COMPARISON OF AIR RAVE DEMAND FORECASING MEHODS Ružica Škurla Babić, M.Sc. Ivan Grgurević, B.Eng. Universiy of Zagreb Faculy of ranspor and raffic Sciences Vukelićeva 4, HR- Zagreb, Croaia skurla@fpz.hr, ivan.grgurevic@fpz.hr Zvonimir Majić, B.Eng. Pliva Croaia d. Prilaz baruna Filipovića 25, HR- Zagreb, Croaia zvonimir.majic@pliva.com ABSRAC Accurae forecass of fuure passenger demand are essenial o effecive revenue managemen sysem. he sea invenory conrol leans on predicions abou he bookings o come o opimally allocae aircraf seas among he various booking classes. Forecasing for airline revenue managemen sysems is inherenly difficul because of complex naure of air ravel demand which is highly sochasic. he problem is furher complicaed because of usually grea number of origin desinaion pairs, each wih is own seasonal and weekly effecs, he economic environmen and exernal facors like compeiion or special evens. he paper describes general problem of forecasing airline demand and compares radiional mehods of forecasing (moving averages, exponenial smoohing, ec.) agains neural neworks as a forecasing mehod. All he mehods are compared on he basis of sandard error measures. INRODUCION Airlines forecas air ravel demand in order o harmonize he complex se of heir aciviies ha will adequaely mach supply on he air ranspor marke. For some business funcions, he decisions are made based on he long-erm raffic forecass and hen we speak abou sraegic decision-making. his, firs of all, includes flee planning, planning and evaluaion of he fligh neworks and invesmen aciviies. acical and operaive decision-making is assised by he mid- and shor-erm forecass ha are developed for he periods of six monhs o somewha more han one year, leaning on differen forecasing mehodologies and differen levels of demand aggregaion. Precise forecasing of he air raffic demand is especially imporan for efficien funcioning of he airline revenue managemen sysems ha conrol he availabiliy of ravel seas in differen booking classes wih he goal of maximizing expeced revenues. he demand forecasing module wihin he airline revenue managemen sysem generaes he inpu daa for he opimizaion module, i.e. daa on he expeced demand a he level of he price class of each fligh. he esimaes in pracice sugges ha he reducion of forecasing error by weny percen resuls in he increase of he overall revenue on fligh by percen []. he airline revenue managemen mainly relies on he hisorical booking daa of similar flighs in order o esimae fuure demand. he majoriy of quaniaive mehods ha are described furher in he ex can be described as sandard, i.e. mehods ha are used for forecasing in general. On he oher hand, pick-up forecasing mehods are used exclusively in airline revenue managemen sysems. Insead of averaging he hisorical booking daa, hey calculae he increase, i.e. bookings
incremen in ime inervals beween review poins of he booking process. he increase for he fuure periods is added o he number of on-hand bookings in order o esimae he number of bookings a a cerain momen in he fuure. In his conex one should also menion he forecasing using he mehod of linear regression which brings ino connecion he number of confirmed bookings for a cerain fligh in he ime inervals ha precede he fligh dae and he final number of bookings for ha fligh. 2 IME SERIES FORECASING MEHODS Hisorical daa abou differen phenomena and in differen research areas are usually colleced and analyzed in he form of ime series wih he aim of describing he phenomenon ha is being moniored, explaining is variaions and predicing is movemen in he fuure. he mehods of ime series decomposiion, mehods of moving averages and various smoohing mehods belong o he mehods of ime series analysis wih which i is possible o forecas he level of phenomenon whereas various auo-regression and he respecive models measure he level of saisical relaion beween he members of he series, and are applied for he descripion of phenomena ha do no conain sysemaic componens. he saisical analysis of he movemen of he level of a cerain phenomenon over ime sars from he classical analysis of he ime series ino componens:, rend componen expresses he basic long-erm endency of phenomenon developmen in ime; C, cyclical componen expresses periodical repeaing of cerain values every wo and more years; S, seasonal componen expresses flucuaions around he rend ha are repeaed in he similar way in he period which equals one year or less; e, random, irregular or residual componen expresses non-sysemic influences on he phenomenon developmen, and i remains afer removal of he sysemic componens (rend, seasonal and cyclical) [2]. 2. Defining he sample for mehod comparison Furher in he ex, a sample of 96 values (able ) ha represen he monhly recorded demand (in housands) in a single airline during a ime period of eigh years, will be used o show he resuls of he simple ime series forecasing mehods on he analysis. Alhough he demand forecasing module of airline revenue managemen uses daa on he number of booking requess a micro level (exac fligh and booking class), sill he sample defined in such a way, is suiable for elaboraion of several mehods of analysis and demand forecasing since i conains he rend and seasonal componens. he resuls have been calculaed by using he sofware package Zaiun ime Series (V.2..). able : Sample saisical characerisics characerisic value number of se elemens 96 minimal value 86 maximal value 38 range 294 mean value 87,28 median 74 firs quarile 29 hird quarile 236,5 sandard deviaion 7,32339 2
he mehods will be compared by he mos commonly used forecasing errors: Mean Absolue Error - MAE MAE y ŷ, () Mean Square Error - MSE 2 MSE y ŷ, (2) where is he number of daa ha is used in he esimae, y observed value of he series a momen, and forecased value of he series a momen. ŷ 2.2 Decomposiion mehods Mehods of ime series decomposiion base he forecasing on he separaion of he basic componens from he ime series. For each componen he forecasing of he fuure values is performed by forward exrapolaion, and hen by combining he separae forecass he overall forecas is obained. In applying he addiive model i is assumed ha he seasonal and irregular componen are independen of he rend, ha he ampliude of seasonal variaions does no change over ime and ha he annual average of seasonal flucuaions equals zero. he general form of addiive model is:. (3) he muliplicaive model of ime series decomposiion relies on he assumpions ha he seasonal componen ampliude is direcly proporional o he rend level and ha he irregular componen variance is direcly proporional o he value of sysemic componens. he general model of he muliplicaive model muliplies he rend componen wih he coefficiens of seasonal, cyclic and residual coefficien and has he form: Z I I I, (4) Z C S C S e e For he sample defined in he previous secion, he analysis of he ime series elemens adapaion expecedly indicaes he presence of he linear rend whose equaion is:, (5) Z 2. 394 x 74. 79 and R-square value is.8267. Graph shows he linear rend and he forecased demand values resuling from he inroducion of he values for x for he nex 2 monhs in he rend equaion. R-square value or deerminaion coefficien is a number beween and which indicaes how well he esimaed linear rend values correspond o acual daa. he linear rend is mos reliable when is R-square value is exacly or near. 3
passengers carried (in housands) 4 3 2 y = 2.394x + 74.79 R 2 =.826 2 24 36 48 6 72 84 96 acual values linear rend forecased values ime Graph : inear rend and forecased values for 2 monhs he firs sep in he ime series decomposiion is he removal of he rend componen. he forecas values resuling from he decomposiion models can be seen in Graph 2 and he calculaed seasonaliy coefficiens are presened in able 2. passengers carried (in housands) 45 3 5 2 24 36 48 6 72 84 96 ime acual values forecased values (muliplicaive model) forecased values (addiive model) Graph 2: Acual and forecased values of decomposiion models able 2: Seasonaliy coefficiens of muliplicaive and addiive model of decomposiion 2 3 4 5 6 7 8 9 2 Is,95,876,25,97,984,28,235,24,53,97,788,888 s -7,7-22,45 5,34-4,78-3,62 23,88 33,67 44,,63-4,2-37,45-7,33 2.3 Smoohing mehods he smoohing echniques are used for shor-erm forecasing, in he series wih sligh variaions. Random or unpredicable influences of ime series are smoohed and he las smoohed value is aken as he forecas value for he fuure periods. 4
2.3. Moving average mehods In saisics, he moving averages represen a series of daa ha have been calculaed as simple or weighed averages of subses of he basic se of daa. he mehod of simple moving average is he simples and easily applicable smoohing echnique. he precise ime series values for a cerain period are subsiued by he average of he respecive value and several adjacen values (M values). he moving average mehod will reac fas o major changes in he demand if M is small. On he oher hand, small M resuls in esimaes ha are excessively sensiive o shorerm random deviaions of he values. In pracice M ranges beween 2 and 5, and he very selecion depends on he characerisics of he available daa, lengh of he ime inervals, and smoohing objecive. Graph 3 shows he moving averages for he defined ime series sample. I is obvious ha for a ime series values wih rend and seasonaliy, his mehod fails o be suiable. passengers carried (in housands) 4 35 3 25 2 5 3,3 33,8 288 274 5 2 24 36 48 6 72 84 96 ime acual values moving averages (M=2) moving averages (M=5) moving averages (M=) 2.3.2 Exponenial smoohing moving averages (M=5) Graph 3: Forecasing by means of moving average mehod he exponenial smoohing mehods belong o he mosly widespread demand forecasing mehods in capaciy managemen sysems hanks o heir simpliciy, robusness, and precision. Simple exponenial smoohing (SES) is he simples exponenial smoohing mehod, defined by he smoohing consan α which mus be beween and. he forecas value for period + is calculaed as he weighed average of he acual and forecas ime Ẑ series value in he previous ime period in which he acual value z is assigned he weigh, and he forecas value Ẑ z Ẑ is assigned he weigh, i.e.: Ẑ. (6) he k-period ahead forecas is given by: Ẑ k Ẑ k,...,k. (7) he recursive formula (6) can be wrien in he following way: Ẑ j j z. j (8) 5
Graph 4 shows he forecas values of he previously defined ime series by he Simple exponenial smoohing mehod. Using he sofware package Zaiun ime Series (V.2..), and he leas square mehod, i was calculaed ha he leas forecas error is given by he value α =.9. passengers carried (in housands) 4 3 2 286,32 283,93 276,55 2 24 36 48 6 72 84 96 ime acual values SES, α=,9 SES, α=,5 SES, α=, Graph 4: Simple exponenial smoohing inear Exponenial Smoohing (ES), known as Hol's Mehod is used o smooh daa ha conain he linear rend. If we use < α < and < β < o denoe he smoohing parameers for and, hen he forecas for inerval + is given by he following formulas: Ẑ, z,. While in simple exponenial smoohing he forecas value is simply equal o he las value of, in his case he recursive expression is given by: Ẑ k k, k,,k (9). () Graph 5 shows he forecas values of he previously defined ime series by he exponenial smoohing mehod wih linear rend (ES). he values for α and β have been calculaed wih Zaiun ime Series (V.2..) sofware, using he leas square mehod. passengers carried (in housands) 5 4 3 2 2 24 36 48 6 72 84 96 ime acual values ES, α=,9, β=, Graph 5: Exponenial smoohing wih linear rend 6
Exponenial smoohing mehod wih rend and seasonaliy (Hol-Winer s mehod-hw) is applicable in case when a series of daa apar from rend conain also he seasonal componen. e < α <, < β < and < γ < be smoohing parameers for, and S. Furhermore, we denoe wih he duraion of he season in monhs, e.g. in case of monhly variaions, =2, in case of half-a-year variaions =6. Depending on he characerisics of he ime series, he mehod is available in wo versions: muliplicaive and addiive. In muliplicaive version, he forecas for inerval + k has been se by he following expression [3]:, () Ẑ k ( k )S k, k,,k where he hree componens of forecas values are: z S S z S. For he addiive version he following expressions hold: Ẑ A k S, k,,k k k (2) (3) where he hree forecas value componens are [3]: A S z S A A A z A S. Graph 6 shows he forecas values of he previously defined ime series using he Hol- Winer s mehod, muliplicaive and addiive version. he values α, β and γ have been seleced by he leas square mehod using sofware package Zaiun ime Series (V.2..). (4) passengers carried (in housands) 5 4 3 2 2 24 36 48 6 72 84 96 ime acual values HW muliplicaive model HW addiive model Graph 6: Hol-Winer s mehod muliplicaive and addiive versions 7
For he calculaion of he iniial values, and S i is necessary o have available values z, z 2,, z, ha is, daa for a leas one year, and for he iniial rend value one should know also he values of z + o z 2, i.e. daa for he second year. Iniial values can be calculaed in he following way [3]: aken: - for he average of he firs year is aken: z - for he average of he difference in he averages of he firs and second year is z 2 z (5) (6) - he seasonaliy facor is calculaed for k=, 2,, for muliplicaive version: S k 2 k for addiive version: S z k k z k k 2 (7) (8) 2.4 Neural nework forecasing he neural neworks consis of wo or more layers or groups of processing elemens called neurons. he nework processing capabiliy is he consequence of he connecions among hese unis, and i is achieved hrough he adapaion process or by learning from he se of learning examples. Neurons are conneced ino a nework so ha he oupu of every neuron is he inpu ino one or several oher neurons. he neurons are grouped ino layers. hree basic ypes of layers are he inpu, hidden and oupu ones. Sandard error back-propagaion algorihm includes opimisaion of he error using he deerminisic algorihm of he gradien descen. I calculaes parial derivaions of he qualiy crierion according o nework parameers using recursive procedure which is performed reversely hrough he nework from he oupu o he inpu nework layer. he algorihm is based on he assumpion ha he error derivaion propagaion hrough he nework is linear [4]. he resuls of applying he backpropagaion mulilayer feedfoward neural nework on he previously defined sample using he sofware Zaiun ime Series are presened in able 3 and Graph 7. able 3: Summary of he applied neural nework model value Included observaions 96 Inpu ayer Neurons 2 Hidden ayer Neurons 2 Oupu ayer Neurons Oupu ayer Neurons Sigmoid Funcion earning Rae,5 Momenum,5 Ieraions 8
passengers carried (in housands) 4 3 2 2 24 36 48 6 72 84 96 8 ime acual values approximaed values forecased values Graph 7: Examples of demand forecasing by neural nework wih backpropagaion algorihm 2.5 Comparison of he accuracy of mehods able 4 shows he forecasing error values MAE and MSE for he performed forecasing mehods on he defined sample. I can be seen from he values of he forecasing errors for he mehod of muliplicaive and addiive ime series decomposiion ha he muliplicaive model of decomposiion approximaes more precisely he given ime series han he addiive model. he moving average mehod and he simple exponenial smoohing mehod are no suiable for forecasing he ime series values wih rend and seasonaliy, and he bes resul for such a defined ime series sample is obained by he Hol-Winer s exponenial smoohing mehod, paricularly he muliplicaive version. he obained resuls using he neural nework are comparaive o he muliplicaive model of rend decomposiion, and hey are worse han he resuls obained by Hol-Winer s mehod. able 4: Forecasing errors of he model for a defined ime series MAE MSE decomposiion - - muliplicaive model.6 58. decomposiion - addiive model 2.45 266.59 Moving averages, M=2 22.64 82.67 Moving averages, M=5 28.62 295.49 Moving averages, M= 27.56 389.4 Moving averages, M=5 27.48 2.9 SES, α =,9 8.74 56.4 SES, α =,5 2.46 76.54 SES, α =, 26.79 355.37 ES 9.5 69.59 HW muliplicaive version 7.65 97.56 HW addiive version 9.44 4.3 neural nework.4 79.57 9
3 APPICABIIY OF ARIFICIA NEURA NEWORKS FOR DEMAND FORECASING IN AIRINE REVENUE MANAGEMEN SYSEMS he professional lieraure provides muliple presenaions of using he neural neworks for ime series forecasing wih he resuls ha jusify furher research and developmen of new algorihms. A minor number of sudies has deal wih forecasing by means of neural neworks on ime series wih pronounced seasonaliy, which are relevan wihin he conex of he airline revenue managemen sysems. he resuls of hese sudies lack uniformiy. Whereas some auhors advocae he applicaion of neural neworks on he daa wihou prior de-seasonalisaion, he ohers advocae precisely he opposie [5]. he characerisics of neural neworks ha conribue o heir slow implemenaion in general, including hen he limied use of neural neworks in he forecasing models wihin he aircraf sea invenory conrol sysem are: neural neworks are compuaionally very demanding, oupu of every neuron being he resul of adding several producs and calculaing he non-linear acivaion funcion; neural neworks require large memory space since each neuron has several synapic connecions, whose weigh coefficien has o be sored in he memory; he neural nework memory requiremens grow wih he square of he neuron number; he compuaional speed of he neural nework is deermined by he number of mahemaical operaions of a single neuron, raher han he complee nework since every nework layer has parallel srucure, and every neuron in a layer may be observed as a local processor ha works parallel wih oher neurons [6]. Parallel o he sudies of he srucures of neural neworks and he models of synapic connecions, as well as he developmen of he learning algorihm, he mehods of heir implemenaion ha ensure opimal usage of he neural nework properies have also been sudied. he majoriy of applied neural neworks has been implemened on convenional compuer sysems ha had no been designed exclusively for he implemenaion of neural neworks, for which more adequae soluions are hose ha use parallel srucure of he neural neworks. he real usage of all he good properies of he neural neworks can be expeced only when good hardware is available, specialized for heir implemenaion, so ha he core of he research aciviies in his area is focused on he developmen of specialized elecronic and opical, i.e. opoelecronic implemenaions. Elecronic implemenaions of neural neworks are based on he bus-oriened processes, coprocessors, CCDs (Charge Coupled Device echnology) and VSI (Very arge Scale Inegraed) assemblies, and opical/opoelecronic implemenaions on opical or combined opical and elecronic componens [6]. In heir paper which deals precisely wih forecasing of ranspor demand based on he daa abou he realized ranspor of one airline and which is characerised by he so-called muliplicaive seasonaliy, J. Faraway and C. Chafield emphasise as especially imporan he selecion of: adequae se of inpu variables and weighs; adequae nework archiecure; adequae acivaion funcions ha are no o be equal in he hidden and he oupu layer; adequae numerical procedure for he neural nework model calibraion [7]. Zhang and Kline carried ou a comprehensive research and analyzed 48 models of neural neworks on a large se of daa (756) wih seasonal variaions and concluded ha as a
rule he simple models surpass he complex models and ha he efficiency is mos improved by previous cleaning of daa from he rend and season componens [8]. his paper has presened he applicaion of neural neworks on ime series daa wih rend and, seasonaliy, and he obained resuls indicae good forecas accuracy. 4 CONCUSIONS he dominan demand forecas values for air ravel demand are simple ime series models such as he moving average or he exponenial average smoohing, wih which he known values, i.e. hisorical daa are averaged and adaped o he known seasonal impacs. he sophisicaed ime series models, such as e.g. auoregressive moving average isolae among he hisorical daa he rends and forms of demand flucuaion and exrapolae from hem he marke rend forecass. More complex models, such as e.g. muliple regression, Kalman s filers and neural neworks are seldom used since he specific dynamic of every OD pair and he necessiy for he definiion of he exac ineracion beween he variables on every marke makes he consrucion and upgrading of such models a long-erm one. he neural neworks, alhough being very sophisicaed ools for ime series forecasing wih seasonaliy, leave a lo of room for errors as e.g. non-convergence, convergence ino local minimum or may even resul in unreasonable forecass. By adding hidden layers ino he neural nework model, he number of nework parameers is increased, which may ye lead o wrong forecasing. Apar from he previously menioned reasons, he modes share of neural neworks in airline revenue managemen sysems is no surprising. IERAURE. Pöl, S.: Forecasing is difficul especially if i refers o he fuure, AGIFORS Reservaions and Yield Managemen Sudy Group Annual Meeing Proceedings, Melbourne, Ausralia, 998 2. Bahovec, V., Erjavec, N.: Uvod u ekonomerijsku analizu, Elemen, Zagreb, 29. 3. Inroducion o ime series analysis, hp://www.il.nis.gov/div898/handbook/pmc/secion4/pmc4.hm (5.2.2.) 4. Russell, S.J., Norvig, P.: Arificial Inelligence, a Modern Approach, 2 nd ediion, New York, Prenice Hall, 22. 5. Zhang, G. P., Qi, M.: Neural nework forecasing for seasonal and rend ime series, European Journal of Operaional Research, Vol. 6, No. 2, 25. p. 5-54. 6. Perović, I., Perić, N.: Ineligenno upravljanje susavima, Universiy of Zagreb, Faculy of Elecrical Engineering and Compuaion, Zagreb, 28, p. 5 7. Faraway, J., Chafield, C.: ime series forecasing wih neural neworks: A comparaive sudy using he airline daa, Applied saisics, Vol. 47, No. 2, 998., p. 247 8. Zhang, G.P., Kline, D.: Quarerly ime-series forecasing wih neural neworks, IEEE rans Neural Neworks, Vol. 8, No. 6, 27., p. 8-84.