How do I formulate constraints that check if a parameter is between certain values, using binary variables?

Question

I have $3$ parameters $a_1,a_2,a_3$ and a variable $d$ and $3$ binary variables $b_1,b_2,b_3$ and a "result" variable $s$. How do I model constraints so that:

• If $d$ is between $0$ and $a_1$, then $s=1$
• If $d$ is between $a_1$ and $a_2$, then $s =$ ... complex calculation ...
• If $d$ is between $a_2$ and $a_3$, then $s = 0$

I have found a formulation that is as follows: \begin{align} a_1b_1 + a_2b_2 - d \le 0 \\ -a_1b_1 - a_2b_2 - a_3b_3 + d \le 0 \\ b_1 + b_2 + b_3 = 1 \end{align}

But if use this formulation in my model, the b-variables get calculated incorrectly. Does anyone see the reason why and/or has a better idea?

Answer 1

It looks like your first constraint should instead be $$0b_1 + a_1 b_2 + a_2 b_3 - d \le 0$$ With this change, the logical implications are \begin{align} b_1 = 1 &\implies 0 \le d \le a_1 \\ b_2 = 1 &\implies a_1 \le d \le a_2 \\ b_3 = 1 &\implies a_2 \le d \le a_3 \end{align}

To avoid ambiguous borders, introduce a small tolerance $\epsilon>0$ and impose instead $$0 b_1 + (a_1+\epsilon) b_2 + (a_2+\epsilon) b_3 - d \le 0$$