Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications (Uncertainty and Operations Research) 9789811538902, 9811538905

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Uncertainty and Operations Research

Zhinan Hao Zeshui Xu Hua Zhao

Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications

Uncertainty and Operations Research Editor-in-Chief Xiang Li, Beijing University of Chemical Technology, Beijing, China Series Editor Xiaofeng Xu, Economics and Management School, China University of Petroleum, Qingdao, Shandong, China

Decision analysis based on uncertain data is natural in many real-world applications, and sometimes such an analysis is inevitable. In the past years, researchers have proposed many efficient operations research models and methods, which have been widely applied to real-life problems, such as finance, management, manufacturing, supply chain, transportation, among others. This book series aims to provide a global forum for advancing the analysis, understanding, development, and practice of uncertainty theory and operations research for solving economic, engineering, management, and social problems.

More information about this series at http://www.springer.com/series/11709

Zhinan Hao Zeshui Xu Hua Zhao •

•

Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications

123

Zhinan Hao Command and Control Engineering College Army Engineering University of PLA Nanjing, China

Zeshui Xu Business School Sichuan University Chengdu, China

Hua Zhao Department of General Education Army Engineering University of PLA Nanjing, China

ISSN 2195-996X ISSN 2195-9978 (electronic) Uncertainty and Operations Research ISBN 978-981-15-3890-2 ISBN 978-981-15-3891-9 (eBook) https://doi.org/10.1007/978-981-15-3891-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Uncertainty widely exists in every aspect of the social society. The decision making, characterized by the high level of human thinking activity, is inevitably involved with uncertainty in terms of the cognition. How to describe the uncertainty in the decision making process and provide the effective information aggregation methods are the primary tasks in the uncertain decision making research. One of the prominent techniques to quantify such cognitive uncertainty is the intuitionistic fuzzy set in view of its effectiveness and fine granularity in the descriptions of cognitive uncertainties. Motivated by this powerful uncertainty description tool, various intuitionistic fuzzy multi-attribute decision making methods have been introduced and successfully utilized to solve the practical decision making problems. These research findings also prove the validity and great potentiality of the intuitionistic fuzzy set. The objective of this book is to introduce the recent advances in the research of the intuitionistic fuzzy decision making methods. We shall show them from a new perspective and provide some new approaches. To achieve this, the new theoretical researches and the applications to practical decision making problems, case studies will be introduced in this book. These new advances will help the researchers who have interest in the intuitionistic fuzzy set and stimulate more research interests of this research field. The book is mostly self-contained, and the main prerequisites are provided in the introduction part. It is not necessary to read the chapters sequentially, but their organizations have a clear logic: Chapter 2 introduces a new intuitionistic fuzzy decision making method based on a new distance measure and similarity measure for the intuitionistic fuzzy set. It focuses on how to fully describe the influences of the inner relationships between the alternatives on decision makers’ preferences. The new distance measure and the similarity measure in this chapter can be also served as the basic information measure tools by the decision making methods studied in other chapters. Chapters 3 and 4 mainly present two types of intuitionistic fuzzy dynamic decision making method which are seldom mentioned in the related researches.

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Chapter 3 focuses on the dynamic decision making methods in terms of the emergencies. It hybridizes two powerful tools, the intuitionistic fuzzy set and the Bayesian Network, to solve the dynamic decision making problems in the emergencies. The intuitionistic fuzzy Bayesian Network is established theoretically and proven mathematically. This hybridized model underlies the dynamic intuitionistic fuzzy decision making model in this chapter. Most importantly, it establishes an association among the given uncertain information and provides an inference tool to acquire the unknown decision making information. The major advantage of the introduced decision making method is that it is applicable to solve the dynamic decision making problems, especially the emergency events. Chapter 4 introduces the process-oriented decision making method in the intuitionistic fuzzy environment. To describe the dynamic decision making process more precisely, the new distance measure introduced in Chap. 2 can be utilized in this chapter. This chapter shows the advantages of the process-oriented dynamic decision making method over the traditional decision making methods. The comparisons and analyses prove the validity and reliability of the new dynamic intuitionistic fuzzy decision making method. Moreover, the decision making method in this chapter illustrates the trade-off between decision making time and the quality of the decision results from a new perspective. Chapter 5 looks at a further mechanism that has been discussed in the intuitionistic fuzzy decision making problems, namely taking both the cognitive uncertainty and the statistical uncertainty into account in a single framework. A new extension of the intuitionistic fuzzy set is introduced in this chapter to solve the problems. Its important properties and applications in decision making have also been provided. In addition, this chapter presents a new method to evaluate the reliability of the aggregated results and visualize them, which can help the decision makers make decisions in a more straightforward way. This book reflects the current research progress of the intuitionistic fuzzy decision making methods, aiming at practitioners and researchers’ working in the areas of risk management, decision making under uncertainty, and operational research. It can also be used as the supplementary reading material for the postgraduate and senior undergraduate students in the above research areas. In fact, great numbers of decision making methods under the intuitionistic fuzzy environments have been studied. Most of them are based on the intuitionistic fuzzy aggregation operators. We want to offer the readers a new way to study the decision making method under the intuitionistic fuzzy environment. With this vision in mind, we hope the reader will find this book inspiring. We appreciate the financial support of this book: The Natural Science Foundation of China [No. 71571123]. Nanjing, China July 2019

Zhinan Hao Zeshui Xu Hua Zhao

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1 Intuitionistic Fuzzy Set . . . . . . . . . . . . 1.2 Basic Operation Laws and Aggregation 1.3 Content Design . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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2 The Decision Making Method Based on the New Distance Measure and Similarity Measure . . . . . . . . . . . . . . . . . . . . 2.1 Review of the Related Work . . . . . . . . . . . . . . . . . . . . . 2.2 Distance and Similarity Measures for IFSs . . . . . . . . . . . 2.3 Psychological Distance for IFSs . . . . . . . . . . . . . . . . . . 2.3.1 Psychological Distance . . . . . . . . . . . . . . . . . . . . 2.3.2 Psychological Distance Measure for IFNs . . . . . . 2.3.3 Psychological Distance Measure for IFSs . . . . . . 2.4 Similarity Measure for IFSs Based on the Psychological Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Extended Intuitionistic Fuzzy TOPSIS Method by the New Distance Measure . . . . . . . . . . . . . . . . . . . . 2.5.1 Algorithm of the Extended TOPSIS Method . . . . 2.5.2 Case Study of the Marine Energy Transportation Route Decision Making Problem . . . . . . . . . . . . 2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 The Dynamic Decision Making Method Based on the Intuitionistic Fuzzy Bayesian Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dynamic Decision Making Problems Under the Emergencies . . . . 3.2 Conceptual Framework for Dynamic Intuitionistic Fuzzy Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Intuitionistic Fuzzy Bayesian Network . . . . . . . . . . . . . . 3.3.1 Intuitionistic Fuzzy Event Under the Bayesian Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Probability Transmissibility of the Intuitionistic Fuzzy Bayesian Network . . . . . . . . . . . . . . . . . . 3.4 Dynamic Intuitionistic Fuzzy Decision Making Model . . 3.4.1 Weight Determination Method in the Framework of IFBNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Dynamic IFBN Decision Making Method . . . . . . 3.5 Applications to Mine Emergent Accident . . . . . . . . . . . . 3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Novel Intuitionistic Fuzzy Decision Making Models in the Framework of Decision Field Theory . . . . . . . . . . . . . . . 4.1 Decision Making Methods Based on the Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Intuitionistic Fuzzy Decision Making Model Based on the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Introduction of the DFT . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Integrated Intuition Fuzzy Decision Making Model Based on the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Integrated Intuition Fuzzy Group Decision Making Model Based on the DFT . . . . . . . . . . . . . . . . . . . . . 4.3 The Application of the Intuitionistic Fuzzy Decision Making Model Based on the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Validation and Comparison for the Cognitive Decision Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Application to the Decision Making of the Optimal Investment Country . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 The Decision Making Method Under the Probabilistic and Cognitive Environment . . . . . . . . . . . . . . . . . . . . 5.1 Motivations and Background . . . . . . . . . . . . . . . . . 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Dual Hesitant Fuzzy Set . . . . . . . . . . . . . . . 5.2.2 Basic Operations of DHFEs . . . . . . . . . . . . 5.3 Probabilistic Dual Hesitant Fuzzy Sets . . . . . . . . . . 5.3.1 Concept of PDHFS . . . . . . . . . . . . . . . . . . 5.3.2 Comparison Method of PDHFEs . . . . . . . . 5.3.3 Basic Operations of PDHFEs . . . . . . . . . . .

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5.4 Information Aggregation and Visualization of PDHFSs . . 5.4.1 Basic Aggregation Operator for PDHFSs . . . . . . 5.4.2 Visualization of the PDHFS Based on the Cloud Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application to the Arctic Geopolitics Risk Evaluations . . 5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Chapter 1

Introduction

The intuitionistic fuzzy set (IFS) theory was first proposed by Atanassov (1986), and it has gained scholars’ wide concerns and achieved substantial work. The IFS is a generalization of fuzzy set (FS), distinguished by three parameters: the membership function, the non-membership function and the hesitant degree. As a result, the IFS can describe the fuzzy characteristics of things more comprehensively. A vital objective of the IFS theory is how to acquire the fuzzy information without any loss of information, which is in preparations for the applications to the decision making problems. Accordingly, the basic operation laws and aggregation operators have been studied, such as the addition and multiplication laws (Xu 2007; Xu and Yager 2006), the intuitionistic fuzzy weighted averaging operator (Xu 2007), the intuitionistic fuzzy weighted geometric operator (Xu and Yager 2006) and the improved operators (Xu 2010; Xu and Xia 2011; Xu and Yager 2011; Zhao et al. 2010). The ranking and comparing methods (Bustince et al. 2013a, b; Xu and Yager 2006) have also been studied so as to compare the intuitionistic fuzzy numbers (IFN). These studies constitute the basic intuitionistic fuzzy information processing in the decision making problems. In the following section, we will first introduce the concept of the IFS and its formulation. Then the basic operation laws and aggregation operators about it are also provided. Finally, we present the comparison methods of the IFNs. The concept of the IFN and the relationship with the IFS are also presented. The above-mentioned methods and theory are the preliminary knowledge which will be used in the intuitionistic fuzzy decision making methods in this book.

1.1 Intuitionistic Fuzzy Set Atanassov (1986) extended the fuzzy set and introduced the non-membership function, which gives the decision makers (DMs) more flexibilities. The IFS is defined as follows: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9_1

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1 Introduction

Definition 1.1 (Atanassov 1986) Let X be a fixed set, then the IFS can be described as: A = {x, μA (x), vA (x)|x ∈ X }

(1.1)

where μA (x) is the membership function with μA : X → [0, 1], x ∈ X → μA (x) ∈ [0, 1], and vA (x) is the non-membership function with vA : X → [0, 1], x ∈ X → vA (x) ∈ [0, 1]. The membership function and the non-membership function both comply with the condition 0 ≤ μA (x) + vA (x) ≤ 1, x ∈ X . Meanwhile, the hesitance degree of the IFS A can be expressed as πA (x) = 1 − μA (x) − vA (x), x ∈ X . Also, there is 0 ≤ πA (x) ≤ 1, for all x ∈ X . For convenience, Xu (2007) defined the intuitionistic fuzzy number (IFN), which can be treated as the basic element of the IFS. An IFN takes the form α = (μα , vα ), satisfying the condition: μα ∈ [0, 1], vα ∈ [0, 1], μα + vα ≤ 1

(1.2)

It’s obvious that the largest IFN and the smallest IFN are α + = (1, 0) and α − = (0, 1), respectively. The physical interpretation of the IFN can be explained as the combination of the deterministic information and the indeterminate information from the individual on a given object. Similar to the real number, the IFN is feasible for ranking and comparision. To achieve this target, the score function and the accuracy function are defined, respectively. The score of the IFN can be reflected by the score function s, which takes the form (Chen and Tan 1994) s(α) = μα − vα

(1.3)

Obviously, s(α) ∈ [−1, 1]α = (1, 0). The larger value of the score function, the larger difference between μα and vα . Accordingly, the larger the IFN α. Particularly, s(α) gets the largest value s(α) = 1 when α = (1, 0), and s(α) gets the smallest value s(α) = −1 when α = (0, 1). When the scores of the IFNs are the same, it is difficult to compare them. To solve this problem, the accuracy function (Hong and Choi 2000) is also defined as: h(α) = μα + vα

(1.4)

where h(α) ∈ [−1, 1]. It also holds that h(α)+π (α) = 1

(1.5)

The larger value of h(α), the higher accuracy of the IFN α. The score function and the accuracy function defined above make it possible to compare the IFNs. The comparison method for IFNs is defined as follows:

1.1 Intuitionistic Fuzzy Set

3

    Definition 1.2 (Xu 2007) For two IFNs α1 = μα1 , vα1 and α2 = μα2 , vα2 . Let the score function and the accuracy degrees of them be s(α1 ) = μα1 − vα1 , s(α2 ) = μα2 − vα2 , h(α1 ) = μα1 + vα1 and h(α2 ) = μα2 + vα2 respectively. Then the comparison and ranking principle is: • If s(α1 ) > s(α2 ), then the IFN α1 is larger; • If s(α1 ) = s(α2 ), and (1) If h(α1 ) = h(α2 ), then α1 is equal to α2 , that is, μα1 = μα2. ; (2) If h(α1 ) < h(α2 ), then the α1 is smaller; (3) If h(α1 ) > h(α2 ), then the α2 is smaller.

1.2 Basic Operation Laws and Aggregation Operators The basic operation laws of the IFSs are defined as follows (Atanassov 1986; De et al. 2000): Definition 1.3 (Atanassov 1986) Let X be a fixed set.  The IFS on the set X can  be {x, }. denoted as A = μ = x, μ v ∈ X Let A v ∈ X and (x), (x)|x (x), (x)|x A A  1 A1 A1  A2 = x, μA2 (x), vA2 (x)|x ∈ X be two IFSs. Then it holds that:    (1) A¯ = x, vA (x), μA1 (x) |x ∈ X ;     (2) A1 ∩ A2 = x, min μA1 (x), μA2 (x) , maxvA1 (x), vA2 (x)|x ∈ X ; (3) A1 ∪ A2 = x, max μA1 (x), μA2 (x) , min vA1 (x), vA2 (x) |x ∈ X ;  (4) A1 + A2 = x, μA1 (x) + μA2 (x) − μA1 (x)μA2 (x), vA1 (x)vA2 (x) |x ∈  X ; (5) A1 · A2 = x,A1 (x)μA2 (x), vA1 (x) + vA2 (x) − vA1 (x)vA2 (x) |x ∈ X ; The extended forms are defined as De  et al.(2000):  (6) nA =  x, 1 − (1 − μA (x))n , vA (x)n |x ∈ X  ; (7) An = x, μA (x)n , 1 − (1 − vA (x))n |x ∈ X . The corresponding operations of the IFNs are also defined, which will be used in the remainders.   Definition 1.4 (Xu 2007; Xu and Yager 2006) Let α = (μα , vα ), α1 = μα1 , vα1 and α2 = μα2 , vα2 be three IFNs, then (1) (2) (3) (4) (5) (6) (7)

α¯ = (vα , μα );     α1 ∧ α2 = min μα1 , μα2 , maxvα1 , vα2 ; α1 ∨ α2 = max μα1 , μα2 , min vα1 , vα2 ; α1 ⊕ α2 = (μ1 + μ2 − μ1 μ2 , v1 v2 ); α1 ⊗ α2 = (u1 u2 , v1 + v2 − v1 v2 ); λα = 1 − (1 − μα )λ , vαλ , λ > 0;   α λ = μλα , 1 − (1 − vα )λ , λ > 0.

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1 Introduction

Based on the basic addition and multiplication laws of the IFNs, the subtraction and division operations are deduced (Atanassov 2009, 2012; Chen 2007). Definition 1.5 (Atanassov 2009) The subtraction and division operations for two given IFSs A and B are given as follows respectively: A B = {(x, μA B (x), vA B (x))|xE}; AB = {(x, μAB (x), vAB (x))|xE} (1.6) where ⎧ ⎨ μA (x)−μB (x) if μA (x) ≥ μB (x) and vA (x) ≤ vB (x) and vB (x) > 0 1−μB (x) and vA (x)πB (x) ≤ πA (x)vB (x) μA B = ⎩ 0 others ⎧ ⎨ vA (x) if μA (x) ≥ μB (x) and vA (x) ≤ vB (x) vA B = vB (x) and vB (x) > 0 and vA (x)πB (x) ≤ πA (x)vB (x) ⎩ 1 others ⎧ ⎨ μA (x) if μA (x) ≤ μB (x) and vA (x) ≥ vB (x) and μB (x) > 0 μAB = μB (x) and μA (x)πB (x) ≤ πA (x)μB (x) ⎩ 0 others ⎧ ⎨ vA (x)−vB (x) if μA (x) ≤ μB (x) and vA (x) ≥ vB (x) and μB (x) > 0 1−vB (x) and μA (x)πB (x) ≤ πA (x)μB (x) vAB = ⎩ 1 others By means of the above basic operational laws, two types of commonly used aggregation operators have been developed, i.e., the intuitionistic fuzzy weighted averaging (IFWA) operator (Xu 2007) and the intuitionistic fuzzy weighted geometric (IFWG) operator (Xu and Yager 2006): Definition 1.6 (Xu 2007) Let  be the set of all the IFNs. The IFWA operator is a mapping IFWA : n → , such that IFWAω (α1 , α2 , . . . , αn ) = ω1 α1 ⊕ ω2 α2 ⊕ · · · ⊕ωn αn

(1.7)

T where w = (w1 , w2 , . . . , w n ) is the weight vector of the IFNs αj (j = 1, 2, . . . , n), n satisfying wj ∈ [0, 1] and j=1 wj = 1.

Definition 1.7 (Xu and Yager 2006) The IFWG operator is a mapping IFWG : n → , such that IFWGω (α1 , α2 , . . . , αn ) = α1ω1 ⊗ α2ω2 ⊗ · · · ⊗ αnωn

(1.8)

 where wj ∈ [0, 1], nj=1 wj = 1 and ω = (ω1 , ω2 , . . . , ωn )T is the weight vector of  the IFNs αj (j = 1, 2, . . . , n) with ωj ∈ [0, 1] and nj=1 ωj = 1.

1.2 Basic Operation Laws and Aggregation Operators

5

The IFWA and IFWG operators only consider the weighted information of the IFNs themselves but ignore the importance degrees of their ordered positions. To overcome the deficiency of the IFWA and IFWG operators and take both the weighted intuitionistic fuzzy information as well as the corresponding ordered positions into account, two hybrid aggregation operators: the intuitionistic fuzzy hybrid averaging (IFHA) operator and the intuitionistic fuzzy hybrid geometric averaging (IFHG) operator have also been proposed respectively: Definition 1.8 (Xu 2007) Let  be the set of all the IFNs. The IFHA operator is a mapping IFHA : n → , such that IFHAω,w (α1 , α2 , . . . , αn ) = w1 α˙ σ (1) ⊕ w2 α˙ σ (2) ⊕ . . . ⊕ wn α˙ σ (n)

(1.9)

where w = (w1 , w2 , . . . , ωn )T is the weight vector associated with the IFHA operator with wj ∈ [0, 1] and nj=1 wj = 1. Moreover, ω = (ω1 , ω2 , . . . , ωn )T is the  weight vector for the IFNs αj (j = 1, 2, . . . , n) with ωj ∈ [0, 1] and nj=1 ωj = 1.   α˙ σ (2) , α˙ σ (2) , . . . , α˙ σ (n) is any permutation of the collection of the weighted IFNs α˙ j = nωj αj (j = 1, 2, . . . , n), abiding by α˙ σ (j)  α˙ σ (j+1) (j = 1, 2, . . . , n − 1). n is the total number of the IFNs, which can be also called the balancing coefficient. Definition 1.9 (Xu and Yager 2006) The IFHG operator is a mapping IFHG : n → , such that IFHGω,w (α1 , α2 , . . . , αn ) =

..

α

σ (1)

w1

⊗

..

α

σ (2)

w2

⊗ ... ⊗

..

α

σ (n)

wn (1.10)

  nω where α¨ j = αj j (j = 1, 2, . . . , n), α¨ σ (1) , α¨ σ (2) , . . . , α¨ σ (n) is the permutation of α¨ j = nωj αj (j = 1, 2, . . . , n), such that α¨ σ (j)  α¨ σ (j+1) (j = 1, 2, . . . , n − 1). w = weighting vector associated with the IFHG (w1 , w2 , . . . , ωn )T is the exponential n , . . . , ωn )T is the weight operator, where wj ∈ [0, 1], j=1 wj = 1 and ω = (ω1 , ω2 vector of the IFNs αj (j = 1, 2, . . . , n) with ωj ∈ [0, 1] and nj=1 ωj = 1.

1.3 Content Design This book mainly aims at introducing four intuitionistic fuzzy multi-attribute decision making methods and their developments and applications. The content logic flowchart can be found in detail in Fig. 1.1.

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1 Introduction

Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications

Problems to be solved A more appropriate distance measure for decision making

Dynamic multi-attribute decision making method

Process-oriented dynamic decision making method

Decision making method under the probabilistic and cognitive environment

Solutions Psychological distance measure New similarity measure The decision making method based on the new distance measure

Intuitionistic fuzzy Bayesian network Dynamic weight determination model Improved prospect theory A new dynamic decision making method

The decision field theory(DFT) Intuitionistic fuzzy decision field method The novel decision making method The group intuitionistic decision field decision making method

PDHFS Operation and aggregation laws Information distribution visualization method Reliability analysis method The new decision making method under PDHFS

Applications Marine energy transportation route selections

Mine explosion emergent accident

OBOR investment decision making problem

Arctic geopolitics risk evaluations

Fig. 1.1 Content logic flowchart of this book

References Atanassov KT (2009) Remark on operations “subtraction” over intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets 15(3):20–24 Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 Atanassov KT (2012) On intuitionistic fuzzy sets theory. Springer, Berlin Bustince H, Fernandez J, Kolesárová A, Mesiar R (2013a) Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst 220:69–77 Bustince H, Galar M, Bedregal B, Kolesarova A, Mesiar R (2013b) A new approach to intervalvalued choquet integrals and the problem of ordering in interval-valued fuzzy set applications. IEEE Trans Fuzzy Syst 21(6):1150–1162 Chen SM, Tan JM (1994) Handling multicriteria fuzzy decision-making problems based on vague set-theory. Fuzzy Sets Syst 67(2):163–172 Chen TY (2007) Remarks on the subtraction and division operations over intuitionistic fuzzy sets and interval-valued fuzzy sets. Int J Fuzzy Syst 9(3):169–172 De SK, Biswas R, Roy AR (2000) Some operations on intuitionistic fuzzy sets. Fuzzy Sets Syst 114(3):477–484 Hong DH, Choi CH (2000) Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114(1):103–113 Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15(6):1179–1187 Xu ZS (2010) Choquet integrals of weighted intuitionistic fuzzy information. Inf Sci 180(5):726– 736 Xu ZS, Xia MM (2011) Induced generalized intuitionistic fuzzy operators. Knowl-Based Syst 24(2):197–209

References

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Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J General Syst 35(4):417–433 Xu ZS, Yager RR (2011) Intuitionistic fuzzy Bonferroni means. IEEE Trans Syst Man Cybern Part B Cybern 41(2):568–578 Zhao H, Xu ZS, Ni MF, Liu SS (2010) Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 25(1):1–30

Chapter 2

The Decision Making Method Based on the New Distance Measure and Similarity Measure

The distance measure and the similarity measure are one of the basic aspects of the fuzzy set theory. When an individual makes decisions in our daily life, one of the most effective methods is to evaluate the similarity degrees of the given objects based on their distances. This is also the basic algorithm of the distance-based decision making methods. From this perspective, the accuracy of the distance measure directly influences the reliability of the decision results. The purpose of this section is to reach these goals: (1) we aim at quantifying the context information of the alternative set and their influence on the distance measure; (2) we will introduce the psychological distance under the intuitionistic fuzzy circumstances and provide the detailed calculation process; (3) we will focus on the practical application of the psychological distance to the uncertain decision making problems and discuss its advantages over the methods based on the traditional distance measure. To achieve this target, this chapter is organized as follows: First, the basic concepts and the classical distance measure for the IFSs are presented and then a practical problem concerning the similarity degree of alternatives under the intuitionistic fuzzy environment is discussed. Next, the intuitionistic fuzzy decision making method incorporating the new distance measure is also presented. Their characteristics and limitations are also discussed in details. The main methods presented in this chapter come from Hao et al. (2019).

2.1 Review of the Related Work When an individual makes decisions in daily life, one of the most effective strategies is to evaluate the similarities of the given objects based on their distances. However, the pervasive uncertainty existed in practical problems makes it more difficult to achieve this. Since Atanassov (1986) first proposed the intuitionistic fuzzy set (IFS), it has gained great popularity in the applications to various areas, especially in the decision making field (Feng et al. 2019; Luo et al. 2018; Song et al. 2019). In view of the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9_2

9

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2 The Decision Making Method Based on the New Distance …

significance of the distance in the category research and the decision making process, the distance and the similarity measures have become one of the basic aspects of the fuzzy set theory. Various distance measures for the IFSs have been put forward. The representative works are the Hamming distance measure and the Euclidean distance measure (Atanassov 1999), their geometrical representations (Szmidt and Kacprzyk 2000), the general Hamming and Euclidean distances in Hausdorff metric (Grzegorzewski 2004) and so on. Based on the existing research, the axiom definitions of the distance measure and the similarity measure for the IFSs have also been proposed (Li and Cheng 2002; Wang and Xin 2005). Xu (2007b) developed the weighted distance measure for the IFSs and generalized it. It is obvious that most of the existing distance measures are based on the Hamming distance and the Euclidean distance defined in Atanassov (1999). The extensions of them mainly focus on three aspects: First, taking more information into account, such as the improved forms of the distance measures which take the hesitance degree or the weight information assigned to the attributes into account. Xu and Chen (2008) and Baccour et al. (2013) systematically summarized the existing distance measures and pointed out that most of the distance measures are limited to the aggregation of the belongingness and the non-belongingness information of the IFSs. Meanwhile, some new distance measures integrating the new measures have also been developed for the pattern recognition, such as the distance measures taking the entropy measures (Das et al. 2018; Joshi and Kumar 2019) and the fuzzy measures into account (Dong et al. 2016; Ke et al. 2018). The distance measures above work towards fully utilizing the intuitionistic fuzzy information (or their weighted forms) assigned to the attributes. How the relationships between the attributes influence the distance measure and the preference in the decision results have seldom been studied. The choice behavior research has pointed out that the competition among the alternatives will influence their distances and evaluations (Rooderkerk et al. 2011). For instance, the dominated or inferior ones added in the alternatives will affect their distances and then the preferences, which is also called the attraction effect. The classical Euclidean distance cannot reflect this information in that it fails to capture the interrelationships among the alternatives. Such kind of context information of the alternatives should be considered in the similarity and distance calculation processes. To solve this problem, Huber et al. (1982) proposed the distance based on the dominance and the indifference relationships among the alternatives. A more general solution is the subsequent weighted forms (Rooderkerk et al. 2011; Wedell 1991), which rotate the dominance and indifference vectors in scale or attach the individual’s weight information to the dominance relationships. However, these approaches are limited to two kinds of attributes. Their improved form is the generalized distance function in multi-attribute situations proposed by Berkowitsch et al. (2015). These improved distances have successfully considered the context information of the alternatives and gained great popularity in the cognitive decision making problems (Halamish and Liberman 2017; Huang et al. 2016). However, it is found that past studies mainly focus on dealing with the precise information while seldom take the uncertainty into account. As to the distance defined in the uncertain situation,

2.1 Review of the Related Work

11

the existing distance measures for the intuitionistic fuzzy environment are flexible enough to describe the uncertain information in the practical problems but unable to capture interrelationships among the attributes. Many researchers have noticed this deficiency and started to investigate the solutions under the intuitionistic fuzzy environment. Inspired by the dominance phenomena in the alternative set, Huang et al. (2012, 2013) extended the dominance relation to the intuitionistic fuzzy information systems and the rough set theory so as to eliminate the redundant information and simplify the knowledge representation. The dominance-based ranking methods for IFS are subsequently proposed (Chen and Tu 2014; Yu et al. 2014). These works concern the dominance relationships of the options in the intuitionistic fuzzy environment, which are mainly utilized to simplify the information representation and rank alternatives. However, the quantitative description of this dominance effect and how they influence the distance and similarity degrees of the alternatives under the intuitionistic fuzzy environment have not yet been studied.

2.2 Distance and Similarity Measures for IFSs In this section, we shall review the basic concepts of the distance measure and the similarity measure for IFSs. Let (X ) be the set of all the IFSs over X. The axioms of the similarity and distance measures for IFSs are defined as follows: Definition 2.1 (Li and Cheng 2002). For (X ) and Aj ∈ (X )(j = 1, 2, 3), S : ((X ) × (X ) → [0, 1]) is defined as the similarity measure between the IFSs Aj (j = 1, 2, 3), which satisfies the following conditions: (SP1) 0 ≤ S(A1 , A2 ) ≤ 1; (SP2) For A1 = A2 , it holds S(A1 , A2 ) = 1; (SP3) S(A1 , A2 ) = S(A2 , A1 ); (SP4) If A1 ⊆ A2 ⊆ A3 , then S(A1 , A3 ) ≤ S(A1 , A2 ) and S(A1 , A3 ) ≤ S(A2 , A3 ). Definition 2.2 (Szmidt and Kacprzyk 2000). The Euclidean distance between the IFSs A1 and A2 is defined as:  dE (A1 , A2 ) =

    1/2 n 1  μA1 (xi ) − μA2 (xi ) 2 + vA1 (xi ) − vA2 (xi ) 2   2 1 + πA1 (xi ) − πA2 (xi ) 2

(2.1)

which satisfies (Grzegorzewski 2004): (DP1) 0 ≤ d (A1 , A2 ) ≤ 1; (DP2) For A1 = A2 , there is d (A1 , A2 ) = 0; (DP3) d (A1 , A2 ) = d (A2 , A1 ); (DP4) If A1 ⊆ A2 ⊆ A3 , then d (A1 , A3 ) ≥ d (A1 , A2 ) and d (A1 , A3 ) ≥ d (A2 , A3 ).

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2 The Decision Making Method Based on the New Distance …

Based on these axioms, Xu (2007b) developed a generalized weighted distance, describing the normalized Hamming distance, the normalized Euclidean distance and their improved forms (Szmidt and Kacprzyk 2000) in a single framework: ⎞λ n   λ  1 d3 (A1 , A2 ) = ⎝ ωj 1 + λ2 + λ3 ⎠ 2 j=1 ⎛

(2.2)

            where 1 = μA1 xj − μA2 xj , 2 = vA1 xj − vA2 xj , and 3 = πA1 xj − πA2 xj . Particularly, when λ = 1, it reduces to the weighted Hamming distance, and when λ = 2 , it reduces to the weighted Euclidean distance.

2.3 Psychological Distance for IFSs 2.3.1 Psychological Distance When we calculate the distances among the alternatives, most of the existing methods focus on aggregating the weighted information of various attributes into the distance measures. The competitive relationships among the alternatives, which will influence their distances in the attribute space (Nosofsky and Zaki 2002; Nosofsky 1986), are seldom considered in these distance measures. This is also the difference between the classical Euclidean distance and the psychological distance. The psychological distance is capable of capturing the context information in the alternatives and accounting for the influence on the distances in the attribute space. It has been pointed out that the psychological distance can overcome the counterintuitive phenomenon of the classical distance measures, which is beneficial for improving the decision results under the information overload circumstances and reducing the complexity of the practical decision making problems (Du and Hu 2015; Soderberg et al. 2015; Thomas and Tsai 2012; Williams et al. 2014). To illustrate the counterintuitive phenomenon of the Euclidean distance or the Hamming distance and study the psychological distance, we present some examples of three options expressed by the IFNs. As we all know, an IFN consists of a membership function (μ) and a nonmembership function (v), describing the cognitive uncertainty from the DM with fine granularity. The score function suggests that the large IFNs are those with the high values of μ but the small values of v. From this perspective, these two elements of the IFNs can be treated as two typical attributes so as to evaluate the IFNs. To describe the psychological distance in this attribute space, we build a Cartesian coordinate where the horizontal axis represents the membership function and the vertical axis represents the non-membership function. Let A, B, and C be three IFNs, and their positions in the attribute space are shown in Fig. 2.1a. In this case, the Hamming/Euclidean distance between the options A

2.3 Psychological Distance for IFSs

13

Fig. 2.1 The Hamming/Euclidean distance (a) and the psychological distance (b) for three options A, B, and C under intuitionistic fuzzy environment. The Euclidean/Hamming distance of the options A and B to C are the same but the psychological distances are different. The psychological distance of the option B to the option C is larger in that the option A is dominated by the other two options. The distance on dominance direction is stretched while the distance on the indifference direction remains the same in (b)

and C and that between the options B and C are exactly the same. The options B and A are equivalent compared with the option C. The change from the option C to either the option B or A should be acceptable. However, based on the IFNs’ comparison principle, we know that the option C is superior to the option B in that the option C has higher values of μ but smaller values of v than the option B. In this case, the option B is dominated by the option C (or the option C dominates the option 

B). The direction BC is the dominance direction in this space. The DMs can easily identify the superior or the inferior ones along this direction. As a result, the change from the option C to the option B is unacceptable for the DMs because the option B contains much more uncertainty and is inferior to the option C. On the contrary, the option A is superior to the option C on the non-membership but inferior to the option C on the membership part. As for the option A, the disadvantage over μ is compensated by its advantage over v. Accordingly, the option C and the option A are more similar to each other than either of them to the option B. It appears to be more acceptable to change from the option C to the option A than to the option B from the perspective of the DMs. As a result, the option A and the option C are more 

competitive. The direction CA can be treated as the indifference direction. For the DMs, the psychological distance between the option C and the option A should be smaller than that between the option C and the option B (or the option A and the option B) (Illustrated in Fig. 2.1b). From the above analysis, we can find that the Euclidean distance is incapable of solving this practical problem because it ignores the context information of the alternative sets. This phenomenon has also been fully discussed in the context of the

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2 The Decision Making Method Based on the New Distance …

decision making problems (Berkowitsch et al. 2015; Roe et al. 2001; Rooderkerk et al. 2011). To calculate the psychological distance under the intuitionistic fuzzy environment, we need to quantify the context information in terms of the dominance and indifference degrees and identify how these relationships influence the psychological distances. The most common solution is to calculate the distances along the dominant and indifference directions respectively (Berkowitsch et al. 2015; Huber et al. 1982). In the following, we will introduce the detailed calculations of the intuitionistic fuzzy psychological distance.

2.3.2 Psychological Distance Measure for IFNs The example illustrated in Fig. 2.1 indicates that the distance between the options is changed due to the dominance effect. The attribute space is stretched on the dominance direction while remains the same on the indifference direction. Huber et al. (1982) pointed out that the distance along the dominance direction will be weighted with more attention than that along the indifferent direction, which is in line with the example in Fig. 2.1. In addition, the indifference vector describes the exchange information between options (Rooderkerk et al. 2011), indicating how many units the DMs are willing to give up for changing from one attribute to another. For the case of the IFNs, this exchange ratio reflects the acceptable degrees of the DMs to change from one IFN to the other on the considered attributes. To describe the stretched influence of the dominance and the indifference vector and their roles in the psychological distance calculations, Wedell (1991) provided a rotation method to compare the distance along the dominance and indifference vectors between options in the attribute space. For two attributes with equal weight information in the space, the orthogonal dominance and indifference vectors are shown in the solid line (Fig. 2.2). The inequality of the weights on the attributes will rotate the dominance vector as well as the indifference vector. For instance, the more weight attention drawn on the dimension I, the more the dominance will incline to the dimension I axis, and the steeper the indifference vector will be, respectively. On the contrary, the emphasis on the dimension II will rotate the orthogonal dominance and indifference vector to the dimension II axis. Extremely, when the weight of a certain attribute is infinite, the dominance vector will be parallel to the corresponding axis and the indifference vector will be parallel to the other axis. It should be noted that the two vectors are orthogonal to each other and the orientation of the indifference vector is immaterial. Berkowitsch et al. (2015) also gave a detailed overview of this psychological distance and gave a general distance function in the real number field. Inspired by these works, we will generalize this concept to the IFNs and develop a general psychological distance for the IFNs so as to better understand the relationships between the alternatives in the cognitive decision making problems, particularly under the intuitionistic fuzzy environment.

2.3 Psychological Distance for IFSs

15

Fig. 2.2 Attribute space illustrated by the dominance and indifference vectors

The membership function (μ) and the non-membership function (v) of an IFN can be perceived as two different attributes. Since the membership function and the nonmembership function are different types of indicators for the IFNs, it is necessary to standardize the attributes. For convenience, we transform the attribute v into −v in the attribute space. Also, it is necessary to consider the subjective preference (or weight information) for the IFN’s elements in view of the dominance effect. This subjective preference means that a DM may hold different weights for the information provided by the IFNs (e.g., the membership function values are more credible and the nonmembership values are less reliable). Let the weight vector for the two parts μ and v be w = (wμ , wv )T , where wμ and wv are the weights of μ and v respectively with 0 ≤ wμ ≤ 1, 0 ≤ wv ≤ 1 and wμ +wv =1. Next, we should give the dominance vector and the indifference vector in preparations for calculating the intuitionistic fuzzy psychological distance. As has been discussed before, the dominance vector (ρd ) is orthogonal to the indifference vector (ρi ) so as to calculate the distances along these two directions, noted as ρd ⊥ρi . This property implies that we can get the dominance vector from the indifference vector. Since the indifference vector depicts the exchange information of alternatives on the given attributes, it is necessary to quantify these exchange ratios in terms of the price paid for changing from one attribute to the other. Similar to the indifference vector defined by Berkowitsch (2015), the common indifference vector for the IFN can be defined as:  w    − wμv − wwμv = (2.3) ρi = wμ 1 wμ

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2 The Decision Making Method Based on the New Distance …

This vector describes how many units the DM is willing to give up for changing from one attribute (e.g., the non-membership part) to the other attribute (e.g., the membership part). The algebraic sign marks the vector pointing to the nonmembership attribute axis. Accordingly, the dominance vector can be derived as:     w1 1 (2.4) ρd = ww21 = w2 w1

w1

This vector indicates the dominance direction. Based on the previous analysis, the distances along these two directions gain different scaled effects. The distances along the dominance direction will be stretched while those along the indifference direction will remain the same. Next, we should work on calculating the distances projected in these two directions. To achieve this, a transforming basis based on these two vectors can be defined as (Berkowitsch et al. 2015):

K=

ρd ρi , ρi  ρd 

 (2.5)

where ρi  and ρd  are the Euclidean length of the indifference vector and the dominance vector respectively, each column of K represents the corresponding standardized vector. To depict the progress of the distance transformation along the dominance direction and the indifference direction, the following expression can be also presented: transdist = K−1 distIFN

(2.6)

where the first entry in transdist indicates the distance in terms of the indifference vector and the second entry represents the distance in terms of the dominance vector. This transformed distance reflects the potential gain and loss when the DMs change from one attribute to another attribute among options. Since the distance along the dominance direction will be paid more attention than that along the indifference direction (the dominance effect) (Huber et al. 1982), it is necessary to attach more importance on the distance along the dominance vector. Hence, it is suggested to define a matrix in this transforming process, expressed as: 

1 1 Aw = 1 wd

 (2.7)

The parameter wd plays the role of magnifying the dominance influence so that the difference in the dominance direction gains more attention but the indifference direction remains the same.

2.3 Psychological Distance for IFSs

17

However, it is worth noticing that the original psychological distance doesn’t satisfy the first rule in Definition 2.2. In this chapter, we introduce a novel psychological distance measure of the IFNs as follows:  1 · trans · Aw · trans (2.8) d (A1 , A2 ) = 2 · wd where 2 · wd is a balancing coefficient so as to make this distance measure satisfy the (DP1) of Definition 2.2. Obviously, (2.8) satisfies (DP2-DP3) of Definition 2.2. In what follows, we prove that (2.8) also satisfies (DP4): Let A, B, and C be three IFNs. If A ⊆ B ⊆ C, then μA ≤ μB ≤ μC and and |vA − vC | ≥ |vA − vB |. It vA ≥ vB ≥ vC . We can get |μA − μC | ≥ |μA − μB |  is obvious that the Euclidean distances for AC = distAC · distAC and AB =  distAB · distAB satisfy AC ≥ AB. Let the weight information for the two parts   of the IFN be w = wμ , wv and the weight matrix magnifying the dominance effect   1 1 , then be Aw = 1 wd 

 1 · A · trans · transAB w AB = 2 · wd

d (A, B) =

    1 · ρ −1 distAB · Aw · ρ −1 distAB 2 · wd

and  d (A, C) =

 1 · A · trans · transAC w AC = 2 · wd

    1 · ρ −1 distAC · Aw · ρ −1 distAC 2 · wd

It is easy to get the inequality d (A, C) ≥ d (A, B). Similarly, we can prove d (A, C) ≥ d (B, C). As a result, the distance in (2.8) satisfies (DP4) of Definition 2.2. In particular, the distance defined in (2.8) becomes the normalized Euclidean distance when wμ = wv and wd = 1. Remark The psychological distance defined in (2.8) satisfies the conditions in Definition 2.2. However, it is inappropriate to treat the psychological distance as the classical weighted distances. The classical distances fail to account for the changes in the distances space in practical problems (illustrated in Fig. 2.1). The reason lies in that the traditional distance measures try to aggregate the information on all attributes but ignore the influence of their interrelationships and the context information of the alternative set. The psychological distance, by contrast, incorporates the context information into the distance calculation. The competitions among the alternatives are also reflected through the dominance vector and the indifference vector. These differences make the psychological distance show more advantages over the traditional distances in practical decision making problems.

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2 The Decision Making Method Based on the New Distance …

Fig. 2.3 The positions of the IFNs A, B, and C in the attribute space





Example 2.1 Let A = (0.2, 0.1), B = (0.2, 0.5) and C =(0.6, 0.3) be three IFNs. As mentioned before, we first assign the non-membership with an algebraic sign to make the two attributes consistent. For convenience, we express the three IFNs with the coordinate forms: A = (0.2, −0.1), B = (0.2, −0.5) and C = (0.6, −0.3). Then, we draw their positions in the attribute space (see Fig. 2.3): 



The Euclidean lengths of CA and CB calculated by (2.1) are 0.2, which can be also observed from Fig. 2.3. Let the weight vector for the two attributes be w = (0.5, 0.5)T , which indicates the equal status for the attributes. According to (2.5), the basis matrix K is:  1 1  − √2 √2 K= 1 1 √ 2

√ 2

and the standardized distance vector between the IFNs C and A is:

 −0.4 distCA = 0.2 Similarly,

2.3 Psychological Distance for IFSs

19

Table 2.1 The psychological distances between the options C and A and those between the options C and B under different values of the parameter wd wd = 5

wd = 10

wd = 15

DCA

0.1183

0.0975

0.0894

DCB

0.2145

0.2133

0.2129

DCB − DCA

0. 0962

0. 1158

0.1235

distCB =

−0.4 −0.2



The transformed vector for CA is transCA = K

−1

distCA =

0.4243 −0.1414



and transCB = K −1 distCB =

0.1414 −0.4243



The transformed vector above shows the transformation process in terms of the exchange information. For example, transCA indicates that it needs to move -0.4243 units along the indifference direction and 0.1414 units along the dominance direction to change from the option C to the option A. The psychological distance between the options C and B and that between the options C and A can be calculated by the formula (2.8). To better understand the effect of the dominance weight matrix Aw , we set wd and .. in Aw with three different values 5, 10, 15. The corresponding distances are shown in Table 2.1. The parameter wd determines the weight information of the dominance vector over the indifference one. The larger value of wd will attach the more attention weights on the dominance vector than the indifference vector. The distances in Table 2.1 also show DCA < DCB under different circumstances. Theoretically, the large value of wd will increases the value of the distance between the options. Since we define a balancing coefficient in the formula (2.8), the distance does not always increase with the parameter wd . However, the difference between the distances DCA and DCB still increases with the increase of wd .

2.3.3 Psychological Distance Measure for IFSs The application of the psychological distance to IFNs validates the feasibility of the new psychological distance. Next, we will further extend this distance measure for

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2 The Decision Making Method Based on the New Distance …

IFSs to broaden its application. Before defining this psychological distance measure for IFSs, we should also first check the attributes and transform them into the same type. Generally, we transform the cost type attributes into the benefit type ones. A simple method is to use the complement operators to achieve this (Xu and Cai 2012):  α = (μα , vα ) =

α, α is benefit attribute α, ¯ α is cost attribute

(2.9)

For the intuitionistic fuzzy decision making problems, the attribute information for different alternatives is described by IFNs. To simplify the calculation process, we assume that the two parts of IFNs share the same weight information on each attribute. Then we can double the attributes by considering both parts of IFNs. That is, the attributes’ number is twofold. The weight vector will be extended in the same way. As suggested by Berkowitsch et al. (2015), a simplified strategy to reduce the size of the indifference vector without any loss of information is to compare each attribute with a certain arbitrary one (such as the first attribute). For the case with n attributes, we can get n(n − 1) indifference vectors by comparing each attribute with the first attribute, expressed as: ⎧ wj+1 ⎪ ⎨ − w1 , i = 1 ρij = 1, i = j + 1 (j = 1, . . . , n − 1) ⎪ ⎩ 0, others

(2.10)

These indifference vectors constitute the indifference matrix. In view of the orthogonality of the indifference vector and the dominance vector, the dominance vectors corresponding to these indifference vectors can be gotten by:

ρd =

w1 w2 wn , ,..., w1 w1 w1

T (2.11)

When n = 1, (2.10) becomes (2.3) and (2.11) becomes (2.4), respectively. Based on the indifference vector and the dominance vector, we can get the other parameters related to the intuitionistic fuzzy psychological distance (e.g., the normalized basis B and the transformedvector) with respect to (2.5) and  (2.6). Simiρin−1 ρi1 ρi2 ρd larly, the normalized basis K is K = ρ  , ρ  , . . . , ρ  , ρd  and the transi1

i2

in−1

formed vector can be expressed by transIFS = K−1 distIFS . Then the final normalized psychological distance measure for the IFSs can be expressed with the following form:  1 · transIFS · Aw · transIFS (2.12) d= 2n · wd

2.3 Psychological Distance for IFSs

21

Attributes standardization Distances on the indifference direction

Indifference vectors

Aw

The determination and extension of attribute’s weights Dominance vectors

Psychological distance

Distances on the dominance direction

Fig. 2.4 The general calculation procedures for IFSs

where n is the total number of the attributes (with both the membership and nonmembership parts considered) and 2n · wd is the balancing coefficient ensuring the distance belongs to the interval [0,1]. The parameter transIFS is similar to that in (2.6). The matrix Aw in (2.12) is a n × n diagonal matrix as follows: ⎡

⎤

1

⎥ ⎥ ⎥ ⎦

⎢ 1 ⎢ Aw = ⎢ .. ⎣ .

(2.13)

wd where only the last entry of the diagonal is assigned with more weight to ensure the dominance direction is paid. Similarly, it is easy to prove that the new distance function satisfies (DP1-DP4) of Definition 2.2. Now we can conclude the general steps for calculating the new psychological distance measure for IFSs, illustrated in Fig. 2.4. The standardized attributes and their weight information are prepared for determining the indifference vector and the dominance vector. Then we can calculate the Euclidean distance units along these two directions. A scaled parameter wd larger than 1 in Aw is defined for the dominance direction so as to describe the competitions in the transformation process. After the normalization process, we can finally get the intuitionistic fuzzy psychological distance measure.

2.4 Similarity Measure for IFSs Based on the Psychological Distance The similarity measure is another important measure for IFSs. In the decision making process, the similarity degrees of the alternatives directly influence the DMs’ evaluations and the decision making process. In view of the complementary relationship

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2 The Decision Making Method Based on the New Distance …

between the similarity measure and the distance measure, we can present the mentioned similarity measure for IFSs with a general form ϑA1 ,A2 = f (d (A1 , A2 )), where f represents the decreased function and d (A1 , A2 ) is the distance between two options. Most of the existing researches on the similarity measures for IFSs are based on various distance measures and the common form of the similarity measure is expressed as ϑA,B = 1 − d (A, B), where ϑA,B is the similarity measure and d (A, B) is the distance measure for the alternatives A and B. These classical similarity measures utilize the linear decreased function of the distances to define the similarity measure. The linear function in the existing similarity measures implies that the decrease of the similarity degree is proportional to the change of distances evenly. In addition, the linear form may be ineffective in distinguishing the subtle differences among options. In practical problems, the optimal alternatives sometimes are much more similar, making it more difficult to determine the best choice. To provide a clear contrast of the options and the reliable decision assistance for the DMs, the subtly similar ones should be more evident than the distinctly dissimilar ones or at least at the same level. We need to develop a new similarity measure that can rescale the differences among alternatives and distinguish the optimal one with less difficulty. As a result, the nonlinear form of the decreased function will be more suitable for this purpose. In the following, we will develop such an intuitionistic fuzzy similarity measure integrating the psychological distance discussed above. In the study of the air pollution modeling, the Gaussian Plume Model has been widely used for a continuous point source. It is assumed that the area close to the pollution source presents a high density of pollution material due to the high dispersion. Whereas the area far from the pollution source shows the low density of pollution for the dispersion is not fast enough to reach such places. This dispersion mechanism is similar to the relationships of the similarity and distance measures. The Gaussian function in this model finely simulates the dispersion process. Inspired by this model, we adopt the Gaussian function as the basic decreased function in the construction of the similarity measure. The general form of the Gaussian function is: (x−u)2 1 f (x) = √ e− 2δ2 δ 2π

(2.14)

The parameter δ√12π determines the peak of the function, which can also be interpreted as the height of the point source in the air solution model. The parameter u is the expected value of the function (also the position of the peak) and δ is the variance parameter. Then x − u represents the distance to the source position, and the function f (x) depicts the dispersion distribution in the space. In the field of the similarity measures, the function f (x) can also illustrate the similarity degrees among the options in the alternative sets. However, the formula (2.14) does not always satisfy the first rule in Definition 2.1. To make it applicable to IFSs widely, we use a more general formulation to define the similarity measure:

2.4 Similarity Measure for IFSs Based on the Psychological Distance

ϑA1 ,A2 = ϕ · e−λ·D + C0 2

23

(2.15)

where D is the distance measure between the options A and B. The constant C0 adjusts the value of the function to zero when the distance D is 1. According to the laws of the similarity measure, when D = 0, ϑA1 ,A2 = 1 and when D = 1, ϑA1 ,A2 = 0. Then we can get the relationships of the parameters in (2.15) as follows: !

ϕ=

1

(e−λ −1) C0 = 1 − ϕ

(2.16)

Once the value of the parameter λ is determined, the other parameters’ values can be solved out by (2.16). The formula (2.15) is the general form of the similarity measure based on the Gaussian function. We find that we need to determine three parameters to get the similarity measure in the formula (2.15). In practical problems, we usually hope to use least possible parameters to alleviate the burden of the DMs. As a result, to reduce the computation complexity and make the formula (2.15) more applicable, we develop a simpler approximate method, which is also dynamically suitable for different distance decreasing processes. The simplification can be achieved by the following process. We first set C0 = 0 and ϕ = 1 in the formula (2.16) to make the max value be 1 when the distance is 0. Since we delete the constant parameter C0 , the function value may still be very large when the distance is close to 1. Then we make the parameter λ be λ = 2·wd ·λ, where 2·wd is the balancing coef ficient in (2.15). At present, the exponent is −λ · D2 = −λ · (2 · wd · D2 ) = −λ · D02 and D0 is the original psychological distance without normalization. The similarity measure now will fast approach zero when the distance measure becomes 1. To illustrate this process, we draw a series of curves for the similarity measure under different values of λ in Fig. 2.5.

Fig. 2.5 The curves of the similarity measure with different parameters

24

2 The Decision Making Method Based on the New Distance …

In Fig. 2.5, we find that the smaller the parameter λ is, the less sensitive the similarity curve is to the distance change, indicating the weaker discriminating capability for the similarity measure. The larger λ is, the steeper the similarity curve is. The tiny difference between the options will be enlarged and the vast difference between the options changes slowly in terms of the similarity measure. The small λ provides a dynamic window which expands the influences of the large distance differences and shrinks the influence of the small distance differences for the similarity measure. In addition, if the value of λ is big enough (e.g., larger than 8), the values of the similarity measure at the point 1 are almost close to 0, and the variation in the interval close to 1 is also very small. Hence, the decrease of λ decreases the sensitivity of the similarity curves to the distance variances. The principle for setting this parameter depends on the problem we handle. If the alternative sets are full of much more similar ones, it is suggested to set λ with higher values; For the distinguishable alternative sets, it is suggested to set λ with smaller values. Considering the magnitude of the distance measure for IFSs, we suggest that λ should be larger than 1 to provide a sound discriminable capability. The simplified similarity measure for IFSs in terms of the Gaussian function can be depicted as:

ϑA1 ,A2 = e−(2·wd )·λ·D = e−λ ·D0 2

2

(2.17)

2.5 Extended Intuitionistic Fuzzy TOPSIS Method by the New Distance Measure 2.5.1 Algorithm of the Extended TOPSIS Method As to the decision making methods based on the distance measures, the TOPSIS method is one of the most common and widely used methods. The core idea of this method is to find the alternatives that are nearest to the positive ideal solution but farthest to the negative ideal solutions, that is, to distinguish the similarity of the provided options to the pre-defined ideal one. This similarity information will be further used to rank the option sets. Apparently, the quality and reliability of the distance and similarity measures will directly influence the decision results. Most of the TOPSIS methods under the intuitionistic fuzzy environment in the literature are based on the Euclidean or Hamming distances or their improved forms. However, as has been previously pointed out, the classical distance measures seldom account for the difference between the options and their competitive relationships, which may result in the counterintuitive results. As a result, the decision making approaches based on these distances may also lead to unreliable decision results. The introduced psychological distance for IFSs which takes the context information of the alternative sets into account can overcome such deficiency. In the following, we will incorporate

2.5 Extended Intuitionistic Fuzzy TOPSIS Method ...

25

the new distance and similarity measures into the decision making methods and present an extended intuitionistic fuzzy TOPSIS method. Let α + be the best ideal solution and α − be the worst ideal solution of the alternative set, respectively. For an alternative set described by IFSs, A = {A1 , A2 , . . . , An }. Each alternative Ai contains theintuitionistic information on all attributes, depicted    # " μα1 , vα1 , μα2 , vα2 , . . . , μαm , vαm . As a result, the positive ideal as Ai = alternative α + can be defined as: α+ =



$ % $ % $ % $ % $ % $ % max μαj , min μαj , max μαj , min μαj , . . . , max μαj , min μαj j=1

j=1

j=2

j=2

j=m

j=m

and the negative ideal alternative α − is defined as: a− =



$ % $ % $ % $ % $ % $ % min μαj , max μαj , min μαj , max μαj , . . . , min μαj , max μαj j=1

j=1

j=2

j=2

j=m

j=m

respectively. If an option is closer to the positive ideal one, then such an option will be the optimal choice. On the contrary, the option close to the negative ideal one is of little worth to the DMs. In practice, the best option should be the one closest to the positive ideal option as well as farthest to the negative ideal option. This is the core idea of the TOPSIS method. To fully take advantage of the distance information for the ideal options, in the following, we will extend the TOPSIS method by incorporating the new distance measure. First, we should calculate the distance of each alternative to the given ideal alternative using the psychological distance measure (2.12). The psychological distance between the option i and the positive ideal option is calculated by Sdi+ = dpsy (Ai , Aα+ ) (i = 1, 2, . . . ., n)

(2.18)

Similarly, the psychological distance between the option i and the negative ideal option is calculated by Sdi− = dpsy (Ai , Aα− ) (i = 1, 2, . . . ., n)

(2.19)

where dpsy (·) indicates the psychological distance defined in (2.12). Based on the distances in (2.18) and (2.19), we can obtain the closeness degrees to the ideal situation for each alternative: ηi =

Sdi− , i = 1, 2, . . . , n + Sdi−

Sdi+

(2.20)

The larger value of η means the alternative approaches the ideal option more closely, and the smaller η represents the contrary situation. Besides, η = 1 if and only if the alternative is the positive ideal option itself, and η = 0 if and only if the alternative is the negative ideal option itself. Finally, we can get the priority order of

26

2 The Decision Making Method Based on the New Distance …

the alternatives by ranking the values ηi (i = 1, 2, . . . , n). The detailed procedures for the decision making process are as follows: Algorithm Pseudocode of the extended intuitionistic fuzzy TOPSIS approach Input: A

A1 , A2 ,..., An : Alternative sets

: Attributes’ weight vector

wd : The parameter that magnifies the dominance influence Output: Closeness degrees to the ideal solution Step 1. Standardize the attributes of the alternative sets. Step 2. Initialize the parameter for the dominance effect wd ) and the positive ideal solution (

Step 3. Get the negative ideal solution (

) of the alternatives,

respectively. and the indifference vector

i,

Step 4. Compute the dominance vector

d

Step 5. Prepare the transforming basis

and the magnifying the matrix Aw

respectively.

Step 6. Compute the psychological distance between the target alternative and the positive ideal one ( Sd i ) and that between the target alternative and the negative ideal one ( Sd i ) separately. Step 7. Calculate the closeness degree

i

for the alternative i to the best ideal option.

Step 8. Perform the ranking process and pick up the optimal alternative.

2.5.2 Case Study of the Marine Energy Transportation Route Decision Making Problem In this section, we will illustrate the application of the above decision making method to the marine energy transportation route decision making problem in detail. The marine energy transportation route decision making problem is difficult to handle in that it involves the political, economic, natural and cultural factors simultaneously. Some of these factors are easy to be quantified while some others are only described qualitatively. In the decision making process, the DMs should take all factors into consideration and trade off not only between different options but also various attributes at the same time. In most cases, the comprehensive information available for the decision making is not always possible, especially when the DMs are unable to grasp a full understanding of the given problem. In addition, the

2.5 Extended Intuitionistic Fuzzy TOPSIS Method ...

27

transportation routes are much similar under some circumstances, which will further place more difficulty on the decision making process. Given that the effectiveness of the IFS theory in dealing with the cognitive uncertainty, it is wise to fully utilize the IFS to describe the information in the marine energy transportation decision making problems. To determine the optimal routes catering to the DMs’ demands, we will use the intuitionistic fuzzy psychological distance and the new decision making method to provide the solution, which is also exemplified as a practical validation. Taking the example of the Chinese marine energy transportation routes, there are six main routes currently, including the Middle East route, the Northern African route, the West Africa route, the Latin America route, the Australia route and the South East Asia route. To simplify the problem, the main factors considered in the decision making process are economics, the capacity of routes, the safety of routes, and the probability that a route may be blocked in emergent situations. Assume that the decision making information in terms of IFNs is provided in Table 2.2, we will conduct the analyses of the relationships of the alternatives and the decision making process using the decision making methods of this chapter in the following parts: We first investigate the similarities among the alternative sets in order to analyze the interrelationships of the given four routes. Suppose that the weight vector for the four attributes is ω = (0.20, 0.25, 0.25, 0.30)T and wd = 5, we can calculate the normalized psychological distances between different routes. To illustrate the advantages and differences between the new distance measure and the existing distance measure, we also use the generalized weighted Euclidean distances in (2.2) for comparisons. The results for these two types of distances are listed in Table 2.3. The minimum distance of the psychological distances is 0.0723, the distance between the routes C and D. It indicates that the most similar routes are the West Africa route and the Latin America route. However, the minimum distance of the weighted Euclidean distances is 0.1311, i.e., the distance between the routes B and E. This result means that the Northern African route and the Australia route are Table 2.2 The decision information of the six marine energy routes Economics

Route capacity

Safety

Blocked probability

A: The Middle East route

(0.73, 0.10)

(0.61, 0.30)

(0.38, 0.60)

(0.2, 0.80)

B: The Northern African route

(0.67, 0.20)

(0.54, 0.10)

(0.60, 0.40)

(0.3, 0.65)

C: The West Africa route

(0.37, 0.30)

(0.68, 0.21)

(0.82, 0.18)

(0.40, 0.40)

D: The Latin America route

(0.45, 0.31)

(0.73, 0.25)

(0.80, 0.10)

(0.16, 0.40)

E: The Australia route

(0.80, 0.20)

(0.50, 0.10)

(0.65, 0.30)

(0.10, 0.64)

F: The South East Asia route

(0.70, 0.22)

(0.32, 0.40)

(0.40, 0.35)

(0.25, 0.69)

Note All the attribute values have already been transformed into the same type

28

2 The Decision Making Method Based on the New Distance …

Table 2.3 The new psychological distances and the generalized weighted Euclidean distances (the italic numbers) A

B

C

D

E

F

A

0 0

0.0941 0.1910

0.1968 0.4958

0.2020 0.4623

0.1315 0.2397

0.1129 0.2298

B

0.0941 0.1910

0 0

0.1367 0.3415

0.1553 0.3161

0.0738 0.1311

0.1228 0.2456

C

0.1968 0.4958

0.1367 0.3415

0 0

0.0723 0.1423

0.1255 0.3682

0.1932 0.4484

D

0.2020 0.4623

0.1553 0.3161

0.0723 0.1423

0 0

0.1145 0.3011

0.1988 0.4267

E

0.1315 0.2397

0.0738 0.1311

0.1255 0.3682

0.1145 0.3011

0 0

0.1403 0.2668

F

0.1129 0.2298

0.1228 0.2456

0.1932 0.4484

0.1988 0.4267

0.1403 0.2668

0 0

more similar. Also, the maximum distances of these two types are different from each other. The psychological distances imply that the most dissimilar routes are the Middle East route and the Latin America route, but the most dissimilar routes are the Middle East route and the West Africa route in the case of the generalized weighted Euclidean distances. Based on the calculated distances, we can get the similarity degrees among the alternatives using the new similarity measure. Firstly, we set λ = 3 and the parameter wd remains the same. Secondly, we calculate the similarity degree utilizing (2.17). According to Definition 2.1, we also calculate the traditional similarity degree based on the generalized Euclidean weighted distances calculated by (2.2). The similarity degrees between the options can be directly differentiated by their grid gray degrees in Fig. 2.6. For the results based on the intuitionistic fuzzy psychological distances (Fig. 2.6a), the most similar ones are the West Africa route (C) and the Latin America route (D). In contrast, the most similar routes are the Northern African route (B) and the Australia route (E) in Fig. 2.6b, which is only the second in Fig. 2.6a. We also find that the area corresponding to the routes C and D in Fig. 2.6a nearly shows the similar level of gray degree (e.g., the grid AC and the grid AD; the grid FC and the grid FD). The category in terms of gray level in Fig. 2.6b is not apparent than those in Fig. 2.6a. We also notice that the new similarity measure categories the alternative sets into three types, the much similar ones, the moderate ones, and the most dissimilar ones. The most competitive ones are more apparent than those less relative to each other in terms of their grayscale results, which will be helpful to reduce the difficulty of decision making. Such a distinguishable result is partly ascribed to the adoption of Gaussian function, which provides a dynamic scale window for focusing on different details. On the other hand, this is also partly ascribed to the advantage of utilizing the intuitionistic fuzzy psychological distance

2.5 Extended Intuitionistic Fuzzy TOPSIS Method ...

29

Fig. 2.6 The similarity between different routes based on their distance space a is calculated based on the IF psychological distance and Gaussian function; b is calculated based on the generalized Euclidean weighted distance and the linear function

introduced in this chapter, which allows for the context information of the alternative sets and surmounts the deficiency of the counterintuitive results of the Euclidean distances. Next, we will illustrate the decision making process based on the extended intuitionistic fuzzy TOPSIS algorithm for these transportation routes. Table 2.2 provides

30

2 The Decision Making Method Based on the New Distance …

Table 2.4 The priority degrees for the six given marine energy transportation routes Closeness degrees

A

B

C

D

E

F

0.3666

0.5097

0.7254

0.6806

0.5597

0.3199

comprehensive information for the six transportation routes. Based on the intuitionistic fuzzy information, it is easy to determine the positive ideal option and the negative ideal option respectively. Namely, α + = {(0.80, 0.10), (0.73, 0.10), (0.82, 0.10), (0.40, 0.40)} and α − = {(0.37, 0.31), (0.32, 0.40), (0.40, 0.60), (0.10, 0.80)} Using the psychological distance defined in (2.8), we can calculate the psychological distances between each alternative and the positive ideal route Sdi+ in (2.18) and the negative ideal route Sdi− in (2.19), respectively. Then we can get the closeness degrees to the ideal solutions for the alternative sets by (2.20), which are listed in Table 2.4. The ranking order of the six routes are C > D > E > B > A > F and the optimal alternative route is the West African route. In order to analyze the ranking results thoroughly, we also calculate the ranking order for the routes by means of intuitionistic fuzzy information aggregation techniques. In this chapter, we utilize the widely used intuitionistic fuzzy hybrid aggregation (IFHA) operator (Xu 2007a) to aggregate the intuitionistic fuzzy information of the alternatives and make a decision. The weights for all the attributes remain the same and the operator-associated weights are derived from the normal distribution method (Xu 2005). The detailed calculation procedures are skipped considering the page’s length. The final aggregated result of each alternative and their score function values are listed in Table 2.5. The final priority sequence for the given six routes is D > C > E > B > A > F. Thus, the best choice is the Latin America route. Comparing the two sequences, we observe that the key difference lies in the optimal ranking results. The result based on the introduced method in this chapter takes Table 2.5 The aggregated results of the alternative transportation routes and the corresponding score function values A

B

C

D

E

F

Aggregated values

(0.5199, 0.3941)

(0.5363, 0.2903)

(0.6326, 0.2580)

(0.6252, 0.2359)

(0.5620, 0.2558)

(0.4062, 0.3953)

Score values

0. 1258

0. 2460

0.3746

0.3893

0. 3062

0. 0109

2.5 Extended Intuitionistic Fuzzy TOPSIS Method ...

31

the route C as the optimal choice and the route D is the secondary one, which is contrary to the results derived from the IFHA operator. The other parts of the sequence coincide with each other. In view of the similarity results, the route C is similar to the route D based on our new similarity measure. Meanwhile, the aggregated information of the IFHA operator also implies that the routes C and D contain some similar intuitionistic fuzzy information considering their first and secondary places in the sequences. The intuitionistic fuzzy information aggregation result tests and verifies the accuracy of the new similarity measure from another aspect. As has been discussed before, the IFHA operator deals with the intuitionistic fuzzy information along the dimension of attributes rather than on both the attribute dimension and the alternative dimension. The aggregation operator provides a collective result of all the attributes for a certain option. However, the context information provided by the alternative set in terms of their competitive relationships and its influence on the evaluations of the DMs are ignored. The aggregation operator can tell which alternative is the “excellent” one in terms of IFSs but it may be inefficient to differentiate the optimal alternative catering to the preference of the DMs. In view of this, the reason why the route C ranks first instead of the “excellent” route D is clear. On the other hand, the results of the sequence combined with the similarity results are consistent with those derived by utilizing the IFHA method, especially the last one. For instance, as to the sequence B > A > F, we can directly obtain the similarity measures between each option respectively from Fig. 2.6a and they satisfy the relations ϑB,F < ϑA,F and ϑB,F < ϑA,B .

2.6 Remarks In this chapter, we introduce the psychological distance measure and the new similarity measure for IFSs. The psychological distance measure for IFSs assumes that the different options could be exchanged on the dominance direction to reflect the competitive relationships related to other alternatives. The influence of the context information contained in the alternatives is reflected by the transforming process on the dominance and the indifference vector. The additional parameter wd in the matrix Aw rebuilds the dominant competition between the dominance vector and the indifference vector. Though a balancing parameter is also defined to satisfy the distance measure laws of the IFSs, the psychological distance is different from the general weighted distance measure in the literature. The new similarity measure can fully utilize the advantages of the psychological distance measure and provides a more flexible similarity measure. The new similarity measure can serve as a supplementary method for the existing work. By setting different parameters, the new similarity measure can selectively reflect the similar information of the alternatives in detail. The more attention will be paid to the important ones rather than the inferior ones, and then the alternatives will be more distinguishable. All these characteristics have been integrated into the new decision

32

2 The Decision Making Method Based on the New Distance …

making method introduced in this chapter to improve the decision result’s quality and reduce the selection difficulty for the DMs. The decision making methods presented in this chapter takes a further step in the study of cognitive decision making models under the intuitionistic fuzzy environment, which may play an important role in simulating the influence of context effect on the DM’s decisions.

References Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 Atanassov KT (1999) Intuitionistic fuzzy sets. Springer, Heidlberg Baccour L, Alimi AM, John RI (2013) Similarity measures for intuitionistic fuzzy sets: state of the art. J Intell Fuzzy Syst 24(1):37–49 Berkowitsch NAJ, Scheibehenne B, Rieskamp J, Matthaeus M (2015) A generalized distance function for preferential choices. Br J Math Stat Psychol 68(2):310–325 Chen LH, Tu CC (2014) Dominance-based ranking functions for interval-valued intuitionistic fuzzy sets. IEEE Trans Cybernet 44(8):1269–1282 Das S, Guha D, Mesiar R (2018) Information measures in the intuitionistic fuzzy framework and their relationships. IEEE Trans Fuzzy Syst 26(3):1626–1637 Dong JY, Lin LL, Wang F, Wan SP (2016) Generalized Choquet integral operator of triangular Atanassov’s intuitionistic fuzzy numbers and application to multi-attribute group decision making. Int J Uncertainty Fuzziness Knowl Based Syst 24(5):647–683 Du WS, Hu BQ (2015) Aggregation distance measure and its induced similarity measure between intuitionistic fuzzy sets. Pattern Recogn Lett 60–61:65–71 Feng F, Fujita H, Ali MI, Yager RR, Liu XY (2019) Another view on generalized intuitionistic fuzzy soft sets and related multiattribute decision making methods. IEEE Trans Fuzzy Syst 27(3):474–488 Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst 148(2):319–328 Halamish V, Liberman N (2017) How much information to sample before making a decision? It’s a matter of psychological distance. J Exp Soc Psychol 71:111–116 Hao ZN, Xu ZS, Zhao H, Zhang R (2019) Psychological distance measures for intuitionistic fuzzy information and their application in marine energy transportation route decision making. Tech Rep Huang B, Li HX, Wei DK (2012) Dominance-based rough set model in intuitionistic fuzzy information systems. Knowl-Based Syst 28:115–123 Huang B, Wei DK, Li HX, Zhuang YL (2013) Using a rough set model to extract rules in dominancebased interval-valued intuitionistic fuzzy information systems. Inf Sci 221:215–229 Huang N, Burtch G, Hong YL, Polman E (2016) Effects of multiple psychological distances on construal and consumer evaluation: a field study of online reviews. J Consum Psychol 26(4):474– 482 Huber J, Payne JW, Puto C (1982) Adding asymmetrically dominated alternatives—violations of regularity and the similarity hypothesis. J Consum Res 9(1):90–98 Joshi R, Kumar S (2019) A dissimilarity measure based on Jensen Shannon divergence measure. Int J Gen Syst 48(3):280–301 Ke D, Song Y, Quan W (2018) New distance measure for Atanassov’s intuitionistic fuzzy sets and its application in decision making. Symmetry 10(10) Li DF, Cheng CT (2002) New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recogn Lett 23(1–3):221–225

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Chapter 3

The Dynamic Decision Making Method Based on the Intuitionistic Fuzzy Bayesian Network

The past decades have witnessed the toughest period of disasters in terms of frequency and concurrency, casting great damage to the property and development of our society. The entrance to the risk society puts forwards more requirements for the decision making, especially for the decision making problems under the emergent events. One of the prominent characteristics of the emergent decision making is incomplete or blank information. The DMs have to deal with more uncertainty arising from various reasons and limitations, which are the typical forte of the fuzzy set theory. In this chapter, we will introduce the intuitionistic fuzzy set to the emergent decision making problem. Base on the intuitionistic fuzzy Bayesian Network (IFBN), the dynamic multi-criteria decision-making method is also presented. The main methods and applications in this chapter come from Hao et al. (2018). The main content of this chapter lies in three aspects: First, the conceptual framework for the dynamic intuitionistic fuzzy decision making problems is introduced. Second, the IFBN and the related weight determination method are studied. Third, a novel dynamic intuitionistic fuzzy decision making approach is presented with an illustrative application.

3.1 Dynamic Decision Making Problems Under the Emergencies The emergency decision making, conducting the whole emergent actions, has a pivotal position in the emergency response. Regardless of the importance of the disaster preparedness and response actions for the emergency events, taking an inappropriate emergency decision making approach will result in potential failures of the emergency actions and excessive exposures to the secondary disasters. Most of the emergency decision making problems involve dynamic decision making, which is determined by the intrinsic properties of emergency events. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9_3

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However, one of the inherent characteristics, that is, the dynamic environments, exert more challenges on the DMs. On one hand, the time pressure requires the DMs to make faster decision response; On the other hand, the unpredictable emergency event requests the DMs to handle the uncertainty and vagueness of the decision information more prudently and efficiently. Meanwhile, the evolution of the emergency event and the exterior environment imply that the reliable decision results are based on not only the current situations but also the historical state of affairs. From this perspective, the emergency decision making problem can be treated as a dynamic multi-criteria decision making (MCDM) problem. Noticing that the decision information under emergency situations is always inadequate due to the no-measurable factors, the limited information or the imprecise evaluations of the intangible criteria, most DMs tend to express their options with linguistic variables or fuzzy values instead of crisp numbers. The IFS (Atanassov 1986) provides a powerful and efficient tool to deal with uncertain and vague information, making it practicable for the DMs to overcome the above-mentioned difficulties. The current dynamic fuzzy multicriteria decision-making (DFMCDM) methods mainly involve the human resource evaluations, personnel promotions and other multi-period investment decision making (Bali et al. 2015; Su et al. 2011; Xu and Yager 2008). Few of them consider the emergency decision making problems. The application of the DFMCDM to the emergency circumstances needs to solve two main problems. One is the acquirement of the accurate decision matrix in terms of the given attributes at different times; the other is the determination of the dynamic attribute weights. As for the decision matrix, most DFMCDM methods assume that the decision matrix can be directly acquired from the DMs at the given time (Kou and Lin 2014; Saaty 1977). However, such information is not always available at certain points, and a reasoning process will be necessary. For the attribute weights, the current methods for static situations can be categorized into three groups: the subjective methods, the objective methods, and the hybrid methods. The subjective methods assume that the weights of the attributes are determined regarding the DMs’ subjective preferences or experiences, such as the Analytic Hierarchy Process (AHP) (Kou and Lin 2014; Saaty 1977) and the Delphi method (Hwang and Yoon 1981). The objective methods, on the contrary, determine the weights totally relying on the decision matrix and the solutions of the mathematical models, including the multiple programming method (Srinivas and Shocker 1973; Xu 2010, 2012), the entropy method (Chen and Li 2010; Shannon and Weaver 1947; Xia and Xu 2012), and the ideal point method (Hwang and Yoon 1981; Xu 2007; Zhao and Xu 2016), etc. The subjective methods rely much on the DMs’ judgments and the results are sometimes subjective and untrustworthy. The objective methods have the strong mathematical and theoretical foundations and the final results are impersonal. However, these methods ignore the prior knowledge and experience of the DMs and do not reflect their subjective preferences. As a result, some hybrid methods have been developed to overcome such deficiencies and make the weights more accurate, such as the TOPSIS-oriented hybrid weighting method (Wang and Lee 2009), the entropy-based hybrid weighting method (Li et al. 2015), the PSO-based weighting method (NabaviKerizi et al. 2010) and the sparse ensemble methods (Zhang and Zhou 2011). The

3.1 Dynamic Decision Making Problems Under the Emergencies

37

weight determination methods are almost confined to the static weights and few of them have been applied to dynamic intuitionistic fuzzy decision making. We conclude the reasons as follows: For one thing, the advantages of the subjective methods are the disadvantages of the objective methods and vice versa. It indicates that neither of these two methods will independently solve the dynamic weights for practical problems. For another, the hybrid methods combine the advantages of these two methods but most of them still depend on the multiple programming solutions, which will increase the calculation complexity and decrease the efficiency and timeliness. These deficiencies will be more prominent under emergency decision circumstances. Currently, the dynamic intuitionistic fuzzy decision making methods assume that the weights are subject to some mathematical distribution models over the decision time, e.g., the arithmetic progression and the geometric progression (Xu 2008), the BUM function (Xu and Yager 2008), Poisson distribution (Xu 2011), normal distribution (Xu 2005), the exponential distribution (Sadiq and Tesfamariam 2007), the binomial distribution (Xu and Chen 2007) and the average age methods (Xu and Yager 2008). Su et al. (2016) summarized these distribution function-based methods and developed probability methods to obtain the weights for the weighted arithmetic aggregation operator. The common characteristic shared by these methods is that the dynamic weights of them are some certain determinate time-oriented functions. This assumption simplifies the calculation of the attribute weights but fails to reflect the real situation. The urgent and unexpected emergency events indicate that the generalized patterns of the dynamic weights seldom exist. Besides, we should point out that the above dynamic weights are actually dynamic period weights instead of the attribute weights while the attribute weights still remain static. In reality, the significance of different attributes will dynamically change in accordance with different phases of the emergency events. For instance, at the initial period, the disaster grade is not serious and the related attributes may be treated equally. As the event deteriorates, new issues and situations arise. At this time, the weights of the associated attributes should be increased and others consequently decreased. Besides, the risk attitudes of the DMs will also change. The aim of this chapter is to introduce a dynamic decision making method allowing for the dynamic attribute weights and the risk attitudes of the DMs under emergency circumstances. To depict the dynamic attribute weights of the real circumstances, we need to gain some insights into the causal relationships of the attributes and their evolving regularities. Bayesian network (BN), as a progressively analytical method in the academic research, is suitable for dynamic causal analysis and statistical inference (Constantinou et al. 2016; Jitwasinkul et al. 2016). In view of these characteristics, we shall apply the BN theory to the dynamic weight determination under the intuitionistic fuzzy environment. The dynamic attribute weights derived from the Bayesian-updating model will change in accordance with the plausibility of attributes over time (Weiss-Cohen et al. 2016). Hence, the practicable attribute weights provide the DMs with more accurate information for the final risk decisions. Moreover, this new weight determination approach will serve as a complement to the existing approaches and improve the present situation, where the determination of dynamic weights relies much on distribution functions. As for the risk attitudes in MCDM

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under fuzzy environment, the fuzzy TODIM (an acronym in Portuguese for iterative multi-criteria decision making) technique has become a promising uncertain decision making method (Krohling et al. 2013; Ren et al. 2016; Xu and Zhao 2016). The prospect theory in TODIM can successfully capture the risk preferences of the DMs in the process of decision making, which inspires us to incorporate the prospect theory into the dynamic MCDM to solve the emergency decision making problems. In the following sections, we will present the solutions to the above-mentioned problems in detail.

3.2 Conceptual Framework for Dynamic Intuitionistic Fuzzy Decision Making In the emergency events, one of the most prominent characteristics should be the dynamically evolving environments, including the decision making environment and the exterior situations of the events. It is hard to totally obtain the complete information all the time, especially during the initial period. The fast evolvement and the unexpectedness of the emergency events result in the lack of precise data. Most information available is provided by the experiential knowledge and historical data. In addition, the emergency decision making is also challenged by ambiguity and imprecision. For these reasons, the DMs need to cautiously handle the available information and aggregate them efficiently under uncertain circumstances. As for the information acquisition and empirical knowledge, Bayesian Network (BN) is expert at prediction, diagnosis, causality analysis, and combined-reasoning. The prior distribution model in the BN captures the experiential belief of the unknown objective, and the data inference model describes the causality relationships and the generation mechanism of data, making it an ideal analytical tool even when the information available is imperfect or incomplete (Jitwasinkul et al. 2016). In view of the dilemma where the full grasp of the emergency situations and the complete quantitative descriptions of events are hard to achieve, the IFS theory provides an effective tool to deal with the uncertainty and quantify the emergency. This inspires us to incorporate the IFS into the BN so as to provide an efficient method for dynamic intuitionistic fuzzy decision making problems, especially for the emergency circumstances. The terminology “dynamic” indicates two aspects, i.e., the dynamic attribute weights of each period and the corresponding decision data. When dealing with the attribute weights, as has been pointed out before, the weight information calculated by the assumed distribution functions is actually definitive. The reality is that the importance of attributes changes with the environment and it is unfeasible to define a specific pattern. Meanwhile, the decision information is also dynamically updated by the DMs or from current situations. Most of the existing intuitionistic fuzzy decision making methods accept that the decision information is presented in advance. The objective information on the changing environment and its influence on the preferences of the DMs are often ignored.

3.2 Conceptual Framework for Dynamic Intuitionistic Fuzzy Decision Making Fig. 3.1 The conceptual dynamic intuitionistic fuzzy decision making model

Emergent environments

information

BN (Dynamic attribute weights W(t))

Our work

information BN (Dynamic decision information M(t) )

39

Dynamic intuitionistic fuzzy decision model (Aggregator(M(t),W(t))

Based on the analyses above, we shall introduce a conceptual framework for dynamic intuitionistic fuzzy decision making to better solve the unconventional decision making problems and the emergency decision making problems respectively. The new decision making framework features the practical dynamic information in real situations and integrates the BN into the whole decision making process, illustrated in Fig. 3.1. The BN acts as a bridge between the acquisition of dynamic decision data and the determination of attribute weights. The information provided by the real environments is fed into the BN system where the prior knowledge and the posterior information are soundly integrated and updated. Then the BN will provide the realtime reasoning results in terms of the decision information and the attribute weights. Both the attribute weights and the decision information are the critical elements in the later decision making process. The dynamic intuitionistic fuzzy decision making model is responsible for handling different types of decision information with the proper methods. For instance, the information provided by the BN and the imprecise evaluations of the DMs efficiently aggregated with risk attitude should be fully taken into consideration under emergent environments. Then, by utilizing the appropriate decision making methods, the DMs can finally obtain the optimal result(s). The two fundamental components in this framework are the construction of BN for the decision information or the attribute weights and the intuitionistic fuzzy decision making methods. In the following part, we will discuss the implementation of the BN in determining dynamic attribute weights and the associated decision making method in detail.

3.3 Intuitionistic Fuzzy Bayesian Network The BN is a directed acyclic graph (DAG), consisting of sets of variables and directed links between variables (Holmes and Jain 2008). The theoretical foundations of BN are the graph theory and the Bayes’ theorem. The BN provides an efficient tool for uncertain knowledge representation and comprehensive inference, which has been

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3 The Dynamic Decision Making Method Based on the Intuitionistic …

widely applied to the situation assessment, target recognition, data fusion and so on (Shahriari et al. 2016). As for the emergency decision making problems, to what extent the precious available information is utilized and how to acquire the required decision information strongly influences the decision results. In the initial period of the emergency, the information about the events is usually insufficient and imperfect. The DMs tend to resort to the prior knowledge or the empirical evidence obtained from the historical events or the common experience. With the evolution of the events, the rate of the information acquisition accelerates, which will be further utilized to rectify the prior experience and revise the previous judgements of the DMs. This process is superficially random and unpredictable but intrinsically associated. The former status of the event will influence the succeeding situations. This mechanism can be reflected from the causal relationships among the key factors of emergencies. As a result, the BN is determined as an appropriate method for deducing and updating the uncertain information. We should notice that the information under emergency environment is often incomplete and ambiguous. The lack of sufficient information will lead to more indeterminacy and uncertainty, undermining the accurate judgement and evaluation of the DMs. The IFS, taking all the membership degree, the non-membership degree and the hesitant degree into account in a single framework, provides a more flexible choice for the DMs to express their preferences. Also, the prior knowledge taking the form of probabilities is not always accurate and suitable for all circumstances. The DMs may hold indeterminate and vague attitudes for the prior knowledge. Considering that the decision information is described with the intuitionistic fuzzy set, it is natural to take the form of the intuitionistic fuzzy probabilities in the BN, which is beneficial to fully utilize the fuzzy information and reduce the information loss to the greatest degrees. As a result, it is wise and essential to develop the intuitionistic fuzzy Bayesian Network (IFBN). The classical BN is formally defined as follows (Karl-Rudolf 2007): Let V be a finite set of vertices consisted of random variables and the vertices are connected with the directed edges which represent the conditional dependencies among the variables. Each vertex describes a set of probabilistic events with a finite status of outcomes. The finite set V and the spatial relationships among the elements of V form a directed acyclic graph, that is, the so-called BN. In the intuitionistic fuzzy environment, the vertices depict the fuzzy events and the probabilities of the events take fuzzy forms.

3.3.1 Intuitionistic Fuzzy Event Under the Bayesian Network The intuitionistic fuzzy event (IF-event) and its probability are defined as follows: Definition 3.1 (Ciungu and Rieˇcan 2010; Grzegorzewski 2013) Let X be a set in Rn and A represent the IF-event over X. Let P be the probability measure, and the  as P(A) = [ pmin (A), pmax (A)],  where pmin (A) =  probability of A is defined μ P and p = + π P = 1 − (x)d (A) (μ (x) (x))d A max A A X X X v A (x)d P.

3.3 Intuitionistic Fuzzy Bayesian Network

41

The probability of the IF-event takes the form of a compact interval [ pmin (A), pmax (A)], where pmin (A) and pmax (A) represent the minimal and maximal probabilities of the IF-event, respectively. Definition 3.2 (Grzegorzewski 2013) The independent IF-event is defined as P(A ⊗ B) = P(A) ◦ P(B), where ◦ denotes the arithmetic multiplication of the closed intervals. Correspondingly, the conditional probabilities of A under B can be deduced from P(A|B) ◦ P(B) = P( A ⊗ B). Suppose that P(A) = [ pmin (A), pmax (A)] and P(B) = [ pmin (B), pmax (B)], then P(A) ◦ P(B) = [min{ pmin (A) pmin (B), pmin (A) pmax (B), pmax (A) pmin (B), pmax (A) pmax (B)}, max{ pmin (A) pmin (B), pmin (A) pmax (B), pmax (A) pmin (A), pmax (A) pmax (B)}] = [ pmin (A) pmin (B), pmax (A) pmax (B)] The probabilities of the IF-events also comply with the law of probability and Bayes theorem. We skip these definitions in consideration of space. The detailed definitions and discussions can be found in Grzegorzewski (2013), Grzegorzewski and Mrówka (2002).

3.3.2 Probability Transmissibility of the Intuitionistic Fuzzy Bayesian Network Noting that the probabilities of the IF-events defined in Definition 3.1 are compact intervals and all the decision information is presented with IFNs, we naturally hope that all the data structure should be consistent. Hence, it is suggested to transform the compact interval probabilities of the IF-events to the IFNs. For an interval number, the corresponding IFN can be obtained by A˜ = (a, 1 − b). Since the BN depends on the independent relationships of the factors, we will first prove the independent theorem of the IF-event A = [a, b] in terms of the IFN: Theorem 3.1 The independent IF-event is defined as P( A ⊗ B) = P( A) ⊗ P(B) for the IF-events A and B, where P (A) and P (B) take the form of the IFNs. Proof Let P(A) = (a, 1 − b) and P(B) = (c, 1 − d). Then the interval forms of the IF-events A and B are P  (A) = [a, b] and P  (B) = [c, d], respectively. Based on Definition 3.2, we have P(A ⊗ B) = P  (A) ◦ P  (B) = [ac, (1 − b)(1 − d)] = (ac, 1 − (1 − b)(1 − d)) = P( A) ⊗ P(B)

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(b)

(a) A

(c) A

...

B A

B

...

C

B

C

C

Fig. 3.2 Three typical connection structures in the IFBN (a depicts the diverging connection; b depicts the converging connection; c depicts the serial connection)

Remark The transformation between the arithmetic multiplication of the compact intervals and the multiplication of the IFNs makes it feasible for the possibility operations under intuitionistic fuzzy environments. Hence, the intuitionistic fuzzy probabilities are also applicable to the other laws mentioned above. Theorem 3.2 If all the variables in the nodes take the form of intuitionistic fuzzy probabilities, then the final deduced probabilities are also IFNs. Proof Considering the characteristics of the BN, we can take the three typical connections in the BN: the serial, diverging and converging connections (see Fig. 3.2a–c) for the sake of simplicity and without loss of generality. The proofs for these three typical connections will help us to prove this theorem. Suppose that each node (marked as A, B, C,…) represents an IF-event of the systems and has a discrete status in the IFBN. The proof for each situation will be shown as follows: (a) The diverging connection Assume that the connected nodes are the only ones to be considered and other nodes are conditionally independent. As for the serial connection, the node A is the parent and the nodes B and C are its children. Since all the children nodes of A are independent, we only need to prove one of the children nodes to take the form of intuitionistic fuzzy probability. For convenience, the notation echild represents the evidence from the children nodes while the notation e par ent represents the one from the parent nodes.     Let the probability of the status of the parent node A be P Ae j = μe j , ve j and then the probability of the child node B at a certain status e B will be P(e B ) = P(B|A) ⊗ P(A)             = P e B |e A1 ⊗ P e A1 ⊕ P e B |e A2 ⊗ P e A2 ⊕ · · · ⊕ P e B |e A j ⊗ P e A j =⊕

n 

    P e B |e A j ⊗ P e A j

(3.1)

j=1

where the notation ⊕Σ represents the IFN arithmetic sum. Since the elements in (3.1) take the form of IFNs and the addition operation and the multiplication operation all

3.3 Intuitionistic Fuzzy Bayesian Network

43

abide by the IFN operation laws defined in Definition 1.4, it is easy to prove that the final calculation result P(e B ) also takes the form of IFN. (b) The converging connection As for the diverging connection, let eC represent a status of the evidence in the child node C and {P} (Hao et al. 2017) be the set of all the parent nodes of the node C. The probability of the status eC is: P(eC ) = P(C|A) ⊗ P( A) ⊕ · · · ⊕ P(C|B) ⊗ P(B) = ⊕



P(eC |Pi ) ⊗ P(Pi )

{P}

(3.2) Similarly, the probabilities of all the statuses of the child node C are also IFNs. (c) The serial connection The calculation of the probabilities for the node B in the converging connection is a bit subtle. Let e B represent a status of evidence in the node B.The belief on the par ent and that from the node B can be divided  child as the belief from the parent nodes e , separately (Duda et al. 2001). For convenience, we give the children nodes e proportionality expression for the probability on e B :     P(e B ) ∝ P echild |e B ⊗ P e B |e par ent

(3.3)

  where P echild |e B indicates the  probability of all the children nodes  conditional connected to the node B and P e B |e par ent describes the conditional probability of the node B from all its parent nodes. It is worth noticing that (3.3) only describes the non-normative probability over all the statuses of the node B. The deep analysis for this situation is fully discussed in Duda et al. (2001).   Similar to the situation of the serial connections, the first term P echild |e B can be expanded as:     P echild |e B =P eC1 , eC2 , . . . , eC|C| |e B       =P eC1 |e B ⊗ P eC2 |e B ⊗ . . . ⊗ P eC|C| |e B |C|    P eC j |e B =⊗

(3.4)

j=1

where C j is the jth child node and eC j represents the corresponding probability over the status. The cardinality of the set C represents the number of the elements in the evidence set.   The second term P e B |e par ent , integrating the evidence from all the parent nodes of B, is similar to (3.2) in the situation of the diverging connections. As a result, the final intuitionistic fuzzy probability will be

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⎤ ⎡ ⎤ |C|     P(e B ) ∝ ⎣⊗ P eC j |e B ⎦ ⊗ ⎣⊕ P(e B |e A ) ⊗ P(e A )⎦ ⎡

j=1

(3.5)

{e A }

All the terms take the form of the IFNs and the final probability is an IFN as well, which completes the proof of Theorem 3.2. It also indicates that the intuitionistic fuzzy probabilities or the IFN are transmissible in the IFBN. However, we should notice that the results provided by (3.5) are non-normative. Since the intuitionistic fuzzy probabilities serve as an intermediary to calculate the final attribute weights, we can still utilize the IFBN to determine the attribute weights under intuitionistic fuzzy environments and thereby develop the corresponding dynamic intuitionistic fuzzy decision making approach. The issue of the normalizations of the IFNs will be further discussed in the following sections.

3.4 Dynamic Intuitionistic Fuzzy Decision Making Model 3.4.1 Weight Determination Method in the Framework of IFBNs As has been pointed out before, the current DIFMCDM problem is a bit simple for emergency decision making problems in that all the decision information related to the decision making process is given in advance or assumed to obey some specific mathematical distributions. These properties are inconsistent with the reality and hinder the application of the DFMCDM approach. From this perspective, the current assumption of the DFMCDM problem mentioned above will trigger the unexpected consequences once applied to the complicated decision making environments such as the emergent situations. The existing methods typically rely on the dynamic intuitionistic fuzzy aggregating operators to achieve the collective decision information and decision results. Meanwhile, the reliability and accuracy of the decision matrix and the attribute weights will exert decisive influences on the decisions of the DMs. To develop an excellent DFMCDM approach, it is indispensable to devise an accurate attribute weight determination method. One of the most prominent characteristics of the emergency decision making problems is the dynamic evolving environment. This feature is reflected by two aspects: one is the dynamic decision information which includes the changes of the objective decision data of the given options and the subjective judgements and evaluations for different alternatives, the other is the changing information of the attribute weights. It is suggested to utilize the fuzzy theory to depict the imprecise decision information considering that the evolving and uncertain environments make the data acquisition difficult to achieve. As has been pointed out before, the IFBN can provide a prognosis and diagnosis to predict the future decision data for the DMs. However, the construction of the IFBN for decision data and the attribute weights

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45

simultaneously will increase the complexity of the computation and decrease the efficiency of decision. We assume that the decision information can be conveniently acquired from the DMs or the historical data and mainly focus on how to determine the attribute weights more practicably. The attribute weights in most static decision making methods are time-invariant and their determinations mostly focus on the solutions of the multiple programming models (Xu and Zhao 2016). The DMs rely on the swift gained attribute weights to make scientific decisions, but the computational complexity of the programming model makes it inappropriate for weight’s fast acquisition under dynamic decision making circumstances. The distribution function methods solve out the problems of timeliness and simplicity to some extent. However, such assumptions of the attribute weights violate the practical situations and may bring about the negative impact on the decision outcomes. The attributes related to a specific decision making problem are not isolated and the determination of their priority level is also not groundless, which means that general patterns or specific distributions are unavailable for the determination of the attribute weights. Each attribute has its own hierarchical influencing factors which could be independently related to the attribute with respect to the decision making environment. The changes in these factors will affect the development situations of the events and subsequently exert an effect on the importance of the attributes. This means that the significance of different attributes is evolving with the development of the events. Most of the acquisition methods of the dynamic attribute weights seldom allow for the intrinsic relationships of these influencing factors and their subsequent influence on the determination of the attribute weights. In view of this mechanism, the determination of the attribute weights should take account of the dynamic changes of the relative influencing factors. The IFBN defined above provides an appropriate tool to handle this problem. The hierarchy causal relationships in the IFBN can effectively describe the intrinsic properties of the given attributes. Meanwhile, the IFBN integrates the historical data and the experimental knowledge to update the current information of the object of study and then provides a reliable reasoning and practical prediction. From the perspective of weight determination, the IFBN can be viewed as a hybrid approach integrating the subjective methods and the objective methods together. Noting that the final result provided by the IFBN represents the maximum probability of the state in each node, it is essential to introduce a new method to transform the probability into the weight information. As a result, we will also develop a transformation method based on the normal distribution function. Generally, the attribute weights should be in accordance with their probabilities. That is, the plausibility of the attributes determines the priority relationships. This process should be implemented in two stages. The first stage is the comparisons of different attributes in terms of their maximum probabilities. The ones with high values of probabilities imply higher possibilities of occurrence in practical situations and should be paid with more weight attention. In other words, the priority of the attributes in terms of probabilities contributes to determining the weights. The second stage is a subtle adjusting process where the comparison process is conducted along the same status of the attributes. In the IFBN, the attributes may have different statuses

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featuring their own posterior probabilities. The maximum posterior probabilities determine the final states of the attributes and further influence the importance of the attributes. As a result, the priority over each status should be taken into account to identify the attribute weights. The status with high levels or large potential damages will influence the later evolution of the disasters. Accordingly, more attention should be paid to the corresponding attributes. Based on those analyses, the plausibility of the attributes in the IFBN should be converted into the weight information. We give a Gauss function-based weight determination method to convert the plausibility of attributes. Suppose that the total number of the attributes is n and let l represent the final possible statuses of the given attributes. Let Oi (i = 1, . . . , l) be the

ranking orders of attribute probabilities in terms of the l statuses. O˜ = Oδ(1) , Oδ(2) , . . . , Oδ(l) is the permutation of the ordered sets in terms of the attributes, such that Oδ( j) > Oδ( j+1) ( j = 1, 2, . . . , l − 1). The elements in each attribute set are also in descend. , l) isequal ing orders. The sum of the cardinalities of Oδ( j) ( j = 1, 2, . .   to the ˜ number of attributes. Hence, the attribute weight will be w˜ = f rank O , where   ˜ and f is a decreased function. The rank O˜ is the order of each attribute in O, adoption of the non-linear Gauss function for f is beneficial for discriminating the relationships between the orders and the weights. However, it is necessary to extend the position orders of O˜ to the symmetric interval (−n, −n − 1, . . . , 0, 1, 2, . . . , n) so as to guarantee the monotonically decreasing capability for the positive intervals. This is a bit different from the Gaussian formulation (Xu 2005). Then the ordered weights for each position in O˜ will be −(i − μ¯ n )2 1 e 2σn2 , i = 1, 2, . . . , n ωi =  2π σ p

(3.6)

where μ¯ n and σ p are the expectation and the standard deviation of a sequence of −n, −n −1, . . . , 0, 1, 2, . . . , n. The normalization form of each weight will be n ωi , i = 1, 2, . . . , n. wi = ωi / i=1 T  Finally, the attribute weight vector will be w˜ = wr (O1 ) , wr (O2 ) , . . . , wr (Ol ) , ˜ where r (Ol ) describes the relative position order of each attribute in the set O. For example, suppose that we have set four attributes in the MCDM problems: a1 , a2 , a3 and a4 . There are two status sets O1 and O2 . The priority of them is O1 > O2 . Let O1 and O2 be O1 = {a3 } and O2 = {a2 , a4 , a 1 }, respectively. The ranking order of (a3 ) (a2 ) (a4 ) (a1 ) each attribute will be r = r1 , r2 , r3 , r4 , where the subscription depicts the descending order and the superscript indicates the corresponding attributes. By means of (3.6), we can calculate the ordered weights wi (i = 1, 2, 3, 4). The subscription of wi is consistent with that of the ranking order r . Then the final attribute weight vector for the four attributes is w˜ = (w4 , w2 , w1 , w3 )T .

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3.4.2 Dynamic IFBN Decision Making Method The emergency events generally bring the risk decision making problems. The prospect theory, characterizing the risk preferences of the DMs in decision making, provides a descriptive model for risk decision making. The value function in the prospect theory can reflect the risk attitudes of the DMs in the emergency situations. It has also been successfully integrated into MCDM under risk situations (Krohling et al. 2013; Ren et al. 2016). To combine the advantages of the prospect theory and the IFBN, we will introduce a novel dynamic IFBN decision making method. The core element of the prospect theory is the value function, which is generally defined as follows (Kahneman and Tversky 1979):  v1 (x) =

xα x ≥0 −λ(−x)β x < 0

(3.7)

The value function in (3.7) indicates the attitudes of the DMs for the gain and the loss, where α, β ≥ 0, reflecting the attitudes towards the risk gains and losses (Kahneman and Tversky 1979). The parameter λ in this function is the loss-aversion coefficient (Rieger and Bui 2011), making the S-shape curve for the losses steeper than that for the gains. Similar to the expected utility theory, the weights corresponding to the positive value and the negative value are calculated by a linear value function (Tversky and Kahneman 1992): π + ( p) =

( pγ

pγ + (1 − p γ )γ )1/γ

(3.8)

( pδ

pδ + (1 − p δ )δ )1/δ

(3.9)

and π − ( p) =

respectively, where p is the probability of the positive outcome or the negative outcome. The parameters γ and δ determine the amount of the overweight or underweight and ensure the diverse weights for different outcomes, which are similar to the parameters α and β in (3.7). The experimental results suggest γ = 0.61 and δ = 0.69, implying the risk-aversion attitudes (Tversky and Kahneman 1992). Based on (3.7)–(3.9), the final prospect value of f is V ( f ) = V ( f −) + V ( f +) =

l  j=1

π− j v j (x) +

n  j=l+1

π+ j v j (x) =

n  i=1

πi vi (x)

(3.10)

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The values V ( f + ) and V ( f − ) describe the cumulative risk seeking and risk aversion effects of the DMs, respectively. The value V ( f ), known as the mixed prospect, implies the total marginal benefit for the given event (Tversky and Kahneman 1992). This value also reflects the sensitivity degree for the total losses and gains of the considered event from the perspective of the DM. Another concern is that the classical value function fails to cover the whole riskaverse behaviors of the DMs (Rieger and Bui 2011). To better capture the whole risk-aversion behaviors, Rieger and Bui (2011) defined the following exponential value function:  1 − e−αx , x ≥ 0 (3.11) v2 (x) = −λ(1 − eβx ), x < 0 However, in view of the magnitude of the IFNs, the classical value function in (3.7) may have weak discriminating ability for IFSs. As to the exponential form, it has been pointed out that the logarithmic value function can implement the psychological perceptions of wide ranges of outcomes, which may perform well in the risk-taking situation, especially for gambling decision making problems (Kirby 2011; Kontek 2011; Scholten and Read 2014). Under the intuitionistic fuzzy environment, the value function should have fine discriminating ability in the interval [−1, 1] in view of the magnitudes of the IFNs. This means that the small positive values of x will be assigned with higher prospect values, so does that for the negative values of x, which will be beneficial to the aggregation and compassion process. Since the emergency decision making features the risk pursuit, we will define a natural logarithm-based value function, so as to provide a fine discriminating ability for the prospect values. The new value function is denoted by  v3 (x) =

ln(1 + αx), x ≥0 −λln(1 − βx), x < 0

(3.12)

To depict the difference between these three value functions, we draw their graphs in Fig. 3.3. In general, we suggest the values of the parameters of v1 (x) as α = β = 0.88 and λ = 2.25 (see Fig. 3.3), which are the experimental values (Kahneman and Tversky 1979). This value function approaches to a linear form and the S-shape characteristic is not apparent. The exponential value function can explain more lossaversion behaviors of the DMs (Rieger and Bui 2011). However, considering that the magnitude of the difference between the IFNs is relatively small, we need to discriminate the subtle differences so as to calculate the risk preference. Besides, in view of the complexity and uncertainty of the emergency events, the DMs tend to show more aggressive decision behaviors, such as taking a risk decision to keep the damages to a minimum. This implies that the losses will outweigh the gains with steeper curves. When the parameters of v2 (x) and v3 (x) take the same values, we find that the changes of the logarithmic value function are more apparent than those

3.4 Dynamic Intuitionistic Fuzzy Decision Making Model

49

Fig. 3.3 Three types of value functions of prospect theory. (v1 (x) corresponds to the classical value function defined by Kahneman and Tversky (1979); v2 (x) corresponds to the exponential value function defined by Rieger and Bui (2011); v3 (x) is the value function defined in this chapter. All the functions are restricted to the x-axis interval [−1, 1]. α, β are set 0.88 for v1 (x) and 2.25 for v2 (x) and v3 (x). The three curves for each value function corresponds to λ = 1, λ = 2 and λ = 4)

of the exponential value function. The gains for positive outcomes are larger and the losses for the negative outcomes are more sensitive. More importantly, the loss aversion depicted by the new value function is stronger than that of the exponential value function or the classical value function. Hence, this value function is more suitable for the emergency decision making problems. With the aid of the new value function, we can devise a dynamic decision making approach in the framework of the prospect theory. The general procedure of the new dynamic decision making approach under intuitionistic fuzzy environment is illustrated as follows: Step 1. Set the reference points. Generally, we set the mean values as the t (k) (k = 1, 2, . . . , l) reference points. For the decision information matrices Dm×n at different periods, the reference points of the attributes c j = 1, 2, . . . , m) at ( j  

t(k) t(k) t(k) (k = 1, 2, . . . , l) are obtained by x¯ t(k) = 1/m · d1t(k) j ⊕ d2 j ⊕ . . . ⊕ dm j (k = 1, 2, . . . , l). Step 2. Calculate the dominance degrees of each alternative over others for the given attributes at the periods t(k) (k = 1, 2, . . . , l). The dominance of each alternative at the attribute c j over the reference alternative describes the deviation degrees between each alternative and the reference point at current time. Considering the deficiency and information loss of the subtraction operation for IFSs, we utilize the distance measures to describe the dominance:

50

3 The Dynamic Decision Making Method Based on the Intuitionistic …

xit(k) j

⎧   ⎨ −d x t(k) , x¯ t(k) , x < 0 ij j   = ⎩ d x t(k) , x¯ t(k) , x ≥ 0 ij j

(3.13)

where d is the distance measure of IFSs, and we take the Euclidean distance measure (Szmidt and Kacprzyk 2000):    1 n   d(A1 , A2 ) =  w j 21 + 22 + 23 2 j=1

(3.14)

          where  1 = μ A1 x j − μ A2 x j , 2 = v A1 x j − v A2 x j , and 3 = π A1 x j − π A2 x j . The distance measure can also be other widely used distance measure, such as the Hamming distance measure. These two types of distance measures share the similar mathematical properties and will not influence the final ranking results. However, we should point out that other distance measures such as the distance measure under the Hausdorff metric (Xu and Chen 2008), which does not fully take all the elements of the IFS into account, will influence the dominance result under the same situation, and subsequently, affect the final decision ranking results. Since the main focus of this chapter is not on the distance measure, the influence of different measures on the ranking results will be discussed deeply in our future work. Here we utilize the Euclidean distance measure to represent the dominance difference for the sake of simplicity and generality. Step 3. Calculate the comprehensive prospect values for the alternatives by integrating the information at different times. Firstly, we arrange xit(k) j (k = 1, 2, . . . , l) of each attribute at all the periods in (2) (l) ascending orders as xi(1) j ≤ x i j ≤ . . . ≤ x i j , where (·) is the ordered permutation in {1, 2, . . . ,l}. Secondly, we will obtain the static comprehensive prospect value matrix Vm×n vi j by utilizing the CPT formulation in (3.10): k l       − + V( f ) = V f − + V f + = π(i) v(x) + π(i) v(x) i=1

(3.15)

i=k+1

− + and π(i) are calculated by (3.8) and (3.9) with the empirical values γ = where π(i) 0.61 and δ = 0.68, respectively. The probability parameter p for each attribute can  be set by p kj = wkj / li=1 wij , j = 1, 2 . . . , n; k = 1, 2, . . . , l. The value function v(x) is calculated by (3.12). Step 4. Similar to the technique for order preference by similarity to ideal solution (TOPSIS) approach (Hwang and Yoon 1981), we calculate the closeness coefficient of each alternative. Let V + = max{vi j |1 ≤ i ≤ m} and V − = min{vi j |1 ≤ i ≤ m}. Then the closeness coefficient can be calculated by

3.4 Dynamic Intuitionistic Fuzzy Decision Making Model

  d Vi , V − ϑi = d(Vi , V + ) + d(Vi , V − )

51

(3.16)

    where d Vi , V − = Vi − V − and d Vi , V − = Vi + V − represent the distances to the ideal positive alternative and the ideal negative alternative, respectively. The optimal option is the alternative with the largest closeness coefficient. With the aid of the aggregation operator, we can fully aggregate the intuitionistic fuzzy information provided by the IFBN and the DMs without loss of information. The aggregated result will further serve for the final decision making process. The general procedures of dynamic intuitionistic fuzzy decision making under emergency circumstances are described as follows (see Fig. 3.4). The DMs first offer the attributes to be considered in the decision making problems and then analyze their inner influencing factors and the mutual relationships among them. The causal relationships of the factors facilitate the construction of the IFBN. Meanwhile, the historical information and statistical data related to the events, the given attributes and the corresponding factors are also collected to construct the IFBN. As the realtime data is imported into the IFBN, the dynamic weight information of the attributes will be updated. The IFBN provides both the decision information acquisition and the attribute weight determination. However, we only construct the IFBN model of the decision attributes for the sake of simplification and convenience. It is assumed that the dynamic decision information will be provided by the DMs or acquired from the practical situations. The determination of the attribute weights from the IFBN is the primary focus of this chapter. The IFBN produces the probabilities of different statuses for the given attributes, which will be utilized to calculate the attribute weights. The risk decision making process is responsible for evaluating the alternatives synthetically based on the weight information and the decision data. The evaluation results are then mapped to the prospect values, which reflect the risk preferences of the DMs. By aggregating the dynamic prospect values for the considered alternatives, the DMs can rank the alternatives and then get the optimal one.

3.5 Applications to Mine Emergent Accident In this section, we will discuss the emergency decision making problems of mine accidents utilizing the introduced decision making approach. The mine explosion is one of the most dangerous hazards in mine accidents. The mine explosion greatly threatens the life and safety of the workers and jeopardizes the safety production of the mines. Since the explosion accidents often occur suddenly and unexpectedly, it is not simple to predict accurately the accident and make full preparations and emergency actions in advance. Hence, the simulations of the accidents and the emergency response plans are an essential approach in disaster preparedness and appropriate responses. The quality and effectiveness of the emergency plans will directly affect

52

3 The Dynamic Decision Making Method Based on the Intuitionistic …

AƩribute iniƟlizaƟon

Data collecƟon

IFBN ConstrucƟon

Weight determinaƟon

Incorporate DMs’ data

AƩribute weight determinaƟon

Decision informaƟon acquisiƟon

AƩribute inference

InformaƟon data

Risk decision process

Ranking the prospect values

OpƟmal decision Fig. 3.4 General procedures of the dynamic intuitionistic fuzzy decision making

the later emergency actions, and thus influence the evolution of the disasters and the subsequent damages and losses. As a result, the evaluation and decision of the given emergency plans with simulations is considered necessary for the disaster management of the mine accidents. Assume that there are five emergency plans to be considered for an explosion accident in the coal mine. The DMs set the decision attributes to be the noxious gas concentration (denoted as gas), the smoke and the dust level (denoted as smoke), reducing casualty of current events (denoted as casualty), the feasibility of rescue

3.5 Applications to Mine Emergent Accident

53

operations (denoted as feasibility) and repairing facility damages caused by the emergency (denoted as facility). Based on the general evolving principle and the characteristics of the mine accidents, we can determine the parent influencing factors of the given attributes and the root factors. The ventilation condition in the emergent situations influences the concentrations of the noxious gas, the smoke and dust levels. The collapse size will give rise to the casualties and exert adverse impacts on the following rescue operations. Meanwhile, the ventilation condition, the collapse size and the facility damage in the accident are closely connected with the explosion size of the mine. We should notice that the occurrence of the primary disasters will often induce a series of secondary disasters or collateral disasters. For instance, the toxic and high-temperature flammable gas emitted in the explosion will induce the tunnel fires. This collateral disaster will result in serious damage and loss to the emergency equipment and facility or even threaten the lives and the safety of the workers and rescuers. The mine explosion may destroy the exterior environment such as surface subsidence or dam break, which will lead to the landslide, the rock-mud flow, and so on. The collateral disasters of the mine explosion will finally aggravate the primary disasters. Most importantly, it will bring more difficulty to emergency responses. As a result, the secondary disasters are the essential parts that should be carefully considered in the whole decision making process. We assume that the secondary disasters only influence the scale of the facility damages in our case for the sake of simplicity. Consequently, we can construct the IFBN of the mine accident, as shown in Fig. 3.5. The intuitionistic fuzzy conditional probabilities of the parent and children nodes in the IFBN are calculated in advance and listed in Tables 3.1, 3.2, 3.3 and 3.4. In the initial period, the emergency events occur with fewer damages and destructions, and thus, the available information for the future evolution and prediction is

Explosion size

Ventilation condition

Noxious gas

Smoke and dust

Secondary disaster

Collapse size

Reduce casualty

Rescue operation feasibility

Fig. 3.5 The IFBN of the emergency accident in a mine explosion

Damaged facility repair

54

3 The Dynamic Decision Making Method Based on the Intuitionistic …

Table 3.1 The intuitionistic fuzzy conditional probabilities of the parent nodes Explosion size

Ventilation status Good (I)

Collapse level Bad (II)

Ordinary (I)

Serious (II)

Small (I)

(0.45, 0.30)

(0.60, 0.20)

(0.55, 0.20)

(0.35, 0.26)

Massive (II)

(0.68, 0.30)

(0.50, 0.50)

(0.32, 0.26)

(0.72, 0.14)

Table 3.2 The intuitionistic fuzzy conditional probabilities for the noxious gas and the smoke and dust Ventilation status

Noxious gas level

Smoke and dust level

Acceptable (I)

Dangerous (II)

Acceptable (I)

Dangerous (II)

Good (I)

(0.72, 0.20)

(0.40, 0.30)

(0.34, 0.20)

(0.58, 0.40)

Bad (II)

(0.45, 0.30)

(0.55, 0.20)

(0.80, 0.15)

(0.34, 0.25)

Table 3.3 The intuitionistic fuzzy conditional probabilities of the casualty and the rescue operation difficulty Collapse size

Reduce casualty

Rescue operation feasibility

Low (I)

High (II)

Low (I)

High (II)

Ordinary (I)

(0.67, 0.30)

(0.45, 0.40)

(0.80, 0.10)

(0.35, 0.20)

Serious (II)

(0.53, 0.13)

(0.73, 0.15)

(0.58, 0.20)

(0.50, 0.43)

Table 3.4 The intuitionistic fuzzy conditional probabilities of the facility damage scale

Explosion size Small (I) Massive (II)

Secondary disaster

Damaged facility repair Low (I)

High (II)

Occurrence

(0.54, 0.25)

(0.48, 0.50)

None

(0.85, 0.13)

(0.25, 0.40)

Occurrence

(0.38, 0.41)

(0.72, 0.15)

None

(0.60, 0.39)

(0.42, 0.28)

not sufficient. As the appearance of the new emergent situations and chain damages, the events will turn worse. The chance of the secondary disasters triggered by the original emergent events also increases. In this process, more information referring to the emergency disasters will be available for the DMs and the cognitive indeterminacy on the evolution tendency and pattern will decrease. Suppose that the accident has gone through three periods during the involving process. The three periods correspond to the initial stage, evolving process and the developing phase of the emergent events. In each period, we can acquire the intuitionistic fuzzy probabilities of the exposition size and the secondary disasters, which are shown in Table 3.5. Based on the IFBN defined in Fig. 3.5 and the conditional probabilities in Tables 3.1, 3.2, 3.3, 3.4 and 3.5, we can calculate the final probabilities of different

3.5 Applications to Mine Emergent Accident

55

Table 3.5 The intuitionistic fuzzy probabilities of the parent nodes for different periods Periods

Explosion size

Secondary disaster

Level I

Level II

Occurrence

None

Period I

(0.80, 0.20)

(0.20, 0.10)

(0.30, 0.40)

(0.70, 0.10)

Period II

(0.44, 0.50)

(0.62, 0.31)

(0.55, 0.35)

(0.50, 0.40)

Period III

(0.20, 0.20)

(0.80, 0.15)

(0.72, 0.16)

(0.30, 0.40)

Table 3.6 The possibilities of the most possible status of the given attributes Gas

Smoke

Casualty

Feasibility

Facility

Period I

(0.4842, 0.1448)

(0.5129, 0.1051)

(0.4573, 0.0892)

(0.5185, 0.0612)

(0.5936, 0.0824)

Period II

(0.5220, 0.2696)

(0.5042, 0.2269)

(0.4706, 0.1909)

(0.5255, 0.1558)

(0.5050, 0.3072)

Period III

(0.5442, 0.1524)

(0.5014, 0.1117)

(0.5270, 0.1291)

(0.5266, 0.0687)

(0.5227, 0.1176)

statuses on the given attributes and determine the most possible status in the three periods (Table 3.6). Next, we can obtain the priority orders of the attributes on different periods, thereby calculating the weight vector of the attributes in each period. The ordered sequences for the given three periods are r1 = (5, 4, 2, 3, 1), r2 = (4, 3, 2, 1, 5) and r3 = (5, 3, 4, 1, 2), respectively. Based on (3.6), we can calculate the ordered attribute weights for each period. After that, taking the ordered sequences into account, the final weight vectors of attributes at each period are as follows: w(t1 ) = (0.0531, 0.1909, 0.1091, 0.2848, 0.3621) w(t2 ) = (0.1091, 0.1909, 0.2848, 0.3621, 0.0531) w(t3 ) = (0.1091, 0.0531, 0.2848, 0.1091, 0.3621) In the following procedure, we need to handle the dynamic intuitionistic fuzzy decision information based on the prospect theory. Let the decision evaluation matrices for the five alternative plan schemes in each period be provided by the DMs, respectively (Tables 3.6, 3.7, 3.8 and 3.9). The IFS evaluations on the noxious gas, the smoke, and dust attributes represent the certain and uncertain efficiencies of the given scheme for preventing or eliminating these hazardous factors. Similarly, the IFS values on the casualty and the facility damage attributes depict the fuzzy evaluations of the given schemes on reducing the casualty or the ability to repair the infrastructure facility. The values on the rescue operation attribute depict the fuzzy evaluations on the operation difficulty of the six schemes. We can calculate the reference points for each attribute at the three periods, and then obtain the dynamic dominance matrix of alternatives. Following Steps 1–4, we can conduct the decision making procedures and obtain the optimal alternative. To

56

3 The Dynamic Decision Making Method Based on the Intuitionistic …

Table 3.7 The intuitionistic fuzzy decision matrix D1 at period I Period I

Gas

Smoke

Casualty

Feasibility

Facility

Scheme 1

(0.5,0.3)

(0.6, 0.1)

(0.7, 0.3)

(0.7, 0.1)

(0.8, 0.2)

Scheme 2

(0.7,0.2)

(0.6, 0.2)

(0.4, 0.4)

(0.6, 0.2)

(0.7, 0.3)

Scheme 3

(0.5,0.3)

(0.7, 0.2)

(0.6, 0.3)

(0.4, 0.2)

(0.6, 0.1)

Scheme 4

(0.5,0.4)

(0.8, 0.1)

(0.4, 0.2)

(0.7, 0.2)

(0.7, 0.3)

Scheme 5

(0.7,0.3)

(0.5, 0.4)

(0.6, 0.3)

(0.6, 0.2)

(0.5, 0.1)

Table 3.8 The intuitionistic fuzzy decision matrix D2 at period II Period II

Gas

Smoke

Casualty

Feasibility

Facility

Scheme 1

(0.6, 0.3)

(0.5, 0.2)

(0.6, 0.4)

(0.8, 0.1)

(0.7, 0.3)

Scheme 2

(0.8, 0.2)

(0.5, 0.3)

(0.6, 0.4)

(0.5, 0.2)

(0.6, 0.3)

Scheme 3

(0.6, 0.1)

(0.8, 0.2)

(0.7, 0.3)

(0.4, 0.2)

(0.8, 0.1)

Scheme 4

(0.6, 0.3)

(0.6, 0.1)

(0.5, 0.4)

(0.9, 0.1)

(0.5, 0.2)

Scheme 5

(0.8, 0.1)

(0.6, 0.2)

(0.7, 0.3)

(0.5, 0.2)

(0.7, 0.1)

Table 3.9 The intuitionistic fuzzy decision matrix D3 at period III Period III

Gas

Smoke

Casualty

Feasibility

Facility

Scheme 1

(0.3,0.4)

(0.9, 0.1)

(0.8, 0.1)

(0.5, 0.5)

(0.4, 0.6)

Scheme 2

(0.7, 0.1)

(0.7, 0.3)

(0.4, 0.2)

(0.8, 0.2)

(0.3, 0.1)

Scheme 3

(0.4, 0.1)

(0.5, 0.2)

(0.8, 0.1)

(0.6, 0.2)

(0.6, 0.3)

Scheme 4

(0.8, 0.2)

(0.5, 0.1)

(0.6, 0.4)

(0.7, 0.2)

(0.7, 0.2)

Scheme 5

(0.6, 0.1)

(0.8, 0.2)

(0.7, 0.2)

(0.6, 0.3)

(0.8, 0.1)

illustrate the influence of the parameters in the value function on the final decision results, we set three groups of parameters for the value function. The first group sets experiential values with α = β = 0.88 and λ = 2.25, which are experimental values suggested in Tversky and Kahneman (1992). Additionally, we set λ = 1.25 and α = β = 0.88 in the second group to test the sensitivity to λ. To evaluate the impacts of the parameters α and β, we increase the values of α and β to 1.68 and 1.78 while the parameter λ remains the same value 2.25 in the third group. The final ranking orders of alternatives for the given parameters are shown in Fig. 3.6. The loss-aversion coefficient λ determines to what extent the expectation of losses will prevail over the gains. λ < 1 indicates that the loss is acceptable and can be attenuated while λ ≥ 1 implies that the loss is unbearable and will be amplified. Setting large values of λ indicates that we are to find the alternatives possessing the small losses on all the attributes, that is, the larger λ produces the safer alternative(s) for the DMs. This reflects the loss aversion phenomenon of risk decision. The existing researches have shown that the losses raise more attention than the gains for the DMs under risk situations, and the experimental value for this coefficient is 2.25 (Tversky

3.5 Applications to Mine Emergent Accident

57

Fig. 3.6 The orders of the alternatives on the given three groups (Group 1 α = β = 0.88 and λ = 2.25; Group 2 α = β = 0.88 and λ = 1.25; Group 3 α = 1.68, β = 0.88 and λ = 2.25)

and Kahneman 1992). This is consistent with the finding that the final results are accepted if the gains are at least twice as high as the losses (Kontek 2011). The parameters α and β determine the concave for the gains and the convex for the losses. Tversky also gave the experimental value 0.88 for these two parameters (Tversky and Kahneman 1992). From Fig. 3.6, we find that the different scenarios produce the same optimal one, i.e., Scheme 3. As a result, we should choose Scheme 3 as the final decision result. For Group 1, the values of λ affect the ranking orders of the alternative set. As λ increases, the DM hopes to find the one with fewer losses, which results in the position changes of Scheme 1. The worst alternative Scheme 2 still remains the same. When we change the values of α and β, the ranking sequence also changes subtly. The order of the optimal alternative is not affected but the inferior ones are rearranged. This change also happens in Scheme 1, which implies that the prospect values of Scheme 1 are sensitive to the parameters. From the perspective of practical situations, it corresponds to a plan that much depends on the evolution of exterior environment and the risk preferences of the DMs. The changes of the DMs’ attitudes or the events will influence the evaluation of this plan. As has been analyzed before, the parameters α and β will influence the risk attitudes of the DMs. Such influences are implemented by altering the prospect values on each criterion, which are then accumulated to affect the final results. On the contrary, the loss-aversion coefficient λ, as the name suggests, will directly amplify or diminish the prospect values of alternatives. From this perspective, the parameters α and β exert the subtle changes to the decision results while the parameter λ performs overall influences on the decision results. The ranking results also reflect this mechanism. The changes of λ arrange Scheme 1 from the second place to the last second place. The changes

58

3 The Dynamic Decision Making Method Based on the Intuitionistic …

of α and β subtly insert Scheme 1 to the middle of Scheme 4 and Scheme 5 while the other orders remain the same. Hence, it is expected that the changes of λ will rapidly produce the safest results. If we wish to analyze the detailed differences of alternatives for the final results, it is suggested to subtly adjust α and β with a proper value for λ instead of assigning large values to λ. Furthermore, we also conduct a comparative study between the fuzzy BN, the traditional BN and our method so as to illustrate its advantages. When only the membership degree is considered, the IFS reduces to Zadeh’s fuzzy set. Based on Tables 3.1, 3.2, 3.3, 3.4 and 3.5, we can get the fuzzy probabilities of the factors in the BN taking only the membership function part into account. In view of the IFS probability data provided by Tables 3.1, 3.2, 3.3, 3.4 and 3.5, it is necessary to employ the defuzzification method in preparations for the traditional BN. The probability in the crisp value form is calculated by the mean value of the intervals expressed by the IFS. That is, for an IFS probability, α p = (μ, v), the crisp value is obtained by the mean point of its interval p = u+(1−v) . For convenience, we employ 2 the IFS probability data for the first status for each factor in the BN. Similarly, we can also calculate the final conditional possibilities for the considered attributes and determine their weights according to the procedures defined above. The most possible status and the corresponding probability values obtained by the fuzzy BN and the traditional BN are listed in Table 3.10. Comparing these results with those in Table 3.11, we find that the weight priority information differs a lot. The traditional BN presents the rescue operation feasibility attribute as the most important one for all the three periods. Besides, the priority orders of the three periods change little, especially in Period II and Period III. The crisp values fail to take the fuzzy information and the hard computation method with coarse granularity both result in the insensitivity to the subtle changes of the input information. As to the fuzzy BN results, the priority order for Period I is the same as that deduced from our method. For Period II and Period III, however, the most significant attribute presented by the BN differs greatly. The fuzzy BN deems the reduce casualty attribute as the most important one in the attribute sets for the following two periods, and accordingly, this Table 3.10 The comparative probability results of the most possible status for each attribute under the given three periods Comparative results

Period I Fuzzy BN

Period II BN

Fuzzy BN

Period III BN

Fuzzy BN

BN

Gas

0.4842

0.686

0.5220

0.693

0.5442

0.698

Smoke

0.5129

0.635

0.5042

0.623

0.5014

0.613

Casualty

0.4573

0.69

0.4962

0.691

0.5270

0.692

Feasibility

0.5185

0.799

0.5225

0.787

0.5226

0.778

Facility

0.5936

0.752

0.5050

0.629

0.5227

0.550

Note The italic fonts indicate the second level of the corresponding attributes (e.g., it means the high level of the casualties for the casualty attribute and it means the high damages for the facilities attribute); the normal ones indicate the first level of the corresponding attributes

3.5 Applications to Mine Emergent Accident

59

attribute will be assigned the highest weight for the later decision making process. On the contrary, our method judges the rescue operation feasibility and the damaged facility repair attributes as the most significant factors for Period II and Period III, respectively. Also, we notice that the two attributes with the highest priorities in our method stand in the second position under the fuzzy BN situation. With all the membership degree, the non-membership degree and the indeterminacy degree being taken into account, the IFS BN has fine granularity and recognition capability, and thus is sensitive to the subtle changes in the original information. This may account for the subtle difference and the similarity in the attribute priority orders deduced from the fuzzy BN and our methodology. In a word, the risk decision making method that we discuss in this chapter takes more information into account and can get a more reliable result for dynamic decision making problems. Unlike the dynamic fuzzy decision making method defined by Chen and Tu (2015) in the framework of IFSs, which aims at allocating dynamic indeterminacy information over time, we focus on utilizing the fuzzy information available to dynamically update the parameters related to decision making, especially the dynamic attribute weights. Another typical dynamic decision making method under uncertain environment is the dynamic intuitionistic fuzzy multi-attribute decision making method based on the special operator (Xu and Yager 2008). However, as has been pointed out before, the terminology “dynamic” signifies the information collected at different periods and the corresponding weight information is assigned subjectively or predetermined by special distributions, which may fail to handle the real-time information and reflect the practical causal relationship. Since the dynamic retrieval and renewal of the attribute weights play an influential role in real-time decision making, we develop the IFBN to retrieve and update the practical weights with both the subjective judgements and the objective evaluations considered. The attribute weights play an important role in information fusion and MCDM problems. The accuracy and reliability of the weights will influence the quality of the aggregated results, which will exert an impact on the judgements of the DMs. The methods for determining weights have been fully studied for static decision making problems. However, most of the methods cannot be applied to solve the dynamic decision making problems. Some concerns involve that the computational complexity of the current methods may be not suitable for the dynamic situations in view of the high requirement of timeliness. Other concerns involve whether the current methods and models can describe the real dynamic situations and provide the practical weight information. The existing weight-determination methods for dynamic intuitionistic fuzzy decision making problems depend on the mathematical functions or distributions of time series. The calculated dynamic weights are actually the period weights rather than the dynamic attribute weights changing over time in the real-world situations. It is indisputable that the distribution functions of current methods assure a rapid and simple way to obtain the dynamic weight information of the given attributes. However, we should also realize that the practical situations will never strictly comply with these mathematical functions and distributions, especially when dealing with emergency events. From this perspective, the convenience and conciseness of the current dynamic weight determination methods are also the

60

3 The Dynamic Decision Making Method Based on the Intuitionistic …

defects that impede their application to practical problems. The integrated prospect theory also guarantees the risk-seeking and risk-averse decision behaviors as other uncertain decision making methods (Chen and Tu 2015; Krohling et al. 2013; Wang et al. 2015).

3.6 Remarks The attribute weights play an important role in information fusion and MCDM problems. The accuracy and reliability of the weights will influence the quality of the aggregated results, which will exert an impact on the judgements of the DMs. Most of the existing methods for determining weights cannot be applied to solve the dynamic decision making problems. This chapter introduces the IFBN and the corresponding decision making method to solve this problem. The advantages of the IFBN are apparent. It can not only depict the intrinsic hierarchical structure of the given decision making problems but also provide a dynamic reasoning tool for the complex circumstances. The quantitative depiction of the causal relationships of the attributes ensures the reliability and quality of the final weight information. Consequently, the trustworthy weights of the attributes are beneficial to the dynamic decision results. In spite of the excellence of the studied decision making model in this chapter, some limitations should also be pointed out. First, the probabilities provided by the IFBN are non-normative, which will hinder the efficient process of the information fusion. As a result, it is still imperative to study this issue so as to provide a normalized standard without any loss of information. Secondly, we have utilized the transformation in information processing. One is the weight determination from the IFBN and the other is the dominance calculation from the prospect theory. Since the original input decision matrices and the information are presented with the IFNs, we naturally hope that the whole data in the information processing should take the form of IFNs so as to fully utilize the information. However, in view of the theoretical limitation of the IFS theory (mainly the deficiencies on the normalization and the subtraction laws), we have taken the transformation methods to overcome this deficiency, which are: (1) For the weight-determination problem, we have provided a compromise approach aiming at establishing the mapping relation between the intuitionistic fuzzy probabilities of the attributes and their weight values in the form of real numbers; (2) For the dominance calculation for the prospect theory, we have adopted the widely used distance measures in lots of literature to describe the dominance between the IFSs. It is expected that the new dynamic decision making approach introduced in this chapter will provide a novel solution for the emergency decision making problems and have more potential applications in the future.

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References Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 Bali O, Dagdeviren M, Gumus S (2015) An integrated dynamic intuitionistic fuzzy MADM approach for personnel promotion problem. Kybernetes 44(10):1422–1436 Chen LH, Tu CC (2015) Time-validating-based Atanassov’s intuitionistic fuzzy decision making. IEEE Trans Fuzzy Syst 23(4):743–756 Chen TY, Li CH (2010) Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis. Inf Sci 180(21):4207–4222 Ciungu LC, Rieˇcan B (2010) Representation theorem for probabilities on IFS-events. Inf Sci 180(5):793–798 Constantinou AC, Fenton N, Neil M (2016) Integrating expert knowledge with data in Bayesian networks: preserving data-driven expectations when the expert variables remain unobserved. Expert Syst Appl 56:197–208 Duda RO, Hart PE, Stork DG (2001) Pattern classification. Wiley, New York Grzegorzewski P (2013) On some basic concepts in probability of IF-events. Inf Sci 232:411–418 Grzegorzewski P, Mrówka E (2002) Probability of intuitionistic fuzzy events. Physica-Verlag HD, Heidelberg Hao ZN, Xu ZS, Zhao H, Fujita H (2018) A dynamic weight determination approach based on the intuitionistic fuzzy bayesian network and its application to emergency decision making. IEEE Trans Fuzzy Syst 26(4):1893–1907 Hao ZN, Xu ZS, Zhao H, Zhang R (2017) Novel intuitionistic fuzzy decision making models in the framework of decision field theory. Inf Fusion 33:57–70 Holmes DE, Jain LC (2008) Introduction to Bayesian networks. In: Holmes DE, Jain LC (eds) Innovations in Bayesian networks: theory and applications. Springer, Berlin, pp 1–5 Hwang CL, Yoon K (1981) Methods for multiple attribute decision making. Multiple attribute decision making: methods and applications a state-of-the-art survey. Springer, Berlin, pp 58–191 Jitwasinkul B, Hadikusumo BHW, Memon AQ (2016) A Bayesian belief network model of organizational factors for improving safe work behaviors in Thai construction industry. Saf Sci 82:264–273 Kahneman D, Tversky A (1979) Prospect theory—analysis of decision under risk. Econometrica 47(2):263–291 Karl-Rudolf K (2007) Special models and applications. Introduction to Bayesian statistics. Springer, Berlin, pp 129–192 Kirby KN (2011) An empirical assessment of the form of utility functions. J Exp Psychol Learn Mem Cogn 37(2):461–476 Kontek K (2011) On mental transformations. J Neurosci Psychol Econ 4(4):235–253 Kou G, Lin CS (2014) A cosine maximization method for the priority vector derivation in AHP. Eur J Oper Res 235(1):225–232 Krohling RA, Pacheco AGC, Siviero ALT (2013) IF-TODIM: an intuitionistic fuzzy TODIM to multi-criteria decision making. Knowl-Based Syst 53:142–146 Li GX, Kou G, Peng Y (2015) Dynamic fuzzy multiple criteria decision making for performance evaluation. Technol Econ Dev Econ 21(5):705–719 Nabavi-Kerizi SH, Abadi M, Kabir E (2010) A PSO-based weighting method for linear combination of neural networks. Comput Electr Eng 36(5):886–894 Ren PJ, Xu ZS, Gou XJ (2016) Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl Soft Comput 42:246–259 Rieger MO, Bui T (2011) Too risk-averse for prospect theory? Modern Econ 2(4):691–700 Saaty TL (1977) Scaling method for priorities in hierarchical structures. J Math Psychol 15(3):234– 281 Sadiq R, Tesfamariam S (2007) Probability density functions based weights for ordered weighted averaging (OWA) operators: an example of water quality indices. Eur J Oper Res 182(3):1350– 1368

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Scholten M, Read D (2014) Prospect theory and the “forgotten” fourfold pattern of risk preferences. J Risk Uncertain 48(1):67–83 Shahriari B, Swersky K, Wang ZY, Adams RP, de Freitas N (2016) Taking the human out of the loop: a review of Bayesian optimization. Proc IEEE 104(1):148–175 Shannon CE, Weaver W (1947) The mathematical theory of communication. University of Illinois Press, Urbana Srinivas V, Shocker AD (1973) Linear-programming techniques for multidimensional analysis of preferences. Psychometrika 38(3):337–369 Su Z, Xu ZS, Liu SS (2016) Probability distribution based weights for weighted arithmetic aggregation operators. Fuzzy Optim Decis Making 15(2):177–193 Su ZX, Chen MY, Xia GP, Wang L (2011) An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making. Expert Syst Appl 38(12):15286–15295 Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst 114(3):505–518 Tversky A, Kahneman D (1992) Advances in prospect-theory—cumulative representation of uncertainty. J Risk Uncertain 5(4):297–323 Wang S, Huang G, Baetz BW (2015) An inexact probabilistic-possibilistic optimization framework for flood management in a hybrid uncertain environment. IEEE Trans Fuzzy Syst 23(4):897–908 Wang TC, Lee HD (2009) Developing a fuzzy TOPSIS approach based on subjective weights and objective weights. Expert Syst Appl 36(5):8980–8985 Weiss-Cohen L, Konstantinidis E, Speekenbrink M, Harvey N (2016) Incorporating conflicting descriptions into decisions from experience. Organ Behav Hum Decis Process 135:55–69 Xia MM, Xu ZS (2012) Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Inf Fusion 13(1):31–47 Xu ZS (2005) An overview of methods for determining OWA weights. Int J Intell Syst 20(8):843–865 Xu ZS (2007) Models for multiple attribute decision making with intuitionistic fuzzy information. Int J Uncertain Fuzziness Knowl-Based Syst 15(3):285–297 Xu ZS (2008) On multi-period multi-attribute decision making. Knowl-Based Syst 21(2):164–171 Xu ZS (2010) A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making. Group Decis Negot 19(1):57–76 Xu ZS (2011) Approaches to multi-stage multi-attribute group decision making. Int J Inf Technol Decis Making 10(1):121–146 Xu ZS (2012) Intuitionistic fuzzy multiattribute decision making: an interactive method. IEEE Trans Fuzzy Syst 20(3):514–525 Xu ZS, Chen J (2007, December 2–5) Binomial distribution based approach to deriving time series weights. Paper presented at the IEEE International Conference on Industrial Engineering and Engineering Management, Singapore Xu ZS, Chen J (2008) An overview of distance and similarity measures of intuitionistic fuzzy sets. Int J Uncertain Fuzziness Knowl-Based Syst 16(4):529–555 Xu ZS, Yager RR (2008) Dynamic intuitionistic fuzzy multi-attribute decision making. Int J Approx Reason 48(1):246–262 Xu ZS, Zhao N (2016) Information fusion for intuitionistic fuzzy decision making: an overview. Inf Fusion 28:10–23 Zhang L, Zhou WD (2011) Sparse ensembles using weighted combination methods based on linear programming. Pattern Recogn 44(1):97–106 Zhao H, Xu ZS (2016) Intuitionistic fuzzy multi-attribute decision making with ideal-point-based method and correlation measure. J Intell Fuzzy Syst 30(2):747–757

Chapter 4

Novel Intuitionistic Fuzzy Decision Making Models in the Framework of Decision Field Theory

The intuitionistic fuzzy decision making problems have gained great popularity recently. In this chapter, we will introduce another type of dynamic decision making methods under the intuitionistic fuzzy environment, that is, the process-based decision making method. The main contents come from Hao et al. (2017)

4.1 Decision Making Methods Based on the Aggregation Operators Most of the current decision making methods under intuitionistic fuzzy environment depend on various aggregation operators that provide collective IFNs of the alternatives to be ranked. Such collective information only depicts the overall characteristics of the alternatives but ignores the detailed contrasts among them. Most important of all, the current decision making procedure is not in accordance with the DMs’ decision thoughts. Since the IFS theory was first proposed by Atanassov (1986), various intuitionistic fuzzy decision making methods have been proposed. A vital objective of the existing intuitionistic fuzzy decision making methods is how to acquire the fuzzy information without any loss of information. On that account, many intuitionistic fuzzy information aggregation operators, such as the IFWA operator (Xu 2007a), the IFWG operator (Xu and Yager 2006) and other improved operators (Xu 2010; Xu and Xia 2011; Xu and Yager 2011; Zhao et al. 2010) have successfully achieved the defuzzification goals of IFSs and constituted the theoretical foundations for their applications to multi-criteria decision making (MCDM) problems. Given that the relationships of the attributes and the complexities of the decision making environments, the information aggregation operators above are incompetent to various practical MCDM problems. For instance, the IFWA operator assumes that the loss of one criterion can be compensated by the gain of other criteria, which is © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9_4

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not always practically satisfied, especially when the properties of the criteria are of great difference. As for the IFWG operator, this drawback has been circumvented to some extent. When utilizing the IFWG operator, it evokes a new problem that the prominent criteria would have a decisive effect on the DMs (DMs)’ judgments. Besides, none of these two operators ever consider the interrelationship between the criteria of the alternatives. In consequence, many researchers have improved and developed a series of new aggregation operators to broaden the applications of IFSs. Thereafter, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator (Xu 2007a) and the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator (Xu and Yager 2006) have been defined, which can perceive the significance of the ordered positions of IFNs instead of the IFNs themselves alone. The intuitionistic fuzzy hybrid averaging (IFHA) operator (Xu 2007a) and the intuitionistic fuzzy hybrid geometric (IFHG) operator (Xu and Yager 2006) have also been proposed, which not only take the importance degrees of IFNs for the local criteria into account but also consider those of the ordered positions of the criteria. Meanwhile, some generalized intuitionistic fuzzy aggregation operators have been put forward, such as the generalized intuitionistic fuzzy weighted averaging (GIFWA) operator, the generalized intuitionistic fuzzy ordered weighted averaging (GIFOWA) operator and the generalized intuitionistic fuzzy hybrid averaging (GIFHA) operator (Zhao et al. 2010), each of which has an additional parameter controlling the power of the argument values. Unfortunately, few of the mentioned aggregation operators have considered the interrelationships between the criteria. Xu and Yager (2011) developed the intuitionistic fuzzy Bonferroni mean (IFBM) operator, which has the aptitude for capturing the interrelationships among different attributes. Based on the Choquet integral, the intuitionistic fuzzy correlated averaging (IFCA) operator and the intuitionistic fuzzy correlated geometric (IFCG) operator have been proposed (Xu 2010). Both of them can finely reflect not only the importance degrees of the elements themselves or the ordered positions but also their correlation information. Moreover, Xu and Xia (2011) developed the induced intuitionistic fuzzy correlated averaging (IIFCA) operator in the case of the ordered-inducing IFSs. Recently, in order to develop an idempotent aggregation operator and avoid the methodological shortcomings of the traditional intuitionistic fuzzy aggregation operators, some new intuitionistic fuzzy aggregation operators have been developed in the framework of the Dempster-Shafer theory (Dymova and Sevastjanov 2010, 2012; Xu and Xia 2011). The fuzzy set constructed from the ignorance functions and the corresponding aggregation technology have also been developed (Barrenechea et al. 2014). It is worthwhile to notice that the precision and the characteristics of the aforementioned aggregation operators have a great influence on the decision results. Such influence can be summarized into two aspects: one is the precision of the aggregation operators and the other is the way that the alternatives are compared and deliberated. As for the precision part, the aggregation operators undoubtedly have provided a powerful tool to aggregate the intuitionistic fuzzy criteria of the alternatives without any intermediate defuzzification, and therefore assure the precision and reliability of the final decision results. The development of the intuitionistic fuzzy aggregation

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operators has reflected this intention from another aspect. Reviewing the characteristics of the various intuitionistic fuzzy aggregation operators, we find that the exterior expressions or interior theoretical foundations vary a lot but the essential principle remains the same. They aim at providing the overall intuitionistic fuzzy information for each alternative from different angles. This mechanism in turn brings forward the intuitionistic fuzzy decision making problems. Briefly speaking, the forenamed operators aggregate the intuitionistic fuzzy information of each alternative along the attribute dimension and the aggregated results represent the “average” description of current alternatives. Accordingly, the DMs can compare each option and obtain the optimal result. However, the practical decision making processes are complicated and greatly influenced by both emotional and psychological factors. The traditional processes above reflect how the intuitionistic fuzzy information of the alternatives is collectively fused but fails to reflect the practical reasoning processes intuitively. Additionally, these processes fail to reveal the potential decision behaviors of the DMs. In reality, the DMs take the complex strategies when confronted with the MCDM problems. They habitually evaluate different alternatives in terms of their attributes, which can be represented by the comprehensive aggregated information. This is the necessary procedure but not the whole decision making process. More often, a more detailed advantage and disadvantage analysis based on each attribute of different alternatives should also be conducted. In view of such a reasoning process, the decision making method based on the aggregation operators is apparently an inferior strategy, and the aggregation operator is a one-off calculation process. Once the fuzzy information is completely aggregated, the duty of the operators is also done. When confronted with the strategic decision making problems or emergent decision making problems, the DMs are usually required to fully weigh up the pros and cons of each alternative and the following actions. In this situation, the one-dimension overall aggregated evaluation results are not appropriate since the aggregation process implies a compensation mechanism where the weakness of a certain attribute is compensated by the superiority of other attributes. The current lack of the dynamical comparison methods may make the final results unreasonable in cognitive decision making. As a result, the theoretical development of the aggregation operators is not enough to well improve the reliability of the decision results. Instead, a transformation of the decision making concept is urgent and necessary in solving the intuitionistic fuzzy MCDM problems.

4.2 The Intuitionistic Fuzzy Decision Making Model Based on the DFT To fulfill such cognitive decision goals, especially when the decision making information is uncertain and intuitionistic fuzzy information, it is crucial to build a specialized intuitionistic fuzzy decision making model that not only takes full use of the intuitionistic fuzzy information of the alternatives but also reflects the real decision

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reasoning process. The cognitive decision making model sets a novel example of describing the human’s decision making behavior, aiming at “explaining behavior as it arises from the collaboration of basic psychological processes, rather than through the satisfaction of rational axioms” (Hotaling et al. 2010). So far the decision field theory (DFT) has become one of the most successful cognitive decision making model, and has been widely applied to uncertain decision making, decision making under time pressure, and preference reversal study (Berkowitsch et al. 2014; Busemeyer and Diederich 2002; Busemeyer and Townsend 1993; Roe et al. 2001), regardless of its blank in the field of group decision making. The DFT concentrates on the detailed contrasts among the alternatives in terms of the whole attributes, which accumulate divergently in accordance with competitions. Inspired by this theory, we utilize the DFT to develop a suitable decision making model for the intuitionistic fuzzy MCDM problems. Moreover, such a decision making model will circumvent the demerit of the traditional intuitionistic fuzzy decision making processes and shape a new conception for dealing with intuitionistic fuzzy decision making information.

4.2.1 Introduction of the DFT Busemeyer (1985) pointed out that when confronted with a decision making problem, the DMs usually utilize the following strategy by intuition: The detailed merits and demerits of the options are firstly analyzed and compared, based on which an anticipation and evaluation of the possible consequences of all the options are then considered deeply. It is worth noticing that such an assessment procedure is a complicated and time-consuming process rather than an immediate accomplishment. Hence, the preferences for the alternatives dynamically evolve over decision time. The optimal option is the one whose preference surpasses all the others over the comparing period. To formalize this normal decision making process, Busemeyer proposed the decision field theory (DFT) (Busemeyer and Townsend 1993), which provides a mathematical model for cognitive decision making behaviors under uncertain environment. Figure 4.1 illustrates the basic conceptions of the DFT. The preliminary comparison procedure is conducted for each alternative in terms of the given attributes. Then the weighted differences constitute the expected valence of each option, which is further fed into the decision system. The decision system is responsible for comparing and integrating the valences over the time steps where a momentary preference state for the alternative is calculated. The accumulated preference results stimulate the motor system to produce the final decision result under the given decision making condition. Usually, the decision making rules can be classified into two types: the interior deliberation time restriction and the exterior threshold condition. One of the most significant characteristics of the DFT is that the DM’s preferences for the alternatives dynamically evolve with the decision time, which are also the results of the continual aggregation of the evaluation among the alternative sets (Roe et al. 2001). Let the vector P(t) represent the preference state of all alternatives at

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Attribute1 Positive Contrasts

…...

W1

Option1 AttributeN

…….

Attribute1 Negative Contrasts

Preferences

W1

Decision System

Motor System

Results

WN

…...

Optionm WN

AttributeN

Fig. 4.1 The illustration for the basic conceptions of the DFT

the time t, then the preference state P(t + h) at the next time t + h can be described by the following linear difference equation (Busemeyer and Diederich 2002): P(t + h) = SP + V (t + h)

(4.1)

where h represents an arbitrary small time step and when h → 0, P(t + h) will approximate the diffusion process. The matrix S is the so-called feedback matrix and the vector V is the momentary valence of the alternatives. The formula (4.1) actually interprets the core decision making mechanism of the DFT. The preference state, the valence for the alternatives and the feedback matrix parameters all contribute to this mathematical model. In what follows, the two remaining parameters V and S will be discussed in detail. In the deliberation process, different options will be compared on each attribute based on their expected values, which can be described by the so-called valence. Roe et al. (2001) defined a valence vector as a composition of the three elements: V = CMW (t)

(4.2)

where the matrix M is the decision matrix containing the detailed evaluation information of all the options about the attributes. The vector W is the weight vector of the attributes in accordance with the decision matrix M. In consequence, the expression MW (t) aggregates the weighted averaging information of each alternative, which is similar to the IFWA operator. The parameter C is the contrast matrix procuring the comparison process of the weighted information MW (t). For the decision making problems with n alternatives, 1 , for i = j. the contrast matrix C is computed as Cii = 1 and Cij = n−1

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It is worth noticing that the weight vector in (4.2) evolves over the decision period because of the DM’s shifting attention to different choices. In fact, one of the most crucial assumptions of the DFT is that the attention weights of the attributes dynamically change with time and satisfy the stationary process (Busemeyer 1985). A simplified assumption of this process is that the DM cares for only one attribute at the time t and during the next time step the attention will be shifted to another attribute with some fixed probability. However, it can be proven that the mean and covariance vectors of W(t) are constant over time (Busemeyer and Diederich 2002). Hence, both the valence and the corresponding preference state vector are stochastic. Composed of self-connection and interconnections among the options, the feedback matrix S indicates the memorizing effect and the competitive influences among alternatives. The diagonal elements determine the degrees of the previous preferences and influence the current state of a given alternative. Meanwhile, the off-diagonal elements provide the competitive influences between the alternatives. The setting principle of this competitive strength is determined by the lateral effect (Shepard 1987), which can be calculated by a decreased function of the distances between the alternatives. The feedback matrix S is assumed to be symmetric and the eigenvalues of S should be less than one in magnitude so as to make the decision making model stable (Roe et al. 2001). The exemplary matrix S will be discussed in the subsequent section. At present, the parameters of the core algorithm for the DFT are interpreted completely. The final decision results are concluded when the given decision time reaches or the accumulated preference values reach the threshold criteria.

4.2.2 Integrated Intuition Fuzzy Decision Making Model Based on the DFT The fore-mentioned analysis has pointed out that the current intuitionistic fuzzy decision making theory relies on the information aggregation operator. Nevertheless, most of the operators seldom take the deliberation process into account and evaluate the fuzzy information contrast of different alternatives in detail, which may influence the reliability of the decision results. The theoretical research on the aggregation operators for IFSs has been sufficiently developed, which makes it possible to integrate IFSs into the DFT so as to provide a new decision making model for the uncertain decision making problems. Reviewing the essential mathematical model based on the DFT, we have to handle the following problems under the intuitionistic fuzzy environment. Firstly, it is the way that we aggregate the intuitionistic fuzzy decision matrix M without defuzzification. Since the original information is provided by IFNs, it is natural and reasonable to provide the final evaluation result in the IFN form. For this reason, the decision making model based on the DFT should be adapted in terms of the aggregation operators and the aggregation laws for IFNs. Secondly, the decision making model

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based on the DFT features a contrast comparison between the alternatives on each attribute, which should also be achieved without any loss of information for the decision making problems with IFNs. Finally, the determination of the feedback matrix and the corresponding parameters will also be reconsidered under intuitionistic fuzzy environment. In this section, we will elaborate on the solutions and the integration procedures for the intuitionistic fuzzy decision making model based on the DFT. Fuzzy valence. Similar to the original valence in the DFT, the fuzzy valence V f reflects the weighted anticipated values of different alternatives in terms of their contrast comparison: V f = C ⊗ M  ⊗ W (t)

(4.3)

where M  is the decision matrix in fuzzy environment and the other parameters are the same as those in (4.2). The contrast matrix C rebuilds the comparing mechanism between the options, which inevitably involves the subtraction operator. The subtraction for the IFS is defined as follows: Definition 4.1 (Atanassov 2012). The subtraction operation for two given IFSs A and B is defined as: AB = {(x, μAB (x), vAB (x))|xE}

(4.4)

where μAB

⎧ ⎨ μA (x)−μB (x) if μA (x) ≥ μB (x) and vA (x) ≤ vB (x) 1−μB (x) and vA (x)πB (x) ≤ πA (x)vB (x) = ⎩ 0 others

and vAB

⎧ ⎨ vA (x) if μA (x) ≥ μB (x) and vA (x) ≤ vB (x) = vB (x) and vA (x)πB (x) ≤ πA (x)vB (x) ⎩ 1 others

We notice that the subtraction operation is only valid under a series of strict conditions, they abominably result in the loss of intuitionistic fuzzy information. Hence, the difference between two IFSs interpreted by this operator is partial, which will undoubtedly jeopardize the decision results once they are applied to solve the intuitionistic fuzzy decision making problems. Since the differences expressed by the subtraction results play a key role in the subsequent contrast process, any information loss accidentally or methodologically will result in the unexpected misleading decision results. Considering the methodological imperfection of the subtraction laws for IFSs, it is wiser to develop a new subtraction law or utilize a substitutional operation. Hence, we define a compromise subtraction operation to counteract the deficiency of the original subtraction laws.

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The essential idea of the valence in (4.3) is to extract the complete contrast information between the considered alternative and the other ones. As for the real number decision matrix, the simple subtraction operator can easily achieve the goal. In respect of IFSs, the subtraction operation fails to provide full contrast information. The inevitable information loss resulted from the deficiency of the subtraction law for IFSs in turn will make it hard to achieve the comparison process. As a result, the intuitionistic fuzzy decision making model based on the DFT will be infeasible. The differences between the options nevertheless can still be depicted by the distance measures despite its undirected characteristics. Inspired by this idea, we can devise a compromising scheme for the subtraction operation of IFSs to calculate the full fuzzy valence information of each alternative. Definition 4.2 Let α1 and α2 be two IFNs, the contrast subtraction operation can be defined as follows: ⎧ ⎨ d (α1 , α2 ) if α1 > α2 (4.5) α1 contrast α2 = −d (α1 , α2 ) if α1 < α2 ⎩ 0 if α1 = α2 where d (α1 , α2 ) is the distance measure between α1 and α2 defined in (3.14) and the comparisons between the IFNs comply with the order in Definition 1.2. This scheme makes the difference information directed and compensates the information loss of the original subtraction laws for IFSs. In this chapter, we will employ the distance measure of (3.14) with λ = 2. This operation presents the contrast information between the given option and the others in terms of each attribute. A collective contrast description for different alternatives in terms of each attribute is further gained by means of the weighted aggregation operator, which indicates the valences of the alternative sets. Feedback matrix. The feedback matrix in the intuitionistic fuzzy DFT remains the same interpretation. The diagonal elements determine the degrees of the previous preferences and influence the current state of a given alternative. Generally, it is suggested to set the influence strength between zero and one to provide a partial memory and limited decoy effect (Roe et al. 2001). Meanwhile, the off-diagonal elements provide the competitive influences between the alternatives. The negative value will ensure that the superior option surpasses the inferior while the zero values imply none competition existing between the options. The most widely used decreased function is the Gaussian function. One formation to calculate the feedback matrix can be expressed as follows: S = I − ϕ · e−δ · D

2

(4.6)

where I is the identity matrix and D is the distance degree matrix between the alternatives acquired by means of (2.12), ϕ and δ are the parameters that determine the strength of the feedback. Berkowitsch et al. (2014) suggested that δ should be

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limited to the values between 0.01 and 1000, while the value of ϕ should be between 0 and 1. The principle for setting this parameter under intuitionistic fuzzy environment depends on the problem we handle. The value of the parameter δ determines the discriminable capability. If the alternative sets are full of much more similar ones, then it is suggested to set δ with high values, while for the distinguishable alternative sets, it should be set with a lower one. Correspondingly, to provide a competitive influence between the options under intuitionistic fuzzy environment, we suggest that the values of ϕ should be smaller than one. However, there are no general rules for determining the parameter. The perfect values for these two parameters are the ones deduced from the practical experiment and application. Fuzzy preferences. The fuzzy preference contains the preference for each alternative. Since we adopt a compromise scheme based on the distance measures, the fuzzy valence is constituted by real numbers. Apparently, this preference resulted from the dynamic deliberation process updates over time. The positive preference state represents an approach that has a tendency to a certain option while the negative preference state indicates an off-approach tendency at current moment. At length, the largest one in fuzzy preference vector matches the optimal result. Similar to the original decision making model based on the DFT, two primary decision making rules are offered: an inner deliberation time criterion and an exterior threshold standard for the fuzzy preferences. The important decision making problems generally cost more time while the long deliberation time in turn assures the reliability of the decision results. As to the threshold standard, higher threshold will take more decision time, which can be viewed as a higher standard and decision goal for the given decision making problem. On the contrary, the smaller threshold generally takes less deliberating time and gets an immediate result whose precision and reliability are greatly in question. Generally, the threshold values should be set by the variance of the valence in case that the decision making process stops too soon or the excessive calculation wastes time and submerges the appropriate alternative (see Appendix A for this parameter). Truth is that the threshold differs from different decision making problems whereas the sufficient deliberation time will always be rewarded with a reliable result. From this aspect, the decision making rules should be adjusted according to the practical decision making problems. In the normal situations, the time decision making rules should be easier to be utilized. In respect of emergent circumstances where the time is precious, the threshold decision making criterion is then recommended. The probability of choosing options in the choice set is discussed in detail in Appendix A.

4.2.3 Integrated Intuition Fuzzy Group Decision Making Model Based on the DFT One of the most important applications of the IFSs is the intuitionistic fuzzy group decision making problems. The traditional decision making model with the DFT seldom touches the group decision making problems. However, it is more common and

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indispensable for group decision making problems in reality. Even though the decision making model with the DFT provides an excellent formalized cognitive decision making tool, it is still too complicated and difficult to simulate the group decision making process simultaneously and comprehensively. Fortunately, the theoretical development of the intuitionistic information aggregation operators effectuates the integration of different DMs without information loss. Motivated by this conception, we develop an intuitionistic fuzzy group decision making technology by using the DFT. It is suggested to aggregate intuitionistic fuzzy information from different DMs along each attribute and obtain a collective group intuitionistic fuzzy decision making matrix. This procedure can be depicted as follows: As for a group decision making problem with m considered attributes (G = {G1 , G2 , . . . , Gm }) and l DMs (E = {e1 , e2 , . . . , el }), we can determine the importance degrees of the DMs in advance, expressed by the weight vector ξ =  l T (ξ1 , ξ2 , . . . , ξl ) ξ ≥ 0, j=1 ξj = 1 . Similarly, the weight vector for the attributes    can be denoted as ω = (ω1 , ω2 , . . . , ωm )T ω ≥ 0, m j=1 ωj = 1 . Let the intu  itionistic fuzzy decision matrix from different DMs be Rk = rij(k) , where m×n   (k) rij(k) = μ(k) (i = 1, 2, . . . , m; j = 1, 2, . . . , n), rij(k) represents the intuitionij , vij istic fuzzy information on the attribute Gj of kth DM for the alternative Yi , μ(k) ij and

vij(k) represent the membership function and the non-membership function for the alternative Yi on the attribute Gj respectively, satisfying (k) (k) (k) μ(k) ij ∈ [0, 1], vij ∈ [0, 1], μij + vij ≤ 1, i = 1, 2, . . . , m;

j = 1, 2, . . . , n; k = 1, 2, . . . , l. And then, the group intuitionistic fuzzy decision matrix is acquired by means of the IFHA operator (1.9) or the IFHG operator (1.10). The next critical issue of the group decision making problems is determining the priority weights for the attributes. The weight information of the considered attributes may be given in advance. Nevertheless, it is more often ambiguous to determine the attribute weights rigorously. One reason accounting for this is that the priority for the assorted or even the related attributes are hard to be resolved. This also explains why the fuzzification of the weight information is more and more urgent and popular. Another crucial reason is that the selected decision making attributes will be judged and weighted differently and diversely in practical situation, where the DMs may be experts from various domains. In most cases, we have to accept such a fact that only a maximum compromise consensus is achieved for the weight information of the attributes. This also implies the difference between the individual decision making problems and the group decision making problems. Generally, the trade-off weighted results for the attributes can be constructed as follows Xu and Cai (2012):

4.2 The Intuitionistic Fuzzy Decision Making Model Based …

73

⎧ ⎪ ⎪ ⎨

{ωi ≥ ωj } δi > 0 {ωi − ωj ≥ δi }

= ⎪ 0 ≤ δi ≤ 1 {ωi ≥ δi ωj } ⎪ ⎩ {δi ≤ ωi ≤ δi + εi } 0 ≤ δi < δi + εi ≤ 1

(4.7)

Based on the group intuitionistic fuzzy decision matrix and the weight information rules, an optimization model for the weight ω is devised to gain the optimal weight information for all the attributes. The detailed theoretical principles and calculation procedures for (M-4.1) can refer to Xu’s work (2007b): ⎧ n  ⎪ max(s (ω)) = ωj sij ⎪ i ⎪ ⎪ j=1 ⎨ (M-4.1) s.t. ω =

⎪ n ⎪  ⎪ ⎪ ⎩ ω ≥ 0, ωj = 1 j=1

where sij = s(dij ) = μij − vij , i = 1, 2, . . . , m; j = 1, 2, . . . , n. At present, we totally obtain the group intuitionistic fuzzy decision matrix and the priority weights for the attributes. The valence vector for the alternatives and the feedback matrix can be subsequently acquired, so does the preference state of each option. Based upon the comparison method, we can determine the optimal alternative in the preference vector. Hereafter, the choice probability can also be calculated. In summary, the implementation procedure for the intuitionistic fuzzy decision making model based on the DFT is illustrated in Fig. 4.2. The DMs first analyze the decision making problems and set the attributes to be evaluated in advance. Then an intuitionistic fuzzy assessment for all the alternatives on each attribute is conducted. If all the DMs provide their individual intuitionistic fuzzy decision evaluation matrix, then an essential aggregation for the group information will also be implemented. Besides, as for the group decision making problems, it is more often to solve the weight optimization model and get the weights for the attributes. Based on the intuitionistic fuzzy decision matrix, we can acquire the difference measures between the options, which will further be fed into the intuitionistic fuzzy DFT for feedback matrix. Next, a weighted aggregated contrast information comparison over respective attributes among the alternative sets constitutes the valence vector for the alternatives. The self-influence and inner-competition among the alternatives with respect to their discrepancy are mapped into the feedback matrix. Subsequently, the preference state for each alternative updates and accumulates deliberating over time, describing the DMs’ growing and decaying preferences for the alternatives dynamically. This deliberation process proceeds until the DMs believe a proper deliberation time is spent or the preferences reach the threshold values given in advance. When the deliberation process ceases, the final choice is produced for the DMs. The deliberation time and the threshold criteria rely on the realistic problems we handle. Meanwhile, the determination of these two rules is also closely related with

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4 Novel Intuitionistic Fuzzy Decision Making Models …

Fig. 4.2 The general implementation procedure for the intuitionistic fuzzy decision making model based on the DFT

Attributes determination Attributes evaluation Group Intuitionistic Fuzzy matrix

Intuitionistic Fuzzy matrix

Attributes weight W

Decision results

Distance measures D Model Parameters (C,S)

IFDFT

Decision rules (t/θ )

Valence vector V(t) Preference state P(t)

the attribute type and amount as well as the inner valence characteristics. However, the consensus for the generalized rules for the threshold values is especially unavailable. This is also the reason why we sometimes sacrifice the precious time for the reliability of the decision results. In any event, the motive of the intuitionistic fuzzy decision making model based on the DFT endeavors to refine the deliberation mechanism of the intuitionistic fuzzy decision making problems, where the current decision making methods depend on the merely disposable one-dimensional information aggregation of the alternatives.

4.3 The Application of the Intuitionistic Fuzzy Decision Making Model Based on the DFT 4.3.1 Validation and Comparison for the Cognitive Decision Behavior The practical decision making problems are partly influenced by the context that provided by the set of the options. This means that the decision results sometimes may violate the results derived from the rational decision making model. The DFT has successfully explained three typical decision phenomena in the practical decision

4.3 The Application of the Intuitionistic Fuzzy Decision Making …

75

making problems, such as the attraction effect (Huber and Puto 1983; Simonson and Tversky 1992; Wedell 1991), the similarity effect (Sjoberg 1977; Tversky 1972) and the compromise effect (Simonson 1989). Such decision phenomena reflecting the decision psychology of the DMs also occur in the intuitionistic fuzzy decision making problems. In this section, we introduce three typical decision scenarios to test the validity and effectiveness of the intuitionistic fuzzy decision making model based on the DFT. Similarity effect. The similarity effect implies that the introduction of a competitive option to the dissimilar choice sets will reduce the choice probability of the similar one rather than the dissimilar one (Tversky and Simonson 1993). To illustrate such a decision behavior, we exemplify the case of the decision making problems for selecting the investment countries. Let the alternative sets A, B and C represent three different countries to be determined. E and S represent the economy and safety properties of the given countries. The country A and the country B are two dissimilar options in the choice space (see Table 4.1). If the weights for the two attributes are equal, the choice probabilities for the option A and the option B are equal too. To make the comparing Tprocess for binary choices more apparently, we let W = WE WQ = (0.49 0.51) be the weight vector of the attributes. In this situation, the option C is dominant in the economy attribute but dominated in the safety attribute. Hence, the competition between these two options is fierce. Meanwhile, the option C is dissimilar with neither the option A nor the option C. The competition between the option B and the other two options is relatively weaker. For convenience, the feedback matrix S can be manually set as: ⎛

⎞ 0.95 −0.001 −0.033 S = ⎝ −0.001 0.95 −0.001 ⎠ −0.033 −0.001 0.95 The prediction for the probabilities of choosing different options is shown in Fig. 4.3. For binary choices, the probabilities for choosing the option A and the option B are P(A|{A, B}) = 0.5148 and P(B|{A, B}) = 0.4852 after a full deliberation time. When the option C is introduced, the probability of choosing the country A will decrease (P(A|{A, B, C}) = 0.3869, P(B|{A, B, C}) = 0.2684), which is a result of the so-called similarity effect. The choice probability of the country B decreases more apparently in this situation, which indicates that the DM pays more attention to the two similar alternatives while the dissimilar alternative B is out of consideration to some extent. The decision results acquired by the aggregation operators are listed in Table 4.2. Table 4.1 The alternative sets for the investment countries represented by IFNs for the similarity effect

Similarity effect

Economy

Safety

Counrty A

(0.2, 0.4)

(0.7, 0.2)

Country B

(0.7, 0.2)

(0.2, 0.4)

Country C

(0.15, 0.3)

(0.73, 0.23)

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4 Novel Intuitionistic Fuzzy Decision Making Models … Similarity Effect

0.7

PA3 PB3 PA2 PB2

Choice Probability

0.6

0.5

0.4

0.3

0.2

0.1

50

100

150

200

250

300

350

400

450

500

Deliberation Time

Fig. 4.3 The intuitionistic fuzzy DFT predictions for the similarity effect. (PA2 and PB2 indicate the probabilities of choosing the option A and the option B in the binary choice set. PA3 and PB3 indicate the probabilities of choosing the option A and the option B in the ternary choice set)

Table 4.2 Decision results acquired by the aggregation operators under the similarity effects

Aggregated IFNs

IFHA

IFWA

Country A

(0.5053, 0.2848)

(0.5149, 0.2809)

Country B

(0.5053, 0.2848)

(0.5053, 0.2848)

Country C

(0.4837, 0.2968)

(0.4915, 0.2948)

For binary choices situation, the decision results obtained by the IFHA operator imply that the country A and the country B are equally important while those obtained by the IFWA operator indicates that the country A is superior to the county B, which also accords with the intuitionistic fuzzy DFT decision results. For the decision making problems with ternary choices, the decision results derived by the IFHA and IFWA operators are A = B > C and A > B > C, respectively. From the traditional intuitionistic fuzzy decision results, we can find the considered competitive alternative won’t exert an influence on the original decision results of the binary choices sets. The research work on decision psychology points out that such rational results are impractical (Busemeyer and Townsend 1993; Simonson and Tversky 1992; Tversky and Simonson 1993). Both the IFHA and the IFWA operators fail to predict and reflect the similarity effect in cognitive decision behaviors, and the improved IFHA operator is not that sensitive to the tiny change of the attribute weights as the traditional IFWA operator does.

4.3 The Application of the Intuitionistic Fuzzy Decision Making … Table 4.3 The alternative sets for investment countries represented by IFNs for the attraction effect

Attraction effect

77

Economy

Safety

Counrty A

(0.2, 0.4)

(0.7, 0.2)

Country B

(0.7, 0.2)

(0.2, 0.4)

Country C

(0.15, 0.45)

(0.65, 0.25)

As a result, the intuitionistic decision making model based on the DFT can not only fully utilize the intuitionistic information of the alternatives but also successfully predict the decision behaviors with respect to the similarity effect. Attraction effect. The attraction effect means that the new introduced alternative, which is dominated by an original choice set, will boost the choosing probability of the dominant alternative (Huber and Puto 1983; Simonson and Tversky 1992; Wedell 1991). To illustrate this decision behavior, we adjust the alternative C in Table 4.1 and get the new intuitionistic fuzzy decision making information for the country investment problems (Table 4.3). The option C is similar to the option A and completely dissimilar to the option B, but it is dominated by the option A. The feedback matrix is the same as that in similarity effect and the decision results are shown in Fig. 4.4. The decision results for binary choice set are the same as the aforementioned ones. For the ternary choice set, the considered alternative C is dominated by the alternative A on both attributes. The intuitionistic fuzzy DFT successfully predicts Attarction Effect

0.7

PA3 PB3 PA2 PB2

Choice Probability

0.6

0.5

0.4

0.3

0.2

0.1

50

100

150

200

250

300

350

400

450

500

Deliberation Time

Fig. 4.4 The intuitionistic fuzzy DFT predictions for the attraction effect. (PA2 and PB2 indicate the probabilities of choosing the option A and the option B in the binary choice set. PA3 and PB3 indicate the probabilities of choosing the option A and the option B in the ternary choice set)

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4 Novel Intuitionistic Fuzzy Decision Making Models …

the change of the choice probability for this situation. The probability of choosing the county A increases largely in contrast with the decreasing choosing probability of the county C in the ternary choice set (.., P(B|{A, B, C}) = 0.2542). Since it is hard to discriminate and determine the best one between the option A and the option B, the introduction of a competitive option C dominated by the option A will make the DM tend to weigh up these two options, and correspondingly the probability of choosing the alternative A will be larger than that in the binary choice set. The decision results derived by the IFHA and IFWA operators are A = B > C and A > B > C respectively. Similarly, the result of the IFWA operator is consistent with our decision making model while the IFHA operator is not sensitive to the tiny difference between the weights. It is worth noticing that none of these two aggregation operators consider the context effect in the choice set and predict the decision behavior result from the attraction effect. Compromise effect. The compromise effect is actually an empirical finding but also a quite natural phenomenon in decision making problems. It depicts the subtle and interesting transformation of decision behaviors of the DMs under the situation where a new alternative is introduced between the current two competing extreme options (Simonson 1989; Simonson and Tversky 1992). Briefly speaking, the DMs attempt to resolve the optimal one between two competitive alternatives. However, when the third compromise option is added into the two extreme alternatives, the DMs always prefer to choose the new compromise one. To illustrate such a decision behavior, we take the country C as the third alternative between the two extremely different alternatives (see Table 4.4). Since the option C can be treated as the compromise alternative between the country A and the county B, the competitions between the country C and the other two countries are relatively equal. As a result, the feedback matrix can be adjusted to: ⎛ ⎞ 0.95 −0.001 −0.033 S = ⎝ −0.001 0.95 −0.033 ⎠ −0.033 −0.033 0.95 The probabilities of choosing different countries are shown in Fig. 4.5. If the available choices are limited to the option A and the option B, then it is embarrassing to distinguish the most optimal one especially when the given attributes are weighted equally. However, when the compromise route C is added in the option sets, the situation will be completely different. Our decision making model predicts the probability of selecting the option A, B and C as P(A|{A, B, C}) = 0.1909, P(B|{A, B, C}) = Table 4.4 The alternative sets for investment countries represented by IFNs for the compromise effect

Compromise effect

Economy

Safety

Counrty A

(0.2, 0.4)

(0.7, 0.2)

Country B

(0.7, 0.2)

(0.2, 0.4)

Country C

(0.5, 0.3)

(0.5, 0.3)

4.3 The Application of the Intuitionistic Fuzzy Decision Making …

79

Compromise Effect

0.7

PA3 PB3 PC3

0.6

Choice Probability

0.5

0.4

0.3

0.2

0.1

0

50

100

150

200

250

300

350

400

450

500

Deliberation Time

Fig. 4.5 The intuitionistic fuzzy DFT predictions for the compromise effect

0.1742 and P(C|{A, B, C}) = 0.6349. For all the practical purposes, the DM under such circumstances favors selecting the country B as the optimal one, which is actually also a compromise process of a comprehensive comparison between different countries on their attributes. In this case, the probability of choosing country A is also larger than that of the country B. However, if we adopt the traditional intuitionistic fuzzy decision making model based on the aggregation operators, we can get the same results as aforementioned (A = B > C for the IFHA operator and A > B > C for the IFWA operator). Rationally speaking, such a rank result is methodologically correct. However, under emergent circumstances or strategic situations, the aggressive decision pattern will result in unpredictable consequences. To make a reasonable and reliable decision results, the DM actually takes full consideration of the context of the choice sets and make comparisons between the alternatives rather than simply rely on the one-off information aggregation techniques. In view of the above analysis, the decision making model based on the intuitionistic fuzzy DFT can not only fully take advantage of the intuitionistic fuzzy information of the alternatives but also successfully predict the common decision behaviors in reality. Most important of all, it is capable of describing all three phenomena above within a single theory.

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4 Novel Intuitionistic Fuzzy Decision Making Models …

4.3.2 Application to the Decision Making of the Optimal Investment Country Recently, China has launched the “One Belt, One Road” (OBOR) conceptual investment strategies, which is consisted of the New Silk Road Economic Belt and the 21st-Century Maritime Silk Road. The OBOR will call together almost 80 market countries and developing countries and account for nearly sixty percent of the global totals. All provinces of China actively participate in this strategy, raising a new hot oversea investing trend. Firms and enterprises are actively seeking infrastructure investment and construction opportunity in the member countries of OBOR. A most successful case is the expansion of Chinese high-speed trains’ oversea expansion, which achieves a series of deals in Turkey, Indonesia and Russian. However, in spite of the excellent investment platforms that the OBOR provides, the investors should carefully evaluate the risk of the investing countries and carefully determine their next actions. The decision making problems under such circumstances are literally challenging in that lots of uncertain culture and humane property factors are involved, which are also not only tough but also ambiguous to quantify. Fortunately, the intuitionistic fuzzy theory provides a potential quantified and descriptive method for these decision making problems. Considering the aforementioned merits of the intuitionistic fuzzy DFT, we will exemplify its application to the OBOR investing country selection problems. The imperfect transportation and industrial infrastructure of the emerging market countries in southern Asia will be the potential investing focus. An imperative and inevitable concern for the investors and the enterprises is that most of the above countries are troubled by domestic political stability, financial risks and other safety problems, which will put the interests of the investors at risk. Most of the information available for these attributes is uncertain and is hard to quantify, which hinders the evaluation and the decision making process for the DMs. The intuitionistic fuzzy decision making theory provides an effective tool to deal with the vagueness and uncertainty. Suppose that the enterprise decides to invest one of the five countries of southern Asia, Sri Lanka, Vietnam, Indonesia, Myanmar and Thailand. After a comprehensive survey and evaluation, the key properties related to the investing strategy includes the safety environment, the country’s political stability, the local infrastructure conditions, and the credit risks. Based on the data and information provided by the report from the Economist Intelligence Unit (2015), the intuitionistic fuzzy evaluations of the aforementioned four countries in terms of the considered attributes can be given in Table 4.5. To describe the various weighted attributes, we provide an attribute weight matrix, as shown in Table 4.6. In the intuitionistic fuzzy decision making model based on the DFT, we let ϕ and δ of (4.6) be 0.05 and 10, respectively, and accordingly, the feedback matrix S can be easily calculated by means of  the distance measures between the alternatives. The 1 i=j . We set the threshold value to 3.5 and contrast matrix C is Cij = −0.25 i = j i=1,..,5 j=1,..,5

4.3 The Application of the Intuitionistic Fuzzy Decision Making …

81

Table 4.5 The intuitionistic fuzzy decision matrix of the given five countries Country

Safety

Political stability

Local infrastructure

Credit risks

Sri Lanka

(0.60, 0.20)

(0.37, 0.10)

(0.55, 0.18)

(0.51, 0.11)

Vietnam

(0.48, 0.27)

(0.52, 0.15)

(0.44, 0.10)

(0.41, 0.20)

Indonesia

(0.30, 0.35)

(0.68, 0.25)

(0.36, 0.20)

(0.60, 0.15)

Myanmar

(0.58, 0.12)

(0.32, 0.30)

(0.25, 0.13)

(0.55, 0.25)

Thailand

(0.72, 0.15)

(0.60, 0.15)

(0.30, 0.15)

(0.48, 0.40)

Table 4.6 Four weighted situations for the given attributes Situations

Safety

Political stability

Local infrastructure

Credit risks

W1

0.40

0.11

0.38

0.11

W2

0.35

0.18

0.07

0.40

W3

0.10

0.11

0.38

0.41

W4

0.25

0.38

0.12

0.25

run the simulations 6000 times. The corresponding preference probabilities for each alternative under different circumstances are listed in Table 4.7. The optimal choices under the four weight situations are Indonesia, Sri Lanka, Indonesia and Sri Lanka, respectively. To make the decision results more accurate and reliable, it is often suggested to adopt the group decision strategy. Suppose that four DMs give their evaluations of the given four counties in terms of five attributes: safety, political stability, local infrastructure level, credit risks, and the regulation and law environment (see Tables 4.8, 4.9, 4.10 and 4.11). The importance degrees of the four DMs can be expressed by the weight vector ξ = (0.3, 0.2, 0.3, 0.2)T . Employing the IFHA operator in (1.9), we can aggregate all the individual intuitionistic fuzzy decision matrices Gk (k = 1, 2, 3, 4) into the collective intuitionistic fuzzy decision matrix G (see Table 4.12). It is worth noticing that an essential issue of the group decision making problems is the determination of the weight information for the decision attributes. Assume that the DMs discuss over all the attributes related to the decision targets and finally reach a consensus for the attributes’ weight information, depicted by a group of linear inequality : Table 4.7 The preference probability of the four countries under different circumstances Probability

Sri Lanka

Vietnam

Indonesia

Myanmar

Thailand

W1

0.2066

0.1734

0.3124

0.1269

0.1807

W2

0.3342

0.1789

0.1996

0.1385

0.1487

W3

0.2233

0.2126

0.2570

0.1052

0.2019

W4

0.2492

0.1876

0.2269

0.1439

0.1924

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4 Novel Intuitionistic Fuzzy Decision Making Models …

Table 4.8 The group intuitionistic fuzzy decision matrix (G1 ) of the four routes Options

Properties Safety

Political stability

Local infrastructure

Credit risks

Regulations and laws

Sri Lanka

(0.4, 0.5)

(0.5, 0.2)

(0.6, 0.2)

(0.8, 0.1)

(0.5, 0.3)

Vietnam

(0.6, 0.2)

(0.7, 0.2)

(0.3, 0.4)

(0.5, 0.1)

(0.6, 0.2)

Indonesia

(0.7, 0.3)

(0.7, 0.1)

(0.5, 0.5)

(0.3, 0.2)

(0.7, 0.3)

Myanmar

(0.3, 0.4)

(0.85, 0.1)

(0.6, 0.1)

(0.4, 0.3)

(0.9, 0.1)

Table 4.9 The group intuitionistic fuzzy decision matrix (G2 ) of the four routes Options

Properties Safety

Political stability

Local infrastructure

Credit risks

Regulations and laws

(0.5, 0.3)

(0.6, 0.1)

(0.7, 0.3)

(0.7, 0.1)

(0.8, 0.2)

Vietnam

(0.7, 0.2)

(0.6, 0.2)

(0.4, 0.4)

(0.6, 0.2)

(0.7, 0.3)

Indonesia

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.3)

(0.4, 0.2)

(0.6, 0.1)

Myanmar

(0.5, 0.4)

(0.8, 0.1)

(0.4, 0.2)

(0.7, 0.2)

(0.7, 0.3)

Sri Lanka

Table 4.10 The group intuitionistic fuzzy decision matrix (G3 ) of the four routes Options

Properties Safety

Political stability

Local infrastructure

Credit risks

Regulations and laws

Sri Lanka

(0.6, 0.3)

(0.5, 0.2)

(0.6, 0.4)

(0.8, 0.1)

(0.7, 0.3)

Vietnam

(0.8, 0.2)

(0.5, 0.3)

(0.6, 0.4)

(0.5, 0.2)

(0.6, 0.3)

Indonesia

(0.6, 0.1)

(0.8, 0.2)

(0.7, 0.3)

(0.4, 0.2)

(0.8, 0.1)

Myanmar

(0.6, 0.3)

(0.6, 0.1)

(0.5, 0.4)

(0.9, 0.1)

(0.5, 0.2)

Table 4.11 The group intuitionistic fuzzy decision matrix (G4 ) of the four routes Options

Properties Safety

Political stability

Local infrastructure

Credit risks

Regulations and laws

Sri Lanka

(0.3, 0.4)

(0.9, 0.1)

(0.8, 0.1)

(0.5, 0.5)

(0.4, 0.6)

Vietnam

(0.7, 0.1)

(0.7, 0.3)

(0.4, 0.2)

(0.8, 0.2)

(0.3, 0.1)

Indonesia

(0.4, 0.1)

(0.5, 0.2)

(0.8, 0.1)

(0.6, 0.2)

(0.6, 0.3)

Myanmar

(0.8, 0.2)

(0.5, 0.1)

(0.6, 0.4)

(0.7, 0.2)

(0.7, 0.2)

4.3 The Application of the Intuitionistic Fuzzy Decision Making …

83

Table 4.12 The collective intuitionistic fuzzy group decision matrix (G) of the four routes Options

Properties Safety

Political stability

Local infrastructure

Credit risks

Regulations and laws

Sri Lanka

(0.448, 0.373)

(0.569, 0.161)

(0.650, 0.260)

(0.724, 0.177)

(0.566, 0.354)

Vietnam

(0.713, 0.186)

(0.626, 0.237)

(0.440, 0.375)

(0.560, 0.162)

(0.512, 0.213)

Indonesia

(0.548, 0.195)

(0.688, 0.173)

(0.638, 0.324)

(0.398, 0.200)

(0.679, 0.195)

Myanmar

(0.514, 0.336)

(0.682, 0.100)

(0.535, 0.274)

(0.669, 0.192)

(0.701, 0.197)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

W1 ≤ 0.3 0.1 ≤ W2 ≤ 0.2 0.2 ≤ W3 ≤ 0.5 W5 ≤ 0.6

= ⎪ − W2 ≥ W4 − W5 W 3 ⎪ ⎪ ⎪ ⎪ W4 ≥ W1 ⎪ ⎪ ⎪ ⎪ W − W1 ≤ 0.1 ⎪ 3 ⎪ ⎩ 0.1 ≤ W4 ≤ 0.3 n Utilizing the model (M-4.1), satisfying j Wj = 1 (j = 1, 2, 3, 4, 5). we could obtain the optimal weight information W = (0.2113, 0.1801, 0.2315, 0.2618, 0.1153)T for the attributes. The parameters in feedback matrix are set to ϕ = 0.12 and δ = 18, respectively. As a result, the feedback matrix is: ⎛

0.8500 ⎜ −0.1173 S=⎜ ⎝ −0.1054 −0.1363

−0.1173 0.8500 −0.1205 −0.1205

−0.1054 −0.1205 0.8500 −0.1205

⎞ −0.1363 −0.1205 ⎟ ⎟ −0.1205 ⎠ 0.8500

Meanwhile, the threshold can be set as 3.8 for this scenario. After 6000 times’ simulations, the preference probability vector for the alternatives is P = (0.2306, 0.2045, 0.2983, 0.2666)T . As a result, the priority order of the investment is Indonesia, Myanmar, Sri Lanka and Vietnam. The optimal country is Indonesia, which is consistent with the risk evaluation result of The Economist Intelligence Unit. Meanwhile, the rank result derived by the IFWA operator is Myanmar, Sri Lanka, Indonesia and Vietnam. As has been analyzed in the case study in Sect. 4.1, the decision mechanism is different from the intuitionistic fuzzy decision making models based on the traditional aggregation operators. From this aspect, it is not appropriate to judge which result is right or wrong. The results of the decision making model offer a decision assistant support for practical problems. We can only evaluate the result and judge which one is more suitable. The intuitionistic fuzzy DFT model is better for cognitive decision making problems and can well depict the practical

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4 Novel Intuitionistic Fuzzy Decision Making Models …

decision behaviors of the DMs, and thus is suitable for such kind of strategic decision making problems. Especially considering the political risk of Sri Lanka and the great infrastructure risk of Myanmar in reality, our optimal result is reliable for the DMs.

4.4 Remarks In this chapter, we have reviewed the development of the intuitionistic fuzzy aggregation operators and analyzed the mechanism of the current intuitionistic fuzzy decision making problems. To solve the deficiencies of the existing methods, this chapter introduces the intuitionistic fuzzy DFT and discusses its applications. The advantages of the intuitionistic fuzzy DFT are apparent in three aspects: Firstly, it has been pointed out that the dependence on the one-dimension aggregation of the intuitionistic fuzzy information alone will influence the reliability and accuracy of the optimal results. From another aspect, such methodological demerit is also a compromise strategy for the limitation and restriction of the partial subtraction and division operational laws. The developed intuitionistic fuzzy decision making model in the framework of DFT shifts the conception of intuitionistic fuzzy decision making from single information aggregation to dynamic comparing and reasoning process. The intuitionistic fuzzy information is fully utilized in each comparison process instead of a mere information source for the aggregation operators. Secondly, the context offered by the given alternative sets is seldom considered in the current intuitionistic fuzzy decision making problems. Hence, the decision results sometimes may violate the results derived from the rational decision theory. Most of the decision paradigms under the intuitionistic fuzzy environment are reliable only if the DMs are rational. Such deficiencies will impair the decision quality and application value of intuitionistic fuzzy decision theory under the emergent and strategic circumstances. The effectiveness of the IFSs for describing uncertain and vague information and the superiority of the DFT in dealing with cognitive decision making problems are combined simultaneously to provide a more reliable decision result. Finally, the group decision making model with uncertain weight information is also developed to make the decision results more scientific and reliable. In the framework of the intuitionistic fuzzy DFT, the final decision making criteria are not merely a reflection of the overall aggregation of all the intuitionistic information with respect to their attributes. Instead, the reasoning and deliberating processes are conducted simultaneously. The accumulated preferences provided by the competition among the alternatives are finally utilized to determine the optimal option. The demerit of one-time information aggregation and decision making are circumvented in the new decision making models. Most importantly, the new decision making models can predict and explain the common decision behaviors in cognitive decision making problems, which is seldom discussed and studied under intuition fuzzy environment. The results of the validation and comparison have revealed that the traditional intuitionistic fuzzy decision theory is incapable of discriminating the context information in choice sets. The influence of the considered competitive alternative is

4.4 Remarks

85

completely ignored, not to mention the prediction and explanation of the attraction effect, the similarity effect, and the compromise effect. Another interesting discovery is that the improved aggregation operator IFHA is less sensitive to the changes in the weight information and may perform worse than the simple IFWA operator in the decision making process. Such superiority makes the new intuitionistic fuzzy DFT capable of dealing with the emergent decision making problems, major policy decision problems, and other uncertain cognitive decision making problems.

References Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 Atanassov KT (2012) On intuitionistic fuzzy sets theory. Studies in Fuzziness and Soft Computing, vol 283. Springer Barrenechea E, Fernandez J, Pagola M, Chiclana F, Bustince H (2014) Construction of intervalvalued fuzzy preference relations from ignorance functions and fuzzy preference relations application to decision making. Knowl-Based Syst 58:33–44 Berkowitsch NAJ, Scheibehenne B, Rieskamp J (2014) Rigorously testing multialternative decision field theory against random utility models. J Exper Psychol-General 143(3):1331–1348 Busemeyer JR (1985) Decision making under uncertainty: a comparison of simple scalability, fixedsample, and sequential-sampling models. J Exper Psychol Learn Memory Cogn 11(3):538–564 Busemeyer JR, Diederich A (2002) Survey of decision field theory. Math Soc Sci 43(3):345–370 Busemeyer JR, Townsend JT (1993) Decision field theory: a dynamic-cognitive approach to decision making in an uncertain environment. Psychol Rev 100(3):432–459 Dymova L, Sevastjanov P (2010) An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect. Knowl-Based Syst 23(8):772–782 Dymova L, Sevastjanov P (2012) The operations on intuitionistic fuzzy values in the framework of Dempster-Shafer theory. Knowl-Based Syst 35:132–143 Hao ZN, Xu ZS, Zhao H, Zhang R (2017) Novel intuitionistic fuzzy decision making models in the framework of decision field theory. Inf Fusion 33:57–70 Hotaling JM, Busemeyer JR, Li JY (2010) Theoretical developments in decision field theory: comment on Tsetsos, Usher, and Chater. Psychol Rev 117(4):1294–1298 Huber J, Puto C (1983) Market boundaries and product choice: Illustrating attraction and substitution effects. J Consum Res 31–44 Roe RM, Busemeyer JR, Townsend JT (2001) Multialternative decision field theory: a dynamic connectionist model of decision making. Psychol Rev 108(2):370–392 Shepard RN (1987) Toward a universal law of generalization for psychological science. Science 237(4820):1317–1323 Simonson I (1989) Choice based on reasons: the case of attraction and compromise effects. J Consum Res 158–174 Simonson I, Tversky A (1992) Choice in context: Tradeoff contrast and extremeness aversion. J Market Res 29(3):281 Sjoberg L (1977) Choice frequency and similarity. Scand J Psychol 18(2):103–115 Tversky A (1972) Elimination by aspects—theory of choice. Psychol Rev 79(4):281–299 Tversky A, Simonson I (1993) Context-dependent preferences. Manag Sci 39(10):1179–1189 Unit TEI (2015) Prospects and challenges on China’s “one belt, one road”: a risk assessment report. http://www.eiu.com/public/topical_report.aspx?campaignid=OneBeltOneRoad. Accessed 13 Mar 2016 Wedell DH (1991) Distinguishing among models of contextually induced preference reversals. J Exper Psychol Learn Memory Cogn 17(4):767

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Xu ZS (2007a) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15(6):1179–1187 Xu ZS (2007b) Models for multiple attribute decision making with intuitionistic fuzzy information. Int J Uncertain Fuzziness Knowl-Based Syst 15(3):285–297 Xu ZS (2010) Choquet integrals of weighted intuitionistic fuzzy information. Inf Sci 180(5):726– 736 Xu ZS, Cai XQ (2012) Intuitionistic fuzzy information aggregation. Springer, Berlin Xu ZS, Xia MM (2011) Induced generalized intuitionistic fuzzy operators. Knowl-Based Syst 24(2):197–209 Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J General Syst 35(4):417–433 Xu ZS, Yager RR (2011) Intuitionistic fuzzy Bonferroni means. IEEE Trans Syst Man Cybern Part B Cybern 41(2):568–578 Zhao H, Xu ZS, Ni MF, Liu SS (2010) Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 25(1):1–30

Chapter 5

The Decision Making Method Under the Probabilistic and Cognitive Environment

In the previous chapters, we have presented some novel multi-attribute decision making methods under the intuitionistic fuzzy environment, which also indicate the excellence of the intuitionistic fuzzy set in the decision making problems. The randomness and imprecision widely exist in the real-world problems. In many practical decision making problems, it is also frequent to deal with these two types of uncertainty simultaneously. Although the IFS is expert at describing the cognitive uncertainty of the DMs, it is slightly insufficient in these cases. To describe the aleatory and epistemic uncertainty in a single framework and take more information into account, in this chapter, we will introduce a new extension of the IFS and the related decision making methods (Hao et al. 2017).

5.1 Motivations and Background Uncertainty widely exists in daily practical problems. When dealing with the decision making problems, such as risk evaluations, intelligent computations or engineering applications. One of the key issues is building mathematical models for the indeterminate problems. Even though we have the chance to get abundant data in the big-data time, it is still difficult to eliminate the uncertainty completely. Very often in practice, such massive data sets available are comprised of uncertain information. As a result, the modeling of uncertainty has become a hot academic issue. For the risk evaluation and decision making problems, one of the common problems is the epistemic uncertainty resulting from incomplete knowledge about the system. To describe this cognitive indeterminacy and vagueness, Zadeh (1965) proposed the concept of fuzzy set. Later, the fuzzy sets have been extended to different types for various applications, including the intuitionistic fuzzy sets (IFS) (Atanassov 1986) and the type-2 fuzzy set (T2FS) (Zadeh 1975). The IFS is equipollent to the IVFS from the mathematical point of view, which introduces membership function, non-membership function, and hesitance function to describe the vagueness © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9_5

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and uncertainty. The T2FS (or its extension type-n fuzzy set Miyamoto 2005) treats the membership function as a fuzzy set. From this perspective, the IFS, where the membership part contains a set of crisp interval values, is a special case of the T2FS. These extensions focus on how to describe the uncertainty degree comprehensively with the general membership degree. However, it has been pointed out that the difficulty in establishing membership degree lies in handling a set of possible values instead of margin errors or some possibility distribution values. Then Torra (2010) proposed another extension of the fuzzy set—the hesitant fuzzy set (HFS), in which the membership degree consists of several possible values reflecting the epistemic certainty but the epistemic uncertainty degree is ignored. Accordingly, Zhu et al. (2012) proposed an extension of the HFS–dual hesitant fuzzy set (DHFS), where both the membership and non-membership degrees contain a set of possible values. Furthermore, all the fuzzy set, the IVFS and the IFS can be treated as the particular cases of the DHFS (Zhu and Xu 2014; Zhu et al. 2012). The DHFS, by comparison, is able to reflect the gradual epistemic uncertainty to the ill-known objects more granularly. The fuzzy set and its extensions are the efficient tools to represent the uncertain information provided by the DMs. However, they are weak in modeling the aleatory uncertainty in terms of the statistical uncertainty. Probabilistic approaches are prevalent in handling the problems involving both aleatory and epistemic uncertainties but are not always effective in the situations where the considered problems are subject to the epistemic uncertainty (Skalna et al. 2015). The coexistence of randomness and imprecision in the real-world problems makes the researchers attempt to incorporate the probability theory into the fuzzy set theory. These works can be roughly organized into three categories, (i) introducing the probabilistic theory and making it as a knowledge representation tool in the fuzzy set, (ii) integrating the probability information into the fuzzy aggregation process, and (iii) the hybrid propagation methods generating random fuzzy values. The representative work of the first case is the fuzzy set in the framework of Dempster-Shafer theory (Dymova and Sevastjanov 2012; Sevastjanov and Dymova 2015; Yager and Alajlan 2015; Yen 1990). This unified theory facilitates the representation of various types of uncertain information in a formal mathematical framework (Yager and Alajlan 2015). However, the belief interval in this case was treated non-probabilistic and might lead to probabilistic information loss. In the second case, the immediate probability conception was introduced into the fuzzy decision making (Merigo 2010; Wei and Merigó 2012). The immediate probability transforms the probability information to the attitudinal characters of the DMs and is treated as parts of the weighted information in the aggregation operator. In the third case, the fuzzy values are generated by stochastic simulations combined with nonlinear programming (Baudrit et al. 2006; Li et al. 2009; Skalna et al. 2015). Note that the cloud model (Li et al. 2009) assumed that the membership values were generated by incorporating random values with stable bias, we classify this theory into the third case as well. To be more specific, these methods above utilize the probabilistic distributions to produce the corresponding fuzzy values for practical applications. A detailed summary of these methods is listed in Table 5.1.

5.1 Motivations and Background

89

Table 5.1 The commonly used solutions to fuzzy sets under probabilistic environment Fuzzy sets theory under probabilistic fuzzy environment

Principles

Fuzzy values in the framework of Dempster-Shafer theory

Utilize the belief intervals to represent the probability information and incorporate the belief intervals into the components of the fuzzy set

Fuzzy sets with immediate probabilities

Probabilities are treated as parts of the weight information and only valid to the aggregation operators

Hybrid data propagation method (Fuzzy random variable)

Probabilities are utilized in the stochastic simulation with nonlinear programming so as to generate the fuzzy random values

Cloud model

The membership function is replaced by probability distributions (such as Gaussian distributions and power-law distributions) with stable random bias

Most of the research attempts to utilize the fuzzy random variables to describe the probability information in the epistemic uncertainty. However, the probabilities in these methods are regarded as fuzzy components partly and will be lost after the fuzzy operations. It is also unrealistic to acquire the probabilistic distributions completely due to the lack of knowledge in reality (Yang 2001). These problems will partly or wholly make the probability-based methodologies invalid. On the other hand, the lack of fuzzy aggregation operators under probabilistic circumstances may also lead to the loss of probabilistic information. In the fuzzy linguistic decision making field, Pang et al. (2016) proposed a new probabilistic fuzzy linguistic term set under the probability environment. The probabilistic linguistic term set (PLTS) successfully solves the two problems mentioned above and aggregates the probability information in the decision making process without loss of information. The PLTSs have been successfully applied to solve the decision making problems involving multi-granular information (Zhai et al. 2016) and preference consistency (Zhang et al. 2016). To solve real-world problems involving both aleatory uncertainty and epistemic uncertainty, we first introduce a new probabilistic fuzzy set. Then we will study the information aggregation and visualization techniques of the new probabilistic dual hesitant fuzzy set (PDHFS) for the purpose of its applications. The new method will be utilized to solve the uncertain risk evaluation problems. Based on the above analysis, we shall aim at proposing the new extension of the IFS, that is, the PDHFS and investigating its operation laws and properties. The information aggregation and the visualization methods will also be presented in preparations for the related decision making method in the subsequent sections.

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5.2 Preliminaries 5.2.1 Dual Hesitant Fuzzy Set Since the IFS takes two functions to describe the membership and non-membership degrees, Zhu and Xu (2014) and Zhu et al. (2012) proposed the DHFS to describe the hesitation for both the membership degree and the non-membership degree. The motivation of DHFS lies not in handling the interval of possibilities (IVFS or IFS) or the possibilities distribution (T2FSs) but in describing the epistemic degree of certainty or uncertainty with a set of possible values. The DHFS is defined as follows: Definition 5.1 (Zhu et al. 2012). Let X be a fixed set, the DHFS on X is defined as: D = {x, h(x), g(x)x ∈ X }

(5.1)

where h(x) and g(x) are two sets of several values in [0, 1], representing the possible membership and non-membership degrees for x ∈ X . Also, there is 0 ≤ γ , η ≤ 1, 0 ≤ γ + + η+ ≤ 1

(5.2)

in which γ ∈ h(x), η ∈ g(x), γ + ∈ h+ (x) = ∪γ ∈h(x) max{γ } and η+ ∈ g + (x) = ∪η∈g(x) max{η}. The DHFS is composed of dual hesitant fuzzy elements (DHFEs), which is denoted by d (x) = (h(x), g(x)) (d = (h, g) for short). A DHFS with more than two elements h(x) and g(x) is then called a typical DHFE (Zhu and Xu 2014). Apparently, if h = ∅ and g = ∅, then the DHFS reduces to the HFS; if h and g have only one element, then the DHFS reduces to the IFS.

5.2.2 Basic Operations of DHFEs The basic operations of DHFEs are summarized as follows (Zhu et al. 2012): (1) The complement of the DHFE: ⎧  {{η}, {γ }}, if g = ∅ and h = ∅ ⎪ ⎪ ⎨  γ ∈h,η∈g dc = γ ∈h {{1 − γ }, {∅}}, if g = ∅ and h  = ∅ ⎪ ⎪ ⎩  η∈g {{∅}, {1 − η}}, if h = ∅ and g  = ∅

(5.3)

5.2 Preliminaries

91

(2) ⊕-union:   d1 ⊕ d2 = hd1 ⊕ hd2 , gd1 ⊗ gd2 =



γd1 ∈hd1 ,ηd1 ∈gd1 ,γd2 ∈hd2 ,ηd2 ∈gd2

   γd1 + γd2 − γd1 γd2 , ηd1 ηd2 (5.4)

(3) ⊗-intersection:   d1 ⊗ d2 = hd1 ⊗ hd2 , gd1 ⊕ gd2 =

γd1 ∈hd1 ,ηd1 ∈gd1 ,γd2 ∈hd2 ,ηd2 ∈gd2



   γd1 γd2 , ηd1 + ηd2 − ηd1 ηd2 (5.5)

(4) nd = (5) d n =





γd ∈hd ,ηd ∈gd



 γd ∈hd ,ηd ∈gd

 1 − (1 − γd )n , (ηd )n (n is a positive integral)

(5.6)

 (γd )n , 1 − (1 − ηd )n (n is a positive integral)

(5.7)

All the results of the above operations are also DHFEs.

5.3 Probabilistic Dual Hesitant Fuzzy Sets To describe the probabilistic information in the fuzzy set and aggregate them without loss of information, we will present the concept of probabilistic dual hesitant fuzzy set (PDHFS) and investigate its basic properties, comparison methods, and operation laws in this section.

5.3.1 Concept of PDHFS As has been discussed before, the DMs or the experts in the real-word problems may give several possible values when expressing their opinions. Besides, some items of these evaluation value sets may repeat among the experts. For instance, one expert may provide a set of values {0.1, 0.2, 0.5} and another may provide a set of values {0.1, 0.5} for the membership degree of a given case. The frequency or probabilistic information is lost in the HFSs or the DHFSs. To fully describe the limited but precious information provided by the experts and reduce the epistemic uncertainty as much as possible, in the following, we will give the concept of the PDHFS: Definition 5.2 Let X be a fixed set, a PDHFS on X is defined by the following mathematical symbol:

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P = {x, h(x)|p(x), g(x)|q(x)|x ∈ X }

(5.8)

The components h(x)|p(x) and g(x)|q(x) are two sets of some possible elements where h(x) and g(x) represent the hesitant fuzzy membership and nonmembership degrees to the set X of x, respectively; p(x) and q(x) are the corresponding probabilistic information for these two types of degrees. Also, there is 0 ≤ γ , η ≤ 1, 0 ≤ γ + + η+ ≤ 1

(5.9)

and pi ∈ [0, 1], qj ∈ [0, 1],

#h

i=1

pi = 1,

#g

qi = 1

(5.10)

i=1

 + where γ ∈ h(x), η ∈ g(x), γ + ∈ h+ (x) = ∈ g + (x) = γ∈h(x) max{γ }, η  max{η}, p ∈ p(x) and q ∈ q(x). The symbols #h and #g are the total i i h∈g(x) numbers of the elements in the components h(x)|p(x) and g(x)|q(x), respectively. Particularly, if all the probability values in p are equal to those in q, then the PDHFS reduces to the DHFS. Similarly, if g(x) = ∅ (there is also q(x) = ∅) and the probabilistic values of p(x) are equal, then the PDHFS reduces to the HFS. For the sake of simplicity and convenience, we call the pair P = h(x)|p(x), g(x)|q(x) as the probabilistic dual hesitant fuzzy element (PDHFE), denoted by P = (h|p, g|q) for short. To illustrate the PDHFS more straightforwardly, we provide an example to depict the difference between the PDHFS and the DHFS: Example 5.1 Take the evaluation of a car on the safety attribute as an example. An expert provides a DHFS {{0.4, 0.6}, {0.1, 0.2, 0.3}} on the safety due to his/her hesitation for this judgment. However, he/she is more confident in the value 0.6 for the membership degree set and the value 0.2 for the non-membership degree set. The DHFS fails to reflect the full information from this expert. As a result, we will utilize the PDHFS to present his/her evaluations, denoted by a PDHFE {{0.4|0.2, 0.6|0.8}, {0.1|0.2, 0.2|0.6, 0.4|0.2}}. In practical problems, the ignorance of the probabilistic information always exists  #g (Pang et al. 2016; Yang 2001), which means #h i=1 qi ≤ 1 in i=1 pi ≤ 1 and Definition 5.2. Such ignorance should be estimated so that the probability sets in the PDHFS can be treated as a probability distribution. Hence, we introduce the concept of the generalized PDHFS and discuss the estimation of the possible ignorance.  Definition 5.3 Let P = {x, h(x)|p, g(x)|q|x ∈ X }. If #h ≥ 2, #g ≥ 2, #h i=1 pi ≤ 1 and .., where #h and #g are the total numbers of the elements in the sets h(x) and g(x), respectively, then P is called a generalized PDHFS. For convenience, we also denote the generalized PDHFS as P.

5.3 Probabilistic Dual Hesitant Fuzzy Sets

93

The conditions #h ≥ 2 and #g ≥ 2 will distinguish the DHFS in PDHFS from the HFS (Zhu and Xu 2014). Most importantly, the partial ignorance is also considered in this definition. The ignorance can be estimated based on the premise that the missing membership degrees or the non-membership degrees in p(x) and q(x) may reappear in the normalized PDHFS (Pang et al. 2016). As a result, we provide a simple solution #g  by assigning the partial ignorance 1 − #h i=1 qi to each membership i=1 pi and 1 − degree and each non-membership degree, respectively. The normalized form of a generalized PDHFS is denoted by P˙ = {x, h(x)|˜p(x), g(x)|˜q(x)|x ∈ X }

(5.11)

where the elements in p˜ (x) and q˜ (x) are calculated by p˜ i = pi / #q qi / i=1 qi , respectively.

#h i=1

pi and q˜ i =

5.3.2 Comparison Method of PDHFEs The PDHFE still cannot avoid the issue that fuzzy numbers can only be partially ordered in theory. However, the comparison of PDHFEs is essential if we tend to apply this theory to decision making, risk analysis or optimization problems. Hence, we define the score function and the deviation function of the PDHFE, making it possible to rank the PDHFEs. Definition 5.4 Let P = (h|p, g|q) be a PDHFE, then we define the score function of the PDHFE as: s=

#h

γi · pi −

i=1 γ ∈h

#g

ηj · qj

(5.12)

j=1 η∈g

For two PDHFEs Pl (l = 1, 2), if s(P1 ) > s(P2 ) then the PDHFE P1 is superior to P2 , denoted as P1 > P2 ; If S(P1 ) < S(P2 ) then the PDHFS P1 is inferior to P2 , denoted as P1 < P2 ; If S(P1 ) = S(P2 ), then it is improper to conclude that the two PDHFEs are identical. In this case, to distinguish the PDHFEs, we define another indicator, named as the deviation degree: Definition 5.5 Let P = (h|p, g|q) be a PDHFE, the deviation degree of a PDHFE is defined as: ⎛ σ =⎝

#h

(γi − s)2 · pi +

i=1 γ ∈h

where s is the score function of the PDHFE.

#g

j=1 η∈g

⎞1/2 (ηi − s)2 · qi ⎠

(5.13)

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5 The Decision Making Method Under the Probabilistic …

As reflected by (5.13), the deviation degree describes the distance from the average value in the PDHFE. It can be interpreted as the consistency indicator of the PDHFE, describing how each element accords with others. The smaller values of σ imply higher consistency while the larger values of σ indicate lower consistency. Based on the score function and the deviation degree of the PDHFE, we can develop a method for comparing two PDHFEs: Definition 5.6 Let Pl (l = 1, 2) be two PDHFEs, s(P1 )(l = 1, 2) and δ(Pl )(l = 1, 2) are the score function and the deviation degree, respectively. Then • If s(P1 ) > s(P2 ), then the PDHFE P1 is superior to P2 , denoted by P1 > P2 ; On the contrary, there is P1 < P2 . • If s(P1 ) = s(P2 ), then (1) If δ(P1 ) < δ(P2 ), then the PDHFE P1 is superior to P2 , denoted by P1 > P2 ; (2) If δ(P1 ) > δ(P2 ), then the PDHFE P1 is inferior to P2 , denoted by P1 < P2 ; (3) If δ(P1 ) = δ(P2 ), then the PDHFE P1 is equal to P2 , denoted by P1 ∼ P2 ;

5.3.3 Basic Operations of PDHFEs Based on the operation laws of the DHFSs (Zhu et al. 2012), we develop some basic operation laws of the PDHFEs and investigate their properties in preparation for the applications to the practical problems. Definition 5.7 The complement of a given PDHFE P = (h|p, g|q) is defined as follows: ⎧     ⎪ ⎪ γ ∈h,η∈g η|qη , γ |pγ , if g = ∅ and h = ∅ ⎨     Pc = (5.14) (1 − γ )|pγ , {∅} , if g = ∅ and h = ∅ γ ∈h ⎪ ⎪   ⎩   η∈g {∅}, (1 − η)|qη , if h = ∅ and g  = ∅ Remark For the second condition where g = ∅ and h = ∅, if all possibilities of the elements in p(x) are equal, then the complement P c will reduce to the complement of a hesitant fuzzy element (HFE) defined by Torra (2010). Similarly, for the third condition, if all the possibilities in q(x) are the same, this operation can be also regarded as a special case of the complement of the HFE. Furthermore, if the possibility values in p(x) are equal to those in q(x), then it will become the complement of the DHFE defined in (5.3). The complement of the PDHFE is involutive, denoted by (P c )c = P. Definition 5.8 Let P, P1 and P2 be three PDHFEs, P = (h|p, g|q), P1 =  h1 |ph1 , g1 |qg1 and P2 = h2 |ph2 , g2 |qg2 . Then

5.3 Probabilistic Dual Hesitant Fuzzy Sets

P1 ⊕ P2 = P1 ⊗ P2 =





γ1 ∈h1 ,η1 ∈g1 ,γ2 ∈h2 ,η2 ∈g2

γ1 ∈h1 ,η1 ∈g1 ,γ2 ∈h2 ,η2 ∈g2

95

   (γ1 + γ2 − γ1 γ2 )|pγ1 pγ2 , (η1 η2 )|qη1 qη2

(5.15)     (γ1 γ2 )|pγ1 pγ2 , (η1 + η2 − η1 η2 )|qη1 qη2 (5.16)

Also, we define some other operation laws as follows: λP = Pλ =

γ ∈h,η∈g



     1 − (1 − γ )λ |pγ , ηλ |qη 

γ ∈h,η∈g

    γ λ |pγ , 1 − (1 − η)λ |qη

(5.17) (5.18)

where λ ≥ 0. {{0.5|0.8, 0.6|0.2}, {0.2|0.4, 0.4|0.6}}, Example 5.2 Let P = {{0.6|0.8, 0.4|0.2} , {0.2|0.7, 0.3|0.3}}, P1 = and P2 = {{0.7|0.6, 0.5|0.4}, {0.1|0.2, 0.2|0.8}} be three PDHFEs. Suppose that λ = 3, then P1 ⊕ P2 = {{0.88|0.48, 0.82|0.12, 0.80|0.32, 0.70|0.08} , {0.02|0.14, 0.03|0.06, 0.04|0.56, 0.06|0.24}}} P1 ⊗ P = {{0.42|0.48, 0.28|0.12, 0.30|0.32, 0.20|0.08} , {0.28|0.14, 0.37|0.06, 0.36|0.56, 0.44|0.24}}} λP = {{0.875|0.8, 0.936|0.2}, {0.008|0.4, 0.064|0.6}} P λ = {{0.125|0.8, 0.216|0.2}, {0.488|0.4, 0.784|0.6}}.   h1 |ph1 , g1 |qg1 , P2 = P1 =  Theorem 5.1 Consider three PDHFEs h2 |ph2 , g2 |qg2 ,and P3 = h3 |ph3 , g3 |qg3 ,where pand qare the possibility sets for the membership degrees and the non-membership degrees, respectively. Suppose that λ, λ1 , λ2 ≥ 0, then (1) (2) (3) (4) (5) (6) (7)

P1 ⊕ P2 = P2 ⊕ P1 (P1 ⊕ P2 ) ⊕ P3 = P1 ⊕ (P2 ⊕ P3 ) λ(P1 ⊕ P2 ) = λP1 ⊕ λP2 P1 ⊗ P2 = P2 ⊗ P1 (P1 ⊗ P2 ) ⊗ P3 = P1 ⊗ (P2 ⊗ P3 ) (P1 ⊗ P2 )λ = P1λ ⊗ P2λ P λ1 +λ2 = P λ1 ⊗ P λ2

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5 The Decision Making Method Under the Probabilistic …

Proof (1) P1 ⊕ P2 =





γ1 ∈h1 ,η1 ∈g1 ,γ2 ∈h2 ,η2 ∈g2

   (γ2 + γ1 − γ1 γ2 )|pγ1 pγ2 , (η2 η1 )|qη1 qη2

= P2 ⊕ P1 (2) (P1 ⊕ P2 ) ⊕ P3 =

 ⎫ ⎧  γ1 + (γ2 + γ3 − γ2 γ3 ) ⎪ ⎪ ⎨ |pγ1 pγ2 pγ3 ,⎬ −γ1 (γ2 + γ3 − γ2 γ3 ) γ1 ∈h1 ,η1 ∈g1 , ⎪ γ2 ∈h2 ,η2 ∈g2 , ⎪  ⎩ ⎭ γ3 ∈h3 ,η3 ∈g3 (η1 (η2 η3 ))|qη1 qη2 qη3



= P1 ⊕ (P2 ⊕ P3 ) (3) λ(P1 ⊕ P2 )    = 1 − (1 − γ1 )λ (1 − γ2 )λ |pγ1 pγ2 , η1λ η2λ |qη1 qη2      λ = 1 − (1 − γ1 ) |pγ1 , η1λ |qη1 γ1 ∈h1 ,η1 ∈g1      λ 1 − (1 − γ2 ) |pγ2 , η2λ |qη2 ⊕ γ2 ∈h2 ,η2 ∈g2

= λP1 ⊕ λP2 (4) P1 ⊗ P2 =



γ1 ∈h1 ,η1 ∈g1 ,γ2 ∈h2 ,η2 ∈g2





 (γ2 γ1 )|pγ2 pγ1 , (η2 + η1 − η1 η2 )|qη2 qη1

(5) (P1 ⊗ P2 ) ⊗ P3

 = P2 ⊗ P1

 ⎧ ⎫ (γ1 (γ2 γ3 ))|pγ1 pγ2 pγ3 , ⎪ ⎪ ⎨   ⎬ = η1 + (η2 + η3 − η2 η3 ) γ1 ∈h1 ,η1 ∈g1 , |qη1 qη2 qη3 ⎪ γ2 ∈h2 ,η2 ∈g2 , ⎪ ⎩ ⎭ −η1 (η2 + η3 − η2 η3 ) γ3 ∈h3 ,η3 ∈g3 = P1 ⊗ (P2 ⊗ P3 )

(6) (P1 ⊗ P2 )λ  =

 

 (γ1 γ2 )|pγ1 pγ2 ,

(η1 + η2 − η1 η2 )|qη1 qη2     λ γ1 |pγ1 , η1 |qη1 = γ1 ∈h1 ,η1 ∈g1     λ γ2 |pγ2 , (1 − (1 − η2 ))|qη2 ⊗ γ1 ∈h1 ,η1 ∈g1 ,γ2 ∈h2 ,η2 ∈g2

=

P1λ

γ2 ∈h2 ,η2 ∈g2 ⊗ P2λ

λ 

5.3 Probabilistic Dual Hesitant Fuzzy Sets

97

(7) P λ1 +λ2  λ1 +λ2     γ = |pγ , 1 − (1 − η)λ1 +λ2 |qη γ ∈h,η∈g  λ1 λ2     γ γ |pγ , 1 − (1 − η)λ1 (1 − η)λ1 |qη = γ ∈h,η∈g

= P λ1 ⊗ P λ2 {{0.5|0.8, 0.6|0.2}, {0.4|1}} and P2 = = Example 5.3 Let P1 {{0.2|1}, {0.4|0.6, 0.7|0.4}} be two PDHFEs. Suppose that λ = 2. For the operation (3) in Theorem 5.1, we have λ(P1 ⊕ P2 ) = {{0.84|0.8, 0.8976|0.2} , {0.0256|0.6, 0.0784|0.4}}, λP1 = {{0.75|0.8, 0.84|0.2}, {0.16|1}}, λP2 = {{0.36|1}, {0.16|0.6, 0.49|0.4}} and λP1 ⊕ λP2 = {{0.84|0.8, 0.8976|0.2} , {0.0256|0.6, 0.0784|0.4}}. There is λ(P1 ⊕ P2 ) = λP1 ⊕ λP2 . For the operation (5.6), we have (P1 ⊗ P2 )λ = {{0.01|0.8, 0.0144|0.2}, {0.8704|0.6, 0.9676|0.4}} P1λ ⊗ P2λ = {{0.01|0.8, 0.0144|0.2}, {0.8704|0.6, 0.9676|0.4}}. Then (P1 ⊗ P2 )λ = P1λ ⊗ P2λ .

    Theorem 5.2 Let P = (h|p, g|q), P1 = h1 |ph1 , g1 |qg1 and P2 = h2 |ph2 , g2 |qg2 be three PDHFEs and λ ≥ 0.Then (1) (2) (3) (4)

((P)c )λ = (λP)c c λ(P)c = (P)λ c c (P1 ) ⊕ (P2 ) = (P1 ⊗ P2 )c (P1 )c ⊗ (P2 )c = (P1 ⊕ P1 )c

Proof Since the complement defined in Definition 5.7 relies on the status of the sets h and g, we will prove this theorem under the following three possible situations. (1) If g = ∅ and h = ∅, then 

(P)c

λ

λ(P)c =

=





γ ∈h,η∈g γ ∈h,η∈g

λ   {{η|q}, {γ |p}} = λ

γ ∈h,η∈g

{{γ |p}, {η|q}}

c

= (λP)c

     c 1 − (1 − η)λ |q , γ λ |p = (P)λ

⎞c ⎛     (P1 )c ⊕ (P2 )c = ⎝ γ1 ∈h1 ,η1 ∈g1 , γ1 γ2 |pγ1 pγ2 , 1 − (1 − η1 )(1 − η2 )|qη1 qη2 ⎠ = (P1 ⊗ P2 )c γ2 ∈h2 ,η2 ∈g2

⎞c ⎛     (P1 )c ⊗ (P2 )c = ⎝ γ1 ∈h1 ,η1 ∈g1 1 − (1 − γ1 )(1 − γ2 )|pγ1 pγ2 , η1 η1 |qη1 qη2 ⎠ = (P1 ⊕ P1 )c γ2 ∈h2 ,η2 ∈g2

(2) If g = ∅ and h = ∅, then

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5 The Decision Making Method Under the Probabilistic … 

(P)c

λ

λ(P)c =

=

γ ∈h

γ ∈h

    (1 − γ )λ |p , {∅} =

    1 − γ λ |p , {∅} =

(P1 )c ⊕ (P2 )c = (P1 )c ⊗ (P2 )c =

 

 c  = (λP)c 1 − (1 − γ )λ |p , {∅}

γ ∈h

 γ ∈h

  c  c = (P)λ γ λ |p , {∅}



  c  = (P1 ⊗ P2 )c (γ1 γ2 )|pγ1 pγ2 , {∅}



   c = (P1 ⊕ P1 )c 1 − (1 − γ1 )(1 − γ2 )|pγ1 pγ2 , {∅}

γ1 ∈h1 ,γ2 ∈h2 γ1 ∈h1 ,γ2 ∈h2

(3) If h = ∅ and g = ∅, then  c λ = (P) λ(P)c =

   {∅}, 1 − ηλ |q =

 η∈g

γ ∈h

    1 − γ λ |q , {∅} =

(P1 )c ⊕ (P2 )c = (P1 )c ⊗ (P2 )c =



 η1 ∈g1 ,η2 ∈g2



 η1 ∈g1 ,η2 ∈g2

η∈g

γ ∈h

c  {∅}, ηλ |q = (λP)c



 c   c = (P)λ γ λ |q , {∅}



c {∅} = (P1 ⊗ P2 )c 



c {∅} = (P1 ⊕ P1 )c 

, 1 − (1 − η2 )(1 − η1 )|qη1 qη2 , 1 − (1 − η1 )(1 − η2 )|qη1 qη2

Example 5.4 Let P = {{0.5|0.8, 0.6|0.2}, {0.4|1}} and λ = 2. Then we have (1) ((P)c )λ = ({{0.4|1}, {0.5|0.8, 0.6|0.2}})2 = {{0.16|1}, {0.75|0.8, 0.84|0.2}} (λP)c = ({{0.75|0.8, 0.84|0.2}, {0.16|1}})c = {{0.16|1}, {0.75|0.8, 0.84|0.2}} Obviously, ((P)c )λ = (λP)c . (2) λ(P)c = 2{{0.4|1}, {0.5|0.8, 0.6|0.2}} = {{0.64|1}, {0.25|0.8, 0.36|0.2}}  λ c (P) = ({{0.25|0.8, 0.36|0.2}, {0.64|1}})c = {{0.64|1}, {0.25|0.8, 0.36|0.2}} c  Similarly, there is λ(P)c = (P)λ . The PDHFS shows its advantages over other methods in Table 5.1 from the following aspects: Firstly, instead of being treated as the intermediate or substitutional variables in the Dempster-Shafer theory or the immediate probability method, the probability information and the fuzzy information enjoy the equal position in the PDHFS. Secondly, the probability information is processed independently in the fuzzy aggregation process. Both the fuzzy information and the probability information are soundly aggregated without loss of information. Thirdly, the PDHFS does not need the probabilistic distribution to generate random variables or run statistic simulations, which will reduce the computational complexity and increase the efficiency in practical applications. In summary, the PDHFS successfully describes the epistemic uncertainty and the statistical uncertainty in a single framework, providing an efficient way to handle the uncertainty in real-world problems.

5.4 Information Aggregation and Visualization of PDHFSs

99

5.4 Information Aggregation and Visualization of PDHFSs When we apply the PDHFSs to decision making or risk evaluation, a major problem is how to fuse and analyze the information provided by the experts or the DMs. Another problem involves the visual representation of the aggregated results. Even though we have developed the comparison method for the PDHFSs in Sect. 5.3, the crisp ranking result fails to fully represent the uncertainty in the aggregated values, let alone tell apart the problematic results involving prejudicial judgments. Hence, it is also necessary to give an approach to visually describe and evaluate the aggregated values taking the intrinsic indeterminacy of the PDHFS into account. In what follows, we will elaborate on the solutions to these problems in detail.

5.4.1 Basic Aggregation Operator for PDHFSs A major problem in the information science is the technology utilized for data fusion. It is a common practice to devise some appropriate aggregation operators to fuse the information available under fuzzy circumstances. Based on Definition 5.8 and the operation laws defined above, we introduce a basic aggregation operator under probabilistic fuzzy environment as follows:   Definition 5.9 Let Pi = hi |phi , gi |qgi (i = 1, 2, . . . , n) be n PDHFEs and the respective weight  information be wj (j = 1, 2, . . . , n) with wj ∈ [0, 1](j = 1, 2, . . . , n) and nj=1 wj = 1. Then the probabilistic dual hesitant fuzzy weighted averaging (PDHFWA) operator is defined as: n

PDHF W A(P1 , P2 , . . . , Pn ) = ⊕ wi Pi

(5.19)

i=1

Particularly, the PDHFWA operator reduces to a probabilistic dual hesitant fuzzy averaging (PDHFA) operator when all weights are the same. Based on the basic operation laws of PDHFSs, it is easy to prove that the PDHFWA operator has the following form: PDHF W A(P1 , P2 , . . . , Pn ) ⎧⎧⎛ ⎫ ⎧ ⎫⎫ ⎞ n n n w  n ⎨⎨ ⎬ ⎨ ⎬⎬   j ⎝1 − = pγi , ηj qηj (1 − γi )wi ⎠ ⎩⎩ ⎭ ⎩ ⎭⎭ γ1 ∈h1 ,γ2 ∈h2 ,...γn ∈hn , η1 ∈g1 ,η2 ∈g2 ,...ηn ∈gn

i=1

i=1

j=1

(5.20)

j=1

    γpl |pγpl , ηpl |qηpl and Pl∗ = Theorem 5.3 (Monotonicity).Let Pl =     γPl∗ |pγP∗ , ηPl∗ |qηP∗ (l = 1, 2, . . . , n)be two groups of PDHFSs. For each elel l ment in the Pand P ∗ ,there are γpl ≤ γPl∗ and ..while the probabilities are the same, i.e.pγpl = pγP∗ and qηpl = qηP∗ . Then l

l

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5 The Decision Making Method Under the Probabilistic …

  PDHF W A(P1 , P2 , . . . , Pl ) ≤ PDHF W A P1∗ , P2∗ , . . . , Pl∗ Proof For any j, there are γhi ≤ γh∗i and ηgj ≥ ηgj∗ . For the terms in the aggregated results, we have 1−



1 − γpj

wj

≤1−

 wj    1 − γpj∗ and ηPwj ≥ ηPw∗j l

l

Accordingly,     w   1− 1 − γpj j ηpl ≤ pγpl − ηPwj l wj       1 − γpj∗ 1− ηPw∗j ηPl∗ pγP∗ − l

l

  Then we have PDHF W A(P1 , P2 , . . . , Pl ) ≤ PDHF W A P1∗ , P2∗ , . . . , Pl∗ with equality if and only if γpl = γPl∗ and ηpl = ηPl∗ . ⎧     ⎫ ⎪ ⎬ ⎨ min hpl |min phpl ,⎪    and P + = Theorem 5.4 (Boundedness).Let P − =   ⎪ ⎭ ⎩ max gpl |max qgp ⎪ l           . Then max hpl |max phpl , min gpl |min qgpl     PDHF W A P − ≤ PDHF W A(P1 , P2 , . . . , Pn ) ≤ PDHF W A P +     Proof For each element in the PDHFS, we have min hpl ≤ γpl ≤ min hpl ,        min gpl ≤ ηpl ≤ max gpl , min phpl ≤ pγ ≤ max phpl , and min qgpl ≤   qη ≤ max qgpl . Then n 

w  w    wl   ! ! 1− 1 − γpl l ≥1 − 1 − min hpl l =1 − 1 − min hpl l=1 = min hpl n   w    wl   ! wl !  ηpl ≥ min gpl l = min gpl l=1 = min gpl n  wl  wl    wl   ! ! l=1 ≤1 − 1 − max hpl =1 − 1 − max hpl = max hpl 1− 1 − γpl n   w    wl   ! wl !  ηpl ≤ max gpl l = max gpl l=1 = max gpl For the probabilities: 

and

pγ ≥



      qη ≥ min phpl , min qgpl

5.4 Information Aggregation and Visualization of PDHFSs



pγ ≤



101

      qη ≤ max phpl , max qgpl

⎧   ⎫   wl   ⎪ ⎬ ⎨ min phpl ,⎪ | 1 − 1 − min h p l  −      PDHF W A P =    w ⎪ ⎪ ⎭ ⎩ max gpl l | max qgpl               min phpl max qgpl = min hpl | , max gpl | ⎧   ⎫   wl   ⎪ ⎬ ⎨ 1− 1 − max hpl ,⎪ | max phpl        PDHF W A P + =  w  ⎪ ⎪ ⎭ ⎩ min qgpl min gpl l |               = max hpl | , min gpl | max phpl min qgpl Then there are     w      1 − γpl l · min phpl 1− pγ ≥ min hpl ·     w      1 − γpl l · max phpl 1− pγ ≤ max hpl · and      qη ≤ max gpl · max qgpl        ηpwll · min qgpl qη ≥ min gpl ·



ηpwll ·



According  to the score function, we have PDHF W A(P1 , P2 , . . . ,−Pn ) ≥ PDHF W A P − with equality if and only if the PDHFS P is the  same as P . Similarly, PDHF W A(P1 , P2 , . . . , Pn ) ≤ PDHF W A P + with equality  if and only if the PDHFS P is the same as P +. As a result, PDHF W A P − ≤ PDHF W A(P1 , P2 , . . . , Pn ) ≤ PDHF W A P + . Utilizing the aggregation operator, we provide a general information fusion procedure under the PDHF environment (Fig. 5.1). The gathered information may contain ignorance in terms of probabilities. So the normalization process is recommended to estimate the ignorance. Next, if the information is provided by groups of experts, the PDHF matrices should be aggregated by all the experts on each criterion utilizing a proper aggregation operator. Then, we can get the comprehensive information matrix, which also takes the form of the PDHFSs. The comprehensive matrix describes the overall assessment from all the individuals on each criterion. This matrix will be normalized and further fed into a chosen aggregation operator to integrate the information on all the criteria. After the normalization, we will get the aggregated vectors, whose elements take the form of PDHFEs. Each component of the aggregated vectors reflects the final assessment for each study subject taking the information of all the criteria into account. The normalization conducted in each aggregation process

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5 The Decision Making Method Under the Probabilistic …

Fig. 5.1 General information fusion process for PDHFSs

Information gathering Normalization

Group of experts and specialists

Fusion for groups

Normalization

(Comprehensive) information matrix

Fusion for individuals

Normalization

Aggregated vectors

Defuzzification/ reprocess

plays different roles in different phases. In the initial period, the normalization aims at estimating and eliminating the partial ignorance of the probabilities. In the following aggregation process, the normalization is responsible not only for estimating the potential ignorance, but also for preventing the probability results from becoming too small to distinguish. Based on the aggregated vectors, we can take different processing strategies according to different requirements of the applications. For example, the aggregated vectors in the PDHF forms will be further defuzzified for the purpose of ranking under different decision making scenarios. Meanwhile, these vectors may be also reprocessed in preparations for the fuzzy computation, fuzzy reasoning, visualizations, or some other special purposes.

5.4.2 Visualization of the PDHFS Based on the Cloud Model The PDHFS contains fuzzy and probabilistic information simultaneously. The ranking methods defined in Definition 5.6 represent the average information of a PDHFE. However, we still cannot evaluate the uncertainty or certainty degrees for a given aggregated PDHFE. As is often the case, the aggregated results from several PDHFEs

5.4 Information Aggregation and Visualization of PDHFSs

103

will lead to plenteous elements in the aggregated PDHFE. This is also the deficiency of the aggregation operators extended from HFSs. The practical group decision making problems will exacerbate this problem in view of considerable numbers of PDHFEs provided by groups of experts. As a result, it is essential to estimate the information distribution of the values in the PDHF aggregation results regarding to the uncertainty or certainty degrees. This, in turn, will improve the quality of the final evaluation result. To solve this problem, we will first define the entropy concept of the PDHFSs and then develop the fuzzy cloud model in preparations for the visual analysis method for the PDHFSs. Zhao and Xu (2015) studied the entropy of the DHFSs and gave the basic definition. Inspired by this, we will give the entropy concept of the PDHFSs and then investigate its properties. Since the PDHFE is the basic element of a PDHFS, we only need to study the entropy measure of the PDHEs. We first need to rearrange the items of the membership part and the non-membership part in ascending orders for the convenience of calculations. The reordered PDHFE is denoted by P = {hδ(i) |phδ(i) }, {g δ(j) |qg δ(j) }, where hδ(i) (i = 1, 2, . . . , #h) is the ith smallest element in h, and g δ(j) is the jth smallest element in g. To operate correctly, we should make the numbers of the elements in h and g identical. If #h − #g > 0, then we will add #h − #g smallest elements of g(x) to g(x). The added items are the smallest ones in g and their probabilities are all zero; If #h − #g < 0, then we will add #h − #g smallest elements of h(x) to h(x). Also, the probabilities of these added ones are zeros.   Let ξ δ(i) = hδ(i) · phδ(i) − g δ(i) · qg δ(i) and ζ δ(i) = 1 − hδ(i) · phδ(i) + g δ(i) · qg δ(i) , where i = 1, 2, · · · , l and l = max(#h, #g), then the entropy of a PDFHE is defined as follows: Definition 5.10 The entropy measure E of a PDHFE P is real-valued function in terms of ξ and ζ , satisfying the following conditions: i. ii. iii. iv. v.

0 ≤ E(P) ≤ 1; E(P) = 0 if and only if ξ = 1 and ζ = 0; E(P) = 1 if and only if ξ = 0 and ζ = 1; E(P c ) = E(P) E monotonically decreases with ξ and monotonically increases with ζ .

These axiomatic requirements in Definition 5.10 are similar to the entropy of HFSs (Xu and Xia 2012). Many entropy functions are formulated and justified by the fact that different types of functions are useful for different purposes (Farhadinia 2013; Hu et al. 2016; Wei et al. 2016; Xu and Xia 2012; Zhao et al. 2015). Zhao and Xu (2015) gave some different entropy functions for the DHFSs. For the sake of simplicity and generality, we formulate the entropy of the PDHFE with a bivariate β α function e(x, y) = 1 − x +(1−y) (x, y ≥ 0; α, β > 0). The function e(x, y) satisfies 2 all the requirements in Definition 5.10. Then the entropy of a PDFHE is calculated by:

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5 The Decision Making Method Under the Probabilistic …

Fig. 5.2 The entropy function e(x, y) for the concave and convex situations

1  δ(i) δ(j)  e ξ ,ζ l i=1 l

E(P) =

(5.21)

The parameters α and β in e(x, y) determine the concavity and convexity of the entropy surface. If 0 < α, β < 1, then the entropy function is concave in the space; If α, β > 1, then the entropy function is convex in the space. For the concave situation, the entropy values have abrupt changes when x approximates 1 or y approximates 0. The changes are relatively small and smooth in other areas of the space. However, for the convex situation, the steep changes happen when x approximates 0 or y approximates 1 (See Fig. 5.2). As a result, we suggest adjusting these two parameters based on the value distributions of ξ and ζ. For the following calculations, we set α = β = 0.5. Utilizing the score function, the deviation degree and the entropy function defined above, we can visualize the aggregated PDHFSs by the cloud model. The cognitive cloud model quantifies the randomness and fuzziness simultaneously in a single framework, which will help to analyze the aggregated results comprehensively. Three fixed parameters are used to describe the cloud model, Ex (Expectation), En (Entropy) and He (Hyper-entropy) (Li et al. 2009). Parameter Ex represents the cloud drop belonging to a concept in the universe. En represents the uncertain degree of such concept. He indicates the uncertainty of En. Inspired by this theory, we use the score function, the deviation degree, and the entropy to describe the cloud model C(s, σ, E) under the probabilistic dual hesitant fuzzy environment. The parameter s is the score function of the PDHFS, which also represents the expectation of the given information; The parameter σ is the deviation degree of the PDHFS, illustrating the divergence from the average status; The parameter E is the entropy of the PDHFS. The entropy E represents the information measure of the PDHFSs and can be treated as a more specific indicator for the deviation degree. In the following, we give an algorithm to generate the Gaussian cloud model of the PDHFSs:

5.4 Information Aggregation and Visualization of PDHFSs

105

 Step 1. Generate   N random variables Exi (i = 1, 2, . . . , N ) from a normal distribution  2 Exi = N δ, E ; Step 2. Generate variables Exi (i = 1, 2, . . . , N ) from another normal   N random distribution N s, Exi2 ; −(

xi −s)2 2Exi2

; Step 3. Calculate the determinacy degree yi = e Step 4. Draw the cloud droplets and their determinacy degrees (xi , yi ). These cloud droplets are illustrated in Fig. 5.3. A typical cloud is composed of many cloud droplets. Each droplet represents the determinacy degree of expectation taking both randomness and fuzziness into account. Since the score value’s range is from -1 to 1, then the droplets locating in this area are valid and meaningful. From the algorithm above, the cloud model is actually a stochastic normal distribution. From this aspect, the drops located in the interval [s − 0.67δ, s + 0.67δ] account for 50% contribution to the universe of discourse. So we define this area as the trustful area, which can represent the general judgement information of the aggregated values. The motivation that we utilize the cloud model is to rebuild the information distribution of the aggregated results and evaluate its consistency, which will be beneficial to the quality of the risk evaluation results. The data provided by different experts may contain uncertain information and bias information. Hence, we should identify the useful information and judge the problematic information. We assume that the general judgements of the experts obey the normal distribution and accordingly can be described by a Gaussian cloud model. The expectation s represents the most trustful information while the deviation and the entropy depict the randomness and the fuzziness in this cognitive process. Most of the droplets are around the area of

Fig. 5.3 The illustration of a normal cloud model C(s, σ, E)

106

5 The Decision Making Method Under the Probabilistic …

expectation s and a few drops may deviate far away from the expectation point, which can be treated as the inaccurate information provided by the experts. The thickness of the cloud droplets represents the uncertainty and indeterminacy information of the experts. From the width and thickness of the cloud, we can evaluate the performance of the aggregated result. The cloud with large width and diffuse droplets contains more uncertainty and the information is less trustful.

5.5 Application to the Arctic Geopolitics Risk Evaluations Recently the Arctic has drawn more and more attention from the whole world. The new commercial shipping routes linking Europe and Asia and the rich natural resources set off a new upsurge of the Arctic exploitation. More and more countries and organizations show their interests in Arctic territorial claims, bringing political conflicts and potential risks for investors. To help the investors grasp the investment opportunity and manage the risks of the Arctic, we conduct a geopolitical risk evaluation for the Arctic area utilizing the algorithm introduced in this chapter, which will also serve as the illustration and validation for our methodology. We first take into account the countries adjacent to the Arctic, such as Canada, Russia, the USA, Norway and Denmark. China, aiming at seeking a seat at the Arctic exploitation through large overseas investments, is also considered. Next, we give four key factors with regards to the risk in resources exploitation and utilization: potential military conflicts (MC), diplomatic disputes (DD), dependence on energy imports (EI) and control over marine routes (MR). A major challenge in the evaluation process is the quantification of those intangible factors, which may contain both aleatory and epistemic uncertainties. Besides, the fuzzy set theory has been successfully applied to the risk assessment and management (Zhang et al. 2009). As a result, we suggest that the DMs utilize the PDHFSs to express their preferences on these factors. Suppose that the assessment data of the given countries over each criterion are provided by three domain experts, listed in Tables 5.2, 5.3 and 5.4. The experts’ assessments usually differ from each other because of the diversity of their research areas, authority, and experience. We assume that the importance degrees of the three experts are described by the weight vector ω = (0.2, 0.3, 0.5)T . Also, the weight information of the given four criteria is assigned by w = (0.3, 0.2, 0.3, 0.2)T . Based on the procedures presented in Fig. 5.2, we can fuse the PDHFEs of the three different domain experts. After the aggregation process, we can obtain a comprehensive assessment matrix for the six countries on all the criteria by aggregating all the experts’ assessments. For each criterion  eval " of Ci (i = 1, 2,# 3, 4), the overall ˜ C2 M ˜ C3 M ˜ C4 = PDHF W A MC(k) ˜ = M ˜ C1 M uation information is calculated by M i (k = 1, 2, 3), where k is the numerical order of the domain experts. The detailed

5.5 Application to the Arctic Geopolitics Risk Evaluations

107

Table 5.2 The assessment matrix M1 of the six countries on the given criteria by the domain expert G1 Criteria

MC

DD

EI

MR

The USA

{{0.7|0.2, 0.6|0.2, 0.5|0.6}, {0.2, 1}}

{{0.7|1}, {0.25|1}}

{{0.2|1}, {0.2|1}}

{{0.7|0.5, 0.6|0.5}, {0.3|1}}

Canada

{{0.1|1}, {0.4|1}}

{{0.3|1}, {0.7|1}}

{{0.7|1}, {0.3|0.5, 0.2|0.5}}

{{0.3|1}, {0.3|1}}

Russia

{{0.6|1}, {0.35|1}}

{{0.56|1}, {0.2|1}}

{{0.1|1}, {0.7|1}}

{{0.2|0.6, 0.4|0.4}, {0.4|1}}

Denmark

{{0.05|0.7, 0.2|0.3}, {0.5|1}}

{0.3|0.5, 0.2|0.5}, {0.6|0.5, 0.5|0.5}}

{{0.8|1}, {0.15|1}}

{{0.2|1}, {0.6|1}}

China

{{0.15|1}, {0.8|1}}

{{0.5|1}, {0.5|1}}

{{0.8|0.6, 0.6|0.4}, {0.15|1}}

{{0.12|1}, {0.7|0.9, 0.6|0.1}}

Norway

{{0.08|1}, {0.6, 1}}

{{0.1|0.6, 0.3|0.4}, {0.7|1}}

{{0.3|1}, {0.65|1}}

{{0.5|1}, {0.2|0.3, 0.4|0.7}}

Table 5.3 The assessment matrix M2 of the six countries on the given criteria by the domain expert G2 Criteria

MC

DD

EI

MR

The USA

{{0.5|1}, {0.5|1}}

{{0.2|1}, {0.4|0.8, 0.6|0.2}}

{{0.7|0.4, 0.4|0.6}, {0.3|0.7, 0.2|0.3}}

{{0.6|0.7, 0.7|0.3}, {0.25|1}}

Canada

{{0.3|0.5, 0.5|0.5}, {0.4|1}}

{{0.1|1}, {0.6|0.6, 0.8|0.4}}

{{0.4|0.8, 0.3|0.2}, {0.5|0.3, 0.4|0.7}}

{{0.2|0.3, 0.3|0.7}, {0.6|1}}

Russia

{{0.1|0.1, 0.2|0.9}, {0.5|1}}

{{0.3|0.5, 0.2|0.5}, {0.3|0.5, 0.2|0.5}}

{{0.2|1}, {0.7|0.6, 0.5|0.4}}

{{0.5|1}, {0.4|1}}

Denmark

{{0.2|1}, {0.7|0.1, 0.6|0.9}}

{{0.1|1}, {0.7|1}}

{{0.2|1}, {0.6|1}}

{{0.2|0.8, 0.1|0.2}, {0.3|0.4, 0.2|0.6}}

China

{{0.2|1}, {0.7|1}}

{{0.45|1}, {0.5|1}}

{{0.8|0.9, 0.6|0.1}, {0.11|1}}

{{0.3|1}, {0.2|1}}

Norway

{{0.4|0.4, 0.5|0.6}, {0.5|1}}

{{0.3|0.4, 0.4|0.6}, {0.7|1}}

{{0.3|1}, {0.6|1}}

{{0.2|1}, {0.6|1}}

results of the comprehensive matrices are listed in Appendix B. To rank and compare different countries, we should further aggregate the comprehensive matrices to get the aggregated vectors, which are also listed in Appendix B. Then we can calculate the score function of each country by (5.12). The results are shown in Table 5.5. The score values show that the USA has the highest risk and Norway has the lowest risk. The ranking order of the six countries in terms of their potential risk is

108

5 The Decision Making Method Under the Probabilistic …

Table 5.4 The assessment matrix M3 of the six countries on the given criteria by the domain expert G3 Criteria

MC

DD

EI

MR

The USA

{{0.4|1}, {0.5|1}}

{{0.9|1}, {0.1|1}}

{{0.3|1}, {0.5|0.4, 0.6|0.6}}

{{0.6|1}, {0.3|1}}

Canada

{{0.75|1}, {0.2|1}}

{{0.4|1}, {0.6|1}}

{{0.2|0.7, 0.4|0.3}, {0.2|1}}

{{0.3|1}, {0.6|1}}

Russia

{{0.6|0.6, 0.8|0.4}, {0.1|1}}

{{0.5|1}, {0.2|1}}

{{0.1|1}, {0.8|1}}

{{0.2|0.7, 0.4|0.3}, {0.6|1}}

Denmark

{{0.2|1}, {0.7|1}}

{{0.5|0.6, 0.7|0.4}, {0.1|1}}

{{0.3|0.3, 0.5|0.7}, {0.2|0.5, 0.5|0.5}}

{{0.1|0.6, 0.3|0.4}, {0.6|1}}

China

{{0.3|0.7, 0.4|0.3}, {0.4|0.6, 0.5|0.4}}

{{0.6|1}, {0.2|0.5, 0.1|0.5}}

{{0.7|1}, {0.2|1}}

{{0.1|0.45, 0.3|0.55}, {0.5|0.5, 0.65|0.5}}

Norway

{{0.2|0.2, 0.1|0.8}, {0.7|1}}

{{0.2|1}, {0.8|1}}

{{0.2|0.8, 0.3|2}, {0.6|1}}

{{0.35|1}, {0.5|0.5, 0.6|0.5}}

Table 5.5 The scores of the given six countries Countries

The USA

Canada

Russia

Denmark

China

Norway

Scores

0.5298

0.0948

0.0499

0.3269

0.1682

-0.3508

the USA > Denmark > China > Canada > Russia > Norway. If we divide the risk into six levels, then risk division map for these countries can be shown in Fig. 5.4. To evaluate the performance of the aggregated results and the reliability of the assessment results, we calculate the entropy for each country and their deviation degrees by (5.13) and (5.21). Based on the cloud generating algorithm, we draw the clouds of the aggregated results for each country (see Fig. 5.5). From the cloud shapes, we can see that the cloud droplets converge to the expectations s for the USA and Denmark, indicating the aggregated results of these two countries are of less uncertainty. The PDHF assessments provided by each expert are more consistent and trustworthy for these two countries. The thickness and width of clouds for Canada, Russia and China are much more similar, but the convergent degrees of the droplets are smaller than those of the USA and Denmark. It implies that the uncertainty and consistency of the PDHF information provided by each expert for those countries are at equal levels. Besides, all these droplets locate in the valid areas and their tails are relatively sparse, implying the vagueness and uncertainty of their source information are acceptable and the aggregated results are also reliable. However, the cloud for Norway is significantly different from all the others. Although the thickness of the cloud is acceptable, the droplets are too scattered horizontally. This distribution means that the major problem for the aggregated result lies in not the fuzziness of its evaluation values but in the information consistency among the experts. That is to say,

5.5 Application to the Arctic Geopolitics Risk Evaluations

109

Fig. 5.4 The risk division map for the six countries

despite the aggregated results and the calculated score function, the comparison result for Norway seems questionable. Such a result also indicates that the information provided by the individuals is not adequate or some experts may give improper judgments for this country. It is necessary to conduct a new round of assessment or make some readjustment for the evaluation of Norway so as to guarantee a more reliable risk evaluation result.

5.6 Remarks This chapter studies a novel concept named PDHFS to describe the aleatory uncertainty and epistemic uncertainty in a single framework. To apply the PDHFS to the practical decision making problems, the definition, general formulation, basic operation laws, and aggregation operators are studied in detail. A visual analysis method based on the entropy and the cloud model is also discussed in this chapter, which aims at estimating the information distribution of the aggregated results and improving the reliability of evaluation results. The application of the PDHFS in this chapter exemplifies the introduced decision making approach and proves its validity. The information provided taking different

110

5 The Decision Making Method Under the Probabilistic …

Fig. 5.5 Clouds of the aggregated results for the given six countries (Droplets N = 1000)

forms (multiple types of fuzzy sets under probabilistic environment) has been successfully aggregated. This case also shows the potential application of the PDHFSs in multi-type data fusion. It is necessary to point out that the PDHFS is a new area and this chapter just starts the pilot research. One of the urgent issues is the estimation of ignorance. Currently, we take the strategy that the ignorance is equally assigned to each element in a PDHFE in this chapter. This may be too simple for practical applications. Some prior knowledge or weight information can be utilized to assign and estimate the ignorance appropriately. As a result, it is suggested to make a further study on the normalization process under probabilistic situations and focus more on dealing with the ignorance of the probabilities.

References Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 Baudrit C, Dubois D, Guyonnet D (2006) Joint propagation and exploitation of probabilistic and possibilistic information in risk assessment. IEEE Trans Fuzzy Syst 14(5):593–608 Dymova L, Sevastjanov P (2012) The operations on intuitionistic fuzzy values in the framework of Dempster-Shafer theory. Knowl-Based Syst 35:132–143 Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144

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Hao ZN, Xu ZS, Zhao H, Su Z (2017) Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl-Based Syst 127:16–28 Hu JH, Zhang XL, Chen XH, Liu YM (2016) Hesitant fuzzy information measures and their applications in multi-criteria decision making. Int J Syst Sci 47(1):62–76 Li DY, Liu CY, Gan WY (2009) A new cognitive model: Cloud model. International Journal of Intelligent Systems 24(3):357–375 Merigo JM (2010) Fuzzy decision making with immediate probabilities. Comput Ind Eng 58(4):651–657 Miyamoto S (2005) Remarks on basics of fuzzy sets and fuzzy multisets. Fuzzy Sets Syst 156(3):427–431 Pang Q, Wang H, Xu ZS (2016) Probabilistic linguistic term sets in multi-attribute group decision making. Inf Sci 369:128–143 Sevastjanov P, Dymova L (2015) Generalised operations on hesitant fuzzy values in the framework of Dempster-Shafer theory. Inf Sci 311:39–58 Skalna I, R˛ebiasz B, Gaweł B, Basiura B, Duda J, Opiła J, Pełech-Pilichowski T (2015) Advances in fuzzy decision making, vol. 333. Springer International Publishing Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539 Wei CP, Yan FF, Rodriguez RM (2016) Entropy measures for hesitant fuzzy sets and their application in multi-criteria decision-making. J Intell Fuzzy Syst 31(1):673–685 Wei GW, Merigó JM (2012) Methods for strategic decision-making problems with immediate probabilities in intuitionistic fuzzy setting. Scientia Iranica 19(6):1936–1946 Xu ZS, Xia MM (2012) Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. Int J Intell Syst 27(9):799–822 Yager RR, Alajlan N (2015) Dempster-Shafer belief structures for decision making under uncertainty. Knowl Based Syst 80:58–66 Yang JB (2001) Rule and utility based evidential reasoning approach for multiattribute decision analysis under uncertainties. Eur J Oper Res 131(1):31–61 Yen J (1990) Generalizing the Dempster-Shafer theory to fuzzy-sets. IEEE Trans Syst Man Cybernet 20(3):559–570 Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353 Zadeh LA (1975) Concept of a linguistic variable and its application to approximate reasoning. Informat Sci 8(3):199–249 Zhai YL, Xu ZS, Liao HC (2016) Probabilistic linguistic vector-term set and its application in group decision making with multi-granular linguistic information. Appl Soft Comput 49:801–816 Zhang GQ, Ma J, Lu J (2009) Emergency management evaluation by a fuzzy multi-criteria group decision support system. Stoch Env Res Risk Assess 23(4):517–527 Zhang YX, Xu ZS, Wang H, Liao HC (2016) Consistency-based risk assessment with probabilistic linguistic preference relation. Appl Soft Comput 49:817–833 Zhao N, Xu ZS (2015) Entropy measures for dual hesitant fuzzy information. In: Tomar G (ed) 2015 Fifth international conference on communication systems and network technologies. IEEE, New York, pp 1152–1156 Zhao N, Xu ZS, Liu FJ (2015) Uncertainty measures for hesitant fuzzy information. Int J Intell Syst 30(7):818–836 Zhu B, Xu ZS (2014) Some results for dual hesitant fuzzy sets. J Intell Fuzzy Syst 26(4):1657–1668 Zhu B, Xu ZS, Xia MM (2012) Dual hesitant fuzzy sets. J Appl Math 22(1):100–121

Appendix A

The appendix discusses the statistical characteristics of the intuitionistic fuzzy DFT in Chap. 4 and the mathematical formula of probability for options at fixed time. The weight vector W (t) is assumed to change over time and comply with the stationary stochastic process. In this paper, we assume that the weight vector is independent and identically distributed (iid). Hence, the expectation and covariance of the weights can be calculated, that is, E(W (t)) = w · h

(A.1)

   Cov(W (t)) = E (W (t) − wh)(W (t) − wh) = ψ · h

(A.2)

where ψ can be calculated by ψ = diag(w) − w · w . Since the time step h is set in advance and h will close to 0 to describe the continuous process, the mean vector and the covariance vector are constant across time. The valence and the preference can be viewed as the linear transformation of the weight vector, which implies that the valence and the preference are also stochastic process. A compromise subtraction law is adopted to calculate the valence vector,  the valence vector V = C ⊗ M ⊗ W (t) degenerates to V = K · W (t), where K is the contrast matrix in the form of real numbers. Then the mean of the valence vector is    (A.3) E(V (t)) = E C ⊗ M ⊗ W (t) = E(K · W (t)) = μ · h and the covariance matrix is    Cov(V (t)) =Cov C ⊗ M ⊗ W (t) = K · Cov(W (t)) · K =K · (ψ · h) · K = h · 

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9

(A.4)

113

114

Appendix A

As for the preference vector, the accumulated expression of P(t) can be rewritten as: P(t) =S P(t − h) + V (t) = S P(nh − h) + V (nh) =

n−1 

Si V (nh − i h) + Sn P(0)

(A.5)

i=0

The eigenvalues of the feedback matrix are assumed to be less than one in magnitude, which guarantee the stableness of the system. The expectation is calculated as: ξ (t) = E( P(t)) = E(S P(t − h) + V (t))   n−1  i n S V (nh − i h) + S P(0) =E i=0

  = (I − S)−1 I − Sn μ · h + Sn P(0)

(A.6)

The eigenvalues of S are less than one in magnitude, which implies that as t → ∞, ξ (t) → (I − S)−1 μ · h. This formula provides an asymptotic analysis for the feedback matrix’s influence on the mean preference. Accordingly, the covariance matrix of the preference is Ω(t) = Cov( P(t))   = E ( P(t) − E( P(t)))( P(t) − E( P(t))) =

n−1 

Si (h · Φ)Si



(A.7)

i=0

as t → ∞, i → ∞, then Si → 0. As a result, Ω(t) → Ω(∞). In view of the iid assumption of the weight vector and the multivariate central limit theorem, we can also conclude that the preference P(t) converges into a Gaussian distribution with the mean ξ (t) and the variance-covariance matrix Ω(t). The small time step h will ensure the approximation of the diffusion process. Most importantly, the preference state will also converge to the Gaussian more rapidly. The probability of choosing a certain  option i of the choice  set at time T can be expressed as Probt=T Pi (T ) = max P j (T ), j = 1, . . . , n . For the binary choice set, the probability of choosing option A at time T is: Probt=T (A|{A, B}) = Probt=T (P1 (T ) − P2 (T ) > 0) ∞√ 2 = ∫ 2πλe−(x−δ) /2λ cdx 0

(A.8)

Appendix A

115

where x = P1 (T ) − P2 (T ), δ = E(P1 (T ) − P2 (T )) = ξ1 (T ) − ξ2 (T ) and λ = D(P1 (T ) − P2 (T )) = ϕ11 (t) + ϕ22 (T ) − 2ϕ21 (T ). For the ternary choice, the probability of choosing the option A over the other options is calculated by Probt=T (A|{A, B, C}) = Probt=T (P1 (T ) − P2 (T ) > 0 ∧ P1 (T ) − P3 (T ) > 0)  √ (A.9) = ∫ 2π |Λ|0.5 e−(X−δ) Λ(X−δ)/2 dX X >0



1 −1 0 . 1 0 −1 It is worth noting that if the alternatives of the choice set are large, the integral will be complicated to calculate. A more simplified solution is to calculate the total probability of choosing a certain option over others in the N times simulations. For the threshold criteria decision rule, the threshold is generally set with respect to the covariance of the valence in case of the deliberation process might stop too early or sustain too long. For more analysis and mathematical discussions, please refer to the existing works (Berkowitsch et al. 2014; Busemeyer and Diederich 2002; Busemeyer and Townsend 1993). where X = L P(t), δ = Lξ (T ), Λ = LΩ(T )L  and L =

References

Berkowitsch NAJ, Scheibehenne B, Rieskamp J (2014) Rigorously testing multialternative decision field theory against random utility models. J Exper Psychol-General 143(3):1331–1348 Busemeyer JR, Diederich A (2002) Survey of decision field theory. Math Soc Sci 43(3):345–370 Busemeyer JR, Townsend JT (1993) Decision field theory: a dynamic-cognitive approach to decision making in an uncertain environment. Psychol Rev 100(3):432–459

Appendix B

This appendix provides the calculated comprehensive matrices and the aggregated values in Chap. 5. For convenience of presentation, we provide the calculated values of the comprehensive PDHFS matrix separately (Tables B.1, B.2, B.3 and B.4). The criteria abbreviations in each table are the same as those defined in Sect. 5.5. Based on the comprehensive matrices, we can get the aggregated fuzzy vectors in the form of the PDHFS. Similarly, the components of the vectors are provided separately for convenience of reading. The superscripts indicate the index of the countries in the above tables. For example, h (1) represents the final membership degrees of the USA over all criteria. The membership degree h(x) and the nonmembership degree g(x) and their corresponding probabilities are shown as follows: Table B.1 The value of h(x) in the comprehensive matrix on each criterion h(x)

MC

DD

EI

MR

USA

{0.5301, 0.4877, 0.4523}

{0.7893}

{0.3850, 0.2935}

{0.6331, 0.6000, 0.6536, 0.6224}

Canada

{0.5817, 0.6089}

{0.3185}

{0.4373, 0.4196, 0.5126, 0.4974}

{0.2811, 0.3000}

Russia

{0.5296, 0.5405, 0.6674, 0.6751}

{0.4853, 0.4714}

{0.121}

{0.2718, 0.3320, 0.3693, 0.4215}

Denmark

{0.1577, 0.2000}

{0.3779, 0.3525, 0.5181, 0.4984}

{0.5063, 0.5827}

{0.1515, 0.1312, 0.2517, 0.2338}

China

{0.2379, 0.2945}

{0.5442}

{0.7551, 0.6984, 0.7186, 0.6536}

{0.1499, 0.2503}

Norway

{0.1708, 0.2005, 0.1930, 0.2219}

{0.1931, 0.2517, 0.2176, 0.2744}

{0.2211, 0.2517, 0.2714, 0.3000}

{0.3737}

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 Z. Hao et al., Several Intuitionistic Fuzzy Multi-Attribute Decision Making Methods and Their Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-3891-9

117

118

Appendix B

Table B.2 The probabilistic information ph(x) corresponding to h(x) in the comprehensive matrix ph(x)

MC

DD

EI

MR

USA

{0.2000, 0.2000, 0.6000}

1

{0.4000, 0.6000}

{0.3500, 0.3500, 0.1500, 0.1500}

Canada

{0.5000, 0.5000}

1

{0.5600, 0.1400, 0.2400, 0.06000}

{0.3000, 0.7000}

Russia

{0.06000, 0.5400, 0.04000, 0.3600}

{0.5000, 0.5000}

1

{0.4200, 0.2800, 0.1800, 0.1200}

Denmark

{0.7000, 0.3000}

{0.3000, 0.3000, 0.2000, 0.2000}

{0.3000, 0.7000}

{0.4800, 0.1200, 0.3200, 0.08000}

China

{0.7000, 0.3000}

1

{0.5400, 0.3600, 0.06000, 0.04000}

{0.4500, 0.5500}

Norway

{0.08000, 0.1200, 0.3200, 0.4800}

{0.2400, 0.1600, 0.3600, 0.2400}

{0.5400, 0.3600, 0.06000, 0.04000}

1

Table B.3 The value of g(x) in the comprehensive matrix on each criterion g(x)

MC

DD

EI

MR

USA

{0.3798}

{0.1737, 0.1884}

{0.3429, 0.3162, 0.3757, 0.3464}

{0.2893}

Canada

{0.2828}

{0.6284, 0.6656}

{0.2713, 0.2402, 0.2595, 0.2297}

{0.4874}

Russia

{0.2009}

{0.2169, 0.2000}

{0.7483, 0.6996}

{0.4899}

Denmark

{0.6328, 0.6136}

{0.2526, 0.2392}

{0.2285, 0.3614}

{0.5223, 0.4816}

China

{0.6158, 0.7286}

{0.3162, 0.2236}

{0.1628}

{0.4605, 0.4397, 0.5250, 0.5013}

Norway

{0.6249}

{0.5698, 0.6242}

{0.5926}

{0.3939, 0.4850, 0.4315, 0.5313}

Table B.4 The probabilistic information qg(x) corresponding to g(x) in the comprehensive matrix qg(x)

MC

DD

EI

MR

USA

1

{0.8000, 0.200}

{0.2800, 0.1200, 0.4200, 0.1800}

1

Canada

1

{0.6000, 0.400}

{0.1500, 0.1500, 0.3500, 0.3500}

1

Russia

1

{0.5000, 0.500}

{0.6000, 0.4000}

1

Denmark

{0.1000, 0.900}

{0.5000, 0.500}

{0.5000, 0.5000}

{0.04000, 0.06000}

China

{0.6000, 0.400}

{0.5000, 0.5000}

1

{0.4500, 0.05000, 0.4500, 0.05000}

Norway

1

{0.4000, 0.6000}

1

{0.03000, 0.2700, 0.07000, 0.6300}

Appendix B

119

⎧ ⎫ ⎨ 0.587, 0.5762, 0.5676, 0.5695, 0.5582, 0.5492, 0.5798, 0.5688, ⎬ h (1) = 0.5601, 0.562, 0.5505, 0.5414, 0.5917, 0.581, 0.5725, 0.5744, ⎩ ⎭ 0.5632, 0.5544, 0.5846, 0.5737, 0.5651, 0.567, 0.5556, 0.5466   0.4382, 0.4494, 0.433, 0.4443, 0.4619, 0.4727, 0.4569, 0.4678, (2) h = 0.4412, 0.4524, 0.436, 0.4473, 0.4648, 0.4755, 0.4598, 0.4706 ⎫ ⎧ 0.3695, 0.3739, 0.4317, 0.4357, 0.3661, 0.3705, 0.4287, 0.4327, ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0.3802, 0.3846, 0.4414, 0.4454, 0.3769, 0.3813, 0.4385, 0.4424,⎪ h (3) = ⎪ 0.3873, 0.3917, 0.4478, 0.4517, 0.3841, 0.3884, 0.4449, 0.4488, ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0.39780.4021, 0.4573, 0.4611, 0.3946, 0.3989, 0.4544, 0.4582 ⎫ ⎧ 0.3236, 0.334, 0.3182, 0.3286, 0.3573, 0.3672, 0.3521, 0.3621, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.3569, 0.3668, 0.3517, 0.3617, 0.3889, 0.3983, 0.384, 0.3935, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.3204, 0.3309, 0.315, 0.3255, 0.3543, 0.3642, 0.3491, 0.3591, ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0.3539, 0.3638, 0.3487, 0.3587, 0.386, 0.3955, 0.3811, 0.3906, ⎪ (4) h = ⎪ ⎪ 0.3404, 0.3505, 0.3351, 0.3453, 0.3733, 0.3829, 0.3682, 0.3779,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.3729, 0.3825, 0.3678, 0.3775, 0.4041, 0.4132, 0.3993, 0.4085,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.3373, 0.3475, 0.332, 0.3422, 0.3703, 0.3800, 0.3652, 0.3750, ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0.3699, 0.3796, 0.3648, 0.3746, 0.4013, 0.4105, 0.3965, 0.4057   0.5000, 0.5114, 0.4678, 0.4800, 0.4788, 0.4907, 0.4452, 0.4579, (5) h = 0.5124, 0.5235, 0.4810, 0.4929, 0.4917, 0.5033, 0.4590, 0.4713 ⎫ ⎧ 0.2348, 0.2432, 0.241, 0.2493, 0.2463, 0.2545, 0.2524, 0.2605, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.2395, 0.2478, 0.2457, 0.2539, 0.2509, 0.2591, 0.2570, 0.2650, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.2440, 0.2522, 0.2501, 0.2582, 0.2553, 0.2634, 0.2613, 0.2693, ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0.2486, 0.2568, 0.2547, 0.2628, 0.2599, 0.2679, 0.2658, 0.2738, ⎪ h (6) = ⎪ 0.2500, 0.2582, 0.2561, 0.2642, 0.2612, 0.2693, 0.2672, 0.2752,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.2546, 0.2627, 0.2606, 0.2687, 0.2658, 0.2738, 0.2717, 0.2796, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.2590, 0.2670, 0.2649, 0.2729, 0.2701, 0.278, 0.2759, 0.2838, ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0.2635, 0.2715, 0.2695, 0.2774, 0.2745, 0.2824, 0.2804, 0.2882 ⎧ ⎫ ⎪ ⎨ 0.0280, 0.0280, 0.0840, 0.0420, 0.0402, 0.1260, 0.0280, 0.0280,⎪ ⎬ ph (1) = 0.0840, 0.0420, 0.0420, 0.1260, 0.0120, 0.0120, 0.0360, 0.0180, ⎪ ⎪ ⎩ ⎭ 0.0180, 0.0540, 0.0120, 0.0120, 0.0360, 0.0180, 0.0180, 0.0540   0.0840, 0.0840, 0.0210, 0.0210, 0.0360, 0.0360, 0.0090, 0.0090, ph (2) = 0.1960, 0.1960, 0.0490, 0.0490, 0.0840, 0.0840, 0.0210, 0.0210

120

Appendix B

⎫ ⎧ 0.0126, 0.1134, 0.0084, 0.0756, 0.0126, 0.1134, 0.0084, 0.0756,⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0.0084, 0.0756, 0.0056, 0.0504, 0.0084, 0.0756, 0.0056, 0.0504,⎪ ph (3) = ⎪ 0.0054, 0.0486, 0.0036, 0.0324, 0.0054, 0.0486, 0.0036, 0.0324,⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0.0036, 0.0324, 0.0024, 0.0216, 0.0036, 0.0324, 0.0024, 0.0216 ⎫ ⎧ 0.0302, 0.0130, 0.0302, 0.0130, 0.0202, 0.0086, 0.0202, 0.0086,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0706, 0.0302, 0.0706, 0.0302, 0.0470, 0.0202, 0.0470, 0.0202,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0076, 0.0032, 0.0076, 0.0032, 0.0050, 0.0022, 0.0050, 0.0022, ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0.0176, 0.0076, 0.0176, 0.0076, 0.0118, 0.0050, 0.0118, 0.0050, ⎪ ph (4) = ⎪ 0.0202, 0.0086, 0.0202, 0.0086, 0.0134, 0.0058, 0.0134, 0.0058, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0470, 0.0202, 0.0470, 0.0202, 0.0314, 0.0134, 0.0314, 0.0134, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0050, 0.0022, 0.0050, 0.0022, 0.0034, 0.0014, 0.0034, 0.0014,⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0.0118, 0.0050, 0.0118, 0.0050, 0.0078, 0.0034, 0.0078, 0.0034   0.1701, 0.0729, 0.1134, 0.0486, 0.0189, 0.0081, 0.0126, 0.0054, ph (5) = 0.2079, 0.0891, 0.1386, 0.0594, 0.0231, 0.0099, 0.0154, 0.0066 ⎫ ⎧ 0.0104, 0.0156, 0.041, 0.0622, 0.0069, 0.0104, 0.0276, 0.0415, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0156, 0.0233, 0.0622, 0.0933, 0.0104, 0.0156, 0.0415, 0.0622,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0069, 0.0104, 0.0276, 0.0415, 0.0046, 0.0069, 0.0184, 0.0276, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0.0104, 0.0156, 0.0415, 0.0622, 0.0069, 0.0104, 0.0276, 0.0415,⎬ ph (6) = ⎪ ⎪ ⎪ 0.0012, 0.0017, 0.0046, 0.0069, 0.0008, 0.0012, 0.0031, 0.0046,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0017, 0.0026, 0.0069, 0.0104, 0.0012, 0.0017, 0.0046, 0.0069,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0008, 0.0012, 0.0031, 0.0046, 0.0005, 0.0008, 0.0020, 0.0031,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0.0012, 0.0017, 0.0046, 0.0069, 0.0008, 0.0012, 0.0031, 0.0046 g (1) = {0.2983, 0.3032, 0.2911, 0.2959, 0.3066, 0.3116, 0.2992, 0.3041} g (2) = {0.3654, 0.3696, 0.3523, 0.3563, 0.3605, 0.3647, 0.3476, 0.3516} g (3) = {0.3617, 0.3559, 0.3545, 0.3488}  g

(4)

=

g

=



0.3674, 0.3640, 0.3634, 0.3600, 0.4215, 0.4176, 0.4169, 0.4131 

(5)

0.3734, 0.3699, 0.3693, 0.3659, 0.4284, 0.4244, 0.4237, 0.4198,

0.3412, 0.3588, 0.3183, 0.3348, 0.3380, 0.3555, 0.3154, 0.3317,



0.3502, 0.3684, 0.3268, 0.3437, 0.3470, 0.3650, 0.3238, 0.3405

g (6) = {0.5505, 0.5607, 0.5739, 0.5845, 0.5607, 0.5710, 0.5845, 0.5952}

Appendix B

121

qg(1) = {0.0224, 0.0056, 0.0096, 0.0024, 0.0336, 0.0084, 0.0144, 0.0036} qg(2) = {0.0900, 0.0600, 0.0900, 0.0600, 0.2100, 0.1400, 0.2100, 0.1400} qg(3) = {0.3000, 0.3000, 0.2000, 0.2000}  qg(4) = qg(5) =

0.01000, 0.0900, 0.0100, 0.0900, 0.0100, 0.0900, 0.0100, 0.0900,



0.0150, 0.1350, 0.0150, 0.1350, 0.0150, 0.1350, 0.0150, 0.13500   0.1350, 0.0900, 0.1350, 0.0900, 0.0150, 0.0100, 0.0150, 0.0100, 0.1350, 0.0900, 0.1350, 0.0900, 0.0150, 0.0100, 0.0150, 0.0100

qg(6) = {0.0120, 0.0180, 0.1080, 0.1620, 0.0280, 0.0420, 0.2520, 0.3780}