Consensus Building in Group Decision Making: Searching the Consensus Path with Minimum Adjustments

207 Pages·2016·3.75 MB·English

by  Yucheng Dong

Yucheng Dong · Jiuping Xu
Consensus
Building in
C
Group Decision
B
l
d Making
G
Searching the Consensus Path with
Minimum Adjustments
Consensus Building in Group Decision Making
Yucheng Dong Jiuping Xu
(cid:129)
Consensus Building
in Group Decision Making
Searching the Consensus Path
with Minimum Adjustments
123
YuchengDong Jiuping Xu
Business School Business School
SichuanUniversity SichuanUniversity
Chengdu,Sichuan Chengdu,Sichuan
China China
ISBN978-981-287-890-8 ISBN978-981-287-892-2 (eBook)
DOI 10.1007/978-981-287-892-2
LibraryofCongressControlNumber:2015950014
SpringerSingaporeHeidelbergNewYorkDordrechtLondon
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Preface
The group decision making (GDM) can be seen as a task to find a collective
solution to a decision problem in the situations where a group of experts express
their opinions regarding multiple alternatives. In essence, the GDM reflects the
internalrelationsbetweentheindividualsandgroupandcanbedefinedasageneral
model as follows:
Fðop1;op2;...;opnÞ¼opc;
wherefop1;op2;...;opngaretheopinionsofnindividuals,opcistheopinionofthe
group, and F is the aggregation function implying the group decision rule to
aggregate individual opinions into a collective one.
The GDM is in the important and core position in the economics and man-
agement science. From a macro perspective, modern society is essentially a GDM
system.Intherationalanddemocraticsociety,thereexittwobasicGDMmethods:
one is “voting”, which is usually used in politics, and the other is “market mech-
anism”, which is applied in economic issues. Microscopically, there are numerous
GDM problems, and people are often faced with the need to work with others in
group settings.
ThehistoryoftheGDMstudiescanbetrackedbacktodesignofvotingmethod
(Lull, 1282, 1287; Borda, 1781; Condorcet, 1785). In the last 60 years, the
researches of the GDM have gotten great progresses, and some famous theories
have been proposed, e.g., social choice theory (Arrow, 1951) and prospect theory
(Kahneman and Tversky, 1979, 1992). The social choice theory dates from
Condorcet's formulation of the voting paradox, and provides a theoretical frame-
work for analysis of combining individual opinions, preferences, interests, or
welfares to reach a collective decision or social welfare in some sense. The Arrow
Impossibility Theorem, the Gibbard-Satterthwaite Strategy-proofness Impossibility
TheoremandtheSenLibertarianImpossibilityTheoremconstructthefootstoneof
social choice theory. The prospect theory is a descriptive model based on psy-
chology, and tries to model real-life choices, rather than optimal decisions, as
normativemodelsdo.The prospecttheorystates that people make decisions based
v
vi Preface
on the potential value of losses and gains rather than the final outcome, and that
people evaluate these losses and gains using certain heuristics. The social choice
theory and the prospect theory both provide exciting insights for us to better
understand and investigate GDM problems.
ConsensusisanimportantareaofresearchintheGDMandisdefinedasastate
of mutual agreement among individuals of a group, where all opinions have been
heardandaddressedtothesatisfactionofthegroup.Aconsensusreachingprocess
is a dynamic and an iterative process composed by several rounds where the
individuals express, discuss, and modify their opinions until to make a decision.
Integrating the consensus reaching process into the GDM offers some advantages:
(1) More effective implementation. When individuals’ opinions and concerns are
taken into account, they are more likely to actively participate in the implementa-
tion of the obtained solution, and (2) Building connection among the individuals.
Using consensus as a decision tool means taking the time to find unity on how to
proceed before moving forward, which promotes communication among
individuals.
Intheconsensusreachingprocess,individualsoftenneedtoadjusttheiropinions
to improve the consensus level among individuals. In this book, we propose a
challenge for analysts: how to minimize the adjustment amounts in the consensus
reaching process, which can be described as an optimization-based model.
Xn
min dðopk;opkÞ;
opk k¼1
where fop1;op2;...;opng are the individuals’ original opinions,
fop1;op2;...;opng are the individuals’ adjusted opinions with a consensus, and
dðopk;opkÞðk ¼1;2;...;nÞ is to measure the adjustment amounts associated with
the individual k. We investigate the optimization-based model to search the con-
sensuspathwithminimumadjustmentsunderdifferentGDMcontexts.Particularly,
inChap.2weclarifythebasicideaoftheconsensuswithminimumadjustments(or
cost), and investigate the consensus model with minimum adjustments (or cost)
under the utility preferences and aggregation functions. Then, in Chap. 3 we pro-
pose two consensus models for the GDM with preference relations: the
iteration-based consensus model and the LP-based consensus model. Next, we
investigate the consensus models with minimum adjustments under the 2-tuple
linguistic context and the hesitant linguistic context in Chap. 4. Subsequently, in
Chap. 5 we propose two consensus models for the GDM with heterogeneous
preference representation structures: the direct consensus model and the prospect
theory-based consensus model. Finally, Chap. 6 presents two multiple attribute
consensusruleswith minimumadjustments:thedistance-basedconsensus ruleand
the count-based consensus rule. Based on the distance-based and count-based
consensusrules,wedevelopaninteractiveconsensusreachingprocessformultiple
attribute group decision making.
Preface vii
We believe that the optimal adjusted opinions, obtained by the methodology
presentedin thebook, can provide abetterdecision aidwhich individuals useas a
reference to modify their individual opinions.
We want to express special thanks to Professor Yinfeng Xu, Professor Yihua
Chen, Professor Zhi-Ping Fan, Professor Wei-Chiang Hong, Professor Enrique
Herrera-Viedma,andProfessorFranciscoHerrera, for theircontributions andgreat
support to this book. We also want to express our sincere thanks to the colleagues
and students in our group, Hengjie Zhang, Haiming Liang, Cong-Cong Li, Xia
Chen, Yuzhu Wu, Nan Luo, Guiqing Zhang, and Bowen Zhang, who have done
much workinthis field andmade anumberof corrections.This bookissupported
byGrants(Nos.70425005,71171160,71571124)fromNSFofChina,andaGrant
(No. skqx201308) from Sichuan University.
Chengdu¸ China Yucheng Dong
July 2015 Jiuping Xu
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Preference Representation Structure and Aggregation
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Preference Representation Structure . . . . . . . . . . . . . . . . 2
1.1.2 Aggregation Function. . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Consensus Reaching Process . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Literature Review Regarding Consensus . . . . . . . . . . . . . 6
1.2.2 General Consensus Framework . . . . . . . . . . . . . . . . . . . 7
1.2.3 The Core Problem in the Consensus Reaching Process . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Consensus with Utility Preferences . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Basic Idea of the Consensus with Minimum Adjustments . . . . . . 17
2.1.1 Consensus with Minimum Adjustments or Cost . . . . . . . . 17
2.1.2 Internal Aggregation Function . . . . . . . . . . . . . . . . . . . . 19
2.2 Consensus Under Aggregation Function . . . . . . . . . . . . . . . . . . 21
2.2.1 Minimum Cost Consensus Model. . . . . . . . . . . . . . . . . . 21
2.2.2 Maximum Expert Consensus Model . . . . . . . . . . . . . . . . 31
2.3 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Consensus Based on IR and DR Rules . . . . . . . . . . . . . . 42
2.3.2 Comparison Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Consensus with Preference Relations. . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Integrating Individual Consistency into Consensus . . . . . . . . . . . 49
3.2 Consensus with Multiplicative Preference Relations . . . . . . . . . . 50
3.2.1 Prioritization and Aggregation Methods. . . . . . . . . . . . . . 51
3.2.2 Consistency and Consensus in Multiplicative
Preference Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.3 Iteration-Based Consensus Model. . . . . . . . . . . . . . . . . . 54
3.3 Consensus with Additive Preference Relations . . . . . . . . . . . . . . 66
ix
x Contents
3.3.1 Consistency and Consensus in Additive Preference
Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.2 LP-Based Consensus Model. . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Consensus Under Linguistic Context. . . . . . . . . . . . . . . . . . . . . . . 77
4.1 Consensus Under the 2-tuple Linguistic Context. . . . . . . . . . . . . 77
4.1.1 Several Symbolic Linguistic Computational Models . . . . . 77
4.1.2 The Consensus Operator . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.3 Properties of the Operator . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Consensus Under Hesitant Linguistic Context . . . . . . . . . . . . . . 95
4.2.1 Hesitant Consensus Problem . . . . . . . . . . . . . . . . . . . . . 95
4.2.2 Hesitant Consensus Measure . . . . . . . . . . . . . . . . . . . . . 97
4.2.3 Minimizing the Adjusted Simple Terms . . . . . . . . . . . . . 98
4.2.4 Properties of the Hesitant Model . . . . . . . . . . . . . . . . . . 118
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5 Consensus with Heterogeneous Preference Representation
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1 Direct Consensus Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1.1 Direct Consensus Framework. . . . . . . . . . . . . . . . . . . . . 127
5.1.2 Direct Selection Process . . . . . . . . . . . . . . . . . . . . . . . . 130
5.1.3 Direct Consensus Process . . . . . . . . . . . . . . . . . . . . . . . 134
5.1.4 Properties of the Direct Model. . . . . . . . . . . . . . . . . . . . 143
5.2 Prospect Theory Based Consensus Model . . . . . . . . . . . . . . . . . 147
5.2.1 Prospect Theory and Preference-Approval Structures. . . . . 147
5.2.2 Prospect Theory Based Consensus Framework. . . . . . . . . 149
5.2.3 Selection Process with Reference Points . . . . . . . . . . . . . 150
5.2.4 Consensus Process with Reference Points . . . . . . . . . . . . 153
5.2.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3 Consensus with Minimum Adjustments Under Prospect
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.3.1 Minimum Adjustments with Reference Points . . . . . . . . . 166
5.3.2 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 168
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6 Consensus in Multiple Attribute Decision Making . . . . . . . . . . . . . 173
6.1 Consensus Problem with Multiple Attributes . . . . . . . . . . . . . . . 173
6.2 Multiple Attribute Consensus Rules . . . . . . . . . . . . . . . . . . . . . 176
6.2.1 Distance-Based Consensus Rule. . . . . . . . . . . . . . . . . . . 176
6.2.2 Count-Based Consensus Rule. . . . . . . . . . . . . . . . . . . . . 180
6.3 Multiple Attribute Consensus Reaching Process . . . . . . . . . . . . . 183
6.3.1 The Interactive Consensus Reaching Process . . . . . . . . . . 183
6.3.2 Convergence Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 185
Contents xi
6.3.3 Mixing Use of Multiple Attribute Consensus Rules . . . . . 190
6.4 Numerical and Comparison Analysis. . . . . . . . . . . . . . . . . . . . . 191
6.4.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.4.2 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200