Modelling for Cruise Two-Dimensional Online Revenue Management System

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1 Modellng for Crue Two-Dmenonal Onlne Revenue Management Sytem Bngzhou L Modellng for Crue Two-Dmenonal Onlne Revenue Management Sytem Bngzhou L Shool of Management, Xamen Unverty, Chna, lbngzhou260@63.om do: 0.456/dta.vol4.ue6.7 Abtrat To olve the rue two-dmenonal revenue management problem and develop uh an automated ytem under unertan envronment, a tat model whh a tohat nteger programmng frtly ontruted to mamze the total epeted revenue from all nd of rue produt. Four method an be appled to olve the above model, whh are hane ontraned programmng, robut optmzaton, determnt programmng, and bd-pre ontrol. n the hane ontraned programmng method, the tohat ontrant are onverted nto determnt equvalent form. n the robut optmzaton method, the model tranformed nto a goal programmng formulaton wth a enaro-baed derpton of problem data. n the determnt programmng method, the tohat demand varable dretly replaed wth the mean value or epeted value of demand. n the bd-pre ontrol, the rule for aeptng rue produt are propoed. Further, to onder tme-varable demand and nreae the proft, a dynam apaty alloaton model for rue two-dmenonal revenue management put forward by applyng Marov Deon Proe. Then the aept/reet optmal pole for a boong requet of rue produt are obtaned. The onluon are a follow: () the apaty n the rue lne ndutry two-dmenonal, that, the number of abn and lfeboat eat an both affet avalablty of rue produt; (2) the demand for all nd of rue produt unertan and the unertanty an be oped wth four oluton method; (3) the haratert of the rue ndutry and tme-varable demand have to be norporated nto the tat and dynam apaty alloaton model to mamze the epeted revenue of the rue lne. Keyword: Crue revenue management, Two-dmenonal, Demand unertanty, Cutomer type. ntroduton Revenue management the method and prate of ontrollng the avalablty and prng of produt or erve n dfferent boong lae to mamze epeted revenue or proft (MGll and van Ryzn, 999) []. t wor by attendng to the partular, not the general (Orn, 2003) [2]. The rue lne ndutry grow very fat n reent year. The rue lne ndutry ha typal haratert for applyng revenue management, nludng fed apaty, perhable nventory, maret egmentaton, advaned boong, hgh fed ot and low varable ot, large demand flutuaton. But one of the mportant dfferene between the rue lne ndutry and other travel ndutre that the apaty ha two dmenon, whh are the number of abn and lfeboat eat. There are dfferent type of utomer, uh a famle and ngle. Thee haratert are a lttle mlar a the ontaner hppng ndutry [3]-[4]. f the rue produt defned to be the attrbute ombnaton of a abn ategory, fare la, and number of guet, then the rue lne fae the followng problem: under the envronment of demand unertanty and the ontrant of two-dmenonal apaty, how to alloate the lmted apaty to all nd of rue produt, or how many boong requet for eah nd of rue produt hould be aepted to mamze the total revenue? To olve th problem, th paper propoe two type of model: tat and dynam model. n a tat model, boong lmt are et for eah rue produt n the begnnng of the boong proe. Whenever reerved boong lmt for a produt reahed, aoated produt loed. A dynam model et the boong lmt for eah rue produt aordng to the atual boong throughout the entre boong proe. Untl now, there are only a few lterature about rue two-dmenonal revenue management. Behn (2006) formulate a determnt lnear program onderng the lfeboat eat apaty ontrant, but th model a mplfed veron whh aume that eah produt ont of eatly one abn 72

2 nternatonal ournal of Dgtal Content Tehnology and t Applaton Volume 4, Number 6, September 200 and at leat two guet [5]. and Mazzarella (2007) preent an effetve oluton for rue nventory applaton that norporate a neted la alloaton (NCA) model whh a modfed veron of EMSR (Belobaba, 989) and a dynam la alloaton (DCA) model whh adapted from the method of Lee and Herh (993) [6]-[8]. 2. Stat model 2.. Model ontruton The rue lne ell produt =,, and =,,. Here, repreent any produt ontng of eatly one abn and a ngle guet and repreent any produt ontng of eatly one abn and at leat two guet. All tay are of the ame length. Let D denote the demand of produt =,,, a tohat varable, and D denote the demand of produt =,,, a tohat varable too. f defned a the fare for a ngle guet, f d a the fare for a double oupany guet and f a a any addtonal guet(regardle of age) n the abn beyond the frt two double oupany guet. the number of requet of the produt =,, that are aepted and the number of requet of the produt =,, that are aepted. and are the deon varable. b the number of guet of produt. Other notaton are a follow: a : f produt ue abn type, a = ; otherwe, a = 0. a : f produt ue abn type, a = ; otherwe, a = 0. C : the mamum avalable apaty for type- abn, =,, K.. C : the mamum avalable lfeboat eat apaty. The ba model formulated a the followng:.t. d a = = ma{ f [2f f (b 2) ]} = = a a C, =,...,K () b C (2) = = D, =,..., (3) D, =,..., (4) N {0}, =,..., (5) N {0}, =,..., (6) (Model ) The obetve funton to mamze the revenue of the rue lne when there a tohat demand of utomer from dfferent maret egment, nludng ngle and famle. The frt ontrant how that the total number of reervaton doe not eeed the apaty of the mth type of abn. The eond ontrant how that the total number of guet doe not eeed the lfeboat eat apaty. The thrd and fourth ontrant mean that the number of requet of produt that are aepted hould be not more than the demand to prevent vaant abn. A D and D are randomzed varable, the thrd and fourth one denote unertan ontrant. The ffth and th ontrant are the non-negatve nteger ontrant. Therefore, Model a tohat nteger programmng model Model oluton Chane ontraned programmng A there are randomzed varable D and D n the ontrant, aordng to the thought of hane 73

3 Modellng for Crue Two-Dmenonal Onlne Revenue Management Sytem Bngzhou L ontraned programmng, we et the onfdene level a α and α for the thrd and fourth ontrant of Model [9]. The thrd ontrant an be onverted nto the followng hane ontrant: Pr{ D } α (7) The fourth ontrant an be onverted nto the followng hane ontrant: Pr{ D } α (8) Aordng to the method of Lu and Zhao (2003) [0], the eventh and eghth hane ontrant an be repetvely onverted nto determnt equvalent form: - Mα = up{m M = (α )} - Mα = up{m M = ( α )} (9) - Where () the revere funton of dtrbuton funton of the demand varable D, and - () the revere funton of dtrbuton funton of the demand varable D. Then Model now tranformed nto Model 2 a the followng:.t Robut optmzaton d a = = ma{ f [2f f (b 2) ]} = = a a C, =,...,K b C = = K α K α M = up{m M = (α )} - α M = up{m M = (α )} - α N {0}, =,..., N {0}, =,..., (Model 2) Mulvey, Vanderbe and Zeno(995) and Ba,Carpenter and Mulvey(997) gve the defnton of robut optmzaton that t a novel approah to ntegrate goal programmng formulaton wth a enaro-baed derpton of problem data to olve tohat programmng problem []-[2]. Th approah an meaure the tradeoff between oluton robutne (meaure whether the oluton optmal) and model robutne (meaure whether the oluton feable). Baed on the method of Yu and L(2000) [3], Model an be tranformed nto Model 3 a the followng: S S S ma pπ λ p π pπ 2δ = = = S p ω (D- 2ε ) ω (D - 2ε ) = = = 74

4 nternatonal ournal of Dgtal Content Tehnology and t Applaton Volume 4, Number 6, September 200.t. S π pπ δ 0, =,...S = = = D ε 0, =,...,; =,...,S D ε 0, =,...,; =,...,S a a C, =,...,K b C = = ma{d }, =,...,; =,...,S ma{d }, =,...,; =,...,S δ 0, ε 0, ε 0, =,...,; =,...,; =,...,S N {0}, =,..., N {0}, =,..., d a = = (Model 3) π = f [2f f (b 2) ], =,...,S n Model 3, =,, S repreent one of all enaro where unertan data appear. p mean the realzaton probablty of the enaro, uh that p 0 and S = p =. The parameter λ a r trade-off fator for the deon-maer of the rue lne, whle the parameter ω and ω are the penalty weght for the feablty robutne. n the obetve funton, the frt term the epeted revenue and the eond term abolute devaton of the revenue, repetvely. We an regard thee two term a a meaurement of oluton robutne trade-off together. The thrd term of the obetve funton denote the mean abolute devaton for the ontrant volaton, and we an regard t a a meaurement of model robutne trade-off Determnt lnear programmng For thoe problem wth tohat varable, a tradtonal oluton method determnt programmng. The man dfferene between determnt programmng and tohat programmng how to treat wth the tohat demand. Determnt programmng dretly replae the tohat demand varable wth the mean value or epeted value of demand. The determnt lnear programmng model a follow:.t. d a = = ma{ f [2f f (b 2) ]} = = a a C, =,...,K b C = = D, =,..., D, =,..., N {0}, =,..., N {0}, =,..., Where D = E[D ] and D = E[D ]. Here, E[ ] denote the mean or epeted value. (Model 4) 75

5 Bd-pre ontrol Modellng for Crue Two-Dmenonal Onlne Revenue Management Sytem Bngzhou L n bd-pre ontrol, threhold or bd pre are et for the reoure (abn and lfeboat eat) and a rue produt old only f the offered fare eeed the um of the threhold pre of all the reoure needed to upply the produt [3]. The dual varable an be ued a bd pre. The requet for produt aepted f f μ μ (0) The varable l μ repreent the dual aoated wth the th apaty ontrant and μ l repreent the dual aoated wth the lfeboat eat apaty. For eample, uppoe a ngle peron requet a abn from the th abn type. The guet pay 2000 dollar. The peron requet aepted f The requet for produt aepted f 2000 μ μ f l l μ b μ () For eample, uppoe a four-peron famly requet a abn from the th abn type. The frt two guet pay 200 dollar eah and the addtonal famly member pay 500 dollar eah. The famly requet aepted f 3400 μ 4μ 3. Dynam model Conder a rue lne wth abn apaty C (a vetor, C K = (C,...,C ) ) and lfeboat eat apaty C. All tay are of the ame length. The boong horzon dvded nto T perod and at mot one boong requet arrve eah perod. Tme perod are numbered n revere hronologal order and the begnnng of the boong horzon perod t = T, and the rue hp depart n perod t = 0. We aume that eah boong requet belong to one of produt and, and that utomer arrval are ndependent aro tme perod. Suppoe p t the probablty that produt- boong requet arrve n perod t, p t the probablty that a produt- boong requet arrve n perod t and P = P P 0t t t = = the probablty that no boong requet arrve n perod t. Eah rue produt ha two dmenon, that, abn type and number of guet and t unt nome r or r. a d a There are two relatonhp: r = f and r = f ( 2)f. When a requet arrve, the rue lne ha b K to dede whether to aept or reet the requet. We defne C = (C,...,C ), A= a a, where A a matr. The mean of other notaton are the ame a n the tat model. Ue u(,) t a the epreon of the value funton whh denote the mamum epeted revenue that an be obtaned from perod untl the tme of departure of the rue hp, when the rue abn apaty left and the rue lfeboat eat apaty left. The obetve of the rue lne to mamze the epeted revenue u T (C C,C ) over the whole boong horzon. The value funton an be omputed reurvely through the Bellman optmalty equaton. We an ontrut the dynam programmng model for apaty alloaton of rue two-dmenonal revenue management a the followng: t t t- t- = t t- t- = u (, ) = p ma{r u ( -A, -),u (, ) 0t t- l p ma{r u ( -A, -b ),u (, ) (2) p u (, ) t=,2,...,t. 76

6 nternatonal ournal of Dgtal Content Tehnology and t Applaton Volume 4, Number 6, September 200 The boundary ondton are u(,) 0 = 0 (3) u (,0) = 0, t =,2,...,T. (4) t t u ((0,0,...,0), ) = 0, t =,2,...,T. (5) The optmal poly for apaty alloaton aept a boong requet for rue produt n perod when the tate (, ) f and only f r u (, )- u ( -A, -) (6) t- t- The optmal poly for apaty alloaton aept a boong requet for rue produt n perod when the tate (, ) f and only f r u (, )- u ( -A, -b ) (7) t- t- (6) and (7) mean that f and only f the epeted nremental revenue from the requet not le than the epeted drop n future revenue from aeptng a requet for produt or, an optmal poly aept t. f there are many rue produt, dynam tohat program, the dynam model may be omputatonally ntratable and the probablt mathematal programmng model an be ued to appromate dynam tohat program. We an ue a Lagrangan relaaton (LR) method to olve the probablt model [4]. 4. Conluon and future tudy The man onern of th paper that the apaty n the rue lne ndutry two-dmenonal and the demand for all nd of rue produt unertan. n th paper, we frt put forward a tat model whh onder two-dmenonal apaty ontrant, demand unertanty and dfferent utomer type nludng famle and ngle. Th tat model a tohat nteger programmng model and we an apply four method to olve t whh are hane ontraned programmng, robut optmzaton, determnt programmng, and bd-pre ontrol. Seond, we propoe a dynam apaty alloaton model for rue two-dmenonal revenue management. Th dynam model a drete Marov Deon Proe model and t obetve to gan an aeptng or reetng optmal poly that mamze the epeted revenue from rue produt boong. How to olve th dynam model well and fat one of future tudy dreton. n addton, the model and method from th paper need further empral tetng. 5. Referene [] MGll, van Ryzn G, Revenue Management: Reearh Overvew and Propet, Tranportaton Sene, vol.33, no.2, pp , 999. [2] Orn E, The Emergng Role of Funton Spae Optmzaton n Hotel Revenue Management, ournal of Revenue and Prng Management, vol.2, no.2, pp.72-74, [3] Xao B, Yang W, Revenue Management wth Multple Capaty Dmenon, Worng Paper, Shool of Bune, Long land Unverty, New Yor, [4] L Bngzhou, A Stohat Model for Dynam Capaty Alloaton of Contaner Shppng Two-Dmenonal Revenue Management, n Proeedng of 2008 nternatonal Conferene on Serve Sytem and Serve Management, E nde, un [5] Behn N, A Crue Shp Not a Floatng Hotel, ournal of Revenue and Prng Management, vol.5, no.2, pp.35-42, [6] L, Mazzarella, Applaton of Modfed Neted and Dynam Cla Alloaton Model for Crue Lne Revenue Management, ournal of Revenue and Prng Management, vol.6, no., pp.9-32,

7 Modellng for Crue Two-Dmenonal Onlne Revenue Management Sytem Bngzhou L [7] Belobaba P P, Applaton of a Probablt Deon Model to Arlne Seat nventory Control, Operaton Reearh, vol.37, no.2, pp , 989. [8] Lee T C, Herh M, A Model for Dynam Arlne Seat nventory Control wth Multple Seat Boong, Tranportaton Sene, vol.27, no.3, pp , 993. [9] Chane A, Cooper W W, Chane-Contraned Programmng, Management Sene, no., pp , 959. [0] Lu B, Zhao R, Wang G, Unertan Programmng and t Applaton, Tnghua Unverty Publher, Beng, pp.79-0, [] Mulvey M, Vanderbe R, Zeno S A, Robut Optmzaton of Large-Sale Sytem, Operaton Reearh, vol.43, no.2, pp ,995. [2] Ba D, Carpenter T, Mulvey M, Mang a Cae for Robut Optmzaton Model, Management Sene, vol.43, no.7, pp , 997. [3] Tallur K, van Ryzn G, An Analy of Bd-Pre Control for Networ Revenue Management, Management Sene, vol.44, no., pp , 998. [4] ang H. A Lagrangan Relaaton Approah for Networ nventory Control of Stohat Revenue Management wth Perhable Commodte, ournal of the Operatonal Reearh Soety, no.59, pp ,