May 2015
Intermediate to advanced
466 pages
10h 33m
English
Content preview from Optimization
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273Integer Programming
b c=
=
b
b
b
c
c
c
m n
1
2
1
2
,
=
, x
x
x
x
n
1
2
If some of the cost coefcients (x
i
) are negative, they can be put in the stan-
dard form by the substitution
x y y
i i i
= − ∈1 0 1, { , }
(10.4)
For example, the following problem
Minimize
f(x) = x
1
− x
2
subject to
−2x
1
− 3x
2
≤ −5
is written in standard form as
Minimize
f(x) = x
1
+ y
2
subject to
−2x
1
+ 3y
2
≤ −2
x y
1 2
0 1, { , }∈
Let us explain Balas’ method through an example. Consider the following
zero-one integer programming problem (Bricker 1999).
Minimize
f(x) = 4x
1
+ 8x
2
+ 9x
3
+ 3x
4
+ 4x
5
+ 10x
6
274 Optimization: Algorithms and Applications
subject to
4x
1
− 5x
2
− 3x
3
− 2x
4
− x
5
+ 8x
6
≤ −8
−5x
1
+ 2x
2
+ 9x
3
+ 8x
4
− 3x
5
+ 8x
6
≤ 7
8x
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