## 6 Numerical Experiments
### Comparison against the Exact Optimization Algorithm  
The first set of numerical experiments focused on a comparative analysis of the developed SAEA, AEA, DPCEA, and EA algorithms against the exact optimization algorithm. The **BSPSR** mathematical model was coded in the GAMS environment and solved using CPLEX, which is widely used in operations research for solving large scale mixed integer linear programming models to the global optimality. The computational time of CPLEX was restricted to 7200 s, the relative optimality gap was restricted to 0.01%, while the rest of settings remained default. CPLEX was executed for all 30 small size problem instances (i.e., instances 1-30), described in Section 6.1 of the manuscript. The SAEA, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  
* Notes: 1—the crossover probability values were changed with an increment of 0.10 within the AEA algorithm;
* 2—the mutation probability values were changed with an increment of 0.01 within the AEA algorithm.  
\end{table} 
Table 3: The parameter tuning analysis results for the developed solution algorithms. Figure 8: Examples of piecewise functions for the crossover and mutation probabilities within DPCEA. AEA, DPCEA, and EA algorithms were launched for all the developed small size problem instances as well. A total of ten replications were executed to estimate the average values of the objective function and computational time for each algorithm and each one of the small size problem instances. The comparative analysis results are reported in Table 4, which provides the following information for each small size problem instance: (1) instance number; (2) number of vessels; (3) number of berths; (4) average objective function value for each solution algorithm; and (5) average computational time (denoted as CPU) for each solution algorithm. The optimality gap for algorithm \(a\) (\(\Delta_{a}\)) was calculated based on the following relationship: \(\Delta_{a}=\frac{f_{a}-f_{CPLEX}}{f_{CPLEX}}\), where \(f_{a}\)--average objective function value, provided by algorithm \(a\); \(f_{CPLEX}\)--the optimal objective function value, provided by CPLEX. The estimated optimality gap values are presented in Figure 9 for each algorithm and each one of the small size problem instances. The optimality gap analysis results highlight that CPLEX was able to provide the global optimal solution only for 9 small size problem instances (1-8 and 10) with up to 8 vessels calling for service at the MCT. The latter can be explained by the fact that consideration of the additional spatial requirements in berth scheduling significantly increased complexity of the mathematical model. For example, in the study by [PERSON] [10], the mathematical model, which did not consider the additional spatial requirements, was solved using CPLEX for the problem instances with up to 12 vessels within 7200 s. Based on the optimality gap analysis results, the SAEA, AEA, DPCEA, and EA algorithms obtained the global optimal solution for the problem instances that were solved by CPLEX within the time limit imposed (i.e., the optimality gaps of the developed solution algorithms are equal to \"0\" for problem instances 1-8 and 10). For the rest of small size problem instances (i.e., problem instances 9 and 11-30), CPLEX was not able to solve the **BSPSR** mathematical model to the global optimality within the time limit imposed (see Table 4). Furthermore, the developed solution algorithms returned superior objective function values as compared to the objective function values, obtained by CPLEX after 7200 s. Therefore, the optimality gaps for problem instances 9 and 11-30 appear to be negative (see Figure 9). The SAEA, AEA, DPCEA, and EA algorithms outperformed CPLEX on average by 4.33%, 4.26%, 4.19%, and 4.12% over the generated small size problem instances, respectively. The averaged computational time comprised 13.49 s, 12.65 s, 11.05 s, and 7.86 s for the SAEA, AEA, DPCEA, and EA algorithms, respectively.