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 mohagheg@umd.edu 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 26 0 00 0 0 AB H M H M zx x subject xx s s s s = + ++ = = ≥≥ to 50 0 x subject =+ +≤ 01 5 2 0 0 0 5 0 2 2 60 60 0 . H H M M Row Ba x z s s s sc ar s iV −− = = = = = = = +− = Rewrite objective 15 20 as the equation 15 20 0. 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: 01 5 2 0 0 15 0 5 0 2 2 60 60 0 . AB H H M M Row Ba zx x z xx s s s sc ar s iV −− = = ++ = = = = 0 11 52 0 00 0 1 10 5 0 5 0 2 6 0 6 1 2 0 . 0 H M H M Row Basi x s s z c s Vr 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 26 0 0 AB H M zx x xx s s −− = ++ = =
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Engineering Management & Systems Engineering The George Washington University LP Canonical Form = LP Canonical Form = LP Standard Form + Jordan Canonical Form LP Standard Form + Jordan Canonical Form -3 2 0 0 1 -3 3 1 0 -4 2 0 1 0 0 1 1 1 -15 -20 = = 60 50 x A x B x 4 x 3 -z = 0 The simplex method starts with an LP in LP canonical form (or it creates canonical form using eros.) z is not a decision variable s M s H
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Engineering Management & Systems Engineering The George Washington University
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Zahra_Mohaghegh_Jan29_module3_P - Engineering Management...

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