## 4 Examples
### Three-Bay Two-Story Framework
#### 4.1.1 Parameter Tuning in PFAM for the Three-Bay Two-Story Framework
From the sensitivity analyses performed in this example, PFAM initially seems efficient and robust for the parameter values in Table 15. Nevertheless, these values were checked in another problem to study PFAM's behavior and establish another level of robustness, namely that of the algorithm against changes in the problem. Furthermore, parameters \(nt\) and \(k_{max}\) in Table 15 clearly depend on the problem. This is because each problem has a different number of variables, which affects the size of the design space. If the design space is large, sufficiently high \(nt\) values should be considered to ensure that the fireflies can capture enough information in the exploration phase. Likewise, sufficiently high \(k_{max}\) values must be taken into account so the algorithm can evolve from the exploration phase to the exploitation phase. The problem in which the connections between members of the structure are rigid, using only the necessary design variables \(X_{1}\), \(X_{2}\), \(X_{3}\), and \(X_{4}\) (see Tables 2 and 3), has been considered in the same example to illustrate these ideas. With rigid connections and initial constant values \(\beta_{0}=1\), \(m=2\), \(\gamma_{0}=10\), \(\alpha_{0}=1\), and \(t_{c}=0.1\), the values of the \(nt\) and \(k_{max}\) parameters were modified. The accuracy results are shown in Table 16, and the standard deviation results are shown in Table 17. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & \multicolumn{3}{c}{\(k_{max}\)} \\ \cline{3-7}  & & **100** & **200** & **300** & **400** \\ \hline \multirow{6}{*}{\(nt\)} & 20 & 0.285222 & 0.247785 & 0.261767 & 0.259278 \\ \cline{2-7}  & 30 & **1.000000** & 0.963353 & **1.000000** & **1.000000** \\ \cline{2-7}  & 40 & 0.973976 & 0.981145 & 0.974113 & 0.958629 \\ \cline{2-7}  & 50 & 0.978822 & 0.973921 & 0.900450 & 0.977659 \\ \cline{2-7}  & 60 & **1.000000** & **1.000000** & 0.903428 & **1.000000** \\ \cline{2-7}  & 70 & **1.000000** & **1.000000** & 0.978588 & 0.975847 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Three-bay two-story framework with rigid connections. Sensitivity analysis for \(nt\) and \(k_{max}\). Accuracy. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{\(k_{max}\)} \\ \cline{3-6}  & & **100** & **200** & **300** & **400** \\ \hline \multirow{6}{*}{\(nt\)} & 20 & 7397.80 & 8252.30 & 7543.02 & 8751.77 \\ \cline{2-7}  & 30 & **0.00** & 1055.17 & **0.00** & **0.00** \\ \cline{2-7}  & 40 & 741.13 & 533.04 & 737.13 & 1197.06 \\ \cline{2-7}  & 50 & 600.14 & 742.73 & 1770.93 & 633.85 \\ \cline{2-7}  & 60 & **0.00** & **0.00** & 1848.24 & **0.00** \\ \cline{2-7}  & 70 & **0.00** & **0.00** & 606.91 & 473.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Three-bay two-story framework with rigid connections. Sensitivity analysis for \(nt\) and \(k_{max}\). Standard deviation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{\(k_{max}\)} \\ \cline{3-6}  & & **100** & **200** & **300** & **400** \\ \hline \multirow{6}{*}{\(nt\)} & 20 & 7397.80 & 8252.30 & 7543.02 & 8751.77 \\ \cline{2-7}  & 30 & **0.00** & 1055.17 & **0.00** & **0.00** \\ \cline{2-7}  & 40 & 741.13 & 533.04 & 737.13 & 1197.06 \\ \cline{2-7}  & 50 & 600.14 & 742.73 & 1770.93 & 633.85 \\ \cline{2-7}  & 60 & **0.00** & **0.00** & 1848.24 & **0.00** \\ \cline{2-7}  & 70 & **0.00** & **0.00** & 606.91 & 473.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Three-bay two-story framework with EEP2 semi-rigid connections. Optimum values of PFAM parameters. In Table 16, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 17. The shaded areas in both figures correspond to the lowest efficiency (accuracy below 0.9) and robustness (standard deviation above 1000) values. The values of parameters \(nt=30\) and \(k_{max}=100\) gave good results and had a low computational cost, so they were chosen for the following sensitivity calculation.