Coldstart: Fleet Assignment at Delta Air Lines

R A D H I K A S U B R A M A N I A N Delta Air Lines. Inc.

1030 Delta BoulevardAtlanta. Georgia 30320

RICHARD P. SCHEFF, JR. Delta Air Lines, Inc.

JOHN D . QUILLINAN Delta Airlines, inc.

D. STEVE WIPER Delta Air Lines, Inc.

ROY E. MARSTEN Georgia Institute of TechnologyAtlanta, Georgia 30332

Delta Air Lines flies over 2,500 domestic flight legs every day,using about 450 aircraft from 10 different fleets. The fleet as-signment problem is to match aircraft to flight legs so that seatsare fllled with paying passengers. Recent advances in mathe-matical programming algorithms and computer hardv^are makeit possible to solve optimization problems of this scope for thefirst time. Delta is the first airline to solve to completion one ofthe largest and most difficult problems in this industry. Use ofthe Coldstart model is expected to save Delta Air Lines $300million over the next three years.

D elta Air Lines has over 2,500 domes- It has been said that an airline seat is the

tic flight departures every day. This most perishable commodity in the world,

includes flights to Canada and Mexico but Each time an airliner takes off with an

excludes the other international routes. empty seat, a revenue opportunity is lost

Delta has about 450 aircraft available to fly forever. So the schedule must be designed

these flights. The aircraft are divided by to capture as much business as possible,

aircraft type into fleets or groups, of which maximizing revenues with as little direct

Delta has 10. The intricate pattern that the operating cost as possible.

Delta aircraft fly along the route system is An airline combines the worst of both

called the schedule. The schedule is the worlds. The capital intensive quality of a

very heartbeat of an airline. manufacturing environment is combined

Copyright © 1994. The Institute of Management Sciences TRANSPORTATION—AIR0091-2102/94/2401/0104(01.25 PROGRAMMING, LINEAR—APPLICATIONS

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with the low-profit environment of retailsales. Airlines are capital, fuel, and laborintensive. Survival and success depend onthe ability to operate flights along theschedule as efficiently as possible. Theprofit-and-loss curve in the airline businesshas historically followed the peaks andvalleys of the general economy. In thisdynamic environment, where profits havehistorically been low and costs historicallyhigh, how well an airline plans and imple-ments its schedule can very well determineits future. A small adjustment in theschedule may result in millions of dollarsof additional revenue or in millions of dol-lars of losses. In planning schedules, con-tinuous refinement is not a luxury; it is arequirement.

Both the size of the fleet and the numberof different types of aircraft have an expo-nential impact on schedule planning. Fleetsof different types of aircraft make up thetotal Delta fleet. We use the phrase fleetingthe schedule to express assigning a particu-lar set of aircraft to a particular set of mar-kets.

The Coldstart project addressed theproblem of fleeting the schedule. The basictrade-off is that if the airline uses too smalla plane, it will leave potential passengersbehind, while if it uses too large a plane, itwill suffer the greater expense of the largerplane to transport empty seats. The goal isto have the right plane in the right place atthe right time, but the many constraints onthe way that planes can actually be oper-ated make this difficult to accomplish.

The Coldstart model is a large-scalemixed-integer linear program that assignsfleet types to flight legs so as to minimize acombination of operating and passenger

"spill" costs, subject to a variety of opera-tional constraints, the most important ofwhich is the number of aircraft available ineach fleet. The model deals with a singleday, which is assumed to be part of a re-peating cyclic schedule. In practice, thereare exceptions to the daily schedule, partic-ularly on the weekends. At present, sched-ulers handle these exceptions manually.We are currently building an extension ofthe Coldstart model to handle the excep-tions.

Our model assigns fleet types, not indi-vidual aircraft tail numbers, to the flightlegs. Actual aircraft are routed after themodel is solved to ensure that the solutionis operational. Because of the hub-and-spoke nature of operations and large fleetsizes, it is always possible to obtain a feasi-ble tail routing from the assignment rec-ommended by the model.

The name Coldstart was inspired by anearlier schedule-planning tool, known in-formally as Warmstart, that was used byDelta Air Lines. Warmstart took a fleetedschedule that had been produced by theplanners and tried to improve it using localswap heuristics. It could not, however,move very far away from its starting point,and this left the schedulers with most ofthe burden of coming up with a good as-signment manually. The most seriousdrawback of the Warmstart system wasthat its improvements were so local that apoor input schedule would result in a pooroutput. Coldstart, being an optimizationmodel, does not require an initial fleeting.

Until very recently, optimizing the fleetassignment for an airline as large as Deltawould not have been possible. Today,however, improvements in mathematical

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programming algorithms and computerhardware make it possible to solve optimi-zation problems of this scope for the firsttime. Delta is the first airline to take ad-vantage of this new opportunity.Modeling the Fleet Assignment

The fundamental mathematical structureupon which the fleet assignment model isbuilt is a time-space network. This networkhas a time line for each aircraft fleet ateach city. This time line is circular and rep-resents a repeating 24-hour period. Alonga given time line, for example, the one for757 in Boston, a node represents eachpoint in time at which at least one eventoccurs. An event is the arrival or departureof a flight. Segments of the time line thatconnect these nodes are referred to asground arcs. A flight leg is a single hop, that

is, one takeoff and one landing. A flightleg is represented in the network by a setof sky arcs, one for each fleet that could beassigned to fly that leg. For example, as-signing a 757 to the flight leg that leavesAtlanta at 6:21 AM and arrives in Boston at8:45 AM is represented by a sky arc begin-ning at the 6:21 AM node on the (757,

(767. ATL)

(767 SJU)

Figure 1: Time line network for a very simpleairline with only one fleet (767) that servesonly two cities, Atlanta (ATL) and San Juan(SJU). Each arrival and departure is labeledwith a time (24-hour clock).

ATL) time line and ending at the 8:45 AMnode on the (757, BOS) time line. Hencethe time lines for a given fleet at differentcities are connected by sky arcs; however,the time lines for different fleets are notconnected. Figure 1 provides a simple illus-tration of how the time lines for the 767-200 fleet at Atlanta (767, ATL) and SanJuan (767, SJU) are connected.

These (fleet, city) time lines allow us toexpress conservation of flow equations forthe aircraft. For each sky arc, we define abinary assignment variable that takes thevalue one if that fleet is assigned to thatleg or the value zero otherwise. For eachground arc, we define an integer variablethat counts the number of planes of thattype on the ground at that city during thattime interval. At each node, there is a con-servation equation. For example, for the6:21 AM node on the (757, ATL) time line,the equation says

(number of 757s on the ground in ATLjust before 6:21 AM) - (number of 757sthat depart from ATL at 6:21 AM)+ (number of 757s that arrive at ATLat 6:21 AM) = (number of 757s on theground in ATL just after 6:21 AM).

The equations are presented in detail inthe appendix. For a development of a simi-lar model, see Berge and Hopperstad[1993] and Hane et al. [1993],

The conservation equations make up thebulk of the model constraints. There is oneconstraint for each fleet at each node,where each flight results in two nodes.Hence, if there are 10 fleets and 2,500flights per day, and if every fleet could flyevery leg (which is not true due to opera-tional limitations), and if all departure andarrival times for a given fleet at a given

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city were unique (also not true), then wewould have 2,500 X 2 X 10 - 50,000 con-servation equations. The actual number forDelta's daily model is closer to 30,000.

In the Coldstart model, note that a planearriving at a city can be paired with anylater departure. This formulation is farmore powerful and realistic than the fleetassignment model of Abara [1989], In themodel discussed by Abara, each feasibleturn and aircraft combination represents adecision variable. To limit tbe size of hismodel, each arrival could be paired with atmost the next five departures. Abara's for-mulation is further limited in that it doesnot allow the use of the model to be ex-tended to other schedule-related applica-tions like fleet planning and route develop-ment. This ability to pair an arrival withany later departure is very important at thehubs, where there can be as many as 65arrivals and 65 departures at a complex. (Acomplex is a set of arrivals and departuresthat connect to each other, a typical fea-ture of the hub-and-spoke operation.)

For a given flight leg, as many as 10 dif-ferent binary variables represent assigningthat leg to the 10 different fleets. These as-signments are mutually exclusive, so thereis a multiple-choice constraint that saysthat the sum of these zero/one variablesmust be exactly one. There is one suchcover constraint for each of the 2,500 flightlegs. These cover constraints are expressedas "special ordered sets, type 3"[Druckerman, Silverman, and Viaropulos1991], which considerably reduces the timefor solution of the mixed-integer program.

Certain pairs of flight legs must be as-signed to the same fleet to provide one-stop, through service. For example, the

plane tbat flies the 1:41 PM flight from At-lanta to Boston, arriving at 4:10 PM, mustalso fly the 5:25 PM flight from Boston toMontreal. Thus these two legs must be as-signed to the same fleet. This is easilymodeled by an equation of the form "X,- XT = 0" for each fleet. The typical dailymodel involves about 250 such requiredhookups, giving rise to about 1,500 con-straints.

The number of planes in each fleet,which is tbe scarce resource in the assign-

An airline seat is the mostperishable commodity in theworld.

ment problem, is obviously limited. Tocapture this in the model, we build a sizeconstraint for each fleet. The size con-straint for 757s includes every 757 that isin the air at midnight Atlanta time andalso counts the 757s on the ground at mid-nigbt Atlanta time. To prevent getting aninfeasible solution, we allow tbe model touse extra planes but at a very high cost.This is preferable to getting an infeasibilitythat may be hard to diagnose.

Aircraft need to be maintained at peri-odic intervals. Short maintenance can bedone while a plane is sitting on the groundduring the day. Longer maintenance mustbe performed at night. Part of the input tothe Coldstart model is a list of overnightmaintenance requirements. For example,one 757 must have a 12-hour maintenancecheck every night. For each sucb require-ment, there is a list of maintenance baseswhere the check can be done. The modelmust then insure tbat a 757 is available in

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one of these maintenance bases for 12hours. To model this, we introduced a newset of maintenance arcs. A maintenance arcrepresents a maintenance opportunity. Itbegins at an evening arrival node at amaintenance city and ends at a morningdeparture node that is at least 12 hourslater {for a 12-hour check). There are sev-eral such opportunities for each require-ment, perhaps at several alternative cities.There is then a multiple choice constraintover the opportunities that correspond toeach requirement. A typical model hasabout 30 maintenance requirements. Animportant modeling extension to the majormaintenance requirement is that the modelcan select the best city at which to performa certain maintenance. This feature is veryuseful in analysis of the schedule and inthe planning mode.

The individual fleets are grouped intopilot aggregates that can be flown by thesame pilots. For example, the Boeing 757,767-200, and 767-300 fleets can all beflown by the same pilots. For each pilotaggregate, there is a limit on the number offlying hours per day that can be assignedto that set of pilots. Each assignment vari-

Airlines are capital, fuel, andlabor intensive.

able must appear in the flying hours con-straint for the pilot aggregate that it be-longs to, with a coefficient equal to thenumber of flying hours for that leg. Theseconstraints, like the cover constraints, havethe effect of coupling the fleets together.

There are other crew-related considera-tions that must be built into the model.

Just as planes need maintenance, pilotsneed rest. A fleet assignment that is goodfrom the point of view of aircraft schedul-ing may be very bad from the point ofview of crew scheduling. For example,suppose there is only one 757 flight in andout of Boise each day and no flights by anyother fleet in the same pilot aggregate. Ifthis flight arrives at 11:00 PM and leaves at7:00 AM, and if the arriving crew has tohave an overnight rest break of at least10V2 hours, then this crew will have to re-main in Boise until 7:00 AM the day after—32 hours after its arrival. This is very ex-pensive in terms of crew costs. What thismeans for fleet assignment is that it is notgood to have too few flights by a pilot ag-gregate into a city, or, equivalently, to havetoo many different pilot aggregates servinga city. In particular, late arrivals should bepaired with midday departures so that acrew can fly out of the city soon after theirrest break. To model this, we add con-straints that result in an objective functionpenalty if the number of midday depar-tures by a pilot aggregate is less than thenumber of late arrivals by the same pilotaggregate. We also try to find "10:30 op-portunities" (IOV2 hours are the requiredtime for a legal crew break). A 10:30 op-portunity is a pairing of an evening or latenight arrival by a certain pilot aggregateand a departure by the same aggregate atleast IOV2 hours later, such that there is atleast one plane of that aggregate on theground at that city for the whole 10'/? +hour period. We find these legal crewbreak opportunities by building crew timelines [Johnson 1992] for fleet aggregates atcertain busy cities. A crew arriving at anairport can leave from a coterminal after a

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legal break. For example, LaGuardia, JohnF. Kennedy, and Newark airports form theNew York area coterminal. The crew con-straints deal with coterminal sets of cities.

The operational restrictions are codedinto the input data for Coldstart. Certainaircraft types are operationally constrainedfrom serving specific flight segments orlegs in the Delta schedule. These restric-tions are due to aircraft performance limits,such as takeoff or landing weight limits atspecific airports, or the range capability ofthe aircraft. The range capability of the air-craft is determined by its fuel capacity, thealtitudes of the airports being served, andthe distance between two airports. Not allaircraft are equipped with overwater navi-gational equipment, life rafts, and so forth;segments requiring such equipment mustbe covered by one of the specific fleettypes with this equipment.

Other restrictions coded into the inputdata include airport restrictions, such asstage 3 noise restrictions, and general arri-val and departure curfews. Aircraft are cer-tified to meet various noise standards, suchas stage 2 or 3, in accordance with FederalAviation Rules and Internationa! CivilAviation Organization standards. Noisecertification standards are complex; theyinvolve a three-point evaluation scheme(that includes noise levels measured attakeoff, sideline, and approach), computa-tion of an effective perceived noise level,and noise limits based on maximum take-off weight. A stage 3 certified aircraft isgenerally quieter than a stage 2 certifiedaircraft of the same weight. Airports, suchas Seattle, Washington, and San Francisco,restrict departures and arrivals between 11o'clock in the evening and 6 o'clock in the

morning to aircraft meeting stage 3 noiserequirements. Orange County, Californiaand Washington, DCs National airportsare airports that allow only certain aircrafttypes at all. Orange County, Cahfornia al-lows only certain aircraft types, which in-clude 737-300, 737-400, and 757, to arriveor depart during the airport's operations;Washington, DCs National airport allowsonly 757 departures and MD88 arrivals be-tween 10 o'clock in the evening and 6:49in the morning.

While some noise restrictions are ac-counted for in the input data, noise restric-tions that are enforced as a percentage oflandings and takeoffs that can be stage 2aircraft are modeled by a blending con-straint. In San Diego, California, for exam-ple, at most 25 percent of the departuresmay be stage 2 aircraft.

A very serious modeling difficulty thatwe overcame has to do with the arrivaltime of each flight leg. Using the time-space network to balance the flow of air-

We spent an immense amountof effort fine tuning andcleaning our data.

craft means that a plane that arrives at10:50 AM, for instance, can leave on anylater departure. This implies that by arrivaltime we do not mean the time when theplane lands on the runway, or even the timewhen it pulls up to the gate. What wereally mean is the time when it is ready togo out again. The time-space network hasto be built with ready times rather thanwith scheduled arrival hmes. The time thatan aircraft takes to be ready for the next

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takeoff depends on whether it is continu-ing v̂ 'ith the same flight number or iscbanging flight numbers. For a continuingflight, we assume that the aircraft needsless time to get ready for the next takeoff.Tbe difference in these ready times is onlyabout five to 10 minutes, but since theschedule is so tight, this difference makes abig impact on the overall fleeting. Theplanners, at their discretion, will son^e-times allow an aircraft ready time to go be-low the minimum if this means a goodconnection somewhere further down in theschedule. This tightening of the scheduleto allow for high revenue connections is avery important modeling feature. We havemodeled this by adding fiight arcs formissed connections with a cost propor-tional to the time shaved from the usualready time. The variables corresponding tothese arcs go into the cover constraint forthat flight leg.

Another related issue that affects the ar-rival time by a few minutes is the fact thatthe different fleets fly at different speeds.We have grouped the fleets into speedclasses (different from the grouping intopilot aggregates). Flying time is more com-plex than just distance divided by speed.For example, it depends on the direction,season, and fleet type. Another factor thataffects arrival time is the taxi time, whichvaries by fleet type and airport. To modelthese effects, we build the sky arcs differ-ently depending on nominal aircraft speedand make small adjustments to the sched-uled arrival and departure times to avoidviolating the arrival and departure banksof complexes at the hub cities. Tbe modelgenerator retimes the flights to avoid infea-sible connections and to make feasible

connections based on the various aircraftspeeds. Hence the resulting assignment isfeasible from an operational standpoint.

Last, we modeled pilot training as a con-straint. Pilot training requires the availabil-ity of specific aircraft types at specific citiesfor fixed periods of time daily.The Objective Function

We have used three objective functionsin Coldstart thus far. The primary objec-tive bas been cost minimization. In the costminimization objective, the goal is to mini-mize the sum of the operating cost, spillcost, and any applicable penalties. The op-erating cost by fleet type is extracted fromthe accounting ledger for each leg. The op-erating cost consists of several componentsthat can vary by fleet type and includescrew (pilot and flight attendant) cost, fuelcost, landing fees, and a maintenance bur-den. Spill cost is estimated, and dependsupon the demand for a leg, aircraft capac-ity, recapture rate, and the revenue of thelost passenger.

Because some constraints in the modelare not hard and fast but rather soft, weincorporated a set of bonuses and penaltiesin the objective function to help make thesolutions operationally feasible. For exam-ple, a penalty exists for flying a wide-bodyfleet into a city that is not currently servedby wide-bodies, and a bonus exists forflying a fleet into a city that is a crew basefor its pilot aggregate.

Spill is the number of passengers notcarried because aircraft capacity is insuffi-cient. It is the unconstrained demand lostdue to insufficient aircraft capacity on a leg[Swan 1983, 1992a, 1992b]. Spill is causedby the truncation of the demand distribu-tion beyond the aircraft capacity (Figure 2).

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7 2 7 S P I L L

LL

Figure 2: The figure shows a normally distributed passenger demand with a mean of 125 andstandard deviation of 45. The area under the demand curve to the right of a vertical line at thecapacity of an aircraft represents full flights with unsatisfied demand. Spill for a Boeing 727with a capacity of 148 and spill for a Boeing 757 with a capacity of 182 are represented by theareas to the right of the vertical lines at 148 and 182, respectively.

The variation in daily passenger demandcan be attributed to differences in day-of-the-week demand, seasonality, cyclic ef-fects, and randon:\ variation. High variabil-ity can result in large passenger spills. Af-ter finding the unconstrained mean andstandard deviation, one can calculate theexpected spill for any size plane.

Spilled passengers are either recapturedon other Delta flights or lost to competi-tors. Spill becomes a cost to an airlinewhen spilled passengers are lost to com-petitors or other modes of transportation.

Using a n\arket-share model, we estimatethe percentage of spilled passengers thatare recaptured on other Delta flights on aleg-by-leg basis. This percentage is calledthe recapture rate. The loss rate for spill issimply one less the recapture rate.

The product of the expected spill andthe loss rate for spill gives the lost spilledpassengers. We convert this into dollars bymultiplying it by an estimate of the aver-age revenue per spilled passenger on thatleg. The resulting lost spilled revenue is thespill cost.

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The spill cost and any applicable penal-ties and bonuses (which are negative pen-alties) are added to the operating cost toestimate the total cost of operating a par-ticular fleet type on a leg. In addition tominimizing the cost, we have the flexibilityto run the model for other objective func-tions. For fleet-planning purposes, we canchange the objective function to minimizethe total number of planes used to fly theschedule. We introduced a third objectivefunction for route-development purposes.If we change the objective function fromcost minimization to profit maximization,we can modify the cover constraints to de-termine when to add new service or dropexisting flights.Solution Technique

The typical size of the daily Coldstartmodel is about 40,000 constraints and60,000 variables. There are roughly 20,000binary variables (assignment of fleets tolegs and maintenance variables) and40,000 general integer variables. Hence wehave a large mixed-integer programmingproblem to solve. Our solution strategy isto use the OBI interior point code [Lustig,Marsten, and Shanno 1991, 1992] to solvethe problem as a linear program; fix someor all of the binary variables that are at 1.0in the LP solution; use these fixed variablesto reduce the size of the problem; andsolve the resulting smaller mixed-integerproblem with the OSL mixed integer pro-gramming code [Dnickerman, Silverman,and Viaropulos 1991].

Before attempting to solve the linearprogram, OBI uses general algebraic re-duction techniques to reduce the problemsize. First it uses the "loneiy-plus/Ionely-minus" reduction [Lustig and Marsten

1993]. Lonely-plus works as follows. Sup-pose we have an equation with just onepositive coefficient, several negative coeffi-cients, and a nonnegative right-hand side.Suppose further that all of the variablesmust be nonnegative. Then the variablewith the lonely-plus can never be negative,and we can use this equation to substituteit out of the model. Thus the model is re-duced by one equation and one variable.The reductions obtained in this way in-clude some very natural node aggregationsthat could be done at the matrix-genera-tion stage. Suppose that on a (fleet, city)time line we find a sequence of nodes thatrepresent only arrivals, followed by a se-quence of nodes that represent only depar-tures. Then any of these arrivals couldconnect to any of these departures, and allof these nodes can be aggregated into asingle node. Figure 3 illustrates the conceptof node aggregation. This eliminates sev-eral nodes and hence several balance equa-tions and several integer variables for the

osoo(a)

(b)

0646

Figure 3: Node aggregation in the time linenetwork; (a) shows the actual network; (b)shows a mathematically equivalent network.A plane that arrives at 0800 or 0815 can besent out on either of the 0846 departures.

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intervening ground arcs. The lonely-plus/lonely-minus reduction also finds andeliminates the required hookup constraints,which are of the form, Xi — X2 = 0. Thishas the effect of combining two separatevariables into a single variable. Other re-ductions that are found do not have a sim-ple interpretation in terms of the network.

The second model reduction used byOBI is to eliminate dependent rows. In us-ing the interior-point method, we do notintroduce artiflcial variables for the equal-ity constraints. This means that there maybe dependent rows, and in fact there arequite a few among the conservation-of-flow constraints. We detect these by per-forming a Gaussian reduction on the coef-ficient matrix. Finally, OBI uses the stan-dard sort of model reductions described byBrearley, Mitra, and Williams [1975], Themodel to be solved ends up being about12,000 rows and 30,000 variables. (For ex-ample, the December 1992 schedule re-sulted in a model with 41,325 rows and62,088 variables and was reduced to12,811 rows and 33,475 variables.)

Computational experiments described inHane et al. [1993] show convincingly thatthe interior-point method dominates thesimplex method for this class of problems.For example OBI uses the interior-pointmethod to solve the December 1992 sched-ule in 45 iterations, taking 43 minutes onan IBM RS/6000 (Model 530) work sta-tion. The OSL primal-simplex code solvesthe same model in 356,854 iterations, tak-ing 19 hours on the same work station.OBI uses the predictor-corrector version ofthe primal-dual interior-point method[Lustig, Marsten, and Shanno 1992]. (TheOSL predictor-corrector, primal-dual bar-

rier code [Druckerman, Silverman, andViaropulos 1991] is very similar to OBI butdoes not perform as well on the fleet as-signment models because the OSL prepro-cessor does not remove dependent rowsand does not do the lonely-plus/lonely-minus reduction if the variable heing sub-stituted out has more than one other non-zero coefficient.)

The LP solution of the daily model typi-cally assigns a unique fleet to about 80percent of the flight legs, or about 2,000 ofthe 2,500 legs. This leaves about 500 legswith two or more fleets assigned at a frac-tional level. If all of the binary variablesthat are at one in the LP solution are fixed,then the same kind of algebraic model re-ductions discussed above will reduce theproblem down to about 4,000 rows and6,000 variables. Near optimal solutions tomodels of this size can be found within anhour or two of branch-and-bound searchby the OSL MIP code [Druckerman,Silverman, and Viaropulos 1991]. By near-optimal, we mean within 0.1 percent ofthe LP objective value of the full model.Of course, the reduced problem may be in-feasible because of the fixed variables. Wehave developed heuristics to fix some butnot all of the X-variables that are at one inthe LP solution so as to prevent infeasibil-ity. Fortunately, infeasibility is rarely aproblem when we are solving the large all-fleet models—so rare that we have not en-countered it. Infeasibility occurs more of-ten when variables are fixed for two orthree fleet models, and in this case, OSLcan solve the whole problem from scratchwithout help from OBI.Implementation and Operational Impact

We developed Coldstart to take advan-

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tage of the economic analysis that hadbeen done for an earlier system that madesmall fleeting changes to improve profit.The earlier system, Warmstart, was essen-tially a local swapper and would look forinterchangeable paths of two to four flightsthat could be switched to improve profit.The following example illustrates the typeof swap that the system produced (alltimes are in Eastern Standard Time):Atlanta to Birmingham 1100 1140,Birminghan:\-Dallas/Ft. Worth

1230 1330, fleet: 72S;Atlanta to Jackson 1100 1210,Jackson-Dallas/Ft. Worth

1245 1330, fleet: 757.Clearly these two paths could be inter-

changed if the end result was desirable,because the situation before and after theswap is identical. The rest of the schedulecan be ignored as far as this swap is con-cerned.

While the scope of the changes that canbe made with this procedure is muchsmaller and the applicability is limited, theeconomic analysis is essentially the same.The question of the cost of a particulartype of aircraft on a particular segmentmust be identified and then used to im-prove the schedule.

Delta used the Warmstart system forover a year for fleeting decisions—andthus the economic assumptions and fore-casts. Once it had decided to use this anal-ysis for fleeting decisions, the leap to ac-cepting Coldstart recommendations was asmaller one.

Once Delta started to use the localswapper, its shortcomings became appar-ent. These weaknesses spawned the effortto generate a more global and far-reaching

fleeting system. Delta formed the opera-tions research team initially to work on theColdstart model.

Once Coldstart was operational, the de-cision to use it was very easy. The sched-ule planners analyzed the difference be-tween their initial schedules and the rec-ommended Coldstart fleetings. They feltthat the solutions Coldstart produced wereclearly superior. The planners can now de-termine analytically which flights need tobe upgraded and which downgrades willbe costly, and they can assess a scheduleonce the fleeting is done. The problemprior to Coldstart was making desirablefleetings operationally feasible.

The economic analysis necessary to runthe fleet optimization makes estimating thebenefits fairly simple. Tbe cost of a fleetingis determined by summing the segmentcosts for the assigned fleets. The costs be-tween fleetings can then be compared. Inthe meantime, the operating departmentsprovide feedback on other unmodeledcosts that may be affected by the changes,including crew, maintenance, and stationcosts.

Delta has two scheduling groups whichwork on each of the six to eight scheduleperiods in a year. One of the groups, theplanning group, works on a schedule be-ginning eight to nine months before itstarts and provides a tentative schedule tothe operating departments seven monthsahead of time. This schedule is used to setstaffing levels at the pilot bases and to pro-vide a good starting solution for the cur-rent schedules group. The other schedulinggroup, the current group, starts work onthe final version of the schedule about fourmonths early using the planning version as

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a starting point. While the group maymake changes, they may not exceed the pi-lot staffing levels at any of the crew basesthat were set as a result of the planningschedule. Since they work several monthscloser to the actual schedule date, the cur-rent group can take advantage of competi-tive changes, growing or declining mar-kets, or other changes in strategy to refleetto better advantage. In addition, there aregenerally some changes to the actual legsin the schedule between the time the plan-

Coldstart is the first ORapplication of this magnitudedeveloped and implementedinside Delta.

ning group issues a schedule and the cur-rent group does. The available fleet countmay change also.

Both the planning and current schedul-ing groups use the Coldstart model exten-sively, although the planning group gener-ally has more opportunity to make sweep-ing changes. The current group, in additionto having to keep within the crew baseconstraints, must schedule all tbe weekendcancellations and other exceptions to theassumption of the repeating 24-hour cycle.Because of the reduced lead time, the cur-rent group generally will have a more ac-curate demand forecast; it will call forsome refleetings from the initial schedulefrom planning.

Both groups tend to use Coldstart in asimilar fashion. Initially, they include allthe fleets in a run to get a good starting so-lution. Invariably, there are assignmentsthat are undesirable for one of a number

of reasons that cannot easily be capturedin the model. This first run takes from oneto three hours since it includes all 10 fleetsand 2,500 flight segments. From this solu-tion, the schedule planners can identify theproblems and work them out individuallyor in groups. They generally solve theseproblems quickly as subproblems involvingfrom two to five fleets, often in five min-utes or less. This tuning of the initial solu-tion will generally continue until the dead-line for issuing the schedule.

Coldstart has changed the basic task of aschedule planner at Delta. While themodel performs all the schedule changes,and very fast, the task of the planner in-volves more analysis of these changes.Planners spend a lot more effort in analyz-ing the various cost numbers that are usedto drive the model. With this tool, they cantest various scenarios in a short amount oftime and choose the best one. As a result,the flight schedule at Delta has changedconsiderably since the model went into ef-fect.

Because the old method of fleeting is nolonger used, benchmarks are not available.While the quesfion, "how much better isthis Coldstart fleeting than what we wouldhave achieved manually," was importantinitially; a better quesfion now is, "howmuch better can a Coldstart solution be-come?" We will always be changing themodel, and we can use an infinite numberof parameter setfings; this is the key to fu-ture enhancement. Coldstart allows theuser to produce two different schedules us-ing different constraints or parameters andallows him or her to answer these types ofquestions with the same comparative pro-cedure.

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Planners can use Coldstart in a numberof vk̂ ays. Initially, they can use it to get afleeting that is close to the desired one.They accomplish this by doing one ormore runs using all the fleets. Once that isaccomplished, they work on segments thatneed to be refleeted because of factors thatcannot be modeled. Coldstart can be usedto solve smaller subproblems—usually thebest and fastest way to address individualproblems. For example, a planner maywish to upgrade a flight from Atlanta toDallas/Fort Worth from a 76S with 254seats to an LIO with 302 seats. The plan-ner can pull the 76S and LIO fleets out asif they were a separate airline, target thedesired upgrade and see what refleetingsare necessary to accomplish the change.This can often be done in just a minute orso, depending on the size of the subprob-lem. A desirable solution to a targetedflight can be accepted and put into theschedule base if desired. Because the sys-tem is so flexible, it can be used to solvesmall fast subproblems and thus accom-plish any number of desired goals.

The schedules department sends outschedule proposals to the operating de-partments and then gets responses specify-ing problems and requesting changes. Ittakes these responses and revises theschedule to best address the problems andthe needs of the operating departments.The targeted subproblem approach withminimal changes works well in the latestages of the schedule development pro-cess. Once schedules has issued scheduleproposals, it must keep the number ofchanges before the schedule is finalized toa minimum. This keeps the number of re-fleetings small while accomplishing the

necessary changes.Extensions

The Coldstart model has grown in threedifferent directions beyond its original pur-pose. These are fleet planning, route devel-opment, and transition planning. Imple-menting these extensions was easy becauseof the way the initial model was built.Fleet planning is a natural extension offleet assignment. Instead of treating thefleet sizes as fixed, it is easy to model thepossibilities of acquiring additional aircraftor retiring existing aircraft. In addition tooperating cost, some measure of the own-ership cost must be included in the model.The success of the model as a fleet plan-ning tool depends mainly on the objectivecost figures. At Delta we have spent an im-mense amount of effort fine tuning andcleaning our data for these runs.

If the objective of the model is changedfrom cost minimization to profit maximiza-tion, then the optimizer can be allowed tochoose which legs to fly, rather than beingrequired to fly all legs. This means that themodel can be used for developing routesby considering the addition of new legs orthe deletion of existing legs.

A transition problem arises whenever aschedule change occurs, major or minor.For example, the schedules changed onSunday, April 4, 1993 when daylight sav-ing time began. On the night of April 3,the planes were al! in their correct posi-tions for continuing the winter schedule.But these were not the correct positions forbeginning the spring schedule! One solu-tion would be to ferry planes, empty, inthe dead of night, to their correct positions.Instead, Delta alters the assignment offleets to legs during the last day of the old

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schedule and the first day of the newschedule, sometimes having to cancel afew flights. This is a very time-consuming,manual planning job. We have developedand implemented a separate version of theColdstart model to solve this problem. Theobjective function tries to minimize thenumber of fleeting changes required, withpreference given to changes involving thesame pilot aggregate or seating capacity.

The model has been built so that theuser can select any subset of constraintsthat are applicable for his or her run. Webave taken extensive effort to ensure thatfor a given run the model is never infeasi-ble. This includes infeasibility due to fleetsize, turn times, maintenance require-ments, noise constraints, and pilot blockhours.Financial Impact of the Coldstart Project

When we began the Coldstart project inSeptember 1991, we were not sure that wewould be able to model and solve such avery large and complex system. The fol-lowing factors were essential to the successof this project. First, we had someone withexperience in the Delta fleet scheduling de-partment working full time as part of thedevelopment team. He made sure that themodel produced operationally feasible so-lutions. Second was the effort devoted tomodeling the details of the system, particu-larly the turn time versus through time is-sue, including all of the many complica-tions surrounding almost missed connec-tions and aircraft speed. Third was state-of-the-art algorithms and software. OBIand OSL are both cutting-edge codes, andOBI was improved to meet the computa-tional challenge posed by these models. Fi-nally, the availability of powerful and in-

expensive work stations put the equivalentof three completely dedicated mainframesat our disposal for development andtesting.

Since the September 11, 1992 schedule,which was the first schedule that usedColdstart extensively, the model has beenused for all of Delta's schedules. The re-fleetings from the September 11, 1992schedule saved an estimated $55,000 perday over the schedule that would havebeen used. The planning group first usedColdstart for the December 15, 1992schedule and obtained an estimated sav-ings of more than $100,000 per day. Allother schedules have been worked exclu-sively with Coldstart, which eliminates thebenchmark comparison to the schedule asit would have been developed under theprevious methodology. However, with ad-ditional experience and confidence. Deltahas used the model much more heavily indeveloping subsequent schedules. The sav-ings in the June 1, 1993 to August 31,1993 schedule have been estimated at$220,000 per day. The savings from themodel have increased from schedule toschedule as the planners gain more confi-dence and accept more and more of themodel's recommendations. The success ofthis process has been mainly due to thewillingness at Delta to understand and im-plement the model, sometimes evenchanging the way operations have tradi-tionally been done at Delta.

A substantial percentage of the cost sav-ings results from reduced direct operatingcost. This amount is much more easilytracked than the reduced spilled revenue.It is a straightforward process to estimatethe direct operating cost of two fleetings

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SUBRAMANIAN ET AL.

over the same set of legs. Even here, there

are some pitfalls. For example, a fleeting

that allows you to save pilot costs and re-

duce the pilot head count by 100 will not

actually save that money in the short term

unless you lay off pilots, which Delta has

not done. In the long run, the savings will

be realized as attrition occurs or as Delta is

able to increase its level of service without

adding pilots.

Even when a schedule has been flown, it

is very difficult to estimate how much

more or less revenue would have been

generated with a different fleeting. There is

no record of spill or recapture even after it

has occurred. Estimating this part of the

objective function from four to eight

months out is difficult. In addition to esti-

mating the unconstrained demand, one

must estimate the unconstrained variance

for each flight. Since most flights have

truncation in the historical data, statistical

analysis must be used to estimate these pa-

rameters. We have used the techniques de-

scribed by Swan [1983, 1992a, 1992b] in

conjunction with techniques developed by

Delta to estimate these key values. To this

point. Delta has been pleased with both

the cost savings and the revenue genera-

tion from the model.

Coldstart is the first operations research

application of this magnitude that has

been developed and implemented inside

Delta Air Lines. Its success ensures that

Delta will be an eager user of operations

research techniques in the future.

APPENDIXWe present here an algebraic formula-

tion of the basic model, which does not in-clude maintenance, pilot training, pilothours, crew breakout, crew 10:30 rest, andnoise constraints.

The Basic Fleet Assignment ModelSets:CITY^—a set of cities, indexed by /,FLEET—a set of aircraft types, indexed by

k,LEG—a set of flight legs, indexed by /, andTIME—a set of times, {OOOO, 0001, . . . ,

2359}, indexed by (.Parameters:cost(k,t) = cost if an aircraft of fleet k flies

leg / ( x if fleet k cannot fly leg /),turi}{k,i) ^ turn time for fleet k in city i; the

minimum time required between the ar-rival and subsequent departure of thesame aircraft,

size{k) = number of aircraft available infleet k,

origQ) = origin city of leg /, an element ofCITY,

desl{l) = destination city of leg /, an ele-ment of CITY,

depart{l) ^ departure time of leg /, an ele-ment of TIME, and

arrive{l) ^ arrival time of leg /, an elementof TIME.

Derived Sets:NODES(iO - {f 6 TIMEU - depart{l) for

some / E LEG such that orig{/) ^ /, or /^ arrive{l)+turn{k,i) for some / G LEGsuch that desl{l) = \].

NODES(fc,0 is the set of times when an ar-rival or departure of an aircraft of type kmay happen at city (. For any ( GNODES(fc,i), we use 1+ to denote the nexttime, and t- to denote the previous time,with the circular assumption that 2359+ -OOOO and OOOO- = 2359.lNTO{k,iJ) = {/ G LUG\dest{l) - /, ar-

rive{l)+turn{k,i) - t], Vk G FLEET, i GCITY, and t G NODES{/r,/).

OUTOF(^,(,0 - {/ G LEG I ong{/) = i, de-part{l)=t}yk e FLEET, i G CITY, and t

GNODES(A,()-count—air{k) = {{kj)\cost{kj) < x̂ and ar-

rive{l)+turn(K desl{l)) < depart{l)]count-ground{k) - {(fc,/,OU e NODES(jt,/)

and t+ < t]cou?it^air{k) captures the set of legs where

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DELTA

an aircraft of fleet k may be in the air atmidnight.

count_ground{k) captures the set of citieswhere an aircraft of fleet k may be sit-ting on the ground at midnight.

HOOK c LEG X LEG(/,,/.) e HOOK means that dest{U)

^ ot'tgOi) and that the same fleet must bechosen for both /i and I2.

Variables:X;; = 1 if fleet k is assigned to leg /;

= 0 otherwise.V^ e FLEET and I G LEG, such thatcost{k,l) < oj.

Vt,, = number of aircraft of fleet k on theground at city i from time t to timef+.VA e FLEET, ( e CITY, and ie NODES(A:,/).

Zi = number of aircraft of fleet k that areused.

Constraints:BALANCE{ii:,U):

2 X,, - 2 X,,

- 0,

"ik G FLEET, i G CITY,

and i G NODES(fc,/).

COVER(/):

2 Xu - 1, V/GLEG.

SIZE(fc):

- z , - 0 ,

VJt G FLEET.

0 < Z, < s;2

Xu, - X;,, - 0, V/r G FLEET,

(/,,/2) E HOOK.

Objective:

Minimize COST - cost{Jt,/)*XikEFLEET.leLEG

Additional constraints can be added formaintenance requirements, crew consider-ations, pilot hours, pilot training, noise re-strictions, and others.

ReferencesAbara, J. 1989, 'Applying integer linear pro-

gramming to the fleet assignment problem,"Interfaces, Vol. 19, No. 4, pp. 20-28.

Berge, M. E. and Hopperstad, C. A. 1993, De-mand driven dispatch: A method for dynamicaircraft capacity assignment, models and al-gorithms," Operations Research, Vol. 41, No.Lpp. 153-168.

Brearly, A. L.; Mitra, G.; and Williams, H. P.1975, "Analysis of mathematical program-ming problems prior to applying the simplexmethod," Mathematical Programming. Vol. 8,No. 1, pp. 54-83.

Druckerman, J.; Silverman, D.; and Viaropulos,K. 1991, "IBM optimization subroutine li-brary, guide and reference, release 2," docu-ment number SC230519 02, IBM, Kingston,New York.

Hane, C. A.; Barnhart, C; Johnson, E. L.; Mar-sten, R. E.; Nemhauser, G. L.; and Sigis-mondi, G. 1993, "The fleet assignment prob-lem: Solving a large-scale integer program,"working paper. School of Industrial and Sys-tems Engineering, Georgia Institute of Tech-nology, Atlanta, Georgia.

Johnson, E. L. 1992, IBM Corporation, personalcommunication.

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Lustig, I. J.; Marsten, R. E.; and Sbanno, D. E.1991, "Computational experience with a pri-maLdual interior point method for linear pro-gramming," Linear Algebra and Its Applica-tions. Vol. 152, pp. 191-222.

Lustig, 1. J.; Marsten, R. E.; and Shanno, D. E.1992, "On implementing Mehrotra's predic-tor-corrector interior point method for linearprogramming," SIAM Journal of Optimization,Vol. 2, No. 3, pp. 435-449.

Lustig, I. J.; Marsten, R. E.; and Shanno, D. F.forthcoming, "Interior point methods for lin-ear programming: Computational state of theart," ORSA Journa! on Computing.

Mehrotra, S. 1992, "On the implementation ofa (primal-dual) interior point method," SIAM

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Journal of Opfimizatiort, Vol. 2, No. 3, pp.575-601.

Swan, W. M. 1983, "Traffic losses at high loadfactors," Proceedings of the 23rd AnnualAGIFORS Symposium, Olive Branch,Mississippi.

Swan, W. M. 1992a, "The value of yield man-agement information for routing and schedul-ing," presentation to AGIFORS Reservationsand Yield Management Study Group.

Swan, W. M. 1992b, "Spill model revisited,"working paper, Boeing Commercial AirplaneCompany, Seattle, Washington.

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