decision making

Amulti-criteriainterval-valuedintuitionisticfuzzygroupdecisionmaking

withChoquetintegral-badTOPSIS

ChunqiaoTan

SchoolofBusiness,CentralSouthUniversity,Changsha410083,China

articleinfo

Keywords:

Multi-criteriagroupdecisionmaking

Interval-valuedintuitionisticfuzzyts

Fuzzymeasures

Geometricaggregationoperator

Choquetintegral

TOPSIS

abstract

AnextensionofTOPSIS,amulti-criteriainterval-valuedintuitionisticfuzzydecisionmakingtechnique,to

agroupdecisionenvironmentisinvestigated,whereinter-dependentorinteractivecharacteristics

broadviewofthetech-

niquesud,first,someoperationallawsoninterval-valuedintuitionisticfuzzyvaluesareintroduced.

Badontheoperationallaws,ageneralizedinterval-valuedintuitionisticfuzzygeometricaggregation

operatorispropodwhichisudtoaggregatedecisionmakers’opinionsingroupdecisionmakingpro-

tion,oquetintegral-badHammingdistance

betweeninterval-valuedintuitionisticfuzzyvaluesisdefiingtheinterval-valuedintuitionis-

ticfuzzygeometricaggregationoperatorwithChoquetintegral-badHammingdistance,anextensionof

TOPSISmethodisdevelopedtodealwithamulti-criteriainterval-valuedintuitionisticfuzzygroupdeci-

y,anillustrativeexampleisudtoillustratethedevelopedprocedures.

Óhtsrerved.

uction

TOPSIS(TechniqueforOrderPreferencebySimilaritytoIdeal

Solution),developedbyHwangandYoon(1981),isaclassicalap-

proachtomulti-attributeormulti-criteriadecisionmaking

(MADM/MCDM)racticalandufultechnique

forrankingandlectionofanumberofexternallydetermined

icprincipleisthat

thechonalternativeshouldhavetheshortestdistancefromthe

positive-idealsolutionandthefarthestdistancefromthenegative-

xistsalargeamountofliteratureinvolving

OPSIS,theperformance

ratingsandtheweightsofthecriteriaaregivenascrispvalues.

Undermanyconditions,crispvaluesareinadequatetomodel

real-worldsituationsbecauhumanjudgmentandpreference

areoftenambiguousandcannotbeestimatedwithexactnumeri-

lvetheambiguityfrequentlyarisingininforma-

tionfromhumanjudgmentandpreference,fuzzyttheory

(Zadeh,1965)hasbeensuccessfullyudtohandleimprecision

(oruncertainty)uzzynumbers

wereappliedtoestablishaprototypefuzzyTOPSIS(Chen&Hwang,

1992;Negi,1989),manyworksonfuzzyTOPSIShavebeeninves-

tigated(Chen,2000;Chu&Lin,2009;Jahanshahloo,Hosinzadeh

Lotfi,&Izadikhah,2006;Kuo,Tzeng,&Huang,2007;Mahdavi,

Mahdavi-Amiri,Heidarzade,&Nourifar,2008;Wang&Chang,

2007;Wang&Elhag,2006;Wang&Lee,2007,2009;Yeh&Deng,

2004;Yeh,Deng,&Chang,2000).AsanextensionofZadeh’s

fuzzytwhobasiccomponentisonlyamembershipfunction,

Atanassov(1986)introducedtheintuitionisticfuzzyts(IFS),

characterizedbyamembershipfunctionandanon-membership

function,Accordingly,IFShasbeenproventobeaverysuitabletool

tobeudtodescribetheimprecioruncertaindecisioninforma-

workhasbeendonetodevelopandenrichtheIFS

theory(Atanassov,1999;Bustince,Herrera,&Montero,2007).As

ageneralizationofthefuzzyts,IFShasreceivedmoreandmore

attentionandhasbeenappliedtothefi

fuzzyTOPSIShasbeenextendedtoIFS(Ashtiani,Haghighirad,

Makui,&Montazer,2009;Boran,Gen,Kurt,&Akay,2009;Chen

&Tsao,2008;Li,Wang,Liu,&Shan,2008).Later,Atanassovand

Gargov(1989)introducedtheconceptofinterval-valuedintuition-

isticfuzzyts(IVIFS)

fundamentalcharacteristicoftheIVIFSisthatthevaluesofits

membershipfunctionandnon-membershipfunctionareintervals

sov(1994)definedsomeopera-

ly,TanandZhang(2006)prented

anovelmethodformultipleattributedecisionmakingbadon

(2007)

developedsomegeometricaggregationoperators,suchasthe

interval-valuedintuitionisticfuzzyweightedgeometricaveraging

(IIFWGA)operatorandtheinterval-valuedintuitionisticfuzzyor-

deredweightedgeometricaveraging(IIFOWGA)operatorandgave

anapplicationoftheIIFWGAandIIFOWGAoperatorstomultiple

attributegroupdecisionmakingwithinterval-valuedintuitionistic

(2009)appliedIIFWGAaggregationfunc-

tionstodealingwithdynamicmultipleattributedecisionmaking

0957-4174/$-efrontmatterÓhtsrerved.

doi:10.1016/.2010.08.092

E-mailaddress:chunqiaot@

ExpertSystemswithApplications38(2011)3023–3033

ContentslistsavailableatScienceDirect

ExpertSystemswithApplications

journalhomepage:/locate/eswa

wherealltheattributevaluesareexpresdinintuitionisticfuzzy

numbersorinterval-valuedintuitionisticfuzzynumbers.

However,theaggregationprocessarebadontheassump-

tionthatthecriteria(attribute)orpreferencesofdecisionmakers

areindependent,andtheaggregationoperatorsarelinearopera-

torsbadonadditivemeasures,whichischaracterizedbyan

independenceaxiom(Keeney&Raiffa,1976;Wakker,1999).For

realdecisionmakingproblems,thereisaphenomenonthatthere

existssomedegreeofinter-dependentorinteractivecharacteris-

ticsbetweencriteria(Grabisch,1995;Grabisch,Murofushi,&

Sugeno,2000).Andforadecisionproblem,decisionmakersinvited

usuallycomefromsameorsimilarfivesimilarknowl-

edge,onmakers’subjective

ndencephenomena

amongthecriteriaandmutualpreferentialindependenceof

1974,Sugeno(1974)introduced

theconceptofnon-additivemeasure(fuzzymeasure),whichonly

ost

effectivetooltomodelinginteractionphenomena(Grabisch,

1996;Ishii&Sugeno,1985;Kojadinovic,2002;Roubens,1996)

anddealwithdecisionproblems(Grabisch,1995,1997;Grabisch

etal.,2000;Onisawa,Sugeno,Nishiwaki,Kawai,&Harima,

1986).Areviewonanalyzingdecisionmakerbehaviorusingfuzzy

measuretheorycanbeeninLiginlalandOw(2006).Ingroup

decisionmakingproblems,aggregationofdecisionmakers’opin-

ionsisveryimportanttoappropriatelyperformevaluationprocess.

Toovercomethislimitationofaboveaggregationoperator,inthis

paper,badonfuzzymeasurewefirstshalldevelopageneralized

interval-valuedintuitionisticfuzzygeometricaggregationoperator

foraggregatingallindividualdecisionmakers’opinionsunder

interval-valuedintuitionisticfuzzygroupdecisionmakingenvi-

ingthisoperatorwithTOPSISonChoquetinte-

gral-badHammingdistance,amulti-criteriainterval-valued

intuitionisticfuzzygroupdecisionmakingisinvestigated,where

interactionsphenomenaamongthedecisionmakingproblemare

considered.

Inordertodothis,thepaperisorganizedasfollows:InSection

2,ion3,weintroduceinterval-

valuedintuitionisticfuzzytandsomeoperationallawsoninter-

val-valuedintuitionisticfuzzyvalues,InSection4,badonthe

operationallaws,ageneralizedinterval-valuedintuitionisticfuzzy

geometricaggregationoperatorispropod,andsomeofitsprop-

ion5,accordingtodefinitionof

Choquetintegral,weinvokethewell-knownHammingdistance

todefinetheChoquetintegral-badHammingdistancebetween

ingthe

generalizedinterval-valuedintuitionisticfuzzygeometricaggrega-

tionoperatorwithChoquetintegral-badHammingdistance,an

extensionofTOPSISisdevelopedtodealwithamulti-criteriainter-

val-valuedintuitionisticfuzzygroupdecisionmakingproblems

whereinter-dependentorinteractivecharacteristicsamongcrite-

Section6,anexampleisgiventoillustratetheconcreteapplication

ofthemethodandtodemonstrateitsfeasibilityandpracticality.

ConclusionsaremadeinSection7.

easure

Fortraditionaladditiveaggregationoperators,suchasthe

weightedarithmeticmeanorOWA(Yager,1988)operator,each

criteriai2N(Ndenotesacriteriat)isgivenaweightw

i

2[0,1]

reprentingtheimportanceofthiscriteriaduringthedecision

process,andthesunofallw

i

(i=1,2,...,n)

doesnotdefi

decisionproblems,sincethereareofteninter-dependentor

interactivephenomenaamongcriteria,theoverallimportanceofa

criterioni2Nisnotsolelydeterminedbyitlfi,butalsobyallother

criteriaT,ethatw(i)denotestheimportancedegreeof

i,wemayhavew(i)=0,suggestingthatelementisunimportant,

butitmayhappenthatformanysubtsT#N,w(T[i)ismuch

greaterthanw(T),suggestingthatiisactuallyanimportantele-

1974,Sugeno(1974)introducedthecon-

ceptoffuzzymeasure(non-additivemeasure),whichonlymake

ldecision

makingproblems,fuzzymeasuredefineaweightonnotonlyeach

criteriabutalsoeachcombinationofcriteria,andthesunofevery

w

i

(i=1,2,...,n)isudasapowerful

toolfordescribingtheinteractionamongthecriteriainat.

Defi={x

1

,x

2

,...,x

n

}beauniverofdiscour,P(X)

measureonXisatfunction

l:P(X)?[0,1],satisfyingthefollowingconditions:

(1)l(/)=0,l(X)=1.

(2)IfA,B2P(X)andA#Bthenl(A)6l(B).

IftheuniversaltXisinfinite,itisnecessarytoaddanextra

axiomofcontinuity(Wang&Klir,1992).However,inactualprac-

tice,itinoughtoconsiderthefiniteuniversalt.l(S)canbe

viewedasthegradeofsubjectiveimportanceofdecisioncriteria

,inadditiontotheusualweightsoncriteriatakenp-

arately,weightsonanycombinationofcriteriaarealsodefined.

Thismakespossiblethereprentationofinteractionbetweencri-

j

={x

j

,x

j+1

,...,x

n

}(16j6n)erac-

tionamongthecriteriainE

j

canbedescribedbyemployingl(E

j

)to

expressthedegreeofimportanceofE

j

.Thatis,thedegreeofimpor-

tanceofE

j

ivaluatedbysimultaneouslyconsideringx

j

,x

j+1

,...,

andx

n

.Hence,lcanbecalledanimportancemeasure(Wang,

Wang,&Klir,1998),andl(E

j

)canbealsoemployedtoexpress

thediscriminatorypowerofE

j

.Intuitively,wecouldsaythefollow-

ingaboutanyapairofcriteriatsA,B2P(X),AB=/:AandBare

consideredtobewithoutinteraction(ortobeindependent)if

l(A[B)=l(A)+l(B),

Bexhibitapositivesynergeticinteractionbetweenthem(orare

complementary)ifl(A[B)>l(A)+l(B),whichiscalledasuper-

xhibitanegativesynergeticinteraction

betweenthem(orareredundantorsubstitutive)if

l(A[B)

Inordertodeterminesuchfuzzymeasure,wegenerallyneedto

find2nÀ2valuesforncriteria,onlyvaluesl(/)andl(X)areal-

wayqualto0and1,valuationmodelob-

tainedbecomesquitecomplex,andthestructureisdifficultto

dtheproblemswithcomputationalcomplexityand

practicalestimations,k-fuzzymeasureg,aspecialkindoffuzzy

measure,waspropodbySugeno(1974),whichsatisfiesthefol-

lowingadditionalproperty:

gðA[BÞ¼gðAÞþgðBÞþkgðAÞgðBÞ;ð1Þ

wherek>À1forallA,B2P(X)andAB=/.However,thereare

-

stance,linearmethods(Marichal&Roubens,1998),quadratic

methods(Grabisch,1996;Grabisch&Nicolas,1994),heuristic-

badmethods(Grabisch,1995)andgeneticalgorithms(Wang

etal.,1998)areavailableintheliterature.

InEq.(1),k=0indicatesthatthek-fuzzymeasuregisadditive

measure.k–0indicatesthatthek-fuzzymeasuregisnon-additive

>0,then

g(A[B)>g(A)+g(B),whichimpliesthatgisasuper-additivemea-

<0,theng(A[B)

meterktheinteractionbetweencri-

teriacanbereprented.

/ExpertSystemswithApplications38(2011)3023–3033

IfXisafinitet,then[n

i¼1

x

i

¼-fuzzymeasuregsatis-

fiesfollowingEq.(2)

gðXÞ¼gð[

n

i¼1

x

i

Þ¼

1

k

Qn

i¼1

½1þkgðx

i

Þ