# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 6 Conclusions  
A new multi-objective evolutionary algorithm called the Multi-Objective Five-Elements Cycle Optimization algorithm (MOFECO) for solving MOO problems is proposed in this paper and the validity of the MOFECO has been verified by using the test function sets ZDT, DTLZ, WFG and MaF. The main idea of MOFECO is that, at each iteration, we divide the initial population into \(q\) independent cycles and each cycle contains \(L\) elements, where \(L\times q\) represents the population number and each element represents an individual in the population. In each cycle, the force applied to each element by the other four elements related to the mass of each element, while the mass is represented by the objective functions. Therefore, the value of the force indirectly reflects the pros and cons of an individual and judges whether the individual is updated. In our paper, in case of an update, the local update or the global update is selected according to the current local-global probability value \(P_{\mathrm{s}}\). Then, the combined mutation based on mutation probability \(P_{m}\) is performed, that is, the uniform distribution mutation operator is used in the early iteration, and the Cauchy distribution mutation operator is used in the middle of the iteration, and the Gauss distribution mutation operator is used in the late iteration. Next, as the offspring individuals are stored in the child population, the child population is combined with the parent population and the next generation is selected from them by using a fast non-dominated sorting and crowded distance calculation method, and then generate the  
Figure 31: Convergence process of MOFECO on DTLZ6.  
Figure 30: Convergence process of MOFECO on DTLZ2.  
current non-dominated solution set. The operations above are repeated until the maximum number of iterations is reached and then the optimal Pareto solution set is output as the result of optimization.  
In this paper, we compared the different values of \(L\) and \(q\), different update conditions, the adaptive update probability \(P_{s}\) and the different mutation methods as parameters and conditions that affect the performance of MOFECO algorithm. In addition, this paper compared the performance of MOFECO with five popular MOEAs NSGA-II, MOPSO, PESA-II, KnEA and NSLS on 15 test problems, and discussed the convergence process of MOFECO on DTLZ2 and DTLZ6. The results demonstrate that the proposed MOFECO significantly outperforms MOPSO and PESA-II in convergence, diversity and distribution, and performs better than all five comparative algorithms in convergence and diversity. The main weakness of MOFECO is that its distribution of the obtained Pareto solution set is slightly worse than the one obtained by NSLS. It is certain that there is no guarantee that the MOFECO algorithm will always show better characteristics on all test problems. The strengths and weaknesses of the algorithm should be further studied based on the characteristics of the test problems in future research.  
Conceptualization, C.Y. and M.L.; methodology, C.Y.; software, C.Y.; validation, C.Y.; formal analysis, C.Y.; writing-original draft preparation, C.Y.; writing-review and editing, C.Y., M.L., Z.M.; visualization, C.Y.; supervision, M.L., Z.M.  
This research received no external funding.  
The authors declare no conflict of interest.