## 6 Numerical Experiments
### Comprehensive Evaluation of the Algorithms
#### 6.4.2 Convergence Patterns  
The analysis of algorithmic convergence patterns plays an important role in evaluation of the algorithmic performance, as it allows monitoring changes in the objective function values from the population initialization step until termination of the algorithm. The convergence patterns were retrieved for all the developed solution algorithms and large size problem instances. Figure 10 illustrates convergence patterns of the SAEA, AEA, DPCEA, and EA algorithms for the first replication of problem instances 49-60 (12 large size problem instances with the largest number of vessels, calling for service at the MCT). Note that similar tendencies in the algorithmic convergence patterns were observed for the rest of replications and problem instances. Based on the convergence patterns, it can be noticed that all the developed solution algorithms started the search process with the same solution. The latter can be explained by the fact that the SAEA, AEA, DPCEA, and EA algorithms deploy the same population initialization mechanism (described in Section 4.3 of the manuscript). Application of the self-adaptive parameter control for the crossover and mutation probabilities allowed the SAEA algorithm discovering the promising domains of the search space much faster as compared to the AEA, DPCEA, and EA algorithms. The AEA algorithm was able to move along the search space more efficiently as compared to the DPCEA algorithm, as it was adjusting the crossover and mutation probabilities based on feedback from the search. Furthermore, the analysis of algorithmic convergence patterns indicates that setting the constant values for the crossover and mutation probabilities (i.e., the parameter tuning strategy, adopted within the EA algorithm) significantly slows down the search process and generally worsens the objective function value at termination. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & **Number of** & **Number of** & \multicolumn{2}{c}{**SAEA vs. EA**} & \multicolumn{2}{c}{**SAEA vs. DPCEA**} & \multicolumn{2}{c}{**SAEA vs. AEA**} \\ \cline{3-8}  & **Vessels** & **Berths** & **F-Statistic** & \(\boldsymbol{p}\)**-Value** & **F-Statistic** & \(\boldsymbol{P}\)**-Value** & **F-Statistic** & \(\boldsymbol{p}\)**-Value** \\ \hline
31 & 55 & 4 & 87.333 & \(2.51\times 10^{-8}\)  
The scope of numerical experiments also includes a detailed analysis of the convergence speed for the developed solution algorithms. Similar to [PERSON] et al. (2016), the convergence speed was assessed based on the convergence rate value for each solution algorithm. The convergence rate of algorithm \(a\) (\(CR_{a}\)) was estimated based on the number of fitness function evaluations until finding its minimum value (\(NF_{a}^{*}\)) and the total number of fitness function evaluations (\(NF_{a}^{TOT}\)) using the following relationship [PERSON] et al. (2016):  
\[CR_{a}=\frac{NF_{a}^{*}}{NF_{a}^{TOT}} \tag{19}\]  
The convergence rate values were calculated for all the developed solution algorithms, performed replications, and large size problem instances. Results are reported in Table 7, including the following information for each large size problem instance: (1) instance number; (2) number of vessels; (3) number of berths; (4) average convergence rate value over ten replications (denoted as \(avg(CR)\)) for each solution algorithm; and (5) convergence rate standard deviation over ten replications (denoted as \(std(CR)\)) for each solution algorithm. Figure 10: The algorithmic convergence patterns for problem instances 49–60.