Question

The following is a Linear Programming problem:

Suppose, I = {set of locations for establishing a hospital} = {1, 2, 3, 4, 5, 6, 7}

x_i is a decision variable which equals 1 if a hospital is set up at location i; otherwise, x_i = 0.

The constraint given is that location 3 can not be selected unless both locations 6 and 7 are selected.

According to the course tutor, the proper formulation of this constraint is - 2x₃ < = x₆ + x₇

Is it also correct if I formulate the constraint as x₃ < = x₆ + x_{7 ?}  If both x₆ and x₇ are 1, then according to the constraint, x₃ = 1 and 1<=2 holds true. If x₆ = x₇ = 1 but x₃ = 0 (since we're not obligated to choose location 3), then 0 < = 2 still holds true. This is why I am wondering if both the formulations are correct.

Answer 1

The constraint given is that location 3 can not be selected unless both locations 6 and 7 are selected. These kind of constraints are dependent decisions
No the formulation x3 <= x6+x7 is not correct as for below case :

if only x6 is selected and x7 is not then it gives option for x3 to be selected as 1<=1+0 so, 1 <= 1. So it will create a wrong selection for project 3

2x3 <= x6 + x7 is correct as it will not set x3 = 1 unless x6=x7=1 so that location 3 can not be selected unless both locations 6 and 7 are selected