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1、1/53JJIIJIBackClosezO9MATLABSy1 55yK2/53JJIIJIBackClose55y$?-|,320-V30c“,?x?J?55y?.,?u1947cdI 0,i=1,2,n.-x=(x1,x2,xn)T Rn,c=(c1,c2,cn)T Rn,A=(a1,a2,am)T Rmn,b=(b1,b2,bm)T Rm,K55yK?IO.?min cTx,s.t.Ax=b,(9.2)4/53JJIIJIBackClosex 0,?x 0U?.?55yKzIO.,eK?8I5?4,=max cTx,|Xmax cTx=min(cTx)=z5?4?.?nXi=1ixi,t
2、Cxn+1,?d/=z?nXi=1ixi xn+1=,OK?xn+1 0.n,?nXi=1ixi6,5/53JJIIJIBackClosetCxn+1,?d/=z?nXi=1ixi+xn+1=,OK?xn+1 0.ugdC(=vkK5?C)xi,KCui1ui2,-xi=ui1 ui2.355y?IO.(9.2),?X?A1?(K,L?K?).,3e,?bK(K,3A?1).e8?:(4:)4?.9.1?C Rn48,x C.e3?:x(1),x(2)C9 (0,1),?x=x(1)+(1 )x(2),Kx8C?:4:,=x C:?7xULC:?|.6/53JJIIJIBackClose9.2
3、?C Rn48,d Rn.e?x C,kx+d|0 C,KdC?.eC?dULC?5|,KC?4.PD=x|Ax=b,x 055yK?1.w,D8.,DN.d8?5D.?k.e?nD?.n9.1d Rn55yK(9.2)?1D?vd 0Ad=0.7/53JJIIJIBackCloseyd9.2,d RnD?:?x D,kx+d|0 D,=A(x+d)=b,x+d 0,x D,0,?d/,Ad=0,d 0.y.?e?nx?55yK(9.2)?1?(?,d55y1?Ln.n9.2(Ln)?55yK(9.2)?1D.K8/53JJIIJIBackClose(1)Dkk:x1,x2,xr;(2)Dk4
4、?D.?,eD.,K3k4d1,d2,dt;(3)x D?3Ki R(i=1,2,r)Ki R(i=1,2,t)?x=rXi=1ixi+tXi=1idi,:rPi=1i=1.55yK?8I?5.TaKkNA:.el“?51D?5.keVg.9.355yK(9.2)?X?A?m m?9/53JJIIJIBackClosef?55yK?|.,55yK?d?A?m5?|?f?.?.A?CC.CC.d?Juy,55yKk|.Au?k?C.55y(9.2)UkCmn|.9.155yKmin 2x1+x2s.t.2x1+x2+x3=2,3x1+2x2+x4=3,xi 0,i=1,2,3,410/53JJI
5、IJIBackClosekB1=213 2,B2=213 0,B3=1 00 1?.A?CO:x1,x2;x1,x3;x3,x4.9.455yK(9.2)?1:1).-C0?1)55yK?1).55yk1),55yK(9.2)kCmn1).X,39.1,k(1/7,12/7,0,0)T(0,0,2,3)T?1).A?OB1B3?.e?nd1)?.n9.355yK(9.2)?1)1)?=?A?X?|5.11/53JJIIJIBackClosee?n?1)?A.n9.455yK(9.2)?1)Au1?:.y?x=?xTB,xTN?T=(xTB,0)TK(9.2)?1).?A=?B N?,BA?,=
6、xvBxB=?B N?xB0=b.yb?x1D?:,K3x(1),x(2)D(x(1)6=x(2)9X (0,1),?x=x(1)+(1 )x(2).dx(1),x(2)?15,?Bx(1)B+Nx(1)N=b,Bx(2)B+Nx(2)N=b.duxN=x(1)N+(1 )x(2)N=0,x(1)N 0,x(2)N 0,(0,1),?x(1)N=x(2)N=0.dd?,Bx(1)B=b,Bx(2)B=b,?B12/53JJIIJIBackClose,dkxB=x(1)B=x(2)B,=x=x(1)=x(2),g.l?x1D?:.y?xD?:.eyx7K(9.2)?1).”?x=(x1,xt,0
7、,0)Txi 0(i=1,2,t).-A=(1),(2),(n).Kx1(1)+x2(2)+xt(t)=b.(9.3)XJxK(9.2)?1),K(1),(2),(t)5,=3?k1,k2,kt?k1(1)+k2(2)+kt(t)=0.(9.4)?,?xi ki 0(i=1,2,t).(9.4)?2(9.3)?(x1 k2)(1)+(x2 k2)(2)+(xt kt)(t)=b.13/53JJIIJIBackClosei=1,2,t,-yi=xi+ki,zi=xi ki,-yj=zj=0(j=t+1,n).y,z D(y 6=z)x=12(y+z).x:g.l?x1).y.?e?n?55yn?
8、n.n9.5(55y?n)(1)e55yKk1),K7k1).(2)e55yKk),K7k1).(3)e55yK?1k.,K7k).y(1)?xK(9.2)?1),”?x=(x1,xt,0,0)T,xi 0(i=1,2,t).w,x1(1)+x2(2)+xt(t)=b.(9.5)14/53JJIIJIBackClosee(i)(i=1,2,t)5,K3?ki(i=1,2,t)?k1(1)+k2(2)+kt(t)=0.(9.6)”?kki(i=1,2,t)?.K,31.-=(k1,kt,0,0)T,:=min16i6tnxiki?ki 0o,Ky():=x K(9.2)?1).?8x?8?.eA
9、?Auy()?5,Ky()1).K,-E,?,1),?8y()?8?.Xd?1e?,?K(9.2)?1).(2)?xK(9.2)?).e1),K?15/53JJIIJIBackClose?,y()1,?cTx 6 cTy()=cTx cT.d?5,cT=0.l?,cTy()=cTx,=y()K(9.2)?).aqu(1)?y,?0,y()1),?8x?8?.-EdL,?1).(3)YcTx3k.48D?.?K(9.2)3).?dn9.4n9.5,55yKek),K7k1?:).du55y5y,)88.d,e8I3:?,K3?:?|?,dKk).e?n?55yKk)?.n9.6?55yK(9.2
10、)?1,KTKk16/53JJIIJIBackClose)?4di,kcTdi 0.y?xi(i=1,2,r)di(i=1,2,t)OK(9.2)?1?:4.dn9.21DLD=?x=rXi=1ixi+tXi=1idi?i 0(i=1,2,r),rXi=1i=1;i 0(i=1,2,t)?.d,K(9.2)?d/?min f(,)=rXi=1icTxi+tXi=1icTdi,s.t.rXi=1i=1,i 0,i=1,2,r,17/53JJIIJIBackClosei 0,i=1,2,t.eK(9.2)k),=?Kk),Kki=1,2,t,7kcTdi 0.K,3,i?cTdi 0(i=1,2,
11、t).K?K?dumin f()=rXi=1icTxi,s.t.rXi=1i=1,i 0,i=1,2,r.PeIpvcTxp=min16i6rcTxi,18/53JJIIJIBackCloseK?x D,kcTx=rXi=1icTxi+tXi=1icTdirXi=1icTxp=cTxp.=x=xpK(9.2)?).y.?ln?yL?,55y?)731?:?.9.2X/91)?(?!0?)55yK?X/91)?(.d55y?n,55yK(9.2)ek),K7k1).?55yK(9.2)?1)kk.?,)55yKI1)8I?.X/|5)55y?k?19/53JJIIJIBackClose.X/?g
12、:l1)u,eT1)K?),KU,K,1),Xde?,?K?1).9.2.155yK?X/ekLf0?X/?.9.2)e?55yKmin f(x)=2x1 3x2,s.t.x1+x26 3,2x1+x26 2,4x1+2x26 16,x1,x2 0.20/53JJIIJIBackClose)k,ntCx3,x4,x5=IO/?55yKmin f(x)=2x1 3x2,s.t.x1+x2+x3=3,2x1+x2+x4=2,4x1+x2+x5=16,xi 0,i=1,2,5.Jw,T55yKk|?,A?Cx3,x4,x5.K?1C/,C u|?,C u|?m,?21/53JJIIJIBackClo
13、semin f(x)=2x1 3x2s.t.x3=3+x1 x2,x4=2+2x1 x2,x5=16 4x1 x2,xi 0,i=1,2,5.-Cx1=0,x2=0?1)x0=(0,0,3,2,16)T.A?8If(x0)=0.x0w,K?),?x1x2?,8I?.|55y?n,e#?1),x1x2?.5?31),?C7C.d,3(#?1),ACx1x2(C)?“?5?Cx3,x4x5(C)22/53JJIIJIBackClose?.?x2C.e0?C?(.(C?K?#?)1,=vK5.du3#?1),x1E,C,?0,d,(CI?yx3=3 x2 0,x26 3,x4=2 x2 0,x26
14、2,x5=16 x2 0,x26 16.dw,?x4=0,K5?y.d,(Cx4,=?#?Cx3,x2x5.?55yKzXe?d?K(C u|?,C u|?m):min f(x)=2x1 3x2,23/53JJIIJIBackCloses.t.x2+x3=3+x1,x2=2+2x1 x4,x2+x5=16 4x1,xi 0,i=1,2,5,?d/,kmin f(x)=6 8x1+3x4,s.t.x3=1 x1+x4,x2=2+2x1 x4,x5=14 6x1+x4,xi 0,i=1,2,5.-Cx1=0,x4=0?1)x1=(0,2,1,0,14)T.A?8If(x1)=6 0,x16 1,x
15、2=2+2x1 0,x1 1,x5=14 6x1 0,x1637.d,Cx3.?dC,C u|?,C u|?m,8I?C?d55yK:min f(x)=14+8x3 5x4,s.t.x1=1 x3+x4,25/53JJIIJIBackClosex2=4 2x3+x4,x5=8+6x3 5x4,xi 0,i=1,2,5.-Cx3=0,x4=0?1)x2=(1,4,0,0,8)T.A?8If(x2)=14 0,x16 1,x2=4+x4 0,x1 1,x5=8 5x4 0,x1637.26/53JJIIJIBackClosed,Cx5.?dC,C u|?,C u|?m,8I?C?d55yK:min
16、 f(x)=22+2x3+x5,s.t.x1=2.6+0.2x3 0.2x5,x2=5.6 0.8x3 0.2x5,x4=1.6+1.2x3 0.2x5,xi 0,i=1,2,5.-Cx3=0,x5=0?1)x3=(2.6,5.6,0,1.6,0)T.A?8If(x3)=22 0,K?1)xkf(x)f(x)=cTBb,=xK(9.2)?).d,ke?n.28/53JJIIJIBackClose?x55yK(9.2)?1),A?I.Kx)?N:=cTN cTBN 0.n9.7?N=cTN cTBNu?(?O).e(!C?.?x=?bT,0T?T55yK?1),A?I.e3,j?u?(N)j 0
17、,KxK?).d,I#?1)x(1),?f(x(1)f(x).X/#?1)?xB?Cxi(C)xN?,Cxj(C).(C?K?#?)?29/53JJIIJIBackClose1)?8I?1)?8I?.d(9.7),xN?v(N)j 0,=xj6 minnbkkj?kj 0,1 6 k 6 mo.?im?eIk,K?#?1)x(1).T1)?8If(x(1)=cTx(1)=cTBxB+(N)jx(1)j30/53JJIIJIBackClose=cTBxB+(N)jbikj 0,KO,?)x;K,3C(Cxjj=mink,k N.2,-j=(ij)mi=1L?N?1j?.ej6 0,KO.d,K?
18、)3;K,=3.31/53JJIIJIBackClose3,(eIibiij=minnbkkj?kj 0o.4,ij?,|Ax=b1?C,1j?Cei,ei?Im?1i?,=1.5 9.14?8?#?)?1)?CA?E,?.?X/o(L9.1.L9.1X/Lcjc1cmcm+1cnicBxBbx1xmxm+1xnc1x1b1101,m+11,n1c2x2b2002,m+12,n2.cmxmbm01m,m+1m,nmf(x)=mPi=1cibij00cm+1mPi=1cii,m+1cnmPi=1cii,n32/53JJIIJIBackCloseL9.1?118I?X,?1O1)A?8Iu?.Nw
19、,C?u?.L?1!?OLC938IAX,1n?Lm.?e5?OLC9X?(?)C9X?.X/L?9.1?3(C?,=k=bk/kj.|L9.1N/Xe?:(1)(1),=-L?1?CO?1n?,C?.(2)?5?.e?1?u?K,KA?1).(3)!C?(.C?1?u?(K)A?C,C?A?33/53JJIIJIBackCloseC.(4)1)A?8I,=?1?1?.9.3|X/L5)55yKmin f(x)=2x1 3x2,s.t.x1+x26 6,x1+2x26 8,0 6 x16 4,0 6 x26 3.)k,otCx3,x4,x5,x6,=IO/?55yKmin f(x)=2x1 3
20、x2,34/53JJIIJIBackCloses.t.x1+x2+x3=6,x1+2x2+x4=8,x1+x5=4,x2+x6=3,xi 0,i=1,2,6.X/L),OLL9.2,L()?.duX/L?u?K,?)x=(4,2,0,0,0,1)T.l?,?K?)x=(4,2)T,8If(x)=14.35/53JJIIJIBackCloseL 9.2 9.3?X/Lcj230000icBxBbx1x2x3x4x5x60 x3611100060 x4812010040 x541000100 x630(1)00013f(x)=0j2300000 x3310100130 x42(1)0010220
21、x5410001043x63010001f(x)=9j2000030 x3100110112x121001020 x5200011(2)13x230100013f(x)=13j0002010 x300010.50.502x141000100 x610000.50.513x220100.50.50f(x)=14j0001.50.5036/53JJIIJIBackClose9.2.21)?(X/1),A?.,?,3e,1)N*?.?e50?1)?.355yK(9.1)0,i=1,2,n+m.(9.8),ex=(x1,xn,xn+m)T?1)xi=0(i=n+1,n+2,n+m),Kx55yK(9.
22、8)?1).37/53JJIIJIBackClose?EXe955yK:minf(x)=n+mXi=n+1xi,s.t.x ,(9.9):Rn+mv(9.8)?Nx?8.Juy,55yK(9.1)k1)?=?55yK(9.9)?)3,8I.3d:?X/?,=k)K(9.9),?2)?55yK(9.1).ef0?T?yL.9.4?X/)55yKmin f(x)=3x1+x2+x3,s.t.x1 2x2+x3+x4=11,4x1+x2+2x3 x5=3,(9.10)2x1+x3=1,38/53JJIIJIBackClosexi 0,i=1,2,5.)0,i=1,2,5.w,55yKkB=I,A?C
23、x4,x6,x7.?,|X/).(JL9.3.39/53JJIIJIBackCloseL9.31?X/Lcj0000011cBxBbx1x2x3x4x5x6x70 x4111211000111x6341201103/21x7120100011f(x)461301000 x41032010011x61010011210 x312010001f(x)101001030 x41230012250 x2101001120 x312010001f(x)00000011?1)x(0)=(0,1,1,12,0)T.l?1?X/L,=)K(9.10)?X/L.?,|X/)?K.O(JL9.4.40/53JJI
24、IJIBackCloseL9.41?X/Lcj31100cBxBbx1x2x3x4x50 x4123001241x21010011x3120100f(x)2-100013x141001/32/31x21010011x390012/34/3f(x)-20001/31/3u,K?)x=(4,1,9,0,0)T.A?8If(x)=2.9.355yK?nu?55yK,3?,55yK,TK?K.?!0?k55y?5y41/53JJIIJIBackClose9?n.9.5Rn55yKmin f(x)=cTx,s.t.Ax b,(9.11)x 0,:A=(aij)Rmn;b=(b1,b2,bm)T;c=(c
25、1,c2,cn)T.55yKmax g(y)=bTy,s.t.ATy 6 c,(9.12)y 0K(9.11)?K.K(9.11)?K.42/53JJIIJIBackCloseOPDPDD?K(9.11)K(9.12)?1.e?n55y?KKm?X.n9.8?K(9.11)K(9.12)k1:.K(1)?x DP,?y DDkcTx bTy;(9.13)(2)e:x DP,y DDvcTx=bTy,KxyO?K(9.11)K(9.12)?).y(1)?x DP,y DD,kAx b,5?y 0,x 0,?bTy 6 yTAx=(ATy)Tx 6 cTx.(2)d(1)?x DP,cTxK(9.
26、12)?.eT.3,y DD?,KyK(9.12)?).43/53JJIIJIBackClose,.y.?e?n55yK?n.n9.9(55yn)(1)e55yK(9.11)K(9.12)k),KKk).?,K?8I?.(2)e55yK(9.11)K(9.12)?8I.,K,K1).y(1)b?K(9.11)k)x.du55y5y,d5yK?KKT,x)?344/53JJIIJIBackClose.KFf Rm,Rn?c AT=0,0,Ax b 0,T(Ax b)=0,0,x 0,Tx=0.Jw,DD,?bT=TAx=cTx.dn9.8K(9.12)?).aq/y,eK(9.12)k),KA?
27、.KFfK(9.11)?).(2)dn9.8(1)?.y.?dn9.9(1)?yw,?K?)A?.KFfK?).,.,lX/?KKX5.,ex?K?45/53JJIIJIBackClose),BA?,KBTcK?).d?n?e?(.n9.1055yK(9.11)9K(9.12)k)?DPDD8.e?nL,Xkg5.n9.11K?K?K.|K?,y55yK(9.1)?KXe55yK:max g(z)=bTz,s.t.ATz 6 c.(9.14)55yK?K?/,K?/.d,O?,J?K).46/53JJIIJIBackClose5,?OO?E,5,?1)?I?.d,XJ?Kk?,?gC?,KL)
28、KO5?.9.4AMATLAB)55yKOE?u,r?E?u,AOMATLAB,lEYA5U?p?.5zY.)55yKB,;?OL,l?)S?5?5yKy.y2E 0.MATLAB):c=0 0.1 0.2 0.30.8;A1=-1,0,0,0,0;0,-1,0,0,0;.0,0,-1,0,0;0,0,0,-1,0;0,0,0,0,-1;b1=0,0,0,0,0;A2=1,2,0,1,0;0,0,2,2,1;3,1,2,0,3;52/53JJIIJIBackClose b2=100,100,100;x,fv=linprog(c,A1,b1,A2,b2)?Optimization terminated.x=12.824327.175717.175732.82430.0000fv=16.000053/53JJIIJIBackClose$1TS?,=?)x=(12.8243,27.1757,17.1757,32.8243,0)T,Uo?x=(13,27,17,33,0)T,z=16.=UYAe?13,YBe?27,YCe?17,YDe?33,?I?90?100e,?16m.