chap12_2011_1

chap12_2011_1 - 11/15/2011 Chapter 12 Discrete Optimization...

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11/15/2011 1 Chapter 12 Discrete Optimization Methods 12.1 Solving by Total Enumeration If model has only a few discrete decision variables, the most effective method of analysis is often the most direct: enumeration of all the possibilities. [12.1] Total enumeration solves a discrete optimization by trying all possible combinations of discrete variable values, computing for each the best corresponding choice of any continuous variables. Among combinations yielding a feasible solution, those with the best objective function value are optimal. [12.2]
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11/15/2011 2 Swedish Steel Model with All-or-Nothing Constraints min 16 (75)y 1 +10 (250)y 2 +8 x 3 +9 x 4 +48 x 5 +60 x 6 +53 x 7 s.t. 75y 1 + 250y 2 + x 3 + x 4 + x 5 + x 6 + x 7 = 1000 0.0080 (75)y 1 + 0.0070 (250)y 2 +0.0085x 3 +0.0040x 4 6.5 0.0080 (75)y 1 + 0.0070 (250)y 2 +0.0085x 3 +0.0040x 4 7.5 0.180 (75)y 1 + 0.032 (250)y 2 + 1.0 x 5 30.0 0.180 (75)y 1 + 0.032 (250)y 2 + 1.0 x 5 30.5 0.120 (75)y 1 + 0.011 (250)y 2 + 1.0 x 6 10.0 0.120 (75)y 1 + 0.011 (250)y 2 + 1.0 x 6 12.0 0.001 (250)y 2 + 1.0 x 7 11.0 0.001 (250)y 2 + 1.0 x 7 13.0 x 3 …x 7 0 y 1 , y 2 = 0 or 1 (12.1) Cost = 9967.06 y 1 * = 1, y 2 * = 0, x 3 * = 736.44, x 4 * = 160.06 x 5 * = 16.50, x 6 * = 1.00, x 7 * = 11.00 Swedish Steel Model with All-or-Nothing Constraints Discrete Combination Corresponding Continuous Solution Objective Value y1 y2 x3 x4 x5 x6 x7 0 0 823.11 125.89 30.00 10.00 11.00 10340.89 0 1 646.67 63.33 22.00 7.25 10.75 10304.08 1 0 736.44 160.06 16.50 1.00 11.00 9967.06 1 1 561.56 94.19 8.50 0.00 10.75 10017.94
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11/15/2011 3 Exponential Growth of Cases to Enumerate Exponential growth makes total enumeration impractical with models having more than a handful of discrete decision variables. [12.3] 12.2 Relaxation of Discrete Optimization Models Constraint Relaxations Model ( ° ) is a constraint relaxations of model ( P ) if every feasible solution to ( ) is also feasible in ( ° ) and both models have the same objective function. [12.4] Relaxation should be significantly more tractable than the models they relax, so that deeper analysis is practical. [12.5]
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