## 1 Introduction
They developed a fuzzy bi-level decision-making technique based on the PSO algorithm and a heuristic method. [PERSON] et al. [19] developed a hybrid approach integrating the particle swarm optimization algorithm with genetic operators for solving a fuzzy job shop scheduling problem with an uncertain processing time. [PERSON] et al. [20] considered a permutation flow shop problem with stochastic processing times. They proposed a biased-randomized heuristic algorithm and extended it to a metaheuristic method by adding a variable neighborhood descent framework. Obviously, in the above research, different uncertain parameters were described and modeled by fuzzy and stochastic theories. In other studies, uncertain parameters are usually regarded as bounded uncertainties, which can be defined as a continuous interval or a finite set of discrete values. [PERSON] et al. [21] studied a similar problem but proposed a min-max regret  
Figure 1: Tree-like assembly structure of product A.  
scheduling model with interval scenarios of processing times. They designed both exact and heuristic algorithms to solve this problem. [PERSON] et al. [22] considered a job-shop scheduling problem with an uncertain processing time, described by discrete scenarios. [PERSON] et al. [23] addressed a single-machine scheduling problem with uncertain processing times and sequence-dependent setup times. All the previous studies focused on the classic flow-shop or job-shop scheduling problems under uncertainty and did not consider the assembly process. In fact, the production process with both machining and assembly is very commonly witnessed in real life. Some examples include production flows of molds, ball valves, and various complex electromechanical products [24; 25]. Normally, production scheduling that considers the assembly process is more complicated, and the schedule is more sensitive to the effects of uncertainty. In addition, most of the previous studies define uncertainty based on prior assumptions or artificial experience, rather than from an objective data-driven perspective. With the recently emerging industrial informatics technologies such as the Internet of Things (IoT), sensors, and artificial intelligence, the historical data of the production process can be continuously collected and can be a new source for scheduling optimization [26; 27]. These data have motivated a shift from a priori reasoning and assumptions to a new data-driven paradigm in dealing with uncertainties in the production process, which is also the focus of this research. In the actual production process, fluctuations in the setup time and processing time of the process may cause the completion time to deviate from the expected time, and this can ultimately lead to delays in subsequent processes. Motivated by realistic needs, this paper considers two types of uncertain parameters that have an important impact on the production scheduling of the assembly manufacturing system, namely process processing time and setup time, and from there we aim to develop a data-driven robust scheduling framework. Briefly, the proposed framework consists of statistical analysis and interval scheduling methods used to describe the distribution of uncertainty parameters. On this basis, a PSO algorithm considering assembly constraints is designed to hedge against the risk of system performance degradation in uncertain environments. The rest of this paper is organized as follows. In Section 2, we give the problem description and the notations. In Section 3, considering that the prior distribution of process setup time and processing time is unknown, we design a nonparametric estimation method based on kernel density estimation to obtain the probability density function of these two parameters. The proposed particle swarm optimization algorithm based on interval scheduling is then elaborated in detail, which is followed by a series of computational experiments and discussion. A summary covering the conclusions is presented in the last section.