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4120_lecture4

# 4120_lecture4 - Computational Methods for Management and...

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Computational Methods for Management and Economics Carla Gomes Lecture 4 Reading: 3.3 and 3.4 of textbook

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Outline Sensitivity Analysis (summary) Linear Programming Assumptions Additional Examples

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Sensitivity Analysis   0 2 4 6 8 8 6 4 2 Production rate for windows Production rate for doors Feasible region x*=(2, 6) 10 W D P = 3600 = 300D + 500W P = 3000 = 300D + 500W P = 1500 = 300D + 500W Our objective function is: maximize 3x 1 +c 2 x 2 How does the optimal solution change as c 2 changes? Optimal solution 0 <c 2 <2 x*=(4,3) Optimal solution c 2 <0 x*=(4,0) Max Z = 3 x1+ 5 x2 S.t. x1 ≤4 2 x2 ≤12 3 x1+ 2 x2 ≤18 x1 0; x2 0 D B C A E
Sensitivity Analysis   0 2 4 6 8 8 6 4 2 Production rate for windows Production rate for doors Feasible region x*=(2, 6) Optimal solution c 2 >2 10 W D P = 3600 = 300D + 500W P = 3000 = 300D + 500W P = 1500 = 300D + 500W Our objective function is: maximize 3x 1 +c 2 x 2 How does the optimal solution change as c 2 changes? Multiple Optimal solution c 2 =2 P=18 x* = (2,6) ; x*=(4,3) And any convex combination (CD) Optimal solution 0 <c 2 <2 x*=(4,3) Multiple Optimal solution c 2 =0 P=12 x*=(4,3) and (4,0) and any Convex combination (DE) Optimal solution c 2 <0 x*=(4,0) Max Z = 3 x1+ 5 x2 S.t. x1 ≤4 2 x2 ≤12 3 x1+ 2 x2 ≤18 x1 0; x2 0 D B C A E

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Linear Programming Assumptions: Proportionality Additivity Divisibility Certainty
LP Assumptions Proportionality The contribution of each activity to the value of the objective function Z is proportional to the level of the activity x j as represented by the c j x j term; The contribution of each activity to the left-hand side of each functional constraint is proportional to the level of the activity x j as represented by the term a ij . This assumption implies that all the x terms of the linear equations cannot

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Examples of violation of proportionality assumption Case 1 - violation occurs as a result of e.g., startup costs , associated with product 1.
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4120_lecture4 - Computational Methods for Management and...

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