Yugosav Journa of Operaions Research 4 (004), Number, 55-70 AIRINE SEAT MANAGEMENT WITH ROUND-TRIP REQUESTS Peng-Sheng YOU Graduae Insiue of Transporaion & ogisics Naiona Chia-Yi Universiy, Taiwan psyuu@mai.ncyu.edu.w Received: February 00 / Acceped: February 004 Absrac: Consider a mui-period mui-fare cass airine booking probem reaed o a wo-eg airine nework. Trave requess incude oubound, inbound rip, and round rips. The round-rip refers o a ourney comprising boh oubound and inbound rips. To deveop a dynamic-nesed booking decision-making sysem for he airine nework, his sudy designs a dynamic mode ha enabes he airine reservaions sysem o devise a se of dynamic decision rues for any given booking saus. The booking process is found o be conroed by some se of booking hreshods. Keywords: Airine, invenory, booking, revenue managemen.. INTRODUCTION Since he fare dereguaion of he airine indusry in 978, many airine companies have used discriminaory pricing poicies in order o segmen poenia cusomers ino compeiivey reevan groups in order o maximize revenues. A common approach is o divide a poo of idenica seas in he same cabin of a figh ino severa fare casses hrough differen resricions and charge differen fares (c.f. Beobaba (987)). In circumsances where he capaciy of he aircraf is reaivey fixed and canno be changed in shor noice, and he margina cos of carrying an addiiona passenger proves reaivey ower compared o high fixed coss incurred from passengers wih reserved bookings, airine companies devise booking schemes in order o fi in vacan seas since hose vacan seas upon deparure ime mean os revenues. Airine passengers can roughy be caegorized ino wo groups: reserved passengers and go-show passengers. Reserved passengers book airine seas in advance. They have he righ o board he airpane during deparure ime. Go-show passengers, on
56 P.-S. You / Airine Sea Managemen wih Round-Trip Requess he oher hand, appear on airine couners wihou reservaions and can ony book remaining seas afer he check-in ime of reserved passengers. Airine companies can raise revenues by opening and cosing a variey of fare casses based on reservaion saus. The reservaion saus of cusomers is based on he cusomer's needs as we as he amoun of fare hey wi pay. One of he probems faced by airine companies ies in he sea invenory conro. The sea invenory conro deermines he number of seas sod a differen fare caegories. In pracice, cusomers make reservaions randomy over ime. Such behavior refecs he sochasic naure of airine passengers. I aso prevens airine companies from predeermining fuure booking requess. If airine companies accep bookings of cusomers regardess of fare cass, hey may ose revenues from cusomers wiing o pay higher fares. On he oher hand, if airine companies reec mos of he ower fare booking requess, hey run he risk of fying wih many vacan seas. Reserved passengers who do no show up during deparure ime are caed "noshow passengers". "No-show passengers" fai o use reserved accommodaions for reasons such as missed connecions and raffic ie-ups. They may aso be passengers who suddeny decided o cance heir reservaions prior o deparure ime. The occurrence of canceaions means oss of revenues for he airine companies since companies canno immediaey repace he canceed booking wih anoher cusomer. Faced wih such sochasic behavior and he necessiy of fiing up vacaed seas, airine companies overbook fighs. The chaenge in overbooking fighs ies in he exen o which overbooking poicies shoud be empoyed. Ahough overbooking poicies reduce he ikeihood of aking off wih much vacan seas, hey may aso ead o difficu siuaions when he number of reservaions exceeds he avaiabe seas a he ime of deparure. In such cases, airine companies no ony ose cusomers bu aso have o dea wih he fac ha hey mus offer some form of compensaion o he cusomer. In order o aid he airine's sea invenory conro and overbooking poicy, airine officias empoy he revenue/yied managemen concep. Revenue managemen is described as he appicaion of invenory daa and pricing sraegies o maximize profi from a fixed number of resources (c.f. Weaherford e a., 99). This paper appies he revenue managemen concep o deveop a round rip sea invenory conro probem. Various modes of revenue managemen have been proposed o deermine booking poicies for various ypes of sea invenory conro probems. The sea invenory conro srucure can be caegorized ino he separaed srucure and he nesed srucure. In a separaed srucure, he booking period is regarded as a singe inerva. Airine personne mus se a booking imi for every fare cass a he sar of he booking process. The sum of he booking imis for every fare cass mus be equa o he oa booking capaciy of he figh. The weakness of his srucure ies in he fac ha requess for higher fare casses may be denied even hough seas are si avaiabe in ower fare casses. In a nesed srucure, requess for higher fare casses can be accommodaed if seas are avaiabe in ower fare casses. The nesed srucure can be furher divided ino wo: he saic nesed srucure and he dynamic nesed srucure. In he saic nesed srucure, booking imis are se a he beginning of he booking period. In he dynamic nesed srucure, booking imis are updaed during he booking period depending on he acua booking saus.
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 57 Various approaches have been proposed o se booking imis. Among hem is he one-eg based mehod. This mehod conros each figh eg independeny. I has he advanage of deveoping dynamic nesed booking conro poicies. Anoher approach is he wo-eg based mehod. This mehod conros wo figh egs and generaes dynamic nesed booking conro poicies. Generay, his mehod generaes higher revenues. Is revenues are higher han he sum of revenues from independen figh eg bookings. This is aso based on he fac ha iineraries are composed mosy of one-sop ciy fighs. The approach ha normay produces mahemaica programming modes is he nework-based mehod. In order o ensure maximum revenues, airine companies mus acke he enire nework. Through airine neworks, airine companies offer numerous origina-desinaion-fare casses o cusomers and figh iineraries may comprise muipe-sop ciy fighs. However, such an approach may resu in he inabiiy of airine companies o deveop dynamic-nesed booking poicies since he combinaion of producs sricy increases wih he number of figh egs. Airine sea invenory conro is a form of revenue managemen. The ieraure conains severa wonderfu inroducions o he airine revenue managemen probems (e.g., Beobaba, 988; auenbacher e a., 999; McGi e a., 999;Weaherford e a., 99). Addiionay, various modes have been proposed o deermine booking poicy for di eren ypes of sea invenory conro probems (e.g. Asrup e a., 986;, Beobaba, 989; Brumee, 990; Brumee, 99; Curry, 990; Gerchak e a., 985; ee e a., 99; iewood, 97; Robinson, 995; Womer, 99; You, 999; You, 00). iewood (97) was he firs o consider he revenue managemen probem, using he margina sea revenue approach o opimize booking poicy for a singe eg probem wih wo-fare casses. Meanwhie, Beobaba (989) bui upon iewood's work and proposed a genera mode wih muipe fare casses, assuming ha he booking process o foow he paern of ower before higher, ha is cusomers requesing ower vaue fare was assumed o be booked before cusomers requesing he higher vaue fare. By he same assumpion, Curry (990) deveoped a muipe fare cass mode using he mahemaica programming approach. Meanwhie, Womer (99) addressed a singe eg mui-fare cass mode and inroduced an agorihm for opimizing he booking poicy. Brumee, McGi, Oum, Sawaki, and Treheway (990) dea wih a muipe fare cass probem by formuaing a revenue funcion for boh discree and coninuous probabiiy disribuions of demand, and he condiions of wha exhibied a concave revenue funcion. Many previous researchers assume a ower before higher paern in he booking process (e.g., Beobaba, 989; Brumee e a., 99; Curry, 990; Womer, 99). However, his assumpion is no aways vaid. Robinson (995) considered a reaivey genera case in which he cusomers of any given fare cass remain cusered bu he order of such cusers may no mach ha of he increasing fares. The airines expec o have a funcions in he Compuer-Reservaion-Sysem which is he funcion of revising he booking decision based on he acua booking saus. Thus, research on seing he dynamic-nesed booking sraegy can no be ignored. Using he dynamic approach, Gerchak, Parar and Yee (985) deveop a dynamic-nesed booking sraegy on a singe eg wo-fare cass mode. An imporan oucome of his sudy (985) is ha booking poicy parameers can be reduced o wo ypes of criica vaues: criica booking capaciy and criica decision periods, and hese vaues are imporan in reducing compuaiona ime and eiminaing he need for exensive daa
58 P.-S. You / Airine Sea Managemen wih Round-Trip Requess sorage. Buiding upon Gerchak's work, ee, and Hersh (99) deveoped a dynamic mode for a singe figh wih muipe fare casses and muipe sea bookings. The singe eg airine sea invenory mode above can be appied o deermine a booking sraegy for each rip in an airine nework. However, passengers may simuaneousy reques muipe figh egs across an airine nework. Thus, o maximize revenues he booking sraegy for a rips in an airine nework mus be se simuaneousy. Noaby, arge airine nework booking conros are usuay ineffecive owing o compuaiona barrier, daa overfow, and so on. Therefore, curren compuer echnoogy makes deveoping sma airine nework booking conro sysems preferabe o buiding arger bu ineffecive sysems. This sudy aemps o deveop a booking poicy for a wo-eg airine nework comprising oubound and inbound egs. The rave requess incude oubound rip, inbound rip and round rips, where round rip refers o a ourney invoving oubound and inbound rips. Deveoping a means of aowing round rip requess is he key difference beween his paper and previous ones. The probem is soved by using he dynamic approach o creae a wo eg airine sea invenory conro mode in which demands are modeed as a sochasic process. The proposed dynamic mode ses he booking poicy for each booking cass according o he acua bookings hroughou he enire booking process. This work aims o maximize expeced revenue. I is found herein ha booking poicy can be reduced o a se of criica vaues, incuding he foowing informaion: which fare casses shoud be opened for sae wihin each rip (ha is, wheher o accep a reques for a fare cass in each rip). The res of he paper is organized as foows: Secion ouines a assumpions made and formuae he probem as a dynamic programming mode. Secion anayze he nove mode and deermine he opima booking poicy. The anaysis reveas ha he booking poicy can be conroed by using a se of criica booking vaues, caed booking imi. Secion 4 hen demonsraes he properies of he nove mode using a numerica exampe, and finay, Secion 6 presens concusions.. MODE ASSUMPTIONS AND FORMUATION Suppose an airine company has he righ o fy passengers beween wo ciies and is permied o se ickes o cusomers requesing an oubound rip from ciy A o ciy B, an inbound rip from ciy B o ciy A, or a round rip ha incudes boh oubound rip and inbound rips. Furhermore, assume ha he airine ries o deveop an opima booking poicy for wo schedued fighs, incuding an oubound figh wih a booking capaciy of I and deparure ime of o, and an inbound figh wih a booking capaciy of I and deparure ime of i > o. The oubound, inbound, and round rips are denoed by =, = and =, respecivey. Suppose he airine divides he seas on rip ino ickes a ino ordered ypes, {,,..., } he eas expensive. fare casses and prices x for fare cass in rip. Herein, he fare casses in each rip are cassified, where fare cass is he mos expensive and is
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 59 For convenience, he oa panning horizon is divided ino T decision periods which are sufficieny sma ha no more han one cusomer arrives during each period. Addiionay, he periods are couned in reverse ime sequence and i is assumed ha he deparure imes of he oubound and inbound fighs are a he end of periods and, respecivey. e i, i and i = min{ i, i } denoe he seas avaiabe on he oubound, inbound and round rips, respecivey. e λ represen he probabiiy ha a reques for fare cass in rip wi arrive during decision period wih λ = =. Meanwhie, e i = ( i, i) and e v () i denoe he maximum oa expeced revenue ha can be generaed wihin periods when i seas remain. Eq. () is hen produced v 0 () i = 0. () Noabe, has aso been used o represen he number of periods remaining before deparure ime. e I = (, 0), I = (0,) and I = (,). Then, Eq. () i = λ + λ = = = v () ( I( i 0)) v () i + λ max{ v ( i), x + v ( i I )} I( i ) = () exiss for 0 <, whie Eq. () exiss for >. i = λ + λ = = = = = v () ( I( i 0)) v () i + λ max{ v ( i), x + v ( i I )} I( i ) = = (). DECISION ANAYSIS This secion anayzes he nove mode and deveops he opima booking poicy. A poicy is ermed a booking-imi poicy when a reques for a cerain cass is acceped if and ony if he oa number of reservaions immediaey preceding reservaion requess is ess han he booking-imi vaue for ha cass. The foowing demonsraes ha he opima booking poicy is aso a booking-imi poicy. Firs, Eqs. () and () are rewrien in he foowing simpe form. v x z I i v () i = v + x z I i < () i + λ max{0, ()} i ( ), = = () i λ max{0, ()} i ( ), = (4)
60 P.-S. You / Airine Sea Managemen wih Round-Trip Requess where z () i = v () i v ( i I ). (5) From Eq. (4), booking poicy is ceary dependen on he vaue z () i, and he foowing emmas are needed. To derive he propery of z () i, we need he foowing emmas. emma.. Suppose (i) z () i is nonincreasing in i and (ii) z () i is nonincreasing in i, hen (a) z () i is nondecreasing in i, (b) z () i is nondecreasing in i. Proof: Saemen (a) is equivaen o saemen (b) since z ( i, i ) z ( i, i ) = z ( i, i) z ( i, i). Consequeny, i is ony necessary o verify ha z () i saisfies he inequaiy z ( i, i) z ( i, i). Checking ha he inequaiy is vaid for = according o Eq. (4). Suppose ha he asserion hods for some. (4) is used o express z ( i, i ) z ( i, i ) as z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i ) + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} = max{0, x z ( i, i )} + max{0, x z ( i, i )} + λ (max{0, x z ( i, i )} max{ 0, x z ( i, i )} = = max{0, x z ( i, i )} + max{0, x z ( i, i )} + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )}. emma. produces inequaiies max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i) (6) (7)
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 6 and max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, (, )} + max{0, (, )} max{0, (, )} max{0, (, )} z i i z i i x z i i x z i i x z i i x z i i (, ) (, ), max{0, x z ( i, i)} max{0, x z ( i, i )} max{0, x z ( i, i)} + max{0, x z ( i, i )} z ( i, i) + z ( i, i ) + z ( i, i) z ( i, i ) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i ) z ( i, i) z ( i, i) if x z ( i, i ) z ( i, i) + z ( i, i ) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i) = z ( i, i) z ( i, i) if x z ( i, i ) and x z ( i, i) z ( i, i) + z ( i, i) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i ) z ( i, i) z ( i, i) if x z ( i, i ) and x z ( i, i) oherwise. (8) (9) Subsiuing inequaiies Eqs. (7)-(9) ino (6) obains λ = = z ( i, i ) z ( i, i ) ( )( z ( i, i ) z ( i, i ) 0 (0) which compees he proof. The foowing imporan heory can now be confirmed. Theorem.. (a) z () i is nondecreasing in i, (b) z () i is nondecreasing in i, (c) z () i is nonincreasing in i, (d) z () i is nonincreasing in i,
6 P.-S. You / Airine Sea Managemen wih Round-Trip Requess Proof: Saemen (a) is proved firs. Proving (a) requires firs verifying ha z () i saisfies he inequaiy z ( i, i) z( i, i). Noaby, he saemen for = is immediaey derived from Eq. (4). Incuding an inequaiy in he inducive hypohesis used o esabish he genera formua is hepfu. For, Eq. (4) and emma 5, we have since z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i ) + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} = max{0, x z ( i, i )} + max{0, x z ( i, i )} + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} = = max{0, x z ( i, i )} + max{0, x z ( i, i )} + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} 0 max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} z ( i, i ) z ( i, i ), () () max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} = z ( i, i) z ( i, i) + z ( i, i) z ( i, i) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i ) if x z ( i, i) z ( i, i) + z ( i, i) = z ( i, i) z ( i, i) + z ( i, i) z ( i, i) z ( i, i) z ( i, i) if x z ( i, i), (9)
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 6 and max{0, x z ( i, i)} max{0, x z ( i, i)} max{0, x z ( i, i)} + max{0, x z ( i, i)} z ( i, i) + z ( i, i) + z ( i, i) z ( i, i) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i ) z ( i, i) z ( i, i) if x z ( i, i) z ( i, i) + z ( i, i) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i) z ( i, i) z ( i, i) if x z ( i, i). (4) A simiar approach can be used o demonsrae ha z ( i, i) z ( i, i ), and hus saemen (b) hods. Since saemens (a) and (b) hod, asserions (c) and (d) foow from emma.. Theorem.(a) and (b) impy ha some criica booking capaciies { m ( i )} and { m ( i )} exis such ha x z () i and x z ( i ) if and ony if i m ( i ) and i m ( i ). Consequeny, he booking imi poicy is he opima booking poicy for rips and. Booking-imi poicy for rips and : I. A reques for fare cass in rip and period shoud be acceped if and ony if i { m ()} (Theorem.(c)); II. A reques for fare cass in rip and period shoud be acceped if and ony if i { m ()} (Theorem.(d)). Theorem.. z () i is nonincreasing in i and i Proof: The proof wi be compeed by demonsraing ha z ( i, i ) z ( i, i ) 0. From Eq. (4) we have z ( i, i ) z ( i, i ) 0 and
64 P.-S. You / Airine Sea Managemen wih Round-Trip Requess z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i ) + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} = max{0, x z ( i, i )} + max{0, x z ( i, i )} + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} = = max{0, x z ( i, i )} + max{0, x z ( i, i )} + λ (max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} whie from inequaiies max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i ) + z ( i, i ) z ( i, i ) z ( i, i ) z ( i, i ), (5) (6) max{0, x z ( i, i)} max{0, x z ( i, i )} max{0, x z ( i, i)} + max{0, x z ( i, i )} = z ( i, i) + z ( i, i ) + z ( i, i) z ( i, i ) = z ( i, i) z ( i, i) + z ( i, i ) z ( i, i ) z ( i, i) z ( i, i) if x z ( i, i ) z ( i, i) + z ( i, i) z ( i, i) z ( i, i) + z ( i, i ) z ( i, i ) z ( i, i) z ( i, i) if x z ( i, i ), (7) and
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 65 max{0, x z ( i, i )} max{0, x z ( i, i )} max{0, x z ( i, i )} + max{0, x z ( i, i )} z ( i, i ) z ( i, i ), (8) we have inequaiy λ = = z ( i, i ) z ( i, i ) ( )( z ( i, i ) z ( i, i ). (9) Thus, z () i is nonincreasing in i. Simiary, i can be shown ha z ( i, i) z ( i, i ), hus compeing he proof. Theorem. impies ha some criica booking capaciies { m ( i)} ({ m ( i )}) exis such ha x z ( i ) if and ony if i m ( i) ( i m ( i)). Thus, he opima booking poicy for rip is hus aso a booking-imi poicy. Booking-imi poicy for rip : I. A reques for fare cass in rip in period shoud be acceped if and ony if i { m ( i)} (Theorem.); or II. A reques for fare cass in rip in period shoud be acceped if and ony if i { m ( i )} (Theorem.). Theorem.. z () i is nondecreasing in. Proof: From Eq. (4) we have i i λ = = x z i i I i z () z () = max{0, x z ( i, i )} I( i ) max{0, (, )} ( ), (0) whie emma. produces inequaiy z ( i, i )} 0 due o max{0, x z ( i, i )} I( i ) max{0, x z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i ) 0; max{0, x z ( i, i )} I( i ) max{0, x z ( i, i )} 0 due o z ( i, i ) = z ( i, i ) z ( i, i ) 0; and max{0, x z ( i, i )} 0 due o z ( i, i ) max{0, x z ( i, i )} I( i ) z ( i, i ) z ( i, i ) = z ( i, i ) z ( i, i ) 0, so he righ hand side of Eq. (0) is no ess han 0. Hence, z () i () i 0 as required. z
66 P.-S. You / Airine Sea Managemen wih Round-Trip Requess Theorem. impies ha he dynamic-nesed booking-imi for rip is a piecewise-consan funcions of he ime o figh deparure. Daa sorage can hus be reduced by soring ony he criica-booking-period { σ ( i, i)}. 4. NUMERICA EXAMPE To iusrae he proposed mode and booking poicies, an exampe is given beow. Assume an oubound figh is deparing afer T = 450 periods and an inbound figh is be deparing afer T = 500 periods and ha he maximum booking capaciy for he oubound and inbound fighs is I = 00 and I = 00, respecivey. Furhermore, assume ha he airine has previousy specified = 4 fare casses for boh fighs wih corresponding icke prices respecivey. x and arriving probabiiies λ as ised in Tabes I and II, Tabe I: he revenues x 4 00 00 50 00 500 400 50 00 700 550 450 50 Tabe II: Reques Probabiiies λ = = = / 4 4 4 00:00 0.0 0.0 0.08 0.0 0.0 0.0 0.06 0.0 0.0 0.06 0.05 0.0 0:00 0.0 0.08 0.07 0.06 0.05 0.09 0.08 0.0 0.04 0.08 0.07 0.0 0:00 0.06 0.06 0.05 0.05 0.06 0.08 0.05 0.05 0.0 0.08 0.07 0.05 0:400 0.0 0.05 0.0 0.06 0.07 0.05 0.04 0.06 0.07 0.05 0.04 0.06 40:500 0.07 0.06 0.0 0.05 0.08 0.04 0.0 0.07 0.08 0.04 0.0 0.07 Tabes III, IV and V respecivey is some booking imi vaues for rips, and given = 00. Tabe III dispays he vaues for 50 i = and = 00. Appicaion of he Tabes can be inerpreed as foows.
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 67 Tabe III Booking imi for rip Tabe IV Booking period for rip i 4 i 4 0 0 6 56 90 0 0 5 47 59 0 6 57 9 0 5 48 60 0 6 58 9 0 5 49 6 0 6 47 67 6 50 6 4 0 6 48 68 4 6 50 6 5 0 7 49 69 5 6 5 64 6 0 7 50 70 6 7 5 65 7 0 7 5 7 7 7 5 66 8 8 5 7 8 8 5 67 9 8 5 7 9 8 54 67 0 9 54 74 0 9 54 68 9 55 75 9 54 68 0 56 76 0 54 68 0 57 77 0 54 68 4 58 78 4 4 54 69 5 59 79 5 4 54 69 6 60 80 6 5 55 69 7 6 8 7 5 55 69 8 6 8 8 6 55 70 9 6 8 9 6 55 70 0 6 8 0 7 56 7 4 64 84 7 56 7 4 65 85 8 57 7 5 65 85 8 57 7 4 5 65 86 4 8 58 74 5 5 66 86 5 9 58 75 6 6 66 86 6 9 59 76 7 6 66 87 7 0 59 77 8 7 66 87 8 0 60 78 9 7 66 87 9 0 4 60 79 0 8 66 87 0 4 6 80 8 66 87 4 6 8 9 66 87 4 6 8 9 66 87 4 6 8 4 40 66 87 4 4 64 8 5 40 66 87 5 4 65 84 6 4 66 87 6 5 65 84 7 4 66 87 7 5 66 85 8 4 4 66 87 8 5 66 85 9 4 4 66 87 9 5 67 86 40 5 4 66 87 40 6 67 86 4 5 4 67 87 4 6 68 86 4 6 4 67 87 4 7 68 87 4 6 4 67 88 4 7 69 87 44 7 44 67 88 44 8 69 87 45 7 44 67 88 45 8 70 87 46 8 44 67 89 46 9 70 87 47 9 45 68 89 47 40 7 87 48 0 45 68 89 48 40 7 87 49 45 68 90 49 4 7 88 50 45 68 9 50 4 7 88
68 P.-S. You / Airine Sea Managemen wih Round-Trip Requess Tabe V Booking imi for rip Tabe VI Booking period for rip i 4 i 4 0 00 0 0 0 0 0 0 0 0 4 00 00 00 78 7 6 56 6 00 00 00 405 9 7 6 00 00 00 4 09 8 70 4 00 00 00 4 45 90 76 5 0 00 00 00 5 445 7 00 8 6 9 00 00 00 6 45 5 07 90 7 9 00 00 00 7 46 6 4 96 8 8 00 00 00 8 467 74 0 9 8 00 00 00 9 47 87 07 0 7 00 00 00 0 477 88 4 7 00 00 00 48 9 40 7 6 00 00 00 486 94 46 5 00 00 00 489 97 50 7 4 4 00 00 00 4 49 0 54 5 00 00 00 5 49 06 56 5 6 00 00 00 6 496 0 58 7 7 80 00 00 7 499 4 60 9 8 76 00 00 8 500 9 6 4 9 0 74 00 00 9 500 6 4 0 0 7 00 00 0 500 7 64 44 9 7 00 00 500 0 64 45 8 7 00 00 500 4 65 46 7 7 00 00 500 7 66 46 4 7 7 00 00 4 500 40 66 47 5 6 7 00 00 5 500 4 67 47 6 4 70 00 00 6 500 44 68 48 7 70 00 00 7 500 46 69 48 8 69 00 00 8 500 49 70 49 9 67 00 00 9 500 5 7 49 0 65 00 00 0 500 54 7 50 60 00 00 500 57 7 50 57 00 00 500 59 7 5 55 00 00 500 6 74 5 4 0 54 00 00 4 500 65 75 5 5 0 5 00 00 5 500 68 76 5 6 0 5 00 00 6 500 7 78 5 7 0 5 00 00 7 500 74 79 54 8 0 50 00 00 8 500 77 80 54 9 0 50 00 00 9 500 80 8 55 40 0 49 00 00 40 500 8 8 56 4 0 49 00 00 4 500 86 85 57 4 0 48 00 00 4 500 89 86 58 4 0 48 00 00 4 500 9 88 59 44 0 48 00 00 44 500 95 90 60 45 0 48 00 00 45 500 98 9 6 46 0 47 00 00 46 500 0 94 6 47 0 47 00 00 47 500 04 96 6 48 0 46 00 00 48 500 08 97 64 49 0 46 00 00 49 500 99 65 50 0 45 00 00 50 500 5 0 66
P.-S. You / Airine Sea Managemen wih Round-Trip Requess 69 From Tabe III, if i = 0 exis, a reques for fare casses,, and 4 in rip a period 00 is acceped if and ony if i >, i >, i > 6 and i > 8, respecivey. From Tabe IV, if i = 40 seas remain, a reques for fare casses,, and 4 in rip a period 00 is acceped if and ony if i >, i > 6, i > 67 and i > 86, respecivey. Addiionay, from Tabes V, if i = 0 seas and = 00 periods are avaiabe, a reques for fare casses,, and 4 in rip is acceped if and ony if i >, i > 65, i > 00 and i > 0, respecivey. Besides he booking imi, he criica booking period can aso be used in conroing he booking process for rip. Tabe VI iss some criica booking period vaues for i = 0. If a reques for fare casses,, and 4 in rip appears when i = 0 and i = 40 remain, he reques shoud be acceped if and ony if he arriva ime is 500, 54, 7, 50, respecivey. In his exampe, since he oa number of periods exceeds oa booking capaciies and he booking casses by 5 imes, using he criica booking period o conro he booking process for rip is more efficien han using he booking imi poicy. 5. CONCUSION This invesigaion sudied a sea invenory probem for muipe-fare casses on a simpe airine nework comprising oubound and inbound. Numerous noeworhy modes have dea wih muipe figh eg probem. However, hese have rarey considered round rip requess. Since cusomers can reques round rips in rea ife siuaions, a dynamic mode was proposed herein o dea wih his probem. This sudy aimed o deveop opima booking decisions-making ha aow an airine reservaion sysem o make imey decisions on wheher o accep or reec a reques. The anayica resus demonsrae ha he opima booking poicy is he booking imi poicy, impying ha daa sorage can be reduced. The booking poicy is ha booking for each rip can be conroed using a se of criica booking capaciies. Addiionay, he booking poicy for he round rip can be conroed using a se of criica booking periods, wih he capabiiy of furher reducing daa sorage. The nove mode coud be exended o incude overbookings, no-shows, goshows and canceaions, and such exensions woud be worhy direcions for fuure research. Acknowedgemen: The auhor woud ike o hank reviewer for his/her hepfu commens and suggesions ha greay improved his paper and he Naiona Science Counci of he Repubic of China for financiay supporing his research under Conrac No. NSC9--E-45-00-.
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