## 4 Computational Experiments
### Experimental Results and Analysis  
In the experiment, the value of \(p\) was set as 10, 20, and 30, respectively for both the H-95 and C-668 cases. Thus, we had six instances with different sizes, and all were solved by the proposed MSCE algorithm 51 times. The parameters of the MSCE algorithm were set as per the illustration in Section 3.5. To illustrate the performance of the MSCE algorithm, a location and allocation heuristic (LAH) was also used to solve all instances. The LAH algorithm is a commonly used approach for solving different facility location problems including the \(p\)-median problem, and can provide relatively good solutions [23; 25]. More details of the LAH can be found in [PERSON] et al. [23]. The average results by the MSCE for 51 runs and the results by the LAH for all instances are presented in Figure 2. We can see that the MSCE algorithm outperformed the LAH algorithm for all scenarios in the six different instances. The statistical results for the computational time of the MSCE algorithm in all instances is reported in Table 1. It can be seen that the computational time increased as the size of the instance became bigger. Considering there were 100 scenarios in each instance, the computational time was acceptable for solving the problem. Figure 1: Node distribution for the two practical cases. (**a**) Ninety-five cities in Hunan Province and (**b**) 668 demand points in Changsha city. The GA is also widely used to solve \(p\)-median problems. Here, we utilized the standard framework of the GA and the general operators used in [PERSON] et al. (2016) to solve the problem. No enforcements on the initial population and genetic operators were considered as these would usually further increase the computational time. The population size of GA was 100 and the maximum number of evolution generations was 1000. The GA will also stop if there is no improvement in the solution for 100 generations. The mutation probability was 0.1. The GA was run 51 times for each scenario individually. When we solved the instances for H-95 and C-668 with \(p=10\), we found that the statistical results on the total computational time by GA for all 100 scenarios (as presented in Figure 3) were relatively large. The experiments for the other larger size instances became unacceptable with time. Thus, we report the results for these two instances here. The solution obtained by the GA was very close to that by the MSCE algorithm. We present the relative value of the GA to the MSCE in Figure 4. As shown in Figure 4a, among the results of 100 scenarios for the H-95 case with \(p=10\), where 15% obtained by the GA was better than that of the MSCE; 7% obtained by the GA was worse than that of the MSCE; and the other 78% were equal to each other. Among the results of the 100 scenarios for the C-668 case with \(p=10\), only 5% obtained by GA was better than that of MSCE, while 93% obtained by the MSCE was better than that of the GA. Thus, the MSCE performed much better than the GA in the C-668 case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**M/N**} & \multirow{2}{*}{\(p\)} & \multicolumn{2}{c}{**Computational Time (Seconds)**} \\ \cline{3-4}  & & & **Average** & **Standard Deviation** \\ \hline \multirow{3}{*}{H-95} & \multirow{3}{*}{95/95} & 10 & 122.861 & 37.353 \\  & & 20 & 318.011 & 66.745 \\  & & 30 & 642.455 & 164.929 \\ \hline \multirow{3}{*}{C-668} & \multirow{3}{*}{668/150} & 10 & 531.771 & 115.468 \\  & & 20 & 1211.889 & 370.137 \\ \cline{1-1}  & & 30 & 1985.167 & 452.943 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The computational time of 51 runs for all 100-scenario instances. Figure 2: The computational results for H-95 and C-668. (**a**) H-95 with \(p=10\); (**b**) H-95 with \(p=20\); (**c**) H-95 with \(p=30\); (**d**) C-668 with \(p=10\); (**e**) C-668 with \(p=20\); and (**f**) C-668 with \(p=30\).