Math-heuris c to solve the airline recovery problem considering aircra" and passenger networks

Recebido: 9 de setembro de 2019 Aceito para publicação: 13 de novembro de 2020 Publicado: 30 de julho de 2021 Editor de área: Alexandre Gomes de Barros ABSTRACT The airline recovery problem involves aircra2, crew and passenger networks impacted by disrup5ons. Models to solve the problem consider one or more of these networks, on an integrated way or not. It belongs to the NP-hard class, even considering only one network. This work presents a math-heuris5c to solve the problem integra5ng the aircra2 and the passenger networks. A model to restore the aircra2 network with minimum cost was also developed. These two models compose a framework which permits the airline to obtain the cost impact of including the passenger network in the recovery problem. Both models were tested with real world ROADEF instances using an Intel i7 microcomputer (16Gb of RAM) and a high-performance cluster node (HPC) with 512 GB of RAM. The microcomputer solved instances with up to 85 aircra2 and 276 impacted flights in less than 30 minutes (imposed limit). The faster high-performance server reached solu5ons with minimum gap of 0 to 0.7% for the instances with higher number of flights. Total costs considering aircra2 and passenger networks were very close to the aircra2 network recovery results, showing a cost compensa5on which highlights the importance of solving the recovery problem integra5ng aircra2 and passenger networks.


INTRODUCTION 1.1 Airline Opera onal Plan
The airline operational plan starts long before a light takes off. Its objective is to provide an effective and ef icient service considering customers' demands and costs perspectives. TRANSPORTES | ISSN: 2237-1346 It encompasses (as shown in Figure 1): • The preparation of a Flight Schedule to meet the market demand • An aircraft schedule (also called aircraft rotation) assigning an aircraft (tail assignment) to each light in order to attend the expected demand (passenger schedule), respecting light operational constraints. • A corresponding schedule of quali ied cabin crew and pilots compliant to the regulatory and labor standards Each of these three schedules can be represented by a resource network. They must be synchronized so that the lights operate as planned and as ef iciently as possible.

Airline disrup on
An airline disruption occurs when at least one of the required resources to operate a light is not available and ready at the light's planned departure time. It may affect the departure of one or more lights. Even minor disruptions can create ripple effects across the network and quickly affect the rest of the light network.
Disruptions can be caused for several reasons. The main ones are classi ied as follows (Artigues et al., 2012): • Flight delays: may be caused by delayed boarding of passengers, ground time longer than planned, ground staff strike, or delays in passenger or crew connections • Flight cancellations • Unavailability of an aircraft for a certain period of time, preventing it to be assigned to light operations during that period • Reduction of airport capacity over a certain period of time, due, for example, to weather conditions or to strike of airport personnel, with consequent reduction on the number of landings and takeoffs in the period. The estimated annual cost of disruptions is in the order of millions of U.S. dollars, encompassing : • Aircraft operating costs • Crew overtime and use of reserve crew costs • Loss of passenger revenue and direct costs related to affected passengers, e.g. hotel, transportation and feed costs.

The airline recovery problem
The solution of the airline recovery problem consists in returning the airline back to the planned schedule after an operational disruption. It is carried out by adjusting light plans, aircraft assignments, crew assignments and passenger itineraries within a period of time called recovery period. It is addressed to minimize operational costs and service penalties. Usually, the recovery period is of one day (Zhang et al., 2015), but it really depends on the disruption event (e.g. a snowstorm may impact an airport operation for several days).
The Operational Control Center (OCC) is responsible to restore the airline light schedule back to the original planning condition after the occurrence of a disruption, within the established recovery span. This task is called "Airline Disruption Management" or "Airline Recovery".
The basic steps of the airline recovery process are shown in Figure 2. In the Operation Monitoring step, the OCC checks whether lights, cargo and baggage loading on aircraft and checkin of passengers and crews are operating as expected. If not, an analysis is performed immediately to decide on corrective actions. The required processing time the OCC usually has to solve the recovery problem is of as much as 20 or 30 minutes (Morais et al., 2018).  (Kohl et al., 2007) Figure 3 shows the sequential procedure usually used to solve the airline recovery problem: This means that, after a disruption event, the OCC irst builds the light cancellation and delay plans, then adjusts the aircraft network, synchronizes the crew network with the aircraft network and, inally, adjust the passenger route network.
The priority of this approach is the return to operation of the most expensive assets, aircraft and crew. It does not look to the best result for customers. As a consequence, it may imply high costs related to passenger delays. Therefore, it does not provide optimal solutions to the problem (Artigues et al., 2012).

LITERATURE REVIEW 2.1. Airline recovery categories and approaches
Airline recovery overall approach can be classi ied according to the following categories : • Non-integrated recovery -only one of the three networks -or dimensions -is considered.
• Non-simultaneous integrated recovery -airline recovery considers the three dimensions -aircraft, crews and passengers -but separately, not simultaneously. • Simultaneous integrated recovery -the three dimensions -aircraft, crews and passengers are treated simultaneously, without imposing degrees of importance to them.
• Partially integrated recovery -airline recovery considers two dimensions, simultaneously or not. Among the 60 articles on airline recovery analyzed by Castro et all (2014), 75% belonged to the non-integrated recovery class and only 25% were related to integrated or partially integrated recovery (of which, non-simultaneous integrated recovery: 26%, simultaneous integrated recovery: 13%, and partially integrated recovery: 61%) Operations research tools dominate work on recovery of airline networks. 73% out of 60 papers analyzed by Castro et all (2014), published between 1998 and 2012, used OR techniques. 70% considered minimizing light delays, light cancellations and operating costs, and only 7% considered impacts to passengers.
Regarding the solution methods, 31% used mathematical programming, 29% used network lows. 22% used heuristics and 18% used a column generation method.
Below is a summary of important work related to Non-simultaneous integrated recovery, Simultaneous integrated recovery and Partially integrated recovery. Kohl et al. (2007) present a sequential modeling with a distinct approach for each of the problem dimensions, aiming to minimize operating costs and passenger costs. For the aircraft dimension, a local search heuristic was used; for the crew dimension, a whole programming approach was used with resolution per column generation; and for the passenger dimension, multi-commodity network programming was used. The model considers the cost of delay at the inal destination for the relocated passenger, direct costs for rescheduled passengers (hotel, food, etc.), impact on customer perception and costs related to class upgrade and downgrade (in this case strongly impacting the customers experience and their willingness to ly again with the company). Maher (2015Maher ( , 2016) present a sequential approach using programming and resolution with a column generation method; passenger recovery occurs after the resolution for aircraft and crew dimensions. The passenger cost depends on the number of passengers relocated due to light cancellation and also on the delay costs for passengers relocated to other lights. Jafari and Zegordi (2010) Castro e Oliveira (2009 and  present approaches using multi-agent systems, expert agents, automated negotiation methods in the airline´s Operational Control Center and learning methods. Bratu and Barnhart (2006) present a multi-commodity network low model for solving the partially integrated recovery problem considering aircraft and passenger dimensions. It incorporates operational cost and cost reduction modeling for passenger itineraries delays and cancellations. Abdelghany et al. (2008) propose an integrated model for recovery of aircraft and crew dimensions by using a nonlinear mixed mathematical programming approach to minimize operational costs. Gao (2007) proposes an approach to partially integrated recovery considering aircraft and crew dimensions. It uses a multi-commodity network low modeling with Benders decomposition to minimize operational costs cancellations and delays. Zhang (2008) incorporates ground transportation in a nonlinear programming model for partially integrated recovery, considering aircraft and passenger dimensions to minimize operating costs and passenger costs.

Partially integrated recovery
Eggenberg and Salani (2010) present a solution for partially integrated recovery considering aircraft and crew dimensions. It applies the column generation method for minimizing operating costs. Zegordi and Jafari (2010) propose a partially integrated recovery solution considering aircraft and passenger dimensions. A heuristic based on ant colony, with the objective of minimizing operational costs, light cancellations, delays and costs for passengers is used. Bisaillon et al. (2011) present a solution for the partially integrated recovery considering aircraft and passenger dimensions. A large search in the vicinity heuristic to minimize operational costs and impacts on passengers is used. Jafari and Zegordi (2011) present a mixed integer mathematical programming model for partially integrated recovery considering aircraft and crew dimensions, for minimizing operating costs. Gomes (2014) presents an integer mathematical programming composition with heuristics, to solve the partially integrated recovery with aircraft and crew dimensions. Sinclair et al. (2014) present a large search in the vicinity heuristic for the partially integrated recovery with aircraft and passenger dimensions, based on Bisaillon et al. (2011). Hu et al. (2016) present a GRASP heuristic for the partially integrated recovery considering aircraft and passenger dimensions for minimizing operating costs and number of passengers affected. Arikan et al. (2016) present a model based on mixed integer mathematical programming considering aircraft and passenger dimensions. The inclusion of the aircraft cruise speed control is considered as an alternative course of action. Zhang et al. (2016) present a recovery solution for the aircraft and passenger dimensions using a three-step math-heuristic. It is addressed to treat cancellation and delay of lights for minimizing costs of passengers' accommodation. Marla et al. (2017) present an approach to the Partially integrated recovery considering aircraft and passenger dimensions. It includes the aircraft cruise speed control as an alternative course of action using a network lows model. Morais et al. (2018) present a two-part math-heuristic to solve the aircraft recovery: a mixedinteger programming network low model to obtain a new schedule with minimum light cancellations and delays; and an integer linear programming model to minimize aircraft swaps. It includes disruptions due to aircraft requiring unexpected maintenance.

Time-space networks
The airline planning can be described as a time-space network (Morais et al., 2018), as shown in Figure 4. For each airport there is a set of nodes and each node represents a pair: [event at the airport -departure or arrival -, time of the event]. Nodes associated with the same airport are positioned in the same column, ordered from top to bottom, depending on the time of the event. The offer node for each airport represents the number of aircraft available at the airport at the beginning of the planning cycle. At the end of the planning cycle, the number of aircraft available at the airport characterizes a termination node. Nodes are connected by directed arcs, where the unit low represents an aircraft. All lows start on offer nodes and end on termination nodes. At all nodes low conservation is maintained.
Time-space networks are directed and acyclic graphs (Abdelghany et al., 2004). Each aircraft has a preparation time between one light and another (min_turn_time). This time can be modeled as part of the previous light time (allows simpli ication of modeling), i.e. a light node ends at the earliest possible time of departure of the next light, rather than at the actual landing time. Therefore, a departure node corresponds to the departure of a light at a location l(j) at time t(j) and an arrival node corresponds to the arrival of a light in l(j) at time t(j) -min_turn_time, because t(j) corresponds to the actual arrival time + min_turn_time = (readiness time for the next light).

Modeling flight alterna ves and disrup ons
Flight alternatives can be modeled as inserted arcs (Zhang et al., 2016). For each of the planned lights in the original Flight Schedule, a set of lights related to a delay of a ixed period of time is inserted. A suf ix "-1", "-2", etc. is added to the original light identi ication, according to the considered delay (e.g. ifteen minutes delay alternatives: original Flight234 departuring 15:00, alternative Flight234-1 departuring 15:15, alternative Flight234-2 departuring 15:30).
Disruptions such as light cancellations are modelled by cancelling the original light and all respective inserted arcs. Other disruptions are modeled by introducing resources unavailability constraints. For example, aircraft unavailability are modeled by removing the aircraft from the solution. Airport operational restrictions are modeled as a departure and a landing capacity reduction (Zhang et al., 2016 andMorais et al., 2018).
Finding shorter paths in light networks corresponds to solving major airline optimization problems (Maher, 2016): maximizing pro it on aircraft routes, minimizing crew allocation cost, and minimizing travel cost for passengers.
The resolution of these problems should consider only viable routes, i.e. aircraft routes should consider the aircraft readiness schedule for the next light -not the arrival time -and, in the case of crews, the subset of possible paths given regulatory restrictions.
As already mentioned, time-space networks are directed and acyclic; therefore, there are eficient algorithms to ind shorter paths in these conditions -algorithms of complexity proportional to the number of arcs in the network and that require topological ordering of nodes (Abdelghany et al., 2004).
New optimized light schedule with a minimum delay cost can be obtained by means of a multi-commodity network low model. The model looks for the lowest cost lows in the disrupted time-space network with multiple lows and capacity limitation (Zhang et al., 2016, Morais et al., 2018.

Approaches considering networks interac on
Industry data indicate that recovery approaches considering the three networks on an integrated and simultaneous way permit to recuperate about 95% of the aircraft, crew and passenger networks in the same day of the disruption .
Meanwhile, sequential recovery approaches allow to recover around 80% to 90% of the aircraft and crew networks, but only about 60% of passengers are re-scheduled in the same period of time. So, the solution obtained includes signi icant costs not only for the airline (image losses and direct expenses related to the affected passengers), but also for the passengers -dissatisfaction, personal and inancial impacts .
Recovery approaches that allow the interaction of at least two of the networks of the problem generate better solutions than the sequential approach. (Petersen et al., 2012).
Airline disruption problems are of the NP-hard class; that is, there are no polynomial algorithms for their resolution, regardless of how it is considered -integrated, partially integrated or not integrated (Yu and Qi, 2004). Hence, the need to develop other types of models, such as heuristics, meta-heuristics and math-heuristics, to solve the recovery problem.
Heuristic-related modeling approaches do not necessarily provide the optimal solution to the problem. However, they can derive near-optimal results for NP-hard problems in very short processing times , Caetano e Gualda, 2011, Gomes e Gualda, 2015, Medau e Gualda, 2016 The decision to develop a math-heuristic for solving the recovery problem in this research took in consideration the opportunity to explore the potential of this relatively novel approach to treat NP-hard problems. According to Caserta et al (2010), math-heuristic models combine heuristic techniques with operations research techniques in an iterative or interactive way, offering more ef icient solutions.

DEVELOPED MODELS 3.1. Models purpose
This research aims at providing the airline with tools to decide on the recovery with priority to the aircraft schedule or to both the aircraft and the passenger schedules. To do so, two models were developed: a model for the aircraft network recovery with minimum light schedule cost (Flight schedule cost model) and a Math-heuristic for solving the airline recovery problem considering both the aircraft and the passenger networks. The Flight schedule cost model aims to restore the aircraft schedule with minimum cost for the airline using a sequential approach. The irst step recovers the light schedule, the second step recovers the aircraft rotation and the last step identi ies the disrupted itineraries due either to light cancellations in Step 1, to lights cancelled for lack of aircraft assignement in Step 2, or to lack of feasible connection time in the resultant light schedule after Step 2. Step 1

TRANSPORTES | ISSN: 2237-1346
FCf Set of planned flights and the inserted arcs for each flight f ∈ F. The original flight is an inserted arc with delay cost equal to 0 Step 1 modeling based on disruption problem input Step 1 Fcancel Set of original flights that, as part of the disruption event, are cancelled as an input of the problem; thus, cannot be part of the flight schedule adjustment.
Step 1 modeling based on disruption problem input Step 1 P Set of airports indexed by p Disruption problem input Step 1 M Set of types of movement in the airport ∈ [arrival, departure] indexed by m Disruption problem input Step 1 H Set of all time slots in the time-space network indexed by h representing each time slot inside the recovery window for each hour interval from the interval [0min-59min] Disruption problem input Step 1 FC_INn Set of all inbound flights to node n -original and inserted arcs Disruption problem input Step 1 FC_OUTn Set of all outbound flights from node n -original and inserted arcs Disruption problem input Step 1 N_IN Set of all aircraft entrance nodes Disruption problem input Step 1 N_OUT Set of all aircraft exit nodes Disruption problem input Step 1 A Set of aircraft indexed by a Disruption problem input Step 1 N_INn Number of aircraft entering node n, in the time-space network, at the start of the recovery window Step 1 modeling based on disruption problem input Step 1 N_OUTn Number of aircraft exiting node n, in the time-space network, at the end of the recovery window Step 1 modeling based on disruption problem input Step 1 Capp, m, h Capacity of airport p ∈ P for departure and arrival activities in each time slot h ∈ H Disruption problem input Step 1 and Step 3 (mathheuristic) cost_delayfc Cost due to delaying a flight f -choose an inserted arc Step 1 modeling Step 1 cost_cancelf Cost due to cancelling flight f Step 1 modeling Step 1 FC'f Set of planned flights and the inserted arcs for each flight f ∈ Rf. The original flight is an inserted arc with delay cost equal to 0 Step 1 Step 2 Cchangea,f Cost to change aircraft a from flight f Step 2 modeling based on disruption problem input planned aircraft rotation Step 2

C_not_assign
Cost of having an aircraft not assigned to flights in Xf Step 2 modeling input Step 2 STDa,f Departure time of flight f with aircraft a Step 1 Step 2 ETAa,f Arrival time of flight f with aircraft a Step 1 Step 2 ETAfc Arrival time of inserted arc fc Step 2 Step 3 STDfc Departure time of inserted arc fc Step 2 Step 3

Rf
Flight solution from Step 2, For every rfc ∈ Rf, rfc = 1 if fc is an active flight or 0 otherwise (flight cancelled in Step 1 or cancelled in Step 2 due to not having an aircraft assigned to it). fc is an inserted flight or the original flight (inserted arc with delay = 0) Step 2 Step 3 T Set of itineraries indexed by τ. Each itinerary τ has a sequence of flights that must be operated. An itinerary is affected if there is a cancelled flight or if there is not enough time for passengers to connect between two consecutive flights in the itinerary

Disruption problem input
Step 3 cτ Cost to disrupt an itinerary τ∈T Disruption problem input Step 3 Xf Flight schedule input to Step 3 of math-heuristic, obtained after Step 1 and Step 2. For every xf ∈ Xf, xf = 1, if flight f is confirmed or 0, if f is cancelled in Step 1 -f is an inserted flight or the original flight (inserted arc with delay = 0) Step 2 Step 3 connτ Minimum passenger connection time between two consecutive flights in a passenger itinerary τ∈T Disruption problem input Step 3 yf Binary decision variable, yf = 1 if original flight f is cancelled, 0 otherwise Step 1 xfc Binary decision variable, xfc = 1 if the inserted arc fc is chosen, 0 otherwise Step 1 ra,f Binary decision variable, ra,f = 1 if aircraft a is assigned to flight f ∈ Xf, 0 otherwise Step 2 wa,f Binary decision variable, wa,f = 1 if aircraft a is not assigned to flights in Xf, 0 otherwise Step 2 zτ Binary decision variable, zτ = 1 if itinerary τ∈T is disrupted (not possible to fly it from start to finish due to a cancelled flight in the itinerary or lack of enough time to connect between any two consecutive flights in the itinerary), 0 otherwise Step 3 (Flight schedule cost and math-heuristic) - x'fc Binary decision variable, x'fc = 1 if inserted arc x'fc ∈ FC'f is selected, 0 otherwise Step 3 (math-heuristic) -

TRANSPORTES | ISSN: 2237-1346
To restore the light schedule with minimum cost and to identify affected passengers, the directions proposed by Zhang et al. (2016) to obtain the new light schedule and to identify disrupted itineraries were adopted.
Step 2 of the Flight schedule cost model aims to recover light rotation by minimizing the cost to change an aircraft's original rotation and the cost of no assigned aircraft. The modeling approach is inspired in Morais et al. (2018) and adapted to a single time-space network.
The math-heuristic objective is to recover passenger itineraries that were disrupted due to lack of enough connection time and to identify the overall cost impact considering the delays inserted in the inal light schedule. This objective is accomplished in Step 3 of the math-heuristic by simultaneously considering, in the same mathematical model, two dimensions -aircraft network and passengers network. An optimal solution is searched for both the light schedule delay cost and the disrupted passenger itineraries cost, as shown in Figue 3. It was implemented based on Zhang et al. (2016) modeling approach. Two simpli ications are inherent to the developed models: all aircraft must have the same con iguration and no aircraft maintenance is considered. Therefore, an aircraft failure is taken as a disturbance that removes the aircraft from the solution. These simpli ications allow modeling with a single time-space network (Zhang et al., 2016). Figure 5 shows the basic steps of the Flight schedule cost model and of the Math-heuristic. Their variables are de ined in Table 1.

Flight schedule cost model
This model, composed of three steps, aims to restore the aircraft schedule with minimum cost for the airline and to identify disrupted passenger itineraries (due to any cancelled light in the itinerary or to lack of enough time to connect between any two consecutive lights in the itinerary).

Step 1 -Flight schedule adjustment
This step attempts to recover the original light schedule by delaying or cancelling lights. The problem is solved through a multi-commodity network low model. The mathematical modeling is based on inding the lowest cost lows in the disrupted time-space network with multiple lows and capacity limitation.

Step 2 -Aircraft rotation
This step aims to obtain the new aircraft rotation. It receives as input the solution obtained in Step 1. The objective function aims to minimize the cost of the new aircraft rotation, given by the cost due to changing the original aircraft rotation and the cost of the aircraft with no assigned light: = <@, ∑ ∑ (C BCDEFG , * H , + 5 IJ KLLMNO * P , ) ∈Q ∈R #S (8) Model restrictions to obtain the new aircraft rotation: Each light has an assigned aircraft or not. It is important to note that a light with no aircraft assignment will be cancelled in Step 2: (∑ H , ) + P , = 1, ∀! ∈ T ∈Q (9) Flights cancelled in Stage 1 (whether original or inserted arcs) cannot have assigned aircraft: `(10) The departure time of an aircraft (considering its readiness time) must be later than the aircraft's arrival time after the previous light:

Step 3 -Identifying disrupted passenger itineraries
The objective of this step is to identify the affected itineraries, either by cancellation of lights in the previous steps or by lack of enough time for passengers to accomplish their light connections in the itinerary.
The objective function aims to minimize the amount of disrupted itineraries: = min ∑ ` * a` `bc (12) Model restrictions are as follows. An itinerary is affected if it contains a cancelled light: a`= 1, ∀d P@ ℎ 6, 6, ]ee]f !e@gℎ (13) For each pair of lights on an itinerary, if the departure of a f2 light, consecutive to f1, is later than the arrival of f1 plus the passenger's connection time, then the connection is not lost. Otherwise, the itinerary is lost:

Math-heuris c
The Math-heuristic also has three stages. Stages 1 and 2 are identical to those modeled in item 3.2, but Step 3 is addressed to reduce the loss of itineraries.

Step 3 -Adjustment of #light schedule to reduce disrupted itineraries
The objective of this step is to minimize the number of passengers with affected connections. (18) For each pair of lights on an itinerary, if the departure time of f2 light, consecutive to f1, is later than the time of arrival of f1 plus the connection time of the passenger, then the connection is not lost. Otherwise the itinerary is affected: ∈ k # l − ,,`≥ −m * a` , ∀(! Z , ! W ) n 1, Big m (19) Airport capacity restrictions: ∑ ∈ 2,3,4 ≤ 567 8,9,: , ∀7 ∈ ;, ∀< ∈ =, ∀ℎ ∈ ? (20)

TESTS AND RESULTS
The models were implemented in Python 3.7, C++ (Step 2 only), Gurobi 8.1.1. They were tested thanks to a set of real world instances created by ROADEF for an operations research chalenge in 2009 (Palpant et al, 2009). The following information were considered for each instance: initial programming in terms of the number of lights per day of the recovery window; and numbers and details of the aircraft leet, of the airports and of the affected passenger itineraries during the recovery window. The following disruptions are considered: light delays in minutes, light cancellations and reduction of airport capacity in speci ic time slots during the recovery window. Table 2 and Table 3 show the characteristics of the test instances. The number of lights in Table 2 and Table 3 correspond to one day of operation and the number of lights in the modeling is proportional to the size of the recovery window, since the airline light schedule problem is replicated for each day of the operating window. Table 4 presents results obtained by the Flight schedule cost model. Table 2 present results obtained by the Math-heuristic. The inserted arcs were limited to a maximum delay of two hours in relation to the original light and the upper limit of the recovery window. Two simpli ications are inherent to the developed models: all aircraft must have the same con iguration and no aircraft maintenance is considered -although available in the test instances these information were not considered in the modeling.
Instances A1, A2, A3, A6, A7, and A8 - Table 2 -were run on an Intel i7 machine with 16Gb of RAM and Windows operating system. Instances B1, B2, B3, B4, B6, B7, B8, B9, A4, A9, A5 and A10 -with the characteristics presented in Table 2 and Table 3 -were processed using a highperformance cluster node -HPC -with Intel(R) Xeon(R) CPU E7-2870 @ 2.40GHz and 512 GB of RAM. Constraint (11), in Step 2, generates a number of constraints in the order of | A| * |Xf| 2 , where |A| = number of aircraft and |Xf| the number of impacted lights, i.e., the solution space grows exponentially with the number of aircraft and lights -one of the characteristics of an NP-hard problem (Gao et al., 2009).
In instances "B" the restriction (11) generated the approximate amount of 430,000,000 constraints in the optimizer software, occupying almost all the available memory (512 Gb) of the High Processor Computing equipment during the model loading step for solution execution. Instances B5 and B10 could not be run on the cluster node because they exceeded the available memory capacity during the execution of Step 2 of the models -both have a larger recovery window than the others (approximately half a day of operation more)  A value of the schedule delay per minute of €100 was used (Muligan, 2019). The cost of each passenger itinerary is provided as a parameter in Euros (€).
Cancelled Itineraries Delta Cost in Table 5    Step 2 was limited to a 12 hours execution and the Gurobi Gap % column in Table 4 and Table  5 indicate the gap obtained after the execution time indicated in column ∆t (min) .
The Math-heuristic presented lower costs of affected itineraries and Total Costs very close to the ones obtained by the Flight schedule cost model in the majority of the tests instances.
So, the cost of delays introduced to recover more passengers did not substantially affect the Total Cost for the airline. This is a very important result. It allerts that is worthwhile to compare the aircraft network recovery solution with the corresponding aircraft and passenger one. This allert and the fact that the developed models permit the airline to compare these alternative recovery options represent contributions of this work.
The Math-heuristic was designed to reduce the number of disrupted passenger itineraries due to lack of enough connection time after the aircraft schedule adjustment. This is accomplished by inserting additional delays in the light network beyond those caused by disruption. These solutions should have a lower cost of cancelled itineraries than the light schedule cost model solution. However, the total cost may eventually be higher than that of the light schedule cost solution, depending on the cost of the inserted delays. The Math-heuristic actualy led to reductions on the costs of cancelled passenger itineraries due to lack of enough connection time between lights for sixteen out of eighteen test instances (for two of them no improvements to passenger disruption cost were achieved). The achieved cost reductions represent an indicator of the effectiveness of the proposed Math-heuristic to solve the airline recovery problem considering aircraft and passenger networks. Table 6 shows the number of affected lights during the recovery window for each instance. The solution time of the Flight schedule model and of the Math-heuristic for instances with 85 aircraft and 276 impacted, as presented in instances A1, A2, A3, A6, A7 and A8, is acceptable by the industry (up to 30 minutes).
However, usual enterprise computer (Intel i7 machine with 16Gb of RAM and Windows operating system) could not solve instances with a quantity of aircraft greater than 85 and an impacted light greater than 276 within the industry acceptable timeframe (30 minutes).
Failure to achieve the optimal solution for the larger instances, even using a high-performance HPC-USP cluster node, con irms the NP-hard nature of the problem.
Meanwhile, the high-performance server -HPC allowed to extend the execution time of the model Step 2 to up to 12 hours in order to reach a solution with minimum gap for instances with a high number of lights. The gap ranged from 0% to 0.07%.

CONCLUSIONS
This work presented a math-heuristic to solve the airline recovery problem considering aircraft and passenger networks in partialy integrated, partially interactive and non-simultaneous way. It seeks to reach the best solution for both the airline and the affected passengers. A model to restore the light schedule with minimum cost for the airline was also developed.
These two models permit to compare results of the recovery with and without the passenger dimension. This type of comparison was pursued by this research. It allows the airline to decide on the pros and cons due to integrating or not the passenger dimension in the recovery process This airline recovery problem is of the NP-hard class, that is, there are no polynomial algorithms for its resolution, regardless of how it is considered -integrated, partially integrated or not integrated (Yu and Qi, 2004). Hence, the proposition of the Math-heuristic for solving it.
The math-heuristic modeling was chosen due to the fact that it represents a relatively novel modeling approach addressed to solve NP-hard problems.
The real world instances created by ROADEF for an operations research chalenge in 2009 were used to test the models.
The solution times of the Flight schedule model and of the Math-heuristic for instances with up to 85 aircraft and, 276 impacted lights as in instances A1, A2, A3, A6, A7 and A8, are compatible with the industry requirement (up to 30 minutes).
Instances with a quantity of aircraft greater than 85 and with an impacted light greater than 276 could not be solved in the industry acceptable timeframe (30 minutes) by an usual computer (Intel i7 machine with 16Gb of RAM and Windows operating system).
Failure to achieve the optimal solution for the larger instances, even using a high-performance HPC cluster node, con irms the NP-hard nature of the problem.
However, the high-performance server -HPC allowed to extend the execution time of the model Step 2 to up to 12 hours, in order to reach solutions with minimum gaps. The gap values ranged from 0% to 0.07%.
The applications of the Math-heuristic to the ROADEF instances presented, for 11 out of 18 instances, total costs of recovery with a diference of as much as 1.0 % to the ones obtained by the Flight schedule cost model. For 14 out of 18 instances the difference was of less than 3,7%. To say, the cost of the delays introduced to recover more passengers did not substantially affect the airline recovery cost for the tested instances.
This result permits to realize the importance of solving the recovery problem looking for the best solution for both the airline and the passengers. It justi ies the purpose of this research to develop models to permit the airline to analyse the cost impacts related to including the passenger dimension in the recovery attempt.