# Integer Problem

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1. A manager of a firm's research and development is currently reviewing a set of proposals

prepared by her staff. She must soon decide which proposals to fund. The table below

displays each proposal's net present value (NPV) and, for each of the next three years,

each proposal's requirements for cash and labor:

NPV Cash Requirements Labor Requirements

Proposal ($000,000) ($000) (man-years)

Year 1 Year 2 Year 3 Year 1 Year 2 Year 3

1 5.0 200 300 300 1 2 2

2 2.8 200 200 0 2 2 0

3 4.8 300 300 0 2 2 0

4 6.0 0 300 300 0 1 2

5 5.4 100 500 400 1 3 3

Maximum Availability 700 800 900 4 6 7

(a) Formulate an Integer Linear Program (ILP) to maximize the total NPV of the

approved proposals.

(b) What modifications to the ILP are necessary under each of the following assumptions.

Here, as you consider each assumption, you should ignore the others.

i. At most two of proposals 1, 3, and 5 can be approved.

ii. Exactly one of proposals 1, 2, and 4 must be approved.

iii. Proposal 2 cannot be approved unless proposal 4 is also approved.

iv. Proposal 3 cannot be approved unless both proposals 1 and 5 are also approved.

v. If proposals 2 and 3 are both approved, then proposal 4 must also be approved.

vi. If proposals 1 and 2 and approved, proposal 3 cannot be approved.

2. A certain Job shop has four operators and four machines. Based on the background and

training of the operators and on the nature of the machines the Industrial Engineering

department estimated the cost of assigning each operator to each of the machines as

given below. Determine a minimum cost assignment of the operators to the machines.

Machine

operator A B C D

1 $20 $30 $32 $36

2 28 26 32 20

3 22 18 16 36

4 26 26 22 18

B. Madhu Rao 4

3. Super Sunday, the once-a-year TV extravaganza focused on the activities of the NFL

Super Bowl, has almost become a tradition. CBS has received the TV contract for this

year's festivities. Producers have identified 12 potential camera locations within the

stadium. They have also identified 25 stadium areas that may require camera coverage

during the pregame, game and postgame activities. The camera locations and the

stadium areas the camera can over are given below:

Camera Location Stadium Areas

1 1,3,4,6,7

2 8,4,7,12

3 2,5,9,11,13

4 1,2,18,19,21

5 3,6,10,12,14

6 8,14,15,16,17

7 18,21,24.25

8 2,10,16,23

9 1,6,11

10 20,22,24,25

11 2,4,6,8

12 1,6,12,17

CBS executives are concerned about costs for the production. Consequently, they

set an objective of minimizing the number of camera locations used. In seeking this

objective they want at least one camera to be available to cover each stadium area.

Camera location 9 is the "blimp", and executives have decided that the blimp will be

used because of viewer expectation and fascination with the shots from this location.

Stadium areas 1 and 2 are locker room locations. The viewer interest in football

personalities has led the executives to request that at least two camera locations be

available to cover each of these areas. Formulate a 0-1 programming model to determine

the minimum number of camera's needed for coverage

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###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
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