Internet companies extensively use the practice of drop-shipping, where the wholesaler stocks and owns the

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1 MANAGEMENT SIENE Vol. 52, No. 6, June 26, pp ssn essn nforms do /mnsc INFORMS Supply han hoce on the Internet Sergue Netessne The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 1914, netessne@wharton.upenn.edu Nls Rud INSEAD, 7735 Fontanebleau, France, nls.rud@nsead.edu Internet companes extensvely use the practce of drop-shppng, where the wholesaler stocks and owns the nventory and shps products drectly to customers at retalers request. Under the drop-shppng arrangement, the supply chan benefts from rsk poolng because the nventory for multple retalers s stocked at the same locaton, the wholesaler s. Another more tradtonal channel alternatve on the Internet s one n whch retalers stock and own the nventory. These two supply chan structures, whch predomnate on the Internet, result n dfferent nventory rsk allocaton, stockng decsons, and profts for channel members. Moreover, the two channel alternatves can be combned nto a dual strategy whereby the retaler uses local nventory as a prmary source and reles on drop-shppng as a backup. We model the dual strategy as a noncooperatve game among the retalers and the wholesaler, analyze t, and obtan nsghts nto the structural propertes of the equlbrum soluton to facltate development of recommendatons for practcng managers. Fnally, we characterze stuatons n whch each of three channels s preferable by specfyng approprate ranges of crtcal parameters, ncludng demand varablty, the number of retalers n the channel, wholesale prces, and transportaton costs. Key words: supply chan; drop-shppng; e-commerce; fulfllment; equlbrum; competton Hstory: Accepted by Fangruo hen, supply chan management; receved December 11, 21. Ths paper was wth the authors 1 year and 1 months for 4 revsons. 1. Introducton and Lterature Survey Allance Entertanment orporaton, Ingram Entertanment, and Baker and Taylor are major players n the busness of dstrbutng home entertanment products ncludng Ds, vdeos, games, and books to Internet, as well as to brck-and-mortar, retalers. One of the servces they offer s drop-shppng, or consumer-drect fulfllment, the practce whereby the wholesaler stocks and owns the product and shps t drectly to customers at the retaler s request. Under a drop-shppng arrangement, the retaler serves as a mddleman who acqures customers and accepts orders, whle the wholesaler owns and holds nventory and also fulflls orders. Drop-shppng offers two obvous benefts to the retaler: No nvestment nto fulfllment capabltes s requred, and the retaler takes no nventory rsk (see the example of Spun.com, Forbes 2). Drop-shppng also benefts the wholesaler n that t allows her to charge a hgher wholesale prce and may provde access to a wder customer base (see Scheel 199, p. 42). Fnally, one would expect that the supply chan as a whole would beneft due to rsk poolng and possble economes of scale n transportaton n cases n whch the wholesaler serves multple retalers. The nventon of the Internet, a relatvely cheap electronc medum, greatly reduced the transacton costs of drop-shppng by allowng a seamless ntegraton of retalers and wholesalers. The Internet allows n-stock avalablty and prces to be communcated from the wholesaler to the retaler, and orders can be placed from the retaler to the wholesaler, all n real tme. Before the Internet, such ntegraton would have requred mplementng a rather expensve EDI system and hence was not economcally vable n most channels. As a result of ths ntegraton of retalers and wholesalers, n a recent survey 3.6% of pureplay Internet retalers cted drop-shppng as ther prmary way of fulfllng orders. At the same tme, 44.5% of Internet retalers reled prmarly on stockng nventory nternally (see eretalng World 2), whereas the rest ether outsourced fulfllment to thrd partes or used the fulfllment capabltes of strategc partners. Supply chan arrangements n whch the retalers stock and own nventory have been extensvely analyzed n the operatons lterature. However, the practce of drop-shppng, although almost as popular among Internet retalers, has attracted only lmted attenton. The marketng lterature offers some references to drop-shppng: Although Scheel (199) provdes extensve qualtatve analyss of drop-shppng n the catalog busness, he also notes that the practce s generally perceved as havng somewhat lmted 844

2 Management Scence 52(6), pp , 26 INFORMS 845 potental. The goal of ths paper s to analyze comparatve advantages of nventory ownershp n the tradtonal channel and rsk poolng under drop-shppng Summary of Results We begn by presentng two alternatve models: the tradtonal channel, n whch each retaler stocks nventory locally, and the drop-shppng channel, n whch the wholesaler stocks nventory centrally and shps drectly to customers. We model both channels as two-stage supply chans wth one wholesaler and multple dentcal retalers. In both models the wholesaler charges the retalers a fxed wholesale prce (not necessarly the same). Addtonally, based on our conversatons wth Internet companes, we assume that dfferent channels may have dfferent transportaton costs due to economes of scale and possbly other cost dfferences. When faced wth a choce between the tradtonal and drop-shppng channels, the retaler trades off a hgher wholesale prce aganst nventory rsk, and the wholesaler trades off the lower margn aganst nventory rsk. Moreover, transportaton cost dfferences play a role. By combnng the two strateges, the retaler can frst satsfy demand wth the nternally stocked nventory (purchased at a lower wholesale prce) and then request the wholesaler to drop-shp the rest. For example, BlueLght.com, a major Internet retaler, uses the dual strategy for Ds: It stocks certan quanttes of the top 4 Ds to ncrease proft margns and reles on drop-shppng n case of a stockout (see Apparel Industry Magazne 2) whle utlzng pure drop-shppng for the remanng D ttles. Also, several major retalers have recently nstalled Internet kosks nsde physcal stores (see omputerworld 21); f a customer does not fnd a product on the store shelves, she can order t n store over the Internet and the product wll be shpped drectly to her from a central warehouse. Such an arrangement resembles the dual strategy we analyze. The dual strategy s somewhat smlar to the combnaton of make-to-stock and assemble-to-order practces dscussed n the operatons lterature (see Rud and Zheng 1997 and attan et al. 2) wth one major dfference: Under the make-to-stock/assembleto-order practce, the same company decdes both how much to stock and how much to assemble to order, whereas under the dual strategy consdered n ths paper, the retalers and the wholesaler manage ther nventores compettvely, such that a gametheoretc stuaton arses. We model the dual-strategy problem as a noncooperatve game n whch retalers compete wth the wholesaler for demand and, moreover, they compete wth each other for the wholesaler s nventory allocaton. We analyze equlbra of ths game, derve the structural propertes of players best-response functons, and use them to show the unqueness of the symmetrc equlbrum. One of our results s that n equlbrum under the dual strategy t may be optmal for the wholesaler or for retalers to stock nothng, and thus the dual channel converts nto a pure tradtonal or a pure drop-shppng channel. We are able to dentfy smple condtons that lead to such boundary equlbra. Usng propertes of supermodular games and derved propertes of the best-response functons, we conduct senstvty analyss of the equlbrum to the changes n the wholesale prce, the drop-shppng markup, and transportaton cost parameters. We further compare the tradtonal, drop-shppng, and dual channels usng numercal experments. We fnd that when the number of retalers s large, the benefts of rsk poolng make ether the drop-shppng or the dual channel more attractve than the tradtonal channel to both retalers and the wholesaler. Ths fndng s consstent wth the practcal observaton that drop-shppng s often used for books and electronc products, two segments n whch there are many retalers, most served by a few large wholesalers. In the dual channel we fnd that changes n any cost-related parameters result n the reallocaton of nventory between retalers and the wholesaler. For example, n ths channel, ncreasng the markup on drop-shpped products leads to hgher retal nventory and lower wholesale nventory, whereas n the drop-shppng channel the same acton would lead to hgher wholesale nventory. Surprsngly, the mpact of the transportaton cost dfferental on profts n the dual channel s not always ntutve. In partcular, as the transportaton cost structure becomes more favorable for drop-shppng, the wholesaler s proft ntally drops and the retalers proft ncreases. We fnd that the drop-shppng markup and the transportaton cost dfferental are the man drvers of the choce of channel. We observe that both the dropshppng and the dual channels have the potental to be Pareto-optmal choces, but the tradtonal channel does not. Ths fndng ndcates that most Internet retalers should strve to utlze drop-shppng at least to augment ther nternal nventory. However, n some ndustres wholesalers prefer the tradtonal channel, and hence do not want to stock any nventory for drop-shppng, so ths opton s smply not avalable. In partcular, ths scenaro arses when the drop-shppng markup s relatvely small but transportaton costs favor drop-shppng, so that the wholesaler does not make any money on shppng drectly to customers even though the retaler prefers ths opton. In stuatons wth a large set of problem parameters the dual channel s the Paretooptmal choce, whch s consstent wth the prolferaton of Internet kosks and the ncreasng popularty

3 846 Management Scence 52(6), pp , 26 INFORMS of ths arrangement. Due to the nonntutve mpact of transportaton costs, the dual channel becomes the Pareto-optmal choce when the transportaton costs are much hgher n the drop-shppng channel than n the tradtonal channel. As a result, retalers carry a lot of nventory and rely on drop-shppng for only a small proporton of demand. Agan, ths fndng s consstent wth the exstence of Internet kosks n retal stores that already carry a lot of nventory. Overall, our results conform wth the emprcal fndngs of Randall et al. (26) that no channel opton s unformly preferred over the others and that channel choce depends on companes envronments Relaton to the Lterature Our paper addresses two key ssues: (1) nventory ownershp and stockng decson rghts and (2) rsk poolng. From the nventory ownershp/stockng decson rghts pont of vew, the tradtonal channel n our paper s smlar to that n most of the supply chan lterature because the retaler both owns nventory and decdes on stockng quantty (see, for example, Zpkn 2). In contrast, n the drop-shppng channel both nventory ownershp and stockng decsons rest wth the wholesaler. The only related arrangement we are aware of s vendor-managed nventory (see, e.g., Bernsten et al. 22), n whch the wholesaler s endowed wth stockng decson rghts, but nventory s stll owned by the retaler. A combnaton of vendor-managed nventory wth consgnment transfers both nventory ownershp and stockng decson rghts to the wholesaler, a scenaro somewhat smlar to our drop-shppng model. The mportant dfferences are that under consgnment, nventory s stored at each retaler s locaton so there are no benefts to rsk poolng. Moreover, the structure of transportaton costs s dfferent. In the dual-channel model we consder, each echelon of the supply chan stocks nventory, a settng that parallels the multechelon lterature. Typcally, though, n ths stream of lterature all echelons are managed centrally (whereas n our paper retalers and the wholesaler are separate enttes) and, moreover, sales n each perod are lmted by nventory at the lowest (retal) echelon (whereas n our model the wholesaler s nventory s used as a backup). achon and Zpkn (1999) and Lee and Whang (1999) relax the former assumpton by consderng a supply chan wth one retaler and each echelon locally managed. Bernsten and Federgruen (23), achon (21), and hen et al. (21) further consder decentralzed multechelon supply chans wth multple retalers. Lawson and Porteus (2) relax the second assumpton by addng an opton to expedte the product between stages but stll assume that nventory s managed centrally. We are not aware of any papers that relax both of these assumptons. The dual strategy s also smlar to the practce of subcontractng n case demand exceeds capacty (see Van Meghem 1999 and Plambeck and Taylor 25). In another related paper, achon (22) models advance purchase dscounts, a practce whereby the retaler can prebook nventory before the season at a lower per-unt prce. Hs work s smlar to ours n that both the retaler and the wholesaler are allowed to hold nventory. achon focuses on dentfyng Pareto-optmal prce-only contracts and studes supply chan effcency under such contracts. However, because there s only one retaler, the benefts of rsk poolng are not reflected, and ssues of nventory allocaton among retalers do not arse. Tradtonally, research on rsk poolng focused on comparng the costs and benefts of buldng one central, versus many regonal, warehouses for a sngle company (see Eppen 1979). Van Meghem (1998) and Rud and Zheng (1997) consder dual strateges under rsk poolng by combnng rsk poolng at an extra cost per unt wth the cheaper local nventory. All of these papers focus on the performance of a sngle frm and, therefore, do not compare the benefts of rsk poolng for both the retalers and the wholesaler, whereas we do. Anupnd and Bassok (1999) compare centralzed and decentralzed nventory management n a model wth two retalers and one suppler whle focusng on the effect of market search by customers n case of a stockout at one retaler. They compare the decentralzed model wth the model n whch retalers form an allance to centralze ther nventores. Ther work s smlar to ours n that they consder the mpact of rsk poolng not only on downstream retalers, but also on the upstream suppler. The dfference, however, s that n the case of centralzed nventory management, Anupnd and Bassok (1999) assume that the retalers stll make stockng decsons, whereas n our drop-shppng model the wholesaler makes the stockng decsons, resultng n dfferent channel dynamcs. Furthermore, Anupnd and Bassok do not study the dual strategy that s the man focus of ths paper. Another relevant stream of lterature consders transshpment,.e., the practce whereby a retal locaton that s out of stock of certan goods can receve them from another locaton wth excess nventory. The most relevant papers n ths stream are Rud et al. (21) and especally Dong and Rud (24), where both the retalers and the wholesaler s profts are consdered. Sefert and Thonemann (21) and Sefert et al. (21) model sngle-drectonal transshpments from physcal to Internet retalers. Anupnd et al. (21) consder a very general decentralzed transshpment model n whch multple retalers not only stock nventory nternally, but also stock t at

4 Management Scence 52(6), pp , 26 INFORMS 847 multple jontly owned warehouse locatons (smlar to Anupnd and Bassok 1999). They focus on nventory allocaton mechansms and use cooperatve game theory to defne approprate allocatons and separate ownershp of stock usng the noton of clams. The major dfference between these papers and our model s that under transshpment the wholesaler does not take nventory rsk and cannot satsfy customer demand, so the dstnct features of dropshppng do not apply. A number of marketng papers explctly focus on the ssue of channel choce, but not the ssues of nventory ownershp and nventory rsk that are central to ths paper. A notable excepton s the marketngoperatons nterface paper by Porteus and Whang (1991), where the authors use a prncpal-agent framework to model a sngle frm wth multple marketng managers responsble for promotng ther products and a producton manager responsble for mountng an effort to ncrease the avalable manufacturng capacty. The centralzed decson maker s endowed wth the power to set up an effcent compensaton scheme. Although ths work ncludes some essental elements of our model, lke demand uncertanty and nventory rsk, t s clearly very dfferent n motvaton, modelng assumptons, and goals pursued. The rest of ths paper s organzed as follows. In 2 we present and analyze the tradtonal, dropshppng, and dual-strategy models. Secton 3 contans a detaled analyss and comparson of the three channels. Secton 4 dscusses the mpact of relaxng some of our assumptons and concludes wth a summary of manageral nsghts and mplcatons for practcng managers. All proofs are relegated to the appendx. 2. Supply han Models We assume that there s a sngle product (or, equvalently, that there are multple products, but the problem s separable and solved for each product ndependently). The wholesaler ether dstrbutes the product to the retalers n the tradtonal channel (Model T) or shps drectly to the customers at each retaler s request n the drop-shppng channel (Model D). The two structures are combned under the dual strategy (Model ), wth retalers nventory used as a prmary source of stock. We use a sngleperod model that captures the trade-offs related to nventory rsk whle remanng suffcently transparent to analyss and manageral nsghts. It s possble to extend our results nto a multperod model (see Netessne and Rud 24) usng rather standard tools at the cost of addtonal notaton that makes the nsghts less transparent. We gnore the possble salvage value of leftover nventory, whch can easly be ncorporated at the cost of ntroducng an extra parameter, but wthout any addtonal nsghts. There are n dentcal retalers, ndexed by = 1 n. Retaler faces demand d, and all demands d are random varables wth dentcal dstrbutons and a symmetrc correlaton structure; we therefore drop subscrpt whenever approprate. The symmetry assumpton provdes a bass for comparson among the models whle stll preservng the essence of the problem. We defne D as the total demand for the product,.e., D = d, and f X denotes the densty functon of the random varable X. We denote retaler s proft by and the wholesaler s proft by. Optmal stockng quanttes and profts are marked wth the superscrpt, whereas best responses are marked wth superscrpt Br and bold varables ndcate vectors. Each retaler sells the product at an exogenously gven unt prce r (whch ncludes the shppng charge, f any), and the wholesaler buys the product from the upstream manufacturer at a fxed unt cost c. At the begnnng of the perod, each retaler stocks q T unts of the product n the tradtonal model, the wholesaler centrally stocks Q D unts of the product n the drop-shppng model, and both partes stock q and Q, respectvely, under the dual strategy. Retalers and the wholesaler are all ndependently managed. Retalers pay a wholesale prce w for each unt purchased to stock. Furthermore, retalers may request that the wholesaler drop-shp products. Because under a drop-shppng arrangement the wholesaler takes on the task of dong fulfllment and bears the nventory rsk, a reasonable drop-shppng contract wll have a hgher wholesale prce than when retalers buy the product for ther own stock. In practce, the dfference n wholesale prces s typcally 1% 2% (see Scheel 199, p. 42). Hence, we denote the wholesale prce under drop-shppng by w +, where s the addtonal drop-shppng markup. We assume that the wholesaler exsts n a compettve envronment and thus does not possess prce-settng power (e.g., books or Ds can be purchased from several major dstrbutors). Ths assumpton wll be dscussed n 4. Our analyss also gnores fxed costs, but we provde a dscusson of ther potental mpact on the choce of channel n 4. As descrbed n the ntroducton, the structure of transportaton costs wdely defned (e.g., ncludng pckng and packng) can vary accordng to channels. The total prce quoted to the customer typcally ncludes a separate shppng charge. However, the amount a customer pays for shppng may or may not exceed the actual shppng cost ncurred. For example, n the tradtonal channel the quantty q T s shpped n bulk from the wholesaler to the retaler, and later the quantty sold s shpped to customers (typcally

5 848 Management Scence 52(6), pp , 26 INFORMS n sngle unts). We wll denote by t I the nbound unt transportaton cost, and by t T the outbound unt transportaton cost, both ncurred by the retaler n the tradtonal channel. In contrast, n the drop-shppng channel, the wholesaler shps drectly to customers. In ths case, although the actual transportaton costs are ncurred by the wholesaler, the retaler receves the payment from the customer and transfers a part of ths payment to the wholesaler. We wll let t D be the outbound transportaton cost ncurred by the wholesaler. A part of ths transportaton cost s passed on to the retaler through the drop-shppng markup. For convenence, we defne r = r t T as the net unt revenue n the tradtonal channel, = t D as the net drop-shppng markup, and = t T t D as the transportaton cost dfference of the two channels. The combnaton of parameters, t I, and allows us to account for a varety of practcal stuatons that can arse (e.g., both or nether channel members can make/lose money on transportaton charges, and they can splt costs/benefts arbtrarly). To avod trval stuatons, we assume that c<w w + <r+. One of the key assumptons we make s that demand for each retaler s exogenously determned. The consequence of ths assumpton s that addng a retaler to the channel ncreases the total demand that the channel faces and does not cannbalze demand for the other retalers. Naturally, ths assumpton drves some of our results. A dfferent approach mght be to nclude the effect of cannbalzaton: Whereas addng a retaler to the channel should ncrease total demand, demand for each specfc retaler mght go down. We beleve, however, that our assumpton s a reasonable one for many of the stuatons we have n mnd: Imagne that there are numerous supply chans of all three types (tradtonal, drop-shppng, and dual), and each retaler belongs to one of them. Then, f a retaler desres to swtch between channels, he wll brng hs customer base wthout affectng demand for other retalers ether n the channel that he leaves or n the channel that he jons. Hence, when we analyze the mpact of the number of retalers on a channel s profts, the reader should not thnk of an addton of a retaler as a new entry nto the market; t s merely a shft from one channel to another. Ths assumpton s especally approprate f retalers are very small compared to the total market sze. In our model T there s no explct justfcaton for the exstence of the wholesaler, just as there s no justfcaton for the exstence of the retaler n model D,.e., t s not clear why the retaler or the wholesaler gets a cut of the channel s proft because the only functon each performs s the collecton of the proft margn. In practce, there are many other reasons for the exstence of the retalers and the wholesaler n a partcular channel that we chose to gnore to keep our focus on nventory rsk. Some reasons are access to a unque customer base, aggregaton of dfferent product lnes, contractual oblgatons, etc. (see Randall et al. 22) The Tradtonal hannel In the tradtonal channel, the retalers and the wholesaler s profts are T = remn d q T w + t I q T T = w c nq T Retalers make stockng decsons ndvdually, each maxmzng ts expected proft, leadng to the wellknown newsvendor optmalty condton Pr d < q T = r w t I /r The Drop-Shppng hannel In the drop-shppng channel there s an ssue of allocatng nventory among retalers. Varous allocaton mechansms are possble (achon and Larvere 1999 and Anupnd et al. 21 focus on ths ssue). We assume that the allocaton rule s effcent (.e., nventory s always allocated f there s demand for t) and ncreasng (.e., a retaler never receves less by requestng more). achon and Larvere (1999) provde extensve dscusson of and the motvaton behnd these assumptons. Let A q Q d be the nventory allocaton to the retaler through dropshppng. Then the expected profts of an arbtrary retaler and the wholesaler n the drop-shppng channel can be expressed as D = r + w EA q Q d D = w + E mn D Q D cq D In the drop-shppng channel, the nventory decson s made by the wholesaler, maxmzng her expected proft and leadng to the newsvendor optmalty condton wth respect to total demand and a set of cost/revenue parameters that s dfferent from those of the tradtonal channel Pr D < QD = w + c / w The Dual hannel: ombnng Drop-Shppng and Stockng Decsons A company does not necessarly have to choose between tradtonal and drop-shppng channels. The two pure strateges can also be combned to reap the benefts of both channels. Under such an arrangement, the retaler would stock a certan amount of nventory nternally and use drop-shppng only n cases where demand could not be completely satsfed from hs own nventory. In what follows, we begn by formulatng the game and ntroducng an approprate nventory allocaton rule. We then demonstrate the

6 Management Scence 52(6), pp , 26 INFORMS 849 exstence of equlbrum n ths game and analyze the best-response functons. Ths analyss reveals that an equlbrum soluton can be nteror or boundary. We then conduct a parametrc senstvty analyss of the nteror equlbrum soluton wth respect to problem parameters. Ths s done both to gan nsghts nto the dstrbuton of nventory between the retal and the wholesale echelons and to obtan ntermedate results that help to demonstrate the unqueness of the symmetrc equlbrum n ths game, as shown n the last result n ths secton. Each retaler ndependently and smultaneously sets the stockng level q by purchasng nventory from the wholesaler at unt prce w before the uncertanty n demand s resolved. Once demand s known, each retaler requests drop-shppng at a unt prce w + for any demand n excess of q. In 4.2 we brefly address the stuaton n whch retalers decsons are coordnated. Note also that a varant of the problem n whch the decsons of all retalers and the wholesaler are coordnated has been analyzed by Rud and Zheng (1997). We denote by D = d q + the demand for drop-shppng consstng of the retalers resdual demands. Further, we defne L q Q d = d q + A q Q d, whch s the amount of nventory that the retaler requests but does not get due to nventory shortages. Ths defnton wll be useful shortly n order to show the exstence of the equlbrum. The objectve functons of the players can be expressed as = remn d q + r + w E d q + EL q Q d w + t I q = w c q + w + E mn D Q cq In our stuaton the presence of nventory allocaton adds sgnfcant complexty to the analyss because the allocaton has to be done based on retalers resdual demands rather than on retalers ntal order quanttes. (Extant lterature on nventory allocaton rules s restrcted to the latter stuaton; see achon and Larvere 1999.) To guarantee the exstence of a Nash equlbrum n ths game, we must mpose addtonal restrctons on the allocaton rule as follows. Proposton 1. If L q Q d s convex n q, = 1 n, then the best responses of all players are unque, and hence the set of Nash equlbra s nonempty. Further, f L q Q d s symmetrc across retalers, the subset of symmetrc Nash equlbra s nonempty as well. Notce that assumng convexty of L q Q d for results of Proposton 1 to hold s less restrctve than assumng concavty of A q Q d, whch would also be suffcent for the exstence of the equlbrum. We verfed that the result of Proposton 1 holds for at least two allocaton rules commonly used n the lterature: relaxed lnear and proportonal, whch are defned as follows, respectvely, (see achon and Larvere 1999): L Ln q Q d = D Q + n ( L Prop q Q d = d q + 1 Q ) + D orollary 1. For the relaxed lnear and proportonal allocatons defned above, the set of Nash equlbra s nonempty, and the subset of symmetrc Nash equlbra s nonempty as well. To obtan further nsghts, for the remander of the paper we focus on the relaxed lnear allocaton rule as defned above because of ts analytcal tractablty (see 4.4 for dscusson of other allocaton rules). The downsde of ths assumpton s that t s an approxmaton of the lnear allocaton rule (as reflected by the word relaxed ). Namely, the relaxed lnear allocaton rule may result n negatve nventory allocatons to retalers, especally when the wholesaler holds small amounts of nventory for drop-shppng. Nevertheless, achon and Larvere (1999) show that n ther scenaros the relaxed lnear allocaton s a remarkably good approxmaton. In most practcal stuatons the wholesaler serves many retalers and stocks sgnfcant amounts of nventory, whch suggests that ths approxmaton wll not affect our results sgnfcantly. Under the relaxed lnear allocaton rule, the retaler s and the wholesaler s expected profts are: = remn d q + r + w (E d q + E ) D Q + n w + t I q (1) = w c q + w+ Emn D Q cq (2) Next, we establsh monotoncty propertes of the best-response functons as well as obtan optmalty condtons. Proposton 2. (1) The game s submodular, so the best-response functons q Br q 1 q 1 q +1 q n Q, and Q Br q are decreasng. (2) Each equlbrum pont s found from the followng set of optmalty condtons: Pr d <q + r + w ( Pr d r >q Pr D>Q d >q n ) = r w t I (3) r Pr D<Q = w + c w + (4)

7 85 Management Scence 52(6), pp , 26 INFORMS Proposton 2 ndcates that, n the dual channel, an ncrease n stockng quantty by any player results n a decrease n stockng quantty of any other player, so that ths becomes a game of strategc substtutes. In Lemmas 1 and 2 stated n the appendx, we proceed by carefully analyzng the best-response functons employed by the players. The best-response functons are mplctly defned by the players frst-order condtons, and hence cannot be found n a closed form. Nevertheless, we are able to dentfy several useful structural propertes of the best responses. Note that pure channels always result n symmetrc solutons (n terms of both nventores and profts). Hence, to enable proper comparson of all three channels, we focus on the symmetrc equlbra n the dual channel as well (see 4.5 for dscusson of ths assumpton). Man results of Lemmas 1 and 2 can be summarzed as follows: As a best response, the wholesaler (retalers) may stock nothng f the retalers (wholesaler s) stockng quantty s large enough. These results mply that there mght be boundary equlbra n the game n whch one of the channel members does not stock nventory. We now formally state condtons that are necessary and suffcent for two boundary equlbra to arse and dscuss the ntuton behnd them. (These propostons follow from Lemmas 1 and 2.) Proposton 3. Defne Q = Q max = { Q Pr D < Q = w + c w + { Q Pr D > Q = mn } [ ]} n + ti + r + w 1 If Q max Q, then there exsts one and only one boundary equlbrum n whch the retalers stocks nothng, and the channel members earn the same profts as under the pure drop-shppng strategy. Furthermore, f n + t I / r + w 1, then ths condton s equvalent to n + t I / r + w > c/ w +. We observe that ths boundary equlbrum arses when the tradtonal channel has dsadvantages compared to the drop-shppng channel: (a) when cost/ revenue parameters result n a relatvely hgh margn for the wholesaler and a relatvely low margn for the retaler (e.g., when w s large or when c or r are small), (b) when there are sgnfcant benefts to rsk poolng (.e., when n s large), and (c) when the transportaton cost structure favors the drop-shppng channel (e.g., when t I or are large). (These results are smple to verfy.) As a result, the dual channel converts nto the pure drop-shppng channel. Proposton 4. Defne q = q max = { q Pr d < q = r w nt I + n 1 r + n 1 w + { q Pr d 1 <q d n <q = w + c w + } } If q q max, then there exsts one and only one boundary equlbrum n whch the wholesaler stocks nothng, and the channel members earn the same profts as under the pure tradtonal strategy. Furthermore, f d s are ndependent random varables, then ths condton s equvalent to n w + c r w nt I + n 1 w + r + n 1 w + Notce the usefulness of the condton q q max : It allows us to characterze stuatons n whch the wholesaler s ablty to drop-shp does not add value because the wholesaler, as a best response, wll not stock anythng anyway. Because drop-shppng mght requre sgnfcant nvestment nto sngle-unt pckng and packng capabltes, t s mportant for the wholesaler to understand what condtons justfy such an nvestment. Proposton 4 gves us some ntutve deas about when the dual strategy s not benefcal: (a) when cost/revenue parameters result n a relatvely hgh margn for the retaler and a relatvely low margn for the wholesaler (e.g., when w s small or when c or r are large), (b) when there s lttle beneft to rsk poolng (.e., when n s small), and (c) when the transportaton cost structure favors the tradtonal channel (e.g., when t I or are small). (These results are smple to verfy.) An easy way to check Proposton4stoletn = 1 so that the nequalty that must be satsfed smply becomes w + c / w + r w t I /r. If ths nequalty does not hold, then t wll not hold for any n>1. 1 Results of Propostons 3 and 4 are best llustrated graphcally. Three stuatons are possble (wth parameters defned n 3 unless noted otherwse). In Fgure 1 n = 1 t I = 6, there s a boundary equlbrum n whch the retalers stock nothng, whch happens when Q max Q accordng to Proposton 3. In ths case, the dual channel n the equlbrum becomes a pure drop-shppng channel. In Fgure 2 (n = 1, t I = ), there s an nteror equlbrum n whch the retalers and the wholesaler stock nonzero quanttes. Such a stuaton arses when Q max >Q and q <q max. In Fgure 3 n = 1 t I = 3, there s a dfferent boundary equlbrum n whch the wholesaler stocks nothng, whch occurs f q q max accordng to Proposton 4. In ths case the dual channel n the equlbrum becomes a pure tradtonal channel. Because we are mostly nterested n stuatons when both retalers and the wholesaler stock nventory, for the remander of ths secton we focus on the nteror equlbra by assumng that problem parameters are such that q <q max and Q max >Q. 1 It should be noted that the relaxed lnear allocaton rule results n underestmaton of q for n>1. Hence, f we were to use the exact lnear allocaton rule, the boundary equlbrum would occur for a smaller range of problem parameters than mpled by Proposton 4.

8 Management Scence 52(6), pp , 26 INFORMS 851 Fgure 1 Q max Q Fgure 3 q q max Q Q Equlbrum Q Q Q max 1 * q (Q ) 8 * Q (q ) 8 Equlbrum * q (Q ) 4 2 * Q (q ) q 1 q max q q 1 q max q q Before we can establsh the unqueness of the symmetrc equlbrum n the game, we obtan a couple of other results. These results concern the monotoncty of equlbrum nventores n w,, and t I (.e., ordnal reactons to changes n w,, and t I. Moreover, we are able to compare relatve changes n Q and q as a result of change n w,, and t I or cardnal reactons to changes n w,, and t I. Proposton 5. (1) onsder a collecton of games parametrzed by w. The set of symmetrc equlbra s monotone n w,.e., Q w ncreases n w, whle q w decreases n w and for n 2: q / w Q / w. (2) The set of symmetrc nteror equlbra s monotone n,.e., Q ncreases n, whle q decreases n and for n 2: 1/n q / / Q / 1. Fgure 2 q <q max and Q max >Q Q Q * Q (q ) q* (Q ) 2 q Equlbrum q q max q (3) The set of symmetrc nteror equlbra s monotone n t I,.e., Q t I ncreases n t I, whle q t I decreases n t I and for n 2: 1/n q / t I / Q / t I 1. As we mght expect, the hgher wholesale prce nduces the retalers to stock less and the wholesaler to stock more because the retal margn goes down and the wholesale margn goes up, resultng n reallocaton of nventory between two echelons. An ncrease n the wholesaler s nventory exceeds the correspondng decrease n each retaler s nventory because the wholesaler stocks the product for multple retalers and enjoys the benefts of rsk poolng. The ntuton for transportaton costs s clear as well: hgher values of (or t I ) make drop-shppng preferable, and hence the wholesaler stocks more whle the retalers stock less. We also see that the ncrease n the wholesaler s nventory s larger than the decrease n nventory for each retaler, although t s also smaller than a decrease of nventory by all retalers taken together. It s rather straghtforward to verfy that the approach we use to show parametrc monotoncty n w,, and t I fals to work for the drop-shppng markup because the retalers and wholesaler s payoffs fal to have ncreasng dfferences n. The ntuton behnd such nonmonotoncty s as follows. An ncrease n may have two effects: The wholesaler may ncrease her stockng quantty Q because her proft margn ncreases, and the retaler may ncrease hs stockng quantty q because t s becomng more expensve to drop-shp. However, because players stockng decsons are substtutes n the game, there s also a second-order effect: One compettor s decson to ncrease nventory nduces the other player to reduce nventory. Hence, extendng the senstvty analyss to the drop-shppng markup appears dffcult. We

9 852 Management Scence 52(6), pp , 26 INFORMS can stll, however, observe some relatons wth respect to drop-shppng markup. For example, under the argument above, we would not expect an ncrease n the drop-shppng markup to lead to a smultaneous decrease n both players nventores. Moreover, one would thnk that n a majorty of stuatons, the total nventory n the system would ncrease as ncreases due to the above effects. As the next proposton shows, ths ntuton s correct. Proposton 6. (1) onsder a collecton of games parametrzed by. It s never the case that the retalers and the wholesaler s nventores n any symmetrc equlbrum smultaneously go down n. (2) If the retalers nventores go down and the wholesaler s nventory goes up, then q / < Q /. (3) If the retalers nventores go up and the wholesaler s nventory goes down, then q / n> Q /. We are now ready to prove the last result of ths secton: the unqueness of the symmetrc equlbrum. Although t s not mmedately obvous, the by-products of the last two proofs enable the characterzaton of the relatve slopes of the players best-response curves at the equlbrum, whch ultmately leads to the guarantee of a unque symmetrc equlbrum. Proposton 7. There s a unque symmetrc Nash equlbrum (boundary or nteror) n the retalers-wholesaler game. 3. omparson of hannels The goal of ths secton s to gan nsght nto the ssue of channel choce by channel members as well as to see f/when Pareto-optmal channel choces exst. We begn the analyss by notng that the retaler always prefers the dual strategy to the tradtonal strategy because the retaler obtans another safety stock at the wholesaler s locaton at no extra cost. At the same tme, t s not clear whch channel the wholesaler prefers. From a channel perspectve, the dual strategy mght be a reasonable compromse gven addtonal consderatons (e.g., workng captal constrants or lmted fulfllment capabltes). An observaton can also be made regardng the stockng quanttes. learly, under the dual strategy the retaler wll stock less than he would n the tradtonal channel because now the wholesaler s nventory represents a backup opton. On the other hand, the wholesaler wll not stock as much as she would n the drop-shppng channel because the retaler has frst access to customer demand. It follows that the dual strategy s somewhere between the two pure strateges n terms of stockng quanttes for ndvdual players, whch s also reflected n Propostons 3 and 4. In ths secton we present results of extensve numercal experments n order to compare all three channels. For the numercal experments, we have selected the followng problem parameters (unless otherwse noted): n = 1, r = 2, w = 8, c = 3, = 1, = 5, and t I =. 2 We beleve that these parameters represent a reasonable practcal stuaton: The number of retalers s large enough to acheve sgnfcant benefts of rsk poolng and >, meanng that the wholesaler n the drop-shppng channel can shp more economcally due to economes of scale and negotatng power. Demand s Normal, symmetrcal across retalers wth mean = 1, standard devaton = 5, and all coeffcents of correlaton = j =. We employ the Round-Robn Optmzaton algorthm (see Topks 1998, Algorthm 4.3.1), whch guarantees convergence. To evaluate the probablty terms n (3) and (4), we use Monte arlo ntegraton. We always dvde the wholesaler s nventory and proft by the number of retalers so as to brng all graphs to comparable scale. We begn by analyzng the mpact of the number of retalers on channel choce by plottng the optmal stockng levels as a functon of the number of retalers partcpatng n the channel (Fgure 4). In the dual channel, ncreasng the number of retalers results n nventory reallocaton from retalers to the wholesaler due to the benefts of rsk poolng. However, nventores n all three models become rather nsenstve to the number of retalers once n 1, because most benefts of rsk poolng are already acheved. As expected, the retalers and the wholesaler stock less n the dual model than n the tradtonal and dropshppng models, respectvely. Lookng at the retalers and the wholesaler s profts (Fgures 5 and 6), we see how each addtonal retaler n the drop-shppng channel (almost always) benefts everyone: Not only does the wholesaler s total proft go up, but the proft per retaler goes up as well, and, moreover, as new retalers jon the dual channel each retaler lowers nventory whle ncreasng profts. The only excepton s the wholesaler s proft n the dual channel, whereby ncreasng the number of retalers from one to two dmnshes the wholesaler s proft. Ths effect s the drect outcome of the retal competton for the wholesaler s nventory: When the number of retalers exceeds one, retalers greatly decrease the amount of nventory stocked locally and aggressvely rely on drop-shppng. After the number of retalers reaches 5 to 1, addng retalers has a dmnshng mpact on profts. It s also clear that a larger number of retalers makes drop-shppng and dual channels sgnfcantly more attractve than the tradtonal channel. 2 The mpact of the nbound transportaton cost n the tradton channel t I s qute straghtforward, so we let t I =.

10 Management Scence 52(6), pp , 26 INFORMS 853 Fgure 4 hannel Inventores Fgure 6 Wholesaler s Profts Π T 1 Q D /n q T Inventory Q /n q Wholesaler s proft Π Π D We note also that the dfference between the dropshppng and the dual channels s qute small for both the retalers and the wholesaler, so the number of retalers s not a major drvng force n the channel choce (because n most practcal stuatons n s rather large). Next, we nvestgate the mpact of wholesale prce on channel choce. In the drop-shppng channel the wholesaler s proft and nventory ncrease n w and the retalers proft decreases. Larvere and Porteus (21) have demonstrated that n the tradtonal channel the wholesaler s proft s unmodal n w, whle the retaler s proft and nventory decrease n w. Fnally, for the dual channel some characterzatons were gven n Proposton 5 (that the wholesaler s nventory s monotoncally ncreasng and the retaler s nventory s monotoncally decreasng n w). Fgures 7 n to 9 llustrate these fndngs. Once agan, the tradtonal channel does not appear to be a good alternatve for channel members: Drop-shppng and dual channels domnate the tradtonal channel for most values of w. Remarkably, the wholesaler s (as well as the retalers ) profts are essentally the same under the dual and drop-shppng strateges for any gven wholesale prce: Hence, keepng everythng else fxed, wholesale prce s not a major drvng force n channel choce ether. We now analyze the mpact of drop-shppng markup on channel choce. Numercal experments ndcate (Fgure 1) that nventores n the dual channel are qute senstve to changes n. As dropshppng becomes more expensve, the retalers and the wholesaler, respectvely, ncrease or decreases nventory. Ths outcome ndcates that the drect n Fgure 5 Retaler s Profts Fgure 7 hannel Inventores 1,15 1,1 15 q T Q D /n Retaler s proft 1,5 1, 95 9 π D π Inventory 1 5 q Q /n 85 π T n w

11 854 Management Scence 52(6), pp , 26 INFORMS Fgure 8 1,6 Retaler s Proft Fgure 1hannel Inventores 12 1,4 11 1,2 π 1 9 q T Q D /n Retaler s proft 1, π T π D w Inventory Q /n 2 δ q effect for the retaler (recall the dscusson precedng Proposton 6) domnates: Although seemngly hgher should lead to hgher wholesale nventory, ths does not happen because retalers shft away from dropshppng towards stockng nternally. As a result, the drop-shppng markup has a drastcally dfferent effect on nventory n the two pure channels and n the dual channel. Namely, the wholesaler s nventory n the dual channel decreases whle her nventory n the drop-shppng channel ncreases. Although ths result for the drop-shppng channel s expected, n the dual channel the opposte happens due to reallocaton of nventory from the wholesaler to retalers. Indeed, n the dual channel, retalers nventory ncreases wth the drop-shppng markup: The retalers ncreasngly rely on ther own nventory as t becomes more expensve to drop-shp. Retalers profts decrease (Fgure 11) and the wholesaler s profts ncrease (Fgure 12) as the drop- shppng markup ncreases. Hence, even though n the dual channel retalers try to compensate for hgher markup by stockng more locally, they stll earn less as a result. The opposte s true for the wholesaler: Because the retalers stock more nternally for hgher, the wholesaler makes more money by sellng the product to retalers than she loses due to reduced relance on drop-shppng. To ths end, we note that whle changng n and w does not result n sgnfcant proft dfferences n the drop-shppng and dual channels, we can clearly observe that dfferent values of can result n these two channels dfferng sgnfcantly n terms of profts. We explore ths ssue further at the end of ths secton. Fnally, we nvestgate the mpact of dfferences n transportaton costs. Recall that lower makes the tradtonal channel more effcent, and hgher favors the drop-shppng channel. Whle does not affect stockng quanttes n pure channels, hgher n the Fgure 9 Wholesaler s Profts Fgure 11 Retaler s Proft 1,6 1,2 1,4 Π 1,15 Wholesaler s proft 1,2 1, Π D Π T Retaler s proft 1,1 1,5 1, π D π T π w δ

12 Management Scence 52(6), pp , 26 INFORMS 855 Fgure 12 Wholesaler s Proft Fgure 14 Retaler s Proft 85 1,2 π 8 1,1 75 1, Wholesaler s proft 7 65 Π D Π Retaler s proft π T π D Π T δ τ dual channel leads to ncreasng relance on dropshppng, as we proved n Proposton 5. Fgures 13 to 15 demonstrate these fndngs. In terms of retalers and wholesaler s profts, we see that the transportaton cost s another major dfferentator that sgnfcantly affects the comparson of the drop-shppng and dual channels. As we expect, hgher values of make the drop-shppng and the dual channels preferable for retalers. Surprsngly, the wholesaler s proft n the dual channel s nonmonotone n, although t s mostly decreasng. We explan as follows: For very low values of the wholesaler benefts, because retalers purchase a lot of nventory through the tradtonal channel. As ncreases, the wholesaler actually loses money as she gans n transportaton economes, a nonntutve result that s drven by the fact that retalers stock less and less nternally and rely more on drop-shppng, whch s less proftable to the wholesaler. However, when s very hgh, then retalers stock almost nothng and purchase a lot through drop-shppng, whch s, agan, very proftable for the wholesaler. The precedng analyss sheds some lght on the ssue of channel choce. For convenence, we summarze all our analytcal fndngs as well as conjectures based on numercal experments n Table 1. We do not nclude senstvty analyss wth respect to mean demand and demand uncertanty because ts results are obvous: Hgher mean demand (lower demand uncertanty) benefts all channel members n all models. Although many parameters affect channel choce, the precedng analyss also ndcates that the man drvng forces n channel choce are the drop-shppng markup and the transportaton cost dfferental. Fgure 13 hannel Inventores Fgure 15 Wholesaler s Proft Π 1 q T Q D /n 58 Π T Inventory q Wholesaler s proft Q /n 54 Π D τ τ

13 856 Management Scence 52(6), pp , 26 INFORMS Table 1 The Summary of Parametrc Analyss Fgure 17 Wholesaler s hannel hoce Q D D D q T T T q Q n ± w + + ± ± τ Dual channel δ Because t s hard to envson a complete pcture of channel choce by varyng just one parameter at a tme, n Fgures 16 and 17 we plot the preferred channels for the retalers and the wholesaler, respectvely, as functons of these two parameters, and n Fgure 18 we plot Pareto-optmal choces. In Fgure 16 we see that the channel choce for retalers s qute straghtforward: Retalers prefer the drop-shppng channel when the tradtonal channel has a transportaton cost dsadvantage ( s hgh) whle drop-shppng s not too expensve ( s small). Otherwse, retalers prefer the dual channel. The choce for the wholesaler (Fgure 17) s more complex: Any of the three channels may be preferable. When the drop-shppng markup s small and the tradtonal channel s transportaton cost dsadvantage s relatvely hgh, the pure tradtonal channel s the best opton for the wholesaler, because offerng drop-shppng at ths low value of the dropshppng markup s not proftable; f the wholesaler were to offer drop-shppng, retalers would stock lttle nternally, thus resultng n low proft for the wholesaler. However, as transportaton costs under drop-shppng ncrease ( decreases), the wholesaler (surprsngly) swtches from the pure tradtonal to the dual channel. The reason for ths counterntutve effect s that when transportaton costs n the Tradtonal channel Dual channel Drop-shppng channel drop-shppng channel are hgh, the wholesaler stocks very lttle nventory, and thus retalers stock a lot nternally. Therefore, offerng drop-shppng at low values of does not cannbalze very proftable sales through the tradtonal channel whle addng to the wholesaler s proft. When the drop-shppng markup s hgh, the wholesaler prefers ether the dual or the drop-shppng channel. Interestngly, the dual channel s preferred when the transportaton cost dfference s ether very low or very hgh. Ths behavor s drectly related to the ntuton underlyng Fgure 15, n whch the wholesaler s proft n the dual channel s hghest at the extremes of the transportaton cost dfference parameter. It s also clear that there are lmted opportuntes for Pareto-optmal choces (Fgure 18). Frst, the Fgure 16 Retalers hannel hoce τ 2 Drop-shppng channel δ 2. Fgure τ Pareto-Optmal hannel Drop-shppng channel δ 4 5 Dual channel Dual channel

14 Management Scence 52(6), pp , 26 INFORMS 857 dual channel can be preferred by both partes for negatve values of, somethng that we mght see when the retaler s very large (e.g., Amazon.com) and therefore has better deals on transportaton prces than the wholesaler does. Second, the drop-shppng channel can be preferred by both partes for relatvely large values of and postve, somethng we mght observe when the wholesaler can transport for a lower prce than the retalers can. For many problem parameters, however, the wholesaler and the retalers have dfferent preferences. 4. Dscusson of Modelng Assumptons and Summary 4.1. Endogenous Wholesale Prce and Drop-Shppng Markup One of our assumptons s that the wholesale prce w and the drop-shppng markup are exogenous, although n some cases a powerful wholesaler may be able to set both of these prces. In the tradtonal channel ths problem has been solved by Larvere and Porteus (21). Even n ths relatvely smple problem, addtonal condtons are needed to ensure the unqueness of the soluton. In the drop-shppng channel, the wholesaler would set w + = r + and extract all profts from the channel. A smlar problem arses n the dual channel: The wholesaler has the opportunty to set w, to the hghest possble values and extract most profts (as s evdent from Fgures 9 and 12). learly, ths outcome s not reasonable n the eyes of the retaler. achon (22) consders stuatons n whch the wholesaler does not have complete barganng power by usng the noton of a Pareto-mprovng set of contracts. Another possble approach commonly used n the lterature s to ntroduce the partcpaton constrant on the retaler s (expected) reservaton proft. The dual channel problem s then formulated as follows: max w c r t I t D r+ w w s.t. w U where w and w are profts resultng from the equlbrum n the nventory game and U s the retaler s reservaton proft. We see that the result s a very complex Stackelberg game n whch the wholesaler frst sets prces, and then the Nash nventory game ensues. Below we demonstrate through numercal experments the nsghts resultng from ths game (Fgures 19 to 21). When the reservaton proft constrant s absent, the wholesaler establshes a hgh prce w = r t I (n ths Fgure Optmal Prces Wholesale prce Drop-shppng markup U ,12 1,4 case w = 13 68) and the hghest possble markup = r + w (n ths case = 5 82) so that retalers make no money on drop-shppng. The hgh w makes retalers stock lttle and rely extensvely on drop-shppng. Interestngly, the wholesaler does not fnd t advantageous to set the maxmum possble w, but rather sets the maxmum possble. The ntuton behnd ths decson comes from nventory rsk. If the wholesaler were to set a very hgh w, retalers would not stock anythng. Thus, most of the wholesaler s proft would come from drop-shppng, whch means takng a sgnfcant nventory rsk. As U ncreases, the wholesaler decreases w whle keepng at the hghest possble value (so that w + = constant). In response, retalers ncreasngly shft towards stockng nventory and relyng less on drop-shppng (Fgure 2). Fnally, Fg- Fgure 2Stockng Quanttes U ,12 1,4 q Q /n

15 858 Management Scence 52(6), pp , 26 INFORMS Fgure Π /n Wholesaler s Proft U , 1,2 1,4 ure 21 ndcates that the wholesaler s proft decreases n U, as we would expect. As U ncreases, the wholesaler must reduce her proft by ncreasngly shftng away from drop-shppng. Although an outcome of ths model (namely, that retalers do not make any money on drop-shppng) s not practcal, t s useful to be aware of the wholesaler s ncentves to nflate the drop-shppng markup. In practce, there are transacton costs assocated wth forwardng orders to the wholesaler as well as wth handlng subsequent customer nqures. Therefore, we mght expect that the retaler wll be unwllng to partcpate n the drop-shppng arrangement unless a certan per-transacton proft s ensured oordnated Retalers We have so far focused on the varant of the problem n whch all retalers act ndependently and maxmze ther ndvdual profts. In practce, however, there are stuatons n whch retalers stockng decsons are coordnated because, for nstance, retalers belong to the same chan of stores and the parent company mposes centralzed controls over ndvdual stores nventory polces. When retalers are centrally managed, the objectve functon of the retal frm becomes the sum of ndvdual retalers objectve functons. Although the resultng optmalty condtons are somewhat dfferent, we have verfed that all the equlbrum propertes derved n Propostons 3 to 7 contnue to hold when retalers stockng decsons are centrally managed and the attenton s restrcted to the symmetrc equlbrum. The man dfference s summarzed n the next proposton. Proposton 8. Suppose the relaxed lnear allocaton rule s used. Then, n a symmetrc nteror equlbrum, under the coordnaton of the retalers stockng decsons, the retalers (the wholesaler) stock more (stocks less) than when retalers stockng decsons are not coordnated. The last result can be explaned as follows: When retalers are not coordnated, they compete for the nventory allocaton through drop-shppng. Retalers stockng quanttes are substtutes n ths game: If one retaler decdes to ncrease hs nventory, t wll reduce ths retaler s nventory allocaton and ncrease other retalers nventory allocatons. Hence, other retalers wll stock less and rely more on drop-shppng. As has been shown for other games n whch retalers stockng decsons are substtutes, competng retalers tend to ncrease ther share of the scarce resource (allocated nventory n ths case) by stockng less and relyng more on drop-shppng (see Netessne and Rud 23). The wholesaler s stockng quantty decreases n the retalers stockng quanttes, so the wholesaler responds by stockng more when retalers are not coordnated The Impact of Fxed osts Throughout ths paper, we do not take nto account fxed costs, whch can be qute substantal. We let k T, k D, and k denote the fxed costs of the three channels to the retaler and let K T, K D, and K denote the fxed costs of the three channels to the wholesaler. In the tradtonal channel, k T wll typcally nclude larger costs than k D for physcal fulfllment capabltes (such as warehouses, shppng/packng equpment, and personnel). k D mght have a larger component than k T n terms of expenses assocated wth the nformaton technology that s needed to support order placement/trackng and communcaton wth the wholesaler. Hence, the relatve sze of k T and k D s stuaton dependent. On the other hand, n the tradtonal channel, K T wll typcally nclude smaller costs than K D for physcal fulfllment capabltes as well as for nformaton technology. Hence, t s reasonable to assume that K T K D. In the dual channel, both the retalers and the wholesaler need to spend money to support both channels (.e., everyone must be able to do everythng ), so t can be argued that k > max k T k D and K > max K T K D. From the above dscusson t s clear that, n most reasonable stuatons, larger fxed costs wll make the dual channel less attractve for channel members relatve to pure channels. We note that fxed costs do not have an mpact on order quantty decsons, but when comparng proft expressons n the three channels, the approprate fxed costs must be subtracted from the expressons for varable profts. Overall, fxed cost consderatons should augment our analyss of varable costs n the prevous sectons. However, one must be careful to remember that our analyss was conducted for a sngle product, whle n practce fxed costs wll be ncurred for multple products. Hence,

16 Management Scence 52(6), pp , 26 INFORMS 859 the varety of products that the retaler/wholesaler offers n dfferent channels wll have an mpact on the ultmate proftablty (see Randall et al. 26) The Impact of the Allocaton Rule hoce achon and Larvere (1999) demonstrate that dfferent allocaton rules result n dfferent ncentves n supply chans and therefore lead to dfferent equlbrum nventores. To understand how dfferent allocaton rules mght alter our results, we conducted numercal experments comparng nventory decsons and profts under three allocaton rules: the relaxed lnear (Ln), the lexcographc (Lex), and the unform (Un). Under the assumpton that there are only two retalers n the channel, we are able to obtan analytcal expressons for optmalty condtons for each of these allocaton rules, and we use them to calculate the equlbra of the game. For the lexcographc allocaton rule we assume that each retaler has a 5% chance of obtanng access to the wholesaler s nventory frst and, thus, a 5% chance of obtanng access to the wholesaler s nventory second. Under the unform allocaton rule each retaler s guaranteed access to 5% of the wholesaler s nventory plus any of the other 5% of nventory unused by the other retaler. Note that the unform allocaton rule favors retalers wth small resdual demands (see achon and Larvere 1999), whch s the opposte of the relaxed lnear allocaton rule, whch favors retalers wth hgh resdual demands. The lexcographc allocaton rule s purely random and, n ths sense, falls n-between the other two allocaton rules. Thus, by consderng these three allocaton rules we cover a varety of ncentves that dfferent allocaton rules mght create. In all numercal experments wth the above three allocaton rules, we observe that q Ln q Lex qun and Q Ln QLex QUn, thus confrmng our ntuton. For the vast majorty of the problem parameters we expermented wth, gaps between nventores were rather small and rarely exceeded 1%. Gaps between profts were even smaller, rarely exceedng 3%. The typcal rank orderng of profts was the opposte of the rank orderng of nventores, namely, Ln Lex Un and Ln Lex Un. More mportantly, n all of our experments we observe that nsghts from Propostons 1 7 as well as nsghts from numercal experments usng the relaxed lnear allocaton rule contnue to hold. Thus, we conclude that our fndngs are generally robust to the allocaton rule specfcaton The Impact of the Symmetry Assumpton In the analyss so far we have focused on studyng symmetrc equlbra, although we are unable to prove that only symmetrc equlbra can exst n ths game. In our extensve numercal experments wth symmetrc problem parameters, we never encountered asymmetrc equlbra, and therefore we conjecture that asymmetrc equlbra are unlkely to exst n a problem wth symmetrc parameters. A related queston s whether our nsghts contnue to hold when problem parameters are asymmetrc across retalers, and therefore only asymmetrc equlbra exst. To address ths ssue we conducted extensve numercal experments wth asymmetrc problem parameters and two retalers. We ntroduced asymmetry through revenue parameters r as well as through demand parameters and compared results wth the base case n whch r = r 1 + r 2 /2 and = /2 for both retalers. In these experments we dd not encounter multple equlbra. Overall, we found that, as expected, the retaler wth the hgher retal prce or hgher mean demand stocks more nventory, and the mpact of change n mean demand s stronger than the mpact of a (smlar n relatve terms) change n the retal prce. The reason mght be that changes n mean demand affect both each retaler s nventory and hs nventory allocaton by the wholesaler. However, the total retal nventory and nventory held by the wholesaler are largely unaffected by the presence of two asymmetrc retalers. Moreover, we observed that all of our nsghts contnue to apply to the asymmetrc problem as well. Therefore, we beleve that our results are representatve of results for asymmetrc problem parameters as well Summary and Dscusson The practce of drop-shppng, whch has become very popular among Internet retalers, s rather new to the operatons lterature. In ths paper, we further advance the study of drop-shppng by modelng a supply chan wth multple retalers. In addton to the models for the tradtonal and drop-shppng supply chans, we also present a dual strategy that combnes the two channel alternatves: The retaler stocks some nventory nternally and drop-shps whenever local nventory s depleted. Ths s a compettve model, because the retalers and the wholesaler make ther decsons ndependently. We compare all three channel alternatves and fnd that the drop-shppng and dual channels can be vable choces for both the retaler and the wholesaler, and that both of these channels can be Pareto optmal (whereas the dual channel s always preferable to the tradtonal channel for retalers). In summary, we show that the dual channel often offers advantages to both retalers and the wholesaler, the fndng that s n lne wth the predcton of ndustry analysts (see Jupter ommuncatons 2) that n the near future most Internet retalers wll operate usng a combnaton of nternal nventory and drop-shppng. By comparng the supply chan alternatves on the Internet, we provde practcng managers wth specfc gudelnes for

17 86 Management Scence 52(6), pp , 26 INFORMS choosng an approprate channel structure for Internet retalng. Our fndngs are n lne wth the emprcal results of Randall et al. (26), who found that no sngle channel opton s unformly preferred: The choce of channel depends on the characterstcs of the compettve envronment. The man focus of ths paper s on economc nsghts nto manageral decsons of channel choce, so, as wth any economc modelng, n specfyng the model we face the challenge of capturng the relevant manageral trade-offs wthout dffusng nsghts. Hence, we make several smplfyng assumptons. Frst and foremost, we focus on the operatonal sde of busness, namely the allocaton of the rsk assocated wth carryng nventory and the related effects of rsk poolng. Thus, we gnore the other aspects of channel choce that are extensvely descrbed n the marketng lterature (see an excellent account n oughlan et al. 21). We also abstract away ssues of real-tme nventory allocaton and the related problem of nventory ratonng. Thus, our model s meant to ad manageral understandng of nventory ownershp ssues under drop-shppng and should be used wth these and other lmtatons n mnd (see Randall et al. 22 for other practcal consderatons that are hard to ncorporate nto a formal analytcal model). Moreover, both the present work and a paper by Netessne and Rud (24) consder sngle-product stuatons. An mportant open queston s, should the same channel strategy be used for each of the multple products wth dfferent cost/revenue/demand characterstcs? We have already found that demand varablty nfluences the decson to select a partcular channel, and hence our model provdes a framework to classfy dfferent products nto approprate channel structures. Another major area for future research s prcng decsons, the drecton pursued n Byalogorsky and Koengsberg (24) for a sngle product/retaler. Acknowledgments The authors thank Gérard achon for enlghtenng dscussons. Edtors and two anonymous referees provded helpful suggestons that mproved the paper. The authors gratefully acknowledge fnancal support from the Wharton E-busness Intatve. Appendx. Proofs Proof of Proposton 1. It s straghtforward to verfy that the retalers objectve functons are concave n ther respectve decson varables, whch s suffcent to prove the exstence of pure-strategy equlbra. Furthermore, the symmetry of the problem mples that at least one equlbrum s symmetrc. Proof of orollary 1. For the relaxed lnear allocaton we note that D = j d j q j + s convex n q, = 1 n. Furthermore, + s a convex ncreasng functon. The ncreasng convex functon of a convex functon s tself convex. Hence, D Q + /n s convex n q, = 1 n. For the proportonal allocaton, we note that L Prop q Q d s a multplcaton of two functons. It can be shown that a multplcaton of two postve, decreasng, convex functons s tself convex. Because d q + s convex decreasng, t suffces to show that 1 Q / D + s convex decreasng. We note that 1 Q / D = 1 + Q / j d j q j + d q + s convex decreasng n q on d and constant on d, mplyng that 1 Q / D s convex decreasng. Fnally, 1 Q / D + s convex as a convex ncreasng transformaton of a convex functon. Proof of Proposton 2. Frst dervatves are: q ( = r Pr d >q r + w Pr d >q Pr ) D>Q d >q w + t n I = w + Pr D>Q Q c The resultng optmalty condtons follow. To demonstrate the submodularty of the objectve functons, t s suffcent to show that the second-order cross-partal dervatves are negatve (see Topks 1998): 2 = r + w f D d q q >q d j >q j Q j Pr d >q d j >q j /n < 2 = r + w f D d q Q >q Q Pr d >q /n < 2 = w + f D d Q q >q Q Pr d >q < and Result 1) s verfed. Lemma 1. Defne { Q = Q Pr D < Q = w + c } w + { q max = q Pr d 1 <q d n <q = w + c w + Then the wholesaler s best-response functon Q Br q to a set of symmetrc retalers strateges (.e., the wholesaler responds to the smultaneous change n the stockng quantty by all retalers, q = q possesses the followng propertes: (1) Q Br = Q. (2) On the nterval q q max, Q Br q s defned by optmalty condton (4). It s a decreasng functon wth slope 1 Q Br q / q for n = 1, and n Q Br q / q 1 for n 2. (3) On the nterval q q max, Q Br q =. Proof. Result 1 s straghtforward. To see Result 3, we observe that D = d q + has a probablty dstrbuton wth a mass pont at zero. The probablty of ths mass pont s calculated as follows: ( ) Pr D = = Pr d q + = = Pr d 1 <q d n <q From the wholesaler s optmalty condton (4), we know that as long as Pr d 1 <q d n <q w + c / w +, }

18 Management Scence 52(6), pp , 26 INFORMS 861 or n other words as long as q >q max, the wholesaler wll stock nothng, leavng us wth Result 3. Fnally, to see that Result 2 holds, we note that for q q max and for any n we fnd from mplct dfferentaton that Q Br q = f D d >q Q Pr d >q f D Q nf D Q f D Q = n To prove that Q Br/ q 1 for n 2, t s suffcent to show that f D d >q Q Pr d >q f D Q Ths s done by mathematcal nducton. We frst show that the statement holds for n = 2, then assume that t holds for any n and prove that t wll also hold for n + 1. Frst, for n = 2, 2 f D d >q Q Pr d >q f D Q =1 = f D d 1 >q Q Pr d 1 >q +f D d 2 >q Q Pr d 2 >q f D Q = f D d 1 >q Q Pr d 1 >q + f D d 2 >q Q Pr d 2 >q f D d 1 >q Q Pr d 1 >q f D d 1 <q Q Pr d 1 <q = f D d 2 >q Q Pr d 2 >q f D d 1 <q Q Pr d 1 <q = f D d 1 >q d 2 >q Q Pr d 1 >q d 2 >q + f D d 1 <q d 2 >q Q Pr d 1 <q d 2 >q f D d 1 <q d 2 >q Q Pr d 1 <q d 2 >q f D d 1 <q d 2 <q Q Pr d 1 <q d 2 <q = f D d 1 >q d 2 >q Q Pr d 1 >q d 2 >q The last equalty follows from the fact that f D d 1 <q d 2 <q Q Pr d 1 <q d 2 <q = because Q > on the studed nterval. Ths completes the proof for n = 2. For convenence, we denote D n = n =1 d q +. We now assume that the nequalty n queston holds for some n, that s, n f D n d >q Q Pr d >q f D n Q (5) =1 We need an addtonal lemma that can be proven trvally. Lemma. If (5) s true, then for a random varable d n+1 and any n 2 t s also true that n f D n d >q d n+1 <q Q Pr d >q d n+1 <q =1 f D n d n+1 <q Q Pr d n+1 <q Wth ths lemma n mnd, we are fnally ready to show that for n + 1: n+1 f D n+1 d >q Q Pr d >q f D n+1 Q =1 f D n+1 d >q d n+1 >q Q Pr d >q d n+1 >q n+1 = =1 f D n+1 d >q d n+1 <q Q Pr d >q d n+1 <q n+1 + =1 f D n+1 d n+1 >q Q Pr d n+1 >q f D n+1 d n+1 <q Q Pr d n+1 <q n = f D n+1 d >q d n+1 >q Q Pr d >q d n+1 >q =1 + f D n+1 d n+1 >q Q Pr d n+1 >q n + f D n d >q d n+1 <q Q Pr d >q d n+1 <q =1 f D n+1 d n+1 >q Q Pr d n+1 >q f D n d n+1 <q Q Pr d n+1 <q n = f D n+1 d >q d n+1 >q Q Pr d >q d n+1 >q =1 + n f D n d >q d n+1 <q Q Pr d >q d n+1 <q =1 f D n d n+1 <q Q Pr d n+1 <q n f D n+1 d >q d n+1 >q Q Pr d >q d n+1 >q =1 where the frst nequalty follows from lemma above. Ths completes the proof. Lemma 2. Defne { q = q Pr d < q = r w nt } I + n 1 r + n 1 w + q = {q Pr d < q = t I + f > + t I } { Q max = Q Pr D > Q = mn w + [ ]} n + ti + r + w 1 The retalers (symmetrc) best response to the wholesaler s strategy (mplyng that all retalers smultaneously respond to a change n the stockng quantty of the wholesaler) q Br Q possesses the followng propertes: (1) q Br = q. (2) On the nterval Q Q max, q Br Q s defned by the optmalty condton (3). It s a decreasng functon wth slope 1 q Br Q / Q. (3) If Q max <, then on the nterval Q Q max, q Br Q =. Otherwse, lm Q q Br Q = q. Proof. Results 1 and 3 can be obtaned by substtutng Q = and Q = nto (3) and smplfyng. To obtan Result 2 we note that the slope of the retaler s best-response functon can be bounded usng mplct dfferentaton as follows: q Br Q = r + w f D d >q Q Pr d >q /n [ rf d q + r + w f d q + f d D>Q q Pr D>Q /n + j f D d >q d j >q Q Pr d >q d j >q /n ] 1

19 862 Management Scence 52(6), pp , 26 INFORMS [ f D d >q Q Pr d >q f d D>Q q Pr D>Q + ] 1 f D d >q d j >q Q Pr d >q d j >q j [ f D d >q Q Pr d >q f D d >q d j >q Q j ] 1 Pr d >q d j >q + f D d >q Q Pr d >q 1 j = 1 n Note that the sgn of the slope of the best-response functon s negatve. Ths completes the proof. Proof of Proposton 5. To show results n ths proposton, we construct a fcttous game between one wholesaler and one retaler n whch the payoffs are defned as follows (notce the n 2 coeffcent): = remn d q + r + w (E d q + E ) D Q + w + t I q = w c nq + w + E mn D Q cq It s straghtforward to verfy that the optmalty condtons for ths game concde wth the optmalty condtons for the symmetrc equlbrum n the orgnal game. Hence, the sets of equlbra n these two games concde as well, and any equlbrum propertes that we can show for the fcttous game wll carry over to the symmetrc equlbrum of the orgnal game. If we redefne Q = Q, w = w and consder frst dervatves of the fcttous game, n 2 Thus, the set of equlbra q Q ncreases n w. The rest of the proof reles on the Implct Functon Theorem (IFT). When appled to the optmalty condtons of the fcttous game, the IFT can be stated as follows. At the equlbrum we have F = Pr d >q r + w Pr d >q Pr D>Q d >q /n /r w + t I /r = G = Pr D>Q c/ w + = so that q / w 1 Q / w = F q F Q F w G q G Q G w (6) where subscrpts denote partal dervatves. We can fnd each term analytcally as follows: r + w F q = f d q r ( f d q + f d D>Q q Pr D>Q /n + j f D d >q d j >q Q Pr d >q d j >q /n) / r r + w f D d r >q d j >q Q j Pr d >q d j >q /n < F Q = r + w f D d >q Q Pr d >q /r < G q = f D d >q Q Pr d >q < q = r Pr d >q r + + w ( Pr d >q Pr ) D> Q d >q w + t n I G Q = f D Q < F w = Pr d >q /r Pr D>Q d >q / rn 1/r < G w = c/ w + 2 > = w + Pr D> Q + c Q We can verfy that the fcttous game s supermodular because ts cross-partal dervatves are postve: 2 = r + + w f D d q Q >q Q Pr d >q /n > 2 = w + f D d Q q >q Q Pr d >q > To demonstrate the parametrc monotoncty of the set of equlbrum nventores q w Q w n w, we must show that the retaler s payoff functon has ncreasng dfferences n q w and the wholesaler s payoff functon has ncreasng dfferences n Q w (see Topks 1998, Theorem 4.2.2), whch n our case s equvalent to showng that secondorder cross-partal dervatves are nonnegatve (see Topks 1998). Ths s readly verfed: ( 2 q w = Pr d >q Pr ) D> Q d >q + 1 > n 2 Q w = Pr D> Q > From the IFT, we have the followng equalty: q G q w + G Q Q w = G w Usng the establshed fact that G w > and the result from the proof of Lemma 1 that, for n 2, G Q < G q, we have q / w Q / w < G Q < 1 Ths completes the proof of Result 1. To prove Result 2, we note that F = Pr d >q Pr D>Q d >q /n /r < G = so that q / / Q / = G Q/G q. Recall that 1/n G Q /G q 1 for n 2 and the result follows. Result 3 s proven smlarly. Proof of Proposton 6. Smlar to the proof of Proposton 5, we can apply the IFT to the optmalty condtons of the fcttous game. It s readly verfed that (consstent wth the notaton from the proof of Proposton 5) F > and G q

20 Management Scence 52(6), pp , 26 INFORMS 863 G >. Usng the IFT smlar to the proof of Proposton 5 we obtan ( ) ( ) q F Q q + F Q < G q ( q ) + G Q ( Q ) < Also, from the proof of Lemma 1 we have, for any n, G q < n G Q, and from the proof of Lemma 2 we have F q > F Q. Usng these results, we consder three possbltes: (1) q / Q / <. Ths s clearly mpossble because (7) s not satsfed. (2) q / <, Q / >. Then we have q / / Q / <F Q/F q < 1. (3) q / >, Q / <. Then we have q / / Q / >G Q/G q > 1/n. Ths completes the proof. Proof of Proposton 7. We frst establsh that there s at most one nteror equlbrum. By solvng the matrx equaton (6) and nvokng the result of Proposton 5 we obtan: q w = G wf Q + F w G Q < G q F Q F q G Q Note that G w F Q + F w G Q >, and thus t must be that G q F Q F q G Q < or, smlarly, G q F Q < 1 Q q < 1 G Q F q q Q so that at any nteror equlbrum, ( q Q Q q ) q Q < 1 We defne the nverse of q Q as Q q. Our goal s to show that there s only one q such that Q q = Q q.if we consder an auxlary functon q = Q q Q q, we fnd that provng unqueness of the equlbrum s then dentcal to showng that q crosses zero once and only once (because we are assured of the exstence of the equlbrum, we do know that there s at least one crossng). Ths technque s known as the ndex approach (see Vves 2, p. 48). We then consder the dervatve of q w.r.t. q evaluated at the equlbrum: ( ) ( ) ( ) q Q = q Q q q q q q q q ( ) ( ) 1 Q = q Q q > / Q Q q q That s, every tme q crosses zero, t does so from below. learly, because q s a contnuous functon, there can be only one such crossng. Therefore, there can be at most one nteror equlbrum. Furthermore, we note that, f there s a boundary equlbrum as n Fgure 1, then there could not be another equlbrum because, based on Lemmas 1 and 2, the slopes of the best responses are such that, once Q max Q, the best responses wll never cross. Based on the same argument, f there s a boundary equlbrum as n Fgure 3, then there could not be another equlbrum. Proof of Proposton 8. When the retalers stockng decsons are coordnated and the relaxed lnear allocaton (7) rule s used, the total proft of the retal channel can be wrtten as follows: = r E mn d q ( ) + r + w E d q + E D Q + q w + t I = r E mn d q + r + w E mn D Q w + t I q whle the wholesaler s proft s unchanged. To ths end, we consder a fcttous game parametrzed by k n, wth equlbra found from the followng set of optmalty condtons: = r Pr d q >q ( r + w Pr d >q kpr ) D<Q d >q n w+t I = = w+ Pr D>Q Q c = We note that when k = 1, ths system of equatons concdes wth optmalty condtons for the fcttous game ntroduced n the proof of Proposton 5, and thus ts solutons wll concde wth the symmetrc solutons to the problem wth uncoordnated retalers. At the same tme, when k = n t can be verfed that the soluton to ths system of equatons concdes wth the symmetrc soluton when the retalers stockng decsons are coordnated. We now show that the set of equlbra satsfyng ths system of equatons s monotone n k, whch allows us to compare stockng decsons wth and wthout coordnaton at the retal level. It s readly verfed that ths fcttous game s supermodular n q Q, q k and n Q k. Hence, the set of equlbra q Q ncreases monotoncally n k. The result follows. References Anupnd, R., Y. Bassok entralzaton of stocks: Retalers vs. manufacturers. Management Sc. 45(2) Anupnd, R., Y. Bassok, E. Zemel. 21. A general framework for the study of decentralzed dstrbuton systems. Manufacturng Servce Oper. Management Apparel Industry Magazne. 2. Wll BlueLght attract bg sales n cyberspace? (September) Bernsten, F., A. Federgruen. 23. Prcng and replenshment strateges n a dstrbuton system wth competng retalers. Oper. Res Bernsten, F., F. hen, A. Federgruen. 22. oordnatng supply chans wth smple prcng schemes: The role of vendor managed nventores. Workng paper, Duke Unversty, Durham, N. Byalogorsky, E., O. Koengsberg. 24. Lead tme, uncertanty, and channel decson makng. Workng paper, olumba Unversty, New York. achon, G. P. 21. Stock wars: Inventory competton n a twoechelon supply chan wth multple retalers. Oper. Res