## 2 Optimization Model  
### Problem Description and Assumptions  
As previously mentioned, ITRTSP can be described as a two-stage HFSP with transport constraints as shown in Figure 1. In Figure 1, cargoes from \(|\)R\(|\) origins (\"\(\bigtriangleup\)\") need to be transported to a destination p (\"\(\times\)\") through road-rail multimodal transport, undergoing two transshipment operations (operations 2 and 3). Operations 2 and 3 have \(|\)U\(|\) and \(|\)D\(|\) upstream and downstream stations (\"\(\square\)\" and \"\(\diamond\)\") available for selection, respectively. The transport start moment for the cargo at each origin is known, with the earliest one referred to as the start moment of the overall multimodal transport task. Once transport begins, the cargo immediately departs from its origin and is transported by road to an upstream station (operation 1). Upon arrival, the cargo first enters the transshipment queue (\"\(\bigtriangleup\)...\(\bigtriangleup\)\") for waiting (operation 2-1). Once its immediate predecessor activity is completed, it can start loading operation (operation 2-2). Upon completion of loading, it immediately departs and is transported by rail to a downstream station (operation 3), and follows the same operation process (operations 4-1 and 4-2) as at the upstream station. Thereafter, it immediately departs and is transported by road to p (operation 3). The arrows (\"\(\rightarrow\)\") in the operations 1, 3, and 3 represent the transport arcs. The time consumed on the transport arcs varies when cargo selects different upstream and downstream stations. R, U, and D, respectively, denote the sets of origins, upstream stations, and downstream stations, and \(|\cdot|\) denotes the number of elements in the set. When the last cargo arrives at p, the overall multimodal transport task ends, and the arrival moment is referred to as the end moment of the overall multimodal task. The difference between the end moment and the start moment of the multimodal transport task is called makespan. To handle a crisis, all cargoes must be delivered as quickly as possible. Hence, the objective of ITRTSP is to find the optimal allocation of upstream and downstream stations and the transshipment sequence at hubs to minimize the makespan.  
The assumptions of ITRTSP are as follows. (1) The cargo quantity and start moment for each origin are known. (2) Once transport begins, each cargo departs from its origin immediately, disregarding its preparation time. (3) All cargoes need to sequentially complete two road transport operations (operations 1 and 3) and one rail transport operation (operation 3), as shown in Figure 1, and must select one upstream and one downstream station from U and D, respectively, for transshipment (operations 2 and 4). (4) The transport arcs and hubs are continuously available and fault-free. (5) There is an unlimited supply of trucks and trains so that cargoes can be transported immediately after transshipment. (6) In operations 1, 3, and 3, the distances and speeds of all transport arcs are known with no restrictions on flow, the delivery time of each cargo only depends on the ratio of distance to speed. (7) In operations 2 and 4, cargo from the same origin must be allocated to the same upstream (or downstream) station. (8) Simultaneously transshipment at the same hub is prohibited for cargoes from different origins but is allowed at different hubs. (9) In operations 2-1 and 4-1, the transshipment sequence of cargoes from different origins at the same hub can be adjusted arbitrarily, without being restricted by their arrival moments. (10) Upon arrival at a hub, cargo can be transshipped immediately if its immediate predecessor activity is completed. (11) In operations 2-2 and 4-2, transshipment for the cargo from the same origin cannot be interrupted once it starts, and the transshipment time only depends on the ratio of the cargo quantity to the transshipment efficiency.