【正文】 0 x3 6 5 0 1 2 0 0 6/5 4 x2 4 3 1 0 1 0 0 4/3 0 x5 5 4 0 0 1 1 0 5/4 0 x6 1 2 0 0 1 0 1 cjzj 15 0 0 4 0 0 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 0 x5 1/5 0 0 4/5 3/5 1 0 76 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 0 x3 6 5 0 1 2 0 0 6/5 4 x2 4 3 1 0 1 0 0 4/3 0 x5 5 4 0 0 1 1 0 5/4 0 x6 1 2 0 0 1 0 1 cjzj 15 0 0 4 0 0 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 0 x5 1/5 0 0 4/5 3/5 1 0 0 x6 7/5 0 0 2/5 1/5 0 1 77 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 0 x3 6 5 0 1 2 0 0 6/5 4 x2 4 3 1 0 1 0 0 4/3 0 x5 5 4 0 0 1 1 0 5/4 0 x6 1 2 0 0 1 0 1 cjzj 15 0 0 4 0 0 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 0 x5 1/5 0 0 4/5 3/5 1 0 0 x6 7/5 0 0 2/5 1/5 0 1 cjzj 0 0 3 2 0 0 78 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 0 x3 6 5 0 1 2 0 0 6/5 4 x2 4 3 1 0 1 0 0 4/3 0 x5 5 4 0 0 1 1 0 5/4 0 x6 1 2 0 0 1 0 1 cjzj 15 0 0 4 0 0 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 2 0 x5 1/5 0 0 4/5 3/5 1 0 1/3 0 x6 7/5 0 0 2/5 1/5 0 1 7 cjzj 0 0 3 2 0 0 79 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 2 0 x5 1/5 0 0 4/5 3/5 1 0 1/3 0 x6 7/5 0 0 2/5 1/5 0 1 7 cjzj 0 0 3 2 0 0 0 x4 1/3 0 0 4/3 1 5/3 0 80 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 2 0 x5 1/5 0 0 4/5 3/5 1 0 1/3 0 x6 7/5 0 0 2/5 1/5 0 1 7 cjzj 0 0 3 2 0 0 3 x1 4/3 1 0 1/3 0 2/3 0 0 x4 1/3 0 0 4/3 1 5/3 0 81 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 2 0 x5 1/5 0 0 4/5 3/5 1 0 1/3 0 x6 7/5 0 0 2/5 1/5 0 1 7 cjzj 0 0 3 2 0 0 3 x1 4/3 1 0 1/3 0 2/3 0 4 x2 1/3 0 1 1/3 0 1/3 0 0 x4 1/3 0 0 4/3 1 5/3 0 82 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 2 0 x5 1/5 0 0 4/5 3/5 1 0 1/3 0 x6 7/5 0 0 2/5 1/5 0 1 7 cjzj 0 0 3 2 0 0 3 x1 4/3 1 0 1/3 0 2/3 0 4 x2 1/3 0 1 1/3 0 1/3 0 0 x4 1/3 0 0 4/3 1 5/3 0 0 x6 4/3 0 0 2/15 1/5 1/3 1 83 cj 3 4 0 0 0 0 CB XB b x1 x2 x3 x4 x5 x6 θ 3 x1 6/5 1 0 1/5 2/5 0 0 4 x2 2/5 0 1 3/5 1/5 0 0 2 0 x5 1/5 0 0 4/5 3/5 1 0 1/3 0 x6 7/5 0 0 2/5 1/5 0 1 7 cjzj 0 0 3 2 0 0 3 x1 4/3 1 0 1/3 0 2/3 0 4 x2 1/3 0 1 1/3 0 1/3 0 0 x4 1/3 0 0 4/3 1 5/3 0 0 x6 4/3 0 0 2/15 0 1/3 1 cjzj 0 0 1/3 0 2/3 0 此時(shí)對(duì)偶問(wèn)題和原問(wèn)題都達(dá)到可行，所以均達(dá)到了最優(yōu)解 X＝（ 4/,0,1/3,0,4/3) Z=8/3 84 第 7節(jié) 靈敏度分析 前提條件 : ? 原線性規(guī)劃問(wèn)題已取得了最優(yōu)解； ? 每次只討論一種參數(shù)的變化，而參數(shù)之間的變化互不關(guān)聯(lián)。使得 alk＝ 1，其余 aik為 0 48 Minw=5y1+2y2+4y3 . 3y1+ y2+2y3≥4 6y1+3y2+5y3≥10 y1,y2,y3≥0 舉例： Maxw’=5y12y24y3 . 3y1 y22y3≤4 6y13y25y3≤10 y1,y2,y3≥0 Maxw’=5y12y24y3 . 3y1 y22y3+y4=4 6y13y25y3+y5=10 yi≥0(i=1,2,…,5) 49 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 cjzj 5 2 4 0 0 50 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 cjzj 5 2 4 0 0 51 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 52 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 2 y2 10/3 2 1 5/3 0 1/3 53 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 0 y4 2/3 1 0 1/3 1 1/3 2 y2 10/3 2 1 5/3 0 1/3 54 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 0 y4 2/3 1 0 1/3 1 1/3 2 y2 10/3 2 1 5/3 0 1/3 cjzj 1 0 2/3 0 2/3 55 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 0 y4 2/3 1 0 1/3 1 1/3 1 2 2 2 y2 10/3 2 1 5/3 0 1/3 cjzj 1 0 2/3 0 2/3 56 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 0 y4 2/3 1 0 1/3 1 1/3 1 2 2 2 y2 10/3 2 1 5/3 0 1/3 cjzj 1 0 2/3 0 2/3 5 y1 2/3 1 0 1/3 1 1/3 57 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 0 y4 2/3 1 0 1/3 1 1/3 1 2 2 2 y2 10/3 2 1 5/3 0 1/3 cjzj 1 0 2/3 0 2/3 5 y1 2/3 1 0 1/3 1 1/3 2 y2 2 0 1 1 2 1 58 cj 5 2 4 0 0 CB XB b y1 y2 y3 y4 y5 θ 0 y4 4 3 1 2 1 0 0 y5 10 6 3 5 0 1 5/6 2/3 4/5 cjzj 5 2 4 0 0 0 y4 2/3 1 0 1/3 1 1/3 1 2 2 2 y2 10/3 2 1 5/3 0 1/3 cjzj 1 0 2/3 0 2/3 5 y1 2/3 1 0 1/3 1 1/3 2 y2 2 0 1 1 2 1 cjzj 0 0 1/3 1 1/3 此時(shí)對(duì)偶問(wèn)題和原問(wèn)題都達(dá)到可行，所以均達(dá)到了最優(yōu)解 Y＝（ 2/,0,0,0) W’=22/3 W=22/3 59 Minw=2x1+3x2+4x3 . x1+2x2+ x3≥3 2x1 x2+3x3≥4 x1,x2,x3≥0 練習(xí) :用對(duì)偶單純形法求解并求出對(duì)偶變量的最優(yōu)解 Maxw’=2x13x24x3 . x1 2x2x3≤3 2x1 +x23x3≤4 x1,x2,x3≥0 Maxw’=2x13x24x3 . x12x2x3+x4=3 2x1 +x23x3 +x5=4 xi≥0(i=1,2,…,5) Maxz=3y1+4y2 . y1+2y2≤2 2y1 y2≤3 y1+3y2≤4 y1,y2≥0 對(duì)偶問(wèn)題為 60 cj 2 3 4 0 0 CB XB b x1 x2 x3 x4 x5 θ 0 x4 3 1 2 1 1 0 0 x5 4 2 1 3 0 1 1 － 4/3 cjzj 2 3 4 0 0 0 x4 1 0 5/2 1/3 1 1/2 8/5 3 2 2 x1 2 1 1/2 2/3 0 1/2 cjzj 0 4 1 0 1 3 x2 2/5 1 0 1/5 2/5 1/5 2 x1 11/5 0 1 7/5 1/5 2/5 cjzj 0 0 3/5 8/5 1/5 此時(shí)對(duì)偶問(wèn)題和原問(wèn)題都達(dá)到可行， 所以均達(dá)到了最優(yōu)解 Y＝（ 11/,0,0,0) W’=28/5 W=28/5 61 Maxz=3y1+4y2 . y1+2y2≤2 2y1 y2≤3