MIDTERM EXAM Quantitative Methods for Business Spring 2016 1. A payoff table is given as d1 d2 d3 s1 10 14 7 s2 8 15 8 s3 6 2 9 For 5 points each, answer the following questions: a. b. c. d. e. f. What decision should be made by the optimistic decision maker? What decision should be made by the conservative decision maker? What decision should be made under minimax regret? If the probabilities of s1, s2, and s3 are .2, .4, and .4, respectively, then what decision should be made under expected value? What is the EVPI? Draw a decision tree for this problem. 2. If sample information is obtained, the result of the sample information will be either positive or negative. No matter which result occurs, the choice to select option A or option B exists. And no matter which option is chosen, the eventual outcome will be good or poor. For 10 points complete the table. Sample Result Negative Prior Probabilitie s .7 poor Positive States of Nature good .3 good .7 poor .3 Conditional Probabilities P(positive | good) = .8 P(positive | poor) = .10 P(negative | good) = P(negative | poor) = 1 Joint Probabiliti es Posterior Probabilitie s 3. For 5 points each, answer the following questions concerning the forecast listed below. Month Demand 1 20 2 25 3 40 4 30 5 50 6 58 7 65 a. Forecast for month 8 using the three-month moving average. What is the MSE? b. Forecast for month 8 using exponential smoothing with a smoothing constant of .4. What is the MSE? c. From the analysis performed above, which technique is most desirable? 4. For the payoff table, S1 -5000 -15,000 d1 d2 S2 1,000 -2,000 S3 10,000 40,000 The decision-maker will use P(s1)=.15, P(s2)=.5, and P(s3)=.35. For 5 points each, answer the following questions. a. b. What alternative would be chosen according to expected value? For a lottery having a payoff of 40,000 with probability p and -15,000 with probability (1p), the decision-maker expressed the following indifference probabilities. Payoff 10,000 1,000 -2,000 -5,000 c. Probability .85 .60 .53 .50 Let U(40,000)=10 and U(-15,000)=0 and find the utility value for each payoff. What alternative would be chosen according to the expected utility? 2 5. As part of their application for a loan to buy Sunnyside Farm, a property they hope to develop as a bed-and-breakfast operation, the prospective owners have projected: Daily fixed cost (loan payment, taxes, insurance, maintenance) are $200/night. Variable cost per occupied room per night Revenue per occupied room per night $ 25 $ 70 For 5 points each, answer the following questions: a. b. c. Write the expression for total cost. Write the expression for total revenue per day. If there are 12 guest rooms available, can they break even? What percentage of rooms would need to be occupied, on average, to break even? 6. Below is a list of sales. Year Sales 1 20 2 25 3 30 4 33 5 39 6 43 7 51 For 5 points each, answer the following questions: a. b. c. Graph this time series. Does a linear trend appear? Develop the equation for the linear trend component for the time series. Use the linear trend developed in part (b) to prepare a forecast for sales in year 8. 3