# lec18 - 15.053 Tuesday April 24 Integer Programming...

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1 15.053 Tuesday, April 24 Integer Programming Formulations 2

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Quote of the Day What chiefly characterizes creative thinking from more mundane forms are (i) willingness to accept vaguely defined problem statements and gradually structure them, (ii) continuing preoccupation with problems over a considerable period of time, and (iii) extensive background knowledge in relevant and potentially relevant areas. -- Herbert Simon 2
Overview of lecture z First half of lecture: more logical constraints non-linearities z Second half of lecture classic combinatorial problems set covering facility location graph coloring exam scheduling 3

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4 Logical Constraints with non-binary variables x 1 10 or x 2 25 or both Variables: x 1 ,x 2 , x 3 , are non-negative integers Create a variable w 1 with the following properties: If w 1 = 1, then x 1 10; If w 1 = 0, then x 2 25; w 1 = 0 or 1. Assume that x 1 100 . We need an upper bound. x 1 9 + 91(1-w 1 ) x 2 25 - 25w 1 x 2 25 x 2 0 x 1 100 x 1 9 w 1 = 0 w 1 = 1
5 “Or-constraints” in general. ax b or cx d or both Let M 1 and M 2 be integers so that ax M 1 and cx M 2 in any feasible solution. Typically M 1 is very large, and M 2 is “very negative” ax b + (M 1 –b±)(1-w 2 ) cx d + (M 2 –d) w 2 cx d cx M 2 ax M 1 ax b w 2 = 0 w 2 = 1 If w 2 = 1, then ax b; If w 2 = 0, then cx d; w 2 = 0 or 1.

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6 Another example | x 1 –x 2 | 10 Assume x 1 100 and x 2 200. Translation: x 1 2 10 or x 2 1 10 If w 3 = 1, then If w 3 = 0, then x 1 2 10 + x 2 1 Fill in the table Note: x 1 2 -200 x 2 1 -100 x 2 1 x 2 1 x 1 -x 2 x 1 2 w 3 = 0 w 3 = 1
7 IP and piecewise linear functions. f(x) = 2 x if 0 x 3 f(x) = 9 – x if 4 x 7 f(x) = -5 + x if 8 x 9 0 3 7 9 cost x An NLP formulation How do we model the function f(x)? Later we will assume that f(x) is part of the objective function, and we will maximize it. Assume that x is integer valued.

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8 Version 1: Create Binary Variables Step 1. Create three binary variables, w 1 , w 2 , and w 3 If 0 x 3 then w 1 = 1 w 1 + w 2 + w 3 = 1 If 4 x 7 then w 2 = 1 If 8 x 9 then w 3 = 1 w j {0, 1} for j = 1, 2, 3. Almost works, but we need something else.
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lec18 - 15.053 Tuesday April 24 Integer Programming...

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