Reveue Maagemet Wthout Forecastg or Optmzato: A Adaptve Algorthm for Determg Arle Seat Protecto Levels Garrett va Ryz Jeff McGll Graduate School of Busess, Columba Uversty, New York, New York 10027 School of Busess, Quee s Uversty, Kgsto, Otaro, Caada gvaryz@research.gsb.columba.edu jmcgll@busess.queesu.ca We vestgate a smple adaptve approach to optmzg seat protecto levels arle reveue maagemet systems. The approach uses oly hstorcal observatos of the relatve frequeces of certa seat-fllg evets to gude drect adjustmets of the seat protecto levels accordace wth the optmalty codtos of Brumelle ad McGll (1993). Stochastc approxmato theory s used to prove the covergece of ths adaptve algorthm to the optmal protecto levels. I a smulato study, we compare the reveue performace of ths adaptve approach to a more tradtoal method that combes a cesored forecastg method wth a commo seat allocato heurstc (EMSR-b). (Yeld Maagemet; Reveue Maagemet; Arles; Forecastg; Optmzato; Fare Class Allocato; Dstrbuto Free; Adaptve Algorthms; Stochastc Approxmato) Itroducto Moder arles must decde thousads of tmes per day whether or ot to accept dscout seat bookg requests or refuse them the hope of later, hgherfare bookgs. Ther objectve s to maage the opeg ad closg of dscout fare classes such a way that overall expected reveues are maxmzed. Ths reveue maagemet (also yeld maagemet) problem s greatly complcated by such factors as volatle, stochastc demad for ar travel, fluctuatos fare prces, multple-leg passeger terares, ad dverso of passegers to other fare classes or flghts. Whle o operatos research model has succeeded dealg wth all of these complextes, smplfed models ad heurstcs have bee appled wth remarkable success at may arles. (See, for example, Smth et al. 1992). See Etschmaer ad Rothste (1974), Belobaba (1987b), Weatherford ad Bodly (1992), ad McGll ad va Ryz (1999) for overvews ad surveys. Oe fudametal yeld maagemet model cosders a sgle flght leg wth bookg requests arrvg order of bookg class. Ths sgle-leg model ca be aalyzed to determe the structure of optmal bookg polces ad the optmal polcy parameters. Belobaba (1987a, b, 1989), Brumelle ad McGll (1993), Curry (1989), ad Wollmer (1992) provde aalyses of the sgle-leg model whch the bookg fares crease mootocally from low to hgh as the tme of flght departure approaches, ad Robso (1991) geeralzes Brumelle ad McGll s optmalty codtos to the case that fares are omootoc. May yeld maagemet systems (called leg-based) use solutos from ths elemetary model to gude heurstc solutos for more realstc stuatos. Maagemet Scece 2000 INFORMS Vol. 46, No. 6, Jue 2000 pp. 760 775 0025-1909/00/4606/0760$05.00 1526-5501 electroc ISSN
A Adaptve Algorthm for Determg Arle Seat Protecto Levels To llustrate, cosder the smplest verso of the sgle-leg model whch there are oly two fare classes, demad the dscout fare class arrves before hgh-fare demad, ad the hgh-fare demad s statstcally depedet of the dscout demad. It s kow that a optmal oatcpatg polcy has the followg structure: Set a fxed protecto level for the hgh-fare seats. Seats ca the be sold to the dscout class as log as there are more tha seats remag. If hgh-fare demad, X, s modeled as a cotuous radom varable, the optmal protecto level * ca be determed from the ewsvedor type optmalty codto frst proposed ( dscrete form) Lttlewood (1972): P X * r, (1) where r s the rato of the dscout to the hgh fare. Codto (1) stpulates that f the dscout fare s, for example, 60% of the full fare, the the optmal protecto level wll be such that full-fare demad exceeds ts protecto level o 60% of all flghts over the log ru. Note that the frequecy of such fll evets does ot correspod to the rate of lost full-fare bookgs sce some overflow demad ca be accommodated whe dscout seats rema ubooked. The codtos for more tha two fare classes are more complex tha ths but are coceptually smlar. Typcally, applcato of codtos lke (1) requres three steps. Frst, hstorcal demad data are studed to determe sutable models for the demad dstrbutos. Secod, forecastg techques are appled to estmate the parameters of these dstrbutos. Because the bookg lmts themselves or arcraft capacty costrats cause cesorg of the demad data, specal techques must be employed to ucesor the demad data. Thrd, the demad statstcs from the forecast are passed to a optmzato route that solves for protecto levels lke *. The resultg protecto levels are the used to make dvdual accept-dey decsos as reservatos come. I practce, bookgs from smlar flghts are fed back to the forecastg system, ad the process s repeated cyclcally over tme. Note that each of these cycles, bookgs data from the curret departure are beg coverted va the forecastg ad optmzato procedures to updated polcy parameters for the ext departure. Ths rases a terestg questo: Is t possble to drectly update bookg polcy parameters for the ext departure based o smple observato of the performace of the parameters prevous departures, wthout recourse to the complex cycles of forecastg ad optmzato? Such ad hoc adjustmet of protecto levels was commoly used the early days of yeld maagemet (ad stll s today some arles). However, most huma aalysts fd t dffcult to guess at reveue-maxmzg protecto levels. I ths paper, we show how to costruct a smple ad effectve adjustmet scheme by usg propertes of the optmal polcy. Moreover, we show that uder statoary demad codtos, the repeated applcato of our updatg scheme evetually produces optmal bookg polcy parameters. I 1 we propose a smple adaptve updatg scheme that reles o observatos of certa fll evets, whch correspod to subsets of fare classes reachg ther respectve protecto levels. These fll evets are easly determed from bookg records data. Protecto levels are updated based o a multvarate verso of the stochastc approxmato method of Robbs ad Moro (1951), appled to the sgle-leg optmalty codtos of Brumelle ad McGll (1993). We prove 3 that our proposed algorthm coverges (almost surely) to a optmal set of protecto levels, ad we obta bouds o the rate of covergece. Some modfcatos of the algorthm to hadle practcal ssues lke ostatoarty, tegralty, ad bookg lead tmes are dscussed 4. I 5, we report results of a umercal comparso of our adaptve algorthm agast a tradtoal procedure that combes cesored forecastg ad expected margal seat reveue (EMSR) protecto levels. (See Belobaba (1989)). Coclusos are provded 6. 1. Notato, Model Assumptos, ad Optmalty Codtos We let 1(E) deote the dcator fucto of the evet E; that s, 1(E) 1 f evet E occurs ad 1(E) 0 otherwse. The expresso 1 x s abbrevated as x, ad (a.s.) s short for almost surely. Superscrpts Maagemet Scece/Vol. 46, No. 6, Jue 2000 761
A Adaptve Algorthm for Determg Arle Seat Protecto Levels o vectors or o elemets of vectors dex the members of a sequece of vectors; for example, {X 1, X 2,... } s a sequece of demad vectors; whle X s the demad for fare class the -th demad vector. Subscrpts wll dex sequeces of scalar quattes; for example, { 1, 2,... } s a sequece of scalar step szes. Superscrpts o such scalar quattes wll have the usual terpretato as expoetato. Much of our aalyss establshes upper bouds volvg suffcetly large arbtrary costats o the rght-had sdes of equaltes. To avod a prolferato of such costats, we let C deote a geerc, suffcetly large costat. The value of C chages throughout the paper depedg o cotext. For example, a statemet of the form D CE C 2 F ca be replaced wth D C(E F), where C 4 max{c, C 2 }. We cosder a model whch k 1 fare classes book o a sgle-leg, fare-class allocato s ested (descrbed below), low-fare classes book strctly before hgher fare classes, fare-class demads are mutually depedet, ad there are o cacellatos or oshows. Let f deote the fare (or expected cotrbuto) from fare class, where we assume f 1 f 2... f k 1. The demad for fare class s deoted X.We assume {X 1,...,X k 1 } are mutually depedet, the probablty dstrbutos of demad are cotuous, ad that seat capacty s a cotuous quatty. (See 4.2 for a dscusso of tegral demad ad capacty.) The stochastc process {X 1, X 2,... } of demad vectors from successve flghts s assumed to be statoary the sese that the jot probablty dstrbutos of demad rema costat over tme. Ths mples that the same protecto levels are optmal for all flghts the sequece ad that successve flghts are comparable; for example, mdweek morg commuter flghts betwee two specfc ceters hgh seaso. Extesos to ostatoary demad processes are dscussed 4.3. A fxed protecto level polcy for fare classes 1 through k s defed by a statc set of protecto levels gve by the vector ( 1,..., k ), where 1 2... k. (There s o protecto level for the lowest fare class, k 1.) Protecto levels are ested the sese that represets the umber of seats to reserve (protect) for all of fare classes 1, 2,...,. Reservatos for fare class 1 are accepted f ad oly f the umber of seats remag s strctly greater tha the protecto lmt. For the sgle-leg problem uder the assumptos stated above, t s show Brumelle ad McGll (1993) that such fxed protecto level polces are optmal amog all oatcpatg polces. Defe the ested sequece of fll evets: A 1,X X 1 1 ; A 2, X X 1 1, X 1 X 2 2 ; A, X X 1 1, X 1 X 2 2,...,X 1 X }. (2) We refer to these evets as fll evets because, whe the curret vetory of ubooked seats s, all remag seats are sold f ad oly f the evet A (, X) occurs. Let the vector * ( * 1, * 2,..., * k ) deote a optmal set of protecto levels. Brumelle ad McGll (1993) show that, whe all passeger demad dstrbutos are cotuous, a optmal * satsfes r 1 P A *, X, for 1,...,k, (3) where r 1 f 1 /f 1 s the dscout rato of fare 1 relatve to the full fare. The parallel wth Lttlewood s Rule (1) s clear: For example, f dscout fare s 60% of the full fare, optmal protecto levels wll be such that evet A (, X) occurs o 60% of future flghts the log ru. The codtos (3) are depedet of arcraft capacty ad assume cotuty of demad. I practce, protecto levels that exceed arcraft capacty ca be smply trucated to the capacty wthout loss of optmalty. Ths wll be dscussed further below. Also, the cotuous protecto levels obtaed by (3) are geerally good approxmatos for the true optmal protecto levels for teger-valued demad. Ideed, Brumelle ad McGll (1993) show that there wll be at least oe set of fxed, teger-valued, optmal protecto levels whe demad s teger-valued. Moreover, the reveue maagemet problem s kow to be robust to small departures from the optmal protecto levels. (See Brumelle ad McGll (1993) ad Robso 762 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels (1991).) Wollmer (1992) aalyzes teger-optmal protecto levels whe demad forecasts are avalable as dscrete probablty dstrbutos. (See 4.2 for further dscusso of tegralty.) I prcple, t s easy to determe the frequecy of the evets A (, X) from a record of past flghts. No ucesorg of the demad s requred t s oly ecessary to observe f demad reached the protecto levels, ot the degree to whch t exceeded them. There are two mportat exceptos to ths. Frst, f happes to exceed the maxmum umber of seats avalable for sale (usually the physcal capacty plus a overbookg pad ), the the evet X 1... X s ot observable. 1 Secod, f protecto levels are revsed durg the lead tme pror to flght departure t ca easly happe that a ew protecto level exceeds the remag capacty o the arcraft (a problem smlar to the frst oe), or that earler, hgh protecto levels costraed demad durg part of the bookg perod oe or more dscout fare classes. I ths case, total demad s ot observed relatve to (a varat of cesorshp of the demad data). Itally, we wll make the smplfyg assumpto that A (, X) s always observable. We show 4 how the algorthm ad results ca be modfed to allow for the physcal capacty costrat ad bookg lead tmes practcal mplemetatos. The codtos (3) are appealg o practcal grouds because oe ca check the optmalty of protecto levels a seres of departed flghts retroactvely by smply comparg the fracto of flghts o whch A (, X) occurred to the dscout rato r 1. Ths approach has the dstct advatage that t requres o assumptos o the specfc ature of the probablty dstrbutos of the demads. Observg fll evet frequeces therefore provdes a dstrbuto-free test of the optmalty of protecto levels. However, what happes f the observed frequeces do ot equal the dscout rato? Next, we show that these same fll evet codtos ca be used to adap- 1 Oe could stll observe ths evet f rejected sales were recorded, but ths formato s ot avalable to most arles. Rejectos occur at the pot of sale (e.g., the travel aget) ad are ot recorded the reservato system. tvely adjust protecto levels usg a exteso of the classcal stochastc approxmato algorthm of Robbs ad Moro (1951). 2. A Adaptve Algorthm For 1,...,k defe H (, X) r 1 1( A (, X)). The quatty H (, X) wll be egatve f the evet A occurs, ad postve otherwse. If protecto levels are beg adjusted, occurrece of the evet A (all of classes 1 through reached ther protecto levels) suggests that the protecto level should be adjusted upwards. Thus H (, X) ca be vewed as a adjustmet drecto for protecto level. The correspodg adjustmet vector s H(, X) (H 1 (, X),..., H k (, X)). Now defe h ( ) r 1 P( A (, X)), 1,..., k; ad let h( ) (h 1 ( ),..., h k ( )). Note that h( ) EH(, X). Thus, h( ) ca be properly vewed as the expected adjustmet vector for protecto levels gve curret levels. The optmalty Codto (3) stpulates that we should seek a * such that the expected adjustmet for all protecto levels s zero; or, h( *) 0. The Robbs-Moro procedure (geeralzed here for vector quattes) costructs a sequece of parameter estmates, { 1, 2,...,,... }, from a sequece of depedet trals, {X 1, X 2,..., X,... }, usg 1 H, X, (4) where s a sequece of oegatve step szes satsfyg ad 2. (5) (The smplest example of a sutable step-sze sequece s defed by 1/, however, ths smple averagg sequece takes small steps early the procedure, whch ca delay covergece. I the developmet to follow, we use a sequece of the form A/( B), where A ad B are costats chose to effect larger early steps.) The drectos H(, X ) ca be determed after the departure of each flght. If the fll evet A occurs, H r 1 1 0 ad the protecto level s creased by (1 r 1 ); f ot, the H r 1 0, Maagemet Scece/Vol. 46, No. 6, Jue 2000 763
A Adaptve Algorthm for Determg Arle Seat Protecto Levels ad s reduced by r 1. Thus protecto levels are stepped up whe hgh demad s observed ad stepped dow whe low demad s observed, wth the step sze becomg smaller as the algorthm progresses. A key theoretcal ad practcal problem s determg codtos uder whch such a adaptve algorthm wll coverge to optmal protecto levels. It s kow that almost sure covergece to a pot * satsfyg h( *) 0 s guarateed f h( ) s the gradet of a cocave fucto wth a uque maxmum at * or, more geerally, f f * T h 0 for all 0. (6) * 1/ (See Beveste et al. (1990) ad Blum (1954). Loosely speakg, ths codto stpulates that the expected adjustmet vector h( ) always pots to the halfspace cotag *.) However, wth a ested allocato polcy, the expected reveue s ot jotly cocave the protecto levels. More to the pot, the fucto h( ) s ot eve the gradet of the expected reveue fucto; rather, t s a collecto of gradets from a sequece of scalar subproblems, oe for each stage the bookg process durg whch reservatos for a partcular fare class arrve. (See Brumelle ad McGll (1993).) Therefore, we caot assume that the terates satsfy a jot stablty codto lke (6). A secod dffculty s that the Algorthm (4) ca produce vectors that volate the mootocty codto 1 2,..., k requred for the evets A to be approprately ested. To address ths problem, we defe term protecto levels p, where p max j :1 j, (7) ad let p( ) ( p 1 ( ),..., p k ( )). (Note that f 1 2,..., k, the p( ).) The algorthm uses the protecto levels p ( ) to cotrol the avalablty of seats, but motors the evets A (, X) to update the parameters. Because p ( ), ths modfcato preserves our ablty to observe the evet A (, X) whe the compoets of are ot mootocally creasg, ad allows us to treat as ucostraed. We wll show the ext secto that, spte of these dffcultes, reasoable regularty codtos esure that the Procedure (4) does coverge (a.s.) to a value * satsfyg h( *) 0, ad that p( *) * (a.s.). 3. Covergece Proof To prove the covergece of the Iterato (4) we eed several prelmary lemmas. The frst provdes a suffcet codto for the almost sure covergece of a seres of radom varables: Lemma 1. (Lukacs (1975), Theorem 4.2.1) Let {Y } be a sequece of radom varables wth EY for all ad E Y. The Y (a.s.). The followg supermartgale lemma s essetal to may covergece proofs stochastc approxmato. It s due orgally to Robbs ad Segmud (1971) (see also Beveste et al. (1990), p. 344): Lemma 2. Let {, F, F, P} be a probablty space wth a creasg famly of -felds F. Suppose Z, B, C, ad D are fte, oegatve radom varables, adapted to the -feld F, whch satsfy E(Z 1 F ) (1 B ) Z C D. The o the set { B, C }, D (a.s.), ad Z 3 Z (a.s.). We shall requre that rema bouded (a.s.). The followg lemma follows easly from (4) ad the fact that the step szes are oegatve ad decreasg: Lemma 3. If demad X has bouded support for 1, 2,... (.e., X C (a.s.) for 1, 2,... ), the s bouded (a.s.). Fally, we wll eed the followg lemma, whch s adapted from Beveste et al. (1990 Lemma 23, p. 245): Lemma 4. Let A/( B), where A ad B are costats such that A 0 ad B 0, ad let u, where ad are oegatve. Let 0 ad C 0 be arbtrary costats. The there exsts a 0,, ada 0 such that the equalty u 1 1 2 u C 1 s satsfed for all 0. Proof. By substtutg u the above equalty ad rearragg, we obta 764 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels 1 C 2 1. Now, f A/( B), the by takg a Taylor seres expaso, we fd that 1 A B 1 O 2. Substtutg ths expresso to the frst term o the left-had sde above ad smplfyg we obta O 2 A 2 C. Because 0, we ca choose suffcetly large ad 0 suffcetly small so that A(2 (C/ ). The exstece of a 0 satsfyg the codtos of the lemma the follows. We are ow ready to prove our ma covergece result: Theorem 1. Let be defed as (4), ad let be defed as Lemma 4. Suppose the followg assumptos hold: A1) Each X has bouded support (.e., P(X C) 1 for some costat C). A2) There exsts a 0 such that, for all, ( * )h (, * 1,..., * 1 ) * 2. A3) The dstrbutos of the partal sums, X 1... X are Lpschtz cotuous. That s, for all 1,...,k P(X 1... X x) P(X 1... X y) C x y. The, for 1,..., k, 3 * a.s. (8) ad p * *. (9) Furthermore, there exsts a 0 such that for 1, 2,..., k, E * 2 /2 C 1. (10) Proof. The proof s by ducto o the fare classes. Frst, cosder Fare Class 1 ad ote that the evet A 1 (, X) s ot a fucto of, 1. It s easy to see that the sequece { 1 } s a classcal (scalar) Robbs Moro process ad hece coverges (a.s.). Ideed, from A2 we have that for all 0, f 1 * 1 h 1 1 0, 1 * 1 1/ ad H s uformly bouded, so E H 1 (, X) 2 C for all. These codtos together wth the fact that the ga sequece { } satsfes (5) guaratee that 1 3 * 1 (a.s.). (See Beveste et al. 1990 ad Robbs ad Segmud (1971) for proofs.) The fact that p 1 ( *) * 1 follows trvally from (7). Fally, from Beveste et al. (Theorem 22, p. 244) we have that wth A2 ad the ga sequece { } there exsts a costat such that for all 0 1, we have E 1 * 1 2 C. Thus for 1, (8) (10) are all satsfed. Now, suppose (8) (10) hold for fare class ad cosder fare class 1. Defe T 1 * 1 ad Z T 2. The by (4) Z 1 Z 2 T H 1, X 1, X,...,X 1 2 H 1, X 1, X,...,X 1 2. Takg expectatos codtoed o F yelds E Z 1 F Z 2 T h 1 1,,..., 1 2 E H 1, X 1, X,...,X 1 2. Sce H s uformly bouded, we have that E[ H 1 (, X 1, X,..., X 1 ) 2 ] C for all, hece E Z 1 F Z 2 T h 1 1,,..., 1 C 2 2 Z 2 T h 1 1, *,..., * 1 C 2 T h 1 1, *,..., * 1 h 1 1,,..., 1. (11) We ext boud the last term (11). From the defto of h, we have h 1 1, *,..., * 1 h 1 1,,..., 1 E H 1 1,,..., 1 H 1 1, *,..., * 1 E 1 X 1 X 1 1 1 A 1 A * E 1 A 1 A * P 1 A 1 A *. Now defe the evets E j m * j, j X 1 X j max * j, j, Maagemet Scece/Vol. 46, No. 6, Jue 2000 765
A Adaptve Algorthm for Determg Arle Seat Protecto Levels ad observe that f 1( A ( )) 1( A ( *)), the at least oe of the evets E j, j 1,..., must occur. Combg ths observato wth A3, we obta P 1 A 1 A * P E j j 1 Therefore, we have h 1 1 C j * j. j 1, *,..., * 1 h 1 1,,..., 1 C j * j. j 1 Substtutg ths boud (11) ad usg the fact that by A1 ad Lemma 3, T C (a.s.) we obta E Z 1 F Z 2 T h 1 1, *,..., * 1 C 2 C j * j. (12) j 1 We ext show that the last term above s fte (a.s.). Ideed, we have by A1 ad Lemma 3 that E( * ) s bouded. Also, by the ducto hypothess, (10) holds for, ad therefore E * E * 2 1 /2 C, where the last equalty follows from the defto of { } ad the fact that 1/ p coverges for all p 1. Applyg Lemma 1, we coclude that * (a.s.). Sce 2 s bouded, we ca apply A2 ad Lemma 2 to (12) ad coclude that Z 3 Z (a.s.) ad T h 1 ( 1, *,..., * 1 ) (a.s.). We ext show that Z 0. Ideed, f Z 0 the Assumpto A2 esures that there exsts a N such that T h 1 ( 1, *,..., * 1 ) 0 for all N. But ths tur would mply T h 1 ( 1, *,..., * 1 )s ubouded, whch s a cotradcto. Therefore, Z 0 ad (8) s prove for 1. We ext show (9) by cotradcto. Ideed, f (9) s ot true, the sce p ( *) * by the ducto hypothess, we must have * 1 *. Ths tur mples A 1 ( *, X) A ( *, X), whch volates (3) f f 2 f 1. Thus, we must have p 1 ( *) * 1. Fally, to show (10) holds for 1 we apply A2 to (12), whch yelds 2 E Z 1 F 1 2 Z C C j * j. j 1 Ucodtog ad usg the ducto hypothess that (10) holds for we obta, 2 E Z 1 1 2 E Z C C E j * j j 1 1 2 E Z C 2 C E j * j 2 j 1 1 2 E Z C 1 /2. Applyg Lemma 4 to the above equalty, we coclude that there exst costats, 0, ad 0 such that the sequece u /2 satsfes 1 /2 u 1 1 2 u C, for all 0. Takg suffcetly large so that /2 EZ 0 0 ad applyg ducto o we have /2 EZ. Therefore (10) also holds for 1, ad the ducto s complete. Some commets o Assumptos A2 ad A3 are order. A2 requres that the dstrbuto of demad ot be too flat the eghborhood of *. For example, oe ca show that A2 s satsfed f the dstrbutos have a desty that s bouded below by a strctly postve costat the eghborhood of *. Itutvely, A2 s eeded because f the dstrbuto s too flat ear *, the algorthm could stall before reachg *. The Lpschtz codto A3 s satsfed f the dvdual demad dstrbutos are Lpschtz smooth. For 766 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels example, A3 holds f the demad dstrbutos have uformly bouded destes. Both A2 ad A3 are ot overly restrctve. Fally, ote that (10) suggests that the covergece rate decreases wth, the dex of the fare class. That s, lower fare classes have slower covergece tha hgher fare classes. Our umercal results 5 llustrate ths behavor. 4. Practcal Modfcatos to the Basc Algorthm I ths secto, we dscuss four modfcatos to the basc algorthm (4) that address problems metoed earler. 4.1. Capacty Costrat Let c be the leg capacty or maxmum umber of bookgs allowed. The recurso (4) ca the be modfed to 1 m c, H, X, 1,...,k, (13) whch correspods to projectg 1 oto the costrat set [0, c]. Wth ths modfcato, t s ot dffcult to show that the algorthm coverges to a pot ˆ, where ˆ m{ *, c}, 1,...,. 4.2. Itegralty I practce, passeger demad ad seat allocatos are tegral so adjustmets of less tha oe seat are ot feasble. However, wthout fractoal adjustmets the algorthm could become stuck at a ooptmal pot. Oe soluto to ths dlemma s to radomze the choce of protecto levels. For llustrato, we oly cosder the case k 1 (two fare classes). Let be a cotuous parameter ad let p be the actual protecto level used. Let U be a uform [0, 1] radom varable. The at terato, we use protecto level p where p U U. (14) Ths correspods to radomzg the selecto of ad based o the value of. We the redefe the evet A 1 at terato to be A 1 (, X) {X 1 p }. Uder ths scheme, h 1 ( ) s cotuous ad, provded mld codtos o the dscrete demad dstrbuto are met, 2 t satsfes the codtos of Theorem 1. Note ths case that coverges to a * satsfyg P{X 1 * 1{U * *}} r 2, for some teger *, where * satsfes P{X 1 *} r 2 ad P{X 1 * 1} r 2. Therefore, * s the optmal teger protecto level. The polcy, however, radomzes betwee a protecto level * ad * 1, whch results some devato from optmalty. Nevertheless, the smplcty of ths radomzato scheme s attractve. 4.3. Bookg Lead Tmes May arles ope bookgs for flghts 10 moths or more pror to flght departure. However, most bookg actvty occurs over a shorter tme spa, typcally 30 to 60 days before departure. It s ths shorter, effectve bookg lead tme that s relevat to settg seat protecto levels. Stadard arle practce s to fx oe set of protecto levels at the begg of the full bookg perod ad the delay further adjustmets utl effectve bookg begs. Thereafter, readg of bookg levels ad (possble) adjustmet of protecto levels occurs wth creasg frequecy as departure tme approaches. A total of 15 readgs ad adjustmets across the full bookg perod s commo practce. These multple adjustmets are desged to accommodate the ostatoarty of demad ad to corporate recet demad data to curret bookg levels. I ths subsecto we dscuss modfcatos to the basc adaptve scheme ecessary to accommodate the effectve bookg perod. For purposes of ths dscusso, cosder a flght that departs every week ad that receves most bookgs the 10 weeks pror to each departure. A drect, but usatsfactory, mplemetato of algorthm (4) would volve dvdg the flght sequece to 10 terleaved, depedet sequeces, each of whch s updated oce every 10 weeks. I the frst of these sequeces, fll evets from the flght that departs Week 0 would be used to set protecto levels for the flght that wll be departg Week 10, 2 Namely, that the probablty mass fucto s bouded away from zero the eghborhood of the statoary pot. Maagemet Scece/Vol. 46, No. 6, Jue 2000 767
A Adaptve Algorthm for Determg Arle Seat Protecto Levels ad that departure wll be used to set levels for the Week 20 departure, ad so o. The secod through teth sequeces develop a smlar way from the Week 1 through 9 departures. Ths mplemetato s usatsfactory for at least two reasos: Frst, data (o demad, whch s assumed statoary) are ot shared across the separate flght sequeces, ad secod, the protecto levels wth each sequece are adjusted very slowly over tme (oce every 10 weeks). Fortuately, a smple modfcato of the adaptve algorthm ca be used to produce a sgle sequece of protecto levels that use formato from all flghts ad have the same covergece propertes as those determed by the orgal algorthm. To see ths, let k deote the umber of tme uts (e.g., weeks) the effectve bookg lead tme, ad assume a ew flght departs every ut of tme, where tme uts are dexed by as before. I ths case, bookgs for the frst flght beg (effectvely) at 1, the frst flght departs at k, ad the frst complete observatos of fll evets are ot avalable utl tme k 1. Flghts 1, 2,..., k must use protecto levels based o tal guesses or formato exteral to the algorthm. We assume that the tal k protecto levels are detcal, vz 1 2 k. (15) Let X be the demad for the flght, labelled, that begs bookg at tme, ad departs at tme k. Flght uses protecto levels gve by (15) for k, ad for k t uses 1 k H X k, k. (16) Ths adjustmet may seem strage a adjustmet drecto away from a old protecto vector k s beg appled to the curret vector but, t turs out that ths s smply a delayed verso of the orgal recurso (4). Ideed, summg the orgal recurso (4) we obta 1 1 H X,. 1 Smlarly, by summg the modfed recurso (16) ad usg the fact that 1 k from the tal codto (15) we obta k 1 1 H X,. 1 Thus the modfed recurso (16) s equal to k the orgal recurso (4). It therefore follows that Theorem 1 holds for the modfed sequece (16). What about updatg protecto levels durg the bookg process tself? (e.g., a flght that starts bookg Week 1 for departure Week 11 could possbly use dfferet bookg levels each week pror to departure). Ths creates dffculty because fll evets A (, X) o loger have the same terpretato. I partcular, ths case the fact that the bookgs o had for a fare class at the tme of departure are less tha the bookg lmt does ot mea that total demad was less tha the bookg lmt. Such behavor could easly cause the terato (4) to fal, but ths questo deserves further vestgato. 3 4.4. Nostatoary Demad The stochastc approxmato scheme follows a sequece of steps of decreasg sze to coverge to a optmal set of protecto levels. Strctly speakg, ths restrcts the approach to seres of flghts for whch the protecto levels are ot beleved to vary. I realty, however, optmal protecto levels drft over tme as demad ad prcg factors vary. Varatos o stochastc approxmato have bee developed to deal wth such ostatoary systems. I geeral, they track a movg set of optmal parameters wth steps that do ot ted to zero sze. See Beveste et al. (1990). We have coducted smulatos wth ostatoary demad usg a varat of stochastc approxmato that gves more weght to recet fll evet frequeces a maer smlar to expoetal smoothg. Not surprsgly, the method correctly tracks the optmal protecto levels, but wth a lag typcal of (smple) expoetal smoothg. We cojecture that ths type of trackg system wll perform reasoably well, but aga ths topc deserves further vestgato. 3 It s mportat to recogze that ths type of dyamc cesorshp s also a problem covetoal forecastg/optmzato systems. A dssertato by Lee (1990) addresses ths problem. 768 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels 5. Numercal Examples I ths secto we preset umercal examples of the performace of our adaptve algorthm the statoary demad case. For comparso, we solve the same examples wth a represetatve procedure that combes cesored forecastg wth EMSR protecto levels. We compare the covergece of the protecto levels produced by the two methods ad ther reveue performace uder varous startg codtos, load factors, ad demad dstrbutos. These comparsos are based o smulated data a dealzed statoary settg. Whle the examples are useful for llustratg some of the operatg characterstcs of the methods, they caot lead to fal coclusos about relatve merts. Such coclusos ca oly come from trals practce. 5.1. A Combed Forecastg-EMSR Scheme The combed forecastg-emsr scheme costructs a demad forecast based o a correcto to the cesored observatos of demad each fare class. For our tests, we corrected for cesorshp wth a estmate of the survvor fucto S( x) P(X x) based o lfe tables. Detals ca be foud Lawless (1982, 2.2). The lfe table estmator works as follows: Let deote the total umber of observatos (cesored ad ucesored). Let t 1 t 2..., t m be m dstct tervals. (We call [t j, t j 1 ) terval j.) Let j be the umber of observatos wth values t j or more (the umber of at rsk observatos at the start of terval j); let d j be the umber of ucesored observatos that fall terval j (the umber of deaths the terval j); ad let w j be the umber of cesored observatos that fall terval j (the umber of wthdrawals because of cesorg terval j). Defe 0 ad ote that j j 1 d j w j, j 1,..., m. The the stadard lfe table estmate s gve by Ŝ t j t j 1 j 2 1 1 d w /2, j 1,...,m. (Note the approxmato of S(t) s take at the mdpot of terval j.) The dea here s that each term 1 (d / w / 2) s a estmate of the codtoal probablty that demad exceeds t 1 gve that t exceeded t. The deomator, w /2, s a estmate of the umber of samples at rsk durg perod, where w / 2 s a correcto term for the umber of cesored observatos perod (e.g., a cesored observato durg perod s assumed to be at rsk for half the perod o average). I our mplemetato, we mataed 20 tervals (m 21 for each fare class, wth 0 t 1 t 2... t m t m 1, chose so that P (X [t j, t j 1 )) 0.05 for all j 1,..., m. Whle more tervals clearly result a more accurate estmate of the survvor fucto, 20 provded adequate accuracy our case, especally as the values t 1,..., t m were chose to match each dstrbuto. 4 We the used the lfe table estmator to estmate the mea ad stadard devato of the dstrbuto by lear regresso. Specfcally, let ( x) be the stadard ormal dstrbuto ad let 1 ( x) deote ts verse. Defe s j 1 (1 Ŝ((t j t j 1 )/2)). If demad s ormally dstrbuted, the pots (s j, t j ) j 1,..., should le approxmately o a straght le, amely s j at j b. Usg lear regresso, we estmated the slope, â, ad the tercept, bˆ, ad the costructed estmates of the mea, ˆ 1/â, ad stadard devato, ˆ bˆ/â. Ths procedure results potetally based estmates (See Lawless 1982, 2.5.), but t s smple to mplemet ad seemed to perform well our tests. We updated the lfe table ad lear regresso estmates of the mea ad stadard devato after each smulated flght departure. Ths procedure may ot always produce a ubouded estmate of the mea ad stadard devato. I partcular, f all samples are cesored, the the lfe table estmate s 1 for all values of j ad the lear regresso produces a estmate of â 0, whch results a ubouded estmate of the mea ad stadard devato. Ths s qute ormal behavor for cesored demad estmators. Ideed, f all observatos are cesored, the ay reasoable estmator (e.g., 4 Of course, practce oe would ot be able to fe-tue the tervals of tme t 1,...,t m so precsely, sce the demad dstrbuto s ukow. Thus, a wder rage wth more tervals would be requred to esure that the data were adequately covered. Maagemet Scece/Vol. 46, No. 6, Jue 2000 769
A Adaptve Algorthm for Determg Arle Seat Protecto Levels maxmum lkelhood) ether wll be ucomputable or produce a ubouded mea ad/or stadard devato. I such cases, we gored the forecast ad smply mataed the curret protecto levels utl the forecast produced bouded estmates. For settg seat protecto levels, we used a varato of the expected margal seat reveue (EMSR) heurstc (Belobaba 1989), called EMSR-b. Ths s the most commo seat protecto heurstc used practce. EMSR-b works as follows: Gve estmates of the meas, ˆ, ad stadard devatos, ˆ, for each fare class, the EMSR-b heurstc sets protecto level so that f 1 f P(X ), where X s a ormal radom varable wth mea j 1 ˆ j ad varace j 1 ˆ j2, ad f s a weghted average reveue, gve by f j ˆ j j 1 f. ˆ j j 1 The dea behd ths approxmato s to reduce the complexty of the fully ested problem by aggregatg fare classes 1, 2,..., to a sgle-fare class. The, oe treats the problem as a smple, 2-fare-class problem. Aga, we emphasze that ths overall forecastg- EMSR scheme s ot costructed to be the most sophstcated oe possble. Rather, t s teded be represetatve of a basc yeld maagemet system. Both methods could o doubt be refed further. However, such refemets ofte volve dosycratc ad/or ad hoc modfcatos that oly serve to make performace comparsos more complex, cotroversal, ad ultmately less sghtful. Our teto, therefore, s to perform a trasparet test of oe smple approach (the adaptve algorthm) agast aother smple approach (basc cesored forecastg ad EMSR-b). 5.2. Test Problem Scearos Our test problem s a modfcato of Wollmer s (1992) 5-fare-class example, whch we aggregated hs Classes 3 ad 4 to reduce the problem to 4 classes. (Classes 3 ad 4 Wollmer had very smlar fares of $534 ad $520, respectvely.) Wth 4-fare-classes, there are 3 protecto levels to determe. Table 1 Fares, Demad Statstcs, ad Protecto Levels for Numercal Examples Class Fare Mea Std. Dev. -EMSR *-Normal *-Log N 1 $1,050 17.3 5.8 16.7 16.7 15.9 2 $567 45.1 15.0 51.5 44.6 45.7 3 $527 73.6 17.4 131.4 134.0 130.0 4 $350 19.8 6.6.a..a..a. Table 2 Startg Values of Protecto Levels for Numercal Examples 1 2 3 Low 0 15 65 Hgh 35 110 210 The data alog wth optmal ad EMSR-b protecto levels are show Table 1. The protecto level *-Normal s the optmal level whe demad s ormally dstrbuted, whle *-Log N s the optmal protecto level whe demad s log-ormally dstrbuted. To test the covergece of the adaptve algorthm ad the forecastg-optmzato scheme, we purposely started wth protecto levels that were far from optmal, correspodg to hgh ad low startg values (see Table 2). These somewhat extreme values were chose to test the covergece propertes of each algorthm. I practce, oe may have some pror kowledge about demad that ca be used to set better tal protecto levels. At the same tme, usg extreme startg values provdes a good robustess test. Hgh startg values produce less tal cesorg the hgher fare classes, so oe has better observatos of the actual demad dstrbutos. However, reveues may be low due to hgh levels of rejected demad. Low startg values produce severe tal cesorg, whch may adversely affect forecast accuracy. I terms of reveue, low startg values are geerally better f demad s low, but may produce poor reveue performace whe demad s hgh because suffcet capacty s reserved for hgher fare classes. Observg how the algorthms react to these varous factors provdes useful sghts. We also used two demad scearos. The hgh 770 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels demad scearo has a startg vetory of 124 seats, correspodg to a 125% demad factor (rato of expected total demad to capacty) ad approxmately a 95% load factor (rato of average umber of seats sold to capacty) uder optmal protecto levels. The low demad scearo starts wth 164 seats, resultg a demad factor of 95% ad a load factor of approxmately 90% uder optmal protecto levels. (Both scearos are set wth somewhat hgher demad factors tha ormally ecoutered practce to hghlght the reveue mpacts of the dfferet methods.) For each scearo, we ra the two methods parallel o a same sample path of 100 radom departures. For the stochastc approxmato procedure, we used the ga sequece 200/(10 ), whch appeared to provde good performace o a rage of examples. O the same sample path, we also ra the optmal polcy; that s, a polcy that apples the fxed protecto level * o each realzato. Ths provded a bechmark for reveue performace. For smplcty, we also assumed observed demad was cesored by the protecto levels but ot the capacty costrat. Fgure 1 Low Demad, Low Start, Normal Dstrbuto 5.3. Numercal Results The frst set of smulatos used ormally geerated demad wth parameters gve Table 1. Thus, the actual demad dstrbuto s cosstet wth the assumptos made the forecastg ad optmzato procedures. Each method was tested agast the same sample path (.e., the smulatos are coupled). We performed 64 smulatos of each 100-flght sample path for each case ad tracked the protecto levels ad reveue performace over tme. Fgure 1 shows three graphs of the data for the frst case of low demad ad low startg values. To geerate these graphs, we sampled reveue ad protecto level data at every teth terato durg the progresso of the algorthms (e.g., at values 10, 20,... ). Ths samplg was performed for each of the 64 sample paths, ad summary statstcs were the computed to llustrate the typcal evoluto of reveues ad protecto levels over tme. The top graph of Fgure 1 shows the average cumulatve reveue as a percetage of the optmal reveue for the two methods as a fucto of the umber of teratos (flghts). The error bars show the 95% cofdece tervals about these averages. The mddle graph shows the average protecto levels over tme for the stochastc approxmato (SA) procedure. The horzotal dotted les are the optmal protecto levels. The lowest le correspods to * 1, the mddle le to * 2, ad the top le to * 3. The sold les are the correspodg average protecto levels produced by the stochastc approxmato (SA) method. The error bars o the sold les gve the 25th percetle ad 75th percetle values for each protecto level at each terato, whch provdes some sese of the varablty protecto levels across sample paths. The bottom graph shows the detcal plot of protecto levels for the F/EMSR method. Maagemet Scece/Vol. 46, No. 6, Jue 2000 771
A Adaptve Algorthm for Determg Arle Seat Protecto Levels Note from Fgure 1 that, ths frst case, both procedures have early detcal (ad very close to optmal) cumulatve reveue performace, although SA s slghtly better early o. 5 Note that the F/EMSR procedure quckly reaches stable protecto levels; however, the secod ad thrd protecto levels devate from the optmal oes. Ths s cosstet wth the kow ooptmalty of EMSR levels beyod the frst level (see Table 1). The SA procedure takes loger to coverge. I partcular, the thrd protecto level (the top le the graph) s the slowest to coverge. Ths behavor s cosstet wth the bouds o covergece rate developed Theorem 1. Also, whe the protecto levels coverge, there s mmal devato from optmalty, whch s cosstet wth the theoretcal results as well. Fgure 2 shows the results for the same low demad factor, but wth startg protecto levels that are all hgher tha the optmal levels (see Table 2). As Fgure 1, the F/EMSR procedure coverges more quckly tha the SA procedure. However, ths case the faster covergece of the F/EMSR has a more sgfcat mpact o the cumulatve reveue performace: F/EMSR geerates about 2% 3% hgher reveue o average the early teratos. Note also that the absolute reveue performace of both procedures s cosderably worse ths case compared wth the prevous case of low tal protecto levels, especally the early teratos. Wth low demad, overprotectg seats s worse tha uderprotectg them, ad thus errg o the sde of low tal startg protecto levels s preferred. The results are qute dfferet the hgh-demad factor case. Fgure 3 shows the smulato results for low startg protecto values ad hgh load factor. Note as dcated by the error bars the bottom graphs that the F/EMSR procedure s very volatle ad somewhat slow to coverge the early teratos. The reveue effect of ths behavor s qute sgfcat, wth F/EMSR geeratg cumulatve reveues 5 The fact that F/EMSR has lower cumulatve reveues at 10 despte the fact that the protecto levels look close to optmal s due to the behavor of ts protecto levels teratos 1 to 10, whch are ot show Fgure 1. I partcular, F/EMSR requres several teratos to produce a fte forecast. Fgure 2 Low Demad, Hgh Start, Normal Dstrbuto roughly 8% lower tha optmal ad 2% 3% lower tha SA the early teratos. However, the performace ad protecto levels of F/EMSR mprove after about 30 teratos. I cotrast, the SA procedure s cosderably more stable ad t coverges faster the early teratos, whch accouts for ts superor reveue performace. F/EMSR performs badly ths case because the forecastg procedure suffers from the frequet cesorg caused by a combato of low protecto levels ad hgh demad. As metoed above, whe all observatos are cesored the forecastg procedure produces ubouded estmates of the demad meas ad stadard devatos. The F/EMSR proce- 772 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels Fgure 3 Hgh Demad, Low Start, Normal Dstrbuto the rght drecto, thereby reducg the amout of cesorg. Whe the startg protecto levels are hgh ad the demad factor s hgh, the stuato s reversed; F/EMSR performs better tha SA as show Fgure 4, though, compared wth Fgure 3, the absolute performace of both methods s worse overall tha the prevous case (.e., overprotectg s worse tha uderprotectg). I ths case wth hgh protecto levels, there s lttle cesorg of the hgher fare classes ad the forecastg procedure quckly produces good estmates of the meas ad stadard devatos. Thus, by Iterato 10 the F/EMSR s able to acheve ear Fgure 4 Hgh Demad, Hgh Start, Normal Dstrbuto dure therefore operates wth the tal low protecto levels for may teratos utl a few ucesored observatos of demad are obtaed ad bouded estmates ca be computed. Ths behavor suggests a potetal applcato of the SA procedure. Namely, SA may prove useful as a meas of automatcally adjustg protecto levels the early lfe of ew flghts whe very lttle demad formato s avalable ad forecastg s dffcult due to a hgh degree of cesorg. I such cases, the SA method ot oly provdes a robust way to adjust protecto levels, but t also serves to speed up the forecastg method tself by udgg protecto levels Maagemet Scece/Vol. 46, No. 6, Jue 2000 773
A Adaptve Algorthm for Determg Arle Seat Protecto Levels optmal protecto levels. The SA procedure, cotrast, suffers from slower relatve covergece. I summary, the SA procedure appears to have some dstct advatages cases where there s a hgh degree of tal cesorg. However, cases where cesorg s less of a problem, the slower covergece of the SA method s a weakess ad causes t to uderperform F/EMSR. These fdgs are certaly tutve ad gve some sese of the relatve stregths ad weakesses of each approach. We also tested these same four cases wth radom demad draw from a log-ormal dstrbuto wth the same mea ad varace. Because the adaptve algorthm requres o dstrbutoal assumpto, our hypothess was that SA would perform relatvely better ths case. However, the actual smulato results were early detcal to those for the ormal demad case, wth the excepto that the F/EMSR method had slghtly greater devatos from the optmal thresholds. The devatos, however, were ot large eough to sgfcatly affect the overall reveue performace. Of course, t s possble that a dstrbuto that s more extreme tha the log-ormal (e.g., a bmodal dstrbuto) mght troduce sgfcat errors a forecastg ad optmzato procedure that assumes ormalty. 6. Coclusos Adaptve procedures for yeld maagemet are attractve because they are smple ad robust; however, our prelmary umercal studes dcate that the method has mxed performace uderperformg tradtoal forecastg ad optmzato methods whe demad s ot hghly cesored but outperformg tradtoal methods whe demad s heavly cesored. Ths behavor suggests that, for arles wth exstg reveue maagemet systems, adaptve algorthms may be most useful ot to replace, but to augmet, tradtoal forecastg ad optmzato approaches. Thus, for example, a adaptve approach could automate short-ru updatg of protecto levels cases where forecasts are hghly urelable or dramatc chages the market are takg place. Adaptve methods may also be approprate for small or start-up arles that lack the resources requred to develop ad mata a full reveue maagemet system. The adaptve algorthm ca also be used as a smple, smulato-based method for computg optmal protecto levels wth the optmzato stage of a tradtoal forecastg ad optmzato system. Ths approach s qute smlar to Robso s (1991) Mote Carlo method for determg optmal protecto levels, but offers the potetal of greater data effcecy (though most lkely slower covergece). Whle our paper shows how to costruct a adaptve algorthm ad provdes theoretcal guaratees o log-ru performace, more research s eeded. For example, we have ot tested the performace of ths approach wth large umbers of fare classes ad have ot geeralzed the method for ostatoary demad or protecto levels that are modfed durg the bookg process. Moreover, t may be that the shortru, traset performace of the method s more relevat practce tha ts covergece ad log-ru performace. Fally, we beleve that our approach may prove useful for studyg the process of forecastg ad optmzato over tme. Ideed, oe ca vew forecastg ad optmzato as methods of geeratg drectos ad step szes for updatg protecto levels. Stochastc approxmato theory may prove a useful theoretcal framework for studyg the covergece propertes of a wde class of forecastg ad optmzato methods. 6 6 Research supported part by the Natural Sceces ad Egeerg Research Coucl of Caada NSERC OGP0138093. Refereces Belobaba, P. P. 1987a. Ar travel demad ad arle seat vetory maagemet. Upublshed Ph.D. Dssertato. MIT, Cambrdge, MA.. 1987b. Arle yeld maagemet: A overvew of seat vetory cotrol. Tras. Sc. 21 63 73.. 1989. Applcato of a probablstc decso model to arle seat vetory cotrol. Oper. Res. 37 183 197. Beveste, A., M. Metver, P. Prouret. 1990. Adaptve Algorthms ad Stochastc Approxmato. Sprger-Verlag, Berl, Germay. Blum, J. 1954. Multvarate stochastc approxmato methods. A. Math. Statst. 25 737 744. Brumelle, S. L., J. I. McGll. 1993. Arle seat allocato wth multple ested fare classes. Oper. Res. 41 127 137. 774 Maagemet Scece/Vol. 46, No. 6, Jue 2000
A Adaptve Algorthm for Determg Arle Seat Protecto Levels,, T. H. Oum, K. Sawak, M. W. Tretheway. 1990. Allocato of arle seats betwee stochastcally depedet demad. Tras. Sc. 24 183 192. Curry, R. E. 1989. Optmal arle seat allocato wth fare classes ested by orgs ad destatos. Tras. Sc. 24 193 204. Etschmaer, M. M., M. Rothste. 1974. Operatos research the maagemet of the arles. Omega 2 160 175. Lawless, J. F. 1982. Statstcal Models ad Methods for Lfetme Data. Joh Wley ad Sos, New York. Lee, A. O. 1990. Arle reservatos forecastg: Probablstc ad statstcal models of the bookg process. Upublshed Ph.D. Dssertato, MIT, Cambrdge, MA. Lttlewood, K. 1972. Forecastg ad cotrol of passegers. Proc. 12th AGIFORS Sympos., Amerca Arles, New York, 95 117. Lukacs, E. 1975. Stochastc Covergece. Academc Press, New York. McGll, J. I., G. J. va Ryz. 1999. Reveue maagemet: Research overvew ad prospects. Tras. Sc. 33 233 256. Robbs, H., S. Moro. 1951. A stochastc approxmato method. A. Math. Statst. 22 400 407., D. Segmud. 1971. A covergece theorem for o-egatve almost supermartgales ad some applcatos. J. Rustag, ed. Optmzg Methods Statstcs. Academc Press, New York, 235 257. Robso, L. W. 1991. Optmal ad approxmate cotrol polces for arle bookg wth sequetal o-mootoc fare classes. Oper. Res. 43 252 263. Smth, B. C., J. F. Lemkuhler, R. M. Darrow. 1992. Yeld maagemet at Amerca Arles. Iterfaces 22 8 31. Weatherford, L. R., S. E. Bodly. 1992. A taxoomy ad research overvew of pershable-asset reveue maagemet: Yeld maagemet, overbookg ad prcg. Oper. Res. 40 831 844. Wollmer, R. D. 1992. A arle seat maagemet model for a sgle leg route whe lower fare classes book frst. Oper. Res. 40 26 37. Accepted by Lda V. Gree; receved October 1997. Ths paper was wth the authors 9 1 2 moths for 2 revsos. Maagemet Scece/Vol. 46, No. 6, Jue 2000 775