## 3 Simulation and Results
### Simulation Results on Fixed Dimensional Multi Modal Test Function \(F_{14}\) to \(F_{23}\)  
The third group of objective functions, including F\({}_{14}\) to F\({}_{23}\), was selected from the fixed dimensional multimodal type. Complete information on these objective functions is given in Table A3 in Appendix A. This type of objective function was also suitable for evaluating the exploration power of optimization algorithms. The results of optimization of these objective functions using GBGBO and eight other algorithms are presented in Table 4  
Optimization results of the F\({}_{1}\) to F\({}_{23}\) objective functions using the proposed algorithm and eight other optimization algorithms are presented in Tables 2-4. The boxplot of results for each algorithm and objective function are drawn in Figure 2 for further analysis and visual comparison of the performance of the optimization algorithms. Based on the boxplots shown in Figure 2, in the unimodal objective functions type F\({}_{1}\), F\({}_{2}\), F\({}_{3}\), F\({}_{4}\), F\({}_{6}\), and F\({}_{7}\), the superiority of the GBGBO over the other eight algorithms is obvious. The function F\({}_{5}\) of the GWO offers better performance. However, the GBGBO offers acceptable performance with little difference to the GWO. In the functions F\({}_{9}\), F\({}_{10}\), F\({}_{11}\), F\({}_{12}\), and F\({}_{13}\) of the high-dimensional multimodal objective functions type, the GBGBO is the best optimizer among the review algorithms. In the objective function F\({}_{8}\), the GA presented better performance. In the functions F\({}_{14}\), F\({}_{15}\), F\({}_{16}\), F\({}_{18}\), F\({}_{19}\), F\({}_{20}\), F\({}_{21}\), F\({}_{22}\), and F\({}_{23}\) of the fixed-dimensional multi-model objective function type, the proposed algorithm could provide a more efficient quasi-optimal solution with smaller values of standard deviation. The function F\({}_{17}\) in the GWO offers better performance. However, the GBGBO offers acceptable performance with little difference to the GWO.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{E} & \multicolumn{1}{c}{**CA**} & **PSO** & **CSA** & **TBOO** & **CWO** & **COA** & **TSA** & **MFA** & **GGEGO** \\ \hline \multirow{2}{*}{F\({}_{14}\)} & \multirow{2}{*}{_Aure_} & 0.0986 & 2.1758 & 3.9933 & 2.2721 & 3.3748 & 0.998 & 1.9023 & 0.998 & 0.998 \\  & & rad & 1.560\(\times 10^{-15}\) & 7.941\(\times 10^{-16}\) & 7.9481 & 1.908\(\times 10^{-16}\) & 6.465\(\times 10^{-15}\) & 9.036\(\times 10^{-16}\) & 2.658\(\times 10^{-7}\) & 4.275\(\times 10^{-16}\) & 1.44 \(\times 10^{-16}\) \\ \hline \multirow{2}{*}{F\({}_{15}\)} & \multirow{2}{*}{_Aure_} & 5.902\(\times 10^{-2}\) & 0.0535 & 0.0042 & 0.0033 & 0.0003 & 0.0009 & 0.0004 & 0.003 & 0.003 \\  & & rad & 7.201\(\times 10^{-18}\) & 3.4879 \(\times 10^{-19}\) & 2.9002 \(\times 10^{-19}\) & 1.228\(\times 10^{-17}\) & 1.1566\(\times 10^{-18}\) & 3.490\(\times 10^{-18}\) & 9.0425\(\times 10^{-4}\) & 4.0951\(\times 10^{-15}\) & 3.64 \(\times 10^{-20}\) \\ \hline \multirow{2}{*}{F\({}_{16}\)} & \multirow{2}{*}{_Aure_} & -1.0316 & -1.0316 & -1.0316 & -1.0316 & -1.0316 & -1.0316 & -1.0316 & -1.0316 & -1.0316 & -1.0316 \\  & & rad & 7.941\(\times 10^{-16}\) & 3.475\(\times 10^{-16}\) & 3.5980 \(\times 10^{-16}\) & 1.408\(\times 10^{-15}\) & 3.5720 \(\times 10^{-16}\) & 9.501\(\times 10^{-16}\) & 2.6544 \(\times 10^{-16}\) & 9.93 \(\times 10^{-17}\) \\ \hline \multirow{2}{*}{F\({}_{17}\)} & \multirow{2}{*}{_Aure_} & 0.4009 & 0.7584 & 0.3078 & 0.3078 & 0.3078 & 0.4007 & 0.3991 & 0.3079 & 0.3079 \\  & & rad & 4.960\(\times 10^{-17}\) & 4.9600 \(\times 10^{-17}\) & 9.9001 \(\times 10^{-17}\) & 7.460\(\times 10^{-17}\) & 8.6888 \(\times 10^{-17}\) & 2.4625 \(Figure 2: _Cont._