## 5 Results
### Simulation Results of Benchmark Functions
With respect to the other three algorithms, MOPSO has better convergence performance in functions ZDT1, ZDT2 and ZDT3, but its convergence stability is not as good as NSGA-II. MOEA/D has better performance in the \(GD\) metric in functions ZDT4, ZDT6, SCH and FON than MOPSO and NSGA-II. Table 3 shows the results of \(\Delta\) indicator. We can see that MOPSO has achieved the best average value in all eight test problems, which shows that the non-dominated solution generated by MOPSO has the best distribution. However, it can be seen from the standard deviation of some functional indicators that MOPSO is inferior to other algorithms in the \(\Delta\) metric. The distribution performance of NSGA-II in the functions ZDT1, ZDT2 and ZDT3 is second only to that of MOPSO. MOEA/D has obtained better \(\Delta\) values in the functions ZDT4, ZDT6 and FON, while MOPSO has the worst \(\Delta\) performance and has not gained any optimal value. Table 4 is the statistical data of distribution width index \(DW\). It can be seen from the data that MOPSO is the best among the four algorithms except the minimum values of ZDT3, ZDT4, ZDT6 and KUR, and the data stability is also the best. The index performance of NSGA-II is second only to MOPSO. Compared with MOPSO and MOEA/D, it has obtained the best values in functions ZDT1, ZDT2, ZDT3, ZDT4 and SCH, which implies that the solutions obtained by NSGA-II has the wider distribution range. MOEA/D performs well on functions ZDT6 and FON. The distribution width metric DW of MOPSO is still the worst among the four algorithms. In order to show the ability of convergence to Pareto front more clearly, the comparison results of four algorithms for any one run of eight test problems are shown in Figures 5-12. Figure 5: Comparison of the results of four algorithms for solving SCH function. Figure 6: Comparison of the results of four algorithms for solving FON function. Figure 7: _Cont._  
Figure 8: Comparison of the results of four algorithms for solving ZDT1 function. Figure 7: Comparison of the results of four algorithms for solving KUR function. Figure 10: _Cont._  
Figure 9: Comparison of the results of four algorithms for solving ZDT2 function. Figure 11: Comparison of the results of four algorithms for solving ZDT4 function. Figure 10: Comparison of the results of four algorithms for solving ZDT3 function. It can be seen from Figure 5 that the non-dominated solutions generated by IMOPSO are superior to the other three algorithms in both convergence and distribution. The solutions obtained by the other three algorithms can converge to the true PF, but the distribution of the solutions is poor, especially MOPSO and MOEA/D. It can be seen from the figure that many solutions cannot be searched, resulting in the solutions generated by the algorithms not covering the entire Pareto front. The performance of NSGA-II is second only to that of IMOPSO. Figure 6 shows that the non-dominated solutions generated by the four algorithms can converge to the vicinity of the true PF. However, we can also find that the solution set generated by MOPSO and NSGA-II is not uniform enough and does not cover the entire Pareto front, which is also the reason for the poor statistical value of its distribution index. The distribution of the solution set obtained by MOEA/D is second only to that of IMOPSO. The solution set generated by IMOPSO has the best convergence and distribution among the four algorithms. We can intuitively see from Figure 7 that the effect of MOPSO in solving KUR function is very poor. The non-dominated solutions obtained by MOPSO fail to converge to the true PF, and the solution set is not evenly distributed. The non-dominated solutions generated by NSGA-II and MOEA/D converge to the true PF. However, the solutions obtained by MOEA/D fail to cover the entire Pareto front, and the distribution of solutions is not as good as that of NSGA-II. IMOPSO is the best of the four algorithms. It can be seen from Figure 8 that the solutions obtained by IMOPSO and MOPSO converge to the true PF.