# Research on Low-Carbon-Emission Scheduling of Workshop under Uncertainty
## 3 Methods
### Analysis of Test Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Algorithm** & **Average** & **St.dev** \\ \hline \multirow{4}{*}{ZDT1} & MOEA/D & \(1.0569\times 10^{-2}\) & \(8.96\times 10^{-3}\) \\  & NSGA-II & \(4.7751\times 10^{-3}\) & \(1.83\times 10^{-4}\) \\  & NSGA-III & \(3.9120\times 10^{-3}\) & \(1.20\times 10^{-5}\) \\  & NSGA-III-ST & \(\mathbf{3.8896\times 10^{-3}}\) & \(3.89\times 10^{-6}\) \\ \hline \multirow{4}{*}{ZDT2} & MOEA/D & \(1.9229\times 10^{-2}\) & \(3.01\times 10^{-2}\) \\  & NSGA-II & \(4.9055\times 10^{-3}\) & \(1.96\times 10^{-4}\) \\  & NSGA-III & \(3.8716\times 10^{-3}\) & \(4.50\times 10^{-5}\) \\  & NSGA-III-ST & \(\mathbf{3.8098\times 10^{-3}}\) & \(2.69\times 10^{-6}\) \\ \hline \multirow{4}{*}{ZDT3} & MOEA/D & \(2.6567\times 10^{-2}\) & \(1.30\times 10^{-2}\) \\  & NSGA-II & \(7.3206\times 10^{-3}\) & \(7.39\times 10^{-3}\) \\  & NSGA-III & \(6.1249\times 10^{-3}\) & \(2.05\times 10^{-4}\) \\  & NSGA-III-ST & \(\mathbf{6.0997\times 10^{-3}}\) & \(1.82\times 10^{-4}\) \\ \hline \multirow{4}{*}{ZDT4} & MOEA/D & \(2.0121\times 10^{-2}\) & \(1.08\times 10^{-2}\) \\  & NSGA-II & \(5.3536\times 10^{-3}\) & \(8.16\times 10^{-4}\) \\  & NSGA-III & \(9.5565\times 10^{-3}\) & \(1.29\times 10^{-2}\) \\  & NSGA-III-ST & \(\mathbf{4.0198\times 10^{-3}}\) & \(1.67\times 10^{-4}\) \\ \hline \multirow{4}{*}{DTLZ1} & MOEA/D & \(\mathbf{2.0741\times 10^{-2}}\) & \(2.17\times 10^{-4}\) \\  & NSGA-II & \(2.7259\times 10^{-2}\) & \(1.51\times 10^{-3}\) \\  & NSGA-III & \(2.0780\times 10^{-2}\) & \(2.20\times 10^{-4}\) \\  & NSGA-III-ST & \(2.1333\times 10^{-2}\) & \(3.36\times 10^{-3}\) \\ \hline \multirow{4}{*}{DTLZ2} & MOEA/D & \(\mathbf{5.4467\times 10^{-2}}\) & \(1.33\times 10^{-6}\) \\  & NSGA-II & \(6.9125\times 10^{-2}\) & \(2.38\times 10^{-3}\) \\ \cline{1-1}  & NSGA-III & \(5.4480\times 10^{-2}\) & \(7.48\times 10^{-6}\) \\ \cline{1-1}  & NSGA-III-ST & \(5.4668\times 10^{-2}\) & \(6.80\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The test results based on the IGD. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Algorithm** & **Average** & **St.dev** \\ \hline \multirow{4}{*}{ZDT4} & MOEA/D & \(6.9605\times 10^{-1}\) & \(1.21\times 10^{-2}\) \\  & NSGA-II & \(7.1726\times 10^{-1}\) & \(1.70\times 10^{-3}\) \\  & NSGA-III & \(7.1307\times 10^{-1}\) & \(8.60\times 10^{-3}\) \\  & NSGA-III-ST & \(\mathbf{7.1975\times 10^{-1}}\) & \(4.28\times 10^{-4}\) \\ \hline \multirow{4}{*}{DTLZ1} & MOEA/D & \(\mathbf{8.3994\times 10^{-1}}\) & \(1.53\times 10^{-3}\) \\  & NSGA-II & \(8.2031\times 10^{-1}\) & \(5.55\times 10^{-3}\) \\  & NSGA-III & \(8.3962\times 10^{-1}\) & \(1.65\times 10^{-3}\) \\  & NSGA-III-ST & \(8.3884\times 10^{-1}\) & \(9.24\times 10^{-3}\) \\ \hline \multirow{4}{*}{DTLZ2} & MOEA/D & \(\mathbf{5.5947\times 10^{-1}}\) & \(2.65\times 10^{-5}\) \\  & NSGA-II & \(5.3180\times 10^{-1}\) & \(4.47\times 10^{-3}\) \\  & NSGA-III & \(5.5946\times 10^{-1}\) & \(4.55\times 10^{-5}\) \\  & NSGA-III-ST & \(5.5824\times 10^{-1}\) & \(1.23\times 10^{-3}\) \\ \hline \multirow{4}{*}{DTLZ4} & MOEA/D & \(4.0454\times 10^{-1}\) & \(1.35\times 10^{-1}\) \\  & NSGA-II & \(5.0529\times 10^{-1}\) & \(1.13\times 10^{-1}\) \\ \cline{1-1}  & NSGA-III & \(4.6308\times 10^{-1}\) & \(1.28\times 10^{-1}\) \\ \cline{1-1}  & NSGA-III-ST & \(\mathbf{5.5790\times 10^{-1}}\) & \(1.48\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._The convergence of the four algorithms on the HV on the ZDT1 problem is drawn, as shown in Figure 5. It can be seen that the convergence speed of NSGA-III-ST has been greatly improved. Although NSGA-III-ST has no optimal solution among the four algorithms in the DTLZ2 problem, where the Pareto optimal solution set is evenly distributed, the improved algorithm retains the evaluation method based on the reference vector in NSGA-III. It ensures the uniformity of the intersection point with the Pareto front.