Problem Set
(Integer Programming: Formulations)
1. The board of directors of General Wheels Co. is considering six large capital investments. Each
investment can be made only once. These investments differ in the estimated long-run profit (NPV)
that they will generate as well as in the amount of capital required, as shown by the following table
(in units of millions of dollars):
Investment Opportunity
Estimated
1
2
3
4
5
6
15
12
16
18
9
11
38
33
39
45
23
27
Profit
Capital
Required
The total amount of capital available for these investments is $100 million. Investment opportunities
1 and 2 are mutually exclusive, and so are 3 and 4. Furthermore, neither 3 nor 4 can be undertaken
unless one of the first two opportunities is undertaken. There are no such restrictions on investment
opportunities 5 and 6. The objective is to select the combination of capital investments that will
maximize the total estimated long-run profit (NPV).
Formulate a BIP model for this problem.
2. DAC is considering manufacturing of three types of cars: compact, sedan, and SUV. The resources
required for, and the profits yielded by, each type of car are shown in the following table. Currently,
6000 tons of steel and 60000 hours of labor are available. For production of a type of car to be
economically feasible, at least 1000 cars of that type must be produced. Formulate an IP to
maximize DAC’s profit.
Car Type
Resources
Compact
Sedan
SUV
Steel required (in tons)
1.5
3
5
Labor required (in hours)
30
25
40
Profit yielded ($)
2000
3000
4000
3. There are six cities (Cities 1-6) in Joka County. The county must determine where to build fire
stations. The county wants to build the minimum number of fire stations needed to ensure that at
least one fire station is within 15 minutes (driving time) of each city. The times (in minutes) required
to drive between the cities in Joka County are shown in the table below. Formulate an IP that will
tell Joka County how many fire stations should be built and where they should be located.
To
From
City 1
City 2
City 3
City 4
City 5
City 6
City 1
0
10
20
30
30
20
City 2
10
0
25
35
20
10
City 3
20
25
0
15
30
20
City 4
30
35
15
0
15
25
City 5
30
20
30
15
0
14
City 6
20
10
20
25
14
0
4. EGas produces two types of gasoline (Gas 1 and Gas 2) from two types of oil (Oil 1 and Oil 2). Each
gallon of Gas 1 must contain at least 50% Oil 1 and each gallon of Gas 2 must contain at least 60%
Oil 1. Each gallon of Gas 1 can be sold for $12 and each gallon of Gas 2 can be sold for $14.
Currently, 500 gallons of Oil 1 and 1000 gallons of Oil 2 are available. As many as 1500 more gallons
of Oil 1 can be purchased at the following prices: first 5000 gallons at $25 per gallon; next 500
gallons at $20 per gallon; next 500 gallons at $15 per gallon. Formulate an IP that will maximize
EGas’s profit.
5. TSire has a $20000 advertising budget. Tsire can purchase full-page ads in two magazines: Business
Tomorrow (BT) and Family Square (FS). An exposure occurs when a person reads a TSire ad for the
first time. The number of exposures generated by each ad in BT is as follows: ads 1-6 give 10000
exposures each; ads 7-10 give 3000 exposures each; ads 11-15 give 2500 exposures each; while ads
16 onwards give zero exposures each. For example, 8 ads in BT would generate 6*10000 + 2*3000 =
66000 exposures. The number of exposures generated by each ad in FS is as follows: ads 1-4, 8000
exposures each; ads 5-12, 6000 exposures each; ads 13-15, 2000 exposures; ads 16+, 0 exposures
each. Thus, 13 ads in FS would generate 4*8000 + 8*6000 + 1*2000 = 82000 exposures. Each full
page ad in either magazine costs $1000. Assume there is no overlap in the readership of the two
magazines. Formulate an IP to maximize the number of exposures that TSire can obtain with the
limited advertising budget.