# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Comparison with Other Optimization Algorithms  
In this section, the performances of MOFECO were verified by empirically comparing it with three classic MOEAs NSGA-II [5], MOPSO [12], PESA-II [11], and two latest algorithms KnEA [2] and NSLS [15]. For a fair-minded comparison, we adopt the consistent parameters of population size \(N=100\) and termination condition \(T=1000\) for all the six algorithms. The other parameters of MOFECO are shown in Table 2. For the other parameters used in NSGA-II, MOPSO, PESA-II, KnEA and NSLS, they are consistent with the parameters in their original study [2; 5; 11; 12; 15]. The detailed parameters configuration for the five compared algorithms are shown in Table 15. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Functions** & **Combined Mutation** & **Cauchy Mutation** \\ \hline ZDT1 & \(\mathbf{8.70\times 10^{-3}}\) & \(9.10\times 10^{-3}\) \\ ZDT2 & \(\mathbf{7.50\times 10^{-3}}\) & \(8.10\times 10^{-3}\) \\ ZDT4 & \(\mathbf{7.40\times 10^{-3}}\) & \(8.80\times 10^{-3}\) \\ ZDT6 & \(\mathbf{6.80\times 10^{-3}}\) & \(1.74\times 10^{-2}\) \\ DTLZ2 & \(\mathbf{5.57\times 10^{-2}}\) & \(5.63\times 10^{-2}\) \\ DTLZ4 & \(4.33\times 10^{-2}\) & \(\mathbf{3.76\times 10^{-2}}\) \\ DTLZ5 & \(\mathbf{9.70\times 10^{-3}}\) & \(1.13\times 10^{-2}\) \\ DTLZ6 & \(\mathbf{9.40\times 10^{-3}}\) & \(9.60\times 10^{-3}\) \\ DTLZ7 & \(6.50\times 10^{-2}\) & \(\mathbf{5.34\times 10^{-2}}\) \\ WFG2 & \(\mathbf{3.37\times 10^{-1}}\) & \(4.33\times 10^{-1}\) \\ WFG3 & \(\mathbf{1.42\times 10^{-1}}\) & \(3.02\times 10^{-1}\) \\ WFG4 & \(\mathbf{4.26\times 10^{-1}}\) & \(4.84\times 10^{-1}\) \\ MaF1 & \(\mathbf{7.31\times 10^{-2}}\) & \(8.42\times 10^{-2}\) \\ MaF2 & \(6.27\times 10^{-2}\) & \(\mathbf{5.71\times 10^{-2}}\) \\ MaF12 & \(\mathbf{4.31\times 10^{-1}}\) & \(4.52\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: \(SP\) results of the mutation methods of MOFECO algorithms on test functions. (The best result is bolded for each function). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithms** & \multicolumn{4}{c}{**EAreal**} & \multicolumn{4}{c}{**PSO**} & **NSLS Operator** \\ \cline{2-9}  & _proC_ & _proM_ & _disC_ & _disM_ & _div_ & _rate_ & \(\omega\) & _muu_ & _delta_ \\ \hline NSGA-II & 1 & 1 & 20 & 20 & / & / & / & / \\ PESA-II & 1 & 1 & 20 & 20 & 10 & / & / & / \\ KnEA & 1 & 1 & 20 & 20 & / & 0.4 & / & / \\ MOPSO & / & / & / & / & / & 0.4 & / & / \\ NSLS & / & / & / & / & / & / & 0.5 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: The detailed parameter configuration for the five compared algorithms. In the EAreaal operator, _proC_ is the probability of implementing crossover operation, _proM_ is the expectation of polynomial mutation, _disC_ represents the distribution index of simulated binary crossover, and _disM_ represent the distribution index of polynomial mutation. In PESA-II, _div_ is the number of divisions in each objective. In KnEA, _rate_ represents the rate of knee points in the population. In MOPSO, \(\omega\) represents the weight factor, and in the NSLS operator, the \(mu\) is the mean value of the Gaussian distribution and \(delta\) is the standard deviation of the Gaussian distribution. We use the proposed MOFECO algorithm to optimize 15 test functions selected from the ZDT, DTLZ, MaF and WFG series in the aforementioned environment, and the obtained Pareto front by MOFECO and the true Pareto front are shown in Figures 6-29. Figures 10 and 12 shows the true Pareto front of the test function DTLZ2 and DTLZ4, and each of the plots from Figures 10-15 has two subgraphs, respectively representing the front and left or rear view of the optimized Pareto front in problems DTLZ2, DTLZ4, DTLZ5 and DTLZ6. From Figures 6-29, we can see that the obtained Pareto front by using the proposed MOFECO can evenly distribute on the true Pareto front, which shows that the algorithm can converge to the optimal solution set and is effective in solving MOO problems.