## 4 Simulation and Discussion
### Results

Table 4: Optimization results of gravitational search algorithm (GSA) for original and “modified by DM” versions. \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 & \(8.3373\times 10^{-60}\) & **1.1627 \(\times\) 10\({}^{-157}\)** & F\({}_{8}\) & Avg & \(-7408.6107\) & \(\mathbf{-12569.4866}\) & F\({}_{14}\) & Avg & \(2.2721\) & **0.9980** \\  & std & \(4.9436\times 10^{-76}\) & **0** & std & \(3.0505\times 10^{-12}\) & \(\mathbf{1.6269\times 10^{-11}}\) & F\({}_{14}\) & std & \(1.9860\times 10^{-16}\) & **0** \\ \hline F\({}_{2}\) & Avg & \(7.1704\times 10^{-35}\) & \(\mathbf{1.9426\times 10^{-80}}\) & F\({}_{9}\) & Avg & \(10.2485\) & **0** & F\({}_{15}\) & Avg & \(0.0033\) & \(3.9560\times 10^{-4}\) \\  & std & \(6.6936\times 10^{-30}\) & \(\mathbf{1.2562\times 10^{-66}}\) & F\({}_{10}\) & Avg & \(5.5608\times 10^{-15}\) & **0** & F\({}_{14}\) & std & \(1.2218\times 10^{-17}\) & **0** \\ \hline F\({}_{3}\) & Avg & \(2.7531\times 10^{-15}\) & \(\mathbf{1.0076\times 10^{-30}}\) & F\({}_{10}\) & Avg & \(2.02757\) & \(\mathbf{4.7961\times 10^{-15}}\) & F\({}_{16}\) & Avg & \(-1.0316\) & **-1.0316** \\  & std & \(2.6459\times 10^{-31}\) & \(\mathbf{8.5751\times 10^{-46}}\) & F\({}_{10}\) & std & \(2.5641\times 10^{-15}\) & \(\mathbf{1.4111\times 10^{-30}}\) & F\({}_{14}\) & std & \(1.4398\times 10^{-15}\) & **9.9301 \(\times 10^{-19}\)** \\ \hline F\({}_{4}\) & Avg & \(9.4199\times 10^{-15}\) & \(\mathbf{5.6754\times 10^{-71}}\) & F\({}_{11}\) & Avg & \(0.6082\) & **0.0182** & \(\mathbf{F}_{17}\) & Avg & \(0.3978\) & **0.4085** \\  & std & \(2.1167\times 10^{-30}\) & \(\mathbf{2.8775\times 10^{-86}}\) & F\({}_{11}\) & std & \(1.9860\times 10^{-16}\) & \(\mathbf{6.2063\times 10^{-18}}\) & std & \(7.4476\times 10^{-17}\) & **9.9301 \(\times 10^{-17}\)** \\ \hline F\({}_{5}\) & Avg & \(146.4564\) & \(\mathbf{21.4361}\) & F\({}_{12}\) & Avg & \(0.0203\) & \(\mathbf{1.5705\times 10^{-23}}\) & F\({}_{18}\) & Avg & \(3.0009\) & **3** \\  & std & \(1.9065\times 10^{-14}\) & \(\mathbf{2.0654\times 10^{-21}}\) & F\({}_{12}\) & std & \(7.7579\times 10^{-19}\) & \(\mathbf{1.2239\times 10^{-6}}\) & std & \(1.5888\times 10^{-15}\) & \(1.3902\times 10^{-26}\) \\ \hline F\({}_{6}\) & Avg & \(0.4435\) & **0** & F\({}_{13}\) & Avg & \(0.3293\) & \(\mathbf{1.3497\times 10^{-23}}\) & F\({}_{19}\) & Avg & \(-3.8609\) & **-3.6627** \\  & std & \(4.2203\times 10^{-16}\) & **0** & std & \(2.1101\times 10^{-16}\) & \(\mathbf{1.2239\times 10^{-6}}\) & std & \(7.3483\times 10^{-15}\) & **9.9301 \(\times 10^{-45}\)** \\ \hline F\({}_{7}\) & Avg & \(0.0017\) & \(\mathbf{3.4102\times 10^{-4}}\) & & & & & & \\  & std & \(3.87896\times 10^{-19}\) & \(\mathbf{2.4849\times 10^{-27}}\) & & & & & & \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline F\({}_{2}\) & Avg & \(-10.0389\) & \(-10.4029\) \\  & std & \(1.5292\times 10^{-14}\) & \(\mathbf{0}\) \\ \hline F\({}_{23}\) & Avg & \(-9.2905\) & \(-10.5364\) \\  & std & \(1.1916\times 10^{-15}\) & \(\mathbf{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimization results of teaching learning based optimization (TLBO) for original and “modified by DM” versions.