## 6 Results
### Problem 1  
The results in terms of experimental dispersion versus analytical dispersion and of iteration versus accuracy are reported in Figures 5 and 6 for the first case study. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Population Size (P)** & **Selection Rate (SR)** & **Crossover Rate** & **Mutation Rate** \\
**[-]** & **[\%]** & **(CR)** & **(MR)** \\  & & **[\%]** & **[\%]** \\ \hline
1000 & 30 & 50 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Setting of the genetic operators. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Implementation** & \(\delta\) & \(\beta\) \\ \hline
1 & & 0.2 \\
2 & & 0.5 \\
3 & & 1 \\
4 & 1 & 2 \\
5 & & 4 \\
6 & & 10 \\
7 & & 50 \\
8 & 0 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Setting of the parameters \(\delta\) and \(\beta\). It can be observed that, for each level of complexity, the absolute value of accuracy grows while the value of \(\beta\) increases. For example, in the case of \(\beta=0.2\) the maximum accuracy achieved is 0.30, while for \(\beta=50\) an accuracy equal to 0.95 is reached. This result can be explained by an analysis of Equation (10), where the parameter \(\beta\) represents the exponent of the ratio between the analytical dispersion and the experimental mean value. Since this ratio is generally less than one in the case of quite homogeneous datasets, for increasing values of the exponent \(\beta\), the value of the function \(f\) (\(\sigma_{Ylim}\)) will tend towards zero. In other words, the analytical variability will tend to become negligible; so for \(\beta\) tending towards infinity, the accuracy \({R_{N}}^{2}\) will assume the same value as the classical accuracy defined as a function of just the experimental dispersion. Figure 5: Problem 1. Experimental dispersion versus analytical dispersion and iteration versus accuracy for \(m\) in the range 1 to 4 and for: \(\delta=1\) and \(\beta=0.2\), 0.5, 1, 2, 7, 50. The above findings are confirmed by the results obtained and represented in Figures 5 and 6. For small values of \(\beta\), the analytical variance assumes values smaller than those corresponding to large values of \(\beta\). At the same time, a completely opposite trend is observed for the experimental variance, that is, it diminishes as \(\beta\) grows. So it is important to underline how the choice of the parameter \(\beta\) can affect the model performance and that, even if the value of \({R_{N}}^{2}\) increases moving from low to high values of \(\beta\), a suitable trade-off between analytical and experimental dispersions already occurs for values of \(\beta\) quite close to 1. From a deeper analysis of the results, it emerges that, for values of \(\beta\leq 1\), the models with reduced complexity present a higher analytical dispersion than the ones with increasing complexity. Conversely, as the model complexity increases, the experimental dispersion also grows. This effect is more marked for the model with \(m=1\), whose analytical dispersion is significantly higher than that of the models with more terms (\(m>1\)). In the cases with \(\beta>1\), there is a better uniformity between the analytical and experimental dispersions, for different levels of complexity. In the following models, the unit for \(V_{\text{opt}}\) is [kN] while all the mechanical and geometrical quantities are expressed in [N] and [mm], respectively. By considering only the experimental dispersion (\(\delta=0\)), it can be observed that there is no substantial difference in terms of dispersion.