## 3 Results and Discussion
### Genetic Algorithm-Based Optimization  
The aim of this step is to find the best regression model able to fit the experimental data among the infinite possible combinations of input parameters and their powers in terms of the number of cycles until failure (\(N_{GA}\)), as reported in Equation (1). A preliminary investigation aimed at identifying the best number of terms for the optimized regression model. To this end, the number of terms was increased starting from two, i.e., one term plus the constant one, to the number of terms after which any more decrease in the fitness value is obtained. Additionally, three different discrete sets of real values representing all the possible exponents were investigated (see Table 6). In this way, the explored space is discrete and contains the number of exponents to the power of \(N_{T}\)-\(N_{V}\) possible models, where \(N_{V}\) is the number of variables constituting each term (2 in this case). Moreover, in order to evaluate the capability of the optimized models in representing the experimental findings, they were compared with the regression model typically used for statistical investigations, as in the case of two-way ANOVA test, i.e., considering the effect of each single \begin{table}
\begin{tabular}{c c} \hline
**Set** & **Exponents** \\ \hline \(Q_{1}\) & \(\{0,1,2\}\) \\ \(Q_{2}\) & \(\{-2,-1,0,1,2\}\) \\ \(Q_{3}\) & \(\{-2,-1,-0.5,0,0.5,1,2\}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Discrete set of real values representing the possible exponents. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(c\) & \(Ss\), mm/s & \(P\) & \(N\) \\ \hline \(1\) & 18 & 150 & 114,754 \(\pm\) 20,695 \\ \(2\) & 18 & 200 & 224,327 \(\pm\) 39,017 \\ \(3\) & 18 & 250 & 146,136 \(\pm\) 18,011 \\ \(4\) & 20 & 150 & 421,982 \(\pm\) 34,238 \\ \(5\) & 20 & 200 & 364,860 \(\pm\) 33,197 \\ \(6\) & 20 & 250 & 170949 \(\pm\) 16,390 \\ U & - & - & 92,608 \(\pm\) 10,698 \\ \hline \end{tabular}
\end{table} 
Table 5: Combination of parameters (\(c\)) used in the optimization process. The ’U’ label refers to the untreated samples. parameter and their combinations [44]. In this case, the set of possible exponents \(Q_{R}\) can be defined as \(\{0,1\}\), while the statistical regression model is described by the following equation:  
\[N_{R}=k_{0}+k_{1}\cdot Ss+k_{2}\cdot P+k_{3}\cdot Ss\cdot P. \tag{4}\]  
Figure 10 shows the results obtained during the preliminary investigation. In particular, among the different combinations of numbers of terms and sets of exponents, the worst one, i.e., the one with the highest fitness value, is the model with only two terms, as expected. While the best models were obtained with four terms, adding more terms did not lead to a decrease in the fitness value. It is important to note that, for the regression model, the number of terms is set at four, which is the number of possible combinations. The resulting optimized models are described by Equations (5)-(7), while Table 7 reports the relative empirical coefficients, evaluated by means of standard linear regression. \[N_{GA}^{1}=k_{0}^{1}+k_{1}^{1}\cdot P+k_{2}^{1}\cdot Ss^{2}+k_{3}^{1}\cdot Ss ^{2}\cdot P^{2} \tag{5}\]  
\[N_{GA}^{2}=k_{0}^{2}+k_{1}^{2}\cdot\frac{1}{Ss\cdot P^{2}}+k_{2}^{2}\cdot\frac {Ss}{p}+k_{3}^{2}\cdot Ss^{2}\cdot P \tag{6}\]  
\[N_{GA}^{3}=k_{0}^{3}+k_{1}^{3}\cdot\frac{1}{Ss\cdot P}+k_{2}^{3}\cdot\frac{Ss }{\sqrt{P}}+k_{3}^{3}\cdot Ss\cdot\sqrt{P} \tag{7}\]  
As shown in Figure 10, the statistical model is characterized by a fitness value greater, yet more negative, than the optimal value obtained with the genetic algorithm, i.e., \(-0.1052\) against \(-0.0622\). In order to better clarify this result, Figure 11 shows the root mean square error and the standard deviation of the regression models compared with the experimental findings for each combination of parameters (see Table 5).