Chapter 9 Integer Programming to accompany Operations Research: Applications and Algorithms 4th edition by Wayne L. Winston Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

2 9.1 Introduction to Integer Programming An IP in which all variables are required to be integers is call a pure integer programming problem. An IP in which only some of the variables are required to be integers is called a mixed integer programming problem. An integer programming problem in which all the variables must be 0 or 1 is called a 0-1 IP. LP relaxation The LP obtained by omitting all integer or 0-1 constraints on variables is called LP relaxation of the IP.

3 9.2 Formulating Integer Programming Problems Practical solutions can be formulated as IPs. The basics of formulating an IP model

4 Example 1: Capital Budgeting IP Page 478 Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of $16,000; investment 2, an NPV of $22,000; investment 3, an NPV of $12,000; and investment 4, an NPV of $8,000. Each investment requires a certain cash outflow at the present time: investment 1, $5,000; investment 2, $7,000; investment 3, $4,000; and investment 4, $3,000. Currently, $14,000 is available for investment.

5 Example 1: Capital Budgeting IP Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from the four investments.

6 Example 1: Solution Begin by defining a variable for each decision that Stockco must make. The NPV obtained by Stockco is Total NPV obtained by Stocko = 16x x x 3 + 8x 4 Stockco’s objective function is max z = 16x x x 3 +8x 4 Stockco faces the constraint that at most $14,000 can be invested. Stockco’s 0-1 IP is max z = 16x x x 3 +8x 4 s.t. 5x 1 + 7x 2 + 4x 3 +3x 4 ≤ 14 x j = 0 or 1 (j = 1,2,3,4)

7 Fixed-Charge Problems  Suppose activity i incurs a fixed charge if undertaken at any positive level. Let = Level of activity i = 1 if activity i is undertaken at positive level = 0 if activity i is not undertaken at positive level  Then a constraint of the form < must be added to the formulation. It must be large enough to ensure that will be less than or equal to.

8 In a set-covering problem, each member of a given set must be “covered” by an acceptable member of some set. The objective of a set-covering problem is to minimize the number of elements in set 3 that are required to cover all the elements in set 1. Given two constraints ensure that at least one is satisfied by adding an either-or-constraint.

9 M is a number chosen large enough to ensure that both constraints are satisfied for all values of that satisfy the other constraints in the problem. Suppose we want to ensure that > 0 implies. Then we include the following constraint in the formulation: Here, M is a large positive number, chosen large enough so that f < M and – g < M hold for all values of that satisfy the other constraints in the problem. This is called an if-then constraint.

variables can be used to model optimization problems involving piecewise linear functions. A piecewise linear function consists of several straight line segments. The graph of the piecewise linear function is made of four straight-line segments. The points where the slope of the piecewise linear function changes are called the break points of the function. A piecewise linear function is not a linear function so linear programming can not be used to solve the optimization problem.

11 By using 0-1 variables, however, a piecewise linear function can e represented in linear form. Suppose the piecewise linear function f (x) has break points.  Step 1 Wherever f (x) occurs in the optimization problem, replace f (x) by.  Step 2 Add the following constraints to the problem:

12 If a piecewise linear function f(x) involved in a formulation has the property that the slope of the f(x) becomes less favorable to the decision maker as x increases, then the tedious IP formulation is unnecessary. LINDO can be used to solve pure and mixed IPs. In addition to the optimal solution, the LINDO output also includes shadow prices and reduced costs. LINGO and the Excel Solver can also be used to solve IPs.