## 5 Case Study Applications
To reduce this prolonged journey, a constraint is set in case C4d to avoid solutions that exceed a fifteen-hour trip limit. In such a case, the algorithm provides a quite different route planning solution that provides competitive performance values in regard to the main evaluation parameters but with a significant reduction in the maximum trip time. The comparison of the last two cases indicates that the problem holds a large solution space and that quite different routing solutions may lead to comparable fitness values demonstrating the potential of several local optima in such kinds of problems. To gain a better insight into the multi-objective optimization outcomes, several iterations with varying sub-objective weights were conducted for case C3, and the results are depicted in Figure 12. It is interesting to note that in every case, a well-shaped Pareto front is developed with a main characteristic that there is an almost sharp (rather than gradual) adjustment to the ultimately lowest level of each objective parameter. In fact, it appears that each Pareto curve closely fits into two straight lines, indicating the lowest total traveling distance and the lowest cumulative passenger time, respectively. Depending on the hub selection, these lines may be shifted at different levels of performance. For instance, selecting Limnos as the hub island (C3b), despite the efficiency concerning traveling distance, it cannot compete with other solutions in terms of total passenger hours spent on board. The reason is that the island is located well in the northern part of the cluster, and many passengers directed to the south have to make a wide deviation of the straight-line pathway. The automatic hub assignment in C3d has provided comparable results to those of the manual hub selection in the whole parameter range. Figure 10: Optimization results of combined objectives D and PH for case study C3 (one vessel from the central port and one from a hub node) (the colours in the numbers below correspond to those of the lines in the maps). (**a**) C3a: \(\mathrm{D_{1}/D_{2}/D_{tot}}\) = 443/160/603 nm, \(\mathrm{T_{1}/T_{2}/T_{tot}}\) = 17:44/15:38/33:22 h, PH = 16,627 h, \(\mathrm{T_{max-trip}}\) = 20:40 h. (**b**) C3b: \(\mathrm{D_{1}/D_{2}/D_{tot}}\) = 500/85/585 nm, \(\mathrm{T_{1}/T_{2}/T_{tot}}\) = 20:31/15:21/35:52 h, PH = 19,852 h, \(\mathrm{T_{max-trip}}\) = 20:31 h. (**c**) C3c: \(\mathrm{D_{1}/D_{2}/D_{tot}}\) = 486/106/592 nm, \(\mathrm{T_{1}/T_{2}/T_{tot}}\) = 19:30/15:16/34:46 h, PH = 15,438 h, \(\mathrm{T_{max-trip}}\) = 19:30 h. (**d**) C3d: \(\mathrm{D_{1}/D_{2}/D_{tot}}\) =441/170/611 nm, \(\mathrm{T_{1}/T_{2}/T_{tot}}\) =17:30/16:44/34:14 h, PH = 16,894 h, \(\mathrm{T_{max-trip}}\) = 21:45 h.  
In case C4, when minimizing the traveling distance (C4a), the algorithm results in a solution with only two vessels in use and the passengers spending a significant amount of time on board. On the other end, minimizing the total passenger hours (C4b) leads to a rather extended traveling distance plan. Several solutions exist between these two extreme cases that follow the trade-off between the two objectives (Figure 13), which can be developed by adjusting the relative weights in the objective function. A number of solutions almost retain the minimum distance value to the left of the diagram; however, beyond a point, an inclined and rather linear point pattern is developed along the Pareto front describing the relation between traveling distance and passenger hours. The developed solutions in C4c and C4d lie close to the turning point of the Pareto front. It is concluded then that a full weight-related sensitivity analysis is necessary to fully understand the solution's effectiveness. Figure 11: Optimization results for case study C4 (up to two vessels from central ports and one from a hub node) (the colours in the numbers below correspond to those of the lines in the maps).