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

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
1 15.053 Tuesday, April 24 Integer Programming Formulations 2
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 32

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

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online