WAYNE STATE UNIVERSITY Department of Industrial and Manufacturing Engineering May, 2010 PhD Preliminary Examination Candidate Name: 1- Sensitivity Analysis (20 points) Answer ALL Questions Question 1-20 Points Question 2-20 Points Question 3-20 Points Question 4-20 Points Question 5-20 Points David, LaDeana, and Lydia are the sole partners and workers in a company which produces fine clocks. David and LaDeana each are available to work a maximum of 40 hours per week at the company, while Lydia is available to work a maximum of 20 hours per week. The company makes two different types of clocks: a grandfather clock and a wall clock. To make a clock, David (a mechanical engineer) assembles the inside mechanical parts of the clock while LaDeana (a woodworker) produces the hand-carved wood casings. Lydia is responsible for taking orders and shipping the clocks. The amount of time required for each of these tasks is shown below. Each grandfather clock built and shipped yields a profit of $300, while each wall clock yields a profit of $200. The three partners now want to determine how many clocks of each type should be produced per week to maximize the total profit. (a) Formulate a linear programming model for this problem. (b) Use the graphical method to solve the model. (c) Use this sensitivity analysis information to determine whether the optimal solution must remain optimal if the estimate of the unit profit for grandfather clocks is changed from $300 to $375 (with no other changes in the model). (d ) Repeat part (c) if, in addition to this change in the unit profit for grandfather clocks, the estimated unit profit for wall clocks also changes from $200 to $175. (e) Use graphical analysis to verify your answers in parts (c) and ( d ). (f) To increase the total profit, the three partners have agreed that one of them will slightly increase the maximum number of hours available to work per week. The choice of which one will be based on which one would increase the total profit the most. Use the 1
sensitivity analysis information to make this choice. (Assume no change in the original estimates of the unit profits.) (g) Explain why one of the shadow prices is equal to zero. (h) Can the shadow prices given in the sensitivity analysis information be validly used to determine the effect if Lydia were to change her maximum number of hours available to work per week from 20 to 25? If so, what would be the increase in the total profit? (i) Repeat part ( h) if, in addition to the change for Lydia, David also were to change his maximum number of hours available to work per week from 40 to 35. (j) Use graphical analysis to verify your answer in part (i). 2
2- Queuing Theory (20 points) Solve the following independent problems. 2.A. Trucks arrive at a warehouse according to a Poisson process with a mean rate of 4 per hour. Only one truck can be loaded at a time. The time required to load a truck has an exponential distribution with a mean of 10/n minutes, where n is the number of loaders (n = 1, 2, 3,...). The costs are (i) $18 per hour for each loader and (ii) $20 per hour for each truck being loaded or waiting in line to be loaded. Determine the number of loaders that minimizes the expected hourly cost. 2.B. Antonio runs a shoe repair store by himself. Customers arrive to bring a pair of shoes to be repaired according to a Poisson process at a mean rate of 1 per hour. The time Antonio requires to repair each individual shoe has an exponential distribution with a mean of 15 minutes. (a) Consider the formulation of this queueing system where the individual shoes (not pairs of shoes) are considered to be the customers. For this formulation, construct the rate diagram and develop the balance equations, but do not solve further. (b) Now consider the formulation of this queueing system where the pairs of shoes are considered to be the customers. Identify the specific queueing model that fits this formulation. (c) Calculate the expected number of pairs of shoes in the shop. (d) Calculate the expected amount of time from when a customer drops off a pair of shoes until they are repaired and ready to be picked up. 2.C. Consider a queueing system with two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers also arrive according to a Poisson process with a mean rate of 5 per hour. The system has two servers, both of which serve both types of customers. For both types, service times have an exponential distribution with a mean of 10 minutes. Service is provided on a firstcome-first-served basis. (a) What is the probability distribution (including its mean) of the time between consecutive arrivals of customers of any type? (b) When a particular type 2 customer arrives, she finds two type 1 customers there in the process of being served but no other customers in the system. What is the probability distribution (including its mean) of this type 2 customer s waiting time in the queue? 3
3- IP Problem Formulation: (20 points) The Fly-Right Airplane Company builds small jet airplanes to sell to corporations for the use of their executives. To meet the needs of these executives, the company s customers sometimes order a custom design of the airplanes being purchased. When this occurs, a substantial start-up cost is incurred to initiate the production of these airplanes. Fly-Right has recently received purchase requests from three customers with short deadlines. However, because the company s production facilities already are almost completely tied up filling previous orders, it will not be able to accept all three orders. Therefore, a decision now needs to be made on the number of airplanes the company will agree to produce (if any) for each of the three customers. The relevant data are given in the next table. The first row gives the start-up cost required to initiate the production of the airplanes for each customer. Once production is under way, the marginal net revenue (which is the purchase price minus the marginal production cost) from each airplane produced is shown in the second row. The third row gives the percentage of the available production capacity that would be used for each airplane produced. The last row indicates the maximum number of airplanes requested by each customer (but less will be accepted). Fly-Right now wants to determine how many airplanes to produce for each customer (if any) to maximize the company s total profit (total net revenue minus start-up costs). (a) Formulate a model with both integer variables and binary variables for this problem. A more detailed analysis of the various cost and revenue factors now has revealed that the potential profit from producing airplanes for each customer cannot be expressed simply in terms of a start-up cost and a fixed marginal net revenue per airplane produced. Instead, the profits are given by the following table. (b) Formulate a BIP model for this problem that includes constraints for mutually exclusive alternatives. (c) Formulate another BIP model for this model that includes constraints for contingent decisions. 4
4- Decision Analysis (20 points) Dr. Switzer has a seriously ill patient but has had trouble diagnosing the specific cause of the illness. The doctor now has narrowed the cause down to two alternatives: disease A or disease B. Based on the evidence so far, she feels that the two alternatives are equally likely. Beyond the testing already done, there is no test available to determine if the cause is disease B. One test is available for disease A, but it has two major problems. First, it is very expensive. Second, it is somewhat unreliable, giving an accurate result only 80 percent of the time. Thus, it will give a positive result (indicating disease A) for only 80 percent of patients who have disease A, whereas it will give a positive result for 20 percent of patients who actually have disease B instead. Disease B is a very serious disease with no known treatment. It is sometimes fatal, and those who survive remain in poor health with a poor quality of life thereafter. The prognosis is similar for victims of disease A if it is left untreated. However, there is a fairly expensive treatment available that eliminates the danger for those with disease A, and it may return them to good health. Unfortunately, it is a relatively radical treatment that always leads to death if the patient actually has disease B instead. The probability distribution for the prognosis for this patient is given for each case in the following table, where the column headings (after the first one) indicate the disease for the patient. The patient has assigned the following utilities to the possible outcomes: In addition, these utilities should be incremented by -2 if the patient incurs the cost of the test for disease A and by -1 if the patient (or the patient s estate) incurs the cost of the treatment for disease A. Use decision analysis with a complete decision tree to determine if the patient should undergo the test for disease A and then how to proceed (receive the treatment for disease A?) to maximize the patient s expected utility. 5
5- LP(Linear Programming) Problem (20 points) The Audiofile Company produces boomboxes. However, management has decided to subcontract out the production of the speakers needed for the boomboxes. Three vendors are available to supply the speakers. Their price for each shipment of 1,000 speakers is shown below. In addition, each vendor would charge a shipping cost. Each shipment would go to one of the company s two warehouses. Each vendor has its own formula for calculating this shipping cost based on the mileage to the warehouse. These formulas and the mileage data are shown below. Whenever one of the company s two factories needs a shipment of speakers to assemble into the boomboxes, the company hires a trucker to bring the shipment in from one of the warehouses. The cost per shipment is given in the next column, along with the number of shipments needed per month at each factory. Each vendor is able to supply as many as 10 shipments per month. However, because of shipping limitations, each vendor is able to send a maximum of only 6 shipments per month to each warehouse. Similarly, each warehouse is able to send a maximum of only 6 shipments per month to each factory. Management now wants to develop a plan for each month regarding how many shipments (if any) to order from each vendor, how many of those shipments should go to each warehouse, and then how many shipments each warehouse should send to each factory. The objective is to minimize the sum of the 6
purchase costs (including the shipping charge) and the shipping costs from the warehouses to the factories. (a) Draw a network that depicts the company s supply network. Identify the supply nodes, transshipment nodes, and demand nodes in this network. (b) Formulate this problem as a minimum cost flow problem by inserting all the necessary data into this network. Also include a dummy demand node that receives (at zero cost) all the unused supply capacity at the vendors. 7