# Improvement Technique for Group Search Optimization Using Experimental Design Method
## 4 Experimental Results
### Results of Multimodal Functions  
Next, this paper evaluated the performance of the GSO, UD-GSO, and R-GSO using multimodal functions. Table 13 also shows each function's minimum, mean, and S.D. and the \(p\)-value from the t-distribution test between UD-GSO vs. GSO and R-GSO vs. GSO. From Table 13, for the \(f_{4}\) function, the best parameter combination for UD-GSO was the seventh combination in Table 7, with a minimum of \(-\)12,569.4866, a mean of \(-\)12,569.4858, and an S.D. of 0.0013. The best parameter combination for R-GSO was the eighth in Table 10, with a minimum of \(-\)12,569.4866, a mean of \(-\)12,569.4866, a mean of \(-\)12,569.4866, a mean of \(-\)12,569.4866, and an S.D. of 0.0002. The \(p\)-values from the t-distribution test are 5.6422 \(\times\) 10\({}^{-7}\) and 5.6421 \(\times\) 10\({}^{-7}\) for UD-GSO vs. GSO and R-GSO vs. GSO, respectively. For the \(f_{5}\), the best parameter combination for UD-GSO was the ninth combination in Table 7, with a minimum of 8.3805 \(\times\) 10\({}^{-16}\), a mean of 1.5842, and an S.D. of 1.3657. The best parameter combination for R-GSO was the eighth one in Table 10, with a minimum of 2.2971 \(\times\) 10\({}^{-16}\), a mean of 1.1219, and an S.D. of 1.1252. The \(p\)-values from the t-distribution test are 6.1909 \(\times\) 10\({}^{-9}\) and 3.4610 \(\times\) 10\({}^{-9}\) for UD-GSO vs. GSO and R-GSO vs. GSO, respectively. For the \(f_{6}\), the best parameter combination for UD-GSO was the ninth combination in Table 7, with a minimum of 1.1490 \(\times\) 10\({}^{-19}\), a mean of 1.1963 \(\times\) 10\({}^{-14}\), and an S.D. of 6.4158 \(\times\) 10\({}^{-14}\). The best parameter combination for R-GSO was the fourth one in Table 10, with a minimum of 1.1218 \(\times\) 10\({}^{-16}\), a mean of 1.3433 \(\times\) 10\({}^{-13}\), and an S.D. of 3.0473 \(\times\) 10\({}^{-13}\). The \(p\)-values from the \(t\)-distribution test are both 0.0026 for UD-GSO vs. GSO and R-GSO vs. GSO. The results show a significant difference between UD-GSO and R-GSO in terms of the  
eighth one in Table 10 (minimum of 3.0749 \(\times\) 10\({}^{-4}\); mean of 3.6912 \(\times\) 10\({}^{-4}\); and S.D. of 2.3457 \(\times\) 10\({}^{-4}\)). The \(p\)-values from the t-distribution test are 0.0164 and 0.0389 for UD-GSO vs. GSO and R-GSO vs. GSO, respectively. For the \(f_{9}\), the best parameter combination for UD-GSO was the eighth combination in Table 7 (minimum of 3; mean of 3; and S.D. of 1.4060 \(\times\) 10\({}^{-14}\)) and for R-GSO was the tenth one in Table 10 (minimum of 3; mean of 3; and S.D. of 1.0981 \(\times\) 10\({}^{-14}\)). The \(p\)-values from the \(t\)-distribution test are NaN for UD-GSO vs. GSO and R-GSO vs. GSO because the evaluation values are close. For the \(f_{10}\), the best parameter combination for UD-GSO was the third combination in Table 7 (minimum of \(-\)10.1519; mean of \(-\)7.7473; and S.D. of 2.7845) and for R-GSO was the seventh one in Table 10 (minimum of \(-\)10.1530; mean of \(-\)8.1125; and S.D., 2.6089). The \(p\)-values from the \(t\)-distribution test are 0.1346 and 0.0428 for UD-GSO vs. GSO and R-GSO vs. GSO, respectively. The results show a significant difference between UD-GSO and R-GSO in terms of the \(p\)-value of t-distribution in \(f_{8}\) and \(f_{10}\); in terms of mean performance, both UD-GSO and R-GSO are smaller than GSO in \(f_{8}\) and \(f_{10}\), and UD-GSO and R-GSO have the same mean to GSO in \(f_{7}\) and \(f_{9}\); in addition, in terms of standard deviation, R-GSO performs better than UD-GSO in \(f_{7}\) to \(f_{10}\). The best combinations of parameters were selected by the uniform layout because each function's characteristics are inconsistent.