1º Exame de Optimização e Decisão Mestrado Integrado em Engenharia Mecânica 18 de Janeiro de 2010 5º Ano / 1º Semestre No books, notes, or calculators are allowed. 13:00 – 16:00 h Please write your student number in all exam pages. The exam has 8 questions. The value of each question is indicated inside parenthesis. 1.
(4.0) You are given the following problem: 4
Maximize 2 , subject to 16 2
3
(resource 1) 17 (resource 2) 5 and 0,
(resource 3) 0. a. (2.0) Solve this problem. b. (1.5) Obtain the shadow prices for the resources. c. (0.5) Determine how many additional units of resource 1 would be needed to increase the optimal value of by 15. 2. (3.0) The owner of a chain of three grocery stores has purchased five crates Store of fresh strawberries. The estimated probability distribution of potential sales of the strawberries before spoilage differs among the three stores. Crates 1 2 3 Therefore, the owner wants to know how to allocate five crates to the three 0 0 0 0 stores to maximize expected profit. For administrative reasons, the owner 1 5 6 4 does not wish to split crates between stores. However, he is willing to 2 9 11 9 distribute no crates to any of his stores. The table to the right gives the estimated expected profit at each store when it is allocated various 3 14 15 13 numbers of crates. 4 17 19 18 Use dynamic programming to determine how many of the five crates should be assigned to each of the three stores to maximize the total 5 21 22 20 expected profit. 3. (2.5) Formulate and graph a primal problem with two decision variables and two functional constraints that has feasible solutions and an unbounded objective function. Then formulate the dual problem and demonstrate graphically that it has no feasible solutions. 4. (1.5) Consider the following statements about any pure IP problem (in maximization form) and its LP relaxation. Label each of the statements as True or False, and then justify your answer. 1
a. (0.5) The feasible region for the LP relaxation is a subset of the feasible region for the IP problem. b. (0.5) If an optimal solution for the LP relaxation is an integer solution, then the optimal value of the objective function is the same for both problems. c. (0.5) If a noninteger solution is feasible for the LP relaxation, then the nearest integer solution (rounding each variable to the nearest integer) is a feasible solution for the IP problem. 5. (1.5) A machine shop makes two products. Each unit of the first product requires 3 hours on machine 1 and 2 hours on machine 2. Each unit of the second product requires 2 hours on machine 1 and 3 hours on machine 2. Machine 1 is available only 8 hours per day and machine 2 only 7 hours per day. The profit per unit sold is 16 for the first product and 10 for the second. The amount of each product produced per day must be an integral multiple of 0.25. The objective is to determine the mix of production quantities that will maximize profit. Formulate this problem. 6. (2.5) Consider the following convex programming problem: 4 , Minimize subject to 2 and 0. Use simple calculations just to check whether the optimal solution lies in the interval 0
1 or the interval 1
2. Do not actually solve for the optimal solution in order to determine in which interval it must lie. Explain your logic. 7. (2.5) Consider the following payoff table. Each player must make its selection before learning the decision of the other player. Player 2 Strategy 1 2 3 a. (1.5) Without eliminating dominated strategies, use the minimax 1 2 3 1 (or maximin) criterion to determine the best strategy for each player. Does this game have a saddle point? Can you tell what Player 1 2 1 4 0 the value of the game is? 3 3 ‐2 ‐1 b. (1.0) Now identify and eliminate dominated strategies as far as possible, showing the resulting reduced payoff table with no remaining dominated strategies. Make a list of the dominated strategies, showing the order in which you were able to eliminate them. 8. (2.5) You are given the following payoff table (in units of thousands of dollars): State of nature Alternative S1 S2 S3 a. (0.5) Which alternative should be chosen under the A1 220 170 110 maximin payoff criterion? Justify. A2 200 180 150 b. (0.5) Which alternative should be chosen under the maximum likelihood criterion? Justify. Prior probability 0.6 0.3 0.1 c. (0.5) Which alternative should be chosen under Bayes’decision rule? Justify. d. (1.0) Using Bayes’ decision rule, do sensitivity analysis with respect to the prior probabilities of states S1 and S2 (without changing the prior probability of state S3) to determine the crossover point where the decision shifts from one alternative to the other. Define the probability range for each alternative. 2