## 4 Simulation Results and Comparisons
### Example (2): Proposed AUFPSO-NTE in Comparison with FVFP-PSO, FP-PSO, FV-PSO, and PSO
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Function** & \begin{tabular}{c} **Number of** \\ **Dimension (\(D_{n}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Number of** \\ **Particles (S)** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Iterations (I)** \\ \end{tabular} \\ \hline \(f_{5}\) & Rastrigin & 10 & 30 & 300 \\ \(f_{8}\) & Sphere & 10 & 30 & 300 \\ \(f_{9}\) & Ackley & 10 & 30 & 300 \\ \(f_{10}\) & Rosenbrock & 10 & 30 & 300 \\ \(f_{11}\) & Griewank & 10 & 30 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Settings for the number of dimensions, number of particles, and number of iterations in the proposed AUFPSO-NTE, fractional-order position (FP-PSO), fractional-order velocity (FV-PSO), FVFP-PSO, and PSO in Example (2). Table 14 compares the mean values obtained using the proposed AUFPSO-NTE, FVFP-PSO, FP-PSO, FV-PSO, and PSO developed by [PERSON] et al. [36] for \(f_{5}\) and \(f_{8}\)-\(f_{11}\) in 100 independent trials. Table 14 shows that the means obtained by the proposed AUFPSO-NTE were better than those obtained using FVFP-PSO, FP-PSO, FV-PSO, and PSO. Example (3): Comparison of the Proposed AUFPSO-NTE with AFO-FPSO, NCPSO, FO-DPSO, FPSO, APSO, DPSO, HPSO, and PSO  
Example (3) compares the performance of the proposed AUFPSO-NTE with the AFO-DPSO developed by [PERSON] et al. [38], and the NCPSO (new chaos PSO), FO-DPSO (fractional-order Darwinian PSO), FPSO, APSO (Adaptive PSO), DPSO (Darwinian PSO), HPSO (hybrid PSO), and PSO developed by [PERSON] et al. [38]. The AFO-DPSO introduces fractional-order velocity into a Darwinian PSO algorithm and includes a mutation mechanism to overcome premature convergence, NCPSO improves the chaos-PSO algorithm, FO-DPSO is a fractional-order Darwinian PSO, APSO provides adaptive PSO to enable automatic control of parameters, DPSO is Darwinian particle swarm optimization, and HPSO combines the concept of evolutionary computation with PSO. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & \(\Lambda\) & \(\alpha\) & \(\beta\) & \(\gamma\) \\ \hline Rastrigin & 1.3333 & 0.4444 & 0.2222 & 0.8889 \\ [PERSON] & 1.4689 & 0.5306 & 0.2595 & 0.9879 \\ [PERSON] & 1.5538 & 0.3519 & 0.1248 & 0.7688 \\ [PERSON] & 1.2 & 0 & 1.7235 & 1.8864 \\ [PERSON] & 1.5111 & 0.2765 & 0.1136 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The best combinations of the constant coefficients for each benchmark function of the proposed AUFPSO-NTE in Example (2). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Terms**} & \multicolumn{6}{c}{**Algorithms**} \\ \cline{2-7}  & & **AUFPSO-NTE** & **FVFP-PSO** & **FP-PSO** & **FV-PSO** & **PSO** \\ \hline \multirow{2}{*}{\(\omega\)} & min & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\  & max & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 \\ \hline \multirow{2}{*}{\(c_{1}\)} & min & 0 & & & & \\  & max & 2 & 1 & 1 & 1 & 1 \\ \hline \multirow{2}{*}{\(c_{2}\)} & min & 0 & & & & \\  & max & 2 & 1 & 1 & 1 & 1 \\ \hline \multirow{2}{*}{\(\varepsilon\)} & min & & 0.1 & & & 0.1 \\  & max & & 1.2 & 1 & 1.2 & N/A \\ \hline \multirow{2}{*}{\(\zeta\)} & min & & 0.1 & 0.1 & & \\  & max & N/A & 1.2 & 1.2 & 1 & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Parameters settings for the proposed AUFPSO-NTE, FVFP-PSO, FP-PSO, FV-PSO, and PSO in Example (2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **AUFPSO-NTE** & **FVFP-PSO** & **FP-PSO** & **FV-PSO** & **PSO** \\ \hline Rastrigin & 0 & 0 & 3.4182 & 20.3351 & 18.3371 \\ Sphere & \(2.8280\times 10^{-41}\) & \(8.9588\times 10^{-36}\) & \(1.9469\times 10^{-19}\) & 941.4338 & \(8.2506\times 10^{-12}\) \\ Ackley & \(8.8818\times 10^{-16}\) & \(8.4555\times 10^{-15}\) & 0.0299 & 10.4455 & 0.0231 \\ Rosenbrock & \(7.7881\) & \(8.0633\) & \(8.8267\) & \(2.5590\times 10^{5}\) & 56.5664 \\ Griewank & 0 & 0.0013 & 0.3770 & 10.3937 & 0.1041 \\ \hline \hline \end{tabular}
\end{table}