## 5 Results
### Simulation Results of the DWTA Problem  
For three DWTA problems of different scales (small, medium and large), four algorithms, i.e., IMOPSO, MOPSO, NSGA-II and MOEA/D, are used to solve them respectively. Each algorithm runs independently for 30 times. The experimental results are shown in the Table 5.  
It can be seen from the results in Table 5 that the IMOPSO algorithm has achieved the best \(IGD\) values in all three instances, and it indicates that IMOPSO has better performance \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cases** & **Statistic** & **IMOPSO** & **MOPSO** & **NSGA-II** & **MOEA/D** \\ \hline Small & max & 2.0899 & 4.4919 & 3.7781 & 18.5805 \\  & min & 0.1794 & 0.7148 & 0.4456 & 9.3359 \\  & ave & 0.5871 & 1.5536 & 1.4704 & 11.6643 \\  & std & 0.4426 & 0.9613 & 0.8294 & 2.2159 \\ \hline Medium & max & 7.1089 & 11.7417 & 7.1093 & 38.1172 \\  & min & 2.4670 & 2.2153 & 1.3007 & 29.3465 \\  & ave & 3.2486 & 6.4850 & 3.7795 & 32.1274 \\  & std & 1.1570 & 3.0458 & 1.2904 & 2.1213 \\ \hline Large & max & 7.1926 & 12.7778 & 9.8487 & 87.5192 \\  & min & 2.7628 & 3.1651 & 3.0999 & 64.7363 \\  & ave & 4.3028 & 6.2359 & 5.2511 & 73.0749 \\  & std & 1.0646 & 2.0957 & 1.4718 & 6.0825 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the \(IGD\) index of four algorithms. than the other three algorithms in solving multiobjective DWTA problems. With the increase of the scale of DWTA problems, the non-dominated solutions generated by IMOPSO also obtain excellent convergence and distribution. The performance of NSGA-II in solving multiobjective DWTA issues is second only to that of IMOPSO, especially in medium scale, and the statistical values of NSGA-II are very close to that of IMOPSO. MOPSO has little difference with NSGA-II in small-scale and large-scale instance, and its overall performance is second only to NSGA-II. The effect of MOEA/D in solving DWTA problems is very poor. Its index values are far behind the other three algorithms, and this gap is becoming more and more obvious with the increase of the problem scale. This is due to the strictness of the five constraints in the multiobjective DWTA problem discussed in this paper, which leads to the increased difficulty of solving the model. Moreover, MOEA/D needs to normalize the objectives when dealing with multiobjective DWTA problems, so the  
Figure 14: Comparison of four algorithms on small-scaled instance. Figure 15: Pareto front produced by four algorithms on medium-scaled instance. Figure 16: Comparison of four algorithms on medium-scaled instance. Figure 17: Pareto front produced by four algorithms on large-scaled instance. It can be seen by Figures 13 and 14 that IMOPSO performs very well in solving small-scale multiobjective DWTA problem, and the solutions obtained by IMOPSO can converge to the true PF. The solutions obtained by MOPSO failed to converge to the true PF, and the distribution of the solution set is poor, which failed to completely cover the entire Pareto front. NSGA-II is second only to IMOPSO, and its non-dominated solutions mostly converge to Pareto front. MOEA/D is the worst one, which almost obtain no useful Pareto solutions. Figures 15 and 16 show the distribution comparison of solution sets of four algorithms in medium scale. It can be seen that most of the solutions obtained by IMOPSO converge to the true PF, and only a few solutions fail to fall on the Pareto front. The non-dominated solutions generated by NSGA-II have a distance from true PF, and its convergence is far less than that of IMOPSO. Compared with NSGA-II, the solution set obtained by MOPSO is farther from the true PF. MOEA/D is the worst among the four algorithms, which generates very few solutions, and the solution set cannot completely cover the entire Pareto front. Figures 17 and 18 are the comparison results of four algorithms for solving large-scale multiobjective DWTA problem.