## 5 Numerical Experiments
### Results of Search Strategy before Obtaining a Feasible Solution  
Results of the numerical experiments for the search strategy before obtaining a feasible solution are shown in Table 4. In Table 4a-c, the first column lists the instance numbers, the second to the fourth columns present the average, the best and the worst objective function values [min] for each instance, and the fifth column indicates the number of obtained feasible solutions for each trial. We performed 50 trials; thus, the maximum value was 50. Table 4a summarizes the result that searches for all neighborhood solutions of the Or-opt and the CROSS-exchange (the search method 1A-S). From Table 4a, feasible solutions of the mBSSRP were obtained for almost all trials. From Table 4b, the search method that searches for only normal order of the Or-opt (the search method 1B-S) cannot obtain several feasible solutions compared to Table 4a. By contrast, the search method 1C-S (Table 4c) obtained almost the same number of feasible solutions and the average objective function values [min] as the search method 1A-S (Table 4a). To compare the performance of the search methods 1A-S and 1C-S by statistical analysis, we used the \(t\)-test. Table 5 shows the results of the \(t\)-test. In Table 5, the first column lists the instance numbers, the second and the fourth columns present the average objective function values [min] for each instance, the third and the fifth columns show the standard deviations and the sixth column presents the results of the \(t\)-test. In the numerical experiment, the level of significance was 5%. Thus, if \(p\geq 0.05\), there is no significant difference between the performance of the search methods 1A-S and 1C-S. By contrast, if the results show that \(p<0.05\), there is a significant difference between the performance of the search methods 1A-S and 1C-S. From Table 5, only for instance No.1 and No.10, the \(p\)-values are smaller than 0.05. However, for instance No.2\(\sim\)No.9 (Table 5), the \(p\)-values are larger than 0.05. These results indicate that we cannot say that there is a significant difference between the performance of the search methods 1A-S and 1C-S.  
Next, to clarify the difference between the search methods 1A-S and 1C-S, we evaluated the relationship between the time [s] until a feasible solution was obtained and the number of obtained feasible solutions (Figure 8). Figure 8a shows the result of the search method 1A-S (search for all neighborhood solutions of the Or-opt and the CROSS-exchange), and Figure 8b shows the result of the search method 1C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange). The horizontal axis shows time [s] and the vertical axis indicates the number of obtained feasible solutions. We can see that the highest value of the number of obtained feasible solutions takes approximately 70 [s] from Figure 8a. Meanwhile, the highest value of the number of obtained feasible solutions takes approximately 40 [s] from Figure 8b. Thus, although the number of obtained feasible solutions and the average objective function values [min] for the search methods 1A-S and 1C-S were almost the same as shown in Table 4, the search method 1C-S finds a feasible solution approximately 30 [s] faster than the search method 1A-S (Figure 8). To investigate whether there is a significant difference in the average time [s] until a feasible solution was obtained between the search methods 1A-S and 1C-S, we also conducted a statistical analysis by the \(t\)-test. Table 6 shows the results of the \(t\)-test. In Table 6,  
Figure 7: An example of instance. A black square represents a depot, red circles represent pickup ports and blue circles represent delivery ports. the first column lists the instance numbers, the second and the fourth columns present the average time [s] until a feasible solution was obtained for each instance, the third and the fifth columns show the standard deviations and the sixth column presents the results of the \(t\)-test.