WAYNE STATE UNIVERSITY Department of Industrial and Manufacturing Engineering February, 2008

Transcription

1 WAYNE STATE UNIVERSITY Department of Industrial and Manufacturing Engineering February, 2008 PhD Preliminary Examination Candidate Name: 1- Sensitivity Analysis (14 points- 2 each) A manufacturer produces two types of plastic cladding. These have the trade names Ankalor and Beslite. One yard of Ankalor requires 8 lb of polyamine, 2.5 lb of diurethane and 2 lb of monomer. A yard of Beslite needs 10 lb of polyamine, 1 lb of diurethane, and 4 lb of monomer. The company has in stock 80,000 lb of polyamine, 20,000 lb of diurethane, and 30,000 lb of monomer. Both plastics can be produced by alternate parameter settings of the production plant, which is able to produce sheeting at the rate of 12 yards per hour. A total of 750 production plant hours are available for the next planning period. The contribution to profit on Ankalor is $10/yard and $20/yard on Beslite. The company has a contract to deliver at least 3,000 yards of Ankalor. An LP model is developed to find the production plan maximizing the contribution to the firm's profit from this product division. Definition of variables: A = Number of yards of Ankalor produced B = Number of yards of Beslite produced LP model: 1) Maximize 10 A + 20 B subject to 2) 8 A + 10 B 80,000 (lbs. Polyamine available) 3) 2.5 A + 1 B 20,000 (lbs. Diurethane available) 4) 2 A + 4 B 30,000 (lbs. Monomer available) 5) A + B 9,000 (lbs. Plant capacity) 6) A 3,000 (Contract) A 0, B 0 The LINDO solution is: OBJECTIVE FUNCTION VALUE VARIABLE VALUE REDUCED COST A B ROW SLACK OR SURPLUS DUAL PRICES 2) 0 2 3) ) ) ) 0-6 Answer ALL Questions Question 1-14 Points Question 2-20 Points Question 3-21 Points Question 4-25 Points Question 5-20 Points 1

2 RANGES IN WHICH THE BASIS IS UNCHANGED VARIABLE CURRENT COEF ALLOWABLE INCREASE ALLOWABLE DECREASE A 10 6 INFINITY B 20 INFINITY 7.5 ROW RIGHTHAND SIDE RANGES ALLOWABLE ALLOWABLE CURRENT RHS INCREASE DECREASE INFINITY INFINITY INFINITY Row Basis A B SLK1 SLK2 SLK3 SLK4 SLK5 SLK6 1 ART E+05 2 B SLK SLK SLK A Consult the LINDO output above to answer the following questions Suppose that the company can purchase 2000 pounds of additional polyamine for $2.50 per pound. Should they make the purchase? Explain. 2. If the profit contribution from Beslite were to decrease to $12/yard, will the optimal solution change? 3. If the profit contribution from Ankelor were to increase to $15/yard, will the optimal solution change? 4. Suppose that the company could deliver 1000 yards less than the contracted amount of Ankalor by paying a penalty of $5/yard shortage. Should they do so? 5. Regardless of your answer above, suppose that they do deliver 1000 yards less Ankalor. This is equivalent to a. increasing the slack in row 6 by 1000 b. increasing the surplus in row 6 by 1000 c. decreasing the slack in row 6 by 1000 d. decreasing the surplus in row 6 by If the company delivers 1000 yards less of Ankalor, how much Beslite should they deliver? How will the decision to deliver 1000 yards less Ankalor change the quantity of diurethane used during the next planning period? 2

3 2- Queuing Theory (20 points) The Centerville International Airport has two runways, one used exclusively for takeoffs and the other exclusively for landings. Airplanes arrive in the Centerville air space to request landing instructions according to a Poisson process at a mean rate of 10 per hour. The time required for an airplane to land after receiving clearance to so has an exponential distribution with a mean of 3 minutes, and this process must be completed before giving clearance to so to another airplane. Airplanes awaiting clearance must circle the airport. The Federal Aviation Administration has a number of criteria regarding the safe level of congestion of airplanes waiting to lands. These criteria depend on a number of factors regarding the airport involved, such as the number of runways available for landing. For Centerville, the criteria are (1) the average number of airplanes waiting to receive clearance to land should not exceed 1 (2) 95 percent of the time, the actual number of airplanes waiting to receive clearance to land should not exceed 4. (3) For 99 percent of the airplanes, the amount of time spent circling the airport before receiving clearance to land should not exceed 30 minutes (since exceeding this amount of time often would require rerouting the plane to another airport for an emergency landing before its fuel runs out) a) Evaluate how well these criteria are currently being satisfied. (12 points) b) To attract additional business, airport management is considering adding a second runway for landings. It is estimated that this eventually would increase the mean arrival rate to 25 airplanes per hour. Test whether criteria (1) and (2) are satisfied? ( 8 points) 3

4 3- IP Problem Formulation: Dream Team (20 points) Coach Bob is faced with the decision of selecting 7 star players for the Dream Team. He has narrowed his choice down to 10 players. For each player, Bob has collected and rated some statistics (1 being best, and 5 being worst) for the players. In addition, players can only play certain positions of the line up. The positions that each player is allowed to play and the player s assists, scoring, defense and rebound skills are listed in the table below. In order to have a well-rounded team, the coach knows he must fulfill the following requirements: a) At least two members must be able to play guard, at least four members must be able to play forward, and at least two players must be able to play center (some players have to be versatile). b) The average assists, scoring, and rebounding level of the 7 star players must be at worst 4. ( Keep in mind, 1 is best and 5 is worst) c) If player 4 and player 6 both play, then player 5 can not be on the team (Players have compatibility issues!). d) Players 3 and 9 must be selected together because they feel they are most effective when they play together (so either both or neither are selected). e) Either player 4 or player 3 (or both) must be included because they are the ones that bring in the fans. Given these constraints, Coach Bob wants to maximize the total scoring ability of the Dream team. Formulate an IP that will help him choose his starting team. (Do not solve the IP) Player Position Assists Scoring Rebounding Defense 1 G C G-F F-C G-F F-C G-F G-C F G-F

5 4- Decision Analysis (25 points) A recent IME bachelors graduate is trying to choose a post-graduation job. The graduate has two offers: BIG automotive company, an established organization that offers a 2-year contract with an annual salary of 80K, which could (Probability = 20%) provide a bonus of 20K at the end of any year. START UP tier-2 supplier that pays only 40K but, at the end of the year 1, gives the graduate shares in the company if the graduate signs up for the year 2. If the START UP s product works out well by the end of year 2 (probability = 20%), the graduate could then cash in these shares and receive $400K. If the graduate does not sign up for year 2, then the graduate can go to company MID (a tier-1 supplier) and get 50K for that year. Graduate has never taken a course on decision analysis and hence asked for your help on this though decision. a) Graduate wants to know which offer to choose in order to maximize the expected value at the end of 2 years? (Draw the decision tree for this choice, giving all information provided) (10 points) b) The graduate is concerned about the uncertainty associated with START UP, and is thinking about getting some extra information that would help you make his/her choice. Calculate the Expected Value of Perfect Information on the success of START UP s product. (6 points) c) Alternatively, the graduate could buy a market study of START-UP s product before deciding which job to take. By buying this market study, the graduate would get a report that would assess it as either great or poor. You assess that this type of market study is inaccurate, only giving correct results 60% of the time [i.e, probability (study says great if product will be success) = prob (says poor if product will fail) = 0.6]. (9 points) i. Given the initial assessment that the product has 20% chance of success, what is the probability that the market study will report that the product is great. ii. Calculate how the test results would change your initial assessment of success? Prob(Product will be success if Study says Great )=? Prob(Product will be success if Study says Poor )=? iii. What is the expected value of the market study information? Under what condition would you recommend buying the market survey? 5

6 5- LP(Linear Programming) Problem (20 points) Sarah has just graduated from high school. As a graduation present, her parents have given her a car fund of $21,000 to help purchase and maintain a certain three-year-old used car for college. Since operating and maintenance costs go up rapidly as the car ages, Sarah's parents tell her that she will be welcome to trade in her car on another three-yearold car one or more times during the next three summers if she determines that this would minimize her total net cost. They also inform her that they will give her a new car in four years as a college graduation present, so she should definitely plan to trade in her car then. (These are pretty nice parents!) The table gives the relevant data for each time Sarah purchases a three-year-old car. For example, if she trades in her car after two years, the next car will be in ownership year 1 during her junior year, etc. Sarah's Data Each Time She Purchases a Three-Year Old Car Purchase Operating and Maintenance Costs for Ownership Year Trade-in Value at End of Ownership Year Price $12,000 $2,000 $3,000 $4,500 $6,500 $8,500 $6,500 $4,500 $3,000 When should Sarah trade in her car (if at all) during the next three summers to minimize her total net cost of purchasing, operating, and maintaining the cars over her four years of college? a) Formulate this as a shortest-path problem i. Provide the network diagram with description of arcs and nodes as well as values of arc lengths. (7 points) ii. Provide a linear programming model. (Explicitly state the decision variables, objective function and constraints). (7 points) b) One solution approach is to solve using simplex method. Suggest another solution approach and solve with this approach. (6 points) 6