How to express this logical constraint for an ILP?

Question

I am trying to write an ILP for a problem but I have this logical constraint and I'm stuck.

In my model I have:

two binary variables: $x$ and $y$

One Integer variable: $z$

The logical constraint I am trying to write is:

if $x = 1$ and $y = 1$ then $z \le Mx$, else $z = 0$

Thank you for your help.

Answer 1

You always want $$z \ge 0 \tag1$$ You want $x = 0 \implies z \le 0$: $$z \le M x \tag2$$ Similarly, you want $y = 0 \implies z \le 0$: $$z \le M y \tag3$$