A Comparison of Unconstraining Methods to Improve Revenue Management Systems

Transcription

1 A Comparson of Unconstranng Methods to Improve Revenue Management Systems Carre Crystal a Mark Ferguson b * Jon Hgbe c Roht Kapoor d a The College of Management Georga Insttute of Technology 800 West Peachtree Street Atlanta, GA Phone: Fax: Carre.Crystal@mgt.gatech.edu b* Correspondng author The College of Management Georga Insttute of Technology 800 West Peachtree Street Atlanta, GA Phone: Fax: Mark.Ferguson@mgt.gatech.edu c Revenue Analytcs, Inc. 100 Gallera Parkway, Sute 1500 Atlanta, GA Phone: Fax: jhgbe@revenueanalytcs.com d Captal One 1680 Captal One Drve McLean, VA Phone: Fax: Roht.kapoor@captalone.com

2 A Comparson of Unconstranng Methods to Improve Revenue Management Systems Abstract A successful revenue management system requres accurate demand forecasts for each customer segment. The forecasts are used to set bookng lmts for lower value customers to ensure an adequate supply for hgher value customers. The very use of bookng lmts, however, constrans the hstorcal demand data needed for an accurate forecast. Ignorng ths nteracton leads to substantal penaltes n a frm's potental revenues. We revew exstng unconstranng methods and propose a new method that ncludes some attractve propertes not found n the exstng methods. We evaluate several of the common unconstranng methods aganst our proposed method by testng them on ntentonally constraned smulated data. Results show our proposed method outperform other methods n two out of three data sets. We also test the revenue mpact of our proposed method, EM, and no unconstranng on actual bookng data from a hotel/casno. We show that performance vares wth the ntal startng protecton lmts and a lack of unconstranng leads to sgnfcant revenue losses. Keywords: Revenue Management, Truncated Demand, Forecastng, Unconstraned Demand 1

3 1. Introducton Revenue Management has been credted wth mprovng revenues 3%-7% n the arlne, hotel, and car rental ndustres (Cross, 1997). One of the core concepts behnd revenue management s the reservaton of a porton of capacty for hgher value customers at a later date. The amount of capacty to reserve s typcally determned through the calculaton of bookng lmts, whch place restrctons on the amount of capacty made avalable to a lower value segment of customers so as to reserve capacty for a hgher value segment that may arrve n the future. Most bookng lmt calculatons depend on the deducton of a demand dstrbuton for each customer value segment from past demand data that occurred under smlar crcumstances and operatng envronments. In practce, however, true demand data s dffcult to obtan as many frms are unable to record all demand request that arrve after a bookng lmt has been exceeded and capacty for that customer segment has been restrcted. To overcome ths problem, unconstranng methods are used to extrapolate the true demand dstrbuton parameters from truncated demand data collected over prevous sellng opportuntes. Once a frm sells out of capacty for a gven segment, the sales data for that segment represents truncated demand (equal to the bookng lmt) nstead of true demand. Whle there s no perfect way to unconstran sales data, Weatherford and Polt (2002) clam that, n the arlne ndustry, swtchng from one common ndustry method to a better method ncreases revenues 0.5 to 1.0 percent. Snce most frms usng revenue management have low margnal costs, maxmzng revenues translates nto maxmzng operatng profts. Hence, unconstranng methods sgnfcantly mpact revenues, and n turn, profts, and deserve closer research attenton. Despte the sgnfcant mpact unconstranng has on the success of a revenue management applcaton, ths topc has receved much less attenton n the lterature compared to the work on methods for settng and adjustng bookng lmts. Ths s surprsng snce the demand dstrbuton 2

4 parameter estmates represent a prmary nput to most bookng lmt technques, fundamentally lnkng the value of the former wth the qualty of the latter. A frm facng constraned sales data faces three choces: 1) leave the data constraned, 2) drectly observe and record latent demand, or 3) statstcally unconstran the data after the fact. If hstorcal sales data s left constraned, true demand s underestmated, creatng a spral-down effect on total revenue where the frm s expected revenue decreases monotoncally over tme (Cooper et al., 2006). Unfortunately, due n part to the absence of research and teachng n ths area, ths practce s common at frms usng less sophstcated revenue management systems. In 5 we demonstrate how gnorng constraned data mpacts revenue usng actual bookng data from a hotel/casno. Drect observaton nvolves the recordng of latent (unsatsfed) demand. Many hotels for example, record both bookngs (requests that are met) and turndowns (requests that are not met). Care must be taken however, as turndowns may be attrbuted to avalablty (denals) or prce (regrets). The former s consdered latent demand whle the latter s not. To dfferentate between the two, some hotel chans have nvested n systems and tranng for ther reservatons agents to track turndowns, and rely on these drect observatons to unconstran ther sales data. Unfortunately, there are many ssues wth usng turndown data for unconstranng. These ssues nclude multple avalablty nqures from the same customer, ncorrect categorzaton of turndowns by reservatons agents, and customer requests whch arrve through a channel not controlled by the frm (Orkn 1998). The latter provdes the largest hurdle for most ndustres. Drect observatons of demand are not an opton for many ndustres because of ther dstrbuton channels. For example, tradtonally most arlne bookngs have been made va travel agents usng global dstrbuton systems lke Sabre and Worldspan, and no turndown nformaton s collected on these bookngs. Whle arlnes have recently been strvng to ncrease ther drect sales and mprove ther 3

5 customer nformaton, the percentage of total demand collected through frm-owned channels s stll very small. On the other hand, hotels and casnos have hstorcally taken the majorty of ther bookngs through ther own agents, ether at the property tself or through a central reservatons center. The advent of the Internet has compromsed the qualty of hotels turndown data. Whle Internet drect sales s a growng channel for hotels, wth some hotels takng up to 10% of ther bookng through ths channel, most companes have yet to ncorporate turndowns from ther own web ste nto ther total demand pcture, and for good reason. Carroll and Sguaw (2003) pont out that only 20% of hotel customers checkng avalablty va the drect Internet channel actually book ther rooms at the same ste. Along wth the growth n drect Internet sales, sales va thrd party web stes (such as Expeda and Travelocty) have grown at an even faster rate. Most thrd party web stes do not provde any turndown nformaton. The net effect for hotels s an ncreasng proporton of bookngs from channels wth no turndown nformaton, and as a result, hotelers have an ncreasng nterest n alternatve unconstranng methods. Statstcal unconstranng covers a spectrum of optmzaton and heurstc technques that rely only on observed bookngs and a state ndcator (open/closed). The purpose of ths paper s to compare common statstcal unconstranng methods to our proposed forecastng-based method. In addton, we apply the most accurate of these methods to actual hotel bookng data. Prevous studes on unconstranng methods tested a subset of the methods aganst smulated arlne data. Most tradtonal unconstranng methods follow a common methodology. Demand observed over tme s categorzed as constraned or unconstraned, and dstrbuton parameter value estmates are adjusted based on the percentage of data that was constraned. In these unconstranng methods, a demand stream constraned twenty days before the end of the bookng wndow s treated the same as a demand stream constraned one day before. Ths methodology gnores an mportant aspect of the 4

6 revenue management envronment: frms often know the tme perods when demand was constraned. Our proposed method takes advantage of ths nformaton and uses t when calculatng the demand dstrbuton parameter estmates. In addton, our method offers two other advantages over many of the alternatve statstcal methods: 1) It s based on a wdely accepted statstcal forecastng technque (double exponental smoothng) requrng mnmal computatons and 2) t s non-parametrc, requrng no a-pror assumptons about the shape of the bookng curve or the dstrbuton of the fnal total demand. 50 Fgure 1 - Example bookng pace curve Projected Demand Bookngs 25 Bookng lmt reached True Demand Days before expraton of good We llustrate the key concept behnd our proposed method through the example bookng curve shown n Fgure 1. Most tradtonal unconstranng methods only use the fnal observed demand and whether or not the demand was constraned. Our proposed method also uses the tme demand was truncated (30 days before the guest arrval n our example). Our method then uses a forecastng technque (double exponental smoothng) to project total demand that would have been observed n the 5

7 absence of any bookng lmts (the dashed part of the bookng pace curve). Through a smulaton experment, we fnd our method outperforms most of the tradtonal statstcal methods n estmatng the demand dstrbuton parameters of constraned data sets. Compared to the one method t does not always outperform, our method s smpler and works under condtons where the other method does not, such as when all hstorcal data sets are constraned. Snce there s no clear domnance by ether method, we evaluate the mpact on revenue the two methods provde usng actual bookng and revenue data from a leadng hotel/casno. The rest of the paper s organzed as follows. In 2 we revew the lterature, n 3 we defne the proposed method, n 4 we test the method aganst other common methods used n practce, and n 5 we test the two best performng methods on real hotel/casno data and measures ther mpact on the hotel s revenue. Fnally, n 6 we offer conclusons and recommendatons. 2. Lterature Revew Weatherford and Bodly (1992) and McGll and van Ryzn (1999) provde general revews of the broad range of lterature n the revenue management feld. Tallur and van Ryzn (2004b) provde an excellent overvew of the current state-of-the-art n all aspects of revenue management systems. As these studes show, the prmary research focus has been weghted towards the development of overbookng and bookng lmt technques wth lttle focus on unconstranng (also called uncensorng) sales data. We concentrate here on revewng the unconstranng research. Relablty engneers, bomedcal scentsts and econometrcans have used unconstranng procedures for many years to compensate for the early termnaton of experments. Ths parallels how revenue managers termnate demand for a partcular customer segment through the use of bookng lmts. Relevant research n these felds nclude: (Cox, 1972; Kalbflesch and Prentce, 1980; Lawless, 1982; Cox and Oakes, 1984; Schneder, 1986; Nelson, 1990; Lu and Maks, 1996). These methods rely 6

8 heavly on the hazard rate functon to determne the probablty dstrbuton of lfetme data. To our knowledge, van Ryzn and McGll (2000) provde the only use of ths type method n a revenue management framework when they utlze a method based on demand lfetables. We nclude the lfetable method of uncensorng data n our comparson as descrbed n Lawless (1982). Weatherford and Polt (2002) and Zen (2001) compare unconstranng methods usng smulaton and apply the best methods to an arlne s reservaton data to test the revenue mpact of dfferent methods. Sx unconstranng methods are tested: three dfferent averagng methods, a bookng profle method, projecton detruncaton (PD), and expectaton maxmzaton (EM). The averagng methods are the smplest computatonally and therefore are often used n practce. We compare our proposed method (DES) aganst the three best performng methods found n Weatherford and Polt (2002): the most accurate averagng method (referred to as Naïve 3 n Weatherford and Polt, abbrevated to AM n ths paper), EM and PD. Both Weatherford and Polt (2002) and Zen (2001) conclude the EM method outperforms the others and ncreases revenues by 2-12 % n full capacty stuatons. We also fnd the EM method outperforms PD. Of the three best methods that Weatherford and Polt (2002) and Zen (2001) use, only EM s grounded n statstcal theory. Dempster et al. (1977) prove the theory behnd the EM method based on data from a unvarate dstrbuton. The EM method dscussed by Dempster et al. (1977) s essentally the same as the tobt model used n econometrcs (Maddala, 1983). McGll (1995) extends the EM method to a multvarate problem when demand for dfferent classes (segments) of a good are correlated. The PD method closely resembles the EM method, but takes a condtonal medan n place of a condtonal mean. Addtonally, the PD method allows users to change a weghtng constant to obtan more aggressve demand estmates. The tradeoffs nclude ncreased computaton and ncreased rsk of no soluton convergence (Weatherford and Polt, 2002). 7

9 Lu et al. (2002) examne unconstranng demand data through the lens of the hotel ndustry and argue the EM method s unrealstc n applcaton because of ts computatonal ntensty. The authors argue that parametrc regresson models take nto account all relevant nformaton and are computatonally more feasble n real-world applcatons. They develop a parametrc regresson model whch uses bookng curve data, but requres knowledge of the shape of the demand dstrbuton and other specfcs of the demand constrants. Ths knowledge requrement restrcts the general use of ther model, as frms often do not know a pror the shape of the bookng curve. Also, the authors do not provde comparsons between ther proposed parametrc method and the methods dscussed n other papers. We do not nclude ther method n our comparson because we do not assume a known, functonal form for the bookng curve. We do agree, however, wth ther crtcsm regardng the computatonal ntensty of the EM method. Our proposed method s much easer to calculate but, unlke the parametrc models, does not requre knowledge about the shape of the bookng curves. To test the revenue mpact of our new unconstranng method, we must set the protecton levels effectvely. To do so, we use the most common seat protecton heurstc used n practce, EMSR-b (Belobaba 1989). McGll and van Ryzn (1999) gve an explanaton of the EMSR-b method along wth a revew of the bookng lmts problem n general. 3. Proposed Unconstranng Method Our proposed method uses Double Exponental Smoothng (DES) or Holt s Method to forecast the constraned values of a gven data set. DES uses two smoothng constants: one for smoothng the base component of the demand pattern and a second for smoothng the trend component. Armstrong (2001) provdes a good revew of ths method. Below, we descrbe how t may be used to solve the unconstranng problem where demand s constraned only once, pror to the fnal perod. In the 8

10 appendx we provde an example of how ths method can be used to solve the unconstranng problem where demand s constraned multple tmes due to re-openng closed bookng classes Let t represent the tme perods between I, the perod that reservatons are ntally accepted, and B, the perod where demand reaches the bookng lmt (tme s counted backwards from I days before arrval untl B days before arrval). That s t [ I, I 1,... B], I B. After perod B, demand contnues to occur but s unobserved. If demand s never constraned then B = 0. Thus, demand seen equates to the cumulatve demand observed from perods I to B and s always less than or equal to true demand. From Fgure 1, I corresponds to perod 140 and B corresponds to perod 30, after whch demand s unobserved. Now defne: A t = Actual cumulatve demand n perod t F t = The exponentally smoothed base component for perod t T t = The exponentally smoothed trend component for perod t FIT t = The forecast of cumulatve demand ncludng trend for perod t α = Base smoothng constant β = Trend smoothng constant The forecast for the upcomng perod t s FIT t = F t + T t (1) where F t =FIT t+1 +α(a t+1 - FIT t+1 ) and (2) T t = T t+1 +β(f t - FIT t+1 ). (3) The smoothng constants, α and β, are decson varables. For each constraned demand nstance, we use a non-lnear optmzaton routne to select the α and β values that mnmze the sum of the squares of the forecast error: 9

11 B αβ, t= I mn ( A FIT ) t t 2 (4) For the ntal values, F I and T I, we use the actual demand n perod I as our estmate for the base component and the average trend over the avalable observed cumulatve demand as our estmate for the ntal slope component. Snce the problem s not jontly convex n α and β, a non-lnear search algorthm such as tabu search or smulated annealng s needed to fnd the global mnmzers. The forecastng model s then used to project the cumulatve demand over the perods n the data set where demand s constraned,.e. over perods B-1 to 0. It does so by takng the last forecast where t was possble to update wth observed demand, FITB = FB + TB, and projectng the fnal B perods where demand s constraned: FIT0 = FB + BTB. We demonstrate our method usng the bookng curve shown n Fgure 1. Frst, ntal estmates of F 140 and T 140 are calculated usng the cumulatve demand n perod 140 and the average trend between perod 140 and perod 30 (the last perod n whch we observe unconstraned demand). If A 140 =1 and A 30 = 25, then F 140 = 1 and T 140 = (25-1)/(140-30) = Next, we choose an ntal value of 0.1 for both the α and β smoothng coeffcents and perform one-perod ahead forecastng to fnd FIT t for perods t = (the ntal startng values chosen for the smoothng coeffcents are not mportant as long as we use a search algorthm that does not get stuck at local mnmums to fnd the smoothng coeffcents that best ft the data avalable). We then choose the α and β smoothng coeffcents that mnmze ( A FIT ). Let α and β represent the smoothng coeffcents that result from ths t= 30 t t mnmzaton search. These smoothng coeffcents are used to recalculate FIT t ; call the new forecasts FITt F t T t = +, for perods t = 140,,30. Because we know the actual demands that occurred durng days 140 to 30, we calculate FIT 140 to FIT 30 usng the recursve methods of (1), (2) and (3) for each day 10

12 untl we reach FIT = F 30 + T 30, after whch we no longer know true demand and can no longer update 30 our forecast. The remanng 30 perods (t = 29,, 0) are when demand s constraned n our example. Our objectve s to determne the fnal cumulatve demand (f demand was not constraned) at the termnatng perod t = 0. We estmate ths value usng FIT0 = F T 30. The cumulatve demand over the observed and projected components of the bookng curve ( FIT 0 ) s then used as a sngle pont estmate of true cumulatve demand for a partcular sellng occurrence (.e. a gven Thursday nght stay for a gven fare class at a hotel). Call ths ndvdual pont estmate for the th bookng curve X. We repeat ths procedure over each constraned bookng curve n a gven data set (.e. all Thursday nght stays for a gven fare class at a hotel). Thus, f there are n hstorcal bookng curves n the data set, we end up wth a set of pont estmates ( X1, X2,... X ). The fnal demand dstrbuton parameters (mean μ and varanceσ 2 ) are then estmated usng ths set of pont estmates by: n n = 1 μ = n X and n 2 = 1 σ = ( X μ) n 2. (5) The basc model of DES descrbed above s a very general method for forecastng demand and, as presented, does not account for seasonalty, ntermttent demand, and other specfcs that mght be relevant n applcaton. However, DES can be easly adjusted to ncorporate seasonalty (Armstrong, 2001). In Secton 4.2, we provde an alternatve formulaton that can be used when total demand s small and ntermttent. 4. Comparson of Unconstranng Methods In ths secton we compare four of the most common statstcal unconstranng methods that have prevously appeared n the lterature wth our proposed DES method. To compare the performance of 11

13 the dfferent methods, we smulate bookng curves representng true demand and then mpose bookng lmts to create constraned data. We apply fve dfferent unconstranng methods to the constraned data sets and compare the estmated demand parameters aganst the true parameters. We defne the best unconstranng method as the method that estmates the demand dstrbuton parameters closest to the true parameters. To compare the performances of the chosen methods, we frst smulate bookng curves and set bookng lmts to constran the demand data. To test each unconstranng method aganst a broad range of demand scenaros, we smulate three data sets wth 100 bookng curves each and 140 days n each bookng curve. The three data sets represent three common shapes of bookng curves: convex, homogeneous and concave (Lu et al., 2002) as shown n Fgure 2. The 100 bookng curves represent 100 hstorcal demand records (for each shape curve) a hotel or arlne may use to predct future demand. For example, a hotel may keep demand data from ts last 100 Thursday nght stays n order to estmate demand for future Thursday nght stays. Snce most hotels and arlnes observe the majorty of ther reservatons wthn 140 days before the day of arrval or departure, we smulate 140 days of daly demand arrvals for each bookng curve, resultng n 100 ndvdual bookng curves of 140 days each, or 14,000 ndvdual data ponts. For each bookng curve shape, we look at demand observed for all 100 bookng curves smultaneously (some where the total demand was not constraned and others where total demand exceeded the bookng lmts) and use each unconstranng method to estmate the true demand dstrbuton parameters. 12

14 Fgure 2 - Concave, Homogeneous and Convex bookng curves Total bookngs Concave Homogeneous Convex Days before arrval date To construct the bookng curves, we assume arrvals on a gven day are randomly drawn from a Posson dstrbuton. Ths assumpton s common n the lterature and matches closely wth actual data from the hotel and arlne ndustres (Rothsten, 1974; Btran and Mondschen, 1995; Btran and Glbert, 1996; Badnell, 2000; Lu et al., 2002). For the homogeneous bookng curve, we mantan a constant mean arrval rate over all 140 days. For the convex (concave) bookng curves, we ncrement the mean arrval rate from low to hgh (hgh to low) respectvely, so the expected total demand over the 140 day perod s approxmately equal for all three curves (average demand for concave, homogeneous and convex curves s 700, 698, and 696, respectvely). After we create the demand curves, we calculate bookng lmts. A user sees the mnmum of true demand and the bookng lmt. A smple example s shown n Table 1. If daly demand arrvals are Posson and the demands on dfferent days are ndependent, then total demand s agan Posson. Because the mean of the Posson-dstrbuted total demand s suffcently large, the dstrbuton of total demand s approxmately Normal. Thus, we calculate an expected average (μ) and standard devaton 13

15 (σ) of the total demand and generate fve sets of bookng lmts representng varous ranges of constrant levels. For example, a 20% constranng level means that, on average, 20% of the data sets have ther total demand constraned by the bookng lmt. To fnd the bookng lmts at these varous levels, we use the z-score from a standard Normal dstrbuton correspondng to the 20%, 40%, 60%, 80%, and 98% constraned levels, where z represents the number of standard devatons above or below zero for a standard Normal dstrbuton. Thus, to fnd the z-score correspondng to 98%, we fnd the pont where the area under the standard Normal curve equals 0.98, or z = We then set our correspondng bookng lmts usng: μ + z * σ. (See the appendx for more explanaton on how we constructed these curves and set our bookng lmts to constran them) Table 1 - Example of true demand vs. demand observed True Demand Bookng Lmt Demand Seen 98* 105* * * ndcates constraned demand, also called a closed segment We test the fve unconstranng methods across the three bookng curve shapes (homogeneous, convex, and concave) for each of the fve constranng levels to test how each method performs under vared condtons. We chose the unconstranng methods to test from prevous research; the frst three methods are the best performng methods from Weatherford and Polt s (2002) comparson. These methods nclude: 1) an averagng method (AM), called Naïve #3 by Weatherford and Polt, 2) Projecton Detruncaton (PD) and 3) Expectaton Maxmzaton (EM). Medcal and relablty engneerng researchers commonly use lfetables (LT), the fourth method. Addtonally, van Ryzn and McGll (2000) use lfetables n a revenue management context. We provde a short descrpton of each of these methods n the appendx. The ffth method s DES, whch was descrbed n 3. 14

16 4.1 Results of Comparson Overall, the DES, EM, and AM methods outperform the LT and PD methods. Table 2 compares the percentage error for each unconstranng method versus the actual mean of the demand dstrbuton (the percentage errors were smlar but slghtly larger for the estmated varances). DES outperforms all methods for the homogeneous and convex data sets as ts error remans less than 0.5% for all levels of constranng, compared to a maxmum 5% error for the other methods. In the concave data set however, AM and EM outperform DES. The strong performance of AM n the concave data set skews ts average error, and so on average, AM outperforms the other methods over all three curves. Table 2 summarzes the results of the comparson and Fgure 3 graphcally summarzes the mean absolute error over all three curves. Prevous comparsons (Zen, 2001; Weatherford and Polt, 2002) show EM outperformng PD; we confrm ths result. Asde from the accuracy ssues, PD has two dsadvantages compared to EM: t takes more teratons to converge than EM and t requres the choce of a weghtng parameter, τ, creatng an opportunty for varyng results. A τ < 0.5 can lead to better results, but ncreases both the tme to convergence and the chance for no convergence. The AM shows consstent performance across the data sets, wth especally strong performance n the concave data set. Unfortunately, the concave demand pattern s typcally the pattern least mportant n settng bookng lmts. Arlnes often offer cheaper fares to customers bookng at least three weeks n advance. Because of ths, and other smlar restrctons, the lowest valued segment s often forced to follow the concave demand pattern. Due to the fundamental concepts behnd revenue management, estmatng the true demand for the lowest valued segments s typcally less mportant than estmatng the demand for hgher valued segments. The LT method of unconstranng data produces estmates wth errors very close to zero and even outperforms the DES and EM methods n a few of the concave cases. However, ths method requres 15

17 many computatons and a large quantty of hstorcal demand data. In a dynamc envronment such as the travel ndustry, customer demand data changes quckly due to changes n the economc clmate, broader market supply-demand-prce relatonshps, and customer preference. Because of ths, suffcent hstorcal demand s often not avalable for the LT method to produce effectve results. Table 2 - Percentage error between calculated and actual mean for each unconstranng method Bookng Curve Homogeneous Convex Concave Mean absolute error over all 3 Bookng Curves Method Percent of Data Sets Constraned 20% 40% 60% 80% 98% AM -0.12% -0.25% -0.45% -0.44% -1.12% PD 0.23% 0.43% 0.56% -0.53% -2.99% EM -0.06% -0.23% -0.56% -0.22% -0.58% LT -0.17% -1.31% -1.53% 0.20% 0.43% DES 0.00% 0.00% 0.00% 0.00% -0.14% AM -0.16% -0.27% -0.39% -0.41% -0.63% PD 0.26% 1.29% 0.98% -0.91% -2.99% EM -0.08% 0.39% -0.25% -0.85% -1.10% LT 0.13% 0.76% 0.89% 0.55% 5.72% DES 0.00% 0.14% 0.00% -0.14% -0.29% AM -0.01% 0.05% -0.02% -0.17% -0.23% PD 0.24% 0.73% 1.78% -0.35% -3.24% EM -0.08% -0.09% 0.09% -0.19% -0.93% LT -0.21% -0.83% 1.14% -0.11% 2.29% DES 0.29% 0.71% 1.00% 1.86% 3.43% AM 0.09% 0.19% 0.29% 0.34% 0.66% PD 0.24% 0.82% 1.11% 0.60% 3.07% EM 0.07% 0.24% 0.30% 0.42% 0.87% LT 0.17% 0.97% 1.19% 0.29% 2.81% DES 0.10% 0.28% 0.33% 0.67% 1.29% For the homogeneous data pattern, the DES method has neglgble error across the range of constraned data sets due to the hgh predctablty when arrval rates are constant over a gven tme perod. For ths data pattern, the DES method estmates the dstrbuton mean up to 4% closer to the true mean than the next closest method. Smlarly, for the convex data set, the DES method provdes the most accurate estmate over all of the constranng condtons. 16

18 The DES method does not perform as well on the concave data set, although t stll performs wthn a 1% error untl demand s constraned n over 80% of the observatons. DES underperforms on ths demand pattern because bookng segments close farther away from the arrval date for the concave bookng curve, so many more data ponts must be estmated compared to the convex or homogeneous demand patterns. Here, the trend component of DES affects ts accuracy as hgh demand occurrng early n the bookng curve s projected to contnue once the bookng lmt has been met. For ths bookng curve shape, a forecastng method wth a trend that s dampened over tme may perform better. In practce however, an naccuracy n unconstranng demand followng a concave demand pattern s not a great concern, because the concave pattern typcally corresponds to the lowest fare customers, as explaned prevously. Fgure 3 - Average absolute error from true demand for each unconstranng method 4% 3% Naïve 3 PD EM DES Lfe tables Error 2% 1% 0% 20% 40% 60% 80% 98% Data Constraned 17

19 4.2 Performance wth Smaller Demand The frst set of results (Table 2) compares unconstranng methods when total demand averaged 698 unts. However, n many applcatons, total demand s much smaller than 698, so we run a smlar experment wth an average total demand of 19. We call ths the Small Demand data set. Just as before, we ran smulatons on homogeneous, concave, and convex bookng curve shapes, wth 100 trals of 140 days each for each shape. The DES and EM methods provde the most accurate results across a range of constraned data, and the AM provdes the least accurate results. Due to dffcultes of predctng data wth ntermttent demand (many perods wth zero demand), the small demand data has less accurate results than the large demand data. Ths observaton s consstent wth prevous studes; goods wth ntermttent demand are dffcult to forecast and requre specalzed forecastng tools for the most effectve results (Altay and Ltteral, 2005). 70% Fgure 4 - Unconstranng error wth small demand data set Percent Error 60% 50% 40% 30% 20% AM PD EM LT DES 10% 0% 0% 20% 40% 60% 80% 100% Percent Constraned 18

20 Because DES has sgnfcantly hgher error wth the small demand data set than wth the large demand data set, we sought an alternatve formulaton. Croston s forecastng method (Croston, 1972) s a smple exponental smoothng method desgned to accommodate small or ntermttent demand. Ths method forecasts the sze of the non-zero demands and nter-arrval tme between non-zero demands. (See the appendx for explanaton of ths method) In a smple smulaton over 60 trals and 60 days wth a total demand of 12 (based on the smallest observed demand segment of our partner hotel), Croston s method outperforms DES across the range of constrants, as shown n Fgure 5. These results provde evdence that Croston s method may be superor to DES for unconstranng ntermttent demand. When the percentage of days wth zero demand exceeds 10% of the total number of days n the bookng curve, Croston s method begns to outperform DES. Fgure 5 - Unconstranng error wth small demand data set DES vs. Croston s method 35% 30% Percent Error 25% 20% 15% 10% Croston's DES 5% 0% 0% 20% 40% 60% 80% Percent Constraned 19

21 5. Revenue Impact Usng Industry Data In ths secton we compare the potental revenue mpact of a major hotel/casno usng DES, EM, and no unconstranng. Snce unconstranng methods only provde estmated parameters for the demand dstrbuton, we use the EMSR-b (Expected Margnal Seat Revenue) algorthm (Belobaba, 1989) - a wdely accepted method for settng bookng lmts for a basc revenue management system - to translate the demand dstrbuton parameters (and correspondng room rates) nto bookng lmts. We appled the bookng lmts to bookng data from a hotel/casno to calculate the total revenue mpact. Thus, we compare the revenue convergence usng EM, DES, and gnorng unconstranng based on actual (but normalzed) bookng and revenue data from a major hotel/casno. Whle the examples presented n ths secton are very useful for llustratng the effectveness of the methods, they cannot lead to conclusons about ndustry performance. Such conclusons can only be drawn from trals n practce. 5.1 Demand Data We use actual hotel/casno bookng data to test how unconstranng mpacts revenue. We use bookng curve data for 12 consecutve Frday nght stays, unconstraned usng drect observaton of turndowns. Extra care was taken to ensure that all demand was captured for ths data set ncludng demand that occurred after bookng lmts were met. Because of the ncreased cost nvolved n such careful data collecton, we lmted the data collecton perod to 12 weeks and used bootstrappng to create 1000 bookng curve samples from the ntal data. Hotel reservatons vary greatly by day of the week, dependng on the type of hotel. For ths hotel, weekends are the most popular, and therefore, have the hghest constranng rate. In order to control for dfferences n demand between dfferent days of the week, we focus on Frday nght stays durng the 12-week perod. Wthn any Frday nght s bookng data, ths hotel/casno has many dfferent customer segments, wth some customer segments so valuable they are rarely constraned (revenue per nght from the hghest fare customer can be 12 tmes the 20

22 revenue from the lowest fare customer); therefore we focused our unconstranng efforts on the most popular four segments that are constraned. We base many of our smulaton choces on the hotel/casno data characterstcs. In order to understand these characterstcs, we plotted the unconstraned demand data for the four customer segments for 12 consecutve Frday nght stays. The shape of a gven customer segment arrval rate was consstent throughout the 12 week perod, although each of the 4 customer segments had a dstnctly dfferent bookng curve. We present one such dagram n Fgure 6 to show the varety of bookng curve shapes. Based on the smlar arrval rates throughout the 12 week perod, we utlzed bootstrappng to create a suffcent number of dfferent demand realzatons for our smulaton study. Fgure 6 - Cumulatve hotel/casno reservatons for four separate fare classes Number of arrvals Fareclass 1 Fareclass 2 Fareclass 3 Fareclass 4 Days before arrval 60 days pror Bookng Date We created bootstrapped samples as follows. The 12 Frday nght stay bookng curves showed the slope of the curve changed dramatcally at dfferent ntervals before arrval. Based on these slope 21

23 changes, we created multple ntervals wthn the 60 day wndow whch had smlar arrval rates. Pckng randomly (wth replacement) from 12 weeks worth of Frday nght bookng patterns wthn a smlar arrval rate nterval, we used the bootstrap method wth replacement to create 1000 dfferent 60 day bookng curves for each of the four customer segments. The hotel would temporarly close the lower valued segments mdway through the 60 day bookng curve, thus we smulated ths practce of openng and closng the classes multple tmes. We closed a fare class (constraned demand) mdway through the bookng curve, reopened the fare class, and then closed t agan before the actual day of arrval. We ntally closed the fare class f cumulatve demand reached the frst bookng lmt wthn 30 days of acceptng bookngs. We reopened the fare classes wth 30 days left n the bookng perod and closed the fare class agan f demand reached the second bookng lmt. We set protecton levels so that 50% and then 75% of a gven data set would be constraned to test our methodology aganst dfferent constranng levels. To set protecton levels so that 50% of a gven data set would be constraned, we calculated the frst bookng lmt as the average cumulatve demand after 30 days of acceptng bookngs; the second bookng lmt was the average cumulatve demand over the 60 day perod. Smlarly, for 75% constraned data, we used the average and standard devatons at 30 and 60 days to fnd the z-score correspondng to 25% unconstraned data (μ+z*σ). We dd ths for each of the 4 dfferent segments n all 1000 replcatons. Usng both DES and EM, we unconstraned the data sets and compared the dstrbuton parameter estmates for each method aganst the true parameter values. Both methods performed well, wth average errors lsted n Table 3. Just as n prevous trals, all methods performed better wth less constraned data. The methods better predct mean values than standard devatons. Over 1000 nstances, both methods predct the mean wthn 5% of the true mean, showng the methods performed well even when demand was constraned multple tmes n a bookng curve. 22

24 Table 3 Error comparson between EM and DES wth nterrupted arrvals and actual data 50% of Data Sets Constraned 75% of Data Sets Constraned EM 0.84% 1.72% Mean DES 0.84% 4.87% Standard Devaton EM 7.38% 14.72% DES 7.65% 23.86% Whle provdng a useful comparson, such a smple test as the one above does not, however, capture the true mpact the use of an unconstranng method has n practce. Three man ssues are yet to be addressed: 1) n practce a hotel wll never have 1000 hstorcal bookng curves avalable to estmate a demand dstrbuton durng a perod of statonary demand; 2) protecton levels evolve over tme (the study above does not capture the transent nature of demand nformaton updatng or learnng); and 3) there are no prevous studes lnkng the accuracy of an unconstranng method to the revenue mpact of the user. We attempt to address these ssues n the followng study. In the next study, we test the mpact each unconstranng method has on the hotel s total expected revenue. Total expected revenue from a revenue management system s the ultmate ndcator of a system s success, but unconstranng methods only provde estmates for the demand dstrbuton parameters. Thus, we borrow van Ryzn and McGll s (2000) general methodology for translatng protecton levels nto revenue. To test convergence and robustness, we start wth purposefully hgh and 23

25 low protecton levels, smlarly to van Ryzn and McGll (2000). Note our swtch to protecton levels rather than bookng lmts for ths study. Protecton levels are the opposte of bookng lmts,.e. how many unts of capacty at a gven class to protect for hgher fare classes. To llustrate, consder a hotel wth two fare classes, hgh and low, and a total capacty of 100 rooms. If a bookng lmt for the low fare rooms s set at 60, the correspondng protecton level for the hgh fare rooms s = 40 rooms protected for hgh fare customers. Protecton levels must be set at some estmated level for ntal product offerngs and for exstng products when there has been a fundamental shft n the underlyng demand dstrbutons. As more demand s observed over tme, the frm adjusts protecton levels accordngly to ncrease total revenue. The convergence rate to the optmal protecton levels depends on both the startng levels chosen and the unconstranng method used. Thus, we test protecton level convergence and total revenue convergence usng the top performng unconstranng methods (EM and DES) wth two dfferent startng protecton levels low and hgh. To underscore the mportance of unconstranng, we ncluded data wth no unconstranng, labeled (Spral) for the decreasng revenue named after the spral-down effect (Cooper et al., 2006) whch occurs when data s not unconstraned. Cooper et al. (2006) gve the name spral-down effect to the phenomenon of systematc decreased revenue due to ncorrect customer behavor assumptons nherent n many revenue management systems. Specfcally, they show how assumng that customers wll reman n a gven fare class, regardless of avalablty of other (less expensve) fare classes, negatvely mpacts revenue wth each successve calculaton of a protecton level. Our Spral data has a smlar downward spralng revenue pattern wth each successve calculaton of a new protecton level, but ths s due to usng constraned data n forecasts, rather than ncorrect assumptons of customer behavor. Ths downward spral effect has been recognzed as a problem, and can occur when usng constraned data, or when 24

26 usng underestmated unconstraned data (Weatherford & Polt, 2002). No matter the cause or termnology, these research papers show that poor settng of protecton levels leads to ncreasngly poor revenue performance over tme. We compare revenue results between unconstraned data and constraned data to llustrate the mportance of unconstranng. 5.2 Settng Protecton Levels: the EMSR-b Method For settng the protecton levels for the hotel rooms, we use a varaton of the EMSR (Expected Margnal Seat Revenue) heurstc (Belobaba 1989), called EMSR-b. Ths s the most common heurstc used n practce for settng protecton levels. The EMSR-b method does not produce optmal protecton levels under all real world condtons, but s representatve of a basc revenue management system and s suffcent for comparng unconstranng methods. The EMSR-b method works as follows: Gven the estmates of the means, μˆ and standard devatons, σˆ for each customer value segment, the EMSR-b heurstc sets protecton level θ so that f+ 1 = fp( X > θ ), where X s a normal random varable wth 2 mean ˆμ j and varance ˆ σ j j= 1 j= 1, f s the fare for customer value segment and f s weghted average revenue from classes 1,.,, gven by f f ˆ μ ˆ μ =. j j j j= 1 j= 1 In smpler terms, ths rule performs a margnal analyss on the benefts of holdng capacty for a hgher valued customer versus the cost of turnng away the next lower valued customer. To demonstrate the EMSR-b method, consder the data gven n Table 4 representng a hotel wth four fare classes. The nested protecton levels calculated usng the EMSR-b method are gven n the far rght column (note that fractonal values are rounded up). Thus, for ths example, 49 rooms should be reserved for fare class 1 customers (the hghest payng), 125 rooms should be reserved for fare class 1 and 2 customers,

27 rooms should be reserved for fare class 1, 2, and 3 customers, and any remanng rooms can be sold to the lowest fare class customers (class 4). Table 4 Example of protecton levels set usng EMSR-b heurstc Class Fare Mean Varance Protecton level θ Capacty The example above shows how demand dstrbutons can be converted to recommended protecton levels for a revenue management system. However, revenues can only be optmzed f the true demand dstrbuton parameters are known; hence the need for a good unconstranng method. 5.3 Smulaton We test the revenue mpact of the unconstranng methods by applyng protecton levels (based on the EMSR-b method usng the dstrbuton parameter estmates from the unconstranng method) to the ndustry data descrbed n 5.1. As n van Ryzn and McGll (2000), we assume nested fare-class allocatons, low fare classes book strctly before hgh fare classes, demand for each fare class s ndependent, and there are no cancellatons or no shows. Although n realty, low fare classes may not completely book before hgh fare classes, our data plotted n Fgure 6 shows evdence that the majorty of a low fare class wll book before the hgher fare class (especally when the mpact of bookng lmts are consdered). Because of ths behavor, the assumpton of low bookng before hgh s a reasonable approxmaton. It should be noted that standard practce wthn arlne revenue management ncludes updatng protecton levels at set ntervals throughout the bookng perod. Ths allows arlnes to adjust protecton levels to actual demand. We do not ncorporate ths practce nto our smulaton because for hotels 26

28 (especally casnos), a large percentage of the hghest fare class customers book hours before arrval, not days or weeks. Ths late bookng practce makes accurate forecastng (ncludng unconstranng) even more mportant wthn ths applcaton. We defne notaton n the followng manner: X = demand for fare class ; I = nventory (hotel rooms) avalable for fare class ; n = number of fare classes; and R = revenue earned from fare class. We can now calculate total revenue usng: n R = 1 Total Revenue = where (6) Capacty θ 1 for = n n I = Capacty θ 1 mn( I, ) for 1 j 1 j X j < < n =+ n Capacty mn( I, ) for 1 j 1 j X = =+ j and (7) R = f * mn [X, I ]. (8) (See the appendx for an example of ths revenue calculaton wth a partcular demand realzaton) Normalzng the fare class data from our hotel/casno on a scale from $1 - $100, the per-nght expected revenues for the four customer segments are: $25, $35, $62, $100. These expected revenues are based on the total amount a customer n that segment s expected to spend at the hotel/casno per nght, ncludng the revenue from the room rate, food, beverages, shows, and casno. The hotel/casno tracks customer spendng by ssung frequent stay cards whch record each tme the customer makes a transacton. Because estmates for the parameter values of the demand dstrbuton and the correspondng protecton levels, evolve over tme, we smulate ths evoluton n our study. Frst, we splt each of the 4 sets of 1000 bookng curves descrbed n 5.1 nto 10 sets of 100 bookng curves Workng wth the frst set of 100 bookng curves, we estmate ntal protecton levels for each customer segment (two 27

29 ntal startng protecton levels are used for each segment, one lower and the other hgher than the protecton levels calculated wth perfect demand nformaton). These ntal protecton levels are rough estmates of demand and equate to rough demand estmates when a frm offers a new product. Hence, wth hgh protecton levels, we set the protecton levels to approxmately equal average demand plus one standard devaton for the three hghest fare classes. Smlarly, wth low protecton levels, we set the protecton levels to approxmately equal average demand mnus one standard devaton for the three hghest fare classes. We then calculate the revenue the hotel/casno would have receved f they used these ntal protecton levels for each customer segment over all 100 bookng curves n the set. The frst data pont n Fgures 7 and 8 s the percentage dfference n revenues the hotel/casno would have receved usng these ntal protecton levels versus f they had used optmal protecton levels calculated wth the true demand dstrbuton parameters. Next, we appled the ntal protecton levels to the frst 10 bookng curves (bookng curves 1-10 out of the 100 n the set). Thus, some segments of the frst 10 bookng curves were constraned by the calculated protecton levels. We apply each of the unconstranng methods to ths group of 10 constraned bookng curves and calculated new protecton levels for the next group of 10 bookng curves (bookng curves out of the 100 n the set). Based on these new protecton levels, we calculated the revenue generated f these protecton levels were used on all 100 bookng curves n the set. We contnue ths procedure, unconstranng the demand data and readjustng the protecton levels every 10 bookng curves. Ths procedure smulates a hotel manager watchng demand for 10 consecutve Frdays, then adjustng hs protecton levels for the next 10 Frdays, and contnung ths procedure for a total of 100 consecutve Frdays. For robustness, we appled ths methodology to 10 sets of 100 bookng curves to calculate a standard error of our estmates. Fgures 7 and 8 represent average results over the 10 sets along wth upper and 28

30 lower lmts correspondng to a 95% confdence nterval. All of the methods (EM, DES, Spral) use the same data sets (the smulatons are coupled) We compare revenues for each of the unconstranng methods to optmal revenue. We calculate the perfect nformaton calculated revenue by fndng the mean and standard devaton of each set of 1000 bookng curves (one set of 1000 for each of four customer segments). Snce the ntal data was unconstraned, we know the true demand for every bookng curve, and hence know the true mean and standard devaton parameter estmates. We apply EMSR-b usng these parameters to fnd protecton levels, then usng nested protecton levels, apply (6-8) to calculate total revenue. Fgure 7 Revenue acheved usng hgh protecton lmts for EM, DES, and no unconstranng 101% % of perfect nformaton revenue 100% 99% 98% 97% 96% 95% 94% 93% 92% EM_h DES_h Spral_h Intal Number of teratons Fgure 7 shows a convergence to the perfect nformaton calculated revenue for DES and EM methods after startng wth the hgh ntal protecton levels. Here both unconstranng methods converge to the perfect nformaton calculated protecton levels and hence acheve the benchmark. The 29

31 DES and EM methods yeld smlar results, both startng at 95.5% of the perfect nformaton calculated revenues and mprovng to almost 100% after only one teraton. (Ten Frday nght stays equals one teraton.) Hgh startng protecton levels restrct early bookngs n the lower value segments whle savng capacty for the hgh value segments. When hstorcal data s lmted and the dfference n revenue between hgh and low value segments s large, a frm may want to ntally employ hgh protecton levels. Fgure 8 - Revenue acheved usng low protecton lmts for EM, DES, and no unconstranng 102% % of perfect nformaton revenue 100% 98% 96% 94% 92% 90% 88% 86% EM_lo DES_lo Spral_lo Intal Number of teratons Compared to the hgh startng protecton levels n Fgure 7, the low startng protecton levels n Fgure 8 converge to the perfect nformaton calculated revenue for both unconstranng methods at a much slower rate. Because all three of the hghest fare classes are ntally 100% constraned, and the EM method requres at least one unconstraned bookng curve, we cannot use the EM method durng the frst teraton. Instead, we ncrease the protecton lmts by 10% for each group of 10 bookng curves 30