# Improvement Technique for Group Search Optimization Using Experimental Design Method
## 4 Experimental Results
### Results of Unimodal Functions  
In this study, the GSO, UD-GSO, and R-GSO were first performed using the unimodal functions. Table 12 shows the minimum, mean, and standard deviation (S.D.) of each function. In addition, Table 12 also shows the significant differences between UD-GSO vs. GSO and R-GSO vs. GSO, represented by the \(p\)-value from the \(t\)-distribution test. From Table 12, for the \(f_{1}\) function, the best parameter combination for UD-GSO was the ninth combination in Table 7, with a minimum of 1.9555 \(\times\) 10\({}^{-18}\), a mean of 1.2741 \(\times\) 10\({}^{-15}\), and an S.D. of 4.0520 \(\times\) 10\({}^{-13}\). The best parameter combination for R-GSO was the fourth one in Table 10, with a minimum of 4.6597 \(\times\) 10\({}^{-14}\), a mean of 7.1195 \(\times\) 10\({}^{-11}\), and an S.D. of 1.5407 \(\times\) 10\({}^{-11}\). The \(p\)-values from the t-distribution test are 3.6553 \(\times\) 10\({}^{-4}\) and 3.6557 \(\times\) 10\({}^{-4}\) for UD-GSO vs. GSO and R-GSO vs. GSO, respectively. For the \(f_{2}\), the best parameter combination for UD-GSO was the ninth combination in Table 7, with a minimum \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Benchmark Function**} & **Target** & **He et al. [8]** & **UD-GSO** & **R-GSO** \\ \hline \multirow{4}{*}{\(f_{7}\)} & Minimum & & 0.9980 & 0.9980 & 0.9980 \\  & Mean & & 0.9980 & 0.9980 & 0.9980 \\  & S.D. & & 3.6299 \(\times\) 10\({}^{-16}\) & 3.1266 \(\times\) 10\({}^{-16}\) & 2.6562 \(\times\) 10\({}^{-16}\) \\  & \(t\)-test & & N/A & NaN & NaN \\  & \(p\)-value & & & & \\ \hline \multirow{4}{*}{\(f_{8}\)} & Minimum & & 3.0749 \(\times\) 10\({}^{-4}\) & 3.0749 \(\times\) 10\({}^{-4}\) & 3.0749 \(\times\) 10\({}^{-4}\) \\  & Mean & & 5.9644 \(\times\) 10\({}^{-4}\) & 3.4302 \(\times\) 10\({}^{-4}\) & 3.1272 \(\times\) 10\({}^{-4}\) \\  & S.D. & 4.8922 \(\times\) 10\({}^{-4}\) & 1.9042 \(\times\) 10\({}^{-4}\) & 7.6724 \(\times\) 10\({}^{-6}\) \\  & \(t\)-test & & N/A & 0.0164 & 0.0036 \\  & \(p\)-value & & & & \\ \hline \multirow{4}{*}{\(f_{9}\)} & Minimum & & 3 & 3 & 3 \\  & Mean & 3 & 3 & 3 & 3 \\  & S.D. & 3 & 5.3564 \(\times\) 10\({}^{-15}\) & 1.4060 \(\times\) 10\({}^{-14}\) & 1.0981 \(\times\) 10\({}^{-14}\) \\  & \(t\)-test & & & N/A & NaN & NaN \\  & \(p\)-value & & & & \\ \hline \multirow{4}{*}{\(f_{10}\)} & Minimum & & \(-\)10.1532 & \(-\)10.1519 & \(-\)10.1530 \\  & Mean & & \(-\)6.5572 & \(-\)7.7473 & \(-\)8.1125 \\ \cline{1-1}  & S.D. & 3.3262 & 2.7845 & 2.6089 \\ \cline{1-1}  & \(t\)-test & & N/A & 0.1346 & 0.0428 \\ \cline{1-1}  & \(p\)-value & & & & \\ \hline \end{tabular}
\end{table} 
Table 13: Results comparison among GSO [8], UD-GSO, and R-GSO in multimodal functions. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Benchmark Function**} & **Target** & **He et al. [8]** & **UD-GSO** & **R-GSO** \\ \hline \multirow{4}{*}{\(f_{4}\)} & Minimum & & \(-\)12,569.4866 & \(-\)12,569.4866 & \(-\)12,569.4866 \\  & Mean & & \(-\)11,299.2395 & \(-\)12,569.4858 & \(-\)12,569.4866 \\  & S.D. & 1090.4528 & 0.0013 & 0.0002 \\  & \(t\)-test & & N/A & 5.6422 \(\times\) 10\({}^{-7}\) & 5.6421 \(\times\) 10\({}^{-7}\) \\ \hline \multirow{4}{*}{\(f_{5}\)} & Minimum & & 5.9397 & 8.3805 \(\times\) 10\({}^{-16}\) & 2.2971 \(\times\) 10\({}^{-16}\) \\  & Mean & 0 & 16.2189 & 1.5842 & 1.1219 \\ \cline{1-1}  & S.D. & 9.9765 & 1.3657 & 1.1252 \\ \cline{1-1}  & \(t\)-test & & N/A & 6.1909 \(\times\) 10\({}^{-9}\) & 3.4610 \(\times\) 10\({}^{-9}\) \\ \hline \multirow{4}{*}{\(f_{6}\)} & Minimum & & 6.8523 \(\times\) 10\({}^{-10}\) & 1.1490 \(\times\) 10\({}^{-19}\) & 1.1218 \(\times\) 10\({}^{-16}\) \\ \cline{1-1}  & Mean & & 6.8426 \(\times\) 10\({}^{-8}\) & 1.1963 \(\times\) 10\({}^{-14}\) & 1.3433 \(\times\) 10\({}^{-13}\) \\ \cline{1-1}  & S.D. & 1.1370 \(\times\) 10\({}^{-7}\) & 6.4158 \(\times\) 10\({}^{-14}\) & 3.0473 \(\times\) 10\({}^{-13}\) \\ \cline{1-1}  & \(t\)-test & & N/A & 0.0026 & 0.0026 \\ \hline \end{tabular}
\end{table}