# 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} \hline \hline
**Problems** & **MOFECO** & **NSGA-II** & **MOPSO** & **PESA-II** & **KnEA** & **NSLS** \\ \hline ZDT1 & \(8.70\times 10^{-1}\) & \(8.70\times 10^{-1}\) (\(\sim\)) & \(8.58\times 10^{-1}\) (+) & \(8.62\times 10^{-1}\) (+) & \(7.62\times 10^{-1}\) (+) & \(8.66\times 10^{-1}\) (+) \\ ZDT2 & \(5.37\times 10^{-1}\) & \(5.35\times 10^{-1}\) (+) & \(5.28\times 10^{-1}\) (+) & \(5.25\times 10^{-1}\) (+) & \(3.90\times 10^{-1}\) (+) & \(4.95\times 10^{-1}\) (+) \\ ZDT4 & \(8.68\times 10^{-1}\) & \(8.61\times 10^{-1}\) (+) & \(6.88\times 10^{-1}\) (+) & \(8.62\times 10^{-1}\) (+) & \(6.64\times 10^{-1}\) (+) & \(6.15\times 10^{-1}\) (+) \\ ZDT6 & \(4.33\times 10^{-1}\) & \(4.33\times 10^{-1}\) (\(\sim\)) & \(3.95\times 10^{-1}\) (+) & \(4.28\times 10^{-1}\) (+) & \(4.30\times 10^{-1}\) (+) & \(4.33\times 10^{-1}\) (\(\sim\)) \\ DTLZ2 & \(6.70\times 10^{-1}\) & \(7.08\times 10^{-1}\) (-) & \(6.61\times 10^{-1}\) (+) & \(6.94\times 10^{-1}\) (-) & \(7.21\times 10^{-1}\) (-) & \(7.46\times 10^{-1}\) (-) \\ DTLZ4 & \(8.08\times 10^{-1}\) & \(6.92\times 10^{-1}\) (+) & \(6.78\times 10^{-1}\) (+) & \(7.14\times 10^{-1}\) (+) & \(7.09\times 10^{-1}\) (+) & \(7.07\times 10^{-1}\) (+) \\ DTLZ5 & \(1.33\times 10^{-1}\) & \(1.33\times 10^{-1 The \(GD\) values are used to measure the convergence of a Pareto solution set. From the Table 16, it can be seen that the proposed MOFECO performed well on ZDT, DTLZ, MaF and WFG test problems. In terms of \(GD\) indicator, the proposed MOFECO is better than NSGA-II, MOPSO, PESA-II, KnEA and NSLS on more than 10 test functions among the 15 test functions. Specifically, among the four ZDT test problems, MOFECO can achieve the smallest \(GD\) values on the ZDT test problems except for ZDT1 compared with other algorithms. Among the five DTLZ test problems, MOFECO can achieve the smallest \(GD\) values on DTLZ, DTLZ5 and DTLZ6 compared with NSGA-II, MOPSO, PESA-II and KnEA, while MOFECO does not work well on DTLZ4 and DTLZ7. Among the three MaF and three WFG test problems, MOFECO can achieve the smallest \(GD\) values on all the MaF test problems, and the smallest \(GD\) values on MaF2 and MaF12. These experimental results indicate that MOFECO is competitive in solving ZDT, DTLZ, WFG and MaF test problems with better convergence. The \(PD\) values are used to measure the diversity of the Pareto solution set. Generally speaking, large \(PD\) values represent good diversity of solution sets. It can be seen from Table 17 that MOFECO can obtain the largest \(PD\) values on at least 10 test functions compared with the other five algorithms, particularly, it achieves a good performance on all the four ZDT test problems with two objectives in terms of \(PD\). While on the five DTLZ test functions, the PD values obtained by MOFECO on DTLZ2 and DTLZ7 are slightly worse than those obtained by NSGA-II and MOPSO. But on all the five DTLZ test problems, MOFECO can show good performance compared with KnEA and NSLS. Among the four-objective test functions (WFG2, WFG3, WFG4, MaF1, MaF2 and MaF12), MOFECO can achieve the largest \(PD\) values except for WFG3 compared with NSGA-II, MOPSO, PESA-II, KnEA and NSLS. From those empirical results, we can confirm that the proposed MOFECO algorithm is promising in solving MOO problems, and it has advantages in maintaining the diversity of obtained Pareto solution sets, because of its novel characteristics, such as dividing the population into several independent cycles, relating the new solutions generated in the update process to the optimal solution in each cycle, and the differences in the optimal solutions obtained between different cycles, etc. As mentioned earlier, \(HV\) is one of the indicators to measure the comprehensive performance of a Pareto solution set. Generally, the larger the \(HV\) value is, the better the comprehensive performance of an algorithm has. From Table 18, we can find that MOFECO can shows the largest \(HV\) values on all