Transportation Research Part E

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1 Transportaton Research Part E 47 (2011) Contents lsts avalable at ScenceDrect Transportaton Research Part E journal homepage: Arcraft replacement schedulng: A dynamc programmng approach Chaug-Ing Hsu a, *, Hu-Cheh L b, Su-Mao Lu a, Chng-Cheng Chao c a Department of Transportaton Technology and Management, Natonal Chao Tung Unversty, Hsnchu 300, Tawan, ROC b Marketng and Logstcs Management, Ta Hwa Insttute of Technology, Qongln, Hsnchu County 307, Tawan, ROC c Department of Shppng and Transportaton Management, Natonal Kaohsung Marne Unversty, Kaohsung 81143, Tawan, ROC artcle nfo abstract Artcle hstory: Receved 22 March 2007 Receved n revsed form 4 March 2010 Accepted 13 June 2010 Keywords: Dynamc programmng Fleet plannng Arcraft replacement schedule Ths study developed a stochastc dynamc programmng model to optmze arlne decsons regardng purchasng, leasng, or dsposng of arcraft over tme. Grey topologcal models wth Markov-chan were employed to forecast passenger traffc and capture the randomness of the demand. The results show that severe demand fluctuatons would drve the arlne to lease rather than to purchase ts arcrafts. Ths would allow greater flexblty n fleet management and allows for matchng short-term varatons n the demand. The results of ths study provde a useful reference for arlnes n ther replacement decsonmakng procedure by takng nto consderaton the fluctuatons n the market demand and the status of the arcraft. Ó 2010 Elsever Ltd. All rghts reserved. 1. Introducton The ablty to match fleet capacty to passenger demand s one of the crucal factors decdng the proftablty of an arlne. The extent to whch economc cycles nfluence ar transportaton demand s qute apparent. An economc recesson usually accompanes reduced ar demand, resultng n nsuffcent revenue and surplus capacty that further burdens the arlnes wth fleet dle costs, thereby lowerng profts. On the other hand arlnes also suffer a great proft loss under a quck economc recovery, when the fleet capacty may not be able to expand n tme to satsfy the hgh demands, due to the tme lag between orderng, recevng and operatng of extra arcraft. Although arcraft replacement decsons can be made n advance n order to match future demand, the fluctuatng and cyclcal nature of passenger demand complcates the fleet capacty management problem. Decsons about fleet capacty management are classfed under arlne strategc plannng, whch nvolves decsons such as when to purchase, lease or dspose of arcraft. Fleet expansons and reductons are acheved through arcraft purchase, lease or by dsposng of the surplus arplanes. Leasng an arplane gves the arlnes flexblty n capacty management. However, arlnes must pay a rsk premum to leasng companes for bearng the rsks (Oum et al., 2000). Also, the lease cost for an arplane may be very hgh when there s a hgh demand for them n the market. The scrappng and replacng of an exstng arcraft s generally motvated by the physcal deteroraton of the arcraft or the avalablty of newer, more effcent ones. However, the decson to replace can be scheduled n advance to concde when the arlne market s forecasted to gong nto downward trend, thereby reducng the operatng and mantenance costs. How to schedule capacty expanson or reducton decsons n advance s an essental and crtcally mportant task for the arlnes, snce the arcraft fleet must not only serve current but also future demands. Although any partcular replacement decson s necessarly nfluenced by the current fleet composton as well as any possble future demand, t stll has a long-term mpact on the arlne fleet. Under these * Correspondng author. Address: Department of Transportaton Technology and Management, Natonal Chao Tung Unversty, 1001 Ta Hsueh Road, Hsnchu 300, Tawan, ROC. Tel.: ; fax: E-mal address: [email protected] (C.-I. Hsu) /$ - see front matter Ó 2010 Elsever Ltd. All rghts reserved. do: /j.tre

2 42 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) crcumstances, accurate demand forecasts are requred to enable the arlnes to properly schedule ther arcraft replacement decsons n response to the fluctuatng and cyclcal demands. Past studes have nvestgated the ssues n the context of fleet capacty problems, such as decsons on arcraft type, flght frequency (e.g. Kanafan and Ghobral, 1982; Teodorovc and Krcmar-Nozc, 1989) and optmal combnatons of owned and leased capacty (Oum et al., 2000). Researchers have studed fleet management problems at operatonal and tactcal levels n addton to the strategc level (e.g. Powell and Carvalho, 1997; Jn and Kte-Powell, 2000). There s scant lterature avalable on replacement cost n relaton to fleet capacty management over dfferent tme perods, or for revenue loss assocated wth dynamc and cyclcal demand. In ths study, the cost of operatng an arcraft s dependent upon ts status, as defned by type of arcraft, age and total mleage traveled. The fleet s composed of dfferent number and status of purchased and leased arcraft. On the demand sde, ths study employs the Grey topologcal forecastng method combned wth the Markov-chan model to forecast passenger traffc and capture the random and cyclc demand. The decson perods are dentfed accordng to the pattern of the passenger demand cycles over the length of the study perod. For each decson perod, the arlne makes decsons not only on whether and whch arcraft to be replaced wth a purchased or leased one, but also on whether or not to purchase or lease an arcraft as an entrely new addton to the fleet. Ths study ams to determne an optmal replacement schedule for an arlne by consderng the randomness n arlne operatons and the cyclcal demand through the use of stochastc dynamc programmng. Ths study wll also determne the optmal canddate arcraft to be recruted or dsposed of. The stochastc dynamc programmng method s solved wth backward dynamc programmng n whch the mpact of replacement decsons made at a specfc perod under uncertan passenger demand on arlne operaton can be fully consdered. Ths study frst formulates arlne cost functon of a decson perod assumng ndependent decson-makng results between perods. These costs nclude operatng cost, replacement cost and penalty cost. The operatng cost s the cost related to the operaton of the exstng fleet. The replacement costs arse from the replacement decsons made at a specfc perod. In addton, a penalty cost s ntroduced to reflect losses n revenue assocated wth the dfference between the forecasted and realzed passenger demand. The expected cost functon of the perod s further formulated by takng nto consderaton the cost dependent relatonshp between decsons made n neghborng perods and the probabltes of dfferent varatons n the forecasted and realzed passenger demand. Then, the stochastc dynamc programmng model for the replacement schedule can be formulated to determne the optmal replacement schedule by mnmzng the total expected cost of each perod over the study perod. The remander of ths paper s organzed as follows: Secton 2 revews the lterature on fleet capacty and equpment replacement problems. Secton 3 formulates the cost functons based on a sngle perod operaton. Secton 4 provdes the stochastc dynamc programmng model for determnng the optmal schedule of the replacement decsons. A numercal example s provded n Secton 5, to llustrate the applcaton of the models and the effects of changes n key parameters on the optmal solutons. In secton 6, we make our concludng remarks. 2. Lterature revew The fleet capacty of an arlne s the total number of dfferent types of arcraft purchased, leased and scrapped over a perod of tme. Relevant studes have focused manly on choosng the rght type of arcraft, route, and flght frequency usng determnstc mathematcal programmng methods (e.g. Kanafan and Ghobral, 1982; Teodorovc and Krcmar-Nozc, 1989; Yan et al., 2006). We and Hansen (2007) consdered the factors of competton n the decsons on both arcraft sze and servce frequency. They examned the mpact of these decsons on both the cost and the demand of ar transportaton. Equpment replacement problems n ndustres wth hgh captal assets have been wdely dscussed n ndustral engneerng and operatons research lterature (e.g. Hartman, 2004, 1999; Rajagopalan, 1998; Jones et al., 1991). Hartman (2001) examned the effect of probablstc asset utlzaton on the replacement decson makng process, usng dynamc programmng. Powell and Carvalho (1997) dealt wth the mult-commodty fleet problem and formulated the problem as a dynamc control problem. Jn and Kte-Powell (2000) explored the replacement problem for a fleet of shps for a proftmaxmzng operator, assumng a homogenous fleet and unform demand. Wu et al. (2005) addressed a rental fleet-szng problem n the truck-rental ndustry. They combned both operatonal and tactcal decson levels, subject to uncertan customer travel tme and non-statonary customer demand. Oum et al. (2000) developed a model for the arlnes to determne the optmal mx of leased and owned capacty, takng nto consderaton that the demand for ar transportaton s uncertan and cyclcal. The emprcal results suggested that the optmal demand for the arlnes would range between 40% and 60% of ther total fleet. The fnancal status of the arlne and the passenger demand are crtcal factors when t comes to leased and owned capacty decsons. Although the uncertanty n demand has been ncluded and nvestgated n the lterature, research regardng the schedules of the above decsons and ther mpacts on arlne operaton and the total cost over a tme horzon s scant. Furthermore, the cost dependent relatonshps between subsequent perods due to replacement decsons made n prevous perods have not been dscussed yet. When t comes to methods to forecast arlne passenger demand, the mult-regresson model and the tme-seres model are the most wdely employed. Horonjeff and McKelvey (1994) generalzed past lterature and classfed arlne passenger traffc forecastng models nto four categores: judgment predcton, trend projecton and speculaton, market analyss and econometrc modelng method. However, the number of avalable traffc observatons has usually not been large enough

3 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) due to a short accumulaton tme, partcularly cty-par data (Horonjeff and McKelvey, 1994). Collectng a large number of data to develop a conventonal statstcal forecastng model s dffcult. The Grey topologcal forecastng model was developed based on the Grey system theory (Deng, 1985, 1986), and s also called the Grey pattern predcton or system trend predcton model. The Grey theory deals wth systems wth poor nformaton. Other related models have also been used n many applcatons (e.g. Deng and Guo, 1996; Deng, 1999; Hsu and Wen, 1998). Hsu and Wen (1998) appled the Grey theory to forecast arlne passenger traffc. They constructed an mproved GM(1, 1) tme-seres model and showed that the forecasted result from the Grey model s more accurate than those predcted by the regresson model or the ARIMA model. However, there s no lterature avalable that apples the Grey topologcal model for forecastng arlne passenger traffc nfluenced by the economc cycle. The advantage of employng the Grey topologcal model les not only n that t requres only lttle hstorc data to formulate a predcton model, t s also constructed to forecast system data wth pattern development, makng t sutable for pattern or economc cycle forecastng. Ths makes the Grey topologcal forecastng model sutable for predctng arlne passenger traffc, snce nternatonal cty-par ar passenger data s usually not suffcent, and arlne market traffc shows a pattern of beng nfluenced by the economc cycle. Passenger demand forecasts are nherently uncertan because of assumptons about random future demand. Any forecast result nvolves a potental varance or bas. In sum, few have combned the Grey topologcal forecastng model wth the Markov-chan to nvestgate the demand fluctuatons and the stochastc demand realzatons. Ths study ntegrates Grey topologcal forecastng model, Markov-chan model and dynamc programmng method to nvestgate the replacement schedule for an arlne by consderng the randomness n arlne operatons and the cyclcal demand. 3. Cost functon Consder an arlne that operates varous routes, wth R and r representng the set of these routes and a partcular route, respectvely, r e R. Let T be the study perod wth n number of decson perods t, t =0,1,2,..., n. The duraton of the decson perods may vary from each other and from dfferent routes due to dfferent economc cycles. Let s suppose that there are three possble future demand trends forecasted by the Grey topologcal forecastng model, upward, equal and downward, respectvely. Let w represent three possble fluctuatons for the demand, wth w = 1, 2 and 3. We let w = 1 represent a rsng demand; w = 2 a smlar demand; and w = 3 a declnng demand of the perod, as compared wth that of the prevous perod. Let p P t w represent the probablty of the demand fluctuaton labeled as w at perod t. It must be noted that pt w P 0 and 3 w¼1 pt w ¼ 1. In addton, Ft r represents the forecasted passenger demand on route r at perod t. The values of pt w and Ft r are then determned by the Grey topologcal forecastng method combned wth the Markov chan and they are summarzed n Appendx A. Let N Bt and NLt be the number of arcraft assocated wth the replacement decsons made at perod t, where superscrpts B and L represent the arcraft beng purchased and leased, whle the subscrpts q, y and m descrbe the status of an arcraft as ts type, remanng avalable years and mleage traveled, respectvely. The remanng avalable years of an arcraft, y, s determned by ts number of years of maxmum usage, Y, and the age of the arcraft, y 0, such that y = Y y 0. Note that N Bt and NLt are both ntegers. The decson of whether the fleet beng expanded or reduced n terms of arcraft beng recruted or dsposed of, s judged by the value of N Bt and NLt. When the varables NBt and NLt are postve, the arlnes wll decde to expand ther fleet capacty through purchasng and/or leasng, and the numbers of recruted arcraft wth status (q, y, m) are N Bt and NLt, respectvely. Otherwse, the optmal decson wll result n a capacty reducton wth negatve values for N Bt and NLt. Snce the total number of arcraft scrapped cannot be larger than the exstng scale, the followng nequaltes hold: 8 < : E Bt P jnbt j E Lt gym P jnlt gym j f N Bt < 0 N Lt gym < 0 where E Bt and ELt represent the total number of purchased and leased arcraft wth status (q, y, m) at perod t, respectvely, n the arlne fleet. Let S t be the set of all arcraft operated by the arlne durng perod t, S t fe Bt ; ELt ; 8q; y; mg; EBt ; ELt 2 Iþ [f0g. Let d t 1 denote the set of arcraft recruted or dsposed of at perod (t 1) and these arcraft wll be operated at perod t, d t 1 fn Bðt 1Þ ; NLðt 1Þ ; 8q; y; mg; NBt ; NLt 2 I and t =1,2,..., n. Then, the fleet operated at perod t, S t, can be specfcally formulated as follows: S t ¼ S t 1 þ d t 1 t ¼ 1; 2;...; n ð2aþ E Bt ¼ EBðt 1Þ E Lt ¼ ELðt 1Þ þ NBðt 1Þ t ¼ 1; 2;...; n ð2bþ þ NLðt 1Þ t ¼ 1; 2;...; n ð2cþ ð1þ whch show that the fleet capacty and composton of perod t are the result of the replacement decsons made at perod (t 1).

4 44 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) The total fleet capacty of an arlne can change accordng to the dfferent numbers of seats offered by dfferent arcraft types. Let Q q represent the capacty of arcraft type q and let K tr be the total flght frequences on route r offered by the arcraft wth status (q, y, m) durng perod t. Then the total capacty,.e. the number of seats on route r durng perod t, A t r can be formulated as A t r ¼ X X X d tr Q qðe Btr þ ELtr ÞKtr 8r ð3þ 8q 8y 8m where d tr s an ndcator varable; and dtr ¼ 1 for an arcraft wth status (q, y, m) durng perod t servng route r; otherwse, d tr ¼ 0. Moreover, the nequalty P P 8q 8y P8m dtr P 1 8r must hold to ensure every route s beng served by at least one arcraft. In practce, the arlne may set an deal load factor on each route, and then the mnmzed fleet capacty can be obtaned. The realzed fleet capactes on the routes, dependng on the average load factor, must be equal to or larger than the forecasted demand of decson perod t, whch yelds l t r At r P Ft r 8r ð4þ where l t r denotes the average load factor on route r durng perod t. From Eq. (3), Eq. (4) can be further revsed as P P 8q 8y P8m dr q Q qðe Bt þ ELt ÞKtr P Ft r. The fact that not all arcraft can be assgned to a flght due to factors such as mantenance and turnover accounts, should be consdered n the arcraft utlzaton model. Let B r q l t r denote the block tme of type q arcraft on route r, ncludng the tme spent n varous arcraft trp modes, and let u t represent the maxmum possble utlzaton of the arcraft wth status (q, y, m) durng perod t, respectvely. A maxmum possble utlzaton also mples a maxmum possble daly use of the arcraft for a certan perod of tme (Kane, 1990; Teodorovc, 1983). For all arcraft wth dfferent status, the total arcraft utlzaton must be less than or equal to the maxmum possble utlzaton. Ths study uses the relaton of Teodorovc et al. (1994), such as P P P 8r 8q 8y P8m Br q Ktr 6 P P 8q 8y P8m ðebt þ ELt Þut. For a specfc arcraft type wth status (q, y, m), the nequalty P 8r Br q Ktr 6 ut must hold. Any surplus capacty from an arcraft not reachng the maxmum possble operaton tme can be relocated to routes wth an arcraft of nsuffcent capacty. Therefore, an arcraft mght be shared on two routes. The drect operatng costs are all those expenses assocated wth operatng a fleet of arcraft, ncludng deprecaton costs, mantenance costs and flyng costs. The deprecaton costs reflect the reducton n the value of the exstng fleet and can be calculated based on the purchase or lease prce of the arcraft. In some ways, the deprecaton costs depend on the market demand when the arcraft s orgnally purchased or leased. For nstance, when most arlnes forecast an upward trend n future demand, the orgnal purchase or lease cost wll be hgh, resultng n a hgh deprecaton cost. However, snce the total lease expense decreases wth the total duraton of the lease perod, the foregong can be neglected when the leased perod s contracted for a long tme, thereby yeldng a constant average lease cost. Let P represent the average purchase cost for an arcraft wth status (q, y, m) and R td denote the average lease cost for an arcraft wth status (q, y, m) wth a total leased perod d at perod t, respectvely. Then, the deprecaton cost related to the exstng fleet of perod t can be formulated as X X X E Bt P X t g þ X X X E Lt Rtd 8t ð5þ 8q 8y 8m 8q 8y 8m where X t g denotes the average remanng resale rato of the orgnal purchase prce wth an average yearly nterest rate g of perod t. Mantenance cost can be further dvded nto fxed mantenance cost and varable mantenance cost. Fxed mantenance costs ncludes mantenance overhead ncludng the mantenance of the buldng and equpment as well as land rental, none of whch vary wth the number of arcraft. On the other hand, varable mantenance costs change wth the status of the arcraft, and the number of arcraft. Generally speakng, the runnng and preventve mantenance costs ncrease wth the age of the arcraft and the mleage traveled. In addton, there are economes of scale that allow arlnes wth many arcraft of a smlar type n ther fleet to operate more effcently than those wth several dfferent types. The mantenance cost of perod t can then be expressed as M t þ X X X V t ðebt þ ELt Þ ð6þ 8q 8y 8m where M t represents the fxed mantenance cost (overhead) of perod t and V t denotes the varable mantenance cost of the arcraft wth status (q, y, m) durng perod t. The flyng cost related to the total flght frequences on all routes s X X X X 8r 8q 8y 8m b t qr dr q Ktr where b t qr represents the average flyng cost of an arcraft of type q on route r durng perod t. The total drect operatng cost of the arlne for operatng the exstng fleet durng perod t, C t D, can be formulated as C t D ¼ X X X ðe Bt P X t g þ ELt Rtd þ V t ðebt þ ELt ÞÞ þ X X X X b t qr dr q Ktr þ Mt ð8þ 8q 8y 8m 8r 8q 8y 8m ð7þ

5 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) The total ndrect operatng cost as a result of servng passengers at perod t, C t I can be expressed as follows: C t I ¼ X 8r F t r Hr ð9þ where H r denotes the average ndrect cost per passenger on route r. Summng up the total drect and ndrect operatng costs n Eqs. (8) and (9) yelds the total operatng cost of the arlne durng perod t, C t. When dsposng of a purchased arcraft, the arlne wll receve the salvage value of the arcraft, whch s ts remanng value after deprecaton. The salvage value s nversely related to the age and mleage traveled. When termnatng the contract of a leased arcraft, the arlne has to pay a penalty for returnng the arcraft earler than stpulated n the lease contract. The longer the remanng lease perod, the hgher the penalty wll be. Moreover, both salvage value and penalty cost as a result of fleet reducton are dependent upon the demand for arcraft n the market. If most arlnes forecast a boom n demand n the near future, the tendency towards expandng fleet capacty wll be hgh, resultng n a hgher prce for arcraft,.e. lower salvage cost borne by the arlne. Conversely, t costs the arlne a lot of tme and effort to dspose of ther excess capacty when the demand s low, resultng n an ncreased loss of salvage value. Let D t and Zte represent the salvage value and penalty cost of an arcraft wth status (q, y, m) and wth a remanng lease perod e at perod t, respectvely. Let P t and Yt denote the orgnal purchase prce and total deprecaton cost of an arcraft wth status (q, y, m) at perod t, respectvely. The arlne suffers a loss f t dsposes of an arcraft when D t < Pt Y t, whle on the other hand D t > Pt Y would mply a revenue gan. The total replacement cost for dsposng of an arcraft durng perod t can be expressed as P P 8q 8y P8m jnbt jðpt Y t Dt Þ, where jnbt j s the number of purchased arcraft to be dsposed of. On the other hand, the penalty cost for dsposng of an arcraft durng perod t can be expressed as P P 8q 8y P8m jnlt jzt ;e, where jnlt j s the number of arcraft whose lease wll be termnated. Takng nto consderaton both salvage and penalty costs, the replacement cost durng perod t wth demand fluctuatons labeled w can be expressed as U t ¼ X X X a Bt jnbt jðpt Yt Dt ÞþX X X b Lt jnlt jzte ð10þ 8q 8y 8m 8q 8y 8m Indcators a Bt and blt are both bnary varables, and ther relatonshp wth the replacement decsons are as follows: ( a Bt ¼ 1 b Lt gym ¼ 1 f NBt < 0 N Lt gym abt else ¼ 0 < 0 b Lt gym ¼ 0 In the study, the decsons on whether or not, and whch arcraft should be dsposed of depend manly on the sum of operatng cost, replacement cost and penalty cost. However, an arlne that has safety as ts hghest prorty should mmedately dspose of or termnate the lease of any arcraft once ts age and mleage traveled has reached the safety threshold. The utlzaton of an arcraft s only nfluental f the two factors of remanng years and mleage traveled, are wthn the safety parameters. The relatonshp between the optmal canddate arcraft to be dsposed of and ts remanng years as well as ts mleage traveled can be expressed as n o mn Aq 6 1 W t ¼ 0 f n 1 mn ; Gq y m Aq ; Gq y m o > 1 where W t s an ndcator varable; and where Wt ¼ 0 refers to the arcraft wth status (q, y, m) beng dsposed of at perod t, otherwse, W t ¼ 1. And, A q and G q represent, respectvely, the maxmum years of expected servce and the maxmum allowable mleage to be traveled by a type q arcraft. The actual demand may be underestmated, overestmated or be correct, regardless of the demand fluctuaton labeled as w, snce label w represents the cyclcal demand fluctuaton. Let fr t be the actual passenger demand on route r durng perod t. If the actual demand s less than the forecasted result,.e. fr t Ft r < 0, then the arlne bears an ncreased total ndrect operatng cost for servng ther passengers due to the unsold seats. The punshment assocated wth an overestmaton s ncluded n Eq. (9). On the contrary, there wll be unsatsfed passengers for fr t Ft r P 0 due to nsuffcent fleet capacty as determned n accordance wth the forecasted demand. Let I t r represent the average revenue loss assocated wth one unt of nsuffcent seats on route r durng perod t, whch can be estmated by the average fare on the route. The penalty cost functon due to the naccurate forecast on route r at perod t, t, can then be formulated as t r ¼ðf t r Ft r ÞIt r The total penalty cost of the arlne durng perod t s L t ¼ P 8r t r. The total cost durng perod t, Qt, gven by the operatng cost, the replacement cost and the penalty cost can be formulated as follows: Q t ¼ C t þ U t þ L t Note that Q t s ndependent of the fleet operated n prevous perod and depends on the fleet beng operated and the decsons made n perod t. ð11þ ð12þ ð13þ ð14þ

6 46 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) Stochastc dynamc programmng model Secton 3 formulates the cost functon of the arlne for a sngle perod. From Eqs. (2a) (2c), the fleet operated durng perod t s the result of the replacement decsons made at perod (t 1) and n addton, the fleet capacty durng perod t s determned based on the demand forecast at perod (t 1) towards perod t. That s to say, the replacement decsons made and the demand forecast executed at perod (t 1) have a certan level of nvolvement wth the operatng cost, C t, and the penalty cost, L t, of perod t. Smlarly, the demand forecast result for perod (t + 1) served as the reference of the replacement decsons made at perod t, whch resulted n the replacement cost, U t n Eq. (14). These cost dependent relatonshps between decsons made at neghborng perods are explaned as recursons, and are depcted graphcally n Fg. 1 by takng nto consderaton the demand fluctuaton. The crcular node represents the set of arcraft operated durng perod t, S t, whle the square node s the set of arcraft recruted or dsposed of at perod t when the demand of perod (t + 1) s forecasted as label w, d w t, respectvely. As shown n Fg. 1, the resultng fleet S t s the result of the decson made durng perod (t 1) wth respect to dfferent demand fluctuatons labeled w. For a gven perod t, the arlne makes the replacement decsons n accordance wth the forecasted result for perod (t + 1), ncludng the three possble demand trends, the demand of perod (t + 1) forecasted to be upward, equal and downward compared wth the demand of perod t. However, the realzaton of the demand mght fall short of the forecasted result. In other words, the total cost of perod (t + 1), gven by the sum of operatng, replacement and penalty costs, s drectly affected by the decson made at perod t and the forecasted demand for perod (t + 1). As for dynamc programmng, the stage and the state n ths study refer to decson perod t and operatng fleet S t, respectvely. Let C t (S t, d t ) represent the total cost from perod t forward, where d t denotes the replacement decson. Gven S t and t, let d t denote any value of dt that mnmzes C t (S t, d t ), and let C t ðst Þ be the correspondng mnmum value of C t (S t, d t ). Then, C t ðst Þ¼mn d t C t ðs t ; d t Þ¼C t ðs t ; d t Þ ð15þ In order to consder the stochastc feature of future demand even further, the mnmum expected sum from perod t forward, C t (S t, d t ), gven that the fleet and replacement decson n perod t are S t and d t, can be formulated as follows: C t ðs t ; d t Þ¼ Xw¼3 w¼1 p t w ½Q t þ C tþ1 ðs tþ1 ÞŠ ð16þ where C tþ1 ðs tþ1 Þ¼mn d tþ1c tþ1 ðs tþ1 ; d tþ1 Þ s the recursve relatonshp that dentfes the optmal decson for perod (t + 1), gven that the optmal decson for perod (t + 2) has been made. The objectve for the arcraft replacement schedule problem s to determne p =[d 1, d 2,..., d t,..., d T ]soasto 11 d 1 11 S 2 S 0 w=1 w=2 w=3 1 d 0 2 d 0 3 d 0 B1 + N L1 + N B1 N L1 N B1 + N L1 + N B1 N L1 N B1 + N L1 + N B1 N L1 N 1 S 1 2 S 1 3 S 1 w=1 w=2 13 d 1 21 d 1 w=3 w=1 w=2 w=3 w=1 w=2 w=3 12 d 1 22 d 1 23 d 1 31 d 1 32 d 1 33 d 1 12 S 2 13 S t Fg. 1. A stochastc dynamc programmng network.

7 mn E X p t X X s:t: 8q 8y Q t! X 8m N Bt and NLt d r q Q qðe Bt þ ELt ÞKtr P Ft r l t r ð17aþ ð17bþ ntegers 8q; y; m 8t ð17cþ The recursve relatonshp for the problem s C tþ1 ðs tþ1 Þ¼mn d tþ1c tþ1 ðs tþ1 ; d tþ1 Þ. The optmal decson at perod t s found by solvng by backwards nducton startng at t = n and usng Eq. (16) at each step to fnd the optmal decson for the perods. The replacement decsons nclude when, how many, and how many dfferent types of arcraft are to be recruted through lease or purchase, as well as when and whch arcraft, leased and purchased, wth varous statuses are to be dsposed of. Moreover, the duraton of each perod may be dfferent for dfferent cty-pars due to varatons n the length and trend of the economc cycle. The fleet operatng on the routes, dependng on the average load factor, must be equal or larger than the forecasted demand of the perod. Only two routes wth dentcal duraton and number of perods can share an arcraft. 5. Example C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) Ths study further presents a case study to demonstrate applcatons of the models, based on avalable data from EVA Arways (EVA). For the sake of smplfcaton, nne ctes n seven countres were selected from all the ctes currently beng served by EVA. The eght cty-pars (routes) are Tape (TPE) Los Angeles (LAX), Seattle (SEA), San Francsco (SFO), Tokyo (TYO), Hong Kong (HKG), Sngapore (SIN), Bangkok (BKK) and Sydney (SYD). There are 15 wde-body arcraft ncludng 6 Boeng comb, 4 Boeng , 4 Boeng and 1 MD11 flyng on these routes. Tables 1 and 2 lst the basc data of the fleet and the arcraft n the fleet, respectvely. In the present study, the forecast results from the Grey topologcal forecastng model represent the demands on the routes carred by all arlnes on the market. Ths study further estmates the demand carred by EVA based on ther market share. Accordng to Teodorovc and Krcmar-Nozc (1989), the market share of arlne on route r, MS r can be estmated by Ka r MS r ¼ P K a r 8 ð18þ Table 1 Basc data of the fleet. Source: Arcraft, r Number Average capactes (numbers of seat) Average age (year) B comb /6 B /4 B /4 MD /0 Number of purchased and leased arcrafts (purchased/leased) Table 2 Basc data of arcrafts n the fleet. Source: Leased arcraft Average lease cost per month (US$) Contracted lease perod B , / /03 B , / /04 B , / /12 B , / /12 B comb 1,300, / /04 B comb 1,300, / /10 B comb 1,100, / /12 B comb 1,200, / /12 B comb 1,130, / /01 B comb 1,040, / /08 B ,125, / /12 B ,290, / /12 B ,400, / /04 B ,200, / /11 Purchased arcraft Purchase date Total purchase cost (US$) MD /08 3,345,313,186 2,220,935,549 Salvage value at the end of 2000 (US$)

8 48 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) where K r represents the flght frequences of arlne on route r and a s an emprcally obtaned constant whch value s approxmately 1.2 (Teodorovc and Krcmar-Nozc, 1989). Table 3 descrbes the supply parameters for the routes as related to arcraft type, frequences, block tme and fares. The parameter values related to the market shares and the load factors on the dfferent routes are also shown n Table 3, where MS r represents the market share of EVA. For the sake of smplfcaton, the mpacts of the newly developed arcraft types such as Boeng 787 and Arbus A380 on the optmal replacement decsons are not dscussed n the current case study. The study perod n ths study totals eght years, from 2002 to Table 4 shows the optmal replacement decsons made n the frst perod, ncludng the duraton of the decson perods and the fleet compostons. As shown n Table 4, the arlne tends to smplfy the fleet compostons to three types of arcraft, such that each of the routes s served by only one type of arcraft, although wth dfferent numbers of arcraft. Through ths strategy, operatng and mantenance costs are decreased due to the realzaton of economes of scale. Moreover, as shown n Table 4, durng the second perod the arcraft servng these 8 routes are all leased. The reason for ths s that the severe demand fluctuaton encourages arlnes to choose lease arrangements for ther arcraft. Ths allows them to manage ther fleet sze and composton, n as flexble a manner as possble to match the demand. It should be noted that there s a tme lag between purchase/lease and the delvery of these arcraft. Hence, after havng determned the fleet compostons for each perod usng our proposed model, arlnes can then estmate the tme lag usng past experence, and nclude that n ther plan for purchasng, leasng or dsposng of arcraft thereby satsfyng the demand for arcraft n dfferent perods. Although the replacement decsons made for each perod are affected by the forecasted demand, the total expected cost of the arlne s ncreased wth each naccurate forecasted result. Fg. 2 shows the cost of route TPE BKK wth dfferent fluctuatng demands for dfferent perods, where the frst and second number n the parentheses represent label w and the probablty, p t w, respectvely. From left to rght, the costs represent the total expected cost over the study perod, and the expected cost of the frst and second decson perods, respectvely. As shown n Fg. 2, there are three demand forecast results, but wth dfferent probabltes. For each decson perod, there s a mnmzed cost when the forecasted demand s totally matched to the realzed one,.e. w = 2, and whle there are Table 3 Parameter values related to routes. Source: Route, r Arcraft type, q Weekly flght frequences (one-drecton) Block tme (hours) Fare (US$/persontrp) Market share, MS r (%) TPE LAX B , B comb TPE SEA B comb TPE SFO B comb TPE TYO B TPE HKG B comb, B TPE SIN B comb TPE BKK B , MD TPE SYD B Load factor, l t r (%) Table 4 The optmal purchase and replacement decsons made n the frst perod. The frst perod, t = 1 The second perod, t = 2 Route, r Fleet composton Purchase and replacement decsons Fleet composton Duraton Arcraft type Number Arcraft type Number Duraton Arcraft type Number TPE LAX B (leased) B (leased) TPE SEA B comb 2 (leased) B comb 2 (leased) TPE SFO B comb 2 (leased) B comb 1 (leasng) B comb 1 3 (leased) TPE TYO B (leased) B (leasng) B (leased) TPE HKG B (leased) B (dsposng of) B comb 1 (leased) B comb 1 (leased) TPE SIN B comb 1 (leased) B comb 1 (leased) TPE BKK MD11 1 (purchased) MD11 1 (dsposng of) B (leased) B (leased) B (leasng) TPE SYD B (leased) B (leased) Total B comb 6 (leased) B comb 1 (leasng) comb 7 (leased) B (leased) B (leasng) B (leased) (leased) B (leasng) B (leased) MD11 1 (purchased) MD11 1 (dsposng of) B (dsposng of) 1 8% of the capactes from the B comb are shared wth route TPE LAX. 2 11% of the capactes from the B are shared wth route TPE SYD.

9 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) US$ (2, 0.07) (1, 0.45) (3, 0.33) (2, 0.47) (1, 0) US$ (3, 0.53) (2, 0.47) (1, 0) US$ (3, 0.53) (2, 0.47) (1, 0) US$ (2, 0.22) (3, 0.53) (2, 0.07) (1, 0.45) (1, 0.42) US$ (3, 0.33) (2, 0.47) (1, 0) US$ (3, 0.53) (2, 0.47) US$ (1, 0) (3, 0.53) (2, 0.47) (1, 0) US$ (3, 0.53) (2, 0.47) (3, 0.36) US$ (1, 0) (2, 0.07) (3, 0.53) US$ (1, 0.45) (2, 0.47) (1, 0) US$ (3, 0.53) (3, 0.33) (2, 0.47) US$ (1, 0) (3, 0.53) t=1 t=2 t= Tme Fg. 2. The costs of route TPE BKK wth dfferent fluctuated demand at dfferent perods. ncreased costs wth ether an overestmated or an underestmated demand,.e. w = 1 and w = 3. However, the mpact of the forecast results on total cost rely not only on the dfference between forecasted and realzed demand, but also on the probablty that the forecast results n fact occur. As shown n Fg. 2, the hgh probablty that the forecasts are an ncreasng or decreasng demand, p 1 w¼1 and p1 w¼3, respectvely, combned wth the ncreased costs leads to a relatvely hgh expected cost over the study perod. In the present study, the arlne serves the routes entrely wth leased arcraft because that way the arlne s exempt from the hgh deprecaton cost and only needs to pay the lease cost. However, hgh mantenance cost places a heavy fnancal burden on the arlne when the arcraft become older and have hgh mleage. When that happens, the arlne may prefer to purchase rather than lease these older arcraft snce the flexblty of leasng may not compensate for the hgh mantenance cost. Next we perform a senstvty analyss to nvestgate how changes n the age of the arcraft and the average lease cost per year affect the decsons to purchase or lease. Fg. 3 shows the threshold of the purchase and lease decson by comparng varous lease costs and the age of the B comb arcraft. Compared to just leasng or termnatng the lease of an arcraft, the purchase or the dsposal of an arcraft requres a much longer tme. Hence, arlnes tend to lease arcraft rather than purchasng them n order to satsfy short-term fluctuatons n demand. To smplfy the problem, the tme untl recevng a new arcraft that has been purchased or leased s neglected n

10 50 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) ths study because arcraft replacement decsons are made earler n order to match future demand. The benefts of leasng nclude savngs n deprecaton cost and greater flexblty n matchng the demand n the short run. Moreover, the older the arcraft the less the benefts of leasng, and the hgher the costs of leasng. As seen n Fg. 3, the threshold of leasng an arcraft decreases wth the ncrease n the age of the arcraft. Nevertheless, leasng s an optmal alternatve f there s a substantal decrease n lease cost. Also, the effect of lease cost on purchase and lease decsons s margnal for B comb arcraft f they are older than fve years, as shown n Fg. 3. The value of an arcraft deprecates exponentally from the moment the arcraft has been manufactured and s beng operated. In other words, the purchase cost of the arcraft decreases wth ts age, and so does the deprecaton cost. In addton, a low salvage prce s of no consequence to the arlne once the arcraft s scrapped because t s beng replaced or smply because of fleet reducton. When the arlne expands the fleet capacty by addng a used but not aged arcraft, these advantages explan why an arlne may prefer to purchase rather than lease. The results provde a reference for the arlne n ther decson makng process of replacement decsons n accordance wth the arcraft age and n negotatng wth the leasng company for the lease prce of dfferent arcraft. The mantenance cost depends on the status of the arcraft, ncludng type, age and mleage. In ths study, the cost of operatng an arcraft s dependent on ts status, as defned by type, age and total mleage traveled. Gven the passenger demand, an arcraft should be dsposed of and replaced wth a new arcraft when the mantenance requrements become excessve. On the other hand, all thngs beng equal, t pays to keep an arcraft f t s n good condton,.e. low mantenance cost. Fg Average lease cost per year (US$) Lease Purchase Age of arcraft B comb (year) Fg. 3. The threshold of purchase and lease decsons by comparng lease cost and the age of arcraft B comb Varable mantenance cost (US$) Replace Do-nothng Age of arcraft B comb (year) Fg. 4. The threshold of whether or not dsposng of the arcraft by comparng the varable mantenance cost and the age of arcraft B comb.

11 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) shows the threshold of whether or not to dspose of an older arcraft and replace t wth a new one by comparng the mantenance cost and the age of the B comb arcraft. The left-hand and rght-hand sdes of the sold lne n Fg. 4 represent the decsons regardng dsposng of the exstng arcraft and replacng wth a new one, and do-nothng, respectvely. As shown n Fg. 4, the threshold of the replacement ncreased wth the ncreased age of the arcraft. As an arcraft becomes older, the annual deprecaton decreases wth the accumulated deprecaton beng spread over an ncreasng number of servce years. Hence, arlnes are more nclned to retan older arcraft despte ther hgh mantenance cost. As shown n Fg. 4, the threshold for a replacement ncreases wth the age of the arcraft. If the mantenance cost of an arcraft does not exceed the threshold, the arlne should retan the arcraft; whle f the mantenance cost keeps ncreasng, the tendency to dspose of the arcraft wll also ncrease. Wth other words, f the mantenance cost of the arcraft does not exceed the threshold, the result suggests that the arlne keeps the arcraft. However, the tendency towards dsposng of the arcraft s hgh once the arcraft has a hgh mantenance cost. In ths study, the total cost of the arlne s affected by dsturbance of demand fluctuatons. An overestmated demand leads to excess capacty, whle an underestmated demand results n nsuffcent capacty. Varables p t w¼1 ; pt w¼2 and pt w¼3 represent, respectvely the probabltes that the followng occur, the demand of perod t s fluctuated to be ncreasng, the same and decreasng, as compared wth that of perod (t 1). Fg. 5 shows the total expected cost of the routes under dfferent occurrence of forecast results. The X-axs n Fg. 5 represents dfferent crtera regardng the combnatons of the three probabltes, and from left to rght, the X-axs ndcates (1) the orgnal probabltes from the Grey topologcal model and the Markov-chan; (2) three forecast results exst evenly,.e. p t w¼1 ¼ pt w¼2 ¼ pt w¼3 ¼ 0:33; (3) the future demand s exclusvely ncreasng,.e. p t w¼1 ¼ 1; (4) the future demand s exclusvely decreasng,.e. pt w¼3 ¼ 1; and (5) the future demand s the same wth that of the prevous perod,.e. p t w¼2 ¼ 1. As shown n Fg. 5, there s a smlar cost pattern among the routes, where the future demand s the same wth that of the prevous perod shows the lowest,.e. label (5) n X-axs whle the demand beng exclusvely ncreasng and decreasng are the hghest. The total cost when the three forecast results exst equally s moderate between all crtera. The results demonstrate the mportance of stochastc future demand. Accurate demand forecasts wll enable the arlne not only to schedule arcraft replacement decsons n response to fluctuatng and cyclcal demands, but wll also acheve an overall mnmzed cost. The forecasted demand of the route s calculated based on the market share of that route. The market share s postvely affected by the flght frequences provded. An ncreased flght frequency leads to a hgher market share and a hgher passenger demand carred by the arlne. Supposng EVA ntends to ncrease ts market share of route TPE HKG from 5.62% up to 30%, by ncreasng ts flght frequency. Table 5 shows the optmal replacement decsons of route TPE HKG wth market shares of 5.62% and 30%, respectvely. Due to the lmted utlzaton of the arcraft, the total number of arcraft should be ncreased wth the ncreased total flght frequences. As shown n Table 5, the fleet should obtan 3 addtonal leased B comb f the arlne expects an ncrease n market share from 5.62% to 30%. The addtonal costs, such as the costs related to the arcraft and the costs orgnated from the passengers, the total expected cost s substantally ncreased as shown n Table 5. In ths study, the realzed fleet capactes,.e. the numbers of seats on the routes are nfluenced by the load factor, whch s determned based on hstorcal data. Assume the servce performance remans the same under dfferent settngs of the load factor. As Eq. (4) shows, under a constant demand, a larger value of the load factor leads to a lower capacty requrement, thus a lower number of arcraft and lower total expected cost. Supposng EVA decdes to ncrease the average load factor of route TPE SYD from 71% to 80%. Table 6 shows the optmal replacement decsons of route TPE SYD wth the average load factors of 71% and 80%, respectvely. Expected cost of the route (10 9 US$) TPE-BKK TPE-LAX TPE-HKG TPE-SYD TPE-SFO TPE-SEA TPE-SIN TPE-TYO (1) (2) (3) (4) (5) Fg. 5. The total expected cost of the routes under dfferent occurrence of forecast results.

12 52 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) Table 5 The optmal purchase and replacement decsons of route TPE HKG wth market shares of 5.62% and 30%. The frst perod, t = 1 The second perod, t = 2 Total expected cost (US$) Market share Fleet composton Purchase and replacement decsons Fleet composton Arcraft type Number Arcraft type Number Arcraft type Number 5.62% B (leased) B (dsposng of) B comb 1 (leased) 3,460,846,000 B comb 1 (leased) 30% B (leased) B (dsposng of) B comb 4 (leased) 6,453,704,000 B comb 1 (leased) B comb 3 (leasng) Table 6 The optmal purchase and replacement decsons of route TPE SYD wth the average load factors of 71% and 80%. The frst perod, t = 1 The second perod, t = 2 Total expected cost (US$) Average load factor (%) Fleet composton Purchase and replacement decsons Fleet composton Arcraft type Number Arcraft type Number Arcraft type Number 71 B (leased) B a 2 (leased) 450,932, B (leased) B (leased) 433,195,100 a One of whch s shared wth route TPE TYO. Because the load factor s as low as 71%, route TPE SYD requres 2 leased B to provde the servces. One of these arcraft s shared by route TPE TYO wth 89% of ts capacty. As Table 6 shows, there s only 1 leased B requred to serve the route under the ncreased load factor,.e. 80%. Because there are fewer arcraft beng operated, the total expected cost s correspondngly reduced. 6. Conclusons Past studes have nvestgated the equpment replacement problems n the feld of ndustral engneerng and operatons. Other studes have dscussed fleet management problems at both operatonal and tactc levels, n addton to the strategc level. However, there s scant lterature avalable on replacement cost n relaton to fleet capacty management over dfferent tme perods, or for revenue loss assocated wth dynamc and cyclcal demand. Therefore, the contrbuton of ths paper to the lterature s to fll n the above gap. Moreover, the decson on whether to expand a fleet by purchasng new arcraft or lease them, or to reduce a fleet through dsposal of the purchased or leased arcraft are also nvestgated. The applcaton of our proposed dynamc programmng model s llustrated wth a case study nvolvng EVA arlnes. It was found that EVA tends to smplfy ts fleet composton by usng a sngle type of arcraft for each route served. To maxmze capacty utlzaton and reduce any related costs, some arcraft are assgned to two routes. In addton, severe demand fluctuatons have drven EVA to lease rather than purchase ther arcraft. Ths s allowng EVA greater flexblty n fleet management and n matchng short-term varatons n demand. In addton, the total cost for a partcular decson perod can be mnmzed by provdng a perfect match of the forecasted demand wth the actual demand, nstead of overestmated or underestmated forecasts that wll lead to ncreased costs. However, the mpact of forecasted results for total cost vares not only wth the dfference between forecasted and actual demands, but also on the probablty that a demand forecast wll occur. In other words, although an accurate demand forecast avods a penalty cost, the total cost wll stll be hgh f the precse estmaton occurs only rarely. Hence, the total cost for the arlne can only be mnmzed f all the mpacts of the demand fluctuatons and cyclc demands on the arlne s fleet management are fully captured. As a leased arcraft becomes older, the benefts of leasng wll declne further, resultng n a smaller tendency towards leasng the arcraft. Leasng an older arcraft s an optmal alternatve only f there s a substantal reducton n lease cost. In addton, the threshold of the replacement decson ncreases wth the ncrease n age of the arcraft. In other words, f the ncreased mantenance cost of an older arcraft does not exceed the threshold, the arcraft should be retaned and vce versa. The results of ths study provde a useful reference for arlnes n ther arplane replacement decson-makng takng nto account the fluctuatons n market demand and the status of the arcraft. The study perod n the case study s set to be eght years, and nvolves only replacement schedulng for a short run. Future studes can extend the study perod to explore medum- and long-term replacement schedulng. A lmtaton of our study s the fact that t consders only passenger demand whle neglectng the demand for ar cargo, whch makes up a very mportant porton of the demand for ar transport. To get an overall pcture of the actual operaton of an arlne t s worth explorng the replacement schedulng consderng both passenger and ar cargo demands. The case study n ths research s

13 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) focused on a sngle arlne, and the effect of strategc allances wth other arlnes has been neglected. It would be nterestng to examne f arlnes that have formed strategc allances have a dfferent approach to optmzng ther replacement schedulng. Ths study employs the Grey topologcal forecastng method combned wth the Markov-chan model to forecast passenger traffc and to capture the random and cyclc demands. Nevertheless, ar passenger demand s not only affected by the economc stuaton but also by the threat of terrorsm, arplane crashes, and the development of new routes and markets. The mpact of all these ssues must be taken nto account when assessng the fluctuaton n passenger demand when decdng on a replacement schedule. The computatonal dffcultes are of the most challenges when solvng larger scale-nstances of the problem. To make backwards computng possble, at each step the decson functons must be ncluded n the computatons and stored untl the end. Consderable storage capacty s therefore requred, because these functons are, as a rule, obtaned only n tabular form (Bronshten and Semendyayev, 1985). Acknowledgement The authors would lke to thank the Natonal Scence Councl of the Republc of Chna for fnancally supportng ths research under Contract No. NSC E Appendx A. Grey topologcal model and Markov-chan A.1. Grey topologcal model The steps of constructng a Grey topologcal model are descrbed as follows: 1. Plot a seres of X (0) n two-dmensonal X, Y-plane. Every X n X (0) has ts own Y-axs coordnate, whle a Y-axs coordnate may be mapped to several X-axs coordnates. Let k represent the order number of the X-axs coordnates sharng the same Y-axs coordnate and x (0) (k) represent the X-axs coordnates mapped by that Y-axs coordnate. Plot the curve, X (0),nX, Y-plane, usng [k, x (0) (k)]. 2. Accordng to the sequence x (0) (k), fnd the maxmum value Max X (0) and mnmum value mn X (0). Select many reference values f at Y-axs, =1,2,..., m. Note that the doman of the reference value s mn X ð0þ 6 f 6 MaxX ð0þ ; ¼ 1; 2;...; m. 3. Fnd the correspondng Y-axs coordnate of f,asf : fx ð0þ g!fmt ð0þ g. Let mt ð0þ ðkþ represent the k th tangent pont of the horzontal lne, f passng curve X (0). Then, all X-axs coordnates can form a set of P : fðmt ð0þ ðkþ; f Þg! fmt ð0þ ðkþg; k ¼ 1; 2;...; n and mt ð0þ ¼fmt ð0þ ð1þ; mt ð0þ ð2þ;...; mt ð0þ ðn Þg. 4. Every fxed reference value should map to a coordnate set W ð0þ, composed by number of X-axs coordnates, t s mt ð0þ ðkþ ¼W ð0þ ðkþ, and Pðmt ð0þ ðkþ; f Þ¼W ð0þ ðkþ, therefore W ð0þ ¼fW ð0þ ð1þ; W ð0þ ð2þ;...; W ð0þ ðn Þg. And, W ð0þ represents a set of X-axs coordnates, whch map to a fxed reference value f n the Y-axs. 5. Perform an accumulated generatng operaton for set W ð0þ, and obtan a new generatng seres W ð1þ, t s AGO : W ð0þ! W ð1þ. 6. Construct a GM(1, 1) model for each new generatng seres W ð1þ, represented as GM : W ð1þ! W cð1þ. Perform an nverse accumulated generatng operaton to each new GM model and obtan the predctng model IAGO : W cð1þ! W cð0þ. The whole procedure can be represented as follows: GM AGO P f ðfx ð0þ gþ ¼ c W ð1þ ða1þ IAGO GM AGO P f ðfx ð0þ gþ ¼ c W ð0þ ða2þ 7. Every fxed reference value can develop a partcular forecastng model as shown n step 6. Accordng to these forecastng models, fnd the X-axs coordnate correspondng to the fxed reference value f n the Y-axs for =1,2,..., m, then these X-axs coordnates are cw ð0þ 1 ðn 1 þ 1Þ; c W ð0þ 2 ðn 2 þ 1Þ;...; c W ð0þ m ðn m þ 1Þ The forecastng value W cð0þ ðn þ 1Þ represents the dstance from the orgn to the (n + 1)th data n the X-axs. The coordnate s represented as ðw cð0þ ðn þ 1Þ; f Þ n a two-dmensonal plane. By lnkng these coordnates as a curve and the Topologcal forecastng curve, X b ð0þ can be obtaned bx ð0þ ¼fð c W ð0þ ðn þ 1Þ; f Þj ¼ 1; 2;...; mg ða4þ ða3þ A.2. Markov-chan The Markov-chan theory s wdely appled to predct a dynamc random system. A Markov-chan descrbes the states of a system at successve tmes. At these tmes the system may have changed from the state t was n the moment before

14 54 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) to another or remaned n the same state. The changes of state are called transtons. The Markov property means that the condtonal probablty dstrbuton of the state n the future, gven the state of the process currently and n the past, depends only on ts current state and not on ts state n the past. A n-step Markov-chan s composed of a set of n-state and one set of transton probablty. There s only one state at one moment, and any further changes n the system can be determned by the transton probablty n each state at dfferent moments. The transton probablty of each state represents the level of effects ncorporatng every random factor. Therefore the Markov-chan s sutable for forecastng random seres. Ths study combnes the Grey topologcal and Markov-chan models for forecastng arlne passenger demand wth respect to dfferent economcal stuatons. Prevous lterature has forecasted gross natonal product (GNP) based on Grey predctng GM (1, 1) combned wth the Markov-chan model, and the result was shown to be more accurate than GM(1, 1) alone. The mplementaton steps of the Markov-chan model are lsted below. A.2.1. Categorze the states Categorze every moment n the Grey topologcal model nto k states. Let the result of the Grey topologcal forecastng model, W cð0þ n moment be the central pont of every state. Determne a proper percentage P% ofw cð0þ to be the upper and lower bounds of every moment n each state. Then, the boundary of the jth state n moment, E j can be represented as E j 2½A j ; B j Š; j ¼ 1; 2;...; k where A j and B j represent the upper and lower bounds of the jth state n moment, respectvely. Lnkng the boundary of the same states n every moment results n a functon curve whch s nearly parallel wth the curve of the Grey topologcal forecastng model. The zone between every two adjacent curves form a state zone, so we can determne the state n whch every Grey topologcal predctng result wll be at each moment. Classfy those predctng results whch are less than A 1 as state one, those whch are larger than B k as state k. The values of A j, B j, and k can be decded by research subject and the amount of orgnal data. A.2.2. Establsh a matrx of state transton probablty The state transton probablty can be formulated as P ðmþ ab ¼ MðmÞ ab M a where P ðmþ represents the probablty of transton from state a to state b after m steps, ab MðmÞ ab represents the frequency of transton from state a to state b after m steps, M a s the frequency of state a. Due to the unknown transton from the last state to ts next state of the orgnal seres, the data of the last (m 1) steps wll be elmnated when calculatng M a. The state transton probablty matrx, R (m) can be wrtten as ða5þ ða6þ TPE-SEA TPE-TYO TPE-LAX TPE-HKG TPE-BKK TPE-SYD TPE-SIN TPE-SFO Passenger Demand Year Fg. 6. Passenger demand of the routes from 1993 to 2001.

15 2 R ðmþ ¼ P ðmþ 11 ; PðmÞ 12 P ðmþ 21 ; PðmÞ 22 P ðmþ ; k1 PðmÞ k2 ;...; PðmÞ 1k ;...; PðmÞ 2k ;...; PðmÞ kk C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) ða7þ where P ðmþ represents the probablty of transton from state a to state b after m steps. Consderng m = 1, f the forecasted ab data falls n the ath state, then check the ath row of matrx R (1).IfMax b P ð1þ ¼ P ab al, then state L s the most lkely state that the seres transfer to at the next moment. Passenger Demand Passenger Demand Passenger Demand Passenger Demand Year (a) TPE-LAX Year (b) TPE-SEA Year (c) TPE-SFO Year (d) TPE-TYO Fg. 7. Forecasted yearly demand of the routes.

16 56 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) A.2.3. Compose a transton probablty matrx table Determne a number of r as the number of moments from the past to the forecasted moment. From the nearest to furthest moments, the transton steps from the past moments to the forecasted moment are 1, 2,..., r. For all transton probablty matrces of the steps, extract the vector rows from the transton probablty matrx mapped by the begnnng state and compose those as a new transton probablty matrx. By summng up all the vectors n the column, the state of the forecasted moment can be obtaned as the state wth the maxmum value Passenger Demand Passenger Demand Passenger Demand Year (e) TPE-HKG Year (f) TPE-SIN Year (g) TPE-BKK Passenger Demand Year (h) TPE-SYD Fg. 7 (contnued)

17 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) A.2.4. Calbrate the forecasted values After obtanng the transton state of the future moment of the seres by step 3 the upper and lower bounds of the state can be further determned. The forecasted value wll be the average of the upper and lower bounds. Ths study further apples the Grey topologcal model and the Markov-chan model formulated above to forecast the future passenger traffc of the routes n the case study. The mplementaton steps are descrbed as follows: Step 1. Collect hstorcal passenger traffc, construct a passenger demand forecastng model usng the Grey topologcal model. Compare the forecasted result wth the actual data to verfy ts accuracy. Moreover, determne the decson perods for the routes accordng to the economc cycle. Table 7 Comparson between forecasted demand based on Grey Topologcal model and the actual data. Route Average yearly demand from 1993 to 2001 Average dfference (%) Actual Forecasted TPE LAX 994, , TPE SEA 139, , TPE SFO 547, , TPE TYO 1,727,317 1,718, TPE HKG 4,816,613 4,798, TPE SIN 876, , TPE BKK 1,241,295 1,218, TPE SYD 134, , Table 8 The boundares of states and the state of the forecasted demand on route TPE BKK. Year Actual demand Forecasted demand A1 A2 = B1 A3 = B2 B3 State , , , , , , ,060,692 1,100,000 1,034,880 1,078,000 1,122,000 1,166, ,215,534 1,164,000 1,095,091 1,140,720 1,187,280 1,234, ,178,071 1,188,000 1,117,670 1,164,240 1,211,760 1,260, ,147,768 1,213,333 1,141,504 1,189,066 1,237,600 1,287, ,201,234 1,230,000 1,157,184 1,205,400 1,254,600 1,304, ,363,266 1,346,667 1,266,944 1,319,733 1,373,600 1,428, ,626,603 1,550,769 1,458,963 1,519,753 1,581,784 1,645, ,780,378 1,572,000 1,478,937 1,540,560 1,603,440 1,667,577 3 Table 9 Transton probablty matrx of route TPE BKK. Orgnal state State after transton R R R R R R

18 58 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) Table 10 The state transton probablty matrx from year on route TPE BKK. Year State Step State Predcted year: Total Predcted year: Total Predcted year: Total Predcted year: Total Predcted year: Total Predcted year: Total Predcted year: Total

19 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) Table 10 (contnued) Year State Step State Predcted year: Total Step 2. Combne the Grey topologcal model wth the Markov-chan model, to estmate the state transton probablty matrx between the decson perods. Then, the probablty of the forecasted passenger traffc fluctuaton status for every perod can be obtaned from ths state transton probablty matrx. The obtaned probablty wll be the probablty of the forecasted passenger demand fluctuaton combnatons of two adjacent decson perods n the case study. Fg. 6 shows the passenger traffc data from June, 1993 to December, 2001 of the eght routes n the case study. As shown n Fg. 6, although the total passenger traffc of these eght routes dffers from each other, there s an upward trend that began n 1993, reached a peak n 1995 or 1996, and started to fall startng n 1997 and After 1999, the passenger traffc ncreased agan. The results show that the arlne passenger traffc ndeed exhbts an economc cycle trend. Fg. 7 represents the forecasted passenger demand of the routes from January, 2002 to December, 2009, showng the duraton of the decson perods. Table 7 calbrates the forecast results by comparng the forecasted results of the Grey topologcal model wth the actual data from June, 1993 to December, As shown n Table 7, the maxmum dfference between the forecasted demand from the Grey topologcal forecastng model and the actual data s less than 6%. Moreover, the average dfferences on routes TPE LAX, TPE HKG, TPE TYO and TPE BKK are less than 1%. It can be concluded that overall the result from the Grey topologcal forecastng model s accurate. Ths study further combnes the Grey topologcal forecastng results wth the Markov-chan model, to nvestgate the demand fluctuatons and to determne the probablty of the three demand realzatons. Let the forecasted results from the Grey topologcal model be the mddle value and be denoted by X. The boundary values of the three states,.e. upward fluctuatng demand, a demand smlar to that of the prevous perod and a decreasng demand, can be determned accordng to the mddle value, X and a dfference rate of 4%. The boundary values of these three states, A1, A2, A3, and A4 can be expressed as follows: A1 ¼ A2 ð1 0:04Þ A2 ¼ X ð1 0:04=2Þ A3 ¼ X ð1 þ 0:04=2Þ A4 ¼ A3 ð1 þ 0:04Þ ða8aþ ða8bþ ða8cþ ða8dþ For the demand fluctuaton labeled as w = 1, 2 and 3, the future demand may le between A1 and A2, A2 and A3, and A3 and A4 and the realzed demand s 1 (A1 + A2), 1 (A2 + A3) and 1 (A3 + A4), respectvely. Take route TPE BKK as an example. Table shows the boundares of the states and the results of the state of the forecasted demand. The transton probablty of route TPE BKK can be further calculated based on Eqs. (A6) and (A7) and s shown n Table 9. As shown n Table 9, R1, R2, R3, R4 and R5 represent the steps requred for state transferrng to state j. For example, the probablty of transton from state 1 to state 2 by 1 step s Table 10 shows the state transton probablty matrx from 2002 to 2009 on route TPE BKK. Table 11 Transton probablty of states of route TPE BKK. Year (State) State (2) (1) (1) (1) (3) (3) (2) (2)

20 60 C.-I. Hsu et al. / Transportaton Research Part E 47 (2011) Take year 2002 as an example. As shown n Table 10, the largest total probablty of state 2 of 3.5 shows that there s every lkelhood that the forecasted demand s precse wthout fluctuaton. The probabltes of the demand beng overestmated, precsely estmated and underestmated can be further calculated as 1.5/( ), 3.5/( ), and 1.0/ ( ). Then, the probablty of the transton from the current state of the year to dfferent states can be restated as shown n Table 11. References Bronshten, I.N., Semendyayev, K.A., Handbook of Mathematcs. Van Nostrand Renhold Company, New York, USA. Deng, J., Grey System Fundamental Method. Huazhong Unversty of Scence and Technology, Wuhan, Chna (n Chnese). Deng, J., Grey Predcton and Decson. Huazhong Unversty of Scence and Technology, Wuhan, Chna (n Chnese). Deng, J., Guo, H., Grey Predcton Theory and Applcaton. Chan-Hua Publcatons, Chna (n Chnese). Deng, J., Grey System Theory and Applcaton. 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Transportaton Research Part E 42, Glossary of symbols Notaton: Defnton p t w : the probablty of demand fluctuaton labeled w at perod t F t r : the forecasted passenger demand on route r at perod t fr t : the actual passenger demand on route r at perod t N Bt : the number of purchased arcraft wth status (q, y, m) assocated wth the replacement decsons made at perod t N Lt : the number of leased arcraft wth status (q, y, m) assocated wth the replacement decsons made at perod t E Bt : the total number of purchased arcraft wth status (q, y, m) at perod t n the arlne fleet E Lt : the total number of leased arcraft wth status (q, y, m) at perod t n the arlne fleet Q q : the capacty of a q type arcraft K tr : the total flght frequences on route r offered by arcraft wth status (q, y, m) durng perod t d tr : an ndcator varable denotng whether the arcraft wth status (q, y, m) durng perod t s servng route r or not B r q : the block tme of a q type arcraft on route r u t : the maxmum possble operatng tme of an arcraft wth status (q, y, m) durng perod t P t : the average purchase cost for an arcraft wth status (q, y, m) at perod t X t g : the average remanng resale rato of the orgnal purchase prce wth an average annual nterest rate g n perod t R td : the average lease cost for an arcraft wth status (q, y, m) wth a total leased perod d n perod t V t : the varable mantenance cost of the arcraft wth status (q, y, m) durng perod t M t : the fxed mantenance cost (overhead) of perod t b t qr : the average flyng cost of an arcraft of type q on route r durng perod t O t D : the total drect operatng cost for the arlne for operatng the exstng fleet durng perod t O t I : the total ndrect operatng cost as a result of servng passengers at perod t H r : the average ndrect cost per passenger on route r C t : the total operatng cost of the arlne durng perod t A t : the maxmum usage of the arcraft wth status (q, y, m) at perod t G t : the maxmum allowable mleage traveled of an arcraft wth status (q, y, m) at perod t D t : the salvage cost of an arcraft wth status (q, y, m) durng perod t Z te : the penalty cost of an arcraft wth status (q, y, m) and wth a remanng lease perod e durng perod t U t : the replacement cost durng perod t W t : the ndcator varable representng whether the arcraft wth status (q, y, m) should be dsposed of at perod t : the average revenue loss assocated wth one unt of nsuffcent seats on route r durng perod t I t r