# Adaptive Salp Swarm Algorithm for Optimization of Geotechnical Structures
## 4 Model Verification
The [PERSON]'s rank sum test returned the \(p\)-value, the sum of the positive ranks (R+), and the sum of the negative ranks (R\(-\)). Table 6 shows the results of the [PERSON]'s rank sum test for all the benchmark functions. The \(p\)-value is the smallest level of significance for detecting differences. In this study, the level of significance was set at 0.05 (\(\alpha=0.05\)). If the \(p\)-value was smaller than 0.05, it meant that the better result achieved by the best method in each pairwise comparison was statistically significant and was not obtained by chance. However, there was no significant difference between the two examined methods if the \(p\)-value was greater than 0.05. Such a result is indicated with \"NA\" in the \"win\" rows of Table 6. In addition, if the R+ was greater than the R\(-\), the ASSA had a better performance than the alternative technique. Otherwise, the ASSA had a poor performance, and the other approach had a better performance [58]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(F\) & **Index** & **ASSA** & **SSA** & **GA** & **PSO** & **FA** & **MVO** & **TSA** \\ \hline \(F_{1}\) & Mean & \(\mathbf{2.23\times 10^{-227}}\) & \(3.29\times 10^{-7}\) & \(1.95\times 10^{-12}\) & \(4.98\times 10^{-9}\) & \(7.11\times 10^{-3}\) & \(2.81\times 10^{-1}\) & \(8.31\times 10^{-56}\) \\  & Std. & \(\mathbf{0.00}\) & \(5.92\times 10^{-7}\) & \(2.01\times 10^{-11}\) & \(1.40\times 10^{-8}\) & \(3.21\times 10^{-3}\) & \(1.11\times 10^{-1}\) & \(1.02\times 10^{-58}\) \\ \(F_{2}\) & Mean & \(\mathbf{5.96\times 10^{-105}}\) & \(1.911\) & \(6.53\times 10^{-18}\) & \(7.29\times 10^{-4}\) & \(4.34\times 10^{-1}\) & \(3.96\times 10^{-1}\) & \(8.36\times 10^{-35}\) \\  & Std. & \(\mathbf{1.91\times 10^{-104}}\) & \(1.614\) & \(5.10\times 10^{-17}\) & \(1.84\times 10^{-3}\) & \(1.84\times 10^{-1}\) & \(1.41\times 10^{-1}\) & \(9.86\times 10^{-35}\) \\ \(F_{3}\) & Mean & \(\mathbf{3.27\times 10^{-180}}\) & \(1.50\times 10^{3}\) & \(7.70\times 10^{-10}\) & \(14.0\) & \(1.66\times 10^{3}\) & \(43.1\) & \(1.51\times 10^{-14}\) \\  & Std. & \(\mathbf{0.00}\) & \(7.07\times 10^{2}\) & \(7.36\times 10^{-9}\) & \(7.13\) & \(6.72\times 10^{2}\) & \(8.97\) & \(6.55\times 10^{-14}\) \\ \(F_{4}\) & Mean & \(\mathbf{1.56\times 10^{-104}}\) & \(2.44\times 10^{-5}\) & \(91.7\) & \(6.00\times 10^{-1}\) & \(1.11\times 10^{-1}\) & \(8.80\times 10^{-1}\) & \(1.95\times 10^{-5}\) \\  & Std. & \(\mathbf{3.47\times 10^{-105}}\) & \(1.89\times 10^{-5}\) & \(56.7\) & \(1.72\times 10^{-1}\) & \(4.75\times 10^{-2}\) & \(2.50\times 10^{-1}\) & \(4.49\times 10^{-4}\) \\ \(F_{5}\) & Mean & \(\mathbf{2.56\times 10^{-1}}\) & \(1.36\times 10^{2}\) & \(5.57\times 10^{2}\) & \(49.3\) & \(79.7\) & \(1.18\times 10^{2}\) & \(28.4\) \\  & Std. & \(\mathbf{4.78\times 10^{-1}}\) & \(1.54\times 10^{2}\) & \(41.6\) & \(38.9\) & \(73.9\) & \(1.43\times 10^{2}\) & \(8.40\times 10^{-1}\) \\ \(F_{6}\) & Mean & \(\mathbf{3.76\times 10^{-7}}\) & \(5.72\times 10^{-7}\) & \(3.15\times 10^{-1}\) & \(6.92\times 10^{-2}\) & \(6.94\times 10^{-3}\) & \(2.02\times 10^{-2}\) & \(3.67\) \\  & Std. & \(\mathbf{1.23\times 10^{-7}}\) & \(2.44\times 10^{-7}\) & \(9.98\times 10^{-2}\) & \(2.87\times 10^{-2}\) & \(3.61\times 10^{-3}\) & \(7.43\times 10^{-3}\) & \(3.35\times 10^{-1}\) \\ \(F_{7}\) & Mean & \(\mathbf{2.71\times 10^{-6}}\) & \(8.82\times 10^{-5}\) & \(6.79\times 10^{-4}\) & \(8.94\times 10^{-2}\) & \(6.62\times 10^{-2}\) & \(5.24\times 10^{-2}\) & \(1.80\times 10^{-3}\) \\  & Std. & \(\mathbf{2.33\times 10^{-6}}\) & \(6.94\times 10^{-5}\) & \(3.29\times 10^{-3}\) & \(2.06\times 10^{-2}\) & \(4.23\times 10^{-2}\) & \(1.37\times 10^{-2}\) & \(4.62\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of different methods in solving unimodal test functions.