运筹学清华大学出版社(胡运权着)课后答案(....pdf

1、运筹学教程第二版运筹学教程第二版 习题解答习题解答运筹学教程运筹学教程School of ManagementSchool of Managementpage page 2 26 January 20116 January 20111.1 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(21212121xxxxxxstxxZ0,124322.23max)2(21212121xxxxxxstxxZ85105120106.max)3(212121xxxxstxxZ0,23222.65max)4(21212121xxxx

2、xxstxxZ运筹学教程运筹学教程School of ManagementSchool of Managementpage page 3 36 January 20116 January 2011是一个最优解无穷多最优解3,31,10,422664.32min)1(2121212121ZxxxxxxxxstxxZ该问题无解0,124322.23max)2(21212121xxxxxxstxxZ运筹学教程运筹学教程School of ManagementSchool of Managementpage page 4 46 January 20116 January 201116,6,108510

3、5120106.max)3(21212121ZxxxxxxstxxZ唯一最优解该问题有无界解0,23222.65max)4(21212121xxxxxxstxxZ运筹学教程运筹学教程School of ManagementSchool of Managementpage page 5 56 January 20116 January 20111.2 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(xxxxxxxx

4、xstxxxZ运筹学教程运筹学教程School of ManagementSchool of Managementpage page 6 66 January 20116 January 2011.,0,2321422245243min)1(43214321432143214321无约束xxxxxxxxxxxxxxxxstxxxxZ0,232142222455243max64241321642413215424132142413214241321xxxxxxxxxxxxxxxxxxxxxxxstxxxxxZ运筹学教程运筹学教程School of ManagementSchool of Mana

5、gementpage page 7 76 January 20116 January 2011无约束321321321321,0,0624322min)2(xxxxxxxxxstxxxZ0,6243322max43231214323121323121323121xxxxxxxxxxxxxxstxxxxZ运筹学教程运筹学教程School of ManagementSchool of Managementpage page 8 86 January 20116 January 20111.3 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。6,1,0031024893631223m

6、ax)1(6153214321321jxxxxxxxxxxxstxxxZj)4,1(,0322274322325min)2(432143214321jxxxxxxxxxstxxxxZj运筹学教程运筹学教程School of ManagementSchool of Managementpage page 9 96 January 20116 January 20116,1,0031024893631223max)1(6153214321321jxxxxxxxxxxxstxxxZj基可行解x1x2x3x4x5x6Z03003.503001.5080300035000.7500022.252.25运

7、筹学教程运筹学教程School of ManagementSchool of Managementpage page 10106 January 20116 January 2011)4,1(,0322274322325min)2(432143214321jxxxxxxxxxstxxxxZj基可行解x1x2x3x4Z00.5205001152/5011/5043/5运筹学教程运筹学教程School of ManagementSchool of Managementpage page 11116 January 20116 January 20111.4 分别用图解法和单纯形法求解下述线性规划

8、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运筹学教程运筹学教程School of ManagementSchool of Managementpage page 12126 January 20116 January 20110,24261553.2max)2(21212121xxxxxxstxxZ运筹学教程运筹学教程School of ManagementSchool of Managementpage page 13136 January 20116 January 2011l.5 上题(1

9、)中若目标函数变为max Z=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Cjcd00CB基bx1x2x3x4dx23/2015/14-3/4cx1110-2/1410/35j00-5/14d+2/14c 3/14d-10/14c运筹学教程运筹学教程School of ManagementSchool of Managementpage page 14146 January 20116 January 2011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、/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。运筹学教程运筹学教程School of ManagementSchool of Managementpage page 15156 January 20116 January 2011式中1c1 3,4c2 6,-1a11 3,2a12 5,8b1 12,2a21 5,4a22 6,10b2 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运筹学教

11、程School of ManagementSchool of Managementpage page 16166 January 20116 January 2011最优值上界为210,14421221.63max21212121xxxxxxstxxZ解上界对应的模型如下c,b取大a取小运筹学教程运筹学教程School of ManagementSchool of Managementpage page 17176 January 20116 January 2011最优值下界为6.40,1064853.4max21212121xxxxxxstxxZ解下界对应的模型如下 c,b取小a取大运筹学

12、教程运筹学教程School of ManagementSchool of Managementpage page 18186 January 20116 January 2011l.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(3231321321jxxxxxxxxstxxxZj运筹学教程运筹学教程School of ManagementSchool of Managementpage page 19196 January 20116 January 20116,0,54,590,623824.32min)2(32

13、12121321321ZxxxxxxxxxxstxxxZ最优解之一该题是无穷多最优解。运筹学教程运筹学教程School of ManagementSchool of Managementpage page 20206 January 20116 January 2011517,0,1,59,524,1,042634334max)3(43214213212121ZxxxxjxxxxxxxxxstxxZj该题是唯一最优解运筹学教程运筹学教程School of ManagementSchool of Managementpage page 21216 January 20116 January 20

14、11该题无可行解。3,1,052151565935121510max)4(321321321321jxxxxxxxxxxstxxxZj运筹学教程运筹学教程School of ManagementSchool of Managementpage page 22226 January 20116 January 20111.8 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al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=

15、2,c=4,d=-2,g=1,h=0,f=3,i=5,e=2,l=0,a=3,j=5,k=-1.5运筹学教程运筹学教程School of ManagementSchool of Managementpage page 23236 January 20116 January 20111.9 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

16、()1()2()1(运筹学教程运筹学教程School of ManagementSchool of Managementpage page 24246 January 20116 January 20111.10 线性规划问题max ZCX,AXbX0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*00XXCXXCXXCCXCXCXCZXCXCXCXZX的最优解故是的最优解故是运筹学教程运筹学教程School of ManagementSchool of Managementpage page 25256 January 20116 January 20111.11 考虑线性规划问题0,)(75232)(24.42min432143214214321xxxxiixxxxixxxstxxxxZ模型中为参数要求(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运筹学教程School of Manage