## 4 Mathematical Formulation  
The problem is modelled as a three-index formulation with decision variables, speed variables and time constraints regarding the problem definition, as shown in Figure 3. The mathematical formulation is described below. Our mathematical model is inspired by a three-index formulation of the MDVRP [23] in which the customer capacity constraints are relaxed. Unlike the formulations of the  
Figure 3: The multidepot point-to-point network problem with dynamic scheduling at a departure station. (**a**) Route selection for multiple vehicles at the same departure station; (**b**) time–space networks for multiple vehicles at the same departure station for dynamic scheduling. Figure 2: Step-by-step graphical representation of the proposed point-to-point network route; (**a**) given network; (**b**) single vehicle, single depot and multiple destinations; (**c**) multiple vehicles, single depot and multiple destinations; (**d**) Single vehicle, multiple depots and multiple destinations; (**e**) multiple vehicles, multiple depots and multiple destinations; (**f**) multiple vehicles, multiple depots and multiple destinations. routing problem and the TSP, the mathematical model incorporates Muiller-Tucker-Zelim constraints [24] for subtour elimination and considers the travel speed of the vehicles and the constraints on their arrival, departure and waiting times. In addition, route selection constraints are also to each vehicle at each depot in order to select routes that minimise the cost function. The following assumptions are made for our proposed formulation:  
* Each vehicle must leave from and return to the same vertiport;
* All the vehicles need to travel to others vertiport locations;
* Vehicles in the same vertiports must travel by selecting different routes;
* Vehicles in the same vertiports cannot depart at the same time, meaning that the second vehicle needs to travel after the first vehicle after waiting some time;
* Vehicles cannot arrive at the same time to a vertiport, meaning that the second vehicles need to arrive after the first vehicles after waiting some time; and
* All vehicles can travel at varying speeds defined by vehicle specifications. The following notation is used in this formulation:  
* Set of vertiports \(i,j\)
- Index of vertiports \(K\)
- Set of all vehicles \(k\)
- Index of vehicles \(VD\)
- Set of vertiports, \(VD\) = \(V\) \(K_{VD}\)
- Set of vehicles belonging to each vertiport
* Binary variable that is equal to one if vehicle \(k\) travels from \(i\) to \(j\) and zero otherwise. \(Td_{ijk}\geq 0\)
- Departure time of vehicle \(k\) from vertiport \(i\), where \(i\in\ V\), \(k\in K\) \(T_{ik}\geq 0\)
- Arrival time of vehicle \(k\) at vertiport \(i\), where \(i\in\ V\), \(k\in K\) \(T_{mini,i}\in\mathbb{R}^{+}\)
- Waiting time of vehicle \(k\) at depot \(i\), where \(i\in\ V\), \(k\in K\) \(V_{k}\geq 0\)
- Speed of vehicle \(k\), where \(k\in K\)
* Distance between \(i\) and \(j\), where \(i\in\ V\), \(j\in\ V\) \(V_{max}\) \(V_{mini}\in\mathbb{R}^{+}\)
- Maximum and minimum speeds of vehicles \(T_{mini,i}T_{mini,i}\)
- Maximum and minimum waiting times \(nK_{VD}\)
- Number of vehicles at each vertiport \(nV_{VD}\)
- Number of vertiports The objective function is written as follows: Objective: \[\textbf{min}\ \sum_{j}^{V}\sum_{i}^{V}\sum_{k}^{K}c_{ij}x_{ij,k}\qquad\forall\ \ k\in K,\ \ i\in V,\ \ j\in V\] (1)  
The objective is to minimise the total routing cost of all vehicles. The cost function can be selected depending on the user-defined cost function from one vertiport to another. The following constraints are required to complete the model. \[\sum_{i}^{V}x_{i,j,k}=1,\qquad\forall\ \ k\in K,\ \ j\in V \tag{2}\]  
Constraint (2) allows each vehicle to enter all vertiports only once. \[\sum_{i}^{V}x_{i,j,k}=1,\qquad\forall\ \ k\in K,\ \ i\in V. \tag{3}\]  
Constraint (3) allows all vehicles to leave from all vertiports only once.