## 2 Problem Description and Model Construction  
### Description of the Hybrid Flow-Shop Scheduling Problem in Parallel Sequential Mode  
The background of this paper is multi-equipment work center production scheduling in an MTO production mode, and multi-equipment work center production scheduling can be directly borrowed from the HFSP. Therefore, this paper investigates an HFSP that considers auxiliary time (setup time and handling time) and is based on parallel sequential movement.  
The HFSP based on the parallel movement model is described as follows. \(n\) workpieces are machined in the same order on \(i\) work centers. Each work center has \(m_{h}\) machining machines (\(m_{h}\geq 1\), \(h=1\), \(\dots\), \(i\)). \(r\) is the number of types of workpieces, and \(n_{a}\) is the number of type \(a\) workpieces, with \(n_{1}+n_{2}+\dots\)\(n_{r}=n\). Workpieces are transported to different work centers with corresponding handling times, and the equipment has setup times for processing different kinds of workpieces. The equipment must be continuous during machining (i.e., the next workpiece should be machined immediately after the current workpiece is machined), and no idle time is allowed. One or more performance indicators is optimized by sequencing the workpieces and assigning equipment to them. The Gantt chart is shown in Figure 1. This is a machine with \(n=6\), \(i=3\), workpieces of type 1 numbered 1 and 2, and workpieces of type 2 numbered 3, 4, 5, and 6. The S notation in the figure indicates the setup time. It can be seen that the workpieces are processed continuously on each machine. The end machining time of workpiece 3 at work center 1 is \(t_{1}\), and the machining start time at work center 2 is \(t_{2}\). This is due to the handling time of workpiece 3 from work center 1 to work center 2. Note: All machining machines do not require an adjustment time when machining a workpiece for the first time.  
Figure 1: HFSP based on parallel sequential movement pattern.  
This paper is based on the following assumptions to help understand the issue:  
1. The workpiece may not be halted after it has begun being processing on the equipment, nor may it be stopped before inserting other workpieces.
2. Each machine is in good working order, and there is no downtime.
3. The handling time includes the time spent handling the workpiece from the processing equipment to the handling equipment and from the handling equipment to the processing equipment.
4. The handling equipment has an unlimited carrying capacity.
5. For each workpiece on different pieces of equipment in multiple work centers, the machining time, setup time, and handling time between work centers are known.
6. All workpieces are processed on different equipment, and no nonconforming products are produced.  
Figure 2 shows the layout of a multi-device work center, where each work center is equivalent to a process (stage). It can be seen that there are multiple pieces of machining equipment in each work center, and this machining equipment can be completely related (i.e., the workpiece is processed at the same time by all equipment in the current work center) or unrelated (i.e., the workpiece is not necessarily processed at the same time by all equipment in the current work center). There is a certain distance between the different work centers, and the workpiece needs to be moved with the help of handling equipment. The handling equipment can be vehicles, handling robots, etc., and the number of pieces of handling equipment is greater than or equal to 1. The production side should reasonably arrange the production and processing scheduling plan according to the customer's order delivery time and the efficiency of the enterprise in order to reduce the manufacturing cycle, improve the utilization of the processing equipment, and deliver on time.