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Zahra_Mohaghegh_Jan29_module3_P

Zahra_Mohaghegh_Jan29_module3_P - Engineering Management...

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Engineering Management & Systems Engineering The George Washington University EMSE 102 EMSE 102 - - 202 202 Survey of Operations Research Methods Survey of Operations Research Methods Zahra Mohaghegh [email protected] Post-Doctoral Research Associate Center For Risk and Reliability University of Maryland January29, 2008 Module_3
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Engineering Management & Systems Engineering The George Washington University Simplex Algorithm Simplex Algorithm (Cont.) (Cont.) CREDITS: PARTS OF THIS LECTURE HAVE BEEN DEVELOPED BY PROFESSORS ABELEDO AND CAMPOS-NONEZ
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Engineering Management & Systems Engineering The George Washington University Outline Outline How simplex algorithm can be used to find optimal solutions to LPs: Standard Form LP The Rationale of the Simplex Algorithm The Simplex Algorithm Alternative Optimal Solutions Unbounded Problems / Multiple optimal solutions How two computer packages (LINDO & LINGO) can be used to solve LPs Explaining how the information on the outputs of the software is related to the discussion of simplex algorithm
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Engineering Management & Systems Engineering The George Washington University The Simplex Algorithm The Simplex Algorithm Step 1. Convert LP to canonical form. Step 2. Obtain a bfs (if possible). Step 3. Determine whether the current bfs is optimal. Step 4. If current bfs is not optimal, determine which nonbasic variable should become basic ( enters ) and which basic variables should become nonbasic ( leaves ). Step 5. Use ERO’s to find the new bfs. Go to step 3.
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Engineering Management & Systems Engineering The George Washington University Keeping Track: The Simplex Tableau Keeping Track: The Simplex Tableau max 15 20 to 50 2 60 0 0 0 0 A B A B H A B M A B H M z x x subject x x s x x s x x s s = + + + = + + = max 15 20 to 50 2 60 0 0 A B A B A B A B z x x subject x x x x x x = + + + 0 15 20 0 1 50 50 2 2 60 60 0 . A B A B H H A B M M Row Ba z x x z x x s s x x s s c ar s i V = = + + = = + + = = = + = Rewrite objective 15 20 as the equation 15 20 0. A B A B z x x z x x 1 2 0
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Engineering Management & Systems Engineering The George Washington University A Simplex Tableau A Simplex Tableau Use coefficients only. So, for example becomes: 0 15 20 0 1 50 50 2 2 60 60 0 . A B A B H H A B M M Row Ba z x x z x x s s x x s s c ar s i V = = + + = = + + = = 0 1 15 20 0 0 0 0 1 1 1 0 50 50 0 1 2 0 1 60 6 1 2 0 . 0 A B H M H M Row Basi z x x s s z c s V r s a = = = = = =
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Engineering Management & Systems Engineering The George Washington University Canonical Form Canonical Form A system of linear equations is in canonical form if each equation has a variable with a coefficient of 1 in that equation and coefficient 0 in all others. A system in canonical form allows computing a basic solution very easily from the right hand side values. 15 20 50 2 60 0 A B A B H A B M z x x x x s x x s = + + = + + =
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