Vol. 1, No. 3, May June 01, pp. 18 33 ISSN 109-1478 EISSN 1937-96 1 103 018 DOI 10.1111/j.1937-96.011.0169.x 011 Producton and Operatons Management Socety Quck Response under Competton Yen-Tng Ln, Al Parlaktürk Kenan-Flagler Busness School, The Unversty of North Carolna at Chapel Hll, Chapel Hll, North Carolna 799, USA, lnyt@emal.unc.edu, pturk@unc.edu W e consder a manufacturer servng two competng retalers that sell ther products over a sngle sellng season. The retalers place ther regular orders before the season starts. In addton to ths ntal order, quck response (QR) provdes a retaler wth an addtonal replenshment opportunty after demand uncertanty s resolved. The manufacturer determnes the unt prce for QR replenshment. We characterze the retalers orderng, and the manufacturer s prcng decsons n equlbrum when none, only one, and both of the retalers have QR ablty. We study how the proftablty of the manufacturer, the retalers, and the channel depend on QR and competton. We fnd t may be optmal for the manufacturer to offer QR to only one of the ex ante dentcal retalers when demand varablty s suffcently, but not overly hgh. The manufacturer may also fnd t optmal to offer QR to both or none of the retalers, dependng on demand varablty. Fnally, whle QR ablty s always attractve for a retaler when competton s gnored, we fnd QR may prove detrmental when ts mpact on competton s taken nto account. Key words: quck response; competton; prcng; supply chan Hstory: Receved: October 009; Accepted: March 011, after revsons. 1. Introducton Quck response (QR) s an operatonal lever that ams to provde better response to varatons n demand. One of ts benefts s to enable n-season replenshment through lead tme reducton. The success of QR has receved much attenton (Hammond and Kelly 1990), and ts benefts have been studed extensvely n lterature (e.g., Fsher and Raman 1996, Iyer and Bergen 1997). Naturally, more and more frms have adopted QR to gan a compettve edge. For example, a burgeonng Brtsh apparel retaler, Prmark, uses QR for faster product turnover, and t fetched 10.1% market share, whle the market leader Marks & Spencer garnered 11.4% market share n the Unted Kngdom n 008 (Vckers 008). Nevertheless, the growng popularty of QR has also ntensfed competton, whch can potentally dmnsh the value of QR. For nstance, after ts domestc success wth quck response, the Japanese retaler Unqlo nvaded the U.S. market n 006 (Alexander 009), and a smlar move s taken by the Brtsh retaler Topshop whose New York flagshp opened n 009 (Resto 010). The effect of competton on QR, however, has receved less attenton and not been fully understood, and t s our man area of focus n ths study. Despte the extensve studes on the benefts of QR for retalers (e.g., Cachon and Swnney 009, Caro and Martínez-de-Albénz 010), there has been less focus on the value of QR to a manufacturer. When should a manufacturer offer QR? What s ts optmal 18 supply chan structure? Should a manufacturer servng competng retalers offer QR? Indeed, t s not uncommon to see a manufacturer servng competng customers. For example, Hot Kss, a Calforna-based manufacturer serves junor fashon retalers Hot Topc and dela s as well as upscale department stores lke Dllard s and Nordstrom (Bhatnagar 006). Hot Kss acheves quck response by takng advantage of local producton n Calforna. Smlarly, Makalot, a leadng Tawanese apparel manufacturer serves Kohl s, Target, JC Penney, and Gap. In addton to regular delveres, Makalot also provdes faster n-season delveres to ts clents, and acheves quck response by flexble capacty allocaton and mproved nformaton sharng wth ts clents. 1 In the footwear ndustry, Yue Yuen, a major sportswear manufacture that provdes a shorter lead tme than ts compettors, supples brand names lke Nke, Puma, and Addas (Taylor 008). In ths study, we model a supply chan wth a sngle manufacturer supplyng homogeneous products to two competng retalers. The retalers sell ther products n a consumer market wth a sngle sellng season. Pror to the sellng season, the manufacturer sets the QR prce for QR replenshment, and then each retaler places a regular (ntal) order at an exogenous wholesale prce. We allow the manufacturer to determne ths prce n secton 6.1. After observng the actual demand, each retaler wth QR ablty places a second order at the QR prce. Fnally, the sellng season starts and the retalers compete n the consumer
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety 19 market, followng Cournot quantty competton. Quantty-based competton s approprate for ndustres wth long supply lead tme (e.g., apparel and footwear); n these ndustres, prce competton s less lkely because t requres nstant adjustment of producton quantty (Feng and Lu 010). We consder three scenaros, wth zero, one, and both retalers havng QR ablty, respectvely. We derve the equlbrum for each of these scenaros and ther comparson leads to a number of nterestng results. As a result of nterplay between demand varablty and retal competton, we fnd that the manufacturer may fnd t optmal to offer QR to only one of the ex ante symmetrc retalers, rather than both of them. When a retaler attans QR ablty, the tendency s to reduce the ntal order quantty and use the QR order to fulfll any addtonal demand. The manufacturer s value of QR therefore depends on the trade-off between the ntal order loss and addtonal QR proft gan. As demand varablty decreases, the expected sze of QR orders and therefore the manufacturer s QR proft decreases as well. Furthermore, due to ntensfyng effect on retal competton, the manufacturer s QR proft from offerng QR to the second retaler s less than that of the frst. Thus, when demand varablty s suffcently small, although the manufacturer s QR proft from the frst retaler outweghs the proft loss n ts regular orders, ts QR proft from the second retaler s nsuffcent to compensate the proft loss n ts regular order. Thus, t s more advantageous for the manufacturer to offer QR exclusvely to one of the retalers. When demand varablty s suffcently large, the manufacturer offers QR to both of the retalers. Moreover, the total channel proft can also be maxmzed wth only one retaler wth QR opton nstead of both, as retal competton hnders the value of havng a second retaler wth QR opton. When retal competton s gnored, QR always benefts a retaler. Surprsngly, however, we fnd that n the presence of retal competton havng QR ablty can be detrmental to a retaler when demand varablty s suffcently small. When competng aganst a compettor wthout QR ablty, the compettor ncreases ts order quantty to compensate for ts lack of QR ablty by orderng a hgh amount, threatenng to deflate the prce. Ths n turn forces the retaler wth QR opton to reduce ts ntal order. When demand varablty s small, the beneft of usng QR to match addtonal demand s nsgnfcant. Consequently, ganng QR ablty hurts the retaler due to potental loss from the ntal order. In contrast, when the demand varablty s suffcently large, QR benefts the retaler. Smlarly, when competng aganst a compettor who already has QR ablty, not havng QR ablty enables a retaler to force ts compettor to reduce ts ntal order quantty. When demand varablty s small, commttng to such a threat as a result of not havng QR ablty domnates the beneft of reducng msmatch between supply and demand usng QR ablty. We demonstrate that our results can contnue to hold for a number of extensons by: () Allowng the manufacturer to set the wholesale prce endogenously; () Consderng alternatve sequence of events such as allowng the QR prce to be set after retalers place ther regular orders or after demand uncertanty s resolved; () Consderng normally dstrbuted demand through numercal studes; and (v) Studyng the outcomes when the manufacturer has lmted capacty for fulfllng QR orders. Overall, our results demonstrate how retal competton changes the value of QR, and provde manageral nsghts to a manufacturer s QR offerng decson as well as a retaler s QR adopton decson. These extensons also yeld some addtonal results. Specfcally, when the QR prce s determned after the retalers place ther ntal orders, the manufacturer may fnd t optmal not to offer QR to any of the retalers. In addton, when there s lmted QR capacty, the manufacturer may always fnd t optmal to offer QR to only one of the retalers due to capacty lmt even when demand varablty s suffcently hgh. The remander of ths artcle s organzed as follows. In secton, we present our lterature revew. Secton 3 descrbes our model. Secton 4 derves the equlbrum. Secton dscusses the value of QR both from the manufacturer s and retalers perspectve. Secton 6 presents several extensons to the base model. Secton 7 offers our concludng remarks. We present the monopoly retaler benchmark n Appendx A, and all proofs appear n the onlne appendx.. Lterature Revew Understandng the value of QR has attracted growng attenton n academc crcles snce the early 1990s. Here, we frst summarze the studes that consder QR n the monopoly settng. In ther semnal paper, Fsher and Raman (1996) show how early sales nformaton can be used to mprove demand forecasts and better manage producton decsons. Iyer and Bergen (1997) evaluate the effect of lead tme reducton enabled by QR n a two-level supply chan, and fnd that QR benefts the retaler whle t may be detrmental to the manufacturer. Eppen and Iyer (1997) examne the value of backup agreements. Under ths agreement, a retaler can place an addtonal order usng QR up to a certan percentage of ts ntal order at the orgnal cost, but any addtonal order n excess of that fracton s charged a hgher cost. They show that backup agreements can beneft both the retaler and the manufacturer. Cachon and Swnney (009)
0 Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety dentfy the suffcent condtons under whch QR benefts a retaler when t faces strategc consumers. Fsher et al. (001) propose a heurstc that determnes both orderng quanttes and n-season replenshment tme for a catalog retaler, fndng ths procedure offers the potental to double that retaler s proft. These studes do not consder the effect of competton, and they treat the prce for QR replenshment as exogenously determned. By contrast, we study the effect of competton on the value of QR, allowng the manufacturer to set endogenously the prce for QR replenshment. Whle the above studes are restrcted to monopoly, recently, Caro and Martínez-de-Albénz (010), L and Ha (008), and Krshnan et al. (010) have examned the compettve value of QR. Caro and Martínez-de- Albénz (010) and L and Ha (008) focus on retaler competton lke our study whereas Krshnan et al. (010) focus on manufacturer competton. Specfcally, Caro and Martínez-de-Albénz (010) and L and Ha (008) consder duopoly retalers competng for spll-over demand where consumers seek the other retaler only when ther frst choce retaler runs out of stock. In contrast, we adopt Cournot competton, n whch nventory competton has a drect mpact on the retal prce. In addton, both Caro and Martínez-de-Albénz (010) and L and Ha (008) treat the retalers cost for QR replenshment as exogenous, whereas we allow the manufacturer to set that prce. Ths allows us to study vertcal nteractons between the manufacturer and the retalers n addton to horzontal retal competton. Such varances n modelng approaches also lead to dfferent results. Assumng dentcal prces for all replenshment opportuntes, Caro and Martínez-de-Albénz (010) demonstrate that QR always benefts a retaler. Smlarly, L and Ha (008) fnd that a frm always benefts from havng reactve capacty that enables replenshments after better demand nformaton s observed. In contrast, we fnd QR may hurt a retaler when demand varablty s too small. Krshnan et al. (010) consder a manufacturer sellng ts product through a retaler who also carres a competng product from another manufacturer. The retaler can exert sales effort to swtch demand from one product to another. Therefore, ther model studes the competton between two manufacturers products sold by a sngle retaler. In contrast, we consder the competton between two retalers sellng products suppled by a sngle manufacturer. Krshnan et al. (010) fnd that QR can hurt the manufacturer s sales because t reduces the retaler s commtment to promote the product. Quck response enables addtonal order placement after better demand nformaton becomes avalable, and there exsts a rch lterature studyng how frms can make use of updated demand nformaton n ther procurement decsons. Gurnan and Tang (1999) analyze a stuaton n whch a retaler can place a second order when t receves better demand nformaton. But at the tme of the frst order, the prce for the second order s uncertan. Weng (004) consders a sngle-buyer sngle-manufacturer channel n whch the manufacturer s able to dctate ts prce for the buyer s second order. He presents a quantty dscount scheme that coordnates the channel. Mlner and Kouvels (00) study the effect of demand characterstcs on the value of supply chan flexblty, whch s characterzed by the tmng or quantty flexblty for the second orderng opportunty. Donohue (000) shows that a buy-back contract can acheve channel coordnaton for a supply chan wth one manufacturer supplyng a sngle retaler. Cvsa and Glbert (00) examne a manufacturer s trade-off between offerng early and delayed purchases. In ther model, a retaler places an order ether before or after demand uncertanty s realzed, whereas we allow a retaler to place orders both before and after the uncertanty s resolved. In addton, Cachon (004), Dong and Zhu (007), and Erhun et al. (008a) examne the mpact of push, pull, and advance-purchase dscount contracts. Although these models ncorporate the dea of makng use of updated demand nformaton n procurement decson, only Mlner and Kouvels (00), Erhun et al. (008a), and Cvsa and Glbert (00) study the value of ths ablty. Moreover, only Cvsa and Glbert (00) consder the effect of competton. Models wth multple orderng decsons, albet wthout demand nformaton updates, are also of partcular nterest to operatons management. Martínezde-Albénz and Smch-Lev (007) and Erhun et al. (008b) examne the effect of multple procurement opportuntes before an uncertan sellng season starts. Both fnd more frequent procurement decreases double margnalzaton whle ncreasng proftablty for all supply chan partcpants. Anand et al. (008) present a two-perod model wth dentcal determnstc demand curves and endogenous wholesale prces. They remove classcal motvatons and hghlght strategc mplcatons for carryng nventory. Ths model s extended by Kesknocak et al. (008) to ncorporate capacty lmtatons. Works that hghlght competton wth multple orderng opportuntes nclude Hall and Porteus (000), Netessne et al. (006), and Lu et al. (007). All of these study products sold n multple perods, whereas we are concerned wth a short lfe cycle product that s sold over a sngle perod. Furthermore, these works do not study the effect of mproved demand nformaton on nventory decsons. In addton to usng quck response, researchers have dentfed a number of operatonal strateges to
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety 1 cope wth demand uncertanty. For example, a frm that produces and sells ts product drectly to consumers may nvest n reactve capacty to allow for addtonal producton after better demand nformaton s obtaned (e.g., L and Ha 008, Raman and Km 00). Although both QR and reactve capacty enable a second replenshment opportunty, the second replenshment s lmted by the capacty level set beforehand n the case of reactve capacty. Delayed product dfferentaton provdes frms wth another nstrument to respond to demand uncertanty (e.g., Anand and Grotra 007, Lee and Tang 1997). It allows a frm to confgure an ntermedate good nto dfferent products after demand uncertanty s resolved, whereas QR consders a frm s ablty to order addtonal nventory. Fnally, spot tradng s also another commonly used strategy and ts value s studed by Mendelson and Tunca (007). Whle spot tradng allows retalers to trade among themselves, n our model QR only allows them to buy addtonal unts from the manufacturer. Mendelson and Tunca (007) show spot tradng can adversely affect a frm, and smlarly we fnd that QR ablty can be harmful n a compettve envronment. 3. The Model Frst we ntroduce the demand model, followed by detaled descrptons of the frms decsons. We consder a manufacturer supplyng homogeneous products to two competng retalers, ndexed by = 1,. All frms are rsk neutral and seek to maxmze ther ndvdual expected profts. The retalers sell ther products n an uncertan consumer market wth a lnear demand curve: p ¼ A X ¼1 X ; where p s the clearng prce, X s the quantty sold by retaler, and A s the demand state that takes values m + v and m v wth equal probabltes, that s, P(A = m + v) = P(A = m v) = 0., where m s the mean demand, and v s a measure of demand varablty. We also dscuss what happens when A s normally dstrbuted n secton 6.4. The dstrbuton of A s publc nformaton. We assume 0 < v < m to avod non-postve demand state. We refer to A = m + v as hgh market, and smlarly A = m v as low market. There are two types of retalers: slow (S) and fast (F). They dffer n ther orderng opportuntes. A slow retaler has only one orderng opportunty: t places ts regular order before the demand uncertanty s resolved. In addton to ths ntal order, a fast retaler has QR ablty to place a second order after the demand uncertanty s resolved. Each retaler places ts regular order Q at a wholesale prce c w per unt, and each fast retaler places ts QR order q at a prce c q per unt. We assume the order quanttes are publc nformaton, whch s common n models of nventory competton (e.g., L and Ha 008, Netessne et al. 006, Olsen and Parker 008). The products are sold n a sngle sellng season. We assume that the salvage value of the products s nsgnfcant, and the retalers sell out all of ther nventory n the sellng season, that s, X = Q + q. Ths s a common assumpton n lterature (e.g., Anand et al. 008, Chod and Rud 00, Goyal and Netessne 007). Therefore, a retaler s proft p s gven as follows. Note that q = 0 for a slow retaler, and t only chooses Q. 0 1 p ¼ @ A X ðq j þ q j ÞÞðQ þ q A c w Q c q q ; ¼ 1; : j¼1 ð1þ As Equaton (1) shows, retaler s proft conssts of three parts: the frst part represents retaler s revenue; the second, ts cost for the ntal order; and the last part captures ts cost for the QR order. The wholesale prce c w for regular orders s exogenously determned. However, the manufacturer determnes ts unt prce c q for QR orders. Ths mmcs the stuaton n whch many other manufacturers are able to delver the products when gven suffcently long lead tme, determnng that the wholesale prce c w s dctated by competton. By contrast, few other manufacturers are able to offer quck response as t requres addtonal capabltes. Ths makes t possble to dctate ts QR prce c q. Note that we also study what happens when the wholesale prce c w s set endogenously n secton 6.1. The manufacturer s producton cost for regular orders s normalzed to zero. Because mplementng QR requres addtonal costs (e.g., overtme expenses and more costly transportaton methods) however, the manufacturer ncurs a cost premum d > 0 per unt for QR replenshments. We assume d < v to elmnate trval cases n whch the QR cost d s so hgh, QR s never used. Thus, gven the retalers order quanttes, the manufacturer s proft p M s calculated as: p M ¼ c w X ¼1 Q!þðc q dþ X ¼1 q!: ðþ To avod an addtonal trval case, we assume c w < m. When c w m the product s not feasble (.e.,
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety no unt wll be sold). Ths can be seen clearly from Equaton (3). Fgure 1 shows the order of events: Frst, the manufacturer announces the QR prce c q. Retalers then place ther regular orders smultaneously for delvery before the begnnng of the sellng season. The demand state A s revealed completely to the retalers. Next, each fast retaler places ts QR order, whch wll also be delvered before the sellng season. Fnally, the sellng season ensues durng whch the retalers sell ther nventory, and profts are realzed. 4. Competton We consder three competton scenaros, denoted by SS (two competng slow retalers), FS (one fast retaler vs. one slow retaler), and FF (two competng fast retalers). In ths secton, we solve for the frms subgame perfect Nash equlbrum (SPNE) strateges n each scenaro. We wll compare these scenaros to characterze the value of QR n the next secton. 4.1. SS Scenaro (Two Competng Slow Retalers) We consder the SS scenaro as a benchmark. In ths scenaro, none of the retalers has QR ablty, that s, each retaler can place only a sngle order that must be decded pror to the resoluton of demand uncertanty. Consequently, ths problem reduces to a sngle stage standard Cournot duopoly model (Trole 1988). In ths scenaro, retaler s expected proft s gven by E½p Š, where E s the expectaton wth respect to the demand ntercept A and p s gven n Equaton (1) wth q 1 = q = 0. It s straghtforward to show that the unque equlbrum s gven by: Fgure 1 The Sequence of Events Q ¼ m c w ; ¼ 1; : ð3þ 3 4.. FS Scenaro (a Fast Retaler vs. a Slow Retaler) We now study competton between a fast (1) and a slow retaler (): In ths scenaro, as descrbed by the sequence of events gven n Fgure 1, QR ablty allows the fast retaler to place an addtonal order after demand uncertanty s revealed. In the followng, we derve the frms equlbrum decsons by applyng backward nducton. In the last stage game, the demand state A s revealed to the retalers. The fast retaler determnes ts QR order quantty q 1 to maxmze ts proft p 1 that s gven by Equaton (1). It s straghtforward to show that p 1 s concave n q 1, and, followng the frst order condton, retaler 1 s optmal QR order quantty s gven by: q 1 ¼ A c q Q þ Q 1 ; ð4þ where A s the demand state, c q s the unt QR orderng cost, and (x) + = max(0,x). As Equaton (4) shows, retaler 1 places ts QR order followng a base-stock polcy and the base-stock level decreases n both the QR prce and the competng retaler s regular order quantty. In the second stage game, the retalers determne ther regular order quanttes to maxmze ther expected profts E½p Š. The followng lemma characterzes the retalers equlbrum regular and QR order decsons. LEMMA 1. There exsts a unque equlbrum for the retalers regular order quantty game n the FS scenaro. The retalers equlbrum actons are descrbed below and the equlbrum regular order quanttes are gven n Appendx B. () For h FS c q : Q ¼ Q 1 0, and retaler 1 does not place a QR order for any market outcome. () For h FS c q \ h FS : Q [Q 1 0, and retaler 1 places a QR order only n a hgh market. () For c q \h FS : Q [Q 1 ¼ 0, and retaler 1 places QR orders n both hgh and low market outcomes, where h FS ¼ c w þ v and h FS ¼ mn c w ; 3 7 m þ 4 7 c w v; m v : 7 A hgher QR prce, c q, reduces the attractveness of QR ablty. As a result, when c q s suffcently hgh, as n case () of Lemma 1, QR s never used and thus the retalers behavor s dentcal to that of the SS scenaro. On the other hand, QR s used only n a hgh market for h FS c q \ h FS. In ths case, the slow retaler places a larger regular order than ts fast compettor to compensate for the lack of QR opton. Fnally, when c q \h FS, the QR prce s extremely low, and the fast retaler reles only on QR for nventory replenshment, t does not place a regular order. In the frst stage game, the manufacturer sets the QR prce, c q, to maxmze ts expected proft E½p M Š.We characterze the manufacturer s optmal c q n the followng proposton:
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety 3 PROPOSITION 1. b FS ¼ ( Let pffffff 18mþ 1 ð3m vþdþ 36 for v 3 m þ d m 6 ðv dþ otherwse : () When c w < b FS, the manufacturer sets c q ¼ c w þ v þ d, the retalers order Q 1 ¼ 3 10 ðm c wþ 1 4 ðv dþ; Q ¼ m c w. () When c w b FS, the manufacturer sets c q ¼ mnð 8c w 3m þ v; 3m þ 8c w þ 7ðv þ dþ 14 Þ, the retalers order Q 1 ¼ 0; Q ¼ð 4m 48c w 7ðv dþ 48 Þ þ. In both cases, the fast retaler places a QR order only n a hgh market, and ts QR order quantty s gven by Equaton (4). When c w b FS, the wholesale prce s extremely hgh and ths results n a trval case, where the fast retaler never places a regular order, whereas when c w < b FS both retalers place a regular order. In comparson to the SS scenaro, Equaton (3) and Proposton 1 show the fast retaler chooses a smaller regular order quantty because t has a second replenshment opportunty. 4.3. FF Scenaro (Two Competng Fast Retalers) The FF scenaro concerns competton between two fast retalers. Here both retalers can place a QR order after the market uncertanty s resolved, as shown n Fgure 1. We derve the frms equlbrum decsons by applyng backward nducton. In the last stage game, the retalers determne ther QR order quanttes. It s straghtforward to show that each retaler s proft, as gven n Equaton (1), s concave n ts QR order quantty q. Therefore, retaler s best response QR order quantty, q BR, can be derved usng the frst order condton: q BR ða; Q ; Q j ; q j Þ¼ A c q q j Q þ j Q ; where, = 1, and j = 3. Wthout loss of generalty, we assume that retaler places a larger regular order, that s, Q Q j. Let q FF be retaler s equlbrum QR order quantty n the FF scenaro. By usng the fact that the equlbrum should satsfy q BR ða; Q ; Q j ; q FF j Þ¼q FF, we obtan the followng equlbrum QR order quantty par: ( ðq FF ; q FF j Þ¼ ða c q 3 Q ; A c q 3 Q j Þ; f A c q 3Q ð0; ð A c q Q Q j Þ þ Þ; otherwse. ðþ Thus, a retaler places a QR order only when ts regular order quantty Q relatve to the demand A s suffcently small. In the second-stage game, the retalers determne ther regular order quanttes smultaneously to maxmze ther expected profts pror to observng the actual demand state. The followng lemma descrbes the retalers equlbrum actons: LEMMA. There exsts a unque equlbrum for the retalers regular order quantty game n the FF scenaro. In equlbrum, Q 1 = Q and they are gven n Appendx B. The retalers equlbrum actons are gven below: () For h FF c q : the retalers do not place a QR order for any market outcome. () For h FF c q \ h FF : the retalers place QR orders only n a hgh market. () For c q \h FF : Q 1 = Q = 0, and the retalers place QR orders n both hgh and low markets, where h FF ¼ c w þ v and h FF ¼ mnðc w ; m vþ: Note that Lemma s structurally smlar to Lemma 1. In the FS scenaro, Lemma 1 establshes the slow retaler ntally orders more than ts fast counterpart due to asymmetrc QR ablty. In contrast, Lemma shows the retalers choose equal regular order quanttes n the FF scenaro as both of them have symmetrc QR ablty. In the frst-stage game, the manufacturer sets ts QR prce to maxmze ts expected proft E½p M Š. The followng proposton summarzes the equlbrum. PROPOSITION. b FF ¼ ( Let ffffffffffffffffffffffffffffffffffffff pffff v þmd vd for v ð 1Þðm dþ : 4 otherwse p mþ p ffff mþ ðm vþdþ () When c w < b FF, the retalers order Q 1 ¼ Q ¼ m c w 3 1 6 ðv dþ, the manufacturer sets c q ¼ c w þ v þ d, and the retalers place QR orders only n a hgh market. () When c w bp FF ffffff, the retalers choose Q 1 = Q = 0, a. for v ð 1Þðm dþ, the manufacturer sets c q ¼ m þ d, and the retalers place QR orders n both hghp and ffff low markets. b. for v[ð 1Þðm dþ, the manufacturer sets c q ¼ m þ v þ d, and the retalers place QR orders only n a hgh market. In all cases retalers QR order quanttes are gven by Equaton (). Retaler behavor n the FF scenaro s smlar to that of the fast retaler n the FS scenaro they place regular orders only when the wholesale prce, c w,s not extremely hgh. In ths case, the retalers place QR orders f the market turns out to be hgh, but do not place any QR order f the market turns out to be
4 Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety low. Also, Equaton (3) and Proposton show a fast retaler n the FF scenaro chooses a smaller regular order quantty due to QR, n comparson to the SS scenaro. Wth a sold understandng of the frms equlbrum actons, we proceed to evaluate the value of QR.. The Value of QR Here we study the mpact of havng QR ablty on the proftablty of all channel partcpants. Ths allows us to address numerous questons of manageral nterest, ncludng: Should the manufacturer offer QR ablty to all, some, or none of the retalers? How does retal competton affect the value of QR? Does QR mprove the performance of the supply chan as a whole? What s the mpact of demand uncertanty? Secton.1 consders the monopoly retaler benchmark. Sectons.,.3, and.4 consder duopoly competton and explore the value of QR for the manufacturer, the retalers, and the whole channel..1. Monopoly Retaler Benchmark To tease out the effect of competton, we frst consder a monopoly retaler. We wll contrast monopoly and duopoly results to understand the effect of retal competton. When the manufacturer serves a monopoly retaler, the frms prcng and orderng decsons are descrbed n Appendx A. Let a R and a M be the expected equlbrum profts for the monopoly retaler and the manufacturer, respectvely, when the retaler s type a, where a = F,S stands for fast and slow. The followng proposton summarzes the effect of QR on the proftablty of the manufacturer, the retaler, and the channel: PROPOSITION 3. () F M [S M. () F R [S R. () F M þ F R [S M þ S R. Proposton 3 shows that QR ncreases the proftablty of the manufacturer, the monopoly retaler, and the entre channel. Ths s ntutve, because both the manufacturer and the retaler can always match ther no-qr proft. The manufacturer can nullfy QR optons by settng a suffcently hgh QR prce c q. Smlarly, the monopoly retaler utlzes QR only f t wll ncrease proftablty... Impact of QR on the Manufacturer s Equlbrum Proft Next we turn our attenton back to duopoly retalers. For example, how many retalers should receve QR offers from the manufacturer? Most strkngly, we fnd that offerng QR ablty to only one of the ex ante symmetrc retalers may be the optmal choce. Let ab M show the manufacturer s expected equlbrum proft when retalers 1 and are types a and b, where a,b = F,S. We defne thresholds for demand varablty parameter v to llustrate our results n ths secton, these thresholds are dsplayed n Table C1 n Appendx C. The followng proposton dentfes the supply chan confguraton that maxmzes the manufacturer s proft: PROPOSITION 4. () For v v M, FS M () For v > v M, FF M [FS M [SS FF M [SS M. M. Fgure llustrates the optmal scenaro for the manufacturer as Proposton 4 descrbes for m = 1 and d = 0.. Note that the shape of v M boundary n the fgure depends on c w Rb FS,b FF followng Propostons 1 and. A retaler wth QR opton decreases ts regular order as seen n Propostons 1 and. Furthermore, the expected sze of the QR order decreases as demand varablty gets smaller. The manufacturer exchanges loss from regular orders for gan from QR orders whch ncreases n demand varablty. When demand varablty s hgh, as n case (), the manufacturer prefers offerng QR to both retalers. When t s small, however, as n case (), surprsngly, the manufacturer s better off by offerng QR ablty to only one of the retalers as opposed to both of them, because the FS scenaro generates a larger proft for the manufacturer from regular orders than the FF scenaro. In ths case, such profts outwegh the addtonal QR proft for the manufacturer n the FF scenaro. Due to retal competton, the manufacturer s QR proft from the addton of a second fast retaler (FS to FF) s smaller than that of the frst (SS to FS). Thus, even when QR proft from the frst fast retaler Fgure 1.0 0.9 0.8 0.7 0.6 v The Scenaros that Maxmze the Manufacturer s Proft for m = 1, d = 0. FF FS 0. 0.0 0. 0.4 0.6 0.8 v M c w
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety (SS to FS) outweghs the proft loss n ts regular orders, QR proft from the second (FS to FF) may not be suffcent to compensate the proft loss n ts regular order. Fnally, the FS scenaro always yelds a hgher proft than the SS scenaro as the manufacturer sets the QR prce endogenously: t can always nullfy QR opton through prcng. In sum, the manufacturer does not always beneft from offerng QR to both of the retalers. Ths s n contrast to the monopoly benchmark n secton.1, where the manufacturer always benefts from offerng QR to the monopoly retaler. Our results n Propostons 3 and 4 demonstrate the manufacturer s optmal polcy crtcally depends on () the competton n retal market (monopoly vs. duopoly), () the demand varablty, () ts wholesale prce for regular orders (dctated by the level of competton n the supply market), and (v) the cost premum for QR replenshments..3. Impact of QR on the Retalers Equlbrum Profts Turnng to the mpact of QR on retaler equlbrum profts, we now explore the value of QR for a retaler under competton. Let ab be retaler s expected equlbrum proft when retalers 1 and are types a and b respectvely, where = 1, and a,b = F,S. The followng proposton descrbes a retaler s value of QR as well as the mpact of ganng QR ablty on the compettor s proftablty. It shows havng QR ablty can be detrmental to a retaler whle beneftng ts compettor. (All of the threshold values used n ths secton are provded n Table C1 n Appendx C.) PROPOSITION. () FS 1 \SS 1 f and only f v\v S 1, and FF 1 \SF 1 f and only f v\v F 1, furthermore vs 1 vf 1. () FS [SS () FS 1 \FS, and FF [SF f and only f v < v FS. f and only f v\v F. Contrary to basc ntuton, Proposton () demonstrates havng QR ablty can hurt a retaler regardless of ts compettor s type when demand varablty s suffcently small (Fgure 3), due to the mpact of QR ablty on the compettor s actons. For ntuton, consder a fast retaler, Retaler A (who has QR opton), competng aganst a slow retaler, Retaler B (who does not). Acqurng QR opton can be harmful to Retaler A n ths case, because the slow compettor, Retaler B, can credbly threaten to deflate the prce by orderng a hgh amount to compensate ts lack of QR opportunty. Deflaton of the prce forces Retaler A to reduce ts regular order quantty. When demand varablty s low, there s lttle to be ganed from a QR order, and thus, Retaler A s loss due to regular orders domnates, makng QR ablty harmful. By the same token, when demand varablty s hgh, msmatch between supply and demand s also hgh, and Retaler B benefts from havng QR ablty even f ths means gvng up forcng the fast compettor to reduce ts regular order quantty. Note that Proposton () also shows v S 1 vf 1. For QR to be benefcal, a hgher level of demand varablty s requred when competng aganst a slow compettor compared to a fast compettor. In other words, a retaler whose compettor already has QR opton s more lkely to beneft from havng QR opportunty compared to a retaler whose compettor does not have QR opton. Ignorng compettve factors, our monopoly benchmark and exstng work show that QR always benefts the retaler (e.g., Fsher et al. 1997, Iyer and Bergen 1997). In contrast, Proposton demonstrates how competton can actually make QR unattractve to a retaler. In addton, part () of Proposton shows when a retaler gans QR ablty, t can actually beneft ts slow compettor. In partcular, a slow compettor always fares better as t enjoys a larger order quantty over the fast retaler. Fast compettor only fares better f demand varablty s small. Lkewse, f both frms have QR opportunty, the competton n a hgh market s ntensfed and ths makes the fast compettor fare worse when demand varablty s hgh. In 1.0 v Fgure 3 The Boundares Gven n Proposton for m = 1 and d = 0. FS scenaro FF scenaro v 1.0 0.9 v S 1 0.9 v F 1 0.8 Π FS 1 > Π SS 1 0.8 Π FF 1 > Π SF 1 0.7 0.7 0.6 Π FS 1 < Π SS 1 0. 0.0 0. 0.4 0.6 0.8 1.0 0.6 Π FF 1 < Π SF 1 c w 0. 0.0 0. 0.4 0.6 0.8 1.0 c w
6 Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety addton, part () of Proposton compares the retalers profts n the FS scenaro, showng the slow retaler acheves a hgher proft only when the demand varablty s suffcently low. Comparng Propostons 4 and also reveals that when a retaler s gven QR opton, ths can beneft all supply chan members. In partcular, all of the frms are strctly better off n the FS scenaro than n the SS scenaro when v S 1 \v. In the next proposton, we descrbe what happens when both of the retalers gan QR ablty smultaneously: PROPOSITION 6. FF [ SS for = 1,. Proposton 6 shows that both retalers reap greater benefts f both gan QR ablty smultaneously. When they all have QR opportunty, no retaler can threaten to place a hgher regular order quantty. One mght wonder what the equlbrum would be f retalers choose to adopt QR themselves rather than havng t dctated to them by the manufacturer. Ths s studed n detal n Appendx S1. We fnd that the equlbrum s always symmetrc, ether both (FF) or none (SS) of the retalers choose to adopt QR. Specfcally, when demand varablty s low, none of the retalers adopt QR (SS), when demand varablty s hgh, both of them adopt QR (FF), and when demand varablty s moderate both SS and FF scenaros can be equlbra..4. Impact of QR on the Channel s Equlbrum Proft Next, we analyze whch channel confguraton, namely the number of fast retalers, s the most proftable for the entre channel. Let ab C be the expected channel proft n equlbrum, that s, the total expected proft acheved by the manufacturer and both of the retalers n equlbrum when retalers 1 and are types a and b respectvely, a,b = F,S: ab C ¼ ab M þ X ¼1 ab : The followng proposton compares the expected channel profts across the three scenaros. PROPOSITION 7. FF C FS C [ maxðff C ; SS maxðfs C ; SS C Þ for v v C, and C Þ otherwse. Propostons 4 and demonstrate QR ablty benefts the manufacturer and the retalers when the demand varablty s suffcently hgh but can be detrmental when t s low. Proposton 7 s n agreement. Ths s ntutve, snce the channel proft s the sum of the manufacturer s and retalers profts. Proposton 7 shows the channel proft s maxmzed wth two fast retalers when demand varablty s suffcently hgh, otherwse the channel mght be better off wth only one fast retaler. Overall, the expected channel proft can be maxmzed by grantng QR optons exclusvely to a sngle retaler. In contrast to the monopoly benchmark where havng a QR retaler always benefts the entre channel, retal competton extends the optmal channel confguraton to a contnuum: the total channel proft may be maxmzed by havng one or two retalers wth QR ablty. 6. Extensons We now consder a number of extensons to our base model that suggest our key nsghts contnue to hold n varous settngs, and llustrate the robustness of our results. 6.1. Endogenous Wholesale Prce Frst, we extend the base model gven n secton 3 by allowng the manufacturer to dctate the wholesale prce at the begnnng of the tmelne. 3 Specfcally, now t chooses the wholesale prce c w to maxmze ts expected proft n equlbrum. In the followng, we present the optmal wholesale prce the manufacturer would choose, then dscuss the value of QR. LEMMA 3. Suppose the manufacturer can dctate the wholesale prce, t chooses c w ¼ m to maxmze ts expected proft n all scenaros (SS, FS, and FF). Knowng the manufacturer s choce of the wholesale prce, we are able to derve the frms equlbrum profts n each scenaro. Comparng these profts across scenaros reveals the frms value of QR as the followng proposton summarzes. PROPOSITION 8. () FS M [FF M f and only f v\v M, and SS M \ maxðfs M ; FF M Þ. () a. FS 1 \SS 1 f and only f v\v S 1, and FF 1 \SF 1 f and only f v\v F 1, furthermore vs 1 [vf 1. b. FS [SS ; FF [SF f and only f v\v C. () FS C [FF C f and only f v\v C, and SS C \ maxðfs C ; FF C Þ. The threshold values v M, v F 1, vs 1, and v C are provded n Appendx D. Proposton 8 shows our results n secton contnue to hold when the manufacturer s able to choose the wholesale prce n addton to the QR prce. Specfcally, Proposton 8() extends Proposton 4, showng the manufacturer s optmal polcy s to offer QR to only one of the retalers when demand varablty s low. Proposton 8(a) and (b) echo Proposton.
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety 7 They demonstrate how QR ablty can hurt a retaler when demand varablty s suffcently low, and ganng QR can actually beneft the competng retaler. Fnally, Proposton 8() mmcs Proposton 7 showng that the total channel proft can be maxmzed by havng only one QR-enabled retaler when demand varablty s small. 6.. Alternatve Sequence of Events In our base model, the QR prce s set at the begnnng of the tmelne before the retalers place ther regular orders. Here, we dscuss two alternatve models wth regard to tmng of the QR prce and analyze the value of QR for the manufacturer, the retalers, and the channel as a whole. Specfcally, we consder the followng models: (E1): The QR prce s set after the regular orders are placed, but before the realzaton of demand uncertanty. (E): The QR prce s set after the demand uncertanty s resolved. The remanng events are the same as our base model. Models E1 and E actually yeld dentcal equlbrum outcomes n our setup. Ths s because of the bnary nature of demand dstrbuton. In partcular, n equlbrum, a fast retaler places a QR order only n a hgh market. Therefore, the manufacturer always sets the QR prce for a hgh market, and the tmng of the QR prce (whether before or after demand realzaton) becomes rrelevant. We mpose an addtonal assumpton, c w d, n ths subsecton. If ths assumpton s volated, t demonstrates the manufacturer s chosen QR prce would be smaller than the wholesale prce (.e., c q < c w ). 4 Thus, retalers always place QR orders regardless of the demand outcome, whch s nconsstent wth practce. Furthermore, relaxng ths assumpton creates a regon wth no pure-strategy equlbrum n the FS scenaro, whch would complcate our analyss. The followng proposton summarzes the frms equlbrum actons for the models E1 and E. PROPOSITION 9. For the models E1 and E: () The FS scenaro has a unque equlbrum n whch Q 1 Q and a. For v e 1, the fast retaler does not place a QR order for any market outcome. b. For v > e 1, the fast retaler places a QR order only n a hgh market and t does not place a QR order n a low market. () The FF scenaro has a unque equlbrum only for v e 1 and v e, but there does not exst a purestrategy equlbrum for e 1 < v < e. When the equlbrum exsts, Q 1 = Q and a. for v e 1, the retalers do not place a QR order for any market outcome. b. for v e, the retalers place QR orders only n a hgh market and they do not place any QR order n a low market. The threshold values e 1 and e are gven n Appendx D. Note that the SS scenaro n E1 and E models s same as our base model QR s not offered and thus QR prce s not relevant. When the retalers have QR ablty, Proposton 9 shows QR s only used n a hgh market as n the base model. Notce, however, a purestrategy equlbrum n the FF scenaro for e 1 < v < e does not exst, because havng the QR prce set after the regular orders are placed results n pecewse concave proft functons for the retalers. Retaler proft functons may contan multple maxma, whch leads to dscontnuty n the retalers best response functons. Buldng on Proposton 9, we characterze the value of QR for the manufacturer, retalers, and the entre channel n Appendx E. These are formally stated n Propostons 11 13 n that appendx. We fnd that our results of the base model contnue to hold even when the QR prce s determned after retalers place ther regular orders. In partcular, the profts of the manufacturer and the entre channel can stll be maxmzed by grantng QR ablty to only one of the retalers, rather than both of them (Propostons 11 and 13). Furthermore, havng QR ablty can stll be detrmental to a retaler whle benefttng the opponent (Proposton 1). We also fnd addtonal results. In models E1 and E, the manufacturer may fnd t optmal not to offer QR at all when the demand varablty s too low (Proposton 11). In contrast, n our base model, the QR prce s set at the begnnng of the tmelne and the manufacturer enjoys the frst mover advantage, consstently offerng QR to at least one of the retalers (Proposton 4). When the QR prce s set after retalers place regular orders, the manufacturer loses the frst mover advantage, and ths reduces the value t can extract from the retalers due to QR. Smlarly, the total channel proft can also be maxmzed wth no QR-enabled retaler at all (Proposton 13). Fnally, we compare retalers proftablty n our base and E1 and E models n Proposton 14 n Appendx E. We fnd that demand varablty s the key factor; competng fast retalers are better off n models E1 and E f and only f demand varablty s suffcently small. 6.3. Lmted QR Capacty In ths secton, we study what occurs when the manufacturer has lmted QR capacty to grant. Specfcally, we assume the manufacturer can fulfll at most
8 Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety k unts usng QR. When the retalers total QR order quantty exceeds the manufacturer s QR capacty, the manufacturer allocates ts capacty evenly among the retalers. Any unused capacty by one retaler can be reallocated to the other retaler. In addton to the assumptons for the base model, we further restrct our analyss to k < (m d)/6 to ensure the QR capacty s ndeed lmted and bnds n both FS and FF scenaros. Moreover, gven any QR capacty level k, we focus only on c w < m k/3 to elmnate the unrealstc scenaro n whch retalers do not place a regular order due to a hgh wholesale prce. We derve SPNE for FF and FS scenaros and subsequently examne the value of QR. The followng proposton summarzes the effect of lmted capacty on the value of QR. PROPOSITION 10. When the manufacturer has a total capacty k for QR replenshment: () Manufacturer: a. FF M [ maxðfs M ; SS M Þ for c w\ ~w M and v[~v M. b. FS M [ maxðff M ; SS M Þ otherwse. () Retalers: a. FS 1 [SS 1 f and only f c w [ ~w S 1 and v[~v S 1 ; FF 1 [SF 1 f and only f c w [ ~w F 1 and v[~vf 1. b. The mposton of QR capacty lmt ncreases (weakly) the regular order sze of a fast retaler. () Channel: a. FS C [ maxðff C ; SS C Þ for c w[ ~w C. b. FF C [ maxðfs C ; SS C Þ otherwse. Note that all of the threshold values are summarzed n Appendx D. Imposng QR capacty lmt nduces a fast retaler to ncrease (weakly) the sze of ts regular order. We fnd our key nsghts contnue to hold for ths extenson. Ths extenson also yelds an addtonal nsght. In our man model wthout the capacty lmtaton, the manufacturer prefers havng two fast retalers when demand varablty s suffcently hgh (Proposton 4). Wth lmted QR capacty, Proposton 10() mples havng only one fast retaler maxmzes the manufacturer s proft when the wholesale prce s suffcently hgh (.e., c w ~w M ). Ths result shows that the QR capacty lmt can be also another reason for not offerng QR opton to both of the retalers. Intutvely, gven a hgh wholesale prce, a fast retaler wth QR opton decreases ts ntal order and reles more heavly on ts QR order. In ths case, however, the manufacturer does not have suffcent capacty to satsfy QR orders of two fast retalers. Thus, the manufacturer s better off by offerng QR opton to only one of the retalers, whch allevates the reducton n ther ntal order quanttes. 6.4. Numercal Study: Normally Dstrbuted Demand In ths secton, we use computatonal studes to explore an alternatve demand dstrbuton. Specfcally, we allow the demand ntercept A to follow a truncated normal dstrbuton wth mean 1 and standard devaton r. We consder all combnatons of the followng parameters: c w f0:1; 0:; 0:3; 0:4; 0:g; d f0:0; 0:04; 0:06; 0:08g; r f0:1; 0:; 0:3; 0:4; 0:g: For each parameter combnaton (c w,d,r), we numercally search for the frms equlbrum decsons n the SS, FS, and FF scenaros, that s, when there are zero, one, and two fast retalers respectvely, and ths generates a total of 300 nstances for our study. We defne the value of QR (VQR) for a retaler as the percentage ncrease n ts proft after adoptng QR. Specfcally, t s gven by Fb 1 Sb 1 Sb 1 100%; b ¼ F; S; ð6þ where Fb 1 s retaler 1 s equlbrum proft when the compettor s type s b. Smlarly, we defne the value of QR for the manufacturer and the entre channel as the percentage ncrease n ther profts compared to the SS scenaro, whch s gven by Fb SS SS 100%; b ¼ F; S and ¼ M; C; ð7þ where Fb and SS are the equlbrum profts of the manufacturer (M) or the channel (C) n the scenaros Fb and SS respectvely. We fnd our key results contnue to hold n our numercal studes. Table 1 reports our fndngs for c w = 0., 0.3 and d = 0.0, whch s representatve, and other parameter combnatons consdered n our studes also yeld smlar results. As expected, Table 1 shows the value of QR for the retalers and the manufacturer ncreases n the demand standard devaton r. For c w = 0.3, the manufacturer prefers havng only one fast retaler (FS)whenr 0.3, and two fast retalers (FF) otherwse. In other words, the manufacturer s optmal polcy s to offer QR to only one of the retalers when the demand varablty s not suffcently hgh. Smlarly, Table 1(a) also demonstrates the total channel proft can be maxmzed wth only one QRenabled retaler when demand varablty s not suffcently hgh (r 0.4 for c w = 0.3). Furthermore, Table 1(a) shows the manufacturer and entre channel are always better off offerng QR to at least one of the
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety 9 Table 1 Value of QR for c w = 0.,0.3 and d = 0.0 (a) Manufacturer and Channel VQR (%) n Equaton (7) (b) Retaler VQR (%) n Equaton (6) Manufacturer Channel Compettor type c w = 0. c w = 0.3 c w = 0. c w = 0.3 c w = 0. c w = 0.3 r FS FF FS FF FS FF FS FF S F S F 0.1 10.71 0.3 9.43 0.33 3.6 0.3.1 0.1 30.9 10.33 31.1 11.3 0. 11. 3. 10.83 3.8 4.68.30 4.3 1.4 9.7 4.37 9.4 6.7 0.3 14.88 9.0 13.31 6. 7.8.1 7.8 3.3.4.6 11.1.83 0.4 16.30 16.10 1.31 19.74 16.48 13.97 14.33 13.31 11.40 4.81.9.4 0. 3.74 6.87.4.3.80 3.87.43.79 8.84 13.46 30.4 1.74 retalers. Moreover, Table 1(b) confrms havng QR ablty can hurt a retaler f the demand varablty s not suffcently hgh: Adoptng QR hurts a retaler when r 0.. 7. Conclusons In ths study we examne the value of QR under retal competton. For ths purpose, we consder a market served by two competng retalers and compare the equlbrum profts for the manufacturer, the retalers, and the entre supply chan as a whole, when QR s avalable to one, both, or none of the retalers. We allow the manufacturer to set the prces for regular and QR replenshments. We also consder a hgher cost for mplementng QR, thereby quantfyng the trade-off between benefts and addtonal costs of QR. We demonstrate offerng QR ablty to a retaler may harm the manufacturer when the demand varablty s not suffcently hgh. In partcular, we fnd a manufacturer may fnd t attractve to offer QR to only one of the ex ante symmetrc retalers. Ths happens because a retaler reduces ts regular (ntal) order quantty when t can place a QR order. Furthermore, when the demand s not suffcently volatle, offerng QR can generate nsuffcent QR proft to balance the loss that results from a retaler s reducton n ts regular order. Moreover, the manufacturer s addtonal QR proft gan from offerng QR to the second retaler s less than that from the frst retaler, as a consequence of retal competton. Therefore, the manufacturer does not necessarly beneft from havng two retalers wth QR ablty. The total channel proft can also be maxmzed wth only one retaler wth QR ablty, nstead of two, when demand varablty s not suffcently hgh. We also hghlght the potental strategc perl of QR ablty for a retaler n the presence of retal competton. As expected, QR ablty benefts a monopoly retaler wth better response to varaton n demand. However, retal competton undermnes the value of QR, and obtanng QR ablty can actually harm a retaler when the demand varablty s low and we explctly characterze when ths happens. We recognze our model has several lmtatons. We assume retalers who am to maxmze ther expected profts are rsk neutral. Unlke a regular order, a QR order faces no demand rsk, thus t has a lower rsk than a regular order. A rsk-averse retaler wll be more nclned to use QR to decrease ts demand rsk. We expect a rsk-averse retaler to ncrease ts allocaton of QR order (and hence decrease ts allocaton of regular order), makng QR more valuable than our model predcts. Quantfyng the mpact of rsk-averson on the value of QR could be a frutful avenue for future work. Furthermore, our model assumes QR lead tme s relatvely short compared to the sellng season. However, ths lead tme can be sgnfcant and also later arrvng unts may suffer from drops n sales prce over the sellng season. These factors wll degrade the attractveness of QR and frms wll shft ther allocatons from QR to regular orders. Thus, when such factors are accounted, we expect the outcome to fall between our fast (QR) and slow (no QR) frm scenaros. Nonetheless, our model cannot fully address these extensons, and t would be worthwhle to generalze our settng to a mult-perod model to allow for long QR lead tme and declnng prces and study ther mpact. We study a sngle suppler servng two retalers. Whle ths s not uncommon n practce (ntroducton provdes some examples), we recognze other supply chan scenaros are possble; for example, a retaler havng multple supplers, or each retaler havng a dstnct suppler and so on, and some of our results may not apply to these scenaros. Thus, future work can study the mpact of supply chan confguraton on the value of QR consderng varous scenaros. Furthermore, our numercal study n secton 6.4 suggests that our results can contnue to hold for other more general demand dstrbuton functons; however, showng ths extenson analytcally would be worthwhle for future work. We also note that, n practce, retalers may not observe each other s order
30 Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety quanttes. In ths case, the manufacturer s prcng would provde a sgnal about order quanttes and retalers would choose ther best actons accordngly. Addtonally, our model assumes the manufacturer ncurs an dentcal unt QR cost d for each retaler, makng t ndfferent between them. In practce, however, due to geographc dsperson, one retaler may actually result n a hgher expedtng cost, and thus the manufacturer may prefer offerng QR to the less costly retaler. Fnally, generalzng our duopoly model to olgopoly retalers s another possble extenson. We expect wth many compettors, reactons to a retaler s ganng of QR ablty may not be as strong, thus, a retaler may be more lkely to beneft from QR n an olgopoly. Acknowledgment The authors thank the senor edtor and the referees for many helpful comments and suggestons. Appendx A: Monopoly Retaler Benchmark Let q H and q L be the QR order quanttes for the monopoly retaler n the hgh and low markets respectvely. The followng lemma characterzes the supply chan partcpants equlbrum strateges: LEMMA 4. () When the monopoly retaler does not have QR ablty, the unque equlbrum order quantty for the retaler s Q ¼ m c w. () When the monopoly retaler can place a QR order, there exsts a unque equlbrum as follows: (a) For c w c F :Q¼ m v c w þ c q, c q ¼ c w þ v þ d, q H > 0 and q L = 0. (b) For c w >c F : Q = 0, q H 0, q L 0 and 1. For v m d :c q ¼ m þ d.. For m d \v: c q = m v. Table C1 Crtcal Threshold Values Appendx B: Addendum to Lemmas LEMMA 1: Ths lemma descrbes the retalers equlbrum actons after c q s chosen n the FS scenaro. The followng descrbes ther equlbrum regular order quanttes: () For h FS c q :Q 1 ¼ Q ¼ m c w. 3 () For h FS c q \ h FS : ðq 1 ; Q Þ¼ 3m v 8c w þ c q ; ðm c wþ 10 for c w a 1 ; ðq 1 ; Q Þ¼ 0; 3m v 4c w þ c q 6 for a 1 <c w a ; (Q 1,Q ) = (0,0) otherwse, where a 1 ¼ 3m v þ c q and a ¼ 3m v þ c q 8. 4 () For c q \h FS : ðq 1 ; Q Þ¼ 0; m þ c q c w. LEMMA : Ths lemma descrbes the retalers equlbrum actons after c q s chosen n the FF scenaro. The followng descrbes ther equlbrum regular order quanttes: () For h FF c q :Q 1 ¼ Q ¼ m c w. 3 () For h FF c q \ h FF : Q 1 ¼ Q ¼ c w \ c q þ m v, and Q 1 = Q = 0 otherwse. () For c q \ h FF : Q 1 = Q = 0. m v c w þ c q 3 for Appendx C: Demand Varablty v Threshold Values for the BaseModel Table C1 descrbes the threshold values n secton for c w < mn(b FS,b FF ) and c w mn(b FS,b FF ), where b FS and b FF are gven n Propostons 1 and respectvely. rffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff x 1 ¼ d m þ 8 7 ðm þ 3c w m 3c w Þ x ¼ 1 3 m þ p 4 3 ffff ðc w mþþd 7 Condton Threshold values v M ¼ p ffffffffffffffffffffffffffffffffffffffffffffffff c w ðm c w Þ pffffff þ d v S 1 ¼ p ffffffffff 19 1 ðm c wþþd c w < mn(b FS,b FF ) v F ¼ rffffffffff 19 ðm c w Þþd v F 1 ¼ p ffffff ðm c wþþd pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff c w m 41c v C ¼ w þ 11m p ffffff þ d v FS ¼ p ffffff 3 ðm c wþþd v M = mn(x 1,x ) v S 1 ¼ vf 1 ¼ vfs ¼ mnðb FS ; b FF Þ c w mn(b FS,b FF ) v F s rrelevant n ths case v C = mn(x 3,max(b FS,b FF ))
Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety 31 qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 194m þ 7 18ðd vþþ 6ð7m 36mðv dþþ61ðv dþ Þ x 3 ¼ 416 Appendx D: Demand Varablty v Threshold Values for the Extensons Table D1 Endogenous wholesale prce v M ¼ p m ffff þ d v S 1 ¼ p ffffff 19 m 1 þ d v F 1 ¼ p ffff p m þ d v C ¼ ffffff 19 p ffff m þ d Table D Alternatve sequence of events e 1 = d c w ^v F ¼ 13447c p w þ 7ð0 ffffffffffffffffffff 9161 661Þm p 187 140 ffffffffffffffffffff þ d 9161 ¼ 13m 168c w þ 1d ^v a C 1 ¼ m 4c w þ 8 p ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff c w þ mc w 8m p 47 47 ffffff þ d ^v 1 M ¼ 7 p ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 3c w ð107c w 3mÞ 71c w 8 176 þ d ^vb C ¼ 88m 688c pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff w þ 77 3496c w 07mc w þ 601m þ d 86 ^v M ¼ 11 p ffffffff y 1 49m 13 47 904c w ^v c C 07 ¼ 4091m 890c p w þ 1848 ffffffff y þ d 361 ^v S 1 ¼ 13c p w þ 7ð3 33 ffffffffffffff 36 Þm p 31 ffffffffffffff þ d y 1 ¼ 363m 41844mc w 16607c w 36 6003Þ pffffffffff ^v S ð161 33 14 Þm 30cw ¼ p 3ð48 þ 11 ffffffffff þ d y ¼ ð 63m þ 11970mc w 487c w 14 Þ Þ ð^v 1 C ; ^v C Þ¼ ð^va C ; ^va C Þ; for c w ^w C ð^v b C ; ^vc C Þ; otherwse ; where ^w C s gven by the soluton to ^v b C ¼ ^vc C Lmted QR capacty ~v M 8 pffffffffffffffffffffffffffffff pffffffffffffffffffffffffffffff c w ðm c w Þ pffff m 4m þd for c w 4k >< 4, qffffffffffffffffffffffffffffffffffffffffffffffffff ¼ kþ k 4c pffffffffffffffffffffffffffffff wðm c w Þ m 4m 1 þd for 4k 4 \c w ~w M, >: rrelevant for ~w M \c w. pffffffffffffffffffffffffffffffffffffffffff ~w M ¼ m m 1k ~w S 1 1k ¼ m p ffffff 19 ~w F 1 1k ¼ m p ffff 4 ~v S 1 ¼ ffffff p 19 1 ðm c wþþd ~v F 1 ¼ ffff p ðm c wþþd 8 pffffffffffffffffffffffffffffffffffffffffffffffffff m 36m 0ðv dþ >< 8 for d\v pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 3k þ d, ~w C ¼ m 1 4m þ164k 38kðv dþþ13ðv dþ 8 for 3k þ d\c w k þ d, >: 11m 41 for k þ d\v.
3 Producton and Operatons Management 1(3), pp. 18 33, 011 Producton and Operatons Management Socety Appendx E: Value of QR n Models E1 and E Here, we dscuss the value of QR to the manufacturer, retalers, and the entre channel for the extended models E1 and E descrbed n Secton 6.. Snce a pure-strategy equlbrum may not exst n FS scenaro of these extended models (see Proposton 9), n ths secton we only compare FS to FF scenaros for v e 1 and v e n whch a pure-strategy equlbrum exsts n both scenaros. The followng proposton characterzes the value of QR for the manufacturer. PROPOSITION 11. For the models E1 and E: () SS M [ maxðfs M ; FF M Þ for v\^v1 M. () FS M maxðss M ; FF M Þ for ^v1 M v\^v M. () FF M maxðss M ; FS M Þ for If ^v M v. The threshold values ^v 1 M and ^v M are gven n Appendx D. The next proposton characterzes retalers value of QR. It ndcates havng QR ablty can stll be detrmental to a retaler and t can beneft ts rval. PROPOSITION 1. For the models E1 and E: () FS 1 \SS 1 f and only f v\^v S 1 () FS, and FF 1 [SF 1. [SS f and only f v\^v S, and FF [SF f and only f v\^v F. The threshold values ^v S 1 and ^vs are gven n Appendx D. In the base model, we show that QR ablty can hurt a retaler regardless of ts compettor s type (fast or slow). In contrast, when the regular orders are placed at the begnnng of the tmelne, retalers become the frst mover, ncreasng the value extractable from QR. As a result, Proposton 1 ndcates that ganng QR ablty s now always benefcal to a retaler when ts compettor already has QR ablty. Nevertheless, QR ablty can stll be harmful to a retaler aganst a compettor who does not have QR opton. In addton, ganng QR ablty can stll beneft a competng retaler. The thrd proposton addresses the effect of QR on the channel proftablty for the models E1 and E. It shows the channel proft can stll be maxmzed wth only one fast retaler and the demand varablty s the key determnant. PROPOSITION 13. For the models E1 and E: () FF C [maxðfs C ; SS C Þ for ^v1 C \v. () FS C maxðff C ; SS C Þ for ^v C \v ^v1 C. () SS C maxðff C ; FS C Þ for v ^v C. The threshold values ^v 1 C ; ^v C are gven n Appendx D. Dfferent from our base model, Proposton 13 shows the total channel proft can also be maxmzed wth no QR-enabled retaler at all. Ths result reflects the effect of the retalers gan of frst mover advantage: placng regular orders before the QR prce s set. The frst mover advantage encourages the excess use of QR. When demand varablty s suffcently low, there s lttle value to QR and t does not justfy the cost for the entre channel. Fnally, we compare the retalers profts n our base model and alternatve E1 and E models. Let ab ;B show retaler s equlbrum proft n the base model when retalers 1 and are types a and b, where a,b = F,S and = 1,. Smlarly, let ab ;E be retaler s equlbrum proft n the alternatve models. Recall that E1 and E models result n the same outcome. PROPOSITION 14. () FF ;B \FF ;E f and only f v\ 4c w m þ pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 13 4 ð788c w 376c w m þ 63m Þ 6 for = 1,. () FS 1;B \FS 1;E and FS ;B [FS ;E. Note that snce QR opton s not used n SS scenaro, our base and E1 and E models do not dffer. In models E1 and E, the QR prce s set after ntal orders are placed. Therefore, when choosng ther ntal order quantty, retalers take nto account the mpact on the QR prce whereby a larger ntal order quantty results n a lower QR prce. When the demand varablty v s hgh, the QR opton s more valuable, thus a retaler ndeed orders larger ntal order quanttes to receve a lower QR prce. However, n the FF scenaro ncreased order quanttes of both retalers results n more ntense competton makng the retalers worse off. In contrast, when the demand varablty v s low, the QR opton s less valuable, retalers do not have a strong ncentve to order a large quantty ntally, and they enjoy the frst mover advantage, whch makes them better off compared to the base scenaro. In the FS scenaro, the fast retaler enjoys a hgher proft n E1 and E models due to ts frst mover advantage. Thus, not surprsngly, the slow retaler s worse off n n E1 and E models. Notes 1 Chou, L., Personal ntervew wth the presdent of Makalot, February 009. Furthermore, quantty-based competton keeps the problem tractable, thus, t s commonly used n compettve models n the operatons lterature (e.g., Anand and Grotra 007, Goyal and Netessne 007, Ha et al. 011, Mendelson and Tunca 007). 3 It does not matter whether the wholesale prce s set frst or smultaneously wth QR prce, because both prces are determned by the manufacturer and there s no other event happenng n between these decsons.
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