Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets [1 ed.] 9811515204, 9789811515200

Table of contents :
Preface
Contents
1 A Possibility Degree Method for Interval-Valued Intuitionistic Fuzzy Multi-attribute Group Decision Making
Abstract
1.1 Introduction
1.2 A New Ranking Method of IVIFS from the Probability Viewpoint
1.2 A New Ranking Method of IVIFS from the Probability Viewpoint
1.2.1 New Ranking Method for Intervals from the Probability Viewpoint
1.2.1 New Ranking Method for Intervals from the Probability Viewpoint
1.2.1 New Ranking Method for Intervals from the Probability Viewpoint
1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree
1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree
1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree
1.2.3 Comparative Analysis with Score and Accuracy Functions for IVIFNs
1.3 Ordered Weighted Average Operator and Hybrid Weighted Average Operator for IVIFNs
1.3.1 Weighted Average Operator for IVIFNs
1.3.1 Weighted Average Operator for IVIFNs
1.3.2 Proposed Ordered Weighted Average Operator for IVIFNs
1.3.2 Proposed Ordered Weighted Average Operator for IVIFNs
1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs
1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs
1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs
1.4 MAGDM Problem and Method with IVIFS
1.4.1 Problem Description for MAGDM with IVIFS
1.4.2 Determination of the Weights of DMs
1.4.2 Determination of the Weights of DMs
1.4.3 Group Decision Making Method
1.5 An Air-Condition System Selection Example and Comparison Analysis
1.5.1 An Air-Condition System Selection Problem and the Analysis Process
1.5.2 Comparison Analysis of the Obtained Results
1.6 Conclusions
References
2 A New Method for Atanassov’s Interval-Valued Intuitionistic Fuzzy MAGDM with Incomplete Attribute Weight Information
Abstract
2.1 Introduction
2.2 Preliminaries
2.2.1 Interval Objective Programming
2.2.2 Orderings of Intervals
2.2.3 Atanassov’s Intuitionistic Fuzzy Set and Atanassov’s Interval-Valued Intuitionistic Fuzzy Set
2.3 A Novel Method for MAGDM with AIVIFVs and Incomplete Attribute Weight Information
2.3.1 Presentation of the Problems
2.3.2 Determine the DMs’ Weights with Respect to Different Attributes
2.3.2.1 Calculate the Similarity Degree Based on an Extended TOPSIS
2.3.2.2 Calculate Proximity Degree Using the Distance Measure
2.3.2.3 Obtain the Weights of DMs with Respect to Different Attributes
2.3.3 Converting Individual Decision Matrices into a Collective Interval Matrix
2.3.4 Construct Multi-objective Interval-Programming for Deriving Attribute Weights
2.3.5 Decision Process and Algorithm for MAGDM with AIVIFVs
2.4 A Real-World R & D Project Selection Example and Comparison Analyses
2.4.1 A Real-World R & D Project Selection Problem and the Solution Process
2.4.2 Comparison with the Extended TOPSIS Method
2.4.3 Comparison with Barrenechea et al.’s Method
2.5 Conclusions
References
3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming Method for Hybrid Multi-criteria Group Decision Making with Interval-Valued Intuitionistic Fuzzy Truth Degrees
Abstract
3.1 Introduction
3.2 Basic Concepts
3.2.1 Concepts of Interval-Valued Intuitionistic Fuzzy Sets and Distances
3.2.2 Definition and Distance for Trapezoidal Fuzzy Numbers and Triangular Fuzzy Numbers
3.2.3 Linguistic Variables
3.3 Hybrid MCGDM Problems Considering Alternative Comparisons with IVIF Truth Degrees
3.3.1 Hybrid MCGDM Problems with IVIF Truth Degrees and Incomplete Weight Information
3.3.2 Normalization Method
3.4 IVIF Mathematical Programming Method for Hybrid MCGDM
3.4.1 IVIFS-Type Consistency and Inconsistency Measurements
3.4.2 Bi-Objective IVIF Mathematical Programming Model
3.4.3 Goal Programming Approach to Solving Bi-Objective IVIF Mathematical Programming Model
3.4.4 GDM Process and Steps for Solving Hybrid MCGDM
3.5 Empirical Example of Critical Infrastructure
3.5.1 A Critical Infrastructure Evaluation Example and the Analysis Process
3.5.2 Comparison Analysis with the Existing LINMAP Methods
3.6 Conclusions
Appendix 1
Appendix 2
References
4 A Selection Method Based on MAGDM with Interval-Valued Intuitionistic Fuzzy Sets
Abstract
4.1 Introduction
4.2 Preliminaries
4.2.1 Interval-Valued Intuitionistic Fuzzy Set
4.2.2 Gray Relation Analysis
4.3 A Novel Method for MAGDM with IVIFSs and Incomplete Attribute Weight Information
4.3.1 Determine the Weights of Experts by the Extended GRA Method
4.3.2 Integrate Individual Decision Matrices into a Collective Matrix
4.3.3 Identify the Attribute Weights by a New Multi-objective Linear Programming Model
4.3.4 Decision Process and Algorithm for MAGDM Problems with IVIFSs
4.3.5 Decision Support System Framework Based on MAGDM with IVIFSs
4.4 A Cloud Service Selection Problem and Comparison Analysis
4.4.1 A Cloud Service Provider Selection Problem and the Solution Process
4.4.2 Sensitivity Analysis for Parameter
4.4.3 Comparison Analysis with the Method Using the Score Function
4.5 Conclusions
References
5 Aggregating Decision Information into Interval-Valued Intuitionistic Fuzzy Numbers for Heterogeneous Multi-attribute Group Decision Making
Abstract
5.1 Introduction
5.2 Some Basic Concepts
5.2.1 Interval-Valued Intuitionistic Fuzzy Set
5.2.2 Definitions and Distances for Real Number, Interval Number, TFN and TrFN
5.3 Aggregating Heterogeneous Decision Information into IVIFNs
5.3.1 Presentation of Heterogeneous MAGDM Problem
5.3.2 A General Method for Aggregating Heterogeneous Decision Information into IVIFNs
5.3.2.1 Compute the Qsd, Qdd and Qud
5.3.2.2 Calculate Qsi, Qdi and Qui
5.3.2.3 Induce an IVIFN
5.3.3 Concrete Computation Formulas for Aggregating Heterogeneous Information into IVIFNs
5.3.3.1 For Trapezoidal Fuzzy Numbers
5.3.3.2 For Triangular Fuzzy Numbers
5.3.3.3 For Interval Numbers
5.3.3.4 For Real Numbers
5.4 A Novel Approach for Heterogeneous MAGDM Problems
5.4.1 Construct an Intuitionistic Fuzzy Programming Model to Determine the Attribute Weights
5.4.2 Algorithm for Solving Heterogeneous MAGDM
5.5 Comparison Analysis with Existing Methods
5.6 Illustrative Examples
5.6.1 An IT Outsourcing Service Provider Evaluation Example
5.6.1.1 Decision Process Using the Proposed Method
5.6.1.2 Comparison with Extended TOPSIS Method
5.6.2 A Supplier Selection Example
5.7 Conclusions
References
6 A Novel Method for Group Decision Making with Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relations
Abstract
6.1 Introduction
6.2 Multiplicative Consistency of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations
6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs
6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs
6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs
6.3 Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs
6.3.2 Repair Multiplicative Consistency of IV-AIFPRs
6.3.2 Repair Multiplicative Consistency of IV-AIFPRs
6.4 A Novel Method for Group Decision Making with IV-AIFPRs
6.4.1 Determine DMs’ Weights Objectively and Integrate Individual IV-AIFPRs
6.4.1 Determine DMs’ Weights Objectively and Integrate Individual IV-AIFPRs
6.4.2 Derive IVAIF Priority Weights and Rank Alternatives
6.4.2.1 Derive IVAIF Priority Weights of Alternatives
6.4.2.2 A TOPSIS Based Approach to Ranking IVAIF Priority Weights
6.4.2.2 A TOPSIS Based Approach to Ranking IVAIF Priority Weights
6.4.3 A Novel Method for Solving GDM Problems with IV-AIFPRs
6.5 A Practical Example of a Virtual Enterprise Partner Selection and Comparative Analyses
6.5.1 A Practical Example of a Virtual Enterprise Partner Selection
6.5.2 Comparative Analyses
6.5.2.1 Comparison with Liao’s Method
6.5.2.2 Comparison with Other Existing Group Decision Making Methods
6.6 Conclusions
Appendix 1
References
7 Additive Consistent Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relation and Likelihood Comparison Algorithm Based Group Decision Making
Abstract
7.1 Introduction
7.2 Preliminaries
7.2 Preliminaries
7.2 Preliminaries
7.2 Preliminaries
7.2 Preliminaries
7.2 Preliminaries
7.3 A New Likelihood Comparison Algorithm of IVAIFVs
7.3 A New Likelihood Comparison Algorithm of IVAIFVs
7.4 Additive Consistency Analyses for IVAIFPR
7.4.1 Additive Consistency Definition of IVAIFPR
7.4.1 Additive Consistency Definition of IVAIFPR
7.4.1 Additive Consistency Definition of IVAIFPR
7.4.2 Derive the IVAIF Priority Weights from IVAIFPR
7.5 Method for Solving the Group Decision Making Problems with IVAIFPRs
7.5.1 Description for GDM Problems with IVAIFPRs
7.5.2 Determination of DMs’ Weights
7.5.2 Determination of DMs’ Weights
7.5.3 Method for Group Decision Making with IVAIFPRs
7.5.3 Method for Group Decision Making with IVAIFPRs
7.5.3 Method for Group Decision Making with IVAIFPRs
7.6 An Example of ERP System Selection and Comparative Analysis
7.6.1 A Practical Example of ERP System Selection
7.6.2 Comparative Analysis with Existing GDM Methods
7.7 Conclusions
Appendix 1
References
8 A Three-Phase Method for Group Decision Making with Interval-Valued Intuitionistic Fuzzy Preference Relations
Abstract
8.1 Introduction
8.2 Preliminaries
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.1 Some Related Concepts on IFPR
8.2.2 Additive Consistency of IVIFPR
8.2.2 Additive Consistency of IVIFPR
8.2.2 Additive Consistency of IVIFPR
8.2.2 Additive Consistency of IVIFPR
8.2.2 Additive Consistency of IVIFPR
8.2.2 Additive Consistency of IVIFPR
8.3 Determination of the Intuitionistic Fuzzy Priority Weights from an IVIFPR
8.3.1 Extracting a Risk Attitudinal-Based Consistent IFPR from an IVIFPR
8.3.1 Extracting a Risk Attitudinal-Based Consistent IFPR from an IVIFPR
8.3.2 Deriving the Intuitionistic Fuzzy Priority Weights from the Extractive IFPR
8.3.2 Deriving the Intuitionistic Fuzzy Priority Weights from the Extractive IFPR
8.4 A Novel Three-Phase Method for Solving GDM with IVIFPRs
8.4.1 Presentation of Problem for GDM with IVIFPRs
8.4.2 Integrating Individual IVIFPRs to a Collective One
8.4.2 Integrating Individual IVIFPRs to a Collective One
8.4.2 Integrating Individual IVIFPRs to a Collective One
8.4.3 A Three-Phase Method for GDM with IVIFPRs
8.5 An Example of Network System Selection and Comparison Analyses
8.5.1 A Network System Selection Example and the Analysis Process
8.5.2 Comparison Analysis with Xu’s Method
8.5.3 Comparison Analysis with Wang’s Method
8.6 Conclusions
References
9 A Group Decision-Making Method Considering Both the Group Consensus and Multiplicative Consistency of Interval-Valued Intuitionistic Fuzzy Preference Relations
Abstract
9.1 Introduction
9.2 Preliminaries
9.2 Preliminaries
9.2 Preliminaries
9.2 Preliminaries
9.3 A New Ranking Method of IVIFVs
9.3.1 A New Ranking Method of IVIFVs
9.3.1 A New Ranking Method of IVIFVs
9.3.2 Comparison with Existing Ranking Methods of IVIFVs
9.3.2 Comparison with Existing Ranking Methods of IVIFVs
9.3.2 Comparison with Existing Ranking Methods of IVIFVs
9.3.2 Comparison with Existing Ranking Methods of IVIFVs
9.4 Analyses of Group Consensus in GDM
9.4.1 Problem Description for GDM with IVIFPRs
9.4.2 Group Consensus Analysis of GDM
9.4.2 Group Consensus Analysis of GDM
9.4.2 Group Consensus Analysis of GDM
9.4.3 An Iteration Algorithm in Group Consensus Improving Process
9.4.3 An Iteration Algorithm in Group Consensus Improving Process
9.4.3 An Iteration Algorithm in Group Consensus Improving Process
9.4.3 An Iteration Algorithm in Group Consensus Improving Process
9.4.3 An Iteration Algorithm in Group Consensus Improving Process
9.4.4 Statistical Comparative Study on Group Consensus
9.5 Multiplicative Consistency of IVIFPR
9.5.1 Multiplicative Consistency Concept of IVIFPR
9.5.1 Multiplicative Consistency Concept of IVIFPR
9.5.1 Multiplicative Consistency Concept of IVIFPR
9.5.2 Determine the Priority Weights from an IVIFPR
9.6 Method for GDM with IVIFPRs
9.6.1 Determination of Expert Weights Based on the Markov Model
9.6.1 Determination of Expert Weights Based on the Markov Model
9.6.2 Method for GDM with IVIFPRs
9.7 Two Case Studies
9.7.1 Case 1: An Example of a Virtual Enterprise Partner Selection
9.7.1.1 A Practical Example of a Virtual Enterprise Partner Selection
9.7.1.2 Comparative Analyses and Spearman’s Rank-Correlation Test
9.7.1.3 Fitted Error Analysis of the Obtained Results
9.7.2 Case 2: A Practical Example of an Enterprise Resource Planningsystem Selection
9.7.2.1 A Practical Example of an Enterprise Resource Planning System Selection
9.7.2.2 Comparative Analysis with the Method of Liao et al
9.8 Conclusions
Appendix 1
Appendix 2
References

Citation preview

Shuping Wan Jiuying Dong

Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets

Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets

Shuping Wan Jiuying Dong •

Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets

123

Shuping Wan Jiangxi University of Finance and Economics Nanchang, Jiangxi, China

Jiuying Dong Jiangxi University of Finance and Economics Nanchang, Jiangxi, China

ISBN 978-981-15-1520-0 ISBN 978-981-15-1521-7 https://doi.org/10.1007/978-981-15-1521-7

(eBook)

© Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved 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

Preface

Fuzzy set (FS) theory has long been introduced to handle inexact and imprecise data by Zadeh (1965). The drawback of using the single membership value in FS theory is that it cannot be used to express the evidences of support and objection simultaneously in many practical situations. In order to tackle this problem, Atanassov (1986) proposed the intuitionistic fuzzy set (IFS) using two characteristic functions expressing the degree of membership and the degree of non-membership of elements of the universal set to the IFS. It can cope with the presence of vagueness and hesitancy originating from imprecise knowledge or information. Atanassov and Gargov (1989) further generalized the IFS in the spirit of the ordinary interval-valued fuzzy set (IVFS) and defined the concept of an interval-valued intuitionistic fuzzy set (IVIFS), which enhances greatly the representation ability of uncertainty than IFS. Over the last few decades, the theories of IFSs and IVIFSs have received extensive attention from researchers and practitioners, and have been widely applied to various fields. The theories of IFSs and IVIFSs are undergoing continuous in-depth study as well as continuous expansion of the scope of their applications. As such, it has been found that intuitionistic fuzzy decision making, interval-valued intuitionistic fuzzy decision making, intuitionistic fuzzy preference relations, and interval-valued intuitionistic fuzzy preference relations become significantly important. Both IFSs and IVIFSs have broad prospects of application in the fields of multi-attribute decision making (MADM) and multi-attribute group decision making (MAGDM), but pose many interesting yet challenging topics for research. In this book, we will give a thorough and systematic introduction to the latest research achievements on the theories of IVIFSs and their applications to MADM and MAGDM. The main theoretical results obtained in this book include the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs), the ordered weighted average operator and hybrid weighted average operator for IVIFNs based on the Karnik–Mendel algorithms, the asymptotic property of the Atanassov’s interval-valued intuitionistic fuzzy (AIVIF) matrix, the weight of each decision maker (DM) with respect to every attribute, the v

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attribute weights, a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid multi-criteria group decision making (MCGDM) considering alternative comparisons with hesitancy degrees, aggregating decision information into interval-valued intuitionistic fuzzy numbers (IVIFNs), a new consistency index of an Atanassov intuitionistic fuzzy preference relation (AIFPR), a new iterative algorithm to repair the consistency of an interval-valued Atanassov intuitionistic fuzzy preference relation (IVAIFPR) with unacceptable consistency, a linear program to derive interval-valued Atanassov intuitionistic fuzzy (IVAIF) priority weights of alternatives, a TOPSIS (technique for order performance by similarity to an ideal solution) based approach to rank such IVAIF priority weights, the likelihood of IVAIF values (IVAIFVs), the additive consistency of an IVAIFPR, the IVAIF priority weights of an IVAIFPR, the mean and variance of IVIF values (IVIFVs), a ranking method for IVIFVs considering the risk attitude of the expert, the group consensus, an iteration algorithm to improve the group consensus, a new multiplicative consistency of IVIFPR, and a new method for the group decision making (GDM) with IVIFPRs. At the same time, some real decision making problems are analyzed, such as supply chain collaboration, cooperative alliance and stock selection, cloud computing service evaluation, etc. This book is organized into nine chapters that deal with nine different but related issues and are listed below: Chapter 1 defines the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik–Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. The individual overall attribute values of alternatives are obtained by using the weighted average operator of IVIFNs. By using the hybrid weighted average operator for IVIFNs, we can obtain the collective overall attribute values of alternatives, which are used to rank the alternatives. A numerical example is examined to illustrate the effectiveness and flexibility of the proposed method in this chapter. Chapter 2 develops a new method for solving multiple attribute group decision-making (MAGDM) problems with Atanassov’s interval-valued intuitionistic fuzzy values (AIVIFVs) and incomplete attribute weight information. Firstly, we investigate the asymptotic property of the Atanassov’s interval-valued intuitionistic fuzzy (AIVIF) matrix. It is demonstrated that after applying weights an infinite number of times, all elements in an AIVIF matrix will approach the same AIVIFV without regard to the initial values of elements. Then, the weight of each decision maker (DM) with respect to every attribute is determined by considering the similarity degree and proximity degree simultaneously. To avoid weighting an AIVIF matrix too many times, the collective decision matrix is transformed into an interval matrix using the risk coefficient of DMs. Subsequently, to derive the attribute weights objectively, we construct a multi-objective interval-programming model which is solved by transforming it into a linear program. The ranking order of alternatives is generated by the comprehensive interval values of alternatives.

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Finally, an example of a research and development (R&D) project selection problem is provided to illustrate the implementation process and applicability of the method developed in this chapter. Chapter 3 develops a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid multi-criteria group decision making (MCGDM) considering alternative comparisons with hesitancy degrees. The subjective preference relations between alternatives given by each decision maker (DM) are formulated as an IVIF set (IVIFS). The IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals, and real numbers are used to represent the multiple types of criteria values. The information of criteria weights is incomplete. The IVIFS-type consistency and inconsistency indices are defined through considering the fuzzy positive and negative ideal solutions simultaneously. To determine the criteria weights, we construct a novel bi-objective IVIF mathematical programming of minimizing the inconsistency index and meanwhile maximizing the consistency index, which is solved by the technically developed linear goal programming approach. The individual ranking order of alternatives furnished by each DM is subsequently obtained according to the comprehensive relative closeness degrees of alternatives to the fuzzy positive ideal solution. The collective ranking order of alternatives is derived through establishing a new multi-objective assignment model. A real example of critical infrastructure evaluation is provided to demonstrate the applicability and effectiveness of the proposed method. Chapter 4 puts forward a new selection method based on MAGDM with interval-valued intuitionistic fuzzy sets and applies to cloud service selection problem. As the cloud computing develops rapidly, more and more cloud services appear. Many enterprises tend to utilize cloud service to achieve better flexibility and react faster to market demands. In the cloud service selection, several experts may be invited and many attributes (indicators or goals) should be considered. Therefore, the cloud service selection can be regarded as a kind of multi-attribute group decision making (MAGDM) problems. The aim of this paper is to develop a new method for solving such MAGDM problems. In this method, the ratings of the alternatives on attributes in individual decision matrix given by each expert are in the form of Interval-valued Intuitionistic Fuzzy Sets (IVIFSs) which can precisely describe the preferences of experts on qualitative attributes. First, the weights of experts with respect to each attribute are determined by extending the classical Gray Relational Analysis (GRA) into IVIF environment. Then, based on the collective decision matrix obtained by aggregating the individual matrices, the score (profit) matrix, accuracy matrix and uncertainty (risk) matrix are derived. A multi-objective programming model is constructed to determine the attribute weights. Finally, the alternatives are ranked by employing the overall scores and uncertainties of alternatives. The feasibility and effectiveness of the proposed methods are illustrated by a cloud service selection problem and the comparison analysis is constructed. Chapter 5 aggregates decision information into interval-valued intuitionistic fuzzy numbers (IVIFNs) to solve heterogeneous MAGDM problem in which the decision information involves real numbers, interval numbers, triangular fuzzy

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numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs). There are three issues being addressed in this chapter. The first is to propose a new general method to aggregate the attribute value vector into IVIFNs under heterogeneous MAGDM environment utilizing the relative closeness in technique for order preference by similarity to ideal solution (TOPSIS). The second is to construct a multiple objective intuitionistic fuzzy programming model to determine the attribute weights. Borrowing the results of the former two issues, the last is to present a new method to solve heterogeneous MAGDM problem. A comparison analysis with existing method is conducted to demonstrate the advantages of the proposed method. Two examples are provided to verify the practicality and effectiveness of the proposed method. Chapter 6 investigates the group decision making (GDM) problems with interval-valued Atanassov intuitionistic fuzzy preference relations (IV-AIFPRs) and develops a novel method for solving such problems. A new consistency index of an Atanassov intuitionistic fuzzy preference relation (AIFPR) is introduced to judge the consistency of an AIFPR, and then a convergent iterative Algorithm I is designed to repair the consistency of an AIFPR with unacceptable consistency. Subsequently, the consistency and acceptable consistency of an IV-AIFPR are defined through separating an IV-AIFPR into two AIFPRs. Based on Algorithm I, a new iterative Algorithm II is devised to repair the consistency of an IV-AIFPR with unacceptable consistency. Afterward, to determine decision makers’ (DMs’) weights objectively, an optimization model is established by minimizing the deviations between each individual IV-AIFPR and the collective one. This model is skillfully transformed into a linear goal program to resolve sufficiently considering different principles of decision making. A linear programming model is built to derive interval-valued Atanassov intuitionistic fuzzy (IVAIF) priority weights of alternatives. Then, a TOPSIS (technique for order performance by similarity to an ideal solution) based approach is proposed to rank such IVAIF priority weights. Thereby, a method for GDM with IV-AIFPRs is put forward. At length, a practical example of a virtual enterprise partner selection is provided to illustrate the feasibility and validity of the proposed method. Chapter 7 investigates a group decision making (GDM) method based on additive consistent interval-valued Atanassov intuitionistic fuzzy (IVAIF) preference relations (IVAIFPRs) and likelihood comparison algorithm. Firstly, the likelihood of IVAIF values (IVAIFVs) is defined by the likelihood of intervals. Then a likelihood comparison algorithm is designed to rank IVAIFVs. According to the additive consistent interval fuzzy preference relation, we define the additive consistency of an IVAIFPR. Two special interval fuzzy preference relations are extracted from an IVAIFPR. They can be regarded as the lowest and highest preferred matrices of the IVAIFPR, respectively. Using a parametric linear program, the IVAIF priority weights of an IVAIFPR are generated from these two extracted special interval fuzzy preference relations. For the GDM with IVAIFPRs, the group consensus is defined by the distances between the individual IVAIFPRs and the collective one. To derive decision makers’ weights, an optimization model is constructed by maximizing the group consensus and transformed into a linear

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program to resolve. Subsequently, utilizing the IVAIF weighted averaging operator, the collective IVAIFPR is obtained and applied to derive the IVAIF priority weights. The order of alternatives is generated by ranking the IVAIF priority weights. At length, an enterprise resource planning system selection example is analyzed to verify the effectiveness of the proposed method. Chapter 8 develops a new method for solving group decision making (GDM) problems with interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, an additive consistency of an IVIFPR is defined by the additive consistency of intuitionistic fuzzy preference relation (IFPR). Based on the additive consistency definition of IVIFPR, two linear programming models are established to extract the most optimistic and pessimistic consistent IFPRs from an IVIFPR, respectively. Especially, if the feasible regions of these two models are empty, two adjusted programming models are constructed. Afterward, a risk attitudinal-based consistent IFPR is determined considering decision maker’s (DM’s) risk attitude. To derive the intuitionistic fuzzy priority weights from the risk attitudinal-based consistent IFPR, a multi-objective programming model is established and transformed into a linear goal program to resolve. Subsequently, combining DMs’ subjective and objective importance degrees, the comprehensive importance degrees of DMs are generated. Using comprehensive importance degrees as order inducing variables, a new comprehensive importance interval-valued intuitionistic fuzzy induced ordered weighted averaging (CI-IVIF-IOWA) operator is defined to aggregate the individual IVIFPRs into a collective one. Thereby, a three-phase method is proposed for GDM with IVIFPRs. An example of network system selection is examined to illustrate the practicability and effectiveness of the proposed method. Chapter 9 investigates a group decision-making (GDM) method that considers group consensus and multiplicative consistency of interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). First, the mean and variance of IVIF values (IVIFVs) are defined and a ranking method for IVIFVs is proposed considering the risk attitude of the expert. Then, the group consensus is presented by the individual similarity between experts. An iteration algorithm is designed to improve the group consensus. A statistical comparative analysis validates this algorithm. Subsequently, a new multiplicative consistency of IVIFPR is defined based on the multiplicative consistency of interval fuzzy preference relation. Two single-objective programming models are established to extract the most optimistic and pessimistic interval priority weight vectors from an IVIFPR, respectively. In particular, if the feasible domains of these two models are empty, two adjusted programs are constructed to replace the originals. Combining the most optimistic and pessimistic interval priority weights, the IVIF priority weights are generated. Further, expert weights are derived from Markov model and used to derive the collective IVIFPR for generating the IVIF priority weights. Therefore, a new method is proposed to solve the GDM with IVIFPRs. Finally, two cases are analyzed to verify the effectiveness of the proposed method.

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This book is suitable for practitioners and researchers working in the fields of fuzzy mathematics, operations research, information science and management science and engineering, etc. It can help practitioners making scientific and reasonable decision in practice or help researchers swiftly access into the intuitionistic fuzzy decision making field. It can also be used as a textbook for postgraduate and senior undergraduate students. This work was supported by the National Natural Science Foundation of China (71740021 and 11861034), the Natural Science Foundation of Jiangxi Province of China (No. 20192BAB207012), and “Thirteen five” Programming Project of Jiangxi Province Social Science (2018) (No. 18GL13). Last but not least, we would like to acknowledge the encouragement, selfless help and support of all our friends and colleagues. We would like to extraordinarily appreciate our doctoral graduates and co-authors Gai-Li Xu, Feng Wang and Jun Xu for completing and publishing several articles as well as our doctoral students Wen-Chang Zou, Xian-Juan Cheng, Ze-Hui Chen, Ai-Hua Liu and our master students Wen-Bo Cheng Huang, Jia Yan, Qin Zhang, Xue-Lin Zhang, Yun-Qian Gu and Hao Wu for checking and editing the final manuscript. Nanchang, China September 2019

Shuping Wan Jiuying Dong

Contents

1 A Possibility Degree Method for Interval-Valued Intuitionistic Fuzzy Multi-attribute Group Decision Making . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A New Ranking Method of IVIFS from the Probability Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 New Ranking Method for Intervals from the Probability Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 New Ranking Method for IVIFNs Based on the Possibility Degree . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Comparative Analysis with Score and Accuracy Functions for IVIFNs . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Ordered Weighted Average Operator and Hybrid Weighted Average Operator for IVIFNs . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Weighted Average Operator for IVIFNs . . . . . . . . . . . . 1.3.2 Proposed Ordered Weighted Average Operator for IVIFNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Proposed Hybrid Weighted Average Operator for IVIFNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 MAGDM Problem and Method with IVIFS . . . . . . . . . . . . . . . 1.4.1 Problem Description for MAGDM with IVIFS . . . . . . . 1.4.2 Determination of the Weights of DMs . . . . . . . . . . . . . 1.4.3 Group Decision Making Method . . . . . . . . . . . . . . . . . . 1.5 An Air-Condition System Selection Example and Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 An Air-Condition System Selection Problem and the Analysis Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Comparison Analysis of the Obtained Results . . . . . . . . 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 A New Method for Atanassov’s Interval-Valued Intuitionistic Fuzzy MAGDM with Incomplete Attribute Weight Information . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Interval Objective Programming . . . . . . . . . . . . . . . . . . 2.2.2 Orderings of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Atanassov’s Intuitionistic Fuzzy Set and Atanassov’s Interval-Valued Intuitionistic Fuzzy Set . . . . . . . . . . . . . 2.3 A Novel Method for MAGDM with AIVIFVs and Incomplete Attribute Weight Information . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Presentation of the Problems . . . . . . . . . . . . . . . . . . . . 2.3.2 Determine the DMs’ Weights with Respect to Different Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Converting Individual Decision Matrices into a Collective Interval Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Construct Multi-objective Interval-Programming for Deriving Attribute Weights . . . . . . . . . . . . . . . . . . . 2.3.5 Decision Process and Algorithm for MAGDM with AIVIFVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Real-World R & D Project Selection Example and Comparison Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Real-World R & D Project Selection Problem and the Solution Process . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Comparison with the Extended TOPSIS Method . . . . . . 2.4.3 Comparison with Barrenechea et al.’s Method . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming Method for Hybrid Multi-criteria Group Decision Making with Interval-Valued Intuitionistic Fuzzy Truth Degrees . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Concepts of Interval-Valued Intuitionistic Fuzzy Sets and Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Definition and Distance for Trapezoidal Fuzzy Numbers and Triangular Fuzzy Numbers . . . . . . . . . . . . . . . . . . . 3.2.3 Linguistic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hybrid MCGDM Problems Considering Alternative Comparisons with IVIF Truth Degrees . . . . . . . . . . . . . . . . . . . 3.3.1 Hybrid MCGDM Problems with IVIF Truth Degrees and Incomplete Weight Information . . . . . . . . . . . . . . . 3.3.2 Normalization Method . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4 IVIF Mathematical Programming Method for Hybrid MCGDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 IVIFS-Type Consistency and Inconsistency Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Bi-Objective IVIF Mathematical Programming Model . 3.4.3 Goal Programming Approach to Solving Bi-Objective IVIF Mathematical Programming Model . . . . . . . . . . . 3.4.4 GDM Process and Steps for Solving Hybrid MCGDM 3.5 Empirical Example of Critical Infrastructure . . . . . . . . . . . . . . 3.5.1 A Critical Infrastructure Evaluation Example and the Analysis Process . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Comparison Analysis with the Existing LINMAP Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 A Selection Method Based on MAGDM with Interval-Valued Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Interval-Valued Intuitionistic Fuzzy Set . . . . . . . . . . . . . 4.2.2 Gray Relation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A Novel Method for MAGDM with IVIFSs and Incomplete Attribute Weight Information . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Determine the Weights of Experts by the Extended GRA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Integrate Individual Decision Matrices into a Collective Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Identify the Attribute Weights by a New Multi-objective Linear Programming Model . . . . . . . . . . . . . . . . . . . . . 4.3.4 Decision Process and Algorithm for MAGDM Problems with IVIFSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Decision Support System Framework Based on MAGDM with IVIFSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 A Cloud Service Selection Problem and Comparison Analysis . 4.4.1 A Cloud Service Provider Selection Problem and the Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Sensitivity Analysis for Parameter . . . . . . . . . . . . . . . . 4.4.3 Comparison Analysis with the Method Using the Score Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5 Aggregating Decision Information into Interval-Valued Intuitionistic Fuzzy Numbers for Heterogeneous Multi-attribute Group Decision Making . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Some Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Interval-Valued Intuitionistic Fuzzy Set . . . . . . . . . . . . 5.2.2 Definitions and Distances for Real Number, Interval Number, TFN and TrFN . . . . . . . . . . . . . . . . 5.3 Aggregating Heterogeneous Decision Information into IVIFNs 5.3.1 Presentation of Heterogeneous MAGDM Problem . . . . 5.3.2 A General Method for Aggregating Heterogeneous Decision Information into IVIFNs . . . . . . . . . . . . . . . . 5.3.3 Concrete Computation Formulas for Aggregating Heterogeneous Information into IVIFNs . . . . . . . . . . . 5.4 A Novel Approach for Heterogeneous MAGDM Problems . . . 5.4.1 Construct an Intuitionistic Fuzzy Programming Model to Determine the Attribute Weights . . . . . . . . . 5.4.2 Algorithm for Solving Heterogeneous MAGDM . . . . . 5.5 Comparison Analysis with Existing Methods . . . . . . . . . . . . . 5.6 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 An IT Outsourcing Service Provider Evaluation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 A Supplier Selection Example . . . . . . . . . . . . . . . . . . 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 A Novel Method for Group Decision Making with Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relations . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Multiplicative Consistency of Atanassov Intuitionistic Fuzzy Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations . . . . . . . . . . . . 6.2.2 An Iterative Algorithm for Repairing the Consistency of AIFPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Multiplicative Consistency of IV-AIFPRs . . . . . . . . . . . . . . . . 6.3.1 Define and Check Multiplicative Consistency of IV-AIFPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Repair Multiplicative Consistency of IV-AIFPRs . . . . . .

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6.4 A Novel Method for Group Decision Making with IV-AIFPRs . 6.4.1 Determine DMs’ Weights Objectively and Integrate Individual IV-AIFPRs . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Derive IVAIF Priority Weights and Rank Alternatives . . 6.4.3 A Novel Method for Solving GDM Problems with IV-AIFPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 A Practical Example of a Virtual Enterprise Partner Selection and Comparative Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 A Practical Example of a Virtual Enterprise Partner Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Comparative Analyses . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Additive Consistent Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relation and Likelihood Comparison Algorithm Based Group Decision Making . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A New Likelihood Comparison Algorithm of IVAIFVs . . . . 7.4 Additive Consistency Analyses for IVAIFPR . . . . . . . . . . . . 7.4.1 Additive Consistency Definition of IVAIFPR . . . . . . 7.4.2 Derive the IVAIF Priority Weights from IVAIFPR . . 7.5 Method for Solving the Group Decision Making Problems with IVAIFPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Description for GDM Problems with IVAIFPRs . . . . 7.5.2 Determination of DMs’ Weights . . . . . . . . . . . . . . . . 7.5.3 Method for Group Decision Making with IVAIFPRs . 7.6 An Example of ERP System Selection and Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 A Practical Example of ERP System Selection . . . . . 7.6.2 Comparative Analysis with Existing GDM Methods . 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 A Three-Phase Method for Group Decision Making with Interval-Valued Intuitionistic Fuzzy Preference Relations . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Some Related Concepts on IFPR . . . . . . . . . . . . . . . 8.2.2 Additive Consistency of IVIFPR . . . . . . . . . . . . . . .

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8.3 Determination of the Intuitionistic Fuzzy Priority Weights from an IVIFPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Extracting a Risk Attitudinal-Based Consistent IFPR from an IVIFPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Deriving the Intuitionistic Fuzzy Priority Weights from the Extractive IFPR . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 A Novel Three-Phase Method for Solving GDM with IVIFPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Presentation of Problem for GDM with IVIFPRs . . . . . . 8.4.2 Integrating Individual IVIFPRs to a Collective One . . . . 8.4.3 A Three-Phase Method for GDM with IVIFPRs . . . . . . 8.5 An Example of Network System Selection and Comparison Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 A Network System Selection Example and the Analysis Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Comparison Analysis with Xu’s Method . . . . . . . . . . . . 8.5.3 Comparison Analysis with Wang’s Method . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A Group Decision-Making Method Considering Both the Group Consensus and Multiplicative Consistency of Interval-Valued Intuitionistic Fuzzy Preference Relations . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 A New Ranking Method of IVIFVs . . . . . . . . . . . . . . . . . . . . . 9.3.1 A New Ranking Method of IVIFVs . . . . . . . . . . . . . . . 9.3.2 Comparison with Existing Ranking Methods of IVIFVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Analyses of Group Consensus in GDM . . . . . . . . . . . . . . . . . . 9.4.1 Problem Description for GDM with IVIFPRs . . . . . . . . 9.4.2 Group Consensus Analysis of GDM . . . . . . . . . . . . . . . 9.4.3 An Iteration Algorithm in Group Consensus Improving Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Statistical Comparative Study on Group Consensus . . . . 9.5 Multiplicative Consistency of IVIFPR . . . . . . . . . . . . . . . . . . . 9.5.1 Multiplicative Consistency Concept of IVIFPR . . . . . . . 9.5.2 Determine the Priority Weights from an IVIFPR . . . . . . 9.6 Method for GDM with IVIFPRs . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Determination of Expert Weights Based on the Markov Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Method for GDM with IVIFPRs . . . . . . . . . . . . . . . . . .

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9.7 Two Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Case 1: An Example of a Virtual Enterprise Partner Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Case 2: A Practical Example of an Enterprise Resource Planningsystem Selection . . . . . . . . . . . . . . . . . . . . . . . 9.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

A Possibility Degree Method for Interval-Valued Intuitionistic Fuzzy Multi-attribute Group Decision Making

Abstract An interval-valued intuitionistic fuzzy set (IVIFS) is the extension of an intuitionistic fuzzy set (IFS). The ranking of IVIFSs is very important for the interval-valued intuitionistic fuzzy decision making. From a viewpoint of probability, the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) is defined by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik-Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. In the proposed decision method, the individual overall attribute values of alternatives are obtained by using the weighted average operator of IVIFNs. The collective overall attribute values of alternatives are integrated by using the hybrid weighted average operator of IVIFNs. The ranking orders of alternatives are generated according to the collective overall attribute values. A numerical example is examined to illustrate the proposed method and the comparison analysis demonstrates the effectiveness and flexibility of the proposed method in this chapter.




Keywords Multi-attribute group decision making Interval-valued intuitionistic fuzzy set Aggregation operator Karnik-Mendel algorithm




1.1




Introduction

Fuzzy set (FS) theory has long been introduced to handle inexact and imprecise data by Zadeh [56]. The drawback of using the single membership value in FS theory is that the evidence for x 2 X and the evidence against x 2 X are in fact mixed together (Here X is the universe of discourse). In order to tackle this problem, Atanassov [1] proposed the intuitionistic fuzzy set (IFS) using two characteristic functions expressing the degree of membership and the degree of non-membership of elements of the universal set to the IFS. It can cope with the presence of vagueness and hesitancy originating from imprecise knowledge or information. © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_1

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1 A Possibility Degree Method for Interval-Valued Intuitionistic …

IFS has been widely applied to the multi-attribute decision making (MADM) [3, 5, 6, 9–13, 19, 22, 23, 25–30, 32, 33, 36, 47, 50–52, 54, 57] and multi-attribute group decision making (MAGDM) [17, 18, 20, 27, 35, 37, 39, 41]. These achievements can be roughly classified into four types: aggregation operators, similarity (or distance) measures, extension of classic decision making methods and judgment matrix, which are respectively reviewed as follows. In the aspect of aggregation operators, Li [11, 12] and Li et al. [18] proposed the generalized OWA operators with IFSs. Liu and Wang [23] proposed the intuitionistic fuzzy point operators. Xu [39] developed the intuitionistic fuzzy power aggregation operators. Xu and Yager [48] developed some geometric aggregation operators based on IFS, such as the intuitionistic fuzzy weighted geometric operator, the intuitionistic fuzzy ordered weighted geometric operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, and applied them to MADM. Xu [35] used the IFHG operator to propose a MAGDM method with incomplete weight information under intuitionistic fuzzy environment. Yang and Chen [52] defined the quasi-arithmetic intuitionistic fuzzy OWA operators. Zeng and Su [57] proposed the intuitionistic fuzzy ordered weighted distance operator. Wei [27] proposed the induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and applied to MAGDM with intuitionistic fuzzy numbers (IFNs [48]). Zhao et al. [58] developed some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, and applied to MADM with intuitionistic fuzzy information. These aggregation operators for IFNs may be viewed as the extension of the ones for the real numbers, which mainly involve the arithmetic aggregation operators, geometric aggregation operators, power aggregation operators, generalized average operators and induced aggregation operators. In the aspect of similarity (or distance) measures, Li [9] discussed some measures of dissimilarity in intuitionistic fuzzy structures. Xu [32] defined the normalized Hamming and distance between two IFNs and proposed the intuitionistic fuzzy MADM method. Xu [33] defined similarity measures of IFSs based on the geometric distance model, the set-theoretic approach and the matching function, respectively, and applied the similarity measures to the MADM under intuitionistic fuzzy environment. Xu and Yager [49] also defined some new similarity measures of IFSs based on the distance measures. These similarity (or distance) measures of IFSs also can be applied to pattern recognitions and approximate reasoning. In the aspect of extension of classic decision making methods, Li [13] and Li et al. [17] extended the classic LINMAP method to the intuitionistic fuzzy environments. Wu and Chen [28] proposed the ELECTRE multicriteria analysis approach based on IFSs. Xu and Hu [47] constructed the projection models for intuitionistic fuzzy MADM. These decision making methods under intuitionistic fuzzy environment generalize the classic decision making methods, such as TOPSIS, ELECTRE and LINMAP. In the aspect of judgment matrix, Xu [36] defined some concepts, such as consistent intuitionistic preference relation, incomplete intuitionistic preference

1.1 Introduction

3

relation and acceptable intuitionistic preference relation, etc. He also developed an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations, respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Xu and Yager [49] investigated some intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. The intuitionistic fuzzy preference relations enrich the research content of intuitionistic fuzzy decision making theory. Atanassov and Gargov [2] further generalized the IFS in the spirit of the ordinary interval-valued fuzzy set (IVFS) and defined the concept of an interval-valued intuitionistic fuzzy set (IVIFS), which enhances greatly the representation ability of uncertainty than IFS. Recently, IVIFSs were also used in the fields of MADM [4, 14–16, 26, 32, 42, 53] and MAGDM [27, 31, 34, 38, 40, 43–45, 49, 58]. Similar to the IFS, these achievements on IVIFSs mainly focused on the aggregation operators, similarity (or distance) measures, extension of classic decision making methods and judgment matrix, which are respectively reviewed as follows. In the aspect of aggregation operators, Xu [31] defined some operational laws of interval-valued intuitionistic fuzzy numbers (IVIFNs) and then proposed the interval-valued intuitionistic fuzzy weighted arithmetic aggregation operator and interval-valued intuitionistic fuzzy weighted geometric aggregation operator. Xu [38] used the Choquet integral to propose the interval-valued intuitionistic fuzzy correlated averaging operator and the interval-valued intuitionistic fuzzy correlated geometric operator to aggregate interval-valued intuitionistic fuzzy information. Xu and Chen [44] developed some geometric aggregation operators for IVIFNs. Wei [27] proposed the induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator and applied to MAGDM with IVIFNs. Xu and Chen [45] proposed the interval-valued intuitionistic fuzzy Bonferroni means. Zhao et al. [58] developed generalized interval-valued intuitionistic fuzzy weighted averaging operator, generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval-valued intuitionistic fuzzy hybrid average operator, and applied to MADM with interval-valued intuitionistic fuzzy information. These aggregation operators for IVIFNs may be roughly divided into two kinds. One is the aggregation operators with independent attributes, including arithmetic aggregation operators and geometric aggregation operators, the other is the aggregation operators with dependent attributes, such as the interval-valued intuitionistic fuzzy correlated averaging operator and the interval-valued intuitionistic fuzzy correlated geometric operator [38]. In the aspect of similarity (or distance) measures, Xu [32] defined the normalized Hamming distance between two IVIFNs and proposed the interval-valued fuzzy MADM method. Xu [34] defined the Hamming and Euclidean distances, the Hamming and Euclidean distances based on the Hausdorff metric for IVIFSs. The corresponding similarity measures for IVIFSs are also defined and applied to

4

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

pattern recognitions. The similarity measures for IVIFSs also can be applied to MADM with interval-valued fuzzy assessment information. In the aspect of extension of classic decision making methods, Li [14] developed the closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy MADM with incomplete preference information. Li [15] proposed the TOPSIS-based nonlinear-programming methodology for MADM with IVIFSs. Li [16] proposed the linear programming method for MADM with IVIFSs. These decision methods under interval-valued intuitionistic fuzzy environment also generalize the classic decision making methods, such as TOPSIS and LINMAP. The other methods, such as ELECTRE and PROMETHEE, are expected to be applied to MADM and MAGDM with IVIFSs. In the aspect of judgment matrix, Xu and Cai [42] investigated the incomplete interval-valued intuitionistic fuzzy preference relations. Xu and Chen [43] developed the ordered weighted aggregation operator and hybrid aggregation operator for aggregating interval-valued intuitionistic preference information. Interval-valued intuitionistic judgment matrix and its score matrix and accuracy matrix were defined. Some of their desirable properties were investigated in detail. The relationships among interval-valued intuitionistic judgment matrix, intuitionistic judgment matrix and complement judgment matrix, were discussed. Xu and Yager [49] investigated some interval-valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. In most the existing aggregation operators for IFSs and IVIFSs, the weighted vector of the aggregated arguments usually takes the form of real numbers. However, with ever increasing complexity in many real decision situations, there are often some challenges for the decision maker (DM) to provide precise weight information due to time pressure, lack of knowledge (or data) and the DM’s limited expertise about the problem domain. In other words, usually weights are not necessary the real numbers and may be the intervals, IFNs or IVIFNs. Recently, Chen et al. [4] proposed the intuitionistic fuzzy weighted average operator and interval-valued intuitionistic fuzzy weighted average operator based on the traditional weighted average method and the Karnik-Mendel algorithms [8]. Although they sufficiently considered the different forms of the weighted vector, such as real numbers, intervals, IFNs and IVIFNs, the proposed intuitionistic fuzzy weighted average operator and interval-valued intuitionistic fuzzy weighted average operator can only weight the arguments themselves rather than the ordered positions of the arguments. As we all know, both the arguments themselves and the ordered positions of the arguments are very important in the process of information aggregation. To overcome this drawback, this chapter first proposes a new ranking method of IVIFNs based on the possibility degree from the probability viewpoint and then develops the ordered weighted average operator and hybrid weighted average operator for IVIFNs. Finally, combining the interval-valued intuitionistic fuzzy weighted average operator and hybrid weighted average operator, a new MAGDM method with IVIFSs is proposed.

1.1 Introduction

5

The rest of the paper is arranged as follows. Section 1.2 develops a new ranking method of IVIFNs based on the possibility degree. Section 1.3 proposes the ordered weighted average operator and hybrid weighted average operator for IVIFNs. Section 1.4 constructs the MAGDM model with IVIFSs and proposes the corresponding decision making method. An air-condition system selection example is examined to illustrate the proposed method in Sect. 1.5. The comparison analysis with other method is also conducted in Sect. 1.5. Concluding remark is made in Sect. 1.6.

1.2

A New Ranking Method of IVIFS from the Probability Viewpoint

Atanassov and Gargov [2] gave the definition of IVIFS as follows: Definition 1.1 [2] An IVIFS A in the universe of discourse X is defined as A ¼ f\x; lA ðxÞ; mA ðxÞ [ jx 2 Xg; where lA ðxÞ½0; 1 and mA ðxÞ½0; 1 denote respectively the membership degree interval and the non-membership degree interval of x to A, with the condition: sup lA ðxÞ þ sup mA ðxÞ  1; 8x 2 X: Since IVIFS is composed of two ordered interval pairs, Xu [31, 32] called them interval-valued intuitionistic fuzzy numbers (IVIFNs) and simply denoted by G ¼ ð½a; b; ½c; dÞ, where ½a; b½0; 1, ½c; d½0; 1 and a þ b  1:

1.2.1

New Ranking Method for Intervals from the Probability Viewpoint

Since the IVIFN consists of two intervals, we first give the ranking method for two intervals. Let x ¼ ½a; b½0; 1 and y ¼ ½c; d½0; 1 be two intervals. Let X be a random variable with uniform probability distribution, which is defined on the closed interval ½a; b: Since the arbitrary element z 2 ½a; b has the equal possibility for the interval x ¼ ½a; b, the arbitrary basic random event has equal occurrence probability for the random variable X: Thus, we can view the intervals x and y as the random variables with uniform distribution defined on the corresponding intervals. The occurrence possibility for the fuzzy event fx  yg can be transformed into the occurrence probability for the random event fx  yg . Maybe there is a constant deviation or constant multiplier between the occurrence possibility and probability. The main reason and explanation come from [55].

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

6

Yoon [55] proposed two kinds of methods for the transformation between the membership function and probability density function. One is the proportional probability distribution as follows: f1 ðxÞ ¼ c1 lA ðxÞ where f1 ðxÞ is the occurrence probability for the random event A, lA ðxÞ is the value of membership function, c1 is a proportional constant satisfying the condition that the area under the continuous probability function is equal to one. The other is the uniform probability distribution as follows: f2 ðxÞ ¼ lA ðxÞ þ c2 where c2 is a uniform constant satisfying the probability function requirement. The probability is based on probability distribution while the possibility is based on possibility distribution (membership function). There only exists a difference of constant multiplier c1 or constant c2 for the transformation between the membership function and probability density function. The difference of constant multiplier c1 or constant c2 does not influence the ranking order for the intervals. Therefore, we can use the occurrence probability for the random event fx  yg to define the possibility of fx  yg . Without loss of generality and for simplicity, the random variable x is supposed to be independent of y: Then, the jointly density function for the 2-dimensional random vector ðx; yÞ is ( f ðx; yÞ ¼

1 ðbaÞðdcÞ ;

ðx; yÞ 2 ½a; b  ½c; d;

0;

else:

The marginal density functions for x and y are as follows: ( fx ðxÞ ¼

1 ðbaÞ ;

x 2 ½a; b;

0;

else

1 ðdcÞ ;

y 2 ½c; d;

0;

else,

and ( fy ðyÞ ¼

respectively. Because the location relations between x ¼ ½a; b and y ¼ ½c; d include the following six cases, we can calculate the occurrence probability for the fuzzy (or random) event fx  yg denoted by Pðx  yÞ under different cases.

1.2 A New Ranking Method of IVIFS from the Probability Viewpoint Fig. 1.1 a\b  c\d

7

y d

C

D

c

y=x

A

B

a

b

b

a

c

d

x

(1) Case 1: a\b  c\d as Fig. 1.1 shown. The intersection between rectangle ½a; b  ½c; d and x  y is null, so Pðx  yÞ ¼ 0:

ð1:1Þ

(2) Case 2: a  c\b\d or a\c\b  d as Fig. 1.2 shown. The intersection between rectangle ½a; b  ½c; d and x  y is triangle ABC, so Zb Pðx  yÞ ¼ c

2 fx ðxÞ4

Zx

3 fy ðyÞdy5dx ¼

ðb  cÞ2 : 2ðb  aÞðd  cÞ

ð1:2Þ

c

(3) Case 3: a  c\d\b or a\c\d  b or a  c\d  b as Fig. 1.3 shown. The intersection between rectangle ½a; b  ½c; d and x  y is trapezoid ABCD, so

Fig. 1.2 a  c\b\d or a\c\b  d

y y=x

d b

C

c

A

a

a

c

B

b

d

x

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

8

Fig. 1.3 a  c\d\b or a\c\d  b or a  c\d  b

y b

y=x

d

C

D

c

A

B

a a

Zd Pðx  yÞ ¼

2 6 fy ðyÞ4

c

Zb

c

d

x

b

3 2b  d  c 7 : fx ðxÞdx5dy ¼ 2ðb  aÞ

ð1:3Þ

y

(4) Case 4: c  a\b\d or c\a\b  d as Fig. 1.4 shown. The intersection between rectangle ½a; b  ½c; d and x  y is trapezoid ABCD, so 2 3 Zb Z x b þ a  2c Pðx  yÞ ¼ 4 fy ðyÞdy5fx ðxÞdx ¼ : 2ðd  cÞ a

ð1:4Þ

c

(5) Case 5: c  a\d\b or c\a\d  b as Fig. 1.5 shown. The intersection between rectangle ½a; b  ½c; d and x  y is pentagon ABCDE, so

Fig. 1.4 c  a\b\d or c\a\b  d

y d

y=x

b

C

a

D

c

A

B

a

b

c

d

x

1.2 A New Ranking Method of IVIFS from the Probability Viewpoint Fig. 1.5 c  a\d\b or c\a\d  b

9

y b

y=x

d

D

a

E

c

Pðx  yÞ ¼

Zb fy ðyÞ

c

¼

B

A c

Za

C

a Zd

fx ðxÞdxdy þ a

d

b

Zb fy ðyÞ

a

fx ðxÞdxdy y

2bd þ 2ac  2bc  a2  d 2 : 2ðb  aÞðd  cÞ

ð1:5Þ

(6) Case 6: c\d  a\b, as Fig. 1.6 shown. The intersection between rectangle ½a; b  ½c; d and x  y is rectangle ABCD, so Zd Pðx  yÞ ¼

2 b 3 Z fy ðyÞ4 fx ðxÞdx5dy ¼ 1:

c

ð1:6Þ

a

Fig. 1.6 c\d  a\b

y b

y=x

a d c c

d

D

C

A

B

a

b

x

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

10

Theorem 1.1 Let x ¼ ½a; b and y ¼ ½c; d be two intervals. Define the possibility degree of x  y as Pðx  yÞ: Then, the following properties hold: 0  Pðx  yÞ  1, 0  Pðy  xÞ  1; Pðx  yÞ ¼ 1 if and only if d  a: Similarly, Pðy  xÞ ¼ 1 if and only if b  c; Pðx  yÞ ¼ 0 if and only if b  c: Similarly, Pðy  xÞ ¼ 0 if and only if d  a; Pðx  xÞ ¼ 0:5; Pðx  yÞ þ Pðy  xÞ ¼ 1; Pðx  yÞ  0:5 if and only if a þ b  c þ d. Especially, Pðx  yÞ ¼ 0:5 if and only if a þ b ¼ c þ d; (vii) Let z ¼ ½e; f  be an interval. If Pðx  yÞ  0:5 and Pðy  zÞ  0:5, then Pðx  zÞ  0:5: (i) (ii) (iii) (iv) (v) (vi)

Proof Obviously, the properties (i)–(v) are right. We prove (vi) in the following. If Pðx  yÞ  0:5, then the location relations between x ¼ ½a; b and y ¼ ½c; d must be Case 3 or Case 4. For Case 3, by Eq. (1.3), we have Pðx  yÞ ¼ 2bdc 2ðbaÞ  0:5, so þ a2c a þ b  c þ d: For Case 4, by Eq. (1.4), we have Pðx  yÞ ¼ b2ðdcÞ  0:5, so a þ b  c þ d: If a þ b  c þ d, then the location relations between x ¼ ½a; b and y ¼ ½c; d must be Case 3 or Case 4. For Case 3, by Eq. (1.3), we have Pðx  yÞ ¼ 2bdc 2ðbaÞ

þ a2c  0:5: For Case 4, by Eq. (1.4), we have Pðx  yÞ ¼ b2ðdcÞ  0:5:

Thus, the property (vi) holds. In a similar way, the property (vii) can be proved, which completes the proof of Theorem 1.1. In order to rank intervals ~ai ¼ ½ai ; bi ði ¼ 1; 2; . . .; nÞ, similar to Xu [29], we can construct the matrix of possibility degree as P ¼ ðPij Þnn , where Pij ¼ Pð~ai  ~aj Þði ¼ 1; 2; . . .; n; j ¼ 1; 2; . . .; nÞ: Then, the ranking vector x ¼ ðx1 ; x2 ; . . .; xn ÞT is derived as follows: xi ¼

n X j¼1

! n Pij þ  1 =ðnðn  1ÞÞði ¼ 1; 2; . . .; nÞ: 2

ð1:7Þ

~i ¼ ½ai ; bi : The larger the value of xi , the bigger the corresponding interval a Xu and Da [46] defined the possibility degree of x  y as follows (see Definition 2.3 of [46]): 



  da ;0 ;0 : P(x  yÞ ¼ max 1  max b  aþd  c Then, they discussed the properties about the possibility degree and formulated these properties as Theorem 2.1. Theorem 2.1 of [46] contains seven properties. It is easy to see that Theorem 1.1 of this chapter is similar to Theorem 2.1 of [46]. In other words, the possibility degrees proposed in this chapter and [46] have the identical properties. However,

1.2 A New Ranking Method of IVIFS from the Probability Viewpoint

11

the former is based on the 2-dimensional random vector. The possibility degree P(x  yÞ is explained as the occurrence probability for the random event fx  yg, which has the intuitive meaning from the probability viewpoint, whereas the latter did not give the corresponding explanation. Example 1.1 [46] Let a~1 ¼ ½3; 5, ~a2 ¼ ½4; 6, ~a3 ¼ ½4; 7 and ~ a4 ¼ ½3; 6 be four intervals. Using the ranking method proposed in this chapter, the matrix of possibility degree is obtained as follows: 0 B P¼B @

0:5

1 8

7 8 11 12 2 3

0:5 2 3 1 3

1 12 1 3

0:5

1 3 2 3 7 9

2 9

0:5

1 C C: A

By Eq. (1.7), the ranking vector is calculated as follows: x1 ¼ 0:1701; x2 ¼ 0:2813; x3 ¼ 0:3218; x4 ¼ 0:2269 Hence, the ranking order is ~a3 a~2 ~a4 ~a1 . The ranking order given by [46] is identical, i.e., ~a3 ~a2 ~a4 ~a1 . This example shows the effectiveness of the ranking method proposed in this chapter.

1.2.2

New Ranking Method for IVIFNs Based on the Possibility Degree

Definition 1.2 Let G1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ and G2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ be two IVIFNs. The possibility degree of G1  G2 can be defined as 1 2

PðG1  G2 Þ ¼ fPð½a1 ; b1   ½a2 ; b2 Þ þ Pð½c2 ; d2   ½c1 ; d1 Þg:

ð1:8Þ

It is easy to verify that the above definition satisfies the following properties: (i) 0  PðG1  G2 Þ  1; (ii) PðG1  G1 Þ ¼ 0:5; (iii) PðG1  G2 Þ þ PðG2  G1 Þ ¼ 1: In order to rank IVIFNs Gi ¼ ð½ai ; bi ; ½ci ; di Þði ¼ 1; 2; . . .; nÞ, similar to Xu [29], we can construct the matrix of possibility degree as P ¼ ðPij Þnn , where Pij ¼ PðGi  Gj Þði ¼ 1; 2; . . .; n; j ¼ 1; 2; . . .; nÞ: Then, the ranking vector x ¼ ðx1 ; x2 ; . . .; xn ÞT is derived as follows:

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

12

xi ¼

n X j¼1

! n Pij þ  1 =ðnðn  1ÞÞði ¼ 1; 2; . . .; nÞ: 2

ð1:9Þ

Thus, the IVIFNs Gi ði ¼ 1; 2; . . .; nÞ can be ranked in descending order in accordance with the values of xi ði ¼ 1; 2; . . .; nÞ: Example 1.2 Let G1 ¼ ð½0:1; 0:2; ½0:3; 0:5Þ, G2 ¼ ð½0:4; 0:6; ½0:2; 0:4Þ, G3 ¼ ð½0:3; 0:5; ½0:1; 0:2Þ and G4 ¼ ð½0:6; 0:8; ½0:0; 0:1Þ be four IVIFNs. Using the ranking method proposed in this chapter, the matrix of possibility degree is obtained as follows: 0 B P¼B @

0:5

1 16

15 16

0:5

1 1

9 16

1

1 0 7 0 C 16 C: 0:5 0 A 1 0:5 0

ð1:10Þ

By Eq. (1.9), the ranking vector is calculated as follows: x1 ¼ 0:1302; x2 ¼ 0:2396; x3 ¼ 0:2552; x4 ¼ 0:3750: Hence, the ranking order is G4 [ G3 [ G2 [ G1 .

1.2.3

Comparative Analysis with Score and Accuracy Functions for IVIFNs

Xu [31] and Xu and Chen [43] defined the score function and accuracy function of an IVIFN G ¼ ð½a; b; ½c; dÞ as follows: 1 2

SðGÞ ¼ ða þ b  c  dÞ; 1 2

HðGÞ ¼ ða þ b þ c þ dÞ:

ð1:11Þ ð1:12Þ

Let G1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ and G2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ be two IVIFNs. Xu [31] and Xu and Chen [43] gave the ranking method as follows: (1) If SðG1 Þ\SðG2 Þ, then G1 \G2 ; (2) If SðG1 Þ ¼ SðG2 Þ, then if HðG1 Þ\HðG2 Þ, then G1 \G2 ; if HðG1 Þ ¼ HðG2 Þ, then G1 ¼ G2 .

1.2 A New Ranking Method of IVIFS from the Probability Viewpoint

13

Applying the above ranking method to Example 2, the score functions for the four IVIFNs are respectively computed by Eq. (1.11) as follows: SðG1 Þ ¼ 0:25; SðG2 Þ ¼ 0:20; SðG3 Þ ¼ 0:25; SðG4 Þ ¼ 0:65: Therefore, the ranking order obtained by [31, 43] is G4 [ G3 [ G2 [ G1 , which coincides with that obtained by this chapter. Nevertheless, it should be pointed out that the ranking method proposed in this chapter has an outstanding advantage, namely, the possibility degrees between any adjacent IVIFNs for the ranking order can also be obtained by the matrix of possibility degree (i.e., Eq. (1.10)) as follows: G4 [ 1 G3 [ 9=16 G2 [ 15=16 G1 : That is to say, the possibility degree of G4 [ G3 is 1, the possibility degree of G3 [ G2 is 169 and the possibility degree of G2 [ G1 is 15 16, which is very useful for the DM to make decision reasonably in some real decision making problems. However, this advantage cannot be reflected in the ranking method of [31, 43].

1.3

Ordered Weighted Average Operator and Hybrid Weighted Average Operator for IVIFNs

Karnik-Mendel algorithms are iterative procedures and converge monotonically and super-exponentially fast [8, 21, 24]. They have successfully been used to deal with the fuzzy weighted average and the linguistic weighted average [4]. Chen et al. [4] proposed the intuitionistic fuzzy weighted average operator and interval-valued intuitionistic fuzzy weighted average operator based on the traditional weighted average method and the Karnik-Mendel algorithms [8]. As stated earlier, these two operators did not consider the ordered positions of the arguments. Consequently, based on Karnik-Mendel algorithms, this section develops the ordered weighted average operator and hybrid weighted average operator for IVIFNs to overcome this drawback.

1.3.1

Weighted Average Operator for IVIFNs

Definition 1.3 (See [4]) Let Gi ði ¼ 1; 2; . . .; nÞ be a collection of the IVIFNs, where Gi ¼ ð½ai ; bi ; ½ci ; di Þ ¼ ½½ai ; bi ; ½1  di ; 1  ci . If

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

14

Pn j¼1 wj Gj Yw ðG1 ; G2 ; . . .; Gn Þ ¼ Pn ; j¼1 wj

ð1:13Þ

where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weight vector, then the function Y is called the weighted average operator for the IVIFNs. The weight vector w ¼ ðw1 ; w2 ; . . .; wn ÞT may take different forms as follows: Case 1: If wj ðj ¼ 1; 2; . . .; nÞ are crisp values, then the weighted average operator Y is calculated as follows: Pn j¼1 wj Gj Y ¼ Pn j¼1 wj " " Pn # " Pn ## Pn Pn j¼1 wj aj j¼1 wj bj j¼1 wj ð1  dj Þ j¼1 wj ð1  cj Þ : Pn Pn Pn ¼ ; Pn ; ; w w w w j j j j j¼1 j¼1 j¼1 j¼1 h i D E ¼ ½y1 ; y1 ; ½y2 ; y2  ¼ ½y1 ; y1 ; ½1  y2 ; 1  y2 

ð1:14Þ

 j , then the Case 2: If wj ðj ¼ 1; 2; . . .; nÞ are intervals, where wj ¼ ½wj ; w weighted average operator Y is calculated as follows: Pn  j Gj j¼1 ½wj ; w Y ¼ Pn  j j¼1 ½wj ; w " Pn # Pn  j  j j¼1 ½aj ; bj ½wj ; w j¼1 ½1  dj ; 1  cj ½wj ; w Pn Pn ¼ ;  j  j j¼1 ½wj ; w j¼1 ½wj ; w h i D E ¼ ½y1 ; y1 ; ½y2 ; y2  ¼ ½y1 ; y1 ; ½1  y2 ; 1  y2  ;

ð1:15Þ

where y1 , y1 , y2 and y2 are calculated by the Karnik-Mendel algorithms [8], respectively.   Case 3: If wj ðj ¼ 1; 2; . . .; nÞ are IFNs, where wj ¼ xj ; qj ¼ ½xj ; 1  qj , satisfying that 0  xj  1, 0  xj þ qj  1 and 0  qj  1, then the weighted average operator Y is calculated as follows: Pn j¼1 Y ¼ Pn

½xj ; 1  qj Gj

j¼1

" Pn

½xj ; 1  qj 

# Pn ½aj ; bj ½xj ; 1  qj  j¼1 ½1  dj ; 1  cj ½xj ; 1  qj  Pn Pn ¼ ; j¼1 ½xj ; 1  qj  j¼1 ½xj ; 1  qj  h i D E ¼ ½y1 ; y1 ; ½y2 ; y2  ¼ ½y1 ; y1 ; ½1  y2 ; 1  y2  ; j¼1

ð1:16Þ

1.3 Ordered Weighted Average Operator and Hybrid …

15

where y1 , y1 , y2 and y2 are calculated bythe Karnik-Mendel algorithms [8], respectively. D E  j ; ½qj ; q j  ¼ Case 4: If wj ðj ¼ 1; 2; . . .; nÞ are IVIFNs, where wj ¼ ½xj ; x h i  j ; ½1  q  j  1, 0  qj  q j ; 1  qj  , satisfying that 0  xj  x j  1 and ½xj ; x j þ q j  1, then the weighted average operator Y is calculated as follows: 0x Pn h

i  j ; ½1  q j ; 1  qj  Gj ½xj ; x i Y¼ P h n  j ; ½1  q j ; 1  qj  j¼1 ½xj ; x Pn " Pn # j ; 1  qj   j j¼1 ½1  dj ; 1  cj ½1  q j¼1 ½aj ; bj ½xj ; x Pn Pn ¼ ;  j j ; 1  qj  j¼1 ½xj ; x j¼1 ½1  q h i D E ¼ ½y1 ; y1 ; ½y2 ; y2  ¼ ½y1 ; y1 ; ½1  y2 ; 1  y2  ; j¼1

ð1:17Þ

where y1 , y1 ,y2 and y2 are calculated by the Karnik-Mendel algorithms [8], respectively.

1.3.2

Proposed Ordered Weighted Average Operator for IVIFNs

Definition 1.4 Let Gi ði ¼ 1; 2; . . .; nÞ be a collection of the IVIFNs, where Gi ¼ ð½ai ; bi ; ½ci ; di Þ ¼ ½½ai ; bi ; ½1  di ; 1  ci . If Pn Uw ðG1 ; G2 ; . . .; Gn Þ ¼

j¼1 P n

wj GrðjÞ

j¼1

wj

;

ð1:18Þ

where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weighted vector which correlates with U, ðrð1Þ; rð2Þ; . . .; rðnÞÞ is a permutation of ð1; 2; . . .; nÞ such that Grðj1Þ  GrðjÞ for any j, then the function U is called the ordered weighted average operator for the IVIFNs. 

 Denote GrðjÞ ¼ ½arðjÞ ; brðjÞ ; ½crðjÞ ; drðjÞ  ¼ ½arðjÞ ; brðjÞ ; ½1  drðjÞ ; 1  crðjÞ  . The weighted vector w ¼ ðw1 ; w2 ; . . .; wn ÞT may take different forms as follows: Case 1: If wj ðj ¼ 1; 2; . . .; nÞ are crisp values, then the ordered weighted average operator U is calculated as follows:

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

16

Pn j¼1 wj GrðjÞ U ¼ Pn j¼1 wj ""Pn # "Pn ## Pn Pn j¼1 wj arðjÞ j¼1 wj brðjÞ j¼1 wj ð1  drðjÞ Þ j¼1 wj ð1  crðjÞ Þ Pn Pn Pn ¼ ; Pn ; ; j¼1 wj j¼1 wj j¼1 wj j¼1 wj ¼ ½½u1 ; u1 ; ½u2 ; u2  ¼ ð½u1 ; u1 ; ½1  u2 ; 1  u2 Þ: ð1:19Þ  j , then the ordered Case 2: If wj ðj ¼ 1; 2; . . .; nÞ are intervals, where wj ¼ ½wj ; w weighted average operator U is calculated as follows: Pn  j GrðjÞ j¼1 ½wj ; w U ¼ Pn  j j¼1 ½wj ; w " Pn # Pn  j  j j¼1 ½arðjÞ ; brðjÞ ½wj ; w j¼1 ½1  drðjÞ ; 1  crðjÞ ½wj ; w Pn Pn ¼ ;  j  j j¼1 ½wj ; w j¼1 ½wj ; w

ð1:20Þ

¼ ½½u1 ; u1 ; ½u2 ; u2  ¼ ð½u1 ; u1 ; ½1  u2 ; 1  u2 Þ; where u1 , u1 , u2 and u2 are calculated by the Karnik-Mendel algorithms [8], respectively.   Case 3: If wj ðj ¼ 1; 2; . . .; nÞ are IFNs, where wj ¼ xj ; qj ¼ ½xj ; 1  qj , satisfying that 0  xj  1, 0  xj þ qj  1 and 0  qj  1, then the ordered weighted average operator U is calculated as follows: Pn U¼

" Pn ¼

½xj ; 1  qj GrðjÞ j¼1 ½xj ; 1  qj 

j¼1 P n

j¼1

½arðjÞ ; brðjÞ ½xj ; 1  qj  Pn ; j¼1 ½xj ; 1  qj 

Pn j¼1

½1  drðjÞ ; 1  crðjÞ ½xj ; 1  qj  Pn j¼1 ½xj ; 1  qj 

#

¼ ½½u1 ; u1 ; ½u2 ; u2  ¼ ð½u1 ; u1 ; ½1  u2 ; 1  u2 Þ; ð1:21Þ where u1 , u1 , u2 and u2 are calculated by the Karnik-Mendel algorithms [8], respectively.  j ; ½qj ; q j i ¼ Case 4: If wj ðj ¼ 1; 2; . . .; nÞ are IVIFNs, where wj ¼ h½xj ; x h i  j ; ½1  q  j  1, 0  qj  q j ; 1  qj  , satisfying that 0  xj  x j  1 and ½xj ; x j þ q j  1, then the ordered weighted average operator U is calculated as 0x follows:

1.3 Ordered Weighted Average Operator and Hybrid …

17

i Pn h   ½x ; x ; ½1  q ; 1  q  GrðjÞ j j j j¼1 j i U¼ P h n   ½x ; x ; ½1  q ; 1  q  j j j j¼1 j Pn " Pn # j ; 1  qj   j j¼1 ½1  drðjÞ ; 1  crðjÞ ½1  q j¼1 ½arðjÞ ; brðjÞ ½xj ; x Pn Pn ¼ ;  j j ; 1  qj  j¼1 ½xj ; x j¼1 ½1  q ¼ ½½u1 ; u1 ; ½u2 ; u2  ¼ ð½u1 ; u1 ; ½1  u2 ; 1  u2 Þ; ð1:22Þ It is easily seen from Definition 1.4 that, the ordered weighted average operator U first reorders the arguments in descending order and then weights these ordered arguments. The ordered weighted average operator U considers only the ordered positions of arguments.

1.3.3

Proposed Hybrid Weighted Average Operator for IVIFNs

Definition 1.5 Let Gi ði ¼ 1; 2; . . .; nÞ be a collection of the IVIFNs, where Gi ¼ ð½ai ; bi ; ½ci ; di Þ ¼ ½½ai ; bi ; ½1  di ; 1  ci . If Pn Zw;v ðG1 ; G2 ; . . .; Gn Þ ¼

j¼1

Pn

wj G0rðjÞ

j¼1

wj

;

ð1:23Þ

where w ¼ ðw1 ; w2 ; . . .; wn ÞT is be the weighted vector which correlates with Z, G0rðjÞ is the jth largest number of IVIFNs G0k ðk ¼ 1; 2; . . .; nÞ, here G0k ¼ nvk Gk , v ¼ ðv1 ; v2 ; . . .; vn ÞT is the weighting vector of Gi ði ¼ 1; 2; . . .; nÞ, satisfying that n P 0  vj  1 ðj ¼ 1; 2; . . .; nÞ and vj ¼ 1, n is the balancing coefficient, then the j¼1

function Z is called the hybrid weighted average operator for IVIFNs. Especially, if vj ¼ 1n ðj ¼ 1; . . .; nÞ, then the hybrid weighted average operator Z is reduced to the ordered weighted average operator U for IVIFNs. Using the operation laws of IVIFNs [31, 43], we have  nvk k G0k ¼ nvk Gk ¼ ½1  ð1  ak Þnvk ; 1  ð1  bk Þnvk ; ½cnv k ; dk 

 k ¼ ½1  ð1  ak Þnvk ; 1  ð1  bk Þnvk ; ½1  dknvk ; 1  cnv k  : Simply denote G0k by

ð1:24Þ

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

18



 G0k ¼ ½a0k ; b0k ; ½c0k ; dk0  ¼ ½a0k ; b0k ; ½1  dk0 ; 1  c0k  ; nvk 0 k where a0k ¼ 1  ð1  ak Þnvk , b0k ¼ 1  ð1  bk Þnvk , c0k ¼ cnv k , dk ¼ dk . 0 Similarly, denote GrðjÞ by

h i 0 0  ¼ ½a0rðjÞ ; b0rðjÞ ; ½1  drðjÞ ; 1  c0rðjÞ  : G0rðjÞ ¼ ½a0rðjÞ ; b0rðjÞ ; ½c0rðjÞ ; drðjÞ

ð1:25Þ

ð1:26Þ

The weighted vector w ¼ ðw1 ; w2 ; . . .; wn ÞT may take different forms as follows: Case 1: If wj ðj ¼ 1; 2; . . .; nÞ are crisp values, then the hybrid weighted average operator Z is calculated as follows: Pn Z¼

j¼1

Pn

wj G0rðjÞ

j¼1

""Pn

wj

0 j¼1 wj arðjÞ Pn j¼1 wj

Pn

0 j¼1 wj brðjÞ Pn j¼1 wj

# " Pn

0 wj ð1  drðjÞ Þ Pn ¼ ; ; ; j¼1 wj

  ¼ ½z1 ; z1 ; ½z2 ; z2  ¼ ½z1 ; z1 ; ½1  z2 ; 1  z2  : j¼1

Pn

wj ð1  c0rðjÞ Þ Pn j¼1 wj

##

j¼1

ð1:27Þ  j , then the hybrid Case 2: If wj ðj ¼ 1; 2; . . .; nÞ are intervals, where wj ¼ ½wj ; w weighted average operator Z is calculated as follows: Pn j¼1

Pn

Z¼

" Pn

 j G0rðjÞ ½wj ; w

j¼1

 j ½wj ; w

# Pn 0 0  j  j ½a0rðjÞ ; b0rðjÞ ½wj ; w j¼1 ½1  drðjÞ ; 1  crðjÞ ½wj ; w Pn Pn ¼ ;  j  j j¼1 ½wj ; w j¼1 ½wj ; w

  ¼ ½z1 ; z1 ; ½z2 ; z2  ¼ ½z1 ; z1 ; ½1  z2 ; 1  z2  ; j¼1

ð1:28Þ

where z1 , z1 , z2 and z2 are calculated by the Karnik-Mendel algorithms [8], respectively. Case 3: If wj ðj ¼ 1; 2; . . .; nÞ are IFNs, where wj ¼ hxj ; qj i ¼ ½xj ; 1  qj , satisfying that 0  xj  1, 0  xj þ qj  1 and 0  qj  1, then the hybrid weighted average operator Z is calculated as follows: Pn Z¼

j¼1

Pn "Pn

½xj ; 1  qj G0rðjÞ

j¼1

½xj ; 1  qj 

# Pn 0 0 ½a0rðjÞ ; b0rðjÞ ½xj ; 1  qj  ð1:29Þ j¼1 ½1  drðjÞ ; 1  crðjÞ ½xj ; 1  qj  Pn Pn ¼ ; j¼1 ½xj ; 1  qj  j¼1 ½xj ; 1  qj 

  ¼ ½z1 ; z1 ; ½z2 ; z2  ¼ ½z1 ; z1 ; ½1  z2 ; 1  z2  ; j¼1

1.3 Ordered Weighted Average Operator and Hybrid …

19

where z1 , z1 , z2 and z2 are calculated by the Karnik-Mendel algorithms [8], respectively.

 j ; ½qj ; q j  ¼ Case 4: If wj ðj ¼ 1; 2; . . .; nÞ are IVIFNs, where wj ¼ ½xj ; x h i  j ; ½1  q  j  1, 0  qj  q j ; 1  qj  , satisfying that 0  xj  x j  1 and ½xj ; x j þ q j  1, then the hybrid weighted average operator Z is calculated as 0x follows: Pn h

i   ½x ; x ; ½1  q ; 1  q  G0rðjÞ j j j j¼1 j h i Z¼ P n   ½x ; x ; ½1  q ; 1  q  j j j j¼1 j Pn "Pn # 0 0 0 0 j ; 1  qj  ð1:30Þ  j j¼1 ½1  drðjÞ ; 1  crðjÞ ½1  q j¼1 ½arðjÞ ; brðjÞ ½xj ; x Pn Pn ¼ ;  j j ; 1  qj  j¼1 ½xj ; x j¼1 ½1  q

  ¼ ½z1 ; z1 ; ½z2 ; z2  ¼ ½z1 ; z1 ; ½1  z2 ; 1  z2  ; where z1 , z1 , z2 and z2 are calculated by the Karnik-Mendel algorithms [8], respectively. From Definition 1.5, we know that, the hybrid weighted average operator Z first weights the given arguments, and then reorders the weighted arguments in descending order and weights these ordered arguments, and finally aggregates all the weighted arguments into a collective one. The hybrid weighted average operator Z reflects the important degrees of both the arguments themselves and the ordered positions of the arguments simultaneously. It generalizes the ordered weighted average operator U of IVIFNs. Remark 1.1 How to objectively determine the weighted vectors correlated with the ordered weighted average operator U and the hybrid weighted average operator Z (i.e., w ¼ ðw1 ; w2 ; . . .; wn ÞT ) is a critical issue. For the real number form of w ¼ ðw1 ; w2 ; . . .; wn ÞT , it can be determined by the method of fuzzy linguistic quantifier [51] or by the normal weighting method [30] (see the review literature [30] in detail). For the other forms of w ¼ ðw1 ; w2 ; . . .; wn ÞT , such as intervals, IFNs, and IVIFNs, we will investigate their determination methods in future.

1.4 1.4.1

MAGDM Problem and Method with IVIFS Problem Description for MAGDM with IVIFS

Suppose there exists m non-inferior decision making alternatives which make up the alternative set A ¼ fA1 ; A2 ; . . .; Am g. Each alternative is assessed on n attributes fa1 ; a2 ; . . .; an g. The weight vector of attributes is v ¼ ðv1 ; v2 ; . . .; vn ÞT . Let fp1 ; p2 ; . . .; pk g be the set of DMs (or experts). The weight vector of DMs is

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

20

denoted by z ¼ ðz1 ; z2 ; . . .; zk ÞT which is to be determined. Assume that GtTij ¼ ½atij ; btij  and GtFij ¼ ½ctij ; dijt  are respectively the membership degree and non-membership degree of alternative Ai 2 A on attribute aj given by DM pt to the fuzzy concept “excellent”. In other words, the evaluation of Ai on aj given by pt is an IVIFN as follows: Gtij ¼ ðGtTij ; GtFij Þ;

ð1:31Þ

where ½atij ; btij ½0; 1; ½ctij ; dijt ½0; 1 and btij þ dijt  1ð1  i  m; 1  j  n; 1  t  kÞ. The MAGDM problem considered in this chapter is how to choose the best alternative from the alternative set A.

1.4.2

Determination of the Weights of DMs

In real-life MAGDM problems, the importance of DMs should be taken into consideration since different DMs play different roles during the process of decision making. How to objectively determine the weights of DMs is a critical issue for the MAGDM. In the following, an approach is investigated to determine the weights of DMs. All the attribute values of alternative Ai given by pt are the IVIFNs Gti1 ; Gti2 ; . . .; Gtin . Integrating these IVIFNs by the interval-valued intuitionistic fuzzy weighted average Y (i.e., Eq. (1.13)), the individual overall attribute value of Ai given by pt is obtained as follows:  Eit ¼ ½ati ; bti ; ½cti ; dit  ¼ Yv ðGti1 ; Gti2 ; . . .; Gtin Þ;

ð1:32Þ

where v ¼ ðv1 ; v2 ; . . .; vn ÞT is the weight vector of attributes. Therefore, for DMs pt and pu , we can obtain their evaluation vectors of all alternatives, ðE1t ; E2t ; . . .; Emt Þ and ðE1u ; E2u ; . . .; Emu Þ, respectively. The similarity degree between evaluation vectors given by pt and pu can be defined as follows: stu ¼ 1 

m 1 X ðjati  aui j þ jbti  bui j þ jcti  cui j þ jdit  diu jÞ; 4m i¼1

ð1:33Þ

then the similarity matrix for the decision group is constructed as follows: S ¼ ðstu Þkk :

ð1:34Þ

Obviously, the higher the similarity degree between pt and other DMs, the bigger the support degree of pt by other DMs. Hence, the greater the weight zt of pt in the decision making. The weight zt can be derived through the similarity matrix.

1.4 MAGDM Problem and Method with IVIFS

21

P Assume that zt ¼ ku¼1 xu stu . Let z ¼ ðz1 ; z2 ; . . .; zk ÞT and X ¼ ðx1 ; x2 ; . . .; xk ÞT be two column vectors of n dimensions. Then, one has z ¼ SX:

ð1:35Þ

Since S is a non-negative symmetric matrix, by the Perron-Frobenius theorem [7], matrix S exists the maximum module eigenvalue k [ 0, and the corresponding eigenvector X ¼ ðx1 ; x2 ; . . .; xk ÞT satisfies that xt [ 0 ðt ¼ 1; 2; . . .; kÞ and kX ¼ SX. From Eq. (1.35), we get kX ¼ SX ¼ z. Therefore, zt ¼ kxt , in Eq. (1.35) is exact the eigenvector of maximum module eigenvalue k. Normalized zt ðt ¼ 1; 2; . . .; kÞ, the weight of pt (still denoted by zt ) is obtained as follows: zt ¼ xt =ðx1 þ x2 þ


þ xk Þ ðt ¼ 1; 2; . . .; kÞ:

ð1:36Þ

For the individual overall attribute values Ei1 ; Ei2 ; . . .; Eik of alternatives Ai , substituting the weight vector v ¼ ðv1 ; v2 ; . . .; vn ÞT of Eq. (1.23) by the weight vector of DMs z ¼ ðz1 ; z2 ; . . .; zk ÞT , and combining the hybrid weighted average operator Z (i.e., Eq. (1.23)), we can be obtain the collective overall attribute value of Ai as follows: Ei ¼ ð½ai ; bi ; ½ci ; di Þ ¼ Zw;z ðEi1 ; Ei2 ; . . .; Eik Þ;

ð1:37Þ

where w ¼ ðw1 ; w2 ; . . .; wk ÞT is the weight vector correlated with the hybrid weighted average operator Z. Remark 1.2 According to the Karnik-Mendel  t t t t ½ai ; bi ; ½ci ; di  of Eq. (1.32) satisfies that

algorithms

[8],

Eit ¼

min fatij g  ati  max fatij g; min fbtij g  bti  max fbtij g

1jn

1jn

1jn

1jn

and min fctij g  cti  max fctij g; min fdijt g  dit  max fdijt g:

1jn

1jn

1jn

1jn

Since ½atij ; btij ½0; 1; ½ctij ; dijt ½0; 1, we have ½ati ; bti ½0; 1 and ½cti ; dit ½0; 1. Thus, Eq. (1.33) can assure 0  stu  1. In some real decision problems, a minority may have more confident judgements, they may assign unduly high or unduly low uncertain preference values to their preferred or repugnant objects. But by using Eqs. (1.32) and (1.33), the similarity degree stu always exists in the closed interval ½0; 1, which can well relieve the influence of these “biased” preference values on the weights of DMs.

22

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

Certainly, if there exist some extreme preference values given by some DMs such that stu is very close to 1 or 0, we can set the thresholds to redefine the similarity degree s0tu as follows: 8 < n1 ; if stu  n1 ; s0tu ¼ stu ; if n1  stu  n2 ; : n2 ; if stu  n2 ; where n1 and n2 are the thresholds (e.g., n1 ¼ 0:1 and n2 ¼ 0:9), which can efficiently ensure the consensus and consistency between DMs and avoid the conflict by heterogeneity among DMs and the possibility that a minority may have more confident judgments.

1.4.3

Group Decision Making Method

In the following we shall utilize the interval-valued intuitionistic fuzzy weighted average operator Y (i.e. Eq. 1.13) and the hybrid weighted average operator Z (i.e. Eq. 1.23) to propose a new MAGDM method with IVIFN information. The detailed steps are summarized as follows: Step 1. DMs use IVIFNs to represent the evaluation information of alternatives; Step 2. Calculate the individual overall attribute value of each alternative by Eq. (1.32); Step 3. Obtain the similarity matrix according to Eq. (1.34); Step 4. Derive the weight vector of DMs from Eq. (1.36); Step 5. Compute the collective overall attribute value of each alternative in terms of Eq. (1.37); Step 6. By the possibility degree ranking method for IVIFNs in Sect. 1.2.2, the ranking order of alternatives can be generated according to the collective overall attribute values.

1.5

An Air-Condition System Selection Example and Comparison Analysis

This section analyzes an air-condition system selection example to illustrate the effectiveness of the proposed method of this chapter. Then, the comparison analyses are conducted to demonstrate the advantages of the proposed method of this chapter.

1.5 An Air-Condition System Selection Example …

1.5.1

23

An Air-Condition System Selection Problem and the Analysis Process

Consider an air-condition system selection problem. Suppose there exist three air-condition systems fA1 ; A2 ; A3 g, four attributes a1 (economical), a2 (function) a3 (being operative) and a4 (longevity) are taken into consideration in the selection problem. Three experts (DMs) fp1 ; p2 ; p3 g participate in the decision making. Using the statistical methods, the membership degrees and non-membership degrees for the alternative Ai on the attribute aj given by expert pt to the fuzzy concept “excellent” can be obtained (i.e., the evaluation of Ai on aj given by pt is an IVIFN), listed in Tables 1.1, 1.2, 1.3. In the following, we will illustrate the decision making process according to the different forms of the weight vector of attributes v ¼ ðv1 ; v2 ; . . .; vn ÞT , respectively. (a) The weight vector of attributes is given in the form of real numbers Suppose that v ¼ ð0:35; 0:28; 0:46; 0:55ÞT . Then, the individual overall attribute value of each alternative can be calculated by Eqs. (1.14) and (1.32) as in Table 1.4.

Table 1.1 IVIFNs given by the expert p1 Attribute

A1

a1 a2 a3 a4

([0.4, ([0.3, ([0.2, ([0.3,

A2 0.8], 0.6], 0.7], 0.4],

[0.0, [0.0, [0.2, [0.4,

0.1]) 0.2]) 0.3]) 0.5])

([0.5, ([0.3, ([0.4, ([0.1,

A3 0.7], 0.5], 0.7], 0.2],

[0.1, [0.2, [0.0, [0.7,

0.2]) 0.4]) 0.2]) 0.8])

([0.5, ([0.6, ([0.4, ([0.6,

0.8], 0.6], 0.8], 0.6],

[0.1, [0.2, [0.0, [0.3,

0.2]) 0.3]) 0.2]) 0.4])

([0.5, ([0.6, ([0.4, ([0.2,

0.8], 0.6], 0.8], 0.5],

[0.1, [0.1, [0.0, [0.2,

0.2]) 0.3]) 0.2]) 0.3])

([0.2, ([0.2, ([0.3, ([0.4,

0.7], 0.8], 0.7], 0.8],

[0.2, [0.1, [0.1, [0.0,

0.3]) 0.2]) 0.2]) 0.2])

0.6], 0.7], 0.8], 0.6],

[0.1, [0.1, [0.1, [0.2,

0.4]) 0.2]) 0.2]) 0.3])

0.6], 0.6], 0.6], 0.7],

[0.1, [0.2, [0.1, [0.1,

0.2]) 0.3]) 0.3]) 0.2])

Table 1.2 IVIFNs given by the expert p2 Attribute

A1

a1 a2 a3 a4

([0.5, ([0.4, ([0.5, ([0.4,

A2 0.9], 0.5], 0.8], 0.7],

[0.0, [0.3, [0.0, [0.1,

0.1]) 0.5]) 0.1]) 0.2])

([0.7, ([0.5, ([0.5, ([0.5,

A3

Table 1.3 IVIFNs given by the expert p3 Attribute

A1

a1 a2 a3 a4

([0.3, ([0.2, ([0.4, ([0.3,

A2 0.9], 0.5], 0.7], 0.6],

[0.0, [0.1, [0.1, [0.3,

0.1]) 0.4]) 0.2]) 0.4])

([0.3, ([0.5, ([0.2, ([0.3,

A3

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

24

Table 1.4 Individual overall attribute values of the alternatives for crisp weight vector of attributes Eit

p1

p2

p3

A1

([0.2933, 0.6037]; [0.1902, 0.3073]) ([0.3037, 0.4982]; [0.2902, 0.4354]) ([0.5226, 0.7506]; [0.0878, 0.2213])

([0.4494, 0.7366]; [0.0848, 0.2018]) ([0.5427, 0.6988]; [0.1561, 0.2841]) ([0.3884, 0.6732]; [0.1335, 0.2762])

([0.3110, 0.6750]; [0.1457, 0.2799]) ([0.3061, 0.6652]; [0.1055, 0.2506]) ([0.2951, 0.6335]; [0.1171, 0.2451])

A2 A3

The similarity matrix for the decision group is constructed by Eq. (1.34) as follows: 0

1 0:8719 S ¼ @ 0:8719 1 0:9085 0:9272

1 0:9085 0:9272 A: 1

Because the maximum module eigenvalue of S is 2.8053, the corresponding eigenvector is X ¼ ð0:5721; 0:5761; 0:5838ÞT , the experts’ weights are obtained from Eq. (1.36) as follows: z1 ¼ 0:3303; z2 ¼ 0:3326; z3 ¼ 0:3371: By using the normal weighting method [30], the correlated weighted vector with the hybrid weighted average operator Z is obtained as w ¼ ð0:243; 0:514; 0:243ÞT . The collective overall attribute values of alternatives are respectively calculated by Eqs. (1.27) and (1.37) as follows: E1 ¼ ð½0:3378; 0:6693; ½0:1442; 0:2708Þ; E2 ¼ ð½0:3604; 0:6295; ½0:1653; 0:3069Þ and E3 ¼ ð½0:3956; 0:6790; ½0:1207; 0:2585Þ: In terms of Eq. (1.8), the matrix of possibility degree for IVIFNs E1, E2 and E3 is obtained as follows: 0

0:5 P ¼ @ 0:3922 0:6134

0:6078 0:5 0:7094

1 0:3866 0:2906 A: 0:5

By Eq. (1.9), the ranking vector is calculated as follows:

1.5 An Air-Condition System Selection Example …

25

x1 ¼ 0:3324; x2 ¼ 0:2805; x3 ¼ 0:3871: Hence, the ranking order of the alternatives is A3 0:6134 A1 0:6078 A2 , which indicates that the possibility degree that A3 is superior to A1 is 0.6134, the possibility degree that A1 is superior to A2 is 0.6078, and the best air-condition system is A3 . (b) The weight vector of attributes is given in the form of intervals With ever increasing complexity in many real decision situations, there are often some challenges for the DM to provide precise and complete preference information due to time pressure, lack of knowledge (or data) and the DM’s limited expertise about the problem domain. Sometimes the DM only can give the approximate scope about the attribute weights, i.e., use the intervals to represent the attribute weights. Suppose that the interval weight vector of attributes is v ¼ ð½0:12; 0:35; ½0:22; 0:30; ½0:18; 0:46; ½0:35; 0:53ÞT . Then, the individual overall attribute value of each alternative can be calculated by Eqs. (1.32) and (1.15) as in Table 1.5. The similarity matrix for the decision group is constructed by Eq. (1.29) as follows: 0

1 0:8893 S ¼ @ 0:8893 1 0:9093 0:9455

1 0:9093 0:9455 A: 1

Because the maximum module eigenvalue of S is 2.8296, the corresponding eigenvector is X ¼ ð0:5708; 0:5785; 0:5827ÞT , the experts’ weights are obtained from Eq. (1.36) as follows: z1 ¼ 0:3296; z2 ¼ 0:3340; z3 ¼ 0:3364: Combining with the correlated weighted vector w ¼ ð0:243; 0:514; 0:243ÞT , the collective overall attribute values of alternatives are respectively calculated by Eqs. (1.37) and (1.27) as follows:

Table 1.5 Individual overall attribute values of the alternatives for interval weight vector of attributes Eit

p1

p2

p3

A1

([0.2704, 0.6333]; [0.0732, 0.2632]) ([0.2390, 0.5413]; [0.2329, 0.4478]) ([0.5080, 0.7589]; [0.0692, 0.2318])

([0.4213, 0.7432]; [0.0754, 0.2215]) ([0.5134, 0.6789]; [0.1267, 0.2543]) ([0.3765, 0.6422]; [0.1119, 0.2532])

([0.3089, 0.6664]; [0.1328, 0.2442]) ([0.3123, 0.6543]; [0.1111, 0.2637]) ([0.2897, 0.6130]; [0.1275, 0.2557])

A2 A3

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

26

E1 ¼ ð½0:3239; 0:6729; ½0:1070; 0:2472Þ; E2 ¼ ð½0:3403; 0:6287; ½0:1476; 0:3102Þ and E3 ¼ ð½0:3840; 0:6593; ½0:1080; 0:2526Þ: In terms of Eq. (1.8), the matrix of possibility degree for IVIFNs E1 , E2 and E3 is obtained as follows: 0

0:5 P ¼ @ 0:3389 0:5223

0:6611 0:5 0:6606

1 0:4777 0:3394 A: 0:5

By Eq. (1.9), the ranking vector is calculated as follows: x1 ¼ 0:3565; x2 ¼ 0:2797; x3 ¼ 0:3638: Hence, the ranking order of the alternatives is A3 0:5233 A1 0:6611 A2 , the best air-condition system is A3 . (c) Weight vector of attributes is given in the form of IFNs Under many conditions, numeric values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information are vague or fuzzy in nature and as such it may not be appropriate to represent them by accurate numeric values. Due to time pressure, lack of knowledge (or data) and the DM’s limited expertise about the problem domain, the DM usually exists some hesitation during the assessment process. IFS seems to be well suited for expressing hesitation of the decision makers. Therefore, the attribute weights may be represented by IFNs, i.e., use the IFNs to represent the attribute weights. Suppose that the intuitionistic fuzzy weight vector of attributes is v ¼ ðh0:12; 0:35i; h0:21; 0:45i; h0:52; 0:31i; h0:48; 0:37i; h0:53; 0:29iÞT , which can be equivalently represented as follows: v1 ¼ h0:12; 0:35i ¼ ½0:12; 0:65; v2 ¼ h0:21; 0:45i ¼ ½0:21; 0:55; v3 ¼ h0:52; 0:31i ¼ ½0:52; 0:69; v4 ¼ h0:53; 0:29i ¼ ½0:53; 0:71: Then, the individual overall attribute value of each alternative can be calculated by Eqs. (1.32) and (1.16) as in Table 1.6. The similarity matrix for the decision group is constructed by Eq. (1.29) as follows:

1.5 An Air-Condition System Selection Example …

27

Table 1.6 Individual overall attribute values of the alternatives for intuitionistic fuzzy weight vector of attributes Eit

p1

p2

p3

A1

([0.2687, 0.6477]; [0.0788, 0.2733]) ([0.2432, 0.5397]; [0.2142, 0.4132]) ([0.5123, 0.7411]; [0.0556, 0.2014])

([0.3876, 0.7566]; [0.0865, 0.2046]) ([0.6003, 0.6702]; [0.1341, 0.2610]) ([0.3543, 0.6243]; [0.1012, 0.2341])

([0.3066, 0.6664]; [0.1296, 0.2221]) ([0.3247, 0.6324]; [0.1024, 0.2573]) ([0.2773, 0.6143]; [0.1174, 0.2378])

A2 A3

0

1 S ¼ @ 0:9611 0:9402

0:9611 1 0:9586

1 0:9402 0:9586 A: 1

Because the maximum module eigenvalue of S is 2.9066, the corresponding eigenvector is X ¼ ð0:5763; 0:5800; 0:5758ÞT , the experts’ weights are obtained from Eq. (1.36) as follows: z1 ¼ 0:3327; z2 ¼ 0:3348; z3 ¼ 0:3324: Combined with the correlated weighted vector w ¼ ð0:243; 0:514; 0:243ÞT , the collective overall attribute values of alternatives are respectively calculated by Eqs. (1.37) and (1.27) as follows: E1 ¼ ð½0:3865; 0:6684; ½0:0196; 0:0550Þ; E2 ¼ ð½0:5139; 0:7176; ½0:0199; 0:0517Þ; E3 ¼ ð½0:2695; 0:6931; ½0:0203; 0:0627Þ: In terms of Eq. (1.8), the matrix of possibility degree for IVIFNs E1 , E2 and E3 is obtained as follows: 0

0:5 P ¼ @ 0:6673 0:3961

0:3327 0:5 0:2759

1 0:6039 0:7241 A; 0:5

By Eq. (1.9), the ranking vector is calculated as follows: x ¼ ð0:3228; 0:3986; 0:2787ÞT : Therefore, the ranking is A2 0:6673 A1 0:6039 A3 , the best is alternative A2 .

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

28

(d) Weight vector of attributes is given in the form of IVIFNs Since the IVIFN utilizes the intervals to represent the membership degree and the non- membership degree, the ability of an IVIFN for capturing vagueness and uncertainty is significantly stronger than that of a IFN. Therefore, the attribute weights may be represented by IVIFNs. Suppose that the interval-valued intuitionistic fuzzy weight vector of attributes is v ¼ ðh½0:10; 0:15; ½0:28; 0:35i; h½0:18; 0:25; ½0:28; 0:47i; h½0:51; 0:62; ½0:19; 0:36i; h½0:62; 0:73; ½0:14; 0:21iÞT , which can be equivalently represented as follows: v1 ¼ h½0:10; 0:15; ½0:28; 0:35i ¼ ½½0:10; 0:15; ½0:65; 0:72; v2 ¼ h½0:18; 0:25; ½0:28; 0:47i ¼ ½½0:18; 0:25; ½0:53; 0:72; v3 ¼ h½0:51; 0:62; ½0:19; 0:36i ¼ ½½0:51; 0:62; ½0:64; 0:81; v4 ¼ h½0:62; 0:73; ½0:14; 0:21i ¼ ½½0:62; 0:73; ½0:79; 0:86: Then, the individual overall attribute value of each alternative can be calculated by Eqs. (1.32) and (1.17) as in Table 1.7. The similarity matrix for the decision group is constructed by Eq. (1.29) as follows: 0 1 1 0:7256 0:9123 S ¼ @ 0:7256 1 0:9740 A: 0:9123 0:9740 1 Because the maximum module eigenvalue of S is 2.7454, the corresponding eigenvector is X ¼ ð0:5541; 0:5693; 0:6073ÞT , the experts’ weights are obtained from Eq. (1.36) as follows: z1 ¼ 0:3202; z2 ¼ 0:3289; z3 ¼ 0:3509: Combined with the correlated weighted vector w ¼ ð0:243; 0:514; 0:243ÞT , the collective overall attribute values of alternatives are respectively calculated by Eqs. (1.37) and (1.27) as follows:

Table 1.7 Individual overall attribute values of the alternatives for interval-valued intuitionistic fuzzy weight vector of attributes Eit

p1

p2

p3

A1

([0.2687, 0.6477]; [0.0788, 0.2733]) ([0.2432, 0.5397]; [0.2142, 0.4132]) ([0.5123, 0.7411]; [0.0556, 0.2014])

([0.3876, 0.7566]; [0.0865, 0.2046]) ([0.6003, 0.6702]; [0.1341, 0.2610]) ([0.3543, 0.6243]; [0.1012, 0.2341])

([0.3066, 0.6664]; [0.1296, 0.2221]) ([0.3247, 0.6324]; [0.1024, 0.2573]) ([0.2773, 0.6143]; [0.1174, 0.2378])

A2 A3

1.5 An Air-Condition System Selection Example …

29

E1 ¼ ð½0:3042; 0:6675; ½0:1073; 0:2382Þ; E2 ¼ ð½0:3307; 0:6375; ½0:1555; 0:3214Þ and E3 ¼ ð½0:3897; 0:6483; ½0:1087; 0:2537Þ: In terms of Eq. (1.8), the matrix of possibility degree for IVIFNs E1 , E2 and E3 is obtained as follows: 0

0:5 P ¼ @ 0:3034 0:5165

0:6966 0:5 0:6894

1 0:4835 0:3106 A: 0:5

By Eq. (1.9), the ranking vector is calculated as follows: x1 ¼ 0:3634; x2 ¼ 0:2690; x3 ¼ 0:3677: Hence, the ranking order of the alternatives is A3 0:5165 A1 0:6966 A2 , the best air-condition system is A3 . The above example analysis shows that the obtained decision results may be different with the different forms of attribute weights. For the real number, interval and IVIFN forms of attribute weights, the ranking orders of the alternatives are A3 0:6134 A1 0:6078 A2 , A3 0:5233 A1 0:6611 A2 and A3 0:5165 A1 0:6966 A2 , respectively. Though the ranking orders are the same (which is only a coincidence), the possibility degree between the adjacent alternatives are remarkably different. For the IFN forms of attribute weights, the ranking order of the alternatives is A2 0:6673 A1 0:6039 A3 , which is not in accordance with that obtained for the first three cases. In the above example analysis, we only consider the weighted vector correlated with the hybrid weighted average operator Z in the form of real numbers. Combined with Eqs. (1.28)–(1.30), the other forms of the correlated weighted weights, such as intervals, IFNs and IVIFNs, also can respectively be used to solve this example in a similar way to the above computational process.

1.5.2

Comparison Analysis of the Obtained Results

To further illustrate the superiority of the proposed method, we use the method proposed in this chapter to solve the example of Wei [27], and then conduct a comparison analysis.

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

30

Suppose that five possible alternatives A1 ; A2 ; . . .; A5 are to be evaluated using the IVIFNs by the three DMs under four attributes (whose weighting vector v ¼ ð0:2; 0:1; 0:3; 0:4ÞT ). The DMs construct the decision matrices as follows [27]: 0

h½0:3; 0:4; ½0:4; 0:5i

B B h½0:3; 0:6; ½0:3; 0:4i B ~ R1 ¼ B B h½0:2; 0:5; ½0:4; 0:5i B @ h½0:4; 0:5; ½0:3; 0:5i h½0:5; 0:6; ½0:2; 0:4i 0 h½0:4; 0:5; ½0:3; :0:4i B B h½0:6; 0:7; ½0:2; 0:3i B ~ 2 ¼ B h½0:3; 0:6; ½0:3; 0:4i R B B @ h½0:7; 0:8; ½0:1; 0:2i

h½0:5; 0:6; ½0:1; 0:3i

h½0:4; 0:5; ½0:3; 0:4i

h½0:4; 0:7; ½0:1; 0:2i

h½0:5; 0:6; ½0:2; 0:3i

h½0:2; 0:3; ½0:4; 0:6i

h½0:3; 0:5; ½0:3; 0:4i

h½0:5; 0:8; ½0:1; 0:2i

h½0:2; 0:5; ½0:3; 0:4i

h½0:6; 0:7; ½0:1; 0:3i

h½0:3; 0:4; ½0:1; 0:3i

h½0:4; 0:6; ½0:2; 0:4i

h½0:1; 0:3; ½0:5; 0:6i

h½0:4; 0:6; ½0:2; 0:4i

1

h½0:6; 0:7; ½0:1; 0:3i

h½0:3; 0:4; ½0:1; 0:2i

h½0:3; 0:4; ½0:2; 0:3i

h½0:3; 0:5; ½0:1; 0:3i

h½0:2; 0:5; ½0:4; 0:5i

C h½0:6; 0:7; ½0:2; 0:3i C C h½0:1; 0:3; ½0:5; 0:6i C C; C h½0:4; 0:7; ½0:1; 0:2i A h½0:6; 0:7; ½0:1; 0:3i 1 h½0:3; 0:4; ½0:3; 0:5i C h½0:5; 0:6; ½0:1; 0:3i C C h½0:4; 0:5; ½0:2; 0:4i C C; C h½0:3; 0:7; ½0:1; 0:2i A h½0:3; 0:4; ½0:5; 0:6i

h½0:2; 0:5; ½0:3; 0:4i B h½0:2; 0:7; ½0:2; 0:3i B ~ 3 ¼ B h½0:1; 0:6; ½0:3; 0:4i R B @ h½0:3; 0:6; ½0:2; 0:4i h½0:4; 0:7; ½0:1; 0:3i

h½0:4; 0:5; ½0:1; 0:2i h½0:3; 0:6; ½0:2; 0:4i h½0:1; 0:4; ½0:3; 0:5i h½0:4; 0:6; ½0:2; 0:3i h½0:5; 0:6; ½0:3; 0:4i

h½0:3; 0:6; ½0:2; 0:3i h½0:4; 0:7; ½0:1; 0:2i h½0:2; 0:6; ½0:2; 0:3i h½0:1; 0:4; ½0:3; 0:6i h½0:2; 0:5; ½0:3; 0:4i

1 h½0:3; 0:7; ½0:1; 0:3i h½0:5; 0:8; ½0:1; 0:2i C C h½0:2; 0:4; ½0:1; 0:5i C C; h½0:3; 0:7; ½0:1; 0:2i A h½0:5; 0:6; ½0:2; 0:4i

h½0:6; 0:7; ½0:2; 0:3i

h½0:4; 0:7; ½0:1; 0:2i

h½0:5; 0:6; ½0:3; 0:4i

h½0:5; 0:6; ½0:1; 0:3i

and 0

respectively. Applying the proposed method in this chapter, the individual overall attribute value of each alternative can be calculated by Eqs. (1.32) and (1.14) and attribute weight vector v ¼ ð0:2; 0:1; 0:3; 0:4ÞT as in Table 1.8. The similarity matrix for the decision group is constructed by Eq. (1.34) as follows: 0

1 0:8033 S ¼ @ 0:8033 1 0:8692 0:8208

1 0:8692 0:8208 A: 1

Table 1.8 Individual overall attribute values of the alternatives Eit

p1

p2

A1 A2 A3 A4 A5

([0.37,0.53]; [0.26, 0.41]) ([0.49,0.65]; [0.21, 0.31]) ([0.19,0.40]; [0.41, 0.52]) ([0.35, 0.61]; [0.20, 0.32]) ([0.49,0.59]; [0.12, 0.32])

([0.27, ([0.50, ([0.42, ([0.41, ([0.27,

p3 0.41]; 0.66]; 0.56]; 0.63]; 0.44];

[0.35, [0.13, [0.20, [0.10, [0.37,

0.50]) 0.27]) 0.37]) 0.21]) 0.48])

([0.29, ([0.39, ([0.17, ([0.25, ([0.39,

0.61]; 0.73]; 0.50]; 0.58]; 0.59];

[0.17, [0.13, [0.19, [0.19, [0.22,

0.31]) 0.24]) 0.42]) 0.37]) 0.38])

1.5 An Air-Condition System Selection Example …

31

Because the maximum module eigenvalue of S is 2.6625, the corresponding eigenvector is X ¼ ð0:5798; 0:5684; 0:5837ÞT , the DM weights are obtained from Eq. (1.36) as follows: z1 ¼ 0:3348; z2 ¼ 0:3282; z3 ¼ 0:3370: Taking the correlated weighted vector w ¼ ð0:25; 0:50; 0:30ÞT with I-IIFOWG [27] as the correlated weighted vector with the hybrid weighted average operator Z, the collective overall attribute values of alternatives are respectively calculated by Eqs. (1.37) and (1.27) as follows: E1 ¼ ð½0:3011; 0:5355; ½0:2406; 0:3854Þ; E2 ¼ ð½0:4634; 0:6776; ½0:1687; 0:2795; E3 ¼ ð½0:2268; 0:4835; ½0:2566; 0:4385Þ; E4 ¼ ð½0:3492; 0:6116; ½0:1667; 0:2955Þ and E5 ¼ ð½0:3752; 0:5465; ½0:2436; 0:3964Þ: In terms of Eq. (1.8), the matrix of possibility degree for IVIFNs E1 , E2 , E3 , E4 and E5 is obtained as follows: 0

0:5 B 0:6174 B P¼B B 0:2957 @ 0:8185 0:5672

0:3826 0:5 0:0083 0:3166 0:0661

0:7043 0:9917 0:5 0:9169 0:7575

0:1815 0:6834 0:0831 0:5 0:1718

1 0:4328 0:9339 C C 0:2425 C C: 0:8282 A 0:5

By Eq. (1.9), the ranking vector is calculated as follows: x1 ¼ 0:4502; x2 ¼ 0:7044; x3 ¼ 0:2716; x4 ¼ 0:6467; x5 ¼ 0:4271: Hence, the ranking order of the alternatives is A2 0:6834 A4 0:8185 A1 0:4328 A5 0:7575 A3 , which indicates that the possibility degree that A2 is superior to A4 is 0.6834, the possibility degree that A4 is superior to A1 is 0.8185, the possibility degree that A1 is superior to A5 is 0.4328, the possibility degree that A5 is superior to A3 is 0.7575, and the best alternative is A2 . Wei [27] assumed that the weight vector of DMs is z ¼ ð0:35; 0:40; 0:25ÞT , and computed the scores of the collective overall preferences of alternatives as follows: S1 ¼ 0:086; S2 ¼ 0:330; S3 ¼ 0:069; S4 ¼ 0:191; S5 ¼ 0:175:

1 A Possibility Degree Method for Interval-Valued Intuitionistic …

32

Thus, the ranking order obtained by Wei [27] is A2 A4 A5 A1 A3 , which is not completely the same as that obtained by this chapter. The difference is the ranking order of alternatives A1 and A5 , the former is A5 A1 while the latter is A1 A5 . Compared with the former, the main advantages of the latter mainly lie in the following: (i) The latter can not only give the ranking order of alternatives but also obtain the possibility degree between the adjacent alternatives simultaneously (as the stated above), which can help the DMs to make decision reasonably and improve the decision results. (ii) The weights of DMs are objectively determined by the eigenvector method in the latter while the former gave the weights of DMs a priori and did not consider the determination of the weights of DMs. The former cannot avoid the subjective randomness of selecting the weights of DMs. (iii) The latter adopts the interval-valued intuitionistic fuzzy weighted average operator Y and the hybrid weighted average operator Z to obtain the collective overall attributes. The attribute weights and the correlated weighted vector with operator Z can take several forms, such as real numbers, intervals, IFNs and INIFNs, whereas in the former the attribute weights and the correlated weighted vector with I-IIFOWG operator are only in the single form of real numbers. Similar to the example analysis of Sect. 1.5.1, we can respectively consider the attribute weights in the forms of real numbers, intervals, IFNs and INIFNs to solve this example. In other words, the latter is more flexible than the former (see the analysis process of Sect. 1.5.1 in detail).

1.6

Conclusions

A new method is proposed to solve the MAGDM problems in which the attribute values are IVIFNs. By using the notion of 2-dimensional random vector, a possibility degree method is developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined on the basis of the Karnik-Mendel algorithms. In the proposed decision making method, the collective overall attribute values of alternatives are derived using the weighted average operator and hybrid weighted average operator of IVIFNs. The ranking order of alternatives is thus generated according to the collective overall attribute values based on the possibility degree ranking method of IVIFNs. Although an air-condition system selection problem is illustrated to demonstrate the applicability and implementation process of the proposed method in this chapter, it is expected to be applicable to decision making problems in many areas, such as supplier selection, risk investment analysis, and so on. The possibility degree ranking method of IVIFNs has the intuitive meaning from the probability viewpoint, which is different from the ranking method based on the

1.6 Conclusions

33

score and accuracy functions. Moreover, the proposed ordered weighted average operator for IVIFNs considers the ordered positions of arguments, and the proposed hybrid weighted average operator for IVIFNs reflects the important degrees of both the arguments themselves and the ordered positions of the arguments simultaneously, which are the salutary extension of the aggregation operators developed in [4]. How to determine the correlated weighted vectors in the forms of intervals, IFNs and IVIFNs, will be studied for future research.

References 1. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 2. K.T. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989) 3. G. Beliakov, H. Bustinc, D. P. Goswami, U. K. Mukherjee, N. R. Pal, On averaging operators for Atanassov’s intuitionistic fuzzy sets. Inf. Sci. 181(6), 1161–1124 (2011) 4. S.M. Chen, L.W. Lee, H.C. Liu, S.W. Yang, Multiattribute decision making based on interval-valued intuitionistic fuzzy values. Expert Syst. Appl. 39, 10343–10351 (2012) 5. T.Y. Chen, C.H. Li, Objective weights with intuitionistic fuzzy entropy measures and computational experiment analysis. Appl. Soft Comput. 11, 5411–5423 (2011) 6. Z.P. Chen, W. Yang, A new multiple criteria decision making method based on intuitionistic fuzzy information. Expert Syst. Appl. 39(4), 4328–4334 (2012) 7. Z.X. Jiang, G.L. Shi, Matrix theory and application (Beijing University of Aeronautics and Astronautics pressing, Beijing, 1998) 8. N.N. Karnik, J.M. Mendel, Centroid of a type-2 fuzzy set. Inf. Sci. 132(1–4), 195–220 (2001) 9. D.F. Li, Some measures of dissimilarity in intuitionistic fuzzy structures. J. Comput. Syst. Sci. 68(1), 115–122 (2004) 10. D.F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets. J. Comput. Syst. Sci. 70(1), 73–85 (2005) 11. D.F. Li, The GOWA operator based approach to multiattribute decision making using intuitionistic fuzzy sets. Math. Comput. Model. 53(5–6), 1182–1196 (2011) 12. D.F. Li, Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets. Expert Syst. Appl. 37(12), 8673–8678 (2010) 13. D.F. Li, Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment. Fuzzy Optim. Decis. Making 7(1), 17–34 (2008) 14. D.F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information. Appl. Soft Comput. 11(4), 3402–3418 (2011) 15. D.F. Li, TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets. IEEE Trans. Fuzzy Syst. 18(2), 299–311 (2010) 16. D.F. Li, Linear programming method for MADM with interval-valued intuitionistic fuzzy sets. Expert Syst. Appl. 37(8), 5939–5945 (2010) 17. D.F. Li, G.H. Chen, Z.G. Huang, Linear programming method for multiattribute group decision making using IF sets. Inf. Sci. 180(9), 1591–1609 (2010) 18. D.F. Li, L.L. Wang, G.H. Chen, Group decision making methodology based on the Atanassov’s intuitionistic fuzzy set generalized OWA operator. Int. J. Uncertainty, Fuzziness Knowl. Based Syst. 18(6), 801–817 (2010)

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1 A Possibility Degree Method for Interval-Valued Intuitionistic …

19. D.F. Li, Y.C. Wang, Mathematical programming approach to multiattribute decision making under intuitionistic fuzzy environments. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 16 (4), 557–577 (2008) 20. D.F. Li, Y.C. Wang, S. Liu, F. Shan, Fractional programming methodology for multi-attribute group decision-making using IFS. Appl. Soft Comput. 9(1), 219–225 (2009) 21. F. Liu, J.M. Mendel, Aggregation using the fuzzy weighted average as computed by the Kernik-Mendel algorithms. IEEE Trans. Fuzzy Syst. 16(1), 1–16 (2008) 22. L. Lin, X.H. Yuan, Z.Q. Xia, Multicriteria muzzy decision-making methods based on intuitionistic fuzzy sets. J. Comput. Syst. Sci. 73(1), 84–88 (2007) 23. H.W. Liu, G.J. Wang, Multi-criteria decision-making methods based on intuitionistic fuzzy Sets. Eur. J. Oper. Res. 179(1), 220–233 (2007) 24. J.M. Mendel, F. Liu, Super-exponential convergence of the Karnik-Mendel algorithms for computing the centroid of an interval type-2 fuzzy set. IEEE Trans. Fuzzy Syst. 15(2), 309– 320 (2007) 25. Z. Pei, L. Zheng, A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets. Expert Syst. Appl. 39(3), 2560–2566 (2012) 26. W.Z. Wang, Comments on “Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment” by Jun Ye. Expert Syst. Appl. 38, 13186–13187 (2011) 27. G.W. Wei, Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Appl. Soft Comput. 10, 423– 431 (2010) 28. M.C. Wu, T.Y. Chen, The ELECTRE multicriteria analysis approach based on Atanassov’s intuitionistic fuzzy sets. Expert Syst. Appl. 38, 12318–12327 (2011) 29. Z.S. Xu, Algorithm for priority of fuzzy complementary judgment matrix. J. Syst. Eng. 16(4), 311–314 (2001) 30. Z.S. Xu, An overview of methods for determining OWA weights. Int. J. Intell. Syst. 20(8), 843–865 (2005) 31. Z.S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis 22(2), 215–219 (2007) 32. Z.S. Xu, Models for multiple attribute decision making with intuitionistic fuzzy information. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 15, 285–297 (2007) 33. Z.S. Xu, Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optim. Decis. Making 6, 109–121 (2007) 34. Z.S. Xu, On similarity measures of interval-valued intuitionistic fuzzy sets and their application to pattern recognitions. J. SE Univ. 23, 139–143 (2007) 35. Z.S. Xu, Multi-person multi-attribute decision making models under intuitionistic fuzzy environment. Fuzzy Optim. Decis. Making 6, 221–236 (2007) 36. Z.S. Xu, Intuitionistic preference relations and their application in group decision making. Inf. Sci. 177, 2363–2379 (2007) 37. Z.S. Xu, A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making. Group Decis. Negot. 19, 57–76 (2010) 38. Z.S. Xu, Choquet integrals of weighted intuitionistic fuzzy information. Inf. Sci. 180, 726– 736 (2010) 39. Z.S. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowl. Based Syst. 24, 749–760 (2011) 40. Z.S. Xu, A method based on distance measure for interval-valued intuitionistic fuzzy group decision making. Inf. Sci. 180, 181–190 (2010) 41. Z.S. Xu, X.Q. Cai, Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information. Int. J. Intell. Syst. 25, 489–513 (2010) 42. Z.S. Xu, X.Q. Cai, Incomplete interval-valued intuitionistic fuzzy preference relations. Int. J. Gen Syst. 38, 871–886 (2009) 43. Z.S. Xu, J. Chen, An approach to group decision making based on interval-valued intuitionistic judgment matrices. Syst. Eng. Theory Pract. 27(4), 126–133 (2007)

References

35

44. Z. S. Xu, J. Chen, On geometric aggregation over interval-valued intuitionistic fuzzy information, in 4th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD’07), vol. 2 (Haikou, China, Aug 24–27, 2007), pp. 466–471 45. Z.S. Xu, J. Chen, A multi-criteria decision making procedure based on interval-valued intuitionistic fuzzy Bonferroni means. J. Syst. Sci. Syst. Eng. 20, 217–228 (2011) 46. Z.S. Xu, Q.L. Da, The uncertain OWA operator. Int. J. Intell. Syst. 17, 569–575 (2002) 47. Z.S. Xu, H. Hu, Projection models for intuitionistic fuzzy multiple attribute decision making. Int. J. Inf. Technol. Decis. Making 9, 267–280 (2010) 48. Z.S. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen Syst. 35, 417–433 (2006) 49. Z.S. Xu, R.R. Yager, Intuitionistic and interval-valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optim. Decis. Making 8, 123–139 (2009) 50. Z.S. Xu, R.R. Yager, Dynamic intuitionistic fuzzy multi-attribute decision making. Int. J. Approximate Reasoning 48, 246–262 (2008) 51. R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Syst. Man Cybern 18, 183–190 (1988) 52. W. Yang, Z.P. Chen, The quasi-arithmetic intuitionistic fuzzy OWA operators. Knowl. Based Syst. 27, 219–233 (2012) 53. J. Ye, Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Expert Syst. Appl. 36, 6899–6902 (2009) 54. J. Ye, Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment. Eur. J. Oper. Res. 205(1), 202–204 (2010) 55. K.P. Yoon, A probabilistic approach to rank complex fuzzy numbers. Fuzzy Sets Syst. 80, 167–176 (1996) 56. L.A. Zadeh, Fuzzy sets. Inf. Control 18, 338–353 (1965) 57. S.Z. Zeng, W.H. Su, Intuitionistic fuzzy ordered weighted distance operator. Knowl. Based Syst. 24, 1224–1232 (2011) 58. H. Zhao, Z.S. Xu, M.F. Ni, S.S. Liu, Generalized aggregation operators for intuitionistic fuzzy sets. Int. J. Intell. Syst. 25, 1–30 (2010)

Chapter 2

A New Method for Atanassov’s Interval-Valued Intuitionistic Fuzzy MAGDM with Incomplete Attribute Weight Information

Abstract This chapter develops a new method for solving multiple attribute group decision-making (MAGDM) problems with Atanassov’s interval-valued intuitionistic fuzzy values (AIVIFVs) and incomplete attribute weight information. Firstly, we investigate the asymptotic property of the Atanassov’s interval-valued intuitionistic fuzzy (AIVIF) matrix. It is demonstrated that after applying weights an infinite number of times, all elements in an AIVIF matrix will approach the same AIVIFV without regard to the initial values of elements. Then, the weight of each decision maker (DM) with respect to every attribute is determined by considering the similarity degree and proximity degree simultaneously. To avoid weighting an AIVIF matrix too many times, the collective decision matrix is transformed into an interval matrix using the risk coefficient of DMs. Subsequently, to derive the attribute weights objectively, we construct a multi-objective interval-programming model that is solved by transforming it into a linear program. The ranking order of alternatives is generated by the comprehensive interval values of alternatives. Finally, an example of a research and development (R & D) project selection problem is provided to illustrate the implementation process and applicability of the method developed in this chapter.




Keywords Multiple attribute group decision-making Atanassov’s interval-valued intuitionistic fuzzy set Risk preference Interval objective programming




2.1




Introduction

For many venture capital companies, investment project selection plays an important role in company development. In real-life project selection problems, companies usually select research and development (R & D) projects with higher ventures and higher income. Before an R & D project is selected, several experts (decision makers, or DMs) from different domain areas will be invited to evaluate the potential R & D projects on several attributes (or indices), such as organizing ability, credit quality, level of research and development, profitability and debt © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_2

37

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2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

servicing ability, to name a few. Thus, R & D project selection problems can be regarded as a kind of multi-attribute group decision-making (MAGDM) problem. MAGDM problems, common tasks in human activities, consist of finding the most preferred alternative from a given alternative set by a group of DMs. Generally, the resolution of a MAGDM problem needs two phases: aggregation and exploration [8] (see Fig. 2.1). In the aggregation phase, the individual decision matrices are aggregated into a collective one. The DM weights play a crucial role in this process. Different vectors of DMs’ weights may result in different collective decision matrices, which can significantly influence the final decision results. In the exploration phase, the collective decision matrix is transformed into comprehensive values of alternatives by which the ranking of alternatives is obtained. In this phase, the weights of attributes are used to aggregate the attribute values of alternatives. It is easy to see that the aggregation methods play a significant role in both the aggregation and exploration phases. In the aggregation phase, it is necessary to employ the aggregation methods by which the individual decision matrices can be aggregated into a collective one after the DM weights are determined. Otherwise, the aggregation process cannot be realized. In the exploration phase, the aggregation method is also needed to transform the collective decision matrix into the comprehensive values of the alternatives after the attribute weights are derived. Finally, the decision methods are used to rank alternatives based on the comprehensive values of the alternatives.

Aggregation phase Individual decision matrices Weights of DMs Aggregation methods Collective decision matrix Exploration phase

Weights of attributes Aggregation methods

Comprehensive values of alternatives Decision methods

Resolution of a MAGDM problem

Fig. 2.1 Process of resolving a MAGDM problem

2.1 Introduction

39

Owing to the uncertainty and vagueness of the objects and the ambiguity of human thinking, it is difficult for DMs to express their preferences over alternatives as precise values. The fuzzy set (FS) initiated by Zadeh [47] is a powerful tool to characterize such uncertainty and fuzziness that have been extensively investigated by many researchers from different disciplines [15, 19, 24–26, 30, 32, 36, 43, 48, 49]. In the field of MAGDM, interval numbers [32, 43, 48], triangular fuzzy numbers [36, 49] and trapezoidal fuzzy numbers [19, 30] are usually used to represent alternative ratings on attributes. However, the FS characterizes the fuzziness by membership degree only. In 1986, Atanassov [1] generalized the FS and introduced Atanassov’s intuitionistic fuzzy set (AIFS), which considers membership degree, non-membership degree and hesitant degree simultaneously. In 1989, Atanassov and Gargov [3] further extended AIFS into Atanassov’s interval-valued intuitionistic fuzzy (AIVIF) set (AIVIFS), with the membership degree and non-membership degree described as intervals. In a famous monograph [22], Pedrycz pointed out that it is not quite justifiable or technically sound to quantify grades of membership and non-membership in terms of a single numeric value. Thus, compared with AIFSs, AIVIFSs with interval-valued membership and non-membership functions [11–13, 15, 50] are more flexible and effective in expressing fuzziness and uncertainty in practical applications. In recent years, MAGDM with AIVIF values (AIVIFVs) has received more and more attention and many achievements have appeared. As shown in Fig. 2.1, the DMs’ weights, attributes’ weights, aggregation methods and decision methods are four key issues for solving MAGDM problems. The former two should be solved first. Then, the aggregation methods are employed to integrate individual decision matrices into a collective one, followed by an integration of the collective matrix into the comprehensive values of the alternatives. Finally, decision methods are used to select the best alternative(s). According to these four issues, existing achievements on MAGDM problems with AIVIFVs are briefly reviewed as follows: (1) Determination of DMs’ weights. Yue and Jia [46] proposed a TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method to determine the weights of DMs by considering individual positive ideal decision (IPID) and three individual negative ideal decisions (INIDs). Meng et al. [20] utilized the distance between one decision matrix given by a DM and those given by other DMs to derive the weights of DMs. Wan and Dong [31] determined the weights of DMs based on a similarity degree matrix. Zhang and Xu [51] constructed a goal-programming model to derive the weights of DMs in the case in which the information about the weights of DMs is incomplete. (2) Determination of attribute weights. When the information about attribute weights is incomplete, Park et al. [21] determined attribute weights by constructing a linear programming model that maximizes score functions of alternatives. By minimizing the inconsistency degree, Wan and Li [33] constructed a linear programming model to estimate the weights of attributes. Subsequently, Wan and Dong [34] built a bi-objective programming model,

40

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

which minimizes the inconsistency degree and maximizes the consistency degree simultaneously, to identify attribute weights. Chen [10] employed a comprehensive concordance index to build linear programming models to derive the weights of attributes. When the weights of attributes are completely unknown, Jin et al. [17] extended the information entropy to the AIVIF environment to determine the weights of attributes. (3) Aggregation methods. After determining the DMs’ weights and the attribute weights, the comprehensive values of the alternatives can be obtained by utilizing different aggregation methods. In these methods, aggregation operators are often adopted. Many different operators were suggested to aggregate collective decision information into comprehensive values of alternatives. For example, Xu [42] developed an AIVIF weighted average operator. Hashemi et al. [14] gave an AIVIF weighted geometric averaging (AIVIFWGA) operator. Zhou et al. [52] proposed a continuous AIVIF aggregation (ACIVIFA) operator. Meng et al. [20] presented an arithmetical AIVIF generalized k-Shapley Choquet operator. (4) Decision methods. Xu [41] introduced the score and accuracy functions of AIVIFSs that are widely used in MAGDM problems [21, 46, 52]. Xu and Shen [39] suggested a new outranking method to rank alternatives. Chen [10] developed a QUALIFLEX (Qualitative Flexible multiple criteria method) to select the best alternative in an AIVIF environment. Tan [28] extended the TOPSIS method by applying the Choquet integral. Ye [44] utilized entropy to define a weighted correlation coefficient by which the alternatives are ranked. Although the aforementioned methods are very effective for solving many MAGDM problems with AIVIFVs, they suffer from some limitations: (1) In methods [20, 31, 46, 51], the weights of each DM are often viewed as the same with respect to different attributes, which may be somewhat unreasonable and unrealistic. In real-world decision problems, every DM is good only at some subjects. Thus, it is more natural and reasonable to allocate different weights for each DM with respect to different attributes. (2) The method in [46] integrated individual decision matrices into a collective one by the DMs’ weights. The weighted collective decision matrix is then calculated by the attribute weights. Due to such weighted integration operations, all the elements in the weighted collective decision matrix are very close to the same AIVIFV, regardless of the initial attribute values. Thus the comprehensive alternative values will be quite close to each other, making it difficult to rank alternatives and select the best one. (3) Existing methods [28, 39, 41, 44] ranked alternatives without considering DMs’ risk preferences. However, different DMs often have diverse attitudes towards risk. For the same MAGDM problem, decision results may vary, as per DMs’ risk preferences. To overcome the above limitations, we propose a new method for MAGDM with AIVIFVs. Firstly, we prove that after weights are applied an infinite number of

2.1 Introduction

41

times, all elements in an AIVIF matrix approach the same AIVIFV regardless of the initial values of elements. Then, a novel approach is developed to determine the DMs’ weights with respect to each attribute. We transform the AIVIF collective decision matrix into an interval decision matrix after adequately considering the DMs’ risk preferences. By maximizing the comprehensive alternative values, a multi-objective interval-programming model is constructed to derive the weights of attributes objectively. Finally, the comprehensive alternative values are computed to rank the alternatives. Compared with existing research, the key features of the proposed method in this chapter are listed as follows: (1) We first study the asymptotic property of an AIVIF matrix by weighted integration operations. The more times an AIVIF decision matrix is weighted, the closer the comprehensive alternative values are to each other. This result indicates that it is not suitable to weight an AIVIF matrix too many times. (2) A novel approach is developed to determine the DMs’ weights with respect to each attribute by considering the similarity degree and the proximity degree simultaneously. The similarity degree measures the similarity between an individual decision matrix and the collective one while the proximity degree measures the proximity between the decision matrix given by one DM and those given by all other DMs. Key characteristics of this approach are: (i) the weights of each DM obtained are different with respect to different attributes; (ii) it considers not only the similarity between the decision matrix provided by an individual DM and that of the group judgment but also the proximity between the decision matrix provided by the individual DM and those provided by all other DMs. (3) The collective AIVIF decision matrix is first transformed into an interval matrix and then weighted with the attributes’ weights. This process can effectively avoid weighting the AIVIF matrix too many times and overcome the drawback in the method [46]. (4) In the process of transforming the collective decision matrix into an interval matrix, different DM’s risk preferences are properly considered, such as pessimistic, neutral, and optimistic preferences. Multi-objective interval-programming models are constructed and solved to determine the attribute weights objectively. The rest of this chapter is organized as follows. Section 2.2 reviews some preliminary information regarding interval programming, orderings of intervals, AIFSs and AIVIFSs. Furthermore, a theorem is presented to analyze the asymptotic property of AIVIF matrices with repeated weighting integration. In Sect. 2.3, a new approach is developed to determine DMs’ weights with respect to each attribute. The attribute weights are objectively derived through constructing a multi-objective interval-programming. Then, a novel method is proposed for the MAGDM with

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

42

AIVIFVs and incomplete weight information. In Sect. 2.4, the proposed method is illustrated with a real-world R & D project selection example and comparison analyses are conducted. The primary conclusions are presented in Sect. 2.5.

2.2

Preliminaries

In this section, we review some basic preliminaries related to interval-programming, orderings of intervals, AIFS and AIVIFS. Moreover, an important theorem on AIVIF matrices is presented.

2.2.1

Interval Objective Programming

Let ~a ¼ ½aL ; aU  ¼ fajaL  a  aU ; a 2 Rg be an interval on the real number set R, where aL and aU are left and right bounds, respectively. mð~ aÞ ¼ 12ðaL þ aU Þ is called the center of interval ~a. For two intervals ~a ¼ ½aL ; aU  and ~b ¼ ½bL ; bU , addition and scalar multiplication are defined as [16]: (1) a~ þ ~b ¼ ½aL þ bL ; aU þ bU ; (2) b~a ¼ ½baL ; baU  ðb [ 0Þ. Ishibuchi and Tanaka [16] first studied the interval objective programming which is reviewed as follows. A maximization interval-programming model is described as maxf~ag ~a 2 X

ð2:1Þ

which is equivalent to the following bi-objective mathematical programming model: maxfaL ; mð~aÞg ~a 2 X

ð2:2Þ

A minimization interval-programming model is described as minf~ag ~a 2 X

ð2:3Þ

which is equivalent to the following bi-objective mathematical programming model:

2.2 Preliminaries

43

minfaU ; mð~aÞg ~a 2 X

ð2:4Þ

where X is a set of constraints that the interval number ~ a should satisfy.

2.2.2

Orderings of Intervals

Denote by Lð½0; 1Þ the set of all closed subintervals of the unit interval. A usual partial order  2 on intervals is given by ½a; b  2 ½c; d , a  c and b  d. The set Lð½0; 1Þ with a partial order  is called a partially ordered set (poset) and denoted by ðLð½0; 1Þ; Þ[4]. Definition 2.1 [4] Let ðLð½0; 1Þ; Þ be a poset. The order  is called an admissible order if (i)  is a linear order on Lð½0; 1Þ; (ii) For all ½a; b; ½c; d 2 Lð½0; 1Þ, ½a; b  ½c; d whenever ½a; b  2 ½c; d. The usual admissible orders are lexicographical orders (Lex1 and Lex2 ) and XY introduced by Xu and Yager [40]. They are described as [5]: (i) ½a; bLex1 ½c; d , a\c or a ¼ c and b  d; (ii) ½a; bLex2 ½c; d , b\d or b ¼ d and a  c; (iii) ½a; bXY ½c; d , a þ b\c þ d or a þ b ¼ c þ d and b  a  d  c. Furthermore, Bustince et al. [5] defined a continuous aggregation function Ka ða; bÞ ¼ a þ aðb  aÞ ¼ ð1  aÞa þ ab. For a; b 2 ½0; 1; a 6¼ b, if the relation a;b on Lð½0; 1Þ is given by ½a; ba;b ½c; d , Ka ða; bÞ\Ka ðc; dÞ or ðKa ða; bÞ ¼ Ka ðc; dÞ and Kb ða; bÞ\Kb ðc; dÞÞ;

ð2:5Þ

then it is an admissible order on Lð½0; 1Þ generated by an admissible pair of aggregation functions ðKa ; Kb Þ. For instance, when a ¼ 0 and b ¼ 1, ½a; ba;b ½c; d is simplified as ½a; bLex1 ½c; d. When a ¼ 1 and b ¼ 0, ½a; ba;b ½c; d is simplified as ½a; bLex2 ½c; d. In real-world decision-making, choosing which order to rank intervals is complicated because both underlying problems and DMs should be considered. For example, if the DMs are optimistic, it may be appropriate to choose the order Lex2 . Conversely, if the DMs are pessimistic, the order Lex1 should be chosen. When the DMs are neutral, we can choose XY . If we do not know which the most appropriate order is, Bustince et al. [5] proposed a Shapley value method to rank intervals with m different orders as follows:

44

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

(i) If we derive the same ranking results for all of the considered orders, then we can finish and choose the best interval accordingly. ~ are ranked in different positions with different orders, (ii) If two intervals, ~a and b, we can determine their final positions by the frequency of one being ranked ~ is identical to higher than the other. If the frequency of ~a being greater than b ~ being greater than ~a, then we calculate their Shapley values. The that of b larger the Shapley value of an interval, the bigger the interval.

2.2.3

Atanassov’s Intuitionistic Fuzzy Set and Atanassov’s Interval-Valued Intuitionistic Fuzzy Set

Definition 2.2 [1] Let X be a non-empty set of the universe. An AIFS A in X is defined as A ¼ f\x; lA ðxÞ; vA ðxÞ [ jx 2 Xg;

ð2:6Þ

where the functions lA : X ! ½0; 1 and vA : X ! ½0; 1 satisfy the condition 0  lA ðxÞ þ vA ðxÞ  1; 8x 2 X. lA ðxÞ and vA ðxÞ denote, respectively, the membership degree and non-membership degree of the element x 2 X to set A. The pair \lA ðxÞ; vA ðxÞ [ is called an Atanassov’s intuitionistic fuzzy (AIF) value (AIFV) and simply denoted by a ¼ \la ðxÞ; va ðxÞ [ . In addition, pA ðxÞ ¼ 1  lA ðxÞ  vA ðxÞ is called the AIF index, representing the degree of indeterminacy of x to A. Atanassov [1, 2] pointed out that an AIFV \lA ðxÞ; vA ðxÞ [ could be converted into an interval number ½lA ðxÞ; 1  vA ðxÞ. Using the product t-norm and its dual t-conorm, Beliakov et al. [6] defined the Atanassov’sintuitionistic Weighted Arithmetic Mean (AIWAM) with respect to a weighting vector w ¼ ðw1 ; w2 ; . . .; wn ÞT as AIWAMw ða1 ; a2 ; . . .; an Þ ¼ \

n X

wi lai ;

i¼1

n X

wi vai [ ;

ð2:7Þ

i¼1

where aj ¼ \lai ; mai [ ðj ¼ 1; 2; . . .; nÞ are n AIFVs, wj 2 ½0; 1 ðj ¼ 1; 2; . . .; nÞ P and nj¼1 wj ¼ 1. Meanwhile, for intervals ½ai ; bj ðj ¼ 1; 2; . . .; nÞ, Bustince et al. [5] claimed that / is an additive aggregation function on ðLð½0; 1Þ; 2 Þ with idempotent element [0,1] if and only if " /ð½a1 ; b1 ; ½a2 ; b2 ; . . .s; ½an ; bn Þ ¼

n X i¼1

w i ai ;

n X i¼1

# wi bi :

ð2:8Þ

2.2 Preliminaries

45

~ in X is defined as Definition 2.3 [3] An AIVIFS A ~ ¼ f\x; l ~A~ ðxÞ; ~vA~ ðxÞ [ jx 2 X g; A

ð2:9Þ

~A~ : X ! Lð½0; 1Þ and ~vA~ : X ! Lð½0; 1Þ, and 0  supð~ where l lA~ ðxÞÞ þ ~A~ ðxÞ and ~vA~ ðxÞ represent the memsupð~vA~ ðxÞÞ  1 for any x 2 X. The intervals l ~ respectively. bership degree and non-membership degree of element x to AIVIFS A, L R L R ~ ~A~ ðxÞ ¼ ½lA~ ðxÞ; lA~ ðxÞ and ~vA~ ðxÞ ¼ ½vA~ ðxÞ; vA~ ðxÞ. Therefore, the AIVIFS A Denote l can be equivalently expressed as ~¼ A

     L  x; lA~ ðxÞ; lRA~ ðxÞ ; vLA~ ðxÞ; vRA~ ðxÞ jx 2 X ;

where 0  lLA~ ðxÞ  lRA~ ðxÞ  1; 0  vLA~ ðxÞ  vRA~ ðxÞ  1 and lRA~ ðxÞ þ vRA~ ðxÞ  1 for any x 2 X. ~A~ ðxÞ ¼ ½~ ~RA~ ðxÞ is called the AIVIF index of element x 2 X, Similarly, p pLA~ ðxÞ; p L R ~A~ ðxÞ ¼ 1  lA~ ðxÞ  vRA~ ðxÞ and p ~RA~ ðxÞ ¼ 1  lLA~ ðxÞ  vLA~ ðxÞ. If l ~A~ ðxÞ ¼ where p L R L R ~ is reduced to an AIFS. l ~ ðxÞ ¼ l ~ ðxÞ and ~v ~ ðxÞ ¼ v ~ ðxÞ ¼ v ~ ðxÞ, then AIVIFS A A

A

A

A

A

The pair ~a ¼ ð~ l~a ðxÞ; ~v~a ðxÞÞ is called an AIVIFV and simply denoted by ~a ¼ ð½a; b; ½c; dÞ, where ½a; b½0; 1, ½c; d½0; 1 and b þ d  1. a2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ and ~ a¼ Definition 2.4 [2, 3] Let a~1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ, ~ ð½a; b; ½c; dÞ be three AIVIFVs, then

(1) Addition : ~a1 þ a~2 ¼ ð½a1 þ a2  a1 a2 ; b1 þ b2  b1 b2 ; ½c1 c2 ; d1 d2 Þ; (2) Complement: ð~aÞc ¼ ð½c; d; ½a; bÞ; (3) Scalar multiplication : k~a ¼ ð½1  ð1  aÞk ; 1  ð1  bÞk ; ½ck ; d k Þ, k [ 0.

where

Definition 2.5 [42] Let ~a ¼ ð½a; b; ½c; dÞ be an AIVIFV. Then 1 sð~aÞ ¼ ða þ b  c  dÞ 2

ð2:10Þ

1 hð~aÞ ¼ ða þ c þ b þ dÞ 2

ð2:11Þ

and

are called the score function and accuracy function of the AIVIFV ~ a, respectively. Definition 2.6 [42] Let ~a1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ and ~ a2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ be two AIVIFVs. Then, If sð~a1 Þ\sð~a2 Þ; then ~a1 \~a2 ; If sð~a1 Þ ¼ sð~a2 Þ; then

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

46

(i) If hð~a1 Þ ¼ hð~a2 Þ, then a~1 ¼ a~2 ; (ii) If hð~a1 Þ\hð~ a2 Þ, then ~a1 \~a2 . Definition 2.7 [42] Let ~aj ¼ ð½aj ; bj ; ½cj ; dj Þ ðj ¼ 1; 2; ::; nÞ be a collection of AIVIFVs. If fw ð~a1 ; ~a2 ; . . .; ~an Þ ¼

n X

wj ~ aj ;

ð2:12Þ

j¼1

then f is called an AIVIF weighted averaging operator of dimension n, where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weight vector of ~aj ðj ¼ 1; 2; . . .; nÞ, satisfying that P wj 2 ½0; 1ðj ¼ 1; 2; . . .; nÞ and nj¼1 wj ¼ 1. The aggregated value by using the operator f in Definition 2.7 is also an AIVIFV, i.e., " fw ð~a1 ; ~a2 ; . . .; ~an Þ ¼

1

n Y j¼1

wj

ð1  aj Þ ; 1 

n Y j¼1

# " ð1  bj Þ

wj

;

n Y

w cj j ;

j¼1

n Y

#! w dj j

:

j¼1

ð2:13Þ There exist three undesirable properties for the operator f in Eq. (2.13). Property 2.1 Whenever one of AIVIFVs ~aj ðj ¼ 1; 2; . . .; nÞ in Eq. (2.13) is taken as ð½1; 1; ½0; 0Þ and its corresponding weight is not 0, we have fw ð~a1 ; ~a2 ; . . .; ~an Þ ¼ ð½1; 1; ½0; 0Þ. Property 2.2 If one of AIVIFVs in Eq. (2.13) is ð½0; 0; ½1; 1Þ and its corresponding weight is not 0, it is not accounted for at all. In other words, if ~aj ¼ ð½0; 0; ½1; 1Þ, then we have fw ð~a1 ; ~a2 ; . . .; ~aj1 ; ~aj ; ~aj þ 1 ; . . .; ~an Þ ¼ fw ð~a1 ; ~a2 ; . . .; ~ aj1 ; ~ aj þ 1 ; . . .; ~ an Þ: Property 2.3 The operator f is not monotonic with respect to the ordering in ~ ¼ ð½ej ; fj ; Definition 2.6. Namely, let ~aj ¼ ð½aj ; bj ; ½cj ; dj Þðj ¼ 1; 2; . . .; nÞ and b j ½gj ; hj Þðj ¼ 1; 2; . . .; nÞ be two sets of AIVIFVs and w ¼ ðw1 ; w2 ; . . .; wn ÞT be the ~ weight vector. If ~aj  b for all j ¼ 1; 2; . . .; n, the inequality j ~ ~ ~ fw ð~a1 ; a~2 ; . . .; a~n Þ  fw ðb1 ; b2 ; . . .; bn Þ may not hold. Property 2.3 can be easily seen from the following Example 2.1. ~ ¼ Example 2.1 Consider a~1 ¼ ð½0; 0; ½1; 1Þ, ~a2 ¼ ð½0:3; 0:6; ½0:2; 0:4Þ, b 1 ~ ð½0; 0; ½1; 1Þ and b2 ¼ ð½0:5; 0:6; ½0:4; 0:4Þ and a weighting vector w ¼ ð0:4; 0:6ÞT .

2.2 Preliminaries

47

~ Þ ¼ 0:15, hð~a2 Þ ¼ 0:75 and hðb ~ Þ ¼ 0:95, we have ~ ~ and Since sð~a2 Þ ¼ sðb a1 ¼ b 2 2 1 ~ ~a2 \b2 . However, by Eq. (2.13), we obtain ~c1 = AIVIFWAw ð~a1 ; ~a2 Þ ¼ ð½0:1927; 0:4229; ½0:3807; 0:5771Þ; ~ ;b ~ Þ ¼ ð½0:3402; 0:4229; ½0:5771; 0:5771Þ: ~c ¼ AIVIFWAw ðb 2

1

2

Since sð~c1 Þ ¼ 0:1711 and sð~c2 Þ ¼ 0:1955, we derive ~c1 [ ~c2 . Thus, ~ ;b ~ fw ð~a1 ; ~a2 Þ [ fw ðb 1 2 Þ. To overcome these undesirable properties, by combining Eq. (2.7) with Eq. (2.8) we develop a new weighted mean operator for AIVIFVs. Definition 2.8 Let ~aj ¼ ð½aj ; bj ; ½cj ; dj Þðj ¼ 1; 2; . . .; nÞ be a collection of AIVIFVs. If " # " #! n n n n X X X X AIVIFWMw ð~a1 ; ~a2 ; . . .; ~an Þ ¼ w j aj ; wj bj ; w j cj ; wj dj ; j¼1

j¼1

j¼1

j¼1

ð2:14Þ aj ðj ¼ 1; 2; . . .; nÞ, satisfying that where w ¼ ðw1 ; w2 ; . . .; wn ÞT is a weight vector of ~ Pn wj 2 ½0; 1ðj ¼ 1; 2; . . .; nÞ and j¼1 wj ¼ 1, then the AIVIFWM is called an AIVIF weighted arithmetic mean (AIVIFWM) operator of dimension n. Specially, when w ¼ ð1=n; 1=n; . . .; 1=nÞT , the AIVIFWM is called the AIVIF arithmetic mean (AIVIFM) operator. ~ ¼ð~gij Þ Definition 2.9 The matrix G mn is called an AIVIF matrix if all entries of ~ matrix G are AIVIFVs, i.e., ~gij ¼ ð½aij ; bij ; ½cij ; dij Þði ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; nÞ. Definition 2.10 Let w ¼ ðw1 ; w2 ; . . .; wn ÞT be a weight vector, satisfying that Pn ~ wj 2 ½0; 1ðj ¼ 1; 2; . . .; nÞ and gij Þmn , if j¼1 wj ¼ 1. For AIVIF matrix G ¼ð~ ~fij ¼ wj ~gij ði ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; nÞ, then F ~ ¼ð~fij Þ mn is called the weighted ~ matrix of G. ~ ð0Þ ¼ð~gð0Þ Þ ~ ðtÞ gðtÞ Þ Theorem 2.1 Let G ij mn be an initial AIVIF matrix and G ¼ð~ ij mn be ð0Þ ~ the t-th weighted matrix by weighting G with a given weighting vector ðtÞ

ðtÞ

ðtÞ

ðtÞ

wðtÞ ¼ ðw1 ; w2 ; . . .; wn ÞT , where 0  wj  1  eðt ¼ 1; 2; . . .; j ¼ 1; 2; . . .; nÞ, P ðtÞ e [ 0 and nj¼1 wj ¼ 1. Then, the following conclusions are true: (i) (ii)

ðtÞ lim a t! þ 1 ij ðtÞ lim c t! þ 1 ij ð0Þ

ðtÞ

¼ lim bij ¼ 0; t! þ 1

ðtÞ

¼ lim dij ¼ 1, t! þ 1

ð0Þ

ð0Þ

ð0Þ

ð0Þ

where g~ij ¼ð½aij ; bij ; ½cij ; dij Þ, j ¼ 1; 2; . . .; nÞ.

ðtÞ

ðtÞ

ðtÞ

ðtÞ

ðtÞ

~gij ¼ð½aij ; bij ; ½cij ; dij Þði ¼ 1; 2; . . .; m;

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

48

ðtÞ

Proof (i) We first prove lim aij ¼ 0. t! þ 1

ðtÞ

ðtÞ ðt1Þ

From Definition 2.10, we have ~gij ¼ wj ~gij ð1Þ ~gij

. Specially, if t ¼ 1, then

ð1Þ ð0Þ wj ~gij .

¼ Using the operations in Definition 2.4, one has ð1Þ

ð0Þ

ð1Þ

ð2Þ

ð1Þ

ð2Þ

aij ¼ 1  ð1  aij Þwj :

ð2:15Þ

Similarly, aij ¼ 1  ð1  aij Þwj :

ð2:16Þ

ð2Þ

ð0Þ

ð1Þ

Plugging Eq. (2.15) into Eq. (2.16), we obtain aij ¼ 1  ð1  aij Þwj utilizing Mathematical Induction, it follows that ðtÞ

ð0Þ

ð1Þ

aij ¼ 1  ð1  aij Þwj ðtÞ lim a t! þ 1 ij ð1Þ ð2Þ ðtÞ lim w wj . . .wj ¼ 0. t! þ 1 j

To

prove

¼ 0,

from

ð2Þ

ðtÞ

wj


wj

Eq. (2.17),

: we

ð2Þ

wj

. By

ð2:17Þ only

need

prove

ðtÞ

Since 0  wj  1  e for 8e [ 0; j ¼ 1; 2; . . .; n; t ¼ 1; 2; . . ., we have ð1Þ

ð2Þ

ðtÞ

0  wj wj . . .wj  ð1  eÞt : For 8e [ 0, one has

lim ð1  eÞt ¼ 0. Based on the Squeeze Theorem in

t! þ 1

Mathematical Analysis, it holds that ð1Þ

ð2Þ

ðtÞ

lim wj wj . . .wj ¼ 0:

ð2:18Þ

t! þ 1

Consequently, taking limits on both sides of Eq. (2.17) and combining ðtÞ ðtÞ Eq. (2.18), we have lim aij ¼ 0. In the same way, we can prove lim bij ¼ 0. t! þ 1

t! þ 1

ðtÞ

ðtÞ

(ii) The conclusion lim cij ¼ lim dij ¼ 1 can be easily proved analogously. t! þ 1

t! þ 1

This completes the proof of Theorem 2.1. Due to the fact that an AIF matrix is a special case of an AIVIF matrix, this theorem is also valid for AIF matrices. In the following, we give an example to illustrate the above Theorem 2.1. Example 2.2 Let us consider the following initial AIVIF matrix:

2.2 Preliminaries

49

0

ð½0:70; 0:75; ½0:15; 0:20Þ B ð0Þ ~ G ¼@ ð½0:65; 0:70; ½0:15; 0:25Þ ð½0:45; 0:55; ½0:15; 0:25Þ ð½0:55; 0:60; ½0:35; 0:40Þ ð½0:10; 0:20; ½0:70; 0:75Þ ð½0:10; 0:20; ½0:50; 0:60Þ

ð½0:65; 0:75; ½0:20; 0:20Þ ð½0:45; 0:55; ½0:25; 0:45Þ ð½0:50; 0:60; ½0:25; 0:35Þ 1 ð½0:40; 0:20; ½0:35; 0:45Þ C ð½0:30; 0:45; ½0:55; 0:65Þ A ð½0:05; 0:10; ½0:70; 0:80Þ

Suppose that the weight vectors are wð1Þ ¼ ð0:10; 0:15; 0:40; 0:35ÞT and wð2Þ ¼ ð0:30; 0:15; 0:25; 0:30ÞT , respectively. By Definitions 2.4 and 2.10, we have 0

ð½0:113; 0:129; ½0:827; 0:851Þ B ð1Þ ~ G ¼@ ð½0:273; 0:307; ½0:657; 0:693Þ ð½0:100; 0:113; ½0:827; 0:871Þ ð½0:041; 0:085; ½0:867; 0:891Þ ð½0:058; 0:077; ½0:827; 0:871Þ ð½0:041; 0:085; ½0:758; 0:815Þ ð1Þ

ð½0:146; 0:188; ½0:786; 0:786Þ ð½0:075; 0:215; ½0:693; 0:756Þ ð½0:086; 0:113; ½0:812; 0:887Þ 1 ð½0:117; 0:189; ½0:811; 0:860Þ C ð½0:099; 0:128; ½0:812; 0:854Þ A ð½0:018; 0:036; ½0:883; 0:925Þ ð1Þ

ð0Þ

ð1Þ

~13 , namely, g ~13 ¼ 0:40 For example, from Definition 2.10, g~13 ¼ w3  g ð½0:55; 0:60; ½0:35; 0:40Þ. According to the operations in Definition 2.4, we obtain ð1Þ

~g13 ¼ ð½1  ð1  0:55Þ0:40 ; 1  ð1  0:60Þ0:40 ; ½ð0:35Þ0:40 ; ð0:40Þ0:40 Þ ¼ ð½0:273; 0:307; ½0:657; 0:693Þ: In the same way, we get 0

ð½0:036; 0:041; ½0:945; 0:953Þ B ð2Þ ~ G ¼@ ð½0:031; 0:036; ½0:945; 0:960Þ ð½0:018; 0:024; ½0:945; 0:959Þ ð½0:077; 0:088; ½0:900; 0:912Þ ð½0:011; 0:022; ½0:965; 0:972Þ ð½0:011; 0:022; ½0:933; 0:950Þ ð2Þ

ð2Þ

ð½0:023; 0:031; ½0:964; 0:964Þ ð½0:013; 0:018; ½0:969; 0:982Þ ð½0:016; 0:020; ½0:970; 0:977Þ 1 ð½0:023; 0:070; ½0:896; 0:920Þ C ð½0:037; 0:061; ½0:939; 0:956Þ A ð½0:005; 0:011; ½0:963; 0:977Þ

ð1Þ

For example, ~g13 ¼ w3  ~g13 , i.e., ð2Þ

~g13 ¼ 0:25  ð½0:273; 0:307; ½0:657; 0:693Þ ¼ ð½0:077; 0:088; ½0:900; 0:912Þ: Clearly, although there are dramatically different elements in the initial matrix ~ ð2Þ are very close to ~ ð0Þ , all the elements in the weighted AIVIF matrix G G ð½0; 0; ½1; 1Þ. Remark 2.1 From Theorem 2.1, if an AIVIF matrix is weighted an infinite number of times, all elements in it will approach the same AIVIFV, ð½0; 0; ½1; 1Þ, without

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

50

regard to the initial element values. Therefore, it is not suitable to apply weights on an AIVIF matrix too many times. Similarly, this observation holds true for an AIF matrix as well. Remark 2.2 In fact, for some AIVIF matrices, weighting them twice can make all elements quite close to the same AIVIFV. For example, the elements of the matrix ~ ð2Þ in Example 2.2 are close to ð½0; 0; ½1; 1Þ. In addition, the elements of the G collective matrix in Table 6 on page 11 in [46] are close to ð½0:23; 0:25; ½0:70; 0:75Þ since this collective matrix was obtained through weighting the individual matrices in Table 2.1 on page 10 in [46] twice (by using the DMs’ weights and attribute weights). Definition 2.11 [32] Let ~aj ¼ ð½aj ; bj ; ½cj ; dj Þ ðj ¼ 1; 2Þ be two AIVIFVs. The distance between them is defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dð~ a1 ; ~ a2 Þ ¼

1 ½ða 4 1

 a2 Þ2 þ ðb1  b2 Þ2 þ ðc1  c2 Þ2 þ ðd1  d2 Þ2 þ ðpl~a1  pl~a2 Þ2 þ ðpr~a1  pr~a2 Þ2 ;

ð2:19Þ where pl~aj ¼ 1  bj  dj and pr~aj ¼ 1  aj  cj ðj ¼ 1; 2Þ. According to Definition 2.11, we define the distance between two AIVIF matrices. ~ k ¼ð½ak ; bk ; ½ck ; d k Þ ðk ¼ 1; 2Þ be two AIVIF matrices. Definition 2.12 Let G ij ij ij ij mn 1 2 ~ ~ The distance between G and G is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m X n u 1 X 2l 2 1r 2r 2 dðG ; G Þ ¼ t ½ða1ij  a2ij Þ2 þ ðb1ij  b2ij Þ2 þ ðc1ij  c2ij Þ2 þ ðdij1  dij2 Þ2 þ ðp1l ij  pij Þ þ ðpij  pij Þ  ~1

~2

4mn

i¼1 j¼1

ð2:20Þ

~ k ðk ¼ 1; 2; 3; 4Þ given by DM ek ðk ¼ 1; 2; 3; 4Þ Table 2.1 AIVIF decision matrices R DM

Alternative

Attribute u1

u2

u3

u4

u5

e1

A1

([0.40,0.90], [0.05,0.10]) ([0.65,0.75], [0.15,0.25]) ([0.30,0.40], [0.05,0.15]) ([0.10,0.50], [0.15,0.30])

([0.50,0.90], [0.05,0.10]) ([0.65,0.70], [0.15,0.15]) ([0.45,0.55], [0.15,0.25]) ([0.20,0.60], [0.10,0.30])

([0.40,0.80], [0.10,0.20]) ([0.60,0.75], [0.15,0.15]) ([0.10,0.50], [0.15,0.15]) ([0.45,0.55], [0.35,0.45])

([0.63,0.96], [0.02,0.04]) ([0.66,0.78], [0.10,0.20]) ([0.42,0.54], [0.25,0.35]) ([0.12,0.57], [0.23,0.43])

([0.10,0.40], [0.30,0.50]) ([0.70,0.80], [0.10,0.20]) ([0.50,0.60], [0.20,0.30]) ([0.20,0.30], [0.50,0.60])

A2 A3 A4

(continued)

2.2 Preliminaries

51

Table 2.1 (continued) DM

Alternative

Attribute u1

u2

u3

u4

u5

e2

A1

([0.10,0.40], [0.40,0.60]) ([0.15,0.25], [0.65,0.70]) ([0.75,0.85], [0.10,0.15]) ([0.60,0.90], [0.10,0.10]) ([0.40,0.75], [0.10,0.15]) ([0.75,0.85], [0.10,0.15]) ([0.35,0.45], [0.05,0.15]) ([0.15,0.55], [0.20,0.45]) ([0.50,0.75], [0.05,0.05]) ([0.55,0.80], [0.10,0.15]) ([0.55,0.60], [0.10,0.30]) ([0.20,0.60], [0.10,0.40])

([0.35,0.95], [0.05,0.05]) ([0.45,0.55], [0.10,0.15]) ([0.50,0.60], [0.25,0.35]) ([0.15,0.55], [0.20,0.40]) ([0.55,0.65], [0.10,0.15]) ([0.55,0.80], [0.10,0.10]) ([0.40,0.50], [0.30,0.40]) ([0.10,0.45], [0.25,0.35]) ([0.20,0.50], [0.15,0.25]) ([0.85,0.95], [0.05,0.05]) ([0.80,0.90], [0.05,0.10]) ([0.60,0.70], [0.10,0.15])

([0.45,0.85], [0.05,0.15]) ([0.55,0.65], [0.10,0.15]) ([0.05,0.45], [0.05,0.35]) ([0.50,0.60], [0.20,0.30]) ([0.10,0.40], [0.10,0.30]) ([0.20,0.25], [0.30,0.40]) ([0.50,0.70], [0.10,0.30]) ([0.85,0.95], [0.05,0.05]) ([0.70,0.85], [0.05,0.15]) ([0.50,0.95], [0.05,0.05]) ([0.20,0.55], [0.05,0.35]) ([0.35,0.60], [0.10,0.25])

([0.58,0.85], [0.05,0.10]) ([0.70,0.88], [0.06,0.12]) ([0.52,0.65], [0.13,0.23]) ([0.37,0.77], [0.13,0.20]) ([0.55,0.90], [0.05,0.05]) ([0.82,0.92], [0.05,0.08]) ([0.38,0.49], [0.12,0.37]) ([0.10,0.62], [0.24,0.37]) ([0.35,0.50], [0.10,0.30]) ([0.20,0.55], [0.40,0.45]) ([0.78,0.83], [0.12,0.15]) ([0.75,0.95], [0.02,0.05])

([0.15,0.35], [0.25,0.45]) ([0.65,0.85], [0.05,0.15]) ([0.55,0.65], [0.15,0.35]) ([0.25,0.45], [0.45,0.55]) ([0.35,0.55], [0.25,0.45]) ([0.65,0.75], [0.15,0.25]) ([0.65,0.85], [0.05,0.15]) ([0.15,0.25], [0.55,0.75]) ([0.15,0.25], [0.45,0.65]) ([0.65,0.75], [0.05,0.15]) ([0.55,0.65], [0.25,0.35]) ([0.35,0.45], [0.45,0.55])

A2 A3 A4 e3

A1 A2 A3 A4

e4

A1 A2 A3 A4

where pklij ¼ 1  bkij  dijk m; j ¼ 1; 2; . . .nÞ.

2.3

and

k k pkr ij ¼ 1  aij  cij ðk ¼ 1; 2; i ¼ 1; 2; . . .;

A Novel Method for MAGDM with AIVIFVs and Incomplete Attribute Weight Information

In this section, we first describe the problems of MAGDM with AIVIFVs and incomplete attribute weight information and then propose a new method for solving these problems.

2.3.1

Presentation of the Problems

For the sake of convenience, let M ¼ f1; 2; . . .; mg, N ¼ f1; 2; . . .; ng and K ¼ f1; 2; . . .; sg. The MAGDM problems concerned in this chapter are described as follows.

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

52

Let A ¼ fA1 ; A2 ; . . .; Am g be the set of m feasible alternatives, U ¼ fu1 ; u2 ; . . .; un g be the set of attributes and E ¼ fe1 ; e2 ; . . .; es g be the set of DMs. Assume that w ¼ ðw1 ; w2 ; . . .; wn ÞT is the attribute weight vector, where P 0  wj  1ðj 2 NÞ and nj¼1 wj ¼ 1. Suppose that ½akij ; bkij  and ½ckij ; dijk  provided by the DM ek are respectively the membership (or satisfaction) degree interval and non-membership (or dissatisfaction) degree interval of the alternative Ai on attribute uj , where ½akij ; bkij ½0; 1 and ½ckij ; dijk ½0; 1 with bkij þ dijk  1. In other words, the rating of alternative Ai on the attribute uj provided by the DM ek is an AIVIFV ~rijk ¼ ð½akij ; bkij ; ½ckij ; dijk Þði 2 M; j 2 N; k 2 KÞ. Thus, the individual decision matrix ~ k ¼ ð~r k Þ . given by DM ek can be denoted as R ij mn

Usually, the attribute weights are required to satisfy the normalization condin P Pm m tions: j¼1 wj ¼ 1 and wj eðj ¼ 1; 2; . . .; mÞ. We denote D0 ¼ w j¼1 wj ¼ 1; wj e; for j ¼ 1; 2; . . .; m:g, where e is a sufficiently small positive number to ensure that the weights obtained are not zeros. Due to the uncertainty of the decision-making environment and the DMs’ limited knowledge, DMs can often only supply partial information about attribute weights. In other words, information on attribute weights is incomplete. Incomplete information structures of attribute weights are often given in the following five basic relations among attributes [18, 32–34]: (1) (2) (3) (4) (5)

A ranking with multiples: wh #hj wj , 0  #hj  1; A weak ranking: wh wj ; A strict ranking: 0\ahj  wh  wj  bhj , 0  ahj ; bhj  1; An interval-valued form: 1j  wj  vj , 0  1j ; vj  1; A ranking of differences: wh  wj  ws  wg .

Denote by D the incomplete information of the attribute importance given by the DMs, which may consist of several or all of the five basic relations in D0 . The focus of this chapter is how to rank alternatives based on individual decision ~ k ðk 2 KÞ and on incomplete information D of the attributes’ importance. matrices R The proposed method is mainly devoted to solving two key issues: (i) Determine the DMs’ weights with respect to different attributes; (ii) Derive attribute weights objectively.

2.3.2

Determine the DMs’ Weights with Respect to Different Attributes

As mentioned previously, every DM is an expert in only some fields. Accordingly, it is more reasonable to assign different values for the weights of every DM on different attributes.

2.3 A Novel Method for MAGDM with AIVIFVs …

53

For each attribute uj , the attribute values of all alternatives given by DM ek can k k k ; ~r2j ; . . .; ~rmj Þ. Let kkj be the weight of DM ek be denoted as an AIVIF vector ~rkj ¼ ð~r1j with respect to attribute uj . To determine kkj , two aspects should be considered simultaneously. One aspect is the similarity degree measuring the similarity between the individual decision matrix given by DM ek and the collective one by the group of DMs. The other aspect is the proximity degree capturing the proximity between the individual decision matrix given by DM ek and those given by all other DMs. The method in [45] only utilized the similarity degree to determine the weights of the DMs and failed to consider the proximity degree. Unfortunately, a greater similarity degree may not guarantee a bigger proximity degree, as shown in Remark 2.4 (see Sect. 2.4.1 for details). With the above analysis in mind, we determine kkj ðj 2 N; k 2 KÞ from two aspects. On the one hand, motivated by the idea of TOPSIS [45], we extend the TOPSIS method to calculate the similarity degree. On the other hand, we calculate the proximity degree using the distance measure.

2.3.2.1

Calculate the Similarity Degree Based on an Extended TOPSIS

(1) Determine the positive ideal decision (PID) vector ~rj on attribute uj . The PID vector on attribute uj is defined as the arithmetic average of all individual AIVIF vectors ~rkj ðk 2 KÞ, i.e., ~rj ¼ ð~r1j ; ~r2j ; . . .; ~rmj Þ, where ~rij ¼ ð½a ij ; b ij ; ½c ij ; dij Þ ¼ AIVIFMw ð~rij1 ; ~rij2 ; . . .; ~rijs Þ ði 2 MÞ:

ð2:21Þ

(2) Determine all the negative ideal decision (NID) vectors on attribute uj . The NID vectors on attribute uj include the individual negative ideal decision (INID) vector, the left individual negative ideal decision (LINID) vector and the right individual negative ideal decision (RINID) vector. Denote the INID, LINID c c c l l l ; ~r2j ; . . .; ~rmj Þ, ~rl r1j ; ~r2j ; . . .; ~rmj Þ and RINID vectors on attribute uj by ~rcj ¼ ð~r1j j ¼ ð~ r r r r and ~rj ¼ ð~r1j ; ~r2j ; . . .; ~rmj Þ, respectively, where rijc ¼ ð½acij ; bcij ; ½ccij ; dijc Þ and acij ¼ c ij ; bcij ¼ dij ; ccij ¼ a ij ; dijc ¼ b ij ;

ð2:22Þ

l l l l k l k l rijl ¼ ð½al ij ; bij ; ½cij ; dij Þ and aij ¼ minfaij g; bij ¼ minfbij g; cij k

k

¼ maxfckij g; dijl ¼ maxfdijk g; k

ð2:23Þ

k

r r r r k l k r rijr ¼ ð½ar ij ; bij ; ½cij ; dij Þ and aij ¼ maxfaij g; bij ¼ maxfbij g; cij k

¼ minfckij g; dijr ¼ minfdijk g k

k

k

ð2:24Þ

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

54

(3) Compute the distances dð~rkj ; ~rj Þ, dð~rkj ; ~rcj Þ, dð~rkj ; ~rl rkj ; ~rr j Þ and dð~ j Þ by Eq. (2.20). (4) Compute the similarity degree. Let us denote by skj the similarity degree between the individual decision matrix given by the DM ek and the collective one by the group of DMs on attribute uj . Then, the similarity degree skj is calculated as skj ¼

dð~rkj ; ~rcj Þ þ dð~rkj ; ~rl rkj ; ~rr j Þ þ dð~ j Þ k ~ k ~c k ~l dð~rj ; rj Þ þ dð~rj ; rj Þ þ dð~rj ; rj Þ þ dð~rkj ; ~rr j Þ

ðj 2 N; k 2 KÞ:

ð2:25Þ

Since skj indicates the similarity degree between vector ~rkj ðk 2 KÞ and the PID vector ~rj , the larger the skj , the greater the weight kkj that should be assigned.

2.3.2.2

Calculate Proximity Degree Using the Distance Measure

Denote the proximity degree between ~rijk and ~rijl by nlkij , which can be computed as nlkij ¼ 1  dð~rijl ; ~rijk Þ;

ð2:26Þ

where dð~rijl ; ~rijk Þ is the distance between ~rijl and ~rijk using Definition 2.11. Compute the average proximity degree cð~rlj ; ~rkj Þ between ~rkj and ~rlj as follows: cð~rlj ; ~rkj Þ ¼

1 m

m X

nlkij :

ð2:27Þ

i¼1

The average proximity degree cð~rlj ; ~rkj Þ describes the degree of closeness between vector ~rkj and vector ~rlj on attribute uj , which represents the proximity degree between the individual information given by DM ek and that given by DM el on attribute uj . For attribute uj , the average proximity degree ckj between DM ek and all other DMs el ðl 2 K; l 6¼ kÞ is computed as ckj ¼

1 s1

s X

cð~rlj ; ~rkj Þ:

ð2:28Þ

l¼1;l6¼k

The larger the ckj , the more proximal the information given by DM ek is to that given by all other DMs, so the bigger the weight of DM ek on attribute uj that should be assigned.

2.3 A Novel Method for MAGDM with AIVIFVs …

2.3.2.3

55

Obtain the Weights of DMs with Respect to Different Attributes

To comprehensively consider the similarity and proximity degrees, we employ a control parameter g (0  g  1) to construct the combined weight  kkj of DM ek on attribute uj as follows: kk ¼ gsk þ ð1  gÞck : j j j

ð2:29Þ

The combined weight is an appealing concept since Eq. (2.29) can tradeoff the similarity degree versus the proximity degree by changing the values of parameter g. In particular, if g ¼ 0, then the weight kkj only depends on the proximity degree; if g ¼ 1, then the weight kk only depends on the similarity degree. In practical j

application, we can take g ¼ 0:5. Normalized combined weights kkj (k 2 K), the weight kkj of DM ek on attribute uj is obtained as kkj ¼ kkj

=

s X

kl ðj 2 N; k 2 KÞ: j

ð2:30Þ

l¼1

2.3.3

Converting Individual Decision Matrices into a Collective Interval Matrix

~ k ¼ ð~r k Þ Using the weights of DMs, individual decision matrices R ij mn (k 2 K) are ~ integrated into a collective decision matrix R ¼ ð~rij Þmn by the AIVIFWM operator (i.e., Eq. 2.14), where s X ~rij ¼ ð½aij ; bij ; ½cij ; dij Þ ¼ kkj~rijk k¼1 " # " #! s s s s X X X X k k k k k k k k ¼ kj aij ; kj bij ; kj cij ; kj dij : k¼1

k¼1

k¼1

ð2:31Þ

k¼1

~ is also an AIVIF matrix. Obviously, the collective decision matrix R To sufficiently consider the DMs’ risk preference, let rij ¼ \lij ; vij [ ¼ \ð1  h)aij þ hbij ; hcij þ ð1  h)dij [ :

ð2:32Þ

~ can be transformed into an AIF matrix Then, the collective decision matrix R R ¼ ðrij Þmn , where h means the risk coefficient of DMs and 0  h  1. When

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

56

0  h\0:5, the DMs are pessimistic and averse to risk; when h ¼ 0:5, the DMs are neutral; when 0.5\h  1, the DMs are optimistic and risk seeking. According to Theorem 2.1 and Remark 2.2, it is not suitable to weight an AIF matrix too many times. Inspired by Wu and Chiclana [37, 38], we directly convert the AIF matrix R into an interval matrix P ¼ ðpij Þmn , where pij ¼ ½lij ; 1  vij :

ð2:33Þ

Then, combining the attribute weight vector w ¼ ðw1 ; w2 ; . . .; wn ÞT with Eqs. (2.8) and (2.32), the comprehensive value of alternative Ai can be obtained as " Ti ¼ ¼

n X

" j¼1 n X

wj lij ;

n X

# wj ð1  vij Þ #

j¼1

wj ðð1  h)aij þ hbij Þ;

j¼1

n X

wj ð1  ðhcij þ ð1  h)dij ÞÞ :

ð2:34Þ

j¼1

For convenience, denote Ti ¼ ½TiL ; TiU , where TiL ¼

n X

wj ðð1  h)aij þ hbij Þ; TiU ¼

j¼1

n X

wj ð1  ðhcij þ ð1  h)dij ÞÞ:

ð2:35Þ

j¼1

If the attribute weights are already known, then the alternatives can be ranked and selected according to the Shapley value method [5] described in Sect. 2.2.2. Subsequently, a multi-objective interval-programming model is constructed to determine the attribute weights objectively.

2.3.4

Construct Multi-objective Interval-Programming for Deriving Attribute Weights

For alternative Ai , the larger the comprehensive value Ti , the better the alternative Ai . It is reasonable to maximize the comprehensive values of all alternatives for the purpose of determining attribute weights. Thus, a multi-objective programming model is constructed as follows: max s:t:

Ti ði 2 M) w2D

ð2:36Þ

Since the objective functions Ti ði 2 MÞ in Eq. (2.36) are all intervals, Eq. (2.36) is a multi-objective interval fuzzy programming model. To solve this programming

2.3 A Novel Method for MAGDM with AIVIFVs …

57

model, according to Eq. (2.2), Eq. (2.36) can be further rewritten as the following multi-objective programming model: max s:t:

fTiL ; mðTi Þg ði 2 MÞ w2D

ð2:37Þ

where mðTi Þ is the center of comprehensive interval value Ti , respectively. There exist many methods to solve multi-objective programming, which is usually transformed into a single objective programming model. To solve Eq. (2.37), let y ¼ minfTiL g and x ¼ minfmðTi Þg. By employing the Max-Min i

i

method, Eq. (2.37) can be transformed into the following bi-objective linear programming model: max y 8 max x ði 2 MÞ > < TiL y s:t: mðTi Þ x ði 2 MÞ > : w2D

ð2:38Þ

Finally, we further convert Eq. (2.38) into the following single objective linear programming model by using the equal weight summation method: max

y þ x

2 8 T y ði 2 MÞ > iL > < s:t: mðTi Þ x ði 2 MÞ > > : w2D

ð2:39Þ

By Eq. (2.35), it yields that " mðTi Þ ¼

1 ðT þ TiU Þ 2 iL

¼

1 2

n X

# wj ðð1  hÞðaij  dij Þ þ hðbij  cij Þ þ 1Þ :

j¼1

Thus, Eq. (2.39) can be rewritten as  y þ x 2 8 max n X > > > wj ðð1  h)aij þ hbij Þ y ði 2 MÞ > > > j¼1 > < n X s:t: 1 > ½ wj ðð1  hÞðaij  dij Þ þ hðbij  cij Þ þ 1Þ x ði 2 MÞ > >2 > j¼1 > > > : w2D ð2:40Þ

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

58

Therefore, the attribute weight vector w ¼ ðw1 ; w2 ; . . .; wm ÞT can be derived by solving Eq. (2.40) for different values of the risk coefficient h. Remark 2.3 To determine attribute weights, this chapter constructs a multi-objective interval programming model that maximizes the comprehensive interval values of all alternatives, whereas Park et al. [21] constructed linear programming model that maximizes score functions of alternatives, Wan and Li [33] constructed linear programming model that minimizes the inconsistency degree, and Jin et al. [17] extended the information entropy to AIVIF environment. Thus, the essential difference is that the present model is a multi-objective interval-programming model which belongs to fuzzy mathematical programming models, while those in previous work are linear programming models or information entropy. In addition, the proposed model sufficiently considers the risk preferences of DMs that were overlooked in previous work.

2.3.5

Decision Process and Algorithm for MAGDM with AIVIFVs

Based on the above analysis, the decision process and algorithmfor MAGDM are summarized as follows: ~ k ¼ ð~r k Þ Step 1. Each DM establishes his/her individual decision matrix R ij mn with AIVIFVs and the group of DMs gives the preference information D on the attributes’ importance. Step 2. Calculate the weight kkj of DM ek with respect to attribute uj ðk 2 K; j 2 NÞ by Eq. (2.30). ~ k ¼ ð~r k Þ ðk 2 KÞ into a collective Step 3. Integrate all individual decisions R ij mn ~ AIVIF matrix R ¼ ð~rij Þmn by Eq. (2.31). ~ into an interval matrix P by Eqs. (2.32)–(2.33). Step 4. Convert the matrix R Step 5. Construct multi-objective interval programming model Eq. (2.36) and transform it into linear programming model Eq. (2.40). Step 6. Solve Eq. (2.40) to obtain the attribute weights. Step 7. Compute the interval comprehensive values Ti ði ¼ 1; 2; . . .; mÞ by Eq. (2.34). Step 8. Rank the alternatives and select the best one using the Shapley value method described in Subsection 2.2.2. The above decision-making process is depicted in Fig. 2.2.

2.4 A Real-World R & D Project Selection Example … Establish the individual decision matrix R k and

59

Individual decision matrices R k (k ∈ K )

acquire the preference information D of attributes importance.

Determine vectors r jk (k ∈ K )

Calculate the weight λ jk of DM ek on attribute u j Calculate the similarity degree Calculate the proximity degree s kj by Eqs. (2.21)-(2.25) γ kj by Eqs. (2.26)-(2.28)

Obtain the collective matrix R by Eq. (2.31)

Calculate combined weight λ jk by Eq. (2.29)

Covert R into an interval matrix P by Eqs. (2.32)-(2.33)

Determine the weight λ jk of DM ek on attribute u j by Eq. (2.30) Construct Eq. (2.36) and transform into Eq. (2.40)

Solve Eq. (2.40) to obtain attribute weight vector

Compute the comprehensive values Ti

w

By Lingo Soft

By Eq. (2.34)

By the Shapley value Rank alternatives and select the best one.

method in Subsection 2.2.2

Fig. 2.2 Decision making process of the MAGDM with AIVIFVs and incomplete attribute weight information

2.4

A Real-World R & D Project Selection Example and Comparison Analyses

In this section, a real-world R & D project selection example is given to illustrate the application of the proposed method. Meanwhile, comparison analyses are also conducted to show the superiority of the proposed method.

2.4.1

A Real-World R & D Project Selection Problem and the Solution Process

A venture capital company desires to invest in an R & D project. After the market research and preliminary screening have been conducted, there are four potential R & D projects fA1 ; A2 ; A3 ; A4 g for further evaluation. To reduce risk and increase profits, the company invites four DMs fe1 ; e2 ; e3 ; e4 g to evaluate these four projects based on five factors (attributes), including organizing ability (u1 ), credit quality (u2 ), level of research and development (u3 ), profitability (u4 ) and debt servicing

60

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

ability (u5 ). Each DM provides the evaluation information for each of the four R & D projects on each attribute, as listed in Table 2.1. After further discussion and negotiation, the group of DMs gives incomplete information on the attributes’ importance as follows: ( ) w 2w ; 0:05  w  w  0:1; w  2w ; w  0:4; 1 1 4 5 3 2 2 D ¼ w 2 D0 : w4  w5 w2  w3 ; w5 0:05; w1 þ w2 þ w3 0:3: Step 1. See Table 2.1. Step 2. Determine the weights of the DMs with respect to each attribute. Take the weight kk1 of DM e1 with respect to attribute u1 as an example. (i) Calculate the similarity degree of DM e1 . The PID vector ~r1 for attribute u1 is calculated by Eq. (2.21) as ~r1 ¼ ð~r11 ; ~r21 ; ~r31 ; ~r41 Þ ¼ ðð½0:3500; 0:7000; ½0:1500; 0:2250Þ; ð½0:5250; 0:6625; ½0:2500; 0:3125Þ ð½0:4875; 0:5750; ½0:0750; 0:1875Þ; ð½0:2625; 0:6375; ½0:1375; 0:3125ÞÞ: Using Eqs. (2.22)–(2.24), INID (~rc1 ), LNID (~rl1 ) and RNID(~rr 1 ) vectors for attribute u1 can be easily identified. Then, by Eq. (2.20), we have dð~r11 ; ~r1 Þ ¼ 0:1507; dð~r11 ; ~rc1 Þ ¼ 0:3878; dð~r11 ; ~rl r11 ; ~rr 1 Þ ¼ 0:3326; dð~ 1 Þ ¼ 0:3127: According to Eq. (2.25), the similarity degree of DM e1 on attribute u1 is obtained as s11 ¼ 0:8727. (ii) Calculate the proximity degree of DM e1 Combining Eq. (2.19) with Eq. (2.26), we get the following proximity degrees: 21 21 21 n21 11 ¼ 0:5772; n21 ¼ 0:5114; n31 ¼ 0:5363; n41 ¼ 0:5832:

According to Eq. (2.27), the average proximity degree between ~r21 and ~r11 is calculated as cð~r21 ; ~r11 Þ ¼

1 4

Likewise, we obtain cð~r31 ; ~r11 Þ ¼ 14 0:8285.

4 X

n21 i1 ¼ 0:5520:

i¼1

P4 i¼1

n31 r41 ; ~r11 Þ ¼ 14 i1 ¼ 0:9038, cð~

P4 i¼1

n41 i1 ¼

2.4 A Real-World R & D Project Selection Example … Table 2.2 Similarity degree, proximity degree and combined weight of DM ek on attribute u1

61 DM e1

DM e2

DM e3

DM e4

Similarity degree sk1

0.8727

0.7700

0.8820

0.8891

Proximity degree fk1

0.7614

0.6008

0.7779

0.7701

0.8171

0.6854

0.8299

0.8296

The combined weight kk1

By using Eq. (2.28), the average proximity degree between DM e1 and the other three DMs is computed as c11 ¼

1 3

4 X

cð~rl1 ; ~r11 Þ ¼ 0:7614:

l¼2

(iii) Determine the weight of DM e1 on attribute u1 Taking g ¼ 0:5 in Eq. (2.29), the combined weight of DM e1 on attribute u1 is obtained as k11 ¼ 0:8171. Analogously, we can acquire the similarity degrees and proximity degrees of other DMs on attribute u1 , as well as the combined weights of the DMs on attribute u1 . These results are listed in Table 2.2. Remark 2.4 From Table 2.2, for attribute u1 , the similarity degrees of DM e3 and DM e4 are 0.8820 and 0.8891, respectively. Obviously, the similarity degree of DM e4 is greater than that of DM e3 . However, the proximity degree of DM e3 is greater than that of DM e4 , because the former is 0.7779 and the latter is 0.7701. Therefore, a greater similarity degree may not guarantee a greater proximity degree. According to Remark 2.4, the idea in Yue [45] that only considered the similarity degree is not sufficient to determine the weights of the DMs. It is more reasonable to consider the similarity degree and the proximity degree simultaneously, as this work does. By normalizing k11 ; k21 ; k31 and k41 , the weights of the four DMs with respect to u1 are obtained as follows: k11 ¼ 0:2584; k21 ¼ 0:2168; k31 ¼ 0:2625; k41 ¼ 0:2624: In the same way, the DMs’ weights on the other attributes can be computed and are shown in Table 2.3.

Table 2.3 Weights of each DM with respect to different attributes

e1 e2 e3 e4

u1

u2

u3

u4

u5

0.2584 0.2168 0.2625 0.2624

0.2603 0.2545 0.2548 0.2305

0.2587 0.2676 0.2230 0.2506

0.2576 0.2674 0.2548 0.2202

0.2574 0.2530 0.2472 0.2724

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

62

Step 3. Obtain the collective decision matrix. By using Eq. (2.31), the collective IVIF matrix is acquired as follows: 0

ð½0:361; 0:713; ½0:139; 0:208Þ B ð½0:542; 0:681; ½0:232; 0:295Þ e ¼B R B @ ð½0:476; 0:563; ½0:074; 0:189Þ

ð½0:405; 0:757; ½0:086; 0:158Þ ð½0:620; 0:745; ½0:102; 0:205Þ ð½0:531; 0:631; ½0:191; 0:279Þ

ð½0:248; 0:626; ½0:139; 0:322Þ ð½0:254; 0:572; ½0:164; 0:304Þ ð½0:422; 0:737; ½0:074; 0:196Þ ð½0:535; 0:814; ½0:053; 0:116Þ ð½0:472; 0:662; ½0:145; 0:181Þ ð½0:610; 0:792; ½0:143; 0:203Þ ð½0:201; 0:544; ½0:087; 0:287Þ

ð½0:516; 0:621; ½0:156; 0:279Þ

ð½0:528; 0:665; 0:180; 0:271Þ ð½0:321; 0:720; ½0:160; 0:270Þ 1 ð½0:187; 0:388; ½0:311; 0:511Þ ð½0:663; 0:788; ½0:088; 0:188Þ C C C ð½0:562; 0:687; ½0:162; 0:288Þ A ð½0:237; 0:362; ½0:488; 0:612Þ

Step 4–Step 6. Determine the vector of the attribute weights. By Eq. (2.40), the linear programming model is constructed as follows: y+ x max { 2 }

⎧(0.361 + 0.352θ ) w ⎪(0.542 + 0.139θ ) w ⎪ ⎪(0.476 + 0.087θ ) w ⎪ ⎪(0.248 + 0.378θ ) w ⎪ s.t . ⎨(0.576 + 0.211θ ) w ⎪(0.623 + 0.101θ ) w ⎪ ⎪(0.644 + 0.101θ ) w ⎪ ⎪(0.463 + 0.281θ ) w ⎪⎩ w ∈ D

1

+ (0.405 + 0.352θ ) w2 + (0.422 + 0.315θ ) w3 + (0.535 + 0.279θ ) w4 + (0.187 + 0.202θ ) w5 ≥ y;

1

+ (0.620 + 0.125θ ) w2 + (0.472 + 0.190θ ) w3 + (0.610 + 0.182θ ) w4 + (0.663 + 0.125θ ) w5 ≥ y;

1

+ (0.531 + 0.100θ ) w2 + (0.201 + 0.343θ ) w3 + (0.516 + 0.105θ ) w4 + (0.562 + 0.125θ ) w5 ≥ y;

1

+ (0.254 + 0.318θ ) w2 + (0.528 + 0.138θ ) w3 + (0.321 + 0.399θ ) w4 + (0.405 + 0.125θ ) w5 ≥ y;

1

+ (0.624 + 0.212θ ) w2 + (0.613 + 0.219θ ) w3 + (0.709 + 0.171θ ) w4 + (0.338 + 0.201θ ) w5 ≥ x;

1

+ (0.707 + 0.115θ ) w2 + (0.646 + 0.113θ ) w3 + (0.704 + 0.121θ ) w4 + (0.738 + 0.113θ ) w5 ≥ x;

1

+ (0.626 + 0.094θ ) w2 + (0.457 + 0.272θ ) w3 + (0.618 + 0.114θ ) w4 + (0.637 + 0.125θ ) w5 ≥ x;

1

+ (0.475 + 0.229θ ) w2 + (0.629 + 0.114θ ) w3 + (0.526 + 0.255θ ) w4 + (0.312 + 0.125θ ) w5 ≥ x;

(2.41)

Solving Eq. (2.41), we derive the attribute weights for different values of risk coefficient h, which are listed in Table 2.4. Step 7. Compute the comprehensive values of the alternatives using Eq. (2.34). Table 2.5 presents the comprehensive values of the alternatives. Step 8. Rank the alternatives and select the best project.

2.4 A Real-World R & D Project Selection Example …

63

Table 2.4 Weights of the attributes for different values of risk coefficient h h

w

h

w

0

(0.14018, 0.28036,0.43927,0.09018,0.05) (0.14203,0.28405, 0.43190,0.09203,0.05) (0.14421,0.28842, 0.42315,0.09421,0.05) (0.14684,0.29368, 0.41264,0.08968,0.05) (0.15016,0.30031, 0.39938,0.10016,0.05) (0.15430,0.30860, 0.38280,0.10430,0.05)

0.6

(0.15980,0.31960, 0.36080,0.10980,0.05) (0.16733,0.33466, 0.33068,0.11733,0.05) (0.17822,0.35644, 0.28712,0.12822,0.05) (0.18000,0.36000, 0.28000,0.13000,0.05) (0.18000,0.36000, 0.28000,0.1300,0.05)

0.1 0.2 0.3 0.4 0.5

0.7 0.8 0.9 1

In terms of the Shapley value method described in Sect. 2.2.2, the ranking orders and the best alternative(s) for different admissible orders are also shown in Table 2.5. Remark 2.5 The numbers of A2 being superior to A1 are the same as those of A1 being superior to A2 when h ¼ 0:7. Therefore, we should compute the Shapley values to compare alternatives A1 and A2 . By employing the Shapley value method (see page 10 in [5]), we obtain the Shapley values of alternatives A1 and A2 as uðA1 Þ ¼ 0:2392 and uðA2 Þ ¼ 0:2228, respectively. Since uðA1 Þ [ uðA2 Þ, the alternative A1 is better than A2 (i.e., A1  A2 ). As shown in Table 2.5, if the DMs are pessimistic (i.e., 0  h\0:5), the best alternative is A2 . If the DMs are neutral (i.e., h ¼ 0:5), the best alternative is still A2 . However, if the DMs are optimistic (i.e., 0:5\h  1), the best alternative changes from A2 to A1 . Clearly, the ranking orders of the alternatives may vary with different risk coefficients. The DMs are able to select the corresponding decision results according to their risk preferences. Therefore, it is very necessary and reasonable to take the risk preference of the DMs into consideration during the decision-making process.

2.4.2

Comparison with the Extended TOPSIS Method

To illustrate the superiority of the proposed method, we compare it with the TOPSIS method given by Yue [45]. Since method [45] was developed under the AIF environment, we first directly extend the method [45] to the AIVIF environment (called extended TOPSIS) and then use this extended TOPSIS to solve the above R & D project selection problem.

([0.407,0.804], [0.545,0.794] [0.378,0.730], [0.378,0.696]) ([0.439,0.815], [0.562,0.800] [0.401,0.744], [0.401,0.708]) ([0.472,0.825], [0.579,0.806] [0.424,0.759], [0.424,0.720]) ([0.504,0.835], [0.597,0.812] [0.447,0.773], [0.447,0.732]) ([0.536,0.845], [0.614,0.819] [0.470,0.788], [0.470,0.744]) ([0.569,0.855], [0.631,0.825] [0.492,0.802], [0.492,0.756]) ([0.601,0.864], [0.649,0.831] [0.515,0.815], [0.515,0.769]) ([0.634,0.874], [0.668,0.837] [0.538,0.828], [0.538,0.781]) ([0.667,0.883], [0.687,0.844] [0.561,0.840], [0.561,0.795]) ([0.700,0.892], [0.703,0.851] [0.579,0.852], [0.589,0.808]) ([0.732,0.901], [0.718,0.858] [0.596,0.865], [0.617,0.820])

0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

T ¼ ðT1 ; T2 ; T3 ; T4 Þ

h

 lex2 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1 A4 A3 A2 A1

 lex1

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A3 A4 A1 A2

A3 A4 A1 A2

Table 2.5 Comprehensive values of alternatives and the ranking results  XY

A4 A3 A2 A1

A4 A3 A2 A1

A4 A3 A2 A1

A4 A3 A2 A1

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

 1=3;2=3

A4 A3 A2 A1

A4 A3 A2 A1

A4 A3 A2 A1

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A4 A3 A1 A2

A1

A1

A1

A1

A2

A2

A2

A2

A2

A2

A2

Best

64 2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

2.4 A Real-World R & D Project Selection Example …

65

Suppose that the attribute weight vectors wk ¼ðwk1 ; wk2 ; . . .; wkn ÞT given by DM ek ðk ¼ 1; 2; 3; 4Þ are as follows: w1 ¼ ð0:15; 0:25; 0:15; 0:3; 0:15ÞT ; w2 ¼ ð0:35; 0:15; 0:25; 0:15; 0:1ÞT ; w3 ¼ ð0:1; 0:2; 0:3; 0:25; 0:15ÞT ; w4 ¼ ð0:2; 0:1; 0:12; 0:35; 0:23ÞT : The weights of the DMs are obtained as k1 ¼ 0:254, k2 ¼ 0:249, k3 ¼ 0:250, k4 ¼ 0:247. The relative closeness of the alternatives is computed as RC1 ¼ 0:940, RC2 ¼ 0:929, RC3 ¼ 0:936, RC4 ¼ 0:940. Thus, the ranking order is A1 A4  A3  A2 and the best alternative is A1 or A4 . That is to say, the extended TOPSIS cannot distinguish alternatives A1 and A4 . Since the venture capital company possesses limited capital, it can select only one project to invest. Further decision is needed to select the best one from projects A1 and A4 . In comparison with the extended TOPSIS method, the method proposed in this chapter has the following advantages: (1) The approach to determining the weights of the DMs in the latter is better than that in the former, because the weights of each DM obtained by the latter are different with respect to different attributes, whereas those obtained by the former are the same. In reality, since each DM may be an expert only in some, but not all subjects (attributes), the weights of each DM with respect to different attributes should be differentiated. (2) The latter objectively derives the attribute weights through constructing multi-objective interval programming, while the former gave the attributes’ weights in advance, which makes it very difficult to avoid subjective randomness. (3) In the latter, the risk preferences of the DMs are sufficiently considered during the decision-making process, which can provide more choices for DMs with different risk preferences. In contrast, the method [45] ignored the risk preferences of the DMs. (4) The latter has a stronger distinguishing power than the former. For example, the latter can always determine the sole best alternative for different risk preferences (see Table 2.5), whereas the former cannot distinguish which is better between alternative A1 and alternative A4 . The main reason is that the former first weighted the individual decision matrices by the weights of the DMs to obtain the collective decision matrix, and then weighted the collective decision matrix by the attribute weights to derive the comprehensive values of the alternatives. Such weighted integration operations may make the comprehensive values of the alternatives too close to each other, which weakens the distinguishing power of the method. This analysis also verifies Theorem 2.1 in Sect. 2.2.3.

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66

2.4.3

Comparison with Barrenechea et al.’s Method

In this subsection, we compare the Barrenechea et al.’s method [7] with the method proposed in this chapter. (1) The focuses of both methods are remarkably different. The former is to construct interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations and apply it to decision problems with a single DM, while the latter is to develop a new method for MAGDM with AIVIFVs and incomplete weight information. In the former, the DM only provides the pair-wise preference relations of the alternatives as a whole, rather than on each attribute, whereas in the latter, each DM gives assessment information on the alternatives on each attribute. The former uses real numbers to represent the preference relations of alternatives, while the latter utilizes the AIVIFVs to express the attribute values of the alternatives. (2) If there is only one DM, the latter can also be used to solve MADM with AIVIFVs. Namely, the latter can solve not only MADM with AIVIFVsbut also MAGDM with AIVIFVs. The former did not consider group decision problems. Therefore, the latter has a wider application scope than the former. (3) The decision-making process in the latter is relatively simple compared to that in the former. On the one hand, to get the comprehensive values of the alternatives, the latter uses only one aggregation operator, (i.e., an AIVIFWM operator), while the former utilized four operators. On the other hand, according to the comprehensive interval values of the alternatives, the former employed three ranking methods for each operator to rank the alternatives, which results in twelve ranking results of alternatives. The latter adopts five methods to rank alternatives and there are only five ranking results of alternatives. Hence, the workload and time cost in the latter are less than those in the former.

2.5

Conclusions

In this study, we investigated AIVIFMAGDM problems with incomplete attribute weight information. A novel method was developed for solving such problems. The primary contributions of this study are summarized as follows: (1) We discussed the asymptotic property of an AIVIF matrix by repeated weighting, which can caution us not to weight an AIVIF matrix an excessive number of times. (2) By considering both the similarity degree and the proximity degree, a new approach was developed to determine the weights of the DMs with respect to every attribute. The proposed approach is more comprehensive and convincing than the existing ones.

2.5 Conclusions

67

(3) The attribute weights were derived objectively through constructing multi-objective interval programming that was transformed into a linear programming model. (4) We considered the DMs’ risk preferences during the decision-making process. The DMs are able to select the corresponding decision results according to their risk preferences, which can provide more choices for the DMs and greatly enhance the decision-making flexibility. Future studies will extend the proposed method to the MAGDM problems with hesitant fuzzy sets [9, 27, 35] or linguistic variables [23, 29].

References 1. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 2. K. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications (Springer-Verlag, Heidelberg, 1999) 3. K. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343– 349 (1989) 4. H. Bustince, J. Fernandez, A. Kolesárová, R. Mesiar, Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst. 220, 69–77 (2013) 5. H. Bustince, M. Galar, B. Bedregal, A. Kolesárová, R. Mesiar, A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy sets applications. IEEE Trans. Fuzzy Syst. 21, 1150–1162 (2013) 6. G. Beliakov, H. Bustince, D.P. Goswami, U.K. Mukherjee, N.R. Pal, On averaging operators for Atanassov’s intuitionistic fuzzy sets. Inf. Sci. 181, 1116–1124 (2011) 7. E. Barrenechea, J. Fernandez, M. Pagola, F. Chiclana, H. Bustince, Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Appl. Decis. Making Knowl. Based Syst. 58, 33–44 (2014) 8. F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets Syst. 97, 33– 48 (1998) 9. N. Chen, Z.S. Xu, Hesitant fuzzy ELECTRE II approach: a new way to handle multi-criteria decision making problems. Inf. Sci. 292, 175–197 (2015) 10. T.Y. Chen, Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis. Inf. Sci. 261, 149–169 (2014) 11. T.Y. Chen, A prioritized aggregation operator-based approach to multiple criteria decision making using interval-valued intuitionistic fuzzy sets: a comparative perspective. Inf. Sci. 281, 97–112 (2014) 12. T.Y. Chen, Multiple criteria decision analysis using a likelihood-based outranking method based on interval-valued intuitionistic fuzzy sets. Inf. Sci. 286, 188–208 (2014) 13. S.M. Chen, T.S. Li, Evaluating students’ answerscripts based on interval-valued intuitionistic fuzzy sets. Inf. Sci. 235, 308–322 (2013) 14. H. Hashemi, J. Bazargan, S.M. Mousavi, B. Vahdani, An extended compromise ratio model with an application to reservoir flood control operation under an interval-valued intuitionistic fuzzy environment. Appl. Math. Model. 38, 3495–3511 (2014) 15. B. Huang, D.K. Wei, H.X. Li, Y.L. Zhuang, Using a rough set model to extract rules in dominance-based interval-valued intuitionistic fuzzy information systems. Inf. Sci. 221, 215– 229 (2013)

68

2 A New Method for Atanassov’s Interval-Valued Intuitionistic …

16. H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48, 219–225 (1990) 17. F.F. Jin, L.D. Pei, H.Y. Chen, L.G. Zhou, Interval-valued intuitionistic fuzzy continuous weighted entropy and its application to multi-criteria fuzzy group decision making. Knowl. Based Syst. 59, 132–141 (2014) 18. D.F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information. Appl. Soft Comput. 11, 3402–3418 (2011) 19. D.F. Li, S.P. Wan, Fuzzy linear programming approach to multiattribute decision making multiple types of attribute values and incomplete weight information. Appl. Soft Comput. 13, 4333–4348 (2013) 20. F.Y. Meng, Q. Zhang, X.H. Cheng, Approaches to multiple-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral with respect to generalized kShapley index. Knowl.-Based Syst. 37, 237–249 (2013) 21. J. H. Park, Il Y. Park, Y. C. Kwun, X. Tan, Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment. Appl. Math. Model. 35, 2544–2556 (2011) 22. W. Pedrycz, Granular Computing: Analysis and Design of Intelligent Systems (CRC Press, Francis Taylor, Boca Raton, 2013) 23. W. Pedrycz, M.L. Song, A granulation of linguistic information in AHP decision-making problems. Inf. Fusion 17, 93–101 (2014) 24. W. Pedrycz, Statistically grounded logic operators in fuzzy sets. Eur. J. Oper. Res. 2, 520–529 (2009) 25. W. Pedrycz, B.J. Park, S.K. Oh, The design of granular classifiers: a study in the synergy of interval calculus and fuzzy sets in pattern recognition. Pattern Recogn. 41, 3720–3735 (2008) 26. W. Pedrycz, G. Vukovich, Feature analysis through information granulation and fuzzy sets. Pattern Recogn. 35, 825–834 (2002) 27. J.J. Peng, J.Q. Wang, J. Wang, X.H. Chen, An extension of ELECTRE to multi-criteria decision-making problems with multi-hesitant fuzzy sets. Inf. Sci. 307, 113–126 (2015) 28. C.Q. Tan, A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet intergral-based TOPSIS. Expert Syst. Appl. 38, 3023–3033 (2011) 29. S.P. Wan, 2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making. Knowl. Based Syst. 45, 31–40 (2013) 30. S.P. Wan, J.Y. Dong, Possibility linear programming with trapezoidal fuzzy numbers. Appl. Math. Model. 38, 1660–1672 (2014) 31. S.P. Wan, J.Y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J. Comput. Syst. Sci. 80, 237–256 (2014) 32. S.P. Wan, D.F. Li, Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees. Omega 41, 925–940 (2013) 33. S.P. Wan, D.F. Li, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with Atanassov’s intuitionistic fuzzy truth degrees. IEEE Trans. Fuzzy Syst. 22, 300–312 (2014) 34. S.P. Wan, J.Y. Dong, Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degree. Inf. Fusion 26, 49–65 (2015) 35. J.Q. Wang, J.T. Wu, J. Wang, H.Y. Zhang, X.H. Chen, Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf. Sci. 288, 55–72 (2014) 36. J. Wu, F. Chiclana, Visual information feedback mechanism an attitudinal prioritization method for group decision making with triangular fuzzy complementary preference relations. Inf. Sci. 279, 716–734 (2014) 37. J. Wu, F. Chiclana, Non-dominance and attitudinal prioritisation methods for intuitionistic and interval-valued intuitionistic fuzzy preference relations. Expert Syst. Appl. 39, 13409– 13416 (2012)

References

69

38. J. Wu, F. Chiclana, A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions. Appl. Soft Comput. 22, 272–286 (2014) 39. J.P. Xu, F. Shen, A new outranking choice method for group decision making under Atanassov’s interval-valued intuitionistic fuzzy environment. Knowl. Based Syst. 70, 177– 188 (2014) 40. Z.S. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen Syst. 35, 417–433 (2006) 41. Z.S. Xu, Intuitionistic preference relations and their application in group decision making. Inf. Sci. 177, 2363–2379 (2007) 42. Z.S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis. 22, 215–219 (2007) 43. G.L. Xu, F. Liu, An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection. Appl. Math. Model. 37, 3929–3943 (2013) 44. J. Ye, Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Appl. Math. Model. 34, 3864–3870 (2010) 45. Z.L. Yue, TOPSIS-based group decision-making methodology intuitionistic fuzzy setting. Inf. Sci. 277, 141–153 (2014) 46. Z.L. Yue, Y. Jia, An application of soft computing technique in group decision making under interval-valued intuitionistic fuzzy environment. Appl. Soft Comput. 13, 2490–2503 (2013) 47. L. A, Zadeh, Fuzzy sets. Inf. Control 8, 338–356 (1965) 48. F. Zhang, J. Ignatius, C.P. Lim, Y.J. Zhao, A new method for deriving priority weights by extracting consistent numerical-valued matrices from interval-valued fuzzy judgment matrix. Inf. Sci. 279, 280–300 (2014) 49. X.F. Zhao, R. Lin, G.W. Wei, Fuzzy prioritized operators and their application to multiple attribute group decision making. Appl. Math. Model. 37, 2075–2086 (2013) 50. Q.S. Zhang, H.Y. Xing, F.C. Liu, J. Ye, P. Tang, Some new entropy measures for interval-valued intuitionistic fuzzy sets based on distances and their relationships with similarity and inclusion measures. Inf. Sci. 283, 55–69 (2014) 51. X.L. Zhang, Z.S. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making. Appl. Soft Comput. 26, 42–56 (2015) 52. L.G. Zhou, Z.F. Tao, H.Y. Chen, J.P. Liu, Continuous interval-valued intiuitionistic fuzzy aggregation operators and their applications to group decision making. Appl. Math. Model. 38, 2190–2205 (2014)

Chapter 3

Interval-Valued Intuitionistic Fuzzy Mathematical Programming Method for Hybrid Multi-criteria Group Decision Making with Interval-Valued Intuitionistic Fuzzy Truth Degrees

Abstract As an important component of group decision making, the hybrid multi-criteria group decision making (MCGDM) is very complex and interesting in real applications. The purpose of this chapter is to develop a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid MCGDM considering alternative comparisons with hesitancy degrees. The subjective preference relations between alternatives given by each decision maker (DM) are formulated as an IVIF set (IVIFS). The IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals and real numbers are used to represent the multiple types of criteria values. The information of criteria weights is incomplete. The IVIFS-type consistency and inconsistency indices are defined through considering the fuzzy positive and negative ideal solutions simultaneously. To determine the criteria weights, we construct a novel bi-objective IVIF mathematical programming model of minimizing the inconsistency index and meanwhile maximizing the consistency index, which is solved by the technically developed linear goal programming approach. The individual ranking order of alternatives furnished by each DM is subsequently obtained according to the comprehensive relative closeness degrees of alternatives to the fuzzy positive ideal solution. The collective ranking order of alternatives is derived through establishing a new multi-objective assignment model. A real example of critical infrastructure evaluation is provided to demonstrate the applicability and effectiveness of this method.




Keywords Multi-criteria group decision making Fuzzy mathematical programming model Interval-valued intuitionistic fuzzy set Linear programming technique for multidimensional analysis of preference




© Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_3




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72

3.1

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

Introduction

With the drastic development of modern information technology and economy, many decision making problems become more and more complex. It is very difficult or unrealistic for a single decision maker (DM) or expert to cope with a complex or important issue. Thus, group decision making (GDM) has attracted intensive concern in the decision analysis area. As an important component of GDM, fuzzy multi-criteria group decision making (MCGDM) (or multi-attribute group decision making (MAGDM)) is an intractable research subject due to the fuzziness and uncertainty of objective things and human thinking [1]. As the generalizations of fuzzy sets [2], the intuitionistic fuzzy (IF) set (IFS) [3] and interval-valued intuitionistic fuzzy (IVIF) set (IVIFS) [4] are the powerful tools to deal with imperfect and imprecise information. A prominent characteristic of IFS and IVIFS is that they assign to each element a membership degree and a non-membership degree. Over the last decades, many researchers have investigated the theories of IFS and IVIFS and applied to various fields, such as decision making [5–8], image fusion [9], and so on. Considering the prioritization relationship over attributes, Yu and Xu [5] defined the prioritized intuitionistic fuzzy aggregation operators. Li and He [6] investigated the intuitionistic fuzzy PRI-AND and PRI-OR aggregation operators and applied to multi-attribute decision making (MADM) problem under intuitionistic fuzzy environment. Zhang [7] claimed that these prioritized aggregation operators ignored the relationship between the values being fused. Thus, he developed a series of generalized intuitionistic fuzzy power geometric operators and employed these aggregation operators to propose some methods for MAGDM with IFSs. Liu et al. [8] constructed a partial binary tree DEA-DA cyclic classification model for decision makers in complex multi-attribute large-group interval-valued intuitionistic fuzzy decision-making problems. Generally, MCGDM involves multiple different qualitative and quantitative criteria, the assessments of these criteria may be expressed with different formats, such as real numbers, intervals, linguistic variables, trapezoidal fuzzy numbers (TrFNs), IFSs, IVIFSs. Such a MCGDM with multiple formats of information is called the hybrid (or heterogeneous) MCGDM, which is very complex and interesting in real applications [10–15]. The aforementioned recent and interesting studies [4–8] seem to be efficient to handle MADM or MCGDM with IFSs or IVIFSs. However, they only considered the assessment information of criteria as single types (i.e., IFSs or IVIFSs) and cannot be used to solve hybrid MCGDM problems. Therefore, incorporating IFSs and IVIFSs into the hybrid MCGDM is of great importance for scientific research and actual application. Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) developed by Srinivasan and Shocker [16] is one of typical methods for solving MCGDM problems. It is based on pair-wise comparisons of alternatives given by DM and generates the best compromise alternative as the solution that has the shortest distance to the ideal solution (IS). The main advantages of LINMAP lie in two aspects: (1) It not only considers the preferences of DM on pair-wise

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73

comparisons of alternatives but also takes the assessments of alternatives on multiple attributes into account; (2) Through constructing linear programming of minimizing the consistency index, LINMAP can derive objectively the attribute weights and IS. Currently, the classic LINMAP [16] has been extended and some fuzzy LINMAP methods [17–21] have been proposed under fuzzy or IF or IVIF environments. Nevertheless, there are some limitations in these existing LINMAP methods [16–21]. First, in the classic LINMAP [16], all decision data are known precisely or given as crisp values. Owing to inherent complexity and uncertainty in real-life decision problems, it is often impractical to require a DM to provide his/her judgment in precise numerical values. The fuzzy sets [2], linguistic variables [22], TrFNs [23], IFSs [3], IVIFSs [4] are usually more adequate or sufficient to model real-life decision problems than real numbers. Second, the classic LINMAP [16] and fuzzy LINMAP methods [17–21] also only considered single type of attributes and cannot deal with hybrid MCGDM. Third, in the classical LINMAP [16] and fuzzy LINMAP [17–21], the DM gives pair-wise comparisons of alternatives in the form of the ordered pairs with crisp truth degrees 0 or 1. However, in the real world, DM is not sure enough in all comparisons and may express his/her opinion with a fuzzy truth degree [24–29]. Although Sadi-Nezhad and Akhtari [24] considered the fuzzy truth degree as a triangular fuzzy number (TFN) and proposed the possibility LINMAP in MAGDM, it still only considered single type of attributes and was not suitable for hybrid MCGDM. Li and Wan [25] represented the fuzzy truth degree as a TrFN and proposed the fuzzy linear programming method of multi-attribute decision making (MADM). The method [25] is only an extension of possibility LINMAP [24]. Li and Wan [26] developed fuzzy heterogeneous MADM method for outsourcing provider selection integrating technique for order preference by similarity to ideal solution (TOPSIS) [30] and LINMAP. Due to time pressure, lack of knowledge (or data), and limited expertise about the problem domain, DMs usually give the pair-wise comparisons of alternatives with some hesitancy degrees [27, 28]. Thus, Wan and Li [27, 28] used IFSs to represent IF truth degrees of alternatives’ comparisons and proposed fuzzy heterogeneous MADM and MAGDM methods. Zhang and Xu [29] developed interval programming method for hesitant fuzzy MAGDM with interval truth degrees on alternatives’ comparisons. It is not difficult to find that the LINMAP has been extended into various forms for solving MADM or MCGDM problems under a variety of different environments. However, these LINMAP-based methods are not appropriate for the MCGDM problems with IVIF truth degrees on alternatives’ comparisons. In fact, since IVIFSs use the intervals to characterize the membership and non-membership degrees, while IFSs use the real numbers to represent the membership and non-membership degrees, IVIFSs have stronger ability and flexibility to express the hesitant fuzzy truth degrees than IFSs. Therefore, it is necessary and natural to extend the LINMAP to suit hybrid MCGDM problems considering comparisons of alternatives with IVIF truth degrees. In this chapter, we firstly use the IVIFSs to capture the hesitant fuzzy truth degrees of alternative comparisons and develop a

74

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new IVIF mathematical programming method for solving hybrid MCGDM with IVIF truth degrees and incomplete weight information. The main motivations of the proposed method in this chapter are summarized as follows: (1) The existing LINMAP [16–21, 24–29] only minimized the inconsistency index, which cannot assure that the consistency index achieves the maximum. In fact, the inconsistency and consistency indexes are of equal importance in GDM. Minimizing the inconsistency index and meanwhile maximizing consistency index should be taken into account, which is the biggest motivation of this chapter. (2) The methods [24–29] considered the fuzzy truth degrees of alternatives’ comparisons, they respectively utilized TFNs, TrFNs, IFSs, and intervals to characterize the fuzzy truth degrees. Nevertheless, IVIFSs have stronger ability and flexibility to express the uncertainty and hesitancy than IFSs since IVIFSs represents the membership and non-membership degrees by two closed intervals of the interval [0,1]. In the real-life hybrid MCGDM, DMs are not always certain about their given preference information and they often have some degree of uncertainty. It is more suitable to utilize IVIFSs to capture the fuzzy truth degrees, which is the second motivation of this chapter. (3) Most of the prior studies [16–29] only adopted the distances of alternatives to fuzzy positive ideal solution (FPIS) to rank alternatives and ignored the fuzzy negative ideal solution (FNIS). According to TOPSIS [30], the FNIS is as important as the FPIS during the process of decision making. To remedy this flaw of ignoring the FNIS, we define the IVIFS-type consistency and inconsistency by the relative closeness degree on the basis of TOPSIS, which can make decision results more reasonable and convincing. (4) The methods [17, 19–21, 24, 28, 29] utilized the social choice functions (such as Borda’s score and Copeland’s function) to obtain the group ranking order of alternatives. It could happen that there exist two or more alternatives with the same Borda’s score, i.e., a total order of the set of alternatives are not guaranteed. This fact leads us to construct a new multi-objective assignment model to derive the collective ranking order of alternatives. This model ensures that different alternatives are ranked in different positions and can effectively overcome this limitation. The rest of this chapter is planned as follows. Section 3.2 reviews some concepts related to IVIFSs, IFSs, TrFNs as well as linguistic variables. In Sect. 3.3, the hybrid MCGDM problems with IVIF truth degrees and incomplete weight information are described. The normalization method is also presented. In Sect. 3.4, a new bi-objective IVIF mathematical programming model is constructed and solved by the developed linear goal programming approach. Hereby, the GDM method is then proposed for solving this sort of hybrid MCGDM problems. Section 3.5 illustrates the proposed method with a real critical infrastructure evaluation example and makes comparison analyses with the existing LINMAP methods. Section 3.6 ends the paper with some concluding remarks.

3.2 Basic Concepts

3.2

75

Basic Concepts

In this section, we first review some basic concepts related to IFSs [3], IVIFSs [4], trapezoidal fuzzy numbers, and triangular fuzzy numbers [23] as well as linguistic variables [22].

3.2.1

Concepts of Interval-Valued Intuitionistic Fuzzy Sets and Distances

Let U ¼ fu1 ; u2 ; . . .; um g be a finite universe of discourse. An IVIFS A in U is an A ðui Þ; ½mA ðui Þ; mA ðui Þ [ jui 2 Ug, where object in the form: A ¼ f\ui ; ½lA ðui Þ; l A ðui Þ and ½mA ðui Þ; mA ðui Þ denote respectively the membership degree ½lA ðui Þ; l interval and the non-membership degree interval of u to A, satisfying the conditions: A ðui Þ þ mA ðui Þ  1 for ui 2 U. Let pA ðui Þ ¼ ½1  l A ðui Þ lA ðui Þ  0, mA ðui Þ  0, l mA ðui Þ; 1  lA ðui Þ  mA ðui Þ, which is called the IVIF index of an element ui in A. A ðui Þ; ½mA ðui Þ; mA ðui Þ [ jui 2 Ug and Definition 3.1 [4] Let A ¼ f\ui ; ½lA ðui Þ; l B ðui Þ; ½mB ðui Þ; mB ðui Þ [ jui 2 Ug be two IVIFSs in U. Then, B ¼ f\ui ; ½lB ðui Þ; l we stipulate: A ðui Þ½lB ðui Þ; l B ðui Þ and ½mA ðui Þ; mA ðui Þ  (1) AB if and only if ½lA ðui Þ; l ½mB ðui Þ; mB ðui Þ for any ui 2 U. Namely, AB iff lA ðui Þ  lB ðui Þ, A ðui Þ  l B ðui Þ, mA ðui Þ  mB ðui Þ and mA ðui Þ  mB ðui Þ. l (2) A ¼ B if and only if AB and BA; (3)

 A ðui Þ þ l B ðui Þ  l  A ðui Þ A þ B ¼ f\ui ; ½lA ðui Þ þ lB ðui Þ  lA ðui ÞlB ðui Þ; l lB ðui Þ; ½mA ðui ÞmB ðui Þ; mA ðui ÞmB ðui Þ [ jui 2 Ug;

(4) kA ¼ f\ui ; ½1  ð1  lA ðui ÞÞk ; 1  ð1  lA ðui ÞÞk ; ½ðmA ðui ÞÞk ; ðmA ðui ÞÞk  [ jui 2 Ug, where k [ 0. of the elements ui 2 U, which satisfy the Let wi ði ¼ 1; 2; . . .; mÞ be the weightsP m normalized conditions: wi 2 ½0; 1 and i¼1 wi ¼ 1. The weighted Minkowski distance between IVIFSs A and B is defined as follows: dq ðA; BÞ ¼ ½

m  q 1X    A ðui Þ  l B ðui Þjq þ jmA ðui Þ  mB ðui Þjq þ jmA ðui Þ  mB ðui Þjq wi ðlA ðui Þ  lB ðui Þ þ jl 4 i¼1

A ðui Þ  p B ðui Þjq Þ1=q : þ jpA ðui Þ  pB ðui Þjq þ jp

ð3:1Þ When q ¼ 1, q ¼ 2 and q ! þ 1, the corresponding d1 ðA; BÞ, d2 ðA; BÞ and d þ 1 ðA; BÞ are called the weighted Hamming, Euclidean and Chebyshev distances, respectively.

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76

If an IVIFS A containsD only one element, E i.e., the cardinality j Aj ¼ 1, then A is A ; ½tA ; tA  for short. If lA ðui Þ ¼ l A ðui Þ ¼ lA ðui Þ usually denoted as A ¼ ½lA ; l A ðui Þ; and mA ðui Þ ¼ mA ðui Þ ¼ mA ðui Þ, then the IVIFS A ¼ f\ui ; ½lA ðui Þ; l  ½mA ðui Þ; mA ðui Þ [ jui 2 Ug is reduced to IFS A ¼ f\ui ; lA ðui Þ; mA ðui Þ [ jui 2 Ug, and the above Eq. (3.1) is reduced to the weighted Minkowski distance between IFSs [31].

3.2.2

Definition and Distance for Trapezoidal Fuzzy Numbers and Triangular Fuzzy Numbers

~ ¼ ðm1 ; m2 ; m3 ; m4 Þ is called a TrFN if its membership function is A quadruple m 8 ðx  m1 Þ=ðm2  m1 Þ ðm1  x\m2 Þ > > < 1 ðm2  x  m3 Þ ð3:2Þ lm~ ðxÞ ¼ ðm4  xÞ=ðm4  m3 Þ ðm3 \x  m4 Þ > > : 0 ðx\m1 or x [ m4 Þ where m1  m2  m3  m4 are real numbers and reflect the fuzziness of evaluation data [23]. ~ ¼ ðm1 ; m2 ; m3 ; m4 Þ is reduced to a TFN m ~ ¼ ðm1 ; m2 ; m4 Þ if A TrFN m ~ ¼ ðm1 ; m2 ; m3 ; m4 Þ is called a normalized positive TrFN if m2 ¼ m3 . TrFN m m1  0 and m4  1. ~ ¼ ðm1 ; m2 ; m3 ; m4 Þ and ~ The distance between two TrFNs m n ¼ ðn1 ; n2 ; n3 ; n4 Þ is defined as follows [18]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ~ ~nÞ ¼ ½ðm1  n1 Þ2 þ 2ðm2  n2 Þ2 þ 2ðm3  n3 Þ2 þ ðm4  n4 Þ2 : dðm; 6

ð3:3Þ

~ ¼ ðm1 ; m2 ; m3 Þ and ~ The distance between two TFNs m n ¼ ðn1 ; n2 ; n3 Þ is defined as follows [19]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ~ ~nÞ ¼ ½ðm1  n1 Þ2 þ ðm2  n2 Þ2 þ ðm3  n3 Þ2 : dðm; 3

3.2.3

ð3:4Þ

Linguistic Variables

In MCGDM, many of the attributes are intangible or non-monetary because they reflect social and environmental impacts. These attributes are not always expressed as crisp numerical values. It is more suitable to provide their preferences by means of linguistic variables [22]. A linguistic variable is a variable whose values are

3.2 Basic Concepts Table 3.1 Relations between linguistic variables and TFNs

77 Linguistic variable

Triangular fuzzy number

s0 s1 s2 s3 s4

(0.8, 0.9, 1) (0.6, 0.7, 0.8) (0.4, 0.5, 0.6) (0.2, 0.3, 0.4) (0, 0.1, 0.3)

= = = = =

Very high High Fair Low Very low

linguistic terms. For example, as evaluating the “vulnerability” of an emergency operating center, linguistic variables like “very high”, “high”, “fair”, “low”, and “very low” are usually used. Such linguistic values can be transformed into positive TFNs. For example, “very low” and “high” can be represented by positive TFNs (0, 0.1, 0.3) and (0.6, 0.7, 0.8), respectively. Let S ¼ fs0 ; s1 ; s2 ; . . .; st1 g be a finite and totally ordered discrete linguistic term set with odd cardinality, where si represents a possible value for a linguistic variable, t is an odd number [22]. For example, S = {s0 = “very high”, s1 = “high”, s2 = “fair”, s3 = “low”, s4 = “very low”}. In this chapter, the transformed relations between linguistic variables and TFNs are listed in Table 3.1 [19].

3.3

Hybrid MCGDM Problems Considering Alternative Comparisons with IVIF Truth Degrees

In this section, we describe the hybrid MCGDM problems and give the normalization method.

3.3.1

Hybrid MCGDM Problems with IVIF Truth Degrees and Incomplete Weight Information

The following symbols are used to characterize the MCGDM problems considered in this chapter. (1) The set of DMs (or experts) ep ðp 2 Q ¼ f1; 2; . . .; qgÞ. (2) The set of alternatives A ¼ fa1 ; a2 ; . . .; an g, where aj represents the j-th alternative ðj 2 N ¼ f1; 2; . . .; ngÞ. (3) The set of criteria C ¼ fc1 ; c2 ; . . .; cm g, where ci represents the i-th criterion ði 2 I ¼ f1; 2; . . .; mgÞ. Let C1 ¼ fc1 ; c2 ; . . .; ci1 g, C2 ¼ fci1 þ 1 ; ci1 þ 2 ; . . .; ci2 g, C3 ¼ fci2 þ 1 ; ci2 þ 2 ; . . .; ci3 g, C4 ¼ fci3 þ 1 ; ci3 þ 2 ; . . .; ci4 g, C5 ¼ fci4 þ 1 ; ci4 þ 2 ; . . .; ci5 g and C6 ¼ fci5 þ 1 ; ci5 þ 2 ; . . .; cm g be six subsets of C, representing criteria whose values are in

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

78

the form of IVFSs, IFSs, TrFNs, linguistic variables, interval values and real numbers, respectively, where 1  i1  i2  i3  i4  i5  m, Ct \ Ck ¼ £ ðt; k ¼ 6 S 1; 2; . . .; 6; t 6¼ kÞ and Ct ¼ C, £ is an empty set. Denote I1 ¼ f1; 2; . . .; i1 g, t¼1

I2 ¼ fi1 þ 1; i1 þ 2; . . .; i2 g, I3 ¼ fi2 þ 1; i2 þ 2; . . .; i3 g, I4 ¼ fi3 þ 1; i3 þ 2; . . .; i4 g,

I5 ¼ fi4 þ 1; i4 þ 2; . . .; i5 g, and I6 ¼ fi5 þ 1; i5 þ 2; . . .; mg.

(4) The fuzzy decision matrices Let the rating of alternative aj on criterion ci given by DM ep be denoted by ypij . If 0ijp ; ½t0ijp ; t0ijp ig, usually denoted by ypij ¼ i 2 I1 , ypij is an IVIFS fhðci ; aj Þ; ½l0ijp ; l 0ijp ; ½t0ijp ; t0ijp i for short; If i 2 I2 , ypij is an IFS fhðci ; aj Þ; l0ijp ; t0ijp ig, usually h½l0ijp ; l

denoted by ypij ¼ hl0ijp ; t0ijp i for short; If i 2 I3 , ypij ¼ ðb0ij1p ; b0ij2p ; b0ij3p ; b0ij4p Þ is a TrFN; if 0p 0p 0p i 2 I4 , ypij ¼ spij is a linguistic variable and transformed into TFN ypij ¼ ðdij1 ; dij2 ; dij3 Þ using Table 3.1; if i 2 I5 , ypij ¼ ½fij0 p ; g0ijp  is an interval; if i 2 I6 , ypij ¼ z0ijp is a real 0ijp þ t0ijp  1, l0ijp  0, t0ijp  0, l0ijp þ t0ijp  1, number, where l0ijp  0, t0ijp  0, l

0  b0ij1p  b0ij2p  b0ij3p  b0ij4p , Namely,

0p 0p 0p 0  dij1  dij2  dij3  1,

8 0p 0p 0p 0p ij ; ½tij ; tij i h½lij ; l > > > > 0 p > > hl ; t0 p i > > 0ijp ij0 p 0 p 0 p < ðbij1 ; bij2 ; bij3 ; bij4 Þ ypij ¼ 0p 0p 0p > ; dij2 ; dij3 Þ ðdij1 > > > 0 p 0 p > > ½fij ; gij  > > : 0p zij

0  fij0 p  g0ijp

and

z0ijp  0.

ði 2 I1 Þ ði 2 I2 Þ ði 2 I3 Þ ði 2 I4 Þ ði 2 I5 Þ ði 2 I6 Þ

When all the DMs provide the assessment information of alternatives on criteria, the fuzzy decision matrices can be elicited as Y p ¼ ðypij Þm n ðp 2 QÞ. (5) The subjective preference relations between alternatives The DM gives the pair-wise comparisons of alternatives with hesitancy degrees. Integrating all the comparisons provided by DM ep , we can formulate the subjective ep ¼ preference relations between alternatives as an IVIFS of ordered pairs X fhðk; jÞ; ~tp ðk; jÞijak p aj with an IVIF truth degree ~tp ðk; jÞ ðk; j 2 NÞ}, where ðk; jÞ expresses an ordered pair of alternatives ak and aj that the DM ep prefers ak to aj (denoted by ak p aj ) with the truth degree ~tp ðk; jÞ, ~tp ðk; jÞ is an IVIFS denoted by ~tp ðk; jÞ ¼ h½l~  ;l ; ½t~tp ðk;jÞ ; t~tp ðk;jÞ i for short, satisfying l~t ðk;jÞ  0, t~tp ðk;jÞ  0 t ðk;jÞ ~tp ðk;jÞ p

and

~tp ðk;jÞ þ t~tp ðk;jÞ  1. l

 a;½b;bi e h½a; X p

Define

the

 h½a; a; ½b; bi

p

cut

set

of

ep X

as

 k; j 2 NÞg, where ~tp ðk;jÞ  a, t~tp ðk;jÞ  b; t~tp ðk;jÞ  bð ¼ fðk; jÞj l~t ðk;jÞ  a; l p

3.3 Hybrid MCGDM Problems Considering Alternative …

79

   1, then the support of X e p is ½a; a½0; 1, ½b; b½0; 1 and a þ b e h½0;0;½1;1i ¼ fðk; jÞj l ~t ðk;jÞ  0; t~t ðk;jÞ  1; t~t ðk;jÞ  1 ð k; j 2 NÞg, sim 0, l X p

ply denoted by

~tp ðk;jÞ h0;1i e . X p

p

p

p

e p are given through pair-wise comparisons The subjective preference relations X between alternatives as a whole rather than each criterion. Usually DMs would not be able to specify pair-wise comparisons of all alternatives and   may only give  e h0;1i  pair-wise comparisons of some alternatives. The symbol  X  means the cardip

e h0;1i , i.e., the number of alternatives’ comparisons, which at nality of the support X p 2 most is equal to Cn ¼ nðn  1Þ=2. (6) Incomplete weight information structures Denote the weight vector of criteria by ¼ ðx1 ; x2 ; . . .; xm ÞT , where xi is the  xP relative weight of criterion ci . Let K0 ¼ xj m i¼1 xi ¼ 1; xi  e for i 2 I , where e [ 0 is a sufficiently small positive number which ensures that the weights generated are not zeros [16]. The incomplete weight information structures can be expressed in the five basic relations among criteria weights, which are denoted by subsets Ks ðs ¼ 1; 2; 3; 4; 5Þ of weight vectors in K0 , respectively (see [27, 32] for details). In reality, usually the preference information structure K of criteria importance may consist of several basic subsets Ks ðs ¼ 1; 2; 3; 4; 5Þ. The hybrid MCGDM considered in this chapter is how to rank the alternatives according to the decision matrices Y p ¼ ðypij Þm n ðp 2 QÞ, the subjective preference e p ðp 2 QÞ between alternatives, and the preference information structure relations X K of criteria importance.

3.3.2

Normalization Method

Due to different dimensions and measurements for different criteria, the criteria values should be normalized to eliminate effect of different physical dimensions and measurements on the final decision. In this chapter, all normalized data including IVIFSs, IFSs, TrFNs, linguistic variables, intervals and real numbers are required in the unit interval [0, 1]. Additionally, all normalized data in this chapter are so that the larger the normalized value the better the alternative whether the criterion is benefit or cost. We normalize the criteria value ypij as follows:

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

80

8 p p p p ij ; ½tij ; tij i h½lij ; l > > > p p > > hlij ; tij i > > < p p p p ðbij1 ; bij2 ; bij3 ; bij4 Þ p rij ¼ p p p > ðdij1 ; dij2 ; dij3 Þ > > p p > > ½f ; g  > > : pij ij zij

ði 2 I1 Þ ði 2 I2 Þ ði 2 I3 Þ ði 2 I4 Þ ði 2 I5 Þ ði 2 I6 Þ

For the benefit criteria,  pij ; ½tpij ; tpij i ¼ h½l0ijp ; l  0ijp ; ½t0ijp ; t0ijp i; hlpij ; tpij i ¼ hl0ijp ; t0ijp i; bpijt ¼ b0ijtp =bmax h½lpij ; l i4 ðt ¼ 1; 2; 3; 4Þ; p 0p max ¼ dijt =di3 ðt ¼ 1; 2; 3Þ; fijp ¼ fij0 p =gmax ; gpij ¼ g0ijp =gmax ; zpij ¼ z0ijp =zmax ; dijt i i i

ð3:5Þ for the cost criteria, pij ; ½tpij ; tpij i ¼ h½t0ijp ; t0ijp ; ½l0ijp ; l 0ijp i; hlpij ; tpij i ¼ ht0ijp ; l0ijp i; h½lpij ; l p p 0p max bpijt ¼ 1  b0ijð5tÞ =bmax i4 ðt ¼ 1; 2; 3; 4Þ; dijt ¼ 1  dijð4tÞ =di3 ðt ¼ 1; 2; 3Þ; ð3:6Þ

fijp ¼ 1  g0ijp =gmax ; gpij ¼ 1  fij0 p =gmax ; zpij ¼ 1  zpij =zmax ; i i i where 0p bmax i4 ¼ max fbij4 g; j2N;p2Q

zmax ¼ max fz0ijp g: i

0p max di3 ¼ max fdij3 g; j2N;p2Q

gmax ¼ max fg0ijp g; i j2N;p2Q

ð3:7Þ

j2N;p2Q

By using Eqs. (3.5)–(3.7), the fuzzy decision matrix Y p ¼ ðypij Þm n is transformed into the normalized fuzzy decision matrix Rp ¼ ðrijp Þm n . Denote all the p p p ; r2j ; . . .; rmj Þ, which fully elements of the jth column in matrix Rp by rpj ¼ ðr1j p characterizes an alternative aj for DM ep . Namely, rj and aj have the same meaning for DM ep and can be interchangeably used.

3.4

IVIF Mathematical Programming Method for Hybrid MCGDM

In this section, we develop a new IVIF mathematical programming method to solve the above hybrid MCGDM.

3.4 IVIF Mathematical Programming Method for Hybrid MCGDM

3.4.1

81

IVIFS-Type Consistency and Inconsistency Measurements

The classic LINMAP [16] utilized the distance of each alternative to ideal solution to assess the alternative. Denote the fuzzy positive ideal solution (FPIS) by r þ ¼ ðr1þ ; r2þ ; . . .; rmþ Þ and the fuzzy negative ideal solution (FNIS) by r ¼ ðr1 ; r2 ; . . .; rm Þ, where riþ and ri are, respectively, the best and worst ratings on criterion ci ði ¼ 1; 2; . . .; mÞ. Namely,

riþ

8 þ þ i ; ½tiþ ; tiþ i h½li ; l > > > þ > > hl ; t þ i > < iþ iþ þ þ ðbi1 ; bi2 ; bi3 ; bi4 Þ ¼ > ðdi1þ ; di2þ ; di3þ Þ > > > > ½f þ ; giþ  > : þi zi

ði 2 I1 Þ ði 2 I2 Þ ði 2 I3 Þ : ði 2 I4 Þ ði 2 I5 Þ ði 2 I6 Þ

and 8     i ; ½ti ; ti i h½li ; l > > >   > hl ; t i > i > <  i   ðbi1 ; bi2 ; bi3 ; bi4 Þ  ri ¼ ðdi1 ; di2 ; di3 Þ > > > >  > ½fi ; g > :  i zi

ði 2 I1 Þ ði 2 I2 Þ ði 2 I3 Þ : ði 2 I4 Þ ði 2 I5 Þ ði 2 I6 Þ

iþ ; ½tiþ ; tiþ i and ri ¼ h½l  More specifically, if i 2 I1 , riþ ¼ h½liþ ; l ;l i ; i are IVIFSs, where

½t t i ; i i

liþ ¼ max flpij g; j2N;p2Q

iþ ¼ max f l lpij g;

l ¼ min flpij g; i

 l lpij g; i ¼ min f

j2N;p2Q

j2N;p2Q

j2N;p2Q

tiþ ¼ min ftpij g;

tiþ ¼ min ftpij g;

j2N;p2Q

p t i ¼ max ftij g; j2N;p2Q

j2N;p2Q

t tpij g; i ¼ max f j2N;p2Q

ð3:8Þ  If i 2 I2 , riþ ¼ hliþ ; tiþ i and ri ¼ hl i ; ti i are IFSs, where

liþ ¼ max flpij g; j2N;p2Q

tiþ ¼ min ftpij g; j2N;p2Q

p l i ¼ min flij g; j2N;p2Q

p t i ¼ max ftij g; j2N;p2Q

ð3:9Þ    If i 2 I3 , riþ ¼ ðbi1þ ; bi2þ ; bi3þ ; bi4þ Þ and ri ¼ ðb i1 ; bi2 ; bi3 ; bi4 Þ are TrFNs, where

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

82

bitþ ¼ max fbpijt g; j2N;p2Q

p b it ¼ min fbijt g ðt ¼ 1; 2; 3; 4Þ;

ð3:10Þ

j2N;p2Q

   If i 2 I4 , riþ ¼ ðdi1þ ; di2þ ; di3þ Þ and ri ¼ ðdi1 ; di2 ; di3 Þ are TFNs, where p g; ditþ ¼ max fdijt j2N;p2Q

p dit ¼ min fdijt g ðt ¼ 1; 2; 3Þ;

ð3:11Þ

j2N;p2Q

If i 2 I5 , riþ ¼ ½fi þ ; giþ  and ri ¼ ½fi ; g i  are intervals, where fi þ ¼ max ffij g; j2N;p2Q

giþ ¼ max fgij g; j2N;p2Q

fi þ ¼ max ffij g; j2N;p2Q

giþ ¼ max fgij g; j2N;p2Q

ð3:12Þ If i 2 I6 , riþ ¼ ziþ and ri ¼ z i are real numbers, where ziþ ¼ max fzij g; j2N;p2Q

z i ¼ min fzij g: j2N;p2Q

ð3:13Þ

According to Eqs. (3.1), (3.3), and (3.4), we can calculate the squares of the Euclidean distances between rijp and riþ as well as ri as follows: 8 1 p þ 2 >  iþ Þ2 þ ðtpij  tiþ Þ2 þ ðtpij  tiþ Þ2 þ ðppij  piþ Þ2 þ ð iþ Þ2 ði 2 I1 Þ lpij  l ppij  p > 4 ½ðlij  li Þ þ ð > > > p p p 2 2 2 þ þ þ 1 > > > 2 ½ðlij  li Þ þ ðtij  ti Þ þ ðpij  pi Þ ði 2 I2 Þ > ½ðdij1  di1þ Þ2 þ ðdij2  di2þ Þ2 þ ðdij3  di3þ Þ2 ði 2 I4 Þ > >3 > p p > 2 2 þ þ 1 > ½ðfij  fi Þ þ ðgij  gi Þ ði 2 I5 Þ > >2 > : ðzp  z þ Þ2 ði 2 I Þ 6 i ij

sijþ p

ð3:14Þ and

sp ij

8 p p 2 2 2 1 p  2  2  2 >   lpij  l tpij  t ppij  p > i Þ þ ðtij  ti Þ þ ð i Þ þ ðpij  pi Þ þ ð i Þ ði 2 I1 Þ 4 ½ðlij  li Þ þ ð > > > p p p 1  2  2  2 > > 2 ½ðlij  li Þ þ ðtij  ti Þ þ ðpij  pi Þ ði 2 I2 Þ > >

> 3 ½ðdij1  di1 Þ þ ðdij2  di2 Þ þ ðdij3  di3 Þ ði 2 I4 Þ > > p p > 1  2  2 > > 2 ½ðfij  fi Þ þ ðgij  gi Þ ði 2 I5 Þ > > : ðzp  z Þ2 ði 2 I Þ 6 ij i

ð3:15Þ where

pij  tpij , ppij ¼ 1  l

p i

l i

¼1



t i .

pij ¼ 1  lpij  tpij , p

piþ ¼ 1  liþ  tiþ ,

and

3.4 IVIF Mathematical Programming Method for Hybrid MCGDM

83

The relative closeness degree of rijp with respect to riþ is defined as follows: þp p bpij ¼ sp ij =ðsij þ sij Þ

ð3:16Þ

þp p p Obviously, 0  sp ij  sij þ sij . Hence, it directly follows that 0  bij  1. If

sijþ p ¼ 0, then bpij ¼ 1. The comprehensive relative closeness degree of the alternative rj ¼ ðr1j ; r2j ; . . .; rmj Þ with respect to the FPIS r þ ¼ ðr1þ ; r2þ ; . . .; rmþ Þ is defined as follows: Dpj ¼

m X

xi bpij :

ð3:17Þ

i¼1

It is clear that the bigger Dpj the better the alternative aj for DM ep . e \0;1 [ , if Dp  Dp , then ak is better than aj , the objective For each ðk; jÞ 2 X j p k ranking order determined by Dpj and Dpk is ak p aj , which is consistent with the subjective preference relation given by DM ep preferring ak to aj . Conversely, if Dpk \Dpj , then the objective ranking order is inconsistent with the subjective preference relation. Thus, there exist some deviations between the objective ranking order and subjective preference relation. To measure such deviations, we introduce the following inconsistency and consistency indexes. e \0;1 [ , an index E e p is defined to measure Definition 3.4 For each ðk; jÞ 2 X p kj inconsistency between the objective ranking order and the subjective preference relation given by DM ep as follows: ep ¼ E kj



~tp ðk; jÞðDpj  Dpk Þ if Dpk \Dpj 0 if Dpk  Dpj :

ð3:18Þ

Obviously, the objective ranking order of ak and aj is consistent with the sube p is defined to be 0. jective preference relation given by DM ep if Dpk  Dpj . Hence, E kj Otherwise, the objective ranking order of ak and aj is inconsistent with the sube p is defined to be jective preference relation given by DM ep if Dpk \Dpj . Thus, E kj ~tp ðk; jÞðDpj  Dpk Þ. The inconsistency index can be rewritten as e p ¼ ~tp ðk; jÞmaxf0; Dpj  Dp g. Hereby, a group inconsistency index is defined as E kj k follows: e¼ E

q X

X

p¼1

ðk;jÞ2 e X p\0;1 [

ep ¼ E kj

q X

X

p¼1

ðk;jÞ2 e X p\0;1 [

½~tp ðk; jÞmaxf0; Dpj  Dpk g:

ð3:19Þ

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

84

Analogously, the consistency measurement can be given in Definition 3.5. e \0;1 [ , an index F e p to measure consistency can Definition 3.5 For each ðk; jÞ 2 X p kj be defined as follows: ep ¼ F kj



~tp ðk; jÞðDpk  Dpj Þ 0

if Dpk  Dpj if Dpk \Dpj ;

ð3:20Þ

e p ¼ ~tp ðk; jÞmaxf0; Dp  Dpj g. Hence, a group consiswhich directly infers that F kj k tency index is defined as: e¼ F

q X

X

p¼1

ðk;jÞ2 X p\0;1 [

e

ep ¼ F kj

q X

X

p¼1

[ ðk;jÞ2 X \0;1 p

e

½~tp ðk; jÞmaxf0; Dpk  Dpj g:

ð3:21Þ

Remark 3.1 Due to the fuzzy truth degrees represented as IVIFSs, the group e and F e defined in this chapter are all inconsistency and consistency indices E IVIFSs, whereas the counterparts in the methods [2, 16–21, 24–27] and [29] are real numbers, TFNs, IFSs, and intervals respectively. This is the most difference between [16–21, 24–29] and this chapter.

3.4.2

Bi-Objective IVIF Mathematical Programming Model

e and consistency index F e respectively reflect the As the group inconsistency index E overall inconsistency and consistency between the objective ranking order and the e and at the same subjective preference relations of the group of DMs, the smaller the E e , the better the model characterizes the DMs’ decision rationales. time the bigger the F Bearing this in mind, we establish the following bi-objective IVIF mathematical programming model to determine the weight vector x ¼ ðx1 ; x2 ; . . .; xm ÞT : e minf Eg eg maxf F

ð3:22Þ

s:t: x 2 K e \0;1 [ , let kp ¼ maxf0; Dp  Dp g, pp ¼ maxf0; Dp  Dp g. For each ðk; jÞ 2 X j j p kj k kj k p p Then, kkj  0, pkj  0, kpkj  Dpj  Dpk , ppkj  Dpk  Dpj and kpkj  ppkj ¼ Dpj  Dpk . Thus, Eq. (3.22) can be converted into the following bi-objective IVIF mathematical programming model:

3.4 IVIF Mathematical Programming Method for Hybrid MCGDM

8 >
=

q P

P e¼ ~tp ðk; jÞkpkj min E > > p¼1 ; : [ ðk;jÞ2 e X \0;1 p 9 8 > > = < q P P p e ~ max F ¼ tp ðk; jÞpkj > > p¼1 ; : [ ðk;jÞ2 e X \0;1 p 8 p e \0;1 [ ; p 2 QÞ k  ppkj ¼ Dpj  Dpk ððk; jÞ 2 X > p > > kjp > p p e \0;1 [ ; p 2 QÞ > ððk; jÞ 2 X < Dk  Dj þ kkj  0 p s:t: Dp  Dp þ pp  0 e \0;1 [ ; p 2 QÞ ððk; jÞ 2 X j p k kj > > > p p e \0;1 [ ; p 2 QÞ > k  0; p  0 ððk; jÞ 2 X > p kj : kj x2K Note Eq. (3.17), we have Dpj  Dpk ¼

m P i¼1

ð3:23Þ

xi ðbpij  bpik Þ. Equation (3.23) can be

written as the following bi-objective IVIF mathematical programming model: 9 8 > > q = < X X p e ~ min E ¼ tp ðk; jÞkkj > > ; : p¼1 ðk;jÞ2 e X p\0;1 [ 9 8 ð3:24Þ > > q = < X X p e¼ ~tp ðk; jÞpkj max F > > ; : p¼1 \0;1 [ e ðk;jÞ2 X p s:t:ðkT ; pT ; xT Þ 2 H k ¼ ðkpkj ÞPq e\0;1 [  and p ¼ ðppkj ÞPq e\0;1 [  be two Xp 1 Xp 1 p¼1 p¼1   Pq  e \0;1 [  p p  1 vector, kkj and pkj are listed in the lexicographic order p¼1  X p

where

e \0;1 [ ðp 2 QÞ and according to the subscripts ðk; jÞ 2 X p  H¼ m P i¼1

ðkT ; pT ; xT Þjkpkj  ppkj ¼

xi ðbpij



bpik Þ þ ppkj

 0; kpkj

m P i¼1

xi ðbpij  bpik Þ;

 0; ppkj

0

m P

xi ðbpik  bpij Þ þ kpkj  0;

 \0;1 [ ~ ððk; jÞ 2 Xp ; p 2 QÞ; x 2 K : i¼1

  P   e \0;1 [  2 qp¼1  X  þ m variables which p P  e \0;1 [  need to be determined, including m weights xi ði ¼ 1; 2; . . .; mÞ, 2 qp¼1  X  p   P   p p q e \0;1 [ , and e \0;1 [  equalities and variables kkj and pkj ðk; jÞ 2 X p p¼1  X p Remark 3.2 In Eq. (3.24) there exist

86

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

   e \0;1 [  X   inequalities (excluding the constraints in K). To determine these p p¼1 P  e \0;1 [  variables, number 2 qp¼1  X  of inequalities should not be much small. In p Pq  e \0;1 [  general, the larger p¼1  X p  (i.e., pair-wise comparisons between alterna-

2

Pq

tives), the more precise and reliable determining the weight vector. Remark 3.3 Considering the IVIF truth degrees, we firstly construct the bi-objective IVIF mathematical programming model, whereas the counterparts constructed in the methods [16–21] are common linear programming, in the methods [24–26] are fuzzy linear programming with TrFNs (or TFNs), in the methods [27, 28] are IF mathematical programming model, in the method [29] is interval programming model. This is another sharp distinction between [16–21, 24– 29] and this chapter. Remark 3.4 In all methods [16–21, 24–29], they merely constructed single objective programming model of minimizing the inconsistency index under the condition in which the inconsistency index is smaller than or equal to the consistency index by a constant h [16–21] or a TrFN ~h ¼ ðh1 ; h2 ; h3 ; h4 Þ [24–26] or an IFS ~h ¼ hl~h ; v~h i [27, 28] or an interval e~a ¼ ½a; a [29]. Since these parameters are given by DMs a priori, it is not easy to avoid the subjective randomness in practical application. Whereas the bi-objective programming Eq. (3.22) not only minimizes the inconsistency and maximizes the consistency index simultaneously, but also does not need such a condition. The constraints kpkj  ppkj ¼ Dpj  Dpk ððk; jÞ 2 e \0;1 [ ; p 2 QÞ in Eq. (3.23) skillfully link with inconsistency and consistency X p

indices. There are not such constraints in [16–21, 24–29]. Therefore, the presented model in this chapter is quite novel and provides essential improvements of previous similar research.

3.4.3

Goal Programming Approach to Solving Bi-Objective IVIF Mathematical Programming Model

Since the objective functions of Eq. (3.24) contain the IVIFSs ~tp ðk; jÞ, we call Eq. (3.24) the bi-objective IVIF mathematical programming model. In the sequel, we develop a new linear goal programming approach to solving this IVIF mathematical programming model. Theorem 3.1 Equation (3.24) can be changed into the following multi-objective linear programming:

3.4 IVIF Mathematical Programming Method for Hybrid MCGDM

8 > < min Z1 > : 8 > < min Z2 > : 8 > < min Z3 > : 8 > < min Z4 > : 8 > < min Z5 > : 8 > < min Z6 > : 8 > < min Z7 > : 8 > < min Z8 > :

¼

¼

¼

¼

¼

¼

¼

¼

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

87

kpkj lnð1  l~t ðk;jÞ Þ p

~tp ðk;jÞ Þ kpkj lnð1  l

kpkj lnðt~tp ðk;jÞ Þ

kpkj lnðt~tp ðk;jÞ Þ

> ;

> ; 9 > = > ;

p

~tp ðk;jÞ Þ ppkj lnð1  l

ppkj lnðt~tp ðk;jÞ Þ

> ; 9 > =

9 > =

ppkj lnð1  l~t ðk;jÞ Þ

ppkj lnðt~tp ðk;jÞ Þ

9 > =

9 > =

ð3:25Þ

> ; 9 > = > ;

9 > = > ; 9 > = > ;

s:t:ðkT ; pT ; xT Þ 2 H Proof See Appendix 1. Normally, the concept of Pareto optimal/efficient solutions is commonly used to define a solution of multi-objective programming model. There exist several solution methods for it. However, in this study we focus on developing a goal programming approach to solving Eq. (3.25). Goal programming approach is concerned with the conditions of achieving pre-specified targets or goals on the various objectives. Given a portfolio of properly established goals, one tries to achieve them as closely as possible. As originally conceived, goal programming approach attempts to minimize the set of deviations from pre-specified multiple goals, which are considered simultaneously but are weighted according to their relative importance.

88

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

þ Let hi be the goal of the objective Zi , d i be the underachievement of goal, di be the overachievement of goal, Pi be the preemptive priority assigned to the objective Zi ði ¼ 1; 2; . . .; 8Þ. Then, by using the goal programming approach, Eq. (3.25) can be changed into linear goal programming model as follows:

( min Z ¼ (

8 X

) Pi diþ

i¼1

þ Z i þ d i  di ¼ hi s:t: ðkT ; pT ; xT Þ 2 H

ði ¼ 1; 2; . . .; 8Þ

ð3:26Þ

On the basis of the above analysis, solving Eq. (3.25) may be involved the following steps: Step 1: Get the goal hi to achieve for each objective Zi from the decision group ði ¼ 1; 2; . . .; 8Þ; Assume that Zi is the optimal value of the single objective programming minfZi g ignoring other objectives in Eq. (3.25), called the ideal value for the objective Zi , then a common approach to getting the goal hi is to set the goal at some proportion of the ideal value, i.e., hi ¼ gZi , where g is the proportion parameter, such as g ¼ 0:8, g ¼ 0:9, and so on. Step 2: Get the preemptive priority Pi assigned to each objective Zi ði ¼ 1; 2;


; 8Þ; This can be done through the expert system in advance. Step 3: Solve Eq. (3.26) by the Simplex method of linear programming model. The optimal solution to Eq. (3.26) will come as close as possible to the stated goals in the specified preference order. After solving Eq. (3.26), we can obtain the criteria weight vector x ¼ ðx1 ; x2 ; . . .; xm ÞT . Then, the comprehensive relative closeness degree Dpj for DM ep is calculated by Eq. (3.17). Remark 3.5 We firstly develop linear goal programming approach to solving bi-objective IVIF mathematical programming. DMs can select different proportion parameters according to their preferences on the ideal values of objectives and thus obtain different optimal solutions, which demonstrate the flexibility of the proposed approach. However, the method [26] transformed the fuzzy linear programming with TrFNs into four-objective linear programming model which is further aggregated into linear programming model with equal weights; the methods [27, 28] transformed the IF mathematical programming into bi-objective programming model which is solved by using equal weights or lexicographic method; the method [29] transformed the interval programming into bi-objective linear programming model which is further aggregated into linear programming model with equal weights. Apparently, the methods [26–29] lack flexibility. This is the notable distinction between [26–29] and this chapter.

3.4 IVIF Mathematical Programming Method for Hybrid MCGDM

3.4.4

89

GDM Process and Steps for Solving Hybrid MCGDM

In the above, the IVIF mathematical programming method is proposed, especially the bi-objective IVIF mathematical programming model (i.e., Eq. 3.22) is constructed to solve the weight vector. The individual ranking order of alternatives given by each DM can be generated according to the comprehensive relative closeness degree Dpj . The multi-objective assignment model is then constructed to derive the collective ranking of alternatives. The decision flowchart of solving hybrid MCGDM is depicted in Fig. 3.1 as follows: Based on the aforesaid analyses, we are now in a position to present the decision steps for solving the hybrid MCGDM problems. Step 1: The DMs identify the evaluation criteria. Step 2: The DM ep gives the IVIFS of ordered pairs for the subjective preference e p ðp 2 QÞ. relations between alternatives by X Step 3: The group of DMs provides the incomplete preference information structure K of criteria importance. Step 4: Elicit the fuzzy decision matrix Y p and obtain the normalized decision matrix Rp by Eqs. (3.5)–(3.7). Step 5: Construct the bi-objective IVIF mathematical programming model (i.e., Eq. 3.24). Step 6: Obtain the weight vector x ¼ ðx1 ; x2 ; . . .; xm ÞT through solving Eq. (3.26) by the linear goal programming approach developed in Sect. 3.4.3. Step 7: Calculate the comprehensive relative closeness degree Dpj of alternatives aj for DM ep using Eq. (3.17). Step 8: The individual ranking matrix X p ¼ ðxpij Þn n of the alternatives is generated for DM ep ðp ¼ 1; 2; . . .; QÞ according to the decreasing order of Dpj ðj 2 NÞ, where  xpij

¼

1; 0;

DM ep ranks alternative aj in the i  th position otherwise

which means that if DM ep ranks the alternative aj in the i-th position i 2 f1; 2; . . .; ng, then xpij ¼ 1, otherwise xpij ¼ 0. For example, x315 ¼ 1 shows that DM e3 ranks the alternative a5 in the first position, i.e., alternative a5 is the best alternative for DM e3 . Step 9: Determine the collective ranking matrix X ¼ ðxij Þn n of the alternatives and select the best alternative. Suppose that the decision group ranks the alternative aj in the i-th position i 2 f1; 2; . . .; ng, i.e.,

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

90

Criteria values in the formats of IVIFSs, IFSs, TrFNs,

Hybrid MAGDM problems

linguistic variables, intervals and real numbers

Form the IVIFSs of subjective preference relations between alternatives with IVIF truth degrees

Q p (p ∈ Q)

Incomplete preference information structure of attribute importance By using Eqs. Elicit the fuzzy decision matrixes and normalize

Define group inconsistent and consistent indexes

Construct

bi-objective

IVIF

mathematical

Decision process

programming and transform into linear programming

Obtain the criteria weight vector through solving goal programming model

Calculate

the

comprehensive

relative

(3.5)-(3.7)

By using Eqs. (3.18)-(3.21)

Transform Eq. (3.22) into Eq. (3.26)

Solve Eq. (3.26)

Using Eq. (3.17)

closeness degree

Generate the individual ranking matrix of the

According to D jp ( j ∈ N)

alternatives for each DM

Determine the collective ranking matrix of the

Multi-objective assignment

alternatives and select the best alternative

model Eq. (3.28)

Fig. 3.1 Decision flowchart of the hybrid MCGDM with IVIFS truth degrees and incomplete weight information

3.4 IVIF Mathematical Programming Method for Hybrid MCGDM

 xij ¼

1; 0;

91

Decision group rank alternative aj in the i  th position otherwise,

which is to be determined. Generally, the individual ranking order of alternatives for each DM should be as close as possible to the group ranking order in GDM. Thus, the multi-objective assignment model can be constructed as follows: min

 n X n  X  p  xij  xij ðp ¼ 1; 2; . . .; qÞ j¼1 i¼1

8 n P > > xij ¼ 1 ðj ¼ 1; 2; . . .; nÞ > > > < i¼1 n s:t: P xij ¼ 1 ði ¼ 1; 2; . . .; nÞ > > > j¼1 > > : xij ¼ 0 or 1 ði; j ¼ 1; 2; . . .; nÞ

ð3:27Þ

Since there is no any preference among the DMs, the above model can be aggregated into the following single objective assignment model: min

q X  n X n  X  p  xij  xij  p¼1 j¼1 i¼1

ð3:28Þ

s:t: same as the constraints of Eq:ð3:27Þ By using the Hungarian method to solve Eq. (3.28), the collective ranking matrix X ¼ ðxij Þn n can be obtained. Thus, the group ranking order of the alternatives can be easily generated and the best alternative from the alternative set A is determined. Remark 3.6 In Step 7, if directly using simple linear weighting method, then the P group comprehensive relative closeness degree Dj ¼ qp¼1 dp Dpj of alternative aj can be obtained to ranking the alternatives, where dp represents the weight of DM ep . However, some alternatives may have the identical group comprehensive relative closeness degrees. Thus, there is no way to distinguish such alternatives by using simple linear weighting method. Remark 3.7 The fuzzy LINMAP methods [17, 19–21, 24, 28, 29] used Borda’s score to determine the group ranking order of alternatives. It is possible that some alternatives have the same Borda’s score which results in that these alternatives also cannot be distinguished by using these fuzzy LINMAP methods. In Eq. (3.28), the P constraints ni¼1 xij ¼ 1 ðj ¼ 1; 2; . . .; nÞ assure that each alternative is ranked in Pn only one position; the constraints j¼1 xij ¼ 1 ði ¼ 1; 2; . . .; nÞ assure that each position can be put by only one alternative. Therefore, Eq. (3.28) can rank different alternatives in different positions, which effectively avoids the drawback that some

92

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

alternatives cannot be distinguished. Even though there may be several optimal solutions of Eq. (3.28), for different optimal solutions, each alternative must be ranked in only one position and each position must be put by only one alternative. The super capability of identification is the prominent advantage of constructed assignment model over other group ranking methods.

3.5

Empirical Example of Critical Infrastructure

To demonstrate the effectiveness of the proposed GDM method, it is applied to a critical infrastructure evaluation example. The comparison analyses are also conducted in this section.

3.5.1

A Critical Infrastructure Evaluation Example and the Analysis Process

Critical infrastructures are the pulsating heart of one country. Evaluating critical infrastructure security under the framework of risk management has taken on increasing importance during the past decades [33]. In order to improve their ability to provide goods and services efficiently and reduce cost effectively, the critical infrastructures within a country must be evaluated to determine the priorities of protection. Generally, the critical infrastructures include telecommunications, electric power systems, natural gas and oil, banking and finance, transportation, water supply systems, government services, and emergency services. As one of the critical infrastructures, emergency services have remarkable significance and their security must not go unanswered. Emergency risk management (ERM) is a process which involves dealing with risks to the community arising from emergency events. Emergency management evaluation, as one of the important parts of ERM, aims to evaluate and improve social preparedness and organizational ability of an emergency operating center (EOC) in identifying, analyze and treat emergency risks to the community arising from emergency events. Jiangxi province of China is a natural disaster prone province. To assess the emergency services, the government of Jiangxi province has organized a committee to identify ECOs. All committee members would be invited to participate in the survey. There are twenty committee members from different areas of specialization that include electrical, water, transportation, information, telecommunication, energy, nuclear and government offices who are surveyed. Those members come from major departments administrating the infrastructures and national security of China who have been in the relative fields for more than 20 years experience. Before conducting the survey, the questionnaire is explained and the committee members are free to express their opinions.

3.5 Empirical Example of Critical Infrastructure

93

Table 3.2 Factors and explanations of evaluating EOCs Factor

Explanation

Equipment and facility

Health and medical equipment and facility Communication equipment Information platform, such as large screen display system, the public security image monitoring system, scheduling system Comprehensive coverage, digital meeting, intelligent building Stimulate department to improve their performance and enhance morale Improve communication problems and selfishness Increase the ability of management and resource reserve ability of EOC department Keep the flexibility to adjust department, including consolidation or decentralization Get new technology and learn new technology of management for equipment and facility Reduce delivery time, delivery cost and time to site Share the risks Reduce the developing and maintaining cost of equipments and facilities Increase the flexibility in finance Whether it is only a weak link Whether the nature of weak links make it difficult to be found The present prevention measures can play much utility Effect of unexpected events occur in the evaluation of weak links

Management

Risk

Vulnerability

The criteria for evaluating EOCs are very important because they obviously influence the decision result. Through the deep investigation and study, we conclude that four dimensions or factors, equipment and facility, management, risk, and vulnerability, should be taken into account during the process of evaluating EOCs. These factors and the corresponding explanations are listed in Table 3.2. It is impossible and unwieldy to incorporate all the factors discussed above into the LINMAP model. Through discussion with other researchers and practitioners, redundant factors are eliminated and similar ones are combined, which yields six independent factors. These factors become the criteria for the proposed decision model, involving health and medical services c1 , communication equipment c2 , resource reserve ability c3 , vulnerability c4 , delivery time c5 , and delivery cost c6 , where the former four criteria are all qualitative criteria. To preserve confidentiality and simplify our problem, we only extract five EOCs to demonstrate the proposed method. The set of alternatives consists of these five EOCs. They are Ganzhou EOC a1 , Jiujiang EOC a2 , Xinyu ECO a3 , Shangrao EOC a4 and Nanchang EOC a5 , respectively. Without loss of generality, we only invite three committee members (i.e., experts e1 , e2 and e3 ) to fill in the questionnaire. They evaluate the five EOCs on the basis of the six criteria. There exist some hesitancy when experts evaluate the health and medical services c1 and communication equipment c2 . Thus, the assessments for c1 and c2 are represented

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

94

by IVIFSs and IFSs, respectively. For resource reserve ability c3 , the experts like to give the lower, upper limits and the most possible intervals, i.e., the assessments of c3 can be expressed with TrFNs. It is more natural and reasonable to utilize linguistic variables to assess the vulnerability c4 . Due to the uncertainty of product process, the experts only can give the approximate scopes for delivery time c5 , i.e., it is better to use the intervals to represent the delivery cost c5 . The assessment for c6 can be represented by crisp numbers. The criteria c1 , c2 , and c3 are all benefit criteria, c4 , c5 and c6 are cost criteria. After brainstorming sessions, site surveys, design review, expert judgment, data eliciting, statistic data analysis and statistical treatment, we get the ratings of all EOCs on every criterion given by each expert listed in Table 3.3.

Table 3.3 Decision matrices of three experts Expert

Alternative

Criteria c1

e1

a1











a2 a3 a4 a5 e2

a1 a2 a3 a4 a5

e3

a1 a2 a3 a4 a5

c2

c3

c4

c5

c6

(4,5,6,7)

(0.8,0.9,1)

[70,80]

115

(2,3,4,5)

(0.4,0.5,0.6)

[88,92]

113

(6,7,8,9)

(0,0.1,0.3)

[80,89]

110

(2,3,5,6)

(0.2,0.3,0.4)

[88,93]

120

(1,3,5,6)

(0.6,0.7,0.8)

[90,95]

117

(3,4,5,6)

(0.6,0.7,0.8)

[74,85]

112

(6,7,8,9)

(0.4,0.5,0.6)

[85,90]

115

(5,6,7,8)

(0,0.1,0.3)

[87,92]

120

(1,2,3,4)

(0.2,0.3,0.4)

[76,89]

118

(2,3,4,5)

(0.8,0.9,1)

[92,95]

109

(2,3,4,5)

(0.4,0.5,0.6)

[80,90]

120

(4,6,8,9)

(0.8,0.9,1)

[78,83]

109

(3,5,7,8)

(0.2,0.3,0.4)

[85,92]

115

(1,2,3,4)

(0.6,0.7,0.8)

[89,95]

114

(3,5,7,8)

(0,0.1,0.3)

[92,94]

106

3.5 Empirical Example of Critical Infrastructure

95

With their comprehensions and judgments, the experts respectively provide the IVIFS preference relations between alternatives as follows: e 1 ¼ fhð2; 1Þ; ~t1 ð2; 1Þi; hð1; 3Þ; ~t1 ð1; 3Þi; hð5; 4Þ; ~t1 ð5; 4Þi; hð2; 5Þ; ~t1 ð2; 5Þi; hð3; 2Þ; ~t1 ð3; 2Þi; hð3; 4Þ; ~t1 ð3; 4Þig; X e 2 ¼ fhð1; 2Þ; ~t2 ð1; 2Þi; hð3; 1Þ; ~t2 ð3; 1Þi; hð4; 5Þ; ~t2 ð4; 5Þi; hð5; 2Þ; ~t2 ð5; 2Þi; hð2; 3Þ; ~t2 ð2; 3Þi; hð4; 3Þ; ~t2 ð4; 3Þig; X e 3 ¼ fhð2; 4Þ; ~t3 ð2; 4Þi; hð1; 4Þ; ~t3 ð1; 4Þi; hð5; 1Þ; ~t3 ð5; 1Þi; hð5; 3Þ; ~t3 ð5; 3Þig; X

where the corresponding IVIF truth degrees as follows: ~t1 ð2; 1Þ ¼ h½0:1; 0:2; ½0:4; 0:6i, ~t1 ð5; 4Þ ¼ h½0:2; 0:3; ½0:4; 0:5i, ~t1 ð3; 2Þ ¼ h½0:5; 0:6; ½0:1; 0:3i, ~t2 ð1; 2Þ ¼ h½0:2; 0:4; ½0:3; 0:4i, ~t2 ð4; 5Þ ¼ h½0:1; 0:3; ½0:5; 0:6i, ~t2 ð2; 3Þ ¼ h½0:2; 0:5; ½0:3; 0:4i, ~t3 ð2; 4Þ ¼ h½0:3; 0:5; ½0:2; 0:3i, ~t3 ð5; 1Þ ¼ h½0:2; 0:5; ½0:2; 0:4i,

~t1 ð1; 3Þ ¼ h½0:2; 0:4; ½0:2; 0:4i, ~t1 ð2; 5Þ ¼ h½0:3; 0:4; ½0:4; 0:5i, ~t1 ð3; 4Þ ¼ h½0:2; 0:4; ½0:3; 0:5i; ~t2 ð3; 1Þ ¼ h½0:5; 0:7; ½0:1; 0:2i, ~t2 ð5; 2Þ ¼ h½0:3; 0:4; ½0:4; 0:5i, ~t2 ð4; 3Þ ¼ h½0:5; 0:6; ½0:2; 0:3i; ~t3 ð1; 4Þ ¼ h½0:4; 0:5; ½0:2; 0:3i, ~t3 ð5; 3Þ ¼ h½0:1; 0:3; ½0:2; 0:5i.

e p ðp ¼ 1; 2; 3Þ are X e h0;1i ¼ fð2; 1Þ; ð1; 3Þ; ð5; 4Þ; ð2; 5Þ; Thus, the supports of X 1 h0;1i e h0;1i ¼ e ¼ fð1; 2Þ; ð3; 1Þ; ð4; 5Þ; ð5; 2Þ; ð2; 3Þ; ð4; 3Þg, and X ð3; 2Þ; ð3; 4Þg, X 2 3 fð2; 4Þ; ð1; 4Þ; ð5; 1Þ; ð5; 3Þg, respectively. e 2 . For It is easily seen that there are some intransitive relations in the set X instance, the experts e2 prefers a1 to a2 and a2 to a3 , whereas he/she prefers a3 to a1 . Moreover, the preference relations of alternatives given by expert e1 are completely contrary to that given by expert e2 . The preference information structure K of criteria importance given by the experts is given as follows: K ¼ fx 2 K0 jx3  2:3x2 ; 0:1  x3  x4  0:25; 0:18  x5  0:35; x3  x4  x1  x2 ; x5  x6 g:

By using Eqs. (3.5)–(3.7), The normalized decision matrices are obtained as in Table 3.4. Combined with the above normalized decision matrices, the bi-objective IVIF mathematical programming is obtained by Eq. (3.24) (see Eq. (3.35) in Appendix 2). According to Eq. (3.25), Eq. (3.35) can be transformed into the multi-objective linear programming model (see Eq. (3.36) in Appendix 2). Solving the single objective linear programming model minfZi g ignoring other objectives in Eq. (3.36), we get the ideal value of each objective as follows: Z1 ¼ 0:1427, Z2 ¼ 0:2820, Z3 ¼ 0:6401, Z4 ¼ 0:4116, Z4 ¼ 0:4116,    Z6 ¼ 0:7154, Z7 ¼ 1:7269, Z8 ¼ 1:0986. The goal programming model corresponding to Eq. (3.36) is constructed by Eq. (3.26) as follows:

a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5

e1

e3

e2

Alternative

Expert











Criteria c1

Table 3.4 Normalized decision matrices of three experts c2











c3 (0.44,0.56,0.67,0.78) (0.22,0.33,0.44,0.56) (0.67,0.78,0.86,1) (0.22,0.33,0.56,0.67) (0.11,0.33,0.56,0.67) (0.33,0.44,0.55,0.67) (0.67,0.78,0.89,1) (0.56,0.67,0.78,0.89) (0.11,0.22,0.33,0.44) (0.22,0.33,0.44,0.56) (0.22,0.33,0.44,0.56) (0.44,0.67,0.89,1) (0.33,0.56,0.78,0.89) (0.11,0.22,0.33,0.44) (0.33,0.56,0.78,0.89)

(0,0.1,0.2) (0.4,0.5,0.6) (0.7,0.9,1) (0.6,0.7,0.8) (0.2,0.3,0.4) (0.2,0.3,0.4) (0.4,0.5,0.6) (0.7,0.9,1) (0.6,0.7,0.8) (0,0.1,0.2) (0.4,0.5,0.6) (0,0.1,0.2) (0.6,0.7,0.8) (0.2,0.3,0.4) (0.7.0.9,1)

c4

[0.16,0.26] [0.032,0.074] [0.063,0.158] [0.021,0.074] [0,0.0526] [0.105,0.221] [0.053,0.105] [0.032,0.084] [0.063,0.2] [0,0.032] [0.053,0.158] [0.1263,0.1789] [0.032,0.105] [0,0.063] [0.011,0.0312]

c5

0.0417 0.0583 0.0833 0 0.0205 0.0667 0.0417 0 0.0167 0.0917 0 0.0917 0.0417 0.05 0.1167

c6

96 3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

3.5 Empirical Example of Critical Infrastructure

( min Z ¼ ( s:t:

8 X

97

) Pi diþ

i¼1 þ Zi þ d  i  di ¼ hi

ð3:29Þ

ði ¼ 1; 2; . . .; 8Þ

Other constraints are the same as that of Eq: ð3.36Þ

By setting the goal at 90% of the ideal value, i.e., the proportion parameter is g ¼ 0:9, then we obtain h1 ¼ 0:1285, h2 ¼ 0:2538, h3 ¼ 0:5761, h4 ¼ 0:3705, h5 ¼ 0:4153, h6 ¼ 0:6438, h7 ¼ 1:5542, h8 ¼ 0:9887. Without loss of generality, assume that the preemptive priorities are P1 P2


P8 . By using Lingo 11.0 to solve Eq. (3.29), the components of the optimal solution can be obtained as follows: k121 ¼ 0:0080, k113 ¼ 0, k154 ¼ 0:0155, k125 ¼ 0, k132 ¼ 0:0612, k134 ¼ 0:0102, k212 ¼ 0:0612, k231 ¼ 0:0852, k245 ¼ 0, k252 ¼ 0:2241, k223 ¼ 0, k243 ¼ 0, k324 ¼ 0, k314 ¼ 0, k351 ¼ 0:0626, k353 ¼ 0:0537, p121 ¼ 0, p113 ¼ 0:0692, p154 ¼ 0, p125 ¼ 0:0665, p132 ¼ 0, p134 ¼ 0, p212 ¼ 0:1850, p231 ¼ 0, p245 ¼ 0:3078, p252 ¼ 0, p223 ¼ 0:2702, p243 ¼ 0:0769, p324 ¼ 0:2087, p314 ¼ 0:1668, p351 ¼ 0, p353 ¼ 0, x1 ¼ 0:05, x2 ¼ 0:345, x3 ¼ 0:15, x4 ¼ 0:0499, x5 ¼ 0:3499, x6 ¼ 0:055. Then, the comprehensive relative closeness degree of each EOC from the FPIS r þ given by the three experts can be computed using Eq. (3.17) as follows: D11 ¼ 0:3611;

D12 ¼ 0:1760;

D13 ¼ 0:4463;

D14 ¼ 0:3694;

D15 ¼ 0:4002;

D21 ¼ 0:3435;

D22 ¼ 0:3515;

D23 ¼ 0:4127;

D24 ¼ 0:4025;

D25 ¼ 0:4181;

D31 ¼ 0:3773;

D32 ¼ 0:4152;

D33 ¼ 0:3643;

D34 ¼ 0:2065;

D35 ¼ 0:3107:

Therefore, the individual ranking order of the five EOCs is generated as follows: a3 a5 a4 a1 a2 for expert e1 ; a5 a3 a4 a2 a1 for expert e2 ; a2

a1 a3 a5 a4 for expert e3 . The individual ranking matrix Xp of the alternatives is generated for each expert ep ðp ¼ 1; 2; 3Þ as follows: 0

0 B0 B X1 ¼ B B0 @1 0 0

0 B1 B X3 ¼ B B0 @0 0

0 0 0 0 1

1 0 0 0 0 1 0 0 0 0

0 0 1 0 0 0 0 1 0 0

1 0 0 0 B0 1C C B 2 B 0C C; X ¼ B 0 @0 0A 0 1 1 0 0 0 0C C 0 0C C; 0 1A 1 0

0 0 0 1 0

0 1 0 0 0

0 0 1 0 0

1 1 0C C 0C C; 0A 0

98

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

According to Eq. (3.28), the corresponding assignment model is constructed as follows: min

3 X 5 X 5   X  p  xij  xij  p¼1 j¼1 i¼1

8 5 P > > > xij ¼ 1 ðj ¼ 1; 2; . . .; 5Þ > > > < i¼1 5 s:t: P > xij ¼ 1 ði ¼ 1; 2; . . .; 5Þ > > > j¼1 > > : xij ¼ 0 or 1 ði; j ¼ 1; 2; . . .; 5Þ

ð3:30Þ

By using Lingo 11.0 to solve Eq. (3.30), the optimal solution (i.e., the collective ranking matrix) is obtained as follows: 0

0 B0 B X¼B B0 @0 1

0 0 0 1 0

0 1 0 0 0

0 0 1 0 0

1 1 0C C 0C C: 0A 0

Thus, the collective ranking order of the five EOC is generated as follows: a5 a3 a4 a2 a1 . The best selection is Nanchang EOC a5 . Since Nanchang is the provincial capital of Jiangxi, its security is of great significance. The government of Jiangxi province has invested a lot of money for the construction of Nanchang EOC a5 . Therefore, EOC a5 is the best ECO, which is in accordance with the real-life situation. As stated earlier, though there are some e 2 and the preferences in X e 1 are completely contrary intransitive relations in the set X e to that in X 2 , the group ranking order of alternatives can also be obtained. This analysis indicates the validation of the proposed method in this chapter. In addition, similar to the case of g ¼ 0:9, we can obtain the corresponding collective ranking orders of alternatives for different values proportion parameter g, which are listed in Table 3.5. It is easily seen from Table 3.5 that choosing different values proportion parameter may lead to different ranking orders of alternatives, which shows the flexibility of decision making process.

Table 3.5 Collective ranking orders for different values proportion parameter

Proportion parameter

Collective ranking order

g ¼ 0:75 g ¼ 0:8 g ¼ 0:85 g ¼ 0:9 g ¼ 0:95

a5 a5 a5 a5 a5

a3

a4

a3

a3

a3

a2

a2

a4

a4

a4

a4

a3

a2

a2

a2

a1

a1

a1

a1

a1

3.5 Empirical Example of Critical Infrastructure

99

Moreover, it should be pointed out that through solving Eq. (3.35) we can obtain e ¼ h½0:1346; 0:2488; ½0:5221; 0:6577i and the consisthe inconsistency index E e tency index F ¼ h½0:3701; 0:5117; ½0:1760; 0:3315i. However, if we only minimize the inconsistency index and do not maximize consistency index in e ¼ h½0:1346; 0:2488; Eq. (3.35), then we can obtain the inconsistency index E e ½0:5221; 0:6577i and the consistency index F ¼ h½0:3112; 0:4001; ½0:1956; 0:3762i. Since h½0:3701; 0:5117; ½0:1760; 0:3315i is bigger than h½0:3112; 0:4001; ½0:1956; 0:3762i, the only minimizing the inconsistency index cannot assure the consistency index achieves the maximum as stated in Introduction.

3.5.2

Comparison Analysis with the Existing LINMAP Methods

In this subsection, we compare the results obtained by the existing LINMAP methods [16–21, 24–29] and the proposed method in this chapter. The existing LINMAP methods [16–21, 24–29] only minimized the inconsistency and did not take maximizing the consistency into consideration. The classic LINMAP [16] and fuzzy LINMAP methods [17–21] overlooked the fuzzy truth degrees on the comparisons of alternatives. (1) The classic LINMAP [16] only considered the real numbers of criteria. If all e h0;1i , C1 ¼ C2 ¼ C3 ¼ C4 ¼ the IVIFSs ~tp ðk; jÞ ¼ h½1; 1; ½0; 0i ðk; jÞ 2 X p C5 ¼ £ and q = 1, then the IVIF linear programming model (i.e., Eq. 3.24) constructed in this chapter is reduced to the common linear programming model in [16]. That is to say, the classic LINMAP is just a special case of the method proposed in this chapter. (2) Li and Yang [17] and Li and Sun [19] transformed linguistic variables into TFNs and proposed fuzzy LINMAP methods for MAGDM. If criteria subsets C1 ¼ C2 ¼ C3 ¼ C5 ¼ C6 ¼ £ and ~tp ðk; jÞ ¼ h½1; 1; ½0; 0i for all e h0;1i , then the linear programming models (i.e., Eq. (19)) constructed ðk; jÞ 2 X p by [17] and Eq. (27) constructed by [19]) are also a special case of Eq. (3.24) constructed in this chapter. (3) Xia et al. [18] transformed linguistic variables into TrFNs and proposed fuzzy LINMAP methods for MADM. If criteria subsets C1 ¼ C2 ¼ C4 ¼ C5 ¼ C6 ¼ £, q = 1 and ~tp ðk; jÞ ¼ h½1; 1; ½0; 0i for all e h0;1i , then the linear programming models (i.e., Eq. (18) constructed ðk; jÞ 2 X p by [18] and Eq. (27) constructed by [19]) are also a special case of Eq. (3.24) constructed in this chapter. (4) Li [20] developed fuzzy LINMAP methods for solving MAGDM with IFSs information. If the criteria subsets C1 ¼ C3 ¼ C4 ¼ C5 ¼ C6 ¼ £, ~tp ðk; jÞ ¼

100

(5)

(6)

(7)

(8)

(9)

(10)

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

e h0;1i , then the mathematical programming model h½1; 1; ½0; 0i for all ðk; jÞ 2 X p (i.e., Eq. (31)) constructed by [20] is still a special case of Eq. (3.24) constructed in this chapter. Wang and Li [21] proposed fuzzy LINMAP methods for solving MAGDM in which the criteria values are in the form of IVIFSs. If C2 ¼ C3 ¼ C4 ¼ C5 ¼ e h0;1i , the mathematical C6 ¼ £ and ~tp ðk; jÞ ¼ h½1; 1; ½0; 0i for all ðk; jÞ 2 X p programming model (i.e., Eq. (4.4)) constructed by [21] is still a special case of Eq. (3.24) constructed in this chapter. Sadi-Nezhad and Akhtari [24] proposed possibilistic LINMAP for MAGDM with TFNs. If C1 ¼ C2 ¼ C3 ¼ C5 ¼ C6 ¼ £ and IVIF truth degrees are changed as triangular fuzzy truth degrees, then the mathematical programming model (i.e., Eq. (19)) constructed by [24] is still a special case of Eq. (3.24) constructed in this chapter. Li and Wan [25] proposed fuzzy linear programming method for MADM. If C1 ¼ C2 ¼ C4 ¼ £, q = 1 and the IVIF truth degrees are changed as trapezoidal fuzzy truth degrees, then the fuzzy programming model (i.e., Eq. (18)) constructed by [25] is still a special case of Eq. (3.24) constructed in this chapter. Li and Wan [26] proposed fuzzy heterogeneous MADM method for outsourcing provider selection. If C1 ¼ C2 ¼ C4 ¼ £, q = 1 and IVIF truth degrees are changed as trapezoidal fuzzy truth degrees, then the mathematical programming model (i.e., Eq. (23)) constructed by [26] is still a special case of Eq. (3.24) constructed in this chapter. Wan and Li [27, 28] respectively developed fuzzy heterogeneous MADM and MAGDM methods using IFSs representing hesitancy truth degrees. However, the solving methods of the constructed IF mathematical programming [27, 28] lack flexibility. If C1 ¼ C2 ¼ C4 ¼ £ and IVIF truth degrees are changed as IF truth degrees, then the mathematical programming models (i.e., Eq. (17) constructed by [27] and Eq. (18) constructed by [28]) are still a special case of Eq. (3.24) constructed in this chapter. Zhang and Xu [29] proposed interval programming method for hesitant fuzzy MAGDM using the intervals to express the fuzzy truth degrees. They transformed bi-objective mathematical programming into single objective programming by the equal weighted summation approach, which also lacks flexibility.

The detailed comparisons with the methods [16–21, 24–29] are listed in Table 3.6. In the above empirical example, suppose that ~tp ðk; jÞ ¼ h½1; 1; ½0; 0i for all ~ \0;1 [ ; p ¼ 1; 2; 3Þ. That is to say, the three experts provide the crisp ððk; jÞ 2 X p preference relations between alternatives as follows: X1 ¼ fð2; 1Þ; ð1; 3Þ; ð5; 4Þ; ð2; 5Þ; ð3; 2Þ; ð3; 4Þg, X2 ¼ fð1; 2Þ; ð3; 1Þ; ð4; 5Þ; ð5; 2Þ; ð2; 3Þ; ð4; 3Þg, and X3 ¼ fð2; 4Þ; ð1; 4Þ; ð5; 1Þ; ð5; 3Þg: These are only crisp sets without fuzzy truth degrees. Subsequently, we extend the fuzzy LINMAP [20] without considering

TrFN, interval and real number TrFN, interval and real number IFS, TrFN, interval and real number

Method [25] Method [26] Method [27]

TrFN TrFN IFS

1

1

1

TFN

P  1

TFN

Method [24]

0 or 1

q  1

Linguistic, real number transformed as IVIFS

Method [21]

0 or 1

Q  1

IFS

Method [20]

0 or 1

Q  1

Linguistic variable transformed as TrFN Linguistic variable transformed as TFN

Method [18] Method [19]

0 or 1

0 or 1

P  1

Linguistic variable transformed as TFN

1

0 or 1

1

Real number

Truth degree

Method [16] Method [17]

Number of DMs

Types of attribute values

Method

Table 3.6 Comparisons with existing methods

Four-objective programming with equal weights (lack flexibility) Bi-objective programming with equal weights (lack flexibility)

Possibilistic method with TrFNs

Possibilistic method with TFNs

Linear programming method

Linear programming method

Linear programming method

Linear programming method

Linear programming method

Linear programming method

Mathematical programming

Euclidean distance Euclidean distance Euclidean distance

Euclidean distance

Euclidean distance

Euclidean distance

Euclidean distance Euclidean distance

Euclidean distance Euclidean distance

Measurement tool

FPIS unknown

FPIS unknown FPIS known

FPIS unknown

FPIS unknown

FPIS unknown

FPIS unknown FPIS unknown

PIS unknown PIS unknown

FPIS or FNIS

(continued)

MADM

MADM

MAGDM with Q matrices MAGDM with Q matrices MAGDM with q matrices MAGDM with one matrix MADM

MAGDM with P matrices MADM

MADM

Solve problem

3.5 Empirical Example of Critical Infrastructure 101

Types of attribute values

IFS, TrFN, interval and real number

HFS

IVIFS, IFS, TrFN, linguistic, interval and real number

Method

Method [28]

Method [29]

This chapter

Table 3.6 (continued) Truth degree IFS

Interval

IVIFS

Number of DMs

Q  1

f  1

q  1 Goal programming approach (with flexibility)

Bi-objective programming with equal weights (lack flexibility)

Bi-objective programming with lexicographic method (lack flexibility)

Mathematical programming

Closeness degree

Euclidean distance

Euclidean distance

Measurement tool

FPIS and FNIS known

FPIS unknown

FPIS unknown

FPIS or FNIS MAGDM with Q matrices MAGDM with f matrices MCGDM with Q matrices

Solve problem

102 3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

3.5 Empirical Example of Critical Infrastructure

103

IVIF truth degrees to solve this example. Then the IVIF linear programming model (i.e., Eq. 3.35) is reduced to the corresponding linear programming model (see Eq. 3.37 in Appendix 2). By using the existing simplex method of the linear programming model, the optimal solution of Eq. (3.37) can be obtained, where its components are as follows: k121 ¼ k154 ¼ k125 ¼ k132 ¼ 0; k113 ¼ 0:0129; k134 ¼ 0:0121; k212 ¼ k231 ¼ k245 ¼ k252 ¼ k243 ¼ 0; k223 ¼ 0:0024; k324 ¼ k314 ¼ k351 ¼ k353 ¼ 0; d1þ ¼ 0:0014; d2þ ¼ 0:0030; d3þ ¼ 0:0080; d4þ ¼ 0;    d 1 ¼ d2 ¼ d3 ¼ 0; d4 ¼ 0:0932; b31 ¼ b32 ¼ b33 ¼ b34 ¼ 0:1565; 1 ¼ 0:15; v1 ¼ v1 ¼ 0; l2 ¼ 0:0393; v2 ¼ 0:008; d41 ¼ 0:0152; l1 ¼ 0:0156; l d42 ¼ 0:0346; d43 ¼ 0:18; f5 ¼ g5 ¼ 0:0498; z6 ¼ 0:16; x1 ¼ 0:15; x2 ¼ 0:05; x3 ¼ 0:28; x4 ¼ 0:18; x5 ¼ 0:18; x6 ¼ 0:16:

The FPIS r þ ¼ ðr1þ ; r2þ ; . . .; r6þ Þ can be calculated, where h i

 1þ ; ½t1þ ; t1þ  ¼ h½0:1041; 1; ½0; 0i; r2þ ¼ l2þ ; t2þ ¼ h0:7868; 0:1601i; r1þ ¼ ½l1þ ; l þ þ þ þ þ þ þ r3þ ¼ ðb31 ; b32 ; b33 ; b34 Þ ¼ ð0:5589; 0:5589; 0:5589; 0:5589Þ; r4þ ¼ ðd41 ; d42 ; d43 Þ ¼ ð0:0842; 0:1925; 1Þ;

r5þ ¼ ½f5þ ; g5þ  ¼ ½0:2766; 0:2766; r6þ ¼ z6þ ¼ 1:

The square of the distance of each ECO from the FPIS r þ given by each expert can be computed by Eq. (3.14) as follows: D11 ¼ 0:2761;

D12 ¼ 0:2761;

D13 ¼ 0:2761;

D14 ¼ 0:2951;

D15 ¼ 0:2862;

D21 ¼ 0:2599;

D22 ¼ 0:2718;

D23 ¼ 0:2717;

D24 ¼ 0:2717;

D25 ¼ 0:2718;

D31 ¼ 0:2754;

D32 ¼ 0:2821;

D33 ¼ 0:2587;

D34 ¼ 0:3156;

D35 ¼ 0:2642:

Therefore, the individual ranking order of the five ECOs is generated as follows: a1 a2 a3 a5 a4 for expert e1 ; a1 a3 a4 a2 a5 for expert e2 ; a3

a5 a1 a2 a4 for expert e3 . The Borda’s scores of the five ECOs can be obtained as in Table 3.7. The collective ranking order of the five ECOs is generated as follows: a3 a1 a2 a5 a4 . Therefore, the best selection is the ECO a3 . Compared the fuzzy LINMAP without considering IVIF truth degrees, the proposed method in this chapter has the following advantages:

Table 3.7 Borda’s scores of the ECOs

a1 a2 a3 a4 a5

e1

e2

e3

Borda’s score

4 4 4 2 1

4 2 3 3 3

2 1 4 0 3

10 7 11 5 7

104

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

(a) The ranking order obtained by the former is remarkably different from that obtained by the latter. Moreover, two alternatives a2 and a5 cannot be distinguished by the former. The main reason is that the former utilized the Borda’s scores to derive the collective ranking orders. As stated in Remark 3.7, if some alternatives have equal score, they cannot be further distinguished. However, the latter constructs the assignment model to generate the collective ranking order which assures that different alternatives have different ranking orders. Therefore, the proposed method in this chapter has higher distinguishing power than the existing LINMAP methods [16–21]. (b) Compared the individual ranking orders obtained by the former and latter, it is not difficult to find that these individual ranking orders are remarkably different. This analysis shows that introducing the fuzzy truth degrees of alternative comparisons and considering the FNIS are of great importance. It is often that DM gives the pair-wise comparisons of alternatives with some hesitancy degrees in actual decision problems. Consequently, it is very natural and reasonable to introduce IVIFSs to represent information of fuzzy truth degrees. The IVIF truth degrees are the suitable tools which can be used to effectively represent such hesitancy degrees. Nevertheless, the existing LINMAP methods [16–21] neglected the DM’s preferences on the comparisons of alternatives with fuzzy truth degrees and the FNIS. (c) The methods [24–29] ignored the hesitancy feature on alternatives’ comparisons, the methods [27, 28] utilized IFSs to represent fuzzy truth degrees, whereas this chapter uses IVIFSs to characterize the fuzzy truth degrees, which has stronger ability to depict the hesitancy degrees of subjective judgments. (d) We develop the linear goal programming approach to solving the bi-objective IVIF mathematical programming with IVIFSs. As Table 3.5 indicates, the ranking orders may be changed by changing the value proportion parameter g, which can give greater flexibility to the DMs in choosing the best solution.

3.6

Conclusions

In GDM, there are often three kinds of evaluation information: criteria values, criteria weights, and pair-wise comparisons of alternatives. In this chapter, we developed a new IVIF mathematical programming method to solve hybrid MCGDM with IVIF truth degrees and incomplete weight information. The main features of the proposed method are outlined as follows: (1) We firstly introduced the IVIFSs to describe pair-wise comparisons of alternatives with hesitancy degrees. Thus, the preference relations between alternatives given by DMs were formulated as an IVIFS of ordered pairs of alternatives. Thereby, we defined the IVIFS-type inconsistency and consistency indices considering the FPIS an FNIS simultaneously, which can effectively overcome the drawbacks of prior studies.

3.6 Conclusions

105

(2) To estimate the criteria weight vector, a novel bi-objective IVIF mathematical programming model of minimizing inconsistency and maximizing consistency was constructed and solved by the proposed linear goal programming approach. It is more flexible than the existing ones because it can provide the DMs with more choices as proportion parameter is assigned different values. (3) The collective ranking order of alternatives was derived through constructing a new multi-objective assignment model which assures that different alternatives are ranked in different positions and thus greatly enhances the distinguishing power. A real example of critical infrastructure evaluation example was analyzed to illustrate the effectiveness of the proposed method in this chapter. The comparison analyses with other existing methods clearly indicated that the existing LINMAP methods are all a special case of the proposed method in this chapter under some conditions. Apparently, the proposed method can also be applied to many GDM problems, such as supply chain management, water source assessment, and weapon system evaluation. In the coming research, we will further discuss the different approaches to solving the constructed IVIF mathematical programming and apply to the field of decision analysis.

Appendix 1 The proof of Theorem 3.1: According to Definition 3.1, the objective functions of Eq. (3.24) are IVIFS as follows: e¼ E

q X p¼1

2

X ðk;jÞ2 e X p\0;1 [

6Y 4 q

Y

p¼1

ðk;jÞ2 e X p\0;1 [

* ~tp ðk; jÞkpkj ¼

2

3

Y 6 41  p¼1

p

ðt~tp ðk;jÞ Þkkj ;

Y

q

kpkj

ðk;jÞ2 e X p\0;1 [

q Y

Y

p¼1

ðk;jÞ2 e X p\0;1 [

ð1  l~t ðk;jÞ Þ ; 1  p

q Y

Y

p¼1

[ ðk;jÞ2 e X \0;1 p

3 + p kkj 7 ðt~tp ðk;jÞ Þ 5

7 ð1  l ~tp ðk;jÞ Þ 5; kpkj

ð3:31Þ and e¼ F

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

2 q 6Y 4 p¼1

Y ðk;jÞ2 e X p\0;1 [

* ~tp ðk; jÞppkj ¼

ppkj

ðt~tp ðk;jÞ Þ ;

2

3

6 41 

q Y

Y

p¼1

[ ðk;jÞ2 e X \0;1 p

q Y

Y

p¼1

ðk;jÞ2 e X p\0;1 [

ppkj

ð1  l~t ðk;jÞ Þ ; 1  p

3

7 ðt~tp ðk;jÞ Þ 5

q Y

Y

p¼1

[ ðk;jÞ2 e X \0;1 p

7 ~tp ðk;jÞ Þ 5; ð1  l ppkj

+

ppkj

ð3:32Þ

106

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

Therefore, using the inclusion relationship of IVIFSs (see Definition 3.1), Eq. (3.24) can be solved by the following bi-objective interval objective programming model: 8 >
=

min 1  ð1  l~t ðk;jÞ Þ p > > ; : p¼1 [ ðk;jÞ2 e X \0;1 p 8 9 > > q < = Y Y p ~tp ðk;jÞ Þkkj min 1  ð1  l > > : ; p¼1 [ ðk;jÞ2 e X \0;1 p 8 9 > > q

> :p¼1 ; [ ðk;jÞ2 e X \0;1 p 8 9 > > q

> :p¼1 ; [ ðk;jÞ2 e X \0;1 p 8 9 > > q < = Y Y p ð1  l~t ðk;jÞ Þpkj max 1  p > > : ; p¼1 [ ðk;jÞ2 e X \0;1 p 8 9 > > q < = Y Y p ~tp ðk;jÞ Þpkj ð1  l max 1  > > : ; p¼1 [ ðk;jÞ2 e X \0;1 p 8 9 > > q

> :p¼1 ; [ ðk;jÞ2 e X \0;1 p 8 9 > > q

> :p¼1 ; [ ðk;jÞ2 e X \0;1 p s:t:ðkT ; pT ; xT Þ 2 H

Eq. (3.33) is equivalent to

ð3:33Þ

Appendix 1

107

max

max

max

max

min

min

min

min

8 > q

:p¼1 ðk;jÞ2 e X p\0;1 [ 8 > q

:p¼1 ðk;jÞ2 e X p\0;1 [ 8 > q

:p¼1 ðk;jÞ2 e X p\0;1 [ 8 > q

:p¼1 ðk;jÞ2 e X p\0;1 [ 8 > q

:p¼1 [ ðk;jÞ2 e X \0;1 p 8 > q

:p¼1 [ ðk;jÞ2 e X \0;1 p 8 > q

:p¼1 [ ðk;jÞ2 e X \0;1 p 8 > q

:p¼1

[ ðk;jÞ2 e X \0;1 p

9 > = p

ð1  l~t ðk;jÞ Þkkj

> ; 9 > = p

p

~tp ðk;jÞ Þkkj ð1  l

> ;

9 > = p

ðt~tp ðk;jÞ Þkkj

> ; 9 > = p

ðt~tp ðk;jÞ Þkkj

> ; 9 > = p

ð3:34Þ

ð1  l~t ðk;jÞ Þpkj

> ; 9 > = p

p

~tp ðk;jÞ Þpkj ð1  l

ðt~tp ðk;jÞ Þ

ppkj

> ;

9 > =

> ; 9 > = p

ðt~tp ðk;jÞ Þpkj

> ;

s:t:ðkT ; pT ; xT Þ 2 H It8is easily seen that the first objective in Eq. (3.34) is equivalent to 9 > > =

> ; : [ ðk;jÞ2 e X \0;1 p 9 8 > > = < Pq P p min Z1 ¼  p¼1 kkj lnð1  l~t ðk;jÞ Þ since 0  l~t ðk;jÞ  1. The other p p > > ; : ðk;jÞ2 e X \0;1 [ p

objectives in Eq. (3.34) can be similarly handled. Thus, Eq. (3.34) can be changed into the following multi-objective linear programming model:

108

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

8 > < min Z1 > : 8 > < min Z2 > : 8 > < min Z3 > : 8 > < min Z4 > : 8 > < min Z5 > : 8 > < min Z6 > : 8 > < min Z7 > : 8 > < min Z8 > :

¼

¼

¼

¼

¼

¼

¼

¼

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

q X

X

p¼1

[ ðk;jÞ2 e X \0;1 p

kpkj lnð1  l~t ðk;jÞ Þ p

~tp ðk;jÞ Þ kpkj lnð1  l

kpkj lnðt~tp ðk;jÞ Þ

kpkj lnðt~tp ðk;jÞ Þ

9 > = > ; 9 > = > ;

ppkj lnð1  l~t ðk;jÞ Þ p

~tp ðk;jÞ Þ ppkj lnð1  l

ppkj lnðt~tp ðk;jÞ Þ

ppkj lnðt~tp ðk;jÞ Þ

s:t:ðkT ; pT ; xT Þ 2 H Thus, the proof of Theorem 3.1 is complete.

9 > = > ; 9 > = > ;

9 > = > ; 9 > = > ;

9 > = > ; 9 > = > ;

Appendix 2

109

Appendix 2 e ¼ h½0:2; 0:4; ½0:3; 0:4ik112 þ h½0:5; 0:7; ½0:1; 0:2ik131 þ h½0:1; 0:3; ½0:5; 0:6ik145 þ h½0:3; 0:4; ½0:4; 0:5ik152 minf E þ h½0:2; 0:5; ½0:3; 0:4ik123 þ h½0:5; 0:6; ½0:2; 0:3ik143 þ h½0:1; 0:2; ½0:4; 0:6ik221 þ h½0:2; 0:4; ½0:2; 0:4ik213 þ h½0:2; 0:3; ½0:4; 0:5ik254 þ h½0:3; 0:4; ½0:4; 0:5ik225 þ h½0:5; 0:6; ½0:1; 0:3ik233 þ h½0:2; 0:4; ½0:3; 0:5ik234 þ h½0:3; 0:5; ½0:2; 0:3ik324 þ h½0:4; 0:5; ½0:2; 0:3ik314 þ h½0:2; 0:5; ½0:2; 0:4ik351 þ h½0:1; 0:3; ½0:2; 0:5ik353 g e maxf F ¼ h½0:2; 0:4; ½0:3; 0:4ip112 þ h½0:5; 0:7; ½0:1; 0:2ip131 þ h½0:1; 0:3; ½0:5; 0:6ip145 þ h½0:3; 0:4; ½0:4; 0:5ip152 þ h½0:2; 0:5; ½0:3; 0:4ip123 þ h½0:5; 0:6; ½0:2; 0:3ip143 þ h½0:1; 0:2; ½0:4; 0:6ip221 þ h½0:2; 0:4; ½0:2; 0:4ip213 þ h½0:2; 0:3; ½0:4; 0:5ip254 þ h½0:3; 0:4; ½0:4; 0:5ip225 þ h½0:5; 0:6; ½0:1; 0:3ip233 þ h½0:2; 0:4; ½0:3; 0:5ip234 þ h½0:3; 0:5; ½0:2; 0:3ip324 þ h½0:4; 0:5; ½0:2; 0:3ip314 þ h½0:2; 0:5; ½0:2; 0:4ip351 þ h½0:1; 0:3; ½0:2; 0:5ip353 g

ð3:35Þ 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 > > > k121  p121 ¼ D11  D12 ; k213  p213 ¼ D23  D21 ; k254  p254 ¼ D24  D25 ; k225  p225 ¼ D25  D22 ; k232  p232 ¼ D22  D23 ; > >  p ¼ D  D ; k  p ¼ D  D ; k  p ¼ D  D ; k  p ¼ D  D k > 34 34 4 3 12 12 2 1 31 31 1 3 45 45 5 4 ; k52  p52 ¼ D2  D5 ; > > 2 2 3 3 3 2 2 2 2 2 2 3 3 3 3 3 3 3 3 > k23  p23 ¼ D3  D2 ; k43  p43 ¼ D3  D4 ; k24  p24 ¼ D4  D2 ; k14  p14 ¼ D4  D1 ; k51  p51 ¼ D1  D35 ; > > > > k3  p3 ¼ D3  D3 ; D1  D1 þ k1  0; D1  D1 þ k1  0; D1  D1 þ k1  0; D1  D1 þ k1  0; D1  D1 þ k1  0; > 53 21 13 54 25 32 3 2 1 1 3 4 2 3 2 53 5 5 5 > > > D13  D14 þ k134  0; D21  D22 þ k212  0; D23  D21 þ k231  0; D24  D25 þ k245  0; D25  D22 þ k252  0; D22  D23 þ k223  0; > > > 2 3 3 3 3 2 2 3 3 3 3 3 3 3 3 1 1 1 > > D  D3 þ k43  0; D2  D4 þ k24  0; D1  D4 þ k14  0; D5  D1 þ k51  0; D5  D3 þ k53  0; D1  D2 þ p21  0; > < 41 D3  D11 þ p113  0; D14  D15 þ p154  0; D15  D12 þ p125  0; D12  D13 þ p132  0; D14  D13 þ p134  0; D22  D21 þ p212  0; s:t: > D21  D23 þ p231  0; D25  D24 þ p245  0; D22  D25 þ p252  0; D23  D22 þ p223  0; D23  D24 þ p243  0; > > > > D34  D32 þ p324  0; D34  D31 þ p314  0; D31  D35 þ p351  0; D33  D35 þ p353  0; > > > > k121  0; k113  0; k154  0; k125  0; k132  0; k134  0; k212  0; k231  0; k245  0; k252  0; k223  0; k243  0; > > 3 3 3 3 > 1 1 1 1 1 1 > k > 24  0; k14  0; k51  0; k53  0; p21  0; p13  0; p54  0; p25  0; p32  0; p34  0; > > > > p212  0; p231  0; p245  0; p252  0; p223  0; p243  0; p324  0; p314  0; p351  0; p353  0; > > > > x3  2:3x2 ; 0:1  x3  x4  0:25; 0:18  x5  0:35; x3  x4  x1  x2 ; x5  x6 > : x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 1; xi  0:05 ði ¼ 1; 2; 3; 4; 5; 6Þ

where D11 ¼ 0:5000x1 þ 0:5943x2 þ 0:7030x3 þ 0:0200x4 þ 0:0700x5 þ 0:0019x6 ; D12 ¼ 0:3333x1 þ 0:2561x2 þ 0:2970x3 þ 0:5000x4 þ 0:0036x5 þ 0:0038x6 ; D13 ¼ 3507x1 þ 0:6724x2 þ 0:9475x3 þ 0:9583x4 þ 0:0179x5 þ 0:0082x6 ; D14 ¼ 0:3563x1 þ 0:7206x2 þ 0:4000x3 þ 0:8371x4 þ 0:0032x5 ; D15 ¼ 0:4653x1 þ 0:9054x2 þ 0:3723x3 þ 0:1629x4 þ 0:0015x5 þ 0:0007x6 ; D21 ¼ 0:2756x1 þ 0:6724x2 þ 0:5000x3 þ 0:1629x4 þ 0:0409x5 þ 0:0051x6 ; D22 ¼ 0:5714x1 þ 0:4432x2 þ 0:9475x3 þ 0:5000x4 þ 0:0081x5 þ 0:0019x6 ; D23 ¼ 0:7027x1 þ 0:5781x2 þ 0:8576x3 þ 0:9583x4 þ 0:0045x5 ; D24 ¼ 0:4432x1 þ 0:8906x2 þ 0:1424x3 þ 0:8371x4 þ 0:0282x5 þ 0:0003x6 ; D25 ¼ 0:8082x1 þ 0:9605x2 þ 0:2970x3 þ 0:0200x4 þ 0:0005x5 þ 0:0101x6 ; D31 ¼ 0:3507x1 þ 0:8125x2 þ 0:2970x3 þ 0:5000x4 þ 0:0169x5 ; D32 ¼ 0:4702x1 þ 0:7206x2 þ 0:8684x3 þ 0:0200x4 þ 0:0323x5 þ 0:0101x6 ; D33 ¼ 0:7244x1 þ 0:5000x2 þ 0:7416x3 þ 0:8371x4 þ 0:0069x5 þ 0:0019x6 ; D34 ¼ 0:3507x1 þ 0:4597x2 þ 0:1424x3 þ 0:1629x4 þ 0:0021x5 þ 0:0028x6 ; D35 ¼ 0:4653x1 þ 0:3684x2 þ 0:7416x3 þ 0:9583x4 þ 0:0006x5 þ 0:0171x6 :

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

110

minfZ1 ¼ 0:1054k121 þ 0:2231k113 þ 0:2231k154 þ 0:3567k125 þ 0:6931k132 þ 0:2231k134 þ 0:2231k212 þ 0:6931k231 þ 0:1054k245 þ 0:3567k252 þ 0:2231k223 þ 0:6931k243 þ 0:3567k324 þ 0:5108k314 þ 0:2231k351 þ 0:1054k353 g minfZ2 ¼ 0:2231k121 þ 0:5108k113 þ 0:3567k154 þ 0:5108k125 þ 0:9163k132 þ 0:5108k212 þ 1:204k231 þ 0:3567k245 þ 0:5108k252 þ 0:6931k223 þ 0:9163k243 þ 0:5108k134 þ 0:6931k324 þ 0:693k314 þ 0:693k351 þ 0:3567k353 g minfZ3 ¼ 0:9163k121 þ 1:609k113 þ 0:9163k154 þ 0:9163k125 þ 2:303k132 þ 1:204k134 þ 1:204k212 þ 2:303k231 þ 0:6931k245 þ 0:9163k252 þ 1:204k223 þ 1:609k243 þ 1:609k324 þ 1:609k314 þ 1:609k351 þ 1:609k353 g minfZ4 ¼ 0:5108k121 þ 0:9163k113 þ 0:6931k154 þ 0:6931k125 þ 1:2040k132 þ 0:6931k134 þ 0:9163k212 þ 1:6094k231 þ 0:5108k245 þ 0:6931k252 þ 0:9163k223 þ 1:2040k243 þ 1:2040k324 þ 1:2040k314 þ 0:9163k351 þ 0:6931k353 g minfZ5 ¼ 0:1054p121 þ 0:2231p113 þ 0:2231p154 þ 0:3567p125 þ 0:6931p132 þ 0:2231p212 þ 0:6931p231 þ 0:1054p245 þ 0:3567p252 þ 0:2231p223 þ 0:6931p243 þ 0:2231p134 þ 0:3567p324 þ 0:5108p314 þ 0:2231p351 þ 0:1054p353 g minfZ6 ¼ 0:2231p121 þ 0:5108p113 þ 0:3567p154 þ 0:5108p125 þ 0:9163p132 þ 0:5108p134 0:5108p212 þ 1:204p231 þ 0:3567p245 þ 0:5108p252 þ 0:6931p223 þ 0:9163p243 þ þ 0:6931p324 þ 0:6931p314 þ 0:6931p351 þ 0:3567p353 g minfZ7 ¼ 0:9163p121 þ 1:609p113 þ 0:9163p154 þ 0:9163p125 þ 2:303p132 þ 1:204p212 þ 2:303p231 þ 0:6931p245 þ 0:9163p252 þ 1:204p223 þ 1:609p243 þ 1:204p134 þ 1:609p324 þ 1:609p314 þ 1:609p351 þ 1:609p353 g minfZ8 ¼ 0:9163p113 þ 0:6931p154 þ 0:6931p125 þ 1:2040p132 þ 0:6931p134 þ 0:9163p212 þ 1:6094p231 þ 0:5108p245 þ 0:6931p252 þ 0:9163p223 þ 1:2040p243 þ 0:5108p121 þ 1:2040p324 þ 1:2040p314 þ 0:9163p351 þ 0:6931p353 g s:t: same as the constraints of Eq: ð35Þ

ð3:36Þ minfz ¼ k121 þ k113 þ k154 þ k125 þ k133 þ k134 þ k212 þ k231 þ k245 þ k252 þ k223 þ k243 þ k324 þ k314 þ k351 þ k353 g 8 1 > n þ n113 þ n154 þ n125 þ n133 þ n134 þ n212 þ n231 þ n245 þ n252 þ n223 þ n243 þ n324 þ n314 þ n351 þ n353  0:0001; > > 21 > > > n1 þ k121  0; n113 þ k113  0; n154 þ k154  0; n125 þ k125  0; n132 þ k132  0; n134 þ k134  0; > > 21 > 2 2 2 2 2 2 2 2 2 2 2 2 > > > > n12 þ k12  0; n31 þ k31  0; n45 þ k45  0; n52 þ k52  0; n23 þ k23  0; n43 þ k43  0; > 3 3 3 3 3 3 3 3 > > > > n24 þ k24  0; n14 þ k14  0; n51 þ k51  0; n53 þ k53  0; > < k1  0; k1  0; k1  0; k1  0; k1  0; k1  0; k2  0; k2  0; k2  0; k2  0; k2  0; k2  0; 21 13 54 25 32 34 12 31 45 52 23 43 s:t: > 1 ; 0  v1  v1 ; l 1 þ v1  x1 ; k324  0; k314  0; k351  0; k353  0; 0  l1  l > > > > > > 0  l2  v2 ; l2 þ v2  x2 ; 0  b31  b32  b33  b34  x3 ; 0  d41  d42  d43  x4 ; > > > > > 0  f5  g5  x5 ; 0  z6  x6 ; > > > > > x3  2:3x2 ; 0:1  x3  x4  0:25; 0:18  x5  0:35; x3  x4  x1  x2 ; x5  x6 ; > > : x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 1; xi  0:05 ði ¼ 1; 2; 3; 4; 5; 6Þ

ð3:37Þ where n121 ¼ 0:05x1  0:1l1  0: l1 þ 0:1m1  0:05m1 þ 0:42x2  0:6l2  0:6m2 þ 0:2222x3  0:7047b31  0:1481b32  0:1481b33  0:0741b34 þ 0:56x4  0:2667d41  0:2667d42  0:2667d43 þ 0:4388x5  0:1263f5  0:1895g5  0:0017x6 þ 0:0333z6 n113 ¼ 0:015x1 þ 0:1 l1  0:15m1 þ 0:05m1  0:24x2 þ 0:6m2 þ 0:321x3  0:0741b31  0:1481b32  0:1481b33  0:0741b34  0:7833x4 þ 0:5333d41 þ 0:5333d42 þ 0:4667d43  0:0326x5 þ 0:0947f5 þ 0:1053g5 þ 0:0052x6  0:0833z6

Appendix 2

111

n154 ¼ 0:075x1 þ 0:1l1 þ 0:15 l1 þ 0:05m1 þ 0:0278m1  0:18x2 þ 0:3l2 þ 0:0062x3  0:037b31  0:4x4 þ 0:2667d41 þ 0:2667d42 þ 0:2667d43 þ 0:0016x5  0:0211f5  0:0211g5  0:0063x6 þ 0:05z6 n125 ¼ 0:015x1  0:05l1  0:05 l1 þ 0:05m1 þ 0:05m1 þ 0:46x2  l2  0:2m2 þ 0:0535x3 þ 0:037b31  0:0741b33  0:037b34 þ 0:24x4  0:1333d41  0:1333d42  0:1333d43  0:0018x5 þ 0:0316f5 þ 0:0211g5  0:0028x6 þ 0:0667z6 n132 ¼ 0:035x1 þ 0:1l1 þ 0:05m1  0:18x2 þ 0:6l2  0:5432x3 þ 0:1481b31 þ 0:2963b32 þ 0:2963b33 þ 0:1481b34 þ 0:2233x4  0:2667d41  0:2667d42  0:2d43  0:0113x5 þ 0:0312f5 þ 0:0842g5  0:0035x6 þ 0:05z6 n134 ¼ 0:125x1 þ 0:15l1 þ 0:1 l1 þ 0:15m þ 0:05m1  0:28x2  0:1l2  0:2m2  0:4835x3 þ 0:1481b31 þ 0:2963b32 þ 0:2222b33 þ 0:1111b34 þ 0:0633x4  0:1333d41  0:1333d42  0:0667d43  0:0115x5 þ 0:0316f5 þ 0:0421g5  0:0069x6 þ 0:1667z6 n212 ¼ 0:035x1  0:1l1  0:2 l1 þ 0:1m1 þ 0:05m1 þ 0:3l2  0:3m2 þ 0:4444x3  0:1111b31  0:2222b32  0:2222b33  0:1111b34  0:24x4 þ 0:1333d41 þ 0:1333d42 þ 0:1333d43  0:0231x5 þ 0:0526f5 þ 0:1158g5 þ 0:0027x6 þ 0:05z6 n231 ¼ 0:055x1 þ 0:15l1 þ 0:25 l1  0:15m1  0:1m1 þ 0:02x2  0:1l2 þ 0:1m2  0:2716x3 þ 0:0741b31 þ 0:1481b32 þ 0:1481b33 þ 0:0741b34 þ 0:4633x4  0:4d41  0:4d42  0:3333d43 þ 0:0259x5  0:0737f5 þ 0:1368g5 þ 0:0044x6  0:1333z6 n245 ¼ 0:15x1  0:35l1  0:15 l1  0:1m1 þ 0:15m1 þ 0:16x2  0:2l2  0:1m2 þ 0:0741x3  0:037b31  0:0741b32  0:0741b33  0:037b34 þ 0:72x4  0:4d41  0:4d42  0:4d43  0:0215x5 þ 0:0632f5 þ 0:1684g5 þ 0:0081x6  0:15z6 n252 ¼ 0:09x1 þ 0:15l1 þ 0:1 l1  0:1m1  0:34x2 þ 0:8l2  0:2m2 þ 0:5432x3  0:1481b31  0:2963b32  0:2963b33  0:1481b34  0:56x4 þ 0:2667d41 þ 0:2667d42 þ 0:2667d43 þ 0:0064x5  0:0526f5  0:0737g5  0:0067x6 þ 0:1z6 n223 ¼ 0:02x1  0:05l1  0:05 l1 þ 0:05m þ 0:05m1  0:02x2  0:2l2 þ 0:2m2  0:1728x3 þ 0:037b31 þ 0:0741b32 þ 0:0741b33 þ 0:037b34  0:2233x4 þ 0:2667d41 þ 0:2667d42 þ 0:2d43  0:0029x5 þ 0:0211f5 þ 0:0211g5  0:0017x6 þ 0:0833z6

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

112

n243 ¼ 0:08x1  0:25l1  0:1 l1  0:05m þ 0:1m1  0:2x2 þ 0:4l2  0:1m2 þ 0:4444x3  0:1481b31  0:2963b32  0:2963b33  0:1481b34  0:0633x4 þ 0:1333d41 þ 0:1333d42 þ 0:0667d43  0:0179x5 þ 0:0316f5 þ 0:1158g5  0:0003x6 þ 0:0333z6 n324 ¼ 0:14x1  0:1l1  0:05 l1  0:2m1  0:1m1 þ 0:08x2 þ 0:4l2  0:4v2  0:5226x3 þ 0:1111b31 þ 0:2963b32 þ 0:3704b33 þ 0:1852b34  0:32x4 þ 0:1333d41 þ 0:1333d42 þ 0:1333d43  0:022x5 þ 0:1263f5 þ 0:1158g5  0:5903x6 þ 0:0833z6 n314 ¼ 0:05l1 þ 0:05x2 þ 0:5l2  0:5m2  0:0741x3 þ 0:037b31 þ 0:074b32 þ 0:074b33 þ 0:037b34  0:2233x4 þ 0:2667d41 þ 0:2667d42 þ 0:2d43  0:0119x5 þ 0:0526f5 þ 0:0947g5 þ 0:0025x6  0:1z6

n351 ¼ 0:05x1  0:05l1 þ 0:05 l1  0:1m1  0:05m1 þ 0:24x2  0:6l2  0:2922x3 þ 0:037b31 þ 0:1481b32 þ 0:2222b33 þ 0:1111b34 þ 0:2233x4  0:2667d41  0:2667d42  0:2d43 þ 0:0133x5  0:04211f5  0:1263g5  0:0136x6 þ 0:2333z6 n353 ¼ 0:03x1  0:1l1  0:1 l1 þ 0:1m þ 0:1m1 þ 0:24x2  0:2l2  0:4m2 þ 0:0633x4  0:1333d41  0:1333d42  0:0667d43 þ 0:0055x5  0:0211f5  0:0737g5  0:0119x6 þ 0:15z6

References 1. S.P. Wan, 2-tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making. Knowl. Based Syst. 45, 31–40 (2013) 2. L.A. Zadeh, Fuzzy sets. Inf. Control 8, 338–356 (1965) 3. K. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 4. K. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989) 5. X.H. Yu, Z.S. Xu, Prioritized intuitionistic fuzzy aggregation operators. Information Fusion 14, 108–116 (2013) 6. B.Q. Li, W. He, Intuitionistic fuzzy PRI-AND and PRI-OR aggregation operators. Inf. Fusion 14, 450–459 (2013) 7. Z.M. Zhang, Generalized Atanassov’s intuitionistic fuzzy power geometric operators and their application to multiple attribute group decision making. Inf. Fusion 14, 460–486 (2013) 8. B.S. Liu, Y.H. Shen, X.H. Chen, Y. Chen, X.Q. Wang, A partial binary tree DEA-DA cyclic classification model for decision makers in complex multi-attribute large-group interval-valued intuitionistic fuzzy decision-making problems. Inf. Fusion 18, 119–130 (2014) 9. P. Balasubramaniam, V.P. Ananthi, Image fusion using intuitionistic fuzzy sets. Inf. Fusion 20, 21–30 (2014)

References

113

10. X. Chen, H.J. Zhang, Y.C. Dong, The fusion process with heterogeneous preference structures in group decision making: a survey. Inf. Fusion 24, 72–83 (2015) 11. D.F. Li, Z.G. Huang, G.H. Chen, A systematic approach to heterogeneous multiattribute group decision making. Comput. Ind. Eng. 59, 561–572 (2010) 12. F. Herrera, L. Martinez, P.J. Sanchez, Managing non-homogenous information in group decision making. Eur. J. Oper. Res. 166, 115–132 (2005) 13. S.J. Chuu, Selecting the advanced manufacturing technology using fuzzy multiple attributes group decision making with multiple fuzzy information. Comput. Ind. Eng. 57(3), 1033–1042 (2009) 14. L. Martinez, J. Liu, D. Ruan, J.B. Yang, Dealing with heterogeneous information in engineering evaluation processes. Inf. Sci. 177(7), 1533–1542 (2007) 15. Z.S. Xu, Multiple-attribute group decision making with different formats of preference information on attributes. IEEE Trans. Syst. Man Cybern. B Cybern. 37(6), 1500–1511 (2007) 16. V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference. Psychometrica 38, 337–342 (1973) 17. D.F. Li, J.B. Yang, Fuzzy linear programming technique for multiattribute group decision making in fuzzy environments. Inf. Sci. 158, 263–275 (2004) 18. H.C. Xia, D.F. Li, J.Y. Zhou, J.M. Wang, Fuzzy LINMAP method for multiattribute decision making under fuzzy environments. J. Comput. Syst. Sci. 72, 741–759 (2006) 19. D.F. Li, T. Sun, Fuzzy LINMAP method for multiple attribute group decision making with linguistic variables and incomplete information. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 15(2), 153–173 (2007) 20. D.F. Li, Linear programming method for multiattribute group decision making using IF sets. Inf. Sci. 180, 1591–1609 (2010) 21. Z.J. Wang, K.W. Li, An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives. Expert Syst. Appl. 39, 13509–13516 (2012) 22. L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning Part I. Inf. Sci. 8(3), 199–249 (1975) 23. D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980) 24. S. Sadi-Nezhad, P. Akhtari, Possibilistic programming approach for fuzzy multidimensional analysis of preference in group decision making. Appl. Soft Comput. 8, 1703–1711 (2008) 25. D.F. Li, S.P. Wan, Fuzzy linear programming approach to multiattribute decision making with multiple types of attribute values and incomplete weight information. Appl. Soft Comput. 13(11), 4333–4348 (2013) 26. D.-F. Li, S.P. Wan, Fuzzy heterogeneous multiattribute decision making method for outsourcing provider selection. Expert Syst. Appl. 41, 3047–3059 (2014) 27. S.P. Wan, D.F. Li, Fuzzy LINMAP approach to heterogeneous MADM considering the comparisons of alternatives with hesitation degrees. Omega 41(6), 925–940 (2013) 28. S.P. Wan, D.-F. Li, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with Atanassov’s intuitionistic fuzzy truth degrees. IEEE Trans. Fuzzy Syst. 22(2), 300–312 (2014) 29. X.L. Zhang, Z.S. Xu, Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives. Comput. Ind. Eng. 75, 217– 229 (2014) 30. C.L. Hwang, K. Yoon, Multiple attributes decision making methods and applications (Springer, Berlin Heidelberg, 1981) 31. E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst. 114(3), 505–518 (2000)

114

3 Interval-Valued Intuitionistic Fuzzy Mathematical Programming …

32. D.F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information. Appl. Soft Comput. 11, 3402–3418 (2011) 33. F. Flammini, A. Gaglione, N. Mazzocca, C. Pragliola, Optimisation of security system design by quantitative risk assessment and genetic algorithm. Int. J. Risk Assess. Manag. 15, 205– 221 (2011)

Chapter 4

A Selection Method Based on MAGDM with Interval-Valued Intuitionistic Fuzzy Sets

Abstract As the cloud computing develops rapidly, more and more cloud services appear. Many enterprises tend to utilize cloud service to achieve better flexibility and react faster to market demands. In the cloud service selection, several experts may be invited and many attributes (indicators or goals) should be considered. Therefore, the cloud service selection can be regarded as a kind of Multi-attribute group decision making (MAGDM) problems. This chapter develops a new method for solving such MAGDM problems. In this method, the ratings of the alternatives on attributes in individual decision matrices are interval-valued intuitionistic fuzzy sets (IVIFSs) which can flexibly describe the preferences of experts on qualitative attributes. First, the weights of experts on each attribute are determined by extending the classical gray relational analysis (GRA) into IVIF environment. Then, based on the collective decision matrix obtained by aggregating the individual matrices, the score (profit) matrix, accuracy matrix and uncertainty (risk) matrix are derived. A multi-objective programming model is constructed to determine the attribute weights. Subsequently, the alternatives are ranked by employing the overall scores and uncertainties of alternatives. Finally, a cloud service selection problem is provided to illustrate the feasibility and effectiveness of the proposed methods.




Keywords Cloud service Multi-attribute group decision making Interval-valued intuitionistic fuzzy set Gray related analysis

4.1







Introduction

Cloud computing [1–4] is the latest computing paradigm that delivers hardware and software resources as virtualization services in which users are free from the burden of worrying about the low-level system administration details. In recent years, cloud computing is developing rapidly and has provided enterprises with many advantages such as flexibility, business agility and pay-as-you-go cost structure. As a result, many enterprises with limited financial and human resources are increasingly © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_4

115

116

4 A Selection Method Based on MAGDM with Interval-Valued …

adopting cloud computing to deliver their business services and products online to extend their business markets. In many domains, multiple cloud services often supply similar functional properties. For example, in Customer Relationship Management (CRM), CRM venders offer functionally-equivalent cloud services, such as Microsoft Dynamic CRM, Salesfore Sales Cloud, SAP Sales on Demand, and Oracle Cloud CRM. However, for enterprises which are lack of cloud computing knowledge, it is difficult to select an appropriate candidate from a set of functionally-equivalent cloud services. Therefore, it is necessary for enterprises to invite several related experts to evaluate the potential candidates from several indicators (attributes), such as payment, performance, reputation, scalability and security. The selection of cloud services has attracted increasing attention and many methods have been presented to guide enterprises in selecting the cloud services. Roughly, these methods may be divided into two categories and briefly reviewed as follows, respectively. The first category is the Multi-attribute Decision Making (MADM) methods. According to the key performance indicators defined by Siegel and Perdue [5], Garg et al. [6] proposed the cloud service ranking framework using the Analytic Hierarchy Process technique (AHP). Menzel et al. [7] utilized the Analytic Network Process (ANP) to develop a Multi-criteria Comparison Method which is used to select Infrastructure-as-a-Service (IaaS). Limam and Boutaba [8] presented a trustworthiness-based service selection method based on the Multiple Attribute Utility Theory (MAUT). By employing the Elimination and Choice Expressing Reality (ELECTRE) method, Silas et al. [9] developed a cloud service selection middleware to help cloud users select desired cloud service. Saripalli and Pingali [10] discussed Simple Additive Weighting (SAW) methods to rank alternatives in a decision problem of cloud service adoption. Zhao et al. [11] suggested a SAW-based service searching and scheduling algorithm to obtain a set of ranked services. The second category is the optimization approaches. Chang et al. [12] designed a dynamic programming algorithm by maximizing the overall survival probability to select cloud storage providers. Sundareswaran et al. [13] selected cloud service with a greedy algorithm method which can make experts retrieve information fast. In order to help service providers to select Software-as-a-Service (SaaS) services with multi-tenants, He et al. [14] explored three types of optimization algorithms, including integer programming, skyline and greedy algorithm, and proposed a quality of service (QoS)-driven optimization framework. By minimizing costs and risks, Martens et al. [15] constructed a scalable mathematical decision model to select cloud service. Yang et al. [16] built a Markov decision process model to guarantee the near-optimal performance in a changing environment by dynamically adjusting the components of a service composition. The aforementioned methods seem to be effective and applicable for selecting cloud services. However, they have following shortcomings. (1) The decision making in methods [13–15] are single Multi-attribute Decision Making (MADM). i.e., only one expert participates in the decision making and

4.1 Introduction

117

gives assessment information of alternatives with respect to several attributes. Since every expert is good at only some fields rather than all fields, the reliability of some information given by the expert is a little doubtful. (2) Current methods [9, 12] are more focused on quantitative attributes measured via precise numerical values, such as response time, storage space and latency time. Nevertheless, in cloud service, some qualitative attributes (such as reputation and security) usually play important roles, but they do not gain enough attention. (3) In existing methods [12, 13, 16], the assessment values (attribute values) are crisp numbers, which is somewhat unrealistic. Due to the inherent vagueness of human preferences as well as the fuzziness and uncertainty of objects, it is more suitable to express the assessment values as fuzzy numbers [17–20]. (4) The attribute weights provided by experts are given a priori in methods [8, 10, 11], which always cannot avoid subjective randomness of the expert’s preference. Furthermore, with increasing complexity in many real decision situations, it is difficult for expert to provide precise and complete preference information due to time pressure and lack of data. One of the reasons leading to the above shortcomings is that the fuzziness and uncertain are not fully considered during the decision making process. The fuzzy set (FS) theory introduced by Zadeh [21] is a very useful tool to describe fuzzy and uncertain information. Based on the FS theory, Atanassov [22] presented the intuitionistic fuzzy set (IFS), which considers the membership (satisfaction) degree, non-membership (dissatisfaction) degree and hesitant degree, simultaneously. Subsequently, Atanassov and Gargov [23] generalized IFS and presented Interval-valued intuitionistic fuzzy set (IVIFS) that describes the membership and non-membership degrees as intervals. Compared with the FS and IFS, IVIFS is more suitable to express the fuzziness and uncertainty and has been widely used in many fields [24–28]. According to IVIFS theory, to overcome the aforementioned shortcomings, we investigate the cloud service selection problems with IVIFSs and develop a novel method. The proposed method has the following key characteristics: (1) The selection of cloud service is regarded as a Multi-attribute Group Decision Making (MAGDM) problem that several experts are invited to evaluate the potential cloud services, whereas it is considered as a single MADM problem in methods [6–11]. With increasing complexity and the limit knowledge owned by single expert, in order to increase the quality of cloud service, it is more reasonable and reliable for enterprises to invite multiple experts to participate in making decision together. (2) The assessment values given by experts are expressed as IVIFSs. Compared with the crisp number, IVIFS is more flexible to measure the qualitative attributes since IVIFS considers membership, non-membership and hesitant degrees which are expressed as intervals. Additionally, it is easier for experts to

4 A Selection Method Based on MAGDM with Interval-Valued …

118

supply assessment values with IVIFSs in the increasing uncertain and complex environment. (3) By extending the classical gray relational analysis (GRA) [29] into IVIF environment, a new approach is proposed to determine the weights of experts. A notable characteristic of the proposed approach is that the obtained weights of each expert are different with respect to different attributes. (4) For MAGDM problems with incomplete information on attributes, a multi-objective programming model is constructed to objectively determine the attribute weights, which can avoid the subjective randomness appearing in the methods [8, 10, 11]. Moreover, it is easier for experts to give partial information on attribute weights than to assign a crisp number to the attribute weights. The rest of this chapter unfolds as follows. Some preliminaries about IVIFSs and the classical GRA method are introduced in Sect. 4.2. In Sect. 4.3, a new method is proposed to solve MAGDM problems with IVIFSs and incomplete attribute weight information. In addition, a framework of decision supporting system (DSS) is constructed. In Sect. 4.4, a cloud service selection example is provided to illustrate the applicability of the proposed method and comparison analysis is conducted. Finally, the conclusions are discussed in Sect. 4.5.

4.2

Preliminaries

In this section, we introduce some basic concepts related to interval-valued fuzzy set (IVIFS) and gray relational analysis (GRA).

4.2.1

Interval-Valued Intuitionistic Fuzzy Set

Definition 4.1 ([23]). Let X ¼ fx1 ; x2 ; . . .; xn g be a non-empty set of the universe. ~ in X is defined as An IVIFS A ~ ¼ fðxi ; ½lL~ ðxi Þ; lR~ ðxi Þ; ½vL~ ðxi Þ; vR~ ðxi ÞÞjxi 2 X g; A A A A A h i h i where lLA~ ðxi Þ; lRA~ ðxi Þ and vLA~ ðxi Þ; vRA~ ðxi Þ denote the intervals of membership

~ respectively, satisfying degree and non-membership degree of element xi 2 A, L R R L R lA~ ðxi Þ þ vA~ ðxi Þ  1, 0  lA~ ðxi Þ  lA~ ðxi Þ  1 and 0  vA~ ðxi Þ  vRA~ ðxi Þ  1 for all xi 2 X. h i pA~ ðxi Þ ¼ 1  lRA~ ðxi Þ  vRA~ ðxi Þ; 1  lLA~ ðxi Þ  vLA~ ðxi Þ is called the interval-valued ~ For any xi 2 X, if lL ðxi Þ ¼ lR ðxi Þ and intuitionistic hesitant degree of IVIFS A. ~ ~ A A ~ is reduced to an IFS. vL~ ðxi Þ ¼ vR~ ðxi Þ, then, A A

A

4.2 Preliminaries

119

Xu [30] called the pair ~a ¼ ðl~a ðxi Þ; v~a ðxi ÞÞ an interval-valued intuitionistic fuzzy number (IVIFN), and denoted an IVIFN by ~a ¼ ð½a; b; ½c; dÞ, where ½a; b½0; 1, ½c; d½0; 1, b þ d  1. Definition 4.2 ([30]). Let a~1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ, ~ a2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ and ~ a¼ ð½a; b; ½c; dÞ be three IVIFNs, then (1) Complement : ð~aÞc ¼ ð½c; d; ½a; bÞ; (2) Addition : ~a1 þ ~a2 ¼ ð½a1 þ a2  a1 a2 ; b1 þ b2  b1 b2 ; ½c1 c2 ; d1 d2 Þ; (3) Scalar multiplication : k~a ¼ ð½1  ð1  aÞk ; 1  ð1  bÞk ; ½ck ; d k Þ; k [ 0. Definition 4.3 ([30]). Let ~aj ¼ ð½aj ; bj ; ½cj ; dj Þ ðj ¼ 1; 2; . . .; nÞ be a collection of IVIFNs. If IVIFWAx ð~a1 ; ~a2 ; . . .; ~an Þ ¼

n X

xj ~ aj ;

ð4:1Þ

j¼1

then the IVIFWA is called an interval-valued intuitionistic fuzzy weighted averT aging (IVIFWA) operator of dimension Pn n, where x ¼ ðx1 ; x2 ; . . .; xn Þ is a weight vector of ~aj with xj 2 ½0; 1 and j¼1 xj ¼ 1. The aggregated value determined by the IVFWA operator is also an IVIFN, i.e., " IVIFWAx ð~a1 ; ~a2 ; . . .; ~ an Þ ¼

1

n Y j¼1

xj

ð1  aj Þ ; 1 

n Y

# " ð1  bj Þ

xj

;

j¼1

n Y j¼1

x cj j ;

n Y

#! x dj j

:

j¼1

ð4:2Þ Definition 4.4 ([30]). Let ~a ¼ ð½a; b; ½c; dÞ be an IVIFN. Then 1 sð~aÞ ¼ ða þ b  c  dÞ 2

ð4:3Þ

1 hð~aÞ ¼ ða þ c þ b þ dÞ 2

ð4:4Þ

and

~, are respectively called the score function and accuracy function of the IVIFN a where sð~aÞ 2 ½1; 1 and hð~aÞ 2 ½0; 1 can be considered as net membership and accuracy degree, respectively. Since sð~aÞ 2 ½1; 1, when many score functions are aggregated with linear weighted summation method, it maybe appears that positive score functions are offset by negative score functions. Therefore, we normalize the score function and make it belong to [0, 1].

4 A Selection Method Based on MAGDM with Interval-Valued …

120

Given a variable y 2 ½1; 1, if we define f ðyÞ ¼

yþ1 ; 2

ð4:5Þ

then f ðyÞ can not only retain the monotonicity of the variable y, but also map y to [0, 1]. Hence, we modify the score function in Definition 4.4 and define a new score function of IVIFN ~a below. Definition 4.5 Let ~a ¼ ð½a; b; ½c; dÞ be an IVIFN. Then 1 s ð~aÞ ¼ ðsð~aÞ þ 1Þ 2

ð4:6Þ

is called a normalized score function, where sð~aÞ ¼ 12 ða  c þ b  dÞ. Obviously, s ð~aÞ 2 ½0; 1. Definition 4.6 Let ~a ¼ ð½a; b; ½c; dÞ be an IVIFN. Then cð~aÞ ¼ 1  hð~aÞ

ð4:7Þ

is called an uncertainty function, where hð~aÞ ¼ 12 ða þ c þ b þ dÞ. Let ~a ¼ ð½a; b; ½c; dÞ be an assessment value of the cloud service x with respect aÞ and the to the attribute (indicator) ~a. Then the normalized score function s ð~ uncertainty function cð~aÞ can be respectively interpreted as the “net profit” and “risk” provided by cloud service x on attribute ~a. Hence, the bigger the s ð~ aÞ and the smaller the cð~aÞ, the better cloud service x. In the following, a new order relationship between IVIFNs is given. ~2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ be two Definition 4.7 Let ~a1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ and a IVIFNs. The order relation between ~a1 and ~a2 is specified as follows: (1) If s ð~a1 Þ\s ð~ a2 Þ; then ~a1 \~a2 ; (2) If s ð~a1 Þ ¼ s ð~a2 Þ; then (i) If cð~a1 Þ ¼ cð~a2 Þ, then a~1 ¼ ~a2 ; (ii) If cð~a1 Þ [ cð~a2 Þ, then a~1 \~a2 . Definition 4.8 Let a~1 ¼ ð½a1 ; b1 ; ½c1 ; d1 Þ, ~a2 ¼ ð½a2 ; b2 ; ½c2 ; d2 Þ be two IVIFNs, the Euclidean distance between ~a1 and ~a2 is defined as follows: dð~a1 ; ~a2 Þ ¼

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða1  a2 Þ2 þ ðb1  b2 Þ2 þ ðc1  c2 Þ2 þ ðd1  d2 Þ2 :

ð4:8Þ

4.2 Preliminaries

4.2.2

121

Gray Relation Analysis

GRA method is an important part of Gray Theory developed by Deng [29]. GRA investigates uncertain relationship between one main factor and all other factors in a system and has been used in a wide variety of decision making environments, such as supplier selection [31], material selection [32], and water protection strategy evaluation [33]. The details of the classical GRA method are presented as follows: (i) Calculate the normalized decision matrix. Let F ¼ ðfij Þmn be a decision matrix. The normalized matrix R ¼ ðrij Þmn is calculated as

rij ¼

8 max f f ij ij i > > < max fij min fij i

fij min fij > > : max f i min f i

i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .n; j 2 cost attributes

i

ij

i

ij

i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .n; j 2 benifite attributes ð4:9Þ

(ii) Generate comparability sequences ri ¼ ðri1 ; ri2 ; . . .; rin Þ ði ¼ 1; 2; . . .; mÞ and a reference sequence is r0 ¼ ðr01 ; r02 ; . . .; r0n Þ. For example, we can take r0j ¼ maxfrij g ðj ¼ 1; 2; . . .; nÞ. i

(iii) Compute the gray relational coefficient between the comparability sequence ri and the reference sequence r0 by the following formula: nðrij ; r0j Þ ¼

d  þ sd þ ði ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; nÞ; dðrij ; r0j Þ þ sd þ

  Where dðrij ; r0j Þ ¼ rij  r0j , d  ¼ min

ð4:10Þ

min dðrij ; r0j Þ, d þ ¼ max

1im 1jn

max

1im 1jn

dðrij ; r0j Þ, and s 2 ½0; 1 is a distinguishing coefficient. Usually, s ¼ 0:5: (iv) Calculate the gray relational grade between ri and r0 , i.e., ni ¼

n X

xj nij ;

ð4:11Þ

j¼1

where x ¼ ðx1 ; x2 ; . . .; xn ÞT is a weight vector satisfying xj 2 ½0; 1 and n P xj ¼ 1. The bigger the ni , the closer the sequence ri to the sequence r0 . j¼1

122

4.3

4 A Selection Method Based on MAGDM with Interval-Valued …

A Novel Method for MAGDM with IVIFSs and Incomplete Attribute Weight Information

In this section, a new method is proposed to handle MAGDM with IVIFSs. The proposed method includes determination of the weights of experts and identification of attribute weights. Let A ¼ fA1 ; A2 ; . . .; Am g be the set of m feasible alternatives, U ¼ fu1 ; u2 ; . . .; un g be the set of attributes and E ¼ fe1 ; e2 ;


; et g be the set of T experts. Assume that x ¼ ðx1 ; x 2 ; . . .; xn Þ is a attribute weight vector, where P n xj 2 ½0; 1 ðj ¼ 1; 2;


; nÞ and j¼1 xj ¼ 1. Let the individual decision matrix ~ k ¼ ð~f k Þ , where ~f k ¼ ð½ak ; bk ; ½ck ; d k Þ is an IVIFN for given by expert ek be F ij mn ij ij ij ij ij h i h i the alternative Ai with respect to attribute uj . In this chapter, akij ; bkij and ckij ; dijk provided by the expert ek are respectively the satisfaction (agreeing) degree interval and dissatisfaction (disagreeing) degree interval of the i-th cloud service Ai with respect to the j-th attribute (indicator) uj .

4.3.1

Determine the Weights of Experts by the Extended GRA Method

Due to the fact that each expert is skilled in some fields rather than all fields, it is more reasonable that the weights of each expert with respect to different attributes should be assigned different values. However, the weights of each expert with respect to different attributes are the same in existing methods [34–37]. Let kkj be the weight of expert ek with respect to attribute uj . Generally, for the attribute uj , the closer the attribute values of all alternatives given by expert ek are to those given by all other t  1 experts, the more similar the information provided by the expert ek is to that implied by the group. Consequently, the weight of expert ek should be assigned a greater value. Bearing this idea in mind, we present a novel method to determine the weights of experts by extending classical GRA method. ~k ~ k ¼ ð~f k Þ Given the decision matrices F ij mn ðk ¼ 1; 2; . . .; tÞ, the elements fij can be normalized as ( ~rijk

¼

j 2 benifite attributes fijk ðfijk Þc j 2 cost attributes

ð4:12Þ

~ k ¼ ð~r k Þ The normalized decision matrices can be denoted by R ij mn ðk ¼ 1; 2; . . .; tÞ. k k k ; ~r2j ; . . .; ~rmj Þ be the reference sequence and all other sequences Let ~rkj ¼ ð~r1j l l l ~rlj ¼ ð~r1j ; ~r2j ; . . .; ~rmj Þ ðl ¼ 1; 2; . . .; t; l 6¼ kÞ be comparability sequences. Then, for

4.3 A Novel Method for MAGDM with IVIFSs and Incomplete …

123

the attribute uj , the gray relational coefficient between ~rjk and ~rjl with respect to alternative Ai is defined as nlkij ¼ nð~rijl ; ~rijk Þ ¼

djk þ sdjk þ dð~rijl ; ~rijk Þ þ sdjk þ

;

ð4:13Þ

where dð~rijl ; ~rijk Þ is the distance between ~rijl and ~rijk (see Eq. (4.8)), djk ¼ min min dð~rijl ; ~rijk Þ, djk þ ¼ max max dð~rijl ; ~rijk Þ and s ¼ 0:5. 1  l  s;l6¼k 1  i  m

1  l  s;l6¼k 1  i  m

Thus, the matrix of gray relational coefficient between ~rijl and ~rijk is constructed as nkj ¼ ðnlkij Þðt1Þm ;

ð4:14Þ

where l ¼ 1; 2; . . .; t; l 6¼ k. The gray relational grade between ~rjk and ~rjl is calculated as gð~rjl ; ~rjk Þ ¼

1 m

m X

nlkij :

ð4:15Þ

i¼1

The gray relational grade gð~rjk ; ~rjl Þ describes the degree of closeness between sequence ~rjk and sequence ~rjl . In other words, gð~rjk ; ~rjl Þ indicates the similarity degree between the information given by expert ek and that given by expert el on attribute uj . For the attribute uj , the average gray relational grade between expert ek and all other experts el ðl 2 D; l 6¼ kÞ is computed as: gkj ¼

1 t1

t X

cð~rjl ; ~rjk Þ:

ð4:16Þ

l¼1;l6¼k

Thus, the larger the gkj , the more similar the information given by the expert ek is to that given by the group. Therefore, the bigger the kkj is. Accordingly, the weight of expert ek with respect to attribute uj , denoted by kkj , can be defined as kkj ¼ gkj =

t X l¼1

glj :

ð4:17Þ

4 A Selection Method Based on MAGDM with Interval-Valued …

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4.3.2

Integrate Individual Decision Matrices into a Collective Matrix

~k ¼ After the weights of experts are obtained, individual decision matrices R ~ ¼ ð~rij Þ ð~rijk Þmn ðk ¼ 1; 2; . . .; tÞ can be integrated into a collective matrix R mn with IVIFWA operator, where ~rij ¼

t X k¼1 "

¼

~rijk kkj

1

t Y k¼1

ð1 

akij Þkj ; 1 k



t Y

# " ð1 

bkij Þkj

k

;

k¼1

t Y k¼1

ðckij Þkj ; k

t Y

#! ðdijk Þkj

k

:

k¼1

ð4:18Þ For convenience, we denote ~rij by ~rij ¼ ð½aij ; bij ; ½cij ; dij Þ:

ð4:19Þ

By employing Eq. (4.4), Eq. (4.6) and Eq. (4.7), the score matrix, accuracy ~ are respectively obtained as follows: matrix and uncertainty matrix of matrix R S ¼ ðsij Þmn ;

ð4:20Þ

H ¼ ðhij Þmn ;

ð4:21Þ

c ¼ ðcij Þmn ;

ð4:22Þ

where sij ¼ s ð~rij Þ, hij ¼ hð~rij Þ and cij ¼ cð~rij Þ. Utilizing the weighted summation method, we can derive the overall score function, accuracy function and uncertainty function of alternative Ai as si ¼

n X

xj sij

ð4:23Þ

xj hij

ð4:24Þ

xj cij :

ð4:25Þ

j¼1

hi ¼

n X j¼1

ci ¼

n X j¼1

If the attribute weights are known in advance, then alternatives can be ranked and selected according to Definition 4.7. In what follows, a new multi-objective linear programming model is constructed to determine the attribute weights.

4.3 A Novel Method for MAGDM with IVIFSs and Incomplete …

4.3.3

125

Identify the Attribute Weights by a New Multi-objective Linear Programming Model

Due to the uncertainty of decision making environment and the limited knowledge possessed by experts, experts only may supply partial information about attribute weights. Namely, the information of the attribute weights is incomplete. Let D be the set of incomplete information on attribute weights. According to Definition 4.7, the bigger the overall score function (i.e., profit function) si and the smaller the overall uncertainty function (i.e., risk function) ci of the alternative Ai , the better the alternative Ai . Therefore, by maximizing the overall score functions and minimizing the overall uncertainty functions, a multi-objective programming is built to objectively determine the weights of attributes. max f s1 ; s2 ; . . .; sm g min fc1 ; c2 ; . . .; cm g

ð4:26Þ

s:t: x 2 D By the Max-min method for solving multi-objective programming [38], Eq. (4.26) can be converted as max f min si g i

min f max ci g i

ð4:27Þ

s:t: x 2 D From the relationship between ci and hi (see Eq. (4.7)), when ci reaches maximum, hi reaches minimum. Accordingly, minimizing the maximum among ci is equivalent to maximizing the minimum among hi . Therefore, Eq. (4.27) can be transformed as max f min si g i

max f min hi g i

ð4:28Þ

s:t: x 2 D Assume that y ¼ min si , x ¼ min hi , we have si y and hi x. Thus, by i

i

employing Eqs. (4.23), (4.24) and (4.28) can be rewritten as

126

4 A Selection Method Based on MAGDM with Interval-Valued …

max y max8x n P > > s wj y i ¼ 1; 2; . . .; m > > < j¼1 ij n s:t: P > hij wj x i ¼ 1; 2; . . .; m > > > : j¼1 x2D

ð4:29Þ

By the linear weighted summation method, Eq. (4.29) can be converted into the following single objective programming model: max8fpy þ (1  p)xg n P > > s wj y i ¼ 1; 2; . . .; m > > < j¼1 ij n s:t: P > hij wj x i ¼ 1; 2; . . .; m > > > j¼1 : x2D

ð4:30Þ

where p 2 ½0; 1 represents the relative importance of the two objects. If 0  p\0:5, then experts are pessimistic and more concern about uncertainty function (i.e., risk) than score function (i.e., profit); If 0:5\p  1, then experts are optimistic and more concerned about profit than risk; If p ¼ 0:5, then experts consider that profit is as important as risk. By solving Eq. (4.30), the vector of attribute weights x ¼ ðx1 ; x2 ; . . .; xm ÞT can be obtained.

4.3.4

Decision Process and Algorithm for MAGDM Problems with IVIFSs

Based on the above analysis, the algorithm and decision process for MAGDM problems are summarized as follows: Step 1: The experts establish the individual decision matrices with IVIFSs and ~ k ¼ ð~r k Þ normalize them into R ij mn ðk ¼ 1; 2; . . .; tÞ and supply the set of information on the attribute weights D. Step 2: Calculate the weight of expert ek by Eqs. (4.12)–(4.17), where k ¼ 1; 2; . . .; t; j ¼ 1; 2; . . .; n. ~ k ¼ ð~r k Þ Step 3: Integrate all normalized individual decision matrices R ij mn ðk ¼ ~ ¼ ð~rij Þ 1; 2; . . .; tÞ into a collective matrix R mn by Eq. (4.18). Step 4: Derive the score matrix S , accuracy matrix H and uncertainty matrix c of ~ by Eqs. (4.19)–(4.22). the matrix R Step 5: Determine the weight vector of attributes x by solving Eq. (4.30).

4.3 A Novel Method for MAGDM with IVIFSs and Incomplete …

127

Step 6: Compute the overall score si and uncertainty ci of alternatives Ai by Eqs. (4.23) and (4.25). Step 7: Rank the alternatives and select the best one according to Definition 4.7.

4.3.5

Decision Support System Framework Based on MAGDM with IVIFSs

As the scale of decision making increases, the procedure solving a MAGDM may be complicated. In this case, a decision supporting system (DSS), which is a class of computer-based information system including knowledge-based systems [39, 40], can be formulated to help experts improve their decision-making level and quality through problem analysis, establishment of models and simulation of decision-making process in a human-computer interaction way. Figure 4.1 depicts a framework of DSS designed in this chapter for MAGDM with IVIFSs. As shown in Fig. , the DSS consists of three modules: User interface, Knowledge base and Model base. Generally, the user interface establishes an interaction between experts and inputs the basic decision information, such as attributes, alternatives and assessment values of alternatives on attributes. The main function of Knowledge base is to help experts perform information transformation and store the corresponding information. For example, the ratings of alternatives on attributes given by experts are transformed into IVIF forms from which individual decision matrices with IVIFSs are constructed and used for the next calculation procedure. Model base involves the methods, such as extended GRA and objective programming as mentioned above. Thus, the ranking of alternatives can be deduced and the optimal decision can be derived by DSS.

4.4

A Cloud Service Selection Problem and Comparison Analysis

In this section, a real cloud service selection problem is given to illustrate the application of the proposed method of this chapter. Meanwhile, the comparison analysis is also conducted to show the superiority of the proposed method of this chapter.

4 A Selection Method Based on MAGDM with Interval-Valued …

128

Description of the problems Knowledge Base Construct IVIF individual decision matrices

Normalize the individual decision matrices Select the reference and comparability sequences Compute the gray relational coefficient

Determination module of experts’ weights

Calculate the gray relational grade Calculate the average gray relational grade User Interface

Decision Group

Derive the experts’ weights by normalizing the average gray relational grade Integrate the individual decision matrices into a collective matrix Compute the score, accuracy and uncertainty matrices Construct a bi-objective programming model

Determination module of attributes’ weights

Obtain the weights of attributes by solving the model Compute the overall score values and uncertainty values

Ranking order module of alternatives

Rank alternatives and select the best one

Model Base

Decision supporting

Fig. 4.1 Framework of interval-valued intuitionistic fuzzy MAGDM decision supporting system

4.4.1

A Cloud Service Provider Selection Problem and the Solution Process

Due to the limited technology and capital, an enterprise itself may be unable to build the cloud platform and tries to seek a cloud service to realize its CRM. After the market research and preliminary screening, there are four potential cloud services for further evaluation, including SAP Sales on Demand ðA1 Þ, Salesfore Sales Cloud ðA2 Þ, Microsoft Dynamic CRM ðA3 Þ and Oracle Cloud CRM ðA4 Þ. Four experts ðe1 ; e2 ; e3 ; e4 Þ are invited to evaluate these cloud services on five indicators (attributes), including performance ðu1 Þ, payment ðu2 Þ, reputation ðu3 Þ, scalability ðu4 Þ, and security ðu5 Þ. In terms of each attribute, each expert has presented his

4.4 A Cloud Service Selection Problem and Comparison Analysis

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(her) evaluation information for four cloud services and the normalized decision matrices are shown in Tables 4.1, 4.2, 4.3, 4.4. The preference relation set of attribute weights information supplied by experts is as follows: 8 9 < x1  2x2 ; 0:05  x2  x4  0:1; x5 2x3 ; = n D ¼ x  0:4; x þ x þ x 0:3; P x ¼ 1; x 0: 1 2 3 j j : 1 ; j¼1

Step 1: See Tables 4.1, 4.2, 4.3, 4.4. Step 2: Calculate the weights of experts. We take the weights of experts on u1 as an example, i.e., kk1 ðk ¼ 1; 2; 3; 4Þ, to illustrate the calculating process of the experts’ weights. The calculating processes for kk1 are as follows: (i) Select the reference sequence and comparability sequences. Selecting ~r11 as a reference sequence and ~r21 ; ~r31 ; ~r41 as comparability sequences, where ðð½0:55; 0:65; ½0:15; 0:25Þ; ð½0:35; 0:45; ½0:25; 0:35Þ; ð½0:55; 0:65; ½0:15; 0:25Þ; ð½0:35; 0:55; ½0:35; 0:45ÞÞ; ð ð½0:45; 0:55; ½0:25; 0:45Þ; ð½0:35; 0:55; ½0:30; 0:40Þ; 2 2 2 2 ¼ ð~r11 ; ~r21 ; ~r31 ; ~r41 Þ¼ ð½0:45; 0:65; ½0:25; 0:35Þ; ð½0:35; 0:45; ½0:35; 0:55ÞÞ; ðð½0:45; 0:75; ½0:15; 0:25Þ; ð½0:45; 0:55; ½0:25; 0:45Þ; 3 3 3 3 ¼ ð~r11 ; ~r21 ; ~r31 ; ~r41 Þ¼ ð½0:25; 0:45; ½0:35; 0:45Þ; ð½0:35; 0:45; ½0:25; 0:45ÞÞ; ð ð½0:65; 0:75; ½0:15; 0:25Þ; ð½0:40; 0:50; ½0:40; 0:50Þ; 4 4 4 4 ¼ ð~r11 ; ~r21 ; ~r31 ; ~r41 Þ¼ ð½0:40; 0:50; ½0:30; 0:40Þ; ð½0:30; 0:40; ½0:40; 0:50ÞÞ;

1 1 1 1 ~r11 ¼ ð~r11 ; ~r21 ; ~r31 ; ~r41 Þ¼

~r21 ~r31 ~r41

~1 Table 4.1 IVIF decision matrix R A1 A2 A3 A4

u1

u2

u3

u4

u5

([0.55, 0.65], [0.15, 0.25]) ([0.35, 0.45], [0.25, 0.35]) ([0.55, 0.65], [0.15, 0.25]) ([0.35, 0.55], [0.35, 0.45])

([0.35, 0.55], [0.35, 0.45]) ([0.15, 0.35], [0.15, 0.35]) ([0.75,0.85], [0.05, 0.15]) ([0.15, 0.25], [0.65, 0.75])

([0.65, 0.75], [0.15, 0.25]) ([0.35, 0.45], [0.45, 0.55]) ([0.55, 0.85], [0.15, 0.15]) ([0.15, 0.25], [0.55, 0.75])

([0.55, 0.75], [0.05, 0.15]) ([0.25, 0.45], [0.45, 0.55]) ([0.45, 0.65], [0.25, 0.35]) ([0.35, 0.45], [0.35, 0.55])

([0.10, 0.40], [0.30, 0.50]) ([0.70, 0.80], [0.10, 0.20]) ([0.50, 0.60], [0.20, 0.30]) ([0.20, 0.30], [0.50, 0.60])

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~2 Table 4.2 IVIF decision matrix R A1 A2 A3 A4

u1

u2

u3

u4

u5

([0.45, 0.55], [0.25, 0.45]) ([0.35, 0.55], [0.30, 0.40]) ([0.45, 0.65], [0.25, 0.35]) ([0.35, 0.45], [0.35, 0.55])

([0.30, 0.40], [0.40, 0.60]) ([0.10, 0.30], [0.30, 0.70]) ([0.60, 0.80], [0.10, 0.20]) ([0.10, 0.20], [0.60, 0.80])

([0.55, 0.65], [0.10, 0.15]) ([0.25, 0.35], [0.35, 0.45]) ([0.65, 0.75], [0.05, 0.15]) ([0.05, 0.15], [0.65, 0.75])

([0.55, 0.65], [0.05, 0.25]) ([0.25, 0.35], [0.55, 0.65]) ([0.45, 0.65], [0.25, 0.35]) ([0.35, 0.45], [0.35, 0.55])

([0.15, 0.35], [0.25, 0.45]) ([0.65, 0.85], [0.05, 0.15]) ([0.55, 0.65], [0.15, 0.35]) ([0.25, 0.45], [0.45, 0.55])

~3 Table 4.3 IVIF decision matrix R A1 A2 A3 A4

u1

u2

u3

u4

u5

([0.45, 0.75], [0.15, 0.25]) ([0.45, 0.55], [0.25, 0.45]) ([0.25, 0.45], [0.35, 0.45]) ([0.35, 0.45], [0.25, 0.45])

([0.35, 0.55], [0.25, 0.35]) ([0.25, 0.45], [0.35, 0.45]) ([0.65, 0.85], [0.05, 0.15]) ([0.15, 0.25], [0.55, 0.75])

([0.60, 0.70], [0.10, 0.20]) ([0.40, 0.50], [0.30, 0.40]) ([0.50, 0.70], [0.10, 0.30]) ([0.10, 0.30], [0.50, 0.70])

([0.55, 0.65], [0.05, 0.25]) ([0.15, 0.25], [0.65, 0.75]) ([0.55, 0.75], [0.15, 0.25]) ([0.25, 0.35], [0.45, 0.65])

([0.35, 0.55], [0.25, 0.45]) ([0.65, 0.75], [0.15, 0.25]) ([0.65, 0.85], [0.05, 0.15]) ([0.15, 0.25], [0.55, 0.75])

~4 Table 4.4 IVIF decision matrix R A1 A2 A3 A4

u1

u2

u3

u4

u5

([0.65, 0.75], [0.15, 0.25]) ([0.40, 0.50], [0.40, 0.50]) ([0.40, 0.50], [0.30, 0.40]) ([0.30, 0.40], [0.40, 0.50])

([0.30, 0.40], [0.30, 0.40]) ([0.10, 0.20], [0.20, 0.30]) ([0.60, 0.70], [0.10, 0.30]) ([0.10, 0.30], [0.60, 0.70])

([0.75, 0.85], [0.05, 0.15]) ([0.35, 0.45], [0.45, 0.55]) ([0.55, 0.85], [0.15, 0.15]) ([0.15, 0.25], [0.55, 0.75])

([0.50, 0.60], [0.10, 0.30]) ([0.20, 0.30], [0.40, 0.60]) ([0.40, 0.50], [0.20, 0.30]) ([0.20, 0.30], [0.40, 0.50])

([0.15, 0.25], [0.45, 0.65]) ([0.65, 0.75], [0.05, 0.15]) ([0.55, 0.65], [0.25, 0.35]) ([0.35, 0.45], [0.45, 0.55])

(ii) Compute the gray relational coefficient matrix. By Eqs. (4.13), (4.14), the gray relational coefficient matrix is derived as 0

n11

0:7122 1:000 ¼ @ 0:9489 0:8739 0:9489 0:7766

(iii) Calculate the gray relational grades. According to Eq. (4.15), we have

1 0:8739 0:9489 0:5115 0:9489 A: 0:6645 0:8739

4.4 A Cloud Service Selection Problem and Comparison Analysis

131

gð~r21 ; ~r11 Þ ¼

4 4 1X 1X n21 r31 ; ~r11 Þ ¼ n31 ¼ 0:8208; i1 ¼ 0:8837; gð~ 4 i¼1 4 i¼1 i1

gð~r41 ; ~r11 Þ ¼

4 1X n41 ¼ 0:8160: 4 i¼1 i1

(iv) Determine the average relational grade. By using Eq. (4.16), the average relational grade between expert e1 and all other three experts is obtained as g11 ¼

t 1X gð~r1l ; ~r11 Þ ¼ 0:8402: 3 l¼2

Similarly, we can get g21 ¼ 0:7754; g31 ¼ 0:8319; g41 ¼ 0:7664 By employing Eq. (4.17), the weights of four experts with respect to u1 are derived as: k11 ¼ 0:2614; k21 ¼ 0:2413; k31 ¼ 0:2588; k41 ¼ 0:2385: The calculating processes for the weights of experts with respect to other attributes are omitted, and the results are shown in Table 4.5. ~ k ¼ ð~r k Þ Step 3: Integrate individual decision matrices R ij mn ðk ¼ 1; 2; 3; 4Þ into a ~ collective decision matrix R ¼ ð~rij Þmn by Eq. (4.18), i.e., 0

ð½0:531; 0:685; ½0:170; 0:288Þ B ð½0:389; 0:514; ½0:292; 0:420Þ B ~¼B R @ ð½0:423; 0:572; ½0:249; 0:353Þ ð½0:338; 0:467; ½0:331; 0:484Þ

ð½0:326; 0:482; ½0:320; 0:440Þ ð½0:153; 0:332; ½0:236; 0:425Þ

ð½0:658; 0:809; ½0:070; 0:191Þ ð½0:568; 0:790; ½0:099; 0:182Þ ð½0:126; 0:251; ½0:599; 0:749Þ ð½0:109; 0:239; ½0:561; 0:736Þ 1 ð½0:535; 0:663; ½0:062; 0:234Þ ð½0:176; 0:381; ½0:308; 0:511Þ ð½0:214; 0:341; ½0:482; 0:628Þ ð½0:665; 0:796; ½0:074; 0:178Þ C C C ð½0:459; 0:636; ½0:209; 0:310Þ ð½0:556; 0:687; ½0:154; 0:289Þ A ð½0:284; 0:385; ½0:385; 0:553Þ

Table 4.5 Weighs of each expert with respect to different attributes

e1 e2 e3 e4

ð½0:639; 0:743; ½0:093; 0:182Þ ð½0:339; 0:439; ½0:375; 0:476Þ

ð½0:249; 0:378; ½0:480; 0:595Þ

u1

u2

u3

u4

u5

0.2614 0.2413 0.2588 0.2385

0.2580 0.2422 0.2536 0.2462

0.2176 0.2758 0.2814 0.2252

0.2392 0.2297 0.2162 0.3149

0.2818 0.2757 0.1752 0.2673

4 A Selection Method Based on MAGDM with Interval-Valued …

132

Step 4: Derive the score matrix, accuracy matrix and uncertainty matrix. By Eqs. (4.4), (4.6) and (4.7), the score matrix, accuracy matrix and uncertainty ~ are computed as: matrix of the collective matrix R 0

0:6898 B 0:5477 B S ¼ B @ 0:5980 0:4975 0 0:8373 B 0:8076 B H¼B @ 0:7983 0:8106 0 0:1625 B 0:1924 B c¼B @ 0:2017 0:1894

1 0:4343 0:8023 C C C 0:7001 A

0:5122 0:4562

0:7766 0:7254 0:4820 0:3586

0:8015 0:2572 0:7837

0:7691 0:6440 0:2629 0:4326 0:8284 0:7473

0:5733

0:8143 0:8370

0:8638 0:8625 0:2163

0:8194 0:8066 0:8221 0:8039 0:1716 0:2527

0:4267 0:1362

0:1857 0:1630 0:1806 0:1934

0:8512 1 0:3119 0:1437 C C C: 0:1571 A

0:1375

0:1779 0:1961

0:1488

0:3879 1 0:6881 0:8563 C C C 0:8429 A

Step 5: Determine the attribute weights. By Eq. (4.29), the following linear programming model is constructed: max8fpy þ ð1  pÞxg 0:6898w1 þ 0:5122w2 þ 0:7766w3 þ 0:7254w4 þ 0:4343w5 y; > > > > 0:5477w 1 þ 0:4562w2 þ 0:4820w3 þ 0:3586w4 þ 0:8023w5 y; > > > > 0:5980w > 1 þ 0:8015w2 þ 0:7691w3 þ 0:6440w4 þ 0:7001w5 y; > > 0:4975w þ 0:2572w þ 0:2629w þ 0:4326w þ 0:3879w y; > > 1 2 3 4 5 > > > 0:8373w þ 0:7837w þ 0:8284w þ 0:7473w þ 0:6881w > 1 2 3 4 5 x; > < 0:8076w1 þ 0:5733w2 þ 0:8143w3 þ 0:8370w4 þ 0:8563w5 x; s:t: 0:7983w1 þ 0:8638w2 þ 0:8194w3 þ 0:8066w4 þ 0:8429w5 x; > > > > > 0:8106w1 þ 0:8625w2 þ 0:8221w3 þ 0:8039w4 þ 0:8512w5 x; > > > > w1  2w2 ; 0:05  w2  w4  0:1; w5 2w3 ; w1  0:4; > > > > w1 þ w2 þ w3 0:3; w3 0:05; > > > > w > : 1 þ w2 þ w3 þ w4 þ w5 ¼ 1; w1 ; w2 ; w3 ; w4 ; w5 0:

ð4:31Þ

Set p ¼ 0:5 and solve Eq. (4.31) with Simplex Method. The main components of the optimal solution for Eq. (4.31) are as follows: y ¼ 0:405; x ¼ 0:779; w1 ¼ 0:3822; w2 ¼ 0:1911; w3 ¼ 0:05; w4 ¼ 0:1411; w5 ¼ 0:2355: Step 6: Compute the overall score and overall uncertainty of each alternative. Utilizing Eqs. (4.23) and (4.25), we can calculate the overall scores and uncertainties of all alternatives which are shown in Table 4.6.

4.4 A Cloud Service Selection Problem and Comparison Analysis Table 4.6 Overall scores, uncertainties and ranking of alternatives

133

Alternative

Score

Uncertainty

Ranking

A1 A2 A3 A4

0.6050 0.5602 0.6759 0.4048

0.2213 0.2213 0.1765 0.1704

2 3 1 4

Step 7: Rank alternatives in term of Definition 4.7. The result of ranking is also listed in Table 4.6. From Table 4.6, it can be seen that alternative A3 is the best one. i.e., Microsoft Dynamic CRM is the best cloud service.

4.4.2

Sensitivity Analysis for Parameter

In above example, we get the computation results by a given weighting coefficient p ðp ¼ 0:5Þ. However, the attribute weights may vary as the value of weighting coefficient p changes, which may result in different decision results. Hence, it is necessary to do the sensitivity analysis for parameter p. The results of sensitivity analysis are depicted in Fig. 4.2. As shown in Fig. 4.2, when the value of parameter p changes from 0 and 1, although the overall scores of four providers change slightly, the rankings among the four cloud services remain unchanged. A3 is first, followed by A1 and followed by A2 and the A4 is ranked in the last all along. Therefore, we can use Eq. (4.30) freely.

0.7

The scores of alternatives

0.65 0.6 0.55 A1 A2 A3 A4

0.5 0.45 0.4 0.35

0

0.1

0.2

0.3

0.4

0.5

p

0.6

0.7

Fig. 4.2 Overall scores of four candidate providers with respect to p

0.8

0.9

1

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134

4.4.3

Comparison Analysis with the Method Using the Score Function

In the above cloud service selection example, if the scores of alternatives are computed with the score function given by Xu [30] (see Eq. (4.3)), then the score matrix is given as 0

0:3795 0:0244 B 0:0954 0:0876 S¼B @ 0:1959 0:6029 0:0050 0:4856

0:5532 0:0361 0:5382 0:4742

0:4508 0:2828 0:2880 0:1349

1 0:1313 0:6047 C C: 0:4002 A 0:2241

The accuracy matrix retains unchanged. Putting the score matrix S and accuracy matrix H into Eq. (4.30), we have the following programming model: max8fpy þ ð1  pÞxg 0:3795w1 þ 0:0244w2 þ 0:5532w3 þ 0:4508w4  0:1313w5 y; > > > > 0:0954w1  0:0876w2  0:0361w3  0:2828w4 þ 0:6047w5 y; > > > > 0:1959w1 þ 0:6029w2 þ 0:5382w3 þ 0:2880w4 þ 0:4002w5 y; > > > 0:005w  0:4856w  0:4742w  0:1349w  0:2241w y; > > 1 2 3 4 5 > > > 0:8373w þ 0:7837w þ 0:8284w þ 0:7473w þ 0:6881w x; > 1 2 3 4 5 > < 0:8076w1 þ 0:5733w2 þ 0:8143w3 þ 0:8370w4 þ 0:8563w5 x; s:t: 0:7983w1 þ 0:8638w2 þ 0:8194w3 þ 0:8066w4 þ 0:8429w5 x; > > > > > 0:8106w1 þ 0:8625w2 þ 0:8221w3 þ 0:8039w4 þ 0:8512w5 x; > > > > w1  2w2 ; 0:05  w2  w4  0:1; w5 2w3 ; w1  0:4; > > > > w1 þ w2 þ w3 0:3; w3 0:05; > > > > w > : 1 þ w2 þ w3 þ w4 þ w5 ¼ 1; w1 ; w2 ; w3 ; w4 ; w5 0:

ð4:32Þ

Still let p ¼ 0:5, by employing the Lingo Soft, we find that Eq. (4.32) has no feasible solution. Thus, the ranking order of alternatives cannot be obtained. This shows that introducing the normalized score function proposed in this chapter is very important.

4.5

Conclusions

In order to stand out in the fierce competition, more and more enterprises begin to select cloud service as one of their development strategy. Cloud service selection can be regarded as a kind of MAGDM. In this chapter, we have studied the cloud service selection problems with IVIFSs and incomplete information on attribute weights. A novel MAGDM method was proposed to solve this kind of GDM problems. There are following three dramatic features in the proposed method.

4.5 Conclusions

135

(1) The assessment values of alternatives on attributes are in the form of IVIFSs which can help experts express their preferences more flexibly. (2) By extending the classical GRA method into IVIF environment, a new approach is presented to determine the weights of experts. Furthermore, the weights of each expert obtained are different on different attributes, which is much closer to the real-world decision situation. (3) A multi-objective programming model is constructed to derive the weights of attributes. The future work of this study is to apply the proposed method to other management areas, such as risk investment, material selection and so on.

References 1. L. Sun, H. Dong, F.K. Hussain, O.K. Hussain, E. Chang, Cloud service selection: State-of-the-art and future research directions. J. Netw. Comput. Appl. 45, 134–150 (2014) 2. S. Ding, C.Y. Xia, Q. Cai, K. Zhou, S. Yang, Qos-aware resource matching and recommendation for cloud computing systems. Appl. Math. Comput. 247(15), 941–950 (2014) 3. A. Goscinski, M. Brock, Toward dynamic and attribute based publication, discovery and selection for cloud computing. Futur. Gener. Comput. Syst. 26(7), 947–970 (2010) 4. A. Jula, E. Sundararajan, Z. Othman, Cloud computing service composition: a systematic literature review. Expert Syst. Appl. 41(8), 3809–3824 (2014) 5. J. Siegel, J. Perdue, Cloud services measures for global use: the service measurement index (SMI), in Proceedings of the Annual SRII Global Conference (SRII) (San Jose, CA, 2012), pp. 411–415 6. S.K. Garg, S. Versteeg, R. Buyya, SMICloud: a framework for comparing and ranking Cloud services, in Proceedings of the 4th IEEE/ACM International Conference on Utility and Cloud on Utility and Cloud Computing (UCC), Melbourne, Australia, pp. 210–218, Dec 2011 7. M. Menzel, M. Schönherr, S. Tai, (MC2)2: criteria, requirements and a software prototype for cloud infrastructure decisions. Softw. Pract. Exp. 43(11), 1283–1297 (2013) 8. N. Limam, R. Boutaba, Assessing software service quality and trustworthiness at selection time. IEEE Trans. Softw. Eng. 36(04), 559–574 (2010) 9. S. Silas, E.B. Rajsingh, K. Ezra, Efficient service selection middleware using ELECTRE methodology for Cloud environments. Inf. Technol. J. 11(07), 868–875 (2012) 10. P. Saripalli, G. Pingali, MADMAC: multiple attribute decision methodology for adoption of Clouds, in Proceedings of the IEEE 4th International Conference on Cloud Computing (CLOUD) (Washington, DC, 2011), pp. 316–323 11. L. Zhao, Y. Ren, M. Li, K. Sakurai, Flexible service selection with user-specific QoS support in service-oriented architecture. J. Netw. Comput. Appl. 35(3), 962–973 (2012) 12. C.W. Chang, P. Liu, J.J. Wu, Probability-based Cloud storage providers selection algorithms with maximum availability, in Proceedings of the 41th International Conference on Parallel Processing (ICPP) (Pittsburgh, PA, USA), pp. 199–208, Sept 2012 13. S. Sundareswaran, A. Squicciarini, D. Lin, A brokerage-based approach for Cloud service selection, in Proceedings of the IEEE 5th International Conference on Cloud Computing (CLOUD) (Honolulu, HI, USA), pp. 558–565, June 2012 14. Q. He, J. Han, Y. Yang, J. Grundy, H. Jin, QoS-driven service selection for multi-tenant SaaS, in Proceedings of the International Conference on Cloud Computing (CLOUD) (Honolulu, HI, USA), pp. 24–29, June 2012

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15. B. Martens, F. Teuteberg, M. Gräuler, Design and implementation of a community platform for the evaluation and selection of Cloud computing services: a market analysis, in ECIS 2011 Proceedings (2011) pp. 215 16. J. Yang, W.M. Lin, W.C. Dou, An adaptive service selection method for cross-Cloud service composition. Concurr. Comput. Pract. Exp. 25(18), 2435–2454 (2013) 17. G.L. Xu, F. Liu, An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection. Appl. Math. Model. 37(6), 3929–3943 (2013) 18. S.P. Wan, D.F. Li, Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees. Omega 41(6), 925–940 (2013) 19. S.P. Wan, J.Y. Dong, Possibility linear programming with trapezoidal fuzzy numbers. Appl. Math. Model. 38(5–6), 1660–1672 (2014) 20. H. Akdag, T. Kalayci, S. Karagöz, H. Zülfikar, D. Giz, The evaluation of hospital service quality by MCDM. Appl. Soft Comput. 23, 239–248 (2014) 21. L.A. Zadeh, Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 22. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 23. K. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989) 24. S.P. Wan, J.Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl. Soft Comput. 29, 153–168 (2015) 25. S.P. Wan, W. Feng, L.L. Lin, J.Y. Dong, An intuitionistic fuzzy linear programming method for logistics outsourcing provider selection. Knowl.-Based Syst. 82, 80–94 (2015) 26. S.P. Wan, J.Y. Dong, Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees. Inf. Fusion 26, 49–65 (2015) 27. S.P. Wan, G.L. Xu, F. Wan, J.Y. Dong, A new method for Atanassov’s interval-valued intuitionistic fuzzy MAGDM with incomplete attribute weight information. Inf. Sci. 316, 329–347 (2015) 28. S.P. Wan, D.F. Li, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with Atanassov’s intuitionistic fuzzy truth degrees. IEEE Trans. Fuzzy Syst. 22(2), 300–312 (2014) 29. J.L. Deng, Gray System Theory (Huazhong University of Science and Technology Press, Wuhan, 2002) 30. Z.S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis. 22(2), 215–219 (2007). (Chinese) 31. G.D. Li, D. Yamaguchi, M. Nagai, A gray-based decision-making approach to the supplier selection problem. Math. Comput. Model. 46(3–4), 573–581 (2007) 32. W.J. Gu, Z.C. Sun, X.Z. Wei, H.F. Dai, A new method of accelerated life testing based on the Gray System Theory for a model-based lithium-ion battery life evaluation system. J. Power Sour. 267, 366–379 (2014) 33. H.B. Kuang, D.M. Kilgour, K.W. Hipel, Gray-based PROMETHEE II with application to evaluation of source water protection strategies. Inf. Sci. 294(10), 376–389 (2015) 34. F.Y. Meng, Q. Zhang, X.H. Cheng, Approaches to multiple-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral with respect to generalized kshapley index. Knowl.-Based Syst. 37, 237–249 (2013) 35. S.P. Wan, J.Y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J. Comput. Syst. Sci. 80(1), 237–256 (2014) 36. Z.L. Yue, Y. Jia, An application of soft computing technique in group decision making under interval-valued intuitionistic fuzzy environment. Appl. Soft Comput. 13(5), 2490–2503 (2013) 37. X.L. Zhang, Z.S. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making. Appl. Soft Comput. 26, 42–56 (2015)

References

137

38. D.F. Li, Multiobjective Many-Person Decision Makings and Games (National Defense Industry Press, Beijing, 2003) 39. S. Cebi, C, Kahraman, Developing a group support system based on fuzzy information axiom. Knowl.-Based Syst. 23(1) 3–16 (2010) 40. G. Qian, H. Wang, X.Q. Feng, Generalized hesitant fuzzy sets and their application in decision support system. Knowl.-Based Syst. 37, 357–365 (2013)

Chapter 5

Aggregating Decision Information into Interval-Valued Intuitionistic Fuzzy Numbers for Heterogeneous Multi-attribute Group Decision Making

Abstract Multi-attribute group decision making (MAGDM) has attracted more and more attention in many fields. Correspondingly, a number of usable methods have been proposed for various MAGDM problems, nevertheless, very few studies focus on the aggregation techniques of intuitionistic fuzzy information. The aim of this chapter is to aggregate decision information into interval-valued intuitionistic fuzzy numbers (IVIFNs) to solve heterogeneous MAGDM problem in which the decision information involves real numbers, interval numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs). There are three issues being addressed in this chapter. The first is to propose a new general method to aggregate the attribute value vector into IVIFNs under heterogeneous MAGDM environment utilizing the relative closeness in technique for order preference by similarity to ideal solution (TOPSIS). The second is to construct a multiple objective intuitionistic fuzzy programming model to determine the attribute weights. Borrowing the results of the former two issues, the last is to present a new method to solve heterogeneous MAGDM problem. A comparison analysis with existing method is conducted to demonstrate the advantages of the proposed method. Two examples are provided to verify the practicality and effectiveness of the proposed method.







Keywords Multi-attribute group decision making Heterogeneous information Aggregation technique Interval-valued intuitionistic fuzzy numbers Intuitionistic fuzzy programming model




5.1




Introduction

As a generalization of Zadeh’s fuzzy sets [1], intuitionistic fuzzy (IF) set (IFS) [2] has better agility in expressing the uncertainty and ambiguity since it can be used to describe the characteristics of affirmation, negation and hesitation simultaneously. However, since sufficient information may be unavailable in practice, it is not easy to use the crisp values to express the membership and non-membership degrees of © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_5

139

140

5 Aggregating Decision Information into Interval-Valued …

IFS. In such case, an interval may be a more suitable measurement to describe the vagueness. So, Atanassov and Gargov [3] proposed the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) by assigning membership and non-membership degrees in the form of intervals rather than real numbers. Because of its advantages in describing uncertainty, many researchers have devoted to the theory of IVIFS and its applications, such as aggregation operators [4–6], entropy measures [7, 8], ranking functions [9, 10], decision making [11–15], to name a few. In recent years, several scholars have presented some useful and valuable techniques for dealing MAGDM problems by aggregating attribute values into intuitionistic fuzzy number (IFN) and interval-valued intuitionistic fuzzy numbers (IVIFNs). Yue [16] and Yue et al. [17] employed Golden Section idea to aggregate crisp values into IFN. Yue et al. [18] proposed the method based on Minimax Criterion to aggregate crisp values into IFN for MAGDM. Yue [19] developed a new useful and practical method for aggregating crisp values into IFN using the idea of mean value. Later, Xu et al. [20] introduced a general aggregation method, which is a generalization of the method [19]. Yue [21] first defined the concepts of the attribute satisfactory interval and the attribute dissatisfactory interval, respectively, according to attribute values. Then, a method for aggregating the obtained attribute satisfactory interval and attribute dissatisfactory interval into an IVIFN was developed for MAGDM problems. Yue and Jia [22] presented a soft computing model in which it aggregates all individual decisions on an attribute into IVIFN for MAGDM whose attribute values are expressed by real numbers. By the idea of mean and standard deviation in statistics, Yue [23] introduced a straightforward and practical algorithm to aggregate interval numbers into IVIFN. Although these aggregation methods [21–23] have some advantages, they suffer from some limitations: (1) The basic elements of an IVIFN are the interval membership degree, interval non-membership degree and interval hesitation degree. However, those methods [21–23] do not consider interval hesitation degree. (2) Existing aggregation methods [21–23] are just designed to deal with the MAGDM problems with real numbers or interval numbers, but cannot be used to solve heterogeneous MAGDM problems in which the attribute values may be triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs). (3) The weights of attributes are given by DMs in advance in those methods [21–23], which cannot avoid the subjectivity of giving the attribute weights. In fact, MAGDM often involves attribute values which may be described by multiple formats, such as real numbers, intervals, TFNs and TrFNs. Such a MAGDM with multiple types of information is called the heterogeneous MAGDM [24, 25]. At present, several methods have been proposed to solve heterogeneous MAGDM [26–32]. Wan and Li [26] and Li and Wan [27, 28] presented the extended fuzzy Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) [33] for solving heterogeneous multi-attribute decision making (MADM) problems. Wan and Li [29] put forward an intuitionistic fuzzy

5.1 Introduction

141

programming method for heterogeneous MAGDM with Atanassov’s intuitionistic fuzzy truth degrees. Further, considering the interval-valued intuitionistic fuzzy truth degrees of alternatives’ comparison, Wan and Li [30] developed interval-valued intuitionistic fuzzy LINMAP method for heterogeneous MADM; Wan and Dong [31] proposed an interval-valued intuitionistic fuzzy mathematical programming method for hybrid MCGDM. Zhang et al. [32] constructed some deviation models to solve heterogeneous MAGDM. It should be pointed out that the aforesaid methods employed the distance of fuzzy numbers to unify the heterogeneous information. During the process of unifying the heterogeneous information, much useful information may be lost. As mentioned previously, IVIFN has powerful ability to capturing uncertainty and ambiguity. It is more reasonable to convert heterogeneous information into IVIFNs in real-life heterogeneous MAGDM problems. For example, in an actual IT outsourcing service provider evaluation, the attributes might be better to be expressed by using multiple information representations, such as real numbers, interval numbers, TFNs and TrFNs, etc. However, these information representations characterize the fuzziness by membership function only, whereas IVIFSs with interval-valued membership and non-membership functions are more flexible and abundant in expressing the imprecise or uncertain decision information. For instance, the experts may give the collective ratings of an alternative on the attributes, product quality and flexibility, as TFN r12 ¼ ð4:66; 6:34; 8:00Þ and TrFN r14 ¼ ð4:18; 5:12; 6:48; 8:02Þ, respectively. However, the ratings for these attributes are divided into two parts: dissatisfaction degree and satisfaction degree, which just are the membership degree and non-membership degree of IVIFN, and there exist some hesitancy when experts evaluate these attributes. So, the attributes, product quality and flexibility, might be more suitable to be expressed by IVIFNs r12 ¼ ð½0:181; 0:283; ½0:103; 0:152Þ and r14 ¼ ð½0:195; 0:241; ½0:125; 0:175Þ, respectively. Therefore, this chapter attempts to convert the decision information into interval-valued intuitionistic information. Aggregating heterogeneous decision information into IVIFNs for heterogeneous MAGDM is of great importance for scientific research and actual applications. However, there exist two major difficulties and challenges for solving such heterogeneous MAGDM problems. The first is how to develop a new general method for aggregating heterogeneous decision information into IVIFNs. The second is how to determine the attribute weights objectively. In this chapter, we develop a new general method for aggregating heterogeneous decision information into IVIFNs. The proposed general aggregating method involves three steps: (i) the extraction of the satisfactory element, dissatisfactory element and uncertain element of each attribute value using the concept of relative closeness of technique for order preference by similarity to ideal solution (TOPSIS) [34]; (ii) the construction of satisfactory interval, dissatisfactory interval and uncertain interval of each attribute according to the statistic method; (iii) the induction of the IVIFN of each attribute by considering membership degree, non-membership degree and hesitation degree. Then, to determine the weights of the attributes objectively, a new multiple objective IF programming model is constructed and transformed into linear

142

5 Aggregating Decision Information into Interval-Valued …

program to solve. Thereby, a new method is proposed to solve the heterogeneous MAGDM problems. Compared with existing research, the primary features of the proposed method are illuminated as follows: (1) The proposed general aggregating method can aggregate different types of decision information into IVIFNs, including real numbers, interval numbers, TFNs and TrFNs, while existing methods [21–23] are just suitable to aggregate decision information of single types (i.e., real numbers or interval numbers). (2) The proposed general aggregating method considers all basic elements of IVIFNs in the aggregation process, thus the induced IVIFNs derived by the developed new linear transformation are more persuasive than those derived by the methods [21–23]. (3) A new multiple objective interval-valued IF programming model is constructed to determine objectively the weights of attributes, which avoids the subjective randomness of giving the weights in methods [21–23]. The structure of this chapter is organized as follows: in Sect. 5.2, we briefly recalls some basic concepts on IVIFSs, interval numbers, TFNs and TrFNs. Section 5.3 describes the heterogeneous MAGDM problems and presents a general method to aggregate heterogeneous decision information into IVIFNs. In Sect. 5.4, the attribute weights are determined objectively by constructing a multiple objective IF programming. Thus, a new method is proposed to solve heterogeneous MAGDM problems. In Sect. 5.5, comparison analyses are made with existing methods. Section 5.6 provides two illustrative examples to verify the effectiveness of the proposed method. Section 5.7 makes some conclusions.

5.2

Some Basic Concepts

In the following, we recall some basic notations of IVIFS, interval numbers, TFNs and TrIFNs.

5.2.1

Interval-Valued Intuitionistic Fuzzy Set

Definition 5.1 [3]. Let X be a non-empty set of the universe. An interval-valued ~ in X is defined as A ~ ¼ f\x; l ~ ðxÞ; v ~ ðxÞ [ jx 2 Xg, where intuitionistic fuzzy set A A A l h lA~ ðxÞ ¼ ½lA~ ðxÞ; lA~ ðxÞ : X ! ½0; 1 and vA~ ðxÞ ¼ ½vlA~ ðxÞ; vhA~ ðxÞ : X ! ½0; 1, such that lhA~ ðxÞ þ vhA~ ðxÞ 2 ½0; 1, for any x 2 X. The interval numbers lA~ ðxÞ and vA~ ðxÞ indicate, respectively, the membership degree and non-membership degree of the ~ The third parameter p ~ ðxÞ ¼ ½pl ðxÞ; ph ðxÞ is called hesitancy index element x in A. ~ ~ A A A

5.2 Some Basic Concepts

143

~ where pl ðxÞ ¼ 1  lh ðxÞ  vh ðxÞ and ph ðxÞ ¼ 1  ll ðxÞ  vl ðxÞ. of x in A, ~ ~ ~ ~ ~ ~ A A A A A A l ~ is reduced to an intuitionistic If lA~ ðxÞ ¼ lhA~ ðxÞ and vlA~ ðxÞ ¼ vhA~ ðxÞ, then IVIFS A fuzzy set [2]. Xu and Chen [4] called ð½llA~ ðxÞ; lhA~ ðxÞ; ½vlA~ ðxÞ; vhA~ ðxÞÞ an interval-valued intuitionistic fuzzy number (IVIFN). For simplicity, an IVIFN will be denoted by ~a ¼ ð½ll ; lh ; ½vl ; vh Þ, where ½ll ; lh ½0; 1; ½vl ; vh ½0; 1; lh þ vh  1:

ð5:1Þ

a2 ¼ ð½ll2 ; lh2 ; ½vl2 ; vh2 Þ be two Definition 5.2 [3] Let a~1 ¼ ð½ll1 ; lh1 ; ½vl1 ; vh1 Þ and ~ IVIFNs, the containment is: ~a1 ~a2 iff

ll1  ll2 ;

lh1  lh2 ;

vl1  vl2 and

vh1  vh2 :

ð5:2Þ

Also, the arithmetic operations are expressed as follows [19]: (1) a~1 þ ~a2 ¼ ð½ll1 þ ll2  ll1 ll2 ; lh1 þ lh2  lh1 lh2 ; ½vl1 vl2 ; vh1 vh2 Þ; (2) k~a ¼ ð½1  ð1  ll Þk ; 1  ð1  lh Þk ; ½ðvl Þk ; ðvh Þk Þ k [ 0: Definition 5.3 [4] For a set of IVIFNs a~j ¼ ð½llj ; lhj ; ½vlj ; vhj Þ ðj ¼ 1; 2; . . .; nÞ, an interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator is defined as IVIFWAð~ a1 ; ~ a2 ; . . .; ~ an Þ ¼

n X

" aj ¼ wj ~

1

n Y

j¼1

j¼1

ð1 

llj Þwj ; 1



n Y

ð1 

j¼1

lhj Þwj ; ½

n Y j¼1

ðvlj Þwj ;

n Y

#! ðvhj Þwj

;

j¼1

ð5:3Þ where w ¼ ðw1 ; w2 ; . . .; wn ÞT is an associated weight vector satisfying wj 2 ½0; 1 P ðj ¼ 1; 2; . . .; nÞ and nj¼1 wj ¼ 1. The aggregated value by the IVIFWA operator is also an IVIFN. Definition 5.4 [4] Let ~a ¼ ð½ll ; lh ; ½vl ; vh Þ be an IVIFN, the score and accuracy degrees of ~a can be defined by 1 Sð~aÞ ¼ ðll þ lh  vl  vh Þ; 2

ð5:4Þ

1 Hð~aÞ ¼ ðll þ lh þ vl þ vh Þ; 2

ð5:5Þ

where Sð~aÞ 2 ½1; 1, Hð~aÞ 2 ½0; 1.

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144

Definition 5.5 [4] Let ~a1 ¼ ð½ll1 ; lh1 ; ½vl1 ; vh1 Þ and ~ a2 ¼ ð½ll2 ; lh2 ; ½vl2 ; vh2 Þ be two IVIFNs, then: (1) If Sð~a1 Þ [ Sð~a2 Þ, then ~a1 [ ~a2 . (2) If Sð~a1 Þ ¼ Sð~a2 Þ, then (i) If Hð~a1 Þ [ Hð~a2 Þ, then a~1 [ ~a2 . (ii) If Hð~a1 Þ ¼ Hð~a2 Þ, then a~1 ¼ ~a2 .

5.2.2

Definitions and Distances for Real Number, Interval Number, TFN and TrFN

Definition 5.6 Let a ¼ a1 and b ¼ b1 be two nonnegative real numbers. The Hamming distance between a and b is defined as follows: dRN ða; bÞ ¼ ja1  b1 j:

ð5:6Þ

Definition 5.7 [35] Let a ¼ ½al ; au  and b ¼ ½bl ; bu  be two nonnegative interval numbers, and k  0. The operation laws are specified as follows: (1) a þ b ¼ ½al þ bl ; au þ bu ; (2) ka ¼ ½kal ; kau : The Hamming distance between two nonnegative interval numbers a ¼ ½al ; au  and a ¼ ½bl ; bu  is defined as follows: 1 dIN ða; bÞ ¼ ðjal  bl j þ jau  bu jÞ: 2

ð5:7Þ

Definition 5.8 [36, 37] Let a ¼ ða1 ; a2 ; a3 Þ and b ¼ ðb1 ; b2 ; b3 Þ be two TFNs, and k  0, then (1) a þ b ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 Þ; (2) ka ¼ ðka1 ; ka2 ; ka3 Þ: Definition 5.9 Let a ¼ ða1 ; a2 ; a3 Þ and b ¼ ðb1 ; b2 ; b3 Þ be two TFNs. The Hamming distance between a and b is defined as follows: 1 dTFN ða; bÞ ¼ ðja1  b1 j þ 2ja2  b2 j þ ja3  b3 jÞ 4

ð5:8Þ

5.2 Some Basic Concepts

145

Theorem 5.1 The Hamming distance dTFN ða; bÞ satisfies the following properties: (i) (ii) (iii) (iv)

0  dTFN ða; bÞ  1; dTFN ða; bÞ ¼ 0 if and only if a ¼ b; dTFN ða; bÞ ¼ dTFN ðb; aÞ; If c ¼ ðc1 ; c2 ; c3 Þ is any TFN, then dTFN ða; cÞ  dTFN ða; bÞ þ dTFN ðb; cÞ.

Proof Obviously, the proposed distance meets (i)–(iii) of Theorem 5.1. We need only to prove (iv). It is easy to see that ja1  c1 j  ja1  b1 j þ jb1  c1 j; 2ja2  c2 j  2ja2  b2 j þ 2jb2  c2 j and ja3  c3 j  ja3  b3 j þ jb3  c3 j: Hence, one has 1 1 ðja1  c1 j þ 2ja2  c2 j þ ja3  c3 jÞ  ðja1  b1 j þ 2ja2  b2 j 4 4 1 þ ja3  b3 jÞ þ ðjb1  c1 j þ 2jb2  c2 j þ jb3  c3 jÞ: 4 Therefore, dTFN ða; cÞ  dTFN ða; bÞ þ dTFN ðb; cÞ. Namely, the proposed distance satisfies (iv) of Theorem 5.1. Definition 5.10 [38]. Let a ¼ ða1 ; a2 ; a3 ; a4 Þ and b ¼ ðb1 ; b2 ; b3 ; b4 Þ be two TrFNs, and k  0, then (1) a þ b ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 ; a4 þ b4 Þ; (2) ka ¼ ðka1 ; ka2 ; ka3 ; ka4 Þ: The Hamming distance between two TrFNs a ¼ ða1 ; a2 ; a3 ; a4 Þ and b ¼ ðb1 ; b2 ; b3 ; b4 Þ is defined as follows [39]: 1 dTrFN ða; bÞ ¼ ðja1  b1 j þ ja2  b2 j þ ja3  b3 j þ ja4  b4 jÞ 4

ð5:9Þ

Remark 5.1 Generally, a TrFN is regarded as the generalized form of real number, interval number and TFN. It is worth mentioning that: (i) if a2 ¼ a3 and b2 ¼ b3 , then TrFNs a and b are reduced to TFNs, dTrFN ða; bÞ is reduced to dTFN ða; bÞ; (ii) if

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146

a1 ¼ a2 , a3 ¼ a4 , b1 ¼ b2 and b3 ¼ b4 , then TrFNs a and b are reduced to interval numbers, dTrFN ða; bÞ is reduced to dIN ða; bÞ; (iii) if a1 ¼ a2 ¼ a3 ¼ a4 and b1 ¼ b2 ¼ b3 ¼ b4 , then TrFNs a and b are reduced to real numbers, dTrFN ða; bÞ is reduced to dRN ða; bÞ.

5.3

Aggregating Heterogeneous Decision Information into IVIFNs

This section describes the heterogeneous MAGDM problem and then proposes a general method for aggregating heterogeneous decision information into IVIFNs.

5.3.1

Presentation of Heterogeneous MAGDM Problem

For convenience, denote M ¼ f1; 2; . . .; mg, N ¼ f1; 2; . . .; ng, and P ¼ f1; 2; . . .; pg. Let fDi jði 2 MÞg be a group of DMs, Sk ðk 2 PÞ be a discrete set of alternatives, and Aj ðj 2 NÞ be a set of attributes. Since there are multiple formats of rating values, the attribute set A ¼ fA1 ; A2 ; . . .; An g is divided into four subsets ^ 1 ¼ fA1 ; A2 ; . . .; Aj1 g, A ^ 2 ¼ fAj1 þ 1 ; Aj1 þ 2 ; . . .; Aj2 g, A ^ 3 ¼ fAj2 þ 1 ; Aj2 þ 2 ; . . .; Aj3 g A ^t \ A ^k ¼ ; ^ 4 ¼ fAj3 þ 1 ; Aj3 þ 2 ; . . .; An g, where 1  j1  j2  j3  n, A and A 4 S ^ t ¼ A, ; is the empty set. The rating values in the A (t; k ¼ 1; 2; 3; 4 ;t 6¼ k) and t¼1

^ e ðe ¼ 1; 2; 3; 4Þ are in the form of real numbers, interval numbers, TFNs subsets A ^ e ðe ¼ 1; 2; 3; 4Þ by and TrFNs, respectively. Denote the subscript sets for subsets A N1 ¼ f1; 2; . . .; j1 g, N2 ¼ fj1 þ 1; j1 þ 2; . . .; j2 g, N3 ¼ fj2 þ 1; j2 þ 2; . . .; j3 g and N4 ¼ fj3 þ 1; j3 þ 2; . . .; ng, respectively. Thus, a group decision matrix of alternative Sk can be expressed as

X k ¼ ðxkij Þmn

D1 ¼ D2 .. . Dm

0 kA1 Ak2 . . . Ank 1 x11 x12 . . . x1n B xk21 xk22 . . . xk2n C B C ðk 2 PÞ B . .. C .. .. @ .. . A . . xkm1 xkm2 . . . xkmn

ð5:10Þ

where xkij is attribute value of the alternative Sk on attribute Aj given by DM Di . Denote the attribute weight vector by w ¼ ðw1 ; w2 ; . . .; wn Þ, where wj represents P the weight of attribute Aj such that wj 2 ½0; 1 ðj 2 NÞ and nj¼1 wj ¼ 1. Practically, DMs may specify some preference relations on weights of attributes according to

5.3 Aggregating Heterogeneous Decision Information into IVIFNs

147

his/her knowledge, experience and judgment [12]. Thus, the attribute weights are incomplete. Let D be the set of incomplete weights. Generally, D may consist of several basic forms [26–29]: (1) (2) (3) (4) (5)

A weak ranking: fwi  wj g; A strict ranking: fwi  wj  ei gðei [ 0Þ; An interval form: fei  wi  ei þ /i gð0  ei  ei þ /i  1Þ; A ranking of differences: fwi  wj  wk  wl g, for j 6¼ k 6¼ l; A ranking with multiples: fwi  ei wj gð0  ei  1Þ.

According to Yue [21–23], we integrate the decision matrices X k ¼ ðxkij Þmn ðk ¼ 1; 2;


; pÞ into a collective decision matrix with IVIFNs to derive the ranking order of alternatives.

5.3.2

A General Method for Aggregating Heterogeneous Decision Information into IVIFNs

For the sake of calculation simplicity, the jth column vector in X k is rewritten as Akj ¼ ðxk1j ; xk2j ; . . .; xkmj Þ

ðk 2 P; j 2 NÞ;

ð5:11Þ

which is the assessment vector of alternative Sk on attribute Aj given by all DMs Di ði ¼ 1; 2; . . .; mÞ. Suppose that Amax , Amid and Amin j j j , respectively, are the largest grade, the middle grade and the smallest grade employed in rating system (for example, in the 10-point scale system for the ratings of real numbers, if attribute Aj is benefit attribute, then Amin ¼ 0, Amid ¼ 5 and Amax ¼ 10; if attribute Aj is cost j j j mid max ¼ 10, A ¼ 5 and A ¼ 0). It is easy to see that the elements attribute, then Amin j j j min max of Akj are always between Aj and Aj . To integrate the decision matrices Xk ¼ ðxkij Þmn ðk ¼ 1; 2; . . .; pÞ into a collective IVIF decision matrix, all the elements in vector Akj need to be aggregated into an IVIFN. In what follows, some definitions including the Quasi-satisfactory degree (Qsd), Quasi-dissatisfactory degree (Qdd) and Quasi-uncertain degree (Qud) of xkij are introduced. Then, using the idea of mean and standard deviation, we construct the Quasi-satisfactory interval (Qsi), Quasi-dissatisfactory interval (Qdi) and Quasi-uncertain interval (Qui) of Akj . Finally, the Qsi, Qdi and Qui are used to generate an IVIFN by the proposed normalized method. The aggregating process is depicted in Fig. 5.1.

5 Aggregating Decision Information into Interval-Valued …

148

Akj = ( x1kj , x2k j ,

k , xmj )

TOPSIS Quasi-dissatisfactory set (Dds) ζ kj = {ζ 1kj , ζ 2k j , , ζ mjk }

Quasi-uncertain set (Qus) ηkj = {η1kj ,η 2k j , ,η mjk }

Quasi-satisfactory set (Qss) ξ kj = {ξ1kj , ξ 2k j , , ξ mjk }

Mean Deviations Quasi-dissatisfactory Interval (Qdi) ζ kj = [ζ kjl , ζ kjh ]

Quasi-satisfactory Interval (Qsi) ξ kj = [ξ kjl , ξ kjh ]

Quasi-uncertain interval (Qui) η kj = [η kjl ,η kjh ]

Linear transformation

α kj = ([ μ , μ ],[vkjl , vkjh ]) l kj

h kj

Fig. 5.1 Process for aggregating IVIFN

5.3.2.1

Compute the Qsd, Qdd and Qud

Let xkij be the rating of alternative Sk on attribute Aj given by DM Di . Due to the complexity, fuzziness and uncertainty inherent in the evaluated attributes, identifying the degrees of satisfactory, dissatisfactory and uncertainty implied in xkij is very difficult. Note the facts that: (1) the distance from xkij to Amin is greater than the j k max distance from xij to Aj , which shows that the DM prefers satisfaction; (2) the is less than the distance from xkij to Amax , which shows that distance from xkij to Amin j j k the DM prefers dissatisfaction; and (3) the distance from xij to Amin is equal to the j k max distance from xij to Aj , which shows that the DM maintains neutrality. Based on the above analysis, some notations are introduced to calculate satisfactory degree, dissatisfactory degree and uncertain degree of each element in Akj . Definition 5.11 Let Akj be a benefit attribute vector, xkij be arbitrary element in Akj . The Qsd, Qdd and Qud of xkij are defined as

nkij ¼

dðxkij ; Amin j Þ Þ þ dðxkij ; Amin dðxkij ; Amax j j Þ

;

ð5:12Þ

fkij ¼ 1  nkij ;

ð5:13Þ

8 dðAmax ;xkij Þ j > if dðxkij ; Amax Þ\dðxkij ; Amin > j j Þ; ;xkij Þ þ dðxkij ;Amid < dðAmax j Þ j k min k dðxij ;Aj Þ k max k gij ¼ dðxk ;Amin Þ þ dðxk ;Amid Þ if dðxij ; Aj Þ [ dðxij ; Amin j Þ; ij j ij j > > :1 k max k min if dðxij ; Aj Þ ¼ dðxij ; Aj Þ;

ð5:14Þ

5.3 Aggregating Heterogeneous Decision Information into IVIFNs

149

respectively, where dðxkij ; Amax Þ is the distance between xkij and the largest grade Amax j j k mid of attribute Aj , dðxkij ; Amid Þ is the distance between x and the middle grade A of j ij j k min k min attribute Aj , dðxij ; Aj Þ is the distance between xij and the smallest grade Aj of attribute Aj . Remark 5.2 For the benefit attribute vector Akj ¼ ðxk1j ; xk2j ;


; xkmj Þ, the Quasi-satisfactory set (Qss) nkj , Quasi-dissatisfactory set (Qds) fkj and Quasi-uncertain set (Qus) gkj of Akj are defined as nkj ¼ fnk1j ; nk2j ; . . .; nkmj g, fkj ¼ ff1j ; f2j ; . . .; fmj g and gkj ¼ fgk1j ; gk2j ; . . .; gkmj g.

5.3.2.2

Calculate Qsi, Qdi and Qui

Note that the membership degree and non-membership degree of an IVIFN are intervals rather than real numbers. Moreover, nkij , fkij and gkij are the Qsd, Qdd and Qud of xkij , which is one element in Akj . In such cases, the ranges of nkij , fkij and gkij may be more appropriate measurements of Qsd, Qdd and Qud of Akj , respectively. For these reasons, it is necessary to construct the Qsi, Qdi and Qui of Akj . Definition 5.12 For the benefit attribute vector Akj , the Qsi ~ nkj , Qdi ~fkj and Qui ~ gkj of alternative Sk on attribute Aj are defined as ~nkj ¼ ½nl ; nh  ¼ ½maxðmðnkj Þ  dðnkj Þ; 0Þ; mðnkj Þ þ dðnkj Þ ðk 2 P; j 2 NÞ; ð5:15Þ kj kj ~fkj ¼ ½fl ; fh  ¼ ½maxðmðfkj Þ  dðfkj Þ; 0Þ; mðfkj Þ þ dðfkj Þ ðk 2 P; j 2 NÞ; ð5:16Þ kj kj ~gkj ¼ ½glkj ; ghkj  ¼ ½maxðmðgkj Þ  dðgkj Þ; 0Þ; mðgkj Þ þ dðgkj Þ ðk 2 P; j 2 NÞ; ð5:17Þ P Pm k k 1 g , respectively, where mðnkj Þ ¼ m1 m i¼1 nij , mðfkj Þ ¼ 1  mðnkj Þ, mðgkj Þ ¼ m ffi i¼1 ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P m 2 m 2 k k 1 1 dðnkj Þ ¼ m1 dðnkj Þ ¼ m1 and i¼1 ðnij  mðnkj ÞÞ , i¼1 ðfij  mðfkj ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P m 2 1 k dðnkj Þ ¼ m1 i¼1 ðgij  mðgkj ÞÞ represent the mean values and standard deviations of Qss nkj , Qds fkj and Qus gkj , respectively. Here we call an ordered pair ð½nlkj ; nhkj ; ½flkj ; fhkj Þ a quasi-IVIFN.

5 Aggregating Decision Information into Interval-Valued …

150

5.3.2.3

Induce an IVIFN

Let akj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 NÞ be the induced IVIFN by the attribute vector Akj . To satisfy the conditions in Eq. (5.1), Yue [21] constructed some linear transformations as follows: . . . . llkj ¼ nlkj skj ; lhkj ¼ nhkj skj ; vlkj ¼ flkj skj ; vhkj ¼ fhkj skj ðk 2 P; j 2 NÞ;

ð5:18Þ

where skj ¼ nlkj þ nhkj þ flkj þ fhkj . The linear transformations of Eq. (5.18) may cause inconsistency in some situations. Example 5.1 Let A ¼ ð½nl1 ; nh1 ; ½fl1 ; fh1 Þ ¼ ð½0:75; 0:82; ½0:13; 0:16Þ and B ¼ ð½nl2 ; nh2 ; ½fl2 ; fh2 Þ ¼ ð½0:67; 0:82;½0:10; 0:17Þ, where A and B satisfy the conditions nl1 þ fl1  1 and nh1 þ fh1  1. By Eq. (5.3), we have sðAÞ ¼ 0:64 and sðBÞ ¼ 0:61. Hence, A [ B. However, by Eq. (5.18), A and B can be converted into A0 ¼ ð½0:403; 0:441; ½0:070; 0:086Þ and B0 ¼ ð½0:381; 0:466; ½0:057; 0:096Þ. Then, we have sðA0 Þ ¼ 0:344 and sðB0 Þ ¼ 0:347. Thus, A0 \B0 , which is not consistent with A [ B. An effective transformation method should consider not only the membership degree and non-membership degree of quasi-IVIFN, but also the hesitation degree. However, the hesitation degree is ignored in Eq. (5.18). Here, some new linear transformations are developed as follows. Definition 5.13 Let akj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 NÞ be the induced IVIFN by the attribute vector Akj . The values of llkj , lhkj , vlkj and vhkj can be computed as follows: . . . . llkj ¼ nlkj wkj ; lhkj ¼ nhkj wkj ; vlkj ¼ flkj wkj ; vhkj ¼ fhkj wkj ; ðk 2 P; j 2 NÞ; ð5:19Þ where wkj ¼ nlkj þ nhkj þ flkj þ fhkj þ 12ðglkj þ ghkj Þ. Apparently, llkj , lhkj , vlkj and vhkj satisfy Eq. (5.1). Let us turn to the aforementioned Example 5.1. By Eq. (5.19), A and B can be converted into A0 ¼ ð½0:389; 0:425; ½0:067; 0:083Þ and 0 B ¼ ð½0:356; 0:436; ½0:053; 0:090Þ. Then, we have sðA0 Þ ¼ 0:332 and sðB0 Þ ¼ 0:324. So, A0 [ B0 , which is consistent with A [ B. Therefore, the new transformations of Eq. (5.19) can overcome the drawback of Eq. (5.18).

5.3 Aggregating Heterogeneous Decision Information into IVIFNs

5.3.3

151

Concrete Computation Formulas for Aggregating Heterogeneous Information into IVIFNs

Without loss of generality, suppose that all the attributes are benefit attributes in this subsection.

5.3.3.1

For Trapezoidal Fuzzy Numbers

If j 2 N4 , then Akj ¼ ðxk1j ; xk2j ; . . .; xkmj Þ is an attribute value vector in the form of ¼ ð0; 0; TrFNs (i.e., xkij ¼ ðakij ; bkij ; ckij ; dijk Þði ¼ 1; 2; . . .; mÞ are TrFNs). Let Amin j þ þ þ þ þ 0; 0Þ, Amax ¼ ðx ; x ; x ; x Þðj 2 N Þ, where x means the largest grade for the 4 j j j j j j ^ attribute Aj 2 A4 of TrFNs. Plugging dTrFN ð
Þ into Eqs. (5.12)–(5.14), it yields that akij þ bkij þ ckij þ dijk

nkij ¼

4xjþ fkij ¼ 1  nkij ;

ð5:20Þ ð5:21Þ

8 4xjþ akij bkij ckij dijk > > < jankj  xjþ j þ jbnkj  xjþ j þ jcnkj  xjþ j þ xjþ ankj bnkj cnkj

if akij þ bkij þ ckij þ dijk [ 2xjþ ;

> > :1

if akij þ bkij þ ckij þ dijk ¼ 2xjþ :

1 2

gkij ¼

;

1 2

1 2

7 2

ankj þ bnkj þ cnkj þ dijk jbfkj 12xjþ j þ jcfkj 12xjþ j þ jdfkj 12xjþ j þ bfkj

þ cfkj þ dfkj þ 12xjþ

if akij þ bkij þ ckij þ dijk \2xjþ ; ð5:22Þ

Subsequently, the Qsi, Qdi and Qui of alternative Sk on attribute Aj can be obtained by Eqs. (5.15)–(5.17). Let akj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 N4 Þ be the induced IVIFN by the attribute vector Akj , the, llkj , lhkj , vlkj and vhkj can be computed through Eq. (5.19). Example 5.2 Let Akj ¼ ðð2; 3; 4; 5Þ; ð2; 5; 6; 8Þ; ð1; 3; 4; 7Þ; ð4; 5; 7; 8Þ; ð2; 3; 6; 8ÞÞ ¼ ð0; 0; 0; 0Þ, Amax ¼ be a TrFN vector in the ten-mark system, then Amin j j ð10; 10; 10; 10Þ. According to Eqs. (5.20)–(5.22), the Qss, Qds and Qus of Akj are calculated respectively as nkj ¼ f0:35; 0:53; 0:38; 0:60; 0:48; g, fkj ¼ f0:65; 0:47; 0:62; 0:40; 0:52g and gkj ¼ f0:70; 0:73; 0:63; 0:73; 0:68g. Subsequently, it follows from Eqs. (5.15)–(5.17) that n~kj ¼ ½0:36; 0:57, ~fkj ¼ ½0:43; 0:64 and ~gkj ¼ ½0:65; 0:74. Then, the induced IVIFN akj ¼ ð½0:134; 0:211; ½0:160; 0:237Þ can be derived by using Eq. (5.19).

5 Aggregating Decision Information into Interval-Valued …

152

5.3.3.2

For Triangular Fuzzy Numbers

If j 2 N3 , then Akj ¼ ðxk1j ; xk2j ; . . .; xkmj Þ is an attribute value vector in the form of ¼ ð0; 0; 0Þ, TFNs (i.e., xkij ¼ ðakij ; bkij ; dijk Þði ¼ 1; 2; . . .; mÞ are TFNs). Let Amin j þ þ þ þ Amax ¼ ðx ; x ; x Þðj 2 N Þ, where x means the largest grade for the attribute 3 j j j j j ^3. Aj 2 A According to Remark 5.1, if bkij ¼ ckij , then TrFN xkij ¼ ðakij ; bkij ; ckij ; dijk Þ reduces to TFN xkij ¼ ðakij ; bkij ; dijk Þ, dTrFN ð
Þ reduces to dTFN ð
Þ. Therefore, the Qsd, Qdd and Qud of TFN xkij ¼ ðakij ; bkij ; dijk Þ also can be derived from those of TrFN xkij ¼ ðakij ; bkij ; ckij ; dijk Þ. Plugging bkij ¼ ckij in Eqs. (5.20)–(5.22), it yields that nkij ¼

akij þ 2bkij þ dijk 4xjþ

;

ð5:23Þ

fkij ¼ 1  nkij ; 8 4xjþ ankj 2bnkj dnkj > > < jankj  xjþ j þ 2jbnkj  xjþ j þ xjþ ankj 2bnkj 1 2

gkij ¼

1 2

7 2

ankj þ bnkj þ cnkj

2jbfkj  xjþ j þ jdfkj  xjþ j þ 2bfkj þ dfkj þ xjþ > > :1 1 2

1 2

1 2

ð5:24Þ

if akij þ 2bkij þ dijk [ 2xjþ ; if akij þ 2bkij þ dijk \2xjþ ; if

akij

þ 2bkij

þ dijk

¼

ð5:25Þ

2xjþ :

Subsequently, the Qsi, Qdi and Qui of alternative Sk on attribute Aj can be obtained by Eqs. (5.15)–(5.17). Let akj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 N3 Þ be the induced IVIFN by the attribute vector Akj , the llkj , lhkj , vlkj and vhkj can be computed through Eq. (5.19). Example 5.3 Let Akj ¼ ðð3; 4; 7Þ; ð4; 6; 9Þ; ð2; 4; 5Þ; ð6; 8; 9Þ; ð3; 4; 5ÞÞ be a TFN vector in the ten-mark system, then Amin ¼ ð0; 0; 0Þ, Amax ¼ ð10; 10; 10Þ. Using j j Eqs. (5.23)–(5.25), the Qss, Qds and Qus of Akj are calculated respectively as nkj ¼ f0:45; 0:63; 0:38; 0:78; 0:40g, fkj ¼ f0:55; 0:37; 0:62; 0:22; 0:60g and gkj ¼ f0:75; 0:68; 0:75; 0:45; 0:80g. Subsequently, it follows from Eqs. (5.15)– ~kj ¼ ½0:55; 0:83. Then, the (5.17) that n~kj ¼ ½0:35; 0:70, ~fkj ¼ ½0:30; 0:65 and g induced IVIFN akj ¼ ð½0:132; 0:259; ½0:113; 0:240Þ can be derived by applying Eq. (5.19).

5.3.3.3

For Interval Numbers

If j 2 N2 , then Akj ¼ ðxk1j ; xk2j ; . . .; xkmj Þ is an attribute value vector in the form of interval numbers (i.e., xkij ¼ ½akij ; dijk ði ¼ 1; 2; . . .; mÞ are interval numbers). Let

5.3 Aggregating Heterogeneous Decision Information into IVIFNs

153

Amin ¼ ½0; 0, Amax ¼ ½xjþ ; xjþ  ðj 2 N2 Þ, where xjþ means the largest grade for the j j ^2. attribute Aj 2 A According to Remark 5.1, if akij ¼ bkij and ckij ¼ dijk , then TrFN xkij ¼ reduces to interval number xkij ¼ ½akij ; dijk , dTrFN ð
Þ reduces to dIN ð
Þ. Therefore, the Qsd, Qdd and Qud of interval number xkij ¼ ½akij ; dijk  also can be derived from those of TrFN xkij ¼ ðakij ; bkij ; ckij ; dijk Þ. Plugging akij ¼ bkij and ckij ¼ dijk in Eqs. (5.20)–(5.22), it yields that ðakij ; bkij ; ckij ; dijk Þ

nkij ¼

akij þ dijk 2xjþ

;

fkij ¼ 1  nkij ;

gkij ¼

8 > >
> :1 if akij þ dijk ¼ xjþ : 1 2

ð5:26Þ

ð5:28Þ

Subsequently, the Qsi, Qdi and Qui of alternative Sk on attribute Aj can be obtained by Eqs. (5.15)–(5.17). Let akj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 N2 Þ be the induced IVIFN by the attribute vector Akj , the llkj , lhkj , vlkj and vhkj can be computed through Eq. (5.19). Example 5.4 Let Akj ¼ ð½4; 10; ½7; 9; ½4; 9; ½6; 10; ½2; 7Þ be a interval number ¼ ½0; 0, Amax ¼ ½10; 10. According to vector in the ten-mark system, then Amin j j Eqs. (5.26)–(5.28), the Qss, Qds and Qus of Akj are calculated respectively as nkj ¼ f0:70; 0:80; 0:65; 0:80; 0:45g, fkj ¼ f0:30; 0:20; 0:35; 0:20; 0:55g and gkj ¼ f0:50; 0:40; 0:58; 0:40; 0:64g. Subsequently, from Eqs. (5.15)–(5.17) it fol~kj ¼ ½0:40; 0:61. Then, the lows that ~nkj ¼ ½0:54; 0:82, ~fkj ¼ ½0:18; 0:46 and g induced IVIFN akj ¼ ð½0:214; 0:329; ½0:070; 0:185Þ can be derived by using Eq. (5.19).

5.3.3.4

For Real Numbers

If j 2 N1 , then Akj ¼ ðxk1j ; xk2j ; . . .; xkmj Þ is an attribute value vector in the form of real numbers (i.e., xkij ¼ akij ði ¼ 1; 2; . . .; mÞ are real numbers). Let Amin ¼ 0, Amax ¼ xjþ j j ^1. ðj 2 N1 Þ, where x þ means the largest grade for the attribute Aj 2 A j

According to Remark 5.1, if akij ¼ bkij ¼ ckij ¼ dijk , then TrFN xkij ¼ ðakij ; bkij ; ckij ; dijk Þ reduces to real number xkij ¼ akij , dTrFN ð
Þ reduces to dRN ð
Þ. Therefore, the Qsd, Qdd

5 Aggregating Decision Information into Interval-Valued …

154

and Qud of real number xkij ¼ akij also can be derived from those of TrFN xkij ¼ ðakij ; bkij ; ckij ; dijk Þ. Plugging akij ¼ bkij ¼ ckij ¼ dijk into Eqs. (5.20)–(5.22), it yields that nkij ¼

akij xjþ

;

ð5:29Þ

fkij ¼ 1  nkij ; 8 þ k xj aij > > < xjþ 1 2

gkij ¼

akij 1 þ 2xj

> > :1

ð5:30Þ

if akij [ 12xjþ ; ð5:31Þ

if akij \12xjþ ; if

akij

¼

xjþ :

1 2

Subsequently, the Qsi ~nkj , Qdi ~fkj and Qui ~gkj of alternative Sk on attribute Aj can be obtained by Eqs. (5.15)–(5.17). Then, let akj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 N1 Þ be the induced IVIFN by the attribute vector Akj , the llkj , lhkj , vlkj and vhkj can be computed through Eq. (5.19). Example 5.5 Let Akj ¼ ð3:4; 5:4; 7:2; 8:1; 5:9Þ be a real number vector in the ten-mark system, then Amin ¼ 0, Amax ¼ 10. According to Eqs. (5.29)–(5.31), the j j Qss, Qds and Qus of Akj are calculated respectively as nkj ¼ f0:34; 0:54; 0:72; 0:81; 0:59g, fkj ¼ f0:66; 0:46; 0:28; 0:19; 0:41g and gkj ¼ f0:68; 0:92; 0:56; 0:38; 0:82g. Subsequently, it follows from Eqs. (5.15)–(5.17) that ~nkj ¼ ½0:42; 0:78, ~fkj ¼ ½0:22; 0:58 and ~g ¼ ½0:46; 0:88. Then, the induced kj IVIFN akj ¼ ð½0:157; 0:292; ½0:082; 0:217Þ can be derived by using Eq. (5.19).

5.4

A Novel Approach for Heterogeneous MAGDM Problems

By the general aggregation method proposed in Sect. 5.3, we can aggregate the decision matrices X k ¼ ðxkij Þmn ðk 2 PÞ into a collective decision matrix with the induced IVIFNs, i.e.,

X ¼ ðrkj Þpn

S1 ¼ S2 .. . Sp

A1 ð½ll11 ; lh11 ; ½vl11 ; vh11 Þ l h B ð½l21 ; l21 ; ½vl21 ; vh21 Þ B B .. @ . 0

ð½llp1 ; lhp1 ; ½vlp1 ; vhp1 Þ

A2 ð½ll12 ; lh12 ; ½vl12 ; vh12 Þ l h ð½l22 ; l22 ; ½vl22 ; vh22 Þ .. . ð½llp2 ; lhp2 ; ½vlp2 ; vhp2 Þ










.. .




An 1 ð½ll1n ; lh1n ; ½vl1n ; vh1n Þ l h l h ð½l2n ; l2n ; ½v2n ; v2n Þ C C; C .. A . ð½llpn ; lhpn ; ½vlpn ; vhpn Þ

ð5:32Þ

5.4 A Novel Approach for Heterogeneous MAGDM Problems

155

where rkj ¼ ð½llkj ; lhkj ; ½vlkj ; vhkj Þ ðk 2 P; j 2 NÞ is an induced IVIFN aggregated by the attribute vector Akj in X k . Once the attribute weight vector w ¼ ðw1 ; w2 ; . . .; wn Þ is completely known, the overall evaluation rk ¼ ð½llk ; lhk ; ½vlk ; vhk Þ of the alternative Sk is computed using Eq. (5.3), where llk ¼ 1 

n Y j¼1

ð1  llkj Þwj ; lhk ¼ 1 

n Y

ð1  lhkj Þwj ; vlk ¼

j¼1

n Y j¼1

ðvlkj Þwj ; vhk ¼

n Y

ðvhkj Þwj :

j¼1

ð5:33Þ Since the attribute weight vector w ¼ ðw1 ; w2 ; . . .; wn Þ is incompletely known, we develop a new IF programming model to determine the weights of attribute in the sequel.

5.4.1

Construct an Intuitionistic Fuzzy Programming Model to Determine the Attribute Weights

A reasonable attribute weight vector w should make the collective overall rating rk ¼ ð½llk ; lhk ; ½vlk ; vhk Þ of alternative Sk ðk 2 PÞ as large as possible. To this end, a multiple objective IF mathematical optimization model can be constructed to determine the attribute weights: maxfrk gðk 2 PÞ s:t: w 2 D

ð5:34Þ

According to Definition 5.2, the bigger ½llk ; lhk  and the smaller ½vlk ; vhk , the larger rk is. It is natural to maximize ½llk ; lhk  and minimize ½vlk ; vhk  for maximizing rk . Let a ¼ ½al ; au  and b ¼ ½bl ; bu  be two nonnegative interval numbers, then a  b if al  bl and au  bu [40]. Thus, rk will be maximized if vlk and vhk are minimized, while llk and lhk are maximized. To solve the Eq. (5.34), it can be transformed into a multiple objective optimization model: maxfllk g maxflhk g minfvlk g minfvhk g s:t: w 2 D

ðk ðk ðk ðk

2 PÞ 2 PÞ 2 PÞ : 2 PÞ

ð5:35Þ

5 Aggregating Decision Information into Interval-Valued …

156

Plugging Eq. (5.33) into Eq. (5.35), Eq. (5.35) can be further rewritten as (

n Q

) ulkj Þwj

max 1  ð1  j¼1 ( ) n Q h wj max 1  ð1  ukj Þ j¼1 ( ) n Q l wj ðvkj Þ min (j¼1 ) n Q l wj ðvkj Þ min

ðk 2 PÞ ðk 2 PÞ ðk 2 PÞ

ð5:36Þ

ðk 2 PÞ

j¼1

s:t: w 2 D Q It is worth mentioning that the objective maxf1  nj¼1 ð1  llkj Þwj g may be Q which is equivalent to equivalent to minf nj¼1 ð1  llkj Þwj g, Pn l l minf j¼1 wj lnð1  lkj Þg since 0  lkj  1 and logarithm (ln) is an increasing Q function. Likewise, the objective maxf1  nj¼1 ð1  lhkj Þwj g is equivalent to Q P minf nj¼1 wj lnð1  lhkj Þg. The objective minf nj¼1 ðvlkj Þwj g may be equivalent to P minf nj¼1 wj lnðvlkj Þg since 0  vlkj  1 and logarithm (ln) is an increasing function. P Q Similarly, the objective minf nj¼1 ðvhkj Þwj g is equivalent to minf nj¼1 wj lnðvhkj Þg. Thus, Eq. (5.36) can be converted into a multiple objective linear programming model: (

n P

)

min Z1k ¼ wj lnð1  llkj Þ j¼1 ( ) n P h wj lnð1  lkj Þ min Z2k ¼ j¼1 ( ) n P l wj lnðvkj Þ min Z3k ¼ j¼1 ( ) n P wj lnðvhkj Þ min Z4k ¼

ðk 2 PÞ ðk 2 PÞ ðk 2 PÞ

ð5:37Þ

ðk 2 PÞ

j¼1

s:t: w 2 D There are several methods to solve a multiple objective programming. Here, we apply membership function-based linear sum approach to solving Eq. (5.37). Suppose that Ztkmax and Ztkmin ðt ¼ 1; 2; 3; 4; k ¼ 1; 2; . . .; pÞ are the maximum and

5.4 A Novel Approach for Heterogeneous MAGDM Problems

157

minimum values of the single objective function Ztk ðt ¼ 1; 2; 3; 4; k ¼ 1; 2; . . .; pÞ ignoring the other objectives in Eq. (5.37). The corresponding linear membership function of the objective function Ztk is defined as follows: 8

> > jk > > aij > > > > ½xjþ  dijk ; xjþ  akij  > > > < ½ak ; d k  ij

ij

ðxjþ  dijk ; xjþ  bkij ; xjþ  akij Þ > > > > > ðakij ; bkij ; dijk Þ > > > þ þ þ þ > k k k > > ðxj  dij ; xj  cij ; xj  bij ; xj > : k k k k ðaij ; bij ; cij ; dij Þ

if if if if if if  akij Þ if if

j 2 N1c j 2 N1b j 2 N2c j 2 N2b j 2 N3c j 2 N3b j 2 N4c j 2 N4b

ð5:40Þ

where xjþ is the largest grade of the attribute Aj , Nic and Nib ði ¼ 1; 2; 3; 4Þ denote ^ i , respectively. the subscript sets of cost and benefit attributes in subset A Step 3: Calculate the Qsd nkij , Qdd fkij and Qud gkij of each attribute value x0kij as follows: (1) If j 2 N1 , then nkij , fkij and gkij can be computed by Eqs. (5.29)–(5.31); (2) If j 2 N2 , then nkij , fkij and gkij can be computed by Eqs. (5.26)–(5.28); (3) If j 2 N3 , then nkij , fkij and gkij can be computed by Eqs. (5.23)–(5.25); (4) If j 2 N4 , then nkij , fkij and gkij can be computed by Eqs. (5.20)–(5.22); Step 4: Calculate the Qsi ~nkj , Qdi ~fkj and Qui ~ gkj of attribute vector Akj by Eqs. (5.15)–(5.17); Step 5: Induce the IVIFN corresponding to the attribute vector Akj through Eq. (5.19) and form a collective IVIF matrix M ¼ ðrkj Þpn (i.e., Eq. 5.32). Step 6: Construct a multiple objective IF programming model (i.e., Eq. 5.34) and convert it into a linear programming model (i.e., Eq. 5.39). Step 7: Determine the attribute weights by solving the linear programming model Eq. (5.39). Step 8: Derive the comprehensive evaluation value rk of each alternative Sk by Eq. (5.3). Step 9: Compute the score sðrk Þ and accuracy degree hðrk Þ of comprehensive assessment rk by using Eqs. (5.4) and (5.5).

5.4 A Novel Approach for Heterogeneous MAGDM Problems

159

Step 10: Rank the alternatives according to the Definition 5.5 and select the best one. The schematic structure of the algorithm is depicted in Fig. 5.2.

Decision goal

S1

Sp

Sk

Distance measure

Distance measure

Aggregating decision information into IVIFN Real numbers Interval numbers

TOPSIS

TIFNs

Qsd, Qss and Qud of xijk

Mean Deviation

Qsi, Qsi and Qui of Akj

Transfor μ kjl , μ kjh , vkjl , vkjh mation of Akj

IVIFN 2 n

IVIFN p1

TrIFNs

IVIFN11

IVIFN1 j

IVIFN1n

IVIFN 21

IVIFN 2 j

IVIFN pj

IVIFN pn

([μ kjl , μ kjh ],[ vkjl , vkjh ])) p×n

Obtain the weights of attributes by constructing a multi-objective IVIF programming model

Making decision DM

Real number attribute

Interval number attribute

TFN attribute

Fig. 5.2 Framework of proposed heterogeneous MAGDM approach

TrFN attribute

160

5.5

5 Aggregating Decision Information into Interval-Valued …

Comparison Analysis with Existing Methods

In this section, we compare the proposed method with existing methods [21–23]. The method [22] aggregated real numbers into IVIFN. The methods [21, 23] aggregated interval numbers into IVIFN. Their objective and idea are the same, that is to aggregate the decision information into IVIFN for GDM problems. Notably, in contrast with existing methods [21–23], the proposed method here can be highlighted from the following aspects: (1) The attribute values in the method [22] are real numbers, and the attribute values in the methods [21, 23] are interval numbers, namely, they only considered the assessment information of attribute as single types (i.e., real numbers or interval numbers); whereas the attribute values in the proposed method may be multiple types (i.e., real numbers, interval numbers, TFNs and TrFNs). Therefore, the proposed method can be applied to solve heterogeneous MAGDM problems while the methods [21–23] cannot. (2) In aggregation stage, the proposed method distinguishes the satisfaction and dissatisfaction by calculating the relative closeness between each attribute value and the ideal satisfaction and dissatisfaction, whereas the methods [21– 23] distinguish the satisfaction and dissatisfaction through 0.5, which results in the methods [21–23] can only handle real number or interval number attribute vectors. Hence, the proposed method has wider application scope than the methods [21–23]. (3) The proposed method considers the satisfaction, dissatisfaction and hesitation index of each attribute value simultaneously, whereas the methods [21–23] just consider the satisfaction and dissatisfaction index. So, the proposed method induces an IVIFN that takes into account as much as possible all the information in the aggregation process. (4) The proposed method objectively determines the attribute weights by constructing a multiple objective IF mathematical programming model, whereas the attribute weights of the methods [21–23] are given in advance. It is very difficult to avoid subjective randomness while giving attribute weights a priori. Table 5.1 presents a comprehensive comparison between existing methods and the proposed method of this chapter.

5.6

Illustrative Examples

In this section, an IT outsourcing service provider evaluation example and a supplier selection example are analyzed to illustrate the effectiveness of the proposed method.

skj in linear transformation

Hesitancy interval

Dissatisfactory interval

Quasi-satisfactory degree (Qsd) Quasi-dissatisfactory degree (Qdd) Quasi-uncertain degree (Qud) Quasi-satisfactory set (Qss) Quasi-dissatisfactory set (Qds) Quasi-uncertain set (Qus) Satisfactory interval None None None None Based on Minimax elements in [0,0.5] Based on Minimax elements in [0.5,1] None

None

None

None

None

Based on Minimax elements in [0,0.5] Based on Minimax elements in [0.5,1] None nlkj þ nhkj þ flkj þ fhkj

Derived from [0.5,1]

Derived from [0.5,1]

nlkj þ nhkj þ flkj þ fhkj

Derived from [0,0.5]

Interval number MAGDM with interval numbers Given in advance

Method [21]

Derived from [0,0.5]

Real number MAGDM with real numbers Given in advance

Attribute weight

Method [22]

Characteristics

Attribute values Solve problem

maxfnlkj þ nhkj þ flkj þ fhkj g

Based on mean and standard deviation of all elements in [0,0.5] Based on mean and standard deviation of all elements in [0.5,1] None

None

None

None

None

Derived from [0.5,1]

Derived from [0,0.5]

Given in advance

Interval number MAGDM with interval numbers

Method [23]

Table 5.1 Comparisons between existing methods and the proposed method

1 l ðg þ ghkj Þ 2 kj

nlkj þ nhkj þ flkj þ fhkj þ

Based on mean and standard deviation of Qss Based on mean and standard deviation of Qds Based on mean and standard deviation of Qus

Comprised of Qud of each element

Comprised of Qdd of each element

Determined by multi-objective IF programming Derived from relative closeness between elements and the ideal satisfaction Derived from relative closeness between elements and the ideal dissatisfaction Derived from relative closeness between elements and the ideal median Comprised of Qsd of each element

TFN, TrFN, interval and real number Heterogeneous MAGDM

The proposed method

5.6 Illustrative Examples 161

5 Aggregating Decision Information into Interval-Valued …

162

5.6.1

An IT Outsourcing Service Provider Evaluation Example

China CNR Corporation Limited (hereinafter “CNR) is a world class enterprise in the industry of railway transport equipment. CNR is mainly engaged in design, manufacture, refurbishment, service and lease of railway transportation equipment. Headquartered in Beijing, CNR has 88,700 employees divided among over 20 subsidiaries, among which are the leading manufacturers in diesel coaches, wagons and locomotives in China. The annual output of CNR is 1100 urban railway vehicles, 460 diesel locomotives, 26,000 freight wagons and 2300 passenger coaches, which share over half of the home market, and are exported to more than 60 countries and regions. To cope with today’s fast growing IT environment, CNR is engaged in decision about an IT outsourcing service project which provides a cost-effective alternative to customer’s IT needs. CNR plans to select the best service providers from four candidates, whose key competencies are evaluated by the following six attributes. Now there are five experts are invited to form a DM team. In the evaluation stage, the DM team considers six attributes for each provider, which are described in Li [27, 28]. These attributes include the research and development capability (A1), the product quality (A2), the technological level (A3), flexibility (A4), delivery time (A5) and price (A6), in which the former four ones are benefit attributes, the latter two ones are cost attributes. Due to the imprecision and subjectivity in human opinions, the research and development capability (A1), product quality (A2) and technological level (A3) are suitably represented by TFNs. The assessments of A4 are represented by TrFNs. The assessments of delivery time (A5) are represented by interval numbers to consider the uncertainties in the product process. The assessments of the providers on (A6) can be represented by real numbers. Experts adopt the ten-mark system to evaluate the four candidates IT outsourcing providers based on the six attributes. The decision information can be expressed by the matrices, which are listed in Table 5.2. The DM team gives the known information on the attributes’ importance as follows: ( D¼

5.6.1.1

w1  0:1; w1  w4  0:1; w2  0:2; w2  w6  0:05; w1 þ w3 þ w4  0:6; w4  w5  0:05; w5  0:05; w6  0:05:

)

Decision Process Using the Proposed Method

Obviously, the decision problem mentioned above is a heterogeneous MAGDM problem involving four different data formats: real numbers, interval numbers, TFNs and TrFNs. To solve this issue, we apply the proposed method to the selection of the IT outsourcing providers below. Step 1: The decision matrix of providers listed in Table 5.2.

5.6 Illustrative Examples

163

Table 5.2 Decision matrix of four alternatives Provider

DM

A1

A2

A3

A4

A5

A6

S1

D1 D2 D3 D4 D5 D1 D2 D3 D4 D5 D1 D2 D3 D4 D5 D1 D2 D3 D4 D5

(5.3,6.5,8.9) (6.5,7.6,8.4) (4.7,6.8,8.2) (5.3,6.1,7.9) (4.1,5.3,7.7) (6.6,7.6,8.8) (5.4,6.4,7.9) (5.2,6.4,8.5) (4.6,6.8,7.8) (4.1,5.2,6.5) (6.6,7.6,8.8) (5.5,5.7,8.8) (4.3,5.5,7.9) (4.4,6.0,7.3) (4.7,5.4,6.5) (5.3,6.4,7.5) (6.3,7.2,8.4) (5.1,6.2,6.7) (4.5,5.7,6.8) (4.3,5.6,7.4)

(3.3,5.0,6.7) (5.0,6.7,8.3) (5.0,6.7,8.3) (3.3,5.0,6.7) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (1.7,3.3,5.0) (5.0,6.7,8.3) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (5.0,6.7,8.3) (8.3,10,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (3.3,5.0,6.7)

(4.5,7.1,8.3) (3.3,4.5,6.7) (5.5,6.1,8.3) (5.5,6.4,8.5) (3.8,6.5,6.7) (4.1,6.7,8.3) (4.5,6.4,7.3) (6.7,8.1,10) (3.3,5.0,7.7) (6.9,8.1,10) (5.1,6.6,8.0) (3.9,6.5,6.7) (6.1,6.7,8.3) (5.7,7,3,10) (4.3,5.0,6.7) (3.3,4.7,6.7) (5.9,6.5,6.7) (4.6,5.5,6.7) (4.3,5.0,6.7) (5.7,7.3,10)

(4.2,5.3,6.5,7.9) (4.4,4.5,6.6,7.8) (4.1,4.4,5.5,8.8) (5.2,6.3,7.5,8.9) (3.0,5.1,6.3,6.7) (4.2,5.3,5.4,6.7) (5.3,6.4,7.4,7.9) (4.2,5.2,6.4,6.5) (5.5,5.6,6.8,6.8) (3.1,4.1,4.2,5.5) (4.5,4.6,5.6,7.8) (5.0,5.5,7.7,8.8) (3.0,4.0,5.0,9.0) (5.4,5.7,6.6,7.8) (5.3,6.8,7.4,8.5) (4.2,4.3,5.4,7.5) (3.2,5.3,6.4,6.7) (3.1,6.1,6.2,7.7) (2.4,4.5,4.7,5.8) (4.2,4.3,6.6,8.6)

[0.7,4.4] [1.1,3.3] [1.1,2.6] [0.0,2.4] [2.3,3.8] [1.3,4.6] [1.2,3.5] [0.4,1.7] [0.6,3.3] [1.6,3.7] [2.0,4.4] [1.4,2.4] [0.0,3.3] [0.3,2.5] [1.4,2.7] [1.6,3.7] [0.5,2.6] [1.0,2.9] [1.6,3.4] [2.1,4.6]

5.6 3.9 2.8 1.8 4.1 4.2 2.7 3.8 2.3 3.1 2.9 3.5 3.8 2.3 3.1 2.9 3.5 4.7 1.7 3.2

S2

S3

S4

Step 2: Owing to the attribute values given on the basis of the ten-mark system, one has 8 8 0 if j 2 N1 10 if j 2 N1 > > > > < < ½0; 0 if j 2 N2 max ½10; 10 if j 2 N2 min ; Aj ¼ : ð5:41Þ Aj ¼ ð0; 0; 0Þ if j 2 N ð10; 10; 10Þ if j 2 N3 > > 3 > > : : ð0; 0; 0; 0Þ if j 2 N4 ð10; 10; 10; 10Þ if j 2 N4 We use Eq. (5.41) to transform the cost attribute value to benefit attribute value. The transformation results are shown in Table 5.3. Step 3: According to Step 3, the Qsd, Qdd and Qud of each attribute value can be obtained. The results are summed up in Table 5.4. Step 4: By Eqs. (5.15)–(5.17), the Qsi ~nkj , Qdi ~fkj and Qui ~ gkj of Akj are presented in Table 5.5. Step 5: By Eq. (5.19), the induced IVIFNs of Akj are also presented in Table 5.5. Thus, the collective IVIF matrix M ¼ ðrkj Þpn is constructed by Eq. (5.32). Step 6: By Eq. (5.34), a multiple objective IF programming model is formed as follows:

5 Aggregating Decision Information into Interval-Valued …

164

Table 5.3 Benefit decision matrix of four alternatives Provider

DM

A1

A2

A3

A4

A5

A6

S1

D1 D2 D3 D4 D5 D1 D2 D3 D4 D5 D1 D2 D3 D4 D5 D1 D2 D3 D4 D5

(5.3,6.5,8.9) (6.5,7.6,8.4) (4.7,6.8,8.2) (5.3,6.1,7.9) (4.1,5.3,7.7) (6.6,7.6,8.8) (5.4,6.4,7.9) (5.2,6.4,8.5) (4.6,6.8,7.8) (4.1,5.2,6.5) (6.6,7.6,8.8) (5.5,5.7,8.8) (4.3,5.5,7.9) (4.4,6.0,7.3) (4.7,5.4,6.5) (5.3,6.4,7.5) (6.3,7.2,8.4) (5.1,6.2,6.7) (4.5,5.7,6.8) (4.3,5.6,7.4)

(3.3,5.0,6.7) (5.0,6.7,8.3) (5.0,6.7,8.3) (3.3,5.0,6.7) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (1.7,3.3,5.0) (5.0,6.7,8.3) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (5.0,6.7,8.3) (8.3,10,10) (5.0,6.7,8.3) (6.7,8.3,10) (5.0,6.7,8.3) (3.3,5.0,6.7)

(4.5,7.1,8.3) (3.3,4.5,6.7) (5.5,6.1,8.3) (5.5,6.4,8.5) (3.8,6.5,6.7) (4.1,6.7,8.3) (4.5,6.4,7.3) (6.7,8.1,10) (3.3,5.0,7.7) (6.9,8.1,10) (5.1,6.6,8.0) (3.9,6.5,6.7) (6.1,6.7,8.3) (5.7,7.3,10) (4.3,5.0,6.7) (3.3,4.7,6.7) (5.9,6.5,6.7) (4.6,5.5,6.7) (4.3,5.0,6.7) (5.7,7.3,10)

(4.2,5.3,6.5,7.9) (4.4,4.5,6.6,7.8) (4.1,4.4,5.5,8.8) (5.2,6.3,7.5,8.9) (3.0,5.1,6.3,6.7) (4.2,5.3,5.4,6.7) (5.3,6.4,7.4,7.9) (4.2,5.2,6.4,6.5) (5.5,5.6,6.8,6.8) (3.1,4.1,4.2,5.5) (4.5,4.6,5.6,7.8) (5.0,5.5,7.7,8.8) (3.0,4.0,5.0,9.0) (5.4,5.7,6.6,7.8) (5.3,6.8,7.4,8.5) (4.2,4.3,5.4,7.5) (3.2,5.3,6.4,6.7) (3.1,6.1,6.2,7.7) (2.4,4.5,4.7,5.8) (4.2,4.3,6.6,8.6)

[5.6,9.3] [6.7,8.9] [7.4,8.9] [7.6,10] [6.2,7.7] [5.4,8.7] [6.5,8.8] [8.3,9.6] [6.7,9.4] [6.3,8.4] [5.6,8.0] [7.6,8.6] [6.7,10] [7.5,9.7] [7.3,8.6] [6.3,8.4] [7.4,9.5] [7.1,9.0] [6.6,8.4] [5.4,7.9]

4.4 6.1 7.2 8.2 5.9 5.8 7.3 6.2 7.7 6.9 7.1 6.5 5.3 8.3 6.8 4.9 8.1 3.9 5.8 7.0

S2

S3

S4

maxfr1 ¼ð½0:221; 0:273; ½0:102; 0:155Þw1 þ ð½0:181; 0:283; ½0:083; 0:185Þw2 þ ð½0:195; 0:257; ½0:113; 0:175Þw3 þ ð½0:195; 0:241; ½0:125; 0:172Þw4 þ ð½0:293; 0:350; ½0:060; 0:118Þw5 þ ð½0:184; 0:291; ½0:082; 0:189Þw6 g maxfr2 ¼ð½0:201; 0:245; ½0:119; 0:164Þw1 þ ð½0:183; 0:344; ½0:051; 0:212Þw2 þ ð½0:215; 0:318; ½0:070; 0:173Þw3 þ ð½0:168; 0:237; ½0:123; 0:192Þw4 þ ð½0:290; 0:351; ½0:060; 0:120Þw5 þ ð½0:227; 0:286; ½0:092; 0:151Þw6 g maxfr3 ¼ð½0:199; 0:261; ½0:106; 0:168Þw1 þ ð½0:254; 0:325; ½0:070; 0:141Þw2 þ ð½0:207; 0:274; ½0:099; 0:166Þw3 þ ð½0:202; 0:257; ½0:113; 0:168Þw4 þ ð½0:259; 0:336; ½0:063; 0:140Þw5 þ ð½0:217; 0:299; ½0:080; 0:162Þw6 g maxfr4 ¼ð½0:204; 0:252; ½0:114; 0:161Þw1 þ ð½0:216; 0:353; ½0:039; 0:177Þw2 þ ð½0:176; 0:255; ½0:108; 0:187Þw3 þ ð½0:172; 0:216; ½0:146; 0:190Þw4 þ ð½0:279; 0:334; ½0:069; 0:125Þw5 þ ð½0:158; 0:280; ½0:088; 0:211Þw6 g s:t: w 2D

ð5:42Þ According to Eq. (5.37), Eq. (5.42) can be transformed into the following multiple objective linear programming model:

D1 D2 D3 D4 D5 D1 D2 D3 D4 D5 D1 D2 D3 D4 D5 D1 D2 D3 D4 D5

S1

S4

S3

S2

DM

Provider

0.69,0.31,0.62 0.75,0.25,0.50 0.66,0.34,0.66 0.64,0.36,0.71 0.57,0.43,0.77 0.58,0.42,0.77 0.66,0.34,0.69 0.67,0.33,0.66 0.64,0.36,0.68 0.53,0.47,0.85 0.77,0.23,0.47 0.67,0.33,0.67 0.59,0.41,0.75 0.59,0.41,0.76 0.55,0.45,0.76 0.64,0.36,0.72 0.73,0.27,0.54 0.60,0.40,0.80 0.57,0.43,0.81 0.58,0.42,0.73

A1 0.50,0.50,1.00 0.67,0.33,0.67 0.67,0.33,0.67 0.50,0.50,1.00 0.83,0.17,0.33 0.83,0.17,0.33 0.67,0.33,0.67 0.83,0.17,0.33 0.67,0.33,0.67 0.33,0.67,0.67 0.67,0.33,0.67 0.83,0.17,0.33 0.83,0.17,0.33 0.67,0.33,0.67 0.67,0.33,0.67 0.94,0.06,0.11 0.67,0.33,0.67 0.83,0.17,0.33 0.67,0.33,0.67 0.50,0.50,1.00

A2

Table 5.4 Qsd, Qdd and Qud of each attribute value 0.66,0.34,0.63 0.48,0.52,0.79 0.66,0.34,0.67 0.68,0.32,0.64 0.57,0.43,0.75 0.64,0.36,0.65 0.61,0.39,0.74 0.83,0.17,0.35 0.53,0.47,0.76 0.83,0.17,0.33 0.66,0.34,0.69 0.57,0.43,0.75 0.70,0.30,0.59 0.77,0.23,0.47 0.53,0.47,0.85 0,49,0.51,0.80 0.64,0.36,0.73 0.56,0.44,0.84 0.53,0.47,0.85 0.77,0.23,0.47

A3 0.60,0.40,0.75 0.58,0.42,0.75 0.57,0.43,0.75 0.70,0.30,0.61 0.53,0.47,0.79 0.54,0.46,0.85 0.68,0.32,0.65 0.56,0.44,0.82 0.62,0.38,0.77 0.42,0.58,0.81 0.56,0.44,0.80 0.68,0.32,0.65 0.53,0.47,0.73 0.64,0.36,0.73 0.70,0.30,0.60 0.53,0.47,0.81 0.54,0.46,0.78 0.58,0.42,0.71 0.43,0.57,0.81 0.59,0.41,0.71

A4 0.74,0.26,0.51 0.78,0.22,0.44 0.82,0.18,0.37 0.88,0.12,0.24 0.69,0.31,0.61 0.71,0.30,0.59 0.76,0.24,0.47 0.89,0.11,0.21 0.81,0.19,0.39 0.78,0.22,0.44 0.68,0.32,0.64 0.81,0.19,0.38 0.84,0.16,0.33 0.86,0.14,0.28 0.79,0.21,0.41 0.73,0.27,0.53 0.84,0.16,0.31 0.81,0.19,0.39 0.75,0.25,0.50 0.67,0.33,0.67

A5

0.44,0.56,0.88 0.61,0.39,0.78 0.72,0.28,0.56 0.82,0.18,0.36 0.59,0.41,0.82 0.58,0.42,0.84 0.73,0.27,0.54 0.62,0.38,0.76 0.77,0.23,0.46 0.69,0.31,0.62 0.71,0.29,0.58 0.65,0.35,0.70 0.53.0.47,0.94 0.83,0.17,0.34 0.68,0.32,0.64 0.49,0.51,0.98 0.81,0.19,0.38 0.39,0.61,0.78 0.58,0.42,0.84 0.70,0.30,0.60

A6

5.6 Illustrative Examples 165

5 Aggregating Decision Information into Interval-Valued …

166

Table 5.5 Qsi ~ nkj , Qdi ~fkj , Qui ~gkj and aggregated IVIFNs of attribute vectors Provider

Attribute

~nkj

~fkj

~gkj

IVIFN

S1

A1

[0.588,0.728]

[0.272,0.412]

[0.549,0.782]

A2

[0.494,0.773]

[0.227,0.506]

[0.455,1.011]

A3

[0.528,0.695]

[0.305,0.472]

[0.632,0.779]

A4

[0.532,0.658]

[0.342,0.468]

[0.657,0.798]

A5

[0.713,0.853]

[0.147,0.287]

[0.294,0.574]

A6

[0.493,0.779]

[0.221,0.507]

[0.464,0.896]

A1

[0.551,0.673]

[0.327,0.449]

[0.654,0.833]

A2

[0.462,0.871]

[0.129,0.538]

[0.352,0.714]

A3

[0.553,0.821]

[0.179,0.447]

[0.362,0.796]

A4

[0.468,0.657]

[0.343,0.532]

[0.700,0.856]

A5

[0.707,0.855]

[0.145,0.293]

[0.291,0.585]

A6

[0.600,0.756]

[0.244,0.400]

[0.488,0.799]

A1

[0.541,0.711]

[0.289,0.459]

[0.570,0.875]

A2

[0.643,0.824]

[0.176,0.357]

[0.352,0.714]

A3

[0.551,0.741]

[0.259,0.449]

[0.522,0.818]

A4

[0.546,0.694]

[0.306,0.454]

[0.623,0.780]

A5

[0.727,0.865]

[0.135,0.273]

[0.269,0.547]

A6

[0.572,0.788]

[0.212,0.428]

[0.424,0.856]

A1

[0.559,0.689]

[0.312,0.441]

[0.622,0.846]

A2

[0.550,0.900]

[0.100,0.450]

[0.200,0.901]

A3

[0.489,0.706]

[0.294,0.511]

[0.578,0.895]

A4

[0.474,0.598]

[0.402,0.526]

[0.713,0.812]

A5

[0.691,0.829]

[0.171,0.309]

[0.342,0.618]

A6

[0.428,0.760]

[0.239,0.572]

[0.484,0.948]

([0.221,0.273], [0.102,0.155]) ([0.181,0.283], [0.083,0.185]) ([0.195,0.257], [0.113,0.175]) ([0.195,0.241], [0.125,0.172]) ([0.293,0.350], [ 0.060,0.118]) ([0.184,0.291], [0.082,0.189]) ([0.201,0.245], [0.119,0.164]) ([0.183,0.344], [0.051,0.212]) ([0.215,0.318], [0.070,0.173]) ([0.168,0.237], [0.123,0.192]) ([0.290,0.351], [0.060,0.120] ([0.227,0.286], [0.092,0.151]) ([0.199,0.261], [0.106,0.168]) ([0.254,0.325], [0.070,0.141] ([0.207,0.274], [0.099,0.166]) ([0.202,0.257], [0.113,0.168]) ([0.259,0.336], [0.063,0.140]) ([0.217,0.298], [0.080,0.162]) ([0.204,0.252], [0.114,0.161]) ([0.216,0.353], [0.039,0.177]) ([0.176,0.255], [0.108,0.187]) ([0.172,0.216], [0.146,0.190]) ([0.279,0.334], [0.069,0.125]) ([0.158,0.280], [0.088,0.211])

S2

S3

S4

5.6 Illustrative Examples

167

minf0:249w1  0:199w2  0:217w3  0:217w4  0:347w5  0:203w6 g minf0:224w1  0:202w2  0:242w3  0:184w4  0:343w5  0:257w6 g minf0:222w1  0:292w2  0:232w3  0:226w4  0:300w5  0:244w6 g minf0:229w1  0:243w2  0:194w3  0:188w4  0:327w5  0:171w6 g minf0:319w1  0:332w2  0:297w3  0:276w4  0:432w5  0:344w6 g minf0:282w1  0:421w2  0:383w3  0:270w4  0:432w5  0:337w6 g minf0:302w1  0:393w2  0:321w3  0:297w4  0:409w5  0:355w6 g minf0:290w1  0:435w2  0:294w3  0:244w4  0:407w5  0:328w6 g minf2:282w1  2:487w2  2:184w3  2:076w4  2:807w5  2:497w6 g minf2:128w1  2:974w2  2:665w3  2:092w4  2:820w5  2:383w6 g minf2:241w1  2:666w2  2:316w3  2:179w4  2:768w5  2:523w6 g minf2:171w1  3:241w2  2:221w3  1:926w4  2:674w5  2:427w6 g minf1:867w1  1:687w2  1:745w3  1:763w4  2:138w5  1:665w6 g minf1:810w1  1:550w2  1:754w3  1:653w4  2:120w5  1:889w6 g minf1:781w1  1:960w2  1:795w3  1:783w4  1:968w5  1:819w6 g minf1:824w1  1:734w2  1:677w3  1:660w4  2:083w5  1:557w6 g s:t: w 2 D ð5:43Þ By using Eq. (5.43), we obtain the maximum and minimum values corresponding to each objective as follows: max min max min max Z11 ¼  0:221; Z11 ¼ 0:254; Z21 ¼ 0:212; Z21 ¼ 0:246; Z31 ¼ 0:244; min max min max min ¼  0:258; Z41 ¼ 0:208; Z41 ¼ 0:237; Z12 ¼ 0:310; Z12 ¼ 0:336; Z31 max min max min max ¼  0:323; Z22 ¼ 0:365; Z32 ¼ 0:332; Z32 ¼ 0:350; Z42 ¼ 0:309; Z22 min max min max min ¼  0:337; Z13 ¼ 2:284; Z13 ¼ 2:402; Z23 ¼ 2:364; Z23 ¼ 2:586; Z42 max min max min max ¼  2:370; Z33 ¼ 2:467; Z43 ¼ 2:331; Z43 ¼ 2:469; Z14 ¼ 1:754; Z33 min max min max min ¼  1:862; Z24 ¼ 1:708; Z24 ¼ 1:814; Z34 ¼ 1:834; Z34 ¼ 1:866; Z14 max min ¼  1:697; Z44 ¼ 1:799: Z44

Based on these ideal values, we construct some membership functions corresponding to the objectives by Eq. (5.38). Combined with these membership functions, a linear programming model is constructed by Eq. (5.39) as follows:

5 Aggregating Decision Information into Interval-Valued …

168

min Z ¼ d1 ðð0:249w1 þ 0:199w2 þ 0:217w3 þ 0:217w4 þ 0:347w5 þ 0:203w6 Þ  0:222Þ=0:033 þ d2 ðð0:224w1 þ 0:202w2 þ 0:242w3 þ 0:184w4 þ 0:343w5 þ 0:257w6 Þ  0:212Þ=0:034 þ d3 ðð0:222w1 þ 0:292w2 þ 0:232w3 þ 0:226w4 þ 0:300w5 þ 0:244w6 Þ  0:248Þ=0:025 þ d4 ðð0:229w1 þ 0:243w2 þ 0:194w3 þ 0:188w4 þ 0:327w5 þ 0:171w6 Þ  0:208Þ=0:029 þ d5 ðð0:319w1 þ 0:332w2 þ 0:297w3 þ 0:276w4 þ 0:432w5 þ 0:344w6 Þ  0:310Þ=0:025 þ d6 ðð0:282w1 þ 0:421w2 þ 0:383w3 þ 0:270w4 þ 0:432w5 þ 0:337w6 Þ  0:323Þ=0:043 þ d7 ðð0:302w1 þ 0:393w2 þ 0:321w3 þ 0:297w4 þ 0:409w5 þ 0:355w6 Þ  0:334Þ=0:025 þ d8 ðð0:290w1 þ 0:435w2 þ 0:294w3 þ 0:244w4 þ 0:407w5 þ 0:328w6 Þ  0:307Þ=0:029 þ d9 ðð2:282w1 þ 2:487w2 þ 2:184w3 þ 2:076w4 þ 2:807w5 þ 2:497w6 Þ  2:283Þ=0:117 þ d10 ðð2:128w1 þ 2:974w2 þ 2:665w3 þ 2:092w4 þ 2:820w5 þ 2:383w6 Þ  2:369Þ=0:217 þ d11 ðð2:241w1 þ 2:666w2 þ 2:316w3 þ 2:179w4 þ 2:768w5 þ 2:523w6 Þ  2:380Þ=0:110 þ d12 ðð2:171w1 þ 3:241w2 þ 2:221w3 þ 1:926w4 þ 2:674w5 þ 2:427w6 Þ  2:319Þ=0:136 þ d13 ðð1:867w1 þ 1:687w2 þ 1:745w3 þ 1:763w4 þ 2:138w5 þ 1:665w6 Þ  1:756Þ=0:108 þ d14 ðð1:810w1 þ 1:550w2 þ 1:754w3 þ 1:653w4 þ 2:120w5 þ 1:889w6 Þ  1:709Þ=0:104 þ d15 ðð1:781w1 þ 1:960w2 þ 1:795w3 þ 1:783w4 þ 1:968w5 þ 1:819w6 Þ  1:844Þ=0:070 þ d16 ðð1:824w1 þ 1:734w2 þ 1:677w3 þ 1:660w4 þ 2:083w5 þ 1:557w6 Þ  1:698Þ=0:101 s:t: w 2 D

ð5:44Þ Step 7: By using Lingo 11.0 to solve Eq. (5.44), the weight vector of attributes is obtained as w ¼ ð0:10; 0:200; 0:12; 0:29; 0:24; 0:05Þ for d ¼ ð161 ; 161 ;


; 161 Þ. Step 8: Use Eq. (5.3) to calculate the comprehensive evaluation value of each provider Sk ðk 2 PÞ, which is shown in Table 5.6. Step 9: Use Eq. (5.4) to obtain the score of the comprehensive evaluation value of each provider, which is also shown in Table 5.6. Step 10: According to Definition 5.5, the ranking order of the providers is S3 S2 S1 S4 and the best provider is S3 .

5.6.1.2

Comparison with Extended TOPSIS Method

Hwang and Yoon [34] firstly proposed the TOPSIS method on the ground of the fact that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). Lots of work has been done about heterogeneous MAGDM problems with extended

Table 5.6 Comprehensive evaluation values, scores and ranking of the four providers Provider

Comprehensive assessment

Score

Ranking

S1 S2 S3 S4

([0.219,0.285],[0.092,0.159]) ([0.214,0.300],[0.080,0.168]) ([0.228,0.295],[0.086,0.155]) ([0.210,0.285],[0.086,0.167])

0.127 0.133 0.141 0.121

3 2 1 4

5.6 Illustrative Examples

169

TOPSIS methods [24, 42, 43]. For well comparison, we extend TOPSIS method to manage appropriately the IT outsourcing service provider evaluation example, and the steps can be summarized as follows. Step 1 and Step 2: Refer to Step 1 and Step 2 in Sect. 5.4.2. Step 3: Form the benefit decision matrices Xk ¼ ð~x0kij Þmn , where 8 0k xij > > > < ½x0k ; x0k  ij ij ~x0kij ¼ ða0kij ; b0kij ; c0kij Þ > > > : ða0k ; b0k ; c0k ; d 0k Þ ij ij ij ij

if if if if

j 2 N1 j 2 N2 j 2 N3 j 2 N4

ð5:45Þ

Step 4: Determine the ideal decision of group decision of alternatives. Denote the positive ideal decision of group decision of alternatives by X þ ¼ ð~x0ijþ Þmn , where

~x0ijþ

8 max xkij > > > k2P  > > > > k k > max xij ; max xij > < k2P k2P   ¼ k k k > max a ; max b ; max c > ij ij ij > k2P k2P > >  k2P  > > > > : max akij ; max bkij ; max ckij ; max dijk k2P

k2P

k2P

k2P

if j 2 N1 if j 2 N2 if j 2 N3

ð5:46Þ

if j 2 N4

and the negative ideal decision of group decision of alternatives by X  ¼ ð~x0 ij Þmn , where

~x0 ij

8 min xkij > > > k2P  > > > > k k > xij ; min xij > < min k2P k2P  ¼ k k k > min aij ; min bij ; min cij > > k2P k2P k2P > >   > > > > : min akij ; min bkij ; min ckij ; min dijk k2P

k2P

k2P

k2P

if j 2 N1 if j 2 N2 if j 2 N3

ð5:47Þ

if j 2 N4

Step 5: Calculate Dkþ and D k , i.e., the separations between the group decision of each alternative and X þ as well as X , respectively. They can be given by the following equations: Dkþ ¼

n X m X j¼1 i¼1

wj disð~x0kij ; ~x0ijþ Þ;

ð5:48Þ

5 Aggregating Decision Information into Interval-Valued …

170

D k ¼

n X m X

wj disð~x0kij ; ~x0 ij Þ;

ð5:49Þ

j¼1 i¼1

respectively, where wj denotes the weight of attribute Aj . Step 6: Calculate the relative closeness (RC) of the alternative Sk by Eq. (5.50) as follows RCk ¼

D k : þ D k

ð5:50Þ

Dkþ

Step 7: Rank the alternatives according to the RCk of the alternative Sk . In the sequel, the extended method mentioned above is applied to the same example in Sect. 5.6.1. The ideal decisions of group decision of alternatives are calculated by Step 4 as shown in Table 5.7. By Step 5 and Step 6, taking the attribute weight vector as w ¼ ð0:10; 0:200; 0:12; 0:29; 0:24; 0:05Þ obtained by Step 7 in Sect. 5.6.1.1, the separations and RC of alternatives are calculated as shown in Table 5.8. By Step 7, the rankings of alternatives are obtained as shown in Table 5.8. From Table 5.8, the ranking of the providers is S3 S2 S1 S4 , which seems to be in line with the ranking obtained by the proposed method in Table 5.6 in Sect. 5.6.1.1. Therefore, the proposed method here is reliable. Based on the comparison analysis between the extended TOPSIS method and the proposed method mentioned above, it is not hard to see that the proposed method of this chapter has the following desirable advantages: (1) The latter utilizes the multi-objective IF programming model to determine the attribute weights, which is more objective and reasonable, whereas the former needs the DM to provide the attribute weights in advance, which is inevitably subjective. Table 5.7 Ideal decisions of group decision of alternatives Provider Xþ

X

DM1 DM2 DM3 DM4 DM5 DM1 DM2 DM3 DM4 DM5

A1

A2

A3

A4

A5

A6

(6.6,7.6,8.9) (6.5,7.5,8.8) (5.2,6.8,8.5) (5.3,6.8,8.5) (4.7,5.6,7.7) (4.3,5.4,7.5) (5.4,5.7,7.9) (4.3,5.5,6.7) (4.4,5.7,6.8) (4.1,5.2,6.5)

(8.3,10,10) (6.7,8.3,10) (6.7,8.3,10) (5.0,6.7,8.3) (6.7,8.3,10) (3.3,5.0,6.7) (5.0,6.7,8.3) (5.0,6.7,8.3) (3.3,5.0,6.7) (1.7,3.3,5.0)

(5.1,7.1,8.3) (5.9,6.5,7.3) (6.7,8.1,10) (5.7,7.3,10) (6.9,8.1,10) (3.3,4.7,6.7) (3.3,4.5,6.7) (4.6,5.5,6.7) (3.3,5.0,6.7) (3.8,5.0,6.7)

(4.5,5.3,6.5,7.9) (5.3,6.4,7.7,8.8) (4.2,6.1,6.4,9.0) (5.5,6.3,7.5,8.9) (5.3,6.8,7.4,8.6) (4.2,4.3,5.4,6.7) (3.2,4.5,6.4,6.7) (3.0,4.0,5.0,6.5) (2.4,4.5,4.7,5.8) (3.0,4.1,4.2,5.5)

[6.3,9.3] [7.6,9.5] [8.3,10] [7.6,10] [7.3,8.6] [5.4,8.0] [6.5,8.6] [6.7,8.9] [6.3,8.4] [5.4,7.7]

7.1 8.1 7.2 8.3 7.0 4.4 6.1 3.9 5.8 5.9

5.6 Illustrative Examples Table 5.8 D+, D−, RCC and ranking order of each provider

171 Provider

D+

D−

RCC

Ranking

S1 S2 S3 S4

5.068 4.905 3.674 5.691

5.230 5.392 6.323 4.606

0.508 0.524 0.632 0.447

3 2 1 4

(2) The collective overall ratings of alternatives in the latter are presented by IVIFNs, which is well-suited to express the hesitancy degree inherent in DMs’ judgments, while the collective overall rating in the former is a relative closeness which is a crisp value and cannot reflect the hesitancy degree inherent in DMs’ judgments.

5.6.2

A Supplier Selection Example

This section will apply another example (adapted from [22]) involving the assessments for three potential suppliers to illustrate the proposed method. A manufacturing company intends to select an appropriate supplier to increase its customer base. After detailed survey, data eliciting and statistical treatment, we get three potential suppliers S1, S2 and S3 as alternatives for further evaluation, with the specifications listed in Table 5.9. This is a GMADM problem with real

Table 5.9 Three evaluation matrices provided by nine DMs Supplier

DM

S1

The production manager (D1) The quality manager (D2) The material manager (D3) The maintenance manager (D4) The development manager (D5) The planning manager (D6) The purchasing manager (D7) The finance manager (D8) A group leader (D9)

Delivery capability (A2)

Cost reduction performance (A3)

Post-sales service (A4)

97

27

55

89

98

28

63

98

98

28

68

90

99

29

45

91

99

29

62

83

100

30

60

70

87

30

56

86

90

40

63

88

92

45

57

79

Product quality (A1)

(continued)

5 Aggregating Decision Information into Interval-Valued …

172 Table 5.9 (continued) Supplier

DM

S2

The production manager (D1) The quality manager (D2) The material manager (D3) The maintenance manager (D4) The development manager (D5) The planning manager (D6) The purchasing manager (D7) The finance manager (D8) A group leader (D9) The production manager (D1) The quality manager (D2) The material manager (D3) The maintenance manager (D4) The development manager (D5) The planning manager (D6) The purchasing manager (D7) The finance manager (D8) A group leader (D9)

S3

Delivery capability (A2)

Cost reduction performance (A3)

Post-sales service (A4)

77

36

52

90

93

43

55

76

79

33

51

83

78

60

69

81

76

32

41

71

77

34

53

79

78

35

55

77

80

38

52

78

78

51

53

80

85

90

80

33

89

89

81

41

62

89

85

52

83

90

88

44

87

72

87

49

73

89

97

32

89

88

77

55

77

91

89

46

90

93

90

53

Product quality (A1)

numbers. All attributes are benefit criteria. To address this issue, we utilize the proposed method to the selection of the supplier below. First, owing to the assessments given with the hundred-mark system, it is easy to ¼ 0 and Amax ¼ 100. According to Step 3 in Sect. 5.4.2, the Qsd, Qdd see that Amin j j and Qud of each attribute value can be calculated and their results are summed up in Table 5.10. Then, by Step 4 in Sect. 5.4.2, the Qsi ~ nkj , Qdi ~fkj and Qui ~ gkj of Akj are presented in Table 5.11. By Step 5 in Sect. 5.4.2, the induced IVIFNs of Akj are

5.6 Illustrative Examples

173

Table 5.10 Qsd, Qdd and Qud of each attribute value Supplier

DM

A1

A2

A3

A4

S1

D1 D2 D3 D4 D5 D6 D7 D8 D9 D1 D2 D3 D4 D5 D6 D7 D8 D9 D1 D2 D3 D4 D5 D6 D7 D8 D9

(0.97,0.03,0.06) (0.98,0.02,0.04) (0.98,0.02,0.04) (0.99,0.01,0.02) (0.99,0.01,0.02) (1.00,0.00,0.00) (0.87,0.13,0.26) (0.90,0.10,0.20) (0.92,0.08,0.16) (0.77,0.23,0.46) (0.93,0.07,0.14) (0.79,0.21,0.42) (0.78,0.22,0.44) (0.76,0.24,0.48) (0.770.23,0.46) (0.78,0.22,0.44) (0.80,0.20,0.40) (0.78,0.22,0.44) (0.85,0.15,0.30) (0.89,0.11,0.22) (0.62,0.38,0.76) (0.83,0.17,0.34) (0.87,0.13,0.26) (0.73,0.27,0.54) (0.89,0.11,0.22) (0.77,0.23,0.46) (0.90,0.10,0.20)

(0.27,0.73,0.54) (0.28,0.72,0.56) (0.28,0.72,0.56) (0.29,0.71,0.58) (0.29,0.71,0.58) (0.30,0.70,0.60) (0.30,0.70,0.60) (0.40,0.60,0.80) (0.45,0.55,0.90) (0.36,0.64,0.72) (0.43,0.57,0.86) (0.33,0.67,0.66) (0.60,0.40,0.80) (0.32,0.68,0.64) (0.34,0.66,0.68) (0.35,0.65,0.70) (0.38,0.62,0.76) (0.51,0.49,0.98) (0.90,0.10,0.20) (0.89,0.11,0.22) (0.89,0.11,0.22) (0.90,0.10,0.20) (0.72,0.28,0.56) (0.89,0.11,0.22) (0.88,0.12,0.24) (0.91,0.09,0.18) (0.93,0.07,0.14)

(0.55,0.45,0.90) (0.63,0.37,0.74) (0.68,0.32,0.64) (0.45,0.55,0.90) (0.62,0.38,0.76) (0.60,0.40,0.80) (0.56,0.44,0.88) (0.63,0.37,0.74) (0.57,0.43,0.86) (0.52,0.48,0.96) (0.55,0.45,0.90) (0.51,0.49,0.98) (0.69,0.31,0.62) (0.41,0.59,0.82) (0.53,0.47,0.94) (0.55,0.45,0.90) (0.52,0.48,0.96) (0.53,0.47,0.94) (0.80,0.20,0.40) (0.81,0.19,0.38) (0.85,0.15,0.30) (0.88,0.12,0.24) (0.87,0.13,0.26) (0.97,0.03,0.06) (0.77,0.23,0.46) (0.89,0.11,0.22) (0.90,0.1,0.20)

(0.89,0.11,0.22) (0.98,0.02,0.04) (0.90,0.10,0.20) (0.91,0.09,0.18) (0.83,0.17,0.34) (0.70,0.30,0.60) (0.86,0.14,0.28) (0.88,0.12,0.24) (0.79,0.21,0.42) (0.90,0.10,0.20) (0.76,0.24,0.48) (0.83,0.17,0.34) (0.81,0.19,0.38) (0.71,0.29,0.58) (0.79,0.21,0.42) (0.77,0.23,0.46) (0.78,0.22,0.44) (0.80,0.20,0.40) (0.33,0.67,0.66) (0.41,0.59,0.82) (0.52,0.48,0.96) (0.44,0.56,0.88) (0.49,0.51,0.98) (0.32,0.68,0.64) (0.55,0.45,0.90) (0.46,0.54,0.92) (0.53,0.47,0.94)

S2

S3

also presented in Table 5.11. In method [22] the weight vector w ¼ ð0:3; 0:2; 0:2; 0:3Þ of attributes is given in advance. By Step 8 in Sect. 5.4.2, the comprehensive evaluation value of each supplier Sk is obtained as shown in Table 5.12. By Step 9 and 10 in Sect. 5.4.2, the scores and suppliers’ ranking are also shown in Table 5.12. The ranking order obtained by the proposed in this chapter is S1 S3 S2 , which is in accordance with that obtained by method [22]. Thus, the proposed method is reliable.

5 Aggregating Decision Information into Interval-Valued …

174

Table 5.11 Qsi ~ nkj , Qdi ~fkj , Qui ~gkj and aggregated IVIFNs of attribute vectors Supplier

Attribute

~ nkj

~fkj

~ gkj

IVIFN

S1

A1 A2 A3 A4 A1 A2 A3 A4 A1 A2 A3 A4

[0.909,1.002] [0.255,0.381] [0.522,0.654] [0.780,0.940] [0.744,0.847] [0.307,0.497] [0.463,0.606] [0.742,0.847] [0.723,0.910] [0.818,0.940] [0.800,0.921] [0.366,0.534]

[0.000,0.091] [0.619,0.745] [0.346,0.478] [0.060,0.220] [0.153,0.256] [0.503,0.693] [0.394,0.537] [0.153,0.258] [0.090,0.277] [0.060,0.182] [0.079,0.201] [0.466,0.634]

[0.000,0.182] [0.510,0.761] [0.713,0.892] [0.120,0.440] [0.305,0.512] [0.646,0.865] [0.779,1.003] [0.307,0.516] [0.179,0.554] [0.120,0.365] [0.159,0.401] [0.730,0.981]

([0.434,0.479],[0.000,0.044]) ([0.097,0.144],[0.235,0.283]) ([0.186,0.233],[0.124,0.171]) ([0.342,0.412],[0.026,0.096]) ([0.309,0.352],[0.063,0.106]) ([0.111,0.180],[0.182,0.251]) ([0.160,0.210],[0.136,0.186]) ([0.308,0.351],[0.064,0.107]) ([0.305,0.385],[0.038,0.117]) ([0.365,0.419],[0.027,0.081]) ([0.351,0.404],[0.035,0.088]) ([0.128,0.187],[0.163,0.222])

S2

S3

Table 5.12 Comprehensive evaluation values, scores and ranking of the three suppliers Supplier

Comprehensive assessment

Score

Ranking

S1 S2 S3

([0.301,0.355],[0.000,0.106]) ([0.244,0.293],[0.091,0.141]) ([0.279,0.343],[0.054,0.125])

0.275 0.152 0.222

1 3 2

5.7

Conclusions

This chapter puts forward a new method for aggregating the attribute information into IVIFNs and applies it to solve heterogeneous MAGDM problem. The primary contributions of the proposed method are summarized as follows: (1) A new general method is developed to aggregate heterogeneous information into IVIFNs. This method can aggregate different types of information (including real numbers, interval numbers, TFNs and TrFNs) into IVIFNs. (2) The attribute weights are determined objectively by constructing a multiple objective IF programming which is transformed into a linear programming model. (3) Combining the proposed aggregating heterogeneous information method with the obtained attribute weights, a new method is presented to solve heterogeneous MAGDM problems. Although the proposed method is developed for solving heterogeneous MAGDM problems, it is also appropriate for the complex multi-attribute large-group decision-making problems. Future research will extend the developed method to heterogeneous MAGDM including IFNs [44–46], hesitant fuzzy elements [47] and hesitant fuzzy linguistic terms [48, 49].

References

175

References 1. L.A. Zadeh, Fuzzy sets. Inf. Control 8, 338–356 (1965) 2. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 3. K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989) 4. Z.S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis. 22, 215–219 (2007) 5. E.K. Zavadskas, J. Antucheviciene, S.H.R. Hajiagha, S.S. Hashemi, Extension of weighted aggregated sum product assessment with interval-valued intuitionistic fuzzy numbers. Appl. Soft Comput. 24, 1013–1021 (2014) 6. T.Y. Chen, A prioritized aggregation operator-based approach to multiple criteria decision making using interval-valued intuitionistic fuzzy sets: A comparative perspective. Inf. Sci. 281, 97–112 (2014) 7. K. Guo, Q. Song, on the entropy for Atanassov’s intuitionistic fuzzy sets: An interpretation from the perspective of amount of knowledge. Appl. Soft Comput. 24, 328–340 (2014) 8. F. Jin, L. Pei, H. Chen, L. Zhou, Interval-valued intuitionistic fuzzy continuous weighted entropy and its application to multi-criteria fuzzy group decision making. Knowl. Based Syst. 59, 132–141 (2014) 9. J. Wu, F. Chiclana, A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions. Appl. Soft Comput. 22, 272–286 (2014) 10. J. Xu, F. Shen, A new outranking choice method for group decision making under Atanassov’s interval-valued intuitionistic fuzzy environment. Knowl. Based Syst. 70, 177– 188 (2014) 11. S.P. Wan, J.Y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J. Comput. Syst. Sci. 80(1), 237–256 (2014) 12. Z. Wang, K.W. Li, J. Xu, A mathematical programming approach to multi-attribute decision making with interval-valued intuitionistic fuzzy assessment information. Expert Syst. Appl. 38(10), 12462–12469 (2011) 13. Z.J. Wang, K.W. Li, An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives. Expert Syst. Appl. 39(18), 13509–13516 (2012) 14. S.P. Wan, G.L. Xu, F. Wang, J.Y. Dong, A new method for Atanassov’s interval-valued intuitionistic fuzzy MAGDM with incomplete attribute weight information. Inf. Sci. 316, 329–347 (2015) 15. Z. Yue, Y. Jia, A group decision making model with hybrid intuitionistic fuzzy information. Comput. Ind. Eng. 87, 202–212 (2015) 16. Z.L. Yue, The comprehensive evaluation of urban environmental quality based on intuitionistic fuzzy set. Math. Practice Theory 38(8), 51–58 (2008) 17. Z. L. Yue, Y. Y. Jia, X. L. Huang, Multiple attribute decision making method based on intuitionistic fuzzy set, in Proceedings of the Second International Conference on Intelligent Information Management Systems and Technology (2007), pp 137–140 18. Z.L. Yue, Y.Y. Jia, G.D. Ye, An approach for multiple attribute group decision making based on intuitionistic fuzzy information. Int. J. Uncertainty, Fuzzi Knowl. Based Syst. 17(3), 317– 332 (2009) 19. Z.L. Yue, Aggregating crisp values into intuitionistic fuzzy number for group decision making. Appl. Math. Model. 38(2), 2969–2982 (2014) 20. J. Xu, S.P. Wan, J.Y. Dong, Aggregating decision information into Atanassov’s intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making. Appl. Soft Comput. 41, 331–351 (2016) 21. Z.L. Yue, An approach to aggregating interval numbers into interval-valued intuitionistic fuzzy information for group decision making. Expert Syst. Appl. 38(5), 6333–6338 (2011)

176

5 Aggregating Decision Information into Interval-Valued …

22. Z.L. Yue, Y.Y. Jia, A method to aggregate crisp values into interval-valued intuitionistic fuzzy information for group decision making. Appl. Soft Comput. 13(5), 2304–2317 (2013) 23. Z.L. Yue, A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information. Appl. Math. Model. 38(2), 683–698 (2014) 24. D.F. Li, Z.G. Huang, G.H. Chen, A systematic approach to heterogeneous multi-attribute group decision making. Comput. Ind. Eng. 59(4), 561–572 (2010) 25. X. Chen, H. Zhang, Y. Dong, The fusion process with heterogeneous preference structures in group decision making: a survey. Inf. Fusion 24, 72–83 (2015) 26. S.P. Wan, D.F. Li, Fuzzy LINMAP approach to heterogeneous MADM considering the comparisons of alternatives with hesitation degrees. Omega Int. J. Manage Sci. 41(6), 925– 940 (2013) 27. J.Y. Dong, S.P. Wan, Virtual enterprise partner selection integrating LINMAP and TOPSIS. J. Oper. Res. 67(10), 1288–1308 (2016) 28. D.F. Li, S.P. Wan, Fuzzy heterogeneous multi-attribute decision making method for outsourcing provider selection. Expert Syst. Appl. 41, 3047–3059 (2014) 29. S.P. Wan, D.F. Li, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multi-attribute group decision making with Atanassov’s intuitionistic fuzzy truth degrees. IEEE Trans. Fuzzy Syst. 22, 300–312 (2014) 30. S.P. Wan, D.F. Li, Fuzzy mathematical programming approach to heterogeneous multi-attribute decision-making with interval-valued intuitionistic fuzzy truth degrees. Inf. Sci. 325, 484–503 (2015) 31. S.P. Wan, J.Y. Dong, Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees. Inf. Fusion 26, 49–65 (2015) 32. X.L. Zhang, Z.S. Xu, H. Wang, Heterogeneous multiple criteria group decision making with incomplete weight information: a deviation modeling approach. Inf. Fusion 25, 49–62 (2015) 33. V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference. Psychometrica 38, 337–342 (1973) 34. C.L. Hwang, K. Yoon, Multiple attributes decision making methods and applications (Springer, Berlin Heidelberg, 1981) 35. Z.L. Yue, TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting. Inf. Sci. 277, 141–153 (2014) 36. C.T. Chen, Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst. 114(1), 1–9 (2000) 37. P.J.M. Van Laarhoven, W. Pedrycz, A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 11(1), 199–227 (1983) 38. J. Lu, G. Zhang, D. Ruan, F. Wu, Multi-objective Group Decision Making: Methods, Software and Applications with Fuzzy Set Techniques (Imperial College Press, London, 2007) 39. H.J. Zimmermann, Fuzzy set theory-and its applications, 2nd edn. (Kluwer Academic Publishers, London, 1991) 40. H. Bustince, M. Galar, B. Bedregal, A. Kolesárová, R. Mesiar, A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy sets applications. IEEE Trans. Fuzzy Syst. 21, 1150–1162 (2013) 41. X.L. Zhang, Z.S. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making. Appl. Soft Comput. 26, 42–56 (2015) 42. R.A. Krohling, V.C. Campanharo, Fuzzy TOPSIS for group decision making: a case study for accidents with oil spill in the sea. Expert Syst. Appl. 38(4), 4190–4197 (2011) 43. X.Z. Sang, X.W. Liu, J.D. Qin, An analytical solution to fuzzy TOPSIS and its application in personnel selection for knowledge-intensive enterprise. Appl. Soft Comput. 30, 190–204 (2015) 44. S.P. Wan, Q.Y. Wang, J.Y. Dong, The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers. Knowl. Based Syst. 52, 65–77 (2013)

References

177

45. S.P. Wan, J.Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl. Soft Comput. 29, 153–168 (2015) 46. S.P. Wan, F. Wang, L.L. Liu, J.Y. Dong, Some new generalized aggregation operators for triangular intuitionistic fuzzy numbers and application to multi-attribute group decision making. Comput. Ind. Eng. 93, 286–301 (2016) 47. B. Farhadinia, Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci. 240, 129–144 (2013) 48. J.M. Merigó, D. Palacios-Marqués, S.Z. Zeng, Subjective and objective information in linguistic multi-criteria group decision making. Eur. J. Oper. Res. 248(2), 522–531 (2015) 49. H. Liao, Z. Xu, X.J. Zeng, J.M. Merigó, Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets. Knowl. Based Syst. 76, 127–138 (2015)

Chapter 6

A Novel Method for Group Decision Making with Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relations

Abstract This chapter investigates the group decision making (GDM) problems with interval-valued Atanassov intuitionistic fuzzy preference relations (IV-AIFPRs) and develops a novel method for solving such problems. A new consistency index of an Atanassov intuitionistic fuzzy preference relation (AIFPR) is introduced to judge the consistency of an AIFPR and then a convergent iterative Algorithm I is designed to repair the consistency of an AIFPR with unacceptable consistency. Subsequently, the consistency and acceptable consistency of an IV-AIFPR are defined through separating an IV-AIFPR into two AIFPRs. Based on Algorithm I, a new iterative Algorithm II is devised to repair the consistency of an IV-AIFPR with unacceptable consistency. Afterwards, to determine decision makers’ (DMs’) weights objectively, an optimization model is established by minimizing the deviations between each individual IV-AIFPR and the collective one. This model is skillfully transformed into a linear goal programming model to resolve sufficiently considering different principles of decision making. A linear programming model is built to derive interval-valued Atanassov intuitionistic fuzzy (IVAIF) priority weights of alternatives. Then, a TOPSIS (technique for order performance by similarity to an ideal solution) based approach is proposed to rank such IVAIF priority weights. Thereby, a method for GDM with IV-AIFPRs is put forward. At length, a practical example of a virtual enterprise partner selection is provided to illustrate the feasibility and validity of the proposed method.




Keywords Group decision making Interval-valued Atanassov intuitionistic fuzzy preference relation Multiplicative consistency Goal programming model




6.1




Introduction

Due to the high complexity of socioeconomic environments, it is not easy for a single decision maker (DM) to consider all important aspects in practical decision making problems. Therefore, group decision making (GDM) has been widely applied in many real-life decision making problems over the last few years. © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_6

179

180

6 A Novel Method for Group Decision Making …

In GDM, different DMs may employ distinct information granules to describe their preferences over alternatives. Pedrycz [27] pointed that information granules can be expressed by diverse preference relations. Two basic preference relations are multiplicative preference relations [31] and fuzzy preference relations [35]. In these two preference relations, DMs’ judgments are represented by exact numerical values. Later, other forms of preference relations appeared, such as interval preference relations [50], linguistic preference relations [5, 10] and linguistic interval preference relations [36]. A common characteristic of these preference relations is that they only describe the membership degree to which one alternative is preferred to the other. As a result, DMs’ hesitancy or uncertainty, which usually exists in decision making, is often overlooked. To solve this issue, Atanassov [1] introduced the Atanassov intuitionistic fuzzy set (AIFS) using the membership, non-membership and hesitant degrees to express DMs’ preferences. In a famous monograph [28], Pedrycz and Chen claimed that the idea of AIFS has contributed massively to decision making methods where AIFSs evaluate the membership degree (positive), non-membership degree (negative) and intuitionistic index (hesitation margin). Based on the AIFS, Szmidt and Kacprzyk [34] proposed Atanassov intuitionistic fuzzy preference relation (AIFPR). Subsequently, Xu and Chen [42] extended the AIFPR to the interval-valued AIFPR (IV-AIFPR) in which DMs’ preferences are expressed as interval-valued Atanassov intuitionistic fuzzy values (IVAIFVs) [4]. Since IVAIFVs utilize intervals to describe the membership and non-membership degrees (see [4]), IV-AIFPRs are more flexible and comprehensive to represent the imprecise or uncertain preferences provided by DMs. In GDM with preference relations, the lack of consistency is inclined to result in misleading conclusions. Therefore, it is critical to study the consistency of preference relations, including how to define, check and repair the consistency. There are fruitful achievements available on the consistency of multiplicative preference relations, fuzzy preference relations and interval preference relations [31, 35, 50]. To check the consistency of linguistic preference relations, Cabrerizo et al. [5] introduced a consistency index by measuring the errors between the preferences by DMs and their corresponding estimated values. Dong et al. [10] investigated the consistency of linguistic preference relations by transforming linguistic preference relations into interval preference relations. As for AIFPRs, Xu et al. [43] proposed a multiplicative consistency definition. According to this definition, Xu and Liao [45] introduced a consistency index and developed an approach to repairing the consistency of an AIFPR. Nevertheless, Liao and Xu [13] pointed out the shortcomings of the multiplicative consistency definition of Xu et al. [43] and thus gave a new multiplicative consistency definition of an AIFPR. By this new consistency definition, Xu et al. [48] constructed a mathematical model to repair the consistency of AIFPRs. Liao et al. [16] put forward an iterative algorithm to repair the consistency of an AIFPR and built a consensus model to improve the consensus among experts. Recently, Liao et al. [18] improved the consensus model constructed in [16] and set up an enhanced reaching model. For IV-AIFPRs, the research on the consistency is very little. Up to now, only Liao et al. [15] presented a multiplicative consistency

6.1 Introduction

181

definition of an IV-AIFPR by extending the multiplicative consistency of fuzzy preference relations. However, when an IV-AIFPR is reduced to an AIFPR, this definition reduces to the consistency defined by Xu et al. [43] which is unreasonable. In this sense, the multiplicative consistency defined by Liao et al. [15] may be also unreasonable. The next important issue in GDM is to fuse the individual preference relations into a collective one. In fusing process, DMs’ weights play an important role whether in homogenous or heterogeneous decision environments [26, 29, 40]. Hence, DMs’ weights should be determined in a reasonable way. For GDM with AIFPRs, Ureña et al. [37] identified DMs’ weights depending on the confidence and consistency degrees of individual AIFPRs; Considering DMs’ risk attitudes, Wan et al. [39] built an Atanassov intuitionistic fuzzy (AIF) program and proposed three different solving methods to obtain DMs’ weights. For the GDM under IV-AIFPR environment, there is no investigation on the determination of DMs’ weights objectively. Existing methods [15, 44, 46] assumed that each DM has equal importance or assigned DMs’ weights in advance, which may lead to the subjective randomness. In GDM, the last and most important issue is how to reasonably determine priority weights of alternatives from the collective preference relation. Generally, it is more sensible that the form of priority weights should be consistent with that of elements in the preference relations. For instance, the priority weights of interval preference relations should be represented by intervals. In this regard, many research results have been achieved. As the components of AIFPRs are Atanassov intuitionistic fuzzy values (AIFVs), the priority weights of AIFPRs should be also expressed by AIFVs. Following this principle, Liao and Xu [13] proposed a fractional programming method for generating the AIF priority weights from AIFPRs. Subsequently, Liao and Xu [14] further developed two different algorithms to get the AIF priority weights of AIFPRs. Based on the aforesaid analyses, it is more reasonable and rational that the priority weights of IV-AIFPRs should be IVAIFVs. Methods [44, 46] directly obtained interval-valued Atanassov intuitionistic fuzzy (IVAIF) priority weights with the geometric/arithmetic average operator, but did not check the consistency of individual IV-AIFPRs or the collective IV-AIFPR. Therefore, how high the reliabilities of the obtained priority weights are unknown. Recently, Liao et al. [15] improved method [44] and constructed individual consistent IV-AIFPRs from their corresponding individual IV-AIFPRs. Thus, the aggregated collective IV-AIFPR is also consistent. However, the constructed IV-AIFPRs may have large deviations from their corresponding initial IV-AIFPRs. Hence, the collective IV-AIFPR may be unable to represent the opinions of the group and the obtained priority weights may be unreasonable. Literature review reveals that the research on the GDM with IV-AIFPRs is still at the starting stage. Although some achievements have been gained, there are some limitations mentioned above. To overcome these limitations, this chapter develops a novel method for solving the GDM with IV-AIFPRs. First, by analyzing the

182

6 A Novel Method for Group Decision Making …

relations between the direct information and indirect information implied in an AIFPR, a consistency index of an AIFPR is introduced to measure the consistency degree of an AIFPR. For an AIFPR with unacceptable consistency, a convergent iterative Algorithm I is designed to repair its consistency. Then, by separating an IV-AIFPR into two AIFPRs and employing the introduced consistency index of an AIFPR, a new multiplicative consistency and an acceptable multiplicative consistency of an IV-AIFPR are respectively defined. To repair the consistency of an IV-AIFPR with unacceptable consistency, a new iterative Algorithm II is devised based on Algorithm I. Afterwards, to determine DMs’ weights, an optimization model is established by minimizing the deviations between each individual IV-AIFPR and the collective one. It is skillfully transformed into a linear goal programming model to resolve considering different decision making principles, including minority, majority and compromise principles. A linear programming model is built to derive the IVAIF priority weights of alternatives from the collective IV-AIFPR. At last, a TOPSIS (technique for order performance by similarity to an ideal solution) based approach is presented to rank alternatives according to these IVAIF priority weights. The major contributions of this chapter are outlined as follows: (1) A new multiplicative consistency and an acceptable multiplicative consistency of an IV-AIFPR are respectively defined. A common notable feature of them is that they can exactly reduce to corresponding consistency of an AIFPR. Therefore, the defined multiplicative consistency and acceptable multiplicative consistency of an IV-AIFPR are reasonable. (2) For an IV-AIFPR with unacceptable consistency, a new algorithm is developed to repair its consistency. When this IV-AIFPR is reduced to an AIFPR, the algorithm is proved to be convergent. (3) A linear goal programming model is set up to determine DMs’ weights objectively by considering different decision making principles, which may avoid the subjective randomness appearing in [15, 44, 46] and make the decision results more flexible. (4) A linear programming model is established to derive the priority weights of an IV-AIFPR. It is worth mentioning that the derived priority weights are IVAIFVs whose form is in accordance with that of an IV-AIFPR. Hence, the derived priority weights are more reasonable. This chapter is arranged into six sections. Section 6.2 introduces a new consistency index to measure the consistency degree of an AIFPR and designs a convergent iterative Algorithm I to repair the consistency of an AIFPR. Section 6.3 defines a new multiplicative consistency and an acceptable consistency of an IV-AIFPR. A new iterative Algorithm II is devised to repair the consistency of an IV-AIFPR with unacceptable consistency. Section 6.4 proposes a new method for solving GDM with IV-AIFPRs. In Sect. 6.5, a practical example of a virtual enterprise partner selection is provided to illustrate the effectiveness of the proposed method. Section 6.6 ends with some conclusions.

6.2 Multiplicative Consistency of Atanassov Intuitionistic …

6.2

183

Multiplicative Consistency of Atanassov Intuitionistic Fuzzy Preference Relations

As a preparation for analyzing the consistency of an IV-AIFPR in the next section, this section first addresses the multiplicative consistency of an AIFPR, including checking and repairing the consistency of an AIFPR. For solving this issue, although Xu et al. [48] and Liao et al. [16] respectively provided different consistency indices and approaches, they suffer from some shortcomings: (1) in method [48], the consistency repairing approach cannot ensure the convergence; (2) in method [16], although the consistency repairing approach is convergent, it is needed to construct the other consistent AIFPR besides the AIFPR itself, which may result in time-consuming. To make up these shortcomings, this section introduces a new multiplicative consistency index for checking the consistency of an AIFPR and designs an iterative algorithm for repairing the consistency of an AIFPR. It should be emphasized that the introduced consistency index and the designed algorithm can respectively check and repair the consistency of an AIFPR only relying on itself. Furthermore, the algorithm is proved to be convergent. In this sense, the proposed method improves methods [16, 48].

6.2.1

A New Multiplicative Consistency Index of Atanassov Intuitionistic Fuzzy Preference Relations

Definition 6.1 [13]. An AIFPR R on X is presented by a matrix R ¼ ðrij Þnn  X  X with rij ¼ ððxi ; xj Þ; lðxi ; xj Þ; vðxi ; xj ÞÞ for all i; j ¼ 1; 2; . . .; n. For convenience, let rij ¼ ðlij ; vij Þ, where lij is the degree to which xi is preferred to xj , vij is the degree to which xi is non-preferred to xj , and pij ¼ 1  lij  vij is interpreted as the hesitancy degree to which xi is preferred to xj . Furthermore, lij and vij fulfill the following conditions: lij ¼ vji ; vij ¼ lji ; lii ¼ vii ¼ 0:5; 0  lij þ vij  1 for all i; j ¼ 1; 2; . . .; n: Remark 6.1 In practical decision making, due to the complexity of decision making problems and the ambiguity of the human thinking, DMs rarely provide their preferences by (0, 1) or (1, 0). Meanwhile, since DMs are often expert in the decision problems they participate in, the completely indeterminacy seldom happens. Therefore, the preference value (0, 0) scarcely appears. Hence, in this chapter, we only discuss the AIFPRs R ¼ ððlij ; vij ÞÞnn satisfying 0\lij ; vij \1 for all i; j ¼ 1; 2; . . .; n.

6 A Novel Method for Group Decision Making …

184

Definition 6.2 [13]. An AIFPR R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ is called multiplicative consistent if the following multiplicative transitivity is satisfied: lij
ljk
lki ¼ vij
vjk
vki ; for all i; j ¼ 1; 2; . . .; n:

ð6:1Þ

Let qij ¼ vij =lij . Then qij [ 0 and qij can be interpreted as the ratio of the degree of alternative xi non-preferred to xj to that of alternative xi preferred to xj . Thus, an equivalent multiplicative consistency definition of an AIFPR can be given below. Definition 6.3 An AIFPR R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ is multiplicative consistent if and only if qij ¼ qik
qkj ; for all i; j ¼ 1; 2; . . .; n:

ð6:2Þ

Employing the ratio of the degree of one alternative non-preferred to the other to that of this alternative preferred to the other, Eq. (6.2) reflects the multiplicative transitivity of an AIFPR in some senses. On the other hand, qij can be considered as the direct information on alternative xi and xj , while qik
qkj can be interpreted as the indirect information on alternative xi and xj obtained by using the intermediate alternative xk . Thus, the direct information on any two alternatives should be equal to the indirect information obtained by using another intermediate alternative if an AIFPR R ¼ ðrij Þnn is multiplicative consistent. Therefore, the deviation between the direct information and indirect information on alternatives can be used to measure the consistency degree of an AIFPR. Bearing this idea in mind, a new multiplicative consistency index of an AIFPR is defined as MCIðRÞ ¼

1 nðn  1Þðn  2Þ

n n X X

n X   ln qij  ln qik  ln qkj :

ð6:3Þ

i¼1 j¼1;j6¼i k¼1;k6¼i;j

Apparently, consistency index MCIðRÞ ¼ 0 if and only if the AIFPR R is multiplicative consistent. The greater MCIðRÞ, the weaker the multiplicative consistency of the AIFPR R. For convenience, let   rijk ¼ ln qij  ln qik  ln qkj :

ð6:4Þ

Since lji ¼ vij and vji ¼ lij for any i; j ¼ 1; 2; . . .; n, Theorem 6.1 can be easily proved. Theorem 6.1 Given an AIFPR R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ, it holds that rijk ¼ rjik ¼ rjki ¼ rkji ¼ rkij ¼ rikj :

6.2 Multiplicative Consistency of Atanassov Intuitionistic …

185

From Theorem 6.1, it can be seen that the deviations related to subscripts i, j and k are not influenced by the orderings of their subscripts. Therefore, Eq. (6.3) can be simplified as MCIðRÞ ¼

1 nðn  1Þðn  2Þ

n n X X

n X

rijk ¼

i¼1 j¼1;j6¼i k¼1;k6¼i;j

1 Cn3

n X

rijk :

ð6:5Þ

i\k\j

Definition 6.4 Let a  0 be a predefined consistency threshold. If MCIðRÞ  a, then the AIFPR R is called acceptable multiplicative consistency. Otherwise, the AIFPR R is called unacceptable multiplicative consistency. Especially, if MCIðRÞ ¼ 0, the AIFPR R is called completely multiplicative consistency. Definition 6.5 [17]. Let Rl ¼ ðrijl Þnn with rijl ¼ ðlijl ; vijl Þðl ¼ 1; 2; . . .; KÞ be a collection of AIFPRs and k ¼ ðk1 ; k2 ; . . .; kK ÞT be a weight vector of Rl ðl ¼ P 1; 2; . . .; KÞ such that kl  0 and Kl¼1 kl ¼ 1. The combined AIFPR, denoted by RC ¼ ðrijC Þnn with rijC ¼ ðlCij ; vCij Þ, can be defined as rijC

¼

ðlCij ; vCij Þ

¼

K Y

kl

ðlijl Þ ;

l¼1

K Y

! ðvijl Þ

kl

:

ð6:6Þ

l¼1

Further, qCij

¼

vCij

.

lCij

¼

K Y l¼1

ðvijl Þ

kl

, K Y

ðlijl Þkl ¼

l¼1

K Y

ðvijl =lijl Þkl ¼

l¼1

K Y

ðqijl Þkl :

ð6:7Þ

l¼1

In [17], Liao and Xu proved that the combined AIFPR (also called the fused AIFPR or collective AIFPR) is acceptable multiplicative consistent if all the individual AIFPRs are acceptable multiplicative consistent. Inspired by [17], using the proposed consistency index (i.e., Eq. (6.5)) which is different from that given by Liao and Xu [17], we also prove this conclusion. Theorem 6.2 If Rl ðl ¼ 1; 2; . . .; KÞ are all acceptable multiplicative consistent, then the combined AIFPR RC is also acceptable multiplicative consistent. In addition, MCI(RC Þ  maxfMCI(Rl Þg: l

Proof According to Eqs. (6.4) and (6.5), we have

ð6:8Þ

6 A Novel Method for Group Decision Making …

186

MCIðRC Þ ¼

1 Cn3

¼

1 Cn3

X 1  i\k\j  n

X

1  i\k\j  n

rCijk    C  ln qij  ln qCik  ln qCkj 

    Y K K K Y Y  kl kl kl  ¼ lnð ðqijl Þ Þ  lnð ðqikl Þ Þ  lnð ðqkjl Þ Þ   l¼1 1  i\k\j  n l¼1 l¼1    K X X  1 ¼ 3 kl ½lnðqijl Þ  lnðqikl Þ  lnðqkjl Þ  Cn   1  i\k\j  l¼1 X

1 Cn3

 ¼

1 Cn3

K X

X

K X   kl lnðqijl Þ  lnðqikl Þ  lnðqkjl Þ

1  i\k\j  n l¼1

kl MCIðRl Þ  maxfMCIðRl Þg: l

l¼1

Namely, MCI(RC Þ  maxfMCI(Rl Þg. The proof of Theorem 6.2 is finished. l

From Theorem 6.2, to obtain a reasonable decision result from the combined AIFPR, it is required that individual AIFPRs provided by DMs should be acceptable multiplicative consistent. Unfortunately, due to the vagueness inherent in the human thinking, it is not easy for DMs to provide such individual AIFPRs in practical GDM with AIFPRs. Thus, how to repair the consistency of AIFPRs is very important in decision making. To circumvent this issue, an iterative algorithm is designed in the sequel.

6.2.2

An Iterative Algorithm for Repairing the Consistency of AIFPRs

Employing the direct information and indirect information of an AIFPR, this subsection designs an iterative Algorithm I for repairing the consistency of an AIFPR with unacceptable consistency. Algorithm I Input: The original AIFPR R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ, the parameter d 2 ½0; 1 which is used to tradeoff the direct information and the indirect information, the maximum number of iterations h and the threshold a  0.  ¼ ðð Output: The repaired AIFPR R lij ; vij ÞÞnn and corresponding consistency  index MCI(RÞ.

6.2 Multiplicative Consistency of Atanassov Intuitionistic … ð0Þ

187

ð0Þ

Step 1: Let Rð0Þ ¼ ððlij ; vij ÞÞnn ¼ R ¼ ððlij ; vij ÞÞnn and h ¼ 0. Compute the consistency index MCI(RðhÞ Þ by Eqs. (6.4) and (6.5). i.e., MCI(RðhÞ Þ ¼

1 Cn3

X

ðhÞ

1  i k\j  n

rijk ;

ð6:9Þ

  .  ðhÞ ðhÞ ðhÞ ðhÞ  ðhÞ ðhÞ ðhÞ where rijk ¼ ln qij  ln qjk  ln qkj  and qij ¼ vij lij . Step 2: If MCI(RðhÞ Þ  a or h [ h , then go to Step 5; Otherwise, go to the next step. ðhÞ ðhÞ Step 3: Find the maximum deviation rstf ¼ max rijk and repair its cor1  i\k\j  n

responding ratio

ðhÞ qst

as

ðh þ 1Þ qst ,

where

ðh þ 1Þ

qst

ðhÞ

ðhÞ ðhÞ

¼ ðqst Þd ðqsf qft Þ1d :

ð6:10Þ

Namely, ðh þ 1Þ

vst

.

ðh þ 1Þ

lst

ðhÞ

ðhÞ ðhÞ

¼ ðqst Þd ðqsf qft Þ1d :

ð6:11Þ

ðhÞ

In order to repair ratio qst by Eq. (6.10) and retain the initial decision inforðhÞ ðhÞ mation as much as possible, it is sensible to only repair vst or lst as follows. ðhÞ

ðh þ 1Þ

(1) Repair vst to derive upper triangular elements of AIFPR Rðh þ 1Þ ¼ ððlij ðh þ 1Þ ÞÞnn , vij

where (

ðh þ 1Þ vij

¼

ðhÞ

ðhÞ ðhÞ

ðhÞ

ðqst Þd ðqsf qft Þ1d lst ; if ði; jÞ ¼ ðs; tÞ ðhÞ

lij ;

if i\j and ði; jÞ 6¼ ðs; tÞ ðh þ 1Þ

lij

ðhÞ

¼ lij ði\jÞ:

ðhÞ

ð6:12Þ ð6:13Þ ðh þ 1Þ

(2) Repair lst to derive upper triangular elements of AIFPR Rðh þ 1Þ ¼ ððlij ðh þ 1Þ ÞÞnn , vij

;

where ðh þ 1Þ

vij ðh þ 1Þ lij

;

ðhÞ

¼ vij ði\jÞ;

8 . < vðhÞ ðqðhÞ Þd ðqðhÞ qðhÞ Þ1d ; if ði; jÞ ¼ ðs; tÞ st st sf ft ¼ : vðhÞ ; if i\j and ði; jÞ 6¼ ðs; tÞ ij

ð6:14Þ ð6:15Þ

6 A Novel Method for Group Decision Making …

188

ðh þ 1Þ

Step 4: Determine the AIFPR Rðh þ 1Þ ¼ ððlij

ðh þ 1Þ

; vij

ÞÞnn , where

ðh þ 1Þ

8 ðh þ 1Þ > ; if i\j < lij if i ¼ j ¼ 0:5; > : vðh þ 1Þ ; if i [ j ji

ð6:16Þ

ðh þ 1Þ

8 ðh þ 1Þ > ; if i\j < vij if i ¼ j ¼ 0:5; > : lðh þ 1Þ ; if i [ j ji

ð6:17Þ

lij

and

vij

Set h ¼ h þ 1 and go to Step 1.  ¼ RðhÞ . Output the repaired AIFPR R  and its consistency index Step 5: Let R  MCI(RÞ. Step 6: End. Theorem 6.3 Let R ¼ ðrij Þnn be an AIFPR, where rij ¼ ðlij ; vij Þ. The set fRðhÞ g is a sequence of matrices generated from Algorithm I. Then it follows that lim MCI(RðhÞ Þ ¼ 0:

h! þ 1

ð6:18Þ

Proof Please see Appendix 1. It is easy to see from Theorem 6.3 that Algorithm I is convergent.

6.3

Multiplicative Consistency of IV-AIFPRs

Due to the complexity of computation on IV-AIFPRs, the research on the multiplicative consistency of an IV-AIFPR is very little. So far as our knowledge, only Liao et al. [15] presented a multiplicative consistency definition by extending the multiplicative consistency of fuzzy preference relations. According to this definition, an iterative algorithm is developed to repair the consistency of an IV-AIFPR. However, some disadvantages still exist. For an example, when an IV-AIFPR is reduced to an AIFPR, this consistency definition just reduces into the consistency of an AIFPR defined in [42]. Unfortunately, Liao and Xu [13] has pointed that the consistency in [42] is unreasonable. Thus, the consistency of an IV-AIFPR defined in [15] is also unreasonable. In addition, while using the iterative algorithm in [43] to repair the consistency of an IV-AIFPR, all the elements of this IV-AIFPR have to be modified, which may result in the loss of information provided by DMs. To make up these disadvantages, a new multiplicative consistency and an acceptable multiplicative consistency of an IV-AIFPR are defined based on the consistency of

6.3 Multiplicative Consistency of IV-AIFPRs

189

an AIFPR analyzed in Sect. 6.2. When an IV-AIFPR reduces to an AIFPR, the defined consistency can degenerate to the consistency of an AIFPR defined in [13] which has been verified to be reasonable. Hence, the defined consistency of an IV-AIFPR is more reasonable. For an IV-AIFPR with unacceptable multiplicative consistency, a novel algorithm is devised to repair its consistency. While repairing the consistency, the devised algorithm modifies only one element of an IV-AIFPR each time. Thus, most decision making information can be retained. Therefore, the proposed method improves method [15].

6.3.1

Define and Check Multiplicative Consistency of IV-AIFPRs

~ on X is presented by a matrix R ~ ¼ ð~rij Þ Definition 6.6 [42]. An IV-AIFPR R nn  ~ðxi ; xj Þ; ~vðxi ; xj ÞÞ for all i; j ¼ 1; 2; . . .; n. For simplicity, X  X with ~rij ¼ ððxi ; xj Þ; l ~ij is the degree range that xi is preferred to xj , ~vij is the let ~rij ¼ ð~ lij ; ~vij Þ, where l ~ij and ~vij fulfill the foldegree range that xi is non-preferred to xj . Furthermore, l lowing conditions: þ þ ~ij ¼ ½l ~ij ¼ ~vji ; ~vij ¼ l ~ji ; l vij ¼ ½v ij ; lij   ½0; 1; ~ ij ; vij   ½0; 1; l

~ii ¼ ~vii ¼ ½0:5; 0:5 and lijþ þ vijþ  1 for all i; j ¼ 1; 2; . . .; n: l þ ~ ¼ ðð~ ~ij ¼ ½l Definition 6.7 For an IV-AIFPR R lij ; ~vij ÞÞnn with l ij ; lij  and þ ~ ~vij ¼ ½v ij ; vij , RðgÞ ¼ ðrij ðgÞÞnn ð0  g  1Þ is called the induced matrix of ~ IV-AIFPR R, where

8 ð1gÞ g > ðlijþ Þg ; ðvijþ Þð1gÞ ðv < ððl ij Þ ij Þ Þ; if i\j if i ¼ j : ð6:19Þ rij ðgÞ ¼ ðlij ðgÞ; vij ðgÞÞ ¼ ð0:5; 0:5Þ; > : ððl þ Þð1gÞ ðl Þg ; ðv Þð1gÞ ðv þ Þg Þ; if i [ j ij ij ij ij ~ It can be easily proved that matrix RðgÞ is an AIFPR for any g 2 ½0; 1. ~ is multiplicative consistent if its induced matrix Definition 6.8 An IV-AIFPR R ~ RðgÞ is multiplicative consistent for any g 2 ½0; 1. Definition 6.8 indicates that a multiplicative consistent IV-AIFPR should include a group of multiplicative consistent AIFPRs. In a similar way, an acceptable multiplicative consistency of an IV-AIFPR is defined below. ~ is called acceptable multiplicative consistency if its Definition 6.9 An IV-AIFPR R ~ induced matrix RðgÞ is acceptable multiplicative consistent for any g 2 ½0; 1.

190

6 A Novel Method for Group Decision Making …

~ is called unacceptable multiplicative consistency and needs to be Otherwise, R repaired. ~ is acceptable multiFrom Definition 6.9, to check whether an IV-AIFPR R plicative consistent or not, it is only needed to check the consistency of its induced matrix, which can be completed by Eqs. (6.4) and (6.5) and Definition 6.4. Nevertheless, it is still difficult to perform because we have to check the consistency ~ of AIFPRs RðgÞ for any g 2 ½0; 1. As such, we provide the following two theorems by which the consistency checking can be completed in a simple way. ~ its induced matrix RðgÞ ~ Theorem 6.4 Given an IV-AIFPR R, has multiplicative consistency or acceptable multiplicative consistency for any g 2 ½0; 1 if and only if ~ ~ both Rð0Þ and Rð1Þ possess such a corresponding property. Proof Necessity can be easily proved. It is only necessary to prove the sufficiency. ~ ~ ¼ ðrij ð1ÞÞnn , From Definition 6.7, it yields that Rð0Þ ¼ ðrij ð0ÞÞnn and Rð1Þ where 8  þ < ðlij ; vij Þ; if i\j rij ð0Þ ¼ ðlij ð0Þ; vij ð0ÞÞ ¼ ð0:5; 0:5Þ; if i ¼ j : ðl þ ; v Þ; if i [ j ij ij

ð6:20Þ

8 þ  < ðlij ; vij Þ; if i\j rij ð1Þ ¼ ðlij ð1Þ; vij ð1ÞÞ ¼ ð0:5; 0:5Þ; if i ¼ j :  þ ðlij ; vij Þ; if i [ j

ð6:21Þ

and

~ Observing Eqs. (6.19), (6.20) and (6.21), it can be easily seen that RðgÞ is a ~ ~ ~ combined AIFPR of Rð0Þ and Rð1Þ. As per Theorem 6.2, RðgÞ has multiplicative ~ ~ consistency or acceptable multiplicative consistency if both Rð0Þ and Rð1Þ have such a corresponding property. The proof of Theorem 6.4 is completed. ~ ~ To facilitate the subsequent discussion, Rð0Þ and Rð1Þ are called the lower and ~ ~ and upper matrices of IV-AIFPR R, and denoted by RL ¼ ððlL ; vL ÞÞ ij

~ U ¼ ððlU ; vU ÞÞ , respectively, where R ij ij nn

8  þ < ðlij ; vij Þ; ðlLij ; vLij Þ ¼ ð0:5; 0:5Þ; : þ  ðlij ; vij Þ;

if i\j if i ¼ j ; if i [ j

ij

nn

ð6:22Þ

6.3 Multiplicative Consistency of IV-AIFPRs

8 þ  < ðlij ; vij Þ; if i\j U ð0:5; 0:5Þ; if i ¼ j : ðlU ; v Þ ¼ ij ij :  þ ðlij ; vij Þ; if i [ j

191

ð6:23Þ

Remark 6.2 From Eqs. (6.22) and (6.23), a given IV-AIFPR can be composed into its lower and upper matrices. In return, if the lower and upper matrices of an unknown IV-AIFPR are given a priori, this IV-AIFPR can be constructed via Eqs. (6.22) and (6.23). Combining Definitions 6.8 and 6.9 with Theorem 6.4, Theorem 6.5 can be easily proved. ~ possesses multiplicative consistency or acceptable Theorem 6.5 An IV-AIFPR R ~ L and multiplicative consistency if and only if both its lower and upper matrices, R ~ U , possess such a corresponding property. R ~ is multiplicative From Theorem 6.5, in order to check whether an IV-AIFPR R consistent or acceptable multiplicative consistent, it is only necessary to judge the corresponding property of its lower and upper matrices, which can be easily completed by Eqs. (6.4) and (6.5) and Definition 6.4. ~ L and R ~ U are multiplicative consistent, it yields from Definition 6.2 that When R   þ þ þ þ þ þ    l ij ljk lki ¼ vij vjk vki ; lij ljk lki ¼ vij vjk vki for all i; j ¼ 1; 2; . . .; n:

ð6:24Þ

Thus, a concise definition of multiplicative consistency of an IV-AIFPR is given below. þ ~ ¼ ðð~ ~ij ¼ ½l Definition 6.10 An IV-AIFPR R lij ; ~vij ÞÞnn with l vij ¼ ij ; lij  and ~  þ ½vij ; vij  is multiplicative consistency if Eq. (6.24) holds.

~ degenerates into an AIFPR, It is worth mentioning that when an IV-AIFPR R Definition 6.10 just reduces to the multiplicative consistency definition of an AIFPR (i.e., Definition 6.2). From this point of view, the multiplicative consistency of an IV-AIFPR defined in this chapter is reasonable.

6.3.2

Repair Multiplicative Consistency of IV-AIFPRs

In real-world group decision making with IV-AIFPRs, it is not easy for DMs to supply acceptable multiplicative consistent IV-AIFPRs, especially when the number of alternatives is large. Consequently, according to Algorithm I designed in Sect. 6.2.2, this subsection devises a new iterative Algorithm II to repair the consistency of an IV-AIFPR with unacceptable multiplicative consistency.

6 A Novel Method for Group Decision Making …

192

~ ¼ ð~rij Þ Let R nn be an IV-AIFPR with unacceptable consistency, which implies that at least one of its lower and upper matrices is unacceptable consistent. Denote U ~ by R ~ L ¼ ðr L Þ ~U the lower and upper matrices of IV-AIFPR R ij nn and R ¼ ðrij Þnn , ~ L and respectively, where r L ¼ ðlL ; vL Þ and r U ¼ ðlU ; vU Þ. Due to the fact that R ij

ij

ij

ij

ij

ij

~ U are both AIFPRs, we can repair their consistency by the iterative Algorithm I R L  ~ ¼ designed in Sect. 6.2.2. Denote the repaired lower and upper matrices by R ~ U ¼ ðr U Þ ðrijL Þnn with rijL ¼ ð lLij ; vLij Þ and R rijR ¼ ð lRij ; vRij Þ, respectively. ij nn with  L ~ U are derived independently, it may be not guaranteed that the ~ and R Since R

~ L and R ~ U via Eqs. (6.22)–(6.23) (see Remark 6.2) is an matrix constructed by R IV-AIFPR. To ensure that the constructed matrix is an IV-AIFPR, a novel algo~U. ~ L and R rithm, i.e., Algorithm II, is proposed to repair R Algorithm II ~ L and R ~ U by Eqs. (6.4) Step 1. Check the multiplicative consistency of AIFPRs R and (6.5). ~U. ~ L or R Step 2. Repair the unacceptable consistency AIFPRs R ~ U is acceptable consistent. In this ~ L is unacceptable consistent, while R Case 1: R L ~ based on Algorithm I. To ensure that case, it is only necessary to repair AIFPR R L  ~ and the acceptable consistent AIFPR R ~U the matrix constructed by the repaired R via Eqs. (6.22)–(6.23) is an IV-AIFPR, Eqs. (6.12)–(6.15) in Algorithm I should be respectively modified as ( ðvLij Þðh þ 1Þ

¼

maxfððqLst ÞðhÞ Þd ððqLsf ÞðhÞ ðqLft ÞðhÞ Þ1d ðlLst ÞðhÞ ; vRst g; ðhÞ vij ;

if ði; jÞ ¼ ðs; tÞ else ð6:25Þ

( ðlLij Þðh þ 1Þ

¼

ðlLij Þðh þ 1Þ ¼ ðlLij ÞðhÞ ;

ð6:26Þ

ðvLij Þðh þ 1Þ ¼ ðvLij ÞðhÞ ;

ð6:27Þ

minfðvLij ÞðhÞ =½ððqLst ÞðhÞ Þd ððqLsk ÞðhÞ ðqLkt ÞðhÞ Þ1d ; lRst g; if ði; jÞ ¼ ðs; tÞ ðlLij ÞðhÞ ;

else ð6:28Þ

. where qLij ¼ vLij lLij ði ¼ 1; 2; . . .; n  1; j ¼ 2; . . .; nÞ.

6.3 Multiplicative Consistency of IV-AIFPRs

193

~ U is unacceptable consistent, while R ~ L is acceptable consistent. In this Case 2: R U ~ case, it is only necessary to repair AIFPR R based on Algorithm I. Similar to Case 1, Eqs. (6.12)–(6.15) should also be accordingly modified to ensure the matrix ~ U and the acceptable consistent AIFPR R ~ L to be an constructed by the repaired R IV-AIFPR, i.e., ( ðvRij Þðh þ 1Þ

¼

minfððqRst ÞðhÞ Þd ððqRsk ÞðhÞ ðqRkt ÞðhÞ Þ1d ðlRij ÞðhÞ ; vLij g; if ði; jÞ ¼ ðs; tÞ ðvRij ÞðhÞ ;

else ð6:29Þ

( ðlRij Þðh þ 1Þ

¼

ðlRij Þðh þ 1Þ ¼ ðlRij ÞðhÞ ;

ð6:30Þ

ðvRij Þðh þ 1Þ ¼ ðvRij ÞðhÞ ;

ð6:31Þ

maxfðvRij ÞðhÞ =½ððqRst ÞðhÞ Þd ððqRsk ÞðhÞ ðqRkt ÞðhÞ Þ1d ; lLij g; if ði; jÞ ¼ ðs; tÞ ðlRij ÞðhÞ

else ð6:32Þ

~ L and R ~ U are both unacceptable consistent. In this case, R ~ L and R ~ U should Case 3: R L U   ~ and R ~ to be an be repaired. To ensure the matrix constructed by the repaired R L ~U ~ IV-AIFPR, we can first repair R according to Algorithm I, and then repair R according to Case 2 in Algorithm II.  ¼ ð ~ ~r ij Þnn , where Step 3. Obtain the repaired IV-AIFPR, denoted by R 8 L R R L ij ; ½vij ; vij Þ; if i\j lij ; l < ð½ þ  þ ~r ij ¼ ðl ~ij ; ~vij Þ ¼ ð½ ð½0:5; 0:5; ½0:5; 0:5Þ; if i ¼ j ð6:33Þ   l ; l ; ½ v ; v Þ ¼ ij ij ij ij : R L L R ð½ lij ; lij ; ½vij ; vij Þ; if i [ j ~ is acceptable multiplicative consistent. Obviously, matrix R

6.4

A Novel Method for Group Decision Making with IV-AIFPRs

This section provides a novel method for solving the GDM problems with IV-AIFPRs. Let fA1 ; A2 ; . . .; An g be a set of feasible alternatives and fd1 ; d2 ; . . .; dm g be a set of DMs whose weight vector is k ¼ ðk1 ; k2 ; . . .; km ÞT , where kk is the weight of DM dk satisfying kk  0 ðk ¼ 1; 2; . . .; mÞ and Pm k¼1 kk ¼ 1. DM dk conducts pair-wise comparisons between alternatives and

6 A Novel Method for Group Decision Making …

194

þ ~ k ¼ ð~rijk Þ ~ijk ¼ ½l elicits an IV-AIFPR R rijk ¼ ð~ lijk ; ~vijk Þ, where l ijk ; lijk  nn with ~ þ  and ~vijk ¼ ½vijk ; vijk . For simplicity, assume that all the individual IV-AIFPRs are acceptable consistent. If some individual IV-AIFPRs are unacceptable consistent, we can repair their consistency by Algorithm II devised in Sect. 6.3.2. To obtain a reasonable decision result, this section is devoted to solving three key issues:

(1) Determine DMs’ weights objectively and integrate individual IV-AIFPRs; (2) Derive the IVAIF priority weights of alternatives; (3) Rank these IVAIF priority weights and select the best alternative.

6.4.1

Determine DMs’ Weights Objectively and Integrate Individual IV-AIFPRs

It is known that DMs’ weights play an important role in fusing individual IV-AIFPRs into a collective one. Hence, how to reasonably determine DMs’ weights is a key issue which needs to be solved. However, existing methods [15, 44, 46] did not cover this issue. In this subsection, a goal programming model is built to derive DMs’ weights objectively. To facilitate the future discussions, denote all the individual IV-AIFPRs with ~ ¼ ð~r Þ r ijk ¼ ð½ Rijk ; ½vLijk ; vRijk  ðk ¼ acceptable consistency by R lLijk ; l k ijk nn with ~ ~G ¼ 1; 2; . . .; mÞ whether they are directly provided by DMs or repaired. Suppose R  ~ ð~r G Þ is the collective IV-AIFPR integrated from individual IV-AIFPRs R k

ij nn

ðk ¼ 1; 2; . . .; mÞ, where G G þ G G þ ~rijG ¼ ð~ lG vG ij ; ~ ij Þ ¼ ð½lij ; lij ; ½vij ; vij Þ m m m m Y Y Y Y kk kk þ kk þ kk l ð lijk Þ ; ½ ðv ðvijk Þ Þ: ¼ ð½ ð ijk Þ ; ijk Þ ; k¼1

k¼1

k¼1

ð6:34Þ

k¼1

  ~k ¼ R ~ t for any When all the individual IV-AIFPRs are the same, i.e., R k; t ¼ 1; 2; . . .m and k 6¼ t, the agreements among DMs achieve full consensus. In ~ for any k ¼ 1; 2; . . .m. It follows that ~G ¼ R this case, we have R k Gþ þ G þ þ  ¼ lijk ; vij ¼ vijk ; vG ¼ vijk ði; j ¼ 1; 2; . . .; nÞ: lG ij ¼ l ijk ; lij ij

ð6:35Þ

~ G and R ~ k ðk ¼ 1; 2; . . .; mÞ are all IV-AIFPRs, Eq. (6.35) can be simSince R plified as Gþ þ G Gþ þ  ¼ lijk ; vij ¼ v ¼ vijk ði ¼ 1; 2; . . .; n  1; i\jÞ: ð6:36Þ lG ij ¼ l ijk ; lij ijk ; vij

6.4 A Novel Method for Group Decision Making with IV-AIFPRs

195

Putting Eq. (6.34) into (6.36) and taking logarithm on both sides of Eq. (6.36), it yields that ln l ijk ¼

m X

þ  kl ln l lijk ¼ ijl ; ln

l¼1 þ ln vijk

m X

ijlþ ; ln v kl ln l ijk ¼

l¼1

¼

m X

m X

kl ln v ijl ;

l¼1

ð6:37Þ

kl ln vijlþ :

l¼1 þ þ  ijk , cijk ¼ ln v vijk . Equation (6.37) can Let aijk ¼ ln l ijk , bijk ¼ ln l ijk and dijk ¼ ln  be rewritten as

aijk ¼

m X l¼1

kl aijl ; bijk ¼

m X

kl bijl ; cijk ¼

l¼1

m X

kl cijl ; dijk ¼

l¼1

m X l¼1

kl dijl

ð6:38Þ

ði ¼ 1; 2; . . .; n  1; i\jÞ: In general, it is utopian to achieve full consensus among DMs because of the different knowledge, experiences and preferences of diverse DMs. In other words, Eq. (6.38) does not always hold. In this case, we turn to seek a weight vector k such that the deviations between the individual preferences and the group’s opinion are minimized. Keeping this idea in mind, an optimization model is built by minimizing the deviations between each individual IV-AIFPR and the collective one, i.e., min s:t:

p  p  p  p #1=p "        m X  m m m m X X X X X         kl aijl  þ bijk  kl bijl  þ cijk  kl cijl  þ dijk  kl dijl  aijk          t¼1 t¼1 k¼1 i\j l¼1 l¼1 t X

kk ¼ 1; kk  0 ðk ¼ 1; 2; . . .; mÞ

k¼1

ð6:39Þ (i) If p ¼ 1, Eq. (6.39) can be converted as       # "        m X  m m m m X X X X X         min kl aijl  þ bijk  kl bijl  þ cijk  kl cijl  þ dijk  kl dijl  aijk          k¼1 i\j l¼1 l¼1 l¼1 l¼1 s:t:

t X

kk ¼ 1; kk  0 ðk ¼ 1; 2; . . .; mÞ

k¼1

ð6:40Þ (ii) If p ¼ þ 1, Eq. (6.39) turns into the following one:                 m m m m X X X X         min maxfaijk  kl aijl ; bijk  kl bijl ; cijk  kl cijl ; dijk  kl dijl g        i;j;k  l¼1 l¼1 l¼1 l¼1 s:t:

t X

kk ¼ 1; kk  0 ðk ¼ 1; 2; . . .; mÞ

k¼1

ð6:41Þ

6 A Novel Method for Group Decision Making …

196

In Eq. (6.40), the objective function is to minimize the sum of all deviations between  ~ k ðk ¼ 1; 2; . . .; mÞ and all the upper triangular elements of the individual IV-AIFPRs R G ~ , which is based on the majority principle. In those of the collective IV-AIFPR R Eq. (6.41), the objective function is to minimize the maximum deviation between all  ~ k ðk ¼ 1; 2; . . .; mÞ and those the upper triangular elements of individual IV-AIFPRs R G ~ , which is based on the minority principle. To comof the collective IV-AIFPR R prehensively consider the majority and minority principles, a general mathematical programming model embracing the above two particular cases (p ¼ 1 and p ¼ þ 1) is constructed through introducing a control parameter n ð0  n  1Þ, i.e.,       ) (         m m m m X X X X         min ð1  nÞ max aijk  kl aijl ; bijk  kl bijl ; cijk  kl cijl ; dijk  kl dijl          i;j;k l¼1 l¼1 l¼1 l¼1       !)          m m m m m XX  X X X X        þn kl aijl  þ bijk  kl bijl  þ cijk  kl cijl  þ dijk  kl dijl  aijk          k¼1 i\j l¼1 l¼1 l¼1 l¼1 (

s:t:

t X

kk ¼ 1; kk  0 ðk ¼ 1; 2; . . .; mÞ

k¼1

ð6:42Þ      P Pm     s ¼ max aijk  m l¼1 kl aijl ; bijk  l¼1 kl bijl ; cijk 

To solve Eq. (6.42), let i;j;k   Pm dijk  Pm kl dijl g. Then, Eq. (6.42) can be converted as k c j; l ijl l¼1 l¼1

      !)         m X  m m m m X X X X X         kl aijl  þ bijk  kl bijl  þ cijk  kl cijl  þ dijk  kl dijl  aijk          k¼1 i\j l¼1 l¼1 l¼1 l¼1    8  m m    P P > aijk  kl aijl   s; bijk  kl bijl   s ði; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; mÞ > >     > > l¼1 l¼1 >    > <  m m    cijk  P kl cijl   s; dijk  P kl dijl   s ði; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; mÞ s:t:     > l¼1 l¼1 > > > > t P > > : kk ¼ 1; kk  0 ðk ¼ 1; 2; . . .; mÞ (

min ð1  nÞs þ n

k¼1

ð6:43Þ Further, Eq. (6.43) is transformed into a linear goal programming model: ( min ð1  nÞs þ n

m X X

) þ ðuijk

þ u ijk

þ þ wijk

þ w ijk

þ þ eijk

þ e ijk

þ þ vijk

þ v ijk Þ

k¼1 i\j

8 þ þ  þ þ   uijk þ u > ijk  s; wijk þ wijk  s; eijk þ eijk  s; vijk þ vijk  s ði; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; mÞ > > > m m > P P > þ  þ >  > > uijk  uijk ¼ aijk  kl aijl ; wijk  wijk ¼ bijk  kl bijl ði; j ¼ 1; 2; . . .; nÞ > > l¼1 l¼1 > > < m m P P þ þ eijk  e kl cijl ; vijk  v kl dijl ði; j ¼ 1; 2; . . .; nÞ s:t: ijk ¼ cijk  ijk ¼ dijk  > l¼1 l¼1 > > m > > P > > kk ¼ 1; kk  0 ðk ¼ 1; 2; . . .; mÞ > > > > k¼1 > > þ  þ  þ  þ  : uijk ; uijk ; wijk ; wijk ; eijk ; eijk ; vijk ; vijk ; s  0ði; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; mÞ

ð6:44Þ

6.4 A Novel Method for Group Decision Making with IV-AIFPRs

197

where þ uijk ¼ ðaijk 

m X

kl aijl Þ _ 0; u ijk ¼ ð

l¼1 m X

¼ð þ ¼ ðcijk  eijk

m X

kl bijl Þ _ 0; w ijk

l¼1

kl bijl  bijk Þ _ 0;

kl cijl Þ _ 0; e ijk ¼ ð

¼ð

þ kl aijl  aijk Þ _ 0; wijk ¼ ðbijk 

l¼1

l¼1 m X

l¼1 m X

m X

m X l¼1

þ kl cijl  cijk Þ _ 0; vijk ¼ ðdijk 

m X

kl dijl Þ _ 0; v ijk

l¼1

kl dijl  dijk Þ _ 0:

l¼1

By solving Eq. (6.44), the vector k ¼ ðk1 ; k2 ; . . .; km ÞT can be obtained. Thus, ~ G can be derived by Eq. (6.34). In addition, an attractive the collective IV-AIFPR R property is derived as below. ~ G has acceptable multiplicative consisTheorem 6.6 The collective IV-AIFPR R  ~ k ðk ¼ 1; 2; . . .; mÞ possess such a property. tency if all individual IV-AIFPRs R ~ G is acceptable multiplicative consistent, it Proof From Theorem 6.5, to prove that R is only needed to prove that its lower and upper matrices, RGL and RGU , are both ~ GL is acceptable multiplicaacceptable multiplicative consistent. First, we prove R tive consistent. Combining Eqs. (6.22), (6.23) and (6.34), it is known that RGL can be regarded L ~ L ðk ¼ 1; 2; . . .; mÞ, where R  ~ is the lower as a combined AIFPR of AIFPRs R k k ~ . Since R ~ ðk ¼ 1; 2; . . .; mÞ are acceptable multiplicative consistent, matrix of R k

k

~ L ðk ¼ 1; 2; . . .; mÞ are acceptable multiplicative consistent. From Theorem 6.2, R k the lower matrix RGL is acceptable multiplicative consistent. Similarly, the upper matrix RGU can be proved to be acceptable multiplicative consistent. This completes the proof of Theorem 6.6.

6.4.2

Derive IVAIF Priority Weights and Rank Alternatives

~ G is an IV-AIFPR whose elements are IVAIFVs, it is As the collective matrix R more logical and natural that the priority weights of alternatives should be IVAIFVs, too. In this subsection, a new approach is presented to derive the IVAIF ~ G . Moreover, a TOPSIS based priority weights from the collective matrix R approach is proposed to rank alternatives according to these IVAIF priority weights.

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6.4.2.1

Derive IVAIF Priority Weights of Alternatives

~ G be x ~ 1; x ~ 2 ; . . .; x ~ n ÞT ~ ¼ ðx Let the priority weight vector of the collective matrix R l l ~ i ¼ ð½xi ; x  i ; ½xvi ; x  vi Þ ði ¼ 1; 2; . . .; nÞ. From Definition 6.8, we can with x investigate the AIF priority weights of induced matrix RG ðgÞ to determine the ~ Suppose the AIF priority weight vector of RG ðgÞ is priority weight vector x. xðgÞ ¼ ðx1 ðgÞ; x2 ðgÞ; . . .; xn ðgÞÞT with xi ðgÞ ¼ ðxli ðgÞ; xvi ðgÞÞ ð0  g  1; ~ i should satisfy following conditions: i ¼ 1; 2; . . .; nÞ. Then x  li ¼ max fxli ðgÞg; xvi ¼ min fxvi ðgÞg and x  vi xli ¼ min fxli ðgÞg; x 0g1

¼ max fxvi ðgÞg:

0g1

0g1

0g1

ð6:45Þ

~ G by RGL ¼ On the other hand, denote the lower and upper matrices of R GU GL GU ¼ ððlGU ððlGL ij ; vij ÞÞnn and R ij ; vij ÞÞnn , whose normalized AIF priority   T weight vectors (see Eq. (9) in [16]) are x ¼ ðx and 1 ; x2 ; . . .; xn Þ l þ þ þ þ þ T  v x ¼ ðx1 ; x2 ; . . .; xn Þ , respectively, where xi ¼ ðxi ; xi Þ and xi ¼ ðxli þ ; xvi þ Þ ði ¼ 1; 2; . . .; nÞ. As RG ðgÞ is the combined AIFPR of RGL and RGU , it is reasonable to consider that xui ðgÞ is between xl and xli þ for any i vþ i ¼ 1; 2; . . .; n. Similarly, xvi ðgÞ should be between xv i and xi , too. Combining Eq. (6.45), one has lþ lþ v v vþ  li ¼ maxfxl  vi xli ¼ minfxl i ; xi g;x i ; xi g; xi ¼ minfxi ; xi g and x v vþ ¼ maxfxi ; xi g:

ð6:46Þ  li þ x  vi  1 ði ¼ 1; 2; . . .; nÞ are imposed to ensure In addition, the inequalities x ~ to be an IVAIF vector. Namely, x lþ v vþ maxfxl i ; xi g þ maxfxi ; xi g  1 ði ¼ 1; 2; . . .; nÞ:

ð6:47Þ

~ it On the basis of the above analyses, to determine the priority weight vector x, is only needed to identify AIF priority weight vectors x and x þ satisfying Eq. (6.47). According to Corollary 1 in method [16], if there exist normalized AIF weight vectors x and x þ such that ( GL ðlGL ij ; vij Þ

¼

ð0:5; 0:5Þ;  l



2xl 2xi j l l l v v v v xl i xi þ xj xj þ 2 xi xi þ xj xj þ 2

;

if i ¼ j ; if i 6¼ j

ð6:48Þ

6.4 A Novel Method for Group Decision Making with IV-AIFPRs

199

and ( GU ðlGU ij ; vij Þ

¼

ð0:5; 0:5Þ;  lþ xli þ xvi þ

 ;

2xlj þ 2xi þ xlj þ xvj þ þ 2 xli þ xvi þ þ xlj þ xvj þ þ 2

;

if i ¼ j if i 6¼ j

ð6:49Þ

then both RGL and RGU are multiplicative consistent AIFPRs. GL Since RGL and RGU are AIFPRs, it is clear that ðlGL ii ; vii Þ ¼ ð0:5; 0:5Þ and GU GU GL GL GU GU ðlii ; vii Þ ¼ ð0:5; 0:5Þ hold. Furthermore, as 0\lij ; vij ; lij ; vij \1, the denominators in Eqs. (6.48) and (6.49) are greater than zero. Thus, Eqs. (6.48) and (6.49) can be respectively simplified as 

l l v v 2xl ¼ lGL i ij ðxi  xi þ xj  xj þ 2Þ; l l l v v 2xj ¼ vGL ij ðxi  xi þ xj  xj þ 2Þ;

if i 6¼ j if i 6¼ j

ð6:50Þ

and (

lþ  xvi þ þ xlj þ  xvj þ þ 2Þ; 2xli þ ¼ lGU ij ðxi lþ l þ  xvi þ þ xlj þ  xvj þ þ 2Þ; 2xj ¼ vGU ij ðxi

if i 6¼ j if i 6¼ j

ð6:51Þ

As stated before, it is hard for DMs to provide multiplicative consistent IV-AIFPRs in practical decision making. Therefore, RGL and RGU may be not multiplicative consistent, which results in that Eqs. (6.50) and (6.51) may be invalid. To derive x and x þ , it is expected that x and x þ satisfy Eqs. (6.50) and (6.51) as much as possible. That is to say, the underlying priority weight vectors x and x þ should minimize the deviations between the values on the left sides and those on the right sides of Eqs. (6.50) and (6.51). For convenience, some deviation variables are introduced as follows: l l l l l  GL v v GL v v bijþ ¼ ð2xl i  lij ðxi  xi þ xj  xj þ 2ÞÞ _ 0; bij ¼ ðlij ðxi  xi þ xj  xj þ 2Þ  2xi Þ _ 0; l l l l l  GL v v GL v v /ijþ ¼ ð2xl j  vij ðxi  xi þ xj  xj þ 2ÞÞ _ 0; /ij ¼ ðvij ðxi  xi þ xj  xj þ 2Þ  2xj Þ _ 0; lþ lþ GU  xvi þ þ xlj þ  xvj þ þ 2ÞÞ _ 0; c  xvi þ þ xlj þ  xvj þ þ 2Þ  2xli þ Þ _ 0; cijþ ¼ ð2xli þ  lGU ij ðxi ij ¼ ðlij ðxi lþ lþ GU  xvi þ þ xlj þ  xvj þ þ 2ÞÞ _ 0; #  xvi þ þ xlj þ  xvj þ þ 2Þ  2xlj þ Þ _ 0; #ijþ ¼ ð2xlj þ  vGU ij ðxi ij ¼ ðvij ðxi

where i 6¼ j. Thus, the smaller the deviation variables, the better the consistency of AIFPRs GL R and RGU . Therefore, combining the normalized AIF priority weights conditions (see Eq. (9) in [16]) with Eq. (6.47), an optimization model is built as follows:

6 A Novel Method for Group Decision Making …

200

min

n X

þ  þ  þ  þ ðb ij þ bij þ /ij þ /ij þ cij þ cij þ #ij þ #ij Þ

i6¼j

8 l l l  þ v v 2x  lGL > ij ðxi  xi þ xj  xj þ 2Þ þ bij  bij ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > i > l l l >  þ GL v v > > > 2xj  vij ðxi  xi þ xj  xj þ 2Þ þ /ij  /ij ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > lþ > þ >  xvi þ þ xlj þ  xvj þ þ 2Þ þ c 2xli þ  lGU > ij ðxi ij  cij ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > > l þ l þ l þ > 2x  vGU ðx  xv þ þ x  xv þ þ 2Þ þ #  # þ ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > j i j i j ij ij ij > < n n P P l l l v v s:t: xi þ x  1; xj  xi ; xi þ n  2  xv ði ¼ 1; 2; . . .; nÞ i j > > j¼1;j6¼i j¼1;j6¼i > > > > n n P P > lþ > vþ > xlj þ  xvi þ ; xli þ þ n  2  xvj þ ði ¼ 1; 2; . . .; nÞ > xi þ xi  1; > > j¼1;j6¼i j¼1;j6¼i > > > > l lþ vþ v > > > maxfxi ; xi g þ maxfxi ; xi g  1 ði ¼ 1; 2; . . .; nÞ > : l v l þ v þ xi ; xi ; xi ; xi [ 0 ði ¼ 1; 2; . . .; nÞ

ð6:52Þ lþ vþ v have oi ¼ maxfxl i ; xi g and 1i ¼ maxfxi ; xi g. Then, we vþ v  oi and xi ; xi  1i . Equation (6.52) can be transformed into a linear programming model:

Let

lþ xl i ; xi

n X

þ  þ  þ  þ ðb ij þ bij þ /ij þ /ij þ cij þ cij þ #ij þ #ij Þ i6¼j 8 l l l  þ v v 2xi  lGL > ij ðxi  xi þ xj  xj þ 2Þ þ bij  bij ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > > l l >  þ GL v v > 2xl > > j  vij ðxi  xi þ xj  xj þ 2Þ þ /ij  /ij ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > > 2xl þ  lGU ðxl þ  xv þ þ xl þ  xv þ þ 2Þ þ c  c þ ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > i j ij i j ij ij > i > > > 2xl þ  vGU ðxl þ  xv þ þ xl þ  xv þ þ 2Þ þ #  # þ ¼ 0 ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ > > i j i j ij ij ij > j < n n P P l l l v v s:t: xi þ x  1; xj  xi ; xi þ n  2  xv ði ¼ 1; 2; . . .; nÞ i j > > j¼1;j6¼i j¼1;j6¼i >

min

> > > n n P P > lþ > vþ > xlj þ  xvi þ ; xli þ þ n  2  xvj þ > xi þ xi  1; > > j¼1;j6¼i j¼1;j6¼i > > > > l lþ vþ v > > > oi þ 1i  1; xi ; xi  oi ; xi ; xi  1i ði ¼ 1; 2; . . .; nÞ > : l v l þ v þ xi ; xi ; xi ; xi ; oi ; 1i [ 0 ði ¼ 1; 2; . . .; nÞ

ði ¼ 1; 2; . . .; nÞ

ð6:53Þ þ , x v , x l and Solving Eq. (6.53), the optimal solutions, denoted by x l i i i can be generated. Putting these optimal solutions into Eq. (6.46), the ~ 2 ; . . .; x ~ n ÞT with x ~i ¼ ~ 1; x ~ ¼ ðx IVAIF priority weight vector is derived as x  li ; ½xvi ; x  vi Þ ði ¼ 1; 2; . . .; nÞ. To rank these IVAIF priority weights ð½xli ; x ~ i ði ¼ 1; 2; . . .; nÞ, a TOPSIS based approach is proposed in the sequel. x þ , x v i

6.4 A Novel Method for Group Decision Making with IV-AIFPRs

6.4.2.2

201

A TOPSIS Based Approach to Ranking IVAIF Priority Weights

Two common methods for ranking IVAIF priority weights are the score-accuracy based method proposed by Xu and Chen [42] and the two criteria method developed by Dymova et al. [11]. Although these two methods seem feasible to rank IVAIF priority weights, there are some flaws: (1) Method [42] cannot compare the IVAIF priority weights whose midpoints of intervals of membership and non-membership degree are respectively equal (see Example 6.1); (2) Method [11] is based on the interval operations which have always been controversial by researchers. For example, according to the interval operation, one has ½0:1; 0:2  ½0:1; 0:2 ¼ ½0:1; 0:1. Intuitively, ½0:1; 0:2  ½0:1; 0:2 ¼ 0 is more sensible. Hence, method [11] may be unconvincing. To remedy these flaws, a TOPSIS based approach is presented to rank IVAIF priority weights. ~ i ¼ ð½xli ; x  li ; ½xvi ; x  vi Þ ði ¼ 1; 2; . . .; nÞ, let the For the IVAIF priority weights x ~ P ¼ ð½xlP ; x  lP ; ½xvP ; x  vP Þ and positive and negative IVAIF priority weights be x l l v v  N ; ½xN ; x  N Þ, respectively, where ~ N ¼ ð½xN ; x x  li g; xlP ¼ maxffxli ; x  li gnfx  lP gg; xvP ¼ minfxvi g; x  vP  lP ¼ maxfx x i

i

 vi gnfxvP gg; ¼ minffxvi ; x

i

i

ð6:54Þ

 lN ¼ minffxli ; x  li gnfxlN gg; x  vN ¼ maxfx  vi g; xvN xlN ¼ minfxli g; x i

i

 vi gnfx  vN gg: ¼ maxffxvi ; x

i

i

ð6:55Þ

~ i to the positive IVAIF priority Then, the comprehensive closeness degree of x weight can be calculated as ~ i; x ~ P ÞÞ þ ð1  qÞdðx ~ i; x ~ N Þ; ~ i Þ ¼ qð1  dðx Tðx

ð6:56Þ

~ i; x ~ P Þ and dðx ~ i; x ~ N Þ can be computed by the distance measure given in where dðx Wan et al. [38], and q 2 ½0; 1 is called optimistic coefficient whose value can be chosen depending on the decision group’s opinions. Especially, when q ¼ 0, DMs ~ i from the are pessimistic and only consider the separation of the priority weight x ~ N . When q ¼ 1, DMs are optimistic and only consider the negative priority weight x ~ i from the positive priority weight x ~ P . When separation of the priority weight x ~i q ¼ 0:5, DMs are neutral and consider the separations of the priority weight x ~ P as well as the negative priority weight x ~N from the positive priority weight x ~ i Þ, IVAIF priority weights simultaneously. According to the descend order of Tðx ~ i ði ¼ 1; 2; . . .; nÞ can be ranked. x

6 A Novel Method for Group Decision Making …

202

To illustrate the proposed TOPSIS based approach, we give the following example. ~ 1 ¼ ð½0:1; 0:5; ½0:3; 0:4Þ and Example 6.1 Consider two IVAIF priority weights x ~ 2 ¼ ð½0:2; 0:4; ½0:25; 0:45Þ. It is noted that the midpoints of intervals ½0:1; 0:5 x and ½0:2; 0:4 are the same (i.e., 0.3). Meanwhile, the midpoints of intervals ½0:3; 0:4 and ½0:25; 0:45 are also the same (i.e., 0.35). ~ 1 Þ ¼ sðx ~ 2Þ ¼ (i) By method [42], it yields that ~a1 ¼ a~2 because of the scores sðx ~ 1 Þ ¼ hðx ~ 2 Þ ¼ 0:65. However, it is obvious that 0:05 and the accuracies hðx ~ 1 is not equal to x ~ 2 intuitively. x ~P ¼ (ii) Employing the proposed TOPSIS based method, we can easily obtain x ~ N ¼ ð½0:10; 0:20; ½0:40; 0:45Þ via Eqs. (6.54) ð½0:40; 0:50; ½0:25; 0:30Þ and x and (6.55). Then, the comparison results are derived by Eq. (6.56) as follows: ~ 1 Þ ¼ 0:26 [ Tðx ~ 2 Þ ¼ 0:24. Hence, x ~1 [ x ~ 2; If q ¼ 0:1, then Tðx ~ 1 Þ ¼ Tðx ~ 2 Þ ¼ 0:50. Hence, x ~1 ¼ x ~ 2; If q ¼ 0:5, then Tðx ~ 1 Þ ¼ 0:74\Tðx ~ 2 Þ ¼ 0:76. Hence, x ~ 1 \x ~ 2; If q ¼ 0:9, then Tðx Example 6.1 indicates that the TOPSIS based approach is more flexible and effective.

6.4.3

A Novel Method for Solving GDM Problems with IV-AIFPRs

By putting the aforesaid algorithms and models together, a novel method is put forward for solving GDM problems with IV-AIFPRs. The main steps are outlined below. ~ U from R ~ k ¼ ð~r k Þ ~ L and R Step 1: Obtain R ij nn ðk ¼ 1; 2; . . .; mÞ by Eqs. (6.22)– k k (6.23). ~ k ðk ¼ 1; 2; . . .; mÞ using Eqs. (6.4) and (6.5) Step 2: Check the consistency of R ~ k are acceptable consistent, go to step 5; and Theorem 6.5. If all IV-AIFPRs R otherwise, go to the next step. Step 3: Pick out the unacceptable consistent IV-AIFPRs.  ~ k from unacceptable conStep 4: Derive new acceptable consistent IV-AIFPRs R sistent IV-AIFPRs by Algorithms I and II. Step 5: Determine DMs’ weights kk ðk ¼ 1; 2; . . .; mÞ by solving Eq. (6.44).  ~ k ðk ¼ 1; 2; . . .; mÞ into a collective one R ~ G by Eq. (6.34). Step 6: Integrate R  þ Step 7: Obtain AIF priority weight vector x and x via Eq. (6.53). ~ i ði ¼ 1; 2; . . .; nÞ of RG via Eq. (6.46). Step 8: Derive the IVAIF priority weights x ~ i Þ ði ¼ 1; 2; . . .; nÞ by Step 9: Compute the comprehensive closeness degree Tðx Eqs. (6.54)–(6.56) and rank alternatives.

6.5 A Practical Example of a Virtual Enterprise …

6.5

203

A Practical Example of a Virtual Enterprise Partner Selection and Comparative Analyses

In this section, a practical example of a virtual enterprise partner selection is provided to show the application of the proposed method. Comparative analyses are carried out to demonstrate the desirable advantages of the proposed method over other methods.

6.5.1

A Practical Example of a Virtual Enterprise Partner Selection

Due to the constant change of today’s market environment, a single real enterprise is unable to adjust itself rapidly to satisfy market demand. In this scenario, virtual enterprise, a dynamic alliance composed of many members (real enterprises), emerges. In a virtual enterprise, members can share cost, risk, technology and key competitiveness, by which win-win among members can be achieved. As an example, in order to develop a new product which cannot be finished by a single enterprise, a real enterprise (also called a key enterprise) will seek other enterprises (also called virtual enterprise partners) based on its demand. Thus, the key enterprise and other ones compose a virtual enterprise. While developing, designing, producing and selling this product, each enterprise gives full play to its advantages and obtains corresponding benefit. Once this process ends, this virtual enterprise disintegrates and each enterprise can seek new members again. In this process, selecting suitable partners is very important for this key enterprise. AHEAD Information Technology Co., LTD (AHEAD for short), a famous software enterprise of China, has been identified to focus on medical information integrating and service since its inception in 2003. AHEAD desires to develop a new-type rural cooperative medical care management information system for facilitating health reimbursement management. This system consists of some software systems and a hardware device integrating chips. AHEAD can develop software systems and only needs to seek a partner to produce the hardware device. After primary screening, four different partners fA1 ; A2 ; A3 ; A4 g chosen from the virtual industry cluster remain for further evaluation. To select the best partner, AHEAD asks three DMs fd1 ; d2 ; d3 g to evaluate these four partners. By conducting pairwise comparisons on four partners, DMs furnish their IV-AIFPRs as

6 A Novel Method for Group Decision Making …

204

0

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ B ð½0:6500; 0:7000; ½0:1400; 0:3000Þ e 1 ¼B R B @ ð½0:1200; 0:1500; ½0:6200; 0:6500Þ ð½0:0400; 0:1000; ½0:8300; 0:9000Þ ð½0:6200; 0:6500; ½0:1200; 0:1500Þ ð½0:7300; 0:7800; ½0:0400; 0:0500Þ

ð½0:1400; 0:3000; ½0:6500; 0:7000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ

ð½0:0400; 0:0500; ½0:7300; 0:7800Þ ð½0:0200; 0:0220; ½0:9000; 0:9500Þ 1 ð½0:8300; 0:9000; ½0:0400; 0:1000Þ ð½0:9000; 0:9500; ½0:0200; 0:0220Þ C C C ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:5500; 0:6000; ½0:1000; 0:2000Þ A ð½0:1000; 0:2000; ½0:5500; 0:6000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ 0 ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:6500; 0:6700; ½0:0500; 0:3300Þ B ð½0:0500; 0:3300; ½0:6500; 0:7000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ e 2 ¼B R B @ ð½0:0360; 0:3500; ½0:6200; 0:6500Þ ð½0:3000; 0:4000; ½0:5000; 0:5500Þ ð½0:0060; 0:1400; ½0:7800; 0:8200Þ ð½0:0800; 0:1800; ½0:7500; 0:8000Þ 1 ð½0:6200; 0:6500; ½0:0360; 0:3500Þ ð½0:7800; 0:8200; ½0:0060; 0:1400Þ ð½0:5000; 0:5500; ½0:3000; 0:4000Þ ð½0:7500; 0:8000; ½0:0800; 0:1800Þ C C C ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:7000; 0:7500; ½0:1000; 0:2000Þ A ð½0:1000; 0:2000; ½0:7000; 0:7500Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:4500; 0:5000; ½0:1000; 0:2000Þ B ð½0:1000; 0:2000; ½0:4500; 0:5000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ e 3 ¼B R B @ ð½0:1100; 0:1700; ½0:5000; 0:6000Þ ð½0:1500; 0:2000; ½0:7500; 0:8000Þ ð½0:1000; 0:1200; ½0:8000; 0:8500Þ ð½0:1000; 0:1500; ½0:6500; 0:7000Þ 1 ð½0:5000; 0:6000; ½0:1100; 0:1700Þ ð½0:8000; 0:8500; ½0:1000; 0:1200Þ ð½0:7500; 0:8000; ½0:1500; 0:2000Þ ð½0:6500; 0:7000; ½0:1000; 0:1500Þ C C C ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:8000; 0:8500; ½0:0500; 0:1000Þ A 0

ð½0:0500; 0:1000; ½0:8000; 0:8500Þ

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ

In what follows, the proposed method of this chapter is employed to solve this example. Step 1: Check the consistency of individual IV-AIFPRs. By Eqs. (6.4) and (6.5), the consistency indices are obtained as ~ L Þ ¼ 0:1738, MCIðR ~ U Þ ¼ 0:4507, MCIðR ~ L Þ ¼ 0:2181, MCIðR ~UÞ ¼ MCIðR 1 1 2 2 L U ~ ~ 0:1638, MCIðR3 Þ ¼ 1:1575, MCIðR3 Þ ¼ 1:9883. Take the consistency threshold ~ U are acceptable consistent, while R ~U, R ~ L, R ~ L and R ~ U are ~ L and R a ¼ 0:2, then R 1 2 1 2 3 3 ~ 1, R ~ 2 and R ~ 3 are unacceptable consistent. According to Theorem 6.5, IV-AIFPRs R all unacceptable consistent. ~ 1, R ~ 2 and R ~ 3. Step 2: Repair the consistency of individual IV-AIFPRs R U ~ ~U ~ As for R1 , since its upper matrix R1 is unacceptable consistent, we can repair R 1 U by modifying vij1 ði\jÞ according to Case 1 in Algorithm II. To retain most decision information supplied by DM d1 , the trade-off parameter d is taken as 0.7.

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U  ~ can be obtained by Thus, the upper triangular elements of the repaired AIFPR R 1 Eqs. (6.25) and (6.26). According to the definition of AIFPRs, the lower triangular U  ~ U can be derived. Thus, AIFPR R ~ can be determined. elements of AIFPR R 1 1  ~ L which is acceptable consistent, the repaired IV-AIFPR R ~ can be Combing R 1

1

obtained via Eq. (6.33), i.e., 0

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:1400; 0:3000; ½0:6500; 0:7000Þ B ð½0:6500; 0:7000; ½0:1400; 0:3000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ e 1 ¼B R B @ ð½0:0860; 0:1500; ½0:6200; 0:6500Þ ð½0:0400; 0:0500; ½0:7300; 0:7800Þ ð½0:0234; 0:1000; ½0:8300; 0:9000Þ ð½0:0094; 0:0220; ½0:9000; 0:9500Þ 1 ð½0:6200; 0:6500; ½0:0860; 0:1500Þ ð½0:8300; 0:9000; ½0:0234; 0:1000Þ ð½0:7300; 0:7800; ½0:0400; 0:0500Þ ð½0:9000; 0:9500; ½0:0094; 0:0220Þ C C C: A ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:5500; 0:6000; ½0:1000; 0:2000Þ ð½0:1000; 0:2000; ½0:5500; 0:6000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ

~ 2 , it can be seen from Step 1 that its lower matrix R ~ L is unacceptable As for R 2 ~ L can be repaired by modconsistent. According to the Case 2 in Algorithm II, R 2 ~ 1 , parameter d is also taken as 0.7 in this ifying vLij2 ði\jÞ. Similar to repairing R  ~ U , the repaired IV-AIFPR R ~ 2 can be repairing process. Combining upper matrix R 2 derived by Eq. (6.33), i.e., 0

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ B ð½0:0500; 0:3300; ½0:6500; 0:7000Þ e 2 ¼B R B @ ð½0:0360; 0:3171; ½0:6200; 0:6500Þ

ð½0:6500; 0:6700; ½0:0500; 0:3300Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:3000; 0:4000; ½0:5000; 0:5500Þ

ð½0:0060; 0:1246; ½0:7800; 0:8200Þ ð½0:0800; 0:1800; ½0:7500; 0:8000Þ 1 ð½0:6200; 0:6500; ½0:0360; 0:3171Þ ð½0:7800; 0:8200; ½0:0060; 0:1246Þ ð½0:5000; 0:5500; ½0:3000; 0:4000Þ ð½0:7500; 0:8000; ½0:0800; 0:1800Þ C C C A ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:7000; 0:7500; ½0:1000; 0:2000Þ ð½0:1000; 0:2000; ½0:7000; 0:7500Þ

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ

~ 3 , its lower and upper matrices R ~ L and R ~ U are both unacceptable As for R 3 3 consistent and can be respectively repaired by modifying their non-membership degrees according to Case 3 in Algorithm II. Plugging the repaired lower and upper  ~ 3 as matrices into Eq. (6.33) gets the repaired IV-AIFPRs R

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0

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ B ð½0:1000; 0:2000; ½0:4500; 0:5000Þ e 3 ¼B R B @ ð½0:0271; 0:0707; ½0:5000; 0:6000Þ

ð½0:4500; 0:5000; ½0:1000; 0:2000Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:1500; 0:2000; ½0:7500; 0:8000Þ

ð½0:0028; 0:0167; ½0:8000; 0:8500Þ ð½0:0094; 0:0254; ½0:6500; 0:7000Þ 1 ð½0:5000; 0:6000; ½0:0271; 0:0707Þ ð½0:8000; 0:8500; ½0:0028; 0:0167Þ C ð½0:7500; 0:8000; ½0:1500; 0:2000Þ ð½0:6500; 0:7000; ½0:0094; 0:0254Þ C C ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:8000; 0:8500; ½0:0500; 0:100Þ A ð½0:0500; 0:1000; ½0:8000; 0:8500Þ

ð½0:5000; 0:5000; ½0:5000; 0:500Þ

By Eqs. (6.4) and (6.5), the new consistency indices are computed as L ~ U Þ ¼ 0:1694, MCIðR ~ L Þ ¼ 0:1727, MCIðR ~ U Þ ¼ 0:1984, MCIðR  ~ Þ ¼ 0:1646 MCIðR 1 2 2 3 L U ~ U Þ ¼ 0:1638. Since the consistency threshold a ¼ 0:2, AIFPRs R  ~ ,R ~ , and MCIðR 3

2

2

L ~ L and R ~ U are all acceptable consistent. As MCIðR  ~ L Þ ¼ MCIðR ~ Þ ¼ 0:1738 and R 1 3 3 1 U U      ~ ~ ~ ~ ~ 3 , are all MCIðR3 Þ ¼ MCIðR3 Þ ¼ 0:1638, the repaired IV-AIFPRs, R1 , R2 and R acceptable consistent. Step 3: Determine the DMs’ weight vector. Setting three particular values 0, 1 and 0.5 for parameter n in Eq. (6.44) corresponding to the minority, majority and compromise principles, different DMs’ weights are respectively derived. The computation results are listed in Table 6.1. Step 4: Derive the IVAIF priority weight vector of alternatives. Using different vectors of DMs’ weights obtained in Step 3 (i.e., k1 , k2 and k3 ), diverse collective IV-AIFPRs can be derived by Eq. (6.34). Solving the linear programming model Eq. (6.53) yields optimal solutions. In virtue of Eq. (6.45), distinct IVAIF priority weight vectors of alternatives corresponding to different collective IV-AIFPRs are obtained and presented in Table 6.2. Step 5: Rank alternatives. Employing the proposed TOPSIS based approach, the ranking results of alternatives are also listed in Table 6.2. As shown in Table 6.2, the first priority weight vector is obtained based on the minority principle ðn ¼ 0Þ. When q 2 ½0:00; 0:70Þ, the best alternative is A1 . When q ¼ 0:70, A1 and A2 are both the best alternatives. While q 2 ½0:70; 1:00, the best alternative changes into A2 . The second one and the third one are generated based on the majority principle ðn ¼ 1Þ and the compromise principle ðn ¼ 0:5Þ, respectively. In these two cases, the best alternative is A1 for any q 2 ½0; 1. These

Table 6.1 DMs’ weight vectors based on different principles

Parameter

Principle

DMs’ weight vector k

n¼0 n¼1 n ¼ 0:5

Minority Majority Compromise

k1 ¼ ð0:3855; 0:2649; 0:3496Þ k2 ¼ ð0:2131; 0:3805; 0:4064Þ k3 ¼ ð0:2183; 0:3899; 0:3918Þ

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Table 6.2 Priority weight vectors of alternatives with different DMs’ weights and ranking results k

~ x

q

[([0.2955, 0.3990], [0.4022, 0.4823]), ([0.3328, 0.3795], [0.4234, 0.5113]), ([0.0550, 0.0833], [0.8731, 0.8786]), ([0.0043, 0.0210], [0.9790, 0.9790])] [([0.3475, 0.4559], [0.3586, 0.4351]), ([0.2895, 0.3096], [0.5161, 0.5813]), ([0.0657, 0.1028], [0.8540, 0.8971]), ([0.0033, 0.0225], [0.9774, 0.9774])] [([0.3443, 0.4534], [0.3601, 0.4356]), ([0.2917, 0.3119], [0.5108, 0.5770]), ([0.0650, 0.1015], [0.8556, 0.8984]), ([0.0033, 0.0221], [0.9778, 0.9778])]

q 2 ½0:00; 0:70Þ q ¼ 0:70 q 2 ð0:70; 1:00

~ TðxÞ ~ 1 Þ [ Tðx ~ 2 Þ [ Tðx ~ 3 Þ [ Tðx ~ 4Þ Tðx ~ 1 Þ ¼ Tðx ~ 2 Þ [ Tðx ~ 3 Þ [ Tðx ~ 4Þ Tðx ~ 2 Þ [ Tðx ~ 1 Þ [ Tðx ~ 3 Þ [ Tðx ~ 4Þ Tðx

Ranking order

k1

q 2 ½0; 1

~ 1 Þ [ Tðx ~ 2 Þ [ Tðx ~ 3 Þ [ Tðx ~ 4Þ Tðx

A1  A2  A3  A4

q 2 ½0; 1

~ 1 Þ [ Tðx ~ 2 Þ [ Tðx ~ 3 Þ [ Tðx ~ 4Þ Tðx

A1  A2  A3  A4

k2

k3

A1  A2  A3  A4 A1 A2  A3  A4 A2  A1  A3  A4

observations verify that different DMs’ weights can result in diverse priority weights and thus influence the choice of the best alternatives. Hence, it is necessary to determine the DMs’ weights objectively.

6.5.2

Comparative Analyses

To reveal the advantage of our proposed method, this subsection conducts comparative analyses with Liao’s method [15] and other existing group decision making methods [6, 7].

6.5.2.1

Comparison with Liao’s Method

First, we use Liao’s method [15] to solve the Example in Sect. 6.5.1 as follows. Step 1. Check the consistency degrees of individual IV-AIFPRs. ð0Þ  ~ ð0Þ ¼ R ~ k ðk ¼ 1; 2; 3Þ. Three consistent IV-AIFPRs R ~ ðk ¼ 1; 2; 3Þ can Let R k k  ð0Þ ðk ¼ ~ be constructed based on the Algorithm 2 in [15], and then integrated R k

1; 2; 3Þ into a collective one. To make the comparison more validly, assume that

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DMs’ weights are the same as those obtained by the proposed method based on the ð0Þ  ~ can be minor principle (i.e. k ). Via Eq. (33) in [15], the collective IV-AIFPR R 1

 ð0Þ are  ð0Þ ðk ¼ 1; 2; 3Þ and R ~ ~ derived. By Eq. (34) in [15], the distances between R k derived as: ð0Þ ~ ð0Þ ; R ~ ð0Þ Þ ¼ 0:1445; dðR ~ ð0Þ ; R ~ ð0Þ Þ ¼ 0:1131; dðR  ð0Þ ; R  ~ ~ Þ ¼ 0:1431: dðR 1 2 3

Similar to Liao’s method, set the consistency threshold s ¼ 0:1. Since ~ ð0Þ Þ [ s ðk ¼ 1; 2; 3Þ, R ~ ð0Þ ; R ~ ð0Þ ðk ¼ 1; 2; 3Þ are all unacceptable consistent dðR k k and need to repair. Step 2. Repair the unacceptable consistent individual IV-AIFPRs.  ð1Þ ðk ¼ ~ Let the iterative pace g ¼ 0:05. The new consistent IV-AIFPRs R k ð1Þ  ~ 1; 2; 3Þ are constructed by Eqs. (35)–(38) in [15]. Integrating all Rk ðk ¼ 1; 2; 3Þ, the collective IV-AIFPR is acquired as 0

e R

ð1Þ

ð½0:5000; 0:5000; ½0:5000; 0:5000Þ B ð½0:0183; 0:3024; ½0:1742; 0:4429Þ B ¼B @ ð½0:0001; 0:0037; ½0:5987; 0:9209Þ ð½0:0000; 0:0003; ½0:9448; 0:9920Þ ð½0:5987; 0:9209; ½0:0001; 0:0037Þ ð½0:8761; 0:9360; ½0:0032; 0:0084Þ

ð½0:1742; 0:4429; ½0:0183; 0:3024Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ

ð½0:0032; 0:0084; ½0:8761; 0:9360Þ ð½0:0000; 0:0001; ½0:9838; 0:9963Þ 1 ð½0:9448; 0:9920; ½0:0000; 0:0003Þ ð½0:9838; 0:9963; ½0:0000; 0:0001Þ C C C ð½0:5000; 0:5000; ½0:5000; 0:5000Þ ð½0:8955; 0:9963; ½0:0010; 0:0095Þ A ð½0:0010; 0:0095; ½0:8955; 0:9963Þ ð½0:5000; 0:5000; ½0:5000; 0:5000Þ

ð1Þ  ð1Þ are respectively com~ ðk ¼ 1; 2; 3Þ and R ~ Thus, the distances between R k puted as ð1Þ  ð1Þ ~ ð1Þ ; R ~ ð1Þ Þ ¼ 0:0637; dðR ~ ð1Þ ; R ~ ð1Þ Þ ¼ 0:0445; dðR  ~ ;R ~ Þ ¼ 0:0582: dðR 1 2 3

~ ð1Þ ; R ~ ð1Þ Þ\s ¼ 0:1 ðk ¼ 1; 2; 3Þ, the consistency repairing process ends. As dðR k Step 3. Compute the priority weights and rank alternatives. ~ ð1Þ with ~r ¼ ð½lL ; lR ; ½vL ; vR Þ. By Eq. (39) in [15], the ~ ¼ ð~rij Þ Let R ij ij ij ij ij 44 ¼ R priority weights of alternatives are obtained as follows: ~r1 ¼ ð½0:6038; 0:8535; ½0:0012; 0:0252Þ; ~r2 ¼ ð½0:8769; 0:9359; ½0:0027; 0:0154Þ; ~r3 ¼ ð½0:0260; 0:0507; ½0:1665; 0:5150Þ; ~r4 ¼ ð½0:0003; 0:0018; ½0:8023; 0:9332Þ:

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Thus, the ranking order of alternatives obtained by Liao’s method is A2  A1  A3  A4 and the best alternative is A2 , which is in accordance with that obtained by the proposed method in minority principle. Compared with Liao’s method, the proposed method in this chapter has the following advantages: (1) The multiplicative consistency of an IV-AIFPR defined by the latter is more reasonable. When an IV-AIFPR reduces to an AIFPR, the multiplicative consistency of an IV-AIFPR defined in the former and latter degenerate into the consistency of an AIFPR defined in [43] and [13], respectively. Liao and Xu [13] has pointed that the consistency they defined is more reasonable than that defined by Xu et al. [43]. In this sense, the consistency defined by the latter can degenerate a more reasonable consistency of an AIFPR when an IV-AIFPR reduces to an AIFPR. Therefore, it is more reasonable. (2) The consistency definition of an IV-AIFPR proposed by the latter is simpler than that proposed by the former. While checking the consistency of an IV-AIFPR, the former needs to employ another consistent IV-AIFPR besides the IV-AIFPR itself, while the latter can carry out the checking only depending on itself (see Definition 6.10). (3) Different from the iterative algorithm proposed by the former, the iterative Algorithms I and II designed by the latter are not only more time-saving but can retain more information. First, the latter performs the consistency repairing process only relying on the IV-AIFPR itself, while the former needs a consistent IV-AIFPR besides IV-AIFPR itself. In this regard, the latter is more time-saving. Second, while repairing an IV-AIFPR, the latter only repairs one element each time (see Step 3 in Algorithm I), whereas the former needs to repair all the elements of initial IV-AIFPR (see Eqs. (35)–(38) in [15]). Hence, the latter can retain more decision information of the initial IV-AIFPR. (4) For determining DMs’ weights, the latter builds a parameterized goal programming model which can derive DMs’ weights objectively. Moreover, different DMs’ weights can be obtained based on diverse decision making principles. Hence, the latter can avoid the subjective randomness and flexibly reflect the principles of decision making. However, the former did not consider the determination of DMs’ weights.

6.5.2.2

Comparison with Other Existing Group Decision Making Methods

This subsection compares the proposed method with other existing group decision making methods [6, 7]. (1) The focuses of these methods are different. Method [7] investigated the reaching consensus among DMs in GDM with fuzzy preference relations, whereas the method proposed in this chapter and method [6] focus on the

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consistency of preference relations. The difference between them is that the proposed method is devoted to the consistency of the IV-AIFPR, while method [6] concentrates on the consistency of the fuzzy preference relation. (2) Method [6] discussed several types of transitivity of the reciprocal preference relation (usually also called the fuzzy preference relation) and concluded that multiplicative transitivity is the most appropriate property for modelling cardinal consistency of the reciprocal preference relation. This chapter works on the multiplicative consistency of the IV-AIFPR. When an IV-AIFPR is reduced to a reciprocal preference relation, the multiplicative consistency definition of the IV-AIFPR proposed in this chapter is just degenerated to that of reciprocal preference relations, which indicates that the multiplicative consistency of the IV-AIFPR defined in this chapter is meaningful and rational. (3) Although method [6] pointed that multiplicative transitivity is most appropriate for modelling consistency of reciprocal preference relations, it did not provide any tool or method for checking multiplicative consistency or repairing the consistency when a reciprocal preference relation is not strictly multiplicative consistent. This chapter not only defines the multiplicative consistency of IV-AIFPRs, but also provides some effective means to check the consistency of an IV-AIFPR (see Theorem 6.5 and Definition 6.10). For an IV-AIFPR with unacceptable consistency, a novel Algorithm II is supplied to repair its consistency.

6.6

Conclusions

This chapter proposed a novel method for solving GDM problems with IV-AIFPRs. The consistency and acceptable consistency of an IV-AIFPR were defined. A common desirable characteristic of these two definitions is that they can reduce to the corresponding consistency and acceptable consistency of an AIFPR when an IV-AIFPR degenerates into an AIFPR. A novel iterative algorithm was designed to repair the consistency of an IV-AIFPR with unacceptable consistency. To determine DMs’ weights, an optimization model was constructed and dexterously transformed into a linear goal programming model to resolve sufficiently considering different principles of decision making, including minority, majority and compromise principles. Afterwards, a linear programming model was built to derive the IVAIF priority weights. Finally, a TOPSIS based approach was put forward to rank alternatives according to these IVAIF priority weights. As a new computing paradigm for processing information, granular computing [19, 20, 22, 25, 30, 32, 33, 41, 49] has received increasing attention in recent years. We will apply granular computing techniques to solve GDM problems in different granular contexts, such as fuzzy set context [9, 12, 23], rough set context [2, 3, 8, 21], multi-granularity linguistic context [47], and interval type 2 fuzzy context [24]. Additionally, it is worth extending the proposed method in this chapter to solve GDM with IV-AIFPRs considering consensus among DMs, web contexts or dynamic contexts.

Appendix 1

211

Appendix 1 Proof In virtue of Eq. (6.9), one has MCIðRðhÞ Þ  0, which shows that the sequence fMCIðRðhÞ Þg has lower bound. Hence, to prove lim MCIðRðhÞ Þ ¼ 0, it only h! þ 1

needs to prove that the set fMCIðRðhÞ Þg is a monotone decreasing sequence, i.e., MCIðRðh þ 1Þ Þ  MCIðRðhÞ Þ for any h.    ðhÞ ðhÞ ðhÞ ðhÞ  For AIFPR RðhÞ , compute deviations rijk ¼ ln qij  ln qjk  ln qkj  for . ðhÞ ðhÞ ðhÞ ðhÞ i\k\j, where qij ¼ vij lij . Let rstf be the maximum deviation, i.e., . ðhÞ ðhÞ ðhÞ ðhÞ ðhÞ rstf ¼ max rijk , and its corresponding ratio qst ¼ vst lst should be 1  i\k\j  n

ðhÞ

ðhÞ

repaired. In other words, vst or lst should be repaired. To prove MCIðRðh þ 1Þ Þ  MCIðRðhÞ Þ, two cases are discussed below. ðhÞ

Case 1: vst is repaired by Eq. (6.12). In this case, it yields that ðh þ 1Þ qij

¼

ðh þ 1Þ vij

.

( ðh þ 1Þ lij

¼

ðhÞ

ðhÞ ðhÞ

ðqst Þd ðqsf qft Þ1d ; if ði; jÞ ¼ ðs; tÞ ðhÞ

qij ;

else

ð6:57Þ

According to Eq. (6.4), one has ðh þ 1Þ

rstf

   ðh þ 1Þ ðh þ 1Þ ðh þ 1Þ  ¼ ln qst  ln qsf  ln qft 

ð6:58Þ ðh þ 1Þ

Plugging Eq. (6.57) into (6.58), it follows that rstf ðhÞ ðhÞ

ðhÞ

  ðhÞ ¼ lnððqst Þd

ðhÞ

ðqsf qft Þ1d Þ  ln qsf  ln qft j. Namely, ðh þ 1Þ

rstf

   ðhÞ ðhÞ ðhÞ ðhÞ ðhÞ  ¼ d ln qst þ ð1  dÞlnqsf þ ð1  dÞlnqft  ln qsf  ln qft     ðhÞ ðhÞ ðhÞ  ðhÞ ¼ dln qst  lnqsf  lnqft  ¼ drstf :

If ði; jÞ 6¼ ðs; tÞ, while k 6¼ s; t; f , one has ðh þ 1Þ

rstk

   ðhÞ ðhÞ ðhÞ ðhÞ ðhÞ  ¼ d ln qst þ ð1  dÞlnqsf þ ð1  dÞlnqft  ln qsk  ln qkf     ðhÞ ðhÞ ðhÞ ðhÞ ðhÞ ðhÞ  ¼ ðd  1Þ ln qst þ ð1  dÞlnqsf þ ð1  dÞlnqft þ ln qst  ln qsk  ln qkf       ðhÞ  ðhÞ ðhÞ ðhÞ  ðhÞ ðhÞ  ðhÞ ðhÞ  ð1  dÞln qst  lnqsf  lnqft  þ ln qst  lnqsk  lnqkt  ¼ ð1  dÞrstf þ rstk ðh þ 1Þ

If ði; jÞ 6¼ ðs; tÞ, then rijk

ðhÞ

¼ rijk for any k 6¼ s; t.

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Consequently, MCIðRðh þ 1Þ Þ

MCIðRðhÞ Þ ¼ C1n3

P 1  i\j  n

ðh þ 1Þ

ðrijk

ðhÞ

 rijk Þ

That is MCIðRðh þ 1Þ Þ  MCIðRðhÞ Þ. ðhÞ

Case 2: lst is repaired by Eq. (6.15). In this case, the proving process is similar to that of Case 1 and omitted here. The proof is completed.

References 1. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 2. M. Antonelli, P. Ducange, B. Lazzerini, F. Marcelloni, Multi-objective evolutionary design of granular rule-based classifiers. Granul. Comput. 1(1), 37–58 (2016) 3. B. Apolloni, S. Bassis, J. Rota, G.L. Galliani, M. Gioia, L. Ferrari, A neuro fuzzy algorithm for learning from complex granules. Granul. Comput. 1, 1–22 (2016) 4. H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez, Z.S. Xu, B. Bedregal, J. Montero, H. Hagras, F. Herrera, B. De Baets, A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 24, 179–194 (2016) 5. F.J. Cabrerizo, E. Herrera-Viedma, W. Pedrycz, A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts. Eur. J. Oper. Res. 230, 624–633 (2013) 6. F. Chiclana, E. Herrera-Viedma, S. Alonso, F. Herrera, Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans. Fuzzy Syst. 17, 14–23 (2009) 7. F. Chiclana, J.M. Tapia Garcia, M.J. Del Moral, E. Herrera-Viedma, A statistical comparative study of different similarity measures of consensus in group decision making. Inf. Sci. 221, 110–123 (2013) 8. D. Ciucci, Orthopairs and granular computing. Granul. Comput. 1 (2016). https://doi.org/10. 1007/s41066-015-0013-y 9. D. Dubois, H. Prade, Bridging gaps between several forms of granular computing. Granul. Comput. 1(2), 115–126 (2016) 10. Y. Dong, E. Herrera-Viedma, Consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic GDM with preference relation. IEEE Trans. Cybern. 45, 780–792 (2015) 11. L. Dymova, P. Sevastjanov, A. Tikhonenko, Two-criteria method for comparing real-valued and interval-valued intuitionistic fuzzy values. Knowl. Based Syst. 45, 166–173 (2013) 12. V. Kreinovich, Solving equations (and systems of equations) under uncertainty: how different practical problems lead to different mathematical and computational formulationsGranul. Comput. 1 (2016). https://doi.org/10.1007/s41066-015-0014-x 13. H.C. Liao, Z.S. Xu, Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency. IEEE Trans. Fuzzy Syst. 22, 1669–1681 (2014) 14. H.C. Liao, Z.S. Xu, Some algorithms for group decision making with intuitionistic fuzzy preference information. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 22, 505–529 (2014) 15. H.C. Liao, Z.S. Xu, M.M. Xia, Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation. J. Intell. Fuzzy Syst. 27, 2969–2985 (2014)

References

213

16. H.C. Liao, Z.S. Xu, X.J. Zeng, J.M. Merigó, Framework of group decision making with intuitionistic fuzzy preference information. IEEE Trans. Fuzzy Syst. 23, 1211–1227 (2015) 17. H.C. Liao, Z.S. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process. Appl. Soft Comput. 35, 812–826 (2015) 18. H.C. Liao, Z.S. Xu, X.J. Zeng, D.L. Xu, An enhanced consensus reaching process in group decision making with intuitionistic fuzzy preference relations. Inf. Sci. 329, 274–286 (2016) 19. L. Livi, A. Sadeghian, Granular computing, computational intelligence, and the analysis of non-geometric input spaces. Granul. Comput. 1(1), 13–20 (2016) 20. P. Lingras, F. Haider, M. Triff, Granular meta-clustering based on hierarchical, network, and temporal connections. Granul. Comput. 1(1), 71–92 (2016) 21. H. Liu, A. Gegov, M. Cocea, Rule-based systems: a granular computing perspective. Granul. Comput. 1 (2016). https://doi.org/10.1007/s41066-016-0021-6 22. V. Loia, G. D’Aniello, A. Gaeta, F. Orciuoli, Enforcing situation awareness with granular computing: a systematic overview and new perspectives. Granul. Comput. 1(2), 127–143 (2016) 23. L. Maciel, R. Ballini, F. Gomide, Evolving granular analytics for interval time series forecasting. Granul. Comput. 1 (2016). https://doi.org/10.1007/s41066-016-0016-3 24. J.M. Mendel, A comparison of three approaches for estimating (synthesizing) an interval type-2 fuzzy set model of a linguistic term for computing with words. Granul. Comput. 1(1), 59–69 (2016) 25. F. Min, J. Xu, Semi-greedy heuristics for feature selection with test cost constraints. Granul. Comput. 1 (2016). https://doi.org/10.1007/s41066-016-0017-2 26. J.A. Morente-Molinera, I.J. Pérez, M.R. Ureña, E. Herrera-Viedma, Building and managing fuzzy ontologies with heterogeneous linguistic information. Knowl. Based Syst. 88, 154–164 (2015) 27. W. Pedrycz, in Granular Computing: Analysis and Design of Intelligent Systems (CRC Press, Francis Taylor, Boca Raton, 2013) 28. W. Pedrycz, S.M. Chen, Granular Computing and Decision-Making: Interactive and Iterative Approaches (Springer, Heidelberg, Germany, 2015) 29. I.J. Pérez, F.J. Cabrerizo, S. Alonso, E. Herrera-Viedma, A new consensus model for group decision making problems with non homogeneous experts. IEEE Trans. Syst. Man Cybern. Syst. 44, 494–498 (2014) 30. G. Peters, R. Weber, DCC: a framework for dynamic granular clustering. Granul. Comput. 1 (1), 1–11 (2016) 31. T.L. Saaty, The Analytic Hierarchy Process (McGraw-Hill, New York, 1980) 32. A. Skowron, A. Jankowski, S. Dutta, Interactive granular computing. Granul. Comput. 1(2), 95–113 (2016) 33. M. Song, Y. Wang, A study of granular computing in the agenda of growth of artificial neural networks. Granul. Comput. 1 (2016). https://doi.org/10.1007/s41066-016-0020-7 34. E. Szmidt, J. Kacprzyk, A consensus-reaching process under intuitionistic fuzzy preference relations. Int. J. Intell. Syst. 18, 837–852 (2003) 35. T. Tanino, Fuzzy preference orderings in group decision making. Fuzzy Sets Syst. 12, 117– 131 (1984) 36. J.M. Tapia García, M.J. Del Moral, M.A. Martínez, E. Herrera-Viedma, A consensus model for group decision making problems with linguistic interval fuzzy preference relations. Expert. Syst. Appl. 39, 10022–10030 (2012) 37. R. Ureña, F. Chiclana, H. Fujita, E. Herrera-Viedma, Confidence-consistency driven group decision making approach with incomplete reciprocal intuitionistic preference relations. Knowl. Based Syst. 89, 86–96 (2015) 38. S.P. Wan, G.L. Xu, F. Wang, J.Y. Dong, A new method for Atanassov’s interval-valued intuitionistic fuzzy MAGDM with incomplete attribute weight information. Inf. Sci. 316, 329–347 (2015)

214

6 A Novel Method for Group Decision Making …

39. S.P. Wan, F. Wang, J.Y. Dong, A novel group decision making method with intuitionistic fuzzy preference relations for RFID technology selection. Appl. Soft Comput. 38, 405–422 (2016) 40. S.P. Wan, F. Wang, J.Y. Dong, A preference degree for intuitionistic fuzzy values and application to multi-attribute group decision making. Inf. Sci. (2016). https://doi.org/10.1016/ j.ins.2016.07.013 41. G. Wilke, E. Portmann, Granular computing as a basis of human-data interaction: a cognitive cities use case. Granul. Comput. 1 (2016). https://doi.org/10.1007/s41066-016-0015-4 42. Z.S. Xu, J. Chen, An approach to group decision making based on interval-valued intuitionistic judgment matrices. Syst. Eng. Theory Pract. 27, 126–133 (2007) 43. Z.S. Xu, X.Q. Cai, E. Szmidt, Algorithms for estimating missing elements of incomplete intuitionistic preference relations. Int. J. Intell. Syst. 26, 787–813 (2011) 44. Z.S. Xu, X.Q. Cai, Incomplete interval-valued intuitionistic fuzzy preference relations. Int. J. Gen Syst 38, 871–886 (2009) 45. Z.S. Xu, H.C. Liao, Intuitionistic fuzzy analytic hierarchy process. IEEE Trans. Fuzzy Syst. 22, 749–761 (2014) 46. Z.S. Xu, X.Q. Cai, Group decision making with incomplete interval-valued intuitionistic preference relations. Group Decis. Negot. 24, 193–215 (2015) 47. Z.S. Xu, H. Wang, Managing multi-granularity linguistic information in qualitative group decision making: an overview. Granul. Comput. 1(1), 21–35 (2016) 48. G.L. Xu, S.P. Wan, F. Wang, J.Y. Dong, Y.F. Zeng, Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations. Knowl. Based Syst. 98, 30–43 (2016) 49. Y.Y. Yao, A triarchic theory of granular computing. Granul. Comput. 1(2), 145–157 (2016) 50. F. Zhang, J. Ignatius, C.P. Lim, Y.J. Zhao, A new method for deriving priority weights by extracting consistent numerical-valued matrices from interval-valued fuzzy judgement matrix. Inf. Sci. 279, 280–300 (2014)

Chapter 7

Additive Consistent Interval-Valued Atanassov Intuitionistic Fuzzy Preference Relation and Likelihood Comparison Algorithm Based Group Decision Making

Abstract This chapter investigates a group decision making (GDM) method based on additive consistent interval-valued Atanassov intuitionistic fuzzy (IVAIF) preference relations (IVAIFPRs) and likelihood comparison algorithm. Firstly, the likelihood of IVAIF values (IVAIFVs) is defined by the likelihood of intervals. Then a likelihood comparison algorithm is designed to rank IVAIFVs. According to the additive consistent interval fuzzy preference relation, we define the additive consistency of an IVAIFPR. Two special interval fuzzy preference relations are extracted from an IVAIFPR. They can be regarded as the lowest and highest preferred matrices of the IVAIFPR, respectively. Using a parametric linear programming model, the IVAIF priority weights of an IVAIFPR are generated from these two extracted special interval fuzzy preference relations. For the GDM with IVAIFPRs, the group consensus is defined by the distances between the individual IVAIFPRs and the collective one. To derive decision makers’ weights, an optimization model is constructed by maximizing the group consensus and transformed into a linear programming model to resolve. Subsequently, utilizing the IVAIF weighted averaging operator, the collective IVAIFPR is obtained and applied to obtain the IVAIF priority weights. The order of alternatives is generated by ranking the IVAIF priority weights. At length, an enterprise resource planning system selection example is analyzed to verify the effectiveness of the proposed method.




Keywords Decision support systems Interval-valued atanassov intuitionistic fuzzy preference relation Additive consistency Group decision making Group consensus




7.1







Introduction

Decision making almost happens in every aspect of life in today’s society. However, with the increasing complexity in decision making problems, group decision making (GDM) has been brought into being because the single decision © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_7

215

216

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

maker (DM or expert) cannot evaluate such problems in the round [1–4]. In GDM problems, once the feasible alternatives are determined, DMs would be invited to provide their opinions on these alternatives. By pairwise comparison over the alternatives, the preference relation (judgment matrix) can be formed by each DM. Now, preference relations have been widely applied to GDM because of the flexible structure and human’s innate ability to make relative comparisons. The classical preference relation consists of fuzzy preference relation [5, 6] and multiplicative preference relation [7], where the preference information on pairwise comparisons is expressed by numerical values. DMs’ hesitations or indeterminacies are ignored in these preference relations. To address this issue, interval fuzzy preference relation (IFPR) [8–12], Atanassov intuitionistic fuzzy preference relation (AIFPR) [13– 17] and interval-valued Atanassov intuitionictic fuzzy (IVAIF) preference relation (IVAIFPR) [17–23] appear one after another. Especially, the elements in an IVAIFPR are described by IVAIF values (IVAIFVs) [17], in which both the membership and non-membership degrees are intervals [15, 24–27]. Thus, the IVAIFPR can capture uncertain information flexibly in the decision making. In addition, by the transformation operators in [28], the IVAIFPR can be transformed into AIFPR and IFPR, respectively. Therefore, the GDM with IVAIFPRs is an important and interesting topic. In general, the GDM with IVAIFPRs contains three key issues. (1) Consistency of IVAIFPRs The consistent IVAIFPR provided by a DM contains no contradiction. Owing to the limitation of DMs’ knowledge, cognition and time pressure, it is nearly impossible for DMs to provide a consistent IVAIFPR in the real-life decision making problems. The inconsistency, as a herald of improper judgment, would lead to incoherent decision results. Thus, the consistency of an IVAIFPR should be checked and improved in decision making. In this process, the concept of the consistent IVAIFPR should be defined firstly. (2) Group consensus reaching process in GDM The group consensus reaching process is used to find a final solution that is supported by all group members despite their different opinions. In this process, the group consensus should be defined and then the consensus of DMs’ opinion needs to be reached. The higher the consensus of the decision group, the more reliable the final decision making result. (3) Selection process of alternatives The aim of the selection process is to find the final solution that is accepted by most of individuals. Some aggregation operators and ranking methods are applied to this process. Up to date, there exists little research on GDM with IVAIFPRs owing to the computational complexity of IVAIFVs and difficult solving process of GDM. Xu and Chen [23] proposed an approach to GDM with IVAIFPRs by aggregation operators. Xu and Yager [17] developed a similarity measure between AIFPRs and extended it to IVAIFPRs. Xu and Cai [22] defined the additive consistent

7.1 Introduction

217

incomplete IVAIFPR and multiplicative consistent incomplete IVAIFPR. Then two algorithms were proposed to extend the given acceptable incomplete IVAIFPRs into complete ones. Different from the consistency in Xu and Cai [22], Xu and Cai [18] defined the IVAIFPR with multiplicative transitivity and presented an algorithm to estimate all the missing elements. Then an approach was proposed to GDM with incomplete IVAIFPRs. Wu and Chiclana [21] developed an attitudinal based expected score function of an IVAIF set and prioritisation method for IVAIFPR. By constructing the score judgment matrix of IVAIFPR, Wu et al. [20] presented an approach for multi-criteria decision making with IVAIFPRs. Motivated by the multiplicative transitivity of fuzzy preference relation, Liao et al. [19] introduced multiplicative transitivity for IVAIFPRs and developed some iterative algorithms to adjust or repair the inconsistent IVAIFPRs. Then, a convergent iterative approach was proposed for GDM with IVAIFPRs. Although these methods are effective to solve decision making problems with IVAIFPRs, they also have some limitations on the aforesaid three key issues of GDM with IVAIFPRs. (1) Methods [17, 21, 23] did not take the consistency of IVAIFPRs into consideration. Although the consistency definitions of IVAIFPRs or incomplete IVAIFPRs are given in methods [18, 19, 22], these definitions are the straightforward extensions of consistency of fuzzy preference relation. They are not based on the IVAIF judgments directly and thus are not easy to be used and may not sufficiently represent the original IVAIF preference information. In particular, when an IVAIFPR is reduced to an AIFPR, the multiplicative consistency of IVAIFPR defined in [19] just degenerates to a consistency definition of AIFPR. Liao and Xu [16] pointed out that the degenerated consistency definition of AIFPR may be too strict for an AIFPR and unreasonable. (2) Existing research [17, 20–22] only focuses on some aspects about IVAIFPRs and neglects the decision methods for solving GDM problems. The GDM methods [18, 23] overlook the group consensus of IVAIFPR. Methods [17, 19] measure the consensus by comparing the threshold with the distance or similarity between the individual IVAIFPR and the collective one. However, it is difficult for DMs to provide a proper threshold in real-world decision making problems. (3) To rank alternatives, existing methods [18, 19, 23] adopt aggregation operators to obtain the collective alternative values, which may result in loss of information. It is more reliable to use the priority weights of IVAIFPR for ranking alternatives. Moreover, during the aggregation process, DMs are allocated to equal weight (importance) a priori, which is a little bit subjective and may run counter to the actual decision situation. In addition, how to rank the priority weights is also an important issue during the decision making process. Bentkowska et al. [9] studied the admissible order of intervals and defined a particular way of obtaining admissible orders. Subsequently, De Miguel et al. [24] introduced a method for constructing linear orders between

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intervals using aggregation functions and adapted it to suit the case of IVAIF sets. Then two different constructions of IVAIF admissible orders are presented. Most of existing ranking methods of IVAIF sets are the particular instances of these admissible orders. To make up these limitations, this chapter focuses on developing a new GDM method with IVAIFPRs. The likelihood of IVAIFVs is firstly defined by extending the likelihood of intervals. A new likelihood comparison algorithm is designed to rank IVAIFVs. Then the additive consistency of an IVAIFPR is defined according to the additive consistency of IFPR. To derive DMs’ weights for GDM, an optimization model is constructed by maximizing the group consensus and transformed into a linear programming model to resolve. The collective IVAIFPR is obtained by aggregating the individual IVAIFPRs using the IVAIF weighted averaging operator. Two special IFPRs are extracted from the collective IVAIFPR. Solving a parametric linear programming model, two interval priority weight vectors are respectively obtained from these two IFPRs. Then the IVAIF priority weights of IVAIFPR are derived to generate the ranking order of alternatives by the likelihood comparison algorithm of IVAIFVs. Compared with existing research, the main innovations of this chapter are highlighted as four aspects: (1) A most widely used method for ranking of IVAIFVs in literature [18, 19, 21– 23] is based on the score function and accuracy function. Such a ranking method only utilizes the endpoints of membership and non-membership of IVAIFVs, which may result in great loss of information and unreasonable results. In this chapter, an IVAIFV is first transformed into an interval by considering the risk attitude of DM. Then, the likelihood of IVAIFVs is defined as the integral of the likelihood of intervals within a full coverage of all possible values of risk attitude parameter. Thus, a likelihood comparison algorithm is designed to rank IVAIFVs. Because of integrating all possible values of risk attitude parameter, the designed algorithm can not only adequately express the uncertainty of IVAIFVs, but also effectively reduce the loss of information. (2) Existing consistency definitions of IVAIFPRs [18, 19, 22] are based on that of fuzzy preference relations. These definitions only consider the consistency of fuzzy preference relations formed independently by the four endpoints of membership and non-membership of IVAIFVs. They may not sufficiently capture the original IVAIF preference information and possibly cause distortion of original information. This chapter defines an additive consistency of an IVAIFPR based on the additive consistent IFPR. This new definition can reduce to the consistencies of IFPR and AIFPR, which is justifiable and reasonable. The IVAIF priority weights are generated from two special IFPRs extracted from the IVAIFPR. Therefore, the new consistency definition of IVAIFPR and the IVAIF priority weights can sufficiently capture the original IVAIF preference information and remarkably decrease distortion of original information. (3) Although methods [17, 19] measure the consensus in GDM, it is difficult for DMs to provide a proper threshold. In this chapter, the group consensus is

7.1 Introduction

219

defined by the distances between the individual IVAIFPRs and the collective one without need of any thresholds. To derive DMs’ weights, an optimization model is constructed by maximizing the group consensus and transformed into a linear programming model to resolve. Generally speaking, DMs’ weights should be determined according to the consensus level of their judgments. The derivation of DMs’ weights well reflects this idea and is more convincing. (4) Using the IVAIF weighted averaging operator, the collective IVAIFPR is obtained and applied to derive the IVAIF priority weights. Utilizing the designed likelihood comparison algorithm of IVAIFVs, the order of alternatives is generated by ranking IVAIF priority weights. Subsequently, a method is proposed to solve GDM with IVAIFPRs. The rest of this chapter is organized as follows. In Sect. 7.2, some definitions about preference relations are reviewed. In Sect. 7.3, a likelihood comparison algorithm is designed to rank IVAIFVs. Section 7.4 is devoted to the consistency analysis for IVAIFPR. The additive consistency of IVAIFPR is defined and applied to obtain the IVAIF priority weights. Section 7.5 proposes a new method for GDM with IVAIFPRs. In Sect. 7.6, an example of enterprise resource planning (ERP) system selection is presented to illustrate the proposed method. Finally, we draw our conclusions in Sect. 7.7.

7.2

Preliminaries

In this section, some definitions about preference relations are reviewed. Definition 7.1 [29] A fuzzy reciprocal preference relation A on the alternative set X ¼ fx1 ; x2 ; . . .; xn g is represented by a fuzzy reciprocal preference matrix A ¼ ðnij Þnn  X  X, where nij denotes the fuzzy preference degree or the intensity of alternative xi over xj , satisfying that 0  nij  1, nij þ nji ¼ 1, nii ¼ 0:5 for all i; j ¼ 1; 2; . . .; n. Definition 7.2 [11] An IFPR R on the alternative set X ¼ fx1 ; x2 ; . . .; xn g is denoted by an interval fuzzy judgment matrix R ¼ ðrij Þnn  X  X, where rij ¼ ½r ij ; rij  is an interval which means that the preference degree of alternative xi over xj is between r ij and rij . Furthermore, r ij and rij fulfill the conditions that 0  r ij ; rij  1, r ij þ rji ¼ r ji þ rij ¼ 1, r ii ¼ rii ¼ 0:5 for all i; j ¼ 1; 2; . . .; n. Definition 7.3 [30] An AIFPR R_ on set X is denoted by an Atanassov intuitionistic fuzzy judgment matrix R_ ¼ ð_rij Þnn  X  X with r_ ij ¼ \lij ; mij [; where r_ ij is called an Atanassov intuitionistic fuzzy value (AIFV), lij and mij are the degrees to which xi is preferred and non-preferred to xi respectively, and pij ¼ 1  lij  mij is represented as the hesitation degree to which which xi is preferred to xj .

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

220

Furthermore, lij and mij satisfy the conditions that 0  lij þ mij  1, lij ¼ mji , mij ¼ lji , lii ¼ mii ¼ 0:5 for all i; j ¼ 1; 2; . . .; n. ~ in Z is defined as A ~ ¼ fðz; ½l ðzÞ; l A~ ðzÞ; Definition 7.4 [31] An IVAIF set A ~ A A~ ðzÞ½0; 1 and ½vA~ ðzÞ; vA~ ðzÞ½0; 1 repre½vA~ ðzÞ; vA~ ðzÞÞjz 2 Z g, where ½lA~ ðzÞ; l sent the membership degree interval and non-membership degree interval of ele~ respectively, satisfying l A~ ðzÞ þ vA~ ðzÞ  1 for any z 2 Z. ment z to IVAIF set A, A~ ðzÞ  vA~ ðzÞ; 1  lA~ ðzÞ  vA~ ðzÞ is an IVAIF index of element Especially, ½1  l z 2 Z. ~a ðzÞ; ½v~a ðzÞ; v~a ðzÞÞ is called an IVAIFV [17] and simply The pair ~a ¼ ð½l~a ðzÞ; l ; ½m; mÞ, where ½l; l ½0; 1, ½m; m½0; 1 and l  þ m  1. denoted by ~a ¼ ð½l; l i ; ½mi ; mi Þði ¼ 1; 2Þ, the normalized Hamming For two IVAIFVs ~ai ¼ ð½li ; l distance between ~ a1 and ~a2 is defined as [32]: 1 4

2 j þ jm1  m2 j þ jm1  m2 j þ jp1  p2 j dð~a1 ; ~a2 Þ ¼ ðjl1  l2 j þ j l1  l 2 jÞ; þ j p1  p i  mi where pi ¼ 1  l 0  dð~a1 ; ~a2 Þ  1.

and

i ¼ 1  li  mi ði ¼ 1; 2Þ. p

It

is

clear

ð7:1Þ that

~ on set X is denoted by an IVAIF judgment Definition 7.5 [23] An IVAIFPR R ~ ij ; ½mij ; mij Þ is an IVAIFV, ½lij ; l ij  matrix R ¼ ð~rij Þnn  X  X, where ~rij ¼ ð½lij ; l and ½mij ; mij  are the degrees to which alternative xi is preferred and non-preferred to ij  ¼ ½1  l ij  vij ; 1  lij  vij  is the hesialternative xj respectively, and ½pij ; p

tancy degree to which alternative xi is preferred to alternative xj . Furthermore, ij  ij   ½0; 1, ½lij ; l and ½mij ; mij  satisfy that ½lij ; l ½mij ; mij   ½0; 1, ii  ¼ ½mii ; mii  ¼ ½0:5; 0:5, ½lii ; l

ji  for all i; j ¼ 1; 2; . . .; n. ½lji ; l

7.3

ij þ mij  1, 0l

ij  ¼ ½mji ; mji , ½lij ; l

½mij ; mij  ¼

A New Likelihood Comparison Algorithm of IVAIFVs

This section proposes a likelihood comparison algorithm to rank IVAIFVs. ~ ¼ fðz; ½l ðzÞ; l A~ ðzÞ; ½vA~ ðzÞ; vA~ ðzÞÞjz 2 Z g, the operator Ck For an IVAIF set A ~ A is defined as follows [28]: ~ ¼ fðz; ½l ðzÞ þ kð1  v ~ ðzÞ  l ðzÞÞ; l A~ ðzÞ þ kð1  vA~ ðzÞ Ck ðAÞ ~ ~ A A A A~ ðzÞÞÞjz 2 Z g; l

ð7:3Þ

~ is an interval-valued fuzzy set. where the parameter k 2 ½0; 1. Obviously, Ck ðAÞ

7.3 A New Likelihood Comparison Algorithm of IVAIFVs

221

; ½m; mÞ is transformed into the interval By Eq. (7.3), an IVAIFV ~a ¼ ð½l; l  þ kð1  m  l Þ, where the parameter k 2 ½0; 1 Ck ð~aÞ ¼ ½l þ kð1  m  lÞ; l  can be regarded as the reflects DM’s risk attitude. When k ¼ 0, Ck ð~aÞ ¼ ½l; l aÞ ¼ ½1  m; 1  m can be lower preferred degree of IVAIFV ~a. When k ¼ 1, Ck ð~ regarded as the upper preferred degree of IVAIFV ~ a. Thus, for an IVAIFV ~a ¼ ð½l; l ; ½m; mÞ, its preferred degree can be rewritten as a pair of closed intervals , ½1  m; 1  m) with l   1  m. (½l; l i ; ½mi ; mi Þði ¼ As mentioned above, the preferred degree of IVAIFVs ~ ai ¼ ð½li ; l þ 1; 2Þ can be transformed into the interval ai ðkÞ ¼ ½a ðkÞ; a i ðkÞ where i i þ kð1  mi  l i Þ and k 2 ½0; 1. Hence, mi  li Þ, aiþ ðkÞ ¼ l a i ðkÞ ¼ li þ kð1   comparing IVAIFVs is converted into comparing the corresponding intervals. The likelihood of intervals a1 ðkÞ [ a2 ðkÞ (i.e., interval a1 ðkÞ is greater than a2 ðkÞ) is proposed by Xu and Da [33] as

Lða1 ðkÞ [ a2 ðkÞÞ ¼ max 1  max

a2þ ðkÞ  a 1 ðkÞ ;0 þ þ  a2 ðkÞ  a 2 ðkÞ þ a1 ðkÞ  a1 ðkÞ

  ;0

ð7:4Þ

In many actual decision making problems, it is difficult for DMs to evaluate their risk attitude accurately, especially using numeric values. It is necessary to consider all possible values of risk attitude, i.e., all the values of k from 0 to 1. Thus, the likelihood of IVAIFVs ~a1 [ ~a2 (i.e., IVAIFV ~a1 is greater than ~ a2 ) can be obtained by synthesizing the likelihood of intervals a1 ðkÞ [ a2 ðkÞ with all the possible values of k. i ; ½mi ; mi Þði ¼ 1; 2Þ, the likelihood of Definition 7.6 For two IVAIFVs ~ai ¼ ð½li ; l IVAIFVs ~a1 [ ~a2 is defined as ^ a1 [ ~a2 Þ ¼ Lð~

Z1 Lða1 ðkÞ [ a2 ðkÞÞdk;

ð7:5Þ

0

where Lða1 ðkÞ [ a2 ðkÞÞ is the likelihood of intervals a1 ðkÞ [ a2 ðkÞ calculated by Eq. (7.4). The detailed computing results of Eq. (7.5) are presented in Appendix 1. ^ a1 [ ~a2 Þ measures the possibility of IVAIFV ~ It is clear that Lð~ a1 being greater than ~a2 . Moreover, the likelihood of IVAIFVs possesses the following desirable properties: ^ a1 [ ~a2 Þ  1; (1) 0  Lð~ ^ (2) Lð~a1 [ ~a1 Þ ¼ 0:5; ^ a2 [ ~a1 Þ ¼ 1. ^ a1 [ ~a2 Þ þ Lð~ (3) Lð~ ^ a1 [ ~a2 Þ is depicted in Fig. 7.1. The likeliThe geometric interpretation of Lð~ ^ hood Lð~a1 [ ~a2 Þ is the area of the region closed by the curve of Lða1 ðkÞ [ a2 ðkÞÞ, the k-axis and the vertical lines k ¼ 0 and k ¼ 1.

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7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

Fig. 7.1 Geometric interpretation for the likelihood of IVAIFVs

1

Lˆ (α 2 > α1 )

L(α1 (λ ) > α 2 (λ ))

1

0

λ

~i ¼ Based on the likelihood of IVAIFVs, an order relation of IVAIFVs a i ; ½mi ; mi Þði ¼ 1; 2Þ is given as: ð½li ; l ^ a1 [ ~a2 Þ [ 12, then ~a1 is greater than ~a2 , denoted by ~ If Lð~ a1 [ ~ a2 ; ^ a1 [ ~a2 Þ ¼ 12, then ~a1 is indifferent to ~a2 , denoted by ~ If Lð~ a1 ~ a2 ; 1 ^ If Lð~a1 [ ~a2 Þ\2, then ~a1 is smaller than ~a2 , denoted by ~ a1 \~ a2 . i ; ½mi ; mi Þði ¼ 1; 2; . . .; nÞ, a likelihood To rank a series of IVAIFVs ~ai ¼ ð½li ; l ^ ¼ ðlij Þ ^ ai [ ~ matrix L aj Þði; j ¼ 1; 2; . . .; nÞ are nn is constructed, where lij ¼ Lð~ calculated by Eq. (7.5). Then the dominance and non-dominance degrees of IVAIFV ~ai are employed to rank IVAIFVs. The non-dominance degree of IVAIFV ~ai denotes the degree to which IVAIFV ~ai is not dominated by the others. It is defined as NDi ¼ minf1  lsji g; j6¼i

ð7:6Þ

where lsji ¼ maxflji  lij ; 0g represents the degree to which ~ ai is strictly dominated by ~aj . The dominance degree of IVAIFV ~ai is defined as Di ¼

1 n1

n X

lij ;

ð7:7Þ

j¼1;j6¼i

~i over the others. which can quantify the dominated degree of IVAIFV a It becomes apparent that the larger both the dominance degree Di and the non-dominance degree NDi , the better the IVAIFV ~ ai . Thus, combined the non-dominance degree with dominance degree, a ranking index of IVAIFV ~ ai is defined as 1 2

RDi ¼ ðDi þ NDi Þ:

ð7:8Þ

Hence, IVAIFVs ~ai ði ¼ 1; 2; . . .; nÞ can be ranked by the descending order in accordance with the values of RDi ði ¼ 1; 2; . . .; nÞ.

7.3 A New Likelihood Comparison Algorithm of IVAIFVs

223

According to the aforementioned analysis, a likelihood comparison algorithm for IVAIFVs can be summarized as follows: Algorithm: A likelihood comparison algorithm for IVAIFVs ˆ Step 1 Calculate the likelihood lij Lˆ ( i j ) by Eq. (7.5) and form the likelihood matrix L = (lij ) n×n . Step 2 Compute the non-dominance degree NDi of IVAIFV

i

(i = 1, 2,

Step 3 Use Eq. (7.7) to calculate the dominance degree Di of IVAIFV Step 4 Determine the ranking index RDi of IVAIFV Step 5 The ranking order of IVAIFVs

7.4

i

(i = 1, 2,

i

(i = 1, 2,

, n) by Eq. (7.6). i

(i = 1, 2,

, n) .

, n) according to Eq. (7.8).

, n) is generated by descending the ranking indices.

Additive Consistency Analyses for IVAIFPR

In this section, the additive consistency of an IVAIFPR is defined based on the additive consistency of IFPR. Thereby, the IVAIF priority weights are derived by two special IFPRs extracted from the IVAIFPR.

7.4.1

Additive Consistency Definition of IVAIFPR

Consistency is a critical criterion to measure the level of agreement among the preference information provided by DMs. An IVAIFPR with low consistency would lead to the unreasonable results in decision making. It is necessary to discuss the consistency of IVAIFPR. ~ ¼ ð~rij Þ ij ; ½mij ; mij Þ, the lowest preferred For an IVAIFPR R rij ¼ ð½lij ; l nn with ~ ij  and the highest preferred degree of degree of pair of alternatives ðxi ; xj Þ is ½lij ; l

ij  and ½1  pair of alternatives ðxi ; xj Þ is ½1  mij ; 1  mij . That is to say, ½lij ; l

mij ; 1  mij  represent the lower bound and upper bound of preferred degrees, respectively. Therefore, the preferred degree of pair of alternatives ðxi ; xj Þ, denoted by ½nij ; gij , ij  and ½1  mij ; 1  mij . In other words, the preferred should lie between ½lij ; l ij   ½nij ; gij   ½1  mij ; 1  mij  where the degree should satisfied the condition ½lij ; l

relation “  ” mean “is not larger than” [34]. Accordingly, nij and gij should satisfy ij  gij  1  mij by Definition 3.6 in Wang et al. [34]. lij  nij  1  mij and l Noticing

that

ij , lij  l

ij  1  mij  1  mij . lij  l

mij  mij

and

ij þ mij  1, 0l

one

has

As a result, an IFPR R ¼ ðrij Þnn ¼ ð½nij ; gij Þnn can be extracted from the ~ ¼ ð~rij Þ , where l  nij  1  mij , l ij  gij  1  mij , nij þ gji ¼ 1 and IVAIFPR R nn ij

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

224

nii ¼ gii ¼ 0:5 for all i; j ¼ 1; 2; . . .; n. Therefore, the additive consistency of an IVAIFPR can be defined as follows. ~ ¼ ð~rij Þ ij ; ½mij ; mij Þ is additive rij ¼ ð½lij ; l Definition 7.7 An IVAIFPR R nn with ~

consistent iff there exists an additive consistent IFPR R ¼ ðrij Þnn with rij ¼ ½nij ; gij  ij  gij  1  mij for all i; j ¼ 1; 2; . . .; n. satisfying lij  nij  1  mij and l ~ ¼ ð~rij Þ ij ¼ lij and mij ¼ mij ¼ mij , the IVAIFPR R Especially, when lij ¼ l nn is reduced to an AIFPR. In this case, Definition 7.7 is degenerated to the additive consistency of an AIFPR below. Definition 7.8 An AIFPR R_ ¼ ð_rij Þnn with r_ ij ¼ \lij ; mij [ is additive consistent iff there exists an additive consistent fuzzy preference relation A ¼ ðnij Þnn satisfying lij  nij  1  mij for all i; j ¼ 1; 2; . . .; n. Since AIFV r_ ij ¼ \lij ; mij [ is usually treated equivalently as interval rij ¼ ½lij ; 1  mij , AIFPR R_ ¼ ð_rij Þnn with r_ ij ¼ \lij ; mij [ can be viewed as IFPR R ¼ ðrij Þnn with rij ¼ ½lij ; 1  mij . Thus, Definition 7.8 just coincides with Definition 5 of additive consistency of IFPR in Zhang et al. [35]. Therefore, Definitions 7.7 and 7.8 generalize the additive consistency of IFPR [35]. This observation indicates the feasibility and reasonability of Definitions 7.7 and 7.8.

7.4.2

Derive the IVAIF Priority Weights from IVAIFPR

After defining the additive consistency of IVAIFPR, the next problem that needs to be addressed is to derive the priority weights from IVAIFPR. Meng et al. [8] claimed that it is natural to derive the priority vector from IFPR based on different consistent fuzzy preference relations. Since it is not sure that a DM has the same uncertainty for all his/her judgments, the priority weights of IVAIFPR should also be obtained on the basis of different consistent IFPRs. As per Definition 7.7, there ~ ¼ ð~rij Þ . To facilexist a great number of IFPRs extracted from an IVAIFPR R nn itate the operation, two special IFPRs are extracted from an IVAIFPR for generating the IVAIF priority weights in what follows. ij   ½nij ; gij   ½1  mij ; 1  mij , two special matrices Rl ¼ Given that ½lij ; l l ~ ¼ ð~rij Þ , respecðr Þ and Rm ¼ ðr m Þ are extracted from the IVAIFPR R ij nn

ij nn

nn

tively, where 8 ij ; if i\j > < ½lij ; l l ½0:5; 0:5; if i¼j rij ¼ > : ½1  l ji ; 1  lji ; if i [ j

ð7:9Þ

7.4 Additive Consistency Analyses for IVAIFPR

8 < ½1  mij ; 1  mij ; rijm ¼ ½0:5; 0:5; : ½m ; m ; ji ji

225

if i\j if i ¼ j if i [ j

ð7:10Þ

It is easy to verify from Definition 7.3 that both Rl and Rm are IFPRs. As far as the elements in the upper triangular part of matrices are concerned, the IFPR ~ because r l ¼ Rl ¼ ðrijl Þnn can be regarded as the lowest preferred matrix of R ij ij  is the minimum of rij ¼ ½nij ; gij  for i\j. Similarly, the IFPR Rm ¼ ðrijm Þnn ½lij ; l ~ because r m ¼ ½1  mij ; 1  mij  is can be regarded as the highest preferred matrix of R ij

the maximum of rij ¼ ½nij ; gij  for i\j. Thus, the matrices Rl and Rm , as two special IFPRs extracted from the IVAIFPR ~ R, should participate in the determination of the priority weights of alternatives. Let þ w ¼ ðw1 ; w2 ; . . .; wn ÞT be an interval priority weight vector, where wi ¼ ½w i ; wi  reflects the importance of alternative xi . The IFPR R ¼ ðrij Þnn ¼ ð½nij ; gij Þnn is additive consistent if there exists a normalized interval priority weight vector w ¼ ðw1 ; w2 ; . . .; wn ÞT satisfying [36]
rij ¼

½0:5; 0:5; if i ¼ j þ þ  ½0:5ðw  w þ 1Þ; 0:5ðw  w þ 1Þ; if i 6¼ j j i i j

ð7:11Þ

In general, it is nearly impractical for DMs to provide the absolutely consistent IVAIFPRs. In this way, it is difficult to extract an additive consistent IFPR from an IVAIFPR. A reasonable method to derive the priority weight vector w ¼ ðw1 ; w2 ; . . .; wn ÞT is to minimize all the deviations between ½0:5ðw i  þ þ  wj þ 1Þ; 0:5ðwi  wj þ 1Þ and ½nij ; gij  ði; j ¼ 1; 2; . . .; n; i\jÞ. The deviations þ þ  between ½0:5ðw i  wj þ 1Þ; 0:5ðwi  wj þ 1Þ and ½nij ; gij  can be introduced as þ þ  dij ¼ j0:5ðw i  wj þ 1Þ  nij j and eij ¼ j0:5ðwi  wj þ 1Þ  gij j, respectively. Motivated by the idea of mathematical programming model with p-metric [5, 37], a minimization deviation model is constructed as follows: min T ¼

n1 X n  X i¼1 j¼i þ 1

p p þ þ  j0:5ðw i  wj þ 1Þ  nij j þ j0:5ðwi  wj þ 1Þ  gij j

1=p

8  Pn þ > < wi þ Pj¼1;j6¼i wj 1 ði ¼ 1; 2; . . .; nÞ s:t: wiþ þ nj¼1;j6¼i w j  1 ði ¼ 1; 2; . . .; nÞ > : þ 0  w i  wi  1 ði ¼ 1; 2; . . .; nÞ ð7:12Þ The constraints of Eq. (7.12) are the conditions of the normalized interval weight vector [12]. The parameter p reflects the importance assigned to the largest deviation. As p increases, more importance is assigned to the largest deviation [37].

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

226

In particular, when p ¼ 1, Eq. (7.12) is rewritten as min

n1 X n   X þ þ  j0:5ðw  w þ 1Þ  n j þ j0:5ðw  w þ 1Þ  g j ij ij i j i j i¼1 j¼i þ 1

8  Pn þ > < wi þ Pj¼1;j6¼i wj 1 ði ¼ 1; 2; . . .; nÞ s:t: wiþ þ nj¼1;j6¼i w j  1 ði ¼ 1; 2; . . .; nÞ > : þ 0  w i  wi  1 ði ¼ 1; 2; . . .; nÞ

ð7:13Þ

When p ! þ 1, Eq. (7.12) is converted into þ þ  min maxfj0:5ðw i  wj þ 1Þ  nij j; j0:5ðwi  wj þ 1Þ  gij jg i;j [ i 8  Pn þ > < wi þ Pj¼1;j6¼i wj 1 ði ¼ 1; 2; . . .; nÞ s:t: wiþ þ nj¼1;j6¼i w j  1 ði ¼ 1; 2; . . .; nÞ > : þ 0  w i  wi  1 ði ¼ 1; 2; . . .; nÞ

ð7:14Þ

The objective function in Eq. (7.13) is to minimize the sum of all deviations dij and eij ði; j ¼ 1; 2; . . .; n; i\jÞ, which is based on the majority principle similar to the model in Wang and Li [36]. This case would lead to a more robust estimation. Different from Eq. (7.13), the objective function in Eq. (7.14) is to minimize the maximum deviation among dij and eij ði; j ¼ 1; 2; . . .; n; i\jÞ, which is based on the minority principle. This case would result in a more sensitive estimation to extreme deviation [5]. Combining the above two particular cases (p ¼ 1 and p ! þ 1), a parametric mathematical programming model is constructed by introducing parameter w as follows: min w

n1 X n X i¼1 j¼i þ 1

þ þ  ðj0:5ðw i  wj þ 1Þ  nij j þ j0:5ðwi  wj þ 1Þ  gij jÞ

þ þ  þ ð1  wÞ maxfj0:5ðw i  wj þ 1Þ  nij j; j0:5ðwi  wj þ 1Þ  gij jg i;j [ i ð7:15Þ 8  Pn þ þ w

1 ði ¼ 1; 2; . . .; nÞ w > j¼1;j6¼i j < i P s:t: wiþ þ nj¼1;j6¼i w j  1 ði ¼ 1; 2; . . .; nÞ > : þ  0  wi  wi  1 ði ¼ 1; 2; . . .; nÞ

where w is a control parameter that trades off majority versus minority principles satisfying 0  w  1. Equation (7.15) can be regarded as an integrated model which includes the above two particular cases (i.e., Eqs. (7.13) and (7.14)).

7.4 Additive Consistency Analyses for IVAIFPR

227

To solve Eq. (7.15), let þ þ  u ¼ maxfj0:5ðw i  wj þ 1Þ  nij j; j0:5ðwi  wj þ 1Þ  gij jg; i;j [ i

1 2 1  þ  þ dij ¼ ðj0:5ðw i  wj þ 1Þ  nij j  ð0:5ðwi  wj þ 1Þ  nij ÞÞ; 2 1 þ  eijþ ¼ ðj0:5ðwiþ  w j þ 1Þ  gij j þ ð0:5ðwi  wj þ 1Þ  gij ÞÞ; 2 1 þ  þ  e ij ¼ 2ðj0:5ðwi  wj þ 1Þ  gij j  ð0:5ðwi  wj þ 1Þ  gij ÞÞ;

dijþ

þ  þ ¼ ðj0:5ðw i  wj þ 1Þ  nij j þ ð0:5ðwi  wj þ 1Þ  nij ÞÞ;

Thus, Eq. (7.15) turns into a parametric linear programming model as follows: min T ¼ w

n1 X n X i¼1 j¼j þ 1

s:t:

ðdijþ þ dij þ eijþ þ e ij Þ þ ð1  wÞu

P 8  Pn þ  wi þ j¼1;j6¼i wjþ 1; wiþ þ nj¼1;j6¼i w > j  1; 0  wi  wi  1 ði ¼ 1; 2; . . .; nÞ > > > < 0:5ðw  w þ þ 1Þ  nij  d þ þ d  ¼ 0 ði; j ¼ 1; 2; . . .; n; i\jÞ j ij i ij þ  > 0:5ðwiþ  w > j þ 1Þ  gij  eij þ eij ¼ 0 ði; j ¼ 1; 2; . . .; n; i\jÞ > > : þ dij þ dij  u; eijþ þ e ij  u ði; j ¼ 1; 2; . . .; n; i\jÞ

ð7:16Þ þ Solving Eq. (7.16), the priority weights wi ¼ ½w i ; wi ði ¼ 1; 2; . . .; nÞ can be obtained from the IFPR R ¼ ðrij Þnn ¼ ð½nij ; gij Þnn . Especially, if T  ¼ 0, R is ~ is additive consistent by Definition 7.7. additive consistent and thus the IVAIFPR R However, there are numerous values of nij and gij associated with IVAIFPR ~ ¼ ð~rij Þ . Two special cases are first taken into account in the sequel. R nn The priority weights could be derived respectively from two special extracted ~ When R ¼ Rl , solving Eq. (7.16) yields the IFPRs Rl and Rm of the IVAIFPR R. þ 0þ  optimal solutions wi ¼ ½wi ; wi  (denoted by w0i ¼ ½w0 i ; wi ) ði ¼ 1; 2; . . .; nÞ. þ When R ¼ Rm , solving Eq. (7.16) yields the optimal solutions wi ¼ ½w i ; wi  00 þ 00 00 (denoted by wi ¼ ½wi ; wi ) ði ¼ 1; 2; . . .; nÞ). In addition, using Eq. (7.11), two additive consistent IFPRs CRl ¼ ðcrijl Þnn and CRm ¼ ðcrijm Þnn corresponding to the extracted IFPR Rl and Rm are derived, respectively, where

crijl


¼

½0:5; 0:5; if i ¼ j 0þ 0þ 0 ½0:5ðw0  w þ 1Þ; 0:5ðw  w þ 1Þ; if i 6¼ j j i i j

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

228

and crijm ¼




½0:5; 0:5; if i ¼ j : 00 þ ½0:5ðw00 þ 1Þ; 0:5ðw00i þ  w00 i  wj j þ 1Þ; if i 6¼ j

Motivated by Wang and Elhag [12], an IVAIFPR should give an IVAIF weight ~ ¼ ð~ ~ 2 ; . . .; w ~ n ÞT be an IVAIF priority weight vector, where estimate. Let w w1 ; w l l ~ i ¼ ð½wi ; w  i ; ½wmi ; w  mi Þ is an IVAIFV, ½wli ; w  li  and ½wmi ; w  mi  are the membership w and non-membership degrees of alternative xi as a fuzzy concept “priority”, respectively. Set 8 l 00 w ¼ 1minfw0 > i ; wi g > < il j1 0  i ¼ jmaxfwi ; w00 w i g 0þ 00 þ m 1 w ¼ ð1  maxfw > i ; wi gÞ > : im j1 0þ 00 þ  i ¼ jð1  minfwi ; wi gÞ w

ð7:17Þ

  0þ 00 þ 00 where j ¼ max maxi¼1;2;...;n fmaxfw0 i ; wi g  minfwi ; wi g þ 1g; 1 . ~ i ¼ ð½wli ; w  li ; ½wmi ; w  mi Þ ði ¼ 1; 2; . . .; nÞ Accordingly, the IVAIF priority weights w ~ are generated from IVAIFPR R. The parameter j can guarantee that the IVAIF ~ i ði ¼ 1; 2; . . .; nÞ satisfy the condition w  li þ w  mi  1. priority weights w

7.5

Method for Solving the Group Decision Making Problems with IVAIFPRs

This section addresses the GDM problems with IVAIFPRs. To derive DMs’ weights, an optimization model is established and transformed into a linear programming model for resolution. Thus, a new method is developed to solve such problems.

7.5.1

Description for GDM Problems with IVAIFPRs

Suppose that there exist n non-inferior alternatives which make up the alternative set X ¼ fx1 ; x2 ; . . .; xn g. Let T ¼ ft1 ; t2 ; . . .; tm g be the set of DMs and x ¼ ðx1 ; x2 ; . . .; xm ÞT be DMs’ weight vector which is unknown and needs to be kij ; ½mkij ; mkij Þ is the preference for each pair of determined. Assume that ~rijk ¼ ð½lkij ; l

alternatives ðxi ; xj Þ provided by DM tk , which can elicit the individual IVAIFPR ~ k ¼ ð~r k Þ R ij nn ði; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; mÞ. Denote the IVAIF priority weight ~ ¼ ð~ ~ 2 ; . . .; w ~ n ÞT , where w ~ i ¼ ð½wli ; w  li ; ½wmi ; w  mi Þ is vector of alternatives by w w1 ; w an IVAIFV and needs to be derived for ranking alternatives.

7.5 Method for Solving the Group Decision Making Problems with IVAIFPRs

229

For the GDM problems, the main task is to find a solution which is accepted by ~ k ¼ ð~r k Þ all DMs. According to the individual IVAIFPR R ij nn provided by DM tk , the individual IVAIF priority weight vector could be obtained by the parametric linear programming model (i.e., Eq. (7.16)) and Eq. (7.17) in Sect. 7.4.2. Then the individual ranking order of alternatives for each DM can be generated based on the designed likelihood comparison algorithm. However, different DMs would derive different ranking orders. In such a way, a final ranking order which is agreed by all DMs cannot be obtained. To circumvent this issue, one reasonable solution is to aggregate all the individual IVAIFPRs into a collective one and then get the final ranking order from the collective IVAIFPR. By using IVAIF weighted averaging ~ ¼ ð~rij Þ operator in Xu and Yager [17], the collective IVAIFPR R rij ¼ nn with ~ ij ; ½mij ; mij Þ is obtained, where ð½lij ; l lij ¼

m X

ij ¼ xk lkij ; l

k¼1

m X

kij ; mij ¼ xk l

k¼1

¼ 1; 2; . . .; n

m X

xk mkij ; mij ¼

k¼1

m X k¼1

xk mkij for all i; j ð7:18Þ

and xk is the weight of DM tk ðk ¼ 1; 2; . . .; mÞ. It is required to determine DMs’ weight vector x before obtaining the collective ~ ¼ ð~rij Þ . Therefore, Sect. 7.5.2 is devoted to determining DMs’ IVAIFPR R nn weights.

7.5.2

Determination of DMs’ Weights

The group consensus reflects the agreement among ratings provided by all individual DMs. The low group consensus would lead to unreasonable decision making result. To incorporate the group consensus into the GDM, an optimization model is constructed to determine DMs’ weights by maximizing the group consensus in this subsection. Firstly, considering the conditions of IVAIFPR in Definition 7.5, the distance ~ k ¼ ð~r k Þ ~ between the individual IVAIFPR R ij nn and the collective IVAIFPR R ¼ ð~rij Þnn is defined as: ~ k ; RÞ ~ ¼ dðR ¼

1 ðn  1Þðn  2Þ

n1 X n X i¼1 j¼i þ 1

1 4ðn  1Þðn  2Þ

dð~rijk ; ~rij Þ

n1 X n X i¼1 j¼i þ 1

ij j þ jmkij  mij j þ jmkij  mij j þ jpkij  pij j þ j ij jÞ; ðjlkij  lij j þ j lkij  l pkij  p

ð7:19Þ

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

230

where dð~rijk ; ~rij Þ is the normalized Hamming distance between IVAIFVs ~rijk and ~rij by kij  mkij , kij ¼ 1  lkij  mkij , ij  mij Eq. (7.1), pkij ¼ 1  l p pij ¼ 1  l and k k ~ ; RÞ ~  1. ij ¼ 1  l  mij . Due to 0  dð~r ; ~rij Þ  1, it follows that 0  dðR p ij

ij

According to the distance between the individual IVAIFPR and the collective one, the group consensus for the decision group can be defined below. Definition 7.9 For a GDM problem, suppose that DMs’ weight vector is x ¼ ~ k ¼ ð~r k Þ ðx1 ; x2 ; . . .; xm ÞT and DM tk provides his/her individual IVAIFPR R ij nn kij ; ½mkij ; mkij Þ ðk ¼ 1; 2; . . .; mÞ. Then the group consensus of such a with ~rijk ¼ ð½lkij ; l decision group is defined as ~ k ; RÞg; ~ C ¼ 1  max fdðR k¼1;2;...;m

ð7:20Þ

~ ¼ ð~rij Þ ~k ~ where R nn is the collective IVAIFPR defined as Eq. (7.18) and dðR ; RÞ is k ~ calculated by Eq. (7.19). ~ and R the distance between R The group consensus measures the level of the agreement among DMs for the final decision. The higher the group consensus, the more reliable the decision making result. Consequently, by maximizing the group consensus, an optimization model is established to derive DMs’ weights as follows: ~ k ; RÞg ~ max C ¼ 1  max fdðR k¼1;2;...;m
Pm k¼1 xk ¼ 1 s:t: xk 0 ðk ¼ 1; 2; . . .; mÞ

ð7:21Þ

r ~ k ; RÞg. ~ ~ k ; RÞ ~  One has dðR Set r ¼ 4ðn  1Þðn  2Þ maxk¼1;2;...;m fdðR 4ðn1Þðn2Þ for all k ¼ 1; 2; . . .; m. Both parameters n and m are constants in the decision making problem. Combining Eqs. (7.18) and (7.19), Eq. (7.21) can be transformed into a mathematical programming model:

min r 8 Pm xk ¼ 1; xk 0 ðk ¼ 1; 2; . . .; mÞ > > < Pk¼1 P n1 Pn k kij  ij j þ jmkij  mij j þ jmkij  mij j þ jpkij  ð1  m lkij  l s:t: i¼1 j¼i þ 1 ðjlij  lij j þ j k¼1 xk l > P P P > m m m : xk mk Þj þ j pk  ð1  xk lk  xk mk ÞjÞ  rðk ¼ 1; 2; . . .; mÞ k¼1

ij

ij

k¼1

ij

k¼1

ij

ð7:22Þ

7.5 Method for Solving the Group Decision Making Problems with IVAIFPRs

231

To solve Eq. (7.22), some variables are introduced as follows: s1ijkþ

¼

s2ijkþ ¼ s3ijkþ

¼

s4ijkþ ¼ s5ijkþ

¼

s5 ijk ¼ s6ijkþ

¼

s6 ijk ¼

1 2 1 2 1 2 1 2 1 2

jlkij



m X

xk lkij j þ lkij



k¼1

j lkij 

m X

jmkij



xk mkij j þ mkij



m X

m X

k¼1

jpkij



1

 kij xk l



k¼1

1 2

jpkij 

1 2

j pkij

1 2

j pkij 

1

m X

kij  xk l

1

xk lkij

m X k¼1

m X

m X k¼1

1 2

1 2 1 2

j lkij 

j þ pkij



jmkij

m X

m X





kij þ  kij j  l xk l

jmkij 

xk mkij j



mkij

þ

xk mkij j  mkij þ

k¼1

1

m X

! 1

!  kij þ 1  xk mkij j  p

kij ; xk l !

xk mkij

 kij xk l

m X



kij  xk l

m X

xk mkij ;

!!

xk mkij

m X



xk mkij

m X

; !!

xk mkij

k¼1

xk lkij 

m X

;

!!

k¼1

xk lkij

; !

k¼1

k¼1

k¼1

m X

k¼1

m X

m X

m X

; !

k¼1

m X

m X

! xk lkij

k¼1

k¼1



þ

m X k¼1

k¼1

 kij jþp

lkij

k¼1

xk mkij j  pkij þ 1  xk mkij

xk lkij j

k¼1

! xk mkij

m X k¼1

!

m X k¼1

xk lkij 

¼

xk mkij ; s4 ijk ¼

m X



k¼1

1

; s3 ijk

k¼1

m X



!

k¼1

k¼1



xk mkij

k¼1 m X

jlkij

!

k¼1

xk mkij j þ mkij 

¼

1 2

kij ; s2 xk l ijk ¼

k¼1 m X

; s1 ijk !

m X

 kij   kij j þ l xk l

k¼1

jmkij 

xk lkij

k¼1

k¼1 m X

!

m X

;

!! xk mkij

:

k¼1

Thus, Eq. (7.22) can be further converted into a linear programming model: min r 8 Pm > k¼1 xk ¼ 1; xk 0 ðk ¼ 1; 2; . . .; mÞ > > > Pn1 Pn > 1þ 2þ 3þ 4þ 5þ 6þ 1 2 3 4 5 6 > > i¼1 j¼i þ 1 ðsijk þ sijk þ sijk þ sijk þ sijk þ sijk þ sijk þ sijk þ sijk þ sijk þ sijk þ sijk Þ  r ðk ¼ 1; 2; . . .; mÞ > > Pm Pm > 2þ k k 2 k k > < s1ijkþ  s1   l ¼ l  x l ; s  s ¼ l  x ði; j ¼ 1; 2; . . .; n; i\j; k ¼ 1; 2; . . .; mÞ k k ijk ijk ij ij ijk k¼1 k¼1 ij ij Pm Pm s:t: 3 þ 4þ 3 k k 4 k k >    s ¼ m  x m ; s  s ¼ m  x ði; j ¼ 1; 2; . . .; n; i\j; k ¼ 1; 2; . . .; mÞ s m > ijk ij ijk ij k¼1 k ij k¼1 k ij ijk > > ijk Pm Pm > 5þ 5 k k k > >   l  s ¼ p  ð1  x  x Þ ði; j ¼ 1; 2; . . .; n; i\j; k ¼ 1; 2; . . .; mÞ m s > ijk ijk ij k¼1 k ij k¼1 k ij > > P Pm > k k : s6 þ  s6 ¼ p kij  ð1  m ijk ijk k¼1 xk lij  k¼1 xk mij Þ ði; j ¼ 1; 2; . . .; n; i\j; k ¼ 1; 2; . . .; mÞ

ð7:23Þ By solving Eq. (7.23), DMs’ weight vector x ¼ ðx1 ; x2 ; . . .; xm ÞT is deter~ ¼ ð~rij Þ mined. Thus, the collective IVAIFPR R nn can be generated using Eq. (7.18). It can be further applied to obtain the IVAIF priority weights. Meanwhile, the group consensus C can be derived by Eq. (7.20).

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

232

7.5.3

Method for Group Decision Making with IVAIFPRs

Based on the above analysis, a new method for GDM with IVAIFPRs is graphically depicted in Fig. 7.2. The concrete steps of the GDM method with IVAIFPRs are summarized below. Step 1. Determine DMs’ weight vector x ¼ ðx1 ; x2 ; . . .; xm ÞT by Eq. (7.23). ~ k ¼ ð~r k Þ Step 2. Integrating the individual IVAIFPRs R ij nn ðk ¼ 1; 2; . . .; mÞ by ~ DMs’ weight vector x, the collective IVAIFPR R ¼ ð~rij Þnn can be obtained using Eq. (7.18). Step 3. According to Eqs. (7.9) and (7.10), establish two special IFPRs Rl ¼ l ~ ¼ ð~rij Þ , respectively. ðr Þ and Rm ¼ ðr m Þ from the IVAIFPR R ij nn

ij nn

nn

Step 4. Let R ¼ Rl . Solving Eq. (7.16) yields the optimal solutions 0þ 0 wi ¼ ½w0 i ; wi  ði ¼ 1; 2; . . .; nÞ. Step 5. Let R ¼ Rm . Solving Eq. (7.16) yields the optimal solutions 00 þ w00i ¼ ½w00 i ; wi  ði ¼ 1; 2; . . .; nÞ. ~ i ði ¼ 1; 2; . . .; nÞ are derived via Step 6. The IVAIF priority weights w Eq. (7.17). Step 7. Using the designed likelihood comparison algorithm for IVAIFVs, the order of alternatives is generated by ranking the IVAIF priority weights ~ i ði ¼ 1; 2; . . .; nÞ. w

input

Individual IVAIFPR R k = (rijk ) n×n given by DM tk

Determine DM' weight vector ω = (ω1 , ω2 ,

(k = 1, 2,

, m) . by Eqs. (7.23)

, ωm ) T .

Obtain the collective IVAIFPR R = (rij ) n×n by aggregating R k = (rijk ) n×n .

by Eq. (7.18)

Establish two IFPRs Rμ = (rijμ ) n×n and Rν = (rijν ) n×n .

method

Derive the optimal solutions

wi′ = [ wi′− , wi′+ ]

for

i = 1, 2,

,n .

by Rμ and Eq. (7.16)

Derive the optimal solutions

wi′′ = [ wi′′− , wi′′+ ]

for

i = 1, 2,

,n

by Rν and Eq. (7.16)

Generate the IVAIF priority weights

Rank the IVAIF priority weights comparison algorithm for IVAIFVs. output

by Eqs. (7.9)-(7.10)

wi = ([ wiμ , wiμ ],[ wνi , wνi ])

wi (i = 1, 2,

for

i = 1, 2,

,n .

by Eq. (7.17)

, n) by the likelihood-based

The order of alternatives is generated by ranking the IVAIF priority weights

by Eqs. (7.5)-(7.8)

wi

Fig. 7.2 Decision making process of the GDM problems with IVAIFPRs

(i = 1, 2,

, n) .

7.5 Method for Solving the Group Decision Making Problems with IVAIFPRs

233

Remark 7.1 Once we have developed a method for solving a GDM problem, the worst-case time requirements should be analyzed by a function of the size of its input (in terms of the O-notation). In this chapter, the complexity of computation of the proposed method depends on the linear programming models, i.e., Eqs. (7.16) and (7.23). Using Karmarkar’s algorithm to solve the linear programming models, time complexity of Eq. (7.16) is Oðn9 Þ, where n is the number of alternatives. Meanwhile, the space complexity of Eq. (7.16) is O(1). Similarly, time complexity of Eq. (7.23) is Oðm4:5 n9 Þ, where m is the number of DMs and n is the number of alternatives. Meanwhile, the space complexity of Eq. (7.23) is O(1). Although the worst-case complexity is a little big, the processing times for solving such linear programming models are very quick since many mature softwares can be applied to solve these models effectively. Remark 7.2 Bustince et al. [38] analyzed the relationships between the different types of fuzzy sets, such as Atanassov intuitionistic fuzzy set, interval-valued fuzzy ; ½m; mÞ can be degenerated set, and IVAIF set. It is clear that an IVAIFV ~a ¼ ð½l; l l  lÞ (when m ¼ m ¼ 0), interval ½l þ kð1  m  into real number l þ að  þ kð1  m  l Þ and AIFV \l þ qð lÞ; l l  lÞ; m þ qðm  mÞ [; respectively, where a; k; q 2 ½0; 1. Accordingly, the IVAIFPR also can be reduced to the corresponding fuzzy preference relations, IFPR and AIFPR. Although the method proposed in this chapter focuses on the GDM with IVAIFPRs, it also can be used to solve the GDM problems with fuzzy preference relations, IFPRs or AIFPRs. Therefore, the proposed method is more flexible and comprehensive.

7.6

An Example of ERP System Selection and Comparative Analysis

In this section, an ERP system selection example is illustrated to demonstrate the applicability of the GDM method proposed in this chapter. Then, the comparative analysis is also conducted to show the superiority of the proposed method.

7.6.1

A Practical Example of ERP System Selection

With the rapid development of information technology, the ERP system as a critical tool can help companies achieve information management effectively. The implement of ERP system could not only need high cost and strong time, but also influence the company’s development. A suitable ERP system which may be not the best in the ERP field, could be the strongest for a company. Now, a company is facing the ERP system selection problem. After experiments and pre-evaluation, there exist four potential ERP systems as alternatives for further evaluation, including Yonyou U9 ðx1 Þ, Sage

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

234

Accpac ERP ðx2 Þ, SAP BusinessOne ðx3 Þ and kingdee K/3 ðx4 Þ. Then, three DMs (or experts) ft1 ; t2 ; t3 g from EPR field are invited to select the most suitable ERP system for this company. They compare each pair of alternatives and provide their ~ k ¼ ð~r k Þ individual IVAIFPRs R ij 44 (k = 1, 2, 3) as follows: 0

ð½0:50; 0:50; ½0:50; 0:50Þ B ð½0:10; 0:20; ½0:60; 0:70Þ R ¼B @ ð½0:60; 0:75; ½0:15; 0:20Þ ð½0:05; 0:15; ½0:70; 0:80Þ ~1

0

ð½0:50; 0:50; ½0:50; 0:50Þ B ~ 2 ¼ B ð½0:40; 0:50; ½0:25; 0:40Þ R @ ð½0:10; 0:25; ½0:60; 0:70Þ ð½0:30; 0:50; ½0:20; 0:35Þ 0

ð½0:50; 0:50; ½0:50; 0:50Þ B ð½0:40; 0:50; ½0:15; 0:30Þ 3 B ~ R ¼@ ð½0:35; 0:45; ½0:25; 0:40Þ ð½0:10; 0:15; ½0:50; 0:55Þ

ð½0:60; 0:70; ½0:10; 0:20Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:20; 0:30; ½0:40; 0:55Þ ð½0:30; 0:40; ½0:20; 0:40Þ

ð½0:15; 0:20; ½0:60; 0:75Þ ð½0:40; 0:55; ½0:20; 0:30Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:50; 0:70; ½0:25; 0:40Þ

1 ð½0:70; 0:80; ½0:05; 0:15Þ ð½0:20; 0:40; ½0:30; 0:40Þ C C; ð½0:25; 0:40; ½0:50; 0:70Þ A ð½0:50; 0:50; ½0:50; 0:50Þ

ð½0:25; 0:40; ½0:40; 0:50Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:10; 0:15; ½0:70; 0:80Þ ð½0:20; 0:35; ½0:30; 0:60Þ

ð½0:60; 0:70; ½0:10; 0:25Þ ð½0:70; 0:80; ½0:10; 0:15Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:10; 0:25; ½0:30; 0:55Þ

1 ð½0:20; 0:35; ½0:30; 0:50Þ ð½0:30; 0:60; ½0:20; 0:35Þ C C; ð½0:30; 0:55; ½0:10; 0:25Þ A ð½0:50; 0:50; ½0:50; 0:50Þ

ð½0:15; 0:30; ½0:40; 0:50Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:10; 0:15; ½0:65; 0:70Þ ð½0:05; 0:10; ½0:80; 0:90Þ

ð½0:25; 0:40; ½0:35; 0:45Þ ð½0:65; 0:70; ½0:10; 0:15Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:05; 0:05; ½0:75; 0:95Þ

1 ð½0:50; 0:55; ½0:10; 0:15Þ ð½0:80; 0:90; ½0:05; 0:10Þ C C: ð½0:75; 0:95; ½0:05; 0:05Þ A ð½0:50; 0:50; ½0:50; 0:50Þ

Step 1: Using Lingo11 software (Implement environment: Windows 7, Intel(R) Core(TM) i5-4590 CPU @ 3.30 GHz. Processing time: 0.00 s) to solve the linear programming model, i.e., Eq. (7.23), DMs’ weights are determined as x1 ¼ 0:4458, x2 ¼ 0:2352, x3 ¼ 0:3190. ~ k by DMs’ weights Step 2: Integrating the individual IVAIFPRs R ~ is generated by Eq. (7.18) as follows: xk ðk ¼ 1; 2; 3Þ, the collective IVAIFPR R 1 0 ð½0:5000; 0:5000; ½0:5000; 0:5000Þð½0:3741; 0:5018; ½0:2663; 0:3663Þð½0:4883; 0:6161; ½0:1915; 0:3020Þð½0:5421; 0:6497; ½0:1012; 0:1970Þ C B ~ ¼ B ð½0:2663; 0:3663; ½0:3741; 0:5018Þð½0:5000; 0:5000; ½0:5000; 0:5000Þð½0:6135; 0:7093; ½0:1235; 0:1853Þð½0:4149; 0:6066; ½0:1967; 0:2925Þ C R @ ð½0:1915; 0:3020; ½0:4883; 0:6161Þð½0:1235; 0:1853; ½0:6135; 0:7093Þð½0:5000; 0:5000; ½0:5000; 0:5000Þð½0:4213; 0:6107; ½0:2624; 0:3868Þ A ð½0:1012; 0:1970; ½0:5421; 0:6497Þð½0:1967; 0:2925; ½0:4149; 0:6066Þð½0:2624; 0:2868; ½0:4213; 0:6107Þð½0:5000; 0:5000; ½0:5000; 0:5000Þ

Step 3: According to Eqs. (7.9) and (7.10), two special IFPRs Rl ¼ ðrijl Þ44 and Rm ¼ ðrijm Þ44 are established, respectively. i.e., 0

½0:5000; 0:5000 B ½0:4982; 0:6259 Rl ¼ B @ ½0:3839; 0:5117 ½0:3503; 0:4579 0

½0:5000; 0:5000 B ½0:2663; 0:3663 Rm ¼ B @ ½0:1915; 0:3020 ½0:1012; 0:1970

½0:3741; 0:5018 ½0:5000; 0:5000 ½0:2907; 0:3865 ½0:3934; 0:5851

½0:4883; 0:6161 ½0:6135; 0:7093 ½0:5000; 0:5000 ½0:3893; 0:5787

1 ½0:5421; 0:6497 ½0:4149; 0:6066 C C ½0:4213; 0:6107 A ½0:5000; 0:5000

½0:6337; 0:7337 ½0:5000; 0:5000 ½0:1235; 0:1853 ½0:1967; 0:2925

½0:6980; 0:8085 ½0:8147; 0:8765 ½0:5000; 0:5000 ½0:2624; 0:3868

1 ½0:8030; 0:8988 ½0:7075; 0:8033 C C ½0:6132; 0:7376 A ½0:5000; 0:5000

7.6 An Example of ERP System Selection …

235

Step 4: Set w ¼ 0:5. When R ¼ Rl , the parametric linear programming model is established by Eq. (7.16) and solved by using Lingo11 software (Implement environment: Windows 7, Intel(R) Core(TM) i5-4590 CPU @ 3.30 GHz. Processing time: 0.00 s). The optimal solutions are derived as w01 ¼ ½0:2143; 0:2798, w02 ¼ ½0:2745; 0:4661,w03 ¼ ½0:0476; 0:2378, w04 ¼ ½0:0164; 0:2081. Step 5: When R ¼ Rm , using Eq. (7.16), the parametric linear programming model is built and solved by using Lingo11 software (Implement environment: Windows 7, Intel(R) Core(TM) i5-4590 CPU @ 3.30 GHz. Processing time: 0.00 s). The optimal solutions are obtained as w001 ¼ ½0:6060; 0:6170, w002 ¼ ½0:3067; 0:3940, w003 ¼ ½0; 0:0763, w004 ¼ ½0; 0. Step 6: By Eq. (7.17), the IVAIF priority weights are generated as follows: ~ 2 ¼ ð½0:2069; 0:2313; ½0:4025; 0:4569Þ; ~ 1 ¼ ð½0:1616; 0:4569; ½0:2888; 0:5431Þ; w w ~ 3 ¼ ð½0; 0:0359; ½0:5747; 0:6965Þ; w ~ 4 ¼ ð½0; 0:0123; ½0:5971; 0:7540Þ: w Step 7: Using the likelihood comparison algorithm for IVAIFVs, the order of ~ i ði ¼ 1; 2; 3; 4Þ. alternatives is obtained by ranking the IVAIF priority weights w (i) Construct the likelihood matrix by Eq. (7.5) as: 0

0:5000 B 0:3352 ^¼B L @ 0 0

0:6648 0:5000 0 0

1:0000 1:0000 0:5000 0:3330

1 1:0000 1:0000 C C 0:6670 A 0:5000

~ i ði ¼ 1; 2; 3; 4Þ by (ii) Compute the non-dominance degrees NDi of IVAIFV w Eq. (7.6): ND1 ¼ 1; ND2 ¼ 0:6705; ND3 ¼ 0; ND4 ¼ 0 ¼ 1: ~ i ði ¼ 1; 2; 3; 4Þ are cal(iii) By Eq. (7.8), the dominance degrees Di of IVAIFV w culated as D1 ¼ 0:8883; D2 ¼ 0:7784; D3 ¼ 0:2223; D4 ¼ 0:1110: (iv) According to Eq. (7.8), the ranking indices are determined as RD1 ¼ 0:9442, RD2 ¼ 0:7245, RD3 ¼ 0:1112, RD4 ¼ 0:0555. (v) By descending ranking indices, the ranking order of IVAIFVs is obtained as ~1 [ w ~2 [ w ~3 [ w ~ 4 . Thus, the order of alternatives is generated as w x1 x2 x3 x4 . Considering that different values of the parameter w in Eq. (7.15) can be set, three special cases are calculated as examples including w ¼ 0, w ¼ 0:5 and w ¼ 1, which represent the minority, compromise and majority principles, respectively. Accordingly, the computation results are listed in Table 7.1.

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

236

Table 7.1 Computation results with different values of parameter w w

RD1

RD2

RD3

RD4

Ranking order

0 0.5 1

0.8343 0.9441 0.7511

0.9284 0.7244 0.9403

0.0765 0.1112 0.0970

0.0906 0.0555 0.0697

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

It can be seen from Table 7.1 that the ranking order of alternatives is not the same for different decision principles of DMs. Based on the minority principle ðw ¼ 0Þ, the ranking order is x2 x1 x4 x3 and the best alternative is x2 . Based on the compromise principle ðw ¼ 0:5Þ, the ranking order is x1 x2 x3 x4 and the best is x1 . Based on the majority principle ðw ¼ 1Þ, the ranking order is x2 x1 x3 x4 and the best is x2 . Therefore, it is necessary to take DMs’ decision principle into account during the decision process.

7.6.2

Comparative Analysis with Existing GDM Methods

In this subsection, we compare the results obtained by methods [19, 39] with the method proposed in this chapter to explain the superiorities of the proposed method. Given that there exists no research on decision making with IVAIFPRs considering additive consistency, Chu et al.’s method with additive consistent AIFPRs ~k ¼ is applied to solve the above example. Firstly, the individual IVAIFPRs R ðk ¼ 1; 2; 3Þ are transformed into the corresponding AIFPRs R_ k ¼ ð_r k Þ ð~r k Þ ij 44

ij 44

kij Þ; 12ðmkij þ mkij Þ [ : Assume that DMs’ weight vector is ð13; 13; 13ÞT , with r_ ijk ¼ \12ðlkij þ l

the threshold values a1 ¼ a2 ¼ 0:90 and the control parameter b ¼ 0:60. Using method in Chu et al. [39], the ranking order of alternatives is generated as x4 x2 x1 x3 , which is different from the results obtained by this chapter. Based on the multiplicative consistency of IVAIFPR, Liao et al. [19] proposed a convergent approach for GDM with IVAIFPRs. In method [19], DMs’ weights are given in advance. Here, suppose that x1 ¼ x2 ¼ x3 ¼ 13. In addition, set the consistency threshold s ¼ 0:1 and parameter g ¼ 0:6. Using Algorithms 2 and 4 in Liao et al. [19] to solve the above ERP example, the ranking order of alternatives is derived as x2 x1 x3 x4 , which is the same as the result in majority principle ðw ¼ 1Þ and different from the results in minority and compromise principles obtained by this chapter. Compared with methods [19, 39], the proposed method of this chapter has the following advantages:

(1) Chu et al.’s method focused on GDM with additive consistent AIFPRs, while this chapter concentrates on GDM with additive consistent IVAIFPRs. The former cannot be used to solve the GDM with IVAIFPRs. Since IVAIFPR

7.6 An Example of ERP System Selection …

(2)

(3)

(4)

(5)

237

generalizes the AIFPR, the proposed method can also be used to solve GDM with AIFPRs. Hence, the proposed method is more flexible and comprehensive. Liao et al. [19] defined the multiplicative consistency of IVAIFPR by extending the multiplicative transitivity of fuzzy preference relation directly. This consistency definition failed to consider the characteristics of IVAIFPR. When the IVAIFPR is reduced to AIFPR, the multiplicative consistency of IVAIFPR in Liao et al. [19] is degenerated to that of AIFPR, which is too strict for AIFPR and not reasonable [16]. In this chapter, the additive consistency of IVAIFPR is defined based on the additive consistency of IFPR. It not only generalizes the additive consistency of IFPR, but also can reduce to the additive consistency of AIFPR. Methods [16, 39] assigned DMs’ weights in advance, which makes it difficult to avoid the subjective randomness. In this chapter, to obtain DMs’ weights, an optimization model is established by maximizing the group consensus. The determination of DMs’ weights incorporates the group consensus into GDM problems and thus is more objective and rational. Liao et al. [16] and Chu et al. [39] discussed the group consensus by comparing the consistency threshold and the deviation between each individual IVAIFPR and the collective one. If the deviations do not satisfy the consistency threshold, it needs to calculate many times to obtain the collective consistent IVAIFPR by iteration. The consistency threshold should be provided a priori, which is difficult for DMs to judge which value is proper for the threshold. This chapter defines the group consensus without need of any thresholds. An optimization model is constructed by maximizing the group consensus and transformed into a linear programming model to resolve. This transformed linear programming model not only maximizes the group consensus, but also determines DMs’ weights to obtain the collective IVAIFPR. Therefore, the defined group consensus can effectively avoid the subjectivity and the obtained collective IVAIFPR is computationally simple. To get the ranking order of alternatives, Liao et al. [19] ranked the collective alternative values by the ranking method in Wang et al. [34]; Chu et al. [39] compared the Atanassov intuitionistic fuzzy priority weights by the ranking method [40]. However, Xu and Yager’s ranking method cannot rank all Atanassov intuitionistic fuzzy values. This chapter ranks the alternatives by comparing the IVAIF priority weights using the devised likelihood comparison algorithm. By solving the parametric linear programming model, the priority weights are generated from two special IFPRs extracted from the IVAIFPR. Adjusting the control parameter can help us consider the different decision principles of DMs properly. Hence, the proposed method can avoid loss information and flexibly generate decision results according to DMs’ decision principles.

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

238

7.7

Conclusions

This chapter discusses the additive consistency of IVAIFPR and develops a new method for GDM problems with IVAIFPRs. According to the transformation between the IVAIFV and the interval, a likelihood of an IVAIFV being greater than another is defined and then a likelihood comparison algorithm is designed to rank IVAIFVs. Subsequently, the additive consistency of IVAIFPR is given according to the additive consistency of IFPR. The IVAIF priority weights of an IVAIFPR are derived based on two special IFPRs extracted from the IVAIFPR. In the GDM method, the group consensus is defined by the distance between the individual IVAIFPR and the collective one. To derive DMs’ weights, a mathematical programming model is constructed by maximizing the group consensus, which is transformed into a linear programming model for resolution. The collective IVAIFPR is obtained by the IVAIF weighted averaging operator and applied to derive the IVAIF priority weights. Utilizing the designed likelihood comparison algorithm, the ranking order of alternatives is generated by comparing IVAIF priority weights. An example is provided to illustrate the effectiveness and superiority of the proposed method. There are still some significant issues for further research. For example, the multiplicative consistency of an IVAIFPR is not discussed. It is unknown how to determine the priority weights from an incomplete IVAIFPR. Once these issues have been addressed, it would be worthwhile to investigate how the current framework can be adapted to handle GDM problems based on multiplicative consistent and incomplete IVAIFPRs.

Appendix 1 ~1 [ a ~2 ” of Definition 7.6 Computing results of likelihood of IVAIFVs “ a i ; ½mi ; mi Þði ¼ 1; 2Þði ¼ 1; 2Þ, the interval ai ðkÞ ¼ For two IVAIFVs ~ai ¼ ð½li ; l þ  ½a mi  li Þ, aiþ ðkÞ ¼ i ðkÞ; ai ðkÞ can be obtained where ai ðkÞ ¼ li þ kð1   i Þ i þ kð1  mi  l l

k 2 ½0; 1.

and

1 l þ kðl  l1 þ m2 m1 Þ l 2 2 2 l þ kðl  1 l þ l l1 þ m1 m1 þ l  l2 þ m2 m2 Þ l 1

2

1

Moreover,

set

f ðkÞ ¼

and gðkÞ ¼ 1  f ðkÞ. Equation (7.4) can be writ-

2

ten as Lða1 ðkÞ [ a2 ðkÞÞ ¼ maxf1  maxff ðkÞ; 0g; 0g 2 þ m1  m2 In addition, denote q ¼ l1  l

Especially, when q 6¼ 0, let m ¼ minf1; maxf h2 6¼ 0, let n ¼ minf1; maxf

l  l1 2 l  l1 þ m2 m1 2

and l  l2 1 l  l2 þ m1 m2 1

ð7:24Þ

1 þ m2  m1 . h2 ¼ l2  l

; 0gg. Similarly, when

; 0gg. It is important to determine the upper

and lower limits of the definite integral. Thus, by the determination of upper and

Appendix 1

239

^ a1 [ ~ lower limits of integral in Eq. (7.5), the computing of Lð~ a2 Þ can be divided in the following eight cases. Case 1: q [ 0 and h2 [ 0 Z1

^ a1 [ ~a2 Þ ¼ Lð~

Zm

Z1

gðkÞdk þ

1dk ¼ 0

maxfm;ng

gðkÞdk þ m

ð7:25Þ

maxfm;ng

Case 2: q [ 0 and h2 \0 (Rn ^ a1 [ ~a2 Þ ¼ Lð~

Rm

m gðkÞdk þ Rm 0 1dk;

0

1dk;

if n [ m if n  m

(Rn m

¼

gðkÞdk þ m;

m;

if n [ m if n  m ð7:26Þ

Case 3: q [ 0 and h2 ¼ 0 ^ a1 [ ~a2 Þ ¼ Lð~

Z1

Zm gðkÞdk þ

m

Z1 1dk ¼

gðkÞdk þ m

ð7:27Þ

m

0

Case 4: q\0 and h2 [ 0 8R R < m gðkÞdk þ 1 1dk; if m [ n n m ^ Lð~a1 [ ~a2 Þ ¼ R 1 : 1dk; if m  n m (Rm n gðkÞdk þ 1  m; if m [ n ¼ 1  m; if m  n

ð7:28Þ

Case 5: q\0 and h2 \0 ^ a1 [ ~a2 Þ ¼ Lð~

minfm;ng Z

Z1 gðkÞdk þ

minfm;ng Z

1dk ¼ m

0

gðkÞdk þ 1  m

ð7:29Þ

0

Case 6: q\0 and h2 ¼ 0 ^ a1 [ ~a2 Þ ¼ Lð~

Zm

Z1 gðkÞdk þ

0

Zm 1dk ¼

m

gðkÞdk þ 1  m 0

ð7:30Þ

240

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

Case 7: q ¼ 0 and h2 6¼ 0 (R1 ^ a1 [ ~a2 Þ ¼ Lð~

0

gðkÞdk;

1;

if l1 \ l2 2 if l1 l

ð7:31Þ

Case 8: q ¼ 0 and h2 ¼ 0 ^ a1 [ ~a2 Þ ¼ gðkÞ Lð~

ðA:9Þ

Therefore, the computing results of likelihood of ~ a1 [ ~ a2 can be obtained.

References 1. S.P. Wan, F. Wang, J.Y. Dong, A preference degree for intuitionistic fuzzy values and application to multi-attribute group decision making. Inf. Sci. 370–371, 127–146 (2016) 2. J.M. Merigó, D. Palacios-Marqués, S. Zeng, Subjective and objective information in linguistic multi-criteria group decision making. Eur. J. Oper. Res. 248, 522–531 (2016) 3. D. Joshi, S. Kumar, Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making. Eur. J. Oper. Res. 248, 183–191 (2016) 4. S.P. Wan, F. Wang, L.L. Lin, J.Y. Dong, An intuitionistic fuzzy linear programming method for logistics outsourcing provider selection. Knowl. Based Syst. 82, 80–94 (2015) 5. Y.J. Xu, F. Herrera, H.M. Wang, A distance-based framework to deal with ordinal and additive inconsistencies for fuzzy reciprocal preference relations. Inf. Sci. 328, 189–205 (2016) 6. H.B. Yan, T. Ma, A group decision-making approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority. Eur. J. Oper. Res. 241, 815–829 (2015) 7. Y.M. Wang, Z.P. Fan, Z.S. Hua, A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations. Eur. J. Oper. Res. 182, 356–366 (2007) 8. F.Y. Meng, X.H. Chen, Y.L. Zhang, Consistency-based linear programming models for generating the priority vector from interval fuzzy preference relations. Appl. Soft Comput. 41, 247–264 (2016) 9. U. Bentkowska, H. Bustince, A. Jurio, M. Pagola, B. Pekala, Decision making with an interval-valued fuzzy preference relation and admissible orders. Appl. Soft Comput. 35, 792– 801 (2015) 10. E. Barrenechea, J. Fernandez, M. Pagola, F. Chiclana, H. Bustince, Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Appl. Decis. Making Knowl. Based Syst. 58, 33–44 (2014) 11. Z.S. Xu, J. Chen, Some models for deriving the priority weights from interval fuzzy preference relations. Eur. J. Oper. Res. 184, 266–280 (2008) 12. Y.M. Wang, T.M.S. Elhag, A goal programming method for obtaining interval weights from an interval comparison matrix. Eur. J. Oper. Res. 177, 458–471 (2007) 13. S.P. Wan, F. Wang, J.Y. Dong, A novel group decision making method with intuitionistic fuzzy preference relations for RFID technology selection. Appl. Soft Comput. 38, 405–422 (2016)

References

241

14. S.P. Wan, G.L. Xu, F. Wang, J.Y. Dong, A new method for Atanassov’s interval-valued intuitionistic fuzzy MAGDM with incomplete attribute weight information. Inf. Sci. 316, 329–347 (2015) 15. S.P. Wan, J.Y. Dong, Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees. Inf. Fusion 26, 49–65 (2015) 16. H.C. Liao, Z.S. Xu, Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency. IEEE Trans. Fuzzy Syst. 22, 1669–1681 (2014) 17. Z.S. Xu, R.R. Yager, Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optim. Decis. Making 8, 123–139 (2009) 18. Z.S. Xu, X.Q. Cai, Group decision making with incomplete interval-valued intuitionistic preference relations. Group Decis. Negot. 24, 193–215 (2015) 19. H.C. Liao, Z.S. Xu, M.M. Xia, Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation. J. Intell. Fuzzy Syst. 27, 2969–2985 (2014) 20. J. Wu, H.B. Huang, Q.W. Cao, Research on AHP with interval-valued intuitionistic fuzzy sets and its application in multi-criteria decision making problems. Appl. Math. Model. 37, 9898– 9906 (2013) 21. J. Wu, F. Chiclana, Non-dominance and attitudinal prioritisation methods for intuitionistic and interval-valued intuitionistic fuzzy preference relations. Expert Syst. Appl. 39, 13409– 13416 (2012) 22. Z.S. Xu, X.Q. Cai, Incomplete interval-valued intuitionistic fuzzy preference relations. Int. J. Gen Syst. 38, 871–886 (2009) 23. Z.S. Xu, J. Chen, Approach to group decision making based on interval-valued intuitionistic judgment matrices. Syst. Eng. Theory Pract. 27, 126–133 (2007) 24. L. De Miguel, H. Bustince, J. Fernandez, E. Induráin, A. Kolesárová, R. Mesiar, Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making. Inf. Fusion 27, 189–197 (2016) 25. S.P. Wan, D.F. Li, Fuzzy mathematical programming approach to heterogeneous multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees. Inf. Sci. 325, 484–503 (2015) 26. S.P. Wan, D.F. Li, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with Atanassov’s intuitionistic fuzzy truth degrees. IEEE Trans. Fuzzy Syst. 22, 300–312 (2014) 27. S.P. Wan, J.Y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J. Comput. Syst. Sci. 80, 237–256 (2014) 28. H. Bustince, Conjuntos Intuicionistas e Intervalo valorados difusos: propiedades y construcci on. Relaciones Intuicionistas Fuzzy, Ph.D. Thesis, Universidad P_ublica de Navarra, (1994) 29. T. Tanino, Fuzzy preference orderings in group decision making. Fuzzy Sets Syst. 12, 117– 131 (1984) 30. Z.S. Xu, Intuitionistic preference relations and their application in group decision making. Inf. Sci. 177, 2363–2379 (2007) 31. K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343– 349 (1989) 32. Z.S. Xu, Intuitionistic fuzzy aggregation and clustering (Springer, Berlin, Heidelberg, 2012) 33. Z.S. Xu, Q.L. Da, A likelihood-based method for priorities of interval judgment matrices. Chin. J. Manage. Sci. 11, 63–65 (2003) 34. Z.J. Wang, K.W. Li, W. Wang, An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights. Inf. Sci. 179, 3026– 3040 (2009) 35. F. Zhang, J. Ignatius, C.P. Lim, Y. Zhao, A new method for deriving priority weights by extracting consistent numerical-valued matrices from interval-valued fuzzy judgement matrix. Inf. Sci. 279, 280–300 (2014)

242

7 Additive Consistent Interval-Valued Atanassov Intuitionistic …

36. Z.J. Wang, K.W. Li, Goal programming approaches to deriving interval weights based on interval fuzzy preference relations. Inf. Sci. 193, 180–198 (2012) 37. P.L. Yu, A class of solutions for group decision problems. Manage. Sci. 19, 936–946 (1973) 38. H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez, Z. Xu, B. Bedregal, J. Montero, H. Hagras, F. Herrera, B. De Baets, A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 24, 179–194 (2015) 39. J. Chu, X. Liu, Y. Wang, K.S. Chin, A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations. Comput. Ind. Eng. 101, 227–242 (2016) 40. Z.S. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen Syst 35, 417–433 (2006)

Chapter 8

A Three-Phase Method for Group Decision Making with Interval-Valued Intuitionistic Fuzzy Preference Relations

Abstract Group decision making (GDM) with interval-valued intuitionistic fuzzy preference relations (IVIFPRs) is a critical issue in decision analysis field. This chapter proposes a new method for GDM problems with IVIFPRs. First, employing the additive consistency of intuitionistic fuzzy preference relation (IFPR), a new definition of additive consistency of an IVIFPR is given. According to this additive consistency definition of IVIFPR, two linear programming models are constructed to extract the most optimistic and pessimistic consistent IFPRs from an IVIFPR, respectively. Moreover, if the feasible regions of these two models are empty, this chapter presents two adjusted programming models. Then, a risk attitudinal-based consistent IFPR is determined considering decision maker’s (DM’s) risk attitude. To derive the intuitionistic fuzzy priority weights from the risk attitudinal-based consistent IFPR, a multi-objective programming model is built and converted into a linear goal programming model for resolution. Afterwards, the comprehensive importance degrees of DMs are obtained through combining DMs’ subjective and objective importance degrees. Taking comprehensive importance degrees as order inducing variables, we develop a new comprehensive importance interval-valued intuitionistic fuzzy induced ordered weighted averaging (CI-IVIF-IOWA) operator to integrate the individual IVIFPRs into a collective one. Subsequently, a three-phase method is put forward for GDM with IVIFPRs. Finally, an example of Network System selection is analyzed to demonstrate the feasibility and effectiveness of the proposed method.




Keywords Interval-valued intuitionistic fuzzy preference relation Additive consistency Group decision making Intuitionistic fuzzy priority weights




8.1




Introduction

With the increasing complexity and significance of decision making problems, more and more decision makers (DMs) take part in the decision making process. Thus, group decision making (GDM) appears [1–3]. It is convenient for DMs to © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_8

243

244

8 A Three-Phase Method for Group Decision Making …

provide their preference information in the form of pairwise comparisons of alternatives in GDM. Preference relations, provided by DMs, are the most common ways to represent judgment information for the GDM problems. Due to the uncertainty of objective things and the fuzziness of human thinking, it is impractical and impossible for DMs to express the preference information by crisp numbers. In some specified situations, there exist some hesitations when DMs make pairwise comparisons of alternatives. The intuitionistic fuzzy set (IFS), initiated by Atanassov [4] in 1989, is a powerful tool to represent such uncertain and fuzzy information which involves membership, non-membership and hesitation degrees simultaneously. Later, Atanassov and Gargov [5] generalized IFS to propose the concept of interval-valued intuitionistic fuzzy set (IVIFS). Since IVIFS utilizes intervals to express the membership and non-membership functions, it is more flexible and effective than IFS in characterizing fuzziness and uncertainty in real-world decision making problems [1, 5]. Consequently, the intuitionistic fuzzy preference relation (IFPR) and interval-valued intuitionistic fuzzy preference relation (IVIFPR) emerged and have been extensively employed to the decision analysis field. As is known to all, the consistency of the preference relation is one of fundamental issues for the decision making. By directly employing the membership and non-membership degrees of intuitionistic fuzzy numbers, Wang [6] proposed the additive consistent IFPR. Gong et al. [7] first presented the definition of multiplicative consistent IFPR through the transformation between an IFPR and its corresponding interval fuzzy preference relation. Analogous to the additive consistent IFPR [6], Xu et al. [8] introduced another definition of multiplicative consistent IFPR. Liao and Xu [9] claimed that the multiplicative consistency in [7] may not be sufficient to depict the original intuitionistic fuzzy preference information of DM and the multiplicative consistency in [8] has some disadvantages and may be too strict for an IFPR. To overcome these drawbacks, Liao and Xu [9] offered a general definition of multiplicative consistent IFPR. According to the operational laws of interval-valued intuitionistic fuzzy numbers (IVIFNs), Xu and Chen [10] proposed the consistent interval-valued intuitionistic judgment matrix. Xu and Cai [11] estimated the missing elements in IVIFPR by utilizing the multiplicative transitivity of IVIFPR. Xu and Cai [12] introduced some new definitions on IVIFPR, such as the additive consistent incomplete IVIFPR, multiplicative consistent incomplete IVIFPR and acceptable incomplete IVIFPR. Inspired by the multiplicative transitivity of fuzzy preference relation, Liao et al. [13] presented three new concepts on IVIFPR including the approximate multiplicative consistent IVIFPR, the perfect multiplicative consistent IVIFPR and the acceptable multiplicative consistent IVIFPR. Literature review reveals that there are abundant achievements on consistency of IFPR. Although the multiplicative consistency of IVIFPR has been discussed by some scholars, there is no investigation on the additive consistency of IVIFPR. To fill this gap, this chapter focuses on the additive consistent IVIFPR and the new method for GDM with IVIFPRs. Recently, GDM with IVIFPRs also has attracted much attention. However, there exist few research fruits on IVIFPRs owing to the complex operations on IVIFS and

8.1 Introduction

245

difficult solving process of IVIFPR. Xu and Chen [10] defined the score matrix and accuracy matrix of IVIFPR. Then an approach to GDM with IVIFPR is provided by using some aggregation operators. Xu and Yager [14] investigated the similarity measure of IVIFSs and studied the consensus analysis in GDM with IVIFPRs. However, they did not propose a decision method for solving such GDM problems. Wu and Chiclana [15] introduced an attitudinal expected score function for IVIFNs and provided non-dominance and attitudinal prioritisation methods for IFPRs and IVIFPRs. Xu [16] defined the compatibility measure between IFPRs and designed a consensus reaching procedure with IFPRs and proposed a method for GDM with IFPRs. Then, he generalized the compatibility measure, a consensus reaching procedure and decision making method to accommodate the situation for GDM with IVIFPRs. Based on multiplicative transitivity of IVIFPR, Xu and Cai [11] devised a procedure to construct an IVIFPR with multiplicative transitivity from an incomplete IVIFPR and then put forward an approach for GDM with incomplete IVIFPRs. Xu and Cai [12] designed two procedures for extending the acceptable incomplete IVIFPRs to the complete IVIFPRs and brought forward an approach to decision making with the incomplete IVIFPR. Liao et al. [13] developed some concepts on multiplicative consistent IVIFPR and then proposed a convergent approach for GDM with IVIFPRs. The aforesaid achievements have made great contributions to decision making or GDM with IVIFPRs. Nevertheless, they still suffer from some flaws which are summarized as the following several aspects. (i) Methods [10, 14, 15] ignored the consistency of IVIFPR. Consistency is a critical criterion which can ensure that DMs are neither illogical nor arbitrary in their pairwise comparisons. An IVIFPR with low consistency should not be applied to decision making directly, otherwise it could lead to the unreasonable results. As mentioned before, there are few research fruits on additive consistent IVIFPR. Thus, it is very urgent to investigate the additive consistency of IVIFPR in the decision making. (ii) Methods [12, 14, 15] researched decision making with single DM (or expert) and failed to consider the function of GDM during the process of decision making. It is very hard for only one DM to make a decision when facing the complex decision making problems. The GDM method is more suitable for solving such complex decision making problems than individual decision making. (iii) Although some methods [10, 11, 13] have been developed to solve the GDM problems with IVIFPRs, they overlooked the determination of DMs’ weights (or importance degrees) in GDM problems. These methods [10, 11, 13] adopted the given or known DMs’ weights to obtain the collective preference, which is not easy to avoid the subjective randomness. Therefore, how to determine DMs’ importance degrees is of great importance for the GDM problems.

246

8 A Three-Phase Method for Group Decision Making …

To circumvent these flaws, this chapter brings forward a new three-phase method for GDM with IVIFPRs which involves Aggregation phase, Extraction phase and Exploitation phase. (1) Aggregation phase: The intention of Aggregation phase is to obtain the collective IVIFPR. The confidence degree of DM is first defined to depict the objective importance degree of DM. The comprehensive importance degrees of DMs are obtained by integrating the subjective and objective importance degrees of DMs. Then using the comprehensive importance degrees as the induced ordering variables, we define a comprehensive importance interval-valued intuitionistic fuzzy induced ordered weighted averaging (CI-IVIF-IOWA) operator to integrate the individual IVIFPRs. (2) Extraction phase: In this phase, two linear programming models are set up to extract the most optimistic and pessimistic consistent IFPRs from an IVIFPR, respectively. In particular, if the feasible regions of these two models are empty, two adjusted programming models are established correspondingly. Combining the most optimistic and pessimistic IFPRs, a risk attitudinal-based consistent IFPR is acquired by considering DMs’ risk preference. (3) Exploitation phase: The purpose of this phase is to derive the intuitionistic fuzzy priority weights and give the ranking order of alternative. To do so, this chapter constructs a multi-objective programming model which minimizes the deviations and the hesitation degrees inherent in intuitionistic fuzzy priority weights. This model is converted into a linear goal programming model. Solving this goal programming model, the intuitionistic fuzzy priority weights are derived from the extracted risk attitudinal-based consistent IFPR. The ranking order of alternatives is then generated according to the new order relations of intuitionistic fuzzy values (IFVs) proposed in this chapter. By comparison with existing research, the proposed method of this chapter has some remarkable differences and prominent features: (1) Some references [17–20] have generalized the IOWA operator to GDM with various types of preference relations, such as fuzzy preference relations, interval fuzzy preference relations or uncertain multiplicative linguistic preference relations. However, there is no investigation on generalizing IOWA operator to accommodate GDM in the context of IVIFPRs. In this chapter, we extend IOWA operator for GDM with IVIFPRs and develop a CI-IVIF-IOWA operator. The CI-IVIF-IOWA operator guided by fuzzy linguistic quantifiers can allow for a better control over the aggregation stage developed in the GDM methods. (2) Some existing research [9, 21–23] investigated how to extract a consistent judgment matrix from preference relation in decision making. Inspired by Zhang et al. [23], this chapter extracts the risk attitudinal-based consistent IFPR from an IVIFPR. The main differences between method [23] and this chapter exist in: (i) The former studied the interval fuzzy preference relation, whereas this chapter concentrates on the IVIFPR. (ii) The former only

8.1 Introduction

247

considered the most optimistic and pessimistic consistent matrices and neglected the DMs’ risk attitude. On the contrary, considering DMs’ risk attitude sufficiently, this chapter generates the risk attitudinal-based consistent IFPR by combining the extracted most optimistic and pessimistic IFPRs. (3) It is very difficult to directly derive the priority weights of alternatives from IVIFPR because of the complexity of operations on IVIFSs. In this chapter, we employ the operator of IVIFS to transform IVIFS into IFS and then dexterously define the consistency of an IVIFPR according to the consistency of an IFPR. Then, to derive the intuitionistic fuzzy priority weights, a multi-objective programming model is constructed and converted into a linear goal programming model to resolve. This multi-objective programming model minimizes the deviations from consistent IFPR and the hesitation degrees of the intuitionistic fuzzy priority weights simultaneously, while the goal programming model built by Wang [6] only minimized the deviations and overlooked the hesitation degrees. The remainder of this chapter is structured as follows. Section 8.2 reviews some concepts of IFPR and defines the additive consistency of IVIFPR. Section 8.3 extracts a risk attitudinal-based consistent IFPR from an IVIFPR by constructing two linear or adjusted programming models. A multi-objective programming model is established to derive the intuitionistic fuzzy priority weights. Section 8.4 obtains a collective IVIFPR by the defined CI-IVIF-IOWA operator and thereby puts forward a novel three-phase method for GDM with IVIFPRs. Section 8.5 offers an example to demonstrate the effectiveness of the method. Section 8.6 ends the paper with some brief concluding remarks.

8.2

Preliminaries

This section reviews the related definitions of IFPRs. Additionally, the additive consistency of IVIFPR is discussed.

8.2.1

Some Related Concepts on IFPR

Definition 8.1 [4]. Let Z ¼ fz1 ; z2 ; . . .; zn g be a non-empty set of universe. An IFS A on Z has the form of A ¼ f\z; lA ðzÞ; mA ðzÞ [ jz 2 Zg, where lA ðzÞ and mA ðzÞ indicate, respectively, the membership and non-membership degrees of the element z to the set A, with the conditions that 0  lA ðzÞ; mA ðzÞ  1 and 0  lA ðzÞ þ mA ðzÞ  1 for all z 2 Z. Let pA ðzÞ ¼ 1  lA ðzÞ  mA ðzÞ. It represents the hesitation margin (hesitation degree) of the element z to the set A.

248

8 A Three-Phase Method for Group Decision Making …

For an IFS A, Xu and Yager [24] called the pair ðlA ðzÞ; mA ðzÞÞ an IFV. For simplicity, denote an IFV by rA ¼ ðlA ; mA Þ, where lA 2 ½0; 1, mA 2 ½0; 1 and lA þ mA  1. Definition 8.2 [23]. Let X ¼ fx1 ; x2 ; . . .; xn g be a finite set of alternatives. An IFPR R on the set X is expressed as an intuitionistic fuzzy judgment matrix R ¼ ðrij Þnn  X  X, where rij ¼ ðlij ; mij Þ is an IFV, lij means the degree to which alternative xi is preferred to alternative xj , mij means the degree to which alternative xi is non-preferred to alternative xj . Let pij ¼ 1  lij  mij , which represents the hesitation degree to which alternative xi is preferred to alternative xj . Moreover, lij and mij meet the conditions: 0  lij þ mij  1, lij ¼ mji , mij ¼ lji , lii ¼ mii ¼ 0:5 for all i; j ¼ 1; 2; . . .; n. Definition 8.3 [6]. An IFPR R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ is said to be an additive consistent IFPR if it fulfills the additive transitivity as follows: lij þ ljk þ lki ¼ lkj þ lji þ lik for all i; j; k ¼ 1; 2; . . .; n

ð8:1Þ

As lij ¼ vji , vij ¼ lji for all i; j ¼ 1; 2; . . .; n, Eq. (8.1) can be rewritten as lij þ ljk þ lki ¼ vij þ vjk þ vki for all i; j; k ¼ 1; 2; . . .; n

ð8:2Þ

Theorem 8.1 [6]. Given an IFPR R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ, R ¼ ðrij Þnn is addititve consistent if and only if (iff) lij þ ljk þ lki ¼ vij þ vjk þ vki for all i\j\k; i; j; k ¼ 1; 2; . . .; n:

ð8:3Þ

Theorem 8.2 [6]. Let R ¼ ðrij Þnn be an IFPR, R ¼ ðrij Þnn is additive consistent iff lij þ ljk  lik ¼ vij þ vjk  vik for all i\j\k; i; j; k ¼ 1; 2; . . .; n:

ð8:4Þ

Theorem 8.2 confines the elements of Eq. (8.4) to the upper triangular part of IFPR R¼ ðrij Þnn , which can simplify the calculation in the decision making. On the basis of the above analysis, it can be concluded that Eqs. (8.1)–(8.4) are equivalent, i.e., any one of Eqs. (8.1)–(8.4) can be treated as the additive consistent condition of an IFPR R ¼ ðrij Þnn . To characterize the additive consistency of an IFPR, Wang [6] introduced the normalized intuitionistic fuzzy priority weight vector. Definition 8.4 [6]. Let w = ðw1 ; w2 ; . . .; wn ÞT be a intuitionistic fuzzy weight vector where wi ¼ ðwli ; wvi Þ is the intuitionistic fuzzy priority weight of alternative xi with the condition that wli ; wvi 2 ½0; 1, wli þ wvi  1 for all i ¼ 1; 2; . . .; n The intuitionistic weight vector w = ðw1 ; w2 ; . .P .; wn ÞT is said to be normalized if Pfuzzy n it satisfies j¼1;j6¼i wlj  wvi and wli þ n  2  nj¼1;j6¼i wvj ði; j ¼ 1; 2; . . .; nÞ.

8.2 Preliminaries

249

Definition 8.5 [6]. An IFPR R ¼ ðrij Þnn with rij ¼ ðlij ; mij Þ is called an additive consistent IFPR if there exists a normalized intuitionistic fuzzy weight vector w = ðw1 ; w2 ; . . .; wn ÞT such that  rij ¼ ðlij ; mij Þ ¼

ð0:5; 0:5Þ; if i ¼ j ð0:5ðwli þ wvj Þ; 0:5ðwvi þ wlj ÞÞ; if i 6¼ j

ð8:5Þ

where wi ¼ ðwli ; wvi Þ indicates the intuitionistic fuzzy priority weight of alternative xi .

8.2.2

Additive Consistency of IVIFPR

~ in Z is symbolized as A ~ ¼ f\z; l ~A~ ðzÞ; Definition 8.6 [5]. An IVIFS A ~A~ ðzÞ ¼ ½lA~ ðzÞ; l A~ ðzÞ ½0; 1 and ~vA~ ðzÞ ¼ ½vA~ ðzÞ; vA~ ~vA~ ðzÞ [ jz 2 Z g, where l ðzÞ ½0; 1 denote the membership degree interval and non-membership degree ~ respectively, with the condition that interval of element z to IVIFS A, ~ can be equivalently rewritten A~ ðzÞ þ vA~ ðzÞ  1 for any z 2 Z. Hence, the IVIFS A l ~ A~ ðzÞ; ½vA~ ðzÞ; vA~ ðzÞÞjz 2 Z g. as A ¼ fðz; ½lA~ ðzÞ; l A~ ðzÞ is called the hesitancy degree interval of ele~A~ ðzÞ ¼ ½pA~ ðzÞ; p Analogously, p A~ ðzÞ  vA~ ðzÞ and p A~ ðzÞ ¼ 1  lA~ ðzÞ  vA~ ðzÞ. ment z 2 Z, where pA~ ðzÞ ¼ 1  l ; ½m; mÞ an interval-valued intuitionistic Yager [25] called the pair ~a ¼ ð½l; l  ½0; 1, ½m; m ½0; 1 and l  þ m  1. fuzzy value (IVIFV), where ½l; l ~ on the set X is characterized by an Definition 8.7 [10]. An IVIFPR R ~ ¼ ð~rij Þ interval-valued intuitionistic fuzzy judgment matrix R nn  X  X, where ~ij ¼ ½lij ; l ij  represents the degree to which alternative ~rij ¼ ð~ lij ; ~mij Þ is an IVIFV, l xi is preferred to alternative xj , ~mij ¼ ½mij ; mij  represents the degree to which alter~ij ¼ ½1  l ij ðxÞ  vij ðxÞ; 1  native xi is non-preferred to alternative xj , and p

lij ðxÞ  vij ðxÞ signifies the hesitation degree to which alternative xi is preferred to

~ij and ~mij satisfy the conditions: l ~ij ¼ ½lij ; l ij  ½0; 1, alternative xj . Moreover, l

~mij ¼ ½mij ; mij  ½0; 1, l ~ii ¼ ~mii ¼ ½0:5; 0:5, 0  l ij þ mij  1, l ~ij ¼ ~mji , ~mij ¼ l ~ji for all i; j ¼ 1; 2; . . .; n. Bustince [26] developed the operator Hs;t of IVIFS which can transform the ~ ¼ fðz; ½l ðzÞ; l A~ ðzÞ; IVIFS into an IFS. The operator Hs;t of IVIFS A ~ A ½vA~ ðzÞ; vA~ ðzÞÞjz 2 Z g is defined as

8 A Three-Phase Method for Group Decision Making …

250

~ ¼ fðz; \ð1  sÞl ðzÞ þ s Hs;t ðAÞ lA~ ðzÞ; ð1  tÞvA~ ðzÞ þ tvA~ ðzÞ [ Þjz 2 Z g; ð8:6Þ ~ A ~ is an IFS. where s; t 2 ½0; 1. Apparently, Hs;t ðAÞ Based on Eq. (8.6), let ij and mij ¼ tij mij þ ð1  tij Þmij for all i\j; i; j ¼ 1; 2; . . .; n; lij ¼ ð1  sij Þlij þ sij l ð8:7Þ where sij ; tij 2 ½0; 1. In terms of Eq. (8.7), we can construct a matrix R ¼ ðrij Þnn , where 8 ij ; tij mij þ ð1  tij Þmij Þ; if i\j < ðð1  sij Þlij þ sij l rij ¼ ðlij ; vij Þ ¼ ð0:5; 0:5Þ; if i ¼ j : ðvji ; lji Þ; if i [ j

ð8:8Þ

As per Definition 8.2, it is easy to verify that such a matrix R ¼ ðrij Þnn is an ij  and mij 2 ½mij ; mij , the matrix R ¼ ðrij Þnn can be IFPR. Because of lij 2 ½lij ; l ~ ¼ ð~rij Þ . regarded as an extracted IFPR from IVIFPR R nn To avoid unreasonable decision results for the decision making problems, it is necessary to discuss the consistency of an IVIFPR. Utilizing the IFPR extracted from an IVIFPR, we give the definition of the additive consistency of an IVIFPR below. ~ ¼ ð~rij Þ Definition 8.8 An IVIFPR R rij ¼ ð~ lij ; ~mij Þ is said to be additive nn with ~ consistent if there exists an additive consistent IFPR R ¼ ðrij Þnn with rij ¼ ðlij ; mij Þ ~ ¼ ð~rij Þ extracted from IVIFPR R nn by Eq. (8.8). ~ ¼ ð~rij Þ ij ¼ lij and mij ¼ mij ¼ mij , IVIFPR R In particular, if lij ¼ l nn degen_ erates to an IFPR R ¼ ð_rij Þ with r_ ij ¼ ðlij ; mij Þ. In this case, it yields by Eq. (8.8) nn

that 8 if i\j < ðlij ; mij Þ; rij ¼ ð0:5; 0:5Þ; if i ¼ j : ðv ; l Þ; if i [ j ji ji Hence, the matrix R ¼ ðrij Þnn with rij ¼ ðlij ; vij Þ extracted by Eq. (8.8) is still an IFPR indeed. ~ ¼ ð~rij Þ ij ¼ 1  mij , then IVIFPR R If lij ¼ 1  mij and l nn is reduced to an ^ ¼ ð^rij Þ ij . In this case, with ^rij ¼ ½l ; l interval fuzzy preference relation R nn

ij

Definition 8.8 reduces to the additive consistency of interval fuzzy preference relation.

8.2 Preliminaries

251

^ ¼ ð^rij Þ ij  Definition 8.9 An interval fuzzy preference relation R rij ¼ ½lij ; l nn with ^ is additive consistent iff there exists an additive consistent fuzzy preference relation ij , sij 2 ½0; 1 and sij þ sji ¼ 1 for all A ¼ ðlij Þnn where lij ¼ ð1  sij Þlij þ sij l i; j ¼ 1; 2; . . .; n.

ij with sij 2 ½0; 1 is equivalent to The expression lij ¼ ð1  sij Þlij þ sij l

ij . Hence, Definition 8.9 is equivalent to Definition 8.5 of additive lij  lij  l consistency of Definitions 8.8 erence relation Definitions 8.8

interval fuzzy preference relation in Zhang et al. [23]. Thus, and 8.9 generalize the additive consistency of interval fuzzy pref[23]. This observation verifies the feasibility and reasonability of and 8.9.

~ ~ ¼ ð~rij Þ rij Þnn is additive consistent if Theorem 8.3 Let R nn be an IVIFPR. R ¼ ð~ ij  þ ½ð1  sjk Þljk þ sjk l jk   ½ð1  sik Þlik þ sik l ik  ¼ ½ð1  sij Þlij þ sij l ½tij mij þ ð1  tij Þmij  þ ½tjk mjk þ ð1  tjk Þmjk   ½tik mik þ ð1  tik Þmik  ði\j\kÞ; ð8:9Þ where 0  sij ; tij  1 ði; j ¼ 1; 2; . . .; n; i\jÞ. It is easy to complete the proof of Theorem 8.3 according to Definition 8.8 and Eq. (8.4).

8.3

Determination of the Intuitionistic Fuzzy Priority Weights from an IVIFPR

This section establishes some programming models to extract the most optimistic and pessimistic IFPRs from an IVIFPR. Then, we introduce a risk attitudinal-based consistent IFPR by considering DMs’ risk attitude. Subsequently, a linear programming model is constructed to derive intuitionistic fuzzy priority weights of alternatives.

8.3.1

Extracting a Risk Attitudinal-Based Consistent IFPR from an IVIFPR

Owing to the computational complexity of IVIFS and IVIFPR, it is quite hard to derive the priority weights of alternatives directly from IVIFPR. Hence, we resort to the IFPR extracted from IVIFPR. As per Definition 8.8, a consistent IFPR could be extracted from a consistent IVIFPR. Then, the intuitionistic fuzzy priority weights

8 A Three-Phase Method for Group Decision Making …

252

of alternatives can be determined from this extracted consistent IFPR. In what follows, how to extract the consistent IFPR from an IVIFPR is addressed. ~ ¼ ð~rij Þ , according to Definition 8.7, it is enough to use Given an IVIFPR R nn ~ ¼ ð~rij Þ only the elements in the upper triangular part of the matrix R nn to extract the consistent IFPR, which can simplify the computation. By using Eq. (8.8), numerous matrices of R ¼ ðrij Þnn can be extracted from an ~ ¼ ð~rij Þ IVIFPR R nn when the parameters sij and tij take different values. Apparently, sij and tij can reflect DMs’ risk preferences on membership and non-membership degrees, respectively. In the upper triangular part of the matrix ~ ¼ ð~rij Þ , the bigger the values of sij and tij , the higher DM’s optimistic degree; R nn the smaller the values of sij and tij , the higher DM’s pessimistic degree. If sij ¼ 1 ij and mij ¼ mij for all i\j (i; j ¼ 1; 2; . . .; n). In this case, and tij ¼ 1, we have lij ¼ l each element rij ¼ ð lij ; mij Þ in the upper triangular part of R ¼ ðrij Þnn can be ~ ¼ ð~rij Þ , which viewed as the upper bound of its corresponding IVIFV in R nn represents the most optimistic judgment for each pairwise comparison from DM. If sij ¼ 0 and tij ¼ 0, one has lij ¼ lij and mij ¼ mij for all i\j (i; j ¼ 1; 2; . . .; n). In

this case, each element rij ¼ ðlij ; mij Þ in the upper triangular part of R ¼ ðrij Þnn can ~ ¼ ð~rij Þ , which be viewed as the lower bound of its corresponding IVIFV in R nn represents the most pessimistic judgment for each pairwise comparison from DM. It follows from Definition 8.8 that, if the extracted IFPR R ¼ ðrij Þnn is con~ ¼ ð~rij Þ sistent, then the IVIFPR R nn is also consistent. Thus, the crucial task is to find appropriate values of sij and tij which make IFPR R ¼ ðrij Þnn consistent as much as possible. Unfortunately, more than one consistent IFPR can be obtained due to the infinite number of possible solutions of sij and tij . Generally speaking, diverse DMs will select different consistent IFPRs depending on their risk preferences. Thus, it is necessary to incorporate DMs’ risk attitude into the decision making process. In what follows, two special extreme cases are considered including the most optimistic and pessimistic cases. To extract the most optimistic consistent IFPR R ¼ ðrij Þnn from an IVIFPR ~ ¼ ð~rij Þ , a linear programming model is set up as follows: R nn P P max k þ ¼ ni¼1 nj¼i þ 1 ðsij þ tij Þ 8 ij  þ ½ð1  sjk Þljk þ sjk l jk   ½ð1  sik Þlik þ sik l ik  ¼ ½tij mij þ ð1  tij Þmij  < ½ð1  sij Þlij þ sij l s:t: þ ½tjk mjk þ ð1  tjk Þmjk   ½tik mik þ ð1  tik Þmik  ði; j; k ¼ 1; 2; . . .; n; i\j\kÞ : 0  sij ; tij  1 ði; j ¼ 1; 2; . . .; n; i\jÞ

ð8:10Þ The constraints in Eq. (8.10) are employed to guarantee the consistency of the IVIFPR R ¼ ðrij Þnn according to Theorem 8.3. After solving Eq. (8.10) by Simplex method, the optimal solutions sij and tij are obtained. Then, the most optimistic consistent IFPR R ¼ ðrij Þnn , denoted by R þ ¼ ~ ¼ ð~rij Þ ðr þ Þ with r þ ¼ ðl þ ; m þ Þ, can be extracted from the IVIFPR R by ij

nn

ij

ij

ij

nn

8.3 Determination of the Intuitionistic Fuzzy Priority Weights from an IVIFPR

253

using Eq. (8.8). Clearly, all the elements in IFPR R þ ¼ ðrijþ Þnn signify the most optimistic judgments on each pair of alternatives. In a similar way, the most pessimistic consistent IFPR R ¼ ðrij Þnn , denoted by  ~ rij Þnn R ¼ ðrij Þnn with rij ¼ ðl ij ; mij Þ, can be extracted from the IVIFPR R ¼ ð~ by solving the following linear programming model: n P n P min k ¼ ðsij þ tij Þ i¼1 j¼i þ 1 8 ij  þ ½ð1  sjk Þljk þ sjk l jk   ½ð1  sik Þlik þ sik l ik  ¼ ½tij mij þ ð1  tij Þmij  < ½ð1  sij Þlij þ sij l s:t: þ ½tjk mjk þ ð1  tjk Þmjk   ½tik mik þ ð1  tik Þmik  ði; j; k ¼ 1; 2; . . .; n; i\j\kÞ : 0  sij ; tij  1 ði; j ¼ 1; 2; . . .; n; i\jÞ

ð8:11Þ Clearly, all the elements in IFPR R ¼ ðrij Þnn signify the most pessimistic judgments on each pair of alternatives. If the feasible regions of the above programming models (i.e., Eqs. (8.10) and (8.11)) are not empty, sij and tij can be obtained. Then the most optimistic consistent IFPR R þ ¼ ðrijþ Þnn and the most pessimistic consistent IFPR R ¼ ðrij Þnn can be obtained. Nevertheless, it cannot ensure that such the models always have the non-empty feasible regions. When the feasible regions of Eqs. (8.10) and (8.11) are empty, no any consistent IFPRs can be extracted from the original IVIFPR. To make up this shortcoming, we can broaden the search space of the consistent IFPR by enlarging the associated IVIFVs to some extent. To achieve this goal, each ~ is enlarged as ~rij ¼ ð½l  ij ; ½mij ; mij Þ in IVIFPR R IVIFV ~rij ¼ ð½lij ; l ij þ ij þ aijþ ; ½mij  b a mij þ bijþ Þ through introducing deviation variables a ij ; l ij , aij , ij ;   þ  þ þ  bij and bij , where aij ; aij ; bij ; bij  0. Thus, Eq. (8.8) can be converted as follows:

8  lij þ aijþ Þ; tij ðmij  b mij þ bijþ ÞÞ; < ðð1  sij Þðlij  aij Þ þ sij ð ij Þ þ ð1  tij Þð rij ¼ ðlij ; vij Þ ¼ ð0:5; 0:5Þ; : ðvji ; lji Þ;

if i\j if i ¼ j if i [ j

ð8:12Þ Consequently, the adjusted programming models corresponding to Eqs. (8.10) and (8.11) are respectively established by Eqs. (8.4) and (8.12) as follows:

254

8 A Three-Phase Method for Group Decision Making …

n P n n P n P P  þ þ max k þ ¼ ðsij þ tij Þ  ða ij þ aij þ bij þ bij Þ i¼1 j¼i þ 1 i¼1 j¼i þ 1 8 ½ð1  sij Þðlij  a Þ þ sij ð lij þ aijþ Þ þ ½ð1  sjk Þðljk  a ljk þ ajkþ Þ > ij jk Þ þ sjk ð > > > > lik þ aikþ Þ ¼ ½tij ðmij  b mij þ bijþ Þ  ½ð1  sik Þðlik  a > ij Þ þ ð1  tij Þð ik Þ þ sik ð > >  þ  > þ ½tjk ðm  b Þ þ ð1  tjk Þðmjk þ b Þ  ½tik ðm  b Þ þ ð1  tik Þðmik þ b þ Þ > jk ik > jk jk ik ik > > > ði; j; k ¼ 1; 2; . . .; n; i\j\kÞ <  lij þ aijþ Þ þ ½tij ðmij  b mij þ bijþ Þ  1; s:t: ½ð1  sij Þðlij  aij Þ þ sij ð ij Þ þ ð1  tij Þð > > > ði; j ¼ 1; 2; . . .; n; i\jÞ > > > > ð1  sij Þðlij  a Þ þ sij ð lij þ aijþ Þ  0 ði; j ¼ 1; 2; . . .; n; i\jÞ > ij > > > > tij ðmij  b Þ þ ð1  tij Þðmij þ bijþ Þ  0 ði; j ¼ 1; 2; . . .; n; i\jÞ > > :  þ ij þ aij ; aij ; bij ; bij  0; 0  sij ; tij  1 ði; j ¼ 1; 2; . . .; n; i\jÞ

ð8:13Þ and n P n n P n P P  þ þ min k ¼ ðsij þ tij Þ þ ða ij þ aij þ bij þ bij Þ i¼1 j¼i þ 1 i¼1 j¼i þ 1 8 ½ð1  sij Þðlij  a lij þ aijþ Þ þ ½ð1  sjk Þðljk  a ljk þ ajkþ Þ > ij Þ þ sij ð jk Þ þ sjk ð > >  > þ  > ½ð1  sik Þðlik  aik Þ þ sik ð lik þ aik Þ ¼ ½tij ðmij  bij Þ þ ð1  tij Þðmij þ bijþ Þ > > > > > mjk þ bjkþ Þ  ½tik ðmik  b mik þ bikþ Þ þ ½tjk ðmjk  b > jk Þ þ ð1  tjk Þð ik Þ þ ð1  tik Þð > > > ði; j; k ¼ 1; 2; . . .; n; i\j\kÞ <  lij þ aijþ Þ þ ½tij ðmij  b mij þ bijþ Þ  1; s:t: ½ð1  sij Þðlij  aij Þ þ sij ð ij Þ þ ð1  tij Þð > > > ði; j ¼ 1; 2; . . .; n; i\jÞ > > þ > > ð1  sij Þðlij  a Þ þ s ð l þ a Þ  0 ði; j ¼ 1; 2; . . .; n; i\jÞ ij > ij ij ij > >  þ > > tij ðmij  b Þ þ ð1  tij Þðmij þ bij Þ  0 ði; j ¼ 1; 2; . . .; n; i\jÞ > > :  þ ij þ aij ; aij ; bij ; bij  0; 0  sij ; tij  1 ði; j ¼ 1; 2; . . .; n; i\jÞ

ð8:14Þ Therefore, if the feasible regions of Eqs. (8.10) and (8.11) are empty, we can respectively solve Eqs. (8.13) and (8.14) to get the optimal solutions. The most optimistic consistent IFPR R þ ¼ ðrijþ Þnn and the most pessimistic consistent IFPR R ¼ ðrij Þnn are then derived according to Eq. (8.12). Next, we use the risk attitude of DMs to integrate the most optimistic consistent IFPR R þ and the most pessimistic consistent IFPR R . Then, an extracted risk attitudinal-based consistent IFPR R ¼ ðrij Þnn with rij ¼ ðl ij ; m ij Þ is introduced as follows:

þ  l ij ¼ hlijþ þ ð1  hÞl ij ; mij ¼ hmij þ ð1  hÞmij for all i; j ¼ 1; 2; . . .; n;

ð8:15Þ

where h indicates the risk attitude of DMs. When 0:5\h  1, DMs are risk-taking; when 0  h\0:5, DMs are risk averse; when h ¼ 0:5, DMs are indifferent to risk.

8.3 Determination of the Intuitionistic Fuzzy Priority Weights from an IVIFPR

255

Remark 8.1 The parameter h in Eq. (8.15) characterizes DMs’ risk attitude. How to determine the risk attitude of DMs is a critical issue in real-life decision making problems, which is overlooked in this chapter. Zhou and Xu [27] developed the optimal discrete fitting and risk preference selection approaches to obtain the risk appetite parameters in the generalized sigmoid scale. Motivated by the idea of Zhou and Xu [27], we will study how to derive the risk attitude reasonably in near future.

8.3.2

Deriving the Intuitionistic Fuzzy Priority Weights from the Extractive IFPR

According to Definition 8.5, the extracted risk attitudinal-based consistent IFPR R ¼ ðrij Þnn is additive consistent if there exists a normalized intuitionistic fuzzy priority weight vector w = ðw1 ; w2 ; . . .; wn ÞT satisfying Eq. (8.5). In real-world decision making, an IFPR cannot always meet Eq. (8.5). Thus, it is natural and reasonable to search a normalized intuitionistic fuzzy priority weight vector w = ðw1 ; w2 ; . . .; wn ÞT such that the deviation of R ¼ ðrij Þnn from an additive consistent IFPR is minimized. Hence, denote the deviation variables by zlij ¼ j0:5ðwli þ wmj Þ  l ij j and zvij ¼ j0:5ðwmi þ wlj Þ  m ij j ði; j ¼ 1; 2; . . .; n; i\jÞ. The smaller the deviation, the stronger the additive consistency of the extracted IFPR R ¼ ðrij Þnn . In the meanwhile, there exists uncertainty in the priority weights represented as IFVs owing to the hesitation degrees inherent in IFVs. The less the uncertainty, the less the unknown information in the intuitionistic fuzzy priority weights and the more reasonable the intuitionistic fuzzy priority weights. For intuitionistic fuzzy priority weight wi ¼ ðwli ; wvi Þ, denote the hesitation degree by wpi ¼ 1  wli  wvi . As a result, to derive the intuitionistic fuzzy priority weights from the extracted risk attitudinal-based consistent IFPR R ¼ ðrij Þnn , it is reasonable to minimize the deviations zlij and zvij as well as the hesitation degree wpi . Thus, a multi-objective programming model is established as follows: min zlij ¼ j0:5ðwli þ wmj Þ  l ij j ði; j ¼ 1; 2; . . .; n; i\jÞ min zvij ¼ j0:5ðwmi þ wlj Þ  m ij j ði; j ¼ 1; 2; . . .; n; i\jÞ min zpi ¼ wpi ði ¼ 1; 2; . . .; nÞ n n P P wlj  wvi ; wli þ n  2  wvj ; wli þ wvi  1; 0  wli ; wvi  1 ði ¼ 1; 2; . . .; nÞ s:t: j¼1;j6¼i

j¼1;j6¼i

ð8:16Þ

256

8 A Three-Phase Method for Group Decision Making …

In order to solve the above model, let 1 2 1 ½j0:5ðwli þ wmj Þ  l ij j  ð0:5ðwli þ wmj Þ  l ij Þ; 2

1 2 1 ½j0:5ðwmi þ wlj Þ  m ij j  ð0:5ðwmi þ wlj Þ  m ij Þ: 2

pijþ ¼ ½j0:5ðwli þ wmj Þ  l ij j þ ð0:5ðwli þ wmj Þ  l ij Þ; qijþ ¼ ½j0:5ðwmi þ wlj Þ  m ij j þ ð0:5ðwmi þ wlj Þ  m ij Þ; p ij ¼

q ij ¼

Since there are no any preferences among all the objectives, by using the equal weight summation method, Eq. (8.16) is transformed into a linear goal programming model: n n P n P P þ þ  min Z ¼ ð1  wli  wvi Þ þ ðp ij  pij þ qij  qij Þ i¼1 i¼1 j¼i þ 1 8 þ

0:5ðwli þ wmj Þ þ p > ij  pij ¼ lij ði; j ¼ 1; 2; . . .; n; i\jÞ > > þ > 

> < 0:5ðwmi þ wlj Þ þ qij  qij ¼ mij ði; j ¼ 1; 2; . . .; n; i\jÞ n n P P s:t: wlj  wvi ; wli þ n  2  wvj ; wli þ wvi  1; 0  wli ; wvi  1 ði ¼ 1; 2; . . .; nÞ > > > j¼1;j6¼i j¼1;j6¼i > > : p ; p þ ; q ; q þ  0 ði; j ¼ 1; 2; . . .; n; i\jÞ ij ij ij ij

ð8:17Þ By solving Eq. (8.17), the intuitionistic fuzzy priority weights wi ¼ ðwli ; wvi Þ ði ¼ 1; 2; . . .; nÞ are derived from the IFPR R ¼ ðrij Þnn , which can be utilized to rank alternatives. Remark 8.2 Xu [28] constructed two linear programming models to develop a method for estimating priority weights from IFPRs. However, this chapter establishes a multi-objective programming model and converts it to a linear programming model for deriving the intuitionistic fuzzy priority weights. The differences between Xu [28] and this chapter exist in two aspects: (i) To obtain the priority weights from IFPR, Xu [28] constructed two programming models by maximizing and minimizing the priority weighs, while this chapter establishes a linear programming model by minimizing the deviations of consistency as well as the hesitation degree; (ii) Xu [28] derived the numerical priority weights from IFPR, while this chapter generates the intuitionistic fuzzy priority weights from the extracted risk attitudinal-based consistent IFPR.

8.4

A Novel Three-Phase Method for Solving GDM with IVIFPRs

In this section, the problem for GDM with IVIFPRs is described. The CI-IVIF-IOWA operator is developed to integrate the individual IVIFPRs. Then a new method is proposed to solve GDM with IVIFPRs.

8.4 A Novel Three-Phase Method for Solving GDM with IVIFPRs

8.4.1

257

Presentation of Problem for GDM with IVIFPRs

Consider a problem of GDM with IVIFPRs, denote the set of DMs by E ¼ fe1 ; e2 ; . . .; em g, the set of alternatives by X ¼ fx1 ; x2 ; . . .; xn g. Assume that each DM ek is able to give his/her preference for each pair of alternatives and then ~ k ¼ ð~r k Þ kij ; ½mkij ; mkij Þ rijk ¼ ð½lkij ; l provides the individual IVIFPR R ij nn with ~ ði; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; mÞ. Let w ¼ ðw1 ; w2 ; . . .; wn ÞT be the normalized intuitionistic fuzzy priority weight vector of alternatives which needs to be determined to rank the alternatives.

8.4.2

Integrating Individual IVIFPRs to a Collective One

i ; ½mi ; mi Þ ði ¼ 1; 2Þ, Xu and Yager [14] defined the For two IVIFVs ~ai ¼ ð½li ; l Hamming distance between ~a1 and ~a2 as follows: 1 4

2 j þ jm1  m2 j þ jm1  m2 j þ jp1  p2 j þ j 2 jÞ hdð~a1 ; ~a1 Þ ¼ ðjl1  l2 j þ j l1  l p1  p

ð8:18Þ i  mi and p i ¼ 1  li  mi ði ¼ 1; 2Þ. where pi ¼ 1  l Analogous to Eq. (8.18), we define the Hamming distance between individual ~l ~ k ¼ ð~r k Þ rijl Þnn as follows: IVIFPRs R ij nn and R ¼ ð~ ~k; R ~ lÞ ¼ hdðR

1 2nðn  1Þ

n X n X i

j¼i þ 1

lij j þ jmkij  mlij j þ jmkij  mlij j þ jpkij  plij j þ j lij jÞ ðjlkij  llij j þ j lkij  l pkij  p

ð8:19Þ ~ l is defined as ~ k and R Then, the similarity degree between R ~k; R ~ l Þ ¼ 1  hdðR ~k; R ~ l Þ: SðR

ð8:20Þ

Therefore, the confidence degree CSk of DM ek is defined as CSk ¼

m X

~k; R ~ l Þ ðk ¼ 1; 2; . . .; mÞ: SðR

l¼1;l6¼k

In the GDM problem, some IVIFPRs provided by some DMs have great similarity if their judgments are almost consistent. Thus, in GDM, the higher the confidence degree of a DM, the more importance should be allocated to such a DM. The confidence degrees can be viewed as the objective importance degrees of DMs.

258

8 A Three-Phase Method for Group Decision Making …

After normalizing CSk ðk ¼ 1; 2; . . .; mÞ, the normalized objective importance P degrees of DMs are generated as CkO ¼ CSk = m k¼1 CSk ðk ¼ 1; 2; . . .; mÞ. In real decision making problems, different DMs have diverse expertise, knowledge, and experiences. Thereby, the subjective importance degrees of DMs Pm S can be given, denoted by CkS ðk ¼ 1; 2; . . .; mÞ with CkS  0 and k¼1 Ck ¼ 1. Hence, DMs’ comprehensive importance degrees are obtained as follows: Ck ¼ hCkO þ ð1  hÞCkS ; ðk ¼ 1; 2; . . .; mÞ;

ð8:21Þ

where h ð0  h  1Þ is a control parameter which can tradeoff the DMs’ subjective and objective importance degrees. In general, take h = 0.5. Aggregation operator is usually used to integrate the individual IVIFPRs into the collective one. To do so, a new CI-IVIF-IOWA operator is developed in the sequel. Definition 8.10 [29] Let R be the set of real numbers, ai ði ¼ 1; 2; . . .; mÞ be the argument variables. An IOWA operator of dimension m is a mapping IOWAx : ðR  RÞm ! R such that IOWAx ð\u1 ; a1 [ ; \u2 ; a2 [ ;


; \um ; am [ Þ ¼

m X

xi aðiÞ ;

ð8:22Þ

i¼1

where x = ðx1 ; x2 ; . . .; xm ÞT isP the associated weighting vector satisfying m 0  xi  1 ði ¼ 1; 2; . . .; mÞ and i¼1 xi ¼ 1, ui is order-inducing variable, and ðð1Þ; ð2Þ; . . .; ðmÞÞ is a permutation of ð1; 2; . . .; mÞ such that uðiÞ  uði þ 1Þ for all i, \uðiÞ ; aðiÞ [ is the 2-tuple with uðiÞ as the i-th highest value of fu1 ; u2 ; . . .; um g. In the GDM problem, the higher the comprehensive importance degree of a DM, the more important the DM. Thus, by using the comprehensive importance degrees as order inducing variables, the CI-IVIF-IOWA operator is introduced to integrate all individual IVIFPRs. Definition 8.11 (CI-IVIF-IOWA operator). Assume that DMs ek ðk ¼ 1; 2; . . .; mÞ provide preferences for each pair of alternatives X ¼ fx1 ; x2 ; . . .; xn g, and form the ~ k ¼ ð~r k Þ kij ; ½mkij ; mkij Þ ðk ¼ 1; 2; . . .; mÞ, rijk ¼ ð½lkij ; l individual IVIFPRs R ij nn with ~ respectively. A CI-IVIF-IOWA operator of dimension m is an IOWA operator where the order inducing values are the comprehensive importance degrees fC1 ; C2 ; . . .; Cm g of DMs. ~ ¼ ð~rij Þ Employing the CI-IVIF-IOWA operator, the collective IVIFPR R nn ij ; ½mij ; mij Þ is acquired as follows: with ~rij ¼ ð½lij ; l lij ¼ CI-IVIF-IOWAx ð\C1 ; l1ij [ ; \C2 ; l2ij [ ;


; \Cm ; lm [Þ ¼ ij

m X

xk lðkÞ ; ij

k¼1

ð8:23Þ

8.4 A Novel Three-Phase Method for Solving GDM with IVIFPRs

259

ij ¼ CI-IVIF-IOWAx ð\C1 ; l 1ij [ ; \C2 ; l 2ij [ ;


; \Cm ; l m l ij [ Þ ¼

m X

ðkÞ

ij ; xk l

k¼1

ð8:24Þ mij ¼ CI-IVIF-IOWAx ð\C1 ; m1ij [ ; \C2 ; m2ij [ ;


; \Cm ; mm ij [ Þ ¼

m X

ðkÞ

xk mij ;

k¼1

ð8:25Þ mij ¼ CI-IVIF-IOWAx ð\C1 ; m1ij [ ; \C2 ; m2ij [ ;


; \Cm ; mm ij [ Þ ¼

m X

ðkÞ

xk mij :

k¼1

ð8:26Þ where ðð1Þ; ð2Þ; . . .; ðmÞÞ is a permutation of ð1; 2; . . .; mÞ such that CðkÞ  Cðk þ 1Þ for all k. ij ; ½mij ; mij Þ can be rewritten as By Eqs. (8.23)–(8.26), ~rij ¼ ð½lij ; l m m m m X X X X ðkÞ ðkÞ ðkÞ ~rij ¼ ð½lij ; l ij ; ½mij ; mij Þ ¼ ð½  xk lðkÞ ; x ; ½ x m ; xk mij Þ: l k k ij ij ij k¼1

k¼1

k¼1

k¼1

ð8:27Þ Yager [25] furnished a procedure to compute the weighting vector associated with an IOWA operator as follows: xk ¼ QðSðkÞ =SðmÞ Þ  QðSðk1Þ =SðmÞ Þ;

ð8:28Þ

where Q is the membership function of the fuzzy linguistic quantifier, P SðkÞ ¼ kl¼1 uðlÞ . The fuzzy linguistic quantifier Q is a basic unit-interval monotone (BUM) function Q : ½0; 1 ! ½0; 1, with the conditions that Qð0Þ ¼ 0, Qð1Þ ¼ 1 and QðxÞ  QðyÞ if x [ y. In the CI-IVIF-IOWA operator, the comprehensive importance degrees of DMs act as the order inducing values. Hence, the weighting vector x ¼ ðx1 ; x2 ; . . .; xm ÞT associated with the CI-IVIF-IOWA operator can be calculated as follows: xk ¼ QðSðkÞ =SðmÞ Þ  QðSðk1Þ =SðmÞ Þ ðk ¼ 1; 2; . . .; mÞ;

ð8:29Þ

P where SðkÞ ¼ kl¼1 CðlÞ and CðlÞ is the l-th largest value in the set fC1 ; C1 ; . . .; Cm g. In general, we can take QðrÞ ¼ r f ðf  0Þ. To ensure that the higher the comprehensive importance degree, the higher the weighting value associated with the CI-IVIF-IOWA operator, i.e., Cð1Þ  Cð2Þ 


 CðmÞ ) x1  x2 


 xm , Chiclana et al. [17] imposed 0  f  1 to the BUM function. Especially, when 1 f ¼ 12, the BUM function QðrÞ ¼ r 2 signifies the linguistic majority “most of”.

8 A Three-Phase Method for Group Decision Making …

260

8.4.3

A Three-Phase Method for GDM with IVIFPRs

On the basis of the above analyses, this section puts forward a three-phase method for GDM with IVIFPRs. The detailed decision steps are presented as follows: (1) Aggregation phase ~ l Þ ðk; l ¼ 1; 2; . . .; m; k 6¼ lÞ by ~k; R Step 1 Compute the similarity degree SðR Eq. (8.20). Step 2 Acquire comprehensive importance degree Ck ðk ¼ 1; 2; . . .; mÞ by Eq. (8.21). ~ by Step 3 Use the CI-IVIF-IOWA operator to derive the collective IVIFPR R Eq. (8.27). (2) Extraction phase ~ If Step 4 Construct models Eqs. (8.10) and (8.11) for the collective IVIFPR R. these two models have non-empty feasible regions, solve them to obtain the values of lij and vij for i\j and derive the most optimistic consistent IFPR R þ and the most pessimistic consistent IFPR R by Eq. (8.8). Then, go to Step 6. Else, go to next Step 5. Step 5 Solve the adjusted programming models constructed by Eqs. (8.13) and (8.14) to derive the values of lij and vij for i\j. Then generate the most optimistic consistent IFPR R þ and the most pessimistic consistent IFPR R by Eq. (8.12). Step 6 Obtain the extracted risk attitudinal-based consistent IFPR R ¼ ðrij Þnn by Eq. (8.15). (3) Exploitation phase Step 7 Employ Eq. (8.17) to derive the intuitionistic fuzzy priority weight wi ¼ ðwli ; wvi Þ of alternative xi from the extracted IFPR R ¼ ðrij Þnn . Step 8 Rank the intuitionistic fuzzy priority weight wi ¼ ðwli ; wvi Þ ði ¼ 1; 2;


; nÞ by method [30] and generate the ranking order of alternatives. The above framework for GDM with IVFPRs is depicted graphically in Fig. 8.1.

8.5

An Example of Network System Selection and Comparison Analyses

This section analyzes a real example of Network System selection and then conducts some comparison analyses with other similar methods.

8.5 An Example of Network System Selection and Comparison Analyses

261

Aggregation phase

Compute the similarity degree S ( R k , R l ) (k , l = 1, 2,3, k ≠ l ) by Eq. (8.20).

Acquire comprehensive importance degree Ck of DM ek by Eq. (8.21).

Use CI-IVIF-IOWA operator (Eq. (8.27)) to obtain the collective IVIFPR R

Extraction phase

Construct two linear programming models by Eqs. (8.10) and (8.11) for the collective IVIFPR R .

If the feasible regions of Eqs. (8.10) and (8.11) are empty

Solve two adjusted linear programming models Eqs. (8.13) and (8.14).

If Eqs. (8.10) and (8.11) have non-empty feasible regions, solve Eqs. (8.10) and (8.11)

Derive the most optimistic and pessimistic consistent IFPR R + and R − .

Obtain the extracted IFPR R * = ( rij* )n × n by Eq. (8.15).

Exploitation phase

Employ Eq. (8.17) to derive intuitionistic fuzzy priority weights wi (i = 1, 2,  , n)

Rank the intuitionistic fuzzy priority weights wi = ( wμi , wvi )

(i = 1, 2,

, n) by method [32] and get the ranking order of alternatives.

Fig. 8.1 Framework of group decision making with IVIFPRs

8.5.1

A Network System Selection Example and the Analysis Process

Jiangxi University of Finance and Economics of China desires to build multi-function building. One of problems that the university is facing is how select a Network System from four feasible plans (alternatives) {x1 ; x2 ; x3 ; x4 } install in this new building. Three DMs E ¼ fe1 ; e2 ; e3 g are invited to take part

a to to in

8 A Three-Phase Method for Group Decision Making …

262

the decision making process. Each DM furnishes his/her preferences with IVIFPR as follows: 0

ð½0:5; 0:5; ½0:5; 0:5Þ

ð½0:5; 0:7; ½0:1; 0:2Þ

ð½0:3; 0:5; ½0:4; 0:5Þ

ð½0:2; 0:5; ½0:3; 0:4Þ ð½0:5; 0:5; ½0:5; 0:5Þ B ð½0:4; 0:5; ½0:2; 0:4Þ ~2 ¼ B R B @ ð½0:4; 0:5; ½0:3; 0:4Þ ð½0:3; 0:4; ½0:2; 0:3Þ 0 ð½0:5; 0:5; ½0:5; 0:5Þ B ð½0:2; 0:4; ½0:3; 0:5Þ ~3 ¼ B R B @ ð½0:3; 0:4; ½0:4; 0:5Þ ð½0:2; 0:3; ½0:4; 0:5Þ

ð½0:5; 0:5; ½0:5; 0:5Þ ð½0:4; 0:5; ½0:2; 0:4Þ ð½0:1; 0:3; ½0:6; 0:7Þ ð½0:2; 0:4; ½0:4; 0:5Þ ð½0:5; 0:5; ½0:5; 0:5Þ ð½0:1; 0:2; ½0:7; 0:8Þ ð½0:2; 0:4; ½0:3; 0:6Þ ð½0:3; 0:5; ½0:2; 0:4Þ ð½0:5; 0:5; ½0:5; 0:5Þ ð½0:1; 0:4; ½0:5; 0:6Þ ð½0:1; 0:2; ½0:7; 0:8Þ

ð½0:2; 0:4; ½0:4; 0:5Þ ð½0:5; 0:5; ½0:5; 0:5Þ ð½0:3; 0:5; ½0:3; 0:5Þ ð½0:3; 0:4; ½0:4; 0:5Þ ð½0:7; 0:8; ½0:1; 0:2Þ ð½0:5; 0:5; ½0:5; 0:5Þ ð½0:1; 0:4; ½0:4; 0:5Þ ð½0:4; 0:5; ½0:3; 0:4Þ ð½0:5; 0:6; ½0:1; 0:4Þ ð½0:5; 0:5; ½0:5; 0:5Þ ð½0:2; 0:3; ½0:4; 0:6Þ

B ð½0:1; 0:2; ½0:5; 0:7Þ ~1 ¼ B R B @ ð½0:4; 0:5; ½0:3; 0:5Þ 0

ð½0:3; 0:4; ½0:2; 0:5Þ

1

ð½0:6; 0:7; ½0:1; 0:3Þ C C C ð½0:3; 0:5; ½0:3; 0:5Þ A ð½0:5; 0:5; ½0:5; 0:5Þ 1 ð½0:2; 0:3; ½0:3; 0:4Þ ð½0:3; 0:6; ½0:2; 0:4Þ C C C ð½0:4; 0:5; ½0:1; 0:4Þ A ð½0:5; 0:5; ½0:5; 0:5Þ 1 ð½0:4; 0:5; ½0:2; 0:3Þ ð½0:7; 0:8; ½0:1; 0:2Þ C C C ð½0:4; 0:6; ½0:2; 0:3Þ A ð½0:5; 0:5; ½0:5; 0:5Þ

~k; R ~ l Þ ðk; l ¼ Step 1 According to Eq. (8.20), the similarity degrees SðR 1; 2; 3; k6¼lÞ are obtained as follows: ~ 1; R ~ 2 Þ ¼ SðR ~ 2; R ~ 1 Þ ¼ 0:5667; SðR ~ 1; R ~ 3 Þ ¼ SðR ~ 3; R ~ 1 Þ ¼ 0:6500; SðR ~ 2; R ~ 3Þ SðR 3 2 ~ Þ ¼ 0:6833: ~ ;R ¼ SðR Step 2 Take h = 0.5 and suppose the subjective importance degrees of DMs are (0.3, 0.2, 0.5)T. According to Eq. (8.21), the comprehensive importance degrees are calculated as C1 ¼ 0:3145; C2 ¼ 0:2601; C3 ¼ 0:4254: Step 3 Set QðrÞ ¼ r 2 . Utilizing the CI-IVIF-IOWA operator, the collective IVIFPR ~ can be derived by Eq. (8.27) as follows: R 1

0

1 ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:41; 0:61; ½0:18; 0:29Þ ð½0:31; 0:48; ½0:39; 0:49Þ ð½0:29; 0:39; ½0:22; 0:45Þ B ð½0:18; 0:29; ½0:41; 0:61Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:35; 0:51; ½0:30; 0:42Þ ð½0:55; 0:69; ½0:12; 0:31Þ C ~¼B C R @ ð½0:39; 0:49; ½0:31; 0:48Þ ð½0:30; 0:42; ½0:35; 0:51Þ ð½0:50; 0:50; ½0:50; 0:50Þ ð½0:33; 0:51; ½0:24; 0:45Þ A ð½0:22; 0:45; ½0:29; 0:39Þ ð½0:12; 0:31; ½0:55; 0:69Þ ð½0:24; 0:45; ½0:33; 0:51Þ ð½0:50; 0:50; ½0:50; 0:50Þ

Step 4 As per Eqs. (8.10) and (8.11), two linear programming models are estab~ respectively. But it is easy to find that both the lished for the collective IVIFPR R, corresponding feasible regions are empty. Therefore, go to Step 5.

8.5 An Example of Network System Selection and Comparison Analyses

263

Step 5 In terms of Eqs. (8.13) and (8.14), two adjusted programming models are ~ Solving these two models, we can obtain constructed for the collective IVIFPR R. all lij and mij ði; j ¼ 1; 2; 3; 4; i\jÞ, respectively. Then by Eq. (8.12), the optimistic consistent matrix R þ and pessimistic consistent matrix R are acquired as follows: 0

Rþ

ð0:5000; 0:5000Þ B ð0:1764; 0:1624Þ B ¼B @ ð0:2848; 0:4862Þ

ð0:2208; 0:7792Þ 0 ð0:5000; 0:5000Þ B ð0:2903; 0:1960Þ B R ¼ B @ ð0:4860; 0:3140Þ ð0:5816; 0:2932Þ

ð0:1624; 0:1764Þ ð0:5000; 0:5000Þ

ð0:4862; 0:2848Þ ð0:5111; 0:2957Þ

ð0:2957; 0:5111Þ ð0:1208; 0:6932Þ ð0:1960; 0:2903Þ ð0:5000; 0:5000Þ

ð0:5000; 0:5000Þ ð0:1570; 0:5140Þ ð0:3140; 0:4860Þ ð0:3459; 0:4236Þ

ð0:4236; 0:3459Þ ð0:3068; 0:1127Þ

ð0:5000; 0:5000Þ ð0:4512; 0:3348Þ

1 ð0:7792; 0:2208Þ ð0:6932; 0:1208Þ C C C ð0:5140; 0:1570Þ A ð0:5000; 0:5000Þ 1 ð0:2932; 0:5816Þ ð0:1127; 0:3068Þ C C C ð0:3348; 0:4512Þ A ð0:5000; 0:5000Þ

Step 6 Suppose that DMs are neutral for risk, i.e., h ¼ 0:5. The extracted IFPR R

can be determined by Eq. (8.15) as follows: 0

ð0:5000; 0:5000Þ B ð0:2334; 0:1792Þ

B R ¼@ ð0:3854; 0:4001Þ ð0:4012; 0:5362Þ

ð0:1792; 0:2334Þ ð0:5000; 0:5000Þ ð0:3597; 0:4285Þ ð0:2138; 0:4030Þ

ð0:4001; 0:3854Þ ð0:4285; 0:3597Þ ð0:5000; 0:5000Þ ð0:3041; 0:4244Þ

1 ð0:5362; 0:4012Þ ð0:4030; 0:2138Þ C C ð0:4244; 0:3041Þ A ð0:5000; 0:5000Þ

Step 7 Employing Eq. (8.17), the intuitionistic fuzzy priority weights of alternatives are derived from the extracted IFPR R as follows: w1 ¼ ð0:2068; 0:7932Þ; w2 ¼ ð0:2580; 0:5302Þ; w3 ¼ ð0:1891; 0:5990Þ; w4 ¼ ð0:0915; 0:8656Þ: Step 8 By using method [30], the similarity measures Lðwi Þ and accuracy degrees Hðwi Þ of wi ði ¼ 1; 2; . . .; nÞ are computed as Lðw1 Þ ¼ 0:2068; Lðw2 Þ ¼ 0:3877; Lðw3 Þ ¼ 0:3309; Lðw4 Þ ¼ 0:1289; Hðw1 Þ ¼ 1:0000; Hðw2 Þ ¼ 0:7882; Hðw3 Þ ¼ 0:7881; Hðw4 Þ ¼ 0:9571: Hence, the order of alternatives is generated as x2  x3  x1  x4 and the best alternative is x2 . In the meanwhile, when h takes different values, the corresponding computation results and ranking orders can be obtained and listed in Table 8.1. In addition, for different BUM functions QðrÞ, the corresponding computation results are presented in Table 8.2. Table 8.2 shows that the ranking order of alternatives may be different for diverse values of parameter h. If 0:5\h  1, i.e., DMs are risk-taking, then the

8 A Three-Phase Method for Group Decision Making …

264

Table 8.1 Similarity, accuracy degrees and ranking orders for different values of parameter h 1 with QðrÞ ¼ r 2 h

Lðw1 Þ

Hðw1 Þ

Lðw2 Þ

Hðw2 Þ

Lðw3 Þ

Hðw3 Þ

Lðw4 Þ

Hðw4 Þ

Ranking order

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.140 0.196 0.207 0.161 0.261 0.207 0.351 0.389 0.414 0.462 0.460

0.837 0.775 0.811 1.000 1.000 1.000 1.000 1.000 1.000 0.974 0.807

0.234 0.311 0.279 0.322 0.291 0.388 0.321 0.341 0.372 0.390 0.441

0.770 0.661 0.777 0.781 0.784 0.788 0.792 0.796 0.799 0.803 0.807

0.297 0.316 0.295 0.313 0.259 0.331 0.239 0.235 0.241 0.235 0.261

0.770 0.886 0.777 0.781 0.784 0.788 0.792 0.796 0.799 0.803 0.807

0.380 0.331 0.273 0.241 0.249 0.129 0.165 0.113 0.048 0.000 0.000

0.913 1.000 0.988 0.825 0.850 0.957 0.900 0.925 0.950 1.000 1.000

x4 x4 x3 x2 x2 x2 x1 x1 x1 x1 x1

 x3  x3  x2  x3  x1  x3  x2  x2  x2  x2  x2

 x2  x2  x4  x4  x3  x1  x3  x3  x3  x3  x3

 x1  x1  x1  x1  x4  x4  x4  x4  x4  x4  x4

ranking orders of alternatives are always x1  x2  x3  x4 . If h ¼ 0:5, i.e., DMs are neutral, then the ranking order of alternatives is x2  x3  x1  x4 . If 0  h\0:5, then the ranking orders vary with the parameter value h. If h 2 ½0; 0:1, the best alternative is x4 . If h ¼ 0:2, the best alternative is x3 . If h 2 ½0:3; 0:4, the best alternative is x2 . These phenomena verify that the parameter h of DMs’ risk attitude indeed plays a vital role in the decision making. It is reasonable and necessary to involve DMs’ risk attitude into GDM with IVIFPRs. Moreover, it is easily observed from Table 8.2 that the ranking orders of alternatives are also different for diverse BUM functions.

8.5.2

Comparison Analysis with Xu’s Method

This subsection utilizes Xu’s method [16] to solve the above example for illustrating the advantages of the method proposed in this chapter. Assume the threshold value d1 ¼ 0:95. For different DMs’ weight vectors, the computation results are obtained by Xu’s method [16] and shown in Table 8.3. To intuitively compare the results obtained by method [16] and the proposed method of this chapter, the ranking orders of alternatives obtained by different methods are depicted graphically in Fig. 8.2. It is easily seen from Fig. 8.2 that the ranking order of alternatives is derived as x2  x1  x3  x4 for different DMs’ weight vectors, which is the same as the order 1 in h ¼ 0:4 with QðrÞ ¼ r 2 and different from other orders obtained by the method in this chapter. The proposed method of this chapter has some advantages over Xu’s method [16] as follows: (1) Xu’s method [16] only considered the compatibility of IVIFPRs in GDM and ignored the consistency of IVIFPR. Consistency is the degree of agreement among the preference values provided by DMs. Lack of consistency will lead to

Existential

Most of

Unit or

QðrÞ ¼ r 0

QðrÞ ¼ r 2

QðrÞ ¼ r 1

1

Quantifier

QðrÞ

0.455

0.207

0.304

Lðw1 Þ

0.906

1.000

0.877

Hðw1 Þ

0.330

0.388

0.254

Lðw2 Þ

0.810

0.788

0.659

Hðw2 Þ

0.160

0.331

0.345

Lðw3 Þ

0.810

0.788

0.841

Hðw3 Þ

Table 8.2 Similarity, accuracy degrees and ranking orders for different BUM functions QðrÞ with h = 0.5

0.099

0.130

0.206

Lðw4 Þ

0.890

0.957

0.741

Hðw4 Þ

Ranking order

x1  x2  x3  x4

x2  x3  x1  x4

x3  x1  x2  x4

8.5 An Example of Network System Selection and Comparison Analyses 265

k ¼ ð13; 13; 13ÞT

([0.3667, 0.4750), ([0.4500, 0.5583), ([0.3500, 0.4583), ([0.2750, 0.4000), 0.5134 0.5673 0.480 0.4401 x2  x1  x3  x4

k

~r1 ~r2 ~r3 ~r4 cð~r1 Þ cð~r2 Þ cð~r3 Þ cð~r4 Þ Order of alternatives (0.3417, (0.2833, (0.3917, (0.4250,

0.4333)] 0.4167)] 0.5083)] 0.5333)]

([0.3800, 0.4875), ([0.4375, 0.5425), ([0.3525, 0.4725), ([0.2625, 0.3825), 0.5262 0.5580 0.4919 0.4246 x2  x1  x3  x4

(0.3200, (0.2875, (0.3775, (0.4475,

k ¼ ð0:3; 0:2; 0:5ÞT 0.4225)] 0.4250)] 0.4850)] 0.5525)]

Table 8.3 Computation results and ranking orders for different DMs’ weight vectors by Xu’s method

([0.3740, 0.4819), ([0.4357, 0.5422), ([0.3551, 0.4699), ([0.2644, 0.3903), 0.5214 0.5571 0.4915 0.4309 x2  x1  x3  x4

(0.3260, (0.2893, (0.3764, (0.4374,

0.4274)] 0.4249)] 0.4880)] 0.5439)]

k ¼ ð0:3145; 0:2601; 0:4254ÞT

266 8 A Three-Phase Method for Group Decision Making …

8.5 An Example of Network System Selection and Comparison Analyses

267

4

Ranking orders

Q(r)=sqrt(r),theta=0 or 0.1 Q(r)=sqrt(r),theta=0.2

3

Q(r)=sqrt(r),theta=0.3 Q(r)=sqrt(r),theta=0.4 or method in [16] Q(r)=sqrt(r),theta=0.5 or method in [25]

2

Q(r)=sqrt(r),theta=0.6,0.7,0.7,0.8,0.9,1 or Q(r)=r1, theta=0.5 Q(r)=r0,theta=0.5

1

1

3

2

4

Alternatives

Fig. 8.2 Ranking orders of alternatives for different methods

the irrational and unreasonable decision results. In this chapter, a consistent IFPR is extracted from IVIFPR according to the additive consistencies of IFPR and IVIFPR. Additionally, the proposed method of this chapter takes the similarity degree between individual IVIFPR and collective one into consideration. (2) Xu [16] overlooked the determination of DMs’ weights. The ranking order of alternatives is the same for different DMs’ weight vectors. However, in this chapter, DMs’ comprehensive importance degrees are generated by combining the objective and subjective importance degrees of DMs. Taking the comprehensive importance degrees of DMs as order inducing variables, the collective IVIFPR is obtained by CI-IVIF-IOWA operator. The weights of the CI-IVIF-IOWA operator are determined by BUM function and comprehensive importance degrees. Therefore, the proposed method is able to effectively avoid subjective randomness during the decision making process.

8.5.3

Comparison Analysis with Wang’s Method

Since Wang method [6] is merely suitable for GDM with IFPRs, the IVIFPRs ~ k ¼ ðð½lk ; l k ; ½mkij ; mkij ÞÞ44 ðk ¼ 1; 2; 3Þ in Sect. 8.5.1 are firstly converted into R ij ij the corresponding IFPRs Rk ¼ ððlkij ; mkij ÞÞ44

kij Þ and with lkij ¼ 0:5ðlkij þ l

mkij ¼ 0:5ðmkij þ mkij Þ. According to model (4.18) in [6], the computation results for

8 A Three-Phase Method for Group Decision Making …

268

Table 8.4 Computation results and orders for different DMs’ weight vectors by Wang’s method k

k ¼ ð13; 13; 13ÞT

k ¼ ð0:3; 0:2; 0:5ÞT

k ¼ ð0:3145; 0:2601; 0:4254ÞT

~1 x ~2 x ~3 x ~4 x ~ 1Þ Sðx ~ 2Þ Sðx ~ 3Þ Sðx ~ 4Þ Sðx Order of alternatives

(0.1050, 0.6736) (0.2193, 0.4903) (0.2000, 0.6820) (0, 0.6525) −0.5686 −0.2710 −0.4819 −0.6525 x2  x3  x1  x4

(0.1005, 0.6764) (0.2148, 0.4925) (0.2001, 0.6845) (0, 0.6502) −0.5759 −0.2778 −0.4845 −0.6502 x2  x3  x1  x4

(0.1026, 0.6752) (0.2169, 0.4915) (0.2000, 0.6834) (0, 0.6512) −0.5726 −0.2746 −0.4833 −0.6512 x2  x3  x1  x4

different DMs’ weight vectors are obtained by Wang’s method and listed in Table 8.4. To visually compare the results obtained by method [6] with those by the method proposed in this chapter, the ranking orders of alternatives for different methods are also depicted in Fig. 8.2. It can be easily seen from Fig. 8.2 that the ranking order of alternatives obtained by method [6] is always x2  x3  x1  x4 for different DMs’ weight vectors. This ranking order is the same as the order 1 obtained by this chapter when h ¼ 0:5 and QðrÞ ¼ r 2 but different from other orders obtained by this chapter. The primary reasons may come from two aspects: (1) To derive the priority weights, method [6] merely considered the deviations from consistent IFPR. This chapter builds a multi-objective programming model by minimizing the deviations and the hesitation degrees of the priority weights. Since large hesitation degrees cannot guarantee the reliability of decision making, the constructed programming model is more appropriate for the actual decision making. (2) Similar to Xu [16], method [6] also neglected the determination of DMs’ weights. On the contrary, this chapter generates DMs’ comprehensive importance degrees by integrating the objective importance degrees with the subjective importance degrees, which is comprehensive and practical. Furthermore, this chapter takes DMs’ risk attitude and BUM function into account. Therefore, DMs have more choices for decision making.

8.6

Conclusions

With the dramatic development of modern economy and technology, GDM has attracted increasing attention. This chapter discussed GDM problems with IVIFPRs. A new three-phase method was put forward for solving such problems. The primary contributions are outlined as follows:

8.6 Conclusions

269

(i) Through extracting IFPR from an IVIFPR, a new additive consistency of an IVIFPR is defined by utilizing the additive consistency of IFPR. (ii) Some programming models are established to extract the most optimistic and pessimistic consistent IFPRs from an IVIFPR. Employing DMs’ risk preference, we integrate these two extracted IFPRs to introduce a risk attitudinal-based consistent IFPR. (iii) To derive intuitionistic fuzzy priority weights from a risk attitudinal-based consistent IFPR, this chapter constructs a multi-objective programming model which is converted into a linear goal programming model to resolve. (iv) For GDM problem, this chapter defines the comprehensive importance degrees of DMs by integrating the subjective and objective importance degrees of DMs. Then, a CI-IVIF-IOWA operator is developed to aggregate individual IVIFPRs into a collective one by taking comprehensive importance degrees as order inducing variables. For an IVIFPR with unacceptable consistency, it is necessary to improve its consistency. However, this chapter fails to consider how to improve the consistency of IVIFPRs, which will be investigated in future. Moreover, the preference relations provided by DMs may be not complete due the influences of various subjective and objective factors. Therefore, GDM with incomplete IVIFPRs is also an interesting and challenging theme that deserves to be studied.

References 1. H. Liao, Z. Xu, X. Zeng, J. Merigó, Framework of group decision making with intuitionistic fuzzy preference information. IEEE Trans. Fuzzy Syst. 23, 1211–1227 (2015) 2. E. Herrera-Viedma, S. Alonso, F. Chiclana, F. Herrera, A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Trans. Fuzzy Syst. 15, 863–877 (2007) 3. S. Wan, D. Li, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with Atanassov’s intuitionistic fuzzy truth degrees. IEEE Trans. Fuzzy Syst. 22, 300–312 (2014) 4. K. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 5. K. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343– 349 (1989) 6. Z.J. Wang, Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations. Appl. Math. Model. 37, 6377–6388 (2013) 7. Z.W. Gong, L.S. Li, F.X. Zhou, T.X. Yao, Goal programming approaches to obtain the priority vectors from the intuitionistic fuzzy preference relations. Comput. Ind. Eng. 57, 1187–1193 (2009) 8. Z.S. Xu, X.Q. Cai, E. Szmidt, Algorithms for estimating missing elements of incomplete intuitionistic preference relations. Int. J. Intell. Syst. 26, 787–813 (2011) 9. H.C. Liao, Z.S. Xu, Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency. IEEE Trans. Fuzzy Syst. 22, 1669–1681 (2014) 10. Z. Xu, J. Chen, Approach to group decision making based on interval-valued intuitionistic judgment matrices. Syst. Eng. Theory Pract. 27, 126–133 (2007)

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11. Z. Xu, X. Cai, Group decision making with incomplete interval-valued intuitionistic preference relations. Group Decis. Negot. 24, 193–215 (2015) 12. Z. Xu, X. Cai, Incomplete interval-valued intuitionistic fuzzy preference relations. Int. J. Gen. Syst. 38, 871–886 (2009) 13. H. Liao, Z. Xu, M. Xia, Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation. J. Intell. Fuzzy Syst. 27, 2969–2985 (2014) 14. Z. Xu, R. Yager, Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optim. Decis. Making 8, 123–139 (2009) 15. J. Wu, F. Chiclana, Non-dominance and attitudinal prioritisation methods for intuitionistic and interval-valued intuitionistic fuzzy preference relations. Expert Syst. Appl. 39, 13409– 13416 (2012) 16. Z. Xu, Compatibility analysis of intuitionistic fuzzy preference relations in group decision making. Group Decis. Negot. 22, 463–482 (2013) 17. F. Chiclana, E. Herrera-Viedma, F. Herrera, S. Alonso, Some induced ordered weighted averaging operators and their use for solving group decision-making problems based on fuzzy preference relations. Eur. J. Oper. Res. 182, 383–399 (2007) 18. H. Chen, L. Zhou, An approach to group decision making with interval fuzzy preference relations based on induced generalized continuous ordered weighted averaging operator. Expert Syst. Appl. 38, 13432–13440 (2011) 19. H. Liu, J. Cai, L. Martınez, The importance weighted continuous generalized ordered weighted averaging operator and its application to group decision making. Knowl. Based Syst. 48, 24–36 (2013) 20. L. Zhou, H. Chen, The induced linguistic continuous ordered weighted geometric operator and its application to group decision making. Comput. Ind. Eng. 66, 222–232 (2013) 21. S. Wan, F. Wang, L. Lin, J. Dong, An intuitionistic fuzzy linear programming method for logistics outsourcing provider selection. Knowl. Based Syst. 82, 80–94 (2015) 22. Z. Xu, Deviation square priority method for distinct preference structures based on generalized multiplicative consistency. IEEE Trans. Fuzzy Syst. 23, 1164–1180 (2015) 23. F. Zhang, J. Ignatius, C. Lim, Y. Zhao, A new method for deriving priority weights by extracting consistent numerical-valued matrices from interval-valued fuzzy judgement matrix. Inf. Sci. 279, 280–300 (2014) 24. Z. Xu, R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen Syst 35, 417–433 (2006) 25. R. Yager, Induced aggregation operators. Fuzzy Sets Syst. 137, 59–69 (2003) 26. H. Bustince, Conjuntos Intuicionistas e Intervalo valorados difusos: propiedades y construcci on. Relaciones Intuicionistas Fuzzy, Thesis, Universidad P_ublica de Navarra, 1994 27. W. Zhou, Z. Xu, Asymmetric fuzzy preference relations based on the generalized sigmoid scale and their application in decision making involving risk appetites. IEEE Trans. Fuzzy Syst. 24, 741–756 (2016) 28. Z. Xu, A method for estimating criteria weights from intuitionistic preference relations. Fuzzy Inf. Eng. 1, 79–89 (2009) 29. R. Yager, D. Filev, Induced ordered weighted averaging operators. IEEE Trans. Syst. Man Cybern. Part B (Cybernetics) 29, 141–150 (1999) 30. X. Zhang, Z. Xu, A new method for ranking intuitionistic fuzzy values and its application in multi-attribute decision making. Fuzzy Optim. Decis. Making 11, 135–146 (2012)

Chapter 9

A Group Decision-Making Method Considering Both the Group Consensus and Multiplicative Consistency of Interval-Valued Intuitionistic Fuzzy Preference Relations

Abstract This chapter develops a group decision-making (GDM) method that considers group consensus and multiplicative consistency of interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). First, the mean and variance of IVIF values (IVIFVs) are defined and a ranking method for IVIFVs is proposed considering the risk attitude of the expert. Then, the group consensus is presented by the individual similarity between experts. An iteration algorithm is designed to improve the group consensus. A statistical comparative analysis validates this algorithm. Subsequently, a new multiplicative consistency of IVIFPR is defined based on the multiplicative consistency of interval fuzzy preference relation. Two single-objective programming models are established to extract the most optimistic and pessimistic interval priority weight vectors from an IVIFPR, respectively. In particular, if the feasible domains of these two models are empty, two adjusted programming models are constructed to replace the originals. Combining the most optimistic and pessimistic interval priority weights, the IVIF priority weights are generated. Further, expert weights are derived from Markov model and used to derive the collective IVIFPR for generating the IVIF priority weights. Therefore, a new method is proposed to solve the GDM with IVIFPRs. Finally, two cases are analyzed to verify the effectiveness of the proposed method.




Keywords Interval-valued intuitionistic fuzzy preference relation Group decision making Multiplicative consistency Group consensus




9.1




Introduction

The preference relation was first introduced in the analytic hierarchy process (AHP) [21]. Owing to the flexible structure and innate ability of humans to make relative comparisons, preference relations have been widely applied to decision-making problems. Several different types of preference relations are now available, such as fuzzy preference relations [4, 11, 28, 36, 38], interval fuzzy © Springer Nature Singapore Pte Ltd. 2020 S. Wan and J. Dong, Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets, https://doi.org/10.1007/978-981-15-1521-7_9

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preference relations (IFPRs) [17, 25, 29, 35, 39], intuitionistic fuzzy preference relations [13, 15, 33, 37] and interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs) [8, 14, 22, 24, 27, 31, 32, 34, 37]. The elements in an IVIFPR are IVIF values (IVIFVs), which adopt intervals to express the degrees of membership, non-membership, and hesitancy [31]. Hence, IVIFPRs are more practical and flexible in representing uncertain preference information. As real numbers, intervals, and intuitionistic fuzzy values are special cases of IVIFVs, the IVIFPR can be transformed into a fuzzy preference relation, IFPR, and intuitionistic fuzzy preference relation, respectively. For example, let e ¼ ð~rij Þ R rij ¼ ð½lij ; lij ; ½mij ; mij Þ be an IVIFPR. If lij ¼ lij ¼ lij nn with IVIFV ~ e is reduced to an intuitionistic fuzzy and mij ¼ mij ¼ mij for i; j ¼ 1; 2; . . .; n, R preference relation R ¼ ðrij Þnn with rij ¼ ðlij ; mij Þ and an IFPR R ¼ ðrij Þnn with e rij ¼ ½lij ; 1  mij , respectively. Furthermore, if lij ¼ 1  mij for i; j ¼ 1; 2;


; n, R

is reduced to an fuzzy preference relation R0 ¼ ðlij Þnn . Although many methods have been proposed for the decision-making problems with the fuzzy preference relations, IFPRs or intuitionistic fuzzy preference relations, they cannot be extended directly to decision-making with IVIFPRs. To avoid partiality caused by an individual subject’s judgment, the group decision-making (GDM) method is usually applied to integrate different opinions to reach the best decision with a common solution. Compared with individual decision-making, GDM can elicit more complete information about the addressed problems and provide more selective alternatives. As IVIFPR is more comprehensive than other types of preference relations, GDM with IVIFPRs turns out to be an important topic in the decision-making field. However, there is little research focused on this topic owing to the complexity of IVIFPRs. To model the uncertain situations where the preferences provided by the expert are not exact numerical values but value ranges, Xu and Chen [31] extended the intuitionistic fuzzy preference relation to first define the concept of IVIFPR and then proposed a method for GDM with IVIFPRs. Xu and Yager [37] defined a similarity measure for the consensus analysis in GDM with IVIFPRs. Xu and Cai [32] developed two procedures for extending the acceptable incomplete IVIFPRs to the complete IVIFPRs. Then an approach is proposed for decision-making based on the incomplete IVIFPR. Xu and Cai [34] defined the multiplicative transitivity of IVIFPR and presented an approach to GDM with incomplete IVIFPRs. Chen et al. [8] determined the relative importance of criteria from an IVIFPR and then developed an approach to tackling multi-criteria GDM with IVIFPRs using some special aggregation operators. Wu and Chiclana [22] constructed an attitudinal score fuzzy preference relation from an IVIFPR and applied it to rank alternatives. By the score judgment matrix of IVIFPR, Wu et al. [24] developed an IVIF AHP-based approach for multi-criteria decision-making. Based on multiplicative transitivity of fuzzy preference relation, Liao et al. [14] defined the consistency of IVIFPR and designed some algorithms to improve the consistency. Moreover, an approach for GDM with IVIFPRs was proposed. Wan et al. [27] defined the

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consistency and acceptable consistency of an IVIFPR by separating the IVIFPR into two intuitionistic fuzzy preference relations. Then a method is developed for solving GDM with IVIFPRs. The aforementioned methods are effective to solve decision-making problems with IVIFPRs. However, they suffer from some limitations as detailed in the following. (1) Some studies [22, 24, 31] proposed the decision-making methods with IVIFPRs. These methods failed to consider the consistency of IVIFPR. Although some studies [14, 32, 34] discussed the consistency of IVIFPR, the consistent or incomplete consistent IVIFPRs in these studies were just simple extensions of consistent fuzzy preference relations. Furthermore, they suffered from some disadvantages. For example, the multiplicative consistency of IVIFPR in [14] cannot derive the multiplicative consistency of fuzzy preference relations because the former is too strict. Wan et al. [27] defined the multiplicative consistent IVIFPR using two intuitionistic fuzzy preference relations separated from the IVIFPR. It is difficult to determine the parameters in the iterative algorithm when repairing the consistency of IVIFPR using the method mentioned in [27]. Both the multiplicative consistent definitions of IVIFPR in the methods in [14, 27] merely considered the endpoints of membership and non-membership degrees of the IVIFPR. They may not sufficiently capture the original IVIF preference information and possibly cause the information to be distorted. (2) The methods in [22, 24, 32] only focused on individual decision-making and are unable to deal with GDM problems. The GDM methods in [8, 31, 34] overlooked the group consensus during the decision-making process. Xu and Yager [37] defined the similarity measure for consensus analysis, but they did not propose the method for GDM. Liao et al. [14] adjusted the group consensus using the deviation between the individual IVIFPR and the collective IVIFPR obtained by an aggregation operator. Apparently, using different aggregation operators would lead to different results. Wan et al. [27] derived expert weights based on the group consensus, which ignored the adjustment of the group consensus. The low level of group consensus would make the results unreasonable. (3) The GDM methods in [8, 14, 31, 34] neglected the determination of expert weights. By minimizing the deviations between the individual IVIFPR and the collective IVIFPR, Wan et al. [27] constructed an optimization model to derive expert weights. The determination of expert weights in [27] was greatly influenced by the aggregation operator that was used to obtain the collective IVIFPR. (4) To rank alternatives, the methods in [14, 31, 32, 34] applied the aggregation operators to obtain the collective values or the expected preference degrees of alternatives, which are prone to causing loss of information. It is more reasonable to use the priority weights to obtain the order of alternatives. In addition, the methods in [8, 31, 32, 34] used the score and accuracy functions to

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rank alternatives, which may lead to some unreasonable results in some circumstances (see Sect. 9.3.2 for more details). The group consensus plays an essential role for GDM with a variety of preference relations. It is preferable that experts reach a high degree of consensus on the results in GDM. Based on the consensus degrees and proximity measures, Cabrerizo et al. [2] proposed a consensus model for GDM with unbalanced fuzzy linguistic information. During the consensus reaching process, Mata et al. [18] built an adaptive consensus support system model for GDM in multi-granular linguistic contexts. Using the consensus measure, Pérez et al. [19] proposed a consensus stage with a feedback mechanism and applied it to mobile decision support systems for GDM based on dynamic decision environments. Cabrerizo et al. [5] analyzed different consensus approaches in fuzzy GDM and proposed two advanced consensus approaches for obtaining the highest consensus degree. Based on the nonparametric Wilcoxon statistical test, Chiclana et al. [6] presented a comparative study of the effect of the application of some different distance functions for measuring consensus in GDM. Herrera-Viedma et al. [12] presented an overview of consensus models based on soft consensus measures where the consensus reaching process was guided by a moderator or a feedback mechanism. Cabrerizo et al. [7] developed a method based on an allocation of information granularity as an important asset to increase the consensus in GDM. Cabrerizo et al. [3] reviewed some consensus approaches and comprehensively analyzed some challenges. Dong et al. [10] proposed a consensus framework for managing non-cooperative behaviors. It is worthwhile mentioning that the above progress on group consensus is not appropriate for GDM with IVIFPRs because IVIFPRs are very different from other types of preference relations. Currently, only Liao et al. [14] have studied consensus for GDM with IVIFPRs. Consensus and consistency are two important issues for GDM. Thus, more attention should be paid to these two issues for GDM with IVIFPRs. To overcome the above-stated limitations, this chapter focuses on a GDM method considering both the group consensus and multiplicative consistency of IVIFPRs. Combining the mean and variance of IVIFVs, a new method is proposed for ranking IVIFVs. The acceptable group consensus is defined for GDM with IVIFPRs and an iteration algorithm is designed to improve the group consensus. A new multiplicative consistency of IVIFPR is defined by the extracted IFPR. Two single-objective programming models or adjusted programming models are respectively established and solved to extract the most optimistic and pessimistic interval priority weight vectors which are applied to generate the IVIF priority weights. Subsequently, expert weights are determined by Markov model and used to obtain the collective IVIFPR. The main contributions and novelties of this chapter are summarized as follows: (1) Based on the relationship between IVIFV and interval, a new ranking method of IVIFVs is developed, which is more reasonable and convincing than existing ranking methods. (2) Using the actual preference transition probability, an iteration algorithm is devised to improve the group consensus, which does not require aggregation operators and control parameters.

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(3) The defined multiplicative consistency of IVIFPR sufficiently captures the original IVIF preference information. Two single-objective programs or adjusted programs are established to generate the IVIF priority weights. (4) Expert weights are objectively determined by Markov model, which is independent on aggregation operators, more robust and reliable. The remainder of this chapter unfolds as follows. Section 9.2 briefly reviews some concepts about IFPRs and IVIFPRs. In Sect. 9.3, a new ranking method is proposed for IVIFVs. Section 9.4 analyzes the group consensus for GDM with IVIFPRs. In Sect. 9.5, the multiplicative consistency of IVIFPR is defined. Then the IVIF priority weights of IVIFPR are determined by constructing some programming models. In Sect. 9.6, expert weights are derived by Markov model. Thereby, a new method is proposed for GDM with IVIFPRs. Two cases are presented in Sect. 9.7. Section 9.8 ends the paper with conclusions.

9.2

Preliminaries

In the section, some concepts about IFPRs and IVIFPRs are reviewed. Definition 9.1 [31] An IFPR R on the alternative set X ¼ fx1 ; x2 ; . . .; xn g is denoted by a judgment matrix R ¼ ðrij Þnn  X  X, where rij ¼ ½rij ; rijþ  is an interval which shows that alternative xi is between rij and rijþ times as important as alternative xj , and rij and rijþ fulfill conditions: 0  rij ; rijþ  1, rij þ rjiþ ¼ rijþ þ rji ¼ 1, rii ¼ riiþ ¼ 0:5 for all i; j ¼ 1; 2; . . .; n. Let w ¼ ðw1 ; w2 ;


; wn ÞT be a normalized interval priority weight vector, which satisfies that [25] þ  0  w i  wi  1; wi þ

n X j¼1;j6¼i

wjþ  1; wiþ þ

n X

w j  1 for i ¼ 1; 2; . . .; n:

j¼1;j6¼i

ð9:1Þ Definition 9.2 [25] An IFPR R ¼ ðrij Þnn is multiplicative consistent if there exists a normalized interval priority weight vector w ¼ ðw1 ; w2 ; . . .; wn ÞT such that 8 ½0:5; 0:5; > # if i ¼ j