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

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Unformatted text preview: Lecture 10: Post-Optimal Analysis February 24, 2009 Lecture 10 Today • Primal-Dual Relationship • Economic Interpretation of Duality • Post-Optimal Analysis Chapters 4.2, 4.3, and 4.5. GE330 1 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 ≥ 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 ≥ , y 2 is unrestricted GE330 2 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), h y 1 y 2 i = h 12 5 i × " 2 / 5- 1 / 5 1 / 5 2 / 5 # GE330 3 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 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...
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This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.

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

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