## 5 Computational Experiments  
A set of computation experiments was performed to study our formulation of the point-to-point network model with different numbers of vehicles and different numbers of depots and locations. Because our formulation is a new type of problem, it cannot be compared with experiments on classical routing and the TSP, owing to the network model used for our proposed problem formulation for UAM operations. Hence, we compared the route cost for each vehicle with the scheduling results for each of the vehicles, as the cost depends on the routing selection constraint in Equation (5). The problem formulation is solved using the GUROBI optimisation solver [26] for different sizes of problems. The problem sizes, CUP time and cost are compared in Table 2. The route selection properties of the proposed formulation are also shown in Figure 4. The cost function considered in these computational experiments involves minimisation of the total travel distance of all vehicles. The computational experiments were run for varying numbers of vertiports (\(nV_{VD}\)) and varying numbers of vehicles at each vertiport (\(nK_{VD}\)). Case 1 involved a total of eight vehicles, and the total cost of travel for all locations was 2607.159 m, with a very low computation time of 0.203 s. However, when the number of vertiports was increased to six, the computation time rose dramatically, and the number of variables and the cost also increased, as all the vehicles travel to all the vertiports. These experiments showed that an increase in the number of vertiports for the same number of vehicles increased the computational time significantly, as evidenced by a comparison of Cases 2 and 4. The characteristics of the mixed-integer linear programming (MILP) algorithm mean that the computational time increases exponentially with an increase in the problem size. The route selection results are shown in Figure 4, and the route and cost for each vehicle are indicated in each image. Each depot has three vehicles, and different routes are selected for these vehicles to minimise the total cost; all the vehicles also travel to all locations while satisfying the routing selection constraint in Equation (5). Depot 0 has vehicles k1, k2 and k3, for which different optimum routes are selected. However, in regard to the total travel distance of the vehicles at the same vertiport, the first vehicle has the longest travel distance, whereas the last vehicle has the second highest travel cost, and the middle vehicle has the shortest travel distance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & \multicolumn{2}{c}{**Problem Size**} & **Total Cost (m)** & **CPU Time (s)** \\ \hline Case 1 & \(nK\)= 8, \(nK_{VD}\)= 2, \(nV_{VD}\) = 4 & 2607.159 & 0.203 \\ Case 2 & \(nK\)= 12, \(nK_{VD}\)= 2, \(nV_{VD}\) = 6 & 7793.7766 & 3.29 \\ Case 3 & \(nK\)= 15, \(nK_{VD}\)= 3, \(nV_{VD}\) = 5 & 9471.775 & 117.82 \\ Case 4 & \(nK\)= 18, \(nK_{VD}\)= 3, \(nV_{VD}\) = 6 & 12,816.8524 & 4137.32 \\ Case 5 & \(nK\)= 14, \(nK_{VD}\)= 2, \(nV_{VD}\) = 7 & 12,000.48011 & 24,895.83 \\ Case 6 & \(nK\)= 21, \(nK_{VD}\)= 3, \(nV_{VD}\) = 7 & 19,623.4244 & 857,601.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational experiments with different problem sizes for the proposed formulation. Figure 4: _Cont_. The results for Case 4 shown in Figure 4 indicate that all the vehicles at a given vertiport select different routes, although vehicles from different vertiports can select the same routes, as evidenced by the routes for k1, k5, k9, k11, k14 and k17, which have the same distance cost, whereas the first and last vehicles from different vertiports select different routes. A solution to the point-to-point network problem can be achieved by applying the proposed optimisation formulation.