**1. Introduction**

The body shop and paint shop for the bus manufacturers are flexible flow shops. The processing flow is divided into multiple stages, and there are multiple parallel machines to process jobs in each stage. Due to the large volume of the buses and the long production cycle, only buffers with a limited number of spaces can be deployed in the production line. At the same time, the bus is not equipped with a chassis for starting the engine in the installation workshop, and they can only be carried on the skids. Therefore, there is usually a buffer between the body shop and paint shop, and the buffer is usually divided into multiple lanes for the ease of scheduling and operation. Each lane has an equal number of spaces for the bus bodies. The bus enters the lane from one side and exits the lane from the other side. The bus body which is waiting for processing will form a waiting queue in each lane. Thus, for multiple lanes, there could exist multiple waiting queues. These waiting queues together are called a multi-queue limited buffer in this paper. When the bus completes the processing in the body shop, it needs to select one of the lanes in the multi-queue limited buffer and enter the corresponding

waiting queue, and the bus is carried by an electric flat carriage to move in the queue. When there is an idle machine in the paint shop, one bus in the multiple waiting queues can be moved out of the buffer to the paint shop for processing. In the actual production line, if the properties such as the model and color are different from those of the previous bus processed by the machine, the cleaning and adjustment of the equipment also have to be done on the machine before proceeding to the next, which results in an extra setup time in addition to the standard processing time. Therefore, these scheduling problems in a bus manufacturer can be characterized as the multi-queue limited buffer scheduling problems in a flexible flow shop with setup times.

The research status of the limited buffers scheduling problem and the research status of the scheduling problem considering the setup times are described below. The scheduling problem with limited buffers is of significant value for practical production scenarios, but is also very challenging in theory. In recent years, the scheduling problem with limited buffers has gained the immense attention of researchers. Zhao et al. [1] designed an improved particle swarm optimization (LDPSO) with a linearly decreasing disturbance term for flow shop scheduling with limited buffers. Ventura and Yoon [2] studied the lot-streaming flow shop scheduling problem with limited capacity buffers and proposed a new genetic algorithm (NGA) to solve the problem. Han et al. [3] studied the hybrid flow shop scheduling problem with limited buffers and used a novel self-adaptive differential evolution algorithm to effectively improve the production efficiency of the hybrid flow shop. Zeng et al. [4] proposed an adaptive cellular automata variation particles warm optimization algorithm with better optimization and robustness to solve the problems of flexible flow shop batch scheduling. Zhang et al. [5] presented a hybrid artificial bee colony algorithm combined with the weighted profile fitting based on Nawaz–Enscore–Ham (WPFE) heuristic algorithm which effectively solved the flow shop scheduling problem with limited buffers.

In recent years, due to the awareness of both the importance of production preparation and the necessity of separating the setup time and processing time, the scheduling problem considering the setup time has gained a lot of attention from the academic and industrial circles. Zhang et al. [6] employed an enhanced version of ant colony optimization (E-ACO) algorithm to solve flow shop scheduling problems with setup times. Shen et al. [7] presented a Tabu search algorithm with specific neighborhood functions and a diversification structure to solve the job shop scheduling problem with limited buffers. Tran et al. [8] studied the unrelated parallel machine scheduling problem with setup times and proposed a branch-and-check hybrid algorithm. Vallada and Ruiz [9] studied the unrelated parallel machine scheduling problem with sequence dependent setup times and proposed a genetic algorithm to solve the problem. Benkalai et al. [10] studied the problem of scheduling a set of jobs with setup times on a set of machines in a permutation flow shop environment, proposing an improved migrating bird optimization algorithm to solve the problem. An et al. [11] studied the two-machine scheduling problem with setup times and proposed a branch and bound algorithm to solve this problem.

The related work in recent years showed that the current research on the limited buffers scheduling problem mainly addressed the shop scheduling problems under a given buffer capacity and mainly highlighted global optimization algorithms. By improving algorithms or combining different algorithms, researchers proposed algorithms with more accuracy in terms of optimization. However, few scholars systematically studied one specific type of limited buffer. At present, the research on scheduling problems with setup times mainly focus on the impact of setup times under the single condition on the processing time. The current studies mainly focus on the improvement in traditional algorithms, while few scholars study the scheduling problems with setup times under the conditions of a dynamic combination of multiple properties (in actual production, setup times are affected by quite a few factors and the combination of multiple factors will produce multiple setup times). Thus, there is no further exploration of the scheduling problem with setup times under the production constraints. The multi-queue limited buffers scheduling problem studied in this paper is more complex than the general limited buffers scheduling problem. This problem requires not only the consideration of the

capacity for limited buffers but also the investigation of (1) the distribution problem when the job enters the lane in the multi-queue limited buffers and (2) the problem of selecting a job from multiple lanes in the multi-queue limited buffers to enter the next stage. On this basis, the research in this paper also considers the influence of setup times on the scheduling process under the dynamic combination of multiple properties, which can greatly increase the complexity of scheduling problems and the uncertainty of scheduling results. The scheduling problems in a flexible flow shop have long proved to be a non-deterministic polynomial hard (NP-hard) problem [12], and thus the multi-queue limited buffers scheduling problem in a flexible flow shop with setup times studied in this paper is also of the NP-hard nature.

As the situations become complicated, more efficient optimization methods needed to be further explored. The compact genetic algorithm (CGA) is a distribution estimation algorithm proposed by Harik [13]. It has great advantages in regard to computational complexity and evolutionary speed, but still the search range is smaller, and it is easy to fall into the local extremum. The scholars overcame its deficiencies mainly by using multiple population and probability distributions [14]. These two approaches, however, could not significantly improve the algorithm's optimization and might lose the advantage of the CGA in the fast search. The probabilistic model and the process of generating new individuals in a probabilistic model are explored to prevent the CGA from being premature and improve the quality and diversity of new individuals. This study proposes an improved compact genetic algorithm (ICGA) that maps the original probabilistic model to a new one through the probability density function of the Gaussian distribution to enhance the algorithm's evolutionary vigor and its ability to jump out of the local extremum, which can better solve the multi-queue limited buffers scheduling problems in a flexible flow shop with setup times.
