# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Comparison with Other Optimization Algorithms
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problems** & **MOFECO** & **NSGA-II** & **MOPSO** & **PESA-II** & **KnEA** & **NSLS** \\ \hline ZDT1 & \(4.59\times 10^{-1}\) & \(4.01\times 10^{-1}\) (\(\ast\)) & \(6.69\times 10^{-1}\) (\(+\)) & \(9.18\times 10^{-1}\) (\(+\)) & \(6.98\times 10^{-1}\) (\(+\)) & \(3.69\times 10^{-1}\) (\(\ast\)) \\ ZDT2 & \(4.54\times 10^{-1}\) & \(4.59\times 10^{-1}\) (\(+\)) & \(7.91\times 10^{-1}\) (\(+\)) & \(1.02\times 10^{0}\) (\(+\)) & \(4.90\times 10^{-1}\) (\(+\)) & \(8.28\times 10^{-1}\) (\(+\)) \\ ZDT4 & \(5.40\times 10^{-1}\) & \(4.65\times 10^{-1}\) (\(\ast\)) & \(8.27\times 10^{-1}\) (\(+\)) & \(1.03\times 10^{0}\) (\(+\)) & \(9.64\times 10^{-1}\) (\(+\)) & \(3.50\times 10^{-1}\) (\(\ast\)) \\ ZDT6 & \(5.08\times 10^{-1}\) & \(5.19\times 10^{-1}\) (\(+\)) & \(1.21\times 10^{0}\) (\(+\)) & \(1.07\times 10^{0}\) (\(+\)) & \(4.44\times 10^{-1}\) (\(\ast\)) & \(5.75\times 10^{-1}\) (\(+\)) \\ DTLZ2 & \(4.86\times 10^{-1}\) & \(5.24\times 10^{-1}\) (\(+\)) & \(3.62\times 10^{-1}\) (\(\ast\)) & \(4.24\times 10^{-1}\) (\(\ast\)) & \(4.63\times 10^{-1}\) (\(\ast\)) & \(1.34\times 10^{-1}\) (\(\ast\)) \\ DTLZ4 & \(5.65\times 10^{-1}\) & \(4.99\times 10^{-1}\) (\(\ast\)) & \(3.41\times 10^{-1}\) (\(\ast\)) & \(4.04\times 10^{-1}\) (\(\ast\)) & \(8.17\times 10^{-1}\) (\(+\)) & \(3.40\times 10^{-1}\) (\(\ast\)) \\ DTLZ5 & \(4.95\times 10^{-1}\) & \(4.79\times 10^{-1}\) (\(\ast\)) & \(6.90\times 10^{-1}\) (\(+\)) & \(9.17\times 10^{-1}\) (\(+\)) & \(8.14\times 10^{-1}\) (\(+\)) & \(2.30\times 10^{-1}\) (\(\ast\)) \\ DTLZ6 & \(5.18\times 10^{-1}\) & \(6.89\times 10^{-1}\) (\(+\)) & \(9.97\times 10^{-1}\) (\(+\)) & \(1.23\times 10^{0}\) (\(+\)) & \(4.00\times 10^{-1}\) (\(\ast\)) & \(2.10\times 10^{-1}\) (\(\ast\)) \\ DTLZ7 & \(6.50\times 10^{-1}\) & \(5.10\times 10^{-1}\) (\(+\)) & \(4.50\times 10^{-1}\) (\(\ast\)) & \(5.10\times 10^{-1}\) (\(\ast\)) & \(3.45\times 10^{-1}\) (\(\ast\)) & \(1.93\times 10^{-1}\) (\(\ast\)) \\ WFG2 & \(4.41\times 10^{-1}\) & \(5.79\times 10^{-1}\) (\(+\)) & \(4.56\times 10^{-1}\) (\(+\)) & \(4.53\times 10^{-1}\) (\(+\)) & \(8.37\times 10^{-1}\) (\(+\)) & \(2.72\times 10^{-1}\) (\(\ast\)) \\ WFG3 & \(6.83\times 10^{-1}\) & \(6.11\times 10^{-1}\) (\(\ast\)) & \(4.93\times 10^{-1}\) (\(\ast\)) & \(5.76\times 10^{-1}\) (\(\ast\)) & \(6.47\times 10^{-1}\) (\(\ast\)) & \(1.77\times 10^{-1}\) (\(\ast\)) \\ WFG4 & \(4.13\times 10^{-1}\) & \(4.43\times 10^{-1}\) (\(+\)) & \(4.90\times 10^{-1}\) (\(+\)) & \(4.63\times 10^{-1}\) (\(+\)) & \(7.51\times 10^{-1}\) (\(+\)) & \(2.65\times 10^{-1}\) (\(\ast\)) \\ MaF1 & \(4.16\times 10^{-1}\) & \(4.77\times 10^{-1}\) (\(+\)) & \(4.22\times 10^{-1}\) (\(+\)) & \(4.52\times 10^{-1}\) (\(+\)) & \(6.33\times 10^{-1}\) (\(+\)) & \(2.68\times 10^{-1}\) (\(\ast\)) \\ MaF2 & \(4.44\times 10^{-1}\) & \(4.89\times 10^{-1}\) (\(+\)) & \(4.65\times 10^{-1}\) (\(+\)) & \(4.76\times 10^{-1}\) (\(+\)) & \(6.32\times 10^{-1}\) (\(+\)) & \(1.48\times 10^{-1}\) (\(\ast\)) \\ MaF12 & \(4.06\times 10^{-1}\) & \(4.57\times 10^{-1}\) (\(+\)) & \(4.15\times 10^{-1}\) (\(+\)) & \(4.12\times 10^{-1}\) (\(+\)) & \(7.52\times 10^{-1}\) (\(+\)) & \(5.67\times 10^{-1}\) (\(+\)) \\ + & / & 9 & 11 & 11 & 10 & 4 \\ - & / & 6 & 4 & 4 & 5 & 11 \\ \(\sim\) & / & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 20: \(S1\) result of the six compared algorithms on test functions. (“+”, “\(\sim\)” and “-” respectively represent that MOFECO is better than, similar to and inferior to the other five algorithms). the four ZDT test functions compared with NSGA-II, MOPSO, PESA-II, KnEA and NSLS. On the five DTLZ test functions, MOFECO can achieve larger or equal \(HV\) values on DTLZ4 to DTLZ6 compared with the other five algorithms, while on DTLZ2 and DTLZ7, MOFECO performs slightly worse. Among the four-objective test functions (WFG2, WFG3, WFG4, MaF1, MaF2 and MaF12), MOFECO has the largest \(HV\) values on at least five test functions compared with MOPSO, PESA-II and NSLS, but it performs worse compared with NSGA-II and KnEA.