## 5 Numerical Experiments
### Results of Search Strategy after Obtaining a Feasible Solution  
Results of the numerical experiments using the six search methods after obtaining the feasible solution are shown in Table 7. In Table 7a-f, 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] in 200 [s] of each trial, when we started the calculation from a feasible solution. From Table 7e,f, the search methods of reducing neighborhood solutions were also effective when the search was started from feasible solutions. The performance showed no significant differences between soft (Table 7e) and hard constraints (Table 7f). In particular, the difference of the average objective function values [min] of the search method 2C-S and the search method 2C-H is only 0.39 [min]. To investigate whether there is a significant difference in performance, the \(t\)-test was used on the average objective function values [min]. Table 8 shows the results of the \(t\)-test. In Table 8, 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. From Table 8, except for instance No.6, the \(p\)-values are larger than 0.05. Thus, we could not find any significant difference between the performance of the search methods 2C-S and 2C-H from the viewpoint of the statistical analysis. Next, to investigate whether there is a difference in searching process between the two search methods 2C-S and 2C-H, we evaluated the transition of the most feasible solutions when solving soft and hard constraints. The results of instance No.1 are shown in Figure 9. From Figure 9, we can find that there is a difference between these two search methods 2C-S and 2C-H. From Figure 9a, it takes a long time (38.82 [s]) to update the better solution for solving the mBSSRP-S. The reason is that to realize an effective search, the weight values of penalty in Equations (3)-(6) must be adjusted to appropriate values. The average transition time to a better solution for other instances is shown in Table 9. In Table 9, the first column lists instance numbers and the second column shows the average transition time [s] to a better solution. From Table 9, the search method 2C-S takes about 30\(\sim\)80 [s] to find a better feasible solution. Meanwhile, from Figure 9b, a solution of the search method 2C-H moves to a better solution as soon as a search is started because it cannot search for an infeasible solution. The results of the \(t\)-test of the average objective function values of the search methods 2C-S and 2C-H showed that there was no significant difference in performance (Tables 7 and 8). However, from Figure 9 and Table 9, the search method 2C-H can find a better feasible solution in a shorter time (approximately 30\(\sim\)80 [s]) than the search method 2C-S. Therefore, as for the search strategy after finding a feasible solution, we concluded that the search method 2C-H (the search for the normal order insertion of the Or-opt and the normal order exchange of the CROSS-exchange with hard constraints) is the best search method. Figure 9: Transition of the best solutions found in case of instance No.1. The red lines show the transition of the best feasible solution from 50 initial feasible solutions, whereas the blue lines show their average. (**a**) Search method 2C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange with soft constraints). (**b**) Search method 2C-H (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange with hard constraints).