An Exact Solution Approach for the Airport
Stand Allocation Problem
Rodrigo Acuna-Agost, Daniel Perez
ORI Operations Research and Innovation Division, Amadeus SAS, France
Email: [email protected]
Julien Guepet
GSCOP, Gronoble INP, Grenoble, France
1
Introduction
Every day planners and dispatcher in airports deal with different decisions related to
the movement of aircraft. These decisions usually involve the use of fixed resources like
runways, stands (parking positions), and passenger gates. Due to the increasing flow of
passengers, these resources are becoming scarce and some airports have experimented a
degradation of the services offered. One clear example is the fact that passengers are more
and more being allocated to remote stands where a bus transfer is necessary to get to the
terminal. This reality not only affects connection times for passengers, but also increases
operation costs and decreases airport revenues, e.g., airlines usually pay lower fees when
their flights are allocated at remote stands.
In this paper we deal with the stand allocation problem. The interested reader is
referred to a complete survey of this problem by Dorndorf (2007). Given a set of aircraft
operations and available stands, the task is to find the more profitable assignment of
aircraft operations to aircraft stands while respecting operational constraints. Aircraft
operations refer to scheduling tasks that need to be physically allocated to a parking
position, e.g., arrival, departure, or parking operations. Controllers can also decide to tow
an aircraft from one stand to another. These towing operations allow a better utilization
of valuable stands (e.g., contact stands); but, on the other hand, an expensive towing
tractor movement is required. Additionally, the assignments should satisfy a number of
constraints, such as the compatibility between operations and stands and the constraints
that prevent adjacency conflicts, e.g., two large aircraft cannot be assigned to adjacent
stands at the same time.
In this paper, an integer programming formulation is presented. Our research involves
the development of several solution methodologies including simple greedy algorithms,
1
metaheuritics, and exact approaches. This paper is focused on an exact method, in other
words, it is able to find and prove optimality.
The proposed solution methods have been evaluated on real life instances from two
large European airports. The results show significant improvements in terms of solution
quality compared to the current practice of the airports, obtaining optimal solutions in a
reasonable CPU times.
2
Mathematical Formulation
In this section, a simplified version of a mathematical formulation of the problem based
on an integer programming model is presented. The model maximizes the preference of
operations to stand (e.g., the airlines and passengers usually prefer contact to remote
stands) while penalizing towing movements.
Let O be the set of aircraft operations, indexed by i; and S be the set of stands,
indexed by j. The unitary benefit of assigning aircraft operation i to stand j is modeled
by the coefficient cij . On the other hand, the decisions of the model can be summarized on
both: assignment of operations to stands and decisions of performing towing movements.
Let α1 ≥ 0 and α2 ≤ 0 be the relative (monetary) contribution of these two aspects to
the global objective respectively. These decisions can be modeled by two binary variables.
Let xij be equal to 1 if operation i ∈ O is assigned to stand j ∈ S, and 0 otherwise. Let
yi be equal to 1 if towing has been performed between operation i and its successor, and
0 otherwise. Therefore, the problem is to solve the following integer programming model:
M ax Z = α1
XX
i∈O j∈S
cij · xij + α2
X
yi
(1)
i∈O
Subject to: Compatibility Constraints + Overlap Constraints + Minimum Buffer Time
Constraints + Towing Constraints + Adjacency Constraints
3
A Viable Exact Approach
The direct solution of the presented model allows, theoretically, to obtain optimal solutions. Thus, our first experimentations used state-of-the-art MIP solvers based on branchand-cut (B&C) algorithms. Nevertheless, tests on real-life instances showed that the
amount of memory and time largely overpass the limits of practical applications. To tackle
this problem, we propose a strategy based on four stages: 1) aggregation of constraints,
2) efficient decomposition of the problem, 3) improved initial solutions and bounds, and
4) perturbation of coefficients. These steps are explained in the following subsections.
3.1
Aggregation of Constraints
One of the main sources of complexity and responsible of the size of the model corresponds
to the constraints that assure not having two overlapped operations assigned to the same
stand. The strategy is to transform several of these constraints into only one through a
fast algorithm. Theoretically, this aggregation is equivalent to find a clique edge cover in
a graph where every node is a variable (xij ) and every edge is a disjunctive constraint of
the type “Overlap Constraint”. The experiments show that this strategy strengthens the
continuous relaxation and reduces the number of nodes in the B&C tree. Thus, it reduces
the memory consumption, which allows a better scalability.
3.2
Decomposition
The goal of a decomposition is to split a big problem in smaller sub-problems that are usually easier to be solved. If the sub-problems are independent, the decomposition provides
an optimal solution and hence the decomposition is exact. Three types of decomposition
have been tested: physical, temporal and hybrid. The physical decomposition divides the
set of stands depending on the values of the coefficients cij . The temporal decomposition
splits the problem in several time slots and the hybrid one combines both strategies. We
have found and proved some conditions of optimality which are often verified in practice
because of the nature of real instances of the problem. This is due, for example, to a clear
division of real stands where it is very easy to identify the most valuable ones. The results
show that the tested decomposition reduces drastically the size of the model without any
loss of quality.
3.3
Initial Solution: Tabu Search and Lagrangian Relaxation
One of the strategies to speed-up B&C methods is to provide an initial solution to the
solver also known as MIP start, advanced start, or warm start. Having an initial solution
from the very beginning of B&C allows the solver to discard portions of the search space
and thus may result in smaller trees saving CPU time. Consequently, we have developed a
metaheuristic approach based on Tabu Search (TS) (Glover, 1986). This method is complemented with a Lagrangian relaxation which provides better bounds than the classical
linear relaxation and also allowing to estimate the quality of the solution of TS. The experiments show that the instances of the problem are solved in just a fraction of the time
the B&C takes. The gap between the solution of TS and the optimal one is estimated to
be less than 3%. These methods also represent an alternative solution with the advantage
of not needing commercial solvers.
3.4
Perturbation of Coefficients
A high symmetry on the generated models is observed affecting significantly the convergence of the B&C algorithm. This is due to the fact that many operations and stands
are equivalent from the point of view of the value of the coefficients they usually have. A
simple way to break this symmetry it is to perturb the value of cij . We have a perturbation procedure that has been proved to respect optimality, i.e., an optimal solution of
the perturbed model is also optimal of the original model. The results show significant
performance improvements with some instances solved in just 3% of the time needed for
the original unperturbed model.
4
Results and Conclusions
The developed approaches have been tested on real-life instances with data provided from
two important European airports with traffic in the order of 1000 flights/day. The results
show both: industrial and mathematical improvements.
First, the provided solutions represent a significant improvement compared to the
current practice on these two airports. For example, one of them could increase passengers
at contact in 1,000,000 passengers per year, minimizing towing costs, and maximizing
aeronautical revenues because of the position and passenger fee charges, i.e., some airport
charge more to the airlines when the passengers are allocate at contact (terminal).
Secondly, the developed strategy has shown to be effective and viable in practice. The
additional effort to boost the B&C algorithms rather than developing heuristic methods
has paid off. The strategy can be summarized as a successfully combination of four steps.
The combined contribution of them is estimated up to 3 orders of magnitude in size
and speed. Finally, and to illustrate the value of optimal solutions rather than heuristic
ones, estimations using an economic model holds that an increase of 1% on quality of the
solutions result in 100K more dollars in more revenues and savings for one of our partner
airports.
References
U. Dorndorf, A. Drexl, Y. Nikulin, and E. Pesch. Flight gate scheduling: State-of-the-art
and recent developments. Omega, 35:326–334, 2007.
F. Glover. Future paths for integer programming and links to artificial intelligence. Comput. Open Res., 13:533–549, 1986.