Practice Midterm IND E 411 Winter 2015 February 2, 2015

Transcription

1 Name: Practice Midterm IND E 411 Winter 2015 February 2, 2015 Instructions: You have 110 minutes. The value of each problem is stated. Partial credit is given for the correct method of solution, so set up each problem clearly and show your work. 34 pts 1. University Toys has developed a brand new product line a series of Engineering Professor Action Figures (EPAFs) featuring likenesses of popular professors at the local engineering school. Management needs to decide how to market the dolls. One option is to immediately ramp up production and simultaneously launch an ad campaign in the university newspaper. This option would cost $1,000. Based on past experience, new action figures either take off and do well or fail miserably. Hence, the prediction is for one of two possible outcomes total sales of 2,500 units or total sales of only 250 units. University Toys receives revenue of $2 per unit sold. Management currently thinks that there is about a 40% chance that the production will do well (sell 2,500 units) and a 60% chance that it will do poorly (sell 250 units). Another option is to test market the product locally. The company could build a few action figures, put up a display in the campus bookstore, and see how they sell without any further advertising. This would require less capital for the production run and no money for advertising. The test market has two possible outcomes, sell 200 units (sell well), or only sell 20 units (sell poorly). The cost for this option is estimated to be $100. University Toys receives revenue of $2 per unit sold for the test market as well. The company has often test marketed toys in this manner. Products that sell well when fully marketed have also sold well in the test market 80% of the time. Products that sell poorly when fully marketed also sell poorly in the test market 60% of the time. (a) Develop a decision analysis payoff table with the decision alternatives, the states of nature, the payoffs and prior probabilities.

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3 (b) Recommend a course of action based on 3 different criteria Maximin payoff Don t produce (payoff 0) Maximum likelihood Don t produce (payoff 0) Bayes decision rule Produce ( expected payoff 1300) (c) Calculate all the posterior probabilities using Bayes Theorem. February 2, 2015 Page 3/11

4 (d) Construct the decision tree and use it to determine the optimal course of action for University Toys. (e) Find EVPI & EVE. What is the maximum amount of money University Toys should be willing to pay to test market the EPAFs? February 2, 2015 Page 4/11

5 21 pts 2. Patty is trying to determine which of two college courses to take. If she takes the Operations Research course, she believes that she has a 10% chance of receiving an A, a 40% chance for a B, and a 50% chance for a C. If Patty takes a statistics course, she has a 70% chance for a B, a 25% chance for a C, and a 5% chance for a D. Patty wants to explore her utility function for grades, and she sets U(A) = 4 and U(D) = 1. Patty is indifferent between the following lotteries for grades, 1 L 1 C and L 2 She is also indifferent between 1 L 3 B and L A D A D (a) If Patty wants to take the course that maximizes the expected utility of her final grade, which course should she take? Show your work! (b) Patty believes that if she spends more time than regular on the statistics course, she might have a chance to get an A. Assuming she still has a 25% chance for a C, and 5% chance for a D, find the break-even probability she gets an A for the statistics course that makes her indifferent between the Operations Research course and the statistics course. February 2, 2015 Page 5/11

6 17 pts 3. A bookstore keeps daily track of the inventory level of a popular book to restock it to a level of 100 copies at the start of each day. The data for the last 30 days provide the following end-of-day inventory position: 1, 2, 0, 3, 2, 1, 0, 0, 3, 0, 1, 1, 3, 2, 3, 3, 2, 1, 0, 2, 0, 1, 3, 0, 0, 3, 2, 1, 2, 2. (a) Let state i indicate the end-of-day inventory position, i {0, 1, 2, 3}. Represent the daily inventory as a Markov chain, and provide the one-step transition probability matrix P. (b) If the end-of-day inventory position today is 0, what s the probability the end-of-day inventory position is 0 the day after tomorrow? February 2, 2015 Page 6/11

7 (c) Write down the equations you need to solve for the steady states probabilities (don t calculate). (d) Determine the expected daily inventory using the steady-state probabilities (π 0, π 1, π 2, π 3) = (0.276, 0.215, 0.271, 0.238). (e) Determine the average number of days between successive zero inventories using the steady-state probabilities given in part (d). February 2, 2015 Page 7/11

8 20 pts 4. A soap company specializes in a luxury type of bath soap. The sales of this soap fluctuate between two levels low and high depending upon two factors: (1) whether they advertise and (2) the advertising and marketing of new products being done by competitors. The second factor is out of the company s control, but it is trying to determine what its own advertising policy should be. The marketing manager s proposal is to advertise when sales are low but not to advertise when sales are high. Advertising in any quarter of a year has its primary impact on sales in the following quarter. Therefore, at the beginning of each quarter, the needed information is available to forecast accurately whether sales will be low or high that quarter and to decide whether to advertise that quarter. The cost of advertising is $1 million for each quarter of a year in which it is done. When advertising is done during a quarter, the probability of having high sales the next quarter is 0.5 or 0.75, depending upon whether the current quarter s sales are low or high. These probabilities go down to 0.25 or 0.5 when advertising is not done during the current quarter. The company s quarterly profits (excluding advertising costs) are $4 million when sales are high but only $2 million when sales are low. (Hereafter, use units of millions of dollars.) (a) Let state 0 indicate the Low level of sales and state 1 indicate the High level of sales during the current quarter, construct the (one-step) transition matrix for each of the following advertising strategies: (1) never advertise (2) always advertise (3) follow the marketing manager s proposal February 2, 2015 Page 8/11

9 (b) The steady states for each of the 3 advertising strategies are given as (1) never advertise: 0 = 2/3, 1 = 1/3; (2) always advertise: 0 = 1/3, 1 = 2/3; (3) follow the marketing manager s proposal: 0 = 1/2, 1 = 1/2. Find the long-run expected average profit (including a deduction for advertising costs) per quarter for each of the three advertising strategies in part (a). Which of these strategies is best according to this measure of performance. (1) never advertise (2) always advertise (3) follow the marketing manager s proposal NEVER ADVERTISE February 2, 2015 Page 9/11

10 18 pts 5. A rat is in the maze pictured below, and moves randomly. The probability that a rat in compartment 1 will move to compartment 2 is 0.3; to compartment 3 is 0.2; and to compartment 4 is 0.5. A rat in compartment 2 will move to compartments 1, 4, or 5 with probabilities of 0.2, 0.6, and 0.2, respectively. A rat in compartment 3 cannot leave that compartment. A rat in compartment 4 will move to 1, 2, 3, or 5 with probabilities of 0.1, 0.2, 0.4, and 0.3, respectively. A rat in compartment 5 cannot leave that compartment. (a) Define the states and provide the one-step transition probability matrix P using this information. Let the state be the compartment the rat is in, 1, 2,..., 5. February 2, 2015 Page 10/11

11 (b) Determine the classes of this Markov chain and, for each class, determine whether it is recurrent or transient. Class 1 {1,2,4} transient Class 2 {3} recurrent Class 3 {5} recurrent (c) Find the probability that a rat ends up in compartment 5 if it was originally in compartment 1. f 15 = 0.3f f 45 f 25 = 0.2f f 45 f 45 = 0.1f f f 15 = (d) If the rat starts in the compartment 2, find the probability that it ends up in compartment 5 f 25 = (e) If the rat starts in the compartment 4, is it more likely to end up in compartment 3 or compartment 5. f 45 = f 43 = COMPARTMENT 3 February 2, 2015 Page 11/11