## 4 Mathematical Formulation
\[\sum_{i}^{V}x_{i,j,k}-\sum_{j}^{V}x_{h,j,k}=0,\qquad\forall\ \ k\in K,\ \ h\in V. \tag{4}\]Constraint (4) is a route continuity constraint, which means that if a vehicle enters a location, it must also leave that location. \[x_{i,j,k+1}\ \geq\ x_{ijk}+1-M\left(1-x_{i,j,k+1}\right),\ \ \ \ \ \forall\ \ i\in V,\ \ j\in V,\ \ k\in K_{VD} \tag{5}\]  
Constraint (5) ensures that different routes are selected for each vehicle at the same vertiport, as illustrated in Figure 3a. UAM1 and UAM2 are at one vertiport (VD); if UAM1 selects a route from VD to depot 1, UAM2 must select another route from the vertiport. This condition is satisfied by multiplying by a large constant number (M), and the route selected by UAM2 will be \(x_{i,j,k+1}=1\), whereas UAM1 is \(x_{i,j,k+1}=0\) from the origin to the other locations. The constant (M) is also used for subtour elimination in the VRP and TSP [23; 24]. \[\sum_{j}^{V}x_{i,j,k}\ \leq\ 1\ \ \ \ \ \ \ \ \ \forall\ \ i\in VD,\ \ k\in K_{VD} \tag{6}\]  
Constraint (6) ensures that each vehicle leaves from the depot no more than once. \[\sum_{i}^{V}x_{i,j,k}\ \leq\ 1\ \ \ \ \ \ \ \forall\ \ j\in VD,\ \ k\in K_{VD} \tag{7}\]  
Constraint (7) ensures that each vehicle arrives at a depot no more than once. Figure 3:0  
Constraint (8) ensures that a vehicle cannot leave from a depot other than its origin. Figure 3:1  
Constraint (9) ensures that a vehicle cannot return to a depot other than its origin. Equations (1)-(9) constitute a point-to-point network problem similar to that shown in Figure 2f. Unlike other problems, such as the TSP and VRP, each location is reached by only one vehicle, and the vehicle cannot travel from one depot to another. In this scenario, all vehicles need to travel to other locations, even those that are depots for other vehicles, which creates a complicated routing problem. For the operation of UAMs with this kind of complex routing, optimum scheduling is required. The following time constraints are therefore introduced to solve the dynamic AADS problem shown in Figure 3b. Figure 3:2  
Figure 3:3  
Figure 3:4  
Constraints (10) and (11) allow UAMs to arrive at different times \(\left(Ta_{j,k}\right)\) when travelling from location \(i\) to location \(j\) by controlling the speed of each vehicle. The arrival time (\(Ta_{j,k}\)) at a node is the departure time \(\left(Td_{i,k}\right)\) from the departure node plus the travel time \(\left(T_{i,j,k}\right)\) between the two nodes, which form a dynamic AADS problem. In this paper, the travel time is defined as the flight time of the UAM, which can vary with the flight airspeed and mission conditions. The travel time (\(Tr_{i,j,k}\)) is calculated using Equation (11) and depends on the variables of speed, departure location and arrival location. The travel time can also be calculated according to the ratio of the distance (\(d_{i,j}\)) and speed (\(V_{i,j,k}\)) of the vehicle, as shown in Equation (12). The arrival time constraint was used for subtour elimination in [23; 25] so that additional subtour elimination was not required. Figure 3:5  
Constraint (13) is the equality constraint, which is used to calculate the departure time of a vehicle from a depot. The departure time of a vehicle is the sum of the arrival time and the waiting time at that depot. In this paper, the waiting time is assumed to be satisfactory for customer requirements, such as battery charging time and customer service time. Figure 3:6  
Constraint (14) also ensures that when two or more vehicles arrive at a vertiport, the second vehicle can only leave after waiting a certain amount of time after the first vehicle has left to provide some separation for safety and that the FIFO [15] rule is satisfied. Figure 3:7  
Figure 3:8  
Figure 3:9  
Constraints (16)-(18) are integrality constraints. Because the cost function, travel time, arrival time and departure time depend on the speed of the vehicle, we also set the speed as a variable in this study.