# A Multi-Agent Optimization Approach for Multimodal Collaboration in Marine Terminals
## 5 Case Study
### Results and Discussion  
The numerical campaign revolved around six core instances. Each instance is characterized by an identifier in the form \(\alpha-\beta-\gamma\), where \(\alpha\) represents the number of vessels approaching the terminal, \(\beta\) represents the number of berths available, and \(\gamma\) represents the level of truck traffic. For example, instance 25-2-low represents an instance with 25 vessels, 2 berths, and low traffic. All numerical results were generated on an Intel Xeon Gold 6226R with 48 CPU cores and 192 GB of RAM. Both algorithms were implemented in Python, with the MOEAs employed using the existing pymoo package [37]. Each instance was executed for 5000 iterations, consisting of 25 generations with a population of 200 for the evolutionary algorithms, and 20 processes each with 250 iterations for the prioritized search. The parameters for exploration, crossover, and mutation for both algorithms have been previously described in Section 4. To assess the robustness of the algorithms against stochastic variations, each instance was solved five times for each algorithm, which was achieved by changing the random seed each time. In Table 6, a comprehensive set of metrics evaluating the performance of each algorithm is presented. Specifically, the table includes Pareto front contributions (PF), hypervolumes (HVs), algorithm runtime in seconds, and iterations to the best Pareto front (ITB). The hypervolume measures the volume of the objective space that is dominated by the obtained set of non-dominated solutions and bounded by a predefined reference point. The reference point is selected just beyond the worst performance observed across all examined algorithms. Specifically, for each instance, it is defined by the maximum values of the Berth Allocation and Truck Arrival Management objectives observed in the proposed fronts across all solution methods. An algorithm with a larger hypervolume over the other indicates a better spread and convergence of the Pareto front approximation toward the true Pareto front. For these metrics, the average, minimum, and maximum values across the five runs are presented. The selection of five runs was made after confirming that the coefficient of variation related to hypervolumes had stabilized after five runs. It is important to note that the true Pareto front is unknown; therefore, the solutions are assessed based on the best results obtained across all solution algorithms. Finally, the combined performance of each algorithm, aggregated from all runs, is reported in the \"Comb.\" row for each instance. The instances are further analyzed based on the traffic level used, beginning with the low-traffic scenarios. For these cases, the performance of the Prioritized Search algorithm was found to be comparable to, or better than that of NSGA-II and SPEA2. For the 20-2-low instance, NSGA-II showed slightly better consistency in identifying the best observed front, reflected in a higher average of Pareto front contributions. However, variability in both Pareto front contributions and hypervolumes remained marginal across all methods. In the 25-2-low instance, none of the algorithms identified the complete best observed Pareto front in a single run. Nonetheless, Prioritized Search achieved the highest number of Pareto front contributions across all runs. Interestingly, its average hypervolume performance was slightly lower than that of SPEA2 and NSGA-II. In the more complex 30-2-low instance, Prioritized Search further solidified its advantage. It consistently delivered more Pareto front contributions, higher hypervolumes, and faster convergence to the best observed front. While NSGA-II and SPEA2 each contributed only six solutions to the front, Prioritized Search contributed twelve, helping to uncover a front of 21 distinct solutions, underscoring the need for arbitration in such a multi-objective context.