# An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms
## 4 Experiments
### Quantitative Performance Evaluations
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & **Test Functions** & **MOPSO** & **IC** & **Cultural** & **Cultural** & \multirow{2}{*}{**NSGA-II**} & **CD** \\  & & & **MOPSO** & **MOQPSO** & **MOPSO** & & & **MOPSO** \\ \hline DTLZ2 & mean & \(6.74\times 10^{-1}\) & \(7.15\times 10^{-1}\) & \(4.27\times 10^{-1}\) & \(2.46\times 10^{-1}\) & \(6.93\times 10^{-1}\) & \(7.20\times 10^{-1}\) \\  & std & \(1.50\times 10^{-2}\) & \(9.74\times 10^{-3}\) & \(2.41\times 10^{-2}\) & \(1.26\times 10^{-2}\) & \(8.35\times 10^{-3}\) & \(8.12\times 10^{-3}\) \\  & \(p\)-value & \(4.76\times 10^{-18}\) & & \(3.83\times 10^{-54}\) & \(1.33\times 10^{-78}\) & \(3.17\times 10^{-13}\) & \(3.85\times 10^{-2}\) \\ \hline DTLZ3 & mean & \(0.00\) & \(7.16\times 10^{-1}\) & \(7.15\times 10^{-1}\) & \(0.00\) & \(0.00\) & \(0.00\) \\  & std & \(0.00\) & \(7.63\times 10^{-3}\) & \(9.61\times 10^{-3}\) & \(0.00\) & \(0.00\) & \(0.00\) \\  & \(p\)-value & \(8.20\times 10^{-108}\) & & \(9.47\times 10^{-1}\) & \(8.20\times 10^{-108}\) & \(8.20\times 10^{-108}\) & \(8.20\times 10^{-108}\) \\ \hline DTLZ4 & mean & \(6.82\times 10^{-1}\) & \(7.29\times 10^{-1}\) & \(5.46\times 10^{-1}\) & \(1.67\times 10^{-1}\) & \(6.65\times 10^{-1}\) & \(7.22\times 10^{-1}\) \\  & std & \(1.56\times 10^{-2}\) & \(7.45\times 10^{-3}\) & \(5.70\times 10^{-2}\) & \(6.76\times 10^{-2}\) & \(3.11\times 10^{-2}\) & \(8.74\times 10^{-3}\) \\  & \(p\)-value & \(1.55\times 10^{-2}\) & \(9.53\times 10^{-2}\) & \(6.11\times 10^{-7}\) & \(9.22\times 10^{-1}\) & \(8.66\times 10^{-4}\) \\ \hline DTLZ5 & mean & \(4.34\times 10^{-1}\) & \(4.39\times 10^{-1}\) & \(4.35\times 10^{-1}\) & \(2.43\times 10^{-1}\) & \(4.38\times 10^{-1}\) & \(4.38\times 10^{-1}\) \\  & std & \(6.21\times 10^{-3}\) & \(4.94\times 10^{-3}\) & \(8.00\times 10^{-3}\) & \(4.96\times 10^{-3}\) & \(5.74\times 10^{-3}\) & \(5.86\times 10^{-3}\) \\  & \(p\)-value & \(1.65\times 10^{-3}\) & & \(3.72\times 10^{-2}\) & \(2.13\times 10^{-77}\) & \(5.64\times 10^{-1}\) & \(3.14\times 10^{-1}\) \\ \hline DTLZ6 & mean & \(4.29\times 10^{-1}\) & \(4.40\times 10^{-1}\) & \(4.21\times 10^{-1}\) & \(2.36\times 10^{-1}\) & \(0.00\) & \(4.39\times 10^{-1}\) \\  & std & \(6.99\times 10^{-3}\) & \(5.97\times 10^{-3}\) & \(7.53\times 10^{-3}\) & \(2.81\times 10^{-2}\) & \(0.00\) & \(6.85\times 10^{-3}\) \\  & \(p\)-value & \(1.43\times 10^{-8}\) & & \(5.64\times 10^{-15}\) & \(3.58\times 10^{-4}\) & \(9.89\times 10^{-102}\) & \(6.28\times 10^{-1}\) \\ \hline DTLZ7 & mean & \(1.27\) & \(1.43\) & \(1.21\) & \(5.44\times 10^{-1}\) & \(1.47\times 10^{-1}\) & \(1.38\) \\  & std & \(2.93\times 10^{-2}\) & \(3.20\times 10^{-2}\) & \(4.10\times 10^{-2}\) & \(8.36\times 10^{-2}\) & \(4.43\times 10^{-2}\) & \(4.25\times 10^{-2}\) \\  & \(p\)-value & \(1.19\times 10^{-28}\) & & \(2.91\times 10^{-31}\) & \(1.98\times 10^{-51}\) & \(5.57\times 10^{-73}\) & \(5.54\times 10^{-6}\) \\ \hline \hline \multicolumn{2}{c}{Better/similar/worse} & \multicolumn{1}{c}{\(12/0/0\)} & \multicolumn{1}{c}{\(9/3/0\)} & \multicolumn{1}{c}{\(11/1/0\)} & \multicolumn{1}{c}{\(11/1/0\)} & \multicolumn{1}{c}{\(5/6/1\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 6: Pareto fronts for DTLZ1: (**a**) the Pareto fronts obtained using normal domination criterion; (**b**) the Pareto fronts obtained using hypercube-based \(\varepsilon\)-domination.