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# IP Models - Contents 7 Modeling Integer and Combinatorial...

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Contents 7 Modeling Integer and Combinatorial Programs 287 7.1 Types of Integer Programs, an Example Puzzle Problem, and a Classical Solution Method . . . . . . . . . . . . . 287 7.2 The Knapsack Problems . . . . . . . . . . . . . . . . . 296 7.3 Set Covering, Set Packing, and Set Partitioning Problems . . . . . . . . . . . . . . . . 302 7.4 Plant Location Problems . . . . . . . . . . . . . . . . . 323 7.5 Batch Size Problems . . . . . . . . . . . . . . . . . . . 328 7.6 Other “Either, Or” Constraints . . . . . . . . . . . . . 330 7.7 Indicator Variables . . . . . . . . . . . . . . . . . . . . 333 7.8 Discrete Valued Variables . . . . . . . . . . . . . . . . 340 7.9 The Graph Coloring Problem . . . . . . . . . . . . . . 340 7.10 The Traveling Salesman Problem (TSP) . . . . . . . . 348 7.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 350 7.12 References . . . . . . . . . . . . . . . . . . . . . . . . . 371 i

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Chapter 7 Modeling Integer and Combinatorial Programs This is Chapter 7 of “Junior Level Web-Book Optimization Models for decision Making ” by Katta G. Murty. 7.1 Types of Integer Programs, an Exam- ple Puzzle Problem, and a Classical Solution Method So far, we considered continuous variable optimization models. In this chapter we will discuss modeling discrete or mixed discrete opti- mization problems in which all or some of the decision variables are restricted to assume values within speci fi ed discrete sets, and com- binatorial optimization problems in which an optimum combina- tion/arrangement out of a possible set of combinations/arrangements has to be determined. Many of these problems can be modeled as LPs with additional integer restrictions on some, or all, of the variables. LP models with additional integer restrictions on decision variables are called integer linear programming problems or just integer programs . They can be classi fi ed into the following types. 287
288 Ch.7. Integer Programs Pure (or, all) integer programs: These are integer programs in which all the decision variables are restricted to assume only integer val- ues. 0 1 pure integer programs: These are pure integer programs, and in addition, all decision variables are bounded variables with lower bound 0, and upper bound 1; i.e., in e ff ect, every decision variable in them is required to be either 0 or 1. Mixed integer programs or MIPs: Integer programs in which there are some continuous decision variables and some integer decision variables. 0 1 mixed integer programs (0-1 MIPs): These are MIPs in which all the integer decision variables are 0 1 variables. Integer feasibility problems: Mathematical models in which it is required to fi nd an integer solution to a given system of linear constraints, without any optimization. 0 1 integer feasibility problems: Integer feasibility problems to fi nd a 0 1 solution to a given system of linear constraints. Many problems involve various yes - no decisions , which can be considered as the 0 1 values of integer variables so constrained. Vari- ables which are restricted to the values 0 or 1 are called 0 1 variables or binary variables or boolean variables . That’s why 0 1 integer programs are also called binary (or boolean) variable optimiza- tion problems .

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