MAT 540 Week 9 Homework - Chapter 5

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1. Rowntown Cab Company has 70 drivers that it must schedule in three 8-hour shifts. However, the demand for cabs in the metropolitan area varies dramatically according to time of the day. The slowest period is between midnight and 4:00 A.M. the dispatcher receives few calls, and the calls that are received have the smallest fares of the day. Very few people are going to the airport at that time of the night or taking other long distance trips. It is estimated that a driver will average \$80 in fares during that period. The largest fares result from the airport runs in the morning. Thus, the drivers who sart their shift during the period from 4:00 A.M. to 8:00 A.M. average \$500 in total fares, and drivers who start at 8:00 A.M. average \$420. Drivers who start at noon average \$300, and drivers who start at 4:00 P.M. average \$270. Drivers who start at the beginning of the 8:00 P.M. to midnight period earn an average of \$210 in fares during their 8-hour shift.

To retain customers and acquire new ones, Rowntown must maintain a high customer service level. To do so, it has determined the minimum number of drivers it needs working during every 4-hour time segment- 10 from midnight to 4:00 A.M. 12 from 4:00 to 8:00 A.M. 20 from 8:00 A.M. to noon, 25 from noon to 4:00 P.M., 32 from 4:00 to 8:00 P.M., and 18 from 8:00 P.M. to midnight.

a. Formulate and solve an integer programming model to help Rowntown Cab schedule its drivers.

b. If Rowntown has a maximum of only 15 drivers who will work the late shift from midnight to 8:00 A.M., reformulate the model to reflect this complication and solve it

c. All the drivers like to work the day shift from 8:00 A.M. to 4:00 P.M., so the company has decided to limit the number of drivers who work this 8-hour shift to 20. Reformulate the model in (b) to reflect this restriction and solve it.

2. Juan Hernandez, a Cuban athlete who visits the United States and Europe frequently, is allowed to return with a limited number of consumer items not generally available in Cuba. The items, which are carried in a duffel bag, cannot exceed a weight of 5 pounds. Once Juan is in Cuba, he sells the items at highly inflated prices. The weight and profit (in U.S. dollars) of each item are as follows:

 Item Weight (lb.) Profit Denim jeans 2 \$90 CD players 3 150 Compact discs 1 30

Juan wants to determine the combination of items he should pack in his duffel bag to maximize his profit. This problem is an example of a type of integer programming problem known as a

“knapsack” problem. Formulate and solve the problem.

3. The Texas Consolidated Electronics Company is contemplating a research and development program encompassing eight research projects. The company is constrained from embarking on all projects by the number of available management scientists (40) and the budget available for R&D projects (\$300,000). Further, if project 2 is selected, project 5 must also be selected (but not vice versa). Following are the resources requirement and the estimated profit for each project.

Project Expense Management Estimated Profit
(\$1,000s) Scientists required (1,000,000s)

 Project Expense Management Estimated Profit (\$1,000s) Scientists required (1,000,000s) 1 50 6 2 105 8 3 56 9 4 45 3 5 90 7 6 80 5 7 78 8 8 60 5

QUESTION 4 & 5 INCLUDED IN THE SOLUTIONS

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