## 3 Definition of Process Constraints for the Balancing Machining Line Problem
To simplify the problem, this study made the assumption that the preparation time (set-up time) is equal at each machine for each operation. It suggests that operations with times \(t1\), \(t2\), \(t4\), \(t5\), and \(t6\) are realized in the same start time. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Task** \\ **Number** \\ \end{tabular} } & \multirow{2}{*}{**Processing Content**} & \multirow{2}{*}{\begin{tabular}{c} **Processing** \\ **Times (s)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Plane It** \\ **Belongs to** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Equipment** \\ **Required** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Cutting Tool Type** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Machining** \\ **Direction of** \\ **Parts** \\ \end{tabular} } \\ \hline
1 & Datum A & \(t1\) & Datum A & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 30\ Milling\ cutter}\) & Bottom surface \\
2 & Datum B & \(t2\) & Datum B & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 10\ combined\ drilling\ \& reaming}\) & Top surface \\
3 & Hole \(\mathrm{\varphi 5}\) (Datum C) & \(t3\) & Datum C & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 5}\) combined drilling \& reaming & Top surface \\
4 & plan 4 & \(t4\) & plan 4 & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 30\ Milling\ cutter}\) & Right surface \\
5 & plan 5 & \(t5\) & plan 5 & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 30\ Milling\ cutter}\) & Left surface \\
6 & The hole \(\mathrm{\varphi 6\ on\ \mathrm{plan\ 5}}\) & \(t6\) & plan 5 & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 6\ drilling\) & Left surface \\
7 & plan 7 on plan A & \(t7\) & plan 5 & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 20\ Milling\ cutter}\) & Left surface \\
8 & Hole \(\mathrm{\varphi 12}\) & \(t8\) & plan 5 & \(\mathrm{Machining\ centre}\) & \(\mathrm{\varphi 12\ drill\) & Left surface \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Task operation procedure, time, operation direction, and tool requirement for the simple bracket part. Figure 3: Process scheduling assignment scheme satisfying the constraints of Figure 2.  
Mathematical Model for Balancing Problem of Machining Production Line (TLBP) with Advanced Station Preparation and Post-Auxiliary Tasks  
In the machining process, multiple steps such as roughing, semi-finishing, and finishing are often required to complete the processing of one size. To simplify the difficulty in solving the problem, the continuous processing steps of a unit of a certain size can be regarded as integrated into a working step. At the same time, this assumption is also consistent with the actual work step combination, such as drilling and reaming combination [36], drilling and boring combination, drilling and tapping combination, and rough and fine milling surface combination. This assumption is also in line with the current situation of the wide application of composite tools such as composite taps and drill reamers. On this basis, the process sequence constraint relationship matrix [31] between each processing element can be proposed as  
\[\text{A}=\left[ \begin{array}{cccc}a_{11}&a_{12}&\dots&a_{1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\end{array}