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(b)
max z = 5 x1 + 6 x2 + 3x3 x1 + 2 x2 + 2 x3 = 5 ⎧ ⎪ − x + 5x − x ≥ 3 ⎪ 1 2 3 s.t. ⎨ ⎪ 4 x1 + 7 x2 + 3x3 ≤ 8 ⎪ ⎩ x1无约束，x2 ≥ 0, x3 ≤ 0
解
min w = 5 y1 + 3 y2 + 8 y3 y1 − y2 + 4 y3 = 5 ⎧ ⎪ 2y + 5y + 7 y ≥ 6 ⎪ 1 2 3 s.t. ⎨ ⎪ 2 y1 − y2 + 3 y3 ≤ 3 ⎪ ⎩ y1无约束, y2 ≤ 0, y3 ≥ 0
2.1 写出下列线性规划的对偶问题 （ a）
min z = 2 x1 + 2 x2 + 4 x3
⎧ x1 + 3 x2 + 4 x3 ≥ 2 ⎪ ⎪ 2 x1 + x2 + 3 x3 ≤ 3 s.t. ⎨ ⎪ x1 + 4 x2 + 3x3 = 5 ⎪ ⎩ x1 , x2 ≥ 0, x3无约束
解
来自百度文库
max w = 2 y1 + 3 y2 + 5 y3 ⎧ y1 + 2 y2 + y3 ≤ 2 ⎪ 3y + y + 4 y ≤ 2 ⎪ 1 2 3 s.t. ⎨ ⎪ 4 y1 + 3 y2 + 3 y3 = 4 ⎪ ⎩ y1 ≥ 0, y2 ≤ 0, y3无约束
(c)
min z = ∑∑ cij xij
i =1 j =1
n ⎧ xij = ai (i = 1," , m) ⎪ ∑ j =1 ⎪ m ⎪ s.t. ⎨ ∑ xij = b j ( j = 1," , n) i =1 ⎪ ⎪ xij ≥ 0 ( i = 1," m; j = 1," , n ) ⎪ ⎩
(d)
max z = ∑ c j x j
j =1
n
⎧ n aij x j ≤ bi ( i = 1, 2," , m1 < m ) ⎪ ∑ j =1 ⎪ n ⎪ a x =b i = m + 1, m + 2," , m ) 1 1 ⎨∑ ij j i ( j =1 ⎪ x j ≥ 0 ( j = 1," , n1 < n ) ⎪ ⎪ x j 无约束 ( j = n1 + 1," , n ) ⎩
解：设对偶变量为 yi ( i = 1," , m1 , m1 + 1," , m )
min w = ∑ bi yi
i =1
m
s.t.
⎧ m ⎪ ∑ aij y i ≥ c j ( j = 1, 2, " , n1 ) i =1 ⎪m ⎪ ⎨∑ aij y i = c j ( j = n1 + 1, " , n) ⎪ i =1 y i ≥ 0 (i = 1, " , m1 ) ⎪ ⎪ y 无约束(i = m + 1, " , m) 1 ⎩ i
m
n
解：设对偶变量为 ui (i = 1, 2," , m) ， v j ( j = 1, 2," , n) ，则所 给问题的对偶问题为
m n
max z = ∑ ai ui + ∑ b j v j
i =1 j =1
ui + v j ≤ cij ⎧ ⎪ s.t. ⎨ ui , v j 无约束 ⎪i = 1, 2," m, j = 1, 2," n ⎩
