how to tackle an integer variable with multipe domains in an integer program

Question

I'm trying to model an integer program whose integer variables may have multipe domains? For example, an integer variable x with range [10,30], [62,70]. How to build this constraint? Any ideas are welcomed and appreciated.

Answer 1

One possibility is to use a set of binary variables to enumerate the options. So you could write $$x=10z_1 + 11z_2 + \dots + 30z_{21} + 62z_{22} + \dots + 70z_{30}$$ together with $$z_1 + \dots + z_{30} = 1,$$ the $z_i$ being binary.

Another option is to use two integer variables and one binary variable. So $$x=x_1 + x_2,$$ $$10z \le x_1 \le 30z$$ and $$62(1-z) \le x_2 \le 70(1-z),$$ with $x_1$ and $x_2$ integer and $z$ binary

Answer 2

You can try with this constraint
$10z + 62(1-z) \le x \le 30z + 70(1-z)$ where $z$ is a binary.
You may need other constraints that will determine which domain to use, there you need to use $z$.