# lec17 - 15.053 z Introduction Thursday, April 19 to Integer...

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1 15.053 Thursday, April 19 z Introduction to Integer Programming Integer programming models

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Quotes of the Day Somebody who thinks logically is a nice contrast to the real world. -- The Law of Thumb “Take some more tea,” the March Hare said to Alice, very earnestly. “I’ve had nothing yet,” Alice replied in an offended tone, “so I can’t take more.” “You mean you can’t take less,” said the Hatter. “It’s very easy to take more than nothing.” -- Lewis Carroll in Alice in Wonderland 2
Integer Programs Integer programs : a linear program plus the additional constraints that some or all of the variables must be integer valued. We also permit “ x j {0,1} ,” or equivalently, “x j is binary This is a shortcut for writing the constraints: 0 x j 1 and x j integer. Integer programs are limited in form in that they can only include: 1. a linear objective 2. linear inequalities and equalities 3. Some or all of the variables can be required to be integer valued. 4. Since it is so easy to model that x j {0, 1}, we permit this type of constraint as well. 3

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4 A 2-Variable Integer program maximize 3x + 4y subject to 5x + 8y 24 x, y 0 and integer z What is the optimal solution? Integer programs are notoriously hard to solve. We illustrate this by pointing out that even a relatively simple integer program offers some challenges.
The Feasible Region 0 1 2 3 4 5 0 1 2 3 4 5 Question: What is the optimal integer solution? What is the optimal linear solution? Can one use linear programming to solve the integer program? The feasible region here consists of the yellow circles only.

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A rounding technique that sometimes is useful, and sometimes not. 0 1 2 3 4 5 0 1 2 3 4 5 Solve LP (ignore integrality) get x=24/5, y=0 and z =14 2/5. Round, get x=5, y=0, infeasible! Truncate, get x=4, y=0, and z =12 Same solution value at x=0, y=3. Optimal is x=3, y=1 , and z =13 Often we will consider the “linear programming relaxation” of an integer program. This relaxation is obtained by dropping the integrality constraints. In this case, if we solve the linear programming relaxation, we get the solution x = 24/5, y = 0. It’s somewhat close to the optimal integer solution. However, in this case, it doesn’t really help us to obtain the optimum integer solution.
Why integer programs? z Advantages of restricting variables to take on integer values More realistic More flexibility z Disadvantages More difficult to model Can be much more difficult to solve In practice, most problems have at least some features that prevent linear programming from being a good enough model. For this reason, integer programming models are much more practical. They can model logical constraints, nonlinear functions, and much more. While it is a robust and excellent modeling tool, it is much harder to solve than linear programs.

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## This note was uploaded on 01/17/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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lec17 - 15.053 z Introduction Thursday, April 19 to Integer...

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