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1,2,2(1.,210007;2.,210096):,[1.FacchinettiG,RicciRC,MuzzioliS.Noteonrankingfuzzytriangularnumbers.InternationalJournalofIntelligentSystems,1998,13:613622;2.,..,1999,19(4):37],,:.,,.:;;:C934:A:1000-5781(2003)01-0067-04PossibilitydegreemethodforrankingintervalnumbersanditsapplicationXUZe-shui1,2,DAQing-li2(1.InstituteofSciences,PLAUniversityofScienceandTechnology,Nanjing210007,China;2.SchoolofEconomicsandManagement,SoutheastUniversity,Nanjing210096,China)Abstract:Thispaperpresentsapossibilitydegreeformulaforthecomparisonbetweentwointervalnum2bers,andhasadetailedstudyonrelationshipsbetweentheformulaandtheonesproposedin[1.FacchinettiG,RicciRC,MuzzioliS.Noteonrankingfuzzytriangularnumbers.InternationalJournalofIntelligentSystems,1998,13:613-622;2.DAQing-li,LIUXin-wang.Intervalnumberslinearpro2gramminganditssatisfactorysolution.SystemsEngineeringTheory&Practice,1999,19(4):3-7].Theirdesirablepropertiessuchascomplementary,transitivity,etc.arestudied.Then,basedontheseformulas,apossibilitydegreemethodforrankingintervalnumbersisgiven.Finally,themethodisuti2lizedtorankthealternativesinuncertainmulti-attributedecisionmaking.Keywords:intervalnumber;possibilitydegree;ranking0,,[36].,.[13,711].[7],;[8,9],,;[1,2]18120032JOURNALOFSYSTEMSENGINEERINGVol.18No.1Feb.,2003:2001-03-01;:2002-10-28.:(79970093);-;.,,.,[1,2],.,,.1a=[-,+]={x|-x+},.,-=+,.1,b,p(b)=1b0b(1)b.2,b,=[-,+],b=[b-,b+],l=+--,lb=b+-b-,p(b)=min{l+lb,mx(+-b-,0)}l+lb(2)b.,bpb.[1,2]:3[1]p(b)=minmx+-b-l+lb,0,1(3)b.4[2]p(b)=mx{0,l+lb-mx(b+--,0)}l+lb(4)b.3,:1=[-,+],b=[b-,b+],(1)0p(b)1.(2)p(b)=1b+-.(3)p(b)=0+b-.(4)()p(b)+p(b)=1,,p()=12.(5)p(b)12++-b++b-.,p(b)=12++-=b++b-.(6)()3,b,c,p(b)12p(bc)12,p(c)12.24.224,(2)Z(3)Z(4).(2)Z(3).(2)p(b)=min{l+lb,mx(+-b-,0)}l+lb=minl+lbl+lb,mx(+-b-,0)l+lb=min1,mx+-b-l+lb,0p(b)=minmx+-b-l+lb,0,1(2)Z(3).(3)Z(4).(3),(2)Z(3),p(b)=1-p(b)=1-minmx+-b-l+lb,0,1=1-min{l+lb,mx(+-b-,0)}l+lb=l+lb-min{l+lb,mx(+-b-,0)}l+lb=mx{0,l+lb-mx(+-b-,0)}l+lbp(b)=mx{0,l+lb-mx(+-b-,0)}l+lb:(3)Z(4).,.N={1,2,,n}.86181i=[-i,+i],iN,.p(ij),pij,i,jN,P=(pij)nn..,,.1:P,,[12]:i=1n(n-1)nj=1pij+n2-1,iN(6)P=(1,2,,n)T,i(iN)i(iN)..2.,3(1,2,3).5A,B,C,D,E,w=(0.4,0.4,0.2)T..[4]1:1ABCDE1[0.214,0.220][0.206,0.225][0.195,0.204][0.181,0.190][0.175,0.184]2[0.166,0.178][0.220,0.229][0.192,0.198][0.195,0.205][0.193,0.201]3[0.184,0.190][0.182,0.191][0.220,0.231][0.185,0.195][0.201,0.211]zj=3i=1rijwi(,rijji),A,B,C,D,E:z1=[0.1888,0.1972],z2=[0.2068,0.2198],z3=[0.1988,0.2070]z4=[0.1874,0.1970],z5=[0.1874,0.1962],(2)zj(j=1,2,3,4,5):P=0.5000.54440.569810.50.99061110.00940.5110.4556000.50.52170.4302000.47830.5(6)P:=(0.1557,0.2995,0.2505,0.1489,0.1454)TP,zj(j=1,,5)z20.9906z31z10.5444z4015217z59p,5A,B,C,D,EB9019906C91A9015444D9015217EB.:[1]FacchinettiG,RicciRG,MuzzioliS.Noteonrankingfuzzytriangularnumbers[J].InternationalJournalofIntelligentSystems,1998,13:613622[2],.[J].,1999,19(4):37[3]YoonK.Thepropagationoferrorsinmultiple-attributedecisionanalysis:Apracticalapproach[J].JournaloftheOperationalRe29620032:searchSociety,1989,40:681686[4]BrysonN,MobolurinA.Anactionlearningevaluationprocedureformultiplecriteriadecisionmakingproblems[J].EuropeanJournalofOperationalResearch,1996,96:379386[5]SajjadM.Incorporatingtheuncertaintyofdecisionjudgementsintheanalytichierarchyprocess[J].EuropeanJournalofOperationalResearch,1991,53:206216[6].[J].,2002,17(2):177181[7],,.[J].,1999,19(5):129133[8]KunduS.Min-transitivityoffuzzyleftnessrelationshipanditsapplicationtodecisionmaking[J].FuzzySetsandSystems,1997,86:357367[9]SenguptaA,PalTK.Oncomparingintervalnumbers[J].EuropeanJournalofOperationalResearch,2000,127:2843[10].[A].SystemsEngineering,SystemsScienceandComplexityResearch[C].Re2searchInformationLtd,HemelHempsteadHp27TD,UnitedKingdom:2000.507511[11]XuZS,DaQL.TheuncertainOWAoperator[J].InternationalJournalofIntelligentSystems,2002,17(6):569575[12].[J].,2001,16(4):311314:(1968),,,,.:.(1945),,,,.:,.(66)[12]SpiessH.Contributionlathorieetautoutilsdeplanificatindesrseauxdetransporturbain[D].DpartementdInformatiqueetdeRechercheOprationnelle,UniversitdeMontral,1984[13]SpiessH,FlorianM.Optimalstrategies:Anewassignmentmodelfortransitnetwork[J].TransportationResearch,1989,23B:83102[14]LastA,LeakSE.TRANSEPT:Abusmodel[J].TrafficEngineeringandControl,1976,17:1417[15]GendreauM.Etudeapprofondiedunmodledquilibrepourlaffectationdespassagersdanslesrseauxdetransportencommun[D].DpartementdInformatiqueetdeRechercheOprationnelle,UniversitdeMontral,1984[16]LamWHK,GaoZY,ChanKS,YangH.Astochasticuserequilibriumassignmentmodelforcongestedtransitnetworks[J].TransportationResearch,1999,33B:118[17]SmithMJ.Existence,uniquenessandstabilityofequilibria[J].TransportationResearch,1979,13B:295304[18]LampkinW,SaalmansPD.Thedesignofroutes,servicefrequenciesandschedulesforamunicipalbusundertaking:Acasestudy[J].OperationsResearchQuarterly,1967,18:375397:(1968),,,.:,.(1946),,,,.:,,,.07181