## 3 Experiment and Application
### Performance Analysis: Classic Benchmark Functions
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**F** & **Result** & **APO** & **GWO** & **WOA** & **PSO** & **DE** & **Rank** \\ \hline \multirow{4}{*}{\(f_{14}\)} & Best & 0.9980 & 0.9980 & 0.9980 & 0.9980 & 0.9980 & \multirow{4}{*}{4} \\ \cline{2-5}  & Worst & 12.6705 & 12.6705 & 10.7632 & 10.7632 & 2.9821 & \\ \cline{2-5}  & Mean & 4.2973 & 4.4937 & 2.0822 & 3.1636 & 1.0641 & \\ \cline{2-5}  & Std & 4.0039 & 4.0086 & 2.1440 & 2.8550 & 0.3622 & \\ \hline \multirow{4}{*}{\(f_{15}\)} & Best & 3.1012 \(\times 10^{-4}\) & 3.0749 \(\times 10^{-4}\) & 3.2572 \(\times 10^{-4}\) & 5.0404 \(\times 10^{-4}\) & 3.0749 \(\times 10^{-4}\) & \\ \cline{2-5}  & Worst & 0.0023 & 0.0204 & 0.0043 & 0.0011 & 0.0204 & \\ \cline{2-5}  & Mean & 5.2403 \(\times 10^{-4}\) & 0.0037 & 0.0012 & 8.2951 \(\times 10^{-4}\) & 0.0018 & \\ \cline{2-5}  & Std & 5.1504 \(\times 10^{-4}\) & 0.0076 & 8.6353 \(\times 10^{-4}\) & 1.5813 \(\times 10^{-4}\) & 0.0051 & \\ \hline \multirow{4}{*}{\(f_{16}\)} & Best & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \\ \cline{2-5}  & Worst & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \\ \cline{2-5}  & Mean & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \(-1.0316\) & \\ \cline{2-5}  & Std & 8.4272 \(\times 10^{-9}\) & 1.7842 \(\times 10^{-8}\) & 2.1111 \(\times 10^{-9}\) & 6.3877 \(\times 10^{-16}\) & 6.7752 \(\times 10^{-16}\) & \\ \hline \multirow{4}{*}{\(f_{17}\)} & Best & 0.3979 & 0.3979 & 0.3979 & 0.3979 & 0.3979 & 0.3979 \\ \cline{2-5}  & Worst & 0.3979 & 0.3979 & 0.3979 & 0.3979 & 0.3979 & \\ \cline{2-5}  & Mean & 0.3979 & 0.3979 & 0.3979 & 0.3979 & 0.3979 & \\ \cline{2-5}  & Std & 9.2877 \(\times 10^{-7}\) & 2.0274 \(\times 10^{-4}\) & 1.4524 \(\times 10^{-5}\) & 0 & 0 \\ \hline \multirow{4}{*}{\(f_{18}\)} & Best & 3.0000 & 3.0000 & 3.0000 & 3.0000 & 3.0000 & \\ \cline{2-5}  & Worst & 30.0000 & 3.0000 & 30.003 & 3.0000 & 3.0000 & \\ \cline{2-5}  & Mean & 6.6000 & 3.0000 & 3.9003 & 3.0000 & 3.0000 & \\ \cline{2-5}  & Std & 9.3351 & 4.3052 \(\times 10^{-5}\) & 4.9301 & 1.6472 \(\times 10^{-15}\) & 2.0301 \(\times 10^{-15}\) & \\ \hline \multirow{4}{*}{\(f_{19}\)} & Best & \(-3.8628\) & \(-3.8628\) & \(-3.8628\) & \(-3.8628\) & \(-3.8628\) & \(-3.8628\) \\ \cline{2-5}  & Worst & \(-3.8549\) & \(-3.8549\) & \(-3.8445\) & \(-3.8628\) & \(-3.8628\) & \\ \cline{2-5}  & Mean & \(-3.8605\) & \(-3.8613\) & \(-3.8611\) & \(-3.8628\) & \(-3.8628\) & \\ \cline{2-5}  & Std & 0.0036 & 0.0027 & 0.0038 & 2.5829 \(\times 10^{-15}\) & 2.7101 \(\times 10^{-15}\) & \\ \hline \multirow{4}{*}{\(f_{20}\)} & Best & \(-3.3220\) & \(-3.3220\) & \(-3.3218\) & \(-3.3220\) & \(-3.3220\) & \\ \cline{2-5}  & Worst & \(-3.0839\) & \(-3.0499\) & \(-3.1627\) & \(-3.2031\) & \(-3.2031\) & \(\approx\) \\ \cline{2-5}  & Mean & \(-3.2599\) & \(-3.2672\) & \(-3.2599\) & \(-3.2784\) & \(-3.2507\) & \\ \cline{2-5}  & Std & 0.0838 & 0.0861 & 0.0651 & 0.0583 & 0.0592 & \\ \hline \multirow{4}{*}{\(f_{21}\)} & Best & \(-10.1532\) & \(-10.1529\) & \(-10.1526\) & \(-10.1532\) & \(-10.1532\) & \\ \cline{2-5}  & Worst & \(-0.8810\) & \(-2.6302\) & \(-2.6020\) & \(-2.6305\) & \(-2.6305\) & \\ \cline{2-5}  & Mean & \(-9.5341\) & \(-8.8857\) & \(-8.2591\) & \(-7.0685\) & \(-9.4850\) & \\ \cline{2-5}  & Std & 2.3522 & 2.3709 & 2.9335 & 3.1409 & 2.0722 & \\ \hline \multirow{4}{*}{\(f_{22}\)} & Best & \(-10.4029\) & \(-10.4025\) & \(-10.4009\) & \(-10.4029\) & \(-10.4029\) & \\ \cline{2-5}  & Worst & \(-10.3988\) & \(-5.0876\) & \(-1.8302\) & \(-2.7519\) & \(-3.7243\) & \\ \cline{2-5}  & Mean & \(-10.4015\) & \(-10.2239\) & \(-7.4542\) & 3.1828 & \(-10.1803\) & \\ \cline{2-5}  & Std & 0.0011 & 0.9701 & 3.0686 & \(-8.2324\) & 1.2193 & \\ \hline \multirow{4}{*}{\(f_{23}\)} & Best & \(-10.5364\) & \(-10.5360\) & \(-10.5363\) & \(-10.5364\) & \(-10.5364\) & \\ \cline{2-5}  & Worst & \(-10.5303\) & \(-2.4217\) & \(-1.6447\) & \(-2.4217\) & \(-2.8711\) & \\ \cline{2-5}  & Mean & \(-10.5348\) & \(-10.2638\) & \(-6.9538\) & \(-9.5065\) & \(-10.0575\) & \\ \cline{2-5}  & Std & 0.0015 & 1.4811 & 3.7470 & 2.3863 & 1.8268 & \\ \hline \hline \end{tabular}
\end{table}