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200244:100026788(2002)0420026206ö21,2,1(1.,150080;2.,066004):ö2,L:;;;:C931.1:AaTheResearchontheProblemofLinearöLinear2FractionalBilevelMultiple2objectiveDecisionMakingTENGChun2xian1,LIZhi2hui2,LILei1(1.InstituteofSystmesEngineering,HarbinUniversityofScienceandTechnology,Harbin150080,Chi2na;2.DepartmentofMathematicsandBasicScience,YanshanUniversity,Qinhuangdao066004,China)Abstract:Inthispaper,thedecisionmakingproblemaboutthelinearölinear2fraction2albilevelmultiple2objectiveprogrammingisdiscussed,thepropertitiesofthesolutionsetandfirst2orderoptimizationconditionareproposed.Keywords:bilevel;programming;multi2objectiveprogramming;decisionmaking;optimizationcondition1L,[1][2]L,[3].,2,L,[4,5]LL22:minx1f1=ATx+BTH(x)xX={xRnûAx=b,xE0}x,y=(y1,y2,,yN)Hi(x)=minAi+c11ix+c12iyiBi+c21ix+c22iyiyiqi(x)qi(x)={yiûyiRni,yiE0,Aix+Biyi=bi}a:2000205219:(69874009)©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.i=i,N,H(x)=(H1(x),,HN(x))TAimin,Bimini,bimiZAmn,bmZARn,BRN.2(LfBM)Z1S={(x,y)ûxX,yiqi(x)i=1,N}(LfBM)Z2Ý(x)=(Ý1(x)Ýi(x)ÝN(x)),Ýi(x),Ýi(x)=yiargminAi+c11ix+c12iyiBi+c21ix+c22iyi,yiqi(x)Ý(x)(LfBM)Z3S3={xRnûxX,Ý(x)Á},S3(LfBM)Z4S1={(x,y)ûxS3,yÝ(x)}S1(Lf(BM),IR=S1.IR=S1[6]ZIRAS.5(xq,yq)(LfBM):1)(xq,yq)IR.2)(x,y)S,ATxq+B1H1(xq)++BNHN(xq)FATx+B1H1(x)++BNHN(x)6YARn,Y1,Y2ARn,Y1Y2,,Y1Y2=Á,Y1Y2=Y,Y;YZ31x0X,Ýi(x0),i=1,N.Ý(x0)Zx0X,yi1,yi2Ýi(x0)i=1,N,Ai+c11ix0+c12iyi1Bi+c21ix0+c22iyi1=Ai+c11ix0+c12iyi2Bi+c21ix0+c22iyi2=minyiqi(x0)Ai+c11ix0+c12iyiBi+c21ix0+c22iyiyiqi(x0)PK[0,1],Kyi1+(1-K)yi2E0Aix0+Biyi1=bi,Aix0+Biyi2=biAix0+Bi(K(yi1)+(1-K)yi2)=KAix0+KBiyi1+[Aix0-(1-K)Biyi2-KAix0]=K(Aix0+Biyi1)+(1-K)(Aix0+Biyi2)=Kbi+(1-K)bi=biAi+c11ix0+c12i[Kyi1+(1-K)yi2]Bi+c21ix0+c22i[Kyi1+(1-K)yi2]=K(Ai+c11ix0+c12iyi1)+(1-K)(Ai+c11ix0+c12iyi2)K(Bi+c21ix0+c22iyi1)+(1-K)(Bi+c21ix0+c22iyi2Ai+c11ix0+c12iyi1Bi+c21ix0+c22iyi1=Ai+c11ix0+c12iyi2Bi+c21ix0+c22iyi2K(Ai+c11ix0+c12iyi1)K(Bi+c21ix0+c22iyi1)=(1-K)(Ai+c11ix0+c12iyi2)(1-K)(Ai+c21ix0+c22iyi2)724ö2©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.=minyiqi(x0)Ai+c11ix0+c12iyiBi+c21ix0+c22iyiK(Ai+c11ix0+c12iyi1)+(1-K)(Ai+c11ix0+c12iyi2)K(Bi+c21ix0+c22iyi1)+(1-K)(Ai+c21ix0+c22iyi2)=minyiqi(x0)Ai+c11ix0+c12iyiBi+c21ix0+c22iyiKyi1+(1-K)yi2Ýi(x0),Ýi(x0)ZÝ(x0)=(Ý1(x0)ÝN(x0)),Ý(x0)ZÝ(x0)Z1S,xX,qi(x)={yiRniûyiE0,Aix+Biyi=bi}Á,i=1,N,IR=S1ZS,S1=IRAS,IR=S1Z[7],,IRZIR,IR,IR1,IR2,IR1IR2=Á,IR1IR2=IR.X1={xûyi,(x,yi)IR1},i=1,NX2={xûyj,(x,yj)IR2},j=1,NijX1X2,X1X2AX.IR,PxX,Ý(x)Á,yÝ(x),(x,y)IR,(x,y)IR1,(x,y)IR2,xX1X2,XAX1X2,X=X1X2ZX,,X1X2Á,x0X1X2,y3,y33,(x0,y3)IR1,(x0,y33)IR2.y3Ý(x0),y33Ý(x0).1,Ý(x0),K[0,1],Ky3+(1-K)y33Ý(x0),(x0,Ky3+(1-K)y33)IR=IR1IR2{(x0,Ky3+(1-K)y33)ûK[0,1]},:{(x0,Ky3+(1-K)y33)ûK[0,1]}AIR(x0,y33)IR1(x0,y3)IR2,IR1IR2Á,Z:SSF,F,FSZ2IRSZIRAS.(x1,y1)S,(x2,y2)S,vK(0,1),(x,y)=K(x1,y1)+(1-K)(x2,y2)IRZxS3,yÝ(x),xE0,Ax=b,xRn,y=(y1,,yj,,yN)y=Ky1+(1-K)y2Ý(x)Pi=1,Nyi=Kyi1+(1-K)yi2Ýi(x)minyiqi(x)Ai+c11ix+c12iyiBi+c21ix+c22iyi=Ai+c11ix+c12iyiBi+c21ix+c22iyi=Ai+c11ix+c12i[Kyi1+(1-K)yi2]Bi+c21ix+c22i[Kyi1+(1-K)yi2]=K(Ai+c11ix1+c12iyi01)+(1-K)(Ai+c11ix2+c12iyi2)K(Bi+c21ix1+c22iyi1+(1-K)(Bi+c21ix2+c22iyi2)8220024©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.=mi=HiZ(x1,y1)|IR,((x2,y2)IR),,y1|Ý(x1),yi01|Ýi0(x1),vi0:1Fi0FN,mi0=Hi0(x1)=minyiqi0(x1)Ai+c11ix1+c12iyiBi+c21ix1+c22iyiAi+c11ix1+c12iyi01Bi+c21ix1+c22iyi01=K(Ai+c11ix1+c12iyi01K(Bi+c21ix1+c22iyi01y2Ý(x2),yi02Ýi0(x2)(1Fi0FN),mi0=Hi0(x2)=minyiqi0(x2)Ai+c11ix2+c12iyiBi+c21ix2+c22iyi=Ai+c11ix2+c12iyi02Bi+c21ix2+c22iyi02=(1-K)(Ai+c11ix2+c12iyi02)(1-K)(Bi+c21ix2+c22iyi02)K(Ai+c11ix1+c12iyi01)+(1-K)(Ai+c11ix2+c12iyi02)K(Bi+c21ix1+c22iyi01)+(1-K)(Bi+c21ix2+c22iyi02)Hi0(x)=mi0yi0Ýi0(x),:K(Ai+c11ix1+c12iyi01)+(1-K)(Ai+c11ix2+c12iyi02)K(Bi+c21ix1+c22iyi02)+(1-K)(Bi+c21ix2+c22iyi02)=mi0=Hi0(x)(vi0:1Fi0FN),IRSZ4x,ZHi(x)=minyiqi(x)Ai+c11ix+c12iyiBi+c21ix+c22iyi,i=1,N(3):1)[8],Bi+c21ix+c22iyi02)(3)[9]Z2Ai+c11ix+c12iyiBi+c21ix+c22iyi,Zyqiqi(xq),Ri=yiqi(x)Ai+c11ix+c12iyiBi+c21ix+c22iyi¸Ai+c11ixq+c12iyqiBi+c21ixq+c22iyqi,i=1,NAi+c11ix+c12iyiBi+c21ix+c22iyi¸Ai+c11ixq+c12iyqiBi+c21ixq+c22iyqiSlater[8],RiyqiK-TZ2,Ai+c11ix+c12iyi¶0.3,(xq,yq)IRKl0,uq¶0,(yq,Kl,uq)K-TýyL(yq,Kl,uq)=KlTýF(yq)+uqTýG(yq)=0(1)924ö2©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.uqTG(yq)=0,uq¶0(2)L(y,K,u)=6Ni=1KiAi+c11ix+c12iyiBi+c21ix+c22iyi+6Nj=1uj(Ajx+Bjyi-bj)F(y)=(F1(y),F2(y),,FN(y))Fi(y)=Ai+c11ix+c12iyBi+c21ix+c22iy,i=1,NGi(y)=Aix+Biy-bi,i=1,NG(y)=(G1(y),G2(y),,GN(y))(SFP)yqminyiRi1FiFNF(y)=6Ni=1Ai+c11ix+c12iyiBi+c21ix+c22iyi(SFP)yqLagrangel(y,K,u)=6Ni=1Ai+c11ix+c12iyiBi+c21ix+c22iyi+6Ni=1KiAi+c11ix+c12iyiBi+c21ix+c22iyi-Ai+c11ix+c12iyqiBi+c21ix+c22iyqi+6Nj=1uj(Ajx+Bjyj-bj).yqÝ(xq),yqi(3),yq(SFP)yqZ2,(SFP)yqRi(i=1,N)K2T,[9],K0i¶0,u0j¶0ýyl(yq,K0,u0)=6Ni=1(1+K0i)ýFi(yqi)+6Nj=1u0j,ýGj(yqj)=0(3)6Nj=1u0jGj(yqj)=0(4)K0i¶0,u0j¶0,i=1,N;j=1,NKli=1+K0i,Kl=(Kl1,Kl2,,KlN)Tuqj=u0j,j=1,N,uq=(uq1,uq2,,uqN)TKl0,uq¶0,Kl,uq,(3),(4)(1),(2)ZK2TZ52,x,2,,,,Z:[1],,1[J]1,1997,17(2):45-49.[2]WangS.Multilevelprogramming:modelsandapplications[A].SNgetal.ManagementScienceandEconom