6417c03 - 1 SIMPLEX ALGORITHM CONTENTS An Application of...

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Unformatted text preview: 1 SIMPLEX ALGORITHM CONTENTS An Application of Simplex Algorithm (Section 2.1) The Simplex Algorithm (Section 2.2) Initialization and Unboundedness (Sections 2.3 and 2.4) Geometry of the Simplex Algorithm (Section 2.5) 2 An Example of LP Maximize 5x 1 + 4x 2 + 3x 3 subject to 2x 1 + 3x 2 + x 3 ≤ 5 4x 1 + x 2 + 2x 3 ≤ 11 3x 1 + 4x 2 + 2x 3 ≤ 8 x 1 , x 2 , x 3 ≥ An alternate formulation : max. ξ = 5x 1 + 4x 2 + 3x 3 subject to w 1 = 5 - 2x 1- 3x 2- x 3 w 2 = 11 - 4x 1- x 2- 2x 3 w 3 = 8- 3x 1- 4x 2- 2x 3 x 1 , x 2 , x 3 , w 1 , w 2 , w 3 ≥ 3 An Example of LP (contd.) max. ξ = 5x 1 + 4x 2 + 3x 3 subject to w 1 = 5 - 2x 1- 3x 2- x 3 w 2 = 11 - 4x 1- x 2- 2x 3 w 3 = 8- 3x 1- 4x 2- 2x 3 x 1 , x 2 , x 3 , w 1 , w 2 , w 3 ≥ Starting solution : x 1 = 0, x 2 = 0, x 3 = 0, w 1 = 5, w 2 = 11, w 3 = 8 ξ = 0 How to improve this solution? How to obtain the solution so that x x x w w w 1 2 3 1 2 3 , , , , , 5 4 3 5 4 3 1 2 3 1 2 3 x x x x x x + + ≥ + + 4 An Example of LP (contd.) Improved Solution : x 1 = 2.5, x 2 = 0, x 3 = 0, w 1 = 0, w 2 = 1, w 3 = .5, ξ = 12.5 How to improve this solution further? max. ξ = 5x 1 + 4x 2 + 3x 3 subject to w 1 = 5 - 2x 1- 3x 2- x 3 w 2 = 11 - 4x 1- x 2- 2x 3 w 3 = 8- 3x 1- 4x 2- 2x 3 x 1 , x 2 , x 3 , w 1 , w 2 , w 3 ≥ The system of equations needs to be rewritten with w 1 , x 2 , x 3 on the right hand side. 5 An Example of LP (contd.) Rewriting of the system of equations (Dictionary): ξ = 12.5 - 2.5x 1- 3.5x 2 + 0.5x 3 x 1 = 2.5 - 0.5w 1- 1.5x 2- 0.5x 3 w 2 = 1 + 2w 1 + 5x 2 w 3 = 0.5 + 1.5w 1 + 0.5x 2- 0.5x 3 x 1 , x 2 , x 3 , w 1 , w 2 , w 3 ≥ Independent Variables : w 1...
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This note was uploaded on 08/27/2011 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.

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6417c03 - 1 SIMPLEX ALGORITHM CONTENTS An Application of...

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