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Unformatted text preview: Chapter 11 Discrete Optimization Models 11.1 Lumpy Linear Programs and Fixed Charges • Lumpy linear problems add “either/or” constraints or objective functions to what is otherwise a linear programs. ILP Modeling of AllorNothing Requirements • Allornothing variable requirements of the form xj = 0 or uj Can be modeled by substituting xj = uj yj , with new discrete variable yj = 0 or 1. [11.1] • The new yj can be interpreted as the fraction of limit uj chosen. Swedish Steel Blending Example min 16 x1+10 x2 +8 x3+9 x4 +48 x5+60 x6 +53 x7 s.t. x1+ x2 + x3+ x4 + x5+ x6 + x7 = 1000 0.0080 x1 + 0.0070 x2 + 0.0085 x3 + 0.0040 x4 6.5 0.0080 x1 + 0.0070 x2 + 0.0085 x3 + 0.0040 x4 7.5 0.180 x1 + 0.032 x2 + 1.0 x5 30.0 0.180 x1 + 0.032 x2 + 1.0 x5 30.5 0.120 x1 + 0.011 x2 + 1.0 x6 10.0 0.120 x1 + 0.011 x2 + 1.0 x6 12.0 (11.1) Cost = 9953.67 x1* = 75, x2* = 90.91, x3* = 672.28, x4* = 137.31 x5* = 13.59, x6* = 0, x7* = 10.91 Swedish Steel Model with AllorNothing Constraints Swedish Steel Model with AllorNothing 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 (11.2) Cost = 9967.06 y1* = 1, y2* = 0, x3* = 736.44, x4* = 160.06 x5* = 16.50, x6* = 1.00, x7* = 11.00 ILP Modeling of Fixed Charges ILP Modeling of Fixed Charges Swedish Steel Model with Fixed Charges Swedish Steel Example with Fixed Charges min 16 x1+10 x2 +8 x3+9 x4 +48 x5+60 x6 +53 x7 + 350 y1+ 350 y2 + 350 y3 + 350 y4 s.t. x1+ x2 + x3+ x4 + x5+ x6 + x7 = 1000 0.0080 x1 + 0.0070 x2 + 0.0085 x3 + 0.0040 x4 6.5 0.0080 x1 + 0.0070 x2 + 0.0085 x3 + 0.0040 x4 7.5 0.180 x1 + 0.032 x2 + 1.0 x5 30.0 0.180 x1 + 0.032 x2 + 1.0 x5 30.5 0.120 x1 + 0.011 x2 + 1.0 x6 10.0 0.120 x1 + 0.011 x2 + 1.0 x6 12.0 0.001 Cost = 11017.06 x1* = 75, x2* = 0, x3* = 736.44, x4* = 160.06 x5* = 13.59, x6* = 1, x7* = 11 y1* = 1, y2* = 0 y3* = 1, y4* = 1 11.2 Knapsack and Capital Budgeting Models • Knapsack and capital budgeting problems are completely discrete. • A knapsack model is a pure integer linear program with a single main constraint. [11.4] Example 11.1 Indy Car Knapsack The mechanics in the Indy Car racing team face a dilemma. Six different features might still be added to this year’s car to improve its top speed. The following table lists their estimated costs and speed enhancements....
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 Summer '11
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