Theory and Approaches of Group Decision Making with Uncertain Linguistic Expressions

226 Pages·2019·3.963 MB·English

by  Hai Wang, Zeshui Xu

Uncertainty and Operations Research
Hai Wang
Zeshui Xu
Theory and Approaches
of Group Decision-
Making with Uncertain
Linguistic Expressions
Uncertainty and Operations Research
Editor-in-chief
Xiang Li, Beijing University of Chemical Technology, Beijing, 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,
researchershaveproposedmanyefficientoperationsresearchmodelsandmethods,
which have been widely applied to real-life problems, such as finance, manage-
ment, manufacturing, supply chain, transportation, among others. This book series
aims to provide a global forum for advancing the analysis, understanding,
development,andpracticeofuncertaintytheoryandoperationsresearchforsolving
economic, engineering, management, and social problems.
More information about this series at http://www.springer.com/series/11709
Hai Wang Zeshui Xu
(cid:129)
Theory and Approaches
of Group Decision-Making
with Uncertain Linguistic
Expressions
123
HaiWang ZeshuiXu
Schoolof Information Engineering Business School
NanjingAudit University SichuanUniversity
Nanjing, Jiangsu,China Chengdu,Sichuan, China
ISSN 2195-996X ISSN 2195-9978 (electronic)
Uncertainty andOperationsResearch
ISBN978-981-13-3734-5 ISBN978-981-13-3735-2 (eBook)
https://doi.org/10.1007/978-981-13-3735-2
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Preface
Duetothecomplexityofproblemsinhandandthelimitationofexperts’cognition,
uncertaintiesaregenerallyinevitableindecisioninformation.Inthefuzzylinguistic
approach, linguistic variables enable a manner to represent uncertain information
which is close to human’s cognition. It is necessary that, in the traditional way of
computingwithwords,theexpertshavetorepresentdecisioninformationbymeans
of certain terms. However, this is quite difficult when facing complex types of
uncertainties. Uncertain linguistic expressions, which include more than one pos-
sible term in a direct or indirect way, are more consistent with people’s language
conventions. In order to enable the use of uncertain linguistic expressions in
decision-making processes, some fundamental theories and approaches have been
developed
Onthebasisoftheexistingmodels,thisbookintroducessomelinguisticmodels
torepresenttwotypesofuncertainlinguisticexpressionswhichconformtonatural
language conventions, i.e., extended hesitant fuzzy linguistic term sets and lin-
guistic terms with weakened hedges, and presents the related fundamental theories
and approaches for group decision-making. Specifically, the book is organized by
five parts as follows:
The first part is formed by Chap. 1. This chapter introduces the background of
computing with words, the focused problems, and some related theory and tech-
niques. A brief overview of this related area, such as the current developments of
models of uncertain linguistic expressions, the group decision-making approaches,
is also given in this chapter.
The second part is Chap. 2. This chapter presents the representational model
of the virtual linguistic terms, extend the model of hesitant fuzzy linguistic term
sets, and then introduces a new technique to model linguistic hedges.
Computational models of these techniques, such as order relations, are also
presented.
The third part goes through Chaps. 3–5. Chapters 3 and 4 focus on the group
decision-making problems with the extended version of hesitant fuzzy linguistic
term sets. Chapter 3 is under the framework of decision matrices, presents an
information fusion based group decision-making approach and a two-phase group
v
vi Preface
decision-making approach. Chapter 4 is based on the framework of preference
relations, presents some new consistency measures, and then employs them to
improve incomplete linguistic preference relations. Group decision-making prob-
lems and preference relations withhedges are focusedin Chap.5. A multigranular
group decision-making approach is introduced and some theoretical aspects of the
new preference relations are also discussed.
ThefourthpartincludesChaps.6and7wheregroupdecision-makingproblems
with multiple types of uncertain linguistic expressions are considered. Two group
decision-making approaches are introduced. The first one considers the aspiration
levels taking the form of uncertain linguistic expressions and the second one pre-
sents descriptive measures for decision makers to understand the effects of uncer-
tain parameters.
The last part includes Chap. 8. A hierarchical model is introduced for the
evaluationofbigdata-basedauditplatforms.Themodelissolvedinthecasewhere
the performances take the forms of multiple types of uncertain linguistic expres-
sions,basedontheuncertainlinguisticexpressionsapproachpresentedinChap.6.
Thisbookcanbeusedasareferenceforengineers,technicians,andresearchers
who are working in the fields of intelligent computation, fuzzy mathematics,
operations research, information science, management science and so on. It could
also serve as a textbook for postgraduate and senior undergraduate students of the
relevant professional institutions of higher learning.
The first author would like to thank Dr. Xiao-Jun Zeng at the University of
Manchesterforhisinsightfulideasandgreatsuggestions.Thisworkwassupported
inpartbytheNationalNaturalScienceFoundationofChinaunderGrant71571123
and Grant 71601092, and the Key University Science Research Project of Jiangsu
Province (No. 16KJA520002, 18KJB413006) and the China Scholarship Council.
Nanjing, China Hai Wang
September 2018 Zeshui Xu
Contents
Part I Introduction
1 Backgrounds and Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Linguistic Decision-Making in Qualitative Setting . . . . . . . . . . . . 3
1.2 Focused Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Novel CWW Models Based on ULEs. . . . . . . . . . . . . . . . 6
1.2.2 Preference Relations Based on ULEs . . . . . . . . . . . . . . . . 6
1.2.3 GDM Approaches Based on ULEs . . . . . . . . . . . . . . . . . . 7
1.2.4 Modelling Complex Problems Under Uncertainties . . . . . . 7
1.3 Recent Advances of the Focused Problems . . . . . . . . . . . . . . . . . 8
1.3.1 Review of Modelling ULEs and Decision-Making
Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Review of Lingustic Hedges. . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Review of Group Decision-Making Approaches
Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 A Summary of the Contributions and Limitations . . . . . . . 20
1.4 Aims and Focuses of This Book . . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Part II Theory and Models of Uncertain Linguistic Expressions
2 Representational Models and Computational Foundations
of Some Types of Uncertain Linguistic Expressions . . . . . . . . . . . . . 35
2.1 Virtual Linguistic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.2 Syntax and Semantics of VLTs . . . . . . . . . . . . . . . . . . . . 37
2.1.3 Computational Model of VLTs. . . . . . . . . . . . . . . . . . . . . 43
2.2 Extended Hesitant Fuzzy Linguistic Term Sets. . . . . . . . . . . . . . . 44
2.2.1 Fuzzy Linguistic Approach and HFLTS . . . . . . . . . . . . . . 45
2.2.2 Representational Model of EHFLTSs . . . . . . . . . . . . . . . . 45
vii
viii Contents
2.2.3 Basic Operations of EHFLTSs and Their Properties. . . . . . 46
2.2.4 A Partial Order of EHFLTSs . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Total Orders of EHFLTSs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Existing Order Relations of EHFLTSs . . . . . . . . . . . . . . . 49
2.3.2 Total Orders of EHFLTSs: A Generation Approach. . . . . . 50
2.4 Linguistic Terms with Weakened Hedges. . . . . . . . . . . . . . . . . . . 55
2.4.1 Respresentational Model of LTWHs . . . . . . . . . . . . . . . . . 56
2.4.2 Linguistic Computational Model Based on LTWHs. . . . . . 63
2.5 A Comparative Analysis on Similar Models of ULEs. . . . . . . . . . 66
2.5.1 Compared with the Existing Techniques of Modeling
Hedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5.2 LTWHs Versus ULTs and HFLTSs . . . . . . . . . . . . . . . . . 67
2.5.3 Compared with Other Techniques. . . . . . . . . . . . . . . . . . . 69
2.6 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Part III Group Decision-Making Based on a Single Type
of Uncertain Linguistic Expressions
3 Group Decision-Making Based on EHFLTSs Under
the Framework of Decision Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 A Framework of Multiple Groups Decision-Making . . . . . . . . . . . 75
3.1.1 Mathematical Description of MGDM . . . . . . . . . . . . . . . . 76
3.1.2 Process of MGDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 A MGDM Approach Based on Information Fusion . . . . . . . . . . . 79
3.2.1 Some Aggregation Operators of EHFLTSs . . . . . . . . . . . . 79
3.2.2 Properties of the Aggregation Operators . . . . . . . . . . . . . . 83
3.2.3 Implementation of the MGDM Processes . . . . . . . . . . . . . 86
3.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2.5 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3 A Two-Phase GDM Approach Based on Admissible Orders . . . . . 92
3.3.1 Defining the EHFLOWA Operator Based on Admissible
Orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.2 The Two-Phase GDM Approach. . . . . . . . . . . . . . . . . . . . 95
3.3.3 Application in Evaluations of Energy Technologies. . . . . . 97
3.3.4 Comparisons and Further Discussions. . . . . . . . . . . . . . . . 100
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Preference Analysis and Applications Based on EHFLTSs. . . . . . . . 107
4.1 Some Consistency Measures of EHFLPRs . . . . . . . . . . . . . . . . . . 108
4.1.1 The Concept of EHFLPRs . . . . . . . . . . . . . . . . . . . . . . . . 108
4.1.2 Preference Relation Graphs . . . . . . . . . . . . . . . . . . . . . . . 109
Contents ix
4.1.3 Additive Consistency for EHFLPRs . . . . . . . . . . . . . . . . . 112
4.1.4 Selective Algorithm for Reducing EHFLPRs to LPRs
Based on Additive Consistency . . . . . . . . . . . . . . . . . . . . 113
4.1.5 Weak Consistency for EHFLPRs . . . . . . . . . . . . . . . . . . . 116
4.1.6 Broken Circle Algorithm for Reducing EHFLPRs
to LPRs Based on Weak Consistency . . . . . . . . . . . . . . . . 117
4.1.7 Comparative Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.2 Improving Incomplete LPRs Based on Consistency Measures
of EHFLPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2.1 Incomplete LPRs and Their Consistency Measures . . . . . . 123
4.2.2 An Interactive Algorithm to Reach Weak Consistency
of Incomplete LPRs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.2.3 A Consistency-Based Interactive Algorithm to Complete
Incomplete LPRs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2.4 The Interactive Algorithm with Self-adaptive Evolution
to Complete Incomplete LPRs . . . . . . . . . . . . . . . . . . . . . 131
4.2.5 An Example Regarding the Evaluation of Energy
Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.6 Comparisions and Discussions . . . . . . . . . . . . . . . . . . . . . 137
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5 Preference Analysis and Group Decision-Making Based
on LTWHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 Multi-granular Linguistic Decision-Making with LTWHs . . . . . . . 141
5.1.1 The Framework of MGLDM Problems . . . . . . . . . . . . . . . 141
5.1.2 Constructing Multi-granular Linguistic Model Based
on Hedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.3 An Approach for MGLDM with LTWHs . . . . . . . . . . . . . 144
5.1.4 An Application of Evaluating the Non-financial
Performance of Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.5 Compared with Similar MGLDM Approaches. . . . . . . . . . 147
5.2 Consistency Measures of Linguistic Preference Relations
with Hedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.2.1 Some Basic Operations and Order Relations
of LTWHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2.2 Linguistic Preference Relations with Weakened Hedges. . . 152
5.2.3 Consistency Measures of LHPRs . . . . . . . . . . . . . . . . . . . 154
5.2.4 Weak Consistency of LHPRs . . . . . . . . . . . . . . . . . . . . . . 156
5.2.5 Additive Consistency of LHPRs. . . . . . . . . . . . . . . . . . . . 158
5.2.6 Consistency Improving of LHPRs. . . . . . . . . . . . . . . . . . . 162
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168