清华大学《运筹学教程》胡运权主编课后习题答案

运筹学教程SchoolofManagementpage113June20201同样适合第三版黄皮版运筹学教程SchoolofManagementpage213June2020运筹学教程（第二版）习题解答安徽大学管理学院洪文运筹学教程SchoolofManagementpage313June20203第一章习题解答1.1用图解法求解下列线性规划问题。并指出问题具有惟一最优解、无穷多最优解、无界解还是无可行解。0,422664.32min)1(21212121xxxxxxstxxZ0,124322.23max)2(21212121xxxxxxstxxZ85105120106.max)3(212121xxxxstxxZ0,23222.65max)4(21212121xxxxxxstxxZ运筹学教程SchoolofManagementpage413June20204第一章习题解答是一个最优解无穷多最优解，3,31,10,422664.32min)1(2121212121ZxxxxxxxxstxxZ该问题无解0,124322.23max)2(21212121xxxxxxstxxZ运筹学教程SchoolofManagementpage513June20205第一章习题解答16,6,1085105120106.max)3(21212121ZxxxxxxstxxZ唯一最优解，该问题有无界解0,23222.65max)4(21212121xxxxxxstxxZ运筹学教程SchoolofManagementpage613June20206第一章习题解答1.2将下述线性规划问题化成标准形式。.,0,,2321422245243min)1(43214321432143214321无约束xxxxxxxxxxxxxxxxstxxxxZ无约束321321321321,0,0624322min)2(xxxxxxxxxstxxxZ运筹学教程SchoolofManagementpage713June20207第一章习题解答.,0,,2321422245243min)1(43214321432143214321无约束xxxxxxxxxxxxxxxxstxxxxZ0,,,,,232142222455243max64241321642413215424132142413214241321xxxxxxxxxxxxxxxxxxxxxxxstxxxxxZ运筹学教程SchoolofManagementpage813June20208第一章习题解答无约束321321321321,0,0624322min)2(xxxxxxxxxstxxxZ0,,,,6243322max43231214323121323121323121xxxxxxxxxxxxxxstxxxxZ运筹学教程SchoolofManagementpage913June20209第一章习题解答1.3对下述线性规划问题找出所有基解，指出哪些是基可行解，并确定最优解。）（6,,1,0031024893631223max)1(6153214321321jxxxxxxxxxxxstxxxZj)4,1(,0322274322325min)2(432143214321jxxxxxxxxxstxxxxZj运筹学教程SchoolofManagementpage1013June202010第一章习题解答）（6,,1,0031024893631223max)1(6153214321321jxxxxxxxxxxxstxxxZj基可行解x1x2x3x4x5x6Z03003.503001.5080300035000.7500022.252.25运筹学教程SchoolofManagementpage1113June202011第一章习题解答)4,1(,0322274322325min)2(432143214321jxxxxxxxxxstxxxxZj基可行解x1x2x3x4Z00.5205001152/5011/5043/5运筹学教程SchoolofManagementpage1213June202012第一章习题解答1.4分别用图解法和单纯形法求解下述线性规划问题，并对照指出单纯形表中的各基可行解对应图解法中可行域的哪一顶点。0,825943.510max)1(21212121xxxxxxstxxZ运筹学教程SchoolofManagementpage1313June202013第一章习题解答0,24261553.2max)2(21212121xxxxxxstxxZ运筹学教程SchoolofManagementpage1413June202014第一章习题解答l.5上题(1)中，若目标函数变为maxZ=cx1+dx2，讨论c,d的值如何变化，使该问题可行域的每个顶点依次使目标函数达到最优。解：得到最终单纯形表如下：Cj→cd00CB基bx1x2x3x4dx23/2015/14-3/4cx1110-2/1410/35j00-5/14d+2/14c3/14d-10/14c运筹学教程SchoolofManagementpage1513June202015第一章习题解答当c/d在3/10到5/2之间时最优解为图中的A点；当c/d大于5/2且c大于等于0时最优解为图中的B点；当c/d小于3/10且d大于0时最优解为图中的C点；当c/d大于5/2且c小于等于0时或当c/d小于3/10且d小于0时最优解为图中的原点。运筹学教程SchoolofManagementpage1613June202016第一章习题解答式中，1≤c1≤3,4≤c2≤6,-1≤a11≤3,2≤a12≤5,8≤b1≤12,2≤a21≤5,4≤a22≤6,10≤b2≤14,试确定目标函数最优值的下界和上界。0,.max21222212112121112211xxbxaxabxaxastxcxcZl.6考虑下述线性规划问题：运筹学教程SchoolofManagementpage1713June202017第一章习题解答最优值（上界）为：210,14421221.63max21212121xxxxxxstxxZ解：上界对应的模型如下（c,b取大，a取小）运筹学教程SchoolofManagementpage1813June202018第一章习题解答最优值（下界）为：6.40,1064853.4max21212121xxxxxxstxxZ解：下界对应的模型如下（c,b取小，a取大）运筹学教程SchoolofManagementpage1913June202019第一章习题解答l.7分别用单纯形法中的大M法和两阶段法求解下列线性规划问题，并指出属哪—类解。该题是无界解。）（3,,1,00222623max)1(3231321321jxxxxxxxxstxxxZj运筹学教程SchoolofManagementpage2013June202020第一章习题解答运筹学教程SchoolofManagementpage2113June202021第一章习题解答517,0,1,59,524,,1,042634334max)3(43214213212121ZxxxxjxxxxxxxxxstxxZj该题是唯一最优解：）（运筹学教程SchoolofManagementpage2213June202022第一章习题解答该题无可行解。）（3,,1,052151565935121510max)4(321321321321jxxxxxxxxxxstxxxZj运筹学教程SchoolofManagementpage2313June202023第一章习题解答1.8已知某线性规划问题的初始单纯形表和用单纯形法迭代后得到下面表格，试求括弧中未知数a∼l值。项目X1X2X3X4X5X46(b)(c)(d)10X51-13(e)01Cj－Zja-1200X1(f)(g)2-11/20X54(h)(i)11/21Cj－Zj0-7jk(l)b=2,c=4,d=-2,g=1,h=0,f=3,i=5,e=2,l=0,a=3,j=5,k=-1.5运筹学教程SchoolofManagementpage2413June202024第一章习题解答1.9若X(1)、X(2)均为某线性规划问题的最优解，证明在这两点连线上的所有点也是该问题的最优解。也是最优解。所以也是可行解，且满足：两点连线上的点对于任何满足：和设XXCXCXaCaXCXaCaXCXCXaaXXXaXbAXXCZXXTTTTTTTT,)1()1(,100max)2()2()2()1()2()1()2()1()2()1(运筹学教程SchoolofManagementpage2513June202025第一章习题解答1.10线性规划问题maxZ＝CX,AX＝b，X≥0，设X0为问题的最优解。若目标函数中用C*代替C后，问题的最优解变为X*，求证(C*-C)(X*-X0)≥00)()())((;0max;0max0***00**0******00XXCXXCXXCCXCXCXCZXCXCXCXZX的最优解，故是的最优解，故是运筹学教程SchoolofManagementpage2613June202026第一章习题解答1.11考虑线性规划问题0,,,)(75232)(24.42min432143214214321xxxxiixxxxixxxstxxxxZ模型中α，β为参数，要求：(1)组成两个新的约束(i)’＝(i)+(ii)，(ii)’＝(ii)一2(i)，根据(i)’，(ii)’以x1,x2为基变量，列出初始单纯形表；运筹学教程SchoolofManagementpage2713June202027第一章习题解答1)(23)(32431xxiixxxiCj→a21-4CB基bx1x2x3x4ax13+2011-12x21-10-10j003-aa-4运筹学教程SchoolofManagementpage2813June202028第一章习题解答(2)在表中，假定β＝0，则α为何值时，x1,x2为问题的最优基变量；解：如果=0，则当3≤a≤4时，x1,x2为问题的最优基变量；(3)在表中，假定α＝3，则β为何值时，x1,x2为问题的最优基。解：如果a=3，则当-1≤≤1时，x1,x2为问题的最优基变量。运筹学教程SchoolofManagementpage2913June202029第一章习题解答1.12线性规划问题maxZ＝CX，AX＝b，X≥0，如X*是该问题的最优解，又λ0为某一常数，分别讨论下列情况时最优解的变化。(1)目标函数变为maxZ＝λCX；(2)目标函数变为maxZ＝(C+λ)X；(3)目标函数变为maxZ＝C/λ*