## 3 Problem Definition
### Mathematical Formulation  
In the developed model, it is assumed that the coverage of points obeys the gradual coverage rule in which by increasing the distance from full coverage radius \((S)\) the coverage value reduces using a partial coverage function as \(f\big{(}d_{ij}\big{)}\) that \(0<f\big{(}d_{ij}\big{)}<1\). The proposed MMCMCLP can be formulated as a mixed integer linear programming (MILP) problem in the following way. Consider the following indices, parameters and decision variables:  
Sets and Parameters: \begin{tabular}{l l} \hline \(i\in I\) & The index and set of potential facilities. \\ \(j\in J\) & The index and set of demand points. \\ \(l\in L\) & The index and set of modules. \\ \(k_{l}\in K_{l}\) & The index and set of size indices for module \(l\) (\(l\in L\)). \\ \(t\in T\) & The index and set of time periods. \\ \(S\) & The maximum full coverage distance. \\ \(S^{\prime}\) & The maximum partial coverage distance (\(S^{\prime}>S\)). \\ \(d_{ij}\) & Distance, traveling time or cost between facility \(i\) and demand point \(j\). \\ \(g_{ij}\) & The level of coverage provided by the facility \(i\) to the demand point \(j\). \\ \(g_{ij}=\left\{\begin{array}{rl}&1\ if\ d_{ij}\leq S\\ &f\Big{(}d_{ij}\Big{)}\ if\ S<d_{ij}\leq St\\ &0\ \ S\prime\ <d_{ij}\end{array}\right.\) \\ \(f\Big{(}d_{ij}\Big{)}\) & Partial coverage function, where \(0<f\Big{(}d_{ij}\Big{)}<1\). \\ \(a_{jlt}\) & The demand of point \(j\) from the service of module \(l\) in period \(t\). \\ \(b_{lk_{l}}\) & The lower level of capacity for module \(l\) with the size indices of \(k_{l}\). \\ \(B_{lk_{l}}\) & The upper level of capacity for module \(l\) with the size indices of \(k_{l}\). \\ \(\alpha_{j}\) & The profit gained by coverage of demand point \(j\). \\ \(\beta_{i}\) & The capacity of facility \(i\). \\ \(h_{ilk_{l}t}\) & The cost of establishing module \(l\) with size indices \(k_{l}\) at facility \(i\) at period \(t\). \\ \(p\) & The number of facilities to be located. \\ \hline \multicolumn{2}{c}{Decision variables:} \\ \hline \(v_{i}\) & Binary variable which equals 1 if a facility is located at point \(i\). \\ \(y_{ilk_{l}t}\) & Binary variable which equals 1 if module \(l\) with size indices of \(k_{l}\) is sited at facility \(i\) at period \(t\). \\ \(x_{ijlt}\) & Coverage amount of demand point \(j\) from module \(l\) of facility \(i\) at period \(t\). \\ \hline \end{tabular}   
Then, the proposed model is as follows:  
\[\text{Max }\sum\limits_{i\in I}\sum\limits_{j\in I}\sum\limits_{l\in I} \sum\limits_{t\in T}\ \alpha_{j}g_{ij}a_{jlt}x_{ijlt}-\sum\limits_{i\in I}\sum\limits_{l\in I} \sum\limits_{k_{l}\in K_{l}}\sum\limits_{l\in T}h_{ilk_{l}t}y_{ilk_{l}t} \tag{1}\]  
\[\sum\limits_{k_{l}\in K_{l}}y_{ilk_{l}t}\leq v_{i}\ \forall i,l,t \tag{2}\]  
\[\sum\limits_{i\in I}x_{ijlt}\leq 1\ \forall j,l,t \tag{3}\]  
\[\sum\limits_{j\in I}a_{jlt}x_{ijlt}\geq\sum\limits_{k_{l}\in K_{l}}b_{lk_{l}} y_{ilk_{l}t}\ \forall i,l,t \tag{4}\]  
\[\sum\limits_{j\in I}a_{jlt}x_{ijlt}\leq\sum\limits_{k_{l}\in K_{l}}B_{lk_{l}} y_{ilk_{l}t}\ \forall i,l,t \tag{5}\]  
\[\sum\limits_{l\in L}\sum\limits_{j\in I}a_{jlt}x_{ijlt}\leq\beta_{i}v_{i}\ \forall i,t \tag{6}\]  
\[\sum\limits_{i\in I}v_{i}\leq p \tag{7}\]  
\[y_{ilk_{l}t}\in\{0,\ 1\},v_{i}\in\{0,\ 1\},x_{ijlt}\geq 0\ \forall i,j,l,k_{l},t \tag{8}\]  
The objective function (1) seeks to maximize the profit gained from the covered allocated demands while simultaneously minimizing the total cost of assigning the modules. It should be noted that the parameter \(\alpha_{j}\) is used to avoid having a bi-objective function structure. Constraints (2) ensure that if there not exists an established facility at site \(i\), no module is allowed to be sited there and only one size of each module can be sited in a facility. Constraints (3) imply that the total coverage of each demand point from all the facilities can be at most 1.