# Ferry Scheduling Optimization Considering Arrival Time Uncertainty and In-Place Time Differences
## 2 Problem Description and Model Construction
The airport vehicle road network is a complex multi-loop structure. To minimize the impact of the ferries on the taxi path of the aircraft, they usually follow fixed routes. In the problem studied in this paper, the ferry travel distance is not considered as an optimization objective. In the case of considering the ferry in-place time difference pairs, the most infrequent use of the ferry is used as the only optimization objective, as shown in Equation (1). Due to the uncertain arrival time of the ferry service, the in-place time of the non-first ferry can slide. In Figure 2, the horizontal axis represents time, and the vertical axis represents the required number of ferries. The rectangular square represents the ferry service. The dashed lines at both ends of the rectangle indicate the possible sliding range. By coordinating the arrival time of ferry services for different flights, the number of ferries used during the decision-making cycle is minimized. At time point \(t_{0}\), coordinating the start time of service \(j=2\) and \(j=3\) can reduce the instantaneous demand for ferries. \[\min\left(\sum_{\forall r\in R}v_{r}\right) \tag{1}\]  
where \(v_{r}\) is used as a statistical decision variable for whether the ferry \(r\) is enabled or not in the decision cycle, as shown in Equation (2). \[v_{r}=\min\left\{\sum_{\forall j\in J}\sum_{\forall m\in\left[0,u_{j}\right] }z_{j,r}^{m},1\right\},\forall r\in R \tag{2}\]  
In Equation (2), when ferry \(r\) does not perform any of the sub-services \(m\) of ferry service \(j\), the number of configured ferries is in surplus in that decision cycle. \[\sum_{\forall r\in R}z_{j,r}^{m}=1,\ \forall m\in\left[0,u_{j}\right),\ j\in J \tag{3}\]The same ferry service \(j\) contains several sub-services \(m\), and each sub-service requires one ferry. From the uniqueness point of view, any ferry sub-service can only be executed by one ferry, as shown in Equation (3). \[t_{i}^{m}+d_{p}+d_{i^{\prime},i^{\prime\prime}}+d_{d}+d_{i^{\prime\prime},i^{ \prime}}+M\big{(}z_{i,r}^{m}+z_{j,r}^{m^{\prime}}-2\big{)}\leq a_{j,r}^{m^{ \prime}}+M\big{(}1-y_{i,j}\big{)} \tag{4}\]  
\[z_{i,r}^{m}+z_{j,r}^{m^{\prime}}-1\leq y_{i,j}+y_{j,i} \tag{5}\]  
\[y_{i,j}+y_{j,i}\leq 1 \tag{6}\]  
\[i\
eq j,\forall i,j\in J,m\in[0,u_{i}),m^{\prime}\in[0,u_{j}) \tag{7}\]  
In Equations (4)-(7), Equation (7) serves as a common value range constraint for the variables in Equations (4)-(6). For any two unequal ferrying services, the decision variable \(y_{i,j}\) takes the value of \(1\) when the same resource \(r\) executes service \(i\) before service \(j\). It is also necessary to satisfy the time \(t_{i}^{m}\) for service \(i\) to start ferrying until all of the work of service \(i\) has been completed and still leave enough transfer time to service \(j\). Among the studies of the ferry scheduling problem, there are several common treatments to simplify the description of the problem. In the first type, it is assumed that each flight needs at most one ferry to provide service. To adapt to the actual problem, \(m-1\) virtual flights are introduced when the number of flights needing ferries is \(u_{j}>1\). It is required that the inbound and outbound times of the introduced virtual flights are the same as shown in Figure 3a [27]. The problem with this method is that the start and end times of the split ferry sub-service and the original ferry service are identical. The second type is based on the time window of the earliest start of the ferry service \(est_{j}\) and the latest start of the ferry service \(lst_{j}\) combined with the number of ferries needed for the flight to equalize the division [10]. The earliest start time and the latest start time defined in this method are both before the arrival time of the ferry service, as shown in Figure 3b. Existing splitting methods for flights requiring multiple ferries mainly simplify the problem by ignoring interrelationships between multiple ferries. These treatments are not conducive to the utilization of ferry resources.