## 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  
For the purpose of the EE-TGA evaluation, we tested the algorithm on ten standard benchmark functions. The details of the functions and their characteristics are described in Table 3 and Table 4, respectively. The parameters of the EE-TGA were set up according to the best results of the parameter tuning in the paper [39] to make an unbiased comparison. The maximum iteration was set to 250 for the problems with a large dimension. In contrast, for the problems with a small dimension, the maximum iteration was set to 50, and we obtained the test results from 30 independent trials of the algorithms. The population size \(N\) was set to 100, and the subpopulations \(N_{1}\), \(N_{2}\), and \(N_{4}\) are set to 20, 20, and 30, respectively. The control parameter \(\lambda\) was set to 0.1, and \(\theta\) had a value of 0.8. Specific control parameters of the EE-TGA were set as shown in Table 1.  
The comparison between the original TGA and the EE-TGA is given in Table 5, where the results of the TGA algorithm were taken from the original TGA paper [39]. We took the best, mean, and standard deviation as performance metrics. In the presented table, the bold values indicated better results. We note that we have also performed tests with the basic TGA and that we got the same results as in [39]. \begin{table}
\begin{tabular}{c l l} \hline \hline
**ID** & **Function Name** & **Function Definition** \\ \hline F1 & Ackley Function & \(f(x)=-20e^{(-0.2\sqrt{\frac{1}{2}\sum_{i=1}x_{i}^{2}})}-e^{(\frac{1}{2}\sum_{ i=1}\cos(2\pi x_{i}))}+20+\epsilon^{(1)}\) \\ F2 & Becker–Lago Function & \(f(x)=(|x_{1}|-5)^{2}+(|x_{2}|-5)^{2}\) \\ F3 & Branin Function & \(f(x)=\left(x_{2}-\frac{5.1x_{i}^{2}}{4\pi^{2}}+\frac{5x_{i}}{\pi}-6\right)^{2} +10\left(1-\frac{1}{8\pi}\right)\cos(x_{1})+10\) \\ F4 & Dekker–Aarts Function & \(f(X)=10^{5}x_{1}^{2}+x_{2}^{2}-\left(x_{1}^{2}+x_{2}^{2}\right)^{2}+10^{-5} \left(x_{1}^{2}+x_{2}^{2}\right)^{4}\) \\ F5 & Rastrigin Function & \(f(x)=\sum_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})+10]\) \\ F6 & Cosine Mixture Function & \(f(x)=0.1\sum_{i=1}^{n}\cos\left(5\pi x_{i}\right)-\sum_{i=1}^{n}x_{i}^{2}\) \\ F7 & Gulf Research Problem & \(f(X)=\sum_{j=1}^{m}\left(e^{-\frac{\left|y_{j}-x_{2}\right|^{x_{3}}}{x_{1}}-t_ {j}}\right)\), \(m=99\) \\  & where \(t_{j}=\frac{j}{100}\), \(y_{j}=25+\sqrt{\left[-50\log\left(t_{j}\right)\right]^{2}}\) \\ F8 & Modified Rosenbrock Function & \(f(X)=100\left(x_{2}-x_{1}^{2}\right)^{2}+\left[6.4\left(x_{2}-0.5\right)^{2}- x_{1}-0.6\right]^{2}\) \\ F9 & Rosenbrock Function & \(f(x)=\sum_{i=1}^{n-1}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]\) \\ F10 & Schwefel 2.26 Function & \(f(x)=\sum_{i=1}^{n}[-x_{i}\sin(\sqrt{|x_{i}|})]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Benchmark function details (EE-TGA simulations). From the results presented in Table 5, it can be clearly seen that EE-TGA obtained better solution quality in most tests by improving the exploitation of the original TGA. Since the most significant performance difference between EE-TGA and the original TGA could be noticed in the \(F9\)benchmark, in Figure 2, we show the convergence speed graph for this problem of the best generated solutions. In the case of this test, the mean values generated by the EE-TGA were much better than the value obtained in the best run by the original TGA.