## 2 Green Scheduling Problem Model of Double Flexible Job Shop
### Mathematical Model of the DFJSP  
The mathematical symbols in the model and their meanings are as follows:  
\(m\): Number of machines;  
\(n\): Number of work pieces;  
\(l\): Number of workers;  
\(i,h\): Workpiece serial number; \(i,h=1,2,\cdots,n\);  
\(r_{i}\): Total number of operations of workpiece \(i\);  
\(j,g\): Operation sequence number, \(j,g=1,2,\cdots,r\);  
\(k\): Machine serial number, \(k=1,2,\cdots,m\);  
\(s\): Worker serial number, \(s=1,2,\cdots,l\);  
\(O_{ij}\): The \(j\)-th process of workpiece \(i\);  
\(C_{ij}\): Completion time of process \(O_{ij}\);  
\(C_{k}\): Completion time of machine \(k\);  
\(P_{ijks}\): Processing time of the process \(O_{ij}\) operated by worker \(s\) on machine \(k\);  
\(L_{ijks}\): Processing cost of the process \(O_{ij}\) operated by worker \(s\) on machine \(k\);  
\(E\,I_{ijks}\): Green index value of the process \(O_{ij}\) operated by worker \(s\) on machine \(k\);  
\(E_{ijks}\): Energy consumption of the process \(O_{ij}\) operated by worker \(s\) on machine \(k\);  
\(N_{ijk}\): Noise generated by the process \(O_{ij}\) on machine \(k\);  
\(R_{ijk}\): Chip recovery rate of the process \(O_{ij}\) processed on machine \(k\);  
\(S_{ijk}\): Safety of the process \(O_{ij}\) processing on machine \(k\);  
\(x_{ijk}=\left\{ \begin{array}{l}1\text{,}\text{theprocess }O_{ij}\text{ is processed on machine }k\\ 0\text{,}\text{other}\end{array} \right.\)  
\(x_{ijks}=\left\{ \begin{array}{l}1\text{,}\text{the process }O_{ij}\text{ is processed on machine }k\text{ and operated by the worker }s\\ 0\text{,}\text{other}\end{array} \right.\)  
\(x_{ijks}=\left\{\  
Based on the research in [5], this paper constructs the following model where \(C_{M}\) represents the maximum completion time target, \(L_{C}\) represents the total labor cost, and \(G\) represents the total green index. Target function:  
\[MinC_{M}=\max_{1\leq k\leq m}(C_{k}) \tag{1}\]  
\[MinL_{C}=\sum_{i=1}^{n}\sum_{j=1}^{r_{i}}\sum_{k=1}^{m}\sum_{s=1}^{l}\Big{(}L_{ ijks}\times x_{ijks}\Big{)} \tag{2}\]  
\[MinG=\sum_{i=1}^{n}\sum_{j=1}^{r_{i}}\sum_{k=1}^{m}\sum_{s=1}^{l}\Big{(}EI_{ ijks}\times x_{ijks}\Big{)} \tag{3}\]  
\[EI_{ijks}=E_{ijks}{}^{\prime}+N_{ijk}{}^{\prime}+\Big{(}1/R_{ijk}\Big{)}{}^{ \prime}+\Big{(}1/S_{ijk}\Big{)}{}^{\prime} \tag{4}\]  
Constraints:  
\[C_{ij}-C_{i(j-1)}\geq\ P_{ijks}x_{ijks},\ \ \ \forall i,j\ i=1,2,\cdots,n;j=2,3, \cdots,r_{i} \tag{5}\]  
\[\sum_{k=1}^{m}\sum_{s=1}^{l}x_{ijks}=1,\forall i,ji=1,2,\cdots,n;j=1,2,\cdots,r_{i} \tag{6}\]  
\[\Big{[}\Big{(}C_{kg}-C_{ij}-t_{hgk}\Big{)}x_{hgk}x_{ijk}\geq 0\Big{]}\ \Big{[}\Big{(}C_{ij}-C_{hg}-t_{ ijk}\Big{)}x_{hgk}x_{ijk}\geq 0\Big{]}, \tag{7}\]  
\[\forall i,j,h,g,ki,h=1,2,\cdots,n;j=1,2,\cdots,r_{i};g=1,2,\cdots,r_{h};k=1,2, \cdots,m;\]  
Here, Equations (1)-(3) represent the objective functions of maximum completion time, total labor cost, and total green index, respectively. Equation (4) defines the green index value \(EI_{ijks}\) of the process \(O_{ij}\) on machine \(k\) operated by worker \(s\). The value of this index is determined by four factors: energy consumption, noise, chip recovery, and safety. The lower the energy consumption and noise value, the better and the higher the chip recovery rate and safety index value, so the reciprocal is taken for calculation. In addition, because the measurement units and meanings of the above four green indicators are different, they cannot be calculated directly. Therefore, it is necessary to normalize the four indicators by the min-max standardization method. The calculation formula of the min-max standardization method is \(x^{\prime}=\frac{x-x_{min}}{x_{max}-x_{min}}\), where \(x^{\prime}\) represents the new value after \(x\) normalization, \(x_{max}\) is the maximum value of \(x\) indicator, and \(x_{min}\) is the minimum value of \(x\) indicator. Equation (5) is a process constraint to ensure that the workpiece is processed according to the specified process.