Formulating the conditional constraint

Question

I want to develop a model extension of capacitated location problem.

The variables are a binary $x_i$ and a continuous $Q_i$. The following condition must be satisfied:

1. if $x_i = 0$, $Q_i$ must be zero.
2. if $x_i = 1$, $Q_i \geq 0$.

How do we formulate it in the integer programming formulation?

Thank you for your answer.

Answer 1

Let $M_i$ be an upper bound on $Q_i$, and impose linear big-M constraints $0 \le Q_i \le M_i x_i$.

Answer 2

I tend to prefer logical constraints to big M unless big M are really needed.

In OPL CPLEX we can write

int n=10;
range r=1..n;

dvar boolean x[r];
dvar float+ Q[r];

subject to
{
  forall(i in r) (x[i]==0) => (Q[i]==0);
}