Lecture 13 - Dual

Lecture 13 - Dual - 540:311DETERMINISTICMODELSIN...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 13: Chapter 6.5‐6.10 Class Mee±ng: Thu April 14 th 10:20‐11:40am Prof. W. Art Chaovalitwongse
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6.5 – Finding the Dual of an LP Associated with any LP is another LP called the dual . Knowledge of the dual provides interesting economic and sensitivity analysis insights. When taking the dual of any LP, the given LP is referred to as the primal. If the primal is a max problem, the dual will be a min problem, or vice versa. Define the variables for a max problem to be z, x 1 , x 2 , ,x n and the variables for a min problem to be w, y 1 , y 2 , , y n . Finding the dual to a max problem in which all the variables are required to be nonnegative and all the constraints are constraints (called normal max problem) is shown on the next slide.
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6.5 – Finding the Dual of an LP max z = c 1 x 1 + c 2 x 2 + + c n x n s.t. a 11 x 1 + a 12 x 2 + + a 1n x n b 1 a 21 x 1 + a 22 x 2 + + a 2n x n b 2 a m1 x 1 + a m2 x 2 + + a mn x n b m x j 0 (j = 1, 2, ,n) min w = b 1 y 1 + b 2 y 2 + + b m y m s.t. a 11 y 1 + a 12 y 2 + + a m1 y m c 1 a 21 y 1 + a 22 y 2 + + a m2 y m c 2 a 1n y 1 + a 2n y 2 + + a mn y m c n y i 0 (i = 1, 2, ,m) Normal max problem It ` s dual Normal min problem It ` s dual
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l Recall z 4.3 – max LPs The Dakota Furniture company manufactures desk, tables, and chairs. The manufacturer of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is given in the table below. Resource
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Lecture 13 - Dual - 540:311DETERMINISTICMODELSIN...

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