chap12_2011_1

chap12_2011_1 - Chapter 12 Discrete Optimization Methods...

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Chapter 12 Discrete Optimization Methods
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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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Swedish Steel Model with All-or-Nothing Constraints min 16 (75)y1 +10 (250)y2 +8 x3+9 x4 +48 x5+60 x6 +53 x7 s.t. 75y1 + 250y2 + x3+ x4 + x5+ x6 + x7 = 1000 0.0080 (75)y1 + 0.0070 (250)y2 +0.0085x3+0.0040x4 6.5 0.0080 (75)y1 + 0.0070 (250)y2 +0.0085x3+0.0040x4 7.5 0.180 (75)y1 + 0.032 (250)y2 + 1.0 x5 30.0 0.180 (75)y1 + 0.032 (250)y2 + 1.0 x5 30.5 0.120 (75)y1 + 0.011 (250)y2 + 1.0 x6 10.0 0.120 (75)y1 + 0.011 (250)y2 + 1.0 x6 12.0 0.001 (250)y2 + 1.0 x7 11.0 (12.1) Cost = 9967.06 y1* = 1, y2* = 0, x3* = 736.44, x4* = 160.06 x5* = 16.50, x6* = 1.00, x7* = 11.00
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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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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]
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12.2 Relaxation of Discrete Optimization Models
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Example 12.1 Bison Booster The Boosters are trying to decide what fundraising projects to undertake at the next country fair. One option is customized T-shirts, which will sell for $20 each; the other is sweatshirts selling for $30.
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