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Unformatted text preview: x2 − 400 w2 + 20 x3 − 300 w3
2 x1 + 4 x2 + 5x3 ≤ 100
x1 + x2 + x3 ≤ 30
10 x1 + 5x2 + 2 x3 ≤ 204
xi ≤ Mwi for i = 1 to 3 xi ≥ 0 and integer for i = 1 to 3.
The constraint xi : Mwi forces wi = 1 whenever xi > 0. The model may look incorrect because it
permits the possibility that xi = 0 and wi = 1. It is true that the IP model allows more feasible
solutions than it should. However, if xi = 0 in an optimal solution, then wi = 0 as well because its
objective value coefficient is less than 0. Because the integer program gives optimal solutions, if xi =
0, then wi = 0. Section 6. Piecewise linear functions.
Integer programming can be used to model functions that are piecewise linear. For example,
consider the following function. ⎧ 2x
⎪
y = ⎨ 9− x
⎪
⎩ −5 + x if 0 ≤ x ≤ 3
if 4 ≤ x ≤ 7
if 8 ≤ x ≤ 9. One can model y in several different ways. Here is one of them. We first define two new variables
for every piece of the curve. ⎧1
⎪
w1 = ⎨
⎪0
⎩ if 0 ≤ x ≤ 3
otherwis...
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This note was uploaded on 03/18/2014 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.
 Spring '07
 JamesOrli
 Management

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