运筹学教程
同样适合 第三版黄皮版
page 1 4 September 2011
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School of Management
运筹学教程
运筹学教程(第二版) 运筹学教程(第二版) 习题解答安徽大学管理学院
洪 文
page 2 4 September 2011
School of Management
运筹学教程
第一章习题解答1.1 用图解法求解下列线性规划问题。 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、 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。 无界解还是无可行解。(1) min Z = 2 x1 + 3 x 2 4 x1 + 6 x 2 ≥ 6 st . 2 x1 + 2 x 2 ≥ 4 x ,x ≥ 0 1 2
( 2)
max Z = 3 x1 + 2 x 2 2 x1 + x 2 ≤ 2 st . 3 x1 + 4 x 2 ≥ 12 x , x ≥ 0 1 2 max Z = 5 x1 + 6 x 2 2 x1 x 2 ≥ 2 st . 2 x1 + 3 x 2 ≤ 2 x ,x ≥ 0 1 2 3
( 3)
max Z = x1 + x 2 6 x1 + 10 x 2 ≤ 120 st . 5 ≤ x1 ≤ 10 5≤ x ≤8 2
( 4)
page 3 4 September 2011
School of Management
运筹学教程
第一章习题解答(1) min Z = 2 x1 + 3 x 2 4 x1 + 6 x 2 ≥ 6 st . 2 x1 + 2 x 2 ≥ 4 x ,x ≥ 0 1 2 1 , Z = 3是一个最优解 3
无穷多最优解， x1 = 1, x 2 =
(2)
max Z = 3 x 1 + 2 x 2 2 x1 + x 2 ≤ 2 st . 3 x 1 + 4 x 2 ≥ 12 x , x ≥ 0 1 2
该问题无解page 4 4 September 2011 4
School of Management
运筹学教程
第一章习题解答( 3) max Z = x1 + x 2 6 x1 + 10 x 2 ≤ 120 st . 5 ≤ x1 ≤ 10 5≤ x ≤8 2
唯一最优解， x1 = 10 , x 2 = 6, Z = 16max Z = 5 x1 + 6 x 2 2 x1 x 2 ≥ 2 ( 4) st . 2 x1 + 3 x 2 ≤ 2 x ,x ≥ 0 1 2 该问题有无界解
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运筹学教程
第一章习题解答1.2 将下述线性规划问题化成标准形式。 将下述线性规划问题化成标准形式。min Z = 3 x1 + 4 x 2 2 x3 + 5 x 4 4 x1 x 2 + 2 x3 x 4 = 2 x + x x + 2 x ≤ 14 2 3 4 st 1 . 2 x1 + 3 x 2 + x3 x 4 ≥ 2 x1 , x 2 , x3 ≥ 0, x 4 无约束 min st x 1 Z = 2 x1 2 x 2 + 3 x 3 x1 + x 2 + x 3 = 4 2 x1 + x 2 x 3 ≤ 6 ≤ 0 , x 2 ≥ 0 , x 3 无约束6
(1)
(2)
page 6 4 September 2011
School of Management
运筹学教程
第一章习题解答minZ = 3x1 + 4x2 2x3 + 5x4 4x1 x2 + 2x3 x4 = 2 x + x x + 2x ≤ 14 (1) 4 st 1 2 3 . 2x1 + 3x2 + x3 x4 ≥ 2 x1, x2 , x3 ≥ 0, x4无约束 max Z = 3 x1 4 x 2 + 2 x3 5 x 41 + 5 x 42 4 x1 + x 2 2 x3 + x 41 x 42 = 2 x + x x + 2 x 2 x + x = 14 2 3 41 42 5 st 1 2 x1 + 3 x 2 + x3 x 41 + x 42 x6 = 2 x1 , x 2 , x3 , x 41 , x 42 , x 6 ≥ 0 page 7 4 September 2011 7
School of Management
运筹学教程
第一章习题解答(2) min st x 1 Z = 2 x1 2 x 2 +
3 x 3 x1 + x 2 + x 3 = 4 2 x1 + x 2 x 3 ≤ 6 ≤ 0 , x 2 ≥ 0 , x 3 无约束
max Z = 2 x1 + 2 x 2 3 x 31 + 3 x 32 x1 + x 2 + x 31 x 32 = 4 st 2 x1 + x 2 x 31 + x 32 + x 4 = 6 x1 , x 2 , x 31 , x 32 , x 4 ≥ 0
page 8 4 September 2011
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运筹学教程
第一章习题解答对下述线性规划问题找出所有基解， 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。 指出哪些是基可行解，并确定最优解。max Z = 3 x1 + x 2 + 2 x 3 12 x1 + 3 x 2 + 6 x 3 + 3 x 4 = 9 8 x + x 4 x + 2 x = 10 1 2 3 5 st 3 x1 x 6 = 0 x j ≥ 0( j = 1, L , 6) , min Z = 5 x1 2 x 2 + 3 x 3 + 2 x 4 x1 + 2 x 2 + 3 x 3 + 4 x 4 = 7 st 2 x1 + 2 x 2 + x 3 + 2 x 4 = 3 x ≥ 0 , ( j = 1, L 4 ) j 9
(1)
(2)
page 9 4 September 2011
School of Management
运筹学教程
第一章习题解答(1) max Z = 3 x1 + x 2 + 2 x 3 12 x1 + 3 x 2 + 6 x 3 + 3 x 4 = 9 8 x + x 4 x + 2 x = 10 1 2 3 5 st 3 x1 x 6 = 0 x j ≥ 0( j = 1, L , 6) ,
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运筹学教程
第一章习题解答max Z = 4 x1 + x2 3x1 + x2 = 3 4x + 3x x = 6 (3) 1 2 3 st x1 + 2 x2 + x4 = 4 x j ≥ 0, j = 1,L,4) ( 该题是唯一最优解： 2 9 17 x1 = , x2 = , x3 = 1, x4 = 0, Z = 5 5 5
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