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9.4 Some Characteristics Of Integer Programs—A Sample Problem 287 subject to: D - J X j = 1 d i j x i j 0 ( i = 1 , 2 , . . . , I ), J X j = 1 x i j = 1 ( i = 1 , 2 , . . . , I ), I X i = 1 x i j y j I ( j = 1 , 2 , . . . , J ), S j - I X i = 1 p i x i j = 0 ( j = 1 , 2 , . . . , J ), J X j = 1 f j ( s j ) B , y 1 + y 2 - 2 y 0 , y 3 + y 4 + 2 y 2 , x i j , y j , y binary ( i = 1 , 2 , . . . , I ; j = 1 , 2 , . . . , J ). Atthispointwemightreplaceeachfunction f j ( s j ) byaninteger-programmingapproximationtocomplete the model. Details are left to the reader. Note that if f j ( s j ) contains a fixed cost, then new fixed-cost variables need not be introduced—the variable y j serves this purpose. The last comment, and the way in which the conditional constraint ‘‘ y j = 0 implies x i j = 0 ( i = 1 , 2 , . . . , I ) ’’ has been modeled above, indicate that the formulation techniques of Section 9.2 should not be applied without thought. Rather, they provide a common framework for modeling and should be used in conjunction with good modeling ‘‘common sense.’’ In general, it is best to introduce as few integer variables as possible. 9.4 SOME CHARACTERISTICS OF INTEGER PROGRAMS—A SAMPLE PROBLEM Whereas the simplex method is effective for solving linear programs, there is no single technique for solving integer programs. Instead, a number of procedures have been developed, and the performance of any particular technique appears to be highly problem-dependent. Methods to date can be classified broadly as following one of three approaches: i) enumeration techniques, including the branch-and-bound procedure; ii) cutting-plane techniques; and iii) group-theoretic techniques. In addition, several composite procedures have been proposed, which combine techniques using several of these approaches. In fact, there is a trend in computer systems for integer programming to include a number of approaches and possibly utilize them all when analyzing a given problem. In the sections to follow, we shall consider the first two approaches in some detail. At this point, we shall introduce a specific problem and indicate some features of integer programs. Later we will use this example to illustrate and motivate the solution procedures. Many characteristics of this example are shared by the integer version of the custom- molder problem presented in Chapter 1. The problem is to determine z * where: z * = max z = 5 x 1 + 8 x 2 ,
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288 Integer Programming 9.5 subject to: x 1 + x 2 6 , 5 x 1 + 9 x 2 45 , x 1 , x 2 0 and integer . The feasible region is sketched in Fig. 9.8. Dots in the shaded region are feasible integer points. Figure 9.8 An integer programming example. If the integrality restrictions on variables are dropped, the resulting problem is a linear program. We will call it the associated linear program . We may easily determine its optimal solution graphically. Table 9.1 depicts some of the features of the problem.
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