Ph.D. Preliminary Examination June 2016 (11:00 am to 4:00 pm) Candidate Name: Answer ALL Questions 1- Decision Analysis (20 points) The executive search being conducted for Western Bank by Headhunters Inc. may finally be bearing fruit. The position to be filled is a key one Vice President for Information Processing because this person will have responsibility for developing a state-of-the-art management information system that will link together Western s many branch banks. Headhunters Inc. feel they have found just the right person, Jane Fonda, who has an excellent record in a similar position for a midsized bank in New York. After a round of interviews, Western s president believes that Jane has a probability of 0.75 of designing the management information system successfully. If Jane is successful, the company will realize a profit of $3 million (net of Jane s salary, training, recruiting costs, and expenses). If she is not successful, the company will realize a net loss of $3.5 million. a. (2 points) Should Western Bank hire Jane Fonda as Vice President for Information Processing? What would be the expected gain/loss? b. (3 points) What is the expected value of perfect information? Headhunters proposed providing a detailed investigative process (including an extensive background check, a battery of academic and psychological tests, etc.) that will further pinpoint Jane s potential for success for an additional fee. This process has been found to be somewhat reliable. A candidate who would successfully design the management information system will pass the test with probability 0.8, and a candidate who would not successfully design the system will fail the test with probability 0.6. c. (2 points) What is the probability that Jane will pass the test? d. (5 points) What is the maximum $ amount you would be willing to pay for this additional investigation? e. (3 points) Suppose that the investigation costs $50,000. What is the optimal policy? What would be the expected gain/loss? f. (5 points) There is an alternative investigation process for a fee of $55,000. This process has been found to be more reliable in identifying unsuccessful candidates. A candidate who would successfully design the management information system will pass the test with probability 0.6, and a candidate who would not successfully design the system will fail the test with probability 0.8. Western s top management needs to decide whether to hire Jane and whether to have Headhunters conduct this investigative process before making this decision. 1
2- Integer Programming Problem (20 points) Dragon Airlines operates a hub at the Detroit International Airport. During the summer, the airline schedules 5 flights daily from Detroit to Atlanta and 7 flights daily from Atlanta to Detroit, according to the following schedule: Flight Leave Detroit Arrive Atlanta Flight Leave Atlanta Arrive Detroit 1 6:00 9:00 A 6:00 9:00 2 7:00 10:00 B 7:00 10:00 3 10:00 13:00 C 10:00 13:00 4 16:00 19:00 D 11:00 14:00 5 21:00 0:00 E 15:00 18:00 F 19:00 22:00 G 21:00 0:00 The flight crews live in Detroit or Atlanta, and each day a crew must fly one flight from Detroit to Atlanta and one flight from Atlanta to Detroit. A crew must return to its home city at the end of each day. For example, if a crew originates in Atlanta and flies a flight to Detroit, it must then be scheduled for a return flight from Detroit back to Atlanta. A crew must have at least 1 hour between flights at the city where it arrives. Some scheduling combinations are not possible; for example, a crew on flight 1 from Detroit cannot return on flights A or B from Atlanta. It is also possible for a flight to ferry one additional crew to a city in order to fly a return flight. The airline wants to schedule its crews in order to minimize the total amount of crew ground time (i.e., the time the crew is on the ground between flights). Excessive ground time for a crew lengthens its workday, is bad for crew morale, and is expensive for the airline. a. (12 points) Formulate an integer programming model to determine a flight schedule for the airline. Define the decision variables, and formulate the objective function and constraints clearly. b. (8 points) Suppose that Dragon Airlines relaxed its restriction that each crew must fly one flight from Detroit to Atlanta and one flight from Atlanta to Detroit such that crews can fly two flights in each direction. For instance, a crew can fly Flights 1, D, 4, and G. As before, each crew must return to its home city at the end of each day. Reformulate the problem to minimize the number of crews for Dragon Airlines. 2
3- Linear Programming Problem (20 points) Web Mercantile sells many household products through an online catalog. The company needs substantial warehouse space for storing its goods. Plans now are being made for leasing warehouse storage space over the next 4 months. Just how much space will be required in each of these months is known. However, since these space requirements are quite different, it may be most economical to lease only the amount needed each month on a month-by-month basis. On the other hand, the additional cost for leasing space for additional months is much less than for the first month, so it may be less expensive to lease the maximum amount needed for the entire 4 months. The space requirement and the leasing costs for the various leasing periods are as follows: Month Required Space Rental Monthly cost per Sq. (in Sq. Ft.) period Ft. leased 1 33000 1 $75 2 19000 2 $55 3 43000 3 $40 4 12000 4 $35 a. (8 points) Web Mercantile wants to determine the least costly rental agreement that will exactly meet its space needs each month and avoid any unused space. Formulate a linear programming model for this problem. Define your decision variables, objective function, and constraints clearly. b. (3 points) Suppose that Web Mercantile relaxed its restriction that it rents exactly the space it needs every month such that it would rent excess space. How would this effect the formulation in part a? How would this effect the optimal cost? c. (9 points) Suppose that the planning horizon is extended to N months. Let D i represent the required space (in sq. ft.) for month i = 1,, N and C i represent the monthly cost of sq. ft. leased for i months, i = 1,, N. Web Mercantile wants to determine the least costly rental agreement that will exactly meet its space needs each month and avoid any unused space. Formulate a linear programming model for this problem. Define your decision variables, objective function, and constraints clearly. You may want to use the summation symbols in your formulation. Clearly show the limits on summation symbols. 3
4- Sensitivity Analysis (20 points) WAYNE STATE UNIVERSITY The Hickory Cabinet and Furniture Company produces tables, sofas, bed frames, chairs, couches, and ottomans at its plant in Greensboro, North Carolina. The plant uses two main resources to make furniture wood and labor. The resource requirements for each piece of furniture, the total resources available weekly and unit profits are as follows: Product Table Sofa Bed frame Chair Couch Ottoman Total Profit ($100) 6 4 4.5 3 5 3 Available Labor (hr.) 1 8 1 2 2 1.5 60 Wood (ft.) 6 1 3 1 2 2 80 a. (1 points) The company wants to know how many pieces of each type of furniture to make per week to maximize profit. Formulate a linear programming model for this problem. b. (2 points) Write the dual of the problem. c. (4 points) Give the optimal solution and objective function values for the primal and dual problems. You may use the attached graph for solving the problem graphically. d. (2 points) What is the range of wood availability that keeps the solution mix in part c optimal? e. (2 points) A very experienced craftswoman, Kitty, has applied for working at the Hickory Cabinet and Furniture Company. She requested $100/hr. Would you be willing to hire her? If so, what would be the effect of hiring her for one hour on the optimal profit? f. (2 points) What is the range for the profit from chairs that keeps the same solution in part c optimal? 4
Suppose that upholstery is also a limited resource that is used in production. In addition, the company agreed to produce at least 35 items in order to have a loan from the local small business association. The problem is reformulated with these two additional constraints and solved with Excel Solver. The sensitivity report is displayed below. Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$4 Table 4 0 6 4.5 3 $D$4 Sofa 0-9.5 4 9.5 1E+30 $E$4 Bed frame 6 0 4.5 0.916666667 0.75 $F$4 Chair 12 0 3 0.785714286 0.375 $G$4 Couch 13 0 5 1 0.578947368 $H$4 Ottoman 0-0.3 3 0.3 1E+30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $I$6 Labor (hr.) 60 1.6 60 5 9.285714286 $I$7 Wood (ft.) 80 1.1 80 30 20 $I$8 Uphostery (yd.) 50 0.9 50 20 21.66666667 $I$9 Count 35-2.2 35 3.421052632 2.5 Answer the following questions with respect to the sensitivity report. g. (3 points) How much upholstery is used in a bed frame? h. (3 points) Business manager proposes to produce love seats, each of which requires 4 hrs. of labor, 2 ft. of wood, one yd. of upholstery. The profit from the love seat is expected to be 7 (in $100). Would you produce love seats? How many? i. (1 point) What would be the change in optimal profit if you purchase 10ft of wood at $50/ft.? 5
5- Queuing Theory (20 points) The Security & Trust Bank employs 1 teller to serve its customers. Customers asking for a deposit arrive according to a Poisson process at a mean rate of 10 per hour, whereas customers asking for a withdrawal request arrive according to a Poisson process at a mean rate of 15 per hour. The customers are served in first-come first-served basis. The transaction time between the teller and customer has an exponential distribution with a mean of 1.5 minutes, regardless of the type of service requested. Suppose that a customer with a deposit request has just arrived at 1:00 pm. a. (1.5 points) What is the probability that the next customer will arrive between 1:05 and 1:10? b. (1.5 points) What is the probability that the next customer will request a deposit service? Given the transaction time is not effected by the type of the service requested; assume that there is only one type of customer, arriving according to a Poisson process at a mean rate of 25 per hour. c. (3 points) Show that this process fits the birth-death process by defining states, specifying the values of the λ n and μ n, and constructing the rate diagram. Management has established the following guidelines for a satisfactory level of service to customers. The average number of customers waiting in line to begin service should not exceed 2. At least 95 percent of the time, the number of customers waiting in line should not exceed 4. For at least 95 percent of the customers, the time spent in line waiting to begin service should not exceed 7 minutes. d. (9 points) Determine how well these guidelines are currently being satisfied. e. (5 points) Management projects that the mean arrival rate will be 45 per hour a year from now. The cost of average waiting time in the queue per minute is estimated as $60 per hour, whereas the cost of a teller is $40 per hour. Determine the optimal number of tellers that minimizes the expected total cost, defined as the sum of expected waiting cost and service cost. 6