## 4 Experimental and Discussions
### Experimental Results  
This section presents the numerical results obtained by MOEA/D, MOEA/D-DE\(+\)PSO [34] and MOEA/D-CPDE on DEB, ZDT problems and the CEC'09 test instances (UF1-UF10). Table 1 shows the results obtained by MOEA/D, MOEA/D-DE\(+\)PSO and MOEA/D-CPDE. The results of MOEA/D-DE\(+\)PSO are taken from literature [34]. It records the IGD-metric values after 100,000 function evaluations in 30 runs. There are five columns in Table 1. The columns refer to best (_i.e._, minimum), mean, standard deviation (std) and worst (_i.e._, maximum) of the _IGD_ values obtained by the algorithms for DEB and ZDT problems. The experimental results indicate that MOEA/D-CPDE performs better than MOEA/D and MOEA/D-DE\(+\)PSO on ZDT1, ZDT2, ZDT4 and ZDT6. The minimum of MOEA/D is the smallest on ZDT3. Table 1 concludes that the cloud differential evolution strategy plays an important role in guiding the population evolution and accelerating the convergence rate of MOEA/D-CPDE for DEB and ZDT problems. Table 2 shows the _IGD_ in terms of minimum, mean, standard deviation and maximum values obtained by MOEA/D, MOEA/D-DE\(+\)PSO and MOEA/D-CPDE for 30 independent times. The results  
of MOEA/D-DE+PSO are taken from literature [34]. Table 2 shows that, in terms of IGD-metric, MOEA/D-CPDE performs better on UF1, UF4-UF7. MOEA/D-DE+PSO performs better on UF2, UF3, UF8 and UF10. MOEA/D-CPDE performs better in terms of the standard deviation and maximum on UF9. MOEA/D-DE+PSO performs better in terms of the minimum and mean best solution on UF9. It may be noted that MOEA/D-CPDE has not performed well on a few functions. UF8-UF10 are three objective functions with complicated _PS_ shapes. The population diversity mechanism in MOEA/D-CPDE is worse for the complicated functions. Therefore, the complicated functions may be very challenging for MOEA/D-CPDE. Because of the poor convergence speed, MOEA/D performs worse for some complicated problems with the given number of function evaluations as the termination criterion. The dynamic use of DE and PSO enable MOEA/D-DE+PSO to get better approximate solutions for some problems. However, the individual produced by DE and PSO comes from two randomly divided subsets. Therefore, MOEA/D-DE+PSO performs worse for some problems. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Function & Min & Mean & Std & Max & Algorithms \\ \hline \multirow{3}{*}{DEB} & 0.005354 & 0.005355 & 4.7946e-7 & 0.005355 & MOEA/D \\  & - & - & - & - & MOEA/D-DE+PSO \\  & 0.005354 & **0.005354** & **5.0741e-7** & 0.005355 & MOEA/D-CPDE \\ \hline \multirow{3}{*}{ZDT1} & 0.003988 & 0.004054 & 5.1587e-5 & 0.004165 & MOEA/D \\  & 0.004012 & 0.004142 & 7.1000e-5 & 0.004359 & MOEA/D-DE+PSO \\  & **0.003966** & **0.004036** & **4.7909e-5** & **0.004088** & MOEA/D-CPDE \\ \hline \multirow{3}{*}{ZDT2} & 0.003799 & 0.003819 & 1.5762e-5 & 0.003864 & MOEA/D \\  & 0.003811 & 0.003874 & 4.2400e-5 & 0.003992 & MOEA/D-DE+PSO \\  & **0.003784** & **0.003808** & **1.1640e-5** & **0.003831** & MOEA/D-CPDE \\ \hline \multirow{3}{*}{ZDT3} & **0.007041** & 0.007084 & 3.4618e-5 & 0.007200 & MOEA/D \\  & 0.008400 & 0.009025 & 5.9532e-4 & 0.012406 & MOEA/D-DE+PSO \\  & 0.007049 & **0.007081** & **2.8241e-5** & **0.007171** & MOEA/D-CPDE \\ \hline \multirow{3}{*}{ZDT4} & 0.125709 & 0.195898 & 6.7636e-2 & 0.256064 & MOEA/D \\  & 0.004141 & 0.007550 & 1.9710e-3 & 0.011584 & MOEA/D-DE+PSO \\  & **0.003953** & **0.003954** & **8.0301e-7** & **0.003956** & MOEA/D-CPDE \\ \hline \multirow{3}{*}{ZDT6} & 0.008892 & 0.013408 & 3.6559e-3 & 0.017854 & MOEA/D \\  & 0.006977 & 0.014575 & 4.0370e-3 & 0.022697 & MOEA/D-DE+PSO \\ \cline{1-1}  & **0.005087** & **0.005970** & **6.5858e-4** & **0.006620** & MOEA/D-CPDE \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The Inverted Generational Distance (_IGD_) values for DEB, ZDT1\(\sim\)ZDT4 and ZDT6 approximated by multiobjective evolutionary algorithm based on decomposition (MOEA/D), MOEA/D-DE+PSO and MOEA/D-CPDE in 30 independent runs.