## 4 Simulation Results and Comparisons
### Example (1): Proposed AUFPSO-NTE in Comparison with FPSO, PSO-FOV, MPSO, and PSO  
The FPSO was the first developed, called PSO-FOV (PSO with the fractional-order velocity), and improved by [PERSON] and coworkers [34; 35], the MPSO introduced a time-varying inertia weight [11], and PSO was first proposed by [PERSON] and [PERSON] [1]. All three algorithms were compared with the proposed AUFPSO-NTE. Table 5 shows the number of dimensions (\(D_{n}\)) that functions \(f_{1}\) to \(f_{7}\) were set to with \(D_{n}=2\), 4, 2, 2, 30, 2, and 4, respectively; the number of particles (\(S\)) was set to \(S=10\); and the number of iterations (\(I\)) was set to 200. Table 6 shows the parameter settings for the proposed AUFPSO-NTE and for the FPSO, PSO-FOV, MPSO, and PSO. In the AUFPSO-NTE, the minimum weight (\(\omega_{\min}\)) and maximum weight (\(\omega_{\max}\)) were set to 0.4 and 0.9, respectively. The minimum cognitive coefficient (\(c_{\min}\)) and maximum cognitive coefficient (\(c_{\max}\)) were set to 0 and 2, respectively. The minimum social coefficient (\(c_{\min}\)) and maximum social coefficient (\(c_{\max}\)) were set to 0 and 2, respectively. In the MPSO, \(\omega_{\min}\) and \(\omega_{\max}\) were set to 0.4 and 0.9, respectively, and \(c_{1}\) and \(c_{2}\) were both set to 2. For FPSO, PSO-FOV and PSO, \(c_{1}\) and \(c_{2}\) were both set to 2. The maximum velocity (\(V_{\max}\)) was defined as the maximum position minus minimum position, and the minimum velocity (\(V_{\min}\)) was defined as negative \(V_{\max}\). Table 7 shows the best combinations of the constant coefficients \(\lambda\), \(\alpha\), \(\beta\), and \(\gamma\) for each benchmark function of the proposed AUFPSO-NTE. For comparisons, the constant coefficient \(\lambda\) and results are obtained by the algorithms developed by [PERSON] and coworkers [34; 35]. Tables 8 and 9 show the constant coefficient \(\lambda\) for each benchmark function of the FPSO [35] and PSO-FOV [34], respectively. In Tables 7-9, \(\lambda\) is the fractional order of the derivative. In Table 7, \(\alpha\), \(\beta\), and \(\gamma\) are coefficients that influence \(\omega\), \(c_{1}\), and \(c_{2}\), respectively. Table 10 shows the performance comparison results for the proposed AUFPSO-NTE and for FPSO [35], PSO-FOV [34], MPSO [11], and PSO [1]. The table shows the best solution, mean, and standard deviation (S.D.) that each algorithm obtained for \(f_{1}\)-\(f_{7}\) in 30 independent trials. In Example (1), all algorithms except PSO obtained the best solutions for \(f_{1},f_{3},f_{4}\), and \(f_{6}\). Only the proposed AUFPSO-NTE and FPSO obtained the best solution for \(f_{5}\)[35]. Additionally, only the proposed AUFPSO-NTE obtained the best solution for \(f_{2}\) and \(f_{7}\). In terms of the mean and S.D., the proposed AUFPSO-NTE outperformed FPSO [35], PSO-FOV [34], MPSO [11], and PSO [1]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Terms**} & \multicolumn{6}{c}{**Algorithms**} \\ \cline{2-7}  & **AUFPSO-NTE** & **FPSO** & **PSO-FOV** & **MPSO** & **PSO** \\ \hline \multirow{2}{*}{\(c_{2}\)} & min & 0 & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} \\  & max & 2 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._ \begin{table}
\begin{tabular}{c c} \hline \hline
**Function** & \(\mathbf{\lambda}\) \\ \hline Bohachevsky 1 & 1.65 \\ [PERSON] & 1.75 \\ Drop wave & 1.43 \\ [PERSON] & 1.98 \\ [PERSON] & 1.99 \\ [PERSON] & 1.99 \\ Rosenbrock’s valley & 1.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The \(\lambda\) parameter values for each benchmark function of FPSO in Example (1). \begin{table}
\begin{tabular}{c c} \hline \hline
**Function** & \(\mathbf{\lambda}\) \\ \hline [PERSON] 1 & 1.65 \\ Colville & 1.75 \\ Drop wave & 1.43 \\ Easom & 1.98 \\ Rastrigin & 1.99 \\ [PERSON] & 1.99 \\ Rosenbrock’s valley & 1.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The \(\lambda\) parameter values for each benchmark function of PSO-FOV in Example (1).