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GE330_lect10 - Lecture 10 Post-Optimal Analysis Lecture 10...

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Lecture 10: Post-Optimal Analysis February 24, 2009
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Lecture 10 Today Primal-Dual Relationship Economic Interpretation of Duality Post-Optimal Analysis Chapters 4.2, 4.3, and 4.5. GE330 1
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Lecture 10 More on Primal-Dual Relations Primal maximize z = 5 x 1 + 12 x 2 + 4 x 3 subject to x 1 + 2 x 2 + x 3 + x 4 = 10 2 x 1 - x 2 + 3 x 3 = 8 x 1 , x 2 , x 3 0 Its dual minimize w = 10 y 1 + 8 y 2 subject to y 1 + 2 y 2 5 2 y 1 - y 2 12 y 1 + 3 y 2 4 y 1 0 , y 2 is unrestricted GE330 2
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Lecture 10 Optimal Tableau (for primal problem, using M-method) Basic x 1 x 2 x 3 x 4 R Solution z 0 0 3 / 5 29 / 5 - 2 / 5 + M 54 4 5 x 2 0 1 - 1 / 5 2 / 5 - 1 / 5 12 / 5 x 1 1 0 7 / 5 1 / 5 2 / 5 26 / 5 Optimal inverse = 2 / 5 - 1 / 5 1 / 5 2 / 5 Optimal dual = original objective coeff. × optimal inverse (order matters), y 1 y 2 = 12 5 × 2 / 5 - 1 / 5 1 / 5 2 / 5 GE330 3
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Lecture 10 Relations in Simplex Table Given a simplex table at any iteration i , we have Column Rule x j -column data = inverse at iteration i × original x j -column Here, “column data” does not include the z -row values. The rule applies to “rhs”-column as well. Shadow Prices at iteration i (dual variables) shadow price = original cost of basis at iteration i × inverse at iteration i Here, order in the basis matter. Shadow price is optimal ONLY if the given table is optimal. Reduced Cost ( z -row data) reduced cost of x j = l.h.s. of j -th dual constraint - r.h.s. of j -th dual constraint The left hand-side of dual constraint is evaluated at the shadow price GE330 4
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Lecture 10 Dual Optimal Solutions Important to sensitivity analysis Change in the right-hand side Change in the objective coefficients Help us analyze additional changes Addition of new operation (constraint in primal) Addition of new activity (variable in primal) We discuss this changes in the light of dual optimal solutions Changes affecting feasibility (right-hand side change or a new constraint) How to recover optimal if the perturbation causes the change in basic optimal solution?
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