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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 12 Dr. Ted Ralphs IE316 Lecture 12 1 Reading for This Lecture • Bertsimas 4.44.6 IE316 Lecture 12 2 More on Complementary Slackness • Recall the complementary slackness conditions, p T ( Ax b ) = 0 , ( c T p T A ) x = 0 . • 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. IE316 Lecture 12 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 T B B 1 ( b + d ) = p T ( b + d ) • This means that the optimal cost changes by p T d . • Hence, we can interpret the optimal dual prices as the marginal cost of changing the right hand side of the i th equation. IE316 Lecture 12 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 the Lecture 10. • 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 are the shadow prices associated with the nonnegativity constraints. IE316 Lecture 12 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
 Operations Research, Linear Programming, Optimization, optimality, dual simplex method

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