## 3 Experiment and Application
### Performance Analysis: Classic Benchmark Functions
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{\(F\)} & **Result** & **APO** & **GWO** & **WOA** & **PSO** & **DE** & **Rank** \\ \hline \multirow{4}{*}{\(f_{1}\)} & Best & 7.8190 \(\times 10^{-145}\) & 1.0102 \(\times 10^{-29}\) & 1.9587 \(\times 10^{-80}\) & 9.3358 \(\times 10^{-6}\) & 2.3626 \(\times 10^{-6}\) & \\ \cline{2-7}  & Worst & 6.9690 \(\times 10^{-108}\) & 1.6995 \(\times 10^{-26}\) & 7.1646 \(\times 10^{-63}\) & 0.0029 & 2.1183 \(\times 10^{-4}\) & 1 \\ \cline{2-7}  & Mean & 2.3236 \(\times 10^{-109}\) & 2.1408 \(\times 10^{-27}\) & 2.8808 \(\times 10^{-64}\) & 2.6064 \(\times 10^{-4}\) & 3.3728 \(\times 10^{-5}\) & \\ \cline{2-7}  & Std & 1.2723 \(\times 10^{-108}\) & 4.1907 \(\times 10^{-27}\) & 1.3261 \(\times 10^{-63}\) & 5.4256 \(\times 10^{-4}\) & 4.4013 \(\times 10^{-5}\) & \\ \hline \multirow{4}{*}{\(f_{2}\)} & Best & 5.5304 \(\times 10^{-93}\) & 1.1204 \(\times 10^{-17}\) & 2.0437 \(\times 10^{-56}\) & 0.0036 & 5.1036 \(\times 10^{-4}\) & \\ \cline{2-7}  & Worst & 2.4478 \(\times 10^{-73}\) & 3.6284 \(\times 10^{-16}\) & 3.4372 \(\times 10^{-46}\) & 0.1201 & 0.0033 & 1 \\ \cline{2-7}  & Mean & 1.3539 \(\times 10^{-74}\) & 9.5431 \(\times 10^{-17}\) & 2.0876 \(\times 10^{-47}\) & 0.0309 & 0.0016 & \\ \cline{2-7}  & Std & 5.2303 \(\times 10^{-74}\) & 8.7339 \(\times 10^{-17}\) & 7.4074 \(\times 10^{-47}\) & 0.0268 & 6.9524 \(\times 10^{-4}\) & \\ \hline \multirow{4}{*}{\(f_{3}\)} & Best & 1.5871 \(\times 10^{-99}\) & 1.1907 \(\times 10^{-63}\) & 4.7969 \(\times 10^{-12}\) & 5.4371 \(\times 10^{-27}\) & 2.2540 \(\times 10^{-35}\) & \\ \cline{2-7}  & Worst & 8.1085 \(\times 10^{-78}\) & 5.6634 \(\times 10^{-50}\) & 0.0293 & 5.0743 \(\times 10^{-22}\) & 1.8224 \(\times 10^{-29}\) & 1 \\ \cline{2-7}  & Mean & 6.0509 \(\times 10^{-79}\) & 2.0580 \(\times 10^{-51}\) & 0.0033 & 3.7279 \(\times 10^{-23}\) & 1.6513 \(\times 10^{-30}\) & \\ \cline{2-7}  & Std & 1.7318 \(\times 10^{-78}\) & 1.0323 \(\times 10^{-50}\) & 0.0068 & 1.1100 \(\times 10^{-22}\) & 3.9570 \(\times 10^{-30}\) & \\ \hline \multirow{4}{*}{\(f_{4}\)} & Best & 1.2424 \(\times 10^{-16}\) & 6.5653 \(\times 10^{-8}\) & 5.3046 & 0.5731 & 5.7123 & \\ \cline{2-7}  & Worst & 0.0312 & 1.9500 \(\times 10^{-6}\) & 89.6219 & 1.5222 & 37.5491 & 2 \\ \cline{2-7}  & Mean & 0.0029 & 5.7114 \(\times 10^{-7}\) & 59.6414 & 1.1226 & 16.8984 & \\ \cline{2-7}  & Std & 0.0078 & 4.6159 \(\times 10^{-7}\) & 29.3820 & 0.2522 & 7.2691 & \\ \hline \end{tabular}
\end{table} 
Table 5: Best, worst, mean, and SD obtained on functions from Table 2.  
The statistical results of the algorithms on low-dimensional benchmark functions are shown in Table 6. Compared with the other four algorithms, APO exhibited the best performance on \(f_{15}\), \(f_{21}\), \(f_{22}\) and \(f_{23}\). For \(f_{16}\) and \(f_{17}\), each algorithm nearly found all the best values. From Table 6, there is almost no difference in the performance on \(f_{18}\), \(f_{19}\) and \(f_{20}\) of each algorithm. In addition, although APO performs moderately on \(f_{14}\), it can also find the optimal value most of the time.