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ADDITIONAL QUESTIONS - 3
1. KURTTutkal produces three types of glue on two different production lines. Each line can be used by
up to seven workers at the same time. Workers are paid $500 per week on production line 1 and $900
per week on production line 2. The cost of production for one week is $1,000 to set up production
line 1 and $2,000 to set up production line 2. During one week on a production line, each worker
produces the number of units of glue shown in the table.
At least 120 units of glue 1, at least 150 units of glue 2 and at least 200 units of glue 3 must be
produced each week. Set up an integer programming formulation that minimizes the total cost of
meeting the weekly demands.
2. Governor Cherry of the state of Cheristan is trying to ensure that the state legislature governs Cherry's
congressional districts. The state consists of 10 cities and the number of registered Republicans and
Democrats (in thousands) in each city is shown in the table. Cherry has five representatives in
Congress. To create congressional districts, cities must be grouped according to the following
constraints:
• All voters in a city must be in the same district.
• Each district must have between 150,000 and 250,000 voters (no independent voters).
Governor Cherry is a Democrat. Assume that every voter always votes for a straight party. Construct
an integer programming formulation to help Governor Cherry maximize the number of Democrats
who will win seats in Congress.
3. Atil's finance company has six assets. The expected selling price (in million dollars) for each asset is
given in the table. If asset 1 is sold in year 2, the firm receives $20 million. To maintain a steady cash
flow, Milkem needs to sell at least $20 million worth of assets in year 1, at least $30 million worth in
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year 2, and at least $35 million worth in year 3. Over the next three years
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Formulate an integer programming model to maximize total revenue from assets sold.
4. ATIL has 10,000 square meters of space to lease in a shopping mall and wants to determine the types
of stores that should be in the mall. The minimum and maximum number of each type of store (along
with the square meters of each type) is given in the table below.
The annual profit for each type of store will of course depend on how many stores of that type there
are. This dependence is given in the table below (all profits are in units of $10,000). Thus, if there are
two department stores in the shopping center, each department store makes a profit of $210,000 per
year. Each store pays 5% of its annual profit as rent to Simon's. Solution Set up an integer
programming formulation that tells Simon's how to maximize the rental income from the mall.
5. A Fire Station currently has seven traditional ladder companies and seven alarm boxes. The two
nearest ladder companies to each alarm box are given in the table. The decision makers want to
maximize the number of traditional ladder companies that can be replaced by tower ladder
companies. Unfortunately, political considerations dictate that a traditional company can only be
replaced if at least one of the two companies closest to each alarm box is still a traditional company
after the replacement.
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Formulate an integer programming model that can be used to maximize the number of traditional
companies that tower companies can replace.
6. An ATIL Gasoline delivery truck consists of five compartments that can carry up to 2,700, 2,800,
1,100, 1,800 and 3,400 gallons of fuel respectively. The company is required to deliver three types of
fuel (super, regular and unleaded) to the customer. Claims, penalties per gallon and the maximum
allowable shortage (no-selling) are given in the table. Only one type of gasoline can be carried in
each compartment of the truck. Formulate an integer programming model for the solution that will
tell ATILBenzin how to load the truck in a way that minimizes shortage costs.
7. The ATILSES radio station is programming radio commercials in 60-second blocks. In this hour, the
station sold commercial time for 15, 16, 20, 25, 30, 35, 40 and 50 second commercials. Formulate an
integer programming model that can be used to determine the minimum number of 60-second
commercial blocks that must be scheduled to fit all commercials of the current hour.
8. You have been assigned to arrange the songs on the cassette version of Madonna's latest album. A
cassette tape has two sides (1 and 2). The total length of the songs on each side of the cassette should
be between 14 and 16 minutes. The length and type of each song are given in the table. The
assignment of the songs to the tape must meet the following conditions:
• There must be exactly two ballads on each side.
• Side 1 must have at least three hit songs.
• Either song 5 or song 6 should be on side 1.
• If songs 2 and 4 are on side 1, song 5 should be on side 2.
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Set up an integer programming formulation to determine if there is a song arrangement that satisfies
these constraints.
9. Algeria is recruiting navy recruits at three draft centers. Each of the recruits then has to be sent to one
of the three naval bases for training. The cost of transferring a conscript from one draft center to one
base is given in the table below.
Each year, 1,000 men are assigned to base 1; 600 to base 2; and 700 to base 3. Base 1 can train 1,000
men a year, base 2 800, and base 3 700. Once trained, the candidates are sent to Algeria's main naval
base (B). They can be transported on a small ship or a large ship. The cost of using a small ship is
$5,000 plus $2 per mile. A small ship can carry up to 200 men to the home base and can visit up to
two bases en route to the home base. Seven small and five large ships are available. The cost of using
a large ship is $10,000 plus $3 per mile. A large ship can visit up to three bases en route to home base
and can carry up to 500 men. The possible "tours" for each type of ship are given in the table below.
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Assume that the assignment of drafts to training bases is done using the transportation method. Then,
construct an integer programming formulation that minimizes the total cost incurred in sending men
from the training bases to the home base. (Hint: let yij be the number of men sent from base j (B) to
the home base in a small ship, yij be the number of men sent from base j to B in a large ship with tour
i, Si be the number of tour i used by a small ship, and Li be the number of tour i used by a large ship.)
10. ATILCopy Company sells photocopiers. One of the most important factors in making sales is
ATILCopy's fast service. ATILCopy sells copiers in six cities: Boston, New York, Philadelphia,
Washington, Providence and Atlantic City. Estimated annual sales of copiers are given in the table,
depending on whether a service representative is located within 150 miles of a city.
Each copier costs $500 to manufacture and sells for $1,000. The annual cost per service agent is USD
80,000. ATILCopy must determine in which market to deploy the service agent. Only Boston, New
York, Philadelphia and Washington are considered as the base of the service agent. The distance
between cities (in miles) is shown in the table. Set up an integer programming formulation that will
help ATILCopy maximize its annual profit.
11. Consider the following puzzle. You will choose 4 three-letter "words" from the list below:
DBA DEG ADI FFD GHI BCD FDF BAI
For each word, you score a point equal to the position in the alphabet where the third letter of the
word appears. For example, DBA gets 1 point, DEG gets 7 points, etc. Your goal is to select four
words that will maximize your total score, subject to the following constraint: The sum of the alphabet
positions of the first letter of each selected word must be at least as large as the sum of the alphabet
positions for the second letter of the word. To solve this problem, construct an integer programming
formulation of an IP.
12. Five jobs must be completed each day in a machine tool plant. The time required to do each job
depends on the machine used to do the job. If a machine is used, a setup time is required. The times
involved are given in the table. The company's objective is to ensure that the set-up and machine
required to complete all jobs
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is to minimize the sum of the runtimes. Construct an integer programming formulation whose
solution will do this.
13. ATILEV sells air conditioners. The annual demand for air conditioners in each region of the country
is as follows: East, 100,000; South, 150,000; Midwest, 110,000; West, 90,000. ATILEV plans to
produce air conditioners in four different cities: New York, Atlanta, Chicago and Los Angeles. The
cost of manufacturing an air conditioner in one city and shipping it to another part of the country is
given in the table below.
A factory can produce up to 150,000 air conditioners per year. The annual fixed cost of running a
factory in each city is given in the table below.
At least 50,000 units of Midwest air conditioning demand must come from New York, or at least
50,000 units of Midwest demand must come from Atlanta. Set up an integer programming
formulation to tell Solution A how to minimize the annual cost of meeting the demand for air
conditioning.
14. The KURTHOMES Project includes both houses and apartments. The development will include up to
10,000 dwellings. The project should include a recreation project: either a swimming-tennis complex
or a sailing marina, but not both. In the case of a marina, the number of houses in the project must be
at least three times the number of apartments in the project. A marina would cost $1.2 million and a
swimming-tennis complex would cost $2.8 million. Developers are proposing that each condo would
be built with an income of $48,000, and each house
With an income of $46,000, he believes it will generate income. Each house (or apartment) costs
$40,000 to build. Set up an integer programming formulation that will help KURTHOMES maximize
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profits.
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15. KURT Publishing sells textbooks to university students. KURT has two sales representatives who can
be assigned to state area A-G. The number of university students (in thousands) in each state is given
in the figure.
Each sales agent must be assigned to two neighboring states. For example, a sales representative can
be assigned to A and B but not to A and D. KURT's goal is to maximize the total number of students
in the states assigned to sales reps. Set up an integer programming formulation whose solution will
tell you where to assign sales representatives.
16. ATILCPU has four production facilities where personal computers are manufactured. ATILCPU can
sell up to 20,000 computers per year at a price of $3,500 per computer. The production capacity for
each plant, the production cost per computer and the fixed cost of running a plant for a year are given
in the table.
Determine how ATILCPU can maximize its annual profit from computer production.
17. One product can be produced on four different machines. A fixed set-up cost, variable production
cost per unit processed and production capacity of each machine are given in the table.
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A total of 2,000 products need to be produced. Set up an integer programming formulation whose
solution will show us how to minimize total costs.
18. The administrator of the CED computer wants to be able to access five different files. These files are
distributed on 10 disks as shown in the table. The amount of storage required by each disk is: disk 1,
3K; disk 2, 5K; disk 3, 1K; disk 4, 2K; disk 5, 1K; disk 6, 4K; disk 7, 3K; disk 8, 1K; disk 9, 2K; disk
10, 2K.
(a) Formulate an IP that determines the set of disks that require the minimum amount of storage for
each file to reside on at least one of the disks. For a given disk, we must either store the entire
disk or store none of the disk; we cannot store part of the disk.
(b) Change your formulation so that if disk 3 or disk 5 is used, disk 2 must also be used
19. A company is considering opening warehouses in four cities: New York, Los Angeles, Chicago and
Atlanta. Each warehouse can ship 100 units per week. The fixed weekly cost of keeping each
warehouse open is $400 for New York, $500 for Los Angeles, $300 for Chicago and $150 for
Atlanta. Region 1 of the country needs 80 units per week, Region 2 needs 70 units per week and
Region 3 needs 40 units per week. The costs of shipping one unit from one plant to one region
(including production and transportation costs) are shown in the table.
Subject to the above information and the restrictions below, we want to meet weekly requests at
minimum cost:
• If the New York warehouse opens, the Los Angeles warehouse must open.
• A maximum of two warehouses can be opened.
• Either the Atlanta or Los Angeles warehouse needs to be opened.
Construct an integer programming formulation that can be used to minimize the weekly costs of
meeting demand.
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