operation research

Applications of Operations Researchin the Air Transport Industry

Cynthia Barnhart Peter Belobaba Amedeo R. OdoniMassachusetts Institute of Technology, Cambridge, Massachusetts 02139

[email protected] [email protected] [email protected]

This paper presents an overview of several important areas of operations research appli-cations in the air transport industry. Specic areas covered are: the various stages ofaircraft and crew schedule planning; revenue management, including overbooking and leg-based and network-based seat inventory management; and the planning and operations ofaviation infrastructure (airports and air trafc management). For each of these areas, thepaper provides a historical perspective on OR contributions, as well as a brief summary ofthe state of the art. It also identies some of the main challenges for future research.

1. IntroductionDuring the one hundred years since the rst ight ofOrville and Wilbur Wright, the air transport indus-try has grown into a major sector of the global econ-omy. Even more importantly, it has become essentialto developing and maintaining cultural and economiclinks among countries and peoples. The airlines alonegenerated more than $300 billion in revenues in 2002,a lean year, and carried about 1.6 billion passengers, anumber expected to grow at an annual rate of 4%5%over the next 20 years according to most forecasts.According to the industry air transport provides28 million direct, indirect, and induced jobs world-wide and carries over 40% of the world trade ofgoods, by value (Collaborative Forum 2003).After spending roughly its rst 40 years trying to

get off the ground, literally at times, the air trans-port industry has grown by leaps and bounds dur-ing the last 60, especially since the advent of the jetage in the late 1950s. Throughout that second period,operations research (OR) has played a critical role inhelping the airline industry and its infrastructure sus-tain high growth rates and make the transition froma novelty that catered to an elite clientele to a serviceindustry for the masses. More than 100 airlines and

air transport associations are currently represented inAGIFORS, the Airline Group of Operational ResearchSocieties, which has been active since 1961. Indeed,it is difcult to think of any single sector, other thanperhaps military operations, with which operationsresearch has been linked more closely. One of the rea-sons is that airline operations and, more generally,the air transport environment provide natural con-texts for the application of OR techniques and models.A second is that the airline industry has consistentlybeen a leader in the use of information technologyand has relied heavily on the intensive use of com-puters over the years.The objective of this paper is to present a histor-

ical perspective on the contributions of operationsresearch to the air transport industry, as well as tooffer an assessment of some of the challenges that willbe confronted next. Any reasonably thorough cover-age of this subject would probably require an entireissue of this journal because the number of OR paperspublished on air transport easily exceeds 1,000 overthe last 50 years. In view of the severe constraintson its length, the scope of the paper will instead beconned to a selected subset of air transport-relatedtopics, where operations research has made some

Transportation Science 2003 INFORMSVol. 37, No. 4, November 2003, pp. 368391

0041-1655/03/3704/03681526-5447 electronic ISSN

BARNHART, BELOBABA, AND ODONIOperations Research in the Air Transport Industry

of its most signicant contributions to date. Exam-ples of important topics that are either not coveredat all or are touched on peripherally include: avia-tion safety and security, airline eet planning, airlinestafng, airline maintenance planning, aircraft load-ing, and decision support tools for the managementof airport operations (e.g., gate assignments). More-over, the specic topics and contributions that arehighlighted are presented in nonquantitative termsand largely reect the authors own interests. In addi-tion to the bibliographic references associated withthese contributions, other survey papers, which pro-vide additional details and references, are cited when-ever possible.Section 2 of the paper deals with the classical

problems of scheduling, routing, and crew assign-ment in the airline industry. This is a context thatis perfectly suited to the use of large-scale, discreteoptimization approaches and, indeed, has motivatedseveral methodological and computational develop-ments in this vibrant area of OR over the years. Sec-tion 3 covers airline revenue management, includ-ing overbooking, ight leg yield management andnetwork revenue maximization. Through a combina-tion of stochastic and optimization models, OR workin this area has generated signicant additional rev-enues for the airlines ever since the late 1980s. More-over, revenue management continues to be a eld inwhich airlines are vying intensively for competitiveadvantage. Section 4 surveys selected applications ofOR to the study, planning, and design of the twomajor pieces of aviation infrastructure, the airportssystem and the air trafc management (ATM) system.Historically the emphasis here has been on stochas-tic models, as the questions addressed have focusedon capacity, delays, and safety under conditions inwhich the probabilistic characteristics of the inputparameters play a dominant role. However, optimiza-tion models, both deterministic and stochastic, havefound use in the intensive recent research on air traf-c ow management, a topic also reviewed brieyin 4. Finally, 5 summarizes the main conclusionsregarding the fundamental challenges faced by futureresearch.

2. Aircraft and CrewSchedule Planning

Schedule planning involves designing future aircraftand crew schedules to maximize airline protabil-ity. This problem poses daunting challenges becauseit is characterized by numerous complexities, includ-ing a network of ights, differing aircraft types, gate,airport slot and air trafc control restrictions, noisecurfews, maintenance requirements, crew work rules,and competitive, dynamic environments in whichpassenger demands are uncertain and pricing strate-gies are complex. Not surprisingly, no single opti-mization model has been solved, or even formulated,to address this complex design task in its entirety.The problems unmanageable size and complexityhas resulted in the decomposition of the overallproblem into a set of subproblems, often dened asfollows:(1) Schedule design: Dening which markets to

serve and with what frequency, and how to scheduleights to meet these frequencies.(2) Fleet assignment: Specifying what size aircraft to

assign to each ight.(3) Aircraft maintenance routing: Determining how

to route aircraft to ensure satisfaction of maintenancerequirements.(4) Crew scheduling: Selecting which crews to assign

to each ight to minimize crew costs.Suboptimal, yet feasible aircraft and crew plans are

constructed by solving the subproblems in order, con-straining the solutions to subsequent problems basedon the solutions to preceding problems. Althoughsmaller and simpler than the overall problem, thesesubproblems are still large-scale and rich in complex-ity. In fact, OR theoreticians and practitioners havebeen developing models and algorithms to solve themfor decades and, in so doing, have had signicant suc-cesses and impacts.In the late 1960s and early 1970s, United Airlines,

American Airlines, British Airways, and Air France,recognizing the competitive advantage that decisiontechnologies could provide, formed OR groups. Thesegroups grew rapidly, developing decision supporttools for a variety of airline applications, includingaircraft and crew schedule planning, and in somecases began offering their services to other airlines

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as well. According to one executive, the scheduleplanning system developed at American Airlines andSabre has generated over $500 million in incremen-tal prots annually (Cook 2000). Most major airlinesaround the world have formed similar groups bynow. Their name and size, as well as their placementin the corporate structure, vary considerably from air-line to airline. Many consulting companies are alsooffering products and services in this general area.Other testimonials on the impact of optimization inthe airline industry can be found in Yu et al. (2003),Butchers et al. (2001), Wiper et al. (1994), Smith et al.(1992), and Patty et al. (1991).Beginning in the 1950s and 1960s, early research on

airline schedule planning focused on approaches toassign aircraft to routes (Ferguson and Dantzig 1956a)and crews to trips (Arabeyre et al. 1969). Since then,researchers have continued to work on these prob-lems, rening and expanding the models to better rep-resent the actual problem faced, or developing moresophisticated algorithms to improve the quality of thesolutions generated.In the following sections, we provide an overview

of progress, highlighting some of the major accom-plishments and describing some of the remainingchallenges.

2.1. Schedule DesignThe ight schedule, specifying the ight legs to beown and the departure time of each ight leg,largely denes the competitive position of an air-line and is thus a key determinant of airline prof-itability. Designing a prot maximizing ight sched-ule, however, is extremely complex. It affects and isaffected by essentially all aircraft and crew schedul-ing decisions of the airline, and competing airlines aswell. No single model has captured all these inter-dependencies, and even if such a model were for-mulated, it surely would be intractable. Moreover, itsinput data requirements are impractical, requiring, forexample, accurate estimates of itinerary-specic pas-senger demands, spill costs, and recapture rates.Notwithstanding this complexity, ight schedule

design and variants of the problem have been ofinterest to researchers for many years, with Simp-son (1966), Chan (1972), Soumis et al. (1980), and

Etschmaier and Mathaisel (1984) describing earlywork. Nonetheless, because of the inability of opti-mization models to adequately capture the scope ofthe design problem, the typical airline practice todayis to build ight schedules manually, with limitedoptimization. With recent research advances, how-ever, this trend is reversing and optimization is begin-ning to play a role.Success has been achieved by dening a simplied

design problem involving only incremental changesto existing ight schedules. Berge (1994), Marstenet al. (1996), and Lohatepanont and Barnhart (2001)develop models and algorithms that select from asubset of candidate ights legs, those that will beadded to or removed from a given (often existing)ight schedule. Their approaches are incremental inthat the changes from one published ight sched-ule to the next are limited. The reported impacts,however, are signicant. Lohatepanont and Barnhartsolve problems at one major airline that contain about800 potential ight legs, 65,000 itineraries, and 165aircraft, resulting in formulations with 30,00060,000rows and 50,00065,000 columns, and solution timesranging from 12 hours to more than 3 days. Theyreport potential improvements in aircraft utilizationand signicant increases in revenue, with an esti-mated impact exceeding $200 million at that airline.A nonincremental, clean-slate approach to airline

schedule design is described in Armacost et al. (2002).They present models and algorithms to generate(near-) optimal ight network designs for express par-cel delivery, and report savings of about 7% in oper-ating costs and potential reductions of 10% in therequired eet size.Another schedule design application involving a

charter airline is described in Erdmann et al. (1999).By exploiting the special characteristics of the prob-lem, they are able to achieve near-optimal solutionsin minutes.Schedule generation, with its important strategic

and nancial implications, represents an importantarea for future research, one rich in opportunity andchallenge. The successes to date are just rst stepsin addressing the myriad of questions surroundingschedule design. Future research is needed to capturethe critical interactions among the various resourcesof the airline, its competitors, and airports.

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2.2. Fleet AssignmentWith the ight schedule determined, the eet assign-ment problem is to nd the cost-minimizing assign-ment of aircraft types to legs in the ight network.Fleeting costs are comprised of:(1) Operating costs: Specied for each ight leg

aircraft type pair, representing the cost of ying thatight leg with that aircraft type.(2) Spill costs: Measuring the revenue lost when

passenger demand for a ight leg exceeds theassigned aircrafts seating capacity.One of the earliest research efforts on airline

scheduling was that of Ferguson and Dantzig (1956a).They developed a linear programming model to allo-cate aircraft to roundtrip routes to minimize operat-ing plus spill costs. In a follow-up paper (Fergusonand Dantzig 1956a), they expanded their approach tohandle uncertain demands. These rst models, whileapplied to simplied ight networks and tiny prob-lems by todays standards, identied key problemattributes and launched decades of research on eetassignment.As a result, some 30 years later, researchers devel-

oped the capabilities to solve eeting problems rep-resentative, both in complexity and size, of thosetruly faced by airlines. Abara (1989) and Hane et al.(1995) formulated the eet assignment problem asa multicommodity network ow problem with sideconstraints. The underlying network (depicted inFigure 1) has:(1) Nodes: Representing the times and locations of

ight leg departures and arrivals.(2) Flight arcs and ground arcs: Each ight arc corre-

sponds to a ight leg with its arrival time adjustedto include the minimum amount of time requiredon the ground for disembarking and embarking pas-sengers, unloading and loading baggage, and refuel-ing. Ground arcs represent idle aircraft on the groundbetween ights.The objective is to ow commodities, that is, air-

craft types, through the network feasibly and withminimum cost. Side constraints enforce the followingrequirements and restrictions:(1) Cover: Each ight leg is assigned to exactly one

aircraft type.

airport A

airport B

count line

flight arc

ground arc

Figure 1 A Timeline Network Involving Two Airports

(2) Aircraft count: Only available aircraft areassigned to the network.(3) Balance: Each aircraft type is assigned to the

same number of ight legs arriving at a station asdeparting that station.Each possible assignment of an aircraft type to a ightleg is represented by a binary variable, with value 1if the aircraft type is assigned to that ight leg, and0 otherwise. The resulting formulation for the eetassignment problem of a major airline contains about20,000 rows and 30,000 columns and can be solvedtypically within minutes.

2.2.1. Impacts and Challenges. Multicommodityow-based eet assignment models are widely usedby the industry and are credited with achievingsignicant cost savings, measuring in the millionsof dollars annually. For example, Rushmeier andKontogiorgis (1997) report realized savings of at least$15 million annually at USAir. Using eet assignmentmodels, Wiper et al. (1994) report annual savingsof $100 million at Delta Airlines, and Abara (1989)reports a 1.4% improvement in operating margins atAmerican Airlines.Beyond providing economic benets, research on

eet assignment problems has led to advanced tech-niques for solving general linear programs. For exam-ple, the node consolidation idea introduced in Haneet al. (1995), which reduced formulation size by morethan 40% for the problems of one major airline, hasbeen generalized and incorporated into commercialsolvers, allowing more efcient solution of large-scaleoptimization problems.



These impressive results notwithstanding, thereremain several critical challenges in eet assign-ment. Many of these challenges stem from modelingassumptions that include:(1) Many eet assignment models assume that the

ight schedules repeat daily, even though most air-lines operate different schedules on the weekend.(2) Most eeting models assume ight leg demand

is known and does not vary by day of week, but his-torical data show that day-to-day demand variationsare present.(3) Flying times and ground times are typically

assumed to be deterministic in eet assignment mod-els; however, congestion on the ground and in the air,weather conditions, and new security practices pro-duce large variations in ight and ground times.(4) Most eet assignment models assume that the

number of spilled passengers, and their associatedspill costs, can be computed at a ight-leg level. Infact, passenger demand, spill, and the revenue asso-ciated with each passenger are itinerary specic, notight-leg specic. As a result, it is possible to estimateleg-specic spill costs only approximately.To improve on leg-based models, researchers intro-

duced itinerary- or origin-destination-(O-D) basedeetassignment approaches (Jacobs et al. 1999; Barnhartet al. 2002a, b). These approaches consider passen-ger fares, demand, spill, and recapture to be itinerary,not ight-leg, specic. Inaccuracies resulting fromthe allocation of fares and demands to ight legsare thus not introduced into these enhanced eetingapproaches. Using their O-D-based eet assignmentapproach, Jacobs et al. (1999) report annual improve-ments of 0.54%0.77% in revenue compared to thoseobtained with a ight-leg-based model. Similarly, ina case study involving a major U.S. airline, Barn-hart et al. (2002a) report that itinerary-based eetassignment improves on leg-based eeting signi-cantly, with estimated savings ranging from 30 mil-lion to over 100 million dollars annually.In a rst step at integrating eet assignment and

schedule design decisions, Rexing et al. (2000) simul-taneously assign an aircraft type to each ight legand select each ight legs departure time, allowingretimings of 520 minutes from the current schedule.

Departure retiming allows additional aircraft assign-ments that can lead to better matches between ightleg demand and assigned capacity. The result, accord-ing to Rexing et al. (2000), is reduced operating costsand improved revenue capture, with savings for onemajor airline of $20$50 million annually.

2.3. Aircraft Maintenance RoutingWith schedule design and eet assignment decisionsmade, the ight network decomposes into subnet-works, each one associated with aircraft of a singletype. The assignment of individual aircraft to ightlegs in a subnetwork occurs in the aircraft mainte-nance routing step. The goal is to determine routings,or rotations, for each aircraft in a eet. A routing isa sequence of ight legs, with the destination of oneight leg the same as the origin of the next leg inthe sequence. A rotation is a routing that starts andends at the same location. Each aircrafts rotation vis-its maintenance stations at regular intervals. Moredetails on the maintenance routing problem are con-tained in Feo and Bard (1989), Gopalan and Talluri(1998), and Clarke et al. (1996b). For restricted mainte-nance routings of three or four days, Gopalan and Tal-luri (1998) and Talluri (1998) describe graph-theoreticapproaches to maintenance routing.In general, the aircraft maintenance routing prob-

lem can be modeled as a network circulation problemwith side constraints. The decision variables corre-spond to sequences (strings) of ight legs, with eachsequence beginning and ending at maintenance sta-tions and satisfying the rules governing the maximumtime between maintenance. If a string is included inthe solution, a single aircraft ies each ight in thesequence and then undergoes maintenance. Side con-straints include cover constraints and count constraints.Cover constraints ensure that each ight leg is con-tained in exactly one selected string, and count con-straints limit the number of assigned aircraft to thenumber available. Additional details are provided inBarnhart et al. (1998a).

2.3.1. Impacts and Challenges. Solving the eetassignment problem rst and then the resultingaircraft routing problems can lead to violationsof aircraft maintenance requirements. To guaran-tee feasible solutions, particularly in low-frequency,



point-to-point networks, researchers have integratedeet assignment and aircraft routing models, as inDesaulniers et al. (1997) and Barnhart et al. (1998a).This has the effect of substantially expanding the air-craft routing model to include multiple eet types,instead of a single aircraft type. The number of con-straints for large airlines is often less than a few thou-sand, but there are many billions of possible variables.For hub-and-spoke networks, however, the increasedsize and complexity of this integrated model is rarelywarranted. Instead, feasible solutions are typicallygenerated using a slightly modied sequential solu-tion approach in which an altered eeting model isrst solved and then the routing model is solved.The modied eet assignment model contains pseudo-maintenance constraints to ensure that sufcient num-bers of aircraft of each type are located at mainte-nance stations periodically. In hub-and-spoke networkswith banks containing many aircraft together on theground at the same time, these maintenance con-straints are often sufcient to ensure that the resultingeet assignment has associated feasible maintenanceroutings.In addition to the possibility of generating infea-

sible solutions, a disadvantage of the sequentialsolution approach to aircraft and crew planning isthat aircraft routing solutions limit possible crewscheduling opportunities, potentially causing crewcosts to increase signicantly. The linkage betweenaircraft routing and crew scheduling occurs becausea crewmember can connect between two ight legsseparated by less than the minimum required connec-tion time only if the same aircraft is assigned to bothlegs. To account for this, Klabjan et al. (2002) swapthe order of the problems and solve the crew pair-ing problem before the maintenance routing problem.This approach has the advantage of generating opti-mal crew solutions, but it does not ensure that forthe optimized crew solution there is a correspond-ing maintenance-feasible solution. To achieve crewoptimality and maintenance feasibility, Cordeau et al.(2000) and Cohn and Barnhart (2003) integrate thebasic maintenance routing and crew pairing models.

2.4. Crew SchedulingCrew scheduling problems with numerous, com-plex rules, and well-dened costs are particularly

amenable to optimization. With so many possibledecisions, it is difcult to nd feasiblelet aloneoptimalsolutions manually. Moreover, crews repre-sent the airlines second highest operating cost afterfuel, so even slight improvements in their utilizationcan translate into signicant savings.For these reasons, airline crew scheduling has gar-

nered considerable attention, with research spanningdecades. Arabeyre et al. (1969) published a surveyof early research activities on the topic. At thattime, because of large problem size and lack ofadvanced techniques and sufcient computing capa-bilities, heuristics were employed to nd improvedcrew solutions.Since then, researchers have continued to work on

the crew scheduling problem, seeking more efcientsolution approaches, expanding models to capturemore of the intricacies of the crew scheduling prob-lem, and integrating the crew problem with otherairline scheduling problems. Recent detailed descrip-tions of the airline crew scheduling problem areincluded in the survey papers by Desaulniers et al.(1998), Clarke and Smith (2000) and Barnhart et al.(2003).Even today, the crew scheduling problem is typ-

ically broken into two sequentially solved subprob-lems, the crew pairing problem and the crew assignmentproblem:(1) The crew pairing problem. The problem generates

minimum-cost, multiple-day work schedules, calledpairings. Regulatory agencies and collective bargain-ing agreements specify the many work rules thatdene how ight legs can be combined to create fea-sible schedules. Work-rule restrictions include limitson the maximum number of hours worked in a day,the minimum number of hours of rest between workperiods, and the maximum time the crew may beaway from their home base. Even with these limi-tations, the number of feasible pairings measures inthe billions for major U.S. airlines. The cost structureof pairings adds further complexity, with cost typ-ically represented as a nonlinear function of yingtime, total elapsed work time, and total time awayfrom base.(2) The crew assignment problem. This problem com-

bines these pairings into equitable and efcient



month-long crew schedules, called bidlines or rosters,assigning them to individual crewmembers. With ros-tering, schedules are constructed for and assignedto specic individuals, taking into account their par-ticular needs and requests. With bidline generation(a practice commonly used in the United States), ageneric set of schedules is constructed and individ-ual employees reveal their relative preferences forthese schedules through a bidding process. The spe-cic assignment of schedules to employees is basedon seniority, with the more senior crewmembers mostoften assigned their preferred schedules. More detailson the bidline and rostering problems can be found inKohl and Karisch (2003), Cappanera and Gallo (2001),Caprara et al. (1998), Christou et al. (1999), Dawidet al. (2001), Day and Ryan (1997), and Gamache et al.(1998).The crew pairing, bidline, and rostering problems

can all be cast as set partitioning problems. While sim-ple in form, the set partitioning model is powerful inthis context. To illustrate, consider the crew pairingproblem.The crew pairing problem has one binary deci-

sion variable for each possible pairing, and its objec-tive is to minimize the cost of the selected pairingssuch that for each ight, exactly one pairing con-taining that ight is chosen. By dening variablesas sequences of ight legs, only feasible pairings areconsidered and explicit formulation of complicatedwork rules is unnecessary. Moreover, nonlinear pair-ing costs can be computed ofine and captured as asingle value. The crew pairing model, is then a set par-titioning problem, with a linear objective function andbinary decision variables. The one drawback to thismodeling approach is that there are potentially manybillions of possible crew pairings, especially in hub-and-spoke ight networks with numerous aircraft athubs at the same time, allowing many possible crewconnections.

2.4.1. Solving Crew Scheduling Problems. Be-cause of the immense size of airline crew schedulingproblems, i.e., thousands of constraints and billionsof variables, researchers initially focused on heuristicmethods to achieve solutions. Many of these heuris-tics generate solutions by considering only a relativelysmall number of pairings. Examples of this strategy

can be found in Arabeyre et al. (1969), Anbil et al.(1991), Gershkoff (1989), and Hoffman and Padberg(1993).While producing improved solutions compared to

those generated earlier, these heuristics still have adrawback; namely, the quality of their solutions can-not be quantied relative to an optimal solution.With advances in optimization theory and comput-ing, however, researchers have built enhanced crewscheduling capabilities and designed optimization-based approaches capable of generating provablyoptimal or near-optimal crew pairing solutions.To generate crew solutions with known optimal-

ity bounds, even when variable enumeration is notpractical, many researchers have turned to branch-and-price. Branch-and-price, surveyed in Barnhart et al.(1998b), is a smart enumeration technique in whichlinear programming bounds are generated at eachnode of a branch-and-bound tree using column gen-eration. Column generation allows very large LPs tobe solved without explicitly enumerating all of thevariables, or columns. Rather than directly solvingthe master problem (the problem with all possiblecolumns), a restricted master problem containing onlya subset of the original columns is solved. The dualsolution to the restricted master problem is used toidentify columns with negative reduced cost. Thesecolumns are added to the restricted master problem;the restricted master problem is then resolved, andthe process is repeated until no negative reduced costcolumns can be found. At that point, the algorithmterminates with an optimal solution to the masterproblem.In the case of the crew pairing problem, negative

reduced cost columns are identied without explic-itly considering each pairing. Instead, all columnsare implicitly considered by solving a pricing prob-lem, often formulated as a multilabel, shortest pathproblem in which some paths correspond to pairings.By exploiting dominance, shortest pathsand henceminimum reduced cost pairingsare identied with-out examining all paths, or pairings. For a detailedexposition of multilabel, shortest path problems, seeDesrochers and Soumis (1988).A particular challenge in branch-and-price algo-

rithms is that a standard branching rule based on



variable dichotomy is often difcult to implement.Instead, a branch-on-follow-ons branching rule is typi-cally used (Ryan and Foster 1981). This rule is basedon pairs of ights, with one ight immediately follow-ing the other. One branching decision requires sched-ules containing either ight to contain both ightsand the other decision disallows this. For applicationsof this branching strategy to airline crew schedul-ing, see Lavoie et al. (1988), Desrosiers et al. (1991),Ryan (1992), Gamache and Soumis (1993), Barnhartet al. (1994), Vance et al. (1997), and Desaulniers et al.(1998).Even with the effective branch-on-follow-ons rule,

the number of branching decisions to prove optimal-ity is typically excessive for crew-scheduling prob-lems. To remedy this, researchers have embeddedheuristics within the branch-and-price optimizationprocess. An example is the variable-xing approachdescribed in Marsten (1994) in which variables withfractional values close to one are sequentially xedto one. Marsten shows that by reducing the num-ber of LPs solved in generating integer solutions,improved solutions can be generated with signi-cantly less computing time and memory. Klabjan et al.(2001) integrate the heuristic consideration of sub-sets of columns with an optimization-based pricingapproach. They produce improved solutions, com-pared to those achieved with branch-and-price, byinitially developing hundreds of millions of columnsand applying random selection and pricing tech-niques to prune the number of columns ultimatelyconsidered.

2.4.2. Impacts and Challenges. Most major air-lines use optimization tools to partially or fullygenerate their crew schedules, with signicant eco-nomic impacts. Already in the 1960s, airlines weresolving crew pairing problems to reduce costs andautomate their burdensome manual planning process(Arabeyre et al. 1969). With advances in optimizationand computing, crew pairing solvers became increas-ingly sophisticated and applicable. Using advancedheuristic techniques, Anbil et al. (1991) achieved costsavings at American Airlines of $20 million dollarsannually, representing an increase in crew utilizationof 1.5%.

Motivated by the potential for even further costsavings, researchers focused on the development ofoptimization-based approaches in the decades span-ning the 1980s and 1990s. Barnhart et al. (2003)reported that using a branch-and-price approach, onemajor U.S. airline was able to produce near-optimalcrew solutions costing $50 million per year less thanthe solutions generated by a state-of-the-art heuristic.In addition to these economic benets, crew

scheduling optimization can be a useful tool in con-tract negotiations, helping airlines to quantify theimpacts of proposed changes in cost structures, bene-ts, and work rules. The economic impact of each pro-posed change can be evaluated by changing selectedinputs to the crew scheduling model, and rerunningthe optimization procedure.While signicant, the effects of optimization on

crew scheduling are nonetheless limited by thesequential schedule planning process. With the ightschedule, eet assignment, and aircraft routing deci-sions xed, the range of crew scheduling possibilitiesis limited. To mitigate the myopic effects of sequentialsolutions, researchers have developed extended mod-els that incorporate crew considerations into some ofthe other subproblems solved. For example, Clarkeet al. (1996a) and Barnhart et al. (1998c) extend eetassignment models to account for some of the down-stream effects on crews. The eet assignment model inthe sequential solution process is then replaced withone of these extended models, resulting in an increasein eet assignment costs that is offset by a greaterreduction in crew costs.

2.5. Ongoing and Future ChallengesNotwithstanding the substantial progress made insolving aircraft and crew scheduling problems inrecent decades, signicant work remains. Researchopportunities, beyond those identied for the sched-ule design, eet assignment, aircraft routing, and crewscheduling problems described above, include:Schedule planning: (1) Integrating decisions involv-

ing schedule design and aircraft and crew routingand scheduling; (2) expanding schedule planningmodels to include pricing and revenue managementdecisions; (3) assessing the systemwide cost and ser-vice impacts of paradigm shifts in airline scheduling,



such as depeaking ight schedules by spreading outaircraft arrival and departure banks; and (4) devel-oping tools, such as the stochastic model and airlineoperations simulator described in Rosenberger et al.(2002), to evaluate airline plans and recovery policiesin a stochastic operating environment.Robust planning: Optimized solutions are rarely exe-

cuted as planned. Crew sickness, mechanical failures,and adverse weather result in necessary changes tothe plan, often leading to signicantly increased costs.Moreover, a nely tuned, optimized solution achievesincreased utilization through the removal of slack,providing crews with less time to connect betweenights and aircraft with less time on the groundbetween ying. Less slack time, although economicalin theory, can translate in practice into less robust-ness and increased costs. To address these issues,researchers have begun to investigate a different plan-ning paradigm, one that considers unplanned, dis-ruptive events and attempts to minimize realized, andnot planned, costs. Current research, such as thatof Ageeva (2000), Rosenberger et al. (2001a), Schae-fer et al. (2001), and Chebalov and Klabjan (2001),attempts to achieve robustness by isolating causes ofdisruption and/or downstream effects. In Lan (2003),robust aircraft routes are generated to place slack judi-ciously, that is, where it is needed to minimize thedisruptive effects of delays on passengers.Operations recovery: Given a plan and one or more

disruptions, the goal of operations recovery is toreturn the airline to its plan quickly (often requir-ing decisions within minutes) and cost effectively.Detailed descriptions of recovery models and algo-rithms are included in Jarrah et al. (1993), Clarke(1997), Thengvall et al. (2000), Rosenberger et al.(2001b), Stojkovic et al. (2002), Bratu (2003), andYu et al. (2003). Although these problems typicallyinvolve multiple airports and many of the airlinesresources, most of the work to date focuses on recov-ering a particular resource, either aircraft, crews, orpassengers. For example, in their Edelman Award-winning work, Yu et al. (2003) report that Conti-nental Airlines saved approximately $40 million in2001 using their crew recovery system. The challengemoving forward is to enhance current capabilities byconsidering integrated passenger, crew, and aircraft

recovery decisions, while rapidly determining low-cost solutions.

3. Airline Revenue ManagementEven with an optimized eet assignment and sched-ule of operations, some ight departures will haveempty seats while others will experience more pas-senger demand than capacity. In an effort to bettermatch the demand for each ight with its capacityand to increase total revenues, airlines practice differ-ential pricing by offering a variety of fare products atdifferent price levels for the same ight. Revenue man-agement is the practice of determining the number ofseats on each ight to be made available at each farelevel, limiting low-fare seats and protecting seats forlater-booking, higher-fare passengers. Given that theoperating costs of a scheduled ight departure are inlarge part xed in the very short run, the goal of rev-enue management is to ll each ight with the maxi-mum possible revenue to maximize operating prot.This section provides a brief review of the role of

operations research in the development of airline rev-enue management (RM) models, with an emphasis onthe works that have most inuenced the state of thepractice in the industry. A much more comprehensivesurvey of OR literature dealing with revenue manage-ment and related problems can be found in McGilland van Ryzin (1999). In addition, Weatherford andBodily (1992) developed a categorization of perish-able asset revenue management problems, of whichthe airline revenue management problem is the best-known example.This review begins with an introduction to the

functions of typical airline RM systems, followed bydescriptions of the types of OR models employed toperform three of the principal techniques of revenuemanagementoverbooking, fare class mix, and O-Dcontrol. Therefore, the focus of this discussion is onthe seat inventory control component of airline rev-enue maximization, as virtually all airline RM systemsassume that the fare structure is determined exoge-nously by a separate airline pricing function.

3.1. The Evolution of Airline RM SystemsThe sheer size and complexity of the airline seatinventory control problem led to the development



of computerized RM systems. A medium-sized air-line might operate 1,000 ight legs per day, using10 booking (fare) classes in its reservations system,and accepting bookings up to 330 days prior to eachdeparture. At any point in time, this airlines seatinventory includes over three million booking limits,which can change with each booking that is accepted.Airline RM systems have evolved in both their

computer database and mathematical modeling capa-bilities over the past 20 years. The rst RM systems,developed in the early 1980s, were designed to collectand store data extracts from computer reservationssystems (CRS). By the mid-1980s, several RM sys-tems offered additional monitoring capabilities thatallowed actual ight bookings to be tracked dynam-ically, relative to an expected or threshold book-ing curve for the ight. By the late 1980s, the moreadvanced airlines began to implement RM systemsthat could perform forecasting and optimization bybooking class for each future ight leg departure, inaddition to having all of the database and bookingmonitoring capabilities of previous systems. It wasinto this third generation of RM systems that ORmodels, some of which had been developed a decadeor more earlier, began to be integrated.The major components of a typical third-generation

RM system are illustrated in Figure 2. Historical book-ing data for the same ight leg and day of the weekare combined with actual booking information foreach future ight departure to generate a forecast oftotal demand by booking class for that departure.

REVENUEDATA

ACTUALBOOKINGS

HISTORICALBOOKING DATA

OVERBOOKINGMODEL

NO-SHOWDATA

FORECASTINGMODEL

RECOMMENDEDBOOKING LIMITS

OPTIMIZATIONMODEL

Figure 2 Third-Generation Airline RM System

These forecasts, together with estimates of the rev-enue value of each booking class, are then fed into anoptimization model that calculates the recommendedbooking limits for the ight departure in question.At the same time, the demand forecasts are fed intoan overbooking model, which also makes use of his-torical information about passenger no-show rates tocalculate an optimal overbooking level for the futureight departure. Both the booking class limits andoverbooking levels calculated by the mathematicalmodels are then presented as recommendations to theRM analyst.The demand forecasts and booking limits are

reviewed by the RM systems at regular intervalsduring the ight booking process, as often as dailyin some cases. Should unexpected booking activityoccur, the system reforecasts demand and reoptimizesits booking limit recommendations. A substantial pro-portion of the revenue gain attributable to fare mixoptimization comes from this dynamic revision ofbooking limits.Most large and medium-sized airlines throughout

the world have implemented third-generation RMsystems, and the benets of such systems have beenwell documented. Effective use of techniques for over-booking and fare class mix alone have been estimatedto generate revenue increases of as much as 4%6%compared to situations in which no seat inventorycontrol tools were applied (Belobaba 1992b, Smithet al. 1992).

3.2. Overbooking ModelsAirlines have been accepting reservations in excessof aircraft capacity for more than two decades, in aneffort to reduce the revenue losses associated with no-shows. With the development of RM systems, over-booking was incorporated into the seat inventorycontrol functions of these systems. The objective ofthe ight overbooking component of revenue man-agement is to determine the maximum number ofbookings to accept for any given future ight depar-ture, trading off the risks and costs of denied board-ings against the potential revenue loss from unsold orspoiled seats.Operations research work on the airline overbook-

ing problem can be traced to the 1960s, well before



overbooking became integrated into RM systems (andbefore RM systems were even required). Notable earlypublished works include those of Simon (1968) andVickrey (1972), as well as articles by Rothstein (1971,1985). Rothsteins 1985 article is a survey of previousOR literature dealing with the airline overbookingproblem, and includes a discussion of the customerservice impacts of inaccurate overbooking and therole of government in regulating denied boardingpenalties.Statistical overbooking models typically represent

no-show rates as Gaussian random variables. Theobjective is to nd the maximum authorized numberof bookings (or AU) that will keep denied board-ings below some airline-specied target level witha desired level of condence. These models provideairline managers with exibility in determining theirown overbooking policy, for example, by increas-ing denied boarding tolerance or reducing statisticalcondence.An extension of the statistical overbooking ap-

proach is the cost-based overbooking model, whichexplicitly accounts for the actual costs associated withdenied boardings and with empty seats (spoilage).The objective is to nd the optimal overbooking levelthat minimizes the total combined costs of deniedboardings and spoilage by performing a search over areasonable range of AU values. The cost-based over-booking model is the current state of the practiceat many airlines. However, this approach representsa static formulation of the overbooking problem, inthat the dynamics of passenger bookings, cancella-tions, and no-shows are not explicitly accounted forin determining an overbooking level.The OR literature contains many additional works

on the airline overbooking problem, some of whichpropose dynamic programming (DP) formulations.Rothsteins (1968) Ph.D. thesis was the rst todescribe such a DP approach, while Alstrup et al.(1986) extended the DP formulation to a two-class,joint overbooking and fare class mix problem. Morerecently, Chatwin (1996) as well as Feng and Xiao(2001) have proposed DP-based approaches that allowfor the incorporation of time-dependent no-showand cancellation rates associated with multiple fareclasses. In practice, few airlines have implemented

such complex DP formulations because of the difcul-ties of providing adequate and accurate inputs in theform of booking and cancellation rates by the time-period before departure.The economic motivation for airline overbooking

is substantial. In the United States, domestic airlineno-show rates average 15%25% of nal predeparturebookings. Given that most airlines struggle to attain a5% operating margin (revenues over costs), the loss of15%25% of potential revenues on fully booked ights(which would occur without overbooking) can repre-sent a major negative impact on prots. As part ofa revenue management system, effective overbookinghas been shown to generate as much revenue gain asoptimal fare class seat allocation, described below.

3.3. Fare Class MixThe second major technique of airline revenuemanagement is the determination of the revenue-maximizing mix of seats available to each booking(fare) class on each future ight-leg departure. Vir-tually all airline RM systems were developed withthe capability to optimize fare class mix as their pri-mary objective. As introduced earlier, RM systemsforecast the expected demand for each fare class oneach future ight-leg departure by applying statisti-cal models to historical booking data for the samefare class on previous departures of the same ight.The forecasting of expected booking demands forfuture ight departures has been addressed in manyOR papers, including Littlewood (1972), LHeureux(1986), Lee (1990), and Curry (1994).These demand forecasts are then used as inputs to a

seat allocation optimization model, which determinesbooking limits to be applied to each of the bookingclasses on the ight departure in question. The vastmajority of airline reservations systems now haveinventory structures based on serial nesting of bookingclasses, as shown in Figure 3. Seats are not allocatedto partitioned classes, but are instead protected forhigher fare classes and nested booking limits are appliedto the lower fare classes. All available seats in theshared inventory are made available to the highestbooking class, in the (unlikely) event that the entirecapacity of the aircraft can be lled with demand forthe highest-priced fare product. This ensures that the



TOTAL AVAILABLE

SEATS

BL1

BL2

BL3

}}

Protected for Class 1 from 2, 3, and 4

Protected for 1+2 from 3 and 4

}Protected for 1+2+3 from 4 BL4

Figure 3 Nested Booking Limits and Class Protection Levels

airline would never turn down a high fare request, aslong as there are any seats still available for the ight.OR work on seat allocation problems for two fare

classes (full-fare and discount fare) can be traced toLittlewood (1972). Littlewoods (1972) rule for protect-ing full-fare seats was extended to multiple nested fareclasses by Belobaba (1987, 1989), who wrote the rstdoctoral dissertation on airline pricing and revenuemanagement. The Expected Marginal Seat Revenue(EMSR) approach for setting RM booking limits wasthen rened to become the EMSRb model (Belob-aba 1992a). These models were the rst to recognizethat the optimality conditions for traditional resource(seat) allocation problems did not result in the max-imum expected revenues in a multiple class, nestedbooking limit environment. Both models are based onheuristic decision rules for nested booking classes andhave become the most commonly used seat inventorycontrol models in airline RM systems. Optimal formu-lations of the multiple nested class problem have alsobeen published by Curry (1990), Brumelle et al. (1990),and Wollmer (1992).The general premise of these fare class mix mod-

els is as follows. Given the forecast demand for eachbooking class, expressed in terms of a mean and stan-dard deviation, along with its associated average fare,the expected marginal revenue of each incrementalseat on a ight leg can be determined. It is equal tothe average fare of the booking class under consider-ation multiplied by the probability that demand will

materialize for that incremental seat. The optimal pro-tection level for a higher-class seat is equal to the num-ber of seats with an expected marginal seat revenuegreater than or equal to the average fare in the nextlower class. Because higher fare classes have accessto unused lower class seats in a nested booking classinventory structure, the problem is to nd seat pro-tection levels for higher classes, and booking limits onlower classes. Simulations and actual airline experi-ence have indicated that use of these decision mod-els for establishing the fare class mix of seats on eachight departure can increase total airline revenues by2%4% (Belobaba 1989).Recent research on the single-leg fare class mix

problem has been rather limited, as both researchersand practitioners have focused on network opti-mization techniques for airline revenue managementinstead, as described below. The most recent pub-lished works involve extensions that include theapplication of DP methods and the joint optimizationof fare class booking limits and overbooking levels(for example, Zhao and Zheng 2001), as well as thejoint optimization of the airline pricing and fare classmix problems, as described in 3.5.

3.4. Network Revenue Management:Origin-Destination Control

Network RM (or O-D Control) represents a major stepbeyond the fare class mix capabilities of most third-generation RM systems, and is currently being pur-sued by the largest and most advanced airlines in theworld. As its name implies, O-D control gives the air-line the capability to manage its seat inventory by therevenue value of the passengers origin-destinationitinerary on the airlines network, not simply accord-ing to the fare class requested on a single ight leg.Because a leg-based RM system cannot distinguishamong itineraries in the same fare class, optimizingfare class mix on each ight leg individually will notensure that total network revenues are being maxi-mized. This is especially true for the large connectinghub networks operated by many airlines, in which asubstantial proportion of passenger itineraries involvemultiple ight legs and a connection at the hub.As a rst step in the implementation of O-D control

strategies, American Airlines in the 1980s developed



what has come to be known as virtual nesting(Smith et al. 1992), an assignment process that mapseach itinerary/fare type to a hidden or virtual valueclass within the airlines own reservations system. Foran itinerary/fare assigned to a given virtual class,the seat availability on a given ight leg is estab-lished by the booking limit for that virtual class. Ini-tial implementations of virtual nesting were based ontotal itinerary fare values, giving preference to longer-haul connecting passengers with higher total fares.The shortcoming of this greedy approach was thatit did not address the need under certain circum-stances to give preference to two local passengersinstead of taking the connecting passenger.OR models have been instrumental in the sub-

sequent development of O-D control strategies thatallowed airlines to manage seat inventories onthe basis of the network revenue value of eachitinerary/fare combination. Network revenue value canbe dened as the total itinerary fare (ticket value)minus the revenue displacement that might occur onconnecting ight legs if the passengers request fora multiple-leg itinerary is accepted. For example, thenetwork revenue value of an $800 total fare on therst leg of a connecting itinerary must be reduced by$300 if acceptance of this passenger would result indisplacement of $300 in total network revenue on thesecond ight leg. The displacement cost associatedwith the last (lowest valued) available seat on a givenight leg is its marginal value to the total networkrevenuethe amount by which the total network rev-enue would decrease if we were to remove (or sell)one seat on the ight leg. If the ight leg consistentlyhas empty seats, the marginal network value of itslast seat would be zero. In the more realistic situa-tion where the ight leg has a signicant probabilityof being full and it carries both local and connectingpassengers, determining the impact on total networkrevenues of removing a seat can require substantiallymore complicated analysis, including network opti-mization models.Various network optimization and heuristic algo-

rithms have been applied to the problem of deter-mining the network revenue value contribution ofeach O-D itinerary and fare product (or ODIF). Asdescribed in Belobaba (1998), several large airlines

make use of leg-based EMSR values to approximatethe network revenue value associated with a givenconnecting itinerary/fare combination. Because theseapproaches do not make use of any formal networkoptimization models, they are referred to as leg-based heuristic models for estimating network rev-enue displacement.The majority of published works devoted to the

O-D seat inventory control problem have proposedmathematical programming approaches to determineoptimal seat allocations for every ODIF over a net-work of connecting ight legs (see, for example,Glover et al. 1982, Dror et al. 1988, and Curry 1990).Williamsons (1992) doctoral thesis provides a compre-hensive review of the literature relevant to the applica-tion of mathematical programming and network owmodels to the O-D seat inventory control problem.More recently, there have been new works propos-ing more sophisticated network models. For exam-ple, Bratu (1998) developed a probabilistic networkconvergence algorithm for estimating the marginalnetwork value of the last available seat on each legin an airline network. Talluri and van Ryzin (1999a)propose a randomized linear programming method,while stochastic dynamic programming is also beingpursued as an alternative algorithm for determiningthe marginal network value of a range of remainingseats on each ight leg to account for changes to themarginal value as bookings are accepted.Given an estimate of down-line revenue displace-

ment it is possible to map ODIFs to virtual classesbased on their estimated network revenue value. Dis-placement adjusted virtual nesting (DAVN) increasesavailability to connecting passengers, while adjust-ment for down-line displacement of revenues ensuresthat two local passengers (with a higher total revenue)will receive preference when two connecting ightsare expected to be heavily booked.The estimated marginal network revenue of the

lowest-valued available seat on a ight leg can alsobe used in a much simpler inventory control mech-anism based on bid price controls. At the time of anODIF request for seat availability, the ODIF fare isevaluated against the sum of the leg bid prices overthe itinerary being requested. As was the case for esti-mates of revenue displacement associated with con-necting passengers, leg bid prices can similarly be



calculated either with network optimization tools orleg-based heuristics such as EMSR approaches. Talluriand van Ryzin (1999b) have done much work onnetwork bid price control and identify the condi-tions under which this approach provides revenueoptimality, while de Boer et al. (2002) compare theperformance of deterministic and stochastic networkformulations for O-D control.Until recently, relatively few airlines had imple-

mented network optimization models for dynamiccalculation of displacement costs and/or bid pricesfor O-D control. Because most reservations systemsand, in turn, third-generation RM systems weredeveloped on the basis of leg/fare class data, mostairlines did not have access to the detailed historicalODIF booking data required by network optimiza-tion models. Use of large-scale network optimizationmodels also raised technical and computational issuesrelated to the solution times and frequency of reop-timization. However, with the development of airlinedatabases designed to capture detailed ODIF histori-cal data, along with advances in both solution algo-rithms and computational speeds, network revenuemanagement has been implemented by over a dozenairlines in different parts of the world.The benets of leg-based revenue management

and incremental benet of O-D controls over leg-based fare class controls have been estimated byseveral researchers through simulation. For example,Williamson (1992) developed a network revenue man-agement simulation approach that allowed differentschemes for optimization and control of seat inven-tories to be tested. An even more realistic approachto simulating the impacts of different RM schemes ina full-scale, competitive airline network environmentis that of the passenger origin-destination simulator(PODS). Developed originally by researchers at Boe-ing (Hopperstad 1997), PODS has been enhanced torealistically simulate large networks in which compet-ing airlines generate RM forecasts and set seat inven-tory controls based on historical (i.e., previouslysimulated) data. At the same time, the simulated pas-sengers in PODS choose among alternative airlines,fares, restrictions, schedules, and seat inventory avail-ability as established by each airlines own RM sys-tem. PODS has been used to simulate the competitive

impacts of RM (Belobaba and Wilson 1997), as well asthe benets of improved forecasting models and theimpacts of RM on airline alliances.The simulations cited here along with others per-

formed by academics and airlines have provided con-sistent estimates of the potential for revenue gains of1%2% from advanced network revenue managementmethods, above and beyond the 4%6% gains real-ized from conventional leg-based fare class control.The potential to realize even 1% in additional revenuethrough network RM is substantial enough that manyof the worlds largest airlines have implemented orare in the process of developing their O-D controlcapabilities. For a large airline with annual revenuesof $5 to $10 billion or more, successful implementa-tion of a network RM system can lead to total revenueincreases of $50 to $100 million per year.

3.5. Future ChallengesDevelopment of OR models for the next genera-tion of airline revenue management is currently anextremely popular topic among academics and prac-titioners alike. The most obvious next steps in thefurther enhancement of airline revenue managementsystems is to integrate the pricing and seat inventorycontrol decisions currently being made with differentdecision support tools and, at many airlines, in dif-ferent parts of the organization. Clearly, the ability torelax the traditional RM assumption that fare struc-tures are given and xed has the potential to furtherincrease the revenue gains of RM. Joint pricing andinventory optimization requires the incorporation ofpassenger choice and demand elasticity models, andpromising OR work in this direction has been pub-lished by Weatherford (1997), Gallego and van Ryzin(1997), and Cote et al. (2003), among others. In arecent Ph.D. dissertation de Boer (2003) examines thisproblem in a network context and presents a varietyof other modeling advances and insights.Looking ahead, it is apparent that information

about the utilization of seat inventories and theresponse of passenger demand to different pricingstrategies can and should provide useful feedbackto eet assignment and even scheduling of airlineight departure times. The integration of airline pric-ing and seat inventory decisions with those of the



scheduling and eet assignment functions is thereforeconsidered to be the ultimate challenge for airlineoperations research. Development of more respon-sive and even dynamic decision support systems thattake into account expected passenger choice behav-ior, as well as the expected response of competitorsin terms of price, schedule, and capacity in determin-ing a prot-maximizing strategy for the airline, willonly become more important with growing competi-tive pressures in the airline industry.

4. Applications to AviationInfrastructure

The infrastructure of the global aviation system con-sists of two principal elements, airports and air trafcmanagement (ATM) systems. Airports can be furthersubdivided into airside facilities (runways, taxiways,aprons, aircraft stands) and landside facilities (pas-senger and cargo buildings, curbside), while ATMsystems are now viewed as being comprised of a tacti-cal subsystemair trafc control (ATC)and a strate-gic oneair trafc ow management (ATFM). Thedesign, development, and operation of all these facil-ities and systems has attracted extensive interest onthe part of operations researchers, usually in responseto ongoing developments in the eld. For example,much of the fundamental work on airside capacitywas performed during the 1960s and early 1970s, thetime when it was rst realized that runways con-stituted an important bottleneck of the air transportsystem. Overall, the body of work on aviation infras-tructure has led to insights and models that haveproved of critical importance in practice and have, insome cases, been adopted by airport and ATM ser-vice providers on a global scale. Because of spacelimitations, this section briey reviews OR applica-tions in airport airside operations and air trafc owmanagementonly two of the four major areas iden-tied above. Surveys of OR models for the analy-sis of passenger terminal operations can be found inTosic (1992) and de Neufville and Odoni (2003). Ofthe many OR-related topics addressed by research onair trafc control, the widely investigated subject ofdetecting and resolving potential conicts betweenairborne aircraft is reviewed well in Kuchar and Yang

(2000). Various other analytical and simulation mod-els on several different aspects of ATC are covered inOdoni et al. (1997).

4.1. Airside OperationsThe runway complexes of major airports are amongthe scarcest resources of todays international airtransport system and, barring a drastic change in thelanding and takeoff requirements of commercial air-craft, will continue to be so in the foreseeable future.New runways are very expensive to build, requiregreat expanses of land, and most importantly haveenvironmental and other impacts that necessitate longand complicated approval processes with uncertainoutcomes. It is not surprising therefore that one of themost mature areas of transportation science dealswith the modeling of runway operations and, moregenerally, airside operations. The products of thiswork include both analytical (mathematical) mod-els and simulation tools.

4.1.1. Analytical Capacity and Delay Models.Analytical models preceded viable simulation tools byabout 20 years. In a landmark paper, Blumstein (1959)dened the capacity of a runway as the expectednumber of movements (landings and takeoffs) thatcan be performed per unit of timetypically onehourin the presence of continuous demand andwithout violating air trafc control separation require-ments. He also presented a model for computing thecapacity of single runways used for arrivals only,for departures only, and for strings of arrivals fol-lowed by strings of departures. Subsequent general-izations included the possibility of inserting depar-tures between successive arrivals, possibly by increas-ing (stretching) the separation between arrivals(Hockaday and Kanafani 1972) and the treatment ofsome of the parameters of Blumsteins (1959) mod-els as random variables, instead of constants (Odoni1972).Extensions to cases involving two or more simulta-

neously operating runways were also developed at anearly stagesee, e.g., Swedish (1981). The complex-ity of multirunway models depends greatly on theextent to which operations on different runways inter-



act because of air trafc control separation require-ments. For example, in the case of a pair of intersect-ing runways, the location of the intersection relativeto the points where takeoffs are initiated or wherelanding aircraft touch down greatly affects the com-bined capacity of the two runways. Similarly, in thecase of two parallel runways, the capacity depends onthe distance between the centerlines of the runways.Approximate capacity analyses for airports with twoparallel or intersecting runways are quite straightfor-ward. Multirunway analytical capacity models alsoprovide good approximate estimates of true capacityin cases involving three or more active runways, aslong as the runway congurations can be decom-posed into semi-independent parts, each consistingof one or two runways. Such models have provedextremely valuable in airport planning, as well as inassessing the impacts of proposed procedural or tech-nological changes on airport capacity.Another topic of intensive study has been the esti-

mation, through the use of queueing models, of thedelays caused by the lack of sufcient runway capac-ity. This is a problem that poses a serious challenge tooperations researchers: The closed-form results devel-oped in the voluminous literature of classical steady-state queueing theory are largely nonapplicableatleast when it comes to the really interesting cases. Thereason is that airport queues are, in general, stronglynonstationary. The demand rates and, in changingweather conditions, the service rates at most majorairports vary strongly over the course of a typicalday. Moreover, the demand rates may exceed capacity( > 1), possibly for extended periods of time, mostoften when weather conditions are less than optimal.This has motivated the development of numer-

ical approaches to the problem of computing air-port delays analytically. In another landmark paper,Koopman (1972) arguedand showed through exam-ples drawn from New Yorks Kennedy and LaGuardiaAirports, at the time among the worlds busiestthatthe queueing behavior of an airport with k runwayequivalents (i.e., k nearly independent servers) canbe bounded by the characteristics of the M(t)/M(t)/kand the M(t)/D(t)/k queueing models, each providingworst-case and best-case estimates, respectively.

Note that this allows for dynamic changes in the ser-vice rates, as well as in the demand rates.Extending the work of Koopman (1972), the

Mt/Ekt/k system was proposed by Kivestu (1976)as a model that could be used to directly com-pute approximate queueing statistics for airportsrather than separately solving the M(t)/M(t)/k andM(t)/D(t)/k models and then somehow interpolatingtheir results. (Note that negative exponential servicetimes (M and constant service times (D) are sim-ply special cases of the Erlang (Ek) family, with k = 1and k = , respectively.) Kivestu (1976) noted that kshould be determined from the relationship ES/S =k, where ES and S denote the expected value

and the standard deviation of the service times andcan be estimated from eld data. He also devel-oped a powerful numerical approximation schemethat computes the (time varying) state probabilitiesfor the Mt/Ekt/k system efciently. Malone (1995)has demonstrated the accuracy and practicality ofKivestus (1976) approach and developed additionalefcient approximation methods, well suited to theanalysis of dynamic aireld queues. Fan and Odoni(2002) provide a description of the application ofKivestus (1976) model to a study of the gridlock con-ditions that prevailed at LaGuardia Airport in 2000and early 2001.Additional (numerical) analytical models for com-

puting airport delays have been developed over thelast few years. Peterson et al. (1995) and Daniel(1995) describe two different models for computingdelays at hub airports, which are characterized bysharp banks or waves of arrivals and departures.Hansen (2002) has used a deterministic model, basedon the notion of cumulative diagrams, to computedelay externalities at Los Angeles International Air-port. Finally, Long et al. (1999) and Malone (1995)present two dynamic queueing network models andtheir application to the study of congestion in theNational Airspace System. Ingolfsson et al. (2002)offer a comprehensive survey and comparison of sev-eral alternative approaches to the analysis of nonsta-tionary queueing systems.Many of the best features of some of the analyt-

ical capacity and delay models just described havebeen integrated recently in a number of new software



packages (Long et al. 1999, Stamatopoulos et al. 2003,EUROCONTROL 2001) that perform both capacityand delay analyses and, in the instance of the lasttwo references, include models of aprons and aircraftstands. After some necessary enhancements and anadequate amount of testing in a variety of airportenvironments, packages of this type may provide air-port planners within a few years with an easy-to-useand very fast set of tools for the study of a host ofairside issues.

4.1.2. Airside Simulations. General-purpose sim-ulation models of airside operations rst becameviable in the early 1980s and have been vested withincreasingly sophisticated features since then. Threemodels currently dominate this eld internationally:SIMMOD, The Airport Machine, and the Total Air-port and Airspace Modeler (TAAM). A report byOdoni et al. (1997) contains detailed reviews (some-what out-of-date by now) of these and several otherairport and airspace simulation models and assessesthe strengths and weaknesses of each. At their cur-rent state of development (and in the hands of expertusers), they can be powerful tools in studying detailedairside design issues, such as guring out the bestway to remove an airside bottleneck or estimating theamount by which the capacity of an airport is reduceddue to the crossing of active runways by taxiingaircraft.Unfortunately, these models are frequently misused

in practice, at great cost to the client organization.This happens when they are applied to the studyof macroscopic issues that can have only approxi-mate answers because of the uncertainty inherent inthe input data. An example is a question that oftenconfronts airport operators: When will airside delaysreach a level that will require a major expansion of anairports capacity (e.g., through the construction of anew runway)? Questions of this type, often requiringa look far into the future, are best answered throughthe approximate analytical models surveyed earlier,which permit easy exploration of a large number ofalternative scenarios and hypotheses. Detailed simu-lation models, by contrast, cannot cope well with themassive uncertainty involved because they requireinputs that are difcult to produce (e.g., a detailedschedule of aircraft movements at the airport for a

typical day 10 or 15 years hence) and lack credibilityunder the circumstances.

4.1.3. Optimizing Airside Operations. The air-side models discussed so far are descriptive in nature.Their objective is to help users understand and pre-dict the operational characteristics of the variousairside facilities under different operating scenarios.A considerable amount of OR work with an optimiza-tion focus also exists, much of it concerned with theeffective use of runway systems.The capacity of a runway is largely determined

by the separation requirements specied by theproviders of ATM services (e.g., the FAA in the UnitedStates). For any pair of consecutive runway opera-tions these requirements depend on the type of air-craft involved. For example, in the United States,when an arriving heavy (H) aircraftdened asone with a maximum takeoff weight (MTOW) greaterthan 255,000 lbsis immediately followed by an ar-riving small (S) aircraft (MTOW < 41,000 lbs), therequired separation between them, at the instantwhen H is about to touch down on the runway, is6 nautical miles (10.9 km). This is because heavyaircraft (wide-body jets) may generate severe waketurbulence, which may be hazardous to other aircraftbehind it. By contrast, when an aircraft of type S isfollowed by one of type H, the required separation is2.5 nautical miles (4.5 km). Note that given a num-ber n of aircraft, all waiting to land on a runway,the problem of determining the sequence of landings,such that the time when the last aircraft lands is min-imized, is a Hamiltonian path problem with n points.However, this is only a static version of a prob-

lem which in truth is a dynamic one: Over time thepool of aircraft available to land changes, as someaircraft reach the runway while new aircraft jointhe arrivals queue. Moreover, minimizing the latestlanding time (or maximizing throughput) shouldnot necessarily be the objective of optimal sequencing.Many alternative objective functions, such as mini-mizing the average waiting time per passenger, arejust as reasonable. A further complication is that thevery idea of sequencing runs counter to the tra-ditional adherence of ATM systems to a rst-come,rst-served (FCFS) discipline. Deviations from FCFSraise concerns among some airside users about the



possibility of systematic discrimination against cer-tain classes of aircraft operators (e.g., general avia-tion) when it comes to runway access. In a dynamicenvironment, this may even result in a compromiseof safety, if some aircraft are indenitely relegated tothe end of the queue as new aircraft show up to land.These observations have led many investigators to

study the runway-sequencing problem with the objec-tive of increasing operating efciency while ensur-ing that all airport users are treated equitably. Dear(1976) and Dear and Sherif (1991) developed the con-cept of constrained position shifting (CPS), i.e., of alimit in the number of positions by which an aircraftcan deviate from its FCFS position in a queue. Forinstance, an aircraft in the 16th position in a FCFSqueue would have to land in one of the positions1418 if the specied maximum position shift (MPS) is 2.Through many numerical examples and for severalreasonable objective functions, Dear (1976) showedthat by setting MPS to a small number, such as twoor three, one can obtain most of the benets of anunconstrained optimized system (e.g., 60%80% ofthe potential improvements). This nding motivatedseveral researchers (e.g., Psaraftis 1980, Venkatakr-ishnan et al. 1992, Bianco et al. 2001) to investigatea number of increasingly complex and realistic ver-sions of the sequencing problem. Two advanced ter-minal airspace automation systems, CTAS and COM-PAS, that have been implemented in the United Statesand in Germany, respectively, incorporate sequenc-ing algorithms based on CPS (Erzberger 1995). How-ever, this feature of CTAS and of COMPAS has notbeen activated, primarily because of concerns about apotential increase in controller workload.Gilbo (1993) and Hall (1999) have gone beyond the

sequencing of arrivals only by considering how avail-able capacity can best be allocated in a dynamic waybetween landings and takeoffs to account for the dis-tinct peaking patterns in the arrival and departurestreams at airports over the course of a day. Pujetet al. (1999) have further examined the issue of opti-mizing the number of aircraft taxiing out during peri-ods of congestion, based on the empirical observationthat departure rates at major airports seem to decreasewhen the number of active aircraft on the taxiwaysystem exceeds a certain airport-specic threshold.

Although still at the theoretical stage, some of thesepromising ideas will eventually nd their way intopractice.

4.2. Air Trafc Flow ManagementThe most advanced OR work on aviation infrastruc-ture to date is undoubtedly associated with air traf-c ow management (ATFM). ATFM took on majorimportance in the United States and Europe duringthe 1980s, when rapid trafc growth made it neces-sary to adopt a more strategic perspective on ATM.Rather than addressing congestion through local mea-sures (e.g., by holding arriving aircraft in the airspacenear delay-prone airports) the goal of ATFM is to pre-vent local system overloading by dynamically adjust-ing the ows of aircraft on a national or regionalbasis. It develops ow plans that attempt to dynam-ically match trafc demand with available capacityover longer time horizons, typically extending from312 hours in the future. The prototypical applica-tion of ATFM is in ground holding, i.e., in intentionallydelaying an aircrafts takeoff for a specied amountof time to avoid airborne delays and excessive con-troller workload later on. Other ATFM tactics includererouting of aircraft and metering (controlling the rate)of trafc ows through specied spatial boundariesin airspace.An important difference in the nature of the ATFM

problem in the United States and in Europe shouldalso be noted. In the United States, ATFM is primar-ily driven by airport capacity constraints, whereas inEurope en route airspace acts as the principal bot-tleneck. Europes Central Flow Management Unit,located in Brussels, currently determines (heuristi-cally) ground delays to ensure that no en route sec-tor capacity constraints are violated. This differencemay, however, become moot in the near future dueto continuing progress in increasing en route airspacecapacity in Europe.OR model development related to ATFM can be

viewed as going through two distinct stages. Therst stage involved problem denition and develop-ment of large-scale mathematical optimization mod-els of an aggregate scope. Attwool (1977) was therst to cast ATFM issues in mathematical terms, while



Odonis (1987) detailed description of the single-airport ground holding problem (GHP) as a dynamicand stochastic optimization problem stimulated muchof the subsequent work. Important advances in mod-eling and solving the GHP are marked by the stochas-tic programming models of Richetta and Odoni(1993), the extension to a multiairport setting byVranas et al. (1994), and the inclusion of en routeconstraints and rerouting options by Bertsimas andStock (1998). Many other interesting papers on vari-ous aspects of optimizing ATFM and GHP appearedin the 1990s. Good reviews of the literature and ofcomputational results can be found in Andreatta et al.(1993) and Hoffman and Ball (2000).The one common characteristic of the models devel-

oped in this rst stage is the implicit assumption of asingle decision-making authority attempting to opti-mize a global objective function: The providers ofATFM services (e.g., the FAA in the United States,Eurocontrol in Europe) are responsible for the allo-cation of ground holding delays among individualights and/or for the rerouting, if necessary, of ights.The objective is to optimize in the aggregate, e.g., byminimizing the overall direct operating costs associ-ated with ground holding and rerouting decisions,summed over all airlines and aircraft. This, however,is an operating philosophy that airlines strongly dis-agree with. They correctly argue that only individ-ual airlines have the information necessary to makedecisions on what is best for their own ights. Asan obvious example, airline A, faced with a periodof delays at a given congested airport, may assignvery high priority to the timely arrival of one of itsights, X, because that ight may be carrying manybusiness-class passengers who will be connecting toother ights or because it carries crews for subsequentights departing from that airport. The assignment ofpriority to ight X has, in fact, little to do with directoperating costs of aircraft and is based on the busi-ness model of airline A and on information that onlyA possesses.In response to such airline concerns, as well as to

various complaints about the limitations of the ATFMsystem during the 1980s and 1990s, the FAA has beenengaged for the past 10 years in developing the Col-laborative Decision-Making (CDM) Program. After an

initial planning period of about ve years, CDM waselded for the rst time in 1998 in connection with theFAAs Ground Delay Programs (GDPs), which go intoeffect whenever long air trafc delays are anticipatedat an airport due to poor weather or other reasons,thus often necessitating ground holding. CDM marksa truly fundamental innovation in the ATM system,possibly the most important one in at least 30 years.The three main elements on which it is based are: (a)a dedicated data communications network (CDM-net), which facilitates the continuous exchange ofinformation between the FAA and the airlines (plusany other CDM participants) about the current andnear-future states of the ATM system; (b) the use of acommon database and a common set of software toolsby all CDM participants; and (c) the partial decentral-ization of decision making. With respect to (c), it isthe FAAs responsibility to forecast the capacity thatwill be available at each part of the ATM system dur-ing the relevant time horizon, as well as to allocatethis capacity among the individual airlines and theother ATM system users. And it is the responsibilityof each individual airline to decide how it will use itsallocated share of capacity at each part of the system.This is a somewhat simplistic description of what, inpractice, is a complicated process that employs sev-eral types of distributed decision-making techniques,such as rationing by schedule (RBS) and schedulecompressionsee Wambsganss (1996) and Vossen et al.(2003) for details.The driver for the adoption and implementation of

the CDM concept was a small, OR-minded team inthe FAA, the U.S. Department of Transportation, andespecially the Metron Corporation (Chang et al. 2001).The CDM Program has already led to major reduc-tions in delays and missed connections for air travel-ers and to documented savings of hundreds of mil-lions of dollars in airline operating costs. ATFM-related OR research has concurrently shifted awayfrom large-scale, aggregate optimization models andtoward real-time decision support tools that assistair trafc managers in the FAA and Airline Opera-tions Centers in taking maximum advantage of themassive, up-to-date information base that CDM hasmade available. It is important to note, however, thatmany of the ideas and formulations developed in the



pre-CDM models can still be adapted to the CDMenvironment, often with little modication. For exam-ple, one of the most critical problems in the plan-ning of GDPs continues to be the determination ofairport acceptance rates (AAR) for several hours intothe future and in the presence of uncertainty aboutairport capacity and air trafc demand. The efcientstochastic integer program developed for this purposeby Ball et al. (2003) can be viewed as a direct descen-dant of the pre-CDM model proposed by Richetta andOdoni (1993).ATFM in the CDM era also provides fertile ground

for much future research because the scope of poten-tial OR analysis and modeling has expanded greatly.Examples of some topics, along with occasional recentreferences, include: identifying (as an airline) ightsthat should be cancelled or delayed (and by howmuch) in connection with GDPs, recovering (as an air-line) from irregular operations (cf. 2) resulting fromGDPs or other ATFM interventions, ensuring equityof access to airports and ATM resources (Vossen et al.2003), collaborative routing of aircraft through con-gested airspace (Ball et al. 2002), introducing barteringand possibly market-based mechanisms in the alloca-tion of airport slots (Vossen and Ball 2001, Hall 1999),and developing efcient simulation environments forthe testing of alternative ATFM strategies.

5. ConclusionOperations research has been one of the principal con-tributors to the enormous growth that the air trans-port sector has experienced during the past 50 years.In the best tradition of OR, the development of mod-els and of solutions has been motivated by issues andproblems encountered in practice and has led, in sev-eral instances, to insights of a general nature and toimportant methodological advances in the OR eld atlarge. At this point, OR models and algorithms arediffused throughout the sector and constitute an inte-gral part of the standard practices of airlines, airports,and ATM service providers.In view of the numerous challenges that it cur-

rently faces, it is safe to expect a continuing cen-tral role for OR in the air transport sectors future.As indicated in this paper, there are many promis-ing topics for future research in each of the areas

examined. At the most fundamental level, and ingeneral terms, the frontiers can be summarized asfollows: Relaxing the boundaries between the successive

stages of aircraft and crew schedule planning, sothat schedule design, eet assignment, aircraft main-tenance routing, and crew scheduling might eventu-ally be performed in an integrated way, rather thansolved sequentially as interrelated, but distinct sub-problems. Including pricing decisions in revenue manage-

ment, instead of treating fares and fare classes asxed, externally specied inputs. Developing fast decision support tools that

increase the safety and efciency of air transport oper-ations by taking advantage of the massive, real-timedata ows in an increasingly info-centric aviationinfrastructure.

ReferencesAbara, J. 1989. Applying integer linear programming to the eet

assignment problem. Interfaces 19 2028.Ageeva, Y. 2000. Approaches to Incorporating Robustness into Airline

Scheduling. Thesis, MIT, Cambridge, MA.Alstrup, J., S. Boas, O. B. G. Madsen, R. Vidal, V. Victor. 1986. Book-

ing policy for ights with two types of passengers. Eur. J. Oper.Res. 27 274288.

Anbil, R., E. Gelman, B. Patty, R. Tanga. 1991. Recent advances increw-pairing optimization at American Airlines. Interfaces 216274.

Andreatta, G., A. R. Odoni, O. Richetta. 1993. Models for theground-holding problem. L. Bianco, A. R. Odoni, eds. Large-Scale Computation and Information Processing in Air Trafc Con-trol. Springer-Verlag, Berlin, Germany.

Arabeyre, J. P., J. Fearnley, F. C. Steiger, W. Teather. 1969. The air-line crew scheduling problem: A survey. Transportation Sci. 3140163.

Armacost, A., C. Barnhart, K. Ware. 2002. Composite variable for-mulations for express shipment service network design. Trans-portation Sci. 36 120.

Attwool, V. W. 1977. Some mathematical aspects of air trafc sys-tems. J. Instit. Navigation 30 394411.

Ball, M. O., A. Futer, R. Hoffman, J. Sherry. 2002. Rationingschemes for en route air trafc management. CDM p

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