## 5 Case Study Applications
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**MODEL**} & \multicolumn{3}{c}{**Traveling Distance (nm)**} & \multicolumn{3}{c}{**Vessels Travel Time**} & \multicolumn{3}{c}{**Islands per**} \\ \hline
**Islands/Routes/** & **Case** & **Performance** & **Total** & **Min** & **Max** & **Passengers/** & **(hh:mm)** & **Total** & **Min** & **Max** & **Min** & **Max** \\ \hline \multirow{4}{*}{49/4/0} & C5a & D & 1666 & 158 & 564 & 29,399 & 7:00 & 23:44 & 73:30 & 9 & 17 \\ \cline{2-11}  & C5b & PH & 1823 & 179 & 669 & 26,347 & 7:46 & 26:37 & 81:40 & 8 & 18 \\ \cline{2-11}  & C5c & D-PH & 1733 & 272 & 565 & 27,437 & 12:05 & 22:45 & 75:01 & 10 & 15 \\ \hline \multirow{4}{*}{49/5/1} & C6a & D & 1508 & 133 & 543 & 28,379 & 6:14 & 21:56 & 70:34 & 7 & 12 \\ \cline{2-11}  & C6b & PH & 1617 & 98 & 586 & 22,920 & 8:11 & 23:31 & 72:41 & 6 & 12 \\ \cline{2-11}  & C6c & D-PH & 1552 & 95 & 685 & 25,134 & 7:00 & 22:45 & 70:05 & 8 & 11 \\ \hline \multirow{4}{*}{49/6/2} & C7a & D & 1383 & 60 & 425 & 26,070 & 5:52 & 17:04 & 69:44 & 3 & 10 \\ \cline{2-11}  & C7b & PH & 1731 & 60 & 518 & 22,591 & 5:52 & 20:30 & 81:30 & 3 & 11 \\ \cline{1-1} \cline{2-11}  & C7c & D-PH & 1462 & 60 & 554 & 23,097 & 5:52 & 21:51 & 74:22 & 3 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of case studies C5–C7. To further elaborate on the model effectiveness, three randomly generated instances (not related to the previous case study with actual islands) with a gradually increased degree of difficulty are tested to evaluate the algorithm performance in small-, medium-, and large-scale problems. The small instance includes 15 islands to be served by 2 lines (routes) that can be initiated from 3 alternative ports, the medium instance includes 50 islands and 6 lines that can be initiated from 8 ports, and the large instance includes 100 islands and 12 lines that can be initiated from 15 ports. The algorithm evaluation is performed at three distinct optimization objectives of minimizing the traveling distance (D), the passenger hours (PH), and the combination of them, respectively. Due to the stochastic process of the optimization  
Figure 17: Optimization results for case study C7b (up to four vessels from central ports and two from a hub node). Figure 18: Traveled distance–cumulative passenger hours point clouds at different weights of the objective function (1) (\(w_{1}>>w_{2}\) in blue, \(w_{1}\approx w_{2}\) in orange, \(w_{1}<<w_{2}\) in green) (**a**) full set of values; (**b**) subset of best values embodied in the circle of (**a**). search, three iterations for each scenario have been performed to provide an indication of the solution variability. The results shown in Table 3 indicate the algorithm's robustness in solving small instances as, in all cases, the optimization results rapidly and closely converge to specific values. As the magnitude and complexity of the problem increase (medium and large-scale instances), although the solution quality in terms of convergence is slightly reduced, the algorithm remains capable of providing acceptable solutions. On the other hand, the required computational time is substantially increased and is in the order of 20 and 80 min, respectively. It is noted, however, that part of the required time is spent on the calculations within the Excel environment.