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chap11_2011

# chap11_2011 - Chapter 11 Discrete Optimization Models 11.1...

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Chapter 11 Discrete Optimization Models

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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 All-or-Nothing Requirements All-or-nothing 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 0.001 (11.1) Cost = 9953.67 x1* = 75, x2* = 90.91, x3* = 672.28, x4* = 137.31 x5* = 13.59, x6* = 0, x7* = 10.91

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Swedish Steel Model with All-or-Nothing Constraints
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 (11.2) 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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ILP Modeling of Fixed Charges
ILP Modeling of Fixed Charges

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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 x2 + 1.0 x7 11.0 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

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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. Proposed Feature, j 1 2 3 4 5 6 Cost (\$000s) 10.2 6.0 23.0 11.1 9.8 31.6 Speed increase (mph) 8 3 15 7 10 12

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Example 11.1 Indy Car Knapsack Speed increase = 25 Cost = 32.8 x1* = 0, x2* = 0, x3* = 1, x4* = 0, x5* = 1, x6* = 0
Example 11.1 Indy Car Knapsack Suppose now that the Indy Car team decides they simply must increase speed by 30 miles per hour to have any chance of winning the next race. Ignoring the budget, they wish to find the minimum cost way to achieve at least that much performance. Min 10.2x1 + 6.0x2 + 23.0x3+ 11.1x4 + 9.8x5 + 31.6x6 s.t. 8 x1 + 3 x2 + 15 x3 + 7 x4 + 10 x5 + 12 x6  30 x1 ,…, x6 = 0 or 1 Cost = 43.0 Speed increase = 33 x1* = 1, x2* = 0, x3* = 1, x4* = 0, x5* = 1, x6* = 0

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Capital Budgeting Models The typical maximize form of a knapsack problem has a single main constraint enforcing a budget.
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chap11_2011 - Chapter 11 Discrete Optimization Models 11.1...

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