## 4 Solution Procedure
### Sub-Gradiant Method
Although the CPLEX could just solve problems up to 300 demand points, the proposed solution procedures were capable of producing bounds with up to 1000 demand points and 100 potential facilities and beyond. Our results indicate that although common LR problems can still generate upper bounds for complex problems like MMCMCLP, Lagrangian decomposition-based approach combined with the lower bound obtained from developed heuristic methods shows better performance and is preferred to solve for both small- and large-scale problems. The most important barrier over-performing MMCMCLP for the real application mentioned as disaster relief facility location, hospitals is data insufficiency for the parameters of demands, capacities and costs. To resolve these barriers there is a need to have access to data sets and also applying well-defined demand prediction methods. In addition, a \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\#} & \multicolumn{2}{c}{**Best LB**} & \multicolumn{2}{c}{**Best UB**} \\ \cline{2-5}  & **LB** & **Method** & **UB** & **Method** \\ \hline
1 & 55 & HLB & 72 & LR2-LB = LR2-HLB \\
2 & 69 & HLB = LR2-LB & 80 & LD-HLB = LD-LB \\
3 & 80 & HLB = LR1-LB & 99 & LD-HLB \\
4 & 114 & HLB & 117 & LR2-LB \\
5 & 1764 & LR2-LB & 2120 & LD-HLB \\
6 & 1907 & LR2-LB & 2280 & LD-HLB = LD-LB \\
7 & 2634 & LR2-LB & 3190 & LD-LB \\
8 & 2906 & HLB & 3456 & LD-HLB = LD-LB \\
9 & 9832 & LR2-LB & 11,013 & LD-LB \\
10 & 10,237 & HLB & 11,095 & LD-HLB \\
11 & 11,403 & HLB & 14,368 & LD-HLB \\
12 & 13,817 & HLB & 15,056 & LD-HLB \\
13 & 12,464 & HLB & 17,699 & LD-HLB \\
14 & 16,096 & HLB & 18,060 & LD-HLB \\
15 & 17,682 & HLB & 23,953 & LD-LB \\
16 & 23,590 & HLB & 25,669 & LD-HLB \\
17 & 20,740 & HLB & 35,123 & LD-LB \\
18 & 29,857 & HLB & 35,937 & LD-HLB \\
19 & 29,623 & HLB & 46,720 & LD-LB \\
20 & 37,469 & HLB & 47,354 & LD-HLB \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Best lower bound and upper bound values and methods. possible future study could be compared using various matheuristics/metaheuristics for this problem, studying the probability of failure and breakdown of modules in different time periods and considering the variable radius coverage for facilities. Conceptualization, R.A.; Mathematical model development, R.A., J.B., and M.B. ; Methodology, T.N. and R.A.; Software, R.A.; Validation, T.N., J.B., and M.B. ; Formal analysis, R.A. and T.N. ; Investigation, T.N. ; Resources, T.N. ; Data Curation, R.A.; Writing--Original Draft Preparation, R.A.; Writing--review and editing, T.N. ; Visualization, R.A.; Supervision, T.N., J.B., and M.B. ; Project administration, T.N. ; Funding acquisition, T.N. All authors have read and agreed to the published version of the manuscript. This research was conducted by the aid of the funding provided by JSPS KAKENHI (A) 18H03826. Not applicable. The data presented in this study are openly available at [[https://github.com/[PERSON]/MMCMCLP](https://github.com/[PERSON]/MMCMCLP)]([https://github.com/[PERSON]/MMCMCLP](https://github.com/[PERSON]/MMCMCLP)). The authors declare no conflict of interest. ## References

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