In this paper, we examine the research and results of dynamic pricing policies and their

Transcription

1 Commissioed Paper A Overview of Pricig Models for Reveue Maagemet Gabriel Bitra Reé Caldetey Sloa School of Maagemet, MIT, Cambridge, Massachusetts 0139 Ster School of Busiess, New York Uiversity, New York, New York 1001 gbitra@mit.edu rcaldet@ster.yu.edu I this paper, we examie the research ad results of dyamic pricig policies ad their relatio to reveue maagemet. The survey is based o a geeric reveue maagemet problem i which a perishable ad oreewable set of resources satisfy stochastic pricesesitive demad processes over a fiite period of time. I this class of problems, the ower (or the seller) of these resources uses them to produce ad offer a meu of fial products to the ed customers. Withi this cotext, we formulate the stochastic cotrol problem of capacity that the seller faces: How to dyamically set the meu ad the quatity of products ad their correspodig prices to maximize the total reveue over the sellig horizo. (Reveue Maagemet; Dyamic Pricig) 1. Itroductio The aim of this paper is to review the growig literature o dyamic pricig policies ad their coectio to reveue maagemet. I geeral terms, the reveue maagemet model that we ivestigate cosiders the problem faced by a seller who ows a fixed ad perishable set of resources that are sold to a pricesesitive populatio of buyers. I this framework, where capacity is fixed, the seller is maily iterested i fidig a optimal pricig strategy that maximizes the reveue collected over the sellig horizo. The motivatio for this work is our strog belief that pricig policies are today, more tha ever before, a fudametal compoet of the daily operatios of maufacturig ad service compaies. The reaso is probably because price is oe of the most effective variables that maagers ca maipulate to ecourage or discourage demad i the short ru. Price is ot oly importat from a fiacial poit of view, but also from a operatioal stadpoit. It is a tool that helps to regulate ivetory ad productio pressures. Airlie compaies ad retail chais are good examples of idustries where dyamic pricig policies are becomig key drivers of the compaies performace. Not surprisigly, pricig models have become icreasigly popular withi the maagemet sciece commuity. Researchers have realized that classical operatioal problems, such as optimal capacity ad ivetory maagemet or cotrollig cogestio i a queueig etwork amog may others, caot be decoupled from marketig activities ad especially pricig decisios. This broad rage of applicatios has geerated a importat volume of work. We believe it is time to survey the field ad to preset the mai results ad their practical implicatios. We do ot attempt, however, a exhaustive review of the vast literature o pricig. Istead, we focus o the work that has bee doe i the cotext of reveue maagemet. The rapid evolutio of iformatio techologies ad the correspodig growth of the Iteret ad e-commerce are sources of ispiratio for a survey o dyamic pricig models for two mai reasos. First, i this electroic world, it is possible to collect valuable iformatio (about demad, ivetory levels, /03/0503/003$ electroic ISSN Maufacturig & Service Operatios Maagemet 003 INFORMS Vol. 5, No. 3, Summer 003, pp. 03 9

2 competitors strategies, etc.) ad process it i real time. This ew reality allows ad forces maagers to act ad react dyamically to chages i the marketplace by adjustig ay variable uder cotrol, especially prices. Furthermore, Iteret-based sellig systems make the logistics of dyamic pricig much easier. The costs associated with relabellig the prices of the products ad iformig customers about these chages have dropped sigificatly i the electroic eviromet whe compared to traditioal brick-admortar busiesses (e.g., Bryjolfsso ad Smith 1999). O the customer side, Iteret price search itermediaries or web aggregators offer customers easy access to better iformatio about product variety ad price lists (e.g., pricelie.com). As a result, ew potetial applicatios for reveue maagemet techiques are emergig i coectio to the Iteret. We cosider it importat to preset the fudametal aspects of dyamic pricig models to a audiece that is curretly workig ad developig e-commerce. From a historical perspective, the iterest i reveue maagemet practices started with the pioeerig research of Rothstei (1971, 1974) ad Littlewood (197) o airlie ad hotel overbookig. However, it was probably after the work of Belobaba (1987a, b; 1989) ad the America Airlies success (Smith et al. 199) that the field really took off. The airlie idustry provided researchers with a cocrete example of the tremedous impact that reveue maagemet tools ca have o the operatios of a compay (e.g., Smith et al. 199). The publicatio of a survey paper by Weatherford ad Bodily (199), where a taxoomy of the field ad a ageda for future work were proposed, was aother symptom of this revival. At this stage, however, much of the work was doe o capacity maagemet ad overbookig with little discussio of dyamic pricig policies. I essece, prices (fares) i these origial models were assumed to be fixed ad maagers were i charge of opeig ad closig differet fare classes as demad evolved. Durig the 1990s, the icreasig iterest i reveue maagemet became evidet i the differet applicatios that were cosidered. Models became idustry specific (e.g., airlies, hotels, or retail stores) with a higher degree of complexity (e.g., multiclass ad multiperiod stochastic formulatios). Furthermore, it was i the last decade that pricig policies really became a active compoet of the reveue maagemet literature (e.g., Gallego ad va Ryzi 1994; Bitra ad Modschei 1997; Feg ad Gallego 1995, 000). Today, dyamic pricig policies i a reveue maagemet cotext is a active field of research that has reached a certai level of maturity. I terms of applicatios, dyamic pricig practices are particularly useful for those idustries havig high start-up costs, perishable capacity, short sellig horizos, ad a demad that is both stochastic ad price sesitive. Succictly, the reveue maagemet problem has bee phrased as sellig the right product to the right customer at the right time. O oe had, the sellers would like to sell their products to those customers who have a high valuatio so that high margis ca be achieved. O the other had, if they wait too log for those high valuatio customers to appear, they might ed the sellig period with usold uits that could have bee sold to low valuatio customers. Clearly, for this trade-off to be otrivial, both perishable capacity ad stochastic demad are eeded. As we will discuss i this paper, it is precisely i this eviromet that dyamic pricig strategies are especially useful to balace utilizatio ad profitability of the available capacity. As we already metioed, the airlie idustry pioeered the use of reveue maagemet techiques i terms of capacity/seat cotrol ad dyamic pricig. Today, reveue maagemet has spread out aturally to other idustries such as retailers (e.g., Bitra ad Modschei 1997, Subrahmaya ad Shoemaker 1996), car retal agecies (e.g., Carol ad Grimes 1995, Geraghty ad Johso 1997), hotels (e.g., Bitra ad Modschei 1995, Bitra ad Gilbert 1996), badwidth ad Iteret providers (e.g., Nair ad Bapa 001), passeger railways (e.g., Ciacimio et al. 1999), cruise lies (e.g., Laday ad Arbel 1991), ad electric power supply (e.g., Schweppe et al. 1987, Smith 1993, Ore ad Smith 1993). Although differet i may respects, these idustries all share the basic properties of the reveue maagemet problems that we cosider i this work, amely, perishable products, fiite sellig horizos, ad price-sesitive ad stochastic demad. 04 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003

3 We coclude this itroductio by positioig this paper with respect to other similar works that have bee published. I terms of goals, our objective is to preset the mai results that have bee reported i the literature durig the last decades o dyamic pricig models. We cocetrate our efforts o uderstadig the mai drivers ad properties of optimal pricig strategies. I this respect, we do ot discuss i detail the somewhat related research that has bee doe i the area of ivetory ad capacity cotrol, although some related results for etwork reveue maagemet problems are preseted i 3... Our work differs from other survey papers, such as Weatherford ad Bodily 199 or McGill ad va Ryzi 1999, because we do ot attempt to provide a taxoomy or a exhaustive eumeratio of all the publicatios i the field. Similar i may aspects to our work is the survey o dyamic pricig models by Elmaghraby ad Keskiocak (00), where a broad view of the field is preseted from a set of differet agles, such as pricig policies for log ad short life-cycle products, combied ivetory ad pricig decisios, or pricig i markets with ratioal customers. However, we preferred to arrow the scope of our work to dyamic pricig models i a reveue maagemet cotext, so that we ca explicitly preset ad discuss the mai results that have bee obtaied. I this regard, we believe that our work provides a helpful summary of this field to those readers (researchers or practitioers) iterested i gettig a geeral overview of the research that has bee doe thus far. Nevertheless, we believe that a good survey should ot oly itroduce the field ad mai results to the ospecialists, but also provide ew isights ad guidace for future research to the experts. For this purpose, we have complemeted our review with some ew results, ad we have icluded a list of potetial ew directios of research i 4. The remaider of this paper is orgaized as follows. I we developa geeric formulatio of the reveue maagemet problem that provides a global view of the differet elemets ad their iterrelatioship. I particular, we preset a geeral pricig problem ad describe how the differet compoets, such as demad attributes, product characteristics, iformatio, ad costraits, affect the formulatio ad its applicability. Next, i 3 we review the literature ad the mai results. We approach this review from differet agles such as determiistic versus stochastic models ad sigle versus multiproducts models. Fially, i 4, we summarize our results ad idetify ope problems ad ew potetial directios of research.. A Geeric Model I this sectio, we describe the reveue maagemet model uder cosideratio. The model that we preset is sufficietly geeral to cover the research that we review i 3 as a special case. Furthermore, some of the elemets of our geeric formulatio i.6 have ot yet bee fully addressed i the literature. I this respect, our motivatio for this apparet excess of geerality is twofold. First of all, we believe that our geeric framework is more appealig to those ospecialist readers iterested i gettig the big picture behid the reveue maagemet problem. Secodly, the cotrast betwee this geeral model ad the specific research preseted i 3 ca be used to idetify potetial research opportuity. I 4, we suggest some ew directios..1. Supply Cosider a seller or market player (e.g., a airlie, hotel, car retal compay, retail store, or Iteret service provider) that has a fixed amout of iitial capacity that is used to satisfy a price-sesitive demad 1 durig a certai sellig period H = 0 T. We model this iitial capacity as a m-dimesioal vector C 0 = c 1 0 c m 0 of resources, where c k 0 is the iitial amout of resource k available. Capacity, i our cotext, is a rather broad cocept that might iclude the umber of rooms i a hotel, available seats for a specific origi-destiatio flight o a give day, or simply the umber of white shirts i stock at a garmet store. Uder the stadard reveue maagemet problem that we cosider, capacity is fixed ad ay strategic cosideratios regardig how to acquire the iitial 1 Demad ca also deped o other variables cotrolled by the seller, like capacity itself. Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer

4 level C 0 have bee excluded. Capacity is essetially give ad the seller is committed exclusively to fidig the best way to sell it. This assumptio is by o meas critical if we cosider that i may idustries capacity is flexible oly i the log ru. Moreover, capacity decisios ad price decisios take place o differet time scales. Issues regardig the size of a hotel or a airplae or the umber of shirts to purchase from a overseas supplier are decided log before demad is realized ad price policies are implemeted. Critical to the reveue maagemet problem are the characteristics of this available capacity ad how it is used to create a set (or meu) of fial products. As we will see shortly, i some cases much of the complexity of the reveue maagemet problem comes from selectig the correct meu of products. From a pricig perspective, two importat attributes of the available capacity are its degree of flexibility ad its perishability. Flexibility measures the ability to produce ad offer differet products usig the iitial capacity C 0. We say that capacity is dedicated if there is a oe-to-oe correspodece betwee capacity ad fial product. For example, a retailer that purchases 500 white t-shirts to sell durig the ext summer seaso has dedicated capacity. O the other had, we say that capacity is flexible if it ca be used to produce differet products or satisfy differet customers eeds. For example, a Iteret provider owig badwidth capacity uses this specific resource to offer a wide rage of products from services to video cofereces. I geeral, flexibility is a cotiuous attribute ragig from highly dedicated (retailig) to highly flexible (the badwidth provider). It should be ituitively obvious that flexibility is a desired feature. I essece, flexible capacity allows the seller to allocate scarce resources efficietly based o observed demad rather tha forecasted demad (productio postpoemet). I practice, however, flexibility is ot always possible. A retailer buyig from a overseas supplier eeds to order moths before the begiig of the sellig seaso. I the hotel idustry, the allocatio of the available space ito luxury, suite, ad stadard rooms is essetially decided whe the hotel is built. From a pricig stadpoit, flexibility icreases the complexity of the problem. As we will discuss later, the actio of sellig a product has associated two quatities: (i) a immediate reveue equal to the price ad (ii) a opportuity cost that is the moetary pealty of usig capacity today that could be used to satisfy future demad. Whe capacity is dedicated, sellig product i does ot affect the ability to supply product j. Thus, the opportuity cost of sellig i ivolves essetially product i ad its demad. However, whe capacity is flexible, sellig product i decreases the resources available to produce product j. This iteractio amog products makes the computatio of the opportuity cost ad the optimal pricig strategy much harder. Perishability relates to the (lack of) ability to preserve capacity over time. For example, a empty seat o a departig flight is a uit of capacity that caot be stocked for a future flight. I geeral, a distictive feature of the reveue maagemet problem is the perishability of the available capacity. A simple way to treat this perishability is makig capacity a time-depedet quatity. For istace, a hotel s uit of resource might be Room 106 o Friday ight, May 10, 00, while a airlie s uit of capacity could be Seat B o flight #143 departig from Bosto to Chicago at 4:00pm o Tuesday, May 14, 00. How much detail is used to defie the uits of capacity depeds o customers prefereces ad the seller s ability to profit from their choice. For example, two ecoomy-class seats A (widow) ad B (aisle) o a give flight could be cosidered two differet resources ad priced differetly if customers have sigificat differeces o their prefereces for widow ad aisle seats. I practice, airlies do ot discrimiate based o this feature ad both seats A ad B are cosidered two uits of the same resource: ecoomy-class seats. Retailers, o the other had, are much more active i this way, chargig differet prices for a blue shirt ad for a red shirt (same model, brad, ad size). From a modelig perspective, perishability icreases the dimesio of the problem, makig capacity, ad therefore fial products, time-depedet quatities. I our dyamic settig, perishability is a iheret property of the model, although it might be 06 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003

5 irrelevat i some cases, e.g., whe capacity is fully ivetoriable ad the sellig horizo goes to ifiity. As time progresses ad resources are cosumed (they are sold or they perish), capacity decreases ad we deote by C t = c 1 t c m t the available capacity at time t... The Product Followig our previous descriptio of capacity, a product i this cotext is a subcollectio of the available resources. Based o Gallego ad va Ryzi s (1997) productio model, we cosider a m matrix A = a ij such that a ij represets the amout of resource i used to produce oe uit of product j. That is, every colum j of A represets a differet product say product A j ad the collectio M = A 1 A is the meu of products offered by the seller. We will cosider for the momet that there are o explicit costs associated to the productio of the fial products. This is, by the way, a commo assumptio i the literature. I may situatios, this assumptio is ot very restrictive because productio costs are egligible, or they are liear ad ca be icorporated directly ito the fial price. Give the available iitial capacity C 0, the first importat decisio of the seller is to defie the meu M of products that will be offered to the ed customers. A aïve approach would be to cosider ay possible subset of C 0 as a product, i.e., M = a m 0 a C 0. However, eve if a demad exists for every coceivable subset of C 0, the task of settig a differet price strategy for every combiatio is computatioally demadig ad hard to implemet. O oe had, maagig a short list of products simplifies the pricig problem. O the other had, a larger list is more suitable for demad-skimmig purposes. The right mix of products should balace this trade-off. For istace, the simplest approach would be to set A = I k, the k k idetity matrix. I this case, every resource is dedicated ad offered as a sigle product. Customers are left with the task of purchasig the appropriate combiatio of each resource depedig o their specific eeds. I this case a miimum set of prices is eeded, oe for each resource. The seller, however, ca try to do better by creatig budles, which are specific subsets of resources that match specific customers eeds. By doig so, the seller is able to target the market ad icrease demad. I this case, a larger set of prices has to be specified with the correspodig icremet i maagemet costs..3. Iformatio Crucial to ay dyamic pricig policy is the kowledge of the system ad its evolutio over time. Realtime pricig ecessarily requires real-time demad data, the available capacity, ad ay other relevat factors (e.g., competitors strategies, weather). Thus, a iformatio system capable of collectig the right iformatio ad makig it available at decisio poits is critical. There is little doubt that oe of the major factors that iflueced the rapid growth of yield maagemet i the airlie idustry was the developmet of electroic iformatio systems capable of gatherig iformatio about demad ad ticket reservatio over the large etwork of travel agecies (e.g., SABRE system for America Airlies, Smith et al. 199). Similarly, as reported by Rama et al. (001), retailers are ivestig large amouts of moey (close to $30 billio a year) to improve IT systems ad reduce the systematic problem of iaccurate ivetory records. I our reveue maagemet settig, short product life cycles ad perishability impose extra pressure to improve the quality ad maagemet of iformatio such as demad forecast ad ivetory positio. For istace, stadard forecast methods rely heavily o demad history that is ot ecessarily available i this short life cycle eviromet, for example, retailers sellig fashioable products (e.g., Fisher ad Rama 1996, Kurawarwala ad Matsuo 1996). Give a iitial capacity C 0, a product meu M, ad demad ad price processes, we defie the observed history H t of the sellig process as the set of all relevat iformatio available upto t. This history should iclude at least the observed demad process ad available capacity, ad it ca also iclude some additioal iformatio such as demad forecasts. Most of the research has focused o the simple but tractable Markovia case where H t = C t,i which oly remaiig capacity is relevat for pricig decisios. However, path-depedet models are especially useful whe demad distributio is ukow ad a learig process is icorporated to improve Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer

6 demad estimates. I geeral, we expect some degree of iformatio asymmetry betwee the seller ad the buyers. Issues regardig the quality of the product or the level of ivetory, for istace, are usually private iformatio held by the seller. O the other had, customers have private iformatio about their product valuatios ad budget. This asymmetry of iformatio ca be modeled usig two subhistories H s t, H b t H t represetig the iformatio available to the seller ad customers, respectively, at time t..4. Demad O the demad side, we divide the set of potetial customers ito differet segmets, each oe havig its ow set of attributes icludig eeds, budget, ad quality expectatios. We defie a d-dimesioal stochastic process N t H t = N 1 t H t N d t H t, where N j t H t is the cumulative potetial demad upto time t from family j give the available iformatio H t. Depedig o the price (ad probably other attributes such as quality) potetial customers will decide whether or ot to purchase the products. Usig Lazear s (1986) termiology, potetial customers are divided ito (i) shoppers which are those customers who search for products but do ot buy because of price or quality cosideratios ad (ii) buyers which are those customers that are effectively willig to buy a product. I geeral, pricig policies should be computed o the bases of both potetial customers ad buyers. However, i most applicatios the seller is oly capable of collectig iformatio about the set of buyers accordig to sales data. To model this purchasig process, we defie a d matrix B P = b ij, where b ij represets the uits of product i M requested by a customer i family j = 1 d; the price process P t = p s s 0 t is described i detail i.5. It is importat to ote that budlig cosideratios are directly liked to Oe exceptio is the catalog idustry, here the seller cotrols the populatio of potetial cosumers accordig to the mailig policy (e.g., Bitra ad Modschei 1996). E-commerce is aother example because iformatio o shoppers (as opposed to buyers) ca be obtaied via the Iteret, by storig the path customers follows o the website. the structure of this matrix B P through its depedece o the product meu M. Combiig the vector of potetial demad N t H t ad the matrix B P, we defie a -dimesioal vector D t P H B P N t H t that represets the effective cumulative demad process i 0 t at the product level. Fially, we provide the seller with the ability to partially serve demad if it is profitable to do so. For istace, retailers do ot display their etire ivetory durig promotio days. I the same way, airlies are able to reject low-fare reservatios (closig a fare) eve if they have available capacity. I light of this, we defie a -dimesioal vector S t that represets the cumulative sales upto time t. Give the demad, sales, ad price processes, the dyamics of the available capacity are govered by the followig coditios: C t = C 0 AS t ad S t D t P H t for all t 0 T (1) I some cotexts, the distictio betwee sales S ad demad D is uecessary. For istace, if the price ca be adjusted cotiuously ad urestrictedly, the seller will prefer to icrease the price rather tha reject customers. I this case, the price is the oly variable that the seller eeds to cotrol. For example, i the yield maagemet literature of seat cotrol, the otio of a ull price (a high price that makes demad equal to zero almost surely) has bee itroduced to model the accept/reject decisio i the cotext of dyamic pricig policies (see 3..). We ote that if the seller is costraied i the way that she/he ca adjust the price (see.5 for some examples of costraits), the the distictio betwee sales ad demad becomes relevat ad the accept/reject decisio is ot ecessarily replicable usig a dyamic pricig strategy. I terms of our assumptios, the use of a pricesesitive demad D t P H t implies that the seller has moopolistic market power over the set of buyers. Competitio might be preset i this formulatio, but it is hidde ad oly the residual demad N t H t faced by the seller is cosidered. We do ot icorporate ay strategic behavior from the customers side, demad might deped o the whole observed history of the sellig process, but we do ot model the 08 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003

7 utility maximizatio process solved by the customers. Demad i this respect is assumed to be give exogeously. Similarly, customers are assumed to be price takers, meaig they observe the price list offered by the seller ad react by buyig or ot buyig some of the products. We will postpoe the discussio of other allocatio mechaisms, such as auctio models, to 4. Certaily, good modelig ad forecastig of demad are key for pricig purposes. The alterative formulatios available i the literature are ulimited especially i the determiistic demad case. The simplest approach is probably to decompose this determiistic demad ito a set of differet factors, each oe addressig a specific aspect of the problem (e.g., Eliashberg ad Jeulad 1986, Kalish 1983, Jai ad Rao 1990): D det t p H t = t p F H t () where t is a estimate of the market size as a fuctio of time, p captures price elasticity effects, ad F H t models the ifluece of the available iformatio o customers purchasig behavior. From microecoomics theory (e.g. Mas-Colell et al. 1995), the otios of cosumers utility, elasticity, ad product substitutio form the bases of our uderstadig ad modelig of p. For example, expoetial demad models are commoly used to model demad i the retail sector (e.g., Smith ad Achabal 1998). That is, p = exp p, where is a measure of demad elasticity per uit of price. Other models usig costat elasticity, p = p, have also bee proposed (e.g., Bitra et al. 1998). A fuctioal form for F H t, for the case H t = C t, was developed ad empirically tested i Smith ad Achabal (1998). O the other had, the modelig of t depeds o the seasoality of demad ad the life cycle of the product. Diffusio models (e.g., Bass 1969) are widely used to model this evolutio of demad. I this framework, a populatio of cosumers of size N gradually purchases the product. The rate at which cosumers buy the product depeds liearly o the umber of previous purchases (by word of mouth or diffusio effects) ad the fractio of iovators existig i the populatio. Iovators are those cosumers who buy the product idepedetly of the other cosumers actios. I Bass s (1969) diffusio model, the rate of purchase at time t is give by d t = pn + q p t q dt N t (3) where p is the fractio of iovators ad q is a measure of the diffusio effect (imitatio). The combiatio of this diffusio model with price has bee proposed i Bass (1980) ad Jeulad ad Dolad (198). The stochastic behavior of the demad has bee added to these determiistic models for discrete ad cotiuous time formulatios. For the discrete time case, the stadard approach is to represet demad as the sum of a determiistic part ad a zero-mea stochastic compoet. Usig the otatio dd t p H for the margial demad i period t, the stochastic additive oise model is give by dd stoc t p H t = dd det p t H t + t p H t (4) }{{} Radom Noise The radom oise, which usually follows a zero-mea ormal radom variable, depeds o price ad time to reflect the chages o demad ucertaity over the life cycle. Aother alterative model is the multiplicative oise model dd stoc t p H t = dd det t p H t t p H t (5) I this case, the expected value of the radom oise is ormally set to oe. Combiatios of the additive ad multiplicative models ca also be used. For the cotiuous time case, the most commo formulatio assumes that demad follows a Poisso process with a determiistic itesity that depeds o price ad time (e.g., Gallego ad va Ryzi 1994, 1997; Bitra ad Modschei 1997; Feg ad Gallego 000), although it is possible to exted the discrete time formulatio above replacig the ormally distributed radom oise by a cotiuous time Wieer process (e.g., Rama ad Chatterjee 1995)..5. Pricig Strategies I our dyamic settig, a pricig policy P = P T = p t t 0 T is a collectio of mappigs Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer

8 p t H t M + where p t i H t is the price of product i M at time t give a curret history H t. Depedig o the applicatio, some coditios have to be imposed to esure that the resultig pricig policy P is cosistet with stadard practices i the idustry. The followig is a list of the most commo costraits that we have come across durig our literature review ad idustrial experiece. Fiite Set of Prices. I may applicatios the seller ca oly select prices from a fiite list of admissible prices, i.e., t = p 1 p Kt (e.g., Chatwi 000; Feg ad Xiao 000a, b; Feg ad Gallego 000). The reasos rage from marketig cosideratios such as customers perceptio of prices ($19.99 versus $0.00) to maagerial aspects because a discrete list of prices is easy to implemet ad cotrol. Maximum Number of Price Chages. Most compaies restrict the umber of price chages durig the sellig horizo (e.g., Feg ad Gallego 1995). I some cases, this restrictio is ot critical because two-price policies have bee show to be asymptotically optimal (e.g., Gallego ad va Ryzi 1994). I practice, compaies restrict the umber of price chages because chagig prices too ofte is difficult ad costly from a operatioal stadpoit. We should metio, however, that for the growig Iteretbased sales systems, the costs of relabellig the prices of products ad those associated with iformig customers about these chages are droppig cosiderably (e.g., Bryjolfsso ad Smith 1999). Markdows, Markups, ad Promotios. It is commo practice i some idustries to eforce a predefied path of the price over time. For istace, retailers usually adopt a markdow policy, or clearace policy that makes the prices of the products decrease mootoically over time (e.g., Bitra ad Modschei 1997). I geeral, these markdow policies are appropriate for those idustries that face customers whose willigess to pay for the product dimiishes over the sellig seaso such as the retailig. O the cotrary, airlie compaies prefer to mark uptheir prices to discrimiate amog travelers ad busiess passegers. I this case, customers willigess to pay icreases over time because the more profitable busiess segmet teds to make last miute travel arragemets. Markdows or markups are rarely advertised, ad customers become aware of these variatios oly through past experieces ad word of mouth. Promotios, o the other had, are discouts that compaies offer at specific momets i time (such as Mother s Day). These discouts are advertised ad reversible. Joit Price Costraits. I some situatios, differet products caot be priced idepedetly. This happes aturally with budles because the price of the budle should deped o the price of the differet compoets. For istace, there are practical issues, such as marketig cosideratios or competitors strategies, that ca force the price of the budle to be at least x% (say 10%) cheaper tha the sum of the price of the compoets. I this case, if product i M is a budle resultig from packig together all the products j B i M, the the budlig costrait o the price is as follows: p t i x p t j for all t 0 T (6) j B i Aother case where joit price costraits arise aturally is whe the same product is offered at differet locatios that have idepedet demad. I this case, it ca be argued that the product i locatio k is differet tha the product i locatio l because they face differet demad processes, ad therefore a differet price ca be set at each locatio. I practice, however, compaies try to avoid this type of geographical discrimiatio because of image ad reputatio issues (Bitra et al. 1998). I this case, the fuctioal costrait that is added to the model is p t k = p t l for all t 0 T (7) Joit price costraits ca also occur over time. For example, i some idustries price is forced to follow a mootoic path. The path might be decreasig, such as permaet markdows i the retail sector (Matrala ad Rao 001), or icreasig as it happes i the airlie idustry. I geeral, most compaies try to avoid pricig policies that may be viewed as ufair by the ed customers. Situatios where two first-class passegers who are seated together after havig paid sigificatly differet prices for their seats ca have a egative impact o customers 10 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003

9 perceptio (especially for the passeger havig the expesive ticket). To miimize this problem, the airlie compay could cosider pricig policies that satisfy the followig coditio: p t p s for all t s 0 T (8) where is a upper boud o the variability of the pricig policy. Cost-Based Pricig. Although capacity is a suk cost i our settig ad i most of the applicatios that we cosider, there is still some tedecy i practice to set prices based o costs. The reaso is probably a mixture of maagers icetives based o margis ad the classical advice from ecoomic theory where margial cost plays a cetral role i pricig decisios. It should be ituitively obvious, however, that oce capacity is determied pricig policies should oly try to maximize the reveue without ay cost cosideratio (except for productio costs that we have assumed are egligible). I some cases, however, there are legal restrictios to price below cost, ad so cost-based pricig is exogeously imposed. A simple, cost-based pricig costrait (that we have come across workig i the apparel idustry) is give by p t 1 + x r, where r is the uit cost of the product ad x is a miimum margi cotributio imposed o the product. Note that if x ad r are fixed, the we ca redefie the price as p t p t 1 + x r. Uder this et margi price formulatio, the cost-based costrait reduces to the oegativity of the pricig policy that is always satisfied i our reveue maximizatio cotext. I geeral, we will deote by the set of all admissible pricig policies, those that satisfy all the relevat costraits..6. Reveue Maagemet Formulatio Give the available capacity C 0, the cumulative demad process N t, the meu of available products M, ad a set of admissible policies, the seller s objective is to fid a pricig strategy P t that maximizes the total reveue collected from sellig the products to the customers. I additio, the seller has the ability to partially serve demad, ad so the sellig process S t is also part of the decisio variables. The problem faced by the seller is to fid the solutio to the followig optimal cotrol problem [ T ] E N p t ds t (9) sup P S 0 subject to: C t = C 0 AS t 0 for all t 0 T (10) 0 S t D t P H t for all t 0 T (11) P ad S t H t (1) We first ote that the model correspods to a reveue maximizatio problem. The objective (9) is simply the expected reveue collected from sellig the products over the available sellig period 0 T.Aswe metioed i.1 all cosideratios associated with acquirig the iitial level of capacity C 0 have bee excluded. Aother key elemet of this formulatio is the implicit risk-eutral behavior of the seller. The seller s objective i (9) is to maximize expected reveue without ay cosideratio o the variability of the resultig output. We made this assumptio to stay i lie with the literature where risk eutrality is by far the most commoly used formulatio. I those situatios, where the seller is costatly solvig this reveue maagemet problem (e.g., airlies maagig thousads of flights a year or retailers sellig thousads of SKUs every seaso) the risk-eutral formulatio is certaily appropriate. Mathematical tractability is aother reaso for this simple modelig of the seller s prefereces. We will retur to this assumptio later i 4, where we discuss the extesio of this stadard reveue maagemet formulatio to the more geeral case of utility maagemet. Fially, we poit out that the sigle source of ucertaity i this formulatio is o the demad side. We coclude this sectio, with a pictorial represetatio (Figure 1) of the geeral reveue maagemet etwork that we cosider. 3. Mai Results ad Related Literature We ow proceed to a systematic review of the research, publicatios, ad mai results o the pricig problem i (9) (1). The goal of this sectio is to Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer

10 Figure 1 The Reveue Maagemet Network uderstad the structure ad properties of a optimal solutio to this geeric problem by examiig the differet models that have bee studied. Certaily, there is o sigle way to approach the task of reviewig the literature ad mai results. From a expositio perspective, we fid it coveiet to start with a basic partitio betwee determiistic ad stochastic models Determiistic Models The determiistic models that we cosider i this sectio assume that the seller has perfect iformatio about the demad process. This is, of course, a major simplificatio especially for those applicatios where demad is hardly predictable at the begiig of the seaso, e.g., ew products or fashio goods. Furthermore, we have argued i the Itroductio that reveue maagemet techiques are particularly useful for idustries facig stochastic demad. There are two importat reasos that explai why we have decided to review determiistic models. First of all, determiistic models are easy to aalyze ad they provide a good approximatio for the more realistic yet complicated stochastic models. Moreover, as we will show shortly, determiistic solutios are i some cases asymptotically optimal for the stochastic demad problem (e.g., Gallego ad va Ryzi 1994, 1997; Cooper 00). The secod reaso is that determiistic models are commoly used i practice. I terms of the literature, determiistic models form the basis of the classic ecoomic model o moopolistic pricig, which is essetially the departig poit of the research that is curretly doe i marketig ad operatios. It is ot i our iterest, however, to review the vast ecoomic literature o pricig that maily focuses o static equilibrium (or steady-state) pricig, where margial cost equals margial reveues. The reader is referred to Nagle (1984) for a comprehesive discussio of the ecoomic literature o pricig theory. As we argued above, determiistic models are good first-order approximatios (asymptotically optimal i some cases) for more sophisticated stochastic models. I particular, they provide valuable isight o how optimal pricig policies deped o the differet parameters of the model Sigle-Product Case. The simplest model i this determiistic settig cosiders the case of a moopolist sellig a sigle product to a pricesesitive demad durig a fixed period 0 T (i.e., M =1). The iitial ivetory is C, demad 1 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003

11 is determiistic with time-depedet ad pricesesitive itesity p t. I additio, the istataeous reveue fuctio r p t = p p t is assumed to be cocave as i most real situatios. The reveue maagemet problem (9) (1) ca be writte i this case as follows: T max p t p t t dt (13) P 0 subject to T 0 p t t dt C (14) This is a stadard problem i calculus of variatios. Let H p t t = p t p t t be the correspodig Hamiltoia fuctio where 0 is the Lagragia multiplier for (14). The optimality coditio (e.g., Gelfad ad Fomi 1963) is give by p t = p t t (15) p pt t where p is the partial derivative of with respect to the price. Let p t = p p p t / p t be the elasticity of demad with respect to price at time t. The, Coditio (15) (together with the fact that 0) asserts that at optimality pt t 1. That is, demad is elastic 3 at the moopolist s optimal price. We ote that the myopic solutio pt m to (13) (14) that maximizes the istataeous rate of retur solves p m t = pm t t (16) p pt m t Therefore, if = 0, i.e., the capacity costrait (14) is ot active, the the optimal strategy p is equal to the myopic strategy p m. O the other had, if (14) is active, the 0 ad the myopic solutio is a lower boud o the optimal strategy. From stadard duality theory, is the shadow price associated with a uit of capacity. Thus, we ca thik of as the opportuity cost of sellig a uit of product ad so ecessarily the optimal strategy must satisfy pt. For the case of a time-homogeeous demad itesity ( p t = p ), a fixed price solutio ca be show to be optimal over the etire sellig period 0 T. To characterize this solutio, let 3 We say that p is elastic at price p if p t = p p p t / p t 1. p m = arg max p p p 0 be the myopic price policy that maximizes the reveue rate ad m = p m be the correspodig demad itesity. Similarly, let p be the solutio to p T = C ad = p be the correspodig demad itesity. 4 The, the followig is a straightforward applicatio of the Karush-Kuh- Tucker (KKT) optimality coditios (e.g., Bazaraa et al. 1993). Propositio 1. Cosider the sigle-product reveue maagemet problem (13) (14) with homogeous demad itesity p ad cocave reveue rate r p = p p. Case 1. Abudat Capacity. If m T C, the the optimal price is p m ad the optimal reveue is equal to p m m T. Case. Scarce Capacity. If m T>C, the the optimal price is p ad the optimal reveue is equal to pc. This result is also show i Gallego ad va Ryzi (1994, Propositio ) ad it is used as a buildig block for costructig heuristics ad bouds for the stochastic couterpart. As a direct corollary of Propositio 1, we have two importat properties of the optimal price strategy: Namely, the optimal price is (i) oicreasig i the iitial capacity C ad (ii) odecreasig i the sellig period T (see Figure ). From Figure, we ca see that the optimal reveue is evidetly a odecreasig fuctio of both the iitial ivetory (C) ad the legth of the sellig seaso (T ). I terms of the iitial ivetory, there is a optimal level C m = m T that maximizes the reveue. Above this threshold, additioal uits of capacity will ot icrease reveue. Maagers should the try to target this optimal value C m whe determiig the iitial level of capacity. O the other had, the optimal reveue is mootoically icreasig i T reflectig the fact that as the sellig horizo icreases the seller faces a larger populatio of potetial buyers ad therefore he ca target the available capacity to those customers havig higher valuatio for the product. I the limit, if p = sup p p > 0, the the seller ca obtai a maximum reveue of pc as T goes to ifiity. Most extesios of this sigle-product determiistic demad problem geeralize some aspect of the 4 Notice that the existece of p is ot guarateed for large C. Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer

12 Figure Optimal Price Strategy for the Sigle-Product Determiistic Case Optimal Price Strategy Optimal Reveue Optimal Reveue Optimal Price Strategy λ m Τ λ m / C Iitial Capacity ( C ) Sellig Horizo ( T ) Note. Both parts of the figure are draw usig a multiomial demad rate p = exp p fuctioal form of the demad process. For example, Smith ad Achabal (1998) studied the case where demad itesity depeds o price as well as o the level of ivetory, i.e., p t C t t. The idea (which aturally arises i the retail sector, for istace) is that demad decreases as the ivetory is depleted. Customers are less likely to fid the product they wat (e.g., i terms of size, color, quality, etc.) whe available ivetory is low. I this settig, the authors derive optimality coditios for the price similar to (15), ad closed-form solutios are reported for the special case of a multiplicative separable demad rate with expoetial price sesitivity, (i.e., p C t = k t y C exp p ). Aother extesive stream of research comig especially from marketig (e.g., Dola ad Jeulad 1981, Jeulad ad Dola 198, Kalish 1983, Mesak ad Berg 1995, Mesak ad Clark 1998, Parker 199, amog may others) cosiders the case of a price-sesitive diffusio model (cf. Bass 1969) to describe the dyamics of the demad. I the reveue maagemet cotext, Feg ad Gallego (000) use a diffusio model to characterize the itesity of the demad process. The Bass (1969) diffusio model is geerally used for durable goods, for which demad at time t depeds o the umber of uits sold prior to t ad the size of the populatio of potetial customers. More specifically, the demad rate t at time t is a fuctio of the curret price p t, the amout sold by that time D t, ad the populatio size N, that is, t = D t = p t D t N (17) t I geeral the diffusio effect, i.e., the depedece of the demad rate o the cumulative sale D t, is ot uiform over time. Upo itroductio, we expect a positive effect (meaig / D 0) because of factors such as word of mouth, improved reputatio, or exclusivity. O the other had, as time passes ad the umber of sold uits icreases, we expect market saturatio ad obsolescece effects to geerate a egative impact o demad (i.e., / D 0). Accordig to Kalish s (1983) results, the evolutio of price over time ca follow three geeric paths: (i) mootoically icreasig if word-of-mouth effects have a positive impact o demad, (ii) uimodal: icreasig at the begiig, reachig a maximum at some itermediate time, ad the decreasig for the rest of the sellig period. This situatio occurs whe there is a positive effect of word of mouth at the begiig followed by demad saturatio. Fially, (iii) the price is mootoically decreasig over the etire horizo if there is a egative effect of peetratio o demad. For a complete review of these sigle-product Bass (1969) diffusio models, we refer the reader to Elmaghraby ad Keskiocak (00, ). I a differet cotext, Raja et al. (199) ad Abad (1996) derive optimal pricig policies for the case where ivetory deteriorates cotiuously ad determiistically over time at a rate proportioal to the ivetory positio. The special cases of liear demad ad expoetial decay are studied i more detail Multiple-Product Case. The case of multiple products ( M ) has received cosiderably less attetio. The reaso is probably because of 14 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003

13 the higher degree of complexity attached to these multiproduct formulatios especially to characterize demad correlatio ad product substitutio effects. I the ecoomics literature, Wilso (1993) studies determiistic, multiproduct models i which the seller objective it to desig a optimal meu of prices ad products. The selectio of a appropriate cosumers choice model such as the multiomial logit or multiomial probit (e.g., Be-Akiva ad Lerma 1985) to characterize customers prefereces becomes a critical compoet of the problem s formulatio (e.g., Talluri ad va Ryzi 001). We otice that i the case whe capacity is dedicated ad the price of product i does ot affect the demad for product j i (idepedet demads), the multiproduct case reduces trivially to a set of discoected sigle-product problems. The iterestig cases arise whe capacity is flexible ad/or demad process depeds o the whole vector of prices (substitute or complemetary products). I geeral, a similar result to Propositio (1) ca be derived i this multidimesioal case. For expositio purposes, we cosider here the simple case of time-homogeous demad processes. I this settig, it is ot hard to show that a fixed-price solutio ca be used without ay sacrifice o performace. Let D i P = i P T be the cumulative demad for product i M give a vector of prices P = p 1 p ( i is the time-homogeous demad rate). Let P = 1 P P be the vector of demad itesities ad T P P be the reveue fuctio (primes ( ) deote vector traspose). I this case, it is coveiet to itroduce for each product i M the iverse demad fuctio P i that represets the price of product i M give a vector of cumulative itesities. We assume the that P is a real-valued fuctio that is cotiuous ad differetiable, such that the reveue fuctio P is strictly cocave. The reveue maagemet problem (9) (1) ca be writte i this case as: max 0 TP (18) subject to TA C (19) This is a multidimesioal, oliear programmig problem that has a uique solutio give the cocavity assumptio o the reveue fuctio. Similar to Propositio (1), two cases characterize the optimal solutio. Let m be the vector of cumulative demads that maximize P. The m is optimal if ad oly if TA m C. If this coditio is ot satisfied, the the optimal solutio is a boudary poit that satisfies the correspodig KKT optimality coditios. The followig propositio characterizes the multiproduct case. Propositio. Cosider the multiproduct reveue maagemet problem (18) (19) with homogeous iverse demad fuctio P, ad cocave reveue fuctio TP. Case 1. Abudat Capacity. If TA m C, the the optimal price is P m = P m. Case. Scarce Capacity. If TA m C, the let be the uique solutio to the followig KKT optimality coditios: P A = 0 T A C = 0 (0) 0 0 where is the gradiet operator with respect to ad is a m-dimesioal vector of Lagragia multipliers. The optimal price i this case is P = P ). Let P be the Jacobia matrix associated to the price vector P. That is, the ij elemet of this matrix is give by P ij = P j / i. Thus, the first KKT coditio above implies that the optimal price vector satisfies: P = A P (1) Similar to the sigle-product case, is the vector of shadow prices associated with the available capacity C, ad A represets the vector of opportuity cost. Therefore, additioal capacity is valuable oly if it is scarce, i.e., 0. It is also importat to otice that for the multiproduct case, it is possible that the optimal price icreases with the level of capacity. For istace, cosider a simple example with two products where (19) is give by two costraits: 1 + C 1 ad 1 C. Suppose that the curret level of capacity is (C 1 = 1 C = 0) ad that the optimal solutio is 1 = 0, = 1. If we icrease C to a ew value C = 1, Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer

14 the uder regular coditios o the reveue fuctio, the ew solutio will satisfy 1 > 0 ad < 1. That is, a icrease i C might iduce a decrease i. Thus, the optimal price of Product will icrease after the icrease of C. This example raises the questio of what the coditios are that will esure that the optimal price is i fact oicreasig i capacity. We partially aswer this questio i the followig propositio based o the work by Topkis (1978) ad Milgrom ad Roberts (1990) o supermodularity ad complemetarity. Propositio 3. Suppose that the iverse demad fuctio is mootoe, that is, if 1, the P 1 P. Suppose, moreover, that the objective P is supermodular ad the sets L C = 0 TA C are sublattices. 5 The, the optimal solutio to (18) (19) is odecreasig i C ad the optimal price is oicreasig i C. The proof follows directly from Theorem 5 i Milgrom ad Roberts (1990). The mootoicity of P is the extesio of the classical dowward slopig demad fuctio coditio to this multidimesioal case. We expect the supermodularity assumptio to hold whe there are product substitutio effects (such as for airlie seats or hotel rooms) because i these cases the margial retur o product i should be icreasig o the price of product j. The requiremet of L C beig a sublattice is more restrictive. This result holds trivially whe capacity is dedicated, i.e., A = I the idetity matrix. Similarly, i the time-homogeous case, a icrease i the sellig period (i.e., T ) ca be iterpreted as a decrease i the level of capacity. 6 Thus, uder the same set of assumptios of Propositio 3, we expect the optimal price to be a icreasig fuctio of T. 3.. Stochastic Models Pricig policies with stochastic demad are more complex ad harder to compute tha their 5 A fuctio f is supermodular if for all x x, f x + f x f mi x x + f max x x. A set L is a sublattice if for all x x L, mi x x L ad max x x L. 6 Notice that i the time-homogeous case the feasible regio TA C is equivalet to A C/T. determiistic couterparts. For istace, i this settig a sigle-price solutio is rarely optimal uless we restrict ourselves to this type of static policy. O the other had, stochastic models are clearly used more appropriately to describe real-life situatios where the paths of demad ad ivetory are upredictable over time ad maagers are forced to react dyamically by adjustig prices as ucertaity reveals itself. The atural way to tackle a problem of this type is by usig stochastic dyamic programmig (SDP) techiques. At every decisio poit durig the sellig seaso, the maager collects all relevat iformatio about the curret ivetory positios ad sales ad establishes the prices at which the products should be sold. With a few exceptios, most of the research has bee doe for the sigle-product case uder Markovia assumptios o the demad process. I this settig, the ivetory levels are the oly relevat iformatio that maagers eed to make pricig decisios Sigle-Product Case. I the sigle-product case ( M =1), we ca assume without ay loss of geerality that the iitial capacity C = C 0 is a scalar represetig the umber of uits of the product that are available at time t = 0. Usig a SDP formulatio, we defie V t C t to be the value fuctio at time t if the ivetory is C t, that is, V t C t is the optimal expected reveue from time t to the ed of the seaso give that the curret ivetory positio at time t is C t. Time t has bee modeled i the literature as either a cotiuous or discrete variable. From a practical perspective, we expect that maagers will revise their price decisios oly at discrete poits i times. However, the explosive growth of the Iteret ad e-commerce make the cotiuous time model much more suitable for practical uses. Sigle-Price Models. The simplest approach to the problem is the sigle-price solutio. I this case, we restrict the pricig policy to be a fixed price durig the etire seaso, i.e., p t = p for all t 0 T. This type of static policy is appropriate for products havig oe or more of the followig characteristics: (i) short sellig period, (ii) high costs of chagig prices, ad/or (iii) legal regulatios that force the price to be fixed. The fixed-price model is simple ad 16 Maufacturig & Service Operatios Maagemet/Vol. 5, No. 3, Summer 003