## 1 Introduction  
Scheduling is one of the core issues in production management, which is very important for improving the production efficiency and resource utilization of the manufacturing system [1]. In real-world manufacturing systems, different types of uncertainties naturally appear in the production process, causing fluctuations in the production performance [2, 3]. In this context, the robustness of a schedule is of great importance in addition to the schedule's quality. With the rapid development of the industrial informatics technologies, a large amount of historical data are now routinely captured and collected in many enterprises. Historical data can be used to analyze the characteristics of uncertainty, and this provides a new view for robust scheduling under uncertainty [4]. The motivation of this study is derived from scheduling problems encountered by many assembly manufacturing enterprises [5, 6], where a variety of different products are obtained through multiple components by multistage machining and assembly processes (shown in Figure 1). Due to the tree-like assembly relationships, higher level components cannot be assembled until all preceding lower levels components are completed. Therefore, compared with a single machining process, fluctuations in the processing time and setup time of any part may cause the waiting of subsequent and parallel components, and this can ultimately lead to the delay of the final product [7]. In order to ensure sufficient lead time for preparing production auxiliary materials and tools, the manager should decide on the complete schedule in advance. In most real-life manufacturing systems, managers employ dispatching rules and optimization methods to generate a baseline schedule, and they then insert extra time redundancy into the schedule to ensure robustness [8; 9]. However, the assessment of uncertainty in this schedule is subjective and largely depends on the experience of the manager [10]. In real-life manufacturing systems, schedules are often confronted with uncertain factors. For example, resource shortages and machine breakdowns can delay a schedule's completion time. In this work, we mainly consider the uncertainty of process setup time and processing time, which are also the two most frequent and influential uncertainties in the assembly production system. To hedge against uncertainties in the production process, traditional optimization methods, such as stochastic programming (SP), fuzzy programming (FP), and robust optimization (RO), have proposed different mathematical expressions of uncertainty [11; 12; 13]. The sources and types of uncertain parameters considered by these methods are different. For example, the FP method assumes that the uncertain parameters belong to a specific perturbation interval under different probabilities, and it establishes a fuzzy set based on the membership function to describe the uncertain scenarios. The SP method establishes a representative scenario tree by enumerating the possible values of uncertain parameters. These methods usually combine a priori reasoning with assumption on the uncertainty to motivate the construction of the optimization models. However, in the real world, the probability distribution of uncertainty is closely related to the specific production environment, and some complex uncertainties are even difficult to describe explicitly. Thus, traditional optimization methods are often criticized for their inflexibility or ineffectiveness in practical applications [14]. In addition to traditional optimization methods, many metaheuristic algorithms have been extensively applied to shop scheduling problems under uncertainty [15]. Among them, the particle swarm optimization algorithm is one of the most commonly used metaheuristic methods because of its flexibility and global optimization capabilities [16; 17]. [PERSON] et al. [18] studied integrated production planning and scheduling with fuzzy startup time and processing time.