# An Enhanced Particle Swarm Optimization (PSO) Algorithm Employing Quasi-Random Numbers
## 3 Results
### Constant Inertia vs. Random Inertia:
 } & \multirow{2}{*}{85} & \multirow{2}{*}{54} & \multirow{2}{*}{48} & \multirow{2}{*}{1.9} & \multirow{2}{*}{1.88} & \multirow{2}{*}{3.9} & \multirow{2}{*}{11.1\%} \\ \cline{2-2} \cline{6-6  
The velocity of each particle in the PSO algorithm at iteration \(n\) is updated according to the following equation:  
\[v_{n+1}^{jj}=\omega*v_{n}^{ij}+c_{1}*r_{1}*\left(P_{n}^{j}-X_{n}^{ij}\right)+c_{2}* r_{2}*\left(G_{n}{}^{j}-X_{n}^{ij}\right) \tag{16}\]  
In the above equation, \(\omega\) is the inertia weight. For constant inertia, \(\omega\) was set to 0.75. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function Type**} & **PSO Sobol** & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**PSO Sobol**} & \multicolumn{1}{c}{**Standard**} & \multicolumn{1}{c}{**Standard**} & \multirow{2}{*}{**Standard**} & \multirow{2}{*}{**Improvement in**} \\  & **Contant Weight** & \multicolumn{1}{c}{**Conventional**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**PSO Sobol**} & \multicolumn{1}{c}{**Conventional**} & \multicolumn{1}{c}{**PSO Sobol**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**Conventional**} \\  & **Inertia** & \multicolumn{1}{c}{**Inertia**} & \multicolumn{1}{c}{**Inertia**} & \multicolumn{1}{c}{**Contant Weight**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**(Conventional**} \\  & & & & & & & & \\ \hline Cigar— & & & & & & & & \\ Continuous & 183 & 96 & 85 & 5.2 & 2.1 & 2.2 & 11.5\% \\ \hline Ellipse— & & & & & & & \\ Continuous & 92 & 63 & 56 & 3.9 & 1.2 & 1.6 & 11.1\% \\ \hline Parabola— & & & & & & & \\ Continuous & 82 & 61 & 52 & 1.91 & 1.3 & 1.9 & 14.8\% \\ \hline Cigar— & & & & & & & \\ Ellipse–Mixed & 147 & 260 & 166 & 11.2 & 44.8 & 16.4 & 36.2\% \\ \hline Ellipse–Mixed & 84 & 140 & 100 & 4.4 & 11.9 & 6.9 & 28.6\% \\ \hline Parabola— & & & & & & & \\ Mixed & 78 & 133 & 86 & 2.5 & 30.1 & 5.7 & 35.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Comparison of performances of random inertial weight-aided Sobol and conventional PSO for the benchmark functions with 15 dimensions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function Type**} & **PSO Sobol** & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**Standard**} & \multicolumn{1}{c}{**Standard**} & \multirow{2}{*}{**Standard**} & \multirow{2}{*}{**Improvement in**} \\  & **Contant Weight** & \multicolumn{1}{c}{**Conventional**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**PSO Sobol**} & \multicolumn{1}{c}{**Conventional**} & \multicolumn{1}{c}{**PSO Sobol**} & \multicolumn{1}{c}{**Random Inertia**} \\  & **Inertia** & \multicolumn{1}{c}{**Inertia**} & \multicolumn{1}{c}{**Inertia**} & \multicolumn{1}{c}{**Contant Weight**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**Random Weight**} & \multicolumn{1}{c}{**(Conventional**} \\  & & & & & & & & \\ \hline Cigar— & & & & & & & & \\ Continuous & 167 & 76 & 67 & 2.4 & 2.6 & 1.8 & 11.8\% \\ \hline Ellipse– & & & & & & & & \\ Continuous & 70 & 35 & 30 & 4.1 & 1.8 & 1.7 & 14.3\% \\ \hline Parabola— & & & & & & & \\ Mixed & 67 & 31 & 29 & 1.2 & 2.3 & 1.6 & 6.5\% \\ \hline Cigar— & & & & & & & & \\ Mixed & 119 & 177 & 136 & 6.1 & 9.2 & 16.1 & 23.2\% \\ \hline Ellipse–Mixed & 62 & 99 & 84 & 4.9 & 8.0 & 17.2 & 15.2\% \\ \hline Parabola— & & & & & & & \\ Mixed & 62 & 89 & 71 & 3.5 & 8.2 & 4.5 & 20.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Comparison of performances of random inertial weight-aided Sobol and conventional PSO for the benchmark functions with 10 dimensions. For random inertia weight, \(\omega\) was set to \(0.5+\frac{rand}{2}\) as per [21]. The comparison results are provided in the table below. The improvement percentage in the table below is the comparison between random inertial weight-aided Sobol and conventional PSO variants.