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1234567,Yyjjyj*jpjjxijjiaijxij/yjxi*imax:Y=nj=1pjyj1s.t.yjyj*0(j=1,2,,n)nj=1xij=nj=1aijyjxi*(i=1,2,,m)1kk+1nmax:Y=nj=k+1pjyj2s.t.nj=k+1xij=nj=k+1aijyjxi*-kj=1xij=xi*-kj=1aijyj*(i=1,2,,m)yjyj*0(j=1,2,,n)yj(j=k+1k+2n)pjaijxi*yj*(j=12k)pjaijxi*yj*(j=12k)121yjSmax:yj=y1j+y2j++ysj3s.t.ai1jy1j+ai2jy2j++aisjysjxi*(i=1,2,,m)yjyj*0(j=1,2,,n)aisjSyjiysjSyjyAyByA,yByASyBTT2T-1(1)yByB*X*-XBZZZZ=122T-1yB*SyA2s-12s-1yAyAyB*:maxyZA=yZ1A+yZ2A++yZsA(z=1,2,,2T-1)4s.t.ss=1xisA(=ss=1aisAyz1A)xi*-xizB(=z(1)ai1By1B+z(2)ai2By2B++z(T)aiTByTB)(i=1,2,,m)yjyj*0(j=1,2,,n):maxyZA(z=1,2,,2T-1)yZAzyB*yAyZsAzyB*SyAxizBzyB*iz1z2zT12TyBznkn-k1T12T1-12T22T2-1kTk2Tk-1k+1Sk+1k+2Sk+2nSnk2T1-12T2-12Tk-1TTkX*-X1~kzz=12TzTzX1~kk3T2Pk+1yk+1+Pk+2yk+2++PnynTT:maxnj=k+1Pjyzj=nj=k+1Pjs(i)s=1yzsj5s.t.nj=k+1xij(=nj=k+1aijyj)xi*-kj=1xizj(=kj=1aizjyj*)(i=1,2,,m)(z=1,2,,T)yjyj*0(j=1,2,,n):maxnj=k+1Pjyzj(z=1,2,,T)6yzjTzkyjj=k+1,nyzsjSjaij=ai1jy1j/yj+ai2jy2j/yj++ais(i)jys(i)j/yj,jj=k+1,nai1jjj=k+1,n12xizjzjj=12kiaizj=z(1)ai1jy1j/yj+z(2)ai2jy2j/yj++z(sj)aizjys(i)j/yjzjj=12kz1z2zsjsjzzsj1x2AbO+abababa[ab]3x115-1-1Bx1x2x2Aax1x2Bbx2x1maxyj7x1s.txij(=aijyj)xi*(i=1,2,,m)15-1-1aij=f(a1j,a2j,,ai-1j,ai+1j,,amj)(i=1,2,,m)mnk,n-kkjTjkkj=1TjT1kk=222=4k3X*-X1ktt=12,,T1kT1kn-knj=k+1SjSk+1nSjjj=k+1,nSk+1nSk+1nT1kT1kSk+1nkT1kn-kSk+1nkSn-kn:maxnj=k+1Pjyt:sjt=12,,T1k,S=12,,Sk+1ns.t.nj=k+1xij(=aisjyj)xi*-kj=1xij(=aitjyj*)8(i=1,2,,m,t=12,,T1k,S=12,,Sk+1n)yjyj*0(j=1,2,,n):maxnj=k+1Pjyt:sjt=12,,T1k,S=12,,Sk+1n97T1kSk+1nmaxnj=k+1Pjyjs.t.nj=k+1xij(=aijyj)xi*-kj=1xij(=aijyj*)(i=1,2,,m)10aij=f(a1j,a2j,,ai-1j,ai+1j,,amj)(i=1,2,,m,j=1,2,,n)fij(ij)aijfij(ij)(i=1,2,,m,j=1,2,,n)yjyj*0ijijij4k+1nkkk+1n4278101112max:Y=nj=k+1Pjyjs.t.nj=k+1xij+si=xi*-kj=1xij(i=1,2,,m)2’yjyj*0(j=1,2,,n):maxnj=k+1Pjyzj=nj=k+1Pjs(i)s=1yzsj(z=1,2,,T)s.t.nj=k+1xij(=nj=k+1aijyj)+si=xi*-kj=1xizj(=kj=1aizjyj*)(i=1,2,,m)(z=1,2,,T)5’yjyj*0(j=1,2,,n):maxnj=k+1Pjyzj(z=1,2,,T)6’:maxnj=k+1Pjyt:sjt=12,,T1k,S=12,,Sk+1ns.t.nj=k+1xij(=aisjyj)+ssi=xi*-kj=1xij(=aitjyj*)8’(i=1,2,,m,t=12,,T1k,S=12,,Sk+1n)yjyj*0(j=1,2,,n):maxnj=k+1Pjyt:sjt=12,,T1k,S=12,,Sk+1n9’maxnj=k+1Pjyjs.t.nj=k+1xij(=aijyj)+si=xi*-kj=1xij(=aijyj*)(i=1,2,,m)aij=f(a1j,a2j,,ai-1j,ai+1j,,amj)10’(i=1,2,,m,j=1,2,,n)fij(ij)aijfij(ij)(i=1,2,,m,j=1,2,,n)yjyj*0sissisi1gg+1m5xaxbxaxbxbxax2x2*E15-1-2x1*x2*x1x20x1*x1E15-1-21C1T+C2T++CsT=2T-12134X1X2y1a1ya2ya2y=1/a1yy2=dx2/dx1x1/x2=-1,dx2/x2=-dx1/x1dlnx2=-dlnx1,lnx2=lnx1-1,x2=1/x1a2y=x2/y=1/x1y,x1=a1ya2y=1/a1yy26121234515-2-115-2-1715-2-2912345678915-2-2815-2-3yOAABBB0ABt15-2-315-2-4P2P30P1ab15-2-4abP2P1P391234123abc1011119849[]1980[]198121934661982254[]1981[]198019346619821984912