# An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms
## 4 Experiments
### Quantitative Performance Evaluations
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Test Functions** & **MOPSO** & **IC** & **Cultural** & **Cultural** & **NSGA-II** & **CD** \\  & & **MOPSO** & **MOQPSO** & **MOPSO** & **NSGA-II** & **MOPSO** \\ \hline ZDT1 & mean & \(5.56\times 10^{-1}\) & \(8.73\times 10^{-1}\) & \(8.42\times 10^{-1}\) & \(8.66\times 10^{-1}\) & \(8.11\times 10^{-1}\) & \(8.74\times 10^{-1}\) \\  & standard deviation (std) & \(9.03\times 10^{-2}\) & \(5.38\times 10^{-3}\) & \(8.01\times 10^{-2}\) & \(7.00\times 10^{-3}\) & \(8.46\times 10^{-3}\) & \(4.76\times 10^{-3}\) \\  & \(p\)-value & \(8.42\times 10^{-2}\) & & \(3.72\times 10^{-2}\) & \(8.82\times 10^{-5}\) & \(6.65\times 10^{-40}\) & \(6.25\times 10^{-1}\) \\ \hline ZDT2 & mean & \(2.88\times 10^{-1}\) & \(5.38\times 10^{-1}\) & \(4.72\times 10^{-1}\) & \(4.20\times 10^{-1}\) & \(1.20\times 10^{-1}\) & \(5.37\times 10^{-1}\) \\  & std & \(1.19\times 10^{-1}\) & \(7.88\times 10^{-3}\) & \(1.07\times 10^{-1}\) & \(1.91\times 10^{-1}\) & \(2.22\times 10^{-2}\) & \(6.43\times 10^{-3}\) \\  & \(p\)-value & \(1.69\times 10^{-16}\) & \(1.40\times 10^{-3}\) & \(1.31\times 10^{-5}\) & \(6.53\times 10^{-16}\) & \(8.25\times 10^{-1}\) \\ \hline ZDT3 & mean & \(1.00\) & \(1.01\) & \(9.39\times 10^{-1}\) & \(7.61\times 10^{-1}\) & \(7.60\times 10^{-1}\) & \(1.01\) \\  & std & \(5.17\times 10^{-3}\) & \(4.01\times 10^{-3}\) & \(1.43\times 10^{-1}\) & \(4.00\times 10^{-1}\) & \(1.31\times 10^{-2}\) & \(4.37\times 10^{-3}\) \\  & \(p\)-value & \(1.72\times 10^{-7}\) & & \(0.0110\) & \(1.35\times 10^{-3}\) & \(2.35\times 10^{-46}\) & \(8.76\times 10^{-1}\) \\ \hline ZDT4 & mean & \(0.00\) & \(8.71\times 10^{-1}\) & \(8.67\times 10^{-1}\) & \(0.00\) & \(3.66\times 10^{-1}\) & \(0.00\) \\  & std & \(0.00\) & \(6.08\times 10^{-3}\) & \(5.18\times 10^{-3}\) & \(0.00\) & \(1.77\times 10^{-1}\) & \(0.00\) \\  & \(p\)-value & \(1.80\times 10^{-118}\) & \(1.59\times 10^{-2}\) & \(1.80\times 10^{-118}\) & \(1.80\times 10^{-118}\) & \(1.80\times 10^{-118}\) \\ \hline ZDT6 & mean & \(4.94\times 10^{-1}\) & \(5.04\times 10^{-1}\) & \(4.65\times 10^{-1}\) & \(5.03\times 10^{-1}\) & \(0.00\) & \(5.05\times 10^{-1}\) \\  & std & \(7.43\times 10^{-3}\) & \(4.88\times 10^{-3}\) & \(1.12\times 10^{-1}\) & \(7.63\times 10^{-3}\) & \(0.00\) & \(5.77\times 10^{-3}\) \\  & \(p\)-value & \(5.97\times 10^{-08}\) & & \(6.31\times 10^{-2}\) & \(8.61\times 10^{-1}\) & \(3.40\times 10^{-10}\) & \(8.76\times 10^{-1}\) \\ \hline DTLZ1 & mean & \(0.00\) & \(1.30\) & \(1.30\) & \(0.00\) & \(0.00\) & \(0.00\) \\  & std & \(0.00\) & \(2.02\times 10^{-3}\) & \(3.07\times 10^{-3}\) & \(0.00\) & \(0.00\) & \(0.00\) \\  & \(p\)-value & \(2.90\times 10^{-156}\) & & \(6.44\times 10^{-1}\) & \(2.90\times 10^{-156}\) & \(2.90\times 10^{-156}\) & \(2.90\times 10^{-156}\) \\ \hline \hline \end{Comparison of the Results Using Hypercube-Based \(\varepsilon\)-Domination and Normal Domination Hypervolume  
To demonstrate the effectiveness of the hypercube-based \(\varepsilon\)-domination criteria introduced in this paper, DTLZ1 and DTLZ7 are chosen as examples. Figures 6 and 7 show the Pareto fronts obtained using two kinds of dominate criterions for DTLZ1 and DTLZ7, respectively. It can be observed that the approximate Pareto fronts found by the improved Cultural MOPSO using hypercube-based \(\varepsilon\)-domination are more well-distributed. The adjacent members of the repository may be deleted based on the hypercube-based domination criterion, because they may locate in the same grid which is not expected in scenes like ZDT4, ZDT6 and DTLZ6. So, for ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, DTLZ5, and DTLZ6, normal domination criterion should be used.