## 6 Conclusions  
In this study, we proposed two search strategies with low computational costs while maintaining the performance of the mBSSRP: the first one is to reduce neighborhood solutions to obtain a feasible solution in a short time before finding a feasible solution of the mBSSRP, and the second one is to change the problem to be solved (mBSSRP or mBSSRP-S) after a feasible solution is obtained and to search good near-optimal solutions in a short time. As the first search strategy before obtaining a feasible solution, we propose two search methods (the search methods 1B-S and 1C-S) and compare their performance with our previous method (the search method 1A-S) [20]. The search method 1A-S searches for all neighborhood solutions of the Or-opt and the CROSS-exchange [20]. The search method 1B-S searches only the normal order insertion of the Or-opt. The search method 1C-S searches for the normal order insertion of the Or-opt and the normal order exchange of the CROSS-exchange. The results of the numerical experiments showed that the search method 1C-S that only searches for the normal order insertion of the Or-opt and the normal order exchange of the CROSS-exchange is the best search method, because the search method 1C-S obtained many feasible solutions within a short time. Next, as the second search strategy after a feasible solution of mBSSRP is obtained, we propose four search methods (2B-S, 2B-H, 2C-S and 2C-H) and compare their performance with our previous methods (2A-S and 2A-H) [20]. The search method 2A-S searches for all neighborhood solutions of the Or-opt and the CROSS-exchange with soft constraints. The search method 2A-H searches for all neighborhood solutions of the Or-opt and the CROSS-exchange with hard constraints. The search method 2B-S searches for insertion in the normal order of the Or-opt with soft constraints. The search method 2C-S searches for the insertion of the normal order of the Or-opt and the exchange of the CROSS-exchange with soft constraints. Finally, the search method 2C-H searches for the insertion of the normal order of the CROSS-exchange with hard constraints. The results of the numerical experiments showed that the search method 2C-H that searches 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, because the search method 2C-H obtained short \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**2C-S**} & \multicolumn{2}{c}{**2C-H**} & \\
**No.** & **Avg. [min]** & **SD** & **Avg. [min]** & **SD** & \(p\) \\ \hline
9 & 237.62 & 5.09 & 238.72 & 4.64 & 0.262 \\
10 & 238.92 & 3.91 & 237.86 & 4.27 & 0.203 \\ \hline Avg. & 237.69 & 4.51 & 238.08 & 4.76 & 0.390 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._ \begin{table}
\begin{tabular}{c c} \hline \hline
**No.** & **Avg. [s]** \\ \hline
1 & 38.82 \\
2 & 31.54 \\
3 & 31.71 \\
4 & 36.01 \\
5 & 34.17 \\
6 & 77.78 \\
68.04 \\
29.10 \\
35.56 \\
10 & 59.94 \\ \hline Avg. & 44.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of the average transition time to a feasible solution for the search method 2C-S.  
tours within a short time. These results indicate that the search method 1C-S reduces the neighborhood solutions in the Or-opt and the CROSS-exchange and that the search space was shifted from soft constraints to hard constraints after a feasible solution is obtained is effective. In this study, we used a dynamically changing method for weight parameters (\(\alpha\) and \(\beta\)) to find good solutions of mBSSRP from the solution of the mBSSRP-S. To adjust these weight parameters \(\alpha\) and \(\beta\), we used an adjustment method (Equations (3)-(6)). Then, from the results of numerical experiments (Figure 9a), it takes a long time to update the better solution for solving the mBSSRP-S (for example, 38.82 [s] for instance No. 1).