## 1 Introduction
Compared to the achievements based on queuing theory, transforming ITRTSP into HFSP has two main advantages. Firstly, models of HFSP can quantitatively describe the dependency between transshipment sequence and waiting time. Secondly, solution methods of HFSP can calculate the minimum waiting time (not the average waiting time) for each cargo in the queue with a given hub allocation and transshipment sequence. In light of this, this study considers the road-rail multimodal transport network structure with transport equipment, transport process constraints, and emergency transport demands (respectively, corresponding to the machine environment, processing characteristics, and objective functions of HFSP [36]) to construct an HFSP model describing ITRTSP and designs corresponding solution methods. Over the past few years, the integrated scheduling of transport and processes has become a significant research branch of HFSP. Researchers have conducted extensive studies using HFSP models and methods in various fields such as manufacturing [37], port loading and unloading operations [38], and logistics distribution [39]. However, from the information currently available, specific studies addressing  
Figure 1: The operation process of road–rail multimodal transport. multimodal transport issues have not yet been discovered. Therefore, this study can be regarded as one of the first attempts to apply HFSP modeling and solution methods to manage congestion in road-rail multimodal transport settings. Generally, HFSP in different application scenarios has significant differences in machine environment, processing characteristics, and objective functions [36]. Correspondingly, there are differences in solving methods as well. Due to numerous achievements in such studies, it is impractical to present them all individually due to space limitations. Here, we only compare our model and solution methods with those in Ref. [40] (see Table 1) to highlight their characteristics. Further details can refer to the comprehensive reviews by [PERSON] et al. [36], [PERSON] et al. [41], [PERSON] et al. [42], and [PERSON] et al. [43]. Ref. [40] constructed a multi-stage HFSP model with dynamic waiting time, considering the effects of buffer capacity, machine allocation, and processing sequence on the waiting time, to minimize makespan. To solve this model, a method was proposed to calculate the waiting time and makespan. A memetic algorithm framework was designed with job sequence as the outer layer and machine allocation as the inner layer. Its modeling and solution ideas provide direct inspiration for our study. According to Table 1, the differences between this paper and Ref. [40] are mainly in the following four aspects. First, in Ref. [40], only transport between two processing stages (corresponding to operation 3 in this paper) is considered, whereas the road-rail multimodal transport also includes road transport before loading and after unloading operations (corresponding to operations 1 and 8). Second, Ref. [40] assumes that there is only one AGV responsible for transport between any two sequential processing stages (i.e., the transport of any two jobs must follow a sequence), whereas the emergency multimodal transport typically involves the temporary requisition of a large number of transport vehicles to support the simultaneous transport of multiple cargoes (i.e., assuming an unlimited quantity and capacity of trucks and trains). Third, since this paper focuses on the influence of transshipment sequence selection on road-rail multimodal transport schedule, only the waiting time before transshipment is considered, while the after is ignored. Fourth, Ref. [40] designs a memetic algorithm framework with job sequence as the outer layer and machine allocation as the inner layer, nevertheless, this paper designs a memetic algorithm framework with hub allocation as the outer layer and transshipment sequence as the inner layer. The following sections extend the work of Ref. [40] by constructing a model for ITRTSP and designing solution methods.