1147743296

TRANSPORTATION SCIENCEVol. 38, No. 1, February 2004, pp. 1932issn 0041-1655 eissn 1526-5447 04 3801 0019

informs doi 10.1287/trsc.1030.0026

2004 INFORMS

Airline Schedule Planning: Integrated Models andAlgorithms for Schedule Design and Fleet Assignment

Manoj LohatepanontOperations Research and Decision Support, American Airlines, MD 5358, DFW Airport, Texas 75261-9616, [email protected]

Cynthia BarnhartCenter for Transportation Studies, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected]

Constructing a protable schedule is of utmost importance to an airline because its protability is criticallyinuenced by its ight offerings. We focus our attention on the steps of the airline schedule planning processinvolving schedule design and eet assignment. Airline schedule design involves determining when and whereto offer ights such that prots are maximized, and eet assignment involves assigning aircraft types to ightlegs to maximize revenue and minimize operating cost. We present integrated models and solution algorithmsthat simultaneously optimize the selection of ight legs for and the assignment of aircraft types to the selectedight legs. Preliminary results, based on data from a major U.S. airline, suggest that signicant benets can beachieved.

Key words : air transportation; airline scheduling; airports; air trafc management; revenue/yield managementHistory : Received: June 2001; revisions received: March 2002, May 2002; accepted: May 2002.

In scheduled passenger air transportation, air-line protability is critically inuenced by the air-lines ability to construct ight schedules contain-ing ights at desirable times in protable markets(dened by origin-destination pairs). To produceoperational schedules, airlines engage in a complexdecision-making process, generally referred to as air-line schedule planning. Because it is impossible tosimultaneously represent and solve the entire air-line schedule planning problem, the many decisionsrequired in airline schedule planning have historicallybeen compartmentalized and optimized in a sequen-tial manner.The schedule planning step typically begins 12months

prior to operation of the schedule and lasts approxi-mately 9 months. It begins with route development, inwhich the airline decides which city pairs it wantsto serve, based primarily on systemwide demandinformation. Most of the time, the schedule planningstep starts from an existing schedule, with a well-developed route structure. Changes are introduced tothe existing schedule to reect changing demands andenvironment; this is referred to as schedule development.There are three major components in the scheduledevelopment step.The rst step of schedule development, schedule

design, is arguably the most complicated step ofall and traditionally has been decomposed into twosequential steps:(1) frequency planning, in which planners determine

the appropriate service frequency in a market; and

(2) timetable development, in which planners placethe proposed services throughout the day, subject tonetwork considerations and other constraints.The purpose of the second step of schedule devel-

opment, eet assignment, is to assign available aircraftto ight legs such that seating capacity on an assignedaircraft matches closely with ight demand. If toosmall an aircraft is assigned to a ight, many poten-tial passengers are turned away, or spilled, resultingin lost revenue. If too big an aircraft is assigned to aight, many empty seats, which can be utilized moreprotably elsewhere, are own.The objective of the third step of schedule develop-

ment, aircraft rotation, is to nd a maintenance feasiblerotation (or routing) of aircraft, given a eeted sched-ule and the available number of aircraft of each type.A maintenance feasible rotation, a sequence of connectedight legs assigned to a specic aircraft respectingthe maintenance rules of the airline and regulatoryagencies, begins and ends at the same location.Subsequent steps in the airline schedule plan-

ning process include revenue management andcrew scheduling. For a detailed overview, interestedreaders are referred to Lohatepanont (2001).Today, advanced technologies and a better under-

standing of the problems have allowed operationsresearchers to begin integrating and globally opti-mizing these decisions. In this paper, we presentintegrated models and algorithms for airline sched-ule design and eet assignment. The schedule design

19

Lohatepanont and Barnhart: Airline Schedule Planning20 Transportation Science 38(1), pp. 1932, 2004 INFORMS

problem involves determining when and where tooffer ights so that prots are maximized, while theeet assignment problem involves assigning aircrafttypes to ight legs to maximize revenue and mini-mize operating costs. Our integrated models simulta-neously optimize the process of selecting ight legs toinclude in the schedule and assigning aircraft types tothese legs. We present our computational experiencesusing data from a major U.S. airline.Our contributions in this paper include:(1) a framework for considering demand and sup-

ply interactions in the context of airline scheduledesign;(2) two new integrated models and solution algo-

rithms for airline schedule design and eet assign-ment, namely,

the integrated schedule design and eetassignment model and

the approximate schedule design and eetassignment model;(3) a proof of concept of our approach, with limited

computational results for problem instances providedby a major U.S. airline.In 1, we describe the nature of air travel demand,

supply, and their interactions in the context of air-line schedule design. We give an overview of ourintegrated models for schedule design and eetassignment in 2. We present integrated models andalgorithms for schedule design and eet assignmenttogether with computational results in 3 and 4.

1. Demand and SupplyInteractions

A crucial element in the construction of an airlineschedule is the interaction between demand and sup-ply. In this section, we review relevant literature onthis issue. As will be seen, such understandings areessential in developing a ight schedule.

Demand. For the purpose of schedule design andeet assignment, a market is dened by an origin anddestination pair. For example, BostonLos Angeles isa market, and Los AngelesBoston is another dis-tinct market, referred to as an opposite market. BostonSan Francisco and BostonOakland are examples ofparallel markets because San Francisco and Oaklandare close enough to each other that passengers areoften indifferent as to which location is preferable.For the purpose of schedule design for a given

airline, we are interested in its unconstrained marketdemand (or more succinctly, market demand), thatis, the maximum demand the airline is able to cap-ture. Unconstrained market demand is allocated toitineraries, sequences of connecting ight legs, in eachmarket to determine unconstrained itinerary demand.The term unconstrained is used to denote that the

quantity of interest is measured or computed withouttaking into account capacity restrictions. For exam-ple, unconstrained revenue is the maximum revenueattainable by an airline, based only on unconstraineddemand and not on capacity offered; while constrainedrevenue is the achievable revenue subject to capacityconstraints.Simpson and Belobaba (1992a) illustrate that the

unconstrained market demand for a carrier is affectedby its ight schedule (with frequency of service beingone critical element). They also show that total marketdemand can change as a result of changes in the ightschedule. Specically, additional demand can be stim-ulated or, more precisely, diverted from other modes oftransportation when the number of itineraries/ights(that is, supply) is increased (given that the demandhas not yet reached the maximum demand) and viceversa. It is also true that supply is a function ofdemand: The carrier purposefully designs its scheduleto capture the largest demands.There are a number of demand forecasting mod-

els available. Most of these models are logit based(Ben Akiva and Lerman 1985), while some are QualityService Index (QSI) based (as described in 3).

Supply. The airline develops its ight networkto compete for market demand. The rst step indeveloping the ight network is to adopt an appro-priate network structure. Unlike other modes oftransportation in which the routes are restricted bygeography, most of the time the route structuresfor air transportation are more exible. Simpsonand Belobaba (1992b) present three basic networkstructures, namely, a linear network, a point-to-point(complete) network, and a hub-and-spoke network.A pure linear network is one in which all ight legs

connect all cities in one large uni- or bi-directionaltour. A pure point-to-point network is a completenetwork, where there are services from every city toevery other city. A hub-and-spoke network containsat least one hub airport. All spoke cities have ightsdeparting from and arriving to the hub airport(s). Thisis the most prevalent network type for most majorU.S. carriers. Its main advantage derives from con-necting opportunities at the hub airport(s), enablingairlines to consolidate demand from several marketsonto each ight. Simpson and Belobaba (1992b) alsonote that the hub-and-spoke network structure createsmore stable demand at the ight-leg level. By mix-ing and consolidating demands from different mar-kets on each ight leg, the hub-and-spoke networkcan reduce variations in the number of passengersat the ight-leg level because markets have differentdemand distributions.

Demand and Supply Interactions. Hub-and-spokenetworks illustrate demand and supply interactions.

Lohatepanont and Barnhart: Airline Schedule PlanningTransportation Science 38(1), pp. 1932, 2004 INFORMS 21

To see this, consider removing a ight leg from aconnecting bank or complex. A bank or complex is aset of ights arriving or departing a hub airport insome period of time. Banks are typically designedwith a set of ight arrivals to the hub followed bya sequence of departures from the hub to facilitatepassenger connections. The removal of a ight from ahub can have serious ramications on passengers inmany markets throughout the network. This occursbecause in addition to carrying local passengers fromthe ights origin to its destination, the removed ightcarries a signicant number of passengers from manyother markets containing that leg. From the view-point of the passengers in those markets, the qual-ity of service is deteriorated because the frequencyof service is decreased. The result can then be thatthe carrier experiences a decrease in its unconstrainedmarket demands in the affected markets. The situa-tion is the opposite when a ight leg is added to thebank. For air transportation, Teodorovic and Krcmar-Nozic (1989) show that ight frequencies and depar-ture times are among the most important factors thatdetermine passengers choices of an air carrier whenthere is a high level of competition.

An Example of Demand and Supply Interactions.In this section, we present an example of how chang-ing the ight schedule of an individual airline canaffect demand for that airline. Consider the examplein Figure 1(a), a market with four itineraries. Let therst itinerary be in the morning and the last threeitineraries be in the afternoon.Above each gure, the airlines market demand is

specied for the case when all itineraries in the gureare own. These market demands are unconstrained,which can be allocated to all itineraries of the airlineas shown. Specically, in Figure 1(a), each itineraryhas 100 requests except the second itinerary, whichhas 150 requests. (We can view the second itineraryas a nonstop itinerary and the rest as connectingitineraries.)

B

100100150100

A

B120

190

100A

B

200

100A

(a)Allitinerariesoperate

(b)Oneitineraryisdropped

(c)Twoitinerariesaredropped

xUnconstrainedItineraryDemand

MarketDemand450

MarketDemand410

MarketDemand300

Figure 1 An Example of Demand and Supply Interactions

In Figure 1(b), we assume that one of the optionalights is deleted and, as a result, the last itineraryno longer exists. Based on this new level of ser-vice, the airline market demand reduces from 450requests in Figure 1(a) to 410 requests in Figure 1(b).These 410 requests can be allocated to the remainingthree itineraries as shown in Figure 1(b). Specically,some of the 100 potential customers previously on thedeleted itinerary will go to other airlines, and somewill remain with the airline. Those that remain withthe airline will request other itineraries still in theschedule (Itineraries 13 in this case). We assume that40 and 20 of the 100 requests previously on the fourthitinerary will request the second and third itineraries,respectively. The rst itinerary does not receive anyadditional requests because it is in the morning andthe 100 passengers prefer itineraries that depart closeto the time (the afternoon) of their original itinerary.The second itinerary receives more requests than thethird because the former is nonstop while the latter isconnecting.In Figure 1(c), we assume that two itineraries are

deleted. The same phenomenon occurs. Demand ofthe airline reduces from 450 requests to 300 requestsas a result of deleting two optional ight legs. The lostrequests when two itineraries are deleted more thandouble those when only one itinerary is deleted, illus-trating the nonlinear relationship between demandand supply.

2. Integrated Models for ScheduleDesign and Fleet Assignment: AnOverview

Previous efforts to improve protability of airlineschedules are described in, for example, Soumis et al.(1980), Dobson and Lederer (1993), Marsten et al.(1996), Berge (1994), Erdmann et al. (1999), and for asurvey: Etschmaier and Mathaisel (1984).Marsten et al. (1996) present a framework for

incremental schedule design. In their approach, theyenumerate potential combinations of ight additionsand deletions. Given a schedule, demands are esti-mated using a schedule evaluation model. Then, theeet assignment problem is solved on the givenschedule with the corresponding estimated demands.Revenues are computed based on the passengerstransported, and costs are computed based on ightoperating costs of the eeted schedule. The protsfrom several sets of proposed additions and dele-tions are then compared to identify the best set. Inthis paper, we present an alternative approach, inwhich ight-leg selection and eet assignment aresimultaneously optimized.


2.1. FrameworkIn developing our integrated models for airline sched-ule design and eet assignment, we assume our sched-ule is daily; that is, the schedule repeats everyday.Because conservation of aircraft, or balance, is alwaysmaintained, we can count the number of aircraft in thenetwork by taking a snapshot of the network at a pre-specied point in time, dened as the count time (forexample, 3:00 a.m.) and counting the number of air-craft both in the air and on the ground at stations.Our approach to schedule design is incremental;

that is, we do not attempt to build a schedule fromscratch. Instead a number of modications are intro-duced to a base schedule, that is, a schedule from thecurrent or previous season. This is, in fact, the prac-tice in the industryplanners build a schedule for thenew season by making changes to the current sched-ule. There are a few reasons for this:(1) Building an entirely new schedule requires data

which might or might not be available to the airline;(2) building an improved new schedule from

scratch is operationally impractical and computation-ally difcult, if not intractable;(3) frequently changing network structures require

signicant investment at airport stations (for example,gates, check-in counters); and

XY101 BOS-ORD10:00amXY234 ORD-DFW11:14amXY333 ORD-SFO08:15amXY444 SFO-DEN09:45amXY498 SEA-SFO10:32am . .

XY234 ORD-DFW11:14amXY444 SFO-DEN09:45am . . .

XY101 BOS-ORD10:00amXY333 ORD-SFO08:15amXY498 SEA-SFO10:32am . . .

XY777 BOS-ORD11:00amXY888 ORD-DEN09:04am . . .

XY234 ORD-DFW11:14amXY444 SFO-DEN09:45amXY777 BOS-ORD11:00amXY888 ORD-DEN09:04am . .

.

(a)Baseschedule (b)Listofcandidateflightsfordeletion

(c)Listofcandidateflightsforaddition

(d)Mandatoryflightlist (e)Optionalflightlist

XY101 BOS-ORD10:00amXY234 ORD-DFW11:14amXY333 ORD-SFO08:15amXY444 SFO-DEN09:45amXY498 SEA-SFO10:32amXY777 BOS-ORD11:00amXY888 ORD-DEN09:04am

. . .

(f)Masterflightlist

+

+

Legend:MandatoryFlightXY101 BOS-ORD10:00amOptionalFlightXY234 ORD-DFW11:14am

Figure 2 Master Flight List Construction Diagram

(4) airlines prefer a degree of consistency from oneseason to the next, especially in business markets inwhich reliability and consistency are highly valued bytravelers.By building the next seasons schedule from the

previous one, (1) airlines are able to use historicalbooking and other important trafc forecast data;(2) required planning efforts and time are manage-able; (3) xed investments at stations can be uti-lized efciently; and lastly, (4) consistency can beeasily maintained by introducing a limited number ofchanges to the schedule.Our models take as input a master ight list, that is,

a list of ight legs composed of: (1) mandatory ightsand (2) optional ights.Mandatory ights are ight legs that have to be

included in the ight schedule, while optional ightsare ight legs that may, but need not, be included.Figure 2 depicts the construction of a master ightlist. We take as our starting point a base schedule(Figure 2a), which is often the schedule from the pre-vious season. The potential modications include:(1) Existing ight legs in the base schedule can be

deleted (Figure 2b), and


(2) new ight legs can be added to the base sched-ule (Figure 2c).Flights identied in Figures 2(b) and 2(c) comprise

the optional ight list (Figure 2e). The mandatory listcontains those ight legs in the base schedule notmarked as optional (Figure 2d). The master ight list(Figure 2f) is the combination of the mandatory andoptional ight lists. A master ight list must be pre-pared by planners for input to our integrated models.Apart from a master ight list, another crucial

input is the average unconstrained itinerary demandsassociated with operating all of the ights in themaster ight list. We estimate these demands for agiven schedule using a schedule evaluation model. Thereare several schedule evaluation models used in theindustry, for example, the Sabre Airline ProtabilityModel, and United Airlines Protability ForecastingModel. Details of these models, however, are oftendifcult to obtain due to their proprietary nature.The output of our models is a list of recommended

ight legs to include in a new schedule and an asso-ciated eet assignment.Our models integrate the schedule design and

eet assignment steps in the schedule develop-ment subprocess. Notice, also, that we depart fromthe traditional approach of sequentially performingfrequency planning and timetable development. Inour approach, market service frequency, departuretimes (given a prespecied list of candidate ightdepartures), and eet assignments are all determinedsimultaneously.

2.2. Underlying Assumptions

2.2.1. Demand Issues and Assumptions. Thereare a number of assumptions underlying most eet-ing models, namely:(1) Fare Class Differentiation: Most eeting models

require that different fare class demands be aggre-gated into one fare class. Kniker (1998) shows thatdemands differentiated by fare class can be utilizedin eet assignment models. To our knowledge, how-ever, there have been no attempts to solve fare classdifferentiated eet assignment models on large-scaleproblems.(2) Demand Variation: Because eeting models typi-

cally solve the daily problem (that is, the problem inwhich the schedule is assumed to repeat each dayof the week), a representative daily demand is inputinto the eet assignment model. This requires thatdemands from different days of the week (demandvaries by day of week) must be aggregated into arepresentative demand.(3) Observed vs. Unconstrained Demand: Fleet assign-

ment models take as input some form of uncon-strained demand, that is, the maximum demand forair travel that an airline can experience regardless

of the airlines network capacity. All of the observeddemand data, however, is constrained demand, reect-ing network capacity. Hence, in practice, the demanddistribution for each leg is truncated at the capac-ity of the eet type assigned to that leg. Therefore,using assumptions about demand distributions, theseobserved data must be unconstrained.(4) Demand Recapture: Recapture occurs when pas-

sengers spilled from their desired itineraries areaccommodated on alternate itineraries in the system.Most eet assignment models ignore recapture, partlybecause it is difcult to observe. In this paper, weemploy the approach described in Barnhart et al.(2001) to model recapture in the eeting process.(5) Flight-Leg Independence: Most eet assignment

models assume ight-leg independence, that is, thatdemand on one leg is independent of all other legsand moreover, that ight revenue is leg specic.A signicant proportion of passengers, especially inthe U.S., travel on multileg itineraries, thus creat-ing ight-leg interdependencies. These interdepen-dencies introduce revenue estimation errors in theeet assignment models assuming leg independence.Barnhart et al. (2001) discuss this assumption andits ramications and provide quantitative evidencethat eet assignment models assuming independencemiss an important aspect of the eet assignment prob-lem. To capture ight-leg interdependencies, or net-work effects, they introduce an itinerary-based eetassignment approach, which we use in our integratedschedule planning models.Barnhart et al. (2001) test the effects of several of

the above assumptions using their itinerary-based eetassignment model (IFAM). They show that using aver-age demand data, ignoring the associated uncertainty(or distribution), and accounting for network effects,IFAM produces better assignments than the basic (leg-based) eet assignment models. Their experimentsalso show that IFAM produces relatively consistenteetings over a limited range of recapture values.Hence, IFAM is the core around which we build theintegrated schedule planning model presented in thispaper.

2.3. NotationTo facilitate the discussion of our integrated model,we rst list all notation in this section.

Parameters/DataCAPi: the number of seats available on ight-leg i

(assuming eeted schedule).SEATSk: the number of seats available on aircraft of

eet type k.Nk: the number of aircraft in eet-type k.Nq : the number of ight legs in itinerary q.Dp: the unconstrained demand for itinerary p, that

is, the number of passengers requesting itinerary p.


Qi: the unconstrained demand on leg i when allitineraries are own.farep: the fare for itinerary p.

farep: the carrying cost-adjusted fare for itinerary p.brp: recapture rate from p to r ; the fraction of passen-

gers spilled from itinerary p that the airline succeedsin redirecting to itinerary r .

pi =

{1 if itinerary p includes ight leg i0 otherwise

Dpq : demand correction term for itinerary p as a

result of cancelling itinerary q.

SetsP : the set of itineraries in a market including the

null itinerary, indexed by p or r .PO : the set of itineraries containing optional ight

legs, indexed by q.A: the set of airports, or stations, indexed by o.L: the set of ight legs in the ight schedule,

indexed by i.LF : the set of mandatory ights, indexed by i.LO : the set of optional ights, indexed by i.K: the set of eet types, indexed by k.T : the sorted set of all event (departure or availabil-

ity) times at all airports, indexed by tj . The event attime tj occurs before the event at time tj+1. SupposeT = m; then t1 is the time associated with the rstevent after the count time and tm is the time associ-ated with the last event before the next count time.N : the set of nodes in the timeline network indexed

by k o tj .CLk: the set of ight legs that pass the count time

when own by eet-type k.Ik o t: the set of inbound ight legs to node

k o tj .Oko t: the set of outbound ight legs from node

k o tj .Lq: the set of ight legs in itinerary q.

Decision Variablestrp : the number of passengers requesting itinerary p

that the airline attempts to redirect to itinerary r .

fk i ={1 if ight leg i is assigned to eet type k0 otherwise

Zq =

1 if itinerary q is included in the ight

network0 otherwise

.

yko t+j : the number of eet-type k aircraft that areon the ground at airport o immediately after time tj .yko tj : the number of eet-type k aircraft that are

on the ground at airport o immediately before timetj . If t1 and t2 are the times associated with adjacentevents, then yko t+1 = yko t2 .

3. Integrated Schedule Design andFleet Assignment Model

3.1. Demand Correction TermsIn this section we present, the integrated schedule designand eet assignment model (ISD-FAM). ISD-FAM isbuilt upon the itinerary-based eet assignment model(IFAM) by Barnhart et al. (2001). We assume that mar-kets are independent of one another; that is, demandsin a market do not interact with demands in any othermarkets. This enables us to adjust demand for eachmarket only if the schedule for that market is altered.ISD-FAM adjusts demand as changes are made tothe schedule, using demand correction terms. For exam-ple, the demand correction term Dpq corrects demandon itinerary p when itinerary q is deleted (that is,one or more ight legs in q are deleted) from theight network. Consider the example in Figure 3. Sup-pose that the rst and second itineraries are manda-tory itineraries; that is, they contain only mandatoryights. The third and fourth itineraries are optionalitineraries; that is, each of these itineraries containsat least one optional ight. Suppose that the fourthitinerary in Figure 3(b) is deleted. Then, the increaseof 40 passengers on the second itinerary and 20 pas-sengers on the third itinerary represent the demandcorrection terms on the second and third itineraries,respectively. Mathematically, D24 = +40, D34 = +20,and D14 = 0. Thus, our demand correction terms cancapture the changes in itinerary demands when oneoptional itinerary is deleted in a market. Assumingthat the effect is the same if only the third itineraryis deleted, then D13 = 0, D23 = +40, and D43 =+20. When the third and fourth itineraries are deletedsimultaneously, the total demand increase on the rstitinerary is estimated to be D13 + D14 = 0 and onthe second itinerary is estimated to be D23 + D24 =+80. The resulting market demand is estimated to be330, however, the actual market demand when twoitineraries are deleted is 300. To estimate these newdemands accurately, a second-order degree correction

B

100

100

150100

A B100+20

150+40

100

A

B

150+40+40

100

A

(a)Allitinerariesoperate

(b)Oneoptionalitineraryisdropped

(c)Twooptionalitinerariesaredropped

MandatoryItineraries OptionalItineraries

xUnconstrainedItineraryDemand

Figure 3 Our Approach for Capturing Demand and Supply Interactions


term, D234 = 30, is needed to adjust the demandon the second itinerary when both the third andfourth itineraries are deleted. In general, nth degreecorrection terms might be needed to correct demandexactly when n optional itineraries are deleted simul-taneously from a market. In our current implementa-tion, however, these higher-order correction terms areomitted for tractability.In our ISD-FAM formulation, demand correction

terms, Dpq p P are applied only if itinerary q isdeleted from the ight network. Zq , the itinerary statusvariable, indicates whether or not itinerary q is oper-ated, with Zq equal to one if itinerary q is operatedand zero otherwise. Notice that itinerary q is operatedonly when all ight legs contained in q are operated.

3.2. Objective FunctionIn this formulation, average unconstrained itinerarydemands are computed with a schedule evaluationmodel, for the schedule with all optional ights own.The objective of ISD-FAM is to maximize schedulecontribution, dened as revenue generated less oper-ating cost incurred. The operating cost of a sched-ule, denoted O, can be computed as

iL

kK Ck ifk ifor all eet-ight assignments. The total revenue ofa schedule can be computed from the followingcomponents:(1) Initial unconstrained revenue (R):

R=pP

farepDp (1)

(2) Changes in unconstrained revenue due to marketdemand changes (R):

R= qPO

(fareqDq

pP p =q

farepDpq

) 1Zq (2)

(3) Lost revenue due to spill (S):

S=pP

rP

fareptrp (3)

(4) Recaptured revenue from recapturing spilled passen-gers (M):

M=pP

rP

brp farer trp (4)

Equation (1) is the initial unconstrained revenuefor the schedule containing all optional legs and theassociated unconstrained demands. The term fareq Dqin Equation (2) is the total unconstrained revenueof itinerary q. The term

pP p =q farepD

pq is the

total change in unconstrained revenue on all otheritineraries p (=q) in the same market due to deletionof itinerary q. Recall that Zq equals one if q is ownand zero otherwise. Thus, Equation (2) is the changein unconstrained revenue if itinerary q is deleted.

Equations (3) and (4) measure the changes in rev-enue due to spill and recapture, respectively. Recallthat spill occurs when passengers cannot be accom-modated on their desired itineraries due to insuf-cient capacity. Some of these passengers are redirectedto alternate itineraries within the airline network.Recapture occurs when some number of these redi-rected passengers are accommodated on alternateitineraries of that airline. In our work, the recapturerate brp, the successful rate of redirecting passen-gers to itinerary r when itinerary p is capacitated,is computed based on the Quality of Service Index(QSI) (Kniker 1998). QSI is an industry measure of theattractiveness of an itinerary relative to that of allother itineraries (including competing airlines) in thatmarket. The sum of the QSI values corresponding toall itineraries (including competitors) in a market isequal to one. The sum of the QSI for one airline is anapproximate measure of its market share for that spe-cic market. Let qp denote the QSI value of itinerary p.Let Qm represent the sum of all QSI values in mar-ket m for the particular airline; that is, Qm =

pm qp.

Then, the recapture rate, brp is

brp =10 if p= rqr1Qm+ qr

otherwise (5)

This recapture rate is a measure of the probabilitythat a passenger spilled from itinerary p will acceptitinerary r as an alternative. (For more details, seeBarnhart et al. 2001.)Mathematically, trp is the number of passengers

being redirected from itinerary p to itinerary r andbrpt

rp is the number of passengers who are recaptured

from itinerary p onto itinerary r . We denote tp as spillfrom itinerary p to a null itinerary and assign its asso-ciated recapture rate, bp , a value of one. Passengersspilled from itinerary p onto a null itinerary are notrecaptured on any other itinerary of the airline andare lost to the airline.The contribution maximizing objective function of

ISD-FAM is therefore

MaxRRS+MO (6)Given unconstrained demands for the schedule,

the initial unconstrained revenue (R) is a constantand we can remove it from the objective function.If we reverse the signs of the rest of the terms inEquation (6), we obtain an equivalent cost-minimizingobjective function

MinO+ SM+R (7)Additional cost items that can be included are

passenger-related costs, which are composed of, but


not limited to,(1) passenger carrying costs (for example, meals,

luggage handling) and(2) cost per revenue dollar (for example, reserva-

tion commission).These cost items can be incorporated into the

model by deducting them from the revenue (orfare) obtained from a passenger. Thus, in the model,instead of using farep to denote revenue obtained from

a passenger traveling on itinerary p, we use farep, thenet revenue from a passenger traveling on itinerary p.

3.3. FormulationISD-FAM can be formulated as shown in (8)(20):

MiniL

kK

Ck ifk i+pP

rP

farep brp farer trp

+ qPO

(fareqDq

pP p =q

farep Dpq

) 1Zq (8)

subject to kK

fk i = 1 i LF (9)kK

fk i 1 i LO (10)

yko t +

iIk o tfk i yko t+

iOko t

fk i = 0

k o t N (11)oA

yko tn +

iCLkfk i Nk k K (12)

pP

qPO

pi D

pq 1Zq+

kK

CAPkfk i+rP

pP

pi t

rp

rP

pP

pi b

pr t

pr Qi i L (13)

qPODpq 1Zq+

rP

trp Dp p P (14)

Zq kK

fk i 0 i Lq (15)

Zq

iLq

kK

fk i 1Nq q PO (16)

fk i 01 k K i L (17)Zq 01 q PO (18)

yko t 0 k o t N (19)trp 0 p r P (20)

Constraints (9), the cover constraints for manda-tory ights, ensure that every mandatory ight isassigned to a eet type. Constraints (10) are coverconstraints for optional ights allowing the model tochoose whether or not to y ight i in the schedule.If ight i is selected, a eet type has to be assigned

to it. Constraints (11) ensure conservation of aircraftows. Constraints (12) are count constraints guar-anteeing that only available aircraft are used. Con-straints (13) are capacity constraints ensuring for eachight i that the number of passengers on i does notexceed the capacity assigned to i. Constraints (14)are demand constraints restricting the number of pas-sengers spilled from an itinerary to the demand forthat itinerary. The term

qPO D

pq 1 Zq in con-

straints (14) corrects the unconstrained demand foritinerary p P when optional itineraries q PO aredeleted. Similarly, the term

pP

qPO pi D

pq 1 Zq

in constraints (13) represents corrected demand but atthe ight level. Constraints (15)(16) are itinerary sta-tus constraints that control the 01 variable, Zq , foritinerary q. Specically, constraints (15) ensure that Zqtakes on value zero if at least one leg in q is not own,and constraints (16) ensure that Zq takes on value oneif all legs in q are own.

Passenger Flow Adjustment. Notice that ISD-FAMemploys two mechanisms to adjust passenger ows:(1) demand correction terms and (2) recapture rates.Both mechanisms accomplish the objective of reac-

commodating passengers on alternate itinerarieswhen desired itineraries are not available, but withdifferent underlying assumptions. To illustrate, con-sider itineraries p P , q PO , and r P in a marketm. With demand correction terms, ISD-FAM attemptsto capture demand and supply interactions by adjust-ing the unconstrained demand on alternate itinerariesp P in market m when an optional itinerary q POp = q is deleted, utilizing demand correction termsD

pq s. In so doing, the total unconstrained demand of

the airline in market m is altered by Dq

pP p =q Dpq

when itinerary q is deleted.With recapture rates, ISD-FAM attempts to reallo-

cate passengers on alternate itineraries r P in marketm when an itinerary p P is capacitated, utilizing therecapture rates brps. This mechanism, however, doesnot affect the total unconstrained demand of the air-line in market m.An understanding of this difference is important to

the introduction of our model in the next section.

3.4. Solution ApproachISD-FAM takes as input the master ight list, recap-ture rates, demand data, demand correction terms,and eet composition and size. In theory, using aschedule evaluation model, we could estimate exactlyall correction terms and include all of them in ISD-FAM. Then, by solving ISD-FAM once, an optimalschedule is determined. This strategy is impractical,however, because exponentially many runs of theschedule evaluation model (one run for each possi-ble combination of ight additions and deletions) arenecessary to estimate the correction terms.


Consequently, we adopt the solution algorithm,outlined in Figure 4. Instead of trying to estimate alldemand correction terms at the outset, we use roughestimates of these terms and revise them iteratively asneeded. Specically, we initially obtain demand esti-mates for the full schedule (containing all ights fromthe master ight list) using a schedule evaluationmodel. In Step 1 of our approach in Figure 4, we solveISD-FAM (detailed in 3.4.1) to obtain a eeted sched-ule. In Step 2, a schedule evaluation model is calledto determine the new set of demands for the scheduleresulting from Step 1. Given these demand estimatesand the eeted schedule, in Step 3 we solve the pas-senger mix model (PMM), presented by Kniker (1998).(The PMM formulation, cast as a multicommodityow problem with specialized variables is detailed inthe Appendix.) To solve PMM, column and row gen-eration techniques are used to manage problem size.PMM takes as input a eeted schedule and determin-istic unconstrained itinerary demands and outputs arevenue-maximizing ow of passengers. The objectivefunction value of the model is referred to as PMM rev-enue. It represents a maximum, not expected, schedule

START

STOP

HastheSTOPPING

CRITERIA

beenmet?

(1)SolveISD-FAM(SeeFigure5)

(3)SolvePMMwithdemandfrom

(2)andschedulefrom(1)

(5)Identifyitinerarieswith

inaccuraterevenueestimatesin(1)

(6)Revise

demandcorrectionterms

(2)Calculatenewdemandforthe

resultingschedule

NO

YES

(4)

Figure 4 The Solution Approach for ISD-FAM, Estimating DemandCorrection Terms as Needed

revenue (the underlying assumptions do not alwayshold). In Step 4, we compute:(1) the difference between our current and previous

solution and(2) the closeness of the revenue estimates of Steps 1

and 3.If either of these computed values are beneath

our dened threshold values, the algorithm is termi-nated; otherwise itineraries with inaccurate revenueestimates in the ISD-FAM solution are identied inStep 5 and their associated demand correction termsare updated (in Step 6) using the demand informationobtained from solving the schedule evaluation modelin Step 2. ISD-FAM is then resolved and the procedurerepeats.At every iteration, however, existing demand cor-

rection terms are updated and/or new ones are intro-duced, hence, capturing more exactly demand andsupply interactions. Thus, if higher-order correctionterms are included, the algorithm will converge in anite number of iterations. This, however, can takemany iterations if the models solution is highlysensitive to the values of the demand correctionterms.

3.4.1. Solution Algorithm for ISD-FAM. The al-gorithm for solving ISD-FAM is depicted in Figure 5.We rst construct a restricted master problem (RMP)excluding constraints (14) and including only spillvariables corresponding to null itineraries. Then, theLP relaxation of the RMP is solved using column androw generation. Negative reduced-cost columns cor-responding to spill variables and violated constraints(14) are added to the RMP, and the RMP is resolveduntil the ISD-FAM LP relaxation is solved. Given thesolution to the ISD-FAM LP relaxation, branch-and-bound is invoked to nd an integer solution. Becausecolumn generation at nodes within the branch-and-bound tree is nontrivial to implement using availableoptimization software, we employ a heuristic IP solu-tion approach in which branch-and-bound allows col-umn and row generation only at the root node andafter the integer solution is found. Row (and column)generation occurs after an integer solution is obtainedif any constraints are violated by the current solution.In this case, column and row generation allows thebest passenger ow decisions to be determined, forthe selected ight schedule and eeting.

3.5. Computational ResultsWe perform computational tests for ISD-FAM as out-lined in Figure 6. Using a Period I schedule as ourbase schedule, planners provide a master ight listincluding both mandatory and optional ight legs.Using this master ight list, planners generate a pro-posed schedule for Period II, the planners schedule,


Step1

BuildRestrictedMasterProblem

Step2SolveRMPLP

Step3ViolatedCon.Gen.

Step4ColumnGeneration

No

Step5Branch&Bound

DONE

ISD-FAMLPRelaxation

Wereanyrowsorcolumnsadded?

Yes

Step6FixIntegerVariables

Step7SolveLP

Step8Col./RowGeneration

Wereanyrowsorcolumnsadded?

Yes

NoISD-FAMFeasibility

ISD-FAMIP

Figure 5 The ISD-FAM Solution Algorithm (Step 1 in Figure 4) for Specied Demand Correction Terms

which is compared to the ISD-FAM-generated sched-ule. A schedule evaluation model is used to generatea new set of demands for both the planners scheduleand the ISD-FAM-generated schedule. The PMM isthen solved to obtain PMM revenues for the Period IIschedules. Finally, PMM contributions are computedby subtracting operating costs from PMM revenues.In most cases, PMM contributions for an ISD-FAM-generated schedule differs from the ISD-FAM objec-tive function value, called ISD-FAM contribution. Thedifference between PMM and ISD-FAM contributionsfor a schedule measures the revenue discrepancy result-ing from using demand estimates based on a ightschedule that does not match the resulting ISD-FAMight schedule.We perform our experiments on actual data, includ-

ing the planners schedules, provided by a major U.S.airline. Table 1 shows the characteristics of the net-works in our two data sets. All runs are performedon an HP C-3000 workstation computer with twoGB RAM, running CPLEX 6.5.Tables 2 and 3 show the problem sizes and ISD-

FAM run-times, and solution results for data setsD1 and D2, respectively. For data set D1, ISD-FAM generates a considerably improved schedulecompared to the planners schedules, achieving anincrease in the daily contribution of $561,776. Assum-ing that unconstrained demand is an average of daily

demands and that the airline operates this schedule365 days, these daily improvements translate to over$200 million per year. The revenue discrepancy fordata set D1 is $187,263 per day, suggesting that moreiterations of the algorithm, leading to more accurateestimates of demand, could result in even greaterimprovements.For data set D1, ISD-FAM achieves signicant

improvements through savings in operating costs(from operating fewer ights) rather than by gen-erating higher revenues (Table 4). Additionally, theISD-FAM solution requires signicantly fewer aircraftto operate the schedule. Each aircraft removed fromthe schedule can represent substantial savings to theairline because the airline can increase protabilityby employing the aircraft elsewhere, such as in newmarkets.Although ISD-FAM captures interactions between

demand and supply through demand correctionterms, it suffers from tractability issues for larger sizeproblems. For data set D2, while the LP relaxationcould be solved in 66 hours (46 iterations of col-umn and row generations), ISD-FAM ran for threeplus days without achieving an integer solution. Inthe next section, we explore an approximate scheduledesign model that does not continuously adjustdemand as the schedule changes.


PeriodISchedule

MasterFlightList

Planners'input

ISD-FAM

PeriodIISchedule

ScheduleEvaluation

Model

PMMContribution

Planners

PeriodIISchedule

ScheduleEvaluation

Model

PMMContribution

Planners'Schedule ISD-FAMSchedule

PeriodIIDemand

PeriodIIDemand

PassengerMixModel

PassengerMixModel

ISD-FAMContribution

RevenueDiscrepancy

Figure 6 Testing Methodology

4. Approximate Schedule Design andFleet Assignment Model

4.1. An Approximate Treatment of Demand andSupply Interactions

In this section, we present the fundamental con-cept, model formulation, and solution algorithm of

Table 1 Data Characteristics

No. of Flight Legs No. of No. of No. ofData Set Mand. Opt. Total Itineraries Fleets Aircraft

D1 645 108 753 60,347 4 166D2 588 260 848 67,805 4 166

Table 2 Problem Sizes (RMPs) and Solution Times

Data Set D1 Data Set D2

No. of columns 48,742 65,447No. of rows 30,206 60,910No. of nonzeros 164,676 344,517No. of iterations to solve LP relaxation 34 46LP relaxation solution time 11.1 hrs 66 hrsSolution time 12.7 hrs 3+ days

Solution is not available.

Table 3 Contribution Comparison

ISD-FAM Schedule

DailyPlanners Schedule

($/day) ($/day) Improvement

Data Set D1ISD-FAM contribution N/A 1,908,867PMM contribution 1,159,828 1,721,604 561,776Revenue discrepancy N/A 187,263

Table 4 Resulting Schedule Characteristics

Planners Schedule Data Set D1

No. of ights own 717 668No. of optional ights not own 0 85No. of aircraft used 166 157No. of aircraft not used 0 9

our approximate schedule design and eet assignmentmodel (ASD-FAM). ASD-FAM utilizes recapture ratesto approximately capture the interactions betweendemand and supply. We demonstrate that althoughrecapture rates do not alter total unconstraineddemand, capacity constraints on other itineraries andrecapture rates indirectly dictate the maximum num-ber of passengers the airline can reaccommodatewithin the system.To illustrate, consider Market AB in Figure 7. Sup-

pose that there are two nonstop ights, i and j , andtwo one-leg itineraries, p and r , where p is on i and r ison j . Thus, trp denotes the number of passengers redi-rected from ight-leg i to ight-leg j . Suppose furtherthat the average unconstrained demand on itineraryp is 70 and that there are 20 empty seats availableon ight-leg j . If ASD-FAM deletes ight-leg i (andtherefore itinerary p) from the schedule, 70 passengerspreviously on itinerary p need reaccommodation. Sup-pose that the recapture rate is 0.5. Because there are 20seats available on ight-leg j , ASD-FAM will attemptto redirect 40 passengers from p to r (20 of whom will

j:20seatsavailable

B

i:70extrapassengers

A ir rp p

b t =20

p pb t

=30

NetLost:20+30passengers

Figure 7 An Example of Reduced Effective Market Share Due toModied Recapture Rates


be successfully reaccommodated) and will spill 30 tothe null itinerary. Thus, in this example, the demand iseffectively reduced by 20+ 30= 50 passengers becauseof the deletion of ight-leg i. This example illustratesthat recapture rates and capacity restrictions can causeeffective market demand to be less than or equal to theinitial market demand, even when deleting ight-leg iis assumed to have no direct effect on market demand.Notice, however, that the recapture rates employed inthis manner are for the purpose of shifting demandsfrom deleted itineraries to remaining itineraries. Thus,these recapture rate values are not likely to equal thosein IFAM, which are strictly for the purpose of reaccom-modating passengers when ight capacities are con-strained. For this reason, we refer to recapture ratesemployed in ASD-FAM as modied recapture rates, brp.

4.2. FormulationThe approximate schedule design and eet assign-ment model is formulated as

MiniL

kK

Ck ifk i+pP

rP

farep brp farer trp (21)

subject to kK

fk i = 1 i LF (22)kK

fk i 1 i LO (23)

yko t +

iIk o tfk i yko t+

iOko t

fk i = 0

k o t N (24)oA

yko tn +

iCLkfk i Nk k K (25)

kKCAPkfk i+

rP

pP

pi t

rp

rP

pP

pi b

pr t

pr Qi i L (26)

rPtrp Dp p P (27)

fk i 01 k K i L (28)yko t 0 k o t N (29)

trp 0 p r P (30)

ASD-FAM is a special case of ISD-FAM, in whichall demand correction terms are removed. As a result,the objective function (Equation 21) and constraints(26) and (27) are modied. To offset this, a set of mod-ied recapture rates (brp) is employed in ASD-FAM toapproximate changes in market demand as the ightschedule is altered.

Table 5 Problem Sizes (RMPs) and Solution Times

Data Set D1 Data Set D2

No. of columns 35,633 38,837No. of rows 2,999 3,326No. of nonzeros 55,039 61,655No. of iterations to solve LP relaxation 31 28LP relaxation solution time 23 mins 41 minsSolution time 39 mins 78 mins

4.3. Solution ApproachThe accuracy of ASD-FAM is critically linked tothe modied recapture rates (brp) because they are theonly mechanism through which market demand isadjusted and through which passengers can be reallo-cated. Finding the set of modied recapture rates thataccurately capture supply and demand interactionsmight require several iterations or, even worse, mightbe impossible. In our experiments, we let brp = brp forthe case when all itineraries in a market are operated.The solution algorithm for ASD-FAM is the ISD-

FAM algorithm, outlined in Figure 4, with somemodications. Initially, demand estimates for the fullschedule (containing all ights from the master ightlist) are obtained using a schedule evaluation model.In Step 1, ASD-FAM is solved using the ISD-FAMalgorithm (outlined in 3.4.1) to obtain a eetedschedule. In Step 2, the new set of demands forthe schedule resulting from Step 1 is obtained from aschedule evaluation model. Given these demand esti-mates and the eeted schedule, in Step 3 we solvePMM to obtain the PMM revenue. In Step 4, the stop-ping criteria, as used in ISD-FAM, are evaluated. Ifthe stopping criteria are not met, itineraries with inac-curate revenue estimates in the ASD-FAM solutionare identied in Step 5, and their modied recapturerates are revised (in Step 6) using the demand infor-mation from Step 2. ASD-FAM is then resolved andthe procedure repeats.

4.4. Computational ResultsWe evaluate ASD-FAM on the two medium-sizeschedules used for testing ISD-FAM. (Table 1describes the characteristics of the data sets.) The test-ing procedure is as outlined in Figure 6. A Period Ischedule serves as our basis for constructing a masteright list. ASD-FAM is solved on this master ightlist to obtain an ASD-FAM-generated Period II sched-ule. A schedule evaluation model is used to generatea new set of demands for the ASD-FAM-generatedschedule. PMM is then solved to obtain the PMM con-tributions, which are compared with the ASD-FAMcontributions.The problem sizes and ASD-FAM solution times

for the two problems are reported in Table 5. FromTable 6, it can be seen that ASD-FAM solutions


Table 6 Contribution Comparision

ASD-FAM SchedulePlannersSchedule


Data Set D1ASD-FAM contribution N/A 1,999,094PMM contribution 1,159,828 1,742,349 582,521Revenue discrepancy N/A 256,745

Data Set D2ASD-FAM contribution N/A 2,079,504PMM contribution 1,159,828 1,888,948 729,120Revenue discrepancy N/A 190,556

are signicantly improved compared to the plannerssolutions. In data set D1 and D2, daily improve-ments of $582,521 and $729,120 are achieved, respec-tively. Assuming schedules repeat daily with similardemand for every day of the year, these improve-ments translate into annual improvements of approx-imately $210 and $270 million, respectively. Therevenue discrepancy of $256,745 per day for data setD1 exceeds that of the ISD-FAM solution, as expected.Table 7 presents the characteristics of the resulting

schedules. Similar to ISD-FAM, ASD-FAM achievessignicant improvements through savings in operat-ing costs (from operating fewer ights) rather thanthrough the generation of higher revenues. We alsonote that ASD-FAM solutions require signicantlyfewer aircraft to operate the schedule.Finally, we note that ASD-FAM outperforms ISD-

FAM, a couterintuitive result given that ASD-FAM isan approximation of ISD-FAM. The improved ASD-FAM performance results from its reduced complex-ity (relative to ISD-FAM) and, hence, our ability togenerate more integer solutions.

4.5. Full-Size ProblemsUnlike ISD-FAM that cannot be solved for data set D2,ASD-FAM is solvable for full-size schedules. Table 8


PlannersSchedule Data Set D1 Data Set D2

No. of ights own 717 660 614No. of optional ights not own 0 93 234No. of aircraft used 166 154 148No. of aircraft not used 0 12 18

Table 8 Data Characteristics

No. of Flight Legs No. of No. of No. ofData Set Mand. Opt. Total Itineraries Fleets Aircraft

F1 1,993 1,993 179,965 8 350F2 1,988 1,988 180,114 8 350

Table 9 Contribution Comparison

Planners Schedule ASD-FAM Schedule


Data Set F1ASD-FAM contribution N/A 36,932,000PMM contribution 36,387,000 37,375,000 988,000Revenue discrepancy N/A 443,000

Data Set F2ASD-FAM contribution N/A 36,124,000PMM contribution 35,668,000 36,072,000 404,000Revenue discrepancy N/A 52,000


Data Set F1 Data Set F2

Planners PlannersSchedule ASD-FAM Schedule ASD-FAM

No. of ights own 1,619 1,545 1,591 1,557

Table 11 Problem Sizes (RMPs) andSolution Times

Data Set F1 Data Set F2

Solution time 16 hrs 19 hrs

gives the characteristics of two additional data sets,each accompanied by the associated planners sched-ules. Notice that in both data sets, all ights areoptional. They represent full-size problems at a majorU.S. airline. The runs are performed at the airlineusing their implementation of our prototype ASD-FAM model, on a six-processor computer, with twoGB RAM, running Parallel CPLEX 6.Table 9 summarizes the results on these full-size

problems. In data set F1, ASD-FAM achieves a dailyimprovement of $988,000 over the planners sched-ule. In data set F2, the improvement is smaller at$404,000 per day. In both of these data sets, ASD-FAM produces schedules that operate fewer ightlegs (Table 10).We can clearly appreciate the complexity of these

problems when we look at their solution times,reported in Table 11. Note that these are run-times ona six-processor workstation, running parallel CPLEX.

AcknowledgmentsThis research is supported in part by United Airlines. Theauthors thank the staff at United Airlines, in particularAmit Mukherjee and Jon Goodstein, for their insight andassistance.

Appendix. The Passenger Mix ModelThe passenger mix model (PMM) (Kniker 1998) takes a eeted schedule(that is, each ight leg is assigned one eet type) and unconstrained


itinerary demand as input and nds a ow of passengers over thisschedule maximizing eeting contribution, or equivalently mini-mizing assignment cost. Because the schedule is eeted, ight oper-ating costs are xed and only passenger carrying and spill costsare minimized. The objective of the model, then, is to identify thebest mix of passengers from each itinerary on each ight leg. Thesolution algorithm spills passengers on less protable itineraries toaccommodate passengers on more protable itineraries. Hence, thePMM is

Given a eeted ight schedule and the unconstraineditinerary demands, nd the ow of passengers over the net-work that minimizes carrying plus spill cost, such that 1the total number of passengers on each ight does not exceedthe capacity of the ight, and 2 the total number of pas-sengers on each itinerary does not exceed the unconstraineddemand of that itinerary.

Using previously dened notations, the mathematical formula-tion for PMM is

MinpP

rP

(farep brp farer

)trp (A.1)

subject to

pP

rP

pi t

rp rP

pP

pi b

pr t

pr Qi CAPi i L (A.2)

rP

trp Dp p P (A.3)

trp 0 p r P (A.4)

The objective function minimizes the difference of spill andrecapture. Constraints (A.2) are the capacity constraints. For leg i,the term

rP

pP pi t

rp

pP pi t

pp can be viewed as the number of

passengers who are spilled from their desired itinerary p. For leg i,the term

rP

pP pi b

pr t

pr pP pi bpp tpp is the number of passengers

who are recaptured by the airline. (Note that we assume bpp = 1.)CAPi is the capacity of the aircraft assigned to leg i, and Qi is theunconstrained demand on ight-leg i, namely,

Qi =pP

pi Dp (A.5)

Constraints (A.3) are the demand constraints restricting the totalnumber of passengers spilled from itinerary p to the unconstraineddemand for itinerary p. trp must be greater than zero but need not beinteger because we model the problem based on average demanddata, which can be fractional.

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