I am working with different facility location models giving its single (only one center help a demand zone) and multi-source models (multiple centers can help a demand zone).
My decision variables are $$ x_j = \begin{cases} 1 & \text{if a center is located in } j \in J \\ 0 & \text{otherwise} \end{cases} $$ and $$ y_{ij} = \begin{cases} 1 & \text{if demand zone } i \in I \text{ is assisted by center } j \in J \\ 0 & \text{otherwise} \end{cases} $$ for the single-source model.
For the multi-source one we have $y_{ij} \geq 0$ taking integer values and representing the number of people sent to demand zone $i\in I$ from center $j\in J$.
The single-source model is given by \begin{align} \min_{x,y} \quad & \max_{i,j} \{ d_{ij}y_{ij} \} \\ \text{s.a.: } \quad & \sum_j x_j = N \\ & \sum_j y_{ij} = 1, \: \forall i \\ & y_{ij} \leq x_j , \: \forall i, j \\ & x_j \in \{0,1\}, \: \forall i, j \\ & y_{ij} \in \{0,1\}, \: \forall i, j \end{align} So far I have been able to change all the restrictions for the multi-source model. \begin{align} \sum_j x_j = N \\ \sum_j y_{ij} = w_i, \quad \forall i \\ y_{ij} \leq q_jx_j , \quad \forall i, j \\ x_j \in \{0,1\}, \quad \forall i, j \\ y_{ij} \geq 0, \quad \forall i, j \end{align} where $w_i$ is the demand associated to each zone and $q_j$ is the capacity of each center.
However, I am stuck on modifying the objective function so that the distance is not counted several times when sending more than one person.
Could anyone help me? Thank you!
First note that your capacity constraint $y_{ij}\le q_j x_j$ should instead be $\sum_i y_{ij}\le q_j x_j$ for all $j$.
Now to minimize the maximum distance, introduce constant $M_{ij}=\min(w_i,q_j)$ and binary variable $z_{ij}$, impose constraint $y_{ij}\le M_{ij} z_{ij}$, and minimize $\max_{i,j} \{d_{ij}z_{ij}\}$.