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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 11 Dr. Ted Ralphs IE406 Lecture 11 1 Reading for This Lecture Bertsimas 4.44.6 IE406 Lecture 11 2 More on Complementary Slackness Recall the complementary slackness conditions, p ( Ax b ) = , ( c p A ) x = . If the primal is in standard form, then any feasible primal solution satisfies the first condition . If the dual is in standard form, then any feasible dual solution satisfies the second condition . Typically, we only need to worry about satisfying the second condition, which is enforced by the simplex method. IE406 Lecture 11 3 Dual Variables and Marginal Costs Consider an LP in standard form with a nondegenerate, optimal basic feasible solution x * and optimal basis B . Suppose we wish to perturb the right hand side slightly by replacing b with b + d . As long as d is small enough, we have B 1 ( b + d ) > and B is still an optimal basis. The optimal cost of the perturbed problem is c B B 1 ( b + d ) = p ( b + d ) This means that the optimal cost changes by p d . Hence, we can interpret the optimal dual prices as the marginal cost of changing the right hand side of the i th equation. IE406 Lecture 11 4 Economic Interpretation The dual prices, or shadow prices can allow us to put a value on resources. Consider the simple product mix problem from Lecture 9. By examining the dual variable for the production hours constraint, we can determine the value of an extra hour of production time . We can also determine the maximum amount we would be willing to pay to borrow extra cash. Note that the reduced costs can be thought of as the shadow prices associated with the nonnegativity constraints. IE406 Lecture 11 5 Economic Interpretation of Optimality Consider again the product mix example from the Lecture 9. Using the shadow prices , we can determine how much each product costs in terms of its constituent resources....
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 Fall '08
 Ralphs

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