## 4 Examples
### Three-Bay Two-Story Framework
#### 4.1.1 Parameter Tuning in PFAM for the Three-Bay Two-Story Framework  
To study the performance of PFAM, a simple sensitivity analysis was performed considering different values for a pair of parameters, for example:  
\[\pi_{1,z_{1}};\ \forall z_{1}\in \begin{bmatrix}1&\dots&D_{1}\end{bmatrix}  \tag{64}\]  
\[\pi_{2,z_{2}};\ \forall z_{2}\in \begin{bmatrix}1&\dots&D_{2}\end{bmatrix}  \tag{65}\]  
maintaining the rest of the parameters \(\pi_{3},\ \pi_{4},\ \dots\) with constant values. With each pair of concrete values \(\left(\pi_{1,z_{1}}\quad\pi_{2,z_{2}}\right)\), PFAM was executed a sequence of times with randomly generated initial independent populations. After these independent executions of the algorithm, a measure of its performance was obtained using two metrics [12]:  
1. _Accuracy_, calculated as the ratio between the cost of the optimal design solution (Table 3) and the mean value of the cost obtained from the independent executions, which measures the efficiency of the algorithm; and
2. _Standard Deviation_ of the cost obtained from the independent executions, which measures the dispersion or robustness of the results obtained using the algorithm. Once the parameters with the best metrics were obtained (for example, \(\left(\pi_{1,best}\quad\pi_{2,best}\right)\)), they were selected and became part of the group of parameters that were maintained at a constant value, and the process was repeated, taking into account different values of another pair of parameters (for instance, \(\left(\pi_{3}\quad\pi_{4}\right)\)). With EEP2 semi-rigid connections and the 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. As this was the first sensitivity calculation performed, extensive ranges of both parameters were considered. There were 50 independent runs. The accuracy results are shown in Table 4, and the standard deviation results are shown in Table 5.  
In Table 4, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation (below 1.5) in Table 5. The shaded areas in both tables correspond to the lowest values of efficiency and robustness. The values of  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{3}{c}{**Cabrero and Bayo**} & \multirow{2}{*}{**[PERSON] et al. [33]**} & \multirow{2}{*}{**PFAM**} & \multirow{2}{*}{**\multirow{2}{*}{** \begin{tabular}{} \end{tabular} **}} & \multirow{2}{*}{** \begin{tabular}{} \end{tabular} **} & \multirow{2}{*}{**\ \begin{tabular}{} \end{tabular} **} & \multirow{2}{*}{** \begin{tabular}{} \end{tabular} **} \\ \cline{2-2} \cline{6-9}  & **Rigid** & **Semi-rigid** & **Pinned** & **EEP1** & **EEP2** & **Rigid** & **EEP1** & **EEP2** \\ \hline \(X_{1}\) & HEB140 & HEB140 & HEB120 & HEB140 & HEB140 & HEB180 & HEB120 & HEB120 \\ \(X_{2}\) & HEB160 & HEB160 & HEB140 & HEB140 & HEB140 & HEB140 & HEB160 & HEB160 \\ \(X_{3}\) & IPE270 & IPE270 & IPE330 & IPE240 & IPE240 & IPE240 & IPE300 & IPE300 \\ \(X_{4}\) & IPE200 & IPE200 & IPE240 & IPE200 & IPE180 & IPE180 & IPE240 & IPE240 \\ \(X_{5}\) (\(d_{p}\)) & - & 22 & - & & & - & 16 & 16 \\ \(X_{6}\) (\(t_{ep}\)) & - & 15 & - & & & - & 8 & 15 \\ \(X_{7}\) (\(d_{b}\)) & - & 22 & - & & & - & 20 & 20 \\ \(X_{8}\) (\(t_{ep}\)) & - & 14 & - & & & -  
parameters \(nt=250\) and \(k_{max}=100\) gave good results and had a low computational cost, so they were chosen for the following sensitivity calculation. With EEP2 semi-rigid connections and the constant values \(\beta_{0}=1\), \(nt=250\), \(k_{max}=100\), \(\gamma_{0}=10\), and \(\alpha_{0}=1\), the values of the \(t_{c}\) and \(m\) parameters were modified. As was carried out previously with \(nt\) and \(k_{max}\), and because it was the first sensitivity calculation performed with parameters \(t_{c}\) and \(m\), wide ranges of both parameters were considered. There were 50 independent runs.