CSCI 386 Spring 2005
Exam 3 (due April 20, 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.
1. At the beginning of year 1, a manufacturing company must purchase a new machine. The cost of purchasing this machine at the beginning of year 1 (and at the start of each subsequent year) is given in the file Exam3_1.xls. The cost of maintaining this type of machine, depending on its age, is also given in the file Exam3_1.xls. Assume that there is no trade-in value when a machine is replaced. The goal of this company is to minimize the total cost (which includes the purchase and maintenance costs) of owning and using this type of machine for the next five years. In particular, the company wants to determine the years in which a new machine should be purchased. Formulate and solve a linear network model in the given Excel spreadsheet to assist this company.
2. An accounting firm must perform an audit of a large corporation. The first phase of the audit consists of 10 specific activities. The predecessor(s) and duration for each of these activities are given in the file Exam3_2.xls. Formulate and solve an LP model in the given spreadsheet shell to find the critical path and the minimum number of days required to complete the first phase of this audit.
3. A major pharmaceutical company markets a popular prescription drug to physicians across the country. In particular, this company has two new sales representatives available to assign to physicians located in 7 regions (labeled A through G) in southern Florida. The number of physicians located in each pair of adjacent regions is given in the file Exam3_3.xls. Each of the two new sales representatives must be assigned to one pair of adjacent regions. Furthermore, the company wants to ensure that at most one sales representative covers each of the 7 regions under consideration. The company’s goal is to maximize the total number of physicians that are located in the regions that are covered by the sales representatives. Formulate and solve an integer programming model in Exam3_3.xls to help this company assign the two sales representatives to adjacent regions in Florida.
4. A company manufactures its most popular product at three plants. Four major retail customers demand this product. The cost of producing one unit of the product at each plant and shipping it to each of the customers is given in the file Exam3_4.xls. Moreover, the fixed (setup) cost of production at each plant, capacity of each plant, and the demand of each customer are also provided in this file. The company wants to determine the least-cost method for meeting the demands of their four customers. Formulate and solve an appropriate integer programming model in the file Exam3_4.xls.
5. In preparing for a 10-day space shuttle mission, NASA must decide how many of each of six items should be loaded into the shuttle’s payload bay. The weight and perceived benefit (where a larger benefit value indicates a more useful item) of each of the items are given in the file Exam3_5.xls. The shuttle can safely carry no more than 1200 pounds of these items, given the various pieces of equipment and supplies that have already been loaded into the vehicle. In trying to decide what should be taken on the mission, NASA must ensure the following: (a) if any units of item type 1 are brought on board, then at least one unit of item type 5 must brought on board; (b) if any units of item type 2 are brought on board, then no units of item type 6 can be brought on board; (c) no more than four different types of items must be brought on board; and (d) units of item 3 or item 4 can be brought along but not both of these item types. Formulate and solve an appropriate integer programming model in the file Exam3_5.xls to help NASA decide how many units of each type of item to take on the mission in order to maximize the total benefit to be achieved.
6. USA Airlines has decided that it wants to establish a “hub system” in the United States. Each hub is used for connecting flights to and from cities within 550 miles of the hub. This airline runs flights among the cities listed in the file Exam3_6.xls. Note that each city is assigned to one of four geographical regions (southeastern, northeastern, southwestern, and northwestern), where the geographical assignment is denoted by the color of the font in which each city is listed. Moreover, the distance (in miles) between each pair of cities is also provided in the file Exam3_6.xls. USA Airlines wants to determine the smallest number of hubs that it will need to ensure that at least one hub is placed within 550 miles of each city served by the company. Furthermore, the airline wants to make sure that it does not establish more than two hubs in each of the four geographical regions of the nation. Formulate and solve an IP model in the given spreadsheet shell to help USA Airlines solve its problem.
7. A company is considering manufacturing three types of oak chairs: a small oak chair (Product 1), a medium-sized chair (Product 2), and a large oak chair (Product 3). The resources required and the profits yielded by each type of chair are shown in the file Exam3_7.xls. At present, 8000 board feet of oak and 8000 hours of labor are available for production each month. If any units of a given type of chair are manufactured, production of that type will be economically feasible only if at least 80 units of that type are produced. This company wants to find a monthly production plan that maximizes its total profit. Formulate and solve an IP model in the given spreadsheet shell to help the company find the best production schedule.
8. The Beaver Lumber Company sells 3-foot, 4-foot, 5-foot, and 8-foot wood boards. This company’s customers demand 75 3-foot boards, 50 4-foot boards, 40 5-foot boards, and 30 8-foot boards per week. Beaver Lumber meets its demands by cutting up 20-feet long boards. How can this lumber producer satisfy its customers’ demands with the minimal amount of waste? Assume that all boards share the same width and thickness. Formulate and solve an IP model in the given spreadsheet Exam3_8.xls to help Beaver Lumber find the waste-minimizing way to meet all weekly customer demands for lumber.