4120_lecture14-F10

4120_lecture14-F10 - Computational Methods for Management...

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Computational Methods for Management and Economics Carla Gomes Lecture 14 Reading: 6.5- 6.7 of textbook (slides adapted from: M. Hillier’s, J. Orlin’s, and H. Sarper’s)
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Dual (cont.) Duality and Sensitivity Analysis Economic Interpretation of Duality
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Primal vs. Dual Problem Maximize Z = 3 x 1 + 5 x 2 subject to 1 x 1 ≤ 4 2 x 2 ≤ 12 3 x 1 + 2 x 2 ≤ 18 and x 1 ≥ 0, x 2 ≥ 0. Minimize W = 4 y 1 + 12 y 2 + 18 y 3 subject to 1 y 1 + 3 y 3 ≥ 3 2 y 2 + 2 y 3 ≥ 5 and y 1 ≥ 0, y 2 ≥ 0, y 3 ≥ 0
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Adapting to Other Forms What if our problem is not in standard form? We can always transform it to the standard form and then construct the dual;
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Dual of the dual Note : it is not important which problem we call dual and which problem we call primal given the symmetry property of the primal dual relationships. In general we call primal the model formulated to fit the actual problem.
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Complexity of Simplex Method Primal vs. Dual How long does it take to solve an LP using the simplex method? Several factors but the most important one seems to be the number of functional constraints . Computation tends to be proportional to the cube of the number of functional constraints in an LP. The number of variables is a relatively minor factor (assuming revised simplex method) The density of the matrix of technological coefficients is also a factor – the sparser the matrix (i.e., the larger the number of zeroes) the faster the simplex method; Real world problems tend to be sparse, i.e., “sparcity” of 5% or even 1%, which leads to fast runs.
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Complexity of Simplex Method Primal vs. Dual But the most important one seems to be the number of functional constraints . Computation tends to be proportional to the cube of the number of functional constraints in an LP. Problem A takes 8 times longer than problem B Question: So if problem A has twice as many constraints as problem B how much longer takes to solve problem A in comparison to problem B?
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Primal vs. Dual? So, the size of the problem, may determine whether to use the simplex method on the primal or dual problem. If the primal has a large number of constraints and a small number of variables it is better to apply the simplex method to the dual (since it will have a small number of constraints).
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Dual Simplex Method This method is based on the duality results. It is a mirror image of the simplex method : the simplex method deals with primal feasible solutions (but not dual feasible), moving toward a solution that is dual feasible; the dual method deals with basic solutions in the primal problem that are dual feasible but not primal feasible. It moves toward an optimal solution by striving to achieve primal feasibility as well.
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Duality and Sensitivity Analysis Economic Interpretation of Duality
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Sensitivity Analysis How would changes in the problem’s objective function coefficients or right-hand side values change the optimal solution?
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Dual Variables (Shadow Prices) • y 1 *= 0 dual variable (shadow price) for resource 1
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