MIT15_053S13_iprefguide

MIT15_053S13_iprefguide - IP Reference guide for integer...

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IP Reference guide for integer programming formulations. by James B. Orlin for 15.053 and 15.058 This document is intended as a compact (or relatively compact) guide to the formulation of integer programs. For more detailed explanations, see the PowerPoint tutorial on integer programming. The following are techniques for transforming a problem described logically or in words into an integer program. In most cases, the transformation is the simplest to describe. Unfortunately, simplest is not the same as "best." It is widely accepted that the best integer programming formulations are those that result in fast solutions by integer programming solvers. In general, these are the integer programs for which the linear programming relaxation provides a good bound. Section 1. Subset selection problems. Often, models are based on selecting a subset of elements. For example, in the knapsack problem, one wants to select a subset of items to put into a knapsack so as to maximize the value while not going over a specified weight. Or one wants to select a subset of potential products in which to invest. Or, one has a set of different integers, and one wants to select a subset that sums to a value K. In these cases, it is typical for the integer variables to be as follows: 1 if element i is selected x i = 0 otherwise.± Example: knapsack/capital budgeting. In this example, there are six items to select from. Item 1 2 3 4 5 6 Cost 5 7 4 3 4 6 Value 16 22 12 8 11 19 Problem: choose items whose cost sums to at most 14 so as to maximize the utility. maximize 16 x 1 + 22 x 2 + 12 x 3 + 8 x 4 + 11 x 5 + 19 x 6 Formulation: subject to 5 x 1 + 7 x 2 + 4 x 3 + 3 x 4 + 4 x 5 + 6 x 6 14 x i {0,1} for i = 1 to 6. In general: Maximize the value of the selected items such that the weight is at most b . C i = value of item i for i = 1 to n . a i = weight of item i for i = 1 to n . b = bound on total weight. n maximize c i x i i = 1 n subject to a i x i b i = 1 x i {0,1} for i = 1 to n .
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Covering and packing problems. In some selection problems, each item is associated with a subset of a larger set. The larger set is usually referred to as the ground set . For example, suppose that there is a collection of n sets 5 1 , ", 5 n where for i = 1 to n 5 i is a subset of the ground set {1, 2, 3, ", m }. Associated with each set 5 i is a cost C i . 1 if i S j 1 if set S j is selected Let a ij = Let x j = 0 otherwise. 0 otherwise.± The set paCking problem is the problem of selecting the maximum cost subcollection of sets, no two of which share a common element. The set Covering problem is the problem of selecting the minimum cost subcollection of sets, so that each element i E {1, 2, ", m} is in one of the sets. Maximize n c j x j j = 1 subject to n j = 1 a ij x j 1 for each i {1,..., m } Set Packing Problem x j {0,1} for each j {1,..., n }.
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