## 2 Background
### Facility Location Problem
#### 2.1.1 Mclp  
According to [PERSON] [26], the MCLP functions as a network of nodes and arcs that represent the event's geographic space, in which the nodes are the concentrations of demand for the facilities' services and the arcs reflect the distance between the nodes. Given that the model is applied to a plane, that is, a territorial region, the coverage of an installation using the Euclidean distance is represented by a circle of radius 'S' with the facility in the center [36].  
Thus, a problem that contains multiple demand nodes dispersed in the geographic space can lead to a solution with a large number of facility allocations, which can make the project unfeasible from a budgetary point of view [9]. Therefore, the MCLP can be applied to locate a fixed number of installations on the network in order to maximize the number of demand nodes that are covered within a distance or time specified by the decision maker [37].  
In the following, we present the mathematical structure of the MCLP model proposed by [PERSON] and [PERSON] in [22]. The notation seeks to balance three issues: the coverage; the maximum distance covered to provide the service; and the budget constraint, translated into terms of the number of operations to be allocated.  
The following variables are used in the mathematical formulations of the MCLP models:  
* \(i\) = index for demand points;
* \(j\) = index for candidate locations;
* \(I\) = the set of demand nodes;
* \(J\) = the set of potential operating locations;
* \(N_{i}=\left\{j\ \in\ J\ |\ d_{ij}\ \leq S\right\}\), the set of locations that can cover demand point \(i\);
* \(d_{ij}\) = the shortest distance from \(i\) to \(j\);
* \(S\) = maximum value of the distance between the demand node and the operation node (desired service distance);
* \(a_{i}\) = coefficient reflecting the desirability of covering demand point \(i\);
* \(P\) = number of operations (facilities) to be allocated;
* \(x_{j}\) = 1 if a facility is located at potential location \(j\)--0 otherwise;
* \(y_{i}\) = 1 if demand point \(i\) is covered at least one operation \(j\)--0 otherwise.  
The formulation of the MCLP is:  
\[\text{Maximize}\sum_{i\in I}^{n}a_{i}y_{i} \tag{1}\]  
\[\text{s.\ t.:}\ \sum_{j\in N_{i}}x_{j}\geq y_{i}\qquad\text{ for all }i\in\ I \tag{2}\]  
\[\sum_{j\in J}x_{j}=P \tag{3}\]  
\[x_{j}=(0,1)\qquad\text{ for all }j\in\ J \tag{4}\]  
\[y_{i}=(0,1)\qquad\text{ for all }i\in\ I \tag{5}\]  
The objective in Equation (1) of the MCLP is to maximize the coverage demand within the desired service distance \(S\). The constraint in Equation (2) ensures that a point of demand is only covered (i.e., \(y=1\)) if there is at least one operation established in a candidate location within the set \(N\). The restriction in Equation (3) refers to the number of operations (facilities) to be located, which is limited by \(P\), as defined by the decision-maker (DM).  
Lastly, Equations (4) and (5) indicate that only integer values can be part of the solution (binary programming model). The problem is solved through heuristics and exact methods.  
It is possible to invest resources more efficiently, allocating facilities to meet most of the demand with the available resources. The model presents a complete solution with the identification of facilities and specification of the maximum demand covered by the optimal location.  
However, the MCLP model needs some inputs to be executed [17; 22; 38]. The inputs are the points of demand and the possible candidate locations to allocate a facility. Demand points can be collected according to registered demands (historical series), but candidate locations can be defined in different ways [15; 39], as is the case with clustering methods [40; 41; 42].