## 4 Simulation Results and Comparisons
### Example (4): Comparison of the Proposed AUFPSO-NTE with HAFPSO, GAPSO, HFPSO, FPSO, and PSO  
Example (4) compares the performance of the proposed AUFPSO-NTE with the HAFPSO (hunter-attack fractional-order PSO) developed by [PERSON] et al. [38], GAPSO (genetic algorithm-PSO) [59], HFPSO (hybrid firefly algorithm and PSO) [60], FPSO, and PSO developed by [PERSON] et al. [38]. The HAFPSO introduces the concept of hunter-attack into the FODPSO and the GAPSO is a compound optimizer that introduces the crossover and mutation strategy of GA into PSO. The HFPSO combines the firefly optimization algorithm and PSO. Table 23 shows that the number of dimensions for functions \(f_{5},f_{8},f_{10},f_{11}\), and \(f_{13}\)-\(f_{15}\) was set to \(D_{n}=50\), the number of particles was set to \(S=30\), and the number of iterations was set to \(N=1000\). Parameters \(\omega_{\text{min}}\), \(\omega_{\text{max}}\), \(c_{\text{1 min}}\), \(c_{\text{1 max}}\), \(c_{\text{2 min}}\), and \(c_{\text{2 max}}\) of the proposed AUFPSO-NTE were set as in Example (1). Except for the fractional order value (\(\lambda\)) in HAFPSO being 0.6, the parameter settings for the HAFPSO, GAPSO, HFPSO, FPSO, and PSO were not mentioned. In Table 24, \(\lambda\) is the fractional order of the derivative and \(\alpha\), \(\beta\), and \(\gamma\) are coefficients that influence \(\omega\), \(c_{1}\), and \(c_{2}\), respectively. Table 24 shows the best combinations of the 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} \hline \hline
**Function** & **AUFPSO-NTE** & **AFO-FPSO** & **NCPSO** & **FO-DPSO** & **FO-PSO** \\ \hline Rastrigin & 0 & 0.0017 & 0.0043 & 0.0137 & 0.0232 \\ Sphere & \(1.6599\times 10^{-245}\) & 0.0031 & 0.0597 & 0.4091 & 0.7505 \\ Ackley & \(2.9170\times 10^{-61}\) & \(1.4637\times 10^{-5}\) & \(3.5416\times 10^{-4}\) & \(5.9058\times 10^{-4}\) & 0.0025 \\ Griewank & 0 & 0.0116 & 0.1912 & 0.6574 & 0.8151 \\ DeJong F4 & 0 & 0.0201 & 0.2765 & 0.7344 & 0.8836 \\ \hline \hline \end{tabular}
\end{table} 
Table 21: Variances in the optimum obtained using the proposed AUFPSO-NTE, AFO-FPSO, NCPSO, FO-DPSO, and FO-PSO for 5 benchmark functions in 30 independent trials. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **AUFPSO-NTE** & **APSO** & **DPSO** & **HPSO** & **PSO** \\ \hline Rastrigin & 0 & 0.0173 & 0.0774 & 0.2162 & 0.3488 \\ Sphere & \(1.6599\times 10^{-245}\) & 0.5126 & 1.0068 & 1.6022 & 2.0978 \\ Ackley & \(2.9170\times 10^{-61}\) & 0.0011 & 0.0162 & 0.0200 & 0.9074 \\ Griewank & 0 & 0.6819 & 0.9371 & 1.3658 & 1.6408 \\ DeJong F4 & 0 & 0.8381 & 0.9611 & 1.0130 & 1.7960 \\ \hline \hline \end{tabular}
\end{table} 
Table 22: Variances in the optimum obtained using the proposed AUFPSO-NTE, APSO, DPSO, HPSO, and PSO for 5 benchmark functions in 30 independent trials. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **AUFPSO-NTE** & **APSO** & **DPSO** & **HPSO** & **PSO** \\ \hline Rastrigin & 0 & 0.0173 & 0.0774 & 0.2162 & 0.3488 \\ Sphere & \(1.6599\times 10^{-245}\) & 0.5126 & 1.0068 & 1.6022 & 2.0978 \\ Ackley & \(2.9170\times 10^{-61}\) & 0.0011 & 0.0162 & 0.0200 & 0.9074 \\ Griewank & 0 & 0.6819 & 0.9371 & 1.3658 & 1.6408 \\ DeJong F4 & 0 & 0.8381 & 0.9611 & 1.0130 & 1.7960 \\ \hline \hline \end{tabular}
\end{table} 
Table 23: Settings for the number of dimensions, number of particles, and number of iterations in the proposed AUFPSO-NTE and for HAFPSO, GAPSO, HFPSO, FPSO, and PSO in Example (4). Table 25 shows the performance comparison results obtained using the proposed AUFPSO-NTE, and obtained using HAFPSO [38], GAPSO [59], HFPSO [60], FPSO, and PSO developed by [PERSON] et al. [38] for \(f_{5}\), \(f_{8}\), \(f_{10}\), \(f_{11}\), and \(f_{13}\)-\(f_{15}\) in 100 independent trials. The table shows that the means and S.D. obtained using the proposed AUFPSO-NTE were better than those obtained using HAFPSO [38], GAPSO [59], HFPSO [60], FPSO, and PSO.