## 1 Introduction
### Optimization Model of a Multi-Objective DRHFS Problem
\begin{table}
\begin{tabular}{c c} \hline \hline
**Application** & **Significance** \\ \hline \(f1=\min(C_{\max})\) & minimizing the make span \\ \(f2=\min(TCT)=\min\big{\{}\sum\limits_{f\to 1}^{F}TCT_{f}\big{\}}\) & minimizing the total carbon emissions \\ \(f3=\min(TDT)=\min\big{\{}\sum\limits_{f\to 1}^{F}TDT_{f}\big{\}}\) & minimizing the total tardiness \\ \(TCT_{f}=TC_{f}+TC_{f}+TC_{fR}\) & Total carbon emissions from factory \(f\) \\ \(TDT_{f}=\sum\limits_{i=1}^{n_{f}}\max(0,C_{fi}-d_{fi}))\) & Total tardiness of jobs processed in factory \(f\) \\ \(Y_{i,f}r_{ijk}(S_{fi})+T_{fi,ijk}\leq Y_{i,f}r_{ijk,l,j,k}S_{fi,l+1}\) & Indicates the completion time of \(O_{ij}\) must be earlier \\ \(\forall i,j,l,t^{\prime}and\,l\
eq l^{\prime}\) & than the starting time of \(O_{ij+1}\) \\ \(\sum_{k=1}^{n_{f}}r_{ijk}=1\) & \(\forall f,i,j,k,l,O_{ij}\in U_{i}\) & Indicates any operation can be processed on only one machine \\  & \(\sum_{f=1}^{F}Y_{i,f}=1\) & \(\forall i,f\) \\ \(H\times Y_{i,f}(2-r_{ijk}-r_{gab})+H\times Y_{g,i}(1-Z_{ij,gb})+(S_{fgh}-S_{fi}) \geq T_{fi/k}\) & \(\forall f,i,g,j,h,i\leq g,O_{ij}\in U_{i}\) \\ \(H\times Y_{i,f}(2-r_{ijk}-r_{ijk})+H\times Y_{i,f}(1-Z_{ij,fR})+(S_{fij}-S_{fi}) \geq T_{fi/k}\) & \(\forall f,i,j,k,l\leq f,O_{ij}\in U_{i}\) \\ \(\sum_{f=1}^{F}\sum_{i=1}^{s}\sum_{k=1}^{n_{f}}\sum_{i=1}^{n_{f}}\sum_{j=1}^{N _{f}}Y_{i,f}r_{ijk}(S_{fi}+T_{fi,ijk})\leq C_{\max}\) & \(\forall f,i,j,k,l\) \\ \(TC_{P}=\sum_{i=1}^{s}\sum_{k=1}^{n_{f}}\sum_{i=1}^{n_{f}}\sum_{j=1}^{N_{f}} \sum_{i=1}^{N_{f}}\sum_{i=1}^{N_{f}}Y_{i,f}r_{ijk}R_{P}r_{P/ijk}T_{f/ijk}\) \\ \(TC_{fl}=\sum_{i=1}^{s}\sum_{k=1}^{n_{f}}\sum_{k=1}^{n_{f}}FDT_{fl}T_{f/ijk}\) & \(\forall T,i,j,k,l\) \\ \(TC_{R}=\sum_{i=1}^{s}\sum_{k=1}^{n_{f}}\frac{T_{f,i,i,k}-T_{f,i,k}}{1R_{P/i}}UR _{l/k}F_{R}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Objective functions and constraints. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mathematical Symbols** & **Significance** \\ \hline \(Pl_{fR}\) & Idle power of machine \(k\) at stage \(l\) of factory \(f\) \\ \(T_{fhl}\) & Idle time of machine \(k\) at stage \(l\) of factory \(f\) \\ \(TC_{fp}\) & Total carbon emissions from machines processing of factory \(f\) \\ \(TC_{fl}\) & Total carbon emissions from machines in an idle state of factory \(f\) \\ \(F_{e}\) & Carbon emissions factor of electric energy \\ \(T_{fjk,clone}\) & The finishing time of machine \(k\) at stage \(l\) of factory \(f\) (the finishing time of the \\  & last operation on the machine) \\ \(T_{fjk,open}\) & The start time of machine \(k\) at stage \(l\) of factory \(f\) (the starting time of the \\  & first operation on the machine) \\ \(UR_{f/lk}\) & The usage amount of lubricants on machine \(k\) at stage \(l\) of factory \(f\) \\ \(TR_{f/lk}\) & Effective service time of lubricants on machine \(k\) at stage \(l\) of factory \(f\) \\ \(TC_{fR}\) & Total carbon emissions from lubricants used by machines of factory \(f\) \\ \(F_{R}\) & Carbon emissions factor of lubricants \\ \(S_{fi/j}\) & The starting time of \(O_{ij}\) in factory \(f\) \\ \(C_{fi}\) & The completion time of job \(i\) in factory \(f\) \\ \(C_{ij}\) & The completion time of \(O_{ij}\) \\ \(r_{ijk}\) & \(1\) indicates \(O_{ij}\), which is processed on machine \(k\) at stage \(l\) of factory \(f\), \(0\) indicates \\ \(O_{ij}\) is not processed on machine \(k\) at stage \(l\) of factory \(f\) \\ \(Z_{ij,gb,R}\) & \(1\) indicates \(O_{ij}\) is processed before \(O_{gh}\) on machine \(k\) at stage \(l\), \(0\) indicates \\ \(O_{ij}\) is not processed before \(O_{gh}\) & \(O_{ij}\) is not processed before \(O_{gh}\) \\ \(TCT_{f}\) & Total carbon emissions from factory \(f\) \\ \(TDT_{f}\) & Total tardiness of jobs processed in factory \(f\) \\ \(C_{\max}\) & Make span of all jobs \\ \(TCT\) & Total carbon emissions \\ \(TDT\) & Total tardiness \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont\).