# Improvement Technique for Group Search Optimization Using Experimental Design Method
## 4 Experimental Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Functions** & \(\pi\) & \(\mathbf{s}\) & \(f_{\text{target}}\) \\ \hline \(f_{1}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & 30 & \([-100,100]\) & 0 \\ \(f_{2}(x)=\max_{i}\{|x_{i}|,1\leq i\leq n\}\) & 30 & \([-100,100]\) & 0 \\ \(f_{3}(x)=\sum\limits_{i=1}^{n-1}\left(100\left(x_{i+1}-x_{i}^{2}\right)^{2}+ \left(x_{i}-1\right)\right)^{2}\) & 30 & \([-30,30]\) & 0 \\ \(f_{4}(x)=-\sum\limits_{i=1}^{n-1}\left(x_{i}\text{sin}\left(\sqrt{|x_{i}|} \right)\right)\) & 30 & \([-500,500]\) & \(-12,569.5\) \\ \(f_{5}(x)=\sum\limits_{i=1}^{n-1}\left(x_{i}^{2}-100\cos(2\pi x_{i})+10\right) ^{2}\) & 30 & \([-5.12,5.12]\) & 0 \\ \(f_{6}(x)=\frac{\pi}{n}\left\{10\text{sin}^{2}\left(\pi y_{i}^{6}\right)+ \sum\
olimits_{i=1}^{n-1}\left(y_{i}^{6}-1\right)2\left[1+10\text{sin}^{2} \left(\pi y_{i}^{6}+1\right)\right]+\left(y_{n}^{6}-1\right)^{2}\right\}+ \right.\) & \\ \(\left.\sum\
olimits_{i=1}^{n}u^{6}(x_{i},10,\,100,\,4)\right.\) & 30 & \([-50,50]\) & 0 \\ \(y_{i}^{6}=1+\frac{1}{4}(x_{i}+1),u^{6}\left(x_{i},a^{6},k,m\right)=\left\{ \begin{array}{ll}k\left(xi-a^{6}\right)^{m},x>a^{6}\\ 0,x\in\left[-a^{6},a^{6}\right]\\ k\left(-xi-a^{6}\right)^{m},x\gets a^{6}\end{array}\right.\) & \\ \(f_{7}(x)=\left[\frac{1}{500}+\sum\limits_{j=1}^{25}\frac{1}{i+\sum\limits_{j =1}^{2}\left(x_{i}-a^{6}\right)^{m}}\right]^{-1}\) & 2 & \([-65.536,65.536]\) & 1 \\ \(f_{8}(x)=\frac{11}{2}\left[a_{i}^{6}-\frac{x_{1}\left(\left(f_{i}^{6}\right)^ {2}+y_{i}^{6}\right)^{2}}{\left(y_{i}^{6}\right)^{2}+y_{i}^{6}\right)^{2}+y_{ i}^{6}\right.3+x_{4}}\) & 4 & \([-5,5]\) & 0.0003075 \\ \(f_{9}(x)=\left[1+(x_{1}+x_{2}+1)^{2}\left(19-14x_{1}+3x_{1}^{2}-14x_{2}+6x_{1} x_{2}+3x_{2}^{2}\right)\right]\times\) & 2 & \([-2,2]\) & 3 \\ \(\left[30+(2x_{1}+1-3x_{2})^{2}\left(18-32x_{1}+12x_{1}^{2}+48x_{2}-36x_{1}x_{ 2}+27x_{2}^{2}\right)\right]\) & 4 & \([0,10]\) & \(-10\) \\ \(f_{10}(x)=-\sum\limits_{i=1}^{5}\left[\left(x-a_{i}^{f_{10}}\right)\left(x-a_{i }^{f_{1}}\right)^{T}+c_{i}^{f_{10}}\right]^{-1}\) & 4 & \([0,10]\) & \(-10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Benchmark functions used in this study. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Benchmark Function** & **Target** & **[PERSON] et al. [8]** & **UD-GSO** & **R-GSO** \\ \hline \multirow{4}{*}{\(f_{1}\)} & Minimum & & \(1.4436\times 10^{-7}\) & \(1.9555\times 10^{-18}\) & \(4.6597\times 10^{-14}\) \\  & Mean & & \(1.2757\times 10^{-6}\) & \(1.2741\times 10^{-13}\) & \(7.1195\times 10^{-11}\) \\  & S.D. & & \(1.7324\times 10^{-6}\) & \(4.0520\times 10^{-13}\) & \(1.5407\times 10^{-11}\) \\  & \(t\)-test & & N/A & \(3.6553\times 10^{-4}\) & \(3.6557\times 10^{-4}\) \\  & \(p\)-value & & & \\ \hline \multirow{4}{*}{\(f_{2}\)} & Minimum & & \(0.1631\) & \(0.1372\) & \(0.1845\) \\  & Mean & & \(0.3016\) & \(0.4594\) & \(0.3524\) \\  & S.D. & & \(0.0835\) & \(0.1898\) & \(0.0730\) \\  & \(t\)-test & & N/A & \(1.1263\times 10^{-4}\) & \(0.0289\) \\ \cline{1-1}  & Minimum & & \(9.2178\) & \(1.1797\) & \(6.9507\) \\ \cline{1-1}  & Mean & & \(66.4847\) & \(44.3206\) & \(19.1942\) \\ \cline{1-1}  & S.D. & & \(39.8562\) & \(30.9214\) & \(30.6621\) \\ \cline{1-1}  & \(t\)-test & & N/A & \(0.0482\) & \(2.0700\times 10^{-4}\) \\ \cline{1-1}  & \(p\)-value & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Results comparison among GSO [8], UD-GSO, and R-GSO in unimodal functions.