lecture16 - Complementary Slackness Labor Allocation:...

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e62: lecture 16 10/11/10 Complementary Slackness 1 max cx s . t . Ax b x 0 min yb s . t . yA c y 0 Complementary Slackness Theorem Let x and y be feasible. x and y are optimal iff y ( b ! Ax ) = 0 and ( yA ! c ) x = 0 x * ,y * optimal ( y * A - c ) x * + y * ( b - Ax * )= y * b - cx * =0 y * ( b - Ax * ) = 0 ( y * A - c ) x * ( yA - c ) x y ( b - Ax ) = 0 yb - cx =( yA - c ) x + y ( b - Ax )=0 x, y optimal e62: lecture 16 10/11/10 Labor Allocation: Example 2 E1 E2 E3 A1 A2 M1 T1 T2 energy 1.5 1 2 -0.2 -0.1 -0.5 -0.3 -0.2 agriculture -0.2 -0.4 -0.3 0.5 1 -0.5 -0.1 -0.2 manufacturing -0.1 -0.3 -0.6 -0.1 0 2 0 -0.1 transportation -1.2 -0.8 -0.6 -0.8 -1.5 -0.5 4 5 technologies industries How to allocate labor to meet demand? Production matrix: quantities produced/consumed per unit labor energy agriculture manufacturing transportation e62: lecture 16 10/11/10 Solutions 3 E1 E2 E3 A1 A2 M1 T1 T2 energy 1.5 1 2 -0.2 -0.1 -0.5 -0.3 -0.2 agriculture -0.2 -0.4 -0.3 0.5 1 -0.5 -0.1 -0.2 manufacturing -0.1 -0.3 -0.6 -0.1 0 2 0 -0.1 transportation -1.2 -0.8 -0.6 -0.8 -1.5 -0.5 4 5 technologies b = 3 4 6 2 x * = 0 . 0000 0 . 0000 3 . 2835 0 . 0000 7 . 7747 4 . 1622 0 . 0000 3 . 5427 b =
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This note was uploaded on 07/29/2011 for the course MS&E 111 at Stanford.

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lecture16 - Complementary Slackness Labor Allocation:...

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