## 4 Results and Discussion  
### Optimization Results for One Degree of Freedom  
The first experiment consists of the optimization of one degree of freedom of the double Y-shaped cavity, represented by the angles \(\alpha\) and \(\beta\). For this first level optimization, all DE versions were configured with \(NP=5\) and \(NG=10\). The results of 30 runs obtained with each of the DE versions for the average \((\theta_{max})_{m}\) and the statistical metrics as variance, standard deviation, and root mean square error (RMSE) are presented in Table 4. For the sake of brevity, only the best two groups and the worst two ones are shown in Table 4, sorted in ascending order. The versions of the DE algorithm which reach the best results are DE12 and DE14. They conducted the lowest values for the average of \((\theta_{max})_{m}\), as well as the lowest magnitudes of standard deviation and RMSE. These algorithms only have in common the mutation operator (_rand/1/bin_) and the parameter \(F=1.0\). The \(CR\) parameter does not have an influence on the results of versions ED12 and ED14. The second group with satisfactory results is the one having versions DE22 and DE24. These versions have as mutation operators the parameter _best/1/bin_ and \(F=1.0\). Again, the difference in \(CR\) parameters does not affect the results of these versions of the DE algorithm. The results of the Table 4 indicated that the DE versions with \(F=1.0\) are more efficient to obtain the lowest mean magnitude of \((\theta_{max})_{m}\). On the other hand, the last two groups in Table 4 conducted the worst performance. The versions DE31, DE33, DE41, and DE43 achieved more divergent values from the \((\theta_{max})_{m}\) average, which indicate that the versions obtained several local minimum magnitudes of \((\theta_{max})_{m}\). These versions consider the mutation parameters _best/2/bin_ and _current-to-best/1/bin_, Equations (16) and (17). For this level of investigation, the mutation operator _best/2/bin_ led to the worst performance in comparison with the others. Shapiro-Wilk (SW) and Kruskal-Wallis (KW) statistical tests were applied to determine the statistically significant difference between the 16 versions of DE algorithm results, as can be seen in Table 5. According to the SW test, all algorithm results have a free distribution that allows non-parametric statistical analysis. The results of the KW test presented in Table 5 confirm a significant difference between the results of the algorithm for the first-level optimization of the double Y-shaped cavity, corroborating the findings of Table 4. The next step in the analysis of the algorithm's results is to identify which versions of DE differ from the other. For this purpose, the Dunn test is performed, and the results are illustrated in Figures 4 and 5. The distribution of each algorithm version is shown through the _BoxPlot_ graph with the color representing the classification groups generated by the Dunn test. Figure 4 shows two groups of performance, A and B, which indicates that the algorithms in the same group have similar statistical performance, while algorithms in different groups have statistical performance differences. Figure 4 indicates differences between DE12 and DE31 versions, but it is impossible to distinguish in statistical viewpoint differences between the algorithms DE12, DE14, DE22, DE24, DE41, and DE43, even with some differences in the achievement of global point of minimum magnitude of \((\theta_{max})_{m}\). \begin{table}
\begin{tabular}{c c c c c} \hline
**DE Versions** & **Average (\(\theta_{max}\))\({}_{m}\)** & **Variance (\(\times 10^{-4}\))** & **Standard Deviation (\(\times 10^{-3}\))** & **RMSE (\(\times 10^{-3}\))** \\ \hline DE12; DE14 & 0.05232 & 0.017 & 1.33 & 1.61 \\ DE22; DE24 & 0.05253 & 0.038 & 1.96 & 2.24 \\ DE41; DE43 & 0.05491 & 0.728 & 8.53 & 9.10 \\ DE31; DE33 & 0.05770 & 1.870 & 13.70 & 14.9 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical metrics of the results achieved for \((\theta_{max})_{m}\).