## 2 Description of the Problem  
The automobile PBS system consists of seven areas, including the painting-PBS area, the car-receiving lateral-moving machine, six entrance lanes, one return lane (each lane has ten parking spaces), the car-delivery lateral-moving machine, and the PBS-final assembly car-receiving port. The distances between each lane are equal, with a one-meter gap between two adjacent lanes, and each lane has a width of two meters. The lateral-movingmachine maintains a consistent speed during operation. The PBS receiving transverse vehicle transfer system transports vehicle bodies from the paint shop-PBS exit to the appropriate entry lane, and from the return lane to the appropriate entry lane. The receiving transverse vehicle transfer system delivers the first vehicle body in the current paint shop-PBS exit queue or a vehicle body at the 10 th parking spot in the return lane to any of the 10 parking spots in the entry lane. The PBS delivery transverse vehicle transfer system transports the selected vehicle body from the entry lane to the PBS-assembly reception and the vehicle body requiring a sequence adjustment from the entry lane to the return lane. The delivery transverse vehicle transfer system sends the vehicle to the first parking spot in any entry lane to the first parking spot in the return lane or the PBS-assembly reception. Both the receiving and delivery transverse vehicle transfer systems can be set to idle status for a period of time. Moreover, when any vehicle body moves from one parking spot to the next in the entry lane or return lane, it takes 9 s. During the entry process, when any vehicle body arrives at the paint shop-PBS exit, it is positioned at the center, facing entry lane 4. The receiving transverse vehicle transfer system transports the vehicle body from this position to different entry lanes' 10 th parking spots and returns to the initial position. For entry lanes 1-6, the time consumption is 18, 12, 6, 0, 12, and 18 s, respectively. During the exit process, the delivery transverse vehicle transfer system starts from the central initial position, and transports vehicle bodies from the first parking spot in each entry lane to the assembly-PBS reception, which is at the center and directly facing entry lane 4. For entry lanes 1-6, the time consumption is 18, 12, 6, 0, 12 and 18 s, respectively. Optimizing the scheduling for automobile PBS involves various issues, which often require one consider the time consumed during the entry and exit processes and the time consumed when the car enters and leaves the return lane, and ensuring that the car-receiving lateral-moving machine and car-delivery lateral-moving machine comply with the constraint conditions during their actions. Based on these considerations, an optimized scheduling plan can be developed. Figure 1 shows a schematic diagram of the automobile PBS scheduling structure. Figure 1: Schematic diagram of automobile PBS scheduling structure. In addressing the optimization and scheduling problem in the buffer adjustment area of the automobile painting and assembly workshop, we set out the following research hypotheses: (1) The time taken for the body to move from the painting-PBS exit to the car reception cross-shuttle, from the cross-shuttle to entrance 4, and from the shuttle to the assembly reception point can be ignored. (2) The time taken for the body to be unloaded from the shuttle to parking space 10 or return lane 1, and loaded from parking space 1 or return lane 10 onto the shuttle, can be ignored. (3) When multiple bodies are waiting at parking space 1 and the shuttle is available, the first body to arrive at parking space 1 is given priority for processing according to the FIFO rule. (This refers to the scheduling and allocation of car bodies entering the PBS system, where car bodies that enter the system first are processed and completed by transfer machines in the order of their arrival.