Constrained optimization for location allocation — new supply center selection for maximal coverage of demand points

Question

I am trying to figure out the objective function and associated constraints for optimization of the following problem:

1. There is a set of Demand points $I$, which have to be covered by a set of Supply Points $J$

2. Not all Demand Points need be covered

3. Not all Supply Points need Supply

4. Constraint- If a Supply Point $j$ is selected for supplying, at least four demand points $i$ must be covered by it, or in other words, any supply point supplies to any demand center at all, it must do so for at least four demand centers

5. Constraint- There is a travel time constraint, that all demand centers which are covered, must be covered by a supply point which is not more than 60 minutes in travel time. The information of travel times for the set of $I \times J$ is available.

6. Objective- The objective is to maximize coverage of demand points $I$. Any no. of supply points may supply, as long as the constraints are met.

We are looking to formulate the optimization equations for this problem. We are thinking of assigning decision variables $A_i$ and $B_j$ which take value of $0$ or $1$ for both demand centers and supply points, $1$ if demand center is covered (and 1 if supply center supplies), and 0 if demand center is not covered (and $0$ if supply center does not supply)

Answer 1

In addition to the $A_i$ and $B_j$ binary variables, introduce binary variables $X_{i,j}$ to indicate whether demand $i$ is served by supply $j$, where travel time between $i$ and $j$ is at most $60$ minutes. The problem is to maximize $\sum_i A_i$ subject to linear constraints \begin{align} 4B_j &\le \sum_i X_{i,j} &&\text{for all $j$}\\ X_{i,j} &\le B_j &&\text{for all $i$ and $j$}\\ \sum_j X_{i,j} &= A_i &&\text{for all $i$} \end{align}