The following questions refer to a capital budgeting problem with six projects r

Question

The following questions refer to a capital budgeting problem with six projects represented by binary variables a, b, c, d, e, and f. a. Write a constraint modeling a situation in which two of the projects 1, 3, 5, and 6 must be undertaken. b. Write a constraint modeling a situation in which, if project 3 or 5 is undertaken, they must both undertaken. c. Write a constraint modeling a situation in which project 1 or 4 must be undertaken, but not both. d. Write constraints modeling a situation where project 4 cannot be undertaken unless projects 1 and 3 are also undertaken. e. Revise the requirement in part (d) to accommodate the case in which, when projects 1 and 3 are undertaken, project 4 must also be undertaken.

Explanation / Answer

a. as 2 of the 4 projects will be undertaken, 2 variables will be 1 and 2 will be 0. Hence, the constraint will be: a+c+e+f = 2

b. Now, if project 3 and 5 are both taken they will be 1 and if both are excluded they will be 0. It cannot be that c = 1 and e = 0 or c = 0 and e = 1. So, constraint will be = c-e = 0.

c. as only one of the projects 1,4 will be undertaken, either of a or d will be 1. constraint will be a+d = 1

d. project 4 will be undertaken only if 1 and 3 are also undertaken.

d cannot be more than a and c. In other words, d = 1 only if a = 1 and c = 1. Thus constraints are: d<=a, d<=c

e.The first 2 constraints will be the same as is in "d" above.   d<=a, d<=c. Also if a =1 and c= 1, then d will also be 1 (it cannot be 0 then).

so the 3rd constraint will be: d>=a+c-1

Project Binary variable 1 a 2 b   3 c 4 d 5 e 6 f

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