## 2 Problem Description and Model Construction
### Model Establishment for Multi-Objective Optimization
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Definition** \\ \hline \(\text{t}_{h,j,k}\) & the time that the workpiece \(j\) is processed on the equipment \(k\) at the work center \(h\) \\ \(\text{H}_{h,h+1}\) & handling time from work center \(h\) to \((h+1)\) \\ CT & total handling time \\ ST & total setup time \\ CB\({}_{h,h+1,d}\) & start time of the \(q\)th handling from work center \(h\) to \((h+1)\) \\ CE\({}_{h,h+1,d}\) & end time of the \(q\)th handling from work center \(h\) to \((h+1)\) \\ CB\({}_{h,h+1}\) & handling start time of workpiece \(j\) from work center \(h\) to work center \((h+1)\) \\ CE\({}_{j,h,h+1}\) & handling end time of workpiece \(j\) from work center \(h\) to work center \((h+1)\) \\ TB\({}_{h,j}\) & start time of processing of workpiece \(j\) at work center \(h\) \\ TE\({}_{h,j}\) & end time of processing of workpiece \(j\) at work center \(h\) \\ S\({}_{h,j}\) & setup time of workpiece \(j\) at work center \(h\) \\ S\({}_{h,j,k}\) & setup time of the workpiece of type \(a\) on the equipment \(k\) at the work center \(h\) \\ MB\({}_{h,j,k}\) & start time for machining the \(v\)th workpiece on machine \(k\) at work center \(h\) \\ ME\({}_{h,j,k}\) & end time for machining the \(v\)th workpiece on machine \(k\) at work center \(h\) \\ J\({}_{h,d}\) & the set of workpieces for the \(q\)th handling in work center \(h\) \\ \(p(_{h,h+1})\) & number of handling events from work center \(h\) to \((h+1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of parameters. workpieces. In addition, the number of handling events also has a close relationship with energy consumption; less handling also means less energy consumption of the handling equipment. The objective function is shown in Equation (2). \[f(2)=\min\limits_{h=1}NOH \tag{2}\]  
The constraint condition equations are shown in (3)-(18). \[\sum_{k=1}^{m_{h}}X_{h,j,k}=1,h=1,2,\ldots,i,\ j=1,2,\ldots,n \tag{3}\]  
\[p_{(h,h+1)} =\sum_{j=1}^{n}Y_{h,j},h=1,2,\ldots i-1 \tag{4}\]  
\[\text{CT}=\ \sum_{h=1}^{i-1}p_{(h,h+1)}\times H_{h,h+1} \tag{5}\]  
\[\text{CB}_{h,h+1,q}=\max\Bigl{\{}\text{TE}_{h,j}\Bigr{|}j\in\text{J}_{h,q} \Bigr{\}},\ h=1,2,\ldots,i-1,\ q=1,2,\ldots,\ p_{(h,h+1)} \tag{6}\]  
\[\text{CE}_{h,h+1,q}=\text{CB}_{h,h+1,q}+\text{H}_{h,h+1},\ h=1,2,\ldots,i-1, \ q=1,2,\ldots,\ p_{(h,h+1)} \tag{7}\]  
\[\text{CB}_{h,h+1,q+1}\ \geq\ \text{CB}_{h,h+1,q},\ q=1,2,\ldots,\ p_{(h,h+1)},\ h=1,2,\ldots,i-1 \tag{8}\]  
\[\text{CB}_{j,h,h+1}\geq\text{TE}_{j,h},h=1,2\ \ldots,i,j=\in\{1,2,\ldots,n\} \tag{9}\]  
\[\text{S}_{h,j}=\text{A}_{h,j,k}\times\text{S}_{a,h,k},j=2,3,\ldots,n,h=1,2, \ldots,i,a=1,2,\ldots,r,k=1,2,\ldots,m_{h} \tag{10}\]  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]0  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]1  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]2  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]3  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]4  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]5  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]6  
\[f(2)=\min\limits_{h=1}NOH \tag{2}\]7  
Equation (3) ensures that the workpiece is processed on only one piece of equipment in each work center. Equation (4) represents the number of handling events from work center \(h\) to work center \((h+1)\). Equation (5) represents the total handling time. Equation (6) represents the relationship between the start time of handling and the completion time of the workpiece. The \(q\)th handling start time is equal to the maximum \(\text{TE}_{j,h}\) of the artifacts in the handling artifact set. Equation (7) represents the end time of handling. In the same work center, the end time of the \((q+1)\)th handling is equal to the start time of the \(q\)th handling plus the handling time. Equation (8) represents the sequence of the start time of handling. The start time of the \((q+1)\)th handling from the work center \(h\) to \((h+1)\) is greater than or equal to the start time of the \(q\)th handling. Equation (9) represents the relationship between the start and end times of processing, ensuring that the workpiece has been processed prior to handling.