## 2 Problem Description and Modeling
### Mathematical Model Building  
Workshop energy consumption comprises machine tool energy consumption and transportation energy consumption. In Figure 1, which represents a simplified model of the input power of a machine tool during the machining process [23], the energy consumption of a single piece of equipment can be divided into four states: starting state, processing state, no-load state, and stop state. During equipment start-up and shut-down, there is a significant fluctuation in power, but the duration is short. Frequent start-stop operations can negatively impact the machine's lifespan and processing quality. Typically, a machine performs only one start-stop operation, which is not the primary factor affecting energy consumption. Hence, it is not considered in the model. Consequently, only the processing state and no-load state of a single equipment are taken into account when considering energy consumption. The machining energy consumption is determined by the machining power and machining time of the machine tool. \[E_{c}=\sum_{k}^{m}E_{c}^{k}=\sum_{k}^{m}P_{c}^{k}\times T_{c}^{k} \tag{1}\]  
The no-load energy consumption is generated by the idling state of the machine tool before the workpiece is processed, and is determined by the no-load time and no-load power of the machine tool. \[E_{idle}=\sum_{k}^{m}E_{idle}^{k}=\sum_{k}^{m}P_{idle}^{k}\times T_{idle}^{k} \tag{2}\] \begin{table}
\begin{tabular}{c l l} \hline \hline \multicolumn{2}{c}{**Symbols**} & \multicolumn{2}{c}{**Definitions**} \\ \hline \multirow{7}{*}{Parameters} & \(N\) & Workpieces \\  & \(J\) & Working sequence \\  & \(M\) & Machines \\  & \(O_{i,j}\) & The \(j\)-th process of the \(i\)-th workpiece \\  & \(T_{i,j,k}\) & Processing time of \(O_{i,j}\) on machine \(k\) \\  & & Time for transporting \(N\) from machine tool \(M_{m}\) to machine tool \(M_{k}\) \\  & \(T_{i(j-1),jmk}\) & between operation \(O_{i,j-1}\) and \(O_{i,j}\) of workpiece \(N\) \\  & & Integer variable, takes 0 or 1 if \(O_{i,j}\) is processed on machine \(k\), \\  & & otherwise 0 \\  & \(P_{c}^{k}\) & Machining power of machine tool \(k\) \\  & \(P_{idle}^{k}\) & Standby power of machine tool \(k\) \\  & & Power for transporting \(N\) from machine tool \(M_{m}\) to machine tool \(M_{k}\) \\  & & between operation \(O_{i,j-1}\) and \(O_{i,j}\) of workpiece \(N\) \\  & \(S_{i,j,k}\) & Starting processing time of \(O_{i,j}\) at machine \(k\) \\  & \(F_{i,j,k}\) & \(O_{i,j}\) end processing time on machine \(k\) \\  & \(C_{i}^{i}\) & Completion time for workpiece \(i\) \\  & \(T_{c}^{k}\) & Machining time of machine tool \(k\) \\  & \(T_{idle}^{k}\) & Standby time of machine tool \(k\) \\  & \(E_{k}\) & Total energy consumption of machine tool \(k\) \\  & \(E_{c}^{k}\) & Machining energy consumption of machine tool \(k\) \\  & \(E_{idle}^{k}\) & Standby energy consumption of machine tool \(k\) \\  & \(E_{trans}^{k}\) & Total transportation energy consumption \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbol descriptions. During the entire processing process, workpieces need to be transported from one machine tool to another for processing, necessitating transportation. Assuming a constant transportation power, the transportation time is determined by the distance between the two machine tools. The transportation energy consumption can be calculated as the product of the transportation power and the transportation time. \[E_{trans}=\sum_{i=1}^{n}\sum_{j=1}^{q}T_{i(j-1)\,jmk}\times P_{i(j-1)\,jmk} \tag{3}\]  
The total energy consumption of the workshop is thus obtained as:  
\[E=E_{c}+E_{idle}+E_{trans}=\sum_{k}^{m}P_{c}^{k}\times T_{c}^{k}+\sum_{k}^{m}P _{idle}^{k}\times T_{idle}^{k}+\sum_{i=1}^{n}\sum_{j=1}^{q}T_{i(j-1)\,jmk}\times P _{i(j-1)\,jmk} \tag{4}\]  
A multi-objective mathematical model is established with the maximum completion time\(f_{1}\) and total energy consumption of the workshop\(f_{2}\). \[ \begin{split}\text{min}f_{1}&=C=\max_{1\leq i\leq n }C_{i}\\ \text{min}f_{2}&=E\end{split}