Ann Oper Res (2012) 194:177–187 DOI 10.1007/s10479-010-0809-8
Flight gate scheduling with respect to a reference schedule Ulrich Dorndorf · Florian Jaehn · Erwin Pesch
Published online: 4 November 2010 © Springer Science+Business Media, LLC 2010
Abstract This paper considers the problem of assigning flights to airport gates. We examine the general case in which an aircraft serving a flight may be assigned to different gates for arrival and departure processing and for optional intermediate parking. Restrictions to this assignment include gate closures and shadow restrictions, i.e., the situation where certain gate assignments may cause blocking of neighboring gates. The objectives include maximization of the total assignment preference score, a minimal number of unassigned flights during overload periods, minimization of the number of tows, maximization of a robustness measure as well as a minimal deviation from a given reference schedule. We show that in case of a one period time horizon this objective can easily be integrated into our existing model based on the Clique Partitioning Problem. Furthermore we present a heuristic algorithm to solve the problem for multiple periods.
1 Introduction Gate scheduling is concerned with finding an assignment of flights to terminal or ramp positions, called gates. It is a key activity among airport operations. With the increase of civil air traffic and the corresponding growth of airports in the past decades, the complexity of the task has increased significantly. At large international airports, several hundreds of flights must be handled per day. The task is further complicated by frequent changes of the underlying flight schedule on the day of operations due to factors such as delays or aircraft changes.
This work has been supported by the German Science Foundation (DFG) through the grant “Planung der Bodenabfertigung an Flughäfen” (PE 514/10-1). U. Dorndorf INFORM GmbH, Pascalstr. 23, 52076 Aachen, Germany F. Jaehn · E. Pesch () Institute of Information Systems, University Siegen, 57068 Siegen, Germany e-mail:
[email protected]
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Gates are scarce and expensive resources. Increasing the resource supply involves a timeconsuming and costly re-design of terminal buildings or the ramp and is usually not feasible in the short run. It is therefore of great economic importance for an airport or terminal operator to use the available gates in the best possible way. The main input for gate scheduling is a flight schedule with flight arrival and departure times and additional detailed flight information, including pair-wise links between successive flights served by the same aircraft, the type of aircraft, the number of passengers, the cargo volume, and the origin or destination of a flight, classified e.g., as domestic or international. The information in the flight schedule defines the time frame for processing a flight and the subset of gates to which it can or should be assigned, taking into account, e.g., aircraft-gate size compatibility, access to governmental inspection facilities for international flights, etc. It is worth pointing out—from a practical point of view—one of the most important issues of gate scheduling: a gate schedule should be insensitive to small changes of input data; in other words schedule flexibility is required. Obviously, the input data of any flight gate scheduling problem is subject to uncertainty and may change over time. Input data uncertainty in gate scheduling may have several reasons: (1) flight or gate breakdown, (2) flight earliness or tardiness, (3) emergency flights, (4) severe weather conditions, (5) errors made by staff and many others. For example, a tardy arrival of one aircraft may generate a chain of delayed arrivals for other aircraft which have been assigned to the same gate. In the worst case, this may lead to a “domino effect” and finally require a complete rescheduling, a situation which is highly undesirable. Gate scheduling is a task which arises daily. The schedules are commonly determined day by day. Since it is desirable for passengers and flight crews that a given flight be assigned to the same gate throughout the week, it is, however, worth to understand the flight gate scheduling as a multiple period problem rather than as a recurring one period problem. We will show how the flight gate scheduling problem presented in Dorndorf et al. (2008) can be expanded for solving a multiple period problem. Mathematical models of assigning flights to gates often differ concerning their objectives. Passenger satisfaction, which is mostly measured in terms of the passengers’ walking distance inside the terminal building, is the criterion in the branch and bound by Babic et al. (1984). The number of passengers who have to walk maximum distances is minimized. Mangoubi and Mathaisel (1985) use a similar objective, which also includes walking distances of transfer passengers. Further approaches considering the walking distance have been presented by Bihr (1990), Braaksma (1977), and Zhang et al. (1994). Ding et al. (2004) use the walking distance only as a tie-breaker for the case that their primary objective, the minimization of unassigned flights, is optimal for more than one solution. Their problem is modelled as a quadratic assignment problem, which is then solved heuristically using various meta-heuristics. Drexl and Nikulin (2008) add a third objective of maximizing flight gate preference scores. They search for Pareto optimal solutions using simulated annealing. Dorndorf (2002) simultaneously maximizes flight gate preference scores, minimizes dummy gate assignments, and minimizes the number of tows, which are required if an aircraft is assigned to different gates for arrival and departure procedures. The assignment problem is modelled as a resource constrained project scheduling problem with minimal and maximal time lags and solved using a truncated branch and bound algorithm. Based on the ideas of Dorndorf (2002), Drexl and Nikulin (2006, 2010) focus on the multi-criteria aspect and present Pareto optimal solutions. A special case of this gate assignment problem has been considered by Jaehn (2010). He shows that the problem can be
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solved in polynomial time if the shadow restrictions are independent of the aircraft’s size and if the number of gates is fixed. Dorndorf et al. (2008) present a transformation of the problem to the clique partitioning problem, which is then solved using an ejection chain algorithm. Their approach is the basis for the paper at hand. A detailed survey about flight gate assignment approaches is presented by Dorndorf et al. (2007a). Most of the aforementioned papers put little focus on robustness matters. An issue, which is highly relevant in flight gate assignment. Robustness, which in general is described, e. g., by Scholl (2001), can be achieved introducing buffer time between two flight activities assigned to the same gate, see, e. g., Hassounah and Steuart (1993), Yan and Chang (1998), Bolat (2000), and Yan and Huo (2001). Lim and Wang (2005) consider stochastic arrival and departure times. Their only objective is to minimize the expected number of gate conflicts. They show that their problem is NP-hard and they solve it using a hybrid meta-heuristic. Yan and Tang (2007) present a simulation framework for a gate assignment problem in which the passengers’ waiting time is included to the objective. Surveys about robustness in flight gate scheduling are given by Nikulin (2006) and Dorndorf et al. (2007b). The gate scheduling problem can be considered as the assignment of resources (flights, aircraft, gates) to time slots. However, the specific constraints and multi-objective criteria including certain measures of robustness require different solution approaches as they have been used for timetabling problems and surveyed in Burke and Petrovic (2002) and Qu et al. (2009). The remainder of the paper is organized as follows. A formal definition of the problem and related mathematical model are given in Sect. 2. A heuristic for solving the problem is proposed in Sect. 3. Computational experiments and comparisons are presented in Sect. 4. Finally, a summary appears in Sect. 5.
2 Problem description and mathematical model We are concerned with the question of assigning a number of flights, or activities, to a number of gates or alternatively to a dummy gate. Given real life data from airports that work to capacity, feasible assignments might not exist. Thus, a dummy gate with unrestricted capacity is introduced. A flight activity can be an arrival, a parking or a departure activity of an aircraft. Each activity may be assigned to a different gate which would be handled in real life by towing the aircraft. We define the number of flight activities to be assigned by n and the number of gates without the dummy gate by m. In our model, which is described in more detail by Dorndorf et al. (2008), we represent the n flight activities and the m gates as vertices of a complete graph G = (V , E, W ). V := {1, . . . , n + m} is the set of vertices, E := {{i, j }|i, j ∈ V } is the set of edges, and W is a weight function assigning a real value to each edge of the graph and which will be described below. An equivalence relation on the set of vertices, i. e., a partition of the set of vertices into cliques can be seen as an assignment of flight activities to gates if no two gates representing vertices belong to the same clique. If a vertex representing a flight activity i is in the same clique as a vertex representing a gate k, then flight activity i has to be assigned to gate k. If a flight activity vertex i is not in a clique that includes a gate vertex, then we assign flight activity i to the dummy gate. In Fig. 1 bold edges connect vertices that belong to the same clique. Vertices 6 and 7 symbolize gates. Flight activities 3 and 4 are assigned to gate 7 while flight activities 1, 2 and 5 are assigned to the dummy gate, as they do not belong to any clique containing a gate vertex.
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Fig. 1 Equivalence relation on the set of vertices
In the following we will choose appropriate edge weights for this complete graph in order to find a clique partition satisfying some restrictions and maximizing certain objectives. Our goal is to find such a clique partition with a maximum weight. This is known as the clique partitioning problem (CPP). Using binary variables 1 if vertices i and k belong to the same clique, i, k ∈ {1, 2, . . . , n + m} xik = 0 otherwise, for all edges {i, k} and corresponding edge weights wik , the CPP can be described by the following model (see Dorndorf and Pesch 1994): max
wik · xik
1≤i 800 and m > 100. However, there exist efficient heuristics for solving the CPP. To complete this mathematical model, the edge weights wik have to be determined. As mentioned before, any two gate vertices must not be in the same clique. Therefore all corresponding edge weights are set to −∞. Gate Closures: Gates may be closed for certain flight activities. These restrictions are motivated by aircraft that are too big for some gates, or by other technical factors such as gates that exclusively serve national or international flights. However, flight activities can always be assigned to the dummy gate. If a gate k is closed for flight activity i, we set the edge weight wik := −∞. Overlapping: Each gate, except the dummy gate, can serve at most one flight activity at a time. Therefore any two flight activities must not be assigned to the same gate if they either have an overlapping ground time or their ground time intervals are too close that
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the gate setup time cannot be maintained. For all such pairs of activities i and j the edge weight of the corresponding vertices is set to wij := −∞. In practical applications one can also find so called shadow restrictions, which prohibit the assignment of flight activity pairs to specific gates. A major reason for the shadow restrictions is that two big aircraft cannot be assigned to neighbor gates without touching wing tips. Consequently these restrictions do not apply to the dummy gate and are only relevant for flight activities overlapping in time. It is not possible to include these restrictions into the CPP by using edge weights. However, algorithms for solving the CPP can be modified in such way that only a reduced search space is considered, in which the shadow restrictions are satisfied (see Dorndorf et al. 2008). The objective for the flight gate scheduling problem consists of several weighted objectives. The weights for each objective were determined by airport and airline managers using e. g., priority classifications for aircraft or gates. Therefore we regard the weights as default. These objectives will also be introduced into the graph model: Preference Score: There is a preference score pik for every feasible assignment of a flight activity i to a gate k. These preference scores reflect priorities, e. g., for bigger aircraft or for favored gates, and are determined by airport or gate managers. In general, we assume pik > 0 which implicitly makes assignments to the dummy gate undesirable. All edges of feasible assignments of a flight activity i to a gate k are weighted wik := α1 · pik where α1 > 0 is an appropriate weight for this objective. Tows: As mentioned before, the time an aircraft resides at an airport can be split into an arrival, a parking and a departure activity of the aircraft. If two succeeding activities (i. e., the arrival and the parking activity or the parking and the departure activity of an aircraft) are assigned to different gates, then a tow is necessary. It is desirable to keep the number of tows low, or in other words to keep the number of succeeding activities that are assigned to the same gate high. Each pair of succeeding flight activities assigned to the same gate is awarded in the objective by α2 . All edges connecting two succeeding flight activities i and j are weighted wij := α2 > 0. Robustness: Small flight delays are very common. Therefore it is not only reasonable to force overlapping flight activities to be assigned to different gates, but also flight activities separated by a very small time buffer. If the time buffer tij between the two non successive flight activities i and j is less than the given parameter tmax , the objective will be reduced by α3 · (tmax − tij ), where α3 > 0 again describes the appropriate weight for this objective. Accordingly, the edge weight is set to wij := −α3 · (tmax − tij ) if tij < tmax . The remaining edges (these are edges between flight activity vertices which are not successive and have sufficient buffer time) are weighted 0. Another objective of practical importance is the minimal deviation from a reference schedule: A subset of the flight activities appears on a daily basis, i. e., on the previous day there might have been a flight activity with the same origin and arrival time or the same destination and departure time as today. It is desirable to assign such a flight activity to the same gate as it has been on the day before. If we only have the assignment of the previous day and today’s assignment will not be a reference schedule for the next day, then it is easy to insert the objective of minimal deviation from the previous days’ assignment: Let us denote the set of gates as M := {n + 1, . . . , n + m}. The reference schedule can be denoted with binary variables 1 if flight activity i is referenced to gate k 0 ∀ i, k ∈ V xik := 0 otherwise
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0 Note, that xik = 1 implies that i ∈ V \M and k ∈ M. Only to simplify matters later on, we allow i and k to be in V . The variables describe an assignment of flight activities to gates, whereas it is possible that a flight activity is assigned to more than one gate. Indeed, it would be reasonable to force flight activities to be only assigned to at most one gate:
xik = 1 ∧ xil = 1
⇒
l=k
However, having more than one reference assignment of a flight activity does not complicate the problem. Thus, we analyze the general case. Moreover one might think about situations where two reference assignments might appear, e. g., if a flight activity is generally assigned to one gate but lately it was assigned to another gate. The objective of the one-period reference schedule is 0 α4 · xik · xik , 1≤i 0 as the corresponding objective’s weight. This objective can be added to the objective of 2.1: 0 0 wik · xik + α4 · xik · xik = wik + α4 · xik · xik (2.2) 1≤i
