对方案有偏好的区间直觉模糊多属性决策方法

311Vol.31No.120091SystemsEngineeringandElectronicsJan.2009:10012506X(2009)0120116205:2007-10-06;:2008-02-27:(Z2008JG05):(1973-),,,,E2mail:[e-mail](,402160):,,,,,(IIFWA),,,:;;(IIFWA);:C934:AMethodforinterval2valuedintuitionisticfuzzymultipleattributedecisionmakingwithpreferenceinformationonalternativesWEIGui2wu(Dept.ofEconomicsandManagement,ChongqingUniv.ofArtsandSciences,Chongqing402160,China)Abstract:Withrespecttointerval2valuedintuitionisticfuzzymultipleattributedecisionmakingproblemswithpreferenceinformationonalternativesandincompleteweightinformation,someoperationallawsofinter2val2valuedintuitionisticfuzzynumbers,scorefunctionandaccuracyfunctionofinterval2valuedintuitionisticfuzzynumbersareintroduced.Anoptimizationmodelbasedontheminimizingdeviationmethod,bywhichtheattributeweightscanbedetermined,isestablished.Forthespecialsituationswheretheinformationaboutat2tributeweightsiscompletelyunknown,anotheroptimizationmodelisestablished.Bysolvingthismodel,asim2pleandexactformulacanbeusedtodeterminetheattributeweights.Then,theinterval2valuedintuitionisticfuzzyweightedaveraging(IIFWA)operatorisutilizedtoaggregatetheinterval2valuedintuitionisticfuzzyinfor2mationcorrespondingtoeachalternative.Basedonthethescorefunctionandaccuracyfunctionoftheinterval2valuedintuitionisticfuzzynumber,amethodforrankingthealternativesandselectingthemostdesirableone(s)ispresented.Finally,anillustrativeexampleisgiventoverifythedevelopedapproachandtodemonstrateitspracticalityandeffectiveness.Keywords:interval2valuedintuitionisticfuzzynumber;operationlaw;interval2valuedintuitionisticfuzzyweightedaveraging(IIFWA)operator;preference01965Zadeh[1],1986Atanassov,(intuitionisticfuzzysets,IFS),,[2-3]1993GauBuehrerVague[4],BustinceBurilloVagueAtanassov[5],,Xu[6],(IFWG),(IFOWG)(IFHG),IFHG,1:117Xu[7],(IFAA)(IFWA),IFAAIFWA,Xu[8](IFOWA)(IFHA)Atanassov[9],Atanassov[10]Xu[11],(IIAA)(IIFWA),(IIGA),(IIFWG),Xu[12],(IIFOWA)(IIFHA)Xu[13]()[14][15],,,,,IIFWA,,1Atanassov[2-3],:,1[2-3]X,A={x,A(x),A(x)ó|xX},A(x)A(x)XxXAX[0,1]AX[0,1],0A(x)+A(x)1,PxX,A(x)=1-A(x)-A(x)XxX,A(x)A(x),,Atanassov[9-10],2[9-10]X,XAA={x,A(x),A(x)|xX},AX[0,1]AX[0,1],AxX,0sup(A(x))+sup(A(x))1,AA={x,[LA(x),UA(x)],[LA(x),UA(x)]|xX},([a,b],[c,d]),[a,b][0,1],[c,d][0,1],b+d13[11]a1=([a1,b1],[c1,d1])a2=([a2,b2],[c2,d2]),:(1)a1+a2=([a1+a2-a1a2,b1+b2-b1b2],[c1c2,d1d2]);(2)a1a2=([a1a2,b1b2],[c1+c2-c1c2,d1+d2-d1d2]);(3)a1=([1-(1-a1),1-(1-b1)],[c1,d1]),0;(4)(a1)=([a1,b1],[1-(1-c1),1-(1-d1)]),03,[11]:(1)a1+a2=a2+a1;(2)a1a2=a2a1(3)(a1+a2)=a1+a2,0;(4)(a1a2)=a1a2,0;(5)a1+a2=(a1+a2),0;(6)a1a2=(a1a2),04[11]a=([a,b],[c,d]),S(a)=a-c+b-d2,S(a)[-1,1](1)S(a),a=([a,b],[c,d])5[11]a=([a,b],[c,d]),H(a)=a+b+c+d2,H(a)[0,1](2)H(a),a=([a,b],[c,d])6[11]a1=([a1,b1],[c1,d1])a2=([a2,b2],[c2,d2]),(1)S(a1)S(a2),a1a2;(2)S(a1)=S(a2),H(a1)=H(a2),a1=a2;H(a1)H(a2),a1a2Q7[11]aj=([aj,bj],[cj,bj])(j=1,2,,n),IIFWAQnQ,IIFWA(a1,a2,,an)=nj=1jaj=1-nj=1(1-aj)j,1-nj=1(1-bj)j,nj=1cjj,nj=1djj(3)11831,=(1,2,,n)Taj(j=1,2,,n),j[0,1]nj=1j=1IIFWAn(IIFWA),=(1/n,1/n,,1/n)T,IIFWAIIFAA,IIFAA(a1,a2,,an)=1nnj=1aj8[13]a1=([a1,b1],[c1,d1])a2=([a2,b2],[c2,d2]),,d(a1,a2)=14(|a1-a2|+|b1-b2|+|c1-c2|+|d1-d2|)(4)2IIFWA,,A={A1,A2,,Am},G={G1,G2,,Gn},w={w1,w2,,wn}TW,wjGj,nj=1wj=1wj0,j=1,2,,nAiA(A1,A2,,am)GjG(G1,G2,,Gn),([aij,bij],[cij,dij]),i=1,2,,m,j=1,2,,n,[aij,bij]AiGj,[cij,dij]AiGj,[aij,bij][cij,dij][aij,bij][0,1],[cij,dij][0,1],0bij+dij1,R=(rij)mn=([aij,bij],[cij,dij])mnAi,,i=([i,i],[i,i]),i=1,2,,mIIFWAR=(rij)mn,AiGjri=([ai,bi],[ci,di])=IIFWAw(ri1,ri2,,rin)=1-nj=1(1-aij)wj,1-nj=1(1-bij)wj,nj=1cwjij,nj=1dwjiji=1,2,,m(5),ri,Ai,,,,,,,(M-1)minDi(w)=nj=1d(rij,i)wj,i=1,2,,m=14nj=1wj[|aij-i|+|bij-i|+|cij-i|+|dij-i|]s.t.nj=1w2j=1,wj0,j=1,2,,n,,(M-2)minD(w)=mi=1nj=1d(rij,i)wj=14mi=1nj=1wj[|aij-i|+|bij-i|+|cij-i|+|dij-i|]s.t.nj=1w2j=1,wj0,j=1,2,,n,d(rij,i)AiGjrijAiiD(w)(M-2),LagrangeL(w,)=mi=1nj=1d(rij,i)wj+2nj=1w2j-1(6)(6),9L9wj=mi=1d(rij,i)+wj=09L9=nj=1w2j-1=0w3j=mi=1d(rij,i)nj=1mi=1d(rij,i)2(7)w3jwj=mi=1d(rij,i)nj=1mi=1d(rij,i),j=1,2,,n(8)(4),(8)wj=mi=1(|aij-i|+|bij-i|+|cij-i|+|dij-i|)nj=1mi=1(|aij-i|+|bij-i|+|cij-i|+|dij-i|)j=1,2,,n(9),1:1196[16-17]wiwj;wi-wji;wiiwj;iwii+i;iwj(i+i)wjiwiwj(i+i),wj0;wi-wjwk-wl;jkl,i,i,i,iiHwjGj,nj=1wj=1wj0,j=1,2,,n,,w(M-3)minD(w)=mi=1nj=1d(rij,i)wj=14mi=1nj=1wj[|aij-i|+|bij-i|+|cij-i|+|dij-i|]s.t.wH,nj=1wj=1,wj0,j=1,2,,n,w,:1AiA(i=1,2,,m)GjG(j=1,2,,n),AiGj[aij,bij],[cij,bij],R=(rij)mn=([aij,bij],[cij,dij])mn,Aii=([i,i],[i,i]),i=1,2,,m2,(9)w,4;;3,(M-3),w4(5)Airi5AiriS(ri)=ai-ci+bi-di2,i=1,2,,mS(ri),AiriH(ri)=ai+bi+ci+di2,i=1,2,,m6S(ri)H(ri)Ai73,5A1A5,4G1G4(,,)[24],R=([0.3,0.4],[0.4,0.5])([0.5,0.6],[0.1,0.3])([0.4,0.5],[0.3,0.4][0.4,0.6],[0.2,0.4])([0.5,0.6],[0.3,0.4])([0.4,0.7],[0.1,0.2])([0.5,0.6],[0.2,0.3])([0.6,0.7],[0.2,0.3])([0.2,0.5],[0.4,0.5])([0.2,0.3],[0.4,0.6])([0.3,0.5],[0.3,0.4])([0.1,0.3],[0.5,0.6])([0.4,0.5],[0.3,0.5])([0.5,0.8],[0.1,0.2])([0.2,0.5],[0.3,0.4])([0.4,0.7],[0.1,0.2])([0.5,0.6],[0.2,0.4])([0.6,0.7],[0.1,0.2])([0.3,0.4],[0.1,0.3])([0.7,0.8],[0.1,0.2])Ai1=([0.3,0.4],[0.3,0.5]),2=([0.5,0.6],[0.