## 4 Examples
### Analysis and Discussion of the Results
In Table 47, base \(be\) gives the same TNPD previously obtained when it is raised to the number of variables in each case. Thus, when calculating the logarithms, a single value, which was the average of the four values of \(be\) included in Table 47 and is equal to 17.554979964, was used for the base of \(be\). Table 48 includes the minimum populations \(nt\) (\(minpop\)) that yielded the most precise results with minimal computational cost with \(k_{max}=100\) for each case (see Tables 4, 5, 16, 17, 31, 32, 38 and 39). Likewise, the TNPD logarithms have been included for each case using the same base \(be=17.554979964\). The third column of Table 48 shows that the logarithm calculated in each case was close to the number of variables \(nv\) considered (4, 12, 20, and 40). This is evident from looking at the calculations performed. Figure 33 shows \(minpop\) on the ordinate axis and \(nv\) on the abscissa axis. Likewise, the Least-Square Reg \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **TNPD** & **log\({}_{17.554979964}\)(TNPD)** & \(nv\) & _Minpop_ \\ \hline Three-bay two-story framework with & 186,624 & 4.24 & 4 & 30 \\ rigid connections & 3.86983526 \(\cdot\) 10\({}^{13}\) & 10.92 & 12 & 250 \\ \hline Three-bay ten-story framework with & 2.26379694 \(\cdot\) 10\({}^{26}\) & 21.18 & 20 & 450 \\ rigid connections & 1.40168340 \(\cdot\) 10\({}^{47}\) & 37.89 & 40 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 48: Calculation of logarithms over TNPD, \(nv\), and \(minpop\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **TNPD** & \(be\) \\ \hline Three-bay two-story framework with & \(\begin{array}{c}(ne=18)\\ =18^{2}\cdot 24^{2}=186,624\approx 20.784609691^{4}=\\ =(be=20.784609691)\end{array}\) & 20.784609691 \\ \hline Three-bay two-story framework with & \(\begin{array}{c}18^{2}\cdot 24^{2}\cdot 10^{4}\cdot 12^{4}=\\ =3.86983526\cdot 10^{13}\approx\\ \approx 13.561499319^{12}\end{array}\) & 13.561499319 \\ \hline Three-bay ten-story framework with & \(\begin{array}{c}18^{10}\cdot 24^{10}=\\ =2.26379694\cdot 10^{26}\approx\\ \approx 20.784609691^{20}\end{array}\) & 20.784609691 \\ \hline Three-bay two-story framework with & \(\begin{array}{c}18^{10}\cdot 24^{10}\cdot 10^{10}\cdot 12^{10}=\\ =1.40168340\cdot 10^{47}\approx\\ \approx 15.089201156^{40}\end{array}\) & 15.089201156 \\ \hline \hline \end{tabular}
\end{table} 
Table 47: TNPD and \(be\). Figure 33 shows the good fit obtained (R\({}^{2}\) = 0.9997496). The calculations carried out using logarithms remind us of the concept of entropy \(S\), which is a measure of the disorder or randomness of the system. In certain fields of knowledge, when information \(I\) is extracted from a system, entropy \(S\) is reduced [37; 38; 39]. Thus, the entropy \(S\) of a design space with volume TNPD, and the information \(I\) that PFAM and PCPFAM extract from it, are related (\(S\geq S-I\geq 0\)). In this context, _minpop_ could be defined as the minimum or sufficient number of fireflies that PFAM or PCPFAM requires in the design process to provide enough information for a valid design. Since an algorithm can be defined as an information management tool, if the information _minpop_ that the algorithm works with is sufficient, the difference \(S-I\) is a small enough value to obtain good metrics. Finally, using the formula of the Least-Square Regression Line included in Figure 33, a simple expression resolving the uncertainty in the choice of \(nt\) is proposed. The estimated minimum population _estimpop_, as a function of \(nv\), can be obtained using the following expression:  
\[estimpop=3^{3}.(nv-3);\ \forall\ nv>3 \tag{66}\]  
Expression (66), applied to values of \(nv\) = 4, 12, 20, and 40, provides the values of _estimpop_ included in Figure 34. In addition, Figure 34 includes values of _minpop_ and the Least-Square Regression Line.