## 3 Experiment and Application
### Performance Analysis: Classic Benchmark Functions
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\boldsymbol{F}\) & **Result** & **APO** & **GWO** & **WOA** & **PSO** & **DE** & **Rank** \\ \hline \multirow{4}{*}{\(f_{5}\)} & Best & 0.0078 & 26.0434 & 27.1465 & 25.3115 & 19.0580 & \\ \cline{2-7}  & Worst & 28.7807 & 28.7485 & 28.8434 & 188.8556 & 181.2647 & \\ \cline{2-7}  & Mean & 26.6971 & 27.2864 & 28.3636 & 85.1733 & 56.5542 & \\ \cline{2-7}  & Std & 7.2574 & 0.7167 & 0.4466 & 40.5425 & 40.1468 & \\ \hline \multirow{4}{*}{\(f_{6}\)} & Best & 1.2702 \(\times 10^{-6}\) & 1.3209 \(\times 10^{-4}\) & 0.7374 & 7.9911 \(\times 10^{-6}\) & 3.6153 \(\times 10^{-6}\) & \\ \cline{2-7}  & Worst & 4.3859 \(\times 10^{-5}\) & 1.5119 & 3.1981 & 4.1143 \(\times 10^{-4}\) & 1.8978 \(\times 10^{-4}\) & \\ \cline{2-7}  & Mean & 1.3972 \(\times 10^{-5}\) & 0.6602 & 1.9556 & 9.7234 \(\times 10^{-5}\) & 4.7963 \(\times 10^{-5}\) & \\ \cline{2-7}  & Std & 9.8769 \(\times 10^{-6}\) & 0.3692 & 0.6576 & 8.7218 \(\times 10^{-5}\) & 4.8581 \(\times 10^{-5}\) & \\ \hline \multirow{4}{*}{\(f_{7}\)} & Best & 1.9336 \(\times 10^{-5}\) & 4.8037 \(\times 10^{-4}\) & 6.4622 \(\times 10^{-5}\) & 0.0804 & 0.0360 & \\ \cline{2-7}  & Worst & 0.0077 & 0.0034 & 0.0194 & 0.3469 & 0.1085 & \\ \cline{2-7}  & Mean & 8.5533 \(\times 10^{-4}\) & 0.0019 & 0.0050 & 0.1729 & 0.0797 & \\ \cline{2-7}  & Std & 0.0015 & 7.0049 \(\times 10^{-4}\) & 0.0057 & 0.0584 & 0.0158 & \\ \hline \multirow{4}{*}{\(f_{8}\)} & Best & \(-12569\) & \(-7250.9\) & N/A & \(-8727.2\) & \(-7317.7\) & \\ \cline{2-7}  & Worst & \(-12150\) & \(-3727.9\) & N/A & \(-2629.0\) & \(-4691.7\) & \\ \cline{2-7}  & Mean & \(-12529\) & \(-6129.3\) & N/A & \(-4978.3\) & \(-5946.6\) & \\ \cline{2-7}  & Std & 106.3406 & 767.8156 & N/A & 1569.8 & 667.0457 & \\ \hline \multirow{4}{*}{\(f_{9}\)} & Best & 0 & 5.6843 \(\times 10^{-14}\) & 0 & 33.8862 & 1.4607 \(\times 10^{2}\) & \\ \cline{2-7}  & Worst & 0 & 16.9267 & 5.6843 \(\times 10^{-14}\) & 83.6992 & 2.1638 \(\times 10^{2}\) & \\ \cline{2-7}  & Mean & 0 & 4.0526 & 1.8948 \(\times 10^{-15}\) & 58.1507 & 1.8134 \(\times 10^{2}\) & \\ \cline{2-7}  & Std & 0 & 4.5006 & 1.0378 \(\times 10^{-14}\) & 14.4306 & 16.9916 & \\ \hline \multirow{4}{*}{\(f_{10}\)} & Best & 8.8818 \(\times 10^{-16}\) & 5.7732 \(\times 10^{-14}\) & 8.8818 \(\times 10^{-16}\) & 0.0016 & 5.7932 \(\times 10^{-4}\) & \\ \cline{2-7}  & Worst & 7.9936 \(\times 10^{-15}\) & 1.3589 \(\times 10^{-13}\) & 7.9936 \(\times 10^{-15}\) & 1.5175 & 0.0226 & \\ \cline{2-7}  & Mean & 2.6645 \(\times 10^{-15}\) & 1.0309 \(\times 10^{-13}\) & 3.8488 \(\times 10^{-15}\) & 0.1757 & 0.0024 & \\ \cline{2-7}  & Std & 2.2372 \(\times 10^{-15}\) & 1.7645 \(\times 10^{-14}\) & 2.6526 \(\times 10^{-15}\) & 0.4313 & 0.0040 & \\ \hline \multirow{4}{*}{\(f_{11}\)} & Best & 0 & 0 & 0 & 1.2377 \(\times 10^{-6}\) & 1.5597 \(\times 10^{-5}\) & \\ \cline{2-7}  & Worst & 0 & 0.0384 & 0.1508 & 0.0246 & 0.0340 & \\ \cline{2-7}  & Mean & 0 & 0.0030 & 0.0086 & 0.0071 & 0.0032 & \\ \cline{2-7}  & Std & 0 & 0.0090 & 0.0333 & 0.0081 & 0.0072 & \\ \hline \multirow{4}{*}{\(f_{12}\)} & Best & 8.3024 \(\times 10^{-8}\) & 0.0063 & 0.0295 & 1.3634 \(\times 10^{-7}\) & 5.3389 \(\times 10^{-6}\) & \\ \cline{2-7}  & Worst & 0.0065 & 0.0939 & 0.7203 & 0.1037 & 8.3340 \(\times 10^{2}\) & \\ \cline{2-7}  & Mean & 2.1901 \(\times 10^{-4}\) & 0.0373 & 0.1473 & 0.0104 & 62.5083 & \\ \cline{2-7}  & Std & 0.0012 & 0.0186 & 0.1722 & 0.0316 & 1.9621 \(\times 10^{2}\) & \\ \hline \multirow{4}{*}{\(f_{13}\)} & Best & 1.9591 \(\times 10^{-6}\) & 0.2159 & 0.6067 & 1.8128 \(\times 10^{-6}\) & 1.0309 \(\times 10^{-5}\) & \\ \cline{2-7}  & Worst & 4.1407 \(\times 10^{-5}\) & 1.1442 & 1.9571 & 0.0178 & 7.4778 \(\times 10^{2}\) & \\ \cline{2-7}  & Mean & 1.1372 \(\times 10^{-5}\) & 0.6458 & 1.3359 & 0.0057 & 26.6912 & \\ \cline{2-7}  & Std & 1.0199 \(\times 10^{-5}\) & 0.2486 & 0.3731 & 0.0058 & 1.3623 \(\times 10^{2}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._