Optimal sequencing of aircrafts take-off and landing at a busy airport Andrea D’Ariano, Paolo D’Urgolo, Dario Pacciarelli and Marco Pranzo Abstract — This paper studies the problem of sequencing aircraft take-off and landing operations at congested airports. We introduce and analyze alternative detailed formulations and solution algorithms for scheduling arrival and departure times of the aircrafts, such that the delay with respect to the scheduled times is minimized. The aircraft scheduling problem (ASP) is viewed as an extension of the job shop scheduling problem with additional real-world constraints and formulated by using alternative graphs. Two alternative formulations model the required time separation among aircrafts in air segments and runways according to safety regulations and differ for the level of detail used to represent the holding circles. Scheduling rules, heuristic and exact methods are implemented and tested on practical size instances of the Fiumicino airport, the busiest airport in Italy. We show that two versions of an innovative branch and bound algorithm are always able to find good solutions in a few seconds and often improve the best solution computed by the scheduling heuristics. Optimality is proved in less than two minutes for more than half of the instances. I. INTRODUCTION Fiumicino airport (FCO) is one of the world’s busiest airports by passenger traffic and the main airport in Italy. Passenger traffic increased of 73% between 1996 and 2006 [5] and a further increase of traffic is forecasted in the coming years. With the increase in air traffic, airports are becoming a major bottleneck in Air Traffic Control (ATC) operations. Aviation authorities are thus seeking methods to better use the existing airport infrastructure and to better manage aircraft movements in the vicinity of airports, improving aircraft punctuality while maintaining the required level of safety. In this context, the optimization of take-off/landing operations is a key factor to improve the performance of the entire ATC system. However, the real-time air traffic management is still mainly under the control of human controllers whose computer support is most often limited to a graphical representation of the current aircraft position and speed. ATC decisions can be broadly divided into: • Routing decisions, where a route for each aircraft has to be chosen from its current position to its destination. • Scheduling decisions, where routes are considered fixed, and feasible aircraft sequencing and timing have to be determined in each airway, such that safety regulations are satisfied and traversing times are minimized. This work is partially supported by the Italian Ministry of Research, Grant number RBIP06BZW8, project FIRB “Advanced tracking system in intermodal freight transportation”. A. D’Ariano (Corresponding Author), P. D’Urgolo and D. Pacciarelli are with the Dipartimento di Informatica e Automazione,Universit`a degli Studi Roma Tre, via della vasca navale, 79 - 00146 Roma, Italy {a.dariano, durgolo, pacciarelli }@dia.uniroma3.it M. Pranzo is with the Dipartimento di Ingegneria dell’Informazione, Universit` a degli Studi di Siena, via Roma, 56 - 53100 Siena, Italy pranzo@dii.unisi.it One of the main objectives of routing decisions is to bal- ance the use of critical resources, e.g., alternative runways. The objective of real-time scheduling decisions is typically the delay minimization. We refer to the latter problem as the Aircraft Scheduling Problem (ASP). We deal with the development of a reliable decision support system for aircraft scheduling at a busy airport. In this paper, the routing problem is solved off-line in a preliminary step and the main focus is on real-time scheduling decisions with fixed routes. We now briefly recall the general procedure for take- off/landing operations in the proximity of airports. For each Terminal Maneuvering Area (TMA), landing aircrafts move along predefined routes from an entry fix to a runway following a standard descent profile. During all the approach phases, a minimum separation distance between every pair of consecutive aircrafts must be guaranteed. This standard separation depends on the type and relative positions of the two aircrafts (at the same or different altitude). Separations between aircrafts are mandatory and recognized by inter- national aviation regulations. By considering the different aircraft speeds, this safety distance can be translated in a Separation Time Interval (STI). Similarly, departing aircrafts leave the runway flying towards the assigned exit fix along an ascent profile, respecting separation standards. The runway can be occupied by only one aircraft at a time. Holding circles are used to stack the aircrafts until they can be guided into the landing sequence. A departing aircraft can leave a holding circle only after the traversing of (or a multiple of) half length of the circle. Departing aircrafts are supposed to take-off within their respective assigned time slots. An aircraft is late whenever it is not able to accomplish the departing procedure within its assigned time slot. A landing aircrafts is late when landing after its scheduled arrival time. A conflict occurs whenever two or more aircrafts do not respect the minimum required distance at the same air segment or runway. Any potential conflict between landing and departing aircrafts must be detected and solved in real-time by human air traffic controllers by rescheduling aircraft movements [2], i.e., by solving an instance of the ASP. The ASP has been the subject of many papers (see, e.g., the reviews in [2], [10]). Two cases of ASP can be distinguished: the static and the dynamic ASP. In the static ASP one wants to sequence landing/departing aircraft when all the information is known in advance. Beasley et al. [3] present a mixed-integer zero-one formulation for the static ASP in the single and multiple runways cases. The problem is then solved using exact and heuristic algorithms. Ernst et al. [7] tackle the static ASP of aircraft landings by