duality_theory - IOE 310-DUALITY THEORY ' $ 1 DUALITY...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/02/2008 for the course IOE 310 taught by Professor Saigal during the Spring '08 term at University of Michigan.

Page1 / 24

duality_theory - IOE 310-DUALITY THEORY ' $ 1 DUALITY...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online