Problem #1
Greydog Bus Company operates buses between Boston and Washington, D.C. A bus trip between these two cities takes 6 hours. Federal law requires that a driver rest for four or more hours between trips. A driver's workday consists of two trips: one from Boston to Washington and one from Washington to Boston. The following table gives the departure times for the buses. Greydog's goal is to minimize the total downtime for all drivers. How would Greydog assign crews to trips? Note: It is permissible for a driver's "day" to overlap midnight. For example, a Washington-based driver can be assigned to the Washington-Boston 3 P.M. trip and the Boston-Washington 6 A.M. trip.
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Trip
|
Departure Time
|
Trip
|
Departure Time
|
|
Boston 1
|
6 A.M.
|
Washington 1
|
5:30 A.M.
|
|
Boston 2
|
7:30 A.M.
|
Washington 2
|
9 A.M.
|
|
Boston 3
|
11:30 A.M.
|
Washington 3
|
3 P.M.
|
|
Boston 4
|
7 P.M.
|
Washington 4
|
6:30 P.M.
|
|
Boston 5
|
12:30 A.M.
|
Washington 5
|
12 midnight
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Problem #2
At the beginning of year 1, a new machine must be purchased. The cost of maintaining a machine i years old is given in the table below.
|
Age at beginning of year
|
Maintenance Cost for Next Year (5)
|
|
0
|
38,000
|
|
1
|
50,000
|
|
2
|
97,000
|
|
3
|
182,000
|
|
4
|
304,000
|
The cost of purchasing a machine at the beginning of each year is given in the table below.
|
Year
|
Purchase Cost (5)
|
|
1
|
170,000
|
|
2
|
190,000
|
|
3
|
210,000
|
|
4
|
250,000
|
|
5
|
300,000
|
There is no trade-in value when a machine is replaced. Your goal is to minimize the total cost (purchase plus maintenance) of having a machine for five years. Determine the years in which a new machine should be purchased.
Problem #3
For the network shown in the figure below, find the maximum flow from source to sink.
Problem #4
The city of Smalltown consists of five subdivisions. Mayor John Lion wants to build telephone lines to ensure that all subdivisions can communicate with each other. The distances between the subdivisions are given it the figure below. What is the minimum length of telephone line required? Assume that no telephone line can be built between subdivisions I and 4.
Problem #5
Jessica Williams, manager of Kitchen Appliances for the Midtown Department Store, feels that her inventory levels of stoves have been running higher than necessary. Before visiting the inventory policy for stoves, she records the number sold each day over a period of 25 days, as summarized below.
|
Number
|
2
|
3
|
4
|
5
|
6
|
|
Sold
|
|
|
|
|
|
|
Number of
|
4
|
7
|
8
|
5
|
1
|
|
Days
|
|
|
|
|
|
(a) Use these data to estimate the probability distribution of daily sales.
(b) Calculate the mean of the distribution obtained in part (a).
(c) Describe how uniform random numbers can be used to simulate daily sales.
(d) Use the uniform random numbers 0.4476, 0.9713, and 0.0629 to simulate daily sales over 3 days. Compare the average with the mean obtained in part (b).
(e) Formulate a spreadsheet model for performing a simulation of the daily sales. Perform 300 replications and obtain the average of the sales over the 300 simulated days.