## 4 Results and Discussions
### Relaxed Crossover  
The crossover used so far has equal weights as in Equation (2) and is compared to the random weight crossover of Equation (3). Having random weights is a form of relaxation in the sense that small and large weights make the crossover result similar to one from the parent cloud. The algorithms with equal and random weight crossovers are denoted WBGEA_1t and WBGEA_1t_rc, respectively.  
The left plots of Figures 18-20 show that the random weight crossover is more competitive than the equal weight crossover for all the considered test functions, (but on \(F_{45}\), see Table 3).  
The superiority of the random weight crossover is related to its ability to produce clouds close to one from their parent when the weight \(\epsilon\) is close to 0 or 1. This allows us to not always destroy the previously selected designs. Such an explanation follows the same line as the mutation independence principle mentioned in Section 4.1.1: because the evolutionary operators may not always lead to better solutions, in particular when applied to already selected high-performing solutions, it is useful to be able to, sometimes, diminish their effect due to randomness.  
Figure 19: Relaxed Crossover. 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.  
Figure 20: Relaxed Crossover. The test function is the MinDist function, \(F_{inDist}\). (**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 evolution of the population diversities of WBGEA_1t_rc and WBGEA_1t on the right of Figures 18-20 shows a faster decrease in diversity with the random weight crossover in the wind-farm case, which may be explained by a faster convergence to good solutions. No clear difference in diversity evolution is observed for functions \(F_{inert}\) and \(F_{minDist}\).  
Absence of CrossoverVariants of the algorithms where no crossover takes place are tested under the names WBGEA_1t_nc and Ref_alg_nc, i.e., they are equipped solely with mutations. We observe on the left of Figures 18-20 that the absence of crossover improves the performances of both algorithms, WBGEA_1t and Ref_alg, on all the test functions, with the exception of Ref_alg on \(F_{inert}\). The single Wasserstein mutation (WBGEA_1t_nc) makes the best algorithm for the wind-farm test function. The effect of the crossover on the diversity, as reported on the right of Figures 18-20, is different between the uniform crossover on the sequence encoding and the Wasserstein-based crossover. The uniform crossover, through its point shuffling effect, is a provider of diversity. Removing it from the algorithm decreases diversity. The Wasserstein-based crossover replaces pairs of clouds by their barycenter, therefore reducing diversity. Removing it from the algorithm increases diversity, which is visible after about 100 iterations in the wind-farm case. Often, this effect is not as clear with \(F_{inert}\), presumably because removing the crossover from WBGEA_1t makes it converge to the optimum, which contributes to reducing diversity.