**1. Introduction**

With rapid development of information, advanced manufacturing, and artificial intelligence technology, Germany proposed "Industry 4.0" national strategy, which promotes progress in manufacturing industry and provides producing solution schemes for many complex industrial systems [1–3]. In the production of steel casting, assembly of heavy machinery, and other industries, a scheduling problem with non-deterministic polynomial hard (NP-hard) attributes [4–8], such as the flexible flow shop scheduling problem with the characteristics of customized production and parallel processing, was experienced [9,10]. During the actual production of the large equipment manufacturing workshop, only the buffer with limited capacity can be set in the workshop, owing to physical factors such as the workshop space and storage equipment capacity. When the required capacity of the production task fluctuates in the workshop or the takt time between stages is inconsistent, the buffer capacity might reach the upper limit. Consequently, the completed workpieces that are

ready to enter the limited buffer cannot enter the buffer for temporary storage; they are stagnant on their processing workstation while waiting for available space in the limited buffer, which effectuates production blockage, thereby delaying the production process [11–14]. In the actual production of a large equipment manufacturer, it is uncommon to configure the limited buffer with redundant capacity or to temporarily adjust the capacity of limited buffer to avoid the production blockage. The manufacturer often sets up the public buffer in the workshop to dynamically receive workpieces that cannot enter the limited buffer between the stages. This is equivalent to dynamically expanding the capacity of the limited buffer, which can reduce the production blockage and ensure fluent production process. In the actual workshop, the public buffer is often located at a designated position not adjacent to the workstation due to physical factors such as production space. Thus, the transit time between the workstation and public buffer cannot be neglected. It creates the transfer scheduling problem of the workpiece between limited buffer and public buffer, which further increases the uncertainty of the scheduling result and the difficulty of resolving the scheduling problem [15]. Consequently, it is necessary to explore an effective scheduling method for the flexible flow shop scheduling problem with public buffer (FFSP–PB) due to its role in reducing the production blockage and improving the utilization of production resources [16,17]. Therefore, it is of great theoretical and engineering value to solve this problem.

The scheduling problem with the public buffer is derived from the limited buffer scheduling problem, which is closely related to actual engineering. The buffer between stages in the limited buffer scheduling problem is set to the upper limit of the capacity. Once the buffer capacity reaches the upper limit, the workpiece cannot enter this buffer [18]. Presently, the problem is systematically studied worldwide. Zhang et al. [19] proposed two rapidly generating heuristic algorithms with minimum makespan as the criterion. Rooeinfar et al. [20] proposed a novel optimization model and two types of solutions to resolve the uncertain flexible flow shop scheduling problem with limited buffers. Jiang et al. [21] developed an effective multi-objective optimization algorithm in the framework of a multi-objective evolutionary algorithm based on decomposition to solve the hybrid flow shop scheduling problem with limited buffer according to energy orientation. Zeng et al. [22] set forth a two-stage algorithm based on neighborhood search to resolve this issue in accordance with the job shop scheduling problem with limited output buffers.

Investigators have carried out relevant research on the production blockage caused by limited buffer, while studying the scheduling problem with limited buffers. Since the production blockage decreases the production efficiency of enterprises and delays the production process, solving the problem of production blockage of limited buffer in flexible flow shop has been under intensive focus in recent years. Ribas et al. [23] proposed an iterative greedy algorithm to solve the problem of parallel blocking flow shop and distributed blocking flow shop by minimizing the total delay time of the workpiece. Chang [24] established a multi-state manufacturing system (MMS) model to study the reliability of parallel production line manufacturing system with limited-capacity buffer stations to avoiding blockage and starvation. Johri [25] studied the blockage and starvation of limited buffer by linear programming, thereby proposing a method by increasing the selectivity of buffer space to resolve the problem of capacity loss caused by the small capacity buffer.

The above literature demonstrates that the current research on the limited buffer scheduling problem is mainly focused on the various types of workshops, with emphasis on the improvement of global optimization algorithm. However, only a few investigators have addressed the issue of solving the production blockage led by limited buffer stresses on adjusting the production plan and buffer space via alleviating the production blockage by setting the public buffer in the workshop. However, they have not explored the impact of transit time on the production process caused by the movement of the workpiece between public buffer and limited buffer among the stages. Herein, the flexible flow shop scheduling problem with public buffer was more complicated than the generally limited buffer scheduling problem. It considered not only the restricted capacity of limited buffer but also the transfer scheduling problem of workpieces among workstation, limited buffer, and public buffer.

As the research problems become more complicated, it is necessary to explore a global optimization algorithm that can effectively solve these complex problems. The majority of the swarm intelligence algorithms are based on a random search with slow optimization rate, which renders finding the global optimal value challenging. The Hopfield neural network (HNN) algorithm based on non-linear control theory has great advantages in global optimization speed and avoids the shortcomings of the random search of swarm intelligence algorithm [26]. However, the issues pertain small optimization range, easy to fall into, and difficult to break out of the local extremum. Therefore, the idea of a simulated annealing algorithm is introduced to HNN algorithm. During each generation training, neuron input adds random disturbance and Metropolis acceptance criteria controls whether the energy function value which is generated by disturbance input in the next generation of optimization. Thus, the HNN algorithm allows to accept the solution with poor fitness during asymptotic convergence, further changing the optimizing direction of HNN algorithm, expanding its optimizing range and enhancing the ability to jump out of local extremum. By comparison with analysis of groups of simulation schemes, combining the methods of the simulated annealing algorithm-based Hopfield neural network (SAA–HNN) algorithm and local scheduling rules to control the moving process of workpieces between public buffer and the limited buffer can be verified for the efficiency of solving the FFSP–PB. The SAA–HNN algorithm is applied to solve the flexible flow shop scheduling problem with public buffer by extending the application field of neural network algorithm.

With the rise of Industry 4.0, customized production mode has become more popular among manufacturing enterprises, and the varieties of products that cater to customers' needs have diversified. It is difficult for enterprises to control the takt, which leads to production blockage. This increases the importance of public buffer setting on the production line for relieving the production blockage and stabilizing the operation of the whole production line. Because intelligent transportation equipment such as automated guided vehicle (AGV) is widely used in the actual production workshop, it is more convenient for work in progress (WIP) to transport back and forth between the public buffer and the limited buffer, which plays the role of the public buffer. Therefore, the relevant scheduling optimization technology for automatic production lines with public buffer has an extensive application prospect, which would improve the intellectualization of the manufacturing automation technology.

#### **2. FFSP–PB Mathematical Model**

#### *2.1. Problem Description*

As shown in Figure 1, the FFSP–PB could be described as follows: *m* stages in the workshop and the processing queues of *n* workpieces need to be processed in order with *m* processing stages. At least one of the *m* stages consists of two or more parallel workstations, and the processing times of the workpiece are the same on different parallel workstations at one stage. A buffer with limited capacity is set up between stages; if the limited buffer capacity between stages reaches the upper limit, production blockage is likely to occur. In order to alleviate production blockage, a public buffer is set up in the production workshop, and this area provides services for all stages. If the capacity of the limited buffer between the stages reaches the upper limit, the workpiece can be moved into the public buffer for temporary storage. All workpieces are processed online from the first stage, completing all stages sequentially. If the capacity of the limited buffer between stages reaches the upper limit, the newly completed workpiece is transferred to the public buffer. Under certain conditions, the workpiece during transfer can also be returned to the limited buffer. If the workpiece in the limited buffer is moved to the workstation of the next stage for processing, the limited buffer will have an available space. Subsequently, the workpiece in the public buffer that should have accessed the limited buffer is transferred back to the limited buffer. The transit time between limited buffer and public buffer cannot be ignored. Under preconditions of the online sequence of the workpiece, the standard processing time for transferring the workpiece, the standard processing time of the workpiece at each stage and the online sequence is optimized by global optimization method, and the movement of the workpiece

among the workstation, the limited buffer, and the public buffer is controlled by the local scheduling rules. Thus, the scheduling results of the processing workstations, the start time, and the completion time of all workpieces at each stage and the information of the transport process are obtained.

**Figure 1.** Mathematical model of flexible flow shop with limited buffer and public buffer.
