## 5 Case Study Applications
(**a**) C4a: \(\rm D_{1}/D_{2}/D_{3}/D_{tot}=0/347/228/575\) nm, \(\rm T_{1}/T_{2}/T_{3}/T_{tot}=0.00/14:11/21:56/36:07\) h, PH = 18,683 h, \(\rm T_{max-trip}\) = 29:33 h. (**b**) C4b: \(\rm D_{1}/D_{2}/D_{3}/D_{tot}=390/233/97/270\) nm, \(\rm T_{1}/T_{2}/T_{3}/T_{tot}=15:18/9:23/9:18/34:09\) h, PH = 9863 h, \(\rm T_{max-trip}\) = 15:18 h. (**c**) C4c: \(\rm D_{1}/D_{2}/D_{3}/D_{tot}=153/233/261/617\) nm, \(\rm T_{1}/T_{2}/T_{3}/T_{tot}=\)6:00/9:23/22:24/37:47 h, PH = 12,012 h, \(\rm T_{max-trip}\) = 26:16 h. (**d**) C4d: (max trip time \(\leq\) 15h), \(\rm D_{1}/D_{2}/D_{3}/D_{tot}=\)290/22/53/615 nm, \(\rm T_{1}/T_{2}/T_{3}/T_{tot}=\)11:34/11:04/5:04/27:48 h, PH = 12,138 h, \(\rm T_{max-trip}\) = 11:34 h.  
Figure 12: Distance and passenger time trade-off diagram for case C3. Figure 14 presents the Pareto curves developed after several runs of all scenarios examined in the case study. The results indicate that routing only one vessel from the central port to all island nodes (case C1) takes too long (in terms of both distance and time spent on board) to serve the whole cluster. Instead, routing two vessels originating from the central ports and without any intermediate hub (case C2) can better allocate island service with a significant impact on passenger cumulative time but not on the traveling distance. The introduction of a hub line along with a mainline from the central port (Case C3) leads to observably improved solutions (compared to those of C1) in terms of both decision parameters. Finally, the employment of additional vessels (case C4) can improve the effectiveness but not in all regards. In particular, the total passenger time is reduced but not the total distance traveled (compared to C3). In fact, if the goal is distance minimization, this can be equally well facilitated by a two- or three-vessel configuration. The decision, though on the appropriate number of vessels, does not solely rest on the optimization result but is also related to constraints regarding ship availability and capacity in conjunction with the travel demand. To obtain a wider picture of the solution space between the two main decision parameters (i.e., total distance traveled D and cumulative passenger hours PH), Figure 14 presents the point clouds of all solutions from the execution of the algorithm at different levels of parameter weighting. Solving a particular weighting case study leads to a footprint-like  
Figure 14: Pareto fronts for different case scenarios. Figure 13: Pareto front for scenario C4. point cloud between D and PH, indicating a roughly linear relationship between them. As the weights \(\mathrm{w_{1}}\) and \(\mathrm{w_{2}}\) of Equation (1) are altered (after scalarizing the two parameters), the produced point clouds are moved to an up-left and down-right direction (Figure 15a). A full-range Pareto front is formed based on individual points corresponding to different decision parameter weighting (Figure 15b). Regarding computational requirements, the time needed for algorithm convergence increases with the magnitude and complexity of the problem. Figure 16 depicts the typical convergence curves for the four cases examined (C1 to C4). In general, the problem complexity increases with the number of vessels and the potential hub nodes. This is why the convergence speed drops from the simplest case (C1) to the most demanding one (C4). Accordingly, the convergence time in a PC with common characteristics (Intel(r) Core(tm) i7-7500, CPU 2.90 GHz, RAM 8.00 Gb) progressively increases and ranges from about a minute (case C1) up to seven minutes for the most demanding case (C4). Figure 16: Convergence curves for optimizing traveling distance in cases C1 to C4. Figure 10.0 Traveled distance–cumulative passenger-hours point cloud across different runs: (**a**) full set of values; (**b**) subset of best values embodied in the circle of (**a**) (different colors correspond to different values of \(\mathrm{w_{1}}\) and \(\mathrm{w_{2}}\) of the objective function).