Integration methods for aircraft scheduling and trajectory optimization at a busy terminal manoeuvring area

Abstract

This paper deals with the problem of efficiently scheduling take-off and landing operations at a busy terminal manoeuvring area (TMA). This problem is particularly challenging, since the TMAs are becoming saturated due to the continuous growth of traffic demand and the limited available infrastructure capacity. The mathematical formulation of the problem requires taking into account several features simultaneously: the trajectory of each aircraft should be accurately predicted in each TMA resource, the safety rules between consecutive aircraft need to be modelled with high precision, the aircraft timing and ordering decisions have to be taken in a short time by optimizing performance indicators of practical interest, including the minimization of aircraft delays, travel times and fuel consumption. This work presents alternative approaches to integrate various modelling features and to optimize various performance indicators. The approaches are based on the resolution of mixed-integer linear programs via dedicated solvers. Computational experiments are performed on real-world data from Milano Malpensa in case of multiple delayed aircraft. The results obtained for the proposed approaches show different trade-off solutions when prioritizing different indicators.

Appendices

Appendix A

Trade-off solutions are investigated for the numerical example of Fig. 5. Some characteristics are illustrated regarding the solutions computed by the following integrated approaches: maximum delay minimization, lexicographic <MD, LFB>, combined <MD, TTT> and hybrid <LFB, MD>.

Figure 10 presents the graph $\mathcal{G}(F,S)$ of an optimal solution, in terms of maximum delay minimization, for the alternative graph G(N, F, A) of Fig. 6. The start time of each operation is reported nearby the corresponding node and is depicted by using the same colour of the node. In $\mathcal{G}(F,S)$, exactly one arc is selected for each alternative pair. The following aircraft scheduling decisions have been taken: J1 and J2 do not perform airborne holding circles, while J4 performs holding circles for a total of 180 time units in the airborne holding resource; J1 precedes J4 at the entrance in and at the exit from resources 4, 10, 13 and 15; the aircraft sequencing order at the entrance in and at the exit from resource 15 is J2, J1 and J4; J3 (J2) precedes J1 (J4) on resource 16 (17).

Fig. 10

An optimal solution for the maximum delay minimization approach

On the right bottom of Fig. 10, we report the value of each performance indicator. The value of maximum delay minimization is 180. This is due to the additional time required by J4 in the airborne holding resource, in order to satisfy the minimum separation time constraints with J1 (J2) on the landing air segments (runway). The value of the other indicators is not optimized by the current approach.

Figure 11 presents the graph $\mathcal{G}(F,S)$ of an optimal solution for the lexicographic <MD, LFB> approach. The graph presents the same aircraft sequencing (i.e. the selection S) of the solution depicted in Fig. 10. However, the aircraft timing (i.e. the start time related to some nodes in N) is different. The latter difference is due to the landing fuel burn minimization process. When comparing the solutions of Figs. 10 and 11 in terms of the four indicators (as shown in the right bottom tables), the maximum delay (primary indicator in both cases) is the same, while the landing fuel burned (secondary indicator of the lexicographic <MD, LFB> approach) is strongly reduced in the solution of Fig. 11. The remaining two indicators are considerably better in the solution of Fig. 10, since the landing fuel burn minimization (by increasing the travel time) deteriorates the performance of the travel time indicators.

Fig. 11

Optimal solution for the lexicographic <MD, LTT> approach

Figure 12 shows the graph $\mathcal{G}(F \cup \{(n,0)\},S)$ of an optimal solution for the combined <MD, TTT> approach. This graph shows the following aircraft scheduling decisions: J4 and J2 do not perform airborne holding circles, while J1 performs holding circles for a total of 180 time units in the airborne holding resource; J4 precedes J1 at the entrance in and at the exit from resources 4, 10, 13 and 15; the aircraft sequencing order at the entrance in and at the exit from resource 15 is J2, J4 and J1; J3 (J2) precedes J1 (J4) on resource 16 (17).

Fig. 12

Optimal solution for the combined <MD, TTT> approach

The right bottom table of Fig. 12 reports the solution performance in terms of the four indicators. The value of maximum delay minimization (primary indicator) is 180. This is due to the additional time required by J1 in the airborne holding resource, in order to satisfy the minimum separation time constraints with J4 and J2 (J3) on the landing air segments (runway). The value of total travel time (secondary indicator) is 2789. The latter value is better than in the solutions of Figs. 10 and 11. The primary and secondary indicators minimized by this combined approach have a positive effect on the landing travel time indicator that is considerably better than in the previously discussed solutions. On the other hand, as it is expected, the combined <MD, TTT> approach deteriorates the performance of landing fuel burned indicator.

Figure 13 shows the connected graph $\mathcal{G}(F',S)$ related to an optimal solution for the hybrid <LFB, MD> approach, based on the alternative graph $G'(N, F', A)$ of Fig. 7. This solution is characterized by the following aircraft scheduling decisions: J1 and J2 do not perform airborne holding circles, while J4 performs holding circles for a total of 180 time units in the airborne holding resource; J1 precedes J4 at the entrance in and at the exit from resources 4, 10, 13, and 15; the aircraft sequencing order at the entrance in and at the exit from resource 15 is J2, J1, and J4; J3 (J2) precedes J1 (J4) on resource 16 (17). The aircraft sequencing decisions are thus the same as in the solution of Fig. 10.

Fig. 13

An optimal solution for the hybrid <LFB, MD> approach

The right bottom of Fig. 13 reports the value of the optimal solution of the investigated hybrid approach when evaluated with the four performance indicators considered. While the selection of the alternative arcs in $\mathcal{G}(F',S)$ is the same as for the graph $\mathcal{G}(F,S)$ of Fig. 10, the difference between the arc sets F and $F'$ accounts for the difference among all the values of the four indicators. The performance indicator primarily optimized by the hybrid <LFB, MD> approach is landing fuel burned. This is the optimal value achievable for the proposed illustrative example. As for the secondary indicator of this hybrid approach, maximum delay is taken into consideration and the resulting value is 475. The latter value is clearly worst that the optimal value for $\mathcal{G}(F,S)$, since the set $F'$ presents larger processing times than the set F. In fact, J1 and J4 would suffer at least a delay equal to 295 in any scheduling solution related to $G'(N, F', A)$. In the optimal aircraft scheduling solution $\mathcal{G}(F',S)$, J4 is required to perform some holding circles in order to satisfy the minimum separation time constraints with J1 and J2, thus collecting an additional delay equal to the overall time spent in the airborne holding resource.

Appendix B

See Figs. 14 and 15.

Fig. 14

Boxplot on the variance of the average results of Table 4

Fig. 15

Boxplot on the variance of the average results of Table 5

Appendix C

See Fig. 16.

Fig. 16

Pareto optimal points for an ASP–TMA instance of MXP TMA. The coordinates in (a) are (MD, TTT, LTT), while the coordinates in (b) are (MD, TTT, LFB)