1 PRODUCTION AND OPERATIONS MANAGEMENT Vol. 17, No. 5, September October 2008, pp ssn essn POMS do /poms Producton and Operatons Management Socety Coordnatng a Supply Chan System wth Retalers Under Both Prce and Inventory Competton Xuan Zhao School of Busness andeconomcs, WlfrdLaurer Unversty, Waterloo, Ontaro N2J 3C5, Canada, We nvestgate a supply chan system wth a common suppler sellng to downstream retalers who are engagedn both prce andnventory competton. We establsh the exstence andunqueness of the purestrategy Nash equlbrum for the retaler game and study how a suppler can coordnate the system to acheve the best performance. Our man conclusons are as follows: Frst, a buyback contract can be usedto coordnate retalers competng on both prce andnventory n a sense that optmal retal prces andnventory levels arse as the Pareto-domnant equlbrum. Wth symmetrc retalers, the system optmum arses as the unque symmetrc equlbrum. Second, the partcular type of competton experenced by retalers (prce versus nventory competton) affects the characterstcs of the contract. Specfcally, strong prce competton leads to a coordnaton mechansm wth a postve buyback rate, where the suppler subsdzes retalers for leftover nventores; however, strong nventory competton leads to a negatve buyback rate, where retalers are punshed for overstockng. Usng a lnear expecteddemandfuncton, we further explore the mpact of system parameters on the coordnaton contract andthe compettve equlbrum. We also fndthat the performance of the suppler s optmal contract s asymptotc to the system optmal coordnaton contract as competton becomes ferce. Key words: prcng; nventory; competton; coordnaton Hstory: Receved: June 2005; Revsed: July 2006, March 2007, May 2007, and September 2007; Accepted: September 2007 by Suresh Seth. 1. Introducton Advances n nformaton technology and the ncreasng prevalence of global trade have made multple andalternatve dstrbuton channels ubqutous. A car may now be purchasedthrough a dealershp or onlne. A watch may be purchasedn a department store, n a sngle-brandboutque, or over the Internet. Increasng the number of dstrbuton channels nevtably ncreases nterchannel competton, andconsequently, ths nterrelaton between multple channels gves rse to complextes n managng and coordnatng supply chans. Ths paper wll nvestgate the followng questons: How can a suppler coordnate a multchannel supply chan system wth compettve retalers n order to acheve the best performance of the entre supply chan system? How does the coordnaton contract relate to the nature of competton at the market level? How does the suppler s ncentve to use the coordnaton contract relate to the exstng degree of competton? Competton can be modeled n many ways. In ths paper, we assume that retalers compete not only on prce but also on nventory, n the sense that unsatsfedcustomers look aroundandmght swtch (spll over) to other stores that stock smlar products. Ths consumer behavor s very common andcan be llustratedby the followng example (Voce of Small Busness 2002): Talbot s [clothng retalers], for nstance, has a telephone n each of ts outlets so that customers can speak wth representatves famlar wth n-store andcatalog merchandse. Customers can order drectly through these representatves when a store doesn t have a specfc tem n stock. Also, Sears andother large retalers long have had dedcated n-house catalog representatves who work wth customers to locate tems that are not avalable n the store Smlarly, accordng to a survey provded by P&G, approxmately 50% of ts customers swtch to another retaler after a stockout (Terney 2004). Ths consumer swtchng behavor results n the so-called nventory competton between stores. How does the combned prce andnventory competton affect the prcng and stockng behavor of retalers, andhow does t nfluence a suppler s coordnaton strategy? Ths paper wll focus on one well known and wdely studed contract: the buyback contract, whch s commonly usedn some ndustres. Car manufacturers, for example, take back unsold old-model cars from dealershps and recast them nto new ones; smlar observatons apply to the publshng ndustry. In a noncompettve context, studes n channel coordnaton suggest that double margnalzaton between a retaler and a suppler leads to overchargng or understockng at the retaler level. For ths reason, specal ncentve arrangements are needed n order to algn the manufacturer andthe retaler. A buyback contract 532
2 Zhao: Coordnatng a Supply Chan System wth Retalers Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety 533 can coordnate the retaler whose decson relates to procurement only andcan acheve system optmal performance, but t cannot acheve the frst-best result f the retaler also sets the retal prce (e.g., Emmons and Glbert 1998). Much of the marketng andeconomcs lterature has consdered coordnaton contracts n the case of retalers makng only prcng decsons under determnstc demand (Shapro 1989). In ths paper, however, we wll nvestgate how a suppler should set buyback contracts n a supply chan system where retalers facng stochastc demand compete on both prce andt nventory. The man fndng of ths paper s that buyback contracts can perfectly coordnate the supply chan system: the retal prces andnventores that acheve optmal supply chan performance arse as the Paretodomnant equlbrum for the system and as the unque Pareto-domnant equlbrum f the system optmum s unque andprofts are transferrable among supply chan members. In the case of symmetrc retalers, the system optmal retal prces and safety stocks arse as the unque symmetrc equlbrum for the retaler game under the coordnaton contract. Interestngly, the nature of the coordnaton buyback contract s closely relatedto the nature of the competton. For example, f nventory competton outweghs prce competton n the market, the buyback contract wll stpulate that the retaler must pay the suppler for overstockng, n contrast to the common buyback arrangement. Conversely, f prce competton outweghs nventory competton n the market, the buyback contract wll stpulate that the suppler must buy back extra nventores from the retaler. Usng a lnear demand functon for the determnstc porton of demand, we further explore the propertes of the equlbrum andthe coordnaton contract. We fndthat the wholesale prce slghtly ncreases but the buyback rate decreases wth the spll rate and that both terms ncrease wth the degree of prce competton. In our numercal study, we examne the suppler s optmal buyback contract andcompare t wth the system optmal contract. We fndthat the two contracts become extremely close as the degree of the prce compettveness andthe degree of spll ncrease. In other words, the value of coordnaton actually decreases wth the degree of both types of competton. Along wth these man fndngs, we wll also demonstrate that retal-prce-mantenance contracts andquantty-forcng (servce-level-mantenance) contracts can also acheve optmal system performance. The paper s organzedas follows. Secton 2 revews the related lterature. Secton 3 ntroduces detals of the model. Secton 4 provdes the man results of the paper usng the general model. Secton 5 uses the specal case of a lnear demand functon to further examne the propertes of the equlbrum and the coordnaton contract, and 6 provdes numercal results. In 7, other contracts (such as servce-levelmantenance andretal-prce-mantenance contracts) are dscussed. We conclude n Related Lterature The feldof channel coordnaton has attractedconsderable attenton n the past decade. Cachon (2003) gves an excellent revew of the lterature anddscusses a varety of supply chan contracts. Kouvels et al. (2006, p. 456) provde a more recent dscusson on supply chan research, commentng that [coordnaton and contracts are] partcularly mportant n a compettve envronment n whch entre supply chans are competng for customers. Here, we lmt ths revew to works that consder coordnaton and contractng n a compettve envronment. The frst stream of research consders ssue of contractng for supply chans where a manufacturer sells products through both drect (manufacturer-owned) and ndrect (compettor-owned) channels, so the manufacturer s both a suppler anda compettor of the retaler. Tsay andagrawal (2004) analyze the neffcency of such a system and provde recommendatons about how to adjust the manufacturer retaler relatonshp. Boyac (2005) explores the mpact of channel neffcency on stockng decsons and develops contract schemes to coordnate the supply chan. Cattan et al. (2006) analyze prcng strateges to ease the channel tensons as a result of the manufacturer enterng nto the market andcompetng wth ts supply chan partner. Dfferent from ths lterature, apart from consderng multple compettve retalers, we also allow retalers to compete n both prce and nventory. There has been very lttle research that consders coordnaton and contractng for supply chans wth a manufacturer sellng to multple compettve retalers. van Ryzn andmahajan (1999), for nstance, examne the performances of vendor and retaler management nventory under horzontal nventory competton among retalers. They fndthat the two schemes tend to acheve frst-best profts as the nventory competton ncreases. Cachon (2003) consders coordnaton for supply chans wth multple newsvendors. To llustrate the trade-offs, he has desgned two models: The frst model assumes that the uncertan total demands allocatedto each retaler accordng to the rato of ts nventory to total system nventory, wth the retal prce beng fxed. The externalty between retalers n ths model s that each has an ncentve to stock hgher n order to be assgned a hgher demand. To mtgate ths effect, the suppler can ether charge
3 Zhao: Coordnatng a Supply Chan System wth Retalers 534 Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety a wholesale prce greater than the margnal cost or use a buyback contract to adjust the retalers ncentves. The secondmodel assumes that, n the case of market-clearng prces, hgher nventory levels leadto lower prces. The externalty between retalers n ths model drves them to establsh lower nventory levels to ensure hgher market prces. Both the buyback contract andthe retal-prce-mantenance contract can acheve coordnaton n ths settng. Anupnd and Bassok (1999) consder a system wth only nventory competton andfndthat a wholesale prce plus a holdng-cost subsdy contract can coordnate the system. In ths paper, we wll provde a more general model n the sense that t captures the externaltes between retalers ntroduced by both nventory competton andprce competton. Usng a novel approach, Zhao andatkns (2008) establsh the exstence of purestrategy Nash equlbrum for the retaler game wth combnedcompetton under an exogenous wholesale prce contract. They also examne the propertes of the equlbrum andnvestgate the mpact of competton on retalers behavor. Dfferent from them, we focus on explorng how a manufacturer/suppler shoulddesgn the contract to algn the competng retalers n a way that creates the best performance for the whole system, andwe wll nvestgate the propertes that coordnaton contracts should have under competton as well. Bernsten andfedergruen (2007) consder coordnaton mechansms for supply chans under prce and servce competton. They treat servce competton n the same way as prce competton, wth both affectng the determnstc porton of the demand. In contrast, we assume that a proporton of a retaler s unsatsfeddemandswtches to ts compettors, affectng the stochastc porton of a retaler s demand. 3. Model Detals Consder a suppler sellng to N olgopoly retalers n a market. Before the sellng season, the suppler announces the wholesale prce w andbuyback rate to each retaler for = 1toN. These may be dfferent for each retaler, but the dfferences wll be based manly on product characterstcs or costs of producton. Retalers then make decsons about orderng quanttes andprces smultaneously before they know the demand. Products are delvered mmedately, demands are then realzed, and fnally, profts are collected The Retaler Game The General Model. Followng the lterature that consders the prce-senstve newsvendor model (e.g., Petruzz and Dada 1999, Agrawal and Seshadr 2000), we use L p + to model a retaler s drect demand, where p = p 1 p 2 p N. The frst part, L p, represents the determnstc porton of the demand, havng L p = def dl p /dp 0, L j p = def dl p /dp j 0. The secondpart,, represents prcendependent uncertan demand, havng a general probablty densty functon (PDF) f andcumulatve dstrbuton functon (CDF) F. Retalers need to decde on retal prces p andsafety stocks y before knowng the demand. So the total order of a retaler s Y = L p + y, where y s usually vewedas a hedge of uncertan demand, determnng the avalablty or servce level. We buldon the above model by assumng that a proporton j of lost sales from retaler j wll swtch to. By dong ths, we have ncorporatednventory competton between retalers, as some of the demand lost by j wll be stolen by f has enough nventory (a smlar assumpton s usedby Netessne andrud 2003 andthe lterature theren). As a result, each retaler s total demand conssts of three parts: L p + + j j j y j +, wth the thrdpart, j j j y j +, representng ndrect stochastc demand of, andcapturng the fact that a proporton of lost sales from a compettor wll swtch to retaler. We use D s = + j j j y j + to smplfy the notaton; the assocatedpdf s f D s, andthe CDF s F D s. So a retaler s problem s to make a decson about p and y to maxmze = p w L p w y + p E mn D s y + E y D s +. By collectng terms, we have = d w y + p E mn D s y (1) where d = p w L p. We assume that retalers strategy sets are compact: p y w p p max 0 y y max, where p max and y max are large enough numbers andnever restrct the players (Cachon and Netessne 2004). It can be shown that the game wth decsons p y s equvalent to the game wth decsons p Y. There are two extreme cases for our general model. We call a game an nventory competton game f L j p = 0 and L p = L p, anda game a prce competton game f j = 0 for all j and D s = A Specal Case. To explore the comparatve statcs concernng the equlbrum andthe coordnaton contract, we return to a specal case where the determnstc porton of demand has a lnear form;.e., L p = a b + p + j p j p. Ths lnear demandform has been commonly usedto study ssues related to operatons/marketng nterfaces (Tsay andagrawal 2000, Boyac andray 2003). When t s not possble to derve manageral nsght wth a general pure-strategy Nash equlbrum, we sometmes use a symmetrc equlbrum concept,
4 Zhao: Coordnatng a Supply Chan System wth Retalers Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety 535 whch s commonly usedby economsts for complex models (Fudenberg and Trole 1991, Trole 1988). A game s symmetrc n the sense that the players have dentcal parameters and dentcal strateges and payoffs (Cachon andnetessne 2004). In other words, payoff functons are permutable. We use = j for all j when a symmetrc game s consdered. In a symmetrc equlbrum, all of the players choose the same actons The Suppler s Problem Our prmary nterest s to nvestgate whether the heterogenety among retalers s a barrer for coordnaton when retalers compete on both prce andnventory. Therefore, our goal s to fnda suppler contract that maxmzes the total supply chan proft ( c = c, where c = d c y + p E mn D s y and c s the producton cost of product ). A suppler s problem s to set (w ) so that the system optmalty soluton arses as an equlbrum outcome. A suppler mght be nterestedn usng a system optmal contract for several reasons: she mght want to bulda long-term relatonshp wth the retaler, or she mght care most about the product beng delvered to the market effcently. It couldalso be the case that the suppler wouldfollow the recommendaton of a parent frm whose nterest s n the performance of the total supply chan. So the goal of the suppler s to make the total supply chan perform n the best possble way. However, we wll stll nvestgate the contract that maxmzes the suppler s own proft andthen compare the two. 4. Results wth the General Model 4.1. Equlbrum Analyses wth Fxed Contract Parameters Gven the contract parameters w and, we frst study the compettve retaler game under both prce andnventory competton. We start by nvestgatng the exstence of the pure-strategy Nash equlbrum of the game, andthen we ntroduce some results concernng the unqueness of the equlbrum Exstence of Nash Equlbrum n Pure Strateges. Two condtons are needed for establshng the exstence of a Nash equlbrum (Theorem 1): (A) 2 d/ p2 < 0 and 3 d/ p3 0. (B) The dstrbuton of D s has an ncreasng falure def rate (IFR r D s = f D s / 1 F D s ). (Ths s guaranteed but not restrctedby the assumpton that s have IFR dstrbutons and are ndependent.) Theorem 1. Assume (A) and (B) hold. () There exsts a pure-strategy Nash equlbrum. () The best response of retaler s the unque soluton of / p = d / p + E mn D s y = 0 (2) / y = w + p Pr D s >y = 0 (3) The proof s smlar to Zhao andatkns (2008), where a novel methods provdedandassumptons are justfed. We nclude the proof n the Onlne Supplement, avalable at supplements, for the convenence of the reader. An mmedate result from ths theorem s that a symmetrc pure-strategy Nash equlbrum exsts n the game (Theorem 2, Cachon andnetessne 2004) wth symmetrc retalers. If we consder the extreme case of the nventory competton game where L p = L p, the above condtons are not needed. Proposton 1. The duopoly nventory competton game s supermodular, and there exsts a pure-strategy Nash equlbrum Suffcent Condtons for Unqueness of the Nash Equlbrum. Theorem 2. Suffcent condtons for unqueness are 2 d / p2 > j 2 d / p p j +1 and j j + 1/ w r D s 0 < 1, where r D s 0 s the falure rate at y = 0. A smlar result s provedn Zhao andatkns (2008); we nclude the proof n the Onlne Supplement for the convenence of the reader. The condtons are restrctve. However, f we consder dentcal retalers, the condtons can be relaxed. Condton (C) below s commonly usedto establsh the unqueness of a Nash equlbrum for a determnstc prcng game by economsts (Vves 1999): (C) 2 d / p2 + j 2 d / p p j < 0. Proposton 2. Wth N dentcal retalers and condtons (B) and (C), there exsts a unque symmetrc equlbrum System Optmal Coordnatng Contracts After studyng prcng and nventory equlbrum under exogenous contract parameters, we nvestgate supply chan coordnaton. In ths secton, the key queston to be addressed s: n a decentralzed supply chan system wth both prce andnventory competton between ndependent retalers, how should a suppler set a contract to acheve system optmal prces andsafety stocks as an equlbrum andmaxmze supply chan-wde profts? In an ntegratedsystem, the common suppler owns all retalers: c = c, where c = d c y + p E mn D s y. Netessne and Rud (2003) study a smlar system but wth no consderaton of prcng decsons; analytcally they fnd that the objectve functon mght not be jontly quasconcave n stock quanttes, but numercally wth reasonable demand varablty they do not fnd multple solutons. Wth 2N decson varables on both prces and stocks, c
5 Zhao: Coordnatng a Supply Chan System wth Retalers 536 Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety mght not be jontly quasconcave ether, but numercally no multple optmal soluton s found(see 6). However, t can be shown (as n part () of Theorem 1) that a global optmal soluton of c s nteror andsolves c / p = L p + p c L p + E mn D s y + p j c j L j p = 0 (4) j c / y = c + p Pr D s >y p j j Pr D s j <y j >y = 0 (5) j By comparng ths wth (2) and(3), we dentfy the dfference between a global optmum and a compettve equlbrum andthe source of neffcency of the compettve equlbrum. Subtractng (2) and(3) from (4) and(5), we have / p = c / p j p j c j L j p w c L p (6) / y = c / y + j p j Pr D s j <y j >y j w c Pr D s <y (7) By comparng the frst-order condtons of the compettve retalers game andthe jont optmzaton problem, we can deconstruct the mpact of vertcal conflct (double margnalzaton) and horzontal conflct (competton) on prcng andnventory decsons. The secondterm of Equaton (6) reflects the horzontal externalty causedby prce competton, whereas the secondterm of (7) represents the horzontal externalty of nventory competton. Both of them dstort competng retalers decsons regardng prce and safety stock. A compettve retaler tends to set lower p than the system optmum because an ncrease n p benefts rval retalers, resultng from the nature of demand substtutablty. A competng retaler also has a hgher level of safety stock y because a decrease n y results n more spll demand to the compettor. The thrdterms of both equatons reflect the vertcal externaltes ntroduced by double margnalzaton. A decentralzed retaler tends to set hgher p than the system optmum because a lower p hurts the retaler but boosts demandandbenefts the suppler. Smlarly, a decentralzed retaler has a lower level of safety stock y than the system optmum f the buyback rate s small. To fully coordnate ndependent and compettve retalers, the suppler should desgn the contract so that the vertcal and horzontal conflcts balance each other;.e., the sums of the secondandthrdterms become zero for both equatons. Under some condtons on the parameters, a set of wholesale prces (wth = 0) can coordnate the system, but the restrctve parameter condtons make the coordnatng wholesale-prce-only contract less nterestng. Let ( p c y c ) be a global optmum that maxmzes the total supply chan proft. Theorem 3 gves a system optmal coordnatng buyback contract w that acheves the best supply chan performance n a sense that the system optmal retal prces andnventory levels, p c y c, arse as a Pareto-domnant equlbrum for the system. Note that an equlbrum X s sadto Pareto-domnate Y f every player prefers X to Y (Varan 1992),.e., X Y for all, ncludng the suppler andall of the retalers. Theorem 3. There exsts a unque w wth w <p c and w c that perfectly coordnates the supply chan system,.e., p c y c arses as an equlbrum of the retaler game, where w = c j p c j c j L j p c /L p c and (8) = pc pc w / Pr Ds <yc (9) Furthermore, p c y c s a Pareto-domnant equlbrum for the suppler retaler game. If payoffs are transferrable under coordnaton, then t s the unque Pareto-domnant equlbrum f the global optmum for the supply chan system s unque. Note that, n general, there mght be multple equlbra under w. However, Schellng s (1960) theory of focal pont suggests that n some real-lfe stuatons players may be able to coordnate on a partcular equlbrum by usng nformaton that s abstractedaway by the strategc form. For example, f players are able to communcate to one another before the game s played, then t s reasonable that players wll coordnate on the Pareto-domnant equlbrum, a commonly usedfocal pont of a strategc form game (Fudenberg and Trole 1991). If p c y c s the unque global optmum for the total supply chan system, and f a transfer payment can be arrangedamong supply chan members under coordnaton, then p c y c arse as the unque Pareto-domnant equlbrum for the suppler retaler game. The transfer payment could be enforcedunder the request of the parent frm or the suppler as a requste for the suppler to use the coordnaton contract w nsteadof the suppler s optmal contract, n the format of lump-sum payments between supply chan members. However, a detaled nvestgaton on the arrangement of the transfer payments s outsde the scope of ths paper. In the case of symmetrc retalers, accordng to Proposton 2, the symmetrc system optmal retal prces andsafety stocks, p c y c, wll arse as the unque symmetrc equlbrum for the retaler game.
6 Zhao: Coordnatng a Supply Chan System wth Retalers Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety 537 See Mahajan andvan Ryzn (2001) for an argument that a symmetrc equlbrum s more focal than an asymmetrc equlbrum. Proposton 3. Wth prce competton only, > 0 and w >c ; wth nventory competton only, < 0 and w = c. Bernsten and Federgruen (2007), consderng both prce andservce competton, foundthat perfect coordnaton can be acheved f a wholesale prce s combnedwth a backloggng penalty, andthe sgn of the penalty can be ether postve or negatve. Cachon (2003) has postve buyback rates for both models. In contrast, n our contract, may be a subsdy (postve) or a penalty (negatve) for the retalers; nterestngly, whether t s a subsdy or a penalty depends on the partcular type of competton takng place prce or nventory competton. Wth prce competton only, the system-coordnaton contract shouldhave the suppler chargng wholesale prces greater than the producton cost and payng retalers for resdual nventores. The dea s that prce competton nduces retalers to set lower prces andlower levels of safety stock. By chargng hgher wholesale prces, the suppler forces the retalers to set hgher prces, thus offsettng the damagng effect of prce competton on retal prces. At the same tme, by offerng to buy back or subsdze leftovers, the suppler encourages the retalers to set hgher levels of safety stock. Under the theoretcal extreme case of only nventory competton, the coordnaton contract has the suppler set the wholesale prce at a level equal to the margnal cost of producton n order to avod double margnalzaton, chargng fees for leftover nventores to reduce the effect of overstockng under nventory competton. Under ths contract, the suppler does not earn revenue through drect procurement; nstead, the suppler collects revenue f the retalers have leftovers, endng up wth a postve proft. Because the retalers have to pay the suppler for overstockng, t seems that ths wholesale-prce-plus-leftoverpenalty contract wouldrequre the suppler to undertake more nventory montorng; ths has been made possble wth the current advances n nformaton technology. We wll dscuss ths more later. 5. Results wth the Specal Case of a Lnear Demand Functon To further explore the propertes of a compettve equlbrum andthe coordnaton contract, n ths secton, we consder a specal case where the determnstc porton of demand has a lnear form,.e., L p = a b + p + j p j p, andwe focus on a symmetrc equlbrum. We frst explore the propertes of an equlbrum for any buyback contract, and then we examne the propertes of the coordnaton buyback contract (w ) Comparatve Statcs for an Equlbrum Under w How does the equlbrum change wth the contract parameters andthe competton parameters? Usng (p y ) to denote a symmetrc equlbrum, the followng propostons address ths. Proposton 4. For a symmetrc equlbrum, () dp /d > 0 and dy /d > 0. () The only stuaton that cannot occur s dp /dw < 0 and dy /dw > 0. Ths result actually holds for not only the lnear demand case, as shown n the Onlne Supplement. A hgher buyback rate set by the suppler nduces hgher retal prces andsafety stocks from the retalers. So consumers are less lkely to face a stockout, but at a cost of payng more. As shown n the prevous lterature, a hgher wholesale prce (double margnalzaton) leads to two opposte effects: a hgher retal prce (Spengler 1950) but a lower safety stock (Cachon 2003). When retalers compete on both prce and nventory, the two effects work aganst each other, resultng n three possble outcomes. It s straghtforwardto see that the system optmal ( p c y c ) s nsenstve to the contract terms. Usng a methodsmlar to that n Proposton 4, we have that ncreaseddemandspll leads to a hgher equlbrum retal prce andsafety stock, but ncreased prce competton drves the equlbrum prce and safety stock down, echong Zhao and Atkns (2008) Propertes of the Coordnaton Contract w How does the coordnaton contract w change wth the degree of prce competton andthe spll rate ( )? An analytcal result concernng s not tractable; we numercally study ths n 6. Proposton 5 addresses the mpact of. Proposton 5. w ncreases wth, and so does. As prce competton becomes fercer, the coordnaton buyback contract has a hgher wholesale prce andbuyback rate. The ntuton s that as prce competton becomes stronger, retalers tendto reduce prces (whch, n turn, woulddamage the proft of the entre supply chan) but that a hgher w prevents the prce reducton. A hgher wholesale prce w usually results n a hgher buyback rate as compensaton. Proposton 6. For the symmetrc prce competton game, under coordnatng contract w, / c = 1 N 1 / b +.
7 Zhao: Coordnatng a Supply Chan System wth Retalers 538 Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety As the number of retalers ncreases, a retaler s proft share under coordnaton decreases, and the proft share of the suppler ncreases. The same result s observedas the prce competton becomes stronger (a larger ). In general, however, / c does not have such a nce form; an arbtrary dvson of channel proft has to rely on a fxedsde payment between retalers andthe suppler. Proposton 3 examnes the contract terms under two extreme cases, but what happens when prce and nventory compettons coexst? It turns out that the sgn of s determned by the relatve strength of prce competton anddemandspll (to be confrmed n 6). Ths pont becomes clearer f we use a lnear demandfuncton andconsder a symmetrc equlbrum, where p c yc s ndfferent to. From (8) and (9), f p c Pr Ds >yc <c j pj c c j / b +, then > 0. Note that the rght-handsde of ths nequalty ncreases wth. If prce competton s very strong (a relatvely large ), then t s more lkely that > 0 wll holdtrue, andthe contract wll requre the suppler to pay the retalers for leftover nventores. On the other hand, f prce competton s very weak (a relatvely low ), then the reverse of the above nequalty s more lkely to hold, whch means that the contract wll requre the retalers to pay the suppler for the leftover nventores ( < 0). Buyback contracts wth postve buyback rates are commonly mplemented n dfferent ndustres. For example, n the publshng andmusc ndustres, retalers usually return unsoldproducts; n computer andapparel, the suppler usually subsdzes unsold products. For the stuatons where a negatve buyback rate s necessary (where retalers pay back), the mplementaton of suppler s nspecton mght be needed. Inspecton rghts of the suppler are usually wrtten clearly n a suppler contract. For example, quotedfrom 4.6 of HardDsk Drve Supply Agreement by and between Quantum Corporaton andtvo Inc. (Sample Busness Contracts 1998): Quantum may cause, at Quantum s expense except as below provded, an ndependent, certfed publc accountant selectedby Quantum andacceptable to TVo n ts reasonable dscreton to examne such true and complete books of account at all reasonable tmes TVo shall promptly pay to Quantum all amounts due to Quantum hereunder, plus nterest thereon 6. Numercal Results The purpose of the numercal experments s to gan some nsghts nto questons that cannot be answered analytcally. We use normal, unform, andexponental dstrbutons and dfferent sets of parameters; the results obtanedare consstent. Here we represent the results concernng a symmetrc equlbrum when a normal dstrbuton, a symmetrc lnear demand functon, andthe followng parameter setup are used: a = 200, b = 12, c = 6, = 100, = 0 b/2 b 2b 3b 4b (ncorporatng a broadrange of the degree of prce competton), = (resultng n coeffcent of varaton 0.2 and0.5, respectvely), = , and N = 3. Frst, how do the terms of the coordnaton contract change wth system parameters such as the degree of prce competton ( ) andthe spll rate ( )? Consstent wth Proposton 5, both w and ncrease wth prce competton factor. Furthermore, w slghtly ncreases wth, but decreases wth (Fgures 1 and2). The buyback rate s usedto encourage retalers to stock more. So f a market has a hgher spll rate, then a lower buyback rate shouldbe usedto acheve coordnaton. Both w and are qute nsenstve to the coeffcent of varaton for the stochastc porton of demand. Second, how do the system optmal retal prces and safety stocks (.e., the equlbrum under coordnaton) change wth system parameters such as,, andthe coeffcent of varaton? From Fgures 3 and4, we see that both decsons ncrease wth the spll rate but are nsenstve to. The former decreases but the later ncreases wth the coeffcent of varaton. How do the suppler s and retalers profts under coordnaton change wth the parameters? From Fgures 5 and6 we see that the suppler s profts ncrease wth both and. The retalers profts, on the other hand, decrease wth but ncrease very slghtly wth. So most of the proft gan obtanedfrom coordnaton goes to the retaler when prce competton Fgure 1 w Fgure 2 β System Optmal Buyback Contract System Optmal Buyback Contract
8 Zhao: Coordnatng a Supply Chan System wth Retalers Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety 539 Fgure 3 p System Optmal Retal Prce 14.9 Fgure 4 y System Optmal Safety Stock Fgure 7 w Suppler s Optmal Buyback Contract Fgure 8 β Suppler s Optmal Buyback Contract 5 0 s weak; however, most of the proft gan belongs to the suppler when prce competton s strong. As expected, low demand uncertanty leads to hgh profts for both the suppler andthe retalers. The next queston s, how much does ths coordnaton contract dffer from the suppler s optmal contract? The system optmal buyback contracts perfectly coordnate the suppler and the compettve retalers. However, f the suppler needs to negotate wth retalers or wth ts parent frm for compensaton (n the case that the suppler uses a system optmal Fgure 5 P(S) 2,800 2,100 Suppler s Proft Under System Optmal Buyback Contract 1, Fgure 6 P(R) Retaler s Proft Under System Optmal Buyback Contract 100 contract under the nterference of the parent frm), how much wouldthe suppler expect? To answer ths queston, we needto examne the buyback contracts that maxmze the suppler s proft that s, the w and that maxmze w c L p + y E y D s +. The wholesale prce of the suppler s optmal buyback contract ncreases wth, but at a slower rate than the wholesale prce under a system optmal contract does (Fgure 7). It also ncreases wth. The buyback rate under the suppler s optmal contract ncreases wth but decreases wth spll rate (Fgure 8). An nterestng observaton s that the buyback rate s always postve, so the suppler wll never let the retalers pay for overstockng. Both contract terms decrease wth the coeffcent of varaton of the stochastc porton of demand. In Fgure 9, the suppler s effcency loss under the system optmal contract dmnshes dramatcally as the degree of prce competton grows. (Effcency loss of the suppler s proft = (suppler s proft under her optmal contract suppler s proft under coordnaton contract)/suppler s proft under her optmal contract.) When = 3b, the suppler loses only approxmately 6% by usng a coordnaton contract. On the other hand, the effcency gan of the total supply chan proft (Fgure 10) under a system optmal buyback contract drops dramatcally (approachng zero) as the degree of prce competton ncreases. (Effcency gan of the total supply chan proft = c s / c, where s s the total supply chan proft when the suppler chooses w and to maxmze her own
9 Zhao: Coordnatng a Supply Chan System wth Retalers 540 Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety Fgure 9 Suppler s effcency loss Fgure 10 Effcency gan of coordnaton Effcency Loss of the Suppler Proft Under System Optmal Contract Effcency Gan of Total Supply Chan Proft Under System Optmal Contract proft.) Ths result conveys the manageral nsght that system-wde coordnaton provdes less value to supply chans f the market s hghly compettve. Both terms also slghtly decrease wth the degree of spll but are qute nsenstve to demand varablty. 7. Extensons: Dscusson on Other Contracts Next we dscuss two other types of contract. One s the retal-prce-mantenance contract (prce celng or prce floor); the other s the quantty-forcng (servcelevel-mantenance) contract. When retal-prce mantenance was legal n North Amerca, t was wdely used n preventng dscount stores from free rdng. In an envronment where retal-prce mantenance s permtted, the suppler charges retalers wholesale prces w p = c + j j pj c Pr Ds j <yj c >y c andsets retal prces equal to the ntegratedlevel of retal prces. Then the dstorton from horzontal and vertcal externaltes can be adjusted. To see ths, when w = w p, the sum of the secondandthrdterms n (7) s zero, andthe retalers wll choose safety stocks equal to the centralzedsolutons. Quantty-forcng contracts (or servce-level-mantenance contracts) are another wdely used strategy. The suppler charges wholesale prces w s = c j p c j c j L j p c /L p c andsets a quota equal to the centralzedsafety stock. The quota can be seen as a requredservce level forcedon the retalers as a prerequste of the partnershp. Ths scheme can perfectly coordnate the system: as wth w s, Equaton (6) becomes / p = c / p, (.e., prce dstorton under combned competton can be fully adjusted). So the retalers wll set retal prces that are essentally the same as the centralzedretal prces. Retal-prce-mantenance contracts have been used n a wde range of markets and have often been dscussedn the economcs lterature n the context of vertcal control. Wnter (1993) provdes detaled examples andjustfcatons. Servce-level-mantenance contracts (Mathewson andwnter 1984) are wdely observedn the automoble ndustry, where car supplers usually set certan levels of quota n ther contracts wth car dealers, who are then advsed to stock at least a mnmum number of those supplers cars for a gven sales season. 8. Conclusons and Future Research We have studed a two-echelon supply chan system wth a common suppler sellng to retalers engagedn both prce andnventory competton. Each retaler s total stochastc demand depends on ther own retal prce and nventory level, as well as those of ther compettors. In such a system, ncentve conflct exsts not only n the vertcal dmenson of the supply chan (between the suppler anda retaler), but also n the horzontal dmenson,.e., between competng retalers. How can a suppler coordnate such a multchannel supply chan system n order to acheve the best performance of the entre system? How does the vertcal control arrangement (coordnaton contract) relate to the nature anddegree of horzontal competton between retalers? We frst address these research questons wth the general model. We provde suffcent condtons on the exstence andunqueness of a pure-strategy Nash equlbrum, andwe fndthat there exsts a unque optmal contract that perfectly coordnates the system, n the sense that the system optmal retal prces and safety stocks arse as the Pareto-domnant equlbrum, a focal pont that s usually usedby economsts to predct the players behavor when multple equlbra exst. If a transfer payment s allowedunder coordnaton, then the Pareto-domnant equlbrum s unque gven that there s a unque global optmum for the whole supply chan system. Furthermore, f we consder reasonably homogenous retalers, then the system optmal retal prces andsafety stocks arse as the unque symmetrc equlbrum of the retaler game. The nteracton between vertcal externaltes (double margnalzaton) andhorzontal externaltes (prce-nventory competton) enrches the context of coordnaton. When both types of competton coexst, a coordnaton contract has a wholesale prce greater than the producton cost. However, the buyback rate,, can be ether postve or negatve, dependng on the relatve strength of the two types of competton.
10 Zhao: Coordnatng a Supply Chan System wth Retalers Producton and Operatons Management 17(5), pp , 2008 Producton and Operatons Management Socety 541 The buyback rate s postve when prce competton s stronger, whch encourages retalers to stock hgher than the safety stock levels under competton. Alternatvely, t s negatve when nventory competton s stronger, whch dscourages retalers from stockng hgher-than-optmal levels. In addton to the tradtonal functon of the buyback rate, whch s usedto motvate the retaler s ncentve for procurement, the buyback rate n ths context s also usedto adjust the compettve behavor of the retalers. To further explore the problem, we then consder a specal case for the determnstc porton of demand. In ths specal case, we fndthat retalers tendto set hgher safety stocks but also charge more f the upstream suppler promses a hgher buyback rate. However, f the suppler charges a hgher wholesale prce, retalers wll not respondby decreasng retaler prces but ncreasng safety stocks smultaneously. For the optmal contract, we fndthat both the wholesale prce andbuyback rate ncrease wth the degree of prce competton factor. Numercally, the former ncreases but the latter decreases wth the proporton of demand spll. Both terms are qute nsenstve to the coeffcent of varaton of the stochastc porton of demand. By comparng the system optmal contract wth the suppler s optmal contract, we fndthat the contract terms become reasonably close as competton ntensfes. As a result, the performances of the two contracts are asymptotc to each other as competton grows. The coordnaton of supply chan systems has been an mportant focus of research n the area of operatons management. Whereas most research has examneda sngle-monopoly supply chan, n ths paper we have consdered supply chans wth downstream retalers competng n the same market. Two man nsghts that supply chan managers can derve from ths are the followng: Frst, the partcular type of competton n the market (prce competton versus nventory competton) nfluences the desgn of contracts between the suppler andretalers. Second, as prce competton becomes fercer, t s not necessary to mplement the system optmal contract, because the suppler s optmal contract wll not damage the system-wde performance of the supply chan. There are many avenues for future research. One possblty s to desgn a coordnaton contract that has ncer propertes such as beng free of system parameters and provdng arbtrary proft dvson. A buyback contract can coordnate a system wth arbtrary proft dvson f the system has only stochastc demand, but not for a hybrd demand system wth both stochastc and determnstc demand components. It wll be nterestng but also challengng to explore a smple contract form havng those nce propertes to coordnate such a complcated system. Another possblty wouldbe to consder a sequental retaler game rather than a smultaneous prce andnventory retaler game. If an open-loop equlbrum concept (Fudenberg and Trole 1991) s used, then the results of ths paper holdtrue. However, wth a closed-loop equlbrum concept, the analyss wll be much more complcated, f tractable. Fnally, rather than assumng a fxedspll rate, a further step wouldbe to see how t relates to retal prce by explctly modelng consumer swtchng behavor. However, ths consderaton could make the model ntractable. In fact, our study of the marketng lterature (e.g., Anderson et al. 2006, Bell and Ftzsmons 2000, Campo et al. 2000) on consumer behavor after a stockout leads us to conclude that when consumers face a stockout at the retaler they chose ntally, prce appears to drop sharply down the herarchy of decson crtera they use when swtchng to another retaler. 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