## 4 Results and Discussions
### Comparison of Wasserstein and Sequence-Based Operators
#### 4.2.2 Wasserstein vs. Classical Evolutionary Operators  
In order to better understand the role of the operators, we now invert the mutations of WBGEA_1t and Ref_alg, and the two new obtained algorithms are denoted by Wass_gauss and Ref_was  
Figure 14: Wasserstein-Based vs. Reference Evolutionary Algorithm. The test function is the MinDist function, \(F_{minDist}\). **(Left)**: mean \(\pm\) std. deviation of the best so far solution. (Right): mean \(\pm\) std. deviation of the population diversity. Statistics calculated from 20 independent runs.  
Figure 13: Wasserstein vs. Classical Evolutionary Operators. The test function is the wind-farm layout problem averaged over 4 directions, \(F_{4d}\). **(Left)**: mean \(\pm\) std. deviation of the best so far solution. (Right): mean \(\pm\) std. deviation of the population diversity. Statistics calculated from 20 independent runs.  
Figure 15: Wasserstein-Based vs. Reference Evolutionary Algorithm. The test function is the inertia function, \(F_{inert}\). **(Left)**: mean \(\pm\) std. deviation of the best so far solution. **(Right)**: mean \(\pm\) std. deviation of the population diversity. Statistics calculated from 20 independent runs.  
The left plots of Figures 13-17 illustrate that in general, the Wasserstein crossover (Equation (2)) outperforms the uniform crossover on sequences (Algorithm 5) when the mutation is the same. Compare Ref_alg to Wass_gauss, which both have a Gaussian mutation, and Ref_wass to WBGEA_1t, which both have an a Wasserstein mutation: the versions with the Wasserstein crossover perform best on all the test functions except \(F_{inert}\). By comparing Ref_wass to Ref_alg, the same graphics demonstrate that, when combined to the uniform crossover, the Wasserstein-based mutation is better than the Gaussian mutation on the wind farm and \(F_{inert}\) tests. The performances of WBGEA_1t and Wass_gauss on the left plots of Figures 13-17 show that the alternating Wasserstein mutation (Algorithm 2) outdoes the Gaussian mutation (Algorithm 6) when the crossover is Wasserstein-based only on the wind-farm problem.  
The right plots of Figures 13-17 illustrate also that the presence of the classical operators, the uniform crossover on sequences and the Gaussian mutation in the algorithms Ref_alg, Ref_wass and Wass_gauss result in large population diversities on all the test functions. On the contrary, WBGEA_1t, which is based uniquely on Wasserstein operators, has vanishing diversities.  
The good results achieved by the algorithms based on Wasserstein operators on the wind farm can be explained by the geometrical properties of their optimal designs: the good solutions for such problems are expected to possess more restrictive geometrical properties, such as regular alignments. On the contrary, the optimal design of \(F_{inert}\) has points grouped at the four corners of the domain, which, as will soon be seen, is very favorable to the uniform crossover on sequences. \(F_{inDist}\), with its good designs made of points spread over the domain, has an intermediate status made of some alignments for the highest performance solutions but is otherwise sensitive to point local positions.