Selling to the Newsvendor with a forecast update: Analysis of a dual purchase contract q,qq

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1 Euopean Jounal of Opeational Reseach 182 (2007) Poduction, Manufactuing and Logistics Selling to the Newsvendo with a foecast update: Analysis of a dual puchase contact q,qq Özalp Öze a, *, Onu Uncu a, Wei Wei b a Depatment of Management Science and Engineeing, Stanfod Univesity, 314 Teman Engineeing, Stanfod, CA 94305, United States b Mogan Stanley, 20 Cabot Squae, Canay Waf, London, United Kingdom Received 5 Mach 2005; accepted 13 Septembe 2006 Available online 13 Decembe Abstact We conside a supply chain in which a manufactue sells to a pocue-to-stock etaile facing a newsvendo poblem with a foecast update. Unde a wholesale pice contact, the etaile waits as long as she can and optimally places he ode afte obseving the foecast update. We show that the etaile s wait-and-decide stategy, induced by the wholesale pice contact, hindes the manufactue s ability to (1) set the wholesale pice and maximize his pofit, (2) hedge against excess inventoy isk, and (3) educe his pofit uncetainty. To mitigate the advese effect of wholesale pice contact, we popose the dual puchase contact, though which the manufactue povides a discount fo odes placed befoe the foecast update. We chaacteize how and when a dual puchase contact ceates stict Paeto impovement ove a wholesale pice contact. To do so, we establish the etaile s optimal odeing policy and the manufactue s optimal picing and poduction policies. We show how the dual puchase contact educes pofit vaiability and how it can be used as a isk hedging tool fo a isk avese manufactue. Though a numeical study, we povide additional manageial insights and show, fo example, that maket uncetainty is a key facto that defines when the dual puchase contact povides stict Paeto impovement ove the wholesale pice contact. Ó 2006 Elsevie B.V. All ights eseved. Keywods: Supply chain; Contacting; Advance puchase; Newsvendo model; Foecast updating; Pocuement 1. Intoduction In many supply chains, the upsteam fim (the manufactue) must adjust to the downsteam fim s (the etaile s) demand though continuous efinement of poduction pocesses and delivey systems. A well-known example fom the appael industy is quick esponse, a seies of pocess impovement initiatives that enable q The aticle was peviously titled as Analysis of a Dual Puchase Contact. qq The pape was pesented duing the 2003 INFORMS Annual Meeting in Atlanta session SD 40, the seminas at Conell Univesity and Stanfod Univesity. The authos ae gateful fo the stimulating discussions. * Coesponding autho. Tel.: addesses: ooze@stanfod.edu (Ö. Öze), onu@stanfod.edu (O. Uncu), wei.wei@moganstanley.com (W. Wei) /$ - see font matte Ó 2006 Elsevie B.V. All ights eseved. doi: /j.ejo

2 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) faste esponse to etaile odes. Such initiatives allow the etaile to wait until the last possible minute, placing he ode afte a bette foecast becomes available. The pactice of last minute odeing is not unique to the appael industy. A 2002 suvey shows that 57% of industial buyes fom automative to electonics inceased just-in-time puchases and educed buying based on long tem foecasts (Ansbey, 2002). In this pape, we show that last minute odeing and its advese effect on manufactues ae the esult of a wholesale pice contact, which is widely used due to it s simplicity. Often the pocuement contact between a manufactue and a etaile is negotiated well in advance of the sales season, actual odes and poduct delivey. Last minute odeing coupled with ealy commitment to a wholesale pice futhe squeezes the manufactue s pofit. Ansbey (2002, Wall Steet Jounal) epots anecdotal evidence to conclude that... [etailes] ae holding onto thei cash as long as they can... waiting until the last possible moment to ensue that evey ode they place will lead to pofits...[those] manufactues eceiving last minute odes have difficulty in justifying equipment puchases. many manufactues invest in expensive equipment...to justify the cost is to un [the equipment] constantly... the system woks best when [etaile] places big odes well in advance... The above obsevations suggest that the consequence of last minute odeing (due to the wholesale pice contact) to the manufactue is theefold. Fist, the manufactue may not be able to chage the wholesale pice that maximizes his pofit. The negotiation to set the wholesale pice often takes place well in advance of the sales season. Hence, the maket potential fo the poduct is uncetain duing this negotiation. This uncetainty hindes the manufactue s ability to enfoce the optimal wholesale pice. Second, last minute odeing negatively affects the manufactue s ability to invest in new technology o justify equipment puchases. Unde a wholesale pice contact, the etaile delays he odeing decision as much as she can and does not commit to buying any poduct pio to obtaining a final foecast update, hence bette demand infomation. The etaile allows just enough time to the manufactue to poduce he odes. The etaile s wait-and-see stategy, howeve, equies the manufactue to invest in poduction equipment in the face of uncetain pofits. Thid, the manufactue often initiates pat of his poduction pio to eceiving the etaile s ode when his costs ae deceasing in poduction lead time. In this case, the manufactue also faces excess inventoy isk. In this pape, we povide a new contact fom with which the manufactue can (1) push inventoy to the etaile, known also as channel stuffing, (2) ceate a stict Paeto impovement ove the wholesale pice contact while inheiting the wholesale pice contact s simplicity, and (3) educe the manufactue s pofit vaiability. To do so, we popose a dual puchase contact that induces a etaile to place two consecutive odes; befoe and afte obtaining the final foecast update. A dual puchase contact specifies two pices: a pe unit advance puchase pice w a fo odes placed pio to the foecast update and the pe unit wholesale pice w fo odes placed afte the foecast update (and hence close to the sales season). The etaile often obtains the foecast update afte a majo tade show conducted close to the sales season as in the appael industy (see, fo example, Zaa case study 1 ). In this case, the advance puchase pice can be chaged fo each unit odeed pio to this tade show. Fist, we study the wholesale pice contact with which a pocue-to-stock etaile pays a manufactue w pe unit odeed. The manufactue poduces to satisfy the etaile s ode in full. The shot poduction lead time enables the etaile to wait and impove he foecast befoe finalizing he odeing decision. The manufactue has the capability to poduce at a cheape cost if the time pessue to build poducts is low. In othe wods, the manufactue can build at a cheape cost pio to eceiving final odes fom the etaile. We chaacteize this manufactue s optimal advance poduction quantity and the wholesale pice. We show that the manufactue optimally poduces in two batches. We chaacteize the optimal quantity fo the fist batch that is built to stock pio to eceiving an ode fom the etaile. Afte obtaining the foecast update, the etaile places an ode and the manufactue poduces the second batch if needed. We also show that the manufactue chages a highe wholesale pice when the poduct s maket potential is high and when poduction is costly. 1 Faiman, N, Singh, M., Zaa. Teaching case, Columbia Business School.

3 1152 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) Second, we study the dual puchase contact that induces the etaile to place an ode befoe obseving the foecast update and an additional ode afte the foecast update. We show that a discount fo advance odes; i.e., w a < w, induces the etaile to place an ode pio to obtaining the foecast update (o befoe the tadeshow). In paticula, the etaile optimally follows an ode-up-to policy. In addition, we show that a lowe advance puchase pice o a highe wholesale pice induces the etaile to place a lage ode befoe the foecast update and a smalle ode afte the foecast update. Given the etaile s best esponse, we chaacteize the manufactue s optimal advance poduction policy. We also show how a dual puchase contact enables the manufactue to push inventoy to the etaile and incease his expected pofit. Thid, we investigate scenaios though which the manufactue sets the dual puchase contact paametes. In the fist scenaio, the manufactue sets only the advance puchase pice while the wholesale pice is exogenous. Fo example, the global compute memoy pices fo DRAMs ae set by a spot maket. Lage etailes and DRAM manufactue agee on a wholesale pice based on the pevailing maket pice (Billington, 2002). Note, howeve, that the manufactue can decide whethe to povide a discount and the size of the discount fo an advance puchase at his own discetion. In the pesent pape, we establish the manufactue s optimal contact picing stategy. We also chaacteize the conditions unde which the manufactue ceates stict Paeto impovement ove the wholesale pice contact by using the dual puchase contact. In the second scenaio, the manufactue sets the wholesale pice in addition to the advance puchase pice. This case is obseved when the manufactue is the dominant paty who dictates the contact tems. Fo example, PS2 game consoles ae manufactued by Sony and the wholesale pice is set exclusively by Sony. 2 Fo this case, we show that the manufactue always pefes the dual puchase contact ove the wholesale pice contact. Ou analytical esults togethe with a numeical study show that maket uncetainty is a key facto that defines when the dual puchase contact povides stict Paeto impovement ove the wholesale pice contact. Next we show that the dual puchase contact impoves a manufactue s pofit even when he does not have advance poduction capability. Note that when the manufactue cannot poduce at a cheape cost fo ealy odes, he has no eason to (hence he does not) build any inventoy to stock. Supisingly, the dual puchase contact impoves even a build-to-ode manufactue s pofit. Finally, we study the impact of the dual puchase contact on a isk avese manufactue. With a wholesale pice contact, the manufactue s pofit is uncetain befoe the etaile commits to puchase any quantity. Reducing the esulting pofit volatility is often moe impotant than inceasing the expected pofit fo a manufactue when the manufactue cannot divesify his financial isk. This is often the case when he invests in specialized equipment to build a single poduct o when he elies heavily on a single etaile to sell his poduct. We show that the manufactue s pofit volatility can be loweed with a dual puchase contact. We chaacteize the theshold isk avesion level above which any isk avese manufactue would pefe a dual puchase contact ove a wholesale pice contact. We also show that the optimal advance puchase discount inceases with the manufactue s isk avesion facto. Ou esults detemine when a dual puchase contact, a simple pice only contact, ceates stict Paeto impovement ove a wholesale pice contact. Given the numbe of supplies, customes, and poducts a fim has to manage, pice-only contacts will continue to be the most common contacts in pactice because of thei simplicity. Hence, the dual puchase contact is a simple yet a poweful mechanism that inceases pofits while still being amenable to eal applications. 2. Liteatue eview Supply chain liteatue studying the inteaction between two fims often focuses on channel coodinating contacts, such as buy-back contacts, quantity flexibility contacts and evenue shaing contacts (Cachon, 2003). All these contacts involve tems othe than the pice. As Aow (1985) and Laiviee and Poteus (2001) point out, such contacts incu administative costs that ae not explicitly included in thei coesponding models. Many pactitiones also point out the difficulties associated with administeing complex contacts 2 Kleindofe and Wu, 2003 povides seveal othe examples of custom and commodity poducts fom vaious industies. A manufactue of a highly specialized o customized poduct is likely to set the wholesale pice.

4 (Billington and Kupe, 2003). Hence, we focus on a pice-only contact, the dual puchase contact, because of its implementability and study whethe it achieves stict Paeto impovement ove a wholesale pice contact when the supply chain obtains a foecast update. To do so, in Section 4 we extend the esults of Laiviee and Poteus (2001) to account fo a foecast update and advance poduction capability at the manufactue. Next in Section 5, we fully develop the dual puchase contact and chaacteize the manufactue s and etaile s optimal decisions. A numbe of papes examine the impact of ode timing on pofit impovements in a supply chain established by a wholesale pice contact. Cachon (2004) addesses inventoy isk shaing with a newsvendo model. The etaile can ode afte the demand ealization (in which case the manufactue faces inventoy isk) o befoe demand ealization (in which case the etaile faces inventoy isk). Iye and Begen (1997) and Feguson et al. (2005) conside the impact of foecast updating on the ode timing. Taylo (2006) addesses simila issues without a foecast update. Howeve, unlike the above papes, the etaile sets the selling pice. The autho also investigates the affect of etaile sales effot and infomation asymmety. None of the above authos conside the possibility of sequential decisions, two poduction modes, and the impact of foecast on poduction and odeing decisions both befoe and afte the foecast update. Donohue (2000) consides two poduction modes and focuses on etun option to achieve channel coodination. The pesent papes focus is on pice-only contacts, the optimal pices, and stict Paeto impovement ove the wholesale pice contact. Anothe steam of liteatue focuses on foecast infomation asymmety. Cachon and Laiviee (2001) and Öze and Wei (2006) stuctue contacts that enable cedible foecast infomation shaing between a manufactue and a etaile. Note, howeve, that the manufactue in the pesent pape does not need to obseve the foecast update fo his decisions. Hence, the fims in the pesent pape do not face an incentive poblem due to foecast update infomation. Fo a discussion on asymmetic infomation models in supply chains, we efe the eade to Chen (2003). A final goup of eseaches study the effect of advance odeing whee supply chain coodination is not an issue; i.e., eithe the etaile o the manufactue is the only decision make (Wheng and Pala, 1999; Bown and Lee, 1998; Gallego and Öze, 2001; Tang et al., 2003; Ehun et al., 2003). None of the above papes conside the effect of isk avesion. Eeckhoudt et al. (1995) study the compaative static of changes in pice and cost paametes fo a single isk avese newsvendo. Chen and Fedeguen (2001) conduct a mean-vaiance analysis of basic inventoy models and extend this analysis to include infinite hoizon inventoy models. Neithe Eeckhoudt et al. no Chen and Fedeguen conside the effect of isk avesion in the supply chain context. We study the effect of isk avesion in the supply chain context, and show the value of a dual puchase contact fo a isk avese manufactue. We oganize the est of the pape as follows. In Section 3, we pesent the demand model. In Section 4, we study the wholesale pice contact and chaacteize the etaile s optimal odeing policy, the manufactue s optimal poduction policy and his optimal wholesale pice. In Section 5, we chaacteize optimal policies unde the dual puchase contact. In Section 6, we show why the dual puchase contact impoves supply chain efficiency. In Sections 7 and 8, we chaacteize the effect of a dual puchase contact when the manufactue has only one poduction mode and when he is isk avese, espectively. In Section 9, we povide additional manageial insights though a numeical study. In Section 10, we conclude with possible futue eseach diections. 3. The demand model Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) Conside a supply chain with a manufactue and a etaile. The etaile buys a poduct fom the manufactue pio to a sales season. The manufactue poduces and satisfies all odes placed by the etaile in exchange fo a payment based on the contact tems. Next, the maket demand D is ealized and the etaile satisfies as much custome demand as possible fom available inventoy on hand. The etaile s sales pice is a fixed unit pice >0. Demand is of the fom D = X + whee X and ae both uncetain. Befoe the selling season stats, the etaile leans X, which can be intepeted as a foecast update and is possibly obtained afte a majo tade show o maket eseach. Pio to this maket eseach, X is a continuous andom vaiable with a cdf and a pdf of F(Æ) and f(æ), espectively. We assume that the suppot of f(æ) is[,l H ] whee l H > P 0. We

5 1154 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) Table 1 Glossay of notation Cost paametes c a : pe unit poduction cost befoe the foecast update is ealized c: pe unit poduction cost afte the foecast update is ealized w a : pe unit advance puchase pice w: pe unit wholesale pice : pe unit etail pice Uncetainty elated notation X: andom vaiable epesenting the foecast update l: ealization of foecast update X, l H : lowest and highest suppot of X F(Æ), f(æ): cdf and pdf of X : andom vaiable epesenting the maket uncetainty G(Æ), g(æ): cdf and pdf of Decision vaiables x: etaile s advance ode quantity x * : etaile s optimal advance ode quantity y: etaile s total ode quantity y * (l): etaile s optimal total ode quantity given the foecast update y cs (l): centalized system s optimal total ode quantity z: manufactue s advance poduction quantity z * : manufactue s optimal advance poduction quantity z cs : centalized system s optimal advance poduction quantity w * : manufactue s optimal wholesale pice w a : manufactue s optimal advance puchase pice use l to denote the ealization of X. The vaiable epesents the esidual maket uncetainty and is ealized afte the sales season. We model as a continuous, mean-zeo andom vaiable. Hence, the mean demand befoe obtaining the foecast update is l EX. We denote the cdf and pdf of with G(Æ) and g(æ), espectively. The suppot of g(æ) is[a,b) whee 6 a < b 6 1. This suppot ensues nonnegative demand. We assume that G(Æ) has an inceasing failue ate (IFR). Distibutions such as nomal, gamma, and exponential have IFRs. Fo an easy efeence, Table 1 summaizes the notation used thoughout the pape. The contactual ageement between the fims govens thei actions and the esulting pofits. We stat ou analysis with the wholesale pice contact. 4. Wholesale pice contact The sequence of events unde the wholesale pice contact is summaized in Fig. 1. (1) Paties agee on a wholesale pice w and sign the contact. Hee, we do not assume a paticula pocess by which the paties set the wholesale pice. We will study possible pocesses late in Section 4.3. (2) The manufactue decides how much to poduce in advance at a pe unit cost c a, i.e., befoe eceiving the etaile s ode. (3) The etaile obtains the foecast update l, which is the ealization of X, and decides how much to ode fom the manufactue. (4) The manufactue poduces an additional batch at a pe unit cost c, if necessay, and satisfies the etailes ode. At this stage poduction lead time equiement is shot because the poduction is initiated Fig. 1. Sequence of events unde the wholesale pice contact.

6 close to the sales season. Hence, the manufactue pe unit poduction cost is highe than his cost in the ealie poduction stage; i.e., c a < c. Finally, (5) maket uncetainty is ealized and the etaile satisfies demand fom on-hand inventoy at a fixed pice. We assume that w 2 [c,]; othewise it is neve pofitable fo the manufactue to poduce o the etaile to place any ode. Note that thee decisions ae made in a sequel: the wholesale pice, the manufactue s advance poduction quantity and the etaile s ode quantity. We chaacteize the optimal decisions by using a backwad induction algoithm; i.e., solve fo the last decision fist. The manufactue s poduction decision does not affect the etaile s odeing policy. Hence, the etaile s ode decision depends only on the wholesale pice. Theefoe, fist we solve the etaile s odeing decision fo a given wholesale pice. The manufactue s advance poduction decision depends closely on the etaile s ode decision and the wholesale pice. Hence, next we solve the manufactue s poduction poblem. The picing decision affects all othe decisions. Theefoe, we solve fo the optimal wholesale pice last The etaile s poblem Given an ode quantity y and the foecast update l, the etaile s expected pofit is P ðy; lþ ¼E ½minðy; l þ ÞŠ wy: ð1þ The etaile maximizes this newsvendo poblem and detemines he optimal ode quantity y ðlþ l þ G 1 w : ð2þ 4.2. The manufactue s poblem The manufactue can save fom poduction cost by initiating pat of his poduction befoe the foecast update. At the time of the advance poduction decision, howeve, the etaile s total ode is unknown. Hence, the manufactue tades off between excess inventoy and lowe poduction cost. Let z P 0 units be the manufactue s advance poduction quantity. Since the etaile odes only afte the foecast update, the manufactue builds all z units to stock. Given the advance poduction quantity and the etaile s optimal ode quantity, manufactue s expected pofit unde a wholesale pice contact is P m ðw; zþ we X y ðx Þc a z ce X ðy ðx ÞzÞ þ : ð3þ Next we chaacteize the manufactue s advance poduction policy. We defe all the poofs and some of the technical lemmas to Appendix A. Theoem 1 1. The manufactue optimally poduces z 1 cca ¼ F c þ G 1 w units in advance. 2. z * is deceasing in c a and z * 2 (y * ( ), y * (l H )). This theoem chaacteizes the manufactue s advance poduction policy. Pat 1 chaacteizes the optimal advance poduction quantity. Fom Eq. (2), the manufactue knows that the etaile will ode at least y * ( ) units. Pat 2 shows that the optimal poduction quantity is inceasing in the cost saving fom advance poduction. When the advance poduction cost is lowe than c, the manufactue optimally poduces moe than the minimum possible etaile ode. Hence, he faces excess inventoy isk. In paticula, when c a < c, the manufactue optimally stocks moe than y * ( ) Contact picing decision Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) So fa we have chaacteized the etaile s optimal ode size and the manufactue s optimal advance poduction quantity fo a given wholesale pice w. This scenaio is possible when the poduct is a commodity and the fims take the wholesale pice as given. Next, we study the scenaio in which the manufactue sets the

7 1156 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) wholesale pice as a Stackelbeg leade. The etaile esponds by setting he ode size. This scenaio is possible, fo example, when the manufactue builds a custom poduct and is the only poduce of this poduct. To solve fo the manufactue s optimal wholesale pice, we substitute the optimal advance poduction quantity z * chaacteized in Theoem 1 Pat 1 to Eq. (3) and solve fo w ag max w Pm ðw; z Þ: The next theoem chaacteizes the manufactue s optimal wholesale pice. Theoem 2 1. The manufactue s expected pofit in Eq. (3) is unimodal in w. Hence, the manufactue s optimal wholesale pice is w ¼ ð1 Gðv ÞÞ; whee v is the solution to ð1 GðvÞÞ 1 2. w * is inceasing 3 in l. 3. w * is inceasing in c a. gðvþðl þ vþ 1 GðvÞ c a ¼ 0: Pat 1 chaacteizes the manufactue s optimal wholesale pice. Pat 2 shows that the manufactue s optimal wholesale pice is highe when expected demand is high. Pat 3 eveals that educing the advance poduction cost enables the manufactue to educe his optimal wholesale pice. These esults show that the manufactue would be chaging a highe wholesale pice when the maket potential fo the poduct is high and when it is costly to poduce the poduct. The manufactue, fo example, can use a new technology o a pocess such as quick esponse initiatives to educe the cost of poduction. The above theoem shows that the benefit of such impovements would be shaed with the etaile though educed wholesale pices. 5. Dual puchase contact This contact specifies two pices, an advance puchase pice w a and the egula wholesale pice w. The etaile pays w a fo each unit that she odes befoe obseving the foecast update and w afte the foecast update. The sequence of events unde the dual puchase contact is summaized in Fig. 2. (1) Paties agee on w and w a. Hee, we do not assume a paticula pocess by which contact paametes ae set. Late, in Section 5.3, we addess how the contact tems ae set. (2) The etaile decides how much to ode in advance of obtaining the foecast update and pays w a pe unit of advance ode. (3) The manufactue decides how much to poduce in advance at a pe unit poduction cost c a. (4) The etaile obtains the foecast update l and decides how much moe to ode and pays w pe unit. (5) The manufactue poduces an additional batch at a pe unit pice c, if necessay, and satisfies the etaile s total ode. (6) Maket uncetainty is ealized and the etaile satisfies demand fom on-hand inventoy as much as possible at a pe unit pice. Note that unde this contact fou decisions ae made in a sequel: the dual puchase contact pices, the etaile s advance puchase quantity, the manufactue s advance poduction quantity, and the etaile s egula ode quantity. We solve fo the optimal decisions by using a backwad induction algoithm. The manufactue s poduction decision does not affect the etaile s ode quantity. Hence, fist we solve fo the etaile s poblem. The manufactue s advance poduction decision depends closely on the etaile s ode decision. Hence, second we solve the manufactue s poblem. The picing decision affects all othe decisions. Hence, we solve fo the contact-picing poblem last. 3 We use the tems inceasing and deceasing in the stong sense; i.e., inceasing means stictly inceasing.

8 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) Fig. 2. Sequence of events unde the dual puchase contact The etaile s poblem Given a dual puchase contact, the etaile has two oppotunities to place odes: befoe and afte the foecast update. While the etaile saves on pocuement cost by placing an advance ode, she obtains bette infomation by placing an ode afte the foecast update. Thus, the etaile faces a tadeoff between lowe cost and bette infomation. Let x be the etaile s ode quantity befoe the foecast update and y be the etaile s total ode quantity afte the foecast update. To maximize he pofit, the etaile solves the following dynamic optimization poblem: max xp0 P ðxþ ¼E X p ðx; X Þw a x; ð4þ whee p ðx; lþ ¼max ½minðy; l þ ÞŠ wðy xþ: ypx ð5þ When w a > w, the optimal advance puchase quantity is zeo because the etaile can puchase at a cheape pice afte obtaining the foecast update. This case is equivalent to the wholesale pice contact. Hence, in this section we conside w a 6 w. The following theoem summaizes the etaile s optimal policy. Theoem 3 1. The solution to Eq. (5) is max(y * (l),x), whee y 1 w ðlþ l þ G, and it is deceasing in w. 2. P (x) is concave in x and lim jxj!1 P (x) = 1. Hence, the objective function in Eq. (4) has a maximize x *, which is the etaile s optimal advance puchase quantity. This theoem shows that the etaile s optimal ode policy is an ode-up-to policy. Befoe obseving the foecast update, the etaile odes up to x * units. Afte the foecast update, if x * 6 y * (l), she odes up to y * (l). Theefoe, the etaile s additional ode quantity afte the foecast update is (y * (l) x * ) +. Notice that ode-up-to level y * (l) is the same as the optimal ode quantity unde the wholesale pice contact given in Eq. (2). Hence, when the manufactue offes a dual puchase contact, the etaile odes at least as much as what she would ode unde a wholesale pice contact. This esult implies that the dual puchase contact is a mechanism that induces the etaile to place additional odes. Next we futhe chaacteize the etaile s advance puchase quantity. Theoem 4 h 1. When w a 2 [c a,s(w)] whee sðwþ 1 R i l H Gðy ðl H ÞlÞf ðlþdl < w, the etaile odes only befoe the foecast update. In this case, x * is the solution to the fist ode condition: Z lh ½1 Gðx lþf ðlþdlšw a ¼ 0: ð6þ x * is deceasing in w a and x * P y * (l H ). When w a = s(w), we have x * =y * (l H ). 2. When w a 2 (s(w),w), the optimal advance puchase quantity x * is the solution to the following fist ode condition:

9 1158 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) " 1 w Z # xg 1 w ð Þ w 1 F x G þ ð1 Gðx lþþf ðlþdl w a ¼ 0: ð7þ x * is deceasing in w a, inceasing in w and x * 2 (y * ( ),y * (l H )). 3. When w a =w, x * can take any value between [0,y * ( )]. The etaile s total puchase quantity and he expected pofit unde the dual puchase contact and unde the wholesale pice contact ae equal. Hence, without loss of geneality, x * =0. 4. x * > y * ( ) if and only if w a < w. Pat 1 shows that when the advance puchase pice w a is below the theshold s(w), the advance puchase quantity is lage than y * (l H ). This esult, togethe with Theoem 3 Pat 1, implies the following. The etaile odes only befoe obseving the foecast update. Intuitively, the discount is deep enough to offset the etaile s gain fom waiting and obseving the foecast update befoe placing his ode. Pat 2 shows that when the advance puchase pice w a 2 (s(w), w), the optimal advance puchase quantity x * 2 (y * ( ),y * (l H )). Thus, given a discount, the etaile odes moe than y * ( ) units in advance and exactly (y * (l) x * ) + units afte obseving the foecast update. A highe wholesale pice induces a lage advance puchase quantity. Pat 3 shows that when w a = w, the etaile odes nothing in advance. Pat 4 states that a discount induces an advance puchase of x * > y * ( ) The manufactue s poblem Given the etaile s optimal esponse as chaacteized by Theoems 3 and 4, the etaile places pat of he ode befoe the foecast update. Hence, the manufactue s expected pofit unde a dual puchase contact can be witten as ( P m dp ðzþ ¼ w ax c a z cðx zþ þ ; w a 2½c a ; sðwþš; ð8þ w a x þ we X ðy ðx Þx Þ þ c a z ce X ðmaxfx ; y ðx Þg zþ þ ; w a 2ðsðwÞ; wš: To decide on the advance poduction quantity, the manufactue solves max zp0 Pm dp ðzþ: Next we chaacteize the advance poduction quantity z dp. Theoem 5 1. The pofit function P m dpðzþ is stictly concave in z. 2. When w a 2 [c a,s(w)], the optimal advance poduction quantity is z dp ¼ x. When w a 2 (s(w), w], we have þ G 1 w þ G 1 w z dp ¼ max x ; F. Fo all wa, we have z dp 1 cca c P F 1 cca c This theoem fully chaacteizes the manufactue s poduction policy. Pat 2 shows that the manufactue s optimal advance poduction is at least as much as the etaile s advance ode quantity. This makes intuitive sense because the manufactue s advance poduction cost is c a < c. Pat 2 also chaacteizes the manufactue s minimum advance poduction quantity; i.e., he optimally poduces at least F 1 cca c þ G 1 w units in advance Contact picing decision. Hee we solve fo the optimal contact paametes w a and w. We conside two possible scenaios. In the fist scenaio, the manufactue and the etaile take the wholesale pice w as given. Howeve, the manufactue decides whethe to offe the advance puchase pice w a. If he does, he maximizes his pofit by deciding on w a. In most supply chains, the fims establish a elationship, hence a supply chain, fo the poduct by ageeing

10 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) on a wholesale pice long befoe the actual poduction is initiated o any foecast update is obtained. This pice could also be set by the maket. Neithe the manufactue no the etaile may be able to enfoce the wholesale pice that maximizes thei own pofit. Howeve, the manufactue can offe an advance puchase pice on his own discetion. In the second scenaio, the manufactue as the Stackelbeg leade sets both w a and w. To obtain the optimal advance puchase pice, the manufactue solves the following poblem: w a agmax w a2½c a;wšp m dp ðw a; z dp Þ; ð9þ whee P m dp is defined in Eq. (8). Next we chaacteize the solution. h i Theoem 6. Let w (1 G(v )) whee v satisfies ð1 Gðv ÞÞ 1 gðv Þð þv Þ 1Gðv Þ ¼ When w 6 w, the optimal advance puchase pice is w a ¼ w. 2. When w > w, the optimal advance puchase pice is w a < w. The fist pat states that if the wholesale pice is low; i.e., w < w, then the manufactue should not offe an advance puchase discount and hence evet back to the wholesale pice contact (simply by setting w a = w). In this case, the etaile has no incentive to ode in advance (ecall fom Theoem 4 Pat 3). Howeve, when the wholesale pice is lage than the theshold w, it is optimal fo the manufactue to povide a discount fo odes placed pio to the foecast update. Theoem 7. The dual puchase contact with ðw a ; wþ inceases the manufactue s pofit ove the wholesale pice contact if and only if w > w. The etaile always has the option not to place any ode befoe the foecast update. Theefoe, the etaile is always bette off with a dual puchase contact ove a wholesale pice contact when w a < w. Hence, this theoem states that the manufactue can stictly impove his expected pofit as well as the etaile s expected pofit when he sets and offes the advance puchase pice. Now conside the scenaio in which the manufactue sets both w a and w. To do so, he solves P dp max w a;w ½Pm dp ðw a; w; z dp ÞŠ; ð10þ whee P m dp ðw a; w; z dpþ is as defined in Eq. (8). We have the following esult. Theoem 8. When the manufactue sets the wholesale pice, the optimal dual puchase contact always inceases his pofit ove the optimal wholesale pice contact. Hence, the manufactue always pefes the dual puchase contact when he sets both pices. Whethe the dual puchase contact also impoves etaile s pofit ove the wholesale pice contact is not analytically conclusive when the manufactue sets both pices. Howeve, in most of ou numeical expeiments in Section 9, the etaile s pofit was also highe unde dual puchase contact. In bief, Theoem 7 states that when the manufactue sets the advance puchase pice, the dual puchase contact impoves both the manufactue s and the etaile s pofits ove the wholesale pice contact, hence enabling stict Paeto impovement. Theoem 8 states that when the manufactue sets the advance puchase pice and the wholesale pice, the dual puchase contact always impoves the manufactue s pofit ove the optimal wholesale pice contact. 6. Supply chain efficiency This section povides the eason why the manufactue can achieve highe pofits with the dual puchase contact. To do so, we conside the centalized supply chain, fo which no intenal payment needs to be exchanged. In paticula, we compae the esulting optimal pofit of the centalized supply chain with the total supply chain pofit of the decentalized system; i.e., the sum of the manufactue s and the etaile s optimal

11 1160 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) expected pofits. When the centalized supply chain pofit and decentalized total pofit ae equal, we say the system is coodinated. The diffeence in pofits is a measue of decentalized supply chain s efficiency. The centalized system detemines fist the advance poduction quantity and the additional poduction quantity afte obtaining the foecast update. Let z be the centalized system s advance poduction quantity and y be the total poduction quantity afte the foecast update. The centalized system solves the following dynamic pogam: max zp0 Pcs ðzþ ¼E X p cs ðz; X Þc a z; whee p cs ðz; lþ ¼max E ½minðy; l þ ÞŠ cðy zþ: ypz The following theoem chaacteizes the centalized system s poduction policy. Theoem 9 ð11þ ð12þ 1. The solution to (12) is max(z,y cs (l)) whee y cs 1 c ðlþ ¼l þ G fo all l 2 [ll,l H ]. 2. P cs (z) is concave in z and lim jzj!1 P cs (z) = 1. Hence, the objective function in Eq. (11) has a maximize z cs, which is the centalized system s optimal advance poduction quantity. This theoem chaacteizes the centalized system s poduction policy to be a poduce-up-to policy. The centalized system poduces z cs befoe the foecast update. Afte the foecast update, if z cs 6 y cs (l), the centalized system poduces up to y cs (l). Next we chaacteize z cs. Theoem 10 h 1. When c a 2 [0,s cs (c)] whee s cs ðcþ 1 R i l H Gðy cs ðl H ÞlÞf ðlþdl < c, the centalized system poduces only befoe the foecast update. In this case, z cs is the solution to the fist ode condition: Z lh 1 Gðz lþf ðlþdl c a ¼ 0: ð13þ z cs is deceasing in c a and z cs P y cs (l H ). When c a = s cs (c), we have z cs =y cs (l H ). 2. When c a 2 (s cs (c), c), optimal advance poduction quantity z cs is the solution to the following fist ode condition: " 1 c Z # zg 1 ð c Þ c 1 F z G þ ð1 Gðz lþþf ðlþdl c a ¼ 0: ð14þ z cs is deceasing in c a, inceasing in c and z cs 2 (y cs ( ),y cs (l H )). 3. When c a =c,z cs can take any value between [0,y cs ( )]. Hence, without loss of geneality, z cs =0. 4. z cs > y cs ( ) if and only if c a < c. 5. z cs > F þ G 1 w fo all ca < c. 1 cca c The advance poduction and the total poduction quantity detemine supply chain pofit. Theefoe, to achieve channel coodination in a decentalized system, the advance poduction quantity must be equal to z cs and the total poduction quantity must be equal to max(z cs,y cs (l)) fo all l 2 [,l H ]. Fist conside the wholesale pice contact. It is well known that the wholesale pice contact does not coodinate even the supply chain without a foecast update, esulting in supply chain inefficiency (Pastenack, 1985). This contact also does not coodinate the supply chain discussed hee. To obseve this, note that the manufactue s advance poduction quantity unde this contact is less than the centalized system s advance poduction quantity (compae Theoem 1 Pat 1 with Theoem 10 Pat 5). Next conside the dual puchase contact. Can a dual puchase contact coodinate, if not, impove the efficiency of the supply chain? Theoem 11. Thee always exists a dual puchase contact unde which supply chain pofit is geate than the pofit unde a given wholesale pice contact. In othe wods, fo any w, thee exists w a such that

12 E X P ðy ðx Þ; X ÞþP m ðw; z Þ < P ðx ÞþP m dp ðz dpþ, whee each pofit function is defined in (1), (3),(4), and (8), espectively. This esult shows that a dual puchase contact mitigates the inefficiency caused by a wholesale pice contact. The wholesale pice contact causes double maginalization when w > c (Pastenack, 1985). The dual puchase contact mitigates the advese effect of double maginalization by poviding an additional oppotunity to ode at a discounted advance puchase pice w a < w. Theefoe, by offeing an advance puchase pice the manufactue can incease the supply chain efficiency, etain pat of the savings and leave the emaining to the etaile. 7. Without advance poduction Next we investigate whethe the stict Paeto impovement ove the wholesale pice contact is due to the advance poduction capability. In othe wods, can the manufactue and the etaile be bette off with a dual puchase contact than with a wholesale pice contact even when c a > c? To answe this question, fist note that when c a > c, the optimal advance poduction amount is zeo unde both contacts because the manufactue can poduce at a cheape cost afte obtaining etaile s total ode. Hence, the manufactue poduces in full afte the etaile places he ode. Note that the etaile s ode policy emains the same as befoe. She follows the optimal odeing policy chaacteized in the pevious sections fo both contacts. Hence, the only decision we need to analyze is the manufactue s picing decision. We stat with the wholesale pice contact. The manufactue s expected pofit unde the wholesale pice contact can be witten as P m ðwþ ¼ðwcÞE X y ðx Þ: ð15þ The next theoem chaacteizes the manufactue s optimal. Theoem The manufactue s pofit in Eq. (15) is unimodal in w. Hence, if he can, the manufactue sets the wholesale pice to w ¼ ð1 Gðv ÞÞ; whee v is the solution to ð1 GðvÞÞ 1 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) gðvþðl þ vþ 1 GðvÞ c ¼ 0: We denote the solutions of the above equations with w L and v L when the pobability of facing the wost possible foecast ealization is one (Pob{X = } = 1), and hence l ¼. 2. w * is inceasing in l and w P w L. 3. w * is inceasing in c. Fom Pat 1, if the manufactue knows with cetainty that the etaile will face the wost possible sales season; i.e., if X = with pobability one, then he optimally sets the wholesale pice equal to w L. This wholesale pice is the lowest pice that the manufactue would offe. Pat 2 shows that the manufactue optimally offes a highe wholesale pice when the foecast update is expected to be high. Pat 3 shows that the manufactue optimally chooses a highe wholesale pice if his egula poduction cost inceases. Without advance poduction, the manufactue s expected pofit unde the dual puchase contact simplifies to the following: P m dp ðw aþ¼ ðw a cþx ; w a 2½c; sðwþš; ðw a cþx þðwcþe X ðy ðx Þx Þ þ ð16þ ; w a 2ðsðwÞ; wš: Theoem 13. When c a > c, the dual puchase contact with w a inceases the manufactue s pofit ove the wholesale pice contact if and only if w > w L. This theoem shows that stict Paeto impovement unde the dual puchase contact is achievable even fo a manufactue that does not have advance poduction capability. In paticula, if the manufactue can set

13 1162 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) the advance puchase pice, he can ceate a stict Paeto impovement ove any wholesale pice contact by offeing the advance puchase pice as long as w > w L. The stict Paeto impovement in this case is also due to having two ode oppotunities. The discounted advance puchase pice educes the advese affect of double maginalization. 8. Manufactue s isk attitude So fa, we have shown how a dual puchase contact inceases expected pofits. Ou next aim is to chaacteize how the dual puchase contact also educes the manufactue s pofit volatility and how it affects a isk avese manufactue. Unde a wholesale pice contact, the manufactue s pofit is uncetain pio to the foecast update. This uncetainty in pofit pojection often discouages capital investment. The manufactue can, howeve, educe this uncetainty by using a dual puchase contact. Theoem 14. Unde a dual puchase contact whee c a > c, the vaiance of the manufactue s pofit is Va dp (w a ) (w c) 2 Va(y * (X) x * ) +, which is less than o equal to the vaiance of the manufactue s pofit unde the wholesale pice contact. This vaiance equals zeo when w a 2 [c,s(w)] and it is inceasing in w a when w a 2 (s(w), w]. Note that the vaiance depends on w a though the etaile s optimal advance puchase quantity x *. A dual puchase contact educes the manufactue s pofit volatility and, hence, enables isk hedging. If a manufactue is isk avese, then he can detemine the optimal advance puchase pice by tading off expected pofit and pofit vaiance. The mean-vaiance tadeoff is widely used in potfolio theoy and can be econciled with the expected utility appoach by using a quadatic utility function. Conside utility function uðxþ ¼ax 1 2 bx2 (a >0,b P 0,x 6 a/b). When x is andom, let x EðxÞ. The expected utility is E½uðxÞŠ ¼ ax 1 2 bx2 1 bvaðxþ. 2 Clealy, fo all feasible x s with the same expected value, the optimal one must have minimum vaiance. Altenatively, note that maximizing expected utility is equivalent to maximizing the cetainty equivalent, which can u 00 ðxþ u 0 ðxþ VaðxÞ (Luenbege, 1998, p. 256). Fo utility functions with constant abso- be appoximated as c x þ 1 2 lute isk avesion k, wehave u00 ðxþ ¼k. u 0 ðxþ Specifically, the manufactue solves the following optimization poblem: max P m a ðw aþp m dp ðw aþkva dp ðw a Þ: ð17þ w a Note that, given the etaile s esponse, the manufactue s objective function is ( P m a ðw aþ¼ ðw a cþx ; w a 2½c; sðwþš; ðw a cþx þðwcþe X ðy ðx Þx Þ þ kðw cþ 2 Va X ðy ðx Þx Þ þ ; w a 2ðsðwÞ; wš: ð18þ The coefficient k eflects the manufactue s isk attitude. When k = 0, the manufactue is isk neutal as in the pevious section. When k > 0, the manufactue is isk avese and a lage k implies that the manufactue is moe isk avese. When k < 0, the manufactue is isk seeking. Theoem 15. Let k wcy ð Þgðy ð Þ Þ. 2ðwcÞ 2 ðl Þ 1. When k > k, the manufactue pefes the dual puchase contact with ðw a ; wþ to a wholesale pice contact with w. The optimal advance puchase pice w a is deceasing in k. 2. When k 6 k, the manufactue pefes the wholesale pice contact with w to a dual puchase contact with (w a,w). The theoem states that when the manufactue s isk avesion facto is above the theshold k, he always pefes a dual puchase contact. The theoem also chaacteizes the isk avesion theshold above which any manufactue would pefe the dual puchase contact ove the wholesale pice contact. Futhemoe,

14 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) to incease the advance puchase quantity and hence educe his pofit volatility, a manufactue with highe isk avesion offes a geate discount. 9. Numeical examples The pupose of this section is to illustate some of ou esults and numeically compae the pofits unde the dual puchase contact to the pofits unde the wholesale pice contact. Ou base case is c a = 2.5, c = 3 and = 10. The foecast update X follows the tuncated nomal distibution on [8, 24] with mean 16 and standad deviation X. The maket uncetainty also follows the tuncated nomal distibution on [8,8] with mean 0 and standad deviation. The tuncated nomal distibution has IFR. Conside the picing scenaio (discussed in Section 5.3) in which the manufactue sets the advance puchase pice w a while w is exogenously set by the maket. We have shown that the manufactue achieves stict Paeto impovement ove the wholesale pice contact by offeing a dual puchase contact as long as w > w (Theoem 7). Hence, the set of wholesale pices fo which the dual puchase contact povides stict Paeto impovement is lage when w is smalle. Table 2 povides the manufactue s and the etaile s expected pofits unde vaious paamete settings. Both fims expected pofits ae highe unde the dual puchase contact. Hence, the manufactue ceates a stict Paeto impovement o a mutually beneficial poposition by intoducing the dual puchase contact. Table 2 also illustates the sensitivity of w with espect to the vaiability in the foecast update X and maket uncetainty. In these expeiments, we keep the suppots and means of X and fixed while vaying the value of X and. Note that w deceases as inceases. Theefoe, the dual puchase contact achieves stict Paeto impovement fo a lage set of wholesale pices when the maket uncetainty is moe vaiable. Note also that w depends on (Theoem 6) but not on X. In summay, the maket uncetainty plays a citical ole in defining when the manufactue should conside using a dual puchase contact instead of a wholesale pice contact. Table 2 also illustates the pecentage incease in total supply chain pofit by switching to the dual puchase contact (see the columns titled as % ). Notice that the pecentage impovement is highe when the atio X / is low. This atio epesents the infomative value of the foecast update. A low X / value implies that most of the demand uncetainty emains to be esolved afte the foecast update. Theefoe, ou expeiments suggest that the dual puchase contact impoves the supply chain moe when the foecast update is not vey infomative. If the manufactue knows that the tade show o the maket eseach conducted to obtain a foecast update is likely to be not infomative, then he should highly conside offeing the dual puchase contact. Conside next the picing scenaio in which the manufactue sets both w a and w. When the manufactue offes the wholesale pice contact, he sets w to maximize his pofit in Eq. (3). His optimal wholesale pice w * is Table 2 Paeto impovement when w = 9,8,7, and 6 and c a = 2.5, c =3, =10 Wholesale w = 9 Dual puchase ðw a ; 9Þ % Wholesale w = 8 Dual puchase ðw a ; 8Þ % X / w P m P P m dp P w a P m P P m dp P w a w =7 ðw a ; 7Þ % w =6 ðw a ; 6Þ %

15 1164 Ö. Öze et al. / Euopean Jounal of Opeational Reseach 182 (2007) Table 3 Paeto impovement when manufactue sets w X X / Wholesale Dual puchase % P m P w * P m dp P (w a,w) * c a =2,c =3, = (5.29, 5.62) (5.28, 5.58) (5.40, 5.72) (5.27, 5.62) (5.26, 5.58) (5.38, 5.72) c a = 2.5, c =3, = (5.48, 5.58) (5.45, 5.54) (5.61, 5.82) (5.48, 5.56) (5.45, 5.56) (5.60, 5.88) 7.04 c a =2,c =3, = (8.56, 9.16) (8.43, 8.98) (8.43, 8.98) (8.54, 9.18) (8.42, 9.00) (8.42, 9.00) 7.80 c a = 2.5, c =3, = (8.81, 9.00) (8.65, 8.88) (8.64, 9.12) (8.80, 9.00) (8.62, 8.94) (8.63, 9.24) 5.80 chaacteized in Theoem 2 Pat 1. When he offes the dual puchase contact, he sets both w a and w to maximize his pofit. He solves the optimization poblem in Eq. (10). In Table 3, we compae the pofits esulting fom the manufactue s optimal choices. As Theoem 8 poves, the manufactue s pofit is always highe unde the dual puchase contact. In all the expeiments, the etaile s pofit is also highe unde the dual puchase contact. The pecentage impovement in supply chain pofit tends to incease when X / is low. Compaing the c a = 2.5, c = 3 and the c a =2,c = 3 cases, we note that the pecentage impovement is highe when the cost diffeence between the advance poduction and nomal poduction is high. Intuitively, the dual puchase contact allows the supply chain to enjoy low poduction costs by poviding the etaile incentive to place most of he need though advance odes. 10. Conclusion We study a supply chain in which the etaile obseves a foecast update and the manufactue poduces to satisfy the etaile s ode. Unde a wholesale pice contact, the etaile places odes only afte obseving the foecast update. Peseving the simplicity, we study anothe pice-only contact, the dual puchase contact, which povides the etaile with incentive and flexibility to ode both befoe and afte the foecast update. We show that the dual puchase contact inceases supply chain pofit as well as the etaile s pofit, and identify conditions unde which it also inceases the manufactue s pofit and hence ceates a stict Paeto impovement. The analytical esults togethe with the numeical examples suggest that the dual puchase contact is