## 4 Simulation Experiments
### Performance Measures  
In this paper, \(SP\), \(GD\), \(IGD\), and \(\Omega\) are selected as evaluation indicators [33,41]. It is very important to note that we do not know the Pareto optimal solution or the best known Pareto solutions for the considering problem. Thus, we regard the Pareto optimal solutions of all test algorithms as the Pareto optimal solutions.  
The \(SP\) indicator is used to evaluate the uniformity of the non-inferior solution distributed on the Pareto front. The minimum value of \(SP\) is 0. The smaller the value, the more uniform of the non-inferior solution distribution. The formula is shown in Equation (21).  
\[SP=\sqrt{\frac{1}{|PF|-1}\sum_{i=1}^{|PF|}{(\overline{d}-d_{i})}^{2}} \tag{21}\]  
where \(i,j=1,\ldots,|PF|i\
eq j\). \(\overline{d}\) is the mean of all \(d_{i}\).\(|PF|\) is the number of non-inferior solutions on the Pareto front.  
The \(GD\) indicator is used to measure the approximation between the Pareto front obtained by this algorithm and the optimal Pareto front. The minimum value of \(GD\) is 0. The smaller the value,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of jobs & An integer follows uniform distribution between [10,20] \\ Number of stages & An integer follows uniform distribution between [5,10] \\ Number of re-entries & An integer follows uniform distribution between [1,2] \\ Number of factories & 2.3 \\ Crossover probability (NSGA-II) & 0.8 \\ Mutation probability (NSGA-II) & 0.3 \\ Number of cycles of algorithms & 100 \\ Number of populations & 50 \\ Archive size (MOGWO) & 50 \\ Gamma (MOGWO) & 2 \\ Alpha (MOGWO) & 0.1 \\ ArGrid (MOGWO) & 10 \\ Beta (MOGWO) & 4 \\ Processing power of machines at each stage & An integer follows uniform distribution between [5, 8] \\ Idle power of machines at each stage & An integer follows uniform distribution between [1, 5] \\ Effective service time of lubricants on the machines at each stage & An integer follows uniform distribution between [4, 6] \\ The amount of lubricants used on the machines at each stage & An integer follows uniform distribution between [0.2, 0.4] \\ Carbon emissions factor of electric energy & 0.6747 kgCO\({}_{2}\)/kw-h \\ Carbon emissions factor of lubricants & 2.85 kgCO\({}_{2}\)/L \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters and values of algorithms.  
the closer the non-inferior solution on the Pareto front to the optimal Pareto front. The formula is shown in Equation (22).  
\[GD=\frac{\sqrt{\sum_{i=1}^{|P|}d_{i}^{2}}}{|P|} \tag{22}\]  
where \(|PF|\) is the number of non-inferior solutions on the Pareto front. \(d_{i}\) is the Euclidean distance between the i-th solution on the Pareto front and the nearest solution on the optimal Pareto front.  
The \(IGD\) indicator is more comprehensive than the \(GD\) indicator, which can evaluate the convergence and diversity of the non-dominant solutions simultaneously. The smaller the \(IGD\) value, the better the performance of the algorithm is. The formula is shown in Equation (23).  
\[IGD=\frac{\sum_{x\in PF}dist(x,PF)}{|PF^{*}|} \tag{23}\]  
where \(dist(x,PF)\) is the Euclidean distance of a solution on the optimal Pareto front and the nearest solution on the obtained Pareto front. \(|PF|\) represents the number of non-inferior solutions on the optimal Pareto front.  
\(\Omega\) indicator is used to calculate the percentage of the non-inferior solution set obtained by an algorithm to the optimal Pareto solution, which reflects the dominant performance of the algorithm. The formula is as shown in Equation (24).  
\[\Omega_{L_{k}}=\frac{|P(\cup\limits_{l}L_{i})\backslash P(\cup\limits_{i\
eq k }L_{i})|}{|P(\cup\limits_{i}L_{i})|}\times 100\% \tag{24}\]  
where \(L_{i}\) represents the non-inferior solution set obtained by the i-th algorithm,\(|P(O)|\) represents the number of non-dominated solutions in the set \(O\). \(\Omega_{L_{k}}\) takes a value between [0, 1], the larger it is, and the better the dominance of the Pareto front obtained by the algorithm.