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duality_theory

# duality_theory - IOE 310-DUALITY THEORY \$ 1 DUALITY THEORY...

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IOE 310-DUALITY THEORY 1 & \$ % DUALITY THEORY OF LPs What is a Dual? Writing the dual. Economic meaning of the Dual. The Duality Theorems and their importance.

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IOE 310-DUALITY THEORY 2 & \$ % FINDING THE DUAL OF AN LP Normal Duals max z = c 1 x 1 + c 2 x 2 + · · · + c n x n a 11 x 1 + a 12 x 2 + · · · + a 1 n x n b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n b 2 . . . . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n b m x 1 0 x 2 0 · · · x n 0 and in matrix notation: maxmize z = c T x Ax b x 0
IOE 310-DUALITY THEORY 3 & \$ % Its dual is min w = b 1 y 1 + b 2 y 2 + · · · + b m y m a 11 y 1 + a 21 y 2 + · · · + a m 1 y m c 1 a 12 y 1 + a 22 y 2 + · · · + a m 2 y m c 2 . . . . . . . . . . . . . . . a 1 n y 1 + a 2 n y 2 + · · · + a mn y m c n y 1 0 y 2 0 · · · y m 0 and in matrix notation: minimize w = b T y A T y c y 0

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IOE 310-DUALITY THEORY 4 & \$ % The matrix A , and vectors b and c are the data of the model. THE PRIMAL maxmize z = c T x Ax b x 0 THE DUAL minimize w = b T y A T y c y 0
IOE 310-DUALITY THEORY 5 & \$ % FINDING THE DUAL OF AN LP EXAMPLE OF A NORMAL DUAL: max z = 60 x 1 +30 x 2 +20 x 3 8 x 1 +6 x 2 + x 3 48 4 x 1 +2 x 2 +1 . 5 x 3 20 2 x 1 +1 . 5 x 2 +0 . 5 x 3 8 x 1 0 x 2 0 x 3 0 Its dual is

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IOE 310-DUALITY THEORY 6 & \$ % min w = 48 y 1 +20 y 2 +8 y 3 8 y 1 +4 y 2 +2 y 3 60 6 y 1 +2 y 2 +1 . 5 y 3 30 y 1 +1 . 5 y 2 +0 . 5 y 3 20 y 1 0 y 2 0 y 3 0
IOE 310-DUALITY THEORY 7 & \$ % FINDING THE DUAL OF A MIN PROBLEM EXAMPLE OF A NORMAL DUAL (The diet problem ): min w = 50 y 1 +20 y 2 +30 y 3 +80 y 4 400 y 1 +200 y 2 +150 y 3 +500 y 4 500 3 y 1 +2 y 2 6 2 y 1 +2 y 2 +4 y 3 4 y 4 10 2 y 1 +4 y 2 + y 3 +5 y 4 8 y 1 0 y 2 0 y 3 0 y 4 0 with its dual:

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IOE 310-DUALITY THEORY 8 & \$ % max z = 500 x 1 +6 x 2 +10 x 3 +8 x 4 400 x 1 +3 x 2 +2 x 3 +2 x 4 50 200 x 1 +2 x 2 +2 x 3 +4 x 4 20 150 x 1 +4 x
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