MIT15_053S13_iprefguide

Or one wants to select a subset of potential products

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Unformatted text preview: d 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 ⎪ xi = ⎨ ⎪0 ⎩ if element i is selected otherwise. Example: knapsack/capital budgeting. In this example, there are six items to select from. Item Cost Value 1 5 16 2 7 22 3 4 12 4 3 8 5 4 11 6 6 19 Problem: choose items whose cost sums to at most 14 so as to maximize the utility. maximize Formulation: 16 x1 + 22 x2 + 12 x3 + 8 x4 + 11x5 + 19 x6 5x1 + 7 x2 + 4 x3 + 3x4 + 4 x5 + 6 x6 ≤ 14 subject to xi ∈{0,1} In general: for i = 1 to 6. Maximize the value of the selected items such that the weight is at most b. Ci = value of item i for i = 1 to n. ai = weight of item i for i = 1 to n. b = bound on total weight. n maximize ∑c x i =1 ii n subject to ∑a x i =1 ii ≤ b xi ∈{0,1} for i = 1 to n. Covering and packing problems. In some selection problems, each item is associated with a subset of a lar...
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This note was uploaded on 03/18/2014 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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