Consider a budget allocation problem where $30 million is available for a number

Q: Consider a budget allocation problem where $30 million is available for a number

a) Project 1 Project 2 Project 3 Project 4 Project 5 Decision Var. X1 X2 X3 X4 X5 Changing Cells 1 0 1 1 1 Budget ($ Millions) 6 18 10 9 4 Expected Utility 18 16 12 25 14 Constraints Cost 29

ID: 457042 • Letter: C

Question

Consider a budget allocation problem where $30 million is available for a number of projects listed below. The investment required in each project along with the expected return in terms of utility is also given. (a) The problem is to find which projects should be financed in order to maximize the total expected utility not exceeding the budget limitation. Formulate the problem in integer programming. (b) Now suppose there are additional conditions in selecting the projects. Write the constraints for these conditions. Any two of the first four projects must be undertaken. Projects 1 and 3 must be taken simultaneously or not taken at all. Project-1 will be undertaken only if project-3 is undertaken but project-3 is not conditional on project-1 (that means, you can have project-3 without project-1, but you cannot have project-1 unless project-3 is undertaken).

Explanation / Answer

Project 1

Project 2

Project 3

Project 4

Project 5

Decision Var.

Changing Cells

Budget ($ Millions)

Expected Utility

Constraints

Cost

Constraint 2 (Binary)

Bin

Constraint 3 (Binary)

Bin

Constraint 4 (Binary)

Bin

Constraint 5 (Binary)

Bin

Constraint 6 (Binary)

Bin

Obj. Function

1) Any two of the first four projects must be undertaken

Constraint X1 + X2 + X3 + X4 = 2

2) Project 1 and 3 must be taken simultaneously or not taken at all

Constraint X1 + X3 = 2 &

X1 + X3 = 0

3) Project 1 will be taken only if Project 3 is undertaken

Constraint X3 = 1 &

X1 + X3 = 2

4) Project 3 is not conditional on Project 1

Constraint X3 =1 &

X1 + X3 =1 (or)

X1 + X3 =2