addressing the pushback time allocation problem at heathrow

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Addressing the Pushback Time Allocation Problem atHeathrow AirportJason A. D. Atkin, Geert De Maere, Edmund K. Burke, John S. Greenwood

To cite this article:Jason A. D. Atkin, Geert De Maere, Edmund K. Burke, John S. Greenwood (2013) Addressing the Pushback Time AllocationProblem at Heathrow Airport. Transportation Science 47(4):584-602. http://dx.doi.org/10.1287/trsc.1120.0446

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Vol. 47, No. 4, November 2013, pp. 584602ISSN 0041-1655 (print) ISSN 1526-5447 (online) http://dx.doi.org/10.1287/trsc.1120.0446

2013 INFORMS

Addressing the Pushback Time AllocationProblem at Heathrow Airport

Jason A. D. Atkin, Geert De MaereAutomated Scheduling, Optimisation and Planning Research Group, School of Computer Science,

The University of Nottingham, Nottingham, NG8 1BB, United Kingdom{[email protected], [email protected]}

Edmund K. BurkeDepartment of Computing and Mathematics, University of Stirling, Stirling FK9 4LA, Scotland, United Kingdom,

[email protected]

John S. GreenwoodNATS CTC, Whiteley, Fareham, Hampshire, PO15 7FL, United Kingdom

This paper considers the problem of allocating pushback times to departing aircraft, specifying the timeat which they will be given permission to push back from their allocated stand, start their engines, andcommence their taxi to the runway. The aim of this research is to first predict the delay (defined as the waitingtime at the stand or runway) for each departure, then to use this to calculate a pushback time such that anappropriate amount of the delay is absorbed at the stand, prior to starting the engines. A two-stage approachis used, where the feasibility of the second stage (pushback time allocation) has to be considered within thefirst stage (takeoff sequencing). The characteristics of this real-world problem and the differences between itand similar problems are thoroughly discussed, along with a consideration of the important effects of thesedifferences. Differences include a nonlinear objective function with a nonconvex component; the integration oftwo sequence dependent separation problems; separations that can vary over time; and time-slot extensions.Each of these factors has contributed to the design of the solution algorithm. Results predict significant fuel-burn benefits from absorbing some of the delay as stand hold, as well as delay benefits from indirectly aidingthe runway controllers by reducing runway queue sizes. A system for pushback time allocation at LondonHeathrow has been developed by NATS (formerly National Air Traffic Services) based upon the algorithmdescribed in this paper.

Key words : optimisation; stand holding; scheduling; runway scheduling; departure operations; sequencedependent separations

History : Received: December 2009; revisions received: February 2011, July 2012; accepted: July 2012. Publishedonline in Articles in Advance December 13, 2012.

1. IntroductionLondon Heathrow is the busiest two-runway air-port in the world, with more than 650 departuresand 650 arrivals per day. Despite this, various con-straints upon the departure system (detailed in BAAHeathrow 2007) mean that only one runway is usedfor takeoffs at any time of the day. Efficient take-off sequencing is extremely important (Idris et al.1999; Atkin et al. 2007), however, the demand forthe runway at busy times can far exceed the possiblethroughput, thus delays must accumulate regardlessof the efficiency of the takeoff sequencing, causinginconvenience for passengers, incurring costs for air-lines, and resulting in unnecessarily high fuel burnand pollutant emissions. In this paper, the term delayrefers to the difference between the takeoff time anaircraft could achieve if it was alone in the departuresystem (and had no allocated takeoff time window),

and the actual takeoff time it will be able to achievegiven the queues for the runways and any take-off time restrictions. These delays are experienced aswaiting time either at the runway or the stands. Theprimary aim of this research is to decrease the amountof delay at the runway (with the engines running) andabsorb it at the stand (prior to starting the engines) byallocating a delayed pushback time. This will simul-taneously simplify the task for the runway controller,who has to sequence the takeoffs, by reducing thecongestion at the runway queues, thereby potentiallyenabling delay reductions for all concerned.

This paper considers a real problem at airports,with all of the awkward constraints and objectivesthat this can imply. The system that it describes hasbeen implemented for London Heathrow Airport andintegrated into the systems there. The implementa-tion of TSAT (target start-up approval time) systemsat other airports is already being considered.

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The remainder of this paper is structured as fol-lows: The problem that is under consideration is intro-duced in 2, where the important real-world elementsand problem characteristics are explained. Modelsfor the takeoff sequencing and pushback time allo-cation problems are presented in 3, consisting oftwo linked subproblems. The solution method thathas been implemented for Heathrow is presentedin 4, where the design decisions are also discussed.Section 5 presents experimental results, demonstrat-ing the potential benefits of the system, the trade-offbetween the solution time and solution quality, andthe effects of the algorithm parameters. Finally, thepaper ends with conclusions that can be drawn aboutthe potential benefits of a pushback time allocationsystem.

2. Problem DescriptionThe majority of aircraft at Heathrow are parked atstands1 around the piers of the five main terminals.Each terminal has multiple piers and these piers formcul-de-sacs between them, as illustrated in Figure 1.Airlines declare a target off-block time (TOBT) foreach aircraft, specifying the time at which it willbe ready to push back from the stand and start itsengines. An aircraft pushing back can block other air-craft from pushing back from nearby stands or frombeing able to pass in order to exit the cul-de-sac,delaying these other aircraft. For example, an aircraftpushing back from stand X in Figure 1 would usuallyblock those at stands labelled 1 from leaving thecul-de-sac, be delayed by the pushback of any aircraftfrom the stands labelled 2, and block the pushbackof those at stands labelled 3.

Controllers use these TOBTs to determine poten-tial pushback times for aircraft,2 given the state ofthe airport as a whole. After engine start-up, air-craft are directed out of the cul-de-sac and aroundthe taxiways to a holding area near the end of thecurrent departure runway, where control is passed toa runway controller. These holding areas consist ofqueues with interchange points, allowing the takeoffsequence to be modified considerably, as explained inAtkin et al. (2007).

2.1. Runway SequencingSequence dependent separations apply between air-craft at takeoff, both to allow the wake vortices fromprevious takeoffs to safely dissipate (the separationsare greater when a lighter category aircraft follows

1 The locations where aircraft are loaded/unloaded are namedstands. These may be at gates in the terminals or at remote loca-tions accessed by bus.2 These are actually named TSATs (target start-at times), because ofthe conceptual single start-up and pushback operation at Heathrow.

1 1 1

11 1

2

2

X

3

1, 3 2, 3

2, 31, 3

Taxi

way

s

Cul-de-Sac

Terminal pier

Term

inal

Terminal pier

Figure 1 Illustration of a Cul-de-Sac Between the Stands SurroundingTerminal Piers

a heavier category aircraft) and to ensure that safeen route separations are attained (the separations aregreater when the following aircraft follows the sameor a similar departure route as the preceding air-craft, and may be increased further if the followingaircraft has a faster speed category or decreased ifthe following aircraft has a slower speed category).The minimum required separation between any twoaircraft can be determined from their weight cate-gories, departure routes, and speed groups, and isalways at least one minute. Eight departure routes,four speed groups and up to four weight classes usu-ally need to be considered. Full separation rules forHeathrow can be found in Atkin (2008).

The underlying takeoff sequencing problem hasmany similarities to the asymmetric cumulative trav-elling salesman problem (Bianco, Mingozzi, andRicciardelli 1993) and to machine scheduling prob-lems with sequence dependent setup or processingtimes (Bianco, DellOlmo, and Giordani 1999; Pinedo2008, 4.4). However, the separations are not onlyasymmetric but they also do not obey the triangleinequality, so consideration of adjacent departures (oronly the previous city/job in the similar problems) isnot sufficient.

2.2. Flow Control Measures: CTOTs and MDIsEuropean airspace is often more congested than theairport infrastructure at many European airports, andflow control measures are often applied at the airportrunways to control the airspace congestion.

2.2.1. CTOTs. The main long-term control mea-sure is the calculated takeoff time (or calculatedtime of takeoff), commonly named CTOT. Theseare 15 minute takeoff windows that are appliedby the EUROCONTROL Central Flow ManagementUnit3 (CFMU) to aircraft that will pass through busyairspace sectors. CTOTs apply to around 30%40%of Heathrow departures. Aircraft that cannot meet

3 Homepage: http://www.cfmu.eurocontrol.int/cfmu/public/subsite_homepage/homepage.html.

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windows need to request a new CTOT, and maynecessitate a considerable additional delay (poten-tially an hour or more at busy times), so this shouldbe avoided. Extensions to these windows can some-times be obtained4 in order to reduce the number ofCTOT renegotiations that are necessary. The aim insequencing is, therefore, to hit CTOTs if possible andextensions if not.

2.2.2. MDIs. Increased minimum departure inter-vals (MDIs) are a shorter term flow control measure,applied to control the workload of the local en routecontrollers by applying increased minimum separa-tions (of up to 10 minutes or more in some cases)between takeoffs along specific departure routes, toreduce the number of flights that the en route con-trollers will have to deal with. Some are planned inadvance (a three minute MDI applies for much ofthe day on the Dover route) and others are appliedtactically, as required, potentially requiring late rese-quencing of aircraft. MDIs often increase the delay foraircraft on the affected departure routes.

2.3. Takeoff SequencingThere is a dedicated runway controller position in thecontrol tower for each of the runways at Heathrow.The departure runway controller is responsible forattaining a high-quality takeoff sequence given theaircraft that are available at the holding area (orthat will be available soon). Given the separationrules and the takeoff time slots, the runway controllerwill attempt to maintain a high runway throughput,by finding sequences that maintain low separations,while ensuring that CTOT time slots are attained(where possible) and maintaining a degree of fairnessbetween the waiting aircraft. The three main objec-tives (reducing total delay, equity of delay, and reduc-ing the number of CTOT extensions or renegotiationsneeded) are in conflict, as discussed in Atkin, Burke,and Greenwood (2010), where the tradeoff betweenthem was investigated.

2.4. Pushback Time AllocationThe complexity of the sequencing operation meansthat it is usually performed once aircraft have reachedthe holding areas at the runway, rather than whenthey are on the taxiways or at the stands. Con-sequently, aircraft are currently released from thestands as soon as practical to ensure the maximalpool of aircraft at the holding area from which the

4 At the time when the data used in this paper was collected, con-trollers were able to tactically apply five-minute extensions, with upto four five-minute CTOT-window extensions being permitted perhour and no more than 20 in a single day, without having to contactthe CFMU in each instance. Five-minute extensions are assumedthroughout this paper, although current operations (in 2010) allowairlines to request 10-minute extensions.

runway controller can select the next aircraft. How-ever, the characteristics of the takeoff sequencingproblem mean that delay can vary greatly betweenaircraft, particularly because of CTOTs and inequityof demand across departure routes. The aim of thesystem described in this paper is to change the cur-rent behaviour, without adversely affecting the run-way throughput, by predicting what a good runwaycontroller would do, predicting the delays for aircraft,and allocating pushback times to absorb more of thesedelays prior to engine start-up.

A collaborative decision making5 (CDM) systemhas been implemented at London Heathrow (EURO-CONTROL Experimental Centre 2005), enabling andencouraging airlines, ground handling agents, air-port operators, and air traffic controllers to shareinformation so that all are able to make better-informed decisions. For the first time, this has pro-vided the necessary information from airlines aboutwhen aircraft will be ready to push back (TOBT) soonenough to be used in takeoff time prediction whileaircraft are still at the stands.

If the runways were running in mixed mode (forarrivals and departures on the same runway), thesequencing problem would be easier. When alternat-ing departures and arrivals, there is necessarily atwo or more minute gap between departures. Becauserequired separations are often two minutes or less(in the absence of MDIs), it is much easier to find take-off sequences that attain these two-minute separations(i.e., optimal throughput) on each runway than it is tofind sequences that attain as many one-minute sepa-rations as possible on a single runway. Takeoff timeprediction can then often become a case of allocatingaircraft to generic takeoff slots according to any ear-liest takeoff time and CTOT. This is the basis of thepushback time allocation methods that are used at air-ports such as Munich (Munich Airport CDM Group2007). Of course, even though the takeoff sequenc-ing task for the departures may be easier, the overalltask of the runway controller is not, because there areadditional coordination issues between arrivals anddepartures.

Jung et al. (2011) use a similar approach forTSAT allocation, first predicting a takeoff sequenceusing a dynamic programming approach, then usingthis to allocate TSATs, by deducting the taxi timesallowing some slack, although no pushback timeinterdependencies are considered. Their sequencingalgorithm utilises the precedence constraints from therunway queues (significantly restricted in comparisonto Heathrow) and a simplified objective function isused: maximising runway throughput.

5 European airport collaborative decision making website: http://www.euro-cdm.org/.

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Departure metering may be a sensible alternativeapproach for many airports, and has been used suc-cessfully in the United States (Brinton et al. 2011). Theaim is to allocate pushback times to reduce conges-tion on the airport surface and at the runways, underthe assumption that the selection of aircraft at therunway is not so important, although Brinton et al.(2011) suggested that separate allocation groups maybe needed to handle MDIs and that flight-specificvirtual queue concepts may provide somewhat largerbenefits 0 0 0 0 However, there is a potential for anadverse reduction in runway throughput at Europeanairports unless such metering considers the takeoffsequence,6 because the characteristics of the aircraft atthe holding area (weight classes, speed groups, depar-ture routes, and any takeoff windows) can greatlyaffect the throughput.

2.5. Taxi Times and Cul-de-Sac ContentionEarliest takeoff times are required in order to predicttakeoff times, so taxi time predictions are very impor-tant when predicting sequences at the stands. Theusual concept of a taxi time refers to the total dura-tion from pushback to takeoff and can vary greatlydepending on the runway queue length and the con-gestion on the airport surface (Idris et al. 2002). Thistotal taxi time can differ greatly between aircraft,because it is dependent upon the takeoff sequencethat is adopted, as well as upon the taxiing speed.At Heathrow, the taxi time can be divided into threesubcomponents: delays near the stands, travel timearound the taxiways, and queues at the runways.The solution method in this paper explicitly mod-els both the delays by the stands (cul-de-sac delays)and the runway delays (from takeoff sequencing), andrequests from the CDM system the taxi duration fromcul-de-sac to holding area.

Figure 2 provides a stylised layout of LondonHeathrow, showing the relative positions of the termi-nals, runways, runway holding areas where aircraftare resequenced for takeoff (labelled R.H.A., whichhave differing sizes and structures, see Atkin 2008),and the taxiways (the unlabelled lines around the ter-minals). Note that the diagram has been greatly sim-plified by omitting the many entrances to the holdingareas from the taxiways, the many runway crossingpoints, and the entrances to the cul-de-sacs from thetaxiways. Furthermore, it is possible to taxi straightthrough terminal 5, reaching the stands from eitherthe north or south side.

Once aircraft have left the cul-de-sacs, the struc-ture of Heathrow means that the travel times on thetaxiways are relatively predictable. Because takeoffs

6 See for example, http://www.eurocontrol.int/eec/public/standard_page/EEC_News_2008_2_DM.html, accessed July 27, 2012.

Northern runway

Southern runway

R.H.A.

R.H.A.

R.H.A.

R.H.A.

R.H.A.27L

Terminal 4

09R

Terminals1, 2 and 3

09L

Terminal5

R.H.A.

27R

Figure 2 Stylised Diagram of the Layout of London HeathrowShowing the Positions of the Terminals, Runways, HoldingAreas, and Primary Taxiways

and landings take place in the same direction buton different runways, aircraft enter and leave therunways (respectively) at opposite sides of the air-port. The vast majority of the traffic will, therefore,flow in the same direction (heading either north andeast, south and east, or south and west dependingon the runways that are in use). In addition, themajority of the taxiways are paired, providing twoparallel routes for most traffic, reducing the numberof delays by slower traffic, providing potential wait-ing positions out of the main flow, and preventinglarge delays from the occasional traffic in the otherdirection or from arrivals waiting to enter cul-de-sacs.This component of the taxi time is, therefore, rela-tively predictable7 and is handled by the introductionof slack into the schedule and by ensuring that thesystem finds solutions quickly enough that it can bereexecuted as the situation changes. Indeed, currentresearch such as Ravizza et al. (2012) is consideringtaxi time prediction and has found that taxi timesat Zurich and Arlanda are remarkably predictableonce sufficient factors have been taken into account(Chen et al. 2011), and similar results were found forHeathrow.

Taxi times may be less predictable, or the runwaysequence less important, at some airports (for instancewhere mixed mode is being used or multiple runwaysare available), justifying a more detailed considera-tion of the ground movement when takeoff sequenc-ing. For example, Deau, Gotteland, and Durand (2009)considered the combined problem for Roissy Charlesde Gaulle airport, despite the ground movement prob-lem being both challenging and interesting in its ownright (Atkin, Burke, and Ravizza 2010).

7 The one exception is for aircraft that travel between the north-ern runway and terminal 4, where a runway crossing is required.A significantly increased slack should be allowed for these aircraft,but is only needed when 27R is being used for departures.

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2.6. Characteristics of the ProblemThe important characteristics of this problem and thedifferences from previously considered problems areconsider in this section.

2.6.1. Cul-de-Sac Delays Matter. As will be dis-cussed in 5.5, the cul-de-sac delays can affect theability to achieve a predicted takeoff sequence and,thus, cannot be ignored.

2.6.2. Separation Rules Must Consider MoreThan Wake Vortices. Downstream constraints (routeseparations and CTOTs) at Heathrow increase thedelays far more than the wake vortex separations(Atkin et al. 2009), however most previous airportsequencing research considered only wake vortexseparations. Because there are only between threeand six weight categories, previous methods havegrouped aircraft by weight class to reduce the com-plexity of the problem (Psaraftis 1980). Includingdeparture routes and speeds increases the number ofgroups considerably (combinations of weight, route,and speed), compromising the effectiveness of thesegrouping methods.

2.6.3. Route Separation Structures Are Unhelp-ful. Wake vortex separations have a useful structureto the separation matrix so that (ignoring time win-dows and equity) it is always better to sequence air-craft by weight class, from lighter to heavier aircraft.Departure route separations are not so usefully struc-tured. Table 1 shows simplified separations for threeroutes (taken from the 27L separation table in Atkin2008 and ignoring the effects of speed groups), wherethe row and column specify the routes of the leadingand trailing aircraft, respectively, and the value in thetable body gives the required minimum separation inseconds.

Good schedules need to alternate departure routes(e.g., routes 1121113) and (unlike wake vortex sep-arations) must consider more than the immediatelyprevious aircraft (e.g., routes 31113).

2.6.4. Total Delay and Equity Are in Conflict.There is an obvious conflict in Table 1 between totaldelay and equity of delay for departure separations,because the optimal throughput of one aircraft perminute requires four-minute or greater separations onroute 3 (e.g., repeating routes 3111211). The three-minute separations on route 3 apply to reduce the

Table 1 Simplified Example Departure Separations

Trailing route

Preceding route 1 2 3

1 120 60 602 60 120 1203 60 120 180

flow on the route because it is so busy, and delaysalready accumulate on this route. Increasing the gapsto four minutes soon makes the delays for theseaircraft unacceptable. Equity of delay is extremelyimportant in this problem, but large delays are stilloften unavoidable on route 3.

2.6.5. Large Position Shifts Are Often Necessary.Alternative simplification approaches have utilised amaximum position shift approach (Dear and Sherif1989, 1991; Balakrishnan and Chandran 2010) andrecommended permitting only low maximum posi-tion shifts. Unfortunately, CTOTs and inequity of loadacross departure routes means that large positionshifts are unavoidable for the departure sequencingproblem, as discussed in Atkin, Burke, and Green-wood (2010). Furthermore, it was shown in Atkinet al. (2006) that consideration of more aircraft oftenresults in a better takeoff sequence (within a 20-minutewindow).

2.6.6. Time Windows Are Soft. Hard time win-dows around landing times have been used in thepast to control inequity, from which partial landingsequences can be inferred to greatly simplify the prob-lem (Beasley et al. 2000). In addition, these are usuallydefined as offsets from an ideal time, so that the ordermatches the objective function (Beasley et al. 2000).Conversely, in the departure problem, the CTOT endtimes are only soft constraints, with extensions beingpossible, so they cannot be used to directly prune thesearch tree. Furthermore, CTOTs are not usually in thesame order as earliest takeoff times, so it is not alwaysideal for aircraft to take off in time-slot order. Thismeans that partial sequences of aircraft are more sen-sitive to movement in time because they may be goodonly as long as it is possible for all aircraft to fit withintheir windows. This problem is even more apparentif time-dependent separations are considered.

2.6.7. Earlier Takeoff Times Are Important. Be-cause partial subsequence costs can vary greatly ifthe times are advanced or delayed, it is important toanchor them in time if an accurate cost is required.Takeoff times for earlier aircraft are necessary in orderto predict takeoff times for later aircraft, so searchmethods that utilise partial sequences have to start atthe first takeoff.

2.7. Scope of This PaperThis paper considers the basic operations at Heathrowand presents a model for the pushback time allocationproblem as well as a solution method for solvingit. A comparison of the takeoff sequencing elementagainst a previously designed metaheuristic approachfor sequencing at the holding areas and the real con-troller performance was provided in Atkin, Burke,and Greenwood (2009), showing comparable resultsbetween the two systems even with the consideration

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of the cul-de-sac delays in this case, so it is notrepeated here. Similarly, the tuning of the objectivefunction toward controller preferences was addressedin Atkin, Burke, and Greenwood (2010), and is alsonot addressed here. In contrast, this paper consid-ers the characteristics of the problem in more detail,discusses the algorithm design motivations, and pro-vides an assessment of the effects of the variousalgorithm parameters, such as the window size, thenumber of passes, the minimum and ideal runwayhold values, and the optional use of a presequencingalgorithm.

2.7.1. Steady State Operating Behaviour isAssessed. For simplicity of explanation and reasonsof space, this paper considers the steady state oper-ations and does not evaluate the effects of the dailyperturbations that have to be handled in practice.Many perturbations (e.g., bad weather conditions,runway changeover, temporary mixed mode opera-tions, runway closures, departure route or airport clo-sures, and the application of MDIs) can easily behandled within the takeoff time prediction stage bydelaying the takeoff time according to the conditionsin effect at the time. Being able to handle such eventswas an important algorithm design decision, how-ever this paper concentrates upon the core algorithm,because an analysis of how well each of these is han-dled would be very lengthy.

2.7.2. The Static Problem is Considered. Thispaper considers the static problem and assumes thatall information is known in advance and is unchang-ing, so that an analysis can be made of the algorithmwithout requiring a consideration of what informa-tion was available when. Beyond the scope of thispaper, we note that experiments considering thedynamic situation have shown that this solver canhandle it as a series of static problems, with threenotable differences, as follows:

1. Once a pushback time has been issued there isa commitment that the aircraft will be permitted topush back by that time, thus the concept of a fixedpushback time and latest pushback time has to exist inthe dynamic problem, even when the associated take-off time is still flexible. (This requires only a minorchange to the algorithm described here and does notinvalidate any of the discussion or results in thispaper.)

2. There may need to be a preference for previousdecisions to be readopted rather than changed, requir-ing slight modifications to the objective function inorder to penalise changes.

3. In the dynamic problem, it is necessary to han-dle situational changes (e.g., MDIs or runway clo-sures) after aircraft have left their stands, in whichcase the pushback has been fixed (and passed) but

the takeoff time predictions are still flexible. The pres-ence of these released aircraft may affect the controllerbehaviour and this may need to be reflected in theobjective function that is used to predict the take-off sequence. Note that, although the pushback timesfor these released aircraft cannot be delayed, the newpredicted takeoff times will (when appropriate) resultin aircraft that are still at the stands being held forlonger, to absorb the additional delays.None of these three additions change the core algo-rithm that is expressed here, merely modifying itslightly, so they have been omitted from this paper inthe interests of clarity and brevity.

3. Problem ModelThe pushback time allocation problem is dividedinto two subproblems: determining a good takeoffsequence and allocating stand holds that allow thesequence to be achieved but absorb more of the delayat the stand. This section starts by introducing theinput data and decision variables for the problem. Thevarious constraints are then introduced and, finally,the objectives for the two stages of the problem aregiven, along with an explanation for why a two-stageproblem decomposition is appropriate.

3.1. Input ValuesA significant amount of input data is both availableand utilised for this problem. This is represented inthe mathematical model by the constants as shown inTable 2.

3.2. Cul-de-Sac Times and SeparationsThe concept of a cul-de-sac time is introduced hereas the time at which an aircraft will set off from thecul-de-sac and head for the taxiways, having pushedback from its stand and started its engines. Cul-de-sacseparations are easier to describe with cul-de-sactimes than pushback times but pushback times can bedetermined from cul-de-sac times (see Equation (1),explained later).

The MSij separations are designed to estimate thetimes for which aircraft will normally be blocked inthe cul-de-sacs and are used to enforce separationsbetween cul-de-sac times:

If i and j are not in contention (e.g., they are indifferent cul-de-sacs) then MSij =MSji = 0.

If pushback is not delayed (or can take placesimultaneously), but j is blocked from leaving untili moves, then MSij will be small, giving the blockedtime.

If j cannot even commence pushback until i hasmoved, then MSij > pdj and MSij pdj will be thedelay after i starts to leave before j can commencepushback.In fact, controllers can sometimes change the push-back behaviour to reduce certain delays (for example,

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Table 2 Definitions of Constants

Value Meaning

eptj The earliest pushback time for aircraft j . This is usually equal tothe TOBT that the airline issued. In the dynamic case thiscould be modified to limit changes to declared pushbacktimes.

pdj The predicted pushback and engine start duration for aircraft j .This will usually depend on the aircraft size and number ofengines but could also potentially depend upon the airline(e.g., for differing operating procedures). For the experimentsin this paper, pdj was assumed to depend only on the type ofaircraft j .

MSij The minimum cul-de-sac separation between aircraft i andaircraft j ; the (sequencedependent) minimum time thatmust elapse after aircraft i has started to leave the cul-de-sac(following the completion of the pushback and engine startoperation) before aircraft j can also start to leave thecul-de-sac.

ecj The earliest time at which aircraft j can take off and meet itsallocated CTOT time slot. If j has no CTOT then ecj = .

lcj The latest time at which aircraft j can take off and meet itsallocated CTOT time. Because time slots are 15 minutes andall times are stored internally in seconds, lcj = ecj + 900.Aircraft can take off later than this by using time-slotextensions. If j has no CTOT then lcj = .

etj The earliest takeoff time for aircraft j , for the purpose ofmeasuring the delay for j and for determining afirst-come-first-served takeoff sequence. This ignores anyCTOT window or runway queue and assumes no runway hold.

asj The position of aircraft j in the first-come-first-served takeoffsequence implied by the earliest takeoff times (etj ) for eachaircraft.

MRHj The minimum runway hold for aircraft j . This is the minimumduration that the solver should assume that the aircraft willneed after arriving at the runway queue before it can take off.

IRHj The ideal runway hold for aircraft j , such that any delay beyondthis value should be absorbed as stand hold.

by requesting a longer pushback, further into thecul-de-sac, to get out of the way of another pushback)but that behaviour is not assumed here.

3.3. Input FunctionsThere are also a number of functions that can be calledwhen desired, to request important input values thatdepend on the current airport situation or time of day.These are defined in Table 3.

3.4. Decision VariablesTable 4 lists the various decision variables that areused.

3.5. ConstraintsThe various constraints upon the decision variablesfor each aircraft j , or pair of aircraft i and j , are shownin the following, then discussed in more detail:

ptj = ctj pdj1 (1)ctj eptj + pdj1 (2)

ctj cti +MSij i1 j s.t. csj > csi1 (3)

Table 3 Definitions of Input Functions

Function Meaning

TD4j5 The predicted taxi duration from the cul-de-sac to the holdingarea for aircraft j . This value will be provided by the CDMplatform, which could utilise either the start time or endtime of the taxi operation in order to improve predictions. Inthe experiments in this paper, taxi durations are assumed tobe constant for each aircraft, depending on the aircraft type,allocated stand, and destination runway, but this is notassumed by the algorithm.

RS4i1 j5 The required runway separation (in seconds) between leadingaircraft i and trailing aircraft j . This can be a function oftime (allowing separations to depend on the time of day) ora constant that depends on the pair of aircraft involved andthe runway.

ttj ctj + TD4j5+MRHj1 (4)ttj ecj1 (5)

ttj tti +RS4i1 j5 i1 j s.t. tsi < tsj1 (6)ctj ttj TD4j5MRHj1 (7)etj = eptj + pdj + TD4j50 (8)

3.5.1. Determining Pushback Times. Becausethere is no advantage from pushing back earlyand blocking the cul-de-sac, and there are obviousdisadvantages from doing so, it can be assumed thataircraft j will only pushback when it will be able tostart its engines and leave the cul-de-sac immediately.This means that a solution system can deal withcul-de-sac times rather than pushback times, anddetermine the resulting pushback times afterwardusing Equation (1).

Table 4 Definitions of Decision Variables

Variable Meaning

ptj The pushback time for aircraft j . This is the ultimate output ofthe system.

cti , ctj The cul-de-sac time for aircraft i and j , respectively. Given thedefinition for the cul-de-sac time, ctj ptj + pdj for allaircraft j .

csi , csj The unique position number of aircraft i and j (respectively)in the cul-de-sac sequence. This determines the order inwhich aircraft leave the cul-de-sacs and enter the taxiways;4csi > csj 5= 4cti ctj 5. Sequence dependentseparations apply at the cul-de-sacs, so the sequence isnecessary in order to determine the required cul-de-sacseparations.

tti , ttj The predicted takeoff times for aircraft i and j , respectively.tsi , tsj The unique position number of aircraft i and j (respectively)

in the takeoff sequence; 4tsi > tsj 5= 4tti ttj 5. Thesequence is necessary to determine the requiredseparations at takeoff.

ict j The ideal cul-de-sac time for aircraft i. Given a predictedtakeoff time, the ideal cul-de-sac time can be determined toallocate an appropriate amount of the predicted delay asstand hold.

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3.5.2. Cul-de-sac Time Constraints. In this mo-del, the allocated cul-de-sac time must obey the fol-lowing two constraints:

1. Cul-de-sac times cannot be earlier than the ear-liest pushback time permits (Inequality 2).

2. All minimum cul-de-sac separations must beattained from aircraft that have previously left the cul-de-sac (Inequality 3).

3.5.3. Takeoff Time Prediction. Takeoff time pre-diction takes advantage of the fact that aircraft willtake off as soon as they can, given the three con-straints upon their takeoff times, as follows:

1. Aircraft must be able to physically reach the run-way (Inequality 4). An aircraft must have sufficienttime after setting off from the cul-de-sac (ctj ) to taxito the runway (TD4j5), traverse the holding area, andline up for takeoff (expressed by a minimum run-way hold value MRHj , which may include some slacktime). Note that stands are often closer to one runwaythan another, so taxi times and earliest takeoff timesmay need to be recalculated if aircraft are allocated todifferent runways. This is not a problem for the solu-tion system described in this paper, which allocatestakeoff times to aircraft one at a time in takeoff order.

2. Aircraft can take off no earlier than the start ofthe CTOT time-slot (Inequality 5).

3. Minimum runway separations must be main-tained between the current takeoff and all previoustakeoffs (Inequality 6). These may be time dependent,in which case constraints 4 and 5 have to be consid-ered before RS4i1 j5.

3.5.4. Second Stage Cul-de-Sac Time Constraints.Once takeoff times are known, the aim is to finda set of achievable cul-de-sac times for the aircraftsuch that the cost of deviations from ideal times,as measured by some objective function, are min-imised. The earliest cul-de-sac time can be determinedfrom Inequality 2, as in the takeoff sequencing stage.The latest cul-de-sac time that could achieve the pre-dicted takeoff time can be determined using Inequal-ity 7 (which is merely a reformulation of Inequality 4),where TD4j5 specifies here the predicted taxi durationif aircraft j arrived at the runway at time ttj MRHj .Together the earliest and latest cul-de-sac times spec-ify a time window for each aircraft. Although all tim-ings are maintained to one-second accuracy, allocatedpushback times must be on minute boundaries, thusso should cul-de-sac times (because pushback dura-tions are also specified in minutes). There are, there-fore, usually very few possibilities for each cul-de-sactime within its time window.

3.5.5. The Earliest Takeoff Time. An earliesttakeoff time, etj , is needed to measure the delay. Forthese experiments, it was calculated using Equation(8), by assuming that aircraft j has no cul-de-sac delay

or runway queue. Where a runway change may beinvolved, the differing taxi times have to be consid-ered, for example by taking the time for the runwaywith minimum taxi time, however this is beyond thescope of this paper, as discussed in 2.7.

3.6. Takeoff Sequencing Objective FunctionThe objective function for the takeoff sequencing stagewas designed around controller preferences and wasadapted with the benefit of controller feedback aboutthe value or weaknesses of various resulting takeoffsequences. The objective function for takeoff sequenc-ing can be expressed by Formula 9, where values W1to W3 are constant weights that determine the relativepriorities of the three components, relating (respec-tively) to the objectives of meeting takeoff time-slots,controlling delay and controlling positional inequityacross aircraft:

Nj=1

4W1C4ttj1 lcj5+W24ttj etj5 +W34tsj asj5251 (9)

C4ttj1 lcj5

=

0 if ttj lcj 4i5114ttj lcj5+2 if lcj < ttj lcj + 300 4ii5134ttj lcj5+4 if ttj > lcj + 300 4iii50

(10)

The first term (C4ttj1 lcj5) is a function to deter-mine the cost of takeoff time-slot compliance and canbe expressed by Equation (10). The values 1 to 4are constant weights that are used to determine therelative importance of using CTOT extensions andmissing extensions. Terms (i)(iii) apply, respectively,to aircraft that are predicted to take off within theirtime slot, within an extension, or that even miss anextension. Values of 1 = 1, 2 = 101000, 3 = 10,and 4 = 110001000 were used for the experimentsdescribed in this paper: 2 is much larger than 1to prefer schedules where fewer CTOT extensions areused, and 4 is much larger than 2 to prefer sched-ules that use multiple extensions over those whereaircraft miss extensions.

The second term in Formula 9 denotes a cost forthe delay, as measured by the deviation betweenthe predicted and earliest takeoff times, raised to apower of (1), a constant that determines the bal-ance between minimising total delay and minimisingequity of delay across aircraft.

The third term in Formula 9 measures the posi-tional inequity in the sequencing, applying a penaltyequal to the sum of the squares of the positionaldeviations between the takeoff sequence and the first-come-first-served sequence.

An evaluation of the effects of the weights was pre-sented in Atkin, Burke, and Greenwood (2010), wherethey were observed to provide significant tuning

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flexibility, even when applied only within the rollingwindow. The delay, time-slot compliance, and equityobjectives were seen to be in conflict in the bestsequences and a nonlinear cost for delay was seen tobe useful in order to promote equity of delay. The val-ues = 105, W1 = 100, W2 = 10, and W3 = 5 were usedfor the experiments in this paper.

3.7. TSAT Allocation Objective FunctionIdeal cul-de-sac times (ictj for aircraft j) can be deter-mined for aircraft such that any hold above a certainthreshold (denoted by IRHj , the ideal runway holdfor j) is absorbed at the stand rather than the run-way, using Equation (11). The objective of the push-back time allocation algorithm is to find an allocationof cul-de-sac times ctj for each aircraft j , such thatInequalities 2, 3, and 7 are satisfied, and deviationsfrom ideal times (as measured by Formula 12), arelow. Formula 12, has been designed to apply a nonlin-ear penalty to deviations, preferring a larger numberof smaller deviations rather than fewer larger onesand preferring aircraft to push back earlier (increasingthe holding area delay) rather than later (reducing theslack available for absorbing taxi delays):

ictj =max4eptj +pdj1ttj IRHj tdj51 (11)Nj=1

(1004max401ctj ictj55101 +4max401ictj ctj55101

)0

(12)

The IRHj values may be constants or may dependon the takeoff time (allowing for increased pool sizesat the runway at certain times of day). They may alsodepend on the aircraft types, for example, allowingmore slack for heavier aircraft.

3.8. Consideration of the Two-StageDecomposition

The pushback time allocation problem is solved intwo separate stages, resulting in two simpler subprob-lems to solve. If there were to be some benefit fromsolving the problem in a single stage then the result ofcul-de-sac time allocation would have to have a posi-tive influence on the takeoff sequencing stage in someway. However, without a known takeoff time, the aimfor the cul-de-sac sequencing stage is not obvious.

One possibility would be to prefer sequences thatwould increase the fuel-burn benefits of the system bymaximising the stand hold times. In this case, giventwo sequences with identical total delays ( IRHj )and an objective of absorbing delay over a thresholdvalue (IRHj ) at the stand, the schedule where one air-craft had the lowest delay (as low as possible below IRHj ) would be preferred, resulting in schedulesthat are both less robust to delays and less equitable

in terms of the division of the delay between aircraft.Obviously, this is a highly undesirable property.

A second possibility could be to bias against sched-ules that have aircraft with total delay lower than theIRHj values (to increase schedule robustness towarddelays) and/or against those that have aircraft withcul-de-sac delays because of contention, because theseare more likely to be unable to achieve ideal runwayholds. Although appearing useful, this would intro-duce undesirable biases into the takeoff sequencingstage, and would bias against pushing back aircraft onsimilar stands at similar times. This would potentiallypenalise larger airlines or those with larger aircraft,which require longer pushback durations and conse-quent cul-de-sac delays. At the very least, the pres-ence and effects of any introduced biases would haveto be fully understood and justified from a regulatorypoint of view. Although not practical at the moment(there is a requirement to avoid biases between air-lines), this would be an interesting area for futureresearch.

3.9. Modelling AssumptionsA number of assumptions were made for this modeland they are listed as follows:

1. Normal pushback behaviour is assumed (ratherthan permitting long pushbacks, as previously dis-cussed), to determine the required cul-de-sac separa-tion values.

2. The use of cul-de-sacs by arrivals is ignored,because arrival times for landing aircraft will not cur-rently be known early enough to consider these whendetermining pushback times. Under current opera-tions, such conflicts are often resolved by the arrivalswaiting outside of the cul-de-sac until the departureshave left and the stands become available.

3. These results assume that aircraft can be held onthe stands indefinitely, but this may not always bepossible, even though the majority of stand holds arerelatively low. Stand contention can be handled byapplying a latest pushback time to aircraft, however,the objective function for takeoff sequencing mustthen ensure that stand contention does not prioritisethese aircraft.

4. Solution MethodInequality 4 shows that cul-de-sac sequencing mustbe considered within the takeoff sequencing pro-cess in order to ensure that predicted takeoff timesare achievable. The solution method presented heredecomposes the problem into two separate stages.In the first stage, a takeoff time and a feasible (butnot usually optimal) cul-de-sac time is determined foreach aircraft. The cul-de-sac times are then optimisedin the second stage, given the takeoff times for thefirst stage.

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4.1. Overview of the Solution MethodStage 1. The first stage can be summarised byAlgorithm 1 and will result in a predicted takeoffsequence, with predicted takeoff times and feasiblecul-de-sac times for all aircraft. See details in 4.2.

Algorithm 1 (Stage 1. Takeoff sequence prediction)1. Heuristically generate an initial takeoff sequence

(4.2.1)2. Apply a rolling window through the current

takeoff sequence (4.2.2):3. for each window position do4. Optimise the sequence of aircraft within the

window using a branch-and-bound algorithm(4.2.3). Aircraft are added to a subsequenceone at a time, in a depth-first manner

5. for each aircraft added to the subsequence do6. Determine a takeoff time and a feasible

cul-de-sac time for the aircraft (4.2.4)7. Calculate a lower bound for the cost of

remaining aircraft (4.2.5)8. Prune any subsequence that is provably

suboptimal, record any complete sequencethat improves upon the best found sofar, or add the next aircraft to anincomplete subsequence

9. end for10. end for.

Each element is detailed in a separate section, asindicated. Step 6 involves the most complex taskbecause it is not always simple to find a feasiblecul-de-sac time. In some cases it will involve solving asimplified version of the cul-de-sac sequencing prob-lem, using the decomposition described in 4.3.

Stage 2. The second stage involves finding acul-de-sac time for each aircraft (from which a push-back time can be determined) such that cul-de-sacseparations are maintained and the total cost of devi-ation between the allocated and ideal cul-de-sac times(determined from the takeoff times, Equation (11)), asmeasured by Formula 12, is minimised.

This stage is much simpler than the first and hasthe following steps:

1. Decompose the problem according to cul-de-saccontention (4.3), using the takeoff times from the firststage.

2. Determine optimal cul-de-sac times for each air-craft in the contention group (4.4) using a pruneddepth-first search.

4.2. Takeoff Sequence PredictionThe takeoff sequencing stage has the followingcomponents.

4.2.1. Heuristic Presequencing. An initial takeoffsequence is generated using either a first-come-first-served sequence (ordering aircraft in increasing value

of etj ) or a heuristic prioritisation method to allowfor movement to achieve CTOTs, ordering aircraft inincreasing value of hj , where hj = ecj if ecj > etj +300 (i.e., the CTOT requires a delay), hj = lcj if lcj csi i Sj(Equation (15))

4. Calculate an upper bound (ubtj ) for the earliesttakeoff time that will meet the cul-de-sacsequencing constraints by assuming that pct1jis adopted as the cul-de-sac time (Equation (16))

5. if ubtj = lbtj (i.e., lbtj can be achieved by leavingthe cul-de-sac last) then

6. adopt ctj = pct1j and ttj = lbtj and end thealgorithm

7. end if8. Determine the set S2j Sj of aircraft that must

leave the cul-de-sac before j to avoid delayingtheir takeoff time, as defined by Equation (17).if S2j = then go to step 13

9. Let pct2j be the earliest cul-de-sac time to leaveafter every aircraft i S2j (Equation (18))

10. if 4pct2j + tdj +MRHj5= ubtj then11. adopt ctj = pct2j and ttj = pct2j + tdj +MRHj

and end the algorithm12. end if13. Solve the cul-de-sac sequencing problem to find

minimal ttj and a feasible ctj14. Adopt the resulting cul-de-sac and takeoff times

for j and end the algorithm.

Algorithm 2 defines the algorithm that is used to cal-culate the cul-de-sac and takeoff times. The following

equations define the sets and decision variables ituses.

lbtj = max(eptj + pdj + TD4j5+MRHj1 ecj1

maxi81000N 92 tsi 0}1 (14)

pct1j = max(eptj + pdj1max

iSj4cti +MSij5

)1 (15)

ubtj = max(lbtj1 pct1j + TD4j5+MRHj

)1 (16)

S2j ={i Sj tti < ectj +MSji + tdi +MRHi

}1 (17)

pct2j = maxiS2j

4ecti +MSij50 (18)

Algorithm 2 is invoked whenever an aircraft j isadded to the partial takeoff sequence, to determine atakeoff time (ttj ) as well as a feasible cul-de-sac time(ctj ) that will allow it to achieve that takeoff time. Theaim is to find an achievable takeoff time without hav-ing to solve the full cul-de-sac sequencing problem,if possible. Three attempts are made to find a take-off time and feasible cul-de-sac time, increasing thecomplexity each time.

Attempt 1: As long as accumulated delays are highenough then the cul-de-sac sequencing is irrelevantfor this j , because any cul-de-sac delays will beabsorbed within the normal delay awaiting takeoff.Attempt 1 (steps 27) involves determining whetherthe new aircraft, j , can just leave the cul-de-sac afterall earlier takeoffs with which it has a positive cul-de-sac separation requirement (set Sj ) have done so.If this is possible then the takeoff time is adopted andthe earliest cul-de-sac time that maintains cul-de-sacseparations from the aircraft in Sj is adopted as thefeasible cul-de-sac time.

Attempt 2: In low delay situations, cul-de-sac con-tention is often between just a pair of aircraft. Thesecond attempt (steps 812) involves a considerationof each aircraft i Sj , to see whether j could leave thecul-de-sac before i or not. If j leaving the cul-de-sacprior to i would prevent i from achieving its takeofftime, then j must leave after i and the lower bound forthe takeoff time of j may be tightened (i.e., the takeofftime for j may be delayed because of the cul-de-saccontention with i). If this new takeoff time is achiev-able by leaving the cul-de-sac after all aircraft in Sjthen the takeoff time and cul-de-sac time are adoptedas with Attempt 1.

Attempt 3: If the lower bounds cannot be achievedthen cul-de-sac resequencing is required to find theearliest takeoff time for j (step 13). All earlier takeoffsthat are in cul-de-sac contention with j (as defined

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in 4.3) are identified and are added to a cul-de-sacsequence one at a time in a pruned depth-first search.Each aircraft has an earliest (Inequality 2) and latest(Inequality 7) cul-de-sac time, defining a time win-dow for its potential cul-de-sac times. As each aircraftis added to the cul-de-sac sequence, it is allocatedthe earliest cul-de-sac time that it can achieve giventhe separations from aircraft earlier in the sequence(Inequality 3) and its own cul-de-sac window. As air-craft are added, the earliest times for the remainingaircraft are updated according to the minimum sep-arations from the current aircraft, and subsequencesare pruned if any aircraft are left without valid cul-de-sac times. The search can end as soon as a fullsequence is found that minimises ttj . This includesany sequence that achieves a cul-de-sac time equal topct2j (Equation (18)) or a takeoff time equal to lbtj(Equation (13)).

4.2.5. Lower Bounds. If aircraft remain to beadded to the current window, then a lower bound Lis determined for the cost of the remaining aircraft.The objective function (Equation (9)) has three compo-nents and a lower bound is determined for each com-ponent independently then summed to give a lowerbound for the overall cost.

The first-come-first-served sequence is utilisedfor both the equity and delay objectives, determiningtakeoff times by considering the earliest takeoff timefor each aircraft and assume one-minute separationsbetween aircraft. (Note that the delay component ofthe objective function is convex and real separationsare at least one minute.)

The nonconvexity of the CTOT cost has to beconsidered here. The earliest takeoff time of eachof the remaining aircraft is considered in isolation,assuming it takes off next in the sequence. All costsfor missed CTOTs or missed extensions are summedand added to the lower bound.

4.3. Decomposition Into Contention GroupsLet ecti and ectj denote the start of the cul-de-sac timewindows for aircraft i and j , respectively (see Inequal-ity 2), and let lcti and lctj denote the end of the cul-de-sac time windows (from Inequality 7). Two aircraft,i and j , are said to be in contention in the cul-de-sacif they have a strictly positive minimum cul-de-sacseparation requirement (i.e., MSij +MSji > 0) and theircul-de-sac time windows are close enough that theircul-de-sac times can conflict, i.e., 4lcti + MSij > ectj )and 4lctj +MSji > ecti5.

Cul-de-sac contention is important because cul-de-sac sequencing (step 13 of Algorithm 2) only hasto consider aircraft that are in contention with eachother. It is also used to decompose the second stageproblem (4.4) into mutually independent subprob-lems consisting of only those aircraft that are in con-tention with at least one other aircraft in the set.

4.4. Stage 2: Pushback Time Allocation forAircraft in a Contention Group

Algorithm 3 (Stage 2 algorithm, executed when anaircraft j is added)

1. Tighten the window for j as it is added to thesequence, adjusting ectj so that ectj ecti +MSij for all aircraft i already in the sequence

2. Windows for the aircraft that have not yetbeen added are also tightened in the samemanner, because any remaining aircraft mustcome after aircraft j in the cul-de-sacsequence, i.e., ectk ectj +MSjk for allaircraft k that have not yet been added

3. Adjust the end time of thewindow for j so that lctj lctk MSjk for allaircraft k that have not yet been added

4. Adjust the end times of the windows of allaircraft i already in the sequence so thatlcti lctj MSij for all aircraft i already in thesequence

5. if any window ceases to exist (i.e., ectl > lctl) forany aircraft l then prune the partial sequence

6. Determine a lower bound for the sequence costof each aircraft i (from Equation (12)), asmax41004max401 ecti icti5510114max401 icti lcti551015

7. if the sum of the lower bound costs for allaircraft (those in the sequence and those notyet added) exceeds the cost of the bestsequence found so far then prune the partialsequence

8. if all aircraft have been added then9. Place all aircraft as early as possible within

their time windows10. for each aircraft, from the last in the sequence

to the first do11. Increase the cul-de-sac time in one-minute

increments while doing so reduces thevalue of the objective function. Includingany cost for any pushing of later aircraft(to maintain the separations), which canonly raise the cost

12. end for13. Determine the sequence cost from the

resulting cul-de-sac times. Maintain the costand details of the best sequence found

14. end if.

A search is executed in this stage to find optimalcul-de-sac times for the aircraft based on the predictedtakeoff times from the first stage. The aircraft fromthe current contention group are added into a takeoffsequence one at a time in a depth-first search. Algo-rithm 3 is executed as each aircraft is added, to prunesuboptimal sequences or to optimise the times once acomplete sequence has been generated.

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An initial earliest (ectj , from Inequality 2) and latest(lctj , from Inequality 7) cul-de-sac time is determinedfor each aircraft j . These times are then modifiedthroughout the execution of the search, potentiallynarrowing the time windows. As more and more air-craft are added, the windows for each aircraft mayget progressively tighter and are used for pruning thesearch.

The developed implementation calculates andmaintains slack values for each pair of aircraft atstep 9, comparing the current and minimum separa-tions between them so that delays (and cost changes)can be enforced on later aircraft when earlier aircraftare delayed.

Despite the nonlinear nature of the objective func-tion and the sequence dependent separations, theproblem decomposition results in subproblems thatare usually very small (see 5.6) and can be solvedvery quickly. Finding optimal cul-de-sac times fora known sequence (steps 912) is also extremelyfast because the objective function is convex, thecul-de-sac time windows will usually be small, cul-de-sac times must lie on minute boundaries, and theobjective function (Formula 12) can be decomposedto consider the contributions of individual aircraft.

5. ResultsExperiments were conducted to determine thefollowing:

1. The effectiveness of the developed algorithmin terms of its ability to find high-quality takeoffsequences within a reasonable execution time for real-istically sized problems.

2. The potential benefits of stand holding atHeathrow.

3. Whether the minimum and ideal runway holdvalues have the expected effects.

4. Whether the cul-de-sac delays have to be consid-ered within takeoff sequencing.

Experiments were performed using 36 data setscontaining real recorded takeoff data covering sixhalf-day periods and consisting of 55 sequential take-offs each. All aircraft in each data set took off fromthe same runway, either 27R or 27L, and standardseparation rules (as described in Atkin 2008) wereassumed to be in operation. The takeoff times ofthe first five aircraft in each data set were fixed forthese experiments, providing the short-term historyrequired for comparability, leaving 50 aircraft (overan hour of takeoffs) free for sequencing. The sup-plied data included standard taxi times (for standgroup/destination runway pairs) and cul-de-sac con-tention information as well as pushback times andtakeoff times. It was assumed that there had beenno stand hold, so the recorded pushback time wasassumed to be the earliest pushback time possible.

Although there is no intention to provide predictedtakeoff times to controllers, the output of the take-off sequencing element will be used in this paper todetermine the effectiveness of the sequencing algo-rithm. The delays and CTOT compliances that wereobtained from the various experiments can then becompared against each other and against those for thereal takeoff times.

5.1. Manual and First-Come-First-ServedSequences

Table 5 shows the total number of CTOT time-slotsthat were missed across all 36 test data sets and theaverage delay for aircraft, in seconds, for the realand first-come-first-served sequences. The first row ofTable 5 shows the real controller performance and thesecond row shows the predicted performance fromapplying the takeoff time prediction method (utilisingInequalities 4, 6, and 5 and assuming no cul-de-sacdelays) to the real controller sequence and assessingthe predicted takeoff times. The third row is exactlythe same as the second except that the delay for thestart of CTOT (Inequality 5) is not enforced.

The details for some CTOT renegotiations wereobviously missing from the supplied data, so a num-ber of aircraft actually took off earlier than theirrecorded CTOT would have allowed. Comparisonof the second and third rows makes this apparent, andthe third row provides a closer approximation of whatthe controllers actually achieved. Nevertheless, ratherthan attempting to guess the reallocated CTOTs, theoriginal CTOTs have been kept in the data, to presentthe solution system with the original problem to seehow well it can do even without any CTOT advance-ments. Obviously, this will enforce a delay upon theseaircraft while they await the CTOT start time, but thesystem will be observed to perform well regardless ofthis handicap.

The predicted times have far greater delays than thereal times, but the discrepancy is lower when the startof CTOT is ignored. The reasons for these high pre-dicted delays are discussed in Atkin et al. (2007) and

Table 5 CTOT Compliance and Delay (s) for Manual andFirst-Come-First-Served Sequences

Total number Mean delay perType of sequence of CTOTs missed aircraft (s)

Real times for manual sequence 66 50502Predicted times, manual sequence, 89 79809

CTOT start enforcedPredicted times, manual sequence, 87 78708

CTOT start relaxedFirst-come-first-served, 205 1123204

CTOT start enforcedFirst-come-first-served, 134 1100504

CTOT start relaxed

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are due to controllers being able to safely reduce cer-tain separations at their discretion. Predicted takeofftimes tend to be later than may actually be achievedand the discrepancies can accumulate. A live systemfor the dynamic problem has to handle this by pro-viding feedback to realign predictions with reality orby assuming a reduction in some or all separations toprevent the misalignment of predictions. This realign-ment is beyond the scope of this paper, which willconsequently underestimate benefits from the systemimplementation than risk overestimating them.

The fourth row of Table 5 shows the results of tak-ing the first-come-first-served takeoff sequence andapplying the takeoff time prediction system. The fifthrow shows the results when the earliest CTOT timeconstraint is ignored. These results show that the first-come-first-served sequence is extremely poor, andhence a comparison of algorithm performance againsta first-come-first-served sequence would not be par-ticularly enlightening.

5.2. The Effects of the Window Size andNumber of Passes

Experiments were performed using a minimum run-way hold of one minute, an ideal hold of 10 minutes,and a maximum hold of 30 minutes. Table 6 showsthe results of applying the rolling window approachwith various window sizes, numbers of passes, andpresequencing algorithms (Basic shows the resultswithout presequencing). The better results are showngraphically in Figure 3. The points for the basicalgorithm and the type 3 presequencing algorithm

1Size:444445446447448449450

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2 2 2 22 2 2 2 2 2 3 44 4 42 3

3 3 3 33 3 3 333 4 1

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8 8 8 877776666555544444 4 4

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11

Figure 3 Graph of Mean Delay per Aircraft with Varying WindowSizes and Number of Passes

(discussed later) have been linked with lines to illus-trate the relationship between the points for each win-dow size.

These results show that the solution system stillstarts to find solutions that improve upon thecontroller-produced schedules as soon as six aircraftat a time are taken into consideration, or earlier ifmultiple passes of the window are permitted or pre-sequencing is implemented, despite the pessimistictakeoff time prediction system. Because a runwaycontroller is unlikely, at present, to be able to utilisethe amount of information that an automated systemcan, this result is less surprising than it may seem.

Type 1 presequencing involved heuristically pre-sequencing the aircraft in order to move aircraftwith CTOTs closer to their likely takeoff positions, asexplained in 4.2.1. Comparison of the results for thebasic algorithm and the type 1 presequencing showsthat the presequencing was very useful for improv-ing the delay and CTOT compliance until the windowsize and number of passes were large enough to easilyaccommodate the moves required to meet the CTOTs.

5.3. Execution TimeEach experiment was executed three times for eachdata set, using a single-threaded version of the algo-rithm running on a desktop PC with a 2.4 GHz CPU.8

The mean and maximum runtimes of the algorithmacross the various data sets are shown in Table 7and key mean results are illustrated in Figure 4. Thisdeterministic algorithm gives the same delay andCTOT results, with extremely similar execution times,when reapplied to the same data set, but the execu-tion times can vary greatly between the data sets.

Both the mean and maximum runtimes for the basicalgorithm increased rapidly as the window size wasincreased, but subsequent passes of the rolling win-dow can be observed to usually be much faster thanthe initial pass, implying that there is an advantage tobe gained from having a better initial takeoff sequenceon the subsequent passes. One potential improvementcould be to change the order of consideration of air-craft. Unfortunately, it is not obvious which aircraft tochoose to add first, because CTOTs, delay, and equityobjectives conflict (see Atkin, Burke, and Greenwood2010), partial sequences are very time dependent andthe results in Figure 3 show distinct benefits from con-sidering more aircraft for each position when seek-ing a good takeoff sequence, so a greedy, myopicapproach is unlikely to be successful.

8 We note that the algorithm is CPU bound and that the memoryrequirements are extremely low. A faster CPU has been observedto give lower runtimes, but these were achievable with relativelyinexpensive equipment. We also note that the branch-and-boundelement could be parallelised for multicore computers, if desired.

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Table 6 Mean CTOT Compliance and Delay (s) for Varying Window Size and Passes

Total CTOTs missed Mean delay per aircraft (s)

Presequencing PresequencingWindow Windowsize passes Basic 1 2 3 Basic 1 2 3

1 1 205 151 16 13 1123204 1103509 47306 465052 1 97 77 19 18 80009 72903 49500 490002 2 63 60 19 18 68707 65009 49606 491012 3 53 55 19 18 66206 63403 49606 491002 4 52 54 19 18 65602 63004 49606 491073 1 56 45 15 13 61500 58504 47905 472063 2 30 26 15 13 54404 53309 47804 472003 3 22 21 15 13 52509 51506 47708 472003 4 21 20 15 13 51806 51000 47708 472004 1 33 28 14 13 54203 52808 46807 465044 2 20 17 14 13 49709 49100 46509 463054 3 17 14 14 13 48302 48105 46505 463024 4 15 14 14 13 47803 47703 46505 463025 1 27 22 13 13 50902 49401 46502 457075 2 15 13 13 13 47306 46505 46207 457015 3 13 13 13 13 46502 45707 46202 454085 4 13 13 13 13 46207 45701 46201 455096 1 21 20 13 13 48104 47507 45807 455056 2 13 13 13 13 45903 45803 45300 451036 3 13 13 13 13 45109 45009 45208 451026 4 13 13 13 13 45102 45007 45300 451047 1 19 15 13 13 46703 46109 45404 450067 2 13 13 13 13 44908 44807 45206 449077 3 13 13 13 13 44608 44700 44908 447047 4 13 13 13 13 44600 44608 44803 447048 1 17 15 13 13 45707 45507 45108 448058 2 13 13 13 13 44508 44600 44807 446038 3 13 13 13 13 44409 44501 44708 445078 4 13 13 13 13 44405 44407 44604 44507

The algorithm was instead provided with a bet-ter initial sequence by applying a two-pass five-aircraft version of the sequencing algorithm as apresequencing stage. Type 2 and 3 presequencinginvolves the application of this algorithm to the first-come-first-served sequence and the heuristically pro-duced (type 1 presequencing) sequence, respectively.The effects can be observed by comparing type 1 pre-sequencing with type 3 presequencing and the basicresults with type 2 presequencing. This presequencingconsiderably reduced the execution time for windowsizes over 5, and the reductions were greater for thelarger window sizes. For smaller window sizes it alsoimproved the CTOT compliance and delay, althoughthe delay was slightly increased for some number ofpasses with window sizes 7 and 8.

Comparison of the results for one-pass of a two orthree aircraft window with type 2 or 3 presequenc-ing against the results for the two-pass five-aircraftalgorithm with no or type 1 presequencing (respec-tively) shows that the two or three aircraft rollingwindow actually takes a better initial sequence andmakes it worse in these cases rather than improvingit. This illustrates that locally good solutions often do

not create globally good solutions and provides fur-ther evidence that a greedy myopic branch selectionapproach is unlikely to perform well.

Given these results, a three-pass seven-aircraftrolling window algorithm was adopted for theremaining experiments, with presequencing type 3.This had a worst-case runtime of just over one minuteand a mean runtime of only 13.8 seconds. Moreover,the larger windows did not perform much better andthe dynamic nature of the full problem (see 2.7) islikely to restrict the accuracy of information, reducingthe benefits from larger windows.

5.4. The Effect of the Minimum and IdealRunway Hold

Despite the predictability of taxi durations atHeathrow (see 2.5), there will always be some levelof uncertainty and this is handled by introducingslack into the system using ideal and minimum run-way hold duration parameters.

Experiments were performed with differing idealand minimum runway hold values. In each case themaximum runway hold was set to be 30 minutes. Theresults are shown graphically in Figure 5. Points have

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Table 7 Mean and Maximum Execution Time for Varying Window Size and Number of Passes Over 36 Data Sets

Mean runtime (s) Maximum runtime (s)

Presequencing PresequencingWindow Windowsize passes Basic 1 2 3 Basic 1 2 3

1 1 002 001 109 108 107 008 407 4002 1 001 001 201 200 008 005 501 4032 2 002 002 201 200 006 005 502 4032 3 002 002 201 200 006 200 502 4042 4 003 002 202 201 007 006 502 4043 1 002 002 201 200 004 006 502 4043 2 003 003 202 201 006 005 503 4053 3 004 004 204 203 007 202 505 4073 4 006 006 205 204 009 009 507 4094 1 004 004 203 202 008 008 505 4084 2 008 007 206 205 104 104 509 5024 3 101 100 209 208 201 208 605 5074 4 104 104 301 300 207 207 609 6025 1 102 101 207 206 206 205 607 6035 2 200 109 304 303 409 403 805 8005 3 207 206 401 400 608 603 1002 9065 4 305 303 408 407 805 709 1200 11056 1 305 302 400 308 908 907 1102 10026 2 507 505 508 506 1608 1708 1701 15086 3 706 703 706 704 2303 2400 2303 21016 4 903 900 904 901 2904 3001 2903 26057 1 1104 1000 705 700 3706 3903 3109 27037 2 1702 1504 1206 1108 6300 6604 5708 47057 3 2202 2002 1704 1603 9001 9307 8304 68037 4 2608 2409 2200 2007 11603 11907 10905 88088 1 4300 3407 1707 1703 20701 19606 9101 101018 2 5803 5000 3101 3002 30405 30001 15401 159028 3 7009 6207 4400 4206 33302 34407 22804 233058 4 8303 7502 5608 5409 40801 42102 30102 30603

888877776666555544443333222211110

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Size:Passes: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 2

Window size and number of passes3 4 1 2 3 4 1 2 3 4 1 2 3 41

Figure 4 Graph of Mean Algorithm Runtime per Data Set for VaryingWindow Sizes and Number of Passes

been plotted for five different values of the minimumrunway hold (from one to five minutes). The resultsfor each value of the minimum runway hold parame-ter have been connected, to illustrate the relationshipbetween them. The horizontal lines in Figure 5 showthe mean total hold time (runway + stand hold) forthe aircraft for each value of the minimum runwayhold, illustrating that it is independent of the idealrunway hold value. For any given minimum runwayhold, the vertical distance between the point for theideal runway hold value and the horizontal line forthe appropriate total runway hold illustrates the meanstand hold that was obtained.

These results show that both the total hold and themean runway hold increase as the minimum runwayhold is increased (because schedules that require alower delay in the holding area are no longer achiev-able) and that the stand hold decreases as the idealrunway hold is increased, even though the total holdis unaffected.

As the minimum runway hold is increased, thelikelihood of a controller being able to achieve theplanned takeoff sequence (allowing for deviationsbetween actual and predicted taxi times) is increased,

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One minute minimum runway holdTwo minute minimum runway holdThree minute minimum runway hold

900840780720660600540480420Ideal holding area delay (seconds)

360300240180120600

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Total hold with four minute minimum runway hold

Total hold with three minute minimum runway hold

Total hold with two minute minimum runway hold

Total hold with one minute minimum runway hold

Four minute minimum runway holdFive minute minimum runway holdKey:

Figure 5 Graph of Holding Area Delay by Ideal and Maximum Runway Hold

but the value of the planned takeoff sequences isdecreased and the controller is more likely to beable to find a better sequence, utilising lower runwayholds than MHRj would allow.

Increasing the ideal runway hold increases theaverage number of aircraft queued in the holdingarea, thus, rather than increasing the likelihood thatthe predicted take-off sequence will be achievable, itincreases the chance that the controller would haveaircraft available from which to create an alternative,but potentially as good, takeoff sequence. An unnec-essarily large ideal runway hold would reduce theamount of the delay that is absorbed at the stand andwould risk unnecessarily congesting the holding area.

Major potential fuel-burn benefits can be observedfrom Figure 5. Because aircraft actually need lessthan a minute in the holding area, the one or twominute minimum holding area delay results are prob-ably the most realistic. With a (probably excessive)two minute minimum runway hold, these resultsindicate a potential reduction in mean holding areadelay (and idle engine running time) of more than50% (from 505.4 seconds to 249.1 seconds), even withthese pessimistic takeoff times. Even with an idealrunway hold of 10 minutes, the mean runway holdtime was predicted to be reduced by more than 27%

(to 366.4 seconds), resulting in a considerable reduc-tion in fuel burn.

5.5. The Effects of Cul-de-Sac ContentionAlgorithm 2 aims to reduce the number of times thatcul-de-sac sequencing has to be considered, and wasfound to be extremely effective. On average, across alldata sets, the effects of cul-de-sac delays on the takeofftime had to be considered only around 150,000 timesper data set. Approximately two-thirds of the timeconsidering two-aircraft contention was sufficient toprevent the need for cul-de-sac resequencing. Of theremaining cases, the takeoff time was only delayedin an average of 30 cases per data set beyond thedelays that were already calculated for two-aircraftcontention. In all other cases, the aircraft could beresequenced to avoid the contention.

However, when experiments were performed with-out taking cul-de-sac contention into considerationwithin the takeoff sequencing stage, the majority ofcases presented infeasible problems to the pushbacktime algorithm. Further experimentation revealed thatsome infeasible second stage problems were still gen-erated even when two-aircraft contention was takeninto account (i.e., three or more aircraft in mutualcontention were causing the delays). Thus, cul-de-sac

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delays cannot be ignored within the takeoff sequenc-ing stage.

5.6. The Second StageThe second stage algorithm took much less than onesecond (almost always less than 150 milliseconds) toallocate pushback times to all aircraft in the data setfor all problems where a window size of 2 or morewas used. The largest problem size that had to beconsidered was eight aircraft, except with a windowsize of 1, when the first-come-first-served sequenc-ing was so poor that the second stage problems weremuch harder to solve and took up to 1.7 seconds insome cases without presequencing, and 0.7 secondswith heuristic (type 1) presequencing. Thus, the sec-ond stage problems were solved extremely quicklycompared with the takeoff sequencing element of theproblem.

6. ConclusionsThis paper considered the problem of allocating push-back times to aircraft at Heathrow Airport. The prob-lem involves first predicting the delays that aircraftare likely to have by finding an achievable takeoffsequence that controllers will value, then allocatingappropriate stand holds to aircraft in order to absorbdelay at the stands while not preventing aircraft fromachieving these takeoff times. The problem and itscharacteristics were explained extensively, and expla-nations were given both for why the problem differsfrom many similar existing problems and the difficul-ties that many solution methods have with it.

A solution algorithm was described. The perfor-mance of this algorithm has previously been com-pared against an earlier system in Atkin, Burke, andGreenwood (2009) and the tuning of the objectivefunction toward controller preferences was consid-ered in Atkin, Burke, and Greenwood (2010), how-ever this is the first paper to provide full details ofthe algorithm, to explain the design decisions, and toconsider the ways in which the algorithm parametersaffect the performance.

This paper considers the static pushback time allo-cation problem. The necessary enhancements for thedynamic problem were outlined in 2.7 and are rela-tively minor but have interesting and complex effectsthat have been left for future explanation and discus-sion. Following evaluation of the algorithm outputsby controllers, this system has been implemented forLondon Heathrow. The current tuning phase involvesthe validation of the various assumptions discussedin 3.9 as well as the handling of any lack of pre-dictability in taxi times, pushback times, ready times(TOBTs), and the reduced separations that controllerscan use at their discretion.

This system considers two separate sequencingproblems, ensuring in the first that a feasible solution

is possible for the second. The takeoff sequencingsubproblem was observed to be the hardest elementof the TSAT allocation problem to solve. The solu-tion algorithm attempts to avoid having to solvethe cul-de-sac sequencing problem within the takeoffsequencing stage, but this is sometimes unavoidable.The second (pushback time allocation) problem canbe solved extremely quickly once takeoff times areknown, because the problems to solve are small andvery constrained.

The primary conclusion is that the solution methodexecutes quickly enough to provide real-time decisionsupport for controllers, even given the larger-than-realistic problems that were considered here. Mul-tiple passes, heuristic presequencing, and a smallerwindow presequencing stage were all observed toimprove the results.

The secondary conclusion is that there are largepotential benefits from applying a pushback time allo-cation system. Much of the delay for aircraft couldbe absorbed at the stands instead of the holding area,reducing the amount of waiting time with the enginesrunning by about 50% for reasonable parameter val-ues. Results also showed that even the applicationof a stand delay to only those aircraft with a largeexpected delay (for example, over 10 minutes) couldreduce the mean hold time at the runway by morethan 27%. This has obvious benefits for fuel burnand the associated financial and envi

addressing the pushback time allocation problem at heathrow

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