## 4 Simulation Experiments
### Performance Testing and Results Analysis  
In the experimental stage, each algorithm runs 10 times independently, taking the average value as the final result, and the optimal values of each indicator are shown in bold. The mean value (Avg) and the minimum value (Min) or the maximum value (Max) of each indicator are selected for comparative analysis. For \(SP\), \(GD\), and \(IGD\) indicators, the smaller the value is, the better the algorithm is. Then, its minimum value (Min) is selected. For the \(\Omega\) indicator, the larger the value is, the better the dominance of the algorithm is. Then, its maximum value (Max) is selected. It can be seen from Table 6 that the IMOGWO algorithm is superior to the MODGWO, MOGWO, and NSGA-II algorithms in the uniformity, convergence, diversity, and dominance of the solutions on the Pareto front for 2 factories cases. In order to more clearly determine whether there are significant differences between algorithms, SPSS Statistics 17 was used in this paper to conduct the Wilcoxson symbolic rank test, and the results are shown in Table 7.  
Table 6 shows the experimental results of the test cases with 2 factories. For the dominant proportion of 'Avg' in \(SP\) indicator, the IMOGWO algorithm is 59%, the MODGWO algorithm is 8%, the MOGWO algorithm is 33%, and the NSGA-II algorithm is 0%. For its dominant proportion of 'Min', the IMOGWO algorithm is 25%, the MODGWO algorithm is 33%, the MOGWO algorithm is 25%, and the NSGA-II algorithm is 17%. For the dominant proportion of the 'Avg' in the \(GD\) indicator, the IMOGWO algorithm is 83%, the MODGWO algorithm is 17%, and the NSGA-II algorithm is 0%. For its dominant proportion of 'Min', the IMOGWO algorithm is 83%, the MODGWO algorithm is 8.5%, the NOGWO algorithm is 8.5%, and the NSGA-II algorithm is 0%. For the dominant proportion of 'Avg' in the \(IGD\) indicator, the IMOGWO algorithm is 100%, the MODGWO algorithm is 0%, the MOGWO algorithm is 0%, and the NSGA-II algorithm is 0%. For its dominant proportion of 'Min', the IMOGWO algorithm is 83%, the MODGWO algorithm is 83%  
8.5%, the MOGWO algorithm is 8.5%, and the NSGA-II algorithm is 0%. For the dominant proportion of 'Avg' in \(\Omega\) indicator, the IMOGWO algorithm is 100%, the MODGWO algorithm is 0%, the MOGWO algorithm is 0%, and the NSGA-II algorithm is 0%. For its dominant proportion of 'Max,' the IMOGWO algorithm is 75%, the MODGWO algorithm is 8%, the MOGWO algorithm is 0%, and the NSGA-II algorithm is 17%. As can be seen from 'Mean' in the last line of Table 6, the IMOGWO algorithm is superior to the other three compared algorithms in \(SP\), \(GD\), \(IGD\), and \(\Omega\) indicators. Table 7 summarizes the results of the Wilcoxson symbolic rank test with 2 factories, from which it can be seen that, all relevant p-values are less than 0.05 for \(IGD\) and \(\Omega\) indicators. For \(SP\) and \(GD\) indicators, the p-values are less than 0.05 for all except when IMOGWO compares with the MOGWO algorithm aiming at one problem. It can be concluded that the proposed IMOGWO is significantly superior to the other compared algorithms at 95% confidence level for \(IGD\) and \(\Omega\) indicators. For \(SP\) and \(GD\) indicators, there is no significant difference between IMOGWO and MOGWO for 01 problem, but compared with other algorithms, IMOGWO is significantly superior to them. The box-plots of the four evaluation indicators in Figure 5 further verifies the conclusion. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**IMOGWO**} & \multicolumn{2}{c}{**MODGWO**} & \multicolumn{2}{c}{**MOGWO**} & \multicolumn{2}{c}{**NSGA-II**} \\ \cline{3-10}
**Test Example** & \multirow{2}{*}{**Avg**} & \multicolumn{2}{c}{**Min**} & \multirow{2}{*}{**Avg**} & \multicolumn{2}{c}{**Min**} & \multirow{2}{*}{**Avg**} & \multicolumn{2}{c}{**Min**} & \multirow{2}{*}{**Avg**} & \multicolumn{2}{c}{**Min**} \\  & & & & & & & & **(Max)** & & **(Max)** \\ \hline Problem & \(SP\) & 0.2525 & 0.0553 & 0.2965 & 0.1485 & **0.1717** & 0.