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HW2 - MS&E111 Introduction to Optimization Prof Amin Saberi...

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MS&E111 Spring 2006 Introduction to Optimization May 1, 2006 Prof. Amin Saberi Homework Assignment 2: Due Wednesday May 10 1. Consider the following linear program (LP). max x 1 - x 2 subject to - 2 x 1 - 3 x 2 ≤ - 4 - x 1 + x 2 ≤ - 1 x 1 , x 2 0 a) Plot the feasible region of the primal and show that the primal objective value goes to infinity. b) Formulate the dual, plot the feasible region of the dual, and show that it is empty. 2. Consider the LP max - x 1 - x 2 subject to - x 1 - 2 x 2 ≤ - 3 - x 1 + 2 x 2 4 x 1 + 7 x 2 6 x 1 , x 2 0 a) Solve this problem in Excel using solver. After solver finds an optimal solution, ask it to generate the sensitivity report. b) Find the dual of the above LP. Read the shadow prices from the sensitivity report, and verify that it satisfies the dual and gives the same dual objective value as the primal.
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3. a) Consider the problem of feeding an army presented in Section 3.1.2. Provide a dual linear program whose solution gives sensitivities of the cost of an optimal diet to nutritional requirements. Check to see whether the sensitivities computed by solving this dual linear program match the sensitivities given by Solver when solving the primal. b) Suppose a pharmaceutical company wishes to win a contract with the army to sell digestible capsules containing pure nutrients. They sell three types of capsule, with 1 grams of carbohydrates, 1 gram of fiber, and 1 grams of protein, respectively. The army requires that the company provide a price for each capsule independently, and that substituting the nutritional value of any food item with capsules should be no more expensive than buying the food item. Explain the relation between the situation faced by the pharmaceutical company and the dual linear program.
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