Research Article Airline Overbooking Problem with Uncertain No-Shows

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1 Applied Matheatics, Aticle ID 34217, 7 pages Reseach Aticle Ailine Ovebooking Poble with Uncetain No-Shows Chunxiao Zhang, 1,2 Congong Guo, 2 and Shenghui Yi 2 1 Tianjin Key Laboatoy fo Civil Aicaft Aiwothiness and Maintenance, Civil Aviation Univesity of China, Tianjin 33, China 2 College of Science, Civil Aviation Univesity of China, Tianjin 33, China Coespondence should be addessed to Chunxiao Zhang; cxzhang@cauc.edu.cn Received 5 Decebe 213; Accepted 14 Febuay 214; Published 16 Mach 214 Acadeic Edito: Adeei Oluyinka Adewui Copyight 214 Chunxiao Zhang et al. This is an open access aticle distibuted unde the Ceative Coons Attibution License, which peits unesticted use, distibution, and epoduction in any ediu, povided the oiginal wok is popely cited. This pape consides an ailine ovebooking poble of a new single-leg flight with discount fae. Due to the absence of histoical data of no-shows fo a new flight, and vaious uncetain huan behavios o unexpected events which causes that a few passenges cannot boad thei aicaft on tie, we fail to obtain the pobability distibution of no-shows. In this case, the ailines have to invite soe doain expets to povide belief degee of no-shows to estiate its distibution. Howeve, huan beings often oveestiate unlikely events, which akes the vaiance of belief degee uch geate than that of the fequency. If we still egad the belief degee as a subjective pobability, the deived esults will exceed ou expectations. In ode to deal with this uncetainty, the nube of no-shows of new flight is assued to be an uncetain vaiable in this pape. Given the chance constaint of social eputation, an ovebooking odel with discount faes is developed to axiize the pofit ate based on uncetain pogaing theoy. Finally, the analytic expession of the optial booking liit is obtained though a nueical exaple, and the esults of sensitivity analysis indicate that the optial booking liit is affected by flight capacity, discount, confidence level, and paaetes of the uncetainty distibution significantly. 1. Intoduction Ovebooking is a stategy that ailines accept the booking esevations of custoes oe than flight capacity in ode to ake up fo the vacancy loss caused by no-shows who do not show up fo check-in without canceling thei booking equests befoe the flight takes off o ae late fo thei scheduled flights. A lot of exaples can eflect that ovebooking stategy contibutes to huge pofits fo ailines. Sith et al. [1] estiated that 15 pecent of seats on sold-out flights would be lost if ovebooking wee not pacticed and that the benefit of ovebooking at Aeica in 199 exceeded $225 illion. Ovebooking can educe the waste of seats and axiize ailines pofits, but it also bings potential isk. When thenubeofaivalpassengesexceedsflightcapacity,it ight cause that soe aival passenges cannot boad flight (denied boading); thus, ailines need to copensate this pat of passenges (denied-boading copensation), which lead to losses on both social eputation and pofits of ailines. Ovebooking is one of the oldest pobles and ost effective evenue anageent pactices and was officially sanctioned and published by the Aeican Civil Aeonautics Boadin1965[2]. In 1958, Beckann [3]establishedasingle peiod static odel on ovebooking, and a booking-liit policy, which instucted the anage to set a cetain liit to accept esevations up to the liit, was poved to be optial. Rothstein [4] foulated the poble as a nonhoogenous Makoviandecisionpocessandsolveditbyadynaic pogaing ethod in Alstup et al. [5] in 1986 as well poposed a dynaic pogaing foulation of a poble with two cabins, in which the teinal conditions allowed fo upgadinganddowngading. Chatwin [6] in 1993 consideed a ultipeiod ovebooking poble elating to a single-leg flight and a single fae class, gave conditions that ensue a booking-liit policy to be optial, and descibed the continuous odel with stationay faes and efunds as a bith and death pocess. Robet [7] in 25, on the assuption that passenges aiving obeyed the binoial distibution,

2 2 Applied Matheatics established a single-fae-class ovebooking odel and then extended to the conditions of ultiple-fae classes and dynaic. In the pevious eseaches on ovebooking poble, the aival ate of custoes is an ipotant eleent and is usuallyassuedtobeaandovaiable,andpobability theoy plays a geat ipotant chaacte in the optiization of the ovebooking stategy. We will conside an ailine ovebooking poble of a new single-leg flight with discount fae in this pape. Due to the absence of deand data fo a new flight and vaious uncetain huan behavios o unexpected events which causes that a few passenges cannot boad thei aicaft on tie, we have to invite soe doain expets to povide belief degee to estiate the distibution of no-shows. Howeve, huan beings often oveestiate unlikely events, which akes the vaiance of belief degee uch geate than that of the fequency (Kahnean and Tvesky [8]). In this case, if we egad the belief degee as a subjective pobability, the deived esults will exceed ou expectations. In ode to deal with the involved huan uncetainty, uncetainty theoy was founded by Liu [9] in 27 and was efined by Liu [1]in21basedonnoality, duality, subadditivity, and poduct axios. Nowadays, uncetainty theoy has becoe a banch of axioatic atheatics fo odelling huan uncetainty, and a lot of applications can be found in vaious fields such as uncetain pogaing [11], uncetain statistics [1], uncetain isk analysis [12], uncetain eliability analysis [12], uncetain logic [13], uncetain infeence [14], uncetain pocess [15], uncetain calculus [15], and uncetain finance [16]. Based on the theoy of uncetain enewal pocess, Yao and Ralescu [17] investigated the uncetain age eplaceent policy and obtained the long-un aveage eplaceent cost. Zhang and Guo [18] applied the uncetain enewal pocess to the odeing poble of spae pats fo aicafts assuing the inteaival ties to be uncetainty vaiables. With uncetainty theoy, Liu and Yao [19] pesented an uncetain expected value odel fo ultilevel pogaing with equivalent cisp fo. In this pape, we conside the ailine ovebooking poble of new flight in uncetain envionent and assue the nube of no-shows as an uncetain vaiable. The est of this pape is oganized as follows. Section 2 ecalls soe basic concepts and popeties about uncetainty theoy which will be used thoughout the pape. In Section 3, webuild the odel of ailine ovebooking in uncetain envionent andgetsoetheoes.weapplyouodeltopactical ovebooking syste, and sensitivity analysis is given in Section 4. Finally, conclusions ae dawn in Section Peliinaies Let Γ be a nonepty set. A collection L of Γ is called a σalgeba if (a) Γ L; (b) ifλ L, thenλ c L; and (c) ifλ 1,Λ 2,..., L, thenλ 1 Λ 2 L. Each eleent Λ in the σ-algeba L is called an event. Uncetain easue is a function fo L to [, 1]. In ode to pesent an axioatic definition of uncetain easue, it is necessay to assign to each event Λ anubem{λ} which indicates the belief degee that the event Λ will occu. In ode to ensue that the nube M{Λ} has cetain atheatical popeties, Liu [9]poposedthefollowingaxios. Axio 1 (noality axio). Γ. M{Γ} = 1 fo the univesal set Axio 2 (duality axio). M{Λ} + M{Λ c }=1fo any event Λ. Axio 3 (subadditivity axio). Fo evey countable sequence of events Λ 1,Λ 2,...,wehave M { i=1 Λ i } i=1 M {Λ i }. (1) Definition 1 (Liu [9]). The set function M is called an uncetain easue if it satisfies the noality, duality, and subadditivity axios. The tiplet (Γ, L, M) is called an uncetainty space. Besides, in ode to povide the opeational law, Liu [2] defined the poduct uncetain easue on the poduct σalgebel as follows. Axio 4 (poduct axio). Let (Γ k, L k, M k ) be uncetainty spaces fo k=1,2,... Then the poduct uncetain easue M is an uncetain easue satisfying M { i=1 Λ k }= k=1 M k {Λ k }, (2) whee Λ k ae abitaily chosen events fo L k fo k = 1, 2,...,espectively. Definition 2 (Liu [9]). An uncetain vaiable is a easuable function ξ fo an uncetainty space (Γ, L, M) to the set of eal nubes; that is, fo any Boel set B of eal nubes, the set is an event. {ξ B} = {γ Γξ(γ) B} (3) Definition 3 (Liu [9]). The uncetainty distibution Φ of an uncetain vaiable ξ is defined by fo any eal nube x. Φ (x) = M {ξ x}, (4) Definition 4 (Liu [1]). Let ξ be an uncetain vaiable with egula uncetainty distibution Φ(x). Then the invese function Φ 1 (α) is called the invese uncetainty distibution of ξ. Theoe 5 (Liu [9]). Let ξ be an uncetain vaiable with uncetainty distibution Φ. If the expected value exists, then + E[ξ]= (1 Φ(x)) dx Φ (x) dx. (5)

3 Applied Matheatics 3 Uncetain pogaing is a type of atheatical pogaing involving uncetain vaiables [9]. Assue that x is a decision vecto, ξ is an uncetain vecto, and f(x, ξ) is an uncetain objective function. Liu [11]poposedthefollowing uncetain pogaing odel: in x E[f(x, ξ)] subject to : M {g j (x, ξ) } α j, j=1,2,...,p, whee M{g j (x, ξ) } α j (j = 1,2,...,p) ae a set of chance constaints. It is natually desied that the uncetain constaints g i (x,ξ) (j=1,2,...,p) hold with confidence levels α 1,α 2,...,α p. Theoe 6 (Liu [11]). Assue that the constaint function g(x, ξ 1,ξ 2,...,ξ n ) is stictly inceasing with espect to ξ 1,ξ 2,...,ξ k and stictly deceasing with espect to ξ k+1,ξ k+2,...,ξ n.ifξ 1,ξ 2,...,ξ n ae independent uncetain vaiables with uncetainty distibutions Φ 1,Φ 2,...,Φ n, espectively, then the chance constaint holds if and only if g(x,φ 1 1. (6) M {g (x, ξ 1,ξ 2,...,ξ n ) } α (7) (α),...,φ 1 k (α),φ 1 k+1 3. Matheatical Foulation (1 α),...,φ 1 n (1 α)) Fistofall,soenotationsaeadeasfollowsfothe atheatical foulation: (8) :Bookingliitofanewflight n:capacityofanaicaft ξ: The nube of no-shows with uncetainty distibution Φ g: Pice of a new flight ticket :Totalcostofaflight p: Penaltycostwhichispaidtoeachcowed-out custoe who is denied boading because the nube of aival passenges exceeds flight capacity β:adiscountoffae, β 1 s(, ξ): Pofit ate function of the new flight when booking liit is and the nube of no-shows is ξ j: The axiu peissible nube of cowed-out custoes. We suppose that an ailine is planning to open up a new single-leg flight with discount faes and a booking liit should be decided to atch the new flight. Due to the absence of histoical data, the ailine cannot obtain the pobability distibution of no-shows. In this case, the nube ξ of no-shows is assued to be a positive uncetain vaiable, and its uncetainty distibution Φ can be obtained by the belief degee of the invited expets. We suppose that the ailine povides no efunds fo the no-shows, and copensates those cowed-out custoes. Ou objective function is the pofitatethatisthepofitdividedbythetotalcostofanew flight; then it is expessed as gβ { 1, if ξ n s (, ξ) = { gβ ( ξ n) p 1, if ξ n. { Since the nube ξ of no-shows is an uncetain vaiable, the pofit ate function s(, ξ) is an uncetain vaiable as well. As s(, ξ) cannot be diectly axiized, we ay axiize its expected value; that is, (9) ax E [s (, ξ)]. (1) Thepuposeofdecision-akeofailineistofindthe optial booking liit that axiizes the pofit. Howeve, if thenubeofaivalpassengesexceedsflightcapacitywhen the flight takes off, it will cause that soe aival passenges could not boad flight, which lead to not only pofit losses but also bad social eputation. Fo educing the negative ipact on the ailine, a chance constaint is given that the uncetain easue of the event that the nube of cowdedout custoes is less than j is geate than a given confidence level of α;thatis, M { n ξ j} α. (11) In ode to find the optial booking liit,wepesent the following theoes. Theoe 7. Let ξ be a positive uncetain vaiable with an uncetainty distibution Φ.Giventhat gβ { 1, if ξ n s (, ξ) = { gβ ( ξ n) p 1, if ξ n, { (12) the uncetain vaiable s(, ξ) has an uncetainty distibution Ψ (x), if x< gβ ( n) p 1 { Φ( x+ gβ + n), = p if gβ ( n) p 1 x< gβ { 1, if x gβ 1. { 1 (13) Poof. It is easy to know that s(, ξ) gβ/ ( n)p/ 1; theefoe, Ψ (x) = M {s (, ξ) x} =, (14)

4 4 Applied Matheatics fo any x (, gβ/ ( n)p)/ 1.Ifx [gβ/ ( n)p/ 1, gβ/ 1),then Ψ (x) = M {s (, ξ) x} = M { gβ ( ξ n) p = M {ξ x+ gβ p =Φ( x+ gβ p Since s(, ξ) gβ/ 1,wehave 1 x} + n} + n). (15) Ψ (x) = M {s (, ξ) x} =1, (16) fo any x [gβ/ 1, + ).Thetheoeisveified. Theoe 8. Let ξ be a positive uncetain vaiable with an uncetainty distibution Φ.Given that then gβ { 1, if ξ n s (, ξ) = { gβ ( ξ n) p 1, if ξ n, { (17) E[s(, ξ)] = gβ p n Φ (x) dx 1. (18) Poof. The expected value of the uncetain vaiable s(, ξ) is E[s(, ξ)] + = [1 Ψ(x)] dx Ψ (x) dx gβ/ ( n)p/ 1 = 1dx gβ/ 1 + [1 Φ gβ/ ( n)p/ gβ/ 1 dx ( x+ gβ p gβ/ 1 dx= 1dx + n)]dx gβ/ 1 Φ( x+ gβ + n)dx gβ/ ( n)p/ 1 p = gβ p n Φ (x) dx 1. The theoe is veified. (19) Since the constaint function ( n ξ j)is stictly deceasing with espect to the uncetain vaiable ξ,the chance constaint (11) can be conveted to cisp foula accoding to Theoe 6;then n+j+φ 1 (1 α). (2) Theefoe, the uncetain pogaing odel is developed as follows: ax 4. Application [ gβ p n Φ (x) dx 1] subject to : n+j+φ 1 (1 α). (21) In this section, we will pesent a nueical case to illustate the afoeentioned odel. Suppose that the ovebooking poble is consideed fo a new single-leg flight of an ailine. Assue that the nube ξ of no-shows is a positive uncetain vaiable with linea uncetainty distibution L(a, b) (a ) (Liu [21]); that is,, if x a {(x a) Φ (x) = { (b a), if a x b { 1, if x b. (22) Then, the invese uncetainty distibution of linea uncetain vaiable L(a, b) (Liu [21]) is Φ 1 (α) = (1 α) a αb. (23) Fo the uncetain pogaing odel (21), we can obtain that the optial booking liit with discount fae is gβ (b a) ( +a+n) (n+j+αa+(1 α) b), { p = if gβ p { { n+j+αa+(1 α) b, if gβ > p. (24) We conside two failia types of aicaft A32-2 and B747-3 with capacity of 15 and 416, espectively. We give that a = and the pice of the flight ticket g = RMB 96 yuan. Accoding to the geneal povision set by the ailine, we suppose that the penalty cost p is 3%ofthefacevalueof the ticket and p will be counted by RMB 2 yuan if it is less than RMB 2 yuan; that is, p=ax {1.3gβ, gβ + 2}. (25) Fo instance, if the uppe liit of no-shows b = 8,the peissible axiu of cowed-out passages j=5,aicaft capacity n = 15, the chance constaint α =.95,thediscount β =.4, then the optial booking liit is 155. Figue 1 shows the optial booking liit changing with discount β fo b = 8, j = 5. On the whole,

5 Applied Matheatics β β α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 (a) n = 15 (b) n = 416 Figue 1: Optial booking liit unde the changes of α and β when b=8,j= β β α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 (a) n = 15 (b) n = 416 Figue 2: Optial booking liit unde the changes of α and β when b=8,j=8. with a gadually educed slope, the optial booking liit inceases in discount β till β eaches a cetain value. The optial booking liit does not incease in the confidence level α, and,whenβ.4, α has no effect on.bythe copaison of Figues 1(a) and 1(b), we can see that the incease of caused by the incease of flight capacity n is distinct. Figue 2 shows the optial booking liit changing with discount β fo b=8, j=8.copaedwithfigue 1, due to a little incease of j, inceases still in β and n,while the vaiation tend of with β has nealy nothing to do with α. As can be seen fo the copaison between Figues 1 and 2, the tuning point of the slope of the optial booking liit cuve inceases in j.

6 6 Applied Matheatics b b α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 (a) n = 15 (b) n = 416 Figue 3: Optial booking liit unde the changes of α and b when β=.8,j= b b α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 α = 1. α =.95 α =.9 α =.85 α =.8 α =.75 (a) n = 15 (b) n = 416 Figue 4: Optial booking liit unde the changes of α and b when β =.8, j = 8. Figues 3 and 4 show the optial booking liit changing with the paaete b fo j = 5 and j = 8, espectively. The optial booking liit inceases in b, and, when b eaches a cetain value, the slope becoes salle.inthecaseofα = 1.,whenb is bigge than a cetain value, eains the sae. We also obseve that still does not incease in the confidence level α, and the close α gets to 1, the less sensitive is to b.theflightcapacityn still has a geat influence on. AscanbeseenfothecopaisonbetweenFigues3 and 4, the tuning point of the slope of the optial booking liit cuve inceases in j.

7 Applied Matheatics 7 5. Conclusions Ovebooking poble is an ipotant stategy fo ailine evenue anageent. Fo a new flight, thee is no histoical date to obtain the pobability distibution of the nube of no-shows; theefoe, stochastic ethod is not suitable. This pape intoduced an uncetain vaiable to descibe thenubeofno-shows.anailineovebookingodel with discount fae was developed with chance constaint, anditwasconvetedtoacisppogaingbasedon uncetain pogaing theoy. Though a nueical case, we obtained the analytic expession of the optial booking liit to axiize the pofit ate function of the new flight. The esults of sensitivity analysis indicated that flight capacity, discount, confidence level, and paaetes of the uncetainty distibution significantly affected the optial booking liit. Howeve, thee is still a lot of wok to be done in the futue eseach. The dynaic natue of the booking pocess can be taken into account. Futheoe, andoness and uncetainty ay coexist in pactical situation; thus an uncetain ando ovebooking odel ight be ou futue eseach. Conflict of Inteests The authos declae that thee is no conflict of inteests egading the publication of this pape. [1] B. Liu, Uncetainty Theoy: A Banch of Matheatics fo Modeling Huan Uncetainty, Spinge, Belin, Geany, 21. [11] B. Liu, Theoy and Pactice of Uncetain Pogaing,Spinge, Belin, Geany, 2nd edition, 29. [12] B. Liu, Uncetain isk analysis and uncetain eliability analysis, Uncetain Systes,vol.4,no.3,pp ,21. [13] B. Liu, Uncetain logic fo odeling huan language, Jounal of Uncetain Systes,vol.5,no.1,pp.3 2,211. [14] B. Liu, Uncetain set theoy and uncetain infeence ule with application to uncetain contol, Uncetain Systes, vol. 4, no. 2, pp , 21. [15] B. Liu, Fuzzy pocess, hybid pocess and uncetain pocess, Uncetain Systes,vol.2,no.1,pp.3 16,28. [16] B. Liu, Towad uncetain finance theoy, Uncetainty Analysis and Application,vol.1,aticle1,213. [17] K. Yao and D. A. Ralescu, Age eplaceent policy in uncetain envionent, Ianian Fuzzy Systes, vol. 1, no. 2, pp.29 39,213. [18] C. Zhang and C. Guo, An odeing policy based on uncetain enewal theoy with application to aicaft spae pats, [19] B. Liu and K. Yao, Uncetain ultilevel pogaing: algoith and applications, [2] B. Liu, Soe eseach pobles in uncetainty theoy, Jounal of Uncetain Systes,vol.3,no.1,pp.3 1,29. [21] B. Liu, Uncetainty Theoy, Uncetainty Theoy Laboatoy, 4th edition, 213. Acknowledgents ThiswokissuppotedbytheFundaentalReseachFunds fo the Cental Univesities unde Gant no. ZXH212K5 and the Undegaduate Taining Pogas fo Innovation and Entepeneuship no. IECAUC1337. Refeences [1] B. C. Sith, J. F. Leikuhle, and R. M. Daow, Yield anageent at Aeican Ailines, Intefaces, vol. 22, no. 1, pp. 8 31, [2]K.T.TalluiandG.vanRyzin,The Theoy and Pactice of Revenue Manageent, Kluwe, Boston, Mass, USA, 24. [3] M. J. Beckann, Decision and tea poble in ailine esevations, Econoetica,vol.26,no.1,pp ,1958. [4] M. Rothstein, An ailine ovebooking odel, Tanspotation Science, vol. 5, pp , [5] J.Alstup,S.Boas,O.B.G.Madsen,andR.V.V.Vidal, Booking policy fo ights with two types of passenges, Euopean Jounal of Opeational Reseach, vol. 27, no. 3, pp , [6] R.E.Chatwin,Optial ailine ovebooking [Ph.D. dissetation], Depatent of Opeations Reseach, Stanfod Univesity, Palo Alto,Calif,USA,1993. [7] L. P. Robet, PicingandRevenueOptiization, Stanfod Univesity Pess, Palo Alto, Calif, USA, 25. [8] D. Kahnean, A. Tvesky, and Pospect theoy:, An analysis of decision unde isk, Econoetica, vol. 47, no. 2, pp , [9] B. Liu and Uncetainty Theoy Spinge, Belin, Geany, 2nd edition, 27.

8 Advances in Opeations Reseach Advances in Decision Sciences Applied Matheatics Algeba Pobability and Statistics The Scientific Wold Jounal Intenational Diffeential Equations Subit you anuscipts at Intenational Advances in Cobinatoics Matheatical Physics Coplex Analysis Intenational Matheatics and Matheatical Sciences Matheatical Pobles in Engineeing Matheatics Discete Matheatics Discete Dynaics in Natue and Society Function Spaces Abstact and Applied Analysis Intenational Stochastic Analysis Optiization