# How to model a logical indicator when two inequalities hold in Integer Programming?

I have an IP program where $$\forall i \in I, j \in J$$ my decision variables are $$x_{i,j}$$. I have two sets of inequalities (one inequality for every $$i,j$$ pair) that are of interest which are $$a_{i,j} x_{i,j} \geq 1 \ \ \ \ \forall i \in I, j\in J$$ and $$\sum_{j \in J} b_{i,j}x_{i,j} \leq b_{i,j}- 1 \ \ \ \ \forall i \in I, j\in J$$

I now want to introduce a logical indicator variable $$\delta_{i,j}$$ which equals 1 if and only if both of the inequalities are true and 0 else. How could I implement this idea by introducing appropriate constraints for my Integer Progamming model? I was thinking about defining a variable $$\delta^1_{i,j}$$ and $$\delta^2_{i,j}$$ which equals 1 iff the first and second inequality are true respectively and then maybe combine these two auxiliry logical variables into a single variable $$\delta_{i,j}$$, but I couldn't figure out exactly how to make this work.

Your general approach is a good one. Use Boolean variable that captures whether an inequality holds to define $$\delta_{i,j}^1$$ and $$\delta_{i,j}^2$$. Then, use Express boolean logic operations in zero-one integer linear programming (ILP) to define $$\delta_{i,j}$$ in terms of $$\delta_{i,j}^1$$ and $$\delta_{i,j}^2$$ (via a logical-AND operation).