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 a * Jon Hgbe b Roht Kapoor a a The College of Management Georga Insttute of Technology 800 West Peachtree Street Atlanta, GA b Manugstcs, Inc 2839 Paces Ferry Road, SE, Sute 1000 Atlanta, GA *Correspondng Author: Tel: (404) , Fax: (404) , Mark.Ferguson@mgt.gatech.edu

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. These 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 methods used to unconstran hstorcal demand data aganst our proposed method by testng them on ntentonally constraned smulated data. Results show our proposed method along wth the Expectaton Maxmzaton (EM) method perform the best. 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 that 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. 2005). 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. Many hotel chans have nvested n systems and tranng for ther reservatons agents n order to track turndowns, and reled on these drect observatons to unconstran ther sales data. Unfortunately, there are many ssues wth usng turndown data for unconstranng ncludng multple avalablty nqures from the same customer, ncorrect categorzaton of turndowns by reservatons agents, and the fact that only small potons of customer request arrve through a channel 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 these 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 however has compromsed the qualty of ther 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 that check avalablty va the drect Internet channel actually book ther rooms at that 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 for whch no turndowns are collected, and as a result, hotelers have 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 some of the most common statstcal unconstranng methods that have appeared n the lterature and compare them to our proposed forecastng-based method. In addton, we apply the most accurate of these methods to real hotel bookng data. Prevous studes on unconstranng methods have only tested a subset of methods aganst smulated arlne data. Most tradtonal unconstranng methods follow a smlar methodology. They label the demand observed over tme for a gven secton of the bookng perod nto constraned and unconstraned categores, then adjust the parameter value estmates for the demand dstrbuton based on the percentage of data that was constraned. These unconstranng methods treat all truncated demand the 4

6 same, a demand stream that s constraned twenty days before the end of the bookng wndow s treated the same as one that s constraned one day before. Ths methodology gnores an mportant aspect of the revenue management envronment: frms often know the tme when demand s 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 alternatves. It s based on a wdely accepted statstcal forecastng technque (double exponental smoothng) requrng mnmal computatons and 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 fact that demand was truncated and the fnal demand observed was 25, the bookng lmt for ths partcular scenaro. Our proposed method also uses the fact that demand was truncated 30 days before the plane departure or the 5

7 guest arrval. Our method then uses double exponental smoothng to project the total demand that would have been observed n the absence of any bookng lmts (the dashed part of the bookng pace curve). Through a smulaton experment, we fnd that our method outperforms the majorty 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 that the two methods provde usng actual bookng and revenue data from a leadng hotel/casno. The rest of the paper s organzed as follows. 2 revews the lterature, 3 defnes the proposed method, 4 tests the method aganst other common methods used n practce, 5 tests the two best performng methods on real hotel/casno data and measures ther mpact on the hotel s revenue. Fnally, 6 concludes the paper. 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. 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 s smlar to the way revenue managers termnate demand for a partcular customer value 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 6

8 methods heavly rely on the use of 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. Lawless (1982) explans the lfetable method of uncensorng data and we nclude t n our comparson. For more tradtonal revenue management unconstranng methods, 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 usng an nferor method. Sx unconstranng methods are tested: three dfferent averagng methods, bookng profle (BP), 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): a smple 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 that the EM method outperforms the others and ncreases revenues by 2-12 % n full capacty stuatons. We also fnd the EM method provdes the most accurate parameter estmates although there are many condtons where our proposed method s more accurate. Of the three best methods that Weatherford and Polt (2002) and Zen (2001) use, only EM and PD are 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 7

9 obtan more aggressve demand estmates. The tradeoffs nclude ncreased computatons and the rsk of no soluton convergence (Weatherford and Polt, 2002). Lu et al. (2002) examne unconstranng demand data through the lens of the hotel ndustry and argue that 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 about the computatonal ntensty requred 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. Tallur and van Ryzn (2004b) provde an excellent treatment of all aspects of revenue management systems. 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 8

10 (2001) provdes a good revew of ths method. We descrbe how t may be used to solve the unconstranng problem below. 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). That s t [ I, I 1,... B], I B. After perod B, demand contnues to occur but s unobserved. If demand s not 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 our example gven n Fgure 1, I corresponds to perod 140 and B corresponds to perod 30, after whch demand s unobserved. Now defne: A t = Actual demand n perod t F t = The exponentally smoothed forecast for perod t T t = The exponentally smoothed trend for perod t FIT t = The forecast 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 ) (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 alpha and beta values that mnmze the sum of the squares of the forecast error: 9

11 B αβ, t= I mn ( A ) 2 t FIT (4) t 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 data set 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. We then use our forecastng model to project the total demand over the number of perods that the data set s constraned. We do so by usng (1) to forecast demand over perods B-1 to 0. Referrng back to Fgure 1, we calculate the α and β parameters over the sold porton (perods 140 to 30) of the demand curve and project the trend component over the dotted porton (perods 29 to 0) of the demand curve. The cumulatve demand over the observed and projected components of the bookng curve 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 rate program 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 rate program 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 n ). The fnal demand dstrbuton parameters (mean µ and varanceσ 2 ) are then estmated usng ths set of pont estmates by: 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, openng and closng of bookng lmts, and other specfcs that mght be relevant n applcaton. However, DES can be easly adjusted to ncorporate these specfc characterstcs (Armstrong, 2001). 10

12 4. Comparson of Unconstranng Methods In ths secton, we present our methodology for comparng fve of the most common statstcal unconstranng methods and the ensung results. To compare the performance of 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. The unconstranng method that estmates the demand dstrbuton parameters closest to the actual true parameters s judged the best method. To compare the performances of the chosen methods, we frst smulate bookng curves and set bookng lmts to constran the 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) that 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 see the great 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 looked at the total demand seen for all 100 bookng curves smultaneously, (some where the total demand was not constraned and others where total demand exceeded the bookng lmts), and used each unconstranng method to estmate the true demand dstrbuton parameters. 11

13 Fgure 2 - Concave, Homogeneous and Convex bookng curves Total bookngs Concave Homogeneous Convex Days before arrval date To construct the bookng curve, 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 that the expected total demand over the 140 day perod s the same for all three curves. After we create the demand curves, we calculate bookng lmts. The mnmum of true demand and the bookng lmt s the demand seen by the user. A smple example s shown n Table 1. 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 12

14 We use the fact that f daly demand arrvals are Posson and the demands on dfferent days are ndependent, then total demand s agan Posson. Snce 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 (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: BookngLmt = µ + z * σ (6) We test the fve unconstranng methods across the three bookng curve shapes (homogeneous, convex, and concave) for each of the fve bookng lmts to test how each method performs under vared condtons. We chose unconstranng methods from prevous research; the frst three methods are the best performng methods from Weatherford and Polt s (2002) comparson. These nclude: 1) an averagng method (AM), called Naïve #3 by Weatherford and Polt, 2) Projecton Detruncaton (PD) and 3) Expectaton Maxmzaton (EM). The fourth method, lfetables (LT), s commonly used n medcal studes and relablty engneerng and s used n a revenue management context n van Ryzn and McGll (2000). A short descrpton of each of these methods s provded n the appendx. The ffth method s DES, whch was descrbed n 3. 13

15 4.1 Results of Comparson Overall, the EM and DES methods outperform the other three unconstranng methods. Table 2 shows the percentage error each unconstranng method produces compared to 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, EM outperforms DES on average. Table 2 summarzes the results of the comparson and Fgure 2 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. As seen n Table 2, the averagng method s the worst performng, although n some nstances the dfference between t and the others s small. Thus, under some condtons, practtoners may be justfed n forgong the ncreased accuracy of the other methods for the computatonal smplcty of the averagng method. Table 2 - Percentage error between calculated and actual mean for each unconstranng method Bookng Curve Homogeneous Convex Method Percent of Data Sets Constraned 20% 40% 60% 80% 98% AM -0.57% -1.29% -2.15% -2.87% -4.30% 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 -0.72% -1.01% -1.87% -2.73% -4.17% 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% 14

16 Concave Mean absolute error over all 3 Bookng Curves AM -0.71% -1.43% -1.86% -2.57% -4.57% 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 0.67% 1.24% 1.96% 2.72% 4.35% 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% The lfetable 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 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 lfetable method to produce effectve results. 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 provdes an estmate for the dstrbuton mean that s up to 4% closer to the true mean than the next closes method. Smlarly, for the convex data set, the DES method also provdes the most accurate estmate n all the constranng condtons. 15

17 Fgure 3 - Average absolute error from true demand for each unconstranng method Error 5% 4% 3% 2% AM PD EM DES LT 1% 0% 20% 40% 60% 80% 98% Data Constraned 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. The method underperforms on ths demand pattern because bookng segments close farther away from the arrval date for concave data, 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. 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 16

18 management, errors n estmatng the true demand for the lowest valued segments are typcally less costly than are errors n estmatng the demand for hgher valued segments. 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 40 days each for each shape. Results show that the DES and EM methods are the most accurate unconstranng method across a range of percentage constraned data, and the averagng method s the worst performng method. We notce the magntude of error s much hgher wth the small demand data set as compared to the large demand data set due to dffcultes of predctng data wth ntermttent demand (many perods wth zero demand). 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). 17

19 Fgure 4 - Unconstranng error wth small demand data set 70% Percent Error 60% 50% 40% 30% 20% AM PD EM LT DES 10% 0% 0% 20% 40% 60% 80% 100% Percent Constraned 5% Fgure 5 - Unconstranng error wth large demand data set Percent Error 4% 3% 2% AM PD EM LT DES 1% 0% 0% 20% 40% 60% 80% 100% Percent Constraned 18

20 Snce DES has sgnfcantly hgher error wth the small demand data set than wth the large demand data set, an alternatve formulaton was sought. 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. 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 6. These results provde evdence that Croston s method may be superor to DES for unconstranng when demand s ntermttent. We have found that the condtons where Croston s method begns to outperform DES s when the percentage of days wth zero demand exceeds 10% of the total number of days n the bookng curve. Fgure 6 - 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 versus EM versus 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. The bookng lmts are then appled 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 the mpact that unconstranng has on 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 20

22 they are rarely constraned (revenue per nght from the hghest fare customer can be 12 tmes the revenue from the lowest fare customer); therefore we focused our unconstranng efforts on the most popular four segments that are constraned. Bootstrappng was performed as follows. Frst, observng the 12 Frday nght stay bookng curves, t was apparent that at dfferent ntervals before arrval, the slope of the curve changes dramatcally. Based on these slope 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 use the bootstrap method to create 1000 dfferent 60 day bookng curves for each of the four customer segments. Observng the orgnal hotel bookng data showed that 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 bookng class (constraned demand) mdway through the bookng curve, reopened the bookng class, and then closed t agan before the actual day of arrval. We dd ths for each of the 4 dfferent segments n all 1000 replcatons. We set protecton levels so that 50% of a gven data set would be constraned, then agan so that 75% of a gven data set would be constraned, to test our methodology aganst dfferent constranng levels. Usng both DES and EM, we unconstraned these 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 perform 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 perform well even when demand s constraned multple tmes n a bookng curve. 21

23 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% Next we testthe mpact each unconstranng method would have on the hotel s revenue. Total revenue from a revenue management system s the ultmate ndcator of a system s success. Unconstranng methods however, 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 low protecton levels, smlarly to van Ryzn and McGll (2000). Protecton levels must be set at some estmated level for ntal product offerngs because a frm wll have very lttle dea of demand for a new product (protecton levels are the opposte of bookng lmts,.e. how many unts of capacty at a gven class to protect for hgher fare classes). As more demand s observed over tme, the frm adjusts protecton levels accordngly to ncrease total revenue. The convergence rate to 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 two best 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 spral-down effect whch occurs when data s not unconstraned. 22

24 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. EMSR-b 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 mean j= 1 ˆµ j and varance j= 1 revenue, gven by: ˆ σ 2 j, f s the revenue for customer value segment and f s weghted average f j= 1 = f j= 1 ˆ µ j ˆ µ j j (7) 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. However, protecton levels 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. Total revenue s calculated by multplyng the customer value of 23

25 each segment by the number of reservatons that would have been sold n that segment; the mn of the protecton level and the total demand. Mathematcally, ths s shown as: 4 = 1 [ θ ] Revenue = V * Mn, x (8) where V = customer value for segment x = total demand for segment θ = bookng level for segment 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. Customer spendng s tracked over tme by ssung frequent stay cards that are recorded each tme the customer makes a transacton. Snce 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 estmated ntal protecton levels for each customer segment (two ntal startng protecton levels are used for each segment, one lower and the other hgher than the optmal protecton levels). We then calculated the revenue the hotel/casno would have receved f they used these ntal protecton levels for each 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 24

26 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 appled 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. Ths procedure contnued, unconstranng the demand data and readjustng the protecton levels every 10 bookng curves, untl all 100 bookng curves n the set were used. 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 n order to calculate a standard error of our estmates. The data plotted n Fgures 7 and 8 represent an average over the 10 sets along wth upper and lower lmts correspondng to a 95% confdence nterval. All of the methods (EM, DES, Spral) were tested aganst the same data sets (the smulatons are coupled) Revenues for each of the unconstranng methods are compared to optmal revenue. We calculate the optmal 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 knew the true demand for every bookng curve, and hence knew the true mean and standard devaton parameter estmates. We appled EMSR-b usng these parameters to fnd protecton levels, then usng nested protecton levels, appled (8) to calculate total revenue. 25

27 Fgure 7 Revenue acheved usng hgh protecton lmts for EM, DES, and no unconstranng 101% % of optmal Revenue 100% 99% 98% 97% 96% 95% DES_h Spral_h EM_h 94% Intal Number of Iteratons Fgure 7 shows a convergence to the optmal revenue for DES and EM methods after startng wth the hgh ntal protecton levels. Here both unconstranng methods allow the EMSR-b method to quckly converge to optmal protecton levels and hence acheve optmal revenues. The DES and EM methods yeld smlar results, both startng at 95.5% of optmal revenues and mprovng to close to100% after only 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. 26

28 Fgure 8 - Revenue acheved usng low protecton lmts for EM, DES, and no unconstranng 102% % of optmal Revenue 100% 98% 96% 94% 92% 90% DES_lo Spral_lo EM_lo 88% Intal Number of Iteratons Compared to the hgh startng protecton levels n Fgure 7, the low startng protecton levels n 8 converge to the optmal revenue for both unconstranng methods at a much slower rate, as all three of the hghest fare classes are ntally 100% constraned. Because of ths, we were unable to use the EM method durng the frst teraton snce the EM method requres at least one unconstraned bookng curve. Instead, we ncreased the protecton lmts by 10% for each group of 10 bookng curves untl at least one bookng curve was unconstraned, at whch pont we could begn usng the EM method. Whle ths practce may seem arbtrary (justfably so), t s representatve of technques commonly appled n practce. The DES method does not suffer from such a lmtaton, thus t outperforms the EM method durng the early stages of the low startng protecton levels case. Once the EM method can be used, revenue quckly converges to greater than 99% of optmal revenue. DES performs better n early teratons, then converges to just above 98% of optmal revenues, slghtly tralng the EM method s performance. 27

29 Both Fgures 7 and 8 llustrate that unconstraned data may lead to a loss n revenue. When startng wth low protecton levels, Fgure 8 shows the no unconstranng (Spral) data never mproves past the 89% of revenue ntally acheved. Ths compares unfavorably wth the two unconstranng methods whch steadly mprove as more demand s observed. Fgure 7 shows that when usng hgh ntal protecton levels, falng to unconstran data causes revenue to decrease every tme protecton lmts are recalculated as the hstorcal data becomes more and more constraned. Ths shows graphcally the spral-down effect descrbed by Cooper et al. (2005). One may conjecture from a comparson of Fgures 7 and 8 that, n the absence of hstorcal demand data, t s always better to start wth hgh protecton levels versus low snce the revenue converges to the optmal much faster n Fgure 7 than n Fgure 8. Such a generalzaton s ncorrect however, as ths phenomenon s an artfact of our choce of revenues for each customer class. For ths hotel, the dfference n revenues between the hghest fare class ($100) and the next hghest ($62) s much larger than the dfference between the two lowest fare classes ($35 and $25). If these dfferences had been reversed (say fare classes of $100, $90, $63, and $25), startng wth low ntal protecton levels would converge to the optmal revenue much faster than startng wth hgh protecton levels. We note a few addtonal observatons from our bootstrappng results. Frst, when demand has sudden shfts or a large number of constraned days, all statstcal methods become less accurate. In practce, we recommend qualtatve adjustments to the demand data or the numercal forecast for these stuatons. Second, n any stuaton, statstcal unconstranng should be supplemented by a physcal constraned count, where possble, to check the valdty of unconstraned forecasts. For a hotel, ths physcal count could nclude reservaton agents and bellhops keepng a manual tally of the number of people turned away. A store mght better promote ran checks of sold out tems and keep track of how many people ask for the sold out good. Thrd, EM becomes more accurate for larger sample szes, but 28

30 often performs poorly for small sample szes. In these cases, DES may be a better soluton untl more hstorcal data s avalable. Fourth, a small frm wth a lmted IT budget may not have the resources to afford a sophstcated statstcal program to run the EM method. Such a frm may fnd t more useful to use the straghtforward DES method to estmate total demand. Lastly, f data s fully constraned, EM does not work, and an alternate method must be used. Ether an alternatve unconstranng method or a rule of thumb adjustment to the protecton level s needed untl unconstraned demand s observed. 6. Conclusons/ Recommendatons Ths paper examnes the often overlooked but essental topc of unconstranng sales data. True demand dstrbuton parameters are a crtcal ngredent to revenue management systems; unfortunately the data avalable s often constraned. Ignorng the constraned data problem results n sgnfcant reductons n revenue and observng demand after t exceeds capacty s often mpractcal, thus statstcal unconstranng methods are often used to estmate the parameters of the demand dstrbuton. We propose a new unconstranng method (DES) based on a common forecastng model that, unlke tradtonal statstcal unconstranng methods, takes nto account the pont n tme on the hstorcal demand bookng curves that demand was constraned. We fnd that our proposed DES method and the EM method perform smlarly well and outperform the alternatves. When lttle hstorcal data s avalable or all demand sets are constraned, then DES s an better choce than EM. Also, f sophstcated statstcal software packages are not avalable, DES provdes a better alternatve than the averagng method. We test the revenue mpact of DES, 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. In our example, startng wth hgh(low) ntal protecton levels converge to optmal lmts more quckly(slowly). Low ntal protecton levels can 29

31 lead to completely constraned classes, forcng frms usng the EM method for unconstranng to use other methods untl some unconstraned bookng curves are observed. Both EM and DES take numerous teratons (5 and 3, respectvely) to converge to wthn 2% of optmal wth low ntal protecton levels. On the other hand, both EM and DES converge to wthn 0.3% of optmal after only one teraton when startng wth hgh protecton levels. These results show that both EM and DES are effectve methods for unconstranng and provde much hgher revenues than no unconstranng at all. As s true wth all research, there are lmtatons to our work. In our secton on revenue mpact, we assume ndependent demand for a gven customer value segment. That s, a customer assocated wth value segment 2 wll not turn up as demand n segments 1 or 3, even f value segment 2 s closed. Ths s a very realstc assumpton n a casno applcaton, where customer gamng habts are not dependent upon the rate program they book under, but rather vce versa. However, n many other revenue management applcatons partcularly arlnes - demand for a gven segment often depends on the choce of segments avalable. The ndependence assumpton s commonly used however n most bookng lmt algorthms. To ncorporate consumer choce behavor such as buy-up or buy-down behavor, Tallur and van Ryzn (2004a) present a model that explctly accounts for the probabltes that customers n a gven fare class (value segment) wll purchase from other fare classes f ther preferred fare class s unavalable. To use ths model, however, a frm needs to know the probabltes that customers n all classes wll buy-up, or buy-down, probabltes that are rarely known n practce. Further development of such models along wth unconstranng methods to accommodate them s a promsng area of future research. Addtonally, our DES method does not account for seasonalty or prce promotons. Lke most other forecastng methods, hstorcal data should be decomposed nto components of promoton effects, seasonalty, and compettve effects before DES s appled. For further nformaton on these adjustments, see Armstrong (2001). 30

32 References Altay, N. and Ltteral, L A Comparatve Study of Intermttent Demand Forecastng Models, workng paper, Robns School of Busness, Unversty of Rchmond. Armstrong, J. Scott (ed.) Prncples of Forecastng: A Handbook for Reserachers and Practtoners. Norwell, MA: Kluwer Academc Publshers. Badnell, R.D An Optmal, Dynamc Polcy for Hotel Yeld Management, European Journal of Operatonal Research 121: Belobaba, P.P Applcaton of a Probablstc Decson Model to Arlne Seat Inventory Control, Operatons Research 37 (2): Btran, G.R. and S. M. Glbert Managng Hotel Reservatons wth Uncertan Arrvals, Operatons Research 44: Btran, G.R. and S. V. Mondschen An Applcaton of Yeld Management to the Hotel Industry Consderng Multple Day Stays, Operatons Research 43: Carroll, B. and J. Sguaw The Evoluton of Electronc Dstrbuton: Effects on Hotels and Intermedares, Cornell Hotel and Restaurant Admnstraton Quarterly 44(4) Cooper, W.L, T. Homem-de-Mello, A.J. Kleywegt Models of the Spral-Down Effect n Revenue Management, Workng Paper, Department of Mechancal Engneerng, Unversty of Mnnesota. Cox, D.R Regresson Models and Lfe Tables (wth dscusson), Journal of the Royal Statstcal Socety. B34 (2): Cox, D.R. and D. Oakes Analyss of Survval Data. New York, Chapman & Hall. Cross, R. G Revenue Management. New York, Broadway books. Croston, J.D Forecastng and Stock Control for Intermttent Demands, Operatonal Research Quarterly 23 (3): Dempster, A.P., N.M. Lard, and D.B. Rubn Maxmum Lkelhood from Incomplete Data va the EM Algorthm, Journal of the Royal Statstcal Socety, Seres B. 39 (1): Kalbflesch, J.D. and R.L. Prentce The Statstcal Analyss of Falure Tme Data. New York, John Wley. Lawless, J.F Statstcal Models and Methods for Lfetme Data. New York, John Wley. Lu, P.H. and V. Maks Cuttng-tool relablty assessment n varable machnng condtons, IEEE Transactons on Relablty. 45(4):