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Zahra_Mohaghegh_March 4

Zahra_Mohaghegh_March 4 - Engineering Management Systems...

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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 [email protected] Post-Doctoral Research Associate Center For Risk and Reliability University of Maryland March 4, 2008

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2 Engineering Management & Systems Engineering The George Washington University Sensitivity Analysis Sensitivity Analysis (Cont.) (Cont.)
3 Engineering Management & Systems Engineering The George Washington University Outline Outline Graphical Sensitivity Analysis Shadow Prices Some Important formula Sensitivity Analysis Duality LINDO examples

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4 Engineering Management & Systems Engineering The George Washington University Outline Outline Graphical Sensitivity Analysis Shadow Prices Some Important formula Sensitivity Analysis Duality LINDO examples
5 Engineering Management & Systems Engineering The George Washington University A Graphical Introduction to Sensitivity A Graphical Introduction to Sensitivity Analysis Analysis Sensitivity analysis is concerned with how changes in an LP’s parameters affect the optimal solution. The optimal solution to the Giapetto problem was z = 180, x 1 = 20, x 2 = 60 (Point B in the figure to the right) and it has x 1 , x 2 , and s 3 (the slack variable for the demand constraint) as basic variables. How would changes in the problem’s objective function coefficients or the constraint’s right-hand sides change this optimal solution?

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6 Engineering Management & Systems Engineering The George Washington University If the isoprofit line is flatter than the carpentry constraint, Point A(0,80) is optimal. Point B(20,60) is optimal if the isoprofit line is steeper than the carpentry constraint but flatter than the finishing constraint. Point C(40,20) is optimal if the slope of the isoprofit line is steeper than the slope of the finishing constraint. X1 X2 20 40 50 60 80 20 40 60 80 100 finishing constraint Slope = -2 carpentry constraint Slope = -1 demand constraint Feasible Region A D Isoprofit line z = 120 Slope = -3/2 C B Giapetto Problem
7 Engineering Management & Systems Engineering The George Washington University Changing the objective function coefficient : The current bases remain optimal as long as the current optimal solution is the last point in the feasible region to make contact with isoprofit lines , as we move in the direction of increasing z (for a max problem) If the current basis remains optimal , then the values of the decision variables remain unchanged, but the optimal z- value may change. X1 X2 20 40 50 60 80 20 40 60 80 100 finishing constraint Slope = -2 carpentry constraint Slope = -1 demand constraint Feasible Region A D Isoprofit line z = 120 Slope = -3/2 C B Giapetto Problem

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8 Engineering Management & Systems Engineering The George Washington University X1 X2 20 40 50 60 80 20 40 60 80 100 finishing constraint, b1 = 100 carpentry constraint demand constraint Feasible Region A D Isoprofit line z = 120 C B finishing constraint, b1 = 120 finishing constraint, b1 = 80 Changing the right hand side of a constraint : Find the constraints that are
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