Information Acquisition for Capacity Planning via Pricing and Advance Selling: When to Stop and Act?

Transcription

1 OPERATIONS RESEARCH Vol. 58, No. 5, Sepember Ocober 2010, pp issn X eissn informs doi /opre INFORMS Informaion Acquisiion for Capaciy Planning via Pricing and Advance Selling: When o Sop and Ac? Tamer Boyacı Desauels Faculy of Managemen, McGill Universiy, Monreal, Quebec H3A 1G5, Canada, amer.boyaci@mcgill.ca Özalp Özer School of Managemen, The Universiy of Texas a Dallas, Richardson, Texas 75080, oozer@udallas.edu This paper invesigaes a capaciy planning sraegy ha collecs commimens o purchase before he capaciy decision and uses he acquired advance sales informaion o decide on he capaciy. In paricular, we sudy a profi-maximizaion model in which a manufacurer collecs advance sales informaion periodically prior o he regular sales season for a capaciy decision. Cusomer demand is sochasic and price sensiive. Once he capaciy is se, he manufacurer produces and saisfies cusomer demand (o he exen possible) from he insalled capaciy during he regular sales period. We sudy scenarios in which he advance sales and regular sales season prices are se exogenously and opimally. For boh scenarios, we esablish he opimaliy of a conrol band or a hreshold policy ha deermines when o sop acquiring advance sales informaion and how much capaciy o build. We show ha advance selling can improve he manufacurer s profi significanly. We generae insighs ino how operaing condiions (such as he capaciy building cos) and marke characerisics (such as demand variabiliy) affec he value of informaion acquired hrough advance selling. From his analysis, we idenify he condiions under which advance selling for capaciy planning is mos valuable. Finally, we sudy he join benefis of acquiring informaion for capaciy planning hrough advance selling and revenue managemen of insalled capaciy hrough dynamic pricing. Subjec classificaions: advance selling; capaciy; sochasic demand; dynamic pricing; opimal sopping; demand learning. Area of review: Revenue Managemen. Hisory: Received February 2008; revisions received January 2009, July 2009; acceped Sepember Published online in Aricles in Advance June 8, Inroducion Capaciy invesmens, such as consrucion of semiconducor fabricaion or power plans, share four imporan characerisics. Firs, capaciy invesmens are ofen very expensive and irreversible. The cos of unused capaciy can be only parially recovered (if a all) by salvaging a a minimum value. Second, demand for he capaciy is uncerain a he ime of he capaciy decision. Demand uncerainy is ofen quie significan because he capaciy decision is aken well in advance of he sales season. Third, adjusing capaciy during sales and producion is ofen difficul or impossible. Hence, he amoun of capaciy buil defines how much can be produced and sold during he sales season. Fourh, managemen ofen has some leeway abou he iming of when o build capaciy. The laes ime o insall capaciy is he beginning of he sales period minus he consrucion leadime necessary o build he capaciy. Many sraegic invesmens share hese common characerisics (see, for example, Dixi and Pindyck 1994). In an environmen driven by demand uncerainy, a build i and hey will come sraegy requires a capaciy provider o bear considerable risk in making he expensive capaciy invesmen. In his paper, we explore a differen sraegy. We allow some cusomers o commi o buying prior o he capaciy decision and he provider o build he capaciy laer; i.e., le hem come and build i laer. In his paper, we refer o he capaciy provider as he manufacurer because she also produces and delivers he final produc. Advance selling is a sraegy ha can help he manufacurer enhance her undersanding of he marke poenial for her produc and reduce demand uncerainy. By offering he produc a a ime preceding he regular sales period, he manufacurer can capure some of he marke demand in advance and hereby moderaes overall demand uncerainy. In addiion, he amoun of advance purchase commimens provides he manufacurer wih informaion on he marke demand poenial of he produc and enables her o plan capaciy according o more accurae demand informaion. She also sars collecing revenue earlier. Advance selling sraegy may be aracive o some cusomers as well. By commiing earlier o purchase, cusomers reserve heir 1328

2 Operaions Research 58(5), pp , 2010 INFORMS 1329 producs and herefore are no vulnerable o possible capaciy shorages. Furhermore, cusomers may garner discouns for commiing early (Neslin e al. 1995). When he manufacurer pospones he capaciy invesmen o acquire advance sales informaion, she may face an addiional rade-off due o capaciy building coss. There are always leadimes associaed wih obaining sies, equipmen and oher resources when he manufacurer builds new capaciy. When hese invesmens are delayed, he manufacurer may face a igher deadline for building capaciy, which would resul in a nondecreasing capaciy cos srucure due o various expediing coss (e.g., second shif premium, premium ransporaion). Alernaively, if expediing is no possible (i.e., he leadime is consan), hen he manufacurer runs he risk of compleing he capaciy insallmen afer he sar of he sales season, in which case she may lose some poenial revenue. This can also be cas as a nondecreasing capaciy cos srucure. Advance selling sraegy is commonly used in he service secor. The prime example is he airline indusry. However, as Xie and Shugan (2001) poin ou, advance selling does no require indusry-specific characerisics; i can be used o enhance profis provided ha cusomers can purchase he service a a ime preceding consumpion, which is possible for mos services. New echnology such as elecronic ickes, on-line prepaymens and smar cards enable more service providers o experimen wih advance selling. Cusomers can buy advance ickes o concers, spors evens, or fesivals. They can book hoel rooms, buy railroad ickes, and acquire some oher services in advance. The advance sales informaion is ofen used o plan and manage capaciy. For example, conference organizers offer early regisraion a discouned prices and use his informaion in planning he hoel rooms o reserve, meeing rooms o book, as well as caering. Advance selling is also used in he manufacuring secor, alhough i is no as prevalen as in he service secor. This may be due o he presence of organizaional silos which ofen decouples demand managemen and capaciy planning. Recen advances in informaion echnology and managemen pracices, however, are enabling firms o coordinae acions across funcional areas, such as markeing and operaions. Some manufacurers in high echnology and apparel indusries sared o use advance selling sraegy o beer plan for capaciy and producion. For example, Ericsson, a elecommunicaions equipmen manufacurer, recenly explored his sraegy o improve is long-range forecasing for planning he capaciy of a new facory for is hird-generaion (3G) wireless nework equipmen. Accordingly, he company announced he dae for he launch of is 3G saions. Before securing he capaciy, Ericsson presold 3G wireless base saions o some of is cusomers such as NTTdocomo, he regional cellular phone operaor in Japan. 1 Apparel manufacurers have also been using early sales informaion o decide on producion. To do so, his indusry developed several iniiaives o reduce he cos of excess invenory and shorage. Fisher and Raman (1996) discuss how apparel manufacurers, such as Spor Obermeyer, commi par of producion capaciy o cerain SKUs afer observing some iniial demand. Zara uses early marke sales informaion o preposiion and decide how much sewing capaciy o reserve (Fraiman and Singh 2002). Advance selling is also used in consrucion projecs. Commercial developers sell some unis a an advance sales price before consrucion begins. Revenue from advance sales is used in par o finance he consrucion. This informaion can also be used o decide wheher o purchase addiional land and build more unis. Anoher example is from e-ailers (such as Amazon.com) ha collec pre-orders for cerain iems before heir marke inroducion. As discussed a he ouse, he rade-off beween delaying a decision and proacively acquiring informaion (such as demand) versus deciding early (such as building capaciy up fron) is inheren o many sraegic invesmen decisions. The presen paper akes a sep in he direcion of providing an undersanding of his rade-off in capaciy planning. In summary, our primary objecive is o deermine he effeciveness of advance selling in conjuncion wih a le hem come and build i laer capaciy sraegy. In paricular, we sudy a manufacurer who decides on he level of capaciy o build for a produc ha faces price-sensiive sochasic demand. The manufacurer has one opporuniy o inves in capaciy before he sales season sars. The amoun of capaciy buil defines he upper bound on how much he manufacurer can produce and sell during he sales season. By delaying he capaciy decision and offering advance sales, he manufacurer can miigae he demand uncerainy and obain addiional informaion abou he marke poenial. We also consider he manufacurer s pricing problem. We sudy he case in which he manufacurer deermines advance sales and sales season prices opimally, as well as he case in which hese prices are exogenously specified. For each scenario, we esablish he opimaliy of conrol-band policies ha prescribe he opimal ime o sop collecing advance sales informaion. Under his policy, he manufacurer moniors he prevailing advance sales informaion including he oal number of commimens o dae, and if his quaniy falls wihin he conrol band, i is opimal o sop advance selling and o decide on he capaciy. Oherwise, he manufacurer coninues advance selling. Through an exensive numerical sudy, we compare he opimal expeced profis wih and wihou advance selling, and we show ha an advance selling sraegy can increase expeced profi significanly. We also quanify he value of knowing exacly when o sop advance selling. Our sudy generaes managerial insighs on how he value of informaion acquired hrough advance selling is influenced by operaing and marke characerisics, by quanifying he profi-impac of such characerisics. Consequenly, we idenify he condiions under which advance selling offers he mos value (e.g., when demand uncerainy or

3 1330 Operaions Research 58(5), pp , 2010 INFORMS cos of building capaciy is high). Finally, we sudy he join benefis of acquiring informaion for capaciy planning hrough advance selling and revenue managemen of insalled capaciy hrough dynamic pricing. Modeling and sudying his scenario bridges he revenue and capaciy managemen lieraures. 2. Lieraure Review There is an exensive body of research dealing wih capaciy managemen in differen environmens. When produc lifecycles and sales seasons are long and producion leadimes are shor, adjusmen of he capaciy level over ime could be possible. Examples for such producs are cemen and seel. For such producs, Manne (1967) esablishes opimal expansion policies for a marke wih sochasic growh paerns. Luss (1982) provides a comprehensive review of join capaciy expansion and producion managemen problems. This lieraure assumes seady growh in demand. More recenly, Angelus and Poreus (2002) characerize an opimal policy for simulaneous managemen of capaciy and producion, whereby sochasic demand can reduce over ime. Bradley and Glynn (2002) sudy a coninuous version of a simulaneous capaciy and invenory decision problem. Lovejoy and Li (2002) sudy capaciy decisions for hospial operaing rooms ha consider he objecives of paiens, surgical saff and hospial adminisraion. Van Mieghem (2003) provides an exensive review of he recen capaciy lieraure, in which demand is always modeled as an exogenous process. As poined ou by Van Mieghem (2003), he capaciy lieraure has no paid much aenion o he more realisic demand models which are parially exogenous and parially endogenous. The presen paper akes a sep in his direcion. When he capaciy canno be adjused during he sales horizon and when he capaciy is perishable afer he sales horizon, he firm can increase expeced profi hrough dynamic pricing and revenue managemen. Biran and Caldeney (2003) and Elmaghraby and Keskinocak (2003) provide comprehensive reviews of his opic. The exbook reamen of his lieraure can be found in Talluri and van Ryzin (2004) and Phillips (2005). This lieraure akes capaciy as given and maximizes revenue by adjusing prices over ime based on he level of lef-over capaciy and price-sensiive cusomer arrival raes (e.g., Gallego and van Ryzin 1994, Feng and Gallego 1995, Biran and Mondschein 1997). Carr and Lovejoy (2000) provide an alernaive mehod in which he capaciaed firm selecs a porfolio of demand disribuions from a se of poenial cusomer segmens. All hese auhors also poin ou ha frequen price adjusmens could be cosly and hence should be exercised wih care. In he manufacuring secor, where cusomer relaionships are imporan, selling capaciy and he produc a differen prices is generally no considered o be a relaionship-preserving sraegy. In our basic model, a finie number of price adjusmens are made prior o he capaciy decision, bu once advance selling sops and he capaciy is se, he sales price remains uniform for all remaining cusomers. In his way, we reain focus on he capaciy planning problem, which consiues he core of our sudy. Neverheless, we also consider an exension ha allows insalled capaciy o be sold via dynamic pricing. This exension bridges capaciy planning wih revenue managemen hrough an advance selling sraegy. A recen line of research in dynamic pricing invesigaes he effec of demand parameer learning (Braden and Oren 1994; Aviv and Pazgal 2005a, b; Araman and Caldeney 2009 and he references herein). This line of research focuses on he pricing problem for which he perishable capaciy a he beginning of a regular sales season is given. These auhors exend he pricing model of Gallego and van Ryzin (1994) in various ineresing direcions, in paricular, o accoun for learning demand parameers. They provide mehods o compue opimal (and close-o-opimal) prices for lef-over invenory unil all remaining capaciy is sold. Araman and Caldeney (2009) inroduce he opion of sopping sales before consuming all capaciy if he value funcion is lower han an exogenously specified profi level (which can be inerpreed as he reservaion profi obained from selling an alernaive produc). Unlike hese papers, we do no assume ha capaciy level a he beginning of a regular period is given. Insead, we focus on pricing schedules ha collec commimens o purchase earlier, i.e., during advance sales periods, and we use his informaion o decide on he capaciy a he beginning of he regular sales season. These sudies explicily examine he role of demand parameer learning in a Bayesian conex. In our concluding remarks, we discuss how our proposed model can also accoun for such a learning process. Addiionally, here is a large body of research ha sudies how capaciy can be used effecively hrough managing producion and invenory. In his lieraure, capaciy is an upper bound on producion quaniy and i is exogenously specified (see Aviv and Federgruen 1997, Özer and Wei 2004 and references herein). Wihin his line of research, here is a growing lieraure ha sudies advance demand informaion and is use in capaciy consrained producion and invenory sysems (see Gallego and Özer 2001, Özer and Wei 2004, Hu e al. 2004, Wang and Tokay 2008, Gayon e al. 2009). These papers esablish opimal producion policies for various environmens and show, for example, ha advance demand informaion is a subsiue for capaciy. This lieraure akes advance demand informaion as exogenous o he sysem. For an invenory sysem wih ample capaciy, Weng and Parlar (1999) and Chen (2001) explore he cos and benefi of price incenives o induce ime-and-price-sensiive cusomers o place advance orders. Prasad e al. (2010) consider a newsvendor reailer serving heerogenous cusomers wih uncerain fuure valuaions of a produc, and hey explore he benefis of advance selling a discouned prices.

4 Operaions Research 58(5), pp , 2010 INFORMS 1331 Anoher group of researchers sudies how advance purchase or selling affecs he allocaion of invenory risk wihin a supply chain (Dana 1998, Cachon 2004, Neessine and Rudi 2006, Taylor 2006, Özer e al. 2007, Dong and Zhu 2007 and references herein). Alhough advance selling sraegy forms he common ground, he primary focus of his sream is differen from ours. Modeling invenory decisions wihin a supply chain under exogenous sochasic demand, hese papers focus on allocaion of invenory risk, division of profis, and incenives and coordinaion in a supply chain. In conras, we invesigae how pricing and advance selling can be used o manage demand for he purpose of capaciy planning. The res of he paper is organized as follows. In 3, we describe he basic elemens of our model. In 4 and 5, we esablish he opimaliy of conrol-band policies for offering advance sales for exogenous and opimal pricing sraegies, respecively. In 6, we presen a numerical sudy and generae managerial insighs regarding he value of an advance selling sraegy for capaciy planning. In 7, we provide an exension where insalled capaciy can be sold via dynamic pricing. In 8, we conclude and sugges direcions for fuure research. Proofs of all proposiions are deferred o he appendix. Some exensions are also discussed in he elecronic companion, which is available as par of he online version a hp://or.journal.informs.org/. 3. The Model 3.1. Preliminaries We consider a planning horizon wih T ime unis ha consiss of he regular sales season and he prior capaciy planning period. We do no make any assumpions abou he lengh of each period. The regular sales season is indexed as T. Periods 1 T 1 represen possible advance sales periods, during which he firm can collec advance commimens from cusomers before he capaciy decision is made and before he sar of he regular season, hence he erm advance selling. The manufacurer faces random, price-sensiive sochasic demand in each period. The manufacurer has one opporuniy o inves in capaciy before he regular sales season sars. When he manufacurer sops offering advance sales in period, she builds capaciy a uni cos c, based on he acquired advance sales informaion. The remaining demand is served during he regular sales season. The insalled capaciy defines he upper bound on how much he manufacurer can produce and sell during he sales season. If he manufacurer underinvess in capaciy, she loses poenial sales revenue. If she over-invess in capaciy, she incurs a uni cos c u for unused capaciy a he end of he sales season. The uni producion cos is denoed as c p. Le L denoe he nominal leadime for capaciy consrucion. By expediing he building process, he manufacurer can reduce his leadime o L. Consequenly, he laes ime for he manufacurer o decide on he capaciy level o build is he beginning of period T L For exposiional clariy, we assume L = 0 which means ha he manufacurer can pospone he capaciy decision unil he beginning of he sales season T A posiive L can easily be incorporaed ino he model wihou changing he naure of he resuls. As noed earlier, expediing he building of capaciy ypically resuls in addiional coss, implying ha he capaciy coss c should be nondecreasing. Alhough his is plausible, we do no posi such an assumpion as i is no required in our analysis. Noe ha by delaying he capaciy invesmen decision, he manufacurer can acquire demand informaion hrough advance selling and use his informaion for beer capaciy planning. She also collecs revenue earlier. Hence, she may earn ineres on advance sales. However, he manufacurer may incur addiional coss if his delay resuls in higher consrucion coss. The revenue colleced laer is also discouned. Furhermore, depending on he advance sales prices, she also runs he risk of selling capaciy a a lower profi margin. In he presence of such muliple radeoffs, he manufacurer needs o address wo key quesions: (i) how much advance sales informaion is sufficien o decide on he capaciy? and (ii) given his informaion, wha is he opimal capaciy level? To answer hese quesions, we develop a dynamic programming formulaion of he manufacurer s problem and deermine he opimal ime for ending advance selling and building capaciy. We also consider he manufacurer s pricing decision. In paricular, we sudy wo fundamenal pricing sraegies ha differ in he degree of pricing conrol he manufacurer can exer during he advance sales and he regular sales periods. Firs, we sudy he exogenous pricing scenario in which he manufacurer has a predeermined sequence of prices for he advance sales periods as well as he regular selling season. These prices may represen a mark-up or a markdown srucure (see, for example, Feng and Gallego 1995 or Biran and Mondschein 1997). This prevalen case is modeled by aking an arbirary sequence of prices p 1 p T as given. Nex, we sudy he opimal pricing scenario in which he manufacurer deermines he advance and regular sales prices opimally in addiion o he capaciy decision. For each of he pricing sraegies, he manufacurer s decision process is as follows: A he beginning of each period <T, he manufacurer firs observes he prevailing advance sales informaion and he oal revenue from advance sales. Based on his informaion, she decides o eiher (i) sop advance selling and build capaciy, or (ii) delay he capaciy invesmen for one period and coninue advance selling a an opimally se price (resp., or an exogenously se price depending on he pricing policy we address). If she sops, she deermines he opimal level of capaciy based on he advance sales informaion and he remaining uncerain demand in he marke, and she ses he regular sales price (resp., or akes his price as given). A he beginning of regular sales season, i.e., = T,ifhe capaciy invesmen decision has no already been aken,

5 1332 Operaions Research 58(5), pp , 2010 INFORMS he manufacurer builds he capaciy. As before, she deermines he opimal level of capaciy and ses he regular sales price (resp., or akes i as given). Nex, we describe he demand model, he updaing mechanism, and he manufacurer s expeced profi Demand Model We model marke demand using an iso-elasic, pricesensiive aggregae demand model wih a muliplicaive form of uncerainy. Demand in each period depends on he uncerain poenial marke size, he price p charged in ha period, he cumulaive commimens q, and relevan hisorical informaion 2 available a he beginning of period. The pair q defines he advance sales informaion available a he beginning of period. The acual demand in period has he form D p q = f q p b The price elasiciy b is assumed o be he same across periods for ease of exposiion. The poenial marke size in each period is uncerain. They are independen random variables ha have increasing failure raes (IFR) wih suppor on 0. The nonsaionary funcion f q 0 models he informaion ha he commimens q in relaion o he hisory, provide abou fuure demand. I capures he predicive value of commimens. We call his he marke signal impared by he advance sales informaion regarding he demand poenial of he produc. Based on he level of his signal, fuure demand is scaled up or down. For = 1, q and hence f 1 q We assume ha f q is linear increasing in q for any given and for all >1. 3 Based on he informaion available a ime zero, E f q = 1 for any selecion of pas prices p = p 1 p 1 and for all >1. 4 This propery implies ha a ime zero, before he manufacurer sars o gaher advance sales informaion, she expecs demand in any period o be E p b, which is independen of prices charged in periods s<. This propery ensures ha he manufacurer canno use prices o arificially increase he poenial marke size for he produc. When he manufacurer engages in advance selling, however, he acquired informaion can have predicive value regarding curren and fuure demand. Depending on he demand realizaion, he nex period s marke signal f +1 q can be lower or higher han or equal o f q. We do no impose any assumpion on is evoluion. 5 Our framework and srucural resuls are robus o various forms of marke signal funcion and evoluion. This flexibiliy enables us o model a variey of scenarios as discussed laer. In 6 and 8, we provide specific marke signal funcions, including hose arising from Bayesian learning models, and we show ha hey saisfy he above assumpions. Hence, in he res of he paper, we do no resric ourselves o a specific form bu insead sudy he problem given he above general funcional form. The evoluion of demand is as follows. A he beginning of period, if he manufacurer decides o offer advance sales, he uncerainy is realized as, and accordingly he acual demand d = f q p b is observed. The cumulaive commimens and he hisory are updaed as q +1 = q + d and +1 = p d for some funcion 6 If he manufacurer decides o sop offering advance sales, he remaining cusomers are served during he regular sales season. This remaining demand is a funcion of he curren marke signal f q, he remaining poenial marke size T j= j, he price p charged in he selling season, and is given by X p q f q p b Since IFR propery is closed under convoluions (Barlow and Proschan 1975), is also IFR. Noice also ha is sochasically decreasing in. Noe ha he model accouns for a reducion of fuure demand due o a longer advance sales period. (In oher words, advance sales cannibalizes some porion of fuure demand.) 3.3. The Manufacurer s Expeced Profi When he manufacurer coninues advance selling in period she collecs he revenue p d Consider an arbirary period 1 T, and suppose ha a he beginning of period he manufacurer has decided o sop advance selling and inves in capaciy. The manufacurer already has q commied cusomers a some pas prices. A he beginning of period, he oal revenue obained from advance selling, i.e., 1 k=1 p kd k is deerminisic. The manufacurer is required o serve he commied cusomers, so she would se he capaciy level Q above q o also mee he remaining demand X p q she will face during he selling season a price p. For his reason, i is convenien o wrie Q = q + S where S 0 denoes he surplus capaciy. The manufacurer s expeced (undiscouned) profi a he ime when she sops advance selling and invess in capaciy for a given q and is 1 p S q = p k d k + p S q k=1 c p +c q where (1) p S q = p c p E min X p q S c S c u E S X p q + (2) Noe ha x + max 0 x and he expecaion is aken a ime wih respec o he remaining uncerain marke demand X p q. Noe ha maximizing (1) is equivalen o maximizing (2) when he opimizaion is over he surplus capaciy. Nex, we analyze he manufacurer s problem of acquiring informaion hrough advance selling and pricing for capaciy planning. We sar wih he exogenous pricing case followed by he case where prices are deermined opimally.

6 Operaions Research 58(5), pp , 2010 INFORMS Exogenous Prices Consider an arbirary sequence of prices = p 1 p 2 p T where prices p 1 hrough p T 1 are for advance selling periods and p T is he price for he regular sales season. A he beginning of period, he manufacurer observes he advance sales informaion q and decides wheher or no o coninue advance sales. If he manufacurer decides o sop advance sales, she deermines he opimal surplus capaciy S o build. We assume p T c p + c for T. This assumpion ensures ha he manufacurer makes posiive profi from cusomers who buy during he regular sales season. Oherwise, here is no reason o build capaciy more han he oal commimens q. 7 The manufacurer deermines S by maximizing he profi funcion in Equaion (2), which yields { S min S P X p T q >S = c } + c u p T c p + c u The resuling opimal expeced (undiscouned) ne profi from he remaining cusomers is hen q = p T S q By leing s = S /f q we can rewrie q as q = f q, where = p T c p E min p b T s c s c u E s p b T + (3) The manufacurer s opimal capaciy level is Q = q + S and opimal oal expeced profi is q = p T S q p k d k + q c p + c q 1 = k=1 Nex we formulae a dynamic program o deermine he opimal ime for he manufacurer o sop acquiring advance sales informaion. The sae space is given by he cumulaive commimens q and he hisory. We inroduce an auxiliary sae (N ) in he commimen space o indicae ha he capaciy decision has already been aken; i.e., q = N if he capaciy decision has been made, and q N oherwise. Le u q denoe he manufacurer s acion in period : u c coninue advance sales a price p, u q = u s sop advance sales, se price o p (4) T and capaciy level o Q. A he end of period, he cumulaive commimens and he hisory are updaed as q +d if q N and u q =u c q +1 = N if q N and u q =u s or q =N, p d +1 = if q N and u q = u c if q N and u q = u s or q = N. Le us now inroduce 0 1 as he discoun facor. 8 Revenue is realized a he end of he period when cusomers place advance orders and a he end of he regular sales period when remaining cusomers purchase. The coss are incurred in he sales period. All resuls remain valid if revenues from advance orders are colleced a he ime of delivery and/or capaciy coss are incurred when he invesmen decision is aken. The reward funcion for 1 T 1 is given by p d if q N and u q = u c, T g q = q c p + c q if q N and u q = u s, 0 oherwise and for = T,iisgivenby T q T T c p + c T q T if q T N, g T q T T = 0 oherwise The funcion g q records he revenue from wo sources. The firs source is from cusomers who purchase in period when he manufacurer decides o coninue advance selling (i.e., u q = u c ). This revenue source is from advance purchases. The second source is he expeced revenue from saisfying he remaining marke demand (as much as possible) minus he cos of building capaciy and he cos of producion (i.e., u q = u s ). The manufacurer s problem is o maximize he oal expeced profi discouned o he firs period: [ T ] max E 1 g q u 1 u 2 u T =1 where he expecaion is aken a ime zero over D p q for all. The soluion o his problem is obained by he following funcional equaion: T q T T c p + c T q T if q T N hen u T q T T = u s J T q T T = (forced decision), (5) 0 if q T = N, and for = 1 T 1, we solve max T q c p + c q E p D p q J q = + J +1 q if q N (6) 0 if q = N where he expecaion is aken a ime wih respec o D p q. When he maximum in Equaion (6) is

7 1334 Operaions Research 58(5), pp , 2010 INFORMS aained by T q c p + c q, i is opimal o sop advance selling, se he regular sales price o p T, and se he capaciy level o Q ; oherwise, i is opimal o coninue advance selling a price p. For a clearer represenaion of he above opimal sopping problem, we define V q = J q T q c p + c q and subsiue V q + T q c p + c q for J in Equaions (5), (6) and subrac T q c p + c q from boh sides of hese equaions o obain an equivalen formulaion. The resuling dynamic program for = 1 T 1, is given by V q =max 0 H q + E V +1 q (7) where H q E p T c p + c +1 D p q + T +1 q q T c +1 c q (8) and V T q T T = 0 Under his formulaion, if H q + E V +1 q > 0, i is opimal o coninue advance selling. Noe ha he funcion H q can be inerpreed as he myopic expeced profi ha he manufacurer can make by delaying he capaciy decision one more period and collecing advance sales wihou considering he possible benefi of coninuing advance selling beyond period + 1. The funcion EV +1 q is he addiional expeced profi due o he impac of he coninue decision, i.e., advance selling during fuure profis. To characerize an opimal policy, we define H q H q + E V +1 q and idenify if and when H q crosses he zero line. For = 1 q 1 0, and 1 =, and hence he decision is based on V 1 0. For = 2 T 1 we define L = min q q 0 H q 0 (9) and we se L = if min q q 0 H q 0 = 0 and L = if min q q 0 H q 0 = Similarly, we define U = max q q 0 H q 0 (10) and we se U = if max q q 0 H q 0 = Theorem 1. The following saemens hold for all 1 T : 1. The funcion H q is convex in q for any. 2. A sae-dependen conrol-band policy is opimal; he opimal decision is u if L q U, u q = u c if q <L or q >U. 3. The funcion V q is convex in q for any. Theorem 1 shows ha a sae-dependen conrol-band policy is opimal. Under his policy, given he hisory, he manufacurer opimally sops advance selling when he cumulaive commimens fall beween he conrol bands, i.e., q L U. When he cumulaive commimens are lower han L, i is opimal o coninue acquiring informaion abou fuure marke poenial hrough advance sales. In his case, he benefis of delaying a capaciy decision (e.g., acquiring demand informaion, resolving par of marke uncerainy, and collecing revenue) ouweighs he coss (e.g., higher cos of building capaciy, risk of selling a a lower profi margin laer, and earning a discouned revenue). When he cumulaive commimens are higher han U, he commimens signal a srong, significanly more han expeced, expanding fuure marke poenial. Such a srong marke poenial allows he manufacurer o have one more opporuniy o sell a a differen price poin by posponing he capaciy invesmen decision for anoher period when doing so is no oo cosly. We noe ha he proposiion does no rule ou he cases when he upper hreshold is very large or infinie. Inuiively, when commimens have no predicive value or delaying he capaciy decision is oo cosly, one may expec he upper hreshold o be infiniy. The nex heorem formalizes his observaion and idenifies he condiions under which a hreshold policy is indeed opimal. Theorem 2. The following saemens hold for all for 1 T : 1. Suppose ha he advance commimens have no predicive value (i.e., f q = 1 for all q, and ). If c +1 >c, hen a hreshold policy is opimal, i.e., u s u q = u c if q L if q <L If c +1 = c, hen he opimal policy does no depend on he advance sales informaion, and he funcions H equal consans H for each In his case, he opimal sopping ime is he firs 1 T such ha H = Suppose ha c +1 is sufficienly larger han c for all. Then a sae-dependen hreshold policy is opimal. Theorem 2, shows ha even when commimens carry no predicive value; 9 i.e., f = 1 for all, i can be opimal o engage in advance selling. Recall ha he manufacurer collecs revenue earlier by advance selling. Hence, she earns ineres on advance sales. She also reduces demand uncerainy by inducing cusomers o place early orders, reducing her risk of excess capaciy and shorage. Ye, delaying he capaciy decision is cosly when he consrucion cos is increasing, i.e., c +1 >c. The heorem characerizes when i is opimal for he manufacurer o sop advance selling. I also shows ha if i is opimal o sop advance selling when he commimens exceed he hreshold L, hen i is never opimal o coninue advance

8 Operaions Research 58(5), pp , 2010 INFORMS 1335 selling for oal commimens above his hreshold. So, par 1 of Theorem 2 shows ha in he absence of he predicive value of commimens, here is no reason for he manufacurer o reverse he sopping decision. In his case he manufacurer also does no need o ake he hisory ino accoun. She only needs o know he cumulaive commimens because he cos of building capaciy depends on and hence q. Hence, an opimal sopping policy for advance sales is a hreshold policy and is sae independen. In addiion, if he cos of capaciy does no depend on hen he manufacurer does no need o rack cumulaive commimens eiher. Par 2 of Theorem 2 shows ha a saedependen hreshold policy is opimal when he capaciy cos in period + 1 is sufficienly larger han ha of period for all periods. In shor, he opimal policy is provably a hreshold one when commimens have no predicive value or when capaciy cos increases excessively over ime. 5. Opimal Prices for Advance Sales and Regular Season We sudy a manufacurer who ses he advance sales prices and he price for he regular sales season. We firs characerize he manufacurer s opimal pricing and capaciy decisions and he resuling profis when she sops advance selling. We conclude by characerizing he opimal advance sales prices and he opimal sopping policy. Theorem 3. Suppose ha he manufacurer decides o sop advance selling and o build capaciy in period wih an exising level of cumulaive commimens q and hisory, and ha c p c u Le z = S / f q p b. Then, here exiss a unique opimal regular sales price p s given as ( b p s =p z )(c = b 1 p + c z +c ue z ) + z E z (11) + where z is he unique soluion of P >z = c + c u / p z c p + c u. The resuling opimal surplus capaciy is S = f q z ps b and opimal expeced profi from remaining cusomers is q = f q, where = 1 b ps b 1 z E z + (12) The opimal capaciy level is Q = q + S expeced undiscouned profi q is q = p s S q 1 = and opimal p k d k + q c p + c q (13) k=1 Theorem 3 provides closed-form soluions for boh opimal surplus capaciy and sales price during he regular season. Noe ha he manufacurer uses he marke signal, hence he advance sales informaion, o se he opimal capaciy level. Inuiively, he manufacurer is promped o build opimal surplus capaciy o accoun for he marke signal accordingly. As a resul, he manufacurer s expeced profi from he remaining cusomers already akes ino consideraion he marke signal. Hence, he opimal regular sales prices p s do no need o depend on he advance sales informaion. To elaborae more on he srucure of he opimal prices for he regular selling season, we esablish he following basic properies: Theorem 4. Suppose ha he manufacurer has decided o sop advance selling in period. The opimal regular sales season price saisfies p s >c p + c. Furhermore, p s is increasing in c c p, and c u Consequenly, he opimal capaciy level Q is decreasing in c c p, and c u. This resul shows ha when seing he selling season prices, he manufacurer guaranees herself a posiive margin from sales. This margin ensures ha he opimal surplus capaciy S is posiive. Inuiively, when he manufacurer has he abiliy o se he regular sales price, she would se he price such ha building surplus capaciy is profiable. Furhermore, he higher he coss, he higher he prices charged o cusomers. Higher prices reduce demand, and promp he manufacurer o build less capaciy. To derive he manufacurer s opimal sopping policy and he opimal advance sales prices, we modify he dynamic programming in 4. For <T, he funcional equaion is given by V q { =max 0 maxe p T c p +c +1 D p q p R + T +1 q q } T c +1 c q + E V +1 q (14) { } max 0 max H p q + E V +1 q (15) p R { } max 0 maxr p q (16) p R max 0 H q (17) and V T 0. The expecaions are aken a period wih respec o. R is a convex se of possible advance sales prices for each period. Le p c denoe he opimal advance sales price in period To sae he opimal sopping resul, we define if and when H crosses he zero line. As before, hese poins are L and U which are defined as in (9) and (10), respecively. Theorem 5. The following saemens hold for all 1 T : 1. The funcion R p q is convex in q for any p and. 2. The funcion H q is convex in q for any.

9 1336 Operaions Research 58(5), pp , 2010 INFORMS 3. A sae-dependen conrol-band policy is opimal, i.e., u s if L q U, u q = u c if q <L or q >U. 4. The funcion V q is convex in q for any. This resul shows ha he srucure of he opimal sopping policy for acquiring advance sales informaion remains he same when he manufacurer deermines advance and regular sales prices opimally. The acual hreshold values, however, depend on sales prices. Opimal advance sales prices and hresholds can be deermined numerically, for example, by a backward inducion algorihm. 6. Numerical Sudy We conduc a numerical sudy o illusrae he impac of differen operaing and marke/demand facors on he advance selling policy and on he manufacurer s profi using a specific a marke signal funcion. We base his analysis on he scenario in which he manufacurer ses advance and regular sales prices opimally in addiion o he capaciy A Specific Marke Signal Funcion So far, we have characerized he opimal advance selling policy under a generic marke signal funcion. The manufacurer keeps rack of he oal commimens q and he hisory in order o specify and updae he marke signal and demand. Hence, he dimension of he sae space depends on he funcional form of f q. In cerain cases, i may increase as ime progresses, for example, when one needs o keep rack of all pas prices. Neverheless, when is a scalar represening summary saisics of he advance sales informaion, hen he sae space is given by he wo-uple q, achieving sae-space reducion. Consider he following marke signal funcion, which we use in our numerical experimens: ( ) q f q = 1 + = 1 + q for = 2 T (18) where = 1 j=1 E j p b j and 0 1 is a consan. Noe ha given he pas prices p 1 p 1, he manufacurer s esimae of he demand for each period, based on he informaion available a ime zero, is E p b. Hence, before acquiring any advance sales informaion, i.e., a ime zero, he manufacurer expecs o collec unis of commimens by period. Noe ha when he expecaion is aken a ime zero, E f q = 1 for all and any price pah. Depending on he demand realizaions, however, he acual commimen level q can exceed or fall below he expeced amoun and he marke signal flucuaes around 1. If q exceeds, he manufacurer has colleced more advance sales han she iniially expeced, and vice versa. In consequence, nex period s expeced marke signal can be higher or lower han f q The parameer 0 1 is akin o a smoohing consan in forecasing and defines he exen of correlaion beween he marke signal provided by advance sales and fuure demand. Noe ha as he manufacurer coninues advance selling, he cumulaive commimens and he summary saisics for advance sales informaion are updaed as q +1 = q + d = q + f q p b and +1 = + E p b respecively Numerical Sudy Seup Marke Signal and he Predicive Value of Commimens. We use he marke signal funcion f specified in Equaion (18). The exen of correlaion beween advance purchase commimens and fuure demand is measured hrough he smoohing consan. When = 0, demand in each period is independen of prior commimens, and as 1 he signal provided by he commimens is a very srong indicaor of fuure demand. A srong dependence beween pas and fuure demand could be observed, for example, in fashion producs and he apparel indusry. In conras a low level of would likely apply more o maure consumer producs. Cusomer Time Preferences. We model +1 as an independen random variable wih a disribuion idenical o he disribuion of 1+k for >1 and k 1 1. Noe ha k (k> 1) is a measure of cusomers ime preference for purchasing decisions. When k>0 (resp., k<0), more cusomers prefer o purchase laer (resp., earlier), indicaing a higher (resp., lower) anicipaion in he marke for poenial shorages in capaciy. In oher words, when k>0 he disribuion of he fuure period s marke poenial +1 is sochasically larger han he previous period s. When k = 0, here is no clear ime preference. Hence, we refer o k as he lae purchase endency. We also model as normally disribued random variables wih mean 1 + k 1 / T j=1 1 + k j 1 and sandard deviaion T 1 + k 1 / j=1 1 + k 2 j 1 Hence, he oal marke poenial T =1 is normally disribued wih mean and, and i is independen of k Capaciy Cos. The uni cos of capaciy c is of he form c = c 0 + where c 0 is he base cos of capaciy and ( 0) is he measure of how his cos increases as he sales season is approached. By varying c 0 we invesigae he effecs of overall cos of capaciy, whereas differen values indicae he imporance of ime in building capaciy. Price Sensiiviy and Se of Advance Sales Prices R. The price sensiiviy of cusomers in he model is measured by he parameer b. The se R has n>0 finie number of prices ha are uniformly disribued in he region

10 Operaions Research 58(5), pp , 2010 INFORMS a p s 1 + a ps, where a>0 and ps is defined in Equaion (11), i.e., R = 1 a p s ps 1+a ps. We include he opimal price for he regular selling season p s among he possible prices o be offered during he advance selling period. For pracical moivaions of considering discree prices, see Gallego and van Ryzin (1994) Measures of Ineres In addiion o he opimal policy, he manufacurer can follow wo advance selling sraegies. One exreme is o se he regular sales price and capaciy in period 1 wihou advance selling. We refer o his scenario as he no advance selling scenario, for which he corresponding expeced profi G no is given by T 1 1 0, where 1 0 is defined in Equaion (13). The oher exreme is o coninue advance selling unil he las advance sales period, T 1. We refer o his one as he full advance selling scenario. The corresponding expeced profi G f is obained by policy evaluaion in which he decision u q is forced o be u c insead of choosing he maximum in Equaion (17). The expeced opimal profi G is given by V 1 0 in Equaion (14). The difference beween he opimal sraegy and he firs exreme is he expeced value of advance selling or value of informaion acquisiion. To quanify his value, we repor I no = G G no /G no 100%. The profi difference beween he opimal sraegy and he second exreme is he expeced value of knowing when o sop advance selling or he value of opimal advance selling. To quanify his measure, we repor I f = G G f /G f 100%. Figure 1 illusraes he resuling expeced profis under opimal advance selling, no advance selling, and full advance selling. The resuling percenages are I no = 10 93% and I f = 1 02%. For his example, he expeced profi o advance sell and build capaciy laer is 50 85, which is larger han he profi of sopping and building capaciy of in he firs period. The figure also illusraes he opimal lower hreshold L as a funcion of expeced commimens when = 2. For example, if he manufacurer expecs more commimens a he beginning of period wo (such as 2 = 3) and he quaniy commied so far is low (such as q 2 = 2), hen i is opimal o coninue advance selling. Noe also ha he hreshold increases wih 2. Inuiively, he manufacurer is more likely o coninue advance selling and acquiring informaion when he expeced commimens are high. We noe ha he upper hreshold U was large relaive o he L and he mean demand, or i was infinie in our srucured numerical sudy. However, i is possible o consruc cases where boh L and U are finie, and are relaively close o each oher. For example, when T = 5, c p = 3, c u = 2, c 0 = 1 2, = 0 1, b = 2, = 1 000, = 80, = 0 3, = 1, k = 0, for he pricing policy = , he opimal conrol band for he second period is L 2 = and U 2 = resuling in opimal expeced profi of and I no = 4 06% and I f = 4 78% Effec of he Environmen We quanify he effec of various operaing and marke/demand facors on he opimal advance selling sraegy. In paricular, we invesigae how hese facors affec he opimal expeced profi, he value of advance selling, and he value of knowing when o sop. We use he following parameers in he base scenario: T = 5 = = 100, c u = 2 c p = 3, c 0 = 1 2, = 0 3 k= 0 b= 2 = 0 95 = 0 3 a= 0 1, and n = 7. For his base case, he expeced opimal profi is G = 48 02, he value of advance selling is I no = 4 75%, and he value of knowing when o sop is I f = 4 73%. To have a balanced view, we chose he base parameer se such ha hese wo measures of value are equal. We change one parameer a a ime while keeping he ohers consan. Impac of Overall Marke Uncerainy. We es he effec of he coefficien of variaion of he oal marke demand poenial by varying The resuls are illusraed in Figure 2. The expeced profi decreases Figure 1. Expeced profis and hresholds for T = 5 = 1 000, = 80, c u = 2 c p = 3, c 0 = 1 2, = 0 18 k= 0 b = 2 = 0 95, = 0 3, a = 0 1, and n = Value of knowing when o sop Expeced profi Value of advance selling L G* G f G no Expeced commimen level

11 1338 Operaions Research 58(5), pp , 2010 INFORMS Figure 2. The impac of overall marke demand variabiliy ( ). Expeced profi I no & I f (%) I no I f Demand variabiliy ( ) Demand variabiliy ( ) wih higher demand variabiliy. This is consisen wih known resuls for he radiional newsvendor problem. We also observe ha he value of informaion acquisiion (or advance selling) increases wih marke uncerainy while he value of knowing when o sop decreases. Inuiively, when he marke uncerainy is high, he value, of acquiring informaion hrough advance selling is high. Hence, sopping closer o he las period is more likely o be an opimal policy. In mos cases, he wo measures I f and I no would be complemens. While one is increasing, he oher will be decreasing. These observaions sugges ha advance selling miigaes he adverse effec of demand uncerainy. Impac of Capaciy Cos Srucure. Figure 3 illusraes he effecs of he incremenal cos of capaciy as measured by (i.e., how rapidly cos of building capaciy increases as he regular sales season nears). In general, increasing capaciy consrucion coss reduces profi. Noe also ha he value of informaion acquisiion hrough advance selling is also decreasing wih. Essenially, large penalizes lae consrucion. If he lae consrucion is oo expensive, he opimal soluion is o build capaciy sooner raher han laer and no o acquire informaion. From he above resuls we conclude ha he value of informaion acquisiion is greaer when ime is no a major consrain for he manufacurer in building capaciy. Figure 4 illusraes he effecs of base capaciy cos c 0 (i.e., how expensive capaciy cos is in general). Noe ha he manufacurer s profi naurally decreases as capaciy becomes more cosly, bu he reducion in profis is even higher when he manufacurer does no offer advance sales. Consequenly, he benefi of advance selling acually increases as base capaciy cos increases. These resuls show ha he value of informaion acquisiion is higher when capaciy is more expensive relaive o he penaly cos of lae consrucion. Impac of Predicive Value of Commimens. Figure 5 shows he impac of he predicive value of commimens, as measured by he smoohing parameer Observe ha as increases, he value of advance selling firs increases. However, when is very high, he commimens send srong signals of fuure demand, indicaing a poenial for high mean and variance. High uncerainy reduces he expeced profi, he value of informaion acquisiion, as well as knowing when o sop. Consequenly, advance selling is mos beneficial when he predicive value Figure 3. The impac of incremenal cos of capaciy I no I f Expeced profi I no & I f (%) Rae of capaciy cos increase ( ) Rae of capaciy cos increase ( )

12 Operaions Research 58(5), pp , 2010 INFORMS 1339 Figure 4. The impac of base capaciy cos (c 0 ) I no I f Expeced profi Base capaciy cos (c 0 ) I no & I f (%) Base capaciy cos (c 0 ) Figure 5. The impac of predicive value of commimens Expeced profi I no & I f (%) I no Smoohing consan ( ) I f Smoohing consan ( ) Figure 6. The impac of cusomer ime preferences k. Expeced profi I no & I f (%) I no I f Lae purchase endency (k) Lae purchase endency (k) of commimens is moderae. Noe, however, ha he absolue scale of changes is quie small. Impac of Cusomer Time Preferences. The value of informaion acquisiion is also relaed o wheher cusomers anicipae shorages or no. In he even of poenial shorages in supply, cusomers would be more emped o commi o advance purchases and also o commi earlier in ime. Figure 6 demonsraes he influence of cusomers ime preferences hrough he lae purchase endency parameer k. As more cusomers end o commi earlier (smaller k), he profi and he value of informaion acquisiion increases. Figure 6 suppors he claim ha advance selling yields a higher benefi when here is more anicipaion of capaciy shorages in he marke. Impac of Price Sensiiviy. Figure 7 illusraes he effecs of cusomer price sensiiviy. When cusomers are

13 1340 Operaions Research 58(5), pp , 2010 INFORMS Figure 7. The impac of cusomer price sensiiviy b Expeced profi Price sensiiviy (b) I no & I f (%) I no I f Price sensiiviy (b) more price concerned, he manufacurer s profiabiliy is reduced. This resuls in a lower value of informaion acquisiion. Noe also ha when he cusomers are no price sensiive, acquiring informaion unil he las period becomes more likely. Hence he expeced value of knowing when o sop advance selling decreases. These observaions sugges ha low cusomer price sensiiviy is anoher condiion for maximizing he gains from advance selling. These numerical resuls help idenify ha advance selling coupled wih a le-hem-come-and-build-i-laer approach is a profiable sraegy, in paricular, when (i) demand uncerainy is high, (ii) more cusomers anicipae capaciy shorages in he marke, (iii) building capaciy is expensive bu iming is no a major concern, (iv) commimens have moderae predicive value abou marke poenial, and (v) cusomer price sensiiviy is relaively low. We noe ha, wih hese condiions, one can consruc cases where he value of advance selling I no is arbirarily high Opimal Prices To quanify he value of advance selling and he value of knowing when o sop in he previous secion, we compue, for each period, he opimal advance selling price p c and he opimal regular season price p s when he manufacurer decides o sop advance selling. Figure 8 plos he opimal prices as a funcion of commimens q for a given expeced commimen level = 2 2 when = 2. The base daa se is he same as in Figure 1. Noe ha he opimal marke price p2 s does no depend on he commimens (Theorem 3). The opimal advance selling price p2 c, however, is increasing in he number of commimens. This is because having more commimens suggess a sronger marke and allows he manufacurer o charge higher advance selling prices. Figure 8 also depics he evoluion of he prices over ime for differen commimen levels, q = 0 7 q =, q = 1 3 for each corresponding o low, medium, and high levels of commimen. For = T = 5 he manufacurer is forced o build capaciy. As before, for a given p c is nondecreasing in q (high commimens induce high advance sales prices). Furhermore, boh he advance sales prices and he regular season prices are increasing over ime. Finally, noe ha he advance sales prices are always lower han he regular season prices, implying ha he cusomers commiing o buy earlier are garnering discouns. Figure 8. Opimal advance selling and regular season prices p s q = 0.7 q = Prices p c 2 p s 2 Prices q = Cumulaive commimen level Period ()

14 Operaions Research 58(5), pp , 2010 INFORMS 1341 Figure 9. Opimal advance sales prices as changes for q = 0 7 q =, and q = 1 3. p c = 0.06 = 0.1 = 0.14 = Period () p c Period () p c Period () Figure 9 depics he opimal advance sales prices as he incremenal cos of building capaciy changes for differen commimen levels q = 0 7 q =, and q = 1 3. Noe ha for any given where p c is no repored, i is opimal for he manufacurer o sop advance selling in ha period. I is possible o infer from Figure 9 ha for any capaciy cos srucure, higher levels of commimen, induce he manufacurer o sop advance selling earlier. Generally speaking, increasing he incremenal cos of capaciy resuls in higher advance selling prices. The excepions are due o discreizaion of he advance selling price se R which are se based on he opimal regular season price p s When changes, so does he se of advance prices considered during ha period, which can resul in nonmonoonic resuls. Consequenly, he opimal advance sales prices can display nonmonoonic behavior over ime as well (alhough wihin a fairly resriced range). We noe ha he changes in oher cos or marke parameers have raher predicable effecs on opimal prices, and hence are omied for breviy Compuaional Aspecs, Algorihmical Complexiy, and a Heurisic In deermining he opimal advance selling prices, we search n uniformly disribued prices in he range Figure 10. The impac of price se R. Expeced revenue a = 0.05 a = 0.1 a = Number of prices 1 a p s 1+a ps for each period Figure 10 illusraes he opimal expeced profi as a funcion of he number of prices n and he price range a. We highligh wo observaions. Firs, he expeced profi increases wih eiher facor. Essenially, increasing n and a is equivalen o relaxing he consrain se, which in urn increases he opimal expeced profi. Second, noe ha he marginal reurn on profi is decreasing wih hese facors. For example, when a = 0 1 considering seven or more prices does no change he profi significanly. Hence, here is a decreasing reurn o considering larger price ses, which require significanly more compuaional effor. The algorihm o solve he general case is of complexiy O n T, which implies ha finding opimal advance and regular season prices is a compuaionally expensive ask. For his reason we examined heurisic pricing policies. Recall ha p s is he opimal price for he regular sales season when he manufacurer sops collecing commimens a period. Consider a heurisic advance sales pricing policy under which he manufacurer charges he same price p s even when she coninues o collec commimens in period The compuaional effor required o numerically solve his heurisic is he same as ha of he exogenous pricing case (i.e., O T ). We have esed he performance of his heurisic for differen values of, c 0, and and used regression o compare he heurisic profi o he opimal profi (for deails refer o he elecronic companion o his paper). The resuling ha R 2 was close o 1 for all facors, suggesing ha he heurisic can safely be used o invesigae he impac of parameer changes. The average opimaliy gap across all experimens was also very small (0 535%). 7. Dynamic Pricing o Sell Capaciy Our objecive here is o invesigae he radeoff beween wo sraegies ha miigae he adverse effec of demand uncerainy: (1) informaion acquisiion for capaciy planning hrough advance selling, and (2) revenue managemen of insalled capaciy hrough pricing during he regular sales season. To do so, we sudy a manufacurer who, in addiion o employing advance selling and pricing o deermine he capaciy level, ses prices dynamically o sell he insalled capaciy during he regular sales season. The

15 1342 Operaions Research 58(5), pp , 2010 INFORMS manufacurer firs acquires informaion hrough pricing and advance selling. When i is opimal o do so, she sops collecing advance sales informaion and uses his informaion o build capaciy opimally. During he regular sales season, he manufacurer sells he insalled capaciy hrough dynamically seing prices. Noe ha his selling process combines a sraegic pricing and capaciy decision wih an operaional pricing decision. Hence, he ime scale and periods for a sraegic versus acical level decision could be differen. All acual sales and deliveries ake place in he regular sales season during which he manufacurer is allowed o adjus prices M imes, M In order o solve his problem, we have o embed a second-sage dynamic program ha prescribes he opimal dynamic pricing policy and he resuling profi during he regular sales season for a given level of capaciy and hen deermine he opimal capaciy level. Since he manufacurer has M opporuniies o adjus prices over he regular sales season, le A m m= 1 M denoe he random marke demand poenial she will face in he mh pricing epoch (or subperiod). We assume ha A m s are independenly disribued IFR random variables wih suppor on 0 Recalling ha is he remaining demand poenial he manufacurer serves afer sopping advance selling, for logical consisency i is desirable ha he disribuion of M m=1 A m is he same as he disribuion of Denoing he price charged in he mh epoch as p T m, he random demand during he mh pricing epoch is D T m p T m q = f q A m pt m b Noe ha our original model corresponds o he case when M = 1 For a given pricing policy = p T 1 p T M, he manufacurer s expeced profi from remaining cusomers during he regular sales season is S q M = p T m c p E min S m D T m p T m q = m=1 c u E S M D T m p T m q + M p T m E min S m D T m p T m q m=1 + c p c u E S M D T m p T m q + c p S 1 ( M = f q p T m E min s m A m p b T m m=1 ) + c p c u E s M A m p b T M + c p s 1 f q s where we define S m S for m = 1 and S m+1 = S m A m pt m b + for m>1 and s m S m /f q. The second equaliy holds because oal sales during he regular season plus he remaining capaciy equals he oal capaciy a he beginning of regular sales season, i.e., S 1 = M m=1 min S m D T m + S M D T M + The objecive is o solve s = min P s, where P denoes he se of all policies. Finding s involves solving he following dynamic program: m s m = max p T m p T m E min s m A m p b T m + E m+1 s m A m p b T m + for 1 m<m (19) M s M = max p T M E min s M A m p b T M p T M + c p c u E s M A m p b T M + (20) where he expecaion is wih respec o he random variable A m in each period m Noice ha by definiion s 1 s 1. The manufacurer s opimal ne profi from remaining cusomers when she sops advance selling in period and sells he surplus capaciy S via dynamic pricing is given by [ ( ) ] S f q S c f q + c p f q = f q s c + c p s = f q 1 s 1 c + c p s 1 Opimizing over he capaciy level also, he manufacurer s opimal ne profi from remaining cusomers becomes q = f q s c + c p s = f q 1 s 1 c + c p s 1 (21) Defining s c +c p s we have a similar srucure as in Equaion (3), i.e., q = f q This implies ha all of he preceding resuls regarding he opimal sopping policy for acquiring advance sales informaion hold wih he profi q replaced by is new definiion above. We formalize his in he nex heorem, which we sae wihou proof. Theorem 6. When he manufacurer sells insalled capaciy by dynamically adjusing prices, he opimal sopping policy for advance selling is a sae-dependen conrol-band policy. The above resul esablishes he srucure of he opimal policy. Ye, compuing policy parameers, such as he opimal prices and hresholds, remains a difficul ask. Essenially, he problem is a wo-sage, nesed, sochasic dynamic program wih muliple decision epochs and coninuous, mulidimensional sae spaces. The firs sage is he opimal sopping problem whose soluion depends on he second-sage dynamic program specified in Equaions (19) (20). Even his second-sage problem is a challenging one o solve numerically. Monahan e al. (2004) sudy a

16 Operaions Research 58(5), pp , 2010 INFORMS 1343 Figure 11. Muliple pricing opporuniies o sell capaciy during he regular sales season Expeced profi I no & I f (%) I no I f M similar problem as he second-sage DP and repor ha efficien resuls can only be obained when c p = c u For his case, hey show ha m s m = r m s m n where r m = max z E z A m + + rm+1 E z A m + n z z n ( ) nr b s 1 = 1 c + c p s c + c p s = 1 n ( ) nr b n c + c p 1 c + c p where n = 1 1/b and rm+1 = 0. Hence, q = f q where = 1 n /n nr1 /c + c p b. We use his resul o numerically solve he second-sage dynamic program and embed is soluion o he opimal sopping problem and solve for he opimal advance sales prices and sopping hresholds. Figure 11 illusraes he resuls of a numerical sudy in which A m is normally disribued wih same mean and variance across all m. The parameers of his example are he same as he base se, excep = 40 and = 0 2 Three observaions are worh noing. Firs noe ha he expeced profi is increasing wih he number of pricing opporuniies. The percenage increase in expeced profi beween having M = 7 pricing epochs o having one is 1 25% = / Second, he marginal increase in profi is decreasing. Third, he expeced value of advance selling is decreasing, bu he marginal decrease is also decreasing. This observaion suggess ha dynamic pricing during he regular sales season is only a parial subsiue for dynamic pricing during advance sales periods. Alogeher, hese observaions sugges ha using a small number of price adjusmens or even a single price during he regular sales season is reasonably close o opimal, considering also he fac ha such price adjusmens are cosly due o ransacion coss (e.g., due o adverising new prices). Similar observaions for his second-sage problem are also repored in Gallego and van Ryzin (1994) M 8. Summary and Discussion In his paper, we sudy he sraegy of obaining informaion abou marke poenial hrough advance sales for a beer capaciy decision. In paricular, we consider a manufacurer who collecs revenue and informaion hrough advance selling prior o building capaciy. Using advance sales informaion, he manufacurer ses he capaciy and saisfies he remaining demand during he regular season as much as possible, subjec o he available capaciy. We esablish he opimal pricing sraegy boh for he advance and regular selling seasons and he opimal capaciy o build. We also esablish he opimaliy of a conrol-band policy ha prescribes when o sop collecing advance sales informaion. This policy is also opimal when prices are se exogenously (e.g., as mark-up or mark-down schedules). Through a numerical sudy, we quanify he expeced value of his capaciy planning sraegy under differen marke and operaing condiions. We show ha advance selling coupled wih a le-hem-come-and-build-i-laer approach is a profiable sraegy, in paricular, when (i) demand uncerainy is high, (ii) more cusomers anicipae capaciy shorages in he marke, (iii) building capaciy is expensive bu iming is no a major concern, (iv) commimens have moderae predicive value abou marke poenial, and (v) cusomer price sensiiviy is relaively low. We also show ha he exreme sraegy of collecing full advance selling informaion or no collecing any informaion leads o inferior soluions in comparison o he opimal pricing sraegy. These resuls sugges ha he pracice of advance selling is of mos value for, e.g., high echnology, apparel, and pharmaceuical indusries. For example, elecommunicaion and semiconducor indusries face high capaciy building coss. They ofen inroduce new producs for which he marke uncerainy is also high relaive o commodiy ype producs. Finally, we sudy a scenario in which he manufacurer coninues o sell insalled capaciy hrough dynamic pricing. Modeling his scenario bridges he revenue and capaciy managemen lieraures. We show ha selling capaciy by dynamically adjusing regular sales prices increases he

17 1344 Operaions Research 58(5), pp , 2010 INFORMS expeced profi bu only o a limied exen. Nex, we revisi a few aspecs of our framework. Oher Marke Signal Funcions. In our numerical examples we have uilized a specific form of he marke signal funcion, which smoohed he predicive value of cumulaive commimens over ime. As noed earlier, i is possible o envision a case where nex period s expeced marke signal is he same as he curren level (i.e., E f +1 q = f q ) for any seleced price p. In oher words, he evoluion of he marke signal is a maringale. This would correspond o an exreme case where he curren marke signal is a firm indicaor of fuure demand, such ha he manufacurer expecs he curren signal o susain regardless of he price charged. As an example, consider he following scenario where he manufacurer racks he value f of he marke signal iself and recursively updaes i: f = 1 f 1 + d 1 E 1 p b 1 = 1 f 1 + f 1 1 (22) E 1 In his case, he manufacurer needs o know only f o updae he marke signal and he cumulaive commimens q, o deermine is profi and opimal course of acion. This does no sugges, however, ha f is independen of he q As a maer of fac, subsiuing q q 1 for d and accordingly aking = f 1 q 1 p 1, Equaion (22) can be saed equivalenly as f q = 1 f 1 + q q 1 / E 1 p b 1 which is a linear increasing funcion of cumulaive commimens q. Also, a ime zero, E f q = 1 for any given and price pah. Hence, our srucural resuls on he form of he opimal policy for acquiring advance sales informaion apply o his maringale evoluion model as well. I is also possible o envision a case where he marke signal depends only on he cumulaive commimens q and no on pas prices or oher hisorical informaion. In his case, he linear increasing signal f q would model sricly he word-of-mouh effec creaed by he cumulaive number of early purchasers. This ype of dependency of fuure demand on pas sales is a common feaure in new produc diffusion models (as in Bass 1969). This case may occur when he manufacurer announces he prices privaely o each poenial cusomer wihou revealing hisorical informaion. Residenial real esae developers someimes use such a selling sraegy. Alernaively, such a marke signal funcion could approximaely model a case where early cusomers are relaively insensiive o prices. Our srucural resuls in 4 and 5 would apply o his case as well, wih he added simplificaion ha he opimal conrol-band policies would be sae independen. Connecion wih Demand Learning Models. Alhough here is no formal learning of demand parameers in our demand model, he funcional form of he marke signal makes i applicable o a class of Bayesian models. Bayesian models of demand learning involve a muliple period horizon whereby he demand in each period follows a known disribuion wih an unknown parameer or vecor of parameers (say, ). There is a known prior disribuion of, which is updaed on he basis of a sufficien saisic as ime progresses and demand realizes. For a cerain class of a conjugae family of disribuions, is cumulaive pas sales q or a funcion of i, and is effec on he demand disribuion can be facored ou as a scaling funcion in he same spiri as our marke signal funcion f This approach was firs used by Scarf (1960) o solve he dynamic invenory managemen problem efficienly, and i was laer exended by Azoury (1985). Nex we provide some specific examples o illusrae he applicabiliy of our framework and resuls in his seing. We refer he reader o Azoury (1985) for a full accoun of he required condiions, deails, and oher examples. Suppose ha he marke demand poenial in each period are independen wih disribuion = k, where k s T =1 k = 1 are known scalars. Consider he case k = 1/T and an exogenous, fixed pricing scheme p = p for = 1 T This implies he demands across periods are independen and disribued idenically. When he disribuion belongs o he Gamma family wih unknown scale parameer ha also has a Gamma prior disribuion, cumulaive sales q is he sufficien saisic for updaing demand (hence = for all ). Furhermore, he Bayes esimae of demand in period can be wrien as f q D, where f q = a + q and a>0 is a known consan, and he disribuion of D only depends on (Scarf 1960, Azoury 1985). Noice ha f q is increasing linear in q. Consequenly, our opimal policy resuls in 4 apply o his scenario. When k and p are nonidenical (bu sill exogenous), defining k = k p b,wehave = k, which are independen bu no longer idenically disribued. In his case, q is no longer a sufficien saisic; i is necessary o know he hisory of pas demand realizaions (hence q s), causing a significan increase in he dimensionaliy of he sae space. 12 When prices are decision variables, i becomes necessary o rack pas prices ha define he scalars k. Neverheless, given he hisory of pas prices and commimens, he scaling funcion can be expressed as a linear funcion of q, which implies ha our sae-dependen opimal policy resuls apply. The deails are deferred o he elecronic companion o his paper. Oher Price Funcions. In a muliplicaive demand environmen, here are alernaives o he iso-elasic price funcion d p = p b. This form faciliaes he derivaion of unique opimal regular sales prices p s in Theorem 3. As shown in Song e al. (2008), he uniqueness is guaraneed when he curvaure of d p given as d p d p /d p 2

18 Operaions Research 58(5), pp , 2010 INFORMS 1345 is increasing in p and is no oo large (see heir Assumpion 2 for deails). A large class of price funcions, including he iso-elasic one, fis ino his caegory. Some oher examples include d p = a bp a > 0 b >0 >0, d p = ae bp a > 0 b>0 d p = a p b a > 0 b>1 and a e bp a > 0 b >0 Hence all he resuls remain valid for more general price funcions. The only complicaion would arise when he insalled capaciy is sold via dynamic pricing because he iso-elasic funcion faciliaes solving he second-sage DP numerically. Oher Pricing Sraegies. The flexibiliy o se advance and regular sales prices opimally would generae he highes profis for he manufacurer. However, as also noed earlier, here are some pracical scenarios in which hese prices are predeermined by he manufacurer or he marke. For example, Biran and Mondschein (1997) discuss a pricing sraegy wih announced premiums or discouns. Under his pricing sraegy only he iniial price p 1 is a decision; he remaining prices are p = p 1, where >1 for he announced premium (mark-up pricing) sraegy and <1 for he announced discoun (mark-down pricing) sraegy. Noe ha for any p 1 and, he resuling problem is equivalen o he exogenous pricing case. Hence he opimal iniial p 1 and he resuling profi can be obained by aking one exra sep and searching over p 1 R 1 o maximize expeced profi. The area of informaion acquisiion for capaciy planning offers a ferile avenue for fuure research. This paper akes a firs sep oward addressing pricing sraegies o acquire demand informaion used for capaciy planning. There are oher possible research direcions. One possibiliy is o explore he impac of advance selling when muliple producs can be produced given a flexible capaciy or when produc subsiuion is a possibiliy. For example, Neessine e al. (2002) show ha one can gain significan benefis if he capaciy decision incorporaes he possibiliy of upward subsiuion, i.e., saisfying cusomer demand by a beer produc. We consider he impac of pricing and advance selling sraegy implemened prior o he capaciy decision, whereas hey consider he impac of a subsiuion sraegy implemened afer capaciy is se. An ineresing research avenue is o invesigae he join effec of boh. We leave hese for fuure research. 9. Appendix. Proofs We use he noaion f and F o denoe he pdf and cdf of disribuion Proof of Theorem 1. The proof is based on an inducion argumen. Before he inducive proof we firs show ha H q is linear in q for all To do so, we subsiue q defined in (3) o Equaion (8) and rearrange erms o derive H q =f q p T c p +c +1 E p b T + T E f +1 q T c +1 c q (23) The expecaion of f +1 is wih respec o D p q. The hird erm is linear. As f q is linear, he firs erm is also linear in q Since q +1 = q + f q p b by same reason, E f +1 q is also linear in q. This proves he lineariy of H q. To iniiae he inducive argumen, noe for = T 1 ha H T 1 q T 1 T 1 = H T 1 q T 1 T 1 which is linear (and hence convex) in q T 1, proving par 1 for T 1. If H T 1 q T 1 T 1 is increasing in q T 1, hen i can cross zero eiher once or no a all. In he former case, L T 1 T 1 = and U T 1 T 1 <, and i is opimal o coninue advance selling if q T 1 >U T 1 T 1 In he laer case, L T 1 T 1 = U T 1 T 1 = and i is opimal o coninue advance selling. If H T 1 q T 1 T 1 is decreasing in q T 1, i can again hi zero eiher once or no a all. In eiher case, we have L T 1 T 1 < and U T 1 T 1 =, and i is opimal o coninue advance selling if q T 1 L T 1 T 1 and sop oherwise. Noicing ha he funcion max 0 x is increasing convex, and recognizing ha increasing convex ransformaion of a convex funcion is sill convex, V T 1 q T 1 T 1 = max 0 H T 1 q T 1 T 1 is convex, proving par 3 for = T 1. Suppose for an inducion argumen ha par 1 is rue for some + 1 <T 1 This implies H can cross zero a mos wice and hose poins are precisely defined by L and U Hence, for q L U +1 +1, we have H +1 q +1 < 0, which implies i is opimal o sop advance selling. Oherwise, i is opimal o coninue advance selling, proving par 2 for + 1. Since max 0 x is increasing convex and increasing convex ransformaion of a convex funcion is sill convex, V +1 q +1 is convex in q, proving par 4 for + 1. To conclude he inducion argumen, we show ha par 4 for + 1 implies par 1 for. Noe ha E V +1 q is convex because (i) he updae q +1 = q + f q p b is increasing convex in q, (ii) he composiion of increasing convex funcion is convex, and (iii) convexiy is preserved under expecaion. Since H q is linear, he sum H q is also convex, proving par 1 for. This concludes he inducion argumen. Proof of Theorem 2. To prove par 1, noe ha when f q 1 q and, H q in Equaion (8) is independen of and is given as H q = p T c p + c +1 E p b T +1 T c +1 c q Hence, if c +1 >c for all hen H q is sricly decreasing and linear in q. Then i is easy o esablish inducively ha H q and V q are also independen of and sricly decreasing convex funcions of q. Hence U = and L < for all, proving he opimaliy of saeindependen hreshold policy.

19 1346 Operaions Research 58(5), pp , 2010 INFORMS When c +1 = c he las erm in H q also drops. Consequenly, H q, H q and V q are all independen of q and equal consans H, H, and V, respecively. Then i is easy o verify ha he opimal sopping ime for acquiring advance sales informaion is he firs 1 T such ha H = 0 To prove par 2, recall from Theorem 1 and is proof ha he srucure of he policy is driven by H q given in Equaion (23), which is linear in q. For any,, and c, for a sufficienly large c +1 he erm in is decreasing because, defined in (3), does no depend on c +1. The second erm is also decreasing in q because +1 is negaive for large c +1 (he higher he cos of building capaciy, he lower will be opimal profi from remaining cusomers), while E f +1 q does no depend on c +1. The las erms are clearly decreasing. As a resul, H q is decreasing in q. This means ha when c +1 is sufficienly larger han c, H q would be decreasing in q Nex, one can esablish inducively ha H q and V q are decreasing convex funcions of q because his propery is preserved under max 0 f x operaor. Hence, U = for all, proving he opimaliy of saedependen hreshold policy. Proof of Theorem 3. Le p denoe he selling season price se in period, and p s is opimal value. Clearly, p s maximizes p S q defined in (2) over p Subsiuing y S / f q in (2), we ge p S q = f q ˆ p y where ˆ p y = p c p c y p c p + c u E y p b + (24) Hence, maximizing (2) for a given commimen q and hisory boils down o maximizing (24). Recall ha since IFR propery is closed under convoluions (Barlow and Proschan 1975), is also IFR, meaning is failure rae h x = f x / 1 F x is increasing. Since is IFR, i also has increasing generalized failure rae (IGFR), i.e., xh x is also increasing (Lariviere and Poreus 2001). Song e al. (2008) sudy he opimal ordering and pricing problem for a newsvendor wih order-up-o level y, reail price p purchase cos w, and salvage value b (p >w> b 0). They esablish (in Theorem 1) he exisence of a unique opimal y p pair under muliplicaive demand for a large class of demand funcions ha includes he iso-elasic funcion when he disribuion of he underlying uncerainy is IGFR. They derive he opimal pair y p sequenially, i.e., hey firs deermine he unique opimal price p y for a given y and hen derive he unique opimal y and resuling p Observe ha (24) is equivalen o he sandard newsvendor funcion wih p = p, w = c p + c, and b = c p c u. Hence, heir resul applies as long as p > c p + c >c p c u 0 Noe ha he second inequaliy is immediaely saisfied. Consequenly, when c p c u and he firs inequaliy holds, (24) has a unique opimal y ps Nex we show ha he firs inequaliy is rue for a candidae opimal socking level and price pair and hence he pair is also he unique maximizer. To do so, i is convenien o conduc he socking facor ransformaion (Peruzzi and Dada 1999) z = y /p b and hence wrie (24) equivalenly as ˆ p z = p b p c p c z p c p + c u E z + (25) For a fixed z, aking he derivaive of (25), we obain afer some manipulaion ˆ p z p = b 1 p b+1 z E z + { b (c b 1 p + c } z + c u E z ) p + z E z + (26) Seing he derivaive o zero and solving for p yields he unique opimal price ( b p z = )(c b 1 p + c ) z + c u E z + z E z + as in (11) because p z >c p + c for all z 0. Subsiuing p z in (25) resuls in ˆ z =ˆ p z z = 1 b p z b 1 z E z + (27) Taking he derivaive of (27) we ge ˆ z z { =p z b 1 F z p z c p +c u c } +c u 1 F z Hence z is he unique soluion of p z c p + c u c + c u / 1 F z = 0 as saed in he heorem, and p s = p z. By definiion S = f q z ps b, so ha Q = q + S Subsiuing z in (27) resuls in given by (12) and q = f q Hence, q = p s S q = 1 k=1 p kd k c p + c q + f q Proof of Theorem 4. Noe ha p s >c p + c follows direcly from (11) in Theorem 3, since b/ b 1 >1 and z / z E z + >1 In order o prove he remaining resuls, i is convenien o change he order of opimizaion of he profi funcion ˆ p z given by (25). Suppose ha p is fixed. Le z p denoe he corresponding opimal socking facor, which is given uniquely by he equaion P >z = c + c u p c p + c u (28)

20 Operaions Research 58(5), pp , 2010 INFORMS 1347 Subsiuing z p ino ˆ p z, we can wrie ˆ p z p p = ˆ p z p + ˆ p z z z p a z = z p Noing ha he second erm in ˆ p z p / p equals zero a z = z p i follows from (26) ha he opimal price p s is given as he unique soluion o ( b ){c b 1 p + c } z p +c u E z p + p z p E z p + =0 (29) Furhermore, for p <p s, he lef-hand side of (29) is posiive and for p >p s,iisnegaive. We firs characerize he change in he opimal socking facor (28) and hen he resuling change in (29). This enables us o prove he impac on opimal price p s. From he join effec, we can hen esablish he effec on opimal capaciy level Q = q + f q z ps b. The proof for each cos elemen (c p c c u ) follows idenical logic and inermediae seps. For his reason, in order o avoid repeiion, we provide a deailed proof for only one of hem, namely he producion cos c p Taking derivaives wih respec o c p,forafixedp,wehave z p / c p = c + c u / p c p + c u 2 f z p < 0 Le Õ p = c p + c z p + c u E z p + z p E z p + Noe from (29) ha p s is increasing in c p if Õ p / c p > 0 Õ p c p =1+ c +c u z p F z p E z p + z p E z p + 2 ( c +c =1 u p c p +c u =1 1 F z p 2 f z p z p c p ) 2 z p F z p E z p + f z p z p E z p + 2 z p F z p E z p + z p E z p + 2 The las equaliy is from (28). Noice ha 1 F z p 2 z p F z p E z p + f z p < 1 F z p 2 f z p < 1 F z p < 1 z p E z p + 2 F z p z p E z p + The firs inequaliy is eviden. The second one is obained by bounding he funcion z E z + using he fac ha is IFR. In paricular, z E z + = z 0 1 F u du 1 F z F f z z Consequenly, Õ p / c p > 0 and hence p s is increasing in c p Wih his resul, using implici differeniaion on (29), i is easy o verify ha p s/ c p > 1 which immediaely implies (from (28)) ha z / c p < 0 Since p s is increasing and z is decreasing, Q is decreasing in c p. Proof of Theorem 5. Similar o he proof of Theorem 1, firs noe ha H p q in Equaion (15) is linear in q for a given p and. The res of he proof is based on an inducion argumen. For = T 1, noe ha R T 1 p T 1 q T 1 = H T 1 p T 1 q T 1 which is linear in q, proving par 1 for T 1. This implies ha if H T 1 p T 1 q T 1 is increasing (resp., decreasing) in q hen R T 1 p T 1 q T 1 is also increasing (resp., decreasing) in q. Nex we show ha his implies H T 1 q T 1 is also increasing (resp., decreasing) in q. Define q 1 <q 2 and le p 1 arg max p R T 1 p q 1 T 1 and p 2 arg max p R T 1 p q 2 T 1. When R T 1 p q T 1 is decreasing in q, we have H T 1 q 2 T 1 = R T 1 p 2 q 2 T 1 < R T 1 p 2 q 1 T 1 < R T 1 p 1 q 1 T 1 = H T 1 q 1 T 1. Hence, H T 1 q T 1 is also decreasing in q. When R T 1 p q T 1 is increasing in q, we have H T 1 q 1 T 1 = R T 1 p 1 q 1 T 1 < R T 1 p 1 q 2 T 1 < R T 1 p 2 q 2 T 1 = H T 1 q 2 T 1. Hence H T 1 q T 1 is also increasing in q. This implies pars 3 and 4 for T 1 along he same argumens as in he proof of Theorem 1. Nex, suppose for an inducion argumen ha Par 1 is rue for. This implies par 2 because convexiy is preserved under maximizaion (Poreus 2002, p. 226). Hence, given H q can cross zero a mos wice, and hose poins are given precisely as L and U From convexiy, i follows also ha H q 0onL q U, in which case i is opimal o sop advance selling. Oherwise, i is opimal o coninue advance selling, proving par 3. Noing ha he funcion max 0 x is increasing convex, and increasing convex ransformaion of a convex funcion is sill convex, V q = max 0 H q is also convex, proving par 4 for. To complee he proof, we show ha par 4 for implies par 1 for 1. Noe ha q = q 1 + f 1 q p 1 b is linear increasing in q 1 Hence, E V q is also convex in q 1. Since H 1 p 1 q 1 1 is linear in q 1, he sum R 1 p 1 q 1 1 is also convex in q 1, proving par 1 for 1 and concluding he inducion argumen. 10. Elecronic Companion An elecronic companion o his paper is available as par of he online version ha can be found a hp://or.journal.informs.org/. Endnoes 1. During Sepember 2002, Chriser Lundberg from Ericsson presened an advance selling sraegy for long range forecasing during he Ericsson Supply Chain Academy Conference in Sweden.