## 4 Solution Procedure
### Sub-Gradiant Method
\begin{table}
\begin{tabular}{c c c c|c c c c|c c c c} \hline \hline \multirow{2}{*}{\#} & \multicolumn{3}{c}{**LR1**} & \multicolumn{6}{c}{**LR2**} & \multicolumn{3}{c}{**LD**} \\ \cline{2-13}  & **LB** & **UB** & **Time** & **Gap** & **LB** & **UB** & **Time** & **Gap** & **LB** & **UB** & **Time** & **Gap** \\ \hline
9 & 2476 & 14,072 & 183 & 4.68 & 9832 & 11,630 & 672 & 0.18 & 9147 & 11,013 & 2356 & 0.20 \\
10 & 2653 & 13,937 & 361 & 4.25 & 9095 & 11,806 & 2640 & 0.30 & - & 11,102 & 693 & - \\
11 & 2122 & 18,532 & 205 & 7.73 & - & 15,282 & 930 & - & - & 14,429 & 713 & - \\
12 & 3453 & 18,746 & 317 & 4.43 & 12,310 & 16,000 & 1301 & 0.30 & 11,134 & 15,141 & 790 & 0.36 \\
13 & 7047 & 21,305 & 404 & 2.02 & - & 19,036 & 1941 & - & - & 17,748 & 1620 & - \\
14 & 6088 & 21,228 & 766 & 2.49 & 12,534 & 19,117 & 1898 & 0.53 & 8007 & 20,443 & 287 & 1.55 \\
15 & 7808 & 28,253 & 543 & 2.62 & - & 25,545 & 2157 & - & - & 23,953 & 1548 & - \\
16 & 6616 & 29,053 & 844 & 3.39 & 18,372 & 26,282 & 2815 & 0.43 & 12,742 & 30,472 & 229 & 1.39 \\
17 & 11,110 & 41,389 & 711 & 2.73 & - & 37,517 & 4466 & - & - & 35,123 & 4818 & - \\
18 & 11,345 & 41,046 & 1784 & 2.62 & - & 37,768 & 8020 & - & - & 35,970 & 5243 & - \\
19 & 14,002 & 54,636 & 1329 & 2.90 & - & 50,177 & 4759 & - & - & 46,720 & 4870 & - \\
20 & 17,823 & 54,471 & 2101 & 2.06 & - & 50,043 & 21,466 & - & - & 47,399 & 6919 & - \\ Average & & 795 & 3.5 & & & 4422 & 0.34 & & & 2507 & 0.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Calculation results to compare the bounds for LR1, LR2, and LD. \begin{table}
\begin{tabular}{c c|c c c|c c c|c c c} \hline \hline \multirow{2}{*}{\#} & \multicolumn{3}{c}{**Heuristic**} & \multicolumn{3}{c}{**LR1**} & \multicolumn{3}{c}{**LR2**} & \multicolumn{3}{c}{**LD**} \\ \cline{2-13}  & **HLB** & **UB** & **Time** & **Gap** & **UB** & **Time** & **Gap** & **UB** & **Time** & **Gap** \\ \hline
9 & 7945 & 11,673 & 3703 & 0.47 & 11,617 & 585 & 0.46 & 11,118 & 451 & 0.40 \\
10 & 10,237 & 11,870 & 4817 & 0.16 & 11,808 & 787 & 0.15 & 11,095 & 775 & 0.08 \\
11 & 11,403 & 15,334 & 7016 & 0.34 & 15,283 & 617 & 0.34 & 14



bounds of two studied approaches are equal for 3 problems and LD-LB was superior only in 5 cases. According to the results, we suggest using HLB to compute lower bounds and LD-HLB to compute the upper bounds. The GAMS source codes of MMCMCLP are available at [[https://github.com/Alizadehroqayeh/MMCMCLP](https://github.com/Alizadehroqayeh/MMCMCLP)]([https://github.com/Alizadehroqayeh/MMCMCLP](https://github.com/Alizadehroqayeh/MMCMCLP)). ## 6 Conclusions

In this paper, a new extension of the maximal covering location problem is presented, which has some important features such as the possibility of having different facilities that are composed of capacitated modules, the possibility of choosing between the different size of modules commensurate to demanded services, capacity constraints for facilities, multi-periodic demands for points, and gradual coverage for facilities. The proposed model has many applications such as locating the facilities in post-disaster situations, supermarkets, and hospitals. The idea of using the modularity concept for the facilities resulted in less cost, more flexibility, and it can provide sustainability from environmental and economic aspects. A mathematical model has been proposed for MMCMCLP, which can fit different real-life problem characteristics. Our conducted sensitivity analysis approves the accuracy of the proposed model as it can generate logical solutions by investigating various changes of parameters. After examining the possible relaxations for the MMCMCLP, different methods are proposed to solve the relaxed and decomposed problems and obtained various upper and lower bounds. Accordingly, a heuristic method is proposed that is able to generate higher lower bounds for the optimal solutions.