# Multi-Objective Ship Route Optimisation Using Estimation of Distribution Algorithm
## 5 Results and Discussion
### Pareto Front Evolution (The Criterion Space Perspective)  
The Pareto front for [_Probl_1] in consecutive steps of the algorithm is shown in Figure 8. The colour used illustrates the quality of the front measured by the adapted Hypervolume (HV) metric [52]. With subsequent steps of the algorithm, it can be observed that the generated Pareto front is more and more numerous and its quality, measured by the value of the HV metric, is higher. At the same time, it can be noticed that from the very beginning of the algorithm's operation, the front is very uniform and smooth, and there are no solutions that deviate significantly from it. Figure 9 shows the same process but for [_Prob2_].  
Figure 8: [_Probl_1]: Pareto front (the criterion space) in consecutive steps of the algorithm. The colour of the Pareto front illustrates the value of the adapted Hypervolume (HV) metric. (**top left**) step is 1, Pareto set size is 50, and HV is 0.0. (**top right**) step is 30, Pareto set size is 267, and HV is 0.61. (**bottom left**) step is 100, Pareto set size is 438, and HV is 0.82. (**bottom right**) step is 200, Pareto set size is 512, and HV is 1.0.  
Figure 7: [_Probl_2]: Pareto set (the design space) in consecutive steps of the algorithm. (**top left**) step is 1, Pareto set size is 59. (**top right**) step is 30, Pareto set size is 458. (**bottom left**) step is 100, Pareto set size is 772. (**bottom right**) step is 200, Pareto set size is 907. The colour illustrates the relative speed (in m/s) of the ship.  
Figure 10 shows the Pareto front size and quality for [_Prob1_], measured by the adapted HV metric value, in consecutive steps of the algorithm. The graphs present two opposite views of the same process: the left graph emphasises the Pareto front size (with the colour representing HV metric value), whilst the right one highlights the HV metric value (the colour illustrates the front size). As the algorithm proceeds, the solutions on the Pareto front become more numerous and cover the front better and better. Because of this, the quality of the Pareto front measured by the HV metric increases. These results confirm the observations from Figure 8. At the same time, it should be emphasised that the algorithm maintains a high diversity of solutions on the Pareto front, which contributes to obtaining very good results, confirmed by the high value of the HV metric.  
Figure 11 shows the Pareto front size and quality for [_Prob2_]. As in the case of [_Prob1_] (Figure 10), in the consecutive steps of the algorithm, the solutions on the Pareto front become more numerous and cover the front better and better.  
Figure 10: [_Prob1_]: Pareto front size and quality (measured by the adapted HV metric value) in consecutive steps of the algorithm. (**left**) Pareto front size in consecutive steps, and the colour illustrates the HV metric value. (**right**) HV metric value, and the colour illustrates the Pareto front size.  
Figure 9: [_Prob2_]: Pareto front (the criterion space) in consecutive steps of the algorithm. The colour of Pareto front illustrates the value of adapted Hypervolume (HV) metric. (**top left**) step is 1, Pareto set size is 59, and HV is 0.0. (**top right**) step is 30, Pareto set size is 458, and HV is 0.75. (**bottom left**) step is 100, Pareto set size is 772, and HV is 0.96. (**bottom right**) step is 200, Pareto set size is 907, and HV is 1.0.