Additional problems Integer Programming (2)



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Additional problems - Integer Programming (2)

1. The manager of State University's DED computer wants to be able to access five different files. These file are scattered on ten disks as shown in the table below. The amount of storage required by each disk is as follows: 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 a set of disks requiring the minimum amount of storage such that each file is 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 a disk.


  1. Modify your formulation so that if disk 3 or disk 5 is used, then disk 2 must also be used.





1

2

3

4

5

6

7

8

9

10


File 1

File 2


File 3

File 4


File 5

X

X

X



X
X
X

X
X



X

X


X
X

X

X



X
X


X

X


X

X


X
X

2. The Lotus Point Condo Project will contain both homes and apartments. The site can accommodate up to 10,000 dwelling units. The project must contain a recreation project either swimming - tennis complex or sailboat marina, but not both. If a marina is built, the number of homes in the project must be at least triple the number of apartments in the project. A marina will cost $1.2 million, and a swimming - tennis complex will cost $2.8 million. The developers believes that each apartment will yield revenues with an NPV $48,000, and each home will yield revenues with an NPV $46,000. Each home (or apartment) costs $40,000 to build. Formulate an IP to help Lotus Point maximize profits.



  1. A product can be produced on four different machines. Each machine has a fixed setup-cost, variable production costs per-unit processed, and a production capacity given in the table below. A total of 2000 units of the product must be produced. Formulate an IP whose solution will tell us how to minimize total costs.


Machine


Fixed

Cost


Var. Cost per unit

Capacity


1

2

3



4

$1000

920


800

700


$20

24

16



28

900

1000


1200

1600



  1. A 34

    B 29

    E 56

    D 21

    C 42

    G 71

    F 18


  1. WSP Publishing sells textbooks to college students. WSP has two sales reps available to assign to the A-G state area. The number of college students (in thousands) in each state is given in the figure below. Each sales rep must be assigned to two adjacent states. For example, a sales rep could be assigned to A and B but not to A and D. WSP’s goal is to maximize the number of total students in the states assigned to the sales reps. Formulate an IP model whose solution will tell you where to assign sales reps.

5. 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 weekly fixed 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 requires 80 units per week, region 2 requires 70 units per week, and region 3 requires 40 units per week. The costs (including production and shipping costs) of sending one unit from a plant to a region are shown in the table below. We want to meet weekly demands at minimum cost, subject to the preceding information and the following restrictions:
1 If the New York warehouse is opened, then the Los Angeles warehouse must be opened.
2 At most two warehouses can be opened.
3 Either the Atlanta or the Los Angeles warehouse must be opened.
A. Formulate an IP that can be used to minimize the weekly costs of meeting demand.
B. Now add the following constraint: If the Atlanta warehouse is opened the Chicago warehouse must not be opened.

From


To

Region 1


Region 2

Region 3

NY

20

40

50

LA

48

15

26

Chicago

26


35


18


Atlanta

24

50

35

6. A product can be produced on four different machines. Each machine has a fixed setup cost (incurred only if the machine is used), a variable production cost per unit processed, and a production capacity (see the table below). A total of 2000 units of the product must be produced.



  1. Formulate an IP whose solution will tell us how to minimize total costs.

  2. Assume there are two products that must be produced on the same four machines. A machine can be dedicated to one product only. If 1000 units of the second product must be produced, reformulate your IP model.




Machine

Fixed Cost

Variable cost/unit

Capacity

1

2

3



4

$1000

920


800

700


$20

24

16



28

900

1000


1200

1600

7. “Fruit computer” produces two types of computers: Pear computers, and Apricot computers. A total of 1200 hours of labor and 3000 chips are available. A Pear computer requires 1 hour and 2 chips, and an Apricot computer requires 2 hours and 5 chips. A ‘Pear’ sells for $400, and an ‘Apricot’ sells for $900. To produce the ‘Pears’ “Fruit” needs to buy equipment at a cost of $5000, and similarly, for the production of the ‘Apricots’ the equipment costs $7000.


  1. Formulate the IP model that maximizes “Fruit” profit.

  2. For the production of the ‘Pear’ to be economically viable, at least 100 computers must be produced. Add this constraint to your model.

8. You have been assigned to arrange the songs on the cassette version of Rocky Rock latest album. A cassette tape has two sides (1 and 2). The songs on each side of the cassette must total between 14 and 16 minutes in length. The length and type of each song are given in the table below. The assignment of songs to the tape must satisfy the following conditions:


a. Each side must have exactly two ballads



    1. Side 1 must have at least 3 hit songs

    2. Either song 5 or song 6 must be on side 1

    3. If song 2 and 4 are on side 1, then song 5 must be on side 2.




  1. Write the constraints that describe the above restrictions and requirements.

  2. Only 3 of the 4 conditions mentioned above really need to met. Make the necessary changes in the model.


B = Ballad; H = Hit

Song: 1 2 3 4 5 6 7 8

TYPE B H B H B H B&H

Length 4 5 3 2 4 5 3 4


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