Applying consistent fuzzy preference relations to partnership selection.docx

资源描述

Applying consistent fuzzy preference relations to partnership selection.docx

《Applying consistent fuzzy preference relations to partnership selection.docx》由会员分享，可在线阅读，更多相关《Applying consistent fuzzy preference relations to partnership selection.docx（17页珍藏版）》请在冰豆网上搜索。

Applying consistent fuzzy preference relations to partnership selection.docx

Applyingconsistentfuzzypreferencerelationstopartnershipselection

Tien-ChinWanga,1,

andYueh-HsiangChenb,c,

aDepartmentofInformationManagement,I-ShouUniversity,Kaohsiung840,Taiwan

bDepartmentofInformationEngineering,I-ShouUniversity,Kaohsiung840,Taiwan

cDepartmentofInformationManagement,Kao-YuanUniversityofTechnology,Kaohsiung821,Taiwan

Received28January2005;

accepted19July2005.

Availableonline9September2005.

Abstract

Partnershipselectionhasbeenimportanttotheformationofavirtualenterprise.Basedonthefuzzypreferenceprogramming（FPP）methodproposedbyMikhailov[Fuzzyanalyticalapproachtopartnershipselectioninformationofvirtualenterprises.Omega2002;30:

393–401],thisinvestigationpresentsaconsistentfuzzypreferencerelationsmethodtoselectpartners.Humanthoughtsarefullofuncertainty,sothedecision-makerscannotmakeexactpairwisecomparisons.TheFPPmethodsolvesthisproblemusinganintervalvalueinsteadofSaaty's1–9scale.Inthisstudy,theFPPmethodisreviewed,andthentheconsistentfuzzypreferencerelationsmethodiselucidated.Finally,thepresentedmethodisappliedtotheexampleaddressedbyMikhailov[Fuzzyanalyticalapproachtopartnershipselectioninformationofvirtualenterprises.Omega2002;30:

393–401].Thisstudyrevealsthattheproposedmethodyieldsconsistentdecisionrankingsfromonlyn-1pairwisecomparisons—thesamenumberasinMikhailov'sresearch.Briefly,thepresentedconsistentfuzzypreferencerelationsmethodisaneasyandpracticalwaytoproviderankingsofpartnershipinmakingdecision.

Keywords:

AHP;FPP;Consistency;Consistentfuzzypreferencerelations

ArticleOutline

Introduction

FPPmethod

Consistentfuzzypreferencerelations

Illustrativeexample

Conclusions

References

1.Introduction

AccordingtoMikhailov's[1]researchonpartnershipselection,allrelationalissuescanbeaddressedbytraditionalAHPmethod.However,thetraditionalAHPmethodisproblematicinthatitusesanexactvaluetoexpressthedecisionmaker'sopinioninacomparisonofalternatives.However,inreality,ahumanpreferencemodelisuncertain.Inthepartnershipselectionprocess,thedecisionmakerisgenerallyunsureofhispreferencesbecauseinformationaboutthefuturepartnersandtheirperformanceisincompleteanduncertain.Someofthedecisioncriteriaaresubjectiveandqualitative,sothedecisionmakercannoteasilyexpressthestrengthofhispreferencesorprovideexactpairwisecomparisons.MikhailovutilizestheintervalvaluestoexpressthecomparisonsanddevelopstheFuzzyPreferenceProgramming（FPP）methodtocalculatetheweightofeverylevel,whichiscalledalocalpriority,tosolvetheproblem.HethenusedtheAHPmethodtodeterminetheglobalprioritiesbyaggregatingallofthelocalpriorities.Finally,heappliedthemethodinanexampletorankthealternativesandselections.

TheFPPmethodprovidessomeadvantages,includingaconsistencyindicator,simplicityofcomputation,highprecisionandpreservationofranks.BasedontheFPPmethod,thispaperappliesthefuzzypreferencerelationsmethodproposedbyHerrera-Viedmaetal.[2]toselectthevirtualpartners.Themethodconstructsthedecisionmatricesofpairwisecomparisonsusinganadditivetransitivity.Onlyn-1comparisonsarerequiredtoensureconsistencyforalevelwithncriteria.Themethodissimplyandpracticallyprovidesrankingchoicesindecision-makingproblems.

Thispaperisorganizedasfollows.Section2reviewstheFPPmethod.Section3reviewstheconsistentfuzzypreferencerelations.Section4presentsanillustrativeexample,andSection5drawstheconclusions.

2.FPPmethod

MikhailovproposedtheFPPmethod[1],[2],[3]and[4]toderivepriorityvectorsfromasetofcrisporintervalcomparisons.Theassessmentoftheprioritiesisanoptimizationproblem,maximizingthedecision-maker'ssatisfactionwithaspecificcrisppriorityvector.

ConsidertheintervalpairwisecomparisonswithncriteriaatthesamelevelinahierarchyX1,X2,…,Xn.aij=（lij,uij）isassumedtorepresentthedecision-maker'sintervaljudgmentsofcriteriaXitoXj.Whentheintervaljudgmentsareconsistent,severalpriorityvectors,whosecomponentratiossatisfytheinequalitieslij

wi/wj

uij,i=1,2,…,n-1,j=2,3,…,n,j>i.Nopriorityvectorsimultaneouslysatisfiesallintervaljudgmentsintheinconsistentcase.However,avectorthatsatisfiesalljudgments“aswellaspossible”canbereasonablysought,indicatingthatasufficientlygoodsolutionvectormustapproximatelybeconsistentwithalljudgments,or

（1）

where

denotesthestatement“fuzzylessorequalto”.

Theinequality

（1）canbetransformedintoasetoftwosingle-sidefuzzyconstraintstohandleeasily,

（2）

Theabovesetofn×（n-1）fuzzyconstraintscanbepresentedinamatrixformas,

（3）

wherethematrix

m=n×（n-1）.

Thekthrowof

（2）,forwhich

k=1,2,…,m,representsafuzzylinearconstraintandisdefinedbyalinearmembershipfunctionofthetype

（4）

wheredkisatoleranceparameter,whichrepresentstheadmissibleintervalofapproximatesatisfactionofthecrispinequalityRkw

0.Themembershipfunction（4）representsthedecision-maker'ssatisfactionwithaspecificpriorityvector,withrespecttothekthsingle-sideconstraint

（2）.Thevalueofthemembershipfunctionμk（Rkw）iszerowhenthecorrespondingcrispconstraintRkw

0isstronglyviolated;itriseslinearlyandtakespositivevaluesoflessthanonewhentheconstraintisapproximatelysatisfied.Ittakesvaluesofgreaterthanonewhentheconstraintisfullysatisfied.

Letμk（Rkw）,k=1,2,…,mbethemembershipfunctionsofthefuzzyconstraints

onthe（n-1）dimensionalsimplex

Definition 2.1

Thefuzzyfeasiblearea

onthesimplexTn-1isafuzzyset,

（5）

Thefuzzyfeasibleareaisdefinedasanintersectionofallfuzzyconstraintsonthesimplex.Ifthetoleranceparametersoftheirfuzzysetsare“largeenough”,thenanon-emptyfuzzyfeasibleareacanbeobtained.Therefore,anon-emptyfeasiblearea

onthesimplexTn-1isaconvexfuzzyset.

Theconvexfuzzyfeasiblearea

representstheoverallsatisfactionofthedecision-makerwithaspecificcrisppriorityvector.Assumingthatthedecision-makerisinterestedinthebestpossiblesolution,apriorityvectorthatmaximizeshisoveralldegreeofsatisfactioncanbereasonablydetermined.

Definition 2.2

Themaximizingsolutionisavectorw*,whichcorrespondstothemaximumfuzzyfeasiblearea

（6）

Thefuzzyfeasiblearea

isaconvexsetandallfuzzyconstraintsaredefinedasconvexsets,soatleastonepointw*isalwayspresentonthesimplexthathasamaximumdegreeofmembershipin

Theproblemoffindingthemaximizingsolutionistransformedtoalinearprogrambyintroducingavariableλ;measuringthedegreeofmembershipofthefuzzyfeasiblearea

andusingEqs.（4）and（6）;

（7）

TheoptimalsolutiontoEq.（7）isavector（w*,λ*）whosefirstelementrepresentsthepriorityvectorthathasamaximumdegreeofmembershipinthefuzzyfeasiblearea,whilethesecondelementrepresentsthevalueofthatmaximumdegree,

.Thevalueofλ*representsthedegreeofsatisfactionandisanaturalindicatoroftheinconsistencyofthedecision-maker'sjudgments,andsocanberegardedasaconsistencyindex.Whentheintervaljudgmentsareconsistent,λ*isone.Forinconsistentjudgments,λ*isbetweenoneandzero,dependingonthedegreeoftheinconsistencyandthetoleranceparametersdk.

3.Consistentfuzzypreferencerelations

Herrera-Viedmaetal.proposedtheconsistentfuzzypreferencerelations[2]forconstructingthedecisionmatricesofpairwisecomparisonsbasedonadditivetransitivity.Fuzzypreferencerelationsenableadecision-makertogivevaluesforasetofcriteriaandasetofalternatives.Thevaluerepresentsthedegreeofthepreferenceforthefirstalternativeoverthesecondalternative.Twomajorkindsofpreferencerelationsapply—

（1）multiplicativepreferencerelations,and

（2）fuzzypreferencerelations.

（1）Multiplicativepreferencerelations[5]:

AmultiplicativepreferencerelationRintermsofasetofalternativesAisrepresentedbyamatrixR

A×A,R=（rij）,whererijisthepreferenceratioofalternativeaitoaj.Saatysuggestsmeasuringrijusingaratioscale,andthedefined1–9scale.Herein,rij=1representstheabsenceofadifferencebetweenaiandaj;rij=9denotesthataiismaximallybetterthanaj.Inthiscase,thepreferencerelationRistypicallyassumedtobeamultiplicativereciprocal,

{1,…,n}.

（2）Fuzzypreferencerelations[6],[7]and[8]:

AfuzzypreferencerelationPonasetofalternativesAisafuzzysetontheproductsetA×Awithmembershipfunctionμp:

A×A→[0,1].Thepreferencerelationisrepresentedbythen×nmatrixP=（pij）,where

.Herein,pijisthepreferenceratioofalternativeaito

meansthatnodifferenceexitsbetweenaiandaj,pij=1indicatesthataiisabsolutelybetterthanaj,and

indicatesthataiisbetterthanaj.Inthiscase,thepreferencematrixPisgenerallyassumedtobeanadditivereciprocal,pij+pji=1,

i,j

{1,…,n}.

Herrera-Viedmaetal.[2]proposedaconsistentadditivepreferencerelation.Thesepropositionsaredescribedasfollows.

Proposition 3.1

Considerasetofalternatives,X={x1,…,xn},associatedwithareciprocalmultiplicativepreferencerelationA=（aij）foraij

[1/9,9].Then,thecorrespondingreciprocalfuzzypreferencerelation,P=（pij）withpij

[0,1]associatedwithAisgivenas

isconsideredbecauseaijisbetween

展开阅读全文