Zahra_mMohaghegh_Feb05

Zahra_mMohaghegh_Feb05 - Engineering Management &...

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1 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 February 05, 2008
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2 Engineering Management & Systems Engineering The George Washington University The Simplex Algorithm The Simplex Algorithm (cont.) (cont.) CREDITS: PARTS OF THIS LECTURE HAVE BEEN DEVELOPED BY PROFESSORS CAMPOS-NONEZ (GWU) AND ORLIN (MIT)
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3 Engineering Management & Systems Engineering The George Washington University In this session. .. In this session. .. ± Review of simplex algorithm ± Minimization problems ± Alternative optimal solution ± Unbounded LPs ± Degenerate LPs and Convergence of the Simplex Algorithm ± Finding an Initial Basic Feasible Solution ± The Big M Method ± The Two-Phase Method ± Using LINDO and LINGO packages
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4 Engineering Management & Systems Engineering The George Washington University Input: Canonical Simplex Tableau (bfs identified and LP in canonical form). Most Negative Entry * (Entering) SIMPLEX Tableau Minimum Ratio* (Leaving) All entries in row 0 are non-negative => optimal solution found Pivoting Max LP Formulation Standard Form LP All column entries negative or zero => unbounded LP Non-basic variables with zero coefficient in Row 0 => Multiple Optimal Solutions *Tie-breaks = any rule works Simplex Recap Simplex Recap
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5 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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6 Engineering Management & Systems Engineering The George Washington University Basic variable & Basic variable & nonbasic nonbasic variable variable ± Given a linear system in canonical form , a variable that appears with a coefficient of 1 in a single equation and a coefficient of 0 in all other equations is called a basic variable (BV) . If there is more than one such variable in the same equation, then any one of these variables (but only one) can be selected to be basic. ± Any variable that is not a basic variable is called a nonbasic variable (NBV) .
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7 Engineering Management & Systems Engineering The George Washington University Basic Solution Basic Solution ± To find a basic solution to AX=b: 1. Choose a set of basic variable (BV) and nonbasic variable (NBV). For a system AX=b of m linear equations in n variables (n>=m), there may be different sets of m basic variables and n-m nonbasic variables.
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Zahra_mMohaghegh_Feb05 - Engineering Management &...

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