Discuss the below:
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