## 4 Simulation and Discussion
### Results
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Unimodal**} & \multicolumn{3}{c}{**Multimodal**} & \multicolumn{3}{c}{**Fixed-Dimension Multimodal**} \\  & \multicolumn{2}{c}{**Original**} & \multicolumn{2}{c}{**DM**} & \multicolumn{2}{c}{**Original**} & \multicolumn{2}{c}{**DM**} & \multicolumn{2}{c}{**Original**} & \multicolumn{2}{c}{**DM**} \\ \hline F\({}_{1}\) & Avg & \(1.7740\times 10^{-5}\) & \(\mathbf{6.746\times 10^{-218}}\) & F\({}_{8}\) & Avg & \(-6908.6558\) & \(\mathbf{-12516.1893}\) & F\({}_{14}\) & Avg & \(2.1735\) & \(\mathbf{0.9980}\) \\  & std & \(6.4396\times 10^{-21}\) & \(\mathbf{0}\) & std & \(1.0168\times 10^{-12}\) & \(\mathbf{9.3549\times 10^{-39}}\) & F\({}_{14}\) & std & \(7.9441\times 10^{-16}\) & \(\mathbf{5.645\times 10^{-19}}\) \\ \hline F\({}_{2}\) & Avg & \(0.3411\) & \(\mathbf{2.9565\times 10^{-11}}\) & F\({}_{9}\) & Avg & \(5.70613\) & \(\mathbf{1.5631\times 10^{-39}}\) & F\({}_{15}\) & Avg & \(0.0535\) & \(\mathbf{0.0034}\) \\  & std & \(7.4476\times 10^{-17}\) & \(\mathbf{1.0736\times 10^{-126}}\) & F\({}_{10}\) & Avg & \(6.3552\times 10^{-15}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(3.8789\times 10^{-19}\) & \(\mathbf{2.0849\times 10^{-18}}\) \\ \hline F\({}_{3}\) & Avg & \(589.4920\) & \(\mathbf{1.9627\times 10^{-70}}\) & F\({}_{10}\) & Avg & \(2.1546\) & \(\mathbf{4.7784\times 10^{-14}}\) & F\({}_{16}\) & Avg & \(-1.0316\) & \(\mathbf{-1.0316}\) \\  & std & \(7.1179\times 10^{-13}\) & \(\mathbf{5.0357\times 10^{-86}}\) & F\({}_{10}\) & std & \(7.9441\times 10^{-16}\) & \(\mathbf{1.1289\times 10^{-29}}\) & \(\mathbf{F}_{16}\) & std & \(3.4755\times 10^{-16}\) & \(\mathbf{4.4685\times 10^{-21}}\) \\ \hline F\({}_{4}\) & Avg & \(3.9634\) & \(\mathbf{1.5239\times 10^{-97}}\) & F\({}_{11}\) & Avg & \(0.0462\) & \(\mathbf{0.0214}\) & \(\mathbf{F}_{17}\) & Avg & \(0.7854\) & \(\mathbf{0.4076}\) \\  & std & \(1.9860\times 10^{-16}\) & \(\mathbf{5.8112\times 10^{-113}}\) & F\({}_{11}\) & std & \(3.1031\times 10^{-18}\) & \(\mathbf{3.1031\times 10^{-23}}\) & \(\mathbf{F}_{17}\) & std & \(4.9650\times 10^{-17}\) & \(\mathbf{0}\) \\ \hline F\({}_{5}\) & Avg & \(50.26245\) & \(\mathbf{2.3706\times 10^{-13}}\) & F\({}_{12}\) & Avg & \(0.4806\) & \(\mathbf{1.5705\times 10^{-23}}\) & \(\mathbf{F}_{18}\) & Avg & \(3\) & \(3\) \\  & std & \(1.5888\times 10^{-14}\) & \(\mathbf{1.9911\times 10^{-28}}\) & F\({}_{12}\) & Avg & \(1.8619\times 10^{-16}\) & \(\mathbf{1.2239\times 10^{-6}}\) & \(\mathbf{1.2239\times 10^{-6}}\) & \(\mathbf{1.3674\times 10^{-15}}\) & \(\mathbf{2.5818\times 10^{-19}}\) \\ \hline F\({}_{6}\) & Avg & \(2025\) & \(\mathbf{0}\) & F\({}_{13}\) & Avg & \(0.5084\) & \(\mathbf{1.3497\times 10^{-23}}\) & \(\mathbf{F}_{19}\) & Avg & \(-3.8627\) & \(\mathbf{-3.8627}\) \\  & std & \(0\) & \(\mathbf{0}\) & std & \(4.9650\times 10^{-17}\) & \(\mathbf{1.2239\times 10^{-6}}\) & \(\mathbf{1.36}\) & \(\mathbf{8.9371\times 10^{-15}}\) & \(\mathbf{9.0364\times 10^{-21}}\) \\ \hline F\({}_{7}\) & Avg & \(0.1134\) & \(\mathbf{0.0110}\) & & & & & & \(-3.2619\) & \(\mathbf{-3.2744}\) \\  & std & \(4.3444\times 10^{-17}\) & \(\mathbf{1.2412\times 10^{-27}}\) & & & & & \(2.9790\times 10^{-16}\) & \(3.9720\times 10^{-21}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimization results of particle swarm optimization (PSO) for original and “modified by DM” versions.