## 5 Conclusions
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Function** & **Terms** & **AUFPSO-NTE** & **HAFPSO** & **GAPSO** & **HFPSO** & **PSO** & **PSO** \\ \hline \multirow{3}{*}{Rastrigin} & Mean & 0 & \(2.18\times 10^{-2}\) & \(6.76\times 10^{1}\) & \(8.83\times 10^{1}\) & \(7.40\times 10^{1}\) & \(7.90\times 10^{1}\) \\  & S.D. & 0 & \(2.83\times 10^{-2}\) & \(1.84\times 10^{1}\) & \(3.08\times 10^{1}\) & \(2.04\times 10^{1}\) & \(1.86\times 10^{1}\) \\ \hline \multirow{3}{*}{Sphere} & Mean & \(1.89\times 10^{-121}\) & \(2.15\times 10^{-9}\) & \(3.67\times 10^{-3}\) & \(1.43\times 10^{-5}\) & \(7.51\times 10^{-3}\) & \(4.57\times 10^{-5}\) \\  & S.D. & \(1.75\times 10^{-120}\) & \(3.54\times 10^{-9}\) & \(6.11\times 10^{-3}\) & \(8.05\times 10^{-6}\) & \(3.20\times 10^{-2}\) & \(1.61\times 10^{-4}\) \\ \hline \multirow{3}{*}{[PERSON]} & Mean & \(4.88\times 10^{1}\) & \(1.00\times 10^{2}\) & \(8.14\times 10^{1}\) & \(1.53\times 10^{2}\) & \(1.16\times 10^{2}\) & \(1.06\times 10^{2}\) \\  & S.D. & \(1.06\times 10^{-1}\) & \(5.57\times 10^{1}\) & \(4.16\times 10^{1}\) & \(5.93\times 10^{1}\) & \(5.56\times 10^{1}\) & \(4.86\times 10^{1}\) \\ \hline \multirow{3}{*}{Griewank} & Mean & 0 & \(1.27\times 10^{-2}\) & \(1.60\times 10^{-2}\) & \(9.88\times 10^{-1}\) & \(3.64\times 10^{-2}\) & \(5.81\times 10^{-2}\) \\  & S.D. & 0 & \(1.54\times 10^{-2}\) & \(2.06\times 10^{-2}\) & \(6.49\times 10^{-2}\) & \(6.61\times 10^{-2}\) & \(8.96\times 10^{-2}\) \\ \hline \multirow{3}{*}{Schwefel P1.2} & Mean & \(6.98\times 10^{-123}\) & \(1.03\times 10^{3}\) & \(3.99\times 10^{1}\) & \(1.15\times 10^{3}\) & \(6.32\times 10^{2}\) & \(2.22\times 10^{3}\) \\  & S.D. & \(6.49\times 10^{-122}\) & \(4.83\times 10^{2}\) & \(3.53\times 10^{1}\) & \(1.09\times 10^{3}\) & \(4.32\times 10^{2}\) & \(8.84\times 10^{2}\) \\ \hline \multirow{3}{*}{Quartic} & Mean & \(1.34\times 10^{-4}\) & \(1.34\times 10^{-2}\) & \(4.46\times 10^{-2}\) & \(3.70\times 10^{-2}\) & \(6.45\times 10^{-2}\) & \(5.23\times 10^{-2}\) \\  & S.D. & \(9.80\times 10^{-5}\) & \(3.91\times 10^{-3}\) & \(1.17\times 10^{-2}\) & \(1.28\times 10^{-2}\) & \(2.12\times 10^{-2}\) & \(1.90\times 10^{-2}\) \\ \hline \multirow{3}{*}{Salomon} & Mean & \(7.99\times 10^{-3}\) & \(6.40\times 10^{-1}\) & \(7.85\times 10^{-1}\) & \(1.15\) & \(1.27\) & \(1.13\) \\  & S.D. & \(2.72\times 10^{-2}\) & \(7.91\times 10^{-2}\) & \(1.38\times 10^{-1}\) & \(1.99\times 10^{-1}\) & \(3.60\times 10^{-1}\) & \(2.74\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 25: Performance results obtained using the proposed AUFPSO-NTE in comparison with HAFPSO, GAPSO, HFPSO, FPSO, and PSO for 7 benchmark functions in 100 independent trials in Example (4). **Author Contributions:** Formal analysis, P.-Y.Y. and F.-I.C. ; funding acquisition, J.-T.T. and J.-H.C.; methodology, J.-T.T. and J.-H.C.; software, P.-Y.Y. and [PERSON]-I.C. ; supervision, [PERSON] and [PERSON]; validation, P.-Y.Y. and F.-I.C. ; writing--original draft, P.-Y.Y. and F.-I.C. ; writing--review and editing, [PERSON] and [PERSON]. **Funding:** This research was funded in part by the Ministry of Science and Technology, Taiwan, R.O.C., grant numbers MOST 105-2221-E-992-304-MY3, MOST107-2221-E-992-086-MY3, and MOST107-2221-E-153-005-MY2, and in part by the \"Intelligent Manufacturing Research Center\" (iMRC) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan, R.O.C. And The APC was funded by MOST107-2221-E-992-086-MY3. **Conflicts of Interest:** The authors declare no conflict of interest.