一种信息不完全确定的多准则语言群决策方法王坚强

第22卷第4期Vol.22No.4控制与决策ControlandDecision2007年4月Apr.2007:2005-12-24;:2006-03-02.:(05YB74);(70631004);(06FJ4126).:(1963)),,,,,.:1001-0920(2007)04-0394-05王坚强(,410083):针对权系数信息不完全确定且方案的准则值为确定语言等级或位于两个连续语言等级之间,甚至缺失的群决策问题,提出一种群体语言决策方法.该方法利用证据推理算法得到方案属于各语言等级的信任度,利用二元语义对方案进行语言集结;然后结合决策者和准则权重的不完全确定信息及方案与理想方案的二元语义间的距离构建非线性规划模型,利用遗传算法求解所得模型,计算得到各方案的排序.实例计算表明了该方法的可行性和有效性.:群决策;证据推理;二元语义;语言评价:C934:AGroupmulti-criterialinguisticdecision-makingmethodwithincompletecertaininformationWANGJian-qiang(SchoolofBusiness,CentralSouthUniversity,Changsha410083,China.E-mail:[e-mail])Abstract:Togroupdecision-makingproblemsthatthedecision-makers.weightsandthecriteria.sweightsareincompletelycertain,andthedecision-makers.preferenceinformationtakestheformoflinguisticgrade,orisbetweentwocontinuouslinguisticgrade,orisnotgiven,anewmethodbasedonevidentialreasoning(ER)andtwo-tupleisproposed.ByusingtheERalgorithm,thebeliefdegreesofeachalternativebelongingtoeachlinguisticgradeareobtained,andthentransformedintotwo-tuplewhichisalsotheresultofaggregatingeachalternative.Thenonlinearprogrammingmodelisconstructedbasedontheincompletecertaininformationonthedecisionmakersweightsandcriteriaweights,andthedistancesbetweenalternativesandidealalternative.Theoptimalweightsondecision-makerandcriteriaaregainedbyusinggeneticalgorithmstosolvethenonlinearprogrammingmodel.Thedistancesbetweenalternativesandidealalternativearecomputedandtherankofthealternativesisobtained.Anexampleshowsthefeasibilityandeffectivenessoftheproposedmethod.Keywords:Groupdecision-making;Evidentialreasoning,Two-tuple;Linguisticassessment1引言,.,.[1-10].,3.1[1],,.2[2-7],OWALOWA.,,,,,.3[8-10],,.,,,.,,DOI10.13195/j.cd.2007.04.36.wangjq.007第4期王坚强:一种信息不完全确定的多准则语言群决策方法.,.,.2二元语义及相关算子H={H0,H1,,,H2t},H:1):ij,HiHj;2):/Neg0,i+j=2t,Hi=Neg(Hj);3):i\j,Max(Hi,Hj)=Hi,Min(Hi,Hj)=Hj.(Hi,Ai).:Hi;AiHi,AiI[-0.5,0.5)[8].1[8]HiIH,::HyH@[-0.5,0.5),(Hi)=(Hi,0).2[8]H={H0,H1,,,H2t},HI[0,2t],H$,$:[0,2t]yH*[-0.5,0.5),$(H)=(Hn,An)=Hn,n=Round(H),An=H-n.Round/0.3[8]H={H0,H1,,,H2t},(Hi,Ai),$-1,HI[0,2t],$-1:H@[-0.5,0.5)y[0,2t],$-1(Hi,Ai)=i+Ai.4(Hi,Ai)(Hj,Aj),ij,(Hi,Ai):(Hj,Aj),/:0/0.i=j,3[8]:1)AlAk,(Hi,Al):(Hj,Ak);2)Al=Ak,(Hi,Al)=(Hj,Ak),/=0/0;3)AlAk,(Hi,Al);(Hj,Ak),/;0.3信息不完全确定的多准则语言群决策方法mA={a1,a2,,,am},sDM={D1,D2,,,Ds},K=(K1,K2,,,Ks),Esi=1Ki=1.q,C={C1,C2,,,Cq},DpClXpl,Cpl=(cplij)m@q.:cplijDpHClai,,,,.3.1,,,,,AHP,ANP,CNP.,;;,.,3[11]:1){X:A1X\b,X0,b\0};2){X:A1X[b,X0,b\0};3){X:A1X=b,X0,b\0}.:X=(X1,X2,,,Xn)T,A1l@n.8,Dp7p.3.2,Hn+A.AI(0,1),HnHn+1,HnA,Hn+11-A..1)DpcpijHn,aiCjHnT(p,i,j,n)=1.2)DpcpijHn+A,aiCjHnT(p,i,j,n)=1-A,Hn+1T(p,i,j,n+1)=A.3)Dpcpij,DpaiCj,aiCjHnT(p,i,j,n)=0(n=0,1,,,2t).aiCjHn395控制与决策第22卷B(i,j,n)=Esp=1Kp@T(p,i,j,n),n=0,1,,,2t,KpDp.:Step1aiHn.B(i,j,n),K1,K2,,,Ks,Xpj(j=1,2,,,q;p=1,2,,,s),.h,mn,j(ai)=XjB(i,j,n),n=0,1,,,2t;j=1,2,,q.XjCj.,,,Xj=Esp=1Xpjs.mh,j(ai)=1-E2tn=0mn,j(ai),j=1,2,,,q.Hn[12],mn,J(j+1)(ai)=KJ(j+1)(ai)mn,J(j)(ai)mn,j+1(ai)+mh,J(j)(ai)mn,j+1(ai)+mn,J(j)(ai)mh,j+1(ai)];h,mh,J(j+1)(ai)=KJ(j+1)(ai)mh,J(j)(ai)mh,j+1(ai).mn,J(1)(ai)=mn,1(ai),n=0,1,,,2t,mh,J(j)(ai)=mh,1(ai),KJ(j+1)(ai)=(1-E2tx=0E2tb=0bXxmx,J(j)(ai)mb,j+1(ai))-1.aiHn[12]Bn(ai)=1-Bh(ai)1-mh,J(q)(ai)mn,J(q)(ai),Bh(ai)=Eqj=1Xj(1-E2tn=0B(i,j,n)).Bn(ai)Bh(ai)X=(K1,K2,,,Ks,X11,X21,,,Xs1,X12,X22,,,Xs2,,,X1q,X2q,,,Xsq),Bn(ai,X)Bh(ai,X).Step2.Step1ai:B0(ai,X),B1(ai,X),B2(ai,X),,,B2t(ai,X),B0(ai,X)+B1(ai,X)+,+B2t(ai,X)+Bh(ai,X)=1.,H0,H1,,,H2t(H0,0),(H1,0),,,(H2t,0).H(i,X)=B0(ai,X)*$-1(H0,0)+B1(ai,X)*$-1(H1,0)+,+B2t(ai)*$-1(H2t,0).ai.$-1,H(i,X)=B0(ai,X)*0+B1(ai,X)*1+B2(ai,X)*2+,+Bn(ai,X)*n+,+B2t(ai,X)*2t.(1)H(i,X)[Hmin(i,X),Hmax(i,X)],Hmin(i,X)=B1(ai,X)*1+B2(ai,X)*2+,+B2t(ai,X)*2t,Hmax(i,X)=Hmin(i,X)+Bh(ai,X)*2t,Hmax(i,X)I[0,2t].H(i,X)1$,H(i,X)$(H(i,X))=(Hn,Ai)=Hn,n=Round(H(i,X)),Ai=H(i,X)-n.(2)Step3.H2t,I+.I+Bn(I+)=0,n=0,1,,,2t-1,B2t(I+)=1.(1)H(I+)=2t.$,I+(H2t,0).aiI+(di,ai)=$(|$-1($(H(i,X)))-$-1(H2t,0)|).(3)|$-1($(H(i,X)))-$-1(H2t,0)|=(|$-1($(Hmin(i,X)))-$-1(H2t,0)|+|$-1($(Hmax(i,X)))-$-1(H2t,0)|)/2.aiI+,.$$-1,(3)396第4期王坚强:一种信息不完全确定的多准则语言群决策方法1D1a1a2a3a4a5D2a1a2a3a4a5D3a1a2a3a4a5C1H2H0.5H3.5H1H0H2H0H0H0.5H0H2H1.5H0.5H0.5H0.5C2H3.5H2H3--H4H2H0.5H0.5H1.5H2.5H2H2H0.5H1.5C3H0.5H1H2H1H1.5H4H3.5H2H2H2H3.5H2H2H1.5H2.5C4H3-H3H2H3.5H3.5H3.5H2H2H1H3.5H3.5H2.5H2H2.5C5H4-H2.5H0.5H2H4H2.5H2H3H2H3.5H2.5H1.5H1.5H2(di,ai)=$(|(H1(i,X)-2t|),(4)H1(i,X)=Hmin(i,X)+Bh(ai,X)*t.H1(i,X)[2t,(4)(di,ai)[$(2t-H1(i,X)).(5),ai,:min(di,ai)=$(2t-H1(i,X)),s.t.Esi=1Ki=1,(K1,K2,,,Ks)I8;Esi=1Xpj=1,p=1,2,,,s,(Xp1,Xp2,,,Xpq)I7p,p=1,2,,,s;X=(K1,,,Ks,X11,,,Xs1,X12,,,Xs2,,,Xq1,,,Xsq).(6)$[9],(2)(6)mind+(ai,X)=2t-E2tn=0n*Bn(ai,X)-t*Bh(ai,X),s.t.(6).(7),,.(7)minEmi=1d+(ai,X)=Emi=1(2t-E2tn=0n*Bn(ai,X)-t*Bh(ai,X)),s.t.(6).(8)Step4.Bn(ai,X)Bh(ai,X),(8),,[11].Step5.(8)X,(5),aiI+(di,ai).,(di,ai),A.4数值算例,D1,D2,D33,a1,a2,a3,a4,a55,C1,C2,C3,C4,C55,H={,,,,}={H0,H1,H2,H3,H4}.1,/-0.3:K1K2K3,K3I[0.15,0.25],K1I[0.33,0.5],K2I[0.3,0.4].:X15X11X12X13X14,0.15[X15[0.35,0.1[X12[0.25,0.08[X14[0.15,X21=X22=X23=X24=X25=0.2,X31X35X33X32\X34,0.2[X31[0.35,0.1[X32[0.2,0.1[X33[0.25,0.05[X34[0