9.1 Introduction to Integer Programming

Recall that we defined Integer Programming problems in our discussion of the Divisibility As sumption in Section 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 , we begin with necessary definitions and some introductory comments about IPs. In Section , we explain how to formulate Integer Programming models. We also dis cuss how to solve IPs on the computer with UNDO, LINGO, and Excel Solver. In Sections , we discuss other methods used to solve IPs. 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 = 3x\ + 2x2. x, + .v2 ^ 6 (1)..v,, x2 2= 0, xu x2 Integer is a pure Integer Programming problem.

TABLE 1 ables the optimal combination to obtain a higher NPV than any combination that includes invest ment 1. If, as in Chapter 3, fractional investments were allowed, the optimal solution to (9) would be x\ = .v> = 1, .v3 = 0.50, xA = 0, z = 544,000, and investment 1 would be used. This simple ex

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