Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital (Uncertainty and Operations Research) 9811602425, 9789811602429

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Uncertainty and Operations Research

Xiaoli Tian Zeshui Xu

Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital

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

Xiaoli Tian · Zeshui Xu

Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital

Xiaoli Tian Liulin Campus Southwestern University of Finance and Economics Chengdu, China

Zeshui Xu Business School Sichuan University Chengdu, China

ISSN 2195-996X ISSN 2195-9978 (electronic) Uncertainty and Operations Research ISBN 978-981-16-0242-9 ISBN 978-981-16-0243-6 (eBook) https://doi.org/10.1007/978-981-16-0243-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 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

As the development of computer techniques, information updating has sped up and the decision-making environments have been more and more complex. It brings huge challenges for Decision-Makers (DMs) to make a complete rational choice under such complex decision-making circumstance. Hence, the researches about behavioral decision-making with bounded rationality have become more and more popular among researchers around the world. Prospect Theory (PT) is an important theory to explain the bounded rational decision-making under uncertainty, and it has been widely used in behavioral decision making. In addition, fuzzy information has been constantly developed and extended since it was proposed due to its advantage in describing the real perceptions of DMs for decision-making objects. Thus, the Multi-Attribute Decision-Making (MADM) with fuzzy information has been rapidly constructed and widely applied to various fields. Based on the analysis above, in this book, we focus on introducing the MADM methods under different fuzzy circumstances with prospect framework, in which the PT is used to portray the bounded rational characteristics of DMs and the fuzzy information is adopted to depict the real perceptions of them for alternatives in the decision-making process. Also, these methods will be used to solve the decision-making problems in investment field. The detailed contexts of this book are summarized as follows: (1)

Considering the net certain level represented by the shortfall between the membership and non-membership of intuitionistic fuzzy information, the score function of intuitionistic fuzzy information takes the place of the variable of weighting function in PT. Furthermore, the average of evaluation information under each attribute is adopted as the decision-making reference point. According to this, the prospect value of each alternative is calculated. In this book, the detailed steps of the decision-making method with PT under intuitionistic fuzzy circumstance [1] are given in Chap. 2. Then, an illustrative example for investors to select an optimal alternative is conducted to show the feasibility and effectiveness of the given method. Also, a comparative analysis is carried out between this method and the TOPSIS with intuitionistic fuzzy information to show its advantages.

v

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(2)

(3)

(4)

Preface

The QUALIFLEX is a pairwise comparison method for alternatives with respect to each attribute under all possible permutations. Moreover, the linguistic term is a very general way used by DMs to express their real perceptions. In particular, the probabilistic linguistic information, including the probability of each linguistic term, can simulate the vague perceptions of the DMs well. It is common for the DMs to have different risk attitudes for gain and loss when making their decisions under uncertainty, which is well explained by PT. Hence, PT has been integrated into the QUALIFLEX. Then, in this book, a QUALIFLEX based on PT (named as prospect QUALIFLEX) with probabilistic linguistic information [2] is introduced in Chap. 3. In order to show the advantages of this prospect QUALIFLEX, an extended QUALIFLEX with probabilistic linguistic information [2] is given in this chapter as well. The feasibility and validity of those methods have been verified by a numerical example in venture capital. The comparative and simulated analyses show that the former method with prospect framework is more appropriate than the latter one because of the inherent psychological behaviors of the DMs and its excellent ability in identifying the similar alternatives. In this book, the idea of PT has been integrated into the ranking methodPROMETHEE [3] in Chap. 4 as well. Additionally, considering the universality and flexibility of the linguistic information in daily life, the hesitant fuzzy linguistic information is adopted as the basic line of evaluation information of the method. Also, the advantages of group decision-making have been considered in this book. Therefore, a group PROMETHEE based on PT under hesitant fuzzy linguistic circumstance3 is given. Moreover, in order to show its feasibility and availability, other related methods [3] have been introduced in this book as well, such as the extended PROMETHEE and TODIM with hesitant fuzzy linguistic information. The advantages and disadvantages of those methods have been verified by the comparative analysis from an illustrative example, by the sensitive analysis and by the simulation analysis, respectively. Consensus is an important and essential issue, which is deserved to be studied in group decision making. In this book, the PT has been introduced to explore the psychological characteristics of DMs in consensus problem, and the probabilistic hesitant fuzzy preference information is used as the basic line of the evaluation information. Therefore, a consensus model based on the PT under probabilistic hesitant fuzzy preference circumstance [4] is introduced in Chap. 5. To explain the advantages of this model, other consensus models [4] are also given, such as the consensus process based on PT with hesitant fuzzy preference information, the consensus process based on expected theory with probabilistic hesitant fuzzy preference information and hesitant fuzzy preference information correspondingly. Moreover, the idea of variance to measure the fluctuation of data [4] has been introduced in the consensus model to measure the consensus degree of DMs. Then, the measurement and adjustment of consensus based on priority vector [4] is given. Furthermore, these methods are applied to solve a decision-making problem so as to demonstrate

Preface

(5)

vii

their feasibility. Also, the comparative analysis [4] is used to explore the advantages and disadvantages of those methods. Obviously, the method based on PT with probabilistic hesitant fuzzy preference information is better due to the fact that PT reflects the behaviors of DMs in the decision-making process and that the probabilistic hesitant fuzzy preference information includes more original information. Finally, 1000 sets of random decision-making information [4] are produced to demonstrate the difficult degree of reaching consensus among the four methods, and it demonstrates that the consensus with PT is more difficult to achieve than with expected theory, which means that the former one is more precise in reaching consensus and making decisions. According to PT, the improvement of the conventional TODIM [5] is shown in Chap. 6. Because the classical TODIM could not fully reflect the different risk attitudes of DMs and ignores the transformed weighting function in the decision-making process explained by PT. Hence, an improved TODIM which comprehensively considers those behaviors is introduced.

Due to the advantage of probabilistic hesitant fuzzy information in describing the different hesitancy degrees of DMs among several possible hesitancy values, the improved TODIM with probabilistic hesitant fuzzy information [6] is introduced in Chap. 7. The improved TODIM with hesitant fuzzy information [6] is introduced, too. In order to show the advantages of this improved TODIM, the classical TODIM has been given under both probabilistic hesitant fuzzy circumstance and hesitant fuzzy circumstance [6]. Those four methods are used to analyze the investment decisionmaking problem. Then, the comparative analysis about the difference of the results is presented [6]. Also, the sensitive analysis about the parameters in those methods and the simulation analysis with 1000 decision-making information are used to show the advantages of the improved TODIM with probabilistic hesitant fuzzy information [6]. When the improved TODIM is compared with the classical TODIM, the former one includes the transformed weighting function to reflect the real perceptions of DMs is more appropriate. When the TODIM under probabilistic hesitant fuzzy circumstance is compared with TODIM under hesitant fuzzy circumstance, the former one can reflect more evaluation information and it is more flexible. From the perspective of theory, they are presented above. There are also applications for those MADM methods. From the perspective of practical application, they are used to solve the decision-making problems of investors. It not only considers the bounded rational characteristics of investors in the decision-making process, but also reflects the vague perceptions of investors for the objects caused by the limited ability of them, the asymmetric information, etc. It provides an effective way for investors to solve the decision-making problems. Moreover, in this book, the decision-making indices for both the initial selection of project and the sequential decision-making of funding the project or not have been given. In general, PT is adopted as the basic theory to describe the decision-making behaviors of DMs under uncertain decision-making circumstance, and the different types of fuzzy information are used to describe the decision-making information, a set of MADM methods have been introduced based on PT with different fuzzy

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Preface

information. Those methods in this book can be helpful for not only perfecting the decision-making framework in the MADM field but also expanding the research strategy of behavioral science. Moreover, it can promote the integration of crosscurricular interests between fuzzy decision-making and behavioral decision making and play a fundamental role to build the more scientific and effective decision-making theories. This book is suitable for the engineers, technicians and researchers in the fields of fuzzy mathematics, operations research, behavioral sciences, management science and engineering, etc. It can also be used as a textbook for postgraduate and senior-year undergraduate students of the relevant professional institutions of higher learning. This work was supported in part by the National Natural Science Foundation of China under Grant 71771155. Chengdu, China

Xiaoli Tian Zeshui Xu

References 1.

2.

3.

4.

5. 6.

Tian XL, Xu ZS, Jing G, Herrera-Viedma E (2018) How to select a promising enterprise for venture capitalists with prospect theory under intuitionistic fuzzy circumstance? Appl Comput 67:756–763 Tian XL, Xu ZS, Wang XX, Gu J, Alsaadi FE (2019) Decision Models to Find a Promising Start-up Firm with QUALIFLEX under Probabilistic Linguistic Circumstance. Int J Inf Technol Decis Making 18(4):1379–1402 Tian XL, Xu ZS,Gu J (2019) Group decision-making models for venture capitalists: the PROMETHEE with hesitant fuzzy linguistic information. Technol Econ Dev Econ 25(5):743– 773 Tian XL, Xu ZS, Fujita H (2018) Sequential funding the venture project or not? A prospect consensus process with probabilistic hesitant fuzzy preference information. Knowl-Based Syst 161:172–184. Tian XL, Xu ZS, Gu J (2019) An extended TODIM based on cumulative prospect theory and its application in venture capital. Informatica 30(2):413–429. Tian XL, Niu ML, Ma JS, Xu ZS (2020) A novel TODIM with probabilistic hesitant fuzzy information and its application in green supplier selection. Complex: 2540798.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Development of Bounded Rationality . . . . . . . . . . . . . . . . . . . 1.1.2 Development of Fuzzy Information . . . . . . . . . . . . . . . . . . . . . 1.1.3 Importance of Research About Fuzzy Decision Making with PT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Corresponding Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 PT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 TODIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Intuitionistic Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Probabilistic Hesitant Fuzzy Information . . . . . . . . . . . . . . . . 1.2.5 Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . . . . . 1.2.6 Probabilistic Linguistic Information . . . . . . . . . . . . . . . . . . . . 1.3 Aim and Focus of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 4 5 6 7 9 11 13 14

2 Intuitionistic Fuzzy MADM Based on PT . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Decision-Making Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Decision-Making Attributes Used by VCs . . . . . . . . . . . . . . . 2.2.2 Selecting Process and Results Derived by IFPT . . . . . . . . . . . 2.2.3 Selecting Process and Results Derived by TOPSIS . . . . . . . . 2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 18 21 22 23 25 27 28

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Procedure of P-QUALIFLEX with Probabilistic Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Procedure of the Extended QUALIFLEX with Probabilistic Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2

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3.3.1 Results of P-QUALIFLEX with Probabilistic Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Results of the Extended QUALIFLEX with Probabilistic Linguistic Information . . . . . . . . . . . . . . . . 3.4 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Comparison of P-QUALIFLEX with Extended QUALIFLEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Comparison of P-QUALIFLEX with TODIM . . . . . . . . . . . . 3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Group PROMETHEE Based on PT with Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 GP-PROMETHEE with Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 G-PROMETHEE with Hesitant Fuzzy Linguistic Information . . . . . 4.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Decision-Making Background . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Results of the GP-PROMETHEE with Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Results of the G-PROMETHEE with Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Results of TODIM with Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Probabilistic Hesitant Fuzzy Preference Information . . . . . . . . . . . . . 5.2 Consensus Model Based on PT with P-HFPs . . . . . . . . . . . . . . . . . . . 5.2.1 Prospect Consensus Measure with P-HFPs . . . . . . . . . . . . . . . 5.2.2 Procedure of Reaching Prospect Consensus and Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Sequential Decision-Making Attributes . . . . . . . . . . . . . . . . . . 5.3.2 Results of Prospect Consensus with P-HFPs . . . . . . . . . . . . . 5.3.3 Results of the Expected Consensus Process with P-HFPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Results of Prospect Consensus with HFPs . . . . . . . . . . . . . . . 5.3.5 Results of the Expected Consensus with HFPs . . . . . . . . . . . 5.3.6 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simulated Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 40 41 41 43 46 47 49 51 54 56 56 58 62 63 67 73 74 77 79 79 80 81 84 86 86 88 96 100 103 103 106

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5.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6 An Improved TODIM Based on PT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Procedure of the Improved TODIM . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Decision-Making Background . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Results of the Improved TODIM . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Results of the Classical TODIM . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Comparative Analysis Between the Improved and the Classical TODIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Procedure of the Improved TODIM with Probabilistic Hesitant Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Procedure of the Improved TODIM with Hesitant Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Illustrative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Screening Process of the Improved TODIM with Probabilistic Hesitant Fuzzy Information . . . . . . . . . . . . 7.3.2 Screening Process of the Extended TODIM with Probabilistic Hesitant Fuzzy Information . . . . . . . . . . . . 7.3.3 Screening Process of the Improved TODIM with Hesitant Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Screening Process of the Extended TODIM with Hesitant Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Comparative Analysis with the TOPSIS Method . . . . . . . . . . 7.4.2 Sensitivity Analysis Based on the Parameter Values . . . . . . . 7.5 Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 113 113 114 117 118 119 119 121 121 123 125 125 126 129 130 133 134 134 137 144 147 148 149 149 151 152

Chapter 1

Introduction

In this chapter, we concentrate on the background, the importance of research about fuzzy decision making with prospect theory, the preliminaries used in our book and its focus.

1.1 Background Decision making happens anytime and anywhere in our daily life, such as the allocation of public resource, the determination of the subway lines, the selection of vehicle, etc. Those decision-making problems are widely distributed in various fields, including management, economics, engineering, military, etc. Hence, it is important to discuss the decision-making methods which are widely used in decision-making processes. Decision maker (DM), as the most important part in the decision-making process, has a significant effect on the decision-making result. However, under the complex decision-making circumstance, it is hard for the DMs to make decisions according to the complete rational hypothesis in the early classical decision-making theory. With the development of decision science, the decision-making methods considering the bounded rationality of DMs receive much attention from both researches and practitioners. As one of the most famous research results in bounded rational field, prospect theory (PT) [1, 2] has been widely discussed. It combines the advantages of both psychics and economics to describe the psychological characteristics of DMs in decision-making processes. The DMs prefer to consider each alternative from serval different aspects when make their decisions. Then, multi-attribute decision-making (MADM) has been widely used. It distinguishes the best alternative through integrating the obtained evaluation information of each attribute. Moreover, fuzzy information is an effective

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_1

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1 Introduction

way to express the perceptions of DMs. Hence, in this book, we focus on the fuzzy MADM methods with bounded rational psychological characteristics of DMs.

1.1.1 Development of Bounded Rationality According to the expectancy-value theory [3], expected utility theory [4], subjective expected utility theory [5, 6], etc., decision making in early times relies on the future expectation of the alternatives. Those theories built on the assumptions that the alternatives are given and do not change, and all the possible results and their corresponding probabilities are known. The aim of decision making is to make the utility or value to the max one with the known possible results and their corresponding probabilities. These theories promote the development of both economics and management. However, the precondition of those theories is that the DMs are completely rational, which ignores the irrational factors of them. According to the book “Motivation and Personality” [7], written by a psychologist named Maslow, the needs of people come from the lower one to the higher one and they will not pursue higher one until the lower one is satisfied. Hence, the assumption that people always pursued the max utility/value is not reasonable all the time. In reality, the decision making under complete rational framework of the expected utility has some problems according to the Allais Paradox [8], Ellsberg Paradox [9], etc. Hence, the decision making with bounded rationality has been developed, such as PT [1, 2], regret theory [10, 11], overconfidence theory, overreaction theory, etc. Among them, PT [1, 2] is the most widely used and studied one. It is a great breakthrough in depicting the DMs’ irrational behavioral decision under uncertainty. It aims to describe the asymmetric attitudes for gain and loss, and the risk aversion, the transformed weighting function. In this book, we focus on introducing the decisionmaking methods with PT to better simulate the psychological characteristics of DMs in their decision-making processes.

1.1.2 Development of Fuzzy Information Since the fuzzy set with the membership of element was proposed by Zadeh [12], it has been widely extended to depict the evaluation information of DMs under different decision-making situations. Intuitionistic fuzzy set (IFS) [13] was proposed to describe not only the membership but also the non-membership, and then the interval-valued IFS (IVIFS) [14] was given to include the interval number as the membership/non-membership. However, they cannot describe the hesitant situation when the DM is hesitant between several possible evaluation values. Hence, hesitant fuzzy set (HFS) [15] was born. Then, the interval-valued HFS (IVHFS)

1.1 Background

3

[16], the dual HFS (DHFS) [17], the interval-valued intuitionistic HFS (IVIHFS) [18], the probabilistic HFS (P-HFS) [19, 20], etc. have been gradually developed by researchers. Due to the fact that the above fuzzy information is used to describe the quantitative data, the qualitative evaluation information such as “good”, “bad”, “general” is a common way adopted by DMs in their decision-making processes. Hence, the linguistic variable was given [21–23], and it has been extended in various forms. Other techniques to model complex linguistic expressions, such as linguistic 2-tuple [24], virtual linguistic [25], hesitant fuzzy linguistic term set (HFLTS) [26], double hierarchy hesitant fuzzy linguistic term set (DHHFLTS) [27], probabilistic linguistic term set (PLTS) [28], etc., have been developed. The rapid extensions and development of FSs has enriched the expressions for DMs and promoted the improvement of MADM methods.

1.1.3 Importance of Research About Fuzzy Decision Making with PT PT is well known for its ability in dealing with the psychological behaviors of the DMs under uncertainty. It has been widespread in behavioral decision making which is one of the hot topics in the decision-making field in recent years. Meanwhile, TODIM is a famous MADM method derived by the idea of PT. However, the classical PT and TODIM are built on crisp numbers. As the development of economy and society, the decision-making objects and circumstances become more and more complicated. The crisp numbers do not satisfy the needs of the DMs to express their complex and uncertain perceptions for the evaluated objects. Hence, the MADM methods based on PT with FSs and its extensions are necessary to be explored. Since the combination of FSs and PT in MADM methods, it receives much attention from researchers. According to the data in Fig. 1.1 derived from Web of Science, the number of researches for PT/TODIM with fuzzy information has quickly Number of researches 200 150 100 50 0 2010

2011

2012

2013

2014

2015

2016

2017

2018

Fig. 1.1 The number of researches on PT/TODIM with fuzzy information

2019

4

1 Introduction

increased in recent three years, which also demonstrates that the combination of the idea of PT with FSs has been more and more popular.

1.2 Corresponding Preliminaries The basic knowledge of this book is given in this chapter, including the decisionmaking framework-PT [1, 2], decision-making information (such as intuitionistic fuzzy information, probabilistic hesitant fuzzy information, hesitant fuzzy linguistic information, probabilistic linguistic information, etc.) and the corresponding operational rules.

1.2.1 PT PT [1, 2] is a great innovation to describe behavioral decision-making of individuals under bounded rationality through the prospect value V (xi ) which is the product of the value function v(xi j ) and the weighting function w( p j ). A MADM problem can be abstracted as a decision matrix X obtained from DMs, which includes all the available alternatives A = {A1 , A2 , . . . , An } and all the attributes C = {c1 , c2 , . . . , cm }. For convenience, let N = {1, 2, . . . , n} and M = {1, 2, . . . , m}. V (xi ) =

m 

v(xi j )w( p j )

(1.1)

j=1

 v(xi j ) =

w( p j ) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−λ(x0 − xi j )β xi j − x0 < 0 xi j − x0 ≥ 0 (xi j − x0 )α pδj

1

[ pδj +(1− p j )δ ] /δ γ pj

(1.2)

xi j − x0 < 0

1/ xi j − x0 ≥ 0 γ [ p j +(1− p j )γ ] γ

(1.3)

where xi j is the value of the i-th alternative Ai over the attribute c j ; x0 is the reference point; p j is the probability/weighting value of the attribute c j ; β, λ, α, δ and γ are the parameters obtained from experimental economics.

1.2 Corresponding Preliminaries

5

1.2.2 TODIM The classical TODIM [29] is applied to MADM through the measurement of relative dominance degree for each alternative over the others. The ranking result is presented from the comparison of the relative dominance degree of each alternative over the others and based on which the DMs will find the optimal one. The classical TODIM method involves the following steps: Step 1. Obtain the original decision-making information, including the evaluation information X and the vector ω of attribute weights: c1 · · · cm ⎤ A1 x11 · · · x1m m  ⎢ ⎥   X = ... ⎣ ... . . . ... ⎦ = xi j n×m , ω = (ω1 , ω2 , . . . , ωm ), ωj = 1 ⎡

An

(1.4)

j=1

xn1 · · · xnm

where xi j is the value of the j-th attribute for the alternative Ai from DMs; ω j is the original weight of the j-th attribute.     Step 2. Standardize the decision matrix X = xi j n×m into G = gi j n×m , i ∈ N , j ∈ M.  gi j =

xi j , c j is benefit atrribute −xi j , c j is cost atrribute

(1.5)

Step 3. Calculate the relative weight ω jr by (1.6):  ω jr = ω j ωr , r, j ∈ M

(1.6)

where ω j and ωr are the original weights of the attributes c j and cr correspondingly and ωr = max(ω j | j ∈ N ); cr is called a reference attribute. Step 4. Determine the dominance of the alternative Ai over each alternative Ai  (i, i  ∈ N ) depending on (1.7): ψ(Ai , Ai  ) =

m  j=1

where

ϕ j (Ai , Ai  ),∀(i, i  )

(1.7)

6

1 Introduction

⎧   ⎪ ⎪ m ⎪   (gi j − gi  j ) ω  ji ⎪ i f gi j > gi  j ⎪ ω ji  ⎪ ⎪ ⎪ j=1 ⎨ ϕ j (Ai , Ai  ) = 0  i f gi j = gi  j ⎪  ⎪   ⎪ m ⎪  ⎪ −1 ( ⎪ ω ji  )(gi  j − gi j ) ⎪ ⎪ ω ji  i f gi j < gi  j λ ⎩ j=1

(1.8)

ϕ j (Ai , Ai  ) explains the contribution of the attribute c j to the function ψ(Ai , Ai  ) when comparing the dominance of the alternative Ai with the alternative Ai  . The parameter λ shows the attenuation factor of the losses. Three cases will be presented in (1.8): ➀ if gi j > gi  j , then it states a gain; ➁ if gi j < gi  j , then it describes a loss; ➂ if gi j = gi  j , then it depicts a nil, that is, it represents neither gain nor loss. Step 5. Obtain the overall value of the alternative Ai on the basis of (1.9): n 

(Ai ) =

i  =1

maxi {

ψ(Ai , Ai  ) − mini {

n 

i  =1

n 

ψ(Ai , Ai  )}

i  =1

ψ(Ai , Ai  )} − mini {

n 

i  =1

,i ∈ N

(1.9)

ψ(Ai , Ai  )}

Step 6. Rank the overall value (Ai ), i ∈ N , and based on which, the best alternative is then found. The bigger the overall value (Ai ) is, the better the alternative Ai will be.

1.2.3 Intuitionistic Fuzzy Information Intuitionistic fuzzy information was represented by intuitionistic fuzzy set (IFS) [13]. As a kind of more functional information, IFS was developed from fuzzy set [12] which is expressed as: F = {< xi , μ(xi ) > |xi ∈ X }

(1.10)

where μ(xi ) : X → [0, 1] is the membership of F, μ(xi ) ∈ [0, 1]. In reality, not only the membership is given, but also the non-membership is needed. Then, IFS [13] was developed as: I = {< xi , μ(xi ), μ(xi ) > |xi ∈ X }

(1.11)

where μ(xi ) and μ(xi ) express the membership and non-membership of x to I. There is μ(xi ), μ(xi ) ∈ [0, 1] and 0 ≤ μ(xi ) + μ(xi ) ≤ 1. Moreover, when 0 ≤ μ(xi ) + μ(xi ) < 1, μ(xi ) = 1 − μ(xi ) − μ(xi ) is called hesitancy degree.

1.2 Corresponding Preliminaries

7

For the sake of convenience, α = (μα , μα ) is denoted as intuitionistic fuzzy number (IFN) [30]. Then, the corresponding score function ρ(α) and accuracy function σ (α) are: ρ(α) = μα − μα

(1.12)

σ (α) = μα + μα

(1.13)

For the IFNs α1 = (μα1 , μα1 ) and α2 = (μα2 , μα2 ), there is: (1) (2) (3)

if there is ρ(α1 ) > ρ(α2 ), then α1 > α2 ; if there is ρ(α1 ) < ρ(α2 ), then α1 < α2 ; if there is ρ(α1 ) = ρ(α2 ), then (1) (2) (3)

if there is σ (α1 ) > σ (α2 ), then α1 < α2 ; if there is σ (α1 ) < σ (α2 ), then α1 > α2 ; if there is σ (α1 ) = σ (α2 ), then α1 = α2 .

Yager extended the IFS to the Pythagorean fuzzy set (PFS) [31] and the general q-rung orthorpair fuzzy set (q-ROFS) [32]. For the PFS, there is: I  = {< xi , μ(xi ), μ(xi ) > |xi ∈ X }

(1.14)

where μ(xi ) and μ(xi ) are membership and non-membership of I  respectively. Moreover, there is μ(xi ), μ(xi ) ∈ [0, 1] and 0 ≤ μ2 (xi ) + μ2 (xi ) ≤ 1. For the q-ROFS, there is: I  = {< xi , μ(xi ), μ(xi ) > |xi ∈ X }

(1.15)

where μ(xi ) and μ(xi ) are membership and non-membership of I  respectively. Moreover, there is μ(xi ), μ(xi ) ∈ [0, 1] and 0 ≤ μq (xi ) + μq (xi ) ≤ 1. It degrades to IFS when q = 1 and it degrades to PFS when q = 2.

1.2.4 Probabilistic Hesitant Fuzzy Information The probabilistic hesitant fuzzy information was presented by probabilistic hesitant fuzzy set (P-HFS) [19, 20]. The P-HFS is the extension of HFS [15] which has been widely used in MADM [33]. Also, the DHFS has been extended [17, 34]. However, the P-HFS assigns a probability to each piece of hesitant fuzzy value, which makes a good expression of original perception of decision makers (DMs) for alternatives. Let A = {A1 , A2 , . . . , Ai } (i ∈ N , N = {1, 2, . . . , n}) be the alternative set and X be a fixed set, a P-HFS on X is expressed as:

8

1 Introduction

H = {< xi , h xi ( pxi ) > |xi ∈ X }

(1.16)

where h xi (·) is named as probabilistic hesitant fuzzy element (P-HFE). It is a set of some values in [0, 1] and it includes all the possible membership degrees for  x ∈ X to the set H. Moreover pxi shows the probability of h xi (·), and pxi = 1. For convenience, h xi ( pxi ) is simply symbolized as h( p): h( p) = {h t ( p t )|t = 1, 2, . . . , #h( p) }, where #h( p) is the number of possible membership #h( p) #h( p) degrees and t=1 p t = 1. It is interested that t=1 p t < 1 is a common situation in real decision-making circumstance, and it is reasonable to assign #h( p) the incomplete probability information 1 − t=1 p t to each h t ( p t ) averagely #h( p) ˙ p) = [35]. Therefore, when t=1 p t < 1, the corresponding P-HFE willbe h(  #h( #h( p) t ˙ p) t t t  t t ˙ p) } and ˙ = 1, where p˙ = p {h ( p˙ ) t = 1, 2, . . . , # h( t=1 p t=1 p , (t = 1, 2, . . . , #h( p)) [35]. Then, the comparison of P-HFEs is introduced. As is well known, score function, deviation function and distance measure are the general indexes used for the differential analysis between P-HFEs. According to [35], the score function ρ(h( p)) and the deviation function σ (h( p)) are defined as: 

#h( p)

ρ(h( p)) =

p × h (p ) t

t

t=1



t

#h( p) 

σ (h( p)) =

t=1

(1.17)

t=1

#h( p)

( p × (h ( p ) − ρ(h( p))) ) t

pt

t

t

2

#h( p) 

pt

(1.18)

t=1

It is worthy to be noted that the score function and the deviation function have the similar means as the mean value and the variance in statistics. Example 1.1 Let h( p) = {0.78(0.2), 0.7(0.5), 0.65(0.1)} be a piece ˙ p) = of evaluation information, then, its corresponding P-HFE is h( ˙ {0.78(0.25), 0.7(0.625), 0.65(0.125)}. The score function is s(h( p)) = 0.78 × ˙ p)) = 0.25+0.7×0.625+0.65×0.125 = 0.71375 and the deviation function is σ (h( 0.25×(0.78−0.71375)2 +0.625×(0.7−0.71375)2 +0.125×(0.65−0.71375)2 = 0.001723. 0.25+0.625+0.125 To compare the relationship of two P-HFEs h 1 ( p) and h 2 ( p), the comparative rules are given as: (1) (2) (3)

if there is ρ(h 1 ( p)) > ρ(h 2 ( p)), then h 1 ( p) > h 2 ( p); if there is ρ(h 1 ( p)) < ρ(h 2 ( p)), then h 1 ( p) < h 2 ( p); if there is ρ(h 1 ( p)) = ρ(h 2 ( p)), then the deviation function is used to analyze the relationship of the two P-HFEs:

1.2 Corresponding Preliminaries

(1) (2) (3)

9

if there is σ (h 1 ( p) > σ (h 2 ( p)), then h 1 ( p) < h 2 ( p); if there is σ (h 1 ( p) < σ (h 2 ( p)), then h 1 ( p) > h 2 ( p); if there is σ (h 1 ( p) = σ (h 2 ( p)), then h 1 ( p) = h 2 ( p).

Moreover, distance measure is an important and essential tool to find the difference of P-HFEs. However, the same number of possible membership degrees is the precondition of distance measure between P-HFEs. If #h 1 ( p) is smaller than #h 2 ( p), then the number of #h 2 ( p) − #h 1 ( p) possible membership degrees should be added to h 1 ( p). Due to the special background of this book that pursuing huge revenue with high risk is the fundamental characteristic of VC, we should add the biggest membership degree with the probability of zero to h 1 ( p). It is obvious that such adding rules do not change the values of score function and deviation function. Next, the ordered P-HFS satisfies the following prerequisites: (1)

(2)

if there is p t h t ( p t ) < p t+1 h t+1 ( p t+1 ), then the P-HFS is an ascending order one. On the other hand, if there is p t h t ( p t ) > p t+1 h t+1 ( p t+1 ), then it is a descending order one. if there is p t h t ( p t ) = p t+1 h t+1 ( p t+1 ), then the corresponding order is determined by the probability of the possible membership degrees p t and p t+1 . That is, (1) (2)

if there is p t < p t+1 , then the P-HFS is an ascending order one; if there is p t > p t+1 , then the P-HFS is a descending order one.

Based on the ordered P-HFS and the definition of Hamming distance measure, the Hamming distance between the ordered P-HFEs h 1 and h 2 is (1.19): d(H1 , H2 ) =

#h 1 ( p)    1  p t h t ( p t ) − p t h t ( p t ).#h 1 ( p) = #h 2 ( p) 1 1 1 2 2 2 #h 1 ( p) t=1

(1.19)

For the sake of convenience, all the following P-HFSs are the ordered and #h  ( p ) normalized ones and t=1 p t = 1.

1.2.5 Hesitant Fuzzy Linguistic Information Language, as the most common and prevalent expression in our daily life, is an important and popular tool for DMs to deal with the qualitative decision-making problem. For instance, when a venture capitalist is invited to evaluate the product or the service of a start-up firm, he/she tends to use the word such as “bad”, “general” or “good”. When he/she is evaluating the management team of a start-up firm, the sentence such as “the management ability is general” or “the management ability is strong” may be used. Hence, since the linguistic variable was proposed by Zadeh

10

1 Introduction

[21–23], it has been extended in various forms. Other techniques to model complex linguistic expressions such as linguistic 2-tuple [24], virtual linguistic [25], hesitant fuzzy linguistic [26], etc., have been developed. The hesitant fuzzy linguistic information is expressed by the HFLTS which was proposed by Rodríguez et al. [26] to deal with the qualitative MADM problems under uncertain situations in which the DMs are hesitant between several linguistic terms to evaluate the alternatives. For example, when the DMs are asked to evaluate the product or the service of a start-up firm, they are more likely to adopt the word such as “at least good”, “medium”, “a little bad”, and so on. In such cases, it is easy for us to transform those linguistic expressions into the HFLTSs. Let S = {st |t = −ς , . . . , −1, 0, 1, . . . , ς } be a subscript-symmetric linguistic term set [25]. Then, the HFLTS was defined as [36]:  HS (x) = (x, h S (x)|x ∈ X )={< x, stl (x)stl (x) ∈ S, l = 1, 2, . . . , #h S (x) > |x ∈ X }

(1.20) where x is a piece of evaluation information and X is the set of evaluation information. h S (x) is called hesitant fuzzy linguistic element (HFLE) and #h S (x) is the number of linguistic terms in HFLE. For the sake of convenience, h S (x) is shortened as h S . For instance, let S = {s−3 = ver y bad, s−2 = bad, s−1 = a little bad, s0 = medium, s1 = a little good, s2 = good, s3 = ver y good} be the linguistic term set. The evaluation information “at least good” is translated to the HFLTS as: H = (x, h S (x)) = {< x, (s2 , s3 ) >}, because ‘at least good’ means ‘good’ or ‘very good’ which is represented by s2 or s3 . Please refer to [26, 37, 38] for more details. It is necessary for us to introduce the comparison rules for any two HFLTSs [39, 40], which is important to the practical application. At first, the score function and the deviation function of h S should be presented [39]: ρ(h S ) = σ (h S ) =

1 #h S

1  st = s 1 stl ∈h S l #h S #h S

 stl ,stg ∈h S

(stl − stg )2 = s

1 #h S

#h S

(1.21)

l=1 tl

 stl , stg ∈h S

(tl −tg )2

(1.22)

Next, the comparison rules [39] for two HFLTSs h S and h S will be: (1) (2) (3)

if there is ρ(h S ) > ρ(h S ), then h S  h S ; if there is ρ(h S ) < ρ(h S ), then h S ≺ h S ; if there is ρ(h S ) = ρ(h S ), in this situation, the deviation function can be introduced for further comparison: (1) (2) (3)

if there is σ (h S ) > σ (h S ), then h S ≺ h S ; if there is σ (h S ) < σ (h S ), then h S  h S ; if there is σ (h S ) = σ (h S ), then h S ≈ h S .

1.2 Corresponding Preliminaries

11

Furthermore, #h S = #h S is the precondition of the distance measure of any two HFLTSs h S and h S . If there is #h S = #h S , then the linguistic terms should be added to the shorter one. For instance, if there is #h S < #h S , then, let h + S = max{sl |sl ∈ h S } and h − = min{s |s ∈ h } be the maximum and minimum linguistic terms in h S l l S S respectively. Then, according to [41], the added linguistic term to h S is: − + − sˆ = C 2 (ξ, h + S , 1 − ξ, h S ) = ξ h S ⊕ (1 − ξ ) h S

(1.23)

where ξ (0 ≤ ξ ≤ 1) is the parameter that reflects the risk preferences of the − DMs and C 2 (ξ, h + S , 1 − ξ, h S ) is the convex combination of two linguistic terms [42]. Because the particularity of decision-making problem in venture capital, in this book, it is assumed that ξ =1. In other words, we should add the max linguistic term to the shorter one. For instance, if there are two HFLTSs h 1 = {s0 , s1 , s2 , s3 } and h 2 = {s0 , s1 , s2 }, then, we should add the linguistic term s2 to h 2 , and get the extended h 2 = {s0 , s1 , s2 , s2 }. After that, the distance measure between h 1 and h 2 is exhibited. There are many kinds of distance measures for the HFLTSs such as the Hamming distance, the Euclidean distance, the Hausdorff distance and their extensions. Since it is not the main focus of this book, then the Hamming distance for the HFLTSs is adopted as [36]: d(h S , h S )

 #h  1 S tl − tl  = #h S l=1 2ς + 1

(1.24)

1.2.6 Probabilistic Linguistic Information The probabilistic linguistic information takes the form of probabilistic linguistic term set (PLTS) [28]. It assigns probability to each possible linguistic term that does not lose the original evaluation information of the DMs. Let S = {st |t = −ς , . . . , −1, 0, 1, . . . , ς } be the subscript-symmetric linguistic term set [25]. Then, the definition of PLTS is [43]: #L( p)   L( p) = {L k ( p k ) L k ∈ S, p k ≥ 0, k = 1, 2, . . . , #L( p), p k ≤ 1}

(1.25)

k=1

where L k ( p k ) expresses the k-th linguistic term L k with the probability of p k and the subscript of t k ; #L( p) is the number of linguistic terms in L( p). There are two ways to explain the evaluation information, taking the linguistic term set S = {s−3 , s−2 , s−1 , s0 , s1 , s2 , s3 } (s−3 = ver y bad, s−2 = bad, s−1 = a little bad, s0 = medium, s1 = a little good, s2 = good, s3 = ver y good) and the PLTS L( p) =

12

1 Introduction

{s0 (0.3), s1 (0.45), s2 (0.2)} as an example. One way is from the perspective of group decision making (GDM). All the experts give their opinions about an alternative. Among them, 30% of the experts use the “medium”, and 45% of them think that “a little good” is suitable, then 20% of them give the “good”, while 5% of them give up to show their opinions. Another way is from the perspective of single decision making. When an expert is asked to evaluate an alternative. He/she is not sure among several linguistic term. He/she thinks that there is 30% possibility to be “medium”, and 45% possibility to be “a little good”, then 20% possibility to be “good”, while 5% uncertainty. If L k ( p k ) in L( p) is arranged as descending/ascending order on the basis of the value of t k p k , for this situation, then L( p) is named as an ordered PLTS. Moreover, if t 1 p 1 = t 2 p 2 , then the sequence of L 1 ( p 1 ) and L 2 ( p 2 ) is determined by the values of p 1 and p 2 . #L( p) In addition, if k=1 p (k) < 1 in L( p), then the associated PLTS L  ( p) was defined [28] as: #L( p)   L  ( p) = {L k ( p k ) L k ∈ S, p k ≥ 0, k = 1, 2, . . . , #L( p), p k = 1}

(1.26)

k=1

where t¯ =

= L( p) k=1

k

t p

k

 = L( p) k=1

pk .

The score function ρ(L( p)), the deviation function σ (L( p)) and the distance measure d(L 1 ( p), L 2 ( p)) are the basic tools used to compare the superiority and the degree of difference between PLTSs: ρ(L( p)) = st

where t =

#L( p) k=1

(1.27)

 k

t p

k

#L( p)

pk .

k=1



#L( p)

σ (L( p)) = (

k=1

2 1/ 2

( p (t − t)) ) k

k

#L( p) 

pk

k=1

Suppose that L 1 ( p) and L 2 ( p) are two PLTSs, the comparative rules are: (1) (2) (3)

if there is ρ(L 1 ( p)) > ρ(L 2 ( p)), then L 1 ( p)  L 2 ( p); if there is ρ(L 1 ( p)) < ρ(L 2 ( p)), then L 1 ( p) ≺ L 2 ( p); if there is ρ(L 1 ( p)) < ρ(L 2 ( p)), then (1) (2) (3)

if there is σ (L 1 ( p)) > σ (L 2 ( p)), L 1 ( p) ≺ L 2 ( p); if there is σ (L 1 ( p)) < σ (L 2 ( p)), L 1 ( p)  L 2 ( p); if there is σ (L 1 ( p)) = σ (L 2 ( p)), L 1 ( p) ≈ L 2 ( p).

(1.28)

1.2 Corresponding Preliminaries

13

According to Eqs. (1.27) and (1.28), the superiority between the PLTSs can be obtained. Before introducing the distance between PLTSs, the basic operations should be discussed first. If there is #L 1 ( p) = #L 2 ( p), then, the linguistic term should be added to make #L 1 ( p) = #L 2 ( p). Suppose #L 1 ( p) < #L 2 ( p), the number of linguistic terms #L 2 ( p) − #L 1 ( p) should be added to L 1 ( p). According to the different risk attitudes of DMs, the added linguistic terms will be different. If the DMs are risk aversion, the smallest ones should be added to L 1 ( p), whereas, if the DMs are risk seeking, the biggest ones should be added. In another situation, if the DMs are risk neutral, then the median ones should be added. At the same time, whatever the risk attitudes of the DMs are, the probability of all the added linguistic terms will be zero. In this book, the decision-making problem of VCs, who are dedicated to acquiring high return under high uncertain circumstance, are discussed. It is obvious that VCs are risk seeking. Therefore, the biggest ones with the probability of zero are added to PLTSs. Taking Eq. (1.26) and the added rules of linguistic terms for PLTSs into consideration, the PLTSs are acquired as L 1 ( p) and L 2 ( p). Then, referring to the distance for HFLTSs [36], the distance measure for the PLTSs L 1 ( p) and L 2 ( p) is introduced as: 

d(L 1 ( p),

L 2 ( p))

#L 1 ( p)  1 tk tk = ( p1k × 1 − p2k × 2 )  #L 1 ( p) k=1 τ τ

(1.29)

where t1k is the subscript of the k-th linguistic term with the probability of p1k in L 1 ( p) and such explanation suits t2k and p2k as well. It is worth to mention that the distance here is the relative concept to measure the difference between L 1 ( p) and L 2 ( p). Hence, it could be positive or negative or zero.

1.3 Aim and Focus of This Book The aim of this book is to call the attention to the effect of phycological states of DMs in their decision-making process. The focus of this book is to introduce the fuzzy MADM methods based on PT which is used to describe the unsymmetrical risk attitudes of DMs and the perceived transformed weighting function. In this book, an intuitionistic fuzzy MADM method based on PT is shown in Chap. 2. Also, a QUALIFLEX based on PT with PLTS is introduced in Chap. 3 and in order to demonstrate its advantages, the extended QUALIFLEX with PLTS is given. Considering the generality of GDM, in Chap. 4, a group PROMETHEE based on PT with HFLTS is introduced. At the same time, an extended group PROMETHEE with HFLTS is shown in this chapter and used for comparative analysis. Consensus is an important issue in GDM. Hence, a consensus model based on PT with probabilistic hesitant fuzzy preference relation is considered in this book of Chap. 5. Furthermore, consensus models based on PT with hesitant fuzzy preference relation and based on

14

1 Introduction

expected theory with probabilistic hesitant fuzzy preference relation and with hesitant fuzzy preference relation are also shown in this chapter as well to demonstrate the advantages of former one. In addition, TODIM is recognized as a MADM method based on PT, but it could not deal with the unsymmetrical risk attitudes. In this book, TODIM is improved to fully reflect the idea of PT. Hence, an improved TODIM is shown in Chap. 6 and it is compared with the classical one. Furthermore, the improved TODIM with probabilistic hesitant fuzzy circumstance and hesitant fuzzy circumstance has been given in Chap. 7. Finally, some conclusions and future studies are presented in Chap. 8.

References 1. Kahneman D, Tversky A (1979) Prospect theory-analysis of decision under risk. Econometrica 47(2):263–291 2. Tversky A, Kahneman D (1992) Advances in prospect-theory-Cumulative representation of uncertainty. J Risk Uncertainty 5(4):297–323 3. Wigfield A, Eccles JS (2000) Expectancy-value theory of achievement motivation. Contemp Educ Psychol 25(1):68–81 4. Hey JD, Orme C (1994) Investigating generalizations of expected utility-theory using experimental-data. Econometrica 62(6):1291–1326 5. Becker R (2003) Educational expansion and persistent inequalities of education- Utilizing subjective expected utility theory to explain increasing participation rates in upper secondary school in the Federal Republic of Germany. Eur Sociol Rev 19(1):1–24 6. Zambrano E (2005) Testable implications of subjective expected utility theory. Games Econ Behav 53(2):262–268 7. Maslow AH (1954) Motivation and personality. Harper, New York 8. Mongin P (2019) The Allais paradox: what it became, what it really was, what it now suggests to us. Econ Philos 35(3):423–459 9. Chow CC, Sarin RK (2001) Comparative ignorance and the Ellsberg Paradox. J Risk Uncertainty 22(2):129–139 10. Bell DE (1982) Regret in decision making under uncertainty. Oper Res 30:961–981 11. Loomes G, Sugden R (1982) Regret theory: an alternative theory of rational choice under uncertainty. Econ J 92:805–824 12. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353 13. Atanassov KT, Rangasamy P (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 14. Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy-sets. Fuzzy Sets Syst 31(3):343–349 15. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539 16. Chen N, Xu ZS, Xia MM (2013) Interval-valued hesitant preference relations and their applications to group decision making. Knowl Based Syst 37:528–540 17. Zhu B, Xu ZS, Xia MM (2012) Dual Hesitant fuzzy sets. J Appl Math, 13 18. Zhang ZM (2013) Interval-valued intuitionistic hesitant fuzzy aggregation operators and their application in group decision-making. J Appl Math, 33 19. Zhu B, Xu ZS (2018) Probability-hesitant fuzzy sets and the representation of preference relations. Technol Econ Dev Econ 24(3):1029–1040 20. Zhu B (2014) Decision method for research and application based on preference relation. Southeast University, Nanjing 21. Zadeh LA (1975) Concept of a linguistic variable and its application to approximate reasoning.1. Inform Sci 8(3):199–249

References

15

22. Zadeh LA (1975) Concept of a linguistic variable and its application to approximate reasoning. Inform Sci 8(4):301–357 23. Zadeh LA (1975) Concept of a linguistic variable and its application to approximate reasoning. Inform Sci 9(1)::43–80 24. Herrera F, Martinez L (2000) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752 25. Xu ZS (2005) Deviation measures of linguistic preference relations in group decision making. Omega-Int J Manag Sci 33(3):249–254 26. Rodriguez RM, Martinez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20(1):109–119 27. Gou XJ, Liao HC, Xu ZS, Herrera F (2017) Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: a case of study to evaluate the implementation status of haze controlling measures. Inform Fusion 38:22–34 28. Pang Q, Wang H, Xu ZS (2016) Probabilistic linguistic linguistic term sets in multi-attribute group decision making. Inform Sci 369:128–143 29. Gomes LFAM, Lima MMPP (1991) TODIM: basic and application to multicriteria ranking of projects with environmental impacts. Paris 16(January):113–127 30. Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15(6):1179– 1187 31. Yager RR (2013) Pythagorean fuzzy subsets. IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), Edmonton, Canada, New York: IEEE, 57–61 32. Yager RR (2017) Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst 25(5):1222–1230 33. Yu DJ, Zhang WY, Xu YJ (2013) Group decision making under hesitant fuzzy environment with application to personnel evaluation. Knowl Based Syst 52:1–10. 34. Yu DJ, Li DF (2014) Dual hesitant fuzzy multi-criteria decision making and its application to teaching quality assessment. J Intell Fuzzy Syst 27(4):1679–1688 35. Zhang S, Xu ZS, He Y (2017) Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inform Fusion 38:1–11 36. Liao HC, Xu ZS, Zeng XJ (2014) Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inform Sci 271:125–142 37. Rodriguez RM, Martinez L, Herrera F (2013) A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets. Inform Sci 241:28–42 38. Rodriguez RM, Labella A, Martinez L (2016) An Overview on fuzzy modelling of complex linguistic preferences in decision making. Int J Comput Intell Syst 9:81–94 39. Liao HC, Xu ZS, Zeng XJ (2015) Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making. IEEE Trans Fuzzy Syst 23(5):1343–1355 40. Wei CP, Rodriguez RM, Martinez L (2018) Uncertainty measures of extended hesitant fuzzy linguistic term sets. IEEE Trans Fuzzy Syst 26(3):1763–1768 41. Wei CP, Ren ZL, Rodriguez RM (2015) A hesitant fuzzy linguistic TODIM method based on a score function. Int J Comput Intell Syst 8(4)701–712 42. Delgado M, Verdegay JL, Vila MA (1993) On aggregation operations of linguistic labels. Int J Intell Syst 8(3):351–370 43. Zhang YX, Xu ZS, Wang H, Liao HC (2016) Consistency-based risk assessment with probabilistic linguistic preference relation. Appl Soft Comput 49:817–833

Chapter 2

Intuitionistic Fuzzy MADM Based on PT

The IFS [1] is an effective tool to comprehensively express the uncertain perceptions of DMs for the objects from the perspective of support and opposition. Hence, the extensions of IFS have been popular, such as PFS, q-ROFS, etc. Moreover, PT is famous for its ability in depicting the bounded rational psychological characteristic of DMs under uncertainty. Hence, the decision-making approaches under the PT with fuzzy information, such as interval-valued intuitionistic fuzzy information [2], triangular intuitionistic fuzzy information [3], trapezoidal fuzzy information [4], trapezoidal intuitionistic fuzzy information [5], and so on, have already been studied. Most of those existing researches use the original probability as the weight to integrate the value function. Only few of them considered the impact of nonlinear pattern on the decision-making results and used the weighting function to integrate the value function. However, none of them takes the uncertain psychological state of DMs into account as a component of the weighting function of the PT. The uncertain psychological state of DMs will affect the final result of decision-making. Therefore, in this chapter, such bounded rational psychology will be discussed under intuitionistic fuzzy circumstance, which is named as intuitionistic fuzzy prospect theory (IFPT) [6]. The IFPT considers the score function of the intuitionistic fuzzy value (IFV) as the variable in the weighting function that exhibits such a psychological characteristic of DMs under intuitionistic fuzzy circumstance well and also the value function of the PT has been modified in the IFPT that adapts to the real decision-making situation of DMs. Therefore, it is an effective approach to select an optimal alternative for DMs.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_2

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18

2 Intuitionistic Fuzzy MADM Based on PT

2.1 Decision-Making Procedure Let A = {A1 , A2 , . . . , An } be the set of alternatives, C = {c1 , c2 , . . . , cm } be the set of attributes, (N = {1, 2, . . . , n}, M = {0, 1, . . . , m}, i ∈ N , j ∈ M), X be the set of evaluation information and xi j ∈ X , where xi j denotes the evaluation information given by the DM for the i-th alternative Ai under the j-th attribute c j and it has the characteristic of IFN, that is, ∂i j = (μ∂ (xi j ), μ∂ (xi j )) = (μi j , μi j ) [7]. Then, the prospect f i j = (xi j , ∂i j ), a binary function, describes the situation that the outcome xi j has the membership degree μi j and the non-membership degree μi j . Furthermore, μi j = 1 − μi j − μi j shows the fuzzy degree of the outcome xi j , 0 ≤ μi j ≤ 1, 0 ≤ μi j ≤ 1 and 0 ≤ μi j + μi j ≤ 1. Let C = {c1+ , c2+ , . . . , cm+ , cm− +1 , cm− +2 , . . . , cm− } be the set of attributes, where + c denotes the attribute which the perceived value is no less than zero relative to the reference point, that is, the evaluation information is bigger than or at least equal to the reference point, and c− means that attribute which the perceived value is below zero relative to the reference point, that is, the evaluation information is smaller than the reference point. In general, there are two kinds of attributes: cost and benefit one. For the cost attribute, the smaller evaluation value means the better alternative, whereas, the bigger evaluation value means the better alternative for benefit attribute. For the convenience of calculation, they should be unified to one kind. In this chapter, the cost attribute is transformed to the benefit one through adding the negative sign to the cost evaluation value. For the alternative Ai with the attribute c j , the prospect value can be represented as f i j = (xi j , ∂i j ). Let f i+ represent the positive outcomes of the alternative Ai with the attributes c1+ , c2+ , . . . , cm+ and f i− shows the negative outcomes of the alternative Ai for cm− +1 , cm− +2 , . . . , cm− accordingly. Then, the intuitionistic fuzzy prospect value (IFPV) V ( f i ) of the alternative Ai is given as follows: V ( f i ) = V ( f i+ ) + V ( f i− ) V ( f i+ ) =

gm + 

(2.1)

g + (ρi j + )v(xi j + ) + β + πi j + v(xi j + )

(2.2)

(g − (ρi j − )v(xi j − ) + β − πi j − v(xi j − ))

(2.3)

j + =g1

V(

f i− )

=

lm −  j − =l1

where β + and β − are the parameters of the fuzzy terms in the positive and negative prospects respectively. They are denoted as the random numbers here which come from a normal distribution. Moreover, v(·) is the value function, which is usually expressed as a two-part power function:

2.1 Decision-Making Procedure

19

 v(xi j )

=

λ(xi j + )α i f xi j + ≥ 0 −λ(−xi j − )β i f xi j − < 0

(2.4)

where α and β are the risk attitude parameters; λ and λ are the gain coefficient and loss aversion coefficient respectively; xi j + and xi j − are the variables related to the reference point. As we all know, the value of the reference point will affect the value of xi j and V ( f i ). Therefore, it is necessary for us to choose the reference point carefully [8]. In this chapter, the average value of all enterprises for the same attribute is regarded as the reference point: xi j

  n 1  = xi j − xi j n i=1

(2.5)

where xi j inherits the property of xi j and has the characteristic of the IFV ∂i j = (μi j , μi j ). In addition, the weighting function g(·) is an important part to show the nonlinear perceptions of DMs. According to [9], it adopts the score function instead of the unidimensional probability to be the independent variable in weighting function. It is assumed that DMs make their decisions based on the net certain level represented by the shortfall between the membership and the non-membership in the score function ρi j of the IFV ∂i j rather than the probability in the existing literature. Moreover, this assumption has been demonstrated to be reasonable through the shape of the weighting function resulted by experiment in [9] and in the classical one [10]. It is used to reveal the subjective weights of attribute values in the forms as shown below [11]: g + (ρi j + ) = g − (ρi j − ) =

η(ρi j +

η(ρi j + + 1)γ , x + ≥ 0 + 1)γ + (1 − ρi j + )γ i j

(2.6)

η(ρi j −

η(ρi j − + 1)δ , x + < 0 + 1)δ + (1 − ρi j − )δ i j

(2.7)

where η, γ and δ are the constant parameters acquired from experiments; Both g + (·) and g − (·) belong to [0, 1] which satisfy the following two constraints: g + (−1) = g − (−1) = 0, and g + (1) = g − (1) = 1. Both of them are non-linear and strictly increasing functions in the interval [−1, 1] . The independent variable ρi j of the weighting function is [12]: ρi j = μi j − μi j

(2.8)

When there is V ( f i ) = V ( f i  ), how to find the optimal one from the alternatives according to the equations above? In this situation, the accuracy function σi j is

20

2 Intuitionistic Fuzzy MADM Based on PT

introduced to replace ρi j to deal with this problem. Then, σi j will be the independent variable of the weighting function to calculate the IFPVs and based on which to find the optimal one. On the basis of Ref. [13], σi j is defined as: σi j = μi j + μ¯ i j

(2.9)

where μi j and μi j were introduced earlier. From the introduction above, it is known that there are several parameters in the aforementioned equations. Those parameter values come from experimental economics. Generally, a group of individuals are asked to finish the questionnaire which is about two-outcome gamble crossed with probability. Then, the parameter values are estimated according to the data obtained from questionnaire. The detailed steps of this procedure [6] are shown as follows: Step 1. Get xi j and ∂i j correspondingly. The DMs may give the alternative’s assessment values xi j corresponding to the decision-making factor/attribute c j , which is uncertainly expressed as (xi j , μi j (xi j ), μi j (xi j )), then we transfer cost attributes into benefit ones if necessary. Step 2. Determine xi j according to Eq. (2.5). Step 3. Acquire the weighting value g(ρi j ) depending on (2.6) and (2.7). Step 4. Obtain v(xi j ) based on Eq. (2.4) by xi j . Step 5. Calculate the IFPV of the alternative expressed as V ( f i ) by aggregating v(xi j ) and g(ρi j ) on the basis of (2.1) to (2.3). Step 6. Rank V ( f i ) to find the maximum one which is the promising enterprise accordingly. Step 7. If there are equal IFPVs V ( f i ) = V ( f i  ) as the maximum ones, then we should turn to Step 8; otherwise, end. Step 8. If there are alternatives Ai and Ai  , where V ( f i ) = V ( f i  ), then we should replace ρi j in Eqs. (2.6) and (2.7) with σi j and follow Steps 3 to 6 above. Hereby the optimal alternative between Ai and Ai  will be found. Step 9. End. So far, we have already given a detailed procedure of evaluating the special enterprise and selecting a promising one. A more visual one is shown in Fig. 2.1. As our brief introduction of IFPT, on the one hand, the weighting function with the IFVs can exactly depict the uncertain circumstance of DMs’ decisions. On the other hand, the value function confirms to the psychological states of DMs properly. Because uncertainty lies in the entire processes of VCs’ decisions. An example to demonstrate the use of this approach in the VC field will be presented in the next chapter.

2.2 Illustrative Example

21

Sets of alternatives and attributes Obtain the intuitionistic fuzzy information under each attribute Acquire weighting value

Get the gain or loss relative to reference point Determine parameters Value function

Weighting function

Calculate the IFPV

Is there any equal IFPVs

Accuracy function Yes

No Rank and selection Fig. 2.1 Visual procedure of IFPT

2.2 Illustrative Example Before the selecting procedure, the choice of values for parameters should be discussed. In the classical PT, Tversky and Kahneman [10] estimated that α = β = 0.88, λ = 2.25, γ = 0.61, and δ = 0.69. However, in the experiment conducted by Abdellaoui [14], they concluded that α = 0.89, β = 0.92, γ = 0.6 and δ = 0.7. Moreover, Gonzalez and Wu [11] gave the different forms of value function and weighting function. They pointed out that the parameter in the value function was 0.49 while the parameters in the weighting function were 0.77 and 0.44. It is obvious that the values of those parameters depend on the forms of equations, the experiment background, the statistical approach used to analyze the data obtained from experiment. A lot of literature has studied the values of those parameters [10, 11, 14, 15]. Considering the particularity of the background that our book cares about and the forms of equations that this chapter adopts, the values of the parameters come from the experiment that is conducted by Gu et al. [9]. In this case, the parameters are given as α = 0.93, β = 0.52, λ = 2.25, λ¯ = 1.27, η = 1.08, γ = 0.53, and δ = 0.59, because the experiment circumstance [9] used to obtain them is very close to the background of our case. Also, the numbers β + = 0.62 and β − = 0.49

22

2 Intuitionistic Fuzzy MADM Based on PT

are generated randomly, which satisfy the normal distribution. Then, an illustrative example is presented to demonstrate the superiority of IFPT. In this chapter, an illustrative example in venture capital field has been discussed to demonstrate the advantage of the method in Sect. 2.1. Hundreds of enterprises have been provided by entrepreneurs to VCs as available alternatives. The key challenge for VCs is that how to pick out an optimal one from thousands of alternatives. In particular, when the VCs abandon most of enterprises that perform not as well as expected, determining which one is to be funded from the rest becomes particularly difficult for them. After preliminary screening of enterprises, there are only 3 alternatives left for VCs to select, which are denoted as Ai (i = 1, 2, 3) respectively. Moreover, the attributes used to select an optimal alternative is important for VCs to make their decisions. Hence, in the following, they are discussed firstly [6].

2.2.1 Decision-Making Attributes Used by VCs It is critical for us to know the existing researches about attributes used by VCs to make a better understanding of their decisions and to give an appropriate method for them as decision aid. A majority of literature has indicated that VCs primarily concentrated on the projects’ management team, the potential finance of projects, the market conditions and service or product the project offered when they decided to invest their limited capital [16–21]. In addition, Riquelme and Rickards [22] pointed out that managerial experience was a general factor accepted by all VCs to evaluate a project. Furthermore, Franke et al. [23] discovered that project with the member of management team who had experience of the interrelated industry or had a background of crossed education got the support of venture capital more easily. Whereas, acquiring huge revenue is the ultimate goal of venture capital. Thus, the market with great potential profits and high risk is popular among venture capital, such as software which is the new technology developed rapidly in recent years and biotechnology which has attracted more and more attention in the past few years. Moreover, from an 11-years period of funded enterprise data in a venture capital firm, Petty and Gruber [24] found that VCs considered more about products in the final decision as time goes by. However, the management team, the finance situation, the market conditions and the service or product offered by project are the four general attributes accepted by VCs in the decision-making process [6]. They are explained as follows: (1)

Management team (c1 )

It is clearly recognized by not only investors but also the other important stakeholders that a creative and passionate management team will drive the start-ups to the road of success. Furthermore, educational background or experience in the related industry of entrepreneur or management team and their excellent ability are the decisive factors in the investigation of management team. The VCs prefer the entrepreneurs

2.2 Illustrative Example

23

with higher emotional quotient and intelligence quotient, with independent thought and an open mind, etc. Hence, it is obvious that the management team of the project shows a significant role for VCs in their investment decision-making processes [21]. (2)

Financial situation (c2 )

According to CB Insight,1 lack of capital support leads almost 30% of start-ups to become failing. Although the VCs who provide capital for the project concentrate more on the potential finance of it and acquiring huge profits is the ultimate goal of them, they investigate the current financial situation of the project as well. Also, the pay-back period, return on asset, etc., are considered in the investment decisionmaking process. Market condition (c3 )

(3)

Market demand is dynamic for providing product or service, and it is one of the key factors for the success of start-ups. The reason for the failure of more than forty percent start-ups is lack of effective market demand according to the CB Insight. Moreover, market prospect, market growth rate, market competition level, etc. are the important aspects in the VCs’ decision-making processes. Service or product (c4 )

(4)

When VCs have chosen a target market, they prefer to investigate whether the product or service provided by the optional start-up project is competitive in the aimed market. Also, the acceptability of customers for product or service is an important aspect for VCs in the decision-making process because the customers who consume such product or service are the basic source of earnings. In our brief retrospect and explanation, it is known that VCs pay very close attention to management team of project, the financial situation, the market conditions and the service or product offered by the project and the decision-making process of IFPT is shown in Sect. 2.2.2 [6].

2.2.2 Selecting Process and Results Derived by IFPT Step 1. After being carefully assessed by the VCs, the detailed information of the three enterprises are listed in Table 2.1.2 Step 2. In order to obtain the weighting function g(ρi j ), the first thing we should do is to calculate xi j according to Eq. (2.5) and Table 2.1. The results are shown in Table 2.2. Step 3. Compute the weighting value in accordance with Eqs. (2.6) and (2.7) on the strength of ρi j . The results are shown in Table 2.3. 1 It

is a famous data analysis corporate (http://www.gamelook.com.cn/2014/10/185579). outcome of each attribute ranges from 0 to 100 and has a characteristic of IFV.

2 The

24

2 Intuitionistic Fuzzy MADM Based on PT

Table 2.1 Intuitionistic fuzzy assessment information from VCs c1

c2

c3

c4

A1

(91, (0.59, 0.2))

(84, (0.68, 0.22))

(77, (0.61, 0.27))

(84, (0.79, 0.1))

A2

(76, (0.69, 0.21))

(93, (0.8, 0.12))

(86, (0.71, 0.16))

(85, (0.91, 0.05))

A3

(83, (0.78, 0.16))

(88, (0.59, 0.27))

(92, (0.72, 0.18))

(79, (0.69, 0.16))

Table 2.2 Results of xi j xi j

c1

c2

c3

c4

A1

7.67

−4.33

−8.00

A2

−7.33

4.67

1.00

2.33

A3

−0.33

−0.33

7.00

−3.67

1.33

Table 2.3 Results of weighting values g(ρi j ) g(si j )

c1

c2

c3

c4

A1

0.63

0.66

0.62

0.73

A2

0.67

0.72

0.68

0.81

A3

0.72

0.61

0.67

0.68

Step 4. Calculate v(xi j ) according to Eq. (2.4) by virtue of xi j in Table 2.2, and it is shown in Table 2.4. Step 5. Obtain V ( f i+j ) and V ( f i−j ) by using g(ρi j ) in Table 2.3 and v(xi j ) in Table 2.4 on the basis of Eqs. (2.2) and (2.3). Then, we integrate them as V ( f i ) of the enterprise Ai with Eq. (2.1), shown in Table 2.5. Step 6. In Table 2.5, we know V ( f 2 ) > V ( f 1 ) > V ( f 3 ). Then, we get A2  A1  A3 . Moreover, A2 is the optimal enterprise for VCs. Step 7. End. In order to demonstrate the effectiveness of the selecting approach given in this chapter, the classical TOPSIS approach [25] has been analyzed in the same case. By comparing the ranking results of the two approaches, much more important implication will be revealed. Table 2.4 Results of attribute values v(xi j ) v(xi j )

c1

c2

c3

c4

A1

8.44

−4.82

−6.63

1.66

A2

−6.34

5.32

1.27

2.79

A3

−1.27

−1.27

7.76

−4.42

2.2 Illustrative Example

25

Table 2.5 Prospect value of Ai V ( fi )

A1

A2

A3

−0.23

2.86

−0.99

2.2.3 Selecting Process and Results Derived by TOPSIS The TOPSIS indicates that the promising enterprise has the shortest distance from the positive-ideal one and the farthest distance from the negative-ideal one. If we want to obtain the promising enterprise by using the TOPSIS, then we should follow the steps below [6]: Step 1. Compute the normalized decision matrix according to (2.10) (Table 2.6):   n  ri j = xi j / xi2j

(2.10)

i=1

Step 2. Determine the weighted normalized decision matrix according to (2.11) (Table 2.7): vi j = g(ρi j )ri j

(2.11)

where g(ρi j ) is the weighting value calculated by (2.6) and (2.7). Step 3. Obtain the positive-ideal and negative-ideal ones in accordance with (2.12) and (2.13) respectively:

Table 2.6 Normalized decision-making matrix ri j

c1

c2

c3

c4

A1

52.22

46.08

40.17

49.25

A2

39.91

56.48

50.10

50.43

A3

47.6

50.57

57.34

43.57

Table 2.7 Weighted value matrix vi j

c1

c2

c3

c4

A1

35.80

30.42

24.95

35.77

A2

26.61

40.80

33.84

40.84

A3

34.16

31.10

38.53

29.81

26

2 Intuitionistic Fuzzy MADM Based on PT

Table 2.8 Ideal values c1

c2

c3

c4

A+

35.80

40.80

38.53

40.84

A−

26.61

30.42

24.95

29.81

A+ = {v1+ , v2+ , . . . , vm+ } =





  max vi j  j ∈ B  , min vi j  j ∈ C 

(2.12)

A− = {v1− , v2− , . . . , vm− } =





  min vi j  j ∈ B  , max vi j  j ∈ C 

(2.13)

i

i

i

i

where B  expresses the benefit attribute while C  expresses the cost attribute. (Table 2.8) Step 4. Acquire the separation degree of each enterprise from the positive-ideal one according to (2.14) with the Euclidean distance:    m (vi j − v+j )2 Di+ = 

i = 1, 2, . . . , n

(2.14)

j=1

Similarly, the separation degree of each enterprise from the negative-ideal one can be calculated on the basis of (2.15) (Table 2.9):    m − Di =  (vi j − v−j )2

i = 1, 2, . . . , n

(2.15)

j=1

Step 5. Get the relative closeness of the enterprise Ai with respect to the ideal enterprise A+ according to (2.16):

Ri+ = Di− /(Di− + Di+ )

i = 1, 2, . . . , n

(2.16)

Step 6. Rank the orders of Ri+ (i = 1, 2, 3) obtained in Step 5, and then the largest one is the optimal enterprise. Table 2.9 Separation of alternative to the ideal one

A1

A2

A3

Di+

17.83

10.31

14.77

Di−

10.95

17.56

15.55

2.2 Illustrative Example Table 2.10 Closer degree of alternative to the ideal one

27

Ri+

A1

A2

A3

0.38

0.63

0.51

As introduced above, Ri+ are listed in Table 2.10. It can be seen in the table that > R3+ > R1+ . Then, we know A2  A3  A1 . According to the above analysis, A2 is consistently recognized as the optimal enterprise among the three alternatives with both the given IFPT and the TOPSIS. But there is little difference in ranking results for A1 and A3 with those two approaches. This is because the given method considers more about the psychological states of VCs and the uncertain circumstance of venture capital, and it is more consistent with the decision-making circumstance in reality. Furthermore, the method in Sect. 2.1 is an effective application of the IFPT in the venture capital field. The fundamental difference between this method and that of Gu et al. [9] lies in that an approach with the detailed steps for VCs to select a promising enterprise has been shown in this chapter while a theoretical framework called IFPT was just presented in Ref. [9]. On one hand, the method in Sect. 2.1 is based on this framework and proposes the detailed steps. On the other hand, it enriches the application of this method, and both the psychological states of VCs and the uncertain situation in the venture capital field have been contained in this method. It is more consistent with the real decision-making circumstance of venture capital. R2+

2.3 Remarks The DMs are confronted with the decision-making problems all the time under uncertain circumstances which will produce psychological discomforts. It motivates them to take actions such as decision-making strategy or model to transfer the uncertain situations to the relative certain ones. While, in the decision-making process, the DMs make their decisions with intuition and psychological states involuntarily. So, it is important to simulate the psychological characteristics of DMs’ in their decisionmaking processes. The main goal of this chapter is to give such an appropriate decision-making method that considering the uncertainty which comes from both the psychological characteristics of DMs and the decision-making circumstances. The detailed decision-making procedure is given. The effectiveness and feasibility of the given approach is exemplified by the illustrative example to venture capital field. This approach is of not only theoretical importance but also practical guiding importance. It not only serves as an appropriate alternative for VCs in decisionmaking, but also enriches the application of IFPT which is a theory based on the PT. The given approach and the related analysis can also be extended to other uncertain decision-making situations and play the role of demonstration for future research

28

2 Intuitionistic Fuzzy MADM Based on PT

in other fields. Furthermore, this method may inspire more scholars to integrate overconfidence, regret, disappointment and other behavioral factures with uncertain information, and it is also a good way for the integration of behavior finance and uncertain information. Further researches may pay more attention to the investment behaviors of VCs, and from this prospective, the development of behavioral finance will be advanced. There are some advantages of the method in Sect. 2.1, but the shortcomings may still exist in it. Perhaps, more outcomes of the application of the IFPT will be emerged as we wish.

References 1. Atanassov KT, Rangasamy P (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 2. Meng FY, Tan CQ, Chen XH (2015) An approach to Atanassov’s interval-valued intuitionistic fuzzy multi-attribute decision making based on prospect theory. Int J Comput Intell Syst 8(3):591-605 3. Chen ZS, Chin KS, Ding H, Li YL (2016) Triangular intuitionistic fuzzy random decision making based on combination of parametric estimation, score functions, and prospect theory. J Intell Fuzzy Syst 30(6):3567-3581 4. Wang JQ, Sun T, Society IC (2008) Fuzzy multiple criteria decision making method based on prospect theory. Proceedings of the international conference on information management, innovation management and industrial engineering, Los Alamitos: IEEE Computer Society, vol 1, pp 288-291 5. Li XH, Wang FQ, Chen XH (2014) Trapezoidal intuitionistic fuzzy multiattribute decision making method based on cumulative prospect theory and dempster-shafer theory. J Appl Math 8 6. Tian XL, Xu ZS, Jing G, Herrera-Viedma E (2018) How to select a promising enterprise for venture capitalists with prospect theory under intuitionistic fuzzy circumstance? Appl Soft Comput 67:756-763 7. Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J General Syst 35(4):417–433 8. Dolan P, Robinson A (2001) The measurement of preferences over the distribution of benefits: the importance of the reference point. Europ Econ Rev 45(9):1697–1709 9. Gu J, Wang ZJ, Xu ZS, Chen XZ (2018) A decision-making framework based on the prospect theory under an intuitionistic fuzzy environment. Technol Econ Dev Econ 24(6):2374-2396 10. Tversky A, Kahneman D (1992) Advances in prospect-theory-cumulative representation of uncertainty. J Risk Uncertainty 5(4):297–323 11. Gonzalez R, Wu G (1999) On the shape of the probability weighting function. Cognitive Psychol 38(1):129–166 12. Chen SM, Tan JM (1994) Handling multicriteria fuzzy decision-making problems based on vague set-theory. Fuzzy Sets Syst 67(2):163–172 13. Hong DH, Choi CH (2000) Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114(1):103–113 14. Abdellaoui M (2000) Parameter-free elicitation of utility and probability weighting functions. Manag Sci 46(11):1497–1512 15. Tversky A, Fox CR (1995) Weighing risk and uncertainty. Psychol Rev 102(2):269–283 16. Tyebjee TT, Bruno AV (1984) Venture capital-Investor and investee perspectives. Technovation 2(3):185–208 17. Macmillan IC, Zemann L, Subbanarasimha PN (1987) Criteria distinguishing successful from unsuccessful ventures in the venture screening process. J Bus Vent 2(2):123-137

References

29

18. Hisrich RD, Jankowicz AD (1990) Intuition in venture capital decisions-An exploratory-study using a new technique. J Bus Vent 5(1):49–62 19. Mason C, Stark M (2004) What do investors look for in a business plan? A comparison of the investment criteria of bankers, venture capitalists and business angels. Int Small Bus J 22(3):227–248 20. Carpentier C, Suret JM (2015) Angel group members’ decision process and rejection criteria: A longitudinal analysis. J Bus Vent 30(6):808–821 21. Widyanto HA, Dalimunthe Z (2015) Evaluation criteria of venture capital firms investing on indonesians’ SME. Soc Sci Electron Publishing 22. Riquelme H, Rickards T (1992) Hybrid conjoint-analysis-An estimation probe in new venture decisions. J Bus Vent 7(6):505–518 23. Franke N, Gruber M, Harhoff D, Henkel J (2008) Venture capitalists’ evaluations of startup teams: Trade-offs, knock-out criteria, and the impact of VC experience. Entrepreneurship Theory Pract 32(3):459-483 24. Petty JS, Gruber M (2011) “In pursuit of the real deal” A longitudinal study of VC decision making. J Bus Vent 26(2):172–188 25. Bell DE (1982) Regret in decision making under uncertainty. Oper Res 30:961–981

Chapter 3

QUALIFLEX Based on PT with Probabilistic Linguistic Information

The fundamental principle of QUALIFLEX is to treat the cardinal and ordinal information in a correct way and to take all the possible rankings of alternatives into account. The focus of QUALIFLEX is the pairwise comparison of alternatives with respect to each attribute under all possible permutations. The optimal permutation is recognized through the comprehensive concordance/discordance index, and the best alternative will be identified according to it. It is very useful in handling the MADM problems with a few alternatives and numerous attributes. The conventional QUALIFLEX was proposed by Paelinck [1, 2]. Given that the DMs may change their opinions in a time period and the classification of all alternatives according to each attribute may be imprecise, the sensitivity analysis has been implemented to improve the efficiency of QUALIFLEX [3]. Sometimes, it is difficult for the DMs to give accurate evaluation values owing to the complexity of the decision-making objects and the uncertain decision-making circumstance. Therefore, based on the development of FS, the extension of QUALIFLEX with fuzzy information has been popular. The existing researches about those extensions are given in Table 3.1. From the literature in Table 3.1, it is clear that different types of fuzzy information have been applied to the QUALIFLEX as its extensions. But seldom of them consider the effect of psychological behaviors of the DMs on the decision results. Hence, to fully simulate the psychological behaviors of the DMs, a prospect QUALIFLEX (P-QUALIFLEX) [22] is given in this chapter based on the extended one as well. To sum up, under probabilistic linguistic circumstance, a P-QUALIFLEX to portray the psychological behaviors of DMs and an extended QUALIFLEX are given respectively.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_3

31

32

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

Table 3.1 Extensions of QUALIFLEX with different fuzzy information Type of fuzzy information

References

Intuitionistic fuzzy information/interval-valued intuitionistic fuzzy information

[4–6]

Pythagorean fuzzy information

[7]

Hesitant fuzzy information/Interval-valued hesitant fuzzy information

[8, 9]

Probabilistic hesitant fuzzy information

[10]

Probability multi-valued neutrosophic set

[11]

Interval type-2 fuzzy information

[12, 13]

Interval type-2 trapezoidal fuzzy information

[14, 15]

Hesitant trapezoidal fuzzy information

[16]

Single-valued neutrosophic information

[17]

Hesitant 2-tuple linguistic model information

[18]

Hesitant fuzzy linguistic information

[19, 20]

Neutrosophic linguistic information

[21]

Probabilistic linguistic information

[22, 23]

3.1 Procedure of P-QUALIFLEX with Probabilistic Linguistic Information As the popularity of behavioral science, the effect of DMs’ psychological behaviors on the decision-making catches much attention from the whole society. Therefore, integrating this common phenomenon into QUALIFLEX is essential as well. In this chapter, a P-QUALIFLEX with probabilistic linguistic information [22] is given to simulate the behavioral characteristics of DMs in the decision-making process. Following the steps below, the promising alternative will be recognized. Let A = {A1 , A2 , . . . , An } be the set of the optional alternatives, C = {c1 , c2 , . . . , cm } be the set of the attributes for the alternatives. For convenience, N = {1, 2, . . . , n}, M = {1, 2, . . . , m}, and L( p) represents the standardized PLTSs mentioned in Sect. 1.2.6. Step 1. Get the evaluation matrix from DMs as the probabilistic linguistic information and standardize them as PLTSs: ⎡

L 11 ( p) L 12 ( p) ⎢ L 21 ( p) L 22 ( p) ⎢ Z=⎢ . .. ⎣ .. . L n1 ( p) L n2 ( p)

⎤ · · · L 1m ( p) · · · L 2m ( p) ⎥ ⎥ ⎥ .. .. ⎦ . . · · · L nm ( p)

(3.1)

where L i j ( p) is the evaluation information for Ai over c j , and i ∈ N , j ∈ M. The standardized process includes transforming the cost attributes to the benefit ones and Eq. (1.26) in Sect. 1.2.6.

3.1 Procedure of P-QUALIFLEX with Probabilistic Linguistic Information

33

Moreover, for benefit attributes, the bigger the value is, the better the attribute will be, whereas, it is reverse for the cost attributes. Thus, before the calculation process,it is necessary to unify those attributes. If there is cost attribute L i jc ( p) = {L k ( p k ) L k ∈ S, p k ≥ 0, k = 1, 2, . . . , #L( p)} with the probability of p k and the subscript of t k , it is converted into benefit ones through adding the symbol “-” in front of t k with the unchanged p k . For example, the evaluation information of cost attribute L i jc ( p) = {s−2 (0.2), s−1 (0.3), s0 (0.5)} will be transformed into the benefit one as L i jb ( p) = {s0 (0.5), s1 (0.3), s2 (0.2)}. Step 2. List all the possible permutations of the alternatives. According to the theory of permutation and combination, there are n! number of possible permutations for the n alternatives. Let Hl = (· · · , Ai , . . . , Ai  , . . .) denote the l th permutation. Then, l = 1, 2, . . . , n!, Ai , Ai  ∈ A and the alternative Ai is superior or at least equal to Ai  . Step 3. Determine the prospect value between alternatives. v Hj l (Ai  , Ai ) = v(d(L i  j ( p), L i j ( p))) ⎧ ⎨ (−λ(−d(L i  j ( p), L i j ( p)))β ) L i  j ( p) ≺ L i j ( p) = 0 L i  j ( p) ≈ L i j ( p) ⎩ L i  j ( p)  L i j ( p) (d(L i  j ( p), L i j ( p)))α

(3.2)

Step 4. Calculate the psychological weight of each attribute according to (3.3).

wj(pj) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

pδj δ [ p j +(1− p j )δ ]1/δ

L i  j ( p) ≺ L i j ( p)

0

L i  j ( p) ≈ L i j ( p)

pj γ [ p j +(1− p j )γ ]1/γ

L i  j ( p)  L i j ( p)

γ

(3.3)

Step 5. Work out the prospect concordance/discordance index for each attribute based on (3.4): I jHl (Ai  , Ai ) = w j v(d(L i  j ( p), L i j ( p))) ⎧ pδj β ⎪   ⎪ ⎨ [ pδj +(1− p j )δ ]1/δ × (−λ(−d(L i j ( p), L i j ( p))) ) L i j ( p) ≺ L i j ( p) = 0 L i  j ( p) ≈ L i j ( p) γ ⎪ ⎪ pj ⎩ α × (d(L i  j ( p), L i j ( p))) L i  j ( p)  L i j ( p) γ [ p +(1− p )γ ]1/γ j

j

(3.4) where d(L i  j ( p), L i j ( p)) is the distance among Ai  and Ai over c j . Step 6. Aggregate the weighted prospect concordance/discordance index between alternatives in each permutation.

34

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

I Hl (Ai  , Ai ) =

m

I jHl (Ai  , Ai )

(3.5)

j=1

Step 7. Collect the comprehensive prospect concordance/discordance index of each permutation according to (3.6). I Hl =

I Hl (Ai  , Ai )

(3.6)

∀Ai ,∀Ai  ∈X

Step 8. Rank I Hl obtained in Step 7 and confirm the biggest I Hl as the optimal permutation. Then, the process of the given P-QUALIFLEX is shown in Fig. 3.1 [22]. Fig. 3.1 Procedure of P-QUALIFLEX

3.2 Procedure of the Extended QUALIFLEX with Probabilistic Linguistic Information

35

3.2 Procedure of the Extended QUALIFLEX with Probabilistic Linguistic Information In order to demonstrate the feasibility and effectiveness of P-QUALIFLEX, an extended QUALIFLEX with probabilistic linguistic information [22] is shown as follows: Step 1. Acquire the evaluation matrix and standardize it as Step 1 in Sect. 3.1. Step 2. Identify all the possible permutations for alternatives. Step 3. Calculate the concordance/discordance index. I jHl (Ai  ,

#L i j ( p)

ti  j ti j 1 − pi j × ) Ai ) = d(L i  j ( p), L i j ( p)) = ( pi  j × #L i j ( p) k=1 τ τ

(3.7) where I jHl (Ai  , Ai ) expresses the concordance/discordance index for each pair alternatives (Ai  , Ai ) over the attribute c j in the l-th permutation Hl . Furthermore, d(L i  j ( p), L i j ( p)) represents the Hamming distance between L i  j ( p) and L i j ( p). There are three cases used to exhibit the concordance/discordance of the signed distance-based ranking orders Ai and Ai  in the l-th permutation Hl : (1)

(2) (3)

If d(L i  j ( p), L i j ( p)) > 0, then it means that Ai  is superior to Ai with the attribute c j in the ranking result. In other words, the sequence for Ai  and Ai in Hl is concordant; If d(L i  j ( p), L i j ( p)) = 0, then it shows that Ai  is equal to Ai with the attribute c j in the ranking result, that is to say, Ai  and Ai rank at least the same place; If d(L i  j ( p), L i j ( p)) < 0, then it indicates that Ai  is worse than Ai with the attribute c j in the ranking result, i.e., the sequence for Ai  and Ai in Hl is discordant.

Step 4. Determine the weighted concordance/discordance index. I Hl (Ai  , Ai ) =

m

ω j I jHl (Ai  , Ai )

(3.8)

j=1

where ω j is the weight of the j-th attribute,

m 

ω j = 1 and I Hl (Ai  , Ai ) shows the

j=1

concordance/discordance index for each pair alternatives (Ai  , Ai ) in Hl . Step 5. Obtain the comprehensive concordance/discordance index based on (3.6). Step 6. Rank the comprehensive concordance/discordance index I Hl .

36

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

If I Hl is the biggest one, then Hl is the optimal permutation. At the same time, the best alternative will be found. As mentioned above, the given QUALIFLEX with probabilistic linguistic information can portray the linguistic expression perfectly and find the promising alternative accurately. In addition, it does not lose any information of DMs, especially when the possible linguistic terms provided by DMs have unequal importance degree or weight. However, the psychological behaviors of DMs are ignored in this method and the DMs play an important role in the decision-making process exactly. Hence, it is necessary to take into account the DMs’ psychological characteristics such as risk seeking for losses and risk aversion for gains in the decision-making models. Then, P-QUALIFLEX is suitable for this situation. In the next chapter, an illustrative example is used to show the advantages of those two methods.

3.3 Illustrative Example An example of choosing the optimal start-up firm for VCs is used in this chapter. Four alternatives are represented by A1 , A2 , A3 , A4 correspondingly. Generally, after a long-time exploration, Widyanto and Dalimunthe [24] found that the attributes such as management team, market condition, service quality or product superiority, and financial situation are the main concerns of VCs in their decision-making processes. Furthermore, the market condition is another important factor, including market potential, market growth rate, market competition level, etc. In addition, service quality or product superiority is the core competitiveness of attracting customers, however, the customers are the fundamental sources of the earnings and also the most important resources in the modern business models. Hence, it catches great attention from VCs. The current and the possible future financial situations are also considered by VCs. To sum up, those four aspects are also adopted in this chapter, and they are expressed as c1 , c2 , c3 , c4 correspondingly. According to the description of the four attributes, all of them are benefit ones and they are associated with the same linguistic term sets: S = {s−3 , s−2 , s−1 , s0 , s1 , s2 , s3 }, where s−3 = ver y bad, s−2 = bad, s−2 = bad, s0 = medium, s1 = a little good, s2 = good, s2 = good. Then, five experienced and senior investors are invited to discuss the above alternatives according to the four attributes. After heated discussion among investors and the reference about the attributes’ weights [24], the original weights of those attributes among the four alternatives are given as follows: ω1 = p1 = 0.36, ω2 = p2 = 0.19, ω3 = p3 = 0.31, ω4 = p4 = 0.14. Then, the feasibility and effectiveness of the given methods will be shown in Sects. 3.3.1 and 3.3.2 respectively.

3.3 Illustrative Example

37

3.3.1 Results of P-QUALIFLEX with Probabilistic Linguistic Information According to the analysis in Sect. 3.1 and the problem description above, the selection processes are as follows [22]: Step 1. The probabilistic linguistic evaluation matrix is obtained as: c1 c2 ⎛ x1 {s0 (0.3), s1 (0.45), s2 (0.2)} ⎜ {s−1 (0.24), s0 (0.66)} x ⎜ Z= 2⎜ x3 ⎝ {s−1 (0.18), s0 (0.53), s1 (0.19)} x4

{s0 (0.5), s1 (0.46)}

c3

c4

{s−1 (0.11), s0 (0.89)} {s0 (0.7), s1 (0.27)} {s0 (0.4), s1 (0.6)} {s−1 (0.19), s0 (0.81)} {s0 (0.63), s1 (0.37)} {s−1 (0.13), s0 (0.87)} {s0 (0.14), s1 (0.33), s2 (0.4)} {s1 }

⎞ {s0 (0.39), s1 (0.61)} ⎟ {s0 (0.74), s1 (0.26)} ⎟ ⎟ {s0 (0.68), s1 (0.32)} ⎠ {s0 (0.51), s1 (0.46)}

In this chapter, the above evaluation information combines the characteristics of both GDM and single decision making mentioned in Sect. 1.2.6. It means that all the investors express their opinions and discuss with each other. Finally, they give their agreeable evaluation matrix above. For example, for the alternative x1 with the attribute c1 , all the investors discussed with each other and gave {s0 (0.3), s1 (0.45), s2 (0.2)} which means that s0 appears with the probability of 0.3, s1 appears with the probability of 0.45 and s2 appears with the probability of 0.2. Then, the original probabilistic linguistic evaluation matrix is standardized as: c1 c2 ⎛ x1 {s0 (0.32), s1 (0.47), s2 (0.21)} ⎜ {s−1 (0.27), s0 (0.73)} x ⎜ Z= 2⎜ x3 ⎝ {s−1 (0.2), s0 (0.59), s1 (0.21)} x4

{s0 (0.52), s1 (0.48)}

c3

c4

{s−1 (0.11), s0 (0.89)} {s0 (0.72), s1 (0.28)} {s0 (0.4), s1 (0.6)} {s−1 (0.19), s0 (0.81)} {s0 (0.63), s1 (0.37)} {s−1 (0.13), s0 (0.87)} {s0 (0.16), s1 (0.38), s2 (0.46)} {s1 }

⎞ {s0 (0.39), s1 (0.61)} ⎟ {s0 (0.74), s1 (0.26)} ⎟ ⎟ {s0 (0.68), s1 (0.32)} ⎠ {s0 (0.53), s1 (0.47)}

Step 2. All the possible permutations with those four start-up firms are shown in Table 3.2. Because only four start-up firms remain for comparisons, the number of the possible permutations is 4! = 24. Step 3. The prospect value according to the value function of Eq. (3.2) is in Table 3.3, taking H1 as an example as well. The parameters in the value function of Eq. (3.2) follow the experiment conducted by Tversky and Kahneman [25]: α = 0.88, β = 0.88, λ = 2.25. Step 4. The psychological weight of each attribute according to Eq. (3.3) is shown in Table 3.4. The parameters in the weighting function of Eq. (3.3) accord with the experiment conducted by Tversky and Kahneman [25]: γ = 0.61, δ = 0.69. Step 5. The prospect concordance/discordance index for each attribute on the basis of Eq. (3.4) is listed in Table 3.5.

38

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

Table 3.2 All the possible permutations H1 = (A1  A2  A3  A4 )

H7 = (A2  A1  A3  A4 )

H13 = (A3  A2  A1  A4 )

H19 = (A4  A2  A3  A1 )

H2 = (A1  A2  A4  A3 )

H8 = (A2  A1  A4  A3 )

H14 = (A3  A2  A4  A1 )

H20 = (A4  A2  A1  A3 )

H3 = (A1  A3  A2  A4 )

H9 = (A2  A3  A1  A4 )

H15 = (A3  A1  A2  A4 )

H21 = (A4  A3  A2  A1 )

H4 = (A1  A3  A4  A2 )

H10 = (A2  A3  A4  A1 )

H16 = (A3  A1  A4  A2 )

H22 = (A4  A3  A1  A2 )

H5 = (A1  A4  A2  A3 )

H11 = (A2  A4  A1  A3 )

H17 = (A3  A4  A2  A1 )

H23 = (A4  A1  A2  A3 )

H6 = (A1  A4  A3  A2 )

H12 = (A2  A4  A3  A1 )

H18 = (A3  A4  A1  A2 )

H24 = (A4  A1  A3  A2 )

Table 3.3 The prospect values between start-up firms under H1 Index

Attributes c1

c2

c3

c4

1 vH j (A1 ,

A2 )

0.165

−0.344

0.106

0.082

1 vH j (A1 ,

A3 )

0.129

−0.244

0.094

0.070

1 vH j (A1 ,

A4 )

0.066

−0.440

−0.348

0.037

1 vH j (A2 , A3 )

−0.106

0.057

−0.039

−0.039

1 vH j (A2 ,

A4 )

−0.361

−0.238

−0.542

−0.118

1 vH j (A3 ,

A4 )

−0.167

−0.305

−0.518

−0.088

Table 3.4 Weight of each attribute between alternative under H1 Weight

Attributes c1

c2

c3

c4

A2 )

0.422

0.213

0.362

0.158

A3 )

0.422

0.213

0.362

0.158

A4 )

0.422

0.213

0.356

0.158

A3 )

0.415

0.217

0.356

0.155

A4 )

0.415

0.213

0.356

0.155

1 wH j (A3 , A4 )

0.415

0.213

0.356

0.155

1 wH j (A1 , 1 wH j (A1 , 1 wH j (A1 , 1 wH j (A2 , 1 wH j (A2 ,

3.3 Illustrative Example

39

Table 3.5 Prospect concordance/discordance indices between alternatives under H1 Index

Attributes c1

c2

c3

c4

I jH1 (A1 ,

A2 )

0.070

−0.073

0.038

0.013

I jH1 (A1 ,

A3 )

0.055

−0.052

0.034

0.011

I jH1 (A1 ,

A4 )

0.028

−0.094

−0.124

0.006

I jH1 (A2 ,

A3 )

−0.044

0.012

−0.014

−0.006

I jH1 (A2 , A4 )

−0.150

−0.051

−0.193

−0.018

I jH1 (A3 ,

−0.070

−0.065

−0.184

−0.014

A4 )

Step 6. The weighted prospect concordance/discordance indices between start-up firms in each permutation are worked out based on Eq. (3.5) (Table 3.6). Step 7. The comprehensive prospect concordance/discordance indices of each permutation according to Eq. (3.6) are calculated (Table 3.7). Step 8. Ranking I Hl obtained in Step 7 and the biggest one is the optimal permutation (Table 3.8). It is confirmed that H24 is the best permutation and that H24 is the worst one. In other words, A4  A1  A3  A2 , i.e., A4 is the best start-up firm, whereas, A1 is the worst one. Table 3.6 The weighted concordance/discordance indices of H1 I H1 (A1 , A2 )

I H1 (A1 , A3 )

I H1 (A1 , A4 )

I H1 (A2 , A3 )

I H1 (A2 , A4 )

I H1 (A3 , A4 )

0.048

0.048

−0.184

−0.052

−0.411

−0.332

Table 3.7 Comprehensive prospect concordance/discordance indices of all the permutations H1 = −0.884

H5 = 0.195

H9 = −1.411

H13 = −1.357

H17 = −0.552

H21 = −0.069

H2 = −0.402

H6 = 0.259

H10 = −1.203

H14 = −1.149

H18 = 0.270

H22 = 0.213

H3 = −0.831

H7 = −1.166

H11 = −0.476

H15 = −1.075

H19 = −0.123

H23 = 0.404

H4 = −0.234

H8 = −0.684

H12 = −0.720

H16 = −0.478

H20 = 0.126

H24 = 0.457

40

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

3.3.2 Results of the Extended QUALIFLEX with Probabilistic Linguistic Information In order to demonstrate the advantages of P-QUALIFLEX, the example in this chapter is the same as in Sect. 3.3.1 [22]. Step 1. The standardized evaluation matrix is the same as Step 1 in Sect. 3.3.1. Step 2. All the possible permutations are already exhibited in Table 3.2. Step 3. The concordance/discordance index on the basis of Eq. (3.7) is exhibited in Table 3.9, taking H1 as an example. Step 4. The weighted concordance/discordance index for H1 is given in Table 3.10 based on Eq. (3.8). Step 5. The comprehensive concordance/discordance index under all permutations are presented in Table 3.11. Step 6. The ranking results of the comprehensive concordance/discordance indices are listed in Table 3.12.

Table 3.8 Ranking results of all the permutations 1

2

3

4

5

6

7

8

9

10

11

12

H24

H23

H6

H22

H5

H20

H21

H19

H4

H18

H2

H11

13

14

15

16

17

18

19

20

21

22

23

24

H16

H17

H8

H12

H3

H1

H15

H14

H7

H10

H13

H9

Table 3.9 Concordance/discordance indices between alternatives under H1 Index

Attributes c1

c2

c3

c4

A2 )

0.129

−0.118

0.078

0.058

A3 )

0.098

−0.08

0.068

0.048

A4 )

0.046

−0.157

−0.12

0.023

A3 )

−0.031

0.038

−0.01

−0.01

I jH1 (A2 , A4 )

−0.125

−0.078

−0.198

−0.035

I jH1 (A3 ,

−0.052

−0.103

−0.188

−0.025

I jH1 (A1 , I jH1 (A1 , I jH1 (A1 , I jH1 (A2 ,

A4 )

Table 3.10 Weighted concordance/discordance indices of H1 I H1 (A1 , A2 )

I H1 (A1 , A3 )

I H1 (A1 , A4 )

I H1 (A2 , A3 )

I H1 (A2 , A4 )

I H1 (A3 , A4 )

0.056

0.048

−0.047

−0.008

−0.126

−0.100

3.3 Illustrative Example

41

Table 3.11 Comprehensive concordance/discordance indices of all the permutations H1 = −0.178 H5 = 0.275

H9 = −0.387

H2 = 0.023

H10 = −0.292 H14 = −0.275 H18 = 0.090

H6 = 0.292

H3 = −0.161 H7 = −0.291 H11 = 0.005 H4 = 0.091

H13 = −0.370 H17 = −0.023 H21 = 0.178 H22 = 0.291

H15 = −0.257 H19 = 0.161

H23 = 0.370

H8 = −0.090 H12 = −0.091 H16 = −0.005 H20 = 0.257

H24 = 0.387

Table 3.12 Ranking results of all the permutations 1

2

3

4

5

6

7

8

9

10

11

12

H24

H23

H6

H22

H5

H20

H21

H19

H4

H18

H2

H11

13

14

15

16

17

18

19

20

21

22

23

24

H16

H17

H8

H12

H3

H1

H15

H14

H7

H10

H13

H9

where H24 is recognized as the best permutation, i.e., A4  A1  A3  A2 . Therefore, A4 is the best start-up firm among the four, whereas, A1 is the worst one among them.

3.4 Comparative Analysis In this chapter, the comparative analysis is presented [22]. Firstly, a comparison is processed between the P-QUALIFLEX and the extended QUALIFLEX, including the results from the numerical example and the simulated analysis. Secondly, in order to show the advantage of the P-QUALIFLEX, a comparison is also conducted between P-QUALIFLEX and the extended TODIM which is an outranking method based on PT.

3.4.1 Comparison of P-QUALIFLEX with Extended QUALIFLEX As shown in Tables 3.8 and 3.12, the ranking results of the two methods are analyzed firstly (Table 3.13). According to Table 3.13, both the P-QUALIFLEX and the extended QUALIFLEX recognize H24 and H9 as the best and the worst permutations respectively. Also, the ranking results of all the possible permutations are the same. Hence, a more detailed analysis of those two methods from the perspective of simulation is conducted. Firstly, we stochastically generate 1000 probabilistic linguistic evaluation values for the alternatives. Then, operating the procedure of the two methods with the generated 1000 evaluation values. The ranking results from 763 evaluation values

42

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

Table 3.13 Ranking results of the given methods Serial number

1

2

3

4

5

6

7

8

9

10

11

12

P-QUALIFLEX

H24

H23

H6

H22

H5

H20

H21

H19

H4

H18

H2

H11

QUALIFLEX

H24

H23

H6

H22

H5

H20

H21

H19

H4

H18

H2

H11

Serial number

13

14

15

16

17

18

19

20

21

22

23

24

P-QUALIFLEX

H16

H17

H8

H12

H3

H1

H15

H14

H7

H10

H13

H9

QUALIFLEX

H16

H17

H8

H12

H3

H1

H15

H14

H7

H10

H13

H9

are the same, while different ranking results are shown by the remaining evaluation values. This difference is just from the ranking results. Furthermore, the difference of the maximal and minimum comprehensive concordance/discordance indices of 1000 decision-making data is presented to explain the difference of the P-QUALIFLEX and the extended QUALIFLEX. Based on Fig. 3.2, with the same evaluation information, the difference of the maximal and minimum comprehensive prospect concordance/discordance indices is bigger than the difference of the maximal and minimum comprehensive concordance/discordance indices in most cases. This indicates that the P-QUALIFLEX 4 3 2 1 0 -1 -2 -3

0

200

400

600

800

1000

Fig. 3.2 The differences of the maximal and minimum comprehensive concordance/discordance indices (red line) and comprehensive prospect concordance/discordance indices (blue line)

3.4 Comparative Analysis

43

increases the difference of the comprehensive concordance/discordance indices of the permutations. And then, it is more effective in identifying the similar alternatives. According to the above analysis, the P-QUALIFLEX has more advantages than the extended QUALIFLEX from three aspects: (1) The P-QUALIFLEX fully considers the risk attitudes of DMs, such as risk seeking for losses and risk aversion for gains. Those risk attitudes are demonstrated to be the universal phenomena of DMs in their decision-making processes, and thus, it is reasonable and suitable. (2) The P-QUALIFLEX includes the weighting function which is a general law for DMs to use such a transformed weighting to make their decisions. (3) The P-QUALIFLEX integrates the risk attitudes of DMs and the weighting function into the ranking results well. Also, it comprehensively considers the psychological behaviors of DMs. As revealed by the well-known PT, the DMs indeed show up those psychological behaviors in their decision-making processes. Therefore, under the probability linguistic circumstance, the P-QUALIFLEX with psychological behaviors of DMs has more advantages than the extended QUALIFLEX.

3.4.2 Comparison of P-QUALIFLEX with TODIM The P-QUALIFLEX is an effective method to record the DMs’ psychological behaviors on the basis of PT, however, TODIM is also an outranking method derived from PT. In order to show the advantages of the given P-QUALIFLEX, a comparative analysis between the P-QUALIFLEX and the TODIM is conducted in this chapter, including the ranking results and the sensitivity of them. Firstly, the procedure of TODIM with probabilistic linguistic information [22] is presented as follows: Step 1. Acquire the probabilistic linguistic information as shown in Step 1 of Sect. 3.3.1. Step 2. Determine the relative weight ω jr according to the original attributes’ weights: ω jr =

ωj ωr

(3.9)

where r, j ∈ M, ωr = max(ω j | j ∈ M) and cr is called a reference attribute. Then, ω1r = 1, ω2r = 0.528, ω3r = 0.861, ω4r = 0.389. Step 3. Calculate the dominance of Aa over Ab (a, b ∈ N ): ψ(Aa , Ab ) =

m

j=1

where

ϕ j (Aa , Ab ), a = b

(3.10)

44

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

Table 3.14 The calculated results of dominance ψ(A1 , A2 )

0.111

ψ(A2 , A1 )

−0.626

ψ(A3 , A1 )

−0.578

ψ(A3 , A4 )

0.026

ψ(A1 , A3 )

0.127

ψ(A2 , A3 )

−0.244

ψ(A3 , A1 )

−0.001

ψ(A3 , A4 )

0.652

ψ(A1 , A4 )

−0.495

ψ(A2 , A4 )

−1.124

ψ(A3 , A4 )

−1.031

ψ(A3 , A4 )

0.578

ϕ j (Aa j , Abj ) =



⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

ω jr m  ω jr

d(L a j , L bj )

L a j  L bj

j=1

m ⎪ ⎪  ⎪ ⎪ ω jr ⎪ ⎪ ⎩ −1 j=1 λ

ω jr

0

L a j ≈ L bj

(3.11)

(−d(L a j , L bj )) L a j ≺ L bj

The parameter λ is the attenuation factor of the losses and it is the same with in the P-QUALIFLEX. Hence, there is λ = 2.25. Then, the results are listed in Table 3.14. Step 4. Identify the overall value of the start-up firm Ai  :  n  n b=1 ψ(Ai  , Ab ) − mini b=1 ψ(Ai  , Ab )  n  n   i  , i, b ∈ N (Ai  ) =  , A b ) − mini  , Ab ) maxi ψ(A ψ(A i i b=1 b=1 (3.12) Then, (x1 ) = 0.383, (x2 ) = −0.989, (x3 ) = −0.562, (x4 ) = 1.965. Step 5. Rank the overall value (Ai  ), i  ∈ N . Ai  will be the promising start-up firm if max((Ai  ), i  ∈ N ). According to the results in Step 4, we have A4  A1  A3  A2 . The ranking results derived by the P-QUALIFLEX and the TODIM are the same. Hence, the feasibility of the P-QUALIFLEX is demonstrated. Additionally, the TODIM is an outranking method according to the overall values of all start-up firms, whereas, there is possibility that the overall values of two start-up firms are the same. Hence, it is difficult to distinguish the ranking of those two start-up firms. However, the P-QUALIFLEX has given the ranking of all the start-up firms in advance, and an optimal ranking will be found anyway. From this perspective, the P-QUALIFLEX is superior to the TODIM. As narrated in Sect. 3.1 that some parameters exist in the given P-QUALIFLEX. However, the determination of the values of the parameters is not random and they are always following a certain rule. Hence, all of them are obtained from the experimental economics. A lot of researchers have given the ways of parameter estimation on the basis of different backgrounds and disparate cases. Since there exists the same parameter λ in both the P-QUALIFLEX and the TODIM, on the one hand, the stability of P-QUALIFLEX is shown through the comparison of the ranking results from both the P-QUALIFLEX and the TODIM with the change of λ. In Table 3.15, the ranking results of the start-up firms derived by the two methods are presented.

3.4 Comparative Analysis

45

Table 3.15 The ranking results Value of parameter

Methods TODIM

P-QUALIFLEX

λ = 2.25

A4  A1  A3  A2

A4  A1  A3  A2

λ = 2.15

A4  A1  A3  A2

A4  A1  A3  A2

λ = 2.05

A4  A1  A3  A2

A4  A1  A3  A2

λ = 1.85

A4  A1  A3  A2

A4  A1  A3  A2

λ = 1.75

A4  A1  A3  A2

A4  A1  A3  A2

λ = 1.55

A4  A1  A3  A2

A4  A1  A3  A2

λ = 1.25

A4  A1  A3  A2

A4  A1  A3  A2

According to Table 3.15, when λ changes, the ranking results of start-up firms derived by both P-QUALIFLEX and TODIM do not change. On the one hand, this indicates that the P-QUALIFLEX is as stable as the TODIM. On the other hand, it is necessary to discuss the stability of the P-QUALIFLEX according to the effect of different values of other parameters on the ranking results. Three representative results of parameter estimation are chosen to verify the parameters’ effect on the ranking results of the alternatives. Firstly, the parameters employed in the aforementioned numerical example in Sect. 3.1 come from Ref. [25]. It is the original parameter estimation and the most classical one. Also, it has been widely used in various application fields. Then, the parameters in Ref. [26] serve as an option in the comparative analysis. Finally, the parameter estimation conducted by Chinese researcher has been chosen as well [27]. Because Zeng [27] examined the psychological behaviors of DMs in China, whereas, the experiment of other two references involved in the comparative analysis of Table 3.16 has taken place in abroad with the graduate students [28] and the undergraduate or Ph.D. students [26]. The ranking results with the parameters in those three references are shown in Table 3.16. It is easy to discover that the ranking results between the parameters in Refs. [25, 26] are identical. However, there is a little difference in the ranking results of the start-up firms for using the parameters in Refs. [25, 27]. Although the numerical Table 3.16 Ranking results of alternatives with different parameters in the P-QUALIFLEX Serial number

1

2

3

4

5

6

7

8

9

10

11

12

Parameters in Ref. [25] H24 H23 H6

H22 H5 H20 H21 H19 H4

H18 H2

H11

Parameters in Ref. [26] H24 H23 H6

H22 H5 H20 H21 H19 H4

H18 H2

H11

Parameters in Ref. [27] H24 H23 H22 H6

H5 H20 H21 H19 H18 H4

H2

H11

Serial number

17

23

24

13

14

15

16

18

19

20

21

22

Parameters in Ref. [25] H16 H17 H8

H12 H3 H1

H15 H14 H7

H10 H13 H9

Parameters in Ref. [26] H16 H17 H8

H12 H3 H1

H15 H14 H7

H10 H13 H9

Parameters in Ref. [27] H16 H17 H12 H8

H3 H1

H15 H14 H10 H7

H13 H9

46

3 QUALIFLEX Based on PT with Probabilistic Linguistic Information

example of this chapter occurs in China, it is reasonable and feasible to use the classical and widespread parameters [25] in the computing process owing to the internationalization of VC market and the fusion of culture. In this chapter, not only an extended QUALIFLEX has been presented, but also a P-QUALIFLEX which fully reflects the psychological behaviors of the DMs has been given. Although the ranking results of the start-up firms derived by the P-QUALIFLEX, the extended QUALIFLEX and the TODIM, are the same A4  A1  A3  A2 , the significances of those methods are different. Moreover, the same results just come from a special numerical case in this chapter. According to the simulated analysis, the P-QUALIFLEX is more effective in identifying the similar alternatives. Also, both the risk attitudes of the DMs and the weighting function are comprehensively considered in the P-QUALIFLEX, while such psychological behaviors of the DMs are not included in the extended QUALIFLEX. Even if the TODIM considers the partial risk attitudes of the DMs, it ignores the important transformed weighting function of the DMs. To sum up, from the perspective of the DMs’ psychological behaviors, the P-QUALIFLEX is superior to the other two and is suitable for uncertain decision making.

3.5 Remarks The ordering methods are always the important ways to solve the optimizing problems among all alternatives. The QUALIFLEX is one of those ordering methods, which is dedicated to finding an optimal sequence under all the possible permutations. It is a difficult work to deal with so many possible permutations by hand calculation, especially with a lot of alternatives, but exhausting all the possible permutations is a most radical way to pick out the optimum alternative. Thanks for the development of computer sciences, when there are large numbers of alternatives, we could handle numerous possible permutations by virtue of computer program. For the four alternatives in the simulated analysis, the results of 24 possible permutations with 1000 evaluation values only need 2.41 s. Six alternatives with 720 possible permutations and 1000 evaluation values also have been tested. The results only need 4.24 s. Furthermore, under the highly uncertain decision-making environment of DM, probabilistic linguistic information does a good job to simulate the ideas of DMs. To sum up, in this chapter, the extended QUALIFLEX provides a convenient way for the VCs to evaluate the start-up firms and to find a promising one among them. Nevertheless, it is more important to consider the effect of psychological behaviors of DMs on decision-making. In other words, because the different risk attitudes for gains and losses are the general phenomena of DMs, the P-QUALIFLEX under the probabilistic linguistic circumstance is more significant than the extended QUALIFLEX. It will draw the researchers’ attention to behavioral decision making and play a decisive role in applying psychological behavior to the ordering methods.

3.5 Remarks

47

Meanwhile, the application of the given methods to the VC field is not only useful to improve the performance of VC but also helpful to gain the desired capital for promising start-up firm.

References 1. Paelinck JHP (1976) Qualitative multiple criteria analysis, environmental protection and multiregional development. Papers Region Scie Assoc 36(1):59–74 2. Paelinck JHP (1978) QUALIFLEX: a flexible multiple-criteria method. Econ Lett 1(3):193–197 3. Alinezhad A, Esfandiari N (2012) Sensitivity analysis in the QUALIFLEX and VIKOR methods. J Optim Industr Eng 4. Chen TY, Tsui CW (2012) Intuitionistic fuzzy QUALIFLEX method for optimistic and pessimistic decision making. Adv Inform Sci Serv Sci 4(14):219–226 5. Chen TY (2014) Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihoodbased comparison approach for multiple criteria decision analysis. Inf Sci 261:149–169 6. Chen TY (2013) Data construction process and qualiflex-based method for multiple-criteria group decision making with interval-valued intuitionistic fuzzy sets. Int J Inf Technol Decis Mak 12(3):425–467 7. Zhang XL (2016) Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inf Sci 330:104–124 8. Zhang XL, Xu ZS (2015) Hesitant fuzzy QUALIFLEX approach with a signed distance-based comparison method for multiple criteria decision analysis. Expert Syst Appl 42(2):873–884 9. Zhang Z (2017) Multi-criteria decision-making using interval-valued hesitant fuzzy QUALIFLEX methods based on a likelihood-based comparison approach. Neural Comput Appl 28(7):1835–1854 10. Li J, Wang JQ (2017) An Extended QUALIFLEX method under probability hesitant fuzzy environment for selecting green suppliers. Int J Fuzzy Syst 19(6):1866–1879 11. Peng HG, Zhang HY, Wang JQ (2018) Probability multi-valued neutrosophic sets and its application in multi-criteria group decision-making problems. Neural Comput Appl 30(2):563– 583 12. Dincer H, Yuksel S (2019) IT2-Based fuzzy hybrid decision making approach to soft computing. IEEE Access 7:15932–15944 13. Dincer H, Yuksel S, Korsakien R, Raisiene AG, Bilan Y (2019) IT2 hybrid decisionmaking approach to performance measurement of internationalized firms in the baltic states. Sustainability 11(1):296 14. Chen TY, Chang CH, Lu JFR (2013) The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making. Europ J Oper Res 226(3):615–625 15. Wang JC, Tsao CY, Chen TY (2015) A likelihood-based QUALIFLEX method with interval type-2 fuzzy sets for multiple criteria decision analysis. Soft Comput 19(8):2225–2243 16. Zhang XL, Xu ZS, Liu MF (2016) Hesitant trapezoidal fuzzy QUALIFLEX method and its application in the evaluation of green supply chain initiatives. Sustainability 8(9) 17. Tian C, Zhang WY, Zhang S, Peng JJ (2019) An extended single-valued neutrosophic projection-based qualitative flexible multi-criteria decision-making method. Mathematics 7(1):16 18. Xue YX, You JX, Zhao X, Liu HC (2016) An integrated linguistic MCDM approach for robot evaluation and selection with incomplete weight information. Int J Product Res 54(18):5452– 5467 19. Tian ZP, Wang J, Wang JQ, Zhang HY (2016) A likelihood-based qualitative flexible approach with hesitant fuzzy linguistic information. Cognitive Comput 8(4):670–683

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20. Dong JY, Chen Y, Wan SP (2018) A cosine similarity based QUALIFLEX approach with hesitant fuzzy linguistic term sets for financial performance evaluation. Appl Soft Comput 69:316–329 21. Tian ZP, Wang J, Wang JQ, Zhang HY (2017) Simplified neutrosophic linguistic multicriteria group decision-making approach to green product development. Group Decision Negoti 26(3):597–627 22. Tian XL, Xu ZS, Wang XX, Gu J, Alsaadi FE (2019) Decision models to find a promising startup firm with QUALIFLEX under probabilistic linguistic circumstance. Int J Inform Technol Decision Making 18(4):1379–1402 23. Feng XQ, Liu Q, Wei CP (2019) Probabilistic linguistic QUALIFLEX approach with possibility degree comparison. J Intell Fuzzy Syst 36(1):719–730 24. Widyanto HA, Dalimunthe Z (2015) Evaluation criteria of venture capital firms investing on indonesians’ SME. Soc Sci Electron Publishing 25. Tversky A, Kahneman D (1992) Advances in prospect-theory-Cumulative representation of uncertainty. J Risk Uncertainty 5(4):297–323 26. Abdellaoui M (2000) Parameter-free elicitation of utility and probability weighting functions. Manage Sci 46(11):1497–1512 27. Zeng JM (2007) An experimental test on cumulative prospect theory. J Jinan Univ 28(1):44–472 28. Tian XL, Xu ZS, Jing G, Herrera-Viedma E (2018) How to select a promising enterprise for venture capitalists with prospect theory under intuitionistic fuzzy circumstance? Appl Soft Comput 67:756–763

Chapter 4

Group PROMETHEE Based on PT with Hesitant Fuzzy Linguistic Information

The PROMETHEE is an outranking method based on the relative preferences. It is a family of methods developed to solve different kinds of ranking problems. For example, the PROMETHEEs I [1] and II [2] were introduced for partial and complete rankings of alternatives correspondingly. The PROMETHEE III is a ranking method on the basis of intervals. Later, it was extended to the continuous case that is called PROMETHEE IV [3]. The PROMETHEE V was given by Brans and Mareschal [4], which is efficient in the problem with segmentation constraints. The PROMETHEE VI was built to represent the human brain [5]. The evaluation information in the conventional PROMETHEE has been expressed as crisp numbers, which could not deal with the complexity of objective affairs. In order to improve the effectiveness of the PROMETHEE, FS was introduced into the PROMETHEE to portray the uncertain context [6–8]. Taking both the support and opposition information into account, the intuitionistic fuzzy PROMETHEE [9, 10], the interval-valued intuitionistic fuzzy PROMETHEE [11] and interval type-2 fuzzy PROMETHEE [8] were developed, but they could not deal with the qualitative linguistic information. Therefore, considering that some attributes are hard for the DMs to evaluate by crisp numbers or by intuitionistic fuzzy ways, the PROMETHEE model with fuzzy linguistic information was built to handle qualitative decisionmaking situations [12, 13]. Later, based on the decision-making situation that the DM may be hesitant in several possible values, the hesitant fuzzy information was introduced to the PROMETHEE with the purpose of describing hesitant situations [14]. Since then, more explorations were made such as the fuzzy AHP PROMETHEE [15], the combination of ANP and PROMETHEE [16], the fuzzy mathematical programming PROMETHEE [17], GDSS PROMETHEE [18], etc. Although the PROMETHEE has been expanded to various forms, the main idea of the conventional PROMETHEE, which is to select the promising alternative according to the net flow, still exists in those extensions. Seldom of those extensions consider the impact of psychological state of DMs on the decision-making results. In Ref. [19], both PT and thermodynamic method were used to construct © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_4

49

50

4 Group PROMETHEE Based on PT with Hesitant …

the prospect decision-making matrix under HFLTS circumstance, and then built the PROMETHEE procedure. However, in their model, only the subscript of the linguistic term is used to get the prospect value as the subscript of HFLTS and it ignores the difference between the original HFLTS. According to the above analysis, the existing researches about PROMETHEE are concentrated on extending it to fuzzy circumstance and use various types of fuzzy information to depict the different uncertainty. The uncertainty in the decisionmaking process not only comes from the decision-making objects but also is derived from the vague perceptions of DMs for the objects. Therefore, PT, which is used to portray the psychological state of DMs, is adopted as the framework in decisionmaking models in this chapter. Although Ref. [20] considered the intuitionistic fuzzy preference and intuitionistic fuzzy weighting, it ignores the impact of psychological states of DMs on the final decision. The DMs are individuals who are quite different in gender, experience, education, and so on, which leads to different risk preferences. Therefore, in consideration of risk preferences of the DMs, a generalized PROMETHEE was built [21]. Although the risk preferences on gain and loss are depicted via modifying the preference function with the parameters of the PT, the transformed weight is not considered in this generalized model. Another research [22] took the deviation of the value function of the PT instead of the original evaluation values as the independent variable of the preference function in PROMETHEE. Both of those two researches attempted to integrate PT into PROMETHEE and the different risk attitudes on gain and loss have been considered indeed, but none of them take into account the transformed weight which is the important part of PT. Moreover, the evaluation information was expressed as crisp numbers in those two papers, which is hard for the DMs to express their opinions accurately. Additionally, according to the investigation [23], VCs only interested 20% of proposals among the 3631 proposals during 11 years. But due to the lack of capacity, 9% of those interesting deals were not accessed by the VCs, and again they lost some opportunities. Thus, as the deepening of the social division of labor and the specialization of individuals, GDM which takes the advantage of each group member can avoid such situations and recognize opportunities easily. It has been a process of solving the decision-making problems in most cases. To sum up, in this chapter, a GDM method which integrates the merit of both the PT and the HFLTS named GP-PROMETHEE is given to well portray the psychological factors and the evaluation information of the DMs [24]. Moreover, an extended group PROMETHEE and TODIM with HFLTSs [24] are also shown as the comparative methods to show the superiority of GP-PROMETHEE.

4.1 GP-PROMETHEE with Hesitant Fuzzy Linguistic Information

51

4.1 GP-PROMETHEE with Hesitant Fuzzy Linguistic Information Generally, there are two kinds of attributes in the decision-making processes: cost attributes and benefit attributes. The smaller the value of the cost attribute is, the better it will be. On the contrary, the bigger the value of the benefit attribute is, the better it will be. Therefore, for the sake of convenience, we should unify those different attributes. In this chapter, we transform the cost attributes to the benefit ones. For the benefit attribute, b HS = b HS . While for the cost attribute, c HS = c HS , where c HS is the inverse operation of the HFLTS with c HS = {s−i |si ∈ c HS }. Algorithm 4.1 [24] Input: The original decision-making information, including the evaluation information, the attributes weighting and the parameters. Output: The optimal alternative. Step 1. Obtain the original evaluation information from the DMs under hesitant fuzzy linguistic circumstance. ⎛

k ··· x11 ⎜ .. . . k X =⎝ . . k ··· xn1

⎞ k x1m .. ⎟, ω = {ω , ω , · · · , ω } 1 2 m . ⎠

(4.1)

k xnm

where n is the number of alternatives; m is the number of attributes; xikj is the kth DM’s evaluation information for the alternative Ai over the attribute c j . For convenience, let N = {1, 2, . . . , n}, M = {1, 2, . . . , m} and Z = {1, 2, . . . , z}. Then, i ∈ N , j ∈ M and k ∈ Z . Step 2. Translate the original evaluation information into the hesitant fuzzy linguistic judgment matrix, and then convert the cost attributes into the benefit ones. ⎞ k k · · · H1m H11 ⎟ ⎜ Hk = ⎝ ... . . . ... ⎠ k k · · · Hnm Hn1 ⎛

(4.2)

    where Hikj = (xikj , h S (xikj )xikj ∈ X ) = {< xikj , [stl (xikj )stl (xikj ) ∈ S, l = 1, 2, · · · ,   # Hikj ] > xikj ∈ X }, which denotes the possible evaluation values from the kth DMs for the ith alternative over the jth attribute, that is, the kth DM expresses his/her evaluation information on xi j with Hikj ; Moreover, Hikj is the standardized evaluation information according to the laws in the first paragraph of Sect. 4.1. For instance, Hikj = Hikjb when c jb is the benefit attribute, whereas, Hikj = Hikjc when c jc is the cost attribute.

52

4 Group PROMETHEE Based on PT with Hesitant …

Step 3. Calculate the deviation of prospect value between the alternatives Ai and Ai  over the attribute c j based on the hesitant fuzzy linguistic matrix: V jk (xik , xik ) = v j (xik , xik )w( p j ) ⎧ k k k β k ⎪ ⎨ −λ(d j (h i j , h i  j )) h i j ≺ h i  j v j (xik , xik ) = 0 h ikj ≈ h ik j ⎪ ⎩ (d (h k , h k ))α h k  h k j ij i j ij i j ⎧ ωδj ⎪ h k ≺ h ik j ⎪ ⎪ ⎨ [ωδ +(1−ω )δ ]1/δ i j j j γ w(ω j ) = ωj ⎪ ⎪ ⎪ 1/ other wise ⎩ γ γ [ω j +(1−ω j ) ] γ

(4.3)

(4.4)

(4.5)

where ∀i, i  ∈ N , ∀ j ∈ M and ∀k ∈ Z ; the relationship between h ikj and h ik j is determined by (1.21) and (1.22); d(h ikj , h ik j ) is the distance between the alternatives Ai and Ai  over the attribute c j , and it is acquired from (1.24); ω j is the importance degree of c j given by the DMs, in other words, it is the original weight of the attribute c j . Step 4. Work out the DMs’ preferences according to the preference function: P jk (xik , xik ) = F j (V jk )

(4.6)

where V jk is acquired from (4.3) and it is the independent variable of the preference function F(·). There are six types of preference functions [3]. The type V with linear preference and indifference area has been chosen as the representation of preference function of the DMs in this chapter: ⎧ −1 ⎪ ⎪ ⎪ ⎪ V jk +q j ⎪ ⎪ − ⎪ ⎪ ⎨ q j −q j F j (V jk ) = 0 ⎪ ⎪ ⎪ V jk −q j ⎪ ⎪ ⎪ ⎪ q j −q j ⎪ ⎩ 1

if V jk ≤ −q j if −q j < V jk ≤ −q j if −q j < V jk ≤ q j

(4.7)

if q j < V jk ≤ q j if q j < V jk

where q j is the indifference threshold and q j is the strict preference threshold under c j . Step 5. Aggregate the global preference index of the DMs for Ai related to Ai  : G k (xik , xik ) =

m

j=1

P jk (xik , xik )

(4.8)

4.1 GP-PROMETHEE with Hesitant Fuzzy Linguistic Information

53

Step 6. Calculate the positive outranking flow and the negative outranking flow of the DMs correspondingly: φ k+ (xik ) =

1 k k k G (xi , xi  ) n − 1 i  ∈N

(4.9)

φ k− (xik ) =

1 k k k G (xi  , xi ) n − 1 i  ∈N

(4.10)

Step 7. Get the net outranking flow of the DMs: φ k (xik ) = φ k+ (xik ) − φ k− (xik )

(4.11)

Step 8. Integrate the net outranking flow of each DM: ψ(xi ) =

z

πk φ k (xik )

(4.12)

k=1

where πk is the kth DM’s weight and it is calculated by the entropy method. As we all know that the entropy is used to measure the degree of disorder in a system. The higher disorder of the system is, the larger of the entropy value will be. While, for a GDM problem, if a DM shows a similar assessment for each alternative, then the entropy value of this DM will be larger, and his/her weight will be smaller. Thus, there is: πk =

1 − ek z  z− ek

(4.13)

k=1

where ek is the entropy value of the kth DM, and it can be obtained by (4.14) and (4.15):  ⎞ ⎛     n k k ⎟  ϕ ϕ 1 ⎜ i i ⎟  ⎜ (4.14) ek = − ∗ ln  n n ⎠ ln n i=1 ⎝    ϕk  ϕik  i  i=1

m 

ϕik =

j=1

i=1

#h ikj l=1

tl

#h ikj

m

(4.15)

54

4 Group PROMETHEE Based on PT with Hesitant …

The promising alternative will be found out according to the ranking order of ψ(xi ). The bigger ψ(xi ) is, the better the alternative Ai will be. If there are equal ψ(xa ) and ψ(xa  ) (ψ(xa ) = ψ(xa  )), then we should turn to Step 9. Otherwise, we should go to Step 10. Step 9. If ψ(xa ) = ψ(xa  ), then Eq. (4.17) should take the place of (4.15) in (4.14). In this situation, the entropy weight of the kth DM is obtained by (4.16) and (4.17). ⎞  ⎛     n k k ⎟

 ⎜ 1 ⎜ ϕi ∗ ln  ϕi ⎟ (4.16) ek = − n n ⎠  ln n i=1 ⎝  k k   ϕi  ϕi  m 

ϕik =

j=1

i=1 

i=1

stl ,stg ∈h ikj #h ikj

(tl −tg )2

m

(4.17)

Step 10. End. Until now, the detailed steps of the GP-PROMETHEE have been shown. To be understood easily, a visual procedure has been drawn up as well (see Fig. 4.1) [24]. In order to show the superiority of GP-PROMETHEE, an extended GPROMETHEE without considering the prospect framework [24] will be given in the next chapter.

4.2 G-PROMETHEE with Hesitant Fuzzy Linguistic Information Algorithm 4.2 [24]. Input: The original decision-making information, including the evaluation information, the attributes weighting and the parameters. Output: The optimal alternative. Step 1. Obtain the original evaluation information from the VCs under hesitant fuzzy linguistic circumstance. Step 2. Transform the evaluation matrix into the HFLTSs and convert the cost attributes into the benefit ones. Step 3. Identify the deviation between the alternatives on the basis of (4.18):

4.2 G-PROMETHEE with Hesitant Fuzzy Linguistic Information

Informa on input

Behavioral characteris cs of venture capitalists

55

Discover the background and decision problem

Iden fy the alterna ve start-up firms and a ributes

Obtain the original evalua on matrix from venture capitalists

Transform the evalua on matrix into standard HFLTSs

Calculate the devia ons of prospect values

Determine the weight of each venture capitalist using score func on

Decision process

Work out the preference between start-up firms

Determine the weight of each venture capitalist using Eq. (22)

Aggregate the global preference index

Compute the net out ranking flow of each venture capitalist

Integrate the net out ranking flow of all venture capitalists

if

( xa )

( xa )

Order the global net out ranking flow

Affirm the promising start-up firm

Fig. 4.1. The visual procedure of the GP-PROMETHEE for the VCs

⎧ #h ikj ⎪ ⎪ 1  |til −ti  l | k ⎪ k k k ⎪ −d j (h i j , h i  j ) = − #h k h i j ≺ h ik j ⎪ 2ς+1 ⎪ ij ⎨ l=1 D kj (h ikj , h ik j ) = 0 h ikj ≈ h ik j ⎪ ⎪ #h ikj ⎪ ⎪ ⎪ 1  |til −ti  l | k k k ⎪ h ikj  h ik j ⎩ d j (h i j , h i  j ) = #h k 2ς+1 ij

(4.18)

l=1

where ∀i, i  ∈ N , ∀ j ∈ M and ∀k ∈ Z ; ω j is the importance degree of c j given by the DMs, in other words, it is the original weight of the attribute c j . Moreover, the relationship between h ikj and h ik j is determined by (1.21) and (1.22). d(h ikj , h ik j ) is the distance between xi and xi  over the attribute c j , and it is acquired from (1.24). When there is h ikj ≺ h ik j , the deviation between them is the opposite number of the distance. In contrast, when there is h ikj  h ik j , the deviation between them is the number of distance. Actually, the deviation between h ikj and h ik j is a relative concept. It means that, for h ik j , the deviation of h ikj and h ik j is negative when h ikj ≺ h ik j and is positive when h ikj  h ik j .

56

4 Group PROMETHEE Based on PT with Hesitant …

Step 4. Determine the preference according to the preference function: P jk (xik , xik ) = F jk (D kj )

(4.19)

where D kj (·) is the independent variable of the preference function F(·). Step 5. Aggregate the global preference index: G k (xik , xik ) =

m

ω j P jk (xik , xik )

(4.20)

j=1

where ω j is the weight of the jth attribute. Then, following Steps 6–10 in Sect. 4.1, the ranking results will be acquired and also the optimal alternative will be recognized [24].

4.3 Illustrative Example After giving the detailed steps of the GDM models, the feasibility and effectiveness of the given models are presented in this chapter through an example. Meanwhile, a TODIM, which is an outranking method based on PT, is used in the same example for the sake of showing the stability of the GP-PROMETHEE. Then, the comparative analyses are presented [24].

4.3.1 Decision-Making Background In this chapter, an example that numerous VCs are preoccupied by selecting the promising start-up firm to gain extra revenue and bear high risk at the same time is provided. Owing to the merge of DiDi and Uber and also the intervention of the government, the form of online ordering car has been out of the public eye gradually. Since the year of 2015, the bike-share program has received much attention from all the society in China including the VCs. It is a new structure of business which satisfies individuals’ vehicle needs for short distance with lower cost and convenient cycling instead of walk. The idea of ‘Green travel, enjoy cycling’ has been accepted by more and more individuals. Furthermore, the government actively advocates citizens to choose a healthy and lower carbon way to travel which is beneficial not only to decrease the emission of carbon dioxide but also to relieve the urban traffic jam problems. Hence, as the most popular business program at present, the promising company with bike-share program is worthy of pursuing by the VCs.

4.3 Illustrative Example

57

There are several competitive start-up firms in the bike-share market such as ‘ofo’, ‘Xiaoming bike’, ‘mobike’, ‘U-Bicycle’, ‘Panda bike’, etc. After preliminary investigation by the interviewees, four competitive bike-share companies remain to be comprehensive evaluation. Moreover, A1 , A2 , A3 and A4 have been used to represent those alternative bike-share companies correspondingly. After furious discussion in the evaluation process, the consistent opinion among the DMs is reached. They have agreed to make their decisions according to the following attributes: ➀ c1 , the management team of the bike-share company (such as the educational background of the team member, the experience of the team member, the effort level of the team member, etc.); ➁ c2 , the market potential of the bike-share program; ➂ c3 , the service quality of the bike-share company (such as customer satisfaction, the market acceptance, etc.); ➃ c4 , the financial situation of the bike-share company (including the potential revenue, the asset liquidity, the uncertain investment cost, etc.). In this chapter, those four attributes are adopted as the gist of decision-making for the VCs. As has been mentioned above, they are hard to be quantified as crisp numbers. Therefore, using the hesitant fuzzy linguistic information to portray the perceptions of the VCs is suitable. Although the attributes are correlated with different contexts so that different attributes will be depicted with different linguistic term sets. Fortunately, the contexts of those four attributes associated in this VC environment are the same. Thus, the linguistic term sets of the four attributes are shown as: SC j = {s−3 = very bad, s−2 = bad, s−1 = a little bad, s0 = medium, s1 = a little good, s2 = good, s3 = very good}, where c j is the jth attribute. The VCs express their evaluation information with hesitant fuzzy linguistic forms. For example, the kth venture capitalist thinks that the management team of ‘mobike’ is ‘at least good’. Next, the evaluation k k k k = (x11 , h S (x11 )) = {}, information is translated to the HFLTSs: H11 because ‘at least good’ means ‘good’ or ‘very good’ which is represented by s2 or s3 correspondingly (please refer to [25] for more detailed explanations). Considering the limited capacity of each person and the advantage of GDM, three senior investors have been invited to evaluate the remaining bike-share firms ( A1 , A2 , A3 , A4 ) from the aspects of four attributes (c1 , c2 , c3 , c4 ). By adopting the attributes of six countries/regions discussed [26] and the suggestions of the interviewees, the attribute weights are: ω1 = 0.535 = p1 , ω2 = 0.086 = p2 , ω3 = 0.23 = p3 and ω4 = 0.149 = p4 . Also, the indifference threshold is supposed to be zero under all attributes. It means that any tiny discrepancies between bike-share firms will produce relative preferences. The strict preference threshold expresses that if the independent variable is greater than the strict preference threshold, the preference degrees of the VCs to a particular bike-share firm will be the values of the preference function. That is to say, the smaller the strict preference threshold for a specific attribute is, the more sensitivity the VCs for this attribute will be of. According to the attributes that the VCs care about, the strict preference thresholds of those attributes are supposed to be q 1 = 0.2, q 2 = 0.3, q 3 = 0.25 and q 4 = 0.6. Afterwards, following the steps, the promising bike-share company will be found [24].

58

4 Group PROMETHEE Based on PT with Hesitant …

4.3.2 Results of the GP-PROMETHEE with Hesitant Fuzzy Linguistic Information Step 1. The original evaluation information (linguistic expressions) from the VCs under hesitant fuzzy linguistic circumstance will be collected. The VCs know that their linguistic expressions conform to the linguistic term sets Sc j . Taking the 1st venture capitalist as an example, his/her evaluation information is shown in Table 4.1 [24]. Step 2. As the example stated in Sect. 1.2.5, the linguistic expressions are represented by the corresponding linguistic terms in Sc j . According to the transformation function E [25], the linguistic expressions in Table 4.1 are transformed into HFLTSs. The transformed rules of E are: (1). (2). (3). (4).

E(s j ) = {s j /s j ∈ S}; E(less than s j ) = {s j  /s j  ∈ S and s j  ≤ s j }; E(at least s j ) = {s j  |s j  ∈ S and s j  ≥ s j }; E(between s j and s j  ) = {s j  |s j  ∈ S and s j ≤ s j  ≤ s j  }.

For instance, for the linguistic expression “Between medium and good”, there is a transformation function E(between medium and good) = {medium, a little, good, good}, then, the corresponding HFLTS is H = {s0 , s1 , s2 } (more details about transformation function please refer to [25]. Thereby, the original evaluation information (linguistic expressions) provided by all the VCs such as in Table 4.1 are transformed into the standardized hesitant fuzzy linguistic evaluation matrices as H1 , H2 and H3 respectively: Table 4.1 The original evaluation information from the 1st venture capitalist Attributes start-up firms

c1

c2

c3

A1

Between a little good and good

Between medium and good

Between medium Between medium and a little good and a little good

c4

A2

Between medium and a little good

Between bad and a little bad

Between a little bad and a little good

Between a little bad and medium

A3

At least good

Between medium and good

A little good

Between a little good and good

A4

Between medium and a little good

Between a little bad and medium

Between a little bad and medium

Between a little bad and a little good

4.3 Illustrative Example

59

⎛

c1

A1 {s1 , s2 } A2 ⎜ 1 ⎜ {s0 , s1 } H = A3 ⎝ {s2 , s3 } A4 {s0 , s1 }

c2

c3

c4

⎞ {s0 , s1 , s2 } {s0 , s1 } {s0 , s1 } {s−2 , s−1 } {s−1 , s0 , s1 } {s−1 , s0 } ⎟ ⎟ {s0 , s1 , s2 } {s1 } {s1 , s2 } ⎠ {s−1 , s0 } {s−1 , s0 } {s−1 , s0 , s1 }

c1

c2

c3

c4

⎞ A1 {s0 , s1 , s2 } {s0 , s1 } {s−1 , s0 , s1 } {s0 , s1 } A ⎜ {s0 , s1 } {s−2 , s−1 } {s−1 , s0 } {s−1 , s0 } ⎟ ⎟ H2 = 2 ⎜ ⎝ A3 {s0 , s1 } {s0 , s1 } {s0 , s1 , s2 } ⎠ {s2 } A4 {s−2 , s−1 , s0 } {s−1 , s0 } {s−1 , s0 } {s1 } ⎛

⎛

c1

A1 {s1 , s2 } ⎜ {s0 , s1 } A H3 = 2 ⎜ A3 ⎝ {s1 } A4 {s0 , s1 }

c2

c3

{s0 , s1 , s2 } {s0 , s1 } {s0 , s1 } {s−1 , s0 , s1 } {s0 , s1 , s2 } {s1 } {s−1 , s0 } {s−1 , s0 , s1 }

c4

⎞ {s1 , s2 } {s−1 , s0 } ⎟ ⎟ {s0 , s1 } ⎠ {s0 , s1 }

Step 3. The deviations of the prospect values are calculated according to (4.3)– (4.5) and the results are listed in Table 4.2. V jk (xi , xi ) = 0 (i ∈ N ) are not shown in the above table, and similarly, all P jk (xi , xi ) = 0 and G kj (xi , xi ) = 0 (i ∈ N ) are not listed in the following tables. Step 4. The VCs’ preferences between the bike-share firms under all attributes are worked out according to (4.6) and (4.7). Please see the results in Table 4.3. Step 5. The global preference indexes of the VCs are presented in Table 4.4 based on (4.8). Step 6. The positive outranking flow and the negative outranking flow of the VCs are determined by (4.9) and (4.10) correspondingly. Step 7. Table 4.5 gives the net outranking flow from VCs based on (4.11). Step 8. The weights of VCs are obtained according to (4.13)–(4.17), which are listed in Table 4.6. Then, Table 4.7 exhibits the net outranking flow of each venture capitalist for all start-up firms. According to Table 4.7, it is known that ψ(x3 ) > ψ(x1 ) > ψ(x4 ) > ψ(x2 ). Thus, A3  A1  A4  A2 , in other words, A3 is considered as the best start-up firm among the four, whereas, A2 is the worst one. Step 9. End.

0.031

−0.356

V jk (x4 , x3 )

−0.081

−0.081

0.040

0.146

0

0.066

0.146

−0.193

0

0.079

−0.063

−0.133

0

−0.133

−0.356

0.040

0

0.066

−0.193

0.079

−0.193

0.079

c2

Venture capitalist 1

c1

V jk (x4 , x1 ) V jk (x4 , x2 )

V jk (x2 , x4 ) V jk (x3 , x1 ) V jk (x3 , x2 ) V jk (x3 , x4 )

V jk (x2 , x1 ) V jk (x2 , x3 )

V jk (x1 , x2 ) V jk (x1 , x3 ) V jk (x1 , x4 )

Deviations

Attributes

−0.137

0.016

−0.162

−0.061

−0.043

0.064

0.075

0.041

−0.033

−0.161

−0.088

0.029

−0.088

0.041

c4

−0.113

0.072

0.050

0.027

0.019

−0.113

−0.079

0.050

−0.062

0.035

c3

Table 4.2 Deviations of the prospect values from VCs

− 0.193

0.043

0.055

0.079

0.113

0.079

−0.105

−0.276

−0.073

−0.099

0.012

−0.099

0.049

0.057

0

−0.024

−0.116

−0.116

0.049

−0.135

0.057 0

0.030

c2

−0.193

c1

Venture capitalist 2

−0.113

0

−0.043

0.050

0.050

0.035

0

−0.113

−0.043

0.019

−0.079

0.019

c3

−0.113

0

−0.088

0.053

0.053

0.016

0

−0.113

−0.088

0.041

−0.033

0.041

c4

−0.105

0

−0.193

0.043

0.043

−0.105

0

−0.105

−0.193

0.079

0.043

0.079

c1

−0.081

−0.063

−0.081

0.040

0.012

0

0.031

−0.024

−0.024

0.040

0

0.012

c2

Venture capitalist 3

−0.113

0

−0.079

0.050

0.050

0.027

0

−0.113

−0.079

0.035

−0.063

0.035

c3

0

0.041

−0.088

0

0.041

−0.088

−0.088

−0.088

−0.161

0.041

0.041

0.075

c4

60 4 Group PROMETHEE Based on PT with Hesitant …

0.104

−1

P jk (x4 , x3 )

−0.271

−0.271

0.134

0.729

0

0.219

0.729

− 0.966

0

0.396

−0.210

−0.443

0

−0.443

−1

0.134

−0.966

0.396

0

0.219

c2

P jk (x4 , x1 ) P jk (x4 , x2 )

P jk (x2 , x4 ) P jk (x3 , x1 ) P jk (x3 , x2 ) P jk (x3 , x4 )

P jk (x2 , x1 ) P jk (x2 , x3 )

−0.966

P jk (x1 , x2 ) P jk (x1 , x3 ) P jk (x1 , x4 )

0.396

c1

Preferences

Venture capitalist 1

Attributes

−0.229

0.026

−0.648

−0.102

−0.172

0.107

0.125

0.068

−0.056

−0.269

−0.146

0.048

−0.146

0.068

c4

−0.454

0.288

0.201

0.109

0.077

−0.454

−0.317

0.201

−0.246

0.141

c3

Table 4.3 Prospect preference values from VCs

−0.966

0.215

0.277

0.396

0.566

0.396

−0.525

−1

−0.330

0.040

−0.330

0.163

0.191

0

−0.080

−0.387

−0.387

0.163

−0.676 −0.367

0

0.191

c2

−0.966

0.151

c1

Venture capitalist 2

−0.454

0

−0.172

0.201

0.201

0.141

0

−0.454

−0.172

0.077

−0.317

0.077

c3

−0.188

0

−0.146

0.088

0.088

0.026

0

−0.188

−0.146

0.068

−0.056

0.068

c4

−0.525

0

−0.966

0.215

0.215

−0.525

0

−0.525

−0.966

0.396

0.215

0.396

c1

−0.271

−0.210

−0.271

0.134

0.040

0

0.104

−0.080

−0.080

0.134

0

0.040

c2

Venture capitalist 3

−0.454

0

−0.317

0.201

0.201

0.109

0

−0.454

−0.317

0.141

−0.246

0.141

c3

0

0.068

−0.146

0

0.068

−0.146

−0.146

−0.146

−0.269

0.068

0.068

0.125

c4

4.3 Illustrative Example 61

62

4 Group PROMETHEE Based on PT with Hesitant …

Table 4.4 Global preference index VCs Global preferences

z1

VCs z2

z3

Global preferences

z1

z2

z3

G k (x1 , x2 )

0.824

0.487

0.702

G k (x3 , x1 )

0.573

0.563

− 0.562

G k (x1 , x3 )

− 1.359

− 1.339

0.037

G k (x3 , x2 )

1.274

1.046

0.524

G k (x

G k (x

1 , x4 )

0.779

− 0.369

0.739

3 , x4 )

1.257

0.848

G k (x2 , x1 )

− 1.873

− 1.073

− 1.632

G k (x4 , x1 )

− 1.793

− 0.371

− 1.701

G k (x2 , x3 )

− 2.165

− 2.029

− 1.205

G k (x4 , x2 )

− 0.043

0.255

− 0.142

− 0.042

G k (x

− 2.148

− 1.938

− 1.249

G k (x

2 , x4 )

− 0.189

− 0.605

4 , x3 )

0.551

Table 4.5 Net out ranking flow VCs Net out ranking flow

z1

φ k (x1k ) φ k (x2k ) φ k (x3k ) φ k (x4k )

z2

z3

1.112

−0.113

1.791

−2.094

−1.831

−1.321

2.926

2.587

0.977

−1.943

−0.643

−1.447

Table 4.6 Weights of the VCs VCs Weight

z1

z2

z3

πk

0.353

0.556

0.091

Table 4.7 Net out ranking flow Alternatives

A1

A2

A3

A4

The net out ranking flow

0.493

−1.877

2.559

−1.175

The ranking results

2

4

1

3

4.3.3 Results of the G-PROMETHEE with Hesitant Fuzzy Linguistic Information In Sect. 4.3.2, the ranking results of the GP-PROMETHEE with hesitant fuzzy linguistic information are presented. In this chapter, the G-PROMETHEE with hesitant fuzzy linguistic information [24] is used to analyze the same illustrative example above for the sake of comparing the advantages of the two methods easily. Step 1. For ease of comparisons, the original evaluation information under hesitant fuzzy linguistic circumstance is shown in Table 4.1.

4.3 Illustrative Example

63

Step 2. Considering the situation in Step 1, the standardized evaluation matrix follows Step 2 in Sect. 4.3.2 as well. Step 3. The differences between the start-up firms are shown in Table 4.8. D kj (h ikj , h ikj ) = 0 (i ∈ N ) are not presented in Table 4.8, by the way, for i ∈ N , P jk (xi , xi ) = 0 and G kj (xi , xi ) = 0 are not exhibited in the following tables as well. Step 4. The preferences between the start-up firms under each attribute are listed in Table 4.9 on the basis of (4.19). Step 5. The global preference indices of each venture capitalist between the startup firms are acquired in Table 4.10. Step 6. The positive and negative outranking flows are derived by (4.9) and (4.10). Step 7. The net outranking flows are calculated according to (4.11) and the results are shown in Table 4.11. Step 8. The weights of VCs are obtained according to (4.13)–(4.17), which are listed in Table 4.6. Then, Table 4.12 exhibits the net outranking flow of each venture capitalist for all start-up firms. In Table 4.12, it is shown that ψ(x3 ) > ψ(x1 ) > ψ(x4 ) > ψ(x2 ), that is, A3  A1  A4  A2 . Step 9. End.

4.3.4 Results of TODIM with Hesitant Fuzzy Linguistic Information The GP-PROMETHEE is used to simulate the psychological states of VCs through net outranking flow, however, the TODIM [27, 28] is another developed outranking method based on PT to consider the behavioral decision making of DMs. Therefore, in this chapter, the results of the TODIM with hesitant fuzzy linguistic information [24] are calculated, and it is used to show the better results of the GP-PROMETHEE from the comparison of the TODIM and the GP-PROMETHEE. Step 1. Obtain the decision information from experts as shown by Steps 1 and 2 in Sect. 4.3.2. Step 2. Calculate the relative weight ω jr according to the attributes’ weights in Sect. 4.3.1. ω jr =

ωj ωr

(4.21)

where r, j ∈ M, ωr = max(ω j | j ∈ M) and cr is called a reference attribute. Then, ω1r = 1, ω2r = 0.16, ω3r = 0.43, ω4r = 0.28. Step 3. Determine the kth venture capitalist’s dominance of the project Ai over Ai  (i, i  ∈ N ):

D kj (H1k j , D kj (H1k j , D kj (H1k j , D kj (H2k j , D kj (H2k j , D kj (H2k j , D kj (H3k j , D kj (H3k j , D kj (H3k j , D kj (H4k j , D kj (H4k j , D kj (H4k j ,

H2k j ) H3k j ) H4k j ) H1k j ) H3k j ) H4k j ) H1k j ) H2k j ) H4k j ) H1k j ) H2k j ) H3k j )

Deviations

0

0.333

0.190

−0.190

0.143

−0.190

0.286

0.286

−0.143

0

−0.286

0.048

−0.143

0

0.143

−0.143

−0.333

−0.286

−0.238

0.048

−0.214

−0.095

−0.048

0.238

0.286

0.143

−0.048

−0.286

−0.143

0.095

−0.143

0.143

c4

−0.143

0.214

0.143

0.071

−0.095

−0.333

−0.143

0.143

−0.071

0.095

c3

0.190

0

0.333

c2

0.143

−0.143

0.143

c1

Venture capitalist 1

Attributes

Table 4.8 Deviations of the start-up firms from VCs

−0.143

0.071

0.095

0.143

0.214

0.143

−0.071

−0.214

−0.238

0.048

−0.238

0.238

0.286

0

−0.048

−0.286

−0.286

0.238

−0.095 −0.048

0

0.286

c2

−0.143

0.048

c1

Venture capitalist 2

−0.143

0

−0.048

0.143

0.143

0.095

0

−0.143

−0.048

0.048

−0.095

0.048

c3

−0.190

0

−0.143

0.190

0.190

0.048

0

−0.190

−0.143

0.143

−0.048

0.143

c4

0.143

0.071

0.143

− 0.071

0

− 0.143

0.071

0.071

− 0.071

0

− 0.071

− 0.143

c1

0.190

0

0.048

− 0.190

− 0.143

− 0.190

0.190

0.048

0

0.143

− 0.048

− 0.048

c2

Venture capitalist 3

−0.143

0

−0.095

0.143

0.143

0.071

0

−0.143

−0.095

0.095

−0.071

0.095

c3

0

0.143

−0.143

0

0.143

−0.143

−0.143

−0.143

−0.286

0.143

0.143

0.286

c4

64 4 Group PROMETHEE Based on PT with Hesitant …

0.476

−1

P jk (x4 , x3 )

−0.635

−0.635

0.635

1

0

1

1

−0.714

0

0.714

−0.476

−1

−1

0

−1

−0.714

0.635

0

−0.714

0.714

1

0.714

c2

Venture capitalist 1

c1

P jk (x4 , x1 ) P jk (x4 , x2 )

P jk (x2 , x4 ) P jk (x3 , x1 ) P jk (x3 , x2 ) P jk (x3 , x4 )

P jk (x2 , x1 ) P jk (x2 , x3 )

P jk (x1 , x2 ) P jk (x1 , x3 ) P jk (x1 , x4 )

Preferences

Attributes

−0.397

0.079

−0.857

−0.159

−0.190

0.397

0.476

0.238

−0.079

−0.476

−0.238

0.159

−0.238

0.238

c4

−0.571

0.857

0.571

0.286

0.190

−0.571

−0.381

0.571

−0.286

0.381

c3

Table 4.9 Preference values between the start-up firms

−0.714

0.357

0.476

0.714

1

0.714

−0.357

−1

−0.794

0.159

−0.794

0.794

0.952

0

−0.159

−0.952

−0.952

0.794

−0.476 −0.238

0

0.952

c2

−0.714

0.238

c1

Venture capitalist 2

−0.571

0

−0.190

0.571

0.571

0.381

0

−0.571

−0.190

0.190

−0.381

0.190

c3

−0.317

0

−0.238

0.317

0.317

0.079

0

−0.317

−0.238

0.238

−0.079

0.238

c4

−0.357

0

−0.714

0.357

0.357

−0.357

0

−0.357

−0.714

0.714

0.357

0.714

c1

−0.635

−0.476

−0.635

0.635

0.159

0

0.476

−0.159

−0.159

0.635

0

0.159

c2

Venture capitalist 3

−0.571

0

−0.381

0.571

0.571

0.286

0

−0.571

−0.381

0.381

−0.286

0.381

c3

0

0.238

−0.238

0

0.238

−0.238

−0.238

−0.238

−0.476

0.238

0.238

0.476

c4

4.3 Illustrative Example 65

66

4 Group PROMETHEE Based on PT with Hesitant …

Table 4.10 Global preference indices obtained by the G-PROMETHEE VCs Global preferences

z1

VCs z2

z3

Global preferences

z1

z2

z3

G k (x1 , x2 )

2.333

1.619

1.730

G k (x3 , x1 )

1.238

1.175

−0.310

G k (x1 , x3 )

−1.238

−1.175

0.310

G k (x3 , x2 )

3.048

2.841

1.325

G k (x

G k (x

1 , x4 )

2.079

0.746

1.968

3 , x4 )

2.889

2.397

G k (x2 , x1 )

−2.333

−1.619

−1.730

G k (x4 , x1 )

−2.079

−0.746

−1.968

G k (x2 , x3 )

−3.048

−2.841

−1.325

G k (x4 , x2 )

0.365

0.516

−0.238

−2.889

−2.397

−1.563

G k (x

2 , x4 )

−0.365

−0.516

0.238

G k (x

4 , x3 )

1.563

Table 4.11 Net out ranking flows VCs Net out ranking flow

z1

φ k (x1k ) φ k (x2k ) φ k (x3k ) φ k (x4k )

z2

z3

2.116

0.794

2.672

−3.831

−3.317

−1.878

4.783

4.275

1.720

−3.069

−1.751

−2.513

Table 4.12 Net out ranking flow Alternatives

A1

A2

A3

A4

The net out ranking flows

1.432

−3.367

4.221

−2.285

The ranking results

2

4

1

3

ψ k (xi , xi  ) =

m

ϕ kj (xi , xi  )

(4.22)

j=1

where ⎧  ω jr ⎪ d(h ikj , h ik j ) h ikj  h ik j ⎪ m  ⎪ ⎪ ω jr ⎪ ⎪ j=1 ⎨ ϕ kj (h ikj , h ik j ) = 0 h ikj ≈ h ik j  ⎪ ⎪ m  ⎪ ⎪ ω jr ⎪ ⎪ ⎩ −1 j=1 d(h k , h k ) h k ≺ h k ij i j ij i j λ ω jr

(4.23)

The parameter λ is the attenuation factor of the losses and there is also the parameter λ in the GP-PROMETHEE. Then, the results are listed in Table 4.13.

4.3 Illustrative Example

67

Table 4.13 Dominance among projects from each venture capitalist VCs Dominances ψ k (x

VCs

z1

z2

z3

Dominances

z1

z2

z3

1 , x2 )

0.740

0.567

0.695

ψ k (x

3 , x1 )

0.551

0.509

−0.469

ψ k (x1 , x3 )

−0.913

−0.767

0.094

ψ k (x3 , x2 )

0.948

0.845

0.587

ψ k (x1 , x4 )

0.705

0.206

0.698

ψ k (x3 , x4 )

0.929

0.769

0.505

ψ k (x

2 , x1 )

−1.826

−1.580

−1.462

ψ k (x

4 , x1 )

−1.597

−1.151

−1.612

ψ k (x2 , x3 )

−2.166

−1.944

−1.279

ψ k (x4 , x2 )

−0.007

0.259

−0.427

ψ k (x2 , x4 )

−0.719

−0.493

−0.324

ψ k (x4 , x3 )

−1.977

−1.822

−1.174

Step 4. Aggregate all the VCs’ dominances of Ai over Ai  (i, i  ∈ N ): ψ(xi , xi  )=

z

πk ψ k (xi , xi  )

(4.24)

k=1

where πk is the weights of VCs obtained from (4.13)–(4.15). Step 5. Identify the overall value of the start-up firm Ai : n 

(xi ) =





n 

ψ(xi , xi  ) − minb ψ(xi , xi  ) i  =1    ,i ∈ N n n   ψ(xi , xi  ) − minb ψ(xi , xi  )

i  =1



maxb

i  =1

(4.25)

i  =1

Then, there is (x1 ) = 0.718, (x2 ) = 0, (x3 ) = 1, (x4 ) = 0.177. Step 6. Rank all the overall values (xi ) (i ∈ N ). Ai will be the optimal start-up firm if max( (xi ), i ∈ N ). If there is (xa ) = (xa  ), then, we use (4.13), (4.16) and (4.17) to get the weights of VCs and repeat the above steps 4–6 until the optimal project is got. According to the results in Step 5, there is A3  A1  A4  A2 .

4.3.5 Comparative Analysis In this chapter, the comparative analysis is based on the results of illustrative example and on the sensitive analysis of the parameters [24].

68

4 Group PROMETHEE Based on PT with Hesitant …

4.3.5.1

Comparative Analysis Based on the Results of Illustrative Example

On the one hand, it is necessary to analyze the feasibility and advantage of the given methods between the GP-PROMETHEE and the G-PROMETHEE. On the other hand, in order to explore the advantage of the GP-PROMETHEE in simulating the VCs’ behaviors, it is necessary to compare the applicability between the existing TODIM and the GP-PROMETHEE. Hence, the comparative analysis is conducted from those two aspects. (1) Comparison of the results of GP-PROMETHEE and G-PROMETHEE As Table 4.14 shows [24], the ranking results of the two methods are identical in this case. Both of them consider A3 as the best bike-share firm to be invested and A2 as the worst one. But the values of the net outranking flow are different under those two methods. This is mainly because GP-PROMETHEE considers the impact of risk attitudes of DMs on the decision results while the G-PROMETHEE just extends the classical one to the hesitant fuzzy linguistic circumstance without including the psychology of DMs. (2) Comparison of the results of GP-PROMETHEE and TODIM According to the results of Sects. 4.3.2 and 4.3.4, the ranking results of those two methods are the same. It reveals that GP-PROMETHEE is feasible. However, the final comprehensive values of start-up firms are different, which are listed in Table 4.15 [24]. It is obvious that the difference of the value of each start-up firm derived from GP-PROMETHEE is bigger than from TODIM. The value from TODIM is always within [0, 1] which is determined by TODIM itself. Hence, GP-PROMETHEE is superior than TODIM, especially in distinguishing the similar alternatives. Table 4.14 Net outranking flow of the two methods Net out ranking flow Methods

ψ(x1 )

ψ(x2 )

ψ(x3 )

ψ(x4 )

GP-PROMETHEE

0.493

−1.877

2.559

−1.175

G-PROMETHEE

1.432

−3.367

4.221

−2.285

Table 4.15 Comprehensive value of the two methods Value Method

ψ(x1 )

ψ(x2 )

ψ(x3 )

ψ(x4 )

GP-PROMETHEE

0.493

−1.877

2.559

−1.175

Extended TODIM

0.718

0

1

0.177

4.3 Illustrative Example

4.3.5.2

69

Comparative Analysis Based on the Sensitivity of Parameters

There are parameters in the given methods. In order to explain the impact of parameters on the decision results, sensitive analysis of those parameters is presented in this chapter [24]. (1) Comparison of sensitivity of GP-PROMETHEE and G-PROMETHE As has been mentioned before, q and q are the parameters in PROMETHEE, it is necessary to discuss the effect of those parameters on the ranking results of the given methods. Hence, Table 4.16 shows the ranking results of them with different parameters [24].

Table 4.16 The ranking results Methods Values of parameters

The GP-PROMETHEE The G-PROMETHEE

q 1 = 0, q 2 = 0, q 3 = 0, q 4 = 0

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

A3  A1  A4  A2

q 1 = 0.2, q 2 = 0.3, q 3 = 0.25, q 4 = 0.6 q 1 = 0.05, q 2 = 0.05, q 3 = 0.05, q 4 = 0.05 q 1 = 0.2, q 2 = 0.3, q 3 = 0.25, q 4 = 0.6 q 1 = 0.1, q 2 = 0.1, q 3 = 0.1, q 4 = 0.1 q 1 = 0.2, q 2 = 0.3, q 3 = 0.25, q 4 = 0.6 q 1 = 0.15, q 2 = 0.15, q 3 = 0.15, q 4 = 0.15 q 1 = 0.2, q 2 = 0.3, q 3 = 0.25, q 4 = 0.6 q 1 = 0.2, q 2 = 0.2, q 3 = 0.2, q 4 = 0.2 q 1 = 0.2, q 2 = 0.3, q 3 = 0.25, q 4 = 0.6 q 1 = 0, q 2 = 0, q 3 = 0, q 4 = 0 q 1 = 0.4, q 2 = 0.75, q 3 = 0.28, q 4 = 0.32 q 1 = 0.05, q 2 = 0.05, q 3 = 0.05, q 4 = 0.05 q 1 = 0.3, q 2 = 0.78, q 3 = 0.36, q 4 = 0.4 q 1 = 0.27, q 2 = 0.2, q 3 = 0.4, q 4 = 0.9 q 1 = 0.34, q 2 = 0.68, q 3 = 0.32, q 4 = 0.45

70 2

4 Group PROMETHEE Based on PT with Hesitant … Alternative 1

-1

Alternative 2

1.5 -2 1 -3

0.5 0 -0.5

0

0.5

-4 -0.5

0

0.5

4.3.2 6

4.3.3

Alternative 3

-0.5

5

-1

4

-1.5

3

-2

2

-2.5

1 -0.5

0

0.5

-3 -0.5

Alternative 4

0

0.5

Fig. 4.2 When q changes, the net outranking flow of each alternative with two methods

It is easily recognized that the ranking results of the two methods do not change under different parameters, which indicates the strong stability of the given two methods. Then, we give the effect of the changes of parameters on the value of each start-up firm with those two methods, shown in Fig. 4.2. Where horizontal axis shows q 1 = 0.2 × (1 + t), q 2 = 0.3 × (1 + t), q 3 = 0.25 × (1 + t), q 4 = 0.6 × (1 + t) (t ∈ [−0.3, 0.5]), and vertical axis denotes the net outranking flow of each alternative with the two methods.

4.3 Illustrative Example

71

Alternative 1

1.5

Alternative 2

0 -1

1

-2 0.5

0

-3

0

0.05

0.1

0.15

0.2

0

4

-0.5

3

-1

2

-1.5

1

-2

0

0

0.05

0.1

0.15

0

4.3.2 4.3.3

Alternative 3

5

-4

0.2

-2.5

0

0.05

0.1

0.15

0.2

Alternative 4

0.05

0.1

0.15

0.2

Fig. 4.3 When q changes, the net outranking flow of each alternative with two methods

Where horizontal axis shows q 1 = q 2 = q 3 = q 4 ∈ [0, 0.19], and vertical axis denotes the net outranking flow of each alternative with the two methods. According to Figs. 4.2 and 4.3, when q or q changes, the net outranking flow obtained from the two methods shows a trend of change in the same direction, but the fluctuation ranges are different. The biggest difference between those two methods lies that GP-PROMETHEE considers the impact of psychological states of DMs on the decision results. Hence, the GP-PROMETHEE is superior to the G-PROMETHEE. For example, many experiments [29] has demonstrated that the investors do exhibit different risk attitudes toward gain and loss. Moreover, risk seeking for loss and risk aversion for gain of DMs are especially obvious under the highly uncertain investment circumstance. Such different risk attitudes have been well revealed through the value function. Furthermore, in any decision-making methods, the weight affects the decision results though its structure and strength while it is demonstrated that the DMs usually adopt the transformed weighting function as the weight [29]. From this aspect, the GP-PROMETHEE using the weighting function in PT as the weights of attributes is superior to the G-PROMETHEE which only uses the original weight. To sum up, the GP-PROMETHEE based on PT not only integrates the value function of PT to simulate the different risk attitudes for

72

4 Group PROMETHEE Based on PT with Hesitant …

Fig. 4.4. When λ changes, the net out ranking flow and comprehensive dominance degree derived from the two methods

gain and loss of DMs but also considers the transformed weighting function of DMs. Therefore, GP-PROMETHEE has more advantage in reality. (2) Comparison of sensitivity of GP-PROMETHEE and TODIM As shown in the results of Sects. 4.3.2 and 4.3.4, the rankings of the projects in those two methods, the GP-PROMETHEE and the TODIM, are the same. Then, in order to compare the two methods, the stability of them is tested since there is the parameter λ which has the same meaning. The results are presented in Table 4.17 [24]. According to Table 4.17, when λ changes, the ranking results of both GPPROMETHEE and TODIM do not change, but the final value are different in those two methods. The effect of the changes of parameter on the overall values is given in Fig. 4.4. In TODIM, the overall value of the worst or best start-up firm has always been 0 or 1, whereas, the overall values of the other projects change between 0 and 1. The change of λ only brings little changes for the overall values of projects. In GPPROMETHEE, the values of net outranking flow of projects are not limited within 0 and 1. It is much easier for DMs to recognize the best project among the similar ones. Therefore, the GP-PROMETHEE is better than the TODIM under hesitant fuzzy linguistic circumstance.

4.3 Illustrative Example

73

Table 4.17 Ranking results Methods Value of parameter

TODIM

GP-PROMETHEE

λ = 2.25

A3  A1  A4  A2

A3  A1  A4  A2

λ = 2.15

A3  A1  A4  A2

A3  A1  A4  A2

λ = 2.05

A3  A1  A4  A2

A3  A1  A4  A2

λ = 1.85

A3  A1  A4  A2

A3  A1  A4  A2

λ = 1.79

A3  A1  A4  A2

A3  A1  A4  A2

λ = 1.75

A3  A1  A4  A2

A3  A1  A4  A2

λ = 1.55

A3  A1  A4  A2

A3  A1  A4  A2

λ = 1.25

A3  A1  A4  A2

A3  A1  A4  A2

The advantages of the GP-PROMETHEE over the G-PROMETHEE and the TODIM are comprehensively shown by the comparative analysis. They are summarized as follows: (1) The GP-PROMETHEE considers both the VCs’ risk attitudes such as risk seeking for loss and risk aversion for gain and the transformed weight, whereas, the other methods usually ignore the transformed weight. (2) Although there are parameters in the GP-PROMETHEE, it is stable and effective.

4.4 Simulation Analysis Section 4.3 gives the ranking results from the three methods with the same group of evaluation information and the results are the same with those methods. In order to analyze the difference of those three methods deeply, 1000 groups of evaluation information are generated by Matlab and apply them to the three methods. The results are shown in Table 4.18. According to Table 4.18, 733 groups of evaluation information get the same ranking results under the three methods. When adopting the methods-GPPROMETHEE and G-PROMETHEE, 807 groups of evaluation information get the same ranking results while it declines to 743 when using GP-PROMETHEE and TODIM. Most of the evaluation information get the same results which indicates the Table 4.18 Ranking results of 1000 data Ranking result Methods

Number of data derived the same ranking results

Number of data derived the different ranking results

Sections 4.3.2, 4.3.3 and 4.3.4

733

267

Sections 4.3.2 and 4.3.3

807

193

Sections 4.3.2 and 4.3.4

743

257

74

4 Group PROMETHEE Based on PT with Hesitant …

Fig. 4.5 Ranking results of alternative 1 with 200 group of data

feasibility of the given method. However, some of the evaluation information get the different ranking results which explains the differences of those methods. Then, the ranking results of 200 sets of data are presented in Figs. 4.5, 4.6, 4.7 and 4.8 with the different methods.

4.5 Remarks The biggest difference between the GP-PROMETHEE and the G-PROMETHEE lies in that the former one introduces the effect of behaviors of the DMs on the decisionmaking process. From the theoretical aspect, the GP-PROMETHEE is better than the G-PROMETHEE. For example, numerous laboratory experiments have shown that the different attitudes of the DMs are indeed reflected in the aspects of gains and losses [29]. Moreover, the DMs’ attitudes with respect to risk aversion for gain or risk seeking for loss is more prominent under the highly uncertain circumstance. This different attitude for gain or loss has been considered in the GP-PROMETHEE as the value function. In addition, whatever the methods are, the attributes weights affect

4.5 Remarks

Fig. 4.6 Ranking results of alternative 2 with 200 group of data

Fig. 4.7 Ranking results of alternative 3 with 200 group of data

75

76

4 Group PROMETHEE Based on PT with Hesitant …

Fig. 4.8 Ranking results of alternative 4 with 200 group of data

the decision making via its structure and strength, whereas, the transformed weight adopted by DMs has been confirmed [29]. Therefore, the GP-PROMETHEE using the weighting function rather than the simple weight is better than the G-PROMETHEE. On account of the different risk attitudes for gain and loss, the weighting function enhances the negative value or the positive value of the net outranking flow respectively under the GP-PROMETHEE. To sum up, the GP-PROMETHEE that combines the ranking method (PROMETHEE) with the behavioral theory (the PT which includes the value function to simulate the risk attitudes of the VCs and the weighting function to portray the transformed probability perceived by the VCs) exhibits more advantages in the decision-making process. The main focus of this chapter lies as follows: ➀ It considers the psychological state of DMs in the MADM method and integrates the PT into PROMETHEE. It will promote the development of behavior decision and may encourage the decisionmaking method to consider psychological state such as regret, overconfidence, overreaction, etc. ➁ It extends PROMETHEE to the hesitant fuzzy linguistic circumstance. ➂ It applies the given methods to the venture capital field which enriches the application of them and provides a new way for VCs to solve their problems. ➃ It introduces the HFLTS to depict the evaluation information of VCs and offers a new direct to solve finance problems under fuzzy circumstance.

References

77

References 1. Brans JP (1982) L’ingénièrie de la décision; Elaboration d’instruments d’aide à la décision. La méthode PROMETHEE. In: L’aide à la décision: Nature, Instruments et Perspectives d’Avenir 2. Brans JP, Vincke PH (1985) A preference ranking organization method (the PROMETHEE method for multiple criteria decision-making). Manage Sci 31(6):647–656 3. Brans JP, Vincke P, Mareschal B (1986) How to select and how to rank projects—the PROMETHEE method. Eur J Oper Res 24(2):228–238 4. Brans JP, Mareschal B (1992) PROMETHEE-V-MCDM problems with segmentation constraints. Infor 30(2):85–96 5. Brans JP, Mareschal B (1995) The PROMETHEE VI procedure: How to differentiate hard from soft multicriteria problems. J Decis Syst 4(3):213–223 6. Mateo JRSC (2012) Fuzzy PROMETHEE. Springer, London 7. Li WX, Li BY (2010) An extension of the Promethee II method based on generalized fuzzy numbers. Expert Syst Appl 37(7):5314–5319 8. Chen TY (2014) A PROMETHEE-based outranking method for multiple criteria decision analysis with interval type-2 fuzzy sets. Soft Comput 18(5):923–940 9. Liao HC, Xu ZS (2014) Multi-criteria decision making with intuitionistic fuzzy PROMETHEE. J Intell Fuzzy Syst Appl Eng Technol 27(4):1703–1717 10. Krishankumar R, Ravichandran KS, Saeid AB (2017) A new extension to PROMETHEE under intuitionistic fuzzy environment for solving supplier selection problem with linguistic preferences. Appl Soft Comput 60:564–576 11. Chen TY (2015) IVIF-PROMETHEE outranking methods for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets. Fuzzy Optim Decis Making 14(2):173–198 12. Liang RX, Wang JQ, Zhang HY (2018) Projection-based PROMETHEE methods based on hesitant fuzzy linguistic term sets. Int J Fuzzy Syst 20(7):2161–2174 13. Chen CT, Hung WZ, Cheng HL (2011) Applying linguistic PROMETHEE method in investment portfolio decision-making. Int J Electr Bus Manag 9(2):139–147 14. Mahmoudi A, Sadi-Nezhad S, Makui A, Vakili MR (2016) An extension on PROMETHEE based on the typical hesitant fuzzy sets to solve multi-attribute decision-making problem. Kybernetes 45(8):1213–1231 15. Peko I, Gjeldum N, Bilic B (2018) Application of AHP, fuzzy AHP and PROMETHEE method in solving additive manufacturing process selection problem. Tehnicki Vjesnik-Technical Gazette 25(2):453–461 16. Samanlioglu F, Ayag Z (2016) Fuzzy ANP-based PROMETHEE II approach for evaluation of machine tool alternatives. J Intell Fuzzy Syst 30(4):2223–2235 17. Fernandez-Castro AS, Jimenez M (2005) PROMETHEE: an extension through fuzzy mathematical programming. J Oper Res Soc 56(1):119–122 18. Mareschal B, Brans JP, Macharis C (1998) The GDSS PROMETHEE procedure: a PROMETHEE-GAIA based procedure for group decision support. J Decis Syst 7:283–307 19. Liao HC, Wu D, Huang YL, Ren PJ, Xu ZS, Verma M (2018) Green logistic provider selection with a hesitant fuzzy linguistic thermodynamic method integrating cumulative prospect theory and PROMETHEE. Sustainability 10(4):16 20. Liao HC, Xu ZS (2014) Multi-criteria decision making with intuitionistic fuzzy PROMETHEE. J Intell Fuzzy Syst 27(4):1703–1717 21. Shih HS, Chang YT, Cheng CP (2016) A generalized PROMETHEE III with risk preferences on losses and gains. Int J Inform Manag Sci 27(2):117–127 22. Lerche N, Geldermann J (2014) Integration of prospect theory into the outranking approach PROMETHEE. In: Lübbecke M, Koster A, Letmathe P, Madlener R, Peis B, Walther G (eds) Operations research proceedings 2014. Operations research proceedings (GOR (Gesellschaft für Operations Research e.V.)). Springer, Cham, pp 363–368 23. de Treville S, Petty JS, Wager S (2014) Economies of extremes: lessons from venture-capital decision making. J Oper Manag 32(6):387–398

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24. Tian XL, Xu ZS, Gu J (2019) Group decision-making models for venture capitalists: the PROMETHEE with hesitant fuzzy linguistic information. Technol Econ Dev Econ 25(5):743– 773 25. Rodriguez RM, Martinez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20(1):109–119 26. Abdellaoui M (2000) Parameter-free elicitation of utility and probability weighting functions. Manage Sci 46(11):1497–1512 27. Wei CP, Ren ZL, Rodriguez RM (2015) A hesitant fuzzy linguistic TODIM method based on a score function. Int J Comput Intell Syst 8(4):701–712 28. Wang L, Wang YM, Rodríguez RM, Martínez L (2017) A hesitant fuzzy linguistic model for emergency decision making based on fuzzy TODIM method. In: 2017 IEEE international conference on fuzzy systems. IEEE, New York 29. Mattos F, Garcia P (2011) Applications of behavioral finance to entrepreneurs and venture capitalists: decision making under risk and uncertainty in futures and options markets. Advances in Entrepreneurial Finance, Springer, New York

Chapter 5

Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

There are always bifurcation points among the DMs in GDM problems. If the DMs ignore those bifurcation points and just simply integrate each DM’s opinion, it may lead unreasonable decision-making result. Hence, such bifurcation points should not be ignored in the decision-making process. Consensus model is an effective way for DMs to decrease the bifurcation points among them though adjusting their evaluations. In this chapter, we will introduce a consensus model considering the risk attitudes of DMs [1].

5.1 Probabilistic Hesitant Fuzzy Preference Information When it is hard for the DMs to give evaluation information for each alternative, it is easy for them to give the preference information for each pair of alternatives. Thus, preference relation is a useful and effective tool to express such preference perceptions of DMs. Moreover, the DMs are irresolute in some uncertain situations and express their preferences with several values. Meanwhile, the hesitancy degree for each possible preference value may be different as well. By means of such a different hesitancy degree, a probability has been assigned to each possible hesitant preference value. This is named as P-HFP (probabilistic hesitant fuzzy preference), represented by P-HFS. When the probabilities of all possible hesitant preference values are the same, the P-HFP can be considered as the hesitant fuzzy preference (HFP). Hence, acting as the upgrading of the HFP, the P-HFP includes more original information perceived by the DMs and it will improve the quality of decision-making through such an expression. The P-HFP is expressed by P-HFS [2] mentioned in Sect. 1.2.4. Suppose that  A = {A1 , A2 , . . . , Ai , . . . , Ai  , . . . , An } i, i  ∈ N , N = {1, 2, . . . , n} is a fixed set of alternatives, then, the P-HFP matrix is:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_5

79

80

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

··· A1 A2 h 11 ( p11 ) h 12 ( p12 ) · · · ⎜ h 21 ( p21 ) h 22 ( p22 ) · · · ⎜ ⎜ .. .. .. ⎝ . . . An h n1 ( pn1 ) h n2 ( pn2 ) · · ·

A1 H = A2 .. .

⎛

An ⎞ h 1n ( p1n ) h 2n ( p2n ) ⎟ ⎟ = (h ii  ( pii  ))n×n ⊂ X × X (5.1) ⎟ .. ⎠ .

h nn ( pnn )

   where h ii  ( pii  ) = h iit  piit |t = 1, 2, . . . , #h ii  ( pii  ) is named as probabilistic hesitant fuzzy preference element (P-HFPE) which shows the set of all the possible the corresponding probability of preference for Ai related to Ai  , and  pii  represents  h ii  ( pii  ). Furthermore, h ii  ( pii  ) i, i  ∈ N satisfies the following conditions:   t+1 t+1 (1) For all i < i  , i, i  ∈ N , there is piit  h iit  ( piit  ) > piit+1  h ii  ( pii  ) or t+1 t+1 t+1 t+1 t+1 t t t t t t pii  h ii  ( pii  ) < pii  h ii  ( pii  ), if and only if pii  h ii  ( pii  ) = pii  h ii  ( piit+1  ), or piit  < piit+1 piit  > piit+1   . The former one is named as descending ordered P-HFP while the latter one is called ascending ordered P-HFP. In this chapter, all the P-HFPs are the ascending one unless there is especial explanation.

 t  #h ii  ( pii  )−t+1 #h ii  ( pii  )−t+1 #h  ( p  )−t+1 t pi  i = 1, piit  = pii  ii ii and (2) h ii  pii  + h i  i #h ii  ( pii  ) = #h i  i ( pi  i ). (3) 0 ≤ h iit  ( piit  ) ≤ 1, 0 ≤ piit  ≤ 1, h ii ( pii ) = 0.5. Example 5.1 [1] Consider a decision-making problem with two alternatives and an expert is invited to give his/her preference information as:  H = (h ii  ( pii  ))n×n

A = 1 A2

A1 A2  0.5 (0.76(0.5), 0.6(0.3), 0.9(0.2)) (0.1(0.2), 0.4(0.3), 0.24(0.5)) 0.5

Because there is 0.18=0.6×0.3=0.9×0.2 and 0.3>0.2, according to the above rules, the orders of those two possible preferences are: 2 3 )=0.6(0.3) and h 312 ( p12 )=0.9(0.2). For h ii  ( pii  )(i h h h h  , jk p ii  , jk  , jk p ii  , jk ⎪     ii ii ii , jk ii , jk ii , jk ii , jk ⎪

⎨ t 0 h iit  , jk piit  , jk = h ii  , jk piit  , jk v t (h iit  , jk ) = ⎪



β



⎪ ⎪ t t t t ⎩ −λ h t  h iit  , jk piit  , jk < h ii  , jk piit  , jk ii , jk p ii  , jk − h ii  , jk pii  , jk

(5.4)

82

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

  wt piit  , jk =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  pt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 



piit  , jk

δ

δ

δ 1/δ + 1− piit  , jk ii  , jk

γ piit  , jk

piit  , jk

γ

γ 1/γ + 1− piit  , jk

  t h iit  , jk ( piit  , jk ) < h ii  , jk piit  , jk (5.5) other wise



where h iit  , jk piit  , jk is the t-th possible preference of the alternative i related to the alternative i  under the j-th attribute from the k-th DM,

t t t whereas, h ii  , jk ( pii  , jk ) is the corresponding reference point of h ii  , jk piit  , jk . In this chapter, the reference point is  derived from the corresponding   = = h ii  , jk pii  , jk n×n , where h ii  , jk pii  , jk reference matrix H jk      t     t |t = 1, 2, . . . , #h ii  , jk pii  , jk . Hence, h ii  , jk piit  , jk = 0.5t #h  1( p  ) ii , jk

ii , jk

V (h ii  , jk ) is the prospect preference value of the i-th alternative related to the i  -th alternative under the j-th attribute from the k-th DM.   For a P-HFP matrix H jk , there is a priority vector ω jk = ω1, jk , ω2, jk , . . . , ωn, jk that satisfies the programming model [1]: ⎡ ⎤ #h ii  ,qk  2 n  n     ω 1 i, jk ⎣ ⎦. − V t h iit  , jk Q jk = min  , jk  , jk #h ω + ω ii i, jk i t=1 i=1 i  =1 ⎧ n  ⎪ Model I . ⎪ ⎪ ωi, jk = 1 ⎨ s.t. i=1 ⎪ ⎪ ⎪ ωi, jk ≥ 0 ⎩ i, i  ∈ N Based on the optimal solutions of Model I, the priority vector matrix of the attribute c j for all DMs is given as: d1 d2 dk dz ⎤ ω 1, j1 ω1, j2 · · · ω1, j z A1 ⎢ ⎥ ω j = A2 ⎢ ω2, j1 ω2, j2 · · · ω2, j z ⎥ ⎢ . ⎥ . . .. ω .. ⎦ Ai ⎣ .. i, jk An ωn, j1 ωn, j2 · · · ωn, j z ⎡

(5.6)

n×z

Inspired by the idea of variance in statistics, it is used to measure the consensus degree of DMs by calculating the fluctuation degree of priority vector among DMs. Hence, for the optimal solutions of all the DMs under the j-th attribute, the variance of the optimal priority vector [1] is presented as: z %

  σ ωi, j =

k=1

  z & 2 % ωi, jk − ωi, jk z k=1

z

(5.7)

5.2 Consensus Model Based on PT with P-HFPs

83

The variance function σ (ωi, j ) measures the volatile degree of the optimal priority vector ωi, jk for the i-th alternative under the attribute c j from each DM. The smaller the value of σ (ωi, j ) is, the more centralized ωi, jk among the DMs will be. In other words, the smaller σ (ωi, j ) expresses that the smaller volatility of the priority vector ωi, jk for the i-th alternative under j-th attribute from all DMs will be and the more easily for the DMs to reach group consensus. On the contrary, the bigger the value of σ (ωi, j ) is, the more difficult for the DMs to reach consensus with the i-th alternative under the j-th attribute. Therefore, if the value σ (ωi, j ) ≤ θ (θ is the predefined acceptable threshold of group consensus), then the group consensus is achieved. On the contrary, if σ (ωi, j ) > θ , then the group consensus is not accepted. When the group consensus has not been accepted, we should find the 2  %z ωi, jk for all k ∈ Z . If there biggest'deviation of the priority vector( ωi, jk − k=1 2  %z ωi, jk  , k ∈ Z for all DMs, then we should correct the k  is max ωi, jk  − k=1 th DM’s preference information of the i-th alternative to the others under the j-th attribute. Then, we repeat the above operations until σ (ωi, j ) ≤ θ (i ∈ N , j ∈ M). After reaching the group consensus, we can aggregate the overall priority vector of the i-th alternative under the j-th attribute. Then, the aggregated priority vector is obtained by (5.8) [1]: i,q =

z 

ζk ωi,qk

(5.8)

k=1

where ζk is the weight of the k-th DM. According to the idea that the more the DM is sure about his/her judgment the more stability of each P-HFE and the of the choice,   smaller the value of σ h ii  , jk pii  , jk and the number of hesitant #h ii  , jk ( pii  , jk ) will be, then the bigger the DM’s weight should be. Hence, a method is used to calculate the weight of each DM by (5.9) and (5.10) [1]. ζk =

1 − ek z % z− ek

(5.9)

k=1

ek =

m 

i Eqs. (5.3)–(5.5). In addition, because there exists h 112,11 p12,11  1    1 h 12,11 p 12,11 = 0.5, according to Eqs. (5.4) and (5.5), there is v1 h 112,11 =  1    α  1  1 h 112,11 p12,11 − h 12,11 p 112,11 = = (0.79 − 0.5)0.88 = 0.3364 and w p12,11 γ 1 ( p12,11 ) 0.60.61 = 0.4742 respectively. Then, based on  1 1 γ = 1 0.61 1/ 0.61 ]  ( p12,11 )γ +(1− p12,11 )γ /  [0.60.61 +(1−0.6)   Eq. (5.3), there is V 1 h 112,11 = v1 h 112,11 w h 112,11 = 0.3364 × 0.4742 = 0.1594.

d3

d2

d1

d3

d2

d1

(0.56(0.7), 0.45(0.18), 0.38(0.12)) 0.5

(0.62(0.12), 0.55(0.18), 0.44(0.7))

0.5

(0.3(0.02), 0.35(0.18), 0.24(0.8))

)

)

*

0.5

(0.76(0.8), 0.65(0.18), 0.7(0.02))

*

0.5

0.5

(0.3(0.2), 0.25(0.2), 0.21(0.6))

(0.76(0.51), 0.55(0.2), 0.4(0.29)) 0.5

0.5

(0.6(0.29), 0.45(0.2), 0.24(0.51))

0.5

(0.58(0.22), 0.44(0.3), 0.31(0.48)) *

(0.69(0.48), 0.56(0.3), 0.42(0.22))

0.5

0.5

(0.42(0.07), 0.38(0.32), 0.27(0.61)) *

(0.73(0.61), 0.62(0.32), 0.58(0.07))

*

0.5

c3 )

)

)

(0.79(0.6), 0.75(0.2), 0.7(0.2))

0.5

c1 )

Table 5.1 Transformed probabilistic hesitant fuzzy preference information

)

)

0.5

*

0.5

0.5

*

*

*

*

(continued)

(0.55(0.62), 0.47(0.28), 0.41(0.1))

0.5

*

(0.68(0.6), 0.55(0.22), 0.42(0.18))

0.5

(0.88(0.78), 0.83(0.16), 0.76(0.06))

0.5

(0.54(0.58), 0.4(0.21), 0.3(0.21))

(0.59(0.1), 0.53(0.28), 0.45(0.62))

0.5

(0.7(0.21), 0.6(0.21), 0.46(0.58))

0.5

(0.58(0.18), 0.45(0.22), 0.32(0.6))

0.5

0.5

(0.92(0.84), 0.83(0.11), 0.71(0.05))

(0.95(0.89), 0.9(0.1), 0.86(0.01))

(0.24(0.06), 0.17(0.16), 0.12(0.78))

0.5

(0.14(0.01), 0.1(0.1), 0.05(0.89))

0.5

(0.29(0.05), 0.17(0.11), 0.08(0.84))

c4 * )

)

)

c2 )

90 5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

(0.82(0.6), 0.73(0.3), 0.71(0.1)) 0.5

0.5

(0.29(0.1), 0.27(0.3), 0.18(0.6))

0.5

(0.19(0.08), 0.25(0.25), 0.12(0.67))

)

)

(0.87(0.69), 0.75(0.23), 0.71(0.08)) 0.5

(0.29(0.08), 0.21(0.23), 0.13(0.69))

0.5

(0.36(0.1), 0.29(0.22), 0.11(0.68))

*

0.5

(0.89(0.68), 0.71(0.22)), 0.64(0.1)

0.5

(0.22(0.06), 0.14(0.26), 0.08(0.68))

0.5

(0.92(0.68), 0.86(0.26), 0.78(0.06))

0.5

*

(0.88(0.67), 0.75(0.25), 0.81(0.08))

*

0.5

0.5

(0.31(0.19), 0.24(0.3), 0.2(0.51))

c7 )

)

)

(0.8(0.51), 0.76(0.3), 0.69(0.19))

0.5

c5 )

*

*

)

* )

0.5

(0.56(0.09), 0.62(0.4), 0.45(0.51))

0.5

(0.59(0.27), 0.48(0.2), 0.37(0.53))

0.5

(0.59(0.1), 0.55(0.34), 0.48(0.56))

c6 )

Note The probabilistic hesitant fuzzy preference information is the transformed ones based on Example 1.1

d3

d2

d1

d3

d2

d1

Table 5.1 (continued)

0.5

(0.55(0.51), 0.38(0.4), 0.44(0.09))

0.5

(0.63(0.53), 0.52(0.2), 0.41(0.27))

0.5

(0.52(0.56), 0.45(0.34), 0.41(0.1))

*

*

*

5.3 Illustrative Example 91

d3

d2

d1

d3

d2

d1

(0.0449, −0.0389, −0.0661) 0

(0.0315, 0.0178, −0.1112)

0

(−0.0340, −0.1023, −0.4600)

0

(0.1856, 0.0467, 0.0197)

0

0

(−0.1403, −0.1707, −0.3922)

)

)

(0.1301, 0.0187, −0.0952) 0

(0.0413, −0.0414, −0.3165)

0

(0.0296, −0.0620, −0.2304)

0

(0.0952, 0.0268, −0.0663)

0

(−0.0334, −0.1186, −0.3239)

0

(0.1315, 0.0509, 0.0169)

0

c3 )

)

)

(0.1594, 0.0770, 0.0633)

0

c1 )

Table 5.2 Prospect preference matrix

*

*

*

*

*

*

)

)

0

(0.0224, 0.0141, −0.0857)

0

(0.0648, 0.0352, −0.0669)

0

0

(0.1048, 0.0196, −0.0589)

0

(0.2523, 0.0882, 0.0441)

0

(0.3461, 0.0832, 0.0225)

0

(0.3000, 0.0736, 0.0333)

0

(0.0348, −0.0323, −0.0460)

0

(0.0272, −0.0785, −0.1444)

(0.0268, −0.0438, −0.2578)

0

(−0.0858, −0.1910, −0.6255)

0

(−0.0363, −0.1709, −0.8495)

0

(−0.0635, −0.1528, −0.7415)

c4 )

)

)

c2 )

(continued)

*

*

*

*

*

*

92 5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

d3

d2

d1

d3

d2

d1

(0.1739, 0.0873, 0.0472) 0

(−0.0970, −0.2022, −0.4277)

0

(−0.1194, −0.1950, −0.5435)

0

(0.2197, 0.0858, 0.0595)

0

0

(−0.1301, −0.2253, −0.3590)

)

)

(0.2198, 0.0939, 0.0423) 0

(−0.0847, −0.2114, −0.5444)

0

(−0.0679, −0.1550, −0.5631)

0

(0.2275, 0.0692, 0.0330)

0

(−0.0915, −0.2751, −0.6011)

0

(0.2428, 0.1206, 0.0471)

0

c7 )

)

)

(0.1476, 0.0973, 0.0590)

0

c5 )

Table 5.2 (continued)

*

*

*

*

*

*

)

)

0

(0.0149, 0.0573, −0.0742)

0

0

(0.0305, −0.1364, −0.0302)

0

(0.0724, 0.0083, −0.0831)

0

(0.0145, −0.0570, −0.0460)

(0.0363, −0.0185, −0.1767)

0

(0.0224, 0.0243, −0.0354)

c6 )

*

*

*

5.3 Illustrative Example 93

94

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

Table 5.3 The optimal priority vector matrix under each attribute ) ) * 0.6672 0.6414 0.5003 0.7275 0.7514 ω1 = ω2 = 0.3328 0.3586 0.4997 0.2725 0.2486 ) ) * 0.6126 0.5531 0.5617 0.5567 0.4619 ω3 = ω4 = 0.3874 0.4469 0.4383 0.4433 0.5381 ) * ) 0.6697 0.7038 0.6725 0.4834 0.5261 ω5 = ω6 = 0.3303 0.2962 0.3275 0.5166 0.4739 ) * 0.7297 0.6859 0.6994 ω7 = 0.2703 0.3141 0.3006

0.7145

*

0.2855 0.5009

*

0.4991 0.4777

*

0.5223

Table 5.4 The variance of optimal priority vector from all investors Variance σ (ω j )

Attributes c1

c2

c3

c4

c5

c6

c7

0.5381%

0.0234%

0.0689%

0.1514%

0.0239%

0.0466%

0.0335%

Note Due to the specialty of the problem in this chapter, only two alternatives are considered and ω1, jk + ω2, jk = 1. Hence, by combining Eq. (5.7), it is easy to understand σ (ω1, j ) = σ (ω2, j ). Here, it is expressed as σ (ω j ) = σ (ω1, j ) = σ (ω2, j ) in Table 5.4

  In the same way, V h 21,11 = (−0.1403, −0.1707, −0.3922) shows the hesitant prospect preference of A1 over A2 . V (h 11,11 ) = V (h 22,11 ) = 0 explains the indifferent prospect preference when compared with itself. Step 3. According to Model I, the optimal priority vector matrix ω j = [ωi, jk ]2×3 is derived from Matlab and the results are listed in Table 5.3. The optimal priorities ωi, jk from the three experts for the project A1 under the attribute c1 are ω1,11 = 0.6672, ω1,12 = 0.6414, and ω1,13 = 0.5003 respectively. Step 4. The variance of optimal priority vector is worked out: Taking σ (ω1 )

= σ (ω2,1 ) = 0.5381% for an example, z &   % ω1,11 + ω1,12 + ω1,13 /3 = referring to Step 3, there is ωi, jk z = k=1 & (0.6672 + 0.6414 + 0.5003) 3 = 0.6030. Then, based on Eq. (12), σ (ω1,1 ) = & 2 %z %z 2 2 2 k=1 (ω1,1k − k=1 ωi, jk z ) = (0.6672−0.6030) +(0.6414−0.6030) +(0.5003−0.6030) = 0.5381%. z

=

σ (ω1,1 )

3

The other variances σ (ω j ) = σ (ω1, j ) = σ (ω2, j ) ( j ∈ M) are obtained in the same way. Based on the consensus threshold θ = 0.5% in Step 1, the evaluation information not reach the group consensus because there exists σ (ω1 ) = for   the attribute c1 could σ ω1,1 = σ ω2,1 = 0.5381% > 0.5%. Furthermore, according to Step 4 in

5.3 Illustrative Example

95

Table 5.5 The weight of each investor

Weight

Investors

ζk

d1

d2

d3

0.3501

0.3251

0.3248

+ 2 

& 2 %z % Sect. 5.2.2, max ωi,qk − k=1 ωi,qk z , k ∈ Z = ωi,q3 − 3k=1 ωi,q3 z = 1.0540%. Therefore, the investor d3 should adjust his/her evaluation information. After careful investigation of the two alternatives, the 3-th investor’s prior evaluation information for the attribute c1 is updated as: 

0.5 (0.63(0.74), 0.48(0.21), 0.42(0.05)) (0.58(0.05), 0.52(0.21), 0.37(0.74)) 0.5



Then, the prospect preference value is: 

0 (0.0932, −0.0190, −0.0272) (0.0143, 0.0085, −0.2311) 0



Next, the priority vector under the attribute c1 from the investor d3 is ω1, j3 = 0.5425 and ω2, j3 = 0.4575. Finally, the variance of optimal priority under the attribute c1 from all investors is smaller than the consensus threshold and the group consensus has reached. Then, turn to Step 5. Step 5. The weight of each investor is obtained in Table 5.5. Here, we show the computational process of ζ1 = 0.3501. At first, for the expert d1 with the preference of A1 over A2 , the score of the evaluation information based on Eq. (1.17) +% under the attribute c1 is ρ(h 12,11 ( p12,11 )) = %#h( p) t #h( p) t 0.6×0.79+0.2×0.75+0.2×0.7 t t = 0.764, t=1 p12,11 × h 12,11 ( p12,11 ) t=1 p12,11 = 0.6+0.2+0.2 and then, the variance of the evaluation information according to Eq. (1.18) is 



σ h 12,11 p12,11



=

#h( p)

t p12,11

t=1

×



h t12,11



t p12,11







− s h 12,11 p12,q1

= 0.6 × (0.79 − 0.764)2 + 0.2 × (0.75 − 0.764)2

+ +0.2 × (0.7 − 0.764)2 (0.6 + 0.2 + 0.2)

2

,#h( p) 

t p12,11

t=1

= 0.001264

   In the same way, the under the other attributes  variances    are: σ  h 12,21 p12,21 = = 0.003359, 0.002680,  σ h12,31 p12,31  σ h 12,41 p12,41  =  0.010201,   σ h 12,51 p12,51 = 0.001697, σ h 12,61 p12,61 = 0.001665, σ h 12,71 p12,71 = 0.001536. According to Eq. (5.10), there is:

96

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

e1 =

7 

i< j σ



     h 12,q1 p12,q1 × #h 12,q1 p12,q1

q=1

= 0.001264 × 3 + 0.002680 × 3 + 0.003359 × 3 + 0.010201 × 3 + 0.001697 × 3 + 0.001665 × 3 + 0.001536 × 3 = 0.0672. Similarly, + e2 = 0.1338, e3 = 0.1348. So, on the basis of Eq. (5.9), there is % ζ1 = (1 − e1 ) (3 − 3k=1 ek ) = 0.3501. Steps 6–8. The overall priority value of each alternative is: 1 = 1.1223 and 2 = 0.6977. Hence, the promising alternative is A1 and they will invest the project sequentially. Using Eq. (5.8) to calculate i,q , the aggregated priority of the project A1 under the % attribute c1 is 1,1 = 3k=1 ζk ω1,1k = 0.3501×0.6672+0.3251×0.6414+0.3248× 0.5425 = 0.6183. Similarly, for the project A1 under other attributes, 1,2 = 0.7310, 1,3 = 0.5767, 1,4 = 0.5078, 1,5 = 0.6817, 1,6 = 0.4954, 1,7 = 0.7056. %  Then, referring to Eq. (5.11), 1 = 7j=1 w j 1, j = 0.26×0.6183+0.11×0.7310+ 0.14 × 0.5767 + 0.12 × 0.5078+0.25 × 0.6817 + 0.07 × 0.4954 + 0.05×0.7056 = 1.1223.

5.3.3 Results of the Expected Consensus Process with P-HFPs Compared to the bounded rationality which is well presented in PT, the expected value thinks that the decision is made under the condition of complete rationality. Therefore, in this chapter, a group consensus is given according to the expected preference information [1]. The original evaluation information follows Table 2. Then, the expected value is the product of original evaluation value and the corresponding probability. Thus, the expected preference information is expressed in Table 5.6 [1]. Compared with PT for revealing the bounded rationality of DMs through the different attitudes for gains or losses and the transformed weighting function, the expected theory supposes that DMs are totally rational. Hence, in this chapter, an expected consensus procedure with P-HFP is given to make a comparison. Algorithm 5.2 [1]

   Input: The P-HFP matrix H jk = h ii  , jk pii  , jk n×n where j ∈ M, k ∈ Z , i, i  ∈ N , and the predefined group consensus threshold θ . Output: The best project Ai∗ .

d3

d2

d1

Investors

Attributes

d3

d2

d1

Investors

Attributes

)

)

)

c4

)

)

)

c1

(0.341, 0.1316, 0.041)

0

(0.059, 0.1484, 0.279)

0

(0.147, 0.126, 0.2668)

0

(0.3132, 0.084, 0.063)

0

(0.1044, 0.099, 0.192)

0

(0.408, 0.121, 0.0756)

0

(0.0744, 0.099, 0.308)

0

(0.392, 0.081, 0.0456)

0

(0.006, 0.063, 0.192)

*

0

(0.608, 0.117, 0.014)

0

(0.06, 0.05, 0.126)

*

0

(0.474, 0.15, 0.14)

0

Table 5.6 Expected decision-making matrix

*

*

*

*

)

)

)

c5

)

)

)

c2

(0.029, 0.081, 0.108)

0

0

0

(0.492, 0.219, 0.071)

0 *

(0.5896, 0.1875, 0.0648)

0

*

(0.6864, 0.1328, 0.0456)

0

*

(0.408, 0.228, 0.1311)

(0.0152, 0.0625, 0.0804)

0

(0.0589, 0.072, 0.102)

0

0

(0.7728, 0.0913, 0.0355)

(0.8455, 0.09, 0.0086)

(0.0144, 0.0272, 0.0936)

0

(0.06, 0.05, 0.126)

0

(0.0145, 0.0187, 0.0672)

0

*

*

*

)

)

)

c6

)

)

)

c3

(0.0504, 0.248, 0.2295)

0

0

*

*

0

*

*

*

*

(continued)

(0.2805, 0.152, 0.0396)

0

(0.3339, 0.104, 0.1107)

0

(0.2912, 0.153, 0.041)

(0.1593, 0.096, 0.1961)

0

0

(0.3312, 0.168, 0.0924)

0

(0.4453, 0.1984, 0.0406)

(0.3876, 0.11, 0.116)

(0.059, 0.187, 0.2688)

0

(0.174, 0.09, 0.1224)

0

(0.1276, 0.132, 0.1488)

0

(0.0294, 0.1216, 0.1647)

0

5.3 Illustrative Example 97

d3

d2

d1

Investors

Attributes

)

)

)

c7

(0.6003, 0.1817, 0.0568) 0

(0.0232, 0.0483, 0.0897)

0

(0.036, 0.0638, 0.0748)

*

0

(0.0652, 0.1562, 0.064)

0

(0.0132, 0.0364, 0.0544)

0

(0.6256, 0.2236, 0.0468)

0

Table 5.6 (continued)

*

*

98 5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

5.3 Illustrative Example

99

Step 1. Obtain the P-HFP   matrices from DMs, H jk = (h ii  , jk ( pii  , jk ))n×n j ∈ M, k ∈ Z , i, i  ∈ N and the consensus threshold  θ.  Step 2. Get the expected decision-making matrix V jk , j ∈ M, k ∈ Z , i, i  ∈ N according to Eq. (5.12):    = V  (h ii  , jk ) n×n V jk

(5.12)



  = V 1 (h ii1  , jk ), V 2 (h ii2  , jk ), . . . , V t (h iit  , jk ) where V  h ii  , jk   t = 1, 2, . . . , #h ii  , jk is HFPE and it is calculated by Eq. (5.13): V t (h ii  , jk ) = h iit  , jk × piit  , jk

(5.13)

Step 3. Solve model II, and get the optimal priority vector matrix ω j . For the P-HFP matrix H jk , the corresponding probabilistic hesitant fuzzy expected matrix is calculated by Eq. (5.12). Then, there is the vector ω jk = (ω1, jk , ω2, jk , . . . , ωn, jk ) satisfying ⎡ the model II [1]: ⎤ #h ii  , jk  2 n n   ω 1 i, jk ⎣ Q jk = min − V t (h iit  , jk ) ⎦  , jk  , jk #h ω + ω ii i, jk i  t=1 i=1 i =1 ⎧ n % Model II . ⎪ ⎪ ⎨ ωi, jk = 1 s.t. i=1 ωi, jk ≥ 0 ⎪ ⎪ ⎩ i, j ∈ N Step 4. Calculate the variance of the alternative Ai under the attribute c j according to Eq. (5.7) with the results in model II. If there is σ (ωi, j ) ≤ θ , then the consensus is reached for the alternative Ai under the attribute c j . Otherwise, we find the expert who contributes most to the non-consensus results according to Sect. 5.2.1 and let the expert correct his/her evaluation information. Then, we repeat the above steps until reaching the consensus. Step 5. Get the weight of each DM based on Eqs. (5.9) and (5.10). Step 6. Obtain the priority vector i, j of the alternative Ai under the attribute c j according to Eq. (5.8). Step 7. Collect the overall priority vector of each project grounded on (5.11). Step 8. Rank the overall priority vector obtained in Step 7 and select the biggest one as the optimal alternative. Step 9. End. Comparing Algorithms 5.1 and 5.2, the most difference of them lies in Step 2 which is used to calculate the prospect preference matrices and expected preference matrices respectively. Hence, in this chapter, not all the decision-making procedure with illustrative example is shown and only the important results are given as follows:

100

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

In order to be comparable, the illustrative example is the same as in Sect. 5.3.2. Based on Eq. (5.12), the expected decision-making matrix is [1]. According to Model II, the optimal priority vector matrix is presented in Table 5.7. All the variances of the optimal priority vector σ (ωi, j ) (i ∈ N , j ∈ M) are smaller than the predefined threshold θ j , which indicates that all the experts reach consensus under each attribute and no one needs to change his/her mind. Then, the overall priority values are 1 = 1.0305 and 2 = 0.7895, where 1 > 2 shows that the project will continuously gain the capital from VC company.

5.3.4 Results of Prospect Consensus with HFPs Actually, probabilistic hesitant fuzzy information is the upgrade of hesitant fuzzy information. It considers each possible hesitant value with different probabilities, whereas, hesitant fuzzy information regards each possible hesitant value with the same probability. Such different probabilities include more evaluation information, and they are more consistent with the real decision-making situations. Thus, in terms of theory, the prospect consensus with P-HFPs is superior to the prospect consensus with HFPs. From the perspective of practice, in this chapter, a prospect consensus process with HFPs [1] is given and used to demonstrate the superiority of the prospect consensus with P-HFPs. The possible hesitant preference values are the same in Table 5.1, but the probability of them is the same (the average value according to #h ii  , jk ( pii  , jk )). For example, the original evaluation information with P-HFP for the attribute c1 from the investor d1 is [1]: 

0.5 (0.79(0.6), 0.75(0.2), 0.7(0.2)) (0.3(0.6), 0.25(0.2), 0.21(0.2)) 0.5



Then, we transform it into HFP as [1]: 

0.5 (0.79, 0.75, 0.7) (0.3, 0.25, 0.21) 0.5



According to Table 5.1, the transformed HFPs are shown in Table 5.8 [1]. Based on Model I in Sect. 5.2.1, the optimal priority vector is calculated as [1] (Table 5.9). It is easy to recognize that for all the attributes, just the variance of the optimal priority vector σ (ω1 ) = 0.7089% > θ = 0.5% and the others are smaller than the predefined group consensus threshold. Moreover, according to Sect. 5.2.1, the expert d3 is the person needed to adjust his/her evaluation information. In order to be comparable, the adjusted information is transformed from Step 4 in Sect. 5.3.2.

Table 5.7 Optimal priority vector under each attribute ) * ) * 0.588 0.5797 0.5062 0.6332 0.648 0.6216 ω1 = ω2 = 0.412 0.4203 0.4938 0.3668 0.352 0.3784 ) ) * * 0.589 0.614 0.594 0.4951 0.5162 0.4907 ω5 = ω6 = 0.411 0.386 0.406 0.5049 0.4838 0.5093 ω7 =

ω3 =

0.5614 0.5305 0.5379

*

0.368 0.3915 0.3871

0.4386 0.4695 0.4621 ) * 0.632 0.6085 0.6129

)

ω4 =

)

0.4651 0.5133 0.4955

0.5349 0.4867 0.5045

*

5.3 Illustrative Example 101

102

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

Table 5.8 Transformed HFPs d1

d2

d3

d1

d2

d3

d1

d2

d3

d1

d2

d3

c1 )

0.5

(0.79, 0.75, 0.7)

(0.3, 0.25, 0.21)

0.5

)

0.5

(0.76, 0.65, 0.7)

(0.3, 0.35, 0.24)

0.5

)

*

)

(0.56, 0.45, 0.38)

(0.62, 0.55, 0.44)

0.5

0.5

(0.73, 0.62, 0.58)

(0.42, 0.38, 0.27)

0.5

)

0.5

(0.69, 0.56, 0.42)

(0.58, 0.44, 0.31)

0.5

)

0.5

(0.76, 0.55, 0.4)

(0.6, 0.45, 0.24)

0.5

c5 )

0.5

(0.8, 0.76, 0.69)

(0.31, 0.24, 0.2)

0.5

)

(0.19, 0.25, 0.12)

0.5

0.5

(0.82, 0.73, 0.71)

(0.29, 0.27, 0.18)

0.5

0.5

(0.92, 0.86, 0.78)

(0.22, 0.14, 0.08)

0.5

0.5

(0.89, 0.71, 0.64)

(0.36, 0.29, 0.11)

0.5

0.5

(0.87, 0.75, 0.71)

(0.29, 0.21, 0.13)

0.5

0.5

(0.92, 0.83, 0.71)

(0.29, 0.17, 0.08)

0.5

0.5

(0.95, 0.9, 0.86)

(0.14, 0.1, 0.05)

0.5

)

(0.88, 0.83, 0.76)

(0.24, 0.17, 0.12)

0.5

c4 )

*

)

0.5

(0.68, 0.55, 0.42)

(0.58, 0.45, 0.32)

0.5

0.5

(0.54, 0.4, 0.3)

(0.7, 0.6, 0.46)

0.5

)

*

)

*

)

*

*

*

*

*

*

0.5

(0.55, 0.47, 0.41)

(0.59, 0.53, 0.45)

0.5

c6 )

*

*

0.5

*

*

(0.88, 0.75, 0.81)

c7 )

*

*

0.5

)

)

c2 )

0.5

c3 )

)

*

0.5

(0.52, 0.45, 0.41)

(0.59, 0.55, 0.48)

0.5

0.5

(0.63, 0.52, 0.41)

(0.59, 0.48, 0.37)

0.5

0.5

(0.55, 0.38, 0.44)

(0.56, 0.62, 0.45)

0.5

*

*

*

*

5.3 Illustrative Example

103

Table 5.9 Optimal priority vector matrix ) * 0.6635 0.6377 0.4734 ω1 = 0.3365 0.3623 0.5266 ) * 0.6005 0.5388 0.5459 ω3 = 0.3995 0.4612 0.4541 ) * 0.6653 0.7017 0.6673 ω5 = 0.3347 0.2983 0.3327 ) * 0.7243 0.6622 0.6882 ω7 = 0.2757 0.3378 0.3118



) ω2 =

*

0.2950 0.2478 0.2925 )

ω4 =

0.5345 0.4410 0.4824

*

0.4655 0.5590 0.5176 )

ω6 =

0.7050 0.7522 0.7075

0.4701 0.5146 0.4687

*

0.5299 0.4854 0.5313

0.5 (0.63, 0.48, 0.42) (0.58, 0.52, 0.37) 0.5



Then, we should run the above operation to replace the optimal priority vector ω1,13 = 0.4734 and ω2,13 = 0.5266 as ω1,13 = 0.5048 and ω2,13 = 0.4952. Furthermore, with the re-evaluated preference information, there is σ (ω j ) < θ j ( j ∈ M), which indicates that the consensus reaches. Finally, the overall priority vectors are 1 = 1.0992 and 2 = 0.7208. A1 is regarded as the best alternative.

5.3.5 Results of the Expected Consensus with HFPs In this chapter, the expected consensus procedure with HFPs [1] is used to analyze the same example. According to the analysis in Sects. 5.3.3 and 5.3.4, the transformed HFPs is given in Table 5.8, and the HFP matrix is obtained by Eq. (5.12). Furthermore, the optimal priority vector matrix is obtained by Model II and shown in Table 5.10 [1]. According to the variance of optimal priority vector, group consensus has been reached and the overall priority value is 1 = 1.0060 and 2 = 0.8140. A1 is regarded as the best alternative.

5.3.6 Comparative Analysis The results of prospect consensus with P-HFPs and with HFPs have already been calculated in Sects. 5.3.2 and 5.3.4 respectively, whereas, the results of expected consensus with P-HFPs and with HFPs have been exhibited in Sects. 5.3.3 and 5.3.5 correspondingly. They are collected in Table 5.11 [1].

104

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

Table 5.10 Optimal priority vector ) * 0.5822 0.5678 0.4878 ω1 = 0.4178 0.4322 0.5122 ) * 0.5478 0.5189 0.5233 ω3 = 0.4522 0.4811 0.4767 ) * 0.5833 0.6044 0.5844 ω5 = 0.4167 0.3956 0.4156 ) * 0.6178 0.5822 0.5967 ω7 = 0.3822 0.4178 0.4033

) ω2 =

*

0.3933 0.3656 0.3922 )

ω4 =

0.5167 0.4711 0.4922

*

0.4833 0.5289 0.5078 )

ω6 =

0.6067 0.6344 0.6078

0.4867 0.5057 0.4856

*

0.5133 0.4933 0.5144

Table 5.11 Overall priority values with different methods 1

2

The deviation between 1 and 2

Prospect consensus with P-HFP (Sect. 5.3.2)

1.1223

0.6977

0.4246

Expected consensus with P-HFP (Sect. 5.3.3)

1.0305

0.7895

0.2410

Prospect consensus with HFP (Sect. 5.3.4)

1.0992

0.7208

0.3784

Expected consensus with HFP (Sect. 5.3.5)

1.0060

0.8140

0.1920

The overall priority values Methods

All the results from the four group consensus methods have shown A1 as the best choice, which indicates the feasibility of the given method. However, the processes of reaching group consensus are different. Group consensus with prospect values has not been reached when the original evaluation information was used in Sects. 5.3.2 and 5.3.4, whereas, group consensus with expected values has reached when the same original evaluation information was adopted in Sects. 5.3.3 and 5.3.5. Hence, the prospect consensus is stricter than the expected consensus. It means that the group consensus method under the condition of prospect values is superior to that under the condition of expected values. Furthermore, the group consensus in both Sects. 5.3.2 and 5.3.4 has been reached after the adjustment of evaluation information. But the overall priority values from prospect consensus with P-HFPs are different from prospect consensus with HFPs. Also, the overall priority values from expected consensus with P-HFPs are different from expected consensus with HFPs. Therefore, group consensus with P-HFPs, whatever the prospect consensus or expected consensus, has more advantages in exhibiting more original evaluation information from investors. All in all, the given prospect consensus with P-HFPs is feasible and effective.

5.3 Illustrative Example

105

Alternative 1 0.77 0.72 0.67 0.62 0.57 0.52 0.47 0.42 VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC VC 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 C1

C2

C3 5.3.2

C4 5.3.3

5.3.4

C5

C6

C7

5.3.5

Fig. 5.2 Optimal priority value under each attribute of the VCs. Note 5.3.2 represents the result from the method in Sect. 5.3.2, and so does the others

The overall priority values of the best choice and the worst choice among the four methods are different, so does the deviation between the value of the best one and the worst one. But it is easy to see that all the four methods consider A1 as the best choice. The detailed comparisons of each method from every venture capitalist under each attribute is shown in Fig. 5.2 [1]. Figure 5.2 presents the optimal priority values ω1, jk of A1 which are calculated by Model I and Model II. For ω1, jk > 0.5 with the method in Sect. 5.3.2, the investor d k prefers the alternative A1 under the attribute cj . In this situation, the optimal priority values from prospect consensus with P-HFPs in Sect. 5.3.2 are bigger than the expected consensus with P-HFPs in Sect. 5.3.3 for the investor d k with the attribute cj . The same conclusions are presented for the HFPs from the comparative analyses in Sect. 5.3.4 and in Sect. 5.3.5. On the contrary, for ω1, jk < 0.5 with the method in Sect. 5.3.2, the investor d k prefers the alternative A2 under the attribute cj . The results from prospect consensus with P-HFPs in Sect. 5.3.2 is smaller than the expected consensus with P-HFPs in Sect. 5.3.3 for the investor d k with the attribute cj . The same conclusion is found for the HFPs from the comparison results in Sect. 5.3.4 and in Sect. 5.3.5. Those analyses indicate that the prospect consensus expands the degree of preference or aversion. This reflects the different risk attitudes of investors for gains and losses. Therefore, prospect consensus is more suitable in depicting the decision making under uncertainty. Although there is difference of optimal priority values between the prospect consensus process and the expected consensus process, all of the two frameworks

106

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

regard A1 as the best alternative. In addition, due to the constraint condition of Model I ω1, jk + ω2, jk = 1, the bigger ω1, jk means the smaller ω2, jk . Hence, the figure of the optimal priority value ω2, jk from A2 has not been exhibited in this chapter. In a word, by combining Table 5.11 and Fig. 5.2, we know that the prospect consensus shows more differences between the optimal project and the worst project than the expected consensus as well. This character is particularly useful in the selection of similar projects. Due to the background of the example used in this chapter, there are only two alternatives. Hence, in order to explain the advantage of the given methods, a simulated analysis is given in Sect. 5.4 based on the four methods with five alternatives.

5.4 Simulated Analysis In this chapter, we will increase the number of alternatives and experts, and then, we calculate the consensus result with 1000 random data using the four consensus models in Sects. 5.3.2–5.3.5 respectively. Suppose that there are five optional alternatives and 4 experts, that is, n = 5 and z = 4. Then, using the Matlab to generate 1000 groups of evaluation information from each expert for each alternative. According to the steps in Sects. 5.3.2–5.3.5, we will get the variance of the optimal priority vector, that is the value of (5.7). They are presented in Table 5.12 to show the consensus situation of those models. According to Table 5.12, with the same kind of evaluation information, when comparing the model in Sect. 5.3.2 with the model in Sect. 5.3.3 (probabilistic hesitant fuzzy circumstance) and comparing the model in Sect. 5.3.4 with the model in Sect. 5.3.5 (hesitant fuzzy circumstance), we find that the number of data reaching consensus with prospect framework is smaller than the number of data with expected framework. It indicates that consensus model with PT considering the different risk attitudes of DMs enhances the risk preference of DMs. The consensus model with prospect framework is stricter than with expected framework in reaching consensus. However, with the same decision-making framework, when comparing the model in Sect. 5.3.2 with the model in Sect. 5.3.4 (consensus with prospect framework), Table 5.12 Consensus situation of models in Sects. 5.3.2–5.3.5 Number of data The number of data reach consensus

The number of data does not reach consensus

Alternative

6.3.2

6.3.3 6.3.4 6.3.5 6.3.2

6.3.3 6.3.4 6.3.5

Alternative 1

951

961

39

61

39

Alternative 2

993

1000 979

1000 7

0

21

0

Alternative 3

986

1000 973

1000 14

0

27

0

Alternative 4

992

1000 983

1000 8

0

17

0

Alternative 5

995

1000 976

1000 5

0

24

0

939

961

49

5.4 Simulated Analysis

107

the P-HFP takes different importance for each preference value, which makes the preference to be more centralized, while the HFP takes each preference value as the same importance, which makes the expression to be relative decentralized. Hence, the prospect consensus with P-HFP is easy to satisfy the consensus, that is, the number of data reaching consensus in Sect. 5.3.2 is bigger than in Sect. 5.3.4. When comparing the model in Sect. 5.3.3 with the model in Sect. 5.3.5 (consensus with expected framework), there is no difference in the number of data reaching consensus because consensus with expected framework is easy to reach. Table 5.12 has shown the results of reaching consensus. Then, the following figures will give us a more clearly way to show the consensus results, please see Figs. 5.3, 5.4, 5.5 and 5.6. According to Figs. 5.3, 5.4, 5.5, 5.6 and 5.7, for the five alternatives with the same decision-making information (whatever with the P-HFP or with the HFP), the variance of the optimal priority vector from prospect framework is bigger than from expected framework. From this perspective, the prospect consensus with different risk attitudes for gains and losses is difficult to reach than the expected consensus. In other words, the prospect consensus has higher demand than the expected consensus. For the five alternatives with the same decision-making framework (whatever with the prospect framework or with the expected framework), all the variance of the optimal

Fig. 5.3 Variance of the optimal priority vector for Alternative 1

108

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

Fig. 5.4 Variance of the optimal priority vector for Alternative 2

Fig. 5.5 Variance of the optimal priority vector for Alternative 3

5.4 Simulated Analysis

Fig. 5.6 Variance of the optimal priority vector for Alternative 4

Fig. 5.7 Variance of the optimal priority vector for Alternative 5

109

110

5 Prospect Consensus with Probabilistic Hesitant Fuzzy Preference Information

priority vector with P-HFP is smaller than with HFP, which indicates that the P-HFP makes the preference to be more centralized than HFP. Hence, the consensus model with P-HFP is easy to reach consensus than with HFP. Additionally, the P-HFP gives the different preference valued with different probabilities, which accords with the real decision-making situation.

5.5 Remarks In this chapter, the PT has been integrated into the consensus model and the different attitudes for gains and losses of DMs has been considered in the model. Moreover, the P-HFP is used to describe the evaluation information of DMs. Hence, a prospect consensus with P-HFP has been built, which not only reflects the bounded rational psychology of DMs in the decision-making process but also including the uncertain evaluation information of DMs. In order to demonstrate the advantages of prospect consensus model with P-HFP, the prospect consensus with HFP and the expected consensus with P-HFP/HFP are given to make comparison. The main focus of this chapter is as follows: (1) The PT is integrated into the given consensus models with P-HFP and HFP. (2) The consensus models with P-HFP and HFP based on expected theory are given. (3) Using the idea of variance to measure the consensus degree and giving the method to adjust the non-consensus situation.

References 1. Tian XL, Xu ZS, Fujita H (2018) Sequential funding the venture project or not? A prospect consensus process with probabilistic hesitant fuzzy preference information. Knowl-Based Syst 161:172–184 2. Abdellaoui M (2000) Parameter-free elicitation of utility and probability weighting functions. Manage Sci 46(11):1497–1512 3. Zhou W, Xu ZS (2017) Group consistency and group decision making under uncertain probabilistic hesitant fuzzy preference environment. Inf Sci 414:276–288 4. Bergemann D, Hege U, Peng L (2011) Venture capital and sequential investments. Cowles Foundation Discussion Papers 148(1):92 5. Arkes HR, Blumer C (1985) The psychology of sunk cost. Organ Behav Hum Decis Process 35(1):124–140 6. Felix EGS, Pires CP, Gulamhussen MA (2014) The exit decision in the European venture capital market. Quantitative Financ 14(6):1115–1130 7. Chaplinsky S, Gupta-Mukherjee S (2016) Investment risk allocation and the venture capital exit market: Evidence from early stage investing. J Bank Finance 73:38–54 8. Guler I (2003) A study of decision making, capabilities and performance in the venture capital industry. Unpublished doctoral dissertation, University of Pennsylvania, Philadelphia, PA 9. Guler I (2007) Throwing good money after bad? Political and institutional influences on sequential decision making in the venture capital industry. Adm Sci Q 52(2):248–285 10. Zhang S, Xu ZS, He Y (2017) Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inform sFusion 38:1–11

Chapter 6

An Improved TODIM Based on PT

Although the classical TODIM considers relative importance of attributes, this method neither provides an appropriate way to determine the weights of attributes nor comprehensively expresses the real perceptions for gains or losses of DMs. Generally, there are two significant hurdles when the classical TODIM is applied in decisionmaking environment. First, the weight determination of attributes is presented as an objective probability in the classical TODIM, which is accused of a deviation from decision-making practice by Mattos and Garcia [1]. According to their opinions, the weight of attribute should be a transformed weighting function, driving from PT to improve the efficiency of decision-making for DMs. Second, although the gain or loss instead of the final states of wealth is incorporated in the classical TODIM, it is inconsistent with the perceived gain or loss in the value function of the prominent PT. The real perceptions of gain or loss are well captured by the value function of PT. Applying the transformed weighting function and the value function of the prominent PT into the improved TODIM can not only make the method more suitable for decision-making environment but also increase the accurateness of decisions for DMs. Tian et al. gave a deep modification of the classical TODIM [2], which incorporates prospect function (the product of the transformed weighting function and the value function described in Sect. 1.2.1 to respectively identify the weights of attributes and describe the different risk attitudes for gain and loss of DMs) as the relative dominance. Compared with the classical TODIM, this improved TODIM [2] is more appropriate for DMs’ decisions in both accurate and efficient perspectives. The improved TODIM is described step by step in Sect. 6.1 [2].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_6

111

112

6 An Improved TODIM Based on PT

6.1 Procedure of the Improved TODIM Suppose that there are the alternatives A = {A1 , A2 , . . . , An } N = {1, 2, . . . , n} and the attributes C = {c1 , c2 , . . . , cm } M = {1, 2, . . . , m}. Then, the procedure of the improved TODIM [2] is given as follows: Step 1. Identify the decision matrix and attribute values from DMs described as follows: c1 · · · cm ⎛ ⎞ A1 x11 · · · x1m m  ⎜ ⎟ X = ... ⎝ ... . . . ... ⎠ = (xi j )n×m , w = (w1 , w2 , · · · , wm ), w j = 1 (6.1) An

xn1 · · · xnm

j=1

Step 2. Work out the transformed probability of the alternative Ai to Ai  , i, i  ∈ M and i = i  according to (6.2) or (6.3). When xi j − xk j ≥ 0, the transformed weight is acquired by (6.2): γ

γ

1

πii+ j (w j ) = w j /(w j + (1 − w j )γ ) γ

(6.2)

Otherwise, the transformed weight is calculated by (6.3): πii− j (w j ) = wδj /(wδj + (1 − w j )δ ) δ

1

(6.3)

where γ and δ are the parameters defined in Sect. 1.2.1. Step 3. Determine the relative weight πii  j ∗ for Ai to Ai  by (6.4): πii  j ∗ = πii  j (w j )/πii  r (wr ) , r, j ∈ M, ∀(i, i  )

(6.4)

where πii  j (w j ) and πii  r (wr ) are all acquired by (6.2) or (6.3) for the alternatives Ai to Ai  depending on the value of xi j − xi  j ; while πii  j (w j ) represents the transformed weight of the jth attribute for the alternative Ai ; πii  r (wr ) refers to the transformed weight of reference attribute for the alternatives Ai to Ai  satisfying πii  r (wr ) = max(πii  j (w j )| j ∈ M ). Step 4. Calculate the relative prospect dominance of the alternatives Ai over Ai  under the jth attribute with (6.5):

6.1 Procedure of the Improved TODIM

ϕ j ∗ (Ai , Ai  ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

113

πii  j ∗ (xi j − xi  j )α /

m

j ∗ =1

πii  j ∗

if xi j > xi  j

0 if xi j = xi  j ⎪ m ⎪

⎪ ⎪ πii  j ∗ )(xi  j − xi j )β /πii  j ∗ if xi j < xi  j ⎩ −λ(

(6.5)

j ∗ =1

where α, β, and λ are the parameters defined in Sect. 1.2.1. Step 5. Aggregate the relative prospect dominance of the alternatives Ai over Ai  under all the attributes depending on (6.6): ψ(Ai , Ai  ) =

m 

ϕ ∗j (Ai , Ai  ), ∀(i, i  )

(6.6)

j ∗ =1

Step 6. Obtain the overall prospect dominance of the alternative Ai based on (1.9). Step 7. Rank the overall prospect dominance (Ai ), i ∈ N and based on which the optimal alternative is then found. The bigger the overall prospect value (Ai ) is, the better the alternative Ai will be. According to the steps above, this improved TODIM includes the transformed weighting function and the proper value function in PT, which is more consistent with reality theoretically. Then, an example [2] is shown in the next chapter to demonstrate the practical effectiveness of the given method.

6.2 Illustrative Example 6.2.1 Decision-Making Background In this chapter, an example that how to select a promising project from numerous ones for VCs is provided to discuss the advantage of the improved TODIM. This improved TODIM modifies the unidimensional weight as a form of weighting function and takes the real perceptions for gain or loss into consideration. Although it is theoretically reasonable, the practical importance will be demonstrated through the example of selecting the promising project [2]. As the overexploitation of natural resources by human and the enhanced awareness of sustainable development, new energy has caused much attention from both governments and customers. For instance, the governments subsidize the manufacturers of new energy automobile through reduction of rates and encourage customers to buy them through price support. Thus, the industry of new energy has great prospect and has already attracted many investors. After preliminary investigation, four VC projects (thermal power A1 , wind power generation A2 , hydroelectric power A3 , solar

114 Table 6.1 The attributes used by VCs in decision-making

6 An Improved TODIM Based on PT Aspects

Attributes

Management team

The familiar degree of target market (c1 ) The effort level (c2 ) The ability of evaluating and reacting to the risk (c3 ) The ability of leadership (c4 ) The related experience and acquired relevant performance (c5 ) The explicit plan (c6 )

Service or product

Realized the initial functioning prototype (c7 ) Accepted by market (c8 ) The degree of being protected (c9 )

Finance

At least 10 times revenue acquired with 10 years (c10 )

Market

Significant growth (c12 )

Easily cashability (c11 )

photovoltaics A4 ) remain to be further investigated. First, we draw on the previous research [3–5] to find out an appropriate evaluation attributes system used by VCs in the selection process as shown in Table 6.1 [2]. According to Widyanto and Dalimunthe [3], the weights of attributes are calculated as: ωC = (ωc1 , ωc2 , . . . , ωc12 ) = (0.0975, 0.0918, 0.0764, 0.0977, 0.0869, 0.0847, 0.0855, 0.0754, 0.0738, 0.0735, 0.0768, 0.08).1 Then, the promising project is obtained through using the improved TODIM step by step.

6.2.2 Results of the Improved TODIM The decision-making procedure of the improved TODIM is shown as follows [2]. Step 1. Some senior investors gave the evaluation information in Table 6.2.2 From now on, Step 1 has already been finished. Next, we take the alternative A1 as an example to calculate its overall prospect dominance.

1 This reference introduced investigated data about attributes used by VCs in numerous countries and

we comprehensively aggregated those data as the weights of attributes in this chapter. We believe that those comprehensive weights are reasonable. 2 The VCs give each attribute of each alternative a value. Furthermore, the value ranges from 0 to 100. For the benefit attribute, the higher of the value is, the better of the alternative will be. It is contrary for the cost attribute.

6.2 Illustrative Example

115

Table 6.2 The evaluation matrix c1

c2

c3

c4

c5

c6

c7

c8

c9

c10

c11

c12

A1

88

92

80

73

88

72

83

68

96

71

77

79

A2

79

82

87

90

69

83

79

74

86

82

84

90

A3

81

69

91

76

82

74

80

78

81

80

88

90

A4

91

78

90

75

80

65

82

80

79

83

81

94

Table 6.3 The transformed weight for each attribute c1

c2

c3

c4

c5

c6

c7

c8

c9

c10

c11

c12

π12 j

0.18

0.18

0.14

0.17

0.17

0.15

0.17

0.14

0.16

0.14

0.15

0.15

π13 j

0.18

0.18

0.14

0.17

0.17

0.15

0.17

0.14

0.16

0.14

0.15

0.15

π14 j

0.17

0.18

0.14

0.17

0.17

0.17

0.17

0.14

0.16

0.14

0.15

0.15

Note Here, γ = 0.61, δ = 0.69 in (1.3). The values of them come from the experiment conducted by Tversky and Kahneman (1992) and they are accepted by most researchers

Table 6.4 The relative weight for each attribute c1 ∗

c2 ∗

c3 ∗

c4 ∗

c5 ∗

c6 ∗

c7 ∗

c8 ∗

c9 ∗

c10∗

c11∗

c12∗

π12 j

1.0

0.97

0.79

0.91

0.94

0.84

0.94

0.78

0.87

0.77

0.79

0.81

π13 j

1.0

0.97

0.79

0.91

0.94

0.84

0.94

0.78

0.87

0.77

0.79

0.81

π14 j

0.94

1.00

0.81

0.94

0.97

0.96

0.97

0.80

0.90

0.79

0.81

0.83

Step 2. The transformed weight π1i  j is calculated according to (1.3), which depends on the relative value of the alternative A1 to the others under all attributes and it can be seen in Table 6.3. Step 3. From the transformed weight obtained in Step 2, the relative weight π1i  j ∗ of the alternative A1 to the others under each attribute is worked out according to Eq. (6.4), as shown in Table 6.4. Step 4. The relative prospect dominance of the alternative A1 over the others for each attribute will be determined according to Eq. (6.5) and the result is exhibited in Table 6.5. Step 5. The relative prospect dominance of the alternative A1 over the others based on Eq. (6.6) is acquired and shown in Table 6.6. Step 6. The overall prospect dominance of each alternative is calculated on the basis of repeating Steps 2–5 and Eq. (1.9). The results are exhibited in Table 6.7. Step 7. It is known that (A4 ) > (A2 ) > (A3 ) > (A1 ), so there is A4 > A2 > A3 > A1 . The alternative A4 is recognized as the best option among the four alternatives, whereas, A1 is regarded as the worst one. The results above rely on the degrees of risk attitudes of VCs, that is to say, the results depend on the values of the parameters α, β, λ, γ and δ, but the great difference between the given method and the classical TODIM lies in the prospect function which is the product of disparate weighting function and the value function.

0.53

−67.62

ϕ j ∗ (A1 , A3 )

ϕ j ∗ (A1 , A4 )

0.95

1.47

0.71

c2 ∗

−226.11

−245.73

−165.09

c3 ∗

Note The α = 0.88, β = 0.88, λ = 2.25 in Eq. (6.5)

0.66

ϕ j ∗ (A1 , A2 )

c1 ∗ 0.44 0.57

−47.26

1.21

c5 ∗

−67.48

−310.52

c4 ∗

Table 6.5 The relative prospect dominance for each attribute

0.50

−51.49

−230.78

c6 ∗

0.09

0.24

0.31

c7 ∗

−267.49

−227.68

−145.25

c8 ∗

1.01

0.91

0.64

c9 ∗

−271.68

−210.77

−251.48

c10∗

−100.63

−244.93

−164.55

c11∗

−314.21

−239.00

−239.00

c12∗

116 6 An Improved TODIM Based on PT

6.2 Illustrative Example

117

Table 6.6 The relative prospect dominance

ψ(A1 , A2 )

ψ(A1 , A3 )

ψ(A1 , A4 )

−1503.13

−1283.48

−1291.88

Table 6.7 The overall prospect dominance

(A1 )

(A2 )

(A3 )

(A4 )

0

0.94

0.89

1

6.2.3 Results of the Classical TODIM In this chapter, the classical TODIM [6] is processed for the sake of comparing it with the improved one. In order to compare those two methods more conveniently, the decision-making information in Tables 6.1 and 6.2 is adopted here as well. Then, the overall dominance of each alternative is calculated depending on the steps in Sect. 1.2.2 [2]. Step 1. According to the attributes in Table 6.1 and the decision-making matrix in Table 6.2, it is known that there is X = (xi j )n×m = G = (gi j )n×m because all the attributes are the benefit ones. The attribute weights are also calculated via Widyanto and Dalimunthe [3] as shown in Sect. 6.2.2. Step 2. The relative weight of each attribute is obtained by Eq. (1.6) and it is shown in Table 6.8. Step 3. The dominance of each alternative Ai over each alternative Ai  (i, i  ∈ N ) shown in Table 6.9 depending on Eq. (1.7). Step 4. The overall value of each alternative exhibits in Table 6.10 based on Eq. (1.9). Step 5. From Table 6.10, it is known that (A4 ) > (A2 ) > (A3 ) > (A1 ), that is to say, A4  A2  A3  A1 . The ranking result explains that the alternative A4 is the best choice and A2 is the suboptimal one, however, A1 is the worst option. Table 6.8 The relative weight c1 ω jr

c2

c3

c4 c5

0.998 0.939 0.782 1

c6

c7

c8

c9

c10

c11

Table 6.9 Dominance of each alternative over the others ψ(Ai , Ak )

A1

c12

0.889 0.867 0.875 0.772 0.755 0.752 0.786 0.818

A2

A3

A4

A1

0

−29.41

−25.96

−25.99

A2

−17.47

0

−14.50

−19.85

A3

−18.06

−17.71

0

−16.33

A4

−16.66

−17.47

−12.57

0

118

6 An Improved TODIM Based on PT

Table 6.10 Overall dominance of each alternative

(Ai )

A1

A2

A3

A4

0

0.85

0.84

1

Table 6.11 The results of the two methods The overall dominance

(A1 )

(A2 )

(A3 )

(A4 )

The improved TODIM

0

0.94

0.89

1

The classical TODIM

0

0.85

0.84

1

6.2.4 Comparative Analysis Between the Improved and the Classical TODIM From the results of Sects. 6.2.2 and 6.2.3, it is known that the ranking results from the two methods are the same. The alternatives A4 and A2 are recognized as the promising project and the unworthy one respectively in both the improved TODIM and the classical TODIM. The overall dominance derived from the two methods is shown in Table 6.11, and then, a comparative analysis between the improved TODIM and the classical TODIM is provided in this chapter [2]. Although the ranking results of the four projects from the two methods are consistent, the great difference between the given method and the classical TODIM lies in the disparate weighting function and value function. From Table 6.11, it is easy to see that the overall dominance of them are different between the two methods as well. The main reason for such a difference ranking is: the evaluating information of the improved TODIM is presented as prospect values which are the products of value functions (nonlinear gains or losses) and the transformed weights of attributes for VC projects, whereas, the evaluating information of the classical TODIM comes from the product of linear gain or loss and the objective probability that could not reflect the psychological perception of VCs for projects. From theoretical terms, the improved TODIM accompanied with the value function and the transformed weighting function confirms to the real decision-making situation is more reasonable as the aid for investors. From practical terms, the improved TODIM increases the difference between the alternatives. For instance, the difference of overall dominance between A2 and A3 with the improved TODIM is larger than with the classical one. This is very useful, especially for the choice among the similar alternatives. To sum up, the improved TODIM is feasible and suitable for investors to make their decisions.

6.3 Remarks

119

6.3 Remarks The traditional decision-making methods have focused on the decision making with assumption of perfect rationality. However, these previous methods seldom consider the irrational characteristics of DMs, which are always significant to the evaluation information and the DMs’ decision making. Although the TODIM is a useful tool to simulate the irrational parts of DMs, it could not overall reflect the DMs’ psychological states explained by PT. Hence, the classical TODIM method has been improved on the basis of prospect value in PT for the sake of comprehensively handling the irrational decision-making of DMs in Sect. 6.1 [2], the superiority of the improved TODIM has been demonstrated by the comparative analysis in Sect. 6.2. Although the improved TODIM is well applied in VC, it only concerns on the VC problem here and ignores the application of DMs’ psychology in other fields. Furthermore, we believe that this book may provide inspiration for follow-up research of decision-making methods under the framework of bounded rationality. Meanwhile, we will focus on extending the decision-making method under fuzzy decision-making circumstance with bounded rationality of DMs in the future.

References 1. Mattos F, Garcia P (2011) Applications of behavioral finance to entrepreneurs and venture capitalists: decision making under risk and uncertainty in futures and options markets. Advances in Entrepreneurial Finance, Springer, New York 2. Tian XL, Xu ZS, Gu J (2019) An extended TODIM based on cumulative prospect theory and its application in venture capital. Informatica 30(2):413–429 3. Widyanto HA, Dalimunthe Z (2015) Evaluation criteria of venture capital firms investing on indonesians’ SME. Social Science Electronic Publishing 4. Félix EGS, Nunes JC, Pires CP (2014) Which criteria matter most in the evaluation of venture capital investments? CEFAGE-UE Working Papers 21(3):505–527 5. Monika, Sharma AK (2015) Venture capitalists’ investment decision criteria for new ventures: a review. In: Ramesh A, Prakash G (eds) Operations management in digital economy. Elsevier Science BV, Amsterdam, pp 465–470 6. Gomes LFAM, Lima MMPP (1991) TODIM: basic and application to multicriteria ranking of projects with environmental impacts. Paris 16(January):113–127

Chapter 7

An Improved TODIM with Probabilistic Hesitant Fuzzy Information

However, the fuzzy information of the DMs could not be well expressed by the method in Sect. 6.1. Hence, Tian et al. [1] proposed the improved TODIM with the probabilistic hesitant fuzzy information and it is introduced in this chapter.

7.1 Procedure of the Improved TODIM with Probabilistic Hesitant Fuzzy Information Based on the analysis above, this chapter presents a procedure of the improved TODIM with probabilistic hesitant fuzzy information, which is based on the idea of the original PT. Let A = {A1 , A2 , . . . , An } be a finite set of alternatives, and C = {c1 , c2 , . . . , cm } be the attributes. The procedure is given as follows [1]: Step 1: Obtain the original evaluation information matrix and the weights of the corresponding attributes as: ⎛

h 11 ( p11 ) ⎜ .. ⎜ Y =⎝ . h n1 ( pn1 )

⎞ · · · h 1m ( p1m ) ⎟ .. .. ⎟ = (h i j ( pi j ))n×m , ω= (h ω ( pω ), h ω ( pω ), . . . , h ω ( pω )) m m 1 1 2 2 . ⎠ . · · · h nm ( pnm )

(7.1) where i ∈ N , j ∈ M; h i j ( pi j ) is the evaluation information of the alternative Ai over the attribute c j ; h ω ( pω ) is the weighting information of c j . Step 2: Normalize the evaluation information matrix according to Sect. 1.2.4.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_7

121

122

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

⎞   ) · · · h 1m ( p1m ) h 11 ( p11 ⎟ ⎜ .. ..       .. Y = ⎝ ⎠ = (h i j ( pi j ))n×m , ω = (ω1 , ω2 , . . . , ωm ) (7.2) . . . ⎛

  ) · · · h nm ( pnm ) h n1 ( pn1

where

m  j=1

pi j = 1 (i ∈ N ),

m  j=1

ωj = 1; ωj =

ωj

m 

ωj

; ωj =

#h ω j ( pω j )



t=1

pωt j h tω j ( pωt j ).

j=1

Step 3: Work out the transformed weight function πii  j (ωj ) according to the weighting function of PT. ⎧ γ ⎪ ωj ⎪ +      ⎪ π (ω ) = ⎪  1 , h i j ( pi j ) ≥ h i  j ( pi  j ) ⎪ γ  γ γ ⎨ ii j j (ω j + (1 − ω j ) ) πii  j (ωj ) = ⎪ ωδj ⎪ ⎪ −  ⎪ π (ω ) = h i j ( pi j ) < h i  j ( pi  j )  ⎪ 1 , ⎩ ii j j (ωδ + (1− )δ ) δ j

(7.3)

j

where the comparison between h i j ( pi j ) and h i  j ( pi  j ) is determined by (1.17) and (1.18). Step 4: Acquire the relative weight πii  j ∗ for the alternatives Ai to Ai  from (6.4). Step 5: Calculate the relative prospect dominance ϕ j ∗ (Ai , Ai  ) of the alternative Ai over Ai  under the attribute c j . When c j is the benefit attribute, the relative prospect dominance is ϕ Bj∗ (Ai , Ai  ): ⎧ πii  j ∗ (d(h  ∗ ( p ∗ ),h   ∗ ( p ∗ )))α ij ij i j i j ⎪ h i j ∗ ( pi j ∗ ) > h i  j ∗ ( pi  j ∗ ) m ⎪  ⎪ ⎪ π  ∗ ii j ⎪ ⎨ j ∗ =1 B  ϕ j ∗ (Ai , Ai ) = 0 h i j ∗ ( pi j ∗ ) = h i  j ∗ ( pi  j ∗ ) (7.4) ⎪ m  ⎪ ⎪ −λ( πii  j ∗ )(d(h i j ∗ ( pi j ∗ ),h i  j ∗ ( pi  j ∗ )))β ⎪ ⎪ j ∗ =1 ⎩ h i j ∗ ( pi j ∗ ) < h i  j ∗ ( pi  j ∗ ) π  ∗ ii j

When c j is the cost attribute, the relative prospect dominance is ϕ Cj∗ (Ai , Ai  ): ⎧ m  −λ( πii  j ∗ )(d(h i j ∗ ( pi j ∗ ),h i  j ∗ ( pi  j ∗ )))β ⎪ ⎪ ∗ j =1 ⎪ ⎪ h i j ∗ ( pi j ∗ ) > h i  j ∗ ( pi  j ∗ ) ⎪ πii  j ∗ ⎨ h i j ∗ ( pi j ∗ ) = h i  j ∗ ( pi  j ∗ ) (7.5) ϕ Cj∗ (Ai , Ai  ) = 0 ⎪ πii  j ∗ (d(h i j ∗ ( pi j ∗ ),h i  j ∗ ( pi  j ∗ )))α ⎪ ⎪ ⎪ h i j ∗ ( pi j ∗ ) < h i  j ∗ ( pi  j ∗ ) m ⎪  ⎩ πii  j ∗ j ∗ =1

where α, β and λ are the parameters of PT; d(h i j ∗ ( pi j ∗ ), h i  j ∗ ( pi  j ∗ )) is the corresponding distance calculated by (1.19). Step 6–8: Repeat Steps 5–7 in Sect. 7.1 to find the best alternative.

7.2 Procedure of the Improved TODIM with Hesitant Fuzzy Information

123

7.2 Procedure of the Improved TODIM with Hesitant Fuzzy Information According to Chap. 5, we can easily understand that hesitant fuzzy information is used to describe the situation that the DMs hesitate between several different values, and each hesitation value is equally important. In fact, it also can be expressed as a special form of probabilistic hesitant fuzzy information. According to the definition of hesitant fuzzy information, it regards each HFE in it as equal importance. Hence, we use the equal probability of each hesitant fuzzy element to represent the hesitant fuzzy information here. For convenience, the equal probability can be ignored. Hence, a HFS can be denoted as: H = {< xi , h xi > |xi ∈ X } and h xi = {h txi t = 1, 2, . . . , #h xi } is called HFE. In order to show the advantages of the method in Sect. 7.1, in the chapter, the procedure of the improved TODIM with hesitant fuzzy information [1] is given to make the comparative analysis. Step 1: Obtain the original evaluation information matrix and the weights of the corresponding attributes as: ⎛

h 11 · · · ⎜ Y = ⎝ ... . . . h n1 · · ·

⎞ h 1m .. ⎟ = (h ) i j n×m , ω= (h ω1 , h ω2 , . . . , h ωm ) . ⎠

(7.6)

h nm

where i ∈ N , j ∈ M; h i j is the evaluation information of the alternative Ai over the attribute c j ; h ω j is the weighting information of c j . Step 2: Normalize the weight information: ωj =

ωj m  ωj

(7.7)

j=1

where ω j = ρ(h ω j ) and ρ(h ω j ) is the score function of the HFE h ω j , and there is: ρ(h) =

#h 1  t h #h t=1

(7.8)

Step 3: Get the transferred weight function πii  j (ωj ): ⎧ γ ⎪ ωj ⎪ +    ⎪ π  (ω ) = γ ⎪ 1 , hi j ≥ hi  j ⎪  γ γ ⎨ ii j j (ω + (1 − ω ) ) j j πii  j (ωj ) = δ ⎪ ω ⎪ j ⎪ −    ⎪ ⎪ ⎩ πii  j (ω j ) = (ωδ + (1 − ω )δ ) 1δ , h i j < h i  j j j

(7.9)

124

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

where the comparison of the HFEs h i j and h i  j is decided by the score function (7.8) and the deviation function (7.10) of HFEs. 1 σ (h) = #h

 ∀h t ∈h

(h t − ρ(h))2

(7.10)

The detailed comparative rules are represented as follows: (1) (2) (3)

If ρ(h i j ) > ρ(h i  j ), then h i j > h i  j ; If ρ(h i j ) < ρ(h i  j ), then h i j < h i  j ; If ρ(h i j ) = ρ(h i  j ), then (1) (2) (3)

If σ (h i j ) > σ (h i  j ), then h i j < h i  j ; If σ (h i j ) < σ (h i  j ), then h i j > h i  j ; If σ (h i j ) = σ (h i  j ), then h i j = h i  j .

Step 4: Calculate the relative weight πii  j  based on (7.3). Step 5: Compute the relative prospect dominance ϕ j ∗ (Ai , Ai  ) of the alternative Ai over Ai  under the attribute c j . When c j is the benefit attribute, the relative prospect dominance is ϕ Bj∗ (Ai , Ai  ): ⎧ πii  j  (d(h   ,h    ))α ij i j ⎪ h i j  > h i  j  m ⎪  ⎪ πii  j  ⎪ ⎪ ⎨ j  =1 ϕ Bj (Ai , Ai  ) = 0 h i j  = h i  j  ⎪ m  ⎪   β ⎪ ⎪ ⎪ λ( j  =1 πii  j  )(d(h i j  ,h i  j  ))  ⎩ − h i j  < h i  j  π  

(7.11)

ii j

When c j is the cost attribute, the relative prospect dominance is ϕ Cj∗ (Ai , Ai  ): ⎧ m  λ( πii  j  )(d(h i j  ,h i  j  ))β ⎪ ⎪ j  =1 ⎪ ⎪ − h i j  > h i  j  ⎪ πii  j  ⎨ h i j  = h i  j  ϑ Cj (Ai , Ai  ) = 0   α ⎪ π (d(h ,h ))   ⎪ ii j i j i j ⎪ ⎪ h i j  < h i  j  m ⎪  ⎩ πii  j 

(7.12)

j  =1

where α, β and λ are the parameters of PT; d(h i j  , h i  j  ) is the corresponding distance calculated by (7.13). #h i j 1  t |h − h it  j |, #h i j = #h i  j d(h i j , h j ) = #h i j t=1 i j i

Step 6–8: Repeat Steps 5–7 in Sect. 6.1 to find the best alternative.

(7.13)

7.3 Illustrative Analysis

125

7.3 Illustrative Analysis In this chapter, an illustrative example is given to show the advantages and disadvantages of the methods in Sects. 7.1 and 7.2 [1].

7.3.1 Screening Process of the Improved TODIM with Probabilistic Hesitant Fuzzy Information Step 1: To better distinguish the probability in the evaluation information from the membership degree, the evaluation information is enlarged by 100 times. Then, the evaluation information is given as follows: c1 ⎛ {55(0.22), 68(0.51), 73(0.27)} A1 ⎜ A2 ⎜ ⎜ {62(0.28), 77(0.63), 79(0.09)} Y = ⎜ A3 ⎜ ⎝ {63(0.32), 71(0.48), 77(0.2)} A4 {67(0.49), 72(0.44), 75(0.07)}

c2

c3

c4

{60(0.45), 66(0.39),70(0.16)} {62(0.69),68(0.21), 71(0.1)} {64(0.66), 72(0.32), 77(0.02)}

⎞

⎟ {68(0.29), 77(0.68),80(0.03)} {60(0.18),73(0.21),85(0.61)} {77(0.6),88(0.36), 80(0.04)} ⎟ ⎟ ⎟ {66(0.39),71(0.52), 77(0.09)} {68(0.59),74(0.32), 79(0.09)} {71(0.53),78(0.22), 81(0.25)}⎟ ⎠ {62(0.58),69(0.3), 74(0.12)} {67(0.61),71(0.26), 78(0.13)} {68(0.36), 73(0.49), 79(0.15)}

ω = ({0.34(0.68), 0.40(0.32)}, {0.09(0.39), 0.11(0.61)}, {0.19(0.56), 0.22(0.44)}, {0.21(0.43), 0.27(0.57)})

Step 2: Normalize the evaluation matrix and get the normalized weight information at the same time as: c1 ⎛

c2

c3

c4

⎞ A1 ⎜ ⎟ ⎜ ⎟ {80(0.04),88(0.36), 77(0.60)} {79(0.09), 62(0.28), 77(0.63)} {80(0.03),68(0.29), 77(0.68)} {60(0.18),73(0.21),85(0.61)} A2 ⎜ ⎟ Y = ⎜ ⎟ ⎟ A3 ⎜ {77(0.2), 63(0.32), 71(0.48)} {77(0.09),66(0.39), 71(0.52)} {79(0.09),74(0.32), 68(0.59)} {78(0.22),81(0.25), 71(0.53)} ⎝ ⎠ A4 {75(0.07), 72(0.44), 67(0.49)} {74(0.12),69(0.3), 62(0.58)} {78(0.13),71(0.26), 67(0.61)} {79(0.15), 68(0.36), 73(0.49)} {55(0.22), 73(0.27), 68(0.51)} {70(0.16),66(0.39), 60(0.45)} {71(0.1),68(0.21), 62(0.69)} {77(0.02), 72(0.32), 64(0.66)}

ω = (0.395, 0.112,0.224, 0.269) Step 3: Calculate the transformed weights and the results are shown in Table 7.1. Step 4: Obtain the relative weights as Table 7.2: Step 5: Work out the relative prospect dominance of the alternative A1 over the others for each attribute and they are presented in Table 7.3. Step 6: Obtain the prospect dominance degrees of the alternative Ai over the others (Table 7.4). Step 7: The overall prospect dominance of each alternative is calculated and the results are exhibited in Table 7.5. Step 8: Since (A2 ) > (A3 ) > (A4 ) > (A1 ), A2  A3  A4  A1 . The project A2 should be selected.

126

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Table 7.1 Transformed weights Weight

Attribute c1

π12 j (ωj ) π13 j (ωj ) π14 j (ωj ) π21 j (ωj ) π23 j (ωj ) π24 j (ωj )

Weight c2

c3

c4

0.389

0.183

0.275

0.306

0.389

0.183

0.275

0.306

0.389

0.183

0.275

0.306

0.368

0.197

0.275

0.301

0.368

0.197

0.275

0.301

0.368

0.197

0.275

0.301

Attribute c1

c2

c3

c4

π31 j (ωj ) π32 j (ωj ) π34 j (ωj ) π41 j (ωj ) π42 j (ωj ) π43 j (ωj )

0.368

0.197

0.275

0.301

0.389

0.183

0.275

0.306

0.389

0.197

0.275

0.301

0.368

0.197

0.275

0.301

0.389

0.183

0.275

0.306

0.368

0.183

0.275

0.306

Relative weight

Attribute c1

c2

c3

c4

31 j ∗

0.82

Table 7.2 Relative weights Relative weight

Attribute c1

c2

c3

c4

π

12 j ∗

1

0.47

0.707

0.789

π

1

0.537

0.749

π13 j ∗

1

0.47

0.707

0.789

π32 j ∗

1

0.47

0.707

0.789

π14 j ∗

1

0.47

0.707

0.789

π34 j ∗

1

0.508

0.708

0.775

π21 j ∗

1

0.537

0.749

0.82

π41 j ∗

1

0.537

0.749

0.82

π23 j ∗

1

0.537

0.749

0.82

π42 j ∗

1

0.47

0.707

0.789

π24 j ∗

1

0.537

0.749

0.82

π43 j ∗

1

0.497

0.747

0.834

7.3.2 Screening Process of the Extended TODIM with Probabilistic Hesitant Fuzzy Information Steps 1–2: The normalized evaluation matrix is transformed in the same way as Step 1 and Step 2 in Sect. 7.3.1. Step 3: Calculate the relative weight of each attribute based on (1.6) (Table 7.6). Step 4: Get the relative dominance ϕ j (Ai , Ai  ) of the alternative Ai over Ai  under the attribute c j as follows. The result is exhibited in Table 7.7, and ϕ j (Ai , Ai ) = 0 is not shown in the table. When c j is the benefit attribute, the relative dominance is: ⎧  ω jr ⎪ ⎪  d(h i j ( pi j ), h i  j ( pi  j )) h i j ( pi j ) > h i  j ( pi  j ) m ⎪ ⎪ ωjr ⎪ ⎪ ⎨ j=1 ϕ Bj (Ai , Ai  ) = 0 h i j ( pi j ) = h i  j ( pi  j ) m ⎪ ⎪   ⎪ ⎪ ω jr ⎪ ⎪ ⎩ − 1 j=1  d(h  ( p  ), h   ( p   )) h  ( p  ) < h   ( p   ) ij ij i j i j ij ij i j i j λ ω jr

When c j is the cost attribute, the relative dominance is:

(7.14)

−9.25

−36.46

1.80

2.12

2.58

ϕ j ∗ (A1 , A3 )

ϕ j ∗ (A1 , A4 )

ϕ j ∗ (A2 , A1 )

ϕ j ∗ (A2 , A3 )

ϕ j ∗ (A2 , A4 )

−37.23

c1

Attribute

ϕ j ∗ (A1 , A2 )

Relative dominance

1.07

1.16

1.70

−63.05

−55.75

−139.33

c2

0.98

1.49

0.92

−25.14

−32.14

−36.21

c3

Table 7.3 Relative prospect dominance under each attribute c4

1.78

2.23

1.04

−41.38

−50.83

−33.35

ϕ j ∗ (A4 , A3 )

ϕ j ∗ (A4 , A2 )

ϕ j ∗ (A4 , A1 )

ϕ j ∗ (A3 , A4 )

ϕ j ∗ (A3 , A2 )

ϕ j ∗ (A3 , A1 )

Relative dominance 0.45

1.95

−53.35

1.76

−40.28

−43.89

c1

Attribute 0.68

−32.87

−88.08

0.77

0.40

−94.96

c2

0.82

−24.37

−38.39

0.64

0.62

−58.35

c3

1.59

−26.89

−57.11

1.29

0.84

−71.57

c4

7.3 Illustrative Analysis 127

128

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Table 7.4 Prospect dominance degree ψ(A1 , A2 )

−246.12

ψ(A2 , A1 )

5.46

ψ(A3 , A1 )

3.53

ψ(A4 , A1 )

4.46

ψ(A1 , A3 )

−147.97

ψ(A2 , A3 )

7.00

ψ(A3 , A2 )

−268.77

ψ(A4 , A2 )

−236.93

ψ(A1 , A4 )

−166.03

ψ(A2 , A4 )

6.41

ψ(A3 , A4 )

−38.42

ψ(A4 , A3 )

−82.19

Table 7.5 Overall prospect dominance

(A1 )

(A2 )

(A3 )

(A4 )

0

1

0.44

0.42

Table 7.6 Relative weight ω1r

ω2r

ω3r

ω4r

1

0.28

0.57

0.68

Table 7.7 Relative dominance degree Relative dominance

Attributes c1

c2

c3

c4

Relative dominance

ϕ j (A1 , A2 ) −1.88 −4.85 −2.02 −1.87 ϕ j (A3 , A1 )

Attributes c1 0.76

c2 0.73

c3 0.95

c4 1.44

ϕ j (A1 , A3 ) −0.85 −2.88 −1.89 −2.38 ϕ j (A3 , A2 ) −2.06 −3.90 −2.65 −2.88 ϕ j (A1 , A4 ) −1.86 −3.09 −1.64 −2.11 ϕ j (A3 , A4 ) −1.95

0.55

0.82

1.01

ϕ j (A2 , A1 )

1.67

1.23

1.01

1.13 ϕ j (A4 , A1 )

0.78

0.82

1.28

ϕ j (A2 , A3 )

1.83

0.99

1.33

1.74 ϕ j (A4 , A2 ) −2.30 −3.74 −2.09 −2.54

ϕ j (A2 , A4 )

2.05

0.95

1.05

1.53 ϕ j (A4 , A3 )

1.65

1.74 −2.16 −1.63 −1.67

m ⎧   ⎪ ω jr ⎪ ⎪ j=1 1 ⎪ ⎪ −λ d(h i j ( pi j ), h i  j ( pi  j )) h i j ( pi j ) > h i  j ( pi  j )  ⎪ ω jr ⎨ ϕ Cj (Ai , Ai  ) = 0 h i j ( pi j ) = h i  j ( pi  j )   ⎪ ⎪ ω jr ⎪ ⎪ d(h i j ( pi j ), h i  j ( pi  j )) h i j ( pi j ) < h i  j ( pi  j ) ⎪ m ⎪ ⎩  ωjr

(7.15)

j=1

where λ is the attenuation factor of the loss; d(h i j ( pi j ), h i  j ( pi  j )) is the distance of h i j ( pi j ) and h i  j ( pi  j ). Step 5: The dominance of the alternative Ai over the others for each attribute is determined by (6.6) and the results are exhibited in Table 7.8. ψ j (Ai , Ai ) = 0 is not shown in the table. Step 6: The overall dominance of each alternative is calculated (Table 7.9). Step 7: Since (A2 ) > (A3 ) > (A4 ) > (A1 ), there is A2  A3  A4  A1 . The project A2 should be selected.

7.3 Illustrative Analysis

129

Table 7.8 Dominance degree ψ j (A1 , A2 )

−5.76

ψ j (A2 , A1 )

3.82

ψ j (A3 , A1 )

3.14

ψ j (A4 , A1 )

3.75

ψ j (A1 , A3 )

−5.11

ψ j (A2 , A3 )

4.91

ψ j (A3 , A2 )

−7.59

ψ j (A4 , A2 )

−6.93

ψ j (A1 , A4 )

−5.61

ψ j (A2 , A4 )

4.63

ψ j (A3 , A4 )

−0.12

ψ j (A4 , A3 )

−1.56

Table 7.9 Overall dominance

(A1 )

(A2 )

(A3 )

(A4 )

0

1

0.40

0.39

7.3.3 Screening Process of the Improved TODIM with Hesitant Fuzzy Information Step 1: Obtain the evaluation matrix and the weight information under hesitant fuzzy circumstance. c2 c3 c1 ⎛ x1 {55, 68, 73} {60, 66, 70} ⎜ x ⎜ {62, 77, 79} {68, 77, 80} Y = 2⎜ x3 ⎝ {63, 71, 77} {66, 71, 77} x4 {67, 72, 75} {62, 69, 74}

c4 {62, 68, 71} {64, 72, 77}

⎞

{60, 73,85} {77, 80, 88} ⎟ ⎟ ⎟ {68, 74, 79} {71, 78, 81}⎠ {67, 71, 78} {68, 73, 79}

ω = ({0.34, 0.40},{0.09, 0.11},{0.19, 0.22},{0.21, 0.27}) Step 2: Normalize the evaluation matrix according to benefit and cost attributes. The four attributes all belong to the benefit attributes, so the evaluation matrix does not need to change. Besides, the normalized weight information is based on (7.7). Step 3: Get the transformed weights by (7.9) as shown in Table 7.10. Step 4: Obtain the relative weight of Ai over Ai  based on (7.3). Table 7.10 Transformed weights Weight

Attributes

Weight

Attributes

c1

c2

c3

c4

c1

c2

c3

c4

π12 j (ωj )

0.394

0.179

0.275

0.302

π31 j (ωj )

0.372

0.195

0.276

0.298

π13 j (ωj )

0.394

0.179

0.275

0.302

π32 j (ωj )

0.394

0.179

0.276

0.302

π34 j (ωj ) π41 j (ωj ) π42 j (ωj ) π43 j (ωj )

0.394

0.195

0.276

0.298

0.372

0.195

0.276

0.298

0.394

0.179

0.275

0.302

0.372

0.179

0.275

0.302

π14 j (ωj ) π21 j (ωj ) π23 j (ωj ) π24 j (ωj )

0.394

0.179

0.275

0.302

0.372

0.195

0.276

0.298

0.372

0.195

0.275

0.298

0.372

0.195

0.276

0.298

130

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Step 5: Calculate the relative prospect dominance ϕ ∗j (Ai , Ai  ) of the alternative Ai over Ai  under the attribute c j . When c j is the benefit attribute, the relative prospect dominance is calculated by (7.11); Otherwise, the relative prospect dominance is calculated by (7.12). The results are exhibited in Table 7.11. Step 6: Calculate the dominance degrees of the alternative Ai over the others by (6.6). Step 7: Obtain the overall dominance degrees according to (1.9) (Table 7.12). Step 8: Since (A2 ) > (A3 ) > (A4 ) > (A1 ), then A2  A3  A4  A1 . Thus, A2 should be selected.

7.3.4 Screening Process of the Extended TODIM with Hesitant Fuzzy Information Step 1: Obtain the evaluation matrix and the weight information under hesitant fuzzy circumstance. They are the same as the information shown in Step 1 of Sect. 7.3.3. Step 2: Calculate relative weights and the results show in Table 7.13. ω jr =

ρ(h ω j ) ωj = ωr ρ(h ωr )

(7.16)

where j, r ∈ M, ρ(h ω j ) is the score function of HFE h calculated by (7.8). Step 3: Get the relative dominance ϕ j (Ai , Ai  ) of the alternative Ai over Ai  under the attribute c j as follows, and the results are exhibited in Table 7.14. When c j is the benefit attribute, the relative dominance is: ⎧ ω jr ⎪ d(h i j , h i  j ) hi j > hi  j ⎪ m  ⎪ ⎪ ω jr ⎪ ⎪ j=1 ⎨ ϕ Bj (Ai , Ai  ) = 0  hi j = hi  j ⎪ ⎪ m  ⎪ ⎪ ω jr ⎪ ⎪ ⎩ − 1 j=1 d(h i j , h i  j ) h i j < h i  j λ ω jr

(7.17)

When c j is the benefit attribute, the relative dominance is: m ⎧  ⎪ ω jr ⎪ ⎪ j=1 ⎪ 1 ⎪ − d(h i j , h i  j ) h i j > h i  j ⎪ ω jr ⎨ λ C ϕ j (Ai , Ai  ) = 0 hi j = hi  j  ⎪ ⎪ ω jr ⎪ ⎪ d(h i j , h i  j ) h i j < h i  j m ⎪  ⎪ ω jr ⎩ j=1

(7.18)

−37.91

−27.06

−31.77

1.89

0.86

1.27

ψ j (A1 , A3 )

ψ j (A1 , A4 )

ψ j (A2 , A1 )

ψ j (A2 , A3 )

ψ j (A2 , A4 )

c1

Attribute

ψ j (A1 , A2 )

Relative dominance

0.91

0.54

1.26

−37.94

−69.83

−106.24

c2

Table 7.11 Relative prospect dominance degree c3

1.05

−38.44

1.34

−38.82

−50.00

−52.20

c4

1.69

1.08

2.10

−18.07

−39.45

−68.83

ψ j (A4 , A3 )

ψ j (A4 , A2 )

ψ j (A4 , A1 )

ψ j (A3 , A4 )

ψ j (A3 , A2 )

ψ j (A3 , A1 )

Relative dominance 1.35

0.70

−25.47

1.58

−13.98

−17.27

c1

Attribute 0.83

−37.21

−76.61

0.45

0.44

−45.29

c2

1.00

0.37

0.99

1.28

−14.48

−41.09

c3

1.20

−24.25

−55.39

0.55

0.74

−35.35

c4

7.3 Illustrative Analysis 131

132

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Table 7.12 Overall prospect dominance

(A1 )

(A2 )

(A3 )

(A4 )

0

1

0.86

0.56

Table 7.13 Relative weight ω1r

ω2r

ω3r

ω4r

1

0.27

0.55

0.65

Table 7.14 Relative dominance degree Relative dominance

Attribute c1

c2

c3

c4

Relative dominance

ϕ j (A1 , A2 ) −1.89 −4.18 −2.48 −2.83 ϕ j (A3 , A1 )

Attribute c1

c2

1.42

0.81

c3

c4

1.22

1.22

ϕ j (A1 , A3 ) −1.56 −3.29 −2.42 −2.07 ϕ j (A3 , A2 ) −1.21 −2.57

1.06 −1.94

ϕ j (A1 , A4 ) −1.71 −2.33 −2.10 −1.33 ϕ j (A3 , A4 ) −1.07

0.57

0.61

0.94

ϕ j (A2 , A1 )

1.72

1.03

0.57

1.06

0.78

ϕ j (A2 , A3 )

1.10

0.63 −2.10

1.15 ϕ j (A4 , A2 ) −1.51 −3.47 −2.17 −2.51

ϕ j (A2 , A4 )

1.37

0.85

1.48 ϕ j (A4 , A3 )

1.25 1.09

1.67 ϕ j (A4 , A1 )

1.56

0.97 −2.33 −1.21 −1.58

where d(h i j , h i  j ) is the distance of hesitant fuzzy information h i j and h i  j . Step 4: Compute the dominance degrees of the alternative Ai over the others by (6.6) (Table 7.15). Step 5: Acquire the overall dominance degrees according to (1.9) (Table 7.16). Step 8: Since (A2 ) > (A3 ) > (A4 ) > (A1 ), then A2  A3  A4  A1 . Thus, the project A2 should be selected. Table 7.15 The dominance degree ψ j (A1 , A2 )

−11.39

ψ j (A2 , A1 )

5.67

ψ j (A3 , A1 )

4.67

ψ j (A4 , A1 )

3.97

ψ j (A1 , A3 )

−9.35

ψ j (A2 , A3 )

0.78

ψ j (A3 , A2 )

−4.67

ψ j (A4 , A2 )

−9.65

ψ j (A1 , A4 )

−7.47

ψ j (A2 , A4 )

4.80

ψ j (A3 , A4 )

1.05

ψ j (A4 , A3 )

−4.15

Table 7.16 Overall dominance

(A1 )

(A2 )

(A3 )

(A4 )

0

1

0.74

0.47

7.3 Illustrative Analysis

133

7.3.5 Analysis In this chapter, we summarize the results of above decision-making processes and show them in Table 7.17 [1]. By comparing these results of the four methods, the preponderance of combining the improved TODIM with probabilistic fuzzy information is fully illustrated. From Table 7.17, four kinds of methods have the same ranking results, A2  A3  A4  A1 . Obviously, A2 is considered as the optimal choice and A1 is the worst choice. However, there are huge differences in the overall prospect dominance or the overall dominance obtained by the four methods. Based on Table 7.17 and Fig. 7.1, compared with Method 1 (Sect. 7.3.1) and Method 2 (Sect. 7.3.2), as well as Method 3 (Sect. 7.3.3) and Method 4 (Sect. 7.3.4), the difference of the overall prospect dominance or the overall dominance degree between the alternatives A3 and A4 which uses the improved TODIM based on PT is greater than the extended one. This is because the different risk attitudes of the DMs concerning gain and loss which are considered in PT. Compared with Method 1 Table 7.17 The results of 4 methods Methods

Overall prospect dominance

(A1 )

(A2 )

(A3 )

(A4 )

Improved TODIM with probabilistic hesitant fuzzy information

0

1

0.44

0.42

Extended TODIM with probabilistic hesitant fuzzy information

0

1

0.40

0.39

Improved TODIM with hesitant fuzzy information

0

1

0.86

0.56

Extended TODIM with hesitant fuzzy information

0

1

0.74

0.47

1 0.8 0.6 0.4

Method_4 Method_3 Method_2

0.2 0 A_1

Method_1

A_2

Method_1 A_3

Method_2

A_4 Method_3

Fig. 7.1 Overall prospect dominance or overall dominance

Method_4

134

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

and Method 3, as well as Method 2 and Method 4, it is easier to discover that the overall prospect dominance degree and the overall dominance degree obtained from the methods which adopt probabilistic hesitant fuzzy information and concern different degrees of hesitant preferences is smaller than those obtained from hesitant fuzzy information. The result illustrates that probabilistic hesitant fuzzy information is good at expressing the DMs’ evaluation information and the different preferences for hesitation values. On the other hand, from the perspective of GDM, probabilistic hesitant fuzzy information can reflect more details of different DMs. For example, five experts are asked to evaluate the market potential of a project, three of them consider the market potential of the project to be 72 points, while one expert considers it to be 80, and another considers it to be 82, then the evaluation information obtained from the experts by using probabilistic hesitant fuzzy information can be expressed as {72(0.6), 80(0.2), 82(0.2)}. While it can be only expressed as {72, 80, 82} by using hesitant fuzzy information. It is distinct that more experts tend to 72 points, however, hesitant fuzzy information does not reflect this situation in GDM problems. Therefore, probabilistic hesitant fuzzy information is more comprehensive and effective in illustrating decision-making information.

7.4 Comparative Analysis To better illustrate the effectiveness of the given method, the comparative analysis with the TOPSIS method, the sensitivity analysis and simulation analysis are carried out [1]. The results of analysis strongly support the superiority of the given method.

7.4.1 Comparative Analysis with the TOPSIS Method Analysis in Sect. 7.3.5 focuses on the comparisons of the different extensions of TODIM with different fuzzy information. In this chapter, to illustrate the advantages of TODIM with probabilistic hesitant fuzzy information, we compare it with TOPSIS under probabilistic hesitant fuzzy environment. It offers a more convincing analysis with the method which is not concerning the psychological factor of DMs. Motivated by [2, 3], probabilistic hesitant fuzzy information is extended to TOPSIS. By applying the example in Sect. 7.3, the extended TOPSIS is used to obtain the best alternative. First, we obtain the normalized evaluation matrix which is the same as the matrix of Step 2 in Sect. 7.3.1. Second, we find the positive ideal alternative A+ which is the alternative with the closest distance to ideal solution and the negative ideal alternatives A− which is farthest to ideal solution by (7.19) and (7.20) as shown in Table 7.18:

   

+ + + +

(7.19) A = h 1 , h 2 , . . . , h j = max h i j i = 1, 2, . . . , n i

7.4 Comparative Analysis

−

A =



135

− h− 1 , h2 ,

h −j

...,



 =



max h i j i = 1, 2, . . . , n i

(7.20)

Then, we use the following (7.21) and (7.22) to compute the distances between alternatives and the ideal solution, where the distances measures are obtained by (7.13) (Table 7.19). Di+ =

m 

w j d(h i j ( pi j ), h i+ )

(7.21)

w j d(h i j ( pi j ), h i− )

(7.22)

j=1

Di− =

m  j=1

We can easily obtain the relative closeness coefficients of each alternative by (7.23). They are listed in Table 7.20. Hence, the ranking of the alternatives obtained from the TOPSIS with probabilistic hesitant fuzzy information is A2  A4  A3  A1 . Ci∗ =

Di+

Di− , i = 1, 2, . . . , n + Di−

(7.23)

The superiority of the improved method can be seen from the results in Tables 7.17 and 7.20. The ranking result obtained from the TOPSIS is different from the one obtained from the improved TODIM with probabilistic hesitant fuzzy information, and the ranking result of the middle two alternatives ( A3 and A4 ) is different in those two methods. This distinction can be attributed to the following reasons, which are also the advantages of the improved method. First, the improved TODIM has identified more information of DMs. It not only involves the transformed weight, but also considers the difference between every two alternatives, instead of focusing on the difference between the alternative and positive ideal solution or negative ideal solution shown in TOPSIS. In addition, TOPSIS with probabilistic fuzzy information does not reflect the psychological factors of DMs, while probabilistic fuzzy information describes the uncertain decision-making process and all participants have made limited rational choices. This defect is fully compensated in the TODIM by considering the risk attitudes for gain and loss, which makes the results more accurate, objective and more consistent with practical experience. By addressing the comparison of the existing method, the demand of combining probabilistic hesitant fuzzy information with TODIM has also been fully demonstrated. The overall prospect dominances of TODIM and the relative closeness coefficients of the TOPSIS with probabilistic hesitant fuzzy information are much smaller than the results of TODIM with hesitant fuzzy information. It is strongly proven that the probabilistic hesitant fuzzy information has discerned and reflected more information of DMs. Therefore, probabilistic hesitant fuzzy information is an effective tool to express a wider range of uncertain information in the decision-making process.

{79(0.09), 62(0.28), 77(0.63)}

{55(0.22), 73(0.27), 68(0.51)}

A+

A−

c1 {70(0.16),66(0.39), 60(0.45)}

{80(0.03),68(0.29), 77(0.68)}

c2

Table 7.18 The positive ideal solution and negative ideal solution c3 {71(0.1),68(0.21), 62(0.69)}

{60(0.18),73(0.21),85(0.61)}

c4 {77(0.02), 72(0.32), 64(0.66)}

{80(0.04),88(0.36), 77(0.60)}

136 7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

7.4 Comparative Analysis

137

Table 7.19 Distances between the alternatives and the idea solution A1

A2

A3

A4

Di+

6.5980

0.0000

9.1503

8.5449

Di−

0.0000

6.5980

4.0679

5.6463

Table 7.20 The relative closeness coefficients Ci∗

A1

A2

A3

A4

0.00

1.00

0.31

0.40

7.4.2 Sensitivity Analysis Based on the Parameter Values To better illustrate the advantages of the improved TODIM with probabilistic hesitant fuzzy information, this part conducts a sensitivity analysis of the improved TODIM and the extended TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information, respectively. Both two comparative analyses fully illustrate the advantages of the improved TODIM based on PT. Moreover, this chapter presents the comparative analysis to illustrate the superiority of probabilistic hesitant fuzzy information by comparing it with hesitant fuzzy information in a fixed method. To better show the contrast, the fuzzy information is extended to the improved TODIM and to the extended TODIM [1]. Both two analyses show the capacity of probabilistic hesitant information in reflecting more evaluation information of DMs.

7.4.2.1

Sensitivity Analysis of the Improved TODIM and the Extended TODIM with the Same Fuzzy Information

In the beginning, we analyze the difference of sensitivity between the improved TODIM and the extended TODIM with the same fuzzy information. According to the results, we recognize that no matter how the parameter λ changes, there is no significant changes for the ranking results from each method. Therefore, the overall prospect dominances are used to show the strength of the improved TODIM. (1)

Sensitivity analysis of the improved TODIM and the extended TODIM with probabilistic hesitant fuzzy information

Since λ is the only common parameter in the improved TODIM and the extended TODIM with probabilistic hesitant fuzzy information, the fluctuation of overall prospect dominance or overall dominance degree can be easily observed by changing the value of λ (1.25 ≤ λ ≤ 2.25) which is shown in Fig. 7.2. Overall prospect dominance and the overall dominance degrees of the first and the last alternative remain unchanged which is naturally determined by the TODIM itself, and they are 1 and 0 separately when the ranking result is unchanged. Subsequently,

138

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Fig. 7.2 Sensitivity analysis with probabilistic hesitant fuzzy information. Note Here 1.25 ≤ λ ≤ 2.25, α = β = 0.88, δ=0.69, γ = 0.61

making the alternatives A3 and A4 as an analysis group, when λ varies, the fluctuation range of overall prospect dominance from the improved TODIM is smaller than the overall dominance obtained from the extended TODIM, which indicates that the improved TODIM is more stable. Besides, the overall prospect dominance value and the overall dominance value obtained from the two methods shows a reverse trend. The main reason is that λ is proportional to the dominance function in the improved TODIM, and it is also proportional to the overall prospect dominance. In the extended TOIDM, the dominance function is affected by reciprocal form of λ, so λ is inversely proportional to the dominant function. This kind of reverse trend also can be found in the following analysis under hesitant fuzzy environment. (2)

Sensitivity analysis of the improved TODIM and the extended TODIM with hesitant fuzzy information

The changes of overall prospect dominance or overall dominance degree in Fig. 7.3 are obtained by altering λ (1.25 ≤ λ ≤ 2.25) in the improved TODIM and the extended TODIM with hesitant fuzzy information, According to Fig. 7.3, it is apparent that the overall prospect dominance or the overall dominance values of

7.4 Comparative Analysis

139

Fig. 7.3 Sensitivity analysis with hesitant fuzzy information. Note Here 1.25 ≤ λ ≤ 2.25, α = β = 0.88, δ=0.69, γ = 0.61

the alternative A1 and the alternative A2 stay constant when the parameter λ varies. At the same time, the fluctuation of overall prospect dominance from the improved TODIM is smaller than the overall dominance from the extended TODIM which also demonstrates that the improved TODIM is more stable.

7.4.2.2

Sensitivity Analysis of the Improved TODIM and the Extended TODIM Based on Different Types of Fuzzy Information

This part presents two groups of comparative analyses to illustrate the advantages of probabilistic hesitant fuzzy information in expressing the perceptions of the DMs. The first one is the improved TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information. The second one is the extended TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information. However, the ranking results of each alternative keep unchanged when the parameters change.

140

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Subsequently, the overall prospect dominances or the overall dominances are used to show the advantages of probabilistic hesitant fuzzy information. (1)

Sensitivity analysis of the improved TODIM with probabilistic hesitant fuzzy information and compared it with hesitant fuzzy information

Since many parameters are used in the improved TODIM, this part presents the changes of overall prospect dominance degree of each alternatives when the parameters change, which are shown in Figs. 7.4, 7.5, 7.6, 7.7 and 7.8. Figure 7.4 presents the fluctuation of overall prospect dominance from the improved TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information separately by changing the parameter λ (1.25 ≤ λ ≤ 2.25). We can clearly see that for the alternatives A3 and A4 , the changes of overall prospect dominance obtained by probabilistic hesitant fuzzy information are smaller than the one obtained by hesitant fuzzy information, which indicates the stability of probabilistic hesitant fuzzy information.

Fig. 7.4 Sensitivity analysis of the improved TODIM by changing λ. Note Here 1.25 ≤ λ ≤ 2.25, α = β = 0.88, δ=0.69, γ = 0.61

7.4 Comparative Analysis

141

Fig. 7.5 Sensitivity analysis of improved TODIM by changing α. Note Here 0.68 ≤ α ≤ 1.21, β = 0.88, δ=0.69, γ = 0.61, λ = 2.25

Figure 7.5 presents the fluctuation of overall prospect dominance from the improved TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information separately by changing the parameter α (0.68 ≤ α ≤ 1.21). It is obvious that, for the alternatives A3 and A4 , the changes of overall prospect dominance obtained by probabilistic hesitant fuzzy information are smaller than the one obtained by hesitant fuzzy information, which also indicates that the probabilistic hesitant fuzzy information is more stable. Figure 7.6 presents the fluctuation of overall prospect dominance from the improved TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information separately by changing the parameter β (0.68 ≤ β ≤ 1.02). It is clear that, for the alternatives A3 and A4 , the changes of overall prospect dominance obtained by probabilistic hesitant fuzzy information and by hesitant fuzzy information have obvious differences. For the alternative A4 , the change trend of the methods with two different types of information goes the same direction, however, the greater fluctuation happens to the method with hesitant fuzzy information. For the alternative A3 , the change trend of the two methods goes to the opposite direction.

142

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Fig. 7.6 Sensitivity analysis of improved TODIM by changing β. Note Here 0.68 ≤ β ≤ 1.02, α = 0.88, δ=0.69, γ = 0.61, λ = 2.25

Figure 7.7 presents the fluctuation of overall prospect dominance from the improved TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information separately by changing the parameter δ (0.36 ≤ δ ≤ 0.84). For the alternatives A3 and A4 , significant fluctuation can be observed in the overall prospect dominance which is obtained by probabilistic hesitant fuzzy information, while small changes happen to the one obtained from hesitant fuzzy information. For both the alternatives A3 and A4 , the overall prospect dominances obtained from two methods tend to change in the same direction. Figure 7.8 presents the fluctuation of overall prospect dominance of the improved TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information separately by changing the parameter γ (0.55 ≤ γ ≤ 0.721). For the alternatives A3 and A4 , the overall prospect dominances obtained from these two kinds of methods are almost unchanged. No matter how A3 and A4 change, the overall prospects tend to change in the same direction.

7.4 Comparative Analysis

143

Fig. 7.7 Sensitivity analysis of the improved TODIM by changing δ. Note Here 0.36 ≤ δ ≤ 0.84, α = β = 0.88, γ = 0.61, λ = 2.25

(2)

Sensitivity analysis of the extended TODIM with probabilistic hesitant fuzzy information and compared with hesitant fuzzy information

Since there is only one common parameter λ in the extended TODIM, this chapter considers the changes of overall prospect dominance by changing λ. Figure 7.9 presents the fluctuation of the overall dominance when changing the parameter λ (1.25 ≤ λ ≤ 2.25) in the extended TODIM with probabilistic hesitant fuzzy information and hesitant fuzzy information. For the alternatives A3 and A4 , the overall dominances obtained by hesitant fuzzy information change significantly, while the one obtained by probabilistic hesitant fuzzy information is nearly unchanged, and it also continues to decrease when increasing the value of the parameter λ. Besides, regardless of the alternative A3 or A4 , the overall dominances obtained from the two methods tend to change in the same direction.

144

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Fig. 7.8 Sensitivity analysis of the improved TODIM by changing γ . Note Here 0.55 ≤ γ ≤ 0.721, α = β = 0.88, δ = 0.69, λ = 2.25

In summary, the improved TODIM is more stable and effective (Figs. 7.2 and 7.3) with the same type of fuzzy information. The overall prospect dominances and the overall dominances obtained by probabilistic hesitant fuzzy information change slightly than the ones obtained by hesitant fuzzy information (Figs. 7.4 and 7.9). The results illustrate that the probabilistic hesitant fuzzy information is steadier and contains more information of the DMs. Figures 7.5, 7.6, 7.7 and 7.8 present the changes of the overall prospect dominance when changing other parameters with different types of fuzzy information from the improved TODIM.

7.5 Simulation Analysis After sensitivity analysis of parameters based on one sample, we present the analysis results of 1000 sets of decision samples which are randomly generated by MATLAB software. The ranking results are shown in Table 7.21.

7.5 Simulation Analysis

145

Fig. 7.9 Sensitivity analysis of extended TODIM by changing λ. Note Here 1.25 ≤ λ ≤ 2.25, α = β = 0.88, γ = 0.61, δ = 0.69

From Table 7.21, the ranking results of 1000 sets of random data by using different kinds of methods are presented. 393 sets of data have the same ranking results for these four methods. With probabilistic hesitant fuzzy information, 633 sets of data have the same ranking results by using the improved TODIM and the extended TODIM. However, with hesitant fuzzy information, 654 sets of date are observed to have the same ranking results by using the improved TODIM and the extended TODIM. The number of ranking results with hesitant fuzzy information is bigger than that of the ranking results with probabilistic hesitant fuzzy information, because the latter one includes more information and it is more difficult to get the same ranking result. 622 sets of data have the same ranking results by using the improved TODIM with probabilistic hesitant fuzzy information and the improved TODIM with hesitant fuzzy information. 679 sets of data have the same ranking results by using the extended TODIM with probabilistic hesitant fuzzy information and the extended TODIM with hesitant fuzzy information.

146

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

Table 7.21 Ranking results of each method with 1000 sets of random samples Methods

Results The number of the same ranking result

The number of the different ranking result

Improved TODIM with 393 probabilistic hesitant fuzzy information & Extended TODIM with probabilistic hesitant fuzzy information & Improved TODIM with hesitant fuzzy information & Extended TODIM with hesitant fuzzy information

607

Improved TODIM with probabilistic hesitant fuzzy information & Extended TODIM with probabilistic hesitant fuzzy information

633

367

Improved TODIM with 654 hesitant fuzzy information & Extended TODIM with hesitant fuzzy information

346

Improved TODIM with probabilistic hesitant fuzzy information & Improved TODIM with hesitant fuzzy information

622

378

Extended TODIM with 679 probabilistic hesitant fuzzy information & Extended TODIM with hesitant fuzzy information

321

According to the results, there are large number of random data with the same ranking result, which shows the feasibility and applicability of the given methods. On the other hand, there are still some existing sets of data sample with different ranking results which indicates the differences between these methods. This difference can be explained by the following two points: (1) Compared with hesitant fuzzy information, probabilistic hesitant fuzzy information contains more original decision information. The former one is just a special form of probabilistic hesitant fuzzy information when the probability is equal, and the later one is more general. (2) Compared with the extended TODIM, the improved TODIM based on PT rewrites dominance function in TODIM and makes it more in line with the actual decision-making environment, which contains more details about risk attitudes for gains and losses of DMs.

7.6 Remarks

147

7.6 Remarks The TODIM is the first MCDM based on PT which shows the risk aversion attitude and risk preference through the dominance function. According to the classical TODIM, the relative weight is calculated by the one-dimensional weight, however, according to the original PT, it considers that the DMs adopt the non-linear transformed weight function in the decision-making process. Without a doubt, the classical TODIM ignores the effect of the transformed weighting function on decision-making result. Besides, it is easy to recognize from PT that the multiply value of relative weight and the perceived gain or loss value are regarded as the overall preferences of the DMs in the classic TODIM. While the different risk attitudes for gain and loss are mainly reflected by the value function in the classic PT, and it takes the product of the value function and the weight function as a decision reference. It can be concluded that the preferences of DMs in the classical TODIM are inconsistent with those of the classical PT. Based on which, this chapter shows an improved TODIM according to PT and rewrites the dominance function of the classical TODIM. Besides, the values of the weight function are used to calculate the relative weights, and the different risk attitudes for gain and loss are expressed by the value function. At the same time, it is considered that the DMs are more likely to express their perceptions as the form of probabilistic hesitant fuzzy information under the highly uncertain circumstance. That is the reason why we show a probabilistic hesitant fuzzy TODIM based on a new perspective of PT. In order to illustrate the feasibility and effectiveness of the given method, this chapter also presents the improved TODIM with hesitant fuzzy information and the TOPSIS with probabilistic fuzzy information. The most important part of this chapter is that an improved TODIM based on PT with probabilistic hesitant fuzzy information is given, and this method replaces one-dimensional weight with weight function to calculate the relative weight value separating the value function from the relative weight. This chapter presents the process of reconstructing the relative dominance function of the classical TODIM based on PT and integrates probabilistic hesitant fuzzy information into the TODIM. Moreover, this chapter also gives the improved TODIM with the hesitant fuzzy information and TOPSIS with the probabilistic fuzzy information to show the advantages of the improved TODIM with probabilistic hesitant fuzzy information. Furthermore, an illustrative example, parameter sensitivity analysis and simulation analysis are carried out to show the advantages of the improved methods and the differences between these methods and the existing ones.

148

7 An Improved TODIM with Probabilistic Hesitant Fuzzy Information

References 1. Tian XL, Niu ML, Ma JS, Xu ZS (2020) A Novel TODIM with probabilistic hesitant fuzzy information and its application in green supplier selection. Complexity: 2540798 2. He Y, Xu ZS (2019) Multi-attribute decision making methods based on reference ideal theory with probabilistic hesitant information. Expert Syst Appl 118:459–469 3. Dagdeviren M, Yavuz S, Kilinc N (2009) Weapon selection using the AHP and TOPSIS methods under fuzzy environment. Expert Syst Appl 36(4):8143–8151

Chapter 8

Conclusions

The fuzzy decision-making methods based on PT has been presented in this book. This chapter will give some conclusions of this book, including the summary of this book and the future studies in the decision-making field with bounded rationality of DMs.

8.1 Summary Decision making is to find an optimal alternative among the optional ones. The MADM method is an important way to decompose the complex decision-making problems in several aspects and give the evaluation information from each aspect. Then, integrating the evaluation information and finding an optimal alternative is the final purpose of MADM. Additionally, fuzzy information is an effective tool to describe the uncertain perceptions of DMs for the alternatives. Hence, in recent years, fuzzy MADM methods have been rapidly developed and acknowledge by many scholars and practitioners. However, numerous existing fuzzy MADM methods are constructed based on the perfectly rational hypothesis, which is hard to deal with the DMs’ bounded rational emotion in real decision-making situations. Therefore, it is important to comprehensively present the fuzzy MADM methods with bounded rationality of DMs, which is the aim of this book. The main focus of this book is listed below: (1)

(2)

Systematically analyze the existing literature about fuzzy MADM with bounded rationality using the database such as Web of Science, Google Scholar, and so on. Then, the focus of this book is found, which is the foundational work for us to present fuzzy MADM based on PT. Considering the evaluation information from the supportive way and opponent way, an intuitionistic fuzzy MADM method based on PT has been given in

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Tian and Z. Xu, Fuzzy Decision-Making Methods Based on Prospect Theory and Its Application in Venture Capital, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-16-0243-6_8

149

150

(3)

(4)

(5)

8 Conclusions

Chap. 2 to solve the decision-making problem with intuitionistic fuzzy information and used to select the best project for VCs. According to the experiment conducted by Gu et al. [1], the DMs rely on the net certain value to make their decisions. Hence, the score function of IFS is used as the variable in the weighting function. The decision-making procedure of intuitionistic fuzzy MADM based on PT is given in Chap. 2. The QUALIFLEX is a flexible method to solve the MADM problem with a few alternatives. Moreover, the linguistic information is a very general way used by DMs to express their real perceptions. In particular, the probabilistic linguistic information, including the probability of each linguistic term, can simulate the vague perceptions of the DMs well. Therefore, in Chap. 3, two QUALIFLEX methods (P-QUALIFLEX and extended QUALIFLEX) with probabilistic linguistic information are presented. Firstly, considering the different risk attitudes for gain and loss of DMs when making their decisions under uncertainty, which is well explained by PT, a P-QUALIFLEX with probabilistic linguistic information is given. Then, in order to demonstrate the useful of the P-QUALIFLEX, an extended QUALIFLEX with probabilistic linguistic information based on the classical QUALIFLEX is shown in this chapter. The feasibility and validity of the given methods have been verified by a numerical example in venture capital. The comparative and simulated analysis shows that the former method with prospect framework is more appropriate than the latter one because of the inherent psychological behaviors of the DMs and its excellent ability in identifying the similar alternatives. Furthermore, the ranking results derived from the P-QUALIFLEX do not change with the different values of parameters. It reveals that the P-QUALIFLEX is stable and reliable. The PROMETHEE is one of the well-known ranking method. Also, the hesitant fuzzy linguistic term set is a suitable tool to simulate DMs’ evaluation information. Additionally, as the deepening of social division of labor and specialization of individuals, GDM is famous for improving the decision-making quality. Moreover, in the decision-making process, DMs exhibit behavioral characteristics which is depicted well by PT that DMs are risk averse for gain and risk seeking for loss and rely on the transformed probability to make their decisions rather than unidimensional probability. Thus, a GP-PROMETHEE with hesitant fuzzy linguistic information is given for DMs to make a better decision. Then, the given method is applied to rank start-up firms and the comparative analyses are made as well. It confirms that the GP-PROMETHEE is better in describing the common behavioral characteristics of DMs and in enhancing the quality of evaluation than the extended G-PROMETHEE. Consensus is one of the hot topics in GDM. Considering the psychological behavior of DMs in the GDM process, PT is introduced into the consensus model. In the decision-making process, because of the complex and uncertain decision-making circumstance, it is hard for the DMs to give their evaluation information for each alternative as crisp numbers, but it may be easy for them to provide their preferences over each pair of alternatives. Motivated by which,

8.1 Summary

(6)

151

the P-HFPs are adopted by the DMs in their evaluation processes. the DMs may hesitate to express their evaluation information and the hesitant degree for each evaluation value is different. Hence, in this book, a prospect consensus with P-HFPs has been introduced. It has been applied to a practical case, and an expected consensus under rational framework has also been used to analyze the case. The superiority of prospect consensus with P-HFPs has been correspondingly demonstrated in this book through the comparative analyses of the expected consensus with P-HFPs, the prospect consensus with HFPs and the expected consensus with HFPs. An improved TODIM is introduced in Chap. 6 to comprehensively reflect the psychological characteristics of DMs according to PT. The original weight is replaced with the weighting function of PT and the perceived value of the dominance is modified based on PT, because the general psychological phenomena of DMs explained in PT are verified by many experiments and recognized by researchers. Hence, the improved TODIM not only integrates the advantages of PT in considering the psychological factors of DMs but also retains the superiority of the classical TODIM in relative dominance. Finally, the improved TODIM is demonstrated to capture the psychological factors of DMs well from the illustrative example. Furthermore, the improved TODIM is extended under probabilistic hesitant fuzzy circumstance and hesitant fuzzy circumstance in Chap. 7. To show the advantages of it, an improved TODIM with hesitant fuzzy information is also presented. Then, the comparative analyses are carried out.

8.2 Future Studies The main focus of this book is to present the MADM with fuzzy information based on the framework of PT and its application in venture capital field. Although there are some performances, it is not enough. There are many aspects deserved to be explored in the future. (1)

(2)

Although the MADM methods are established under different fuzzy circumstances, it only deals with a special fuzzy information. However, there are several attributes which should be considered in the decision-making process and each attribute has its own characteristics. For example, when the DMs are invited to evaluate the comprehensive ability of the team of senior executives of a project, they may use “strong”, “very strong”, etc., while when they evaluate the possible return of a project, they may use “10%”, “15%”, etc. Hence, only one kind of fuzzy information could not fully reflect the uncertain evaluation of DMs. Under this situation, heterogeneous information should be used. Thus, the MADM method based on PT with heterogeneous information should be further discussed in the future research. As time goes by, the decision-making objectives and decision-making circumstance will be changed, and the existing MADM method based on PT could

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(3)

(4)

(5)

(6)

8 Conclusions

not adapt the dynamic changes. Hence, dynamic MADM based on PT with FSs should be considered in the future. In this book, the unsymmetrical risk attitudes of the DMs for gain and loss are introduced into the MADM methods with fuzzy information. There are some advantages of the given MADM methods, but the effect of the DMs’ psychological behaviors such as regret, overconfidence, etc., on decision making is not included in the given methods. Those psychological behaviors are very common in the decision-making process. Hence, much more attention will be paid to the effect of the DMs’ psychological behaviors on decision making in the future work. We believe that this book may provide inspiration for the follow-up researches of decision-making methods under the framework of bounded rationality. The decision-making under big data becomes more and more popular as the development of internet. More research should focus on how to integrate psychological factors of the DMs into fuzzy MADM with large-scale group because the impact path of psychological behaviors on decision making is greatly different from the general GDM. Large number of alternatives is another inevitable result in the Internet Age. For example, if a customer wants to buy the earphones, there are numerous products for him/her to choose in different e-commerce platforms. Moreover, it is more difficult for DMs to make a rational decision when there are so many choices. Hence, the psychological behaviors of DMs under large-scale alternatives is more notable. Then, the research about MADM with large amounts of alternatives under the framework of bounded rationality is another worthy work in the future. Although this book is concentrated on the application of the given MADM methods in the venture capital field, the related analysis can also be extended to other uncertain decision-making situations and play the role of demonstration for future research in other fields.

Reference 1. Gu J, Wang ZJ, Xu ZS, Chen XZ (2018) A decision-making framework based on the prospect theory under an intuitionistic fuzzy environment. Technol Econ Dev Econ 24(6):2374–2396