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Unformatted text preview: Integer Programming Recall that we defined integer programming problems in our discussion of the Divisibility As- sumption in Section 3.1. Simply stated, an integer programming problem (IP) is an LP in which some or all of the variables are required to be non-negative integers. In this chapter (as for LPs in Chapter 3), we find that many real-life situations may be formu- lated as IPs. Unfortunately, we will also see that IPs are usually much harder to solve than LPs. In Section 9.1, we begin with necessary definitions and some introductory comments about IPs. In Section 9.2, we explain how to formulate integer programming models. We also dis- cuss how to solve IPs on the computer with LINDO, LINGO, and Excel Solver. In Sections 9.39.8, we discuss other methods used to solve IPs. 9.1 Introduction to Integer Programming An IP in which all variables are required to be integers is called a pure integer pro- gramming problem. For example, max z 3 x 1 2 x 2 s.t. x 1 x 2 6 (1) x 1 , x 2 0, x 1 , x 2 integer is 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. For example, max z 3 x 1 2 x 2 s.t. x 1 x 2 6 x 1 , x 2 0, x 1 integer is a mixed integer programming problem ( x 2 is not required to be an integer). An integer programming problem in which all the variables must equal 0 or 1 is called a 01 IP. In Section 9.2, we see that 01 IPs occur in surprisingly many situations. The following is an example of a 01 IP: max z x 1 x 2 s.t. x 1 2 x 2 2 (2) 2 x 1 x 2 1 x 1 , x 2 0 or 1 Solution procedures especially designed for 01 IPs are discussed in Section 9.7. A nonlinear integer programming problem is an optimization problem in which either the objective function or the left-hand side of some of the constraints are nonlinear functions and some or all of the variables must be integers. Such problems may be solved with LINGO or Excel Solver. Actually, any pure IP can be reformulated as an equivalent 01 IP (Section 9.7). The concept of LP relaxation of an integer programming problem plays a key role in the solution of IPs. D E F I N I T I O N The LP obtained by omitting all integer or 01 constraints on variables is called the LP relaxation of the IP. For example, the LP relaxation of (1) is max z 3 x 1 2 x 2 s.t. x 1 x 2 6 (1 ) x 1 , x 2 and the LP relaxation of (2) is max z x 1 x 2 s.t. x 1 2 x 2 2 (2 ) s.t. 2 x 1 x 2 1 x 1 , x 2 Any IP may be viewed as the LP relaxation plus additional constraints (the constraints that state which variables must be integers or be 0 or 1). Hence, the LP relaxation is a less constrained, or more relaxed, version of the IP. This means that the feasible region for any IP must be contained in the feasible region for the corresponding LP relaxation. For any IP that is a max problem, this implies that Optimal z-value for LP relaxation optimal z-value for IP (3) This result plays a key role when we discuss the solution of IPs.This result plays a key role when we discuss the solution of IPs....
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- Fall '10