Integrated Robust Airline Schedule Development

Transcription

1 Available online at.sciencedirect.com Procedia Social and Behavioral Sciences 20 (2011) th EWGT & 26 th MEC & 1 RH Integrated Robu Airline Schedule Develoment Luis Cadarso a Ángel Marín a* a E.T.S.I. Aeronáuticos Universidad Politécnica de Madrid Pza. Cardenal Cisneros 3 Madrid Sain Abract In air transortation airline rofitability is influenced by the airlines ability to build flight schedules. In order to generate oerational schedules airlines engage in a comlex decision-making rocess referred to as airline schedule lanning. U to no the generation of flight schedules has been searated and otimized sequentially. The schedule design has been traditionally decomosed into to sequential es. The frequency lanning and the timetable develoment. The urose of the second roblem of schedule develoment fleet assignment is to assign available aircraft tyes to flight legs such that seating caacity on an assigned aircraft matches closely ith flight demand and such that cos are minimized. Our ork integrates these lanning hases into one single model in order to roduce more economical solutions and create feer incomatibilities beteen the decisions. We roose an integrated robu aroach for the schedule develoment e. We design the timetable ensuring that enough time is available to erform assengers flight connections making the syem robu avoiding misconnected assengers. An alication of the model for a simlified IBERIA netork is shon Published by Elsevier Ltd. Selection and/or eer-revie under resonsibility of the Organizing Committee. Keyords: Robu airline schedule; timetable lanning; fleet assignment. 1. Introduction The airline schedule lanning roblem is defined as the sequence of decisions that need to be made to make a flight schedule oerational. Given the high level of cometition in the airline indury effective decision making is crucial to the rofitability of an airline. This is the motivation for this aer in hich e focus on the integration of the decision making rocess. Our goal is to achieve simultaneous rather than sequential solution; a simultaneous solution ill generate more economical solutions and create feer incomatibilities beteen the decisions. Moreover ith the integration of the different lanning rocess hases a greater robuness degree may be achieved obtaining smoother solutions hich in case of incident may be recovered in an easier ay. There are three major comonents in the schedule develoment e. The fir e the schedule design is arguably the mo comlicated e of all. Traditionally the schedule design has been decomosed into to sequential es. Fir the frequency lanning in hich lanners determine the aroriate service frequency in a market; and the second one the timetable develoment in hich lanners lace the roosed services throughout the day subject to netork consideration and other conraints. The urose of the second e of schedule * Corresonding author. Tel.: ; fax: address: angel.marin@um.es Published by Elsevier Ltd. Selection and/or eer-revie under resonsibility of the Organizing Committee doi: /j.sbsro

2 1042 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) develoment fleet assignment is to assign available aircraft tyes to flight legs such that seating caacity on an assigned aircraft matches closely ith flight demand and such that cos are minimized. Then the netork is decomosed into different netorks each one corresonding to a articular fleet tye. Given these netorks the assignment of individual aircraft to flight legs is done in the aircraft maintenance routing e the third e. Cre scheduling involves the rocess of identifying sequences of flight legs and assigning both the cockit and cabin cres to these sequences. Designing an airline netork is an extremely comlex task due to the huge number of variables affecting the design i.e. assenger demand ground facilities and caacity cometition etc. These issues are not alays easily modeled and usually result in huge models. In hub-and-soke netorks connecting assengers are very common. In order to fly from one soke to a different one a connection mu be erformed in the netork hub. The time needed to accomlish these connections is not ell knon and it deends on different asects as congeion delays etc. Thus robuness ill be introduced through assengers itineraries roviding them enough connection time. Hoever as connection time increases assengers dissatisfaction and lo resource utilization may also increase. Therefore robuness is introduced avoiding misconnected assengers but accounting for assengers cos and fleet utilization. This robuness criterion may have different imact over different asects. Fir of all increasing connection time means that assengers ill have to erform longer connections increasing their dissatisfaction due to the longer connection time. Hoever the robability of being misconnected ill be ameliorated. Another issue might be fleet utilization. Designing the timetable roviding robu itineraries ithout accounting for the fleet ill surely rovide a loer utilization of it. Nevertheless this hassle may be controlled by facing the roblem in an integrated ay accounting for fleet resources as e ill roose later in the aer. An imortant hase in airline lanning is the cre scheduling. In this aer e do not address this roblem. Hoever the ne aroach e roose ill affect this lanning hase. As e ill rovide longer connection time for assengers it may also occur the same for cres. This could result in the necessity of having more cres to oerate the schedule. The roblem resented in this aer consis of determining the schedule design and the fleet assignment simultaneously. For this urose average demand values ill be used and for each market demand its disaggregation in time ill be considered. In sulys side a sace-temoral grah ill be used. As different fleet tyes ill be considered the sace-temoral grah mu be relicated for each fleet tye. 2. State of the Art Soumis et al. (1980) consider the roblem of selecting assengers that ill fly on their desired itinerary ith the objective of minimizing sill cos. Flight schedules are otimized by adding and droing flights. They consider the assenger flo rocess by assigning dissatisfaction cos for unattractive itineraries and assenger sill cos for loading good itineraries. Armaco et al. (2002) describe a ne aroach for solving the exress shiment service netork design roblem. They transform conventional formulations to a ne formulation using hat they term comosite variables. The formulation relies on to key ideas: fir they cature aircraft routes ith a single variable and second ackage flos are imlicitly built into the ne variables the comosite variables. Lan et al. (2006) consider assengers ho miss their flight legs due to insufficient connection time. They develo a ne aroach to minimize assenger misconnections by retiming the dearture times of flight legs ithin a small time indo. They resent comutational results using data from a major U.S. airline and shoing that misconnected assengers can be reduced ithout significantly increasing oerational cos. Jiang and Barnhart (2009) roose a dynamic scheduling aroach that reotimizes elements of the flight schedule during the assenger booking rocess. They recognize that demand foreca quality for a articular dearture date imroves as it aroaches; thus they redesign the flight schedule at regular intervals using information from both revealed booking data and imroved forecas. Lately researchers have focused on determining incremental changes to flight schedules roducing a ne schedule by alying a limited number of changes to the exiing schedule. Lohateanont and Barnhart (2004) in their incremental otimization aroach select flight legs to include in the flight schedule and simultaneously otimize aircraft assignments to these flight legs. Garcia (2004) extends the revious model and rooses a combination beteen it and a decision time indo model. Kim and Barnhart (2007) consider the roblem of

3 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) designing the flight schedule for a charter airline. Exloiting the netork ructure of the roblem they develo exact and aroximate models and solutions and comare their results using data rovided by an airline. Esinoza et al. (2008) resent an integer multicommodity netork flo model ith side conraints for on-demand air transortation services. Cadarso and Marín (2010) roose a multiobjective integrated robu aroach for the schedule design hase deciding jointly flight frequencies and timetable. The objectives are assengers satisfaction and oerator cos. They design the timetable ensuring that enough time is available to erform flight connections making the syem robu avoiding misconnected assengers. In this aer a ne integrated aroach to solve the schedule develoment is resented. U to no it has been solved sequentially. This sequential aroach rovides local otimum for each subroblem but not for the overall lanning. Moreover it could be that there are not enough resources to oerate a given timetable. In such a case lanners mu face an iterative rocess until they obtain a feasible lanning but robably not being the overall otimum. The aroach roosed in this aer ill rovide a global otimum to the schedule develoment hase overcoming the mentioned deficiencies. In addition robuness ill be introduced for assengers connections. This ill robably mean longer connection times but as e are also accounting for fleet resources it does not necessarily mean loer fleet utilization. 3. Problem Descrition In this aer the schedule design and fleet assignment roblems are treated in an integrated ay. Frequencies and dearture times mu be determined for every itinerary attending each market. Moreover fleet tyes mu be assigned to every flight leg. Given the eimated demand for travel an airline ishes to determine the flight schedule hich maximizes its rofit hile taking into account the satisfaction of its cuomers. In this syem to agents interact: the aircraft flo in the hysical netork and the assengers using flight legs to travel. The netork is built considering the airorts associated to the demand to be met. It is formed by the airorts and all the feasible airay or sections alternatives linking them. The airorts are defined by the oerations that can be erformed ithin them. They are characterized by available caacity (i.e. slots) for landing and taking off determined by airorts oerators. The sections are the links beteen the airorts. Each section is characterized by an origin airort and a deination airort. When a section is flon it ill be called flight leg; a flight leg is defined by an origin deination and a dearture time that is a flight leg is defined by the air () here s is an element of the sections set S and t in {01...T-1} is the dearture time from the origin of S. The set of all ossible flight legs is F = S x {01...T-1}. In this ay e ill consider that the time is discretized by artitioning the lanning eriod T into intervals of equal length ith arting oints 01...T-1. The intervals length ill be taken as the time unit. For this ork the unconrained demand is characterized by the origin o and deination d airorts. Each air (od) is mentioned as the market. For each air the demand of assengers d is assumed fixed and knon datum. This demand is disaggregated in time. Hoever dearture time is not a fixed value assengers ill accet ithout additional co a dearture time from a set of desired dearture times in each market (T ). In this ay it is suosed that assengers have a feasible set of dearture times. For each demand assengers from origin to deination are considered in all ossible itineraries i that may be classified by air as I. Each itinerary is defined by a set of sections that connect different airorts. It can be comosed of one section or more than one including in this la case intermediate os at different airorts. In this ay assenger dissatisfaction ith connecting time ill be considered. As connecting time gros assenger dissatisfaction rises u. Hoever if connecting time is not enough assengers may be misconnected. In order to avoid misconnected assengers as much as ossible robu itineraries ill be introduced. As cometition effects are not considered in this roblem every flight leg ill robably be croded because the unconrained market demand is considered. Hoever this situation is far from the real one. Due to cometing airlines unconrained demand ill be divided beteen different flight legs. In order to reresent this issue the caacity offered for each flight leg ill not be the entire one but the one obtained by an average load factor. This

4 1044 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) average load factor ill be obtained from the airline data. Besides cometition ithin the same airline ill be avoided by imosing a searation time beteen flight legs oerating the same origin and deination. 4. Robuness As mentioned before robuness is introduced through assengers in itineraries ith more than one flight here a connection is mandated. Adding more slack for connection can be good for connecting assengers but can result in reduced roductivity of the fleet; the challenge then is to determine here to add this slack so as to maximize the benefit to assengers ithout getting orse the netork oeration (Lan et al. (2006)). Every connection is characterized by the minimum time required to erform it. This time varies from airort to airort and it can also vary along the day. In this ay in itineraries ith more than one flight every assenger is mandated a minimum connection time (mct) for flight connections. Hoever this time ill not be alays enough to erform the connection. We assume that the number of disruted assengers deends on the available time to erform the flight connection. Once flights arrival (at) and dearture (dt) times are knon the available connection time (ct) is also knon. From airlines hiorical data disruted assengers number variability ith connection time may be knon. Assigning a atiical diribution to misconnected assengers the robability of getting misconnected assengers deending on connection time can be calculated. The exonential diribution has been chosen that is the number of misconnected assengers ill decrease exonentially ith the available excess connection time. If the available connection time is negative every connecting assenger ill be misconnected. The robability diribution of having misconnected assengers is as follos: ectt itt 0 ( ) itt e et c 0 f ect here itt deends on the itinerary connection characteriics and is chosen adjuing the robability diribution to hiorical data; it is suosed that once the connection characteriics are knon the assigned gates ill be robably knon due to hiorical availability. ect reresents the available excess connection time ( ect ct mct ). 0 t is a location arameter to fit the diribution. In this ay given the available excess connection time ( ect ) the robability of having misconnected assengers ( rob ectt itt 0 itt e ect 0 rob itt ) is: Once misconnected assengers are knon they mu be removed from the remaining flights of the itinerary so extra caacity arises in those flights making ossible to accommodate other assengers in it in case of disruted assengers. 5. Integrated Robu Airline Scheduling Model In the roosed model the entire demand satisfaction is not enforced. The neglected demand is enalized in the objective function. Passengers transfer ossibility is considered that is for every assenger itinerary the ossibility of intermediate os in the flight are taken into account. Itineraries comosed of u to to flight legs are considered. In order to avoid full flights an average load factor is introduced. These full flights may occur because cometing airlines are not considered in the model formulation. To overcome ith this deficiency the average load factor ill determine the maximum attainable assenger demand in each flight leg.

5 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) We suose that the schedule ill be eriodic that is the schedule ill reeat after the lanning eriod ends. For this urose e mu take care about airlanes location at the end of the lanning eriod. Its location mu be the necessary one to reeat the schedule. The folloing notation is introduced to exlain the Integrated Robu Airline Scheduling Model (IRASM): Sets: P ( ): fleet tyes set. Tt (): eriods set. Ss (): sections set. Each section is defined by an origin and a deination. W ( ): markets set. Markets are defined by the origin deination and the dearture time ( o d at ). Kk ( ): airorts set. Ii (): itineraries set. I2() i Ii (): itineraries set comosed of more than one section. I : itineraries set attending market. W i : markets set attended by itinerary i. T : eriods set for each market. I 1 s : itineraries set containing section s as fir section. I 2 s : itineraries set containing section s as second section. AS k : sections set arriving at airort k. DS k : sections set dearting from airort k. TS T( t) { t / t t mct } it it si1s itineraries ith more than one flight leg. CT : count time. t Parameters: : feasible dearture time set for the second flight leg in c : oerating co in section inance () ith fleet tye. q : assenger caacity in each fleet tye. b t: 1 if flight leg () is flying during time eriod t. N : fleet size for each fleet tye. t t : initial and final eriods in the lanning eriod. i f itt : assengers dissatisfaction due to transshiments times in itinerary (it) ith more than one section dearting the second one during t. dc : co er disruted assenger from market. dc i : co er disruted assenger in itinerary i due to lack of time to erform transshiments.

6 1046 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) qa kt : maximum airlane arrival caacity of airort k during each time eriod t. qd kt : maximum airlane dearture caacity of airort k during each time eriod t. q : maximum airlane caacity in each section s and eriod time t. s : minimum searation time beteen to consecutive deartures of section inances s (in eriods). d : assenger unconrained demand for each market. : section inance () tri time. We include the section tri time duration deendent on dearture time; this is due i.e. to congeed airorts or eather conditions hich may obey to slo don the airlane. mct : minimum connection time for each itinerary i dearting during time eriod t. it rob itt : likelihood of assengers from itinerary i dearting during t being misconnected in their flight connection ith the second flight dearting during t. alf : average load factor for each fleet tye. Variables: z : =1 if section s dearts during t eriod ith fleet tye; 0 otherise. yt kt : integer variable. It reresents grounded material of tye at airort k and during time eriod t. h : integer variable. Passengers in itinerary i and market dearting during t eriod. it sh : integer variable. Passengers using itinerary i I2( i) itt dearting during eriod time t and the second section during eriod time t. This auxiliary variable reresents assengers using a flight leg after a transfer. d : integer variable. Disruted assengers from market Objective Function Min z c z dc d dc rob sh sh i itt itt itt itt ssttp W ii2 ttt ii2 ttt The objective (z) function (1) accounts for oerator and assengers cos. The fir term reresents oerating cos the second one incurred cos due to disruted assengers that is sill cos and the folloing one cos due to lack of time to erform transshiments. These cos are considered as oerators cos. The la term in the objective function reresents assengers dissatisfaction cos ith flight connections. Oerating cos are the cos the comany incurs due to the oeration of flight legs. We calculate these cos as the roduct of the number of block hours in each section s dearting during eriod t ith a determined fleet tye cbh ). We comute these cos in (2). NBH ) by the co er block hour for each fleet tye ( ( Disruted assengers are assengers that the comany does not attend due to lack of caacity or high dissatisfaction cos. We can consider them as sill cos that is lo revenue. These cos can be comuted as the diance the assengers ould go if they ere attended by the sill co. (1) c cbh NBH (2) E m rob sh (3) itt itt itt

7 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) K t t itt sň I1s (4) We minimize the number of disruted assengers due to misconnections. In this ay e introduce the robuness criterion defined above. The number of exected misconnected assengers ( Em itt ) is calculated in (3). Passengers cos (4) are comosed of the dissatisfaction. This term measures hether the itinerary is comosed of more than one flight leg or not. The conant K may be calibrated through assengers surveys and transform the time units into cos units. The objective function is subject to the folloing grous of conraints: 5.2. Passengers Conraints iitt hit dd W (5) h sh i I 2 t T (6) it itt W i t TSit h 1 rob sh alf q z s S t T (7) it itt it t ii1 sw i ii 2 T P Conraints (5) ensure the assenger demand allocation through available itineraries in the netork; they account for disruted assengers. Grou of conraints (6) ensures that assengers in to sections itineraries remain in their tri during the second section; they also consider the average necessary time for erforming transshiment. Conraints (7) ensure that there are enough active sections or flight legs to satisfy the assengers flo; misconnected assengers are removed from the flight leg Flight Leg and Airort Conraints P z 1 sst T (8) t TP tt tt t s sask b z q sst T (9) t P tt z 1 sst T (10) t t sdsk P P z qa k K t T (11) kt z qdkt k K t T (12) Conraints (8) ensure that only one unique fleet tye can be assigned to each flight leg. Conraints (9) are section caacity conraints; they ensure that the number of aircraft in a section at each eriod is loer than a maximum number; this caacity may deend on air navigation syems and regulations. Grou of conraints (10) ensures that the same flight leg does not deart until a secified time has been sent; in this ay cometition beteen flights from the same airline is avoided. Conraints (11)-(12) are airort caacity conraints that try to

8 1048 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) sare the deartures and arrivals at airorts at each eriod; this is mandated by the available slots in the airort to land or take off. Deending on the time these slots may vary in cos. These cos are included in the oerating cos Fleet Flo Caacity and Symmetry Conraints yt 1 z yt z k K t T P (13) kt kt sask tt sdsk tt b z yt N t CT P (14) t kt t t T kk ytkt ytkt k K P (15) i f Block of conraints (13) are the flo conservation equations for each airort and fleet tye. Conraints (14) are the fleet caacity conraints; e mu count the necessary aircraft to erform the schedule and comare it to the available ones. Conraints (15) ate that the netork mu be symmetric in order to reeat the same schedule once the lanning eriod has ended Variable Dominion z 01 sst T P (16) ytkt k K t T P (17) d W (18) h i It T W (19) it shitt i I2 t t T (20) Conraints (16)-(20) define the variable dominion. We have defined assenger variables as integer ositive variables. Hoever as the demand number is an average value these variables dominion may be relaxed to ositive yt variables. In addition variables z variables kt being grounded airlanes at airorts are defined as a sum of binary ; in this ay its definition dominion might be relaxed to ositive variables as ell. 6. Comutational Exeriments As a roof of the model e have done some comutational exerience. We have imlemented a simlified version of IBERIAs air netork (Figure 1): the Sanish netork. It is a ure hub-and-soke netork ith 23 different airorts. There are three different fleet tyes available for this udy case: 23 A-319 ith 141 seats each 35 A-320 ith 171 seats each and 19 A-321 ith 200 each. Some of these data are ublicly available in IBERIA s eb age (IBERIA). The lanning eriod e have considered is 24 hours. We require the lanning to be eriodic that is the fleet diribution mu be equal at the beginning and the ending of the lanning eriod. The model size for this udy case is shon in Table 1. In this case time has been discretized into eriods of 15 minutes. We have considered every otential flight leg beteen each soke and the hub. Our runs have been erformed on a Personal Comuter ith an Intel Core2 Quad Q9950 CPU at 2.83 GHz and 8 GB of RAM running under Windos 7 64Bit and our rograms have been imlemented in GAMS 23.2/Clex 12.

9 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) Figure 1: Hub-and-Soke netork As it as exlained above robuness is achieved through assengers that mu erform flight connections. In order to demonrate that a more robu schedule is obtained using the roosed aroach a comarison is made ith a non-robu Integrated Airline Scheduling Model (IASM). The IASM is the same model exlained above but removing robuness asects that is the objective functions term in (1) enalizing misconnected assengers and the terms in conraints (7) accounting for misconnected assengers in flight legs caacity. Table 1. IRASM size: number discrete and continuous variables conraints and non-zero elements Discrete Variables Continuous Variables Conraints Non-zeroes Table 2. Comarison beteen non-robu (IASM) and robu (IRASM) models IASM IRASM Misconnected Passengers (%) Neglected Passengers (%) Dissatisfaction Cos (x 10 3 ) Oerating Cos (x 10 3 ) Objective Function (x 10 3 ) In Table 2 a summary of the results is resented. In the fir ro misconnected assengers are comared for the non-robu and robu cases. The ercentage of exected misconnected assengers is shon; this ercentage is calculated ith resect to the total number of attended assengers. For the robu case the ercentage is sensitively reduced. In the second ro the ercentage of neglected assengers is resented for both cases. Robuness is achieved through the reduction in the exected number of misconnected assengers. Hoever this reduction is not for free it has a rice: the robuness rice. Passengers dissatisfaction cos are ritten in the third ro. One mu note that these cos are greater in the robu case (IRASM) that is in the robu aroach here misconnected assengers number has been reduced assengers dissatisfaction has been increased. This is due to the fact that ameliorating the number of misconnected assengers means increasing connection time. Therefore suosing that assengers dissatisfaction cos increase linearly ith connection time e have that a trade-off mu be found beteen these cos. Oerating cos are shon in the fourth ro. Oerating cos are similar for both cases. In the la ro the objective function value is shon. In the robu case (IRASM) objective functions value is greater than the no robu one (IASM). The comutational times in seconds for IASM and IRASM in the roosed netork ere and resectively. 7. Conclusions A ne robu aroach has been roosed to solve the airline scheduling roblem here schedule design and fleet assignment roblems are jointly solved. In addition assengers flos are obtained through different itineraries in the netork. Robuness has been introduced through itineraries ith more than one flight leg. When an intermediate o

10 1050 Luis Cadarso and Ángel Marín / Procedia Social and Behavioral Sciences 20 (2011) mu be erformed assengers need some undetermined time to accomlish it. This undetermined time is catured through atiical diribution and it is introduced into the model to reresent exected misconnected assengers. In this ay the exected cos that the oerator ould incur due to misconnected assengers are reduced. The model has been teed in a IBERIAs simlified netork. Comutational results sho ho robuness may be achieved. Hoever this robuness has a rice. The robu aroach has been comared ith a no robu aroach shoing the rice of the achieved robuness. Further research may include the introduction of cometition effects in the model formulation. In this ay the obtained market share ould deend on cometing airlines. Acknoledgments This research as suorted by roject grant TRA C02-01 by the "Minierio de Ciencia e Innovación Sain". The authors also ish to thank to the anonymous revieers for their comments on this aer. References Armaco A. Barnhart C. and Ware K. (2002). Comosite Variable Formulations for Exress Shiment Service Netork Design. Transortation Science 36:1-20. Cadarso L. and Marín A. (2010). Robu Airline Scheduling ID1240. General Proceedings of the XVI Pan-American Conference of Traffic and Transortation Engineering and Logiics (PANAM 2010). Esinoza D. Garcia R. Goycoolea M. Nemhauser G. L. Savelsbergh M. W. P. (2008). Per-Seat On-Demand Air Transortation Part I: Problem Descrition and an Integer Multicommodity Flo Model. Transortation Science 42(3): Garcia FA. (2004). Integrated Otimization Model for Airline Schedule Design: Profit Maximization and Issues of Access for Small Markets. Deartment of Civil and Environmental Engineering and the Engineering Syems Massachusetts Initute of Technology Cambridge MA. IBERIA. htt://gruo.iberia.es Jiang H. and Barnhart C. (2009). Dynamic Airline Scheduling. Transortation Science 43: Kim D. Barnhart C. (2007). Flight Schedule Design for a Charter Airline. Comuters & Oerations Research 34: Lan S. Clarke J.P. and Barnhart C. (2006). Planning for Robu Airline Oerations: Otimizing Aircraft Routings and Flight Dearture Times to Minimize Passenger Disrutions. Transortation Science 40: Lohateanont M. and Barnhart C. (2004). Airline Schedule Planning: Integrated Models and Algorithms for Schedule Design and Fleet Assignment. Transortation Science 38: Soumis F. Ferland JA. and Rousseau JM. (1980). A Model for Large Scale Aircraft Routing and Scheduling Problems. Transortation Research 14: