## 5 Experimental results and discussion
### Performance evaluation of EE-TGA and \(E^{3}\)-FA on standard unconstrained benchmarks
#### 5.1.1 Performance evaluation of EE-TGA
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**ID** & **Modality** & **Dimension** & **Input Domain** & **Global Minimum** \\ \hline F1 & Multimodal & 30 & [\(-\)30,30] & \(f(x^{*})=0\) at \(x^{*}=(0,...,0)\) \\ F2 & Multimodal & 2 & [\(-\)30,30] & \(f(x^{*})=0\) at \(x^{*}=\pm 5\) \\ F3 & Multimodal & 2 & \(-5\leq x_{1}\leq 10\) & \(f(x^{*})=0.39788735772973816\) at \\  & & & \(0\leq x_{2}\leq 15\) & \(x_{i}^{*}\approx(-\pi,12.275),(\pi,2.275),(9.42478,2.475)\) \\ F4 & Multimodal & 2 & [\(-\)20,20] & \(f(x^{*})=-24777\) at \(x_{i}^{*}=(0,\pm 15)\) \\ F5 & Multimodal & 10 & [\(-\)5.12,5.12] & \(f(x^{*})=0\) at \(x^{*}=(0,...,0)\) \\ F6 & Unimodal & 2,4 & [\(-\)1,1] & \(f(x^{*})=n/10\) at \(x^{*}=n\) \\ F7 & Unimodal & 3 & [0,100] & \(f(x^{*})=0\) at \(x^{*}=(50,25,1.5)\) \\ F8 & Unimodal & 2 & [\(-\)5,5] & \(f(x^{*})=0\) at \\  & & & \(x^{*}=\{(0.341307503353524,0.116490811845416),(1,1)\}\) \\ F9 & Unimodal & 10 & [\(-\)30,30] & \(f(x^{*})=0\) at \(x^{*}=(1,...,1)\) \\ F10 & Unimodal & 10 & [\(-\)500,500] & \(f(x^{*})=-418.9829\times n\) at \(x^{*}=(420.9687)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Benchmark function characteristics (EE-TGA simulations). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & \multirow{2}{*}{_f(x*)_} & \multicolumn{4}{c}{**TGA**} & \multicolumn{4}{c}{**EE-TGA**} \\ \cline{2-9}  & & \(Best\) & \(Mean\) & \(StdDev\) & \(Best\) & \(Mean\) & \(StdDev\) \\ \hline F1 & 0 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ F2 & 0 & \(1.07\times 10^{-8}\) & \(3.70\times 10^{-7}\) & \(5.84\times 10^{-7}\) & **0.00** & **3.04 \(\times\) 10\({}^{-9}\)** & **1.47 \(\times\) 10\({}^{-9}\)** \\ F3 & 0.39789 & 0.39789 & 0.39789 & 0.00 & 0.39789 & 0.39789 & 0.00 \\ F4 & -24777 & \(-\)24775.72 & \(-\)24775.66 & 0.0944586327 & \(-\)**24777** & **-24776.97** & **0.0139247** \\ F5 & 0 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ F6 & 0.2 & 0.2 & 0.2 & 0.00 & 0.2 & 0.2 & 0.00 \\ F7 & 0 & \(1.52\times 10^{-5}\) & \(7.05\times 10^{-2}\) & \(2.60\times 10^{-1}\) & **5.36 \(\times\) 10\({}^{-6}\)** & **1.50 \(\times\) 10\({}^{-4}\)** & **5.72 \(\times\) 10\({}^{-2}\)** \\ F8 & 0 & 0.00 & \(2.23\times 10^{-3}\) & \(3.58\times 10^{-3}\) & 0.00 & **0.00** & **0.00** \\ F9 & 0 & 0.56 & 0.8231 & 0.3419 & **0.00** & **0.2159** & **0.1510** \\ F10 & \(-\)418.9829 & \(-\)418.9702 & \(-\)418.9292 & 0.23796 & \(-\)**418.9829** & **-418.9829** & **0.00** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparative analysis of the tree growth algorithm (TGA) and the enhanced tree growth algorithm (EE-TGA). The parameter configuration of \(E^{3}\)-FA was conducted based on the suggested control parameter values in [31]. The simulation of \(E^{3}\)-FA was conducted in 100 independent runs, and the maximum number of iterations was set to 100. The population size was set to 40, \(\alpha\) to 0.2, the light absorption coefficient (\(\gamma\)) to 1, and \(\beta_{0}=1\). The \(E^{3}\)-FA specific parameters were set as shown in Table 1.  
Sample plots of the benchmark functions are depicted in Figure 3.  
The comparison of the FA and \(E^{3}\)-FA is presented in Table 8, where the results of FA were taken from [82]. However, after conducting the simulations of the benchmark functions with FA, we got the same results as in [82]. As in the experiment with EE-TGA, we took the best, mean, and standard deviation as the performance metrics. The bold values in Table 8 refer to the best values.