## 1 Introduction  
A product manufacturing site involves various processes, machines, robots, and logistics. With the development of smart factory technology, many elements at manufacturing sites have become unmanned. In an unmanned line, a robot takes charge of materials input/return into/from the machines, and material supply is ensured using a scheduling logic that determines when and which materials to send to the machines. Therefore, efficient scheduling is essential for the timely supply of materials to the machines. Because robots are expensive pieces of equipment, they cannot be deployed in large numbers, and one robot should be able to handle as many machines as possible in one line. Material scheduling in such a multiprocessing line is a hybrid flow-shop scheduling problem because the processes are sequentially performed and several machines process in each stage. This study is aimed at dealing with material scheduling for supplying materials to machines in an unmanned multi-processing line, which is a hybrid flow-shop problem. This problem can be defined as follows. First, the raw material to be processed sequentially is placed in a buffer. Subsequently, the raw material is removed from the buffer, placed into the machine of the first process, and returned to the buffer when the machine completes processing. Following this, the first-treated material is placed into the machine of the next required process. Accordingly, the material is sequentially processedby all processes. The scheduling problem to be addressed here refers to making the decision of into which machine to place a material in and when, so that the total processing time to complete the processing of all materials is optimized. This study makes the following assumptions. The processes of a line must be sequential, and the order cannot be changed. Only one material can be processed at a time by a machine. If a material is loaded into a machine, a new material cannot be loaded to the machine until the first material is unloaded into the buffer. Several machines are used in each process, and one machine deals with one process. Figure 1 shows an example of an unmanned multi-processing line consisting of \(k\) processes, \(C1\), \(C2\),..., \(Ck\). We assumed that machines in the manufacturing process are positioned on both sides of the central rail. One robot transfer unit (RTU) moves along the central rail and takes charge of loading material into and unloading material from the machines. This means that the position of supplying the material changes in real time. As shown in Figure 2, the RTU is equipped with a multi-joint robot arm to pick up materials and buffers for materials in processing and already processed, respectively. The buffer sizes may vary considering the space in the RTU. The RTU takes a try of raw materials from the tray buffer and returns the tray of completed materials to the tray buffer after all materials are processed. The problem covered in this study is the scheduling problem for a multi-process unmanned line. As mentioned above, there can be many machines in one line, which suggests that the size of the state and action of the environment is high, i.e., the environmental space is high-dimensional. Material scheduling is a type of hybrid flow-shop scheduling problem. Heuristic methods have been proposed for hybrid flow-shop scheduling using dispatching rules [1] and various meta-heuristic methods [2; 3; 4; 5; 6; 7]. However, these methods have the disadvantages of being difficult in their application to high-complexity problems, difficult in finding optimal solutions, and having a long calculation time. Reinforcement-learning-based approaches are also used for flow-shop scheduling. However, studies that have used these approaches have ignored practical factors, including  
Figure 1: Example configuration of multi-processing line. It consists of machines, an RTU moving on a rail, buffers, and tray buffers to which the material is supplied.