Lecture 20110401

Lecture 20110401 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques ecture 04/01 Lecture 04/01 Duality Miao Song Dept of Industrial & Manufacturing Systems Engineering
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genda Agenda Duality Rules for creating a dual LP Weak duality and strong duality Complementary slackness py 4/1/2011 2
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rimal and Dual Primal and Dual Primal LP min z = –52x 1 – 30x 2 – 20x 3 s.t. 2x 1 + 4x 2 + 5x 3 + x 4 = 100 x x x 30 max 52x 1 + 30x 2 + 20x 3 s.t. 2x 1 + 4x 2 + 5x 3 100 x x 30 x 1 + x 2 + x 3 + x 5 = 30 10x 1 + 5x 2 + 2x 3 + x 6 = 204 x ,x ,x ,x ,x ,x 0 x 1 + x 2 + x 3 10x 1 + 5x 2 + 2x 3 204 x ,x ,x 0 1 2 3 4 5 6 max 100p 1 + 30p 2 + 204p 3 Dual LP 1 2 3 min 100p 1 + 30p 2 + 204p 3 s.t. 2p 1 + p 2 + 10p 3 –52 4p 1 + p 2 + 5p 3 –30 p + p + 2p 0 s.t. 2p 1 + p 2 + 10p 3 52 4p 1 + p 2 + 5p 3 30 p + p + 2p 0 5p 1 p 2 2p 3 20 p 1 0 p 2 0 5p 1 p 2 2p 3 20 p 1 ,p 2 ,p 3 0 4/1/2011 3 p 3 0
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ore on Primal and Dual More on Primal and Dual Primal LP is feasible to the primal min z = –52x 1 – 30x 2 – 20x 3 s.t. 2x 1 + 4x 2 + 5x 3 + x 4 = 100 x x x 30 x is feasible to the primal p is feasible to the dual Which of these two alues is larger? x 1 + x 2 + x 3 + x 5 = 30 10x 1 + 5x 2 + 2x 3 + x 6 = 204 x ,x ,x ,x ,x ,x 0 values is larger? –52x 1 – 30x 2 – 20x 3 100p 1 + 30p 2 + 204p 3 1 2 3 4 5 6 max 100p 1 + 30p 2 + 204p 3 Dual LP What is the optimal solution to the dual? Shadow Prices! s.t. 2p 1 + p 2 + p 3 –52 4p 1 + p 2 + 5p 3 –30 p + p + 2p 0 What do we know about the optimal values of the primal and dual? 5p 1 p 2 2p 3 20 p 1 0 p 2 0 Opt value of primal Opt value of dual Opt value of primal = 4/1/2011 4 p 3 0 Opt value of dual
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ules for Creating a Dual LP Rules for Creating a Dual LP Primal LP (min) Dual LP (max) Objective Coefficients RHS Coefficients RHS Coefficients Objective Coefficients Column Coefficients Row Coefficients Row Coefficients Column Coefficients Primal LP (min) Dual LP (max) i-th Constraint b i i-th Variable 0 i-th Constraint b i i-th Variable 0 i-th Constraint = b i i-th Variable free - ariable 0 - onstraint j th Variable j th Constraint c j j-th Variable 0 j-th Constraint c j j-th Variable free j-th Constraint = c j 4/1/2011 5
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xample Example min 2x 1 –3x 2 + 5x max 10p 1 + 7p 2 + 15p 3 s.t. 2x 1 + 4x 2 + x 3 10 x 1 + x 2 + x 3 7 3 s.t. 2p 1 + p 2 + 3p 3 2 4p 1 + p 2 + 5p 3 –3 p 1 p 2 3x 1 + 5x 2 + 2x 3 = 15 x 1 0, x 2 0, x 3 free p 1 + p 2 + 2p 3 = 5 p 1 0, p 2 0, p 3 free p 3
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Lecture 20110401 - IMSE2008 Operational Research Techniques...

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