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Unformatted text preview: Integer Programming In Chap. 3 you saw several examples of the numerous and diverse applications of linear programming. However, one key limitation that prevents many more applications is the assumption of divisibility (see Sec. 3.3), which requires that noninteger values be permissible for decision variables. In many practical problems, the decision variables actually make sense only if they have integer values. For example, it is often necessary to assign people, machines, and vehicles to activities in integer quantities. If requiring integer values is the only way in which a problem deviates from a linear progr amming formulation, then it is an intege r programming (IP) problem. (The more complete name is integer lin ear programming, but the adjective linear normally is dropped except when this problem is contrasted with the more esoteric integer nonlinear programming problem, which is beyond the scope of this book.) The mathematical model for integer programming is the linear programming model (see Sec. 3.2) with the one additional restriction that the variables must have integer values. If only some of the variables are required to have integer values (so the divisibility assumption holds for the rest), this model is referred to as mixed in teger 494 programming (MIP). When distinguishing the all-integer problem from this mixed 495 case, we call the former pure integer programming. 12 1/ Prototype For example, the Wyndor Glass Co. problem presented in Sec. 3.1 actually would have been an IP problem if the two decision variables x 1 and x 2 had represe nted the total number of units to be produced of products 1 and 2, respectively, instead of the production rates. Because both products (glass doors and wood-framed windows) nec essarily come in whole units, x 1 and x 2 would have to be restricted to integer values. There have been numerous such applications of integer programming that in volve a direct extension of linear programming where the divisibility assumption must be dropped. However, another area of application may be of even greater importance, namely, problems involving a number of interrelated yes-or-no decisions. In such decisions, the only two possible choices are yes and no. For example, should we undertake a particular fixed project? Should we make a particul ar fixed investment? Should we locate a facility in a particular site? With just two choices, we can represent such decisions by decision variables that are restricted to just two values, say 0 and 1. Thus the jth yes- or-no decision would be represented by, say, x 1 such that Ti if decision j is yes, X jo if decision j is no. Such variables are called binary variables (or 01 variables). Consequently, IP prob lems that contain only binary variables sometimes are called binary integer program ining (BIP) problems (or 01 integer programming problems). Section 12.1 presents a miniature version of a typical BIP problem. Additional formulation possibilities with binary variables are discussed in Sec. 12.2, and Sec. 12.3formulation possibilities with binary variables are discussed in Sec....
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- Fall '09