## 4 Simulation Experiments
### Case Analysis  
This paper takes the test case of Sproblem-02-01 in two factories as an example to make a specific analysis. There are 15 jobs in this case, and the number of re-entry is 1. It is assumed that there are eight stages in each production line, including oxidation, deposition, injection, metallization, lithography, etching, polishing, and cleaning. The related parameters of the first factory are shown in Table 10, and those of the second factory are shown in Table 11. The Pareto optimal fronts obtained by each algorithm for the test case are shown in Figure 7. Figure 7a is a three-dimensional graph of three optimization objectives. The Pareto front obtained by the IMOGWO algorithm is located at the lower right of the Pareto fronts obtained by the other three algorithms. This is closer to the real Pareto front, followed by NSGA-II, MOGWO, and MODGWO, which has the worst convergence. Among the 17 solutions of obtained optimal Pareto front, 15 are obtained by IMOGWO, one is obtained by MODGWO, and one is obtained by NSGA-II. The two-dimensional diagram of make span and total carbon emissions is shown in Figure 7b, the two-dimensional diagram of total carbon emissions and total tardiness is shown in Figure 7c, the two-dimensional diagram of make span and total tardiness is shown in Figure 7d. We can see from them that there are contradictions among the optimization indicators. Improvements in one indicator can lead to deterioration of at least one other indicator. Table 12 lists part of the solutions and their corresponding scheduling schemes. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Indicator**} & \multicolumn{3}{c}{**Sproblem01-06(01 Problem)**} & **Sig (\(p\) \textless{} 0.05)** & \multicolumn{3}{c}{**Sproblem01-06(11 Problem)**} & **Sig (\(p\) \textless{} 0.05)** \\ \cline{2-11}  & **MODGWO** & **MOGWO** & **NSGA-II** & **MODGWO** & **MOGWO** & **NSGA-II** \\ \hline \multirow{3}{*}{_SP_} & P & Sig & P & Sig & P & Sig & P & Sig & P & Sig \\  & 0.000 & Y & 0.866 & N & 0.006 & Y & 0.000 & Y & 0.732 & N & 0.874 & N \\ GD & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.017 & Y & 0.001 & Y \\ IGD & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.003 & Y & 0.000 & Y \\ \(\Omega\) & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.000 & Y & 0.000 & Y \\ \hline \hline \end{tabular}
\end{table} 
Table 9: [PERSON] signed rank test (three factories). Figure 6: Box-plots of four evaluation indicators for three factories’ test cases. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Number of parallel machines at each stage & [1, 1, 1, 1, 1, 2, 2, 1], A total of 10 machines \\ Processing time of each operation & An integer follows uniform distribution between [1, 10] \\ Processing power of machines at each stage & [6, 7, 5, 6, 6, 5] (Unit: kw/h) \\ Idle power of machines at each stage & [5, 3, 4, 2, 1, 2, 2] (Unit: kw/h) \\ Effective service time of lubricants on the machines at each stage & [5, 1, 4, 8, 4, 9, 5, 5, 3, 4, 4, 4, 9, 5] (Unit: h) \\ The amount of lubricants used on the machines at each stage & [0.34, 0.27, 0.25, 0.37, 0.22, 0.28, 0.34, 0.37] (Unit: L) \\  & [69, 5520, 66.6540, 49.2660, 89.8380, 89.8380, 84.0420, \\ Delivery time of each job & 73.8990, 56.5110, 6.52050, 73.8990, 56.5110, 92.7360, \\  & 76.7970, 73.8990, 71.0010] (Unit: min) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Relevant parameters in factory 1.