## 3 Problem Description and Model  
The flexible job shop scheduling problem in rolling production mode (FJSP-RPM) can be defined as: a batch of workpieces need to be processed in the current scheduling cycle of the workshop, including \(n\) workpieces (J1, J2,..., In), which need to be processed on a total of m machines (M1, M2,..., Mm). The quantity of workpieces exceeds the equipment's production capacity. \(O_{ij}\) represents the \(j\)th process of workpiece \(i\), \(\Omega_{ij}\) represents the optional equipment set of the \(j\)th process of workpiece \(i\), each process can be processed on any one of \(m\) devices with processing capacity, and equipment \(k\) can process more different workpieces process. Due to different performance characteristics of the equipment, the processing time for the same process varies among different devices. Each workpiece has a delivery deadline, where \(d_{i}\) represents the delivery deadline for workpiece \(i\). If the delivery time exceeds the delivery deadline, the manufacturer will face the risk of reputation loss and fines. The goal of job shop scheduling is to determine the processing sequence of the workpiece and the processing machine of the process, so that in the current scheduling cycle \(T\), the equipment utilization rate is the highest, and the total tardiness of the workpiece is the lowest. The workshop executes processing according to the scheduling plan. When the scheduling cycle \(T\) is reached, the unfinished workpieces and new workpieces that need to be processed are re scheduled according to the goal of maximizing equipment utilization and minimizing total delay to obtain a new scheduling plan. The assumptions that need to be met during the workpiece processing include:  
1. All workpieces and machines can be put into use at time zero, and all workpieces are released at time zero. 2. Each machine can only complete one process at a time, and each process can only be executed by one machine at any given time. 3. The processing of workpieces must be strictly carried out in the order of the process, and no steps are allowed to be skipped. 4. Once processing begins, it must not be interrupted and must continue until the process is completed. It is not allowed to split working hours or be occupied by other workpieces. 5. Stop running the machine after completing the last process. 6. The transfer time of the workpiece during the processing is ignored. 7. The workpieces are independent of each other and there is no priority relationship. 8. Not considering dynamic factors such as machine malfunctions, order changes, and urgent order insertion. The symbols and their meanings used in the model are shown in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(n\) & Number of workpieces \\ \(m\) & Number of machines \\ \(i\) & Workpiece index \\ \(j\) & Process index \\ \(k\) & Machine index \\ \(h_{i}\) & The total number of processes for the \(i\)th workpiece \\ \(O_{ij}\) & The \(j\)th machining process of workpiece \(i\) \\ \(t_{ij}\) & The machining time of the \(j\)th process of the \(i\)th workpiece \\ \(s_{ij}\) & The start time of the \(j\)th process for the \(i\)th workpiece \\ \(c_{ij}\) & The end time of the \(j\)th process of the \(i\)th workpiece \\ \(C_{i}\) & The completion time of the \(i\)th workpiece \\ \(D_{i}\) & The delivery date of the \(i\)th workpiece \\ \(T\) & Rolling production scheduling cycle \\ \(T_{0}\) & Scheduling start time \\ \(x_{ijk}\) & If the \(j\)th process of workpiece \(i\) is processed on machine \(M_{k}\), \(x_{ijk}=1\); otherwise, it is 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and their meanings in the model. The objective of the job shop scheduling is to minimize total delay time and maximize equipment utilization through optimal allocation of manufacturing equipment and sequencing of operations for pending workpieces. Based on the problem description above, we formulate a multi-objective scheduling model structured as:  
Objective 1.