## 4 Numerical Experiments
### Results of the Proposed Method
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Objective Function** & **CPU-Time [s]** \\ \hline
1 & 45,346 * & 1.31 \\
2 & 32,530 * & 0.21 \\
3 & 63,209 * & 1.37 \\
4 & 27,522 * & 0.43 \\
5 & 45,971 * & 1.20 \\
6 & 32,828 * & 1.44 \\
7 & 38,623 * & 1.82 \\
8 & 40,304 * & 1.20 \\
9 & 41,640 * & 0.76 \\
10 & 31,825 * & 0.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal or near-optimal solution obtained by the Gurobi optimizer. Symbol “-” means that no feasible solutions can be found by the Gurobi optimizer 9.1.1 within 12 h. Symbol “** means that an optimal solution can be obtained using the Gurobi optimizer 9.1.1 within 12 h. (**a**) \(n=10\). (**b**) \(n=30\). (**c**) \(n=50\). The searching processes used in Method 1 and Method 2 when solving mBSSRP-S are shown in Figure 6. The horizontal axis shows the number of iterations and the vertical axis shows the objective function value (Equation (19)). In Figure 6, it is observed that feasible mBSSRP solutions are not obtained (black circle) with 200 iterations in Method 1 and with 400 iterations in Method 2. The parameters \(\alpha(t)\) and \(\beta(t)\) were adjusted using Equations (20) and (21) in Method 1, and Equations (22) and (23) in Method 2. After these phases, as observed in Figure 6a, Method 1 finds several feasible mBSSRP solutions with 1000 iterations. However, Method 2 finds more feasible solutions than Method 1 according to Figure 6b. When the penalty value is zero, Method 2 does not update the penalty weight because the current value is appropriate for conducting a search on feasible mBSSRP solutions. Once feasible mBSSRP solutions are obtained, Method 2 conducts repeated searches for more feasible solutions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & & \\ \hline
**No.** & **Ave.** & **Best** & **Worst** & **\# of Feasible** \\ \hline
1 & - & - & - & 0 \\
2 & - & - & - & 0 \\
4 & - & - & - & 0 \\
5 & - & - & - & 0 \\
7 & - & - & - & 0 \\
8 & 1.90 & 0.00 * & 15.95 & 32 \\
9 & - & - & - & 0 \\
10 & - & - & - & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of exploring feasible solution space for solving the mBSSRP. Ave., Best and Worst represent the average, best and worst gaps calculated using Equation (24). Symbol “-” means that no feasible solutions can be obtained. The feasible solutions of the instances 3 and 6 (\(n=30\)) cannot be found using the latest version of the Gurobi optimizer 9.1.1. For \(n=50\), negative values of the best gaps mean that the method can find better solutions than that found by the Gurobi optimizer 9.1.1 within 12 h. Symbol “_” means that an optimal solution can be obtained using the Gurobi optimizer 9.1.1 within 12 h. (**a**) \(n=30\). (**b**) \(n=50\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**(a)**} \\ \hline
**No.** & **Ave.** & **Best** & **Worst** & **\# of Feasible** \\ \hline
1 & 2.41 & 2.41 & 2.41 & 1 \\
2 & 2.51 & \(-\)0.41 & 9.21 & 13 \\
3 & 4.36 & \(-\)0.69 & 10.71 & 7 \\
4 & 5.53 & 1.60 & 9.92 & 9 \\
5 & \(-\)0.57 & \(-\)1.68 & 0.66 & 3 \\
6 & \(-\)0.23 & \(-\)0.23 & \(-\)0.23 & 1 \\
7 & 3.58 & 1.19 & 6.31 & 3 \\
8 & - & - & - & 0 \\
9 & 3.06 & \(-\)0.63 & 5.26 & 17 \\
10 & 0.50 & \(-\)3.14 & 5.13 & 19 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c} \hline
**(b)** & **(b)** \\ \hline
**No.** & **Ave.** & **Best** & **Worst** & **\# of Feasible** \\ \hline
1 & 1.27 & \(-\)1.61 & 5.87 & 47 \\
2 & 2.57 & \(-\)0.50 & 11.92 & 50 \\
3 & 3.01 & \(-\)2.11 & 13.31 & 49 \\
4 & 3.59 & \(-\)0.33 & 11.13 & 48 \\
5 & 0.83 & \(-\)3.92 & 4.80 & 50 \\
6 & 3.23 & \(-\)0.23 & 8.62 & 50 \\
7 & 1.81 & \(-\)0.69 & 8.06 & 50 \\
9 & 4.33 & \(-\)0.64 & 11.24 & 50 \\
10 & 1.08 & \(-\)3.44 & 9.78 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the method using fixed values of weight parameters for \(n=50\). Symbol “-” means that no feasible solutions can be obtained.