# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Comparison with Other Optimization Algorithms
Overall, the experimental results show that proposed MOFECO can obtain better \(HV\) values on 9 test functions approximately among the 15 test functions compared with the other five algorithms, which confirms that the proposed MOFECO has a good characteristic in terms of the comprehensive performance of the Pareto solution set. \(SP\) and \(SI\) are indicators for measuring the distribution and distribution breadth of a solution set. The smaller the \(SP\) and \(SI\) values are, the better the distribution and breadth a solution set has. It is clear from Table 19 that the MOFECO has better \(SP\) values on all the four ZDT test functions except for ZDT1 compared with NSGA-II, MOPSO and PESA-II but on the ZDT6 test function, MOFECO has the smallest \(SP\) values compared with the other five algorithms. Among the five tri-objective test functions of DTLZ, the MOFECO performs better on metric \(SP\) on at least four test functions compared with NSGA-II, MOPSO and KnEA, while it performs slightly worse in NSLS algorithm. On the four-objective test functions (WFG2, WFG3, WFG4, MaF1, MaF2 and MaF12), MOFECO achieves the smallest \(SP\) values on at least four test functions compared with all the other five algorithms and MOFECO can even obtain the smallest \(SP\) values on all the 6 four-objective test functions compared with NSGA-II and KnEA. From these empirical results, we can confirm that the proposed MOFECO has good performance on solving MOO problems with the objective numbers of 2 to 4, and can obtain better distribution of Pareto solution sets. According to Table 20, we can see that on the four bi-objective test functions of ZDT, the \(SI\) values obtained by MOFECO are revealed to be superior to those obtained by MOPSO, PESA-II and KnEA. While on the five tri-objective test functions of DTLZ, MOFECO performs worse compared with NSLS algorithm, the possible reason of which is that in NSLS, a new method, which combines the non-dominated sorting and the farthest candidate approach, was chosen to generate a new population for improving diversity, while in MOFECO, we only use the crowded comparison mechanism presented in NSGA-II to maintain diversity. Among the six four-objective test functions of the WFG and MaF series, MOFECO shows better \(SI\) values for at least five test functions compared with NSGA-II, MOPSO, PESA-II and KnEA. These empirical results show that the proposed MOFECO has superiority in solving MOO problems and can obtain solution sets with better spread on Pareto front. But when comparing with the NSLS algorithm, MOFECO performs slightly worse. In order to more visually show the performance of the MOFECO algorithm, a detailed convergence process increasing the number of iterations is discussed. Here we choose two test problems DTLZ2 and DTLZ6 to show the convergence process. Experimental results are shown in Figures 30 and 31. The abscissa represents the number of iterations and the ordinate represents the average objective function value of the non-dominated solutions in each iteration. As can be seen from Figures 30 and 31, the population obtained by MOFECO converges rapidly in every objective of DTLZ2 and DTLZ6, which reflect the good convergence of the proposed MOFECO algorithm. Overall, the proposed MOFECO performs better than NSGA-II, MOPSO, PESA-II and KnEA on most of the test functions used in the above experiments in terms of metrics \(GD\), \(PD\), \(HV\), \(SP\) and \(SI\), which indicates that MOFECO algorithm has better convergence and diversity in solving MOO problems. But MOFECO performs slightly worse on \(SI\) indicator compared with NSLS, which will be one of the problems we need to investigate in the follow-up research.