EMGT 501

1. EMGT 501 Final Exam Due: December 14 (Noon), 2004

2. Note • Summarize your solutions in a condensed way. • Send your PPS at toshi@nmt.edu . • Answer on PPS that has a series of slides. • Do not discuss the Final Exam with other people.

3. (1) You are given the opportunity to guess whether a coin is fair or two-headed, where the prior probabilities are 0.5 for each of these possibilities. If you are correct, you win $5; otherwise, you lose $5. You are also given the option of seeing a demonstration flip of the coin before making your guess. You wish to use Bayes’ decision rule to maximize expected profit.

4. Develop a decision analysis formulation of this problem by identifying the alternative actions, states of nature, and payoff table. • What is the optimal action, given that you decline the option of seeing a demonstration flip? • Find EVPI. • Calculate the posterior distribution if the demonstration flip is a tail. Do the same if the flip is a head. • Determine your optimal policy. • Now suppose that you must pay to see the demonstration flip. What is the most that you should be willing to pay?

5. (2) Consider the following blood inventory problem facing a hospital. There is need for a rare blood type, namely, type AB, Rh negative blood. The demand D (in pints) over any 3-day period is given by

6. Note that the expected demand is 1 pint, since E(D)=0.3(1)+0.2(2)+0.1(3)=1. Suppose that there are 3 days between deliveries. The hospital proposes a policy of receiving 1 pint at each delivery and using the oldest blood first. If more blood is required than the amount on hand, an expensive emergency delivery is made. Blood is discarded if it is still on the shelf after 21 days. Denote the state of the system as the number of pints on hand just after a delivery. Thus, because of the discarding policy, the largest possible state is 7.

7. Construct the (one-step) transition matrix for this Markov chain. • Find the steady-state probabilities of the state of the Markov chain. • Use the results from part (b) to find the steady-state probability that a pint of blood will need to be discarded during a 3-day period. (Hint: Because the oldest blood is used first, a pint reaches 21 days only if the state was 7 and then D=0.) • Use the results from part (b) to find the steady-state probability that an emergency delivery will be needed during the 3-day period between regular deliveries.

8. (3)A maintenance person has the job of keeping two machines in working order. The amount of time that a machine works before breaking down has an exponential distribution with a mean of 10 hours. The time then spent by the maintenance person to repair the machine has an exponential distribution with a mean of 8 hours.

9. Show that this process fits the birth-and-death process by defining the states, specifying the values of the and , and then constructing the rate diagram. • Calculate the . • Calculate , , , and . • Determine the proportion of time that the maintenance person is busy. • Determine the proportion of time that any given machine is working.

10. (4)Consider the EOQ model with planed shortage, as discussed in our class. Suppose, however, that the constraint S/Q=0.8 is added to the model. Derive the expression for the optimal value of Q

11. (5) A market leader in the production of heavy machinery, the Spellman Corporation, recently has been enjoying a steady increase in the sales of its new lathe. The sales over the past 11 months are shown below. Month Sales Month Sales 1 2 3 4 5 6 530 546 564 580 598 570 7 8 9 10 11 614 632 648 670 691

12. Find a linear regression line that fits the data set. • Use both Least Squares (LS) and Least Absolute • Value (LAV) methods. • (b) Show the formulation for LAV regression. • (c) Forecast the amount of sales for the 12th month, • based upon LS regression.