## 1 Introduction
### Developing the VAOMP Model Considering the Capacity Criterion (VAOCOMP)  
Capacity will be considered as one of the most important factors in each facility. If capacity is ignored in these facilities, the number of allocations for each facility would be unknown, and sometimes there would be a facility with a large number of allocations or sometimes one without any allocation in the problem. In the real world, every facility has a specific capacity for servicing its allocations, so the VAOMP model, which does not consider capacity, has been developed with a capacity criterion to make the problem more realistic. In this newly developed model, which we call the Vector Assignment Ordered Capacitated Median Problem (VAOCOMP), all the conditions of the VAOMP model are established, such as the assignment of one demand to multiple facilities, levels of order, etc. Meanwhile, each facility has a particular capacity, which ensures that each facility will not operate over its capacity. The mathematical function of the model developed will be:  
\[\textit{Minimize}\ Z=\sum\
olimits_{k=1}^{n}\lambda_{k}w_{k} \tag{1}\]  
\[\sum\limits_{k=1}^{n}w_{k}=\sum\limits_{i\ \in\ I}\sum\limits_{j\ \epsilon}\sum \
olimits_{l=1}^{L}a_{i}\theta_{il}a_{ij}x_{ij}^{l}c_{ij}^{l}\ \textit{for each}\ i\ \epsilon\ I \tag{2}\]  
The variables and parameters of this model are:  
* I is the set of demands
* I is the set of facilities
* \(a_{i}\) the population or the number of demands in each building block or at location i
* k = 1, 2,... n is an index for the relative rank of service time, one assigned to each demand. \(\lambda_{k}\) the weight on the kth rank of service
* \(d_{ij}\) the distance or time (or generalized cost) between i and j. L the maximum number of levels of closeness being considered in the model for any demand (in this research is equal to 1)
* \(\theta_{il}\) the fraction of the time demand at i is served by its lth closest facility
* \(C_{j}\) capacity of each facility or fire station that is equal to 50,000 people
* \(c_{ij}^{l}= \begin{cases}1&\text{if demandli assigns to facility j as the lth closest open facility}\\ &0\ \textit{otherwise}\end{cases} \) The constraints of this model are: \[w_{k}\geq\sum\limits_{j\ \epsilon}\sum\limits_{l=1}^{L}a_{i}\theta_{il}a_{ij}x_{ ij}^{l}-M\big{(}1-u_{i}^{k}\ \big{)}\ \textit{for each}\ i\ \epsilon\ I,k=1,2,\ldots n\] (3)  
\[\sum\limits_{k=1}^{n}u_{i}^{k}=1\ \textit{for each}\ i\ \epsilon\ I \tag{4}\]  
\[\sum\limits_{i\ \in\ I}u_{i}^{k}=1\textit{for each}\ k=1,\ 2,\ \ldots,\ n \tag{5}\]  
\[w_{k}\leq w_{k+1}\ \textit{for each}\ k=1,\ 2,\ \ldots,\ n-1 \tag{6}\]  
\[\sum\limits_{j\ \in\ I}x_{ij}^{l}\leq\ y_{j}\ \textit{for each}\ i\ \epsilon\ I,\ l=1,2,\ldots L \tag{7}\]\[\sum_{L=1}^{L}x_{ij}^{l}\leq\ y_{j}\ for\ each\ i\ e\ I,\ j\ e\ J \tag{8}\]  
\[\sum_{q\ e\ C_{ij}}\ x_{iq}^{l}+\ \sum_{s=1}^{1}x_{ij}^{s}\ \geq\ y_{j}\ for\ each\ i\ e\ I,\ j\ e\ J,\ l=1,\ 2,\ \ldots,\ L \tag{9}\]  
\[P_{2}\ \leq\sum_{j\ e\ J}y_{j}\ \leq P_{1} \tag{10}\]  
\[0\ \leq\ x_{ij}^{l}\ \leq 1\ y_{j}\ and\ u_{i}^{k}\ [0,1] \tag{11}\]  
\[\sum_{j=1}\sum_{k=1}^{n}c_{ij}^{l}=L\ for\ each\ i\ e\ I \tag{12}\]  
\[\sum\limits_{k=1}^{n}u_{i}^{k}=1\ \textit{for each}\ i\ \epsilon\ I \tag{4}\]0  
Before presenting the VAOCMP model, it is necessary to obtain the cost matrix between all the points of demand and the selected facilities by Origin-Destination Cost Matrix (OD Cost Matrix) analysis in GIS. The OD Cost Matrix is a matrix that specifies the cost between each facility and each demand point. In this research, the cost is defined as the arrival time for fire engines to the demand point. Then, all the arrays in the OD Cost Matrix are ranked according to the lowest cost (or time), so it can be determined which facility is closest to which demand, and which demand will be the closest demand to each facility. Based on the number of allocation levels, or 1 and then by considering the capacity at each level of the order, it will be determined which facility will be served by which demand.