Table Of ContentStudies in Fuzziness and Soft Computing
Deng-Feng Li
Decision and
Game Theory
in Management
with Intuitionistic
Fuzzy Sets
Studies in Fuzziness and Soft Computing
Volume 308
Series Editor
J. Kacprzyk, Warsaw, Poland
For furthervolumes:
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Deng-Feng Li
Decision and Game Theory
in Management with
Intuitionistic Fuzzy Sets
123
Deng-FengLi
Fuzhou
People’s Republic ofChina
ISSN 1434-9922 ISSN 1860-0808 (electronic)
ISBN 978-3-642-40711-6 ISBN 978-3-642-40712-3 (eBook)
DOI 10.1007/978-3-642-40712-3
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To my wife, Wei Fei
and to my son, Wei-Long Li
Foreword
Fuzzydecisionandgametheoryhavebeenextensivelystudiedandachievedgreat
success in the last two or three decades. The fuzzy set theory introduced by
L. A. Zadeh in 1965 is an important tool to deal with fuzziness appearing in
management decision and game problems. The concept of fuzzy sets is a cor-
nerstoneofthefuzzysettheory.Zadehusedsingleindex(i.e.,membershipdegree
or function) to define the fuzzy set. Namely, the single membership degree (or
function) was used to express two opposite aspects (or states) offuzziness, fuzzy
concept,and/orfuzzyphenomenasimultaneously.Inotherwords,Zadehusedonly
a membership function, which assigns to each element x of the universe of dis-
course a real number lðxÞ from the unit interval [0, 1] to indicate the degree of
belongingness to the fuzzy set under consideration. The degree of nonbelong-
ingness is just automatically equal to 1(cid:2)lðxÞ. Thus, the fuzzy set is not able to
express the neutral state, i.e., neither supporting nor opposing. That is to say, the
fuzzysetisonlyabletodescribefuzzinessof‘‘thisandalsothat.’’Thelimitofthe
concept offuzzy sets brings forward new research topics and challenges in com-
plexmanagementsituations.Inreality,however,ahumanbeingwhoexpressesthe
membership degree of a given element in a fuzzy set very often does not express
corresponding nonmembership degree as the complement to 1. As a result,
K. T. Atanassov in 1983 introduced the concept of an intuitionistic fuzzy set,
which is characterized by two functions expressing the degree of belongingness
and the degree of nonbelongingness, respectively. The intuitionistic fuzzy set has
two indices (i.e., membership and nonmembership degrees or functions), which
can be used to express three states of fuzzy concepts and/or fuzzy phenomena:
support, opposition, and neutrality. Namely, the intuitionistic fuzzy set is able to
describe fuzziness of ‘‘neither this nor that.’’ So far as I know, however, there is
less investigation on decision and games with intuitionistic fuzzy sets.
Decisionandgameswithintuitionisticfuzzysetsareremarkablydifferentfrom
fuzzy decision and games. The reason is that the former uses the intuitionistic
fuzzy set with two indices to describe fuzziness in actual decision and game
problems, whereas the latter uses the fuzzy set with single index to handle fuzz-
iness. On the other hand, in decision and games with intuitionistic fuzzy sets the
mutual conflicting two indices (i.e., the vector consisting of membership and
nonmembership degrees or functions) have to be compared, whereas in fuzzy
decision and games only single index (i.e., membership degree or function) is
vii
viii Foreword
compared. As a result, the models and methods of fuzzy decision and games
cannot be straightforwardly extended to decision and games with intuitionistic
fuzzy sets. Therefore, we need to establish a new system of theories and methods
for decision andgames withintuitionistic fuzzysets,which naturally includes the
systemoftheoriesandmethodsforfuzzydecisionandgames.Inotherwords,the
latterisaspecialcaseoftheformer.Thus,wemaybelievethatdecisionandgames
with intuitionistic fuzzy sets are an extension and a perfection offuzzy decision
and games.
ThisbookisanacademicmonographbasedontheSCIandEIcitingmorethan
50 papers published in international famous journals by myself and co-authors in
recent years. The intuitionistic fuzzy set is used to describe and expressfuzziness
appearing in real decision and game problems, especially hesitancy degree in
judgmentanddecisionprocessofdecisionmaker(s)orplayer(s).1Payingattention
to such a decision factor is one of the salient signs in which modern decision
science differs from Bayes statistic decision. This book includes two closely rel-
ativepartsofdecisionandgameswithintuitionisticfuzzysets.Itisdividedinto11
chapters. Chapter 1 mainly describes the concept of intuitionistic fuzzy sets and
operations, distances and similarity degrees between intuitionistic fuzzy sets,
representation theorems and extension principles of intuitionistic fuzzy sets,
definitions of trapezoidal intuitionistic fuzzy numbers and triangular intuitionistic
fuzzy numbers as well as algebraic operations. Chapter 2 mainly expatiates
the intuitionistic fuzzy weighted averaging operator, the intuitionistic fuzzy
ordered weighted averaging (OWA) operator, the intuitionistic fuzzy hybrid
weighted averaging operator, and the intuitionistic fuzzy generalized
hybrid weighted averaging operator as well as the intuitionistic fuzzy generalized
hybrid weighted averaging method of multiattribute decision-making with intui-
tionisticfuzzysets.Chapter 3isdevotedtothelinear weighted averagingmethod
and the linear programming method of multiattribute decision-making with both
weightsandattributeratingsexpressedbyintuitionisticfuzzysets,theTOPSISand
theoptimumseekingmethodformultiattributedecision-makingwithintuitionistic
fuzzy positive and negative ideal-solutions and weights known, and the LINMAP
for multiattribute decision-making with an intuitionistic fuzzy positive ideal-
solution and weights unknown as well as the fraction mathematical programming
method and the linear programming method for intuitionistic fuzzy multiattribute
decision-making with intuitionistic fuzzy weights unknown. Chapter 4 mainly
discusses the interval-valued intuitionistic fuzzy continuous ordered weighted
averaging (COWA) operator, the interval-valued intuitionistic fuzzy generalized
hybrid weighted averaging operator, the interval-valued intuitionistic fuzzy con-
tinuous hybrid weighted averaging operator and the multiattribute decision-mak-
ing methods based on the interval-valued intuitionistic fuzzy generalized hybrid
1 Theterms‘‘decisionmaker’’and‘‘player’’havethesamemeaningsandmaybeinterchange-
ablyused.Customarily,however,theterm‘‘decisionmaker’’isusedindecisiontheorywhilethe
term‘‘player’’isusedingametheory.
Foreword ix
weighted averaging operator and the interval-valued intuitionistic fuzzy COWA
operator as well as the TOPSIS-based mathematical programming methods of
interval-valued intuitionistic fuzzy multiattribute decision-making with weights
unknown. Chapter 5 mainly proposes the concepts of weighted value and ambi-
guity of trapezoidal intuitionistic fuzzy numbers and hereby establishes the
weighted value and ambiguity-based ranking method of trapezoidal intuitionistic
fuzzy numbers and the multiattribute decision-making method with trapezoidal
intuitionisticfuzzynumbers.Chapter6isdevotedtotheTOPSISformultiattribute
group decision-making with intuitionistic fuzzy positive and negative ideal-solu-
tions and weights known and the LINMAP for multiattribute group decision-
making with an intuitionistic fuzzy positive ideal-solution and weights unknown.
Chapter 7 expatiates the concept of solutions of matrix games with payoffs of
intuitionistic fuzzy sets and their properties and existence, and hereby establishes
the linear and nonlinear programming methods of matrix games with payoffs of
intuitionistic fuzzy sets. Chapter 8 mainly establishes the multiobjective pro-
gramming models of matrix games with payoffs of interval-valued intuitionistic
fuzzy sets and the linear and nonlinear programming methods. Chapter 9 mainly
proposesthecut-set-basedmethod,theweightedmean-area-basedmethodandthe
weightedvalueandambiguity-basedlexicographicmethod formatrixgameswith
payoffs of trapezoidal intuitionistic fuzzy numbers. Chapter 10 mainly discusses
theconceptofsolutionsofmatrixgameswithgoalsofintuitionisticfuzzysetsand
hereby establishes auxiliary linear programming models and the linear program-
mingmethod.Chapter11isdevotedtotheconceptofsolutionsofbi-matrixgames
with payoffs of intuitionistic fuzzy sets and the bilinear programming method of
parametricbi-matrixgamesderivedfromthenewlyintroducedtotalorderrelation
on the basis of the defuzzification (or linear ranking) function of intuitionistic
fuzzysets.Theaimofthisbookistodevelopandestablishanewresearchfieldof
decisionandgamesinmanagementwithintuitionisticfuzzysets.Itriedmybestto
ensure that the models and methods developed in this book are of practicability,
maneuverability, and universality.
This book is addressed to people in theoretical researches and practical appli-
cationsfromdifferentfieldsanddisciplinessuchasdecisionscience,gametheory,
management science, fuzzy system theory, applied mathematics, optimizing
designofengineeringandindustrialsystem,expertsystem,andsocialeconomyas
well as artificial intelligence. Moreover, it is also addressed to teachers, post-
graduates, and doctors in colleges and universities in different disciplines or
majors: decision analysis, management, operation research, fuzzy mathematics,
systems engineering, project management, industrial engineering, applied mathe-
matics, hydrology and water resources, and so on.
First of all, I extraordinarily appreciate the selfless help and support of the
Academician (Chinese Academy of Engineering) and Prof. Zhong-Tuo Wang.
ThanksalsogotoProfessorsZhao-HanSheng,Guo-HongChen,Zhi-GangHuang,
Zu-PingZhu,andZhen-PengTang.Specialthanksareduetomydoctoralgraduate
andco-authorJiang-XiaNanforcompletingandpublishingseveralarticlesaswell
x Foreword
asmydoctoral graduatesFang-XuanHong,Jia-CaiLiu,Dian-QingYang,andJie
Yangforcheckingandvalidatingthecomputationalresultsinthefinalmanuscript.
This book was supported by the Key Program of National Natural Science
FoundationofChina(No.71231003),theNationalNaturalScienceFoundationof
China (Nos. 71171055, 71101033, and 71001015), the Program for New Century
Excellent Talents in University (the Ministry of Education of China, NCET-10-
0020) and the Specialized Research Fund for the Doctoral Program of Higher
Education of China (No. 20113514110009) as well as ‘‘Science and Technology
Innovation Team Cultivation Plan of Colleges and Universities in Fujian
Province’’.
Iwouldliketoacknowledgetheencouragementandsupportofmywifeaswell
as the understanding of my son.
Lastbutnotleast,Iwouldliketoacknowledgetheencouragementandsupport
of all my friends and colleagues.
Ultimately, I should claim that I am fully responsible for all errors and omis-
sions in this book.
December 18, 2012 Deng-Feng Li
Contents
1 Intuitionistic Fuzzy Set Theories. . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Intuitionistic Fuzzy Sets and Operations . . . . . . . . . . . . . . . . 2
1.2.1 Concepts of Intuitionistic Fuzzy Sets
and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Operations Over Intuitionistic Fuzzy Sets . . . . . . . . . 9
1.2.3 Concepts of Cut Sets for Intuitionistic Fuzzy
Sets and Properties. . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Distances and Similarity Degrees Between Intuitionistic
Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Definition of Similarity Degrees Between
Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Definition of Distances Between Intuitionistic
Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Representation Theorems of Intuitionistic Fuzzy Sets. . . . . . . 24
1.5 Extension Principles of Intuitionistic Fuzzy Sets
and Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.1 Extension Principles of Intuitionistic Fuzzy Sets . . . . 28
1.5.2 Algebraic Operations over Intuitionistic
Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Definitions of Intuitionistic Fuzzy Numbers
and Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.1 Trapezoidal Intuitionistic Fuzzy Numbers
and Algebraic Operations . . . . . . . . . . . . . . . . . . . . 36
1.6.2 Triangular Intuitionistic Fuzzy Numbers
and Algebraic Operations . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Intuitionistic Fuzzy Aggregation Operators and Multiattribute
Decision-Making Methods with Intuitionistic Fuzzy Sets . . . . . . . 47
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Intuitionistic Fuzzy Aggregation Operators and Properties. . . . 48
2.2.1 The Intuitionistic Fuzzy Weighted
Averaging Operator . . . . . . . . . . . . . . . . . . . . . . . . 48
xi
