## 5 Experimental Results and Analysis
### Comparison with DE, PSO, and Three State-of-the-Art DE Variants
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Function** & **Metrics** & **HE-DEPSO** & **DEPSO** & **SHADE** & **CoDE** & **DE** & **PSO** \\ \hline \(F_{3}\) & Std & **0.0000** & **0.0000** & **0.0000** & 2.7390 \(\times 10^{-11}\) & 4.8441 \(\times 10^{-4}\) & 6.2297 \(\times 10^{2}\) \\  & Rank & **1** & **1** & **1** & 2 & 3 & 4 \\  & Significance & & \(\approx\) & \(\approx\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{4}\)} & Mean & **0.0000** & 2.3838 \(\times 10^{-14}\) & 2.0805\(\times 10^{0}\) & 2.3103 \(\times 10^{-9}\) & 1.0409 \(\times 10^{-3}\) & 5.8071 \(\times 10^{1}\) \\  & Std & **0.0000** & 2.8513 \(\times 10^{-14}\) & 6.7544 \(\times 10^{-1}\) & 1.2601 \(\times 10^{-9}\) & 6.4578 \(\times 10^{-4}\) & 6.9704 \(\times 10^{1}\) \\  & Rank & **1** & 2 & 5 & 3 & 4 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{5}\)} & Mean & **2.8439 \(\times 10^{0}\)** & 4.0119 \(\times 10^{0}\) & 3.8489 \(\times 10^{0}\) & 8.5594 \(\times 10^{0}\) & 2.9710 \(\times 10^{1}\) & 3.0728 \(\times 10^{1}\) \\  & Std & **7.2094 \(\times 10^{-1}\)** & 1.6940 \(\times 10^{0}\) & 1.0401 \(\times 10^{0}\) & 1.4246 \(\times 10^{0}\) & 3.9355 \(\times 10^{0}\) & 1.2073 \(\times 10^{1}\) \\  & Rank & **1** & 3 & 2 & 4 & 5 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{6}\)} & Mean & **0.0000** & 4.6321 \(\times 10^{-3}\) & 2.7530 \(\times 10^{-4}\) & 4.6895 \(\times 10^{-3}\) & 4.5748 \(\times 10^{-5}\) & 1.1970 \(\times 10^{1}\) \\  & Std & **0.0000** & 2.5790 \(\times 10^{-2}\) & 5.5026 \(\times 10^{-4}\) & 2.4954 \(\times 10^{-3}\) & 2.2095 \(\times 10^{-5}\) & 9.6167 \(\times 10^{0}\) \\  & Rank & **1** & 4 & 3 & 5 & 2 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{7}\)} & Mean & **1.2678 \(\times 10^{1}\)** & 2.1067 \(\times 10^{1}\) & 1.3953 \(\times 10^{1}\) & 2.0678 \(\times 10^{1}\) & 3.9143 \(\times 10^{1}\) & 3.0657 \(\times 10^{1}\) \\  & Std & **7.0313 \(\times 10^{-1}\)** & 6.7134 \(\times 10^{0}\) & 7.6551 \(\times 10^{-1}\) & 2.2560 \(\times 10^{0}\) & 3.7389 \(\times 10^{0}\) & 8.6791 \(\times 10^{0}\) \\  & Rank & **1** & 4 & 2 & 3 & 6 & 5 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{8}\)} & Mean & **2.2853 \(\times 10^{0}\)** & 4.2045 \(\times 10^{0}\) & 3.8070 \(\times 10^{0}\) & 8.6884 \(\times 10^{0}\) & 2.6037 \(\times 10^{1}\) & 2.4833 \(\times 10^{1}\) \\  & Std & 8.1114 \(\times 10^{-1}\) & 1.9346 \(\times 10^{0}\) & **7.3957 \(\times 10^{-1}\)** & 1.8765 \(\times 10^{0}\) & 5.5290 \(\times 10^{0}\) & 1.1493 \(\times 10^{1}\) \\  & Rank & 1 & 3 & 2 & 4 & 6 & 5 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{9}\)} & Mean & **0.0000** & 7.3346 \(\times 10^{-15}\) & **0.0000** & **0.0000** & **0.0000** & 6.8233 \(\times 10^{1}\) \\  & Std & **0.0000** & 2.8391 \(\times 10^{-14}\) & **0.0000** & **0.0000** & **0.0000** & 1.9812 \(\times 10^{2}\) \\  & Rank & **1** & 2 & **1** & **1** & **1** & 3 \\  & Significance & & \(\approx\) & \(\approx\) & \(\approx\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{10}\)} & Mean & 2.4499 \(\times 10^{2}\) & 8.9742 \(\times 10^{2}\) & **1.5812 \(\times 10^{2}\)** & 7.4438 \(\times 10^{2}\) & 1.4730 \(\times 10^{3}\) & 9.1356 \(\times 10^{2}\) \\  & Std & 1.1864 \(\times 10^{2}\) & 1.6890 \(\times 10^{2}\) & **7.6565 \(\times 10^{1}\)** & 1.1745 \(\times 10^{2}\) & 1.7509 \(\times 10^{2}\) & 3.3829 \(\times 10^{2}\) \\  & Rank & 2 & 4 & **1** & 3 & 6 & 5 \\  & Significance & & \(+\) & \(-\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{11}\)} & Mean & **2.5687 \(\times 10^{-7}\)** & 8.6658 \(\times 10^{-1}\) & 2.3208 \(\times 10^{0}\) & 1.2502 \(\times 10^{0}\) & 6.7909 \(\times 10^{0}\) & 8.1368 \(\times 10^{1}\) \\  & Std & **1.4103 \(\times 10^{-6}\)** & 6.1559 \(\times 10^{-1}\) & 4.7869 \(\times 10^{-1}\) & 4.6467 \(\times 10^{-1}\) & 9.9142 \(\times 10^{-1}\) & 8.3317 \(\times 10^{1}\) \\  & Rank & **1** & 2 & 4 & 3 & 5 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{12}\)} & Mean & 8.1966 \(\times 10^{1}\) & **1.1535 \(\times 10^{1}\)** & 3.5298 \(\times 10^{1}\) & 2.7652 \(\times 10^{2}\) & 5.3458 \(\times 10^{2}\) & 8.7444 \(\times 10^{7}\) \\  & Std & 7.0615 \(\times 10^{1}\) & 2.8045 \(\times 10^{1}\) & **1.4666 \(\times 10^{1}\)** & 6.4786 \(\times 10  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline
**Function** & **Metrics** & **HE-DEPSO** & **DEPSO** & **SHADE** & **CoDE** & **DE** & **PSO** \\ \hline \multicolumn{6}{c}{Significance} & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{16}\)} & Mean & **1.4450\(\times\)10\({}^{0}\)** & 1.6357\(\times\)10\({}^{0}\) & 4.5365\(\times\)10\({}^{0}\) & 4.3009\(\times\)10\({}^{0}\) & 1.9028 \(\times\)10\({}^{1}\) & 1.2496 \(\times\)10\({}^{2}\) \\  & Std & **3.2202 \(\times\)10\({}^{-1}\)** & 3.2942\(\times\)10\({}^{0}\) & 2.2215\(\times\)10\({}^{0}\) & 2.5242\(\times\)10\({}^{0}\) & 1.6484 \(\times\)10\({}^{1}\) & 1.0780 \(\times\)10\({}^{2}\) \\  & Rank & **1** & 2 & 4 & 3 & 5 & 6 \\  & Significance & & \(-\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{17}\)} & Mean & **7.2016\(\times\)10\({}^{0}\)** & 1.2075 \(\times\)10\({}^{1}\) & 2.1001 \(\times\)10\({}^{1}\) & 2.0112 \(\times\)10\({}^{1}\) & 2.8168 \(\times\)10\({}^{1}\) & 5.7710 \(\times\)10\({}^{1}\) \\  & Std & 3.0221\(\times\)10\({}^{0}\) & 8.5311\(\times\)10\({}^{0}\) & 3.2211\(\times\)10\({}^{0}\) & 2.8326\(\times\)10\({}^{0}\) & **1.8397\(\times\)10\({}^{0}\)** & 2.4017 \(\times\)10\({}^{1}\) \\  & Rank & **1** & 2 & 4 & 3 & 5 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{18}\)} & Mean & **3.8770 \(\times\)10\({}^{-1}\)** & 8.2949 \(\times\)10\({}^{0}\) & 1.9823 \(\times\)10\({}^{1}\) & 2.0117 \(\times\)10\({}^{1}\) & 2.0542 \(\times\)10\({}^{1}\) & 2.7241 \(\times\)10\({}^{4}\) \\  & Std & 1.4583 \(\times\)10\({}^{-1}\) & 6.7399 \(\times\)10\({}^{0}\) & 1.9348 \(\times\)10\({}^{0}\) & 3.4569 \(\times\)10\({}^{-2}\) & **1.0532 \(\times\)10\({}^{-1}\)** & 1.8731 \(\times\)10\({}^{4}\) \\  & Rank & **1** & 2 & 3 & 4 & 5 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{19}\)} & Mean & **9.7895 \(\times\)10\({}^{-2}\)** & 5.4625 \(\times\)10\({}^{-1}\) & 1.0349 \(\times\)10\({}^{0}\) & 5.9210 \(\times\)10\({}^{-1}\) & 1.2208 \(\times\)10\({}^{0}\) & 5.0978 \(\times\)10\({}^{3}\) \\  & Std & **1.2871 \(\times\)10\({}^{-1}\)** & 3.7564 \(\times\)10\({}^{-1}\) & 1.5715 \(\times\)10\({}^{-1}\) & 4.4716 \(\times\)10\({}^{-2}\) & 2.0310 \(\times\)10\({}^{-1}\) & 1.1584 \(\times\)10\({}^{4}\) \\  & Rank & **1** & 2 & 4 & 3 & 5 & 6 \\  & Significance & & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{20}\)} & Mean & **5.2046\(\times\)10\({}^{0}\)** & 6.9310 \(\times\)10\({}^{0}\) & 1.7787 \(\times\)10\({}^{1}\) & 2.0000 \(\times\)10\({}^{1}\) & 2.5137 \(\times\)10\({}^{1}\) & 5.9825 \(\times\)10\({}^{1}\) \\  & Std & 3.1153 \(\times\)10\({}^{0}\) & 6.8469 \(\times\)10\({}^{0}\) & 3.9314 \(\times\)10\({}^{0}\) & **1.4932 \(\times\)10\({}^{-4}\)** & 1.9501 \(\times\)10\({}^{0}\) & 4.0129 \(\times\)10\({}^{1}\) \\  & Rank & **1** & 2 & 3 & 4 & 5 & 6 \\  & Significance & & \(\approx\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{21}\)} & Mean & 1.7676 \(\times\)10\({}^{2}\) & **1.0695 \(\times\)10\({}^{2}\)** & 1.5575 \(\times\)10\({}^{2}\) & 1.2554 \(\times\)10\({}^{2}\) & 1.9598 \(\times\)10\({}^{2}\) & 1.9045 \(\times\)10\({}^{2}\) \\  & Std & 4.6029 \(\times\)10\({}^{1}\) & **2.0972 \(\times\)10\({}^{1}\)** & 4.2795 \(\times\)10\({}^{1}\) & 4.8087 \(\times\)10\({}^{1}\) & 5.7606 \(\times\)10\({}^{1}\) & 6.2148 \(\times\)10\({}^{1}\) \\  & Rank & 4 & **1** & 3 & 2 & 6 & 5 \\  & Significance & & \(-\) & \(-\) & \(-\) & \(+\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{22}\)} & Mean & 1.0000 \(\times\)10\({}^{2}\) & 9.7428 \(\times\)10\({}^{1}\) & 9.9647 \(\times\)10\({}^{1}\) & **7.4280 \(\times\)10\({}^{1}\)** & 1.0070 \(\times\)10\({}^{2}\) & 1.2711 \(\times\)10\({}^{2}\) \\  & Std & **0.0000** & 1.5939 \(\times\)10\({}^{1}\) & 2.1997 \(\times\)10\({}^{0}\) & 4.4524 \(\times\)10\({}^{1}\) & 5.7442 \(\times\)10\({}^{-1}\) & 5.6792 \(\times\)10\({}^{1}\) \\  & Rank & 4 & 2 & 3 & **1** & 5 & 6 \\  & Significance & & \(+\) & \(+\) & \(\approx\) & \(\approx\) & \(+\) \\ \hline \multirow{6}{*}{\(F_{23}\)} & Mean & **3.0613 \(\times\)10\({}^{2}\)** & 3.0845 \(\times\)10\({}^{2}\) & 3.0859 \(\times\)10\({}^{2}\) & 3.0272 \(\times\)10\({}^{2}\) & 3.3600 \(\times\)10\({}^{2}\) & 3.4479 \(\times\)10\({}^{2}\) \\  & Std & 1.4783 \(\times\)10\({}^{0}\) & 2.1842 \(\times\)10\({}^{0}\) &