EMGT 501 Fall 2005 Final Exam Due Day: Dec 12 (Noon)

Note: (a) Do not send me after copying your computer results of QSB. Answer what are your decision variables, formulation and solution, only. See my HW answer on my HP. (b) Put your mailing address so that I will be able to return your exam result via US postal service. (c) Answer on a PPS series of slides. (d) Do not discuss on the exam with other students. (e) Return your answer attached to your e-mail.

Question 1 (20%) Consider the birth-and-death process with all Display the rate diagram for the B-D process and balance equations. Calculate P0 and P1. Then give a general expression for Pn in terms of P0 for n = 2, 3, … Consider a queuing system with two severs that fits this process. What is the mean arrival rate for this queuing system? What is the mean service rate for each server when it is busy serving customers?

Question 2 (15%) Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 5 minutes with each customer. Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times.

(b) Service goals dictate that an arriving customer should not wait for service more than an average of 5 minutes. Is this goal being met? If not, what action do you recommend? (c) If the consultant can reduce the average time spent per customer to 4 minutes, what is the mean service rate? Will the service goal be met?

Question 3 (20%) A real estate investor has the opportunity to purchase land currently zoned residential. If the county board approves a request to rezone the property as commercial within the next year, the investor will be able to lease the land to a large discount firm that wants to open a new store on the property. However, if the zoning change is not approved, the investor will have to sell the property at a loss. Profits (in thousands of dollars) are shown in the following payoff table.

Rezoning Approved Rezoning Not Approved State of Nature Rezoning Approved Rezoning Not Approved Decision Alternative s1 s2 Purchase, d1 600 -200 Do not purchase, d2 0 0 If the probability that the rezoning will be approved is 0.5, what decision is recommended? What is the expected profit? The investor can purchase an option to buy the land. Under the option, the investor maintains the rights to purchase the land anytime during the next three months while learning more about possible resistance to the rezoning proposal from area residents. Probabilities are as follows.

Let H = High resistance to rezoning L = Low resistance to rezoning What is the optimal decision strategy if the investor uses the option period to learn more about the resistance from area residents before making the purchase decision. (c) If the option will cost the investor an additional $10,000, should the investor purchase the option? Why or why not? What is the maximum that the investor should be willing to pay for the option?

Question 4 (15%) An individual was interested in determining which of two stocks to invest in, Central Computing Company (CCC) or Software Research, Inc. (SRI). The criteria thought to be most relevant in making the decision are the potential yield of the stock and the risk associated with the investment. The pair-wise comparison matrixes for this problem are Criterion Yield Risk Yield 1 3 Risk 1/3 1 Yield CCC SRI CCC 1 5 SRI 1/5 1 Risk CCC SRI CCC 1 1/4 SRI 4 1

Compute the priorities for each pair-wise comparison matrix. Determine the overall priority for the two investments, CCC and SRI. Which investment is preferred based on yield and risk?

Question 5 (20%) The management of a chain of fast-food restaurants wants to investigate the relationship between the daily sales volume (in dollars) of a company restaurant and the number of competitor restaurants within a 1-mile radius. The following data have been collected. Number of Competitors Within 1 Mile 1 2 3 4 5 Sales ($) 3800 3500 3200 2900 2700 2500 2100 2000

Develop the L-2 (least-squares) regression that relates daily sales volume to the number of competitor restaurants within a 1-mile radius. (b) Do the same task by using L-1 (least absolute value) regression. (c) Use the estimated regression equations developed in part (a) and (b) to forecast the daily sales volume for a particular company restaurant that has four competitors within a 1-mile radius. (d) Compare the above results.

Question 6 (10%) Data collected from selected major metropolitan areas in the eastern United States show that 10% of individuals living within the city limits move to the suburbs during a one-year period while 5% of individuals living in the suburbs move to the city during a one-year period. Answer the following questions assuming that this process is modeled by a Markov process with two states: city and suburbs.

Prepare the matrix of transition probabilities. Compute the steady-state probabilities. In a particular metropolitan area, 35% of the population lives in the city, and 65% of the population lives in the suburbs. What population changes do your steady-state probabilities project for this metropolitan area?