equitable and efficient coordination in traffic flow management - mit

Vol. 46, No. 2, May 2012, pp. 262–280ISSN 0041-1655 (print) � ISSN 1526-5447 (online) http://dx.doi.org/10.1287/trsc.1110.0393

© 2012 INFORMS

Equitable and Efficient Coordination inTraffic Flow Management

Cynthia BarnhartDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, [email protected]

Dimitris BertsimasSloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,

[email protected]

Constantine CaramanisDepartment of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712,

[email protected]

Douglas FearingOperations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,

[email protected]

When air traffic demand is projected to exceed capacity, the Federal Aviation Administration implementstraffic flow management (TFM) programs. Independently, these programs maintain a first-scheduled, first-

served invariant, which is the accepted standard of fairness within the industry. Coordinating conflicting pro-grams requires a careful balance between equity and efficiency. In our work, we first develop a fairness metricto measure deviation from first-scheduled, first-served in the presence of conflicts. Next, we develop an integerprogramming formulation that attempts to directly minimize this metric. We further develop an exponentialpenalty approach and show that its computational performance is far superior and its tradeoff between delayand fairness compares favorably. In our results, we demonstrate the effectiveness of these models using histor-ical and hypothetical scenarios. Additionally, we demonstrate that the exponential penalty approach exhibitsexceptional computational performance, implying practical viability. Our results suggest that this approachcould lead to system-wide savings on the order of $25 to $50 million per year.

Key words : traffic flow management; ground holding programs; equitable flight delayHistory : Received: June 2009; revisions received: March 2010, March 2011; accepted: August 2011. Published

online in Articles in Advance February 8, 2012.

1. IntroductionThe Federal Aviation Administration (FAA) and theairline industry face tremendous challenges fromunexpected weather-induced reductions in systemcapacity and resulting delays. The U.S. Congress JointEconomic Committee estimates that during calendaryear 2007, 2.75 million hours of flight delays led toapproximately $25.7 billion in costs to the U.S. econ-omy—$12.2 billion in increased airline operating costs,$7.4 billion in passenger time lost, and $6.1 billion incosts to related industries (Joint Economic Committee2008). To put this in perspective, the Air Transporta-tion Association’s 2008 Annual Report lists total prof-its for U.S. airlines of approximately $3 billion for the2006 operating year and $5 billion for the 2007 operat-ing year (Air Transportation Association 2008).

Delays over the national air space are projected tooutpace increases in overall traffic. The FAA’s 2004Airport Capacity Benchmark Report demonstratesthat many major U.S. airports regularly operate at or

near peak capacity (e.g., Hartsfield-Jackson AtlantaInternational, New York LaGuardia, and ChicagoO’Hare; U.S. Department of Transportation 2004). Aswith any queueing system, there is a nonlinear rela-tionship between delay and changes in demand whenoperating under these conditions (de Neufville andOdoni 2003). Increasing capacity by building addi-tional runways and airports is logistically complexbecause of cost, space limitations, and environmentalregulations. Additionally, projects of this type oftentake a decade or more to plan and complete. It isthus critical to consider tools to improve operationalefficiency. In §1.4 we detail the contributions of thispaper. But first, to put them in context, we providea brief discussion of the existing traffic flow manage-ment tools and a nonexhaustive literature review.

1.1. Traffic Flow ManagementTraffic flow management (TFM) refers to a set ofstrategic practices utilized by the FAA to ensure safe

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Barnhart et al.: Equitable and Efficient Coordination in TFMTransportation Science 46(2), pp. 262–280, © 2012 INFORMS 263

operations while attempting to minimize costs asso-ciated with delay. TFM activities occur on the dayof operations and generally affect a significant subsetof airline traffic (e.g., all flights into a major airport).According to data publicly available from the U.S.Bureau of Transportation Statistics, we estimate thatTFM activities account for approximately 30% of allair transportation delays (see calculations in §4.5 fordetails). Based on factors such as number of runways,runway configuration, scheduled personnel coverage,and weather forecasts, the FAA determines maximumcapacities for resources in the U.S. air transporta-tion system. These resources include arrival runways,departure runways, and air sectors in the NationalAirspace System (NAS). TFM programs are initiatedonly when significant imbalances between demandand capacity are expected, as in the midst of a severestorm. Minor to moderate inconsistencies betweencapacity and demand are resolved through localizedair traffic control (ATC) techniques (e.g., speed adjust-ments, vectoring, or airborne queueing). Since the airtraffic controllers’ strike in 1981, the primary tool usedfor TFM has been the ground delay program (GDP).In a GDP, the FAA controls the arrival rate into areduced-capacity airport by coordinating departuretimes for affected flights. The goal is to allow eachaircraft to proceed safely to its destination with min-imal airborne delay. Airspace flow programs (AFPs),first introduced in 2006, are operated much like GDPs.The FAA uses an AFP to control the arrival rate into aflow constrained area (FCA), for example, a reduced-capacity air segment of the NAS. The papers byde Neufville and Odoni (2003), and Ball et al. (2007)provide further details regarding the TFM problemand its extensions. To understand the prevalence ofthese programs, we list in Table 1 the number of daysin 2007 that the corresponding numbers of GDPs andAFPs were enacted (e.g., there were only 16 dayswith no GDPs or AFPs). Thus, on approximately 40%of the days during 2007, at least one GDP and atleast one AFP was in place. Although the number ofGDPs varies significantly, the number of AFPs rarelyexceeds two.

Table 1 Number of Days in 2007 and Corresponding Numbers of TFMPrograms of Each Type

Number of GDPsNumberof AFPs 0 1 2 3 4 5 6 7 8 9 10 11 Total

0 16 24 31 44 26 28 15 13 8 2 2 1 2101 5 10 13 7 5 11 5 6 3 0 0 1 662 2 15 13 14 7 16 5 9 3 2 1 0 873 0 0 0 0 0 1 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 1 0 0 0 1

Total 23 49 57 65 38 56 25 28 15 4 3 2 365

GDPs and AFPs are used in concert with a three-stage, collaborative approach to decision making.In the first stage, the FAA allocates arrival slotsto airlines by applying the ration-by-schedule (RBS)method for each TFM program. Arrival slots are allo-cated according to the original schedule ordering, asdescribed in detail in the following section. Althoughfairness is a subjective criterion, the RBS approachis generally considered fair within the airline indus-try because it maintains a first-scheduled, first-servedinvariant. In the second stage, airlines respond to theschedule disruption. Each airline is allowed to makechanges to the schedule within the context of theslots allocated to it. For instance, an airline can swaparrival slots for two of its own flights as long as theswap does not cause either flight to depart prior toits posted departure time. Airlines can also choose tocancel flights in response to operational constraintson aircraft routing, crew assignments, and so forth.In the third stage, the FAA accepts the changes pro-posed by all airlines. These changes, when combined,constitute a capacity-feasible schedule because eachairline is only allowed to make changes within the setof slots allocated to it. Subsequently, the FAA attemptsto improve the schedule by filling any gaps created bycancellations or operator-announced delays. This pro-cedure, known as compression, is described in detailin Vossen and Ball (2005). After compression, the newschedule proposal is sent to the airlines and the pro-cess is repeated as necessary.

1.2. Coordinating Multiple ProgramsIn RBS, arrival slots for a single resource, either anairport in a GDP or an FCA in an AFP, are allocatedto flights according to the original schedule order. ForFCAs, the scheduled arrival order is based on theestimated arrival times into the FCA (i.e., the sched-uled departure time plus the estimated en route timeto reach the FCA). Once the controlled arrival slotshave been allocated for a resource, each affected flightreceives a corresponding controlled time of departure(CTD) from its origin, converting the allocated arrivalslot into a departure delay at the airport of origin.

When multiple TFM programs are implemented onthe same day, applying RBS for each independentlymay lead to a single flight receiving conflicting CTDs(e.g., from a GDP and one or more AFPs). To resolvethese conflicts, the FAA uses heuristics to determinea CTD for each flight based on the order in whichthe programs are initiated over the course of the day.When AFPs were created in 2007, there was longhistory of successful implementation of GDPs; thus,GDPs were given priority by default. Specifically, aflight already affected by a GDP at the time an AFPis initiated is exempted from the AFP (reducing theAFP capacity for other affected flights). But for a flight

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Barnhart et al.: Equitable and Efficient Coordination in TFM264 Transportation Science 46(2), pp. 262–280, © 2012 INFORMS

C

D AB

LaGuardiaAirport

FCA1

Figure 1 Visual Representation of Flight Routes for Flights A, B, C,and D

already affected by an AFP at the time a GDP is initi-ated, the CTD is modified to correspond to the GDP(Federal Aviation Administration 2005). This conflict-resolution heuristic, referred to as precedence RBS, isthe default behavior in Flight Schedule Monitor, theapplication developed by Metron Aviation that theFAA uses to manage GDPs and AFPs (Metron Avia-tion 2009a). In March 2009, functionality was enabledin Flight Schedule Monitor that allows TFM managersto exempt AFP-affected flights from subsequent GDPs(Metron Aviation 2009b). Thus, the other conflict res-olution approach we consider is a strict exemption-based heuristic in which a flight is given a CTD fromthe first GDP or AFP that affects its schedule andexempted from all future GDPs or AFPs. We refer tothis conflict resolution heuristic as exemption RBS.

Consider the following example based on the fourflight routes displayed in Figure 1 with plannedschedule details listed in Table 2. At 17:00, an AFP isinitiated for FCA1 with a controlled arrival rate of oneflight every five minutes from 18:40 until 19:00. Sub-sequently, at 17:05, a GDP is initiated for New York’sLaguardla Airport (LGA), with arrivals into LGArestricted to one flight every 10 minutes from 18:55until 19:15. Note that the time a TFM program is ini-tiated determines which flights are affected, becauseflights already in the air at the time of initiation areexempted from the program. Performing RBS for eachresource independently leads to the CTDs listed inTables 3 and 4. In this case, flight B receives conflict-ing CTDs (17:15 from the AFP at FCA1 and 17:24 fromthe GDP at LGA). Thus, according to precedence RBS,flight B will be given a CTD of 17:24 (because the GDPat LGA takes precedence). This leads to the controlledschedule listed in Table 5. In Table 6, we provide

Table 2 Planned Departure and Arrival Times

Flight Departure FCA1 LGA

A 17:45 — 18:55B 17:15 18:40 18:56C 18:00 18:45 —D 18:15 18:46 —

Table 3 RBS CTDs for FCA1

Flight Slot CTD

B 18:40 17:15C 18:45 18:00D 18:50 18:19

Table 4 RBS CTDs for LGA

Flight Slot CTD

A 18:55 17:45B 19:05 17:24

the controlled schedule according to the exemptionRBS (under the modified assumption that the GDP isimplemented first).

It is important to note that, in this example, itis impossible to satisfy first-scheduled, first-servedfor each resource and simultaneously minimize sys-tem delay. That is, the controlled schedule musteither deviate from first-scheduled, first-served orincur excess delay. This simple example illustratesthe general tradeoff that exists between fairness andefficiency in the multiresource or network setting.We will consider these examples further in §2.1. Foradditional examples of this type, the reader is encour-aged to review Lulli and Odoni (2007).

1.3. Literature ReviewThe first thorough review of the TFM problem isprovided by Odoni (1987). Subsequently, two leadingresearch paths have emerged and diverged. The firsthas focused primarily on collaboration and equityin the context of single-resource approaches. Theseapproaches are applicable for a single TFM pro-gram or multiple, nonconflicting TFM programs. Thesecond research path has focused primarily on com-putational efficiency in the context of network-wide,

Table 5 Controlled Departure and Arrival TimesAccording to Precedence RBS

Flight CTD FCA1 LGA

A 17:45 — 18:55B 17:24 18:49 19:05C 18:00 18:45 —D 18:19 18:50 —

Table 6 Controlled Departure and Arrival TimesAccording to Exemption RBS

Flight CTD FCA1 LGA

A 17:45 — 18:55B 17:24 18:49 19:05C 18:05 18:50 —D 18:24 18:55 —

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or multiresource, approaches to TFM, in which typ-ically all airports and air sectors are placed underthe FAA’s control. Research into single resourceapproaches has gained more traction within theindustry primarily because of (i) the inclusion of col-laboration and equity considerations and (ii) com-putational tractability (i.e., computations involving afull day of flights for a single resource run quickly).Arguably, it is because of failures in these areas thatresearch into network approaches has gained lesstraction. Indeed, it is only recently that network for-mulations have begun to incorporate equity or collab-oration considerations.

For the single-resource TFM problem, determinis-tic, static-stochastic, and dynamic-stochastic versionsof the problem were first formulated in the early1990s (see Richetta and Odoni 1993, 1994; Terrab andOdoni 1993). More recent research has extended thesemodels to incorporate collaboration and equity (seeBall et al. 2003; Vossen and Ball 2005; Kotnyek andRichetta 2006). Vossen et al. (2003) define a measure ofequity for the single-resource TFM problem and cal-culate the inequity associated with flight exemptions.Chang et al. (2001) describe the collaborative decisionmaking (CDM) approach with updated equity con-siderations that was incorporated into the FAA’s GDPin the late 1990s. In a recent paper, Brennan (2007)describes how the CDM-enhanced GDP approach hasbeen extended to the AFP.

On the multiresource side, Vranas, Bertsimas,and Odoni (1994) developed the first integer pro-gramming formulation for the multi-airport GDP.Bertsimas and Patterson (1998) extended this formula-tion to the full air traffic system using a novel variabledefinition. Subsequent research has focused primar-ily on computational efficiency and the incorporationof rerouting constraints (see Andreatta, Brunetta, andGuastalla 2000; Hoffman and Ball 2000; Bertsimas,Lulli, and Odoni 2011). The discussion of inequitiesinherent in a network formulation of the TFM prob-lem by Lulli and Odoni (2007) provides a critical back-drop for our work. Bertsimas, Farias, and Trichakis(2011) show that under reasonable assumptions thetheoretical price of fairness in TFM is bounded andtypically quite low, which is consistent with ourcomputational results. In recent and related work,Bertsimas and Gupta (2011) consider fairness andcollaboration in the context of the nationwide TFMproblem.

In this paper, we develop integer programming for-mulations for the multiresource TFM problem thatincorporate fairness considerations. Unlike other net-work approaches, instead of including all airports andsectors, we restrict the problem to the coordinationof multiple, conflicting TFM programs. We believethat by considering this restricted problem, our work

will help bridge the gap between the two divergentresearch paths described above.

1.4. ContributionsThe contributions of this paper fall into four cate-gories.

1. Demonstrating the potential inefficiencies of theTFM conflict resolutions approaches utilized in prac-tice. The general TFM scheduling problem is NP-hard,so these inefficiencies are not surprising. Nonetheless,this result is not well understood. Most important, wedemonstrate that these inefficiencies are not just the-oretically plausible but are realized in historical sce-narios.

2. Developing a fairness metric that extends to themultiresource setting, including analysis of the result-ing fairness properties and relationship to currentindustry standards.

3. Developing two optimization approaches forcoordinating TFM programs that balance the tradeoffbetween equity, as measured by catagory 2 above, andefficiency, as measured by aggregate system delay.The latter of these two models is computationallytractable for national-scale TFM problems.

4. Generating computational results and analysisusing large-scale historical instances derived from2007 data obtained from Flight Schedule Monitor, thetool used to manage these programs.

The structure of the paper follows these mainpoints. In §2, we demonstrate the limitations of thecurrent TFM conflict resolution approaches, discussinherent fairness properties, and develop our fairnessmetric. In §3, we develop two integer programmingformulations. Lastly, in §4, we report and discuss ourcomputational results.

The starting point for our formulations is the modeldeveloped in Bertsimas and Patterson (1998) and thefirst-scheduled, first-served concept of fairness inher-ent in RBS, as described in §1.1. RBS has three salientfeatures. First, it is algorithmically trivial to imple-ment and has a linear running time with respect tothe number of flight steps. Thus, the approach canbe scaled to arbitrarily large problems. Second, for anisolated GDP or AFP, the RBS method always leadsto a solution that minimizes system delay (Vossenand Ball 2005). Third, it maintains a first-scheduled,first-served invariant, which is the industry acceptednotion of fairness endorsed by the primary stake-holders, that is, the FAA and the airlines. In partic-ular, as should be apparent based on the examplein §1.2, any multiresource extension of RBS will failon a very important front: it will no longer providedelay-optimality guarantees as in the single-resourcecase. This is to be expected, because fairness may ingeneral come at the expense of increased aggregatedelays. The main modeling contribution of this paper

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is precisely to address this deficiency. Specifically, weseek a formulation for fairness that has the followingproperties:

1. In the single-resource setting, it should reduceto (the accepted standard) RBS, which, as discussedabove, is delay-optimal in this case;

2. Because there will typically be a tradeoff be-tween aggregate system delay and any flight-basedfairness criterion, the formulation should essentiallyconsider a bicriterion approach that enables the studyof the tradeoff curve between the two; and

3. The formulation should compare favorably tothe approaches currently utilized in practice for themultiresource setting.

Using historical TFM scenarios, we demonstrate acomputationally viable optimization formulation thatsatisfies all of these properties. We estimate that thismodel can reduce flight delays by 4% or more onsome of the worst days, resulting in system-widesavings on the order of $25 to $50 million annu-ally. The concepts and modeling approaches devel-oped in our paper readily extend to the nationwideTFM problem, although we focus almost exclusivelyon the problems associated with coordinating GDPsand AFPs. We do so with the goal of having ourwork provide a bridge, both academically and prac-tically, between current approaches and a long-termvision of nationwide TFM. That is, we hope thatour work allows future TFM research, including ourown, to build upon a foundation that has a highlikelihood of being accepted in practice. Additionally,as the frequency and complexity of AFPs increasesbecause of increasing en route congestion, we expectthe inefficiencies we identify with current approachesto be exacerbated, providing further justification foran optimization-based approach.

2. Ration-by-Schedule and FairnessAs discussed in the introduction, understanding andincorporating industry-accepted views of fairness hasbeen a significant roadblock to the implementationof optimization-based techniques for managing TFMprograms. One of the more significant challenges isthat the first-scheduled, first-served concept of fair-ness underlying RBS does not directly extend to thesetting in which a single flight may interact with mul-tiple TFM programs (e.g., a GDP plus one or moreAFPs). With this in mind, we turn our attention todeveloping a measure of overall schedule fairness that(i) is consistent with first-scheduled, first-served in asingle-resource environment and (ii) naturally extendsto the setting in which there are interactions betweenTFM programs.

To provide additional context, we first illustrateproblems with the multiresource RBS approaches uti-lized in practice to resolve conflicts between conflict-ing TFM programs (i.e., GDPs and AFPs). The main

advantage of these approaches is that they are simpleextensions of RBS in the single-resource setting, andthus the resulting schedules are similar to the single-resource RBS schedules. Unfortunately, this simplicitycan also lead to significant costs in terms of efficiencyand therefore total delays. Next, we describe the prop-erties we believe should underlie any measure ofschedule fairness in a multi-resource setting, usingsimple examples to demonstrate the importance andsignificance of these properties. Lastly, we develop arobust measure of schedule fairness that incorporatesthe properties we outline. The purpose of this met-ric is to evaluate the relative fairness of competingscheduling approaches.

2.1. Problems with MultiresourceRation-by-Schedule

One downside of precedence RBS that is not a fac-tor with the exemption alternative is that AFP capaci-ties, specified in terms of controlled arrival rates, maybe (and often are) violated. By examining the con-trolled schedule from the example in §1.2 (Table 5),we see that two flights (B and D) are scheduled toarrive at FCA1 simultaneously even though the con-trolled arrival rate was established at one flight everyfive minutes. It is difficult to measure the impact asso-ciated with this issue because, in practice, AFPs areconstructed in a subjective fashion. That is, the param-eters of each AFP, such as duration and arrival rate,are tweaked until the end result satisfies subjectivecriteria for safety. Additionally, with an AFP trafficflow is controlled through a line or region of air spacethat may be hundreds of miles long; thus, two flightsthat arrive at the same time may be very far apartgeographically. Nonetheless, precedence RBS makes itdifficult, if not impossible, to precisely control traf-fic flow through the air. Additionally, as air trafficcongestion continues to increase, airspace controls areexpected to become more common, exacerbating thisproblem.

A more significant issue with both multiresourceRBS scheduling approaches is that either may leadto inefficient resource utilization. For instance, con-sider the planned flight schedule in Table 2 and theprecedence RBS schedule in Table 5. In the prece-dence RBS schedule, we see that the 18:40 arrival slotfor FCA1 is unused because flights C and D cannotdepart earlier than planned. If we swap the orderof flights A and B into LGA, flight B is then ableto use the 18:40 slot, which allows flights C and Dto depart on time (using the same capacity profileas the precedence RBS schedule). This sequence ofexchanges reduces the total delay from 13 minutes(9 minutes for flight B and 4 minutes for flight D) to10 minutes (all for flight A). Similarly, consider theexemption RBS schedule provided in Table 6. This

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schedule results in 23 minutes of delay (9 minuteseach for flights B and D, and 5 minutes for flight C).As with the precedence RBS example, the 18:40 slotinto FCA1 is unused. If we swap the order of flightsA and B into LGA, flight B is able to use the 18:40 slot,which allows flight C to depart on time and flight Dto depart 4 minutes late, resulting in 14 minutes ofdelay (10 minutes for flight A and 4 minutes for flightB). Note that the exemption RBS schedule results inmore delay because, in contrast to the precedence RBSschedule, no AFP capacity violations are allowed.

The last issue with the two approaches is that theexpected RBS arrival order for certain resources maybe violated based on the resolution of conflictingCTDs. In the precedence RBS example above, flight Bwas originally scheduled to arrive at FCA1 first butwas instead scheduled second after resolution of theconflicting CTDs. Although the RBS order is violatedin this case, it is likely not a fairness issue becauseLGA is a more congested resource along flight B’sroute. On the other hand, consider two flights, thefirst of which passes through a severely constrainedFCA en route to a more mildly constrained arrival air-port, the second just through the FCA. Assuming theGDP was implemented first, under either approachthe GDP-based CTD will take priority for the firstflight, allowing it to avoid the impact of the moresevere AFP. The second flight will be affected onlyby the AFP and thus incur significantly greater, andtherefore inequitable, delays. As should be appar-ent from this example, we can construct scenarios inwhich either multiresource RBS schedule is arbitrar-ily unfair.

2.2. Principles for Measuring FairnessThe challenge to incorporating fairness into the mul-tiresource setting is that the link between originalschedule order and delay optimality breaks downwhen some flights are affected by multiple TFM pro-grams. Thus, in a multiresource setting we need tomake a tradeoff between fairness relative to the origi-nal schedule order and efficiency in terms of total sys-tem delay. To find the appropriate tradeoff, we need amethod of measuring the relative (un)fairness of com-peting schedules.

The concept of fairness is by nature subjective andoften domain specific. Even within air traffic, there aremany plausible ways to measure schedule fairness,each leading to different results. For example, in asingle-resource setting, one measure of fairness isthe number of slots a flight deviates from its initialorder position (e.g., if a flight scheduled to arrivefourth is instead allocated the twelfth arrival slot,we would say that flight’s schedule was unfair byeight positions). Unfortunately, in the multiresourcesetting, using position-based metrics without consid-ering delay can lead to imbalances in the fairness

penalty incurred between resources. Other proposalsinclude measuring schedule fairness by comparingaverage or maximum flight delays between airlines.These types of measures ignore variation in conges-tion along flight routes and thus are also problem-atic in the multiresource setting. In this section, wedescribe properties that we believe are critical to mea-suring fairness in the multiresource setting. Theseproperties are motivated primarily as extensions ofthe successful properties of RBS in the single-resourceenvironment. In the following section, we use theseproperties to obtain a multiresource fairness deviationmetric.

Property 1. The measure of schedule fairness should bedetermined with reference to the original schedule ordering.Because of the success of RBS in the single-resource setting,the concept of first-scheduled, first-served has come to bewidely accepted by airlines and the FAA.

Property 2. The measure of schedule fairness should beapplicable to a single flight as well as the overall schedule.That is, the measure should be able to determine the amountby which each flight’s schedule varies from first-scheduled,first-served.

Property 3. The unit of fairness deviation and its rel-ative magnitude should be consistent between resources.In a single-resource setting, position-based deviation is anaccepted measure of fairness deviation. In the multiresourcesetting, this is confounded due to varying congestion lev-els between resources. An eight-position delay (going fromfourth to twelfth) could mean 30 minutes of delay in a low-capacity airport, but only 10 minutes of delay in a highercapacity one.

Property 4. No flight should expect to receive lessdelay than what would be caused by the most congestedresource along its route. That is, there should be a fairnesspenalty only if a flight incurs more delay than its origi-nal schedule order would indicate for each of the resourcesalong its route.

Property 5. The measure of a flight’s deviation fromthe original schedule should be calculated relative to thetotal delay assigned to the flight (ground delay plusair delay), not intermediate arrival times into controlledresources. This property is relevant if the schedulingapproach allows both ground delays (by assigning CTDs)and en route delays (by mandating air speed reductionsor arrival queuing) to be assigned. In practice, the sched-ule created by the FAA using RBS assumes that a flightwill incur no delays en route and only assigns grounddelays through CTDs. Airborne delays are subsequentlymanaged by air traffic controllers en route or at the arrivalairport. Network TFM models, such as the one describedin Bertsimas and Patterson (1998), consider both of theseproblems simultaneously in order to improve efficiency andpredictability.

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2.3. Time-Order Deviation MetricWith these properties in mind, we now develop ameasure for evaluating the unfairness of a controlledschedule. First, we use features of both the planned(predisruption) schedule and the controlled (postdis-ruption) schedule to determine a fair delay thresholdfor each flight, which we refer to as the maximumexpected delay. Next, we calculate the time-order devi-ation for each flight as the amount by which theflight’s delay in the controlled schedule exceeds thisthreshold.

We first determine the fair delay threshold forthe case in which a flight, f , utilizes just one con-trolled resource. We let j be the flight’s position in theplanned arrival ordering for this resource. For exam-ple, if there is a GDP at LGA and flight f was plannedto be the third arrival into LGA, j would equal three.We define flight f ’s expected delay for the controlledresource as the difference between the jth controlledarrival time for the resource and the planned arrivaltime of the flight. Note that the order of flights couldbe swapped in the controlled schedule; thus, the jthcontrolled arrival for the resource might not be thesame as the jth planned arrival. Continuing the exam-ple of a GDP at LGA, if flight f was planned tobe the third flight to arrive into LGA at 19:15, andthe third controlled arrival into LGA is scheduled for19:20, we would say that flight f has a five-minuteexpected delay into LGA. As long as flight f incursno more than five minutes of delay, we would con-sider the controlled schedule to be fair from flight f ’sperspective.

The case in which a flight, f , utilizes multiple con-trolled resources is a bit more complicated. For eachof the controlled resources flight f is scheduled to uti-lize, we could calculate the expected delay as in theexample above. Each of these expected delay valueswould represent a fair delay threshold, assuming thatflight f utilized no other controlled resources along itsroute. Thus, in the case in which a flight utilizes mul-tiple controlled resources, we set the fair delay thres-hold to the maximum of the expected delay valueacross these controlled resources. As long as flight fincurs no more delay than this maximum expecteddelay, we consider the controlled schedule to be fairfrom flight f ’s perspective. One could argue that thefair delay threshold should be higher than this valuebecause the flight uses multiple controlled resources,but it clearly should not be any lower. Note that thisgeneral definition applies to flights that utilize justone controlled resource because for these cases themaximum expected delay equals the expected delayfor the single resource.

Using these definitions, we define each flight’stime-order deviation as the amount by which its total

delay in the controlled schedule exceeds the max-imum expected delay along its route. In the eventthat the maximum expected delay exceeds the flight’stotal delay, we set the time-order deviation equalto zero. That is, a schedule is not fairer if a flightarrives earlier than expected, even though this mightreduce the overall system delay. Time-order devia-tion can be considered a generalization of deviationfrom the ideal RBS allocation used to measure exemp-tion bias in Vossen et al. (2003). The key differencesare (i) time-order deviation is measured relative tothe most congested resource along a flight’s routeand (ii) time-order deviation is measured relative toa feasible, controlled schedule instead of a potentiallyinfeasible, idealized allocation. Next, we consider anexample in which we calculate the time-order devia-tion for a flight that is affected by a GDP at LGA andan AFP at FCA1.

2.3.1. Time-Order Deviation Example. Considera flight planned to depart from Boston Logan Interna-tional Airport (BOS) at 18:00, arrive at FCA1 at 18:45,and arrive into LGA at 19:15. Prior to schedule dis-ruption, the flight is planned to be the fourth arrivalinto FCA1 and the third arrival into LGA. Based ona GDP at LGA and an AFP at FCA1, the flight issubsequently given a CTD of 18:25 (corresponding to25 minutes of ground delay). To calculate the time-order deviation for this flight, we need to addition-ally know the order of arrivals into FCA1 and LGAin the controlled schedule. Based on the controlledarrival orderings listed in Tables 7 and 8, we can cal-culate the time-order deviation as follows. First, wecalculate the flight’s expected delay into FCA1 as thetime of the fourth controlled arrival into FCA1 (18:55)minus the flight’s planned arrival time into FCA1(18:45), which equals 10 minutes. Next, we calculate

Table 7 Controlled Flight Order for FCA1

Order FCA1 Arrival

1 18:352 18:453 18:504 18:555 19:006∗ 19:05

Table 8 Controlled Flight Order for LGA

Order LGA arrival

1 19:002 19:103 19:204 19:305∗ 19:40

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the flight’s expected delay into LGA as the time of thethird controlled arrival into LGA (19:20) minus theflight’s planned arrival time into LGA (19:15), whichequals five minutes. The referenced arrival times inTables 7 and 8 are highlighted in bold. The maxi-mum expected delay for the flight is the 10 minutesfrom FCA1. In the controlled schedule, the total delayfor the flight is 25 minutes (the difference betweenthe 18:25 CTD and the 18:00 planned departure time).Thus, the time-order deviation for the flight is 15 min-utes (25 minutes of total delay minus 10 minutes ofmaximum expected delay). Thus, we would say that60% of the delay assigned to this flight is unfair asmeasured by time-order deviation. In Tables 7 and 8,the rows corresponding to the controlled arrival timesfor the referenced flight have been marked with anasterisk 4∗5 although they are not used directly in thecalculation of the flight’s time-order deviation.

We define the time-order deviation for a controlledschedule as the sum of the time-order deviations foreach flight represented in the schedule. If we dividethe total time-order deviation by the total delayassigned, the result describes the average percentageof unfair delay assigned to each flight (relative to theindividual fair delay thresholds). The time-order devi-ation of individual flights may vary significantly fromthe average, so in our results in §4.4 we also considerthe distribution of flight delay. As expected, time-order deviation satisfies all of the principles laid outin the previous section. That is, (i) time-order devi-ation is calculated relative to the original scheduleorder, (ii) the measure can be applied to each flightin the controlled schedule, (iii) the unit of measure(i.e., time) is consistent between resources, (iv) themeasure is calculated relative to the most restrictedresource along each flight’s route (i.e., relative to themaximum expected delay), and (v) the measure isbased on the total delay and not intermediate arrivaltimes. Note that for a single controlled resource, orfor a set of independent controlled resources (suchas multiple GDPs), the time-order deviation metricachieves zero if the controlled schedule matches theschedule that results from independent RBS alloca-tions for each controlled resource. For this indepen-dent resource case, among the set of delay-minimalschedules and because of the uniqueness of the first-scheduled, first-served solution, the time-order devia-tion metric achieves zero if and only if the controlledschedule matches the RBS allocations.

3. Optimization ApproachesIn this section, we describe two integer program-ming formulations, the solutions to which describethe ground holding that should be assigned to eachflight. Each formulation allows for the flexible trade-off between a delay term and a fairness term in the

minimization objective. In the first model, the fairnessterm is a convex approximation of the fairness met-ric developed in the previous section. We call this thetime-order deviation approximation (TODA) model.In the second model, we use an exponentially grow-ing delay penalty to enforce fairness. This approachhas considerable computational advantages yet sac-rifices little in terms of fairness achieved accordingto time-order deviation. We refer to this model asthe ration-by-schedule exponential penalty (RBS-EP)model.

In §3.1, we develop the common notation as well asdefine the input data used in both our formulations.In §3.2, we provide the portion of the optimizationformulation that is common to both the TODA andRBS-EP models. Section 3.3 provides the formulationfor the TODA model, and §3.4 the formulation forthe RBS-EP model. We discuss some issues related topractical implementation in §3.5.

3.1. Data and NotationWe consider a set of discretized time intervalsT= 801 0 0 0 1 T − 19, where T represents the end of theday and each interval is defined to have equal dura-tion, typically either five minutes or 15 minutes. Weconsider a set of controlled resources, R, that willtypically include arrival airports (for GDPs) and FCAs(for AFPs). System resources that are not capacity con-trolled provide no binding constraints on the systemand are excluded from R. For each resource, r ∈R,and each time interval, t ∈ T, we specify a capacityof brt , which can be either the allowable arrival rateor the maximum number of flights allowed to occupythe resource during the interval. For GDPs and AFPs,resource capacities are specified in terms of the allow-able arrival rate.

Additionally, we consider a set of flight legs, F. Foreach flight leg, f ∈ F, we define the controlled flightplan to be the sequence of controlled resources sched-uled to be utilized over the course of the flight. Forinstance, consider the flight from Portland Interna-tional Airport (PWM) to New York John F. KennedyAirport (JFK) depicted in Figure 2, with TFM pro-grams in place at FCA1 and JFK. For this flight, thecontrolled flight plan would be a sequence contain-ing FCA1 followed by the arrival resource for JFK.Notationally, we let �f � represent the number of steps

FCA1

JFK

Figure 2 PWM to JFK Flight Path Intersecting Two ControlledResources, FCA1 and JFK

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Table 9 Data Values for PWM to JFK Controlled Flight Plan Based ona 07:00 Scheduled Departure

Scheduled time i r � �

07:35 1 FCA1 31 208:15 2 JFK 39 1

in the controlled flight plan for flight f and use theshorthand I4f 5 to represent the set of step indices811 0 0 0 1 �f �9. For each step in the controlled flight plan,in addition to the resource, r , we must specify theearliest start time, �, and the processing time, �.That is, � ∈ T represents the first time interval atwhich the step can be scheduled and � ∈�+ the num-ber of time intervals the step needs to be processed(i.e., landing time at an arrival airport or dwell timein an occupancy-controlled FCA). Notationally, we letr4f 1 i5, �4f 1 i5, and �4f 1 i5 refer to the appropriate val-ues for step i of the flight plan for flight f . In our for-mulation, �4f 1 i+15−�4f 1 i5 represents the minimumnumber of time intervals between the starts of steps iand i+ 1. Thus, we require �4f 1 i5+ �4f 1 i5 to be lessthan or equal, rather than strictly equal, to �4f 1 i+15.For example, if the resources for two sequential stepsare not geographically adjacent, �4f 1 i+ 15−�4f 1 i5−�4f 1 i5 would represent the travel time between theboundaries of the two resources. In Table 9, we pro-vide sample values for these fields based on the exam-ple described above (see Figure 2), with five minutetime intervals starting at 05:00. In our example, thereferenced flight is scheduled to occupy FCA1 for 10minutes en route to JFK.

For each resource, r , we assume there is a preferredordering of tasks (i.e., flight steps) corresponding tothe original schedule. That is, for resource r we wouldprefer to start the task indexed by j before the taskindexed by j + 1, where each task corresponds to aflight step, 4f 1 i5. Using this notation, we let j4f 1 i5represent the task index of flight step 4f 1 i5 for the cor-responding resource, r4f 1 i5. Additionally, we let 4r1 j5represent the time interval task j would be assignedbased on performing single-resource RBS for r .

Summarizing the above, we have the followingmodel inputs:

T= set of discrete time intervals;R= set of capacity-controlled resources;brt = capacity of resource r over time interval t;F= set of flights;

�f � = number of steps in controlled flight planfor flight f ;

I4f 5= set of step indices in controlled flight planfor flight f ;

r4f 1 i5= resource required by flight step i forflight f ;

�4f 1 i5= earliest start time for flight step i for flightf ;

�4f 1 i5= processing time of flight step i for flightf ;

J 4r5= number of tasks (i.e., flight steps)assigned to resource r ;

J4r5= set of task indices 811 0 0 0 1 J 4r5− 19;j4f 1 i5= the task index of flight step i for flight f ;

andRBS4r1 j5= RBS start interval for task j of resource r .

3.2. Model FoundationIn this section, we describe the components ofthe deterministic, multiresource TFM formulationthat provide the foundation for the two modelswe develop. This formulation is derived from theBertsimas and Patterson (1998) nationwide TFMmodel.

3.2.1. Decision Variables. For both formulations,we use the following variable definitions:

yfit =

11 if flight plan step i for flight f has startedby time t and

0 otherwise0

3.2.2. Constraints. We first ensure that thesequence 6yfi01 0 0 0 1 yfi4T−157, which we refer to as 6yfi7,is monotonically increasing:

yfit ≤yfi4t+15 ∀f ∈F1∀i∈I4f 51∀t∈8010001T −290 (1)

Next, we guarantee that each flight step is sched-uled and that no flight step is scheduled before itsminimum start time:

yfi4T−15 = 1 ∀ f ∈F1 ∀ i ∈I4f 50 (2)

yfi4�4f 1 i5−15 = 0 ∀ f ∈F1 ∀ i ∈I4F 5

s.t. �4f 1 i5 > 00 (3)

We also enforce the appropriate order betweenflight steps in a controlled flight plan as follows:

yf 4i+15t = yfi4t−�4f 1 i+15+�4f 1 i55

∀ f ∈F1 ∀ i ∈I4f 5 \ 8�f �90 (4)

We require strict equality in (4) to disallow alloca-tion of airborne delays. To allow the allocation of air-borne delays, we could replace this with an inequality(i.e., yf 4i+15t ≤ · · · ). The last set of constraints is toensure that resource capacities are not violated:

∑

84f 1 i52 r4f 1i5=r9

4yfit − yfi4t−�4f 1i555≤ brt

∀ r ∈R1 ∀ t ∈T0 (5)

Note that yfit −yfi4t−�4f 1i55 represents whether flight fis performing flight plan step i at time t.

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3.2.3. Objective Function. The delay term in theobjective function of each formulation represents theaggregate costs associated with flight delay, which wemodel as follows. First, we note that the start time offlight plan step i for flight f , s4f 1 i5, can be written as

s4f 1 i5= T −

T−1∑

t=0

yfit0

The total delay for flight f , d4f 5, is equivalent to thedelay accumulated through the last step in the flightplan, �f �, which can be written as

d4f 5= s4f 1 �f �5−�4f 1 �f �50

In the base formulation, the objective is to minimizetotal delay:

min∑

f∈F

d4f 50 (6)

Constraints (4) ensure that the total delay for flight fis equivalent to the delay assigned before the first stepin the controlled flight plan, which allows us to allo-cate all of the flight delay as ground holding.

3.3. Time-Order Deviation Approximation(TODA) Model

Using the notation described in the previous section,we first provide the mathematical definition of time-order deviation. Letting s4r1 j5 represent the start timefor the jth flight step to utilize resource r in the con-trolled schedule, we have

(Maximum Expected Delay)

MED4f 5 ¬ maxi∈I4f 5

{

s4r4f 1i51j4f 1i55−�4f 1i5}

3 and (7)

(Time-Order Deviation)

TOD4f 5¬ 4d4f 5− MED4f 55+0 (8)

In Equation (8), d4f 5 represents the total delayassigned to flight f , as in the objective function forthe base formulation (6). There are two challengesto calculating time-order deviation within a mathe-matical programming model. The first is that to cal-culate expected delay, we need the sorted list ofscheduled start times for each resource. That is, inaddition to maintaining a view of the schedule fromeach flight’s perspective, we also need to maintaina view of the schedule from each resource’s per-spective. We address this by creating schedule vari-ables that maintain a fixed relative order for eachresource and are bound to the original flight-centricschedule variables. The second challenge is that time-order deviation is a nonconvex function because ofthe inner maximum from Equation (7). Thus, time-order deviationcannot be represented directly in a

linear minimization objective. Instead, we approxi-mate time-order deviation by replacing the maximumover all flight steps in Equation (7) with an aver-age over the flight steps that we estimate a prioriwill lead to the most delay. We do so by comput-ing which steps, i, would be assigned the most delaybased on independent RBS allocations performed foreach resource. This gives us an estimate of conges-tion from capacity-demand imbalances, although itignores delays introduced due to interactions betweenresources.

3.3.1. Model Adjustments. We first define theordered auxiliary variables described above:

urjt =

11 if j tasks for resource r have beenscheduled to start by time t and

0 otherwise0

Based on this definition, 4urjt − urj4t−155 will indi-cate when task j of resource r starts in the optimizedschedule. Note that this might or might not be thesame as the start time of the task originally scheduledto occupy position j .

Next, we add the following constraints to the modelto ensure that the variables maintain the definitionabove:

urjt ≤ urj4t+15 ∀ r ∈R1 ∀ j ∈J4r51

∀ t ∈ 801 0 0 0 1 T − 291 and (9)

urj4T−15 = 1 ∀ r ∈R1 ∀ j ∈ 811 0 0 0 1 J 4r590 (10)

Constraints (9) and (10) ensure that the sequence ofordered auxiliary variables 6urj 7 maintains the samemonotonically increasing form as the sequence offlight step variables 6yfi7. We also need to ensure thatthe appropriate order for the auxiliary variables ismaintained; that is, task 4j + 15 cannot start beforetask j :

ur4j+15t ≤ urjt ∀ r ∈R1 ∀ j ∈J4r51 ∀ t ∈T0 (11)

The last, and most important, set of constraintsensures that, by each interval, the number of sched-uled flights according to the ordered auxiliary vari-ables and the flight step variables coincide:

J 4r5∑

j=1

urjt =∑

84f 1 i52 r4f 1 i5=r9

yfit ∀ r ∈R1 ∀ t ∈T0 (12)

That is, constraints (12) ensure that when a flightstep is scheduled within an interval, one of thesequences of ordered auxiliary variables must flipfrom zero to one in that same interval.

With these definitions in mind, we can calculate theexpected delay (ED) for flight step 4f 1 i5 and resourcer4f 1 i5, denoted ED4f 1 i5:

ED4f 1 i5=

T−1∑

�4f 1 i5

41 −ur4f 1 i5j4f 1i5t50 (13)

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The right-hand side measures the number of inter-vals from the earliest start time for flight step 4f 1 i5until the j4f 1 i5th task starts for resource r4f 1 i5.

As discussed in the introduction to this section, weestimate which resources for each flight f will max-imize expected delay by computing which steps iwould be assigned the most delay according to inde-pendent RBS allocations. For flight f , we denote themaximum RBS delay as dRBS

MAX4f 5 and the set of stepsachieving the maximum RBS delay as IMAX4f 5:

dRBS4f 1 i5= RBS4r4f 1 i51 j4f 1 i55−�4f 1 i53

dRBSMAX4f 5= max

I4f 58dRBS4f 1 i593 and (14)

IMAX4f 5= 8i ∈I4f 52 dRBS4f 1 i5= dRBSMAX4f 590

We now have the tools needed to describe the fair-ness term we add to (6) to calculate the approximatetime-order deviation in our TODA objective function:

min∑

f∈F

d4f 5+�

(

d4f 5−∑

i∈IMAX4f 5

ED4f 1 i5

�IMAX4f 5�

)+

0 (15)

Within the sum, the second term represents theapproximate time-order deviation scaled by a factorof � > 0, which controls the tradeoff between systemdelay and approximate time-order deviation. Withinthe approximate time-order deviation term, the innersum calculates the average expected delay acrossthe flight steps that achieved maximum RBS delayaccording to independent RBS allocations. When theset IMAX4f 5 equals the steps that achieve the max-imum expected delay in the optimized schedule,our approximate time-order deviation will equal thetrue time-order deviation as described in §2.3. Oth-erwise, the average expected delay across IMAX4f 5will be less than the maximum expected delay, andthe approximate time-order deviation will be strictlylarger than the true time-order deviation. The 4 · 5+

ensures that we add the approximate time-order devi-ation to our objective only if the total delay exceedsthis average expected delay.

3.4. Ration-by-Schedule Exponential Penalty(RBS-EP) Model

Unlike the TODA model, the RBS-EP model requiresthe introduction of no new variables or constraints tothe foundational model described in §3.2. The onlychange required is modifying the functional form ofthe objective function. The intuition behind the RBS-EP model has two parts. The first is that no flightshould expect to incur less delay than its worst-caseRBS delay, dRBS

MAX4f 5 as defined in Equation (14) in theprevious section. But because of interactions betweenresources, it is unlikely that each flight will be ableto achieve this exactly. So to provide flexibility, wepenalize each interval of delay beyond dRBS

MAX4f 5 by anexponentially increasing amount.

3.4.1. Model Adjustments. One of the nice prop-erties of discrete scheduling models is that we canassociate different objective coefficients with each pos-sible start time for a task. To achieve an exponentiallyincreasing penalty, we need only to determine theappropriate coefficients for each flight and potentialstart interval. Thus, we let cft be the coefficient asso-ciated with the last step of flight f starting at time t:

cft = min{

t −�4f 1 �f �51 dRBSMAX4f 5

}

+

t−�4f 1 �f �5−dRBSMAX4f 5

∑

�=1

��0 (16)

Based on the definition above, we have cft −cf 4t−15 =

�4t−�4f 1 �f �5−dRBSMAX4f 55 assuming t > �4f 1 �f �5 + dRBS

MAX4f 5.Thus, assuming � > 1, the incremental cost of eachadditional interval of delay beyond dRBS

MAX4f 5 increasesexponentially. A sample plot of this cost function isrepresented in Figure 3 for � = 2, dRBS

MAX4f 5 = 4, and�4f 1 �f �5= 0.

With the cost coefficients cft defined as above, theobjective function for the RBS-EP model is

min∑

f∈F

[ T−1∑

t=�4f 1�f �5

cft4yf �f �t − yf �f �4t−155

]

0 (17)

The difference 4yf �f �t −yf �f �4t−155 equals 1 if and onlyif the last step for flight f begins at time t, thusapplying a penalty of cft as desired. In the exponen-tial penalty model, the base of the exponent, � > 100,used in defining cft , implicitly controls the tradeoffbetween aggregate system delay and fairness.

3.5. Integration IssuesAs noted, these computational models build onBertsimas and Patterson (1998) and Andreatta,Brunetta, and Guastalla (2000). Beyond the fairnessconsiderations, however, there are two key differencesin the approach we outline. First, in each of the ref-erenced models, flights can be assigned air delay enroute. Because of the limited number of AFPs imple-mented in practice, and the deterministic nature of

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9

t

c ft

Figure 3 Plot of Delay Cost Function for �= 2, dRBSMAX4f 5= 4, and�4f 1 �f �5= 0

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our formulations, allowing air delay provides littlevalue for the historical scenarios we consider in ourresults. Thus, to simplify exposition of the modelas well as maintain consistency with current prac-tice, we consider only ground delay. Additionally, inthe referenced models, planned aircraft connectionsbetween flight legs are maintained in the controlledschedule (i.e., planned aircraft connections are repre-sented as constraints in the formulation). Our mod-els do not include connectivity constraints betweenflight legs. Note that both of our models could includethese constraints and remain entirely consistent; thus,it is an explicit modeling choice to omit them. Wehave made the decision to exclude constraints of thistype because, again, this change leads to an approachthat is consistent with current practice. Most impor-tant, our models can utilize the same inputs as exist-ing TFM programs, allowing for direct comparisonas well as easier integration. Additionally, becauseof each airline’s ability to swap aircrafts and cancelflights, it is unclear whether strict connectivity con-straints are in the best interest of the airlines. Includ-ing these constraints increases the amount of delayassigned, under the assumption that airlines have lessflexibility to respond than they do in practice. On theother hand, excluding these constraints leaves the fullburden of resolving infeasibilities to the airlines. Webelieve that understanding the impact of aircraft con-nectivity in this context is an important open researchquestion.

Another integration consideration is how theseapproaches fit into the three-stage collaborative deci-sion making (CDM) framework described in §1.1. Thekey feature to note is that the output of our modelscan be easily translated into a slot assignment for thecorresponding programs. In this sense, we maintainthe same output format as that of existing approaches(corresponding to the first stage in the CDM pro-cess). To determine a single program to manage eachflight (for stages two and three of the CDM process),we could simply choose the program that would beassigned under the current approaches. Alternatively,we propose that each flight be assigned to the pro-gram that maximizes the expected delay in the result-ing schedule, as defined in Equation (7) in §3.3. Usingthis approach, each flight would be controlled by theprogram corresponding to the resource that createsthe most congestion along its route.

A level of complexity not considered in our workis the dynamic nature of TFM programs. Overthe course of the day, TFM programs are createdand modified as more precise weather forecasts arerevealed. These dynamics create challenges for anydeterministic allocation approach (including RBS)because they introduce potential inequities and ineffi-ciencies into the system. Effectively addressing these

challenges remains an interesting, important, and, toour knowledge, open research question.

4. Computational ResultsHere we provide computational experiments todemonstrate the practical value of the RBS-EP model.We highlight three key results from our historical sce-narios. The first is that under a conservative com-parison between RBS-EP and current practice, theRBS-EP model improves efficiency, as measured bytotal delays, while maintaining equivalent levels ofequity. The second is that the RBS-EP model closelytracks the tighter TODA approximation of the efficientfrontier between aggregate delay and fairness, calcu-lated according to our time-order deviation metric.Lastly, the RBS-EP model is computationally efficient,allowing for the solution of even complex, national-scale problems within reasonable computing times.

4.1. An Apples-to-Apples ComparisonOne challenge in comparing our optimization-basedapproaches to current approaches is that precedenceRBS allows the specified capacities to be violated (asdiscussed in §2.1). An optimization-based approach,on the other hand, ensures that all resource capac-ity constraints are strictly satisfied. Thus, if thesame capacities are utilized as inputs to both pro-cedures, precedence RBS would typically performbetter because of its ability to arbitrarily exceedFCA capacity constraints (and the inability of ouroptimization-based approaches to do so). In mostcases, exemption RBS does not exhibit this same char-acteristic because exempted flights reduce the effec-tive capacity of future programs.

To level the playing field when comparing to prece-dence RBS, we first perform a precedence RBS alloca-tion. Next, we compress the precedence RBS schedule,attempting to fill any gaps created in the resourceschedules. For resources and time intervals for whichthe resulting allocation exceeds the specified capac-ity, we update the corresponding capacity as an inputinto each optimization-based approach. By updatingthe capacity, we ensure that our optimization-basedapproaches do not exceed the initial capacity any morethan the precedence RBS schedule. For instance, basedon the four flight example in §2.1 and five minutediscretization intervals, we would increase the capac-ity of FCA1 to two flights for the five-minute inter-val from 18:45 through 18:49, keeping the capacity atone for all other intervals. Although this leads to afairer comparison between the two approaches, theplaying field is still tilted toward precedence RBS.Because of the inherent limitations of precedence RBS,we can only perform comparisons for the capacityallocations that directly correspond to a precedenceRBS schedule. Fortunately, as demonstrated in §2.1,

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FCAA05

FCAA08

FCAA06

Figure 4 Depiction of Airspace Boundaries for FCAA05, FCAA06, and FCAA08 (Federal Aviation Administration 2006)

this still leaves some inefficiencies that optimization-based approaches are capable of exploiting.

4.2. Construction of Historical ScenariosTo construct each of our scenarios, we start withflight schedule data that correspond to a single dayof relatively clear weather operations (April 23, 2007).This schedule includes estimated entry and exit timesfor each sector along each flight’s route. The sched-ule data were obtained from Flight Schedule Moni-tor, the TFM decision support tool developed for theFAA by Metron Aviation (Metron Aviation 2009a). Forpurposes of our experiments, we treat this scheduleas representing the Official Airline Guide (i.e., theplanned airline flight schedules). Thus, the definingcharacteristics of each scenario are the set of con-trolled resources and corresponding capacities.

To construct the capacity reduction scenarios, weuse historical TFM program data, also obtained fromMetron Aviation’s Flight Schedule Monitor. Thesedata include reporting times, effective times and dura-tions, and TFM program capacities for each 15-minuteinterval. From these data, we choose 10 represen-tative days on which both GDPs and AFPs wereimplemented. For each of these days, we create twoscenarios, one to reproduce historical behavior and asecond to analyze the hypothetical impact of furtherreductions in FCA capacities. To create the histori-cal scenarios, we reduce all hourly arrival capacitiesby 7.5% relative to the historical data. This reductioncompensates for our clear weather day, April 23, 2007,having fewer flight operations than days during thesummer (when all of the capacity reduction scenar-ios occurred). We calculated this reduction level bycomparing flight frequencies in the Airline On-TimePerformance Database, which includes flights for the20 largest domestic airlines. For the hypothetical sce-narios, we reduce airport capacities by 7.5% and FCAcapacities by 25%, both relative to the historical data.We utilize these hypothetical scenarios to understandhow efficiency improvements might change as enroute congestion increases and AFPs are used more

heavily going forward. Each of the approaches uti-lized in practice is sensitive to the order of programimplementation; thus, we use the historical reportingtimes to determine this order, reproducing historicalbehavior as accurately as possible.

The historical AFPs we utilize affect traffic headinginto the Northeast corridor through one or more ofthe boundary-based flow constrained areas: FCAA05,FCAA06, and FCAA08. Figure 4 depicts each of theseboundaries. Because our schedule data include onlysector entry and exit times, in our scenarios wereplace FCAA08 with FCAA06. Both FCAA06 andFCAA08 are used to address weather in the OhioValley region or in the Washington D.C. airspace (Fed-eral Aviation Administration 2006). In Table 10, wereport scenario details for the 10 days of capacityreductions we overlay on the clear weather flightschedule. The Conflicts column reports the percentageof flights affected by more than one TFM program. Byconstruction, each of these values is the same for thehistorical and hypothetical scenarios described above.For each of the scheduling approaches we test, includ-ing the two multiresource RBS approaches, we dis-cretize time into five-minute intervals.

Table 10 Details on Controlled Resources for Historical andHypothetical Scenarios

Date Flights Airports FCAs Conflicts (%)

5/2/2007 1,858 (2) LGA SFO A06 4035/9/2007 1,572 (2) IAD JFK A05 8076/19/2007 5,191 (8) ATL DCA EWR IAD A05 A06 1506

JFK LGA SFO PHL6/27/2007 4,682 (4) CYYZ JFK LGA A05 A06 705

ORD MDW6/28/2007 3,583 (5) EWR IAD JFK LGA SFO A05 A06 15077/5/2007 2,585 (2) CYYZ EWR A05 A06 5017/16/2007 1,858 (2) LGA SFO A06 4037/18/2007 3,705 (5) EWR JFK LGA PHL SFO A05 A06 12057/27/2007 3,944 (4) EWR LGO ORD SFO A05 A06 6069/27/2007 3,953 (6) ATL CYYZ EWR A05 507

JFK LGA PHL

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4.3. The Tradeoff Between Efficiency and FairnessIn this section, we demonstrate the tradeoff betweenefficiency, as measured by aggregate delay, and fair-ness, as measured by the time-order deviation ofthe resulting schedule for each of the 20 scenariosdescribed in the previous section (10 historical and 10hypothetical). We create tradeoff curves by adjusting� and �, the parameters that control the relative trade-off for each of the optimization-based approachesdeveloped in §3. First, we compare the TODA modelto the RBS-EP model for the less complex scenar-ios. Because the TODA model is not computationallytractable for the more complex scenarios, we evaluateonly the RBS-EP model for these problems.

We employ two computational heuristics to solvethe TODA and RBS-EP optimization problems. First,because we discretize time into five minute intervals,it is computationally intractable to allow each flightto be delayed indefinitely. Doing so would result inwell over a million binary decision variables for someof our instances. Instead, we restrict the amount ofallowable delay on a flight-by-flight basis. Specifically,we allow each flight f to be assigned up to dRBS

MAX4f 5plus an additional 15 or 30 minutes of delay, wheredRBS

MAX4f 5 is the maximum independent RBS delay allo-cation as defined in Equation (14) in §2.3. Althoughthere is suboptimality associated with this approach,the resulting schedule is, by construction, quite fairbecause it is close to the accepted RBS allocation. Sec-ond, we use a greedy integer rounding heuristic toconvert each solution of the linear relaxation into afeasible flight schedule. We do so by greedily schedul-ing flights in order, based on the relaxed start time tothe first step in each flight plan, s4f 115. This heuris-tic ensures that after solving the root node relaxation

7/16/2007 (historical)

0.0

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2.85 2.86 2.86 2.87 2.87

Average flight delay (minutes)

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7/16/2007 (hypothetical)

0.0

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3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11


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TODA RBS-EP

Figure 5 Historical and Hypothetical Precedence RBS Scenarios for 7/16/2007


0.0

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2.86 2.86 2.87 2.87 2.88 2.88 2.89 2.89


Perc

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r (%

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TODA RBS-EP

7/16/2007 (hypothetical)

0.00.51.01.52.02.53.0

3.08 3.10 3.12 3.14 3.16 3.18 3.20 3.22


Perc

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r (%

)

TODA RBS-EP

Figure 6 Historical and Hypothetical Exemption RBS Scenarios for 7/16/2007

during branch-and-bound search, we always have agood feasible solution. This is critically importantin the TFM setting, in which we must be able toguarantee a solution in a relatively short amount oftime (preferably one minute or less).

In Figures 5 and 6, we compare the tradeoff curvesgenerated by the TODA model to the tradeoff curvesgenerated by the RBS-EP model. In these plots, eachpoint represents a schedule generated using a spe-cific value of � or �, plotting average flight delay inthe corresponding schedule against the percentage ofunfair delay as measured by time-order deviation. Foreach model, we allow flights to be assigned up todRBS

MAX4f 5 plus 30 minutes of delay. In general, we findthat the RBS-EP model closely tracks the approximateefficient frontier between fairness and delay as esti-mated by the TODA model. The RBS-EP model doesnot, however, allow us to fully explore the lower endof this curve. For more complex scenarios, determin-ing a baseline according to the TODA model is notcomputationally tractable, although we have verifiedthis general relationship on smaller constructed sce-narios outside of the ones shown here. When thereis more significant network-based congestion, thereis typically a small gap between these two curves.Even though the RBS-EP model does not directly min-imize time-order deviation or its approximation, thereis a fairly consistent trend between an increasing �,the base of the exponential penalty, and a decreasingtime-order deviation of the resulting schedule. That is,by simply adjusting the functional form of the delaypenalty, we have created a model that closely tracksthe more complex time-order deviation metric.

One thing to note in these charts is that a rela-tively large benefit in terms of time-order deviation

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Table 11 Comparison of RBS-EP Model to Precedence RBS for Historical Scenarios

Precedence RBS RBS-EP (Fair, 30) RBS-EP (1.001, 15)

Average Unfair Delay Unfair Delay UnfairScenario flight delay delay (%) � reduction (%) delay (%) reduction (%) delay (%)

5/2/2007 608 000 N/A — — 303 1095/9/2007 309 308 10001 009 102 008 1016/19/2007 1907 005 10501 005 003 008 0066/27/2007 1805 000 N/A — — 304 1046/28/2007 2405 000 N/A — — 202 1037/5/2007 302 504 10001 1202 100 1202 1007/16/2007 209 000 N/A — — 006 0067/18/2007 1705 001 N/A — — 206 1037/27/2007 405 005 N/A — — 707 3009/27/2007 709 001 N/A — — 105 101

Summary 1208 003 004 0015 205 102

is gained by sacrificing a relatively small amount interms of total or average flight delay. This is consistentwith both the computational results that followand the theoretical results developed in Bertsimas,Farias, and Trichakis (2011). In general, a substantialimprovement in equity can be gained with little sac-rifice in terms of efficiency.

In Tables 11, 12, 13, and 14 we summarize theresults of our 40 test instances (10 days × 2 scenarios× 2 multiresource RBS approaches). For each instance,we compare the average flight delay and percentageof unfair delay (as measured by time-order deviation)of the multiresource RBS schedules to schedules gen-erated using two different approaches based on theRBS-EP model. In each table, the summary row pro-vides the results averaged across all affected flightsover the 10 days. In the first approach, we alloweach flight to be delayed up to dRBS

MAX4f 5 plus 30 min-utes. We then choose and report the smallest param-eter value for the exponential penalty base, �, thatleads to a solution at least as fair as the correspond-ing multiresource RBS solution, as measured by time-order deviation. For some instances we are unable to

Table 12 Comparison of RBS-EP Model to Exemption RBS for Historical Scenarios

Exemption RBS RBS-EP (Fair, 30) RBS-EP (1.001, 15)


5/2/2007 702 1300 10001 703 302 608 2095/9/2007 405 000 N/A — — 507 1016/19/2007 2009 2504 10001 408 104 404 1006/27/2007 1900 506 10001 505 205 405 1046/28/2007 2406 000 N/A — — 206 1067/5/2007 403 000 N/A — — 1405 0097/16/2007 209 403 10001 005 009 005 0097/18/2007 1801 509 10001 504 204 502 2077/27/2007 407 707 10001 707 500 507 2089/27/2007 802 305 10001 405 106 401 102

Summary 1304 903 400 106 405 106

find a schedule as fair using the RBS-EP model, typi-cally because the multiresource RBS solution is almostperfectly fair. In these cases, we report N/A for theRBS-EP parameter value and list “—” for the per-centage of delay reduction and percentage of unfairdelay. When summarizing the results, we use the mul-tiresource RBS solution in place of these values. Werefer to this approach as RBS-EP (Fair, 30). In the sec-ond approach, we allow each flight to be delayedonly 15 minutes beyond dRBS

MAX4f 5. For this approach,we report the average flight delay and percentage ofunfair delay for the RBS-EP solution using �= 10001.We refer to this approach as RBS-EP (1.001, 15). Ofthe feasible schedules that minimize total delay, thisapproach selects the one that is most fair accordingto the exponential penalty. Although this schedulemay not be as fair as the multiresource RBS sched-ule, no flight is likely to incur more than 15 min-utes of unfair delay by construction. Across the 20precedence scenarios, this second approach allocates1.8% of unfair delay, on average, as compared to 1.5%for the precedence RBS schedules. For the 20 exemp-tion scenarios, the second approach allocates 2.2% of

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Table 13 Comparison of RBS-EP Model to Precedence RBS for Hypothetical Scenarios

Precedence RBS RBS-EP (Fair, 30) RBS-EP (1.001, 15)


5/2/2007 1303 004 20001 104 000 300 1075/9/2007 505 600 10001 007 102 007 1026/19/2007 2200 207 10251 400 203 400 2046/27/2007 2206 100 10751 507 100 609 2016/28/2007 2707 000 N/A — — 608 3007/5/2007 1407 706 10001 500 004 500 0047/16/2007 301 001 N/A — — 201 1047/18/2007 2406 404 10001 1304 408 1101 3017/27/2007 701 406 10001 1006 406 809 2089/27/2007 900 200 10001 305 108 208 101

Summary 1607 205 503 108 604 203

unfair delay on average, compared to 6.5% for theexemption RBS schedules. Thus, we believe that thisapproach represents a reasonable and fair alternative,especially in an aggregate sense.

For the historical scenarios in Table 12, the daysduring which exemption RBS is the most unfair are6/19/2007, 6/27/2007, 7/18/2007, and 7/27/2007.On each of these days, AFPs are reported earlier in theday than GDPs that represent possible sources of con-flict. When the reverse ordering occurs, on days suchas 6/28/2007 and 7/5/2007, the resulting exemp-tion RBS schedule is extremely fair. Because AFPsaffect a large geographic region, the relative capacityreductions are typically mild when compared to thoseresulting from GDPs. Thus, flights that are affectedfirst by AFPs and exempted from subsequent GDPsare able to skirt the largest source of congestion alongtheir routes. This, in turn, pushes further delays downto flights affected only by GDPs, creating large time-order deviations. From a fairness perspective, thisdemonstrates the sensitivity of exemption RBS to theordering of program implementation. In Table 11, wesee that for the historical scenarios precedence RBS

Table 14 Comparison of RBS-EP Model to Exemption RBS for Hypothetical Scenarios

Exemption RBS RBS-EP (Fair, 30) RBS-EP (1.001, 15)

Average unfair Delay Unfair Delay UnfairScenario flight delay delay (%) � reduction (%) delay (%) reduction (%) delay (%)

5/2/2007 1308 304 10001 602 303 407 1075/9/2007 607 000 N/A — — 503 1036/19/2007 2904 1100 10001 1900 504 1701 3026/27/2007 2400 300 10251 608 203 608 2036/28/2007 2803 000 N/A — — 801 4017/5/2007 1609 000 N/A — — 403 0047/16/2007 302 306 10001 300 207 205 2027/18/2007 2805 407 10001 1104 309 903 2067/27/2007 707 109 10501 409 102 604 2099/27/2007 908 207 10001 706 108 609 100

Summary 1900 404 808 206 906 206

remains quite fair even for the days mentioned above.This is not particularly surprising because precedenceRBS mitigates potential fairness issues by creatingadditional FCA capacity as needed. The costs of thisadditional capacity are likely realized downstreamin terms of interventions en route, which makes thecosts difficult to evaluate. Thus, in terms of calculat-ing the cost benefits in §4.5, we compare our RBS-EPapproaches to exemption RBS.

Under our hypothetical scenarios, in which wefurther reduce the capacities of each AFP, we findthat, in general, significantly greater delays are intro-duced using both the precedence and exemption RBSscheduling approaches. Although true in general, thisstatement does not hold for all scenarios. For somescenarios we are able to achieve a greater percentageof delay reductions in the less constrained historicalscenarios. This demonstrates that the opportunity fordelay reduction is not strictly increasing as capaci-ties decrease, even though this appears to be true inaggregate. In these scenarios, precedence RBS remainsmore fair and allocates fewer delays than does exemp-tion RBS. Nonetheless, even when using precedence

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0200400600800

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Exemption RBS RBS-EP (1.001) Exemption RBS RBS-EP (1.001)

Figure 7 Allocated Flight Delay Distributions with RBS-EP � Value of 10001

RBS as a baseline, we are still able to realize delayreductions ranging from 5.3% for schedules that arejust as fair, to 6.4% if we relax this restriction. Theseresults suggest the importance of implementing anoptimization-based approach to TFM program coor-dination before the prevalence and complexity of AFPutilization increases.

4.4. Flight Delay DistributionIn addition to the summary statistics listed in thetables in the previous section, it is important to con-sider the distribution of delays for affected flights.Airlines typically build slack into their flight sched-ules to preserve connections between aircraft, crews,and passengers. Delay that is less than the plannedslack can be absorbed without schedule modifica-tions. Delay that exceeds the planned slack oftenrequires costly recovery operations. Thus, we need toensure that our approach does not lead to a heavy tailof flight delays (i.e., a larger number of flights incur-ring a large amount of delay).

Consider the flight delay distributions charted inFigure 7 for the historical and hypothetical 6/28/2007exemption RBS scenarios. These figures plot the num-ber of flights that incur at least the specified numberof hours of delay (starting at 45 minutes), compar-ing the exemption RBS schedule to the RBS-EP sched-ule allowing dRBS

MAX4f 5 plus 30 minutes of delay with� = 10001. Although the distributions are similar, theRBS-EP schedules have a longer tail, with manyflights incurring at least 2.5 hours of delay, more thanthe maximum delay assigned in the exemption RBSschedule. Based on the discussion in the precedingparagraph, this is likely a significant issue.


0200400600800

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Exemption RBS RBS-EP (2.001) Exemption RBS RBS-EP (2.001)

Figure 8 Updated Flight Delay Distributions Based on RBS-EP � Value of 20001

Fortunately, the RBS-EP model provides an obviousmechanism for resolving these issues. By increasingthe value of �, the base of the exponential penalty, itputs additional pressure on the tail of the flight delaydistribution. For example, consider the updated chartsin Figure 8, for which we utilize � values of 2.001 foreach of the RBS-EP solutions, as compared to 1.001in the previous charts. By increasing the value of �,we have increased the total delay assigned from 1,429hours to 1,443 hours in the historical scenario and from1,527 hours to 1,581 hours in the hypothetical scenario.In so doing, we have managed to shrink the tails of thedelay distribution, with the resulting schedules stillmore efficient than the exemption RBS schedules. Thistradeoff between aggregate delay and the distributionof delay is another important consideration in choos-ing an appropriate value of � in practice.

4.5. The Value of EfficiencyIn Table 12, we see that exemption RBS allocates atotal of 7,339 hours of delay across the 10 scenarios,whereas the RBS-EP (Fair, 30) approach allocates 7,046hours of delay and the RBS-EP (1.001, 15) approach7,008 hours of delay. Thus, we estimate that the RBS-EP model would lead to an overall delay reduction of4.0% to 4.5% across days with conflicting TFM pro-grams. Next, we would like to determine how muchcost reduction, for airlines, passengers, and relatedindustries, can be attributed to these delay reductions.

As mentioned in the introduction, the U.S.Congress Joint Economic Committee estimates thatarrival delays cost the U.S. economy $25.7 billion in2007 (Joint Economic Committee 2008). The Bureau ofTransportation Statistics estimates that in 2007, 37.7%

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of flight delays resulted from the previous flightarriving late; thus, we estimate that the remaining62.3% of flight delays are due to direct impacts(Research and Innovative Technology Administration2009). These direct impacts, to which we attribute thefull delay costs, led to 73.5 million minutes of delay,of which we estimate that 21.3 million minutes weredue to ground holding programs using the AirlineOn-time Performance Database (U.S. Bureau of Trans-portation Statistics 2007). This represents approxi-mately 30% of the direct impact delay; thus, weattribute 30% of the total delay costs—$3.7 billion inincreased airline operating costs, $2.2 billion in pas-senger time lost, and $1.8 billion in costs for relatedindustries—to TFM programs. Of the delay assignedthrough these initiatives, approximately 13% was ondays during which domestic GDPs and AFPs wereboth implemented. We consider these days our base-line for improvement because the multiresource RBSschedules are delay optimal when there are no con-flicts between TFM programs. A 1% delay reductionon these days would save airlines $4.8 million, pas-sengers $2.9 million, and related industries $2.4 mil-lion annually, a total of just over $10 million annually.Combining this with the above, we estimate thatimplementation of the RBS-EP model for coordinatingTFM programs would lead to annual cost savings onthe order of $25 to $50 million.

It is worth noting that the attribution approachutilized likely underestimates the value in at leasttwo ways. First, by focusing our analysis on directimpact delays we are assuming propagated delaycosts are allocated proportionally among differentroot causes. This likely underestimates the costs asso-ciated with TFM programs because these programstypically result in larger magnitudes of delays that aremore likely to exceed schedule slack, leading to delaypropagation. Second, the Joint Economic Committeeestimates passenger delays by multiplying the num-ber of passengers by the corresponding flight delays.This approach does not include the impact of missedconnections or flight cancellations, both of which areprevalent during TFM initiatives.

4.6. Computational PerformanceTo be implemented in practice, an optimization-basedapproach to coordinating TFM programs must beextremely fast, preferably returning a good solutionwithin a minute or less to support the subsequentCDM procedures. Fortunately, this is not a concernfor either of our RBS-EP approaches, which use lessthan 10 seconds of CPLEX solver time per instance.As a reference, in Table 15 we list the CPLEX solvertimes for the hypothetical exemption RBS scenarios.We allow either 15 or 30 minutes of delay beyonddRBS

MAX4f 5 and compare the solution times for � =

Table 15 CPLEX Times for Hypothetical Exemption RBSScenarios Using RBS-EP Model

CPLEX solver times (sec.)

RBS-EP (15) RBS-EP (30)

Date �= 10001 �= 20001 �= 10001 �= 20001

5/2/2007 0016 0015 0047 00465/9/2007 0012 0012 0045 00446/19/2007 8010 8005 9003 60596/27/2007 2015 3037 3070 70696/28/2007 4019 4054 6058 80577/5/2007 0039 0039 1032 10327/16/2007 0024 0024 0067 00667/18/2007 1034 3054 3068 70687/27/2007 0065 0063 1068 10629/27/2007 0051 0051 2009 2006

Total 17083 21053 29065 37008

10001 and � = 20001. Allowing a smaller amount ofdelay in the model reduces the number of decisionvariables, leading to a roughly 40% improvement inCPLEX times, on average. The performance measure-ments utilize the greedy integer rounding heuristicdescribed in §4.3, with a CPLEX relative optimalitygap of 0.01%. The computational tests are performedon a PC with dual Xeon 3220 Quad-Core processorsand 16 Gigabytes of RAM, running Ubuntu v8.04 andCPLEX v11.2 through the Java interface.

5. ConclusionIn this research, we develop an optimization-basedformulation that could be readily incorporated inpractice by the FAA. Specifically, based on principlesthat have made RBS successful, we have developeda time-order deviation metric for schedule fairnessthat extends to the multiresource setting. This metricallows us to evaluate optimization-based schedulingapproaches relative to each other, but more impor-tantly, it allows us to compare these approaches tothe approaches currently utilized in practice. Usingthis metric, we have demonstrated that in each ofthe approaches used in practice to resolve conflictsbetween TFM programs, an implicit tradeoff is madebetween equity and efficiency. We have also demon-strated that our two formulations, the TODA andRBS-EP models, can improve efficiency while main-taining a consistent level of fairness. Lastly, we haveshown that the RBS-EP model is computationallytractable in practice, even for complex, national-scaleproblems.

Introducing optimization into the FAA’s practiceshas been a significant challenge, as should be appar-ent from the literature review in §1.3. The RBS-EPmodel addresses many of these challenges and shouldthus provide a strong foundation for future research.Our goal is to have the RBS-EP model represent the

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first step in an ongoing sequence of practical enhance-ments to the FAA’s TFM procedures.

AcknowledgmentsThe authors gratefully acknowledge Professor AmedeoOdoni for being a supportive and enthusiastic advisor. Hissuggestions and guidance have heavily influenced athiswork. Additionally, the authors thank Robert Hoffman,Bert Hackney, Jason Burke, and Stephen Augustine ofMetron Aviation for their time and expertise. Lastly, theythank the Associate Editor and three anononymous refereeswhose insightful comments have helped shape the paper.This work has been supported by NASA Next generation(NEXTGEN) and NSF’s Emerging Frontiers in Research andInnovation (EFRI) office.

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