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lecture13-handout - Integer Programming Prof Je Linderoth...

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Integer Programming Prof. Jeff Linderoth October, 27
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ISyE Lecture #13 Outline Outline Adapting to Other Primal Forms (Section 6.4) Sensitivity Analysis (Sections 6.5–6.8) ISyE Lecture #13 2
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ISyE Lecture #13 Outline Summary of Graphs Change b Change c c j Z * b i Z * c j Z * b i Z * Maximization Minimization ISyE Lecture #13 3
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ISyE Lecture #13 Outline Relationship to Complementary Slackness Relationship to Complementary Slackness I Suppose there is slack in the i th primal constraint. I Increasing the RHS would not change the optimal solution. I By complementary slackness, y * i must equal 0 (in the dual). I Using the statement on the previous slide, the optimal objective function changes by y * i , or 0. I Suppose there is no slack in the i th primal constraint. I Increasing the RHS would change the optimal solution. I y * i probably (!) is greater than 0. I Using the statement above, the optimal objective function changes by y * i . I This agrees with our interpretation of the dual values as shadow prices . ISyE Lecture #13 4
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ISyE Lecture #13 Outline Relationship to Complementary Slackness Relationship to Complementary Slackness, cont’d I Now suppose there is slack in the j th dual constraint. I By complementary slackness, x * j = 0 (in the primal). I If we increase c j slightly, we’ll still want to set x * j = 0 . I We argued that for each unit increase in c j , Z * changes by x * j (if optimal basis stays the same). I So Z * increases by 0 when c j increases. I Suppose there is no slack in the j th dual constraint. I x * j > 0 (probably). I If we increase c j by 1, the objective value will go up by x * j . ISyE Lecture #13 5
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ISyE Lecture #13 Outline Other Changes Adding Columns ( P ) max Z = 3 x 1 + 5 x 2 s.t. x 1 4 2 x 2 12 3 x 1 + 2 x 2 18 x 1 , x 2 0 ( D ) min W = 4 y 1 + 12 y 2 + 18 y 3 s.t. y 1 + 3 y 3 3 2 y 2 + 2 y 3 5 y 1 , y 2 , y 3 0 I Optimal solution x * = (2 , 6) , y * = (0 , 3 / 2 , 1) Management Question I Suppose Wyndor Glass would like to start producing shutters I Making a shutter requires 2 hours of time at plant 1, 3 hours at plant 2, and 1 hour at plant 3. Selling will bring a profit of 4 . I Should we make the shutter? ISyE Lecture #13 6
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ISyE Lecture #13 Outline Other Changes Duals to the Rescue I How much would a shutter “cost” Wyndor glass in terms of the (limited) resources it has to expend? 2 y * 1 + 3 y * 2 + y * 3 = 2(0) + 3(3 / 2) + 1(1) = 11 / 2 I However, the shutter only brings in a profit of 4, so we can conclude that we should not consider this new activity—making shutters. ISyE Lecture #13 7
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ISyE Lecture #13 Introduction (Section 11.1) Outline Adapting to Other Primal Forms (Section 6.4) Sensitivity Analysis (Sections 6.5–6.8) ISyE Lecture #13 8
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ISyE Lecture #13 Introduction (Section 11.1) Example Problem Example Problem I California Manufacturing Co. will build a new factory in Los Angeles or San Francisco or both.
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