In some supply chains, materials are ordered periodically according to local information. This paper investigates

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1 MANUFACTURING & SRVIC OPRATIONS MANAGMNT Vol. 12, No. 3, Summer 2010, pp ssn essn nforms do /msom INFORMS Improvng Supply Chan Performance: Real-Tme Demand Informaton and Flexble Delveres Kevn H. Shang Fuqua School of Busness, Duke Unversty, Durham, North Carolna 27708, Sean X. Zhou Systems ngneerng & ngneerng Management, Chnese Unversty of Hong Kong, Shatn, Hong Kong, Chna, Geert-Jan van Houtum School of Industral ngneerng, ndhoven Unversty of Technology, 5600 MB ndhoven, The Netherlands, In some supply chans, materals are ordered perodcally accordng to local nformaton. Ths paper nvestgates how to mprove the performance of such a supply chan. Specfcally, we consder a seral nventory system n whch each stage mplements a local reorder nterval polcy;.e., each stage orders up to a local basestock level accordng to a fxed-nterval schedule. A fxed cost s ncurred for placng an order. Two mprovement strateges are consdered: (1 expandng the nformaton flow by acqurng real-tme demand nformaton and (2 acceleratng the materal flow va flexble delveres. The frst strategy leads to a reorder nterval polcy wth full nformaton; the second strategy leads to a reorder pont polcy wth local nformaton. Both polces have been studed n the lterature. Thus, to assess the beneft of these strateges, we analyze the local reorder nterval polcy. We develop a bottom-up recurson to evaluate the system cost and provde a method to obtan the optmal polcy. A numercal study shows the followng: Increasng the flexblty of delveres lowers costs more than does expandng nformaton flow; the fxed order costs and the system lead tmes are key drvers that determne the effectveness of these mprovement strateges. In addton, we fnd that usng optmal batch szes n the reorder pont polcy and demand rate to nfer reorder ntervals may lead to sgnfcant cost neffcency. Key words: mult-echelon; perodc orderng; polcy comparson; value of nformaton; value of flexble delveres Hstory: Receved: May 6, 2008; accepted: July 28, Publshed onlne n Artcles n Advance November 13, Introducton In producton/dstrbuton systems, materals are often ordered and delvered n fxed tme ntervals. A locaton also usually orders accordng to ts local nventory status. Ths nventory replenshment practce s commonly seen n a supply chan that comprses geographcally dspersed enttes, each wth ts own nformaton systems. For example, MC 2, a leadng manufacturer for database servers n the Unted States, mplements ths type of replenshment polcy for some of ts tems. On fxed days of the week, MC 2 revews nventory levels and places orders wth ts supplers. After recevng the orders, the supplers delver materals to MC 2 va a thrd-party logstcs truck f the requested tems are avalable. One reason for the wdespread use of ths replenshment practce s smplcty t s ust smpler to montor local nventores and to plan personnel and labor for scheduled orderng and delveres. In addton, ths practce s partcularly useful for a retaler who manages multple products and needs to coordnate orders. However, t may create at least two operatonal neffcences. Frst, a locaton cannot order (so as to trgger a shpment f t s not n one of ts order perods. Second, real-tme customer demand nformaton cannot be transmtted mmedately to upstream locatons. Ths nformaton delay may lead to neffcent order decsons at upstream stages. We consder one such supply chan wth fxed reorder ntervals. To facltate our dscusson, we 430

2 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 431 assume that the supply chan partners (who possbly belong to a sngle frm share a common goal of optmzng systemwde performance, but they only have local nformaton at the operatonal plannng level. There s a central planner 1 who determnes all nventory control parameters (e.g., based on summary data n a spreadsheet, and each stage smply generates replenshment orders based on these control parameters and the local demand and nventory nformaton. In ths case, t s mportant to evaluate the current supply chan performance and to decde whether to nvest to mprove the supply chan. Here, we specfcally consder two mprovement strateges expandng the nformaton flow by acqurng realtme demand nformaton or acceleratng the materal flow by provdng flexble delveres. The queston s, whch strategy should be chosen and under what condtons? Ths paper ams to answer ths queston by analyzng a perodc-revew, seral nventory system wth generally dstrbuted one-perod demands. As a base model, each stage mplements a local reorder nterval polcy, referred to as the local s T polcy. That s, a stage orders every T perods accordng to ts local nventory order poston (=nventory on order + nventory on hand local backorders. If the order poston s less than the base-stock level s, the stage orders up to s. We assume that the reorder ntervals follow nteger-rato relatons;.e., the reorder nterval of an upstream stage s an nteger-multple of that of ts mmedate downstream stage. The replenshments are synchronzed; that s, a downstream stage, whenever possble, places an order when ts upstream stage receves a shpment. There s a fxed cost ncurred for each order placed. Delvery lead tmes between stages are constant. There s a lnear holdng cost for each unt held at a stage and a lnear penalty cost for each backorder that occurs at the most downstream stage. The obectve s to mnmze the average total supply chan cost per perod. Note that the aforementoned neffcences appear n ths base system: a stage cannot place an order n a perod f the perod s not an 1 If the entre chan belongs to a sngle frm, the central planner can be the owner. For a chan composed of several ndependent frms, the central planner can be one of these frms, a team of them, or a thrd-party organzaton. We refer the reader to Shang et al. (2009, pp for practcal examples. order perod, and the local nventory order poston of a stage s updated by ts mmedate downstream orders, not by the demand receved at the most downstream stage. For the frst mprovement strategy, each stage can update ts local nventory status accordng to the realtme demand nformaton. It can be shown that the resultng system s equvalent to a system that mplements an echelon s T polcy. The echelon s T polcy operates smlarly to the local polcy except that the echelon nventory order poston (=nventory on order + nventory on hand + nventory at or n transt to all downstream stages backorders at the most downstream stage s montored for each stage. For the second mprovement strategy, where a flexble delvery polcy s mplemented, a stage s allowed to place an order n any perod. An upstream stage wll fll and shp the order mmedately, provded that the upstream stage has stock avalable. From ths perspectve, such a replenshment polcy s more responsve to the downstream orders and hence can speed up the materal flow. Because there s a fxed cost for placng an order, we assume that a stage wll not place an order untl ts local nventory order poston s lower than or equal to a threshold level r and wll order the least nteger multple of batch sze Q to rase the nventory order poston above r. Ths replenshment scheme s essentally the same as the so-called local r nq polces n the lterature. For order coordnaton, we assume that the order batches satsfy nteger-rato relatons: an upstream batch s an nteger multple of ts mmedate downstream one. To answer our research queston, we need to compare the optmal costs of the above mentoned three polces. Axsäter and Roslng (1993 showed that the local r nq polcy s a specal case of the echelon r nq polcy (and that the local polcy s therefore suboptmal. Chen (1998b provded an algorthm to search for the optmal local reorder ponts when batch szes are fxed. It s, however, not clear how to obtan the optmal local batch szes. In Appendx B, we provde an approach to obtan these optmal batch szes. For the echelon s T polcy, van Houtum et al. (2007 provded a method to evaluate the echelon s T polcy. They also constructed an algorthm for fndng the optmal echelon base-stock levels when reorder ntervals are fxed. Fnally, a recent paper by Shang and

3 432 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS Zhou (2009 provded an approach for obtanng the optmal reorder ntervals for the echelon s T polcy. The man techncal contrbuton of ths paper s to analyze the local s T polcy. We frst show the dynamcs of several key nventory varables under the local s T polcy. These dynamcs lead to a smple, bottom-up recurson that can evaluate a gven local s T polcy. More specfcally, by convertng the local nventory varables nto echelon terms, we can evaluate the local polcy as f t were an echelon s T polcy wth modfed system lead tmes. Ths evaluaton procedure can be used further to fnd the optmal local base-stock levels for gven reorder ntervals. Unlke the r nq polcy, the local s T polcy s not a specal case of the echelon s T polcy. Nevertheless, we show that the optmal echelon s T polcy always domnates the optmal local one. 2 Wth ths result, we provde a method to search for the optmal reorder ntervals for the local s T polcy. These analytcal results allow us to assess the value of the mprovement strateges by comparng the aforementoned nventory polces n a numercal study. We fnd that the average (maxmum cost reducton when the system swtches from the local s T polcy to the local r nq polcy s 11.27% (28.32%, whch s sgnfcantly larger than the 5.51% (16.67% acheved when swtchng from the local s T polcy to the echelon s T polcy. Ths suggests that ncreasng the flexblty of delvery s more benefcal than acqurng realtme demand nformaton. In addton, the dfference between the benefts of both mprovement strateges ncreases when demand becomes more varable. Ths result provdes an mportant nsght to avod a management ptfall: when demand becomes more varable, t may not be effectve to nvest n IT systems n order to acqure real-tme demand nformaton; an agle logstcs system s key to achevng supply chan effcency. Our fndng confrms the lessons from other papers, such as Cachon and Fsher (2000 and Gavrnen et al. (1999, n whch the authors found that obtanng demand nformaton may not be as valuable as mprovng physcal system confguratons. 2 An echelon polcy may not always domnate a local one. For example, Axsäter and Juntt (1996 showed that a local polcy may domnate an echelon polcy under some condtons for a twoechelon dstrbuton system. Here, we obtan a smlar nsght for a general-echelon model n whch batch szes and reorder ntervals are determned endogenously (nstead of exogenously as n the above two papers. Of course, our concluson does not consder other nonmodeled factors, such as mplementaton costs, labor costs, and easness of executon, whch tend to favor the (s T polcy. A manager who has to make a choce between the two mprovement polces also has to take these nonmodeled factors nto account. We further use our model to examne the ssue of value of demand nformaton (VOI. The VOI n our model s equvalent to the cost reducton by swtchng from the local polcy to the echelon polcy. In a numercal study, we fnd that there s no sgnfcant beneft acheved by swtchng from the local r nq polcy to the echelon r nq polcy. Ths observaton suggests that addng demand nformaton to a system wth flexble delveres provdes lttle value. Interestngly, Chen (1998b conducted a smlar study wth fxed, exogenous batch szes and reported a slghtly hgher average VOI. For the s T polces, the VOI s hgher when the fxed costs are larger and lead tmes are shorter. Ths mples that the demand nformaton s most benefcal when the supply chan has long reorder ntervals (hgher fxed costs lead to longer reorder ntervals and short lead tmes. In 4, we provde ntutve explanatons for the above observatons. Fnally, we nvestgate whether the reorder ntervals can be nferred by the optmal batch szes and the demand rate. The logc behnd ths queston s that the reorder ntervals and batch szes are equvalent n the determnstc demand model. Thus t s plausble to conecture that effectve reorder ntervals may be nferred by the optmal batch szes n the stochastc demand model. Interestngly, our numercal study suggests that the nferred reorder ntervals are seldom optmal and may lead to sgnfcant cost neffcency when the system only has local nformaton. Lterature Revew Our paper s related to the lterature of nventory control n mult-echelon systems wth r nq and s T polces. The local and echelon r nq polces have been extensvely studed n the lterature. Avalable results nclude polcy evaluaton, optmzaton, and approxmatons. Noteworthy examples nclude

4 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 433 Axsäter (1993, Cachon (2001a, De Bodt and Graves (1985, Chen (1998b, 2000, Chen and Zheng (1998, Shang (2008, and Shang and Song (2007. We refer the reader to Axsäter (2003, Chen (1998a, and Smch- Lev and Zhao (2007 for a revew. The lterature on the s T polcy s relatvely sparse. Naddor (1975 studed the s T polcy for sngle- and mult-tem systems. Cachon (1999 studed the reorder-nterval polcy n a dstrbuton system. He showed that the suppler s demand varance wll declne as the retalers reorder nterval becomes longer. Graves (1996 provded a new approach to evaluate the cost for dstrbuton systems under the so-called vrtual allocaton rule. Van Houtum et al. (2007 studed a seral model and showed that the echelon s T polces are optmal when the reorder ntervals are fxed. They also provded an algorthm to obtan the optmal base-stock levels. Chao and Zhou (2009 extended these results to batch-orderng systems. Feng and Rao (2007 consdered a two-stage system wth echelon s T polces. They provded a heurstc for the polcy parameters. Recently, Shang and Zhou (2009 provded a heurstc and an approach for obtanng the exact optmal s T polcy when fxed order costs are present. Our paper s also related to the lterature on comparsons of nventory polces. For sngle-stage systems, Hadley and Whtn (1963 compared the r nq and s T polces and showed that the r nq polcy s superor to the s T polcy. Rao (2003 studed the s T polcy under the contnuous-tme model and constructed a worst-case cost bound for the optmal s T polcy based on the optmalty result of the r Q polcy. For multlocaton models, Cachon (2001b compared three polces,.e., Q S, Q S T, and S T polces, n a multretaler system. The Q S and S T polces are smlar to our r nq and s T polces, respectvely. Hs study suggested that the polces wth fxed replenshment ntervals may lead to sgnfcant neffcency. Recently, Gürbüz et al. (2007 also studed a multretaler model. They proposed a new nventory polcy and compared t wth the three exstng ones. They dentfed the stuatons under whch each polcy would perform effectvely. Fnally, our paper adds to the lterature on assessng the VOI. The lterature n ths subect s extensve. Most models assume ether two-echelon systems or that batch szes/reorder ntervals are fxed (for example, see Avv and Federgruen 1998, Cachon and Fsher 2000, Chen 1998b, Gavrnen 2002, Gavrnen et al. 1999, Graves 1999, Lee et al. 2000, and Smch-Lev and Zhao Smch-Lev and Zhao (2004 studed a two-stage seral system where a capactated manufacturer supples a retaler. Both stages mplement an s T polcy, but the retaler has a longer reorder nterval than the manufacturer. They found that f the retaler shares the demand nformaton between order perods, the system cost can be reduced sgnfcantly. The remander of ths paper s organzed as follows: 2 ntroduces the base model and the local s T polcy. We also dscuss the two mprovement strateges and the resultng nventory polces. Secton 3 analyzes the local s T polcy, and we show how to evaluate and optmze the local s T polcy. Secton 4 conducts a numercal study to compare these nventory polces, wth an emphass on the effectveness of the mprovement strateges. We also dentfy the condtons under whch each of the strateges s most benefcal. Secton 5 concludes. Appendx A presents the proofs. Appendx B provdes an approach for obtanng optmal batch szes for the local r nq polcy. 2. The Base System and Two Improvement Strateges We consder a centralzed-control, perodc-revew seral nventory system wth N stages (N, the set of postve ntegers, N 2. Customer demand occurs at stage 1. Stage 1 s replenshed by stage 2, stage 2 by stage 3, etc., and stage N by an outsde source wth ample supples. Tme s dvded nto perods of length one, and the perods are numbered Let t t + and t t + denote the tme nterval over perods t t + 1 t+ 1 and perods t t + 1 t+, respectvely. Demands are ndependently and dentcally dstrbuted between perods. Let >0 denote the mean of demand n a perod and D t t + and D t t + denote the cumulatve demand over tme n t t + and n t t +. We use D and D to represent the demand n and n + 1 perods, respectvely, f the tme ndex t s omtted. There are constant lead tmes L between stages + 1 and. At the end of each perod, the echelon holdng cost h ( 0 s ncurred for each unt of

5 434 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS nventory held n echelon, and the backorder cost b (>0 s ncurred for each unt of backorders at stage 1. Denote h = k= h k (1 N, so the local holdng cost for stage s h N. The base model operates under the local s T polcy. Under ths polcy, stage vews stage 1 s order as ts demand and orders every perods. At the begnnng of an order perod, referred to as an order epoch, stage revews ts local nventory order poston (=outstandng orders + on-hand nventory stage s backorders and orders up to the local base-stock level s. There s a fxed cost k (>0 ncurred for placng an order. The orderng tmes between stages are synchronzed. That s, stage orders, whenever possble, when stage + 1 receves a shpment. Ths rule was frst ntroduced by Graves (1996, and we hereafter term ths rule the synchronzed replenshment rule. (It can be shown that the synchronzed replenshment rule s the best for the local s T polcy. A proof s avalable from the authors upon request. In addton, the reorder ntervals satsfy nteger-rato relatons;.e., = n 1, n, for = 2 N. We assume that the replenshment actvtes n a perod occur at the begnnng of the perod. At stage >1, they occur n the followng sequence: (1 an order, f any, from stage 1 s receved; (2 an order s placed wth stage + 1 f the perod s n stage s order perod; (3 a shpment, f any, s receved from stage + 1; and (4 a shpment s sent to stage 1f the perod s n stage 1 s order perod. For stage 1, order placement occurs at the begnnng of stage 1 s order perods, whereas customer demand arrves durng a perod. We assume that the stages perform these events sequentally, from stage 1, stage 2, etc., untl stage N. Costs are evaluated at the end of a perod. The obectve s to mnmze the average total cost per perod. We use a two-stage example (see Fgure 1 to llustrate the system dynamcs under the local s T polcy. In ths example, T 1 = 2, T 2 = 4, L 1 = 1, and L 2 = 1. Defne IOP t = local nventory order poston after recevng downstream orders at the begnnng of perod t, IOP t = local nventory order poston after placng an order at the begnnng of perod t. Fgure 1 The Dynamcs of IOP Under the Local s T Polcy s 2 8 s 1 5 IOP 1 IOP 2 Demand t Note. The trangles represent the order epochs. We assume that the system starts wth IOP 0 = s, for all. In ths example, s 1 = 5 and s 2 = 8. Stage 2 orders at t = These orders wll arrve at t = Based on the synchronzed replenshment rule, stage 1 should order at t = Let us focus on IOP 2 at stage 2. At t = 1 and t = 3, stage 1 places an order. These orders are the demand for stage 2, and thus IOP 2 decreases. At t = 4, IOP 2 4 = 2, so stage 2 places an order of sx unts. These sx unts are the customer demand that occurs n 0 3. Note that stage 1 receves two unts of customer demand n 3 4. Because stage 1 does not order at t = 4, stage 2 does not consder these two unts of demand when t places an order. We wll analyze the local s T polcy n detal n the next secton. For now, let us turn to each of the mprovement strateges Real-Tme Demand Informaton For the frst strategy, the supply chan can nvest n IT systems to make the demand nformaton avalable to all stages. In such a case, a stage can update ts local nventory status accordng to the real-tme demand nformaton. Followng Shang et al. (2009, t can be shown that the resultng system s equvalent to a system n whch stage mplements an echelon s T polcy wth the echelon base-stock level S e = s =1 and the reorder nterval, = 1 N. Notce that the resultng echelon s T polcy also follows the synchronzed replenshment rule. The echelon s T polcy s operated smlarly as the local s T polcy

6 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 435 except that the echelon nventory order poston s montored. More specfcally, let IOP t = echelon nventory order poston after recevng customer demand D t 1 t at the begnnng of perod t and before orderng = t =1 IOP IOP t = echelon nventory order poston after orderng at the begnnng of perod t = =1 IOP t Under the echelon s T polcy, each stage orders n every perods. At an order epoch t, fiop t s less than S e, the stage orders up to Se so that IOP t = S e. The echelon s T polcy has been studed recently. We summarze several key results whch wll be used n the subsequent sectons. Defne = T 1, = 1 N, T = T N, and x = max 0 x. For gven echelon base-stock levels S1 e Se N, the average nventory holdng and backorder cost can be evaluated by the followng recurson (van Houtum et al and Chao and Zhou 2009: Defne G e 1 y T 1 = 1 ( T1 1 T 1 For = 2 N, h 1 y D L 1 + l G e y = 1 1 [ h y D L +l ( ( [ mn {S e 1 y D L + +G e 1 + b + h 1 N y D L 1 + l (1 l 1 1 } 1 ] Here, G e y represents the average nventory holdng and backorder costs of echelon when stage +1 has nfnte stock (so that orders of stage are always mmedately fulflled and stages 1 1 follow an echelon s T polcy wth base-stock levels S1 e Se 1 y. For the average fxed cost, note that k s ncurred only when stage places an order. Thus the probablty of orderng at each stage s order epoch (2. Under the echelon s T pol- and IOP t =IOP t D. t s P IOP t <S e cy, IOP t =S e Thus the probablty of orderng s equal to p = P D >0. Wth ths, the average fxed cost s equal to N =1 k p /T. Consequently, the average total cost per perod s C e S e T = N =1 k p / + G e N Se N T N. For fxed T, the average fxed cost term s a constant. Thus mnmzng C e S e T s equvalent to mnmzng G e N Se N T N. A soluton that mnmzes G e N Se N T N can be obtaned by mnmzng (1 and (2 recursvely. More specfcally, let S1 e T 1 =argmn y G e 1 y T 1. For = 2 N, suppose that S 1 e 1 s known. Substtute S 1 e 1 for S 1 e n (2 and let Se be the mnmzer of the resultng G e y functon;.e., S e = argmn y G e y. Then S e T = S1 e T 1 SN e T N s the optmal echelon base-stock vector. Fnally, to fnd the optmal reorder ntervals for the echelon polcy, one needs to solve the followng problem: mn C e T =C e S e T T T s t =n 1 =2 N We refer the reader to Shang and Zhou (2009 for an algorthm Flexble Delveres For the second strategy n whch a flexble delvery polcy s mplemented, stage s allowed to place an order n any perod, and stage +1 wll mmedately shp the order to stage f stage +1 has suffcent stock. Because there s a fxed cost assocated wth each order placed, we assume that stage wll not place an order untl ts nventory order poston s lower than or equal to a threshold level r and wll order the least nteger multple of batch sze Q to rase the nventory order poston above r. In other words, ths polcy s essentally the same as the local r nq polcy. More specfcally, for any perod t, fiop t r, stage places an order of nq, n, such that IOP t r +1 r +2 r +Q. Here, k s ncurred once for placng an order that may nclude several batches. We assume that the order batches satsfy nteger-rato constrants: Q =q Q 1, q, =2 N. Also, the reorder pont r s assumed to be an nteger multple of ts downstream batch sze Q 1 (Chen 1998b.

7 436 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS Note that under ths polcy, the upstream stage s local nventory order poston s only updated by the downstream stage s orders, not by the real-tme customer demand at stage 1. Axsäter and Roslng (1993 showed that for any local r nq polcy, there exsts an equvalent echelon r nq polcy. Shang and Zhou (2009 developed a scheme to evaluate an echelon r nq polcy wth fxed order costs ncurred for each order placed. We can use these results to evaluate a local r nq polcy. Specfcally, we frst convert a local r nq polcy to the correspondng echelon polcy and then apply Shang and Zhou s recurson to evaluate the cost. Wth fxed batch szes, Chen (1998b developed an approach to fnd the best reorder ponts for the local r nq polcy. It s, however, not known how to fnd the optmal batch szes for the local r nq polcy n the lterature. In Appendx B, we provde an approach to obtan the bounds for the optmal local batch szes. The optmal soluton then can be found by a search over all feasble solutons. 3. The Local s T Polcy: valuaton and Optmzaton Ths secton analyzes the local s T polcy, amng to answer several questons. Frst, how does one evaluate a local s T polcy? Second, how does one obtan the optmal local polcy parameters? Thrd, what s the relatonshp between the local and echelon s T polces? Is the local polcy a specal case of the echelon one? If not, does the echelon polcy domnate the local one? We frst provde a method to evaluate the total cost. It s smpler to evaluate the cost from the echelon perspectve. Recall the defnton of IOP t. We further defne IL t = net echelon nventory level for stage at the begnnng of perod t (after order arrval, IL t = net echelon nventory level for stage at the end of perod t, IP t = echelon nventory n-transt poston for stage at the begnnng of perod t after orderng. Note that the dfference between IOP t and IP t s the outstandng orders for stage that are not yet flled by stage +1. Also, the total cost s determned by the IL t values. Consder the dynamcs of the echelon nventory varables under the local s T polces. Suppose that stage N places an order at perod t. Let the resultng echelon nventory order poston be IOP N t. Because stage N has ample supply, IP N t =IOP N t. Ths order wll arrve at stage N at perod t +L N. Because there wll be no other order perods untl perod t +T N, IP N t wll determne both IL N t+l N + and IL N t+l N + for =0 T N 1. That s, and IL N t+l N + =IP N t D t t+l N + IL N t+l N + =IP N t D t t+l N + Now consder stage =N 1 N 2 1 sequentally. Let b a be the operator that returns the remander of a dvded by b, and a b and L = k= L k the total lead tmes between stage and +1. Accordng to the synchronzed replenshment rule, stage wll order n perods t +L +1 N + /, for =0 T N 1. The resultng stage s echelon nventory order poston and the stage +1 s net echelon nventory level ontly determne stage s echelon ntranst nventory poston. That s, for =0 T N 1, IP (t +L +1 N + ( =mn {IOP t +L +1 N + IL +1 ( t +L +1 N + ach IP wll further determne IL and IL : IL t+l N + =IP (t +L +1 N + IL t+l N + =IP (t +L +1 N + } (3 [ D t +L +1 N + t +L N + + T [ D t +L +1 N + t +L N + + T (4 ] (5

8 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 437 Consequently, f we can determne IOP n (3, we are able to determne IL and IL from the recurson (3 (5. Our next task s to determne these IOP under the local s T polcy. Recall that the local base-stock levels are gven by s 1 s N, and defne S = s =1 for =1 N We call S the nduced echelon base-stock level, as t s determned by the local base-stock levels. Because the system starts wth IOP 0 =s, IOP 0 = =1 IOP 0 = =1 s =S. As we llustrated n 2, under the local s T polcy, stage wll order a quantty equal to the total orders receved from stage 1 snce the last order epoch. However, demand may have occurred at stage 1, of whch stage was not aware when placng the order. Thus the echelon nventory order poston IOP t may not be S under the local s T polcy. To further explan ths, let us frst consder a three-stage example wth T 1 =2, T 2 =T 3 =4, L 1 =L 2 = L 3 =1 (see Fgure 2. If, say, stage 3 orders at t = , then stage 2 wll order at t = and stage 1 wll order at t = because of the synchronzed replenshment rule. In ths example, IOP 1 0 =s 1 =5, IOP 2 0 =s 2 =8, and IOP 3 0 =s 3 = 14, so IOP 1 0 =5 IOP 2 0 =13, and IOP 3 0 =27. Fgure 2(a shows IOP t for the local s T polcy for a gven demand sample path; Fgure 2(b s the correspondng IOP t resultng from the local s T polcy. Frst, let us consder the local polcy at stage 2 at t =4 (Fgure 2(a. At ths order epoch, stage 2 orders four unts to brng ts IOP 2 4 to s 2. These four unts come from the two orders placed by stage 1 at t = 1 and t =3, whch account for the total demand n the tme nterval 0 3. Note that when stage 2 places an order at t =4, stage 1 actually has receved two unts of demand n 3 4. However, because stage 1 cannot order untl t =5, stage 2 does not order two addtonal unts for the demand n 3 4. Thus the echelon nventory order poston IOP 2 4 =S 2 2=11, whch s shown n Fgure 2(b. It s ths nformaton delay that causes the neffcency of the system. A smlar stuaton apples for stage 3 at t =7. At ths order epoch, stage 3 places an order of four unts Fgure 2 (a The Dynamcs of IOP t Under the Local s T Polcy, Where the Trangles Represent Order pochs. (b The Correspondng chelon Inventory Order Poston IOP t (a s 3 = 14 s 2 = 8 s 1 = 5 Demand (b S 3 = 27 S 2 = 13 S 1 = Demand to account for stage 2 s order at t = 4. These four unts, agan, are to account for the demand n 0 3. Because of the order delay, stage 3 does not learn of the demand n 3 7, whch s equal to = 8 unts. Thus the echelon nventory order poston for stage 3 at the order epoch t =7 siop 3 7 =S 3 D 3 7 =27 8=19 We generalze the above observaton n the followng proposton. Defne a notaton, where a b =an b a b and n s the smallest postve nteger such that an b. Proposton 1. For local s T polces, the resultng echelon nventory order poston IOP t at an order epoch t at stage s S D t T 1 L t. We make two remarks here. Frst, the random varable D t T 1 L t s the amount of demand occurred at stage 1 but not yet learned by stage when stage places an order under the local s T polcy. Thus t can be vewed as the amount of delayed demand nformaton. For the specal case

9 438 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS where =1 for all, ths term reduces to zero, whch mples that no demand nformaton s delayed under the classc local base-stock polcy (so the local basestock polcy s equvalent to the echelon base-stock polcy, and vce versa. Second, recall that the echelon nventory order poston IOP t resultng from the echelon s T polcy s always S e. Here, IOP t for the local s T polcy s a random varable. Thus n general, the local s T polcy s not a specal case of the echelon s T polcy. Nevertheless, f the reorder ntervals and the lead tmes of a seral system satsfy T 1 L =0, then we can fnd an equvalent echelon polcy by settng S e =S = =1 s, for all. Wth Proposton 1, we can replace IOP n (3 wth S D t T 1 L t and use (3 (5 to obtan the dstrbuton for IL and IL. For smplcty, we wrte IP / for IP t+l +1 N + /, IL for IL t+l N +, and IL for IL t+ L N +. The nventory dynamcs n (3, (4, and (5 can be smplfed as follows: For =0 T N 1, ( { [ IP T =mn S D T 1 L IL =IP ( IL +1 ( } (6 D L + T (7 IL =IP ( D L + T (8 Because the system starts each cycle when stage N places an order, the system s a regeneratve process wth a cycle length of T N perods. Thus the longrun average nventory holdng and backorder cost per perod s equal to the expected nventory holdng and backorder cost ncurred n a cycle dvded by the cycle length T N. That s, [ G s T = 1 TN 1( N h IL + b+h 1 N IL 1 ] N =0 =1 We next compute the average total fxed cost per perod. Notce that stage wll place an order at an order epoch t when the total orders receved between two consecutve order epochs n perods t and t s greater than zero. Proposton 2. At each order epoch t, the order quantty for stage under the local s T polcy s equal to D t T 1 L t T 1 L. At steady state, the probablty that stage wll place an order at an order epoch s p =P D >0. Thus the average order cost per perod s N =1 k p /, whch s the same as that for the echelon s T polcy. Ths s ntutve: the order quantty for stage under an echelon s T polcy s D t t, whch also covers perods of demand. The only dfference s that the coverage s shfted by T 1 L perods n the local polcy. Consequently, both polces should have the same average fxed order cost per perod n the long run when they have the same replenshment ntervals. Wth these results, the average total cost per perod s C s T = N =1 k p + 1 T N ( N h IL [ TN 1 =0 =1 + b+h 1 N IL 1 ] (9 where s= s 1 s N and T= T 1 T N. Below we provde a convenent recurson to evaluate C s T. The dea s smlar to the bottom-up evaluaton scheme for the echelon s T polcy n (1 and (2. At each teraton, we evaluate the average nventory holdng and backorder costs for echelon, referred to as G y, provded that stage s nduced echelon base-stock level s equal to y and ts downstream stage < follows a local s T polcy wth base-stock level s. Proposton 3. Defne G 1 y T 1 = 1 ( T1 1 h T 1 y D L 1 +l 1 For =2 N, + b+h 1 N y D L 1 +l (10 G y = 1 1 ( [ [h y D T 1 L D L +l

10 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 439 { ( +G 1 (mn S 1 y D 1 L [ D L + l 1 1 } 1 ] (11 where S = =1 s. Then C s T = N =1 k p / + G N S N T N. We now turn to optmzaton. We frst show how to fnd the optmal local base-stock levels for fxed T= T 1 T N. Wth fxed T, the total average fxed cost N =1 k p / s a constant, so we only need to mnmze G N S N T N. We provde an observaton below. Defne G e 1 T 1 = G e 1 T 1. For =2 N, defne G e y = 1 1 [ ( h y D T 1 L +L +l ( { ( [ + G e 1 mn S 1 y D 1 L ] l +L + 1 } 1 1 Note that G e s the echelon cost functon for a seral system wth echelon base-stock levels S e =S = =1 s and wth lead tmes L 1 for stage 1 and L + 1 L for stage, =2 N. Let C e S T be the average total cost per perod for ths modfed system. It can be verfed that G y = G e y =1 2 1 G y = G e y T h +1 T 1 L Ths result mples the followng proposton. Proposton 4. N 1 C s T = C e S T h +1 N T 1 L =3 N Proposton 4 states that the average total cost of a system wth a local s T polcy s equal to that of a system wth modfed lead tmes and operated under the echelon s T polcy wth S e = =1 s mnus a term that s ndependent of the local base-stocklevels. Thus the optmzaton procedure of fndng S e n (1 and (2 can be appled to (10 and (11 to fnd the optmal nduced echelon base-stock levels that mnmze G N S N T N. Call the resultng optmal soluton S 1 T 1 S N T N. The optmal local base-stock levels can be found by settng s 1 T 1 =S 1 T 1, and s =S S 1 1, = 2 N. Let s T = s 1 T 1 s. Based on the precedng observaton, the followng result s mmedate. Proposton 5. For fxed T, s T s the optmal local base-stockvector. We next show how to optmze the reorder ntervals for the local s T polcy. We need one addtonal result for ths purpose. Recall the concluson obtaned from Proposton 1: The local s T polcy s not a specal case of the echelon s T polcy. A natural queston s, whch polcy has a smaller optmal cost? From our dscusson, t s concevable that the local polcy s less effectve because of the delay of demand nformaton. We shall confrm ths conecture below. Proposton 6. (1 S e S = =1 s for =1 N. (2 For fxed T, G N S N T N T N G e N Se T N T N. (3 For fxed T, C s T T C e S e T T. Proposton 6(1 states that the optmal nduced echelon base-stock level for the local polcy s no less than that for the echelon polcy. Ths s ntutve because the local polcy requres more stocks to cover the unobserved demand due to the nformaton delay. Parts (2 and (3 of Proposton 6 state that for a gven T, the optmal cost of the local polcy, excludng and ncludng the fxed orderng cost, respectvely, s hgher than or equal to that of the echelon polcy. Suppose that the optmal reorder ntervals for the local polcy are T. From Proposton 6(3, the optmal cost of the local polcy must be greater than or equal to the optmal cost of the echelon polcy that uses the same T. The latter s clearly greater than or equal to the optmal cost of the echelon polcy. Therefore, we have the followng result: Proposton 7. The best echelon s T polcy always domnates the best local s T polcy.

11 440 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS Proposton 6 leads to an approach to fnd T. Shang and Zhou (2009 proposed an approach to fnd bounds for the optmal echelon reorder ntervals. Then a complete enumeraton s conducted to search for the optmal reorder ntervals. Our approach s based on the same dea. More specfcally, let C h be the cost of a heurstc local polcy, and defne C T =C s T T, C e T =C e S e T T, G e N T N =Ge N Se N T N T N.We can fnd bounds for T from the followng nequaltes. For =1 2 N, C h C T C e T >G e N T N N c T +Gl T + (12 =+1 The second nequalty n (12 follows from Proposton 6(2. The c and G l functons and the last nequalty n (12 were establshed n Shang and Zhou (2009. We provde a bref explanaton below: c s a lower bound for stage s cost. It s a functon of and s ndependent of T. = h D L 1 1, the average nventory n-transt cost. G l =mn y G l y where G l y = 1 h y D L 1 +t t=0 + b+h N y D L 1 +t Shang and Zhou (2009 showed that G l s convex n. The procedure starts from stage N. For =N, quaton (12 mples G l N T N <Ch N From ths nequalty, one can obtan an upper bound, T N, for TN. Clearly, a lower bound s equal to one. Next, for =N 1, quaton (12 becomes c N T N +Gl N 1 T N 1 + N 1 <C h (13 To search for bounds for TN 1 by usng Gl N 1, we need to fnd the mnmum value for c N T N. Unfortunately, c s not convex, so one needs to search over 1 T N to fnd the mnmum cost of c N T N, referred to as c N. Consequently, (13 can be rewrtten as G l N 1 T N 1 <Ch c N N 1 From ths, one can obtan an upper bound T N 1 for TN 1. Agan, we set the lower bound equal to one, and for T N 1 1 T N 1, we can search for the mnmum c N 1 T N 1, refereed to as c N 1. The procedure repeats untl =1. At the end of the procedure, we fnd the soluton bounds for T, =1 N. To fnd the optmal local replenshment ntervals, we only need to search all the feasble solutons 1 =1 N, that satsfy nteger-rato constrants. 4. NumercalStudy The goal of the numercal study s to examne the cost mprovement by mplementng each of the suggested strateges. Ths s accomplshed by comparng the cost mprovement between two pars of the nventory polces: local s T polcy versus echelon s T polcy and local s T polcy versus local r nq polcy. To strengthen the credtablty of our numercal conclusons, we desgn cost parameters based on Wllems (2008, where he collected cost data from 38 supply chans. He classfed supply chan stages nto fve categores: Dst, Manuf, Part, Retal, and Trans. (We refer the reader to Wllems 2008 for the defntons. One observaton s useful: the top three types of stages that have the largest drect cost are the Part, Manuf, and Dst stages. These stages account for about 95% of the total supply chan drect cost on average. More specfcally, the Part stage accounts for about 60%; the Manuf stage, 25%; and the Dst stage, 10%. Because nventory holdng cost s related to the drect cost, we shall use ths observaton as a gudelne to desgn the holdng cost parameters. We consder a three-stage system that represents these top three stages. Wthout loss of generalty, we fx the total supply chan holdng cost h 1 3 at one and assume the followng holdng cost parameters: h 1 h 2 h The frst choce, for example, can be vewed as a supply chan wth Part at stage 3, Manuf at stage 2, and Dst at stage 1. In general, these holdng cost patterns are consstent wth Wllems s (2008 fndng that an upstream stage has a hgher drect cost, whch mples a relatvely hgher holdng cost.

12 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 441 Wth the same sprt that most cost s ncurred at one stage, we fx the total system fxed order cost to 10 and 100 and allocate these total costs nto each stage accordng to (20%, 20%, 60% ratos. Thus the resultng fxed cost parameters are k 1 k 2 k We consder a servce level approxmately equal to 90% and 97.5%, whch corresponds to b 9 39 We assume that the demand follows a stutterng Posson demand process (e.g., Hadley and Whtn 1963, Chen and Zheng 1998, one type of compound Posson demand n whch the arrval process s Posson wth rate and the demand sze follows a geometrc dstrbuton,.e., P U =x = 1 x 1 x =1 2 where U s the demand sze of a customer and 0< 1. (When =1, the demand reduces to Posson. The mean and squared coeffcent of varaton (scv of the total demand n one perod s / and 2 /, respectvely. We set / =5 wth scv =1/3 (.e., = 3/4 and =15/4, representng smaller demand varablty, and scv =1 (.e., =1/3 =5/3, representng hgher demand varablty. Fnally we consder two lead tme settngs, short and long: L 1 L 2 L The precedng system parameters generate a total number of 192 nstances. For each of the nstances, we compute the exact optmal soluton for each of the polces. For smplcty, let C LQ C T, and C LT represent the optmal costs for the local r nq, echelon s T, and local s T polces, respectvely. 3 We defne the percentage cost reducton as below: %= C LT C 100% T LQ C LT Table 1 summarzes the average, maxmum, and mnmum percentage cost reducton n these 192 nstances. 3 We use the optmal echelon batch szes (reorder ntervals and ther correspondng optmal local reorder ponts (base-stock levels as a heurstc soluton to obtan C h when we search for the optmal local r nq ( s T polcy. Table 1 Percentage Cost Reducton by Adoptng Two Strateges Average Maxmum Mnmum Cost reducton (% (% (% Local s T vs. local r nq polcy Local s T vs. echelon s T polcy From the table, the average (maxmum cost reducton when the system swtches from the local s T polcy to the local r nq polcy s LQ =11 27% (28.32%, whch s sgnfcantly larger than the reducton of T =5 51% (16.67% when swtchng to the echelon s T polcy. Ths result suggests that mplementng a more flexble delvery polcy s more benefcal than acqurng the real-tme demand nformaton. It s nterestng to further examne ths concluson under dfferent demand varabltes. Fgure 3 presents LQ and T for a three-stage system wth h 1 h 2 h 3 = , k 1 k 2 k 3 = , b =9, and L 1 = L 2 =L 3 =1 under scv =1/3 2/3 1 4/3, and 5/3. It can be seen that the system ndeed benefts more by adoptng an agle logstcs when demand s more varable. Ths observaton holds true for the other sets of parameters. The above observaton provdes an mportant nsght: when demand becomes more varable, managers usually want to nvest n IT systems to acqure real-tme demand nformaton. However, acqurng demand nformaton alone may not be an effectve strategy; t s necessary to mplement an agle logstcs system to acheve supply chan effcency. Our model and the nventory polces have been used to quantfy the benefts of demand nformaton and flexble delvery. Sectons 4.1 and 4.2 provde a more detaled dscusson on these benefts. Secton 4.3 Fgure 3 % The Impact of Demand Varablty on LQ and T LQ T scv

13 442 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS dscusses the effectveness of nferrng reorder ntervals from the optmal batch szes Value of Demand Informaton We numercally llustrated that t s more benefcal for a system to move from the local s T polcy to the local r nq polcy than to the echelon s T polcy. It s natural to ask how much addtonal beneft can be obtaned by addng demand nformaton to the local r nq polcy. Thus to assess ths addtonal beneft, we need to compare the optmal cost between the local and echelon r nq polces. We defne the VOI under the r nq polcy as VOI Q = C LQ C Q C LQ 100% where C Q represents the optmal cost of an echelon r nq polcy. We refer the reader to Chen and Zheng (1998 or Shang and Zhou (2009 for an approach to fnd the optmal soluton. In our test bed of 192 nstances, the average (maxmum VOI Q s 0.55% (2.71%. Ths result suggests that the addtonal beneft from the demand nformaton s surprsngly mnmal n a system wth flexble delveres. Note that ths concluson s made by comparng the exact optmal local and echelon r nq polces. Numercally, we may obtan a very large VOI f both local and echelon polces are not optmzed. For example, Chen (1998b arbtrarly chose the same batch szes for both local and echelon r Q polces. In hs numercal study for three-stage systems, the average (maxmum VOI Q s 1.61% (7.5%. Based on our numercal experence, the VOI Q tends to be small f we fx both local and echelon batch szes to the optmal batch szes obtaned from the echelon r nq polcy. In our test bed, the average (maxmum VOI then s 0.65% (3.99%. Clearly, the dfference between 0.65% and 0.55% s due to the effect of optmzng local batch szes. Notce that the average T =5 51% obtaned earler s the cost reducton by swtchng from the local s T polcy to the echelon s T polcy. Ths s equvalent to the VOI under the less flexble s T polcy. Thus one nsght learned from ths study s that how much VOI a supply chan can acheve depends on how flexble ts logstcs system s. Fgure 4 (a % % (b Senstvty Analyss: Average Value of Demand Informaton / L scv b VOI Q k VOI L / L scv b k We end ths secton by dentfyng the condtons under whch the system benefts most when acqurng the demand nformaton under each of the two logstcs settngs. For a comparable presentaton, denote VOI T = T. We llustrate some comparatve statcs of VOI Q and VOI T n Fgure 4. For the r nq polcy, the average VOI Q ncreases from 0.41% to 0.68% when the total fxed costs k change from 10 to 100. Also, the average VOI Q tends to ncrease as the backorder cost b ncreases. It can be seen from Fgure 4(a that the system lead tmes and demand varablty have a relatvely smaller mpact on the VOI Q. For the s T polcy, the VOI T manly depends on how much demand nformaton s delayed,.e., D T 1 L for each stage. When the lead tmes L are short and reorder ntervals T <Nare long, such that T 1 >L, the delayed demand nformaton D T 1 L becomes larger. Thus t s more benefcal to provde the demand nformaton under

14 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 443 such condtons. To see ths, we compare one nstance wth short lead tmes L 1 L 2 L 3 = to another nstance wth long lead tmes L 1 L 2 L 3 = ; the other parameters are the same: k 1 k 2 k 3 = h 1 h 2 h 3 = b =39 =3/4 For both cases, the optmal reorder ntervals for the echelon and local polces are and 6 6 6, respectvely. The correspondng VOI T s 11.07% for the short lead tme case and 6.87% for the long lead tme case. Fnally, the VOI T ncreases when k s larger. In our test, when k =10, the average VOI T s 2.19%, and when k =100, the average VOI T ncreases to 8.83%. Ths s because hgher fxed costs ncrease the length of the resultng reorder ntervals, whch n turn delays the demand nformaton more sgnfcantly Value of Flexble Delvery Recall the average LQ s 11.27% n our test bed. Ths s the value of flexble delvery (VOD for short when the system only has local nformaton. For ths reason, let us denote VOD L = LQ. Smlarly, we can obtan the average VOD when the system has full nformaton,.e., the cost reducton of swtchng from the echelon s T polcy to the echelon r nq polcy. Denote the VOD under echelon nformaton as VOD = C T C Q C T 100% It s concevable that the average VOD should be lower than the average VOD L because a more responsve delvery speeds up not only the materal flow but also the demand nformaton flow. Our numercal study confrms ths ntuton: In our test bed, the average VOD =6 76% and VOD L =11 27%. In general, flexble delvery reduces total cost sgnfcantly, regardless of the nformaton settng. We provde some comparatve statcs of VOD and VOD L n Fgure 5 and examne the condtons under whch the system benefts more by swtchng from the s T polcy to the r nq polcy. For the echelon nformaton, we observe that when the lead tmes L are shorter (longer, the fxed costs k are larger Fgure 5 (a % % (b Senstvty Analyss: Average Value of Flexble Delvery / L scv b k VOD VOD L / L scv b k (smaller, the backorder cost b s larger (smaller, or the demand s more (less varable, the VOD tends to be hgher (lower. (See Fgure 5 for the exact cost reducton percentage. We explan the above observaton below. Because both echelon polces have the same demand nformaton, the cost beneft of the r nq polcy depends on how flexble t can retan. Larger fxed costs k wll lead to longer replenshment ntervals T and therefore the s T polcy becomes less flexble. Ths ncreases the beneft of swtchng to the r nq polcy. If L s shorter, the flexblty of r nq polcy becomes more promnent, and therefore the correspondng VOD s larger. The last two observatons are ntutve. When the backorder penalty or demand varance ncreases, the system benefts more from the flexble r nq polcy n that t can better respond to the demand to avod hgher costs. Thus the average VOD s hgher when b or scv s larger.

15 444 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS In the local nformaton scenaro, smlar to the echelon nformaton, we observe that when the fxed costs k are hgh, the lead tmes L are short, the backorder cost b s hgh, or demand s more varable, the VOD L tends to be hgh. The ntuton behnd these observatons s smlar to what we dscussed for the echelon nformaton scenaro. However, the mpact of the fxed costs and lead tmes s consderably larger on VOD L than on VOD. Ths observaton suggests that f the fxed costs of the system are large or the lead tmes are short, the system can mprove sgnfcantly by swtchng to a more flexble delvery polcy even wthout real-tme demand nformaton. Fgure 6 summarzes the cost reducton acheved by each of the nventory polces by swtchng from the local s T polcy. The symbol represents swtchng from one polcy to another. The number next to the arrow represents the average percentage cost reducton. For example, 11.75% s the cost reducton acheved by swtchng system from the local s T polcy to the echelon r nq polcy. Ths 11.75% s the maxmum cost reducton that the supply chan system can acheve. Implementng a more responsve delvery polcy (e.g., the local r nq polcy recovers about 95.91% (=11 27/ % of the maxmum cost reducton. On the other hand, acqurng realtme demand nformaton only recovers about 46.89% (=5 51/ % ffectveness of Inferrng Reorder Intervals from Batch Szes In the determnstc demand model, the reorder nterval and the batch sze Q are equvalent. That s, the reorder nterval can be nferred by the batch sze wth the followng equaton: =Q /, where s the demand rate. In ths secton, we am to examne Fgure 6 Flexble delvery polcy Perodc delvery polcy Cost Comparsons of the Consdered Four Inventory Polces Wth real-tme demand nformaton chelon (r, nq polcy VOD 6.76% chelon (s, T polcy VOI Q 0.55% 11.75% VOI T 5.51% Wthout real-tme demand nformaton Local (r, nq polcy VOD L 11.27% Local (s,t polcy whether effectve reorder ntervals can be nferred by the optmal batch szes Q and the demand rate. We frst explan how to obtan the nferred reorder ntervals. Note that Q / may not be an nteger. If ths s the case, we round Q / nto an nteger, denoted as Q /, where s the standard roundng operator. If the resultng Q / satsfes the nteger-rato constrants, then Q / s the nferred reorder nterval for stage. Otherwse, startng from stage =1 2 N, weset Q+1 / equal to the closest nteger multple of Q /. For example, suppose that Q1 Q 2 Q 3 = and =5. Then Q1 / Q 2 / Q 3 / = , and the nferred reorder ntervals are Let C T (C LT denote the total cost obtaned by usng the nferred echelon (local reorder ntervals and the resultng optmal echelon (local base-stock levels. Let = C C C 100% T LT denote the percentage cost ncrease by usng the nferred reorder ntervals. We observe the followng: (1 There are 23 (36 out of 192 nstances that the nferred echelon (local reorder ntervals are equal to the optmal echelon (local reorder ntervals. Thus from the soluton perspectve, t s n general not effectve to nfer the reorder ntervals from the optmal batch szes. (2 Although the nferred local reorder ntervals generate more optmal solutons than the nferred echelon ones, the average T of 1 16% s lower than the average LT of 4 76%. Hence the nferred reorder ntervals are less effectve when the system only has local nformaton. In some cases, the LT can be very large. For example, for the 12 nstances wth b =39, =1/3, L =1, and 3 =1 k =10, the average LT s 16 75%. (3 For both nformaton settngs, the nferred reorder ntervals tend to be longer than the optmal ones. Ths mples that the optmal batch szes are large n the r nq system. Ths may be explaned as follows: Large batch szes have two opposte effects on the system performance. On the one hand, when batch szes are large, the system can gan the beneft of economes of scale. On the other hand, large batch szes wll make the system order less frequently, whch reduces the responsveness to the demand.

16 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 445 However, the latter effect s less sgnfcant n the r nq polcy because a stage can place an order n any perod. Thus the former effect domnates and leads to larger optmal batch szes n the r nq model. 5. Concludng Remarks Ths paper nvestgates how to mprove the cost performance of a supply chan, n whch each locaton can only access local nformaton and replensh n fxed ntervals. We model ths supply chan by a seral nventory system wth local s T polces. We consder two mprovement strateges: mplementng a flexble delvery polcy (local r nq polcy and acqurng real-tme demand nformaton (echelon s T polcy. The nventory polces resultng from the mprovement strateges are well studed n the lterature. To compare costs n these scenaros, we analyze the local s T polcy. The analyss ncludes polcy evaluaton and optmzaton. We show that the local s T polcy s not a subset of the echelon s T polcy and that the former ncurs a hgher system cost. Our numercal study suggests that mplementng a more flexble delvery polcy s more effectve than acqurng the real-tme demand nformaton. We also dentfy the condtons under whch the system s less benefcal for each of the mprovement strateges. Certanly our conclusons are drawn wthout consderng the potental benefts of mplementng s T polces, such as ease of personnel schedulng and coordnaton, and wthout consderng actual mplementaton and mantenance costs when mplementng these strateges. Managers should therefore consder the nsghts obtaned from our study as well as those unmodeled factors when desgnng the logstcs and nformaton systems of a supply chan. Fnally, we fnd that nferrng reorder ntervals from optmal batch szes may lead to sgnfcant cost neffcency, especally when a supply chan only has local nformaton. Acknowledgments The authors thank Gerard Cachon and the anonymous revewers for helpful comments. Appendx A. Proofs For ease of exposton, throughout the appendx, we relax the nteger assumpton on nventory-related varables. As a result, all the functons arecontnuous and satsfy the condton of Lebnz s rule so that we can exchange the sgns of dervatve and expectaton. Proposton 1. We prove the proposton by nducton. For stage 1, by defnton s 1 =S 1, the result s clearly true. Suppose the result s true for = 1. Let t be an order epoch for stage. Then stage 1 s last order epoch before t s t T 1 L, at whch, by the nductve assumpton, the echelon nventory poston at stage 1 ss 1 D t T 1 L 1 =2 1 L t T 1 L. Furthermore, because stage 1 wll not place any order between tme t T 1 L and t, the echelon nventory order poston at tme t for stage = s [ 1 S 1 D t T 1 L 1 L t T 1 L =2 [ +s D t T 1 L t =S D t 1 L t where the equalty follows from the defnton of S. Ths completes the nducton proof. Proposton 2. The lemma s clearly true for =1. Suppose that t s true for stage 1. For any tme pont t that s an order epoch of stage, the last order epoch before t of stage s t. Note that the order quantty of stage at tme t s the total orders from stage 1 n t t. Because the orders between stages are synchronzed, t +L s an order epoch of stage 1. Moreover, we know n t t, stage 1 wll place / 1 orders. We only need to specfy the tme epoch of the frst order from stage 1 n ths nterval, whch s t 1 L + 1. As a result, the last order from stage 1 n ths nterval s at tme t 1 L. Therefore, by the nductve assumpton, the demand covered n these orders s the demand n the nterval t T 1 L t T 1 L. Proposton 3. The dervaton of the formula for C s T s straghtforward. Below, we gve a proof by nducton for the recurson of G y. For notatonal convenence, we omt n G y n ths proof. The dervaton of the formula for G 1 y s straghtforward. Now let 2 N and suppose that the formula for G y has been proved for =1 1. We derve the formula for G y. Suppose that t s an order epoch of stage. Then =2 [ ] IP t =IOP t =y D t T 1 L t where the frst equalty follows from the assumpton that stage +1 has always suffcent nventory and the last

17 446 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS equalty follows from Proposton 2. Ths leads to the followng echelon costs n the perods t +L t+l + 1: 1 [ ( h IP t D t t+l +l ] 1 = 1 = ( [ ] [h y D t T 1 L t ] D t t+l +l ( [ ]] [h y D t T 1 L t +L +l Next we look at the lower echelon costs, whch may be obtaned va G 1 y and IP 1 t n the perods t = t +L t+l + 1 t+l + / It holds that IP 1 t+l =mn { IOP 1 t+l IL t+l } { [ 1 =mn S 1 D t +L T 1 L t +L } IP t D t t+l { [ 1 =mn S 1 D t +L T 1 L t +L [ } y D t T 1 L t +L { [ 1 } =mn S 1 y D t T 1 L t +L T 1 L [ 1 D t +L T 1 L t +L As a result, the costs for the echelons 1 1 that are drectly determned by IP 1 t+l are equal to { [ 1 G 1 (mn S 1 y D t T 1 L Smlarly, 1 } t +L T 1 L IP 1 t+l + 1 { [ =mn S 1 y D t T 1 L 1 } t +L + 1 T 1 L [ 1 D t +L + 1 T 1 L t +L + 1 and the costs for the echelons 1 1 that are drectly determned by IP 1 t+l + 1 are equal to { [ 1 G 1 (mn S 1 y D t T 1 L and so on. Takng all costs together gves G y = 1 = 1 [ T 1 1 } t +L + 1 T 1 L [ ] h (y D t T 1 L t +L +l { [ + 1 G 1 (mn S 1 y D t T 1 L 1 } t +L T 1 L { [ + 1 G 1 (mn S 1 y D t T 1 L [ T 1 1 } ] t +L + 1 T 1 L + ( h (y D L +l+1+ T 1 L + 1 G 1 ( mn { S 1 y D L + 1 L } + 1 G 1 ( mn { S 1 y D L + 1 L + 1 } + = 1 1 ( ( [h y D L +l+1+ T 1 L { ( +G 1 (mn S 1 y D L + 1 L + l 1 1 }] whch s equvalent to formula (11 for G y. Ths completes the proof. Proposton 6. In ths proof, we use (prme to denote the dervatve of a functon and a b to represent mn a b. Also, because T s fxed, we omt n the cost functons. For part (1, t suffces to show that G y G e y due to ther convexty. We prove ths by nducton on. It should be noted that throughout ths proof, the correspondng S and S e are the optmal base-stock levels for < n G y and G e y. It s trvally true for =1 asge 1 y =G 1 y. Suppose t s true for = 1,.e., G 1 y G e 1 y, whch mples ]

18 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS 447 S 1 S e 1. Then for =, let D denote D L + l/t 1 T 1, and G y = 1 T 1 [ h T + ( ( G 1 S 1 ( ] y D T 1 L D 1 T 1 [ h T + ( ( G 1 S e 1 ( ] y D T 1 L D 1 T 1 [ h T + ( ( G e 1 S e 1 ( ] y D T 1 L D 1 T 1 [ h T + ( ( G e 1 S e 1 ( ] y D = G e y where the frst nequalty follows from that S 1 e S 1 and G 1 y 0 for y<s 1, the second nequalty follows from the nductve assumpton, and the last nequalty follows from the convexty of G e 1 y. For part (2, because T s fxed, t s suffcent to show G N S N G e N Se N. To show ths, t s suffcent to show that, for any, [ ( ( G y G e y D T 1 L ] G e Se (14 where the last nequalty s due to the optmalty of S e. Thus we only need to prove the frst nequalty. We show ths result by nducton. It s clearly true for =1 as D T 1 L =0 and G e 1 y =G 1 y. Suppose t s true for = 1. Then for =, G y = 1 T 1 ( [h T y D ( k=2 T k 1 L k D L +l+1 +G 1 ( S 1 ( y D T 1 L D ] 1 T 1 ( ( [h T y D T k 1 L k D L +l+1 = 1 T 1 ( [h T y D k=2 [ ( + G e 1 (S 1 ( y D T 1 L D ( k=2 [ + G 1( e S 1 D ( 1 ]] D T k 1 L k k=2 T k 1 L k D L +l+1 ( 1 ( ( y D T k 1 L k k=2 k=2 T k 1 L k D ]] 1 T 1 ( [h T y D [ + G e 1 ( k=2 T 1 L D L +l+1 ( ( ]] (S e 1 y D T k 1 L k D k=2 [ ( ( = G e y D T k 1 L k ] k=2 where the frst equalty follows from the defnton of G y ; the frst nequalty s due to the nductve assumpton, the second nequalty s due to the optmalty of S 1 e, and the last equalty follows from the defnton of G e y. Ths completes the nducton, and part (2 s proved. Part (3 follows mmedately from part (2 because under a fxed T, a local and an echelon s T polcy have the same fxed orderng cost. Appendx B. An Approach to Fnd OptmalBatch Szes for Local r nq Polces We only sketch the dea. The detaled analyss can be requested from the authors. Let the optmal local r Q polcy be r Q and the resultng optmal cost be C r Q. Defne C e Q = the total cost obtaned from an echelon r Q polcy wth batch szes Q, and the correspondng best echelon reorder ponts R Q = R 1 Q 1 R N Q N, G e Q = the cost for the echelon stage wth batch szes Q = Q 1 Q and the correspondng best reorder ponts R Q = R 1 Q 1 R Q, c Q = a lower bound to the cost for stage wth batch sze Q, G l Q = a lower bound to G e Q We brefly explan how to obtan the above cost functons from the lterature. For fxed Q, the best echelon reorder ponts R Q and the optmal cost C e Q can be obtaned from Chen (2000. The echelon cost functon G e Q and the lower bound G l Q are derved n Shang and Song (2007. The lower bound c Q to the stage cost s derved n Shang and Zhou (2009. Let C h be a heurstc cost for the local r Q polcy. It can be shown that C h C r Q C e Q >G e N Q N N c Q +Gl Q + (15 =+1 The second nequalty n (15 follows from the fact that a local polcy s a specal case of the echelon r Q polcy (Axsäter and Roslng The rest of the nequalty s from Shang and Zhou (2009. Note that (15 s smlar to (12. Thus we can fnd the bounds for Q by usng the same procedure as the one

19 448 Manufacturng & Servce Operatons Management 12(3, pp , 2010 INFORMS for fndng the bounds for local T. That s, startng from stage N, we can fnd the bounds Q N Q N for QN. Then for Q N Q N Q N, we search for the mnmum c N Q N, referred to as c N. Next, applyng (15 agan by settng =N 1, we fnd the soluton bounds for QN 1 and then c N 1, etc., untl =1. After the soluton bounds are establshed, one can apply the algorthm n Chen (1998b to search for the optmal local reorder ponts for each feasble soluton. The optmal local r nq polcy can be found by evaluatng all feasble solutons and ther correspondng optmal local reorder ponts. (One can evaluate a local r nq polcy by usng the evaluaton scheme for the echelon r nq polcy because the former s a specal case of the latter; see Chen 1998b. References Avv, Y., A. Federgruen The operatonal benefts of nformaton sharng and vendor managed nventory (VMI programs. Workng paper, Washngton Unversty n St. Lous, St. Lous. Axsäter, S xact and approxmate evaluaton of batchorderng polces for two-level nventory systems. Oper. Res Axsäter, S Supply chan operatons: Seral and dstrbuton nventory systems. A. G. de Kok, S. C. Graves, eds. Supply Chan Management: Desgn, Coordnaton, and Operaton, Vol. 11. Handbooks n Operatons Research and Management Scence. lsever, Amsterdam, Axsäter, S., L. Juntt Comparson of echelon stock and nstallaton stock polces for two-level nventory systems. Internat. J. Producton conom Axsäter, S., K. Roslng Installaton vs. echelon stock polces for mult-level nventory control. Management Sc Cachon, G Managng supply chan demand varablty wth scheduled orderng polces. Management Sc Cachon, G. 2001a. xact evaluaton of batch-orderng polces n two-echelon supply chans wth perodc revew. Oper. Res Cachon, G. 2001b. Managng a retaler s shelf space, nventory, and transportaton. Manufacturng Servce Oper. Management Cachon, G., M. Fsher Supply chan nventory management and the value of shared nformaton. Management Sc Chao, X., S. Zhou Optmal polcy for mult-echelon nventory system wth batch orderng and fxed replenshment ntervals. Oper. Res Chen, F. 1998a. On R nq polces n seral nventory systems. S. Tayur, R. Ganeshar, M. Magazne, eds. Quanttatve Models for Supply Chan Management. Kluwer Academc Publshers, Norwell, MA, Chen, F. 1998b. chelon reorder ponts, nstallaton reorder ponts, and the value of centralzed demand nformaton. Management Sc. 44 S221 S234. Chen, F Optmal polces for mult-echelon nventory problems wth batch orderng. Oper. Res Chen, F., Y. S. Zheng Near-optmal echelon-stock r nq polces n mult-stage seral systems. Oper. Res De Bodt, M., S. Graves Contnuous revew polces for a mult-echelon nventory problem wth stochastc demand. Management Sc Feng, K., U. S. Rao chelon-stock R nt control n two-stage seral stochastc nventory systems. Oper. Res. Lett Gavrnen, S Informaton flows n capactated supply chans wth fxed orderng costs. Management Sc Gavrnen, S., R. Kapuscnsk, S. Tayur Value of nformaton n capactated supply chans. Management Sc Graves, S A multechelon nventory model wth fxed replenshment ntervals. Management Sc Graves, S A sngle-tem nventory model for a non-statonary demand process. Manufacturng Servce Oper. Management Gürbüz, M., K. Monzadeh, Y.-P. Zhou Coordnated replenshment strateges n nventory/dstrbuton systems. Management Sc Hadley, G., T. Whtn Analyss of Inventory Systems, Prentce- Hall, nglewood Clffs, NJ. Lee, H., K. So, C. Tang The value of nformaton sharng n two-level supply chan. Management Sc Naddor, Optmal and heurstc decsons n sngle- and mult-tem nventory systems. Management Sc Rao, U. S Propertes of the perod revew R T nventory control polcy for statonary, stochastc demand. Manufacturng Servce Oper. Management Shang, K Note: A smple heurstc for seral nventory systems wth fxed order costs. Oper. Res Shang, K., J. Song Supply chans wth economes of scale: Sngle-stage heurstc and approxmatons. Oper. Res Shang, K., S. Zhou Optmal and heurstc r nq T polces n seral nventory systems wth fxed costs. Oper. Res. Forthcomng. Shang, K., J. Song, P. Zpkn Coordnaton mechansms n decentralzed seral nventory systems wth batch orderng. Management Sc Smch-Lev, D., Y. Zhao The value of nformaton sharng n a two-stage supply chan wth producton capacty constrants. Probab. ngrg. Inform. Sc Smch-Lev, D., Y. Zhao Three generc methods for evaluatng stochastc mult-echelon nventory systems. Workng paper, Massachusetts Insttute of Technology, Cambrdge. van Houtum, G. J., A. Scheller-Wolf, J. Y Optmal control of seral, mult-echelon nventory/producton systems wth fxed replenshment ntervals. Oper. Res Wllems, S Data set-real-world multechelon supply chans used for nventory optmzaton. Manufacturng Servce Oper. Management