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Unformatted text preview: Duality and Sensitivity Analysis IE 220 – Spring 2011 – Prof. Katya Scheinberg Chapter 6 Chapter 6: Duality and Sensitivity Analysis Outline Outline Introduction The Essence of Duality Theory (Section 6.1) Economic Interpretation of Duality (Section 6.2) Primal–Dual Relationships (Section 6.3) Adapting to Other Primal Forms (Section 6.4) Sensitivity Analysis (Sections 6.5–6.8) Chapter 6: Duality and Sensitivity Analysis 2 Chapter 6: Duality and Sensitivity Analysis Introduction Outline Introduction The Essence of Duality Theory (Section 6.1) Economic Interpretation of Duality (Section 6.2) Primal–Dual Relationships (Section 6.3) Adapting to Other Primal Forms (Section 6.4) Sensitivity Analysis (Sections 6.5–6.8) Chapter 6: Duality and Sensitivity Analysis 3 Chapter 6: Duality and Sensitivity Analysis Introduction Introduction I Consider the following LP: maximize Z = 30 x 1 + 20 x 2 subject to x 1 + 2 x 2 ≤ 6 (1) 2 x 1 + x 2 ≤ 8 (2) x 2 ≤ 2 (3) x 1 + x 2 ≤ 1 (4) x 1 , x 2 ≥ I Here is a feasible solution: ( x 1 , x 2 ) = (2 , 1) with Z = 80. I Is it optimal? If not, How far from optimal is it? Chapter 6: Duality and Sensitivity Analysis 4 Chapter 6: Duality and Sensitivity Analysis Introduction Introduction I Consider the following LP: maximize Z = 30 x 1 + 20 x 2 subject to x 1 + 2 x 2 ≤ 6 (1) 2 x 1 + x 2 ≤ 8 (2) x 2 ≤ 2 (3) x 1 + x 2 ≤ 1 (4) x 1 , x 2 ≥ I Here is a feasible solution: ( x 1 , x 2 ) = (2 , 1) with Z = 80. I Is it optimal? If not, How far from optimal is it? I We could answer by solving the LP using simplex. Is there another way? Chapter 6: Duality and Sensitivity Analysis 4 Chapter 6: Duality and Sensitivity Analysis Introduction Lower and Upper Bounds I Since (2 , 1) is a feasible solution with Z = 80, 80 is a lower bound on the optimal objective value. I Why? Chapter 6: Duality and Sensitivity Analysis 5 Chapter 6: Duality and Sensitivity Analysis Introduction Lower and Upper Bounds I Since (2 , 1) is a feasible solution with Z = 80, 80 is a lower bound on the optimal objective value. I Why? I ( 10 3 , 4 3 ) is also a feasible solution, with Z = 380 3 = 126 . 67, so 126.67 is also a LB. I What if I told you that 160 is an upper bound on the optimal objective value? Chapter 6: Duality and Sensitivity Analysis 5 Chapter 6: Duality and Sensitivity Analysis Introduction Lower and Upper Bounds I Since (2 , 1) is a feasible solution with Z = 80, 80 is a lower bound on the optimal objective value. I Why? I ( 10 3 , 4 3 ) is also a feasible solution, with Z = 380 3 = 126 . 67, so 126.67 is also a LB. I What if I told you that 160 is an upper bound on the optimal objective value? I Then ( 10 3 , 4 3 ) is no more than 160 126 . 67 = 33 . 33 units away from optimal....
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 Spring '08
 Groover
 Linear Programming, Optimization, Sensitivity analysis, Dual problem, Duality

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