ⎜ . . . ⎜ . . (u1) (u2) (um) (u1) (u2) (um) xr ⎝(zr1 , zr1 , . . . , zr1 ) (zr2 , zr2 , . . . , zr2 )

ts

⎞

(u1) (u2) (um) · · · (z 1s , z 1s , . . . , z 1s )⎟ ⎟ ⎟ ⎟ (u1) (u2) (um) ⎟ · · · (z 2s , z 2s , . . . , z 2s )⎟ . ⎟ . .. ⎟ . ⎟ . . ⎟ (u1) (u2) (um) ⎠ · · · (zrs , zrs , . . . , zrs )

Similarly, a group of v independent m–polar decision matrices Z (1) , Z (2) , . . . , Z (v) is formed to describe the individual opinions of the experts e1 , e2 , . . . , ev , respectively. 2. Construction of aggregated m–polar fuzzy decision matrix. To seek a group satisfactory solution, the individual opinions are combined via an aggregation operator, namely, m–polar weighted averaging operator. The cumulative decisions represent the collective opinion of the decision making panel regarding the aptitude of the alternatives in reference to decision criteria. The aggregated decision of the decision making panel about the alternative xi with respect to criterion t j can be obtained as follows: (1)

(2)

(v)

z i j = α1 z i j ⊕ α2 z i j ⊕ · · · ⊕ αv z i j ⎞ ⎛ v  v  v        (u1) αu (u2) αu (um) αu ⎠ ⎝ . = 1− ,1 − ,...,1 − 1 − zi j 1 − zi j 1 − zi j u=1

u=1

(4.22)

u=1

The aggregated m–polar fuzzy decision matrix can be represented as follows:

4.4 The m–Polar Fuzzy ELECTRE II Method

(t1 )

⎛

261

(t2 )

m 1 2 (x1 )⎜(z 11 , z 11 , . . . , z 11 )

m 1 2 (z 12 , z 12 , . . . , z 12 )

(· · · )

(ts )

⎞

m 1 2 (z 1s , z 1s , . . . , z 1s )⎟

··· ⎟ ⎜ ⎟ ⎜ ⎜ 1 2 m m m ⎟ 1 2 1 2 ) (z 22 , z 22 , . . . , z 22 ) · · · (z 2s , z 2s , . . . , z 2s )⎟ Z = (x2 )⎜(z 21 , z 21 , . . . , z 21 . ⎟ ⎜ .. .. .. . ⎜ .. ⎟ . . . . . (.) ⎜ ⎟ ⎠ ⎝ (xr ) (zr11 , zr21 , . . . , zrm1 ) (zr12 , zr22 , . . . , zrm2 ) · · · (zr1s , zr2s , . . . , zrms )

3. Assignment of criteria weights. The impact of each criterion on the aptitude of the available alternatives may not be same. Therefore, the decision making panel observes the influence of each criterion on the performance of available alternatives and assigns them m– (u1) (u2) (um) ) be the polar fuzzy numbers accordingly. Let w(u) j = (w j , w j , . . . , w j m–polar fuzzy weight of the criterion t j assigned by the expert eu . To obtain the cumulative m–polar fuzzy weight w j = (w 1j , w 2j , . . . , w mj ) of criterion t j , the aggregation skills of m–polar fuzzy weighted averaging operator are utilized as follows: (1)

w j = α1 w j ⎛

= ⎝1 −

(2)

⊕ α2 w j

v   u=1

(v)

⊕ · · · ⊕ αdv w j

 (u1) αu

1 − wj

,1 −

⎞ v  v      (u2) αu (um) αu ⎠ ,...,1 − 1 − wj 1 − wj .

u=1

(4.23)

u=1

The normalized weights γ j of the decision criterion t j can be obtained by the following formula: s(w j ) , (4.24) γj = s  s(w j ) j=1

where s(w j ) denotes the score of m–polar fuzzy number w j which can be evaluated as follows: m  w hj h=1 . (4.25) s(w j ) = m 4. Construction of aggregated weighted m–polar fuzzy decision matrix. The aggregated m–polar fuzzy weighted of the considered decision-criteria are multiplied to the respective entries of aggregated m–polar fuzzy decision matrix to get the corresponding entries of aggregated weighted m–polar fuzzy decision matrix Y . The entry yi j = (yi1j , yi2j , . . . , yimj ) of the aggregated weighted m–polar fuzzy decision matrix Y can be determined by the following formula:

262

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

yi j = (z i1j .w 1j , z i2j .w 2j , . . . , z imj .w mj ).

(4.26)

The aggregated weighted m–polar fuzzy decision matrix Y = (yi j )r ×s can be represented as follows: ⎛

t1

m 1 2 x1⎜(y11 , y11 , . . . , y11 )

t2 m 1 2 (y12 , y12 , . . . , y12 )

···

ts

⎞

m 1 2 (y1s , y1s , . . . , y1s )⎟

··· ⎟ ⎜ ⎟ ⎜ ⎜ 1 2 m m m ⎟ 1 2 1 2 Y = x2⎜(y21 , y21 , . . . , y21 ) (y22 , y22 , . . . , y22 ) · · · (y2s , y2s , . . . , y2s )⎟ . ⎟ ⎜ .⎜ . . . . ⎟ . .. .. .. .. .⎜ ⎟ ⎠ ⎝ 1 2 m m 1 2 1 2 xr (yr 1 , yr 1 , . . . , yr 1 ) (yr 2 , yr 2 , . . . , yr 2 ) · · · (yr s , yr s , . . . , yrms )

5. Construction of m–polar fuzzy concordance, discordance and indifferent sets. The theory of ELECTRE II methods employs three types of sets, namely, concordance, discordance and indifferent sets to unfold the relations between any pair of alternatives. Accordingly, the proposed m–polar fuzzy ELECTRE II method proceeds on the ground of m–polar fuzzy concordance, discordance and indifferent sets to highlight the superiority, inferiority and similarity between any pair of available alternatives. To unfold the pairwise relations among the alternatives, the m–polar fuzzy information is compared with the help of score degrees as the higher score degree indicates the higher performance. The score degree of the entry yi j of aggregated m–polar fuzzy decision matrix can be evaluated as follows:   m 1  h s(yi j ) = y . (4.27) m h=1 i j The rules for the evaluation of the concordance, discordance and indifferent sets are as follows: (i) An m–polar fuzzy concordance set F pq presents the collection of criteria that indicate the superiority of the alternative x p over the alternative xq . An m–polar concordance set assists to collect the evidence in the favor of an alternative to outrank the inept alternative. An m–polar concordance set F pq is established on the basis of following condition: F pq = { j : s(y pj ) > s(yq j ), p = q; p, q = 1, 2, . . . , r ; j = 1, 2, . . . , s}. (4.28) (ii) The main task of m–polar fuzzy discordance set G pq is to pick up all those criteria that predicts the inferiority of the alternative x p over the alternative xq . The discordance set performs the same job as concordance set but in opposite sense. The condition for the construction of the m–polar discor-

4.4 The m–Polar Fuzzy ELECTRE II Method

263

dance set G pq is given as follows: G pq = { j : s(y pj ) < s(yq j ), p = q; p, q = 1, 2, . . . , r ; j = 1, 2, . . . , s}. (4.29) (iii) An m–polar indifferent set I pq is used to investigate the similar performance of two alternatives x p and xq with respect to any particular criterion. An m– polar indifferent set comprises the subscripts of all those criteria according to which two alternatives exhibit the same performance. Mathematically, the m–polar indifferent set I pq can be established by investigating the following condition: I pq = { j : s(y pj ) = s(yq j ), p = q; p, q = 1, 2, . . . , r ; j = 1, 2, . . . , s}. (4.30) 6. Establishment of m–polar concordance matrix. To measure the amount of superiority of alternative x p over alternative xq , the m–polar concordance index f pq is evaluated in the light of m–polar concordance set F pq and m–polar indifferent set I pq . The normalized weights of the criteria along with the weight of corresponding m–polar concordance and indifferent sets contribute to the evaluation of concordance indices. Let γ c and γ i be the weights of m–polar concordance and indifferent sets, respectively. An m–polar concordance index f pq can be determined as follows: f pq = γ c



 γj

+ γi

j∈F pq



 γj .

(4.31)

j∈I pq

Later, the m–polar concordance matrix F, whose entries represent the m–polar concordance indices with respect to any particular pair of alternatives, is established. Mathematically, the m–polar concordance matrix can be represented as follows: x1 x2 · · · xr −1 xr ⎞ ⎛ x1 ⎜ − f 12 ⎜ ⎜ ⎜ x2 ⎜ f 21 − ⎜ .. F = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜ f (r −1)1 f (r −1)2 ⎜ ⎜ ⎝ xr fr 1 fr 2

· · · f 1(r −1) · · · f 2(r −1) .. .. . . ··· − · · · fr (r −1)

f 1r ⎟ ⎟ ⎟ ⎟ f 2r ⎟ .. ⎟ ⎟ . . ⎟ ⎟ f (r −1)r ⎟ ⎟ ⎟ ⎠ −

7. Construction of m–polar fuzzy discordance matrix. To specify the amount of ineptness of the alternative x p in comparison to alternative xq , m–polar fuzzy discordance index g pq is measured in accordance with m–

264

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

polar fuzzy discordance set G pq . The key notion of Euclidean distance between m–polar fuzzy numbers is employed to determine the m–polar fuzzy discordance index g pq as follows: max d(y pj , yq j )

g pq =

j∈G pq

max d(y pj , yq j )

,

p = q.

(4.32)

j

The Euclidean distance between the m–polar fuzzy numbers y pj and yq j can be computed via the following formula:  d(y pj , yq j ) =

(y 1pj − yq1 j )2 + (y 2pj − yq2 j )2 + · · · + (y mpj − yqmj )2 m

.

(4.33)

An m–polar fuzzy discordance indices are compiled in matrix form to establish m–polar fuzzy discordance matrix G, given by: ⎛

x1

x2

x1 ⎜ − g12 ⎜ ⎜ ⎜ x2 ⎜ g21 − ⎜ .. G = .. ⎜ .. . ⎜ . . ⎜ xr −1⎜g(r −1)1 g(r −1)2 ⎜ ⎜ ⎝ xr gr 1 gr 2

· · · xr −1 · · · g1(r −1) · · · g2(r −1) .. .. . . ··· − · · · gr (r −1)

xr

⎞

g1r ⎟ ⎟ ⎟ ⎟ g2r ⎟ .. ⎟ ⎟ . . ⎟ ⎟ g(r −1)r ⎟ ⎟ ⎟ ⎠ −

8. Establishment of strong and weak outranking relations. The theory of ELECTRE II approach is equipped with two types of outranking relations, namely, strong outranking relation R s and weak outranking relation R w , to utilize each minor detail to exhibit the dominance and to investigate the combined effect of m–polar fuzzy concordance and discordance indices for each pair of alternatives. In order to establish these outranking relations, the m–polar fuzzy concordance and discordance indices are compared with five threshold values, including three concordance levels and two discordance levels. The decision making experts specify three strictly increasing concordance levels, namely, low c− , average c◦ and high c∗ concordance levels, to check out whether the m–polar fuzzy concordance index is larger enough to outrank the compared alternative. On the other hand, two strictly increasing discordance indices, namely, low d ∗ and average d ◦ , are assigned by decision making panel to point out the effectiveness of discordance indices. These concordance and discordance levels must satisfy the conditions 0 < c− < c◦ < c∗ < 1 and 0 < d ∗ < d ◦ < 1.

4.4 The m–Polar Fuzzy ELECTRE II Method

265

Now, a pair of competing alternatives (x p , xq ) is said to satisfy the strong outranking relation or the alternative x p strongly outranks the alternative xq (i.e., x p R s xq ) if one of the following conditions holds: ⎧ ⎨ f pq ≥ c∗ g pq ≤ d ∗ ⎩ f pq > f q p

or

⎧ ⎨ f pq ≥ c◦ g pq ≤ d ∗ . ⎩ f pq > f q p

(4.34)

This relation is said to be strong outranking relation as it satisfy all the strict conditions (larger concordance index and smaller discordance index) for which the alternative x p can be preferred over the alternative xq . On the other hand, the weak relation presents a relatively lenient form of strong outranking relation to examine the weak preference between any pair of alternatives. Thus, a pair of competing alternatives (x p , xq ) is said to satisfy the weak outranking relation or the alternative x p weakly outranks the alternative xq (i.e., x p R w xq ) if the following condition holds: ⎧ ⎨ f pq ≥ c− g pq ≤ d ◦ . ⎩ f pq > f q p

(4.35)

9. Final ranking of the alternatives. To evaluate the ranking of the alternatives, the outranking relations are portrayed in the form of two independent outranking graphs. The strong outranking graph G s and weak outranking graph G w are drawn according to strong and weak outranking relations using set of alternatives as vertex set. For the establishment of strong outranking graph, a directed edge from alternative x p to alternative xq is drawn if x p R s xq and similarly, the arcs in weak outranking graph is drawn using the same rule. Then, the ranking of the alternatives in ELECTRE II method is predicted through the following procedure by deploying the outranking graphs: • Forward ranking. Let A(k) be any collection of the potential alternatives and A be the set of all alternatives. The forward ranking of the alternatives can be obtained by the following iterative scheme: (i) Construct the set V (k) consisting of all those alternatives which have no precedent arc in the strong outranking graph G s . ˜ (ii) Construct the set E(k) consisting of all those arcs of weak outranking graph w G which have both end vertices in the set V (k). ˜ (iii) Now, construct the graph G˜ = (V (k), E(k)) and Construct the set S(k) by identifying all those alternative having no incoming flow (precedent arc) ˜ in G. (iv) The iterative procedure for the specification of forward ranking is elaborated in following step:

266

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

(a) For the first iteration (k=1), take A(1) = A. (b) Follow the steps (i), (ii), (iii) and (iv) to unfold the sets R(k) and S(k). Here, k denotes the number of iteration. (c) Rank the alternatives, appearing in S(k), at kth position, i.e., β F (x p ) = k if x p ∈ S(k). (d) For the next iteration, delete all forwardly rank alternatives and their corresponding arcs from the outranking graphs. For (k + 1)-th iteration A(k + 1) = A(k) − S(k). Repeat this procedure from (a) for the next iteration until A(k + 1) = {}. For the comprehensive elaboration of the iterative procedure, an elegant flow chart diagram is shown by Fig. 4.10. • Reverse ranking. The procedure for the evaluation of reverse ranking is described in following steps: (i) To obtain the reverse ranking β R , the direction of all arcs in the strong and weak outranking graphs are reversed for further proceeding. (ii) The same iterative procedure as used for forward ranking is adopted for

identify the ranking β . (iii) Then, the reverse ranking is determined by employing the following formula:

(4.36) β R (x p ) = 1 + max β (x p ) − β (x p ). x p ∈A

The complete iterative procedure to obtain the reverse ranked is explicated by the dint of Fig. 4.11. • Average ranking. The average ranking β of the alternatives can be obtained by the average of the forward and reverse ranking of the alternatives as follows: β(x p ) =

β F (x p ) + β R (x p ) . 2

(4.37)

The complete procedure of presented m–polar fuzzy ELECTRE II method is encapsulated in the Fig. 4.12.

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant

267

Fig. 4.10 Iterative procedure for the forward ranking

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant The province Fujian is populated province which is situated at the southeastern coastal area of China, having 124000 km2 land area, 136000 km2 sea area and more than 39 million population. Further, the province Fujian shares boundaries

268

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Fig. 4.11 Iterative procedure for the reverse ranking

with the Zhejiang, Jiangxi, Guangdong provinces, Taiwan strait and China sea. The Geographical location of the Fujian province is shown by Fig. 4.13. To meet the power demands, the province was producing nearly 57,700 MW electricity through the coal, hydropower and nuclear power generation in 2018. A major

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant

Fig. 4.12 Flow chart diagram of m–polar fuzzy ELECTRE II method

269

270

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Fig. 4.13 Location of Fujian province

part of this nameplate capacity was produced via coal generation. With the passage of time and increase in energy demands, the nominal capacity of Fujian province was insufficient of fulfill the electricity needs. Therefore, the Chinese government aims to enhance the power generation capacity of existing energy structure including renewable or other energy sources for the establishment of a stable, economical, clean, and safe power supply system. The Chinese government figures out the installment of a nuclear power plant on the east coast of Fujian province as the most suitable solution to optimize the nominal capacity by observing the risks of flooding, irregular water release and disruption of natural river flows of hydropower, high generation cost of solar photovoltaic and intermittent behavior of wind generation. The nuclear power plants are preferred owing to lower life cycle greenhouse gas emissions, abundance of uranium inside earth’s crust and its feasibility to produce massive amounts of power on large scales. As this coastal province is far away from neighbor nations, the slight international influences from the nuclear power plants can be ignored. The major target for the construction of the nuclear power plant is the selection of suitable location. This problem can be treated as a MCGDM problem and can be addressed via our presented m–polar fuzzy ELECTRE II approach.

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant

271

4.5.1 Available Alternatives For the solution of this MCGDM problem, adapted from Chen et al. [19], a panel of three decision makers has been designated for the short listing of the alternatives, specification of the decision criteria and thorough inspection of candidate locations according to decision criteria. The weights of individual experts in the decision making panel are given by the weight vector α = (0.33, 0.33, 0.34). After the initial screening, the following locations, represented by cardinal directions, have been opted as alternatives: • Location x1 : 26◦ 17 43.4

N and 119◦ 48 30.97

E; • Location x2 : 26◦ 30 37.98

N and 119◦ 47 23.93

E; • Location x3 : 26◦ 23 14

N and 119◦ 51 28

E. All these candidate locations have satisfactory geological conditions, stable earth’s crust and less risks of seismic activities.

4.5.2 Selection of m–Polar Criteria The selection of an appropriate location for nuclear power plant depends on different social, economical, environmental, meteorological, geographical and climatic factors. The panel of decision making experts has identified the following factors as decision criteria for this MCGDM problem: t1 : Geographic condition This criterion investigates the physical environment and availability of important resources. The criterion geographic condition depends on cooling water availability, land use and topographical features. Land use refers to the use of land for recreational, agricultural and residential purposes by humans. The higher proportion of free land will lead to less adverse impacts to be triggered to other types of land use. Topographical features refer to the qualities and features of land surfaces of the candidates sites. t2 : Meteorological characteristics The major meteorological characteristics affecting the site selection of a nuclear power plant are average temperature, maximum possible precipitation and maximum possible wind speed. The average temperature is an influential characteristic for location selection to manage the heat exchange of the cooling water tower of nuclear power plant with the external environment. The locations with higher possible precipitation may have a high risk of causing floods. The maximum wind speed of a site can affect the dissipation of radioactive materials or other pollutants which can have harmful effects on human health. t3 : Ground condition The criterion ground condition depends on surface faulting, slope instability and vibratory ground motion. Surface faulting is displacement that reaches the

272

t4 :

t5 :

t6 :

t7 :

t8 :

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

earth’s surface during slip along a fault. Slope instability refers to the failure of slope due to local geomorphic, hydrologic and geologic conditions. The slope instability may cause the mass movement which may result into land sliding. Natural hazards The criterion natural hazards include extreme weather risk, seismic acceleration and seismic fortification intensity. Extreme weather risks account the risks of tropical cyclone, typhoons, hurricanes, floods, tornadoes, or snowstorms. Seismic acceleration and seismic fortification intensity are primary notions to investigate the probability of deformation of ground. Social factor This criterion examines the social impacts of the establishment of nuclear power plants. The social factors include emergency planning, nearby hazardous facilities and population distribution. The effective emergency planning is an important step to reduce the damage caused by any unpleasant accident. The nearby hazardous facilities refer to the closeness to the nearby services in case of emergency to minimize the risk of damages due to human activities and to ensure the safety of nuclear power plant. Population distribution prominently contributes to lower security risk of locals by investigating the effectiveness of emergency plans and distance from densely populated areas. Impact on other activities The criterion, accounting the social effects of the nuclear power plant on other human activities, formulates impact on agricultural activities, impact on industrial activities and impact on tourism. The poles of this criterion measures side effects of the establishment of nuclear power plant the nuclear power plant to the agriculture, industry and tourism of that area. Environmental factor This criterion accounts the impact of the nuclear power plant to the environment and ecological systems of that region. The environmental factors include impact on aquatic environments, impact of plant accidents and regular radioactive impact. The impact on aquatic environment refers to the pollution of water and destruction of aquatic life due to nuclear power plant. The accidents at a nuclear power plant could release dangerous levels of radiation over an area that may increase the environmental pollution. The regular radioactive impact refers to the quantity of regular radioactive materials released by the nuclear power plant to the environment. Economic factor The economic factors affecting the location of a nuclear power plant are cost of constructing a cooling system, construction cost and transportation cost. The candidate location, having least construction and transportation costs, is preferred. All these criteria along with their poles are represented in Fig. 4.14.

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant

273

Fig. 4.14 Decision criteria and their poles Table 4.13 An m–polar fuzzy decision matrix of expert e1 Z (1) t1 t2 t3 x1 x2 x3 x1 x2 x3

(0.84, 0.55, 0.80) (0.88, 0.65, 0.80) (0.80, 0.75, 0.84) t5 (0.80, 0.84, 0.70) (0.76, 0.92, 0.76) (0.70, 0.88, 0.72)

(0.80, 0.84, 0.76) (0.84, 0.80, 0.72) (0.80, 0.86, 0.70) t6 (0.90, 0.76, 0.90) (0.90, 0.72, 0.86) (0.84, 0.72, 0.88)

(0.84, 0.90, 0.80) (0.80, 0.80, 0.90) (0.78, 0.84, 0.84) t7 (0.80, 0.72, 0.80) (0.84, 0.70, 0.72) (0.80, 0.76, 0.80)

t4 (0.86, 0.80, 0.76) (0.84, 0.88, 0.92) (0.88, 0.92, 1.00) t8 (0.70, 0.80, 0.80) (0.70, 0.76, 0.84) (0.72, 0.84, 0.76)

4.5.3 Stepwise Procedure The solution of this MCGDM problem can be obtained in following steps: 1. The m–polar fuzzy decision matrices Z (1) , Z (2) and Z (3) , inscribing the evaluations of the experts e1 , e2 and e3 , are shown by Tables 4.13, 4.14 and 4.15, respectively. 2. The aggregated m–polar fuzzy decision matrix is obtained by cumulating the individual decisions in the light of m–polar fuzzy weighted averaging operator using Eq. 4.22. The entries of the aggregated m–polar fuzzy decision matrix are organized in Table 4.16. 3. The individual decisions of the decision making experts, expressed in terms of m– polar fuzzy numbers, are shown in Table 4.17. Further, the aggregated m–polar

274

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Table 4.14 An m–polar fuzzy decision matrix of expert e2 Z (2) t1 t2 t3 x1 x2 x3 x1 x2 x3

(0.65, 0.76, 0.65) (0.61, 0.84, 0.72) (0.65, 0.80, 0.76) t5 (0.90, 0.90, 0.80) (0.84, 0.92, 0.90) (0.80, 0.90, 0.84)

(0.80, 0.84, 0.84) (0.84, 0.80, 0.78) (0.80, 0.86, 0.80) t6 (0.90, 0.80, 0.84) (0.90, 0.78, 0.80) (0.84, 0.76, 0.84)

(0.90, 0.84, 0.80) (0.84, 0.80, 0.90) (0.80, 0.80, 0.84) t7 (0.65, 0.80, 0.84) (0.65, 0.76, 0.80) (0.73, 0.86, 0.90)

Table 4.15 An m–polar fuzzy decision matrix of expert e3 Z (3) t1 t2 t3 x1 x2 x3 x1 x2 x3

(0.92, 0.72, 0.76) (0.80, 0.84, 0.80) (0.84, 0.76, 0.85) t5 (0.86, 0.90, 0.76) (0.84, 0.92, 0.80) (0.80, 0.92, 0.80)

(0.80, 0.84, 0.76) (0.84, 0.80, 0.72) (0.80, 0.86, 0.70) t6 (0.80, 0.84, 0.90) (0.80, 0.84, 0.86) (0.84, 0.80, 0.88)

(0.90, 0.80, 0.80) (0.84, 0.76, 0.90) (0.80, 0.76, 0.84) t7 (0.70, 0.84, 0.80) (0.76, 0.80, 0.72) (0.76, 0.90, 0.80)

t4 (0.80, 0.80, 0.76) (0.76, 0.88, 0.92) (0.80, 0.92, 1.00) t8 (0.80, 0.76, 0.72) (0.80, 0.76, 0.78) (0.84, 0.80, 0.70)

t4 (0.86, 0.80, 0.76) (0.84, 0.88, 0.92) (0.88, 0.92, 1.00) t8 (0.76, 0.70, 0.80) (0.80, 0.70, 0.92) (0.80, 0.70, 0.76)

Table 4.16 Aggregated m–polar fuzzy decision matrix Z

t1

t2

t3

t4

x1

(0.8363, 0.6888, 0.7441)

(0.8000, 0.8400, 0.7901)

(0.8832, 0.8522, 0.8000)

(0.8425, 0.8000, 0.7600)

x2

(0.7894, 0.7928, 0.7765)

(0.8400, 0.8000, 0.7410)

(0.8278, 0.7872, 0.9000)

(0.8171, 0.8800, 0.9200)

x3

(0.7770, 0.7709, 0.8211)

(0.8000, 0.8600, 0.7376)

(0.7936, 0.8023, 0.8400)

(0.8580, 0.9200, 1.0000)

t5

t6

t7

x1

(0.8591, 0.8832, 0.7567)

(0.8734, 0.8031, 0.8832)

(0.7239, 0.7928, 0.8142)

t8 (0.7567, 0.7562, 0.7765)

x2

(0.8171, 0.9200, 0.8310)

(0.8734, 0.7862, 0.8425)

(0.7622, 0.7572, 0.7494)

(0.7714, 0.7411, 0.8596)

x3

(0.7714, 0.9016, 0.7924)

(0.8400, 0.7627, 0.8680)

(0.7651, 0.8508, 0.8409)

(0.7924, 0.7867, 0.7417)

fuzzy weights, evaluated by Eq. 4.23, and the normalized weights, computed by Eq. 4.24, are shown in Table 4.17. 4. The aggregated weighted m–polar fuzzy decision matrix is obtained by multiplying the aggregated m–polar fuzzy weights of the criteria with the corresponding entries of aggregated m–polar fuzzy decision matrix using Eq. 4.26. The aggregated weighted m–polar fuzzy decision matrix is shown by Table 4.18. 5. The entries of aggregated weighted m–polar fuzzy decision matrix are compared in pairs to figure out the superiority or inferiority of an alternative over the other with the help of score degrees. The score degrees of the corresponding entries of aggregated weighted m–polar fuzzy decision matrix, evaluated by Eq. 4.27, are arranged in Table 4.19. The m–polar fuzzy concordance sets, exhibiting the

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant

275

Table 4.17 Weights of decision criteria Criteria

Expert opinions e1

Aggregated m–polar

Normalized weights

t1

(0.98, 0.95, 1.00)

e2 (1.00, 1.00, 0.96)

e3 (0.94, 0.95, 1.00)

(1.0000, 1.0000, 1.0000)

Fuzzy weights 0.2631

t2

(0.08, 0.10, 0.32)

(0.19, 0.18, 0.21)

(0.25, 0.24, 0.16)

(0.1771, 0.1760, 0.2323)

0.0513

t3

(0.32, 0.45, 0.51)

(0.30, 0.50, 0.24)

(0.21, 0.30, 0.40)

(0.2776, 0.4215, 0.3933)

0.0958

t4

(0.38, 0.51, 0.43)

(0.44, 0.31, 0.42)

(0.25, 0.34, 0.38)

(0.3604, 0.3929, 0.4101)

0.1020

t5

(0.58, 0.33, 0.45)

(0.48, 0.37, 0.56)

(0.39, 0.29, 0.46)

(0.4884, 0.3304, 0.4922)

0.1150

t6

(0.28, 0.52, 0.47)

(0.41, 0.27, 0.39)

(0.25, 0.41, 0.32)

(0.3164, 0.4087, 0.3957)

0.0983

t7

(0.71, 0.67, 0.76)

(0.64, 0.70, 0.63)

(0.70, 0.69, 0.75)

(0.6849, 0.6869, 0.7193)

0.1834

t8

(0.21, 0.40, 0.42)

(0.21, 0.34, 0.35)

(0.43, 0.31, 0.41)

(0.2930, 0.3507, 0.3943)

0.0910

Table 4.18 Aggregated weighted m–polar fuzzy decision matrix Y

t1

x1

(0.8363, 0.6888, 0.7441)

t2 (0.1417, 0.1478, 0.1835)

t3 (0.2452, 0.3592, 0.3146)

t4 (0.3036, 0.3143, 0.3117)

x2

(0.7894, 0.7928, 0.7765)

(0.1488, 0.1408, 0.1722)

(0.2298, 0.3318, 0.3540)

(0.2945, 0.3458, 0.3773)

x3

(0.7770, 0.7709, 0.8211)

(0.1417, 0.1514, 0.1713)

(0.2203, 0.3382, 0.3304)

(0.3092, 0.3615, 0.4101)

t5

t6

t7

x1

(0.4196, 0.2918, 0.3724)

(0.2763, 0.3282, 0.3495)

(0.4958, 0.5446, 0.5857)

t8 (0.2217, 0.2652, 0.3062)

x2

(0.3991, 0.3040, 0.4090)

(0.2763, 0.3213, 0.3334)

(0.5220, 0.5201, 0.5390)

(0.2260, 0.2599, 0.3389)

x3

(0.3768, 0.2979, 0.3900)

(0.2658, 0.3117, 0.3435)

(0.5240, 0.5844, 0.6049)

(0.2322, 0.2759, 0.2925)

Table 4.19 Score degrees Alternatives t1 x1 x2 x3

0.7564 0.7862 0.7897

t2

t3

t4

t5

t6

t7

t8

0.1577 0.1539 0.1548

0.3063 0.3052 0.2963

0.3099 0.3392 0.3603

0.3613 0.3707 0.3549

0.3180 0.3103 0.3070

0.5420 0.5270 0.5711

0.2644 0.2749 0.2669

Table 4.20 An m–polar fuzzy concordance sets x1 x2 x1 x2 x3

– {1, 4, 5, 8} {1, 4, 7, 8}

{2, 3, 7, 6} – {1, 2, 4, 7}

x3 {2, 3, 5, 6} {3, 5, 6, 8} –

dominance of the alternatives over the other alternatives, are determined using Eq. 4.28 which are shown by Table 4.20. Further, the m–polar fuzzy indifferent sets, exhibiting the indifference relation between two alternatives, are determined using Eq. 4.30 which are shown by Table 4.21. The m–polar fuzzy discordance sets, exhibiting the inferiority of the alternatives over the other alternatives, are determined using Eq. 4.29 which are shown by Table 4.22.

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4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

Table 4.21 An m–polar fuzzy indifferent sets x1 x1 x2 x1

– {} {}

x2

x3

{} – {}

{} {} –

Table 4.22 An m–polar fuzzy discordance sets x1 x2 x1 x2 x1

– {2, 3, 7, 6} {2, 3, 5, 6}

x3

{1, 4, 5, 8} – {3, 5, 6, 8}

{1, 4, 7, 8} {1, 2, 4, 7} –

Table 4.23 An m–polar fuzzy concordance matrix F x1 x2 x1 x2 x3

– 0.5711 0.6395

x3

0.4288 – 0.5998

0.3604 0.4001 –

Table 4.24 Euclidean distance measure y11

y21

y31

y12

y22

y32

d

y13

y23

y33

y11

–

–

–

y12

–

–

–

y13

–

–

–

y21

0.0685 –

–

y22

0.0087 –

–

y23

0.0291 –

–

y31

0.0735 0.0296 –

y32

0.0073 0.0074 –

y33

0.0209 0.0151 –

y14

y24

y34

y15

y25

y35

y14

–

–

–

y24

0.0423 –

–

y15

–

–

–

y25

0.0252 –

–

y34

0.0631 0.0226 –

y35

0.0269 0.0173 –

y17

y27

y37

y17

–

–

y18

y28

y38

–

y18

–

–

–

y27

0.034

–

–

y28

0.0193 –

–

y37

0.0303 0.0532 –

y38

0.0117 0.0286 –

y16

y26

y36

y16

–

–

–

y26

0.0101 –

–

y36

0.0118 0.0101 –

6. The m–polar fuzzy concordance indices are determined by Eq. 4.31 using the corresponding m–polar fuzzy concordance sets. An m–polar fuzzy concordance matrix is represented by Table 4.23. 7. The Euclidean distance measures between the entries of aggregated weighted m–polar fuzzy decision matrix, calculated by Eq. 4.33, are organized in Table 4.24. The m–polar fuzzy discordance indices are determined by Eq. 4.32 using

4.5 A Case Study: Selection of Appropriate Location for Nuclear Power Plant Table 4.25 An m–polar fuzzy discordance matrix G x1 x2 x1 x2 x3

– 0.4964 0.3660

Table 4.26 Outranking relations x1 x1 x2 x3

– Rs , Rw Rs , Rw

Table 4.27 Outranking relations x1 Forward ranking Reverse ranking Average ranking

3 3 3

1 – 0.5376

277

x3 1 1 –

x2

x3

– – Rw

– – –

x2

x3

2 2 2

1 1 1

the corresponding m–polar fuzzy discordance sets and Euclidean distances. An m–polar fuzzy discordance matrix is represented by Table 4.25. 8. To check the effectiveness of m–polar fuzzy concordance (discordance) indices and to specify the outranking relation among any pair of feasible alternatives, three concordances levels and two discordance levels are established as follows: (c− , c◦ , c∗ ) = (0.50, 0.55, 0.60), (d ∗ , d ◦ ) = (0.50, 0.60). The outranking relations among the alternatives, obtained by checking the conditions of strong and weak outranking relations given in Eqs. 4.34 and 4.35, are represented by Table 5.20. 9. The forward, reverse and average rankings of the alternatives, obtained by processing outranking graphs in the light of iterative procedures, are shown by Table 5.21. Thus, the Location x3 is most suitable site for the construction of nuclear power plant.

278

4 Extended ELECTRE I, II Methods with Multi-polar Fuzzy Sets

4.6 Comparison Analysis ELECTRE presents a family of outranking methodologies that compare the potential of available alternatives on different notions to examine their relative calibre for the establishment of outranking relations. The crisp outranking relations are established to outrank the inferior alternatives on the availability of enough proofs in the favor of dominant alternative. The well known variants of ELECTRE family includes ELECTRE I, ELECTRE II, ELECTRE III, ELECTRE IV and ELECTRE TRI techniques. This chapter encompasses the m–polar fuzzy ELECTRE I and ELECTRE II approaches for multi-criteria group decision making. The proposed approaches present a advantageous blend of the decision making calibre of ELECTRE methods and potential of m–polar fuzzy sets to address the multi-criteria group decision making problems with ambiguous multi-polar information. In fact, the main edge of these techniques is due to the advance and modern structure of m–polar fuzzy sets that enable them to deal with complex, practical and ambiguous information of multipolar nature, appearing frequently in real life phenomena. The presented methodologies perform the pairwise comparison of alternatives owing to the m-polar fuzzy concordance and discordance sets which are employed to determine m-polar fuzzy concordance and discordance indices. In comparison of both presented approaches, the m–polar fuzzy ELECTRE II approach prevails due to the following reasons: • The theory of ELECTRE II approach, being a modified variant of ELECTRE I, is more authentic, rich and efficient to address the decision making problems. • In m–polar ELECTRE I method, the effectiveness of m–polar concordance and discordance indices are accounted with the help of two threshold values, one concordance level and one discordance level. Whereas, m–polar ELECTRE II method employs five threshold values for this purpose including three concordance levels and two discordance levels. These strictly increasing concordance and discordance levels allow m–polar ELECTRE II method to capture weak preference that does not meet the strict conditions of strong preference. • An m–polar ELECTRE I method specifies the optimal alternative using a single crisp outranking relation. On the other hand, the theory of m–polar ELECTRE II method is privileged to employ two types of outranking relations, namely, strong and weak outranking relations to identify the best alternative using each minor detail for the sake of authentic decision making. • Further, the m–polar ELECTRE I procedure may not provide the complete ranking of the alternatives in the presence of incomparable alternatives. On the other hand, m–polar ELECTRE II method is equipped with an iterative procedure that predicts the ranking of the alternatives using strong and weak outranking graphs. • Although, the m–polar fuzzy ELECTRE II method performs exceptionally to impart the ranking of the alternatives. But it does not deal with pseudo criterion aptly when there is very slightly difference in performance of two alternatives. Further, the evaluation of concordance and discordance indices could be time-taking and exhausting task in the presence of a large number of alternatives and criteria.

4.7 Conclusion

279

4.7 Conclusion The m–polar fuzzy sets, unifying the fuzziness and multi-polarity in a single structure, are considered to be the significant extension of fuzzy sets. Noticing the enhancement of multi-polar information associated with real life phenomena, theory of m– polar fuzzy set is privileged to address the m distinct aspect of such information potently. ELECTRE methods are well known outranking techniques that compare the potential of alternatives in pairs and search for the best alternative through crisp outranking relation. This chapter aims to take advantage of the rich and striking theory of ELECTRE methods within the boundaries m–polar fuzzy sets for the sake of authentic decision making in group decision making scenarios. A remarkable contribution of this chapter is the redesigning of m–polar fuzzy ELECTRE I method to address the MCGDM problem with multi-polar obscure information. An m–polar fuzzy ELECTRE I method focuses on the prime notions of concordance and discordance sets that check out the relative performance of any alternative on the basis of score degrees. The information, assembled in concordance and discordance sets, is processed to establish the outranking relations which provide the optimal alternative via outranking graph. This chapter has delivered the compact procedure of m–polar fuzzy ELECTRE II method to get over the shortcomings of m–polar fuzzy ELECTRE I technique with its iterative ranking scheme. The applicability of both procedures have been demonstrated with the help of two case studies: one for the selection of insulation scheme for the exterior wall and other for the selection of suitable location for the construction of nuclear power plant. A comparison analysis of m–polar fuzzy ELECTRE I and m–polar fuzzy ELECTRE II techniques has been presented.

References 1. Adeel, A., Akram, M., Ahmed, I., Nazar, K.: Novel m–polar fuzzy linguistic ELECTRE-I method for group decision making. Symmetry 11(4), 471 (2019) 2. Adeel, A., Akram, M., Koam, A.N.: Group decision making based on m-polar fuzzy linguistic TOPSIS method. Symmetry 11(6), 735 (2019) 3. Adeel, A., Akram, M., Koam, A.N.A.: Multi-criteria decision making under mHF ELECTRE-I and Hm–polar fuzzy ELECTRE-I. Energies 12(9), 1661 (2019) 4. Akram, M.: m−Polar fuzzy graphs-theory, methods and applications. Stud. Fuzziness Soft Comput. 371, 1–284. Springer (2019). ISBN 978-3-030-03750-5 5. Akram, M., Adeel, A.: Novel hybrid decision making methods based on m–polar fuzzy rough information. Granular Comput. 5, 185–201 (2020) 6. Akram, M., Adeel, A.: TOPSIS approach for MAGDM based on interval-valued hesitant fuzzy N –soft environment. Int. J. Fuzzy Syst. 21(3), 993–1009 (2019) 7. Akram, M., Shumaiza, Alcantud, J.C.R.: An m-polar fuzzy PROMETHEE approach for AHPassisted group decision making. Math. Comput. Appl. 25(2), 26 (2020) 8. Akram, M., Shumaiza, Arshad, M.: Bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE-I methods to diagnosis. Comput. Appl. Math. 39(7), 1–21 (2020) 9. Akram, M., Waseem, N., Liu, P.: Novel approach in decision making with m–polar fuzzy ELECTRE-I. Int. J. Fuzzy Syst. 21, 1117–1129 (2019)

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10. Alghamdi, M.A., Alshehri, N.O., Akram, M.: Multi-criteria decision making methods in bipolar fuzzy environment. Int. J. Fuzzy Syst. 20, 2057–2064 (2018) 11. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1983) 12. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17(4), B141–B164 (1970) 13. Benayoun, R., Roy, B., Sussman, N.: Manual de réference du programme ELECTRE. Note de Synthese et Formation 25, 79 (1966) 14. Brans, J.P., Vincke, P.V.: A preference ranking organization method. Manag. Sci. 31, 647–656 (1985) 15. Chen, J., Li, S., Ma, S., Wang, X.: m-Polar fuzzy sets: an extension of bipolar fuzzy sets. Sci. World J. (2014). https://doi.org/10.1155/2014/416530 16. Chen, S.M., Cheng, S.H., Lan, T.C.: Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf. Sci. 367, 279–295 (2016) 17. Chen, S.M., Niou, S.J.: Fuzzy multiple-attributes group decision making based on fuzzy preference relations. Expert Syst. Appl. 38(4), 3865–3872 (2011) 18. Chen, S.M., Jong, W.T.: Fuzzy query translation for relational database systems. IEEE Trans. Syst. Man Cybern. 27(4), 714–721 (1997) 19. Chen, Y., Li, L., Shen, Y., Liu, B., Wang, D., Chen, S.: Group decision making framework for site selection of coastal nuclear power plants in a linguistic environment: a sustainability perspective. Int. J. Green Energy 18(11), 1161–1172 (2021) 20. Dascal, ˘ I.: An implementation of the ELECTRE II method using fuzzy numbers. Theory Appl. Math. Comput. Sci. 11(1), 14–24 (2021) 21. Duckstein, L., Gershon, M.: Multicriterion analysis of a vegetation management problem using ELECTRE II. Appl. Math. Model. 7(4), 254–261 (1983) 22. Greco, S., Figueira, J. Ehrgott, M.: Multiple Criteria Decision Analysis, p. 37. Springer, New York(2016) 23. Govindan, K., Grigore, M.C., Kannan, D.: Ranking of third party logistics provider using fuzzy Electre II. In: The 40th International Conference on Computers and Indutrial Engineering, pp. 1–5 (2010). https://doi.org/10.1109/ICCIE.2010.5668366 24. Grolleau, J., Tergny, J.: Manuel de réference du programme ELECTRE II. Document de travail 24, SEMA-METRA International, Direction Scientifique (1971) 25. Hatami-Marbini, A., Tavana, M.: An extension of the Electre I method for group decisionmaking under a fuzzy environment. Omega 39(4), 373–386 (2011) 26. Hokkanen, J., Salminen, P., Rossi, E., Ettala, M.: The choice of a solid waste management system using the ELECTRE II decision-aid method. Waste Manag. Res. 13(2), 175–193 (1995) 27. Hwang, C.L., Yoon, K.: Multiple Attributes Decision Making Methods and Applications. Springer, Berlin (1981) 28. Jun, D., Tian-Tian, F., Yi-Sheng, Y., Yu, M.: Macro-site selection of wind/solar hybrid power station based on ELECTRE-II. Renew. Sustain. Energy Rev. 35, 194–204 (2014) 29. Kaya, T., Kahraman, C.: An integrated fuzzy AHP-ELECTRE methodology for environmental impact assessment. Expert. Syst. Appl. 38(7), 8553–8562 (2011) 30. Liu, X., Ma, Y.: A method to analyze the rank reversal problem in the ELECTRE II method. Omega 102, 102317 (2021) 31. Mir, M.S.S., Afzalirad, M., Ghorbanzadeh, M.: A robust fuzzy hybrid MCDM ranking method for optimal selection of lithium extraction process from brine and seawater. Miner. Eng. 169, 106957 (2021) 32. Rouyendegh, B.D., Erkan, T.E.: An application of the fuzzy ELECTRE method for academic staff selection. Hum. Factors Ergon. Manuf. Serv. Ind. 23(2), 107–115 (2013) 33. Roy, B.: The outranking approach and the foundations of ELECTRE methods. In: Readings in Multiple Criteria Decision Aid. Springer, Berlin (1990) 34. Saaty, T.L.: Axiomatic foundation of the analytic hierarchy process. Manag. Sci. 32(7), 841– 855 (1986) 35. Sarwar, M., Akram, M., Shahzadi, S.: Distance measures and δ-approximations with rough complex fuzzy models. Granular Comput. 8, 893–916 (2023)

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Chapter 5

Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

This chapter is designed to deliver another outranking approach based on ELECTRE method to expand the number of multi-criteria group decision making techniques in the literature of fuzzy set theory. The core aim of this chapter is to present the significant procedure of multi-polar (m–polar) fuzzy ELECTRE III method by exploiting the tremendous theory of ELECTRE approach. Further, the dominant features and captivating structure of m–polar fuzzy sets empower the presented approach to address the multi-polar ambiguous information of real-world problems competently. To exhibit the applicability of m–polar fuzzy ELECTRE III method, a case study for the selection of most competent hazardous waste carrier company is presented. Moreover, comparison analysis with existing outranking approaches is conducted to highlight the salient features and superiority of the presented methodology.

5.1 Introduction Multi-criteria decision making (MCDM) or multi-criteria group decision making (MCGDM) defines a process that aggregates data in order to reach a befitting solution meeting the requirements of a problem. Multi-criteria group decision making, being more authentic, is implemented in majority of the real-world decision making scenarios. To ease the process of decision making in complicated problems, a number of techniques are available in the literature to process the exact information for the sake of authentic decision making in the presence of multiple conflicting criteria. In this context, ELECTRE [22] refers to a family of outranking methods for decision making. A common feature of these methods is their working principle as they proceed by pairwise comparison of alternatives to point out the best possible alternative with the help of outranking relations. The credit for the development of these reliable © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_5

283

284

5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

and influential strategies goes to Benayoun et al. [17], who put forward the ELECTRE method in 1966. This was later renamed as ELECTRE I. Further, the evolution of the ELECTRE I method made the way for other significant and modified variants, namely, the ELECTRE II, ELECTRE III, ELECTRE IV and ELECTRE TRI methods. ELECTRE III technique, being an eminent and prepotent variant of ELECTRE family of outranking procedures, compute the ranking of alternatives on the basis of their pairwise comparison in reference to three threshold values. ELECTRE III technique presents a potent procedure to deal with the pseudo criterion where the difference of performance between two alternatives is very small or negligible that cannot be considered significant in terms of preference. Roy [36, 37] is accredited to present the procedure of ELECTRE III method after noticing the deficiencies of ELECTRE I, ELECTRE II and other existing methodologies. Li and wang [30] redesigned the ELECTRE III method by providing an improved ranking system. Leyva-Lopez and Fernandez-Gonzalez [29] extended the ELECTRE III method for the sake of group decision making. Buchanan et al. [16], Papadopoulos et al. [35], Marzouk [32] and Abedi et al. [1] applied the ELECTRE III model to real life problems in the fields of business, industry, engineering and geography, respectively. Beside their significant applications, these techniques were not broad and practical enough to compile the inexact information. As uncertainty, being an essential part of daily life decisions, appears in many practical decision making problems. Thus, the theory of classical set was felt inept to deliberate the imprecise information appearing in practical and complex decision making problems. Later on, Zadeh [49] achieved the milestone to comprehensively present the imprecise opinions by presenting the adaptable and competent model of fuzzy set. Thus, fuzzy sets are regarded as the modern extension of classical sets as they extended the range of characteristic function from two values ({0, 1}) to the uncountable interval ([0, 1]). Montazer et al. [33] and Lupo [31] presented the modified versions of ELECTRE III procedure in fuzzy environment to rank the vendors and international airports in Sicily, respectively. Gao et al. [23] fused the theories of AHP and ELECTRE III strategies to present an approach for fuzzy environment to explore the competitiveness of Quanzhou Port in China. La Fata et al. [28] applied the fuzzy ELECTRE III method to study a case concerning Italian public healthcare. Torkayesh et al. [44] and Noori et al. [34] employed a combination of ELECTRE III technique with fuzzy VIKOR and fuzzy Delphi methods to apply it for solid waste management and water supply plan, respectively. To meet the demands of advanced era, different extensions of fuzzy sets were proposed, including intuitionistic fuzzy sets [12] and Pythagorean fuzzy sets [47, 48]. Many researchers worked on the preeminent theory of ELECTRE III methods for the extended models and unfolded their practical applications accordingly [8, 25, 42, 46]. The structure of these models were designed to capture a single information in terms of satisfaction and dissatisfaction grade. But they can comprise information related to a single aspect at a time. Thus, the appearance of multi-polar information

5.1 Introduction

285

served as a major incentive for the establishment of a broader model, namely, m–polar fuzzy set [18]. The most prominent feature of the presented structure is the modeling of m–polar fuzzy information without the restriction of any relation between these poles. Waseem et al. [45] presented some aggregation operators to cumulate the m–polar fuzzy data. Akram et al. [9, 10] and Adeel et al. [2, 3] explored the theories of ELECTRE, TOPSIS and PROMETHEE methods to develop the more adaptable decision making techniques for m–polar fuzzy environment. For the sake of multi-criteria group decision making, m–polar fuzzy ELECTRE I and m–polar fuzzy ELECTRE II methods are explained in previous chapter. Although, both of these variant of ELECTRE technique perform aptly in decision making scenarios but they are not able to deal with pseudo criteria where the slight performance difference between alternatives is not significant. For the detailed study of pseudo criteria, readers are referred to [16, 38, 43]. Here, we review some basic notions to understand the theory of m–polar fuzzy sets. Definition 5.1 ([18]) An m–polar fuzzy set ξ on a non-empty set X is a mapping ξ : X → [0, 1]m . The membership value of every element x ∈ X is denoted by ξ(x) = ( p1 ◦ ξ(x), p2 ◦ ξ(x), ..., pm ◦ ξ(x)), where pi ◦ ξ : [0, 1]m → [0, 1] is defined as the i-th projection mapping. Definition 5.2 ([45]) The score degree of an m–polar fuzzy number ξ = ( p1 ◦ ξ, p2 ◦ ξ, ..., pm ◦ ξ) is defined as follows:  m  1  s(ξ) = ph ◦ ξ m h=1

(5.1)

Definition 5.3 ([45]) For any three m–polar fuzzy numbers ξ = ( p1 ◦ ξ, p2 ◦ ξ, ..., pm ◦ ξ), ξ1 = ( p1 ◦ ξ1 , p2 ◦ ξ1 , ..., pm ◦ ξ1 ) and ξ2 = ( p1 ◦ ξ2 , p2 ◦ ξ2 , ..., pm ◦ ξ2 ), the elementary operations are defined as follows: 1. ξ1 ⊕ ξ2 = ( p1 ◦ ξ1 + p1 ◦ ξ2 − ( p1 ◦ ξ1 )( p1 ◦ ξ2 ), . . . , pm ◦ ξ1 + pm ◦ ξ2 − ( pm ◦ ξ1 )( pm ◦ ξ2 )), 2. ξ1 ⊗ ξ2 = ( p1 ◦ ξ1 . p1 ◦ ξ2 , p2 ◦ ξ1 . p2 ◦ ξ2 , . . . , pm ◦ ξ1 . pm ◦ ξ2 ), 3. μ(ξ) = (1 − (1 − p1 ◦ ξ)μ , 1 − (1 − p2 ◦ ξ)μ , . . . , 1 − (1 − pm ◦ ξ)μ ), μ > 0 4. ξ μ = ((( p1 ◦ ξ)μ , ( p2 ◦ ξ)μ , ..., ( pm ◦ ξ)μ ), μ > 0. Definition 5.4 ([45]) For a finite collection ξi = ( p1 ◦ ξi , p2 ◦ ξi , ..., pm ◦ ξi ), i = 1, 2, . . . , v, of m–polar fuzzy numbers, the m–polar fuzzy weighted averaging operator is defined as follows:

286

5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

m F W AOα (ξ1 , ξ2 , . . . , ξv ) = α1 ξ1 ⊕ α2 ξ2 ⊕ · · · ⊕ αv ξv ⎛ ⎞ v v v    α α α i i i = ⎝1 − (1 − p1 ◦ ξi ) , 1 − (1 − p2 ◦ ξi ) , . . . , 1 − (1 − pm ◦ ξi ) ⎠ , i=1

i=1

i=1

(5.2) where the weight vector α = (α1 , α2 , . . . , αv ) represents the normalized weights of these m–polar fuzzy numbers.

5.2 An m–Polar Fuzzy ELECTRE III Method This section aims to present the methodology of m–polar fuzzy ELECTRE III technique to address the multi-criteria group decision making problems. The presented technique employs three threshold values, including indifference, preference and veto thresholds, to distinguish the best alternatives in the m–polar fuzzy environment. The prevalent theory of ELECTRE III method implements a simple ranking procedure to derive the complete ranking list. Consider a MCGDM problem which is managed by a panel of v skilled experts E = {e1 , e2 , . . . , ev }. The main job of the decision making panel is to assess the expertise and suitability of r alternatives given by the set A = {x1 , x2 , . . . , xr }. The normalized weights of the decision making experts are given by the weight vector α = (α1 , α2 , . . . , αv ). The panel interprets the calibre of alternatives on the basis of s multi-polar decision criteria T = {t1 , t2 , . . . , ts } which have major impact on the performance of the feasible alternatives and thus influence the final decision. The procedure of m–polar fuzzy ELECTRE III technique is described in the following steps: 1. Construction of independent m–polar fuzzy decision matrices. The decision making panel is keen to examine the caliber of the considered alternatives with respect to all multi-polar decision criteria and present the final decision in terms of m–polar fuzzy numbers. The decision of the expert eu about the alternative xi regarding the criterion t j is given by the m–polar fuzzy number (u1) (u2) (um) (uk) z i(u) of the m–polar fuzzy number j = (z i j , z i j , . . . , z i j ). The kth pole z i j denotes the efficiency of the alternative x with respect to kth attribute of the z i(u) i j criteria t j according to expert eu . Initially , we obtain a collection of v independent decision matrices Z (1) , Z (2) , . . . , Z (v) , one corresponding to each individual in the decision making panel. These m–polar fuzzy decision matrix Z (u) of the expert eu is represented as follows:

5.2 An m–Polar Fuzzy ELECTRE III Method t1

287 t2

⎛ (u1) (u2) (um) (u1) (u2) (um) (z 11 , z 11 , . . . , z 11 ) (z 12 , z 12 , . . . , z 12 ) (u1) (u2) (um) (u1) (u2) (um) ⎜ x2 ⎜(z 21 , z 21 , . . . , z 21 ) (z 22 , z 22 , . . . , z 22 ) Z (u) = . ⎜ . . . . . ⎜ . . . ⎝ (u1) (u2) (um) (u1) (u2) (um) xr (zr 1 , zr 1 , . . . , zr 1 ) (zr 2 , zr 2 , . . . , zr 2 ) x1

···

ts

(u1) (u2) (um) ⎞ · · · (z 1s , z 1s , . . . , z 1s ) (u1) (u2) (um) · · · (z 2s , z 2s , . . . , z 2s )⎟ ⎟ ⎟ . . ⎟ .. . ⎠ . . (u1) (u2) (um) · · · (zr s , zr s , . . . , zr s )

2. Construction of aggregated m–polar fuzzy decision matrix. Next step is the aggregation of independent opinions of all experts to reach a cumulative decision of the panel for the further proceeding. For this purpose, the aggregation skills of m–polar fuzzy weighted averaging operator are employed to get the group satisfactory decision z i j = (z i1j , z i2j , . . . , z imj ) as follows: (1)

(2)

(v)

z i j = α1 z i j ⊕ α2 z i j ⊕ · · · ⊕ αv z i j  v v v       (u1) αu (u2) αu (um) αu = 1− ,1 − ,...,1 − (5.3) 1 − zi j 1 − zi j 1 − zi j u=1

u=1

u=1

The mathematical representation of the aggregated m–polar fuzzy decision matrix is given as follows: t1

x1

Z=

x2 . . . xr

t2

⎛(z 1 , z 2 , . . . , z m ) (z 1 , z 2 , . . . , z m ) 11 12 11 11 12 12 m m 1 2 1 2 , z 21 , . . . , z 21 ) (z 22 , z 22 , . . . , z 22 ) ⎜(z 21 ⎜ .. .. ⎝ . .

···

ts

m ⎞ 1 2 · · · (z 1s , z 1s , . . . , z 1s ) m 1 2 · · · (z 2s , z 2s , . . . , z 2s )⎟ ⎟ .. .. ⎠ . .

(zr11 , zr21 , . . . , zrm1 ) (zr12 , zr22 , . . . , zrm2 ) · · · (zr1s , zr2s , . . . , zrms )

3. Assignment of objective weights to criteria by Shannon’s entropy formula. All the decision criteria may not be equally important. The weight of a decision criteria decides its impact on the performance of alternatives that leads to make a perfect decision. The criteria weights are usually divided in two types: subjective weights (determined on the basis of decision expert through any formula) and objective weight (determined by the elementary data regardless of the decisions of the decision making panel). Among the various techniques of evaluating the objective weights, we have considered frequently implemented shannon’s entropy formula to assign normalize weights to criteria. The concept of entropy, being a measure to evaluate the amount of fuzziness in the data, was presented by Shannon [41] in his information theory. The proper description of this weighting procedure is as follows: (i) Firstly, evaluate the score degrees of all the m–polar fuzzy numbers appearing in aggregated m–polar fuzzy decision matrix. The score degree si j of the m– polar fuzzy number z i j can be determined as follows:

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

1 si j = s(z i j ) = m

m 

 z ihj

.

(5.4)

h=1

(ii) Further, the projection values can be computed in the light of the following formula: si j . (5.5) ki j = r i=1 si j (iii) Evaluate the entropy values corresponding to information of each criterion as follows: r 1  ki j log(ki j ). (5.6) ej = − log r i=1 A higher entropy of a criterion will lead to the lower weight of that criterion. (iv) The degree of divergence of the information regarding each criterion is given by: (5.7) dj = 1 − ej. (v) Finally, the normalized weights are obtained in the light of the following formula: dj . (5.8) w j = s j=1 d j 4. Construction of aggregated weighted m–polar fuzzy decision matrix . To obtain aggregated weighted m–polar fuzzy decision matrix, the aggregated weights of the criteria are multiplied with the corresponding entries of the aggregated m–polar fuzzy decision matrix. An entry yi j = (yi1j , yi2j , · · · , yimj ) of the aggregated weighted m–polar fuzzy decision matrix Y is computed as follows: yi j = (z i1j .w j , z i2j .w j , . . . , z imj .w j ).

(5.9)

The aggregated weighted m–polar fuzzy decision matrix Y = (yi j )r ×s can be represented as follows: ⎛

t1

m 1 2 x1 ⎜(y11 , y11 , . . . , y11 )

Y =

x2 . . . xr

t2

m 1 2 (y12 , y12 , . . . , y12 )

···

ts

⎞

m 1 2 (y1s , y1s , . . . , y1s )⎟

··· ⎟ ⎜ ⎟ ⎜ ⎜ 1 2 m m m ⎟ 1 2 1 2 ⎜(y21 , y21 , . . . , y21 ) (y22 , y22 , . . . , y22 ) · · · (y2s , y2s , . . . , y2s )⎟ . ⎟ ⎜ .. .. .. .. ⎟ ⎜ . . . . ⎟ ⎜ ⎠ ⎝ 1 2 (yr 1 , yr 1 , . . . , yrm1 ) (yr12 , yr22 , . . . , yrm2 ) · · · (yr1s , yr2s , . . . , yrms )

5.2 An m–Polar Fuzzy ELECTRE III Method

289

5. Selection of appropriate threshold values. To identify the level of preference between two alternatives, their respective performance indices are compared with some standard values to determine their effectiveness, such type of standard or reference values are referred as threshold values. As the technique of ELECTRE III is designed to deal the pseudo criterion. Therefore three different types of threshold values, including indifference threshold q j , preference p j and veto ν j threshold values are specified for each criterion t j to unfold the preference relations. These threshold values are given by constant values. There is no hard and fast rule to determine these threshold values. These values are specified by the decisionmakers subjectively after a thorough examination of criteria nature and effectiveness of difference between their values. The selected preference and indifference threshold values must satisfy the following conditions:  y pj > yq j ⇒

y pj + q j (x p ) > yq j + q j (xq ), y pj + p j (x p ) > yq j + p j (xq ),

(5.10)

for all criteria t j , p j ≥ q j . These threshold values are established to deal with pseudo criterion where the smaller difference in the performance of two alternatives is not significant. The indifference threshold q is the largest performance difference that can be considered compatible with the indifference situation. On the other hand, preference threshold p is the smallest performance difference that when exceeded is considered significant of a strict preference in the favor of higher performance alternative. Mathematically, the strict preference relation to deal with pseudo criterion can be checked as follows: s(x p ) > s(xq ) + p(x p ) ⇔ x p Pxq .

(5.11)

Similarly, we can check the weak preference relation with the help of preference threshold as follows: s(xq ) + q(x p )) < s(x p ) < s(xq ) + p(x p ) ⇔ x p Qxq .

(5.12)

Mathematically, the indifference preference relation to deal with pseudo criterion can be written as follows: s(xq ) < s(x p ) < s(xq ) + q(x p ) ⇔ x p Ixq .

(5.13)

6. Evaluation of the concordance indices. Now, the pairwise comparison between the performance of the considered alternatives is made on the basis of score degrees to figure out the preference of an alternative over the other in terms of strict preference, weak preference and indifference preference according to the difference of their performances. The main purpose of this comparison between the alternatives x p and xq is the verification

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

of the assertion that x p outranks the alternative xq , (i.e., x p Sxq ). The concordance index C(x p , xq ) supports the assertion that the alternative x p outranks the alternative xq . The comparison is conducted by subtracting the score degrees of the m–polar fuzzy numbers appearing in aggregated weighted m–polar fuzzy decision matrix for each pair of respective alternatives. The concordance index for the pair (x p , xq ) is evaluated using two different sets RPpq and TQpq that will account all those criteria for which satisfy the condition of strict and weak preference between this particular pair of alternatives, respectively. The set RPpq , representing the collection of the subscripts of all those criteria for which x p Pxq , is defined as follows: RPpq = { j ∈ J : s(yq j ) − s(y pj ) ≤ q j }, where J is the collection of subscripts of all available criteria. The set TQpq , containing the subscripts of all those criteria for which x p Qxq , is defined as follows: TQpq = { j ∈ J : q j ≤ s(yq j ) − s(y pj ) ≤ p j }. The partial concordance index C j (x p , xq ) relative to the criterion t j can be calculated by the dint of sets RPpq and TQpq as follows: ⎧ 1, if j ∈ RPpq ; ⎪ ⎪ ⎨ p − (s(y ) − s(y )) j qj pj , if j ∈ TQpq ; C j (x p , xq ) = ⎪ pj − qj ⎪ ⎩ 0, otherwise,

(5.14)

where p, q = 1, 2, 3, . . . , r and p = q. The partial concordance matrix C j of the criterion t j is given as follows: ⎞ − C j (x1 , x2 ) C j (x1 , x3 ) · · · C j (x1 , xr ) ⎜ C j (x2 , x1 ) − C j (x2 , x3 ) · · · C j (x2 , xr ) ⎟ ⎟ ⎜ ⎜ C j (x3 , x1 ) C j (x3 , x2 ) − · · · C j (x3 , xr ) ⎟ Cj = ⎜ ⎟. ⎟ ⎜ .. .. .. .. .. ⎠ ⎝ . . . . . ⎛

C j (xr , x1 ) C j (xr , x2 ) C j (xr , x3 ) · · ·

−

Finally, the comprehensive concordance index C pq can be determined as follows: C pq =

s  w j C j (x p , xq ). j=1

The comprehensive concordance matrix C can be represented as follows:

(5.15)

5.2 An m–Polar Fuzzy ELECTRE III Method

⎛

− C12 ⎜ C21 − ⎜ ⎜ C = ⎜ C31 C32 ⎜ .. .. ⎝ . . Cr 1 Cr 2

291

C13 C23 − .. . Cr 3

⎞ · · · C1r · · · C2r ⎟ ⎟ · · · C3r ⎟ ⎟. . ⎟ .. . .. ⎠ ··· −

7. Formation of discordance indices. To collect the evidence against the assertion x p Sxq , discordance indices are computed with the help of veto threshold value ν j for the criterion t j . The discordance index D j (x p , xq ) relative to the decision criterion t j can be computed as follows: ⎧ 1, ifs(yq j ) − s(y pj ) ≥ ν j ; ⎪ ⎪ ⎨ 0, ifs(yq j ) − s(y pj ) ≤ p j ; D j (x p , xq ) = s(y ) − s(y ) − p qj pj j ⎪ ⎪ , otherwise, ⎩ νj − pj

(5.16)

where p, q = 1, 2, 3, . . . , r and p = q. The discordance matrix D j relative to criterion t j can be represented as follows: ⎛

− D j (x1 , x2 ) D j (x1 , x3 ) ⎜ D j (x2 , x1 ) − D j (x2 , x3 ) ⎜ ⎜ D j (x3 , x1 ) D j (x3 , x2 ) − Dj = ⎜ ⎜ .. .. .. ⎝ . . . D j (xr , x1 ) D j (xr , x2 ) D j (xr , x3 )

⎞ · · · D j (x1 , xr ) · · · D j (x2 , xr ) ⎟ ⎟ · · · D j (x3 , xr ) ⎟ ⎟. ⎟ .. .. ⎠ . . ··· −

8. Disclosure of credibility index. Finally, the credibility index, a measure of outranking degree x p Sxq , can be evaluated as follows: ⎧ ⎪ ⎨ C(x p , xq ), β(x p , xq ) = C(x p , xq ) ⎪ ⎩



j∈J (x p ,xq )

ifD j (x p , xq ) ≤ C(x p , xq ) ∀ j ∈ J; 1 − D j (x p , xq ) , otherwise, 1 − C(x p , xq )

(5.17)

where J = {1, 2, 3, . . . , r } and J (x p , xq ) = { j ∈ J : D j (x p , xq ) > C(x p , xq )}. 9. Evaluation of final ranking. The traditional methodology of ELECTRE III strategy employs the lengthy outranking approach of ascending distillation and descending distillation, proposed by Belton and Stewart [14], to predicts the final ranking of alternative. Later on, Li and Wang [30] put forward an improved and relatively easier ranking scheme for ELECTRE III approach which ranks the alternatives on the basis of net credibility index. We will employ Li and Wang’s procedure to rank the competitive alternatives which can be explained as follows: (i) Determine the concordance credibility degree of each alternative x p by the dint of the formula:

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

γ + (x p ) =



β(x p , xq ).

(5.18)

xq ∈A, p=q

γ + (x p ) represents the collective outranking calibre of alternative x p with respect to all other alternatives, i.e., the extent to which x p is superior than other available alternatives. (ii) Determine the discordance credibility degree of each alternative x p using the formula:  β(xq , x p ). (5.19) γ − (x p ) = xq ∈A, p=q

γ − (x p ) unfolds the outranked calibre of alternative x p , i.e., the extent to which x p is inferior than other available alternatives. (iii) Evaluate the net credibility degree of an alternative x p as follows: γ(x p ) = γ + (x p ) − γ − (x p ).

(5.20)

Finally, propose the ranking of potential alternative with respect to net credibility index in descending order. The highest net credibility index of an alternative refers to the optimal alternative which should be ranked at prior position. The flowchart of the m–polar fuzzy ELECTRE III procedure is given by Fig. 5.1.

5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm This application, adapted from [15], narrates the case of a private chemical trading company of Turkey which is known for trading polymer emulsions and other special chemicals. The chemical company was established in 1924 with 6000 tons/year manufacturing capacity of polymer emulsions and textile auxiliaries which later reached 200,000 tons/year by 2007. Noticing the toxic effects of harmful chemical on the environmental pollution and human lives the company intends to reconsider the entire production process as well as transportation system considering the significant aspects of energy reduction, need for raw materials, equipment wear and tear and the number of by-products. The trading company also plans to reduce the amount of polyethylene in the manufacturing step to improve the environmental droughts. These improvements will result in better hygiene, environmental safety, wellbeing of staff and other human being and customer satisfaction. The management of large amount of hazardous waste, generated by the chemical company, appears as a challenging issue. As it can effect the environmental standards and it can cause serious harm to surrounding human beings. Due to larger scale of hazardous waste, the chemical company prefers to treat/dispose the waste off-site.

5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm

Construct the aggregated weighted m-polar fuzzy decision matrix

Fig. 5.1 Flowchart of m–polar fuzzy ELECTRE III method

293

294

5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Therefore, the transportation of waste has to be done by some licensed hazardous waste transportation company which keenly follows the government regulations to ensure the service quality, safety, minimal spillage and responsibility. Thus, the selection of a competitive waste management firm, being an important issue to avoid the threats to human health and environment, is considered as a MCGDM problem which is solved by the m–polar fuzzy ELECTRE III method. To derive a trustworthy solution of the described problem, a group of three decision making experts of relevant field has been hired. The details of decision experts are as follows: e1 : Logistics and distribution manager. The logistics manager is an experienced and competent employee of this chemical company since 1990 who checks inventory levels, delivery times and deals with customers and suppliers. e2 : Sales coordination manager. The sales coordination manager is a responsible employee of considered chemical company since 1999, who coordinates with the sales team and promotes suitable sales strategies. e3 : Environment, Health and Safety specialist. The Environment, Health and Safety specialist, being a part of this chemical trading company since 2005, ensures the safety standard for employees and works to improve the production and transportation steps to meet environmental standards. The decision making panel has short-listed five waste carrier firms as alternatives for further exploration. The brief introduction, specialities and shortcoming of the short-listed waste carrier firms are as follows: Firm x1 : Firm x1 was established in 1977 in Istanbul. This company is mainly serving in the railways since 1981. In 2001, x1 started the services to transport hazardous waste with 31 vehicles and now the company is working with 121 vehicles. Firm x1 has the goal to transport 400,000 tons of waste in the near future. Firm x2 : Firm x2 is an international company which offers transportation services from Central Anatolia to Italy and Poland. It has 100 wagons, 50 semitrailers and 250 competent staff members. Firm x3 : Firm x3 is an international transportation firm which delivers its services in 37 countries around the world since 1995. Further, its transportation centers are located in Istanbul, Athens, London, Moscow and Prague and its headquarter is situated in Izmir. It owns a fleet consisting of 150 vehicles on land and 25 sea vessels. Firm x4 : Firm x4 is a local firm founded in 1993 in Artvin, having transportation and management centers in Istanbul and Ankara. For more than 20 years, it works to provide the transportation solutions for routes between Turkey, Azerbaijan, Afghanistan, Turkistan and Pakistan. It delivers the transportation services using 125 semi-trailers and 200 efficient staff members.

5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm

295

Firm x5 : Firm x5 , established in 1995 in Gebze, is known for providing all modes of transportation such as air, sea, land and rail freight with the help of 100 vehicles in land, 10 vessels in sea, 5 planes in air and 30 wagons in rail. It is also able to provide full or partial services in road freight transportation to the commonwealth of independent states countries, the Middle East and especially to Europe. Further, it also provides the facility to transport goods in Turkey. The selection of best waste carrier company depends on several technical, economical, quality and environmental factors. Therefore, the decision making panel has considered following m–polar criteria to assess the performance of the considered waste management firms [15, 24, 26, 27]: t1 : Technical and administrative This criterion accounts for technical capability, experience & qualification and problem solving ability. Technical capabilities refer to the technologies used by the firms to safely transport hazard waste in order to ensure its position and reputation in the industry. The accomplishments, current and recent works done, delivery performance, legal compliance serve as significant indicator to determine the qualification and experience of waste management firm. Problem solving ability is determined by the tools and techniques used in the waste carrier firms to improve its approach to address any emerged problem. t2 : Service capability The criterion service capability depends on service time, quality of service and reputation of service provider. Service time is given by the duration taken by a waste carrier firm to handle a freight from order placement to fulfillment. Quality of service is determined by considering empathy, ease of communication, customer service, performance record, equipment and technology of the hazardous waste carrier firms. Reputation of service provider refers to the positive social repute of the waste carrier firm in the industry. t3 : Economic capability The cost of service, financial stability and payment policy are the key factors to check out economic capability of the hazardous waste carrier firms. The cost of providing the transportation facility is an important factor to opt the waste carrier firm. Financial status of the hazardous waste management firms enables them to survive and prevail over the other competitors to initiate the long-term relationship with the company. Payment policy refers to the feasibility of the payment modes offered by hazardous waste management companies to pay transportation charges in order to facilitate their clients. t4 : Environmental and safety The criterion environmental and safety depends on hygiene and safety, taking care of human health and complying with environmental protection standards. The maintenance of hygiene and establishment of new policies to improve hygiene should be first preference of hazard waste carrier firms to ensure the consistent transportation with safety. Further, the waste carrier companies should take enough measures to save the staff and other surrounding human beings

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

from any type of health risks. Moreover, the policies, working disciplines and standards followed by the hazards waste carrier companies should be consistent with legal environmental standards. t5 : Complementary service The complementary services of the hazardous waste carriers firms include equipment and technology, Dependability and responsiveness and flexibility. The attribute equipment and technology refers to the design, range and availability of equipment and modern technology to support the firm which are essential for the high performance, greater mobility, communication and in time fulfillments of freights. Dependability is given by robustness of the waste transportation firms in terms of its number of failures and the likelihood of its durability in retaining the same performance and efficiency. Responsiveness and flexibility is given by the ability of the waste carrier firms to quickly respond to their clients in case of any change in order or placement of urgent order. These decision criteria along with their poles are represented by Fig. 5.2. The stepwise solution of the described problem can be found in following steps:

Fig. 5.2 Decision criteria and their poles

5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm

297

Table 5.1 m–Polar fuzzy decision matrix of expert e1 Z (1)

t1

x1

(0.50, 0.25, 0.25) (0.60, 0.75, 0.60) (0.25, 0.90, 0.4)

x2

(0.60, 0.25, 0.50) (0.60, 0.25, 0.05) (0.50, 0.25, 0.60) (0.75, 0.25, 0.50) (0.60, 0.90, 0.50)

x3

(0.25, 0.75, 0.60) (0.75, 0.50, 0.25) (0.60, 0.75, 0.90) (0.50, 0.25, 0.05) (0.25, 0.25, 0.60)

x4

(0.25, 0.75, 0.50) (0.60, 0.25, 0.25) (0.75, 0.25, 0.50) (0.40, 0.25, 0.75) (0.90, 0.25, 0.05)

x5

(0.25, 0.75, 0.75) (0.60, 0.40, 0.75) (0.50, 0.75, 0.60) (0.60, 0.25, 0.50) (0.25, 0.50, 0.50)

t2

t3

t4

t5

(0.50, 0.50, 0.90) (0.75, 0.50, 0.40)

1. The individual decisions of all experts, participating in decision making process, are interpreted by m–polar fuzzy numbers. The individual decision matrices of the experts e1 , e2 and e3 are represented by Tables 5.1, 5.2, 5.3, respectively. 2. The independent decisions are merged with the help of m–polar fuzzy weighted averaging operator using Eq. 5.3 in accordance with the experts weights, given by weight vector α = (0.3654, 0.2885, 0.3462). Finally, the aggregated m–polar fuzzy decision matrix, comprising the cumulative decision of all experts, is given by Table 5.7. 3. To evaluate the criteria weights, the score degrees of all entries of the aggregated m–polar fuzzy decision matrix are determined using Eq. 5.4 which are organized in Table 5.4. The projection values, evaluated by Eq. 5.5, are arranged in Table 5.5. Finally, the normalized weights of the criteria are obtained by evaluating the entropy and divergence using Eqs. 5.6, 5.7 and 5.8, given by Table 5.6.

Table 5.2 m–Polar fuzzy decision matrix of expert e2 Z (2)

t1

x1

(0.90, 0.50, 0.75) (0.75, 0.25, 0.50) (0.60, 0.90, 0.75) (0.25, 0.50, 0.25) (0.60, 0.60, 0.90)

x2

(0.75, 0.60, 0.75) (0.25, 0.60, 0.40) (0.40, 0.75, 0.60) (0.50, 0.25, 0.40) (0.50, 0.90, 0.50)

x3

(0.75, 0.50, 0.25) (0.75, 0.60, 0.75) (0.40, 0.25, 0.75) (0.25, 0.05, 0.25) (0.50, 0.05, 0.90)

x4

(0.75, 0.25, 0.75) (0.25, 0.60, 0.60) (0.50, 0.75, 0.90) (0.75, 0.25, 0.75) (0.50, 0.40, 0.05)

x5

(0.25, 0.50, 0.75) (0.05, 0.60, 0.25) (0.75, 0.25, 0.25) (0.25, 0.75, 0.50) (0.05, 0.75, 0.60)

t2

t3

t4

t5

Table 5.3 m–Polar fuzzy decision matrix of expert e3 Z (3)

t1

x1

(0.25, 0.50, 0.90) (0.60, 0.25, 0.25) (0.75, 0.90, 0.25) (0.50, 0.90, 0.05) (0, 0.50, 0.60)

x2

(0.90, 0.50, 0.90) (0.60, 0.75, 0.75) (0.75, 0.75, 0.50) (0.25, 0.50, 0.05) (0.75, 0.60, 0.50)

x3

(0.60, 0.60, 0.60) (0.75, 0.75, 0.60) (0.05, 0.05, 0.90) (0.75, 0.25, 0.05) (0.40, 0.50, 0.50)

x4

(0.90, 0.05, 0.90) (0.60, 0.60, 0.50) (0.25, 0.75, 0.90) (0.60, 0.25, 0.75) (0.75, 0.05, 0.40)

x5

(0.25, 0.75, 0.75) (0.60, 0.75, 0.75) (0.75, 0.50, 0.60) (0.60, 0.60, 0.50) (0.05, 0.75, 0.90)

t2

t3

t4

t5

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Table 5.4 Score degrees t1

t2

t3

t4

t5

x1

0.5956

0.5395

0.6563

0.5873

0.5859

x2

0.6686

0.5229

0.5933

0.4145

0.6586

x3

0.5740

0.6473

0.5728

0.2892

0.4650

x4

0.6496

0.4912

0.6667

0.5317

0.4027

x5

0.5649

0.5831

0.5874

0.5270

0.5127

Table 5.5 Projection values t1

t2

t3

t4

t5

x1

0.1951

0.1938

0.2133

0.2499

0.2232

x2

0.2190

0.1878

0.1928

0.1764

0.2509

x3

0.1880

0.2325

0.1862

0.1231

0.1771

x4

0.2128

0.1764

0.2167

0.2263

0.1534

x5

0.1850

0.2094

0.1909

0.2243

0.1953

Table 5.6 Normalized weights of criteria t1

t2

t3

t4

t5

Entropy

0.9985

0.9971

0.9988

0.9829

0.9909

Divergence

0.0015

0.0029

0.0012

0.0171

0.0091

Normalized weight

0.0472

0.0912

0.0377

0.5377

0.2862

Table 5.7 Aggregated m–polar fuzzy decision matrix t1

t2

t3

t4

t5

x1

(0.6384, 0.4202, 0.7281)

(0.6508, 0.4980, 0.4697)

(0.5723, 0.9000, 0.4965)

(0.4380, 0.7136, 0.6102)

(0.5374, 0.5312, 0.6891)

x2

(0.7839, 0.4563, 0.7655)

(0.5205, 0.5723, 0.4759)

(0.5855, 0.6266, 0.5679)

(0.5534, 0.3482, 0.3419)

(0.6375, 0.8384, 0.5000)

x3

(0.5606, 0.6408, 0.5205)

(0.7500, 0.6312, 0.5606)

(0.3934, 0.4552, 0.8698)

(0.5579, 0.1971, 0.1126)

(0.3824, 0.3022, 0.7103)

x4

(0.7281, 0.4552, 0.7655)

(0.5205, 0.4968, 0.4563)

(0.5534, 0.6266, 0.8200)

(0.5950, 0.2500, 0.7500)

(0.7816, 0.2368, 0.1897)

x5

(0.2500, 0.6947, 0.7500)

(0.4867, 0.6058, 0.6568)

(0.6780, 0.5637, 0.5205)

(0.5205, 0.5606, 0.5000)

(0.1286, 0.6780, 0.7315)

4. The aggregated weighted m–polar fuzzy decision matrix, computed in the light of Eq. 4.7, is shown by Table 5.8. 5. The indifference, preference and veto threshold values with respect to each decision criterion are given by Table 5.9. 6. To identify the pairwise relation among alternatives, the difference between the score degrees of all pair of distinct alternatives are listed in Table 5.10. The partial concordance indices, representing the amount of dominance between each pair of alternatives , are determined using Eq. 5.14. The partial concordance matrix C1 with respect to criterion t1 is represented as follows:

5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm

299

Table 5.8 Aggregated weighted m–polar fuzzy decision matrix t1

t2

t3

t4

t5

x1

(0.0301, 0.0198, 0.0344)

(0.0594, 0.0454, 0.0428)

(0.0216, 0.0339, 0.0187)

(0.2355, 0.3837, 0.3281)

(0.1538, 0.1520, 0.1972)

x2

(0.0370, 0.0215, 0.0361)

(0.0475, 0.0522, 0.0434)

(0.0221, 0.0236, 0.0214)

(0.2976, 0.1872, 0.1838)

(0.1825, 0.2400, 0.1431)

x3

(0.0265, 0.0302, 0.0246)

(0.0684, 0.0576, 0.0511)

(0.0148, 0.0172, 0.0328)

(0.3000, 0.1060, 0.0605)

(0.1094, 0.0865, 0.2033)

x4

(0.0344, 0.0215, 0.0361)

(0.0475, 0.0453, 0.0416)

(0.0209, 0.0236, 0.0309)

(0.3199, 0.1344, 0.4033)

(0.2237, 0.0678, 0.0543)

x5

(0.0118, 0.0328, 0.0354)

(0.0444, 0.0552, 0.0599)

(0.0256, 0.0213, 0.0196)

(0.2799, 0.3014, 0.2689)

(0.0368, 0.1940, 0.2094)

Table 5.9 Threshold values Criteria q t1 t2 t3 t4 t5

0.001 0.001 0.0005 0.02 0.01

Table 5.10 Pairwise comparison t1 s j (x2 ) − s j (x1 ) s j (x3 ) − s j (x1 ) s j (x4 ) − s j (x1 ) s j (x5 ) − s j (x1 ) s j (x1 ) − s j (x2 ) s j (x3 ) − s j (x2 ) s j (x4 ) − s j (x2 ) s j (x5 ) − s j (x2 ) s j (x1 ) − s j (x3 ) s j (x2 ) − s j (x3 ) s j (x4 ) − s j (x3 ) s j (x5 ) − s j (x3 ) s j (x1 ) − s j (x4 ) s j (x2 ) − s j (x4 ) s j (x3 ) − s j (x4 ) s j (x5 ) − s j (x4 ) s j (x1 ) − s j (x5 ) s j (x2 ) − s j (x5 ) s j (x3 ) − s j (x5 ) s j (x4 ) − s j (x5 )

0.0034 –0.0010 0.0026 –0.0014 –0.0034 –0.0044 –0.0008 –0.0048 0.0010 0.0044 0.0036 –0.0004 –0.0026 0.0008 –0.0036 –0.0040 0.0014 0.0048 0.0004 0.0040

p

ν

0.0025 0.0050 0.0015 0.08 0.03

0.004 0.01 0.003 0.1 0.05

t2

t3

t4

t5

–0.0015 0.0098 –0.0044 0.0040 0.0015 0.0113 –0.0029 0.0055 –0.0098 –0.0113 –0.0142 –0.0058 0.0044 0.0029 0.01420 0.0084 –0.0040 –0.0055 0.0058 –0.0084

-0.0023 –0.0031 0.0004 –0.0025 0.0023 –0.0008 0.0027 –0.0002 0.0031 0.0008 0.0035 0.0006 –0.0004 –0.0027 –0.0035 –0.0029 0.0025 0.0002 – –0.0006 0.0029

–0.0929 –0.1603 –0.0299 –0.0324 0.0929 –0.0674 0.0630 0.0605 0.1603 0.0674 0.1304 0.1279 0.0299 –0.0630 –0.1304 –0.0025 0.0324 0.0605 –0.1279 0.0025

0.0208 –0.0346 –0.0524 –0.0210 –0.0208 –0.0554 –0.0732 –0.0418 0.0346 0.0554 –0.0178 0.0136 0.0524 0.0732 0.0178 0.0314 0.0210 0.0418 –0.0136 –0.0314

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

⎛

− ⎜ 1 ⎜ C1 = ⎜ ⎜ 1 ⎝ 1 0.7333

0 − 0 1 0

1 1 − 1 1

0 1 0 − 0

⎞ 1 1⎟ ⎟ 1⎟ ⎟. 1⎠

The partial concordance matrix C2 with respect to criterion t2 is represented as follows: ⎛ ⎞ − 1 0 1 0.25 ⎜ 0.8750 − 0 1 0 ⎟ ⎜ ⎟ ⎜ 1 − 1 1 ⎟ C2 = ⎜ 1 ⎟. ⎝ 0.15 0.525 0 − 0 ⎠ 1 1 0 1 − The partial concordance matrix C3 with follows: ⎛ − 1 ⎜0 − ⎜ 0.7 C3 = ⎜ ⎜0 ⎝1 1 0 1

respect to criterion t3 is represented as ⎞ 1 1 1 1 0 1 ⎟ ⎟ − 0 0.9 ⎟ ⎟. 1 − 1 ⎠ 1 0

The partial concordance matrix C4 with respect to criterion t4 is represented as follows: ⎛ ⎞ − 1 1 1 1 ⎜ 0 − 1 0.2833 0.3250 ⎟ ⎜ ⎟ ⎜ 0.21 − 0 0 ⎟ C4 = ⎜ 0 ⎟. ⎝ 0.835 1 1 − 1 ⎠ 0.7933 1 1 1 − The partial concordance matrix C5 with respect to criterion t5 is represented as follows: ⎛ ⎞ − 0.46 1 1 1 ⎜ 1 − 1 1 1 ⎟ ⎜ ⎟ 0 0 − 1 0.82 ⎟ C5 = ⎜ ⎜ ⎟. ⎝ 0 0 0.61 − 0 ⎠ 0.45 0 1 1 − The comprehensive concordance matrix C, determined by Eq. 5.15, is represented as follows: ⎛ ⎞ − 0.7983 0.9088 0.9528 0.9316 ⎜ 0.4132 − 0.9088 0.5769 0.5459 ⎟ ⎜ ⎟ ⎟. 0.1384 0.2305 − 0.3774 0.4070 C =⎜ ⎜ ⎟ ⎝ 0.5476 0.6705 0.7972 − 0.6226 ⎠ 0.6812 0.6666 0.9088 0.9151 −

5.3 Case Study: Selection of Best Hazardous Waste Carrier Firm

301

7. The partial discordance indices with respect to all considered criteria are computed using Eq. 5.16. The partial discordance matrix D1 with respect to criterion t1 is represented as follows: ⎛

− ⎜0 ⎜ D1 = ⎜ ⎜0 ⎝0 0

0.6 − 1 0 1

0 0 − 0 0

0.0667 0 0.7333 − 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0⎠ −

The partial discordance matrix D2 with respect to criterion t2 is represented as follows: ⎛ ⎞ − 0 0.96 0 0 ⎜0 − 1 0 0.1 ⎟ ⎜ ⎟ 0 0 − 0 0 ⎟ D2 = ⎜ ⎜ ⎟. ⎝0 0 1 − 0.68 ⎠ 0 0 0.16 0 − The partial discordance matrix D3 follows: ⎛ − ⎜ 0.5333 ⎜ D3 = ⎜ ⎜ 1 ⎝ 0 0.6667

with respect to criterion t3 is represented as ⎞ 0 0 0 0 − 0 0.8 0⎟ ⎟ 0 − 1 0⎟ ⎟. 0 0 − 0⎠ 0 0 0.9333 −

The partial discordance matrix D4 with respect to criterion t4 is represented as follows: ⎛ ⎞ − 0 0 0 0 ⎜ 0.645 − 0 0 0⎟ ⎜ ⎟ 1 0 − 1 1⎟ D4 = ⎜ ⎜ ⎟. ⎝ 0 0 0 − 0⎠ 0 0 0 0 − The partial discordance matrix D5 with respect to criterion t5 is represented as follows: ⎛ ⎞ − 0 0 0 0 ⎜ 0 − 0 0 0 ⎟ ⎜ ⎟ 0.23 1 − 0 0 ⎟ D5 = ⎜ ⎜ ⎟. ⎝ 1 1 0 − 0.07 ⎠ 0 0.59 0 0 − 8. The credibility indices, obtained by comparing the partial discordance indices with comprehensive concordance indices in the light of Eq. 5.17, are given as follow:

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Table 5.11 Final ranking Alternatives Concordance credibility degree x1 x2 x3 x4 x5

3.0813 1.0174 0 0.5279 1.5900

⎛

− ⎜ 0.1988 ⎜ β=⎜ ⎜ 0 ⎝ 0 0.6812

Discordance credibility degree

degree

0.8800 0.7983 1.3074 1.2255 2.0054

2.2013 0.2191 –1.3074 –0.6976 –0.4154

0.7983 − 0 0 0

0.3986 0 − 0 0.9088

Net credibility

0.9528 0.2727 0 − 0

Ranking

1 2 5 4 3

⎞ 0.9316 0.5459 ⎟ ⎟ 0 ⎟ ⎟. 0.5279 ⎠

9. The concordance and discordance credibility degrees of all alternatives can be calculated via Eqs. 5.18 and 5.19, respectively. The final ranking is derived from net credibility degree using Eq. 5.20 as shown in Table 5.11. Thus, we can conclude that firm x1 is the best company to carry the hazardous waste of the chemical company.

5.4 Comparative Study In this section, a comparison of the presented m–polar fuzzy ELECTRE III method with the m–polar fuzzy ELECTRE I method and m–polar fuzzy ELECTRE II method is presented to verify the competency and exhibit the authenticity of the proposed technique. For this purpose, the numerical example for the selection of best hazardous waste carrier company is being solved by both m–polar fuzzy ELECTRE I method and m–polar fuzzy ELECTRE II method, respectively to highlight the dominance of m–polar fuzzy ELECTRE III method.

5.4.1 Comparison with m–Polar Fuzzy ELECTRE I Method The stepwise solution of the considered case study by m–polar fuzzy ELECTRE I method is given as follows: 1. The m–polar fuzzy decision matrices of all the honorable decision-makers are shown in Tables 5.1, 5.2 and 5.3.

5.4 Comparative Study

303

Table 5.12 Weights of decision criteria e1

e2

e3

Aggregated m–polar fuzzy weights

Normalized weights

t1

(0.20, 0.31, 0.27)

(0.17, 0.37, 0.21)

(0.27, 0.33, 0.15)

(0.2167, 0.3347, 0.2128)

0.1062

t2

(0.23, 0.45, 0.39)

(0.54, 0.29, 0.13)

(0.25, 0.32, 0.29)

(0.3424, 0.3629, 0.2878)

0.1381

t3

(0.11, 0.25, 0.32)

(0.19, 0.14, 0.23)

(0.21, 0.24, 0.12)

(0.1689, 0.2162, 0.2294)

0.0854

t4

(0.90, 0.95, 0.87)

(0.80, 0.78, 0.95)

(0.84, 0.93, 0.88)

(0.8563, 0.9139, 0.9040)

0.3719

t5

(0.78, 0.81, 0.76)

(0.71, 0.81, 0.64)

(0.54, 0.68, 0.61)

(0.6925, 0.7725, 0.6809)

0.2984

Table 5.13 Aggregated weighted m–polar fuzzy decision matrix t1

t2

t3

t4

t5

x1

(0.1383, 0.1406, 0.1549) (0.2228, 0.1807, 0.1352) (0.0967, 0.1946, 0.1139) (0.3751, 0.6522, 0.5516) (0.3721, 0.4104, 0.4692)

x2

(0.1699, 0.1527, 0.1629) (0.1782, 0.2077, 0.1370) (0.0989, 0.1355, 0.1303) (0.4739), 0.3182, 0.3091) (0.4415, 0.6477, 0.3405)

x3

(0.1215, 0.2145, 0.1108) (0.2568, 0.2291, 0.1613) (0.0664, 0.0984, 0.1995) (0.4777, 0.1801, 0.1018) (0.2648, 0.2334, 0.4836)

x4

(0.1578, 0.1524, 0.1629) (0.1782, 0.1803, 0.1313) (0.0935, 0.1355, 0.1881) (0.5095, 0.2285, 0.6780) (0.5413, 0.1829, 0.1292)

x5

(0.0542, 0.2325, 0.1596) (0.1666, 0.2198, 0.1890) (0.1145, 0.1219, 0.1194) (0.4457, 0.5123, 0.4520) (0.0891, 0.5238, 0.4981)

Table 5.14 Score degrees t1

t2

t3

t4

t5

x1

0.1446

0.1796

0.1351

0.5263

0.4172

x2

0.1618

0.1743

0.1216

0.3671

0.4766

x3

0.1489

0.2157

0.1214

0.2532

0.3273

x4

0.1577

0.1633

0.1390

0.4720

0.2845

x5

0.1488

0.1918

0.1186

0.4700

0.3703

Table 5.15 m–Polar fuzzy concordance sets x1

x2

x3

x4

x5

x1

–

{2, 3, 4}

{3, 4, 5}

{2, 4, 5}

{3, 4, 5}

x2

{1, 5}

–

{1, 3, 4, 5}

{1, 2, 5}

{1, 3, 5}

x3

{1, 2}

{2}

–

{2, 5}

{1, 2, 3}

x4

{1, 3}

{3, 4}

{1, 3, 4}

–

{1, 3, 4}

x5

{1, 2}

{2, 4}

{4, 5}

{2, 5 }

–

2. The aggregated m–polar fuzzy decision matrix, obtained by cumulating the individual decisions of all experts in the decision making panel, is shown by Table 5.7. 3. The criteria weights are evaluated in the light of experts opinions using the m–polar fuzzy number. The individual decisions of the decision experts regarding weights of criteria along with their m–polar fuzzy weights and normalized weights are presented by Table 5.12. 4. The aggregated weighted m–polar fuzzy decision matrix is obtained by the product of m–polar fuzzy weights of the criteria and the aggregated m–polar fuzzy decision matrix, as shown by Table 5.13. 5. The score degrees of the corresponding entries of aggregated weighted m–polar fuzzy decision matrix are organized in Table 5.14.

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5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Table 5.16 m–Polar fuzzy discordance sets x1 x2 x1 x2 x3 x4 x5

– {2, 3, 4} {3, 4, 5} {2, 4, 5} {3, 4, 5}

{1, 5} – {1, 3, 4, 5} {1, 2, 5} {1, 3, 5}

x3

x4

x5

{1, 2} {2} – {2, 5} {1, 2, 3}

{1, 3} {3, 4} {1, 3, 4} – {1, 3, 4}

{1, 2} {2, 4} {4, 5} {2, 5} –

Table 5.17 Euclidean distance measure y11

y21

y31

y41

y51

y12

y22

y32

y42

y52

y11

–

–

–

–

–

y12

–

–

–

–

–

y21

0.0201 –

–

–

–

y22

0.0301 –

–

–

–

y31

0.0506 0.0544 –

–

–

y32

0.0373 0.0491 –

–

–

y41

0.0139 0.007

0.0513 –

–

y42

0.0258 0.0162 0.0562 –

–

y51

0.072

0.0812 0.0491 0.0756 –

y52

0.0503 0.0315 0.0547 0.0409 –

y13

y23

y33

y43

y53

y14

y24

y34

y44

y13

–

–

–

–

–

y14

–

–

–

–

y23

0.0354 –

–

–

–

y24

0.245

–

–

–

y33

0.0764 0.0491 –

–

–

y34

0.3811 0.1438 –

–

y43

0.0548 0.0335 0.0273 –

–

y44

0.2668 0.2202 0.3343 –

y53

0.0433 0.0135 0.0556 0.0422 –

y54

0.1072 0.1401 0.2793 0.2127

y15

y25

y35

y45

y55

y15

–

–

–

–

–

y25

0.1609 –

–

–

–

y35

0.1198 0.2729 –

–

–

y45

0.2556 0.3004 0.2612 –

–

y55

0.1768 0.2341 0.1961 0.3902 –

y54

The m–polar fuzzy concordance sets and m–polar fuzzy discordance sets, determined using the score degrees, are given by Tables 5.15 and 5.16, respectively. 6. The m–polar fuzzy concordance matrix F, evaluated according to m–polar fuzzy concordance sets, is given as follows: ⎛

− ⎜ 0.4046 ⎜ F =⎜ ⎜ 0.2443 ⎝ 0.1916 0.2443

0.5954 − 0.1381 0.4573 0.5100

0.7557 0.8619 − 0.6703 0.8084

0.8084 0.5427 0.4365 − 0.4365

⎞ 0.7557 0.4900 ⎟ ⎟ 0.3297 ⎟ ⎟. 0.5635 ⎠ −

7. The distance measure between the alternatives with respect to all criteria are represented by Table 5.17.

5.4 Comparative Study

305

The m–polar fuzzy discordance matrix G, evaluated according to m–polar fuzzy discordance sets, is given as follows: ⎛

⎞ − 0.6567 0.1328 0.2054 0.4072 ⎜1 − 0.1799 0.7330 0.5985 ⎟ ⎜ ⎟ ⎜ ⎟. 1 − 1 1 G=⎜1 ⎟ ⎝1 ⎠ 1 0.7813 − 1 1 1 0.1991 0.5451 − 8. The m–polar fuzzy concordance level f is evaluated as follows to check the effectiveness of concordance indices: 1  0.5954 + 0.7557 + 0.8084 + 0.7557 + 0.4046 + 0.8619 + 0.5427 5(4) + 0.49 + 0.2443 + 0.1381 + 0.4365 + 0.3297 + 0.1916 + 0.4573 + 0.6703  + 0.5635 + 0.2443 + 0.5100 + 0.8084 + 0.4365 = 0.5122.

f =

The m–polar fuzzy concordance dominance matrix H, obtained by the comparison of concordance level and concordance indices, is shown as follows: ⎛ ⎞ − 1 1 1 1 ⎜0 − 1 1 0⎟ ⎜ ⎟ ⎜ 0 − 0 0⎟ H =⎜0 ⎟. ⎝0 0 1 − 1⎠ 0 0 1 0 − 9. The m–polar fuzzy discordance level g is evaluated as follows to check the effectiveness of discordance indices: 1  0.6567 + 0.1328 + 0.2054 + 0.4072 + 1 + +0.1799 + 0.7330 5(4)  + 0.5985 + 1 + 1 + +1 + 1 + 1 + 1 + 0.7813 + +1 + 1 + 1 + 0.1991 + 0.5451

g=

= 0.7919.

The m–polar fuzzy discordance dominance matrix L , obtained by the comparison of discordance level and discordance indices, is shown as follows: ⎛ ⎞ − 1 1 1 1 ⎜0 − 1 0 1⎟ ⎜ ⎟ ⎜ 0 − 0 0⎟ L=⎜0 ⎟. ⎝0 0 0 − 0⎠ 0 0 1 1 −

306

5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Fig. 5.3 Outranking graph

Table 5.18 Analysis of outranking graph Alternatives Submissive alternatives x1 x2 x3 x4 x5

x2 , x3 , x4 , x5 x3 – – x3

Incomparable alternatives – x4 , x5 x4 x2 , x3 , x5 x2 , x4

10. The m–polar fuzzy aggregated dominance matrix M, representing the outranking relations between alternatives, is shown as follows: ⎛ ⎞ − 1 1 1 1 ⎜0 − 1 0 0⎟ ⎜ ⎟ ⎜ 0 − 0 0⎟ M =⎜0 ⎟. ⎝0 0 0 − 0⎠ 0 0 1 0 − 11. The outranking graph, graphically representing the outranking relations between the alternatives, is given by Fig. 5.3. The complete analysis of the outranking graph is given by Table 5.18. Thus, we conclude firm x1 outranking all other alternatives stands at the most prior position.

5.4 Comparative Study

307

5.4.2 Comparison with m–Polar Fuzzy ELECTRE II Method Now, we apply the procedure of m–polar fuzzy ELECTRE II method to the same numerical example for the sake of comparison. The detailed solution by the m–polar fuzzy ELECTRE II method is provided in following steps: 1. The independent m–polar fuzzy decision matrices of all experts are represented by Tables 5.1, 5.2 and 5.3. 2. The aggregated m–polar fuzzy decision matrix, constructed by applying m–polar fuzzy weighted averaging operator, is shown in Table 5.7. 3. The subjective weights of criteria, determined by experts opinions, are organized in Table 5.12. 4. The aggregated weighted m–polar fuzzy decision matrix is obtained by the product of m–polar fuzzy weights of the criteria and the aggregated m–polar fuzzy decision matrix, as shown by Table 5.13. 5. The m–polar fuzzy concordance, discordance and indifferent sets are represented by Table 5.15, 5.16 and 5.19, respectively. 6. The weights of the m–polar fuzzy concordance and indifference sets are given as follows: 1 (γ c , γ i ) = (1, ). 4 The m–polar fuzzy concordance matrix F, evaluated according to m–polar fuzzy concordance sets, is given as follows: ⎛

− ⎜ 0.4046 ⎜ F =⎜ ⎜ 0.2443 ⎝ 0.1916 0.2443

0.5954 − 0.1381 0.4573 0.5100

0.7557 0.8619 − 0.6703 0.8084

0.8084 0.5427 0.4365 − 0.4365

⎞ 0.7557 0.4900 ⎟ ⎟ 0.3297 ⎟ ⎟. 0.5635 ⎠ −

7. The distance measure between the alternatives with respect to all criteria are represented by Table 5.17. The m–polar fuzzy discordance matrix G, evaluated according to m–polar fuzzy discordance sets and distance measures, is given as follows: ⎛ ⎞ − 0.6567 0.1328 0.2054 0.4072 ⎜ 1 − 0.1799 0.7330 0.5985 ⎟ ⎜ ⎟ 1 − 1 1 ⎟ G=⎜ ⎜1 ⎟. ⎝1 1 0.7813 − 1 ⎠ 1 1 0.1991 0.5451 − 8. The concordance levels, specified by decision-makers, to check the effectiveness of concordance indices are given as follows: (c∗ , c◦ , c− ) = (0.5, 0.6, 0.7).

308

5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Table 5.19 m–Polar fuzzy indifferent sets x1 x2

x3

x4

x5

{} – {} {} {}

{} {} – {} {}

{} {} {} – {}

{} {} {} {} –

Table 5.20 Outranking relations x1 x2

x3

x4

x5

Rs , Rw Rs , Rw – Rw Rs , Rw

Rs , Rw Rw – – –

Rs , Rw – – – –

x1 x2 x3 x4 x5

x1 x2 x3 x4 x5

– {} {} {} {}

– – – – –

Rw – – – –

The discordance levels, serving as the standards to check the effectiveness of discordance indices, are given as follows: (d ∗ , d ◦ ) = (0.6, 0.8). The strong and weak outranking relations among the alternatives are represented by Table 4.26. 9. The strong and weak outranking relations are graphically presented in Fig. 5.4 to find the forward ranking through the iterative procedure. The reverse strong and weak outranking graphs, obtained by reversing the direction of the arcs, are given by Fig. 5.5 to find the reverse ranking. The forward, reverse and final rankings of the available alternatives are tabulated in Table 4.27. Finally, m–polar fuzzy ELECTRE II method suggests firm x1 as the most competent hazardous waste carrier company.

5.4.3 Discussion 1. The numerical example for the selection of most suitable hazardous waste carrier firm is being solved by m–polar fuzzy ELECTRE I method and m–polar fuzzy ELECTRE II method. The results of the both compared techniques and proposed approach are arranged in Table 5.22.

5.4 Comparative Study

309

Fig. 5.4 Strong (a) and weak (b) outranking graphs

Fig. 5.5 Reverse strong (a) and weak (b) outranking graphs Table 5.21 Final ranking x1 Forward ranking Reverse ranking Average ranking

x2

x3

x4

x5

1

2

4

3

2

1

2

4

3

2

1

2

4

3

2

310

5 Enhanced ELECTRE III Method with Multi-polar Fuzzy Sets

Table 5.22 Comparative study Methods Ranking An m–polar fuzzy ELECTRE I – method An m–polar fuzzy ELECTRE x1  x2 ∼ x5  x4  x3 II method An m–polar fuzzy ELECTRE x1  x2  x5  x4  x3 III method (proposed)

2.

3.

4.

5.

Best alternative x1 x1 x1

The existing and presented techniques deliver the same alternative as best choice which demonstrates the authenticity and accuracy of the presented approach. The m–polar fuzzy ELECTRE I and ELECTRE II methods are designed to check the superiority between alternatives on the basis of concordance and discordance sets, in which a slight difference among the performance of alternatives is considered significant. On the other hand, the m–polar fuzzy ELECTRE III approach is designed to treat the pseudo criterion to address all those situation where slight performance difference may not be significant by carefully considering the inexactness of information. Further, m–polar ELECTRE I method may fail to provide the complete ranking list due to incomparable pair of alternatives. Oppositely, the presented variant of ELECTRE method aptly provides the complete ranking list via net credibility degree. Another distinction of the presented m–polar fuzzy ELECTRE III technique is its simple ranking measure whereas in m–polar fuzzy ELECTRE II method an iterative procedure is adopted to determine both forward and reverse rankings. The compared techniques evaluate the criteria weight according to expert opinions. Therefore, any biasness or misconception of the decision-maker could effect the results. To avoid such situation, the presented methodology derives objective weights of criteria by Shannon’s entropy formula using the available data.

5.5 Insights of m–Polar Fuzzy ELECTRE III Method • The first and foremost edge of the presented approach is its capability to address multi-polar information connected with real life complex situations using the decision making skills of ELECTRE theory. Further, ELECTRE III method, being more eminent than the previous variants, has the credit to overcome the limitations of existing techniques owing to its more realistic approach, significant theory, striking methodology and numerous applications. • The m–polar fuzzy ELECTRE III method is an excellent approach to deal with pseudo criterion where the slight performance difference may not be significant

5.6 Conclusion

311

unlike the previous variants that consider any minor superiority or inferiority in terms of concordance or discordance, accordingly. • The presented technique is privileged to figure out the indifference, strong and weak preference relations with the help of three threshold values, including preference, indifference and veto threshold values. • Another merit of the m–polar fuzzy ELECTRE III method is the use of objective weights, evaluated via Shannon’s entropy formula, to minimize the influence of personal perspectives of decision-makers.

5.6 Conclusion An m–polar fuzzy set is accredited to present the several aspects associated with inexact information simultaneously owing to its multi-polar structure and feasibility of fuzzy set theory. This chapter has delivered the procedure of a new and more practical variant of ELECTRE method in order to comply the prominent structure of m–polar fuzzy set with the striking theory and decision making accuracy of ELECTRE methods. The presented m–polar fuzzy ELECTRE III method, being an excellent MCGDM approach, accepts the input data in the form of the m–polar fuzzy numbers, whose aggregation has been done by m–polar fuzzy weighted averaging operator. The Shannons’s entropy formula has been employed to evaluate the criteria weights objectively. Then, the pairwise comparison of the performance of alternatives has been conducted with the help of score degrees in order to find their pairwise relations in accordance with preference, indifference and veto threshold values. After the evaluation of concordance and discordance indices, credibility indices have been considered to exhibit the combined effect of dominance and precedence for any pair of alternatives. Finally, the complete ranking has been derived from net credibility degree. Moreover, to demonstrate the applicability and authenticity of m–polar fuzzy ELECTRE III method, a case study has been delineated for the selection of best firm to carry the hazardous waste of a chemical company of Turkey. To exhibit the authenticity and competency of presented technique, a comparison analysis with m–polar fuzzy ELECTRE I and m–polar fuzzy ELECTRE II methods has been presented.

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30. Li, H.F., Wang, J.J.: An improved ranking method for ELECTRE III. In: 2007 International Conference on Wireless Communications. Networking and Mobile Computing, pp. 6659–6662, IEEE (2007) 31. Lupo, T.: Fuzzy ServPerf model combined with ELECTRE III to comparatively evaluate service quality of international airports in Sicily. J. Air Transp. Manag. 42, 249–259 (2015) 32. Marzouk, M.M.: ELECTRE III model for value engineering applications. Autom. Constr. 20(5), 596–600 (2011) 33. Montazer, G.A., Saremi, H.Q., Ramezani, M.: Design a new mixed expert decision aiding system using fuzzy ELECTRE III method for vendor selection. Expert Syst. Appl. 36(8), 10837–10847 (2009) 34. Noori, A., Bonakdari, H., Morovati, K., Gharabaghi, B.: Development of optimal water supply plan using integrated fuzzy Delphi and fuzzy ELECTRE III methods-Case study of the Gamasiab basin. Expert Syst. 37(5), e12568 (2020) 35. Papadopoulos, A., Karagiannidis, A.: Application of the multi-criteria analysis method Electre III for the optimisation of decentralised energy systems. Omega 36(5), 766–776 (2008) 36. Roy, B.: The outranking approach and the foundations of ELECTRE methods. In: Readings in Multiple Criteria Decision Aid. Springer, Berlin. https://doi.org/10.1007/978-3-642-759352_8 37. Roy, B.: Classement et choix en présence de points de vue multiples. Revue française d’informatique et de recherche opérationnelle 2(8), 57–75 (1968) 38. Roy, B., Figueira, J.R., Almeida-Dias, J.: Discriminating thresholds as a tool to cope with imperfect knowledge in multiple criteria decision aiding: theoretical results and practical issues. Omega 43, 9–20 (2014) 39. Sarwar, M., Akram, M., Shahzadi, S.: Distance measures and δ-approximations with rough complex fuzzy models. Granular Comput. 8, 893–916 (2023) 40. Shahzadi, S., Akram, M.: Intuitionistic fuzzy soft graphs with applications. J. Appl. Math. Comput. 55, 369–392 (2017) 41. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948) 42. Shen, F., Lan, D., Li, Z.: An intuitionistic fuzzy ELECTRE-III method for credit risk assessment. In: Proceedings of the Tenth International Conference on Management Science and Engineering Management, pp. 289–296. Springer, Singapore (2017) 43. Takeda, E.: A method for multiple pseudo-criteria decision problems. Comput. Oper. Res. 28(14), 1427–1439 (2001) 44. Torkayesh, A.E., Fathipoir, F., Saidi-Mehrabd, M.: Entropy-based multi-criteria analysis of thermochemical conversions for energy recovery from municipal solid waste using fuzzy VIKOR and ELECTRE III: case of Azerbaijan region. Iran J. Energy Manag. Technol. 3(1), 17–29 (2019) 45. Waseem, N., Akram, M., Alcantud, J.C.R.: Multi-attribute decision making based on m–polar fuzzy Hamacher aggregation operators. Symmetry 11(12), 1498 (2019) 46. Wu, Y., Zhang, J., Yuan, J., Geng, S., Zhang, H.: Study of decision framework of offshore wind power station site selection based on ELECTRE-III under intuitionistic fuzzy environment: A case of China. Energy Convers. Manag. 113, 66–81 (2016) 47. Yager, R.R.: Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 22(4), 958–965 (2013) 48. Yager, R.R.: Pythagorean fuzzy subsets. In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), pp. 57-61 (2013) 49. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

Chapter 6

Extended ELECTRE IV Method with Multi-polar Fuzzy Sets

This chapter is a continuation to the series of ELECTRE methods as it delineates another variant of the ELECTRE family, namely, multi-polar (m–polar) fuzzy ELECTRE IV method. The presented procedure takes the advantage of outranking principles of the ELECTRE family to specify the optimal alternative but the noticeable edge of the m–polar fuzzy ELECTRE IV method is its unique procedure that does not need the criteria weight for operation. The presented technique competently captures the multi-polarity and fuzziness of the real world problems, simultaneously. Further, the presented variant is competent enough to filter out five different classes of dominance to rank the alternatives. The ranking is achieved by two different distillation procedures to examine the ranking order from both perspectives including ascending and descending distillations. The application of the presented technique is manifested by a real-life case study for the selection of a qualified and efficient contractor for Islamic Azad University-Qazvin Branch Innovation Park project. To verify the accuracy of the presented methodology, comparative study with m–polar fuzzy ELECTRE III method is conducted.

6.1 Introduction ELECTRE is a family of multi-criteria decision making (MCDM) techniques that closely follow the outranking principles to eliminate the non-optimal choices via outranking relations. The adaptable methodology and accurate outcomes of the ELECTRE method led to the expansion of literature by development of back to back modified variants of this family. The common operation of these techniques is to conduct pairwise comparison to investigate them from both aspects, i.e., superiority and inferiority. Although, different variants examine the pairwise precedence via different conceptual structures but their chief goal is to establish outranking relations by employing the corresponding information. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_6

315

316

6 Extended ELECTRE IV Method with Multi-polar Fuzzy Sets

The first step to develop the theory of ELECTRE methods was taken by Benayoun et al. [15] who presented the first variant which is now-a-days known as ELECTRE I method. The prevailing concepts of concordance and discordance sets in this approach motivated other researchers to work in this field. To avoid the partial ranking of ELECTRE I method, Grolleau and Tergny [24] came forward with another improved variant, namely, ELECTRE II method that can competently establish the complete ranking list. Duckstein and Gershon [22], Hokkanen et al. [28], Jun et al. [30], Liu and Ma [34] further extended the theory of ELECTRE II approach with a variety of application in different fields of life. Roy [46, 47] served the literature with another variant, namely, ELECTRE III method, that possesses the more reliable way of comparing the alternatives and proceed by examining three different types of preferences. Many progressive studies have been done on ELECTRE III method [1, 16, 33, 35, 38, 42]. The traditional variants were based on simple outranking relations or corresponding outranking degrees but they can not classify the intensity of outranking relations. To observe the outranking relations in more broader perspective, Roy and Hugonnard [49] proposed the advanced variant of ELECTRE IV method for the first time. This method captivated the attentions of the quality researchers and facilitated the decision makers owing to its eminent properties, striking theory, extra-ordinary efficiency and frequent application. Vallee and Zielniewicz [57] explored the two known variants, namely ELECTRE III and ELECTRE IV methods. Z˙ ak and Kruszynski ´ [61] studied a combination of ELECTRE IV and AHP techniques. Rogers et al. [43–45] deeply studied impact of threshold values in ELECTRE methods. These decision making procedures were designed to formulate the exact information systematically to extract the final decision from available information. Practically, the information or observations can not be expressed as exact numerical values in all situations. The expression of imprecise information was made possible by Zadeh’s fuzzy set theory [59]. Hatami-Marbini and Tavana [27] extended the ELECTRE I method within the fuzzy model. Kaya and Kahraman [31] combined the AHP and ELECTRE methods to capture fuzzy information. Govindan et al. [25] and Sevkli [52] explored the applications of ELECTRE methods in fuzzy environment. Dascal ˘ [21] and Mir et al. [39] also reviewed the benefits of ELECTRE method in practical decision making. Further variants of this family in the fuzzy model were provided by Montazer et al. [39], Lupo [36], Gao et al. [24], La fata et al. [32], Torkayesh et al. [55] and Noori et al. [41]. Akram et al. [11] is accredited to exploit the outranking skills of ELECTRE IV approach in fuzzy environment and demonstrate its application in water supply problem. Mabkhot et al. [37] investigated the TOPSIS and ELECTRE IV approaches with different levels of experts involvement and new weighting schemes. Akram et al. [12] and Zahid et al. [60] also implemented this theory to solve the problems from different fields of life. For detailed study, readers are referred to [4–7, 9, 16, 29, 48, 53, 54]. Besides the inherent imprecision of human decisions, another issue was the multipolarity of the real-world data that appears frequently in practical evaluations and needs to be considered to achieve the desired level of accuracy. The fuzzy sets were designed to capture exactly one aspect of the information via truthness grade. To

6.2 An m–Polar Fuzzy ELECTRE IV Method

317

capture all the aspects of multi-polar information, Chen et al. [19] broadened the structure of fuzzy sets and presented m–polar fuzzy sets that can incorporate the fuzziness and multi-polarity in a single framework. Akram et al. [10] and Adeel et al. [2] developed the ELECTRE-based outranking approaches for m–polar fuzzy information. For deep analysis of preliminary ideas and decision making techniques in m–polar fuzzy model, readers are referred to [3, 8, 58].

6.2 An m–Polar Fuzzy ELECTRE IV Method This section is designed to deliver the procedural steps thoroughly. The presented approach significantly captures the pseudo criteria by considering the performance difference of alternatives in reference to few thresholds that determine the intensity of preference. The final ranking are then derived via distillation processes, accordingly. The presented method operates without criteria weights but it does not mean that all criteria are treated equally important. Firstly, lets portray the mathematical model of a multi-criteria group decision making problem. It essentially depends on three indispensable factors, including a set A = {x1 , x2 , . . . , xr } of r feasible alternatives, a finite set T = {t1 , t2 , . . . , ts } of s multi-polar decision criteria that significantly influence the performance of alternatives and last factor is the panel of v honorable experts E = {e1 , e2 , . . . , ev }. The experts assisting the decision making process may not be equally important so their importance are embedded in the weight vector α = (α1 , α2 , . . . , αv ). The descriptive procedure of m–polar fuzzy ELECTRE IV method is described in the following steps: Step 1: Firstly, the decision making panel assesses the performance of the alternatives and express their evaluations regarding multi-polar criteria in the form of m–polar fuzzy numbers that allow to rate the performance of alternatives regarding each aspect accordingly. As the decision making process is governed by v decision making experts, thus we obtain v independent decision matrices, one corresponding to each individual. The m–polar fuzzy decision matrix Z (u) comprising the evaluations of the expert eu is represented as follows: ⎛

t1

(u1) (u2) (um) x1 ⎜(z 11 , z 11 , . . . , z 11 ) ⎜

Z (u)

t2 (u1) (u2) (um) (z 12 , z 12 , . . . , z 12 )

⎜ ⎜ (u1) (u2) ⎜(z , z , . . . , z (um) ) (z (u1) , z (u2) , . . . , z (um) ) 21 21 22 22 22 = x2 ⎜ ⎜ 21 . ⎜ . . . ⎜ . . . ⎜ . . ⎝(z (u1) , z (u2) , . . . , z (um) ) (z (u1) , z (u2) , . . . , z (um) ) xr r1 r1 r1 r2 r2 r2

···

···

ts

⎞

(u1) (u2) (um) (z 1s , z 1s , . . . , z 1s )⎟ ⎟

⎟ ⎟ (u1) (u2) (um) ⎟ · · · (z 2s , z 2s , . . . , z 2s )⎟ ⎟ . .. ⎟ . ⎟ . . ⎟ (u1) (u2) (um) ⎠ · · · (z , z , . . . , z ) rs

rs

rs

318

6 Extended ELECTRE IV Method with Multi-polar Fuzzy Sets

(u1) (u2) (um) where i jth entry z i(u) j = (z i j , z i j , . . . , z i j ) denotes the decision of expert eu expressing the aptitude of the alternative xi relative to the cri(u) terion t j and kth pole z i(uk) j of z i j corresponds to performance regarding kth attribute of criterion t j . Step 2: Next task is the aggregation of individual assessments to obtain a mutual consensus of all experts. For this purpose, averaging operator appears as a most appropriate tool. Thus, the m–polar fuzzy averaging operator is applied to the ijth entries of all independent matrices to obtain the ijth entry of aggregated m–polar fuzzy decision matrix Z as follows:

(1)

(2)

(v)

z i j = α1 z i j ⊕ α2 z i j ⊕ · · · ⊕ αv z i j ⎛ ⎞ v v v



   (u1) αu (u2) αu (um) αu ⎠ = ⎝1 − ,1 − ,...,1 − . 1 − zi j 1 − zi j 1 − zi j u=1

u=1

(6.1)

u=1

Mathematically, the aggregated m–polar fuzzy decision matrix has the following form: ⎛

t1

m 1 2 x1 ⎜ (z 11 , z 11 , . . . , z 11 )

Z=

x2 . . . xr

t2

m 1 2 (z 12 , z 12 , . . . , z 12 )

···

ts

⎞

m 1 2 (z 1s , z 1s , . . . , z 1s )⎟

··· ⎟ ⎜ ⎟ ⎜ ⎜ (z 1 , z 2 , . . . , z m ) (z 1 , z 2 , . . . , z m ) · · · (z 1 , z 2 , . . . , z m )⎟ ⎜ 21 21 21 22 2s ⎟ 22 22 2s 2s . ⎟ ⎜ .. .. .. ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ 1 2 ⎝(zr 1 , zr 1 , · · · , zrm1 ) (zr12 , zr22 , . . . , zrm2 ) · · · (zr1s , zr2s , . . . , zrms ) ⎠ .. .

Step 3: The thresholds values play a vital role in determination of preferences among alternatives relative to the pseudo criteria. Threshold values are actually the boundary lines to identify the intensity of dominance among any pair of alternatives. The threshold values are classified as indifference q, preference p and veto ν threshold values. These threshold values are selected by the decision makers observing the nature of each criterion and the significance of performance difference for that particular criterion. Further, the threshold values must satisfy the following ordering, i.e., q ≤ p ≤ ν. Now, we explain the significance of these threshold values one by one. The indifference threshold value represents the largest value (performance difference) that can be ignored and the alternatives could be declared indifferent. In other words, it represents the highest performance difference that is insignificant to outrank the competing alternative. Next is preference threshold that identifies the significant amount of performance difference to strictly outrank the inferior alternative. This threshold also differentiates two different levels of preference, i.e., weak preference and strong preference. If the performance

6.2 An m–Polar Fuzzy ELECTRE IV Method

319

difference lies between indifference and preference thresholds then this will be considered as weak preference. The performance difference above the preference threshold is regarded as strong preference. Further, for the specification of preference relations, the alternatives are compared on the basis of their score degrees. The score degree of an mpolar fuzzy number z i j = (z i1j , z i2j , . . . , z imj ) is given by m s(z i j ) =

h h=1 z i j

m

.

(6.2)

Mathematically, an alternative x p is indifferent to its competitor alternative xq if and only if the following condition holds: s(xq ) < s(x p ) ≤ s(xq ) + q(x p ) ⇔ x p Ixq .

(6.3)

Mathematically, an alternative x p is to its competitor alternative xq if and only if the following condition holds: s(xq ) + q(x p )) < s(x p ) ≤ s(xq ) + p(x p ) ⇔ x p Qxq .

(6.4)

Mathematically, an alternative x p is to its competitor alternative xq if and only if the following condition holds: s(x p ) > s(xq ) + p(x p ) ⇔ x p Pxq .

(6.5)

Step 4: Now, the outranking relations are observed in accordance to to specify the intensity of dominance. These relations are determined by the number of criteria specifying any particular relation. Among two alternatives x p and xq , one of the following relations must holds, i.e., x p Pxq , x p Qxq , x p Ixq , x p Exq , xq Px p , xq Qx p , or xq Ix p . Further, this implies that s = NP (x p , xq ) + NQ (x p , xq ) + NI (x p , xq ) + N E (x p , xq ) + NP (xq , x p ) + NQ (xq , x p ) + NI (xq , x p ), here, NP (x p , xq ), NQ (x p , xq ), NI (x p , xq ) and NE (x p , xq ) reflect the number of criteria for which x p is strongly preferable, weakly preferable, indifferent and equal to xq , respectively. On the other hand, NP (xq , x p ), NQ (xq , x p ) and NI (xq , x p ) denote the number of criteria for which xq is strongly preferable, weakly preferable and indifferent to x p , respectively. Now, different dominance classes are defined in the light of the following conditions:

320

6 Extended ELECTRE IV Method with Multi-polar Fuzzy Sets

(i) Quasi dominance Dq The quasi dominance of an alternative x p over the alternative xq is determined via the following conditions: (a) There does not exist any criterion for which alternative xq is either strongly or weakly preferable than alternative x p . (b) The number of criteria for which x p is either indifferent, weakly or strongly preferable to xq must be greater than the number of criteria for which xq is indifferent to x p . Mathematically, these conditions can be interpreted as x p Dq xq ⇔ NP (xq , x p ) + NQ (xq , x p ) = 0 and NI (xq , x p ) < NI (x p , xq ) + NQ (x p , xq ) + NP (x p , xq ) (ii) Canonical dominance Dc An alternative x p is enough superior to outrank the other alternative xq with canonical dominance if and only if the following conditions hold: (a) There does not exist any criterion for which alternative xq is strictly preferable over alternative x p . (b) The number of criteria for which xq is weakly preferable over x p should not be greater than the number of criteria for which x p is strongly preferable over xq . (c) The number of the criteria for which the xq is either indifferent or weakly preferable to x p should be less than the number of criteria for which x p is either indifferent, strong or weakly preferable to xq . Mathematically, these conditions can be interpreted as x p Dc xq ⇔ NP (xq , x p ) = 0 and NQ (xq , x p ) ≤ NP (x p , xq ) and NQ (xq , x p ) + NI (xq , x p ) < NP (x p , xq ) + NQ (x p , xq ) + NI (x p , xq ) (iii) Pseudo dominance D p An alternative x p is dominant enough to outrank the alternative xq with pseudo dominance if and only if the following conditions hold: (a) There does not exist any criterion for which alternative xq is strictly preferable over alternative x p . (b) The number of criteria for which xq is weakly preferable over x p should not be greater than the number of criteria for which x p is weakly or strongly preferable over xq . Mathematically, these conditions can be interpreted as x p D p xq ⇔ NP (xq , x p ) = 0 and NQ (xq , x p ) ≤ NP (x p , xq ) + NQ (x p , xq ).

6.2 An m–Polar Fuzzy ELECTRE IV Method

321

(iv) Sub-dominance Ds An alternative x p is dominant enough to outrank the alternative xq with sub-dominance if and only if the following condition holds (a) There does not exist any criterion for which alternative xq is strictly preferable over alternative x p . Mathematically, this type of dominance can be described as x p Ds xq ⇔ NP (xq , x p ) = 0. (v) Veto dominance Dv The veto dominance between a pair of alternatives (x p , xq ) is given by two alternate conditions: (a) There does not exist any criterion for which alternative xq is strictly preferable over alternative x p . Mathematically, this type of dominance can be described as x p Dv xq ⇔ NP (xq , x p ) = 0. Or In case, if NP (xq , x p ) = 0 then the veto dominance can be determined by the following conditions: (a) If there exist exactly one criterion for which alternative xq is strictly preferable over alternative x p then this criterion should not veto the outranking of xq over x p . (b) The alternative x p must be strictly preferable over alternative xq for at least half of the criteria. This alternate condition of veto dominance can be mathematically described as follows: x p Dv xq ⇔ NP (xq , x p ) = 1 such that s(xq ) − s(x p ) ≤ ν and NP (x p , xq ) ≥

s . 2

Step 5: At this stage, the credibility indices are assigned to each pair of compared alternatives relative to the dominance relation occuring between them. Noticing the intensity of different dominance levels, Vallee and Zielniewicz [57] recommended the following indices subjectively with respect to each dominance relation: a. b. c. d. e. f.

For x p Dq xq , ζ(x p , xq ) = 1. For x p Dc xq , ζ(x p , xq ) = 0.8. For x p D p xq , ζ(x p , xq ) = 0.6. For x p Ds xq , ζ(x p , xq ) = 0.4. For x p Dv xq , ζ(x p , xq ) = 0.2. In case of no relation between x p and xq , ζ(x p , xq ) = 0. The matrix of credibility indices is represented as follows:

322

6 Extended ELECTRE IV Method with Multi-polar Fuzzy Sets

⎛

−

ζ(x1 , x2 )

⎜ ⎜ ⎜ ζ(x2 , x1 ) − ⎜ ⎜ ⎜ ⎜ .. .. ζ=⎜ . . ⎜ ⎜ ⎜ ⎜ ζ(xr −1 , x1 ) ζ(xr −1 , x2 ) ⎜ ⎝ ζ(xr , x1 ) ζ(xr , x2 )

· · · ζ(x1 , xr −1 ) ··· ..

.

··· ···

ζ(x1 , xr )

⎞

⎟ ⎟ ζ(x2 , xr −1 ) ζ(x2 , xr ) ⎟ ⎟ ⎟ ⎟ ⎟ .. .. ⎟. . . ⎟ ⎟ ⎟ − ζ(xr −1 , xr ) ⎟ ⎟ ⎠ ζ(xr , xr −1 ) −

Step 6: The last step is to rank the alternatives using two-way distillation procedures, namely ascending distillation and descending distillation. The starts by considering the maximum credibility index then the qualification are evaluated in accordance to cut-off level. The alternatives with the maximum (descending distillation) or minimum (ascending distillation) qualification are short listed for the further proceeding. Before understanding the distillation process, one should be familiar with the following concepts: • Discrimination threshold The discrimination threshold s(ζλ ) for any value ζλ is defined as follows: s(ζλ ) = α ∗ ζλ + β, where α and β are constants which are usually taken as -0.15 and 0.30, respectively. In ELCTRE IV procedure, the value of discrimination threshold is fixed as 0.1 by taking α = 0 and β = 0.1 to capture each dominance level. • Cut-off level A cut-off level of outranking ζλ is a value closer to the maximum credibility index which is defined as follows: ζλ = ζλ−1 − s(ζλ−1 ). • Outranking at cut-off level An alternative x p is said to outrank its competitor xq at the cut-off level ζλ ζ (x p D Aλ xq ) if and only if the following conditions hold: (i) ζ(x p , xq ) > ζλ . (ii) ζ(x p , xq ) − ζ(xq , x p ) > s(ζ(x p , xq )). • Strength at cut-off level The strength of an alternative x p at the cut-off level ζλ is defined by the sum of all credibility indices of the pair for which x p is outranking the other

6.2 An m–Polar Fuzzy ELECTRE IV Method

323

alternatives at cut-off level ζλ . Mathematically, the strength at level ζλ is defined as follows: ζ ζ(x p , xq ), (6.6) S Aλ (x p ) = xq ∈M ζ

where M = {xq ∈ A|x p D Aλ xq }. • Weakness at cut-off level The weakness of an alternative x p at the cut-off level ζλ is defined by the sum of all the credibility indices of the pair for which x p is outranked by the other alternatives at cut-off level ζλ . Mathematically, the weakness at level ζλ is defined as follows:

ζ

S Aλ (x p ) =

ζ(xq , x p ),

(6.7)

xq ∈M ζ



where M = {xq ∈ A|xq D Aλ x p }. • Qualification at cut-off level The qualification of an alternative x p at the cut-off level ζλ is given by the difference of its strength and weakness at ζλ . The mathematical definition of the qualification of alternative x p at level ζλ is defined as follows: ζ

ζ

ζ

Q Aλ (x p ) = S Aλ (x p ) − S Aλ (x p ).

(6.8)

The steps of the ascending distillation are given as follows: (i) Take b = 0 and A0 = A. (ii) Take ζ0 = max ζ(x p , xq ). x p ,xq ∈Ab

(iii) Take λ = 0, D0 = Ab . (iv) Select the value of ζλ+1 as follows: ζλ+1 =

max

ζ(x p ,xq ) k,

(7.3)

where k is the value of indifference threshold. In this case, two alternatives are indifferent as long as their difference does not exceed the value of k, otherwise a strict preference is achieved. Definition 7.4 Type III: The criterion with linear preference is formulated as follows:  x/q if x ≤ q, P(x) = (7.4) 1 if x > q, where q ∈ [0, 1] is the preference threshold assigned by the decision maker. In this type of criterion, the preference of decision maker increases linearly with x until the difference of alternatives is lower than q. When x is greater than q, a strict preference of an alternative is obtained with respect to that criterion. Definition 7.5 Type IV: The level criterion preference function is characterized as follows: ⎧ if x ≤ l, ⎨0 (7.5) P(x) = 1/2 if l < x ≤ l + m, ⎩ 1 if x > l + m, where l and m represent the preference and indifference thresholds respectively, given by the decision maker and can be chosen from interval [0, 1]. In this case, an indifference occurs only if the difference between two alternatives lies in interval [−l, l]. Definition 7.6 Type V: The criterion with linear preference having indifference area is formulated as follows: ⎧ if x ≤ u, ⎨0 P(x) = (x − u)/v if u < x ≤ u + v, (7.6) ⎩ 1 if x > u + v, where the threshold values u and v lie in interval [0, 1]. In this type of preference function, two alternatives are considered to be completely indifferent until the deviation between these alternatives does not exceed the value of u. The preference increases linearly as long as the deviation equals to u + v and after that value, a strict preference is achieved. Definition 7.7 Type VI: The preference function for the Gaussian criteria is defined as follows:  0 if x ≤ 0, (7.7) P(x) = 2 2 1 − e−x /2σ if x > 0,

348

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

where the value of σ ∈ [0, 1] is assigned by decision maker and represents the distance between the origin and the point of inflexion. The next sections describe the methodology of a new version of the PROMETHEE method. It will allow us to deal with MCDM problems having multi-polar or m–polar uncertainties. In this version, the AHP method calculates the normalized weights of criteria. Thus, we first explore this part of the procedure in Sect. 7.3.

7.3 Analytical Hierarchy Process In the AHP method that is used to calculate the weights of the criteria, the pairwise comparison matrix of criteria is determined by using the Saaty (1–9) preference scale as shown in Table 7.1. Then the consistencies of calculated weights are analyzed by interpreting the consistency index and consistency ratio. The step by step procedure of AHP technique is described as follows: Step 1. Construct the hierarchical structure of problem which contains the main criteria and the sub-criteria to evaluate the alternatives. Step 2. Establish a pairwise comparison of criteria and construct a comparison matrix by using the information provided in Table 7.1. Assume that the decision problem is to be assessed on the basis of n criteria, then the pairwise comparison of criterion i with each criterion j yields a square matrix of order n × n. Each entry ci j of matrix C provides the comparative value of criterion i with respect to criterion j.  In the comparison matrix, the entry ci j = 1 if and only if i = j and c ji = 1 ci j .

Table 7.1 Saaty (1–9) preference scale Scale Definition 1

Equally Important

3

Weakly Important

5

Strongly Important

7

Very Strongly Important

9

Extremely Important

2, 4, 6, 8

Intermediate values between

Explanation Both criteria participate equally to the goal Experience weakly favor of one criterion over another Experience strongly favor of one criterion over another Strong dominance of one criterion over another The preference of a criterion is of the highest possible value When compromise is required adjacent scales

7.3 Analytical Hierarchy Process

⎡ Cn×n

349

c11 c21 .. .

c12 c22 .. .

c13 c23 .. .

··· ··· .. .

c1n c2n .. .

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ =⎢ ⎥. ⎥ ⎢ ⎣ c(n−1)1 c(n−1)2 c(n−1)3 · · · c(n−1)n ⎦ cn1 cn2 cn3 · · · cnn

Step 3. Normalize the comparison values of decision matrix Cn×n by deploying the expression given in Eq. 7.8, and construct a normalized decision matrix Cnor m .  n ci j , i, j = 1, 2, 3, . . . , n, (7.8) ei j = ci j i=1

that is, each normalized entry is obtained by dividing each entry of column j by the sum of entries in column j. In the normalized decision matrix, the sum of entries in each column is 1. Step 4. Calculate the weights of criteria by taking the average value of each row of normalized decision matrix as given in Eq. 7.9. w(i) =

n 

 ei j n.

(7.9)

j=1

As a result, a weight vector W satisfying the condition of normality is obtained in the form of column vector as follows, ⎤ ⎡ w(1) ⎢ w(2) ⎥ ⎥ ⎢ ⎥ ⎢ W = ⎢ w(3) ⎥. ⎢ .. ⎥ ⎣ . ⎦ w(n)

Step 5. Construct the matrix C W . Step 6. Compute the maximum Eigenvalue by using the formula given in Eq. 7.10. λmax

n      ith entry in C W ith entry in W . =1 n

(7.10)

i=1

Step 7. Calculate the consistency index as follows:  C I = (λmax − n) (n − 1).

(7.11)

The greater value of consistency index shows the higher deviation from consistency, whereas the smaller value indicates that the decision maker’s comparative values are possibly consistent and the resulting weights are appropriate to obtain the useful estimations. If the consistency index is zero

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7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.2 Random index for different values of n n 2 3 4 5 6 RI

0

0.58

0.90

1.12

1.24

7

8

9

10

1.32

1.41

1.45

1.49

(that is C I = 0), then the decision maker’s comparisons are considered to be perfectly consistent. Step 8. Determine the consistency ratio by dividing the consistency index to the random index as follows:  C R = C I R I,

(7.12)

where RI is the random index which is defined for different values of n, as shown in Table 7.2. If the value of consistency ratio is less than 0.10 (C R < 0.10), then it is acceptable and the weights are consistent. The comparison matrix will be inconsistent if the value of consistency ratio is greater than 0.10, and the AHP weights may not yields the appropriate and meaningful results.

7.4 m–Polar Fuzzy PROMETHEE Method Akram et al. [9] presented the procedure of a new extension of PROMETHEE technique, named as m–polar fuzzy PROMETHEE method, by combining the technique of PROMETHEE method and m–polar fuzzy information. This version of PROMETHEE method is used to evaluate the MCDM problems having multipolar uncertainties. The strategy of m–polar fuzzy PROMETHEE technique is described as follows: define and identify the problem domain and select an appropriate group of decision makers; construct the decision matrices by taking into account the evaluations of each decision maker; aggregate the decision values and establish an aggregated decision matrix; formulate a score matrix by using the score function; define the preference function according to the nature and type of criteria; find out the multi-criteria preference index of each alternative; determine the partial ordering of alternatives (PROMETHEE I); and finally compute the final ranking of alternatives (PROMETHEE II). Suppose a MCDM problem consisting of l alternatives Rφ , φ = 1, 2, . . . , l, that are assessed by a group of s decision makers Dψ , ψ = 1, 2, . . . , s. The group of decision makers is responsible to evaluate the considering set of feasible alternatives on the basis of n conflicting criteria Qϕ , ϕ = 1, 2, . . . , n. The preference ratings of each alternative with respect to different criteria are given in the form of m–polar fuzzy numbers. The steps to explain the procedure of m–polar fuzzy PROMETHEE method are described as follows.

7.4 m–Polar Fuzzy PROMETHEE Method

351

Step 1. Construct a decision matrix. Assume that the performance of each alternative Rφ on the basis of Qϕ criteria is evaluated by Dψ decision maker and represented in the form of ψ decision matrix. As a result, s decision matrices [tφϕ ]l×n are constructed for s decision makers as follows: ⎡ ψ ψ ⎤ ψ t11 t12 · · · t1n ⎢ ψ ψ ψ ⎥ ⎢ t21 t22 · · · t2n ⎥ ψ ⎥ T = [tφϕ ]l×n = ⎢ ⎢ .. .. .. .. ⎥, ⎣ . . . . ⎦ ψ

ψ

ψ

tl1 tl2 · · · tln

 ψ ψ ψ ψ  where each entry tφϕ = p1 ◦ ζφϕ , p2 ◦ ζφϕ , . . . , pm ◦ ζφϕ is an m–polar fuzzy number. Then, s decision values of each alternative with respect to  conflicting criteria are converted into a single value tφϕ = p1 ◦ ζφϕ , p2 ◦ ζφϕ , . . . , pm ◦ ζφϕ by using the averaging formula such as, 1  ψ  pi ◦ ζφϕ , i = 1, 2, . . . , m. s s

pi ◦ ζφϕ =

(7.13)

ψ=1

Then an aggregated decision matrix T = [tφϕ ]l×n is constructed by using the aggregated decision values, where each entry is again an m–polar fuzzy number. Step 2. Construct the score matrix. Further, the aggregated decision values are transformed into simple crisp values by applying the score function of m–polar fuzzy numbers as given below: m  1  pi ◦ ζφϕ . (7.14) sˆφϕ = m i=1 Then, these crisp real values are used to formulate the score matrix S = [ˆsφϕ ]l×n for the further assessment of alternatives. Step 3. Calculate the deviation of alternatives. Since the preference structure of PROMETHEE method is based on pairwise comparison of alternatives, therefore in this step, the deviation between every pair of alternatives is determined with respect to each criterion by taking the difference of the evaluations of alternatives as follows: dϕ (Rφ , Rσ ) = sˆϕ (Rφ ) − sˆϕ (Rσ ), φ, σ = 1, 2, . . . , l,

(7.15)

where the term dϕ (Rφ , Rσ ) represents the deviation of two alternatives Rφ and Rσ on the basis of criteria ϕ and sˆϕ (Rφ ) and sˆϕ (Rσ ) are the evaluations of alternatives Rφ and Rσ , respectively. Step 4. Define the suitable preference function. The preference of an alternative Rφ with respect to other alternative Rσ

352

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets Pϕ (Rφ , Rσ )

Fig. 7.1 Preference function

l

dϕ (Rφ , Rσ )

under each criterion is evaluated by defining an appropriate and suitable preference function. The choice of preference function depends on the nature and type of criteria and these preferences have a real value between 0 and 1. The preferences with zero and negative values are considered as an indifference of decision makers towards that pair of alternatives on the basis of respective criteria. The preference value closest to 1 shows the strong preference. Regarding above discussion, a decision maker will select a preference function of the following form, Pϕ (Rφ , Rσ ) = Fϕ [dϕ (Rφ , Rσ )],

(7.16)

such that 0 ≤ Pϕ (Rφ , Rσ ) ≤ 1 and Pϕ (Rφ , Rσ ) > 0 ⇒ Pϕ (Rσ , Rφ ) = 0. This function defines the preference of an alternative Rφ over Rσ in the case of a criterion to be maximized and has a shape of the following form as shown in Fig. 7.1. In the case of criteria to be minimized, the preference function can be represented as (7.17) Pϕ (Rφ , Rσ ) = Fϕ [−dϕ (Rφ , Rσ )], where negative sign indicates that the preference function for such criteria should be reversed or the alternate of original function. Step 5. Compute the multi-criteria preference index. Next step is to calculate the multi-criteria preference index on the basis of preference function which is defined by decision makers according to the nature of criteria and the criteria weights that are calculated by using AHP method in the proposed technique. The multi-criteria preference index for each pair of alternatives is defined as the weighted average of the corresponding preference function and can be calculated by using the following expression:  (Rφ , Rσ ) =

n

ϕ=1

w(ϕ)Pϕ (Rφ , Rσ ) n ; φ = σ, φ, σ = 1, 2, . . . , l. (7.18) ϕ=1 w(ϕ)

7.4 m–Polar Fuzzy PROMETHEE Method

353

Since  the criteria weights calculated by AHP method are normalized, that is nϕ=1 w(ϕ) = 1, therefore the above expression can be written as 

(Rφ , Rσ ) =

n  w(ϕ)Pϕ (Rφ , Rσ ).

(7.19)

ϕ=1

This preference index indicates the intensity of the preference of decision maker of an alternative Rφ over Rσ with respect to all criteria and has a numeric value between 0 and 1, such that,  – (Rφ , Rσ ) ≈ 0 shows the weak preference of alternative Rφ over Rσ onthe basis of all criteria; – (Rφ , Rσ ) ≈ 1 shows the strong preference of alternative Rφ over Rσ with respect to all criteria. The multi-criteria preference index shows an outranking relationship between every pair of alternatives corresponding to all criteria which is further used to construct an outranking graph. The vertices of this outranking graph represent the alternatives of considering problem and the arc between any two vertices indicates the relation between alternatives. Step 6. Find out the preference ranking. The preference ordering of alternatives are then achieved by using the outranking relation of alternatives determined in Step 5. Two types of ranking are obtained by using this method, that are partial and complete rankings. The alternatives are partially ranked by considering the incoming and outgoing flows of alternatives which is known as PROMETHEE I, and the complete ranking is attained by using the procedure of PROMETHEE II. The procedures of PROMETHEE I and PROMETHEE II are explained as follows. (a) The partial ranking of alternatives (or PROMETHEE I). The outgoing or leaving flow of an alternative Rφ is formulated as follows: χ+ (Rφ ) =

1   (Rφ , Rσ ); φ = σ, φ, σ = 1, 2, . . . , l, (7.20) l − 1 R ∈R σ

that is, the outgoing flow of alternative Rφ is calculated as the average value of the arcs that are going outward form the node Rφ as shown in Fig. 7.2. The outgoing flow is also known as the positive outranking flow and measures the dominance behavior of an alternative over all other alternatives. On the other hand, the incoming or entering flow of an alternative Rφ is calculated as follows: χ− (Rφ ) =

1   (Rσ , Rφ ); φ = σ, φ, σ = 1, 2, . . . , l, (7.21) l − 1 R ∈R σ

354

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Fig. 7.2 Outgoing flow of Rφ Rσ

(R φ

,R

σ)

Rφ

that is, the incoming flow of alternative Rφ is the average value of the inward arcs of the node Rφ as shown in Fig. 7.3. The incoming flow is also known as the negative outranking flow and shows that how much an alternative is dominated by all other alternatives. The alternative with larger outgoing flow and the smaller incoming flow is considered as the favorable or preferable alternative. The preferences of alternatives on the basis of these positive and negative outranking flows can be computed by using the following expressions, respectively. ⎧ ⎨ Rφ P + Rσ ⇐⇒ χ+ (Rφ ) > χ+ (Rσ ); ∀Rφ , Rσ ∈ R, ⎩

Rφ I + Rσ ⇐⇒ χ+ (Rα ) = χ+ (Rσ ); ∀Rφ , Rσ ∈ R,

(7.22)

⎧ ⎨ Rφ P − Rσ ⇐⇒ χ− (Rφ ) < χ− (Rσ ); ∀Rφ , Rσ ∈ R, ⎩

Rφ I − Rσ ⇐⇒ χ− (Rφ ) = χ− (Rσ ); ∀Rφ , Rσ ∈ R.

Fig. 7.3 Incoming flow of Rφ Rσ

(R σ

,R

φ)

Rφ

(7.23)

7.4 m–Polar Fuzzy PROMETHEE Method

355

The intersection of these two preferences provides the PROMETHEE I parˆ Iˆ, R) ˆ of alternatives as follows: tial ranking ( P, ⎧ ˆ σ (Rφ outranks Rσ ) Rφ PR ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

if Rφ P + Rσ and Rφ P − Rσ , or Rφ P + Rσ and Rφ I − Rσ , or Rφ I + Rσ and Rφ P − Rσ ;

⎪ ⎪ Rφ IˆRσ (Rφ is indifferent to Rσ ) iff Rφ I + Rσ and Rφ I − Rσ ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆ σ (Rφ and Rσ are incomparable) otherwise. Rφ RR (7.24) In PROMETHEE I partial ranking, all alternatives are not comparable, so the complete ranking of alternatives is obtained by proceeding the one more step of PROMETHEE II as follows: (b) Complete ordering of alternatives (or PROMETHEE II). The net outranking flow of alternative Rφ is calculated as follows: χ(Rφ ) = χ+ (Rφ ) − χ− (Rφ ),

(7.25)

which is the difference of positive and negative flows and provides the ˜ I˜) of alternatives in the following PROMETHEE II complete ranking ( P, manner as follows: ⎧ ˜ σ (Rφ outranks Rσ ) iff χ(Rφ ) > χ(Rσ ), ⎨ Rφ PR (7.26) ⎩ ˜ Rφ I Rσ (Rφ is indifferent to Rσ ) iff χ(Rφ ) = χ(Rσ ). Thus, all the alternatives can be compared on the basis of net flow of alternatives. The alternative with greatest net outranking flow is considered as the optimal solution or the most preferable alternative. The procedure of m–polar fuzzy PROMETHEE method is summarized in a flowchart as shown in Fig. 7.4. A series of steps and the number of calculations are involved in this multi-criteria decision making method, in which all steps remain same except the choice of preference function. The preference function is defined according to the nature of criteria or by the choice of experts or analysts. Moreover, the criteria weights that are used to determine the multi-criteria preference index can be calculated by applying some appropriate method of normalized weights or can be taken regarding to the preferences of decision makers.

356

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets Ranking of alternatives with respect to conflicting criteria by a team of experts or a group of decision makers

Goal

Consider a set of feasible alternatives and choose the suitable criteria related to the decision problem

Input

Assign the preference ratings of alternatives with respect to criteria in the form of m-polar fuzzy numbers

Construct the aggregated decision matrix by using the averaging operator

Construct the score matrix by applying the score function of m-polar fuzzy numbers

Select the suitable preference function according to the nature and type of criteria

Choice of preference function

Determine the multi-criteria preference index for every pair of alternatives corresponding to all criteria

Calculate the leaving and entering flows and compute the partial ordering of alternatives

PROMETHEE I

Output

Calculate the net outranking flow and find out the complete ranking of alternatives

PROMETHEE II

Fig. 7.4 Flowchart of m–polar fuzzy PROMETHEE method

7.4.1 Ranking the Sites of Hydroelectric Power Stations The electricity is considered as one of the main necessities or requirements for the economic development of a nation. The shortage of electricity not only affects the households, but also the economy. Due to the high and increasing demand of electricity, every state or country needs to generate their own energy without relying on international sources. There are many ways to convert different types of energies into electrical energy, including windmills, solar power, hydroelectric power and by burning the fossil fuels such as coal, oil or natural gas etc. Hydroelectric power

7.4 m–Polar Fuzzy PROMETHEE Method

357

is a renewable source of energy as it produces electricity by using the energy of flowing water. Moreover, it doesn’t pollute the environment like other power plants that use the coal or natural gas as fuel, therefore it is also known as clean fuel source of energy. Assume that a company wants to plant its own power station to fulfill the requirements of electricity. The suitable location or site is one of the most important factors to plant a hydroelectric power station. After initial screening, a set of seven different sites, R = {R1 , R2 , R3 , R4 , R5 , R6 , R7 }, were selected for further evaluation. A committee of two field experts was appointed to rank these sites on the basis of six criteria (or factors) as follows: Q1 : Infrastructure, Q2 : Nature of land, Q3 : Government incentives, Q4 : Social infrastructure, Q5 : Climate changes, Q6 : Cost. Each factor has been further categorized into three characteristics to make a 3F number as follows: • The factor “Infrastructure” includes the availability of water, storage of water and transportation facilities. • The factor “Nature of land” includes the security level, availability of labor and soil type. • The factor “Government incentives” includes the licensing policies, tax incentives and energy subsidies. • The factor “Social infrastructure” includes the public safety, health care facilities and educational institutes. • The factor “Climate changes” includes the atmospheric pressure, wind velocity and air temperature. • The factor “Cost” includes the construction cost, maintenance cost and transportation cost. On the basis of above discussed structure, the ranking for the sites of hydroelectric power plants by using PROMETHEE method is described.

7.4.2 Criteria Weights by AHP Firstly, the weights of criteria are calculated by using the process of AHP technique. The pairwise comparison of criteria are constructed on the basis of Saaty (1–9) preference scale as given in Table 7.1, and the values are given in Table 7.3. By using the condition of normality, which is given in Eq. 7.8, the normalized matrix C for criteria is constructed as follows,

358

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.3 The pairwise comparison of criteria Q1 Q2 Q3 Q4 Q5 Q6

Q1

Q2

Q3

Q4

Q5

Q6

1 0.20 0.11 0.33 0.20 0.14

5 1 0.20 3 3 0.20

9 5 1 5 7 3

3 0.33 0.20 1 1 0.33

5 0.33 0.14 1 1 0.20

7 5 0.33 3 5 1

⎡

0.51 ⎢ 0.10 ⎢ ⎢ 0.06 C =⎢ ⎢ 0.17 ⎢ ⎣ 0.10 0.07

0.40 0.08 0.02 0.24 0.24 0.02

0.30 0.17 0.03 0.17 0.23 0.10

0.51 0.06 0.03 0.17 0.17 0.06

0.65 0.04 0.02 0.13 0.13 0.03

⎤ 0.33 0.23 ⎥ ⎥ 0.02 ⎥ ⎥. 0.14 ⎥ ⎥ 0.23 ⎦ 0.05

Then the criteria weights are calculated by employing the Eq. 7.9 and the weights are provided in the weight vector W as follows,  T W = 0.45 0.11 0.03 0.17 0.18 0.06 . Next, we need to check the consistency of calculated weights by determining the consistency ratio of the comparison matrix. The small consistencies are negligible and do not cause the serious difficulties. For the consistency check, first step is to construct a matrix C W given as,  T C W = 3.10 0.766 0.181 0.159 1.28 0.327 . Then the maximum Eigenvalue λmax is computed by applying the Eq. 7.10, that is,  3.1 0.766 0.181 1.159 1.28 0.327  + + + + + 0.45 0.11 0.03 0.17 0.18 0.06 = 6.54

λmax = 1/6

The consistency index is C I = 0.108, which is obtained by employing the Eq. 7.11, and the consistency ratio is determined by using the random index, R I = 1.24 (for n = 6). Since the consistency ratio is 0.087, which is less than 0.10, so the given comparison matrix shows the consistent behavior and the calculated wights are appropriate for decision making.

7.4 m–Polar Fuzzy PROMETHEE Method

7.4.3

359

Ranking Through m–Polar Fuzzy PROMETHEE

In this subsection, a new version of an outranking method PROMETHEE, named as m–polar fuzzy PROMETHEE, is applied to rank the sites with respect to six criteria. The types of criteria, which are specified by decision maker on the basis of generalized criteria preference functions, and their corresponding parameters are given in Table 7.4. The evaluations for ranking the sites of hydroelectric power plants through m– polar fuzzy PROMETHEE method by applying the AHP weights of criteria are as follows: Step 1. The decision values of alternatives with respect to multiple and conflicting criteria in the form of 3–polar fuzzy numbers are provided by experts D1 and D2 as shown in Tables 7.5 and 7.6, respectively. Then the aggregated

Table 7.4 Types of criteria and corresponding parameters Criteria

Max or Min

Type of criterion

Parameters

Q1

Max

V

u = 0.02 v = 0.1

Q2

Max

III

q = 0.1

Q3

Min

VI

σ = 0.01

Q4

Max

II

k = 0.01

Q5

Min

IV

l = 0.05 m = 0.1

Q6

Min

I

−

Table 7.5 Decision values of alternatives by decision maker D1 Infrastructure

Nature of land

Government incentives

R1

(0.40, 0.37, 0.50)

(0.64, 0.32, 0.50)

(0.40, 0.60, 0.55)

R2

(0.55, 0.60, 0.45)

(0.75, 0.80, 0.65)

(0.40, 0.45, 0.50)

R3

(0.40, 0.55, 0.60)

(0.50, 0.60, 0.57)

(0.60, 0.70, 0.45)

R4

(0.60, 0.65, 0.50)

(0.35, 0.47, 0.60)

(0.57, 0.43, 0.50)

R5

(0.50, 0.45, 0.50)

(0.70, 0.55, 0.50)

(0.45, 0.60, 0.50)

R6

(0.70, 0.57, 0.63)

(0.50, 0.40, 0.60)

(0.63, 0.55, 0.47)

R7

(0.50, 0.43, 0.60)

(0.65, 0.43, 0.70)

(0.70, 0.83, 0.45)

Social infrastructure

Climate changes

Cost

R1

(0.43, 0.35, 0.50)

(0.57, 0.65, 0.60)

(0.60, 0.47, 0.50)

R2

(0.50, 0.40, 0.45)

(0.70, 0.85, 0.73)

(0.80, 0.70, 0.65)

R3

(0.60, 0.57, 0.63)

(0.45, 0.55, 0.67)

(0.40, 0.67, 0.80)

R4

(0.40, 0.65, 0.50)

(0.60, 0.53, 0.47)

(0.80, 0.50, 0.53)

R5

(0.55, 0.47, 0.63)

(0.50, 0.65, 0.43)

(0.57, 0.60, 0.55)

R6

(0.53, 0.50, 0.65)

(0.70, 0.80, 0.75)

(0.47, 0.55, 0.60)

R7

(0.47, 0.65, 0.70)

(0.65, 0.47, 0.55)

(0.40, 0.50, 0.53)

360

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.6 Decision values of alternatives by decision maker D2 Infrastructure

Nature of land

Government incentives

R1

(0.45, 0.50, 0.47)

(0.65, 0.57, 0.60)

(0.47, 0.40, 0.50)

R2

(0.57, 0.55, 0.60)

(0.70, 0.73, 0.80)

(0.35, 0.43, 0.40)

R3

(0.50, 0.47, 0.53)

(0.57, 0.60, 0.50)

(0.53, 0.34, 0.40)

R4

(0.60, 0.57, 0.65)

(0.63, 0.50, 0.65)

(0.60, 0.57, 0.50)

R5

(0.47, 0.65, 0.70)

(0.70, 0.67, 0.50)

(0.45, 0.57, 0.50)

R6

(0.65, 0.70, 0.60)

(0.75, 0.80, 0.50)

(0.50, 0.47, 0.43)

R7

(0.50, 0.60, 0.57)

(0.60, 0.55, 0.47)

(0.60, 0.65, 0.57)

Social infrastructure

Climate changes

Cost

R1

(0.45, 0.57, 0.50)

(0.60, 0.57, 0.65)

(0.70, 0.53, 0.60)

R2

(0.40, 0.35, 0.60)

(0.70, 0.85, 0.60)

(0.75, 0.80, 0.60)

R3

(0.65, 0.50, 0.53)

(0.50, 0.43, 0.57)

(0.65, 0.57, 0.70)

R4

(0.45, 0.50, 0.57)

(0.57, 0.47, 0.50)

(0.57, 0.60, 0.47)

R5

(0.53, 0.40, 0.47)

(0.60, 0.53, 0.55)

(0.60, 0.53, 0.40)

R6

(0.55, 0.45, 0.50)

(0.40, 0.47, 0.50)

(0.45, 0.50, 0.60)

R7

(0.60, 0.63, 0.57)

(0.43, 0.57, 0.60)

(0.57, 0.53, 0.60)

Table 7.7 Aggregated decision values of alternatives Infrastructure

Nature of land

Government incentives

R1

(0.425, 0.435, 0.485)

(0.645, 0.445, 0.55)

(0.435, 0.50, 0.525)

R2

(0.56, 0.575, 0.525)

(0.725, 0.765, 0.725)

(0.375, 0.44, 0.45)

R3

(0.45, 0.51, 0.565)

(0.535, 0.60, 0.535)

(0.565, 0.52, 0.425)

R4

(0.60, 0.61, 0.575)

(0.49, 0.485, 0.625)

(0.585, 0.50, 0.50)

R5

(0.485, 0.55, 0.60)

(0.70, 0.61, 0.50)

(0.45, 0.585, 0.50)

R6

(0.675, 0.635, 0.615)

(0.625, 0.60, 0.55)

(0.565, 0.51, 0.45)

R7

(0.50, 0.515, 0.585)

(0.625, 0.49, 0.585)

(0.65, 0.74, 0.51)

Social infrastructure

Climate changes

Cost

R1

(0.44, 0.46, 0.50)

(0.585, 0.61, 0.625)

(0.65, 0.50, 0.55)

R2

(0.45, 0.375, 0.525)

(0.70, 0.85, 0.665)

(0.775, 0.75, 0.625)

R3

(0.625, 0.535, 0.58)

(0.475, 0.49, 0.62)

(0.525, 0.62, 0.75)

R4

(0.425, 0.575, 0.535)

(0.585, 0.50, 0.485)

(0.685, 0.55, 0.50)

R5

(0.54, 0.435, 0.55)

(0.55, 0.59, 0.49)

(0.585, 0.565, 0.475)

R6

(0.54, 0.475, 0.575)

(0.55, 0.635, 0.625)

(0.46, 0.525, 0.60)

R7

(0.535, 0.64, 0.635)

(0.54, 0.52, 0.575)

(0.485, 0.515, 0.565)

decision preferences are obtained by applying the averaging operator, and the results are summarized in Table 7.7. Step 2. The score matrix S is constructed by applying the score function of 3–polar fuzzy numbers as follows:

7.4 m–Polar Fuzzy PROMETHEE Method

361

R6

R7

R3 R1

R2

R4

R5

Fig. 7.5 Partial relations of PROMETHEE I

Q1

R1 R2 R3

S=

R4 R5 R6 R7

Q2

⎡ 0.448 0.547 ⎢ ⎢ ⎢ 0.553 0.738 ⎢ ⎢ ⎢ ⎢ 0.508 0.557 ⎢ ⎢ ⎢ ⎢ 0.595 0.533 ⎢ ⎢ ⎢ ⎢ 0.545 0.603 ⎢ ⎢ ⎢ ⎢ 0.642 0.592 ⎣

Q3

Q4

Q5

Q6

0.487 0.467 0.607 0.567 ⎤ ⎥ ⎥ 0.422 0.450 0.738 0.717 ⎥ ⎥ ⎥ ⎥ 0.503 0.580 0.528 0.632 ⎥ ⎥ ⎥ ⎥ . 0.528 0.512 0.523 0.578 ⎥ ⎥ ⎥ ⎥ 0.512 0.508 0.543 0.542 ⎥ ⎥ ⎥ ⎥ 0.508 0.530 0.603 0.528 ⎥ ⎦

0.533 0.567 0.633 0.603 0.545 0.522

Step 3. The score matrix is then used to calculate the difference or deviation of an alternative with respect to other alternatives. The deviation for every pair of alternatives with respect to each criterion is computed by using the Equation 7.15, and the outcomes are shown in Table 7.8. Step 4. Further, the preference degree of every pair of alternatives with respect to each criterion is calculated by using the preference function. In this method, six different types of preference functions are used according to the nature or type of criteria as described in Table 7.4. The results for each type of preference functions for every pair of alternatives are shown in Table 7.9. Step 5. The weighted averages of these preference functions are known as multicriteria preference index of alternatives. The multi-criteria preference index or the total degree of preference for each pair of alternative is calculated by deploying the Eq. 7.19, and the values are given in Table 7.10. Step 6. The whole procedure is concluded in this step and the results for partial and net outranking flows are determined.

362

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.8 Deviation of alternatives with respect to criteria Q1

Q2

Q3

Q4

Q5

R1 R2

−0.105

−0.191

0.065

0.017

−0.131

Q6 −0.15

R1 R3

−0.06

−0.01

−0.016

−0.113

0.079

−0.065

R1 R4

−0.147

−0.014

−0.041

0.045

0.084

−0.011

R1 R5

−0.097

−0.056

−0.025

−0.041

0.064

0.025

R1 R6

−0.194

−0.045

−0.021

−0.063

0.004

0.039

R1 R7

−0.085

−0.02

−0.146

−0.136

0.062

0.45

R2 R1

0.105

0.191

−0.065

−0.017

0.131

0.15

R2 R3

0.045

0.181

−0.081

−0.13

0.21

0.085

R2 R4

−0.042

0.205

−0.106

−0.062

0.215

0.139

R2 R5

0.008

0.135

−0.09

−0.058

0.195

0.175

R2 R6

−0.089

0.146

−0.086

−0.08

0.135

0.189

R2 R7

0.02

0.171

−0.211

−0.153

0.193

0.195

R3 R1

0.06

0.01

0.016

0.113

−0.079

0.065

R3 R2

−0.045

−0.181

0.081

0.13

−0.21

−0.085

R3 R4

−0.087

0.024

−0.025

0.068

0.005

0.054

R3 R5

−0.037

−0.046

−0.009

0.072

−0.015

0.09

R3 R6

−0.134

−0.035

−0.005

0.05

−0.075

0.104

R3 R7

−0.025

−0.01

−0.13

−0.023

−0.017

0.11

R4 R1

0.147

−0.014

0.041

0.045

−0.084

0.011

R4 R2

0.042

−0.205

0.106

0.062

−0.215

−0.139

R4 R3

0.087

−0.024

0.025

−0.068

−0.005

−0.054

R4 R5

0.05

−0.07

0.016

0.004

−0.02

0.036

R4 R6

−0.047

−0.059

0.02

−0.018

−0.08

0.05

R4 R7

0.062

−0.034

−0.105

−0.091

−0.022

0.056

R5 R1

0.097

0.056

0.025

0.041

−0.064

−0.025

R5 R2

−0.008

−0.135

0.09

0.058

−0.195

−0.175

R5 R3

0.037

0.046

0.009

−0.072

0.015

−0.09

R5 R4

−0.05

0.07

−0.016

−0.004

0.02

−0.036

R5 R6

−0.097

−0.011

0.004

−0.022

0.06

0.014

R5 R7

0.012

0.036

−0.121

−0.095

0.002

0.02

R6 R1

0.194

0.045

0.021

0.063

0.004

−0.039

R6 R2

0.089

−0.146

0.086

0.08

0.135

−0.189

R6 R3

0.134

0.035

0.005

−0.05

0.075

−0.104

R6 R4

0.047

0.059

−0.02

0.018

0.08

−0.05

R6 R5

0.097

0.011

−0.004

0.022

−0.06

−0.014

R6 R7

0.109

0.025

−0.125

−0.073

−0.058

−0.006

R7 R1

0.085

0.02

0.146

0.136

−0.062

−0.045

R7 R2

−0.02

−0.171

0.211

0.153

−0.193

−0.195

R7 R3

0.025

0.01

0.13

0.023

0.017

−0.11

R7 R4

−0.062

0.034

0.105

0.091

0.022

−0.056

R7 R5

−0.012

−0.036

0.121

0.095

0.002

−0.02

R7 R6

−0.109

−0.025

0.125

0.073

−0.058

−0.006

7.4 m–Polar Fuzzy PROMETHEE Method

363

Table 7.9 Generalized criteria preference function Q1

Q2

Q3

Q4

Q5

Q6

R1 R2

0.00

0.00

0.00

1.00

1.00

1.00

R1 R3

0.00

0.00

0.03

0.00

0.00

1.00

R1 R4

0.00

0.14

0.15

0.00

0.00

1.00

R1 R5

0.00

0.00

0.06

0.00

0.00

0.00

R1 R6

0.00

0.00

0.04

0.00

0.00

0.00

R1 R7

0.00

0.00

0.88

0.00

0.00

0.00

R2 R1

1.00

1.00

0.34

0.00

0.00

0.00

R2 R3

0.25

1.00

0.48

0.00

0.00

0.00

R2 R4

0.00

1.00

0.67

0.00

0.00

0.00

R2 R5

0.00

1.00

0.56

0.00

0.00

0.00

R2 R6

0.00

1.00

0.52

0.00

0.00

0.00

R2 R7

0.00

1.00

0.99

0.00

0.00

0.00

R3 R1

0.40

0.10

0.00

1.00

0.50

0.00

R3 R2

0.00

0.00

0.00

1.00

1.00

1.00

R3 R4

0.00

0.24

0.06

1.00

0.00

0.00

R3 R5

0.00

0.00

0.01

1.00

0.00

0.00

R3 R6

0.00

0.00

0.002

1.00

0.50

0.00

R3 R7

0.00

0.00

0.82

0.00

0.00

0.00

R4 R1

1.00

0.00

0.00

1.00

0.50

0.00

R4 R2

0.22

0.00

0.00

1.00

1.00

1.00

R4 R3

0.67

0.00

0.00

0.00

0.00

1.00

R4 R5

0.30

0.00

0.00

0.00

0.00

0.00

R4 R6

0.00

0.00

0.00

0.00

0.50

0.00

R4 R7

0.42

0.00

0.67

0.00

0.00

0.00

R5 R1

0.77

0.56

0.00

1.00

0.50

1.00

R5 R2

0.00

0.00

0.00

1.00

1.00

1.00

R5 R3

0.17

0.46

0.00

0.00

0.00

1.00

R5 R4

0.00

0.70

0.03

0.00

0.00

1.00

R5 R6

0.00

0.11

0.00

0.00

0.50

0.00

R5 R7

0.00

0.36

0.77

0.00

0.00

0.00

R6 R1

1.00

0.45

0.00

1.00

0.00

1.00

R6 R2

0.69

0.00

0.00

1.00

1.00

1.00

R6 R3

1.00

0.35

0.00

0.00

0.00

1.00

R6 R4

0.27

0.59

0.04

1.00

0.00

1.00

R6 R5

0.77

0.00

0.002

1.00

0.00

1.00

R6 R7

1.00

0.25

0.79

0.00

0.00

0.00

R7 R1

0.65

0.20

0.00

1.00

0.50

1.00

R7 R2

0.00

0.00

0.00

1.00

1.00

1.00

R7 R3

0.05

0.10

0.00

1.00

0.00

1.00

R7 R4

0.00

0.34

0.00

1.00

0.00

1.00

R7 R5

0.00

0.00

0.00

1.00

0.00

1.00

R7 R6

0.00

0.00

0.00

1.00

0.50

1.00

364

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.10 Multi-criteria preference index R1

R2

R3

R4

R5

R6

R7

R1

−

0.41

0.06

0.08

0.002

0.001

0.03

R2

0.57

−

0.24

0.13

0.13

0.13

0.14

R3

0.45

0.41

−

0.20

0.17

0.26

0.02

R4

0.71

0.51

0.36

−

0.14

0.09

0.21

R5

0.73

0.41

0.19

0.14

−

0.10

0.06

R6

0.73

0.72

0.55

0.42

0.58

−

0.50

R7

0.63

0.41

0.25

0.27

0.23

0.32

−

Table 7.11 Positive and negative outranking flows Alternatives

χ+ (Rφ )

χ− (Rφ )

R1

0.097

0.637

R2

0.223

0.478

R3

0.252

0.275

R4

0.337

0.207

R5

0.272

0.209

R6

0.583

0.150

R7

0.352

0.160

(a) Partial ranking of alternatives (or PROMETHEE I) The outgoing and incoming flows of alternatives are computed by employing the Eqs. 7.20 and 7.21, respectively, and the results are summarized in Table 7.11. Then the partial raking of alternatives is determined by considering the intersection of red pre-orders P + and P − , as follows: ˆ 1 , R3 PR ˆ 1 , R3 PR ˆ 2 , R4 PR ˆ 1 , R4 PR ˆ 2 , R4 PR ˆ 3 , R4 Pˆ R2 PR ˆ 1 , R5 PR ˆ 2 , R5 PR ˆ 3 , R6 PR ˆ 1 , R6 Pˆ R5 , R5 PR ˆ 3 , R6 PR ˆ 4 , R6 PR ˆ 5 , R6 PR ˆ 7 , R7 Pˆ R2 , R6 PR ˆ 2 , R7 PR ˆ 3 , R7 PR ˆ 4 , R7 PR ˆ 5, R1 , R7 PR and the partial relations of PROMETHEE I are shown in Fig. 7.5. (b) Complete ranking of alternatives (or PROMETHEE II) The net outranking flows of alternatives are computed by employing the Eq. 7.25, and the net values are given in Table 7.12. It can be easily seen that the alternative R6 is selected as the most suitable site for planting a hydroelectric power station, and the ordering of alternatives is given as follows: R6 R7 R4 R5 R3 R2 R1 .

7.5 Comparative Analysis Table 7.12 Net flow of alternatives Alternatives

365

χ(Rφ ) −0.540 – 0.255 – 0.023 0.130 0.063 0.433 0.192

R1 R2 R3 R4 R5 R6 R7

7.5 Comparative Analysis Akram et al. [9] compared the proposed approach with previously existing approaches.

7.5.1 With Usual Criterion Preference Function The choice of different types of preference functions for different criteria is one of the main advantages of PROMETHEE method. In Sect. 4.2, six different types of generalized criteria preference functions are considered for all six criteria to chose the most suitable site or location. In this subsection, only the usual criterion preference function is considered for all criteria for the location problem of hydroelectric power plant in order to provide the comparison of net results and to check the authenticity of proposed m–polar fuzzy PROMETHEE method. The same weights are used which were calculated by AHP method in Sect. 7.3, and the steps for the construction of score matrix were same as enumerated in Sect. 7.4, step 4 is proceeded onward. Step 4. The preference degree of each pair of alternatives is computed by considering the usual criterion preference function for all criteria. In the case of criterion to be maximized, a strict preference is achieved only if there is a positive deviation between any pair of alternatives with respect to that criterion. On the other hand, the negative deviation between any pair of alternatives provides a strict preference in the case of criteria to be minimized. The results for the usual criterion preference functions are summarized in Table 7.13. Step 5. The multi-criteria preference index for each pair of alternatives is obtained by applying the Eq. 7.19, and the results are given in Table 7.14. Step 6. The partial and net outranking flows of alternatives are calculated as follows: (a) The outgoing and incoming flows of alternatives are computed by using the Eqs. 7.20 and 7.21, respectively, and the results of these flows are summarized in Table 7.15.

366

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets R6

R4

R3 R1

R2

R5

R7

Fig. 7.6 Partial relations of PROMETHEE I

The intersection of pre-orders P + and P − provides the partial ordering of alternatives or the partial results of PROMETHEE I, which is given as follows: ˆ 1 , R2 PR ˆ 3 , R3 PR ˆ 1 , R4 PR ˆ 1 , R4 PR ˆ 2 , R4 Pˆ R2 PR ˆ 5 , R4 PR ˆ 7 , R5 PR ˆ 1 , R5 PR ˆ 2 , R5 PR ˆ 3 , R5 Pˆ R3 , R4 PR ˆ 1 , R6 PR ˆ 2 , R6 PR ˆ 3 , R6 PR ˆ 4 , R6 Pˆ R7 , R6 PR ˆ 7 , R7 PR ˆ 1 , R7 PR ˆ 2 , R7 PR ˆ 3, R5 , R6 PR and the partial relations of PROMETHEE I are shown in Fig. 7.6. (b) The net outranking flows of alternatives are determined by applying the Eq. 7.25, and the results are given in Table 7.16. It is obvious from the net flows of alternatives that the alternative R1 is chosen as the best suitable site and the ranking of different sites is given as follow: R6 R4 R5 R7 R2 R3 R1 . The final ranking for the sites of hydroelectric power stations are given in Table 7.17, which is obtained by applying different types of preference functions under m–polar fuzzy PROMETHEE method. It can easily be seen that R6 is chosen as the most suitable alternative from both types of functions. Although the ranking of the sites obtained from different preference functions are not same, but the optimal solution remains same which shows that the preference function does not have an impact on the first-ranked alternative.

7.5 Comparative Analysis

367

Table 7.13 Usual criterion preference function Q1

Q2

Q3

Q4

Q5

Q6

R1 R2

0

0

0

1

1

1

R1 R3

0

0

1

0

0

1

R1 R4

0

1

1

0

0

1

R1 R5

0

0

1

0

0

0

R1 R6

0

0

1

0

0

0

R1 R7

0

0

1

0

0

0

R2 R1

1

1

1

0

0

0

R2 R3

1

1

1

0

0

0

R2 R4

0

1

1

0

0

0

R2 R5

1

1

1

0

0

0

R2 R6

0

1

1

0

0

0

R2 R7

1

1

1

0

0

0

R3 R1

1

1

0

1

1

0

R3 R2

0

0

0

1

1

1

R3 R4

0

1

1

1

0

0

R3 R5

0

0

1

1

1

0

R3 R6

0

0

1

1

1

0

R3 R7

0

0

1

0

1

0

R4 R1

1

0

0

1

1

0

R4 R2

1

0

0

1

1

1

R4 R3

1

0

0

0

1

1

R4 R5

1

0

0

1

1

0

R4 R6

0

0

0

0

1

0

R4 R7

1

0

1

0

1

0

R5 R1

1

1

0

1

1

1

R5 R2

0

0

0

1

1

1

R5 R3

1

1

0

0

0

1

R5 R4

0

1

1

0

0

1

R5 R6

0

1

0

0

1

0

R5 R7

1

1

1

0

1

0

R6 R1

1

1

0

1

1

1

R6 R2

1

0

0

1

1

1

R6 R3

1

1

0

0

0

1

R6 R4

1

1

1

1

0

1

R6 R5

1

0

1

1

0

1

R6 R7

1

1

1

0

0

0

R7 R1

1

1

0

1

1

1

R7 R2

0

0

0

1

1

1

R7 R3

1

1

0

1

0

1

R7 R4

0

1

0

1

0

1

R7 R5

0

0

0

1

0

1

R7 R6

0

0

0

1

1

1

368

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.14 Multi-criteria preference index R1 R2 R3 R4 R5 R6 R7

R1

R2

R3

R4

R5

R6

R7

− 0.59 0.91 0.80 0.97 0.97 0.97

0.41 − 0.41 0.86 0.41 0.86 0.41

0.09 0.59 − 0.69 0.62 0.62 0.79

0.20 0.14 0.31 − 0.20 0.82 0.34

0.03 0.59 0.38 0.80 − 0.71 0.23

0.03 0.14 0.38 0.18 0.29 − 0.41

0.03 0.59 0.21 0.66 0.77 0.59 −

Table 7.15 Positive and negative outranking flows Alternatives χ+ (Rφ ) R1 R2 R3 R4 R5 R6 R7

0.132 0.440 0.433 0.665 0.543 0.762 0.525

Table 7.16 Net flow of alternatives Alternatives

χ− (Rφ ) 0.868 0.560 0.567 0.335 0.457 0.238 0.475

χ(Rφ ) − 0.736 − 0.12 − 0.134 0.33 0.086 0.524 0.05

R1 R2 R3 R4 R5 R6 R7

Table 7.17 Final ranking of hydroelectric power plants Alternatives Combination of six preference Usual criterion preference functions function R1 R2 R3 R4 R5 R6 R7

7 6 5 3 4 1 2

7 5 6 2 3 1 4

7.5 Comparative Analysis

369

7.5.2 m–Polar Fuzzy ELECTRE I In this subsection, the location problem of hydroelectric power plant is solved by using the existing MCDM approach m–polar fuzzy ELECTRE I method, and made a comparison of net results. Consider the aggregated decision matrix as given in Table 7.7 and the weights of criteria which were calculated by AHP method. Then the weighted aggregated decision matrix is constructed as given in Table 7.18, and follow the next steps of m–polar fuzzy ELECTRE I method to determine an outranking relation of alternatives in account to make a comparison of these multi-attribute decision making methods. The evaluation of m–polar fuzzy concordance sets Fφϕ , m–polar fuzzy discordance sets Gφϕ , m–polar fuzzy concordance indices f φϕ , m–polar fuzzy discordance indices gφϕ , concordance dominance h φϕ , discordance dominance kφϕ , aggregated dominance lφϕ and outranking relations for this location problem is briefly summarized in Table 7.19. The graph sketched by outranking relations is given in Fig. 7.7 and the set of most favorable alternatives is {R2 , R6 }. It can be easily seen that the alternative R6 is chosen as the best possible location for all these MCDM methods under m–polar fuzzy environment. So, m–polar fuzzy PROMETHEE method can successfully be applied to solve the MCDM problems with m–polar fuzzy information. Different versions of this method not only provide the kernel solution, but also produce the ranking of alternatives in a descending order.

Table 7.18 Weighted aggregated decision matrix Infrastructure Nature of land R1 R2 R3 R4 R5 R6 R7 R1 R2 R3 R4 R5 R6 R7

(0.191, 0.196, 0.218) (0.252, 0.259, 0.236) (0.203, 0.230, 0.254) (0.270, 0.275, 0.259) (0.218, 0.248, 0.270) (0.304, 0.286, 0.277) (0.225, 0.232, 0.263) Social infrastructure (0.075, 0.078, 0.085) (0.077, 0.064, 0.089) (0.106, 0.091, 0.099) (0.072, 0.098, 0.091) (0.092, 0.074, 0.094) (0.092, 0.081, 0.098) (0.091, 0.109, 0.108)

(0.071, 0.049, 0.061) (0.080, 0.084, 0.080) (0.059, 0.066, 0.059) (0.054, 0.053, 0.069) (0.077, 0.067, 0.055) (0.069, 0.066, 0.061) (0.069, 0.054, 0.064) Climate changes (0.105, 0.110, 0.113) (0.126, 0.153, 0.120) (0.086, 0.088, 0.112) (0.105, 0.090, 0.087) (0.099, 0.106, 0.088) (0.099, 0.114, 0.113) (0.097, 0.094, 0.104)

Government incentives (0.013, 0.015, 0.016) (0.011, 0.013, 0.014) (0.017, 0.016, 0.013) (0.018, 0.015, 0.015) (0.014, 0.018, 0.015) (0.017, 0.015, 0.014) (0.020, 0.022, 0.015) Cost (0.039, 0.030, 0.033) (0.140, 0.045, 0.038) (0.032, 0.037, 0.045) (0.041, 0.033, 0.030) (0.035, 0.034, 0.029) (0.028, 0.032, 0.036) (0.029, 0.031, 0.034)

370

7 Extended PROMETHEE Method Under Multi-polar Fuzzy Sets

Table 7.19 m–polar fuzzy ELECTRE I results for selection of hydroelectric power plant Alternatives

Fφϕ

Gφϕ

f φϕ

gφϕ

h φϕ

kφϕ

lφϕ

Outranking relation

(R1 , R2 )

{3, 4}

{1, 2, 5, 6}

0.20

1

0

0

0

Incomparable

(R1 , R3 )

{5}

{1, 2, 3, 4, 6}

0.18

1

0

0

0

Incomparable

(R1 , R4 )

{2, 5}

{1, 3, 4, 6}

0.29

1

0

0

0

Incomparable

(R1 , R5 )

{5, 6}

{1, 2, 3, 4}

0.24

1

0

0

0

Incomparable

(R1 , R6 )

{5, 6}

{1, 2, 3, 4}

0.24

1

0

0

0

Incomparable

(R1 , R7 )

{5, 6}

{1, 2, 3, 4}

0.24

1

0

0

0

Incomparable

(R2 , R1 )

{1, 2, 5, 6}

{3, 4}

0.80

0.144

1

1

1

R 2 → R1

(R2 , R3 )

{1, 2, 5, 6}

{3, 4}

0.80

0.376

1

1

1

R 2 → R3

(R2 , R4 )

{2, 5, 6}

{1, 3, 4}

0.35

0.344

0

1

0

Incomparable

(R2 , R5 )

{1,2,5,6}

{3, 4}

0.80

0.176

1

1

1

R 2 → R5

(R2 , R6 )

{2, 5, 6}

{1, 3, 4}

0.35

0.634

0

1

0

Incomparable

(R2 , R7 )

{1,2,5,6}

{3, 4}

0.80

0.454

1

1

1

R 2 → R7

(R3 , R1 )

{1, 2, 3, 4, 6}

{5}

0.82

0.571

1

1

1

R 3 → R1

(R3 , R2 )

{3, 4}

{1, 2, 5, 6}

0.20

1

0

0

0

Incomparable

(R3 , R4 )

{2, 4, 5, 6}

{1, 3}

0.52

1

1

0

0

Incomparable

(R3 , R5 )

{4, 6}

{1, 2, 3, 5}

0.23

1

0

0

0

Incomparable

(R3 , R6 )

{3, 4, 6}

{1, 2, 3, 5}

0.26

1

0

0

0

Incomparable

(R3 , R7 )

{6}

{1, 2, 3, 4, 5}

0.06

1

0

0

0

Incomparable

(R4 , R1 )

{1, 3, 4, 6}

{2, 5}

0.71

0.275

1

1

1

R 4 → R1

(R4 , R2 )

{1, 3, 4}

{2, 5, 6}

0.65

1

1

0

0

Incomparable

(R4 , R3 )

{1, 3}

{2, 4, 5, 6}

0.48

0.441

0

1

0

Incomparable

(R4 , R5 )

{1, 3, 4, 6}

{2, 5}

0.71

0.509

1

1

1

R 4 → R5

(R4 , R6 )

{3, 6}

{1, 2, 4, 5}

0.09

1

0

0

0

Incomparable

(R4 , R7 )

{1, 6}

{2, 3, 4, 5}

0.51

0.444

1

1

1

R 4 → R7

(R5 , R1 )

{1, 2, 3, 4}

{5, 6}

0.76

0.332

1

1

1

R 5 → R1

(R5 , R2 )

{3, 4}

{1, 2, 5, 6}

0.20

1

0

0

0

Incomparable

(R5 , R3 )

{1, 2, 3, 5}

{4, 6}

0.77

0.688

1

0

0

Incomparable

(R5 , R4 )

{2, 5}

{1, 3, 4, 6}

0.29

1

0

0

0

Incomparable

(R5 , R6 )

{2, 3, 6}

{1, 4, 5}

0.20

1

0

0

0

Incomparable

(R5 , R7 )

{1, 2, 6}

{3, 4, 5}

0.62

1

1

0

0

Incomparable

(R6 , R1 )

{1, 2, 3, 4}

{5, 6}

0.76

0.074

1

1

1

R 6 → R1

(R6 , R2 )

{1, 3, 4}

{2, 5, 6}

0.65

1

1

0

0

Incomparable

(R6 , R3 )

{1, 2, 3, 5}

{3, 4, 6}

0.77

0.146

1

1

1

R 6 → R3

(R6 , R4 )

{1, 2, 4, 5}

{3, 6}

0.91

0.359

1

1

1

R 6 → R4

(R6 , R5 )

{1, 4, 5}

{2, 3, 6}

0.80

0.107

1

1

1

R 6 → R5

(R6 , R7 )

{1, 2, 5, 6}

{3, 4}

0.80

0.308

1

1

1

R 6 → R7

(R7 , R1 )

{1, 2, 3, 4}

{5, 6}

0.76

0.301

1

1

1

R 7 → R1

(R7 , R2 )

{3, 4}

{1, 2, 5, 6}

0.20

1

0

0

0

Incomparable

(R7 , R3 )

{1, 2, 3, 4, 5}

{6}

0.94

0.510

1

1

1

R 7 → R3

(R7 , R4 )

{1, 2, 3, 4, 5}

{1, 6}

0.49

1

0

0

0

Incomparable

(R7 , R5 )

{3, 4, 5}

{1, 2, 6}

0.38

0.5

0

1

0

Incomparable

(R7 , R6 )

{3, 4}

{1, 2, 5, 6}

0.20

1

0

0

0

Incomparable

References

371 R6

R4

R1 R3

R5

R7

R2

Fig. 7.7 Graph representing the outranking relation of alternatives

7.6 Conclusion A MCDA technique has been presented that makes an efficient use of m–polar fuzzy information, and it is named as the AHP-assisted m–polar fuzzy PROMETHEE method. It consists of two parts, namely the calculation of the weights of the criteria and the ranking of the set of feasible alternatives. The normalized weights of the attributes are determined by the AHP technique. Then a novel variation of the PROMETHEE approach produces the ranking of alternatives in the context of m– polar fuzzy numbers. As an application, the combination of six types of generalized criteria preference functions delivered partial and complete rankings of hydroelectric power plants. Moreover, the comparative analysis of net obtained results has been provided by assigning the usual criterion preference function for all criteria. Furthermore, the reliability of this method has been analyzed by applying an existing MCDM approach, such as m–polar fuzzy ELECTRE I method to the same location problem.

References 1. Abdullah, L., Chan, W., Afshari, A.: Application of PROMETHEE method for green supplier selection: a comparative result based on preference functions. J. Ind. Eng. Int. 15(2), 271–285 (2019) 2. Adeel, A., Akram, M., Ahmed, I., Nazar, K.: Novel m–polar fuzzy linguistic ELECTRE-I method for group decision making. Symmetry 11(4), 471 (2019) 3. Adeel, A., Akram, M., Koam, A.N.A.: Group decision making based on m−polar fuzzy linguistic TOPSIS method. Symmetry 11(6), 735 (2019) 4. Adeel, A., Akram, M., Koam, A.N.A.: Multi-criteria decision making under mHF ELECTRE-I and HmF ELECTRE-I. Energies 12(9), 1661 (2019) 5. Akram, M., Adeel, A.: Novel hybrid decision making methods based on mF rough information. Granular Comput. 5, 185–201 (2020)

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6. Akram, M., Adeel, A.: Novel TOPSIS method for group decision making based on hesitant m-polar fuzzy model. J. Intell. Fuzzy Syst. 37, 8077–8096 (2019) 7. Akram, M., Adeel, A., Alcantud, J.C.R.: Multi-criteria group decision making using an m−polar hesitant fuzzy TOPSIS approach. Symmetry 11(6), 795 (2019) 8. Akram, M., Arshad, M.: A novel trapezoidal bipolar fuzzy TOPSIS method for group decision making. Group Decis. Negot. 28(3), 565–584 (2019) 9. Akram, M., Shumaiza, Al-Kenani, A.N.: Multi-criteria group decision making for selection of green suppliers under bipolar fuzzy PROMETHEE process. Symmetry 12(1), 77 (2020) 10. Akram, M., Shumaiza, Arshad, M.: Bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE-I methods to diagnosis. Comput. Appl. Math. 39(1), 1–23 (2020) 11. Akram, M., Waseem, N., Liu, P.: Novel approach in decision making with m–polar fuzzy ELECTRE-I. Int. J. Fuzzy Syst. 21(4), 1117–1129 (2019) 12. Alcantud, J.C.R., Biondo, A.E., Giarlotta, A.: Fuzzy politics I: the genesis of parties. Fuzzy Sets Syst. 349, 71–98 (2018) 13. Alcantud, J.C.R., Cruz-Rambaud, S., Muñoz Torrecillas, M.J.: Valuation fuzzy soft sets: a flexible fuzzy soft set based decision making procedure for the valuation of assets. Symmetry 9, 253 (2017) 14. Alcantud, J.C.R., Khameneh, A.Z., Kilicman, A.: Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information. Inf. Sci. 514, 106–117 (2020) 15. Behzadian, M., Kazemzadeh, R.B., Albadvi, A., Aghdasi, M.: PROMETHEE: a comprehensive literature review on methodologies and applications. Eur. J. Oper. Res. 200(1), 198–215 (2010) 16. Bellman, R.E., Zadeh, L.A.: decision making in a fuzzy environment. Manag. Sci. 4(17), 141– 164 (1970) 17. Benayoun, R., Roy, B., Sussman, B.: ELECTRE: Une méthode pour guider le choix en présence de points de vue multiples. Note de travail, 49, SEMA-METRA International, Direction Scientifique (1966) 18. Brans, J.P., Vincke, P.: A preference ranking organization method (The PROMETHEE method for multiple criteria decision making). Manag. Sci. 31(6), 647–656 (1985) 19. Brans, J.P., Vincke, P., Mareschal, B.: How to select and how to rank projects: the PROMETHEE method. Eur. J. Oper. Res. 24(2), 228–238 (1986) 20. Chang, D.Y.: Applications of the extent analysis method on fuzzy AHP. Eur. J. Oper. Res. 95(3), 649–655 (1996) 21. Charnes, A., Cooper, W., Lewin, A.Y., Seiford, L.M.: Data envelopment analysis theory, methodology and applications. Eur. J. Oper. Res. Soc. 48(3), 332–333 (1997) 22. Chen, S.M., Niou, S.J.: Fuzzy multiple-attributes group decision-making based on fuzzy preference relations. Expert. Syst. Appl. 38(4), 3865–3872 (2011) 23. Chen, S.M., Jong, W.T.: Fuzzy query translation for relational database systems. IEEE Trans. Syst. Man Cybern. 27(4), 714–721 (1997) 24. Chen, J., Li, S., Ma, S., Wang, X.: m-polar fuzzy sets: an extension of bipolar fuzzy sets. Sci. World J. 416530, 1–8 (2014) 25. Entani, T., Inuiguchi, M.: Pairwise comparison based interval analysis for group decision aiding with multiple criteria. Fuzzy Sets and Syst. 274, 79–96 (2015) 26. Junior, F.R.L., Osiro, L., Carpinetti, L.C.R.: A comparison between fuzzy AHP and fuzzy TOPSIS methods to supplier selection. Appl. Soft Comput. 21, 194–209 (2014) 27. Goumas, M., Lygerou, V.: An extension of the PROMETHEE method for decision making in fuzzy environment: ranking of alternative energy exploitation projects. Eur. J. Oper. Res. 123(3), 606–613 (2000) 28. Govindan, K., Kadzinski, M., Sivakumar, R.: Application of a novel PROMETHEE-based method for construction of a group compromise ranking to prioritization of green suppliers in food supply chain. Omega 71, 129–145 (2017) 29. Gupta, P., Inuiguchi, M., Mehlawat, M.K.: A hybrid approach for constructing suitable and optimal portfolios. Expert. Syst. Appl. 38(5), 5620–5632 (2011)

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Chapter 8

Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Abstract In this chapter, some arithmetic and geometric aggregation operators are presented using Dombi and Hamacher t-norms and t-conorms to handle uncertainty in multi-polar (m–polar) fuzzy information, including m–polar fuzzy Dombi weighted averaging operator, m–polar fuzzy Dombi ordered weighted averaging operator, m–polar fuzzy Dombi hybrid averaging operator, m–polar fuzzy Dombi weighted geometric operator, m–polar fuzzy Dombi weighted ordered geometric operator, m–polar fuzzy Dombi hybrid geometric operator, m–polar fuzzy Hamacher weighted average operator, m–polar fuzzy Hamacher ordered weighted average operator, m–polar fuzzy Hamacher hybrid average operator, m–polar fuzzy Hamacher weighted geometric operator, m–polar fuzzy Hamacher ordered weighted geometric operator and m–polar fuzzy Hamacher hybrid geometric operator. Some of the properties such as idempotancy, monotonicity and boundedness are investigated for the presented operators. Moreover, the corresponding algorithms are presented to solve multi-criteria decision making (MCDM) issues, which involve m–polar fuzzy information with m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi weighted geometric operators. To prove the validity and feasibility of the presented models, numerical examples are solved for each presented model, and a comparison is discussed with m–polar fuzzy ELECTRE-I approach. The effectiveness of m–polar fuzzy Dombi aggregation operators is also checked by validitytests. This chapter is due to This chapter is due to [9, 10, 19, 22, 40] .

8.1 Introduction MCDM is playing an efficient role in different domains ranging from engineering to social sciences. MCDM approaches identify how attribute information is to be processed to compute a suitable alternative or to rank the alternatives for support© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_8

375

376

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

ing decision making. It has been broadly applied in different domains, including engineering technology, operation research and management science. Aggregation operators perform an important role in order to combine data into a single form and solve MCDM problems. For example, Yager [49] introduced weighted aggregation operators. Xu [44] proposed some new aggregation operators under Intuitionistics fuzzy sets. Xu and Yager [48] developed certain new geometric aggregation operators and solve some real-world MCDM problems. Hamacher t-conorm and t-norm [26] are the algebraic and Einstein t-conorm and t-norm [16] expanded variants, respectively. For example, Wei et al. [41] developed some bipolar fuzzy Hamacher weighted averaging and geometric aggregation operators. Xu and Wei [47] developed dual hesitant bipolar fuzzy arithmetic and geometric aggregation operators. Garg [25] utilized linguistic prioritized aggregation operators to develop a MCDM method under single-valued neutrosophic environment. Wei et al. [42] presented some hesitant bipolar fuzzy weighted arithmetic and geometric aggregation operators. By combining Hamacher operations and prioritized aggregation operators, Gao et al. [23] proposed dual hesitant bipolar fuzzy Hamacher prioritized weighted aggregation operators and applied the proposed methodologies to an MCDM problem. Liu [32] utilized interval-valued intuitionistic fuzzy numbers with Hamacher aggregation operators and developed multi-criteria methods for group decision making. Jana et al. [28] applied weighted, ordered weighted and hybrid average and geometric aggregation operators for the aggregation of bipolar fuzzy information using Dombi t-conorm and t-norm. They also proposed bipolar fuzzy Dombi prioritized aggregation operators in [29]. He [27] developed hesitant fuzzy Dombi aggregation operators and investigated typhoon disaster assessment using proposed theory. Xu [45] proposed intuitionistic fuzzy power aggregation operators for multi-attribute group decision making. Xiao [43] constructed induced interval-valued intuitionistic fuzzy Hamacher aggregation operators and discussed their application to MCDM. Chen and Ye [20] discussed MCDM problem under Dombi operations in single-valued neutrosophic situation. Garg [24] presented some generalized interactive aggregation operators under Einstein operations in Pythagorean fuzzy environment and discussed a decision making issue. Akram et al. [11] proposed different Pythagorean Dombi fuzzy aggregation operators and studied their applications in MCDM. Shahzadi et al. [36] introduced Pythagorean fuzzy Yager aggregation operators for decision making. Peng and Yang [33] investigated different basic properties of interval-valued Pythagorean fuzzy aggregation operators. Wang et al. [38] introduced some new types of q-rung orthopair fuzzy Hamy mean aggregation operators to handle MCDM situations. Arora and Garg [14] proposed robust aggregation operators with an intuitionistic fuzzy soft environment. Wang and Li [39] developed Pythagorean fuzzy interaction power Bonferroni mean aggregation operators and discussed their applications to MCDM. Chiclana et al. [21] introduced some ordered weighted geometric operators and solved a decision making problem. Liang et al. [31] developed Pythagorean fuzzy Bonferroni mean aggregation operators. Nowadays experts believe that multi-polarity performs a vital role in many practical situations. Due to the presence of multi-polar data in different daily life problems of science and technology, Chen et al. [19] initiated the notion of m–polar fuzzy set

8.2 m–Polar Fuzzy Dombi Aggregation Operators

377

theory as generalization of fuzzy and bipolar fuzzy sets. Khameneh and Kilicman [30] proposed m–polar fuzzy soft weighted aggregation operators and applied these aggregation operators in decision making. In view of the fact that m–polar fuzzy sets have an efficient strength to handle vague data which arises in several real-life problems.

8.2 m–Polar Fuzzy Dombi Aggregation Operators In this section, the concept of Dombi aggregation operators is extended to m–polar fuzzy environment and developed m–polar fuzzy Dombi arithmetic and geometric aggregation operators. Some fundamental properties of these operators are also discussed. Definition 8.1 The accuracy function H of an m–polar fuzzy number ζˆ = ( p1 ◦ ζ, . . . , pm ◦ ζ ) is given by H (ζˆ ) =

 1  (−1)r ( pr ◦ ζ − 1) , m r =1 m

H (ζˆ ) ∈ [0, 1].

Clearly, for an arbitrary m–polar fuzzy number ζˆ , S(ζˆ ), H (ζˆ ) ∈ [0, 1]. Definition 8.2 Let ζˆ1 = ( p1 ◦ ζ1 , . . . , pm ◦ ζ1 ), and ζˆ2 = ( p1 ◦ ζ2 , . . . , pm ◦ ζ2 ) be two m–polar fuzzy numbers. Then 1. 2. 3. 4. 5.

ζˆ1 ζˆ1 ζˆ1 ζˆ1 ζˆ1

< ζˆ2 , if S(ζˆ1 ) < S(ζˆ2 ). > ζˆ2 , if S(ζˆ1 ) > S(ζˆ2 ). = ζˆ2 , If S(ζˆ1 ) = S(ζˆ2 ) and H (ζˆ1 ) = H (ζˆ2 ). < ζˆ2 , if S(ζˆ1 ) = S(ζˆ2 ), but H (ζˆ1 ) < H (ζˆ2 ). > ζˆ2 , if S(ζˆ1 ) = S(ζˆ2 ), but H (ζˆ1 ) > H (ζˆ2 ).

Now some fundamental operations on m–polar fuzzy numbers are described as follows:  1. ζˆ1  ζˆ2 = p1 ◦ ζ1 + p1 ◦ ζ2 − p1 ◦ ζ1 . p1 ◦ ζ2 , . . . , pm ◦ ζ1 + pm ◦ ζ2 − pm ◦  ζ1 . pm ◦ ζ2 ,   2. ζˆ1  ζˆ2 = p1 ◦ ζ1 . p1 ◦ ζ2 , . . . , pm ◦ ζ1 . pm ◦ ζ2 ,   3. α ζˆ = 1 − (1 − p1 ◦ ζ )α ), . . . , 1 − (1 − pm ◦ ζ )α , α > 0,   4. (ζˆ )α = ( p1 ◦ ζ )α , . . . , ( pm ◦ ζ )α , α > 0,   5. ζˆ c = 1 − p1 ◦ ζ, . . . , 1 − pm ◦ ζ , 6. ζˆ1 ⊆ ζˆ2 , ifand only if p1 ◦ ζ1 ≤ p1 ◦ ζ2 , . . . , pm ◦ ζ1 ≤ pm ◦ ζ2 , 7. ζˆ1 ∪ ζˆ2 = max( p1 ◦ ζ1 , p1 ◦ ζ2 ), . . . , max( pm ◦ ζ1 , pm ◦ ζ2 ) ,   8. ζˆ1 ∩ ζˆ2 = min( p1 ◦ ζ1 , p1 ◦ ζ2 ), . . . , min( pm ◦ ζ1 , pm ◦ ζ2 ) .

378

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Theorem 8.1 Let ζˆ1 = ( p1 ◦ ζ1 , . . . , pm ◦ ζ1 ) and ζˆ2 = ( p1 ◦ ζ2 , . . . , pm ◦ ζ2 ) be two m–polar fuzzy numbers, α, α1 , α2 > 0, then 1. 2. 3. 4. 5. 6. 7.

ζˆ1  ζˆ2 = ζˆ2  ζˆ1 , ζˆ1  ζˆ2 = ζˆ2  ζˆ1 , α(ζˆ1  ζˆ2 ) = α(ζˆ1 )  α(ζˆ2 ), (ζˆ1  ζˆ2 )α = (ζˆ1 )α  (ζˆ2 )α , α1 ζˆ1  α2 ζˆ1 = (α1 + α2 )ζˆ1 , ˆ α1 (ζˆ1 )α2 = (ζˆ1 )α1 +α2 , α (ζ1 ) α  (ζˆ1 ) 1 2 = (ζˆ1 )α1 α2 .

Let ζˆ1 = ( p1 ◦ ζ1 , . . . , pm ◦ ζ1 ), ζˆ2 = ( p1 ◦ ζ2 , . . . , pm ◦ ζ2 ) and ζˆ = ( p1 ◦ ζ, . . . , pm ◦ ζ ) be three m–polar fuzzy numbers. Some fundamental Dombi operations of m–polar fuzzy numbers are described as follows:  1 • ζˆ1 ⊕ ζˆ2 = 1 −  p ◦ ζ k  p ◦ ζ k 1/k , . . . , 1 1 1 2 1+ + 1 − p 1 ◦ ζ1 1 − p 1 ◦ ζ2

1 1−  p ◦ ζ k  p ◦ ζ k 1/k , m 1 m 2 1+ + 1 − p ◦ ζ 1 − p ◦ ζ2 m 1 m  1 • ζˆ1 ⊗ ζˆ2 =  1 − p ◦ ζ k  1 − p ◦ ζ k 1/k , . . . , 1 1 1 2 1+ + p 1 ◦ ζ1 p 1 ◦ ζ2

1  1 − p ◦ ζ k  1 − p ◦ ζ k 1/k , m 1 m 2 1+ + p ◦ ζ p ◦ ζ m 1 m 2 

1 1 ˆ • βζ = 1 −   p ◦ ζ k 1/k , . . . , 1 −   p ◦ ζ k 1/k , 1 m 1+ β 1+ β 1 − p ◦ ζ 1 − pm ◦ ζ 1

 1 1 • (ζˆ )β = 1 −   1 − p ◦ ζ k 1/k , . . . , 1 −   1 − p ◦ ζ k 1/k , 1 m 1+ β 1+ β p1 ◦ ζ pm ◦ ζ where k > 0.

8.2.1 m–Polar Fuzzy Dombi Arithmetic Aggregation Operators The m–polar fuzzy Dombi arithmetic aggregation operators are presented as follows:

8.2 m–Polar Fuzzy Dombi Aggregation Operators

379

Definition 8.3 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, a mapping from ζˆ n to ζˆ is called an m–polar fuzzy Dombi weighted averaging operator, which is given by

m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(Θ j ζˆ j ),

(8.1)

j=1

where Θ = (Θ1 , Θ2 , . . . , Θn ) denotes the weights of ζˆ j , ∀ j = 1, . . . , n and Θ j > 0 with nj=1 Θ j = 1. Theorem 8.2 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, an accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Dombi weighted averaging operators is defined as m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(Θ j ζˆ j ),

j=1

 = 1− 1+

 n

Θj



j=1

1 1 . , . . . , 1 −     n k 1/k k 1/k p1 ◦ ζ j pm ◦ ζ j 1+ Θj 1 − p1 ◦ ζ j 1 − pm ◦ ζ j j=1

(8.2) Proof The induction approach is utilized to show it. Case 1. For n = 1, by the Eq. (8.2), we obtain m F DW AΘ (ζˆ1 , , ζˆ2 , . . . , ζˆn ) = Θ1 ζˆ1 = ζˆ1 , (since Θ1 = 1)

 1 1 = 1− , . . . , 1 −  p ◦ ζ k 1/k  p ◦ ζ k 1/k . 1 1 m 1 1+ 1+ 1 − p 1 ◦ ζ1 1 − p m ◦ ζ1 Hence, the Eq. (8.2) satisfies when n = 1. Case 2. Now it is presumed that Eq. (8.2) satisfies for n = t, here t is an arbitrary natural number, then m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆt ) =  = 1−

t 

(Θ j ζˆ j ),

j=1

1 1 , . . . , 1 −    p ◦ ζ k 1/k  p ◦ ζ k 1/k . t t 1 j m j 1+ Θj 1+ Θj 1 − p1 ◦ ζ j 1 − pm ◦ ζ j j=1 j=1

(8.3) For n = t + 1,

380

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆt , ζˆt+1 ) =

t 

(Θ j ζˆ j ) ⊕ (Θt+1 ζˆt+1 ),

j=1

1 1 ⊕ ,...,1 −     t k 1/k k 1/k p1 ◦ ζ j pm ◦ ζ j 1+ Θj 1+ Θj 1 − p1 ◦ ζ j 1 − pm ◦ ζ j j=1 j=1 

1 1 1−    p ◦ζ k 1/k , . . . , 1 −  p ◦ζ k 1/k 1 t+1 m t+1 1 + Θt+1 1 + Θt+1 1 − p1 ◦ ζt+1 1 − pm ◦ ζt+1

 1 1 . ,...,1 − = 1−  t+1      t+1 k 1/k k 1/k p1 ◦ ζ j pm ◦ ζ j 1+ Θj 1+ Θj 1 − p1 ◦ ζ j 1 − pm ◦ ζ j j=1 j=1  = 1−

 t



Therefore, Eq. (8.2) satisfies for n = t + 1. Hence, it is deduced that Eq. (8.2) satisfies for every natural number n. 

Example 8.1 Let ζˆ1 = (0.4, 0.3, 0.8), ζˆ2 = (0.3, 0.5, 0.1), ζˆ3 = (0.7, 0.2, 0.4) and ζˆ4 = (0.5, 0.4, 0.6) be 3–polar fuzzy numbers and Θ = (0.2, 0.3, 0.1, 0.4) be weights related to these 3–polar fuzzy numbers. Then, for k = 3, m F DW AΘ (ζˆ1 , ζˆ2 , ζˆ3 ) =

(Θ j ζˆ j )

j=1

 = 1− 1+  = 1−

3 

 n

Θj

j=1



1 1 ,...,1 −  p ◦ ζ k 1/k  n p1 ◦ ζ j k 1/k m j 1+ Θj 1 − p1 ◦ ζ j 1 − pm ◦ ζ j j=1



1  0.3 3  0.7 3  0.5 3 1/3 , 0.4 3 + 0.3 × + 0.1 × + 0.4 × 1 − 0.4 1 − 0.3 1 − 0.7 1 − 0.5 1 1−   0.3 3  0.5 3  0.2 3  0.4 3 1/3 , 1 + 0.2 × + 0.3 × + 0.1 × + 0.4 × 1 − 0.3 1 − 0.5 1 − 0.2 1 − 0.4

1 1−   0.8 3  0.1 3  0.4 3  0.6 3 1/3 , 1 + 0.2 × + 0.3 × + 0.1 × + 0.4 × 1 − 0.8 1 − 0.1 1 − 0.4 1 − 0.6   1 + 0.2 ×

= (0.5467, 0.4312, 0.7076).

Theorem 8.3 (Idempotent Law) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers, which are equal, i.e., ζˆ j = ζˆ , then m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ .

(8.4)

Proof Since ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) = ζˆ , where j = 1, . . . , n. Then, by Equation (8.2),

8.2 m–Polar Fuzzy Dombi Aggregation Operators m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

381

(Θ j ζˆ j ),

j=1

1 1 , ,...,1 −     n k 1/k k 1/k p1 ◦ ζ j pm ◦ ζ j 1+ Θj 1+ Θj 1 − p1 ◦ ζ j 1 − pm ◦ ζ j j=1 j=1 

1 1 = 1−  p ◦ ζ k 1/k , . . . , 1 −  p ◦ ζ k 1/k , 1 m 1+ 1+ 1 − p1 ◦ ζ 1 − pm ◦ ζ  = 1−

 n



= ( p1 ◦ ζ, p2 ◦ ζ, . . . , pm ◦ ζ ), for k = 1 = ζˆ . Hence, m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ holds if ζˆ j = ζˆ , for all ‘ j’ varies from 1 to n.



Theorem 8.4 (Bounded Law) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a collection of ‘n’

 m–polar fuzzy numbers, ζˆ − = nj=1 (ζ j ) and ζˆ + = nj=1 (ζ j ), then ζˆ − ≤ m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ ζˆ + .

(8.5)

Theorem 8.5 (Monotonic Law) For two collections of m–polar fuzzy numbers ζˆ j and ζˆj , j = 1, 2, . . . , n, if ζˆ j ≤ ζˆj , then m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ m F DW AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.6)

Definition 8.4 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ), j = 1, 2, . . . , n, an m–polar fuzzy Dombi ordered weighted averaging operator is a function mFDOWA : ζˆ n → ζˆ , which is given by m F D O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζˆσ ( j) ),

(8.7)

j=1

n where w = (w1 , w2 , . . . , wn ) denotes the weights and w j ∈ (0, 1] with j=1 w j = 1. ˆ ˆ σ ( j), ( j = 1, 2, . . . , n) represents the permutation, for which ζσ ( j−1) ≥ ζσ ( j) .

Theorem 8.6 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, an accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Dombi ordered weighted averaging operators is defined as m F D O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζˆσ ( j) )

j=1

 = 1− 1+

 n j=1

wj



1 1 . , . . . , 1 −     n k 1/k k 1/k p1 ◦ ζσ ( j) pm ◦ ζσ ( j) 1+ wj 1 − p1 ◦ ζσ ( j) 1 − pm ◦ ζσ ( j) j=1

(8.8)

382

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Example 8.2 Let ζˆ1 = (0.4, 0.5, 0.3, 0.8), ζˆ2 = (0.3, 0.4, 0.1, 0.7) and ζˆ3 = (0.8, 0.7, 0.6, 0.4) be 4–polar fuzzy numbers with weights w = (0.3, 0.1, 0.6). Then, for k = 3, the score values are computed as follows: 0.4 + 0.5 + 0.3 + 0.8 = 0.5, 4 0.8 + 0.7 + 0.6 + 0.4 S(ζˆ3 ) = = 0.625. 4 S(ζˆ1 ) =

S(ζˆ2 ) =

0.3 + 0.4 + 0.1 + 0.7 = 0.375, 4

Since, S(ζˆ3 ) > S(ζˆ1 ) > S(ζˆ2 ), thus ζˆσ (1) = ζˆ3 = (0.8, 0.7, 0.6, 0.4),

ζˆσ (2) = ζˆ1 = (0.4, 0.5, 0.3, 0.8),

ζˆσ (3) = ζˆ2 = (0.3, 0.4, 0.1, 0.7). Then, from Definition 8.4, m F D O W Aw (ζˆ1 , ζˆ2 , ζˆ3 ) =

3  (w j ζˆσ ( j) ), j=1



1 1 = 1−  p ◦ζ k 1/k , . . . , 1 −  p ◦ζ k 1/k ,   n n 1 m σ ( j) σ ( j) 1+ wj 1+ wj 1 − p1 ◦ ζσ ( j) 1 − pm ◦ ζσ ( j) j=1 j=1  1 = 1−   0.8 3  0.4 3  0.3 3 1/3 , 1 + 0.3 × + 0.1 × + 0.6 × 1 − 0.8 1 − 0.4 1 − 0.3 1 1−   0.7 3  0.5 3  0.4 3 1/3 , 1 + 0.3 × + 0.1 × + 0.6 × 1 − 0.7 1 − 0.5 1 − 0.4 1 1−   0.6 3  0.3 3  0.1 3 1/3 , 1 + 0.3 × + 0.1 × + 0.6 × 1 − 0.6 1 − 0.3 1 − 0.1

1−

  1 + 0.3 ×

1

 0.8 3  0.7 0.4 3 + 0.1 × + 0.6 × 1 − 0.4 1 − 0.8 1 − 0.7 = (0.7284, 0.6152, 0.5017, 0.7073).

 3 1/3 ,

Remark 8.1 Note that m–polar fuzzy Dombi ordered weighted averaging operators satisfy properties, namely, idempotency, boundedness and monotonicity as described in Theorems 8.3, 8.4 and 8.5. Theorem 8.7 (Commutative Law) For any two collections of m–polar fuzzy numbers ζˆ j and ζˆj j = 1, 2, . . . , n, we get m F D O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) = m F D O W ζw (ζˆ1 , ζˆ2 , . . . , ζˆn ), where ζˆj is any permutation of ζˆj .

(8.9)

8.2 m–Polar Fuzzy Dombi Aggregation Operators

383

It is noted that m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi ordered weighted averaging operators aggregate weighted m–polar fuzzy numbers and their ordering, respectively. Now it is presented a novel operator called m–polar fuzzy Dombi hybrid averaging operator, which obtain the properties of both m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi ordered weighted averaging operators. Definition 8.5 For a family of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ), j = 1, 2, . . . , n, an m–polar fuzzy Dombi hybrid averaging operator is defined as: m F D H Aw,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζ˜ˆσ ( j) ),

(8.10)

j=1

where w = (w1 , w2 , . . . , wn )T denotes the weights corresponding to the m–polar n ˜ˆ fuzzy numbers ζˆj with the conditions w j ∈ (0, 1], j=1 w j = 1, ζσ ( j) is the jth biggest m–polar fuzzy number, ζˆ˜σ ( j) = (nΘ j )ζˆj , ( j = 1, 2, . . . , n), Θ = (Θ1 , Θ2 , n . . . , Θn ) is a vector having weights, with Θ j ∈ (0, 1], j=1 Θ j = 1. Notice that when w = ( n1 , n1 , . . . , n1 ), m–polar fuzzy Dombi hybrid averaging operator converts into m–polar fuzzy Dombi weighted averaging operator. If Θ = ( n1 , n1 , . . . , n1 ), then m–polar fuzzy Dombi hybrid averaging operator becomes m–polar fuzzy Dombi ordered weighted averaging operator. Thus, m–polar fuzzy Dombi hybrid averaging operator is a generalization for both operators, m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi ordered weighted averaging, which describes the degrees and ordering of m–polar fuzzy numbers.

Theorem 8.8 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, an accumulated score of these m–polar fuzzy numbers using the m–polar fuzzy Dombi hybrid averaging operators is defined as m F D H Aw,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζ˜ˆσ ( j) )

j=1

 = 1− 1+

 n j=1

wj



1 p1 ◦ ζσ ( j) 1 − p1 ◦ ζσ ( j)

k 1/k , . . . , 1 −

1+

1 . pm ◦ ζσ ( j) k 1/k wj 1 − pm ◦ ζσ ( j) j=1

 n



(8.11)

ζˆ2 = (0.2, 0.5, 0.7), ζˆ3 = Example 8.3 Let ζˆ1 = (0.7, 0.3, 0.5), (0.8, 0.2, 0.1) and ζˆ4 = (0.6, 0.7, 0.9) be 3–polar fuzzy numbers with w = (0.2, 0.3, 0.1, 0.4), a weight vector corresponding to given 3–polar fuzzy numbers and a vector Θ = (0.3, 0.1, 0.4, 0.2) having weights. Then, by Definition 8.5, for k = 3

384

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets  ζ˜ˆ1 = 1 −  = 1−



1 + nΘ1



1 1 , . . . , 1 − ,     k 1/k k 1/k p 1 ◦ ζ1 p 3 ◦ ζ1 1 + nΘ1 1 − p 1 ◦ ζ1 1 − p 3 ◦ ζ1

1  1 + 4 × 0.3 ×

0.7 3 1/3 1 − 0.7

1 , 1−   1/3  0.5 3 1 + 4 × 0.3 × 1 − 0.5 = (0.7126, 0.3129, 0.5152). 

,1 −

1   1 + 4 × 0.3 ×

0.3 3 1/3 1 − 0.3

,

Similarly,  ζ˜ˆ2 = 1 −

1  1 + 4 × 0.1 ×

,1 −

1  1 + 4 × 0.1 ×

1  1 + 4 × 0.4 ×

,1 −

1   1 + 4 × 0.4 ×

0.2 3 1/3 1 − 0.2

1 1−    0.7 3 1/3 , 1 + 4 × 0.1 × 1 − 0.7 = (0.1555, 0.4242, 0.6322),

 ζ˜ˆ3 = 1 −



0.8 3 1/3 1 − 0.8

1 1−    0.1 3 1/3 , 1 + 4 × 0.4 × 1 − 0.1 = (0.8239, 0.2262, 0.1150), 



0.5 3 1/3 1 − 0.5

,

0.2 3 1/3 1 − 0.2

,

and  ζ˜ˆ4 = 1 −

1 1    0.6 3 1/3 , 1 −  0.7 3 1/3 , 1 + 4 × 0.2 × 1 + 4 × 0.2 × 1 − 0.6 1 − 0.7

1 1−   0.9 3 1/3 , 1 + 4 × 0.2 × 1 − 0.9 = (0.5820, 0.6842, 0.8931). Then, scores of m–polar fuzzy numbers for k = 3 are calculated as:

8.2 m–Polar Fuzzy Dombi Aggregation Operators

385

0.7126 + 0.3129 + 0.5152 = 0.5136, 3 0.1555 + 0.4242 + 0.6322 S(ζ˜ˆ2 ) = = 0.4040, 3 0.8239 + 0.2262 + 0.1150 = 0.3884, S(ζ˜ˆ3 ) = 3 0.5820 + 0.6842 + 0.8931 S(ζ˜ˆ4 ) = = 0.7198. 3

S(ζ˜ˆ1 ) =

Since, S(ζ˜ˆ4 ) > S(ζ˜ˆ1 ) > S(ζ˜ˆ2 ) > S(ζ˜ˆ3 ), thus ζ˜ˆσ (1) = ζˆ4 = (0.5820, 0.6842, 0.8931), ζˆ˜σ (3) = ζˆ2 = (0.1555, 0.4242, 0.6322),

ζ˜ˆσ (2) = ζˆ1 = (0.7126, 0.3129, 0.5152), ζˆ˜σ (4) = ζˆ3 = (0.8239, 0.2262, 0.1150).

Then, from Theorem 8.8, m F D H Aw,Θ (ζˆ1 , ζˆ2 , ζˆ3 , ζˆ4 ) =  = 1− 1+  = 1−

4 

(w j ζˆσ ( j) ) j=1

1 1 ,...,1 −   p3 ◦ ζσ ( j) k 1/k 4 p1 ◦ ζσ ( j) k 1/k wj 1+ wj 1 − p1 ◦ ζσ ( j) 1 − p3 ◦ ζσ ( j) j=1 j=1

 4



,

1 ,  0.7126 3  0.1555 3  0.8239 3 1/3 0.5820 3 + 0.3 × + 0.1 × + 0.4 × 1 − 0.5820 1 − 0.7126 1 − 0.1555 1 − 0.8239 1 1− ,   0.3129 3  0.4242 3  0.2262 3 1/3  0.6842 3 + 0.3 × + 0.1 × + 0.4 × 1 + 0.2 × 1 − 0.6842 1 − 0.3129 1 − 0.4242 1 − 0.2262

1 , 1−   0.5152 3  0.6322 3  0.1150 3 1/3  0.8931 3 + 0.3 × + 0.1 × + 0.4 × 1 + 0.2 × 1 − 0.8931 1 − 0.5152 1 − 0.6322 1 − 0.1150   1 + 0.2 ×

= (0.7819, 0.5620, 0.8304).

8.2.2 m–Polar Fuzzy Dombi Geometric Aggregation Operators Now different types of Dombi geometric aggregation operators are presented with m–polar fuzzy numbers, namely, m–polar fuzzy Dombi weighted geometric operator, m–polar fuzzy Dombi ordered weighted geometric operator and m–polar fuzzy Dombi hybrid geometric operator. Definition 8.6 For a family of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ), j = 1, 2, . . . , n, a mapping mFDWG : ζˆ n → ζˆ is called an m–polar fuzzy Dombi weighted geometric operator, which is given by

386

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F DW G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(ζˆ j )Θ j ,

(8.12)

j=1

where Θ = (Θ1 , Θ2 , . . . , Θn ) represents the weights, with (0, 1].

n j=1

Θ j = 1, Θ j ∈

Theorem 8.9 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, an accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Dombi weighted geometric operator is defined by m F DW G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =  = 1−

n 

(ζˆ j )Θ j ,

j=1

1 1 , . . . , 1 −    1 − p ◦ ζ k 1/k  1 − p ◦ ζ k 1/k . n n 1 j m j 1+ Θj 1+ Θj p1 ◦ ζ j pm ◦ ζ j j=1 j=1

(8.13) Proof Its proof is identical to Theorem 8.2.



Example 8.4 Let ζˆ1 = (0.2, 0.6, 0.3), ζˆ2 = (0.9, 1.0, 0.7), ζˆ3 = (0.1, 0.8, 0.4) and ζˆ4 = (0.4, 0.7, 0.3) be 3–polar fuzzy numbers with weights Θ = (0.1, 0.5, 0.3, 0.1). Then, for k = 3, m F DW G Θ (ζˆ1 , ζˆ2 , ζˆ3 ) =

3 

(ζˆ j )Θ j ,

j=1

1 1 , = 1− , . . . , 1 −       n n 1 − p1 ◦ ζ j k 1/k 1 − pm ◦ ζ j k 1/k 1+ Θj 1+ Θj p1 ◦ ζ j pm ◦ ζ j j=1 j=1  1 = 1−   1 − 0.2 3  1 − 0.9 3  1 − 0.1 3  1 − 0.4 3 1/3 , 1 + 0.1 × + 0.5 × + 0.3 × + 0.1 × 0.2 0.9 0.1 0.4 1 1−   1 − 0.6 3  1 − 1.0 3  1 − 0.8 3  1 − 0.7 3 1/3 , 1 + 0.1 × + 0.5 × + 0.3 × + 0.1 × 0.6 1.0 0.8 0.7

1 1−    1 − 0.3 3  1 − 0.7 3  1 − 0.4 3  1 − 0.3 3 1/3 , 1 + 0.1 × + 0.5 × + 0.3 × + 0.1 × 0.3 0.7 0.4 0.3 = (0.8589, 0.2582, 0.6052). 

It can be readily shown that the m–polar fuzzy Dombi weighted geometric operator holds the notions given below: Theorem 8.10 (Idempotent Law) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers, which are equal, i.e., ζˆ j = ζˆ , then m F DW G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ .

(8.14)

8.2 m–Polar Fuzzy Dombi Aggregation Operators

387

Theorem 8.11 (Bounded Law) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a collection of

 ‘n’ m–polar fuzzy numbers, ζˆ − = nj=1 (ζ j ) and ζˆ + = nj=1 (ζ j ), then ζˆ − ≤ m F DW G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ ζˆ + .

(8.15)

Theorem 8.12 (Monotonic Law) For two collections of m–polar fuzzy numbers ζˆ j and ζˆj , ( j = 1, 2, . . . , n), if ζˆ j ≤ ζˆj , then m F DW G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ m F DW G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.16)

Definition 8.7 For a family of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ), j = 1, 2, . . . , n, an m–polar fuzzy Dombi ordered weighted geometric operator is a mapping mFDOWG : ζˆ n → ζˆ , which is given as

m F D O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (w j ζˆσ ( j) )

(8.17)

j=1

where w = (w1 , w2 , . . . , wn ) is the weight-vector and w j ∈ (0, 1] with nj=1 w j = 1. σ ( j), j = 1, 2, . . . , n represents the permutation, such that ζˆσ ( j−1) ≥ ζˆσ ( j) . Theorem 8.13 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, an accumulated value of these m–polar fuzzy numbers using an m–polar fuzzy Dombi ordered weighted geometric operator is defined by m F D O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (ζˆσ ( j) )w j j=1

 = 1− 1+

1 1 .  1 − p ◦ ζ k 1/k , . . . , 1 −   n 1 − pm ◦ ζσ ( j) k 1/k 1 σ ( j) wj 1+ wj p1 ◦ ζσ ( j) pm ◦ ζσ ( j) j=1 j=1

 n

(8.18) Example 8.5 Let ζˆ1 = (0.2, 0.4, 0.7), ζˆ2 = (0.3, 0.6, 0.1), ζˆ3 = ˆ (0.8, 0.3, 0.5) and ζ4 = (0.6, 0.4, 0.7) be 3–polar fuzzy numbers and w = (0.3, 0.1, 0.2, 0.4) be a weight vector. Then, score values of m–polar fuzzy numbers for k = 3 is calculated as: 0.2 + 0.4 + 0.7 = 0.4333, 3 0.8 + 0.3 + 0.5 S(ζˆ3 ) = = 0.5333, 3

S(ζˆ1 ) =

Since, S(ζˆ4 ) > S(ζˆ3 ) > S(ζˆ1 ) > S(ζˆ2 ), thus

0.3 + 0.6 + 0.1 = 0.3333, 3 0.6 + 0.4 + 0.7 S(ζˆ4 ) = = 0.5667. 3 S(ζˆ2 ) =

388

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

ζˆσ (1) = ζˆ3 = (0.6, 0.4, 0.7),

ζˆσ (2) = ζˆ3 = (0.8, 0.3, 0.5),

ζˆσ (3) = ζˆ1 = (0.2, 0.4, 0.7),

ζˆσ (4) = ζˆ2 = (0.3, 0.6, 0.1).

Then, from Definition 8.7, m F D O W G w (ζˆ1 , ζˆ2 , ζˆ3 , ζˆ4 ) =

4  (ζˆσ ( j) )w j , j=1

 = 1−  = 1−

1 1 , . . . , 1 − , 1 − p ◦ ζ  1 − p ◦ ζ    4 4 1 3 σ ( j) k 1/k σ ( j) k 1/k 1+ wj 1+ wj p1 ◦ ζσ ( j) p3 ◦ ζσ ( j) j=1 j=1 

1 + 0.3 ×

1  1 − 0.6 3 0.6

+ 0.1 ×

 1 − 0.8 3 0.8

+ 0.2 ×

 1 − 0.2 3 0.2

+ 0.4 ×

 1 − 0.3 3 1/3 , 0.3

1

  1 − 0.4 3  1 − 0.3 3  1 − 0.4 3  1 − 0.6 3 1/3 , 1 + 0.3 × + 0.1 × + 0.2 × + 0.4 × 0.4 0.3 0.4 0.6

1 1−   1 − 0.7 3  1 − 0.5 3  1 − 0.7 3  1 − 0.1 3 1/3 , 1 + 0.3 × + 0.1 × + 0.2 × + 0.4 × 0.7 0.5 0.7 0.1 = (0.7237, 0.5926, 0.8690). 1−

Remark 8.2 Note that m–polar fuzzy Dombi ordered weighted geometric operators satisfy properties, namely, idempotency, boundedness and monotonicity as described in Theorems 8.10, 8.11 and 8.12. Theorem 8.14 (Commutativity Law) For two arbitrary collections of m–polar fuzzy numbers ζˆ j and ζˆj ( j = 1, 2, . . . , n), if ζˆ j ≤ ζˆj , then m F D O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) = m F D O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ),

(8.19)

where ζˆj is any permutation of ζˆj . In Definitions 8.3 and 8.4, it is observed that m–polar fuzzy Dombi weighted geometric and m–polar fuzzy Dombi ordered weighted geometric operators aggregate weighted m–polar fuzzy numbers and their ordering, respectively. Now m–polar fuzzy Dombi hybrid geometric operator is presented, which contains the properties of both m–polar fuzzy Dombi weighted geometric and m–polar fuzzy Dombi ordered weighted geometric operators. Definition 8.8 For a family of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ), j = 1, 2, . . . , n, an m–polar fuzzy Dombi hybrid geometric operator is defined by m F D H G w,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (ζ˜ˆσ ( j) )w j , j=1

(8.20)

8.2 m–Polar Fuzzy Dombi Aggregation Operators

389

where w = (w1 , w2 , . . . , wn ) denotes the weights associated to the m–polar fuzzy n ˜ˆ numbers ζˆj , j = 1, 2, . . . , n, w j ∈ (0, 1], j=1 w j = 1, ζσ ( j) is the jth largest m– polar fuzzy numbers, ζ˜ˆσ ( j) = (nΘ j )ζˆj , ( j = 1, 2, . . . , n), Θ = (Θ1 , Θ2 , . . . , Θn ) is n a vector having weights, with Θ j ∈ (0, 1], j=1 Θ j = 1. Notice that when w = ( n1 , n1 , . . . , n1 ), m–polar fuzzy Dombi hybrid geometric operator becomes m–polar fuzzy Dombi weighted geometric operator. When Θ = ( n1 , n1 , . . . , n1 ), then m–polar fuzzy Dombi hybrid geometric operator converts into m–polar fuzzy Dombi ordered weighted geometric operator. Thus, m–polar fuzzy Dombi hybrid geometric operator is a generalization of m–polar fuzzy Dombi weighted geometric and m–polar fuzzy Dombi ordered weighted geometric operators. With induction technique, one can readily show the next theorem. Theorem 8.15 For a collection of m–polar fuzzy numbers ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) where j = 1, 2, . . . , n, an accumulated score of these m–polar fuzzy numbers using an m–polar fuzzy Dombi hybrid geometric operator is defined as m F D H G w,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  w (ζ˜ˆσ ( j) ) j j=1

 = 1− 1+

 n j=1

wj

1 ,...,1 −  1 − p1 ◦ ζ  σ ( j) k 1/k p1 ◦ ζσ ( j)

1+

1  1 − pm ◦ ζ k 1/k . σ ( j) wj pm ◦ ζσ ( j) j=1

 n

(8.21)

Example 8.6 Let ζˆ1 = (0.4, 0.6, 0.3), ζˆ2 = (0.3, 0.2, 0.9), ζˆ3 = (0.6, 0.3, 0.5) and ζˆ4 = (0.3, 0.5, 0.7) be 3–polar fuzzy numbers and w = (0.4, 0.1, 0.3, 0.2) be an associated weight vector and a vector Θ = (0.5, 0.2, 0.1, 0.2) having weights. By Definition 8.8, for k = 3

1 1  1 − p ◦ ζ k 1/k , . . . , 1 −  1 − p ◦ ζ k 1/k ,   1 1 3 1 1 + nΘ1 1 + nΘ1 p 1 ◦ ζ1 p 3 ◦ ζ1  1 1 = 1−    1 − 0.4 3 1/3 , 1 −  1 − 0.6 3 1/3 , 1 + 4 × 0.5 × 1 + 4 × 0.5 × 0.4 0.6

1 1−   1 − 0.3 3 1/3 , 1 + 4 × 0.5 × 0.3 = (0.6540, 0.4565, 0.7462).

 ˜ζˆ = 1 − 1

Similarly,

390

 ˜ζˆ = 1 − 2

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

1 1   1 − 0.3 3 1/3 , 1 −  1 − 0.2 3 1/3 , 1 + 4 × 0.2 × 1 + 4 × 0.2 × 0.3 0.2

1 1−   1 − 0.9 3 1/3 , 1 + 4 × 0.2 × 0.9 = (0.6842, 0.7878, 0.0935), 

 ζ˜ˆ3 = 1 −

1 1    1 − 0.6 3 1/3 , 1 −  1 − 0.3 3 1/3 , 1 + 4 × 0.1 × 1 + 4 × 0.1 × 0.6 0.3

1 1−   1 − 0.5 3 1/3 , 1 + 4 × 0.1 × 0.5 = (0.3294, 0.6322, 0.4242),

and  ζ˜ˆ4 = 1 −

1 1    1 − 0.3 3 1/3 , 1 −  1 − 0.5 3 1/3 , 1 + 4 × 0.2 × 1 + 4 × 0.2 × 0.3 0.5

1 1−   1 − 0.7 3 1/3 , 1 + 4 × 0.2 × 0.7 = (0.6842, 0.4814, 0.2846).

Then, score values of m–polar fuzzy numbers for k = 3 are given as follows: 0.6540 + 0.4565 + 0.7462 = 0.6189, 3 0.6842 + 0.7878 + 0.0935 S(ζ˜ˆ2 ) = = 0.5218, 3 0.3294 + 0.6322 + 0.4242 S(ζ˜ˆ3 ) = = 0.4620, 3 0.6842 + 0.4814 + 0.2846 S(ζ˜ˆ4 ) = = 0.4834. 3

S(ζ˜ˆ1 ) =

Since, S(ζ˜ˆ1 ) > S(ζ˜ˆ2 ) > S(ζ˜ˆ4 ) > S(ζ˜ˆ3 ), thus

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

ζ˜ˆσ (1) = ζˆ1 = (0.6540, 0.4565, 0.7462), ζ˜ˆσ (3) = ζˆ4 = (0.6842, 0.4814, 0.2846),

391

ζ˜ˆσ (2) = ζˆ2 = (0.6842, 0.7878, 0.0935), ζ˜ˆσ (4) = ζˆ3 = (0.3294, 0.6322, 0.4242).

Then, from Definition 8.7, m F D H ζw,Θ (ζˆ1 , ζˆ2 , ζˆ3 , ζˆ4 ) =

w (ζˆσ ( j) ) j

j=1

 = 1− 1+  = 1−

4 

 4 j=1

wj

1 ,...,1 −  1 − p1 ◦ ζ  σ ( j) k 1/k p1 ◦ ζσ ( j)

1+

 4 j=1

wj

1 ,  1 − p3 ◦ ζ  σ ( j) k 1/k p3 ◦ ζσ ( j)

1

,   1 − 0.6540 3  1 − 0.6842 3  1 − 0.6842 3  1 − 0.3294 3 1/3 1 + 0.4 × + 0.1 × + 0.3 × + 0.2 × 0.6540 0.6842 0.6842 0.3294 1 , 1−   1 − 0.4565 3  1 − 0.7878 3  1 − 0.4814 3  1 − 0.6322 3 1/3 1 + 0.4 × + 0.1 × + 0.3 × + 0.2 × 0.4565 0.7878 0.4814 0.6322

1 , 1−   1 − 0.7462 3  1 − 0.0935 3  1 − 0.2846 3  1 − 0.4242 3 1/3 1 + 0.4 × + 0.1 × + 0.3 × + 0.2 × 0.7462 0.0935 0.2846 0.4242

= (0.5482, 0.5073, 0.8210).

8.3 m–Polar Fuzzy Hamacher Aggregation Operators Hamacher operators are more flexible and can be taken as the generalization of algebraic and Einstein operators. Hamacher [26] introduced an extension of t-norm and t-conorm. In this section, firstly Hamacher operations for m–polar fuzzy numbers via Hamacher t-conorm and Hamacher t-norm are presented, then m–polar fuzzy Hamacher arithmetic and geometric aggregation operators are described. Hamacher product ⊗ and Hamacher sum ⊕ are respectively t-norm and t-conorm, which are given as follows, for all l, t ∈ [0, 1]. lt , λ > 0. λ + (1 − λ)(l + t − lt) l + t − lt − (1 − λ)lt , λ > 0. T ∗ (l, t) = l ⊕ t = 1 − (1 − λ)lt T (l, t) = l ⊗ t =

(8.22) (8.23)

In particular, when λ = 1 in Eqs. (8.22) and (8.23), algebraic t-norm and t-conorm are obtained respectively. T (l, t) = l ⊗ t = lt, ∗

T (l, t) = l ⊕ t = l + t − lt,

(8.24) (8.25)

and when λ = 2 in Eqs. (8.22) and (8.23), Einstein t-norm and t-conorm are obtained respectively as follows:

392

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

lt , λ > 0. 1 + (1 − l)(1 − t) l +t T ∗ (l, t) = l ⊕ t = , λ > 0. 1 + lt T (l, t) = l ⊗ t =

(8.26) (8.27)

Let ζˆ1 = ( p1 ◦ ζ1 , . . . , pm ◦ ζ1 ), ζˆ2 = ( p1 ◦ ζ2 , . . . , pm ◦ ζ2 ) and ζˆ = ( p1 ◦ ζ, . . . , pm ◦ ζ ) be m–polar fuzzy numbers. The following basic Hamacher operations for m–polar fuzzy numbers are defined with λ > 0.  p ◦ ζ + p ◦ ζ − p ◦ ζ . p ◦ ζ − (1 − λ) p ◦ ζ . p ◦ ζ 1 1 1 2 1 1 1 2 1 1 1 2 • ζˆ1 ⊕ ζˆ2 = ,..., 1 − (1 − λ) p1 ◦ ζ1 . p1 ◦ ζ2 pm ◦ ζ1 + pm ◦ ζ2 − pm ◦ ζ1 . pm ◦ ζ2 − (1 − λ) pm ◦ ζ1 . pm ◦ ζ2  1 − (1 − λ) pm ◦ ζ1 . pm ◦ ζ2  p 1 ◦ ζ1 . p 1 ◦ ζ2 ˆ ˆ ,..., • ζ1 ⊗ ζ2 = λ − (1 − λ)( p1 ◦ ζ1 + p1 ◦ ζ2 − p1 ◦ ζ1 . p1 ◦ ζ2 )  p ◦ ζ .p ◦ ζ m

1

m

2

λ − (1 − λ)( pm ◦ ζ1 + pm ◦ ζ2 − pm ◦ ζ1 . pm ◦ ζ2 ) (1 + (λ − 1) p1 ◦ ζ )α − (1 − p1 ◦ ζ )α ,..., (1 + (λ − 1) p1 ◦ ζ )α + (λ − 1)(1 − p1 ◦ ζ )α  α α (1 + (λ − 1) pm ◦ ζ ) − (1 − pm ◦ ζ ) , α>0 (1 + (λ − 1) pm ◦ ζ )α + (λ − 1)(1 − pm ◦ ζ )α  α λ( p1 ◦ ζ ) • (ζˆ )α = ,..., (1 + (λ − 1)(1 − p1 ◦ ζ ))α + (λ − 1)( p1 ◦ ζ )α  α λ( pm ◦ ζ ) , α > 0. (1 + (λ − 1)(1 − pm ◦ ζ ))α + (λ − 1)( pm ◦ ζ )α • α ζˆ =



8.3.1 m–Polar Fuzzy Hamacher Arithmetic Aggregation Operators Definition 8.9 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of m–polar fuzzy numbers where ‘ j’ varies from 1 to n. Then, an m–polar fuzzy Hamacher weighted average operator is a mapping from ζˆ n to ζˆ , which is defined as

m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(Θ j ζˆ j )

(8.28)

j=1

where Θ = (Θ1 , Θ2 , . . . , Θn ) represents the weight vector of ζˆ j , for each ‘ j’ varies from 1 to n, with Θ j > 0 and nj=1 Θ j = 1. Theorem 8.16 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of m–polar fuzzy numbers where ‘ j’ varies from 1 to n. The accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Hamacher weighted average operator is also an

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

393

m–polar fuzzy number, which is given as m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(Θ j ζˆ j ),

j=1

=



 Θ j  n  Θ − j=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j Θ j  Θ , . . . , n   + (λ − 1) nj=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j n

 Θ j n  Θ  − j=1 1 − pm ◦ ζ j j j=1 1 + (λ − 1) pm ◦ ζ j Θ j Θ j . n  n  + (λ − 1) j=1 1 − pm ◦ ζ j j=1 1 + (λ − 1) pm ◦ ζ j n

(8.29) Proof The mathematical induction technique is used to prove it. Case 1. When n = 1, from the Eq. (8.29), we get m F H W AΘ (ζˆ1 , , ζˆ2 , . . . , ζˆn ) = Θ1 ζˆ1 = ζˆ1 , (since Θ1 = 1)  1 + (λ − 1) p1 ◦ ζ1 − (1 − p1 ◦ ζ1 ) = ,..., (1 + (λ − 1) p1 ◦ ζ1 ) + (λ − 1)(1 − p1 ◦ ζ1 )  1 + (λ − 1) pm ◦ ζ1 − (1 − pm ◦ ζ1 ) . (1 + (λ − 1) pm ◦ ζ1 ) + (λ − 1)(1 − pm ◦ ζ1 )

Thus, for n = 1 Eq. (8.29) holds. Case 2. It is supposed that Eq. (8.29) holds for n = s, where s ∈ N(set of natural numbers), then we get m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆs ) =

s 

(Θ j ζˆ j ),

j=1

Θ j s  Θ s   − j=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j =   Θ j  Θ , . . . ,  s + (λ − 1) sj=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j Θ j  s  Θ s   − j=1 1 − pm ◦ ζ j j j=1 1 + (λ − 1) pm ◦ ζ j Θ j Θ j . s  s  + (λ − 1) j=1 1 − pm ◦ ζ j j=1 1 + (λ − 1) pm ◦ ζ j

(8.30) For n = s + 1,

394

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆs , ζˆs+1 ) =

s  (Θ j ζˆ j ) ⊕ (Θk+1 ζˆs+1 ), j=1

Θ  Θ  1 + (λ − 1) p1 ◦ ζ j j − sj=1 1 − p1 ◦ ζ j j =   Θ j  Θ , . . . ,  s + (λ − 1) sj=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j Θ j s  Θ s   − j=1 1 − pm ◦ ζ j j j=1 1 + (λ − 1) pm ◦ ζ j    Θ j ⊕ s s Θj + (λ − 1) j=1 1 − pm ◦ ζ j j=1 1 + (λ − 1) pm ◦ ζ j  Θ s+1 (1 + (λ − 1) p1 ◦ ζs+1 ) − (1 − p1 ◦ ζs+1 )Θs+1 ,..., (1 + (λ − 1) p1 ◦ ζs+1 )Θs+1 + (λ − 1)(1 − p1 ◦ ζs+1 )Θs+1  (1 + (λ − 1) p ◦ ζ )Θs+1 − (1 − p ◦ ζ )Θs+1 

s



j=1

m

s+1

m

s+1

(1 + (λ − 1) pm ◦ ζs+1 )Θs+1 + (λ − 1)(1 − pm ◦ ζs+1 )Θs+1 Θ j s+1  Θ s+1   − j=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j =  Θ j  Θ j , . . . ,  s+1  + (λ − 1) s+1 j=1 1 + (λ − 1) p1 ◦ ζ j j=1 1 − p1 ◦ ζ j Θ j s+1  Θ s+1   − j=1 1 − pm ◦ ζ j j j=1 1 + (λ − 1) pm ◦ ζ j Θ j Θ j . s+1  s+1  + (λ − 1) j=1 1 − pm ◦ ζ j j=1 1 + (λ − 1) pm ◦ ζ j

Therefore, Eq. (8.29) holds for n = s + 1. Thus, it is concluded that Eq. (8.29) holds for any n ∈ N.  Example 8.7 Let ζˆ1 = (0.2, 0.5, 0.7, 0.3), ζˆ2 = (0.8, 0.6, 0.6, 0.4) and ζˆ3 = (0.1, 0.2, 0.4, 0.5) be 4–polar fuzzy numbers with a weight vector Θ = (0.3, 0.5, 0.2) for these 4–polar fuzzy numbers. Then, for λ = 3, m F H W AΘ (ζˆ1 , ζˆ2 , ζˆ3 ) =

3 

(Θ j ζˆ j )

j=1

Θ j  3  Θ 3   − j=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j =   Θ j  Θ , . . . ,  3 + (λ − 1) 3j=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j Θ j 3  Θ 3   − j=1 1 − p4 ◦ ζ j j j=1 1 + (λ − 1) p4 ◦ ζ j Θ j Θ j 3  3  ◦ ζ + (λ − 1) ◦ ζ 1 + (λ − 1) p 1 − p 4 j 4 j j=1 j=1  1 + (2)0.20.3 × 1 + (2)0.80.5 × 1 + (2)0.10.2 − 1 − 0.20.3 × 1 − 0.80.5 × 1 − 0.10.2 =  0.5  0.2  0.3  0.5  0.2  , 0.3  × 1 + (2)0.8 × 1 + (2)0.1 + (2) 1 − 0.2 × 1 − 0.8 × 1 − 0.1 1 + (2)0.2  0.3  0.5  0.2  0.3  0.5  0.2 1 + (2)0.5 × 1 + (2)0.6 × 1 + (2)0.2 − 1 − 0.5 × 1 − 0.6 × 1 − 0.2  0.3  0.5  0.2  0.3  0.5  0.2  , 1 + (2)0.5 × 1 + (2)0.6 × 1 + (2)0.2 + (2) 1 − 0.5 × 1 − 0.6 × 1 − 0.2  0.3  0.5  0.2  0.3  0.5  0.2 1 + (2)0.7 × 1 + (2)0.6 × 1 + (2)0.4 − 1 − 0.7 × 1 − 0.6 × 1 − 0.4  0.3  0.5  0.2  0.3  0.5  0.2  , 1 + (2)0.7 × 1 + (2)0.6 × 1 + (2)0.4 + (2) 1 − 0.7 × 1 − 0.6 × 1 − 0.4  0.3  0.5  0.2  0.3  0.5  0.2  1 + (2)0.3 × 1 + (2)0.4 × 1 + (2)0.5 − 1 − 0.3 × 1 − 0.4 × 1 − 0.5 0.5  0.2  0.3  0.5  0.2   0.3  × 1 + (2)0.4 × 1 + (2)0.5 + (2) 1 − 0.3 × 1 − 0.4 × 1 − 0.5 1 + (2)0.3 = (0.5397, 0.4980, 0.5974, 0.3913).

• When λ = 1, m–polar fuzzy Hamacher weighted average operator reduces into m–polar fuzzy weighted averaging operator as below:

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

m F W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

395

n  (Θ j ζˆ j ) j=1

n n n      Θ  Θ  Θ  = 1− 1 − p1 ◦ ζ j j , 1 − 1 − p2 ◦ ζ j j , . . . , 1 − 1 − pm ◦ ζ j j . j=1

j=1

j=1

(8.31) • When λ = 2, m–polar fuzzy Hamacher weighted average operator reduces into m–polar fuzzy Einstein weighted averaging operator as below: m F E W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(Θ j ζˆ j )

j=1

 Θ j n  Θ  n − j=1 1 − p1 ◦ ζ j j j=1 1 + p1 ◦ ζ j = n  Θ j n  Θ , . . . , + j=1 1 − p1 ◦ ζ j j j=1 1 + p1 ◦ ζ j Θ j n  Θ n  − j=1 1 − pm ◦ ζ j j  j=1 1 + pm ◦ ζ j Θ j n  Θ . n  + j=1 1 − pm ◦ ζ j j j=1 1 + pm ◦ ζ j

(8.32)

Theorem 8.17 (Idempotency Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ), be a family of ‘n’ m–polar fuzzy numbers. If all these m–polar fuzzy numbers are same, in other words, ζˆ j = ζˆ , ∀ j = 1, 2, . . . , n, then m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ .

(8.33)

Proof Since ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) = ζˆ , where ‘ j’ varies from 1 to n.. Then, from Eq. (8.29), the following expression is obtained as

m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(Θ j ζˆ j ),

j=1

 Θ j  n  Θ − j=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j  Θ j  Θ , . . . ,  + (λ − 1) nj=1 1 − p1 ◦ ζ j j j=1 1 + (λ − 1) p1 ◦ ζ j

 =  n

n

 Θ j n  Θ  − j=1 1 − pm ◦ ζ j j j=1 1 + (λ − 1) pm ◦ ζ j    Θ j n n Θj + (λ − 1) j=1 1 − pm ◦ ζ j j=1 1 + (λ − 1) pm ◦ ζ j n

=



(1 + (λ − 1) p1 ◦ ζ )Θ − (1 − p1 ◦ ζ )Θ ,..., (1 + (λ − 1) p1 ◦ ζ )Θ + (λ − 1)(1 − p1 ◦ ζ )Θ  (1 + (λ − 1) p ◦ ζ )Θ − (1 − p ◦ ζ )Θ m

m

(1 + (λ − 1) pm ◦ ζ )Θ + (λ − 1)(1 − pm ◦ ζ )Θ = ( p1 ◦ ζ, p2 ◦ ζ, . . . , pm ◦ ζ ), for λ = 1 = ζˆ .

396

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Hence, m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ holds if ζˆ j = ζˆ , ∀ j = 1, 2, . . . , n.



The following properties, namely, boundedness and monotonicity can be easily followed by Definition 8.9. So, we omit their proofs. Theorem 8.18 (Boundedness Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ), be a fam  ily of ‘n’ m–polar fuzzy numbers, ζˆ − = nj=1 (ζ j ) and ζˆ + = nj=1 (ζ j ), then ζˆ − ≤ m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ ζˆ + .

(8.34)

Theorem 8.19 (Monotonicity Property) Let ζˆ j and ζˆj be two families of m–polar fuzzy numbers. If ζˆ j ≤ ζˆ , then j

m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ m F H W AΘ (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.35)

Definition 8.10 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of m–polar fuzzy numbers where ‘ j varies from 1 to n. An m–polar fuzzy Hamacher ordered ˆ ˆn weighted average operator is a mapping mFHOWA n : ζ → ζ with weight vecT tor w = (w1 , w2 , . . . , wn ) where w j ∈ (0, 1] and j=1 w j = 1. Thus,

m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (w j ζˆσ ( j) )

(8.36)

j=1

where (σ (1), σ (2), . . . , σ (n)) is the permutation of the indices, for which ζˆσ ( j−1) ≥ ζˆσ ( j) , ∀ j = 1, 2, . . . , n. Theorem 8.20 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. The accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Hamacher ordered weighted average operator is also an m–polar fuzzy number, which is given by m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζˆσ ( j) )

j=1

 w j n  w − j=1 1 − p1 ◦ ζσ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j)  w j  w , . . . ,  + (λ − 1) nj=1 1 − p1 ◦ ζσ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j) w j n  w n   − j=1 1 − pm ◦ ζσ ( j) j j=1 1 + (λ − 1) pm ◦ ζσ ( j) w j w j . n  n  1 + (λ − 1) p 1 − p ◦ ζ + (λ − 1) ◦ ζ m m σ ( j) σ ( j) j=1 j=1

 = n

n

(8.37) Example 8.8 Let ζˆ1 = (0.3, 0.6, 0.4, 0.7), ζˆ2 = (0.2, 0.5, 0.3, 0.6) and ζˆ3 = (0.7, 0.6, 0.7, 0.8) be 4–polar fuzzy numbers with a weight vector w = (0.4, 0.3, 0.3) for

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

397

these 4–polar fuzzy numbers. Then, scores and aggregated values of m–polar fuzzy numbers for λ = 3 can be computed as below: 0.3 + 0.6 + 0.4 + 0.7 = 0.5, 4 0.7 + 0.6 + 0.7 + 0.8 S(ζˆ3 ) = = 0.7. 4

S(ζˆ1 ) =

S(ζˆ2 ) =

0.2 + 0.5 + 0.3 + 0.6 = 0.4, 4

Since, S(ζˆ3 ) > S(ζˆ1 ) > S(ζˆ2 ), thus ζˆσ (1) = ζˆ3 = (0.7, 0.6, 0.7, 0.8), ζˆσ (3) = ζˆ2 = (0.2, 0.5, 0.3, 0.6).

ζˆσ (2) = ζˆ1 = (0.3, 0.6, 0.4, 0.7),

Then, from Definition 8.10, m F H O W Aw (ζˆ1 , Aˆ2 , Aˆ3 ) =

j=1

w  w  1 + (λ − 1) p1 ◦ ζσ ( j) j − 3j=1 1 − p1 ◦ ζσ ( j) j = 3  w j w , . . . , 3  + (λ − 1) j=1 1 − p1 ◦ ζσ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j) w j 3  w 3   − j=1 1 − pm ◦ ζσ ( j) j j=1 1 + (λ − 1) pm ◦ ζσ ( j) w j w j 3  3  + (λ − 1) j=1 1 − pm ◦ ζσ ( j) j=1 1 + (λ − 1) pm ◦ ζσ ( j)  1 + (2)0.70.4 × 1 + (2)0.30.3 × 1 + (2)0.20.3 − 1 − 0.70.4 × 1 − 0.30.3 × 1 − 0.20.3 =  0.4  0.3  0.3  0.4  0.3  0.3  , 1 + (2)0.7 × 1 + (2)0.3 × 1 + (2)0.2 + (2) 1 − 0.7 × 1 − 0.3 × 1 − 0.2  0.4  0.3  0.3  0.4  0.3  0.3 1 + (2)0.6 × 1 + (2)0.6 × 1 + (2)0.5 − 1 − 0.6 × 1 − 0.6 × 1 − 0.5 0.3  0.3  0.4  0.3  0.3  ,  0.4  × 1 + (2)0.6 × 1 + (2)0.5 + (2) 1 − 0.6 × 1 − 0.6 × 1 − 0.5 1 + (2)0.6  0.4  0.3  0.3  0.4  0.3  0.3 1 + (2)0.7 × 1 + (2)0.4 × 1 + (2)0.3 − 1 − 0.7 × 1 − 0.4 × 1 − 0.3  0.4  0.3  0.3  0.4  0.3  0.3  , 1 + (2)0.7 × 1 + (2)0.4 × 1 + (2)0.3 + (2) 1 − 0.7 × 1 − 0.4 × 1 − 0.3  0.4  0.3  0.3  0.4  0.3  0.3  1 + (2)0.8 × 1 + (2)0.7 × 1 + (2)0.6 − 1 − 0.8 × 1 − 0.7 × 1 − 0.6  0.4  0.3  0.3  0.4  0.3  0.3  1 + (2)0.8 × 1 + (2)0.7 × 1 + (2)0.6 + (2) 1 − 0.8 × 1 − 0.7 × 1 − 0.6 

3



3  (w j ζˆσ ( j) )

j=1

= (0.4528, 0.5714, 0.5077, 0.7192).

Two particular cases of m–polar fuzzy Hamacher ordered weighted average operator are as follows: • When λ = 1, m–polar fuzzy Hamacher ordered weighted average operator converted into m–polar fuzzy ordered weighted averaging operator as below: m F O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζˆσ ( j) )

j=1 n n n      w  w  w  1 − p1 ◦ ζσ ( j) j , 1 − 1 − p2 ◦ ζσ ( j) j , . . . , 1 − 1 − pm ◦ ζσ ( j) j . = 1− j=1

j=1

j=1

(8.38) • When λ = 2, m–polar fuzzy Hamacher ordered weighted average operator reduces into m–polar fuzzy Einstein ordered weighted averaging operator as below:

398

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F E O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζˆσ ( j) )

j=1

 w j n  w  n − j=1 1 − p1 ◦ ζσ ( j) j j=1 1 + p1 ◦ ζσ ( j) w j n  w , . . . , = n  + j=1 1 − p1 ◦ ζσ ( j) j j=1 1 + p1 ◦ ζσ ( j)  w j n  w n  − j=1 1 − pm ◦ ζσ ( j) j  j=1 1 + pm ◦ ζσ ( j)  w j n  w . n  + j=1 1 − pm ◦ ζσ ( j) j j=1 1 + pm ◦ ζσ ( j)

(8.39)

Theorem 8.21 (Idempotency Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. If all these m–polar fuzzy numbers are same, i.e., ζˆ j = ζˆ , ∀ j = 1, 2, . . . , n, then m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ .

(8.40)

Proof Since ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) = ζˆ , where ‘ j’ varies from 1 to n. Then, from Eq. (8.29), we obtain m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (w j ζˆσ ( j) ), j=1

w  w  1 + (λ − 1) p1 ◦ ζσ ( j) j − nj=1 1 − p1 ◦ ζσ ( j) j w j  w , . . . , = n   + (λ − 1) nj=1 1 − p1 ◦ ζσ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j) w j n  w n   − j=1 1 − pm ◦ ζσ ( j) j j=1 1 + (λ − 1) pm ◦ ζσ ( j) w j w j n  n  + (λ − 1) j=1 1 − pm ◦ ζσ ( j) j=1 1 + (λ − 1) pm ◦ ζσ ( j)  (1 + (λ − 1) p1 ◦ ζ )w − (1 − p1 ◦ ζ )w ,..., = (1 + (λ − 1) p1 ◦ ζ )w + (λ − 1)(1 − p1 ◦ ζ )w  (1 + (λ − 1) pm ◦ ζ )w − (1 − pm ◦ ζ )w (1 + (λ − 1) pm ◦ ζ )w + (λ − 1)(1 − pm ◦ ζ )w = ( p1 ◦ ζ, p2 ◦ ζ, . . . , pm ◦ ζ ), for λ = 1 

n



j=1

= ζˆ .

Hence, m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ holds if ζˆ j = ζˆ , ∀ j = 1, 2, . . . , n.



Theorem 8.22 (Boundedness Property) Let ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ) be

 a family of ‘n’ m–polar fuzzy numbers, ζˆ − = nj=1 (ζ j ) and ζˆ + = nj=1 (ζ j ), then ζˆ − ≤ m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ ζˆ + .

(8.41)

Theorem 8.23 (Monotonicity Property) Let ζˆ j and ζˆj be two families of m–polar fuzzy numbers where ‘ j’ varies from 1 to n. If ζˆ j ≤ ζˆj , then

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

399

m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.42)

Theorem 8.24 (Commutativity Property) Let ζˆ j and ζˆj be two families of m–polar fuzzy numbers, then m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn ) = m F H O W Aw (ζˆ1 , ζˆ2 , . . . , ζˆn )

(8.43)

where ζˆj is an arbitrary permutation of ζˆj . In Definitions 8.9 and 8.10, it is observed that m–polar fuzzy Hamacher weighted average operator and m–polar fuzzy Hamacher ordered weighted average operator weight m–polar fuzzy numbers and ordered arrangement of m–polar fuzzy numbers, respectively. Another operator, namely, m–polar fuzzy Hamacher hybrid averaging operator is presented, which combine the qualities of m–polar fuzzy Hamacher weighted average operator and m–polar fuzzy Hamacher ordered weighted average operator. Definition 8.11 Let ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of m–polar fuzzy numbers where j varies from 1 to n. An m–polar fuzzy Hamacher hybrid averaging operator is given as below: m F H H Aw,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζ˜ˆσ ( j) ),

(8.44)

j=1

where w = (w1 , w2 , . . . , wn ) is the associated-weight vector of the m–polar fuzzy n ˜ˆ numbers ζˆj , where j varies from 1 to n, w j ∈ (0, 1], j=1 w j = 1, ζσ ( j) is the jth biggest m–polar fuzzy number, ζ˜ˆσ ( j) = (nΘ j )ζˆ j , ( j = 1, 2, . . . , n), Θ = n (Θ1 , Θ2 , . . . , Θn ) is the weight vector, with Θ j ∈ [0, 1], j=1 Θ j = 1 and n serves as the balancing coefficient. Note that if w = ( n1 , n1 , . . . , n1 ), then m–polar fuzzy Hamacher hybrid average operator degenerates into m–polar fuzzy Hamacher weighted average operator. When Θ = ( n1 , n1 , . . . , n1 ), then m–polar fuzzy Hamacher hybrid average operator degenerates into m–polar fuzzy Hamacher ordered weighted average operator. Therefore, m–polar fuzzy Hamacher hybrid average operator is an extension of the operators, m–polar fuzzy Hamacher weighted average and m–polar fuzzy Hamacher ordered weighted average, which explains the degrees and ordered arrangements of the given m–polar fuzzy values. Theorem 8.25 Let ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. The accumulated value of these m–polar fuzzy numbers using the m– polar fuzzy Hamacher hybrid average operator is also an m–polar fuzzy numbers, which is given by

400

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F H H Aw,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (w j ζ˜ˆσ ( j) ) j=1

w  w  1 + (λ − 1) p1 ◦ ζ˜σ ( j) j − nj=1 1 − p1 ◦ ζ˜σ ( j) j w j w , . . . , n  ˜ + (λ − 1) j=1 1 − p1 ◦ ζ˜σ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j) w j n  w n   ˜ − j=1 1 − pm ◦ ζ˜σ ( j) j j=1 1 + (λ − 1) pm ◦ ζσ ( j) w j w j . n  n  ˜ ˜ + (λ − 1) j=1 1 − pm ◦ ζσ ( j) j=1 1 + (λ − 1) pm ◦ ζσ ( j)



= n



n

j=1



(8.45) Two particular cases of m–polar fuzzy Hamacher hybrid average operator are as follows: • When λ = 1, m–polar fuzzy Hamacher hybrid average operator converted into m–polar fuzzy hybrid averaging operator as below: m F H Aw,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζ˜ˆσ ( j) )

j=1 n n n      w  w  w  = 1− p1 ◦ ζ˜σ ( j) j , 1 − p2 ◦ ζ˜σ ( j) j , . . . , 1 − pm ◦ ζ˜σ ( j) j . j=1

j=1

j=1

(8.46) • When λ = 2, m–polar fuzzy Hamacher hybrid average operator converted into m–polar fuzzy Einstein hybrid averaging operator as below: m F E H Aw,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(w j ζ˜ˆσ ( j) )

j=1

 w j n  w  n ˜ − j=1 1 − p1 ◦ ζ˜σ ( j) j j=1 1 + p1 ◦ ζσ ( j) = n  w j n  w , . . . , ˜ + j=1 1 − p1 ◦ ζ˜σ ( j) j j=1 1 + p1 ◦ ζσ ( j)  w j n  w n  ˜ − j=1 1 − pm ◦ ζ˜σ ( j) j  j=1 1 + pm ◦ ζσ ( j)  w j n  w . n  ˜ + j=1 1 − pm ◦ ζ˜σ ( j) j j=1 1 + pm ◦ ζσ ( j)

(8.47)

Example 8.9 Let ζˆ1 = (0.8, 0.2, 0.6), ζˆ2 = (0.7, 0.4, 0.6) and ζˆ3 = (0.5, 0.6, 0.7) be 3–polar fuzzy numbers with an associated-weight vector w = (0.4, 0.4, 0.2) for these 3–polar fuzzy numbers and a weight vector Θ = (0.3, 0.2, 0.5). Then, from Definition 8.11, for λ = 3

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

401

  nw  nw    1 − 1 − p ◦ ζ nw1 2 − 1 − p ◦ ζ nw2 1 + (λ − 1) p1 ◦ ζ1 1 + (λ − 1) p2 ◦ ζ2 1 1 2 2 ,  nw nw ,  nw    1 + (λ − 1) 1 − p ◦ ζ 1 2 + (λ − 1) 1 − p ◦ ζ nw2 1 + (λ − 1) p1 ◦ ζ1 1 + (λ − 1) p2 ◦ ζ2 1 1 2 2  nw nw   3 − 1− p ◦ζ 3 1 + (λ − 1) p3 ◦ ζ3 3 3 ,  nw   3 + (λ − 1) 1 − p ◦ ζ nw3 1 + (λ − 1) p3 ◦ ζ3 3 3  1 + 2(0.8)3(0.3) − 1 − 0.83(0.3) 1 + 2(0.2)3(0.2) − 1 − 0.23(0.2) =  3(0.3) 3(0.2)  3(0.3) ,   3(0.2) , 1 + 2(0.8) 1 + 2(0.2) + 2 1 − 0.8 + 2 1 − 0.2  3(0.5)  3(0.5)  1 + 2(0.6) − 1 − 0.6  3(0.5)  3(0.5) , 1 + 2(0.6) + 2 1 − 0.6

ζ˜ˆ1 =



= (0.7512, 0.1174, 0.7986).

Similarly,  1 + 2(0.7)3(0.3) − 1 − 0.73(0.3) 1 + 2(0.4)3(0.2) − 1 − 0.43(0.2) ˜ζˆ = 2  3(0.3) 3(0.2)  3(0.3) ,   3(0.2) , 1 + 2(0.7) 1 + 2(0.4) + 2 1 − 0.7 + 2 1 − 0.4  3(0.5)  3(0.5)  1 + 2(0.6) − 1 − 0.6  3(0.5)  3(0.5) , 1 + 2(0.6) + 2 1 − 0.6 = (0.6470, 0.2373, 0.7986), and  1 + 2(0.5)3(0.3) − 1 − 0.53(0.3) 1 + 2(0.6)3(0.2) − 1 − 0.63(0.2) ˜ζˆ = 3  3(0.3) 3(0.2)  3(0.3) ,   3(0.2) , 1 + 2(0.5) 1 + 2(0.6) + 2 1 − 0.5 + 2 1 − 0.6  3(0.5)  3(0.5)  1 + 2(0.7) − 1 − 0.7  3(0.5) ,  3(0.5) + 2 1 − 0.7 1 + 2(0.7) = (0.4528, 0.3725, 0.8782). Then, scores and aggregated values of m–polar fuzzy numbers for λ = 3 can be computed as below: 0.7512 + 0.1174 + 0.7986 = 0.5557, 3 0.6470 + 0.2373 + 0.7986 S(ζ˜ˆ2 ) = = 0.5609, 3 0.4528 + 0.3725 + 0.8782 S(ζ˜ˆ3 ) = = 0.5678. 3 S(ζ˜ˆ1 ) =

Since, S(ζ˜ˆ3 ) > S(ζ˜ˆ2 ) > S(ζ˜ˆ1 ), thus ζ˜ˆσ (1) = ζ˜ˆ3 = (0.4528, 0.3725, 0.8782), ζ˜ˆσ (3) = ζ˜ˆ1 = (0.7512, 0.1174, 0.7986).

ζ˜ˆσ (2) = ζ˜ˆ2 = (0.6470, 0.2373, 0.7986),

402

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Then, from Definition 8.11, m F H H Aw,Θ (ζˆ1 , ζˆ2 , ζˆ3 ) =

3 

(w j ζˆσ ( j) ) j=1  w j n  w n ˜  − j=1 1 − p1 ◦ ζ˜σ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j) =   w j  w , . . . , n n ˜ + (λ − 1) j=1 1 − p1 ◦ ζ˜σ ( j) j j=1 1 + (λ − 1) p1 ◦ ζσ ( j)     n w j n w ˜  − j=1 1 − pm ◦ ζ˜σ ( j) j j=1 1 + (λ − 1) pm ◦ ζσ ( j)  w j  w n n ˜ ˜ + (λ − 1) j=1 1 − pm ◦ ζσ ( j) j j=1 1 + (λ − 1) pm ◦ ζσ ( j)  1 + (2)0.45280.4 × 1 + (2)0.64700.4 × 1 + (2)0.75120.2 − 1 − 0.45280.4 × 1 − 0.64700.4 × 1 − 0.75120.2 =             , 1 + (2)0.4528 0.4 × 1 + (2)0.6470 0.4 × 1 + (2)0.7512 0.2 + (2) 1 − 0.4528 0.4 × 1 − 0.6470 0.4 × 1 − 0.7512 0.2  0.4  0.4  0.2  0.4  0.4  0.2 1 + (2)0.3725 × 1 + (2)0.2373 × 1 + (2)0.1174 − 1 − 0.3725 × 1 − 0.2373 × 1 − 0.1174             , 1 + (2)0.3725 0.4 × 1 + (2)0.2373 0.4 × 1 + (2)0.1174 0.2 + (2) 1 − 0.3725 0.4 × 1 − 0.2373 0.4 × 1 − 0.1174 0.2             1 + (2)0.8782 0.4 × 1 + (2)0.7986 0.4 × 1 + (2)0.7986 0.2 − 1 − 0.8782 0.4 × 1 − 0.7986 0.4 × 1 − 0.7986 0.2   0.4  0.4  0.2  0.4  0.4    1 + (2)0.8782 × 1 + (2)0.7986 × 1 + (2)0.7986 + (2) 1 − 0.8782 × 1 − 0.7986 × 1 − 0.7986 0.2

= (0.6014, 0.2676, 0.8347).

8.3.2 m–Polar Fuzzy Hamacher Geometric Aggregation Operators In this section, different types of Hamacher geometric aggregation operators with m– polar fuzzy numbers, namely, m–polar fuzzy Hamacher weighted geometric operator, m–polar fuzzy Hamacher ordered weighted geometric operator and m–polar fuzzy Hamacher hybrid weighted geometric operator are presented. Definition 8.12 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of m–polar fuzzy numbers where ‘ j’ varies from 1 to n. An m–polar fuzzy Hamacher weighted geometric operator of is a function mFHWG : ζˆ n → ζˆ , which is defined as follows: m F H W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(ζˆ j )Θ j

(8.48)

j=1

where Θ = (Θ1 , Θ2 , . . . , Θn ) denotes the weight vector, with Θ j ∈ (0, 1], = 1.

n j=1

Θj

Theorem 8.26 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. The accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Hamacher weighted geometric operator is also an m–polar fuzzy number, which is given as

8.3 m–Polar Fuzzy Hamacher Aggregation Operators m F H W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

403

n  (ζˆ j )Θ j , j=1

Θ p1 ◦ ζ j j Θ j  Θ , . . . ,  + (λ − 1) nj=1 p1 ◦ ζ j j j=1 1 + (λ − 1)(1 − p1 ◦ ζ j )  Θ   λ nj=1 pm ◦ ζ j j Θ j Θ j . (8.49) n  n  + (λ − 1) j=1 pm ◦ ζ j j=1 1 + (λ − 1)(1 − pm ◦ ζ j )



=  n



λ

n



j=1



Proof It can be easily followed using mathematical induction.

Example 8.10 Let ζˆ1 = (0.5, 0.7, 0.4), ζˆ2 = (0.8, 0.5, 0.4) and ζˆ3 = (0.3, 0.4, 0.5) be 3–polar fuzzy numbers with a weight vector Θ = (0.3, 0.6, 0.1)T for these 3– polar fuzzy numbers. Then, for λ = 3, m F H W G Θ (ζˆ1 , ζˆ2 , ζˆ3 ) =

3  (ζˆ j )Θ j , j=1

Θ p1 ◦ ζ j j   Θ , . . . , Θ j  + (λ − 1) 3j=1 p1 ◦ ζ j j j=1 1 + (λ − 1)(1 − p1 ◦ ζ j )  Θ   λ 3j=1 p3 ◦ ζ j j Θ j Θ j 3  3  + (λ − 1) j=1 p3 ◦ ζ j j=1 1 + (λ − 1)(1 − p3 ◦ ζ j )    3(0.5)0.3 (0.8)0.6 (0.3)0.1 =  0.3  , 0.6  0.1  1 + 2(1 − 0.5) × 1 + 2(1 − 0.8) × 1 + 2(1 − 0.3) + (2) (0.5)0.3 (0.8)0.6 (0.3)0.1   3(0.7)0.3 (0.5)0.6 (0.4)0.1  0.3  , 0.6  0.1  1 + 2(1 − 0.7) × 1 + 2(1 − 0.5) × 1 + 2(1 − 0.4) + (2) (0.7)0.3 (0.5)0.6 (0.4)0.1    3(0.4)0.3 (0.4)0.6 (0.5)0.1  0.3   0.6  0.1  1 + 2(1 − 0.4) × 1 + 2(1 − 0.4) × 1 + 2(1 − 0.5) + (2) (0.4)0.3 (0.4)0.6 (0.5)0.1 

=  3

λ

3



j=1

= (0.6507, 0.5463, 0.4094).

It can be easily shown that the m–polar fuzzy Hamacher weighted geometric operator holds the following notions: Theorem 8.27 (Idempotency Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. If all these m–polar fuzzy numbers are same, i.e., ζˆ j = ζˆ , ∀ j = 1, 2, . . . , n, then m F H W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ .

(8.50)

Theorem 8.28 (Boundedness Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family

 of ‘n’ m–polar fuzzy numbers, ζˆ − = nj=1 (ζ j ) and ζˆ + = nj=1 (ζ j ), then ζˆ − ≤ m F H W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ ζˆ + .

(8.51)

Theorem 8.29 (Monotonicity Property) Let ζˆ j and ζˆj be two families of m–polar fuzzy numbers. If ζˆ j ≤ ζˆj , then

404

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F H W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ m F H W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.52)

Two particular cases of m–polar fuzzy Hamacher weighted geometric operator are as follows: • When λ = 1, m–polar fuzzy Hamacher weighted geometric operator converted into m–polar fuzzy weighted geometric operator as below: m F W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(ζˆ j )Θ j

j=1

=

n  

n n  Θ   Θ  Θ  p1 ◦ ζ j j , p2 ◦ ζ j j , . . . , pm ◦ ζ j j .

j=1

j=1

(8.53)

j=1

• When λ = 2, m–polar fuzzy Hamacher weighted geometric operator reduces into m–polar fuzzy Einstein weighted geometric operator as below: m F E W G Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =  =  n j=1

n 

(ζˆ j )Θ j

j=1

 Θ  Θ   2 nj=1 pm ◦ ζ j j p1 ◦ ζ j j Θ j , . . . , n  Θ j . Θ j n  Θ j n  2 − p1 ◦ ζ j ) + j=1 p1 ◦ ζ j + j=1 pm ◦ ζ j j=1 2 − pm ◦ ζ j )



2

n

j=1

(8.54) Definition 8.13 Let ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ), be a family of m–polar fuzzy numbers. An m–polar fuzzy Hamacher weighted ordered geometric operator is a mapping mFHOWG : ζˆ n → ζˆ with weight vector w = (w1 , w2 , . . . , wn ), n for which w j > 0 and j=1 w j = 1. Thus,

m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (w j ζˆσ ( j) )

(8.55)

j=1

where (σ (1), σ (2), . . . , σ (n)) is the permutation of the indices ‘ j’ varies from 1 to n, for which ζˆσ ( j−1) ≥ ζˆσ ( j) , ∀ j = 1, 2, . . . , n. Theorem 8.30 Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. The accumulated value of these m–polar fuzzy numbers using the m– polar fuzzy Hamacher weighted ordered geometric operator is also an m–polar fuzzy numbers, which is given by

8.3 m–Polar Fuzzy Hamacher Aggregation Operators n 

m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

405

(ζˆσ ( j) )w j

j=1

w p1 ◦ ζσ ( j) j  w , . . . , w j  + (λ − 1) nj=1 p1 ◦ ζσ ( j) j j=1 1 + (λ − 1)(1 − p1 ◦ ζσ ( j) )    w  λ nj=1 pm ◦ ζσ ( j) j w j w j . n  n  + (λ − 1) j=1 pm ◦ ζσ ( j) j=1 1 + (λ − 1)(1 − pm ◦ ζσ ( j) )

 = n



λ

n



j=1

(8.56) Example 8.11 Let ζˆ1 = (0.5, 0.6, 0.8), ζˆ2 = (0.3, 0.5, 0.6) and ζˆ3 = (0.6, 0.7, 0.8) be 3–polar fuzzy numbers with a weight vector w = (0.2, 0.5, 0.3) for these 3–polar fuzzy numbers. Then, scores and aggregated values of m–polar fuzzy numbers for λ = 3 can be computed as below: 0.5 + 0.6 + 0.8 = 0.6333, 3 0.6 + 0.7 + 0.8 S(ζˆ3 ) = = 0.7. 3 S(ζˆ1 ) =

S(ζˆ2 ) =

0.3 + 0.5 + 0.6 = 0.4667, 3

Since, S(ζˆ3 ) > S(ζˆ1 ) > S(ζˆ2 ), thus ζˆσ (1) = ζˆ3 = (0.6, 0.7, 0.8),

ζˆσ (2) = ζˆ1 = (0.5, 0.6, 0.8),

ζˆσ (3) = ζˆ2 = (0.3, 0.5, 0.6). Then, from Definition 8.13, m F H O W G Θ (ζˆ1 , ζˆ2 , ζˆ3 ) =

3  (ζˆσ ( j) )Θ j , j=1

Θ p1 ◦ ζσ ( j) j =  Θ j   Θ , . . . ,  3 + (λ − 1) 3j=1 p1 ◦ ζσ ( j) j j=1 1 + (λ − 1)(1 − p1 ◦ ζσ ( j) )  Θ   λ 3j=1 p3 ◦ ζσ ( j) j Θ j Θ j 3  3  + (λ − 1) j=1 p3 ◦ ζσ ( j) j=1 1 + (λ − 1)(1 − p3 ◦ ζσ ( j) )    3 (0.6)0.2 (0.5)0.5 (0.3)0.3 =  0.5  0.3  0.2  , × 1 + 2(1 − 0.5) × 1 + 2(1 − 0.3) + 2 (0.6)0.2 (0.5)0.5 (0.3)0.3 1 + 2(1 − 0.6)   3 (0.7)0.2 (0.6)0.5 (0.5)0.3  0.2  , 0.5  0.3  1 + 2(1 − 0.7) × 1 + 2(1 − 0.6) × 1 + 2(1 − 0.5) + 2 (0.7)0.2 (0.6)0.5 (0.5)0.3    3 (0.8)0.2 (0.8)0.5 (0.6)0.3 0.5  0.3   0.2   0.2 0.5 0.3 × 1 + 2(1 − 0.8) × 1 + 2(1 − 0.6) + 2 (0.8) (0.8) (0.6) 1 + 2(1 − 0.8) 

λ

3



j=1

= (0.4512, 0.5885, 0.7394).

Two particular cases of m–polar fuzzy Hamacher weighted ordered geometric operator are as follows:

406

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

• When λ = 1, m–polar fuzzy Hamacher weighted ordered geometric operator reduces into m–polar fuzzy ordered weighted geometric operator as m F O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(ζˆσ ( j) )w j

j=1

=

n   j=1

p1 ◦ ζσ ( j)

w j

,

n 



p2 ◦ ζσ ( j)

w j

,...,

j=1

n  

pm ◦ ζσ ( j)

w j  .

(8.57)

j=1

• When λ = 2, m–polar fuzzy Hamacher weighted ordered geometric operator reduces into m–polar fuzzy Einstein ordered weighted geometric operator as below: m F E O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n 

(ζˆ j )w j

j=1

w p1 ◦ ζσ ( j) j w , . . . , w j n  = n  + j=1 p1 ◦ ζσ ( j) j j=1 2 − p1 ◦ ζσ ( j) ) w    2 nj=1 pm ◦ ζσ ( j) j w j n  w j . n  + j=1 pm ◦ ζσ ( j) j=1 2 − pm ◦ ζσ ( j) ) (8.58) 

2

n



j=1

Theorem 8.31 (Idempotency Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of ‘n’ m–polar fuzzy numbers. If all these m–polar fuzzy numbers are same, in other words, ζˆ j = ζˆ , then m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) = ζˆ .

(8.59)

Theorem 8.32 (Boundedness Property) Let ζˆ j = ( p1 ◦ ζ j , . . . , pm ◦ ζ j ) be a family

 of m–polar fuzzy ‘n’ numbers, ζˆ − = nj=1 (ζˆ1 , ζˆ2 , . . . , ζˆn ) and ζˆ + = nj=1 (ζˆ1 , ζˆ2 , . . . , ζˆn ), then ζˆ − ≤ m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ ζˆ + .

(8.60)

Theorem 8.33 (Monotonicity Property) Let ζˆ j and ζˆj , ( j = 1, 2, . . . , n) be two families of m–polar fuzzy numbers. If ζˆ j ≤ ζˆj , then m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) ≤ m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.61)

Theorem 8.34 (Commutativity Property) Let ζˆ j and ζˆj be two families of m–polar fuzzy numbers. If ζˆ j ≤ ζˆj , then

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

407

m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ) = m F H O W G w (ζˆ1 , ζˆ2 , . . . , ζˆn ).

(8.62)

where ζˆj is any permutation of ζˆj , j = 1, 2, . . . , n. In Definitions 8.12 and 8.13, we observe that m–polar fuzzy Hamacher weighted geometric operator and m–polar fuzzy Hamacher weighted ordered geometric operator weight m–polar fuzzy numbers and their ordered arrangement, respectively. Another operator, namely, m–polar fuzzy Hamacher hybrid averaging operator is presented, which combines the features of these operators. Definition 8.14 Let ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ) be a family of m–polar fuzzy numbers where ‘ j’ varies from 1 to n. An m–polar fuzzy Hamacher hybrid geometric operator is given as below: m F H H G w,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (ζ˜ˆσ ( j) )w j ,

(8.63)

j=1

where (σ (1), σ (2), . . . , σ (n)) is the permutation of (1, 2, . . . , n), for which ζˆσ ( j−1) ≥ ζˆσ ( j) , ∀ j = 1, 2, . . . , n and w = (w1 , w2 , . . . , wn ) is the associated weight vec n tor of the m–polar fuzzy numbers (ζˆ1 , ζˆ2 , . . . , ζˆn ), w j ∈ (0, 1], j=1 w j = 1. ˜ ˜ζˆ 1, 2, . . . , n), σ ( j) is the jth biggest m–polar fuzzy number, ζˆσ ( j) = (nΘ j )ζˆj , ( j = n Θ = (Θ1 , Θ2 , . . . , Θn ) represents the weight vector, with Θ j > 0, j=1 Θ j = 1 and n serves as the balancing coefficient. Note that if w = ( n1 , n1 , . . . , n1 ), then m–polar fuzzy Hamacher hybrid geometric operator degenerates into m–polar fuzzy Hamacher weighted geometric operator. When Θ = ( n1 , n1 , . . . , n1 ), then m–polar fuzzy Hamacher hybrid geometric operator degenerates into m–polar fuzzy Hamacher weighted ordered geometric operator. Therefore, m–polar fuzzy Hamacher hybrid geometric operator is an extension of the operators, m–polar fuzzy Hamacher weighted geometric operator and m–polar fuzzy Hamacher weighted ordered geometric operator, which explain the degrees and ordered arrangements of the given m–polar fuzzy values. The following theorem can be easily proved by using mathematical induction. Theorem 8.35 Let ζˆ j = ( p1 ◦ ζ j , p2 ◦ ζ j , . . . , pm ◦ ζ j ), be a family of m–polar fuzzy numbers. The accumulated value of these m–polar fuzzy numbers using the m–polar fuzzy Hamacher hybrid geometric operator is also an m–polar fuzzy number, which is given by m F H H G w,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (ζ˜ˆσ ( j) )w j j=1

 w  λ nj=1 p1 ◦ ζ˜σ ( j) j w j   w , . . . ,  ˜ + (λ − 1) nj=1 p1 ◦ ζ˜σ ( j) j j=1 1 + (λ − 1)(1 − p1 ◦ ζσ ( j) )  w   λ nj=1 pm ◦ ζ˜σ ( j) j w j w j . n  n  ˜ ˜ ◦ ζ ) + (λ − 1) ◦ ζ 1 + (λ − 1)(1 − p p m m σ ( j) σ ( j) j=1 j=1



= n

(8.64)

408

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Example 8.12 Let ζˆ1 = (0.5, 0.4, 0.7), ζˆ2 = (0.8, 0.5, 0.7) and ζˆ3 = (0.6, 0.7, 0.8) be 3–polar fuzzy numbers with an associated weight vector w = (0.3, 0.4, 0.3) for these 3–polar fuzzy numbers and a weight vector Θ = (0.3, 0.5, 0.2)T . Then, from Definition 8.11, for λ = 3 nΘ   λ p 1 ◦ ζ1 1 ˜ζˆ =  nΘ , nΘ  1 1 + (λ − 1)(1 − p1 ◦ ζ1 ) 1 + (λ − 1) p1 ◦ ζ1 1 nΘ  λ p 2 ◦ ζ1 2 nΘ   nΘ , 1 + (λ − 1)(1 − p2 ◦ ζ1 ) 2 + (λ − 1) p2 ◦ ζ1 2 nΘ   λ p 3 ◦ ζ1 3  nΘ3 , nΘ3  1 + (λ − 1)(1 − p3 ◦ ζ1 ) + (λ − 1) p3 ◦ ζ1  3(0.3) 3(0.5) 3(0.8)3(0.5) =  3(0.3) 3(0.5)  3(0.3) ,   3(0.5) , 1 + 2(1 − 0.5) 1 + 2(1 − 0.8) + 2 0.5 + 2 0.8  3(0.6)3(0.2)  3(0.2)  3(0.2) , 1 + 2(1 − 0.6) + 2 0.6 = (0.5472, 0.6952, 0.7627). Similarly,  3(0.4)3(0.3) 3(0.5)3(0.5) ζ˜ˆ2 =  3(0.3) 3(0.5)  3(0.3) ,   3(0.5) , 1 + 2(1 − 0.4) 1 + 2(1 − 0.5) + 2 0.4 + 2 0.5  3(0.7)3(0.2)  3(0.2)  3(0.2) , 1 + 2(1 − 0.7) + 2 0.7 = (0.4519, 0.3000, 0.8237), and  3(0.7)3(0.3) 3(0.7)3(0.5) , ζ˜ˆ3 =   3(0.3)   3(0.5) , 3(0.3) 3(0.5) + 2 0.7 + 2 0.7 1 + 2(1 − 0.7) 1 + 2(1 − 0.7)  3(0.8)3(0.2)  3(0.2)  3(0.2) , 1 + 2(1 − 0.8) + 2 0.8 = (0.7309, 0.5499, 0.8826). Then, scores and aggregated values of 3–polar fuzzy numbers for λ = 3 can be computed as below:

8.3 m–Polar Fuzzy Hamacher Aggregation Operators

409

0.5472 + 0.6952 + 0.7627 = 0.6684, 3 0.4519 + 0.3000 + 0.8237 S(ζ˜ˆ2 ) = = 0.5252, 3 0.7309 + 0.5499 + 0.8826 = 0.7211. S(ζ˜ˆ3 ) = 3 S(ζ˜ˆ1 ) =

Since, S(ζ˜ˆ3 ) > S(ζ˜ˆ1 ) > S(ζ˜ˆ2 ), thus ζ˜ˆσ (1) = ζ˜ˆ3 = (0.7309, 0.5499, 0.8826), ζ˜ˆσ (3) = ζ˜ˆ1 = (0.4519, 0.3000, 0.8237).

ζ˜ˆσ (2) = ζ˜ˆ2 = (0.5472, 0.6952, 0.7627),

Then, from Definition 8.13, m F H O W G Θ (ζˆ1 , ζˆ2 , ζˆ3 ) =

3 

Θ (ζˆσ ( j) ) j ,

j=1



3



Θj j=1 p1 ◦ ζσ ( j)   Θ , . . . , Θ  j + (λ − 1) 3j=1 p1 ◦ ζσ ( j) j j=1 1 + (λ − 1)(1 − p1 ◦ ζσ ( j) )

 =  3

λ

3





Θj  j=1 p3 ◦ ζσ ( j)  Θ Θ  3  j + (λ − 1) 3 j 1 + (λ − 1)(1 − p p ◦ ζ ) ◦ ζ 3 σ ( j) σ ( j) j=1 j=1 3

λ

   3 (0.7309)0.3 (0.5472)0.4 (0.4519)0.3 =  0.3  , 0.4  0.3  1 + 2(1 − 0.7309) × 1 + 2(1 − 0.5472) × 1 + 2(1 − 0.4519) + 2 (0.7309)0.3 (0.5472)0.4 (0.4519)0.3   3 (0.5499)0.3 (0.6952)0.4 (0.3000)0.3 0.4  0.3   0.3  , × 1 + 2(1 − 0.6952) × 1 + 2(1 − 0.3000) + 2 (0.5499)0.3 (0.6952)0.4 (0.3000)0.3 1 + 2(1 − 0.5499)    3 (0.8826)0.3 (0.7627)0.4 (0.8237)0.3  0.3   0.4  0.3  1 + 2(1 − 0.8826) × 1 + 2(1 − 0.7627) × 1 + 2(1 − 0.8237) + 2 (0.8826)0.3 (0.7627)0.4 (0.8237)0.3 = (0.5700, 0.5184, 0.8173).

Two particular cases of m–polar fuzzy Hamacher hybrid geometric operator are as follows: • When λ = 1, m–polar fuzzy Hamacher hybrid geometric operator is converted into m–polar fuzzy hybrid geometric operator as m F H G w,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (ζ˜ˆσ ( j) )w j j=1

=

n  j=1



p1 ◦ ζ˜σ ( j)

w j

,

n  j=1



p2 ◦ ζ˜σ ( j)

w j

,...,

n  

pm ◦ ζ˜σ ( j)

w j  .

(8.65)

j=1

• When λ = 2, m–polar fuzzy Hamacher hybrid geometric operator is converted into m–polar fuzzy Einstein hybrid geometric operator as

410

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

m F E H G w,Θ (ζˆ1 , ζˆ2 , . . . , ζˆn ) =

n  (ζ˜ˆσ ( j) )w j

n



j=1

w 2 j=1 p1 ◦ ζ˜σ ( j) j = n  w , . . . , w j n  ˜ + j=1 p1 ◦ ζ˜σ ( j) j j=1 2 − p1 ◦ ζσ ( j) ) w    2 nj=1 pm ◦ ζ˜σ ( j) j w j . w j n  n  ˜ + j=1 pm ◦ ζ˜σ ( j) j=1 2 − pm ◦ ζσ ( j) ) 

(8.66)

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators To solve MCDM problems containing m–polar fuzzy data, the m–polar fuzzy Dombi and Hamacher aggregation operators are applied. The following notions are utilized to tackle the MCDM situations having m–polar fuzzy information. Suppose that U = is the universe of attributes. {Y1 , Y2 , . . . , Yk } is a universal set and {T1 , T2 , . . . , Tn } Assume Θ = {Θ1 , Θ2 , . . . , Θn } is a weight-vector with nj=1 Θ j = 1, Θ j ∈ (0, 1],   for all j = 1, . . . , n. Consider Sˆ = (ˆsi j )k×n = p1 ◦ ζi j , p2 ◦ ζi j , . . . , pm ◦ ζi j k×n is an m–polar fuzzy decision-matrix, which represents the membership values evaluated by the experts. Algorithmic methods are presented to handle MCDM problems by m–polar fuzzy Dombi and Hamacher weighted averaging (or m–polar fuzzy Dombi and Hamacher weighted geometric) operators. Algorithm 8.4.1 Selection of a suitable alternative ˆ as an m–polar fuzzy decision-matrix having k objects and n attributes. 1. Input S, Θ = (Θ1 , Θ2 , . . . , Θn ), the vector having weights. 2. Apply the m–polar fuzzy Dombi weighted averaging operators to aggregate the data in m–polar fuzzy decision-matrix Sˆ and calculate the preference values sˆi , where ‘i’ varies from 1 to k for the m–polar fuzzy numbers ζi . sˆi = m F DW AΘ (ζˆi1 , ζˆi2 , . . . , ζˆin ) =

n  (Θ j ζˆi j ) j=1

1 1 , . . . , 1 − = 1−  p ◦ζ k 1/k  p ◦ζ k 1/k .   n n 1 ij m ij 1+ Θj 1+ Θj 1 − p1 ◦ ζi j 1 − pm ◦ ζi j j=1 j=1 

When m–polar fuzzy Dombi weighted geometric operators are used, then

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators sˆi = m F DW G Θ (ζˆi1 , ζˆi2 , . . . , ζˆin ) =

n 

411

(ζˆi j )Θ j ,

j=1

 = 1− 1+

1 1 .  1 − p ◦ ζ k 1/k , . . . , 1 −    n 1 − pm ◦ ζi j k 1/k 1 ij Θj 1+ Θj p1 ◦ ζi j pm ◦ ζi j j=1 j=1

 n

3. Compute the score values S(ˆsi ), i = 1, 2, . . . , k. 4. Rank the objects Yi , i = 1, 2, . . . , k with respect to their scores S(ˆsi ). When the scores of two objects are equal, then apply the accuracy function to find the order of alternatives. 5. The object containing maximum score value in last step will be the decision.

Algorithm 8.4.2 Selection of a suitable alternative 1. Input U , as the universe with k alternatives, T , as the set having n attributes, and Θ = (Θ1 , Θ2 , . . . , Θn ), the weight vector for attributes. 2. Use the m–polar fuzzy Hamacher weighted average operator to evaluate the ˆ determine the preference values information in m–polar fuzzy decision matrix S, sˆi , i = 1, 2, . . . , k of the object Ai . eˆi = m F H W AΘ (ζˆi1 , ζˆi2 , . . . , ζˆin ) = n

Θ j

(Θ j ζˆi j )

j=1

Θ   1 + (λ − 1) p1 ◦ ζi j − nj=1 1 − p1 ◦ ζi j j = n  Θ j Θ , . . . ,   + (λ − 1) nj=1 1 − p1 ◦ ζi j j j=1 1 + (λ − 1) p1 ◦ ζi j Θ j n  Θ n   − j=1 1 − pm ◦ ζi j j j=1 1 + (λ − 1) pm ◦ ζi j Θ j Θ j . n  n  + (λ − 1) j=1 1 − pm ◦ ζi j j=1 1 + (λ − 1) pm ◦ ζi j 



n 

j=1

If an m–polar fuzzy Hamacher weighted geometric operator is applied, then eˆi = m F H W G Θ (ζˆi1 , ζˆi2 , . . . , ζˆin ) =

(ζˆi j )Θ j ,

j=1

Θ p1 ◦ ζi j j = n  Θ , . . . , Θ j   + (λ − 1) nj=1 p1 ◦ ζi j j j=1 1 + (λ − 1)(1 − p1 ◦ ζi j ) Θ    λ nj=1 pm ◦ ζi j j Θ j Θ . n    + (λ − 1) nj=1 pm ◦ ζi j j j=1 1 + (λ − 1)(1 − pm ◦ ζi j ) 

λ

n



n 

j=1

3. Compute the scores S(eˆi ). 4. Rank the objects based on their score values S(eˆi ), i = 1, 2, . . . , k. If two alternatives have same score, then use the accuracy function to rank the objects. 5. The alternative having highest score in the step 4 will be the decision alternative.

412

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Table 8.1 3–polar fuzzy decision matrix . T1 T2 Y1 Y2 Y3 Y4 Y5

(0.7, 0.6, 0.3) (0.9, 0.6, 0.5) (0.4, 0.7, 0.3) (0.5, 0.6, 0.3) (0.9, 0.7, 0.6)

(0.5, 0.7, 0.2) (0.8, 0.5, 0.3) (0.5, 0.4, 0.3) (0.8, 0.6, 0.5) (0.8, 0.4, 0.3)

T3

T4

(0.6, 0.3, 0.7) (0.4, 0.5, 0.8) (0.6, 0.8, 0.4) (0.7, 0.8, 0.2) (0.6, 0.5, 0.7)

(0.9, 0.5, 0.4) (0.5, 0.8, 0.5) (0.6, 0.3, 0.8) (0.9, 0.3, 0.7) (0.4, 0.6, 0.5)

8.4.1 Agriculture Land Selection Agriculture is an essential part of Pakistan’s economic system. This area directly supports the population of the country and accounts for 26% of gross domestic product. The leading agricultural crops include sugarcane, wheat, rice, cotton, vegetables and fruits. A business man wants to invest in agriculture sector and searching for an appropriate land. The options in his brain are Y1 , Y2 , . . . , Y5 . He consults to an expert to get his suggestion about the alternatives based on the following desired parameters: S1 S2 S3 S4

denotes the ‘Location’, denotes the ‘Climate’, denotes the ‘Fertility’, denotes the ‘Price’,

Each parameter has been characterized into three parts to construct a 3–polar fuzzy number. • The ‘Location’ includes near to market, near to water channel, transport availability. • The ‘Climate’ includes temperature, pollution level, humidity level. • The ‘Fertility’ includes soil PH, level of nutrients, water retention capacity of land. • The ‘Price’ includes low, medium, high. The 3-polar fuzzy decision matrix is displayed by Table 8.1. According to the businessman, expert assigns weights to parameters as follows: Θ1 = 0.35, Θ2 = 0.25, Θ3 = 0.30, and Θ4 = 0.10. Clearly, 3j=1 Θ j = 1. To compute the most suitable land regarding agriculture, two operators namely, m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi weighted geometric are used respectively: 1. For k = 3, by applying the m–polar fuzzy Dombi weighted averaging operator, we calculate the values sˆi of the lands Yi , i = 1, 2, . . . , 5 regarding agriculture.

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators

sˆ1 = (0.8107, 0.6224, 0.6109),

sˆ2 = (0.8662, 0.6679, 0.7297),

sˆ3 = (0.5443, 0.7418, 0.6515), sˆ5 = (0.8663, 0.6334, 0.6327).

sˆ4 = (0.8192, 0.7347, 0.5366),

413

2. Find the score values S(ˆsi ) of 3–polar fuzzy numbers sˆi , (i = 1, 2, . . . , 5) of the lands Yi : S(ˆs1 ) = 0.6814, S(ˆs2 ) = 0.7546, S(ˆs3 ) = 0.6459, S(ˆs4 ) = 0.6968, S(ˆs5 ) = 0.7108. 3. Rank the lands using scores S(si ), (i = 1, 2, . . . , 5) obtained from the preference values in the form of 3–polar fuzzy numbers: Y2 > Y5 > Y4 > Y1 > Y3 . 4. Y2 has a high score value, so, it is the best land for agriculture.

In a Similar way, apply an m–polar fuzzy Dombi weighted geometric operator to find an appropriate land. 1. Take k = 3. Apply an m–polar fuzzy Dombi weighted geometric operator to determine the values sˆi of the lands Yi . sˆ1 = (0.4171, 0.6142, 0.7334), sˆ3 = (0.5364, 0.5633, 0.6723),

sˆ2 = (0.5092, 0.4648, 0.6058), sˆ4 = (0.4196, 0.5311, 0.7423),

sˆ5 = (0.4303, 0.5153, 0.6007). 2. Determine the score values S(ˆsi ) of 3–polar fuzzy numbers sˆi of the lands Yi : S(ˆs1 ) = 0.5882, S(ˆs2 ) = 0.5266, S(ˆs3 ) = 0.5907, S(ˆs4 ) = 0.5643, S(ˆs5 ) = 0.5154. 3. Rank the lands using scores S(ˆsi ), (i = 1, 2, . . . , 5) obtained from the preference values in the form of 3–polar fuzzy numbers: Y3 > Y1 > Y4 > Y2 > Y5 . 4. Y3 has high score, so, it is the best land for agriculture.

8.4.2 Performance Evaluation of Commercial Banks Commercial bank is one of the largest essential economic institutions. It can pull in money related streams, offering credit and different monetary administrations. These activities vitally affect national monetary improvements. Hence, commercial banks ought to be assessed by the modern and reliable procedures to rank commercial banks in the financial framework. This research establishes a MCDM model that uses m–polar fuzzy Dombi weighted averaging operator, m–polar fuzzy Dombi weighted geometric operator and m–polar fuzzy ELECTRE-I method under a set of criteria and rank commercial banks. The board of specialists will assess each bank under chosen criteria. After a primer evaluation, six banks {B1 , B2 , B3 , B4 , B5 , B6 } are assessed and ranked to pick the best bank. The banks are evaluated on the basis of four parameters.

414

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Table 8.2 4–polar fuzzy decision matrix . T1 T2 B1 B2 B3 B4 B5 B6

S1 S2 S3 S4

(0.7, 0.6, 0.7, 0.7) (0.9, 0.7, 0.8, 0.7) (0.4, 0.8, 0.5, 0.6) (0.8, 0.7, 0.6, 0.5) (0.7, 0.7, 0.6, 0.7) (0.8, 0.8, 0.7, 0.6)

(0.5, 0.7, 0.6, 0.7) (0.8, 0.9, 0.8, 0.7) (0.5, 0.6, 0.6, 0.4) (0.7, 0.5, 0.4, 0.6) (0.5, 0.6, 0.4, 0.4) (0.7, 0.6, 0.6, 0.5)

T3

T4

(0.5, 0.3, 0.5, 0.5) (0.8, 0.7, 0.7, 0.6) (0.5, 0.6, 0.5 0.4) (0.5, 0.5, 0.6, 0.5) (0.7, 0.5, 0.5, 0.4) (0.8, 0.5, 0.5, 0.4)

(0.5, 0.4, 0.4, 0.6) (0.7, 0.7, 0.8, 0.7) (0.6, 0.6, 0.5, 0.3) (0.4, 0.5, 0.6, 0.5) (0.5, 0.5, 0.4, 0.5) (0.6, 0.6, 0.7, 0.6)

denotes the ‘Net Income’, denotes the ‘Customer Service’, denotes the ‘Non-Financial Performance’, denotes the ‘Potential Attractiveness’,

Each parameter has been characterized into four parts to form a 4–polar fuzzy number. • The ‘Net Income’ includes total equity, operating income, total assets, net interest income. • The ‘Customer Service’ includes accessibility for customers, the evaluation of internet page, the number of new services, the number of new products. • The ‘Non-Financial Performance’ includes support from main stake holders, bank management, employee stability, ownership structure. • The ‘Potential Attractiveness’ includes location, involving environment, strategic dimension, external and internal characteristics. The 4–polar fuzzy decision matrix is represented by Table 8.2. The expert assigns weights to parameters as follows: Θ1 = 0.28, Θ2 = 0.34, Θ3 = 0.22, Θ4 = 0.16. Clearly, 4j=1 Θ j = 1. To select the most efficient bank, two operators namely, m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi weighted geometric operators are used respectively: 1. For k = 3, utilize the m–polar fuzzy Dombi weighted averaging operator to compute the values sˆi for the banks Bi , i = 1, 2, . . . , 6. sˆ1 = (0.6188, 0.5392, 0.6306, 0.6723),

sˆ2 = (0.8617, 0.8639, 0.7895, 0.6875),

sˆ3 = (0.5139, 0.7319, 0.5492, 0.5097),

sˆ4 = (0.7385, 0.6188, 0.5699, 0.5492),

sˆ5 = (0.6551, 0.6323, 0.5227, 0.6112),

sˆ6 = (0.7689, 0.7302, 0.6562, 0.6228).

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators

415

2. Calculate the score values S(ˆsi ) of 4–polar fuzzy numbers sˆi , (i = 1, 2, . . . , 6) for the banks Bi . S(ˆs1 ) = 0.6152,

S(ˆs2 ) = 0.8007,

S(ˆs3 ) = 0.5762,

S(ˆs4 ) = 0.6191,

S(ˆs5 ) = 0.6053,

S(ˆs6 ) = 0.6945.

3. Now rank the banks using scores S(si ), (i = 1, 2, . . . , 6) obtained from the preference values in the form of 4–polar fuzzy numbers: B2 > B6 > B4 > B1 > B5 > B3 . 4. B2 has a high score value, so, it is the best bank.

In a similar way, apply the m–polar fuzzy Dombi weighted geometric operator to determine the most efficient bank. 1. Take k = 3. The m–polar fuzzy Dombi weighted geometric operator is employed to compute the values sˆi for the banks Bi . sˆ1 = (0.4752, 0.6016, 0.4896, 0.4052),

sˆ2 = (0.2182, 0.2723, 0.2361, 0.3343),

sˆ3 = (0.5366, 0.3756, 0.4772, 0.6136),

sˆ4 = (0.4006, 0.5206, 0.5246, 0.4772),

sˆ5 = (0.4487, 0.4429, 0.4952, 0.5604),

sˆ6 = (0.3029, 0.4185, 0.4146, 0.5118).

2. Find the score values S(ˆsi ) of 4–polar fuzzy numbers sˆi for the banks Bi . S(ˆs1 ) = 0.4929,

S(ˆs2 ) = 0.2652,

S(ˆs3 ) = 0.5008,

S(ˆs4 ) = 0.4808,

S(ˆs5 ) = 0.4868,

S(ˆs6 ) = 0.4269.

3. Rank the banks with scores S(ˆsi ), (i = 1, 2, . . . , 6) obtained from the preference values in the form of 4–polar fuzzy numbers: B3 > B1 > B5 > B4 > B6 > B2 . 4. B3 has highest score, so, it is the best bank. The methodology utilized in the applications to find the best alternative is displayed in Fig. 8.1.

An m–polar fuzzy ELECTRE-I approach [3] is applied to the problem (performance evaluation of commercial banks). 1. Table 8.3 describes the 4–polar fuzzy decision-matrix. 2. Tables 8.4 and 8.5 respectively describe the 4–polar fuzzy concordance and discordance values. 3. The 4–polar fuzzy concordance matrix is calculated by ⎞ ⎛ − 0 0.84 0.62 0.62 0.34 ⎜ 1 − 1 1 1 1 ⎟ ⎟ ⎜ ⎜ 0.16 0 − 0.16 0.34 0 ⎟ ⎟. F =⎜ ⎜ 0.38 0 1 − 0.56 0.22 ⎟ ⎟ ⎜ ⎝ 0.66 0 0.66 0.44 − 0 ⎠ 0.66 0 1 0.78 1 − 4. The 4–polar fuzzy concordance-level f = 0.498.

416

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Fig. 8.1 Flowchart of selecting the best option

Table 8.3 4–polar fuzzy weighted decision matrix . T1 T2 B1 B2 B3 B4 B5 B6

(0.196, 0.168, 0.196, 0.196) (0.252, 0.196, 0.224, 0.196) (0.112, 0.224, 0.14, 0.168) (0.224, 0.196, 0.168, 0.14) (0.196, 0.196, 0.168, 0.196) (0.224, 0.224, 0.196, 0.168)

(0.17, 0.238, 0.204, 0.238) (0.272, 0.306, 0.272, 0.238) (0.17, 0.204, 0.204, 0.136) (0.238, 0.17, 0.136, 0.204) (0.17, 0.204, 0.136, 0.136) (0.238, 0.204, 0.204, 0.17)

T3

T4

(0.11, 0.066, 0.11, 0.11) (0.176, 0.154, 0.154, 0.132) (0.11, 0.132, 0.11, 0.038) (0.11, 0.11, 0.132, 0.11) (0.154, 0.11, 0.11, 0.038) (0.176, 0.11, 0.11, 0.038)

(0.08, 0.064, 0.064, 0.096) (0.112, 0.112, 0.128, 0.112) (0.096, 0.096, 0.08, 0.048) (0.064, 0.08, 0.096, 0.08) (0.08, 0.08, 0.096, 0.08) (0.096, 0.096, 0.112, 0.096)

5. The 4–polar fuzzy discordance matrix is computed by ⎞ ⎛ − 1 0.522 0.289 0.7484 1 ⎜ 0 − 0 0 0 0 ⎟ ⎟ ⎜ ⎜ 1 1 − 1 1 1 ⎟ ⎟. G=⎜ ⎜ 1 1 0.413 − 1 0.8303 ⎟ ⎟ ⎜ ⎝ 1 1 0.7010 1 − 1 ⎠ 0.95 0 0 1 0 −

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators Table 8.4 4–polar fuzzy concordance sets j 1 2 3 F1 j F2 j F3 j F4 j F5 j F6 j

– {1, 2, 3, 4} {4} {3, 4} {1, 3, 4} {1, 3, 4}

{} – {} {} {} {}

{1, 2, 3} {1, 2, 3, 4} – {1, 2, 3, 4} {1, 3, 4} {1, 2, 3, 4}

Table 8.5 4–polar fuzzy discordance sets j 1 2 3 G1 j G2 j G3 j G4 j G5 j G6 j

– {} {1, 2, 3} {1, 2} {1, 2} {2}

{1, 2, 3, 4} – {1, 2, 3, 4} {1, 2, 3, 4} {1, 2, 3, 4} {}

{4} {} – {4} {2} {}

417

4

5

6

{1, 2} {1, 2, 3, 4} {4} – {1, 4} {1, 2, 4}

{1, 2} {1, 2, 3, 4} {2} {2, 3} – {1, 2, 3, 4}

{2} {1, 2, 3, 4} {} {3} {} –

4

5

6

{3, 4} {} {1, 2, 3, 4} – {2, 3} {3}

{1, 3, 4} {} {1, 3, 4} {1, 4} – {}

{1, 3, 4} {} {1, 2, 3, 4} {1, 2, 4} {1, 2, 3, 4} –

6. The 4–polar fuzzy discordance level g = 0.65. 7. The 4-polar fuzzy concordance and discordance dominance matrices are given by ⎞ ⎛ − 0 0 1 1 0 ⎜1 − 1 1 1 1⎟ ⎟ ⎜ ⎜0 0 − 0 0 0⎟ ⎟, ⎜ H =⎜ ⎟ ⎜0 0 1 − 1 0⎟ ⎝1 0 1 0 − 0⎠ 1 0 1 1 1 − ⎞ ⎛ − 0 1 1 0 0 ⎜0 − 1 1 1 1⎟ ⎟ ⎜ ⎜0 0 − 0 0 0⎟ ⎟. ⎜ L=⎜ ⎟ ⎜0 0 1 − 0 0⎟ ⎝0 0 0 0 − 0⎠ 0 1 1 0 1 −

418

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Fig. 8.2 Outranking relation of banks

B1

B2

B3

B4

B5 B6

8. The 4–polar fuzzy aggregated dominance matrix is constructed as: ⎞ ⎛ − 0 0 1 0 0 ⎜1 − 1 1 1 1⎟ ⎟ ⎜ ⎜0 0 − 0 0 0⎟ ⎟. ⎜ M =⎜ ⎟ ⎜0 0 1 − 0 0⎟ ⎝0 0 0 0 − 0⎠ 0 0 1 0 1 − 9. Figure 8.2 displays the preference relations between the banks. From the Fig. 8.2, it is clear that B2 is the best option.

8.4.3 Assessment of Health Care Waste Treatments Alternatives A waste management system’s fundamental task is to control, process, store and dispose waste in consistence with national prerequisites and international obligations, taking into account the economic and socio-political factors involved. A suitable technology has to be chosen for each step due to the range of procedures, techniques and equipment available for different steps of a waste management scheme. There is a committee which selects five health care waste treatment alternatives, which are listed as below. A1 : Incineration A2 : Steam Sterilization A3 : Microwaves A4 : Land fill and Dumps A5 : Emulsification. These waste treatments are assessed on the basis of four factors. T1 : Economic Factors T2 : Environmental Factors T3 : Technical Factors T4 : Social Factors

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators Table 8.6 3–polar fuzzy decision matrix of treatments . T1 T2 A1 A2 A3 A4 A5

(0.60, 0.40, 0.50) (0.50, 0.70, 0.30) (0.80, 0.40, 0.60) (0.50, 0.40, 0.40) (0.40, 0.60, 0.50)

(0.80, 0.20, 0.60) (0.60, 0.40, 0.60) (0.40, 0.50, 0.40) (0.30, 0.50, 0.60) (0.40, 0.50, 0.50)

419

T3

T4

(0.20, 0.30, 0.50) (0.40, 0.50, 0.70) (0.30, 0.60, 0.90) (0.30, 0.70, 0.40) (0.50, 0.40, 0.60)

(0.8, 0.7 0.3) (0.4, 0.6, 0.9) (0.2, 0.6, 0.7) (0.7, 0.2, 0.5) (0.2, 0.5, 0.9)

Each factor has been divided into three characteristics to make a 3–polar fuzzy number. • The “Economic Factors” include cost and resources, transport regulations and physical infrastructure. • The “Environmental Factors” include geographical conditions, geological conditions and availability of resources. • The “Technical Factors” include waste characteristics, complexity and maintainability of facilities and state of research and development. • The “Social Factors” include social acceptability, communication and societal responsibilities and social equity. The five possible alternatives are to be evaluated using 3–polar fuzzy numbers by the decision makers under the four attributes. 1. The 3–polar fuzzy decision matrix is given in Table 8.6. 2. The weights assigned by the experts are given as Θ1 = 0.40, Θ2 = 0.20, Θ3 = 0.30, Θ4 = 0.10 where,

4 

Θ j = 1.

j=1

The most suitable health care waste treatment alternatives are selected by using the m–polar fuzzy Hamacher weighted average operator, to develop an approach to MCDM problems with 3–polar fuzzy information, which can be described as following. Step 1.

Assume λ = 3. Use the m–polar fuzzy Hamacher weighted average operator to calculate the performance values ei of the health care waste treatment alternatives. eˆ1 = (0.57433, 0.36685, 0.50257), eˆ2 = (0.48273, 0.58007, 0.57729), eˆ3 = (0.5529, 0.50409, 0.70785), eˆ4 = (0.42756, 0.50339, 0.45310), eˆ5 = (0.41163, 0.51359, 0.59164).

420

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Step 2.

Compute the scores S(eˆi ) of all 3–polar fuzzy numbers eˆi . S(eˆ1 ) = 0.481284, S(eˆ2 ) = 0.54669, S(eˆ3 ) = 0.58803, S(eˆ4 ) = 0.46135, S(eˆ5 ) = 0.50562.

Step 3.

Rank all the health care waste treatment alternatives according to the scores S(eˆi ), 1 ≤ i ≤ 5 of all 3–polar fuzzy numbers, A3 > A2 > A5 > A1 > A4 .

Step 4.

A3 is the best alternative.

If the m–polar fuzzy Hamacher weighted geometric operator is used for selection, the best alternative can be chosen in a similar manner. Step 1.

Suppose λ = 3. Use the m–polar fuzzy Hamacher weighted geometric operator to calculate the performance values eˆi of the health care waste treatment alternatives. eˆ1 = (0.50316, 0.34501, 0.49608), eˆ2 = (0.47699, 0.564823, 0.51952), eˆ3 = (0.45521, 0.49591, 0.65444), eˆ4 = (0.40822, 0.47455, 0.44650), eˆ5 = (0.42129, 0.47285, 0.56738).

Step 2.

Compute the scores S(eˆi ) of all 3–polar fuzzy numbers eˆi . S(eˆ1 ) = 0.44808, S(eˆ2 ) = 0.52044, S(eˆ3 ) = 0.535187, S(eˆ4 ) = 0.44309, S(eˆ5 ) = 0.487175.

Step 3.

Rank all the health care waste treatment alternatives, in accordance with the scores S(ei ), i = 1, 2, 3, 4, 5 of all 3–polar fuzzy numbers, A3 > A2 > A5 > A1 > A4 .

Step 4.

A3 is the best alternative. The m–polar fuzzy ELECTRE-I approach is applied to the same problem.

1. Table 8.7 represents the 3–polar fuzzy decision matrix.

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators Table 8.7 3–polar fuzzy weighted decision matrix of treatments . T1 T2 T3 C1 C2 C3 C4 C5

(0.24, 0.16, 0.20) (0.20, 0.28, 0.12) (0.32, 0.16, 0.24) (0.20, 0.16, 0.16) (0.16, 0.24, 0.20)

(0.16, 0.04, 0.12) (0.12, 0.08, 0.12) (0.08, 0.10, 0.08) (0.06, 0.10, 0.12) (0.08, 0.10, 0.10)

Table 8.8 3–polar fuzzy concordance sets of treatments j 1 2 3 F1 j F2 j F3 j F4 j F5 j

– {1, 2, 3, 4} {1, 3} {3} {1, 3}

{1, 2} – {1, 3} {} {1}

{2, 4} {2, 4} – {2} {2, 4}

Table 8.9 3–polar fuzzy discordance sets of treatments j 1 2 3 G1 j G2 j G3 j G4 j G5 j

– {1, 2} {2, 4} {1, 2, 4} {1, 2, 4}

{1, 2, 3, 4} – {2, 4} {1, 2, 3, 4} {1, 2, 3, 4}

{1, 3} {1, 3} – {1, 3, 4} {1, 3}

421

T4

(0.06, 0.09, 0.15) (0.12, 0.15, 0.21) (0.09, 0.18, 0.27) (0.09, 0.21, 0.12) (0.15, 0.12, 0.18)

(0.08, 0.07, 0.03) (0.04, 0.06, 0.09) (0.02, 0.06, 0.07) (0.07, 0.02, 0.05) (0.02, 0.05, 0.09)

4

5

{1, 2, 4} {1, 2, 3, 4} {1, 3, 4} – {1, 2, 3, 4}

{1, 2, 4} {1, 2, 3, 4} {1, 3} {2} –

4

5

{3} {} {2} – {2}

{1, 3} {1, 3} {2, 4} {1, 2, 3, 4} –

2. Table 8.8 and Table 8.9 represent the 3–polar fuzzy concordance and 3–polar fuzzy discordance sets, respectively. 3. The 3–polar fuzzy concordance matrix is constructed as: ⎛ ⎞ − 0.6 0.3 0.7 0.7 ⎜ 1 − 0.3 1 1 ⎟ ⎜ ⎟ ⎟. 0.7 0.7 − 0.8 0.7 F =⎜ ⎜ ⎟ ⎝ 0.3 0 0.2 − 0.2 ⎠ 0.7 0.4 0.3 1 − 4. The 3–polar fuzzy concordance level f = 0.7805. 5. The 3–polar fuzzy discordance matrix is constructed as: ⎛ ⎞ − 1 1 1 1 ⎜ ⎟ 1 − 1 0 1 ⎜ ⎟ ⎜ G = ⎜ 0.7041 0.28867 − 0.29233 0.18077 ⎟ ⎟. ⎝ 0.91630 ⎠ 1 1 − 1 1 1 1 0.22867 −

422

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets A2

Fig. 8.3 Outranking relation of treatments alternatives A1

A5

A3

A4

6. The 3–polar fuzzy discordance level g = 0.58. 7. According to the concordance level and discordance level, the 3–polar fuzzy concordance dominance and 3–polar fuzzy discordance dominance matriies are constructed as: ⎛ ⎞ − 1 0 1 1 ⎜1 − 0 1 1⎟ ⎜ ⎟ ⎟ H =⎜ ⎜ 1 1 − 1 1 ⎟, ⎝0 0 0 − 0⎠ 1 0 0 1 − ⎞ ⎛ − 0 0 0 0 ⎜0 − 0 1 0⎟ ⎟ ⎜ ⎟ L=⎜ ⎜ 1 1 − 1 1 ⎟. ⎝0 0 0 − 0⎠ 0 0 0 1 − 8. The 3–polar fuzzy aggregated dominance matrix is constructed as: ⎛ ⎞ − 0 0 0 0 ⎜0 − 0 1 0⎟ ⎜ ⎟ ⎟ M =⎜ ⎜ 1 1 − 1 1 ⎟. ⎝0 0 0 − 0⎠ 0 0 0 1 − 9. The following graph shows the preference relation of the health care treatments. Therefore, A3 is the best choice (Fig. 8.3).

8.4.4 Selection of a Best Company for Investment Each investment and investment decision entails certain degree of risk. Regardless of the truth that an investor can never understand for sure, research that focuses on fact and an objective pre-purchase analysis can extend the chances of making a proper decision about whether or no longer to invest in a company. The most obvious factor to consider is the financial performance of the company. The acronym “ESG” collectively refers to economic, social and governance factors. ESG integration is the method of consideration of economic, social and governance factors in

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators Table 8.10 4–polar fuzzy decision matrix of companies . T1 T2 C1 C2 C3 C4 C5

(0.36, 0.45, 0.50, 0.41) (0.52, 0.70, 0.46, 0.56) (0.25, 0.35, 0.40, 0.35) (0.73, 0.81, 0.72, 0.69) (0.64, 0.71, 0.60, 0.50)

423

T3

(0.25, 0.40, 0.61, 0.50) (0.36, 0.57, 0.48, 0.73) (0.24, 0.37, 0.56, 0.50) (0.65, 0.73, 0.66, 0.82) (0.50, 0.60, 0.50, 0.70)

(0.50, 0.42, 0.53, 0.63) (0.64, 0.54, 0.72, 0.60) (0.49, 0.38, 0.42, 0.57) (0.75, 0.81, 0.80, 0.72) (0.70, 0.70, 0.60, 0.50)

the investment cycle. Company information is another important factor in assessing a potential business investment. Suppose that an investor wants to invest in a company. Let {C1 , C2 , C3 , C4 , C5 } be the set of five companies. The investor chooses three characteristics to assess companies which are given as: T1 : Financial Performance T2 : Company Information T3 : ESG Integration Each attribute has been divided into four characteristics to make a 4–polar fuzzy number. • The attribute “Financial Performance” includes tax returns, balance sheets, cash flow projections and current accounts receivables. • The attribute “Company Information” includes company’s history, accomplishments, product or service offerings and business plans. • The attribute “ESG Integration” includes pollution prevention, energy efficiency, regulatory standards and adherence to environmental safety. 1. The 4–polar fuzzy decision matrix is given in Table 8.10. 2. Weights assigned by the investor are given as, Θ1 = 0.45, Θ2 = 0.25, Θ3 = 0.30 where,

3 

Θ j = 1.

j=1

The best company is selected for investment by using the m–polar fuzzy Hamacher weighted average operator. Step 1.

Assume λ = 3. Use the m–polar fuzzy Hamacher weighted average operator to calculate the performance values eˆi of the companies.

424

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

eˆ1 = (0.3766, 0.42864, 0.5378, 0.5034), eˆ2 = (0.5217, 0.6244, 0.5536, 0.61924), eˆ3 = (0.3219, 0.3539, 0.4479, 0.4573), eˆ4 = (0.7179, 0.7921, 0.7332, 0.7363), eˆ5 = (0.6275, 0.6817, 0.5762, 0.5560). Step 2.

Compute the scores S(eˆi ) of all 4–polar fuzzy numbers eˆi . S(eˆ1 ) = 0.46161, S(eˆ2 ) = 0.57974, S(eˆ3 ) = 0.39525, S(eˆ4 ) = 0.74486, S(eˆ5 ) = 0.61035.

Step 3.

Rank all alternatives for investment according to the scores S(eˆi ), 1 ≤ i ≤ 5 of all 4–polar fuzzy numbers, C4 > C5 > C2 > C1 > C3 .

Step 4.

Therefore, C4 is the most suitable company for investment.

If the m–polar fuzzy Hamacher weighted geometric operator is used for selection, the best alternative can be chosen in a similar manner. Step 1.

Assume λ = 3. Use the m–polar fuzzy Hamacher weighted geometric operator to calculate the performance values eˆi of the alternatives. eˆ1 = (0.36652, 0.42819, 0.53575, 0.49436), eˆ2 = (0.51044, 0.61822, 0.53856, 0.61352), eˆ3 = (0.3074, 0.36384, 0.4435, 0.4474), eˆ4 = (0.7159, 0.7901, 0.7289, 0.7315), eˆ5 = (0.62142, 0.6792, 0.5743, 0.5477).

Step 2.

Compute the scores S(eˆi ) of all 4–polar fuzzy numbers eˆi . S(eˆ1 ) = 0.456205, S(eˆ2 ) = 0.570185, S(eˆ3 ) = 0.390535, S(eˆ4 ) = 0.7416, S(eˆ5 ) = 0.60566.

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators Table 8.11 4–polar fuzzy weighted decision matrix of companies . T1 T2 C1

(0.162, 0.2025, 0.225, 0.1845) (0.234, 0.315, 0.207, 0.252) (0.1125, 0.1575, 0.18, 0.1575) (0.3285, 0.3645, 0.324, 0.3105) (0.288, 0.3195, 0.27, 0.225)

C2 C3 C4 C5

(0.0625, 0.18, 0.2745, 0.125) (0.09, 0.1425, 0.12, 0.1825) (0.06, 0.0925, 0.14, 0.125) (0.1625, 0.1825, 0.165, 0.205) (0.125, 0.15, 0.125, 0.175)

Table 8.12 4–polar fuzzy concordance set of companies j 1 2 3 F1 j F2 j F3 j F4 j F5 j

– {1, 3} {} {1, 2, 3} {1, 3}

{2} – {} {1, 2, 3} {1, 2, 3}

{1, 2, 3} {1, 2, 3} – {1, 2, 3} {1, 2, 3}

Table 8.13 4–polar fuzzy discordance set of companies j 1 2 3 G1 j G2 j G3 j G4 j G5 j

Step 3.

– {2} {1, 2, 3} {} {2}

{1, 3} – {1, 2, 3} {} {3}

{} {} – {} {}

425

T3 (0.15, 0.126, 0.159, 0.189) (0.192, 0.162, 0.216, 0.18) (0.147, 0.114, 0.126, 0.171) (0.225, 0.243, 0.24, 0.216) (0.21, 0.21, 0.18, 0.15)

4

5

{} {} {} – {}

{2} {3} {} {1, 2, 3} –

4

5

{1, 2, 3} {1, 2, 3} {1, 2, 3} – {1, 2, 3}

{1, 3} {1, 2, 3} {1, 2, 3} {} –

Rank all companies for investment based on the scores S(eˆi ), 1 ≤ i ≤ 5 of all 4–polar fuzzy numbers, C4 > C5 > C2 > C1 > C3 .

Step 4.

Therefore, C4 is the best alternative. The m–polar fuzzy ELECTRE-I method is aplied to the same problem.

3. Table 8.11 represents the 4–polar fuzzy weighted decision matrix. 4. Tables 8.12 and 8.13 represent the 4–polar fuzzy concordance and 4–polar fuzzy discordance sets, respectively. 5. The 4–polar fuzzy concordance matrix is constructed as:

426

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

⎛

⎞ − 0.25 1 0 0.25 ⎜ 0.75 − 1 0 0.3 ⎟ ⎜ ⎟ 0 − 0 0 ⎟ F =⎜ ⎜ 0 ⎟. ⎝ 1 1 1 − 1 ⎠ 0.75 1 1 0 − 6. The 4–polar fuzzy concordance level f = 0.515. 7. The 4–polar fuzzy discordance matrix is constructed as: ⎞ ⎛ − 0.8800 0 1 1 ⎜ 1 − 0 1 1⎟ ⎟ ⎜ ⎜ 1 − 1 1⎟ G=⎜ 1 ⎟. ⎝ 0 0 0 − 0⎠ 0.9445 0.7953 0 1 − 8. The 4–polar fuzzy discordance level g = 0.6309. 9. The 4–polar fuzzy concordance dominance and 4–polar fuzzy discordance dominance matries are constructed as: ⎛ ⎞ − 0 1 0 0 ⎜1 − 1 0 0⎟ ⎜ ⎟ ⎟ H =⎜ ⎜ 0 0 − 0 0 ⎟, ⎝1 1 1 − 1⎠ 1 1 1 0 − ⎛ ⎞ − 0 1 0 0 ⎜0 − 1 0 0⎟ ⎜ ⎟ ⎟ L=⎜ ⎜ 0 0 − 0 0 ⎟. ⎝1 1 1 − 1⎠ 0 0 1 0 − 10.

The 4–polar fuzzy aggregated dominance matrix is evaluated as: ⎛ ⎞ − 0 1 0 0 ⎜0 − 1 0 0⎟ ⎜ ⎟ ⎟ M =⎜ ⎜ 0 0 − 0 0 ⎟. ⎝1 1 1 − 1⎠ 0 0 1 0 −

11.

The following graph shows the preference relation of the companies. Therefore, C4 is the best company for investment (Fig. 8.4).

8.4.5 Selection of Most Affected Country by Human Trafficking In human trafficking, traffickers use force, coercion, or fraud to lure their victims into commercial sexual exploitation or labor. The people who are vulnerable due to

8.4 MCDM Methods Using Dombi and Hamacher Aggregation Operators

427 C1

Fig. 8.4 Outranking relation of companies C5

C4

C2

C3

different reasons like emotional or psychological susceptibleness, lack of a social welfare system, economic hardships, political instability or natural disaster. Based on latest surveys, it can be easily observed that millions of children, women and men become the part of human trafficking over the world, including Saudi Arabia, China, Russia, Kuwait and Iran. It is also observed that victims of traffickers can be of any gender, age, or nationality. Traffickers use manipulation, or fake promises of high-paying jobs or romantic connection to attract victims into trafficking. The traumatology triggered by the traffickers can be so wonderful that several persons may not recognize themselves as victims. The main attributes or causes of human trafficking are political instability, poverty, debt, natural disasters, demand and addiction. Let U = {C1 = Saudi Arabia, C2 = China, C3 = Russia, C4 = Kuwait, C5 = Iran} be a set of five countries and let T = {T1 , T2 , T3 , T4 } be the set of four attributes, where T1 denotes “Poverty”, T2 denotes “Debt”, T3 denotes “Demand”, T4 denotes “Natural Disaster”. Further characterizations of above attributes are force, fraud or lure. The purpose of this application is to evaluate the above countries Ci ’s, i = 1, 2, . . . , 5 concerning worst in human trafficking with the help of 3–polar fuzzy numbers given by the decision-makers under the attributes T j ’s, j = 1, 2, 3, 4. Let Θ = (0.4, 0.3, 0.3) be the weight vector for the preceding characteristics. The 3-polar fuzzy decision matrix is given in Table 8.14 For illustration, the 3–polar fuzzy number (0.3, 0.6, 0.2) in the top left entry of the 3–polar fuzzy decision matrix means that in the country Saudi Arabia C1 with respect to poverty people which become the part of human trafficking are sub-classified as follows: 30% due to force, 60% due to fraud, 20% due to lure. To compute the most worst country regarding human trafficking, we apply the two operators, namely, m–polar fuzzy Hamacher weighted average and m–polar fuzzy

428

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Table 8.14 3–polar fuzzy decision matrix of countries . T1 T2 C1 C2 C3 C4 C5

(0.3, 0.6, 0.2) (0.6, 0.9, 0.3) (0.7, 0.7, 0.6) (0.5, 0.7, 0.3) (0.8, 0.6, 0.4)

(0.2, 0.5, 0.6) (0.4, 0.8, 0.1) (0.4, 0.3, 0.4) (0.7, 0.8, 0.1) (0.1, 0.2, 0.5)

T3

T4

(0.7, 0.6, 0.1) (0.5, 0.2, 0.5) (0.1, 0.3, 0.4) (0.4, 0.5, 0.6) (0.3, 0.7, 0.4)

(0.5, 0.8, 0.3) (0.7, 0.4, 0.2) (0.4, 0.2, 0.5) (0.7, 0.8, 0.2) (0.2, 0.8, 0.3)

Hamacher weighted geometric, to construct methods to MADM problems with m– polar fuzzy information, which are given as follows: 1. Take λ = 3. The m–polar fuzzy Hamacher weighted average operator is applied to find the preference values eˆi of the countries Ci regarding human trafficking. eˆ1 = (0.3582, 0.6220, 0.3382),

eˆ2 = (0.5580, 0.7653, 0.2404),

eˆ3 = (0.5078, 0.4620, 0.5041), eˆ5 = (0.4732, 0.5595, 0.4115).

eˆ4 = (0.5998, 0.7395, 0.2523),

2. Determine the scores S(eˆi ) of overall 3–polar fuzzy numbers eˆi of the countries Ci involved in human trafficking. S(eˆ1 ) = 0.4395, S(eˆ3 ) = 0.4913,

S(eˆ2 ) = 0.5212, S(eˆ4 ) = 0.5305,

S(eˆ5 ) = 0.4814. 3. Now rank all the countries based on score values S(eˆi ), (i = 1, 2, . . . , 5) based on overall 3–polar fuzzy numbers: C4 > C2 > C3 > C5 > C1 . 4. C4 has high rate human trafficking. Similarly, the m–polar fuzzy Hamacher weighted geometric operator is used to select the affected country. 1. Take λ = 3. Use the m–polar fuzzy Hamacher weighted geometric operator to find the preference values eˆi of the countries Ci regarding human trafficking. eˆ1 = (0.3295, 0.6078, 0.2940), eˆ3 = (0.4577, 0.4087, 0.4959), eˆ5 = (0.3337, 0.4943, 0.4061).

eˆ2 = (0.5447, 0.6800, 0.2146), eˆ4 = (0.5858, 0.7290, 0.2200),

2. Determine the scores S(eˆi ) of overall 3–polar fuzzy numbers eˆi , (i = 1, 2, . . . , 5) of the countries Ci involved in human trafficking.

8.5 Comparison Analysis and Discussion

429

Fig. 8.5 Flowchart of selecting the most worst country affecting by human trafficking

Select different alternatives (countries) for ranking affected by human trafficking

Input

Choose particular attributes related to alternatives

Assign weight vector to attributes according to decision-maker

Compute aggregation values of alternatives by using the m-polar fuzzy Hamacher weighted average and m-polar fuzzy Hamacher weighted geometric operators

Calculate score values

Select the alternative having highest score value

Output

S(eˆ1 ) = 0.4104, S(eˆ3 ) = 0.4541,

S(eˆ2 ) = 0.4798, S(eˆ4 ) = 0.5116,

S(eˆ5 ) = 0.4114. 3. Now rank all the countries based on score values S(ei ), (i = 1, 2, . . . , 5) based on overall 3–polar fuzzy numbers: C4 > C2 > C3 > C5 > C1 . 4. C4 has a high rate of human trafficking. From the above analysis, it is easily seen that although the overall rating values of the alternatives are same by using two operators, respectively, the ranking orders of the alternatives are same. From the above results, the most affected country by human trafficking is C4 . The method used in the application to select the most worst country affecting by human trafficking is explained in Fig. 8.5.

8.5 Comparison Analysis and Discussion This section gives a comparison of the Dombi aggregation operators with m–polar fuzzy Hamacher aggregation operators and an m–polar fuzzy ELECTRE-I model to show their feasibility and practicality. 1. The results of m–polar fuzzy Dombi aggregation operators are compared with m– polar fuzzy Hamacher aggregation operators. The results computed by applying

430

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Table 8.15 Comparison of m–polar fuzzy Dombi and Hamacher aggregation operators in land selection Aggregation S(ˆs1 ) S(ˆs2 ) S(ˆs3 ) S(ˆs4 ) S(ˆs5 ) Ranking Operator Order m–Polar fuzzy Dombi weighted averaging m–Polar fuzzy Dombi weighted geometric m–Polar fuzzy Hamacher weighted average m–Polar fuzzy Hamacher weighted geometric

0.6814

0.7546

0.6459

0.6968

0.7108

Y2 > Y5 > Y4 > Y1 > Y3

0.5882

0.5266

0.5907

0.5643

0.5154

Y3 > Y1 > Y4 > Y2 > Y5

0.5403

0.6287

0.5151

0.5725

0.6327

Y5 > Y2 > Y4 > Y1 > Y3

0.5084

0.5881

0.4908

0.5445

0.5909

Y5 > Y2 > Y4 > Y1 > Y3

both operators in first application of Sect. 8.4 are explained by Table 8.15 and Fig. 8.6. In a similar way, the results computed using both operators in second application of Sect. 8.4 are explained by Table 8.16 and Fig. 8.7. Clearly, the results of m–polar fuzzy Hamacher weighted average (m–polar fuzzy Hamacher weighted average) and m–polar fuzzy Hamacher weighted geometric (m–polar fuzzy Hamacher weighted geometric) operators are different from the m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi weighted geometric operators. The results of m–polar fuzzy Hamacher weighted average and m– polar fuzzy Hamacher weighted geometric operators are same. Therefore, m– polar fuzzy aggregation operators are more generalized and versatile than some existing models to handle with m–polar fuzzy MCDM problems. 2. From the second application of Sect. 8.4, it can be observed that the final rankings by applying the m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi weighted geometric operators are B2 > B6 > B4 > B1 > B5 > B3 and B3 > B1 > B5 > B4 > B6 > B2 , respectively. However, the final score values are not same. When m–polar fuzzy ELECTRE-I method is applied, then the best option is B2 . Clearly, the optimal decision using m–polar fuzzy ELECTRE-I method and m–polar fuzzy Dombi weighted averaging operator is B2 . 3. When a number of m–polar fuzzy numbers are aggregated with the help of m–polar fuzzy Dombi weighted averaging and m–polar fuzzy Dombi weighted

8.5 Comparison Analysis and Discussion

431

0.8

0.7 mFDWA mFDWG mFHWA mFHWG

0.6

0.5

0.4

0.3

0.2

0.1

0 S (ˆ s 1)

S (ˆ s 2)

S (ˆ s 3)

S (ˆ s 4)

S (ˆ s 5)

Fig. 8.6 Comparison of first application in Sect. 8.4 Table 8.16 Comparison of m–polar fuzzy Dombi and Hamacher aggregation operators in bank selection AggregationS(ˆs1 ) S(ˆs2 ) S(ˆs3 ) S(ˆs4 ) S(ˆs5 ) S(ˆs6 ) Ranking Operator Order m–Polar fuzzy Dombi weighted averaging m–Polar fuzzy Dombi weighted geometric m–Polar fuzzy Hamacher weighted average m–Polar fuzzy Hamacher weighted geometric

0.6152

0.8007

0.5762

0.6191

0.6053

0.6945

0.4929

0.2652

0.5008

0.4808

0.4868

0.4269

0.5851

0.7681

0.5342

0.5727

0.5490

0.6355

0.5711

0.7592

0.5267

0.5611

0.5381

0.6262

B2 B6 B4 B1 B5 B3 B1 B5 B4 B6 B2 B6 B1 B4 B5 B2 B6 B1 B4 B5

> > > > > > > > > > > > > > > > > > > >

B3

B2

B3

B3

432

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets 0.9

0.8

mFDWA mFDWG mFHWA mFHWG

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 S (ˆ s 1)

S (ˆ s 2)

S (ˆ s 3)

S (ˆ s 4)

S (ˆ s 5)

S (ˆ s 6)

Fig. 8.7 Comparison of second application in Sect. 8.4

geometric operators, different computations will increase rapidly. But m–polar fuzzy aggregation operators can explain the assessed data more flexibly for decision making. The m–polar fuzzy aggregation operators rank every objects in a given problem in comparison with m–polar fuzzy ELECTRE-I approach [3]. 4. It can be seen from the results of forth application that if the operators m– polar fuzzy Hamacher weighted average or m–polar fuzzy Hamacher weighted geometric are used, respectively, then final ranking is C4 > C5 > C2 > C1 > C3 . However, the final scores are slightly different. If m–polar fuzzy ELECTRE-I approach is used, then the optimal alternative is C4 . 5. m–polar fuzzy ELECTRE-I approach is known as a flexible approach relative to other ELECTRE-I extensions. This approach does not result in a single alternative, but rather in a small subset of favorable alternatives. It is very difficult for the experts to rank all alternatives. 6. If more m–polar fuzzy numbers are involved using m–polar fuzzy Hamacher weighted average (or m–polar fuzzy Hamacher weighted geometric) operators, the number of operations and calculations will increase exponentially. But the proposed method can more flexibly explain the assessment details and maintain the integrity of original decision making data, which makes the final results more closely match realistic decision making issues.

8.6 Conclusion

433

8.5.1 Effectiveness Test To examine the validity of the provided Algorithm 8.4.1, it is verified with test criteria proposed in [37] for the first application. • Test criterion I: If the membership grades of nonoptimal object are changed with worse membership values without effecting criteria, then the optimal object should not change. • Test criterion II: MCDM approach should satisfy transitive property. • Test criterion III: When a designated problem is resolved into different small issues and the similar MCDM technique has been utilized, then the rank order of the objects should be similar to the original ranking order. These test criteria are checked on the MCDM approach under m–polar fuzzy Dombi aggregation operators as below. 1. Effectiveness test by criterion I: Using this test, if the membership grades  of alternative Y3 is changed with Y3 = (0.3, 0.6, 0.2), (0.4, 0.3, 0.2), (0.5, 0.7, 0.3), (0.5, 0.1, 0.6) in the Table 8.1 (that is, 3-polar fuzzy decision matrix), then the new 3-polar fuzzy decision matrix is displayed by Table 8.17. By using m–polar fuzzy Dombi weighted averaging operator, the score values of  the alternative are S(ˆs1 ) = 0.6814, S(ˆs2 ) = 0.7546, S(ˆs3 ) = 0.6459, S(ˆs4 ) =  0.6968, S(ˆs5 ) = 0.7108. Clearly, S(ˆs2 ) > S(ˆs5 ) > S(ˆs4 ) > S(ˆs1 ) > S(ˆs3 ) con sequently the ranking of the objects is Y2 > Y5 > Y4 > Y1 > Y3 . Thus, Y2 is the best alternative. According to above information, the m–polar fuzzy Dombi aggregation operators has been employed, the decision alternative is Y2 which is similar to the original optimal object. Similarly, if the membership grades of alternative Y4  is changed with Y4 = (0.4, 0.5, 0.2), (0.7, 0.5, 0.5), (0.8, 0.7, 0.1), (0.8, 0.2, 0.5) in the Table 8.1 (that is, 3-polar fuzzy decision matrix), then the new 3-polar fuzzy decision matrix is displayed by Table 8.18. By applying the m–polar fuzzy Dombi weighted averaging operator, the ranking order of the alternatives  is Y2 > Y5 > Y1 > Y3 > Y4 . Thus, the optimal alternatives is Y2 which is same as that of the original ranking. Therefore, the Algorithm 8.4.1 is feasible under the test criterion I. 2. Effectiveness test by criteria II and III: Based upon these test criteria, if we dissolve the designated problem (Application 1) into the sub-issues {Y1 , Y2 , Y3 }, {Y2 , Y3 , Y4 }, {Y3 , Y4 , Y5 } and {Y4 , Y5 , Y1 } and the procedure steps of the Algorithm 8.4.1 has been employed, then we obtain the ranking of these smaller issues is Y2 > Y1 > Y3 , Y2 > Y4 > Y3 , Y5 > Y4 > Y3 and Y5 > Y4 > Y1 , respectively. Hence, by uniting above criteria II and III, we obtain the overall ranking order of the alternatives is Y2 > Y5 > Y4 > Y1 > Y3 , which is exactly same as the original ranking order. Therefore, the Algorithm 8.4.1 is feasible under the test criteria II and III.

8.6 Conclusion Aggregation operators are mathematical functions that are used to combine information. Some arithmetic and geometric aggregation operators have been presented

434

8 Aggregation Operators for Decision Making with Multi-polar Fuzzy Sets

Table 8.17 3–polar fuzzy decision matrix . S1 S2 Y1 Y2  Y3 Y4 Y5

(0.7, 0.6, 0.3) (0.9, 0.6, 0.5) (0.3, 0.6, 0.2) (0.5, 0.6, 0.3) (0.9, 0.7, 0.6)

(0.5, 0.7, 0.2) (0.8, 0.5, 0.3) (0.4, 0.3, 0.2) (0.8, 0.6, 0.5) (0.8, 0.4, 0.3)

Table 8.18 3–polar fuzzy decision matrix . S1 S2 Y1 Y2 Y3  Y4 Y5

(0.7, 0.6, 0.3) (0.9, 0.6, 0.5) (0.3, 0.6, 0.2) (0.4, 0.5, 0.2) (0.9, 0.7, 0.6)

(0.5, 0.7, 0.2) (0.8, 0.5, 0.3) (0.4, 0.3, 0.2) (0.7, 0.5, 0.5) (0.8, 0.4, 0.3)

S3

S4

(0.6, 0.3, 0.7) (0.4, 0.5, 0.8) (0.5, 0.7, 0.3) (0.7, 0.8, 0.2) (0.6, 0.5, 0.7)

(0.9, 0.5, 0.4) (0.5, 0.8, 0.5) (0.5, 0.1, 0.6) (0.9, 0.3, 0.7) (0.4, 0.6, 0.5)

S3

S4

(0.6, 0.3, 0.7) (0.4, 0.5, 0.8) (0.5, 0.7, 0.3) (0.8, 0.7, 0.1) (0.6, 0.5, 0.7)

(0.9, 0.5, 0.4) (0.5, 0.8, 0.5) (0.5, 0.1, 0.6) (0.8, 0.2, 0.5) (0.4, 0.6, 0.5)

using Dombi and Hamacher t-norms and t-conorms to handle uncertainty in m–polar fuzzy information. Different operators such as m–polar fuzzy Dombi weighted averaging operator, m–polar fuzzy Dombi ordered weighted averaging operator, m–polar fuzzy Dombi hybrid averaging operator, m–polar fuzzy Dombi weighted geometric operator, m–polar fuzzy Dombi weighted ordered geometric operator, m–polar fuzzy Dombi hybrid geometric operator, m–polar fuzzy Hamacher weighted average operator, m–polar fuzzy Hamacher ordered weighted average operator, m–polar fuzzy Hamacher hybrid average operator, m–polar fuzzy Hamacher weighted geometric operator, m–polar fuzzy Hamacher ordered weighted geometric operator and m– polar fuzzy Hamacher hybrid geometric operator have been presented. Some of the properties including idempotancy, monotonicity and boundedness have been investigated for the presented operators. Moreover, the corresponding algorithms have also been presented to solve MCDM issues, which involve m–polar fuzzy information with m–polar fuzzy Dombi weighted averaging (m–polar fuzzy Hamacher weighted average) and m–polar fuzzy Dombi weighted geometric operators. To prove the validity and feasibility of the presented models, numerical examples have been solved for each presented model, and a comparison has been discussed with m–polar fuzzy ELECTRE-I approach. The effectiveness of m–polar fuzzy Dombi aggregation operators has also been checked by validity tests.

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27. He, X.: Typhoon disaster assessment based on Dombi hesitant fuzzy information aggregation operators. Nat. Hazards 90(3), 1153–1175 (2018) 28. Jana, C., Pal, M., Wang, J.Q.: Bipolar fuzzy Dombi aggregation operators and its application in multiple-attribute decision making process. J. Ambient. Intell. Hum. Comput. 10(9), 3533– 3549 (2019) 29. Jana, C., Pal, M., Wang, J.Q.: Bipolar fuzzy Dombi prioritized aggregation operators in multiple attribute decision making. Soft Comput. 24(5), 3631–3646 (2020) 30. Khameneh, A.Z., Kilicman, A.: m–polar fuzzy soft weighted aggregation operators and their applications in group decision making. Symmetry 10(11), 636 (2018) 31. Liang, D., Zhang, Y., Xu, Z., Darko, A.P.: Pythagorean fuzzy Bonferroni mean aggregation operator and its accelerative calculating algorithm with the multithreading. Int. J. Intell. Syst. 33(3), 615–633 (2018) 32. Liu, P.: Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making. IEEE Trans. Fuzzy Syst. 22(1), 83–97 (2013) 33. Peng, X., Yang, Y.: Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators. Int. J. Intell. Syst. 31(5), 444–487 (2016) 34. Sarwar, M., Akram, M., Shahzadi, S.: Distance measures and δ-approximations with rough complex fuzzy models. Granular Comput. 8, 893–916 (2023) 35. Shahzadi, S., Akram, M.: Intuitionistic fuzzy soft graphs with applications. J. Appl. Math. Comput. 55, 369–392 (2017) 36. Shahzadi, G., Akram, M., Al-Kenani, A.N.: decision making approach under Pythagorean fuzzy Yager weighted operators. Mathematics 8(1), 70 (2020) 37. Wang, X., Triantaphyllou, E.: Ranking irregularities when evaluating alternatives by using some ELECTRE methods. Omega 36(1), 45–63 (2008) 38. Wang, J., Wei, G., Lu, J., Alsaadi, F.E., Hayat, T., Wei, C., Zhang, Y.: Some q-rung orthopair fuzzy Hamy mean operators in multiple attribute decision making and their application to enterprise resource planning systems selection. Int. J. Intell. Syst. 34(10), 2429–2458 (2019) 39. Wang, L., Li, N.: Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making. Int. J. Intell. Syst. 35(1), 150–183 (2020) 40. Waseem, N., Akram, M., Alcantud, J.C.R.: Multi-attribute decision making based on m-polar fuzzy Hamacher aggregation operators. Symmetry 11(12), 1498 (2019) 41. Wei, G., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. Int. J. Fuzzy Syst. 20(1), 1–12 (2018) 42. Wei, G., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Hesitant bipolar fuzzy aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 33(2), 1119–1128 (2017) 43. Xiao, S.: Induced interval-valued intuitionistic fuzzy Hamacher ordered weighted geometric operator and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 27(1), 527–534 (2014) 44. Xu, Z.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15(6), 1179–1187 (2007) 45. Xu, Z.: Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowl.-Based Syst. 24(6), 749–760 (2011) 46. Xu, Z., Da, Q.L.: An overview of operators for aggregating information. Int. J. Intell. Syst. 18(9), 953–969 (2003) 47. Xu, X.R., Wei, G.W.: Dual hesitant bipolar fuzzy aggregation operators in multiple attribute decision making. Int. J. Knowl.-Based Intell. Eng. Syst. 21(3), 155–164 (2017) 48. Xu, Z., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. General Syst. 35(4), 417–433 (2006) 49. Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man Cybern. 18(1), 183–190 (1988) 50. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 51. Zhou, L., Zhao, X., Wei, G.: Hesitant fuzzy Hamacher Aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 26(6), 2689–2699 (2014)

Chapter 9

2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

This chapter is devoted to present the concept of 2-tuple linguistic multi-polar (m–polar) fuzzy sets and to introduce some fundamental operations on them. The following aggregation operators are presented including, the 2-tuple linguistic m– polar fuzzy Hamacher weighted average operator, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator, 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator, 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator, and 2-tuple linguistic m–polar fuzzy Hamacher hybrid geometric operator. Their properties, inclusive of the standard cases of monotonicity, boundedness, and idempotency are investigated. Then an algorithm to solve multicriteria decision making problems is formulated. The influence of the parameters on the outputs is also explored with a numerical simulation. Moreover, a comparative study with existing methods is performed in order to demonstrate the applicability and authenticity of the presented model. This chapter is due to [6].

9.1 Introduction With the large display of different approaches, multi-attribute decision making (MADM) is considered as a collective enterprise that aspires to deal with complex situations in the presence of multiple attributes. The choice of a decision making approach plays a vital role in the selection of the desirable alternatives. Also the representation of the information is crucial, because the formulation of problems with real-valued attribute endowments is considered rare in decision sciences. For this reason, Zadeh [38] introduced the idea of fuzzy sets, a mathematical tool that easily tackles MADM, being a mutated form of crisp set theory. Fuzzy sets has a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_9

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path-breaking structure that allowed to account for vagueness for the first time. But it is not a totally general framework for the mathematical treatment of partial knowledge. To enlarge its scope, Atanassov [12] introduced intuitionistic fuzzy set theory, a simple modification that can tackle uncertain and vague data more precisely. Further motivated through multidisciplinary data, Chen et al. [17] developed the m–polar fuzzy set, that enables the decision makers to manipulate multi-polar information for the purpose of decision making. Many people want to express themselves with common terms like “magnificent”, “superb”, “best”, “better”, “poor” and “worst” to gauge some attributes. These assessments of an object’s properties should then be used in MADM to collect the uncertain data more precisely. Firstly, Herrera and Martínez [24] introduced the idea of 2-tuple linguistic representation, which is the most successful tool to take on linguistics decision making issues. Aggregation operators play a crucial role to convert different datasets into a single result, and deal with collective decision making problems. A very popular tool for aggregating data was introduced by Yager [36] under the name ordered weighted averaging aggregation operators. Yager [37] contributed for Quantifier guided aggregation using ordered weighted averaging aggregation operators. Xu [35] first proposed intuitionistic fuzzy sets based aggregation operators. More sophisticated aggregation operators were developed to improve the accuracy of the subsequent applications. For example, based on algebraic and Einstein t-conorm and t-norm [19], Hamacher t-conorm and t-norm [14] were developed to aggregate data for decision making issues. Aggregation operators based on Hamacher operations produce a transparent result in decision making. Thus inspired by these operators, Wei et al. [34] developed some induced geometric aggregation operators with intuitionistic fuzzy information and showed their applications to group decision making. Further contributions were made by Huang [25] with the introduction of a decision making model using intuitionistic fuzzy Hamacher aggregation operators. Waseem et al. [33] used Hamacher operators to aggregate data in an m–polar fuzzy setting. Later on, Jana and Pal [26] also presented the m–polar fuzzy operators and their application. Akram et al. [2, 10] adapted respective mathematical models to approach decisions in m-polar fuzzy environments. Since, the aggregation operators that collect information in a linguistic form are crucially essential. By using the 2-tuple linguistic tool, one can prevent loss of data and get more transparent results in decision making. Garg [18] gave a framework for linguistic prioritized aggregation operators. The pioneering exploration conducted by Akram et al. [3–5, 8, 9] has revealed a diverse set of innovative decision making techniques that effectively integrate linguistic data. Moreover, to gain a deeper comprehension of linguistic concepts and their practical implementations, readers are encouraged to explore the references [1, 6, 7, 20–23, 27–30]. The main goal of this study is the aggregation of 2-tuple linguistic information by using Hamacher operators and its application in decision making.

9.2 Preliminaries

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9.2 Preliminaries Definition 9.1 ([21, 24]) Let a set S = {si |i = 0, 1, . . . , t} of odd number of linguistic terms, where si indicates the probable linguistic term for the linguistic variables. For instance, a linguistic term set S having seven terms can be described as follows: S ={s0 = none, s1 = very low, s2 = low, s3 = medium, s4 = high, s5 = very high, s6 = perfect}. If si , sk ∈ S, then the set S meets with the following characteristics: (i) (ii) (iii) (iv)

Ordered set: si > sk , if and only if i > k. Max operator: max(si , sk ) = si , if and only if i ≥ k. Min operator: min(si , sk ) = si , if and only if i ≤ k. Negation: Neg(si ) = sk such that k =t−i.

Herrera and Martínez [24], introduced 2-tuple linguistic representation model based on the idea of symbolic translation, which is useful for representing the linguistic assessment information by means of a 2-tuple (si , ρi ). where, • si is a linguistic label for predefined linguistic term set S. • ρi is called symbolic translation and ρi ∈ [− 21 , 21 ). Definition 9.2 ([24]) Let ϕ be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation, ϕ ∈ [0, t], where t is the cardinality of S. Let i = round(ϕ) and ρ = ϕ − i be two values, such that, i ∈ [0, t] and ρ ∈ [− 21 , 21 ) then ρ is called a symbolic translation. Definition 9.3 ([24]) Let S = {si | i = 0, . . . , t} be a linguistic term set and ϕ ∈ [0, t] be a number value representing the aggregation result of linguistic symbolic. Then the function Δ used to obtain the 2-tuple linguistic information equivalent to ϕ is defined as:  1 1 , Δ : [0, t] → S × − , 2 2 

 Δ(ϕ) =

si , i = round(ϕ), ρ = ϕ − i, ρ ∈ [− 21 , 21 ).

(9.1)

Definition 9.4 ([24]) Let S = {si |i = 0, . . . , t} be a linguistic term set and (si , ρi ) be a 2-tuple, there exists a function Δ−1 that restore the 2-tuple to its equivalent numerical value ϕ ∈ [0, t] ⊂ R, where 1 1 Δ−1 : S × [− , ) → [0, t], 2 2 Δ−1 (si , ρ) = i + ρ = ϕ.

(9.2)

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Definition 9.5 ([21, 24]) Consider (sk , ρ1 ) and (sl , ρ2 ) be two 2-tuple linguistic values. Then, 1. 2. (i) (ii) (iii)

For k < l, we have, (sk , ρ1 ) ≤ (sl , ρ2 ). If k = l, then For ρ1 = ρ2 ⇒ (sl , ρ1 ) = (sk , ρ2 ). For ρ1 < ρ2 ⇒ (sl , ρ1 ) < (sk , ρ2 ). For ρ1 > ρ2 ⇒ (sl , ρ1 ) > (sk , ρ2 ).

Chen et al. [17] first considered the notion of m–polar fuzzy sets. The membership grade of m–polar fuzzy set belongs to interval [0, 1]m , and m stands for different division of an attributes. Definition 9.6 ([17]) An m–polar fuzzy set C on a non-empty set X is defined as a mapping C : X → [0, 1]m . The representation of membership value for every element x ∈ X is denoted as: C = ( p1 ◦ C(x), p2 ◦ C(x), . . . , pm ◦ C(x)), where pi ◦ C : [0, 1]m → [0, 1] is the i−th projection mapping. Definition 9.7 ([17]) The score function S of an m–polar fuzzy number, C = ( p1 ◦ C, . . . , pm ◦ C) is defined as:  1  ( pr ◦ C) , m r =1 m

S(C) =

S(C) ∈ [0, 1].

Definition 9.8 ([17]) Let C1 = ( p1 ◦ C1 , . . . , pm ◦ C1 ), and C2 = ( p1 ◦ C2 , . . . , pm ◦ C2 ) be two m–polar fuzzy numbers. Then 1. 2. 3. 4. 5.

C1 C1 C1 C1 C1

< C2 , if S(C1 ) < S(C2 ). > C2 , if S(C1 ) > S(C2 ). = C2 , if S(C1 ) = S(C2 ) and H(C1 ) = H(C2 ). < C2 , if S(C1 ) = S(C2 ), but H(C1 ) < H(C2 ). > C2 , if S(C1 ) = S(C2 ), but H(C1 ) > H(C2 ).

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators Akram et al. [6] defined the concept of 2-tuple linguistic m–polar fuzzy sets. Definition 9.9 ([6]) A 2-tuple linguistic m–polar fuzzy set Ψˆ on a nonempty set Y is defined as follows: Ψˆ = {< y, ((sψ1 (y), ρ1 (y)), (sψ2 (y), ρ2 (y)), . . . , (sψm (y) , ρm (y))) >: y ∈ Y },

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators

441

where (sψi (y), ρi (y)), represents the membership degrees, with the conditions sψi (y) ∈ S, ρi (y) ∈ [−0.5, 0.5), 0 ≤ Δ−1 (sψi (y), ρi (y)) ≤ t, i = 1, 2, . . . , m. For convenience, ξ = ((sψ1 , ρ1 ), (sψ2 , ρ2 ), . . . , (sψm , ρm )), is a 2-tuple linguistic m–polar fuzzy number, where 0 ≤ Δ−1 (sψi , ρi ) ≤ t, i = 1, 2, . . . , m. Definition 9.10 [6] The score function S of a 2-tuple linguistic m–polar fuzzy number ξ = ((sψ1 , ρ1 ), (sψ2 , ρ2 ), . . . , (sψm , ρm )), is defined as follows:  m  t  Δ−1 (sψr , ρr ) , S(ξ) = Δ m r =1 t 

Δ−1 (S(ξ)) ∈ [0, t].

Definition 9.11 ([6]) The accuracy function H of a 2-tuple linguistic m–polar fuzzy number ξ = ((sψ1 , ρ1 ), (sψ2 , ρ2 ), . . . , (sψm , ρm )), is defined as follows:  H(ξ) = Δ

   −1 m Δ (sψr , ρr ) t  −1 , (−1)r m r =1 t

Δ−1 (H(ξ)) ∈ [0, t].

Definition 9.12 ([6]) Let ξ1 = ((sψ11 , ρ11 ), (sψ21 , ρ12 ), . . . , (sψm1 , ρ1m )), and 2 2 ξ2 = ((sψ12 , ρ1 ), (sψ22 , ρ2 ), . . . , (sψm2 , ρ2m )), be two 2-tuple linguistic m–polar fuzzy numbers. Then the operations on 2-tuple linguistic m–polar fuzzy numbers are defined as follows:         ⎞⎞ ⎞ ⎛ ⎛ Δ−1 s 2 , ρ21 Δ−1 s 1 , ρ11 Δ−1 s 2 , ρ21 Δ−1 s 1 , ρ11 ψ1 ψ1 ψ1 ψ1 ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎜ Δ ⎝t ⎝ + − . ⎠⎠ , . . . , ⎟ ⎟ ⎜ t t t t ⎟ ⎜ ⎟ ⎜ ⎟; ⎜ ξ1 ⊕ ξ2 = ⎜ ⎛ ⎛         ⎞⎞ ⎟ ⎟ ⎜ ⎟ ⎜ Δ−1 s 1 , ρ1m Δ−1 s 2 , ρ2m Δ−1 s 1 , ρ1m Δ−1 s 2 , ρ2m ψm ψm ψm ψm ⎜ ⎜ ⎜ ⎟⎟ ⎟ Δ t + − . ⎝ ⎝ ⎝ ⎠⎠ ⎠ t t t t ⎛ ⎛ ⎛ ⎛ ⎛     ⎞⎞     ⎞⎞ ⎞ −1 s , ρ1 −1 s , ρ1 Δ−1 sψ2 , ρ21 ⎟⎟ Δ−1 sψm2 , ρ2m ⎟⎟ ⎟ 1 ψm m ⎜ ⎜ ⎜Δ ⎜ ⎜Δ ψ11 1 1 ⎜ ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎟⎟ ⎟ ξ1 ⊗ ξ2 = ⎜ Δ ⎜t ⎜ . . ⎟⎟ , . . . , Δ ⎜t ⎜ ⎟⎟ ⎟ ; ⎝ ⎝ ⎝ ⎠⎠ ⎝ ⎝ ⎠⎠ ⎠ t t t t ⎛

(1)

(2)

         Δ−1 sψ , ρ1 α  Δ−1 sψm , ρm α  1 Δ t 1− 1− ,...,Δ t 1 − 1 − ,α > 0 ; t t         Δ−1 s , ρ α    Δ−1 s , ρm α  1 ψ ψ m 1 ξα = Δ t ,...,Δ t ,α > 0 ; t t         ,α > 0 ; ξ c = Δ t − Δ−1 sψ , ρ1 , . . . , Δ t − Δ−1 sψm , ρm 

(3) (4) (5)

αξ =

1     ⎞     −1 s , ρ1 Δ−1 s 2 , ρ21    Δ−1 s 1 , ρ1   Δ Δ−1 s 2 , ρ2 ψ1 ψ11 1 ⎠; ⎝ ψm m ψm m 1 2 ≤ ,...,Δ t ≤ Δ t t t t t     ⎞⎞⎞ ⎛ ⎞ ⎛ ⎛ ⎛ −1 s , ρ1 Δ−1 s 2 , ρ21 ⎟⎟⎟ ⎜ ⎜Δ ⎟ ⎜ ⎜ ψ11 1 ψ1 ⎜ ⎜ ⎟⎟⎟ ⎜ Δ⎜ t ⎜max ⎜ , ⎟⎟⎟ , . . . , ⎟ ⎟ ⎜ ⎜ ⎝ ⎠⎠⎠ t t ⎟ ⎜ ⎝ ⎝ ⎟ ⎜  ⎜ ⎛ ⎛ ⎛   ⎞⎞⎞ ⎟   ξ1 ξ2 = ⎜ ⎟; ⎟ ⎜ 1 2 −1 −1 ⎜ s 1 , ρm s 2 , ρm ⎟⎟⎟ ⎟ Δ ⎜ ⎜ ⎜Δ ⎟ ⎜ ψ ψ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ m m ⎟ ⎜ Δ ⎜t ⎜max ⎜ , ⎟ ⎟ ⎟ ⎝ ⎝ ⎝ ⎝ ⎠⎠⎠ ⎠ t t ⎛

(6) ξ ⊆ ξ ⇔

(7)

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9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators ⎛ ⎛ ⎛     ⎞⎞⎞ ⎞ −1 s , ρ1 Δ−1 s 2 , ρ21 ⎟⎟⎟ 1 1 ⎜ ⎜Δ ψ ψ ⎜ ⎜ ⎟ ⎟⎟⎟ ⎜ ⎜ 1 1 ⎜ Δ⎜ , t ⎜min ⎜ ⎟⎟⎟ , . . . , ⎟ ⎜ ⎜ ⎟ ⎠⎠⎠ ⎝ t t ⎜ ⎝ ⎝ ⎟ ⎜ ⎟  ⎜ ⎛ ⎛ ⎛     ⎞⎞⎞ ⎟ ξ1 ξ2 = ⎜ ⎟. ⎜ ⎟ −1 s 1 ⎜ Δ−1 s 2 , ρ2m ⎟⎟⎟ ⎟ 1 , ρm ⎜ ⎜ ⎜Δ ψ ψ ⎜ m m ⎟⎟⎟ ⎟ ⎜ ⎜ ⎜ Δ⎜ , ⎟⎟⎟ ⎟ ⎜t ⎜min ⎜ ⎝ ⎠⎠⎠ ⎠ ⎝ ⎝ ⎝ t t ⎛

(8)

Definition 9.13 Let ξ1 = {(sψ11 , ρ11 ), ..., (sψm1 , ρ1m )} , ξ2 = {(sψ12 , ρ21 ), ..., (sψm2 , ρ2m )} and ξ = {(sψ1 , ρ1 ), ..., (sψm , ρm )} be 2-tuple linguistic m–polar fuzzy numbers. Then, the basic Hamacher operations for 2-tuple linguistic m–polar fuzzy numbers with λ > 0 are defined as follows: (1)

(2)

(3)

ξ1 ⊕ ξ2 =        ⎞⎞ ⎛ ⎛ ⎛ −1  sψ1 , ρ11 Δ−1 sψ2 , ρ21 Δ−1 sψ1 , ρ11 Δ−1 sψ2 , ρ21 Δ Δ−1 (sψ1 , ρ11 ) Δ−1 (sψ2 , ρ21 ) 1 1 1 1 1 1 ⎟⎟ ⎜ + − . − (1 − λ) . ⎜ ⎜ ⎟⎟ ⎜ ⎜ Δ⎜ t t t t t t ⎟⎟ , . . . , ⎜t ⎜   ⎜ ⎜ ⎟⎟ ⎜ ⎝ ⎜ Δ−1 (sψ1 , ρ11 ) Δ−1 (sψ2 , ρ21 ) ⎠⎠ ⎝ ⎜ 1 1 . 1 − (1 − λ) ⎜ t t ⎜ ⎜ ⎜ ⎜ ⎛ ⎛ ⎞⎞ ⎜ ⎜ Δ−1 (sψm2 , ρ2m ) Δ−1 (sψm1 , ρ1m ) Δ−1 (sψm2 , ρ2m ) Δ−1 (sψm1 , ρ1m ) Δ−1 (sψm2 , ρ2m ) Δ−1 (sψm1 , ρ1m ) ⎜ ⎜ ⎜ ⎟⎟ + − . − (1 − λ) . ⎜ ⎜ ⎜ ⎟⎟ t t t t t t ⎜ Δ ⎜t ⎜ ⎟⎟   ⎜ ⎜ ⎜ ⎟⎟ Δ−1 (sψm1 , ρ1m ) Δ−1 (sψm2 , ρ2m ) ⎜ ⎝ ⎝ ⎠⎠ ⎜ . 1 − (1 − λ) ⎝ t t

ξ ⊗ξ = ⎛1 ⎛2 ⎛

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟; ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞⎞

⎞

⎟⎟ ⎜ Δ−1 (s 1 , ρ11 ) Δ−1 (s 2 , ρ21 ) ⎟ ⎜ ⎜ ⎟⎟ ⎜ ψ1 ψ1 ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎜ . ⎟⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎜ t t ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎛ ⎞⎞ ⎟⎟ , . . . , ⎟ −1 (s , ρ1 ) −1 (s , ρ2 ) −1 (s , ρ1 ) Δ−1 (s , ρ2 ) ⎟ ⎜ ⎜ ⎜ Δ Δ Δ ⎟ ⎟ 1 1 2 1 1 1 2 1 ⎟ ⎜ ⎜ ⎜ ψ ψ ψ ψ ⎟ ⎟ ⎜ ⎟⎟ ⎠⎠ ⎟ ⎜ ⎝ ⎝⎜ 1 1 1 1 + − . ⎟ ⎜ ⎝λ + (1 − λ) ⎝ ⎠⎠ ⎟ ⎜ t t t t ⎟ ⎜ ⎟; ⎜ ⎟ ⎜ ⎞⎞ ⎟ ⎜ ⎛ ⎛ ⎟ ⎜ −1 (s 1 ) Δ−1 (s 2) ⎟ ⎜ Δ , ρ , ρ 1 2 m m ⎟⎟ ⎟ ⎜ ⎜ ⎜ ψm ψm ⎟⎟ ⎟ ⎜ ⎜ ⎜ . ⎟⎟ ⎟ ⎜ ⎜ ⎜ t t ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎞⎞ ⎟ ⎛ −1 ⎟⎟ ⎟ ⎜ ⎜ ⎜ Δ−1 (s 2 , ρ2m ) Δ−1 (s 1 , ρ1m ) Δ−1 (s 2 , ρ2m ) Δ (s 1 , ρ1m ) ⎟⎟ ⎟ ⎜ ⎜ ⎜ ψm ψm ψm ψm ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ ⎝λ + (1 − λ) ⎝ ⎠ ⎠ + − . t t t t    ⎞α ⎛ ⎞ ⎛ ⎞α ⎛ ⎞⎞ ⎛ ⎛   Δ−1 sψ , ρ1 Δ−1 sψ , ρ1 1 1 ⎜ ⎟ ⎝ ⎝ ⎠ ⎠ ⎟ ⎟ ⎜ ⎜ − 1 − 1 + λ − 1 ⎜ ⎟ ⎟⎟ ⎜ ⎜ t t ⎜ ⎟ ⎟⎟ ⎜ ⎜ Δ⎜ ⎟ ⎟ ⎜ ⎜   ⎞α   ⎞α ⎟⎟ , . . . , ⎟ t ⎜⎛ ⎜ ⎟ ⎛ ⎜ ⎜ ⎟ −1 −1 ⎟ ⎟ ⎜ ⎜     Δ sψ , ρ1 sψ , ρ1 Δ ⎜ ⎟ ⎝ ⎝⎝ 1 1 ⎜ ⎟ ⎠ + λ − 1 ⎝1 − ⎠ ⎠⎠ 1 + λ − 1 ⎜ ⎟ t t ⎜ ⎟ ⎜ ⎟ ⎟; αξ = ⎜ ⎜ ⎛ ⎛ ⎟   ⎞α ⎛   ⎞α ⎛ ⎞⎞ ⎜ ⎟ −1 −1  Δ sψm , ρm sψm , ρm Δ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎝ ⎠ ⎠ ⎝ ⎟⎟ − 1− 1+ λ−1 ⎜ ⎜ ⎜ ⎟ ⎟⎟ t t ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟,α > 0 ⎟   ⎞α   ⎞α ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎟ ⎛ ⎟⎟ ⎜ ⎜ ⎜ ⎟ −1 −1 ⎟⎟    Δ sψm , ρm sψm , ρm Δ ⎜ ⎝ ⎝ ⎟ ⎝ ⎠ ⎠ + λ − 1 ⎝1 − ⎠ ⎠⎠ ⎝1 + λ − 1 t t

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443

  ⎞α ⎞ ⎛ ⎞⎞ ⎛ ⎛ Δ−1 sψ , ρ1 1 ⎜ ⎟ ⎠ ⎝ ⎟ ⎟ ⎜ ⎜ λ ⎜ ⎟ ⎟⎟ ⎜ ⎜ t ⎜ ⎟ ⎟⎟ ⎜ ⎜ Δ⎜ ⎟ ⎟ ⎜ ⎜   ⎞⎞α   ⎞α ⎟⎟ , . . . , ⎟ t ⎜⎛ ⎜ ⎟ ⎛ ⎛ ⎜ ⎜ ⎟ −1 −1 ⎟ ⎟ ⎜ ⎜   Δ   Δ sψ , ρ1 sψ , ρ1 ⎜ ⎟ ⎝ ⎝⎝ 1 1 ⎜ ⎟ ⎝1 − ⎠⎠ + λ − 1 ⎝ ⎠ ⎠⎠ 1 + λ − 1 ⎜ ⎟ t t ⎜ ⎟ α ⎜ ⎟.   ⎞α ⎛ ξ =⎜ ⎛ ⎛ ⎞ ⎞ ⎟ Δ−1 sψm , ρm ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎠ ⎟⎟ λ⎝ ⎜ ⎜ ⎜ ⎟ ⎟⎟ t ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎟     , α > 0 ⎛ ⎛ ⎞⎞α ⎞α ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟   Δ−1 sψm , ρm   Δ−1 sψm , ρm ⎜ ⎝ ⎝ ⎟ ⎠ ⎠ ⎝ ⎠ ⎝1 + λ − 1 ⎝1 − ⎠⎠ + λ − 1 ⎝ ⎠ t t ⎛

(4)

Example 9.1 Let ξ1 = {(s3 , 0.2), (s2 , 0.5), (s4 , 0.0)} and ξ2 = {(s3 , 0.8), (s4 , 0.0), (s2 , 0.6)} be 2-tuple linguistic 3-polar fuzzy numbers. Then for λ = 3, 1.

ξ1 ⊕ ξ2 = ⎛ ⎛ ⎛ −1 −1 −1 −1 −1 −1 ⎜ Δ (s3 , 0.2) + Δ (s3 , 0.8) − Δ (s3 , 0.2) . Δ (s3 , 0.8) − (1 − 3) Δ (s3 , 0.2) . Δ (s3 , 0.8) ⎜ ⎜ ⎜ ⎜ ⎜ 4 4 4 4 4 4 ⎜   4 ⎜ Δ⎜ ⎜ ⎜ ⎜ Δ−1 (s3 , 0.2) Δ−1 (s3 , 0.8) ⎜ ⎝ ⎝ . 1 − (1 − 3) ⎜ 4 4 ⎜ ⎜ ⎜ ⎛ ⎛ ⎜ ⎜ −1 Δ−1 (s4 , 0.0) Δ−1 (s2 , 0.5) Δ−1 (s4 , 0.0) Δ−1 (s2 , 0.5) Δ−1 (s4 , 0.0) ⎜ ⎜ ⎜ Δ (s2 , 0.5) + − . − (1 − 3) . ⎜ ⎜ ⎜ ⎜ Δ ⎜4 ⎜ 4 4 4 4 4 4   ⎜ ⎜ ⎜ −1 (s , 0.5) Δ−1 (s , 0.0) ⎜ ⎝ ⎝ Δ 4 2 ⎜ . 1 − (1 − 3) ⎜ 4 4 ⎜ ⎜ ⎜ ⎛ ⎛ ⎜ ⎜ −1 −1 −1 −1 Δ (s2 , 0.6) Δ (s4 , 0.0) Δ (s2 , 0.6) Δ−1 (s4 , 0.0) Δ−1 (s2 , 0.6) ⎜ ⎜ ⎜ Δ (s4 , 0.0) + − . − (1 − 3) . ⎜ ⎜ ⎜ ⎜ Δ ⎜4 ⎜ 4 4 4 4 4 4   ⎜ ⎜ ⎜ ⎝ ⎝ ⎝ Δ−1 (s4 , 0.0) Δ−1 (s2 , 0.6) 1 − (1 − 3) . 4 4

⎞⎞ ⎞ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ , ⎟ ⎟⎟ ⎟ ⎠⎠ ⎟ ⎟ ⎟ ⎟ ⎞⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ , ⎟ ⎟⎟ ⎟ ⎠⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ , ⎟ ⎟⎟ ⎟ ⎠⎠ ⎠

={{(s4 , −0.0158), (s4 , 0.000), (s4 , 0.01)}. Thus, ξ1 ⊕ ξ2 is again a 2-tuple linguistic 3-polar fuzzy number. So, the closure law is satisfied. Thus in a similar pattern the closure law is verified for all the above defined Hamacher operations for 2-tuple linguistic m–polar fuzzy numbers. j

j

Definition 9.14 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1 tic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then, a 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator is a mapping 2T LmF H W A : ξ n → ξ , whose domain is the family of 2-tuple linguistic m–polar fuzzy numbers ξ n , which is defined as follows: 2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) =

n 

(φ j ξ j ),

(9.3)

j=1

weight vector representation for ξ j , for each ’ j’, where φ = (φ1 , φ2 , ..., φn ) is the j = 1, 2, ..., n, with φ j > 0 and nj=1 φ j = 1. j

j

Theorem 9.1 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1 tic m–polar fuzzy numbers, where j = 1, 2, ..., n. The assembled value of these

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9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

2-tuple linguistic m–polar fuzzy numbers using the 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator is also a 2-tuple linguistic m–polar fuzzy number, given as follows:  2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) = nj=1 (φ j ξ j ), ⎞⎞ ⎛ ⎛   ⎞φ   ⎞φ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ   1 1 n ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ − n ⎜1 − ⎟ ⎜ ⎟⎟ ⎜ ⎜ ⎜ ⎟ j=1 ⎝1 + (λ − 1) j=1 ⎝ ⎠ ⎠ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ Δ⎜ t⎜ ⎟⎟ , . . . , ⎟ ⎜ ⎜ ⎟     ⎞ ⎞ ⎛ ⎛ φ φ ⎜ ⎜ ⎜ ⎟ j j j j ⎟⎟ −1 −1 ⎟⎟ ⎜ Δ Δ s j , ρ1 s j , ρ1 ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ  ⎟⎟ 1 1 n ⎜ ⎜ ⎜ n ⎜1 + (λ − 1) ⎟ ⎟ ⎟ ⎜ 1 − + (λ − 1) ⎠⎠ ⎜ ⎝ ⎝ j=1 ⎝ ⎟ j=1 ⎠ ⎠ ⎝ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟. =⎜ ⎜ ⎞⎞ ⎟ ⎛ ⎛ ⎞ ⎛     ⎜ ⎟ φ ⎛ ⎞ φ j j j j ⎜ ⎟ −1 −1 Δ s j , ρm Δ s j , ρm ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎟ ⎜   ⎜ ψ ψ ⎜ ⎟ m m n n ⎟⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 1 + (λ − 1) 1 − − ⎝ ⎠ ⎟⎟ ⎟ ⎜ ⎜ j=1 ⎝ j=1 ⎠ ⎜ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ Δ⎜ ⎟⎟ ⎟ ⎜t ⎜     ⎜ ⎟ ⎞ ⎞ ⎛ ⎛ φ φ ⎜ ⎜ j j j j ⎟⎟ ⎟ ⎜ −1 −1 ⎟⎟ ⎟ ⎜ ⎜ Δ Δ s j , ρm s j , ρm ⎜ ⎟⎟ ⎟ ⎜ ⎜ n ⎜  ψ ψ ⎜ ⎟ ⎟ ⎜ m m n ⎟ ⎜ ⎜ ⎝ ⎠ + (λ − 1) j=1 ⎝1 − ⎠ ⎟ ⎠⎠ ⎠ ⎝ ⎝ j=1 ⎝1 + (λ − 1) t t ⎛

(9.4) Proof The mathematical induction is used to prove it. Case 1. Take n = 1, by using Eq. 9.4.   ⎞ ⎛ ⎞ ⎛ ⎛

⎞ ⎞⎞ ⎛ ⎛ Δ−1 s 1 , ρ11 Δ−1 s 1 , ρ11 ψ1 ψ1 ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎝1 + (λ − 1) ⎠ − ⎝1 − ⎠ ⎟⎟ ⎜ ⎜ ⎜ ⎟ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎟ ⎟  ⎞   ⎞ ⎟⎟ , . . . , ⎟ ⎛ ⎜ ⎜ ⎜⎛ ⎟ 1 1 −1 −1 ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ Δ Δ s 1 , ρ1 s 1 , ρ1 ⎜ ⎜ ⎜ ⎟ ⎟ ψ1 ψ1 ⎟ ⎜ ⎟⎟ ⎜ ⎝ ⎝ ⎜1 + (λ − 1) ⎟ ⎠ ⎠ ⎝ ⎠ + (λ − 1) ⎝1 − ⎠ ⎜ ⎟ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ . ⎞ ⎛   ⎛ ⎛ ⎞⎞ ⎟   ⎛ ⎞ 1 ⎜ ⎟ −1 1 Δ−1 s 1 , ρm ⎜ ⎟ Δ s 1 , ρm ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ψm ψm ⎟ ⎟ −⎜ ⎜1 + (λ − 1) ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎠ ⎜ ⎠ ⎝1 − ⎝ ⎜ ⎜ ⎟⎟ ⎟ t t ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎟⎟ ⎟ ⎜ Δ⎜ ⎜t ⎜ ⎛ ⎟⎟ ⎟     ⎞ ⎛ ⎞ ⎜ ⎜ ⎜ ⎟⎟ ⎟ −1 1 −1 1 ⎜ ⎜ ⎜ ⎟⎟ ⎟ , ρ , ρ Δ s s Δ 1 1 m m ⎜ ⎜ ⎜⎜ ⎟⎟ ⎟ ψm ψm ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎝1 + (λ − 1) ⎟⎟ ⎟ + (λ − 1) ⎝1 − ⎠ ⎠ ⎝ ⎝ ⎝ ⎠⎠ ⎠ t t

Thus, Eq. 9.4 is holds for n = 1. Case 2. Next it is supposed that the result is true for n = k, where k ∈ N (N:natural numbers).  2T LmF H W Aφ (ξ1 , ξ2 , ..., ξk ) = kj=1 (φ j ξ j ),

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators ⎞⎞ ⎛ ⎛   ⎞φ   ⎞φ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ   1 1 k ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ − k ⎜1 − ⎟ ⎜ ⎟⎟ ⎜ ⎜ ⎜ ⎟ j=1 ⎝1 + (λ − 1) j=1 ⎝ ⎠ ⎠ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ Δ⎜ ⎟ , . . . , t⎜ ⎟ ⎟ ⎜ ⎜ ⎟     ⎞φ ⎞φ ⎟⎟ ⎛ ⎛ ⎜ ⎜ ⎜ ⎟ j j j j ⎟⎟ ⎜ Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟⎟ ⎜ ψ1 ψ1 k ⎜ ⎜ ⎜ ⎜ k ⎜1 + (λ − 1) ⎟ ⎟ + (λ − 1) ⎟ ⎠⎠ ⎜1 − ⎜ ⎝ ⎝ j=1 ⎝ ⎟ j=1 ⎝ ⎠ ⎠ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟. =⎜ ⎜ ⎟ ⎞ ⎞ ⎛ ⎛ ⎞ ⎛     ⎜ ⎟ φj ⎛ j j ⎞φ j ⎜ ⎟ −1 s −1 s Δ , ρ Δ , ρ ⎜ ⎟ j m ⎟ m j ⎟ ⎟ ⎜ ⎜ ⎜   ⎜ ⎟ ψm ψm ⎜ ⎟ k k ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ 1 + (λ − 1) − j=1 ⎝1 − ⎠ ⎟ ⎟ ⎜ ⎜ j=1 ⎠ ⎝ ⎜ ⎟ t t ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ Δ ⎜t ⎜ ⎟ ⎟ ⎟     ⎜ ⎞φ j ⎞φ j ⎟⎟ ⎟ ⎛ ⎛ ⎜ ⎜ j j ⎜ ⎟ −1 s −1 s ⎟ ⎟ ⎜ ⎜ Δ Δ , ρ , ρ ⎜ ⎟ m m j j ⎟ ⎟ ⎜ ⎜   ψm ψm ⎜ ⎟ ⎟ ⎟⎟ ⎟ ⎜ k ⎜ ⎜ k ⎜ ⎝ ⎠ + (λ − 1) j=1 ⎝1 − ⎠ ⎠⎠ ⎠ ⎝ ⎝ j=1 ⎝1 + (λ − 1) t t ⎛

For n = k + 1,  2T LmF H W Aφ (ξ1 , ξ2 , ..., ξk+1 ) = kj=1 (φ j ξ j ) ⊕ (φk+1 ξk+1 ), ⎞⎞ ⎛ ⎛   ⎞φ   ⎞φ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ψ ψ k ⎜ 1 1 ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎟ − k ⎜ 1 − ⎟ ⎟⎟ ⎜ ⎟ ⎜ ⎜ j=1 ⎝1 + (λ − 1) j=1 ⎝ ⎠ ⎠ t t ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⎜ Δ⎜ , . . . , t⎜ ⎟ ⎟ ⎟ ⎜ ⎜   ⎞φ   ⎞φ ⎟⎟ ⎛ ⎛ ⎜ ⎟ ⎜ ⎜ j j j j ⎟⎟ ⎜ Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎜ ⎜ ψ1 ψ1  ⎜ ⎜ k k ⎟ ⎜ ⎟ + (λ − 1) ⎟ ⎠⎠ ⎜1 + (λ − 1) ⎜1 − ⎝ ⎝ ⎟ ⎜ j=1 ⎝ j=1 ⎝ ⎠ ⎠ t t ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ ⎟ ⎜ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛     ⎞φ ⎟ ⎜ φj ⎛ j j j ⎟ ⎜ −1 s −1 s Δ , ρ Δ , ρ ⎟ ⎜ j m ⎟ m j ⎜ ⎜ ⎟ ⎟ ⎜ k ⎜ k ⎜ ⎟ ⎜ ψm ψm ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ 1 − 1 + (λ − 1) − ⎝ ⎠ ⎜ ⎜ ⎟ ⎟ j=1 j=1 ⎠ ⎝ ⎟ ⎜ t t ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎟ ⎟   ⎞φ   ⎞φ ⎟⎟ ⎟ ⎟ ⎜ ⎛ ⎛ ⎜ ⎜ j j j j ⎟ ⎜ −1 s −1 s ⎜ ⎜ ⎟ ⎟ Δ Δ , ρ , ρ ⎜ m m j j ⎜ ⎜ k ⎜ ⎟ ⎟ k ⎜ ψm ψm ⎟ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ + (λ − 1) j=1 ⎝1 − ⎠ ⎠⎠ ⎠ ⎝ ⎝ j=1 ⎝1 + (λ − 1) t t ⎛

⎞⎞ ⎛ ⎛ ⎞ ⎛ ⎛   ⎞φ   ⎞φ k+1 k+1 Δ−1 s k+1 , ρk+1 Δ−1 s k+1 , ρk+1 1 1 ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ψ1 ψ1 ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜1 + (λ − 1) −⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎝1 − ⎠ ⎠ ⎝ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⎜ Δ⎜ , . . . , ⎟ ⎟ ⎟ ⎜ ⎜t ⎜ ⎛ ⎛   ⎞φ   ⎞φ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ k+1 k+1 k+1 k+1 −1 −1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ Δ Δ s k+1 , ρ1 s k+1 , ρ1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜⎜ ⎟ ⎟ ⎜ ψ1 ψ1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜1 + (λ − 1) ⎟ ⎟ ⎜1 − + (λ − 1) ⎠ ⎠ ⎝ ⎝ ⎟ ⎜ ⎠ ⎠ ⎝ ⎝ t t ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎛ ⎛ ⎞ ⎞ ⎛ ⎞   ⎟ ⎜ φk+1   ⎞φ ⎛ k+1 −1 s k+1 ⎟ ⎜ k+1 −1 Δ , ρ Δ s k+1 , ρm k+1 m ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ψm ψm ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ 1 + (λ − 1) − ⎝1 − ⎠ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎟ ⎜ t t ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎟ ⎟     ⎛ ⎞φk+1 ⎞φk+1 ⎟⎟ ⎟ ⎟ ⎜ ⎜ ⎜ k+1 k+1 −1 s −1 s ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ Δ Δ , ρ , ρ k+1 m k+1 m ⎜ ⎜ ⎜⎜ ⎟⎟ ⎟ ψm ψm ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ 1 + (λ − 1) + (λ − 1) 1 − ⎝ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝⎝ ⎠ ⎠ t t ⎛

445

446

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

⎞⎞ ⎛ ⎛   ⎞φ   ⎞φ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ   k+1 ⎜1 + (λ − 1) 1 1 ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ − k+1 ⎜1 − ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎠ ⎠ j=1 ⎝ j=1 ⎝ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ Δ⎜ ⎟ , . . . , t⎜ ⎟ ⎟ ⎜ ⎜ ⎟     ⎞φ ⎞φ ⎟⎟ ⎛ ⎛ ⎜ ⎜ ⎜ ⎟ j j j j ⎟⎟ ⎜ Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟⎟ ⎜ ψ1 ψ1 k+1 ⎜ ⎜ ⎜ ⎜ k+1 ⎜1 + (λ − 1) ⎟ ⎟ + (λ − 1) ⎟ ⎠⎠ ⎜1 − ⎜ ⎝ ⎝ j=1 ⎝ ⎟ ⎠ ⎠ j=1 ⎝ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟. =⎜ ⎜ ⎟ ⎞ ⎞ ⎛ ⎛ ⎞ ⎛     ⎜ ⎟ φj ⎛ j j ⎞φ j ⎜ ⎟ −1 s −1 s Δ , ρ Δ , ρ ⎜ ⎟ j m ⎟ m j ⎟ ⎟ ⎜ ⎜ ⎜   ⎜ ⎟ ψm ψm ⎜ ⎟ k+1 k+1 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ 1 + (λ − 1) − j=1 ⎝1 − ⎠ ⎟ ⎟ ⎜ ⎜ ⎠ ⎝ j=1 ⎜ ⎟ t t ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ Δ ⎜t ⎜ ⎟ ⎟ ⎟     ⎜ ⎞φ j ⎞φ j ⎟⎟ ⎟ ⎛ ⎛ ⎜ ⎜ j j ⎜ ⎟ −1 s −1 s ⎟ ⎟ ⎜ ⎜ Δ Δ , ρ , ρ ⎜ ⎟ m m j j ⎟ ⎟ ⎜ ⎜   ψm ψm ⎜ ⎟ ⎟ ⎟⎟ ⎟ k+1 ⎜ ⎜ ⎜ k+1 ⎜ ⎝ ⎠ + (λ − 1) j=1 ⎝1 − ⎠ ⎠⎠ ⎠ ⎝ ⎝ j=1 ⎝1 + (λ − 1) t t ⎛

Thus Eq. 9.4 holds for n = k + 1. Conclusively, result holds for any n ∈ N.



Example 9.2 Let ξ1 = {(s3 , 0.2), (s2 , 0.5), (s4 , 0.7), (s1 , 0.3)}, ξ2 = {(s3 , 0.8), (s4 , 0.6), (s2 , 0.6), (s1 , 0.4)} and ξ3 = {(s6 , 0), (s2 , 0.2), (s3 , 0.4), (s1 , 0.5)} be 2tuple linguistic 4-polar fuzzy numbers with a weight vector φ = (0.3, 0.5, 0.2). Then, for λ = 3.  2T LmF H W Aφ (ξ1 , ξ2 , ξ3 ) = 3j=1 (φ j ξ j ), ⎞⎞   ⎞φ   ⎞φ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ1 ψ1   3 3 ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ − ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎜ ⎟ j=1 ⎝1 + (λ − 1) j=1 ⎝1 − ⎠ ⎠ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎟ , . . . , ⎟ ⎟ ⎜ ⎜ ⎜ ⎟   ⎞φ   ⎞φ ⎟⎟ ⎛ ⎛ ⎜ ⎜ ⎜ ⎟ j j j j −1 −1 ⎟ ⎟ Δ Δ s j , ρ1 s j , ρ1 ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ  ⎟⎟ 1 1 3 3 ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ 1 + (λ − 1) 1 − + (λ − 1) ⎠⎠ ⎜ ⎝ ⎝ j=1 ⎝ ⎟ j=1 ⎝ ⎠ ⎠ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ =⎜ ⎜ ⎟ ⎛ ⎛ ⎞ ⎞ ⎛   ⎞φ j   ⎞φ ⎜ ⎟ ⎛ j j j ⎜ ⎟ −1 −1 Δ s j , ρm Δ s j , ρm ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟  3 ⎜ ⎜ ⎟ ψ ψ ⎜ ⎟ m m 3 ⎜ ⎜ ⎟ ⎟ ⎟ − ⎜1 + (λ − 1) ⎜ ⎟ 1 − ⎝ ⎠ ⎜ ⎜ ⎟ ⎟ j=1 ⎝ j=1 ⎠ ⎜ ⎟ t t ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎟ ⎟   ⎞φ   ⎞φ ⎟⎟ ⎟ ⎜ ⎟ ⎛ ⎛ ⎜ ⎜ j j j j ⎜ ⎟ −1 −1 ⎜ ⎜ ⎟ ⎟ Δ Δ s j , ρm s j , ρm ⎜ ⎜ ⎜ 3 ⎜ ⎟⎟ ⎟  ψ ψ ⎜ ⎟ ⎟ ⎜ m m 3 ⎜ ⎜ ⎟⎟ ⎟ 1 + (λ − 1) 1 − + (λ − 1) ⎝ ⎠ ⎠ ⎝ ⎝ j=1 ⎝ ⎝ j=1 ⎠⎠ ⎠ t t ⎛

⎛ ⎛

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators

447

⎛ ⎛    3.2 0.3    3.8 0.5    6 0.2  ⎞ 3.2 0.3  3.8 0.5  6 0.2 ⎞⎞ 1+ 2 × 1+ 2 × 1+ 2 − 1− × 1− × 1− ⎟⎟ ⎜ Δ ⎜6 ⎜ ⎟ 6 6 6 6 6 6 , ⎝ ⎝ ⎜   3.8 0.5    6 0.2    3.2 0.3  ⎟ 3.2 0.3  3.8 0.5  6 0.2  ⎠⎠ ⎜ ⎟ × 1+ 2 × 1+ 2 −2 1− × 1− × 1− 1+ 2 ⎜ ⎟ ⎜ ⎟ 6 6 6 6 6 6 ⎜ ⎟ ⎜ ⎛ ⎛  0.3  0.5  0.2 ⎞⎞ ⎟   4.6 0.5    2.2 0.2    2.5 0.3  ⎜ ⎟ 2.5 4.6 2.2 ⎜ ⎟ × 1 + 2 × 1 + 2 − 1 − × 1 − × 1 − 1 + 2 ⎜ ⎜ ⎜ ⎟⎟ ⎟ 6 6 6 6 6 6 ⎜ Δ ⎝6 ⎝  ⎟ , ⎠ ⎠                   ⎜ ⎟ 0.3 0.5 0.2 0.3 0.5 0.2 2.5 4.6 2.2 4.6 2.2 2.5 ⎜ ⎟ × 1+ 2 × 1+ 2 −2 1− × 1− × 1− 1+ 2 ⎜ ⎟ 6 6 6 6 6 6 ⎜ ⎟ ⎜ ⎟ ⎟ ⎞ ⎞ ⎛ ⎛ =⎜                   4.7 0.3 ⎜ ⎟ 4.7 0.3 2.6 0.5 3.4 0.2 2.6 0.5 3.4 0.2 ⎜ ⎟ × 1 + 2 × 1 + 2 − 1 − × 1 − × 1 − 1 + 2 ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ 6 6 6 6 6 6 ⎜ Δ ⎝6 ⎝  ,⎟ ⎠ ⎠                   0.3 0.5 0.2 0.3 0.5 0.2 ⎜ ⎟ 4.7 2.6 3.4 4.7 2.6 3.4 ⎜ ⎟ 1+ 2 × 1+ 2 × 1+ 2 −2 1− × 1− × 1− ⎜ ⎟ 6 6 6 6 6 6 ⎜ ⎟ ⎜ ⎟ ⎜ ⎛ ⎛  ⎟ ⎞ ⎞                  1.4 0.5 1.5 0.2 1.3 0.3 1.4 0.5 1.5 0.2 1.3 0.3 ⎜ ⎟ ⎜ ⎟ × 1+ 2 × 1+ 2 − 1− × 1− × 1− 1+ 2 ⎟⎟ ⎟ ⎜ ⎜ ⎜ 6 6 6 6 6 6 ⎜ Δ ⎝6 ⎝  ⎟   1.4 0.5    1.5 0.2    1.3 0.3  1.3 0.3  1.4 0.5  1.5 0.2  ⎠⎠ ⎟ ⎜ × 1+ 2 × 1+ 2 −2 1− × 1− × 1− 1+ 2 ⎝ ⎠ 6 6 6 6 6 6 ⎛

= ((s6 , 0.00), (s4 , −0.346), (s3 , 0.5142), (s1 , 0.2)). Remark 9.1 1. For λ = 1, 2-tuple linguistic m–polar fuzzy Hamacher weighted operator reduces to 2-tuple linguistic m–polar fuzzy  weighted average operator as given: 2T Lm F W Aφ (ξ1 , ξ2 , ..., ξn ) = nj=1 (φ j ξ j ),     ⎞ j j Δ−1 s j , ρ1     Δ−1 s j , ρ2       φj φj ψ1 ψ2 ⎟ ⎜   ⎜ Δ t 1− n , Δ t 1 − nj=1 1 − , ..., ⎟ ⎟ ⎜ j=1 1 − t t ⎟.   =⎜ ⎟ ⎜ j ⎟ ⎜ Δ−1 s j , ρm φ    ⎠ ⎝ j n ψm Δ t 1 − j=1 1 − t ⎛

2. For λ = 2, 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator reduces to 2-tuple linguistic m–polar fuzzy Einstein weighted averaging operator as below:  2T Lm F E W Aφ (ξ1 , ξ2 , ..., ξn ) = nj=1 (φ j ξ j ),   ⎞φ   ⎞φ ⎞⎞ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ n ⎜ ⎟ ⎟ ⎟ ⎜ ψ1 ψ1  n ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ − ⎟ ⎟⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎜ j=1 ⎝1 + j=1 ⎝1 − ⎠ ⎠ ⎟⎟ t t ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ Δ ⎜t ⎜ , . . . , ⎟ ⎟ ⎟ ⎜ ⎜ ⎜   ⎞φ   ⎞φ ⎟⎟ ⎛ ⎛ ⎟ ⎜ ⎜ ⎜ j j j j −1 −1 ⎟ ⎟ Δ Δ s j , ρ1 s j , ρ1 ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ψ ψ  ⎟⎟ 1 1 n n ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ 1 + 1 − + ⎠⎠ ⎟ ⎜ ⎝ ⎝ j=1 ⎝ j=1 ⎝ ⎠ ⎠ t t ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. =⎜ ⎟ ⎜ ⎞ ⎞ ⎛ ⎛ ⎛   ⎞φ j ⎟   ⎞φ ⎜ ⎛ j j j ⎟ ⎜ −1 −1 s j , ρm Δ Δ s j , ρm ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ n ⎜ n ⎜ ⎟ ⎜ ψ ψ ⎟ m m ⎟ ⎟ ⎜ ⎜ ⎟ − ⎜1 + ⎟ ⎜ 1 − ⎝ ⎠ ⎟ ⎟ ⎜ ⎜ j=1 ⎝ j=1 ⎠ ⎟ ⎜ t t ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ Δ ⎜t ⎜ ⎟ ⎟   ⎞φ   ⎞φ ⎟⎟ ⎟ ⎟ ⎜ ⎛ ⎛ ⎜ ⎜ j j j j ⎟ ⎜ −1 −1 ⎟ ⎟ ⎜ ⎜ Δ Δ s j , ρm s j , ρm ⎜ ⎟⎟ ⎟ ⎜ ⎜ n ⎜  ψ ψ ⎜ ⎜ ⎟ ⎟ m m n ⎟⎟ ⎟ ⎜ ⎜ 1 + 1 − + ⎝ ⎝ ⎝ ⎠ ⎠ ⎠⎠ ⎠ ⎝ ⎝ j=1 j=1 t t ⎛

⎛ ⎛

j

j

Theorem 9.2 Idempotency: Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 21 tuple linguistic m–polar fuzzy numbers, where j = 1, 2, ..., n. If all these numbers are equal, that is, ξ j = ξ, ∀ ’j’ varies 1 to n: 2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) = ξ.

(9.5)

448

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

ˆ where j = Proof Since 2T Lm F H W Aφ ((sψ j , ρ1 ), (sψ j , ρ1 ), ..., (sψ j , ρ1 )) = ξ, 1 1 1 1, 2, ..., n, then by using Eq. 9.4.  2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) = nj=1 (φ j ξ j ), j

j

j

⎞⎞ ⎛ ⎛   ⎞φ   ⎞φ ⎞ ⎛ ⎛ j j j j Δ−1 s j , ρ1 Δ−1 s j , ρ1 ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ   1 1 n ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ − n ⎜1 − ⎟ ⎜ ⎟⎟ ⎜ ⎜ ⎜ ⎟ j=1 ⎝1 + (λ − 1) j=1 ⎝ ⎠ ⎠ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ Δ⎜ ,..., ⎟ t⎜ ⎟ ⎟ ⎜ ⎜ ⎟     ⎞ ⎞ ⎛ ⎛ φj φ j ⎟⎟ ⎜ ⎜ ⎜ ⎟ j j −1 −1 ⎟⎟ ⎜ s j , ρ1 s j , ρ1 Δ Δ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ψ ψ  ⎟ ⎟ 1 1 n ⎜ ⎜ ⎜ n ⎜1 + (λ − 1) ⎟ ⎟ ⎟ ⎜ + (λ − 1) j=1 ⎝1 − ⎠⎠ ⎜ ⎝ ⎝ j=1 ⎝ ⎟ ⎠ ⎠ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ ⎟ ⎞ ⎞ ⎛ ⎛ ⎞ ⎛     ⎜ ⎟ φ ⎛ ⎞φ j j j j ⎜ ⎟ −1 −1 s Δ s j , ρm Δ , ρ ⎜ ⎟ m j ⎟ ⎟ ⎜ ⎜ ⎟ ⎜   ⎜ ⎟ ψ ψ ⎜ ⎟ m m n n ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ 1 + (λ − 1) 1 − − ⎝ ⎠ ⎟ ⎟ ⎜ ⎜ j=1 j=1 ⎠ ⎝ ⎜ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ Δ⎜ ⎟⎟ ⎟ ⎜t ⎜     ⎜ ⎛ ⎛ ⎜ ⎜ j ⎞φ j j ⎞φ j ⎟⎟ ⎟ ⎜ −1 −1 ⎟⎟ ⎟ ⎜ ⎜ Δ Δ s j , ρm s j , ρm ⎜ ⎟ ⎟ ⎜ ⎜ n ⎜ n ⎜ ψm ψm ⎜ ⎟ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ 1 + (λ − 1) 1 − + (λ − 1) ⎝ ⎝ ⎠ ⎠ ⎟ ⎠ ⎠ ⎠ ⎝ ⎝ j=1 ⎝ j=1 t t ⎛

  ⎞φ ⎛   ⎞φ ⎞ ⎞⎞ ⎛ ⎛ ⎛ Δ−1 sψ , ρ1 Δ−1 sψ , ρ1 1 1 ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎠ ⎝ − 1− 1 + (λ − 1) ⎟ ⎜ ⎜ ⎜ ⎟⎟ t t ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ , . . . , ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎟ ⎟     ⎛ ⎞φ ⎞φ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟⎟ Δ−1 sψ , ρ1 Δ−1 sψ , ρ1 ⎟ ⎜ ⎝ ⎝⎝ ⎠ ⎠ 1 1 ⎠ + (λ − 1) ⎝1 − ⎠ 1 + (λ − 1) ⎟ ⎜ ⎟ ⎜ t t ⎟ ⎜ ⎟ ⎜ =⎜ ⎟ ⎟ ⎜     ⎞ ⎞ ⎛ ⎛ ⎞φ ⎛ ⎞φ ⎛ ⎟ ⎜ Δ−1 sψm , ρm Δ−1 sψm , ρm ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎠ − ⎝1 − ⎠ ⎝1 + (λ − 1) ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎛   ⎞φ   ⎞φ ⎟⎟ ⎟ ⎛ ⎜ ⎜ ⎟ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ Δ−1 sψm , ρm Δ−1 sψm , ρm ⎝ ⎝ ⎝1 + (λ − 1) ⎝ ⎠ + (λ − 1) ⎝1 − ⎠ ⎠⎠ ⎠ t t ⎛

= ((sψ1 , ρ1 ), (sψ1 , ρ1 ), ..., (sψ1 , ρ1 )), f or λ = 1 = ξ. Hence 2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) = ξ holds only if ξ j = ξ where ∀ j = 1, 2, ..., n. 

j

j

Theorem 9.3 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )), be a collection of 2-tuple lin1  guistic m–polar fuzzy numbers, where ’j’ varies from 1 to n, ξ − = nj=1 ξ j and  ξ + = nj=1 ξ j , then ξ − ≤ 2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) ≤ ξ + .

(9.6)



Theorem 9.4 Let ξ j and ξ j , j = 1, 2, ..., n be the two collections of 2-tuple linguis tic m–polar fuzzy numbers. If ξ j ≤ ξ j , then 2T Lm F H W Aφ (ξ1 , ξ2 , ..., ξn ) ≤ 2T Lm F H W Aφ (ξ´1 , ξ´2 , ..., ξ´n ).

(9.7)

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators j

449

j

Definition 9.15 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be the set of 2-tuple linguistic m– 1 polar fuzzy numbers, where j = 1, 2, ..., n. Then, a 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator is a mapping 2T Lm F H O W A : ξ n → ξ with a weight vector w = (w1 , w2 , ..., wn ), w j ∈ (0, 1] and nj=1 w j = 1. Then 2T Lm F H O W Aφ (ξ1 , ξ2 , ..., ξn ) =

n 

(w j ξσ( j) ),

(9.8)

j=1

where, σ( j) = (σ(1), σ(2), ..., σ(n)) is the permutation of the indices, j = 1, 2, ..., n, for which ξσ( j−1) ≥ ξσ( j) , ∀ j = 1, 2, ..., n. j

j

Theorem 9.5 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection for 2-tuple linguis1 tic m–polar fuzzy numbers, where ’j’ varies from 1 to n. Then the assembled value of these 2-tuple linguistic m–polar fuzzy numbers using the 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator is again a 2-tuple linguistic m– polar fuzzy number, given as follows:  2T Lm F H O W Aφ (ξ1 , ξ2 , ..., ξn ) = nj=1 (w j ξσ( j) ), ⎛ ⎛   ⎞w   ⎞w ⎛ ⎛ σ( j) σ( j) j j Δ−1 s σ( j) , ρ1 Δ−1 s σ( j) , ρ1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ψ ψ n n ⎜ ⎜ ⎟ ⎟ 1 1 ⎜ ⎜ ⎜ 1 + (λ − 1) − j=1 ⎜1 − ⎟ ⎟ ⎜ ⎜ ⎜ j=1 ⎜ ⎝ ⎝ ⎠ ⎠ t t ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Δ ⎜t ⎜ ⎛ ⎛ ⎞  ⎞w    ⎜ ⎜ w ⎜ σ( j) σ( j) j j ⎜ ⎜ ⎜ Δ−1 s σ( j) , ρ1 Δ−1 s σ( j) , ρ1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ψ ψ ⎜ ⎜ ⎜ n  ⎜ ⎜ ⎟ ⎟ 1 1 n ⎜ ⎜ ⎜ 1 − 1 + (λ − 1) + (λ − 1) ⎟ ⎟ ⎜ ⎝ ⎝ j=1 ⎜ j=1 ⎜ ⎝ ⎝ ⎠ ⎠ t t ⎜ ⎜ ⎜ =⎜ ⎜ ⎛ ⎛ ⎛    ⎛ ⎜ σ( j)  ⎞w j σ( j) ⎞w j ⎜ Δ−1 s σ( j) , ρm Δ−1 s σ( j) , ρm ⎜ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ n  ψ ψ ⎜ ⎟ m m ⎟ ⎜ ⎜ ⎜ 1 + (λ − 1) − nj=1 ⎜ ⎟ ⎝1 − ⎠ ⎜ ⎜ ⎜ j=1 ⎜ ⎝ ⎠ t t ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜Δ t⎜     ⎛ ⎛ ⎜ ⎜ σ( j) ⎞w j σ( j) ⎞w j ⎜ ⎜ ⎜ Δ−1 s σ( j) , ρm Δ−1 s σ( j) , ρm ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ n ψm ψm ⎜ n ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ + (λ − 1) j=1 ⎝1 − 1 + (λ − 1) ⎝ ⎜ ⎠ ⎠ ⎝ ⎝ j=1 ⎝ t t ⎛

⎞⎞

⎞

⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ , . . . , ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎠⎠ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎞⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎠ ⎠⎠

Proof The proof of the theorem directly follows by the similar arguments as used in Theorem 9.1, above. 

Remark 9.2 1. For λ = 1, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator reduces to 2-tuple linguistic m–polar fuzzy ordered weighted average operator as below:

450

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

2T Lm F O W Aw (ξ1 , ξ2 , ..., ξn ) =

n 

(w j ξσ( j) )

j=1

  ⎞ σ( j) w  −1    j σ( j) , ρ Δ s 1 ψ1 ⎜ Δ t 1 − n , ⎟ ⎜ ⎟ j=1 1 − t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟   ⎜  ⎟ σ( j) w  −1  ⎜ ⎟ j σ( j) , ρ Δ s 2  ψ2 ⎜ ⎟ n = ⎜Δ t 1 − 1− , ..., ⎟ . j=1 ⎜ ⎟ t ⎜ ⎟ ⎜ ⎟   ⎜ ⎟ σ( j) w  ⎟ ⎜  −1  j Δ sψmσ( j) , ρm ⎝ ⎠ n Δ t 1 − j=1 1 − t ⎛

(9.9)

2. For λ = 2, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator reduces to 2-tuple linguistic m–polar fuzzy Einstein ordered weighted average operator as given below:  2T Lm F E O W Aw (ξ1 , ξ2 , ..., ξn ) = nj=1 (w j ξσ( j) ), ⎛ ⎛   ⎞w   ⎞w ⎞⎞ ⎛ ⎛ ⎞ σ( j) σ( j) j j Δ−1 s σ( j) , ρ1 Δ−1 s σ( j) , ρ1 ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ψ ψ n ⎜ ⎜ ⎟ ⎟ 1 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ n 1+ − j=1 ⎜1 − ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ j=1 ⎜ ⎝ ⎝ ⎠ ⎠ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ , . . . , ⎟ ⎜ Δ ⎜t ⎜ ⎞ ⎛ ⎞ ⎛     wj w j ⎟⎟ ⎟ ⎜ ⎜ ⎜ σ( j) σ( j) −1 −1 ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ s σ( j) , ρ1 s σ( j) , ρ1 Δ Δ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ψ1 ψ1 ⎟⎟ ⎟ ⎜ ⎜ ⎜ n n ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ + j=1 ⎜1 − 1+ ⎟ ⎟ ⎠ ⎠ ⎟ ⎜ ⎝ ⎝ j=1 ⎜ ⎠ ⎝ ⎠ ⎝ t t ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. =⎜ ⎟ ⎜ ⎛ ⎞   ⎛ ⎛ ⎞ ⎞ w   ⎛ ⎞w j ⎟ ⎜ σ( j) j σ( j) −1 ⎟ ⎜ −1 s σ( j) , ρm Δ s , ρ Δ ⎟ ⎜ m σ( j) ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜  ψ ψ ⎜ ⎟ m m ⎜ ⎜ n ⎟⎟ ⎟ n ⎜ ⎟ ⎜ 1 + − 1 − ⎜ ⎟ ⎜ ⎜ j=1 ⎝ ⎟⎟ ⎟ ⎝ ⎠ ⎜ j=1 ⎠ t t ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ Δ ⎜t ⎜     ⎞ ⎞ ⎛ ⎛ ⎟ ⎟ w w ⎜ j j ⎟ σ( j) σ( j) −1 −1 ⎜ ⎜ ⎟⎟ ⎟ ⎜ s σ( j) , ρm s σ( j) , ρm Δ Δ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜  ψ ψ ⎜ ⎜ n ⎟ ⎟ ⎟ ⎜ m m n ⎟ ⎟ ⎜1 − ⎜1 + ⎜ ⎜ ⎟⎟ ⎟ + j=1 ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ j=1 ⎝ ⎠⎠ ⎠ t t ⎛

j

(9.10)

j

Theorem 9.6 Idempotency: Consider ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) a collection of 1 2-tuple linguistic m–polar fuzzy numbers, where j = 1, 2, ..., n. For the equality of all these numbers, in other words, ξ j = ξ, where, ∀ j = 1, 2, ..., n, then: 2T Lm F H O W Aw (ξ1 , ξ2 , ..., ξn ) = ξ. j

j

j

(9.11)

Proof Since 2T Lm F H O W Aw ((sψ j , ρ1 ), (sψ j , ρ1 ), ..., (sψ j , ρ1 )) = ξ, where j = 1 1 1 1, 2, ..., n. Then by using Eq. 9.10, n 2T Lm F H O W Aw (ξ1 , ξ2 , ..., ξn ) = j=1 (w j ξ j ),

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators

451

⎛ ⎛   ⎞w   ⎞w ⎛ ⎛ ⎞⎞ ⎞ σ( j) σ( j) j j Δ−1 s σ( j) , ρ1 Δ−1 s σ( j) , ρ1 ⎜ ⎟ ⎟ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ψ ψ   ⎜ ⎟ ⎟ ⎜ 1 1 ⎟⎟ n ⎟ ⎜ ⎜ ⎜ 1 + (λ − 1) − nj=1 ⎜1 − ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ j=1 ⎜ ⎝ ⎠ ⎠ ⎝ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎛   ⎞w ⎟⎟ , . . . , ⎟   ⎞w ⎟ ⎜ ⎜ ⎜ σ( j) σ( j) j j ⎟⎟ −1 s −1 s ⎟ ⎜ ⎜ ⎜ Δ , ρ Δ , ρ σ( j) 1 σ( j) 1 ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ψ1 ψ1 ⎟⎟ ⎟ ⎜ ⎜ ⎜ n n ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ 1 − 1 + (λ − 1) + (λ − 1) ⎜ ⎜ ⎟ ⎟ ⎠⎠ ⎟ ⎜ ⎝ ⎝ j=1 ⎝ j=1 ⎝ ⎠ ⎠ t t ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ =⎜ ⎛  ⎞w j  ⎞⎞ ⎟ ⎛ ⎛   ⎞w ⎛ ⎟ ⎜ σ( j) σ( j) j −1 s ⎟ ⎜ −1 , ρ Δ s σ( j) , ρm Δ σ( j) m ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ n n ψm ψ ⎜ ⎟ m ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 − ⎟ ⎜ 1 + (λ − 1) − ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎠ ⎜ j=1 ⎝ j=1 ⎝ ⎠ t t ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ Δ⎜     ⎛ ⎛ ⎟ ⎟ ⎜t ⎜ ⎜ σ( j) ⎞w j ⎟ σ( j) ⎞w j −1 s −1 s ⎟⎟ ⎟ ⎜ ⎜ ⎜ , ρ , ρ Δ Δ m m σ( j) σ( j) ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜   ψ ψ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ m m ⎟ ⎟ ⎜1 + (λ − 1) ⎟⎟ ⎟ ⎜ ⎜ n + (λ − 1) nj=1 ⎜ ⎝ ⎠ ⎝1 − ⎠ ⎠⎠ ⎠ ⎝ ⎝ j=1 ⎝ t t ⎛

  ⎞w ⎛   ⎞w ⎛ ⎞ ⎛ ⎛ ⎞⎞ Δ−1 sψ , ρ1 Δ−1 sψ , ρ1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ 1 1 ⎟⎟ ⎝1 + (λ − 1) ⎠ − ⎝1 − ⎠ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ t t ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ Δ ⎜t ⎜ ⎛ ⎟⎟ , . . . , ⎟     ⎞ ⎛ ⎞ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ w w −1 s , ρ −1 s , ρ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ Δ Δ ψ 1 1 ⎟ ⎟⎟ ψ1 1 ⎟ ⎜ ⎜ ⎜⎜ ⎟ ⎜ ⎜ ⎝ ⎝ ⎝1 + (λ − 1) ⎟ ⎠⎠ + (λ − 1) 1 − ⎠ ⎝ ⎠ ⎜ ⎟ t t ⎜ ⎟ ⎜ ⎟ =⎜ ⎟ ⎜     ⎛ ⎞w ⎛ ⎞w ⎛ ⎛ ⎞⎞ ⎟ ⎜ ⎟ −1 −1 ⎜ ⎟ sψ , ρm Δ sψ , ρm Δ ⎜ m m ⎝1 + (λ − 1) ⎠ − ⎝1 − ⎠ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ t t ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ Δ⎜ ⎟   ⎞w   ⎞w ⎟ ⎛ ⎜t ⎜ ⎛ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ Δ−1 sψ , ρm Δ−1 sψ , ρm ⎜ ⎝ ⎝⎝ ⎠⎠ ⎟ m m ⎝ ⎠ ⎠ ⎠ ⎝ 1 + (λ − 1) + (λ − 1) 1 − t t ⎛

= ((sψ1 , ρ1 ), (sψ1 , ρ1 ), ..., (sψ1 , ρ1 )), for λ = 1 = ξ. Hence 2T L Lm F H O W Aφ (ξ1 , ξ2 , ..., ξn ) = ξ holds only if ξ j = ξ, ∀ j = 1, 2, ..., n. 

j

j

Theorem 9.7 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )), be a collection of 2-tuple linguis1   tic m–polar fuzzy numbers, where j = 1, 2, ..., n, ξ − = nj=1 ξ j and ξ + = nj=1 ξ j , then ξ − ≤ 2T Lm F H O W Aφ (ξ1 , ξ2 , ..., ξn ) ≤ ξ + .

(9.12)



Theorem 9.8 Let ξ j and ξ j , j = 1, 2, ..., n be two collections of 2-tuple linguistic  m–polar fuzzy numbers . If ξ j ≤ ξ j , then 2T Lm F H O W Aφ (ξ1 , ξ2 , ..., ξn ) ≤ 2T Lm F H O W Aφ (ξ´1 , ξ´2 , ..., ξ´n ).

(9.13)



Theorem 9.9 Let ξ j and ξ j , j = 1, 2, ..., n be two collections of 2-tuple linguistic m–polar fuzzy numbers, then 2T Lm F H O W Aφ (ξ1 , ξ2 , ..., ξn ) = 2T Lm F H O W Aφ (ξ´1 , ξ´2 , ..., ξ´n ), 

where, ξ j is the arbitrary permutation of ξ j .

(9.14)

452

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

Remark 9.3 i. In Definitions 9.14 and 9.15, it is observed that 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator and 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator with 2-tuple linguistic m–polar fuzzy numbers and ordered arrangements of 2-tuple linguistic m–polar fuzzy numbers are discussed respectively. ii. Another operator, namely 2-tuple linguistic m–polar fuzzy Hamacher hybrid averaging operator is presented, which combines the qualities of 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator and 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator. j

j

Definition 9.16 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple lin1 guistic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then, a 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator is defined as: 2T Lm F H H Aw,φ (ξ1 , ξ2 , ..., ξn ) =

n  (w j ξ˜σ( j) ),

(9.15)

j=1

where w = (w1 , w2 , ..., wn ) represents the associated weight vector of the 2-tuple linguistic m–polar fuzzy numbers ξ j , instead of weighting the experts for each ’ j’  varies from 1 to n, with w j ∈ (0, 1] and nj=1 w j = 1, ξ˜σ( j) is the jth biggest 2-tuple linguistic m–polar fuzzy numbers of the ξ j ( j = 1, 2, ..., n) with ξ˜σ( j) = (nφ j )ξσ( j) , ( j = 1, 2, ..., n), φ = (φ1 , φ2 , ..., φn ) is the weight vector for the ordered arguments , with φ j ∈ (0, 1], nj=1 φ j = 1 and n serves as the balancing coefficient. The 2-tuple linguistic weighted average operators integrate the importance of linguistic arguments. where, 2-tuple linguistic ordered weighted aggregation operators increase the worth of ordered positions of the linguistic arguments. There are different techniques to evaluate the weight vectors. For this [37] proposed a interesting approach to evaluate the weight vector. Particularly, the weight values are assigned to linguistic terms according to their importance in real life issues. Remark 9.4 Notice that, if w = ( n1 , n1 , ..., n1 ) is given, then 2-tuple linguistic m– polar fuzzy Hamacher hybrid average operator converts into 2-tuple linguistic m– polar fuzzy Hamacher weighted average operator, when φ = ( n1 , n1 , ..., n1 ), then 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator degenerates into 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator. Therefore, 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator is the generalization of the operators, namely, 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators and 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average, which explains the degrees and ordered arguments of the given 2-tuple linguistic m–polar fuzzy values. j

j

Theorem 9.10 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1 tic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then the assembled value of these

9.3 2-Tuple Linguistic m–Polar Fuzzy Hamacher Aggregation Operators

453

2-tuple linguistic m–polar fuzzy numbers using the 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator is again a 2-tuple linguistic m–polar fuzzy number, which is given as:  2T Lm F H H Aw,φ (ξ1 , ξ2 , ..., ξn ) = nj=1 (w j ξ˜σ( j) ),     ⎛ ⎛ σ( j) σ( j) Δ−1 sψσ( j) , ρ1    Δ−1 sψσ( j) , ρ1  w j  w j  n n 1 1 ⎜ ⎜ − j=1 1 − ⎜ ⎜ j=1 1 + λ − 1 t t ⎜     = Δ⎜ ⎜t ⎜ σ( j) σ( j) ⎝ ⎝ Δ−1 sψσ( j) , ρ1     Δ−1 sψσ( j) , ρ1  w j  w j n n 1 1 + λ−1 j=1 1 + λ − 1 j=1 1 − t t     j) σ( j)   Δ−1 sψσ( j) , ρσ(  w j  w j Δ−1 sψσ( j) , ρm m n  m m n ⎜ ⎜ − j=1 1 − ⎜ ⎜ j=1 1 + λ − 1 t t ⎜ ⎜     Δ ⎜t ⎜ j) σ( j) ⎝ ⎝     Δ−1 sψσ( j) , ρσ(  w j  w j Δ−1 sψσ( j) , ρm m m m n n + λ − 1 1 + λ − 1 1 − j=1 j=1 t t ⎛ ⎛

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ , . . . , ⎟⎟ ⎠⎠

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ . ⎟⎟ ⎠⎠

(9.16)

Proof The proof of the theorem directly follows by the similar arguments as used in Theorem 9.1. 

Remark 9.5 1. For λ = 1, 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator reduces to 2-tuple linguistic m–polar fuzzy hybrid average operator as below:  2T Lm F H Aw,φ (ξ1 , ξ2 , ..., ξn ) = nj=1 (w j ξ˜σ( j) ),   n  Δ−1 s σ( j) , ρσ( j)     wj 1 ψ1 , =Δ t 1− t j=1     j) n  Δ−1 s σ( j) , ρσ( j)   n  Δ−1 s σ( j) , ρσ(  w j     m wj 2 ψ2 ψm , ..., Δ t 1 − . Δ t 1− t t j=1

j=1

(9.17) 2. For λ = 2, 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator reduces to 2-tuple linguistic m–polar fuzzy Einstein Hamacher average operator as given: 2T Lm F E H Aw,φ (ξ1 , ξ2 , ..., ξn ) =

n  (w j ξ˜σ( j) ) j=1

    σ( j) σ( j) Δ−1 sψσ( j) , ρ1 Δ−1 sψσ( j) , ρ1 w j  w j    1 1 ⎜ ⎜ n − nj=1 1 − ⎜ ⎜ j=1 1 + ⎜  t   t  = Δ⎜ ⎜t ⎜ σ( j) σ( j) ⎝ ⎝ Δ−1 sψσ( j) , ρ1 Δ−1 sψσ( j) , ρ1 w j w j    n n 1 1 + . j=1 1 − j=1 1 + t t ⎛ ⎛

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ , . . . , ⎟⎟ ⎠⎠

454

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators     σ( j) σ( j) Δ−1 s σ( j) , ρm Δ−1 s σ( j) , ρm w j  w j    ψ ψ m m ⎜ ⎜ n − nj=1 1 − ⎜ ⎜ j=1 1 + ⎜  t   t  Δ⎜ ⎜t ⎜ σ( j) σ( j) −1 s −1 s ⎝ ⎝ Δ Δ , ρ    w j σ( j) σ( j) , ρm m wj n n ψm ψm + j=1 1 − j=1 1 + t t ⎛ ⎛

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ . ⎟⎟ ⎠⎠

(9.18)

9.4 2-Tuple Linguistic m–Polar Fuzzy Hamacher Geometric Aggregation Operators Akram et al. [6] discussed 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator, and 2-tuple linguistic m–polar fuzzy Hamacher hybrid average geometric operator. j

j

Definition 9.17 ([6]) Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple 1 linguistic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then, a 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator is a mapping, 2T Lm F H W G : ξ n → ξ, whose domain is the set of 2-tuple linguistic m–polar fuzzy numbers, is defined below: 2T Lm F H W G φ (ξ1 , ξ2 , ..., ξn ) =

n  (φ j ξ j )φ j ,

(9.19)

j=1

the weight vector of ξ j , for each ’ j’ varies where φ = (φ1 , φ2 , ..., φn ) represents  from 1 to n, with φ j > 0 and nj=1 φ j = 1. j

j

Theorem 9.11 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a set of 2-tuple linguistic m– 1 polar fuzzy number where ’j’ varies from 1 to n. The assembled value of these 2-tuple linguistic m–polar fuzzy number using the 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator is again a 2-tuple linguistic m–polar fuzzy number, as given below:

9.4 2-Tuple Linguistic m–Polar Fuzzy Hamacher Geometric Aggregation Operators

2T Lm F H W G φ (ξ1 , ξ2 , ..., ξn ) =

=

n

j=1 (φ j ξ j )

φj

455

,

⎧ ⎛ ⎛ ⎛   ⎞φ j ⎪ ⎪ −1 s , ρ j ⎪ ⎪ ⎜ ⎜ j 1 ⎟ ⎜Δ ⎪ ⎪ ⎜ ⎜ ψ ⎜ ⎟ ⎪ n ⎪ 1 ⎜ ⎜ ⎜ ⎟ ⎪ λ j=1 ⎜ ⎪ ⎜ ⎜ ⎟ ⎪ t ⎪ ⎜ ⎜ ⎝ ⎠ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ Δ ⎜t ⎜ ⎪ ⎛ ⎛ ⎛    ⎞φ  ⎞⎞φ ⎪ ⎜ ⎪ ⎜ j j ⎪ j ⎜ ⎜ −1 s , ρ j ⎪ Δ−1 s j , ρ1 ⎟⎟ ⎪ ⎜ ⎜ j 1 ⎟ ⎪ ⎜Δ ⎜ ⎜ ⎪     ⎜ ⎜ ψ ψ ⎪ ⎜ ⎜ n ⎜ ⎜ ⎜ ⎟⎟ ⎟ n ⎪ 1 1 ⎪ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ + λ−1 ⎪ ⎜ ⎜ j=1 ⎜1 + λ − 1 ⎜1 − ⎪ ⎟⎟ ⎟ j=1 ⎜ ⎪ t t ⎝ ⎝ ⎪ ⎝ ⎝ ⎝ ⎠⎠ ⎠ ⎪ ⎪ ⎨

⎫ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎟⎟ , . . . , ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎬ ⎞⎞

⎪ ⎪ ⎛ ⎛   ⎞φ ⎪ ⎛ ⎪ j ⎪ j ⎪ −1 s ⎪ ⎪ ⎜ ⎜ j , ρm ⎟ ⎜Δ ⎪ ⎪  ψ ⎜ ⎜ ⎜ ⎟ ⎪ m n ⎪ ⎜ ⎜ λ j=1 ⎜ ⎟ ⎪ ⎪ ⎜ ⎜ ⎪ ⎝ ⎠ t ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ Δ ⎜t ⎜ ⎪  ⎞φ    ⎛ ⎞ ⎛ ⎛ ⎞ ⎪ φj ⎜ ⎜ ⎪ j ⎪ j j ⎜ ⎜ ⎪ ⎪ Δ−1 s j , ρm ⎟ Δ−1 s j , ρm ⎟⎟ ⎜ ⎜ ⎪ ⎜ ⎜ ⎜     ⎪ ⎜ ⎜ n n ⎪ ψm ψm ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎪ ⎜ ⎜ ⎪ + λ−1 1 + λ − 1 ⎜1 − ⎟⎟ ⎟ ⎪ ⎝ ⎝ j=1 ⎜ ⎪ j=1 ⎜ ⎝ ⎝ ⎝ ⎠⎠ ⎠ t t ⎪ ⎪ ⎩

⎪ ⎞⎞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎪ ⎭

.



Proof The proof can be followed easily by using mathematical induction.

Example 9.3 Consider ξ1 = {(s1 , 0.5), (s2 , 0.3), (s3 , 0.2)}, ξ2 = {(s2 , 0.4), (s1 , 0.6), (s5 , 0.1)} and ξ3 = {(s4 , 0.3), (s2 , 0.1), (s1 , 0.3)} be 2-tuple linguistic 3polar fuzzy numbers with the weight vector φ = (0.2, 0.4, 0.1), take λ = 3. Then, the assembled result can be calculated as below:  2T L3F H W G φ (ξ1 , ξ2 , ξ3 ) = 3j=1 (φ j ξ j )φ j ,

=

⎧ ⎛ ⎛ ⎛   ⎞φ j ⎪ ⎪ −1 s , ρ j ⎪ ⎪ ⎜ ⎜ j 1 ⎟ ⎜Δ ⎪ ⎪ ⎜ ⎜ ψ ⎜ ⎟ ⎪  ⎪ 1 ⎜ ⎜ 3 ⎜ ⎟ ⎪ λ j=1 ⎜ ⎪ ⎜ ⎜ ⎟ ⎪ t ⎪ ⎜ ⎜ ⎝ ⎠ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ t Δ ⎜ ⎜ ⎪ ⎛ ⎛ ⎛    ⎞φ  ⎞⎞φ ⎪ ⎜ ⎪ ⎜ j j ⎪ j ⎜ ⎜ −1 s , ρ j ⎪ Δ−1 s j , ρ1 ⎟⎟ ⎪ ⎜ ⎜ j 1 ⎟ ⎪ ⎜Δ ⎜ ⎪ ⎜ ⎜    ⎜ ψ ψ ⎪ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎪ ⎜ ⎜ 3 1 1 3 ⎪ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ + λ−1 1 + λ − 1 ⎜1 − ⎪ ⎜ ⎪ ⎜ ⎟⎟ ⎟ j=1 ⎜ ⎪ t t ⎝ ⎝ j=1 ⎜ ⎪ ⎝ ⎝ ⎝ ⎠⎠ ⎠ ⎪ ⎪ ⎨

⎫ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎟⎟ , . . . , ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎬ ⎞⎞

⎪ ⎪ ⎛ ⎛  ⎞φ  ⎪ ⎛ ⎪ j ⎪ j ⎪ −1 s ⎪ ⎪ ⎜ ⎜ j , ρm ⎟ ⎜Δ ⎪ ⎪ 3 ψ ⎜ ⎜ ⎜ ⎟ ⎪ m ⎪ ⎜ ⎜ λ ⎜ ⎟ ⎪ ⎪ j=1 ⎝ ⎜ ⎜ ⎪ ⎠ t ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ Δ ⎜t ⎜ ⎪  ⎞φ   ⎞⎞φ  ⎛ ⎛ ⎛ ⎪ ⎜ ⎜ ⎪ j j ⎪ j j ⎜ ⎜ −1 s ⎪ ⎪ Δ−1 s j , ρm ⎟⎟ ⎜ ⎜ ⎪ j , ρm ⎟ ⎜Δ ⎜    ⎜ ⎪ ⎜ ⎜  ⎪ ψ ψ ⎜ ⎜ ⎜ ⎟⎟ ⎟ m m ⎪ 3 3 ⎜ ⎜ ⎪ + λ − 1 1 + λ − 1 1 − ⎜ ⎟⎟ ⎟ ⎪ ⎝ ⎝ j=1 ⎜ ⎪ j=1 ⎜ ⎝ ⎝ ⎝ ⎠⎠ ⎠ t t ⎪ ⎪ ⎩

⎪ ⎞⎞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎪ ⎭

⎞⎞  1.5 0.2  2.4 0.4  4.3 0.1  ⎧ ⎛ ⎛ ⎪ 3 × × ⎪ ⎜ ⎜ ⎟⎟ ⎪ 6 6 6 ⎪ ⎟⎟ , ⎪ Δ⎜ 6⎜ ⎪ ⎝ ⎝                       0.2 0.4 0.1 ⎪ 1.5 2.4 4.3 1.5 0.2 2.4 0.4 4.3 0.1 ⎠⎠ ⎪ ⎪ ⎪ 1+ 2 1− × 1+ 2 1− × 1+ 2 1− +2× × × ⎪ ⎪ 6 6 6 6 6 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎞⎞ ⎛ ⎛ ⎪  2.3 0.2  1.6 0.4  2.1 0.1  ⎪ ⎪ ⎪ ⎪ × × 3 ⎪ ⎟⎟ ⎜ ⎜ ⎪ 6 6 6 ⎪ ⎟⎟ ⎜ ⎜ ⎪ ⎨ Δ ⎝6 ⎝     2.3 0.2      1.6 0.4      2.1 0.1   2.3 0.2  1.6 0.4  2.1 0.1  ⎠⎠ , = 1+ 2 1− × 1+ 2 1− × 1+ 2 1− +2× × × ⎪ 6 6 6 6 6 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎞⎞ ⎛ ⎛ ⎪  3.2 0.2  5.1 0.4  1.3 0.1  ⎪ ⎪ ⎪ ⎪ × × 3 ⎪ ⎟⎟ ⎜ ⎜ ⎪ 6 6 6 ⎪ ⎟⎟ ⎜6 ⎜ Δ ⎪ ⎪ ⎝ ⎝     3.2 0.2      5.1 0.4      1.3 0.1   3.2 0.2  5.1 0.4  1.3 0.1  ⎠⎠ ⎪ ⎪ ⎪ 1+ 2 1− × 1+ 2 1− × 1+ 2 1− +2× × × ⎪ ⎪ ⎪ 6 6 6 6 6 6 ⎪ ⎪ ⎪ ⎪ ⎩

= ((s3 , 0.3399), (s3 , −0.0879), (s5 , −0.4585)).

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

456

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators j

j

Theorem 9.12 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1 tic m–polar fuzzy numbers, where j = 1, 2, ..., n. For the equality of all these numbers, that is, ξ j = ξ, ∀ ’j’ varies from 1 to n, then 2T Lm F H W G φ (ξ1 , ξ2 , ..., ξn ) = ξ. j

(9.20)

j

Theorem 9.13 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1   tic m–polar fuzzy numbers, where j = 1, 2, ..., n and ξ − = nj=1 ξ j , ξ + = nj=1 ξ j , then ξ − ≤ 2T Lm F H W G φ (ξ1 , ξ2 , ..., ξn ) ≤ ξ + .

(9.21)



Theorem 9.14 Let ξ j and ξ j , where ’j’ varies from 1 to n, be a set of 2-tuple lin guistic m–polar fuzzy numbers. If ξ j ≤ ξ j , then 2T Lm F H W G φ (ξ1 , ξ2 , ..., ξn ) ≤ 2T Lm F H W G φ (ξ´1 , ξ´2 , ..., ξ´n ).

(9.22)

Remark 9.6 1. For λ = 1, 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator reduces to 2-tuple linguistic m–polar fuzzy weighted geometric operator as below: 2T Lm F W G φ (ξ1 , ξ2 , ..., ξn ) =

n

j=1 (φ j ξ j )

φj

,

   n  Δ−1 (s j , ρ j )   n  Δ−1 (s j , ρ j )     φj φj 1 2 ψ1 ψ2 ,Δ t , ..., = Δ t t t j=1 j=1  j n  −1   Δ (sψmj , ρm ) φ j  Δ t . t j=1

(9.23)

2. For λ = 2, 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator reduces to 2-tuple linguistic m–polar Einstein weighted geometric operator as below:  2T Lm F E W G φ (ξ1 , ξ2 , ..., ξn ) = nj=1 (ξ j )φ j , ⎛ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ = Δ ⎜t ⎜ ⎜ ⎜ ⎝ ⎝ n j=1

2

n j=1

 Δ−1 (s j , ρ j ) φ j 1 ψ 1

t

j   Δ−1 (s j , ρ j ) φ j Δ−1 (sψ j , ρ1 ) φ j  1 ψ1 1 2− + nj=1 t t

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ , . . . , ⎟⎟ ⎠⎠

9.4 2-Tuple Linguistic m–Polar Fuzzy Hamacher Geometric Aggregation Operators

⎛ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ Δ⎜ ⎜t ⎜ ⎝ ⎝ n

2



n

Δ−1 (sψmj , ρm ) j

j=1

 j  Δ−1 (sψmj , ρm ) φ j 2 − j=1 t

457

⎞⎞

φ j

⎟⎟ ⎟⎟ ⎟⎟  −1 j j φ j ⎟⎟ . Δ (sψm , ρm ) ⎠⎠ n + j=1 t (9.24) t

j

j

Definition 9.18 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple lin1 guistic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then, a 2-tuple linguistic m-polar fuzzy Hamacher ordered weighted geometric operator is a mapping, 2T Lm F H O W  G : ξ n → ξ with the weight vector w = (w1 , w2 , ..., wn ), where w j ∈ (0, 1] and nj=1 w j = 1. Then 2T Lm F H O W G W (ξ1 , ξ2 , ..., ξn ) =

n  φ (w j ξσ(j j) )φ j ,

(9.25)

j=1

where σ( j) = (σ(1), σ(2), ..., σ(n)) represents the permutation of the indices ‘ j’ where j = 2, ..., n, for which ξσ( j−1) ≥ ξσ( j) , ∀ j = 2, ..., n. j

j

Theorem 9.15 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1 tic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then the assembled value of these 2-tuple linguistic m–polar fuzzy numbers using the 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator is also a 2-tuple linguistic m–polar fuzzy number, as given below:  2T Lm F H O W G W (ξ1 , ξ2 , ..., ξn ) = nj=1 (w j ξσ( j) ),

=

⎛   ⎞w j ⎧ ⎛ ⎛ σ( j) ⎪ −1 s ⎪ σ( j) , ρ1 ⎪ ⎟ ⎜Δ ψ1 ⎜ ⎜ ⎪  ⎪ ⎟ ⎜ ⎜ ⎜ ⎪ ⎪ λ nj=1 ⎜ ⎟ ⎜ ⎜ ⎪ ⎪ ⎠ ⎝ t ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ Δ ⎜t ⎜ ⎛ ⎛ ⎛ ⎪   ⎞ ⎞w j   ⎞w j ⎪ ⎪ ⎜ ⎜ ⎪ σ( j) σ( j) −1 s ⎪ Δ−1 sψσ( j) , ρ1 σ( j) , ρ1 ⎪ ⎜ ⎜ ⎟⎟ ⎟ ⎜Δ  ⎜   ⎪ ψ1 ⎜ ⎜ n ⎜ ⎪ 1 ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎪ n ⎪ ⎜ ⎜ j=1 ⎜1 + λ − 1 ⎜1 − ⎟⎟ + λ − 1 ⎟ ⎪ j=1 ⎜ ⎪ ⎝ ⎝ ⎝ ⎠⎠ ⎠ ⎝ ⎝ t t ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎛ ⎛

⎛ −1

⎜Δ  ⎜ λ nj=1 ⎜ ⎝



σ( j)

sψσ( j) , ρm

⎫ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎟⎟ , . . . , ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎞⎞

 ⎞w j

⎞⎞

⎟ ⎟ ⎟ ⎠

⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

⎜ ⎜ m ⎜ ⎜ ⎜ ⎜ t ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ t Δ⎜ ⎞ ⎞ ⎛ ⎛ ⎛     ⎞w j wj ⎜ ⎜ ⎜ ⎜ σ( j) σ( j) −1 s Δ−1 sψσ( j) , ρm σ( j) , ρm ⎜ ⎜ ⎟⎟ ⎟ ⎜Δ    ⎜ ψm m ⎜ ⎜ n ⎜ ⎟⎟ ⎟ ⎜ ⎜ n ⎜ ⎜ j=1 ⎜ ⎟⎟ + λ − 1 ⎟ ⎜1 + λ − 1 ⎜1 − j=1 ⎜ ⎝ ⎝ ⎠⎠ ⎠ ⎝ ⎝ ⎝ t t

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

Remark 9.7 1. For λ = 1, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator reduces to 2-tuple linguistic m–polar fuzzy ordered weighted geometric operator as below: 2T Lm F O W G w (ξ1 , ξ2 , ..., ξn ) =

n

j=1 (ξσ( j) )

wj

,

458

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

   n  Δ−1 (s σ( j) , ρσ( j) )   n  Δ−1 (s σ( j) , ρσ( j) )     wj wj 1 2 ψ1 ψ2 ,Δ t , ..., = Δ t t t j=1 j=1  σ( j) n  −1   Δ (sψmσ( j) , ρm ) w j  Δ t . t j=1

(9.26)

2. For λ = 2, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator reduces to 2-tuple linguistic m–polar fuzzy Einstein ordered weighted geometric operator as below:  2T Lm F E O W G φ (ξ1 , ξ2 , ..., ξn ) = nj=1 (ξ j )w j ,

=

⎛   ⎞w ⎧ ⎛ ⎛ j σ( j) ⎪ −1 s ⎪ ⎪ σ( j) , ρ1 ⎜Δ ⎟ ⎪ ⎜ ⎜ ⎪ ψ ⎜ ⎟ n ⎪ ⎜ ⎜ 1 ⎪ ⎜ ⎟ ⎪ ⎜ ⎜ 2 j=1 ⎜ ⎪ ⎟ ⎪ ⎜ ⎜ t ⎪ ⎝ ⎠ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜t ⎜ Δ ⎪ ⎛ ⎛⎛   ⎞w   ⎞⎞w ⎪ ⎜ ⎜ ⎪ j j ⎪ σ( j) σ( j) ⎜ ⎜ −1 s ⎪ Δ−1 s σ( j) , ρ1 ⎪ ⎜ ⎜ σ( j) , ρ1 ⎪ ⎜Δ ⎟⎟ ⎟ ⎜⎜ ⎪ ⎜ ψ ψ ⎪ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜  ⎪ ⎜ ⎜ n 1 1 ⎪ ⎟⎟ ⎟ ⎜⎜2 − + nj=1 ⎜ ⎪ ⎜ ⎪ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎪ ⎝ ⎝ j=1 ⎜ t t ⎪ ⎝ ⎠⎠ ⎠ ⎝⎝ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎛ ⎛

 ⎞w  j σ( j) Δ−1 s σ( j) , ρm ⎜ ⎟ n ψm ⎜ ⎟ 2 j=1 ⎜ ⎟ ⎝ ⎠ t

⎫ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎟⎟ , . . . , ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎬ ⎞⎞

⎛

⎞⎞

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜t ⎜ Δ⎜  ⎞w  ⎞⎞w   ⎛ ⎛⎛ ⎜ j j σ( j) σ( j) ⎜ ⎜ Δ−1 s σ( j) , ρm Δ−1 s σ( j) , ρm ⎜ ⎜ ⎜ ⎟ ⎜⎜ ⎟ ⎟ ⎜ ⎜  ψm ψm ⎜ ⎜⎜ ⎟⎟ ⎟ n ⎜ ⎜ n + j=1 ⎜ ⎜2 − ⎟⎟ ⎟ ⎝ ⎝ j=1 ⎜ ⎝ ⎝⎝ ⎠⎠ ⎠ t t

j

⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

j

Theorem 9.16 Idempotency: Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 1 2-tuple linguistic m–polar fuzzy numbers, where j = 1, 2, ..., n. For the equality of all these numbers, that is, ξ j = ξ, ∀ j = 1, 2, ..., n, then the monotonicity property is defined as follows: 2T Lm F H O W G w (ξ1 , ξ2 , ..., ξn ) = ξ.

(9.27)

The remaining properties, namely, boundedness, monotonicity and commutativity for 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator are defined as follows: j

j

Theorem 9.17 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )), be a collection of 2-tuple linguis1   tic m–polar fuzzy numbers, where j = 1, 2, ..., n, ξ − = nj=1 ξ j and ξ + = nj=1 ξ j then, ξ − ≤ 2T Lm F H O W G φ (ξ1 , ξ2 , ..., ξn ) ≤ ξ + . 

(9.28)

Theorem 9.18 Let ξ j and ξ j , j = 1, 2, ..., n be a collection of 2-tuple linguistic  m–polar fuzzy numbers. If ξ j ≤ ξ j , then

9.4 2-Tuple Linguistic m–Polar Fuzzy Hamacher Geometric Aggregation Operators

459

2T Lm F H O W G φ (ξ1 , ξ2 , ..., ξn ) ≤ 2T Lm F H O W G φ (ξ´1 , ξ´2 , ..., ξ´n )

(9.29)



Theorem 9.19 Let ξ j and ξ j , j = 1, 2, ..., n be a set of 2-tuple linguistic m–polar  fuzzy numbers. If ξ j ≤ ξ j , then 2T Lm F H O W G φ (ξ1 , ξ2 , ..., ξn ) = 2T Lm F H O W G φ (ξ´1 , ξ´2 , ..., ξ´n ),

(9.30)



where ξ j represents the permutation of ξ j , j = 1, 2, ..., n. j

j

Definition 9.19 ([6]) Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple 1 linguistic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then, a 2-tuple linguistic m–polar fuzzy Hamacher hybrid geometric operator is defined as: 2T Lm F H H G w,φ (ξ1 , ξ2 , ..., ξn ) =

n  (w j ξ˜σ( j) ),

(9.31)

j=1

where w = (w1 , w2 , ..., wn ) represents the associated weight vector of the 2-tuple linguistic m–polar fuzzy numbers ξ j , instead of weighting the experts, for each ’ j’  varies from 1 to n, with w j ∈ (0, 1] and nj=1 w j = 1, ξ˜σ( j) is the jth biggest 2-tuple linguistic m–polar fuzzy numbers of the ξi (i = 1, 2, ..., n) with ξ˜σ( j) = (nφi )ξσ( j) , ( j = 1, 2, ..., n),  φ = (φ1 , φ2 , ..., φn ) is the weight vector for the ordered arguments, with φ j ∈ (0, 1], nj=1 φ j = 1. j

j

Theorem 9.20 Let ξ j = ((sψ j , ρ1 ), ..., (sψmj , ρm )) be a collection of 2-tuple linguis1 tic m–polar fuzzy numbers, where j = 1, 2, ..., n. Then the assembled value of these 2-tuple linguistic m-polar fuzzy numbers by using the 2-tuple linguistic m–polar fuzzy Hamacher hybrid average geometric operator is again a 2-tuple linguistic m-polar fuzzy number, as given below:  2T Lm F H H G w,φ (ξ1 , ξ2 , ..., ξn ) = nj=1 (w j ξ˜σ( j) ),

=

⎛   ⎞w j ⎧ ⎛ ⎛ σ( j) ⎪ −1 s ⎪ σ( j) , ρ1 ⎪ ⎟ ⎜Δ ψ1 ⎜ ⎜ ⎪  ⎪ ⎟ ⎜ ⎜ ⎜ ⎪ ⎪ λ nj=1 ⎜ ⎟ ⎜ ⎜ ⎪ ⎪ ⎠ ⎝ t ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ Δ ⎜t ⎜ ⎛ ⎛ ⎛ ⎪   ⎞ ⎞w j   ⎞w j ⎪ ⎪ ⎜ ⎜ ⎪ σ( j) σ( j) −1 s ⎪ Δ−1 sψσ( j) , ρ1 σ( j) , ρ1 ⎪ ⎜ ⎜ ⎟⎟ ⎟ ⎜Δ    ⎜ ⎪ ψ1 ⎜ ⎜ n ⎜ ⎪ 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎪ n ⎪ ⎜ ⎜ j=1 ⎜1 + λ − 1 ⎜1 − ⎟⎟ + λ − 1 ⎟ ⎪ j=1 ⎜ ⎪ ⎝ ⎝ ⎠⎠ ⎠ ⎝ ⎝ ⎝ t t ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎛ ⎛

⎛ −1

⎜Δ  ⎜ λ nj=1 ⎜ ⎝



σ( j)

sψσ( j) , ρm

⎫ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎟⎟ , . . . , ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎞⎞

 ⎞w j

⎞⎞

⎟ ⎟ ⎟ ⎠

⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

⎜ ⎜ m ⎜ ⎜ ⎜ ⎜ t ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ t Δ⎜ ⎛ ⎞ ⎞ ⎛ ⎛     ⎞w j wj ⎜ ⎜ ⎜ ⎜ σ( j) σ( j) −1 s Δ−1 sψσ( j) , ρm σ( j) , ρm ⎜ ⎜ ⎟⎟ ⎟ ⎜Δ  ⎜   ψm m ⎜ ⎜ n ⎜ ⎜ ⎟⎟ ⎟ ⎜ n ⎜ ⎜ j=1 ⎜ ⎟⎟ + λ − 1 ⎟ ⎜1 + λ − 1 ⎜1 − j=1 ⎜ ⎝ ⎝ ⎝ ⎠⎠ ⎠ ⎝ ⎝ t t

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

Proof In order to proof this theorem, same steps are follows as discussed above in the theorem. 

460

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

Remark 9.8 1. For λ = 1, the 2-tuple linguistic m–polar fuzzy Hamacher hybrid average geometric operator reduces to 2-tuple linguistic m–polar fuzzy hybrid geometric operator as given: 2T Lm F H G w,φ (ξ1 , ξ2 , ..., ξn ) =

n

˜

j=1 (w j ξσ( j) ),

   n  Δ−1 (s σ( j) , ρσ( j) )   n  Δ−1 (s σ( j) , ρσ( j) )     wj wj 1 2 ψ1 ψ2 ,Δ t , ..., = Δ t t t j=1 j=1  σ( j) n  −1   Δ (sψmσ( j) , ρm ) w j  Δ t . t j=1

(9.32)

2. For λ = 2, 2-tuple linguistic m–polar fuzzy Hamacher hybrid average geometric operator converts to 2-tuple linguistic m–polar fuzzy Einstein hybrid geometric operator as given:  2T Lm F E H G w,φ (ξ1 , ξ2 , ..., ξn ) = n (w j ξ˜˜σ( j) ), j=1

=

⎧ ⎛ ⎛ ⎛   ⎞φ j ⎪ ⎪ −1 s , ρ j ⎪ ⎪ ⎜ ⎜ j 1 ⎟ ⎜Δ ⎪ ⎪ ⎜ ⎜ ψ1 ⎜ ⎟ ⎪ n ⎪ ⎜ ⎜ ⎟ ⎪ 2 j=1 ⎜ ⎪ ⎜ ⎜ ⎜ ⎟ ⎪ t ⎪ ⎜ ⎜ ⎝ ⎠ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ Δ ⎜t ⎜ ⎪ ⎛ ⎛    ⎞φ  ⎞φ ⎪ ⎜ ⎪ ⎜ j j ⎪ j ⎜ ⎜ −1 s , ρ j ⎪ Δ−1 s j , ρ1 ⎟ ⎪ ⎜ ⎜ j 1 ⎟ ⎪ ⎜Δ ⎜ ⎪ ⎜ ⎜ ψ1 ψ1 ⎪ ⎜ ⎜ n ⎜ ⎜ ⎟ ⎟  ⎪ n ⎪ ⎟ ⎟ + j=1 ⎜ ⎪ ⎜ ⎜ j=1 ⎜ ⎪ ⎜ ⎜2 − ⎟ ⎟ ⎪ t t ⎝ ⎝ ⎪ ⎝ ⎝ ⎠ ⎠ ⎪ ⎪ ⎨

⎫ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎟⎟ , . . . , ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎬ ⎞⎞

⎪ ⎪ ⎛ ⎛  ⎞φ  ⎪ ⎛ ⎪ j ⎪ j ⎪ ⎪ Δ−1 s j , ρm ⎟ ⎪ ⎜ ⎜ ⎜ ⎪ ⎪ n ψm ⎜ ⎜ ⎜ ⎟ ⎪ ⎪ ⎜ ⎜ 2 j=1 ⎜ ⎟ ⎪ ⎪ ⎜ ⎜ ⎪ ⎝ ⎠ t ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ Δ ⎜t ⎜ ⎪  ⎞φ    ⎛ ⎛ ⎞φ j ⎪ ⎜ ⎜ ⎪ j ⎪ j j ⎜ ⎜ ⎪ ⎪ Δ−1 s j , ρm ⎟ Δ−1 s j , ρm ⎟ ⎜ ⎜ ⎪ ⎜ ⎜ ⎪ ⎜ ⎜ n n ⎪ ψm ψm ⎜ ⎜ ⎟ ⎟ ⎪ ⎜ ⎜ ⎪ + j=1 ⎜ 2− ⎟ ⎟ ⎪ ⎝ ⎝ j=1 ⎜ ⎪ ⎝ ⎝ ⎠ ⎠ t t ⎪ ⎪ ⎩

⎪ ⎞⎞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎟⎟ ⎪ ⎪ ⎠⎠ ⎪ ⎪ ⎪ ⎪ ⎭

.

9.5 Mathematical Approach for MADM Using 2-Tuple Linguistic m–Polar Fuzzy Information Akram et al. [6] handled the MADM issues with 2-tuple linguistic m–polar fuzzy information by applying the 2-tuple linguistic m–polar fuzzy Hamacher average(geometric) operators.

9.5 Mathematical Approach for MADM Using 2-Tuple Linguistic

461

Let A = {A1 , A1 , A2 , ..., Ak } be the set of alternatives and ζ = {ζ1 , ζ2 , ..., ζn } be the set of attributes. Assume φ = ({φ1 , φ2 , ..., φ n }), a weight vector for the set of attributes. where φ j > 0 for j = 1, 2, ..., n and nj=1 φ j = 1. ij

ij

Take R = (ri j )k×n = ((sψi j , ρ1 ), ..., (sψmi j , ρm )k×n be a decision matrix for 2-tuple 1

ij

linguistic m–polar fuzzy information. Here (sψri j , ρr ), r = 1, 2, ..., m represent the membership values given by the decision makers that the alternatives assures with the attributes ζ j , where (sψr , ρr ) ∈ [0, 1], ’r ’ varies from 1 to m. In order to deal with MADM issues, the Algorithm 1 is explained using 2-tuple linguistic m-polar fuzzy Hamacher weighted average (2-tuple linguistic m–polar fuzzy Hamacher weighted geometric) operators. Algorithm 1: Procedure to tackle MADM problems using 2-tuple linguistic m– polar fuzzy Hamacher weighted average operators (or 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric) operators 1. Input: U, the set of discourse having k alternatives. ζ, be the set having n attributes. φ = ({φ1 , φ2 , ..., φn }), weight vector representation. 2. In order to calculate the values in 2-tuple linguistic m–polar fuzzy decision matrix, calculate the preference values pˆi , i = 1, 2, 3, ..., k, of the objects Ai , by using 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator.  pˆi = 2T Lm F H W Aφ (ξi1 , ξi2 , ..., ξin ) = nj=1 (φ j ξi j ), ⎞⎞ ⎛ ⎛   ⎞φ   ⎞φ ⎛ ⎛ ⎞ ij ij j j −1 s Δ−1 s i j , ρ1 i j , ρ1  Δ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟ ⎜ ψ1 ψ  n ⎜ ⎜ ⎜ ⎜ ⎟ 1 ⎟⎟ ⎟ − n ⎜1 − ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟⎟ j=1 ⎝1 + λ − 1 j=1 ⎝ ⎠ ⎠ ⎜ ⎜ ⎜ ⎟ t t ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ,..., ⎟ ⎜ Δ ⎜t ⎜ ⎟ ⎟ ⎟     ⎛ ⎛ ⎞ ⎞ φj φ j ⎟⎟ ⎜ ⎟ ⎜ ij ij −1 −1 ⎜ ⎜ ⎟ ⎟⎟ ⎜ Δ s i j , ρ1 s i j , ρ1 ⎜ ⎜ ⎟  Δ   ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ψ ψ ⎜ ⎜ ⎟ ⎟ ⎟ 1 1 n ⎜ ⎜ ⎜ n ⎟ ⎜ ⎜ ⎟ ⎟ 1− + λ−1 ⎠⎠ ⎜ ⎝ ⎝ j=1 ⎝1 + λ − 1 ⎟ j=1 ⎝ ⎠ ⎠ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. =⎜ ⎟ ⎜ ⎟ ⎜   ⎞φ   ⎞φ ⎛ ⎛ ⎞⎞ ⎟ ⎜ ⎟ ⎛ ⎛ ij ij j j ⎜ ⎟ −1 −1 Δ s i j , ρm s i j , ρm ⎜ ⎟  Δ ⎜ n ⎜ n ⎜ ⎜ ⎜ ⎟⎟ ⎟ ψm ψm ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 1 + λ − 1 1 − − ⎠ ⎠ ⎝ ⎝ ⎜ j=1 j=1 ⎜ ⎜ ⎟⎟ ⎟ t t ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎟ Δ ⎜t ⎜ ⎜ ⎟ ⎟   ⎞φ   ⎞φ ⎟ ⎛ ⎛ ⎜ ⎜ ⎜ ⎟⎟ ⎟ i j i j j j −1 −1 ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ Δ Δ , ρ , ρ s s m m i j i j ⎜     ⎜ ⎜ n ⎜ ⎟ ⎟ ψm ψm ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ n ⎝ ⎝ ⎝ ⎠ + λ−1 ⎠ ⎠⎠ ⎠ ⎝1 − j=1 ⎝1 + λ − 1 j=1 t t ⎛

(9.33) Alternatively, if 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator is applied then:  pˆi = 2T Lm F H W G φ (ξi1 , ξi2 , ..., ξin ) = nj=1 (φ j ξi j )φ j ,

462

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

⎛ ⎛ ⎞⎞   ⎞φ ⎞ ⎛ ij j Δ−1 s i j , ρ1 ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ψ  ⎜ ⎜ ⎜ ⎟ 1 ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟ λ nj=1 ⎜ ⎟⎟ ⎠ ⎝ ⎜ ⎜ ⎜ ⎟ t ⎟⎟ ⎜ ⎟ ⎟⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ Δ ⎜t ⎜ ⎟ , . . . , ⎟ ⎟     ⎞⎞φ ⎞φ ⎟⎟ ⎛ ⎛ ⎛ ⎜ ⎜ ⎜ ⎟ i j i j j j −1 s −1 s ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ , ρ , ρ Δ Δ ⎜ ⎜ ⎜ ⎟ ij 1 ij 1     ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎜ ⎜ ψ1 ψ1 n ⎜ n ⎜ ⎟ ⎟⎟ + λ − 1 ⎟ ⎠⎠ ⎜1 − ⎜ 1 + λ − 1 ⎜ ⎝ ⎝ j=1 ⎜ ⎟ j=1 ⎝ ⎠⎠ ⎠ ⎝ ⎝ ⎜ ⎟ t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. =⎜ ⎟ ⎞ ⎞ ⎛ ⎛ ⎜ ⎟ ⎞ ⎛   φj ⎜ ⎟ ij ⎜ ⎟ −1 s i j , ρm ⎟ Δ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟  ψ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ m n ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ λ ⎟ ⎜ ⎜ j=1 ⎝ ⎟⎟ ⎟ ⎜ ⎜ ⎠ t ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ Δ ⎜t ⎜ ⎟ ⎟   ⎞⎞ψ   ⎞φ ⎟ ⎛ ⎛ ⎛ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ i j i j j j ⎜ ⎟⎟ ⎟ ⎜ ⎜ Δ−1 s i j , ρm Δ−1 s i j , ρm ⎜     ⎟ ⎟ ⎜ ⎜ n ⎜ ⎜ ⎟ ψm ψm ⎜ ⎟⎟ ⎟ ⎟ ⎜ n ⎟ ⎟ ⎜ ⎜ ⎜ ⎠⎠ + λ − 1 ⎠ ⎟⎟ ⎟ ⎜ ⎜ j=1 ⎝1 + λ − 1 ⎝1 − ⎜ ⎟ j=1 ⎝ t t ⎝ ⎠⎠ ⎠ ⎝ ⎝ ⎛

(9.34) 3. Compute the score values S( pˆi ), i = 1, 2, 3, ..., k. 4. By using score values S( pˆi ), i = 1, 2, 3, ..., k, ranking for objects can be obtained. If same score values are used for two alternatives, then in order to rank the objects we move toward accuracy function. Output: An alternative which have the highest score value in Step (4) will be the decided alternative. Now, the above algorithm is elaborated for decision making in the form of flowchart which is given below in Fig. 9.1.

9.6 Best Location for the Thermal Power Station: Case Study [6] Thermal power stations are a source of conversion of heat energy into electricity. Thermal power stations use fossil fuels to generate electricity, which produces pollution. By keeping in view all the circumstances, we consider a selection of the best location for a thermal power station. Best location selection plays a significant role in the economic operation of the thermal power station and the long-lasting development of the region. So, the company selects five possible areas, which considered as alternatives A = {A1 , A2 , A3 , A4 , A5 }. The decision maker selects the best place for location under the following criteria: ζ1 ζ2 ζ3 ζ4

: Infrastructure : Environmental conditions : Social impacts : Governmental policies

where each criterion has divided into three components to form a 3-polar fuzzy set.

9.6 Best Location for the Thermal Power Station: Case Study [6]

463

Fig. 9.1 Flowchart for decision making

• Infrastructures: The development of the infrastructure involves different factors such as the supply of water, extra high cable voltage, gas, roadways, etc. But, here we take three factors such as the availability of coal, availability of water and availability of transportation facilities to make the 3-polar fuzzy set. • Environmental conditions: This factor means the state of the environment, including several different natural resources, take the three-factors of the environmental conditions which are necessary for best location selection of the thermal power station includes ambient temperature, humidity, and air velocities. • Social impacts: This criterion involves the study of the social challenges, which may include beneficial and adverse. Three factors are taken including education facilities, hospital facilities, and health care facilities. • Governmental policies: This criterion includes the governmental policies, which has subdivided into three factors such as licensing policies, institutional finance, and government subsidies. A clear vision of attribute selection in the 3-polar fuzzy environment as shown in Fig. 9.2.

464

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators 1. Coal availability 2. Water availability 3.Transportation facilities

Infrastructures

Selected criteria and their classifications

1.Ambient temperature 2.Humidity

Environmental conditions

3.Air condition

1.Educational facilities 2.Hospital facilities

Social Impacts

3.Health care facilities

Governmental policies 1.Licensing policies 2.Institutional finance 3.Government subsidies

Fig. 9.2 Criteria representation in selected 3-polar environment Table 9.1 Decision matrix for 2-tuple linguistic 3-polar fuzzy information A1

A2

A3

A4

A5

ζ1

((s4 , 0), (s4 , 0), (s4 , 0))

((s5 , 0), (s3 , 0), (s4 , 0))

((s4 , 0), (s3 , 0), (s3 , 0))

((s4 , 0), (s5 , 0), (s3 , 0))

((s5 , 0), (s4 , 0), (s3 , 0))

ζ2

((s4 , 0), (s5 , 0), (s3 , 0))

((s4 , 0), (s5 , 0), (s5 , 0))

((s4 , 0), (s5 , 0), (s3 , 0))

((s3 , 0), (s4 , 0), (s5 , 0))

((s4 , 0), (s6 , 0), (s4 , 0))

ζ3

((s4 , 0), (s5 , 0), (s2 , 0))

((s4 , 0), (s5 , 0), (s6 , 0))

((s4 , 0), (s3 , 0), (s6 , 0))

((s4 , 0), (s4 , 0), (s4 , 0))

((s3 , 0), (s5 , 0), (s3 , 0))

ζ4

((s4 , 0), (s3 , 0), (s5 , 0))

((s6 , 0), (s3 , 0), (s4 , 0))

((s3 , 0), (s4 , 0), (s4 , 0))

((s5 , 0), (s2 , 0), (s3 , 0))

((s3 , 0), (s4 , 0), (s4 , 0))

1. In order to construct the decision matrix, the decision makers describe their preferences for the best location of a thermal power plant in the form of linguistic terms. But, if these linguistic terms are proceeded, then the assembled results may give the same linguistic term against different alternatives. So, to manage this issue, it is translated with zero symbolic translation that converts the linguistic term data into 2-tuple linguistic data in which alternatives can be ranked with the same linguistic term on basis of the symbolic translation. The decision matrix for 2-tuple linguistic 3-polar fuzzy data is given in Table 9.1. 2. The weights recommended by the experts are given as follows: φ=(φ1 ,φ2 ,φ3 ,φ4 )=(0.4, 0.3, 0.1, 0.2). 3. Take λ = 3, to assembled the 2-tuple linguistic m–polar fuzzy values. If we choose λ = 1, then the 2-tuple linguistic m–polar fuzzy Hamacher weighted average(2-tuple linguistic m–polar fuzzy Hamacher weighted geometric) operator reduces to 2-tuple linguistic m–polar fuzzy weighted average (2-tuple linguistic m–polar fuzzy weighted geometric) operator and for λ = 2, the 2-tuple linguistic m–polar fuzzy Hamacher weighted average (2-tuple linguistic m–polar fuzzy Hamacher weighted geometric) operator reduces to 2-tuple linguistic m– polar fuzzy Einstein weighted average (2-tuple linguistic m–polar fuzzy Einstein weighted geometric) operator. So, in case of λ = 3, the 2-tuple linguistic m-polar fuzzy Hamacher weighted average (2-tuple linguistic m–polar fuzzy Hamacher

9.6 Best Location for the Thermal Power Station: Case Study [6]

465

weighted geometric) sustain its own nature. Therefore, λ = 3 is the most suitable value to deal with the selection of most suitable place for thermal power station by using operator 2-tuple linguistic m–polar fuzzy Hamacher weighted average. To find the most suitable place for the thermal power station. The working procedure has described below: Step 1: The 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator is used to evaluate the assembled values, pˆi for the thermal power plant location alternatives as given in Table 9.2. Step 2: The score values S( pˆi ) for all the 2-tuple linguistic 3-polar fuzzy numbers pˆi are presented in Table 9.3. Step 3: Alternatives ranking according to their score values (S( pˆi ), i = 1, 2, . . . , 5) for all the 2-tuple linguistic 3-polar fuzzy numbers are recorded as: A2 > A5 > A3 > A1 > A4 . Step 4: Conclusively, A2 is the best place for thermal power station. If we use the 2-tuple linguistic m-polar fuzzy Hamacher geometric operator, the best alternative can be selected in the same pattern as taken above. The procedure is as follows: Here, λ = 3 is taken; in order to select the most suitable place for thermal power station by using 2-tuple linguistic m-polar fuzzy Hamacher weighted geometric operator.

Table 9.2 Assembled assessment by using Hamacher weighted average operator pˆi 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators pˆ1 pˆ2 pˆ3 pˆ4 pˆ5

((s4 , 0.00000), (s4 , 0.31958), (s4 , −0.1689)) ((s6 , 0.00000), (s4 , −0.0079), (s6 , 0.0000)) ((s4 , −0.1794), (s4 , −0.0499), (s6 , 0.0000)) ((s4 , −0.0039), (s4 , 0.18731), (s4 , −0.1398)) ((s4 , 0.24175), (s6 , 0.00000), (s4 , −0.4686))

Table 9.3 Scores values for all the 2-tuple linguistic 3-polar fuzzy numbers pˆi Scores values 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators S ( pˆ1 ) S ( pˆ2 ) S ( pˆ3 ) S ( pˆ4 ) S ( pˆ5 )

(s4 , 0.05022) (s5 , 0.33068) (s5 , −0.4098) (s4 , 0.01450) (s5 , −0.4089)

466

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

Table 9.4 Assembled assessment by using Hamacher weighted geometric operator pˆi 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operators pˆ1 pˆ2 pˆ3 pˆ4 pˆ5

((s4 , 0.00000), (s4 , 0.18858), (s4 , −0.3401)) ((s5 , −0.18858), (s4 , −0.2284), (s5 , −0.4946)) ((s4 , −0.2076), (s4 , −0.2232), (s3 , 0.48176)) ((s4 , −0.1146), (s4 , −0.0608), (s4 , −0.3258)) ((s4 , 0.08352), (s5 , −0.2833), (s3 , 0.48683))

Table 9.5 Scores values for all the 2-tuple linguistic 3-polar fuzzy numbers pˆi Scores values 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator S ( pˆ1 ) S ( pˆ2 ) S ( pˆ3 ) S ( pˆ4 ) S ( pˆ5 )

(s4 , −0.0505) (s4 , 0.36276) (s4 , −0.3163) (s4 , −0.1671) (s4 , 0.09567)

Step 1: The 2-tuple linguistic m-polar fuzzy Hamacher weighted geometric operator is used to assembled the values pˆi for the best thermal power plant location alternatives selection as given in the Table 9.4. Step 2: The score values S( pˆi ) for all the 2-tuple linguistic 3-polar fuzzy numbers pˆi are given in Table 9.5. Step 3: Alternatives ranking corresponding to their score values (S( pˆi ), i = 1, 2, .., 5), for all the 2-tuple linguistic 3 polar fuzzy numbers are presented as follows:

A2 > A5 > A1 > A4 > A3 . Step 4: Thus, A2 is the best alternative. This calculation shows that A2 is the most suitable location by using the 2-tuple linguistic m–polar fuzzy average and 2-tuple linguistic m–polar fuzzy geometric operators, order of ranking is given in Table 9.6.

9.6 Best Location for the Thermal Power Station: Case Study [6] Table 9.6 Alternative ranking order Operators Ranking 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators 2-tuple linguistic m-polar fuzzy Hamacher weighted geometric operators

467

Best alternative

A2 > A5 > A3 > A1 > A4

A2

A2 > A5 > A1 > A4 > A3

A2

Fig. 9.3 Score values based on average operator

9.6.1 Influence of the Parameter λ on Decision Making Results Akram et al. [6] did the work of rechecking the influence of the parameter λ ∈ [0, 6] on the ranking sequence of alternatives by using 2-tuple linguistic m–polar fuzzy weighted average and 2-tuple linguistic m–polar fuzzy weighted geometric operators. The score and ranking sequence for different values of parameter λ are given in Tables 9.7 and 9.8, calculated by using 2-tuple linguistic m–polar fuzzy weighted average and 2-tuple linguistic m–polar fuzzy weighted geometric operators. In the evaluation of 2-tuple linguistic m–polar fuzzy weighted average operator, the same ranking list is obtained for different values of λ, A2 > A5 > A3 > A1 > A4 . Further, by using the 2-tuple linguistic m–polar fuzzy weighted geometric operator the ranking list is represented as, A2 > A5 > A3 > A1 > A4 . The parameter λ working influences on the MADM problem formed on 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators and 2-tuple linguistic m-polar fuzzy Hamacher weighted geometric operators are given in Tables 9.7 and

468

9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

Fig. 9.4 Score values based on geometric operator

9.8, which reveals that, in the both situations, the most desirable alternative is A2 . Where graphical representation of scores variation by changing the parameter λ based on 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators and 2-tuple linguistic m-polar fuzzy Hamacher weighted geometric operators are represented in Figs. 9.3 and 9.4, by applying Δ−1 on score values. Conclusively, the developed MADM problem formed on 2-tuple linguistic m– polar fuzzy Hamacher weighted average operators and 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operators could not change the agnate ranking

Table 9.7 Scores values based on Hamacher weighted average operator λ

S ( pˆ1 )

S ( pˆ2 )

S ( pˆ3 )

1

(s4 , 0.08293)

(s5 , 0.35560)

(s5 , −0.38618) (s4 , 0.07355)

S ( pˆ4 )

(s5 , −0.38708) A2 > A5 > A3 > A1 > A4

S ( pˆ5 )

2

(s4 , 0.06035)

(s5 , 0.33825)

(s5 , −0.40259) (s4 , 0.03289)

(s5 , −0.40223) A2 > A5 > A3 > A1 > A4

3

(s4 , 0.05022)

(s5 , 0.33068)

(s5 , −0.40980) (s4 , 0.01450)

(s5 , −0.40895) A2 > A5 > A3 > A1 > A4

4

(s4 , 0.04441)

(s5 , 0.32643)

(s5 , −0.41386) (s4 , 0.00392)

(s5 , −0.41276) A2 > A5 > A3 > A1 > A4

5

(s4 , 0.04065)

(s5 , 0.32370)

(s5 , −0.41648) (s4 , −0.00297) (s5 , −0.41521) A2 > A5 > A3 > A1 > A4

6

(s4 , 0.03800)

(s5 , 0.32180)

(s5 , −0.41830) (s4 , −0.00783) (s5 , −0.41692) A2 > A5 > A3 > A1 > A4

Ranking order

9.7 Comparative Analysis

469

Table 9.8 Scores values based on Hamacher weighted geometric operator λ

S ( pˆ1 )

1

(s4 , −0.09709) (s4 , 0.29230)

S ( pˆ2 )

(s4 , −0.37132) (s4 , −0.25235) (s4 , 0.03175)

S ( pˆ3 )

S ( pˆ4 )

S ( pˆ5 )

A2 > A5 > A1 > A4 > A3

2

(s4 , −0.06776) (s4 , 0.33370)

(s4 , −0.33819) (s4 , −0.19843) (s4 , 0.06956)

A2 > A5 > A1 > A4 > A3

3

(s4 , −0.05053) (s4 , 0.36276)

(s4 , −0.31638) (s4 , −0.16710) (s4 , 0.09567)

A2 > A5 > A1 > A4 > A3

4

(s4 , −0.03901) (s4 , 0.38489)

(s4 , −0.30040) (s4 , −0.14627) (s4 , 0.11544)

A2 > A5 > A1 > A4 > A3

5

(s4 , −0.03070) (s4 , 0.40261)

(s4 , −0.28794) (s4 , −0.13131) (s4 , 0.13126)

A2 > A5 > A1 > A4 > A3

6

(s4 , −0.02441) (s4 , 0.41732)

(s4 , −0.27779) (s4 , −0.12000) (s4 , 0.14440)

A2 > A5 > A1 > A4 > A3

Ranking order

orders of the alternatives for different parametric values. Thus, the presented model is reliable and has fewer upshots by λ on multi-attribute decision-work.

9.7 Comparative Analysis Akram et al. [6] compared proposed techniques with existing techniques [26, 33].

9.7.1 Comparison with Existing Techniques This segment contains a comparison survey between the proposed techniques, namely, 2-tuple linguistic m–polar fuzzy Hamacher weighted average, 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operators with existing operators. The presented techniques applicability and versatility are verified by comparison of proposed models with existing ones. The developed operators 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators and 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric, comparison with the existing [26, 33] operators are given in Table 9.9. We observe that according to the existing and presented operators, the best location for a thermal power station is A2 , as in Table 9.9. Besides this, in the presented operator 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator provides the same ranking list, A2 > A5 > A1 > A4 > A3 as compared with the existing [26, 33] operator. But in the case of 2-tuple linguistic m–

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9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

Table 9.9 Comparative analysis of presented operators with existing ones Operators Ranking Best alternative m-polar fuzzy Dombi weighted average [26] m-polar fuzzy Dombi weighted geometric [26] m-polar fuzzy Hamacher weighted geometric [33] m-polar fuzzy Hamacher weighted average [33] 2-tuple linguistic m–polar fuzzy Hamacher weighted average(presented) 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric(presented)

A2 > A5 > A1 > A4 > A3

A2

A2 > A5 > A1 > A4 > A3

A2

A2 > A5 > A1 > A4 > A3

A2

A2 > A5 > A1 > A4 > A3

A2

A2 > A5 > A3 > A1 > A4

A2

A2 > A5 > A1 > A4 > A3

A2

Table 9.10 Characteristics comparison of presented operator with existing structures Operators existing and Fusion of linguist data with Aggregation more flexible by a proposed fuzzy information parameter λ m-polar fuzzy Dombi weighted average [26] m-polar fuzzy Dombi weighted geometric [26] m-polar fuzzy Hamacher weighted average [33] m-polar fuzzy Hamacher weighted geometric [33] 2-tuple linguistic m–polar fuzzy Hamacher weighted average(presented) 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric(presented)

×

×

×



×

×

×

×









polar fuzzy Hamacher weighted average operator, the order list is slightly different. The characteristics comparison of 2-tuple linguistic m–polar fuzzy operators with existing structures is in Table 9.10. The comparison chart 1 of 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator with existing operators is given in Fig. 9.5, and the comparison chart 2 of 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric with existing operators is given in Fig. 9.6.

9.7 Comparative Analysis

471

Fig. 9.5 Comparison chart 1

Fig. 9.6 Comparison chart 2

9.7.2 Discussion • The presented operators 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators (2-tuple linguistic m–polar fuzzy Hamacher weighted geometric) consider the interrelationship among the 2-tuple linguistic and m–polar fuzzy data, which was not the case of the existing operators [26, 33] that only deal with m-polar fuzzy information. Thus, the presented operators accommodate more amount of vagueness and provide more reliable results. • The techniques in [26, 33], albeit designed to take over MADM problems are restricted to deal with m–polar fuzzy information only. They are useless in the presence of linguistic features. So this may potentially become a cause of loss of information which typically leads to undesired results. Presented work produced versatile operators that overcome this limitation of previous methods.

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9 2-Tuple Linguistic Multi-polar Fuzzy Hamacher Aggregation Operators

• The presented operators not only operate with 2-tuple linguistic m–polar fuzzy data, but they also have the flexibility to switch from 2-tuple linguistic m-polar fuzzy to mF format by using the Δ−1 transformation. Thus, the presented method is more flexible and transparent than the existing one. After making the comparison, it has been taken into account that the presented operators can handle 2-tuple linguistic m–polar fuzzy information without any complexity. However our presented operators quickly describe the fusion of 2-tuple linguistic and m–polar fuzzy information The new MADM technique for 2-tuple linguistic m–polar fuzzy Hamacher weighted average operators and 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operators improved resilience in the utilization. Conclusively, the progress made with the help of the m–polar fuzzy operators has produced a malleable tool to tackle 2-tuple linguistic m–polar fuzzy information for MADM problems. The ranking order derived from the presented operators is compatible with the result from existing operators, so our proposal is reliable and valid for MADM. Yet more, it is prominent because no loss of data information occurs as in the linguistic information approach. Thus numeric and linguistic information make the presented operators more remarkable and adaptable.

9.8 Conclusions Many real life situations have a framework that contains 2-tuple linguistic representation with multi-polar data information. As several theoretical frameworks are developed to cover the broader range of complicated situations. Thus the basic need is the selection of the most suitable MADM approach to tackle the complicated situation in decision making. Thus in this chapter, the development of the MADM approach is presented in the presence of 2-tuple linguistic m-polar fuzzy data. To overcome the limitations of classical methods, the MADM problems are investigated, that suitably merges the concepts of 2-tuple linguistic with Hamacher-type operators and m–polar fuzzy numbers. Some aggregation operators are presented that are closely motivated by m–polar fuzzy Hamacher operations, namely, the 2-tuple linguistic m– polar fuzzy Hamacher weighted average operator, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator and 2-tuple linguistic m–polar fuzzy Hamacher hybrid average operator. Moreover, the 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator, 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted geometric operator, and 2-tuple linguistic m–polar fuzzy Hamacher hybrid average geometric operators are discussed. Various properties of these operators, namely, monotonicity, idempotency, and the boundedness are also investigated. So that, the practitioners may select the version that best serves. These aggregation operators are applied to enhance the applicability area of MADM in the m–polar fuzzy environment. In the end, a comparative study of the presented operators has been discussed concerning previously existing works.

References

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Chapter 10

Hybrid Models Based on Multi-polar Fuzzy Soft Sets

The goal of this chapter is to present multi-criteria group decision making models which covers criteria evaluation by different experts. Hybrid models for soft computing, namely multi-polar (m–polar) fuzzy soft expert sets, m−polar fuzzy N −soft sets, and m−polar fuzzy N −soft rough sets are discussed. An m–Polar fuzzy soft expert set is the combination of m–polar fuzzy sets with soft expert sets, which investigates soft expert sets in the m–polar fuzzy environment. An m−Polar fuzzy N −soft set is the combination of m–polar fuzzy sets with N −soft set, which investigates N −soft sets in the m–polar fuzzy environment. An m−polar fuzzy N −soft rough set is the combination of m–polar fuzzy sets with N −soft rough set, which investigates N −soft rough sets in the m–polar fuzzy environment. The characteristics of these hybrid model are explored with the aid of numerical examples. Further, their basic properties are investigated, and the operations of complement, intersection, union, plus, OR, and AND operators are discussed. Well-known real-world problems are solved with the developed hybrid models. The algorithms of the presented models provide their efficiency and cogency. The comparison of these presented models with other mathematical methods is provided. This chapter is due to [6, 7].

10.1 Introduction Multi-criteria group decision making (MCGDM) is playing a vital role to handle the widespread situations where a classification of a set of alternatives is required, and it must be based upon the opinions of different experts regarding different criteria [54, 81]. For instance, the selection of a suitable employee is based upon the human resource (HR) department of an organization according to the opinion of different experts regarding multiple criteria [38]. Thus nowadays, how to tackle the situa© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_10

475

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tions concerning MCGDM is a worthy research area. A lot of researchers have been attracted by this applicable topic, and numerous MCGDM methods have been developed in order to account for the multiplicity of sources of information [10, 11, 39, 57, 62, 84, 88, 89]. Simple classical theories are unable to deal MCGDM methods. To handle such immature, imprecise and vague information, Zadeh introduced the concept of fuzzy set theory [82]. Liu et al. [61] proposed a MCGDM model where the opinions of multiple experts concerning to different criteria are expressed by Pythagorean fuzzy information. Further, a number of extensions have been proposed including intuitionistic fuzzy sets [33] and m–polar fuzzy sets [40]. Atanassov [33] introduced the concept of intuitionistic fuzzy set, an extension of fuzzy set, that together with a membership grade μ, makes use of a non-membership grade ν, subject to the condition μ + ν ≤ 1. Although the mathematical performance of intuitionistic fuzzy sets is quite good, they still fail to capture many situations where the sum of membership and non-membership degrees exceeds 1 (at some option). Yager [78, 79] presented the less stringent concept of Pythagorean fuzzy sets for which the condition μ + ν ≤ 1 imposed by intuitionistic fuzzy sets is relaxed to μ2 + ν 2 ≤ 1. Nevertheless real-world models often contain multi-attribute, multiindex, multi-object, and multi-information data. For example, multi-polar technology, which can be used for the management of large-scale applications of information technology. Multipolar technology can be employed to large control systems as well. With this practical motivation, Chen et al. [40] initiated the theory of m–polar fuzzy sets. In an m–polar fuzzy set, an element considers m membership values belonging to [0, 1] and describes the strength of m different features of the element. Until now, many real-world problems have been solved where data comes from m different poles (m ≥ 2). Particularly when m = 2, the m–polar fuzzy set model produces bipolar fuzzy sets. Akram [2] proposed various new concepts on m–polar fuzzy graphs. The theory of rough sets [71] is a successful mathematical tool for handling imprecise and uncertain information. There are several real life problems in various fields, including physical sciences, life sciences, and social sciences contain uncertainties. Molodtsov [70] pointed out some deficiencies of the aforementioned mathematical theories like fuzzy sets [82] and rough sets [71], especially the fact that these theories cannot handle parameterizations-related situations. In order to overcome these difficulties, the model called soft sets can easily represent the uncertainty regarding different parameters [70]. Soft set theory has proved its significance in several fields because it has attracted the attention of many theoretical and applied investigations. The theory of soft sets is playing a very important role in many fields including, data analysis [90] and decision making [48, 56, 63, 74]. Maji et al. [65] explored certain basic notions of soft set model. Ali et al. [26] proposed further a few novel properties of soft sets and verified De Morgan’s laws for soft sets. In addition, Maji et al. [67] discussed applications of soft sets to decision-making. As reported by Molodtsov [70] that being different mathematical tools, the theories of fuzzy, rough and soft sets handle different types of uncertainties, but one can combine these to produce more compact and precise hybrid models. After that, Maji et al. [66] proposed the idea of fuzzy soft sets. A number of extensions of soft set theory have been reported in various directions, including m–polar fuzzy soft set theory [9], N –

10.1 Introduction

477

soft sets [45], valuation soft sets [20], fuzzy N –soft expert sets [24], and m–polar fuzzy soft rough sets [8]. Hybridization of two or more mathematical methods is employed to solve practical problems having uncertain information in many real fields, including medical sciences [68]. But several hybrid soft-set models can only deal with data in which knowledge is acquired from one expert, and in the presence of environments that require the opinions of two or more experts, these hybrid structures of soft sets become useless. In such situations there is a major difficulty for the users, specifically for those who utilize question sheets in their reports. To tackle this issue, Alkhazaleh and Salleh [28] presented the concept of soft expert sets which consider the opinions of all experts. In continuation of this effort, Alkhazaleh and Salleh [29] developed a new hybrid model called fuzzy soft expert sets by combining fuzzy sets and soft expert sets. Several research works have complemented this powerful idea until today. For instance, Adam and Hassan [1] proposed multi Q–fuzzy soft expert set model and studied an application to decision making. Al-Qudah and Hassan [31] generalized the fuzzy soft expert set model and presented the concept of bipolar fuzzy soft expert sets, utilizing them to solve decision making decision making problems. Bashir and Salleh [34, 35] introduced two novel hybrid models, namely, fuzzy parameterized soft expert sets and possibility fuzzy soft expert sets. Broumi and Smarandache [37] developed intuitionistic fuzzy soft expert sets as an extension of fuzzy soft experts sets. Qayyum et al. [73] introduced the cubic soft expert set model for decision-making. Hassan and Alhazaymeh [52] presented a useful extension of soft expert sets, namely, vague soft expert set model. An inspection of the hybrid soft set models readily shows that the researchers attracted by soft sets and their hybrid models typically worked on either a binary framework of evaluations (either 0 or 1) or else, real numbers between 0 and 1 (see [63, 90]). However, examples abound that daily real situations contain data with a non-binary and discrete structure, which therefore do not belong to those formats. For instance in the aggregation of social opinions, Alcantud and Laruelle [18] specified the characteristics of ternary voting systems and gave examples where ternary opinions must be aggregated. Examples closer to our daily experience are the non-binary estimations that we often find in rating or ranking positions. Rankings of tourist resorts, films, or electronic devices, often take the form of number of dots and stars (like ‘one big dot,’ ‘one star,’ ‘two stars,’ ‘three stars’), which can also be seen in the form of natural numbers (like ‘0’ for one big circle, ‘1’ for one star, ‘2’ for two stars, ‘3’ for three stars). Furthermore, Herawan and Deris [53] developed n binary-valued information system in soft sets where every parameter has its own ranking, when compared to the rating order described in Chen et al. [41]. Instead of ratings as estimations, Ali et al. [27] designed rating systems among the elements of soft sets parameters. Motivated by these practical considerations, Fatimah et al. [45] introduced the notion of N −soft set as a generalization of soft sets with a multinary nature. This model was later combined with other features like hesitancy and fuzziness. Indeed, Akram et al. [4] developed a new powerful hybrid model for group decision-making, namely, hesitant N −soft sets. Akram et al. [5] combined fuzzy sets with N −soft sets, and proposed another useful decision making model

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

called fuzzy N −soft sets. For other useful notions, the readers are addressed to [12, 21, 22, 25, 30, 32, 55, 63, 80].

10.2 m–Polar Fuzzy Soft Expert Sets In this section, prior to presenting the notion of m–polar fuzzy soft expert sets, first of all some previous concepts are recalled. After that, the hybrid method is explained via an illustrative example. Different notions, including subset, equality, union, intersection, complement, and the OR and AND operators are also studied. Definition 10.1 [40] An m–polar fuzzy set on a universe V is a mapping Z : V → [0, 1]m . The degree of membership for every object of V is represented by Z (ϑ) = ( p1 ◦ Z (ϑ), p2 ◦ Z (ϑ), . . . , pm ◦ Z (ϑ)) where pi ◦ Z : [0, 1]m → [0, 1] is defined as the ith projection mapping. The smallest tuple in [0, 1]m is 0 = (0, 0, . . . , 0) and the largest tuple in [0, 1]m is 1 = (1, 1, . . . , 1). Definition 10.2 [28] Let V be an initial universal set, S a set of parameters, and E a set of experts (agents). Let O = {0 = disagree, 1 = agree} be a set of judgments. For any X ⊆ Q, a pair (G, X ) is said to be a soft expert set, where Q = S × E × O and G is a function given by G : X → 2V , where 2V denotes the collection of all subset of V . Let ζ = ( p1 ◦ Z (v), . . . , pm ◦ Z (v)) be an m–polar fuzzy number, where pi ◦ Z (v) ∈ [0, 1], for all i = 1, 2, . . . , m. The definitions of score and accuracy of ζ are given below: Definition 10.3 ([77]) A score function S for m–polar fuzzy numbers is defined by: when ζ = ( p1 ◦ Z (v), . . . , pm ◦ Z (v)) is an m–polar fuzzy number, then S(ζ) =  m 1 ( pi ◦ ζ) . Observe that S(ζ) ∈ [0, 1]. m i=1 Definition 10.4 ([77]) An accuracy function H for m–polar fuzzy numbers is given by: when ζ = ( p1 ◦ Z (v), . . . , pm ◦ Z (v)) is an m–polar fuzzy number, then   m ˆ = 1 (−1)i+1 ( pi ◦ ζ − 1) . Observe that H(ζ) ∈ [−1, 1]. H(ζ) m i=1 Akram et al. [6] introduced the concept of m–polar fuzzy soft expert sets. Definition 10.5 ([6]) Let V be a non-empty universal set, S a set of parameters, and let E be a set of experts. Let O = {0 = disagree, 1 = agree} be the set of judgments of experts regarding objects in V , Q = S × E × O and X ⊆ Q. A pair (γ, X ) is said to be an m–polar fuzzy soft expert set over the soft universe (V, X ) where γ : X → m F V is a mapping given by

10.2 m–Polar Fuzzy Soft Expert Sets

479

γ(x) = γ(x)(v), ∀ v ∈ V. The membership of each element is represented by pi ◦ γ(x) = ( p1 ◦ γ(x)(v), p2 ◦ γ(x)(v), . . . , pm ◦ γ(x)(v)) = pi ◦ γ(x)(v) where pi ◦ γ(x)(v) : [0, 1]m → [0, 1] is the ith projection mapping. Note that γ(x) represents the satisfaction degrees to properties corresponding to m–polar fuzzy soft expert set (V, X ) for the objects of V in γ(x). In other words, an m–polar fuzzy soft expert set (γ, X ) can also be written as follows: (γ, X ) = {(x, p1 ◦ γ(x)(v), p2 ◦ γ(x)(v), . . . , pm ◦ γ(x)(v) : x ∈ X, v ∈ V }. Example 10.1 Assume that an organization wishes to purchase an appropriate hotel according to their needs from the options v1 , v2 , v3 and v4 . To choose the most suitable alternative which meets their maximum requirements, the organization decides to take opinions of three experts about these alternatives. Let V = {v1 , v2 , v3 , v4 } be the set of four hotels, and S = {s1 = location, s2 = design, s3 = technology} the set of parameters where • The parameter ‘Location’ includes near to main road, in the green surrounding, and near to city center. • The parameter ‘Design’ includes classic, modern, and mixture of both. • The parameter ‘Size’ includes small, large, and very large. Let E = {ε1 , ε2 } be a set of experts and Q = S × E × O. Suppose that:   γ(s1 , ε1 , 1) = (v1 , 0.7, 0.5, 0.8 , (v2 , 0.3, 0.4, 0.6 , (v3 , 0.6, 0.1, 0.7 , (v4 , 0.5, 0.4, 0.2 ,   γ(s1 , ε2 , 1) = (v1 , 0.2, 0.1, 0.6 , (v2 , 0.5, 0.1, 0.9 , (v3 , 1.0, 0.7, 0.6 , (v4 , 0.8, 0.6, 0.7 ,   γ(s2 , ε1 , 1) = (v1 , 0.9, 0.2, 0.5 , (v2 , 0.6, 0.4, 0.7 , (v3 , 0.8, 0.7, 0.2 , (v4 , 0.6, 0.4, 0.7 ,   γ(s2 , ε2 , 1) = (v1 , 0.2, 0.4, 0.2 , (v2 , 0.3, 0.8, 0.1 , (v3 , 0.9, 0.4, 0.3 , (v4 , 0.1, 0.1, 0.2 ,   γ(s3 , ε1 , 1) = (v1 , 0.2, 0.1, 0.3 , (v2 , 0.8, 0.5, 0.8 , (v3 , 0.2, 0.9, 1.0 , (v4 , 0.9, 0.7, 0.7 ,   γ(s3 , ε2 , 1) = (v1 , 0.6, 0.4, 0.9 , (v2 , 0.8, 0.9, 0.7 , (v3 , 0.2, 0.1, 0.3 , (v4 , 0.8, 0.4, 0.1 ,   γ(s1 , ε1 , 0) = (v1 , 0.4, 0.6, 0.9 , (v2 , 0.9, 0.6, 0.6 , (v3 , 0.5, 0.1, 0.5 , (v4 , 0.6, 0.1, 0.4 ,   γ(s1 , ε2 , 0) = (v1 , 0.7, 0.4, 0.0 , (v2 , 0.4, 0.4, 0.7 , (v3 , 0.6, 0.1, 0.7 , (v4 , 0.3, 0.2, 0.1 ,   γ(s2 , ε1 , 0) = (v1 , 0.6, 0.1, 0.6 , (v2 , 0.1, 0.8, 0.5 , (v3 , 0.1, 0.1, 0.2 , (v4 , 0.2, 0.5, 0.5 ,   γ(s2 , ε2 , 0) = (v1 , 0.6, 0.9, 0.1 , (v2 , 0.4, 0.7, 0.6 , (v3 , 0.8, 0.1, 0.7 , (v4 , 0.5, 0.3, 0.1 ,   γ(s3 , ε1 , 0) = (v1 , 0.1, 0.4, 0.1 , (v2 , 0.2, 0.8, 0.3 , (v3 , 0.6, 0.2, 0.4 , (v4 , 0.6, 0.8, 0.2 ,   γ(s3 , ε2 , 0) = (v1 , 0.9, 0.6, 0.4 , (v2 , 0.7, 0.4, 0.3 , (v3 , 0.6, 0.4, 0.7 , (v4 , 0.5, 0.9, 0.1 ,

Then, the 3–polar fuzzy soft expert set (γ, X ) is given as follows:

480

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.1 A 3–polar fuzzy soft expert set (γ, X ) v1 v2 (s1 , ε1 , 1) (s1 , ε2 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s3 , ε1 , 1) (s3 , ε2 , 1) (s1 , ε1 , 0) (s1 , ε2 , 0) (s2 , ε1 , 0) (s2 , ε2 , 0) (s3 , ε1 , 0) (s3 , ε2 , 0)

(γ, X ) =

0.7, 0.5, 0.8

0.2, 0.1, 0.6

0.9, 0.2, 0.5

0.2, 0.4, 0.2

0.2, 0.1, 0.3

0.6, 0.4, 0.9

0.4, 0.6, 0.9

0.7, 0.4, 0.0

0.6, 0.1, 0.6

0.6, 0.9, 0.1

0.1, 0.4, 0.1

0.9, 0.6, 0.4

0.3, 0.4, 0.6

0.5, 0.1, 0.9

0.6, 0.4, 0.7

0.3, 0.8, 0.1

0.8, 0.5, 0.8

0.8, 0.9, 0.7

0.9, 0.6, 0.6

0.4, 0.4, 0.7

0.1, 0.8, 0.5

0.4, 0.7, 0.6

0.2, 0.8, 0.3

0.7, 0.4, 0.3

v3

v4

0.6, 0.1, 0.7

1.0, 0.7, 0.6

0.8, 0.7, 0.2

0.9, 0.4, 0.3

0.2, 0.9, 1.0

0.2, 0.1, 0.3

0.5, 0.1, 0.5

0.6, 0.1, 0.7

0.1, 0.1, 0.2

0.8, 0.1, 0.7

0.6, 0.2, 0.4

0.6, 0.4, 0.7

0.5, 0.4, 0.2

0.8, 0.6, 0.7

0.6, 0.4, 0.7

0.1, 0.1, 0.2

0.9, 0.7, 0.7

0.8, 0.4, 0.1

0.6, 0.1, 0.4

0.3, 0.2, 0.1

0.2, 0.5, 0.5

0.5, 0.3, 0.1

0.6, 0.8, 0.2

0.5, 0.9, 0.1



 (s1 , ε1 , 1), {(v1 , 0.7, 0.5, 0.8 , (v2 , 0.3, 0.4, 0.6 , (v3 , 0.6, 0.1, 0.7 , (v4 , 0.5, 0.4, 0.2 } ,   (s1 , ε2 , 1), {(v1 , 0.2, 0.1, 0.6 , (v2 , 0.5, 0.1, 0.9 , (v3 , 1.0, 0.7, 0.6 , (v4 , 0.8, 0.6, 0.7 } ,   (s2 , ε1 , 1), {(v1 , 0.9, 0.2, 0.5 , (v2 , 0.6, 0.4, 0.7 , (v3 , 0.8, 0.7, 0.2 , (v4 , 0.6, 0.4, 0.7 } ,   (s2 , ε2 , 1), {(v1 , 0.2, 0.4, 0.2 , (v2 , 0.3, 0.8, 0.1 , (v3 , 0.9, 0.4, 0.3 , (v4 , 0.1, 0.1, 0.2 } ,   (s3 , ε1 , 1), {(v1 , 0.2, 0.1, 0.3 , (v2 , 0.8, 0.5, 0.8 , (v3 , 0.2, 0.9, 1.0 , (v4 , 0.9, 0.7, 0.7 } ,   (s3 , ε2 , 1), {(v1 , 0.6, 0.4, 0.9 , (v2 , 0.8, 0.9, 0.7 , (v3 , 0.2, 0.1, 0.3 , (v4 , 0.8, 0.4, 0.1 } ,   (s1 , ε1 , 0), {(v1 , 0.4, 0.6, 0.9 , (v2 , 0.9, 0.6, 0.6 , (v3 , 0.5, 0.1, 0.5 , (v4 , 0.6, 0.1, 0.4 } ,   (s1 , ε2 , 0), {(v1 , 0.7, 0.4, 0.0 , (v2 , 0.4, 0.4, 0.7 , (v3 , 0.6, 0.1, 0.7 , (v4 , 0.3, 0.2, 0.1 } ,   (s2 , ε1 , 0), {(v1 , 0.6, 0.1, 0.6 , (v2 , 0.1, 0.8, 0.5 , (v3 , 0.1, 0.1, 0.2 , (v4 , 0.2, 0.5, 0.5 } ,   (s2 , ε2 , 0), {(v1 , 0.6, 0.9, 0.1 , (v2 , 0.4, 0.7, 0.6 , (v3 , 0.8, 0.1, 0.7 , (v4 , 0.5, 0.3, 0.1 } ,   (s3 , ε1 , 0), {(v1 , 0.1, 0.4, 0.1 , (v2 , 0.2, 0.8, 0.3 , (v3 , 0.6, 0.2, 0.4 , (v4 , 0.6, 0.8, 0.2 } ,   (s3 , ε2 , 0), {(v1 , 0.9, 0.6, 0.4 , (v2 , 0.7, 0.4, 0.3 , (v3 , 0.6, 0.4, 0.7 , (v4 , 0.5, 0.9, 0.1 } .

The tabular representation of the 3–polar fuzzy soft expert set (γ, X ) is given in Table 10.1. Note that the first cell in the Table 10.1 describes that for the parameter ‘s1 ’, the expert ‘ε1 ’ shows its positive opinion for the hotel v1 because its ‘location’ is 70% near to main road, 50% in the green surroundings, and 80% near to city center. Definition 10.6 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets over V . Then (γ1 , Y ) is said to be an m–polar fuzzy soft expert subset of (γ2 , Z ) if 1. Y ⊆ Z , 2. for all y ∈ Y, γ1 (y) ⊆ γ2 (y), (i.e., pi ◦ γ1 (y)(v) ≤ pi ◦ γ2 (y)(v), ∀ v ∈ V, 1 ≤ i ≤ m.

10.2 m–Polar Fuzzy Soft Expert Sets

481

Table 10.2 A 3–polar fuzzy soft expert set (γ1 , Y ) v1 v2 (s1 , ε1 , 1) (s1 , ε2 , 1) (s2 , ε1 , 0) (s2 , ε2 , 0) (s3 , ε1 , 0)

0.7, 0.5, 0.8

0.2, 0.4, 0.2

0.6, 0.1, 0.6

0.6, 0.9, 0.1

0.1, 0.4, 0.1

0.3, 0.4, 0.6

0.3, 0.8, 0.1

0.1, 0.8, 0.5

0.4, 0.7, 0.6

0.2, 0.8, 0.3

Table 10.3 A 3–polar fuzzy soft expert set (γ2 , Z ) v1 v2 (s1 , ε1 , 1) (s1 , ε2 , 1) (s2 , ε1 , 1) (s2 , ε1 , 0) (s2 , ε2 , 0) (s3 , ε1 , 0) (s3 , ε2 , 0)

0.8, 0.6, 0.9

0.3, 0.5, 0.4

0.9, 0.2, 0.5

0.7, 0.2, 0.7

0.7, 0.9, 0.3

0.3, 0.5, 0.2

0.9, 0.6, 0.4

0.4, 0.5, 0.7

0.4, 0.9, 0.2

0.6, 0.4, 0.7

0.2, 0.9, 0.6

0.5, 0.8, 0.7

0.3, 0.8, 0.4

0.7, 0.4, 0.3

v3

v4

0.6, 0.1, 0.7

0.9, 0.4, 0.3

0.1, 0.1, 0.2

0.8, 0.1, 0.7

0.6, 0.2, 0.4

0.5, 0.4, 0.2

0.1, 0.1, 0.2

0.2, 0.5, 0.5

0.5, 0.3, 0.1

0.6, 0.8, 0.2

v3

v4

0.7, 0.2, 0.7

1.0, 0.7, 0.6

0.8, 0.7, 0.2

0.3, 0.4, 0.5

0.9, 0.3, 0.8

0.7, 0.5, 0.4

0.6, 0.4, 0.7

0.6, 0.5, 0.4

0.8, 0.6, 0.7

0.6, 0.4, 0.7

0.4, 0.6, 0.7

0.6, 0.5, 0.4

0.7, 0.9, 0.3

0.5, 0.9, 0.1

ˆ 2 , Z ), and is read as (γ1 , Y ) is an m–polar fuzzy soft It is denoted by (γ1 , Y )⊆(γ expert subset of (γ2 , Z ). In other words, (γ2 , Z ) is called an m–polar fuzzy soft expert superset of (γ1 , Y ). Example 10.2 Consider Example 10.1 and suppose that the company takes the judgments of the experts once again. Suppose Y = {(s1 , ε1 , 1), (s1 , ε2 , 1), (s2 , ε1 , 0), (s2 , ε2 , 0), (s3 , ε1 , 0)}, Z = {(s1 , ε1 , 1), (s1 , ε2 , 1), (s2 , ε1 , 1), (s2 , ε1 , 0), (s2 , ε2 , 0), (s3 , ε1 , 0), (s3 , ε2 , 0)}. Clearly Y ⊆ Z . Let (γ1 , Y ) and (γ2 , Z ) be two 3–polar fuzzy soft expert sets over V displayed by Tables 10.2 and 10.3, respectively. ˆ 2 , Z ). Thus (γ1 , Y )⊆(γ Definition 10.7 Let (γ, X ) be an m–polar fuzzy soft expert set on V . Then an agreem–polar fuzzy soft expert set (γ, X )1 is an m–polar fuzzy soft expert subset of (γ, X ), which is given as follows: (γ, X )1 = {γ(x) : x ∈ S × E × {1} ⊂ X }. Example 10.3 Consider again Example 10.1. Then, an agree-3–polar fuzzy soft expert set (γ, X )1 of a 3–polar fuzzy soft expert set (γ, X ) over V is given in Table 10.4.

482

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.4 An agree-3–polar fuzzy soft expert set (γ, X )1 v1 v2 (s1 , ε1 , 1) (s1 , ε2 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s3 , ε1 , 1) (s3 , ε2 , 1)

0.7, 0.5, 0.8

0.2, 0.1, 0.6

0.9, 0.2, 0.5

0.2, 0.4, 0.2

0.2, 0.1, 0.3

0.6, 0.4, 0.9

0.3, 0.4, 0.6

0.5, 0.1, 0.9

0.6, 0.4, 0.7

0.3, 0.8, 0.1

0.8, 0.5, 0.8

0.8, 0.9, 0.7

Table 10.5 Disagree-3–polar fuzzy soft expert set (γ, X )0 v1 v2 (s1 , ε1 , 0) (s1 , ε2 , 0) (s2 , ε1 , 0) (s2 , ε2 , 0) (s3 , ε1 , 0) (s3 , ε2 , 0)

0.4, 0.6, 0.9

0.7, 0.4, 0.0

0.6, 0.1, 0.6

0.6, 0.9, 0.1

0.1, 0.4, 0.1

0.9, 0.6, 0.4

0.9, 0.6, 0.6

0.4, 0.4, 0.7

0.1, 0.8, 0.5

0.4, 0.7, 0.6

0.2, 0.8, 0.3

0.7, 0.4, 0.3

v3

v4

0.6, 0.1, 0.7

1.0, 0.7, 0.6

0.8, 0.7, 0.2

0.9, 0.4, 0.3

0.2, 0.9, 1.0

0.2, 0.1, 0.3

0.5, 0.4, 0.2

0.8, 0.6, 0.7

0.6, 0.4, 0.7

0.1, 0.1, 0.2

0.9, 0.7, 0.7

0.8, 0.4, 0.1

v3

v4

0.5, 0.1, 0.5

0.6, 0.1, 0.7

0.1, 0.1, 0.2

0.8, 0.1, 0.7

0.6, 0.2, 0.4

0.6, 0.4, 0.7

0.6, 0.1, 0.4

0.3, 0.2, 0.1

0.2, 0.5, 0.5

0.5, 0.3, 0.1

0.6, 0.8, 0.2

0.5, 0.9, 0.1

Definition 10.8 Let (γ, X ) be an m–polar fuzzy soft expert set on V . Then a disagree-m–polar fuzzy soft expert set (γ, X )0 is an m–polar fuzzy soft expert subset of (γ, X ), which is given by (γ, X )0 = {γ(x) : x ∈ S × E × {0} ⊂ X }. Example 10.4 Consider Example 10.1 again. Then, a disagree-3–polar fuzzy soft expert set (γ, X )0 of the 3–polar fuzzy soft expert set (γ, X ) over V is given in Table 10.5. Definition 10.9 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets over V . Then (γ1 , Y ) and (γ2 , Z ) are said to be equal m–polar fuzzy soft expert sets if (γ1 , Y ) is an m–polar fuzzy soft expert subset of (γ2 , Z ) and (γ2 , Z ) is an m–polar fuzzy soft soft expert subset of (γ1 , Y ). Definition 10.10 Let (γ, X ) be an m–polar fuzzy soft expert set of V . Then, its complement is denoted by (γ, X )c which is given by (γ, X )c = (γ c , X ) where γ c : X → m F V is a mapping given by γ c (x) = (v, 1 − p1 ◦ γ(x)(v), 1 − p2 ◦ γ(x)(v), . . . , 1 − pm ◦ γ(x)(v)), for all x ∈ X, v ∈ V . Example 10.5 Consider again the 3–polar fuzzy soft expert set (γ, X ) in Example 10.1. Then, its complement is given in Table 10.6. A useful relation between agree-m–polar fuzzy soft expert sets and disagree-m– polar fuzzy soft expert sets is provided in the following proposition.

10.2 m–Polar Fuzzy Soft Expert Sets

483

Table 10.6 Complement of 3–polar fuzzy soft expert set (γ, X )c v1 v2 v3 (s1 , ε1 , 1) (s1 , ε2 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s3 , ε1 , 1) (s3 , ε2 , 1) (s1 , ε1 , 0) (s1 , ε2 , 0) (s2 , ε1 , 0) (s2 , ε2 , 0) (s3 , ε1 , 0) (s3 , ε2 , 0)

0.3, 0.5, 0.2

0.8, 0.9, 0.4

0.1, 0.8, 0.5

0.8, 0.6, 0.8

0.8, 0.9, 0.7

0.4, 0.6, 0.1

0.6, 0.4, 0.1

0.3, 0.6, 1.0

0.4, 0.9, 0.4

0.4, 0.1, 0.9

0.9, 0.6, 0.9

0.1, 0.4, 0.6

0.3, 0.4, 0.6

0.5, 0.9, 0.1

0.4, 0.6, 0.3

0.7, 0.2, 0.9

0.2, 0.6, 0.2

0.2, 0.1, 0.3

0.1, 0.4, 0.4

0.6, 0.6, 0.3

0.9, 0.2, 0.5

0.6, 0.3, 0.4

0.8, 0.2, 0.7

0.3, 0.6, 0.7

0.4, 0.9, 0.3

0.0, 0.3, 0.4

0.2, 0.3, 0.8

0.1, 0.6, 0.7

0.8, 0.1, 0.0

0.8, 0.9, 0.7

0.5, 0.9, 0.5

0.4, 0.9, 0.3

0.9, 0.9, 0.8

0.2, 0.9, 0.3

0.4, 0.8, 0.6

0.4, 0.6, 0.3

v4 0.5, 0.6, 0.8

0.2, 0.4, 0.3

0.4, 0.6, 0.3

0.9, 0.9, 0.8

0.1, 0.3, 0.3

0.2, 0.6, 0.9

0.4, 0.9, 0.6

0.7, 0.8, 0.9

0.8, 0.5, 0.5

0.5, 0.7, 0.9

0.4, 0.2, 0.8

0.5, 0.1, 0.9

Proposition 10.1 Let (γ, X ) be an m–polar fuzzy soft expert set of V . Then 1. ((γ, X )c )c = (γ, X ), 2. ((γ, X )1 )c = (γ, X )0 , 3. ((γ, X )0 )c = (γ, X )1 . Proof It directly follows by Definition 10.10.



Definition 10.11 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets over V . Then their union denoted by (γ1 , Y )  (γ2 , Z ), is an m–polar fuzzy soft expert set (J, N ) where N = Y ∪ Z and for all y ∈ N , ⎧ ⎪ if y ∈ Y − Z , ⎨γ1 (y), J (y) = γ2 (y), if y ∈ Z − Y, ⎪ ⎩ γ1 (y) ∪ γ2 (y) if y ∈ Y ∩ Z , where γ1 (y) ∪ γ2 (y) = (v, max( p1 ◦ γ1 (y)(v), p1 ◦ γ2 (y)(v)), . . . , max( pm ◦ γ1 (y)(v), pm ◦ γ2 (y)(v))), for all (v, p1 ◦ γ1 (y)(v), . . . , pm ◦ γ1 (y)(v)) ∈ γ1 (y), (v, p1 ◦ γ2 (y)(v), . . . , pm ◦ γ2 (y)(v)) ∈ γ2 (y), pi ◦ γ j (y)(v) ∈ [0, 1], i = 1, 2, . . . , m and j = 1, 2. Example 10.6 Suppose a university wishes to hire a professor from the four candidates v1 , v2 , v3 , and v4 . The hiring of an employee is the duty of HR department of the university. To select the best candidate, the HR department decides to take opinions of three experts about these candidates. Let V = {v1 , v2 , v3 , v4 } be the set of four candidates, S = {s1 = experiance, s2 = research abilities, s3 = communication abilities} a set of parameters, E = {ε1 , ε2 , ε3 } a set of experts and Y, Z ⊆ Q = S × E × O, which are given as follows:

484

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.7 A 3–polar fuzzy soft expert set (γ1 , Y ) v1 v2 (s1 , ε1 , 1) (s1 , ε2 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s3 , ε3 , 1) (s3 , ε1 , 0) (s3 , ε2 , 0)

0.3, 0.6, 0.2

0.4, 0.5, 0.2

0.5, 0.2, 0.8

0.7, 0.2, 0.8

0.4, 0.1, 0.2

0.3, 0.2, 0.4

1.0, 0.4, 0.3

0.4, 0.5, 0.6

0.6, 0.2, 0.8

0.4, 0.2, 0.8

0.7, 0.2, 0.6

0.5, 0.8, 0.5

0.6, 0.5, 0.2

0.6, 0.8, 0.2

Table 10.8 A 3–polar fuzzy soft expert set (γ2 , Z ) v1 v2 (s1 , ε1 , 1) (s1 , ε3 , 1) (s2 , ε1 , 1) (s3 , ε2 , 1) (s3 , ε3 , 1) (s2 , ε2 , 0) (s3 , ε1 , 0) (s3 , ε3 , 0)

0.2, 0.5, 0.3

0.3, 0.2, 0.4

0.5, 0.4, 0.3

0.4, 0.6, 0.3

0.3, 0.1, 0.3

0.3, 0.5, 0.4

0.7, 0.4, 0.5

0.5, 0.4, 0.1

0.1, 0.2, 0.7

0.5, 0.3, 0.6

0.6, 0.7, 0.5

0.4, 0.5, 0.3

0.3, 0.5, 0.3

0.1, 0.8, 0.7

0.8, 0.6, 0.5

0.6, 0.5, 0.2

v3

v4

0.7, 0.2, 0.6

0.1, 0.8, 0.4

0.3, 0.5, 0.3

0.2, 0.4, 0.1

0.5, 0.9, 0.2

0.8, 0.3, 0.5

0.6, 0.5, 0.7

0.6, 0.3, 0.1

0.2, 0.1, 0.6

0.8, 0.1, 0.1

0.3, 0.7, 0.4

0.1, 0.2, 0.3

0.7, 0.5, 0.4

0.6, 0.6, 0.4

v3

v4

0.8, 0.2, 0.3

0.3, 0.4, 0.1

0.3, 0.7, 0.4

0.5, 0.3, 0.2

0.5, 0.9, 0.2

0.6, 0.4, 0.2

0.9, 0.4, 0.6

0.5, 0.5, 0.6

0.1, 0.5, 0.3

0.7, 0.6, 0.4

0.6, 0.1, 0.5

0.7, 0.5, 0.3

0.2, 0.5, 0.4

0.5, 0.4, 0.3

0.6, 0.4, 0.7

0.2, 0.7, 0.4

Y = {(s1 , ε1 , 1), (s1 , ε2 , 1), (s2 , ε1 , 1), (s2 , ε2 , 1), (s3 , ε3 , 1), (s3 , ε1 , 0), (s3 , ε2 , 0)}, Z = {(s1 , ε1 , 1), (s1 , ε3 , 1), (s2 , ε1 , 1), (s3 , ε2 , 1), (s3 , ε3 , 1), (s2 , ε2 , 0), (s3 , ε1 , 0), (s3 , ε3 , 0)}.

Suppose that (γ1 , Y ) and (γ2 , Z ) are two 3–polar fuzzy soft expert sets over V , which are respectively given in Tables 10.7 and 10.8. Therefore, the union of the 3–polar fuzzy soft expert set (γ1 , Y ) and (γ2 , Z ) produces the output shown in Table 10.9. Union is commutative and associative: Proposition 10.2 Let (γ1 , X ), (γ2 , Y ) and (γ3 , Z ) be three m–polar fuzzy soft expert sets over V . Then 1. (γ1 , X )  (γ2 , Y ) = (γ2 , Y )  (γ1 , X ), 2. (γ1 , X )  ((γ2 , Y )  (γ3 , Z )) = ((γ1 , X )  (γ2 , Y ))  (γ3 , Z ). Proof 1. From Definition 10.11, (γ1 , X )  (γ2 , Y ) = (J1 , N1 ) with N1 = X ∪ Y and for all x ∈ N1 ,

10.2 m–Polar Fuzzy Soft Expert Sets

485

Table 10.9 Union of the 3–polar fuzzy soft expert sets (J, N ) v1 v2 (s1 , ε2 , 1) (s2 , ε2 , 1) (s3 , ε2 , 0) (s1 , ε3 , 1) (s3 , ε2 , 1) (s2 , ε2 , 0) (s3 , ε3 , 0) (s1 , ε1 , 1) (s2 , ε1 , 1) (s3 , ε3 , 1) (s3 , ε1 , 0)

0.4, 0.5, 0.2

0.7, 0.2, 0.8

1.0, 0.4, 0.3

0.3, 0.2, 0.4

0.4, 0.6, 0.3

0.3, 0.5, 0.4

0.5, 0.4, 0.1

0.3, 0.6, 0.3

0.5, 0.4, 0.8

0.4, 0.1, 0.3

0.7, 0.4, 0.5

0.6, 0.2, 0.8

0.7, 0.2, 0.6

0.6, 0.8, 0.2

0.5, 0.3, 0.6

0.4, 0.5, 0.3

0.1, 0.8, 0.7

0.6, 0.5, 0.2

0.4, 0.5, 0.7

0.6, 0.7, 0.8

0.5, 0.8, 0.5

0.8, 0.6, 0.5

v3

v4

0.1, 0.8, 0.4

0.2, 0.4, 0.1

0.6, 0.5, 0.7

0.3, 0.4, 0.1

0.5, 0.3, 0.2

0.6, 0.4, 0.2

0.5, 0.5, 0.6

0.8, 0.2, 0.6

0.3, 0.7, 0.4

0.5, 0.9, 0.2

0.9, 0.4, 0.6

0.2, 0.1, 0.6

0.3, 0.7, 0.4

0.6, 0.6, 0.4

0.7, 0.6, 0.4

0.7, 0.5, 0.3

0.5, 0.4, 0.3

0.2, 0.7, 0.4

0.6, 0.5, 0.3

0.8, 0.1, 0.5

0.2, 0.5, 0.4

0.7, 0.5, 0.7

⎧ ⎪ if x ∈ X − Y, ⎨γ1 (x), J1 (x) = γ2 (x), if x ∈ Y − X, ⎪ ⎩ γ1 (x) ∪ γ2 (x), if x ∈ X ∩ Y, where γ1 (x) ∪ γ2 (x) = (v, max( p1 ◦ γ1 (x)(v), p1 ◦ γ2 (x)(v)), . . . , max( pm ◦ γ1 (x)(v), pm ◦ γ2 (x)(v))), for all (v, p1 ◦ γ1 (x)(v), . . . , pm ◦ γ1 (x) (v)) ∈ γ1 (x), (v, p1 ◦ γ2 (x)(v), . . . , pm ◦ γ2 (x)(v)) ∈ γ2 (x), pi ◦ γ j (x)(v) ∈ [0, 1], i = 1, 2, . . . , m and j = 1, 2. Similarly, by Definition 10.11, (γ2 , Y )  (γ1 , X ) = (J2 , N2 ) with N2 = Y ∪ X and for all x ∈ N2 , ⎧ ⎪ if x ∈ Y − X, ⎨γ2 (x), J2 (x) = γ1 (x), if x ∈ X − Y, ⎪ ⎩ γ2 (x) ∪ γ1 (x), if x ∈ Y ∩ X, = J1 (x) where γ2 (x) ∪ γ1 (x) = (v, max( p1 ◦ γ2 (x)(v), p1 ◦ γ1 (x)(v)), . . . , max( pm ◦ γ2 (x)(v), pm ◦ γ1 (x)(v))), for all (v, p1 ◦ γ1 (x)(v), . . . , pm ◦ γ1 (x) (v)) ∈ γ1 (x), (v, p1 ◦ γ2 (x)(v), . . . , pm ◦ γ2 (x)(v)) ∈ γ2 (x), pi ◦ γ j (x)(v) ∈ [0, 1], i = 1, 2, . . . , m and j = 1, 2. 2. Its proof directly follows as in part 1.



Definition 10.12 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets over V . Then their intersection denoted by (γ1 , Y )  (γ2 , Z ), is an m–polar fuzzy soft expert set (I, N ) where N = Y ∪ Z and for all y ∈ N ,

486

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.10 Intersection of the 3–polar fuzzy soft expert sets (I, N ) v1 v2 v3 (s1 , ε2 , 1) (s2 , ε2 , 1) (s3 , ε2 , 0) (s1 , ε3 , 1) (s3 , ε2 , 1) (s2 , ε2 , 0) (s3 , ε3 , 0) (s1 , ε1 , 1) (s2 , ε1 , 1) (s3 , ε3 , 1) (s3 , ε1 , 0)

0.4, 0.5, 0.2

0.7, 0.2, 0.8

1.0, 0.4, 0.3

0.3, 0.2, 0.4

0.4, 0.6, 0.3

0.3, 0.5, 0.4

0.5, 0.4, 0.1

0.2, 0.5, 0.2

0.5, 0.2, 0.3

0.3, 0.1, 0.2

0.3, 0.2, 0.4

0.6, 0.2, 0.8

0.7, 0.2, 0.6

0.6, 0.8, 0.2

0.5, 0.3, 0.6

0.4, 0.5, 0.3

0.1, 0.8, 0.7

0.6, 0.5, 0.2

0.1, 0.2, 0.6

0.4, 0.2, 0.5

0.3, 0.5, 0.3

0.6, 0.5, 0.2

0.1, 0.8, 0.4

0.2, 0.4, 0.1

0.6, 0.5, 0.7

0.3, 0.4, 0.1

0.5, 0.3, 0.2

0.6, 0.4, 0.2

0.5, 0.5, 0.6

0.7, 0.2, 0.3

0.3, 0.5, 0.3

0.5, 0.9, 0.2

0.8, 0.3, 0.5

v4 0.2, 0.1, 0.6

0.3, 0.7, 0.4

0.6, 0.6, 0.4

0.7, 0.6, 0.4

0.7, 0.5, 0.3

0.5, 0.4, 0.3

0.2, 0.7, 0.4

0.1, 0.3, 0.1

0.6, 0.1, 0.1

0.1, 0.2, 0.3

0.6, 0.4, 0.4

⎧ ⎪ if y ∈ Y − Z , ⎨γ1 (y), I (y) = γ2 (y), if y ∈ Z − Y, ⎪ ⎩ γ1 (y) ∩ γ2 (y) if y ∈ Y ∩ Z , where γ1 (y) ∩ γ2 (y) = (v, min( p1 ◦ γ1 (y)(v), p1 ◦ γ2 (y)(v)), . . . , min( pm ◦ γ1 (y)(v), pm ◦ γ2 (y)(v))), for all (v, p1 ◦ γ1 (y)(v), . . . , pm ◦ γ1 (y)(v)) ∈ γ1 (y), (v, p1 ◦ γ2 (y)(v), . . . , pm ◦ γ2 (y)(v)) ∈ γ2 (y), pi ◦ γ j (y)(v) ∈ [0, 1], i = 1, 2, . . . , m and j = 1, 2. Example 10.7 Consider Example 10.6 again. The intersection of the 3–polar fuzzy soft expert sets (γ1 , Y ), and (γ2 , Z ) is given by Table 10.10. Intersection is commutative and associative too: Proposition 10.3 Let (γ1 , X ), (γ2 , Y ), and (γ3 , Z ) be three m–polar fuzzy soft expert sets over V . Then 1. (γ1 , X )  (γ2 , Y ) = (γ2 , Y )  (γ1 , X ), 2. (γ1 , X )  ((γ2 , Y )  (γ3 , Z )) = ((γ1 , X )  (γ2 , Y ))  (γ3 , Z ). Proof 1. From Definition 10.12, (γ1 , X )  (γ2 , Y ) = (I1 , N1 ) with N1 = X ∪ Y and for all x ∈ N1 , ⎧ ⎪ if x ∈ X − Y, ⎨γ1 (x), I1 (x) = γ2 (x), if x ∈ Y − X, ⎪ ⎩ γ1 (x) ∩ γ2 (x), if x ∈ X ∩ Y, where γ1 (x) ∩ γ2 (x) = (v, min( p1 ◦ γ1 (x)(v), p1 ◦ γ2 (x)(v)), . . . , min( pm ◦ γ1 (x)(v), pm ◦ γ2 (x)(v))), for all (v, p1 ◦ γ1 (x)(v), . . . , pm ◦ γ1 (x)

10.2 m–Polar Fuzzy Soft Expert Sets

487

Table 10.11 A 3–polar fuzzy soft expert set (γ1 , Y ) v1 v2 (s1 , ε2 , 1) (s2 , ε2 , 1) (s3 , ε2 , 1) (s2 , ε3 , 0)

0.3, 0.7, 0.5

0.4, 0.2, 0.6

0.4, 0.1, 0.2

0.3, 0.6, 0.4

0.3, 0.5, 0.7

0.8, 0.3, 0.1

0.5, 0.8, 0.5

0.6, 0.5, 0.3

v3

v4

0.4, 0.9, 0.6

0.3, 0.2, 0.3

0.5, 0.9, 0.2

0.8, 0.5, 0.5

0.6, 0.3, 0.7

0.3, 0.5, 0.4

0.6, 0.5, 0.7

0.3, 0.2, 0.4

(v)) ∈ γ1 (x), (v, p1 ◦ γ2 (x)(v), . . . , pm ◦ γ2 (x)(v)) ∈ γ2 (x), pi ◦ γ j (x)(v) ∈ [0, 1], i = 1, 2, . . . , m and j = 1, 2. Similarly, by Definition 10.12, (γ2 , Y )  (γ1 , X ) = (I2 , N2 ) with N2 = Y ∪ X and for all x ∈ N2 , ⎧ ⎪ if x ∈ Y − X, ⎨γ2 (x), I2 (x) = γ1 (x), if x ∈ X − Y, ⎪ ⎩ γ2 (x) ∩ γ1 (x), if x ∈ Y ∩ X, = I1 (x) where γ2 (x) ∩ γ1 (x) = (v, min( p1 ◦ γ2 (x)(v), p1 ◦ γ1 (x)(v)), . . . , min( pm ◦ γ2 (x)(v), pm ◦ γ1 (x)(v))), for all (v, p1 ◦ γ1 (x)(v), . . . , pm ◦ γ1 (x) (v)) ∈ γ1 (x), (v, p1 ◦ γ2 (x)(v), . . . , pm ◦ γ2 (x)(v)) ∈ γ2 (x), pi ◦ γ j (x)(v) ∈ [0, 1], i = 1, 2, . . . , m and j = 1, 2. 2. Its proof follows immediately by using similar arguments as in part 1.



Definition 10.13 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets of V . Then the AND operation (γ1 , Y ) AND (γ2 , Z ) is represented by (γ1 , Y ) ∧ (γ2 , Z ) and is given as follows: (γ1 , Y ) ∧ (γ2 , Z ) = (J, Y × Z ), where J (y, z) = γ1 (y) ∩ γ2 (z), ∀ (y, z) ∈ Y × Z . Example 10.8 Consider Example 10.6 again. Let Y = {(s1 , ε2 , 1), (s2 , ε2 , 1), (s3 , ε2 , 1), (s2 , ε3 , 0)}, Z = {(s2 , ε1 , 1), (s2 , ε2 , 1), (s3 , ε1 , 0), (s3 , ε3 , 0)}. Suppose that (γ1 , Y ) and (γ2 , Z ) are two 3–polar fuzzy soft expert sets over V , which are given by Tables 10.11 and 10.12, respectively. Thus, the AND of (γ1 , Y ) and (γ2 , Z ) is computed, as shown in Table 10.13. Definition 10.14 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets over V . Then the OR operation (γ1 , Y ) OR (γ2 , Z ) is denoted by (γ1 , Y ) ∨ (γ2 , Z ) and is given by

488

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.12 A 3–polar fuzzy soft expert set (γ2 , Z ) v1 v2 (s2 , ε1 , 1) (s2 , ε2 , 1) (s3 , ε1 , 0) (s3 , ε3 , 0)

0.5, 0.3, 0.7

0.8, 0.4, 0.2

0.8, 0.2, 0.5

0.5, 0.6, 0.4

0.8, 0.3, 0.7

0.4, 0.6, 0.4

0.4, 0.2, 0.7

0.6, 0.6, 0.2

v3

v4

0.7, 0.9, 0.3

0.7, 0.5, 0.3

0.7, 0.4, 0.6

0.5, 0.3, 0.5

0.5, 0.5, 0.2

0.8, 0.4, 0.3

0.3, 0.7, 0.5

0.7, 0.9, 0.4

Table 10.13 AND operation between the 3–polar fuzzy soft expert sets (J, Y × Z )   (s1 , ε2 , 1), (s2 , ε1 , 1)   (s1 , ε2 , 1), (s2 , ε2 , 1)   (s1 , ε2 , 1), (s3 , ε1 , 0)   (s1 , ε2 , 1), (s3 , ε3 , 0)   (s2 , ε2 , 1), (s2 , ε1 , 1)   (s2 , ε2 , 1), (s2 , ε2 , 1)   (s2 , ε2 , 1), (s3 , ε1 , 0)   (s2 , ε2 , 1), (s3 , ε3 , 0)   (s3 , ε2 , 1), (s2 , ε1 , 1)   (s3 , ε2 , 1), (s2 , ε2 , 1)   (s3 , ε2 , 1), (s3 , ε1 , 0)   (s3 , ε2 , 1), (s3 , ε3 , 0)   (s3 , ε3 , 0), (s2 , ε1 , 1)   (s3 , ε3 , 0), (s2 , ε2 , 1)   (s3 , ε3 , 0), (s3 , ε1 , 0)   (s3 , ε3 , 0), (s3 , ε3 , 0)

v1

v2

v3

v4

0.3, 0.3, 0.5

0.3, 0.3, 0.7

0.4, 0.9, 0.3

0.5, 0.3, 0.2

0.3, 0.4, 0.2

0.3, 0.5, 0.4

0.4, 0.5, 0.3

0.5, 0.3, 0.2

0.3, 0.2, 0.5

0.3, 0.2, 0.7

0.4, 0.4, 0.6

0.3, 0.3, 0.5

0.3, 0.6, 0.4

0.3, 0.5, 0.2

0.4, 0.3, 0.5

0.6, 0.3, 0.4

0.4, 0.2, 0.6

0.8, 0.3, 0.1

0.3, 0.2, 0.3

0.3, 0.5, 0.2

0.4, 0.2, 0.2

0.4, 0.3, 0.1

0.3, 0.2, 0.3

0.3, 0.4, 0.3

0.4, 0.2, 0.5

0.4, 0.2, 0.1

0.3, 0.2, 0.3

0.3, 0.5, 0.4

0.4, 0.2, 0.4

0.6, 0.3, 0.1

0.3, 0.2, 0.3

0.3, 0.5, 0.4

0.4, 0.1, 0.2

0.5, 0.3, 0.5

0.5, 0.9, 0.2

0.5, 0.5, 0.2

0.4, 0.1, 0.2

0.4, 0.6, 0.4

0.5, 0.5, 0.2

0.6, 0.4, 0.3

0.4, 0.1, 0.2

0.4, 0.2, 0.5

0.5, 0.4, 0.2

0.3, 0.5, 0.5

0.4, 0.1, 0.2

0.5, 0.6, 0.2

0.5, 0.3, 0.2

0.6, 0.5, 0.4

0.3, 0.3, 0.4

0.6, 0.3, 0.3

0.7, 0.5, 0.3

0.3, 0.2, 0.2

0.3, 0.4, 0.2

0.4, 0.5, 0.3

0.7, 0.5, 0.3

0.3, 0.2, 0.3

0.3, 0.2, 0.4

0.4, 0.2, 0.3

0.7, 0.4, 0.5

0.3, 0.2, 0.4

0.3, 0.6, 0.4

0.6, 0.5, 0.2

0.5, 0.3, 0.5

0.3, 0.2, 0.4

(γ1 , Y ) ∨ (γ2 , Z ) = (O, Y × Z ), where O(y, z) = γ1 (y) ∪ γ2 (z), ∀ (y, z) ∈ Y × Z . Example 10.9 Consider Example 10.8 again. Then, the OR operation of the 3–polar fuzzy soft expert sets (γ1 , Y ) and (γ2 , Z ) is computed, as shown in Table 10.14. The interaction of the AND/OR operations with complementarity is explored in the next result: Proposition 10.4 Let (γ1 , Y ) and (γ2 , Z ) be two m–polar fuzzy soft expert sets over V . Then 1. ((γ1 , Y ) ∧ (γ2 , Z ))c = (γ1 , Y )c ∨ (γ2 , Z )c , 2. ((γ1 , Y ) ∨ (γ2 , Z ))c = (γ1 , Y )c ∧ (γ2 , Z )c . Proof It can be easily proved by Definitions 10.13 and 10.14.



10.2 m–Polar Fuzzy Soft Expert Sets

489

Table 10.14 OR operation between 3–polar fuzzy soft expert sets (O, Y × Z )   (s1 , ε2 , 1), (s2 , ε1 , 1)   (s1 , ε2 , 1), (s2 , ε2 , 1)   (s1 , ε2 , 1), (s3 , ε1 , 0)   (s1 , ε2 , 1), (s3 , ε3 , 0)   (s2 , ε2 , 1), (s2 , ε1 , 1)   (s2 , ε2 , 1), (s2 , ε2 , 1)   (s2 , ε2 , 1), (s3 , ε1 , 0)   (s2 , ε2 , 1), (s3 , ε3 , 0)   (s3 , ε2 , 1), (s2 , ε1 , 1)   (s3 , ε2 , 1), (s2 , ε2 , 1)   (s3 , ε2 , 1), (s3 , ε1 , 0)   (s3 , ε2 , 1), (s3 , ε3 , 0)   (s3 , ε3 , 0), (s2 , ε1 , 1)   (s3 , ε3 , 0), (s2 , ε2 , 1)   (s3 , ε3 , 0), (s3 , ε1 , 0)   (s3 , ε3 , 0), (s3 , ε3 , 0)

v1

v2

v3

v4

0.5, 0.7, 0.7

0.8, 0.5, 0.7

0.7, 0.9, 0.6

0.6, 0.5, 0.7

0.8, 0.7, 0.5

0.4, 0.6, 0.7

0.7, 0.9, 0.6

0.8, 0.4, 0.7

0.8, 0.7, 0.5

0.4, 0.5, 0.7

0.7, 0.9, 0.6

0.6, 0.7, 0.7

0.5, 0.7, 0.5

0.6, 0.6, 0.7

0.5, 0.9, 0.6

0.7, 0.9, 0.7

0.5, 0.3, 0.7

0.8, 0.3, 0.7

0.7, 0.9, 0.3

0.5, 0.5, 0.4

0.8, 0.4, 0.6

0.8, 0.6, 0.4

0.7, 0.5, 0.3

0.8, 0.5, 0.4

0.8, 0.2, 0.6

0.8, 0.3, 0.7

0.7, 0.4, 0.6

0.3, 0.7, 0.5

0.5, 0.6, 0.6

0.8, 0.6, 0.2

0.5, 0.3, 0.5

0.7, 0.9, 0.4

0.5, 0.3, 0.7

0.8, 0.8, 0.7

0.7, 0.9, 0.3

0.6, 0.5, 0.7

0.8, 0.4, 0.2

0.5, 0.8, 0.5

0.7, 0.9, 0.3

0.8, 0.5, 0.7

0.8, 0.2, 0.5

0.5, 0.8, 0.7

0.7, 0.9, 0.6

0.6, 0.7, 0.7

0.5, 0.6, 0.4

0.6, 0.8, 0.5

0.5, 0.9, 0.5

0.7, 0.9, 0.7

0.5, 0.6, 0.7

0.8, 0.5, 0.7

0.8, 0.9, 0.5

0.5, 0.5, 0.4

0.8, 0.6, 0.4

0.6, 0.6, 0.4

0.8, 0.5, 0.5

0.8, 0.4, 0.4

0.8, 0.6, 0.5

0.6, 0.5, 0.7

0.8, 0.5, 0.6

0.3, 0.7, 0.5

0.5, 0.6, 0.4

0.6, 0.6, 0.3

0.8, 0.5, 0.5

0.7, 0.9, 0.4

Associative and distributive properties are the target of the next array of results: Proposition 10.5 Let (γ1 , X ), (γ2 , Y ) and (γ3 , Z ) be three m–polar fuzzy soft expert sets over V . Then 1. 2. 3. 4.

(γ1 , X ) ∧ ((γ2 , Y ) ∧ (γ3 , Z )) = ((γ1 , X ) ∧ (γ2 , Y )) ∧ (γ3 , Z ), (γ1 , X ) ∨ ((γ2 , Y ) ∨ (γ3 , Z )) = ((γ1 , X ) ∨ (γ2 , Y )) ∨ (γ3 , Z ), (γ1 , X ) ∧ ((γ2 , Y ) ∨ (γ3 , Z )) = ((γ1 , X ) ∧ (γ2 , Y )) ∨ ((γ1 , X ) ∧ (γ3 , Z )), (γ1 , X ) ∨ ((γ2 , Y ) ∧ (γ3 , Z )) = ((γ1 , X ) ∨ (γ2 , Y )) ∧ ((γ1 , X ) ∨ (γ3 , Z )).

Proof 1. By Definition 10.13, (γ1 , X ) ∧ ((γ2 , Y ) ∧ (γ3 , Z )) = (J, X × (Y × Z )), where J (x, (y, z) = γ1 (x) ∩ (γ2 (y) ∩ γ3 (y)), for all (x, (y, z) ∈ X × (Y × Z ). From the Proposition 10.3, we get γ1 (x) ∩ (γ2 (y) ∩ γ3 (z)) = (γ1 (x) ∩ γ2 (y)) ∩ γ3 (z). Hence, (γ1 , X ) ∧ ((γ2 , Y ) ∧ (γ3 , Z )) = ((γ1 , X ) ∧ (γ2 , Y )) ∧ (γ3 , Z ). The proofs of parts 2 to 4 are similar to the arguments proving part 1.



490

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

10.3 Mathematical Approach for MCGDM with m–Polar Fuzzy Information In order to handle any MCGDM situation involving information with an m–polar fuzzy structure, Akram et al. [6] used m–polar fuzzy soft expert set model. For this purpose, the next algorithm for MCGDM based on m–polar fuzzy soft expert sets is described as follows: Algorithm 1. Input: V , a universal set having n objects S, a set of parameters E, a set of experts having judgments: agree means ‘1’ and disagree means ‘0’ (γ, Q), an m–polar fuzzy soft expert set in the form of a table where Q ⊆ S × E × O. 2. Separate agree-m–polar fuzzy soft expert set and compute score values for each m–polar fuzzy number in the agree-m–polar fuzzy soft expert set table by Definition 10.3. 3. Separate disagree-m–polar fuzzy soft expert set and compute score values for each m–polar fuzzy number in the disagree-m–polar fuzzy soft expert set table by Definition 10.3. 4. Find the final score table by subtracting the accumulated score values of disagreem–polar fuzzy soft expert set from scores of agree-m–polar fuzzy soft expert set. 5. List all the objects si , i = 1, 2, . . . , n according to the score values S(si ) obtained in step 4. If the final score values are same for two or more object, then use the accuracy function and start the computations again from step 2 to determine the exact ranking of the objects. Output: The object having highest score will be selected as decision. In the applications in the next section, this method will be used to determine a suitable option. The flowchart of the procedure is given in Fig. 10.1.

10.4 Applications

491

Fig. 10.1 Flowchart of the strategy of solution in Sect. 10.3

10.4 Applications Akram et al. [6] solved two real-world applications by the MCGDM approach.

10.4.1 Selection of a Suitable Site for a Dam A dam is a structure which prevents the flow of water, and accumulates it in a reservoir. The location of a dam must be computed by examining all possible alternatives by

492

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

the comparison of merits and demerits very carefully so that maximum advantage can be acquired at lowest level of cost, and risk regarding the entire project. It is normal that the selection of suitable alternative from possible location may vary based upon the needs of the project. If the dam is designed for irrigation, water supply and flood control, it is better that the desired location is possibly close to the place that will benefit, i.e., in the lower or central reaches of the river, with the viewpoint of guaranteeing a steady impact. However, if the dam is designed for electricity purpose, interest is concentrated on the upper reaches, in positions along with a huge water head, although it is far away from the place of demand. Since dam is a big project acquiring a number of funds, utmost care shall be taken during the selection of a site for dam. Incorrect decision may guide extreme cost and problems in the construction and maintenance. Always keeping in mind the site selection for dam, the factors that should be considered are: topography, geology and suitable foundation, environmental conditions, metrological and hydrological factors, availability of suitable construction material, construction and maintenance cost, location of spill way etc. Consider a team of engineers are trying to find a suitable place for a dam according to the instruction of their superiors from the alternatives v1 , v2 , . . . , v5 . For these given options they agree to consider some main parameters for the selection of best alternative which are: Topography, reservoir quality, construction and maintenance cost, and geology. These parameters can be further characterized as follows: • The parameter “Topography”, is a basic general feature of the site which includes soil profile, ground profile, plant profile, valley profile. • The parameter “Reservoir Quality” for the site includes large storage capacity, shape of the reservoir, hydrological condition, water tightness of reservoir. • The parameter “Construction and Maintenance Cost” includes large, very large, very very large, extremely large. • The parameter “Geology” of the site includes bearing capacity of soil, bearing capacity of rocks, foundation settlements, permeability of foundation soil. Let V = {v1 , v2 , v3 , v4 , v5 } be a set of five alternatives (sites) and S = {s1 = Topography, s2 = Reservoir Quality, s3 = Construction and Maintenance Cost, s4 = Geology} be a set of parameters. Let E = {ε1 , ε2 , ε3 } be the set of experts and Q = S × E × O. Then the experts give their report in the form of an 4–polar fuzzy soft expert set as displayed in Table 10.15. The agree-4–polar fuzzy soft expert set is displayed in Table 10.16. By Definition 10.3, the score value for each 4–polar fuzzy number is calculated in Table 10.17. The disagree-4–polar fuzzy soft expert set is displayed in Table 10.18. By Definition 10.3, the score value for each 4–polar fuzzy number is calculated in Table 10.18. Now the final score values can be easily obtained from the agree and disagree score values in Tables 10.17 and 10.19, respectively, which are given in Table 10.20.

10.4 Applications

493

Table 10.15 A 4–polar fuzzy soft expert set (γ, Q)

v1

(s1 , ε1 , 1)

0.5, 0.6, 0.3, 0.4 0.4, 0.2, 0.5, 0.3 0.7, 0.5, 0.6, 0.8 0.6, 0.4, 0.1, 0.3 0.7, 0.9, 0.3, 0.5

v2

v3

v4

v5

(s1 , ε2 , 1)

0.1, 0.8, 0.4, 0.7 0.9, 0.2, 0.3, 0.7 0.5, 0.4, 0.7, 0.8 0.2, 0.6, 0.7, 0.6 0.8, 0.4, 0.5, 0.3

(s1 , ε3 , 1)

0.5, 0.2, 0.4, 0.7 0.3, 0.4, 0.6, 0.5 0.4, 0.7, 0.2, 0.6 0.8, 0.9, 0.5, 0.4 0.1, 0.4, 0.6, 0.7

(s2 , ε1 , 1)

0.2, 0.3, 0.4, 0.7 0.7, 0.5, 0.7, 0.9 0.2, 0.5, 0.1, 0.7 0.4, 0.6, 0.7, 0.3 0.8, 0.4, 0.5, 0.6

(s2 , ε2 , 1)

0.3, 0.5, 0.7, 0.3 0.2, 0.5, 0.4, 0.7 0.9, 0.7, 0.4, 0.5 0.2, 0.1, 0.3, 0.5 0.6, 0.5, 0.3, 0.4

(s2 , ε3 , 1)

0.5, 0.6, 0.5, 0.7 0.4, 0.2, 0.1, 0.6 0.7, 0.3, 0.2, 0.4 0.6, 0.3, 0.5, 0.7 0.8, 0.3, 0.5, 0.1

(s3 , ε1 , 1)

0.1, 0.3, 0.4, 0.7 0.5, 0.3, 0.7, 0.4 0.8, 0.5, 0.4, 0.6 0.2, 0.3, 0.6, 0.4 0.4, 0.5, 0.3, 0.8

(s3 , ε2 , 1)

0.4, 0.1, 0.6, 0.4 0.3, 0.5, 0.7, 0.6 0.7, 0.5, 0.2, 0.6 0.6, 0.3, 0.5, 0.5 0.6, 0.2, 0.5, 0.3

(s3 , ε3 , 1)

0.3, 0.4, 0.7, 0.6 0.4, 0.5, 0.8, 0.3 0.5, 0.3, 0.4, 0.6 0.7, 0.3, 0.6, 0.5 0.4, 0.6, 0.3, 0.7

(s4 , ε1 , 1)

0.3, 0.5, 0.7, 0.4 0.4, 0.7, 0.3, 0.2 0.7, 0.5, 0.4, 0.3 0.8, 0.6, 0.5, 0.4 0.5, 0.2, 0.5, 0.6

(s4 , ε2 , 1)

0.8, 0.5, 0.6, 0.4 0.5, 0.6, 0.7, 0.8 0.4, 0.5, 0.3, 0.2 0.4, 0.1, 0.5, 0.3 0.6, 0.4, 0.3, 0.6

(s4 , ε3 , 1)

0.6, 0.4, 0.3, 0.7 0.4, 0.3, 0.2, 0.3 0.5, 0.7, 0.4, 0.3 0.7, 0.3, 0.5, 0.8 0.4, 0.3, 0.6, 0.4

(s1 , ε1 , 0)

0.3, 0.5, 0.2, 0.3 0.3, 0.2, 0.4, 0.2 0.6, 0.4, 0.5, 0.7 0.5, 0.3, 0.2, 0.4 0.6, 0.8, 0.2, 0.4

(s1 , ε2 , 0)

0.2, 0.7, 0.3, 0.6 0.8, 0.3, 0.5, 0.4 0.4, 0.3, 0.6, 0.7 0.3, 0.5, 0.6, 0.5 0.7, 0.3, 0.4, 0.2

(s1 , ε3 , 0)

0.4, 0.4, 0.5, 0.6 0.2, 0.5, 0.7, 0.6 0.3, 0.6, 0.1, 0.5 0.7, 0.6, 0.3, 0.6 0.3, 0.5, 0.2, 0.6

(s2 , ε1 , 0)

0.1, 0.2, 0.6, 0.5 0.3, 0.4, 0.6, 0.8 0.1, 0.4, 0.0, 0.6 0.3, 0.5, 0.5, 0.2 0.7, 0.3, 0.4, 0.4

(s2 , ε2 , 0)

0.4, 0.3, 0.6, 0.2 0.1, 0.4, 0.5, 0.6 0.8, 0.6, 0.3, 0.7 0.3, 0.2, 0.4, 0.4 0.5, 0.3, 0.5, 0.3

(s2 , ε3 , 0)

0.4, 0.5, 0.4, 0.6 0.3, 0.5, 0.2, 0.4 0.6, 0.2, 0.6, 0.3 0.7, 0.2, 0.4, 0.6 0.6, 0.2, 0.3, 0.4

(s3 , ε1 , 0)

0.4, 0.2, 0.3, 0.5 0.3, 0.1, 0.5, 0.2 0.5, 0.3, 0.7, 0.5 0.3, 0.2, 0.5, 0.3 0.3, 0.6, 0.2, 0.7

(s3 , ε2 , 0)

0.5, 0.2, 0.7, 0.3 0.2, 0.4, 0.6, 0.5 0.6, 0.4, 0.1, 0.5 0.5, 0.2, 0.4, 0.3 0.4, 0.1, 0.3, 0.2

(s3 , ε3 , 0)

0.2, 0.3, 0.6, 0.5 0.3, 0.4, 0.6, 0.2 0.4, 0.2, 0.3, 0.5 0.4, 0.5, 0.3, 0.4 0.5, 0.4, 0.2, 0.5

(s4 , ε1 , 0)

0.2, 0.4, 0.6, 0.3 0.3, 0.6, 0.2, 0.1 0.6, 0.4, 0.3, 0.2 0.7, 0.5, 0.4, 0.3 0.4, 0.1, 0.4, 0.5

(s4 , ε2 , 0)

0.7, 0.4, 0.5, 0.3 0.4, 0.5, 0.6, 0.7 0.3, 0.4, 0.2, 0.3 0.5, 0.4, 0.4, 0.2 0.5, 0.3, 0.2, 0.5

(s4 , ε3 , 0)

0.5, 0.3, 0.2, 0.6 0.3, 0.2, 0.1, 0.2 0.4, 0.6, 0.3, 0.2 0.6, 0.2, 0.4, 0.7 0.3, 0.2, 0.5, 0.3

Table 10.16 An agree-4–polar fuzzy soft expert set (γ, Q)1

v1

(s1 , ε1 , 1)

0.5, 0.6, 0.3, 0.4 0.4, 0.2, 0.5, 0.3 0.7, 0.5, 0.6, 0.8 0.6, 0.4, 0.1, 0.3 0.7, 0.9, 0.3, 0.5

v2

v3

v4

v5

(s1 , ε2 , 1)

0.1, 0.8, 0.4, 0.7 0.9, 0.2, 0.3, 0.7 0.5, 0.4, 0.7, 0.8 0.2, 0.6, 0.7, 0.6 0.8, 0.4, 0.5, 0.3

(s1 , ε3 , 1)

0.5, 0.2, 0.4, 0.7 0.3, 0.4, 0.6, 0.5 0.4, 0.7, 0.2, 0.6 0.8, 0.9, 0.5, 0.4 0.1, 0.4, 0.6, 0.7

(s2 , ε1 , 1)

0.2, 0.3, 0.4, 0.7 0.7, 0.5, 0.7, 0.9 0.2, 0.5, 0.1, 0.7 0.4, 0.6, 0.7, 0.3 0.8, 0.4, 0.5, 0.6

(s2 , ε2 , 1)

0.3, 0.5, 0.7, 0.3 0.2, 0.5, 0.4, 0.7 0.9, 0.7, 0.4, 0.5 0.2, 0.1, 0.3, 0.5 0.6, 0.5, 0.3, 0.4

(s2 , ε3 , 1)

0.5, 0.6, 0.5, 0.7 0.4, 0.2, 0.1, 0.6 0.7, 0.3, 0.2, 0.4 0.6, 0.3, 0.5, 0.7 0.8, 0.3, 0.5, 0.1

(s3 , ε1 , 1)

0.1, 0.3, 0.4, 0.7 0.5, 0.3, 0.7, 0.4 0.8, 0.5, 0.4, 0.6 0.2, 0.3, 0.6, 0.4 0.4, 0.5, 0.3, 0.8

(s3 , ε2 , 1)

0.4, 0.1, 0.6, 0.4 0.3, 0.5, 0.7, 0.6 0.7, 0.5, 0.2, 0.6 0.6, 0.3, 0.5, 0.5 0.6, 0.2, 0.5, 0.3

(s3 , ε3 , 1)

0.3, 0.4, 0.7, 0.6 0.4, 0.5, 0.8, 0.3 0.5, 0.3, 0.4, 0.6 0.7, 0.3, 0.6, 0.5 0.4, 0.6, 0.3, 0.7

(s4 , ε1 , 1)

0.3, 0.5, 0.7, 0.4 0.4, 0.7, 0.3, 0.2 0.7, 0.5, 0.4, 0.3 0.8, 0.6, 0.5, 0.4 0.5, 0.2, 0.5, 0.6

(s4 , ε2 , 1)

0.8, 0.5, 0.6, 0.4 0.5, 0.6, 0.7, 0.8 0.4, 0.5, 0.3, 0.2 0.4, 0.1, 0.5, 0.3 0.6, 0.4, 0.3, 0.6

(s4 , ε3 , 1)

0.6, 0.4, 0.3, 0.7 0.4, 0.3, 0.2, 0.3 0.5, 0.7, 0.4, 0.3 0.7, 0.3, 0.5, 0.8 0.4, 0.3, 0.6, 0.4

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.17 Score values for agree-4–polar fuzzy soft expert set (γ, Q)1 v1 v2 v3 v4 (s1 , ε1 , 1) (s1 , ε2 , 1) (s1 , ε3 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s2 , ε3 , 1) (s3 , ε1 , 1) (s3 , ε2 , 1) (s3 , ε3 , 1) (s4 , ε1 , 1) (s4 , ε2 , 1) (s4 , ε3 , 1)  a j = i vi j

0.45 0.50 0.45 0.40 0.45 0.575 0.375 0.375 0.50 0.475 0.575 0.50 5.625

0.35 0.525 0.45 0.70 0.45 0.325 0.475 0.525 0.50 0.40 0.65 0.30 5.650

0.65 0.60 0.475 0.375 0.625 0.40 0.575 0.50 0.45 0.475 0.35 0.475 5.950

v5

0.35 0.525 0.65 0.50 0.275 0.525 0.375 0.475 0.525 0.575 0.325 0.575 5.675

0.60 0.50 0.45 0.575 0.45 0.425 0.50 0.40 0.50 0.45 0.475 0.425 5.750

v4

v5

Table 10.18 Disagree-4–polar fuzzy soft expert set (γ, Q)0

v1

(s1 , ε1 , 0)

0.3, 0.5, 0.2, 0.3 0.3, 0.2, 0.4, 0.2 0.6, 0.4, 0.5, 0.7 0.5, 0.3, 0.2, 0.4 0.6, 0.8, 0.2, 0.4

v2

v3

(s1 , ε2 , 0)

0.2, 0.7, 0.3, 0.6 0.8, 0.3, 0.5, 0.4 0.4, 0.3, 0.6, 0.7 0.3, 0.5, 0.6, 0.5 0.7, 0.3, 0.4, 0.2

(s1 , ε3 , 0)

0.4, 0.4, 0.5, 0.6 0.2, 0.5, 0.7, 0.6 0.3, 0.6, 0.1, 0.5 0.7, 0.6, 0.3, 0.6 0.3, 0.5, 0.2, 0.6

(s2 , ε1 , 0)

0.1, 0.2, 0.6, 0.5 0.3, 0.4, 0.6, 0.8 0.1, 0.4, 0.0, 0.6 0.3, 0.5, 0.5, 0.2 0.7, 0.3, 0.4, 0.4

(s2 , ε2 , 0)

0.4, 0.3, 0.6, 0.2 0.1, 0.4, 0.5, 0.6 0.8, 0.6, 0.3, 0.7 0.3, 0.2, 0.4, 0.4 0.5, 0.3, 0.5, 0.3

(s2 , ε3 , 0)

0.4, 0.5, 0.4, 0.6 0.3, 0.5, 0.2, 0.4 0.6, 0.2, 0.6, 0.3 0.7, 0.2, 0.4, 0.6 0.6, 0.2, 0.3, 0.4

(s3 , ε1 , 0)

0.4, 0.2, 0.3, 0.5 0.3, 0.1, 0.5, 0.2 0.5, 0.3, 0.7, 0.5 0.3, 0.2, 0.5, 0.3 0.3, 0.6, 0.2, 0.7

(s3 , ε2 , 0)

0.5, 0.2, 0.7, 0.3 0.2, 0.4, 0.6, 0.5 0.6, 0.4, 0.1, 0.5 0.5, 0.2, 0.4, 0.3 0.4, 0.1, 0.3, 0.2

(s3 , ε3 , 0)

0.2, 0.3, 0.6, 0.5 0.3, 0.4, 0.6, 0.2 0.4, 0.2, 0.3, 0.5 0.4, 0.5, 0.3, 0.4 0.5, 0.4, 0.2, 0.5

(s4 , ε1 , 0)

0.2, 0.4, 0.6, 0.3 0.3, 0.6, 0.2, 0.1 0.6, 0.4, 0.3, 0.2 0.7, 0.5, 0.4, 0.3 0.4, 0.1, 0.4, 0.5

(s4 , ε2 , 0)

0.7, 0.4, 0.5, 0.3 0.4, 0.5, 0.6, 0.7 0.3, 0.4, 0.2, 0.3 0.5, 0.4, 0.4, 0.2 0.5, 0.3, 0.2, 0.5

(s4 , ε3 , 0)

0.5, 0.3, 0.2, 0.6 0.3, 0.2, 0.1, 0.2 0.4, 0.6, 0.3, 0.2 0.6, 0.2, 0.4, 0.7 0.3, 0.2, 0.5, 0.3

Clearly, from the Table 10.20, the final score for the object v5 is maximum because z 5 = 1.075 is the highest final score value. Thus, v5 is the most suitable site for constructing the dam according to the team of three experts (engineers).

10.4.2 Country Most Affected by Human Trafficking Every country in the world is affected by human trafficking in various degrees. Despite that, the major causes behind this crime are basically the same throughout

10.4 Applications

495

Table 10.19 Score values for disagree-4–polar fuzzy soft expert set (γ, Q)0 v1 v2 v3 v4 (s1 , ε1 , 0) (s1 , ε2 , 0) (s1 , ε3 , 0) (s2 , ε1 , 0) (s2 , ε2 , 0) (s2 , ε3 , 0) (s3 , ε1 , 0) (s3 , ε2 , 0) (s3 , ε3 , 0) (s4 , ε1 , 0) (s4 , ε2 , 0) (s4 , ε3 , 0)  a j = i vi j

0.325 0.450 0.475 0.350 0.375 0.475 0.350 0.425 0.400 0.375 0.475 0.400 4.875

Table 10.20 Final score table  a j = i vi j a1 a2 a3 a4 a5

= 5.625 = 5.650 = 5.950 = 5.675 = 5.750

0.275 0.500 0.500 0.525 0.400 0.350 0.275 0.425 0.375 0.300 0.550 0.200 4.675

bj = b1 b2 b3 b4 b5

0.550 0.500 0.375 0.275 0.600 0.425 0.500 0.400 0.350 0.375 0.300 0.375 5.025

 i

vi j

= 4.875 = 4.675 = 5.025 = 4.950 = 4.675

v5

0.350 0.475 0.550 0.375 0.325 0.475 0.325 0.350 0.400 0.475 0.375 0.475 4.950

0.500 0.400 0.400 0.450 0.400 0.375 0.450 0.250 0.400 0.350 0.375 0.325 4.675

z j = aj − bj z1 z2 z3 z4 z5

= 0.777 = 0.975 = 0.925 = 0.725 = 1.075

the world for all kinds of modern day slavery, including child trafficking, sex trafficking and labor trafficking. There are some common factors that make people wish to migrate in search of better conditions like political instability, civil unrest, natural disasters, dangers from conflict or instability, internal armed conflict, oppression, poverty, lack of social or economic opportunities, lack of human rights, militarism, etc. The instability and movement of people rise their susceptibility to victimization and abuse by trafficking and forced labor. Public dispute and conflict may direct to massive movements of people, leaving street children and orphans incredibly susceptible to trafficking. Wealth and poverty are related terms which direct to trafficking and migration of victims due to different poverty level. On the other hand, the speedy growth of media and broadcast, including the Internet, across the world may have expanded the wish to migrate to developed countries. Some more factors that facilitate human trafficking are: corrupt government officials, porous borders, lack of legitimate economic opportunities, lack of adequate legislation and of political will and commitment to enforce existing legislation or mandates, involvement of international organized criminal networks or groups and lack of safe migration

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

options. People of every nationality, age or gender can be victimized by traffickers. Based upon most recent analysis, experts agree that a number of men, children and women affected by this crime are from many countries across the world, including New Zealand, China, Russia, Pakistan, India and Iran. The trauma employed by the traffickers can be so undetectable that a lot of people may not identify themselves as victims. Experts say that no one can describe human trafficking by an exact number. Every year experts make this report by registered cases and some other parameters. So, to find a country most affected by human trafficking is an uncertain problem. Assume that an organization decides to give this task to four experts to rank six countries regarding human trafficking. Let V = {v1 = India, v2 = New Zealand, v3 = Pakistan, v4 = Russia, v5 = China, v6 = Iran} be a set of six countries and let S = {s1 , s2 , s3 , s4 } be the set of four criteria setting by the experts, which involves different causes of human trafficking given as follows: • The criteria “s1 ” includes poverty, lack of education and lack of legitimate economic opportunities. • The criteria “s2 ” includes debt, political instability, social factors and cultural practices. • The criteria “s3 ” includes force, conflict and natural disaster and lack of safe migration options. • The criteria “s4 ” includes demand for sex, cheap labor and lack of human rights for vulnerable groups. The evaluation of the experts in the form of a 3–polar fuzzy soft expert set is displayed in Table 10.21. The aim of this application is to analyze the countries vi ’s, i = 1, 2, . . . , 6 regarding human trafficking using 3–polar fuzzy numbers provided by the experts according to parameters s j ’s, j = 1, 2, 3, 4. The agree-3–polar fuzzy soft expert set is displayed in Table 10.22. By Definition 10.3, the score for each 3–polar fuzzy number is calculated in Table 10.22. The figures are shown in Table 10.23. The disagree-3–polar fuzzy soft expert set is displayed in Table 10.24. By Definition 10.3, the score for each 3–polar fuzzy number is calculated in Table 10.24. The figures are shown in Table 10.25. Now the final score values can be easily obtained from the agree and disagree score values in Tables 10.23 and 10.25, respectively, as given in Table 10.26. Clearly, from the Table 10.26, the final score for the object v4 is maximum because z 4 = 1.3000 is the highest final score value. Thus, v4 is the country most affected by human trafficking according to the experts.

10.4 Applications

497

Table 10.21 A 3–polar fuzzy soft expert set (γ, Q)

v1

(s1 , ε1 , 1)

0.6, 0.4, 0.5 0.5, 0.4, 0.7 0.7, 0.3, 0.4 0.4, 0.5, 0.3 0.6, 0.5, 0.5 0.4, 0.3, 0.6

v2

v3

v4

v5

v6

(s1 , ε2 , 1)

0.6, 0.4, 0.7 0.3, 0.5, 0.4 0.5, 0.2, 0.4 0.6, 0.3, 0.3 0.7, 0.6, 0.5 0.6, 0.5, 0.4

(s1 , ε3 , 1)

0.6, 0.4, 0.7 0.4, 0.5, 0.3 0.7, 0.5, 0.4 0.6, 0.5, 0.4 0.6, 0.5, 0.4 0.7, 0.5, 0.4

(s2 , ε1 , 1)

0.5, 0.4, 0.7 0.4, 0.5, 0.7 0.5, 0.3, 0.2 0.4, 0.3, 0.5 0.6, 0.5, 0.3 0.6, 0.4, 0.3

(s2 , ε2 , 1)

0.7, 0.4, 0.5 0.5, 0.3, 0.7 0.3, 0.2, 0.6 0.4, 0.7, 0.6 0.6, 0.5, 0.4 0.7, 0.3, 0.5

(s2 , ε3 , 1)

0.8, 0.6, 0.5 0.2, 0.5, 0.7 0.3, 0.7, 0.4 0.4, 0.7, 0.5 0.4, 0.7, 0.6 0.4, 0.5, 0.3

(s3 , ε1 , 1)

0.6, 0.4, 0.7 0.5, 0.2, 0.6 0.7, 0.4, 0.6 0.4, 0.5, 0.3 0.9, 0.6, 0.8 0.6, 0.9, 0.7

(s3 , ε2 , 1)

0.8, 0.6, 0.4 0.5, 0.7, 0.6 0.7, 0.3, 0.6 0.7, 0.3, 0.4 0.6, 0.3, 0.4 0.5, 0.3, 0.7

(s3 , ε3 , 1)

0.7, 0.5, 0.8 0.3, 0.4, 0.6 0.6, 0.4, 0.7 0.5, 0.4, 0.2 0.9, 0.7, 0.5 0.7, 0.5, 0.4

(s4 , ε1 , 1)

0.4, 0.6, 0.5 0.7, 0.5, 0.8 0.6, 0.4, 0.5 0.7, 0.3, 0.6 0.9, 0.3, 0.5 0.8, 0.7, 0.4

(s4 , ε2 , 1)

0.6, 0.4, 0.7 0.7, 0.5, 0.4 0.4, 0.7, 0.8 0.7, 0.9, 0.8 0.4, 0.7, 0.8 0.6, 0.3, 0.5

(s4 , ε3 , 1)

0.8, 0.6, 0.7 0.6, 0.7, 0.5 0.8, 0.6, 0.5 0.4, 0.7, 0.8 0.7, 0.6, 0.4 0.6, 0.7, 0.5

(s1 , ε1 , 0)

0.5, 0.3, 0.6 0.7, 0.5, 0.4 0.6, 0.5, 0.3 0.3, 0.4, 0.2 0.5, 0.4, 0.4 0.3, 0.2, 0.5

(s1 , ε2 , 0)

0.5, 0.3, 0.6 0.4, 0.4, 0.3 0.4, 0.1, 0.3 0.5, 0.2, 0.2 0.6, 0.5, 0.4 0.5, 0.4, 0.3

(s1 , ε3 , 0)

0.5, 0.3, 0.4 0.3, 0.4, 0.2 0.6, 0.4, 0.3 0.5, 0.4, 0.3 0.5, 0.4, 0.3 0.5, 0.3, 0.2

(s2 , ε1 , 0)

0.4, 0.3, 0.5 0.3, 0.4, 0.5 0.4, 0.2, 0.1 0.5, 0.2, 0.3 0.5, 0.3, 0.4 0.5, 0.3, 0.2

(s2 , ε2 , 0)

0.6, 0.3, 0.4 0.4, 0.2, 0.6 0.2, 0.1, 0.5 0.3, 0.6, 0.4 0.5, 0.3, 0.3 0.6, 0.2, 0.4

(s2 , ε3 , 0)

0.7, 0.4, 0.3 0.1, 0.4, 0.6 0.2, 0.6, 0.3 0.3, 0.6, 0.4 0.3, 0.6, 0.5 0.3, 0.6, 0.2

(s3 , ε1 , 0)

0.5, 0.3, 0.6 0.4, 0.1, 0.5 0.6, 0.3, 0.5 0.3, 0.4, 0.2 0.8, 0.5, 0.7 0.5, 0.8, 0.6

(s3 , ε2 , 0)

0.7, 0.5, 0.3 0.4, 0.6, 0.5 0.6, 0.2, 0.5 0.6, 0.2, 0.3 0.5, 0.2, 0.3 0.4, 0.2, 0.6

(s3 , ε3 , 0)

0.6, 0.4, 0.7 0.2, 0.3, 0.5 0.5, 0.3, 0.6 0.4, 0.3, 0.1 0.8, 0.6, 0.4 0.6, 0.4, 0.3

(s4 , ε1 , 0)

0.3, 0.5, 0.4 0.6, 0.4, 0.7 0.5, 0.3, 0.4 0.6, 0.2, 0.5 0.7, 0.2, 0.4 0.6, 0.5, 0.2

(s4 , ε2 , 0)

0.5, 0.3, 0.6 0.6, 0.3, 0.2 0.3, 0.5, 0.6 0.6, 0.7, 0.5 0.6, 0.8, 0.5 0.5, 0.2, 0.6

(s4 , ε3 , 0)

0.7, 0.5, 0.8 0.5, 0.8, 0.4 0.7, 0.5, 0.4 0.3, 0.6, 0.7 0.6, 0.5, 0.3 0.5, 0.6, 0.4

Table 10.22 An agree-3–polar fuzzy soft expert set (γ, Q)1

v1

(s1 , ε1 , 1)

0.6, 0.4, 0.5 0.5, 0.4, 0.7 0.7, 0.3, 0.4 0.4, 0.5, 0.3 0.6, 0.5, 0.5 0.4, 0.3, 0.6

v2

v3

v4

v5

v6

(s1 , ε2 , 1)

0.6, 0.4, 0.7 0.3, 0.5, 0.4 0.5, 0.2, 0.4 0.6, 0.3, 0.3 0.7, 0.6, 0.5 0.6, 0.5, 0.4

(s1 , ε3 , 1)

0.6, 0.4, 0.7 0.4, 0.5, 0.3 0.7, 0.5, 0.4 0.6, 0.5, 0.4 0.6, 0.5, 0.4 0.7, 0.5, 0.4

(s2 , ε1 , 1)

0.5, 0.4, 0.7 0.4, 0.5, 0.7 0.5, 0.3, 0.2 0.4, 0.3, 0.5 0.6, 0.5, 0.3 0.6, 0.4, 0.3

(s2 , ε2 , 1)

0.7, 0.4, 0.5 0.5, 0.3, 0.7 0.3, 0.2, 0.6 0.4, 0.7, 0.6 0.6, 0.5, 0.4 0.7, 0.3, 0.5

(s2 , ε3 , 1)

0.8, 0.6, 0.5 0.2, 0.5, 0.7 0.3, 0.7, 0.4 0.4, 0.7, 0.5 0.4, 0.7, 0.6 0.4, 0.5, 0.3

(s3 , ε1 , 1)

0.6, 0.4, 0.7 0.5, 0.2, 0.6 0.7, 0.4, 0.6 0.4, 0.5, 0.3 0.9, 0.6, 0.8 0.6, 0.9, 0.7

(s3 , ε2 , 1)

0.8, 0.6, 0.4 0.5, 0.7, 0.6 0.7, 0.3, 0.6 0.7, 0.3, 0.4 0.6, 0.3, 0.4 0.5, 0.3, 0.7

(s3 , ε3 , 1)

0.7, 0.5, 0.8 0.3, 0.4, 0.6 0.6, 0.4, 0.7 0.5, 0.4, 0.2 0.9, 0.7, 0.5 0.7, 0.5, 0.4

(s4 , ε1 , 1)

0.4, 0.6, 0.5 0.7, 0.5, 0.8 0.6, 0.4, 0.5 0.7, 0.3, 0.6 0.9, 0.3, 0.5 0.8, 0.7, 0.4

(s4 , ε2 , 1)

0.6, 0.4, 0.7 0.7, 0.5, 0.4 0.4, 0.7, 0.8 0.7, 0.9, 0.8 0.4, 0.7, 0.8 0.6, 0.3, 0.5

(s4 , ε3 , 1)

0.8, 0.6, 0.7 0.6, 0.7, 0.5 0.8, 0.6, 0.5 0.4, 0.7, 0.8 0.7, 0.6, 0.4 0.6, 0.7, 0.5

498

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.23 Score values for agree-3–polar fuzzy soft expert set (γ, Q)1 v1 v2 v3 v4 (s1 , ε1 , 1) (s1 , ε2 , 1) (s1 , ε3 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s2 , ε3 , 1) (s3 , ε1 , 1) (s3 , ε2 , 1) (s3 , ε3 , 1) (s4 , ε1 , 1) (s4 , ε2 , 1) (s4 , ε3 , 1) aj = i vi j

0.5000 0.5667 0.5667 0.5333 0.5333 0.6333 0.5667 0.6000 0.6667 0.5000 0.5667 0.7000 6.9333

0.5333 0.4000 0.4000 0.5333 0.5000 0.4667 0.4333 0.6000 0.4333 0.6667 0.5333 0.6000 6.1000

0.4667 0.3667 0.5333 0.5333 0.3667 0.4667 0.5667 0.5333 0.5667 0.5000 0.6333 0.6333 5.9667

v5

v6

0.4000 0.4000 0.5000 0.4000 0.5667 0.5333 0.4000 0.4667 0.3667 0.5333 0.8000 0.6333 6.0000

0.5333 0.6000 0.5000 0.4667 0.5000 0.5667 0.7667 0.4333 0.7000 0.5667 0.6333 0.5667 6.8333

0.4333 0.5000 0.5333 0.4333 0.5000 0.4000 0.7333 0.5000 0.5333 0.6333 0.4667 0.6000 6.2667

v4

v5

v6

Table 10.24 Disagree-3–polar fuzzy soft expert set (γ, Q)0

v1

(s1 , ε1 , 0)

0.5, 0.3, 0.6 0.7, 0.5, 0.4 0.6, 0.5, 0.3 0.3, 0.4, 0.2 0.5, 0.4, 0.4 0.3, 0.2, 0.5

v2

v3

(s1 , ε2 , 0)

0.5, 0.3, 0.6 0.4, 0.4, 0.3 0.4, 0.1, 0.3 0.5, 0.2, 0.2 0.6, 0.5, 0.4 0.5, 0.4, 0.3

(s1 , ε3 , 0)

0.5, 0.3, 0.4 0.3, 0.4, 0.2 0.6, 0.4, 0.3 0.5, 0.4, 0.3 0.5, 0.4, 0.3 0.5, 0.3, 0.2

(s2 , ε1 , 0)

0.4, 0.3, 0.5 0.3, 0.4, 0.5 0.4, 0.2, 0.1 0.5, 0.2, 0.3 0.5, 0.3, 0.4 0.5, 0.3, 0.2

(s2 , ε2 , 0)

0.6, 0.3, 0.4 0.4, 0.2, 0.6 0.2, 0.1, 0.5 0.3, 0.6, 0.4 0.5, 0.3, 0.3 0.6, 0.2, 0.4

(s2 , ε3 , 0)

0.7, 0.4, 0.3 0.1, 0.4, 0.6 0.2, 0.6, 0.3 0.3, 0.6, 0.4 0.3, 0.6, 0.5 0.3, 0.6, 0.2

(s3 , ε1 , 0)

0.5, 0.3, 0.6 0.4, 0.1, 0.5 0.6, 0.3, 0.5 0.3, 0.4, 0.2 0.8, 0.5, 0.7 0.5, 0.8, 0.6

(s3 , ε2 , 0)

0.7, 0.5, 0.3 0.4, 0.6, 0.5 0.6, 0.2, 0.5 0.6, 0.2, 0.3 0.5, 0.2, 0.3 0.4, 0.2, 0.6

(s3 , ε3 , 0)

0.6, 0.4, 0.7 0.2, 0.3, 0.5 0.5, 0.3, 0.6 0.4, 0.3, 0.1 0.8, 0.6, 0.4 0.6, 0.4, 0.3

(s4 , ε1 , 0)

0.3, 0.5, 0.4 0.6, 0.4, 0.7 0.5, 0.3, 0.4 0.6, 0.2, 0.5 0.7, 0.2, 0.4 0.6, 0.5, 0.2

(s4 , ε2 , 0)

0.5, 0.3, 0.6 0.6, 0.3, 0.2 0.3, 0.5, 0.6 0.6, 0.7, 0.5 0.6, 0.8, 0.5 0.5, 0.2, 0.6

(s4 , ε3 , 0)

0.7, 0.5, 0.8 0.5, 0.8, 0.4 0.7, 0.5, 0.4 0.3, 0.6, 0.7 0.6, 0.5, 0.3 0.5, 0.6, 0.4

10.5 Comparative Analysis Akram et al. [6] discussed the advantages of the presented model and its comparison with some existing models to prove its practicality and validity. (1) Advantages of the Proposed Model Based on the study of last two decades, one can easily see a lot of development in the hybrid-soft-set models, but many of them handle soft information only with n experts, n ≥ 1. These models are restricted to sort data in m–polar fuzzy

10.5 Comparative Analysis

499

Table 10.25 Score values for disagree-3–polar fuzzy soft expert set (γ, Q)0 v1 v2 v3 v4 (s1 , ε1 , 1) (s1 , ε2 , 1) (s1 , ε3 , 1) (s2 , ε1 , 1) (s2 , ε2 , 1) (s2 , ε3 , 1) (s3 , ε1 , 1) (s3 , ε2 , 1) (s3 , ε3 , 1) (s4 , ε1 , 1) (s4 , ε2 , 1) (s4 , ε3 , 1) bj =  i vi j

0.4667 0.4667 0.4000 0.4000 0.4333 0.4667 0.4667 0.5000 0.5667 0.4000 0.4667 0.6667 5.7000

0.5333 0.3667 0.3000 0.4000 0.4000 0.3667 0.3333 0.5000 0.3333 0.5667 0.3667 0.5667 5.0333

Table 10.26 Final score table  a j = i vi j a1 a2 a3 a4 a5 a6

= 6.9333 = 6.1000 = 5.9667 = 6.0000 = 6.8333 = 6.2667

0.4667 0.2667 0.4333 0.2333 0.2667 0.3667 0.4667 0.4333 0.4667 0.4000 0.4667 0.5333 4.8000

bj = b1 b2 b3 b4 b5 b6

 i

vi j

= 5.7000 = 5.0333 = 4.8000 = 4.7000 = 5.7000 = 5.0000

0.3000 0.3000 0.4000 0.3333 0.4333 0.4333 0.3000 0.3667 0.2667 0.4333 0.6000 0.5333 4.7000

v5

v6

0.4333 0.5000 0.4000 0.4000 0.3667 0.4667 0.6667 0.3333 0.6000 0.4333 0.6333 0.4667 5.7000

0.3333 0.4000 0.3333 0.3333 0.4000 0.3667 0.6333 0.4000 0.4333 0.4333 0.4333 0.5000 5.0000

z j = aj − bj z1 z2 z3 z4 z5 z6

= 1.2333 = 1.0667 = 1.1667 = 1.3000 = 1.1333 = 1.2667

environment with more than one expert. Motivated by these settings, a novel hybrid soft expert model, namely, m–polar fuzzy soft expert set is discussed. In the presented model, the judgments of all experts can be considered in an m–polar fuzzy environment. The developed model is very feasible to handle uncertain and vague m–polar fuzzy information. Specifically, if the given data based on m–polar fuzziness and collected with the help of different experts. (2) Comparative Analysis. Several researches have been done as generalization of different hybrid soft set models like fuzzy soft sets and bipolar fuzzy soft sets. Since the soft expert set is a natural extension of the soft set model. In a similar way, the notion of fuzzy soft expert set model is a generalization of soft expert set model. The presented method is an extension of soft set, soft expert set, fuzzy soft expert set and bipolar fuzzy soft expert set models. Clearly, when m=1, 2, m–polar fuzzy soft expert set degenerates into fuzzy soft expert sets [29] and bipolar fuzzy soft expert sets [31], respectively. By solving the application discussed in Alkhazaleh and Salleh [29] with the developed m–polar fuzzy soft

500

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.27 Comparison between different values of ‘m’ in Application 10.4.1 Cases v1 v2 v3 v4 v5 m=1

0.3

1.6

1.4

0.4

0.9

m=2

0.55

1.0

1.35

0.45

1.15

m=3

0.5334

0.8333

1.0

0.7

1.2333

Proposed 4FSES (m = 4)

0.777

0.975

0.925

0.725

1.075

Table 10.28 Comparison between different values ‘m’ in Application 10.4.2 Cases v1 v2 v3 v4 v5 v6 m=1

1.2

0.7

1.2

1

1

1.4

m=2

1.25

0.8

1.1

1.15

1.1

1.3

Proposed 3FSES (m = 3)

1.2333

1.0667

1.1667

1.3000

1.1333

1.2667

Rankings v2 v5 v3 v3 v2 v4 v5 v2 v1 v5 v3 v4

> v3 > > v4 > > v5 > > v1 > > v3 > > v4 > > v2 > > v1 >

Rankings v6 v3 v5 v6 v4 v5 v4 v1 v5

> v1 > v4 > v2 > v1 > v3 > v2 > v6 > v3 > v2

= = > = > >

expert set approach, we get the same results. Further, the applications Sects. 10.4.1 and 10.4.2 for different values of ‘m’ which are respectively displayed in Tables 10.27 and 10.28 are discussed. From Tables 10.27 and 10.28, it can be easily seen that in both applications for different values of ‘m’, decision object or ranking order may be different (see Figs. 10.2 and 10.3). Both the quantitative and qualitative comparative analysis of the developed m–polar fuzzy soft expert sets for different values of ‘m’ and with certain existing mathematical methods are given in Tables 10.27, 10.28 and 10.29. In addition, the presented model is also an extension of m–polar fuzzy soft set theory [9] regarding number of experts. Altogether, the presented technique is more feasible and reliable to solve real-world problems, particularly, when the judgments of different experts are considered in the form of m–polar fuzzy information.

10.5 Comparative Analysis

Fig. 10.2 Comparison under different values of m in the application described in Sect. 10.4.1

Fig. 10.3 Comparison under different values of m in the application described in Sect. 10.4.2

501

502

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.29 Comparison with existing hybrid models Hybrid models q = Number of Application scope experts Soft sets [70]

q=1

Lowest

m–polar fuzzy soft sets [9] Soft expert sets [28]

q=1

Larger than soft sets

q≥1

Higher than soft sets

Fuzzy soft expert sets [29] Intuitionistic fuzzy soft expert sets [37] Bipolar fuzzy soft expert set [31] Presented m–polar fuzzy soft expert sets

q≥1 q≥1 q≥1 q≥1

Type of data Discrete data in binary form Data in mF soft form

Discrete data in binary form Larger than soft expert Data in fuzzy soft form sets Higher than fuzzy soft Data in intuitionistic expert sets fuzzy soft form Higher than fuzzy soft Data in bipolar fuzzy expert sets soft form Highest Data in mF soft expert form

10.6 m–Polar Fuzzy N–Soft Sets In this section, the concept of m–polar fuzzy N –soft sets is presented, which needs some previous definitions inclusive of the new notion of sub–m–polar fuzzy set. Definition 10.15 [45] Let V be a universe, Z a set of attributes and C ⊆ Z . Let M = {0, 1, . . . , N − 1} be a set of ordered grades where N ∈ {2, 3, . . . }. A triple (F, C, N ) is called an N −soft set over V if F is a mapping from C to 2V ×M , with the property that for each ck ∈ C and v ∈ V there exists a unique (v, rck ) ∈ V × M such that (v, rck ) ∈ F(ck ), v ∈ V, rck ∈ M. Akram et al. [7] presented the concept of m–polar fuzzy N –soft sets in 2019. Definition 10.16 [7] Let V be a universal set, Z a set of parameters and C ⊆ Z . Define a mapping ξ from C to M(V ). Then, a pair (ξ, C) is called an m–polar fuzzy soft set on V , which is defined by (ξ, C) = { v, ( p1 ◦ Cck (v), . . . , pm ◦ Cck (v) | v ∈ V, ck ∈ C}. Definition 10.17 Let V be a universal set. Any m–polar fuzzy set on a subset V  of V is called sub−m–polar fuzzy set on V . The family of all sub−m–polar fuzzy sets over V is denoted by M (V ). Definition 10.18 An m–polar fuzzy N –soft set is a triple ( f, D, m) where D = (F, C, N ) is an N −soft set on V and f maps any attribute in C with an m–polar fuzzy set A on F(ck ), which is a convenient subset of V × M and ck ∈ C. Therefore, the domain of f is of course C, and its codomain is M (V × M), the family of all sub−m–polar fuzzy sets over V × M.

10.6 m–Polar Fuzzy N –Soft Sets

503

Table 10.30 Information extracted from the related data V c1 c2 v1 v2 v3 v4 v5 v6

       

     ◦  

c3

c4

         

         

Example 10.10 In Australia star rating for hotels is awarded by AAA (Accommodation Association of Australia) Tourism and its range is from 1 to 5 stars. AAA Tourism is represented by the State’s automotive clubs, including RACV, RAC, RAA and RACT. AAA inspectors collectively visit hotels and inspect the facilities and levels of comfort and luxury, allowing them to identify emerging trends in the industry. Let V = {v1 , v2 , v3 , v4 , v5 , v6 } be a universe of six hotels in Australia and let C = {c1 , c2 , c3 , c4 } ⊆ Z be the set of attributes, which gives grades to hotels based on location, services and parking area. A 3–polar fuzzy 6–soft set is obtained from Table 10.30, where Five stars represent ‘luxury’, Four stars represent ‘excellent’, Three stars represent ‘very good’, Two stars represent ‘good’, One star represents ‘regular’, Circle represents ‘bad’. The set of ordered grades M = {0, 1, 2, 3, 4, 5} can be easily associated with stars as follows: 0 stand for ‘◦’, 1 stand for ‘’, 2 stand for ‘’, 3 stand for ‘  ’, 4 stand for ‘  ’, 5 stand for ‘    ’. From Definition 10.15, tabular form of a 6−soft set is given in Table 10.31. Therefore, a 3–polar fuzzy 6–soft set ( f, D, 3) is defined as follows: ( f, D, 3) =

504

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.31 Tabular representation of 6−soft set (F, C, 6) c1 c2 v1 v2 v3 v4 v5 v6

3 3 1 2 5 4

1 4 5 0 2 3

Table 10.32 A 3–polar fuzzy 6–soft set ( f, D, 3) c1 c2 v1 v2 v3 v4 v5 v6

3, (0.3, 0.5, 0.7)

3, (0.1, 0.6, 0.8)

1, (0.4, 0.3, 0.5)

2, (0.2, 0.7, 0.5)

5, (0.9, 0.5, 0.7)

4, (0.4, 0.8, 0.7)

1, (0.4, 0.2, 0.6)

4, (0.3, 0.4, 0.8)

5, (0.5, 0.6, 0.8)

0, (0.4, 0.3, 0.4)

2, (0.2, 0.6, 0.5)

3, (0.4, 0.1, 0.9)

c3

c4

2 3 1 4 5 4

3 2 4 3 3 4

c3

c4

2, (0.7, 0.5, 0.3)

3, (0.9, 0.4, 0.6)

1, (0.1, 0.5, 0.8)

4, (0.8, 0.7, 0.7)

5, (1.0, 1.0, 1.0)

4, (0.4, 0.7, 0.9)

3, (0.4, 0.6, 0.9)

2, (0.2, 0.4, 0.8)

4, (0.7, 0.6, 0.9)

3, (0.6, 0.8, 0.1)

3, (0.4, 0.5, 0.7)

4, (0.5, 0.3, 0.9)

⎧ ⎫

⎪ f (c1 ) = (v1 , 3), 0.3, 0.5, 0.7 , (v2 , 3), 0.1, 0.6, 0.8 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ , 1), 0.4, 0.3, 0.5 , (v , 2), 0.2, 0.7, 0.5 , (v ⎪ ⎪ 3 4 ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (v , 5), 0.9, 0.5, 0.7 , (v , 4), 0.4, 0.8, 0.7

, ⎪ ⎪ 5 6 ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ f (c2 ) = (v1 , 1), 0.4, 0.2, 0.6 , (v2 , 4), 0.3, 0.4, 0.8 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ , 5), 0.5, 0.6, 0.8 , (v , 0), 0.4, 0.3, 0.4 , (v ⎪ ⎪ 3 4 ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎨ (v5 , 2), 0.2, 0.6, 0.5 , (v6 , 3), 0.4, 0.1, 0.9 , ⎬

⎪ f (c3 ) = (v1 , 2), 0.7, 0.5, 0.3 , (v2 , 3), 0.9, 0.4, 0.6 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ , 1), 0.1, 0.5, 0.8 , (v , 4), 0.8, 0.7, 0.7 , (v ⎪ ⎪ 3 4 ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (v , 5), 1.0, 1.0, 1.0 , (v , 4), 0.4, 0.7, 0.9

, ⎪ ⎪ 5 6 ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f (c ) = (v , 3), 0.4, 0.6, 0.9 , (v , 2), 0.2, 0.4, 0.8 , ⎪ ⎪ 4 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ , 4), 0.7, 0.6, 0.9 , (v , 3), 0.6, 0.8, 0.1 , (v ⎪ ⎪ 3 4 ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ (v5 , 3), 0.4, 0.5, 0.7 , (v6 , 4), 0.5, 0.3, 0.9 . ⎭ The 3–polar fuzzy 6–soft set ( f, D, 3) can be represented by Table 10.32. Thus, ( f, D, 3) is a 3–polar fuzzy 6–soft set based on location, services and parking area of the hotels. For example, in the Table 10.32, 3, (0.3, 0.5, 0.7) means that the hotel v1 is 30% suitable with the location, 50% suitable with the services and 70% suitable with parking area, where 3 is the evaluation grade of v1 according to attribute c1 . Note that the criterion of assigning grades corresponding to each parameter based

10.6 m–Polar Fuzzy N –Soft Sets

505

upon the membership values of the poles, for example, according to parameter c1 grades are assigned based on average of membership values of all three poles for each object vi , i = 1, 2, . . . , 6. And the criterion of assigning grades may vary from one parameter to other. Remark 10.1 1. An m–polar fuzzy 2–soft set can be naturally associated with an m–polar fuzzy soft set. An m–polar fuzzy 2–soft set f : C → M (V × {0, 1}) is identified with an m–polar fuzzy soft set (ξ, C) by Definition 10.16. 2. If m = 1 is chosen, then a fuzzy 2−soft set f : C → M (V × {0, 1}) can be identified with a fuzzy soft set G : C → F (V ) defined by G(ck ) = {(v, μ(v)) | (v, 1), μ(v) ∈ f (ck ), ck ∈ C}, where F (V ) is the family of all fuzzy subsets of V . 3. Any m–polar fuzzy N –soft set over a universe V can be identified as an m–polar fuzzy N + 1–soft set. For example, from Table 10.32, the 3–polar fuzzy 6–soft set ( f, D, 3) in Example 10.10 can be identified as 3–polar fuzzy 7–soft set over V . In 3–polar fuzzy 7–soft set, let us assume that a 6 grade is given, which is not required. 4. Grade 0 ∈ M in Definition 10.15 represents the lowest score. It does not mean that there is incomplete information or no assessment. Example 10.11 Let V = {s1 , s2 , s3 } be a set of subjects and let C = {c1 , c2 , c3 } be a set of students, candidate c1 selects subject s1 , candidate c2 selects subject s3 , candidate c3 selects subject s3 . Let M = {0, 1}. Therefore, a fuzzy 2−soft set is given as follows: ⎫ ⎧  f (c1 ) = (s1 , 1), 0.7 ,  (s2 , 0), 0.1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (s3 , 0), 0.3 , ⎪ ⎪ ⎪ ⎪  ⎬ ⎨ f (c2 ) = (s1 , 0), 0.2 ,  (s2 , 0), 0.3 , ( f, D, 1) = ⎪ ⎪ ⎪ ⎪  (s3 , 1), 0.9 , ⎪ ⎪ ⎪ f (c3 ) = (s1 , 0), 0.1 , (s2 , 0), 0.0 , ⎪ ⎪ ⎪ ⎪ ⎪  ⎭ ⎩ (s3 , 1), 0.6 . This fuzzy 2−soft set ( f, D, 1) can be identified with fuzzy soft set (ξ, C), which is defined as follows: ⎧   ⎫ ⎨ ξ(c1 ) = (s1 , 0.7), ⎬ (ξ, C) = ξ(c2 ) = (s3 , 0.9), ⎩ ⎭ ξ(c3 ) = (s3 , 0.6) . Definition 10.19 An m–polar fuzzy N –soft set ( f, D, m) is said to be efficient where D = (F, C, N ) is an N −soft set, if f (ck ) = (v j , N − 1), 1 for some ck ∈ C, vj ∈ V . Example 10.12 The 3–polar fuzzy 6–soft set defined in Example 10.10 is efficient.

506

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.33 The weak complement of 3–polar fuzzy 6–soft set ( f c , D1 , 3)

c1

c2

c3

c4

v1

4, (0.7, 0.5, 0.3)

4, (0.6, 0.8, 0.4)

3, (0.3, 0.5, 0.7)

2, (0.6, 0.4, 0.1)

v2

4, (0.9, 0.4, 0.2)

2, (0.7, 0.6, 0.2)

2, (0.1, 0.6, 0.4)

1, (0.8, 0.6, 0.2)

v3

5, (0.6, 0.7, 0.5)

3, (0.5, 0.4, 0.2)

4, (0.9, 0.5, 0.2)

1, (0.3, 0.4, 0.1)

v4

4, (0.8, 0.3, 0.5)

5, (0.6, 0.7, 0.6)

3, (0.2, 0.3, 0.3)

2, (0.4, 0.2, 0.9)

v5

1, (0.1, 0.5, 0.3)

4, (0.8, 0.4, 0.5)

0, (0.0, 0.0, 0.0) 2, (0.6, 0.5, 0.3)

v6

2, (0.6, 0.2, 0.3)

2, (0.6, 0.9, 0.1)

3, (0.6, 0.3, 0.1)

3, (0.5, 0.7, 0.1)

Definition 10.20 Let ( f, D, m) be an efficient m–polar fuzzy N –soft set, where D = (F, C, N ) is an N −soft set and f maps any attribute in C with an mF set A on F(ck ). Then its minimized efficient m–polar fuzzy N –soft set on V is denoted by ( fl , D1 , m) where D1 = (Fl , C, L) and is defined as L = max Fl (ck )(v j ) + 1, ( p1 ◦ j,k

A(v j , rck ), . . . , pm ◦ A(v j , rck )) = 1 for some (v j , rck ), Fl (ck )(v j ) = F(ck )(v j ), ∀ ck ∈ C, v j ∈ V. Proposition 10.6 Every efficient m–polar fuzzy N –soft set corresponds with its minimized efficient m–polar fuzzy N –soft set. Proof Its proof follows directly from the Definitions 10.19 and 10.20.



Definition 10.21 Let ( f 1 , D1 , m) and ( f 2 , D2 , m) be two m–polar fuzzy N –soft sets on a universe V where D1 = (F1 , R, N1 ), D2 = (F2 , S, N2 ) are N −soft sets. Then ( f 1 , D1 , m) and ( f 2 , D2 , m) are said to be equal if and only if f 1 = f 2 , D1 = D2 . Definition 10.22 Let ( f, D, m) be an m–polar fuzzy N –soft set on V , where D = (F, C, N ) is an N −soft set. Then its weak complement is denoted by ( f c , D1 , m) where D1 = (F c , C, N ) is an arbitrary N −soft set and f c maps any attribute in C with an m–polar fuzzy set A on F c (ck ), F c (ck ) ∩ F(ck ) = ∅, for all ck ∈ C and f c (ck ) is given as follows: f c (ck ) = { (v j , rck ), 1 − ( p1 ◦ A(v j , rck ), . . . , pm ◦ A(v j , rck )) | (v j , rck ) ∈ V × M}. Example 10.13 Reconsider Example 10.10, the weak complement of the 3–polar fuzzy 6–soft set ( f, D, 3) in Example 10.10 is denoted by ( f c , D1 , 3) and is given in Table 10.33. Definition 10.23 For an m–polar fuzzy N –soft set ( f, D, m) where D = (F, C, N ) is an N −soft set and f maps any attribute ck ∈ C with an m–polar fuzzy set A on F(ck ), the top weak complement ( f > , D, m) of ( f, D, m) is defined as follows:

10.6 m–Polar Fuzzy N –Soft Sets

507

Table 10.34 The top weak complement of 6–polar fuzzy N –soft set ( f > , D, 3)

c1

c2

c3

c4

v1

5, (0.7, 0.5, 0.3)

5, (0.6, 0.8, 0.4)

5, (0.3, 0.5, 0.7)

5, (0.6, 0.4, 0.1)

v2

5, (0.9, 0.4, 0.2)

5, (0.7, 0.6, 0.2)

5, (0.1, 0.6, 0.4)

5, (0.8, 0.6, 0.2)

v3

5, (0.6, 0.7, 0.5)

0, (0.5, 0.4, 0.2)

5, (0.9, 0.5, 0.2)

5, (0.3, 0.4, 0.1)

v4

5, (0.8, 0.3, 0.5)

5, (0.6, 0.7, 0.6)

5, (0.2, 0.3, 0.3)

5, (0.4, 0.2, 0.9)

v5

0, (0.1, 0.5, 0.3)

5, (0.8, 0.4, 0.5)

0, (0.0, 0.0, 0.0) 5, (0.6, 0.5, 0.3)

v6

5, (0.6, 0.2, 0.3)

5, (0.6, 0.9, 0.1)

5, (0.6, 0.3, 0.1)

5, (0.5, 0.7, 0.1)

⎧ f (ck ) = (v j , N − 1), 1 − ( p1 ◦ A(v j , ⎪ ⎪ ⎪ ⎪ ⎪ rck ), . . . , pm ◦ A(v j , rck )) , ⎪ ⎪ ⎪ ⎨ i f rck < N − 1, ( f > , D, m) = ⎪ f (ck ) = (v j , 0), 1 − ( p1 ◦ A(v j , rck ), ⎪ ⎪ ⎪ ⎪ ⎪ . . . , pm ◦ A(v j , rck )) , ⎪ ⎪ ⎩ i f rck = N − 1. Example 10.14 Reconsider Example 10.10, the top weak complement of the 3– polar fuzzy 6–soft set ( f, D, 3) in Example 10.10 denoted by ( f > , D, 3) and is given in Table 10.34. Definition 10.24 For an m–polar fuzzy N –soft set ( f, D, m) where D = (F, C, N ) is an N −soft set and f maps any attribute ck ∈ C with an m–polar set A on F(ck ), the bottom weak complement ( f < , D, m) of ( f, D, m) is defined as follows: ⎧ ⎪ ⎪ f (ck ) = (v j , 0), 1 − ( p1 ◦ A(v j , rck ), ⎪ ⎪ ⎪ . . . , pm ◦ A(v j , rck )) , ⎪ ⎪ ⎪ ⎨ i f rck > 0, ( f < , D, m) = ⎪ f (ck ) = (v j , N − 1), 1 − ( p1 ◦ A(v j , ⎪ ⎪ ⎪ ⎪ ⎪ rck ), . . . , pm ◦ A(v j , rck )) , ⎪ ⎪ ⎩ i f rck = 0. Example 10.15 Reconsider Example 10.10, the bottom weak complement of the 3–polar fuzzy 6–soft set ( f, D, 3) in Example 10.10 denoted by ( f < , D, 3) and is given in Table 10.35. Definition 10.25 Let V be a universal set of objects and let ( f 1 , D1 , m) and ( f 2 , D2 , m) be two m–polar fuzzy N –soft sets where D1 = (F1 , R, N1 ), D2 = (F2 , S, N2 ) are N −soft sets on V . Then their restricted intersection is denoted by ( f 1 , D1 , m) ∩ R ( f 2 , D2 , m) and is defined as (g, D1 ∩R D2 , m) where D1 ∩R D2 = (G, R ∩ S, min(N1 , N2 )) for all ck ∈ R ∩ S, v j ∈ V , (v j , rck ), z ∈ g(ck ) ⇐⇒

508

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.35 The bottom weak complement of 3–polar fuzzy 6–soft set ( f < , D, 3)

c1

c2

c3

c4

v1

0, (0.7, 0.5, 0.3)

0, (0.6, 0.8, 0.4)

0, (0.3, 0.5, 0.7)

0, (0.6, 0.4, 0.1)

v2

0, (0.9, 0.4, 0.2)

0, (0.7, 0.6, 0.2)

0, (0.1, 0.6, 0.4)

0, (0.8, 0.6, 0.2)

v3

0, (0.6, 0.7, 0.5)

0, (0.5, 0.4, 0.2)

0, (0.9, 0.5, 0.2)

0, (0.3, 0.4, 0.1)

v4

0, (0.8, 0.3, 0.5)

5, (0.6, 0.7, 0.6)

0, (0.2, 0.3, 0.3)

0, (0.4, 0.2, 0.9)

v5

0, (0.1, 0.5, 0.3)

0, (0.8, 0.4, 0.5)

0, (0.0, 0.0, 0.0) 0, (0.6, 0.5, 0.3)

v6

0, (0.6, 0.2, 0.3)

0, (0.6, 0.9, 0.1)

0, (0.6, 0.3, 0.1)

Table 10.36 A 3–polar fuzzy 5–soft set ( f 1 , D1 , 3) c1 v1 v2 v3 v4 v5 v6

3, (0.7, 0.5, 0.3)

3, (0.9, 0.4, 0.2)

4, (0.6, 0.7, 0.5)

4, (0.8, 0.5, 0.5)

1, (0.1, 0.5, 0.3)

2, (0.6, 0.4, 0.3)

Table 10.37 A 3–polar fuzzy 4–soft set ( f 2 , D2 , 3) c1 v1 v2 v3 v4 v5 v6

2, (0.8, 0.7, 0.6)

1, (0.6, 0.5, 0.4)

3, (0.8, 0.8, 0.8)

2, (0.4, 0.9, 0.8)

0, (0.1, 0.2, 0.1)

3, (0.8, 0.6, 0.9)

0, (0.5, 0.7, 0.1)

c2

c3

4, (0.6, 0.8, 0.5)

2, (0.7, 0.6, 0.2)

1, (0.5, 0.4, 0.2)

4, (0.6, 0.7, 0.6)

3, (0.8, 0.4, 0.5)

2, (0.5, 0.9, 0.1)

3, (0.3, 0.5, 0.7)

2, (0.4, 0.6, 0.4)

4, (0.9, 0.5, 1.0)

0, (0.2, 0.3, 0.3)

0, (0.0, 0.0, 0.0)

1, (0.6, 0.3, 0.1)

c3

e

3, (0.9, 0.7, 0.6)

3, (0.8, 0.5, 0.9)

2, (0.6, 0.3, 0.4)

3, (0.8, 0.6, 0.7)

3, (0.7, 0.6, 0.8)

2, (0.4, 0.6, 0.5)

2, (0.4, 0.8, 0.5)

3, (0.7, 0.8, 0.6)

3, (0.8, 0.9, 0.5)

1, (0.6, 0.5, 0.2)

1, (0.6, 0.3, 0.4)

2, (0.7, 0.6, 0.2)

rck = min(rc1k , rc2k ), z = min( pi ◦ A(v j , rc1k ), pi ◦ B(v j , rc2k )), if (v j , rc1k ), pi ◦ A(v j , rc1k ) ∈ f 1 (ck ) and (v j , rc2k ), pi ◦ B(v j , rc2k ) ∈ f 2 (ck ), for all 1 ≤ i ≤ m, A and B are m–polar fuzzy sets on F1 (ck ) and F2 (ck ), respectively. Example 10.16 Suppose that ( f 1 , D1 , 3) and ( f 2 , D2 , 3) are the 3–polar fuzzy 5– soft set and 3–polar fuzzy 4–soft set given in by Tables 10.36 and 10.37, respectively, where D1 = (F1 , R, 5), D2 = (F2 , S, 4) are N −soft sets on V . Then their restricted intersection ( f 1 , D1 , 3) ∩ R ( f 2 , D2 , 3) = (g, D1 ∩R D2 , 3) is defined by Table 10.38. Definition 10.26 Let V be a universal set and let ( f 1 , D1 , m) and ( f 2 , D2 , m) be two m–polar fuzzy N –soft sets where D1 = (F1 , R, N1 ), D2 = (F2 , S, N2 ) are N −soft sets on V . Then their extended intersection is denoted by ( f 1 , D1 , m) ∩ E

10.6 m–Polar Fuzzy N –Soft Sets

509

Table 10.38 Restricted intersection (g, D1 ∩R D2 , 3) c1 v1 v2 v3 v4 v5 v6

c3

2, (0.7, 0.5, 0.3)

1, (0.6, 0.4, 0.2)

3, (0.6, 0.7, 0.5)

2, (0.4, 0.5, 0.5)

0, (0.1, 0.2, 0.1)

2, (0.6, 0.4, 0.3)

3, (0.3, 0.5, 0.6)

2, (0.4, 0.5, 0.4)

2, (0.6, 0.3, 0.4)

0, (0.2, 0.3, 0.3)

0, (0.0, 0.0, 0.0)

1, (0.4, 0.3, 0.1)

Table 10.39 Extended intersection (h, D1 ∩E D2 , 3) v1

c1

c2

c3

e

2, (0.7, 0.5, 0.3)

4, (0.6, 0.8, 0.5)

3, (0.3, 0.5, 0.6)

2, (0.4, 0.8, 0.5)

v2

1, (0.6, 0.4, 0.2)

2, (0.7, 0.6, 0.2)

2, (0.4, 0.5, 0.4)

3, (0.7, 0.8, 0.6)

v3

3, (0.6, 0.7, 0.5)

1, (0.5, 0.4, 0.2)

2, (0.6, 0.3, 0.4)

3, (0.8, 0.9, 0.5)

v4

2, (0.4, 0.5, 0.5)

4, (0.6, 0.7, 0.6)

0, (0.2, 0.3, 0.3)

1, (0.6, 0.5, 0.2)

v5

0, (0.1, 0.2, 0.1)

3, (0.8, 0.4, 0.5)

0, (0.0, 0.0, 0.0) 1, (0.6, 0.3, 0.4)

v6

2, (0.6, 0.4, 0.3)

2, (0.5, 0.9, 0.1)

1, (0.4, 0.3, 0.1)

2, (0.7, 0.6, 0.2)

( f 2 , D2 , m) and is defined as (h, D1 ∩E D2 , m), where D1 ∩E D2 = (H, R ∪ S, max(N1 , N2 )), and h(ck ) is given as follows: ⎧ ⎪ f 1 (ck ), i f ck ∈ R − S, ⎪ ⎪ ⎪ ⎪ i f ck ∈ S − R, ⎪ ⎪ f 2 (ck ), ⎪ ⎪ ⎪ (v j , rck ), z such that rck = min(rc1k , rc2k ), ⎪ ⎪ ⎪ ⎨ z = min( pi ◦ A(v j , rc1k ), pi ◦ B(v j , rc2k )) h(ck ) = ⎪ wher e (v j , rc1k ), pi ◦ A(v j , rc1k ) ∈ f 1 (ck ) ⎪ ⎪ ⎪ ⎪ ⎪ and (v j , rc2k ), pi ◦ B(v j , rc2k ) ∈ f 2 (ck ), ⎪ ⎪ ⎪ ⎪ ∀ 1 ≤ i ≤ m, A and B ar e m F sets on ⎪ ⎪ ⎪ ⎩ F1 (ck ) and F2 (ck ), r espectively. Example 10.17 Suppose that ( f 1 , D1 , 3) and ( f 2 , D2 , 3) are the 3–polar fuzzy 5–soft set and 3–polar fuzzy 4–soft set represented by Tables 10.36 and 10.37, respectively, where D1 = (F1 , R, 5), D2 = (F2 , S, 4) are 5−soft set and 4−soft set on V , respectively. Then their extended intersection ( f 1 , D1 , 3) ∩ E ( f 2 , D2 , 3) = (h, D1 ∩E D2 , 3) is given in Table 10.39. Definition 10.27 Let V be a universe and let ( f 1 , D1 , m) and ( f 2 , D2 , m) be two m–polar fuzzy N –soft sets over V where D1 = (F1 , R, N1 ), D2 = (F2 , S, N2 ) are N −soft sets on V . Then their restricted union is denoted by ( f 1 , D1 , m) ∪ R

510

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.40 Restricted union (w, D1 ∪R D2 , 3) v1 v2 v3 v4 v5 v6

c1

c3

3, (0.8, 0.7, 0.6)

3, (0.9, 0.5, 0.4)

4, (0.8, 0.8, 0.8)

4, (0.8, 0.9, 0.8)

1, (0.1, 0.5, 0.3)

3, (0.8, 0.6, 0.9)

3, (0.9, 0.7, 0.7)

3, (0.8, 0.6, 0.9)

4, (0.9, 0.5, 1.0)

3, (0.8, 0.6, 0.7)

3, (0.7, 0.6, 0.8)

2, (0.6, 0.6, 0.5)

( f 2 , D2 , m) and is defined as (w, D1 ∪R D2 , m) where D1 ∪R D2 = (W, R ∩ S, max(N1 , N2 )) for all ck ∈ R ∩ S, v j ∈ V , (v j , rck ), z ∈ w(ck ) ⇐⇒ rck = max(rc1k , rc2k ), z = max( pi ◦ A(v j , rc1k ), pi ◦ B(v j , rc2k )) if (v j , rc1k ), pi ◦ A(v j , rc1k )

∈ f 1 (ck ) and (v j , rc2k ), pi ◦ B(v j , rc2k ) ∈ f 2 (ck ), for all 1 ≤ i ≤ m, A and B are mF sets on F1 (ck ) and F2 (ck ), respectively. Example 10.18 Suppose that ( f 1 , D1 , 3) and ( f 2 , D2 , 3) are the 3–polar fuzzy 5– soft set and 3–polar fuzzy 4–soft set represented by Tables 10.36 and 10.37, respectively. Then their restricted union ( f 1 , D1 , 3) ∪ R ( f 2 , D2 , 3) = (w, D1 ∪R D2 , 3) is given Table 10.40. Definition 10.28 Let V be a universal set and let ( f 1 , D1 , m) and ( f 2 , D2 , m) be two m–polar fuzzy N –soft sets on V where D1 = (F1 , R, N1 ), D2 = (F2 , S, N2 ) are N −soft sets on V . Then their extended union is denoted by ( f 1 , D1 , m) ∪ E ( f 2 , D2 , m) and is defined as (y, D1 ∪E D2 , m), where D1 ∪E D2 = (Y, R ∪ S, max(N1 , N2 )) and y(ck ) is given as follows: ⎧ ⎪ f 1 (ck ), i f ck ∈ R − S, ⎪ ⎪ ⎪ ⎪ i f ck ∈ S − R, ⎪ ⎪ f 2 (ck ), ⎪ ⎪ ⎪ , r ), z

such that rck = max(rc1k , rc2k ), (v ⎪ j ck ⎪ ⎪ ⎨ z = max( pi ◦ A(v j , rc1k ), pi ◦ B(v j , rc2k )) y(ck ) = ⎪ where (v j , rc1k ), pi ◦ A(v j , rc1k ) ∈ f 1 (ck ) ⎪ ⎪ ⎪ ⎪ ⎪ and (v j , rc2k ), pi ◦ B(v j , rc2k ) ∈ f 2 (ck ), ⎪ ⎪ ⎪ ⎪ ∀ 1 ≤ i ≤ m, A and B are m F sets on ⎪ ⎪ ⎪ ⎩ F1 (ck ) and F2 (ck ), respectively. Example 10.19 Suppose that ( f 1 , D1 , 3) and ( f 2 , D2 , 3) are the 3–polar fuzzy 5– soft set and 3–polar fuzzy 4–soft set represented by Tables 10.36 and 10.37, respectively. Then their extended union ( f 1 , D1 , 3) ∪ E ( f 2 , D2 , 3) = (y, D1 ∪E D2 , 3) is given in Table 10.41.

10.6 m–Polar Fuzzy N –Soft Sets

511

Table 10.41 Extended union (y, D1 ∪E D2 , 3) v1

c1

c2

c3

e

3, (0.8, 0.7, 0.6)

4, (0.6, 0.8, 0.5)

3, (0.9, 0.7, 0.7)

2, (0.4, 0.8, 0.5)

v2

3, (0.9, 0.5, 0.4)

2, (0.7, 0.6, 0.2)

3, (0.8, 0.6, 0.9)

3, (0.7, 0.8, 0.6)

v3

4, (0.8, 0.8, 0.8)

1, (0.5, 0.4, 0.2)

4, (0.9, 0.5, 1.0)

3, (0.8, 0.9, 0.5)

v4

4, (0.8, 0.9, 0.8)

4, (0.6, 0.7, 0.6)

3, (0.8, 0.6, 0.7)

1, (0.6, 0.5, 0.2)

v5

1, (0.1, 0.5, 0.3)

3, (0.8, 0.4, 0.5)

3, (0.7, 0.6, 0.8)

1, (0.6, 0.3, 0.4)

v6

3, (0.8, 0.6, 0.9)

2, (0.5, 0.9, 0.1)

2, (0.6, 0.6, 0.5)

2, (0.7, 0.6, 0.2)

Definition 10.29 Let V be a universe, ( f, D, m) an (m, N )−soft set where D = (F, C, N ) is an N −soft set on V . Let 0 < S < N be a threshold. An m–polar fuzzy soft set related with ( f, D, m) and S, denoted by ( f S , C), is given as follows:  ( f , C) = S

f S (ck ) = (v j , 1), f S (ck ) = (v j , 0),

i f rck ≥ S, other wise.

Specifically, ( f 1 , C) is said to be bottom m–polar fuzzy soft set associated with ( f, D, m) and ( f N −1 , C) is said to be top m–polar fuzzy soft set. Example 10.20 Consider the 3–polar fuzzy 5–soft set ( f 1 , D1 , 3) where D1 = (F1 , R, 5) is a 5−soft set, represented by Table 10.36. From Definition 10.29, we have 0 < S < 5. The possible m–polar fuzzy soft sets associated with feasible thresholds and ( f 1 , D1 , 3) are given as follows: ⎫ ⎧ 1  f 1 (c1 ) = (v1 , α), (v2 , α), (v ⎪ 3 , α), ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ (v4 , α), (v5 , α), (v6 , α) , ⎪ ⎪ ⎪ ⎪  ⎬ ⎨ 1 f (c ) = (v , α), (v , α), (v , α), 2 1 2 3 1 1  ( f 1 , R) = (v4 , α), ⎪ ⎪ ⎪ ⎪  (v5 , α), (v6 , α) , ⎪ ⎪ 1 ⎪ ⎪ f (c ) = (v , α), (v , α), (v , α), ⎪ ⎪ 1 2 3 1 3 ⎪ ⎪  ⎭ ⎩ (v4 , β), (v5 , β), (v6 , α) . ⎫ ⎧ 2  f 1 (c1 ) = (v1 , α), (v2 , α), (v ⎪ ⎪ ⎪  3 , α), ⎪ ⎪ ⎪ ⎪ ⎪ (v4 , α), (v , β), (v , α) , ⎪ ⎪ 5 6 ⎪ ⎪  ⎬ ⎨ 2 f 1 (c2 ) = (v1 , α), (v2 , α), (v , α), 3 2  ( f 1 , R) = (v4 , α), ⎪ ⎪ ⎪ ⎪  (v5 , α), (v6 , α) , ⎪ ⎪ ⎪ ⎪ , α), ⎪ ⎪ f 12 (c3 ) = (v1 , β), (v2 , β), (v 3 ⎪ ⎪  ⎭ ⎩ (v4 , β), (v5 , β), (v6 , β) .

512

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

⎫ ⎧ 3  f 1 (c1 ) = (v1 , α), (v2 , α), (v ⎪ 3 , α), ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ (v4 , β), ⎪ ⎪ ⎪ ⎪  (v5 , β), (v6 , β) , ⎬ ⎨ 3 f 1 (c2 ) = (v1 , α), (v2 , β), (v , β), 3 3  ( f 1 , R) = ⎪ ⎪ ⎪ ⎪ 3 (v4 , α),  (v5 , α), (v6 , α) , ⎪ ⎪ ⎪ ⎪ f (c ) = (v , β), (v , β), (v , α), ⎪ ⎪ 3 1 2 3 ⎪ ⎪ 1  ⎭ ⎩ (v4 , β), (v5 , β), (v6 , β) . ⎫ ⎧ 4  f 1 (c1 ) = (v1 , α), (v2 , β), (v ⎪ 3 , α), ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ (v4 , β), (v5 , β), (v6 , β) , ⎪ ⎪ ⎪ ⎪  ⎬ ⎨ 4 f (c ) = (v , β), (v , β), (v , β), 2 1 2 3 4 1  ( f 1 , R) = (v4 , α), ⎪ ⎪ ⎪ ⎪  (v5 , β), (v6 , β) , ⎪ ⎪ 4 ⎪ ⎪ f (c ) = (v , β), (v , β), (v , α), ⎪ ⎪ 1 2 3 1 3 ⎪ ⎪  ⎭ ⎩ (v4 , β), (v5 , β), (v6 , β) . where α = (1.0, 1.0, 1.0), β = (0.0, 0.0, 0.0). Definition 10.30 An m–polar fuzzy N –soft set over V is said to be a null m–polar fuzzy N –soft set, denoted by (o, C, m), if for all ck ∈ C, (O, C, N ) is the N −soft set ¯ on V with the null mF set O¯ of V where O(v) = 0 for all v ∈ V . An m–polar fuzzy N –soft set on V is called a whole m–polar fuzzy N –soft set, denoted by (w, C, m), if for all ck ∈ C, (W, C, N ) is the N −soft set on V with the whole m–polar fuzzy set I¯ of V where I¯(v) = 1 for all v ∈ V .

10.7

m–Polar Fuzzy N–Soft Rough Sets

Akram et al. [7] proposed the concept of m−polar fuzzy N −soft rough sets., which relies on some definitions that are proceeded to state and exemplify. Definition 10.31 Let ( f, D, m) be an m–polar fuzzy N –soft set on V , where D = (F, C, N ) is an N −soft set. Then, an m–polar fuzzy N –soft subset λ of (V × M) × Z is called an m–polar fuzzy N –soft relation over (V × M) × Z , which is defined as follows: λ=

 ((v j , r jk ), z k ), pi ◦ λ((v j , r jk ), z k ) | (v j , r jk ) ∈  V × M, z k ∈ Z , 1 ≤ i ≤ m ,



where λ : (V × M) × Z → [0, 1]m . If V = {v1 , v2 , . . . , v j }, Z = {z 1 , z 2 , . . . , z k }, M = {r jk | j, k = 1, 2, . . . }, then tabular representation of an m–polar fuzzy N –soft relation λ over (V × M) × Z is given in Table 10.42. Example 10.21 Let V = {v1 , v2 , v3 } be a universe, Z = {z 1 , z 2 , z 3 } a set of attributes and M = {0, 1, 2, 3, 4}. A 3–polar fuzzy 5–soft relation λ : V × M → Z is given Table 10.43.

10.7 m–Polar Fuzzy N –Soft Rough Sets

513

Table 10.42 An m–polar fuzzy N –soft relation λ z1 ··· v1 v2 .. . vj

r11 , pi ◦ λ((v1 , r11 ), z 1 )

r21 , pi ◦ λ((v2 , r21 ), z 1 )

.. . r j1 , pi ◦ λ((v j , r j1 ), z 1 )

···

r1k , pi ◦ λ((v1 , r1k ), z k )

r2k , pi ◦ λ((v2 , r2k ), z k )

.. . r jk , pi ◦ λ((v j , r jk ), z k )

··· ..

.

···

Table 10.43 A 3–polar fuzzy 5–soft relation λ z1 v1 v2 v3

zk

1, (0.6, 0.3, 0.1)

1, (0.5, 0.3, 0.2)

0, (0.3, 0.2, 0.1)

z2

z3

3, (0.4, 0.7, 0.6)

2, (0.5, 0.2, 0.8)

2, (0.3, 0.4, 0.8)

2, (0.4, 0.6, 0.2)

4, (0.6, 0.9, 0.6)

3, (0.7, 0.3, 0.5)

Definition 10.32 Let V be a universal set, Z a set of attributes. For an arbitrary m–polar fuzzy N –soft relation λ on (V × M) × Z , a tuple (V, Z , N , λ) is called an m–polar fuzzy N –soft approximation space. For each P ∈ M (Z ), we define the lower N −soft approximations of P (λ N (P)) and upper N −soft approximations of N P (λ (P)), respectively, as follows:     (v j , r jk ), Pλ (v j , r jk ) | v j ∈ V,

λ N (P) =

j,k

j,k

j,k

j,k

 r jk ∈ M ,     N λ (P) = (v j , r jk ), Pλ (v j , r jk ) | v j ∈ V, r jk

 ∈M ,

where Pλ (v j ,



r jk ) = inf

j,k

Pλ (v j ,

 j,k

z k ∈T

  1 − pi ◦ Pλ ((v j , r jk ), z k ) ∨  pi ◦ P(z k ) ,

 r jk ) = sup pi ◦ Pλ ((v j , r jk ), z k ) ∧ pi ◦ z k ∈T

 P(z k ) .

514

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.44 A 3–polar fuzzy 5–soft relation λ z1 v1 v2 v3 v4

2, (0.6, 0.3, 0.1)

2, (0.5, 0.3, 0.2)

1, (0.3, 0.2, 0.1)

2, (0.4, 0.3, 0.6)

z2

z3

3, (0.4, 0.7, 0.6)

3, (0.5, 0.2, 0.8)

3, (0.3, 0.4, 0.8)

2, (0.5, 0.1, 0.4)

2, (0.4, 0.6, 0.2)

4, (0.6, 0.9, 0.6)

3, (0.7, 0.3, 0.5)

0, (0.3, 0.1, 0.0)

N

for all 1 ≤ i ≤ m. The pair (λ N (P), λ (P)) is referred to as an m–polar fuzzy N – N soft rough set of P with respect to (V, Z , N , λ), and λ N , λ : M (Z ) → M (V × M) are referred to as lower and upper m–polar fuzzy N –soft rough approximation operators. Example 10.22 Let V = {v1 , v2 , v3 , v4 } be a universe of TV shows and let Z =  z 1 , z 2 , z 3 be a set of parameters and M = {0, 1, 2, 3, 4}. Consider a 3–polar fuzzy 5–soft relation λ : V × M → Z is given in Table 10.44: Suppose that P is a 3−polar fuzzy subset of Z , given as follows:  P = (z 1 , 0.3, 0.1, 0.7), (z 2 , 0.3, 0.6, 0.4),  (z 3 , 0.5, 0.6, 0.1) . From Definition 10.32, the lower and upper N −soft approximations are given as follows: Pλ (v1 , 2) = (0.4, 0.6, 0.4), Pλ (v2 , 2) = (0.5, 0.6, 0.4), Pλ (v3 , 1) = (0.5, 0.6, 0.5), Pλ (v4 , 0) = (0.5, 0.7, 0.6), Thus,

Pλ (v1 , 3) = (0.4, 0.6, 0.4), Pλ (v2 , 4) = (0.5, 0.6, 0.4), Pλ (v3 , 3) = (0.5, 0.4, 0.4), Pλ (v4 , 2) = (0.3, 0.1, 0.6),

 λ N (P) = (v1 , 2), 0.4, 0.6, 0.4 , (v2 , 2), 0.5, 0.6, 0.4 ,  (v3 , 1), 0.5, 0.6, 0.4 , (v4 , 0), 0.5, 0.7, 0.6 ,  N λ (P) = (v1 , 3), 0.4, 0.6, 0.4 , (v2 , 4), 0.5, 0.6, 0.4 ,  (v3 , 3), 0.5, 0.4, 0.4 , (v4 , 2), 0.3, 0.1, 0.6 . N

Hence, the pair (λ N (P), λ (P)) is referred to as a 3–polar fuzzy 5–soft rough set. Definition 10.33 Let (V, Z , N , λ) be an m–polar fuzzy N –soft approximation space. For any P ∈ M (Z ), the complement of m–polar fuzzy N –soft rough N N set (λ N (P), λ (P)) of P about (V, Z , N , λ), denoted by (∼ λ N (P), ∼ λ (P)), is given as follows:

10.7 m–Polar Fuzzy N –Soft Rough Sets

515

Table 10.45 A 3–polar fuzzy 5–soft relation λ z1 v1 v2 v3 v4

2, (0.6, 0.3, 0.1)

2, (0.5, 0.3, 0.2)

1, (0.3, 0.2, 0.1)

2, (0.4, 0.3, 0.6)

∼ λ N (P) =



(v j , N −



z2

z3

3, (0.4, 0.7, 0.6)

3, (0.5, 0.2, 0.8)

3, (0.3, 0.4, 0.8)

2, (0.5, 0.1, 0.4)

2, (0.4, 0.6, 0.2)

4, (0.6, 0.9, 0.6)

3, (0.7, 0.3, 0.5)

0, (0.3, 0.1, 0.0)

r jk ), ¬Pλ (v j , N −



j,k

j,k

j,k

j,k

 r jk ) |

 v j ∈ V, r jk ∈ M ,     N ∼ λ (P) = (v j , N − r jk ), ¬Pλ (v j , N − r jk ) | v j ∈ V, r jk

 ∈M ,

where ¬Pλ (v j , N −

 j,k

 r jk ) = sup pi ◦ Pλ ((v j , r jk ), z k )∧ z k ∈T

 (1 − pi ◦ P(z k )) ,   r jk ) = inf (1 − pi ◦ Pλ ((v j , r jk ), ¬Pλ (v j , N − j,k

z k ∈T

 z k )) ∨ (1 − pi ◦ P(z k )) .

  Example 10.23 Let V = {v1 , v2 , v3 , v4 } be a universe and let Z = z 1 , z 2 , z 3 be a set of parameters and M = {0, 1, 2, 3, 4}. Consider a 3–polar fuzzy 5–soft relation λ : (V × M) × Z → [0, 1]m is given in Table 10.45. Suppose P is a 3−polar fuzzy subset of Z defined as follows:   P = (z 1 , 0.5, 0.2, 0.7), (z 2 , 0.3, 0.5, 0.6), (z 3 , 0.9, 0.1, 0.4) . By Definition 10.33, ¬Pλ (v1 , 3) = (0.5, 0.6, 0.4), ¬Pλ (v2 , 3) = (0.5, 0.9, 0.6), ¬Pλ (v3 , 4) = (0.3, 0.4, 0.5), ¬Pλ (v4 , 5) = (0.5, 0.3, 0.4), Thus,

¬Pλ (v1 , 2) = (0.5, 0.5, 0.4), ¬Pλ (v2 , 1) = (0.4, 0.8, 0.4), ¬Pλ (v3 , 2) = (0.3, 0.6, 0.4), ¬Pλ (v4 , 3) = (0.6, 0.8, 0.4),

516

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

 ∼ λ N (P) = (v1 , 3), 0.5, 0.6, 0.4 , (v2 , 3), 0.5, 0.9, 0.6 ,  (v3 , 4), 0.3, 0.4, 0.5 , (v4 , 5), 0.5, 0.3, 0.4 ,  N ∼ λ (P) = (v1 , 2), 0.5, 0.5, 0.4 , (v2 , 1), 0.4, 0.8, 0.4 ,  (v3 , 2), 0.3, 0.6, 0.4 , (v4 , 3), 0.6, 0.8, 0.4 . N

Hence, the pair (∼ λ N (P), ∼ λ (P)) is referred to as a complement of 3–polar N fuzzy 5–soft rough set (λ N (P), λ (P)). Theorem 10.1 Let (V, Z , N , λ) be an m–polar fuzzy N –soft approximation space. Then the lower m–polar fuzzy N –soft rough approximation operators λ N and the N upper m–polar fuzzy N –soft rough approximation operators λ , satisfy the following  properties for all P, Q ∈ M (Z ). 1. 2. 3. 4. 5. 6. 7. 8.

N

λ N (P) =∼ λ (∼ P), P ⊆ Q ⇒ λ N (P) ⊆ λ N (Q), λ N (P ∪ Q) ⊇ λ N (P) ∪ λ N (Q), λ N (P ∩ Q) = λ N (P) ∩ λ N (Q), N λ (P) =∼ λ N (∼ P), N N P ⊆ Q ⇒ λ (P) ⊆ λ (Q), N N N λ (P ∪ Q) = λ (P) ∪ λ (Q), N N N λ (P ∩ Q) ⊆ λ (P) ∩ λ (Q),

where ∼ P denotes the complement of P. Proof Its proof follows directly by Definitions 10.32 and 10.33.



Proposition 10.7 Let (V, Z , N , λ) be an m–polar fuzzy N –soft approximation  space and P, Q ∈ m (Z ). Then the lower and upper N −soft approximations of P and Q satisfies the following properties.   1. ∼ λ N (P) ∪ λ N (Q) = λ N (∼ P) ∩ λ N (∼ Q),   2. ∼ λ N (P) ∪ λ N (Q) = λ N (∼ P) ∩ λ N (∼ Q),   3. ∼ λ N (P) ∪ λ N (Q) = λ N (∼ P) ∩ λ N (∼ Q),   4. ∼ λ N (P) ∪ λ N (Q) = λ N (∼ P) ∩ λ N (∼ Q),   5. ∼ λ N (P) ∩ λ N (Q) = λ N (∼ P) ∪ λ N (∼ Q),   6. ∼ λ N (P) ∩ λ N (Q) = λ N (∼ P) ∪ λ N (∼ Q),   7. ∼ λ N (P) ∩ λ N (Q) = λ N (∼ P) ∪ λ N (∼ Q),   8. ∼ λ N (P) ∩ λ N (Q) = λ N (∼ P) ∪ λ N (∼ Q). Proof Its proof follows directly from the Definitions 10.32 and 10.33.



10.7 m–Polar Fuzzy N –Soft Rough Sets

517

Table 10.46 A 3–polar fuzzy 5–soft relation λ z1 v1 v2 v3 v4

2, (0.6, 0.3, 0.1)

2, (0.5, 0.3, 0.2)

1, (0.3, 0.2, 0.1)

2, (0.4, 0.3, 0.6)

z2

z3

3, (0.4, 0.7, 0.6)

3, (0.5, 0.2, 0.8)

3, (0.3, 0.4, 0.8)

2, (0.5, 0.1, 0.4)

2, (0.4, 0.6, 0.2)

4, (0.6, 0.9, 0.6)

3, (0.7, 0.3, 0.5)

0, (0.3, 0.1, 0.0)

Definition 10.34 ([74]) Let V = {v1 , v2 , v3 , . . . , vn } be a universe of objects. A comparison table is a square table containing equal number of rows and columns labeled by the objects of the universe and the entries c jk where c jk = the number of parameters for which the membership value of v j exceeds or equals the value of vk . Definition 10.35 Let V be a universe, Z a set of attributes and P ∈ M (Z ). Then, the ring sum operation about the lower N −soft approximation λ N (P) and the upper N N −soft approximation λ (P) is defined as follows: N

λ N (P) ⊕ λ (P) =

 (v, r z + r z − r z .r z ), pi ◦ Pλ (v)+ pi ◦ Pλ (v) − pi ◦ Pλ (v) × pi ◦  Pλ (v) | (v, r z ), pi ◦ Pλ (v) ∈ λ N (P), (v, r z ), pi ◦ Pλ (v) ∈  N λ (P), v ∈ V, 1 ≤ i ≤ m .

Example 10.24 Reconsider  10.22, let V = {v1 , v2 , v3 , v4 } be a universe  Example of TV shows and let Z = z 1 , z 2 , z 3 be a set of parameters and M = {0, 1, 2, 3, 4}. Consider a 3–polar fuzzy 5–soft relation λ : V × M → Z as given in Table 10.46. Suppose that P is a 3−polar fuzzy subset of Z , given as follows:  P = (z 1 , 0.3, 0.1, 0.7), (z 2 , 0.3, 0.6, 0.4),  (z 3 , 0.5, 0.6, 0.1) . Using Definition 10.32, the lower and upper N −soft approximations of P are given as follows:  λ N (P) = (v1 , 2), 0.4, 0.6, 0.4 , (v2 , 2), 0.5, 0.6, 0.4 ,  (v3 , 1), 0.5, 0.6, 0.4 , (v4 , 0), 0.5, 0.7, 0.6 ,  N λ (P) = (v1 , 3), 0.4, 0.6, 0.4 , (v2 , 4), 0.5, 0.6, 0.4 ,  (v3 , 3), 0.5, 0.4, 0.4 , (v4 , 2), 0.3, 0.1, 0.6 .

518

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

From Definition 10.35, the ring sum operation about the lower N −soft approximation N λ N (P) and the upper N −soft approximation λ (P) is given as follows:  λ N (P) ⊕ λ N (P) = (v1 , −1), 0.64, 0.84, 0.64 , (v2 , −2), 0.75, 0.84, 0.64 , (v3 , 1), 0.75, 0.76, 0.64 ,  (v4 , 2), 0.65, 0.1, 0.84) .

10.8 Applications Akram et al. [7] explains the decision making mechanisms that operate on the presented models. Therefore, the respective algorithms for problems that are characterized by m–polar fuzzy N –soft sets and m–polar fuzzy N –soft rough sets. In order to prove their implementability and feasibility, presented models are applied to real situations that are fully developed.

10.8.1 Selection of a Restaurant Stars are used by reviewers for ranking things, including hotels, restaurants, TV shows and resorts. Buying a suitable restaurant is a difficult task due to the variation in ratings and reviews on one website to another for the same restaurant. The rating of a restaurant depend on different parameters and reviewers. Suppose that a businessman (Mr. Ali) wants to purchase a suitable restaurant from the alternatives v1 , v2 , v3 , v4 . Let V = {v1 , v2 , v3 , v4 } be a universe of four restaurants and let C = {c1 , c2 , c3 } ⊆ Z be a set of attributes, which gives grades to restaurants based on location, meal options, services and parking area. A 4–polar fuzzy 6–soft set can be obtained from Table 10.47, where Five stars represent ‘luxury’, Four stars represent ‘excellent’, Three stars represent ‘very good’, Two stars represent ‘good’. The set of ordered grades M = {0, 1, 2, 3, 4, 5} can be easily associated with stars as follows: 2 stands for ‘’, 3 stands for ‘  ’, 4 stands for ‘  ’, 5 stands for ‘    ’.

10.8 Applications

519

Table 10.47 Information obtained from the related data V c1 c2 v1 v2 v3 v4

     

   

Table 10.48 Tabular representation of 6−soft set (F, C, 6) c1 c2 v1 v2 v3 v4

3 3 5 4

Table 10.49 A 4–polar fuzzy 6–soft set ( f, D, 4) c1 v1 v2 v3 v4

3, (0.5, 0.4, 0.7, 0.6)

3, (0.5, 0.6, 0.4, 0.7)

5, (0.9, 0.8, 0.7, 0.9)

4, (0.9, 0.6, 0.7, 0.8)

2 5 3 3

0.5 0.5 0.9 0.9

     

c3 2 2 4 3

c2

c3

2, (0.6, 0.5, 0.7, 0.6)

5, (0.8, 0.9, 0.7, 0.9)

3, (0.3, 0.7, 0.6, 0.9)

3, (0.8, 0.5, 0.8, 0.4)

2, (0.5, 0.8, 0.6, 0.9)

2, (0.6, 0.8, 0.7, 0.7)

4, (0.9, 0.8, 0.9, 0.7)

3, (0.7, 0.6, 0.8, 0.9)

Table 10.50 Tabular representation of 1st pole p1 c1 c2 v1 v2 v3 v4

c3

0.6 0.8 0.3 0.8

c3 0.5 0.6 0.9 0.7

Tabular form of a 6−soft set is given in Table 10.48. Therefore, a 4–polar fuzzy 6–soft set ( f, D, 4) is given in Table 15. Thus, ( f, D, 4) is a 4–polar fuzzy 6–soft set based on location, meal option, services and parking area of the restaurants. For example, in the Table 10.49, 3, (0.5, 0.4, 0.7, 0.6) means that the restaurant v1 is 50% suitable with the location, 40% suitable with the meal options, 70% suitable with services and 60% suitable with the parking area, and 3 is the evaluation grade of v1 according to attribute c1 . The tabular representation of the first pole is given in Table 10.50.

520

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.51 Comparison table for the 1st pole . v1 v2 v1 v2 v3 v4

3 3 2 3

1 3 2 3

Table 10.52 Membership score table of the 1st pole . Grades Row sum (e) 3 sum ( i=1 rci ) v1 v2 v3 v4

7 10 12 10

5 8 9 11

v3

v4

1 1 3 2

0 1 2 3

Column sum ( f )

V1 = e − f

11 9 7 6

−6 −1 2 5

Table 10.53 Tabular representation of the 2nd pole p2 c1 c2 v1 v2 v3 v4

0.4 0.6 0.8 0.6

Table 10.54 Comparison table for the 2nd pole . v1 v2 v1 v2 v3 v4

3 3 3 2

c3

0.5 0.9 0.7 0.5

1 3 2 1

0.8 0.8 0.8 0.6

v3

v4

1 2 3 0

2 3 3 3

The comparison table for the 1st pole, is given in Table 10.51. 3 rci ) is The membership score for each restaurant with sum of grades ( i=1 obtained by subtracting the column sum from the row sum of Table 10.51, which is displayed in Table 10.52. Similarly, the remaining three poles are represented in tabular form and the membership scores are calculated with the help of comparison tables for each pole, respectively, as given in Tables 10.53, 10.54, 10.55, 10.56, 10.57, 10.58, 10.59, 10.60 and 10.61.

10.8 Applications

521

Table 10.55 Membership score table of the 2nd pole . Grades Row sum (g) 3 sum( i=1 rci ) v1 v2 v3 v4

7 10 12 10

7 11 11 6

Column sum (h)

V2 = g − h

11 7 6 11

−4 4 5 −5

Table 10.56 Tabular representation of the 3rd pole p3 c1 c2 v1 v2 v3 v4

0.7 0.4 0.7 0.7

0.7 0.7 0.6 0.8

Table 10.57 Comparison table for the 3rd pole . v1 v2 v1 v2 v3 v4

3 2 2 3

2 3 1 3

Table 10.58 Membership score table of the 3rd pole . Grades Row sum (c) 3 sum ( i=1 rci ) v1 v2 v3 v4

7 10 12 10

8 6 8 11

Table 10.59 Tabular representation of the 4th pole p4 c1 c2 v1 v2 v3 v4

0.6 0.7 0.9 0.8

c3

0.6 0.9 0.9 0.4

0.6 0.7 0.9 0.8

v3

v4

2 1 3 2

1 0 2 3

Column sum (d)

V3 = c − d

10 9 8 6

−2 −3 0 5

c3 0.9 0.7 0.7 0.9

522

10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.60 Comparison table for the 4th pole . v1 v2 v1 v2 v3 v4

3 2 2 2

1 3 3 2

Table 10.61 Membership score table of the 4th pole . Grades Row sum (n) 3 sum( i=1 rci ) v1 v2 v3 v4

7 10 12 10

7 8 10 8

Table 10.62 Final score table with grades . Grades V2 3 sum V1 ( i=1 rci ) v1 v2 v3 v4

7 10 12 10

−6 −1 2 5

−4 4 5 −5

v3

v4

1 2 3 1

2 1 2 3

Column sum (o)

V4 = n − o

9 9 7 8

−2 −1 3 0

V3

V4

Final 4 Score ( i=1 Vi )

−2 −3 0 5

−2 −1 3 0

−14 −1 10 5

The final score for each restaurant is obtained by adding the membership scores V1 , V2 , V3 , V4 of all alternatives as shown in Table 10.62. It is clear from the calculations that the maximum score is 10 by v3 with maximum grades. Therefore, Mr. Ali will select the restaurant v3 to buy. The following Algorithm 10.8.1 for the selection of a suitable restaurant is presented as follows: Algorithm 10.8.1 The algorithm for the selection of an alternative in an m–polar fuzzy N –soft set 1. 2. 3. 4. 5. 6.

Input V as a universe with n elements. Input C ⊆ Z as a set of attributes. Consider an m–polar fuzzy N –soft set in tabular form. Determine the comparison table for all 3−poles. Compute the information score for all poles. Compute the final score by adding the scores of all poles.

10.8 Applications

523

7. Find the biggest score with maximum grades, if it occurs in kth row, then we will choose option vk as output, 1 ≤ k ≤ n. If the set of optimal choices in the last step of Algorithm 10.8.1 contain more than one value, that is, va = vb , where 1 ≤ a = b ≤ n, then any one of them can be chosen. For clarification, step 4 produces Tables 10.51, 10.54, 10.57 and 10.60; step 5 produces Tables 10.52, 10.55, 10.58 and 10.61; and Table 10.62 is the result of step 6. The Algorithm 10.8.1 is applied to another real situation.

10.8.2 Selection of a Hotel Choosing a suitable hotel on a tour to stay is a very difficult task due to different star rating for the same hotel on different websites based on different attributes, for example, star rating of the hotel “The Nishat Hotel” is different on various websites including, http://www.agoda.com and http://www.booking.com. Suppose that a tourist (Mr. Nabeel) wants to select a suitable hotel from the alternatives q1 , q2 , q3 , q4 . Let V = {q1 , q2 , q3 , q4 } be a universe of four hotels and let C = {c1 , c2 , c3 } ⊆ Z be a set of attributes, which gives grades to hotels based on location, price and services. A 3–polar fuzzy 6–soft set can be obtained from Table 10.63, where Five stars represent ‘luxury’, Four stars represent ‘excellent’, Three stars represent ‘very good’, Two stars represent ‘good’, One star represents ‘regular’. The set of ordered grades M = {0, 1, 2, 3, 4, 5} can easily be identified with stars as follows: 1 stands for ‘’, 2 stands for ‘’, 3 stands for ‘  ’, 4 stands for ‘  ’, 5 stands for ‘    ’. Tabular form of a 6−soft set is given by Table 10.64. Therefore, a 3–polar fuzzy 6–soft set ( f, D, 3) is given in Table 10.65. Thus, ( f, D, 3) is a 3–polar fuzzy 6–soft set based on location, price and services of the hotels. For example, in the Table 10.65, 4, (0.7, 0.6, 0.5) means that the hotel q1 is 70% suitable with the location, 60% suitable with the price and 50% suitable

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Table 10.63 Information obtained from the related data V c1 c2 q1 q2 q3 q4

         

     

Table 10.64 Tabular representation of 6−soft set (F, C, 6) c1 c2 q1 q2 q3 q4

4 4 5 4

Table 10.65 A 3–polar fuzzy 6–soft set ( f, D, 3) c1 q1 q2 q3 q4

4, (0.7, 0.6, 0.5)

4, (0.5, 0.6, 0.7)

5, (0.8, 0.8, 0.7)

4, (0.7, 0.5, 0.6)

3 1 4 2

0.7 0.5 0.8 0.7

   

c3 1 2 3 2

c2

c3

3, (0.6, 0.7, 0.6)

1, (0.7, 0.4, 0.5)

4, (0.7, 0.7, 0.8)

2, (0.5, 0.6, 0.7)

1, (0.5, 0.7, 0.6)

2, (0.7, 0.6, 0.7)

3, (0.9, 0.7, 0.7)

2, (0.7, 0.6, 0.7)

Table 10.66 Tabular representation of the 1st pole p1 c1 c2 q1 q2 q3 q4

c3

0.6 0.7 0.7 0.5

c3 0.5 0.7 0.9 0.7

with services, where 4 is the evaluation grade of q1 according to attribute c1 . The tabular form of the 1st pole is given in the Table 10.66. Now, the comparison table for the 1st pole is given in Table 10.67. 3 rci ) is obtained The membership score for each hotel with sum of grades ( i=1 by subtracting the column sum from the row sum of Table 10.67, which is given in Table 10.68. Similarly, the remaining two poles are presented in tabular form and the membership scores are calculated with the help of comparison tables for each pole, respectively, see Tables 10.69, 10.70, 10.71, 10.72, 10.73 and 10.74.

10.8 Applications

525

Table 10.67 Comparison Table for the 1st pole . q1 q2 q1 q2 q3 q4

3 2 3 2

1 3 3 2

Table 10.68 Membership score Table of the 1st pole . Grades Row sum (c) 3 sum ( i=1 rci ) q1 q2 q3 q4

8 7 12 8

6 8 12 7

q3

q4

0 1 3 0

2 2 3 3

Column sum (d)

V1 = c − d

10 9 4 10

−4 −1 8 −3

Table 10.69 Tabular representation of 2nd pole p2 c1 c2 q1 q2 q3 q4

0.6 0.6 0.8 0.5

Table 10.70 Comparison table for the 2nd pole . q1 q2 q1 q2 q3 q4

3 1 3 0

3 3 3 2

Table 10.71 Membership score table of the 2nd pole . Grades Row sum (e) 3 sum ( i=1 rci ) q1 q2 q3 q4

8 7 12 8

c3

0.7 0.4 0.7 0.6

11 6 12 5

0.7 0.6 0.7 0.6

q3

q4

2 0 3 0

3 3 3 3

Column sum ( f )

V2 = e − f

7 11 5 11

4 −5 7 −6

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.72 Tabular representation of the 3rd pole p3 c1 c2 q1 q2 q3 q4

0.5 0.7 0.7 0.6

0.6 0.5 0.8 0.7

Table 10.73 Comparison Table for the 3rd pole . q1 q2 q1 q2 q3 q4

3 2 3 3

1 3 3 2

Table 10.74 Membership score table of the 3rd pole . Grades Row sum (g) 3 sum( i=1 rci ) q1 q2 q3 q4

8 7 12 8

4 9 12 9

Table 10.75 Final score table with grades . Grades V1 3 sum ( i=1 rci ) q1 q2 q3 q4

8 7 12 8

c3

−4 −1 8 −3

0.6 0.7 0.7 0.7

q3

q4

0 2 3 1

0 2 3 3

Column sum (h)

V3 = g − h

11 9 6 8

−7 0 6 1

V2

V3

Final 3 Score ( i=1 Vi )

4 −5 7 −6

−7 0 6 1

−7 −6 21 −8

The final score for each hotel with grades is obtained by adding the membership scores V1 , V2 , V3 of all the alternatives. It is clear from the Table 10.75, that the maximum grade is 12 and maximum score is 21 scored by q3 . Therefore, Mr. Nabeel will select the hotel q3 to stay.

10.8 Applications

527

Table 10.76 Information obtained from the related data V z1 y1 y2 y3 y4

         

z2      

10.8.3 Selection of a Resort Stars are a very subjective thing. It’s usually referring to the service and amenities. Choosing a resort is very difficult task due to different star ratings for the same resort on different websites, for example, the resort named as“Komandoo Maldives Resort” has different star ratings on different websites including, http://www.expedia.com and http://www.tripadvisor.com. Suppose that a family plans to spend their vacations in a resort in Maldives from the alternatives y1 , y2 , y3 , y4 . They want to select the best resort to spend vacations with comfort. It is difficult for family members to agree at one option. The natural environment, cost and social environment are the leading parameters for selecting a resort. Let V = {y1 , y2 , y3 , y4 } be a set of resorts and let Z = {z 1 , z 2 } be a set of attributes which gives grades to these resorts based on natural environment, social environment and cost. A 3–polar fuzzy 6–soft relation can be obtained from Table 10.76, where Five stars represent ‘luxury’, Four stars represent ‘excellent’, Three stars represent ‘very good’. The set of ordered grades M = {0, 1, 2, 3, 4, 5} can be easily associated with stars as follows: 3 stands for ‘  ’, 4 stands for ‘  ’, 5 stands for ‘    ’. Assume that the “attractiveness of the resort” by composing a 3–polar fuzzy 6– soft relation λ over (V × M) × Z , is explained and given in Table10.77. Thus λ : (V × M) × Z → [0, 1]m is a 3–polar fuzzy 6–soft relation based on natural environment, social environment, and cost of the resorts. For example, ((y1 , 5), z 1 ), 0.9, 0.8, 0.8 means that the resort y1 has grade 5 include, 90% natural environment, 80% social environment and 80% cost.

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.77 A 3–polar fuzzy 6–soft relation λ z1

z2

5, (0.9, 0.8, 0.8)

4, (0.7, 0.8, 0.7)

4, (0.8, 0.6, 0.8)

4, (0.7, 0.8, 0.7)

y1 y2 y3 y4

3, (0.6, 0.6, 0.6)

4, (0.6, 0.7, 0.6)

5, (0.8, 0.9, 0.5)

5, (0.7, 0.8, 0.7)

Now suppose that they provide the optimal decision object P as follows:   P = (z 1 , 0.9, 0.8, 0.7), (z 2 , 0.7, 0.6, 0.8) . From Definition 10.32, Pλ (y1 , 3) = (0.7, 0.6, 0.7), Pλ (y2 , 4) = (0.7, 0.6, 0.7), Pλ (y3 , 4) = (0.7, 0.8, 0.5), Pλ (y4 , 4) = (0.7, 0.6, 0.7),

Pλ (y1 , 5) = (0.9, 0.8, 0.7), Pλ (y2 , 4) = (0.7, 0.8, 0.7), Pλ (y3 , 5) = (0.8, 0.6, 0.7), Pλ (y4 , 5) = (0.7, 0.6, 0.7). N

By Definition 10.32, the operators λ N (P) and λ (P), respectively, are given as follows:  λ N (P) = (y1 , 3), 0.7, 0.6, 0.7 , (y2 , 4), 0.7, 0.6, 0.7 ,  (y3 , 4), 0.7, 0.8, 0.5 , (y4 , 4), 0.7, 0.6, 0.7 ,  N λ (P) = (y1 , 5), 0.9, 0.8, 0.7 , (y2 , 4), 0.7, 0.8, 0.7 ,  (y3 , 5), 0.8, 0.6, 0.7 , (y4 , 5), 0.7, 0.6, 0.7 . By Definition 10.35,  N λ N (P) ⊕ λ (P) = (y1 , −7), 0.97, 0.92, 0.91 , (y2 , −8), 0.91, 0.92, 0.91, (y3 , −11), 0.94, 0.92, 0.85 ,  (y4 , −11), 0.91, 0.84, 0.91 . N

Thus, y1 is the resort containing optimal values in the set λ N (P) ⊕ λ (P). Therefore, they will select the resort y1 . The following Algorithm 10.8.2 is organized for the selection of a suitable resort.

10.8 Applications

529

Algorithm 10.8.2 The algorithm for the selection of an alternative in an m–polar fuzzy N –soft rough set 1. 2. 3. 4. 5. 6. 7.

Input V as a universe with n objects. Input Z as a set of attributes. Consider an m–polar fuzzy N –soft relation λ on (V × M) × Z .  Input P ∈ m (Z ) as an optimal decision according to needs of decision makers. N Determine λ N (P) and λ (P) by Definition 10.32. N 10.35. Compute the choice set Z = λ N (P) ⊕ λ (P) by Definition  pi ◦ Z (yk , r ), then the Find k for which pi ◦ Z (yk , r ) ≥ Q, where Q = 1≤k≤n

optimal decision will be yk , which can be chosen as output. If the set of optimal choices in the last step of Algorithm 10.8.2 contains more than one value, that is, ya = yb , where 1 ≤ a = b ≤ n, change the m–polar fuzzy subset P and repeat the Algorithm 10.8.2 until the optimal decision is only one. The Algorithm 10.8.2 is applied to another real situation.

10.8.4 Selection of a Laptop The selection of a laptop is a difficult task due to the variation in ratings and reviews on one website to another for the same laptop, for example, star rating of the laptop “Lenovo ThinkPad X1 Carbon 2017” is different on various websites including, http://www.cnet.com and http://www.computershopper.com. Since every person has different requirements about a laptop such as, design, technology, price etc. Suppose a computer programmer (Mr. Raheel) wishes to purchase a most suitable laptop from the alternatives r1 , r2 , r3 , r4 . The design, technology and price are the leading parameters for selecting a laptop. Let V = {r1 , r2 , r3 , r4 } be a universe of laptops, Z = {z 1 , z 2 , z 3 } a set of attributes, which gives grades to these laptops based on design, technology and price. A 3–polar fuzzy 6–soft relation can be obtained from Table 10.78, where Five star represents ‘luxury’, Four star represents ‘excellent’, Three star represents ‘very good’, Two star represents ‘good’, One star represents ‘regular’, The set of ordered grades M = {0, 1, 2, 3, 4, 5} can easily be identified with stars as follows:

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

Table 10.78 Information extracted from the related data V z1 z2 r1 r2 r3 r4

     

z3

     

   

Table 10.79 Tabular representation of 3–polar fuzzy 6–soft relation λ z1 z2 r1 r2 r3 r4

1, (0.2, 0.6, 0.1)

4, (0.4, 0.5, 0.7)

5, (0.7, 0.8, 0.3)

3, (0.5, 0.6, 0.4)

3, (0.3, 0.4, 0.7)

3, (0.4, 0.5, 0.5)

4, (0.8, 0.9, 0.4)

3, (0.6, 0.7, 0.1)

z3 2, (0.7, 0.3, 0.2)

2, (0.7, 0.4, 0.1)

2, (0.6, 0.2, 0.6)

3, (0.8, 0.5, 0.3)

1 stands for ‘’, 2 stands for ‘’, 3 stands for ‘  ’, 4 stands for ‘  ’, 5 stands for ‘    ’. Assume that Raheel describes the “attractiveness of the laptop” by a 3–polar fuzzy 6–soft relation λ : (V × M) × Z → [0, 1]m , as given in Table10.79. Thus λ over (V × M) × Z is the 3–polar fuzzy 6–soft relation based on design, technology and price of the laptops. For example, ((r1 , 1), z 1 ), 0.2, 0.6, 0.1) means that the laptop r1 contains suitability, 20% for design, 60% for technology and 10% for price. Suppose that Raheel provides the optimal decision object P as follows:  P = (z 1 , 0.5, 0.6, 0.7), (z 2 , 0.7, 0.6, 0.9),

 (z 3 , 0.9, 0.6, 0.8) .

From Definition 10.32, Pλ (r1 , 1) = (0.7, 0.6, 0.8), Pλ (r2 , 2) = (0.6, 0.6, 0.7), Pλ (r3 , 2) = (0.5, 0.6, 0.7), Pλ (r4 , 3) = (0.5, 0.6, 0.7),

Pλ (r1 , 3) = (0.7, 0.6, 0.7), Pλ (r2 , 4) = (0.7, 0.5, 0.7), Pλ (r3 , 5) = (0.7, 0.6, 0.6), Pλ (r4 , 3) = (0.8, 0.6, 0.4). N

Now 3–polar fuzzy 6–soft rough approximation operators λ N (P), λ (P), respectively, are given as follows:

10.9 Discussion

531

 λ N (P) = (r1 , 1), 0.7, 0.6, 0.8 , (r2 , 2), 0.6, 0.6, 0.7 ,  (r3 , 2), 0.5, 0.6, 0.7 , (r4 , 3), 0.5, 0.6, 0.7 ,  N λ (P) = (r1 , 3), 0.7, 0.6, 0.7 , (r2 , 4), 0.7, 0.5, 0.7 ,  (r3 , 5), 0.7, 0.6, 0.6 , (r4 , 3), 0.8, 0.6, 0.4 . From Definition 10.35,  λ N (P) ⊕ λ N (P) = (r1 , 1), 0.91, 0.84, 0.94 , (r2 , −2), 0.88, 0.8, 0.91 , (r3 , −3), 0.85, 0.84, 0.88 ,  (r4 , −3), 0.9, 0.84, 0.82) . Therefore, Raheel will choose the laptop r1 to buy because r1 contains the optimal values in the set λ N (P) ⊕ λ N (P).

10.9 Discussion Like soft set theory in the recent past, the theory of N −soft sets is now emerging as a powerful mathematical tool to handle the uncertainty of situations with either binary or multinary data in discrete form. When N = 2, an N −soft set degenerates into a soft set. Thus N −soft set theory has a wider range of practical applications as compared to soft set theory. Recently, Akram et al. [5] developed a new decision makinghybrid model called fuzzy N −soft sets and illustrated it through real life applications to decision-making. The fuzzy N −soft set model adds fuzziness on top of the requirements of N −soft sets. But there are many situations where data comes from multiple poles (agents). For example, weighted games; a company that decides to manufacture an item or product; a country that elects its leader; or a group of friends that plan to visit a country. Our presented models, namely, m–polar fuzzy N –soft sets and m–polar fuzzy N –soft rough sets, enable us to deal with such type of data more precisely than existing models in the literature (e.g., m–polar fuzzy soft sets [9] and the aforementioned models). Since an m–polar fuzzy N –soft set is a generalization of both fuzzy N −soft sets [5] and m–polar fuzzy soft sets [9] (v., Remark 10.1), the applicability of the m–polar fuzzy N –soft set model is wider than the applicability of fuzzy N −soft set and m–polar fuzzy soft set models. Further, m– polar fuzzy N –soft rough sets describe the rough approximations of m–polar fuzzy N –soft sets. In an m–polar fuzzy N –soft rough set model, the decision is based on an m–polar fuzzy N –soft relation and an optimal decision object. But in an m– polar fuzzy N –soft set model, the optimal decision is totally based on a comparison table approach. Therefore it does not seem feasible to contrast these models that are intended to capture alternative forms of inputs.

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10 Hybrid Models Based on Multi-polar Fuzzy Soft Sets

10.10 Conclusion Hybrid models involving soft set theory have become notable mathematical tools that enable to study different kinds of multipolar information, and to produce ranking orders among the objects under analysis. These hybrid models are playing a very important role to capture information and approach MCGDM, because they are capable of condensing relevant data more precisely than individual models. In this chapter, the motivation is based upon the positive features of bipolar fuzzy soft expert sets [31], m–polar fuzzy soft sets [9] and N −soft sets [45]. The advantages of presented hybrid models, such as m–polar fuzzy soft expert sets, m–polar fuzzy N –soft sets and m–polar fuzzy N –soft rough sets are encapsulated. In addition, the presented hybrid models are explained with examples. Moreover, some basic operations are investigated, including OR, AND, complement, intersection and union. Further, numerical examples are described as the application of a methodology that makes full use of the proposed hybrid models. It is shown that the natural decision making procedures can raise to optimal solutions in each of these settings. At the end, a comparative analysis and discussion under different values of ‘m’ are described with respect to certain existing mathematical methods.

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Index

A Absolute m−polar fuzzy soft set, 24 Accuracy function, 377, 440, 441, 478 Aggregated m–polar fuzzy decision matrix, 287, 318 Aggregated m–polar hesitant fuzzy dominance matrix, 185 Aggregated weighted m–polar decision matrix, 257 Aggregated weighted m–polar fuzzy decision matrix, 243, 288, 333 Agree-m–polar fuzzy soft expert set, 481 AHP method, 348 Analytical network process, 345 AND operation, 26, 487 Average ranking, 266 Average value, 202

B Bottom weak complement, 507 Bounded Law, 381

C Canonical dominance Dc , 320 Commutative Law, 382 Comparative analysis, 365 Comparison table, 517 Complement, 482 Complement of m-polar fuzzy N -soft rough set, 514 Complete lattice, 4

Composition, 52 Comprehensive concordance index, 290, 334 Concordance credibility degree, 291 Concordance indices, 112, 289 Concordance matrix, 112 Concordance set, 111 Consistent, 15 Credibility index, 291, 329 Crisp soft relations, 64 Cut-off level, 322

D Decision matrix, 111 Definable, 12 Degree of divergence, 288 Deviation degree, 169 Disagree-m–polar fuzzy soft expert set, 482 Discordance credibility degree, 292 Discordance dominance matrix, 113 Discordance indices, 112, 291, 334 Discordance set, 112 Discrimination threshold, 322 Distillation process, 322 Divergence, 333 Dominance classes, 319 Dominance levels, 319 Dominance matrix, 113

E Efficient, 505

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Akram and A. Adeel, Multiple Criteria Decision Making Methods with Multi-polar Fuzzy Information, Studies in Fuzziness and Soft Computing 430, https://doi.org/10.1007/978-3-031-43636-9_1

537

538 ELECTRE-II method, 344 Entropy, 333 Equal m-polar fuzzy soft expert sets, 482 Euclidean distance, 35, 245 Extended intersection, 508 Extended union, 510

F Financial soundness, 326 Forward ranking, 265 Fuzzy soft class, 23 Fuzzy soft set, 23

G Group decision making, 87

H Hamming distance, 35 Health and safety, 327 Hesitant m–polar fuzzy concordance indices , 217 Hesitant m–polar fuzzy concordance set , 217 Hesitant m–polar fuzzy discordance indices, 217 Hesitant m–polar fuzzy discordance set, 217 Hesitant m−polar fuzzy element, 193 Hesitant m−polar fuzzy set, 192

I Idempotent Law, 380 Inconsistent, 15 Independent m–polar fuzzy decision matrices, 286 Intersection, 24, 485

L Lattice, 3 Level criterion preference function, 347 Linear preference, 347 Linear preference having indifference area, 347 Linguistic ELECTRE-I approach, 110 Linguistic term set, 106, 439 Linguistic variables, 106 Lower and upper approximation operators, 7 Lower and upper approximations, 7 Lower and upper m-polar fuzzy N –soft rough approximation operators, 514

Index M Management capability , 326 Matrix of credibility indices, 321 Mediation mass assignment, 12 Mediation value, 9 Medical diagnosis of anemia, 42 Medical diagnosis of dengue fever, 43 Minimized efficient m–polar fuzzy N –soft set, 506 Monotonic Law, 381 m–polar fuzzy aggregated dominance matrix, 247 m–polar fuzzy averaging operator, 318 m–polar fuzzy concordance dominance matrix, 245 m–polar fuzzy concordance index, 244, 245 m–polar fuzzy concordance matrix, 244 m–polar fuzzy concordance set, 243, 369 m–polar fuzzy decision matrix, 317 m–polar fuzzy discordance dominance matrix, 246 m–polar fuzzy discordance index, 244 m–polar fuzzy discordance set, 243, 244, 369 m–polar fuzzy Dombi hybrid averaging operator, 383 m–polar fuzzy Dombi ordered weighted averaging operator, 381 m–polar fuzzy Dombi ordered weighted geometric operator, 387 m–polar fuzzy Dombi weighted averaging operator, 379 m–polar fuzzy Dombi weighted geometric operator, 385 m–polar fuzzy ELECTRE I, 369 m–polar fuzzy ELECTRE-I method, 89 m–polar fuzzy ELECTRE II, 259 m–polar fuzzy ELECTRE III, 286 m–polar fuzzy ELECTRE III technique, 330 m–polar fuzzy ELECTRE IV method, 317 m–polar fuzzy Hamacher hybrid averaging operator, 399 m–polar fuzzy Hamacher hybrid geometric operator, 407 m–polar fuzzy Hamacher ordered weighted average operator, 396 m–polar fuzzy Hamacher weighted average operator, 392 m–polar fuzzy Hamacher weighted geometric operator, 402 m–polar fuzzy Hamacher weighted ordered geometric operator, 404

Index m–polar fuzzy linguistic TOPSIS method, 137 m−polar fuzzy linguistic variable, 107 m-polar fuzzy N -soft approximation space, 513 m-polar fuzzy N -soft relation, 512 m-polar fuzzy N -soft rough set, 514 m–polar fuzzy number, 9, 241, 319 m−polar fuzzy objective decision information system, 14 m–polar fuzzy PROMETHEE method, 350 m–polar fuzzy rough set, 46 m−polar fuzzy soft approximation space, 59 m−polar fuzzy soft class, 24 m-polar fuzzy soft expert set, 478 m-polar fuzzy soft expert subset, 480 m-polar fuzzy soft expert superset, 481 m−polar fuzzy soft relation, 59 m−polar fuzzy soft rough set, 60 m−polar fuzzy soft set, 23, 502 m−polar fuzzy soft subset, 24 m–polar fuzzy weighted averaging operator, 241 m–polar hesitant fuzzy concordance dominance matrix, 185 m–polar hesitant fuzzy concordance indices, 183 m–polar hesitant fuzzy concordance set, 183 m–polar hesitant fuzzy discordance dominance matrix, 185 m–polar hesitant fuzzy discordance indices, 185 m–polar hesitant fuzzy discordance set, 184 m−polar hesitant fuzzy set, 160 Multi-person decision making, 87 Multi-polar fuzzy set, 3 Multipolar information, 6 N Normalized Euclidean distance, 35 Normalized Hamming distance, 35 Normalized weights, 261, 288, 333 Normal m−polar fuzzy set, 12 N −soft set, 500 Null m-polar fuzzy N –soft set, 512 Null m−polar fuzzy soft set, 24 O Order relation, 4 OR operation, 26, 487 Outranking at cut-off level, 322 Outranking graphs, 330

539 Outranking relations, 247

P Parameter-free rough accuracy degree, 13 Parameter-free rough degree, 12, 13 Partial concordance index, 290, 334 Pawlak approximation space, 7 Preference function for the Gaussian criteria, 347 Preference relations, 328 PROMETHEE I and II, 345 Pseudo dominance D p , 320

Q Quasi dominance Dq , 320 Quasi-criterion preference function, 346

R Reputation, 327 Restricted intersection, 507 Restricted union, 509 Reverse ranking, 266 Ring sum operation, 65, 517 Rough m−polar fuzzy set, 7

S Score degree, 243, 319 Score function, 168, 346, 441, 478 Shannon’s entropy formula, 287 σ -level cut set, 64 σ -lower level boundary, 9 σ -upper level boundary, 9 (σ, τ )-related accuracy degree, 11 (σ, τ )-related rough degree, 11 Significantly similar, 36 Similarity measure, 36 Soft expert set, 478 Soft m–polar fuzzy approximation space, 72 Soft m–polar fuzzy rough set, 73 Soft set, 22 Soft universe, 22 Steepness measure, 36 Strength at cut-off level, 322 Strongly preferable, 319 Strong outranking relation, 264 Strong σ -level cut set, 64 Sub-dominance Ds , 321 Sub−m-polar fuzzy set, 502 Symbolic translation, 439

540 T Technical ability, 326 Threshold values, 289, 318 Top weak complement, 506 2-tuple linguistic information, 439 2-tuple linguistic m–polar fuzzy Hamacher hybrid averaging operator, 452 2-tuple linguistic m–polar fuzzy Hamacher ordered weighted average operator is a mapping, 449 2-tuple linguistic m–polar fuzzy Hamacher weighted average operator, 443 2-tuple linguistic m–polar fuzzy Hamacher weighted geometric operator, 454 2-tuple linguistic m-polar fuzzy Hamacher hybrid geometric operator, 459 2-tuple linguistic m-polar fuzzy Hamacher ordered weighted geometric operator, 457

Index 2-tuple linguistic m–polar fuzzy set, 440

U Union, 24, 483 Usual criterion preference function, 346

V Veto dominance Dv , 321 VIKOR technique, 344

W Weak complement, 506 Weakly preferable, 319 Weakness at cut-off level, 323 Weak outranking relation, 264 Whole m-polar fuzzy N -soft set, 512