4120_lecture13-F10 - Computational Methods for Management...

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Computational Methods for Management and Economics Carla Gomes Lecture 13 Reading: 6.1- 6.4 of textbook
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Outline Duality
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Duality Every maximization LP problem in the standard form gives rise to a minimization LP problem called the dual problem Every feasible solution in one yields a bound on the optimal value of the other If one of the problems has an optimal solution, so does the other and the two optimal values coincide These results have very interesting economic interpretations
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Bounds on the optimal value Let x1 = the number of doors to produce x2 = the number of windows to produce Maximize Z = 3 x1 + 5 x2 subject to x1 ≤ 4 2 x2 ≤ 12 3 x1 + 2 x2 ≤ 18 and x1 ≥ 0, x2 ≥ 0. Lower bound (LB) on Z* Z* ≥ LB any feasible solution Upper bound (UB) on Z* Z* ≤ UB – how do we come up with upper bounds for Z* for a maximization LP problem?
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“Guessing” upper bounds Let x1 = the number of doors to produce x2 = the number of windows to produce Maximize Z = 3 x1 + 5 x2 subject to x1 ≤ 4 2 x2 ≤ 12 3 x1 + 2 x2 ≤ 18 and x1 ≥ 0, x2 ≥ 0. 1 – multiplying the 3 rd costraint by 3 9 x1 + 6 x2 ≤ 54 Z = 3 x1 + 5 x2 ≤ 9 x1 + 6 x2 ≤ 54 Z *≤ 54 2 – multiplying the 3 rd constraint by 2.5 7.5 x1 + 5 x2 ≤ 45 Z = 3 x1 + 5 x2 ≤ 7.5 x1 + 5 x2 ≤ 45 Z *≤ 45 3 – multiplying the 2 nd constraint by 2 and add it to the 3 rd 3 x1 + 6 x2 ≤ 42 Z = 3 x1 + 5 x2 ≤ 3 x1 + 6 x2 ≤ 42 Z *≤ 42 4 – multiplying the 2 nd constraint by 1.5 and add it to the 3 rd 3 x1 + 5 x2 ≤ 36 Z = 3 x1 + 5 x2 ≤ 3 x1 + 5 x2 ≤ 36 Z *≤ 36
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A principled way of finding upper-bounds on Z* Dual Problem Let x1 = the number of doors to produce x2 = the number of windows to produce Maximize Z = 3 x1 + 5 x2 subject to x1 ≤ 4 y1 2 x2 ≤ 12 y2 3 x1 + 2 x2 ≤ 18 y3 and x1
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This note was uploaded on 02/25/2011 for the course AEM 4120 taught by Professor Gomes,c. during the Fall '08 term at Cornell University (Engineering School).

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4120_lecture13-F10 - Computational Methods for Management...

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