# Ferry Scheduling Optimization Considering Arrival Time Uncertainty and In-Place Time Differences
## 2 Problem Description and Model Construction  
The flight transit process can be divided into two parts, inbound and outbound. Both inbound and outbound flights at remote locations require ferry service. The inbound ferry service relies on the base time of the flight's estimated time of arrival (ETA). The outbound ferry service relies on the base time of the flight's estimated time of departure (ETD). With the flight inbound and outbound times determined, the start time of the first ferry service for the flight can be determined as well as the in-place time. A complete ferry service consists of four components: the necessary in-place time, the passenger boarding time, the transfer time and the passenger deboarding time. For the inbound ferry service, the base time is ETA. The relative relationship between the above components of the inbound ferry service is shown in Figure 1a. The first ferry is in place well in advance of the flight arrival, and passengers are transferred from the airplane to the ferry when the inbound ferry service arrives. Passenger boarding ends when the last passenger on board is reached or when the number of passengers on board reaches the capacity limit. At the same time, the ferry transfers passengers from the aircraft to the terminal, and the passengers get off. This is the complete service process performed for one ferry. The base time of the departure ferry is ETD. Through the ferry service standard of each airline company, it can be known how far in advance the ferry service of each flight has to start, thereby determining the start time of the outbound ferry, as shown in Figure 1b. When the ferry service requires multiple ferries, the first ferry's arrival time and start time are based on the arrival time of the ferry service. The arrival and service time of the non-first ferry is determined by the designation of the previous ferry. Figure 1: Process of ferry service for arriving flight. The ferry scheduling problem considering the in-place time can be described as follows. During a decision cycle, there are \(n\) flights parked at remote parking stands that need to perform an inbound or outbound process. Each flight has a ferry service, and all ferry services in the decision cycle are defined as the set \(J\). Let \(j\) be the index of the set of ferry services, and there are \(\forall j\in J\) and \(j=0,1,2,...,n-1\). Due to the large difference in the number of passengers carried by different flights, the number of ferries required \(u_{j}\) by different ferry services \(u_{j}\) is not equal. Define \(m\) as the index of a sub-service of ferry service \(j\). Then, for service \(j\), there is \(\forall m\in\left[0,u_{j}\right)\). The set of ferry cars is defined as \(R\) such that \(\forall r\in R\). When the decision variable \(z_{j,r}^{m}\) takes the value of 1, it indicates that the \(m\)th sub-service of ferry service \(j\) is executed by ferry car \(r\) and otherwise is 0. Correspondingly, the decision variable \(a_{j,r}^{m}\) denotes the time at which ferry \(r\) is in place to execute the \(m\)th sub-service of ferry service \(j\). The decision variable \(t_{j}^{m}\) denotes the start time of the \(m\)th sub-service of the ferry service \(j\). The complete ferry service contains four components: the necessary in-place time, the passenger boarding time \(d_{p}\), the transfer time between the airside and the terminal and the passenger deboarding time \(d_{d}\). For the inbound ferry service, the initial position of the ferry \(j^{\prime}\) is the stop. The target position \(j^{\prime\prime}\) is the terminal. And \(d_{j^{\prime},j^{\prime\prime}}\) is used as the necessary transfer time between the aircraft position and the terminal. Setting \(i\) as a ferry service different from \(j\), \(d_{i,j}\) is used to denote the necessary transfer time between different ferry services. Since the path of the ferry transferring between any two points is determined, there is \(d_{i,j}=d_{j,j}\). The other parameters and decision variables of the model are represented as follows.