## 4 Solution Procedure
### Sub-Gradiant Method
Three modules of the red triangle have been moved to the other facility in the second time period as the demand values change in order to cover more demands. An example of this kind of module can be the ambulances that are dispatched to the other aid center in the second period of the relief operation. Having an illustrative example, the case study of the humanitarian logistics problem in Japan by [50] is solved using the MMCMCLP model. As mentioned in the assumptions of the model and as a real-life fact, when a disaster hits a region the modules dispatch to that area to provide service for demands requests until the end of the period and in the next period as the new disaster occurs, they leave toward the newly affected area for the new mission. In the case study in [50], the threat scenarios are designed having three time periods and disaster hits the south-central (R1) of Japan in the first time period, the second disaster occurs in the north-central part (R2) in the second time period and in the third period, it is the central part (R3) that is affected by a disaster and needs the modules assignment. Solving the problem with MMCMCLP, the solution of the problem for these threats the first area (R1) affected by the disaster could be covered 65.8%, the coverage for the second area (R2) was 49.6% and in the last affected area (R3) the coverage of demand points was 82.7%. The decisions of locating facilities are determined before disasters happen and when a disaster happens in any region, the government or any responsible organization can dispatch the limited modules (trucks, helicopters, medical services, mobile kitchens, shelter tents, etc.) to the located facilities to start service operations there. When the modules fulfill their operations, they can be dispatched to be assigned to the other facilities of other affected regions according to the demand requests. #### 5.1.2 Sensitivity Analysis for the Capacity of Facilities

There are two kinds of capacity constraints in the model, one related to the module capacity and the other for the facility capacity that are very important constraints to shape the feasible region. The parameter values for \(\beta_{i}\) should be selected reasonably. The values all parameters are fixed and the values of \(\beta_{i}\) change from large values to small values as Table 2. The results show that up to the interval of (80, 100) the objective values are fixed at 85.8 and the objective value starts decreasing for the small values, which means the capacity constraints of the facilities become active afterward. In this regard, the number of the assigned modules and the covered demands decrease as well. Figure 2 depicts the results for various capacities of facilities to provide a schematic evaluation. For all test problems, the column \"Obj\" contains the objective values. The column \"Y\" shows the total number of all modules 'kinds allocated to facilities in all time periods and the column \"X\" contains the total number of covered demands. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \# & \(\beta_{i}\) & Obj & Y & X \\ \hline
1 & (260,300) & 85.81 & 11 & 135.6 \\
2 & (160,200) & 85.81 & 11 & 136.22 \\
3 & (100,120) & 85.81 & 11 & 135.14 \\
4 & (80,100) & 85.81 & 11 & 135.46 \\
5 & (60,80) & 76.26 & 10 & 124.24 \\
6 & (50,70) & 65.6 & 8 & 95.53 \\
7 & (40,60) & 63.5 & 6 & 70.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational results for different values of facility capacity. Figure 1: Illustrative example to depict the modules assignment and facilities location. #### 5.1.3 Sensitivity Analysis for the Number of Facilities to be Located

Another parameter that has an effect on the value of the objective function is the number of facilities to be located. It is expected that having more facilities, more demand points would be covered. By increasing the number of facilities, we expect the objective value would increase as well. Table 3 shows the results when there are changes in the number of facilities to be located.