Modelling Airline Network Routing and Scheduling under Airport Capacity Constraints

Transcription

1 Modeing Airine Network Routing and Scheduing under Airort Caacity Constraints Antony D. Evans *, Andreas Schäfer, Lynnette Dray Institute for Aviation and the Environment, University of Cambridge, Cambridge, UK A fight routing and scheduing mode is under deveoment that redicts airine routing and scheduing under airort caacity constraints. It consists of severa comonents describing different asects of the air transort system, incuding assenger demand, airine cometition, fight deay, and airine cost. These modes are integrated into a fight routing and scheduing mode in which an objective function is defined to maximize airine system rofit within a routing network, subject to constraints. This framework aows the reationshis between fares, assenger demand, infrastructure caacity constraints, fight deays, fight frequencies, and routing network to be simuated. In this aer the integrated fight routing and scheduing mode is first aied to a series of sime theoretica routing networks to iustrate its caabiities. With increasing airort caacity constraints the resuts show an increase in average fares, a decrease in O-D assenger demand, and a shift in fight routing away from the most constrained airorts. The mode is then aied to a network of airorts in the United States with 005 ouation, income and airort caacity inuts. With further deveoment the mode is to be aied to forecasting air traffic system growth, incuding network and schedue changes resuting from increasing deays, in the Aviation Integrated Modeing (AIM) roject under deveoment at the University of Cambridge. I. Introduction ORLDWIDE demand for air trave has shown significant growth over the ast five decades. Between 1960 W and 005 wordwide schedued assenger air trave grew from 109 biion assenger-km traveed to 3.7 triion an average growth rate of over 8% er year [1,]. Forecasts for future growth are aso high the Intergovernmenta Pane on Cimate Change (IPCC) forecast a growth rate between 1990 and 015 of 5% er year [1], which corresonds to that of both the Airbus Goba Market Forecast from 007 to 06 [3] and the Boeing Current Market Outook from 006 to 06 [4]. By 050 conservative estimates redict a % growth in assenger kiometres traveed over 005 eves [5], whie more aggressive estimates redict an increase of an order of magnitude [6]. Associated with such growth in demand for air trave is a growth in air traffic (number of aircraft movements) to serve that demand, as modeed by Hancox and Lowe [7], Bhadra et a. [8,9], and Reynods et a. [10]. This growth in air traffic is exected to roduce a significant environmenta imact, as reorted by the IPCC [1] and Cairns et a. [11], incuding air quaity and noise imacts, and goba cimate change. Growth in air traffic is aready constrained by environmenta restrictions (articuary noise). An emerging constraint is air traffic system caacity, given widesread oca community resistance and environmenta restrictions to airort caacity exansion. Airort and airsace caacity aready constrain fight oerations at many major airorts in the United States and Euroe. In the United States, average arriva deays at 71 airorts were greater than 10 minutes in 005, with Newark Liberty Internationa Airort exeriencing an average arriva deay of 3 min [1]. In Euroe, average arriva deays at 4 airorts were greater than 10 minutes in 006, with Istanbu Internationa Ataturk Airort and London Luton Airort exeriencing average arriva deays of 19 and 18 minutes resectivey [13]. As shown by Reynods et a. [10], if airort caacity in the United States ony grows as described in the US Federa Aviation Administration (FAA) Oerationa Evoution Pan [14], and airines continue to increase air traffic to match the rojected growth in demand, average arriva deays for the 50 busiest airorts in the system woud be * PhD Student, Deartment of Architecture, Scrooe Terrace, Cambridge, CB 1PX, UK. AIAA Member. University Lecturer, Deartment of Architecture, Scrooe Terrace, Cambridge, CB 1PX, UK. Post-doctora Researcher, Institute for Aviation and the Environment, Scrooe Terrace, Cambridge, CB 1PX, UK. AIAA Member.

2 over hours er fight. However, as suggested by Kostiuk et a. [15] and Reynods et a. [10], these deays are unikey to occur in reaity as airines woud adjust their oerations, incuding minimising the negative imacts of the increasing costs associated with deays and the reduction in assenger demand. The airine resonse to deays woud ater the structure of future air traffic growth, as airines may use differenty sized aircraft, and avoid the most congested airorts and routes. In order to generate a ausibe forecast of air traffic growth in a caacity constrained system it is therefore essentia to quantify how assengers and airines are ikey to resond to caacity constraints. Airine resonses to deay incude avoiding congested hubs and gateways (dearture routes), using secondary airorts, moving fights to off-eak times, broadening the range of dearture times, and reducing fight frequency whie increasing aircraft size [15]. The effects of each of these resonses can be examined using a scenario-based aroach as imemented by Long et a. [16], where resonses are aied exogenousy and their effects examined. However, assenger and airine resonses are highy interreated, with airines resonding to changes in assenger demand by adjusting fares and schedues. In a highy cometitive market such as the airine industry in the United States and Euroe, airines are aso highy constrained in how they can resond without osing market share (often eading to ower rices and higher fight frequencies than is otima, as demonstrated by Schier [17] and Carsson [18]). Airines do, however, have the fexibiity to adjust their routing networks, and thus change routes aong which assengers are fown from their true origin to their utimate destination. This aer describes an integrated fight routing and scheduing mode to redict changes in assenger demand, fares, fight frequencies, and airine routing under airort caacity constraints. Other airine resonses to deays, incuding moving fights to off-eak times and broadening the range of dearture times oerated are not examined in this aer. The modeing aroach is described in detai in Section II, foowed by its aication to a series of sime theoretica networks of airorts in Section III. These incude: a sime hub and soke network aowing anaysis of the effect of caacity constraints on hub and soke routing; a network with two hubs airorts aowing anaysis of the effect of caacity constraints on the distribution of traffic between hubs; and a network incuding a sime muti-airort system aowing anaysis of the effect of caacity constraints on the distribution of traffic within a muti-airort system. Section IV describes the aication of the mode to a network of airorts in the United States with 005 ouation, income and airort caacity inuts, aowing the mode s redictive caabiity to be identified and discussed. Concusions and recommendations for further deveoments are discussed in Section V. II. Modeing Aroach The fight routing and scheduing mode deveoed integrates a number of sub-modes, incuding a assenger demand mode, an airine cometition mode, a fight deay mode, and an airine cost mode. By integrating these sub-modes the imact of fight deays on fare, fight frequency, and routing network can be examined. The integration of the sub-modes within the fight routing and scheduing mode is resented in Figure 1 beow, and is described in detai foowing a descrition of each sub-mode in the foowing sections. Segment Fight Frequency 0 C. Fight Deay Mode Average Deay Trave Time Cacuator Trave Time A. Passenger Demand Mode B. Airine Cometition Mode D. Airine Cost Mode Oerating Cost O-D Demand Fare Fight Frequency Constraint E. Network Otimisation Segment Fight Frequency Convergence? Yes No Figure 1. Mode integration

3 A. Modeing Passenger Demand True origin/utimate destination (O-D) assenger demand between cities i and j is modeed using a sime oneequation gravity-tye mode, as aied by Reynods et a. [10]: α γ δa ε = ( + θ + ) τ ij Bij Dij ( IiI j ) ( Pi Pj ) e e Farei, j 1 T i, j Ct (1) where I is the greater metrooitan area er caita income; P is the greater metrooitan area or equivaent ouation; A and B are binary variabes indicating whether one or both cities in the air have quaities which might increase visitor numbers (e.g. a major tourist destination or caita city); Fare is assenger airfare between the cities averaged over a itineraries; θ 1 is the assenger vaue of trave time; T is the trave time between the cities averaged over a itineraries; C t covers a other assenger costs associated with the tri, such as getting to and from the airort, and is the same for a airorts; and the exonents give the easticity of demand to each of the exanatory variabes (i.e. % change in demand resuting from a % change in each exanatory variabe). The exression in brackets reresents the generaized cost to a assenger of air trave between the cities, and it is through this exression that it is ossibe to incude the demand-reducing effect of increased fares as we as that of increased trave time. Using demand data for the United States in 000 [19,0,1], the coefficients (exonents) in equation 1 are estimated searatey for short-hau, medium-hau and ong-hau journeys. A estimated coefficients are significant at the 95% confidence eve, with R vaues ranging from 0.46 to 0.83 (the ower vaue is for short-hau routes for which the ack of cometing modes in this mode formuation has a strong effect). The mode described in equation 1 does not cature some assenger demand effects that are significant in some word regions, such as assenger mode choice. This simification is, however, necessary for the comutationa efficiency required of the mode for integration in the framework resented in Figure 1. B. Modeing Airine Cometition As described in Section II.A above, assenger demand is a function of fare and trave time. Fares are defined according to airine economics. In a cometitive market airines tyicay comete using fare and fight frequency. This means that in order to increase their market share airines may reduce fares or increase fight frequencies on a given O-D market. However, most airines wi resond in a simiar way, subject to their cost constraints. Such rice reductions and increases in fight frequency wi continue as ong as airines are sti making a rofit. Airine microeconomics and game theory can be aied to estimate the equiibrium to which average fares and fight frequencies sette. Schier [17] and Carsson [18] describe a formuation of the equiibrium average fare and fight frequency as a function of assenger vaue of time, airine costs and assenger demand, derived by defining the fight schedue as an address (or satia) mode, and soving a two stage game to maximise airine rofit. In the first stage of the game airines simutaneousy choose fight frequencies, and in the second stage, after having observed the other airines chosen frequencies, the airines simutaneousy choose fares. This formuation is resented in equations and 3 beow. Equation 3 ony aies to markets served by more than two airines. θ Fare = C + () nx nx θ = D n C n f where Fare reresents the equiibrium average fare of airine for the O-D city-air market examined; C reresents the margina assenger cost of airine ; θ reresents the assenger vaue of schedue deay ; n reresents the number of airines serving the market; x reresents equiibrium fight frequency of airine on the market, D reresents O-D assenger demand on the market; and C f reresents average cost er fight of airine. The formuation described by Schier [17] and Carsson [18] was modified to mode variabe assenger demand as described by equation 1. The resuts for average fare and fight frequency are resented in equations 4 and 5 beow as a function of the same arameters described for equations 1, and 3. K refers to the exression before the generaized cost term in equation 1: ( I I α γ δa ij εbij ) ( P P ) e e. i j i j (3) Schedue deay refers to the time difference between when a assenger wants to fy, and when the fight which he/she chooses to fy on dearts. In this aer assenger vaue of schedue deay is assumed to be equa to assenger vaue of trave time. 3

4 Fare K = ( τθ θ + θ Tnx C nx C nx ) 1 1 τ ( Fare + θ T + C ) ( Fare C ) x Kτ nx t t ( τθ1 + θ1 θtnx Ctnx + C nx ) + nx ( τc θ + θ θ T + θ C + θ TC nx + C C nx ) t nx 1 ( ) ( ) τ Fare T Fare C Fare + θt + C t + θ + x x nx = ( ) ( ) ( ) τ Fare K τ θ K Fare + θ T + C + Fare + θ T + C Fare C ( ) C f + t x nx θ 1 t nx t (4) (5) Equation 5 incudes the derivative of fare with resect to fight frequency ( Fare x ), which can be derived from equation 4, and the derivative of average trave time with resect to fight frequency ( T x ), which was found to be sma and is therefore ignored. Equation 5 cannot be soved anayticay, but can be soved iterativey. The formuation described above does not mode some imortant characteristics of airine and assenger behaviour. Most notaby, the mode maximises rofit to sove for a singe fare, and therefore does not account for rice discrimination (yied management), which is emoyed extensivey in the airine industry. Instead the mode estimates average fares and fight frequencies by O-D market. The mode aso does not distinguish between different assenger routings on the same O-D market. Fares, assenger vaue of time, and cost easticities may, however, differ quite significanty for different routes in the same market articuary between direct fights and connecting fights. Finay, the address mode aied aso does not cature assenger references to fy at certain times of day (articuary the eary morning and evening), and therefore ignores the increased demand at these times. These simifications are, however, necessary for the comutationa efficiency required of the mode for integration in the framework resented in Figure 1. Equations 4 and 5 are aied to estimate equiibrium average fares and fight frequencies for O-D city airs. However, fight deays resuting from an increase in fight frequency imact average cost er fight through an increase in bock time, and O-D assenger demand through an increase in trave time. Modeing of fight deay is therefore aso essentia to estimate the effects of airort caacity constraints on assenger demand, fares, and fight frequencies. C. Modeing Fight Deay The imact of airort caacity constraints on airine routing and scheduing is modeed using a raid airort deay mode simiar to that emoyed by Reynods et a. [10], and described in detai by Evans []. In this mode fight deays, both on the ground and in the air, are estimated as a function of fight frequencies and airort caacity constraints, and are added to gate dearture deays due to mechanica faiures and ate arrivas, which are assumed to remain at current eves (assuming schedue adding increases to maintain schedue reiabiity). Deays due to airort caacity constraints are estimated using queuing theory, aying the cumuative diagram aroach and cassica steady state simifications described by de Neufvie and Odoni [3]. Runway dearture deays are distributed between the taxiway and the gate according to a taxi-out threshod cacuated for each airort from historica deay data. Simiary, deays due to destination airort caacity constraints are distributed between the air and ground according to an airborne hoding threshod cacuated for each airort from historica deay data, and above which deay is assumed to be roagated ustream to the dearture gate. The deays estimated by this mode increase assenger trave time (T in equation 1), and increase airine cost er fight C f and cost er assenger C in equations 4 and 5. D. Modeing Airine Costs Airine costs modeed incude aircraft oerating costs, aircraft servicing costs, traffic servicing costs, assenger servicing costs, reservation and saes costs, and other indirect and system overhead costs, as described by Beobaba [4]. Aircraft oerating costs incude fue and oi costs, crew costs, maintenance costs, aircraft renta, dereciation and amortization costs, and enroute airsace charges. With the excetion of fue and oi costs, these costs can be 4

5 inut directy from US DOT Form41 data [1] for modeing in the United States. Fue costs are cacuated indeendenty as a function of fue rice and aircraft fue burn in each fight hase: ground ide, taxi, take-off, cimb-out, cruise, descent, aroach and anding. Fue burn rates are estimated according to the EUROCONTROL Base of Aircraft Data (BADA) [5] and the ICAO Aircraft Engine Emissions Databank [6]. A other costs are inut directy from DOT Form41 data [1] for modeing in the United States. These incude aircraft servicing costs, covering handing aircraft on the ground and anding fees; traffic servicing costs, covering the rocessing of assengers, baggage and cargo at airorts; assenger servicing costs, covering meas, fight attendants and in-fight services; reservation and saes costs, covering airine reservations and ticket offices, incuding trave agency commissions; and other indirect and system overhead costs, covering advertising and ubicity exenses and genera and administrative exenses. Airine costs are modeed er fight and er assenger, as required by the airine cometition modeed described in Section II.B. Costs er fight incude a aircraft oerating costs and aircraft servicing costs, with the excetion of the roortion of the fue burn that can be attributed to assengers directy. This fue burn, aong with traffic servicing costs, assenger servicing costs, reservations and saes costs, and other indirect and system overhead costs are modeed er assenger. E. Modeing Fight Routing and Scheduing The assenger demand mode, the airine cometition mode, and the fight deay mode are integrated into the airine routing and scheduing mode, which is based on maximisation of system rofit to otimise fight frequencies and routing using arge scae inear rogramming methods, simiar to the aroach used by Harsha [7]. The airine revenue term is based on assengers fown by itinerary and average fares on those itineraries. The cost terms are based on airine cost er fight mutiied by the fight frequency on a given fight segment, airine cost er assenger mutiied by assengers fown by itinerary, and airine si cost mutiied by tota si **. This objective function is resented in equation 6 beow. As described above, assenger demand is a function of fare and trave time, and is estimated by the assenger demand mode described in Section II.A. Average fares are a function of airine costs and fight frequency, and are estimated by the airine cometition mode described in Section II.B. Airine costs are a function of aircraft tyes fown and bock time redicted by the fight deay mode, as estimated by the airine cost mode described in Section II.D. Fight bock times, and consequenty assenger trave time, is a function of fight deays, which are estimated by the fight deay mode described in Section II.C. max Farei, j Pi, j C f n m n k m, n xm, n, k C i P j i, j Sii, j SiCosti, j (6),,, i, j Itini, j m, n, k i, j Pi, j i, j where Fare i, j reresents the average fare between O-D city air i and j; P i,j reresents assenger demand between O-D city air i and j, on itinerary ; C f m,n,k reresents average cost er fight on the fight segment between airorts m and n, for aircraft tye k; n m,n reresents the number of airines oerating between airorts m and n, and x m,n,k reresents average number of fights er day on the fight segment between airorts m and n, using aircraft tye k, over a airines oerating on the route; C i,j reresents average cost er assenger between O-D city air i and j; Si i,j resents tota sied assengers between O-D city air i and j; and SiCost i,j is the cost to the airine er sied assenger between O-D city air i and j. It is assumed that si cost er assenger is equa to average fare between the O-D city air. Cities i and j are served by one or more airorts m and n resectivey. The objective function is constrained by a system of inear equations describing airine routing and scheduing requirements and imitations, incuding tota O-D demand D i,j between O-D city air i and j estimated by the demand mode (equation 1), average O-D fares Fare i, j and tota O-D fight frequencies (n i,j x i,j ) between O-D city air i and j estimated by the airine cometition mode. A seat constraint aso imits the number of assengers served on each fight segment to be ess than or equa to the number of seats avaiabe; a simified baance constraint imits the number of fights of each aircraft tye dearting an airort in a day to equa the number of fights of that aircraft tye arriving, and vice versa; and a feet constraint imits the tota hours oerated by each aircraft tye to be ess than or equa to that avaiabe in the existing feet. The feet constraint may be reaxed in order to estimate what feet requirements exist, or may be integrated with an aircraft stock mode. In this aer the feet constraint is not aied. It is aso assumed that airines have the otion to route assengers directy between their origin and destination, or through a singe hub. Mutie connections are not modeed. ** Si is the tota number of assengers who want to fy but cannot obtain a reservation due to insufficient caacity rovided by the airines. 5

6 Because the modes integrated into the fight routing and scheduing mode are not a inear, making the objective function and constraints non-inear, the fight routing and scheduing mode is soved by iteration in the foowing variabes: O-D assenger demand D i,j, average O-D fares Fare i, j, fight frequencies x n,m, average fight arriva deays, and average cost er fight C f i,j and er assenger C i,j. The iteration rocedure is resented in Figure 1. As described by the figure, given an initia estimate of segment fight frequencies x n,m,0, average fight deays at each airort are estimated using the fight deay mode described in Section II.C. This aows average trave time and average costs er fight and er assenger to be estimated using a sime trave time cacuator and the airine cost mode described in Section II.D. O-D assenger demand D i,j, average O-D fares Fare i, j, and the minimum fight frequency required for cometition can then be estimated using the demand mode and airine cometition mode described in Sections II.A and II.B. The oututs of these modes, and the costs er fight and er assenger estimated by the airine cost mode aow the network otimization to be run, soving the segment fight frequencies required to maximise the rofit function described by equation 6. These fight frequencies can then be used to re-estimate average fight deay. The iteration is reeated unti the system fight frequency converges to within 1 fight er day. The converged resuts yied modeed assenger demand, fight frequency, average fares, and average deays for the system modeed. Because of the simifications in each of the sub-modes integrated into the fight routing and scheduing mode, and the ack of any constraints secific to any one airine, this mode is not caabe of redicting or advising any one airine s resonse to caacity constraints. Instead, the mode is intended to redict genera system resonse we into the future as far as 030 given aternative airort caacity growth scenarios. III. Aication to Sime Theoretica Networks The fight routing and scheduing mode described in Section II was aied to a series of sime routing networks to iustrate its caabiities and faciitate discussion of the mode functionaity. Three sime routing networks are modeed: Three soke airorts equidistant from a centra hub airort, as iustrated in Figure (a). This network aows anaysis of the effect of caacity constraints at a hub airort on the rofit maximising routing through the hub, causing a shift away from a ure hub and soke network. Three soke airorts surrounding two hub airorts, as iustrated in Figure (b). This network aows anaysis of the effect of caacity constraints at a hub airort on the distribution of traffic between that hub and an aternative hub. Four soke airorts, two of which serve the same market (forming a muti-airort system), equidistant from a centra hub airort, as iustrated in Figure (c). This network aows anaysis of the effect of caacity constraints at one airort within a muti-airort system on the distribution of traffic between the airorts in the system. The aication and resuts of the aication of the mode to each of these theoretica networks is described in detai beow. (a) (b) (c) Figure. Sime theoretica networks modeed a) a sime hub and soke routing network; b) a sime hub and soke routing network with two hub airorts; and c) a sime hub and soke routing network with one soke served by a muti-airort system of two airorts. A. Sime Hub and Soke Routing Network The inut data to the mode is resented in Tabe 1. These inuts are hyothetica, but are based on tyica vaues for the air transort system in the United States in 005. Non-network traffic accounts for a fights from airorts outside the network modeed. Revenues and costs from this non-network traffic are not incuded in the 6

7 rofit maximization within the fight routing and scheduing mode, but the traffic is modeed at each airort to ensure that reaistic deays are simuated. In a rea network, however, a fights woud imact airine rofit. Hyothetica ouations and incomes were seected to generate the unconstrained demand between the cities resented in the tabe. Three aircraft tyes are modeed a sma aircraft tye (aying aircraft erformance data for a Boeing B ), a medium aircraft tye (aying aircraft erformance data for a Boeing ), and a arge aircraft tye (aying aircraft erformance data for a Boeing ). Aircraft cost data is extracted directy from the DOT Form41 data [1] for 005. Passenger vaue of time and cost easticities are derived from Reynods et a. [10]. Average gate dearture deay due to mechanica faiures and ate arrivas is extracted from the FAA ASPM database [1] for the 50 busiest airorts in the US air traffic system. Tabe 1. Inut arameters for anaysis of sime hub and soke routing network. Inut Parameters Non-network traffic Hub Airort,000 [fts/day] Soke Airort 500 Unconstrained demand Hub City Soke City 00,000 [ax/yr] Soke City Soke City 100,000 Fue rice [005 US$/kg] 1.73 Sma,13 Aircraft oerating cost (exc. fue) [005 US$/hr] Medium,99 Large 4,597 Voume reated cost [005 US$/ax] 1.39 Passenger vaue of schedue deay [005 US$/hr] 34.4 Passenger vaue of trave time [005 US$/hr] 34.4 Short hau (< 500 mies) Passenger Cost Easticity Medium hau (500 to 1000 mies) Long hau (> 1000 mies) -.07 Extra cost term C t [005 US$] 50 Hub-soke stage ength [mies] 500 Number of airines serving each O-D market 5 Average gate dearture deay due to mechanicas and ate arrivas [min] 4 Average connection time [min] 40 The mode described in Section II was run aying the inuts described in Tabe 1. Three scenarios were soved: aying unconstrained airort caacities at a airorts; aying a medium caacity constraint at the hub airort; and aying a severe caacity constraint at the hub airort. Resuts are resented for each of the scenarios in Tabe and Figure 3 beow. As can be seen in Figure 3a, the routing network that maximises airine rofit in the unconstrained scenario is a ure hub and soke network, with 11 fights er day between the soke airorts and the hub, and no direct fights between the soke airorts. A the assenger demand between the soke cities connects through the hub airort. Because of this connection the cost er assenger between the soke cities is higher than from the hub to the soke, and therefore the average fare between the soke cities is higher than to the hub city ($113 versus $90). Deays are 4 minutes at both the hub and soke airorts, corresonding to the average gate dearture deay due to mechanicas and ate arrivas ony, as airort caacity is unconstrained. O-D assenger demand is greater from the soke cities to the hub than between the soke cities because of the inut ouation and income data. 7

8 Tabe. Resuts for sime hub and soke routing network under varying hub airort caacity. Unconstrained Hub Soke Soke Soke Medium Caacity Constraint at Hub Hub Soke Soke Soke Severe Caacity Constraint at Hub Hub Soke Soke Soke Airort Caacity [ac/hr] unconstrained unconstrained 100 unconstrained 85 unconstrained O-D Pax Demand [ax/yr] 50, ,000 Fight Frequency [fts/day] ,000 (-39,000) 85,000 (-15,000) 194,000 (-56,000) 14,000 (+4,000) Avg. Arriva Deay [min] Avg. Fare [005 US$] (-4) 131 (+41) 1 (+1) 131 (+18) 4 (-7) 169 (+79) (+) 91 (-) Hub Airort Soke Airort Fights Passenger O-D demand 100,000 ax/yr 85,000 ax/yr 14,000 ax/yr 11 fts/day 1 fts/day 7 fts/day fts/day 4 fts/day 50,000 ax/yr 11,000 ax/yr 194,000 ax/yr (a) (b) (c) Figure 3. Resuts for sime hub and soke routing network a) unconstrained airort caacities; b) medium caacity constraint at hub; c) severe caacity constraint at hub. As can be seen in Figure 3b, the routing network that maximises airine rofit under a medium caacity constraint at the hub airort shifts to a artia hub and soke network of 7 fights er day between the hub and soke airorts, but with a singe direct fight er day between the soke airorts. Considering firsty the hub to soke resuts, O-D assenger demand between the hub and soke cities reduces from the 50,000 assengers er annum redicted in the unconstrained scenario, to 11,000 in the caacity constrained scenario. This reduction in O-D assenger demand resuts from an increase in trave time caused by an average arriva deay of 106 minutes at the hub airort and an increase in average fare by $41 to $131. Average fare increases because of an increase in costs from the hub to the soke airort resuting from the increased fue burn associated with the ortion of the 106 minute average arriva deay incurred on active taxiways and in airborne hoding at the hub. Because of the reduced O-D assenger demand on the route, the number of aircraft oerated is reduced by 4 fights er day to 7 fights er day. Considering the resuts for the traffic between the soke airorts, O-D assenger demand between the soke cities reduces from the 100,000 assengers er annum redicted in the unconstrained scenario, to 85,000 in the caacity constrained scenario. This reduction in O-D assenger demand is for the same reasons as the reduction in demand between the hub and soke cities an increase in trave time caused by the average arriva deay of 106 minutes at the hub airort, and an increase in average fare by $18 to $131. The increase in trave time is sti exerienced by the majority of assengers traveing between the soke cities desite the introduction of a direct fight between the soke cities, because the majority of assengers sti connect through the hub airort, where the 106 minutes deay is incurred. Ony the 4 minutes of average deay due to mechanicas and ate arrivas is incurred at the soke airorts as they remain unconstrained. Average fare between the soke cities aso increases for the same reason as between the hub and soke cities athough the increased fue burn is incurred at the hub airort, fare is based on average costs er assenger, and the majority of assengers sti ass through the hub airort. However, it is rofitabe for the airines to oerate a singe fight er day on the direct routes between the soke airorts. 8

9 As can be seen in Figure 3c the routing network that maximises airine rofit under a severey caacity constrained hub airort shifts to a ure oint to oint network with 4 fights er day between the hub and soke airorts, and direct fights er day between the soke airorts. Technicay this woud require some of the 5 airines serving each market to withdraw, but this effect is not modeed. Again considering firsty the hub to soke resuts, O-D assenger demand between the hub and soke cities reduces further to 194,000 assengers er annum in the severey caacity constrained scenario. This effect is a more extreme case of that in the medium caacity constrained scenario, with a significant increase in trave time, caused by a massive average arriva deay of 17 minutes at the hub airort, and an increase in average fare to $169 $79 higher than that in the unconstrained scenario. Average fare increases because of the increased costs of the fue burn associated with the ortion of the 17 minute average arriva deay incurred on active taxiways and in airborne hoding at the hub. Because of the further reduced O-D assenger demand on the route, the number of aircraft oerated is more than haved to ony 4 fights er day. Considering the resuts for the traffic between the soke airorts, a significanty different resut to the medium caacity constrained scenario is observed. O-D assenger demand between the soke cities increases beyond the vaues redicted by the unconstrained scenario to 14,000 assengers er annum. This is because a assengers are fown directy between the soke airorts, with none connecting through the hub. None of the O-D assengers between the soke airorts therefore exerience the 17 minutes of deay at the hub to increase their trave time. Trave time is in fact reduced reative to even the unconstrained scenario, as the direct fight time between the soke airorts is ess that the combined fight time to and from the hub airort, and assengers do not exerience the connection time at the hub. Average fare is reduced to beow the vaues in the unconstrained scenario for the same reason there is no increase in fue burn incurred for assengers fying between the soke airorts, as a assengers fy on the direct fights between the soke airorts, which ony exerience the 4 minutes of average deay due to mechanicas and ate arrivas. fights er day are required to serve a the O-D demand between the soke cities. The sime theoretica network modeed in this exame shows how the routing network seected by an airine to maximise its rofits can change significanty as airorts become caacity constrained articuary a hub airort in this case. It shoud be noted, however, that the caacity constraint at the hub airort must be very severe (an average arriva deay of 17 minutes is we above any average arriva deay currenty exerienced in any airsace system gobay) for airines to shift to a fuy oint to oint network because of the costs associated with deays ony. Airines may shift to this kind of network for other reasons, however, such as increasing demand between soke cities. B. Distribution of Traffic between Hubs The second theoretica routing network modeed a system of three soke airorts surrounding two hub airorts iustrates the caabiity of the mode to distribute traffic between hub airorts with different caacity constraints. The mode takes the same inuts as the sime hub and soke routing network described above in Tabe 1. The hub airorts are ocated 300 mies aart and reative to the soke airorts as shown in Figure 4. Two scenarios were soved an unconstrained scenario in which both hub airorts are unconstrained; and a constrained scenario, in which one of the hub airorts is caacity constrained. Resuts for both scenarios are resented in Figure 4. Hub Airort 6 fts/day 6 fts/day Soke Airort 6 fts/day fts/day 1 ft/day 7 fts/day 5 fts/day 9 fts/day 9 fts/day 13 fts/day 4 fts/day 7 ft/day 7 fts/day 1 ft/day 5 fts/day (a) (b) Figure 4. Resuts for distribution of traffic between two hubs a) both of which are unconstrained; and b) one of which is caacity constrained. 9

10 In the resuts for the unconstrained scenario, resented in Figure 4a, the traffic between the soke airorts is distributed symmetricay between the two hub airorts in a ure hub and soke network. An equa number of fights are schedued to each hub when the soke airort is equidistant from both hub airorts (the to soke airort in Figure 4a), whie the majority of fights are schedued to the coser hub when one hub is coser than the other (the two ower soke airorts in Figure 4a). This atter effect is due to the ower costs associated with fying the shorter distance to the coser hub. In the resuts resented in Figure 4b, where one of the hubs (the right hub) is caacity constrained, whie the other hub remains unconstrained, there is a shift of traffic from the constrained hub to the unconstrained hub. Thus, even though the unconstrained hub is further from one of the soke airorts (the ower right soke airort), a the connecting traffic from this airort is fown through the unconstrained hub. This is because of the increased fue burn costs associated with the deays at the caacity constrained airort, and the increased trave time through the airort, which woud reduce assenger demand. There are sti direct fights from the soke airorts to the caacity constrained airort serving the O-D assenger demand between these cities. Another further effect of the caacity constraint at one hub is the introduction of a direct fight between two of the soke cities (the soke cities furthest from the unconstrained hub), aowing O-D traffic between these cities to by-ass both hubs. This effect is caused by the increased costs associated with trave between these two soke cities either the deayed route through the constrained hub, or the onger route through the unconstrained hub. These increased costs make a direct fight more attractive to the airine economicay. This exame iustrates the caabiity of the mode to mode airine decisions to distribute traffic between hubs, based on the deays exerienced at each hub. It is noted, however, that a number of factors that affect an airine s decision in seecting hubs for connecting fights are not catured in this mode, incuding existing airine resence at the hub airort and incentives rovided by the airort authority. The mode may however indicate at what deay eves it becomes economicay attractive for airines to switch oerations to aternative hubs. C. Distribution of Traffic in a Muti-Airort System The third and fina theoretica routing network modeed a sime hub and soke routing network with one soke served by a muti-airort system of two airorts iustrates the distribution of traffic between airorts in a muti-airort system with different caacity constraints. The mode takes the same inuts as the sime hub and soke routing network described above in Tabe 1. Three scenarios are comared an unconstrained scenario in which a soke airorts are unconstrained; a artiay constrained scenario in which one of the airorts in the mutiairort system is caacity constrained; and a second constrained scenario in which both airorts in the muti-airort system are caacity constrained, but to different degrees. Resuts for a three scenarios are resented in Figure 5. Hub Airort Soke Airort 5 fts/day 5 fts/day 11 fts/day fts/day 8 fts/day 10 fts/day 10 fts/day 11 fts/day 10 fts/day 11 fts/day 10 fts/day (a) (b) (c) Figure 5. Resuts for distribution of traffic in a muti-airort system of two airorts a) both of which are unconstrained; b) one of which is caacity constrained; and c) both of which are caacity constrained, but to different degrees. In the resuts for the unconstrained scenario, resented in Figure 5a, the traffic between the airorts in the mutiairort system is distributed equay between the two airorts, with the rest of the system forming a ure hub and soke network. Traffic is equay distributed because the costs at each airort are identica. In the resuts resented in Figure 5b, where one of the airorts in the muti-airort system is caacity constrained (the eft airort) whie the other remains unconstrained, a the fights from the hub airort are routed to the unconstrained airort. The constrained airort does not serve any traffic in the network. Instead the system forms a ure hub and soke network with the unconstrained airort in the muti-airort system forming a soke in the 10

11 system. A O-D assenger demand from the muti-airort city is routed through the unconstrained airort. This is again because of the increased fue burn costs associated with the deays at the caacity constrained airort, and the increased trave time through the airort, which woud reduce assenger demand. It is noted that because each airort has exogenousy secified non-network traffic, there are sti high deays at the constrained airort, even though no traffic oerates there in the network modeed. In the resuts resented in Figure 5c, where both of the airorts in the muti-airort system are caacity constrained, athough one (the eft airort) is sighty more constrained than the other, the fights from the hub airort to the muti-airort city are distributed between the two airorts. The more constrained airort receives ess traffic than the ess constrained airort because of the higher fue burn costs associated with deays, and the increased trave time through the airort, which woud reduce assenger demand. A the traffic is not routed to the ess constrained airort because, with this increase in traffic deays woud increase, increasing costs. The routing shown in Figure 5c is the otima distribution of fights between the airorts, yieding maximum rofits. This exame iustrates the caabiity of the mode to mode airine decisions to distribute traffic between airorts in a muti-airort system, based on the deays exerienced at each airort. It is noted that a number of factors not catured in this mode aso affect an airine s choice of airorts in a muti-airort system, such as roximity to urban areas, accessibiity, faciities, and existing airine resence at an airort. IV. Aication to Actua Network The fight routing and scheduing mode described in Section II was aso aied to a network of 16 airorts serving the 10 cities with greatest assenger demand in the United States in 005, aowing the mode s redictive caabiity to be identified. The mode inuts resented in Tabe 1 were aied, with the excetion of non-network traffic, unconstrained demand, stage engths, and average gate dearture deay due to mechanicas and ate arrivas, which were modeed according to 005 data from the DOT T100 database [8] and FAA ASPM database [1]. The mode was run and the converged resuts comared to actua 005 data [0,1,8]. The mode deviations from the observed vaues are resented in Tabe 3 beow. Tabe 3. Resuts for mode aication to actua network. Mean O-D ax demand by O-D market Tota O-D system ax demand Mean O-D fare by O-D market Tota system revenue Mean fight frequency by segment Tota system fight frequency Deviation from Observations 005 6% high (max 59%) 1% high 16% ow (max 93%) 11% ow 34% ow (max 760%) 61% ow As shown in Tabe 3, O-D assenger demand is over redicted by 6% on average er O-D market (with a maximum of 59%), and 1% for the whoe system. This resut is rimariy because of an under rediction of average fares by 16% on average er O-D market (with maximum of 93%), which is caused by an under rediction of airine costs. These resuts combine to under redict system revenue by 11%. Airine costs are under redicted because the mode does not cature a constraints on airine oerations. This incudes articuary constraints on airine seection of aircraft tyes, such as requirements to oerate the same aircraft tyes on mutie routes in order to rovide fexibiity to swa equiment or crews as needed, and to reduce maintenance costs (requiring maintenance faciities for ony a few aircraft tyes). These constraints significanty imit airine feet choice. This effect is aso the cause of the significant under rediction of fight frequency in Tabe Cities (and airorts) modeed incude: New York City (JFK, EWR and LGA), Chicago (ORD and MDW), Atanta (ATL), Washington DC (IAD, DCA), Los Angees (LAX), Daas/Fort Worth (DFW and DAL), Houston (IAH and HOU), Detroit (DTW), Phoenix (PHX), and Seatte (SEA). 11

12 3, by 34% on average er route (with a maximum of 760%) and by 61% for the system as a whoe, desite the over rediction of assenger demand. The under rediction of fight frequency is because the majority of aircraft seected by the mode are medium sized aircraft. This aircraft size cass offers ower costs er assenger kiometre than sma aircraft, and demand is high enough to oerate them at high oad factor. However, the majority of aircraft oerated in the 005 US feet are sma, such as the Boeing B737 series aircraft and Airbus A319/30/31 series aircraft. These aircraft are seected by airines because of their fexibiity to oerate on many different routes. Some fight frequency resuts show very high deviations from observed traffic. This is articuary the case on fight segments between airorts within muti-airort systems. This incudes the segment with maximum deviation (760%) which is between Daas/Fort Worth Internationa airort (DFW) and Newark Liberty Internationa airort (EWR), both of which are art of muti-airort systems. These high deviations refect the factors not catured in the mode which have a significant imact on airine choice of airorts in a muti-airort system, incuding articuary existing airine resence at an airort. DFW and EWR are estabished hubs for American and Continenta Airines resectivey. In the modeed resuts the majority of the demand between Daas/Fort Worth and New York City is instead routed through the other airorts in the resective muti-airort systems. The resuts resented indicate that other airine constraints, incuding secificay feet constraints and existing airine resence at airorts, need to be incuded in the mode for more accurate rediction of air traffic in the US air transort system. Further deveoment of the mode is therefore required. V. Concusions A mode is under deveoment that redicts airine routing and scheduing under airort caacity constraints by integrating modes of assenger demand, airine cometition, fight deay, aircraft cost and fight routing and scheduing. The mode is aied to a series of theoretica routing networks, and to a network of airorts in the United States with 005 ouation, income and airort caacity inuts. The aication of the mode to the theoretica routing networks iustrates the caabiity to simuate airine resonses to airort caacity constraints through adjusting their schedues and routing networks to maximise rofit in a cometitive environment. These resonses incude atering the traditiona hub-and-soke network structure, distributing traffic between aternative hubs and aternative airorts within a muti-airort system. The mode has aso been aied to an actua routing network of 10 cities and 16 airorts in the United States in 005. The mode is found to under redict fight frequencies as it seects arger aircraft than are tyicay oerated. This is because the mode does not cature the airine choice of aircraft to rovide fexibiity over routes and crews, and to reduce maintenance costs. The mode wi be further deveoed to more reaisticay simuate airine choice of aircraft tyes and of airorts within muti-airort systems, which wi imrove its forecasting of air traffic growth in the future. This wi aow its aication in the Aviation Integrated Modeing (AIM) roject under deveoment at the University of Cambridge to forecasting air traffic growth, incuding network and schedue changes resuting from caacity constraints. Acknowedgements This work was funded through the AIM grant from the UK Engineering and Physica Sciences Research Counci (EPSRC) and the Natura Environment Research Counci (NERC). Their suort is gratefuy acknowedged. The authors woud aso ike to thank coeagues in the AIM grou and the wider Institute for Aviation and Environment (IAE) at the University of Cambridge, and Professors Cynthia Barnhart and Amedeo Odoni at the Massachusetts Institute of Technoogy for hefu discussions, and eseciay María Vera-Moraes for suying aircraft fue burn rates. References [1] United Nations Intergovernmenta Pane on Cimate Change, IPCC Secia Reort Aviation and the Goba Atmoshere, Working Grou I and III, IPCC, Geneva, Switzerand. [] Internationa Civi Aviation Organisation, 006. Word Tota Traffic , [3] Airbus, Goba Market Forecast , htt:// 007 [Cited 7 June 008] [4] Boeing, Current Market Outook 007, htt:// [Cited 7 June 008]. 1

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