Chapter 9 Integer Programming to accompany Operations Research

9. 1 Introduction to Integer Programming n An IP in which all variables are

9. 2 Formulating Integer Programming Problems n Practical solutions can be formulated as IPs.

Example 1: Capital Budgeting IP n Page 478 n Stockco is considering four investments.

Example 1: Capital Budgeting IP n Formulate an IP whose solution will tell Stockco

Example 1: Solution n Begin by defining a variable for each decision that Stockco

n Fixed-Charge Problems o Suppose activity i incurs a fixed charge if undertaken at

n In a set-covering problem, each member of a given set must be “covered”

n M is a number chosen large enough to ensure that both constraints are

n 0 -1 variables can be used to model optimization problems involving piecewise linear

n By using 0 -1 variables, however, a piecewise linear function can e represented

n If a piecewise linear function f(x) involved in a formulation has the property

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9. 1 Introduction to Integer Programming n An IP in which all variables are required to be integers is call a pure integer programming problem. n An IP in which only some of the variables are required to be integers is called a mixed integer programming problem. n An integer programming problem in which all the variables must be 0 or 1 is called a 0 -1 IP. n The LP obtained by omitting all integer or 0 -1 constraints on variables is called LP relaxation of the IP. 2

9. 2 Formulating Integer Programming Problems n Practical solutions can be formulated as IPs. n The basics of formulating an IP model 3

Example 1: Capital Budgeting IP n Page 478 n 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. 4

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

Example 1: Solution n Begin by defining a variable for each decision that Stockco must make. n The NPV obtained by Stockco is Total NPV obtained by Stocko = 16 x 1 + 22 x 2 + 12 x 3 + 8 x 4 n Stockco’s objective function is max z = 16 x 1 + 22 x 2 + 12 x 3 +8 x 4 n Stockco faces the constraint that at most $14, 000 can be invested. n Stockco’s 0 -1 IP is max z = 16 x 1 + 22 x 2 + 12 x 3 +8 x 4 s. t. 5 x 1 + 7 x 2 + 4 x 3 +3 x 4 ≤ 14 xj = 0 or 1 (j = 1, 2, 3, 4) 6

n Fixed-Charge Problems o 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 o 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. 7

n In a set-covering problem, each member of a given set must be “covered” by an acceptable member of some set. n 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. n Given two constraints ensure that at least one is satisfied by adding an either-or-constraint. 8

n 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. n Suppose we want to ensure that > 0 implies. Then we include the following constraint in the formulation: n 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. n This is called an if-then constraint. 9

n 0 -1 variables can be used to model optimization problems involving piecewise linear functions. n A piecewise linear function consists of several straight line segments. n The graph of the piecewise linear function is made of four straight-line segments. n The points where the slope of the piecewise linear function changes are called the break points of the function. n A piecewise linear function is not a linear function so linear programming can not be used to solve the optimization problem. 10

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

n 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. n LINDO can be used to solve pure and mixed IPs. n In addition to the optimal solution, the LINDO output also includes shadow prices and reduced costs. n LINGO and the Excel Solver can also be used to solve IPs. 12