# Multi-Objective Ship Route Optimisation Using Estimation of Distribution Algorithm
## 5 Results and Discussion
### Pareto Front Quality: The Impact of the Local Search  
To quantify the impact of the local search on the overall performance of the optimisation algorithm, the numerical experiments described in Sections 5.1 and 5.2 were repeated, but this time with the local search turned off. The corresponding results are shown in Figures 12-17.  
The paths forming (in the design space) the Pareto set for [_Prob1_], shown in Figure 12, differ significantly from those obtained from the full algorithm (see Figure 6): they are less numerous and cover different sailing areas (which means less effective use of the tidal stream). The same can be observed in Figures 7 and 13, which correspond to [_Prob2_].  
Figure 11: [_Prob2_]: 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 12: [_Prob1_]: Pareto set (the design space) in consecutive steps of the algorithm without the local search. (**top left**) step is 1, and Pareto set size is 14. (**top right**) step is 30, and Pareto set size is 30. (**bottom left**) step is 100, and Pareto set size is 54. (**bottom right**) step is 200, and Pareto set size is 76. The colour illustrates the relative speed (in m/s) of the ship.  
Figures 14 and 15 present (in the criterion space) the evolution of the Pareto front for [_Prob1_] and [_Prob2_], respectively. Apart from being less numerous (which was mentioned above), the fronts are not as uniform and smooth as in the case of the full algorithm (see Figures 8 and 9).  
Figure 14: [_Prob1_]: Pareto front (the criterion space) in consecutive steps of the algorithm without the local search. The colour of Pareto front illustrates the value of the adapted Hypervolume (HV) metric. (**top left**) step is 1, Pareto set size is 14, and HV is 0.0. (**top right**) step is 30, Pareto set size is 30, and HV is 0.55. (**bottom left**) step is 100, Pareto set size is 54, and HV is 0.78. (**bottom right**) step is 200, Pareto set size is 76, and HV is 1.0.  
Figure 13: [_Prob2_]: Pareto set (the design space) in consecutive steps of the algorithm without the local search. (**top left**) step is 1, and Pareto set size is 10. (**top right**) step is 30, and Pareto set size is 32. (**bottom left**) step is 100, and Pareto set size is 65. (**bottom right**) step is 200, and Pareto set size is 89. The colour illustrates the relative speed (in m/s) of the ship.  
Figures 16 and 17 show the size of the Pareto front and its quality in consecutive steps of the algorithm for [_Prob1_] and [_Prob2_], respectively. As before (see Figures 10 and 11), the front quality is measured by the adapted HV metric value. This perspective gives a more detailed view of the search process--without the local algorithm support, it resembles a typical random search with \"chaotic\" changes in the size of the Pareto front; weaker exploitation of the solution space results in a smaller number of non-dominated solutions found.  
Figure 16: [_Prob1_]: Pareto front size and quality (measured by the adapted HV metric value) in consecutive steps of the algorithm without the local search. (**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 15: [_Prob2_]: Pareto front (the criterion space) in consecutive steps of the algorithm without the local search. The colour of Pareto front illustrates the value of the adapted Hypervolume (HV) metric. (**top left**) step is 1, Pareto set size is 10, and HV is 0.0. (**top right**) step is 30, Pareto set size is 32, and HV is 0.35. (**bottom left**) step is 100, Pareto set size is 65, and HV is 0.75. (**bottom right**) step is 200, Pareto set size is 89, and HV is 1.0.