MAT540 Homework
Week 9
Page 1 of 3
MAT540
Week 9 Homework
Chapter 5
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:
MAT540 Homework
Week 9
Page 2 of 3
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 ...
MAT540 Homework Week 9 Page 1 of 3 MAT540 Wee.docx
1. MAT540 Homework
Week 9
Page 1 of 3
MAT540
Week 9 Homework
Chapter 5
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
2. 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
3. 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:
MAT540 Homework
Week 9
Page 2 of 3
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
4. 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
($1,000s)
Management
Scientists required
Estimated Profit
(1,000,000s)
1 50 6 0.30
5. 2 105 8 0.85
3 56 9 0.20
4 45 3 0.15
5 90 7 0.50
6 80 5 0.45
7 78 8 0.55
8 60 5 0.40
Formulate the integer programming model for this problem and
solve it using the computer.
4. Corsouth Mortgage Associates is a large home mortgage firm
in the southeast. It has a poll of
permanent and temporary computer operators who process
mortgage accounts, including posting
payments and updating escrow accounts for insurance and taxes.
A permanent operator can process
220 accounts per day, and a temporary operator can process 140
accounts per day. On average, the
firm must process and update at least 6,300 accounts daily. The
company has 32 computer
MAT540 Homework
6. Week 9
Page 3 of 3
workstations available. Permanent and temporary operators
work 8 hours per day. A permanent
operator averages about 0.4 error per day, whereas a temporary
operator averages 0.9 error per day.
The company wants to limit errors to 15 per day. A permanent
operator is paid $120 per day wheras
a temporary operator is paid $75 per day. Corsouth wants to
determine the number of permanent
and temporary operators it needs to minimize cost. Formulate,
and solve an integer programming
model for this problem and compare this solution to the non-
integer solution.
5. Globex Investment Capital Corporation owns six companies
that have the following estimated
returns (in millions of dollars) if sold in one of the next 3 years:
Company
Year Sold
(estimated returns, $1,000,000s)
1 2 3
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Creating Print Version
P1Cab Company Schedulinglet Di = # of drivers who start their
8 hour shift in period I (I = 1,2,3,4,5,6)period 112:00:00 AM--
4:00amperiod 412 noon -- 4:00pmperiod 24:00am --
8:00amperiod 54:00pm -- 8:00pmperiod 38:00am -- 12
noonperiod 68:00pm -- midnightperiod 1period 2period 3period
4period 5period 6average fare/ driver 80500420300270210# of
drivers in each period>=>=>=>=>=>=minimum # of
drivers101220253218DVD1D2D3D4D5D6# of
drivers/periodObjective function
P2Denim JeansCD PlayerCompact
discsprofit9015030weight231Denim JeansCD PlayerCompact
discsDVConstraint<=5Objective function
P3Texas Consolidated Electronics Company ProjectExpense
($1,000s)Management Scientists requiredEstimated
Profit(1,000,000s)Project Selection
constraints1$506$0.30210580.8535690.244530.1559070.568050
.4577880.5586050.4Constraints<=<=30040DVProject12Please
include the following constraints in your solutions34Note:
project 5 >= project 256Note: All projects must be integer (1 or
0)78ObjectiveMaximize Profits
P4Mortgage AssociatesLet P = # of permanent operators and T
= # of temporary operatorsPermanent operatorTemporary
operatoraverage pay/operator12075daily # of accounts/per
27. operator220140>=6300#of computers available11<=32average
errors/ day0.40.9<=15PTDecision variablesobjective function
P5Global Investment CapitalYear Sold(Estimated returns in $
1000000)Company1231141823291115318232741621255121622
6212328000>=>=>=202535constraints12310<=120<=130<=140
<=150<=160<=1Decision variables are C15:E20this a 0-1
integer problem. Each decision variable has to be restricted to
have the value 0 or 1Objective function
