Information Measures for Hesitant Fuzzy Sets and Their Extensions

129 Pages·2019·1.678 MB·English

by  Bahram Farhadinia, Zeshui Xu

Uncertainty and Operations Research
Bahram Farhadinia
Zeshui Xu
Information
Measures for
Hesitant Fuzzy
Sets and Their
Extensions
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
Bahram Farhadinia Zeshui Xu
(cid:129)
Information Measures
for Hesitant Fuzzy Sets
and Their Extensions
123
Bahram Farhadinia ZeshuiXu
Quchan University of Technology Business School
Quchan,Iran SichuanUniversity
Chengdu,Sichuan, China
ISSN 2195-996X ISSN 2195-9978 (electronic)
Uncertainty andOperationsResearch
ISBN978-981-13-3728-4 ISBN978-981-13-3729-1 (eBook)
https://doi.org/10.1007/978-981-13-3729-1
LibraryofCongressControlNumber:2018964040
©SpringerNatureSingaporePteLtd.2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
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
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,
Singapore
Preface
Infuzzyset(FS)theory,whichisageneralizationofclassicalsettheoryintroduced
by Zadeh [1], distance, similarity and entropy measures have drawn the attention
ofmanyresearcherswhohavestudiedthesethreeconceptsfromdifferentpointsof
view. For instance, the similarity and distance measures have been widely applied
in many fields such as multiple criteria decision-making [2, 3], group decision-
making [4, 5], grey relational analysis [6], pattern recognition [7, 8], image pro-
cessing[9],andclusteranalysis[10].Sincethesimilarityanddistancemeasuresof
FSs and their extensions [3, 11] have been applied to many real-world situations,
theentropymeasure[12–14]canbenaturallyappliedtosuchfieldsduetoitsclose
relationship with the similarity measure and the distance measure. In recent years,
Torra[15]introducedtheconceptofhesitantfuzzyset(HFS)asanextensionofFS
in which the membership degree of a given element, called the hesitant fuzzy
element(HFE),isdefinedasasetofpossiblevalues.Forinstance,suchasituation
can be found in group decision-making problems. To clarify the necessity of
introducingHFSs,weconsiderasituationinwhichtwodecision-makersdiscussthe
membershipdegreeofanelementxtoasetA.Onewantstoassign0.2,buttheother
0.4.Accordingly,thedifficultyofestablishingacommonmembershipdegreeisnot
becausethereisamarginoferror(asinintuitionisticfuzzysets),orsomepossibility
distribution values (as in type-2 fuzzy sets), but because there is a set of possible
values.
In this book, we give a thorough and systematic introduction to the main
researchresultsinthefieldofinformationmeasuresforHFSsincludingthedistance
measures,thesimilaritymeasures,andtheentropymeasures.Weorganizethisbook
into four chapters that deal with three different but related issues, which are listed
below:
Chapter 1 provides the readers with further background on the HFSs and their
extensions.Wefirstintroduce theHFSasthegeneralizedformofFS,andthenthe
basicoperationallawstogetherwiththedesirablepropertiesofHFSsaregiven.The
otherpartofthischapterdealswiththemainextensionsofHFSswhichareknown
as interval-valued hesitant fuzzy set (IVHFS), dual hesitant fuzzy set (DHFS),
higher-order hesitant fuzzy set (HOHFS), and hesitant fuzzy linguistic term set
v
vi Preface
(HFLTS). Moreover, this chapter reviews the required properties of HFSs/HFEs
and the extensions of HFSs which are used in the next discussions.
Chapter 2 mainly investigates the distance measures for hesitant fuzzy infor-
mation. Distance measures are fundamentally important in various fields such as
decision-making,marketprediction,andpatternrecognition.Themostwidelyused
distance measures for HFSs/HFEs are the Euclidean distance, the Hamming dis-
tance, and the Hausdorff metric. Moreover, a number of other extensions of the
latterdistancemeasureshavebeenintroducedforHFSs/HFEsinthischapter.Based
onthediscussedHFS/HFEdistancemeasureswhichcoverapproximatelythewhole
existing ones in this area, this chapter is trying to mention the advantages and
disadvantagesoftheformulas.Inadditiontothat,theexistingdistancemeasuresfor
HFSsandtheirextensions,includingIVHFSs,DHFSs,HOHFSs,andHFLTSs,are
comprehensively studied. Moreover, this chapter makes the readers more familiar
with some aspects in this regard.
Chapter 3 focuses on the similarity measures for hesitant fuzzy information
where on the basis of the relationship between the similarity measure and the
distance measure, one can get various formulas to obtain the desired measure. In
particular, on the systematic transformation of the distance measure into the simi-
laritymeasureforHFSs/HFEsandviceversa,wecanderivemoreformulasforthe
similarity measures of HFSs/HFEs.
Chapter 4 is devoted to the entropy measure as one of the main subjects of
multiplecriteriadecision-making(MCDM)modelswithhesitantfuzzyinformation.
Onthebasisofexistingworks,wecandividetheHFS/HFEentropymeasuresinto
twocategories:entropymeasuresderivedfromtheotherinformationmeasuresand
entropy measures which are based on axiomatic frameworks. In each category, we
are going to highlight the advantages and disadvantages offormulas to get a good
picture of effectiveness of the proposed entropy measures. Moreover, to enhance
the information in this report, different formulas of entropy measures for IVHFSs
and DHFSs are presented in the sequel. This chapter also provides comprehensive
discussions on the structure of entropy measures for HFLTSs, from three view-
points: distance-based entropy measures for HFLTSs, similarity-based entropy
measures for HFLTSs, and entropy-based entropy measures for HFLTSs.
Needless to say that this book can be used as a reference for researchers and
practitioners working in the fields of fuzzy mathematics, operations research,
informationscience,managementscience,engineering,etc.Itcanalsobeusedasa
textbook for postgraduate and senior-year undergraduate students. Moreover, this
book as a research-based investigation of the authors’ experience serves both the
recent research results and further research directions.
Quchan, Iran Bahram Farhadinia
Chengdu, China Zeshui Xu
September 2018
Preface vii
References
1. L.A.Zadeh,Fuzzysets.Inf.Comput.8,338–353(1965)
2. T.Y. Chen, C.H. Li, Determining objective weights with intuitionistic fuzzy entropy mea-
sures:acomparativeanalysis.Inf.Sci.180,4207–4222(2010)
3. E.Szmidt,J.Kacprzyk,Anewconceptofasimilaritymeasureforintuitionisticfuzzysetsand
itsuseingroupdecisionmaking,inModellingDecisionforArtificialIntelligence,LNAIvol.
3558,ed.byV.Torra,Y.Narukawa,S.Miyamoto(Springer,2005),pp.272–282
4. G.W. Wei, X.R. Wang, Some geometric aggregation operators based on interval-valued
intuitionisticfuzzysetsandtheirapplicationtogroupdecisionmaking,in2007International
ConferenceonComputationalIntelligenceandSecurity(2007),pp.495-499
5. Z.S.Xu,J.Chen,Approachtogroupdecisionmakingbasedoninterval-valuedintuitionistic
judgementmatrices.Sys.Eng.TheoryPract.27,126–133(2007)
6. G.W. Wei, G. Lan, Grey relational analysis method for interval-valued intuitionistic fuzzy
multipleattributedecisionmaking,inFifthInternationalConferenceonFuzzySystemsand
KnowledgeDiscovery(2008),pp.291–295
7. D.F.Li,C.T.Cheng,Newsimilaritymeasuresofintuitionisticfuzzysetsandapplicationto
patternrecognitions.PatternRecognit.Lett.23,221–225(2002)
8. H.B. Mitchell, On the Dengfeng-Chuntian similarity measure and its application to pattern
recognition.PatternRecognit.Lett.24,3101–3104(2003)
9. S.K.Pal,R.A.King,Imageenhancementusingsmoothingwithfuzzysets,IEEETrans.Syst.
ManCybern.11,495–501(1981)
10. J.Yao,M.Dash,Fuzzyclusteringandfuzzymodeling.FuzzySetsSyst.113,381–388(2000)
11. W.Zeng,P.Guo,Normalizeddistance,similaritymeasure,inclusionmeasureandentropyof
interval-valuedfuzzysetsandtheirrelationship.Inf.Sci.178,1334–1342(2008)
12. K.Atanassov,IntuitionisticFuzzySets,TheoryandApplications(Physica-Verlag,Heidelberg,
1999)
13. W.L. Hung, M.S. Yang, Fuzzy entropy on intuitionistic fuzzy sets. Int. J. Intell. Syst. 21,
443–451(2006)
14. A.DeLuca,S.Termini,Adefinitionofnonprobabilisticentropyinthesettingoffuzzysets
theory.Inf.Control20,301–312(1972)
15. V.Torra,Hesitantfuzzysets.Int.J.Intell.Syst.25,529–539(2010)
Contents
1 Hesitant Fuzzy Set and Its Extensions . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Classical Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fuzzy Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Hesitant Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Extensions of Hesitant Fuzzy Set. . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Interval-Valued Hesitant Fuzzy Set. . . . . . . . . . . . . . . . . . 16
1.4.2 Dual Hesitant Fuzzy Set. . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Higher Order of Hesitant Fuzzy Set . . . . . . . . . . . . . . . . . 22
1.4.4 Hesitant Fuzzy Linguistic Term Set . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Distance Measures for Hesitant Fuzzy Sets
and Their Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1 Distance Measures for Hesitant Fuzzy Sets . . . . . . . . . . . . . . . . . 32
2.2 Distance Measures for Interval-Valued Hesitant Fuzzy Sets. . . . . . 47
2.3 Distance Measures for Dual Hesitant Fuzzy Sets . . . . . . . . . . . . . 49
2.4 Distance Measures for Higher Order Hesitant Fuzzy Sets . . . . . . . 52
2.5 Distance Measures for Hesitant Fuzzy Linguistic Term Sets . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 Similarity Measures for Hesitant Fuzzy Sets and Their
Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 Similarity Measures for Hesitant Fuzzy Sets. . . . . . . . . . . . . . . . . 60
3.2 Similarity Measures for Interval-Valued Hesitant Fuzzy Sets. . . . . 61
3.3 Similarity Measures for Dual Hesitant Fuzzy Sets . . . . . . . . . . . . 63
3.4 Similarity Measures for Higher Order Hesitant Fuzzy Sets . . . . . . 64
3.5 Similarity Measures for Hesitant Fuzzy Linguistic Term Sets . . . . 65
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
ix
x Contents
4 Entropy Measures for Hesitant Fuzzy Sets
and Their Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Entropy Measures for Hesitant Fuzzy Sets . . . . . . . . . . . . . . . . . . 70
4.1.1 Entropy Measures Based on Information Measures . . . . . . 70
4.1.2 Entropy Measures Based on Distance Measures. . . . . . . . . 72
4.1.3 Entropy Measures Based on Similarity Measures. . . . . . . . 78
4.1.4 Entropy Measures Based on Hesitant Operations. . . . . . . . 80
4.1.5 Entropy Measures Based on Fuzziness
and Non-specificity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Entropy Measures for Interval-Valued Hesitant Fuzzy Sets . . . . . . 87
4.3 Entropy Measures for Dual Hesitant Fuzzy Sets. . . . . . . . . . . . . . 92
4.4 Entropy Measures for Hesitant Fuzzy Linguistic Term Sets. . . . . . 94
4.4.1 Distance-Based Entropy Measures for Hesitant Fuzzy
Linguist Term Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.2 Similarity-Based Entropy Measures for Hesitant Fuzzy
Linguist Term Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.3 Entropy-Based Entropy Measures for Hesitant Fuzzy
Linguist Term Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Application of Information Measures in Multiple Criteria
Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Application of Distance Measures in Multiple Criteria
Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Application of Similarity Measures in Multiple Criteria
Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Application of Entropy Measures in Multiple Criteria
Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119