## 3 Examples
### Cross-Section under Flexure
The speed-up and efficiency are shown in Figures 8 and 9. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline  & \(\varphi^{\prime}y=\varphi_{y}=\varphi^{\prime}x=\varphi_{x}\) & \(\varphi^{\prime}y=\varphi_{y}\) & \\  & \(n^{\prime}y=n_{y}\) & \(\varphi^{\prime}x=\varphi_{x}\) & \\  & \(n^{\prime}x=n_{x}\) & \(n^{\prime}y=n_{y}\) & \\ \hline  & [PERSON] et al. [35] & **MPFA** & **MPFA** & **MPFA** \\ \hline \(z\)(mm) & - & 222.7 & 203.2 & 242.1 \\ \(\theta\)(rad) & - & 1.1585 & 1.2390 & 1.1597 \\ \(b\)(mm) & - & 400 & 400 & 400 \\ \(h\)(mm) & - & 700 & 700 & 700 \\ \(\varphi^{\prime}y\)(mm) & - & 16 & 12 & 10 \\ \(n^{\prime}y\) & - & 9 & 9 & 4 \\ \(\varphi_{y}\)(mm) & - & 16 & 12 & 16 \\ \(n_{y}\) & - & 9 & 9 & 10 \\ \(\varphi^{\prime}x\)(mm) & - & 16 & 25 & 12 \\ \(n^{\prime}x\) & - & 4 & 3 & 3 \\ \(\varphi_{x}\)(mm) & - & 16 & 25 & 16 \\ \(n_{x}\) & - & 4 & 3 & 5 \\ \hline \(A_{st}\)(mm\({}^{2}\)) & - & 5217.6 & 4978.7 & 3669.4 \\ Relative \(A_{st}\) (\%) & - & 100.0 & 95.4 & 70.3 \\ \hline Cost (\(\ell\)/m) & - & 109.6 & 107.4 & 95.2 \\ Relative cost (\%) & - & 100.0 & 98.0 & 86.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal design solutions with invariant section. ACI318. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline  & \(\varphi^{\prime}y=\varphi_{y}=\varphi^{\prime}x=\varphi_{x}\) & \(\varphi^{\prime}y=\varphi_{y}\) & \\  & \(n^{\prime}y=n_{y}\) & \(\varphi^{\prime}y=\varphi_{x}=\varphi_{x}\) & \\  & \(n^{\prime}y=n_{y}\) & \(n^{\prime}y=n_{y}\) & \\  & \(n^{\prime}x=n_{x}\) & \(n^{\prime}x=n_{x}\) & \\ \hline  & [PERSON] et al. [35] & **MPFA** & **MPFA** & **MPFA** \\ \hline \(z\)(mm) & - & 251.8 & 243.7 & 264.6 \\ \(\theta\)(rad) & - & 0.8095 & 0.9426 & 0.8786 \\ \(b\)(mm) & - & 500 & 485 & 485 \\ \(h\)(mm) & - & 565 & 605 & 615 \\ \(\varphi^{\prime}y\)(mm) & - & 14 & 12 & 10 \\ \(n^{\prime}y  
As in the previous example, the parallelization produces satisfactory results (Figure 8), although the efficiency quickly drops with more than four processors (Figure 9). The mean value, accuracy (reference value of Cost = 90.9, from Table 8), and robustness (standard deviation) from 50 runs considering ACI318 standard and using MPFA with a  
Figure 8: Cross-section under biaxial bending. Speed-up. Figure 9: Cross-section under biaxial bending. Efficiency. The method is able to obtain designs close to the global optimum as shown in Tables 9-11. When the method considers only GSRD (Table 10), the minimum average value of the mean costs, the maximum average value of the accuracies, and the minimum average value of the standard deviations are obtained. Very similar values, but somewhat higher, are obtained when LSRD and GSRD (Table 11) are taken into account together. The performance metrics are also shown in Figures 10-12. It can be observed that the metrics tend to worsen in all cases when the number of subpopulations increases. This tendency is greater when LSRD is used alone and above six subpopulations. The other cases have a lower tendency. The sensitivity of the metrics with respect to the number of subpopulations is reduced when LSRD and GSRD are taken into account at the same time.