A stochastic approach to hotel revenue optimization

Transcription

1 Computers & Operatons Research 32 (2005) A stochastc approach to hotel revenue optmzaton Kn-Keung La, Wan-Lung Ng Department of Management Scences, Cty Unversty of Hong Kong, Kowloon Tong, Hong Kong Abstract Ownng to smlar busness nature, t should be possble to drectly mgrate successful arlne revenue management technques to the hotel doman. However, one of the salent derences between arlnes and hotels s rarely hghlghted the network structure of length of stay or the dsplacement eect. The hotel patrons go from a rst stay-over nght to a last stay-over nght n consecutve nght stays. The arrval demands for mult-nght stays and the lengths of stay are stochastc n nature. In ths paper, we propose a network optmzaton model for hotel revenue management under an uncertan envronment. The network optmzaton s n a stochastc programmng formulaton so as to capture the randomness of the unknown demand (unknown number of arrvals and length of stays). A novel approach of robust optmzaton technques for stochastc programmng s appled to solve the problem. We also dscuss the strateges for hotel management to take nto account of rsk trade-o; derent prcng polces; cancellatons and no-show; early check-outs; extended stay and over-bookng are dscussed. We showed that our proposed model can be moded to adopt these strategc consderatons.? 2003 Elsever Ltd. All rghts reserved. Keywords: Stochastc programmng; Servce operatons 1. Introducton Revenue management (also known as yeld management) s used to nd optmal nventory allocaton and schedulng strateges as well as prce settng for pershable assets so as to maxmze revenue wthn the plannng horzon. Revenue management s rooted n the arlne ndustry, n whch revenue management systems have been appled for over 40 years [1]. Many successful revenue management systems have led to hundreds of mllon of dollars of mprovement n revenue e.g. [2,3]. Other ndustres wth smlar characterstcs to arlnes are n the mdst of developng ther revenue management systems. The hotel busness has the hghest potental for applcaton of revenue Correspondng author. Tel.: ; fax: E-mal address: mskkla@ctyu.edu.hk (K.-K. La) /$ - see front matter? 2003 Elsever Ltd. All rghts reserved. do: /j.cor

2 1060 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) management technques as hotels share very close characterstcs wth arlnes. For example: () both hotel rooms and ar-seats are pershable and cannot be stored for future sale; () capacty s usually xed and the cost of nstant expanson s very hgh (loss of goodwll and hgh costs n movng customers to other compettor hotels); and () advance bookng s allowed (and thus cancellatons, no-shows and overbookng problems exst). It would appear possble to drectly mgrate successful arlne revenue management tools, such as overbookng models [3], nventory allocaton for nested or non-nested models [2] to the hotel doman. However, one of the salent derences between arlnes and hotels s rarely hghlghted the network structure of length of stay [4] or the dsplacement eect [5]. Hotel patrons go from a rst stay-over nght to a last stay-over nght on consecutve nght stays. The arrval demands for mult-nght stays and the lengths of stay are unfortunately stochastc n nature. There s a very rch lterature on revenue management for the arlne ndustry. For the hotel busness, quanttatve tools for solvng revenue management problem are relatvely lmted when the dsplacement eect of mult-nght stays s taken nto consderaton. However, Weatherford [6] reported that takng the length of stay nto account n hotel revenue management can ncrease revenue by as much as 2.94%. In the hotel busness, ths means a savng n of mllons of US dollars each year. To our best knowledge from a revew of the lterature, smulaton s the man tool currently used for the nvestgaton of hotel revenue management problems of mult-nght stays. Weatherford [6] and Brtan and Modschn s [7] used smulaton models wth data from hotels to test ther heurstc approaches of accept/reject decsons for bookng. Baker extended the above studes and compared them wth some more of hs heurstc models for overbookng and allocaton [4]. In ths paper, we propose a network optmzaton model for hotel revenue management under an uncertan envronment. The network optmzaton s n a stochastc programmng formulaton so as to capture the randomness of the unknown demand (unknown number of arrvals and unknown length of stays). A novel approach of robust optmzaton technques s appled to solve the problem. The paper s organzed as follows. Notatons and parameters used n ths paper wll be ntroduced n Secton 2, and the basc mathematcal model formulaton presented n Secton 3. The mportance of stochastc programmng and the soluton scheme for robust optmzaton wll be dscussed n Secton 4. Some numercal llustratve examples are presented n Secton 5. Secton 6 shows some other consderatons for hotel revenue management systems wth manageral mplcatons for parameter settng n our proposed mathematcal models. Fnally, some concludng remarks and future research recommendatons are gven n Secton Notatons and assumptons The followng are the major notatons for parameters and varables used n ths paper: x ;j s the number of bookngs accepted (decson varables) for check-n on day and check out on day j where 0 6 j6 T. ={0; 1; 2;:::;T 1) s tme ndex for check-n and j ={1; 2; 3;:::;T) s ndex for check-out, C s total capacty,

3 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Check-ns on day k Accrual stay-over k-1 k k+1 Check-outs on day k Fg. 1. Flows of check-ns and check-outs for day k. R ;j s revenue ganed per bookng wth check-n on day and check-out on day j, U ;j s bookng demand for check-n on day and check-out on day j. Note that T j=+1 x ;j s number of check-ns on day, j 1 =0 x ;j s number of check-outs on day j. We assume there are no customers stayng before day 0 and all customers have to leave the hotel on or before day T. It s also assumed that any customer who has checked-n has to stay at least one nght. 3. Stochastc network formulaton The check-ns and check-outs can be vewed as the ows n and out of the nodes n a network. We consder a partcular day, day k (k = {1; 2;:::;(T 1)}), n the plannng perod (Fg. 1). The followng equaton models the hotel s occupaton status on day k for k =1; 2; 3;:::;(T 1): =0 j=k+1 x ;j + j=k+1 x k;j x ;k : (1) =0 Wth lmted capacty, we should have the followng constrants for day k =1; 2; 3;:::;(T 1): =0 j=k+1 x ;j + j=k+1 x k;j x ;k 6 C: (2) =0 On day 0, we assume there are no check-outs and no stay-over accrued. We have the followng equaton for day 0: x 0;j 6 C: j=1 (3) We follow the basc dea of the mathematcal model gven n Raesde and Wndle [8].

4 1062 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Max S:t: T 1 =0 j=k+1 R ;j x ;j x ;j + x 0;j 6 C; j=1 j=k+1 x ;j 6 U ;j ; x ;j 0; for all 0 6 j6t: x k;j x ;k 6 C; =0 (4) 4. Stochastc formulaton and robust optmzaton soluton scheme The problem looks lke a lnear nteger programmng problem. Unfortunately, the parameters U ;j n (4) are usually uncertan at the begnnng of plannng perod. Moreover, the revenues may not be xed, as the decson-maker would lke to set derent prcng, whch n turn results n derent demands. One may want to solve ths by replacng the parameters by ther best pont estmator, for nstance, usng expected value E(U ;j ) to replace the uncertan parameter of U ;j. Although sometmes practtoners can obtan reasonably success by usng expected value approaches, a drawback of ths approach s that we may not always guarantee the soluton s feasble. One may then carry out senstvty analyss for some correctve acton. Such an approach s commented as a reactve one [9]. We beleve decson-makers would prefer to use proactve tools to obtan ther solutons. Whle t s mpossble to remove the uncertanty fully, the best way to make decsons under an uncertan envronment s to accept uncertanty rst, and then understand uncertanty and put t nto the plannng decson model. Stochastc programmng tools are based on ths dea. Robust optmzaton [9,10] s one of the proactve approaches used to solve stochastc problems, and t represents an ntegraton of goal programmng and the scenaro-based descrpton of unknown data. We dene the followng measurements of robustness: Denton 4.1 (Soluton robustness): An optmal soluton s soluton robust wth respect to optmalty f t remans close to optmal for any scenaro s. Denton 4.2 (Model robustness): An optmal soluton s model robust wth respect to feasblty f t remans almost feasble for any scenaro s. It s assumed that the decson-maker has a set of scenaros s = {1; 2;:::;S} assocated wth unknown parameters. For each scenaro, the correspondng probablty s P s such that P s 0 and S p s =1. The phlosophy of robust optmzaton s bult on the trade-o between soluton robustness and model robustness.

5 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) We can then transform our formulaton nto a robust optmzaton model as the followng. Max S T 1 p s R s ;jx ;j S p s T 1 S T 1 R s ;j x ;j p s R s ;jx ;j S:t: S =0 j=k+1 T 1 p s x ;j + x 0;j 6 C; j=1 x ;j 6 max{u s ;j}; x ;j 0; j=k+1 for all s ; 0 6 j6 T; w ;j U s ;j x ;j x k;j x ;k 6 C; =0 (5) where and w ;j are non-negatve weghtng parameters. The rst term n the objectve functon s the expected revenue, whle the second term s the mean absolute devaton of the revenue. When the revenue of derent scenaros are wldly spread, t wll result a larger value of mean absolute values, the penalty wll then be ncreased. Together, these two terms can be vewed as a measurement of trade-o of soluton robustness. The parameter can be regarded as a rsk trade-o factor, between expected revenue and devaton, for the decson-maker. The absolute devaton n the thrd term s a model robustness measurement whle the parameters w ;j are the penalty weghts for the constrants volatons. By usng the mean absolute values as penaltes, the model can generate solutons whch are robust n all scenaros. The mean absolute devaton terms, however, ntroduce some complexty owng to ncreasng number of artcal varables when the model s solved usng lnear programmng. Yu and L [11] propose an mprovement n computatonal ecency of the robust optmzaton wth mean absolute devaton lke formulaton (5). The mean absolute value s transformed n a lnear terms by a lnearzaton method below. Theorem 4.1 (Yu and L [11]). A goal programmng Mnmze Z = f(x ) g ; Subject to X F; (6)

6 1064 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) where F s a feasble set and can be lnearzed usng the followng form: Mnmze Z = f(x ) g +2 Subject to g f(x ) 6 0; 0 X F: (7) Proof. The new varable has a postve coecent n objectve functon. The mnmzaton wll thus force the varable to take ts possble mnmum value. Notce that from the assocated constrants, we have g f(x ) and 0. In other words, the mnmum possble value of =mn{g f(x ); 0}. If f(x ) g 0, then the constrant g f(x ) 6 0 wll always be satsed for all values of 0. So wll be forced to take ts possble mnmum value,.e. =0. Hence, Z =Z. On the other hand, f f(x ) g 0, then the mnmum possble value of s g f(x ). Then Z = g f(x )=Z. In other words, the two formulatons (6) and (7) are equvalent. The proof s completed. To apply Theorem 4.1 to our problem (5), we ntroduce a set of non-negatve varables z s and y;j s for all scenaros where s. Our robust optmzaton model s now converted nto the followng: S T 1 S T 1 S T 1 Max p s R s ;jx ;j R s ;jx ;j p s +2z s S:t: S =0 j=k+1 T 1 p s x ;j + x 0;j 6 C; j=1 T 1 j=k+1 p s w ;j [U s ;j x ;j +2y s ;j] x k;j x ;k 6 C; =0 S T 1 R s ;jx ;j p s x ;j y;j s 6 U;j; s x ;j 6 max{u;j}; s x ;j ;z s ;y;j s 0; for all s ; 0 6 j6 T: R s ;jx ;j + z s 6 0; R s ;jx ;j The promnent feature of formulaton (8) s that t s now n a lnear programmng form and ready to be solved by popular lnear modellng packages lke LINDO [12] when the weghtng parameters are assgned by the decson maker. (8)

7 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Table 1 Demands for sngle scenaro example Illustratve examples 5.1. Sngle scenaro (determnstc) We consder rst a sngle scenaro (determnstc) example for a revenue management problem for a busness hotel under certan demand. The plannng horzon s set to be 10 days (startng from a Sunday). The hotel has a maxmum of 400 rooms. For smplcty, the unt rate for each room nght s kept constant at The revenue for any sngle or multple nght stay s lnearly proportonal to the length of stay. Snce the parameters are all determnstc, there s no devaton. The problem s a standard nteger programmng model of formulaton (4). Demands for all pars of (; j), U ;j are forecast as shown n Table 1. The hotel s customers are manly busness travellers. The demand s usually hgher for medan length stays (e.g. 3-nghts or 4-nghts) whle the demand for short stays (1-nght, 2-nghts) are lower. Check-out rates on Thursday and Frday are usually hgh, whereas demands for stays over the weekend are low. Long stays (over 5-nghts) are low as well. Optmal results obtaned are summarzed n Table Multple scenaro example I We now consder the case where a hotel would lke to take derent future demand scenaros nto ts plannng, although the hotel management wll x ther prces (0.84) for all scenaros. In ths case we have no varaton for the revenue coecents. So the only stochastc varable s the demand U ;j. Suppose there are four scenaros wth probablty of 0.1,0.5,0.3 and 0.1, respectvely. The demands U ;j are shown as the followng Tables 3 6 for the four scenaros. For smplcty, all weghts w ;j are set to be equal to 1. The optmal solutons obtaned are summarzed n Table 7.

8 1066 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Table 2 Optmal soluton for sngle scenaro example Table 3 Demands for multple scenaro example I (scenaro 1) Multple scenaro example II In ths subsecton, we consder the stuaton where the hotel management sets derent prce levels to cope wth uncertanty. The derent strategc prce settngs can smulate demands nto derent levels. The prce per room nght for each scenaro s 0.7,0.8,0.9 and 1, respectvely. The demand s forecast as beng the same as n the above example. The rsk trade-o factor s set to be 1. The optmal soluton obtaned s then summarzed n Table Hotel revenue management strateges The dscussons n above sectons manly focus on the computatonal aspect of hotel revenue management. Indeed, n addton to the uncertan data that wll cause derent strateges, management s

9 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Table 4 Demands for multple scenaro example I (scenaro 2) Table 5 Demands for multple scenaro example I (scenaro 3) atttudes wll be also mportant n strategy settng. The weght parameters n the model are used to capture these hotel management consderatons Rsk trade-o factor The derent values of the parameter of rsk trade-o factor represent derent degrees of management s rsk averson. The followng graph presents the relatonshp between the rsk trade-o factor and the expected revenue generated. We can observe from the graph n Fg. 2 that n general, the expected revenue decreases as the rsk tradng-o factor ncreases. When the rsk trade-o factor s very large, the model gves all values of the decson varable as zero and results n zero expected revenue. In other words, f management s very conservatve toward rsk, the model wll suggest that he get rd of all busness rsks by not runnng a busness.

10 1068 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Table 6 Demands for multple scenaro example I (scenaro 4) Table 7 Optmal solutons for multple scenaro example I Another measurement of rsk s the rato of head-counts per room nght. Consder the examples of acceptng a 3-nght stay customer and three 1-nght stay customers. Although the expected revenues for two cases are the same (as we assume the room-nght rate s the same for the whole perod of plannng horzon), the head-count per room-nght rato s hgher n the latter case. Ths means the hotel would not lke to te-up avalable capacty as t would lke to accept more short-stayng customers. We can see from the Fg. 3 that n general, the rato ncreases as the rsk trade-o factor ncreases Penalty weghts for feasblty robustness The penalty weghts for feasblty robustness are other decson controls used by management. For example, f management would lke to accept more busness travellers (probably ther loyal

11 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Table 8 Optmal solutons for multple scenaro example II Expected Revenue (n unt room-nght rate) Value of rsk trade-off factor Fg. 2. Relatonshp between expected revenue and rsk trade-o factor. Head-count per Room-nght rato Value of rsk trade-off factor Fg. 3. Relatonshp between head-count per room-nght rato and rsk trade-o factor. customers) who have 3-nght to 4-nght stays, or the management would lke to accept more ntakes on a partcular day, he can release the correspondng weghts or add more weghts for other stays. We llustrate ths by re-calculatng the example n Secton 5.2 whle decreasng the weghts relatng to checkng-n on day 0 and checkng-out on day 4. By adjustng the weghts, more customers are accepted for such perod (an ncrease from 80 headcounts to 100 headcounts).

12 1070 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) Other consderatons n hotel revenue management system In our model detaled n the above sectons, the demand for any feasble length of stay s an unknown random varable of randomness due to an unknown number of arrvals and unknown lengths of stay. Denote the probablty of U ;j by f u; j (U ;j ). Assume demand for arrval d follows a probablty dstrbuton (e.g. Posson dstrbuton) f d (d ) for day, and length of stay, L ;j, can be assumed to follow another probablty dstrbuton (e.g. Geometrc dstrbuton) f L (L ) for day. It s further assumed that the two dstrbutons are statstcally ndependent. f u; j (U ;j )=f d (d ) f L (L ): (9) We extend our formulated model to cover more realstc cases ncludng, () cancellaton and no-show; () early-check-out; () stay-over-extenson; and (v) over-bookng Cancellaton and no-show The customer arrvng to check-n wll be changed due to a cancellaton or no show. We denote C as number of cancellatons/no-shows for a bookng (orgnally checkng-n on day ) and check-out on day j (0 6 j6t ). Suppose C ; follows a probablty dstrbuton of f c (C ). Our actual demand U ;j (after takng nto account cancellatons/no-shows) should follow the revsed dstrbuton: f d (d ) f d (d ) (1 f c (C )) for all 0 6 j6t: (10) 6.2. Early check out Denote, E ;e;j as the number of early check-outs on day k (checked-n on day ) orgnally checkng-out day on day j (0 6 e j6t). Suppose E ;e;j follows a probablty dstrbuton of f E; e; j (E ;e;j ). The actual length-of-stay wll be changed to follow the dstrbuton f L; j (L ;j ) (1 f E; e; j (E ;e;j )) f L; j (L ;j ): (11) Whle f L; e (L ;e ) (1 + f E; e; j (E ;e;j )) f L; e (L ;e ) for all 0 6 e j6t: (12) 6.3. Extenson of stay Denote V ;j;v as number of extended check-outs (checked-n on day ) from day j to day v. Suppose V ;j;v follows a probablty dstrbuton of f v (V ;j;v ). The actual length-of-stay wll be changed to follow the dstrbuton f L; j (L ;j ) (1 f v (V ;j;v )) f L; j (L ;j ) (13) whle f L; v (L ;v ) (1 + f v (V ;j;v )) f L; v (L ;v ) for all 0 6 j v6t: (14) The early check out and extend stay are consdered as two ndependent events. When a decson maker tres to consder both cases, a jont dstrbuton s needed to model the lkelhood of

13 K.-K. La, W.-L. Ng / Computers & Operatons Research 32 (2005) length-of-stay. The dstrbuton of stay perods can be obtaned by multplyng the relatve probablty of early check out and extended stay ownng to the assumpton of statstcally ndependence Over-bookng Over-bookng s a wdely adopted strategy for arlnes to solve the cancellaton or no shows problems. The overbookng level for a hotel s complcated, however, owng to the complexty of multple-nght-stays as well as early check-outs and extensons of stay. The bookng level under our consderaton s not xed as hotel overbookng s aected by the demand. If demand s low, the hotel can accept a hgh overbookng level. If the hotel s full, t s easer to nd a room for a customer n a sster hotel or another hotel of same class n the cty. On the other hand, f demand were hgh, the hotel would lke to lower overbookng to reduce that rsk. Dene O as the overbookng level for day check-n. Then the lmtaton of bookng capacty wll be C + O for day. IfO s a non-ncreasng functon of demand of arrval on day, O =h(d ). Then the constrants for our model becomes =0 j=k+1 x ;j + j=k+1 x 0;j 6 C + h j=1 x k;j x ;k 6 C + h U 0;j j=1 =0 : U k;j j=k+1 ; (15) (16) 7. Conclusons and future study Ths paper develops a stochastc network optmzaton model for the hotel revenue management problem wth uncertan demand arrvals and uncertan length of stays. A novel approach of robust optmzaton s appled to solve the problem on a scenaro-bass. The decson-maker s rsk averson s consdered n the objectve functon. Mean absolute value s used to measure rsk of the devaton of revenue from ts expected value. A lnearlzaton technque s appled to transform the absolute value nto a lnear form so that wdely avalable lnear modellng packages can be appled drectly to the model. In ths paper, we also dscuss the strateges for hotel management to take nto account of rsk trade-o and derent prcng polces. Other consderatons such as cancellatons and no-show; early check-outs; extended stay and over-bookng are dscussed. We showed that our proposed model can be moded to adopt these strategc consderatons. From a management pont of vew, an area for future studes s to consder the stuaton when rooms are damaged or planned for mantenance. Whle from mathematcally modellng pont of vew, the mean absolute devaton s only a specal form of rsk measurement for soluton robustness n robust optmzaton. The robust optmzaton based on the dea of a trade-o between soluton robustness and model robustness can be appled to capture a hgher degree of measurements such as varance or hgh moments of probablty dstrbutons. A parameter embeddng technque s under studed [13] to transform the hgher order objectve functon nto a b-level programmng model.

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