## 4 Mathematical Models
### Formulations of CSAECRP  
Firstly, we introduce \(x_{i}^{\text{vs}}\), which represents the container amount remained on a liner ship while the liner ship anchored in the \(i\)th port on voyage \(v\) along route \(s\) to complete the unloading operation but yet not start the loading operation. Here, we discuss two scenarios. Scenario 1 is shown in Figure 2: on the \(v\)th voyage along route \(s\), the liner ship calls at port a, b, c, and d in sequence and finally returns to the port a. After the ship finishes the unloading task in port c on voyage \(v\) along route \(s\), the source of containers still carried on the ship is divided into three parts. They are loaded and empty containers transported from port a to port d, from port b to d and b to a (the last port of this voyage), respectively. Hence, \(x_{c}^{\text{vs}}\) can be expressed as \(x_{c}^{\text{vs}}=XC_{ad}^{\text{vs}}+XE_{ad}^{\text{vs}}+XC_{bd}^{\text{vs}}+ XE_{bd}^{\text{vs}}+XC_{bd}^{\text{vs}}+XC_{bd}^{\text{vs}}\). Based on the same route, we discuss scenario 2. As shown in Figure 3, after the liner ship finishes the unloading task at the first port (port a) on voyage \(v\) along route \(s\), the excess containers come from two voyages. They can also be divided into three parts. They include the loaded and empty containers transported from port c on the prior voyage to port b, port d on the previous voyage to port b, and port d on the initial voyage to port c. Accordingly, \(x_{a}^{\text{vs}}\) can be expressed as \(x_{a}^{\text{vs}}\)=XC\({}_{cb}^{(\text{v}-1)s}\)+XE\({}_{cb}^{(\text{v}-1)s}\)+XC\({}_{db}^{(\text{v}-1)s}\)+XE\({}_{db}^{(\text{v}-1)s}\)+XC\({}_{db}^{\text{vs}}\)+XE\({}_{dc}^{\text{vs}}\). The above two situations can be summarized as Formulations (19) and (20). Figure 2: Diagram of the remained amount of containers after the liner ship has completed the unloading operation. The objective function is to maximize the profit of multiple routes, voyages, ports, and cargo, which consist of seven parts. \(Z\) refers to the total revenue of the liner company. The first two parts are the revenue gained by the liner company, in which the first part is revenue from container slot reservations for TS cargo and the second part is revenue for ordinary cargo. The rest are the costs of the liner company, which consist of the transportation cost of loaded containers and empty containers, the leasing cost of empty containers, the storage cost of delayed loaded containers and empty containers, and the fixed operating cost of the ship. The choice of what prices are set for items is justified by [1,7,17]. Constraints (1) and (2) represent the slot reservation quantity limit of the TS and ordinary cargo, respectively. Constraint (3) ensures that the transportation number of TS cargo is equal to their amount arriving at a port on time. Constraint (4) reveals that part of the ordinary cargo arriving at port on schedule can be transported on the current voyage; the rest will be delayed until the next voyage when the liner ship has space. Constraint (5) shows that the origin of ordinary cargo waiting to be transported on the present voyage can be divided into two parts. The first part is the ordinary cargo that is delayed to be transported from the previous voyage; the next part encompasses it arriving at the port on time for a set voyage. Constraint (6) stipulates that the number of ordinary cargoes allowed to be delayed until the next voyage cannot exceed the restriction. Constraint (7) requires TS cargo to be preferentially transported. Constraint (8) guarantees that the priority for a shipment belongs to TS cargo, with ordinary cargo being delayed. The, ships that have available storage can transport the ordinary cargo arriving on this voyage.