## 4 Simulation Testing and Analysis
### Selection of Pareto Optimal Solution Set  
For multi-objective problems, there are multiple solutions in the resulting set of Pareto solutions, which makes it difficult for the decision maker to select a better scheduling solution to process the product. Therefore, in order to facilitate the decision maker to better select an optimal scheduling solution from the Pareto solution set, a combination of AHP and gray correlation decision analysis was used to select the Pareto solution set. First, the weight value of each index was obtained through the analytic hierarchy process. After obtaining the weight value of each objective, the gray correlation decision analysis method was used to obtain the optimal solution under the current weight from a group of optimal solutions. The advantage of combining AHP with gray correlation decision analysis is that it not only combines the subjectivity of the decision maker to assign weights to the goal according to the current conditions, but also quantitatively analyses the data obtained from the optimal solution to select the optimal solution. Calculate the weight value  
In order to find the weight value using the hierarchical analysis method, firstly, the judgment matrix is established by the decision makers such as schedulers by comparing the importance of each index and by quantifying the judgment matrix of each index. The weights of the two indices under the maximum completion time and energy consumption are solved according to the nine-level scale method and in combination with production practice. If the order \(a_{21}\)= 2, then the indices \(f_{2}\) (total energy consumption) are slightly more important than the indices \(f_{1}\) (maximum completion time). The resulting judgment matrix and the weights of each indicator are shown in Table 7.  
The purpose of data normalization is to eliminate the difference between variables due to different dimensions and thus eliminate the influence on the results. The normalization method used here is shown in Formula (11). \[N_{i,j}=\frac{Y_{i,j}-Y_{j}^{\min}}{Y_{j}^{\max}-Y_{j}^{\min}} \tag{11}\]  
where \(N_{i,j}\) is the matrix after \(Y_{i,j}\) normalization and \(Y_{i,j}\) is the raw data, representing the \(j\)-th objective function value of the group \(i\) data, and \(Y_{j}^{\max}\) and \(Y_{j}^{\min}\) are the maximum and minimum values of the \(j\) column of the original matrix \(Y\) respectively. Calculation of gray correlation coefficient  
The gray correlation coefficient \(\gamma_{i,j}\) reflects the degree of association between the \(j\)-th indicator of the \(i\)-th data set and the ideal value. \[\gamma_{i,j}=\frac{N_{j}^{\min}+\rho N_{j}^{\max}}{N_{i,j}+\rho N_{j}^{\max}} \tag{12}\]  
where \(N_{j}^{\min}\) and \(N_{j}^{\max}\) are, respectively, the minimum and maximum values in the index data group after normalization. \(\rho\) is the resolution coefficient, generally taken as 0.5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Indictors** & \(f_{1}\) & \(f_{2}\) & **Weights \(\omega\)** \\ \hline \(f_{1}\) & 1 & 1/2 & 1/3 \\ \(f_{2}\) & 2 & 1 & 2/3 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Index judgement matrix and weight value. (4) Calculation of gray correlation degree  
The gray relational degree is the product of the gray relational coefficient and the corresponding weight. The weight value of each indicator has been obtained from the AHP. The calculation method of the gray relational degree is shown in Formula (13). \[R_{i}=\sum_{j=1}^{m}\gamma_{i,j}w_{j} \tag{13}\]  
By calculation, the data are shown in Equation (14). Where \(Y\) is the raw data, the first and second columns correspond to the maximum completion time and total energy consumption of the workshop, respectively. \(N\) is the normalized matrix corresponding to the original data obtained by data normalization, \(\gamma\) is the gray correlation coefficient matrix, \(\omega\) is the weight matrix of the two objectives obtained by analytic hierarchy process, \(\omega\)\(=[0.33,0.67]\), \(R\) is the gray correlation matrix. \[Y=