For each formulation problem, please define your decision variables clearly. You need to submit your code for solutions.

1. Merrill Lynch is considering investments into 6 projects: A, B, C, D, E and F. Each project has an initial cost, an expected profit rate (one year from now, with the failure rate being taken into consideration) expressed as a percentage of the initial cost, and an associated risk of failure. These numbers are given in the table below:

A

B

C

D

E

F

Initial cost (in M)

1.3

0.8

0.6

1.8

1.2

2.4

Profit rate

10%

20%

20%

10%

10%

10%

Failure risk

6%

4%

6%

5%

5%

4%

1. Provide a formulation to choose the projects that maximize total expected profit, such that Merrill Lynch does not invest more than 4M dollars and its average failure risk is not over 5%. For example, if Merrill Lynch invests only into A and B, it invests only 2.1M dollars and its average failure risk is (6%+4%)/2=5%.
2. Suppose that if A is chosen, B must be chosen. Modify your formulation.

3. Suppose that if C and D are chosen, E must be chosen. Modify your formulation.