# Research on Low-Carbon-Emission Scheduling of Workshop under Uncertainty
## 3 Methods
### Performance Test of the Algorithm  
To better reflect the comprehensive performance of the algorithm, the three indexes Hypervolume (HV), Inverted Generational Distance (IGD), and Averaged Hausdorff Distance (\(\Delta p\)) are used to evaluate the performance of the algorithm [33; 34].  
(a) HV  
The HV calculates the volume covered by the Pareto front in the objective space, a comprehensive index. The larger the index is, the better the diversity and convergence of the obtained Pareto front are [35]. The formula is given by Equation (17).  
\[\mathrm{HV}=\delta\Big{(}U_{i=1}^{|Q|}v_{i}\Big{)} \tag{17}\]  
where \(\delta\) denotes the Leberger measure. \(|Q|\) denotes the numbers of the Pareto front obtained by solving the algorithm. \(v_{i}\) represents the Hypervolume formed by the reference point and the \(i\)th solution in the resulting solution set.  
(b) IGD  
The IGD is an inverse mapping of the convergence metric generation distance, representing the Euclidean distance between the solutions on the true Pareto front and the calculated non-dominated solution set. In addition to reflecting the algorithm's convergence [36], it can also reflect the uniformity and extensiveness of the solution set's distribution. A smaller value indicates that the solved Pareto front is closer to the true one. The formula is given by Equation (18).  
\[\mathrm{IGD}(p^{*},D)=\frac{\sum_{v\in p^{*}}dis(v,D)}{|p^{*}|} \tag{18}\]  
where \(p^{*}\) denotes the true Pareto optimal solution set; \(D\) denotes the non-dominated solution set that is actually solved by the algorithm; and \(dis(v,D)\) means the minimum Euclidean distance from the individual in \(p^{*}\) to the corresponding point in \(D\). \(|p^{*}|\) represents the size of the true Pareto optimal solution.  
\begin{table}
\begin{tabular}{l} Input: \(Best_{0}\), \(SE\), \(func\), \(a_{min}\), \(a_{max}\), \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(\theta\) \\ \hline \hline \multicolumn{2}{l}{Output: _Best_} \\ \hline
1. \(Best_{0}\) \(\leftarrow\) _Random Initialization_ \\
2. repeat \\
3. if \(a<a\) then\({}_{min}\) \\
4. \(\alpha\) \(\rightarrow\) \(a_{max}\) \\
5. end if \\
6. \(Best_{k}\) \(\leftarrow\) expansion \((func\), \(Best_{k}\), \(SE\), \(\beta\), \(\gamma)\) \\
7. \(Best_{k}\) \(\leftarrow\) rotation \((func\), \(Best_{k}\), \(SE\), \(\beta\), \(\alpha)\) \\
8. \(Best_{k}\) \(\leftarrow\) axesion \((func\), \(Best_{k}\), \(SE\), \(\beta\), \(\delta)\) \\
9. \(\alpha\) \(\leftarrow\) \(\theta\) \\
10. until terminal condition \\
11. \(Best\) \(\leftarrow\) \(Best_{k}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Pseudocode for alternating rotations.  
3. \(\Delta p\)  
\(\Delta p\) is used to evaluate the average Hausdorff distance between the solution set obtained by the algorithm solution and the true Pareto front of a given multi-objective optimization problem. It is a comprehensive performance evaluation indicator combining improved GD and IGD indexes. The smaller the indicator is, the better the comprehensive performance of the algorithm is. The improved GD and IGD are denoted as GD\({}_{\text{p}}\) and IGD\({}_{\text{p}}\), which are calculated in Equations (19) and (20), and \(\Delta\)p is calculated in Equation (21).  
\[\text{GD}_{p}(S,P,q)=\big{(}\frac{1}{|S|}\sum_{x\in S}\underset{p\in P}{\text{ \emph{mind}}^{q}(x,p)}\big{)}^{\frac{1}{q}} \tag{19}\]  
\[\text{IGD}_{p}(S,P,q)=\big{(}\frac{1}{|P|}\sum_{p\in P}\underset{x\in S}{\text{ \emph{mind}}^{q}(p,x)}\big{)}^{\frac{1}{q}} \tag{20}\]  
\[\Delta p(S,P,q)=\max\big{(}GD_{p}(S,P,q),IGD_{p}(S,P,q)\big{)} \tag{21}\]  
where \(x\) is the solution; \(S\) is the solution set; \(P\) is the Pareto approximation front; the size of \(q\) affects the size of \(\Delta p\), and the penalty of \(\Delta p\) on discrete points is greater when \(q\) is larger; and \(d\) is the Euclidean distance.