Question

A firm has prepared the following binary integer program to evaluate a number of potential locations for new warehouses. The firm's goal is to maximize the net present value of their decision while not spending more than their currently available capital.

Max 20x1 + 30x2 + 10x3 + 15x4

s.t. 5x1 + 7x2 + 12x3 + 11x4 ≤ 21 {Constraint 1}

x1 + x2 + x3 + x4 ≥ 2 {Constraint 2}

x1 + x2 ≤ 1 {Constraint 3}

x1 + x3 ≥ 1 {Constraint 4}

x2 = x4 {Constraint 5}

Which of the constraints enforces a mutually exclusive relationship?

Answer 1

X1 + X2 <=1, constraint 3 enforces mutually exclusive relationship. Since all the variables are binary they can either be 0 or 1. So if we make a data table pf all possible outputs for X1 and X2 combination:

X1	X2	X1 + X2
0	0	0
0	1	1
1	0	1
1	1	2

As you can see above, only when both of them are 1, the mentioned constraint 3 will be violated, thus both the values cannot be 1 at the same time and hence this constraint makes these two variables mutually exclusive.