# Research on Low-Carbon-Emission Scheduling of Workshop under Uncertainty
## 1 Introduction
### Objective Function  
Based on the problem description and problem assumptions, the scheduling goals are a time-based index (\(TBI\)), sum carbon emissions (\(SCE\)), and robustness index (\(\Delta\delta\)) depreciation, which are shown as \(F_{1}\), \(F_{2}\), and \(F_{3}\). The relationship between the fuzzy completion time and fuzzy delivery time is not only ahead of the time or delay, so the completion time satisfaction (\(AI_{i}\)) is introduced, which is shown as \(AI_{i}=\frac{area\big{(}\widetilde{C_{i}\cap\widetilde{D}_{i}}\big{)}}{area \widetilde{C_{i}}}\). The index value represents the ratio of the cross area of \(C_{i}\), and \(D_{i}\) membership functions to the area of the whole membership function of \(C_{i}\). The Average Weighted Satisfaction (\(AWS\)) and the Production Minimum Satisfaction (\(PMS\)) are introduced to comprehensively evaluate \(AI_{i}\), where \(AWS=\frac{1}{n}\sum_{i=1}^{n}w_{i}AI_{i}\), \(PMS=\min(AI_{1},AI_{2},\ldots,AI_{n})\), and \(w_{i}\) represents the weighting coefficient of \(SD\), which reflects the importance of a customer's requirements for delivery time, and \(\sum_{i=1}^{n}w_{i}=1\). \(TBI=\theta_{1}AWS+\theta_{2}PMS\), where \(\theta_{1}\) and \(\theta_{2}\) represent the weighting coefficients of \(AWS\) and \(PMS\), and \(\theta_{1}+\theta_{2}=1\). Therefore, the objective function \(F_{1}\) is shown in Equation (3). \[F_{1}=min(TBI)=min(\theta_{1}AWS+\theta_{2}PMS) \tag{3}\]  
The model only considers the fuzzy carbon emissions generated by power loss during the machining process \(\widetilde{CE}_{p}\), machine tool no-load process \(\widetilde{CE}_{r}\), coolant consumption production \(\widetilde{CE}_{f}\), and the public electricity consumption \(\widetilde{PCE}\) generated by the operation of public auxiliary facilities. The flow of coolant is the same when different parts are machined on the same machine, and the power of public auxiliary facilities \(P_{sumRp}\) includes the lighting facilities' power \(P_{sumLED}\), management kanban power \(P_{sumBoard}\), and ventilator power \(P_{sumFan}\). \[\widetilde{CE}_{p}=\sum_{i=1}^{n}\sum_{k=1}^{m}\sum_{j=1}^{l}F_{e}x_{ijk}P_{ ik}\widetilde{T_{ijk}} \tag{4}\]  
\[\widetilde{CE}_{r}=\sum_{i=1}^{n}\sum_{k=1}^{m}\sum_{j=1}^{l}F_{e}x_{ijk}P_{ ko}+\widetilde{t_{ijk}} \tag{5}\]  
\[\widetilde{CE}_{f}=\sum_{i=1}^{n}\sum_{k=1}^{m}\sum_{j=1}^{l}\frac{ \widetilde{T_{ijk}}}{T_{f}}F_{c}L_{fk}x_{ijk} \tag{6}\]  
\[P_{sumRP}=P_{sumLED}+P_{sumBoard}+P_{sumFan} \tag{7}\]  
\[\widetilde{PCE}=F_{e}\widetilde{C_{max}}P_{sumRP} \tag{8}\]  
Therefore, the objective function \(F_{2}\) is shown in Equation (9). \[F_{2}=\min(SCE)=\min\big{(}\widetilde{CE}_{p}+\widetilde{CE}_{r}+\widetilde{ CE}_{f}+\widetilde{PCE}\big{)} \tag{9}\]  
The uncertainty of the fuzzy completion time can be determined by the difference in the span between the shortest and longest times, which reflects robustness. So the \(\triangle\delta_{1}\) and \(\triangle\delta_{2}\) are introduced, where \(\triangle\delta_{1}=\mu_{1}(a_{1}+a_{3}-2a_{2})+\mu_{2}a_{2}\), \(\triangle\delta_{2}=\sigma(a_{3}-a_{1})\), \(\mu_{1}\), \(\mu_{2}\), and \(\sigma\) are the non-negative weights, and \(\sum_{i=1}^{2}\mu_{i}=1\). Therefore, \(F_{3}\) is expressed in Equation (10). \[F_{3}=min(\triangle\delta)=min(\triangle\delta_{1}+\triangle\delta_{2}) \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Parameter Meaning** \\ \hline \(T_{f}\) & Effective cycle period of coolant on machine \(M_{k}\) \\ \(L_{fk}\) & Amount of coolant circulating on the machine \(M_{k}\) \\ \(x_{ijk}\) & Boolean value : when the operation \(O_{ij}\) processes on machine \(M_{k}\), the value is 1; otherwise, the value is 0 \\ \(x_{ijR}\) & Boolean value : when the operation \(O_{ij}\) processes on machine \(M_{k}\) at the time \(t\), the value is 1; otherwise, the value is 0 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._The objective function of the model is shown in Equation (11).