# Particle Swarm Optimization Based on a Novel Evaluation of Diversity
## 4 Experiments and Comparisons
### Sensitivity Analysis  
In this article, two parameters (\(\alpha\) and \(N_{\mathrm{s}}\)) are very critical to performance of PSO-ED. In order to examine the influence of the parameters on precision and efficiency, a sensitivity analysis was conducted to each parameter. As shown in Equation (7), \(\alpha\) is a user-defined threshold used to judge whether the swarm is evolving into a better result. A proper value of \(\alpha\) can reflect the evolutionary state of the swarm and help the swarm choose an appropriate evolution strategy. In order to determine an appropriate value of \(\alpha\), we have selected five values (1.0, 0.1, 0.01, 0.001 and 0.0001) and conducted for 30 runs independently for each value on 12 test functions. Note that the other parameters are set as follows: \(N_{\mathrm{s}}=40\), \(N_{total}=FEs/N_{\mathrm{s}}\), \(N_{smin}=0.01N_{total}\), \(N_{smax}=0.1N_{total}\), \(N_{sumax}=0.01N_{total}\), \(v_{ub}=0.01(x_{ub}-x_{lb})\). As shown in Table 3, the statistical values (_Mean_ and _Rank_) are listed and it is obvious that the result is best (smallest average ranking) when \(\alpha=0.01\). \begin{table}
\begin{tabular}{l l r} \hline \hline
**No.** & **Function Name** & \(F_{i}^{\star}=F_{i}(x^{\star})\) \\ \hline
1 & Rotated High Conditioned Elliptic Function & 100 \\
2 & Rotated Cigar Function & 200 \\
3 & Shifted and Rotated Ackley’s Function & 300 \\
4 & Shifted and Rotated Rastrigin’s Function & 400 \\
5 & Shifted and Rotated Schwefel’s Function & 500 \\
6 & Hybrid Function (\(N=3\)) & 600 \\
7 & Hybrid Function (\(N=4\)) & 700 \\
8 & Hybrid Function (\(N=5\)) & 800 \\
9 & Composition Function 1 (\(N=3\)) & 900 \\
10 & Composition Function 3 (\(N=5\)) & 1100 \\
11 & Composition Function 6 (\(N=7\)) & 1400 \\
12 & Composition Function 7 (\(N=10\)) & 1500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The 12 CEC2015 benchmark functions. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Alpha** & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**0.1**} & \multicolumn{2}{c}{**0.01**} & \multicolumn{2}{c}{**0.0001**} \\
**Item** & _Mean_ & _Rank_ & _Mean_ & _Rank_ & _Mean_ & _Rank_ & _Mean_ & _Rank_ \\ \hline F1 & \(9.12\times 10^{4}\) & 4 & \(8.33\times 10^{4}\) & 3 & \(3.03\times 10^{5}\) & 5 & \(8\times 10^{4}\) & 1 & \(8.27\times 10^{4}\) & 2 \\ F2 & \(2.07\times 10^{3}\) & 4 & \(1.96\times 10^{3}\) & 2 & \(2.24\times 10^{2}\) & 1 & \(1.97\times 10^{3}\) & 3 & \(2.29\times 10^{3}\) & 5 \\ F3 & \(2\times 10^{1}\) & 1 & \(2\times 10^{1}\) & 1 & \(2\times 10^{1}\) & 1 & \(2\times 10^{1}\) & 1 \\ F4 & \(2.02\times 10^{2}\) & 5 & \(1.89\times 10^{2}\) & 2 & \(1.42\times 10^{2}\) & 1 & \(1.99\times 10^{2}\) & 4 & \(1.95\times 10^{2}\) & 3 \\ F5 & \(3.92\times 10^{3}\) & 3 & \(3.91\times 10^{3}\) & 2 & \(3.10\times 10^{3}\) & 1 & \(3.95\times 10^{3}\) & 4 & \(3.99\times 10^{3}\) & 5 \\ F6 & \(3.83\times 10^{4}\) & 5 & \(3.05\times 10^{4}\) & 3 & \(3.05\times 10^{4}\) & 3 & \(2.74\times 10^{4}\) & 1 & \(2.95\times 10^{4}\) & 2 \\ F7 & \(1.22\times 10^{1}\) & 3 & \(1.26\times 10^{1}\) & 5 & \(1.04\times 10^{1}\) & 1 & \(1.16\times 10^{1}\) & 2 & \(1.25\times 10^{1}\) & 4 \\ F8 & \(1.52\times 10^{4}\) & 2 & \(1.77\times 10^{4}\) & 3 & \(1.46\times 10^{4}\) & 1 & \(1.77\times 10^{4}\) & 3 & \(2.29\times 10^{4}\) & 5 \\ F9 & \(1.47\times 10^{2}\) & 4 & \(1.34\times 10^{2}\) & 2 & \(1.04\times 10^{2}\) & 1 & \(1.59\times 10^{2}\) & 5 & \(1.46\times 10^{2}\) & 3 \\ F10 & \(7.72\times 10^{2}\) & 2 & \(8.27\times 10^{2}\) & 4 & \(3.35\times 10^{2}\) & 1 & \(1.01\times 10^{3}\) & 5 & \(8.12\times 10^{2}\) & 3 \\ F11 & \(3.30\times 10^{4}\) & 3 & \(3.14\times 10^{4}\) & 2 & \(2.54\times 10^{4}\) & 1 & \(3.32\times 10^{4}\) & 4 & \(3.35\times 10^{4}\) & 5 \\ F12 & \(1\times 10^{2}\) & 1 & \(1\times 10^{2}\) & 1 & \(1\times 10^{2}\) & 1 & \(1\times 10^{2}\) & 1 & \(1\times 10^{2}\) & 1 \\ AR & & 3.08 & 2.50 & & 1.50 & & 2.83 & & 3.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sensitivity analysis for \(\alpha\). Another important parameter is the swarm size \(N_{s}\).