Suppose you want to solve the following LP: max crx Ax 6, but unfortunately it is infeasible (think e.g. about an inventory problem where the demand cannot be satisfied by the warehouse). Let A E Rm.?. Now suppose you can “augment” your LP by buying some more slack in your problem (think e.g. of buying some of the product from Other warehouses).particular, for i = 1, … ,m, if you want to increase the right-band side of the i-th constraint by some value k, you will pay ddy, for some fixed number dy. Suppose moreover that the right-hand side of the i-th CODetrahat can be augmented by at most for i =1 m. Now can you find the optimal augmentation, i.e. the one that maximizes the profit of the Optimal solution of the augmented LP minim the Chet for the augmentation?

Problem 5: Consider the following feasible region: P = = b, x L. 01. Now suppose that you have two objective functions: a ?primary” err and a “secondary” dTx (think of the primary objective function m the main goal of the company, my profit, and of the secondary one as some organization-related issue, my a measure of how fairly the tasks are divided among the employees). Describe an algorithm that finds, among all vectom that maximize the primary objective function, the one that achieves the maximum value of the secondary objective function.

Problem 6: We saw in class that each LP can be tramsformed into an equivalent LP in any of the following two forms below: (1) max crx Ar = 1, x>0 (2) max crx : Ax

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