Promised LeadTime Contracts Under Asymmetric Information


 Robert Wright
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1 OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp issn 3364X eissn infoms doi /ope INFORMS Pomised LeadTime Contacts Unde Asymmetic Infomation Holly Lutze Univesity of Texas at Dallas, Richadson, Texas 9385, Özalp Öze Columbia Univesity, New Yok, New Yok 127, We study the impotant poblem of how a supplie should optimally shae the consequences of demand uncetainty (i.e., the cost of inventoy excesses and shotages) with a etaile in a twolevel supply chain facing a finite planning hoizon. In paticula, we chaacteize a multipeiod contact fom, the pomised leadtime contact, that educes the supplie s isk fom demand uncetainty and the etaile s isk fom uncetain inventoy availability. Unde the contact tems, the supplie guaantees ontime delivey of complete odes of any size afte the pomised lead time. We chaacteize the optimal pomised lead time and the coesponding payments that the supplie should offe to minimize he expected inventoy cost, while ensuing the etaile s paticipation. In such a supply chain, the etaile often holds pivate infomation about his shotage cost (o his sevice level to end customes). Hence, to undestand the impact of the pomised leadtime contact on the supplie s and the etaile s pefomance, we study the system unde local contol with full infomation and local contol with asymmetic infomation. By compaing the esults unde these infomation scenaios to those unde a centally contolled system, we povide insights into stock positioning and inventoy isk shaing. We quantify, fo example, how much and when the supplie and the etaile oveinvest in inventoy as compaed to the centally contolled supply chain. We show that the supplie faces moe inventoy isk when the etaile has pivate sevicelevel infomation. We also show that a supplie located close to the etaile is affected less by infomation asymmety. Next, we chaacteize when the supplie should optimally choose not to sign a pomised leadtime contact and conside doing business unde othe settings. In paticula, we establish the optimality of a cutoff level policy. Finally, unde both full and asymmetic sevicelevel infomation, we chaacteize conditions when optimal pomised lead times take exteme values of the feasible set, yielding the supplie to assume all o none of the inventoy isk hence the name allonothing solution. We conclude with numeical examples demonstating ou esults. Subject classifications: inventoy/poduction: multiitem/echelon/stage; uncetainty: stochastic; opeating chaacteistics; game theoy: mechanism design; finite types; sceening. Aea of eview: Manufactuing, Sevice, and Supply Chain Opeations. Histoy: Received Apil 24; evisions eceived Novembe 25, August 26, Januay 27; accepted Januay Intoduction Conside a supply chain with a supplie and a etaile. Both fims face positive eplenishment lead times. Inventoies at both locations ae managed peiodically ove a finite planning hoizon. Custome demand is satisfied only though the etaile. The objective of each fim is to minimize the odeing, holding, and shotage costs ove the planning hoizon due to demand and supply uncetainty. In such a supply chain, the supplie pefes the etaile to place odes well in advance of his equiement. Howeve, the etaile pefes the supplie to fulfill odes immediately without facing any backlog at the supplie s site. Hence, the supply chain faces an incentive poblem in which both the supplie and the etaile want the othe paty to bea the consequences of demand uncetainty. How should this supply chain s inventoy be managed? To addess the need fo pope shaing of expected inventoy cost, we conside a pomised leadtime contact. Unde this contact, the etaile places advance odes with the supplie. The supplie guaantees shipment of each ode on time and in full afte a pomised lead time. A pomised leadtime contact eliminates the etaile s isk fom uncetain supply, but extends the etaile s foecast hoizon beyond his standad eplenishment lead time. This contact also povides the supplie with advance odes, theeby deceasing he isk fom uncetain demand. A costbenefit analysis of this inteaction, and the esulting inventoy costs, detemine who pays fo the pomised leadtime ageement. Singlesoucing elationships ely on issues such as contingency planning, inventoy, and ontime delivey, so the study of pomised leadtime contacts is both elevant and timely (Hause 23). Billington (22) and Cohen et al. (23) epot that among the most common foms of contacts ae those specifying lead times. Also, because planning systems such as MRP (mateials esouce planning) stipulate a delivey lead time, puchasing agents ae often diven by a delivey lead time when placing odes. These business pocesses equie fims to agee on delivey lead times hence the name pomised lead time. 898
2 Opeations Reseach 56(4), pp , 28 INFORMS 899 In this pape, we chaacteize the optimal pomised lead time and the coesponding payment that the supplie should offe to minimize he expected inventoy cost while ensuing the etaile s paticipation. To do so, we fist chaacteize both fims optimal stocking and eplenishment decisions unde a given pomised leadtime contact. To undestand the impact of the pomised leadtime contact on the supplie s and the etaile s pefomance, we study the supply chain unde thee contol mechanisms: cental contol, local contol with full infomation, and local contol with asymmetic infomation. By compaing the esults unde these diffeent contol and infomation scenaios, we also povide insights into stock positioning and inventoy isk shaing. Unde cental contol, the twostage system s inventoy and infomation is managed by a single decision make. The manage optimally allocates inventoy to minimize the oveall expected inventoy cost. Hence, this system does not face the afoementioned incentive poblem. Clak and Scaf s (196) seminal pape addesses how to optimally manage this seial system. In this pape, we povide closedfom solutions fo allocating inventoy to each stage. These solutions ae new and povide insights into stock positioning though simple speadsheet calculations. They also seve as a compaison point fo a supply chain unde local contol, which is the main focus of this pape. Unde local contol with full infomation, each fim locally contols its own inventoy. The supplie offes the pomised leadtime contact while having full infomation about the etaile s inventoyelated costs. We show that the optimal pomised lead time is deceasing in the etaile s shotage cost o, equivalently, his sevice level to end customes. In othe wods, the supplie optimally faces lage inventoy cost when woking with a etaile who povides high sevice to customes. We show the conditions unde which the supplie s choice of pomised lead time geneates the supply chain optimal pomised leadtime contact. We also chaacteize the optimal stocking levels unde this egime. By compaing these esults to the cental contol case, we quantify how much and when the supplie and the etaile ove o undeinvest in inventoy. Unde local contol, anothe incentive poblem may aise. Specifically, the etaile can influence the supplie s contact design by exaggeating the seveity of his inventoyelated costs fom demand uncetainty. The inventoy holding costs ae elatively easie to assess because they ae mainly based on factos obsevable to the public. Howeve, companies often state shotage cost as a stategic cost neve to be evealed to competitos and supplies. Duing ou inteactions with a telecommunication equipment distibuto and a data stoage technology povide, we obseved that these two fims employ diffeent accounting fomulas that descibe shotage cost. The inputs to these fomulas ae mainly classified unde two categoies of data: maket specific and company specific. The maketspecific factos influencing a etaile s unit shotage cost consist of factos such as custome patience and the amount of goodwill loss. The confidential companyspecific factos stem fom the opinions of executives egading the company s competitive stategy, maket position, and assessment of inventoy isks. These factos ae not obsevable by othe fims. The etaile tanslates both maketspecific and companyspecific factos into shotage cost by using a fomula that is also unobsevable to the othe fims. Hence, the etaile often keeps pivate infomation egading his shotage cost. We efe to this case as asymmetic shotage cost infomation. Thoughout the pape, we use sevice level and shotage cost intechangeably because in ou setting a sevice level implies a shotage cost, and vice vesa. Unde the afoementioned asymmetic infomation case, the supplie may fist ty to obtain the cost infomation, eithe by asking the etaile o by obseving the etaile s eplenishment odes. Howeve, we show that the etaile has an incentive to exaggeate his sevice level when asked fo this infomation. We also show that the supplie cannot lean the etaile s sevice level by obseving etaile odes ove time. Hence, cedible infomation shaing is not possible without a pope incentive mechanism. In this pape, we stuctue a contact mechanism that the supplie can offe to obtain cedible sevicelevel infomation while minimizing he inventoyelated cost ove the planning hoizon. Unde this mechanism, the supplie offes a menu of pomised lead times with coesponding payments to the etaile. The etaile voluntaily chooses a pomised lead time and pays K each peiod to guaantee contact tems. We also compae the esulting inventoy costs and allocation unde this mechanism to those unde the fullinfomation scenaio. This compaison povides insights into the value of knowing the etaile s sevice level though the use of technologies such as RFID (adio fequency identification device) and compliance management systems. 1 We also compae the esulting inventoy allocation to those unde cental contol. Next, we chaacteize when the supplie should optimally fogo establishing a supply chain by inducing the etaile to sign a pomised leadtime contact. This situation aises when the supplie is not willing to incu an inventoy cost that is lage than he esevation cost. In paticula, we show that the supplie optimally induces the etaile to sign the pomised leadtime contact only when the etaile s sevice level to end customes is lowe than a cutoff level. We also show that the cutoff level is inceasing in both the supplie s and the etaile s esevation costs. In addition, we chaacteize the optimality of the cutoff policy. Finally, we chaacteize the conditions unde which optimal pomised lead times unde both full and asymmetic sevicelevel infomation take exteme values of the feasible set, yielding the supplie to assume all o none of the inventoy isk; i.e., an allonothing solution. We conclude with numeical examples demonstating ou esults. The emainde of this pape is oganized as follows. In 2, we eview elated eseach. In 3, we explicitly
3 9 Opeations Reseach 56(4), pp , 28 INFORMS descibe the supply chain model and the pomised leadtime contact. In 4, we fomulate the poblem unde cental contol. In 5, we study the supplie s and the etaile s inventoy decisions unde a pomised leadtime contact. We analyze the supplie s optimal pomised leadtime contact unde local contol with full and asymmetic sevicelevel infomation. In 6, we compae the esulting inventoy levels unde local contol to those unde cental contol. In 7, we study a special case: an allonothing type inventoy iskshaing ageement. In 8, we povide numeical examples. Finally, in 9, we conclude. 2. Liteatue Review Effective inventoy management and shaing the consequences of demand uncetainty among supply chain membes ove a planning hoizon is one of the most fundamental and impotant poblems in opeations management. Howeve, the liteatue on contacting to shae inventoy isk mainly consides a singlepeiod inteaction in a twolevel supply chain (Cachon 23). To the best of ou knowledge, Cachon and Zipkin (1999) is the fist pape that povides contact mechanisms designed fo two fims facing a long inventoyplanning hoizon. In this espect, ou pape is elated to theis. Howeve, we addess a finitehoizon poblem unde infomation asymmety and conside a diffeent contact mechanism. They study an aveage cost citeion unde full infomation, but they also conside competition. In thei pape, the etaile faces shotages due to supply estictions at the supplie. The supplie pays a faction of the penalty cost fo shotages to the etaile. In ou pape, the etaile is guaanteed ample supply. Essentially, the contact mechanism, infomation, and supply chain stuctues studied in ou and thei papes ae diffeent. Lee and Whang (2) and Poteus (2) also study incentive issues in a seial supply chain. Lee and Whang discuss how Clak and Scaf s (196) optimal eplenishment policy fo a centally managed seial system can be implemented by manages located in each stage. Poteus (2) povides a way of implementing the scheme in Lee and Whang (2). Unlike ou pape, both of these papes assume that a cental decision make fist detemines the optimal inventoy contol policy. They also assume that inventoies and cost infomation at each location ae monitoed and shaed among all manages in the supply chain. Full infomation, cental contol, and monitoing ae stong assumptions. Often, supply chain patnes hold pivate infomation. The infomed patne may use this infomation to impove pofits, at the expense of the othe patne. The lessinfomed patne can solve an advese selection, o sceening poblem, to devise an optimal contact mechanism that minimizes infomation ent. Examples of advese selection poblems in the opeations liteatue can be found in Cachon and Laiviee (21), Cobett (21), Ha (21), Öze and Wei (26), and Cachon and Zhang (26). Fo a compehensive suvey, we efe the eade to Chen (23). Within this steam of liteatue, the closest wok to ous is the wok of Cobett (21). The autho consides optimal contacts between a supplie and a etaile following a (Q) inventoy contol policy to minimize aveage cost. The supplie chooses the batch size Q; the etaile detemines the eode point. When the etaile has pivate infomation about his backode penalty cost, the supplie offes an optimal contact mechanism while sceening the etaile s type. Simila to Cobett (21), we study the impact of infomation asymmety egading the backode penalty cost. Most supply chain papes on sceening implicitly assume that the pincipal wants to induce the agent to establish a supply chain by ageeing to the contact tems. In this pape, we also conside the supplie s (the pincipal s) paticipation constaint explicitly as in Cobett and de Goote (2) and Ha (21). These authos chaacteize cutoff level policies. As in Ha (21), we also chaacteize the optimality of the cutoff policy. Incidentally, management consultants also identify the need to segment etailes with diffeing sevicelevel equiements and to adapt supply chain policies to seve each segment with maximum pofitability (Andeson et al. 1997, Gadiesh and Gilbet 1998). Cohen et al. (23) addess this poblem in a sevice pats supply chain. Finally, a goup of eseaches exploes delivey leadtime commitments and quotations unde nonstategic settings. Fo example, BanesSchuste et al. (26) study a twolevel supply chain unde full infomation and nomally distibuted demand. They povide conditions unde which the etaile o supplie should hold the entie inventoy. This esult is elated to the one in 7 fo full infomation. See also Keskinocak et al. (21), Wang et al. (22), and the efeences theein. 3. The Model Hee we descibe thee impotant aspects of the model: the twostage supply chain, the pomised leadtime contact, and the contol mechanism unde diffeent infomation scenaios TwoStage Supply Chain We study the twostage supply chain shown in Figue 1. Uncetain endcustome demand is satisfied though the etaile. Demand D t in each peiod t is modeled by a sequence of nonnegative, independent and identically distibuted (i.i.d.) andom vaiables dawn fom a stationay distibution with c.d.f. F, density f, and finite mean Figue 1. Ample inventoy Twostage supply chain. h s, p s h, p L Stage 1 I Stage 2 D c s Supplie c t ~F( ) Retaile
4 Opeations Reseach 56(4), pp , 28 INFORMS 91. The supplie and the etaile both know the demand distibution. 2 We denote demand ove the cuent and the next n peiods as 3 D n+1 t+n k=t D k. The inventoy at each stage is eviewed peiodically. The sequence of events is as follows. Both the supplie and the etaile eceive thei espective shipments at the beginning of a peiod. The supplie odes fom an upsteam fim with ample inventoy and will eceive these odes L peiods late. Shipments fom the supplie to the etaile take l peiods to aive. The supplie and etaile incu linea cost c s > and c >, espectively, pe item odeed, and no fixed cost fo placing an ode. At the end of the eview peiod, custome demand is ealized. The etaile satisfies demand though onhand inventoy. Unsatisfied demand is backlogged. Backodes of endcustome demand incu a unit penalty cost p pe peiod only at the etaile. The supplie incus eithe an explicit o an implicit shotage cost based on the contol stuctue we specify late. The supplie and the etaile incu unit installation holding cost h s > and h >, espectively, whee h s h fo any inventoy emaining at the end of each peiod. At the end of peiod T, leftove inventoy (espectively, backlog) is salvaged (espectively, puchased) at a linea peunit value of c s and c at each stage, espectively. This sequence and set of assumptions ae the classical ones potayed, fo example, in Veinott (1965) Pomised LeadTime Contact A pomised leadtime contact has two paametes: pomised lead time and coesponding pepeiod lumpsum payment K. Unde this contact, the etaile places each of his odes peiods in advance of his needs. The supplie guaantees to ship this ode, in full, afte peiods. The etaile must wait anothe l peiods (shipping time) befoe eceiving the guaanteed ode. To do so, the supplie aanges an altenate soucing stategy to fill any etaile demand that exceeds the supplie s onhand inventoy. The supplie boows emegency units fom this altenative souce and incus penalty p s pe unit pe peiod until the altenative souce is eplenished. 4 Unde this contact, the supplie is esponsible fo the supply isk fo a finite planning hoizon. The analysis of this inteaction and the esulting inventoy cost detemine the lumpsum payment between the fims. When K is positive, we intepet this tansaction as a payment fom the etaile to the supplie. The effect of a pomised lead time is to shift inventoyelated costs due to demand uncetainty fom the supplie to the etaile. With a pomised lead time, the supplie leans the etaile s ode peiods in advance. On one hand, fo the supplie, a pomised lead time changes the numbe of peiods of demand uncetainty fom L + 1to L + 1. On the othe hand, fo the etaile, a pomised lead time inceases the numbe of peiods of demand uncetainty fom l + 1 to l Note that when =, the etaile demands immediate shipment of all odes, and the supplie s opeation is completely buildtostock. When = L + 1, exceeding the supplie s eplenishment lead time, the supplie builds to ode fo the etaile and does not need to cay any inventoy. Hence, any easonable pomised lead time would satisfy L System Contol Ou base case is the system unde cental contol without a pomised leadtime contact. Fo this case, a cental decision make, who has full infomation about the fims opeations, sets the inventoyodeing policy fo both the supplie and the etaile. In paticula, she decides how to allocate inventoy within the system so as to minimize the total expected inventoy cost. The supplie satisfies the etaile s ode fom inventoy on hand at the supplie s location. Any unsatisfied ode is backlogged. Hence, the etaile faces supply isk. This inventoy contol poblem is the classical Clak and Scaf (196) model discussed in 4. Ou main case is the system unde local contol. This system consists of two distinct decision makes: the supplie and the etaile. Each fim s goal is to minimize its own expected inventoy cost by placing a eplenishment ode based on its onhand and pipeline inventoy and backlog levels hence the name local contol. Unde the fullinfomation scenaio i.e., when the supplie knows the etaile s inventoy odeing, holding, and shotage costs the supplie designs a pomised leadtime contact and offes it to the etaile. If the etaile accepts the contact, both fims ae committed to the pomised leadtime contact tems fo a finite hoizon. The etaile, howeve, often has pivate infomation about his endcustome sevice level. Unde this asymmetic infomation scenaio, the supplie offes a menu of contacts that minimizes he expected inventoy cost. The etaile chooses a pomised leadtime contact that minimizes his expected inventoy cost. Afte the contacting stage, unde both full and symmetic infomation, the sequence of events is as descibed in 3.1, and each fim independently decides how to eplenish its local onhand inventoy to minimize its own expected inventoy cost. Note that a thidpaty decision make who has full infomation may also choose the optimal pomised leadtime contact fo the entie supply chain, while the supplie and the etaile set thei own stocking levels. This thidpaty decision make s contact choice is what will be efeed to as the fistbest contact unde local contol. 5 We addess the local contol system in 5. Unde both contol systems, each fim optimally follows a basestock policy as discussed next. Hence, we descibe the eplenishment decisions as choosing stocking levels. 4. Cental Contol and the SystemOptimal Solution The systemoptimal solution minimizes the total expected, discounted inventoy cost fo a finitehoizon, peiodiceview inventoy contol poblem in seies. Clak and Scaf (196) show that an echelon basestock policy is
5 92 Opeations Reseach 56(4), pp , 28 INFORMS optimal fo this classical seial system. Othe papes povide simple poofs, efficient computational methods, and extensions to the infinitehoizon case (see, fo example, Fedeguen and Zipkin 1984, Chen and Zheng 1994). Fo a stationay seial system, Gallego and Öze (23) show that a myopic basestock policy is optimal. They show how to allocate costs to fims (stages) in a cetain way to obtain optimal basestock levels. Next, we summaize this method. Let x e jt and y e jt be fim j s echelon inventoy position 6 befoe and afte odeing, espectively, in peiod t, whee j = s epesents the supplie and j = the etaile. Also, let L y e t = 1 c y e t + l E [ h h s y e t Dl+1 + p + h y e t Dl+1 ] (1) be the etaile s expected inventoy cost in peiod t whee is the discount facto. Define y m as the smallest minimize of L. This minimize is the etaile s optimal basestock level. Next, define the implicit penalty cost IP m y = L miny m y L y m (2) The supplie s expected cost chaged in peiod t is then L s y e st = 1 c sy e st + L Eh s y e st DL+1 + IP m y e st DL+1 (3) The supplie s optimal echelon basestock level ys m is the smallest minimize of L s. Unde this echelon basestock policy, fim j odes a sufficient amount in each peiod to aise its echelon inventoy position xjt e to the echelon basestock level ym j fo j s. An equivalent optimal policy, known as an installation basestock policy, is given by installation basestock levels Y s = maxys m y m and Y = minys mym. Let x jt and y jt be fim j s installation inventoy position 7 befoe and afte odeing in peiod t, espectively. Fim j odes a sufficient amount in each peiod to aise its installation inventoy position x jt to the installation basestock level Y j fo j s (see, fo example, Axsäte and Rosling 1993, Chen and Zheng 1994 fo a discussion on the equivalence of installation and echelon basestock policies fo a seial system). 5. Local Contol with Pomised LeadTime Contact Unde local contol, the supplie offes a pomised leadtime contact to the etaile. If the etaile accepts, the fims establish an inventoy iskshaing ageement fo T peiods. Next, the fims choose thei espective optimal stocking levels. The emainde of the sequence of events is as descibed in 3.1. We use backwad induction to chaacteize the optimal decisions; i.e., we solve fo the optimal inventoystocking decisions, followed by the contacting decision The Supplie sand the Retaile s Inventoy Poblem Fo a given pomised leadtime contact (K), each fim minimizes its expected inventoy cost ove the next T peiods. The pepeiod lumpsum payment K is independent of the ode quantity and inventoy on hand; hence, it has no effect on the inventoy eplenishment policy. Note also that the etaile is guaanteed ontime shipment of all odes unde a pomised leadtime contact. Hence, each fim independently solves a stationay, peiodiceview inventoy contol poblem with no upsteam supply estictions. The following dynamic pogamming ecusion minimizes the cost of managing each fim s inventoy ove a finite hoizon with T t peiods emaining until temination: J jt x jt = min y jt x jt G j y jt + EJ jt+1 y jt D t fo all t 1T, J jt+1 8 fo j s, whee G s y st = 1 c s y st + Eh s y st D L p s D L+1 y st + and G y t = 1 c y t + Eh y t D l p D l+1+ y t + Fo these stationay poblems, a myopic basestock policy is optimal (Veinott 1965). The optimal basestock levels fo the supplie and etaile ae the minimizes of G s and G, espectively: ( ) Y 1 ps 1 c s = F s L+1 and h s + p s ( ) Y p = F 1 p 1 c l+1+ h + p To have a easonable solution, we assume that p j 1 c j fo fim j. Hence, with pomised lead time, fim j odes up to an optimal basestock level Yj if its installation inventoy position x jt is below this level at the beginning of peiod t. The expected discounted inventoy cost ove T t peiods equals the sum of the discounted singlepeiod costs, i.e., T J s x st Y s = k t G s Y s = G s whee k=t G s c s + E { h s Y s DL+1 + +p s D L+1 Y s +}
6 Opeations Reseach 56(4), pp , 28 INFORMS 93 J x t Y p = T k=t k t G Y p = G p whee G p c + Eh Y p D l p D l+1+ Y p + and 1 T t+1 /1. The pomised lead time yields total expected inventoy cost G s + G p fo the twostage supply chain. We simplify the expessions fo the total expected inventoy cost as follows. If we let p c be the unit penalty cost yielding the desied sevice level with unit vaiable cost c, then SL = p c 1 c /h + p c. By edefining p = h p c 1 c /h +1 c, we ewite SL = p /h + p,soy p = Fl+1+ 1 p /h + p. Similaly, we ewite Ys 1 = FL+1 p s/h s + p s. The tems c s, c, and do not affect the optimization poblems defined late in any stuctual way. Hence, we dop them fom futhe consideation in what follows. Note that fo a given holding cost, the sevice level implies a penalty cost and vice vesa. Hence, we use these two tems intechangeably thoughout the pape. Next, we exploe the popeties of G s and G p that ae necessay in detemining the optimal contact tems late. To pove these popeties, summaized in the next poposition, we conside a logconcave demand distibution F and use two stochastic odeing elationships fo F n, the nfold convolution of demand distibution F. Fist, F n is less than o equal to F n+1 in egula stochastic ode, which implies Fn 1 Fn+1 1 fo all 1. Second, F n is less than o equal to F n+1 in dispesive ode because the convolution of a logconcave density is logconcave (Theoem 2.B.3 of Shaked and Shanthikuma 1994). That is, Fn 1 Fn 1 Fn Fn+1 wheneve < <1. We define G p G p G p 1. Poposition 1. Fo a given pomised lead time and when F is logconcave (a) G p, (b) G p /p >, and (c) G p /p >. Note that Nomal and Elang distibutions, commonly used in inventoy contol, exhibit logconcavity (Bagnoli and Begstom 1989). Fom pats (a) and (b), the etaile s minimum expected inventoy cost inceases with both pomised lead time and backode penalty cost p.given the similaity between the minimum expected inventoy cost functions fo the supplie and the etaile, pat (a) also implies that G s. The supplie s minimum expected inventoy cost deceases with. Fom pat (c), we obseve an impotant popety known as the singlecossing popety (Fudenbeg and Tiole 1991). Essentially, this popety implies the following. A etaile that povides a high sevice level to end customes is moe sensitive to an incease in pomised lead time o, equivalently, benefits moe fom a decease in pomised lead time. This diffeence in sensitivity fo an incease in pomised lead time enables a supplie to sceen the etaile s pivate sevicelevel infomation. This poposition is mainly used in the analysis of the asymmetic infomation case Full Infomation When the supplie has full infomation about the etaile s sevice level, the optimal contact to offe to the etaile is the solution to the following poblem: min K G s K s.t. G p + K U max L+ 1 The supplie chooses a pomised leadtime contact to minimize he expected inventoy cost, while ensuing that the etaile s expected cost does not exceed his esevation cost U max. This cost could be the etaile s expected inventoy cost unde an existing contact o his outside option. Poposition 2. Unde full infomation, the supplie optimally offes f K f, whee (a) K f = U max G p f and (b) f minimizes G s + G p. (c) f deceases 9 as p inceases. (d) G s f K f inceases as p inceases. Pat (a) states that the supplie minimizes he expected inventoy cost by equiing the highest payment K f that the etaile will accept. Pat (b) states that the supplie optimally offes the fistbest pomised lead time. That is, f is the same that would be chosen by a thid paty who optimizes the supply chain s inventoy iskshaing stategy. Note also fom pats (c) and (d) that the supplie optimally offes a shote pomised lead time fo a highsevice etaile, and by doing so inceases he own expected inventoy cost Asymmetic Cost Infomation The etaile s unit penalty cost p is often pivate infomation. Suppose that p lies within a finite set p 1 p N of possible values, whee p 1 < <p N. The etaile s unit penalty cost, his type, takes value p i with pobability i fo N i=1 i = 1. This discete distibution is public infomation. Leaning the Sevice Level. We investigate whethe the supplie can lean the etaile s sevice level, eithe by asking the etaile o by diect obsevation of the etaile s past odes. If so, the supplie can solve fo the fullinfomation contact. Suppose that the supplie asks the etaile to epot his sevice level. Suppose that the etaile epots his penalty cost as p and the supplie offes the coesponding optimal fullinfomation pomised leadtime contact ( f K f ). The etaile s actual expected cost unde this contact is K f + G p f. (4)
7 94 Opeations Reseach 56(4), pp , 28 INFORMS Poposition 3. If p >p, then f f and G p f + K f <G p f + K f. The poposition states that a etaile who exaggeates his sevice level eceives a shote pomised lead time, theeby shifting moe inventoy isk to the supplie. Plus, the etaile s expected cost deceases, so he obtains a lage shae of supply chain benefit unde the contact. Hence, the etaile has eason to exaggeate his sevicelevel infomation. The supplie, theefoe, has eason not to tust the sevicelevel infomation povided by the etaile. A natual question to ask is whethe the supplie can lean the etaile s optimal sevice level ove time by obseving the etaile s odes. This is not possible because the etaile optimally adopts a basestock policy, and his ode in each peiod is equal to the demand in the pevious peiod. Hence, the supplie obseves the demand, but she does not know how much the etaile satisfies fom onhand inventoy and how much he backlogs. Hence, the supplie cannot lean the etaile s sevice level ove time. Next, we popose a mechanism that minimizes the supplie s inventoyelated cost, while enabling cedible infomation shaing. Optimal Menu of Contacts. Without obseving the etaile s sevice level, the supplie can minimize he inventoy cost by designing a menu of pomised leadtime contacts. Accoding to the evelation pinciple (Myeson 1979), the supplie can limit he seach fo an optimal menu of contacts to the class of tuthtelling contacts unde which the etaile finds it optimal to eveal his unit penalty cost. To detemine the optimal menu i aka i N i=1, the supplie solves the following poblem: min i K i N i=1 s.t. N i G s i K i i=1 IR i G p i i +K i U max i 1N IC ij G p i i +K i G p i j +K j j i i L+1 i 1N The individual ationality (IR) constaints in (5) guaantee that the etaile will find an acceptable contact in the menu. The incentive compatibility (IC) constaints ensue that a etaile with sevice level i voluntaily chooses the pomised leadtime contact ( i K i ) designed fo his tue sevicelevel type. We assume that when indiffeent between ( i K i ) and ( j K j ) fo j i, a etaile with sevice level i chooses the fome. Next, we use Poposition 1 to chaacteize an equivalent fomulation fo the feasible egion defined by IR and IC. To simplify the exposition, we define j j i=1 i,, and p p 1. Poposition 4. (a) A menu of contacts is feasible if and only if it satisfies the following conditions fo all i 1N 1: (i) i i+1, and (ii) G p i i + K i = U max N k=i+1 G p k k G p k 1 k and G p N N + K N = U max. (5) (b) The optimization poblem in 5 has the following equivalent fomulation min 1 N N i G s i + G p i i + i 1 / i i=1 G p i i G p i 1 i U max s.t. i is deceasing in i i L+ 1 i = 1N Pat (ai) states that the supplie optimally offes a shote pomised lead time to a etaile with a highe sevice level. Nevetheless, fom pat (aii), a etaile with the highest sevice level incus his esevation cost. A etaile with a sevice level lowe than p N eceives infomation ent, a cost eduction to discouage him fom exaggeating his sevice level. These esults ae simila in natue to those obtained fo othe mechanism design poblems (see, fo example, Lovejoy 26). In pat (b), the summation in the objective function (6) consists of two tems. The fist tem is the total supply chain inventoy cost unde the pomised lead time. The second tem ( i 1 / i G p i i G p i 1 i )] is the cost of the incentive poblem (infomation ent) because of asymmetic sevice infomation. Without this tem, the supplie would offe the fistbest contact ( f i K f i ) to all etaile types. Let i a denote an optimal solution fo the poblem in (6), and let Ki a denote the optimal payment that is obtained as the solution to the equalities in Poposition 4(aii) when i = i a. Poposition 5. (a) K a i is inceasing and a i is deceasing in i. (b) G p i a i + Ka i is inceasing in i. (c) a 1 = f 1 and a i f i fo all i 2N. This poposition chaacteizes additional popeties of the optimal menu of contacts. In paticula, fom pat (a), a etaile with a highe sevice level optimally accepts a lage payment obligation fo a shote pomised lead time. Because the optimal contact tems i a and Ki a ae monotone in i, we can constuct a function K by setting K = Ki a if = i a. This function specifies a payment fo a given pomised lead time. The supplie offes this function as a contact without any mention of sevice levels. The etaile does not need to announce a sevice level eithe. This esult makes the menu of pomised leadtime contacts amenable fo implementation. Pat (c) states that all but the lowest sevicetype etaile gets a pomised lead time shote than the fistbest pomised lead time. Togethe with pat (a), this implies two facts. Fist, the supplie faces moe inventoy isk when the etaile has pivate sevicelevel infomation. Second, the supplie manages moe of this inventoy isk when doing business with a highesevice etaile. (6)
8 Opeations Reseach 56(4), pp , 28 INFORMS Cutoff Policy The supplie may decide not to induce evey etaile type to sign a pomised leadtime contact when she wants to keep he expected inventoy cost below Us max. Next, we chaacteize an optimal cutofflevel policy (Cobett and de Goote 2, Ha 21). Unde this policy, the supplie optimally induces the etaile to sign a pomised leadtime contact if the etaile s backode penalty cost is lowe than a cutoff level p. If the etaile s penalty cost (o equivalently, his sevice level) is highe, the two fims do not sign a pomised leadtime contact, and hence do business unde the oiginal setting. To deive the supplie s optimal cutoff level, we intoduce a null contact with pomised lead time. Ageeing to the null contact is equivalent to no contact being signed, and expected inventoy costs fo the supplie and etaile evet to G max s Us and G max U, espectively. Unde full infomation, the supplie detemines the optimal cutoff level by solving (4) with an IR constaint fo the supplie and a lage feasible set fo the pomised lead time, i.e., G max s K Us L+ 1 (7) Poposition 6. Unde full infomation, the supplie optimally offes the fistbest contact K = f K f if p p and K = othewise, whee p minp G s f p + G p f p = U max s + U max. The poposition shows that it is optimal fo the supplie not to offe a pomised leadtime contact to a etaile with p > p. The poposition also povides an explicit solution fo the cutoff level. Note that the cutoff level p is inceasing in both Us max and U max. Intuitively, the supply chain has to cay moe inventoy to povide high sevice to the end custome. Hence, the supplie sets up a supply chain with a highesevice etaile only when eithe fim can accept a lage expected inventoy cost. Unde asymmetic sevicelevel infomation, the supplie s optimal menu of contacts, allowing fo the null contact possibility, is the solution of the poblem in (5) with IR s N i=1 i G s i K i U max s i L+ 1 i (8) Fo notational convenience, we define sums ove empty sets to be zeo and <. Poposition 7. An optimal menu of contacts c i Kc i N i=1 satisfies the following popeties If c i =, then c j = and K c j = fo j i. In addition, if c i, then j i fo j i. Hence, an optimal cutoff level is defined as p p c, whee c = maxi 1N i c. Let c = when this set is empty. Also, p must satisfy { c } i U max s + U max G s c i G p i c i i=1 { c i 1 G p i c i G p i 1 c i } i=1 Poposition 7 establishes the optimality of a cutoff policy. In othe wods, if it is optimal fo the supplie not to offe a pomised leadtime contact to a etaile with sevice level i, then it is also optimal not to offe a contact to a etaile with a highe sevice level. The optimal menu of contacts consists of i K i = fo i>c. Hence, fo p i > p, the two fims do not sign a pomised leadtime contact, and the supplie and the etaile incu Us max, espectively. U max and 6. Compaing Stocking Levels To undestand the supplie s and etaile s deviation fom a systemoptimal solution, we compae the optimal basestock levels unde cental contol to those unde local contol with a pomised lead time. The following poposition chaacteizes the optimal allocation of inventoy unde cental contol. Poposition 8. Equations 1 and 3 have finite minimizes when p P 1 c s + L+1 c + L 1 l+1 h s / L+l+1. Then, the echelon and installation basestock levels ae given as follows (a) Fo any p P, y m = Fl+1 1l p + h s 1 c / l p + h. (b) When p PH, then ys m 1 =FL+l+2 L+1 l p +h s 1 c 1 c s L h s / L+l+1 p + h, whee H 1 c s + L+1 c + L 1 l+1 h s + L+l+1 h F L+l+2 y m /L+l+1 1 F L+l+2 y m Fo this case, ys m y m. Hence, Y s = and Y = ys m. When p H, then ys m is the unique solution to 1 c s + L h s + L+1 1 c l p + h s 1 F L+1 y y m + L+l+1 p + h y y m F l+1 y uf L+1 u du = Fo this case, ys m >y m. Hence, Y s = ys m y m and Y = y m. Poposition 8 chaacteizes the optimal inventoystocking levels at each location fo a centally contolled system and povides closedfom solutions unde cetain conditions. In paticula, the optimal inventoy allocation depends on how the penalty cost elates to the othe cost paametes. When the penalty cost is vey low, i.e, lowe than P, the poblem does not have a finite solution. When p PH, the penalty cost is high enough to waant holding inventoy only at the etaile. When the penalty cost exceeds the theshold H, the penalty cost is high enough to waant holding inventoy at both locations.
9 96 Opeations Reseach 56(4), pp , 28 INFORMS Poposition 9. Fo any given pomised lead time, the following ae tue (a) Fo p H, let ( ( )) B p + h F l+1 F 1 p 1 c l+1+ p + h ( ) 1 + c p l which is inceasing in. We have { Y Y p when h s B Y >Y p when h s >B (b) Fo p PH, we have Ys >Y s = fo < L + 1 and Ys = Y s = fo = L + 1. Poposition 9 states that if h s is lowe than B, a cental decision make will hold less inventoy at the etaile location than a locally contolled etaile would cay. In othe wods, unde cental contol, when the holding cost at the supplie location is small, the cental manage caies moe inventoy at the supplie location and less at the etaile location. Howeve, unde local contol, the etaile does not conside the supplie s inventoy cost when making his inventoy decision. Hence, thee is a theshold fo h s below which a locally managed etaile caies moe inventoy than a centally contolled one. Note also that this theshold inceases with the pomised lead time. Intuitively, a longe pomised lead time equies a highe optimal inventoy level fo a locally contolled etaile because the etaile faces highe isk fom holding insufficient inventoy. Pat (b) notes cicumstances in which a local decision make holds inventoy at the supplie site, but a cental decision make does not. Simila compaisons ae possible fo othe values of p. These esults show when and how much the supplie and the etaile may unde o oveinvest in inventoy when compaed to a centally contolled supply chain. Note that these compaisons ae tue fo any given. Hence, they apply to both full and asymmetic infomation scenaios. These solutions also enable one to quantify the diffeence in inventoy levels due to decentalization though simple calculations. We povide moe discussion on these compaisons in AlloNothing Inventoy Allocation Hee, we addess a special case in which the optimal pomised lead time is eithe zeo o L + 1. When the optimal pomised lead time is zeo, the supplie assumes all esponsibility fo the additional inventoy isk. When the optimal pomised lead time is L + 1, the supplie assumes none of this isk, essentially offeing maketoode sevice to the etaile. Hence, this outcome is efeed to as an allonothing solution. Poposition 1. Unde full infomation and when G s and G p ae concave in, the supplie optimally offes f K f, whee f L+ 1 and K f = U max G p f. Fo the supplie, concavity means that the incemental eduction in he expected cost inceases as the pomised lead time inceases. Fo the etaile, concavity means that the incemental eduction in his expected cost inceases as the pomised lead time deceases. The optimal inventoy iskshaing stategy fo the supply chain is then one of the exteme points of the feasible set of pomised lead times, i.e., eithe zeo o L + 1. The next poposition shows when the optimality of an allonothing solution also holds unde asymmetic infomation. Poposition 11. Unde asymmetic infomation and when G s, G p, and G p i i G p i 1 i ae concave in, the supplie s optimal menu of contacts consists of allonothing pomised lead times, i.e., i a L+ 1 fo all i. In paticula, let m = maxi 1N i a = L + 1, and if this set is empty, let m =. Then, we have: (a) If m>, then i a = L + 1 and Ki a = U max G p ml+ 1 G p N G p m fo all i m, wheeas i a = and Ki a = U max G p N fo all i>m. (b) Othewise, if m =, then i a = and Ki a = U max G p N fo all i 1N. Poposition 11 shows that a pooling equilibium tuns out to be an optimal outcome. In this equilibium, the etaile does not necessaily eveal his backode penalty cost by choosing a pomised leadtime contact. In paticula, pat (a) states that if the etaile has a penalty cost lowe than o equal to p m, then he would optimally choose the contact with a = L + 1. Othewise, he would select the contact with a =. Pat (a) with m = N and pat (b) togethe show the following. An optimal contact outcome could be a single contact that allocates all additional inventoy isk to the supplie o to the etaile egadless of the etaile s sevice level. Next, we investigate when the concavity conditions of Popositions 1 and 11 ae met. Poposition 12. G s, G p, and G p i i G p i 1 i ae concave in if (i) F is Nomal, o (ii) f n Fn 1 x 1 is concave in n. Any demand distibution satisfying one of these two conditions geneates an allonothing inventoy iskshaing ageement. 8. Numeical Examples To illustate the behavio of system cost and inventoy levels unde pomised leadtime contacts, we pesent the following numeical examples. We set c s =, h s = 1 fo the supplie and c =, h = 3 fo the etaile. The etaile povides eithe a 85% (low) sevice level o a 98% (high) sevice level, stemming fom unit penalty cost p = 17 o p = 147, espectively. Demand is nomally distibuted with mean 5 and standad deviation 1, and the etaile s esevation cost is U max = 15. We conside 27 scenaios that
10 Opeations Reseach 56(4), pp , 28 INFORMS 97 diffe in thei poduction and pocessing lead times, cost of boowing fom an altenative souce, and the likelihood of the etaile s poviding a low vesus a high sevice level. In each scenaio, we make a combination of selections fom the following sets: L l p s Thoughout this section, an allonothing pomised lead time is optimal because demand is nomally distibuted (Poposition 12) The Supplie s Cost Unde the Pomised LeadTime Contact In Figue 2, we plot the supplie s expected inventoy cost as a function of lead times (L l), the supplie s shotage cost p s, and unde both full and asymmetic infomation. We highlight fou obsevations. Fist, the supplie s cost unde full infomation is always lowe than unde asymmetic infomation. Compae, fo example, the cost cuve unde full and asymmetic infomation when L l = 4 2. Second, the supplie s expected cost inceases with he cost of altenative soucing, i.e., he penalty cost p s. Fo example, unde asymmetic infomation when = 8 and L l = 2 4, the supplie s expected cost is 66.37, 67.62, and when p s = 19, 49, and 99, espectively. Thid, the supplie s expected cost is highe with lowe. Fo example, unde asymmetic infomation when p s = 49 and L l = 2 4, the supplie s expected cost is 76.15, 71.89, and when = 2,.5, and.8, espectively. This obsevation indicates that contacting with a lowsevice etaile is moe pofitable than contacting with a highsevice etaile. Ou fouth and final obsevation is the decease in the supplie s expected inventoy cost when L inceases fo a constant L+l. A supplie located close to the etaile incus lowe expected inventoy cost unde a pomised leadtime contact. In othe wods, a supplie that is futhe away fom the etaile faces moe inventoy isk. The following poposition poves this obsevation. Poposition 13. Fo constant system lead time L + l, nomally distibuted demand, and two possible etaile sevice levels, the supplie s expected inventoy cost unde asymmetic infomation inceases as L deceases The Supplie sstocking Level: Local vs. Cental In the est of this section, we compae the inventoy levels unde local contol to those chosen by a cental decision make. In Figue 3, we plot the pecent incease in the supplie s inventoy level unde local contol fom the inventoy level allocated unde cental contol, i.e., Ys Y s/y s 1%. This pecent can be intepeted as ove (espectively, unde) investment in inventoy due to local contol when the pecent is positive (espectively, negative). The pomised lead time is zeo fo this figue. Conside, fo example, the scenaio in which p = 17, L = 2, and l = 4. When p s = 49, the pecentage incease is 4.7%. In othe wods, the supplie caies 4.7% moe inventoy as compaed to a systemoptimal inventoy allocation. Note that the pecent incease is also plotted as a function of the etaile s penalty cost, the shotage cost at the supplie, and the lead times. The following thee obsevations ae woth noting. Figue 2. Supplie s expected inventoy cost. 9 λ =.2 λ =.5 λ =.8 λ =.2 λ =.5 λ =.8 λ =.2 λ =.5 λ =.8 75 Asymmetic info., L = 2, I = Asymmetic info., L = 3, I = 3 Asymmetic info., L = 4, I = 2 Full infomation, L = 2, I = 4 Full infomation, L = 3, I = 3 Full infomation, L = 4, I = p s = 19 p s = 49 p s = 99
11 98 Opeations Reseach 56(4), pp , 28 INFORMS Figue 3. Pecent incease in supplie inventoy: Decentalized vs. centalized, =. Figue 4. Pecent incease in etaile inventoy: Decentalized vs. centalized, =. (%) p s = 19 p s = 49 p s = 99 p = 17, L = 2, I = 4 p = 17, L = 3, I = 3 p = 17, L = 4, I = 2 p = 147, L = 2, I = 4 p = 147, L = 3, I = 3 p = 147, L = 4, I = 2 Fist, the pecent incease gows as the unit penalty cost p s inceases. Note that the points at which these lines coss zeo, if they do, povide the supplie s imputed penalty costs that equate the inventoy allocation to the supplie s site unde local and cental contol. In othe wods, fo those imputed shotage costs, the supplie chooses the systemoptimal inventoy level. Second, as the penalty cost p inceases o, equivalently, as the sevice level povided to end customes inceases, the pecent incease in supplie inventoy due to local contol deceases. In othe wods, the supplie s oveinvestment in inventoy declines because unde cental contol, moe inventoy is held at the supplie s location (wheeas unde local contol, p does not affect the supplie s stocking level). The supplie may even undeinvest in inventoy when the etaile s shotage cost is high. Fo example, when p s = 49, p = 147, L = 4, and l = 2, the pecent incease is.7%. The supplie allocates.7% less inventoy, hence, undeinvests when compaed to the systemoptimal inventoy allocation decision. Finally, the pecent incease in supplie inventoy deceases as the supplie s lead time L inceases, wheeas supply chain lead time L + l emains constant. In othe wods, the supplie s oveinvestment in inventoy declines as the supplie is located close to the end custome. This is mainly because the systemoptimal solution stats to allocate moe inventoy to the supplie s site as she gets close to the custome The Retaile sstocking Level: Local vs. Cental Next, we illustate how the etaile s inventoy level changes due to local contol. In Figue 4, we plot the pecent incease in the etaile s inventoy level unde local contol fom the inventoy level unde cental contol, i.e., Y Y /Y 1% fo =. Note in Figue 4 that the etaile caies less inventoy than a systemoptimal inventoy allocation. The etaile s pecent undeinvestment in inventoy deceases as eithe his sevice level p, o his pocessing lead time l inceases (%) p = 147 p = 17 L = 2, I = 4 L = 3, I = 3 L = 4, I = 2 (while keeping L+l constant). An incease in eithe p o l equies the etaile to cay moe inventoy to potect this second stage against lage demand uncetainty and moe expensive shotage cost. Unde cental contol, the manage has the option of holding some of this additional inventoy at the supplie location. A etaile unde local contol does not have this option. Hence, the etaile inceases his inventoy level moe damatically than a cental decision make would have inceased the inventoy allocation to the second stage. As a esult, the etaile s inventoy investment unde local contol appoaches the level unde cental contol as l o p inceases. 9. Conclusion We intoduce a contact fom that educes a supplie s uncetainty egading demand and eliminates a etaile s uncetainty egading inventoy availability: a pomised leadtime contact. The contact and the model consideed in this pape fomalize what is aleady common in pactice, that is, specifying lead times as pat of a contact between two supply chain membes (Billington 22, Cohen et al. 23). In this study, by combining supply chain contacting, classical inventoy contol, and mechanism design, we discove the following. When the supplie has full infomation about the etaile s inventoy cost paametes, the optimal pomised leadtime contact geneates the optimal inventoy iskshaing stategy fo the supply chain. Uncetainty egading what sevice level the etaile povides to end customes geneates conflict between the etaile and the supplie. In designing a mechanism to accommodate this uncetainty, the supplie offes shote lead times and infomation ent. Altogethe, this pactice eats at the supplie s contact benefits until, at some point, the supplie does not find it pofitable to offe contacts to etailes with all possible sevice levels. In paticula, this obsevation helps establish the optimality of a cutoff policy. Intuitively, etailes having a high sevice level may epesent a maket segment that is unpofitable fo the supplie to seve unde a pomised leadtime contact. We also show that a centally contolled supply chain holds moe inventoy at
12 Opeations Reseach 56(4), pp , 28 INFORMS 99 the etaile s location than the etaile would cay independently when the holding cost at the supplie location is sufficiently high. Such compaisons show when and how much the supplie and the etaile unde o oveinvest in inventoy as compaed to an integated system managed by a cental decision make. The models can be used to quantify this diffeence in inventoy investment and how this diffeence changes with the system paametes, such as lead times. We also chaacteize conditions unde which the supplie should assume all o none of the inventoy isk. In paticula, the optimal pomised leadtime contact stipulates eithe buildtostock ( = ) o buildtoode ( = L + 1) sevice to the etaile unde eithe full o asymmetic infomation when the singlepeiod cost functions ae concave in the pomised lead time. A supplie woking with multiple independent etailes can conside offeing these two contacts and constuct a potfolio of etailes with pomised lead times zeo and L + 1. We leave this topic to futue eseach. We assume that the supplie boows inventoy fom an altenative souce to satisfy etaile odes on time. Altenatively, if the supplie simply puchases emegency units fom an outside souce o expedites emegency units though ovetime esouces, the supplie pays a onetime unit emegency cost p s above he nomal poduction cost. This scenaio esults in the supplie facing the equivalent of a lostsales inventoy contol poblem: sales ae lost fo the supplie s nomal poduction pocess. Fo a peiodiceview poblem unde lost sales and positive lead times, Nahmias (1979) and efeences theein offe a twotem myopic heuistic, which can be used to extend the pomised leadtime esults to this altenate case. The concept of isk shaing though pomised leadtime contacts is a fetile avenue fo futue eseach. Some issues to exploe include nonstationaities in cost o demand paametes, the impact of possible contact enegotiation, altenative infomation asymmeties, and the situation whee the etaile takes the lead in contact development. 1. Electonic Companion An electonic companion to this pape is available as pat of the online vesion that can be found at infoms.og/. Appendix. Poofs Poof of Poposition 1. To pove pat (a), ecall that is the mean demand fo a single peiod. The etaile s minimum expected inventoy cost can be witten as G p = p l h + p l+1+ p /h +p uf l+1+ u du (9) Hence, G p [ = p h + p l+1+ p /h +p l+ p /h +p uf l+1+ u du ] uf l+ u du We divide this expession by h + p to obtain G p [ l+1+ = uf h + p l+1+ u du l+ ] uf l+ u du whee = p /h + p. To show that G p, we show G p /h + p. To do so, we fist show that G p /h + p is concave in. Note that 2 / 2 G p /h + p = f l+1+ Fl+1+ 1 f l+ Fl+ 1/f l+1+fl+1+ 1 f l+fl+ 1. Because F has a logconcave density, F l+ is less than o equal to F l+1+ in dispesive ode. Hence, fom Equation (2.B.7) in Shaked and Shanthikuma (1994), we have f l+1+ Fl+1+ 1 f l+fl+ 1 fo all in ( 1). Hence, 2 / 2 G p /h + p. Note that G p /h + p is continuous in 1 and equals zeo fo 1. Togethe with concavity, this implies G p /h + p. To pove pat (b), we note that G p p = l l+1+ uf l+1+ udu 1 F 1 l+1+ Fo =, /p G p = l + 1 +, and fo = 1 G p /p =. Because 2 G p /p = 1 /f l+1+ F 1 l+1+ < we have G p /p >. To pove (c), we note that G p p [ = l+1+ uf l+1+ u du l+ ] uf l+ u du 1 F 1 1 l+1+ Fl+ Fo =, G p /p =, and fo = 1, G p /p =. To pove G p /p >, we show that it is deceasing in. To do so, note that 2 G p p [ fl+ Fl+ 1 = 1 f l+1+f 1 f l+ Fl+ 1f l+1+fl+1+ 1 l+1+ ]
13 91 Opeations Reseach 56(4), pp , 28 INFORMS The backeted tem on the ighthand side is positive because F l+ is less than o equal to F l+1+ in dispesive ode. Poof of Poposition 2. Note that G p + K = U max at optimality. Othewise, one can incease K and educe the objective function value without violating the constaint, contadicting the optimality of K. To pove pat (b), substitute U max G p fo K in the objective function. To pove pats (c) and (d), conside p < p and let f and f be the optimal pomised lead times, espectively. Note fom pat (b) that G s f + G p f G s f + G p f and G s f + G p f G s f + G p f. By adding these two inequalities, we obtain G p f G p f G p f G p f. By Poposition 1(c), f f. To pove pat (d), fom pats (a) and (b) and Poposition 1(b), we have G s f K f = G s f + G p f U max G s f + G p f U max <G s f + G p f U max = G s f K f. Poof of Poposition 3. The fist statement follows fom Poposition 2(c). To pove the second statement, we have G p f + K f <G p f + K f = U max = G p f + K f. The inequality is fom Poposition 1(b), and the equalities ae fom Poposition 2(a), concluding the poof. Poof of Poposition 4. To pove pat (a), we fist show that IR and IC imply the conditions stated in the fist pat of the lemma. We begin by simplifying the IR constaint in (5). Suppose that i<j. We have G p i i + K i G p i j + K j <G p j j + K j U max. The inequalities ae due to IC, Poposition 1(b), and IR, espectively. Hence, IR is edundant fo all i N. Fo i<j, we sepaate IC into upwad constaints G p i i +K i G p i j +K j and downwad constaints G p j j + K j G p j i + K i. By adding these two constaints, we have G p j j G p i j G p j i G p i i. Hence, Poposition 1(c) implies j i, showing pat (ai). When i<j<k, we claim that IC ij and IC jk imply the upwad IC ik constaint. Assume fo a contadiction that G p i i + K i >G p i k + K k, so K k <K i + G p i i G p i k. The constaint IC jk implies that K j K k + G p j k G p j j. Using the upwad IC ij constaint, we find that G p i i + K i G p i j + K j K k + G p j k G p j j + G p i j < K i + G p i i G p i k + G p j k G p j j + G p i j. By eaanging tems, we have G p j j G p i j <G p j k G p i k, which implies j < k due to Poposition 1(c). Howeve, this contadicts pat (ai), which we poved above. This esult implies that the set of upwad constaints educes to adjacent upwad constaints G p i i +K i G p i i+1 +K i+1 fo i 1N 1. Next, we show that the emaining upwad constaints ae binding and the downwad constaints ae edundant at optimality. To do so, we use an induction agument. Fist, we show that G p i i +K i = G p i i+1 +K i+1 fo i = 1. Assume fo a contadiction that G p K 1 < G p K 2. Poposition 1(b) and IR fo i = 2 imply that K 1 <K 2 +G p 1 2 G p 1 1 <K 2 +G p 2 2 G p 1 1 U max G p 1 1. Hence, one can incease K 1 and lowe the objective function value without violating othe constaints, contadicting optimality. Hence, IC fo i = 1 must be binding. Next, assume fo an induction agument that G p i i + K i = G p i i+1 + K i+1 fo all i 1j 1, whee j 2. We show that these binding upwad constaints and i+1 i imply downwad IC constaints G p j j + K j G p j i + K i fo all i<j. Note that K i+1 = K i + G p i i G p i i+1, and fo j > i iteatively solve fo K j = K i + j 1 k=i G p k k G p k k+1. Poposition 1(c) implies G p j j G p j 1 j G p j j 1 G p j 1 j 1, and adding this inequality to the identity fo K j yields G p j j + K j K i + j 2 k=i G p k k G p k k+1 + G p j j 1. Poposition 1(c) also implies the sequence of inequalities G p j m 1 j m 1 G p j m 1 j m G p j j m 1 G p j j m j i 1 m=1. Iteatively adding these to K j + G p j j K i + j 2 k=i G p k k G p k k+1 + G p j j 1 yields G p j j + K j G p j i +K i ; hence, downwad constaints ae edundant fo all i<j. To conclude the induction agument, we show that the upwad adjacent constaint fo j is also binding. Assume fo a contadiction that it is not, i.e., G p j j + K j < G p j j+1 + K j+1. Poposition 1(b) and IR fo j + 1 imply that K j <K j+1 + G p j j+1 G p j j <K j+1 + G p j+1 j+1 G p j j U max G p j j. Hence, one can incease K j and lowe the objective function value without violating any constaint. This contadicts optimality. Hence, all emaining upwad IC ae binding at optimality, and they can be eplaced by IC G p i i + K i = G p i i+1 + K i+1 fo i 1N 1 Note also that G p N N + K N = U max at optimality. Othewise, one can incease K N by > and all othe K i by the same amount, theeby educing the objective function value without violating any constaints. This contadicts optimality. Hence, IR constaints can be eplaced by IR G p N N + K N = U max Note that IC and IR imply pat (aii). Convesely, we show that the two conditions (i) and (ii) imply IR and IC. Note that condition (ii) implies IR because G p k k > G p k 1 k fom Poposition 1(b). Next, we inductively show that (i) and (ii) imply upwad IC constaints. Condition (ii) implies K i = K i+1 + G p i i+1 G p i i fo all i 1N 1.
14 Opeations Reseach 56(4), pp , 28 INFORMS 911 Assume fo an induction agument that G p i i + K i G p i j + K j fo some j>i. This inequality holds fo i = 1 and j = 2. We have G p i j + K j = G p i j + K j+1 + G p j j+1 G p j j + G p i j+1 Gp i j+1 = G p i j+1 + K j+1 + { G p i j Gp i j+1 G p j j G p j j+1 } The tem in backets is negative fom Poposition 1(c) and condition (i) j j+1. Hence, we have G p i i + K i G p i j + K j G p i j+1 + K j+1, concluding the induction agument. Note that (ii) also implies IC. Above we also showed that (i) and IC imply downwad IC constaints. This concludes the poof fo pat (a). To pove pat (b), we substitute fo K i and use pat (a) to obtain (6). Poof of Poposition 5. Pat (a) follows fom Poposition 4(a). In paticula, i a must be deceasing in i to be a feasible solution. Now assume fo a contadiction that Ki+1 a <Ka i. Togethe with Poposition 4(aii), this assumption implies Ki+1 a = U max G p i+1i+1 a N k=i+2 G p kk a G p k 1k a<umax G p ii a N k=i+1 G p kk a G p k 1k a = Ka i. Howeve, by canceling tems, this inequality yields G p ii a < G p ii+1 a, which contadicts Poposition 1(a). Hence, Ki+1 a Ka i. To pove pat (b), note that Poposition 4(aii) defines the etaile s esulting expected cost. Fom Poposition 1(b), each tem in N k=i+1 G p k k G p k 1 k is positive, which implies pat (b). To pove pat (c), note that i+1 a i a. Hence, exactly one of the following is tue: (i) i a f i, (ii) i+1 a f i <i a, (iii) j+1 a f i <j a fo some j i + 1N 1, o (iv) f i <N a. To pove i a f i, we will show that the othe possibilities lead to a contadiction. Fist, assume fo a contadiction that i+1 a f i <i a. Then, N a a i+1 f i i 1 a a 1 is a feasible solution to the poblem in (6). Fom optimality, we have N { k G s a k + G p k a k k ik=1 + k 1 G p k a k G p k 1 a k } + { i G s a i + G p i a i + i 1 G p i a i G p i 1 a i } N k ik=1 { k G s a k + G p k a k + k 1 G p k a k G p k 1 a k } + { i G s f i + G p i f i + i 1 G p i f i G p i 1 f i } (1) Note also that optimality of f i unde full infomation implies i G s f i +G p i f i i G s a i +G p ii a. Adding the last two inequalities yields G p ii a G p i 1i a G p i f i G p i 1 f i which contadicts Poposition 1(c). Theefoe, we cannot have i+1 a f i <i a. Second, assume that the statement j+1 a f i <j a fo some j i+1n 1 is tue. Then, N a a j+1 f i j 1 a a 1 is a feasible solution to the poblem in (6). Fom optimality, we have j G s a j + G p j a j + j 1 G p j a j G p j 1 a j j G s f i + G p j f i + j 1 G p j f i G p j 1 f i Optimality of f i unde full infomation implies j G s f i + G p i f i j G s a j + G p i a j Adding the last two inequalities and eaanging tems yields j G p j a j G p i a j + j 1 G p j a j G p j 1 a j j G p j f i G p i f i + j 1 G p j f i G p j 1 f i (11) Because a j >f i, p j >p j 1, and p j >p i, Poposition 1(c) implies the following two elationships: G p j a j G p j 1 a j >G p j f i G p j 1 f i G p j a j G p i a j >G p j f i G p i f i Theefoe, the left side of (11) is geate than the ight side, and we have a contadiction. Hence, we cannot have j+1 a f i <j a fo some j i + 1N 1. Finally, we assume that the statement f i <N a is tue. Then, f i N a 1 a 1 is a feasible solution to the poblem in (6). Aguments simila to those in the pevious case yield a contadiction with Poposition 1(c). Hence, it must be tue that i a f i. To pove 1 a = f 1, note that f 1 2 aa N is a feasible solution to the poblem in (6) because 1 a f 1. Optimality of 1 aa N yields 1G s a 1 + G p 11 a 1 G s f 1 + G p 1 f 1 fom Equation (1) with i = 1. Optimality of f 1 unde full infomation yields 1 G s f 1 + G p 1 f 1 1 G s a 1 + G p 11 a. Hence, G s a 1 + G p 11 a = G s f 1 +G p 1 f 1, so1 a = f 1 is an optimal solution. Poof of Poposition 6. At optimality, we have K = G p p. Hence, K = when =. Then, U max
15 912 Opeations Reseach 56(4), pp , 28 INFORMS poblem (4) with (7) is equivalent to min G s p + G p p U max s.t. G s p + G p p U max L+ 1 + Us max Poposition 2(a) and 2(d) togethe imply that G s f p + G p f p inceases with p. Hence, thee exists a theshold p such that fo p p, contact f K f is feasible and optimal, wheeas fo p > p, the null contact ( ) is optimal. Poof of Poposition 7. Fist, we show that if i c = fo some i 1N 1, then j c = fo all j>i. Assume fo a contadiction that j c. Adding upwad constaints G p ii c + Kc i G p ij c + Kc j and the downwad constaints G p jj c + Kc j G p ji c + Kc i, we have G p jj c G p ij c G p ji c G p ii c. Fom Poposition 1(b), G p jj c G p ij c >. Hence, G p ji c G p ii c >. Howeve, this contadicts i c = because G c i U max, which implies >. Hence, it must be tue that j c =. Note that the above esult also implies that if j c fo some j 1N 1, then i c fo all i<j. Hence, fo such i<j pais, adding upwad constaints and downwad constaints and eaanging tems as above yields G p jj c G p ij c G p ji c G p ii c. This inequality, togethe with Poposition 1(c), implies j c i c. Now let c = maxi 1N i c at optimality. Note that if this set is empty, the poblem is tivial. In othe wods, if i c = fo all i, then all IR and IC constaints educe to Ki c. Hence, at optimality, Ki c =. Othewise, the supplie can incease the Ki c s to zeo and educe the objective function value. If the set is not empty, the est of the poof follows exactly the same aguments of the poof of Poposition 4 fo i c. The diffeences in the poofs ae fo the bounday cases when i>c. In what follows, we will only povide the details fo those cases. Fo example, IR constaints fo j>c, e.g., G p j + Kj c U max, imply that Kj c fo all j>c. Conside i>c. Upwad IC constaints G p i + Ki c G p i + Kj c imply Ki c Kj c fo any j>i. Downwad IC constaints G p i + Ki c G p i + Kk c imply Kc i Kk c fo any k<i and k>c. Hence, we can eplace all upwad and downwad constaints and IR fo i>c with Ki c = K. The set of IR constaints fo the etaile with i<c ae edundant. In addition, IR fo i = c is also edundant because G p cc c + Kc c G p cc+1 c + K = U max + K U max. The fist and second inequalities ae fom IC and Kj c = K fo j>c, espectively. The upwad IC constaints can be educed to adjacent upwad IC constaints fo all i<c.foi c<j, note that G p ii c+kc i G p ij c+k = G p ik c +K fo all k>j.fok i c, we also have G p ii c + Kc i G p ik c + Kc k <G p kk c + Kc k G p kj c + K = G p ij c + K. The inequalities ae due to IC ik, Poposition 1(b), and IC kj, espectively. The equality is because j c = fo j>c. Hence, all upwad IC constaints can be educed to adjacent upwad IC constaints. The adjacent IC ae binding and the downwad IC constaints ae edundant fo all ij c. Fo i c< c + 1 <j, we show that IC c+1i implies IC ji. We have G p jj c + K = G p c+1c+1 c + K G p c+1i c + Ki c <G p ji c + Kc i. The fist equality is due to j c = c+1 c =. The inequalities ae due to IC and Poposition 1(b), espectively. Next, we show that downwad IC c+1i fo i<c is also edundant. Note that G p c+1c+1 c + K G p c+1c c + Kc c G p c+1c c + Ki c + G p ci c G p cc c G p c+1c c + Kc i + G p c+1i c G p c+1c c = G p c+1i c + Kc i. The fist and second inequalities ae due to IC c+1c and IC ci, espectively. The last inequality is fom Poposition 1(c) and i c c c. Next, note that IC cc+1 must be binding. Assume fo a contadiction that G p cc c+kc c <G p cc+1 c +K. This implies Kc c max <K+ U G p cc c. Then, the supplie can incease Kc c max to K +U G p cc c without violating any constaints, which contadicts optimality. Note also that IC cc+1 binding implies IC c+1c.wehaveg p c+1c+1 c + K = G p cc+1 c + K = G p cc c + Kc c <G p c+1c c + Kc c. The fist and second equalities ae fom c+1 c = and the binding IC cc+1, espectively. The inequality is due to Poposition 1(b). Fom binding upwad adjacent IC, we have Ki c = Ki+1 c G p ii c G p ii+1 c fo all i<c, and Kc c = K + U max G p cc c, and Kc j = K fo all j>c. The expessions fo Kc c and Ki c fo all i<c togethe imply Ki c = K +U max G p ii c c k=i+1 G p kk c G p k 1k c fo all i<c. Assume fo a contadiction that K<. Then, the supplie can incease K to zeo, and hence incease Ki c fo all i>c by the same amount, without violating any constaints. This contadicts optimality. Theefoe, Kj c = K = fo all j > c. Hence, G p cc c + Kc c = U max and G p ii c + Kc i = U max c k=i+1 G p kk c G p k 1k c fo all i<c. Iteatively solving fo Kc i and plugging it into the objective function yields the optimal solution c i=1 ig s c i + G p ii c U max + i 1 / i G p ii c G p i 1i c + N i=c+1 ius max Us max. The inequality is due to the IR s. This inequality implies the condition on the cutoff level pesented in the statement of the poposition. Poof of Poposition 8. To pove pat (a), fom Equation (1), we have /yl y = 1 c l p + h s + l p + h F l+1 y, which is inceasing in y. Because p P implies l p + h s 1 c, we have /yl = 1 c l p + h s. Note also that lim y /yl y = 1 c + l h h s >. Hence, thee exists a positive finite y such that /yl y =, and it is the y m given in pat (a).
16 Opeations Reseach 56(4), pp , 28 INFORMS 913 To pove pat (b), fom Equation (3), we have y L sy = 1 c s + L h s + L+1 y EIPm y D L+1 (12) Fom Equation (2), we have y IPm y = y L y y y m y>y m By the definition of y m, the above function is continuous ove all x. We define the fist deivative of L s y in two egions. When y y m, then y u ym fo all u. Hence, fom Equation (12), we have y L sy = 1 c s + L h s + L+1 y L y uf L+1 u du = 1 c s + L h s + L+1 1 c l p + h s + L+l+1 p + h F L+l+2 y (13) which is inceasing in y. When y>y m,wehave/yipm y u = fo u< y y m. Hence, fom Equation (12), we have y L sy = 1 c s + L h s + L+1 y y m y L y uf L+1 u du = 1 c s + L h s + L+1 1 c l p + h s 1 F L+1 y y m + L+l+1 p + h y y m F l+1 y uf L+1 u du (14) which is inceasing in y ove the egion y y m because 2 /y 2 L s y = L+l+1 p + h f y y m l+1 y u f L+1 u du >. Because it is also inceasing in the othe egion, /yl s y is inceasing in y fo all y. Because p P, wehave/yl s = 1 c s + L h s + L+1 1 c l p + h s, and lim y /yl s y = 1 c s + L h s >. Hence, /yl s y cosses the zeo line only once, and ys m is defined as the unique point whee it cosses zeo. When p H, wehave/yl s y m. Theefoe, ys m y m and it is obtained by setting Equation (13) equal to zeo; i.e., /yl s ys m =. Solving fo ym s yields the closedfom solution in pat (b). Note that ys m y m. When p >H,wehave/yL s y m<. Hence, ym s y m and it is the y that sets Equation (14) equal to zeo. Note also that ys m >y m. The installation basestock levels ae obtained fom thei definition; i.e., Y = minys mym and Y s = maxys m ym. Poof of Poposition 9. To pove pat (a), note that Y = y m when p >H. We set Y = Fl+1 1l p + h s 1 c / l p + h = Fl+1+ 1 p 1 c /p + h = Y. Applying F l+1 to both sides and eaanging tems yields the theshold B. Note that when h s B, Y Y p. Note also that B is inceasing in because F l+1+ x F l+1+ x fo any x and. Pat (b) follows immediately fom Poposition 8(b) and the definition of Ys. Poof of Poposition 1. The poof follows fom concavity and Poposition 2(a) and 2(b). Poof of Poposition 11. Fist, ecall fom Poposition 5(c) that 1 a = f 1. Hence, 1 a = L + 1 because G s + G p 1 is concave. Second, conside the two tivial cases. Note that L a 2 a N a fom Poposition 5(a). Hence, if 1 a =, then i a = fo all i. Similaly, if N a = L + 1, then i a = L + 1 fo all i. Pats (a) and (b) and the expession fo Ki a follow fom Poposition 4(a). Next, conside 1 a = L+1 and N a L+1. Note that each tem in the objective function (6) is concave in, that is, W i i G s + G p i + i 1 G p i G p i 1 is concave in. Let i max be the smallest maximize. Next, we show that i a i 1 a a i+1 fo i 2 N 1. Assume fo a contadiction that i 1 a >a i > i+1 a max. We conside thee cases: i <i a, i max >i a, and i max = i a max.ifi <i a, then /W i < fo all i aa i 1. Theefoe, the supplie can incease i a to i 1 a and educe he expected cost without violating the monotonicity constaint in (6), contadicting the optimality of i a. If i max >i a, then /W i > fo all i+1 a a i. Theefoe, the supplie could decease i a to i+1 a and educe he expected cost without violating the monotonicity constaint in (6), contadicting the optimality of i a max.ifi = i a, then i i+1 a a i 1 yields a lowe expected cost fo the supplie without violating monotonicity, again contadicting the optimality of i a. Theefoe, fo all i 2N 1, we must have i a i 1 a a i+1. Simila aguments lead to N a N a 1. The above esult implies that thee exists a finite stictly inceasing sequence k 1 k 2 k n (whee k n N ) fo some n such that 1 a = 2 a = =a k 1 = L + 1; k a 1 +1 = = k a 2 k2 and k a n 1 +1 = 2 a = = k a n 1 = N a = k n. In othe wods, thee ae n segments. Next, we show that in fact n 2, i.e., thee can be at most two such segments. Conside the segment k j fo some j 2N 1. Note that W kj kj k j k=k j 1 +1 kg s k j +G p k kj +
17 914 Opeations Reseach 56(4), pp , 28 INFORMS k 1 G p k kj G p k 1 kj is also concave in kj. Let k max j be the maximize of W kj kj. Following the same aguments as above, one can show that kj kj 1 kj+1. Hence, the n segments can be educed to n 1 segments, that is, thee exists a new finite stictly inceasing sequence k 1 k 2 k n 1. This agument can be applied iteatively until the sequence of numbes is educed to two numbes, and hence two segments. Theefoe, thee exists an m such that i a = 1 a = L + 1 fo i m and i a = N a fo i>m. The expession fo Ki a follows fom Poposition 4(a). Poof of Poposition 12. Let G n be a geneal fom of the optimal expected cost function, whee n is the numbe of peiods of uncetainty, i.e., n = L+1 fo the supplie o n = l fo the etaile. Then, fom (9), G n = pn h + p Fn 1p/h+p uf n u du fo geneic holding and penalty costs h and p. To pove pat (i), when F is Nomally distibuted, the optimal basestock level is Y = Fn 1 p/h + p = n + z n, whee z = 1 p/h + p. The esulting optimal cost is G n = h + pz n. Hence, G n depends on only though n, which is concave in fo both n = L + 1 and n = l Note also that we have G p i G p i 1= h +p i p i /h +p i h + p i 1 p i 1 /h + p i 1 l The tem in backets is the diffeence between optimal inventoy cost functions with l = = and penalty costs p i and p i 1, espectively. Fom Poposition 1(b), this tem is positive. Hence, G p i G p i 1 is concave in. To pove pat (ii), we define n G n = G n G n 1. To show that G n is concave in n, we show n G n> n+1 G n. By algeba, this inequality holds when n+1 p/h+p + uf n+1 u du 2 n 1 p/h+p n p/h+p uf n u du uf n 1 u du > (15) Letting = p/h + p, we note that the left side of (15) equals zeo fo 1. The second deivative of the left side of (15) with espect to equals 1/f n+1 Fn+1 1 2/ f n Fn 1 + 1/f n 1 Fn 1 1, which must be negative fo (15) to be tue fo all 1. That is, f n Fn 1 1 must be concave in n. Next, we show that G p i G p i 1 is concave in. To do so, we show that /p G p 2G p 1 + G p 2. Fom (9), the above is equivalent to [ l+1+ p /h +p uf l+1+ u du 2 + l+ p /h +p l+ 1 p /h +p uf l+ u du ] ( ) h uf l+ 1 u du + h + p [ ( ) ( F 1 p l+1+ 2F 1 p l+ h + p h + p ( + F 1 p l+ 1 h + p ) )] (16) The limit of the left side of (16) as = p /h + p goes to one equals l l + + l + 1 =. Hence, (16) holds as long as the deivative of the left side with espect to is negative. This deivative equals 1 1/f l+1+ Fl /f l+ Fl /f l+ 1Fl+ 1 1, which is negative when f n Fn 1 1 is concave in n. Poof of Poposition 13. The poof is defeed to the online companion that can be found at infoms.og/. Endnotes 1. Recent infomation technologies such as compliance management systems and RFID have been cited as enables fo accounting of sevice and qualityelated activities acoss the supply chain (Lee and Öze 25). 2. In 5, we show that the etaile follows a stationay basestock policy, so the etaile odes in each peiod to ecove the pevious peiod s custome demand. As a esult, the supplie obseves the same demand steam as the etaile. 3. We dop the subscipt t fom the definition of D n+1 fo bevity. 4. This altenative souce may be a finishedgoods inventoy caied eithe by anothe fim o fo anothe etaile. It may also epesent poduction o pocessing esouces lent by anothe opeation. To ou knowledge, this altenative souce concept appeaed fist in Lee et al. (2). See also Gaves and Willems (2), who assume guaanteed sevice between evey stage in a supply chain. 5. We note the diffeence between the centalized (fistbest) solution concept used in supply chain contacting liteatue and the cental contol concepts used in multiechelon inventoy liteatue. In the liteatue, these nomenclatues ae aely discussed togethe. This is due to the fact that most papes eithe study coodination issues in supply chain contacting fo a singlepeiod poblem (as in Cachon 23), o they study cental and local contol in multiechelon inventoy poblems unde full infomation (as in Axsäte and Rosling 1993). 6. Inventoy at fim j and downsteam plus the pipeline inventoy to fim j minus custome backodes. 7. Inventoy on hand at fim j plus its pipeline inventoy minus the backode due to downsteam location s ode (o custome demand fo j = ). 8. Using a simila agument in Veinott (1965), the inventoy contol poblem with linea salvage value fo each fim is conveted into an equivalent poblem with zeo salvage. 9. We use the tem deceasing and inceasing in the weak sense, so deceasing means noninceasing.
18 Opeations Reseach 56(4), pp , 28 INFORMS 915 Acknowledgments The authos ae thankful to the anonymous associate edito and the efeees fo thei helpful suggestions. The discussions duing thei pesentations at the 23 INFORMS annual meeting in Atlanta, Stanfod Univesity, Texas A&M, Univesity of Califonia Santa Cuz, Univesity of Illinois UbanaChampaign, Univesity of Tennessee Knoxville, Univesity of Texas Dallas, L. M. Eicsson, and Hewlett Packad wee also beneficial. The authos also gatefully acknowledge financial suppot fom L. M. Eicsson and the Stanfod Alliance fo Innovative Manufactuing. The pevious vesion of this pape was titled Pomised Leadtime Contacts and Renegotiation Incentives Unde Asymmetic Infomation. The listing of authos is alphabetical. Refeences Andeson, D. L., F. F. Bitt, D. J. Fave The seven pinciples of supply chain management. Supply Chain Management Rev. (Sping). Axsäte, S., K. Rosling Notes: Installation vs. echelon stock policies fo multilevel inventoy contol. Management Sci Bagnoli, M., T. Begstom Logconcave pobability and its application. Woking Pape 1989D, Depatment of Economics, Univesity of Califonia, Santa Babaa, CA. BanesSchuste, D., Y. Bassok, R. Anupindi. 26. Optimizing delivey lead time/inventoy placement in a twostage poduction/distibution system. Eu. J. Ope. Res Billington, C. 22. HP cuts isk with potfolio appoach. Puchasing.com (Febuay 21). Cachon, G. 23. Supply chain coodination with contacts. S. Gaves, T. de Kok, eds. Handbooks in Opeations Reseach and Management Science, Vol. 11, Chapte 6. NothHolland, Amstedam. Cachon, G., M. Laiviee. 21. Contacting to assue supply: How to shae demand foecasts in a supply chain. Management Sci Cachon, G. P., F. Zhang. 26. Pocuing fast delivey: Sole soucing with infomation asymmety. Management Sci Cachon, G., P. H. Zipkin Competitive and coopeative inventoy policies in a twostage supply chain. Management Sci Chen, F. 23. Infomation shaing and supply chain coodination. S. Gaves, T. de Kok, eds. Handbooks in Opeations Reseach and Management Science, Vol. 11, Chapte 7. NothHolland, Amstedam. Chen, F., Y. Zheng Lowe bounds fo multiechelon stochastic inventoy systems. Management Sci Clak, A., H. Scaf Optimal policies fo a multiechelon inventoy poblem. Management Sci Cohen, M. A., V. Deshpande, Y. Wang. 23. Managing supply chains with diffeentiated sevice equiements Models and applications. T. P. Haison, H. L. Lee, J. J. Neale, eds. The Pactice of Supply Chain Management: Whee Theoy and Application Convege, Chapte 11. Kluwe s Intenational Seies in Opeations Reseach and Management Science. Kluwe Academic Publishing, Boston. Cobett, C. J. 21. Stochastic inventoy systems in a supply chain with asymmetic infomation: Cycle stocks, safety stocks, and consignment stock. Ope. Res. 49(4) Cobett, C. J., X. de Goote. 2. A supplie s optimal quantity discount policy unde asymmetic infomation. Management Sci. 46(3) Fedeguen, A., P. Zipkin Computational issues in an infinite hoizon, multiechelon inventoy model. Ope. Res Fudenbeg, D., J. Tiole Game Theoy. MIT Pess, Cambidge, MA. Gadiesh, O., J. L. Gilbet Pofit pools: A fesh look at stategy. Havad Bus. Rev. 76(3) Gallego, G., Ö. Öze. 23. Optimal eplenishment policies fo multiechelon inventoy poblems unde advance demand infomation. Manufactuing Sevice Ope. Management. 5(2) Gaves, S., S. Willems. 2. Optimizing stategic safety stock placement in supply chains. Manufactuing Sevice Ope. Management 2(1) Ha, A. 21. Supplieetaile contacting: Asymmetic cost infomation and cutoff level policy fo etaile paticipation. Naval Res. Logist Hause, L. M. 23. Riskadjusted supply chain management. Supply Chain Management Rev. 7(6) Keskinocak, P., R. Ravi, S. Tayu. 21. Scheduling and eliable leadtime quotation fo odes with availability intevals and leadtime sensitive evenues. Management Sci. 47(2) Lee, H., J. Whang. 2. Decentalized multiechelon supply chains: Incentives and infomation. Management Sci. 45(5) Lee, H. L., Ö. Öze. 25. Unlocking the value of RFID. Poduction Ope. Management. Fothcoming. Lee, H. L., K. C. So, C. S. Tang. 2. The value of infomation shaing in a twolevel supply chain. Management Sci. 46(5) Lovejoy, W. S. 26. Optimal mechanisms with finite agent types. Management Sci. 52(5) Myeson, R. B Incentive compatibility and the bagaining poblem. Econometica Nahmias, S Simple appoximations fo a vaiety of dynamic lead time lostsales inventoy models. Ope. Res. 27(5) Öze, Ö., W. Wei. 26. Stategic commitments fo an optimal capacity decision unde asymmetic foecast infomation. Management Sci. 52(8) Poteus, E. 2. Responsibility tokens in supply chain management. Manufactuing Sevice Ope. Management 2(2) Shaked, M., J. G. Shanthikuma Stochastic Odes and Thei Applications. Academic Pess, Boston. Veinott, A Optimal policy fo a multipoduct, dynamic, nonstationay inventoy poblem. Management Sci. 12(3) Wang, Y., M. Cohen, Y.S. Zheng. 22. Diffeentiating custome sevice on the basis of delivey leadtimes. IIE Tans
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