# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Parameter Analysis and Comparison Experiments of MOFECO
#### 5.3.4 Comparison of Mutation Methods  
The mutation operator has an important influence on the convergence and diversity of evolutionary algorithms. Based on the analysis of the mutation operator in Reference [25], it is found that the local search ability of Gaussian mutation is strong and the global search ability of uniform mutation is strong, while the local and global search capabilities of Cauchy mutation are centered. In this section, we choose two different mutation operators for comparative experiments and analysis. Method 1 combined Gaussian mutation, Uniform mutation and Cauchy mutation as mutation operator, which is shown in Equation (15); And method 2 uses only the Cauchy mutation, as shown in Equation (27), where the parameters \(P_{s}\), \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) are set as Table 2. The comparative experiments results are shown in Tables 12-14, which give average \(GD\), \(IGD\) and \(SP\) values obtained by the above two mutation operators and the bold part of the table represents the optimal values. \[XM_{ij}(k+1)=XC_{ij}(k+1)+C(0,\sigma_{2}) \tag{27}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Functions** & \(\mathbf{P_{s}}\) **Is Nonlinear Variation** & \(\mathbf{P_{s}}\) **Is Linear Variation** & \(\mathbf{P_{s}}\) **= 0.5** \\ \hline ZDT1 & \(\mathbf{8.70\times 10^{-3}}\) & \(9.30\times 10^{-3}\) & \(1.11\times 10^{-2}\) \\ ZDT2 & \(\mathbf{7.50\times 10^{-3}}\) & \(7.50\times 10^{-3}\) & \(7.90\times 10^{-3}\) \\ ZDT4 & \(\mathbf{7.40\times 10^{-3}}\) & \(9.10\times 10^{-3}\) & \(1.07\times 10^{-2}\) \\ ZDT6 & \(6.80\times 10^{-3}\) & \(6.60\times 10^{-3}\) & \(\mathbf{6.50\times 10^{-3}}\) \\ DTLZ2 & \(5.57\times 10^{-2}\) & \(5.71\times 10^{-2}\) & \(\mathbf{5.25\times 10^{-2}}\) \\ DTLZ4 & \(\mathbf{4.33\times 10^{-2}}\) & \(5.74\times 10^{-2}\) & \(4.88\times 10^{-2}\) \\ DTLZ5 & \(\mathbf{9.70\times 10^{-3}}\) & \(1.04\times 10^{-2}\) & \(1.01\times 10^{-2}\) \\ DTLZ6 & \(\mathbf{9.40\times 10^{-3}}\) & \(9.60\times 10^{-3}\) & \(9.50\times 10^{-3}\) \\ DTLZ7 & \(6.50\times 10^{-2}\) & \(\mathbf{5.83\times 10^{-2}}\) & \(8.18\times 10^{-2}\) \\ WFG2 & \(\mathbf{3.37\times 10^{-1}}\) & \(3.94\times 10^{-1}\) & \(4.06\times 10^{-1}\) \\ WFG3 & \(\mathbf{1.42\times 10^{-1}}\) & \(2.92\times 10^{-1}\) & \(2.05\times 10^{-1}\) \\ WFG4 & \(\mathbf{4.26\times 10^{-1}}\) & \(4.88\times 10^{-1}\) & \(4.54\times 10^{-1}\) \\ MaF1 & \(\mathbf{7.31\times 10^{-2}}\) & \(8.14\times 10^{-2}\) & \(5.71\times 10^{-2}\) \\ MaF12 & \(\mathbf{4.31\times 10^{-1}}\) & \(4.67\times 10^{-1}\) & \(4.42\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: \(SP\) result of the local-global update probability \(P_{s}\) of MOFECO algorithms on test functions. (The best result is bolded for each function). Figure 5: Linear variation of \(P_{s}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Functions** & **Combined Mutation** & **Cauchy Mutation** \\ \hline ZDT1 & \(\mathbf{5.40\times 10^{-3}}\) & \(5.60\times 10^{-3}\) \\ ZDT2 & \(\mathbf{5.40\times 10^{-3}}\) & \(6.40\times 10^{-3}\) \\ ZDT4 & \(\mathbf{5.76\times 10^{-3}}\) & \(1.80\times 10^{-2}\) \\ ZDT6 & \(4.70\times 10^{-3}\) & \(\mathbf{4.60\times 10^{-3}}\) \\ DTLZ2 & \(\mathbf{7.49\times 10^{-2}}\) & \(7.54\times 10^{-2}\) \\ DTLZ4 & \(\mathbf{1.21\times 10^{-1}}\) & \(3.83\times 10^{-1}\) \\ DTLZ5 & \(\mathbf{6.70\times 10^{-3}}\) & \(7.00\times 10^{-3}\) \\ DTLZ6 & \(\mathbf{6.40\times 10^{-3}}\) & \(6.60\times 10^{-3}\) \\ DTLZ7 & \(\mathbf{2.88\times 10^{-1}}\) & \(4.30\times 10^{-1}\) \\ WFG2 & \(5.11\times 10^{-1}\) & \(4.95\times 10^{-1}\) \\ WFG3 & \(\mathbf{1.60\times 10^{-1}}\) & \(3.01\times 10^{-1}\) \\ WFG4 & \(\mathbf{7.89\times 10^{-1}}\) & \(7.97\times 10^{-1}\) \\ MaF1 & \(\mathbf{1.21\times 10^{-1}}\) & \(1.21\times 10^{-1}\) \\ MaF2 & \(9.58\times 10^{-2}\) & \(8.34\times 10^{-2}\) \\ MaF12 & \(\mathbf{7.43\times 10^{-1}}\) & \(7.63\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: \(GD\) result of the mutation methods of MOFECO algorithms on test functions.