Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 9575719, 12 pages

https://doi.org/10.1155/2017/9575719

## Two-Stage Heuristic Algorithm for Aircraft Recovery Problem

School of Economics & Management, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Cheng Zhang

Received 26 January 2017; Revised 31 May 2017; Accepted 10 July 2017; Published 24 August 2017

Academic Editor: Aura Reggiani

Copyright © 2017 Cheng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study focuses on the aircraft recovery problem (ARP). In real-life operations, disruptions always cause schedule failures and make airlines suffer from great loss. Therefore, the main objective of the aircraft recovery problem is to minimize the total recovery cost and solve the problem within reasonable runtimes. An aircraft recovery model (ARM) is proposed herein to formulate the ARP and use feasible line of flights as the basic variables in the model. We define the feasible line of flights (LOFs) as a sequence of flights flown by an aircraft within one day. The number of LOFs exponentially grows with the number of flights. Hence, a two-stage heuristic is proposed to reduce the problem scale. The algorithm integrates a heuristic scoring procedure with an aggregated aircraft recovery model (AARM) to preselect LOFs. The approach is tested on five real-life test scenarios. The computational results show that the proposed model provides a good formulation of the problem and can be solved within reasonable runtimes with the proposed methodology. The two-stage heuristic significantly reduces the number of LOFs after each stage and finally reduces the number of variables and constraints in the aircraft recovery model.

#### 1. Introduction

Scheduling is greatly significant to the airline industry because of its direct impact on the cost and service level. Accordingly, numerous optimization methods have been developed [1–3] to generate high-quality schedules, which can efficiently utilize costly resources (aircraft, crews, passengers, etc.). However, over the course of nominal airline operations, disruptions, such as severe weather conditions, unplanned maintenance, and air space control, frequently make airlines suffer from significant loss. According to FlightStats, nearly 700,000 flights are operated in North America in October, and the on-time performance is only 85.73% [4]. Disruptions have a profound influence on costs and profits. According to Airlines for America [5], the per-minute delay cost in 2015 was $65.43. American Airlines operated 725,984 flights in 2015, and the average delay was 12.31 min [6]. The projected delay cost for American Airlines in 2015 was over $500,000,000.

When disruptions occur, airline controllers in the operations control center need to recover the schedule in a timely manner to mitigate the cost. The controllers usually make the decisions manually. Passengers, crews, and aircraft are, in particular, sequentially recovered. A duty manager will then consolidate different solutions to make the final recovery decision. However, the scale of disruptions can easily grow to the extent that even experts cannot fulfill this task with a viable outcome within a reasonable time. For example, on EST November 27, 2013, “Atlanta tallied the highest number of delayed flights (439 departure delays as of 7:15 p.m. ET) because of the air traffic congestion, followed by Philadelphia (291), Charlotte (262), Chicago O’Hare (201), LaGuardia (160), and Newark (130)” [7]. Therefore, efficient recovery solutions are an urgent demand. In fact, how to recover from a schedule failure is always one of the major concerns in an airline operational stage.

The recovery problem is typically solved in a sequential way because of its complexity (large problem size and stringent time limit; see also Section 2). The aircraft recovery problem is solved in the first stage, with the objective of determining the recovered aircraft routings while respecting the maintenance requirements, airport curfew constraints, and flow balance constraints [8–14]. After aircraft recovery, the scheduled and preserved crews are reassigned to the uncancelled flight legs. The crew recovery problem usually adopts math programming methodologies [15–20] with the objective of minimizing the overall deviation from the original schedule. The passenger itineraries are rearranged in the final stage. Solving the problem sequentially will produce suboptimal solutions. Hence, integrating two or more problems has been gradually considered in the recent years [21–26].

The problem focused on in this research is only the aircraft recovery problem (ARP). The scheduled flights assigned to a list of aircraft of the same fleet family are considered. The disruptions include airport closure and unplanned maintenance. Herein, a flight can either be delayed or cancelled or the scheduled aircraft can be changed when a flight disruption occurs.

The remainder of the paper is organized as follows: an overview of the relevant literature is provided in Section 2; the mathematical model for the aircraft recovery problem is provided in Section 3; a heuristic for the route preselection and an aggregated mathematical model will be described in detail in Section 4; and a computational study for the test scenarios is presented in Section 5, where the results show that optimal solutions can be obtained for two small cases and near-optimal solutions for the other larger scenarios within 30 min. A summary of the findings is provided as conclusions.

#### 2. Literature Review

Various methodologies have been developed for the aircraft recovery problem in the literature. A detailed overview of the relevant studies is presented in this section.

##### 2.1. Aircraft Recovery

Pioneer studies on aircraft recovery focused on retiming and reassigning flights when one or more aircraft are unavailable [8]. In Teodorović and Guberinić’s study, the authors created a network in which the nodes were flights, and the arcs represented delays. The flight routes were heuristically generated based on the network. A branch-and-bound method was used to find a solution.

Jarrah et al. [27] adopted two recovery options for the aircraft recovery problem, that is, cancellation and retiming. Moreover, two minimum network flow models were proposed separately: one model for the cancellation and the other for retiming. However, the cancellation and the delay cannot be simultaneously considered in one problem.

Yan and Yang [11] proposed four models to formulate flight cancellation, ferrying, and delay for a single fleet recovery problem. These models were solved by a simplex method and a Lagrangian relaxation with subgradient methods. Yan and Tu [28] improved the work of Yan and Yang [11] to handle disruptions of multifleet and multistop flights. Yan and Lin [29] extended this work to handle the airport closure problem.

Thengvall et al. [12] proposed a network flow model that incorporated a protection arc to minimize the deviation from the original schedule. User preferences were also included in the objective function, such that the concerns of controllers can be reflected in the recovery solution. Meanwhile, Thengvall et al. [30] proposed three multicommodity models to formulate the hub closure problem.

Bard et al. [13] derived an integer minimum cost network flow model from the time-band network to minimize cancellation and delay cost. Rosenberger et al. [31] proposed a set partitioning model to minimize the total cost of flight cancellation and delay. The model attempted to determine the optimal combination of aircraft routes generated using a connection network. A nondisrupted aircraft is heuristically selected as a candidate to swap with the disrupted aircraft to reduce the problem scale.

Heuristics have also been developed to solve the aircraft recovery problem. Teodorovic and Stojkovic [9] formulated the problem as a goal programming model and developed a greedy heuristic to solve it. This heuristic aimed to find a new combination of aircraft routings, which minimized flight cancellations and passenger delays. Argüello et al. [32] presented a greedy randomized adaptive search procedure in response to groundings and delays. The neighborhood of the incumbent solution was constructed, and a new incumbent solution will be randomly selected from a restricted candidate list. Andersson [33] applied two metaheuristics to solve the problem: a tabu search and a simulated annealing approach. These two heuristics aimed to provide a set of ranked solutions for the controllers to select from. The computational results showed that the tabu search stands out in terms of both solution quality and method robustness. Liu et al. [34] presented a multiobjective genetic algorithm to solve daily short-haul recovery problems. A hybrid evolutionary algorithm employing an adaptive evaluated vector was developed. Furthermore, an inequality-based multiobjective genetic algorithm was used to search for Pareto solutions.

##### 2.2. Integrated Recovery

Although sequentially solving the airline recovery problem can generate an optimal solution for each stage, it usually produces a suboptimal solution for the entire problem. Therefore, integrated recovery methodologies were developed to solve the problem.

Abdelghany et al. [23] presented a rolling horizon framework incorporated with a greedy optimization approach for the joint aircraft and crew recovery problem. Two models were integrated in the framework, namely, a schedule simulation model and a resource assignment optimization model. The simulation model predicted possible disrupted flights, while the optimization model combined various recovery options to minimize flight delays and cancellations.

Eggenberg et al. [24] introduced a constraint-specific approach that simultaneously considered the aircraft, crews, and passengers. A different recovery network was generated for each kind of resource to reduce the problem scale. A set partitioning model was then created to embed the resources in one recovery scheme. Subsequently, a column generation approach was used to solve the model.

Bisaillon et al. [35] developed a large neighborhood search (LNS) algorithm to solve the problem combining aircraft, crew, and passenger recovery. The algorithm consisted of three stages: first, the aircraft schedule is recovered by delaying, cancelling, or reassigning; second, flights that violate constraints are repaired, and cancelled passenger itineraries are reassigned to the existing flight schedule; and third, a local search is performed to improve the solution. Sinclair et al. [36] proposed a mixed integer model for the integrated aircraft and passenger recovery problem and solved it by combining LNS heuristic and column generation.

Meanwhile, Petersen et al. [25] employed mathematical programming techniques for the integrated aircraft, crew, and passenger recovery problem. The Benders decomposition was employed to decompose the problem and obtain solutions in a reasonable time. The master problem is a schedule recovery problem with linking variables passing into the other three subproblems: aircraft, crew, and passenger recovery problem. Both Benders cuts and column generation have been adopted to solve these decomposed problems. Maher [37] studied accurate solution techniques for the recovery problem integrating schedule, crew, and aircraft recovery stages. Consequently, a framework for the column-and-row generation was presented as an alternative to the Benders decomposition.

Because of the large problem size and the stringent time limit, the integrated problem was either solved by heuristically decomposing the integrated model into submodels or by solving sequential stages first and then heuristically iterating between stages to achieve the final solutions. The aircraft recovery problem among the submodels or stages is usually solved in the first step. Therefore, the solution quality of the aircraft recovery is of great importance in both the sequential and the integrated problems. The contribution of this paper is the presentation of a two-stage heuristic which, when applied before a set partitioning model, can solve large-scale real-life aircraft recovery problems with a much-improved solution within reasonable times. The solution costs of the algorithm herein are compared with those of the searching heuristic used in a large-scale problem. Furthermore, the recovery cost can be significantly reduced with the increment of the problem scale. In addition, the effectiveness and the efficiency of the proposed method are validated using real-life test instances provided by domestic and international airlines.

#### 3. Problem Definition and Basic Formulation

Two types of disruptions were considered in this paper: aircraft disruptions and airport disruptions. Aircraft disruptions refer to the unavailability of aircraft during a certain period, which is mainly caused by mechanical failures. Airport disruptions refer to airport closures usually incurred by severe weather conditions. When these two disruptions happen, the flights assigned to the disrupted aircraft cannot be operated. Moreover, no flights are allowed to depart from or arrive at the closed airport. Given the original aircraft schedule and a set of disruptions, the objective of the ARP is to create a new combination of aircraft routings during the recovery period to minimize the total cost. The following constraints should be respected during the recovery procedure: a flight is either assigned to an aircraft or cancelled at any time and the disrupted aircraft must undergo the required maintenance.

##### 3.1. LOF Generation

LOFs were used as the basic variables in the proposed model to formulate the ARP. An LOF is a sequence of flights flown by an aircraft within one day. The detailed LOF generation procedure is presented as follows.

*General Procedure and Basic Constraints*. For the generation of LOFs from flight legs, the flight connection network is first constructed (Figure 1). In the network, each node is labeled with the flight leg it represents and the origin and destination stations of that flight. Two flight nodes can then be connected by an arc if they satisfy the time and space constraints: the arrival station of the first is identical with the departure station of the second and the time difference between the flights is within a fixed time frame []. Note that can be negative in the disruption recovery network, which means that the flights will be delayed to satisfy the minimum turn time. With the flight connection network, the depth-first search is applied to generate possible LOFs.