## 3 Solution Method
### Memetic Algorithm for Solving ITRTRSP  
From Section 2.1, WTC can solve for \(EM_{i}\) based on any given transshipment sequence at an upstream station. Therefore, as long as all possible transshipment sequences are traversed, a better sequencing scheme can be obtained theoretically (at least not inferior to the FCFS rule). However, since the two-stage HFSP is an NP-hard problem [45], its global optimum solution is usually difficult to obtain within a limited decision time when the solution space size is large. In such a case, heuristics is usually a more reasonable option. In fact, the FCFS rule is one of the important heuristics for solving the sequencing problem. However, further searching for feasible transshipment sequences based on hub allocation will make the solution space size grow exponentially. If the heuristics are not properly designed, the result may be inferior to the FCFS rule and adversely affect the road-rail multimodal transport scheme. Thus, special research on heuristics is needed to solve ITRTSP. As mentioned above, ITRTSP can be transformed into a two-stage HFSP with transport constraints. Generally speaking, a solution to an HFSP mainly consists of machine allocation and processing sequence [9]. Existing heuristic methods [40; 46; 47; 48] usually first encode the solution as processing sequences of jobs and then use machine allocation rules to decode the given processing sequence and calculate the makespan. For example, in the memetic algorithm framework (referred to as Framework 1) designed by [PERSON] et al. [40], algorithms such as NEH, IG, and GA are for the outer layer to generate and optimize processing sequences; while algorithms such as IMPT, ECTT, and GAMT are for the inner layer to find better machine allocation schemes for the processing sequences generated by the outer layer. The memetic algorithm integrates the strengths of global and local search, allowing for the exploration of multiple high-quality regions within the solution space [40; 49]. This approach leads to more precise identification of optimal solutions. Due to its effectiveness in handling complex problems, this paper also designs nested solution methods based on the memetic algorithm framework. The difference is that the methods in this paper first encode multimodal transport schemes as hub allocations, and then decode them by selecting an appropriate transshipment sequence for each; that is, the memetic algorithm framework in this paper is constructed with hub allocation as the outer layer and transshipment sequence selection as the inner layer (referred to as Framework 2, see Figure 3). As the genetic algorithm (GA) is widely used due to its superior global search performance in minimizing makespan in HFSP [50; 51], the nested genetic algorithm (abbreviated as GA_GA) is used to illustrate the solution process of Framework 2. The detailed calculation process of GA_GA is as follows. Step 1: Set parameters. Firstly, input the data, including quantities and transport start moments of all cargoes, the road and rail transport times, and the transshipment efficiencies at upstream and downstream stations. Then, set the inner and outer layer algorithms parameters, including maximum genetic generations, population sizes, crossover and mutation probabilities, etc. Step 2: Construct an initial outer layer population. Generate an initial population of N chromosomes as the current outer layer population. Each chromosome consists of \(2\,|\,R\,|\,\) genes; the first \(\,|\,R\,|\,\) genes are sequenced by the origin numbers, with the \(i\)-th gene indicating \(u\) chosen for \(i\). The second \(\,|\,R\,|\,\) genes are also sequenced by the origin numbers, with the \(\,|\,R\,|\,+i\)-th gene indicating \(d\) chosen for \(i\).