Firm ABC has received 12 project proposals number from 1 to 12. But it cannot do

Question

Firm ABC has received 12 project proposals number from 1 to 12. But it cannot do all the projects. An OR professional is hired to develop an ILP to help the firm to choice which projects to be conducted. Let binary variable X_j represents project proposal J, where x_1 = 1 means project 1 is selected; and x_j = 0 means project is not selected. Derive (linear) constraints that guarantee the following restrictions are met. If project 3 is selected, then both projects and 7 must also be selected. If project 4 is selected, then none of projects 5, 8 and 10 can be selected. You must choose at least 3 of the 6 projects 1, 2, 6, 8, 11, and 12,

Explanation / Answer

a) The variable defining project 3 will be x3, project 6 will be x6 and project 7 will be x7

x3 + x6 <=1

x3 + x7 <= 1

If x3 is 1, then x6 and x7 can only choice as 0 in order to satisfy the constraint, if x3 is zero, then x6 and x7 can be either 0 or 1

b) The variable defining project 4 will be x4, project 5 will be x5, project 8 will be x8 and project 10 will be x10

x4 + x10 <=1

x4 + x5 <= 1

x4 + x8 <= 1

c)

3 <= (x1 + x2 + x6 + x8 + x11 + x12) <=5

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