The Air Traffic Flow Management Problem with Time Windows

This paper defines a set of temporal intervals, called time windows, which are defined prior to flight departure and constitute milestones to be met during the flight execution. The size of the time windows is variable as it reflects all known constraints, such as punctuality at destination, runway capacities or congested en-route areas that the flight will cross. Once a time window is defined, all the air traffic actors are committed to guarantee that flight operations, e.g. enter an airspace sector, depart from or arrive at an airport, are executed within the time window. We propose a two-step approach based on a mixed integer programming formulation. The first step determines a set of time windows such that the overall cost of delay is minimized. Then in the second step we choose the set of optimal time windows which also maximizes the overall time window size. In such a way, we provide to all air traffic stakeholders the largest degree of flexibility to perform their operations under the constraint that the minimum achievable delay is kept constant. We also gain information on the critical flights of the system: if the optimal width of a time window is equal to its minimum available value, any disruption that may cause the flight not to meet it may produce undesired downstream effects. Our preliminary computational experience based on small-scale random instances confirms that the flexibility granted to flights increases with the capacity while the system delay simultaneously decreases. We also show that when there is no congestion a non negligible share of small size time windows may exist, thus indicating the existence of bottlenecks and critical flights.