# Integer programming problems

Integer programming (ip) modelsInteger programming graphical solutionComputer solution of integer programming problemsAttached are the problems. please show work on different sheet and attached. thank you
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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 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)
1
50
6
0.30
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
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:
Year Sold
(estimated returns, \$1,000,000s)
Company
1
2
3
1
\$14
\$18
\$23
2
9
11
15
3
18
23
27
4
16
21
25
5
12
16
22
6
21
23
28
To generate operating funds, the company must sell at least \$20 million worth of assets in year 1, \$25
million in year 2, and \$35 million in year 3. Globex wants to develop a plan for selling these companies
during the next 3 years to maximize return.
Formulate an integer programming model for this problem and solve it by using the computer.
Cab Company Scheduling
let Di = # of drivers who start their 8 hour shift in period I (I = 1,2,3,4,5,6)
period 1
12:00:00 AM–4:00am
period 4
12 noon — 4:00pm
period 2
4:00am — 8:00am
period 5
4:00pm — 8:00pm
period 3
8:00am — 12 noon
period 6
8:00pm — midnight
average fare/ driver
# of drivers in each period
period 1
80
period 2
500
period 3
420
period 4
300
period 5
270
>=
10
>=
12
>=
20
>=
25
>=
32
minimum # of drivers
DV
# of drivers/period
Objective function
D1
D2
D3
D4
D5
period 6
210
>=
18
D6
profit
weight
Denim Jeans
90
CD Player
150
Compact discs
30
2
Denim Jeans
3
CD Player
1
Compact discs
<= 5 DV Constraint Objective function Texas Consolidated Electronics Company Project 1 2 3 4 5 6 7 8 Expense (\$1,000s) Management Scientists required Estimated Profit \$50 105 56 45 90 80 78 60 6 8 9 3 7 5 8 5 \$0.30 0.85 0.2 0.15 0.5 0.45 0.55 0.4 <= 300 <= 40 (1,000,000s) Constraints DV Project 1 2 3 4 5 6 7 8 Objective Maximize Profits Please include the following constraints in Note: project 5 >= project 2
Note: All projects must be integer (1 or 0)
Project Selection
constraints
include the following constraints in your solutions
project 5 >= project 2
All projects must be integer (1 or 0)
Mortgage Associates
Let P = # of permanent operators and T = # of temporary operators
average pay/operator
daily # of accounts/per operator
#of computers available
average errors/ day
Decision variables
objective function
Permanent operator
120
220
1
0.4
Temporary operator
75
140
1
0.9
P
T
>=
<= <= 6300 32 15 Global Investment Capital Company 1 2 3 4 5 6 Year Sold (Estimated returns in \$ 1000000) 1 2 3 14 18 23 9 11 15 18 23 27 16 21 25 12 16 22 21 23 28 constraints 1 1 2 3 4 5 6 2 3 <= <= <= <= <= <= 1 1 1 1 1 1 Decision variables are C15:E20 this a 0-1 integer problem. Each decision variable has to be restricted to have the value 0 or 1 Objective function he value 0 or 1 ... Purchase answer to see full attachment

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