# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Parameter Analysis and Comparison Experiments of MOFECO
#### 5.3.2 Comparison of Conditions for Judging Whether to Update  
As we described in Section 4 that an individual \(X_{ij}(k)\) corresponds to \(m\) objective function values and each objective function value corresponds to a force value, so that every \(X_{ij}(k)\) received \(m\) force values \(F_{ij\_r}(k)\),(\(r\in\{1,\ 2,\ \ldots,\ m\}\). We know that the values of objective function affect the force \(F_{ij\_r}\), and judging whether the individual retains or updates is mainly according to the values of \(F_{ij\_r}(k)\). Here, we select three update methods for comparison, which are described as follows:  
(1) When individual \(X_{ij}(k)\) satisfies \(F_{ij\_u1}(k)>0\) and \(F_{ij\_u2}(k)>0\),(\(u1,\ u2\in\{1,\ 2,\ \ldots,\ m\}\)), the individual retains; Otherwise, updates. And \(u_{1}\),\(u_{2}\) are two random numbers in the set \(\{1,\ 2,\ \ldots,\ m\}\). (2) When individual \(X_{ij}(k)\) satisfies \(\forall u,F_{ij\_u}(k)>0\),(\(u\in\{1,\ 2,\ \ldots,\ m\}\), the individual retains; Otherwise, updates. (3) When individual \(X_{ij}(k)\) satisfies \(\exists u,F_{ij\_u}(k)>0\),(\(u\in\{1,\ 2,\ \ldots,\ m\}\), the individual retains; Otherwise, updates. The comparison experiments are carried out on the test function sets ZDT, DTLZ, MaF and WFG series for the above three update methods, which are denoted method 1, method 2 and method 3 here for simplicity. The results are shown in Tables 6-8, where average \(GD\), \(IGD\) and \(SP\) values obtained by the above three methods are presented, and the bold part of the table represents the optimal value. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Functions** & \(\mathbf{L=4/q=25}\) & \(\mathbf{L=5/q=20}\) & \(\mathbf{L=10/q=10}\) & \(\mathbf{L=20/q=5}\) \\ \hline ZDT1 & \(4.87\times 10^{-1}\) & \(\mathbf{4.59\times 10^{-1}}\) & \(5.11\times 10^{-1}\) & \(4.66\times 10^{-1}\) \\ ZDT2 & \(5.30\times 10^{-1}\) & \(\mathbf{4.54\times 10^{-1}}\) & \(5.12\times 10^{-1}\) & \(5.96\times 10^{-1}\) \\ ZDT4 & \(\mathbf{4.70\times 10^{-1}}\) & \(5.40\times 10^{-1}\) & \(1.15\times 10^{0}\) & \(1.04\times 10^{0}\) \\ ZDT6 & \(\mathbf{4.58\times 10^{-1}}\) & \(5.08\times 10^{-1}\) & \(1.08\times 10^{0}\) & \(1.18\times 10^{0}\) \\ DTLZ2 & \(4.90\times 10^{-1}\) & \(\mathbf{4.86\times 10^{-1}}\) & \(5.07\times 10^{-1}\) & \(6.07\times 10^{-1}\) \\ DTLZ4 & \(7.04\times 10^{-1}\) & \(\mathbf{5.65\times 10^{-1}}\) & \(1.08\times 10^{0}\) & \(1.05\times 10^{0}\) \\ DTLZ5 & \(\mathbf{4.26\times 10^{-1}}\) & \(4.95\times 10^{-1}\) & \(5.51\times 10^{-1}\) & \(5.33\times 10^{-1}\) \\ DTLZ6 & \(\mathbf{5.10\times 10^{-1}}\) & \(5.18\times 10^{-1}\) & \(5.26\times 10^{-1}\) & \(6.52\times 10^{-1}\) \\ DTLZ7 & \(\mathbf{6.18\times 10^{-1}}\) & \(6.50\times 10^{-1}\) & \(7.31\times 10^{-1}\) & \(8.49\times 10^{-1}\) \\ WFG2 & \(5.11\times 10^{-1}\) & \(\mathbf{4.41\times 10^{-1}}\) & \(5.11\times 10^{-1}\) & \(7.17\times 10^{-1}\) \\ WFG3 & \(\mathbf{5.48\times 10^{-1}}\) & \(6.83\times 10^{-1}\) & \(5.57\times 10^{-1}\) & \(9.54\times 10^{-1}\) \\ WFG4 & \(4.20\times 10^{-1}\) & \(\mathbf{4.13\times 10^{-1}}\) & \(4.39\times 10^{-1}\) & \(5.16\times 10^{-1}\) \\ MaF1 & \(4.59\times 10^{-1}\) & \(\mathbf{4.16\times 10^{-1}}\) & \(4.63\times 10^{-1}\) & \(4.99\times 10^{-1}\) \\ MaF2 & \(4.88\times 10^{-1}\) & \(\mathbf{4.44\times 10^{-1}}\) & \(5.30\times 10^{-1}\) & \(5.92\times 10^{-1}\) \\ MaF12 & \(4.25\times 10^{-1}\) & \(\mathbf{4.06\times 10^{-1}}\) & \(4.35\times 10^{-1}\) & \(5.30\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: \(SI\) result of comparison of \(L\) and \(q\) in MOFECO algorithms on test functions. (The best result is bolded for each function).