## 4 Simulation Results and Comparisons
### Example (2): Proposed AUFPSO-NTE in Comparison with FVFP-PSO, FP-PSO, FV-PSO, and PSO  
Example (2) compared the performance of the proposed AUFPSO-NTE with the standard PSO and with three modifications of PSO proposed by [PERSON] et al. [36]: particle swarm optimization with the fractional-order velocity and the fractional-order position (FVFP-PSO), particle swarm optimization with the fractional-order position (FP-PSO), and particle swarm optimization with the fractional-order velocity (FV-PSO). Table 11 shows that the number of dimensions for functions \(f_{5}\) and \(f_{8}\)-\(f_{11}\) was set to \(D_{n}=10\), the number of particles was set to \(S=30\), and the number of iterations was set to \(N=300\). Table 12 shows the parameter settings for the proposed AUFPSO-NTE, FVFP-PSO, FP-PSO, FV-PSO, and PSO. Parameters \(\omega_{\text{min}}\), \(\omega_{\text{max}}\), \(c_{1\text{min}}\), \(c_{1\text{max}}\), \(c_{2\text{min}}\), and \(c_{2\text{max}}\) of the proposed AUFPSO-NTE were set as in Example (1). In FVFP-PSO, FP-PSO, and FV-PSO, the fractional-order velocities and positions were affected by factors \(\varepsilon\) and \(\zeta\). Notably, factors \(\omega\), \(\varepsilon\), and \(\zeta\) underwent time-varying evolution in FVFP-PSO, FP-PSO, and FV-PSO. Table 12 also shows that \(c_{1}\) and \(c_{2}\) were 1. In Table 13, \(\lambda\) is the fractional order of the derivative and \(\alpha\), \(\beta\), and \(\gamma\) are coefficients that influence \(\omega\), \(c_{1}\), and \(c_{2}\), respectively. Table 13 shows the best combinations of constant coefficients \(\lambda\), \(\alpha\), \(\beta\), and \(\gamma\) for each benchmark function of the proposed AUFPSO-NTE. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Terms**} & **AUFPSO-NTE** & **FPSO** & **PSO-FOV** & **MPSO** & **PSO** \\ \hline \multirow{2}{*}{Bohachevsky 1} & Best & 0 & 0 & 0 & 0 & 0.1118 \\  & Mean & 0 & \(5.6621\times 10^{-16}\) & 0.0138 & 0.0275 & 5.1798 \\  & S.D. & 0 & \(1.8519\times 10^{-15}\) & 0.0754 & 0.1048 & 6.5905 \\ \hline \multirow{2}{*}{Colville} & Best & \(7.7920\times 10^{-5}\) & \(1.3908\times 10^{-3}\) & \(4.0173\times 10^{-3}\) & 0.2002 & 11.7514 \\  & Mean & 1.6906 & 2.4786 & 2.8318 & 5.9281 & 223.7255 \\  & S.D. & 1.8018 & 2.0702 & 2.8068 & 9.8226 & 462.9368 \\ \hline \multirow{2}{*}{Drop wave} & Best & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-0.9803\) \\  & Mean & \(-1\) & \(-0.9763\) & \(-0.9617\) & \(-0.9660\) & \(-0.7892\) \\  & S.D. & \(1.9387\times 10^{-11}\) & 0.0310 & 0.0318 & 0.0324 & 0.1580 \\ \hline \multirow{2}{*}{Easom} & Best & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-0.9103\) \\  & Mean & \(-1\) & \(-1\) & \(0.9998\) & \(-0.9998\) & \(-0.0721\) \\  & S.D. & \(7.6828\times 10^{-11}\) & \(6.8674\times 10^{-6}\) & \(9.0521\times 10^{-4}\) & \(1.0572\times 10^{-3}\) & 0.1996 \\ \hline \multirow{2}{*}{Rastrigin} & Best & 0 & 0 & 94.1343 & 0.0151 & 24.6608 \\  & Mean & \(5.1159\times 10^{-14}\) & 0.1044 & 153.9817 & 133.8380 & 223.4087 \\  & S.D. & \(6.5665\times 10^{-14}\) & 0.3025 & 27.1046 & 77.8643 & 75.7597 \\ \hline \multirow{2}{*}{Michalewiczes} & Best & \(-1.8409\) & \(-1.8409\) & \(-1.8409\) & \(-1.8409\) & \(-1.8388\) \\  & Mean & \(-1.8409\) & \(-1.8409\) & \(-1.8409\) & \(-1.8409\) & \(-1.8388\) \\  & S.D. & \(2.9272\times 10^{-10}\) & \(9.4381\times 10^{-8}\) & \(1.2377\times 10^{-7}\) & \(3.9510\times 10^{-7}\) & \(4.2741\times 10^{-7}\) \\ \hline \multirow{2}{*}{Rosenbrock’s valley} & Best & \(4.0000\times 10^{-34}\) & \(1.2791\times 10^{-21}\) & \(4.8507\times 10^{-16}\) & \(3.9505\times 10^{-14}\) & 0.0177 \\  & Mean & \(3.9129\times 10^{-22}\) & \(1.3965\times 10^{-4}\) & \(2.5900\times 10^{-3}\) & \(2.8691\times 10^{-3}\) & 4.2239 \\ \cline{1-1}  & S.D. & \(1.6158\times 10^{-21}\) & \(6.4890\times 10^{-4}\) & 0.0089 & 0.0157 & 7.7636 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Superior results obtained by the proposed AUFPSO-NTE in comparison with FPSO, PSO-FOV, MPSO, and PSO in 30 independent trials in Example (1).