## 3 SCLP and MCLP  
In this section, the mathematical formulations of SCLP and MCLP are firstly presented, because these models will be utilized in the formulation of the hybrid covering location problem. After that, the assumptions and formulation of the hybrid covering location problem will be presented in the next section. Set covering location problem (SCLP) and maximal covering location problem (MCLP) were introduced in 1971 and 1974 by [PERSON] et al. [18] and [PERSON] [26], respectively. In the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Paper** & **Model** & **Period** & **Coverage Type** & **Facility Type** & **Data Modeling** & **Capacity Constraint** & **Decision Levels** \\  & **SM,H** & **S,M** & **B,G,C** & **S,M** & **D,S,E,R** & **C,N,M** & **S,T,O** \\ \hline [PERSON] et al. [18] & S & S & B & S & D & N & S \\ [PERSON] et al. [9] & S & M & B & S & D & N & S \\ Eisel and Marinaor [51] & S & S & G & S & D & N & S \\ [PERSON] et al. [52] & S & S & G & M & D & N & S \\ Church and ReVelle [26] & M & S & B & S & D & N & S \\ [PERSON] et al. [53] & M & S & C,C & S & D & N & S \\ [PERSON] et al. [54] & M & S & B & M & D & N & S \\ [PERSON] et al. [5] & M & S & B & S & R & N & S \\ [PERSON] and [PERSON] [49] & M & S & B & M & D & C & S \\ [PERSON] et al. [42] & M & S & GC & S & D & N & S \\ [PERSON] and Jayaswal [8] & M & M & B & S & S & C & S \\ [PERSON] [50] & M & M & G & M & D & C & S \\ [PERSON] et al. [45] & M.S & S & B & S & F & N & S \\ [PERSON] et al. [44] & MS & S & B & M & D & N & S \\ Proposed MCLP & H & M & G & M & D & C,M & S,T \\ \hline \hline \end{tabular} Model: S (set covering location problem or SCLP), M (maximal covering location problem or MCLP), and H (Hybrid). Period: S (Single period) and M (Multi-period). Coverage Type: B (Binary), G (Gradual), and C (Cooperative). Facility type: S (Single) and M (Multiple/Modular). Data Modeling: D (Deterministic), S (Stochastic), F (Fuzzy), and R (Robust). Capacity Constraints: C (Capacitated), N (Noncapacitated), and M (Module capacity). Decision level: S (Strategic), T (Tactical), and O (Operational).  
\end{table} 
Table 1: Related papers in the literature (classified based on modeling ideas). original set covering location problem, a facility can serve all demand points that are within a given coverage distance from the facility. The problem is finding the minimum number of facilities to ensure that all demand points can be served. In this model, there are no capacity constraints for the facilities. The demand points are represented by the set of \(j\in J\), and the possible locations of the facilities are given by the set of \(i\in I\). \(\mathit{dis}_{ij}\) is the distance between potential facility \(i\) and demand point \(j\). Having the coverage distance \(\rho\), one can generate the possible facilities that can cover demand points in a binary parameter \(\sigma_{ij}\) that gets the value 1 if \(\mathit{dis}_{ij}\leq\rho\) and 0, otherwise. The decision variable \(\chi_{i}\) is a binary variable that is 1 if a facility is located in \(i\) and 0, otherwise. The mathematical model for original SCLP is:  
\[\min\sum_{i\in I}\chi_{i} \tag{1}\]  
\[ \begin{array}{c}\sum\limits_{i\in I}\sigma_{ij}\chi_{i}\geq 1\\ \forall\ j\in J\\ \chi_{i}\in\{0,1\}\\ \forall\ i\in I\end{array}  \tag{2}\]  
The objective function (1) minimizes the number of located facilities, and the constraints (2) implies that every demand point \(j\) needs to be served by at least one facility. If there is a budget or resource limitation and one desires to locate \(\varphi\) predefined facilities with the objective of maximizing the covered demand points, the problem is called the maximal covering location problem, for which, in addition to the introduced variable for SCLP, another decision variable is needed. \(\gamma_{j}\) is a binary variable that is 1 if demand point \(j\) is covered and 0, otherwise. The mathematical model of MCLP is:  
\[\max\sum_{j\in J}\gamma_{j} \tag{3}\]  
\[