# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Parameter Analysis and Comparison Experiments of MOFECO
#### 5.3.3 Comparison of Local-Global Update Probability \(P_{s}\)  
The parameter \(P_{s}\) means the probability that inferior solutions to be displaced by local or global optimal solutions. Large \(P_{s}\) is inclined to local update while small \(P_{s}\) is inclined to global update. We designed 3 groups of experiments where \(P_{s}\) either varies nonlinearly with the number of iterations, or varies linearly with the number of iterations, or is simply fixed. The nonlinear variation rule is shown in Equation (14) and Figure 4, and the linear variation rule is shown in Equation (26) and Figure 5:  
\[P_{s}=P_{s_{min}}+(P_{s_{max}}-P_{s_{min}})\times\frac{k}{T} \tag{26}\]  
where \(P_{s_{\min}}=0.2\) and \(P_{s_{\max}}=0.8\) are the same as in Equation (14) and in the fixed case \(P_{s}\) is set to 0.5. The complete experiment results are listed in Tables 9-11, which show average \(GD\), \(IGD\) and \(SP\) values obtained by the above three case, and the bold part of the table represents the optimal values. We can see from Tables 9-11 that when the local-global probability \(P_{s}\) changes nonlinearly, the obtained Pareto solution set has smaller \(GD\), \(IGD\) and \(SP\) values than the cases of linearly changing of \(P_{s}\) and the fixed \(P_{s}\). The advantage of choosing nonlinear variation \(P_{s}\) lies in that it maintains a small value for a long time in the initial period of iteration, and a large value for a long time in the late iteration. Thereby increasing the global search time in the early stage of iteration and the local search time in the late stage of iteration can well balance the global and local search ability of MOFECO. Therefore, in the proposed algorithm, we choose a nonlinearly changing local-global probability \(P_{s}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Functions** & **Method 1** & **Method 2** & **Method 3** \\ \hline ZDT1 & \(\mathbf{8.70\times 10^{-3}}\) & \(8.70\times 10^{-3}\) & \(1.03\times 10^{-2}\) \\ ZDT2 & \(\mathbf{7.60\times 10^{-3}}\) & \(7.60\times 10^{-3}\) & \(8.80\times 10^{-3}\) \\ ZDT4 & \(\mathbf{7.40\times 10^{-3}}\) & \(7.40\times 10^{-3}\) & \(1.40\times 10^{-2}\) \\ ZDT6 & \(\mathbf{6.80\times 10^{-3}}\) & \(6.80\times 10^{-3}\) & \(3.06\times 10^{-2}\) \\ DTLZ2 & \(\mathbf{5.57\times 10^{-2}}\) & \(5.86\times 10^{-2}\) & \(5.73\times 10^{-2}\) \\ DTLZ4 & \(\mathbf{4.33\times 10^{-2}}\) & \(5.01\times 10^{-2}\) & \(5.67\times 10^{-2}\) \\ DTLZ5 & \(\mathbf{9.70\times 10^{-3}}\) & \(9.90\times 10^{-3}\) & \(1.18\times 10^{-2}\) \\ DTLZ6 & \(\mathbf{9.40\times 10^{-3}}\) & \(9.70\times 10^{-3}\) & \(9.90\times 10^{-3}\) \\ DTLZ7 & \(6.50\times 10^{-2}\) & \(\mathbf{5.90\times 10^{-2}}\) & \(6.29\times 10^{-2}\) \\ WFG2 & \(\mathbf{3.37\times 10^{-1}}\) & \(4.21\times 10^{-1}\) & \(3.63\times 10^{-1}\) \\ WFG3 & \(\mathbf{1.42\times 10^{-1}}\) & \(3.12\times 10^{-1}\) & \(3.01\times 10^{-1}\) \\ WFG4 & \(\mathbf{4.26\times 10^{-1}}\) & \(4.80\times 10^{-1}\) & \(4.99\times 10^{-1}\) \\ MaF1 & \(\mathbf{7.31\times 10^{-2}}\) & \(8.49\times 10^{-2}\) & \(9.19\times 10^{-2}\) \\ MaF2 & \(\mathbf{6.27\times 10^{-2}}\) & \(7.01\times 10^{-2}\) & \(6.60\times 10^{-2}\) \\ MaF12 & \(\mathbf{4.31\times 10^{-1}}\) & \(4.52\times 10^{-2}\) & \(4.81\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: \(SP\) result of the three update conditions of MOFECO algorithms on test functions. (The best result is bolded for each function).