4. Consider a budget allocation problem where $30 million is available for a num

Question

4. 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. Project Number Cost $ million Expected Utilit 18 16 12 25 14 Project After-school program Road securit 18 10 3 4 Cri me reduction Road extension Child care facilit (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 progra mming. (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

(a) Integer program is following

Decision variables: Xi be a binary variable such that Xi = 1, if project i is undertaken.

Max 18X1+16X2+12X3+25X4+14X5

s.t.

6X1+18X2+10X3+9X4+4X5 <= 30

(b) The additional constraints for the given conditions are following:

i) X1+X2+X3+X4 >= 2

ii) X1 - X3 = 0

iii) X3 - X1 >= 0

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