Integrated airline scheduling
Introduction
Airlines commence their tactical planning with schedule generation where the timetable of profitable legs is devised—see, e.g. [1]. A leg is a flight from a specific origin to a destination at a given departure time. Based on the timetable, airlines proceed onto solving the airline scheduling problem months before the day of operation. Since this problem is considered computationally intractable [2], it is typically decomposed into its constituent stages, and the full problem is solved in a sequential manner where the output of one stage is the input of the next [1]. In this manner, airlines initially solve the fleet assignment (FA) stage, deciding which fleet should fly each scheduled leg, while using the available aircraft and maximizing revenue. After this, the maintenance routing (MR) stage is solved, ensuring that aircraft will be periodically scheduled for maintenance, by devising individual aircraft routes. The obtained routes are used in the crew pairing (CP) stage, which devises the series of legs crew have to fly, while respecting labor rules and minimizing crew costs.
The knowledge of aircraft routes is important for CP in order to determine whether crew should remain on the same aircraft for their following leg, rather than using precious time commuting within the airport to board a different aircraft. The time needed for commuting is known as sit-time or connection, and a connection shorter than the minimum sit-time allowed is known as a short-connection. When the maintenance is guaranteed—i.e. maintenance can always be done during the night—the MR is relaxed to the aircraft routing problem, which devises aircraft routes without maintenance constraints. In this case plane-count constraints can be added in the CP to count the number of available aircraft on the ground at any time to make sure that the aircraft routing problem will be feasible with the set of forced turns [3], [4].
Although the sequential procedure reduces computational complexity, because of the interdependence of each stage, the resulting solution is suboptimal. Even worse, in some cases feasible problems might not be solvable. As an example, in [5] after solving the FA problem with the approximate MR considerations of [6], it was not possible to find MR solutions to an otherwise feasible scheduling problem. In order to circumvent the previous obstacles, over the past years models considering several stages simultaneously have been proposed. The aim of these semi-integrated models was to achieve better quality results, with the ultimate goal being to integrate all the stages. Several models and a solution methodology that achieve complete integration of all stages are presented in this paper for the first time. Before going into detail about this paper's contributions, a summary of related work on semi-integrated airline scheduling and relevant solution methods is initially presented.
The first exact semi-integration attempt included FA and aircraft routing, where the departure times of legs were flexibly defined within time-windows [7]. An integration of FA with MR was presented by [5]. Both papers solved their models by employing column generation in a branch-and-bound tree, known as branch-and-price. Furthermore, time-windows were integrated with FA, resulting in important savings [8]. This model was solved using a specialized algorithm which iteratively added “beneficial” time-windows. Additionally, CP was integrated with time windows and plane-count constraints [3]. That model was also solved with a specialized algorithm selecting a “good” subset of pairings and strong branching during the branch-and-bound algorithm.
Although plane-count constraints assist integrating aircraft routing with CP, MR feasibility is not always guaranteed, a fact speculated by [4], [9] and also verified by the experiments of the present paper in Section 5.2.4. For this reason, in [10] an exact model combining MR and CP was introduced. The great number of constraints in that model was handled by Benders decomposition, where MR was the master problem. Furthermore, MR was integrated with CP by [11], where it was argued that since crew short-connections are involved one could reduce the problem size by identifying key MR decisions. To be more specific, many routes that are maintenance and short-connection feasible are equivalent, and could be represented by a unique route. Additionally, from the collection of these unique routes there exists a maximal subset that covers most possible legs. Thus, the integrated model was solved using column generation where the columns generated had to be unique and maximal. Concerning the efficiency of the previously mentioned methodology the interested reader is also referred to the theoretical and experimental arguments of [12]. The latter paper also presented an integration of MR and CP, extending the work of [10] with the inclusion of special constraints for robust planning. In that paper it was also proved that it is more efficient reversing the decomposition of [10] by having CP on the Benders master problem. In [9] a model merging time-windows, MR, and CP was solved by Benders decomposition, leading to important cost savings. For the integrated MR and CP with Benders decomposition one typically employs artificial variables to obtain an always feasible primal Benders problem, and avoid feasibility cuts stemming from extreme rays. As [13] stressed, the choice of these artificial variables has an impact on the dual subproblem polyhedron and therefore on the generated cuts.
The closest attempt to integrate all airline scheduling stages was that of [4], integrating FA and CP with plane-count constraints, and not considering MR. They did, however, report important profit increase due to this integration. The authors demonstrated two solution methodologies, in the first they used Benders decomposition and in the second a combination of Lagrangian relaxation and column generation. As mentioned, however, at the beginning of the previous paragraph, the model of [4] is typically not MR feasible.
Finally, in the present paper novel methods are used to speed-up Benders decomposition and column generation, and are described respectively in [14], [15].
This paper contributes to airline scheduling optimization since:
- •
fully integrated models are presented for the first time, and a novel, generic, and efficient algorithm is devised for their solution;
- •
based on realistic data of European and North American airlines, computational results are reported, comparing the integrated with the best known methods from the literature, proving that the integrated significantly reduces costs;
- •
although different integrated models are introduced, one of them outperforms the rest, showing that there is probably a unique choice amongst them;
- •
novel plane-count constraints are presented, tighter than those of [4], with the further advantage that, instead of adding them, one can incorporate them in models and maintain the number of total constraints invariant;
- •
some of the integrated formulations include plane-count constraints, and the computational results demonstrate that these constraints alone cannot ensure MR feasibility.
Section snippets
The original integrated model and Benders decomposition
This section introduces the most efficient integrated model of the performed experiments presented in Section 5. The integration consists of fleet assignment (FA), maintenance routing (MR), and crew pairing (CP) problems. Concerning (FA) there are models considering passenger demand of specific legs [16], as well as passenger itineraries [17]. However, the former modeling is used in the present paper for simplicity. Moreover, concerning MR the model considered here is very similar to the one
The Magnanti–Wong method
The Benders subproblem (11)–(14) has a set partitioning structure, making it degenerate, and as a result it has multiple dual solutions. Hence, it is possible to choose among different cuts (23) that might actually have different strengths [28]. One therefore needs to define a relation comparing the strength of cuts corresponding to different dual values . Given two cuts and , it is said that the first dominates the second if and only if
Antagonistic models and methods
This section discusses airline scheduling methods and formulations to be compared with the original integrated approach introduced in Section 2 and 3. More specifically, semi-integrated and sequential methods known from the literature are presented, followed by the introduction of alternative integrated formulations.
Computational experiments
The algorithms presented in the previous sections were implemented in order to evaluate the benefits of the integrated methodology. Thus, the resource-constrained shortest path algorithms were implemented in , and the rest of the algorithms in ECLPS v5.8 [34]. ECLPS is a constraint logic programming language of Cisco Technology Inc., and includes a column and cut generation library that can use different linear solvers [35]; in the present paper ILOG CPLEX 9.030 [36] was chosen. The
Conclusions and future work
In this paper, the previously thought intractable integrated airline scheduling problem was solved for realistic instances of European and North American airlines. These airlines had different network structures, allowing a wide evaluation. The largest solved instance scheduled 700 legs for 6 fleets, and succeeded in reducing overall operating costs by 24 million US dollars in comparison to the best known method from the literature. Even the largest airlines have short-, medium- or long-haul
Acknowledgements
The author is indebted to Dr. Jonathan Lever, Prof. Mark Wallace, Dr. Andy Eremin, Dr. Wilhelm Cronholm, Dr. Olli Kamarainen, and Imogen Rivers for their support in different stages of the project. The author would also like to thank the anonymous referees for their helpful comments.
References (37)
- et al.
A computational study of Benders decomposition for the integrated aircraft routing and crew scheduling problem
Computers & Operations Research
(2005) - et al.
Crew pairing at Air France
European Journal of Operational Research
(1997) - et al.
Time constrained routing and scheduling
- et al.
A stochastic programming approach for supply chain network design under uncertainty
European Journal of Operational Research
(2005) - et al.
Applications of operations research in the airline industry
Transportation Science
(2003) - et al.
Integrated airline schedule planning
- et al.
Airline crew scheduling with time windows and plane-count constraints
Transportation Science
(2002) - Sandhu R, Klabjan R. Integrated airline planning, in: Proceedings 44th AGIFORS Annual Symposium, Singapore,...
- et al.
Flight string models for aircraft fleeting and routing
Transportation Science
(1998) - et al.
Maintenance and crew considerations in fleet assignment
Transportation Science
(1996)