CSCI 386 Spring 2005
Exam 1 (due Feb 14, 2005)
Please complete your exam independently and turn in a hard copy of your exam and a disk which contains the excel files of your exam. Please use a different disk from the disk you use for your homework.
1. A recent college graduate is trying to save for his retirement. He believes that he will most likely be able to place $1800 in his retirement account at the beginning of each of the next 40 years. His goal is to retire after 40 years of work with a total of $1,000,000 in his account. What is the lowest expected annual interest rate that will ensure that this individual reaches his financial goal after 40 years of saving? Answer this question by formulating and analyzing an appropriate model in the file Exam1_1.xls.
2. The file Exam1_2.xls contains data on the price and demand of a particular product for each month of the previous year. The product manager would like to estimate the relationship between price and demand of this product. Which of the following functional relationships yields the lowest MAPE: linear function, power function, or exponential function? Show all of your work and state your conclusion in Exam1_2.xls.
3. A sailboat manufacturer is determining its production schedule for the next 6 months. Assume that it costs this company $950 to manufacture each sailboat. At the end of each month, a holding cost of $200 per unit left in inventory is incurred. Monthly demands for sailboats must be met on time (that is, no backlogging of demand is allowed) and are projected to be as follows: 250 in month 1; 275 in month 2; 265 in month 3; 256 in month 4; 269 in month 5; and 278 in month 6. Assume that at the beginning of month 1, 120 completed sailboats are in inventory. Furthermore, this company can produce up to 270 sailboats per month. Finally, assume that only half of the sailboats manufactured during any month can be used to meet the current month’s demand. Using the shell provided in the file Exam1_3.xls, formulate and solve a linear programming model to find a cost-minimizing production schedule for the given planning horizon.
4. A computer manufacturer is developing a production schedule for the next four quarters. Demands for this manufacturer’s laptop computer are forecasted to be 1200 in quarter 1; 2100 in quarter 2; 1500 in quarter in 3; and 600 in quarter 4. Each quarter, up to 1400 laptops can be produced with regular-time labor at a cost of $1200 per unit. During each quarter, an unlimited number of laptops can be produced with overtime labor at a cost of $1800. Of all units produced, 2% are defective and cannot be used to meet demand. After each quarter’s demand is satisfied and defective items are accounted for, a holding cost of $300 per computer is assessed. All demands must be met on time, and the units produced in one quarter can be used to meet demand for the current quarter as well as for future quarters. Assume that at the beginning of quarter 1 no computers are in inventory. Using the shell provided in the file Exam1_4.xls, formulate and solve a linear programming model to find the production schedule that minimizes total cost over the given planning horizon.
5. Giapetto’s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hours of finishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for train is unlimited, but at most 40 soldiers are bought each week. Giapetto wants to maximize weekly profit. Formulate a mathematical model of Giapetto’s situation that can be used to maximize Giapetto’s weekly profit, and solve the model graphically.
6. The Berwick nuclear power plant employs 33 skilled technicians. Each technician works five days per week. The plant’s labor requirements fluctuate with the day of the week, so the number of skilled technicians required each day depends on the day of the week: Saturday, 18; Sunday, 27; Monday, 19; Tuesday, 29; Wednesday, 28; Thursday, 18; Friday, 23. Plant management wants to schedule skilled technicians so as to minimize the number whose days off are not consecutive. Use linear programming to determine how to accomplish this goal. Construct and solve your linear programming model in the file Exam1_6.xls.
7. During the next 4 quarters, a company faces the following demands for its product: quarter 1, 400 units; quarter 2, 800 units; quarter 3, 200 units; and quarter 4, 600 units. During any quarter a worker can produce up to 60 units of the product. Each worker is paid $10,000 per quarter. Workers can be hired or fired at a cost of $6,000 per worker fired and $8,000 per worker hired. The cost of holding a unit of the product in inventory for one quarter is $200. Demand can be backlogged at a cost of $150 per unit per quarter. At the beginning of quarter 1, this company has 9 workers on its production staff. During any quarter, at most 2 workers can be hired. All demand must be met by the end of quarter 4. Finally, the raw material used to manufacture one unit of the product costs $600. Using the shell provided in the file Exam1_7.xls, formulate and solve a linear programming model to find a cost-minimizing aggregate plan (i.e., including both worker and production planning) for the given planning horizon.
8. Delta Consulting is trying to adjust the size of its staff of professional consultants to meet the expected needs of its clients over the next four quarters. The following numbers of consulting hours will be needed in the coming year: 36000 hours in quarter 1; 42000 hours in quarter 2; 35000 hours in quarter 3; and 39000 hours in quarter 4. At the beginning of quarter 1, Delta employs 65 experienced consultants. Assume that each of Delta’s experienced consultants can work up to 625 hours each quarter. Experienced consultants are paid $20,000 per quarter. To meet demands for services in future quarters, new consultants must be hired and trained from time to time. It takes one quarter to completely train a new consulting professional. During the training period, a trainee must be supervised for 200 hours by an experienced consultant. Given their limited knowledge and skills, trainees are only paid $10,000 per quarter. Historical data suggest that approximately 4% of the company’s experienced consultants quit by the end of each quarter. Delta would like to determine a plan that minimizes the total cost of meeting its demands for consulting services over the upcoming four quarters. Use the shell provided in the file Exam1_8.xls to formulate and solve a linear programming model for this workforce planning problem.