Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets 9819905680, 9789819905683

Table of contents :
Foreword
Preface
Contents
About the Authors
List of Figures
List of Tables
1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations
1.1 Introduction
1.2 Bipolar Fuzzy Sets
1.3 Multi-criteria Decision Making Methods
1.4 Bipolar Fuzzy TOPSIS Method
1.5 Bipolar Fuzzy ELECTRE I Method
1.6 Comparative Analysis
1.7 Bipolar Fuzzy Extended TOPSIS Method
1.8 Conclusion
References
2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers
2.1 Introduction
2.2 Bipolar Fuzzy Numbers
2.3 Bipolar Fuzzy Linguistic Variables
2.4 Ranking of Bipolar Fuzzy Numbers
2.5 (α,β)-Cut of Bipolar Fuzzy Numbers
2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers
2.7 Trapezoidal Bipolar Fuzzy Information System
2.8 Conclusion
References
3 VIKOR Method with Trapezoidal Bipolar Fuzzy Sets
3.1 Introduction
3.2 Trapezoidal Bipolar Fuzzy VIKOR Method
3.3 Comparative Analysis with Trapezoidal Bipolar Fuzzy TOPSIS Method
3.4 Comparative Analysis with Fuzzy VIKOR Method
3.5 Conclusion
References
4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets
4.1 Introduction
4.2 Complex Bipolar Fuzzy Sets
4.3 Structure of Complex Bipolar Fuzzy VIKOR Method
4.4 Application
4.5 Comparative Analysis
4.6 Merits of the Presented Method
4.7 Conclusion
References
5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method
5.1 Introduction
5.2 Bipolar Fuzzy ELECTRE II Method
5.3 Comparative Study and Discussion
5.3.1 Comparison with Bipolar Fuzzy TOPSIS Method
5.3.2 Comparison with Bipolar Fuzzy ELECTRE I Method
5.3.3 Comparison with Fuzzy ELECTRE II Method
5.4 Insights and Limitations of the Proposed Method
References
6 Extended PROMETHEE Method with Bipolar Fuzzy Sets
6.1 Introduction
6.2 Bipolar Fuzzy PROMETHEE Method
6.2.1 Preference Function
6.2.2 Structure of Bipolar Fuzzy PROMETHEE Method
6.3 Comparative Study
6.4 Conclusion
References
7 Enhanced Decision Making Method with Two-Tuple Linguistic Bipolar Fuzzy Sets
7.1 Introduction
7.2 The 2-Tuple Linguistic Bipolar Fuzzy Sets
7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators
7.4 An Approach to MAGDM Problem with 2-Tuple …
7.5 Numerical Example
7.6 Comparative Analysis
7.7 Advantages of the Proposed Strategies
7.8 Conclusions
References
Index

Citation preview

Forum for Interdisciplinary Mathematics Editors-in-Chief Viswanath Ramakrishna, University of Texas, Richardson, USA Zhonghai Ding, University of Nevada, Las Vegas, USA Editorial Board Ashis SenGupta, Indian Statistical Institute, Kolkata, India Balasubramaniam Jayaram, Indian Institute of Technology, Hyderabad, India P. V. Subrahmanyam, Indian Institute of Technology Madras, Chennai, India Ravindra B. Bapat, Indian Statistical Institute, New Delhi, India

The Forum for Interdisciplinary Mathematics is a Scopus-indexed book series. It publishes high-quality textbooks, monographs, contributed volumes and lecture notes in mathematics and interdisciplinary areas where mathematics plays a fundamental role, such as statistics, operations research, computer science, financial mathematics, industrial mathematics, and bio-mathematics. It reflects the increasing demand of researchers working at the interface between mathematics and other scientific disciplines.

Muhammad Akram · Shumaiza · José Carlos Rodríguez Alcantud

Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets

Muhammad Akram Department of Mathematics University of the Punjab Lahore, Pakistan

Shumaiza Department of Mathematics University of the Punjab Lahore, Pakistan

José Carlos Rodríguez Alcantud Faculty of Economics and Business University of Salamanca Salamanca, Spain

ISSN 2364-6748 ISSN 2364-6756 (electronic) Forum for Interdisciplinary Mathematics ISBN 978-981-99-0568-3 ISBN 978-981-99-0569-0 (eBook) https://doi.org/10.1007/978-981-99-0569-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

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

Foreword

The growing necessity of dealing with decisions, and its increasing complexity and inherent uncertainty, demands powerful models of intelligent decision, which are almost exclusively in the form of modeling activities, through assessments that have greatly contributed to the understanding of the complex interactions between different dimensions of the specific application area of decision making. To capture these interactions, the structure of these models has inevitably grown and become significantly complex. But, the usual lack of data availability in different cases and situations implies the necessity of tools to improve the treatment of the existing uncertainty in the problems. Multi-criteria decision making (MCDM) is a well-known branch of decision theory that aims at achieving multiple, and usually conflicting, objectives in decision problems. Usually, these problems are also defined under high uncertain contexts. Therefore, the wide application of MCDM approaches to real-world applications demanding the treatment of uncertainty has led to the development of fuzzy MCDM as a paradigm to deal with uncertainty in MCDM problems. Nowadays, this is a huge field of research with a settled scientific community, that has developed multiple extensions, approaches, and tools applied to a large and growing number of real-world applications. This book is focused on one of the most interesting extensions of fuzzy sets theory such as it is the bipolar fuzzy sets to support the management of non-probabilistic nature of uncertain, incomplete, imprecise, or ambiguous information in MCDM problems. Students and practitioners interested in the field of MCDM under uncertainty will enjoy the introduction, analysis, and comparison of multiple fuzzy bipolar MCDM models introduced in this book and easily understand the importance and usefulness of these models in many real-word decision situations under uncertainty. Due to its interest, organization, and analysis, this book will open many and new possibilities not only to do new research on the topic but also to develop new applications to real-world problems. Muhammad Akram, Shumaiza, and José Carlos Rodríguez

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Alcantud have been able to provide an accessible, but rigorous, introduction to the main existing bipolar fuzzy MCDM methods available in the literature and present new methods too. This book should therefore be useful reading for anyone who wants to learn more about bipolar fuzzy MCDM. Also, for those fuzzy and MCDM researchers who want to learn more about other fuzzy extensions and its application to MCDM methods. Luis Martínez University of Jaén Jaén, Spain

Preface

This monograph contributes to the theoretical and practical development of multicriteria decision making (MCDM). As a subfield of operations research, its goal is to produce optimal results when several conflicting goals and criteria must be considered. The main purpose of MCDM is to build a systematic strategy for the “optimization” of feasible options, and why some alternatives can be declared “optimal”. This is a difficult and controversial task when goals and options are objective and precisely stated. However, this is not always the case. We often have to make decisions in uncertain environments, and this inconvenience leads to more complex situations. We consider models that can flexibly accommodate this lack of certainty, and then explore some valuable strategies for making decisions under multiple criteria. Fuzzy set theory originated in the seminal work of Lotfi Zadeh in 1965. It is considered a way to represent uncertainty and ambiguity in real-world systems. An extension of fuzzy set theory remains one of the most important ways to represent the non-probabilistic nature of uncertain, incomplete, imprecise, or ambiguous information. In 1994, Zhang pioneered the idea of YinYang bipolar fuzzy sets (bipolar fuzzy sets), a bipolar rational method for dealing with two-sided information. In a bipolar fuzzy set, each member is associated with two components, the first of which is in the interval (0, 1] representing membership values for a particular property of the fuzzy set, and the other in the interval [–1, 0) representing the membership value of the counter attribute to the associated fuzzy set. Since the past two decades, many researchers have studied and applied bipolar fuzzy sets in different directions. This book is based on the author’s multiple papers, which have been published in various scientific journals. This book may be useful for researchers, computer scientists, and social scientists. Let’s briefly introduce its contents: In Chaps. 1 and 2, we present certain decision making methods, namely, bipolar fuzzy TOPSIS method, bipolar fuzzy extended TOPSIS method, ELECTRE I method, and TOPSIS method under trapezoidal bipolar fuzzy numbers. In Chap. 3,

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we present a multiple-attribute group decision making method called trapezoidal bipolar fuzzy VIKOR method. For illustration, the proposed technique is applied to two group decision making problems, namely, the selection of waste treatment methods and the site to plant a thermal power station. A comparison of this method with the trapezoidal bipolar fuzzy TOPSIS method is also presented. The theory presented in Chap. 4 incredibly provides a multi-skilled framework for the bipolar fuzzy modeling of inconsistent human interpretations. This remarkable model addresses the bipolar abstruseness of two- dimensional inexact data. Moreover, some fundamental and elementary concepts relevant to the presented theory are introduced. The presented methodology is exclusively designed for the identification of compromise solution possessing maximum group utility and minimum individual regret of the opponent by the quantification of their weighted proximity from ideal solutions. The methodology and structural modeling of the presented complex bipolar fuzzy VIKOR method are demonstrated with the help of an informative flowchart. The potential application for the selection of constructive diagnostic technology is scrutinized to provide computerized imaging modalities in the healthcare centers which demonstrate the accountability of the presented technique. In Chap. 5, we present a multiple-criteria decision making model, namely bipolar fuzzy ELECTRE II method by combining the bipolar fuzzy set with ELECTRE II technique and use to solve the problems having bipolar uncertainty. The implementation of the proposed method is presented by numerical examples. A comparative analysis of the proposed ELECTRE II method is also presented with TOPSIS and ELECTRE I under bipolar fuzzy environment by solving the problem of business location. In Chap. 6, a version of the PROMETHEE method using bipolar fuzzy information, named as, the bipolar fuzzy PROMETHEE method is presented. A numerical example such as the selection of green suppliers by using the bipolar fuzzy PROMETHEE is performed on the basis of the usual criterion preference function, to explain the procedure of the proposed method. The comparable results are derived by using the combination of linear and level preference functions. The results obtained by using different types of preference functions are same that represent the authenticity of the proposed bipolar fuzzy PROMETHEE method. Chapter 7 presents the 2-tuple linguistic bipolar fuzzy sets, a strategy for dealing with uncertainty that incorporates a 2-tuple linguistic term into bipolar fuzzy set. The generalized 2-tuple linguistic bipolar fuzzy Heronian mean operator and generalized 2-tuple linguistic bipolar fuzzy weighted Heronian mean operator are presented. Further, 2-tuple linguistic bipolar fuzzy geometric Heronian mean operator and 2tuple linguistic bipolar weighted geometric Heronian mean operator are discussed along with some of their desirable properties. An approach to multi-attribute group

Preface

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decision making based on the presented aggregation operators under the 2-tuple linguistic bipolar fuzzy framework is studied. Lahore, Pakistan Lahore, Pakistan Salamanca, Spain

Muhammad Akram Shumaiza José Carlos Rodríguez Alcantud

Contents

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Multi-criteria Decision Making Methods . . . . . . . . . . . . . . . . . . . . . . . 1.4 Bipolar Fuzzy TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Bipolar Fuzzy ELECTRE I Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Bipolar Fuzzy Extended TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . 1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 7 16 25 26 31 32

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Bipolar Fuzzy Linguistic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Ranking of Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 (α, β)-Cut of Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Trapezoidal Bipolar Fuzzy Information System . . . . . . . . . . . . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 39 42 44

3 VIKOR Method with Trapezoidal Bipolar Fuzzy Sets . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Trapezoidal Bipolar Fuzzy VIKOR Method . . . . . . . . . . . . . . . . . . . . 3.3 Comparative Analysis with Trapezoidal Bipolar Fuzzy TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 71

51 58 63 64

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3.4 Comparative Analysis with Fuzzy VIKOR Method . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 88 89

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Complex Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Structure of Complex Bipolar Fuzzy VIKOR Method . . . . . . . . . . . . 4.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Merits of the Presented Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 96 98 104 113 118 119 120

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Bipolar Fuzzy ELECTRE II Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Comparative Study and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Comparison with Bipolar Fuzzy TOPSIS Method . . . . . . . . . 5.3.2 Comparison with Bipolar Fuzzy ELECTRE I Method . . . . . 5.3.3 Comparison with Fuzzy ELECTRE II Method . . . . . . . . . . . 5.4 Insights and Limitations of the Proposed Method . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 125 141 142 143 145 147 147

6 Extended PROMETHEE Method with Bipolar Fuzzy Sets . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bipolar Fuzzy PROMETHEE Method . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Preference Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Structure of Bipolar Fuzzy PROMETHEE Method . . . . . . . . 6.3 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 153 154 155 170 172 173

7 Enhanced Decision Making Method with Two-Tuple Linguistic Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The 2-Tuple Linguistic Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 An Approach to MAGDM Problem with 2-Tuple Linguistic Bipolar Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 180 184 193 194

Contents

7.6 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Advantages of the Proposed Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

About the Authors

Muhammad Akram is Professor at the Department of Mathematics, University of the Punjab, Lahore, Pakistan. He earned his Ph.D. in fuzzy mathematics from the Government College University, Lahore, Pakistan. His research interests include fuzzy numerical methods, fuzzy graphs, fuzzy algebras, and fuzzy decision support systems. He has published 10 monographs and 485 research articles in international peer-reviewed journals. According to reports from Stanford University, Dr. Akram is ranked in the top 2% of scientists for years 2020, 2021 and 2022 in the world in artificial intelligence and image processing. He has been on the editorial of 15 international academic journals and reviewer/referee for 155 international journals, including Mathematical Reviews (USA) and Zentralblatt MATH (Germany). Under his supervision, 17 Ph.D. students have completed their research and he is currently guiding five more towards the same. Shumaiza is a research scholar at the University of Punjab, Lahore, Pakistan, from where she also has earned her M.Phil. degree in Mathematics. She has won the HEC indigenous scholarship for her Ph.D. program and has published 15 research articles in renowned international peer-reviewed journals. Her area of research is fuzzy systems. José Carlos Rodríguez Alcantud is Full Professor at the Department of Economics and Economic History, University of Salamanca, Spain, since 2010, and has been heading the BORDA Research Unit since 2015. He received his Ph.D. in Mathematics in 1996 from the University of Santiago de Compostela, Spain and M.Sc. in Mathematics in 1991 from the University of Valencia, Spain. His research interests include social choice theory, mathematical economics, and fuzzy theory and decision support systems. He has published over 200 papers on these topics and has supervised 3 Ph.D. students on their research. According to reports from Stanford University, Dr. Alcantud is ranked in the top 2% of scientists for years 2020, 2021 and 2022.

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List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 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. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5

Hierarchical structure of multi-criteria decision making methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hierarchical system for multi-criteria decision making . . . . . . . . The steps of bipolar fuzzy TOPSIS method . . . . . . . . . . . . . . . . . The flowchart of bipolar fuzzy ELECTRE I method . . . . . . . . . . Bipolar fuzzy decision graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation of diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation for P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation for P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation for P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of bipolar fuzzy number . . . . . . . . . . . . Graphical representation of trapezoidal bipolar fuzzy number (v., Definition 2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of triangular bipolar fuzzy number (v., Definition 2.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy number representing voltage supply in Example 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Membership functions for linguistic values in Example 2.3 . . . . Trapezoidal bipolar fuzzy numbers in Example 2.4 . . . . . . . . . . . An (α, β)-cut of a bipolar fuzzy number . . . . . . . . . . . . . . . . . . . . Using Cantor’s diagonal sweeping to list the rational numbers contained in [0, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using diagonal sweeping to list the rational numbers contained in [−1, 0] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using diagonal sweeping to list the elements in Sα . . . . . . . . . . . Satisfaction and dissatisfaction degree for linguistic values . . . . . The VIKOR position: ideal and compromise solutions . . . . . . . . Linguistic variables for cost criteria . . . . . . . . . . . . . . . . . . . . . . . . Linguistic variables for benefit criteria . . . . . . . . . . . . . . . . . . . . . The framework for selecting the waste treatment strategy . . . . . . The framework to select the site for thermal power station . . . . .

5 6 10 19 21 23 25 26 26 38 38 39 40 42 44 45 48 49 49 53 69 75 75 76 80 xix

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Fig. 3.6 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 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. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6

List of Figures

The flowchart of trapezoidal bipolar fuzzy TOPSIS method . . . . Ideal and compromise solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of complex bipolar fuzzy VIKOR method . . . . . . . . . . Specification of inspected problem . . . . . . . . . . . . . . . . . . . . . . . . Comparative study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of outranking relations between alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of outranking relations between alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The steps of the bipolar fuzzy TOPSIS model . . . . . . . . . . . . . . . The steps of the bipolar fuzzy ELECTRE I method . . . . . . . . . . . Graph representing the outranking relation of alternatives . . . . . . Linguistic variables for criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . Outgoing flow of Sα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incoming flow of Sα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Framework of bipolar fuzzy PROMETHEE method . . . . . . . . . . Partial PROMETHEE I relations . . . . . . . . . . . . . . . . . . . . . . . . . . PROMETHEE diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial PROMETHEE I relations . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the net flows of different preference functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of developed MAGDM approach . . . . . . . . . . . . . . . . . Influence of attribute weight w on alternative ranking (s = 2, t = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative’s score  j ( j = 1, 2, 3, 4, 5) with G2TLBFWHM operator when s, t ∈ [0, 10] . . . . . . . . . . . . Influence of parameters on alternatives ranking . . . . . . . . . . . . . . Alternative’s score  j ( j = 1, 2, 3, 4, 5) with 2TLBFWGHM operator when s, t ∈ [0, 10] . . . . . . . . . . . . Influence of parameters on alternative ranking . . . . . . . . . . . . . . .

85 94 105 108 118 136 141 142 144 147 156 160 160 162 168 169 172 173 195 200 203 204 205 206

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 Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table 2.6

Profit and loss of products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k-matrix format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ratings of the alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Bipolar fuzzy concordance sets . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy discordance sets . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy concordance sets . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy discordance sets . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy ELECTRE I results for detection of skin diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy ELECTRE I results for medical diagnosis of P1 ,P2 and P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decision criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Bipolar fuzzy positive and negative ideal solutions . . . . . . . . . . Distance measures and relative closeness degree . . . . . . . . . . . . Bipolar fuzzy variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy linguistic variable . . . . . . . . . . . . . . . . . . . . . . . . . The k-matrix format of the input of our problem . . . . . . . . . . . . The linguistic ratings and aggregated value of alternatives for C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The linguistic ratings and aggregated value of alternatives for C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The linguistic ratings and aggregated value of alternatives for C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 7 11 11 12 12 14 14 20 20 22 22 23 24 29 30 30 30 31 31 40 41 52 54 54 54 xxi

xxii

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 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

List of Tables

The linguistic ratings and aggregated value of alternatives for C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The linguistic ratings and aggregated value of alternatives for C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importance weights of each criteria and aggregate weights . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Closeness coefficients of alternatives . . . . . . . . . . . . . . . . . . . . . Comparison of the proposed method with other MCDM methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trapezoidal bipolar fuzzy information system to evaluate alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J1 for k = 1, (α, β) = (1, 0) and J2 for k = 2, (α, β) = (0, −1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted fuzzy preeminence relation . . . . . . . . . . . . . . . . . . . . . Total preeminence degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rating of alternatives in terms of linguistic values . . . . . . . . . . . Trapezoidal bipolar fuzzy information system to evaluate alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J1 f or (α, β) = (1, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy preeminence relation . . . . . . . . . . . . . . Total preeminence degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution of different authors towards VIKOR method and bipolar information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic variables and values for cost type criteria . . . . . . . . . Linguistic variables and values for benefit type criteria . . . . . . . Performance ratings by decision makers (linguistic terms) . . . . Performance ratings by decision makers (bipolar fuzzy numbers) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decision matrix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projection values of criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy value, divergence and weights of criteria . . . . . . . . . . . Values f β∗ and f β− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of Sα , Rα and Qα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ranking of alternatives by Sα , Rα and Qα . . . . . . . . . . . . . . Linguistic variables and values for cost type criteria . . . . . . . . . Linguistic variables and values for benefit type criteria . . . . . . . Performance ratings by decision makers (linguistic terms) . . . . Performance ratings by decision makers (bipolar fuzzy numbers) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55 56 56 57 58 60 61 61 61 62 62 62 63 63 70 76 76 77 78 78 79 79 79 79 80 80 81 81 81 82 82

List of Tables

Table 3.18 Table 3.19 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 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 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 5.8

xxiii

Decision matrix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projection values of criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy value, divergence and weights of criteria . . . . . . . . . . . Values f β∗ and f β− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of Sα , Rα and Qα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ranking of alternatives by Sα , Rα and Qα . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Closeness coefficient to BFPIS . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic terms for rating of diagnostic technologies . . . . . . . . Linguistic assessment of diagnostic technologies . . . . . . . . . . . Complex bipolar fuzzy decision matrix of Y1 . . . . . . . . . . . . . . . Complex bipolar fuzzy decision matrix of Y2 . . . . . . . . . . . . . . . Complex bipolar fuzzy decision matrix of Y3 . . . . . . . . . . . . . . . Aggregated complex bipolar fuzzy decision matrix . . . . . . . . . . Specified linguistic variables for importance of criteria . . . . . . . Prominence of conflicting criteria . . . . . . . . . . . . . . . . . . . . . . . . Weightage of criteria relative to their significance . . . . . . . . . . . Normalized weights of decisive criteria . . . . . . . . . . . . . . . . . . . Score matrix M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Best and worst values relative to decision-criteria . . . . . . . . . . . ˜ ˜ ∗s from D Distance of D s ................................ Normalized Euclidean distance measures . . . . . . . . . . . . . . . . . . Values of S,R and Q for diagnostic technologies . . . . . . . . . . . Hierarchical ranking of diagnostic technologies . . . . . . . . . . . . Bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized weightage of criteria . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy weighted decision matrix . . . . . . . . . . . . . . . . . . . Ideal solutions relative to decision-criteria . . . . . . . . . . . . . . . . . Euclidean distance measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative closeness degree of each technology . . . . . . . . . . . . . . Ranking of diagnostic technologies . . . . . . . . . . . . . . . . . . . . . . . Comparative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy weighted distances . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ranking results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy weighted distances . . . . . . . . . . . . . . . . . . . . . . . . Outranking relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ranking results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 83 84 84 84 86 86 87 87 108 109 109 110 110 110 111 111 111 111 112 112 112 112 113 113 114 114 115 115 116 116 117 117 131 134 135 136 136 139 140 141

xxiv

Table 5.9 Table 5.10 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 7.1 Table 7.2 Table 7.3 Table 7.4 Table 7.5 Table 7.6 Table 7.7 Table 7.8 Table 7.9 Table 7.10 Table 7.11

List of Tables

Weighted bipolar fuzzy decision matrix . . . . . . . . . . . . . . . . . . . Bipolar fuzzy ELECTRE I results for selection of business location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic variables and values of criteria . . . . . . . . . . . . . . . . . . Performance ratings by decision makers (linguistic terms) . . . . Performance ratings by decision makers (bipolar fuzzy numbers) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviation of alternatives with respect to criteria . . . . . . . . . . . . Usual criterion preference function . . . . . . . . . . . . . . . . . . . . . . . Projection values of criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy value, divergence, and weights of criteria . . . . . . . . . . . Multi-criteria preference index . . . . . . . . . . . . . . . . . . . . . . . . . . Outgoing and incoming flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net flow of suppliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preference functions corresponding to criteria . . . . . . . . . . . . . . Linear and level criteria preference function . . . . . . . . . . . . . . . Multi-criteria preference index . . . . . . . . . . . . . . . . . . . . . . . . . . Outgoing and incoming flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net flow of suppliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final ranking of suppliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-tuple linguistic bipolar fuzzy decision matrix F 1 provided by first expert D1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-tuple linguistic bipolar fuzzy decision matrix F 2 provided by second expert D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-tuple linguistic bipolar fuzzy decision matrix F 3 provided by third expert D3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregated 2-tuple linguistic bipolar fuzzy decision matrix by operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fused assessment values by G2TLBFWHM and 2TLBFWGHM operators . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of attribute weight w on alternative ranking utilizing G2TLBFWHM operator . . . . . . . . . . . . . . . . . . . . . . . . Influence of attribute weight w on alternative ranking utilizing 2TLBFWGHM operator . . . . . . . . . . . . . . . . . . . . . . . . Score values by varying s and t based on the G2TLBFWHM operator . . . . . . . . . . . . . . . . . . . . . . . . . . Score values by varying s and t based on the 2TLBFWGHM operator . . . . . . . . . . . . . . . . . . . . . . . . . . Ranking by varying s and t based on G2TLBFWHM and 2TLBFWGHM operators . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 146 163 164 165 166 166 167 167 167 168 168 169 170 171 171 171 172 173 181 196 196 197 198 199 199 199 201 202 202

List of Tables

Table 7.12 Table 7.13 Table 7.14 Table 7.15 Table 7.16 Table 7.17

xxv

The outcomes utilizing 2TLBFWA operator . . . . . . . . . . . . . . . The outcomes utilizing 2TLBFWG operator . . . . . . . . . . . . . . . The outcomes utilizing 2TLBFWHM operator . . . . . . . . . . . . . The outcomes utilizing 2TLBFWDHM operator . . . . . . . . . . . . The outcomes utilizing 2TLBFWMSssM operator . . . . . . . . . . The outcomes utilizing 2TLBFWDMSM operator . . . . . . . . . .

208 208 208 208 209 209

Chapter 1

TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

In this chapter, we present various methods for solving multi-criteria decision making problems involving bipolar measurements with positive and negative values. They are attractive because we focus on two fundamental methods in the broad field of multi-criteria decision making, namely, Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) and Elimination and Choice Translating Reality (ELECTRE). In particular, we focus on their valuable implications for medical diagnosis of patients, when data are given in a bipolar fuzzy or bipolar single-valued neutrosophic format. We also study a bipolar fuzzy extended TOPSIS method based on entropy weights. This variation incorporates the capability of bipolar information into the TOPSIS principle in order to address the interactions between criteria, and also measure aggregate values on a bipolar scale. In practical problems, this method can be used to quantify the effects and side effects of medical treatments. Moreover, we conduct a comparative analysis of the TOPSIS and ELECTRE I methods based on bipolar fuzzy information proposed in this chapter. This chapter is based on [12, 17, 56, 67, 68].

1.1 Introduction Decision making is basically the procedure for the specification of the best option(s) among a set of feasible alternatives. A particular instance is multi-criteria decision making, which is concerned with structuring and formulating optimization problems that take account of multiple criteria. Numerous multi-criteria decision making models have been created and put into use in a variety of sectors, including engineering, economics, management, business, and information technology [66]. 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 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Akram et al., Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-99-0569-0_1

1

2

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

determine the conclusion. The advancement of computer technology has complemented the development of decision making approaches, that straightforwardly deal with complicated and large sources of information or data. This chapter focuses on the implementation of two acclaimed approaches when the information available is in a bipolar fuzzy form. First, TOPSIS is a well-known multi-criteria decision making technique that was introduced by Hwang and Yoon [36]. It evaluates the performance of the alternatives with the aid of optimal solutions, known as positive and negative ideal solutions. The positive solutions, being the best values, and the negative solution, being the worst value, contribute to the identification of favorable alternative. In accordance to main principles, the nominated object should therefore be the furthest away from the negative ideal solution as well as the closest from the positive ideal solution. TOPSIS is arguably the frequently adapted 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 defines appropriate and competent procedures which combines mathematical efficiency and a computational ease of the problem-solving mechanism in a systematic 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, Bellman and Zadeh [21] were first to exploit the decision making aptitude of several techniques in fuzzy environment. To deal with uncertainty and vagueness in a multi-criteria scenario, Chen [24] utilized an extended TOPSIS method to opt for the most capable supplier using fuzzy information. This overcame the limitation that in classical TOPSIS methods, the aggregate values and weights are precisely determined. Many researchers developed several TOPSIS methods in related lines, including fuzzy TOPSIS methods [31, 43, 48, 53], intuitionistic fuzzy TOPSIS methods [18, 23, 35, 39, 41, 42, 56], bipolar fuzzy TOPSIS methods [12, 13, 17], interval-valued fuzzy TOPSIS methods [20, 25, 30, 52], and a Pythagorean fuzzy TOPSIS method [10]. As an alternative approach to multi-criteria decision making, the ELECTRE method owes its popularity due to the utilization of outranking relations. The significant approach of ELECTRE method was put forward by Benayoun et al. [22] in 1966. This original variant was renamed as ELECTRE I method after the development of different versions of ELECTRE methods [50] beyond the original formulation, inclusive of ELECTRE II, ELECTRE III, ELECTRE IV, and ELECTRE-TRI. These methods are used as the preference models for outranking relation on the set of actions, obtained via processing the concordance and discordance sets after examining the alternatives in sense of superiority and inferiority. In other settings, several contributions are worth mentioning. Roy [51] also contributed to set up the foundations of ELECTRE methods. To address uncertain, imprecise, and linguistic information, Hatami-Marbini and Tavana [34] proposed an excellent extension of ELECTRE I method by using the outranking principles for group decision making scenarios. Aytac et al. [19] used a fuzzy ELECTRE I method for evaluating the cater-

1.2 Bipolar Fuzzy Sets

3

ing firm alternatives. Wu and Chen [62] developed intuitionistic fuzzy ELECTRE method and Chen et al. [26] proposed an ELECTRE I method based on hesitant fuzzy sets. And recently, Akram et al. [11] have presented an ELECTRE I approach with hesitant Pythagorean fuzzy information.

1.2 Bipolar Fuzzy Sets Zhang [67, 68] is accredited to develop the theory of bipolar fuzzy sets (YinYang bipolar fuzzy sets) to model the double-sided information using the space [−1, 0] × [0, 1]. For some applications of bipolar fuzzy sets, the readers are suggested to [1–9, 37, 38, 46, 47, 49, 54, 55, 57–61, 63, 64]. Definition 1.1 ([68]) A bipolar fuzzy set A on a non-empty set X can be represented p p by an object of the form A = {(x, μ A (x), μnA (x)) | x ∈ X }, where μ A : X → [0, 1] n and μ A : X → [−1, 0] is a pair of functions. p

The positive membership value μ A (x) ∈ (0, 1] represents the satisfaction value of an element x to a certain property corresponding to bipolar fuzzy set A, and the negative membership value μnA (x) ∈ [−1, 0) represents the satisfaction value of an element x to some counter property of bipolar fuzzy set A. The membership value (0, 0) represents that the selected attribute is irrelevant to the object. The hesitancy value cannot be computed in a bipolar fuzzy set since positive membership value(truth value) and negative membership value (falsity value) have independent values. Example 1.1 Suppose a company is manufacturing different products that can be grouped together, such as X = {P1 , P2 , P3 , P4 , P5 , P6 }. These products can be classified in accordance to their profit and loss and this volatility for each product will change with time. Table 1.1 states the profit and loss probabilities of all products. Table 1.1 shows that the product P1 gains 60% profit and 40% loss on the average. The profit and loss of the products are represented by positive and negative behaviors of the products, that is, two-sided behavior. The corresponding bipolar fuzzy set is given as A = {(P1 , 0.6, −0.4), (P2 , 0.8, −0.5), (P3 , 0.9, −0.1), (P4 , 0.7, −0.2), (P5 , 0.5, −0.6), (P6 , 0.6, −0.4)}. Table 1.1 Profit and loss of products Product Profit P1 P2 P3 P4 P5 P6

0.6 0.8 0.9 0.7 0.5 0.6

Loss −0.4 −0.5 −0.1 −0.2 −0.6 −0.4

4

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

A good example of a bipolar fuzzy concept is political acceptance (0, 1] or rejection [−1, 0). Remark 1.1 An intuitionistic fuzzy set theory is a generalization of fuzzy set theory introduced by Atanassov [18] in 1983. The concept of bipolar fuzzy set is helpful when there are some ambiguities in assigning membership values as depicted in the following proficient examples. 1. Bipolar fuzzy set evaluation is actually appropriate when we distinguish irrelevant elements from opposite ones. For example, when we want to show the effects and side effects of a drug, we can use a bipolar fuzzy representation. The side effects of the drug are represented by the negative part of the bipolar fuzzy set. Since the degree of non-membership of an element does not correspond to the negative effect of the element, the negative effect of an element cannot be represented by an intuitionistic fuzzy set. 2. The bipolar fuzzy sets possess the most suitable structure to model the profit and loss of a product. Since loss is an implicit counter property of profit and is seen as a negative impact of the product. In an intuitionistic fuzzy set, this situation can be seen as “profitable” and “unprofitable”. Obviously, the “no-profit” property is not equivalent to “loss”, that is, intuitionistic fuzzy sets cannot be used to model counter property (losses). Therefore, bipolar fuzzy sets are not equivalent to intuitionistic fuzzy sets.

1.3 Multi-criteria Decision Making Methods Many decision making scenarios can be split into different groups based on particular characteristics, informational sources, and representations of preferences. 1. Single criteria decision making deals with all those situations where alternatives are examined in accordance to a single reference (criterion) to define the decision problem. The solution of the problem is exclusively and directly derived from the initially available information in such cases. 2. In decision making processes, an expert compares the considered alternatives xi (i = 1, 2, . . . , n) to figure out the preference relations that are subsequently optimized [15]. Sometimes utilities can be associated with preferences, and if these have a poor structure, other types of tools may be used (e.g., weak utilities [14] or multi-utilities [16]). 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) by governing the performances of all the feasible alternatives under certain characteristics. This type of decision making based on the judgements of multiple experts is called group decision making or multi-person decision making.

1.3 Multi-criteria Decision Making Methods

5

4. In reality, the complexity and ill-structures of the real-world decision making problems cannot be covered using the information regarding a single attribute or viewpoint to approach the optimum decision. In fact, oversimplification of the original problem for adjustment in such unidimensional structure can lead to unreliable and inappropriate decisions. An obvious and comprehensive solution to this problem is the simultaneous consideration of all the key factors that are relevant for the problem. Multi-criteria decision making appears as a discipline in its own right, that makes the decisions by selecting the most efficient one from several competent alternatives, subjected to decision criteria or attributes that may be either concrete or vaguely defined. Multi-criteria decision making competently finds the optimal alternative in domains where the identification of a best alternative is quite complex. This section reviews the main streams of investigation process in multi-criteria decision making theory and its implementation thoroughly. Multi-criteria decision making methods have been used in different applications. The frequently practiced multi-criteria decision making methods are described in Fig. 1.1. Dubois and Prade [32] summarized the procedures of multi-criteria decision making 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. 1.2. 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.

MCDM Methods

AHP

ELECTRE I

ELECTRE

ELECTRE II

TOPSIS

ELECTRE III

ELECTRE IV

VIKOR

PROMETHEE

PROMETHEE I

Fig. 1.1 Hierarchical structure of multi-criteria decision making methods

PROMETHEE II

6

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations Overall objective

Goal

Aspect

Dimension 1

Criteria

C11

Alternatives

A1

···

Dimension j

···

C1r

···

Cj1

···

Cjs

Ai

Dimension k

···

Ck1

···

···

Ckt

An

Fig. 1.2 Hierarchical system for multi-criteria decision making

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 [40] 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 Multi-criteria Decision Making Problem: The multi-criteria decision making problems fall into two classes. One class refers to those types of problems where the ratings and criteria weights are known exactly and given by crisp numbers. The other class includes all those problems where the initial ratings and criteria weights are evaluated on subjective judgment using inexact and imprecise information which are usually interpreted via linguistic terms, interval numbers, fuzzy numbers, or intuitive fuzzy numbers. The initial step for the evaluation of each multi-criteria problem (individual or group decision) is the construction of a decision making matrix (or matrices). In such matrices, the performance values of the alternatives regarding different criteria can be expressed by real, intervals numbers, fuzzy numbers, or qualitative labels. Let D = {1, 2, . . . , k} be the set of decision makers or experts. The multi-criteria problem can be expressed in k-matrix format as given in Table 1.2. where

1.4 Bipolar Fuzzy TOPSIS Method Table 1.2 k-matrix format C1 A1 A2 .. . Am

k x11 k x21

.. . k xm1

7

C2

...

Cn

k x12 k x22

... ... .. .

k x1n k x2n .. . k xmn

.. . k xm2

...

• A1 , A2 , . . . , Am are the available alternatives from which the decision makers have to select the best one. • C1 , C2 , . . . , Cn are criteria having influence on the performance of alternatives. • xikj is the judgement of k-decision maker about alternative Ai with respect to the criterion C j ( xikj may be numerical, interval or fuzzy number). In this way for m alternatives and n criteria, decision matrix X k = (xikj ) is constructed by arranging all the performance values xikj , representing the performance 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 and expert, given by the set of weights, must satisfy the condition of normality, i.e., their sum should be one. Let W k = [w1k , w2k , . . . , wnk ] be the weight vector assigned to criteria by k decision maker, where w kj ∈ W k corresponds to the importance of criterion C j relative to k decision maker and w1k + w2k + · · · + 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.

1.4 Bipolar Fuzzy TOPSIS Method The methodology of TOPSIS method, proposed by Hwang and Yoon [36], originated from the idea of compromise solution to figure out the most favorable alternative by considering the distance from ideal solutions. The object closest to the positive ideal solution (optimal solution) and furthest from the negative ideal solution (inferior solution) is considered to be most preferable. Alghamdi et al. [17] first proposed bipolar fuzzy TOPSIS method. Basically the foundation of bipolar fuzzy TOPSIS method is that the selected alternative should have “shortest distance” from the positive ideal solution and “farthest distance” from negative ideal solution. Consider a set of alternatives, such as X = {x1 , x2 , . . . , xr }

8

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

which are assessed by the decision makers with respect to a set of criteria, such as C = {c1 , c2 , . . . , ck }. The stepwise strategy of bipolar fuzzy TOPSIS method is elaborated as follows: 1. The considering alternatives are assessed on the basis of k criteria. The resulting values of alternatives with respect to each criterion form a bipolar fuzzy decision matrix as ⎡ ⎤ x11 x12 · · · x1k ⎢ x21 x22 · · · x2k ⎥ ⎢ ⎥ D = [xi j ]r ×k = ⎢ . . .. ⎥ . . . ⎣ . . ··· . ⎦ xr 1 xr 2 · · · xr k For each entry xi j = (μi j , νi j ), μi j ∈ [0, 1] illustrates satisfaction degree (or benefit) of alternative xi under criterion c j and νi j ∈ [−1, 0] defines the dissatisfaction degree (or cost) of alternative xi under criterion c j . If μi j = 1, then the alternative xi represents the highest degree of performance in terms of satisfaction regarding criterion c j and in the case of νi j = −1, the alternative xi exhibits the maximum performance in opposite sense. 2. The decision makers assign the weight values to all criteria that accomplish the condition of normality, i.e., for the weight vector W = (w1 w2 . . . wk ), k  w j = 1. j=1

3. A weighted bipolar fuzzy decision matrix is obtained via the multiplication of the decision matrix to weight vector as ⎡

D ⊗ W = [qi j ]r ×k

q11 q12 ⎢ q21 q22 ⎢ =⎢ . . ⎣ .. .. qr 1 qr 2

⎤ · · · q1k · · · q2k ⎥ ⎥ . ⎥, · · · .. ⎦ · · · qr k

where each entry qi j = (m i j , n i j ) is calculated as m i j = w j μi j and n i j = w j νi j , i = 1, 2, . . . , r , j = 1, 2, . . . , k. 4. The bipolar fuzzy positive ideal solution (BFPIS) and bipolar fuzzy negative ideal solution (BFNIS) are computed as B F P I S = {(μ1 + , ν1 + ), (μ2 + , ν2 + ), . . . , (μk + , νk + )}, B F N I S = {(μ1 − , ν1 − ), (μ2 − , ν2 − ), . . . , (μk − , νk − )}, where

μ j + = max{m i j }, i

min{n i j } j = 1, 2, . . . , k. i

ν j + = max{n i j } i

and

μ j − = min{m i j }, i

(1.1) (1.2) νj− =

1.4 Bipolar Fuzzy TOPSIS Method

9

5. The Euclidean distance of alternative xi , i = 1, 2, . . . , r, from ideal solutions is calculated by using Formulae (1.3) and (1.4).

k

1  d(xi , B F P I S) =  ((m i j − μ j + )2 + (n i j − ν j + )2 ), 2 j=1

k

1  ((m i j − μ j − )2 + (n i j − ν j − )2 ). d(xi , B F N I S) =  2 j=1

(1.3)

(1.4)

6. The relative closeness degree of respective alternative to bipolar fuzzy positive ideal solution is represented as ρi and calculated by using Formula (1.5). The alternative having the maximum value of relative closeness degree is chosen as the most favorable alternative. ρi =

d(xi , B F N I S) , d(xi , B F P I S) + d(xi , B F N I S)

i = 1, 2, . . . , r.

(1.5)

According to the discussion, the procedure of bipolar fuzzy TOPSIS method is compressed in Algorithm 1.4.1. The geometrical representation of bipolar fuzzy TOPSIS method is shown in Fig. 1.3. Algorithm 1.4.1 Bipolar fuzzy TOPSIS method 1. Input the set of r alternatives X = {x1 , x2 , . . . , xr }, and k criteria C = {c1 , c2 , . . . , ck }. 2. Construct the bipolar fuzzy decision matrix D = [di j ]r ×k = [(μi j , νi j )]r ×k , where di j is the degree of membership of each alternative xi corresponding to criteria cj. k 3. Input the weight vector [w1 w2 . . . wk ]T such that w j = 1. j=1

4. Construct the weighted bipolar fuzzy decision matrix D ⊗ W = [(m i j , n i j )]r ×k = [(w j μi j , w j νi j )]r ×k . 5. Compute the bipolar fuzzy positive and negative ideal solutions by using Eqs. (1.1) and (1.2), respectively. 6. Evaluate the Euclidean distance of respective alternative from the ideal solutions using Formulae (1.3) and (1.4), respectively. 7. Compute the relative closeness degree of alternative xi , 1 ≤ i ≤ r, by Eq. (1.5). 8. Rank the available choice according to the values of the relative closeness degrees ρi s in descending manner. The alternative, having highest value of ρi , appears at the most prior position. Example 1.2 (Selection of Smartphone) Suppose a person wants to choose one smartphone among five smartphones, which are advertised for sale in different electronics stores. In order to buy his favorite smartphone, he focused on the following

10

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Fig. 1.3 The steps of bipolar fuzzy TOPSIS method

Identify the alternatives and criteria

Construct the bipolar fuzzy decision matrix

Evaluate criteria weights by decision maker

Construct weighted bipolar fuzzy decision matrix

Compute BFPIS and BFNIS

Determine the distance of each alternative from BFPIS and BFNIS

Compute the relative closeness degree of each alternative to BFPIS

Rank the alternatives with respect to closeness degree

features: Color, Memory, Elegancy, and Camera zoom. As we all know, every feature means the advantage and disadvantage of designing a smartphone. Therefore, these smartphones were chosen as replacements, and the features specified as criteria for this multi-criteria decision making issue. Let us specify the considering smartphones by x1 , x2 , x3 , x4 , x5 and criteria by c1 , c2 , c3 , c4 . 1. The preference value of each smartphone xi on the basis of criterion c j is represented in Table 1.3, which is a bipolar fuzzy decision matrix. Each positive membership value μi j illustrates the percentage of satisfaction degree and νi j shows the percentage of dissatisfaction of using smartphone xi with respect to each criterion c j . 2. If W = [0.25 0.3 0.25 0.2] is the weight vector given by decision maker then weighted bipolar fuzzy decision matrix is given in Table 1.4.

1.4 Bipolar Fuzzy TOPSIS Method

11

Table 1.3 Ratings of the alternatives D c1 x1 x2 x3 x4 x5

(0.5, −0.25) (0.2, −0.8) (0.33, −0.25) (0.65, −0.6) (1, −0.5)

c2

c3

c4

(0.8, −0.7) (0.9, −0.4) (0.75, −0.4) (0.3, −0.75) (0.4, −0.35)

(0.3, −0.1) (0.6, −0.3) (0.25, −0.7) (0.8, −0.35) (0.2, −0.6)

(0.6, −0.6) (0.55, −0.5) (0.3, −0.1) (0.65, −0.7) (0.25, −0.65)

Table 1.4 Weighted bipolar fuzzy decision matrix D⊗W c1 c2 x1 x2 x3 x4 x5

(0.125, −0.0625) (0.05, −0.2) (0.0825, −0.0625) (0.1625, −0.15) (0.25, −0.125)

(0.24, −0.21) (0.27, −0.12) (0.225, −0.12) (0.09, −0.225) (0.12, −0.105)

c3

c4

(0.075, −0.25) (0.15, −0.075) (0.0625, −0.175) (0.2, −0.0875) (0.05, −0.15)

(0.12, −0.12) (0.11, −0.1) (0.06, −0.02) (0.13, −0.14) (0.05, −0.13)

3. The bipolar fuzzy positive and negative ideal solutions are computed as follows: BFPIS = [(0.1625, −0.0625), (0.27, −0.105), (0.2, −0.075), (0.13, −0.02)], BFNIS = [(0.05, −0.2), (0.09, −0.225), (0.05, −0.25), (0.05, −0.14)]. 4. The Euclidean distance of all alternatives from ideal solutions are computed using Formulae (1.3) and (1.4) as follows: d(x1 , B F P I S) = 0.187,

d(x1 , B F N I S) = 0.163,

d(x2 , B F P I S) = 0.143, d(x3 , B F P I S) = 0.20,

d(x2 , B F N I S) = 0.344, d(x3 , B F N I S) = 0.186,

d(x4 , B F P I S) = 0.186, d(x5 , B F P I S) = 0.201,

d(x4 , B F N I S) = 0.188, d(x4 , B F N I S) = 0.188.

5. Using Formula (1.5), the relative closeness degrees of alternatives are computed as ρ1 = 0.466, ρ2 = 0.706, ρ3 = 0.482, ρ4 = 0.503, ρ5 = 0.483. Rank the alternatives by considering the descending order of the relative closeness degrees. x2  x4  x5  x3  x1 . Hence x2 is the best choice.

12

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Example 1.3 (Detection of Skin Disease) Suppose that a patient has a skin allergy and is recommended to a dermatologist for the proper examination of the disease. He has the following symptoms: c1 = Pain or itching, c2 = Red and inflamed skin, c3 = Discolored patches, c4 = Pus-filled pimples or blisters, c5 = Swelling, c6 = Oily and waxy patches. These symptoms are taken as the criteria of considered multi-criteria decision making problem. After reading the history of patient and formulating the physical biopsy, the dermatologist arrived at the result that patient may have one of the following diseases: x1 = Acne, x2 = Eczema, x3 = Psoriasis, x4 = Seborrheic Dermatitis. These diseases are considered as the alternatives for this multiple-criteria decision making problem. Now the following steps are performed in order to make the exact diagnosis of skin disease. 1. The bipolar fuzzy decision matrix is established in Table 1.5, whose each entry xi j shows the degree of suffering and safety of the patient for the existence of disease xi , i = 1, . . . , 4 from symptom c j , j = 1, . . . , 6. 2. The weights W = [0.3 0.3 0.1 0.15 0.05 0.1] are assigned to the symptoms which represent the effect of each symptom c j to human body. 3. The weighted bipolar fuzzy decision matrix is given in Table 1.6. 4. The bipolar fuzzy positive and negative ideal solutions are computed as follows: B F P I S = [(0.24, −0.09), (0.21, −0.03), (0.07, −0.03), (0.09, −0.045), (0.035, −0.02), (0.08, −0.01)], B F N I S = [(0.12, −0.18), (0.06, −0.27), (0.01, −0.09), (0.015, −0.135), (0.01, −0.04), (0.03, −0.06)].

5. The Euclidean distance of each disease from ideal solutions is calculated as

Table 1.5 Bipolar fuzzy decision matrix D c1 c2 c3 (0.4, −0.6) (0.6, −0.4) (0.8, −0.3) (0.4, −0.4)

x1 x2 x3 x4

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

(0.1, −0.9) (0.2, −0.6) (0.7, −0.3) (0.4, −0.6)

c4

c5

c6

(0.6, −0.3) (0.5, −0.5) (0.1, −0.7) (0.1, −0.9)

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

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

Table 1.6 Weighted bipolar fuzzy decision matrix D⊗W

c1

c2

c3

c4

c5

c6

x1

(0.12, −0.18)

(0.06, −0.27)

(0.01, −0.09)

(0.09, −0.045)

(0.035, −0.02)

(0.03, −0.06)

x2

(0.18, −0.12)

(0.21, −0.03)

(0.02, −0.06)

(0.075, −0.075) (0.015, −0.03)

(0.05, −0.06)

x3

(0.24, −0.09)

(0.18, −0.03)

(0.07, −0.03)

(0.015, −0.105) (0.01, −0.035)

(0.04, −0.05)

x4

(0.12, −0.12)

(0.15, −0.18)

(0.04, −0.06)

(0.015, −0.135) (0.015, −0.04)

(0.08, −0.01)

1.4 Bipolar Fuzzy TOPSIS Method

13

d(x1 , B F P I S) = 0.23958,

d(x1 , B F N I S) = 0.08591,

d(x2 , B F P I S) = 0.08037, d(x3 , B F P I S) = 0.08420, d(x4 , B F P I S) = 0.16988,

d(x2 , B F N I S) = 0.21911, d(x3 , B F N I S) = 0.22674, d(x4 , B F N I S) = 0.11537.

6. The relative closeness degree (similarity) of each disease to positive ideal solution is given as ρ(x1 ) = 0.26394, ρ(x2 ) = 0.73163, ρ(x3 ) = 0.72921, ρ(x4 ) = 0.40445. The diseases are ranked by considering the descending order of relative closeness degrees of alternatives and the results are obtained as x2  x3  x4  x1 . Therefore, the patient is suffering from the disease Eczema which describes the given symptoms. Example 1.4 (Medical Diagnosis of Patients) Medical or clinical diagnosis is the procedure of discovering a disease, and it describes the signs and symptoms of an affected person. The procedure is usually subjected to the laboratory tests rather than a physical autopsy of the patient. In order to make a correct medical diagnosis, mostly we need to perform physical observation as well as laboratory tests on the patient. In this multi-criteria decision making problem, a medical diagnosis is explained by using bipolar fuzzy TOPSIS method. Suppose that a doctor(decision maker) examines three patients P1 , P2 , P3 having the same symptoms to diagnose the diseases. Five diseases, x1 = Malaria, x2 = Tuberculosis, x3 = Throat disease, x4 = Typhoid, and x5 = Viral fever, are considered as alternatives and symptoms, c1 = Cough, c2 = Flu, c3 = Headache, c4 = Temperature, and c5 = Throat pain, are considered as criteria for examination. 1. The bipolar fuzzy decision matrix is given in Table 1.7, where each entry xi j in the matrix represents the degree of suffering and safety of the patient for the occurrence of disease xi , i = 1, . . . , 4 from symptom c j , j = 1, . . . , 6. 2. The weight vectors of patients are assigned which represent the effect of symptom on the human body. W P1 = [ 0.2 0.25 0.1 0.35 0.1 ], W P2 = [ 0.25 0.2 0.05 0.1 0.4 ], W P3 = [ 0.25 0.15 0.15 0.35 0.1 ].

3. The weighted bipolar fuzzy decision matrix is computed in Table 1.8. 4. The bipolar fuzzy positive and negative ideal solutions for P1 are computed as follows: B F P I S = [(0.18, −0.046), (0.2, −0.075), (0.057, −0.024), (0.235, −0.032), (0.07, −0.034)], B F N I S = [(0.06, −0.12), (0.093, −0.168), (0.007, −0.07), (0.158, −0.256), (0.009, −0.053)].

(1.6)

The bipolar fuzzy positive and negative ideal solutions for P2 are calculated as follows: B F P I S = [(0.215, −0.085), (0.166, −0.05), (0.034, −0.013), (0.09, −0.017), (0.292, −0.052)], B F N I S = [(0.098, −0.14), (0.114, −0.126), (0.014, −0.037), (0.043, −0.067), (0.068, −0.24)].

(1.7)

14

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Table 1.7 Bipolar fuzzy decision matrix D c1 c2 P1

P2

P3

x1 x2 x3 x4 x5 x1 x2 x3 x4 x5 x1 x2 x3 x4 x5

(0.47, −0.58) (0.9, −0.23) (0.87, −0.36) (0.3, −0.6) (0.57, −0.45) (0.39, −0.56) (0.79, −0.34) (0.86, −0.5) (0.69, −0.43) (0.7, −0.35) (0.5, −0.47) (0.89, −0.17) (0.6, −0.45) (0.73, −0.52) (0.67, −0.35)

(0.5, −0.67) (0.63, −0.3) (0.8, −0.43) (0.37, −0.6) (0.75, −0.3) (0.57, −0.63) (0.83, −0.4) (0.73, −0.38) (0.83, −0.25) (0.67, −0.3) (0.63, −0.4) (0.8, −0.25) (0.4, −0.67) (0.54, −0.6) (0.69, −0.3)

c3

c4

c5

(0.35, −0.7) (0.5, −0.24) (0.21, −0.57) (0.07, −0.6) (0.57, −0.35) (0.43, −0.7) (0.67, −0.53) (0.27, −0.73) (0.5, −0.41) (0.63, −0.25) (0.54, −0.5) (0.76, −0.3) (0.37, −0.54) (0.43, −0.6) (0.5, −0.45)

(0.67, −0.09) (0.6, −0.23) (0.45, −0.73) (0.57, −0.29) (0.5, −0.17) (0.8, −0.3) (0.83, −0.23) (0.43, −0.67) (0.9, −0.3) (0.87, −0.17) (0.75, −0.35) (0.9, −0.25) (0.45, −0.5) (0.69, −0.35) (0.64, −0.3)

(0.27, −0.53) (0.57, −0.34) (0.09, −0.39) (0.63, −0.4) (0.7, −0.52) (0.25, −0.54) (0.17, −0.6) (0.73, −0.13) (0.47, −0.25) (0.36, −0.5) (0.47, −0.57) (0.4, −0.6) (0.78, −0.15) (0.53, −0.4) (0.5, −0.35)

Table 1.8 Weighted bipolar fuzzy decision matrix D⊗W

c1

c2

c3

c4

c5

P1

x1

(0.094, −0.116)

(0.125, −0.168)

(0.035, −0.07)

(0.235, −0.032)

(0.027, −0.053)

x2

(0.18, −0.046)

(0.158, −0.075)

(0.05, −0.024)

(0.21, −0.081)

(0.057, −0.034)

x3

(0.174, −0.072)

(0.2, −0.108)

(0.021, −0.057)

(0.158, −0.256)

(0.009, −0.039)

x4

(0.06, −0.12)

(0.093, −0.15)

(0.007, −0.06)

(0.2, −0.102)

(0.063, −0.04)

x5

(0.114, −0.09)

(0.188, −0.075)

(0.057, −0.035)

(0.175, −0.06)

(0.07, −0.052)

x1

(0.098, −0.14)

(0.114, −0.126)

(0.022, −0.035)

(0.08, −0.03)

(0.1, −0.216)

x2

(0.198, −0.085)

(0.166, −0.08)

(0.034, −0.027)

(0.083, −0.023)

(0.068, −0.24)

x3

(0.215, −0.125)

(0.146, −0.076)

(0.014, −0.037)

(0.043, −0.067)

(0.292, −0.052)

x4

(0.173, −0.108)

(0.166, −0.05)

(0.025, −0.021)

(0.09, −0.03)

(0.188, −0.1)

x5

(0.175, −0.088)

(0.134, −0.06)

(0.032, −0.013)

(0.087, −0.017)

(0.144, −0.2)

x1

(0.125, −0.118)

(0.095, −0.06)

(0.081, −0.075)

(0.263, −0.123)

(0.047, −0.057)

x2

(0.223, −0.043)

(0.12, −0.038)

(0.114, −0.045)

(0.315, −0.088)

(0.04, −0.06)

x3

(0.15, −0.113)

(0.06, −0.101)

(0.056, −0.081)

(0.158, −0.175)

(0.078, −0.015)

x4

(0.183, −0.13)

(0.081, −0.09)

(0.065, −0.09)

(0.242, −0.123)

(0.053, −0.04)

x5

(0.168, −0.088)

(0.104, −0.045)

(0.075, −0.068)

(0.224, −0.105)

(0.05, −0.035)

P2

P3

1.4 Bipolar Fuzzy TOPSIS Method

15

The bipolar fuzzy positive and negative ideal solutions for P3 are calculated as follows: B F P I S = [(0.223, −0.043), (0.12, −0.038), (0.114, −0.045), (0.315, −0.088), (0.078, −0.015)], B F N I S = [(0.125, −0.13), (0.06, −0.101), (0.056, −0.09), (0.158, −0.175), (0.04, −0.06)].

(1.8)

5. The Euclidean distance of each disease from bipolar fuzzy positive and negative ideal solutions for P1 are given as follows: d(x1 , B F P I S) = 0.1253, d(x2 , B F P I S) = 0.050, d(x3 , B F P I S) = 0.1789,

d(x1 , B F N I S) = 0.1723, d(x2 , B F N I S) = 0.1907, d(x3 , B F N I S) = 0.1243,

d(x4 , B F P I S) = 0.1532, d(x5 , B F P I S) = 0.0750,

d(x4 , B F N I S) = 0.1204, d(x5 , B F N I S) = 0.1839.

The Euclidean distance of each disease from bipolar fuzzy positive and negative ideal solutions for P2 are as calculated as follows: d(x1 , B F P I S) = 0.212, d(x2 , B F P I S) = 0.209,

d(x1 , B F N I S) = 0.047, d(x2 , B F N I S) = 0.104,

d(x3 , B F P I S) = 0.066, d(x4 , B F P I S) = 0.089,

d(x3 , B F N I S) = 0.227, d(x4 , B F N I S) = 0.163,

d(x5 , B F P I S) = 0.153,

d(x5 , B F N I S) = 0.114.

The Euclidean distance of each disease from bipolar fuzzy positive and negative ideal solutions for P3 are computed as follows: d(x1 , B F P I S) = 0.1118, d(x2 , B F P I S) = 0.0416, d(x3 , B F P I S) = 0.1653,

d(x1 , B F N I S) = 0.0940, d(x2 , B F N I S) = 0.1766, d(x3 , B F N I S) = 0.0472,

d(x4 , B F P I S) = 0.1132, d(x5 , B F P I S) = 0.0926,

d(x4 , B F N I S) = 0.0847, d(x5 , B F N I S) = 0.0988.

6. The relative closeness degrees of each alternative to positive ideal solution for P1 is given as ρ(x1 ) = 0.5790, ρ(x2 ) = 0.7923, ρ(x3 ) = 0.4099, ρ(x4 ) = 0.4401, ρ(x5 ) = 0.7103. For patient P2 , the relative closeness degrees are as follows: ρ(x1 ) = 0.1815, ρ(x2 ) = 0.3323, ρ(x3 ) = 0.7747, ρ(x4 ) = 0.6468, ρ(x5 ) = 0.4270. For patient P3 , the relative closeness degrees are computed as follows: ρ(x1 ) = 0.4568, ρ(x2 ) = 0.8093, ρ(x3 ) = 0.2221, ρ(x4 ) = 0.4280, ρ(x5 ) =

16

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

0.5162. The ordering of alternatives according to the relative closeness degree for P1 , P2 , and P3 are as follows: For P1 , x2  x5  x1  x4  x3 . For P2 , x3  x4  x5  x2  x1 . For P3 , x2  x5  x1  x4  x3 . Hence it is concluded that the patients P1 and P3 are suffering from Tuberculosis, whereas patient P2 has Throat disease.

1.5 Bipolar Fuzzy ELECTRE I Method Roy [50] and Benayoun et al. [22] initiated the ELECTRE methods on the basis of outranking relations. Different variants of ELECTRE method have been presented in order to capture different types of problem to obtain the kernel solution by considering the nature of the criteria (true or pseudo), and the preferential information (weights, concordance index, discordance index, veto effect). ELECTRE I method investigates the pairwise relations of the alternatives on the basis of concordance and discordance sets that finally establish the outranking relations. Concordance and discordance indices actually measure the satisfaction and dissatisfaction of the decision maker to choose one alternative over the other. Consider a multi-criteria decision making problem where the optimal option has to be selected from set of alternatives is X = {x1 , x2 , . . . , xr } in accordance to criteria set C = {c1 , c2 , . . . , ck }. Since the evaluations of decision maker are expressed via bipolar fuzzy sets, the definition of concordance and discordance sets depend on the priority with respect to both negative and positive components. Alghamdi et al. [17] proposed bipolar fuzzy ELECTRE I method. The detailed procedure is elaborated in the following steps: 1. Following steps 1−4 of Algorithm 1.4.1, assessments of the experts about the aptitude of alternatives with respect to criteria are assembled in bipolar fuzzy decision matrix D = [(μi j , νi j )]r ×k . Further., the weightage of the criteria are m w j = 1, which given by weight vector W = [w1 w2 . . . wk ], 0 ≤ w j ≤ 1, j=1

contribute to compute the bipolar fuzzy weighted decision matrix [qi j ]r ×k . 2. Bipolar fuzzy evaluations are investigated to define the concepts of concordance and discordance sets relying on both membership grades of an object. The mathematical definitions of the bipolar fuzzy concordance sets Cαβ and bipolar fuzzy discordance sets Dαβ are given by Eqs. (1.9) and (1.10), respectively. Cαβ = { j | tα j ≥ tβ j , 1 ≥ j ≤ k}, α = β α, β = 1, 2, . . . , r,

(1.9)

Dαβ = { j | tα j < tβ j , 1 ≤ j ≤ k}, α = β α, β = 1, 2, . . . , r,

(1.10)

where ti j = m i j + n i j , i = 1, 2, . . . , r, j = 1, 2, . . . , k.

1.5 Bipolar Fuzzy ELECTRE I Method

17

3. The bipolar fuzzy concordance indices are determined in Eq. (1.11). cαβ =



wj.

(1.11)

j∈Cαβ

Consequently, the r × r bipolar fuzzy concordance matrix is constructed in Eq. (1.12). ⎡ ⎤ − c12 . . . c1r ⎢ c21 − . . . c2r ⎥ ⎢ ⎥ (1.12) C =⎢ . . . . ⎥. ⎣ .. .. . . .. ⎦ cr 1 cr 2 . . . − 4. The bipolar fuzzy discordance indices are determined in Eq. (1.13).

1 [(m α j 2

− m β j )2 + (n α j − n β j )2 ]

1 [(m α j 2

− m β j )2 + (n α j − n β j )2 ]

max

dαβ =

j∈Dαβ

max j

.

(1.13)

Thus, the r × r bipolar fuzzy discordance matrix is constructed in Eq. (1.14). ⎡

− ⎢ d21 ⎢ D=⎢ . ⎣ ..

d12 − .. .

... ... .. .

⎤ d1r d2r ⎥ ⎥ .. ⎥ . . ⎦

(1.14)

dr 1 dr 2 . . . −

5. The sufficiency of the concordance and discordance indices to outrank the alternatives has to be examined with respect to threshold values, known as, concordance and discordance levels. The bipolar fuzzy concordance level c and bipolar fuzzy discordance level d can be evaluated via the Formula (1.15). c=

r  r  1 cαβ , r (r − 1) α=1,

d=

β=1, α=β β=α

r  r  1 dαβ . r (r − 1) α=1,

(1.15)

β=1, α=β β=α

6. The bipolar fuzzy concordance dominance matrix is defined in Eq. (1.16). ⎡

− ⎢ l21 ⎢ L=⎢ . ⎣ ..

l12 − .. .

⎤ . . . l1r . . . l2r ⎥ ⎥ , . . .. ⎥ . . ⎦

lr 1 lr 2 . . . −

(1.16)

18

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

where the lαβ values are defined as  lαβ =

1, if cαβ ≥ c, 0, if cαβ < c.

(1.17)

7. The bipolar fuzzy discordance dominance matrix is defined in Eq. (1.18). ⎡

− ⎢ p21 ⎢ P=⎢ . ⎣ ..

p12 − .. .

... ... .. .

⎤ p1r p2r ⎥ ⎥ .. ⎥ , . ⎦

(1.18)

pr 1 pr 2 . . . −

where the pαβ values are defined as  pαβ =

1, if dαβ ≤ d, 0, if dαβ > d.

(1.19)

8. The bipolar fuzzy aggregated dominance matrix is defined in Eq. (1.20). ⎡

− ⎢ z 21 ⎢ Z =⎢ . ⎣ ..

z 12 − .. .

... ... .. .

⎤ z 1r z 2r ⎥ ⎥ .. ⎥ , . ⎦

(1.20)

zr 1 zr 2 . . . −

where z αβ = lαβ pαβ . 9. The final stage is to derive the ranking from the outranking values z αβ ’s. That is, an arrow will be drawn from xα to xβ , if and only if z αβ = 1. Consequently, there are three possible cases. (a) (b) (c)

A unique arrow is drawn from xα into xβ . Two possible arrows are drawn between xα and xβ . No arrow exists between xα and xβ .

The cases (a), (b), and (c) describe the preference, indifference, and incomparability of the alternative xα to xβ , respectively. Based on the above explanation, the general procedure of bipolar fuzzy ELECTRE I is summarized in Algorithm 1.5.1 and Fig. 1.4.

1.5 Bipolar Fuzzy ELECTRE I Method Fig. 1.4 The flowchart of bipolar fuzzy ELECTRE I method

19

Figure out of alternatives and selection criteria

Construction of a bipolar fuzzy decision matrix

Evaluation of criteria weights by decision maker

Determine weighted bipolar fuzzy decision matrix

Construct concordance sets

Construct discordance sets

Calculate the concordance dominance indices

Evaluate the discordance dominance indices

Determine the aggregated dominance indices

Draw the decision graph

20

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Algorithm 1.5.1 Bipolar fuzzy ELECTRE I method 1. Steps 1 to 4 are same as described in Algorithm 1.4.1. 2. Determine the bipolar fuzzy concordance and bipolar fuzzy discordance sets that are defined in Eqs. (1.9) and (1.10), respectively. 3. Compute bipolar fuzzy concordance and discordance indices by using Eqs. (1.11) and (1.13). 4. Construct bipolar fuzzy concordance matrix by using Eq. (1.12). 5. Establish the bipolar fuzzy discordance matrix by using Eq. (1.14). 6. Calculate bipolar fuzzy concordance and bipolar fuzzy discordance levels as specified in Formulae (1.15). 7. Compute the bipolar fuzzy concordance dominance matrix (1.16) by using Formula (1.17). 8. Determine the bipolar fuzzy discordance dominance matrix (1.18) by using Formula (1.19). 9. Evaluate the bipolar fuzzy aggregate dominance matrix (1.20) by using Equation z αβ = lαβ pαβ , 1 ≤ α, β ≤ r . 10. Portray the outranking graph according to the values of z αβ ’s, for the investigation of pairwise relations to eliminate the less favorable alternatives and specify the optimal one(s). Example 1.5 Consider the problem of smartphone selection as discussed in Example 1.2. The steps 1-4 of Algorithm 1.5.1 have already been done in it. Start the calculations from step 5. 5. The bipolar fuzzy concordance sets Cαβ and the bipolar fuzzy discordance sets Dαβ are determined in Tables 1.9 and 1.10, respectively.

Table 1.9 Bipolar fuzzy concordance sets β 1 2 C1β C2β C3β C4β C5β

− {2, 3, 4} {2, 3, 4} {3} {1, 3}

{1} − {1, 4} {1, 3} {1}

Table 1.10 Bipolar fuzzy discordance sets β 1 2 D1β D2β D3β D4β D5β

− {1} {1} {1, 2, 4} {2, 4}

{2, 3, 4} − {2, 3} {2, 4} {2, 3, 4}

3

4

5

{1} {2, 3} − {3} {1, 3}

{1, 2, 4} {2, 4} {1, 2, 4} − {1, 2}

{2, 4} {2, 3, 4} {2, 4} {3, 4} −

3

4

5

{2, 3, 4} {1, 4} − {1, 2, 4} {2, 4}

{3} {1, 3} {3} − {3, 4}

{1, 3} {1} {1, 3} {1, 2} −

1.5 Bipolar Fuzzy ELECTRE I Method

21

Fig. 1.5 Bipolar fuzzy decision graph

x1

x4

x2

x5

x3

6. The bipolar fuzzy concordance matrix C and bipolar fuzzy discordance matrix D are computed as follows: ⎡

− ⎢ 0.75 ⎢ C =⎢ ⎢ 0.75 ⎣ 0.25 0.5

0.25 − 0.45 0.5 0.25

0.25 0.55 − 0.25 0.5

0.75 0.5 0.75 − 0.55

⎤ 0.5 0.75 ⎥ ⎥ 0.5 ⎥ ⎥, 0.45 ⎦ −

⎡

⎤ − 1 1 1 0.956 ⎢ 0.822 − 1 0.592 1 ⎥ ⎢ ⎥ 1 ⎥ D=⎢ ⎢ 0.3659 0.94 − 0.95 ⎥. ⎣ 0.735 1 1 − 1 ⎦ 1 0.706 0.619 1 −

7. Thus the bipolar fuzzy concordance level is c¯ = 0.5 and bipolar fuzzy discordance level is d¯ = 0.872. 8. According to bipolar fuzzy concordance and discordance levels, the bipolar fuzzy concordance dominance matrix and bipolar fuzzy discordance dominance matrix are evaluated as follows: ⎤ ⎡ ⎤ ⎡ − 0 0 0 0 − 0 0 1 1 ⎢1 − 0 1 0⎥ ⎢1 − 1 1 1⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ P =⎢ L = ⎢ 1 0 − 1 1 ⎥, ⎢ 1 0 − 0 0 ⎥. ⎣1 0 0 − 1⎦ ⎣0 1 0 − 0⎦ 0 1 1 0 − 1 0 1 1 − 9. The bipolar fuzzy aggregate dominance matrix is given as follows: ⎡

− ⎢1 ⎢ Z =⎢ ⎢1 ⎣0 0

0 0 − 0 0 − 0 0 0 1

⎤ 0 0 1 0⎥ ⎥ 0 0⎥ ⎥. − 0⎦ 0 −

10. According to nonzero values z αβ ’s of matrix Z , the decision graph is constructed in Fig. 1.5. Hence, the most favorable alternatives are x2 and x5 . Example 1.6 Consider the problem of skin diseases as discussed in Example 1.3. The weighted bipolar fuzzy decision matrix is given in Table 1.6. The steps 1–4 of Algorithm 1.5.1 have already been done in it. Start the calculations from step 5. 5. The bipolar fuzzy concordance sets Cαβ and bipolar fuzzy discordance sets xαβ are computed in Tables 1.11 and 1.12, respectively.

22

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Table 1.11 Bipolar fuzzy concordance sets β 1 2 C1β C2β C3β C4β

– {1, 2, 3, 6} {1, 2, 3, 6} {1, 2, 3, 6}

{4, 5} – {1, 3, 6} {3, 6}

Table 1.12 Bipolar fuzzy discordance sets β 1 2 D1β D2β D3β D4β

– {4, 5} {4, 5} {4, 5}

{1, 2, 3, 6} – {2, 4, 5} {1, 2, 4, 5}

3

4

{4, 5} {2, 4, 5, 6} – {5, 6}

{4, 5} {1, 2, 4, 5} {1, 2, 3, 4, 5} –

3

4

{1, 2, 3, 6} {1, 3} – {1, 2, 3, 4}

{1, 2, 3, 6} {3, 6} {6} –

6. The bipolar fuzzy concordance matrix C and bipolar fuzzy discordance matrix D are computed as follows: ⎡ ⎤ ⎡ ⎤ − 0.2 0.2 0.2 − 1 1 1 ⎢ 0.8 − 0.6 0.8 ⎥ ⎢ 0.1185 − 0.992 0.361 ⎥ ⎥ ⎥ C =⎢ D=⎢ ⎣ 0.8 0.5 − 0.9 ⎦ , ⎣ 0.3579 1 − 0.3697 ⎦. 0.8 0.2 0.15 − 0.92 1 1 − 7. The bipolar fuzzy concordance level is c = 0.5125 and bipolar fuzzy discordance level is d = 0.7599. 8. The bipolar fuzzy concordance dominance matrix L and bipolar fuzzy discordance dominance matrix P are as follows: ⎡ ⎤ ⎡ ⎤ − 0 0 0 − 0 0 0 ⎢1 − 1 1⎥ ⎢1 − 0 1⎥ ⎥ ⎥ L=⎢ P=⎢ ⎣ 1 0 − 1 ⎦, ⎣ 1 0 − 1 ⎦. 1 0 0 − 0 0 0 − 9. The bipolar fuzzy aggregated dominance matrix Z is computed as follows: ⎡

− 0 ⎢1 − Z =⎢ ⎣1 0 0 0

⎤ 0 0 0 1⎥ ⎥. − 1⎦ 0 −

(1.21)

The whole procedure can be summarized in Table 1.13. 10. The graph showing the outranking relations is given in Fig. 1.6, which shows that x2 and x3 are the most favorable alternatives.

1.5 Bipolar Fuzzy ELECTRE I Method

23

Table 1.13 Bipolar fuzzy ELECTRE I results for detection of skin diseases Alternatives compared

Cαβ

Dαβ

cαβ

dαβ

lαβ

pαβ

z αβ

Outranking relations

(x1 , x2 )

{4, 5}

{1, 2, 3, 6}

0.2

1

0

0

0

Incomparable

(x1 , x3 )

{4, 5}

{1, 2, 3, 6}

0.2

1

0

0

0

Incomparable

(x1 , x4 )

{4, 5}

{2, 3, 6}

0.2

1

0

0

0

Incomparable

(x2 , x1 )

{1, 2, 3, 6}

{4, 5}

0.8

0.1185 1

1

1

x2 → x1

(x2 , x3 )

{2, 4, 5, 6}

{1, 3}

0.6

0.992

1

0

0

Incomparable

(x2 , x4 )

{1, 2, 4, 5}

{3, 6}

0.8

0.361

1

1

1

x2 → x4

(x3 , x1 )

{1, 2, 3, 6}

{4, 5}

0.8

0.3579 1

1

1

x3 → x1

(x3 , x2 )

{1, 3, 6}

{2, 4, 5}

0.5

1

0

0

Incomparable

(x3 , x4 )

{1, 2, 3, 4, 5}

{6}

0.9

0.3697 1

1

1

x3 → x4

(x4 , x1 )

{1, 2, 3, 6}

{4, 5}

0.8

0.92

1

0

0

Incomparable

(x4 , x2 )

{3, 6}

{1, 2, 4, 5}

0.2

1

0

0

0

Incomparable

(x4 , x3 )

{5, 6}

{1, 2, 3, 4}

0.15

1

0

0

0

Incomparable

0

x1

Fig. 1.6 Outranking relation of diseases

x3

x2

x4

Example 1.7 Consider the problem of Example 1.4, the evaluations of bipolar concordance set, bipolar discordance set, bipolar concordance indices, bipolar discordance indices, bipolar concordance dominance indices, bipolar discordance dominance indices, bipolar aggregated dominance indices, and outranking relations for patients P1 , P2 , and P3 are given in Table 1.14. The graphs sketched by using outranking relations are represented in Figs. 1.7, 1.8, and 1.9, respectively. It is clear from graphs that the most preferred alternatives for P1 , P2 , and P3 are {x2 }, {x3 , x4 }, and {x2 }, respectively.

24

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Table 1.14 Bipolar fuzzy ELECTRE I results for medical diagnosis of P1 , P2 and P3 Patients

Alternatives compared

Cαβ

Dαβ

cαβ

dαβ

lαβ

pαβ

z αβ

Outranking relations

P1

(x1 , x2 )

{4}

{1, 2, 3, 5}

0.35

1

0

0

0

Incomparable

(x1 , x3 )

{3, 4, 5}

{1, 2}

0.55

0.405

1

1

1

x1 → x3

(x1 , x4 )

{1, 2, 3, 4}

{5}

0.9

0.490

1

1

1

x1 → x4

(x1 , x5 )

{4}

{1, 2, 3, 5}

0.35

1

0

0

0

Incomparable

(x2 , x1 )

{1, 2, 3, 5}

{4}

0.65

0.496

1

1

1

(x2 , x3 )

{1, 3, 4, 5}

{2}

0.75

0.293

1

1

1

x2 → x1 x2 → x3

(x2 , x4 )

{1, 2, 3, 4, 5}

{}

1

0

1

1

1

x2 → x4

(x2 , x5 )

{1, 3, 4, 5}

{2}

0.75

0.378

1

1

1

x2 → x5

(x3 , x1 )

{1, 2}

{3, 4, 5}

0.45

1

0

0

0

Incomparable

(x3 , x2 )

{2}

{1, 3, 4, 5}

0.25

1

0

0

0

Incomparable

(x3 , x4 )

{1, 2, 3}

{4, 5}

0.55

1

1

0

0

Incomparable

(x3 , x5 )

{1}

{2, 3, 4, 5}

0.2

1

0

0

0

Incomparable

(x4 , x1 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x4 , x2 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x4 , x3 )

{4, 5}

{1, 2, 3}

0.45

0.775

0

0

0

incomparable

(x4 , x5 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x5 , x1 )

{1, 2, 3, 5}

{4}

0.65

0.589

1

1

1

x5 → x1

(x5 , x2 )

{2}

{1, 3, 4, 5}

0.25

1

0

0

0

Incomparable

(x5 , x3 )

{2, 3, 4, 5}

{1}

0.8

0.318

1

1

1

x5 → x3

(x5 , x4 )

{1, 2, 3, 4}

{5}

0.9

0.114

1

1

1

x5 → x4

(x1 , x2 )

{5}

{1, 2, 3, 4}

0.4

1

0

0

0

Incomparable

(x1 , x3 )

{3, 4}

{1, 2, 5}

0.15

1

0

0

0

Incomparable

(x1 , x4 )

{}

{1, 2, 3, 4, 5} 0

1

0

0

0

Incomparable

(x1 , x5 )

{}

{1, 2, 3, 4, 5} 0

1

0

0

0

Incomparable

(x2 , x1 )

{1, 2, 3, 4}

{5}

0.6

0.351

1

1

1

x2 → x1

(x2 , x3 )

{1, 2, 3, 4}

{5}

0.6

1

1

0

0

incomparable

(x2 , x4 )

{1, 3, 4}

{2, 5}

0.4

1

0

0

0

incomparable

(x2 , x5 )

{1, 2}

{3, 4, 5}

0.45

1

0

0

0

Incomparable

(x3 , x1 )

{1, 2, 5}

{3, 4}

0.85

0.207

1

1

1

x3 → x1

(x3 , x2 )

{5}

{1, 2, 3, 4}

0.4

0.203

0

1

0

Incomparable

(x3 , x4 )

{1, 5}

{2, 3, 4}

0.65

0.522

1

1

1

(x3 , x5 )

{1, 5}

{2, 3, 4}

0.65

0.318

1

1

1

x3 → x4 x3 → x5

(x4 , x1 )

{1, 2, 3, 4, 5}

{}

1

0

1

1

1

x4 → x1

(x4 , x2 )

{2, 4, 5}

{1, 3}

0.7

0.184

1

1

1

x4 → x2

(x4 , x3 )

{2, 3, 4}

{1, 5}

0.35

1

0

0

0

incomparable

(x4 , x5 )

{2, 5}

{1, 3, 4}

0.6

0.184

1

1

1

x4 → x5

(x5 , x1 )

{1, 2, 3, 4, 5}

{}

1

0

1

1

1

x5 → x1

(x5 , x2 )

{3, 4, 5}

{1, 2}

0.55

0.440

1

1

1

x5 → x2

(x5 , x3 )

{2, 3, 4}

{1, 5}

0.35

1

1

0

0

Incomparable

(x5 , x4 )

{1, 3, 4}

{2, 5}

0.4

1

0

0

0

Incomparable

P2

(continued)

1.6 Comparative Analysis

25

Table 1.14 (continued) Patients P3

Alternatives compared

Cαβ

Dαβ

cαβ

dαβ

lαβ

pαβ

z αβ

Outranking relations Incomparable

(x1 , x2 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

(x1 , x3 )

{2, 3, 4}

{1, 5}

0.65

0.445

1

1

1

x1 → x3

(x1 , x4 )

{2, 3, 4}

{1, 5}

0.65

1

1

0

0

Incomparable Incomparable

(x1 , x5 )

{4}

{1, 2, 3, 5}

0.35

1

0

0

0

(x2 , x1 )

{1, 2, 3, 4}

{5}

0.9

0.062

1

1

1

x2 → x1

(x2 , x3 )

{1, 2, 3, 4}

{5}

0.9

0.328

1

1

1

x2 → x3

(x2 , x4 )

{1, 2, 3, 4}

{5}

0.9

0.250

1

1

1

x2 → x4

(x2 , x5 )

{1, 2, 3, 4}

{5}

0.9

0.290

1

1

1

(x3 , x1 )

{1, 5}

{2, 3, 4}

0.35

1

0

0

0

x2 → x5 Incomparable

(x3 , x2 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x3 , x4 )

{3, 5}

{1, 2, 4}

0.25

1

0

0

0

Incomparable

(x3 , x5 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x4 , x1 )

{1, 5}

{2, 3, 4}

0.35

0.558

0

1

0

Incomparable

(x4 , x2 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x4 , x3 )

{1, 2, 3, 4}

{5}

0.9

0.358

1

1

1

x4 → x3

(x4 , x5 )

{4}

{1, 2, 3, 5}

0.35

1

0

0

0

Incomparable

(x5 , x1 )

{1, 2, 3, 5}

{4}

0.65

0.819

1

0

0

incomparable

(x5 , x2 )

{5}

{1, 2, 3, 4}

0.1

1

0

0

0

Incomparable

(x5 , x3 )

{1, 2, 3, 4}

{5}

0.9

0.357

1

1

1

x5 → x3

(x5 , x4 )

{1, 2, 3, 4, 5}

{}

1

0

1

1

1

x5 → x4

x4

Fig. 1.7 Outranking relation for P1

x5

x1 x2

x3

1.6 Comparative Analysis Next, we compare the bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE I methods, and show their comparative advantages over other fuzzy TOPSIS and fuzzy ELECTRE I methods. 1. Bipolar fuzzy sets are more applicable for analysis because of their tendency to cover double-sided knowledge related to the signs and symptoms of diseases. For the element xi j = (μi j , νi j ), positive membership values μi j ∈ [0, 1] and negative membership values νi j ∈ [−1, 0] represent satisfaction and dis-

26

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations x4

Fig. 1.8 Outranking relation for P2

x1

x2

x3

x5

Fig. 1.9 Outranking relation for P3

x3

x1

x5 x4

x2

satisfaction grades of respective alternative regarding to the specific criterion. Further, μi j = 1 and νi j = −1 represent two extreme cases where the alternative i exhibits highest satisfaction behavior and dissatisfaction behavior to criterion j. We have applied the bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE I techniques in the medical field to diagnose the diseases timely by observing the symptoms as there exist two-sided information in medical systems. 2. Crisp or fuzzy sets are able to cover only one-sided information about the severity and existence of symptoms or more simply they can carry the information only through the positive membership degree of alternative under criteria. Their structures lack the dissatisfaction grade of considering actions under respective criteria in order to elaborate the counter part of the information. Thus, we apply the bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE I techniques for diagnosis instead of fuzzy TOPSIS and fuzzy ELECTRE I for more accurate outcomes.

1.7 Bipolar Fuzzy Extended TOPSIS Method The latest research inventions in medical sciences have captivated the human attentions. Due to misunderstanding symptoms and their interrelationships, dental treatments are often misdiagnosed in the existing methods. The wrong medical treatments and the improper use of metallic instruments could also be a cause of disease. It is however important to diagnose the disease and detect the disadvantages of various medical treatments. Bipolar fuzziness can be used to detect the diseases caused by

1.7 Bipolar Fuzzy Extended TOPSIS Method

27

various treatments. In the existing bipolar fuzzy TOPSIS methods, weights to alternatives are chosen arbitrarily according to the choice of decision makers. Sarwar et al. [56] proposed bipolar fuzzy extended TOPSIS method. In this method, this technique has been extended using entropy weights which are independent of the choice of decision makers. Teeth diseases are very common nowadays due to the consumption of unhygienic food and sweets. The tooth decay is one of the most common teeth diseases affecting millions of people. Our mouth is a ground of good and bad bacteria. People are not used for thinking teeth as living organs and tissues. A number of treatments are available to prevent cavities, gum disease, toothaches, and missing teeth, etc. But, on the other hand, it is roughly estimated that almost 70% of the human diseases are indirectly or directly due to intervention in the teeth structures. This includes teeth filling with various metallic and non-metallic materials, root canal, dental bridges, teeth implant, metal braces, crowns and caps, dentures, gum surgery, teeth veneers, and composite bridges, etc. Bipolar fuzzy sets can be used to detect the diseases caused by various treatments considering the benefits of the treatment. If x1 , x2 , . . . , xn are the alternative materials and c1 , c2 , . . . , ck are the side effects corresponding to each alternative, then the procedure for the selection of a suitable treatment with minimum danger is described in Algorithm 1.7.1. Algorithm 1.7.1 Bipolar fuzzy extended TOPSIS method based on entropy weights 1. Input: The r number of alternative materials x1 , x2 , . . . , xr . 2. Input: The side effects c1 , c2 , . . . , ck corresponding to each alternative. 3. Input: The bipolar fuzzy decision matrix D = [xi j ]r ×k where xi j = (μi j , νi j ) depicts the membership grade of alternative xi based on criteria c j . The positive degree of membership represents the degree of side effect c j for implementing xi and negative degree of membership represents the degree of benefit of material xi . 4. Determine the criteria weight (information entropy) of each side effect c j , 1 ≤ j ≤ k, by using Formula (1.22). E n (c j ) = −τ

r    |νi j | ln |νi j | + (1 − μi j ) ln(1 − μi j ) , 1 ≤ j ≤ k (1.22) i=1

where τ > 0 is a constant such that 0 ≤ E n (c j ) ≤ 1. 5. Calculate the degree of divergence div j of each side effect c j by using Eq. (1.23). div j = 1 − E n (c j ),

1 ≤ j ≤ k.

(1.23)

6. Calculate the entropy weights w j corresponding to each criteria c j as given in Eq. (1.24). div j wj = k , 1 ≤ j ≤ k. (1.24) div j j=1

28

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

7. Construct the weighted bipolar fuzzy decision matrix D ⊗ W = [qi j ]r ×k where for each 1 ≤ i ≤ r , qi j is defined in Eq. (1.25). qi j = (m i j , n i j ) = w j (μi j , νi j )

1 ≤ j ≤ k.

(1.25)

8. Calculate the bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution respectively, as described with bipolar fuzzy vectors in Eqs. (1.26) and (1.27). B F P I S = {(μ1 + , ν1 + ), (μ2 + , ν2 + ), . . . , (μk + , νk + )}, −

−

−

−

−

(1.26)

−

B F N I S = {(μ1 , ν1 ), (μ2 , ν2 ), . . . , (μk , νk )},

(1.27)

where μ j + = max{m i j }, ν j + = max{n i j } and μ j − = min{m i j }, ν j − = min{n i j } i

i

i

i

j = 1, 2, . . . , k. 9. Compute the distance measures of every alternative xi from bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution by using Formulae (1.28) and (1.29), respectively.

k

 d(xi , B F P I S) =  ((μi j − μ j + )2 + (νi j − ν j + )2 ),

(1.28)

j=1



k

 d(xi , B F N I S) =  ((μi j − μ j − )2 + (νi j − ν j − )2 ).

(1.29)

j=1

10. Determine the degree of relative closeness for each alternative xi using Eq. (1.30). d(xi , B F N I S) , 1 ≤ i ≤ r. (1.30) ρi = d(xi , B F P I S) + d(xi , B F N I S) 11. Arrange all the alternatives in descending order according to relative closeness degree. The procedure of Algorithm 1.7.1 is explained with an example of teeth replacement alternatives. Example 1.8 (Replacement of a Missing Teeth) A person may face the problem of missing teeth due to gum disease, tooth decay, injury, by birth or inherited disorder. With the passage of time, missing teeth can cause various health issues including poor nutrition, food bite, and chewing problems. One missing teeth can weaken the jaw structure. People often visit dentists for replacement of missing teeth. There are various alternative replacement options for a missing teeth such as traditional bridge, denture, cantilever bridge, resin-bonded, and teeth implant. The benefits and side effects of teeth replacements vary in patients. Every person has different skin and teeth sensitivities therefore, the side effects of dental treatments are not constant

1.7 Bipolar Fuzzy Extended TOPSIS Method

29

in patients. Consider the example of a person X who wants a teeth replacement treatment.The solution of this multi-criteria decision making is given in following steps: 1. The treatments including traditional bridge, cantilever bridge, resin-bonded, removable denture, and teeth implant are considered as alternatives for this multicriteria decision making problem. 2. The description of degree of membership of each criteria (side effect) is shown in Table 1.15. 3. The side effects and benefits of alternative treatments with respect to person X are represented in Table 1.16. 4. By taking τ = 0.1, the calculations of entropy wights are given in Table 1.17. 5. The weighted bipolar fuzzy decision matrix is given in Table 1.18. 6. The bipolar fuzzy positive and negative ideal solutions are given in Table 1.19. 7. The distance measures and relative closeness degree of each alternative measure are given in Table 1.20. 8. Arranging the alternatives in decreasing order according to relative closeness degree, we conclude that traditional bridge is a best option for replacement of missing teeth. Remark 1.2 Multi-criteria decision making refers to process of making ideal decision having highest level of achievement from a set of alternatives examined related to several conflicting criteria. TOPSIS methods operate effectively to specify favorable alternatives in multi-criteria decision making problems. Fuzziness, intuitionistic fuzziness, and neutrosophic sets have been utilized successfully in TOPSIS methods

Table 1.15 Decision criteria Criteria Affordable Bone infection

Positive degree of membership Negative degree of membership Replacement option is affordable Infection causes jawbone loss

Damage to natural teeth

Damage to abutment healthy teeth

Long lasting

Restoration can collapse and need to replace early Treatment causes toothache with the passage of time Tooth decay under the replacement Replacement causes gums infections and diseases

Toothache Risk of teeth decay Gum disease

Replacement option is expensive Prevents the risk of jawbone loss Prevents the future decay and shifting of healthy adjacent teeth Treatment will last long Treatment prevents risk of toothache Good dentistry prevents teeth loss Prevents the risk of gums infection

30

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

Table 1.16 Bipolar fuzzy decision matrix Alternatives Disadvantages and benefits Affordable

Bone infection

Damage to Long lasting Toothache natural teeth teeth decay

Risk of

Gum disease

Traditional bridge

(0.4, −0.6)

(0.5, −0.6)

(0.8, −0.6)

(0.7, −0.7)

(0.7, −0.5)

(0.9, −0.8)

(0.7, −0.5)

Cantilever bridge

(0.3, −0.7)

(0.5, −0.6)

(0.6, −0.7)

(0.8, −0.6)

(0.7, −0.5)

(0.8, −0.8)

(0.7, −0.5)

Resinbonded

(0.2, −0.8)

(0.1, −0.8)

(0.9, −0.4)

(0.4, −0.7)

(0.7, −0.3)

(0.7, −0.3)

(0.8, −0.3)

Removable denture

(0.4, −0.6)

(0.8, −0.1)

(0.8, −0.3)

(0.5, −0.6)

(0.5, −0.6)

(0.8, −0.1)

(0.3, −0.8)

Teeth implant

(0.9, −0.3)

(0.5, −0.7)

(0.4, −0.9)

(0.4, −0.9)

(0.7, −0.4)

(0.3, −0.8)

(0.7, −0.6)

Table 1.17 Entropy weights Calculated Disadvantages and benefits values Affordable Bone infection

Damage to Long natural lasting teeth

Toothache

Risk of Gum teeth decay disease

c1

c2

c3

c4

c5

c6

c7

E n (c j )

0.267

0.272

0.293

0.285

0.352

0.261

0.32

div j

0.733

0.728

0.707

0.715

0.648

0.739

0.68

wj

0.148

0.147

0.143

0.14

0.131

0.149

0.137

Table 1.18 Weighted bipolar fuzzy decision matrix Criteria

Alternatives Traditional bridge

Cantilever bridge

Resin-bonded bonded

Removable denture

Teeth implant

c1

(0.059, −0.089) (0.044, −0.104) (0.030, −0.118) (0.059, −0.089) (0.133, −0.044)

c2

(0.074, −0.088) (0.074, −0.088) (0.015, −0.118) (0.118, −0.015) (0.074, −0.103)

c3

(0.114, −0.086) (0.086, −0.100) (0.129, −0.057) (0.114, −0.043) (0.057, −0.129)

c4

(0.098, −0.098) (0.112, −0.084) (0.056, −0.098) (0.070, −0.084) (0.056, −0.126)

c5

(0.092, −0.066) (0.092, −0.066) (0.092, −0.039) (0.066, −0.079) (0.092, −0.052)

c6

(0.134, −0.119) (0.119, −0.119) (0.104, −0.045) (0.119, −0.015) (0.045, −0.119)

c7

(0.1, −0.069)

(0.014, −0.069) (0.11, −0.041)

(0.041, −0.11)

(0.014, −0.082)

for solving multi-criteria decision making problems. But in many cases, the given information is bipolar in nature. A bipolar fuzzy TOPSIS method [17] is suitable to opt for the objects by exploring the available choices. But in bipolar fuzzy extended TOPSIS method [56], personal opinions of decision makers can affect the evaluation

1.8 Conclusion

31

Table 1.19 Bipolar fuzzy positive and negative ideal solutions c1 μ+ j ν+ j μ− j ν− j

c2

c3

c4

c5

c6

c7

0.133

0.118

0.129

0.112

0.092

−0.044

−0.015

−0.043

−0.084

−0.039

−0.015

−0.069

0.015

0.057

0.056

0.066

0.045

0.014

−0.118

−0.129

−0.126

−0.079

−0.119

0.03 −0.118

0.134

0.11

−0.11

Table 1.20 Distance measures and relative closeness degree Calculated values

Traditional bridge

Cantilever bridge

Resin-bonded bonded

Removable denture

Teeth implant

x1

x2

x3

x4

x5

d(xi , B F P I S) 2.3018

2.2634

2.0521

1.9615

2.2648

d(xi , B F N I S) 2.0314

1.9862

1.8164

1.7372

2.002

ρi

0.5326

0.4695

0.4697

0.4692

0.4688

of criteria weights. The chosen weights may be irrelevant of given information which can affect the outcomes of decision making. Therefore, it is more appropriate to compute weights according to the given information. In our method, we have discussed the process for calculating entropy weights from given bipolar fuzzy information. It gives more suitable decisions as compared to the previous methods discussed in literature.

1.8 Conclusion The credit for the utilization of fuzzy set theory in decision making goes to Bellman and Zadeh [21]. Since then, many researchers have worked within fuzzy set theory to determine the optimal solution for decision making problems. Bipolar fuzzy information, covering the two opposite parts of the human opinions, contribute significantly to facilitate the human beings in decision making. Positive information describes the possibility of occurrence of an event or any property of an object, whereas other-side information represents the strength of counter property. Therefore, bipolar fuzzy set theory has the edge to target the fields of information technology, artificial intelligence information, and decision sciences. This chapter has designed three algorithms which help the practitioner to make his decisions in a bipolar context. We have first developed a bipolar fuzzy TOPSIS method. It evaluates the distances of each alternative to bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution and then ranks all alternatives in decreasing manner. The best alternative is clearly identified. Further, we have proposed a bipolar fuzzy ELECTRE I method whose major distinction is the introduction of “outranking relations” to eliminate the infe-

32

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

rior choices. ELECTRE I, a well-known multi-criteria decision making method, has a track record of successful implementation to real-world applications. Although, it has a more clear view of preferable alternatives by the elimination of less favorable instances, it is sometimes inept to specify the optimal alternative. The bipolar fuzzy extended TOPSIS method is also developed by using the entropy weights of criteria and the best alternative is chosen. Additional Reading The readers are suggested to [27–29, 33, 44, 45, 65] for definitions of additional terms and applications not included in this chapter.

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1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations

45. Mahmood, T., Rehman, U.U., Ahmmad, J., Santos-Garcia, G.: Bipolar complex fuzzy Hamacher aggregation operators and their applications in multi-attribute decision making. Mathematics. 10, 23 (2022) 46. Mehmood, M.A., Akram, M., Alharbi, M.G., Bashir, S.: Solution of fully bipolar fuzzy linear programming models. Math. Prob. Eng. 31 (2021) 47. Mehmood, M.A., Akram, M., Alharbi, M.G., Bashir, S.: Optimization of L R-type fully bipolar fuzzy linear programming problems. Math. Prob. Eng. 36 (2021) 48. N˘ad˘aban, S., Dzitac, S., Dzitac, I.: Fuzzy TOPSIS: a general view. Procedia Comput. Sci. 91, 823–831 (2016) 49. Rashmanlou, H., Samanta, S., Pal, M., Borzooei, R.A.: Bipolar fuzzy graphs with categorical properties. Int. J. Comput. Intell. Syst. 8(5), 808–818 (2015) 50. Roy, B.: Classement et Choix en Presence de Points de vue Multiples (la methode Electre). Revue Francaise d’Informatique et de Recherche Opérationnelle. 8(1), 57–75 (1968) 51. Roy, B.: The outranking approach and the foundations of ELECTRE methods. Theor. Decis. 31(1), 49–73 (1991) 52. Roszkowska, E.: Multi-criteria decision making models by applying the TOPSIS method to crisp and interval data. Multiple Criteria Decision Making/University of Economics in Katowice 6, 200–230 (2011) 53. Roszkowska, E., Wachowicz, T.: Application of fuzzy TOPSIS to scoring the negotiation offers in ill-structured negotiation problems. Eur. J. Oper. Res. 242(3), 920–932 (2015) 54. Saqib, M., Akram, M., Shahida, B., Allahviranloo, T.: Numerical solution of bipolar fuzzy initial value problem. J. Intell. Fuzzy Syst. 40, 1309–1341 (2021) 55. Saqib, M., Akram, M., Shahida, B.: Certain efficient iterative methods for bipolar fuzzy system of linear equations. J. Intell. Fuzzy Syst. 39(3), 3971–3985 (2020) 56. Sarwar, M., Akram, M., Zafar, F.: Decision making approach based on competition graphs and extended TOPSIS method under bipolar fuzzy environment. Math. Comput. Appl. 23(4), 68 (2018) 57. Sarwar, M., Akram, M.: Bipolar fuzzy circuits with applications. J. Intell. Fuzzy Syst. 34(1), 547–558 (2018) 58. Singh, P.K., Kumar, A.Ch.: Bipolar fuzzy graph representation of concept lattice. Inf. Sci. 288, 437–448 (2014) 59. Singh, P.K., Kumar, A.Ch.: A note on bipolar fuzzy graph representation of concept lattice. Int. J. Comput. Sci. Math. 5(4), 381–393 (2014) 60. Wei, G., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Bipolar fuzzy Hamacher Aggregation operators in multiple attribute decision making. Int. J. Fuzzy Syst. 20, 1–12 (2018) 61. Wei, G., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Hesitant bipolar fuzzy aggregation operators in multiple attribute decision making. Int. J. Fuzzy Syst. 33, 1119–1128 (2017) 62. Wu, M.C., Chen, T.Y.: The ELECTRE multicriteria analysis approach based on Atanassov’s intuitionistic fuzzy sets. Expert Syst. Appl. 38(10), 12318–12327 (2011) 63. 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, 155–164 (2017) 64. Yang, H.L., Li, S.G., Guo, Z.L., Ma, C.H.: Transformation of bipolar fuzzy rough set models. Knowl.-Based Syst. 27, 60–68 (2012) 65. Yu, P.L.: A class of solutions for group decision problems. Manag. Sci. 19(8), 936–946 (1973) 66. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 67. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 68. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998)

Chapter 2

TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

In this chapter, we consider bipolar fuzzy numbers for the expression of bipolar, non-exact data. In order to impose some structure on them, we focus our attention on trapezoidal bipolar fuzzy numbers and the particular case of triangular bipolar fuzzy numbers. However some experts opt for submitting linguistic expressions of their assessments in similar contexts. To incorporate these opinions, we present bipolar fuzzy linguistic variables. As to theoretical developments, we establish certain characterizations of (α, β)-cuts, strong (α, β)-cuts, and level sets of a bipolar fuzzy number. We also explain the derivation of a ranking function for the set of all bipolar fuzzy numbers to formulate particular expressions for the ranking of trapezoidal and triangular bipolar fuzzy numbers. By using the concept of double upper lower dense sequence, we are able to endow the set of bipolar fuzzy numbers with a total ordering. With the assistance of these tools, we bridge the gap with applications. We explore a group decision making method based on a trapezoidal bipolar fuzzy TOPSIS technique for the analysis, structuring, and solution of the decisions. Besides, we present theoretical comparisons of trapezoidal bipolar fuzzy TOPSIS method with other well-known existing methods such as fuzzy TOPSIS, bipolar fuzzy TOPSIS, and bipolar fuzzy ELECTRE-I. The content of this chapter is based on research performed in [3, 4].

2.1 Introduction A fuzzy number is described as a generalization of a regular, real number in the sense that does not refer to one single value. A fuzzy number is defined to be a connected set of possible values, where each possible value has its own weight between 0 and 1. These weights help to define the membership function of the fuzzy number. A fuzzy number is actually a special case of a convex, normalized fuzzy set of the real line with a specific structure. As fuzzy logic is an extension of two-valued logic (which uses absolute truth and falsehood only, and nothing in between both), fuzzy number is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Akram et al., Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-99-0569-0_2

35

36

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

an extension of real numbers in the same way. Calculations with fuzzy numbers allow the incorporation of uncertainty on both parameters, properties, geometry, and initial conditions. As it is only natural, fuzzy numbers (introduced by Zadeh [32]) enable the researcher to deal with imprecise numerical quantities in a practical way. Their success boosted additional works by Zadeh [30, 31, 33–35]. In these contributions, he set forth various new concepts regarding fuzzy numbers, linguistic variables, and their applications to real-world problems [20, 24]. Several authors have investigated the properties of fuzzy numbers including their operations [17, 18] and procedures for ranking alternatives defined by fuzzy numbers [11]. In order to model even more cases of uncertain information, more general concepts like triangular intuitionistic fuzzy numbers [28] and interval-valued fuzzy numbers [19] have been studied. In particular, a linguistic computational model based on bipolar information (i.e., that allows for bipolar linguistic imprecision) is proposed in this chapter. A number of researchers proposed extensions of TOPSIS for group decision making [12, 14–16, 21, 26, 27]. Alghamdi et al. [9] and Akram et al. [1–3] introduced multi-criteria decision making methods in bipolar fuzzy environment. To identify the most suitable alternative, we explore the efficiencies of the bipolar fuzzy TOPSIS approach in practical decision making. The proposed bipolar fuzzy TOPSIS approach operates by collecting the initial data, including the ratings of various alternatives in reference to criteria and criteria weights, in the form of linguistic terms that are later expressed by trapezoidal bipolar fuzzy numbers. It is a commonly accepted view in modern physics that the universe is either an equilibrium or a quasi-equilibrium. Further, without bipolar fusion/binding, interaction, and oscillation there would be no electromagnetic field, no memory scanning, no human cognition, no logical reasoning, no brain, no equilibrium, and no universe. In this context, linguistic terms are primary tools to represent the double-sided judgements within the bipolar fuzzy logic for authentic decision making [36]. For the sake of comparison and derive the ranking of alternatives, we define a ranking function on the set of all bipolar fuzzy numbers.

2.2 Bipolar Fuzzy Numbers A fuzzy number is described as a fuzzy set given by a fuzzy interval in the real line R. The ambiguous boundary of this interval confirms that fuzzy number is also a fuzzy set. Generally, a fuzzy interval is represented by two end points a1 and a3 and a peak point a2 as A = [a1 , a2 , a3 ]. The α-cut operation can be defined on fuzzy numbers. The α-cut interval for a fuzzy number A is expressed as Aα = [a1(α) , a3(α) ] (cf. [22]). The concept of a bipolar fuzzy number was first introduced by Akram and Arshad [3].

2.2 Bipolar Fuzzy Numbers

37

Definition 2.1 ([3]) A bipolar fuzzy number A = I, K  = [ι1 , ι2 , ι3 , ι4 ], [κ1 , κ2 , κ3 , κ4 ] is a bipolar fuzzy subset of a real line R with satisfaction degree λ I and dissatisfaction degree λ K satisfying the following postulates: 1. λ I is a piecewise continuous function from real line to interval [0, 1], and λ K is a piecewise continuous function from real line to interval [−1, 0], 2. λ I (x) = 0, for all x ∈ (−∞, ι1 ], λ K (x) = 0, for all x ∈ (−∞, κ1 ], 3. λ I (x) is strongly increasing on [ι1 , ι2 ] and λ K (x) is strongly decreasing on [κ1 , κ2 ], λ K (x) = −1, for all x ∈ [κ2 , κ3 ], 4. λ I (x) = 1, for all x ∈ [ι2 , ι3 ], 5. λ I (x) is strongly decreasing on [ι3 , ι4 ], and λ K (x) is strongly increasing on [κ3 , κ4 ], λ K (x) = 0, for all x ∈ [κ4 , ∞). 6. λ I (x) = 0, for all x ∈ [ι4 , ∞), For convenience, the satisfaction degree and dissatisfaction degree can be defined as ⎧ L ⎧ L ⎪ ⎪ ⎪ λ I (x), if x ∈ [ι1 , ι2 ], ⎪ λ K (x), if x ∈ [κ1 , κ2 ], ⎨ ⎨ 1, if x ∈ [ι2 , ι3 ], −1, if x ∈ [κ2 , κ3 ], λ (x) = λ I (x) = λ IR (x), if x ∈ [ι3 , ι4 ], K λ KR (x), if x ∈ [κ3 , κ4 ], ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, otherwise. 0, otherwise, where λ LI (x) : [ι1 , ι2 ] → [0, 1], λ IR (x) : [ι3 , ι4 ] → [0, 1], λ LK (x) : [κ1 , κ2 ] → [−1, 0], λ KR (x) : [κ3 , κ4 ] → [−1, 0]. λ LI(x) and λ LK (x) denote left membership functions for λ I (x) and λ K (x), respectively. Similarly λ IR(x) and λ KR (x) , respectively, denote right membership functions for λ I (x) and λ K (x). Graphically, a bipolar fuzzy number can be represented as in Fig. 2.1, where λ I and λ K respectively represent satisfaction and dissatisfaction degrees. Definition 2.2 ([3]) A bipolar fuzzy number A = I, K  = [ι1 , ι2 , ι3 , ι4 ], [κ1 , κ2 , κ3 , κ4 ] is a trapezoidal bipolar fuzzy number denoted by (ι1 , ι2 , ι3 , ι4 ), (κ1 , κ2 , κ3 , κ4 ), if its satisfaction degree λ I and dissatisfaction degree λ K are given as ⎧ ⎧ x − ι1 κ1 − x ⎪ ⎪ ⎪ ⎪ , if x ∈ [ι , ι ] 1 2 ⎪ι −ι ⎪ κ − κ , if x ∈ [κ1 , κ2 ] ⎪ ⎪ ⎪ ⎪ 1 1 ⎨ 2 ⎨ 2 1, if x ∈ [ι2 , ι3 ] −1, if x ∈ [κ2 , κ3 ] λ I = ι4 − x λK = x − κ4 ⎪ ⎪ ⎪ ⎪ , if x ∈ [ι3 , ι4 ] , if x ∈ [κ3 , κ4 ] ⎪ ⎪ ⎪ ι 4 − ι3 ⎪ κ4 − κ3 ⎪ ⎪ ⎩ ⎩ 0, otherwise 0, otherwise. Definition 2.3 ([3]) A bipolar fuzzy number A = I, K  = [ι1 , ι2 , ι2 , ι3 ], [κ1 , κ2 , κ2 , κ3 ] is a triangular bipolar fuzzy number denoted by (ι1 , ι2 , ι3 ), (κ1 , κ2 , κ3 ), if its satisfaction degree λ I and dissatisfaction degree λ K are given as ⎧ ⎧ x − ι1 κ1 − x ⎪ ⎪ ⎪ ⎪ , if x ∈ [ι , ι ] , if x ∈ [κ1 , κ2 ] 1 2 ⎪ ⎪ ⎨ ι 2 − ι1 ⎨ κ2 − κ1 x − κ3 λ I = ι3 − x , if x ∈ [ι , ι ] λK = , if x ∈ [κ2 , κ3 ] 2 3 ⎪ ⎪ ⎪ ⎪ ι − ι κ ⎪ ⎪ 2 ⎩ 3 ⎩ 3 − κ2 0, otherwise 0, otherwise

38

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

(ι2 , 1)

(0, 1)

(ι3 , 1)

λI

(0, 0)

(ι1 , 0)

(ι2 , 0) (κ1 , 0) (ι3 , 0) (κ2 , 0) (ι4 , 0) (κ3 , 0)

(κ4 , 0)

λK

(0, −1)

(κ2 , −1)

(κ3 , −1)

Fig. 2.1 Graphical representation of bipolar fuzzy number

(κ2 , 1)

(0, 1)

(κ3 , 1)

λI

(0, 0)

(ι1 , 0)

(ι2 , 0) (κ1 , 0) (ι3 , 0) (κ2 , 0) (ι4 , 0) (κ3 , 0)

(κ4 , 0)

λK

(0, −1)

(κ2 , −1)

(κ3 , −1)

Fig. 2.2 Graphical representation of trapezoidal bipolar fuzzy number (v., Definition 2.2)

2.3 Bipolar Fuzzy Linguistic Variables

(κ2 , 1)

(0, 1)

(0, 0)

(0, −1)

39

(ι1 , 0)

(ι2 , 0) (ι1 , 0) (ι3 , 0) (κ2 , 0)

(κ3 , 0)

(κ2 , −1)

Fig. 2.3 Graphical representation of triangular bipolar fuzzy number (v., Definition 2.3)

The graphical representations of a trapezoidal bipolar fuzzy number and a triangular bipolar fuzzy number are shown in Figs. 2.2 and 2.3, respectively. Example 2.1 Consider the statement “Voltage supply is perfect”, then there must be a state where the voltage supply is not balanced. So, there exists a concept of bipolarity that can be comprehensively represented by the triangular bipolar fuzzy number. When we refer to the term “perfect or balanced voltage supply”, then a voltage domain [0, 460] comes to mind, and it can be considered as a continuous domain. A triangular bipolar fuzzy number can be defined as A = (190, 220, 240), (120, 160, 210) whose graphical representation is given by Fig. 2.4. There the voltage and membership function are drawn on x − axis and y − axis, respectively.

2.3 Bipolar Fuzzy Linguistic Variables A linguistic variable is a variable, whose values are not regular numerical values but whose values are given by words of any language or sentences. For instance, the values of the linguistic variable “age” are given by the set {young, ver y young, not young, . . .}. Akram and Arshad [3] defined the concept of bipolar fuzzy linguistic variables and their representation in the form of trapezoidal bipolar fuzzy numbers. Definition 2.4 ([3]) A bipolar fuzzy variable is given by a triplet (Y, X, R(Y, x)), where Y refer to the name of the variable, X represents numerical domain for the variable Y , x is general presentation of any arbitrary element of X , and R(Y, x)

40

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers y − axis 1

0.5

50

150

100

200

250

300

350

400

450

x − axis

−0.5

−1

Fig. 2.4 Bipolar fuzzy number representing voltage supply in Example 2.1 Table 2.1 Bipolar fuzzy variable Bipolar fuzzy variable Universe of discourse Y

X 1 .. . 20 .. .

Young

25 .. .

50 .. . 100

Bipolar fuzzy set R(Y, x) (0, −1) .. . (0.4, −0.5) (0.5, −0.4) (0.6, −0.2) .. . (0.7, −0.1) (1, 0) (0.9, −0.1) .. . (0.3, −0.7) (0, −0.9) .. . (0, −1)

denotes a bipolar fuzzy set on X, which represents the constraint on values of x imposed by Y.

2.3 Bipolar Fuzzy Linguistic Variables

41

Example 2.2 Let ‘young’ be a variable defined for the numerical domain [0, 100]. The given variable with bipolar fuzzy set R(Y, x), as shown in the Table 2.1, can be considered as a bipolar fuzzy variable. Definition 2.5 A bipolar fuzzy linguistic variable is given by a quintuple (y, L (Y ), X, P(Y ), N (Y )), where 1. 2. 3. 4. 5.

y denotes the name of variable. L(y) is the set of possible linguistic values taken by y. Y is a general name to represent random elements of L(y). X defines the numerical domain of y. P(Y ) : X → [0, 1] and N (Y ) : X → [−1, 0] are semantic rules to associate the linguistic value iof L with mappings P(Y ) and T (Y ) over the domain X .

From Definition 2.5, it can be clearly observed that bipolar fuzzy linguistic variables are variables, whose linguistic values are actually bipolar fuzzy variables or bipolar fuzzy subsets of X. Example 2.3 We consider the variable age with linguistic values {young, ver y young, ver y ver y young}, where these linguistic values are bipolar fuzzy variables on the domain X = [0, 100], as shown in Table 2.2. Then the variable “age” can be considered as a bipolar fuzzy linguistic variable. In the present chapter, bipolar fuzzy numbers are used to represent the linguistic values in place of bipolar fuzzy sets. For example, we consider the word “distance”

Table 2.2 Bipolar fuzzy linguistic variable Bipolar fuzzy y L(y) linguistic variable Age Y young

X

1 .. . 18 20 .. . 25 .. . 30 .. . 40 50 100

(0, −1) .. . (0.7, −0.1) (0.8, −0.1) .. . (1, 0) .. . (0.6, −0.4) .. . (0.5, −0.4) (0.3, −0.9) (0, −1)

VY very young

VVY very very young

(0, −1) .. . (0.7, −0.1) (1, 0) .. . (0.9, −0.1) .. . (0.7, −0.3) .. . (0.6, −0.6) (0, −0.8) (0, −1)

(0, −1) .. . (0.9, −0.1) (1, −0.9) .. . (0.8, −0.1) .. . (0.5, −0.6) .. . (0.2, −0.8) (0.1, −1) (0, −1)

42

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

y − axis Large

1 V ery small

V ery largel

Small

x − axis 5

15

10

20

25

30

35

40

45

50

55

60

65

75

Small

V ery largel −1

70

80

85

90

95

100

V ery small

Large

Fig. 2.5 Membership functions for linguistic values in Example 2.3

as a bipolar fuzzy linguistic variable with linguistic values defined as small, very small, large, and very large. Let X = [0, 100] be the universe of discourse. Each linguistic value can be expressed using trapezoidal bipolar fuzzy numbers as shown in Fig. 2.5. In Fig. 2.5, V S = V er y small = (0, 0, 0, 10) (90, 100, 100, 100), S = Small = (20, 30, 30, 40) (70, 80, 80, 90), L = Large = (60, 70, 80, 90) (20, 30, 40, 50), V L = V er y large = (80, 90, 90, 100) (0, 10, 10, 20).

2.4 Ranking of Bipolar Fuzzy Numbers We proceed to describe a procedure for the comparison of bipolar fuzzy numbers. Therefore let H = {h 1 , h 2 , . . . , h n } be a set of bipolar fuzzy numbers. Then a ranking function R f from H to real line R is a mapping satisfying the following conditions: for any distinct h i , h j ∈ H, 1. If R f (h i ) < R f (h j ), then h i ≺ h j . 2. If R f (h i ) = R f (h j ), then h i ∼ h j . 3. If R f (h i ) > R f (h j ), then h i h j . We shall refer to the ranking function that for any bipolar fuzzy number h i = (Ii , K i ) = (ιi1 , ιi2 , ιi3 , ιi4 ), (κi1 , κi2 , κi3 , κi4 ), is defined as [m(ιik ) + σ(Ii )] − [|m(κik )| + σ(K i )], k = 1, 2, 3, 4, where m(ιik ) and m(κik ) denote the mean of ιik and mean of κik respectively, σ(Ii ) represents the area of Ii and σ(K i ) represents the area of K i . The areas can be calculated by taking the mode value of integration of left and right membership 1 0 functions separately and then adding it into d x or | (−1)d x | . 0

−1

2.4 Ranking of Bipolar Fuzzy Numbers

43

Therefore, for any bipolar fuzzy numbers h i and h j , we declare that if [m(ιik ) + σ(Ii )] − [|m(κik )| + σ(K i )] < [m(ι jk ) + σ(I j )] − [|m(κ jk )| + σ(K j )], then h i < h j ; if [m(ιik ) + σ(Ii )] − [|m(κik )| + σ(K i )] = [m(ι jk ) + σ(I j )] − [|m(κ jk )| + σ(K j )], then h i = h j ; and if [m(ιik ) + σ(Ii )] − [|m(κik )| + σ(K i )] > [m(ι jk ) + σ(I j )] − [|m(κ jk )| + σ(K j )], then h i > h j . If σ(Ii ), σ(K i ) ≥ 1, for each i, then the ranking function for bipolar fuzzy numbers can also be defined as m(ιik )σ(Ii ) − |m(κik )|σ(K i ), k = 1, 2, 3, 4.

(2.1)

Derivation of an expression for the ranking of trapezoidal bipolar fuzzy number and triangular bipolar fuzzy number 1 From Definition 2.2, λ LI (x) = ιx−ι 2 −ι1 κ −x λ KR (x) = κ44−κ3 . It is easy to see that

ι2 σ(I ) =

= ι1

λ IR (x) =

ι3 λ LI (x)d x

ι1 ι2

and

+

λ LK (x) =

ι4 dx +

ι2

x − ι1 dx + ι 2 − ι1

ι4 −x , ι4 −ι3

λ IR (x)d x ι3

ι3

ι4 dx +

ι2

ι3

ι4 − x dx ι 4 − ι3

1 = (−ι1 − ι2 + ι3 + ι4 ). 2 Similarly,   κ3   b4   κ2       L σ(K ) =  λ K (x)d x  +  1 d x  +  λ KR (x)d x  κ1

κ2

κ2

κ3

= κ1

x − κ1 dx + κ2 − κ1

κ3

κ4 1 dx +

κ2

1 = (−κ1 − κ2 + κ3 + κ4 ). 2

κ3

κ4 − x dx κ4 − κ3

x−κ1 , κ2 −κ1

44

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

y − axis 1

h1 = (20, 40, 50, 70), (40, 60, 70, 90) = h2 = (30, 45, 45, 60), (50, 65, 65, 70) =

x − axis

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

−1

Fig. 2.6 Trapezoidal bipolar fuzzy numbers in Example 2.4

Hence the ranking function for trapezoidal bipolar fuzzy numbers is given by expression (2.2).



−κ1 − κ2 + κ3 + κ4 −ι1 − ι2 + ι3 + ι4 ] − m(K ) + [ ] , m(I ) + [ 2 2

(2.2)

ι 1 + ι2 + ι3 + ι4 κ 1 + κ2 + κ3 + κ4 where m(I ) = and m(K ) = . Similarly, after 4 4 simplification, the ranking function for triangular bipolar fuzzy numbers is given by expression (2.3).



ι3 − ι1 κ 3 − κ1 m(I ) + [ ] − m(K ) + [ ] , 2 2 where m(I ) =

(2.3)

ι 1 + ι2 + ι3 κ 1 + κ2 + κ3 and m(K ) = . 3 3

Example 2.4 Let h 1 = (20, 40, 50, 70), (40, 60, 70, 90) and h2 = (30, 45, 45, 60), (50, 65, 65, 70) be two trapezoidal bipolar fuzzy numbers as shown in Fig. 2.6. Using expressions (2.2) and (2.3), we calculate R f (h 1 ) = −20 and R f (h 2 ) = −12.5, whereas by Equation (2.1), R f (h 1 ) = −600 and R f (h 2 ) = 50. In both cases we observe R(h 1 ) < R(h 2 ), hence we declare that the comparison h 1 ≺ h 2 holds true.

2.5 (α, β)-Cut of Bipolar Fuzzy Numbers Since bipolar fuzzy numbers are specific cases of bipolar fuzzy sets, the (α, β)-cuts of bipolar fuzzy numbers can be defined with arguments similar to those given for bipolar fuzzy sets:

2.5 (α, β)-Cut of Bipolar Fuzzy Numbers

45

(0, α)

(0, 0)

aoα 1

τ1β

oα 4

τ4β

ζ α,β

(0, β)

Fig. 2.7 An (α, β)-cut of a bipolar fuzzy number

Definition 2.6 ([4]) Let ζ = I, K  = [ι1 , ι2 , ι3 , ι4 ], [κ1 , κ2 , κ3 , κ4 ] be a bipolar fuzzy number. For α ∈ [0, 1] and β ∈ [−1, 0], the (α, β)-cut of ζ is denoted by ζ (α,β) and defined as, ζ (α,β) = {x ∈ R | λ I (x) ≥ α, λ K (x) ≤ β}. It is clear that for any α, β = 0, the (α, β)-cut of a bipolar fuzzy number is a closed interval. It is obtained by taking the intersection of two closed intervals, namely, the α-cut of I and the β-cut of K . The graphical representation for (α, β)-cut of a bipolar fuzzy number is given as in Fig. 2.7. In the particular case of a trapezoidal bipolar fuzzy number, any (α, β)-cut can be β β represented as ζ (α,β) = [ια1 , ια4 ] ∩ [κ1 , κ4 ] where ια1 = ι1 + α(ι2 − ι1 ), ια4 = ι4 − β β α(ι4 − ι3 ), κ1 = κ1 − β(κ2 − κ1 ), κ4 = κ4 + β(κ4 − κ3 ). Similarly, for triangular bipolar fuzzy numbers, any (α, β)-cut can be represented as ζ (α,β) = [ια1 , ια3 ] ∩ β β β [κ1 , κ3 ], where ια1 = ι1 + α(ι2 − ι1 ), ια3 = ι3 − α(ι3 − ι2 ),κ1 = κ1 − β(κ2 − κ1 ), β κ3 = κ3 + β(κ3 − κ2 ). Example 2.5 Let ζ = (2, 4, 5, 6), (−3, 4, 5, 7) be a trapezoidal bipolar fuzzy number. For α = 0.5 and β = −0.7, −0.7 0.5 ζ (0.5,−0.7) = [ι0.5 , κ−0.7 ]. 1 , ι4 ] ∩ [κ1 4

46

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

Definition 2.7 Let ζ = I, K  = [ι1 , ι2 , ι3 , ι4 ], [κ1 , κ2 , κ3 , κ4 ] be a bipolar fuzzy number. For any α ∈ [0, 1] and β ∈ [−1, 0], the strong (α, β)-cut of ζ is denoted by ζ (α,β)+ and defined as ζ (α,β)+ = {x ∈ R | λ I (x) > α, λ K (x) < β}. Definition 2.8 Let P be a bipolar fuzzy set on the universal set Q. A special bipolar fuzzy set in Q related to P (α,β) is denoted by (α,β) P, and defined as  (α,β) P

=

(α, β), x ∈ P (α,β) . (0, 0), x ∈ / P (α,β)

Definition 2.9 Let P be a bipolar fuzzy set on the universal set Q. A special bipolar fuzzy set related to P (α,β)+ is denoted by (α,β)+ P, and it is defined as  (α,β)+ P

=

(α, β), x ∈ P (α,β)+ . (0, 0), x ∈ / P (α,β)+

The following decomposition theorems [8] will show the representation of an arbitrary bipolar fuzzy set in terms of special bipolar fuzzy sets: Theorem 2.1 (First Decomposition Theorem) Let P be a bipolar fuzzy set on a crisp set Q, then P= (α,β) P. (α,β)∈[−1,0]×[0,1] p

Proof For each particular x ∈ Q, denote (μ P (x), μnP (x)) = ( p, q) = P(x), where p is satisfaction degree for the belonging-ness of x in P and q is dissatisfaction degree for the belonging-ness of x in P. Clearly, ⎛ ⎝

(α,β)∈[−1,0]×[0,1]

⎞ (α,β) P

⎠ (x) = sup

inf

(α,β) P(x)

α∈[0,1] β∈[−1,0]



= max



sup

inf

α∈[0, p] β∈[−1,0]

= sup

inf

α∈[0, p] β∈[−1,0]

(α,β) P(x),

sup

inf

α∈[ p,1] β∈[−1,0]

(α,β) P(x)



= max = sup

sup

inf

α∈[0, p] β∈[q,0]

inf

α∈[0, p] β∈[q,0]

(α,β) P(x)

 (α,β) P(x), sup

inf

α∈[0, p] β∈[−1,q]

(α,β) P(x)

(α,β) P(x) = ( p, q).



This completes the proof.

Theorem 2.2 (Second Decomposition Theorem) Let P be a bipolar fuzzy set on a crisp set Q, then P= (α,β) P, (α,β)∈L(P)

where L(P) = {(α, β) | P(x) = (α, β), x ∈ Q}.

2.5 (α, β)-Cut of Bipolar Fuzzy Numbers

47

Example 2.6 Let Q = {q1 , q2 , q3 , q4 , q5 , q6 , q7 } be a universe of discourse. Let E=

 (0.5, −0.7) (0.3, −0.8) (0.2, −0.5) (0.7, −0.1) (0.5, −0.5) (0, −0.8) (1, 0)  , , , , , , q1 q2 q3 q4 q5 q6 q7

be a bipolar fuzzy set on Q, then  (0.5, −0.7) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) , , , , , , , q1 q2 q3 q4 q5 q6 q7   (0, 0) (0.3, −0.8) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) , = , , , , , , q1 q2 q3 q4 q5 q6 q7   (0.2, −0.5) (0.2, −0.5) (0.2, −0.5) (0, 0) (0.2, −0.5) (0, 0) (0, 0) , = , , , , , , q1 q2 q3 q4 q5 q6 q7   (0, 0) (0, 0) (0, 0) (0.7, −0.1) (0, 0) (0, 0) (0, 0) , = , , , , , , q1 q2 q3 q4 q5 q6 q7   (0.5, −0.5) (0, 0) (0, 0) (0, 0) (0.5, −0.5) (0, 0) (0, 0) , = , , , , , , q1 q2 q3 q4 q5 q6 q7   (0, 0) (0, −0.8) (0, 0) (0, 0) (0, 0) (0, −0.8) (0, 0) , = , , , , , , q1 q2 q3 q4 q5 q6 q7   (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (1, 0) . = , , , , , , q1 q2 q3 q4 q5 q6 q7 

(0.5,−0.7) P = (0.3,−0.8) P (0.2,−0.5) P (0.7,−0.1) P (0.5,−0.5) P (0,−0.8) P (1,0) P

Thus, we conclude that P =

 (α,β)∈[−1,0]×[0,1]

(α,β) P

=

 (α,β)∈L(P)

(α,β) P.

Definition 2.10 Let S be a subset of interval [0, 1]. It is said to be a dense set in [0, 1] if for each point x◦ ∈ [0, 1] and any > 0, ∃ δ ∈ S such that |x◦ − δ| < . Definition 2.11 Let S be a subset of interval [0, 1], then S is said to be upper dense in [0, 1] if for each point x◦ ∈ [0, 1] and any > 0, ∃ δ ∈ S such that δ ∈ [x◦ , x◦ + ). S is lower dense in [0, 1] if for each point x◦ ∈ [0, 1] and any > 0, ∃ δ ∈ S such that δ ∈ (x◦ − , x◦ ]. Theorem 2.3 If S is a dense set in the interval [0, 1] and 0 ∈ S, then S is lower dense in [0, 1]. Theorem 2.4 If S is a dense set in the interval [0, 1] and 1 ∈ S, then S is upper dense in [0, 1]. Definition 2.12 Let S be a subset of interval [−1, 0]. It is said to be a dense set in [−1, 0] if for each point x◦ ∈ [−1, 0] and any > 0, ∃ δ ∈ S such that |x◦ − δ| < . Definition 2.13 Let S be a subset of interval [−1, 0]. It is said to be an upper dense set in [−1, 0] if for each x◦ ∈ [−1, 0] and any > 0, ∃ δ ∈ S such that δ ∈ [x◦ , x◦ + ). S is lower dense in [−1, 0] if for each point x◦ ∈ [−1, 0] and any > 0, ∃ δ ∈ S such that δ ∈ (x◦ − , x◦ ]. Theorem 2.5 If S is a dense set in interval [−1, 0] and −1 ∈ S, then S is lower dense in [−1, 0].

48 Fig. 2.8 Using Cantor’s diagonal sweeping to list the rational numbers contained in [0, 1]

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

0/4

1/4

2/4

3/4

0/3

1/3

2/3

3/3

0/2

1/2

2/2

0

0/1

1/1

0

0

Theorem 2.6 If S is a dense set in interval [−1, 0] and 0 ∈ S, then S is upper dense in [−1, 0]. From Theorems 2.3 and 2.4, it is observed that any upper lower dense sequence in [0, 1] is actually a dense sequence in interval [0, 1] with real numbers 0 and 1. Similarly from Theorems 2.5 and 2.6, any upper lower dense sequence in [−1, 0] is a dense sequence in interval [−1, 0] with real numbers −1 and 0. Definition 2.14 Let Sα = {αi | i = 1, 2, . . .} be given upper lower dense set in [0, 1] and let Sβ = {β j | j = 1, 2, . . .} be upper lower dense set in [−1, 0], then the double upper lower dense sequence is of the form S(α,β) = {(αi , β j ) : i, j = 1, 2, . . .} ⊆ [−1, 0] × [0, 1]. Example 2.7 Let S1 = { qp | p ≤ q p, q ∈ Z+ ∪ {0}} be the set of rational numbers in [0, 1]. Rational numbers are countable and already Cantor proved that a sequence Sα = {αi : i = 1, 2, . . .} in which every element of S1 appears (many times) can be obtained by sweeping out the numbers in S1 as in Fig. 2.8. Clearly, Sα = {0, 1, 21 , 13 , 41 , 23 , 15 , . . .} is an upper lower dense set in [0, 1]. Example 2.8 Let S2 = {− qp | p ≤ q p, q ∈ Z+ ∪ {0}} be the set of rational numbers in [−1, 0]. A sequence Sβ = {βi : i = 1, 2, . . .} in which every element of S2 appears (many times) can be obtained by sweeping out the numbers in S2 as in Fig. 2.9. , − 14 , − 23 , − 15 , . . .} is an upper lower dense set in Clearly, Sβ = {0, −1, − 12 , −1 3 [−1, 0]. Example 2.9 In Examples 2.7 and 2.8, let S(α,β) = {(αi , β j ) | αi ∈ Sα , β j ∈ Sβ }, then S(αi ,β j ) being the Cartesian product of two countable sets is countable . It forms a double upper lower dense sequence in [−1, 0] × [0, 1] as illustrated in Fig. 2.10. S(α,β) = {(0, 0), (1, 0), (0, −1), (0, −1/2), (1, −1), (1/2, 0), (1/3, 0), . . .}. Since

2.5 (α, β)-Cut of Bipolar Fuzzy Numbers

49

−0/4

−1/4

−2/4

−3/4

−0/3

−1/3

−2/3

−3/3

−0/2

−1/2

−2/2

0

−0/1

−1/1

0

0

Fig. 2.9 Using diagonal sweeping to list the rational numbers contained in [−1, 0]

(0, −1/3)

(1, −1/3)

(1/2, −1/3)

(1/3, −1/3)

(0, −1/2)

(1, −1/2)

(1/2, −1/2)

(1/3, −1/2)

(0, −1)

(1, −1)

(1/2, −1)

(1/3, −1)

(0, 0)

(1, 0)

(1/2, 0)

(1/3, 0)

Fig. 2.10 Using diagonal sweeping to list the elements in Sα

an element with membership (0, 0) cannot be contained in a bipolar fuzzy set, we can consider S(α,β) = {(1, 0), (0, −1), (0, −1/2), (1, −1), (1/2, 0), (1/3, 0), . . .} excluding (0, 0) as a double upper lower dense sequence, which can be used to identify any given bipolar fuzzy set/number as described in the following Theorem 2.7: Theorem 2.7 (Third decomposition theorem for bipolar fuzzy sets) Let P be a bipolar fuzzy subset of Q and S be a given double upper lower dense sequence in D = [−1, 0] × [0, 1], then

50

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers



P=

(α,β) P.

(α,β)∈S⊆D

Proof Let P be a bipolar fuzzy set on Q and S be given double upper lower dense sequence in D. Claim: P = (α.β) P. Since S ⊆ D, therefore (α,β)∈S



(α.β) P

⊆

(α,β)∈S



(α.β) P

= P.

(α,β)∈D

On the other hand, it is to be shown that p

p

μ P (x) ≤ sup μ(α,β) P (x) and μnP (x) ≥ inf μn(α,β) P (x),

x ∈ Q.

(α,β)∈S

(α,β)∈S

p

Let (μ P (x), μnP (x)) = ( p, q) for each x ∈ Q, then p

μ P (x) =

p

sup

(α,β)∈[−1,0]×[0,1]

μnP (x) =

μ(α,β) P (x) = p

inf

(α,β)∈[−1,0]×[0,1]

p

sup

μ(α,β) P (x)

inf

μ(α,β) P (x).

(α,β)∈[0, p]×[−1,0]

μ(α,β) P (x) =

p

(α,β)∈[0,1]×[b,0]

For each α ∈ [0, p] and β ∈ [q, 0], (since Sα is upper dense sequence in [0, 1] and Sβ is lower dense sequence in [−1, 0]), there exists (lα , lβ ) ∈ S such that p

p

μ(α,β) P (x) ≤ μ(lα ,l μn(α,β) P (x) ≥ μn(lα ,l

β)

β

P (x)

p

≤ sup μ(α,β) P (x) (α,β)∈S

(x) ≥ inf μn(α,β) P (x). )P (α,β)∈S

By taking supremum with respect to α ∈ [0, a] and infimum with respect to β ∈ [b, 0] p

μ P (x) =

p

μ(α,β) P (x) ≤ sup μ(α,β) P (x)

inf

μn(α,β) P (x) ≥ inf μn(α,β) P (x).

(α,β)∈[0, p]×[−1,0]

μnP (x) =

(α,β)∈[0,1]×[q,0]

This completes the proof.

p

sup

(α,β)∈S

(α,β)∈S



Total Ordering of Bipolar Fuzzy Numbers. Let S = {(αi , β j ) | i, j = 1, 2, . . .} be a given double upper lower dense sequence in D for any given bipolar fuzzy number ζ. Since S is countable and so has one-one correspondence with set of natural numbers. Let μk (αi , β j ) represents the kth pair in the sequence for this oneone correspondence. Since the (α, β)-cut of a bipolar fuzzy number is acquired by taking the intersection of α-cut of satisfaction degree of ζ and β-cut of dissatisfaction degree of ζ, denote it by [ak , bk ]. Let Jk = ak + bk , where k = 1, 2, . . . using these

2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers

51

countably many parameters, a relation on the set of all bipolar fuzzy numbers can be defined as explained in Definition 2.15. Definition 2.15 Let ζ1 and ζ2 be two bipolar fuzzy numbers. For a given double upper lower dense sequence S in [−1, 0] × [0, 1], use Jk (ζ1 ) and Jk (ζ2 ) to signify above mentioned J for ζ1 and ζ2 , respectively. ζ1 = ζ2 if and only if their (α, β)(α ,β ) (α ,β ) cuts for each pair (αi , β j ) are equal, i.e., ζ1 i j = ζ2 i j , for all i, j = 1, 2, . . . and ζ1 < ζ2 if and only if there exists an integer k◦ > 0 such that Jk◦ (ζ1 ) ≺ Jk◦ (ζ2 ), Jk (ζ1 ) = Jk (ζ2 ) for all integers k < k◦ , where k > 0. Theorem 2.8 The relation  is a total ordering on the set of all bipolar fuzzy numbers. The proof is similar to the proof of Theorem 3.8 in [23]. The ranking of bipolar fuzzy numbers can be achieved through total orderings. Example 2.10 Let ζ= (0.6, 0.65, 0.67, 0.7), (0.6, 0.7, 0.75, 0.8) and ζ2 = (0.6, 0.65, 0.67, 0.75), (0.2, 0.3, 0.35, 0.4) be two bipolar fuzzy numbers. For k = 1, (α,β) = [0.65, 0.67] and J1 (ζ1 ) = J1 (ζ2 ) = 1.32. For k = (α, β) = (1, 0), we have ζ1 2, J2 (ζ1 ) = 1.45 and J2 (ζ2 ) = 0.65, i.e., J2 (ζ1 ) > J2 (ζ2 ). Hence ζ1 ζ2 . Example 2.11 Let ζ1 = (0.2, 0.3, 0.4, 0.5), (0.7, 0.8, 0.85, 0.9) and ζ2 = (0.3, 0.35, 0.4, 0.5), (0.2, 0.4, 0.45, 0.5) be two bipolar fuzzy numbers. For k = 1 (α, β) = (1, 0), we have J1 (ζ1 ) = 0.7 and J1 (ζ2 ) = 0.75, i.e J1 (ζ1 ) < J1 (ζ2 ). Hence ζ1 ≺ ζ2 .

2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers In this section, a multi-attribute group decision making method, proposed by Akram and Arshad [3], based on the TOPIS method is presented, which uses bipolar fuzzy information. This strategy allows us to use bipolar fuzzy numbers instead of bipolar fuzzy sets to deal with alternatives and criteria. If A1 , A2 , . . . , Am are the alternatives corresponding to criteria C1 , C2 , . . . , Cn , then the steps for decision making model are given as follows: 1. The bipolar fuzzy multi-criteria decision making problem is expressed in a kmatrix format as given in Table 2.3. where xikj = (ιik1j , ιik2j , ιik3j , ιik4j ), (κik1j , κik2j , κik3j , κik4j ) i = 1, 2, . . . , m j = 1, 2, . . . , n are trapezoidal bipolar fuzzy numbers representing the bipolar fuzzy linguistic terms with domain as interval [0, 1]. Wk = [w1k w2k . . . wnk ] represents the weight vector and w jk represents fuzzy values. Here xikj is the performance rating of ith alternative Ai for jth criterion C j and, w jk is weight of the jth criterion assigned by the kth decision maker Dk , k = 1, 2, . . . , p.

52

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

Table 2.3 The k-matrix format of the input of our problem C1 C2 ... k x11 k x21

A1 A2 .. . Am

.. . k xm1

k x12 k x22

.. . k xm2

... ... .. . ...

Cn k x1n k x2n .. . k xmn

2. The aggregate performance rating xi j = (ιi1j , ιi2j , ιi3j , ιi4j ), (κi1j , κi2j , κi3j , κi4j ) of alternative Ai for criterion C j assessed by p decision makers can be evaluated as ιi1j

κi1j =

p p p p 1  k1 2 1  k2 3 1  k3 4 1  k4 = ι ,ι = ι ,ι = ι ,ι = ι , p k=1 i j i j p k=1 i j i j p k=1 i j i j p k=1 i j

p p p p 1  k1 2 1  k2 3 1  k3 4 1  k4 κi j , κi j = κi j , κi j = κi j , κi j = κ . p k=1 p k=1 p k=1 p k=1 i j

Similarly, the aggregated importance weights w j can be calculated as w j = p 1  w jk . p k=1

3. The weighted bipolar fuzzy decision matrix is given as E = [ei j ]m×n , where ei j = xi j w j = (ηi1j , ηi2j , ηi3j , ηi4j ), (oi1j , oi2j , oi3j , oi4j ). 4. The bipolar fuzzy positive ideal solution (BFPIS) A∗ and bipolar fuzzy negative ideal solution (BFNIS) A− are identified as A∗ ={e1∗ , e2∗ , . . . , en∗ } = {(max ei j | j ∈ BC ), (min ei j | j ∈ CC ) | i = 1, 2, . . . , m},

A− ={e1− , e2− , . . . , en− } = {(min ei j | j ∈ BC ), (max ei j | j ∈ CC ) | i = 1, 2, . . . , m},

where BC and CC represent the collection of cost and benefit type criteria. 5. The distance di+ of each alternative from BFPIS solution is calculated in Formula (2.4)       1 n (ηi1j − η 1j ∗ )2 + (ηi2j − η 2j ∗ )2 + (ηi3j − η 3j ∗ )2 + (ηi4j − η 4j ∗ )2  . + (oi1j − o1j ∗ )2 + (oi2j − o2j ∗ )2 + (oi3j − o3j ∗ )2 + (oi4j − o4j ∗ )2 2 j=1 (2.4) Similarly, the distance di− of each alternative from BFNIS can be calculated using Formula (2.5).

2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers

53

      1 n (ηi1j − η 1j )2 + (ηi2j − η 2j )2 + (ηi3j − η 3j )2 + (ηi4j − η 4j )2  . 2 j=1 + (oi1j − o1j )2 + (oi2j − o2j )2 + (oi3j − o3j )2 + (oi4j − o4j )2

(2.5)

6. The closeness coefficient of any alternative Ai from BFPIS is formulated by the formula d− Ci = + i − f or i = 1, . . . , m. di + di The alternative Ai is much closer to A∗ when Ci approaches to 1. The ranking can be derived by ordering the alternatives in descending order of Ci , i = 1, . . . , m. Example 2.12 (Selection of the best project proposal for construction company) Consider that a construction company has to select the best project proposal from the available proposals {Ai | i = 1, 2, 3, 4, 5, 6} with attributes {C j | j = 1, 2, 3, 4, 5}. Three attributes including product quality C2 , sustainability C3 and innovations C4 are benefit criteria, whereas two attributes including cost C1 and risk C5 are cost criteria. Two decision makers D1 and D2 are assigned the job to make suitable decision by analyzing the whole situation keenly. Assume that the decision makers assign the rating from the bipolar fuzzy linguistic values set S1 = {Good = G, Medium Good = M G, Fair = F, Medium Poor = M P, Poor = P} for benefit criteria and the rating to cost criteria are assigned from the set S2 = {Low = L , Medium Low = M L , Fair = F, Medium H igh = M H, H igh = H }. The graphical representation of these bipolar fuzzy numbers are given in Fig. 2.11. The linguistic values are chosen from the numerical domain [0, 1]. Furthermore, the decision makers evaluate the criteria weights subjectively by employing fuzzy values as shown in Table 2.9. The graphical illustration of the considered trapezoidal bipolar fuzzy numbers is shown via Fig. 2.11 where Good = G = (0.8, 0.9, 1, 1), (0.0, 0.1, 0.2, 0.2) = H = High, Medium Good = M G = (0.6, 0.7, 0.7, 0.8), (0.1, 0.2, 0.3, 0.3) = M H = Medium High, Fair = F = (0.4, 0.5, 0.6, 0.7), (0.5, 0.5, 0.6, 0.7) = F = y − axis Medium Poor

1

Good

F air

P oor

Medium Good

x − axis 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Good P oor −1 Medium Good

F air

Medium P oor

Fig. 2.11 Satisfaction and dissatisfaction degree for linguistic values

1

54

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

Table 2.4 The linguistic ratings and aggregated value of alternatives for C1 Criteria C1

Alternatives

Decision makers

Aggregate ratings

D1

D2

A1

L

F

(0.20, 0.25, 0.35, 0.45), (0.60, 0.65, 0.70, 0.80)

A2

ML

ML

(0.05, 0.01, 0.02, 0.20), (0.30, 0.35, 0.40, 0.40)

A3

MH

MH

(0.30, 0.35, 0.35, 0.40), (0.05, 0.01, 0.15, 0.15)

A4

MH

ML

(0.35, 0.45, 0.50, 0.60), (0.35, 0.45, 0.55, 0.55)

A5

F

F

(0.20, 0.25, 0.30, 0.35), (0.25, 0.25, 0.30, 0.35)

A6

H

H

(0.40, 0.45, 0.50, 0.50), (0.0, 0.05, 0.10, 0.10)

Table 2.5 The linguistic ratings and aggregated value of alternatives for C2 Criteria C2

Alternatives

Decision makers

Aggregate ratings

D1

D2

A1

MG

G

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A2

G

MG

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A3

G

G

(0.40, 0.45, 0.50, 0.50), (0.00, 0.05, 0.10, 0.10)

A4

G

MG

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A5

MG

MG

(0.30, 0.35, 0.35, 0.40), (0.05, 0.01, 0.15, 0.15)

A6

G

G

(0.40, 0.45, 0.50, 0.50), (0.0, 0.05, 0.10, 0.10)

Table 2.6 The linguistic ratings and aggregated value of alternatives for C3 Criteria C3

Alternatives

Decision makers

Aggregate ratings

D1

D2

A1

F

G

(0.60, 0.70, 0.80, 0.85), (0.25, 0.30, 0.40, 0.45)

A2

G

F

(0.60, 0.70, 0.80, 0.85), (0.25, 0.30, 0.40, 0.45)

A3

G

MG

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A4

G

MG

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A5

MG

G

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A6

MG

G

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

Fair, Medium Poor = M P = (0.1, 0.2, 0.3, 0.4), (0.6, 0.7, 0.8, 0.8) = M L = Medium Low, Poor = P = (0.0, 0.0, 0.1, 0.2), (0.7, 0.8, 0.8, 0.9) = L = Low. 1. The Linguistic assessments of the decision makers are represented by Tables 2.4, 2.5, 2.6, 2.7, and 2.8. 2. The aggregated weights of the criteria across two decision makers are given in Table 2.9. The aggregated values of alternatives, corresponding to considered conflicting criteria according to the judgements of decision makers, are given in Tables 2.4, 2.5, 2.6, 2.7, and 2.8. 3. The weighted bipolar fuzzy decision matrix for criteria C1 , C2 , and C3 is given in Table 2.10 and for criteria C4 and C5 in Table 2.11.

2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers

55

Table 2.7 The linguistic ratings and aggregated value of alternatives for C4 Criteria C4

Alternatives

Decision makers

Aggregate ratings

D1

D2

A1

MG

MG

(0.30, 0.35, 0.35, 0.40), (0.05, 0.01, 0.15, 0.15)

A2

MG

MG

(0.30, 0.35, 0.35, 0.40), (0.05, 0.01, 0.15, 0.15)

A3

P

MP

(0.25, 0.35, 0.45, 0.55), (0.55, 0.60, 0.70, 0.75)

A4

G

G

(0.40, 0.45, 0.50, 0.50), (0.00, 0.05, 0.10, 0.10)

A5

G

MG

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

A6

G

MG

(0.70, 0.80, 0.85, 0.90), (0.05, 0.15, 0.25, 0.25)

Table 2.8 The linguistic ratings and aggregated value of alternatives for C5 Criteria C5

Alternatives

Decision makers

Aggregate ratings

D1

D2

A1

F

ML

(0.25, 0.35, 0.45, 0.55), (0.55, 0.60, 0.70, 0.75)

A2

L

ML

(0.05, 0.10, 0.20, 0.30), (0.65, 0.75, 0.80, 0.85)

A3

H

H

(0.40, 0.45, 0.50, 0.50), (0.00, 0.05, 0.10, 0.10)

A4

L

L

(0.00, 0.00, 0.05, 0.10), (0.35, 0.40, 0.40, 0.45)

A5

ML

L

(0.05, 0.10, 0.20, 0.30), (0.65, 0.75, 0.80, 0.85)

A6

F

ML

(0.25, 0.35, 0.45, 0.55), (0.55, 0.60, 0.70, 0.75)

Table 2.9 Importance weights of each criteria and aggregate weights Decision makers Criteria C1 C2 C3 C4 D1 0.05 D2 0.1 Aggregated weights 0.075

0.3 0.25 0.275

0.25 0.25 0.25

0.25 0.3 0.275

C5 0.15 0.1 0.125

4. The bipolar fuzzy ideal solution is calculated and represented by A∗ and A− as A∗ ={(0.075, 0.015, 0.023, 0.030), (0.045, 0.053, 0.060, 0.060), (0.220, 0.248, 0.275, 0.275), (0.000, 0.028, 0.055, 0.055), (0.150, 0.175, 0.200, 0.213), (0.063, 0.075, 0.100, 0.113), (0.119, 0.220, 0.234, 0.248), (0.014, 0.041, 0.069, 0.069), (0.006, 0.013, 0.025, 0.038), (0.081, 0.094, 0.100, 0.106)}, A ={(0.060, 0.068, 0.075, 0.075), (0.00, 0.075, 0.015, 0.015), −

(0.193, 0.220, 0.023, 0.248), (0.014, 0.041, 0.069, 0.069), (0.175, 0.200, 0.0213, 0.225), (0.013, 0.004, 0.063, 0.063), (0.220, 0.248, 0.0275, 0.275), (0.000, 0.028, 0.055, 0.055), (0.100, 0.113, 0.125, 0.125), (0.000, 0.013, 0.025, 0.025)}.

56

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

Table 2.10 Weighted bipolar fuzzy decision matrix C1 A1

C2

(0.015, 0.019, 0.026, 0.034), (0.193, 0.220, 0.023, 0.248), (0.15, 0.175, 0.2, 0.213), (0.045, 0.049, 0.053, 0.060)

A2

A4

(0.014, 0.041, 0.069, 0.069)

(0.175, 0.2, 0.0213, 0.225), (0.013, 0.004, 0.063, 0.063)

(0.000, 0.028, 0.055, 0.055)

(0.045, 0.053, 0.053, 0.060), (0.193, 0.220, 0.234, 0.248), (0.175, 0.2, 0.0213, 0.225), (0.014, 0.041, 0.069, 0.069)

A2 A3 A4 A5 A6

(0.013, 0.004, 0.063, 0.063)

(0.030, 0.038, 0.045, 0.053), (0.138, 0.165, 0.179, 0.206), (0.175, 0.2, 0.0213, 0.225), (0.083, 0.096, 0.124, 0.138)

(0.013, 0.004, 0.063, 0.063)

(0.060, 0.068, 0.075, 0.075), (0.22, 0.248, 0.275, 0.275),

(0.175, 0.2, 0.0213, 0.225),

(0.0, 0.075, 0.015, 0.015)

(0.013, 0.004, 0.063, 0.063)

(0.0, 0.028, 0.055, 0.055)

Table 2.11 Weighted bipolar fuzzy decision matrix Alternatives Criteria C4 A1

(0.063, 0.075, 0.1, 0.113)

(0.008, 0.015, 0.023, 0.023)

(0.038, 0.038, 0.045, 0.053) A6

(0.063, 0.075, 0.1, 0.113)

(0.045, 0.053, 0.053, 0.060), (0.22, 0.248, 0.275, 0.275),

(0.008, 0.015, 0.023, 0.023) A5

(0.014, 0.041, 0.069, 0.069)

(0.075, 0.015, 0.023, 0.030), (0.193, 0.220, 0.023, 0.248), (0.15, 0.175, 0.2, 0.213), (0.045, 0.053, 0.060, 0.060)

A3

C3

(0.165, 0.193, 0.193, 0.220), (0.075, 0.055, 0.083, 0.083) (0.165, 0.193, 0.193, 0.220), (0.075, 0.055, 0.083, 0.083) (0.014, 0.028, 0.055, 0.0825), (0.1788, 0.206, 0.220, 0.234) (0.220, 0.248, 0.0275, 0.275), (0.000, 0.028, 0.055, 0.055) (0.193, 0.220, 0.234, 0.248), (0.014, 0.041, 0.069, 0.069) (0.193, 0.220, 0.234, 0.248), (0.014, 0.041, 0.069, 0.069)

C5 (0.031, 0.044, 0.056, 0.069), (0.069, 0.075, 0.088, 0.094) (0.006, 0.013, 0.025, 0.038), (0.081, 0.094, 0.100, 0.106) (0.100, 0.113, 0.125, 0.125), (0.000, 0.013, 0.025, 0.025) (0.000, 0.000, 0.013, 0.025), (0.088, 0.100, 0.100, 0.113) (0.006, 0.013, 0.025, 0.038), (0.081, 0.094, 0.100, 0.106) (0.031, 0.044, 0.056, 0.069), (0.069, 0.075, 0.088, 0.094)

5. The distance of each alternative from ideal solutions and the closeness coefficient of each alternative are computed in Table 2.12. 6. The ordering of alternatives, derived by closeness coefficients, is A6 , A2 , A1 , A5 , A3 , and A4 . Thus, A6 is most suitable project. A comparative analysis with other decision making methods. We now present theoretically comparison of the proposed method with already existing decision making techniques, i.e., TOPSIS and ELECTRE-I methods dealing with fuzzy and bipolar fuzzy information. In fact, we solved the considered problem by various multiple criteria decision making methods. A basic list should contain the Analytical Hierarchy

2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers

57

Table 2.12 Closeness coefficients of alternatives Alternatives Bipolar fuzzy positive Bipolar fuzzy negative Closeness coefficient ideal solution ideal solution di+ di− Ci A1 A2 A3 A4 A5 A6

0.2035 0.1934 0.4463 0.2296 0.2232 0.1780

0.2632 0.2874 0.4019 0.1987 0.2858 0.3100

0.5640 0.5978 0.4738 0.4639 0.5615 0.6352

Process (AHP), ELimination Et Choix Traduisant la REalite (ELECTRE), Technique for the order of preference to ideal solution (TOPSIS), and Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE). The TOPSIS method is considered to be one of the efficient multiple criteria decision making techniques that handles decision making problems effectively as it yields excellent rank inversion. Further, TOPSIS method is a utility-based strategy that directly determines each possible alternative by initial inputs, arranged in decision matrix. Fuzzy TOPSIS is frequently adopted to solve decision problems in real world. In the fuzzy TOPSIS method, the decision is made on the basis of linguistic evaluation rather than the exact numerical values. Although the TOPSIS technique is the most effective tool in fuzzy environments, it only deals with true membership of elements. However, the proposed trapezoidal bipolar fuzzy TOPSIS approach is regarded as a modified version of TOPSIS method as it addresses the multi-criteria decision making problems, whose input is in the form of trapezoidal bipolar fuzzy information. Bipolar logic in trapezoidal bipolar fuzzy TOPSIS approach is beneficial as it can resist the intrinsic difficulties experienced in the examination of ‘crisp’ values with the help of two parameters namely, chance of occurrence and chance of absence. The proposed approach encounters inherent uncertainty of the decision making problems using the significant notion of bipolar fuzzy linguistic variable. The bipolar linguistic values or linguistic terms appear as the elementary tools of bipolar fuzzy logic. To solve the decision making problems, bipolar fuzzy linguistic variables provide a direct correspondence between linguistic values and their corresponding bipolar fuzzy numbers which are usually taken as trapezoidal bipolar fuzzy numbers. In the existing literature, bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE-I methods, introduced by Alghmadi et al., are counted as the earliest developed bipolar fuzzy multi-criteria decision making methods based on TOPSIS and ELECTRE-I methods. The structures of their techniques were not broad enough to use the linguistic imprecision and the concept of triangular or trapezoidal bipolar fuzzy numbers. Thus, these methods cannot be practically implemented to those cases where the decision maker focuses on linguistic assessment based on bipolar judgemental thinking. A brief comparison

58

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers

Table 2.13 Comparison of the proposed method with other MCDM methods Fuzzy TOPSIS Bipolar fuzzy Bipolar fuzzy Trapezoidal bipolar TOPSIS ELECTRE-I fuzzy TOPSIS Group decision making model based on fuzzy linguistic preferences Linguistic terms are induced by fuzzy linguistic variables

Numerical computational model based on bipolar membership

Numerical computational model based on bipolar membership

Rating to the alternatives are assigned by linguistic values Fuzzy membership assigns to each linguistic value, a triangular fuzzy number The decision matrix comprises triangular fuzzy number

Bipolar fuzzy values are used to assign ratings to alternatives

Bipolar fuzzy values are used to assign ratings to alternatives

The decision matrix comprises bipolar fuzzy values

The decision matrix comprises bipolar fuzzy values

Group decision making model based on bipolar fuzzy linguistic preferences Linguistic terms are induced by bipolar fuzzy linguistic variables Rating to the alternatives are assigned by linguistic values Bipolar membership assigns to each linguistic value, a trapezoidal bipolar fuzzy number The decision matrix comprises trapezoidal bipolar fuzzy number

of the proposed method with fuzzy TOPSIS and bipolar fuzzy TOPSIS method and bipolar fuzzy ELECTRE-I method is elaborated in the Table 2.13.

2.7 Trapezoidal Bipolar Fuzzy Information System An Information system is a decision model that helps to organize and analyze data and to take rapid decisions in the selection of a feasible alternative from given alternatives under certain evaluation criteria. In an information system, preeminence or dominance relation totally depends on the ranking of evaluated data or information. In this section, trapezoidal bipolar fuzzy information system is defined and decision making in trapezoidal bipolar fuzzy information system using preeminence value (based on preeminence relation on the set of objects) is described. Definition 2.16 ([4]) An information system I = (X, C R, D, λ) with D =  D , where Dc is domain for criterion c is called trapezoidal bipolar fuzzy c c∈C R information system (TrBFIS) if D is a set of trapezoidal bipolar fuzzy numbers. We denote λ(x, c) ∈ Dc by λ(x, c) = [ι1 , ι2 , ι3 , ι4 ], [κ1 , κ2 , κ3 , κ4 ], where oi , τi ∈ [0, 1].

2.7 Trapezoidal Bipolar Fuzzy Information System

59

Definition 2.17 A trapezoidal bipolar fuzzy information system I = (X, C R, D, λ) together with W = {wc | c ∈ C R} is called weighted trapezoidal bipolar fuzzy information system (WTrBFIS) and is denoted by I = (X, C R, D, λ, W ). Definition 2.18 ([4]) Let c ∈ C R be a criterion and u i , u j ∈ X. If J1 (u i ) > J1 (u j ) or J1 (u i ) = J1 (u j ), J2 (u i ) > J2 (u j ) or Jk◦ (u i ) > Jk◦ (u j ) and Jk (u i ) = Jk (u j ), for all k < k◦ , then u i >c u j which indicates that u i is better than u j with respect to criterion c. Also u i =c u j indicates that u i is equal to u j if λ(u i , c) = λ(u j , c), with respect to criterion c. Definition 2.19 Let I = (X, C R, D, λ, W ) be a weighted trapezoidal bipolar fuzzy information system and C ⊆ C R. Let G C (u i , u j ) = {c ∈ BC | u i >c u j }, L C (u i , u j ) = {c ∈ CC | u i c u j or u j >c u i or u i =c u j for all c ∈ C (C ⊆ C R) and for all u i , u j ∈ X. 2. Numerate G C (u i , u j ) using G C (u i , u j ) = {c ∈ BC | u i >c u j }, L C (u i , u j ) using L C (u i , u j ) = {c ∈ CC | u i S(Ψ2 ), then Ψ1 Ψ2 (Ψ1 is superior to Ψ2 ); 3. If S(Ψ1 ) = S(Ψ2 ), then • If A(Ψ1 ) < A(Ψ2 ), then Ψ1 ≺ Ψ2 (Ψ1 is inferior to Ψ2 ); • If A(Ψ1 ) > A(Ψ2 ), then Ψ1 Ψ2 (Ψ1 is superior to Ψ2 ); • If A(Ψ1 ) = A(Ψ2 ), then Ψ1 ∼ Ψ2 (Ψ1 is equivalent to Ψ2 ).

4.2 Complex Bipolar Fuzzy Sets In this section, the theoretical framework of an advanced complex bipolar fuzzy set is presented along with its fundamental operational laws. In order to determine the closeness and remoteness of any two complex bipolar fuzzy numbers, a pioneer normalized Euclidean distance measure is also presented.

4.2 Complex Bipolar Fuzzy Sets

97

Definition 4.6 Let Z be a universe of discourse. A complex bipolar fuzzy set K over the universe Z is an object of the form: K = {< z, r p (z)eiξ

p

(z)

, −r n (z)eiξ

n

(z)

> | z ∈ Z },

where r p (z) ∈ [0, 1] represents the amplitude term of positive membership grade, r n (z) ∈ [0, 1] represents the amplitude term of negative membership grade, and ξ p (z), ξ n (z) ∈ [0, 2π] represent the phase terms of positive and negative membership degrees of the complex bipolar fuzzy set K , respectively. The negative sign (−) with complex-valued negative membership grade represents some counter-property p n of a complex bipolar fuzzy set. The pair (r p eiξ , −r n eiξ ) can be referred as complex bipolar fuzzy number.

p p n  p n Definition 4.7 Let Ψ = r p eiξ , − r n eiξ and Ψ = r eiξ , − rn eiξ ( = 1, 2) be any three complex bipolar fuzzy numbers and  > 0 be any real number. Then, the operations based on these three complex bipolar fuzzy numbers can be explicated as follows: 

ξ p  ξ p  ξ p  ξ p 

ξn  ξn  

p 1 2 1 2 1 2 p p p 1. Ψ1 ⊕ Ψ2 = r1 + r2 − r1 r2 ei2π 2π + 2π − 2π π , −r1n r2n ei2π 2π 2π ;  n  n  n  n    

ξ p  ξ p 

ξ1 ξ2 ξ1 ξ2  i2π 1 2 2π + 2π − 2π 2π p p 2. Ψ1 ⊗ Ψ2 = r1 r2 ei2π 2π 2π , − r1n + r2n − r1n r2n e ;     i2π 1− 1− ξ p   n  i2π ξn 



2π 2π ; ,− r e e 3. Ψ = 1 − 1 − r p  

ξ p 





   

ξn . 4. Ψ  = (r p ) ei2π 2π , − 1 − 1 − r n ) ei2π 1− 1− 2π

p p n  Definition 4.8 Let Ψ = r eiξ , − rn eiξ ( = 1, 2, . . . , f ) be finite collection of complex bipolar fuzzy numbers and  be the corresponding weightage of Ψ such f  that  ∈ [0, 1] and  = 1. Then, the complex bipolar fuzzy weighted aver=1

aging operator can be characterized as follows: Complex bipolar fuzzy weighted averaging operator=1 Ψ1 ⊕ 2 Ψ2 ⊕ · · · ⊕  f Ψ f         i2π 1− f 1− ξp  f  f  

n  i2π 2π p =1 1− 1 − r e ,− = e r =1



f  =1



ξn    2π

.

=1

(4.4)

p iξ p

p iξ p n n n n iξ iξ Definition 4.9 Let Ψ1 = r1 e 1 , − r1 e 1 and Ψ2 = r2 e 2 , − r2 e 2 be any two complex bipolar fuzzy numbers. Then, the normalized Euclidean distance measure between Ψ1 and Ψ2 can be delineated as follows:

d(Ψ1 , Ψ2 ) =

  p  r − r p 2 + r n − r n 2 +  1 2 1 2

1 4π 2

4



2 

p p 2 ξ1 − ξ2 + ξ1n − ξ2n

. (4.5)

98

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

Definition 4.10 The score function for complex bipolar fuzzy number, Ψ =

p iξ p n r e  , − rn eiξ can be quantified as follows: S(Ψ ) =

    ξp ξn 1  p r − rn + 1 +  −  + 1 , 2 2π 2π

(4.6)

where S(Ψ ) ∈ [0, 2]. Definition 4.11 The accuracy function for complex bipolar fuzzy number, Ψ =

p iξ p n r e  , − rn eiξ can be evaluated as follows:     ξp ξn  1  p n  A(Ψ ) = r + r + + , 2 2π 2π

(4.7)

where A(Ψ ) ∈ [0, 2].

p p

p p n n Definition 4.12 Let Ψ1 = r1 eiξ1 , − r1n eiξ1 and Ψ2 = r2 eiξ2 , − r2n eiξ2 be any two complex bipolar fuzzy numbers, then the comparison of these two complex bipolar fuzzy numbers is characterized as follows: 1. If S(Ψ1 ) < S(Ψ2 ), then Ψ1 ≺ Ψ2 (Ψ1 is inferior to Ψ2 ); 2. If S(Ψ1 ) > S(Ψ2 ), then Ψ1 Ψ2 (Ψ1 is superior to Ψ2 ); 3. If S(Ψ1 ) = S(Ψ2 ), then • If A(Ψ1 ) < A(Ψ2 ), then Ψ1 ≺ Ψ2 (Ψ1 is inferior to Ψ2 ); • If A(Ψ1 ) > A(Ψ2 ), then Ψ1 Ψ2 (Ψ1 is superior to Ψ2 ); • If A(Ψ1 ) = A(Ψ2 ), then Ψ1 ∼ Ψ2 (Ψ1 is equivalent to Ψ2 ).

4.3 Structure of Complex Bipolar Fuzzy VIKOR Method In this section, a pioneer multi-criteria optimization technique, namely, complex bipolar fuzzy VIKOR method is presented to address the multi-criteria group decision making problems fruitfully on the basis of complex bipolar fuzzy information. The proposed cognitive strategy is proficiently designed for the identification of compromise solution with maximum group utility and minimum individual regret in the light of “acceptable advantage” and “acceptable stability” of decision mechanism. This systematic technique focuses on the hierarchical categorization of feasible alternatives in the existence of conflicting criteria by precisely quantifying their weighted proximity from ideal values. Specification of Multi-criteria Group Decision Making Problem Consider a decision making problem comprising a set of f feasible alternatives A = {A1 , A2 , A3 , . . . , A f } from which the most proficient alternative has to be determined by critically analyzing their competency and adeptness relative to some particular conflicting criteria. Let t conflicting criteria C = {C1 , C2 , C3 , . . . , Ct } are

4.3 Structure of Complex Bipolar Fuzzy VIKOR Method

99

identified by the appointed decision makers for the holistic appraisement of considered multi-criteria group decision making problem. Let a board of g decision making experts Y = {Y1 , Y2 , Y3 , . . . , Yg } is authorized to examine the proceedings of decision making problem incorporating selection of rational criteria, consignment of their weightage and evaluation of the functionality of considered alternatives. The normalized weight vector λ = (λ1 , λ2 , λ3 , . . . , λg ) along with the conditions g  λk ∈ [0, 1] and λk = 1, interprets the expertise and reliability of decision makers k=1

according to the nature and requirement of inspected problem. Methodology of Complex Bipolar Fuzzy VIKOR Strategy For the analytical scrutinization of afore-mentioned multi-criteria group decision making problem equipped with complex bipolar fuzzy information, the developed complex bipolar fuzzy VIKOR strategy is presented to specify the most efficient alternative in the presence of conflicting criteria. The structural framework of complex bipolar fuzzy VIKOR method can be exemplified in the following steps: 1. Construction of individual decision matrices of experts. The authorized decision experts of the panel precisely inspect the proficiency of each alternative regarding decision-criteria on the basis of their expertise. The subjective interpretations of experts are represented in terms of linguistic terms which are further symbolized in the form of complex bipolar fuzzy numbers. These complex bipolar fuzzy assessments of decision makers relative to their linguistic evaluations are organized systematically in the form of g independent complex bipolar fuzzy decision matrices D (k) = (D(k) hs ) f ×t as follows: ⎛

D (k)

n (k) n (k) (Ω11 )(k) , (Ω11 )  (Ω12 )(k) , (Ω12 )  · · · (Ω1t )(k) , (Ω1tn )(k)  p

p

p

⎞

⎜ ⎟ ⎜ ⎟ p (k) p (k) p (k) n (k) n (k) n (k) ⎟ ⎜ (Ω21 ) , (Ω )  (Ω ) , (Ω )  · · · (Ω ) , (Ω )  22 2t 21 22 2t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ ⎟, .. .. . . . . ⎜ ⎟ . . . . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ p p p n n n (k) (k) (k) (k) (k) (k) ⎝(Ω f 1 ) , (Ω f 1 )  (Ω f 2 ) , (Ω f 2 )  · · · (Ω f t ) , (Ω f t ) ⎠

where the subscript h (h = 1, 2, . . . , f ) corresponds to the investigated alternative Ah , subscript s (s = 1, 2, . . . , t) refers to the identified decision-criterion Cs and superscript k (k = 1, 2, . . . , g) reflects the assessments of decision makers Yk . The p n (k) p (k) p (k) i(ξhs )(k) n (k) n )(k) ei(ξhs )  of D (k) is the complex , −(rhs entry D(k) hs = (Ωhs ) , (Ωhs )  = (rhs ) e bipolar fuzzy numbers representing the interpretations of decision maker Yk to describe the competency of Ah alternative relative to criterion Cs . 2. Determination of aggregated complex bipolar fuzzy decision matrix. In group decision making conundrum, the individual assessments of all decision makers are assembled into a cumulative decision which is acceptable for all the appointed

100

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

experts of the decision making board. This leads to the formulation of aggregated ˜ hs can be ˜ hs ) f ×t , whose entries D complex bipolar fuzzy decision matrix D˜ = (D evaluated by deploying the following complex bipolar fuzzy weighted averaging operator: (g) (1) (2) (3) ˜ D hs = λ1 Dhs ⊕ λ2 Dhs ⊕ λ3 Dhs ⊕ · · · ⊕ λg Dhs

  g  n (k)    g   ξ p (k) λ  ξhs λk    k hs   i2π 1− 1− g   λ i2π 2π 2π

p (k) λk  n (k) k e k=1 k=1 ,− rhs 1− 1 − rhs , e k=1 k=1



=

g  

(4.8) where h = 1, 2, . . . , f and s = 1, 2, . . . , t. The mutual interpretations of all the nominated decision makers can be expressed in the following aggregated complex bipolar fuzzy decision matrix: ⎛˜ ˜ 1t ⎞ ˜ 12 · · · D D11 D ⎟ ⎜ ⎟ ⎜ ⎜D ˜ 2t ⎟ ˜ 21 D ˜ 22 · · · D ⎟ ⎜ ⎟ ⎜ D˜ = ⎜ ⎟, ⎟ ⎜ . . . . .. . . .. ⎟ ⎜ .. ⎟ ⎜ ⎠ ⎝ ˜ ft ˜ f1 D ˜ f2 ··· D D ˜ hs = Ω˜ p , Ω˜ n  = ˜r p ei ξ˜hsp , −˜r n ei ξ˜hsn  of aggregated complex where the entry D hs hs hs hs (g) (2) bipolar fuzzy decision matrix is the accumulated value of D(1) hs , Dhs , . . . , Dhs , symbolizing a collaborative perspective of all the experts about a feasible alternative relative to conflicting decision-criterion. 3. Quantification of normalized weights of decisive criteria. In multiple-criteria analysis, the relative prominence of specified criteria may not be equivalent corresponding to the judgments of all experts of the decision making board. By investigating the motive and specification of the inspected multi-criteria group decision making problem, the authorized decision makers subjectively determine the importance of these criteria with the aid of linguistic variables which can be further represented in the form of complex bipolar fuzzy numbers. These independent reflections of all the experts regarding significance of each criterion are accumulated to construct the complex bipolar fuzzy weightage vector of decisive criteria, by utilizing the following complex bipolar fuzzy weighted averaging operator: (g) (1) (2) (3) Υ¨s == λ1 Υs ⊕ λ2 Υs ⊕ λ3 Υs ⊕ · · · ⊕ λg Υs  g  n (k)    g   ξ p (k) λ  λk    ξs k s    i2π 1− 1− i2π g  g

  2π 2π

p (k) λk   k=1 k=1 1 − rs e ,− rsn (k) λk e = 1− , k=1 k=1

(4.9)

4.3 Structure of Complex Bipolar Fuzzy VIKOR Method

101

¨n p p ¨ where Υ¨s = Ω¨ s , Ω¨ sn  = ¨rs ei ξs , −¨rsn ei ξs  represents the aggregated weightage of specified criterion Cs , evaluated by the assemblage of Υs(1) , Υs(2) , Υs(3) , . . . , (g) Υs . The normalized weightage Ψs of each decisive criterion can be formulated by employing accumulated complex bipolar fuzzy weight Υ¨s of that criterion, defined as follows: p



  ξ¨sp  p r¨s + ¨ p 2π ¨ n p ξ ξs r¨s + r¨sn + 2πs 2π Ψs = p    ξ¨s  , p t  r¨s 2π + ¨p p ξ¨ n ξs r¨s + r¨sn s=1 + s 2π

(4.10)

2π

where Ψs ∈ [0, 1] and satisfy the normality constraint

t 

Ψs = 1. The weightage

s=1

Ψs is enumerated by the accumulative perspective of all designated experts of the panel, illustrating the relative importance of each identified criterion in the considered problem. 4. Construction of score matrix. The quantified complex bipolar fuzzy entries of aggregated complex bipolar fuzzy decision matrix are compared by translating them into crisp values with the assistance of score function in order to determine the best and worst values of considered alternatives regarding decision-criteria. As a result of defuzzification, the score matrix M = (Shs ) f ×t is constructed, whose entries Shs can be computed as follows: Shs

 p  n   ξ˜hs ξ˜hs 1 p n r˜hs − r˜hs + 1 + − +1 , = 2 2π 2π

(4.11)

where h = 1, 2, . . . , f and s = 1, 2, . . . , t. The fabricated score matrix can be represented in the following decorum: ⎛

S11 S12 · · · S1t

⎞

⎜ ⎟ ⎜ ⎟ ⎜ S21 S22 · · · S2t ⎟ ⎜ ⎟ ⎜ ⎟ M =⎜ ⎟, ⎜ .. .. . . .. ⎟ ⎜ . . . ⎟ . ⎜ ⎟ ⎝ ⎠ Sf1 Sf2 ··· Sft where Shs indicates the computed score degree of complex bipolar fuzzy entry Dhs of aggregated complex bipolar fuzzy decision matrix which is further utilized to locate the best and worst values of given alternatives for the investigated problem. 5. Identification of best and worst values. Our next target is to specify the best and worst values relative to each decision-criterion by analyzing the nature of

102

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

that criterion in the scrutinized multi-criteria group decision making problem. Let C X represents the collection of all cost-type criteria and C B indicates the ˜ ∗s corresponding to collection of all benefit-type criteria. Then, the best value D decision-criterion Cs can be evaluated as follows: ˜ ∗s = D

⎧ ˜ qs : S(D ˜ qs ) = max S(D ˜ hs ), if Cs ∈ C B , ⎨D 1≤h≤ f

˜ qs ) = min S(D ˜ hs ), if Cs ∈ C X . ˜ qs : S(D ⎩D

(4.12)

1≤h≤ f

˜ Analogously, the worst value D s with respect to decision-criterion C s can be computed as follows: ˜ D s =

⎧ ˜ qs ) = min S(D ˜ hs ), if Cs ∈ C B , ˜ qs : S(D ⎨D 1≤h≤ f

˜ qs ) = max S(D ˜ hs ), if Cs ∈ C X , ˜ qs : S(D ⎩D

(4.13)

1≤h≤ f

˜ ∗s = (Ω˜ sp )∗ , (Ω˜ sn )∗  = (˜rsp )∗ ei(ξ˜sp )∗ , −(˜rsn )∗ ei(ξ˜sn )∗  symbolizes the best where D value in the sense that it minimizes all the cost-type criteria and maximizes ˜p  p ˜ ˜ p  ˜n  all the benefit-type criteria, whereas D rs ) ei(ξs ) , s = (Ωs ) , (Ωs )  = (˜ ˜n  −(˜rsn ) ei(ξs )  represents the worst value which maximizes the cost-type criteria but minimizes all the benefit-type criteria of the scrutinized problem. 6. Computation of normalized Euclidean distance measures. After the evaluation of ideal values, the normalized Euclidean distance measures of best values from the worst values and the respective complex bipolar fuzzy entries of aggregated complex bipolar fuzzy decision matrix are quantified, which are further employed for the formulation of group utility measure and individual regret measure of the ˜ ˜ ∗s from D alternatives. These normalized Euclidean distance measures of D s and ˜ Dhs relative to decision-criterion can be computed as follows:     p  (˜r )∗ − (˜r p ) 2 + (˜r n )∗ − (˜r n ) 2 + 1 (ξ˜ p )∗ − (ξ˜ p ) 2 + (ξ˜ n )∗ − (ξ˜ n ) 2  s s s s s s s s 2 ∗  4π ˜ ˜ d(Ds , Ds ) = , 4

(4.14)

    p  (˜r )∗ − (˜r p )2 + (˜r n )∗ − (˜r n )2 + 1 (ξ˜ p )∗ − (ξ˜ p )2 + (ξ˜ n )∗ − (ξ˜ n )2  s s s s hs hs hs hs 4π 2 ˜ ∗s , D ˜ hs ) = d(D , 4

(4.15) ˜ ∗s and D ˜ where h = 1, 2, . . . , f and s = 1, 2, . . . , t. Here, D s represent the best ˜ hs indicates the and worst values regarding selected decision-criterion while D accumulated complex bipolar fuzzy entry of aggregated complex bipolar fuzzy decision matrix of the investigated problem. 7. Enumeration of S, R, and Q. The next step is to evaluate the group utility measure and individual regret measure of each alternative in order to identify the most preferable feasible choice. The effectuality and credibility of each inspected

4.3 Structure of Complex Bipolar Fuzzy VIKOR Method

103

alternative is explicated on the basis of these group utility measure S and individual regret measure R, which can be determined as follows: t 

Ψs

˜ ∗s , D ˜ hs ) d (D , ˜ ∗s , D ˜ d (D s )

(4.16)

Rh = max Ψs

˜ ∗s , D ˜ hs ) d (D , ˜ ∗s , D ˜ d (D s )

(4.17)

Sh =

s=1

s

where Ψs corresponds to the normalized weightage of criterion Cs , illustrating its relative importance in the considered problem. Eventually, both group utility measure and individual regret measure are merged in a single expression to evaluate the ranking measure Q of each alternative, which is defined as follows:     Rh − R∗ Sh − S ∗ + (1 − ) , (4.18) Qh =  S − S∗ R − R∗ where S ∗ = min Sh , h

S  = max Sh , h

R∗ = min Rh , R = max Rh . h

h

The solution determined by S ∗ corresponds to the maximum group utility. Similarly, the solution evaluated by R∗ reflects the minimum individual regret of the opponent. The parameter  ∈ [0, 1] symbolizes the weight for the strategy of maximum group utility, while (1 − ) indicates the weight for minimum individual regret. Generally, ( = 0.5) is prioritized to obtain a compromise solution embracing both features of maximum group utility and minimum individual regret of the opponent. The value of  is interpreted by scrutinizing the nature and specification of the examined decision making problem, as the desired compromise solution can be identified with “voting by majority” ( > 0.5), with “consensus” ( = 0.5) or with “veto” ( < 0.5). 8. Hierarchical ranking of alternatives. After the quantification of S, R and Q, the alternatives are sorted by organizing them in an ascending order by the dint of group utility measure, individual regret measure and ranking measure. These evaluated hierarchical classifications of inspected alternatives render three worthwhile canonical-orderings, which are further utilized for the identification of compromise solution. The alternative possessing a minimum value of Q will be specified as the feasible alternative to the contemplated problem. 9. Determination of compromise solution. The last step of this technique is to specify the required compromise solution of the investigated multi-criteria group decision making problem. Let A(1) be the alternative possessing minimum value of

104

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

Q. Then, the alternative A(1) will be interpreted as the ideal compromise solution if it suffices the following two conditions: C 1: Acceptable advantage. According to this condition, the alternative A(1) must comply the following relation: Q(A(2) ) − Q(A(1) ) ≥ D Q, where A(2) represents the alternative which is ordered at second position in 1 and f indicates the total number of the hierarchical list of Q, D Q = f −1 scrutinized alternatives. C 2: Acceptable stability in decision making. This condition enumerates that alternative A(1) will also be prioritized best relative to S or/and R. This proposed compromise solution must be persistent and stable within the decision making process of inspected problem. • The alternatives A(1) , A(2) , . . . , A(η) will be the members of the proposed set of compromise solutions if condition C 1 does not hold, where A(η) is determined by the following expression: Q(A(η) ) − Q(A(1) ) < D Q, for maximum value of η. • The alternatives A(1) and A(2) will constitute the identified set of compromise solutions of the investigated problem in the case if only condition C 2 is not satisfied. The structural framework of the presented complex bipolar fuzzy VIKOR method is presented with the assistance of explicative flowchart, as illustrated in Fig. 4.2.

4.4 Application In this section, an interesting multi-criteria group decision making problem related to selection of the most proficient medical diagnostic technology is presented by implementing complex bipolar fuzzy VIKOR method to authenticate the effectuality and feasibility of the presented decision making technique. Example 4.1 (Selection of medical diagnostic technology) Diagnostic medical imaging is a series of different techniques used to create visual representation of various internal structures of the human body for monitoring abnormalities and for the identification of relevant medical treatments. It has revolutionized the medical industry by providing essential digital imaging tools and attracted the attention of many scientists and practitioners to reveal more about the anatomy of the human body. The field of medical imaging plays an important role in diagnosis, appropriate evaluations, and documenting courses of many health issues as well as in the interpretation

4.4 Application

105 Establish a panel of decision making experts and allocate them normalized weights

Specify the relative decision-criteria

Specify the list of alternatives

Scrutinize each alternative by individual appraisement of decision making experts

Complex bipolar fuzzy decision matrices

D (1)

D (2)

...

D (g)

Construct the aggregated complex bipolar fuzzy decision matrix Formulate the score matrix

Evaluate the normalized weights of decision-criteria

˜ ∗ and worst D ˜ values Determine the best D s s Compute the normalized Euclidean distance measures ˜ ∗ from D ˜ and D ˜ hs of D s

s

Quantify the value of Rh

Quantify the value of Sh

Eumerate the value of Qh Categorize the alternatives in ascending order relative to S, R and Q

Is No

Q(A(2) ) − Q(A(1) ) ≥ DQ?

Yes

The solution set comprises A(1) , A(2) , . . . , A(η) , where A(η) is determined by Q(A(η) ) − Q(A(1) ) < DQ

Is No

A(1) is also prioritized best with respect to S or R?

The solution set comprises both A(1) and A(2)

Yes Identify the alternative A(1) as optimal compromise solution

Fig. 4.2 Flowchart of complex bipolar fuzzy VIKOR method

106

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

of responses to treatment. Nowadays, the advanced diagnostic technologies with better image resolution have become a key component of digital health. The selection of competent medical diagnostic equipment is one of the most tricky decision making problems due to the accessibility of a wide range of scientific imaging modalities with various innovative features and specifications in global markets. To accomplish the current need, suppose that a prestigious healthcare center is intended to purchase an effective diagnostic imaging technology for the advancement of their holistic medical facilities. The universe of six diagnostic equipments A = {A1 , A2 , A3 , . . . , A6 } is considered after analyzing their multiple potentialities, where A1 : Positron emission tomography scanner: This ideal medical equipment uses a particular type of injectable radioactive tracers to clearly visualize the metabolic processes and various physiological activities. It is configured for proper identification of heart diseases, cancer, and multiple neurological disorders. A2 : Fluoroscopic scanner: This special imaging device produces a continuous moving image of tissues, organs, and other internal structures of the human body on a video screen. The non-invasive procedures of this machine guide doctors to diagnose ailments and injuries related to bones, joints, and digestive system. A3 : X-ray film scanner: This powerful diagnostic equipment consists of two main components, the X-ray generator and an image detection system. Its wide beam of electromagnetic waves are used to capture the structure of bones, stomach, intestines and for the detection of other pathological changes in lungs. A4 : Magnetic resonance imaging scanner: This high-quality imaging technology employs strong magnetic fields and radio waves to produce detailed crosssectional images of the internal organs and tissues of the body. It is designed to visualize the anomalies of joints, brain, spinal cord, and certain heart problems. A5 : Computerized tomography scanner: This medical equipment uses a versatile combination of rotating X-ray machines and computers to produce 3-D clear images of several visceral parts of the human body. It is widely used to monitor spine, blood vessels, tumors, brain, soft tissues, and sundry internal organs. A6 : Thermographic scanner: This diagnostic machine has two parts, the infrared camera and a standard computer which is used for detecting skeletal, muscular, and nervous system irregularities. It usually captures the heat emitted from various parts of the body in the form of an isothermal distribution map. To investigate this complicated multi-criteria group decision making problem, the supreme authorities of healthcare center establish a board of three decision making experts Y = {Y1 , Y2 , Y3 } to critically evaluate the proficiency of each medical technology for the development of their endowed health services, where Y1 : Medical specialist, Y2 : Technical specialist, Y3 : Finance specialist. The relative significance and credibility of appointed experts are illustrated by the normalized weight vector λ = (0.4635, 0.2343, 0.3022). All the experts of the

4.4 Application

107

authorized board determine the pragmatic factors C = {C1 , C2 , C3 , C4 } influencing the selection process of diagnostic equipment with the assistance of their substantiated mutual decision, where C1 : Exposure of Radiation: The medical technologies are the greatest source of ionizing radiations which badly affect DNA and internal healthy tissues of patients. The imaging equipment with low radiation exposure is prioritized in order to minimize the potential health risks for both patients and health specialists. C2 : Investment and maintenance cost: The investment and maintenance cost of diagnostic machines have a great impact on the progressive development of medical centers. The imaging technologies with minimum total cost can save money, strengthen the operational efficiency and increase the profitability of healthcare centers. C3 : Energy consumption: The consumption of energy plays an indispensable role in the selection of productive medical imaging equipment. The energy-saving diagnostic technologies are preferable as they reduce the overall computational expenditures and improve the efficiency of their different types of significant operations. C4 : Temporal and spatial resolution: The diagnostic imaging technologies are characterized by its temporal and spatial resolution as they are the most important parameters of imaging modalities. The high-quality temporal and spatial resolution enable imaging devices to capture microscopic structures of the human anatomy. The main purpose of this study is to identify the optimum digital imaging technology in the light of cost-benefit analysis, required for the provision of computerized medical facilities in the considered healthcare center. The ideal selection of constructive medical equipment with best image resolution will provide assistance to the medical practitioner in competent monitoring of various abnormalities and in the identification of substantial treatment modalities by clearly visualizing the anatomy of the human body which in turn significantly accelerates the profitable development of healthcare centers. The framework for the specification of the investigated multi-criteria group decision making problem is illustrated in Fig. 4.3. The procedure for selecting the most bountiful medical imaging equipment using presented complex bipolar fuzzy VIKOR method is demonstrated in the following elaborative steps: 1. The board of decision making experts rigorously analyze the credibility of six pragmatic diagnostic technologies with respect to each identified decisioncriterion and exhibit their judgements by the dint of specific linguistic variables, as presented in Tables 4.1 and 4.2. 2. The individual linguistic reflections of all decision makers regarding proficiency of inspected medical technologies are encapsulated in the form of complex bipolar fuzzy decision matrices. The independent complex bipolar fuzzy decision matrices D (1) , D (2) , …, D (g) of appointed experts are summarized in Tables 4.3, 4.4 and 4.5, respectively.

108

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets Medical specialist

Decision making experts

Technical specialist

Finance specialist

Positron emission tomography scanner

Fluoroscopic scanner

X-ray film scanner

Identification of inspected problem

Medical diagnostic technologies

Magnetic resonance imaging scanner Computerized tomography scanner Thermographic scanner

Exposure of Radiation

Investment and maintenance cost Identified decision-criteria Energy consumption:

Temporal and spatial resolution

Fig. 4.3 Specification of inspected problem Table 4.1 Linguistic terms for rating of diagnostic technologies Linguistic terms Abbreviation Complex bipolar fuzzy numbers Remarkably good Very very good Very good Good Fairly good Satisfactory Fairly bad Bad Very bad Very very bad Remarkably bad

RG VVG VG G FG S FB B VB VVB RB

0.99ei1.86π , −0.02ei0.05π  0.83ei1.63π , −0.17ei0.08π  0.79ei1.44π , −0.24ei0.16π  0.68ei1.21π , −0.36ei0.68π  0.62ei1.17π , −0.39ei0.82π  0.55ei1.00π , −0.41ei0.85π  0.37ei0.86π , −0.62ei1.13π  0.29ei0.62π , −0.73ei1.24π  0.24ei0.18π , −0.78ei1.45π  0.19ei0.09π , −0.84ei1.62π  0.06ei0.05π , −0.97ei1.85π 

4.4 Application

109

Table 4.2 Linguistic assessment of diagnostic technologies Experts Technologies

Criteria

Y1

Y2

A1

C1

FG

G

B

C2

FB

VVB

G

C3

G

S

VVG

C4

S

VVG

VG

C1

S

VB

VG

C2

VB

FG

RB

C3

RG

VG

S

C4

VVB

FB

RG

C1

RB

VG

S

C2

VG

B

FG

C3

FB

RG

VB

C4

VVG

S

FB

C1

RG

VVG

RG

C2

VB

VVB

VB

C3

VVB

RB

RB

C4

VG

VVG

VG

C1

VB

G

G

C2

RB

VVG

VVB

C3

G

FB

FG

C4

RG

VB

B

C1

VG

S

VVB

C2

VVB

RB

FB

C3

S

FB

VG

C4

FB

RG

G

A2

A3

A4

A5

A6

Y3

Table 4.3 Complex bipolar fuzzy decision matrix of Y1 D (1) C1

C2

C3

C4

A1

0.62ei1.17π , −0.39ei0.82π  0.37ei0.86π , −0.62ei1.13π  0.68ei1.21π , −0.36ei0.68π  0.55ei1.00π , −0.41ei0.85π 

A2

0.55ei1.00π , −0.41ei0.85π  0.24ei0.18π , −0.78ei1.45π  0.99ei1.86π , −0.02ei0.05π  0.19ei0.09π , −0.84ei1.62π 

A3

0.06ei0.05π , −0.97ei1.85π  0.79ei1.44π , −0.24ei0.16π  0.37ei0.86π , −0.62ei1.13π  0.83ei1.63π , −0.17ei0.08π 

A4

0.99ei1.86π , −0.02ei0.05π  0.24ei0.18π , −0.78ei1.45π  0.19ei0.09π , −0.84ei1.62π  0.79ei1.44π , −0.24ei0.16π 

A5

0.24ei0.18π , −0.78ei1.45π  0.06ei0.05π , −0.97ei1.85π  0.68ei1.21π , −0.36ei0.68π  0.99ei1.86π , −0.02ei0.05π 

A6

0.79ei1.44π , −0.24ei0.16π  0.19ei0.09π , −0.84ei1.62π  0.55ei1.00π , −0.41ei0.85π  0.37ei0.86π , −0.62ei1.13π 

3. The subjective interpretations of all decision makers of the authorized panel are accumulated by employing the complex bipolar fuzzy weighted averaging operator, as defined in Eq. (4.8), in order to represent their collaborative decision in the form of aggregated complex bipolar fuzzy decision matrix. Then, their accumulated outcomes relative to the performance of imaging technologies are presented in Table 4.6.

110

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

Table 4.4 Complex bipolar fuzzy decision matrix of Y2 D (2) C1

C2

C3

C4

A1

0.68ei1.21π , −0.36ei0.68π  0.19ei0.09π , −0.84ei1.62π  0.55ei1.00π , −0.41ei0.85π  0.83ei1.63π , −0.17ei0.08π 

A2

0.24ei0.18π , −0.78ei1.45π  0.62ei1.17π , −0.39ei0.82π  0.79ei1.44π , −0.24ei0.16π  0.37ei0.86π , −0.62ei1.13π 

A3

0.79ei1.44π , −0.24ei0.16π  0.29ei0.62π , −0.73ei1.24π  0.99ei1.86π , −0.02ei0.05π  0.55ei1.00π , −0.41ei0.85π 

A4

0.83ei1.63π , −0.17ei0.08π  0.19ei0.09π , −0.84ei1.62π  0.06ei0.05π , −0.97ei1.85π  0.83ei1.63π , −0.17ei0.08π 

A5

0.68ei1.21π , −0.36ei0.68π  0.83ei1.63π , −0.17ei0.08π  0.37ei0.86π , −0.62ei1.13π  0.24ei0.18π , −0.78ei1.45π 

A6

0.55ei1.00π , −0.41ei0.85π  0.06ei0.05π , −0.97ei1.85π  0.37ei0.86π , −0.62ei1.13π  0.99ei1.86π , −0.02ei0.05π 

Table 4.5 Complex bipolar fuzzy decision matrix of Y3 D (3) C1

C2

C3

C4

A1

0.29ei0.62π , −0.73ei1.24π  0.68ei1.21π , −0.36ei0.68π  0.83ei1.63π , −0.17ei0.08π  0.79ei1.44π , −0.24ei0.16π 

A2

0.79ei1.44π , −0.24ei0.16π  0.06ei0.05π , −0.97ei1.85π  0.55ei1.00π , −0.41ei0.85π  0.99ei1.86π , −0.02ei0.05π 

A3

0.55ei1.00π , −0.41ei0.85π  0.62ei1.17π , −0.39ei0.82π  0.24ei0.18π , −0.78ei1.45π  0.37ei0.86π , −0.62ei1.13π 

A4

0.99ei1.86π , −0.02ei0.05π  0.24ei0.18π , −0.78ei1.45π  0.06ei0.05π , −0.97ei1.85π  0.79ei1.44π , −0.24ei0.16π 

A5

0.68ei1.21π , −0.36ei0.68π  0.19ei0.09π , −0.84ei1.62π  0.62ei1.17π , −0.39ei0.82π  0.29ei0.62π , −0.73ei1.24π 

A6

0.19ei0.09π , −0.84ei1.62π  0.37ei0.86π , −0.62ei1.13π  0.79ei1.44π , −0.24ei0.16π  0.68ei1.21π , −0.36ei0.68π 

Table 4.6 Aggregated complex bipolar fuzzy decision matrix D˜

C1

A1

0.55ei1.04π , −0.46ei0.88π  0.45ei0.84π , −0.56ei1.05π  0.71ei1.33π , −0.29ei0.37π  0.71ei1.33π , −0.28ei0.29π 

A2

0.59ei1.03π , −0.40ei0.58π  0.31ei0.45π , −0.70ei1.36π  0.93ei1.64π , −0.08ei0.15π  0.79ei1.23π , −0.25ei0.52π 

A3

0.47ei0.81π , −0.53ei0.82π  0.66ei1.22π , −0.36ei0.42π  0.74ei1.19π , −0.29ei0.58π  0.68ei1.34π , −0.30ei0.30π 

A4

0.98ei1.82π , −0.03ei0.05π  0.22ei0.15π , −0.79ei1.48π  0.12ei0.06π , −0.90ei1.73π  0.80ei1.49π , −0.22ei0.13π 

A5

0.52ei0.83π , −0.51ei0.96π  0.39ei0.68π , −0.61ei0.85π  0.60ei1.12π , −0.41ei0.81π  0.89ei1.49π , −0.13ei0.29π 

A6

0.62ei1.07π , −0.39ei0.47π  0.22ei0.35π , −0.79ei1.49π  0.61ei1.13π , −0.38ei0.54π  0.80ei1.37π , −0.23ei0.46π 

C2

C3

C4

4. The identified linguistic variables corresponding to the importance of selected decision-criteria are represented in Table 4.7. All the designated experts of the panel examine the relative significance of each criterion and rank them with the assistance of their allocated linguistic terms and complex bipolar fuzzy numbers, as given in Tables 4.8 and 4.9, respectively. The assigned complex bipolar fuzzy weightage of conflicting criteria relative to their prominence in the inspected problem are aggregated by utilizing the complex bipolar fuzzy weighted averaging operator which are further normalized in the light of Eq. (4.10), as highlighted in Table 4.10. 5. The score degree of each accumulated complex bipolar fuzzy entry of aggregated complex bipolar fuzzy decision matrix is evaluated, as defined in Eq. (4.11) in order to transform these entries into crisp numerical values. The estimated results are further illustrated in the form of score matrix, as presented in Table 4.11. 6. The selected decision-criteria such as investment cost and energy consumption are the cost-type criteria, whereas exposure of radiation and spatial resolution are

4.4 Application

111

Table 4.7 Specified linguistic variables for importance of criteria Linguistic variables Abbreviation Complex bipolar fuzzy numbers Extremely significant Highly significant Moderately significant Average Moderately insignificant Highly insignificant Extremely insignificant

Table 4.8 Prominence of conflicting criteria C/Y Y1 C1 C2 C3 C4

0.98ei0.84π , −0.12ei0.03π  0.77ei0.68π , −0.21ei0.33π  0.61ei0.72π , −0.39ei0.28π  0.55ei0.47π , −0.43ei0.59π  0.39ei0.27π , −0.62ei0.73π  0.24ei0.31π , −0.78ei0.69π  0.13ei0.08π , −0.92ei0.86π 

ES HS MS A MI HI EI

HS MS A ES

Y2

Y3

ES A HS A

A HS ES MS

Table 4.9 Weightage of criteria relative to their significance Y2

Y3

C1

Criteria Y1 0.77ei0.68π , −0.21ei0.33π 

0.98ei0.84π , −0.12ei0.03π 

0.55ei0.47π , −0.43ei0.59π 

C2

0.61ei0.72π , −0.39ei0.28π 

0.55ei0.47π , −0.43ei0.59π 

0.77ei0.68π , −0.21ei0.33π 

C3

0.55ei0.47π , −0.43ei0.59π 

0.77ei0.68π , −0.21ei0.33π 

0.98ei0.84π , −0.12ei0.03π 

C4

0.98ei0.84π , −0.12ei0.03π 

0.55ei0.47π , −0.43ei0.59π 

0.61ei0.72π , −0.39ei0.28π 

Table 4.10 Normalized weights of decisive criteria Criteria Aggregated complex bipolar fuzzy weights C1 C2 C3 C4

0.84ei0.66π , −0.22ei0.22π  0.65ei0.65π , −0.33ei0.35π  0.84ei0.64π , −0.24ei0.20π  0.89ei0.72π , −0.23ei0.11π 

Normalized weights 0.2540 0.2180 0.2540 0.2740

identified as benefit-type criteria. The best and worst values of medical technologies are determined by using Eqs. (4.12) and (4.13), corresponding to the nature of each decisive criterion and embedded the quantified results in Table 4.12. 7. The normalized Euclidean distance between the best and worst values regarding each identified decision-criterion is evaluated by using Eq. (4.14), as presented in Table 4.13. Similarly, Eq. (4.15) is deployed to calculate the distance of each

112

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

Table 4.11 Score matrix M M C1 A1 A2 A3 A4 A5 A6

1.08676 1.20840 0.96221 1.91587 0.97125 1.26116

C2

C3

C4

0.89378 0.57348 1.35188 0.38524 0.84918 0.42975

1.44915 1.79679 1.37755 0.18989 1.17194 1.26102

1.47591 1.45020 1.44540 1.62834 1.67995 1.51234

Table 4.12 Best and worst values relative to decision-criteria ˜ ∗s ) Criteria Best value(D 0.98ei1.82π , −0.03ei0.05π  0.22ei0.15π , −0.79ei1.48π  0.12ei0.06π , −0.90ei1.73π  0.89ei1.49π , −0.13ei0.29π 

C1 C2 C3 C4

˜ Worst value(D s ) 0.47ei0.81π , −0.53ei0.82π  0.66ei1.22π , −0.36ei0.42π  0.93ei1.64π , −0.08ei0.15π  0.68ei1.34π , −0.25ei0.30π 

˜ ∗s from D ˜ Table 4.13 Distance of D s Distance ˜ ∗s , D ˜ d(D s )

C1

C2

C3

C4

0.477847

0.486216

0.802605

0.126639

˜ ∗, D ˜ h3 ) d(D 3

˜ ∗, D ˜ h4 ) d(D 4

0.629176 0.802225 0.593019 0.005094 0.490454 0.535482

0.122576 0.115740 0.141788 0.074752 0.007040 0.085593

Table 4.14 Normalized Euclidean distance measures ˜ ∗, D ˜ h1 ) ˜ ∗, D ˜ h2 ) Distances d(D d(D 1 2 h h h h h h

=1 =2 =3 =4 =5 =6

0.415694 0.358287 0.480503 0.002363 0.473374 0.334879

0.261411 0.101661 0.487017 0.005592 0.241018 0.052210

entry of aggregated complex bipolar fuzzy decision matrix from the best value relative to selected criterion and encapsulated these distances in Table 4.14. 8. The group utility measure and individual regret measure of investigated medical technologies are quantified by using Eqs. (4.16) and (4.17), respectively. Further, the ranking measure is evaluated by deploying Eq. (4.18) and taking the weight of strategy as 0.5. All these computed measures relative to each considered imaging equipment are assembled in Table 4.15. 9. All the medical technologies of the scrutinized multi-criteria group decision making problem are hierarchically categorized in an ascending order corresponding

4.5 Comparative Analysis

113

Table 4.15 Values of S , R and Q for diagnostic technologies Technologies S R A1 A2 A3 A4 A5 A6

0.802492 0.740325 0.968221 0.167111 0.530132 0.556068

0.265208 0.253880 0.306777 0.161736 0.251622 0.185190

Table 4.16 Hierarchical ranking of diagnostic technologies Ranking 1 2 3 4 S R Q

A4 A4 A4

A5 A6 A6

Q

A6 A5 A5

A2 A2 A2

0.753263 0.675410 1.000000 0.000000 0.536440 0.323617

5

6

A1 A1 A1

A3 A3 A3

to their enumerated group utility measure, regret measure, and ranking measure. All these systematic rankings of the inspected technologies are demonstrated evidently in Table 4.16. 10. The diagnostic technology A4 is prioritized best having least value of ranking measure that suffices the following two conditions of the proposed complex bipolar fuzzy VIKOR method: • Q(A6 ) − Q(A4 ) = 0.323617 − 0.000000 = 0.323617 1 ≥ 6−1 = 0.2, where medical equipment A6 is ordered at second position with respect to ranking measure. • The diagnostic technology A4 is also prioritized as the best option corresponding to both group utility measure and regret measure. Thus, we conclude that magnetic resonance imaging technology (A4 ) will be spotlighted as the most productive diagnostic imaging technology possessing maximum group utility and minimum individual regret of the opponent.

4.5 Comparative Analysis In this section, a comparative analysis of presented complex bipolar fuzzy VIKOR strategy with existing multi-criteria decision making technique, namely, bipolar fuzzy TOPSIS method, is elaborated. An empirical application named selection of medi-

114

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

cal diagnostic technology is presented by implementing the bipolar fuzzy TOPSIS method to substantiate the rationality and versatility of our established decision making technique. Now the pragmatic application Example 4.1 is scrutinized by applying the systematic methodology of bipolar fuzzy TOPSIS method, proposed by Akram et al. [5]. 1. The first step is to construct the bipolar fuzzy decision matrix by analyzing the proficiency of each technology relative to identified criterion. Since bipolar fuzzy TOPSIS method is a simple multi-criteria decision making technique, therefore, the bipolar fuzzy entries of bipolar fuzzy decision matrix D = (Dhs )6×4 are obtained by omitting the phase terms of aggregated complex bipolar fuzzy p n  of bipodecision matrix. The formulated bipolar fuzzy entries Dhs = rhs , rhs lar fuzzy decision matrix regarding capabilities of diagnostic technologies are summarized in Table 4.17. 2. The selected decision-criteria of the inspected problem are ranked on the basis of their relative importance by assigning them weights Ψs , which satisfy the t  conditions Ψs ∈ [0, 1] and Ψs = 1. The allocated normalized weights of each s=1

specified criterion corresponding to their significance are presented in Table 4.18. 3. The bipolar fuzzy weighted decision matrix is constructed by multiplying the bipolar fuzzy decision matrix to the weightage vector of decision-criteria. The ˆ hs = ˆr p , rˆ n  of bipolar fuzzy weighted decision matrix Dˆ = (D ˆ hs )6×4 entries D hs hs are evaluated in the following manner: p

p

rˆhs = Ψs rhs ,

n n rˆhs = Ψs rhs ,

where h = 1, 2, . . . , 6 and s = 1, 2, . . . , 4. Then, the computed bipolar fuzzy entries of the bipolar fuzzy weighted decision matrix are further encapsulated in Table 4.19.

Table 4.17 Bipolar fuzzy decision matrix D C1 C2 A1 A2 A3 A4 A5 A6

0.55, −0.46 0.59, −0.40 0.47, −0.53 0.98, −0.03 0.52, −0.51 0.62, −0.39

0.45, −0.56 0.31, −0.70 0.66, −0.36 0.22, −0.79 0.39, −0.61 0.22, −0.79

Table 4.18 Normalized weightage of criteria Ψ /C C1 C2 Ψs

0.2540

0.2180

C3

C4

0.71, −0.29 0.93, −0.08 0.74, −0.29 0.12, −0.90 0.60, −0.41 0.61, −0.38

0.71, −0.28 0.79, −0.25 0.68, −0.30 0.80, −0.22 0.89, −0.13 0.80, −0.23

C3

C4

0.2540

0.2740

4.5 Comparative Analysis

115

Table 4.19 Bipolar fuzzy weighted decision matrix Dˆ C1 C2 0.141, −0.116 0.150, −0.102 0.118, −0.135 0.247, −0.008 0.131, −0.129 0.156, −0.100

A1 A2 A3 A4 A5 A6

0.099, −0.123 0.067, −0.154 0.145, −0.078 0.049, −0.173 0.086, −0.134 0.048, −0.173

C3

C4

0.180, −0.074 0.236, −0.022 0.189, −0.075 0.031, −0.229 0.153, −0.106 0.155, −0.097

0.197, −0.078 0.220, −0.069 0.188, −0.085 0.220, −0.061 0.248, −0.038 0.222, −0.064

4. The next step is to identify the bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution corresponding to each selected criterion, which are quantified as follows: p + n + P = { P+ s =  (r s ) , (r s )  | s = 1, 2, 3, 4 }, p − n − N = { N− s =  (r s ) , (r s )  | s = 1, 2, 3, 4 },

where (rsp )+ = max rˆhs ,

n (rsn )+ = max rˆhs ,

(rsp )−

(rsn )−

p

h

=

p min rˆhs , h

h

n = min rˆhs . h

The identified bipolar fuzzy positive ideal solution P and bipolar fuzzy negative ideal solution N relative to decision-criteria are highlighted in Table 4.20. 5. The Euclidean distance of each diagnostic technology from the identified bipolar fuzzy positive ideal solution as well as the bipolar fuzzy negative ideal solution are evaluated for the computation of relative closeness degree. These distance measures of technologies are formulated by employing the formulas as given below

Table 4.20 Ideal solutions relative to decision-criteria Criteria Bipolar fuzzy positive ideal solution (P ) C1 C2 C3 C4

0.247, −0.008 0.048, −0.173 0.031, −0.229 0.248, −0.038

Bipolar fuzzy negative ideal solution (N ) 0.118, −0.135 0.145, −0.078 0.236, −0.022 0.188, −0.085

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4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

d(Ah , P) =

d(Ah , N ) =

  4 2 %

n   $ p p + 2  + (ˆrhs ) − (rsn )+  s=1 (ˆrhs ) − (rs ) 2   4 2 %

n   $ p p − 2  + (ˆrhs ) − (rsn )−  s=1 (ˆrhs ) − (rs ) 2

,

(4.19)

.

(4.20)

The computed Euclidean distance measures of each technology by using Eqs. (4.19) and (4.20) are summarized in Table 4.21. 6. The next step is to determine the relative closeness degree (Υh ) of each imaging equipment in order to measure the proximity of each alternative from bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution, which is evaluated as follows:

Υ (Ah ) =

d(Ah , N ) , h = 1, 2, . . . , 6. d(Ah , P) + d(Ah , N )

(4.21)

The formulated relative closeness degree of each diagnostic technology on the basis of their Euclidean distance measures from bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution are provided in Table 4.22. 7. After the quantification of Υh , the diagnostic technologies are ranked in descending order by the assistance of their corresponding relative closeness degree, as presented in Table 4.23. Hence, it is concluded that magnetic resonance imaging technology (A4 ) will be selected as the best diagnostic technology.

Table 4.21 Euclidean distance measures Technologies d(Ah , P ) A1 A2 A3 A4 A5 A6

d(Ah , N )

0.19827 0.23013 0.22999 0.02518 0.17504 0.15955

0.07414 0.08708 0.05019 0.26244 0.11516 0.13198

Table 4.22 Relative closeness degree of each technology Technologies A1 A2 A3 A4 Relative closeness degree (Υ )

0.272161

0.274518

0.179121

0.912466

A5

A6

0.396820

0.452707

4.5 Comparative Analysis

117

Table 4.23 Ranking of diagnostic technologies Technologies A1 A2 A3 Ranking

4

6

5

A4

A5

A5

2

1

3

Discussion A comparison of presented strategy is presented with contemporary MCDM technique, namely, bipolar fuzzy TOPSIS method to demonstrate the sustain-ability and authenticity of presented complex bipolar fuzzy VIKOR method. The methodological implications of both compared and developed techniques are persuasively highlighted in Table 4.24. Both presented and compared strategies prioritize A4 as the most proficient medical diagnostic technology for the provision of computerized imaging modalities in the considered healthcare center which vindicates the rationality and enforceability of decision making specialities of presented technique in the heuristic multi-criteria group decision making problems. An explicative bar chart is presented in Fig. 4.4 in order to picturize the comparative results of both compared and presented multi-criteria decision making methodologies regarding hierarchical ranking of diagnostic technologies, which depicts the plausibility and effectuality of presented strategy. The hierarchical ranking of the inspected diagnostic technologies is same in both compared and presented strategies. Actually, the compared method just handles the imprecision of only one-dimensional information while presented technique models both aspects of two-dimensional ambiguous information. Both compared and presented techniques interpret the similar final outcome which reflects the feasibility of the presented strategy. Presented technique has an edge over the compared technique as it identifies the compromise solution by critically analyzing the weighted distances of alternatives from ideal solutions but the compared method determines the optimal solution without considering the importance and weightage of these distances which may lead to the inaccurate decisions when some identified decision-criteria conflict with other ones. This remarkable feature of presented strategy stimulates it as the most generalized and versatile decision making technique for real-life multi-criteria group decision making problems.

Table 4.24 Comparative analysis Methods Proposed complex bipolar fuzzy VIKOR method Bipolar fuzzy TOPSIS method [5]

Ranking

Best technology

A4 A6 A5 A2 A1 A3

A4

A4 A6 A5 A2 A1 A3

A4

118

4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

Marvellous Remarkable Good Bad

Satisfactory

Hierarchical Ordering

Optimal

Complex bipolar fuzzy VIKOR method Bipolar fuzzy TOPSIS method [5]

S1

S2

S3

S4

S5

S6

Medical Diagnostic Technologies

Fig. 4.4 Comparative study

The presented complex bipolar fuzzy VIKOR strategy has great potential to convincingly address the uncertainty of bipolar fuzzy information by omitting the phase terms of complex bipolar fuzzy numbers and display the same outcomes as derived from compared technique. However, contemporary bipolar fuzzy TOPSIS method fails to capture the periodicity of complex bipolar fuzzy information because its methodology is confined to deal with only one-dimensional data which may deduce inconsistent end-results. This incredible flexibility of presented strategy makes it more powerful as compared to the existing technique.

4.6 Merits of the Presented Method 1. Presented complex bipolar fuzzy VIKOR technique robustly provides a versatile and most proficient mathematical framework for the determination of compromise solution by the accomplishment of its remarkable principle which maximizes the group utility and minimizes the individual regret of the opponent after holistically scrutinizing the rational set of pragmatic alternatives on the basis of identified conflicting criteria. 2. The presented multi-criteria optimization strategy has an incredible peculiarity that its feasible methodology integrates the outstanding modeling specialities of complex bipolar fuzzy model with the ground-breaking theory of VIKOR method. Another major contribution of the presented technique lies in its dynamic competencies to address the periodicity and vagueness of given data at the same time in a persuasive decorum.

4.7 Conclusion

119

3. The structural framework of presented technique dramatically assists appointed decision makers to investigate the capabilities of plausible alternatives by focusing on two independent dominant factors of benefit-type and cost-type conflicting selected criteria of the inspected problem. This cost-benefit assessment of the developed strategy renders more precise and accurate end-results as compared to existing multi-criteria group decision making strategies. 4. In the modern era, a wide range of empirical decision making problems in the domains of artificial intelligence, computational psychiatry, medical sciences, and quantum computing require complex bipolar assessments in order to evaluate the inspected alternatives on the basis of bipolar decision factors. Therefore, presented strategy is especially designed for addressing both polarity and fuzziness of inconsistent periodic data at a time in order to meet the future goals productively. 5. The feasibility and decision making expertise of presented strategy are not only restricted for the modeling of two-dimensional imprecise data but also this technique displays the similar authentic results when applied to one-dimensional obscure data inclusive of bipolar fuzzy data by omitting their respective phase terms. Thus, presented multi-criteria group decision making strategy presents such a multi-skilled and vigorous decision making tool that amazingly captures both traditional and two-dimensional vague information with precision.

4.7 Conclusion In this Chapter, the most generalized and incredible theory of complex bipolar fuzzy sets has been presented. Its outstanding representational potentials have admirably extrapolated the contemporary approaches by addressing both fuzziness and polarity of ambiguous data simultaneously. Further, a competent decision making technique, known as complex bipolar fuzzy VIKOR method has been presented, which integrates remarkable specialities of complex bipolar fuzzy model and ground-breaking theory of VIKOR method. In the complex bipolar fuzzy VIKOR method, the subjective linguistic interpretations of all the appointed experts regarding proficiency of alternatives and priorities of conflicting criteria have been holistically accumulated using the novel complex bipolar fuzzy weighted averaging operator. Another remarkable contribution of this study is the accomplishment of the presented framework on real-life pragmatic application for the selection of expedient diagnostic technology which demonstrates the accountability and potentiality of our presented strategy. The compatibility and persistency of comparative results have been deliberated with the assistance of an explicative bar chart. Finally, the functionality of the presented strategy has drastically been analyzed to throw light on its merits and dominance over the existing approaches. The presented methodology has an edge over the contem-

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4 Extended VIKOR Method with Complex Bipolar Fuzzy Sets

porary decision making strategies as it robustly determines the feasible solution that admirably satisfies all the identified conflicting criteria of an optimization problem. Additional Reading The readers are suggested to [4, 6, 8–10, 17, 23, 30, 32, 41, 46] for definitions of additional terms and applications not included in this chapter.

References 1. Akram, M., Ali, M., Allahviranloo, T.: A method for solving bipolar fuzzy complex linear systems with real and complex coefficients. Soft Comput. 26, 2157–2178 (2022) 2. I Akram, M., Ali, M., Allahviranloo, T.: Solution of complex bipolar fuzzy linear system. In: Allahviranloo, T., Salahshour, S., Arica, N. (eds.) Progress in Intelligent Decision Science. IDS 2020. Advances in Intelligent Systems and Computing, vol 1301. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-66501-2-73 3. Akram, M., Arshad, M.: A novel trapezoidal bipolar fuzzy TOPSIS method for group decisionmaking. Group Decis. Negot. 28(3), 565–584 (2019) 4. Akram, M., Sarwar, M., Dudek, W.A.: Graphs for the analysis of bipolar fuzzy information. Studies in Fuzziness and Soft Computing, vol. 401. Springer, Berlin (2021) 5. Akram, M., Shumaiza, Arshad, M.: Bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE-I methods to diagnosis. Comput. Appl. Math. 39(7), 1–21 (2020) 6. Akram, M., Shumaiza, Smarandache, F.: Decision-making with bipolar neutrosophic TOPSIS and bipolar neutrosophic ELECTRE-I. Axioms 7(2), 33 (2018) 7. Alghamdi, M.A., Alshehri, N.O., Akram, M.: Multi-criteria decision making methods in bipolar fuzzy environment. Int. J. Fuzzy Syst. 20(6), 2057–2064 (2018) 8. Alkouri, A.M., Salleh, A.R.: Complex intuitionistic fuzzy sets. AIP Conf. Proc. 1482(1), 464– 470 (2012) 9. Alsolame, B., Alshehri, N.O.: Extension of VIKOR method for MCDM under bipolar fuzzy set. Int. J. Anal. Appl. 18(6), 989–997 (2020) 10. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 11. Baidya, J., Garg, H., Saha, A., Mishra, A.R., Rani, P., Dutta, D.: Selection of third party reverses logistic providers: an approach of BCF-CRITIC-MULTIMOORA using archimedean power aggregation operators. Complex Intell. Syst. 7(5), 2503–2530 (2021) 12. Bazzazi, A.A., Osanloo, M., Karimi, B.: Deriving preference order of open pit mines equipment through MADM methods: application of modified VIKOR method. Expert Syst. Appl. 38(3), 2550–2556 (2011) 13. Benayoun, R., Roy, B., Sussman, N.: Manual de reference du programme electre. Note De Synthese Et Formation. 25, 79 (1966) 14. Brans, J.P., Vincle, P.V.: A preference ranking organization method. Manag. Sci. 31, 647–656 (1985) 15. Cristobal, ´ J.R.S.: Multi-criteria decision-making in the selection of a renewable energy project in Spain: the VIKOR method. Renew. Ener. 36(2), 498–502 (2011) 16. Fei, L., Deng, Y., Hu, Y.: DS-VIKOR: a new multi-criteria decision-making method for supplier selection. Int. J. Fuzzy Syst. 21, 157–175 (2019) 17. Gupta, P., Mehlawat, M.K., Grover, N.: Intuitionistic fuzzy multi-attribute group decisionmaking with an application to plant location selection based on a new extended VIKOR method. Inf. Sci. 370–371, 184–203 (2016) 18. Hu, J., Zhang, X., Yang, Y., Liu, Y., Chen, X.: New doctors ranking system based on VIKOR method. Int. Trans. Oper. Res. 27(2), 1236–1261 (2020) 19. Hwang, C.L., Yoon, K.: Multiple Attributes Decision Making: Methods and Applications. Springer, Berlin Heidelberg (1981)

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43. Zeng, S., Chen, S.M., Kuo, L.W.: Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method. Inf. Sci. 488, 76–92 (2019) 44. Zeng, Q.L., Li, D.D., Yang, Y.B.: VIKOR method with enhanced accuracy for multiple criteria decision making in healthcare management. J. Med. Syst. 37, 9908 (2013) 45. Zhang, W.R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of the IEEE Conference Fuzzy Information Processing Society Biannual Conference, pp. 305–309 (1994) 46. Zhang, W.-R.: YinYang Bipolar Relativity. IGI Global (2011)

Chapter 5

Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

This chapter aims to deliver a new multiple-criteria decision making model, namely, the bipolar fuzzy ELECTRE II method, by combining the traits of bipolar fuzzy sets with the ELECTRE II technique. The proposed approach can successfully address the imprecision of bipolar fuzzy information. This method examines the strong and weak outranking relations between alternatives using bipolar fuzzy strong, median, and weak concordance as well as discordance sets and an indifferent set. Further, an optimization technique, based on the maximizing deviation, is applied to evaluate the criteria weights. The ranking list is derived through strongly and weakly outrank graphs in finite number of iterations. The practical scope is highlighted by presenting two applications for the selection of optimal business location and the choice of a best supplier. A comparative analysis of the ELECTRE II method with existing multipleattribute decision making methods including TOPSIS and ELECTRE I methods under the common bipolar fuzzy environment is presented. And it is performed in the framework of the problem of business location. This chapter basically owes to [39].

5.1 Introduction In previous chapters, we have argued that multiple-criteria decision making techniques have proliferated due to their utilization in multiple fields. They include AHP [36], TOPSIS [29], VIKOR [32], ELECTRE [16] and PROMETHEE [17]. Their targets may be either to rank the alternatives or to find the kernel solution. The classical multi-criteria decision making methods accept the initial ratings of alternatives with respect to criteria only in the form of exact numerical values. Bellman and Zadeh [15] introduced a decision making model for imprecise and obscure data using the concept of fuzzy sets. After that, many researchers put their efforts to develop new approaches to utilize the imprecise data for reliable decisions. A series of outranking © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Akram et al., Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-99-0569-0_5

123

124

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

methods is present in the literature to rank the alternative according to their performances. The ELECTRE method and its derivatives have played a significant role in the cause of trustworthy decisions, as explained in Chap. 1. To recap, ELECTRE was put forward Benayoun et al. [16] in 1966. Later, Roy [33] expanded this theory in 1968 and renamed this variant as ELECTRE I. The further impressive research in this field yields the modern variants, namely, ELECTRE II, ELECTRE III, ELECTRE IV, ELECTRE IS, and ELECTRE TRI. Hatami-Marbini and Tavana [27] worked on ELECTRE I method using fuzzy information for group of decision making. Akram et al. [4, 6, 9] expanded the theory of ELECTRE methods by presenting new methodologies within the bipolar neutrosophic, Pythagorean fuzzy and hesitant Pythagorean fuzzy environments. Nowadays many applications of the ELECTRE methods have been given, including business management [23], genetic research [1], energy technology [14], assessment for coal gasification [42], and many more [18, 19, 40]. The ELECTRE I method excels among the outranking approaches, but it does not deliver the ranking of the alternatives. The solution, addressing the ineptness of ELECTRE I approach, was presented by Roy and Bertier [34] who established a modified version named ELECTRE II method. The ELECTRE II method ranks the alternatives with the assistance of three concordance and two discordance threshold levels. The performance of the alternatives with respect to each criterion is given via crisp and precise values. With this method, Duckstein and Gershon [25] gave a multicriteria reasoning of the vegetation management. Haung and Chen [28] demonstrated the application of ELECTRE II method in differentiation methodology. In relation to our targets, Govindan et al. [26] improved the ELECTRE II method so that it could be used to rank alternatives defined by uncertain information in the sense of Zadeh’s fuzzy set theory [43]. Later, Devadoss and Rekha [24] worked on ELECTRE II method for intuitionistic model that can process the ambiguous information in terms of both membership and non-membership values [12]. Wang et al. [41] put forward another possibility-based version ELECTRE II method using linguistic fuzzy variables. Liao et al. [31] and Chen and Xu [20] further developed new ELECTRE II-based techniques within the hesitant fuzzy environment [13]. Some other ELECTRE II methods have been discussed elsewhere [7, 8, 10, 37]. In previous chapters, we explained how Zhang [44] proposed the idea of bipolar fuzzy set in 1994. Its influence on multi-criteria decision making has also been discussed. To summarize, we have mentioned that Alghamdi et al. [11] adapted TOPSIS and ELECTRE I methods to the bipolar fuzzy environment. Akram and Arshad [2] explored trapezoidal bipolar fuzzy TOPSIS method. Akram et al. [5] implemented the bipolar fuzzy approaches for medical diagnosis. Then, Shumaiza et al. [38] contributed to the expanding literature on bipolar fuzzy information. The VIKOR method for Pythagorean fuzzy sets has been recently approached by Khan et al. [30], whereas in the same environment, Akram et al. [3] continued to develop competent decision making approaches. The goal of this chapter is to formalize the design of an ELECTRE II method within the context of bipolar fuzzy information. The bipolar fuzzy ELECTRE II

5.2 Bipolar Fuzzy ELECTRE II Method

125

method that arises is established by the concept of bipolar fuzzy strong, median, and weak concordance as well as discordance sets and indifferent set that later systematically derive the outranking relations. The proposed procedure evaluates normalized criteria weights by employing the concept of deviation. Lastly, a convenient iterative scheme is applied to extract the final ranking from strongly and weakly outrank graphs. We are keen to illustrate the implementation of this methodology with numerical examples in the context of the selection of optimal business location and the choice of a best supplier. We shall also perform a comparative analysis of the ELECTRE II method whereby it is tested against existing multiple-attribute decision making methods (TOPSIS and ELECTRE I methods) in the bipolar fuzzy framework. The comparison is applied to the case of the problem of business location.

5.2 Bipolar Fuzzy ELECTRE II Method We go on to propose a new multi-criteria decision making technique called bipolar fuzzy ELECTRE II method. It will take advantage of the bipolar fuzzy set within the methodology established by ELECTRE II method. In these steps, the proposed technique will find a solution to the multi-criteria decision making problem defined in the context of bipolar fuzzy information. 1. Establish a decision matrix. Consider a multi-criteria decision making problem involving the bipolar fuzzy information which basically aims to investigate a set of r alternatives P = {P1 , P2 , . . . , Pr } according to s (possibly conflicting) criteria K = {K 1 , K 2 , . . . , K s }. The preference values of the alternatives Pm , m = 1, 2, . . . , r based on criteria K n , n = 1, 2, . . . , s are assembled in decision matrix as ⎡ ⎤ 11 12 . . . 1r ⎢ 21 22 . . . 2r ⎥ ⎢ ⎥ L=⎢ . . . . ⎥. ⎣ .. .. . . .. ⎦ s1 s2 . . . sr Each entry mn = (μmn , νmn ) represents a bipolar fuzzy value where μmn ∈ [0, 1], νmn ∈ [−1, 0] represent the grades of satisfaction and dissatisfaction, respectively. Furthermore, the weight vector ω = (ω1 , ω2 , . . . ωs ), represents the criteria weight satisfying the condition of normality , i.e., ωn ∈ [0, 1] and s  ωn = 1. All criteria may not have same impact on the alternatives. Theren=1

fore, their individual impact is evaluated via an optimization method relying on maximizing deviation technique. Thus, the formal formula for the evaluation of normalized criteria weight is given by Eq. (5.1) as

126

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method r  r 

ωn =

|μmn − μkn | + |νmn − νkn |

m=1k=1 s r  r  

.

(5.1)

|μmn − μkn | + |νmn − νkn |

n=1 m=1k=1

2. Concordance, indifference, and discordance sets. The basic idea of the ELECTRE II method is to determine the preference relations for each pair of alternatives evaluated on the basis of conflicting and multiple criteria. This preference relationship, also known as the ranking relationship, is the key to the ranking method. Outranking methods are developed on concordance and discordance principles which help to investigate the outranking relationships among considered alternatives. The alternatives with bipolar fuzzy evaluations can be distinguished based on their satisfaction and dissatisfaction grades. The higher the satisfaction grade for a pair of alternatives (Pα , Pβ ) represents the preference of alternative Pα to Pβ with respect to some criteria n. The bipolar fuzzy concordance sets are further assorted in bipolar fuzzy strong, median and weak concordance sets by examining the membership and non-membership grades of bipolar fuzzy evaluations. Similarly, the bipolar fuzzy discordance sets are further classified as bipolar fuzzy strong, median, and weak discordance sets by exploring the membership and non-membership grades of bipolar fuzzy assessments. The bipolar fuzzy concordance set for a pair of alternatives (Pα , Pβ ), {α, β = 1, 2, . . . , r, α = β}, contains that criteria such that the alternative Pα possess larger membership grade as compared to Pβ , that is, the alternative Pα is more favorable to Pβ and whose different categories are described as follows. (i) The bipolar fuzzy strong concordance set K αβ is characterized as K αβ = {n | μαn ≥ μβn , ναn < νβn }.

(5.2)

(ii) The bipolar fuzzy median concordance set K αβ  is characterized as K αβ  = {n | μαn > μβn , ναn = νβn }.

(5.3)

(iii) The bipolar fuzzy weak concordance set K αβ  is characterized as K αβ  = {n | μαn ≥ μβn , ναn > νβn },

(5.4)

where μαn and μβn represent the satisfaction grades of alternatives Pα and Pβ , {α, β = 1, 2, . . . , r, α = β}, respectively. In the same way, ναn and νβn show dissatisfaction grades of alternatives Pα and Pβ , {α, β = 1, 2, . . . , r, α = β}, respectively. These bipolar fuzzy concordance sets define the possible stages or levels to which the alternative Pα is superior to Pβ . The value of dissatisfaction serves as an indicator to place alternatives within the different concordance sets,

5.2 Bipolar Fuzzy ELECTRE II Method

127

i.e., K αβ and K αβ  . In strong concordance set, representing the highest form of superiority captures all those alternatives for which the non-membership value of αth alternative is strictly less than the βth alternative with respect to the criteria n which differentiate it from the median concordance set. Similarly, the greater non-membership value of weak concordance set shows that the median concordance set represents a more strict form of dominance than weak concordance set. To exhibit the similar performance between two alternatives Pα , Pβ , the bipolar fuzzy indifferent set Iαβ is characterized as Iαβ = {n | μαn = μβn , ναn = νβn }.

(5.5)

The definition of the presented indifferent set depicts that both of the alternatives Pα and Pβ are indifferent or equivalent to each other. Discordance sets are actually the complementary subsets of concordance sets to capture the counter behavior. The bipolar fuzzy discordance set for any pair of alternatives (Pα , Pβ ), {α, β = 1, 2 . . . , r, α = β}, comprises the criteria for which alternative Pα is inferior than Pβ . The mathematical conditions are described as follows: (i) The bipolar fuzzy strong discordance set Dαβ is characterized as Dαβ = {n | μαn < μβn , ναn > νβn }.

(5.6)

(ii) The bipolar fuzzy median discordance set Dαβ  is characterized as Dαβ  = {n | μαn < μβn , ναn = νβn }.

(5.7)

(iii) The bipolar fuzzy weak discordance set Dαβ  is characterized as Dαβ  = {n | μαn < μβn , ναn < νβn }.

(5.8)

It is obvious that for all bipolar fuzzy discordance subsets the alternative Pα possess smaller membership value than Pβ which represents the inferiority of alternative Pα to Pβ . 3. Bipolar fuzzy concordance matrix. Next task is the evaluation of concordance indices for every pair of alternatives according to respective concordance sets. These indices are building blocks or elements of the concordance matrix. The bipolar fuzzy concordance matrix F = ( f αβ )r ×r is represented by, ⎡

− f 21 .. .

f 12 − .. .

⎢ ⎢ ⎢ F =⎢ ⎢ ⎣ f (r −1)1 f (r −1)2 fr 1 fr 2

⎤ . . . f 1(r −1) f 1r . . . f 2(r −1) f 2r ⎥ ⎥ .. .. ⎥ . .. . . . ⎥ ⎥ . . . − f (r −1)r ⎦ . . . fr (r −1) −

128

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

The bipolar fuzzy concordance index f αβ ∈ [0, 1] α, β = 1, 2 . . . , r, α = β, is computed by employing the summation of the normalized weights ωn ∈ [0, 1] related to the nth criteria of the respective concordance and indifferent sets. Thus, the formula for the computation of bipolar fuzzy concordance index f αβ is given as     ωn + ωc  × ωn + ωc × ωn + ω = × ωn , f αβ = ωc × n∈K αβ

n∈K αβ 

n∈K αβ 

n∈Iαβ

(5.9) where ωc , ωc , ωc , and ω = denote weights of the bipolar fuzzy strong, median and weak concordance and indifferent sets, respectively, which are assigned by decision expert. 4. Bipolar fuzzy discordance matrix. Nextly discordance indices for each pair of alternatives have to be determined according to the corresponding discordance sets. These indices are building blocks or elements of the discordance matrix. The bipolar fuzzy discordance matrix T = (tαβ )r ×r can be established as ⎡

− t21 .. .

t12 − .. .

⎢ ⎢ ⎢ T =⎢ ⎢ ⎣ t(r −1)1 t(r −1)2 tr 1 tr 2

⎤ . . . t1(r −1) t1r . . . t2(r −1) t2r ⎥ ⎥ .. .. ⎥ . .. . . . ⎥ ⎥ . . . − t(r −1)r ⎦ . . . tr (r −1) −

The bipolar fuzzy discordance index tαβ ∈ [0, 1] α, β = 1, 2 . . . , r, α = β, being contrary to concordance index f αβ , represents the measure that how much alternative Pα is worse than the alternative Pβ . Thus, the mathematical formula to calculate the bipolar fuzzy discordance index tαβ is given by Eq. (5.10) as max

tαβ =

n∈Dαβ ∪D

αβ

 ∪D

αβ





ωd × d(ωn αn , ωn βn ), ω  × d(ωn αn , ωn βn ), ω  × d(ωn αn , ωn βn ) d

max d(ωn αn , ωn βn ) n

d

,

(5.10) where ωd , ωd  , and ωd  denote weights of bipolar fuzzy strong, median and weak discordance sets, respectively, which are assigned by decision expert. Further, d(ωn αn , ωn βn ) represents the weighted distance among bipolar fuzzy values of alternatives Pα and Pβ for some criteria n. 5. Construction of outranking relations. This step is designed to carefully inspect the outranking relationships among alternatives based on of concordance and discordance indices. Firstly, it is the job of decision maker to appropriately assign the concordance and discordance threshold values. The strong outranking relation R s and weak outranking relation R w are analyzed by comparing these threshold levels with the concordance and discordance indices. Consider the strictly decreasing concordance thresholds f ∗ , f ◦ and f − , say, high, average, and low concordance levels, respectively, belong to closed unit interval. Further, consider the strictly increasing discordance thresholds t ∗ and t ◦ , say, low and average discordance levels, respectively, within closed unit inter-

5.2 Bipolar Fuzzy ELECTRE II Method

129

val. In accordance to defined threshold values, the alternative Pβ is found to be strongly outranked by alternative Pα , that is Pα R s Pβ if and only if one or both of the following sets of conditions hold. ⎧ ⎧ ⎨ f αβ ≥ f ◦ , ⎨ f αβ ≥ f ∗ , ◦ tαβ ≤ t , or tαβ ≤ t ◦ , ⎩ ⎩ f αβ > f βα , f αβ > f βα .

(5.11)

The alternative Pβ is found to be weakly outranked by alternative Pα , that is Pα R w Pβ , if and only if the following conditions hold. ⎧ ⎨ f αβ ≥ f − , tαβ ≤ t ∗ , ⎩ f αβ > f βα .

(5.12)

6. Construction of outranking graphs. To determine the ranking list, the strongly G s = (P, E s ) and the weakly outrank graphs G w = (P, E w ) are portrayed in accordance to strong R s and weak outranking relationships R w , respectively. In these outranking graphs, set of alternatives P serves as vertex set and E s and E w represent the set of corresponding arcs of strong and weak outranking relations. These graphs are then exploited by using a specific procedure of iteration to  establish two different rankings, namely forward ranking λ and reverse ranking  λ and finally, the average ranking λ yields the final output or results. 

(a) Forward ranking λ Let P = {P1 , P2 , . . . , Pr } be a set of alternatives and P(x) be any subset of P. The steps for forward ranking are described as follows: (1) Explore the strong outranking graph G s and enclose all the vertices having no incoming or precedent arc in the set V (x). (2) Now identify the arcs from E s having both extremities in V (x) by examining s weak outranking graph G w and enclose them in set E . (3) Construct the set B(x) comprising all vertices with no precedent arc in the s graph (V (x), E ). B(x) coincides to the set of non-dominated solutions at iteration x.  (4) Evaluate the forward ranking λ via following the iterative scheme. (i) Initially, put x = 1 and set P(1) = P. (ii) Specify the sets P(x) and B(x) by following the steps 1, 2, 3 above and 4(iv).  (iii) Rank the alternative Pm by λ (Pm ) = x, ∀ Pm ∈ B(x). (iv) Remove all forwardly ranked alternative and their adjacent arcs from the graphs G s and G w , also set P(x + 1) = P(x) − B(x). Repeat the procedure from step 4(ii) by taking x = x + 1 until P(x) = {}. 

(b) Reverse ranking λ The steps of reverse ranking are as follows:

130

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

(1) Obtain the reverse strongly and weakly outrank graph by reversing the direction of all the arc in strong and weak outranking graphs, respectively. In this process of reversing arcs, the set of alternatives does not change. (2) A ranking γ(Pm ) is accomplished from these reversed graphs by applying the iterative procedure same as for forward ranking. (3) The exact order for reverse ranking is obtained by Eq. (5.13) as 

λ (Pm ) = 1 + max γ (Pm ) − γ (Pm ). Pm ∈P

(5.13)

(c) Average ranking λ All alternatives are finally ordered by employing the formula as 

λ(Pm ) =



λ (Pm ) + λ (Pm ) . 2

(5.14)

The proceedings of bipolar fuzzy ELECTRE II technique is briefly summarized in Algorithm 5.2.1. Algorithm 5.2.1 Bipolar fuzzy ELECTRE II method 1. Establish a bipolar fuzzy decision matrix by arranging the preferences of all alternatives based on conflicting criteria. Also Evaluate the normalized weights of criteria by Eq. (5.1). 2. Compute the bipolar fuzzy concordance (strong, median and weak), indifference and discordance (strong, median and weak) sets by employing Eqs. (5.2)–(5.8), respectively. 3. Determine the bipolar fuzzy concordance indices that lead to the establishment of bipolar fuzzy concordance matrix. 4. Calculate the weighted distances among every pair of alternatives by employing the bipolar fuzzy Euclidean distance formula. 5. Evaluate the bipolar fuzzy discordance indices to establish the bipolar fuzzy discordance matrix. 6. Determine the strong and weak outranking relationships by employing the Eqs. (5.11) and (5.12), respectively. 7. Portray the strongly and weakly outrank graphs by exploring the respective strong and weak outranking relations. Finally order the alternatives by thorough analysis of these graphs by iterative procedure. Example 5.1 (Selection of an appropriate business location) Selection of the appropriate location is one of initial and vital issues for the establishment and growth of any business. The location of a business can address the management cost, total sale, and net profit. Consider the case of a businessman who aims to select an appropriate location for his business. After initial analysis, five different locations P1 , P2 , P3 , P4 , and P5 are shortlisted for through exploration. Furthermore, these locations are investigated by a set of five criteria K = {K 1 , K 2 , K 3 , K 4 , K 5 } representing the availability

5.2 Bipolar Fuzzy ELECTRE II Method Table 5.1 Bipolar fuzzy decision matrix L K1 K2 P1 P2 P3 P4 P5

(0.35, −0.7) (0.6, −0.25) (0.2, −0.5) (0.47, −0.65) (0.8, −0.3)

(0.9, −0.33) (0.4, −0.7) (0.28, −0.6) (0.55, −0.27) (0.3, −0.15)

131

K3

K4

K5

(0.5, −0.6) (0.65, −0.8) (0.37, −0.6) (0.25, −0.5) (1.0, −0.4)

(0.43, −0.72) (0.9, −0.5) (0.33, −0.6) (0.55, −0.4) (0.7, −0.32)

(0.6, −0.35) (1.0, −0.25) (0.8, −0.4) (0.75, −0.6) (0.3, −0.8)

of labor (K 1 ), price (K 2 ), safety (K 3 ), government economic incentives (K 4 ), and transport costs (K 5 ). 1. The assessment values of the decision maker are assembled in bipolar fuzzy decision matrix, as displayed in Table 5.1, where each entry mn in the matrix represents the positivity and negativity of an alternative Pm , m = 1, 2, . . . , 5 for criterion K n , n = 1, 2, . . . , 5. Further, the criteria weights, determined by Eq. (2.1), are given by the vector ω=[ 0.1964 0.2223 0.1979 0.1656 0.2178 ] whose sum is one. 2. The bipolar fuzzy concordance sets K αβ , K αβ  , K αβ  are enumerated by using Eqs. (2.2)–(2.4), respectively. (i) The bipolar fuzzy strong concordance sets K αβ are given as

K αβ

P1 P1 ⎛ − P2 ⎜{3} = P3 ⎜ ⎜{5} P4 ⎝{5} P5 ∅

P2 ∅ − ∅ ∅ {1}

P3 {1,4} {2,3} − {1} ∅

P4 {2,3} {3,4} {3} − ∅

P5 {2} ⎞ {2,4}⎟ ⎟ ∅ ⎟. {2} ⎠ −

(ii) The bipolar fuzzy median concordance sets K αβ  are given as

K αβ 

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜∅ P4 ⎝ ∅ P5 ∅

P2 ∅ − ∅ ∅ ∅

P3 {3} ∅ − ∅ ∅

P4 ∅ ∅ ∅ − ∅

P5 ∅⎞ ∅⎟ ⎟ ∅ ⎟. ∅⎠ −

(iii) The bipolar fuzzy weak concordance sets K αβ  are given as

132

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

K αβ 

P1 P1 ⎛ − P2 ⎜{1,4,5} = P3 ⎜ ⎜ ∅ P4 ⎝ {1,4} P5 {1,3,4}

P2 P3 P4 {2} {2} ∅ − {1,4,5} {1,5} ∅ − {5} {2} {2,4} − {3} {1,2,3,4} {1,3,4}

P5 {5}⎞ {5}⎟ ⎟ {5}⎟. {5}⎠ −

The indifferent sets Iαβ are constructed using Eq. (5.5) as

Iαβ

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜∅ P4 ⎝ ∅ P5 ∅

P2 ∅ − ∅ ∅ ∅

P3 ∅ ∅ − ∅ ∅

P4 ∅ ∅ ∅ − ∅

P5 ∅⎞ ∅⎟ ⎟ ∅ ⎟. ∅⎠ −

The bipolar fuzzy discordance sets Dαβ , Dαβ  , Dαβ  are established by applying Eqs. (5.6)–(5.8). (i) The bipolar fuzzy strong discordance sets Dαβ are given as

Dαβ

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜{1,4} P4 ⎝{2,3} P5 {2}

P2 {3} − {2,3} {3,4} {2,4}

P3 {5} ∅ − {3} ∅

P4 {5} ∅ {1} − {2}

P5 ∅⎞ {1}⎟ ⎟ ∅ ⎟. ∅⎠ −

(ii) The bipolar fuzzy median discordance sets Dαβ  are given as

Dαβ 

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜{3} P4 ⎝ ∅ P5 ∅

P2 ∅ − ∅ ∅ ∅

P3 ∅ ∅ − ∅ ∅

P4 ∅ ∅ ∅ − ∅

P5 ∅⎞ ∅⎟ ⎟ ∅ ⎟. ∅⎠ −

(iii) The bipolar fuzzy weak discordance sets Dαβ  are given in matrix Dαβ  as

5.2 Bipolar Fuzzy ELECTRE II Method

Dαβ 

133

P1 P2 P3 P4 P5 P1 ⎛ − {1,4,5} ∅ {1,4} {1,3,4} ⎞ ∅ {2} {3} ⎟ P2 ⎜ {2} − ⎟ = P3 ⎜ ⎜ {2} {1,4,5} − {2,4} {1,2,3,4}⎟. P4 ⎝ ∅ {1,5} {5} − {1,3,4} ⎠ − P5 {5} {5} {5} {5}

3. The decision maker assigns weights to bipolar fuzzy strong, median, weak concordance sets and indifferent sets, as shown in Eq. (5.15). The bipolar fuzzy concordance indices f αβ , {α, β = 1, 2, . . . , 5, α = β}, obtained by Eq. (5.9), are arranged in bipolar fuzzy concordance matrix F = ( f αβ )5×5 .



3 2 1 = ωc , ωc , ωc , ω = 1, , , . 4 4 4

(5.15)

P1 P2 P3 P4 P5 P1 ⎛ − 0.1112 0.6216 0.4202 0.3312⎞ P2 ⎜0.4878 − 0.7101 0.5706 0.4968⎟ ⎟ F = P3 ⎜ . − 0.3068 0.1089⎟ ⎜0.2178 0 ⎠ ⎝ P4 0.3988 0.1112 0.3904 − 0.3312 − P5 0.280 0.2954 0.3911 0.280 For example, bipolar fuzzy concordance index f 13 is calculated as f 13 = ωc × {ω1 + ω4 } + ωc × {ω3 } + ωc × {ω2 } 3 2 = 1 × {0.1964 + 0.1656} + × 0.1979 + × 0.2223 4 4 = 0.6216.

(5.16) (5.17)

4. Table 5.2 comprises the Euclidean distance between any two alternatives with respect to all criteria. For example, the Euclidean distance between 11 and 21 with respect to criteria K 1 is calculated as 

1 × ω1 [(μ11 − μ21 )2 + (ν11 − ν21 )2 ] 2  1 × 0.1964[(0.35 − 0.6)2 + (−0.7 − (−0.25))2 ] (5.18) = 2 = 0.1613. (5.19)

d(ω1 11 , ω1 21 ) =

Similarly, d(ω2 12 , ω2 22 ) = 0.2047, d(ω3 13 , ω3 23 ) = 0.0786, ω4 24 ) = 0.1493, d(ω5 15 , ω5 25 ) = 0.1361 and others.

d(ω4 14 ,

134

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

Table 5.2 Bipolar fuzzy weighted distances 11

21

11

−

21

−

31

31

41

51

12

22

0.1613 0.0783 0.0407 0.1887 12

−

0.2074 0.2255 0.1184 0.2088

−

0.1478 0.1318 0.0646 22

−

−

0.0521 0.1518 0.1864

−

−

−

0.0968 0.1982 32

−

−

−

0.1422 0.1502

41

−

−

−

−

0.1507 42

−

−

−

−

0.0925

51

−

−

−

−

−

−

−

−

−

−

13

23

33

43

53

14

24

34

44

54

13

−

0.0786 0.0409 0.0847 0.1964 14

−

0.1493 0.0449 0.0983 0.1389

23

−

−

0.1082 0.1573 0.1672 24

−

−

0.1665 0.1047 0.0774

33

−

−

−

0.0491 0.2079 34

−

−

−

0.0856 0.1335

43

−

−

−

−

0.2380 44

−

−

−

−

0.0489

53

−

−

−

−

−

−

−

−

−

−

15

25

35

45

55

15

−

0.1361 0.0680 0.0962 0.1785

25

−

−

0.0825 0.1419 0.2938

35

−

−

−

0.0680 0.2113

45

−

−

−

−

0.1625

55

−

−

−

−

−

52

54

32

42

52

5. The decision maker assigns weights to bipolar fuzzy strong, median, weak discordance sets, as shown in Eq. (5.20). The bipolar fuzzy discordance indices tαβ , {α, β = 1, 2, . . . , 5, α = β}, obtained by Eq. (5.10), are arranged in bipolar fuzzy discordance matrix T = (tαβ )5×5 .



3 2 ωd , ωd  , ωd  = 1, , . 4 4 P1 P2 P3 P4 P5 P1 ⎛ − 0.3891 0.3016 0.8125 0.4521⎞ − 0 0.4825 0.2845⎟ P2 ⎜ 0.5 ⎟ T = P3 ⎜ ⎜0.5002 0.7321 − 0.6807 0.4922⎟. ⎝ 1 1 0.3453 − 0.5 ⎠ P4 1 0.6344 0.5002 0.3887 − P5 For instance, the bipolar fuzzy discordance index t14 is evaluated as

(5.20)

5.2 Bipolar Fuzzy ELECTRE II Method

135

Table 5.3 Outranking relation P1

− Rs , Rw 0 0 0

P1 P2 P3 P4 P5

P2

P3

0 − 0 0 0

Rs ,

Rw

Rs ,

Rw

− Rs , Rw Rs

P4

P5

0 Rs , Rw 0 − 0

Rs , Rw Rs , Rw 0 Rs , Rw −



max ωd × d(ω5 15 , ω5 45 ), ωd  × d(ω1 11 , ω1 41 )ωd  × d(ω4 14 , ω4 44 ) max d(ωn αn , ωn βn ) n∈J

max 1 × 0.0962, 24 × 0.0407, 24 × 0.0983

(5.21) = max 0.0407, 0.1184, 0.0847, 0.0983, 0.0962 max{0.0962, 0.0204, 0.0492} = (5.22) max{0.0407, 0.1184, 0.0847, 0.0983, 0.0962} 0.0962 = (5.23) = 0.8125. 0.1184

t14 =

6. To check the outranking relations, decision maker picks the threshold values as ( f ∗ , f ◦ , f − ) = (0.5, 0.3, 0.2), (t ∗ , t ◦ ) = (0.5, 0.7). The strong and weak outranking relations, assessed by Eqs. (5.11) and (5.12), respectively, are summarized in Table 5.3. 7. The strongly and weakly outrank graphs are represented by Fig. 5.1 by portraying the strong and weak outranking relations, respectively. The outranking graphs are exploited to specify the average ordering of alternatives via an iterative approach. After the thorough analysis of outranking graphs, we obtain the following the for  ward ranking λ , reverse ranking λ and average ranking λ, as shown in Table 5.4. Finally, the alternatives can be ranked in the following order: P2 P4 P1 P5 P3 . That is, P2 stands out as the optimal choice. Example 5.2 (Selection of a supplier) Supplier selection appears as the fundamental part of any business as it refers to the identification, assessment, and collaboration with competent suppliers for smooth services. Supply chain network plays a significant role in the smooth performance and exceptional services of international companies. A competent and responsible supplier is keen to deliver services or products timely, maintain quality with accurate cost. Consider that a company is desired to collaborate with a supplier to manage its business. After initial screening, a set of five suppliers T = {P1 , P2 , P3 , P4 , P5 } is considered for complete investigation which are assessed according to a set of five criteria K = {K 1 , K 2 , K 3 , K 4 , K 5 } depicting

136

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method P1

P1

P2

P2

P3

P4

P3

P4

P5

P5

(b) Weakly outrank graph Gw

(a) Strongly outrank graph Gs

Fig. 5.1 Graphical representation of outranking relations between alternatives Table 5.4 Ranking results 

Forward ranking λ  Reverse ranking λ Average ranking λ

P1

P2

P3

P4

P5

2 2 2

1 1 1

4 5 4.5

2 1 1.5

3 4 3.5

K3

K4

K5

(0.6, −0.3) (0.9, −0.1) (0.6, −0.3) (0.6, −0.2) (0.3, −0.6)

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

(0.8, −0.1) (0.5, −0.5) (0.7, −0.4) (0.8, −0.1) (0.7, −0.3)

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

Table 5.5 Bipolar fuzzy decision matrix L K1 K2 P1 P2 P3 P4 P5

(0.8, −0.1) (0.9, −0.1) (0.8, −0.3) (0.9, −0.4) (0.4, −0.6)

the total cost of opportunity (K 1 ), experience in market (K 2 ), storage and handling facilities (K 3 ), quality and safety (K 4 ), and specific methods of delivery (K 5 ). 1. The assessment values of the decision maker are assembled in bipolar fuzzy decision matrix, as displayed in Table 5.5, where each entry mn consists of the positivity and negativity of an alternative Pm , m = 1, 2, . . . , 5 for criteria K n , n = 1, 2, . . . , 5. Further, the criteria weights, determined by Eq. (5.1), are given by the vector ω = [ 0.2372 0.2249 0.0685 0.1760 0.2934] whose sum is one. 2. The bipolar fuzzy concordance sets K αβ , K αβ  , K αβ  are enumerated using Eqs. (5.2)–(5.4), respectively.

5.2 Bipolar Fuzzy ELECTRE II Method

137

(i) The bipolar fuzzy strong concordance sets K αβ are given as

K αβ

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜{1} P4 ⎝{1} P5 ∅

P2 ∅ − ∅ {1} ∅

P3 ∅ ∅ − {1} ∅

P4 {2} ∅ {2} − ∅

P5 ∅ ⎞ ∅ ⎟ ⎟ {4}⎟. ∅ ⎠ −

(ii) The bipolar fuzzy median concordance sets K αβ  are given as

K αβ 

P1 P1 ⎛ − P2 ⎜{1} = P3 ⎜ ⎜ ∅ P4 ⎝ ∅ P5 ∅

P2 ∅ − ∅ ∅ ∅

P3 {3} {3} − ∅ {3}

P4 {5} ∅ ∅ − {5}

P5 ∅⎞ ∅⎟ ⎟ ∅ ⎟. ∅⎠ −

(iii) The bipolar fuzzy weak concordance sets K αβ  are given as

K αβ 

P1 P1 ⎛ − P2 ⎜{2,5} = P3 ⎜ ⎜ {5} P4 ⎝ {2} ∅ P5

P2 {4} − {4,5} {4} {4}

P3 P4 P5 {1,4} {3} {1,2,4}⎞ {1,2} {1,2,3,5} {1,2,5}⎟ ⎟ − {3,5} {1,2,5}⎟. {2,4} − {1,2,4}⎠ {4} {3} −

The indifferent sets Iαβ are computed by employing Eq. (5.5) as

Iαβ

P1 P1 ⎛ − P2 ⎜ {3} = P3 ⎜ ⎜ {2} P4 ⎝ {4} P5 {3,5}

P2 {3} − ∅ ∅ {3}

P3 {2} ∅ − ∅ ∅

P4 P5 {4} {3,5}⎞ ∅ {3} ⎟ ⎟ ∅ ∅ ⎟. − ∅ ⎠ ∅ −

The bipolar fuzzy discordance sets Dαβ , Dαβ  , Dαβ  are established by applying Eqs. (5.6)–(5.8), respectively.

138

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

(i) The bipolar fuzzy strong discordance sets Dαβ are given as

Dαβ

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜∅ P4 ⎝ ∅ P5 ∅

P2 ∅ − ∅ ∅ ∅

P3 ∅ ∅ − ∅ ∅

P4 {1} ∅ {1} − ∅

P5 ∅⎞ ∅⎟ ⎟ ∅ ⎟. ∅⎠ −

(ii) The bipolar fuzzy median discordance sets Dαβ  are given as

Dαβ 

P1 P1 ⎛ − P2 ⎜ ∅ = P3 ⎜ ⎜{3} P4 ⎝{5} P5 ∅

P2 {1} − {3} ∅ ∅

P3 ∅ ∅ − ∅ ∅

P4 ∅ ∅ ∅ − ∅

P5 ∅ ⎞ ∅ ⎟ ⎟ {3}⎟. {5}⎠ −

(iii) The bipolar fuzzy weak discordance sets Dαβ  are given as

Dαβ 

P1 P1 ⎛ − P2 ⎜ {4} = P3 ⎜ ⎜ {4} P4 ⎝ {3} P5 {1,2,4}

P2 P3 P4 {2,5} {5} ∅ − {4,5} {4} {1,2} − {4} {2,3,5} {3,5} − {1,2,5} {1,2,5} {1,2,4}

P5 ∅ ⎞ {4}⎟ ⎟ ∅ ⎟. {3}⎠ −

3. The decision maker assigns weights to bipolar fuzzy strong, median, weak concordance sets and indifferent sets, as shown in Eq. (5.24). The bipolar fuzzy concordance indices f αβ , {α, β = 1, 2, . . . , 5, α = β}, obtained by Eq. (5.9), are arranged in bipolar fuzzy concordance matrix F = ( f αβ )5×5 .



3 2 1 = ωc , ωc , ωc , ω = 1, , , . 4 4 4 P1 P1 ⎛ − P2 ⎜0.4542 F = P3 ⎜ ⎜0.4401 P4 ⎝0.3937 P5 0.0905

P2 0.1051 − 0.2347 0.3252 0.1051

P3 0.3142 0.2824 − 0.4377 0.1394

P4 0.5232 0.412 0.4059 − 0.2543

P5 0.4095⎞ 0.3949⎟ ⎟ 0.5538⎟. 0.3110⎠ −

(5.24)

5.2 Bipolar Fuzzy ELECTRE II Method

139

Table 5.6 Bipolar fuzzy weighted distances 11

21

11

−

21

−

31 41 51

31

41

51

12

22

0.0344 0.0689 0.1089 0.2205 12

−

0.1209 0

−

0.0770 0.1033 0.2435 22

−

−

0.1209 0.1060 0.2619

−

−

−

0.0487 0.1722 32

−

−

−

0.0335 0.1423

−

−

−

−

0.1855 42

−

−

−

−

0.1677

−

−

−

−

−

−

−

−

−

−

13

23

33

43

53

14

24

34

44

54

13

−

0

0.0185 0.0414 0

14

−

0.1483 0.0938 0

23

−

−

0.0185 0.0414 0

24

−

−

0.0663 0.1483 0.0839

33

−

−

−

0.0262 0.0185 34

−

−

−

0.0938 0.0297

43

−

−

−

−

0.0414 44

−

−

−

−

0.0663

53

−

−

−

−

−

−

−

−

−

−

15

25

35

45

55

15

−

0.2233 0.2762 0.0383 0

25

−

−

0.0542 0.2452 0.2233

35

−

−

−

0.2991 0.2761

45

−

−

−

−

0.0383

55

−

−

−

−

−

52

54

32

42

52

0.0335 0.1423

0.0663

For instance, bipolar fuzzy concordance index f 12 is determined as f 12 = ωc × {ω4 } + ω = × {ω3 } 1 2 = × 0.1760 + × 0.0685 4 4 = 0.1015. 4. Table 5.6 comprises the Euclidean distance between any two alternatives with respect to all criteria. For example, the Euclidean distance between 11 and 21 with respect to criteria K 1 is calculated as 

1 × ω1 [(μ11 − μ21 )2 + (ν11 − ν21 )2 ] 2  1 × 0.2372[(0.8 − 0.9)2 + (−0.1 − (−0.1))2 ] = 2 = 0.0344.

d(ω1 11 , ω1 21 ) =

Similarly, d(ω2 12 , ω2 22 ) = 0.1209, d(ω3 13 , ω3 23 ) = 0, d(ω4 14 , ω4 24 ) = 0.1483, d(ω5 b15 , ω5 b25 ) = 0.2233 and others.

140

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

5. The decision maker assigns weights to bipolar fuzzy strong, median, weak discordance sets, as shown in Eq. (5.25). The bipolar fuzzy discordance indices tαβ , {α, β = 1, 2, . . . , 5, α = β}, obtained by Eq. (5.10), are arranged in bipolar fuzzy discordance matrix T = (tαβ )5×5 .



3 2   ωd , ωd , ωd = 1, , . 4 4

(5.25)

P1 P2 P3 P4 P5 P1 ⎛ − 0.5002 0.5 1 0 ⎞ P2 ⎜0.3321 − 0.2746 0.3026 0.1604⎟ ⎟ T = P3 ⎜ ⎜0.1698 0.5004 − 0.1628 0.0503⎟. ⎝ P4 0.2635 0.5 0.5002 − 0.1547⎠ P5 0.5002 0.5380 0.5 0.5003 − For instance, the bipolar fuzzy discordance index t12 is determined as t12 = = = =



max ωd  × d(ω1 11 , ω1 21 ), ωd  × d(ω2 12 , ω1 22 ), ωd  × d(ω5 15 , ω4 25 ) max d(ωn αn , ωn βn ) n∈J 3

max 4 × 0.0344, 24 × 0.1209, 24 × 0.2233

max 0.0344, 0.1209, 0, 0.1483, 0.2233 max{0.0258, 0.0605, 0.1117} max{0.0344, 0.1209, 0, 0.1483, 0.2233} 0.1117 = 0.5002. 0.2233

6. The threshold values assigned by the decision maker are as follows: ( f ∗ , f ◦ , f − ) = (0.4, 0.3, 0.2), (t ∗ , t ◦ ) = (0.3, 0.5). Table 5.7 comprises the strong and weak outranking relations, assessed in the light of Eqs. (5.11) and (5.12).

Table 5.7 Outranking relation P1 P2 P3 P4 P5

P1

P2

P3

P4

P5

− Rs Rs , Rw 0 0

0 − 0 0 0

0 Rw − 0 0

0 Rs 0 − 0

Rs , Rw Rs , Rw Rs , Rw Rs , Rw −

5.3 Comparative Study and Discussion

141

P1

P1 P2

P3

P2

P3

P4

P5

P4

P5

(a) Strongly outrank graph Gs

(b) Weakly outrank graph Gw

Fig. 5.2 Graphical representation of outranking relations between alternatives Table 5.8 Ranking results 

Forward ranking λ  Reverse ranking λ Average ranking λ

P1

P2

P3

P4

P5

3 3 3

1 1 1

2 2 2

2 3 2.5

4 4 4

7. The strongly and weakly outrank graphs are represented in Fig. 5.2 by portraying the strong and weak outranking relations, respectively. The outranking graphs are exploited to specify the average ordering of alternatives via an iterative approach. After the thorough analysis of outranking graph, we obtain the following the for  ward ranking λ , reverse ranking λ and average ranking λ, as shown in Table 5.8. According to average ranking, these five alternatives are ranked as P2 P3 P4 P1 P5 .

5.3 Comparative Study and Discussion This section presents a comparison of bipolar fuzzy ELECTRE II technique with existing multi-criteria decision making techniques, presented by Alghamdi et al. [11] in a bipolar fuzzy setting. We apply these models to the numerical problem solved in Example 5.1 as “selection of business location” to analyze multiple decision making approaches. The theoretical comparison of the methodology of bipolar fuzzy ELECTRE II model is also discussed with fuzzy ELECTRE II technique, which was presented by Govindan et al. [26].

142

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

5.3.1 Comparison with Bipolar Fuzzy TOPSIS Method A flowchart of steps of the bipolar fuzzy TOPSIS method, introduced by [11], is given in Fig. 5.3. This subsection adopts the bipolar fuzzy TOPSIS method to solve the problem of merchant location. The formation of the bipolar fuzzy decision matrix and the computation of the numerical weights are same as in the bipolar fuzzy ELECTRE II method. The initial decision matrix is represented by Table 5.1, and the weighted bipolar fuzzy decision matrix is given in Table 5.9.

Fig. 5.3 The steps of the bipolar fuzzy TOPSIS model Identification of alternatives and criteria

Construction of a bipolar fuzzy decision matrix

Calculation of weights by maximizing deviation method

Compute the weighted bipolar fuzzy decision matrix

Compute bipolar fuzzy positive ideal solution (BFPIS) and bipolar fuzzy negative ideal solution (BFNIS)

Calculate the distance of each alternative from BFPIS and BFNIS

Calculate the relative closeness degree of each alternative to BFPIS

Rank the alternatives with respect to the descending order of relative closeness degrees

5.3 Comparative Study and Discussion

143

Table 5.9 Weighted bipolar fuzzy decision matrix K1

K2

K3

K4

K5

P1 (0.0687, −0.1375) (0.2001, −0.0734) (0.0990, −0.1187) (0.0712, −0.1192) (0.1307, −0.0762) P2 (0.1178, −0.0491) (0.0889, −0.1556) (0.1286, −0.1583) (0.1490, −0.0083) (0.2178, −0.0545) P3 (0.0393, −0.0982) (0.0622, −0.1334) (0.0732, −0.1187) (0.0546, −0.0994) (0.1742, −0.0871) P4 (0.0923, −0.1277) (0.1223, −0.0600) (0.0495, −0.0990) (0.0911, −0.0662) (0.1634, −0.1307) P5 (0.1571, −0.0589) (0.0667, −0.0333) (0.1979, −0.0792) (0.1159, −0.0530) (0.0653, −0.1742)

The bipolar fuzzy positive ideal solution (BFPIS) and bipolar fuzzy negative ideal solution (BFNIS) are evaluated as follows. B F P I S = [(0.1571, −0.0491), (0.2001, −0.0333), (0.1979, −0.0792), (0.1490, −0.0530), (0.2178, −0.0545)], B F N I S = [(0.0393, −0.1375), (0.0622, −0.1556), (0.0495, −0.1583), (0.0546, −0.1192), (0.0653, −0.1742)].

Furthermore, the Euclidean distance of all alternatives from bipolar fuzzy positive ideal solution and bipolar fuzzy negative ideal solution is computed as D(P1 , B F P I S) = 0.153, D(P2 , B F P I S) = 0.143,

D(P1 , B F N I S) = 0.150, D(P2 , B F N I S) = 0.185,

D(P3 , B F P I S) = 0.196, D(P4 , B F P I S) = 0.161, D(P5 , B F P I S) = 0.168,

D(P3 , B F N I S) = 0.110, D(P4 , B F N I S) = 0.132, D(P5 , B F N I S) = 0.189.

The relative closeness degrees of the alternatives are found to be C(P1 ) = 0.495, C(P2 ) = 0.564, C(P3 ) = 0.359, C(P4 ) = 0.451, C(P5 ) = 0.529.

The arrangement of alternatives in decreasing order of revised closeness index yields the following ranking: P2 P5 P1 P4 P3 and thus the location P2 is best choice having largest closeness degree.

5.3.2 Comparison with Bipolar Fuzzy ELECTRE I Method A flowchart of steps of the bipolar fuzzy ELECTRE I method is presented in Fig. 5.4. To address the same problem of business location via the bipolar fuzzy ELECTRE I technique, consider the weighted bipolar fuzzy decision matrix established in Table 5.9, and follow the next steps of bipolar fuzzy ELECTRE I model to establish

144 Fig. 5.4 The steps of the bipolar fuzzy ELECTRE I method

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

Identification of alternatives and criteria

Construction of a bipolar fuzzy decision matrix

Calculation of weights by using maximizing deviation method

Compute weighted bipolar fuzzy decision matrix

Construct the bipolar fuzzy concordance sets

Construct the bipolar fuzzy discordance sets

Calculate the concordance dominance indices

Calculate the discordance dominance indices

Compute the aggregated dominance indices

Determine the outranking relation of alternatives

5.3 Comparative Study and Discussion

145

an outranking relation of alternatives in account to make a comparison of different decision making techniques. The computations of bipolar fuzzy concordance sets Cαβ , bipolar fuzzy discordance sets Dαβ , bipolar fuzzy concordance indices cαβ , bipolar fuzzy discordance indices dαβ , concordance dominance gαβ , discordance dominance h αβ , aggregated dominance qαβ , and outranking relationships are summarized in Table 5.10. The graph of outranking relationships is drawn in Fig. 5.5, and the set of best fitted alternatives is {P2 , P5 }. It is obvious that the alternative P2 appears as the optimal location by all compared approaches under bipolar fuzzy environment. Thus, the proposed bipolar fuzzy ELECTRE II method can be successfully applied to solve the multi-criteria decision making problems with bipolar fuzzy information. The competency of the proposed method is not limited to provide the solution of problem but it also delivers the complete ranking list.

5.3.3 Comparison with Fuzzy ELECTRE II Method 1. Bipolar fuzzy sets approach the two sided information or bipolar reasoning of human thinking, and play an important role in human decision making. Information that is awarded as possible or true is represented by positive information, while negative information is represented by information deemed impossible or possibly false. As we discussed in our numerical example, a number of real-life problems are evaluated on the basis of bipolar or two-sided information (rather than one-sided information). We used bipolar fuzzy sets, where the preference rating of alternatives comprises of two parts of membership grades, such as positive and negative membership values. A positive membership grade for an alternative represents the benefit or satisfactory behavior of that alternative relative to a particular criterion, and a negative membership value shows the dissatisfaction or cost value of the alternative. We have applied the methodology of bipolar fuzzy ELECTRE II model to determine ordering of alternatives or to take the best possible action. 2. Fuzzy ELECTRE II method is successfully used to evaluate numerical problems with only one-sided information, that is, the actions considered are ordered only according to the satisfaction value of positive membership or alternatives. However, we cannot give information about the degree of dissatisfaction. Therefore, we provide the bipolar fuzzy ELECTRE II method to determine an ordering of alternatives as an improvement over other successful versions of the ELECTRE II model.

Cαβ

{2, 3} {2, 3} {2, 3, 5} {2, 5} {1, 4, 5} {1, 2, 3, 4, 5} {1, 3, 4, 5} {4,5} {1, 4, 5} {} {3, 5} {5} {1, 4} {2} {1, 2, 4} {2, 5} {1, 3, 4} {1, 2, 3, } {1, 2, 3, 4} {1, 3, 4}

Alternatives compared

( P1 , P2 ) ( P1 , P3 ) ( P1 , P4 ) ( P1 , P5 ) ( P2 , P1 ) ( P2 , P3 ) ( P2 , P4 ) ( P2 , P5 ) ( P3 , P1 ) ( P3 , P2 ) ( P3 , P4 ) ( P3 , P5 ) ( P4 , P1 ) ( P4 , P2 ) ( P4 , P3 ) ( P4 , P5 ) ( P5 , P1 ) ( P5 , P2 ) ( P5 , P3 ) ( P5 , P4 )

{1, 4, 5} {1, 4, 5} {1, 4} {1, 3, 4} {2, 3} {} {2} {1, 2, 3} {2, 3} {1, 2, 3, 4, 5} {1, 2, 4} {1, 2, 3, 4} {2, 3, 5} {1, 3, 4, 5} {3, 5} {1, 3, 4} {2, 5} {4, 5} {5} {2, 5}

Dαβ 0.4202 0.4202 0.6380 0.4401 0.5798 1 0.7777 0.3834 0.5798 0 0.4157 0.2178 0.3620 0.2223 0.5843 0.4401 0.5599 0.6166 0.7822 0.5599

cαβ 1 0.5893 0.5187 1 0.5514 0 0.0336 1 1 1 0.4312 1 1 1 1 0.8237 0.1332 0.8744 0.2973 1

dαβ

Table 5.10 Bipolar fuzzy ELECTRE I results for selection of business location

0 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 1 1 1

gαβ 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0

h αβ 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0

qαβ Incomparable Incomparable P 1 → P4 Incomparable P 2 → P1 P 2 → P3 P 2 → P4 Incomparable Incomparable Incomparable Incomparable Incomparable Incomparable Incomparable Incomparable Incomparable P 5 → P1 Incomparable P 5 → P3 Incomparable

Outranking relations

146 5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

References Fig. 5.5 Graph representing the outranking relation of alternatives

147 P1

P5

P2

P3

P4

5.4 Insights and Limitations of the Proposed Method To summarize, some insights of the proposed bipolar fuzzy ELECTRE II method are as follows: 1. The bipolar fuzzy ELECTRE II method described in this chapter is an extension of other existing versions of the ELECTRE II method. It also ranks the alternatives, and it deals with bipolar fuzzy information. This includes the fuzzy case as a particular instance. 2. To minimize the personal influence of the decision expert, an optimization technique based on the maximizing deviation method is employed to enumerate the normalized weights of the criteria. 3. The proposed procedure predicts the ranking by employing two outranking relations and an iterative procedure to get more satisfactory outcomes. We emphasize that the solution for two numerical examples by this method are fully explained. In addition to the discussion above, we are aware that this method also has limitations. Some of them are given below 1. This method can exceptionally address the two counter aspects of the information, but this model is inept to capture m-sided or m-polar information that abundantly appears in real life scenarios. 2. A noticeable limitation of this approach is the high influence of decision maker’s personal choice that can affect the transparency. Additional Reading The readers are suggested to [21, 22, 35, 45] for definitions of additional terms and applications not included in this chapter.

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5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method

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Chapter 6

Extended PROMETHEE Method with Bipolar Fuzzy Sets

A new version of the PROMETHEE method using bipolar fuzzy information is presented in this chapter. Consequently, it is named as the bipolar fuzzy PROMETHEE method. The proposed method accepts the initial assessments via bipolar fuzzy linguistic terms expressed by trapezoidal bipolar fuzzy numbers. A ranking function is defined to assign the crisp values to bipolar fuzzy numbers in case of any comparison. The entropy formula is deployed to compute the normalized weights of attributes. To explain the application of this method, we present a numerical exercise concerning the selection of green suppliers in which each criterion is treated as usual criterion. In comparison, the results are obtained by the combination of linear and level preference functions. When we use different types of preference functions, we check that the results are the same. This coincidence speaks to the authenticity of our bipolar fuzzy PROMETHEE method. This chapter is based on [4].

6.1 Introduction In this competitive era, organizations and companies need to upgrade their technical skills and services to meet the requirements of customers and succeed in order to enhance their profit and repute. From the other perspective, end-consumers are looking for high-quality, budget friendly, and conveniently accessible products with short delivery time. Thus, supply chain must be strong and actively operational in order to satisfy the customers. Therefore, selection of suppliers appears as the hectic and vital decision for supply chain management to captivate and serve more customer. In addition, the manufacturers must adhere to strict environmental policies due to increasing public awareness towards global warming, which causes strong pressure from the consumers, organizations, and governments. For these reasons, in relation with green packaging, green supplier selection must be carefully considered. Recently, many researchers have explored this problem. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Akram et al., Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-99-0569-0_6

151

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

The green supplier selection problem is posed in terms of multiple conflicting criteria, rather than being a single-objective problem. In this respect, the multi-criteria decision analysis techniques provide the elementary tools to explore this problem adequately. multi-criteria decision analysis methods serve competently to opt for the most favorable supplier for any business. In point of fact, many researchers have benefited from several decision making approaches to identify efficient suppliers in different scenarios. Some significant work in this field has been done by Awasthi et al. [12], Handfield et al. [27], Mousakhani et al. [34] and Yeh and Chuang [42]. In a more elaborate approach, Hou and Yanrong [29] explored the advantages of entropy weight information to determine the weights of the criteria, in combination with TOPSIS technique to look for a suitable supplier under green environmental restrictions. Yazdani et al. [41] studied the supply chain management via quality function deployment and SWARA models. We have mentioned in previous chapters that in the last few decades, a number of multi-criteria decision analysis techniques and their innovative extensions have been proposed, including AHP [36], PROMETHEE [18], ELECTRE [16], VIKOR [35], TOPSIS [28]. Although they had been designed to investigate problems having precise information and crisp numerical data, Bellman and Zadeh [15] inaugurated the fuzzy multi-criteria decision analysis methods that were soon followed by more sophisticated models. To improve the accuracy, many multi-criteria group decision analysis methods have been considered to thoroughly study the supply chain management with imprecise information. For example, Chiou et al. [21] employed fuzzy AHP method, Sanayei et al. [37] applied fuzzy VIKOR method, Kannan et al. [30] worked with fuzzy TOPSIS approach, Awasthi and Kannan [13] integrated the nominal group technique and VIKOR method, Hamdan and Cheaitou [26] utilized the combination of fuzzy AHP and TOPSIS techniques, Shumaiza et al. [39] applied the bipolar fuzzy ELECTRE II technique to the supplier selection problem within different scenarios. Ziemba et al. [46] introduced the method of criteria selection and the calculation of weights in the process of web projects evaluation. Liu et al. [33] proposed multioperators based on intuitionistic fuzzy sets, which they used for the evaluation of urban infrastructure-based projects. This chapter is concerned with PROMETHEE technique, an outranking approach within the multi-criteria decision analysis method that was introduced by Brans and Vincke [19] in 1985, either to obtain a partial ranking (PROMETHEE I) or a complete ranking (PROMETHEE II) of a set of alternatives, as a function of multiple attributes or criteria. Moreover, Brans and Mareschal [17] put forward two further extensions of this method: PROMETHEE III, in which a ranking is produced that is based on intervals, and PROMETHEE IV, which is a continuous case of ranking. Exploring the decision making caliber of the PROMETHEE method, Abdullah et al. [1] and Govindan et al. [24] applied this technique to supplier selection. Furthermore, Behzadian et al. [14], Goumas and Lygerou [23], and Krishankumar et al. [31] also demonstrated the applications of PROMETHEE method in practical life.

6.2 Bipolar Fuzzy PROMETHEE Method

153

Chen [20] and Feng et al. [22] considered enhanced PROMETHEE methods. Ziemba [47] proposed a new variant, namely, NEAT F-PROMETHEE technique, using trapezoidal fuzzy numbers (for other uses, see [10] and the references therein). Concerning the informational basis of our model, Zhang [45] was first to capture the double-sided information via bipolar fuzzy set theory. Akram and Arshad [2] broadened the literature by presenting the bipolar fuzzy numbers and linguistic variables. Alghamdi et al. [11], Akram et al. [2], and Shumaiza et al. [40] proposed multi-criteria decision making methods based on TOPSIS, ELECTRE I, and VIKOR methods for bipolar fuzzy environment. This chapter is motivated by the fact that existing versions and extensions of the PROMETHEE technique enable us to examine problems posed with information in the form of crisp or fuzzy values, but they disregard the problems with bipolar uncertainties. Since the crisp and fuzzy sets only provide us one-dimensional information, we can only encode the information about the satisfaction degree of the alternatives. With the support of these sets, we are unable to provide extra information about the dissatisfaction degree of the alternatives. Therefore, this chapter illustrates the procedure of bipolar fuzzy PROMETHEE method to resolve the complex decision making situations carrying the two-sided information. The bipolar fuzzy linguistic evaluations in terms of bipolar fuzzy numbers for the numeric preferences of alternatives [2] are parameterized. This chapter borrows the information theory [32, 38] to compute the criteria weights. The bipolar fuzzy evaluations are converted to crisp values using the ranking function [2]. The significant contributions of this decision making method are as follows: 1. This chapter mainly presents significant extension of PROMETHEE methods for bipolar fuzzy information to capture both sides of human judgements using the trapezoidal bipolar fuzzy numbers to enhance the accuracy. 2. To minimize the personal influence of the decision makers, the objective weights for the criteria are derived by Shannon entropy weighting technique. 3. Respective partial and complete rankings of the alternatives are determined by the PROMETHEE I or the PROMETHEE II variations of the strategy. 4. Furthermore, a demonstrative example for the selection of green suppliers is described to highlight the validity and authenticity of the propose methods.

6.2 Bipolar Fuzzy PROMETHEE Method We go on to give basic definitions, concepts, and terms related to bipolar fuzzy numbers. In addition, the process of the bipolar fuzzy PROMETHEE method is discussed in detail.  on the real line R is of the form B = Definition B     6.1 ([2]) A bipolar fuzzy number + Y, Z = (ϑ1 , ϑ2 , ϑ3 , ϑ4 ), (δ1 , δ2 , δ3 , δ4 ) such that the satisfaction degree ΩB  (x) −  (x) of B are defined as and dissatisfaction degree ΩB 

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

⎧ l+ Ω (x), ⎪ ⎪ ⎨ B 1, + ΩB (x) = Ωr + (x), ⎪ ⎪ ⎩ B 0,

if x ∈ [ϑ1 , ϑ2 ] if x ∈ [ϑ2 , ϑ3 ] , if x ∈ [ϑ3 , ϑ4 ] otherwise

⎧ l− Ω (x), ⎪ ⎪ ⎨ B 1, − ΩB  (x) = Ωr − (x), ⎪ ⎪ ⎩ B 0,

if x ∈ [δ1 , δ2 ] if x ∈ [δ2 , δ3 ] , if x ∈ [δ3 , δ4 ] otherwise

l+ r+ where ΩB  (x) : [ϑ1 , ϑ2 ] → [0, 1] and ΩB  (x) : [ϑ3 , ϑ4 ] → [0, 1] represent the left + l− and right membership degrees of ΩB  (x), respectively. Similarly, ΩB  (x) : [δ1 , δ2 ] → r− [−1, 0] and ΩB  (x) : [δ3 , δ4 ] → [−1, 0] represent the left and right membership − degrees of ΩB (x), respectively.      = Y, Z = (ϑ1 , ϑ2 , ϑ3 , ϑ4 ), (δ1 , δ2 , Definition 6.2 ([2]) A bipolar fuzzy number B  δ3 , δ4 ) is called trapezoidal bipolar fuzzy number if the satisfaction and dissatisfaction degrees are defined as ⎧ x−ϑ1 ⎧ −(x−δ1 ) , if x ∈ [ϑ1 , ϑ2 ] , if x ∈ [δ1 , δ2 ] ⎪ ⎪ ϑ2 −ϑ1 ⎪ ⎪ ⎨ ⎨ δ2 −δ1 1, if x ∈ [ϑ2 , ϑ3 ] −1, if x ∈ [δ2 , δ3 ] + − ΩB . , ΩB ϑ4 −x −(δ4 −x)  (x) =  (x) = ⎪ ⎪ if x ∈ [ϑ , ϑ ] , if x ∈ [δ3 , δ4 ] 3 4 ⎪ ⎪ ⎩ ϑ4 −ϑ3 ⎩ δ4 −δ3 0, otherwise 0, otherwise.   Definition 6.3 ([2])  Consider a trapezoidal bipolar fuzzy number B = (ϑ1 , ϑ2 , ϑ3 , ϑ4 ), (δ1 , δ2 , δ3 , δ4 ) which can be converted into a real number by applying the ranking function as





−δ1 − δ2 + δ3 + δ4 −ϑ1 − ϑ2 + ϑ3 + ϑ4 − m(Z ) + , m(Y ) + 2 2 ϑ1 + ϑ2 + ϑ3 + ϑ4 δ1 + δ2 + δ3 + δ4 where m(Y ) = and m(Z ) = are the mean 4 4 values of respective sets.

6.2.1 Preference Function An appropriate and relevant preference function is an essential requirement for PROMETHEE technique. The deviation between each pair of alternatives for each criterion is defined by a preference function. Different types of preference functions are defined and used by Brans et al. [18, 19]. In this chapter, the preference function is used to make the most of the PROMETHEE method by completing the conditions that will be applied to the case study. Referring to Step 5 of the PROMETHEE algorithm, three different types of preference functions are applied, defined as follows. Definition 6.4 Type I: The usual criterion preference function is defined as

H (d) =

0, if d ≤ 0 , 1, if d > 0

(6.1)

6.2 Bipolar Fuzzy PROMETHEE Method

155

where d shows the deviation or difference between alternatives. In usual criterion preference function, two alternatives a and b are indifferent if and only if f (a) = f (b). Definition 6.5 Type II: The linear criterion preference function is defined as ⎧ ⎨ 0, if d < 0 H (d) = md , if 0 ≤ d ≤ m , ⎩ 1, if d > m

(6.2)

where preference value m is specified by decision makers. For deviation value d lying in the interval (0, m], the preference value increases linearly. For higher values of d, obtain a strict preference for the alternatives for that criterion. Definition 6.6 Type III: The level criterion preference function is defined as ⎧ ⎨ 0 if d ≤ n H (d) = 21 if n < d ≤ m + n , ⎩ 1 if d > m + n

(6.3)

where m and n represent preference value and indifference value. In this preference function, the indifference between alternatives belongs to the interval [−n, n]. The horizontal standard preference function always applies to benefit types or qualitative standards.

6.2.2 Structure of Bipolar Fuzzy PROMETHEE Method In this subsection, a continuation of an outranking technique using bipolar fuzzy information, known as the bipolar fuzzy PROMETHEE method, is proposed for evaluating multiple-attribute group decision making problems. The proceeding for this technique is as follows: identify the problem area and choose an appropriate and best suited group of decision experts; define linguistic terms or variables and their corresponding values; assemble a decision matrix for each decision expert and then compute the aggregated decision values; by using entropy weight method, [38] calculate the normalized weight of attributes; defines a suitable preference function for pairwise comparison of alternatives, finds the multi-criteria preference index of alternatives; establishes a partial ranking of alternatives (PROMETHEE I); and compute the full ranking of the alternatives (PROMETHEE II). Consider a multi-criteria decision making problem in which p alternatives Sα ; α = 1, 2, . . . , p, has to be explored on the basis of q conflicting criteria Qβ ; β = 1, 2, . . . , q. A panel of r decision makers Dϕ ; ϕ = 1, 2, . . . , r , is hired to rate the alternatives with respect to different criteria. The preference values of alternatives Sα with respect to each criterion Qβ are used to construct a decision matrix

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

V ery poor

1

0

0.1

−1

V ery good

P oor

0.2

Good

M edium poor

0.3

0.4

M edium good

M edium good

F air

0.5

0.6

F air

0.7

M edium poor

Good

0.8

P oor

V ery good

0.9

1

V ery poor

Fig. 6.1 Linguistic variables for criteria

ϕ

L = [αβ ] for every decision maker Dϕ . The steps of the bipolar fuzzy PROMETHEE technique are as follows:. 1. Identify the linguistic variables. Decision makers express their opinions regarding alternatives with respect to different criteria by linguistic terms. Therefore, the description of the linguistic terms and their values is the earliest task of this procedure. This methodology employs a combination of seven linguistic values expressible in trapezoidal bipolar fuzzy numbers, as represented in Fig. 6.1. The corresponding values of these numbers are picked from the interval [0,1]. 2. Establish a decision matrix. Each expert Dϕ is assigned the job to evaluate the alternatives respective of conflicting criteria. This step leads to the establishment of r decision matrices consisting of the bipolar fuzzy judgements of r decision makers. The decision matrix of the expert Dϕ is given by ⎡

ϕ

L = [αβ ] p×q

ϕ

11 ⎢ ϕ21 ⎢ =⎢ . ⎣ .. ϕ  p1

ϕ

12 ϕ 22 .. . ϕ  p2

... ... .. . ...

ϕ ⎤ 1q ϕ ⎥ 2q ⎥ .. ⎥ , . ⎦ ϕ  pq

 ϕ  ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ where each entry αβ = [ϑαβ1 , ϑαβ2 , ϑαβ3 , ϑαβ4 ], [δαβ1 , δαβ2 , δαβ3 , δαβ4 ] , α = 1, 2, . . . , p, β = 1, 2, . . . , q, ϕ = 1, 2, . . . , r, shows a trapezoidal bipolar fuzzy number. The aggregated bipolar fuzzy decision values of alternatives based on conflicting   criteria Qβ , denoted by αβ = [ϑαβ1 , ϑαβ2 , ϑαβ3 , ϑαβ4 ], [δαβ1 , δαβ2 , δαβ3 , δαβ4 ] , are enumerated by employing the averaging operator as follows:

6.2 Bipolar Fuzzy PROMETHEE Method ϑαβ1 =

157

1 ϕ 1 ϕ 1 ϕ 1 ϕ ϑαβ1 , ϑαβ2 = ϑαβ2 , ϑαβ3 = ϑαβ3 , ϑαβ4 = ϑαβ4 , ϕ ϕ ϕ ϕ

δαβ1 =

r

r

r

r

ϕ=1

ϕ=1

ϕ=1

ϕ=1

r 1

r 1

r 1

ϕ

ϕ

δαβ1 , δαβ2 =

ϕ=1

ϕ

ϕ

δαβ2 , δαβ3 =

ϕ=1

ϕ

ϕ

δαβ3 , δαβ4 =

ϕ=1

1 ϕ δαβ4 . ϕ r

ϕ=1

(6.4) These aggregated values are arranged to form an aggregated decision matrix as follows: ⎤ ⎡ 11 12 . . . 1q ⎢ 21 22 . . . 2q ⎥ ⎥ ⎢ L = [αβ ] p×q = ⎢ . . . . ⎥. ⎣ .. .. . . .. ⎦  p1  p2 . . .  pq

3. Rank the bipolar fuzzy numbers. At this step, the ranking function is employed to convert the bipolar fuzzy evaluations into crisp values as follows: 

 −ϑαβ1 − ϑαβ2 + ϑαβ3 + ϑαβ4 ϑαβ1 + ϑαβ2 + ϑαβ3 + ϑαβ4 + − 4 2



 δαβ1 + δαβ2 + δαβ3 + δαβ4 −δαβ1 − δαβ2 + δαβ3 + δαβ4 + . 4 2 (6.5)



tαβ =

These crisp values form a simple decision matrix T = [tαβ ] p×q for further proceeding. 4. Determine the deviation by pairwise comparison. The deviation between the alternatives Sα and Sσ with respect to criterion Qβ can be computed by Eq. (6.6) as (6.6) dβ (Sα , Sσ ) = tβ (Sα ) − tβ (Sσ ), α, σ = 1, 2, . . . , p, where dβ (Sα , Sσ ) represents the difference or deviation between two alternatives Sα and Sσ relative criterion Qβ . The terms tβ (Sα ) and tβ (Sσ ) present the crisp values of alternative Sα and Sσ , respectively, with respect to criterion β. 5. Define the preference function. A preference function Pβ (Sα , Sσ ) = Fβ [dβ (Sα , Sσ )] actually exhibits the intensity of preference of alternative Sα over alternative Sσ on the basis of criterion Qβ , according to preference function Fβ and its value belongs to closed unit interval. The zero value of preference value exhibits the indifference relation among two alternatives on the basis of that specific criterion. Moreover, the preference value closer to 1 represents the stronger preference among the considered alternatives. This preference value is mathematically evaluated as follows: (i) Pβ (Sα , Sσ ) = 0 presents the indifference between Sα and Sσ , or no preference of Sα over Sσ ; (ii) Pβ (Sα , Sσ ) ∼ 0 denotes the weak preference of Sα over Sσ ;

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

(iii) Pβ (Sα , Sσ ) ∼ 1 indicates the strong preference of Sα over Sσ ; (iv) Pβ (Sα , Sσ ) = 1 highlights the strict form of preference of Sα over Sσ . 6. Determine the normalized weights. The weight value of a criterion corresponds to its relative influence in that problem. Several approaches are available in the literature to determine the weight values which may be completely or partially unknown for decision makers. If all the criteria are equally significant, then they will be of same weights. In this methodology, we evaluate the normalized weights by entropy information. In order to compute the weight values by entropy measure, we first evaluate the projection values P(αβ) for each criterion Qβ (β = 1, 2, . . . , q), by normalization as P(αβ) =

tαβ . p  tαβ

(6.7)

α=1

These projection values helps to enumerate the entropy value E(β) for each criterion as follows: p  E(β) = −c P(αβ) log(P(αβ)), (6.8) α=1

where c = (log( p))−1 is a constant. Afterwards, the degree of divergence div(β) of the intrinsic information for each criterion is given by formula, div(β) = 1 − E(β), β = 1, 2, . . . , q.

(6.9)

The value of divergence div(β) represents the inherent contrast intensity of criterion Qβ . The greater value of div(β) depicts the more importance of criterion Qβ for the considering problem. Finally, the weights are obtained as follows: w(β) =

div(β) , q  div(β)

(6.10)

β=1

such that w(β) > 0 and

q  β=1

w(β) = 1.

7. Calculate the multi-criteria preference index. The multi-criteria preference index of alternatives is determined using the preference function and weight of  the respective criteria. The multi-criteria preference index is determined by taking the weighted average of the preference value Pβ , q 



(Sα , Sσ ) =

β=1

w(β)Pβ (Sα , Sσ ) q  β=1

; α = σ, α, σ = 1, 2, . . . , p. w(β)

(6.11)

6.2 Bipolar Fuzzy PROMETHEE Method

159

Due to normality of weights, Eq. (6.11) becomes 

(Sα , Sσ ) =

q 

w(β)Pβ (Sα , Sσ ); α = σ, α, σ = 1, 2, . . . , p.

(6.12)

β=1

 The multi-criteria preference index ∈ [0, 1] such that  (i) (Sα , Sσ ) ≈ 0 exhibits the weak preference of alternative Sα over Sσ with respect to all criteria;  (ii) (Sα , Sσ ) ≈ 1 highlights the strong preference of alternative Sα over Sσ with respect to all criteria. These preference indices are the basic elements to check the outranking relations on the set S of alternatives which can be visualized via outranking graph. The alternatives are represented bynodes and between any two nodes Sα and Sσ ,  there are two arcs with values (Sα , Sσ ) and (Sσ , Sα ), that have no particular relation. 8. Find the preference order. The ordering (may be partial or complete) of alternatives is then obtained by employing the outranking relationship of alternatives. The partial ranking of the alternatives is derived by the PROMETHEE I, whereas the complete ranking is obtained by the additional step of PROMETHEE II. (i) Ordering the alternatives by partial ranking or PROMETHEE I The outgoing or leaving flow of each alternative Sα is computed by taking the summation of the values of outward arcs of the alternative Sα as follows: ξ + (Sα ) =

1  (Sα , Sσ ); α = σ, α, σ = 1, 2, . . . , p. p − 1 S ∈S

(6.13)

σ

The outgoing flow of Sα is graphically shown in Fig. 6.2. The dominance behavior of alternative Sα over all other alternatives is measured by this positive outranking flow. Similarly, the incoming or entering flow of alternative Sα is calculated by taking the summation of the values of inward arcs of alternative Sα as follows: ξ − (Sα ) =

1  (Sσ , Sα ); α = σ, α, σ = 1, 2, . . . , p. p − 1 S ∈S

(6.14)

σ

The entering flow of Sα is graphically shown in Fig. 6.3. The negative outranking flow measures dominance of other alternatives over alternative Sα . The alternative with the higher value of ξ + (Sα ) and the lower value of ξ − (Sα ) is selected as the most favorable alternative. The leaving or entering flows are used to evaluate the preferences as described in Eqs. (6.15) and (6.16), respectively.

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

Fig. 6.2 Outgoing flow of Sα

Fig. 6.3 Incoming flow of Sα

⎧ ⎨ Sα P + Sσ ⇐⇒ ξ + (Sα ) > ξ + (Sσ ); ∀Sα , Sσ ∈ S, ⎩

Sα I + Sσ ⇐⇒ ξ + (Sα ) = ξ + (Sσ ); ∀Sα , Sσ ∈ S,

(6.15)

⎧ ⎨ Sα P − Sσ ⇐⇒ ξ − (Sα ) < ξ − (Sσ ); ∀Sα , Sσ ∈ S, ⎩

Sα I − Sσ ⇐⇒ ξ − (Sα ) = ξ − (Sσ ); ∀Sα , Sσ ∈ S.

(6.16)

The PROMETHEE I partial ordering (P1 , I1 , R1 ) is then obtained by taking the intersection of above mentioned two preferences as ⎧ Sα P1 Sσ (Sα outranks Sσ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

if Sα P + Sσ and Sα P − Sσ , or Sα P + Sσ and Sα I − Sσ , or Sα I + Sσ and Sα P − Sσ ;

. ⎪ + − ⎪ I S (S is indifferent to S ), iff S I S and S I S ; S ⎪ α 1 σ α σ α σ α σ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Sα R1 Sσ (Sα and Sσ are incomparable), otherwise. (6.17)

6.2 Bipolar Fuzzy PROMETHEE Method

161

(ii) Ordering the alternatives by complete ranking or PROMETHEE II The difference of positive and negative outranking flows of alternative Sα is known as the net flow of that particular alternative which is computed as ξ(Sα ) = ξ + (Sα ) − ξ − (Sα ).

(6.18)

A complete ordering of alternatives is produced by net flows of alternatives without any incomparability. The PROMETHEE II complete ranking (P2 , I2 ) is given in Eq. (6.19). ⎧ ⎨ Sα P2 Sσ (Sα outranks Sσ ) ⎩

iff ξ(Sα ) > ξ(Sσ ), (6.19)

Sα I2 Sσ (Sα is indifferent to Sσ ) iff ξ(Sα ) = ξ(Sσ ).

Thus, all the alternatives are able to be compared on the basis of net flows ξ(Sα ). The alternative having greatest value of net flow is considered as the best-fitted alternative. The framework of the process of bipolar fuzzy PROMETHEE method is presented in Fig. 6.4. This sketch comprises the goal of selection procedure, the economical and environmental criteria, the alternatives for assessment, the choice of preference functions, and net flows of PROMETHEE I and PROMETHEE II in the form of partial and complete ranking, respectively. This multi-criteria outranking approach is described by few computational steps, in which all steps remain same other than the defining of preference function and the determination of normalized weights. An appropriate method or technique can be adopted to compute the normalized weights of criteria according to choice of decision values or the preferences of decision experts. The decision maker is designated to specify the preference function for each criterion respective of its nature. The choice of preference function for different criteria is a critical step as it can influence the final decision. Example 6.1 (Green Supplier Selection) The challenges of the development and survival for national and international companies or manufacturers have been increased by the advancement of economic globalization. A magnificent supply chain helps the companies to stand out in the competitive markets of business sector. Therefore, supplier selection is a key factor to develop an impressive, low cost, fast and operational chain to supply the product to the end-users. That’s why supplier selection is a trendy field for the researchers globally. The economical development being directly linked with ecological environment is necessary for human survival so this is one of the main reasons for development of manufacturing companies. The manufacturer is also looking for green supplier in order to improve green supply chain. For this case study, criteria and alternatives are adopted from the work of Abdullah et al. [1] and Gurel et al. [25]. Four alternate suppliers S1 , S2 , S3 , and S4 are considered for through exploration.

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

Description of the multiSelection of suitable alternative on the basis of conflicting criteria evaluated by a group of decision makers

criteria group decision making (MCGDM) problem

Perference ratings are assigned on the basis of q confliction criteria Q1 , Q2 , Q3 , . . . , Qq

Evaluated by D1

Evaluated by D2

...

Evaluated by Dr

Aggregation of decision values

Construct the aggregated bipolar fuzzy decision matrix

Construct a crisp decision matrix by using the ranking function of bipolar fuzzy numbers

Usual criterion preference function

Partial ranking by Usual criterion

Complete ranking by Usual criterion

Linear and level criteria preference functions

Partial ranking by the combination of linear and level criteria

Complete ranking by the combination of linear and level criteria

Choice of suitable preference function

PROMETHEE I

PROMETHEE II

Fig. 6.4 Framework of bipolar fuzzy PROMETHEE method

S1 S2 S3 S4

= MVG Food Marketing Sdn Bhd, = CF Org Noodle Sdn Bhd, = Hexa Food Sdn Bhd, = SCS Food Manufacturing Sdn Bhd.

A panel of three experts is assigned the job to assess these companies for the choice of green supplier regarding seven criteria, denoted by Q1 , Q2 , Q3 , Q4 , Q5 , Q6 , and Q7 , whose details are provided as follows:

6.2 Bipolar Fuzzy PROMETHEE Method

163

Q1 = Cost of products (comprises the maintenance, transportation, inventory, purchasing, holding, security, etc.), Q2 = Quality of products (is maintained by techniques, principles, and practices of companies), Q3 = Service provided (such as low costs, high productivity, quick response, minimum wastage, no damage, etc.), Q4 = Delivery (at the correct time, at the right place, and in good condition), Q5 = Pollution control (is an important criterion as pollution is obtained as a byproduct of energy use in the production procedures), Q6 = Environmental management system (and environmental dimension have been recently added in assessment procedures), Q7 = Green packaging (is a type of packaging which defines to protect the environment by using environmental friendly material). The computations of PROMETHEE method using usual criterion preference function are described and computed in the following steps: 1. The panel decided to express its assessments for the alternatives with respect to different criteria via a set of seven linguistic terms {Very good, Good, Medium good, Fair, Medium poor, Poor, Very poor}, as shown in Fig. 6.1. Table 6.1 comprises all linguistic terms along with their corresponding bipolar fuzzy numbers. 2. Table 6.2 presents the linguistic preferences of the experts regarding all alternatives. The preference ratings of these linguistic terms in the form of trapezoidal bipolar fuzzy numbers are used, which are defined in Table 6.1, and the results are given in Table 6.3. The task for the aggregation of bipolar fuzzy numbers is performed using Eq. (6.4) to form the aggregated decision matrix as given by Table 6.4. For example, the aggregated decision value of supplier S1 with respect to criterion Q1 is computed by arithmetic mean as 1 [0.8, 0.7, 0.7] = 0.73, 3 1 [0.0, 0.1, 0.1] = 0.07, 3

1 [0.9 + 0.8 + 0.8] = 0.83, 3 1 [0.0 + 0.2 + 0.2] = 0.13, 3

1 [1.0 + 0.8 + 0.8] = 0.87, 3 1 [0.1 + 0.3 + 0.3] = 0.23, 3

1 [1.0, 0.9, 0.9] = 0.93, 3 1 [0.2, 0.3, 0.3] = 0.27. 3

Table 6.1 Linguistic variables and values of criteria Linguistic variable Abbreviation Bipolar fuzzy number V er y good Good Medium good Fair Medium poor Poor V er y poor

VG G MG F MP P VP

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

164

6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

Table 6.2 Performance ratings by decision makers (linguistic terms) D1 Q1 Q2 Q3 Q4 Q5 Q6 Q7

D2

D3

S1

S2

S3

S4

S1

S2

S3

S4

S1

S2

S3

S4

VG MG MG MG F F MG

MG G G VG F MG MG

G G VG VG F F F

G VG MG G G G G

G VG G MG F MG F

VG MG G G G G G

G G VG VG F MG G

G G MG VG G F G

G VG VG G F F F

G G MG MG MG G G

MG G G MG G G F

VG G G G G F G

3. In this step, a simple decision matrix, having real entries, is obtained by applying the ranking function of bipolar fuzzy numbers. Eq. (6.5) is applied to the entries of Table 6.4 to construct the matrix T . S1

S2

⎡ 0.635 0.555 Q2 ⎢ 0.665 0.455 ⎢ Q3 ⎢ 0.555 0.455 T = Q4 ⎢ ⎢ 0.370 0.555 ⎢ Q5 ⎢ 0.000 0.277 ⎣ 0.100 0.455 Q6 Q7 0.100 0.455 Q1

S3

0.455 0.525 0.740 0.665 0.177 0.277 0.177

S4

0.635 ⎤ 0.635 ⎥ ⎥ 0.370 ⎥ ⎥ . 0.635 ⎥ ⎥ 0.525 ⎥ 0.177 ⎦ 0.177

For instance, t11 is the performance value of supplier S1 on the basis of criterion Q1 which is computed as follows: 

 −0.73 − 0.83 + 0.87 + 0.93 0.73 + 0.83 + 0.87 + 0.93 + t11 = 4 2 



−0.07 − 0.13 + 0.23 + 0.27 0.07 + 0.13 + 0.23 + 0.27 + − 4 2 = (0.84 + 0.12) − (0.175 + 0.15) = 0.635.

4. Now, the deviation between alternatives with respect to all criteria is enumerated using decision matrix T and Eq. (6.6). The deviation values are enclosed in Table 6.5. 5. Next task is the specification of suitable preference function for each criterion accordingly. In this problem, the usual criterion preference function is chosen as characterized in Definition 6.4. The preference values are organized in Table 6.6.

6.2 Bipolar Fuzzy PROMETHEE Method

165

Table 6.3 Performance ratings by decision makers (bipolar fuzzy numbers) S1 S2 S3 S4 D1 Q1 (0.8, 0.9, 1.0, 1.0), (0.5, 0.6, 0.7, 0.8), (0.7, 0.8, 0.8, 0.9), (0.7, 0.8, 0.8, 0.9),

(0.0, 0.0, 0.1, 0.2)

(0.2, 0.3, 0.4, 0.5)

(0.1, 0.2, 0.3, 0.3)

(0.1, 0.2, 0.3, 0.3)

Q2 (0.5, 0.6, 0.7, 0.8), (0.7, 0.8, 0.8, 0.9), (0.7, 0.8, 0.8, 0.9), (0.8, 0.9, 1.0, 1.0),

(0.2, 0.3, 0.4, 0.5)

(0.1, 0.2, 0.3, 0.3)

(0.1, 0.2, 0.3, 0.3)

(0.0, 0.0, 0.1, 0.2)

Q3 (0.5, 0.6, 0.7, 0.8), (0.7, 0.8, 0.8, 0.9), (0.8, 0.9, 1.0, 1.0), (0.5, 0.6, 0.7, 0.8),

(0.2, 0.3, 0.4, 0.5)

(0.1, 0.2, 0.3, 0.3)

(0.0, 0.0, 0.1, 0.2)

(0.2, 0.3, 0.4, 0.5)

Q4 (0.5, 0.6, 0.7, 0.8), (0.8, 0.9, 1.0, 1.0), (0.8, 0.9, 1.0, 1.0), (0.7, 0.8, 0.8, 0.9),

(0.2, 0.3, 0.4, 0.5)

(0.0, 0.0, 0.1, 0.2)

(0.0, 0.0, 0.1, 0.2)

(0.1, 0.2, 0.3, 0.3)

Q5 (0.4, 0.5, 0.5, 0.6), (0.4, 0.5, 0.5, 0.6), (0.4, 0.5, 0.5, 0.6), (0.7, 0.8, 0.8, 0.9),

(0.4, 0.5, 0.5, 0.6)

(0.4, 0.5, 0.5, 0.6)

(0.4, 0.5, 0.5, 0.6)

(0.1, 0.2, 0.3, 0.3)

Q6 (0.4, 0.5, 0.5, 0.6), (0.5, 0.6, 0.7, 0.8), (0.4, 0.5, 0.5, 0.6), (0.7, 0.8, 0.8, 0.9),

(0.4, 0.5, 0.5, 0.6)

(0.2, 0.3, 0.4, 0.5)

(0.4, 0.5, 0.5, 0.6)

(0.1, 0.2, 0.3, 0.3)

Q7 (0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8), (0.4, 0.5, 0.5, 0.6), (0.7, 0.8, 0.8, 0.9),

(0.2, 0.3, 0.4, 0.5)

(0.2, 0.3, 0.4, 0.5)

(0.4, 0.5, 0.5, 0.6)

(0.1, 0.2, 0.3, 0.3)

D2 Q1 (0.7, 0.8, 0.8, 0.9), (0.8, 0.9, 1.0, 1.0), (0.7, 0.8, 0.8, 0.9), (0.7, 0.8, 0.8, 0.9),

(0.1, 0.2, 0.3, 0.3)

(0.0, 0.0, 0.1, 0.2)

(0.1, 0.2, 0.3, 0.3)

(0.1, 0.2, 0.3, 0.3)

Q2 (0.8, 0.9, 1.0, 1.0), (0.5, 0.6, 0.7, 0.8), (0.7, 0.8, 0.8, 0.9), (0.7, 0.8, 0.8, 0.9),

(0.0, 0.0, 0.1, 0.2)

(0.2, 0.3, 0.4, 0.5)

(0.1, 0.2, 0.3, 0.3)

(0.1, 0.2, 0.3, 0.3)

Q3 (0.7, 0.8, 0.8, 0.9), (0.7, 0.8, 0.8, 0.9), (0.8, 0.9, 1.0, 1.0), (0.5, 0.6, 0.7, 0.8),

(0.1, 0.2, 0.3, 0.3)

(0.1, 0.2, 0.3, 0.3)

(0.0, 0.0, 0.1, 0.2)

(0.2, 0.3, 0.4, 0.5)

Q4 (0.5, 0.6, 0.7, 0.8), (0.7, 0.8, 0.8, 0.9), (0.8, 0.9, 1.0, 1.0), (0.8, 0.9, 1.0, 1.0), Q5 Q6 Q7 D3 Q1 Q2 Q3 Q4 Q5 Q6 Q7

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

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

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

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

166

6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

Table 6.4 Aggregated decision matrix Q1 Q2 Q3 Q4 Q5 Q6 Q7

S1

S2

S3

S4

(0.73, 0.83, 0.87, 0.93),

(0.67, 0.77, 0.83, 0.90),

(0.63, 0.73, 0.77, 0.87),

(0.73, 0.83, 0.87, 0.93),

(0.07, 0.13, 0.23, 0.27)

(0.10, 0.17, 0.27, 0.33)

(0.13, 0.23, 0.33, 0.37)

(0.07, 0.13, 0.23, 0.27)

(0.70, 0.80, 0.90, 0.93),

(0.63, 0.73, 0.77, 0.87),

(0.70, 0.80, 0.80, 0.90),

(0.73, 0.83, 0.87, 0.93),

(0.07, 0.10, 0.20, 0.30)

(0.13, 0.23, 0.33, 0.37)

(0.10, 0.20, 0.30, 0.30)

(0.07, 0.13, 0.23, 0.27)

(0.67, 0.77, 0.83, 0.90),

(0.63, 0.73, 0.77, 0.87),

(0.77, 0.87, 0.93, 0.97),

(0.57, 0.67, 0.73, 0.83),

(0.10, 0.17, 0.27, 0.33)

(0.13, 0.23, 0.33, 0.37)

(0.03, 0.07, 0.17, 0.23)

(0.17, 0.27, 0.37, 0.43)

(0.57, 0.67, 0.73, 0.83),

(0.67, 0.77, 0.83, 0.90),

(0.70, 0.80, 0.90, 0.93),

(0.73, 0.83, 0.87, 0.93),

(0.17, 0.27, 0.37, 0.43)

(0.10, 0.17, 0.27, 0.33)

(0.07, 0.10, 0.20, 0.30)

(0.07, 0.13, 0.23, 0.27)

(0.40, 0.50, 0.50, 0.60),

(0.53, 0.63, 0.67, 0.77),

(0.50, 0.60, 0.60, 0.70),

(0.70, 0.80, 0.80, 0.90),

(0.40, 0.50, 0.50, 0.60)

(0.23, 0.33, 0.40, 0.47)

(0.30, 0.40, 0.43, 0.50)

(0.10, 0.20, 0.30, 0.30)

(0.43, 0.53, 0.57, 0.67),

(0.63, 0.73, 0.77, 0.87),

(0.53, 0.63, 0.67, 0.77),

(0.50, 0.60, 0.60, 0.70),

(0.33, 0.43, 0.47, 0.57)

(0.13, 0.23, 0.33, 0.37)

(0.23, 0.33, 0.40, 0.47)

(0.30, 0.40, 0.43, 0.50)

(0.43, 0.53, 0.57, 0.67),

(0.63, 0.73, 0.77, 0.87),

(0.50, 0.60, 0.60, 0.70),

(0.70, 0.80, 0.80, 0.90),

(0.33, 0.43, 0.47, 0.57)

(0.13, 0.23, 0.33, 0.37)

(0.30, 0.40, 0.43, 0.50)

(0.10, 0.20, 0.30, 0.30)

Table 6.5 Deviation of alternatives with respect to criteria S1 S2 S1 S3 S1 S4 S2 S1 S2 S3 S2 S4 S3 S1 S3 S2 S3 S4 S4 S1 S4 S2 S4 S3

Q1

Q2

Q3

Q4

Q5

Q6

Q7

0.08 0.18 0.0 −0.08 0.1 −0.08 −0.18 −0.1 −0.18 0.0 0.08 0.18

0.21 0.14 0.03 −0.21 −0.07 −0.18 −0.14 0.07 −0.11 −0.03 0.18 0.11

0.1 −0.185 0.185 −0.1 −0.285 0.085 0.185 0.285 0.37 −0.185 −0.085 −0.37

−0.185 −0.295 −0.265 0.185 −0.11 −0.08 0.295 0.11 0.03 0.265 0.08 −0.03

−0.277 −0.177 −0.525 0.277 0.1 −0.248 0.177 −0.1 −0.348 0.525 0.248 0.348

−0.355 −0.177 −0.077 0.355 0.178 0.278 0.177 −0.178 0.1 0.077 −0.278 −0.1

−0.355 −0.077 −0.077 0.355 0.278 0.278 0.077 −0.278 0.0 0.077 −0.278 0.0

6. The projection values regarding all criteria, evaluated by Eq. (6.7), are enclosed in Table 6.7. For example, P(11) is the projection value of criterion Q1 regarding to the supplier S1 and is determined as follows: P(11) =

0.635 = 0.279. 0.635 + 0.555 + 0.455 + 0.635

The entropy value and the degree of divergence for each criterion, evaluated by deploying Eqs. (6.8) and (6.9), are given in Table 6.7. The results of entropy

6.2 Bipolar Fuzzy PROMETHEE Method

167

Table 6.6 Usual criterion preference function S1 S2 S1 S3 S1 S4 S2 S1 S2 S3 S2 S4 S3 S1 S3 S2 S3 S4 S4 S1 S4 S2 S4 S3

Q1

Q2

Q3

Q4

Q5

Q6

Q7

1 1 0 0 1 0 0 0 0 0 1 1

1 1 1 0 0 0 0 1 0 0 1 1

1 0 1 0 0 1 1 1 1 0 0 0

0 0 0 1 0 0 1 1 1 1 1 0

0 0 0 1 1 0 1 0 0 1 1 1

0 0 0 1 1 1 1 0 1 1 0 0

0 0 0 1 1 1 1 0 0 1 0 0

Table 6.7 Projection values of criteria P (αβ) Q1 Q2 Q3 S1 S2 S3 S4

0.279 0.243 0.20 0.279

0.292 0.20 0.230 0.279

0.262 0.215 0.349 0.175

Q4

Q5

Q6

Q7

0.166 0.249 0.299 0.285

0.0 0.283 0.181 0.536

0.099 0.451 0.275 0.175

0.11 0.501 0.195 0.195

Table 6.8 Entropy value, divergence, and weights of criteria E (β)

div(β) w(β)

Q1

Q2

Q3

Q4

Q5

Q6

Q7

0.99 0.01 0.02

0.99 0.01 0.02

0.98 0.02 0.04

0.98 0.02 0.04

0.72 0.28 0.50

0.90 0.10 0.18

0.88 0.12 0.21

values, degrees of divergence, and weights of criteria are respectively shown in Table 6.8. 7. The multi-criteria preference indices, calculated by Eq. (6.12), are arranged in Table 6.9. 8. This step concludes the whole procedure the partial as well as net flows of alternatives. (i) Positive and negative flows of alternatives (PROMETHEE I) The outranking and incoming flows, representing the overall superiority and inferiorities of the alternatives, are evaluated by Eqs. (6.13) and (6.14) and the outcomes are summarized in Table 6.10. The partial ordering of suppliers is then obtained by taking the intersection of preorders P + and P − , which is

168

6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

Table 6.9 Multi-criteria preference index Suppliers S1 S2 S1 S2 S3 S4

− 0.93 0.97 0.93

0.08 − 0.10 0.58

Table 6.10 Outgoing and incoming flow Suppliers ξ + (Sα ) S1 S2 S3 S4

0.060 0.757 0.443 0.683

S3

S4

0.04 0.91 − 0.54

0.06 0.43 0.26 −

ξ − (Sα ) 0.943 0.253 0.497 0.25

Fig. 6.5 Partial PROMETHEE I relations

given as follows: S2 P1 S1 , S2 P1 S3 , S3 P1 S1 , S4 P1 S1 , S4 P1 S3 , and the partial results of PROMETHEE I are illustrated in Fig. 6.5. According to positive outranking flow, the supplier S2 possess the highest outgoing flow that highlights the dominance of S2 over other suppliers. Oppositely, the supplier S4 is found to be less dominated by all other suppliers with minimum incoming flow value. These results are insufficient to compare all suppliers for establishment of complete ranking list. For this reason, net outranking flow helps by presenting the collective performance of suppliers to obtain the complete ranking. (ii) Net flow of alternatives (PROMETHEE II) The net flow of each alternative, obtained by Eq. (6.18), is provided by Table 6.11. These results can be justified by sketching the PROMETHEE Diamond which is a chart used to elaborate the results of PROMETHEE I and PROMETHEE II methods simultaneously. Figure 6.6 manifests the ranking positions of suppliers or complete ranking (PROMETHEE II).

6.2 Bipolar Fuzzy PROMETHEE Method Table 6.11 Net flow of suppliers Suppliers S1 S2 S3 S4

169

ξ(Sα ) −0.883 0.504 −0.054 0.433

Fig. 6.6 PROMETHEE diamond

According to ranking list, given in Table 6.11, and the positions of suppliers in Fig. 6.6, the supplier S2 appears to be the most competent one. The final ordering of alternatives is as follows: S2 S4 S3 S1 .

170

6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

6.3 Comparative Study The PROMETHEE technique has the advantage of selecting or choosing different preference functions for all criteria. In the previous section, the usual criteria preference function was used for all criteria and the ranking of green suppliers was obtained. In this section, the level and linear criteria preference functions are applied to green supplier selection for a comparative analysis of the results of the PROMETHEE technique. The preference function is chosen based on the type or nature of the criterion. For example, the linear preference function is chosen as the criterion Q1 (product cost) because linear functions are best suited for quantitative criteria. The level criterion preference function is best suited for the return or qualitative criterion, i.e., Q2 (product quality). The formation of the decision matrix and the calculation of the standard numerical weights are the same as those determined in Sect. 6.1, so we proceed to step 5. 5. In this step, the combination of level and linear preference functions is selected to evaluate the preference values for conflicting criteria. The details of the preference function for each criterion are provided by Table 6.12. In this numerical example, the indifference and preference threshold values for linear preference function and level function are considered as 0.05 and 0.1, respectively. The preference values, obtained by Eqs. (6.5) and (6.6), are shown in Table 6.13. 6. The normalized weights given in Table 6.8, which are calculated in Step 6 of Sect. 6.1, are used in account to make a comparison. 7. The multi-criteria preference indices, evaluated using Eq. (6.12), are summarized in Table 6.14. 8. The partial and net flows are determined in this step. (i) The numerical values of positive and negative outranking flows of each suppliers, enumerated in the light of Eqs. (6.13) and (6.14), are shown in Table 6.15. Table 6.12 Preference functions corresponding to criteria

Criteria

Preference function

Cost of products (Q1 ) Quality of products (Q2 ) Services (Q3 ) Delivery (Q4 ) Pollution control (Q5 ) Environmental management system (Q6 ) Green packaging (Q7 )

Linear Level Level Level Level Level Level

6.3 Comparative Study

171

Table 6.13 Linear and level criteria preference function S1 S2 S1 S3 S1 S4 S2 S1 S2 S3 S2 S4 S3 S1 S3 S2 S3 S4 S4 S1 S4 S2 S4 S3

Q1

Q2

Q3

Q4

Q5

Q6

Q7

1 1 0 0 1 0 0 0 0 0 1 1

1 0.5 0 0 0 0 0 0.5 0 0 1 0.5

0.5 0 1 0 0 0.5 1 1 1 0 0 0

0 0 0 1 0 0 1 0.5 0 1 0.5 0

0 0 0 1 0.5 0 1 0 0 1 1 1

0 0 0 1 1 1 1 0 0.5 0.5 0 0

0 0 0 1 1 1 0.5 0 0 0.5 0 0

Table 6.14 Multi-criteria preference index Suppliers S1 S2 S1 S2 S3 S4

S3

S4

0.06 − 0.07 0.56

0.03 0.66 − 0.53

0.04 0.41 0.13 −

Suppliers

ξ + (Sα )

ξ − (Sα )

S1 S2 S3 S4

0.0433 0.667 0.355 0.6083

0.8433 0.23 0.4067 0.1933

− 0.93 0.865 0.735

Table 6.15 Outgoing and incoming flow

The partial ordering of suppliers is then obtained by taking the intersection of preorders P + and P − , which is given as follows: S2 P1 S1 , S2 P1 S3 , S3 P1 S1 , S4 P1 S1 , S4 P1 S3 , and the partial results of PROMETHEE I are illustrated in Fig. 6.7. (ii) Table 6.16 presents the net outranking flow of each alternative, derived by applying Eq. (6.18). Thus, supplier S2 , possessing the highest net flow, is found to be the most efficient supplier under the combination of linear and level preference functions. The final ordering is as follows: S2 S4 S3 S1 .

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6 Extended PROMETHEE Method with Bipolar Fuzzy Sets

Fig. 6.7 Partial PROMETHEE I relations

Table 6.16 Net flow of suppliers

Suppliers S1 S2 S3 S4

ξ(Sα ) −0.80 0.437 −0.0517 0.415

In an outranking method has been presented to attain the partial and complete ranking of suppliers under green environment. Different preference functions have been applied to make the calculations conducive for the purpose of making comparison of PROMETHEE behavior. The usual criterion preference function is formally applied to obtain the net flows and ordering of alternatives. This choice is regarded as the simplest function, and it is also easy to apply. After that, the combination of linear and level preference functions is used to find out the net outranking flows and ranking of alternatives. The final rankings of suppliers by PROMETHEE method that we obtain with the utilization of these preference functions are given in Table 6.17. From inspection, we observe that supplier S2 is chosen irrespective of the functions. Furthermore, the results and net flows of suppliers using different preference functions are precisely presented in Fig. 6.8.

6.4 Conclusion To sum up, PROMETHEE presents a whole class of outranking methodologies that can be conveniently implemented owing to its computations ease. PROMETHEE I technique determines a partial ranking of the alternatives, whereas PROMETHEE II approach maintains a complete ranking of alternatives (in descending order of net flow values). The preference values of alternatives are derived by different types of preference functions. In this chapter, green supplier’s selection has been done to demonstrate the applicability of approaches based on seven types of environmental and economical criteria. We have employed different preference functions to analyze the influence and impact of these preference functions on the recommendation.

References

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Table 6.17 Final ranking of suppliers Suppliers Usual criterion preference function Supplier S1 Supplier S2 Supplier S3 Supplier S4

4 1 3 2

Linear and level criteria preference functions 4 1 3 2

Fig. 6.8 Comparison between the net flows of different preference functions

Clearly, the results show that these preference functions have no significant impact on the final selection or ranking. Besides the benefits, apparent limitations of the proposed approach are the decision of preference functions and its arithmetic functions. The simple subtraction arithmetic function is employed to compute the net flows of the alternatives. We have restricted our study to three types of preference functions. In future, we aim to extend with other types of preferences, or even modified preference functions. Additional Reading The readers are suggested to [3, 5–9, 43, 44] for definitions of additional terms and applications not included in this chapter.

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Chapter 7

Enhanced Decision Making Method with Two-Tuple Linguistic Bipolar Fuzzy Sets

This chapter presents 2-tuple linguistic bipolar fuzzy sets, a new strategy for the modelization uncertainty that incorporates a 2-tuple linguistic term into bipolar fuzzy sets. This model will enable us to provide new insights in decision making. Their operational rules will be used to define the 2-tuple linguistic bipolar fuzzy weighted averaging/geometric operators, which let us combine 2-tuple linguistic bipolar fuzzy numbers. To examine the impact of the correlation between choice variables and decision results, the Heronian mean operator is incorporated into the 2-tuple linguistic bipolar fuzzy environment. The generalized 2-tuple linguistic bipolar fuzzy Heronian mean operator and generalized 2-tuple linguistic bipolar fuzzy weighted Heronian mean operator are introduced first, and their properties are then discussed. Along with some of their properties, the 2-tuple linguistic bipolar fuzzy geometric Heronian mean operator and the 2-tuple linguistic bipolar weighted geometric Heronian mean operator are also discussed. They are then used to define a technique to multi-attribute group decision making (MAGDM) under the 2-tuple linguistic bipolar fuzzy framework. Finally, in this chapter, a numerical demonstration is offered in the context of selecting the best solar cell, and a comparison analysis is performed to demonstrate the superiority of the presented work. Finally in this chapter, a numerical illustration is provided in a setting of selection of the best photovoltaic cell and a comparative analysis is done that indicates the superiority of presented work. This chapter is based on [26].

7.1 Introduction As non-renewable resources including coal, natural gas, and crude oil are exhausted, and degree of pollution rises, solar photovoltaic research is accelerating, particularly in the aftermath of the COVID-19 pandemic. Photovoltaic cells are widely employed in a variety of fields around the world as a cost-effective and dependable energy © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Akram et al., Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-99-0569-0_7

177

178

7 Enhanced Decision Making Method with Two-Tuple …

source. Solar photovoltaic has emerged as a feasible and profitable energy solution for business owners, while the expenditure of acquiring and establishing solar systems has decreased significantly. Different types of photovoltaic cells are available depending on the manufacturing method. As a result, it is vital to assess the photovoltaic technologies on the market. Building a MAGDM tool to assist decision makers is a step in this direction. In previous chapters, we emphasized that people often performed the decision making actions in their daily routines [29]. When individuals collectively select the most suitable action among considering actions resulting in a group decision, when multiple attributes intervene in the process, we have a MAGDM. Individual decision making is mostly different from collective decision making, and so is method [12, 13]. Fuzzy set theory introduced by Zadeh [38] has given way to another point of view on the modeling of criterion satisfaction. Alternatively, fuzzy sets encode the membership degree of objects into a set. Zhang [35] proposed bipolar fuzzy sets as extension of fuzzy sets. They are indicated by two components: a positive membership grade belonging to (0, 1] and a negative membership grade belonging to [−1, 0). Akram and Arshad [1] and Akram et al. [2–8] presented several decision making techniques for bipolar fuzzy information. People like to express their opinions in natural language, therefore, terms like “excellent”, “good”, “moderate”, “poor”, and “very poor” should be used to evaluate certain properties of an object. Therefore, it is significantly important to study information aggregation operators that use language to describe attribute values. The 2-tuple linguistic item is an efficient approach to minimize the loss of informative data and obtain better and authentic evaluation results. This idea was first proposed by Herrera and Martínez [20, 21], and it has since become one of the most critical methods for dealing with language decision problems. Later, several 2-tuple linguistic aggregation operators and decision methods have been developed. The Heronian mean considers an effective and strong aggregation operator that captures the interrelations between the parameters to be aggregated. Originally, Beliakov [16] proved that the Heronian mean is an aggregation operator. Ayub et al. [15] originated a new class of cubic fuzzy Heronian mean Dombi operators, including cubic fuzzy Heronian mean Dombi, cubic fuzzy weighted Heronian mean Dombi, cubic fuzzy geometric Heronian mean Dombi, and cubic fuzzy weighted geometric Heronian mean Dombi aggregation operators. Lin et al. [22] presented the partition geometric Heronian mean operator and the partition Heronian mean operator, some picture fuzzy interactive partition Heronian means, the geometric picture fuzzy interactive partition Heronian mean operator and its weighted form. Deveci et al. [17] extended the compromise solution of the combinatorial method that combines the power Heronian function and the logarithmic method. Pamucar et al. [27] utilized weighted power Heronian and weighted geometric power Heronian functions to enhance traditional weighted aggregation and product evaluation techniques. Liu et al. [23] introduced and studied the neutrosophic cubic Heronian and the neutrosophic cubic weighted Heronian aggregation operator. Akram et al. [9–11] presented new decision making methods based on 2-tuple linguistic Fermatean fuzzy sets. For in-depth study, readers are referred to [14, 19, 24, 30, 32, 33, 36, 37].

7.1 Introduction

179

With these antecedents, we are ready to present our contribution to the topic as stated above. We begin with technical concepts. Definition 7.1 ([35]) Let X be a fixed set. A bipolar fuzzy set A in X is given as p

A = {x, (μ A (x), μnA (x))|x ∈ X },

(7.1)

p

here the positive membership degree function μ A (x) : X → [0, 1] represents the satisfaction degree of an element x to the property and negative membership degree function μnA (x) : X → [−1, 0] represents the satisfaction degree of an element x to some implicit counter property corresponding to a bipolar fuzzy set A, respectively, and, for every x ∈ X . Let J = (μ p , μn ) be a bipolar fuzzy number. p

p

Definition 7.2 Let J = (μ p , μn ), J1 = (μ1 , μn1 ) and J2 = (μ2 , μn2 ) be three bipolar fuzzy numbers, then the basic operations can be defined as follows: 1. 2. 3. 4. 5. 6. 7. 8.

p

p

p

p

J1 ⊕ J2 = (μ1 + μ2 − μ1 μ2 , −|μn1 ||μn2 |); p p J1 ⊗ J2 = (μ1 μ2 , −(|μn1 | + |μn2 |) + |μn1 ||μn2 |); λJ = (1 − (1 − μ p )λ , −|μn |λ ), λ > 0; (J)λ = ((μ p )λ , −1 + |1 + μn |λ ), λ > 0; Jc = (1 − μ p , |μn | − 1); p p J1 ⊆ J2 if and only if μ1 ≤ μ2 and μn1 ≥ μn2 ; p p J1 ∪ J2 = (max{μ1 , μ2 }, min{μn1 , μn2 }); p p J1 ∩ J2 = (min{μ1 , μ2 }, max{μn1 , μn2 }).

Definition 7.3 ([20]) Let S = {˘sj |j = 1, . . . , σ} be a linguistic term set and ϕ ∈ [1, σ] 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 : [1, σ] → S × − , 2 2 ⎧ ⎨s˘j , j = round(ϕ)  Δ(ϕ) = ⎩υ = ϕ − j, υ ∈ − 21 , 21 .

(7.2)

Definition 7.4 ([20]) Let S = {˘sj |j = 1, . . . , σ} be a linguistic term set and (˘sj , υj ) be a 2-tuple, there exists a function Δ−1 that restore the 2-tuple to its equivalent numerical value ϕ ∈ [1, σ] ⊂ R, where   1 1 → [1, σ], Δ−1 : S × − , 2 2 Δ−1 (˘sj , υ) = j + υ = ϕ.

(7.3)

180

7 Enhanced Decision Making Method with Two-Tuple …

Definition 7.5 ([16]) Let ak (k = 1, 2, . . . , n) be a group of non-negative numbers, if n  n  2 √ a j ak (7.4) HM(a1 , a2 , . . . , an ) = n(n + 1) j=1 k= j then HM is known as Heronian mean operator. Based on Heronian mean, Yu [34] introduced the generalized Heronian mean as follows: Definition 7.6 ([34]) Let s, t > 0 and ak (k = 1, 2, . . . , n) be a group of nonnegative numbers. If ⎛ GHMs,t (a1 , a2 , . . . , an ) = ⎝

2 n(n + 1)

n n  

⎞ s+t1 a sj akt ⎠

(7.5)

j=1 k= j

then G H M s,t is called generalized Heronian mean operator. The generalized Heronian mean operator decreases the Heronian mean operator when s = t = 21 . Yu [34] then introduced the geometric Heronian mean operator, which is as follows: Definition 7.7 ([34]) Let s, t > 0 and ak (k = 1, 2, . . . , n) be a group of nonnegative real numbers, then ⎛ GHMs,t (a1 , a2 , . . . , an ) =

n

2 ⎞ n(n+1)

1 ⎝ (sa j + tak )⎠ s + t j=1,k= j

(7.6)

then G H M s,t is called the geometric Heronian mean operator. The terminologies used in this chapter are abbreviated in Table 7.1.

7.2 The 2-Tuple Linguistic Bipolar Fuzzy Sets In this section, by integrating bipolar fuzzy set and 2-tuple linguistic terms, the generalization of bipolar fuzzy set called 2-tuple linguistic bipolar fuzzy set is presented. Further, the 2TLBFWA and 2TLBFWG operators are presented and their desirable properties are discussed. Definition 7.8 Let S = {˘s0 , s˘1 , s˘2 , . . . s˘σ } be a 2-tuple linguistic term set with odd cardinality σ + 1. If ((˘sl , L), (˘sn , N )) is defined for s˘l , s˘n ∈ S and L , N ∈ [− 21 , 21 ) where (˘sl , L) and (˘sn , N ) express independently the membership degree and nonmembership degree by 2-tuple linguistic sets, then 2-tuple linguistic bipolar fuzzy set is defined as follows:

7.2 The 2-Tuple Linguistic Bipolar Fuzzy Sets

181

Table 7.1 Nomenclature of this chapter Notation Description G2TLBFHM G2TLBFWHM 2TLBFGHM 2TLBFWGHM ((˘slk , L k ), (˘sn k , Nk )) S = {˘sj |j = 0, 1, . . . , σ} (˘slk , L k ) (˘sn k , Nk ) s, t  Q w S (Z ) H(Z )

Generalized 2TLBF Heronian mean Generalized 2TLBF weighted Heronian mean 2TLBF geometric Heronian mean 2TLBF weighted geometric Heronian mean 2-tuple linguistic bipolar fuzzy number 2-Tuple linguistic term Positive membership degree of 2-tuple linguistic bipolar fuzzy number Negative membership degree of 2-tuple linguistic bipolar fuzzy number Parameters of Heronian mean operators Alternative Attribute Weight of attribute Score function of 2-tuple linguistic bipolar fuzzy number Accuracy function of 2-tuple linguistic bipolar fuzzy number

Z k = ((˘slk , L k ), (˘sn k , Nk )), where 0 ≤ Δ−1 (˘slk , L k ) ≤ σ, 0 ≤ Δ−1 (˘sn k , Nk ) ≤ σ. Definition 7.9 Let Z = ((˘sl , L), (˘sn , N )) be a 2-tuple linguistic bipolar fuzzy number. Then the score and accuracy functions of Z are defined as:

S(Z ) = Δ

−1  −1  Δ (˘sl , L) Δ (˘sn , N ) σ 1+ − , S(Z ) ∈ [0, σ] (7.7) 2 σ σ



−1  −1  Δ (˘sl , L) Δ (˘sn , N ) H(Z ) = Δ σ + , H(Z ) ∈ [0, σ] σ σ

(7.8)

Definition 7.10 Let Z 1 = ((˘sl1 , L 1 ), (˘sn 1 , N1 )) and Z 2 = ((˘sl2 , L 2 ), (˘sn 2 , N2 )) be two 2-tuple linguistic bipolar fuzzy numbers, then 1. 2. 3. 4. 5.

S(Z 1 ) < S(Z 2 ), then Z 1 < Z 2 ; S(Z 1 ) > S(Z 2 ), then Z 1 > Z 2 ; S(Z 1 ) = S(Z 2 ), H(Z 1 ) < H(Z 2 ), then Z 1 < Z 2 ; S(Z 1 ) = S(Z 2 ), H(Z 1 ) > H(Z 2 ), then Z 1 > Z 2 ; S(Z 1 ) = S(Z 2 ), H(Z 1 ) = H(Z 2 ), then Z 1 = Z 2 .

Definition 7.11 Let Z 1 = ((˘sl1 , L 1 ), (˘sn 1 , N1 )) and Z 2 = ((˘sl2 , L 2 ), (˘sn 2 , N2 )) be two 2-tuple linguistic bipolar fuzzy numbers, then

182

7 Enhanced Decision Making Method with Two-Tuple …

−1  −1  Δ (˘sl ,L 1 ) Δ (˘sl ,L 2 ) 1 2 Δ σ 1− 1− 1− , σ σ  

1. Z 1 ⊕ Z 2 =

2. 3. 4.

Δ−1 (˘sn 1 ,N1 ) Δ−1 (˘sn 2 ,N2 ) ; Δ σ σ σ

−1  −1    Δ (˘sl ,L 1 ) Δ (˘sl ,L 2 ) Δ−1 (˘sn 1 ,N1 ) Δ−1 (˘sn 2 ,N2 ) 1 2 Z1 ⊗ Z2= Δ σ ,Δ σ 1− 1− 1− ; σ σ σ σ ⎛   ⎞

 

−1 λ  Δ−1 (˘sn 1 ,N1 ) λ ⎠, λ > 0; λZ 1 = ⎝ Δ σ 1− 1− Δ (˘sσl1 ,L 1 ) ,Δ σ σ ⎛ ⎞

 λ    −1 λ  −1 ⎠, λ > 0. Z 1λ = ⎝ Δ σ Δ (˘sσl1 ,L 1 ) , Δ σ 1− 1− Δ (˘sσn1 ,N1 )

We now discuss two types of weighted aggregation operators, namely, the 2TLBFWA operator and the 2TLBFWG operator. Definition 7.12 Let Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers. The 2TLBFWA operator is a mapping P n → P such that 2TLBFWA(Z 1 , Z 2 , . . . , Z n ) = ⊕nk=1 wk Z k where w = (w1 , w2 , . . . , wn )T is the weight vector of Z k (k = 1, 2, . . . , n), such that n  wk = 1. wk ∈ [0, 1] and k=1

Theorem 7.1 Let Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2tuple linguistic bipolar fuzzy numbers with weight vector w = (w1 , w2 , . . . , wn )T , n  thereby satisfying wk ∈ [0, 1] and wk = 1 (k = 1, 2, . . . , n). Then, their aggrek=1

gation value by the 2TLBFWA operator is still a 2-tuple linguistic bipolar fuzzy number, and    2TLBFWA(Z 1 , Z 2 , . . . , Z n ) = Δ σ 1 −

n

1−

k=1

Δ−1 (˘slk , L k ) σ

wk 

 ,Δ σ

w n −1 Δ (˘sn k , Nk ) k σ

 .

k=1

(7.9) Proof It is proved that Eq. (7.9) holds by using mathematical induction method for positive integer n. (a) when n = 1,   w1    −1     −1 Δ (˘sl1 ,L 1 ) Δ (˘sn 1 ,N1 ) w1 . w1 Z 1 = Δ σ 1 − 1 − , Δ σ σ σ Thus, Eq. (7.9) holds for n = 1. (b) Suppose that Eq. (7.9) holds for n = m,

2TLBFWA(Z 1 , Z 2 , . . . , Z m ) =

−1 wk 

m  

m   Δ−1 (˘snk ,Nk ) wk Δ (˘slk ,L k ) 1− ,Δ σ Δ σ 1− . σ σ k=1

k=1

7.2 The 2-Tuple Linguistic Bipolar Fuzzy Sets

183

When n = m + 1, by inductive assumption, then 2TLBFWA(Z 1 , Z 2 , . . . , Z m , Z m+1 ) = 2TLBFWA(Z 1 , Z 2 , . . . , Z m ) ⊕ wm+1 Z m+1

−1 wk 

m  

m   Δ−1 (˘snk ,Nk ) wk Δ (˘slk ,L k ) 1− , Δ σ = Δ σ 1− σ σ

⊕

k=1

k=1

wm+1 

−1 wm+1  



−1 Δ (˘slm+1 ,L m+1 ) Δ (˘sn m+1 ,Nm+1 ) . , Δ σ Δ σ 1− 1− σ σ

=



−1 wk 

m+1   m+1   Δ−1 (˘snk ,Nk ) wk Δ (˘slk ,L k ) . Δ σ 1− 1− , Δ σ σ σ k=1

k=1

Therefore, Eq. (7.9) holds for positive integer n = m + 1. Thus, by mathematical induction method, we know that Eq. (7.9) holds for any n ≥ 1.  Theorem 7.2 Let Z k = ((˘slk , L k ), (˘sn k , Nk )), Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be two sets of 2-tuple linguistic bipolar fuzzy numbers, then the 2TLBFWA operator has the following properties: 1. (Idempotency) If all Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) are equal, for all k = 1, 2, . . . , n, then 2TLBFWA(Z 1 , Z 2 , . . . , Z n ) = Z .

(7.10)

2. (Monotonicity) If Z k ≤ Z k , for all k, then 2TLBFWA(Z 1 , Z 2 , . . . , Z n ) ≤ 2TLBFWA(Z 1 , Z 2 , . . . , Z n ).

(7.11)

3. (Boundedness) If Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers, and let Z − = (mink (˘slk , L k ), maxk (˘sn k , Nk )) and Z + = (maxk (˘slk , L k ), mink (˘sn k , Nk )), then Z − ≤ 2TLBFWA(Z 1 , Z 2 , . . . , Z n ) ≤ Z + .

(7.12)

Definition 7.13 Let Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers. The 2TLBFWG operator is a mapping P n → P, such that 2TLBFWG(Z 1 , Z 2 , . . . , Z n ) = ⊗nk=1 Z kwk where w = (w1 , w2 , . . . , wn )T is the weight vector of Z k (k = 1, 2, . . . , n), such that n  wk = 1. wk ∈ [0, 1] and k=1

Theorem 7.3 Let Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2tuple linguistic bipolar fuzzy numbers with weight vector w = (w1 , w2 , . . . , wn )T , n  thereby satisfying wk ∈ [0, 1] and wk = 1 (k = 1, 2, . . . , n). Then, their aggrek=1

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7 Enhanced Decision Making Method with Two-Tuple …

gation value by the 2TLBFWG operator is still a 2-tuple linguistic bipolar fuzzy number, and

2TLBFWG(Z 1 , Z 2 , . . . , Z n ) =

 



wk  

n −1 n  Δ (˘slk ,L k ) wk  Δ−1 (˘sn k ,Nk ) ,Δ σ 1 − 1− Δ σ . σ σ k=1

k=1

(7.13) Theorem 7.4 Let Z k = ((˘slk , L k ), (˘sn k , Nk )), Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be two sets of 2-tuple linguistic bipolar fuzzy numbers, then the 2TLBFWG operator has the following properties: 1. (Idempotency) If all Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) are equal, for all k = 1, 2, . . . , n, then 2TLBFWG(Z 1 , Z 2 , . . . , Z n ) = Z .

(7.14)

2. (Monotonicity) If Z k ≤ Z k , for all k, then 2TLBFWG(Z 1 , Z 2 , . . . , Z n ) ≤ 2TLBFWG(Z 1 , Z 2 , . . . , Z n ).

(7.15)

3. (Boundedness) If Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers, and let Z − = (mink (˘slk , L k ), maxk (˘sn k , Nk )) and Z + = (maxk (˘slk , L k ), mink (˘sn k , Nk )), then Z − ≤ 2TLBFWG(Z 1 , Z 2 , . . . , Z n ) ≤ Z + .

(7.16)

7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators We present the concept of the generalized 2-tuple linguistic bipolar fuzzy Heronian mean (G2TLBFHM) operator. Definition 7.14 Let Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers. The G2TLBFHM ia a mapping P n → P such that

G2TLBFHM (Z 1 , Z 2 , . . . , Z n ) = s,t

2 ⊕n ⊕n (Z s ⊗ Z kt ) n(n + 1) j=1 k= j j

 s+t1 (7.17)

where s, t ≥ 0. Theorem 7.5 The aggregated value by using G2TLBFHM operator is also 2-tuple linguistic bipolar fuzzy number, where

7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators

185

G2TLBFHMs,t (Z 1 , Z 2 , . . . , Z n ) ⎞ ⎛ ⎛ ⎛ ⎞ 1 ⎞ 2 s+t

−1 s −1 t  n(n+1) n  Δ (˘sl j ,L j ) Δ (˘slk ,L k ) ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎠, 1− ⎟ ⎜ Δ ⎝σ 1 − σ σ ⎟ ⎜ j=1,k= j ⎟ ⎜ ⎟ ⎞ ⎞ ⎛ ⎛ =⎜ 1 ⎛ ⎞ ⎟ ⎜ 2 s+t

−1 s

t  n(n+1) ⎟ ⎜ n  Δ (˘sn j ,N j ) ⎜ ⎜ ⎜ Δ−1 (˘sn k ,Nk ) ⎟ ⎟ ⎠ ⎠⎠ ⎟ ⎠ ⎝ Δ ⎝σ ⎝1 − ⎝1 − 1− 1− 1 − σ σ j=1,k= j

(7.18) where s, t > 0. Proof Utilizing Definition 7.11,    −1     s  s  Δ (˘sl j , L j ) Δ−1 (˘sn j , N j ) Δ σ ,Δ σ 1 − 1 − , σ σ        t  t  Δ−1 (˘slk , L k ) Δ−1 (˘sn k , Nk ) t ,Δ σ 1 − 1 − . (Z k ) = Δ σ σ σ

(Z j )s =

Thus,

 ⎞   Δ−1 (˘sl j ,L j ) s Δ−1 (˘sl ,L k ) t k , ⎜Δ σ ⎟ σ σ ⎜ ⎟    ⎟ . (Z j )s ⊗ (Z k )t = ⎜ 



s ⎜ ⎟ t −1 −1 Δ (˘sn j ,N j ) Δ (˘sn k ,Nk ) ⎝ ⎠ 1− Δ σ 1− 1− σ σ ⎛

Therefore,   ⎞

−1 s −1 t  n  Δ (˘sl j ,L j ) Δ (˘slk ,L k ) 1− , ⎜Δ σ 1− ⎟ σ σ ⎜ ⎟ j=1,k= j  ⎟ .   (Z sj ⊗ Z kt ) = ⎜



  s t ⎜ ⎟ −1 (˘ n −1  Δ s ,N ) Δ (˘sn k ,Nk ) nj j ⎝ ⎠ 1− 1− 1− Δ σ σ σ ⎛

⊕nj=1 ⊕nk= j

j=1,k= j

Furthermore, 2 ⊕n ⊕n (Z s ⊗ Z kt ) n(n + 1) j=1 k= j j ⎞ ⎛ ⎛ ⎛ ⎞⎞    2

−1 n Δ (˘sl j ,L j ) s Δ−1 (˘sl ,L k ) t n(n+1)  k ⎟ ⎜ Δ ⎝σ ⎝1 − ⎠⎠ , 1− ⎟ ⎜ σ σ ⎟ ⎜ j=1,k= j ⎟. ⎛ ⎛ ⎞ ⎞ =⎜ 2   ⎟ ⎜ 



s t n(n+1) −1 ⎟ ⎜ −1 n Δ (˘sn j ,N j ) Δ (˘sn k ,Nk ) ⎝ Δ ⎝σ ⎝  ⎠⎠ ⎠ 1 − 1− 1− σ σ j=1,k= j

186

7 Enhanced Decision Making Method with Two-Tuple …

 1 s+t 2 ⊕nj=1 ⊕nk= j (Z sj ⊗ Z kt ) n(n + 1) ⎞ 1 ⎞ 2 s+t

−1 s −1 t  n(n+1) Δ (˘sl j ,L j ) Δ (˘slk ,L k ) ⎠ ⎟ 1− ⎠, σ σ

G2TLBFHMs,t (Z 1 , Z 2 , . . . , Z n ) = ⎛

⎛ ⎛ n  ⎜ ⎜ ⎝ ⎜ Δ ⎝σ 1 − ⎜ j=1,k= j ⎜ =⎜ ⎛ ⎜ ⎛ ⎛ ⎜ n  ⎜ ⎜ ⎜ ⎝ Δ ⎝σ ⎝1 − ⎝1 −

j=1,k= j

⎞

⎟ ⎟ ⎟ ⎟ ⎟. ⎞ ⎞ 1 ⎞ ⎟ 2 s+t

−1 s

t  n(n+1) ⎟ Δ (˘sn j ,N j ) Δ−1 (˘sn k ,Nk ) ⎟⎟ ⎟ ⎠ 1− 1− 1− ⎠⎠ ⎠ σ σ

 Example 7.1 Let Z 1 = ((˘s3 , 0.4), (˘s5 , −0.2)), Z 2 = ((˘s4 , 0.3), (˘s2 , −0.5)), Z 3 = ((˘s3 , 0.1), (˘s4 , −0.4)), and Z 4 = ((˘s5 , 0.2), (˘s6 , −0.3)) be four 2-tuple linguistic bipolar fuzzy numbers, and suppose s = 2 and t = 3, then according to Theorem 7.5,

G2TLBFHMs,t (Z 1 , Z 2 , . . . , Z n ) =

 1 s+t 2 ⊕nj=1 ⊕nk= j (Z sj ⊗ Z kt ) n(n + 1)

⎞ ⎛ ⎞ ⎞ 1 ⎛ 2+3 ⎞ 2 ⎛ 3.4 3.4 3.4 4.3 2 3 2 3 ⎟ 4(4+1) ⎜ ⎜ ⎟ (1 − ( 6 ) × ( 6 ) ) × (1 − ( 6 ) × ( 6 ) ) ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ×(1 − ( 3.4 )2 × ( 3.1 )3 ) × (1 − ( 3.4 )2 × ( 5.2 )3 ) ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ 6 6 6 6 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ 4.3 4.3 4.3 3.1 2 3 2 3 , Δ 6 1 − ) × ( ) ) × (1 − ( ) × ( ) ) ×(1 − ( ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ 6 6 6 6 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ×(1 − ( 4.3 )2 × ( 5.2 )3 ) × (1 − ( 3.1 )2 × ( 3.1 )3 ) ⎟ ⎜ ⎜ ⎜ ⎟ 6 6 6 6 ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎠ ⎝ ⎟ ⎜ ⎜ ⎝ ⎟ ⎠ 3.1 )2 × ( 5.2 )3 ) × (1 − ( 5.2 )2 × ( 5.2 )3 ) ⎜ ⎝ ⎟ ⎠ ×(1 − ( ⎜ ⎟ 6 6 6 6 ⎜ ⎟ ⎟ ⎞ ⎞ ⎛ ⎛ =⎜ 1 ⎜ ⎟ ⎞ ⎛ 2 ⎜ ⎟ 2+3 ⎛ ⎞ 4.8 )2 × (1 − 4.8 )3 ) × (1 − (1 − 4.8 )2 × (1 − 1.5 )3 ) ⎜ ⎜ ⎜ ⎟ 4(4+1) ⎟ ⎟ (1 − (1 − ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ 6 6 6 6 ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ 4.8 3.6 4.8 5.7 ⎟ ⎟ ⎜ ⎟ ⎟ 2 3 2 3 ×(1 − (1 − 6 ) × (1 − 6 ) ) × (1 − (1 − 6 ) × (1 − 6 ) ) ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ 1.5 1.5 1.5 3.6 2 × (1 − 3 ) × (1 − (1 − 2 × (1 − 3) ⎟ ⎜ Δ ⎜6 ⎜1 − ⎜1 − ⎜ ×(1 − (1 − ⎟ ⎟ ⎟ ⎟ ) ) ) ) 6 6 6 6 ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ 1.5 5.7 3.6 3.6 ⎜ ⎜ ⎜ ⎟ 2 3 2 3 ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ×(1 − (1 − 6 ) × (1 − 6 ) ) × (1 − (1 − 6 ) × (1 − 6 ) ) ⎠ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎝ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎠ ⎝ ⎝ ⎝ ⎝ ⎠ 3.6 5.7 5.7 5.7 2 3 2 3 ⎠ ⎠ ×(1 − (1 − 6 ) × (1 − 6 ) ) × (1 − (1 − 6 ) × (1 − 6 ) ) ⎛

= ((˘s3 , 0.7626), (˘s3 , 0.5081)).

Property 7.1 (Idempotency) Let all Z k (k = 1, 2, . . . , n) are equal, i.e., Z k = Z for all k, then G2TLBFHMs,t (Z 1 , Z 2 , . . . , Z n ) = Z . Proof Since Z k = Z = ((˘sl , L), (˘sn , N )), then G2TLBFHMs,t (Z 1 , Z 2 , . . . , Z n ) ⎞⎞ ⎛ ⎛ ⎛ ⎞ 1 ⎛ s+t ⎛ ⎛ ⎞s  ⎞ 2  ⎟⎟ ⎜ ⎜ t ⎜ n(n+1) Δ−1 (˘sl ,L j ) n Δ−1 (˘sl ,L k ) ⎟ ⎟⎟ ⎜ ⎜ ⎜⎜  j k ⎟ ⎟⎟ , ⎜ Δ ⎜σ ⎜⎜1 − ⎝1 − ⎝ ⎠ ⎠ ⎟⎟ ⎜ ⎜ σ σ ⎜ ⎠ ⎝ ⎜ ⎝ ⎝ ⎠⎠ j=1,k= j ⎜ ⎜ =⎜ ⎛ ⎛ 1 ⎜ ⎞ s+t ⎛ ⎜    −1 s   t  2 ⎜ ⎜ ⎜ n(n+1) n Δ (˘sn j ,N j )  Δ−1 (˘sn k ,Nk ) ⎜ ⎜ ⎜ ⎟ ⎜ ⎜ Δ ⎜σ ⎜1 − ⎝1 − 1− 1− 1− ⎠ σ σ ⎝ ⎝ ⎝ j=1,k= j

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . ⎞⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎠⎠ ⎠

7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators

187

⎞⎞ ⎛ ⎛ ⎞ 1  s −1 t  s+t

−1 Δ (˘sl ,L) Δ (˘sl ,L) ⎟⎟ ⎜ ⎟ ⎜ Δ⎜ ⎠⎠ , ⎟ ⎜ ⎝σ ⎝1 − 1 − σ σ ⎟ ⎜ ⎜ ⎛ ⎞⎟ =⎜ ⎟. 1  ⎟ ⎜ s

t  s+t

⎜ ⎜ −1 ⎟⎟ Δ−1 (˘sn ,N ) ⎝ Δ ⎝σ 1 − 1 − Δ (˘sn ,N ) 1− ⎠⎠ σ σ ⎛

=





 



−1 −1 s ,N ) Δ (˘sl ,L) n . , Δ σ 1 − 1 − Δ (˘ Δ σ 1− 1− σ σ

  = (˘sl , L), (˘sn , N ) = Z .

 Property 7.2 (Monotonicity) Let Z k (k = 1, 2, . . . , n) and Z k (k = 1, 2, . . . , n) be two sets of 2-tuple linguistic bipolar fuzzy numbers. If Z k ≥ Z k for all k, then G2T L B F H M s,t (Z 1 , Z 2 , . . . , Z n ) ≥ G2T L B F H M s,t (Z 1 , Z 2 , . . . , Z n ).     Proof Let Z = (˘sl j , L j ), (˘sn j , N j ) and Z  = (˘slj , L j ), (˘sn j , N j ) be two sets of 2-tuple linguistic bipolar fuzzy numbers. Since (˘sl j , L j ) ≥ (˘slj , L j ) Δ−1 (˘sl j , L j ) ≥ Δ−1 (˘slj , L j )  s s  t t  −1  Δ (˘sl j , L j ) Δ−1 (˘slk , L k ) Δ−1 (˘sl j , L j ) Δ−1 (˘slk , L k ) ≥ σ σ σ σ

n 

1−

j=1,k= j

1−

n  j=1,k= j

Δ−1 (˘sl j ,L j ) σ

s 

 Δ−1 (˘slk ,L k ) t σ

2  n(n+1)

≤

n  j=1,k= j

2

−1 s    n(n+1) Δ (˘sl j ,L j ) Δ−1 (˘slk ,L k ) t ≥1− 1− σ σ

2

−1   s −1   t  n(n+1) Δ (˘sl ,L j ) Δ (˘sl ,L k ) j k 1− . σ σ

n  j=1,k= j

2

−1   s −1   t  n(n+1) Δ (˘sl ,L j ) Δ (˘sl ,L k ) j k . 1− σ σ

⎛ ⎛ ⎞ s+t1 ⎞ 2   s t  n(n+1) n −1 −1 Δ (˘ s , L ) Δ (˘slk , L k ) lj j ⎜ ⎠ ⎟ Δ ⎝σ ⎝1 − 1− ⎠≥ σ σ j=1,k= j ⎛ ⎛ ⎞ s+t1 ⎞ 2   −1  s  t  n(n+1)  n  −1  Δ (˘ s , L ) Δ (˘ s , L ) lj j ⎜ lk k ⎠ ⎟ Δ ⎝σ ⎝1 − 1− ⎠. σ σ j=1,k= j Similarly, it is proved that (˘sn j , N j ) ≤ (˘sn j , N j ).



Property 7.3 (Boundedness) Let Z k (k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers, and let Z − = mink Z k , Z + = maxk Z k , then Z − ≤ G2T L B F H M s,t (Z 1 , Z 2 , . . . , Z n ) ≤ Z + .

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Proof According to Property 7.1 G2T L B F H M s,t (Z 1− , Z 2− , . . . , Z n− ) = Z − and

G2T L B F H M s,t (Z 1+ , Z 2+ , . . . , Z n+ ) = Z +

From Property 7.2 Z − ≤ G2T L B F H M s,t (Z 1 , Z 2 , . . . , Z n ) ≤ Z + .  The importance of aggregated arguments is not taken into consideration by the G2TLBFHM operator. Furthermore, in many real-life situations, particularly in MAGDM problem, attribute weights play an integral role in the aggregation process. If it is taken into the consideration of the weight information of aggregated 2-tuple linguistic bipolar fuzzy numbers in the G2TLBFHM operator, the weighted form of G2TLBFHM is given and its definition is described below. To alleviate the limitations of G2TLBFHM operator, the G2TLBFWHM operator is presented as follows: Definition 7.15 Let s, t > 0, Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a colT lection of 2-tuple linguistic bipolar fuzzy numbers, n w = (w1 , w2 , . . . , wn ) is the weight vector of Z k , satisfying wk > 0 and k=1 wk = 1 (k = 1, 2, . . . , n). The G2TLBFWHM operator is defined as follows:   1  n n s t s+t . G2TLBFWHMs,t w (Z 1 , Z 2 , . . . , Z n ) = ⊕ j=1 ⊕k= j w j wk (Z j ) ⊗ (Z k ) (7.19) Theorem 7.6 Let s, t > 0, Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers. Then the aggregated value using by G2TLBFWHM operator is also a 2-tuple linguistic bipolar fuzzy number, and G2TLBFWHMs,t w (Z 1 , Z 2 , . . . , Z n ) ⎞ ⎛ ⎛ ⎛ 1 ⎞⎞

−1 s −1 t w j wk  s+t n  Δ (˘sl j ,L j ) Δ (˘slk ,L k ) ⎟ ⎜ Δ ⎝σ ⎝ 1 − ⎠⎠ , 1− ⎟ ⎜ σ σ j=1,k= j ⎟ ⎜ ⎟. ⎞ ⎞ ⎛ ⎛ =⎜ 1   ⎟ ⎜



  s t w j wk s+t −1 n ⎟ ⎜ −1 Δ (˘sn j ,N j ) Δ (˘sn k ,Nk ) ⎝ Δ ⎝σ ⎝1 − 1 −  ⎠⎠ ⎠ 1− 1− 1 − σ σ j=1,k= j

(7.20)

7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators

189

Proof According to Definition 7.11,   

s  Δ−1 (˘sn j , N j ) (Z j ) = Δ σ ,Δ σ 1 − 1 − , σ     t  t 

−1 Δ−1 (˘slk , L k ) Δ (˘sn k , Nk ) t (Z k ) = Δ σ ,Δ σ 1 − 1 − . σ σ s

Δ−1 (˘sl j , L j ) σ

s 

 





Thus, ⎞ s 

−1  Δ (˘sl j ,L j ) Δ−1 (˘slk ,L k ) t , Δ σ ⎟ ⎜ σ σ



−1 s   ⎟ . (Z j )s ⊗ (Z k )t = ⎜   t ⎠ ⎝ −1 Δ (˘sn j ,N j ) Δ (˘sn k ,Nk ) Δ σ 1− 1− 1− σ σ ⎛

Therefore,     ⎞ s t w j wk  Δ−1 (˘sl j ,L j ) Δ−1 (˘slk ,L k ) , ⎟ ⎜Δ σ 1− 1− σ σ ⎟ ⎜ ⎟        w j wk (Z j )s ⊗ (Z k )t = ⎜ w w s

 j k ⎟. ⎜ t −1 −1 Δ (˘sn j ,N j ) Δ (˘sn k ,Nk ) ⎠ ⎝ 1 − Δ σ 1− 1− σ σ ⎛

Furthermore, ⊕nj=1 ⊕nk= j (w j wk (Z j )s ⊗ (Z k )t ) ⎛    ⎞

−1 s    w j wk n  Δ (˘sl j ,L j ) Δ−1 (˘slk ,L k ) t 1− , ⎜Δ σ 1 − ⎟ σ σ ⎜ ⎟ j=1,k= j ⎜ ⎟.    =⎜

−1 s   w j wk ⎟   n t −1  Δ (˘sn j ,N j ) Δ (˘sn k ,Nk ) ⎝ ⎠ 1− 1− 1− Δ σ σ σ j=1,k= j

1    s+t n n s t G2TLBFWHMs,t w (Z 1 , Z 2 , . . . , Z n ) = ⊕ j=1 ⊕k= j w j wk (Z j ) ⊗ (Z k ) ⎞ ⎛ ⎛  1 ⎞

−1 s −1 t w j wk  s+t n  Δ (˘sl j ,L j ) Δ (˘slk ,L k ) ⎟ ⎜ Δ ⎝σ 1 − ⎠, 1− ⎟ ⎜ σ σ j=1,k= j ⎟ ⎜ ⎟. ⎜ ⎞ ⎞ ⎛ ⎛ =⎜ 1   ⎟



  s t w j wk s+t −1 ⎟ ⎜ n −1  Δ (˘sn j ,N j ) Δ (˘ s ,N ) n k k ⎝ Δ ⎝σ ⎝1 − 1 − ⎠⎠ ⎠ 1− 1− 1 − σ σ

j=1,k= j

 Example 7.2 Let Z 1 = ((˘s3 , 0.4), (˘s5 , 0.2)), Z 2 = ((˘s4 , 0.3), (˘s2 , −0.5)), Z 3 = ((˘s3 , 0.1), (˘s4 , −0.4)), and Z 4 = ((˘s5 , 0.2), (˘s6 , −0.3)) be four 2-tuple linguistic bipolar fuzzy numbers, and suppose s = 2 and t = 3, w = (0.17, 0.32, 0.38, 0.13) then according to Theorem 7.6,

190

7 Enhanced Decision Making Method with Two-Tuple …

   1 s,t G2TLBFWHMw (Z 1 , Z 2 , . . . , Z n ) = ⊕nj=1 ⊕nk= j w j wk (Z j )s ⊗ (Z k )t s+t ⎛ ⎛ 1 ⎞ ⎛ ⎛ 3.4 3 0.17×0.17 × (1 − ( 3.4 )2 × ( 4.3 )3 )0.17×0.32 ⎞⎞ 2+3 2 (1 − ( 3.4 ⎟ ⎜ ⎜ 6 ) ×( 6 ) ) 6 6 ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ×(1 − ( 3.4 )2 × ( 3.1 )3 )0.17×0.38 × (1 − ( 3.4 )2 × ( 5.2 )3 )0.17×0.13 ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ 6 6 6 6 ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ 4.3 4.3 4.3 3.1 2 3 0.32×0.32 2 3 0.32×0.38 × (1 − ( 6 ) × ( 6 ) ) ⎟⎟ ⎟, ⎜ Δ ⎜6 ⎜1 − ⎜ ×(1 − ( 6 ) × ( 6 ) ) ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ×(1 − ( 4.3 )2 × ( 5.2 )3 )0.32×0.13 × (1 − ( 3.1 )2 × ( 3.1 )3 )0.38×0.38 ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ 6 6 6 6 ⎟ ⎜ ⎜ ⎝ ⎝ ⎠⎠ ⎜ ⎝ ⎠ 3.1 )2 × ( 5.2 )3 )0.38×0.13 × (1 − ( 5.2 )2 × ( 5.2 )3 )0.13×0.13 ⎜ ×(1 − ( ⎜ 6 6 6 6 ⎜ ⎛ ⎛ ⎜ ⎛  ⎛    =⎜ 5.2 3 0.17×0.17 × 1 − (1 − 5.2 )2 × (1 − 1.5 )3 0.17×0.32 2 ⎜ ⎜ ⎜ 1 − (1 − 5.2 ⎜ ⎜ ⎜ 6 ) × (1 − 6 ) 6 6 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  ⎜ ⎜ ⎜ ⎜    ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × 1 − (1 − 5.2 )2 × (1 − 3.6 )3 0.17×0.38 × 1 − (1 − 5.2 )2 × (1 − 5.7 )3 0.17×0.13 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 6 6 6 6 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  ⎜ 0.32×0.32   ⎜ ⎜ ⎜ ⎜ ⎜ 3.6 3 0.32×0.38 2 ⎜ 6 ⎜1 − ⎜1 − ⎜ × 1 − (1 − 1.5 )2 × (1 − 1.5 )3 × 1 − (1 − 1.5 ⎜ Δ⎜ 6 6 6 ) × (1 − 6 ) ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜    ⎜  ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1.5 2 5.7 3 0.32×0.13 × 1 − (1 − 3.6 )2 × (1 − 3.6 )3 0.38×0.38 ⎜ ⎜ ⎜ × 1 − (1 − 6 ) × (1 − 6 ) ⎜ 6 6 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝     ⎝ ⎝ ⎝ 5.7 3 0.38×0.13 × 1 − (1 − 5.7 )2 × (1 − 5.7 )3 0.13×0.13 2 × 1 − (1 − 3.6 6 ) × (1 − 6 ) 6 6

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ ⎞ ⎟ ⎞⎞ 1 ⎟. 2+3 ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎠⎠ ⎠⎠ ⎠

= ((˘s3 , 0.2487), (˘s2 , 0.9455)).

The G2TLBFWHM operator has the following features, which are easily shown. Property 7.4 (Idempotency) Let all Z k (k = 1, 2, . . . , n) are equal, i.e., Z k = Z for all k, then G2T L B F W H M s,t w (Z 1 , Z 2 , . . . , Z n ) = Z . Property 7.5 (Monotonicity) Let Z k (k = 1, 2, . . . , n) and Z k (k = 1, 2, . . . , n) be two sets of 2-tuple linguistic bipolar fuzzy numbers. If Z k ≥ Z k for all k, then s,t    G2TLBFWHMs,t w (Z 1 , Z 2 , . . . , Z n ) ≥ G2T L B F W H M w (Z 1 , Z 2 , . . . , Z n ).

Property 7.6 (Boundedness) Let Z k (k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers, and let Z − = mink Z k , Z + = maxk Z k , then + Z − ≤ G2TLBFWHMs,t w (Z 1 , Z 2 , . . . , Z n ) ≤ Z .

We now present the concept of the 2-tuple linguistic bipolar fuzzy geometric Heronian mean (2TLBFGHM) operator. Definition 7.16 Let Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers. The 2TLBFGHM is a mapping P n → P such that 2TLBFGHMs,t (Z 1 , Z 2 , . . . , Z n ) = where s, t ≥ 0.

 2 1  n ⊗ j=1 ⊗nk= j (s Z j ⊕ t Z k ) n(n+1) s+t

(7.21)

7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators

191

Theorem 7.7 The aggregated value by using 2TLBFGHM operator is also 2-tuple linguistic bipolar fuzzy number, where 2TLBFGHMs,t (Z 1 , Z 2 , . . . , Z n ) ⎛ ⎛ ⎛ 1 ⎞⎞ ⎞ ⎛ ⎞ s+t 2 

−1 s

−1 t  n(n+1) n  Δ (˘sl j ,L j ) Δ (˘slk ,L k ) ⎜ ⎜ ⎜ ⎟ ⎟ ⎠ ⎟ ⎜ Δ ⎝σ ⎝1 − ⎝1 − 1− 1− 1− ⎠⎠ , ⎟ σ σ ⎜ ⎟ j=1,k= j ⎜ ⎟ ⎟ =⎜ ⎞ 1 ⎞⎞ ⎜ ⎛ ⎛⎛ ⎟ 2 s+t

−1 s t  n(n+1) ⎜ ⎟ n −1  Δ (˘sn j ,N j ) ⎜ ⎜ ⎜ ⎟ Δ (˘sn k ,Nk ) ⎟ ⎟ ⎠ ⎠⎠ ⎝ Δ ⎝σ ⎝⎝1 − ⎠ 1− σ σ j=1,k= j

(7.22) where s, t > 0. Example 7.3 Let Z 1 = ((˘s3 , 0.4), (˘s5 , −0.2)), Z 2 = ((˘s4 , 0.3), (˘s2 , −0.5)), Z 3 = ((˘s3 , 0.1), (˘s4 , −0.4)), and Z 4 = ((˘s5 , 0.2), (˘s6 , −0.3)) be four 2-tuple linguistic bipolar fuzzy numbers, and suppose s = 2 and t = 3, then according to Theorem 7.7 we have 2TLBFGHMs,t (Z 1 , Z 2 , . . . , Z n ) =

 2 1  n ⊗ j=1 ⊗nk= j (s Z j ⊕ t Z k ) n(n+1) s+t

⎞⎞ ⎞ ⎛ ⎛ ⎞ 1 ⎛ 2 ⎛ 3.4 )2 × (1 − 3.4 )3 ) × (1 − (1 − 3.4 )2 × (1 − 4.3 )3 ) ⎞ 4(4+1) 2+3 ⎟⎟ ⎟ ⎜ ⎜ ⎜ (1 − (1 − ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ 6 6 6 6 ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ×(1 − (1 − 3.4 )2 × (1 − 3.1 )3 ) × (1 − (1 − 3.4 )2 × (1 − 5.2 )3 ) ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ 6 6 6 6 ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ 4.3 4.3 4.3 3.1 2 3 2 3 ⎟ ⎟⎟ , ⎟ ⎜ Δ ⎜6 ⎜1 − ⎜1 − ⎜ ×(1 − (1 − 6 ) × (1 − 6 ) ) × (1 − (1 − 6 ) × (1 − 6 ) ) ⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎜ ⎜ ×(1 − (1 − 4.3 )2 × (1 − 5.2 )3 ) × (1 − (1 − 3.1 )2 × (1 − 3.1 )3 ) ⎟ ⎜ ⎜ ⎜ 6 6 6 6 ⎟ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎠ ⎝ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎠ ⎝ 3.1 5.2 5.2 5.2 ⎜ ⎝ ⎝ 2 3 2 3 ⎠⎠ ⎟ ×(1 − (1 − 6 ) × (1 − 6 ) ) × (1 − (1 − 6 ) × (1 − 6 ) ) ⎜ ⎟ ⎜ ⎟ ⎟ ⎞ ⎞ ⎛ ⎛ =⎜ 1 ⎜ ⎟ ⎞ ⎛ 2 ⎜ ⎟ 2+3 ⎞ ⎛ 4.8 4.8 4.8 1.5 ⎜⎜ ⎜ ⎟ 2 3 2 3 4(4+1) ⎟ ⎟ (1 − ( 6 ) × ( 6 ) ) × (1 − ( 6 ) × ( 6 ) ) ⎜⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜⎜ ⎜ ⎜ ⎟ 4.8 3.6 4.8 5.7 ⎟ ⎟ ⎟ ⎟ ⎜ 2 ×( 3 ) × (1 − ( 2 ×( 3) ) ) ) ) ×(1 − ( ⎜⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟ ⎜ 6 6 6 6 ⎜⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟ ⎜ ⎜⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟ 1.5 1.5 1.5 3.6 2 3 2 3 ⎜ ⎜Δ ⎜6 ⎜1 − ⎜ ⎟ ⎟⎟ ⎟ ⎜ ×(1 − ( 6 ) × ( 6 ) ) × (1 − ( 6 ) × ( 6 ) ) ⎟ ⎜⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟ ⎜ ⎜⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ ⎟ ⎜ 1.5 5.7 3.6 3.6 ⎜⎜ ⎜ ⎜ ⎟ 2 3 2 3 ⎟⎟ ⎟ ⎜ ×(1 − ( ) × ( ) ) × (1 − ( ) × ( ) ) ⎟ ⎜⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎟ 6 6 6 6 ⎠ ⎝ ⎜⎜ ⎜ ⎝ ⎟ ⎟⎟ ⎠ ⎝⎝ ⎝ ⎠ ⎠⎠ ×(1 − ( 3.6 )2 × ( 5.7 )3 ) × (1 − ( 5.7 )2 × ( 5.7 )3 ) ⎛

6

6

6

6

= ((˘s2 , 0.9433), (˘s4 , 0.6685)).

Property 7.7 (Idempotency) Let all Z k (k = 1, 2, . . . , n) are equal, i.e., Z k = Z for all k, then 2TLBFGHM s,t (Z 1 , Z 2 , . . . , Z n ) = Z . Property 7.8 (Monotonicity) Let Z k (k = 1, 2, . . . , n) and Z k (k = 1, 2, . . . , n) be two sets of 2-tuple linguistic bipolar fuzzy numbers. If Z k ≥ Z k for all k, then 2TLBFGHM s,t (Z 1 , Z 2 , . . . , Z n ) ≥ 2T L B F G H M

s,t

(Z 1 , Z 2 , . . . , Z n ).

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7 Enhanced Decision Making Method with Two-Tuple …

Property 7.9 (Boundedness) Let Z k (k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers, and let Z − = mink Z k , Z + = maxk Z k , then Z − ≤ 2TLBFGHM s,t (Z 1 , Z 2 , . . . , Z n ) ≤ Z + . It can be seen that the importance of aggregated arguments is not taken into consideration by the 2TLBFGHM operator. Furthermore, in many real-life situations, particularly in MAGDM problem, attribute weights play an integral role in the aggregation process. If it is taken into consideration of the weight information of aggregated 2-tuple linguistic bipolar fuzzy numbers in the 2TLBFGHM operator, the weighted form of 2TLBFGHM is produced and its definition is described below. To alleviate the limitations of 2TLBFGHM operator, the 2TLBFWGHM operator is presented, which is as follows: Definition 7.17 Let s, t > 0, Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a colT lection of 2-tuple linguistic bipolar fuzzy numbers, n w = (w1 , w2 , . . . , wn ) is the weight vector of Z k , satisfying wk > 0 and k=1 wk = 1 (k = 1, 2, . . . , n). The 2TLBFWGHM operator is defined as follows: 2TLBFWGHMs,t w (Z 1 , Z 2 , . . . , Z n ) =

 1  n ⊗ j=1 ⊗nk= j (s Z j ⊕ t Z k )w j wk . s+t (7.23)

Theorem 7.8 Let s, t > 0, Z k = ((˘slk , L k ), (˘sn k , Nk ))(k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers. Then, the aggregated value by using the 2TLBFWGHM operator is also a 2-tuple linguistic bipolar fuzzy number, and 2TLBFGWHMs,t w (Z 1 , Z 2 , . . . , Z n ) ⎛ ⎛ ⎛ 1 ⎛ ⎛ ⎛ ⎞⎞s  ⎛  t ⎞w j wk ⎞ s+t Δ−1 (˘sl ,L j ) n Δ−1 (˘sl ,L k ) ⎜  ⎜ ⎜ j k ⎜1 − ⎝1 − ⎝1 − ⎝1 − ⎝ ⎠⎠ 1 − ⎠ ⎠ ⎜ Δ⎜ σ σ σ ⎜ ⎝ ⎝ j=1,k= j ⎜ ⎜ ⎞ =⎜ ⎛ ⎛ 1 ⎜   −1 s  t w j wk ⎞ s+t ⎜ ⎜ −1 n Δ (˘ s ,N ) ⎟ n  Δ (˘ s ,N ) j ⎜ ⎜ ⎝ nk k j ⎟ ⎠ 1− ⎝ Δ ⎝σ 1 − σ σ ⎠

⎞⎞ ⎞ ⎟⎟ ⎟ ⎟⎟ , ⎟ ⎠⎠ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

j=1,k= j

(7.24) Example 7.4 Let Z 1 = ((˘s3 , 0.4), (˘s5 , 0.2)), Z 2 = ((˘s4 , 0.3), (˘s2 , −0.5)), Z 3 = ((˘s3 , 0.1), (˘s4 , −0.4)), and Z 4 = ((˘s5 , 0.2), (˘s6 , −0.3)) be four 2-tuple linguistic bipolar fuzzy numbers, and suppose s = 2 and t = 3, w = (0.17, 0.32, 0.38, 0.13), then according to Theorem 7.8,

7.4 An Approach to MAGDM Problem with 2-Tuple … 1  n w w  s,t ⊗ j=1 ⊗nk= j (s Z j ⊕ t Z k ) j k 2TLBFWGHMw (Z 1 , Z 2 , . . . , Z n ) = s+t ⎛ ⎛ ⎛    ⎛ ⎛  3.4 3 0.17×0.17 × 1 − (1 − 3.4 )2 × (1 − 4.3 )3 0.17×0.32 2 ⎜ ⎜ ⎜ 1 − (1 − 3.4 6 ) × (1 − 6 )  6 6 ⎜ ⎜ ⎜ ⎜ ⎜    ⎜ ⎜ ⎜ 0.17×0.38 ⎜ ⎜ 5.2 3 0.17×0.13 ⎜ ⎜ ⎜ 2 ⎜ ⎜ × 1 − (1 − 3.4 )2 × (1 − 3.1 )3 × 1 − (1 − 3.4 ⎜ ⎜ ⎜ 6 6 6 ) × (1 − 6 )  ⎜ ⎜  ⎜ ⎜ ⎜   ⎜ ⎜ ⎜ ⎜ ⎜ 4.3 2 4.3 3 0.32×0.32 × 1 − (1 − 4.3 )2 × (1 − 3.1 )3 0.32×0.38 ⎜ ⎜ Δ ⎜6 ⎜1 − ⎜ ⎜1 − ⎜ × 1 − (1 − 6 ) × (1 − 6 ) 6 6 ⎜ ⎜ ⎜ ⎜  ⎜    ⎜ ⎜ ⎜ ⎜ ⎜ 4.3 5.2 )3 0.32×0.13 × 1 − (1 − 3.1 )2 × (1 − 3.1 )3 0.38×0.38 2 ⎜ ⎜ ⎜ ⎜ × 1 − (1 − ⎜ ) × (1 − ⎜ ⎜ ⎜ 6 6 6 6 ⎝  ⎝ ⎜ ⎜ ⎜    ⎜ ⎝ ⎝ 5.2 3 0.38×0.13 × 1 − (1 − 5.2 )2 × (1 − 5.2 )3 0.13×0.13 2 ⎜ × 1 − (1 − 3.1 6 ) × (1 − 6 ) 6 6 ⎜ ⎞ =⎜ ⎜ ⎛ ⎛ ⎛ ⎞⎞ 1 ⎜ 2+3 ⎟ (1 − ( 5.2 )2 × ( 5.2 )3 )0.17×0.17 × (1 − ( 5.2 )2 × ( 1.5 )3 )0.17×0.32 ⎜ ⎜ 6 6 6 6 ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ×(1 − ( 5.2 )2 × ( 3.6 )3 )0.17×0.38 × (1 − ( 5.2 )2 × ( 5.7 )3 )0.17×0.13 ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ 6 6 6 6 ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ 1.5 1.5 1.5 3.6 2 3 0.32×0.32 2 3 0.32×0.38 ⎜ Δ ⎜6 ⎜1 − ⎜ ×(1 − ( ⎟ ⎟ ⎟ ) × ( ) ) × (1 − ( ) × ( ) ) ⎜ 6 6 6 6 ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎜ ⎜ ⎜ ×(1 − ( 1.5 )2 × ( 5.7 )3 )0.32×0.13 × (1 − ( 3.6 )2 × ( 3.6 )3 )0.38×0.38 ⎟⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ 6 6 6 6 ⎜ ⎜ ⎜ ⎟ ⎝ ⎠⎠ ⎝ ⎝ ⎝ ⎠ 3.6 5.7 5.7 5.7 2 3 0.38×0.13 2 3 0.13×0.13 ×(1 − ( 6 ) × ( 6 ) ) × (1 − ( 6 ) × ( 6 ) )

193

1 ⎞⎞ ⎞ ⎞⎞ 2+3 ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ , ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎟ ⎠⎠ ⎟⎟ ⎟ ⎠⎠ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

= ((˘s3 , 0.1564), (˘s3 , 0.7362)).

Property 7.10 (Idempotency) Let all Z k (k = 1, 2, . . . , n) are equal, i.e., Z k = Z for all k, then 2TLBFWGHMs,t w (Z 1 , Z 2 , . . . , Z n ) = Z . Property 7.11 (Monotonicity) Let Z k (k = 1, 2, . . . , n) and Z k (k = 1, 2, . . . , n) be two sets of 2-tuple linguistic bipolar fuzzy numbers. If Z k ≥ Z k for all k, then s,t    2TLBFWGHMs,t w (Z 1 , Z 2 , . . . , Z n ) ≥ 2T L B F W G H M w (Z 1 , Z 2 , . . . , Z n ).

Property 7.12 (Boundedness) Let Z k (k = 1, 2, . . . , n) be a collection of 2-tuple linguistic bipolar fuzzy numbers, and let Z − = mink Z k , Z + = maxk Z k , then + Z − ≤ 2T L B F W G H M s,t w (Z 1 , Z 2 , . . . , Z n ) ≤ Z .

7.4 An Approach to MAGDM Problem with 2-Tuple Linguistic Bipolar Fuzzy Information An approach to MAGDM is presented in this section, based on the suggested G2TLBFWHM and 2TLBFWGHM operators. Consider  = {1 , 2 , . . . , m } is a set of alternatives, Q = {Q 1 , Q 2 , . . . , Q n } be the set of attributes, and D = {D1 , D2 , . . . , Dl } be a set of experts. For attribute Q k (k = 1, 2, . . . , n) of alternative  j ( j = 1, 2, . . . , m), the decision maker Dh (h = 1, 2, . . . , l) expresses his assessment by Z hjk = ((˘slhjk , L hjk ), (˘snh jk , N hjk )), which is a 2-tuple linguistic bipolar fuzzy number defined on the 2-tuple linguistic term set s˘ hjk ∈ S = (˘s0 , s˘1 , s˘2 , . . . , s˘σ ). Thus, for each decision maker an individual 2-tuple linguistic bipolar fuzzy assessment matrix can be derived, which can be denoted as F h = (Z hjk )m×n . Let w =

194

7 Enhanced Decision Making Method with Two-Tuple …

(w1 , w2 , . . . , wn )T be the weight vector of attributes, such that wk ∈ [0, 1],

n 

wk =

k=1

1. Let w  = (w1 , w2 , . . . , wl )T be the weight vector of decision makers, satisfying l  wh ∈ [0, 1], wh = 1. The essential steps for addressing the 2TLBF-MAGDM h=1

problem are described below: 1. Utilize the 2TLBFWA operator from Eq. (7.9) and the 2TLBFWG operator from Eq. (7.13) to aggregate all individual 2-tuple linguistic bipolar fuzzy decision matrices F h = (Z hjk )m×n (h = 1, 2, . . . , l) into a collective 2-tuple linguistic bipolar fuzzy decision matrix F = (Z jk )m×n . 2. Aggregate the 2-tuple linguistic bipolar fuzzy evaluation values of alternative  j on all attributes Q k (k = 1, 2, . . . , n) into the overall evaluation value of the alternative  j ( j = 1, 2, . . . , m) by using the G2TLBFWHM operator from Eq. (7.20) and the 2TLBFWGHM operator from Eq. (7.24) to derive the overall preference values of the alternatives  j ( j = 1, 2, . . . , m). 3. Determine the score S(Z j ) of overall assessment value Z j ( j = 1, 2, . . . , m) according to Definition 7.7. 4. By ranking the alternatives  j ( j = 1, 2, . . . , m) based on their score values, select the best alternative. The following flowchart (Fig. 7.1) is presented to better explain the steps of the developed MAGDM approach in this chapter.

7.5

Numerical Example

In this section, a practical example is first given to illustrate the proposed MAGDM method. A decision making process is provided to select the best alternative. Parametric analysis is performed to determine the effect of different s and t values on aggregation and ranking results. Finally, different methods are compared to demonstrate the effectiveness, and the advantages of the proposed method are described. Example 7.5 (Selection of photovoltaic cells for solar plant) The demand for electrical energy is rapidly increasing due to population growth and industrial development. Over the last decade, traditional energy sources such as crude oil, natural gas, and coal were considered to be the main sources of electricity generation. Electricity prices have risen rapidly over the past decade. With limited non-renewable resources and rapidly increasing pollution, the development of solar photovoltaic is expanding. Photovoltaic cells are widely used in different fields around the world as an economical and reliable energy source. Therefore, academics and researchers collaborate with governments and different companies to develop renewable energy standards and reduce CO2 emissions. Photovoltaic cells offer a cost-effective solution to this

7.5 Numerical Example

195

Fig. 7.1 Flowchart of developed MAGDM approach

problem because it is close to the demand area and does not require additional transmission routes. Photovoltaic cells are essential for investors to take appropriate risk appetite measures to ensure smooth project implementation and achieve expected benefits. Renewable energy generation has increased quickly, reaching 181 GW in 2018. Solar photovoltaic systems have the highest capacity at 55%, followed by wind at 28% and hydro at 11%. Implementing this technology provides a tremendous chance to improve system efficiency while lowering expenses. The board of directors of a corporation resolves to cut expenditures to enhance earnings. They remark that energy is a significant expenditure that may be minimized by using solar power. There are five alternatives to photovoltaic cells for their solar power plant: (1) (2) (3) (4)

1 : Mono-crystalline photovoltaic cell; 2 : Poly-crystalline photovoltaic cell; 3 : Thin-film photovoltaic cell; 4 : Amorphous silicon;

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7 Enhanced Decision Making Method with Two-Tuple …

(5) 5 : Copper indium diselenide. They select a photovoltaic cell based on the following attributes: (1) (2) (3) (4)

Q 1 : Heat absorption; Q 2 : Expenditure; Q 3 : Efficiency and reliability; Q 4 : Ability of charge separation.

Furthermore, four attributes have weight vector that is w = (0.23, 0.31, 0.27, n  0.19)T , and wk = 1. Whereas the experts believed that 2-tuple linguistic infork=1

mation is better choice for them. To select the optimal photovoltaic cell, three experts Dh (h = 1, 2, 3) are invited to give their assessments using linguistic term set S = {˘s0 = extremely poor, s˘1 = very poor, s˘2 = poor, s˘3 = fair, s˘4 = good, s˘5 = very good, s˘6 = extremely good}. The weight vector of these experts is w = (0.3, 0.5, 0.2). The assessment values provided by the three experts for each attribute of each alternative are represented in the decision matrix F h = (Z hjk )5×4 (h = 1, 2, 3), as in Tables 7.2, 7.3, and 7.4, respectively. In order to choose the most desirable photovoltaic cell, the G2TLBFWHM and 2TLBFWGHM operators are used to solve the MAGDM problem with 2-tuple linguistic bipolar fuzzy numbers, which involves the following computing steps: 1. Utilizing the 2TLBFWA from Eq. (7.9) and the 2TLBFWG from Eq. (7.13), we fuse all assessment values to get the overall 2-tuple linguistic bipolar fuzzy numbers Z j ( j = 1, 2, 3, 4, 5) of the alternatives. The fused result is shown in Table 7.5. Table 7.2 2-tuple linguistic bipolar fuzzy decision matrix F 1 provided by first expert D1 Q1 Q2 Q3 Q4 1 2 3 4 5

((˘s1 , 0), (˘s6 , 0)) ((˘s4 , 0), (˘s4 , 0)) ((˘s2 , 0), (˘s3 , 0)) ((˘s3 , 0), (˘s5 , 0)) ((˘s7 , 0), (˘s1 , 0))

((˘s3 , 0), (˘s4 , 0)) ((˘s2 , 0), (˘s6 , 0)) ((˘s4 , 0), (˘s3 , 0)) ((˘s2 , 0), (˘s4 , 0)) ((˘s5 , 0), (˘s1 , 0))

((˘s2 , 0), (˘s5 , 0)) ((˘s1 , 0), (˘s7 , 0)) ((˘s3 , 0), (˘s3 , 0)) ((˘s6 , 0), (˘s1 , 0)) ((˘s4 , 0), (˘s1 , 0))

((˘s1 , 0), (˘s5 , 0)) ((˘s4 , 0), (˘s2 , 0)) ((˘s4 , 0), (˘s1 , 0)) ((˘s4 , 0), (˘s3 , 0)) ((˘s5 , 0), (˘s3 , 0))

Table 7.3 2-tuple linguistic bipolar fuzzy decision matrix F 2 provided by second expert D2 Q1 Q2 Q3 Q4 1 2 3 4 5

((˘s2 , 0), (˘s5 , 0)) ((˘s4 , 0), (˘s1 , 0)) ((˘s3 , 0), (˘s2 , 0)) ((˘s2 , 0), (˘s6 , 0)) ((˘s5 , 0), (˘s1 , 0))

((˘s4 , 0), (˘s2 , 0)) ((˘s5 , 0), (˘s2 , 0)) ((˘s3 , 0), (˘s5 , 0)) ((˘s1 , 0), (˘s4 , 0)) ((˘s2 , 0), (˘s2 , 0))

((˘s3 , 0), (˘s5 , 0)) ((˘s6 , 0), (˘s1 , 0)) ((˘s2 , 0), (˘s3 , 0)) ((˘s3 , 0), (˘s5 , 0)) ((˘s3 , 0), (˘s1 , 0))

((˘s4 , 0), (˘s4 , 0)) ((˘s2 , 0), (˘s6 , 0)) ((˘s3 , 0), (˘s4 , 0)) ((˘s5 , 0), (˘s3 , 0)) ((˘s1 , 0), (˘s7 , 0))

7.5 Numerical Example

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Table 7.4 2-tuple linguistic bipolar fuzzy decision matrix F 3 provided by third expert D3 Q1 Q2 Q3 Q4 1 2 3 4 5

((˘s1 , 0), (˘s6 , 0)) ((˘s4 , 0), (˘s4 , 0)) ((˘s2 , 0), (˘s3 , 0)) ((˘s3 , 0), (˘s5 , 0)) ((˘s7 , 0), (˘s1 , 0))

((˘s3 , 0), (˘s4 , 0)) ((˘s2 , 0), (˘s6 , 0)) ((˘s4 , 0), (˘s3 , 0)) ((˘s2 , 0), (˘s4 , 0)) ((˘s5 , 0), (˘s1 , 0))

((˘s2 , 0), (˘s5 , 0)) ((˘s1 , 0), (˘s7 , 0)) ((˘s3 , 0), (˘s3 , 0)) ((˘s6 , 0), (˘s1 , 0)) ((˘s4 , 0), (˘s1 , 0))

((˘s1 , 0), (˘s5 , 0)) ((˘s4 , 0), (˘s2 , 0)) ((˘s4 , 0), (˘s1 , 0)) ((˘s4 , 0), (˘s3 , 0)) ((˘s5 , 0), (˘s3 , 0))

2. Using Eqs. (7.20) and (7.24) to aggregate the 2-tuple linguistic bipolar fuzzy assessment values Z j of alternative  j on all attributes Q k (k = 1, 2, 3, 4), into the overall assessment value Z j of the alternative  j ( j = 1, 2, 3, 4, 5) (take s = 2 and t = 3). The overall assessment values of alternatives  j ( j = 1, 2, 3, 4, 5) are shown in Table 7.6. 3. Compute the score function S(Z j ) of overall assessment value Z j ( j = 1, 2, 3, 4, 5) by employing the Eq. (7.7). The resulting values of score function are given as follows: S H M (1 ) = (˘s3 , 0.2880), S H M (2 ) = (˘s5 , −0.4990), S H M (3 ) = (˘s4 , −0.4510), S H M (4 ) = (˘s3 , 0.4814), S H M (5 ) = (˘s5 , −0.2937). S G H M (1 ) = (˘s4 , −0.4991), S G H M (2 ) = (˘s4 , 0.2817), S G H M (3 ) = (˘s4 , −0.1164), S G H M (4 ) = (˘s3 , 0.4260), S G H M (5 ) = (˘s4 , −0.2311).

4. The alternatives are ranked according to the score index. The ranking of all alternatives is as: 5 > 2 > 3 > 4 > 1 and 2 > 3 > 5 > 1 > 4 utilizing G2TLBFWHM and 2TLBFWGHM operators, respectively. Therefore, 5 or 2 is the best choice. Different attribute weight values have a significant effect on the ranking of alternatives, as demonstrated in Tables 7.7 and 7.8. The score values and ranking of alternatives for different values of weight attributes are given in Table 7.7 based on G2TLBFWHM operator, and graphical representation is shown in Fig. 7.2a. The results clearly show that the alternative 5 is ranked as the best alternative in all cases with a slightest difference in the ranking of all alternatives. As a consequence, in the decision making process the attributes weight can be varied to get appropriate decision results. Similarly the different outcomes by utilizing 2TLBFWGHM operator are shown in Table 7.8 (see Fig. 7.2b). Parameter Influence. Of course, the parameters s and t have a big impact on the rankings of alternatives. Evaluation of score values and ordering of alternatives on the basis of G2TLBFWHM and 2TLBFWGHM operators in this section. A number of values of s and t are fixed and evaluate the aggregated score values. Farther, score values are utilized to order the alternatives. In Tables 7.9 and 7.10, the fractional outputs are assessed by different values of s and t on the basis of G2TLBFWHM and 2TLBFGWHM operators, respectively. These score values are then utilized to

G2TLBFWA operator 1 ((˘s2 , 0.1415), (˘s5 , 0.1857)) 2 ((˘s4 , 0.0000), (˘s2 , −0.1654)) 3 ((˘s2 , 0.2635), (˘s3 , −0.3297)) 4 ((˘s2 , 0.2148), (˘s6 , −0.2148)) 5 ((˘s5 , 0.3747), (˘s1 , 0.0000)) 2TLBFWG operator 1 ((˘s2 , −0.0337), (˘s5 , 0.2337)) 2 ((˘s4 , 0.0000), (˘s2 , 0.3421)) 3 ((˘s2 , −0.0104), (˘s3 , −0.1226)) 4 ((˘s2 , 0.1689), (˘s6 , −0.1689)) 5 ((˘s5 , 0.0018), (˘s1 , 0.0000))

Q1 ((˘s3 , −0.4772), (˘s5 , −0.3238)) ((˘s4 , 0.4274), (˘s2 , 0.0519)) ((˘s2 , −0.0590), (˘s3 , 0.0000)) ((˘s4 , 0.4287), (˘s3 , 0.1090)) ((˘s4 , 0.3675), (˘s1 , 0.0000)) ((˘s2 , 0.4495), (˘s5 , −0.2704)) ((˘s3 , 0.0157), (˘s4 , −0.2879)) ((˘s2 , −0.2383), (˘s3 , 0.0000)) ((˘s4 , 0.0168), (˘s4 , −0.1426)) ((˘s4 , −0.0878), (˘s1 , 0.0000))

((˘s2 , 0.4915), (˘s3 , −0.2382)) ((˘s4 , 0.1628), (˘s3 , 0.1836)) ((˘s3 , 0.1777), (˘s4 , 0.1270)) ((˘s2 , −0.4029), (˘s4 , 0.0000)) ((˘s2 , −0.0488), (˘s4 , 0.3851))

Q3

((˘s3 , 0.0529), (˘s3 , −0.4054)) ((˘s5 , −0.4461), (˘s2 , 0.4915)) ((˘s3 , 0.2182), (˘s4 , −0.1270)) ((˘s2 , −0.1358), (˘s4 , 0.0000)) ((˘s3 , −0.4705), (˘s3 , −0.4646))

Q2

Table 7.5 Aggregated 2-tuple linguistic bipolar fuzzy decision matrix by operators

((˘s3 , −0.2192), (˘s4 , −0.2648)) ((˘s3 , −0.1716), (˘s5 , −0.4641)) ((˘s3 , 0.4641), (˘s3 , 0.2166)) ((˘s3 , −0.0495), (˘s4 , −0.2896)) ((˘s2 , −0.3014), (˘s6 , −0.3618))

((˘s3 , 0.2166), (˘s3 , 0.3973)) ((˘s3 , 0.1010), (˘s3 , 0.4641)) ((˘s4 , −0.4721), (˘s3 , −0.2192)) ((˘s4 , −0.0973), (˘s3 , 0.4968)) ((˘s2 , 0.3582), (˘s4 , 0.0577))

Q4

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7.5 Numerical Example

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Table 7.6 Fused assessment values by G2TLBFWHM and 2TLBFWGHM operators Alternatives Overall assessment values by Overall assessment values by G2TLBFWHM 2TLBFWGHM 1 2 3 4 5

((˘s3 , −0.4392), (˘s4 , −0.0151)) ((˘s4 , −0.1760), (˘s3 , −0.1781)) ((˘s3 , −0.3760), (˘s4 , −0.4739)) ((˘s3 , 0.1383), (˘s4 , 0.1756)) ((˘s4 , −0.3497), (˘s2 , 0.2376))

((˘s3 , −0.1204), (˘s4 , −0.1222)) ((˘s4 , −0.1615), (˘s3 , 0.2750)) ((˘s3 , −0.0786), (˘s3 , 0.1544)) ((˘s3 , −0.0911), (˘s4 , 0.0569)) ((˘s3 , 0.1549), (˘s4 , −0.3829))

Table 7.7 Influence of attribute weight w on alternative ranking utilizing G2TLBFWHM operator Weights

Scores

w1 = 0.1, w2 = 0.2, w3 = 0.3, w4 = 0.4

S1H M S3H M S5H M S1H M S3H M S5H M S1H M S3H M S5H M S1H M S3H M S5H M

w1 = 0.2, w2 = 0.1, w3 = 0.3, w4 = 0.4 w1 = 0.3, w2 = 0.1, w3 = 0.2, w4 = 0.4 w1 = 0.4, w2 = 0.1, w3 = 0.2, w4 = 0.3

= = =

Ranking (˘s3 , 0.3689), S2H M = (˘s4 , 0.2697), (˘s4 , −0.2814), S4H M = (˘s4 , −0.2213), (˘s4 , 0.3436) (˘s3 , 0.2348), S2H M = (˘s4 , 0.2770), (˘s4 , −0.2690), S4H M = (˘s4 , −0.2562),

5 > 2 > 4 > 3 > 1

= = = (˘s5 , −0.4185)

5 > 2 > 4 > 3 > 1

= (˘s3 , 0.2133), S2H M = (˘s4 , 0.2652), = (˘s4 , −0.2444), S4H M = (˘s4 , −0.4446), = (˘s5 , −0.3323)

5 > 2 > 3 > 4 > 1

= (˘s3 , 0.0671), S2H M = (˘s4 , 0.4098), = (˘s4 , −0.3131), S4H M = (˘s3 , 0.3958), = (˘s5 , −0.0206)

5 > 2 > 3 > 4 > 1

Table 7.8 Influence of attribute weight w on alternative ranking utilizing 2TLBFWGHM operator Weights

Scores

Ranking

w1 = 0.1, w2 = 0.2, S1G H M = (˘s4 , −0.3918), S2G H M = (˘s4 , −0.0843), w3 = 0.3, w4 = 0.4 S3G H M = (˘s4 , 0.0048), S4G H M = (˘s4 , −0.2370), S5G H M = (˘s3 , 0.2781)

3 > 2 > 4 > 1 > 5

w1 = 0.2, w2 = 0.1, S1G H M = (˘s3 , 0.4825), S2G H M = (˘s4 , −0.0710), w3 = 0.3, w4 = 0.4 S3G H M = (˘s4 , 0.0133), S4G H M = (˘s4 , −0.3175), S5G H M = (˘s3 , 0.4242)

3 > 2 > 4 > 1 > 5

w1 = 0.3, w2 = 0.1, S1G H M = (˘s3 , 0.4177), S2G H M = (˘s4 , 0.0094), w3 = 0.2, w4 = 0.4 S3G H M = (˘s4 , 0.0358), S4G H M = (˘s3 , 0.4578), S5G H M = (˘s3 , 0.4436)

3 > 2 > 4 > 5 > 1

w1 = 0.4, w2 = 0.1, S1G H M = (˘s3 , 0.2737), S2G H M = (˘s4 , 0.1855), w3 = 0.2, w4 = 0.3 S3G H M = (˘s4 , −0.0331), S4G H M = (˘s3 , 0.2684), S5G H M = (˘s4 , −0.1974)

2 > 3 > 5 > 1 > 4

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Fig. 7.2 Influence of attribute weight w on alternative ranking (s = 2, t = 3)

7.5 Numerical Example

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Table 7.9 Score values by varying s and t based on the G2T L B F W H M operator Parameter values Score functions s = t = 0.5 s=t =1 s=t =2 s = 3, t = 4

S1H M = (˘s2 , 0.3985), S2H M = (˘s4 , −0.3920), S3H M = (˘s3 , −0.3138), S4H M = (˘s2 , 0.4184), S5H M = (˘s4 , −0.3323) S1H M = (˘s3 , −0.1830), S2H M = (˘s4 , 0.0610), S3H M = (˘s3 , 0.1018), S4H M = (˘s3 , −0.1253), S5H M = (˘s4 , 0.1457) S1H M = (˘s3 , 0.1731), S2H M = (˘s4 , 0.4197), S3H M = (˘s3 , 0.4407), S4H M = (˘s3 , 0.3053), S5H M = (˘s5 , −0.4105) S1H M = (˘s3 , 0.4369), S2H M = (˘s5 , −0.3629), S3H M = (˘s4 , −0.3229), S4H M = (˘s4 , −0.3379), S5H M = (˘s5 , −0.0659)

s = 4, t = 3

S1H M = (˘s3 , 0.4359), S2H M = (˘s5 , −0.3434), S3H M = (˘s4 , −0.3369), S4H M = (˘s4 , −0.3654), S5H M = (˘s5 , −0.0288)

s = 5, t = 6

S1H M = (˘s4 , −0.3663), S2H M = (˘s5 , −0.2087), S3H M = (˘s4 , −0.1665), S4H M = (˘s4 , −0.1007), S5H M = (˘s5 , 0.2306)

s = 6, t = 5

S1H M = (˘s4 , −0.3639), S2H M = (˘s5 , −0.2005), S3H M = (˘s4 , −0.1730), S4H M = (˘s4 , −0.1050), S5H M = (˘s5 , 0.2448)

s = 7, t = 8

S1H M = (˘s4 , −0.2402), S2H M = (˘s5 , −0.1187), S3H M = (˘s4 , −0.0700), S4H M = (˘s4 , 0.0445), S5H M = (˘s5 , 0.4174)

s = 8, t = 7

S1H M = (˘s4 , −0.2385), S2H M = (˘s5 , −0.1144), S3H M = (˘s4 , −0.0732), S4H M = (˘s4 , 0.0446), S5H M = (˘s5 , 0.4241)

s = 1, t = 9

S1H M = (˘s4 , −0.3257), S2H M = (˘s5 , −0.2461), S3H M = (˘s4 , −0.1150), S4H M = (˘s4 , 0.0041), S5H M = (˘s5 , 0.1821)

s = 9, t = 1

S1H M = (˘s4 , −0.3293), S2H M = (˘s5 , −0.1503), S3H M = (˘s4 , −0.1849), S4H M = (˘s4 , −0.1453), S5H M = (˘s5 , 0.3953)

s = t = 10

S1H M = (˘s4 , −0.1355), S2H M = (˘s5 , −0.0439), S3H M = (˘s4 , 0.0097), S4H M = (˘s4 , 0.1619), S5H M = (˘s6 , −0.4265)

order the alternatives. The rankings given in Table 7.11 are employed to choose the most favorable alternatives, 5 and 2 are chosen as the most suitable alternatives with respect to G2TLBFWHM and 2TLBFWGHM operators, respectively. The score values of alternatives are changed when the values of s and t are varying at the same time and consequently an irregular change is occurred in the resulting values accordingly, as shown in Tables 7.9 and 7.10 by using G2TLBFWHM and 2TLBFWGHM operators. There is a significant impact of variation of the values of s and t on the ranking outputs of alternatives. Table 7.11 shows that the ranking outputs are relatively stable by changing the values of s and t, and also the most suitable alternative is same. The values of s and t are varying to acquire the bestfitted decision values and so the decision preference can be presented in the decision making process. On the basis of G2TLBFWHM operator, the score values change accordingly, corresponding to different values of s and t for alternatives  j ( j = 1, 2, 3, 4, 5) as presented in Fig. 7.3.

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Table 7.10 Score values by varying s and t based on the 2TLBFWGHM operator Parameter values Score functions s = t = 0.5

S1G H M = (˘s4 , 0.4425), S2G H M = (˘s5 , 0.2342), S3G H M = (˘s5 , −0.2274), S4G H M = (˘s4 , 0.3902), S5G H M = (˘s5 , 0.1811)

s=t =1

S1G H M = (˘s4 , −0.0219), S2G H M = (˘s5 , −0.2048), S3G H M = (˘s4 , 0.3335), S4G H M = (˘s4 , −0.0868), S5G H M = (˘s5 , −0.3859)

s=t =2

S1G H M = (˘s4 , −0.4094), S2G H M = (˘s4 , 0.4340), S3G H M = (˘s4 , −0.0223), S4G H M = (˘s4 , −0.4951), S5G H M = (˘s4 , 0.0061)

s = 3, t = 4

S1G H M = (˘s3 , 0.3385), S2G H M = (˘s4 , 0.1488), S3G H M = (˘s4 , −0.2615), S4G H M = (˘s3 , 0.2222), S5G H M = (˘s3 , 0.4955)

s = 4, t = 3

S1G H M = (˘s3 , 0.3027), S2G H M = (˘s4 , 0.2111), S3G H M = (˘s4 , −0.2736), S4G H M = (˘s3 , 0.1584), S5G H M = (˘s4 , −0.4688)

s = 5, t = 6

S1G H M = (˘s3 , 0.1412), S2G H M = (˘s4 , −0.0319), S3G H M = (˘s4 , −0.4497), S4G H M = (˘s3 , −0.0616), S5G H M = (˘s3 , 0.1477)

s = 6, t = 5

S1G H M = (˘s3 , 0.1198), S2G H M = (˘s4 , 0.0002), S3G H M = (˘s4 , −0.4552), S4G H M = (˘s3 , −0.0884), S5G H M = (˘s3 , 0.1650)

s = 7, t = 8

S1G H M = (˘s3 , 0.0215), S2G H M = (˘s4 , −0.1589), S3G H M = (˘s3 , 0.4237), S4G H M = (˘s3 , −0.2577), S5G H M = (˘s3 , −0.0660)

s = 8, t = 7

S1G H M = (˘s3 , 0.0074), S2G H M = (˘s4 , −0.1413), S3G H M = (˘s3 , 0.4217), S4G H M = (˘s3 , −0.2677), S5G H M = (˘s3 , −0.0561)

s = 1, t = 9

S1G H M = (˘s3 , 0.2199), S2G H M = (˘s4 , −0.1981), S3G H M = (˘s4 , −0.4622), S4G H M = (˘s3 , 0.0421), S5G H M = (˘s3 , 0.0067)

s = 9, t = 1

S1G H M = (˘s3 , 0.0159), S2G H M = (˘s4 , 0.1304), S3G H M = (˘s3 , 0.4816), S4G H M = (˘s3 , −0.2845), S5G H M = (˘s3 , 0.2408)

s = t = 10

S1G H M = (˘s3 , −0.0821), S2G H M = (˘s4 , −0.2708), S3G H M = (˘s3 , 0.3125), S4G H M = (˘s3 , −0.4277), S5G H M = (˘s3 , −0.2376)

Table 7.11 Ranking by varying s and t based on G2TLBFWHM and 2TLBFWGHM operators Parameters G2TLBFWHM operator 2TLBFWGHM operator s s s s s s s s s s s s

= t = 0.5 =t =1 =t =2 = 3, t = 4 = 4, t = 3 = 5, t = 6 = 6, t = 5 = 7, t = 8 = 8, t = 7 = 1, t = 9 = 9, t = 1 = t = 10

5 5 5 5 5 5 5 5 5 5 5 5

> 2 > 2 > 2 > 2 > 2 > 2 > 2 > 2 > 2 > 2 > 2 > 2

> 3 > 3 > 3 > 3 > 3 > 4 > 4 > 4 > 4 > 4 > 4 > 4

> 4 > 4 > 4 > 4 > 4 > 3 > 3 > 3 > 3 > 3 > 3 > 3

> 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1 > 1

2 2 2 2 2 2 2 2 2 2 2 2

> 5 > 5 > 5 > 3 > 3 > 3 > 3 > 3 > 3 > 3 > 3 > 3

> 3 > 3 > 3 > 5 > 5 > 5 > 5 > 1 > 1 > 1 > 5 > 1

> 1 > 1 > 1 > 1 > 1 > 1 > 1 > 5 > 5 > 4 > 1 > 5

> 4 > 4 > 4 > 4 > 4 > 4 > 4 > 4 > 4 > 5 > 4 > 4

7.5 Numerical Example

203

Fig. 7.3 Alternative’s score  j ( j = 1, 2, 3, 4, 5) with G2T L B F W H M operator when s, t ∈ [0, 10]

In Fig. 7.4a the value of parameter s = 3 is fixed, and t is varied from 1 to 10. The value of parameter t = 3 is fixed, and s is varied from 1 to 10 in Fig. 7.4b. Figure 7.4 illustrates the change in score value of each alternative utilizing G2TLBFWHM operator. Based on the 2TLBFWGHM operator, the scores vary accordingly, corresponding to different values of s and t for alternatives  j ( j = 1, 2, 3, 4, 5) as shown in Fig. 7.5. In Fig. 7.6a the value of parameter s = 3 is fixed, and t is varied from 1 to 10. In Fig. 7.6b the value of parameter t = 3 is fixed, and s is varied from 1 to 10. Figure 7.6 illustrates the change in score value of each alternative based on 2TLBFWGHM operator. When the parameters s and t are assigned the same values, the ranking results of the G2TLBFWHM and 2TLBFWGHM operators are not the same, and the score values vary differently. G2TLBFWHM and 2TLBFWGHM operators consider the

204

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Fig. 7.4 Influence of parameters on alternatives ranking

7.5 Numerical Example

205

Fig. 7.5 Alternative’s score  j ( j = 1, 2, 3, 4, 5) with 2T L B F W G H M operator when s, t ∈ [0, 10]

206

7 Enhanced Decision Making Method with Two-Tuple …

Fig. 7.6 Influence of parameters on alternative ranking

7.6 Comparative Analysis

207

relationship between two input arguments. These results indicate that the proposed methods are flexible. In actual decision making, the parameters could be varied based on decision maker’s preferences.

7.6 Comparative Analysis The capacity to evaluate the interrelationships between bipolar fuzzy numbers in 2-tuple languages is a unique feature of the G2TLBFWHM and 2TLBFWGHM operators. To illustrate the efficiency of the proposed operator, a comparative analysis is provided. To verify the effectiveness of the developed method, different methods are used to solve the MAGDM problem mentioned in Sect. 7.5. These methods include 2TLBFWA operator, 2TLBFWG operator, 2-tuple language bipolar fuzzy weighted Hamy mean [18] (2TLBFWHM) operator, 2-tuple language bipolar fuzzy weighted dual Hamy mean [31] (2TLBFWDHM ) operator, 2-tuple language bipolar fuzzy weighted Maclaurin symmetric mean [25] (2TLBFWMSM) operator, and 2-tuple language bipolar fuzzy weighted double Maclaurin symmetric mean [28] (2TLBFWDMSM) operator. Tables 7.12, 7.13, 7.14, 7.15, 7.16, and 7.17 give detailed evaluation results obtained using different MAGDM methods. The presented approach is compared to other techniques such as the 2TLBFWA and 2TLBFWG operators. Tables 7.12, 7.13 represent the 2TLBFWA and 2TLBFWG operators that are unable to provide the interrelationship between the 2-tuple linguistic bipolar fuzzy numbers. In the above example, it should not be considered only the attribute values of each photovoltaic cells, but also the correlations between these attributes while selecting the most optimal photovoltaic cell. Techniques based on 2TLBFWA and 2TLBFWG operators are ineffective in dealing with this issue. Our approach is more appropriate for dealing with this issue than, 2TLBFWA and 2TLBFWG operators as it can capture parameter correlations. The interrelationship between the 2-tuple linguistic bipolar fuzzy numbers is evaluated by the G2TLBFHM, G2TLBFWHM, 2TLBFGHM, and 2TLBFWGHM operators. In addition, 2TLBFWA and 2TLBFWG operators do not have any parameters, but presented operator has two parameters, that make the presented operator more flexible and adaptable. Here, an approach for MAGDM problems is presented based on 2tuple linguistic bipolar fuzzy set, which is an influential approach for demonstrating and indicating decision maker’s assessments. As a result, compared with different approaches, presented approach has some benefits and superiorities. The ranking effects of the above strategies are slightly different, as shown in the above calculations, but still, the best alternative is 2 or 5 . It shows that the G2TLBFWHM and 2TLBFWGHM operators are more effective and appropriate with 2-tuple linguistic bipolar fuzzy numbers for MAGDM problems.

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Table 7.12 The outcomes utilizing 2TLBFWA operator 2TLBFWA 1 2 3 4 5 The order

((˘s3 , −0.2530), (˘s4 , −0.2456)) ((˘s4 , 0.1497), (˘s2 , 0.3458)) ((˘s3 , −0.2482), (˘s3 , 0.1161)) ((˘s3 , 0.1559), (˘s4 , −0.0348)) ((˘s4 , −0.1614), (˘s2 , −0.2589)) 5 > 2 > 3 > 4 > 1

Table 7.13 The outcomes utilizing 2TLBFWG operator 2TLBFWG 1 2 3 4 5 The order

((˘s2 , 0.3982), (˘s4 , 0.1699)) ((˘s4 , −0.4867), (˘s3 , 0.4502)) ((˘s2 , 0.4733), (˘s3 , 0.3939)) ((˘s2 , 0.4698), (˘s4 , 0.4454)) ((˘s3 , −0.1526), (˘s3 , 0.3604)) 2 > 5 > 3 > 1 > 2

Table 7.14 The outcomes utilizing 2TLBFWHM operator 2TLBFWHM 1 2 3 4 5 The order

((˘s6 , 0.1111), (˘s1 0.2620)) ((˘s7 , −0.2399), (˘s1 − 0.3123)) ((˘s6 , 0.0880), (˘s1 − 0.0719)) ((˘s6 , 0.2394), (˘s1 0.3353)) ((˘s7 , −0.4646), (˘s1 − 0.4275)) 2 > 5 > 3 > 4 > 1

Table 7.15 The outcomes utilizing 2TLBFWDHM operator 2TLBFWDHM 1 2 3 4 5 The order

((˘s1 , −0.3159), (˘s7 , −0.2518)) ((˘s1 , 0.0870), (˘s6 , 0.4438)) ((˘s1 , −0.2613), (˘s6 , 0.4293)) ((˘s1 , −0.2369), (˘s7 , −0.1422)) ((˘s1 , −0.0449), (˘s6 , −0.0374)) 5 > 2 > 3 > 1 > 4

Scores values (˘s3 , 0.4963) (˘s5 , −0.0981) (˘s4 , −0.1822) (˘s4 , −0.4046) (˘s5 , 0.0488)

Scores values (˘s3 , 0.1142) (˘s4 , 0.0316) (˘s4 , −0.4603) (˘s3 , 0.0122) (˘s4 , −0.2565)

Scores values (˘s6 , 0.4246) (˘s7 , 0.0362) (˘s7 , −0.4201) (˘s6 , 0.4520) (˘s7 , −0.0185)

Scores values (˘s1 , −0.0321) (˘s1 , 0.3216) (˘s1 , 0.1547) (˘s1 , −0.0473) (˘s1 , 0.4962)

7.7 Advantages of the Proposed Strategies Table 7.16 The outcomes utilizing 2TLBFWMSssM operator 2TLBFWMSM 1 2 3 4 5 The order

((˘s6 , 0.1130), (˘s1 , 0.2595)) ((˘s7 , −0.2398), (˘s1 , −0.3126)) ((˘s6 , 0.0917), (˘s1 , −0.0734)) ((˘s6 , 0.2487), (˘s1 , 0.3320)) ((˘s7 , −0.4588), (˘s1 , −0.4291)) 2 > 5 > 3 > 4 > 1

Table 7.17 The outcomes utilizing 2TLBFWDMSM operator 2TLBFWDMSM 1 2 3 4 5 The order

((˘s1 , −0.3161), (˘s7 , −0.2470)) ((˘s1 , 0.0852), (˘s6 , 0.4470)) ((˘s1 , −0.2622), (˘s6 , 0.4300)) ((˘s1 , −0.2383), (˘s7 , −0.1403)) ((˘s1 , −0.0483), (˘s6 , −0.0121)) 5 > 2 > 3 > 1 > 4

209

Scores values (˘s6 , 0.4267) (˘s7 , 0.0364) (˘s7 , −0.4175) (˘s6 , 0.4584) (˘s7 , −0.0149)

Scores values (˘s1 , −0.0346) (˘s1 , 0.3191) (˘s1 , 0.1539) (˘s1 , −0.0490) (˘s1 , 0.4819)

7.7 Advantages of the Proposed Strategies The aggregation operators are frequently adopted in the decision making scenarios owing to their multi-functionality, exceptional aggregation performance and precise outputs. The notable merits of the proposed approaches are spelled out in the following points: • The 2-tuple linguistic bipolar fuzzy set defines a more practical and competent model to capture the real-life situations quantitatively and qualitatively with minimal data loss. Thus, this preeminent model is fully equipped to represent the ambiguous information more compactly. • The G2TLBFWHM (2TLBFWGHM) operator stands out within the existing literature because of marvelous theoretical base of Heronian mean operators. As the Heronian mean operators are known to interpret the relationships among the influencing attributes. Further, the flexible parameters s and t of the proposed operator enable it to operate efficiently across different preference values. In short, the theoretical background of Heronian mean operators and adjustable parameters expand the application area of the G2TLBFWHM (2TLBFWGHM) operator in aggregation process. • Another notable merit of the proposed approach is the excellent dealing with the interdependence of the multi-input argument. Furthermore, these operators exhibit the general structure due to the parameters s and t. That’s why, many operators

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appear as their special cases corresponding to the different values of the parameters. In a nutshell, the proposed operators present more practical and general approach to be implemented for the aggregation purpose for a variety of real-life scenarios.

7.8 Conclusions In this chapter, the MAGDM problem is investigated by a combination of the concepts of 2-tuple linguistic terms and Heronian mean operators with bipolar fuzzy numbers. We have presented the G2TLBFHM and 2TLBFGHM aggregation operators and their weighted variants, drawing inspiration from the generalized Heronian mean operator and the geometric Heronian mean operator. Their ability to include interdependencies among the arguments is their strongest suit. A numerical example has been given to assess the strategy we have proposed. The effect of the factors s and t on decision making findings and a comparison analysis have been investigated. Regarding restrictions, firstly we want to make clear that our work exclusively aggregates 2-tuple linguistic bipolar fuzzy numbers. Secondly, this presented work only applies to 2-tuple linguistic bipolar fuzzy sets used to evaluate photovoltaic cells. There are more implications to be given for evaluation techniques in allied domains, including management applications, engineering, natural and artificial cognitive systems, and cognitive computation.

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Index

Symbols (α, β)-cut, 44 2-tuple linguistic bipolar fuzzy set, 180 2-tuple linguistic information, 179 2TLBFGHM, 190 2TLBFWA operator, 182 2TLBFWG operator, 183 2TLBFWGHM operator, 192

A Accuracy function, 96 Average ranking, 130

B Bipolar concordance set, 16 Bipolar discordance set, 16 Bipolar fuzzy aggregated dominance matrix, 18 Bipolar fuzzy concordance dominance matrix, 17 Bipolar fuzzy concordance indices, 17 Bipolar fuzzy concordance level, 17 Bipolar fuzzy concordance matrix, 17, 127 Bipolar fuzzy concordance set, 126 Bipolar fuzzy decision matrix, 8 Bipolar fuzzy discordance dominance matrix, 18 Bipolar fuzzy discordance indices, 17 Bipolar fuzzy discordance level, 17 Bipolar fuzzy discordance matrix, 17 Bipolar fuzzy discordance set, 127

Bipolar fuzzy ELECTRE I method, 16 Bipolar fuzzy ELECTRE II, 145 Bipolar fuzzy ELECTRE II method, 125, 130 Bipolar fuzzy linguistic variable, 41 Bipolar fuzzy negative ideal solution, 8 Bipolar fuzzy numbers, 37, 70, 96, 153 Bipolar fuzzy positive ideal solution, 8 Bipolar fuzzy PROMETHEE method, 153 Bipolar fuzzy set, 3, 69, 95, 179 Bipolar fuzzy TOPSIS method, 7 Bipolar fuzzy variable, 39 Bipolar fuzzy VIKOR, 88

C Cartesian product, 48 Complex bipolar fuzzy numbers, 97 Complex bipolar fuzzy set, 97 Complex bipolar fuzzy weighted averaging operator, 97 Countable, 48

D Decision matrix, 72, 125, 156 Dense set, 47 Discordance matrix , 128 Dissatisfaction degree, 37 Divergence value, 73 Double upper lower dense sequence, 48

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Akram et al., Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-99-0569-0

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214 E Entire preeminence value, 59 Entropy values, 73 Euclidean distance, 9 F Forward ranking, 129 Fuzzy ELECTRE II, 145 Fuzzy number, 35 Fuzzy VIKOR, 88 G G2TLBFWHM operator, 188 Generalized Heronian mean operator, 180 Geometric Heronian mean operator, 180 Green supplier selection, 161 Group decision making, 4 H Heronian mean operator, 180 I Information system, 58 Intuitionistic fuzzy set, 4 L Level criterion preference function, 155 Linear criterion preference function, 155 Linguistic value, 41 Linguistic variable, 39, 156 Lower dense, 47 M Multi-person decision making, 4 N Negative ideal solution, 7 Normalized Euclidean distance measure, 96, 97 O Ordered relation, 96 Outrank graph, 129 P Positive ideal solution, 7 Preference function, 154

Index PROMETHEE II, 161 Q Quintuple, 41 R Rank, 157 Ranking function, 42, 71, 154 Rational numbers, 48 Relative closeness degree, 9 Reverse ranking, 129 S Satisfaction degree, 37 Score and accuracy functions, 181 Score function, 96 Semantic rule, 41 Special bipolar fuzzy set, 46 Strong (α, β)-cut, 45 Strong outranking relations, 128 T TOPSIS, 27 Trapezoidal bipolar fuzzy information system, 58 Trapezoidal bipolar fuzzy number, 37, 70, 154 Trapezoidal bipolar fuzzy VIKOR method, 71 Triangular bipolar fuzzy number, 37 U Upper dense, 47 Upper dense set, 47 Usual criterion preference function, 154 V VIKOR method, 67, 68 W Weak outranking relations, 128 Weighted bipolar fuzzy preeminence relation, 59 Weighted trapezoidal bipolar fuzzy information system, 59 Y YinYang bipolar fuzzy sets, 3