## 5 Results
### Simulation Results of Benchmark Functions
For the function KUR, the \(GD\) of IMOPSO has the largest standard deviation, which indicates that IMOPSO  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Problems** & **Statistic** & **IMOPSO** & **MOPSO** & **NSGA-II** & **MOEA/D** \\ \hline SCH & max & 2.4265 \(\times\)\(10^{-3}\) & 1.0986 \(\times\)\(10^{-2}\) & 5.1149 \(\times\)\(10^{-3}\) & 1.2536 \(\times\)\(10^{-2}\) \\  & min & 2.9403 \(\times\)\(10^{-3}\) & 1.4418 \(\times\)\(10^{-4}\) & 4.0847 \(\times\)\(10^{-5}\) & 2.8466 \(\times\)\(10^{-4}\) \\  & ave & 6.0767 \(\times\)\(10^{-4}\) & 3.7421 \(\times\)\(10^{-3}\) & 1.0246 \(\times\)\(10^{-3}\) & 5.0172 \(\times\)\(10^{-3}\) \\  & std & 5.8435 \(\times\)\(10^{-4}\) & 3.4812 \(\times\)\(10^{-3}\) & 1.0683 \(\times\)\(10^{-3}\) & 3.4573 \(\times\)\(10^{-3}\) \\ \hline FON & max & 9.7047 \(\times\)\(10^{-3}\) & 1.6692 \(\times\)\(10^{-2}\) & 1.0742 \(\times\)\(10^{-1}\) & 1.0283 \(\times\)\(10^{-2}\) \\  & min & 7.7566 \(\times\)\(10^{-6}\) & 5.8202 \(\times\)\(10^{-4}\) & 6.1194 \(\times\)\(10^{-5}\) & 1.1283 \(\times\)\(10^{-4}\) \\  & ave & 1.9247 \(\times\)\(10^{-3}\) & 5.1714 \(\times\)\(10^{-3}\) & 1.3052 \(\times\)\(10^{-2}\) & 2.9614 \(\times\)\(10^{-3}\) \\  & std & 2.2364 \(\times\)\(10^{-3}\) & 4.1587 \(\times\)\(10^{-3}\) & 2.0136 \(\times\)\(10^{-2}\) & 2.8547 \(\times\)\(10^{-3}\) \\ \hline KUR & max & 6.5348 \(\times\)\(10^{-3}\) & 1.4637 \(\times\)\(10^{-1}\) & 7.1293 \(\times\)\(10^{-2}\) & 1.8347 \(\times\)\(10^{-1}\) \\  & min & 1.0722 \(\times\)\(10^{-4}\) & 7.1513 \(\times\)\(10^{-5}\) & 7.5517 \(\times\)\(10^{-4}\) & 3.0266 \(\times\)\(10^{-5}\) \\  & ave & 2.5493 \(\times\)\(10^{-3}\) & 2.0829 \(\times\)\(10^{-2}\) & 1.7056 \(\times\)\(10^{-2}\) & 1.1125 \(\times\)\(10^{-2}\) \\  & std & 1.7256 \(\times\)\(10^{-3}\) & 2.7645 \(\times\)\(10^{-2}\) & 1.9176 \(\times\)\(10^{-2}\) & 3.2862 \(\times\)\(10^{-2}\) \\ \hline ZDT1 & max & 4.1474 \(\times\)\(10^{-5}\) & 9.1931 \(\times\)\(10^{-1}\) & 2.2135 \(\times\)\(10^{-3}\) & 1.5789 \(\times\)\(10^{0}\) \\  & min & 0.0000 \(\times\)\(10^{0}\) & 1.1904 \(\times\)\(10^{-4}\) & 0.0000 \(\times\)\(10^{0}\) & 0.0000 \(\times\)\(10^{0}\) \\  & ave & 1.7483 \(\times\)\(10^{-6}\) & 8.7632 \(\times\)\(10^{-2}\) & 4.8683 \(\times\)\(10^{-4}\) & 6.5213 \(\times\)\(10^{-2}\) \\  & std & 7.6347 \(\times\)\(10^{-6}\) & 2.6313 \(\times\)\(10^{-1}\) & 4.9079 \(\times\)\(10^{-4}\) & 2.8732 \(\times\)\(10^{-1}\) \\ \hline ZDT2 & max & 4.8843 \(\times\)\(10^{-3}\) & 1.0000 \(\times\)\(10^{0}\) & 5.4132 \(\times\)\(10^{-3}\) & 9.9673 \(\times\)\(10^{-1}\) \\  & min & 0.0000 \(\times\)\(10^{0}\) & 0.0000 \(\times\)\(10^{0}\) & 0.0000 \(\times\)\(10^{0}\) & 3.9125 \(\times\)\(10^{-4}\) \\  & ave & 1.2353 \(\times\)\(10^{-3}\) & 2.9163 \(\times\)\(10^{-1}\) & 1.5364 \(\times\)\(10^{-3}\) & 3.4862 \(\times\)\(10^{-1}\) \\  & std & 1.0112 \(\times\)\(10^{-3}\) & 4.5072 \(\times\)\(10^{-1}\) & 1.5214 \(\times\)\(10^{-3}\) & 3.9228 \(\times\)\(10^{-1}\) \\ \hline ZDT3 & max & 9.8715 \(\times\)\(10^{-3}\) & 3.7582 \(\times\)\(10^{-1}\) & 2.1613 \(\times\)\(10^{-2}\) & 8.7073 \(\times\)\(10^{-1}\) \\  & min & 4.1056 \(\times\)\(10^{-6}\) & 5.9960 \(\times\)\(10^{-8}\) & 3.5787 \(\times\)\(10^{-6}\) & 1.0889 \(\times\)\(10^{-4}\) \\  & ave & 1.2125 \(\times\)\(10^{-3}\) & 1.8904 \(\times\)\(10^{-2}\) & 5.3247 \(\times\)\(10^{-3}\) & 3.2281 \(\times\)\(10^{-1}\) \\  & std & 2.0499 \(\times\)\(10^{-3}\) & 6.9826 \(\times\)\(10^{-2}\) & 4.6385 \(\times\)\(10^{-3}\) & 2.7195 \(\times\)\(10^{-1}\) \\ \hline ZDT4 & max & 1.0527 \(\times\)\(10^{-5}\) & 2.1396 \(\times\)\(10^{1}\) & 1.0705 \(\times\)\(10^{-5}\) & 1.5036 \(\times\)\(10^{-2}\) \\  & min & 1.0527 \(\times\)\(10^{-5}\) & 6.7712 \(\times\)\(10^{-3}\) & 5.6429 \(\times\)\(10^{-6}\) & 4.2802 \(\times\)\(10^{-6}\) \\  & ave & 1.0527 \(\times\)\(10^{-5}\) & 6.6646 \(\times\)\(10^{0}\) & 1.0759 \(\times\)\(10^{-5}\) & 7.1416 \(\times\)\(10^{-4}\) \\  & std & 4.9426 \(\times\)\(10^{-13}\) & 7.1038 \(\times\)\(10^{0}\) & 2.2420 \(\times\)\(10^{-6}\) & 2.7158 \(\times\)\(10^{-3}\) \\ \hline ZDT6 & max & 5.5332 \(\times\)\(10^{-1}\) & 8.2603 \(\times\)\(10^{0}\) & 5.1950 \(\times\)\(10^{0}\) & 2.2690 \(\times\)\(10^{0}\) \\  & min & 1.0593 \(\times\)\(10^{-8}\) & 9.9859 \(\times\)\(10^{-9}\) & 3.2261 \(\times\)\(10^{-7}\) & 6.0436 \(\times\)\(10^{-10}\) \\  & ave &does not perform well in convergence stability.