# A Multi-Agent Optimization Approach for Multimodal Collaboration in Marine Terminals
## 5 Case Study
### Results and Discussion
It is important to notice that the Prioritized Search algorithm exhibited a clear runtime advantage, being approximately 20-50% faster in runtime, and significantly better convergence to the best solution than the evolutionary algorithms. In Figure 8, the Pareto fronts identified by the solution algorithms and their overall convergence behavior are illustrated. \begin{table}
\begin{tabular}{l l l l l l l l l l l l l} \hline \hline \multicolumn{2}{c}{**Algorithms**} & \multicolumn{4}{c}{**NSGA-II**} & \multicolumn{4}{c}{**SPEA2**} & \multicolumn{4}{c}{**Prioritized Search**} \\ \hline
**Instance** & **Stats** & **PF** & **HV** & **Runtime** & **ITB** & **PF** & **HV** & **Runtime** & **ITB** & **PF** & **HV** & **Runtime** & **ITB** \\ \hline
20-2-low & Avg. & 6.1 & \(3.8\times 10^{3}\) & 1315 & 3178 & 5.8 & \(3.8\times 10^{3}\) & 1318 & 2933 & 5 & \(3.8\times 10^{3}\) & 872 & 2604 \\  & Max & 7 & \(3.8\times 10^{3}\) & 2012 & 4800 & 7.3 & \(3.8\times 10^{3}\) & 2045 & 3800 & 7 & \(3.8\times 10^{3}\) & 1068 & 3540 \\  & Min & 4 & \(3.7\times 10^{3}\) & 894 & 2200 & 3 & \(3.6\times 10^{3}\) & 929 & 2000 & 3 & \(3.6\times 10^{3}\) & 734 & 500 \\  & Comb. & 7 & \(3.8\times 10^{3}\) & - & - & - & 7 & \(3.8\times 10^{3}\) & - & - & - & \(3.8\times 10^{3}\) & - & - \\ \hline
25-2-low & Avg. & 2 & \(2.5\times 10^{4}\) & 1750 & 3933 & 2.2 & \(2.5\times 10^{4}\) & 1483 & 4489 & 3 & \(2.2\times 10^{4}\) & 1252 & 1780 \\  & Max & 2 & \(2.5\times 10^{4}\) & 2571 & 5000 & 2.5 & \(1.5\times 10^{4}\) & 2480 & 5000 & 5 & \(2.5\times 10^{4}\) & 1439 & 4080 \\  & Min & 0 & \(2.4\times 10^{4}\) & 1219 & 2200 & 1 & \(2.4\times 10^{4}\) & 1108 & 3400 & 0 & \(1.8\times 10^{4}\) & 1113 & 320 \\  & Comb. & 2 & \(2.5\times 10^{4}\) & - & - & 2 & \(2.5\times 10^{4}\) & - & - & - & \(6\) & \(2.5\times 10^{4}\) & - & - \\ \hline
30-2-low & Avg. & 2 & \(1.4\times 10^{4}\) & 1902 & 4889 & 1 & \(1.5\times 10^{4}\) & 1856 & 4778 & 3 & \(1.8\times 10^{4}\) & 1387 & 3881 \\  & Max & 7 & \(1.7\times 10^{4}\) & 2945 & 5000 & 4 & \(1.7\times 10^{4}\) & 2959 & 5000 & 5 & \(1.7\times 10^{4}\) & 1518 & 4880 \\  & Min & 0 & \(1.1\times 10^{4}\) & 1326 & 4400 & 0 & \(1.4\times 10^{4}\) &In the medium traffic instances 25-3, 30-3, and 35-3, the Prioritized Search algorithm consistently outperformed both NSGA-II and SPEA2 across all evaluated metrics. In the 25-3 instance, it was the only algorithm to contribute to the Pareto front, providing 16 non-dominated solutions and achieving the highest average hypervolume about 33% higher than the best-performing evolutionary algorithm. The 30-3 instance showed similar trends, with Prioritized Search contributing 10 solutions, compared to none by the evolutionary algorithms, and achieving an average hypervolume approximately 100% higher. In the 35-3 instance, Prioritized Search again dominated, contributing 17 solutions and reaching an average hypervolume that was over 50% higher than NSGA-II and SPEA2. Runtime analysis indicates that in the most complex case, Prioritized Search required about 10% more computation time on average, but reached its best solutions in 25% fewer iterations, suggesting better efficiency per iteration. In contrast, NSGA-II and SPEA2 not only failed to find high-quality trade-offs but also consumed comparable or greater computational resources. This inefficiency is compounded by their zero contributions to the combined Pareto fronts across all instances. Overall, it demonstrates clear advantages in convergence speed, solution quality, and computational efficiency in medium traffic scenarios, as illustrated in Figure 9.  
In the high traffic instances 40-4, 45-4, and 50-4, Prioritized Search again showed a significant advantage over NSGA-II and SPEA2. In all three cases, the evolutionary algorithms failed to contribute any solutions to the combined Pareto fronts, while Prioritized Search contributed 43, 40, and 28 solutions, respectively. This dominance is again reflected in its consistently superior hypervolume values, indicating both better spread and convergence toward the optimal front.