## 4 Numerical Experiments
### Results of the Proposed Method  
To evaluate the performance of the proposed method, we generated 50 initial solutions using the cheapest insertion method for the mBSSRP and the farthest insertion method for the mBSSRP-S. The tabu tenure was set to 20, and the number of iterations was set to 1000. The initial weights were set to \(\alpha(0)=1\) and \(\beta(0)=1\). The increasing \(\lambda\) and decreasing parameters \(\mu\) were set to 1.05 and 0.7, respectively. In evaluating the solutions generated by our method, we used the gap between our solution and the optimal or best solutions (Table 3b,c). The gap is defined by the following equation:  
\[\text{Gap}[\%]=\frac{\text{Obtained best solution}-\text{Optimal or Best solution}}{\text{Optimal solution}}\times 100. \tag{24}\]  
Thus, small gaps indicate good performance. Tables 4-8 present the results. The first column provides the instance numbers and the second to fourth columns present the average gaps obtained after 50 trials of the proposed method with the best and worst gaps as determined from the optimal or best solution. The gaps were calculated using Equation (24). The fifth column states the number of feasible solutions obtained during 50 trials. As observed in Table 4, the method that only searches the feasible solution space does not yield a feasible mBSSRP solution for \(n=30\) and 50. In Table 4a, \"-\" indicates that the feasible solutions are not obtained. Then, for instances 3, 5, 7, 9 and 10 of \(n=50\) (Table 4b), the best gaps take negative values, which means that the method exploring the feasible solution space obtained better solutions than those obtained by the Gurobi optimizer 9.1.1 in 12 h. From Table 4, we cannot obtain feasible solutions for most of the instances. These results indicate that it is difficult to find feasible mBSSRP solutions when only the feasible solution space is explored. Therefore, to find feasible mBSSRP solutions, we solved the mBSSRP-S with fixed values of the weight parameters. Tables 5 and 6 present the results. With the Gurobi optimizer, it takes about 12 h for 30 ports, and more than 12 h for 50 ports to obtain feasible solutions. Such long times are not realistic. Meanwhile, the average calculation times of the proposed method are 14.7 [s] (\(n=30\)) and 56.5 [s] (\(n=50\)). Therefore, the proposed method can obtain better results, in a time shorter than that required by the Gurobi optimizer. As observed in Tables 4a and 5a, the method that explores both feasible and infeasible solution spaces of the mBSSRP obtains many feasible solutions compared to the method that searches only the feasible solution space. In addition, according to Table 5a,b, when the parameters \(\alpha\) and \(\beta\) have large values, the number of feasible solutions increases, and the average gap indicates that the method performs well. Moreover, the method that explores both the feasible and infeasible solution spaces finds optimal solutions for multiple instances. The difference between the best and worst gaps is large for \(n=30\) (Table 3). Tables 4b and 6 indicate that for \(n=50\), we can obtain numerous feasible solutions when the parameters \(\alpha\) and \(\beta\) have large values. Tables 7 and 8 present the results obtained using the method of dynamically adjusted parameters. From Tables 7 and 8, the number of feasible solutions is 50 for almost all instances. The difference between the best gap and the worst gaps is smaller than that of the fixed case, as can be observed in Tables 5 and 6. The results indicate that the weight adjusting method can effectively control the search in the feasible and infeasible solution spaces. However, there is no significant difference between Method 1 and Method 2.