As widely accepted performance measures in supply chain management practice, frequency-based service

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1 MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 6, No., Winer 2004, pp issn eissn informs doi 0.287/msom INFORMS On Measuring Supplier Performance Under Vendor-Managed-Invenory Programs in Capaciaed Supply Chains Ki-Seok Choi Samsung SDS Co., Ld., Yoksam-2Dong, Kangnam-Gu, Seoul, 35-98, Korea, J. G. Dai School of Indusrial and Sysems Engineering, Georgia Insiue of Technology, Alana, Georgia , Jing-Sheng Song The Fuqua School of Business, Duke Universiy, Durham, Norh Carolina 27708, As widely acceped performance measures in supply chain managemen pracice, frequency-based service levels such as fill rae and sockou rae are ofen considered in supply conracs under vendor-managedinvenory (VMI) programs. Using a decenralized wo-pary capaciaed supply chain model consising of one manufacurer and one supplier in a VMI environmen, we demonsrae ha supplier s service level is in general insufficien for he manufacurer o warran he desired service level a he cusomer end. The mehod by which he supplier achieves her service level o he manufacurer also affecs cusomer service level. By developing bounds on he cusomer service level, we show ha he expeced backorders a he supplier should also be aken ino accoun. We sugges a supply conrac ha offers a menu of differen combinaions of supplier s service level and expeced backorders according o a linear funcion. Under his conrac, he manufacurer can conrol he end cusomer service regardless of how he supplier manages her invenory. The supplier has complee flexibiliy on which combinaion of he wo quaniies on he menu o choose according o her own cos funcions. Because i does no require any deailed informaion on supplier s operaional characerisics nor her coss, his kind of conrac is expeced o be easily implemenable. In addiion, we derive an esimae of he cusomer service level in erms of he new measures. Our findings have direc implicaions o supply chain merics in general: The local service levels are insufficien measures o guaranee he sysem wide performance. Alernaive local measures and/or coordinaion mechanisms should be employed o achieve desired sysem performance. Our analysis illusraes a possible way o explore such alernaive measures. Key words: service-level guaranees; supply conracs; supplier performance measures; vendor-managed invenory Hisory: Received: April, 2003; Senior Edior: Kaj Rosling.. Inroducion Frequency-based cusomer service measures, commonly known as service levels, such as fill rae and sockou rae, have always been imporan performance indicaors ha all companies care abou. This is refleced by heir repeaed appearance in business press, company Web sies, and business adverisemens across indusries. (See various examples cied in Sobel See also Kleijnen and Smis 2002.) While he issue of how o manage a cenralized supply chain wih a arge cusomer service level has received much aenion, especially in recen years, o our knowledge how o coordinae differen paries in a decenralized supply chain o achieve a arge end cusomer service level has no been sufficienly addressed in he lieraure. For example, when differen paries are involved in a supply chain, a common belief is ha as long as he immediae upsream pary has a higher service level han he desired downsream pary s arge service level, he downsream pary would be able o manage his operaions o mee his arge. However, his rule of humb does 53

2 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 54 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS no appear o have any heoreical jusificaion. Also, here exiss no guideline as o how much higher he upsream pary s service level should be. This paper explores hese issues using a simple wo-pary capaciaed supply chain consising of one manufacurer and one supplier. We find ha he above-menioned rule of humb is no valid. We provide an example in which he supplier mees her pre-agreed service level such as a low sockou rae, ye he manufacurer is no able o mee his cusomer service level. The mehod by which he supplier achieves her service level o he manufacurer such as increasing invenory, improving producion reliabiliy, or increasing producion capaciy also affecs he cusomer service level. This example also indicaes ha any conrac ha is primarily based on supplier s service level is flawed. Our analysis suggess an alernaive conrac form ha conains a menu of wo easily observable supplier delivery performance measures: he supplier s sockou rae and expeced backorders. When he supplier mees hese performance measures, he manufacurer is guaraneed o mee his cusomer service level. This ype of conrac leaves he supplier complee flexibiliy on which combinaion of he wo quaniies on he menu o choose according o her own cos funcions. Because i does no require any deailed informaion on supplier s operaional characerisics nor her coss, his kind of conrac is expeced o be easily implemenable. Our research was moivaed by he quesions faced by he managers of elecronic manufacuring service providers, such as Solecron and Flexronics, hrough our indusry experiences wih hem. In he las decade or so, in response o increasingly shorer produc life cycles, higher cusomer expecaions, and fierce global compeiion, he elecronics indusry has experienced exensive growh and resrucuring, and is supply chains have become highly decenralized. To improve supply chain efficiency, he indusry has adoped wo successful innovaions. One is he make-o-order (MTO), also known as build-o-order, manufacuring sraegy, led by Dell Compuer in he compuer indusry (see Simchi-Levi e al. 2000). Under MTO, he manufacurer does no keep finished-produc invenory and releases producion orders only afer receiving cusomer orders. This approach eliminaes he risk of wased invesmen in unwaned finishedproduc invenory due o echnological obsolescence or unmached demands. Anoher innovaion is he vendor-managed-invenory (VMI) programs, led by Wal-Mar and Procer & Gamble in he reail indusry and by Campbell Soup in he grocery indusry (see Buzzell and Ormeyer 995 and Fisher 997) and followed by oher indusries (see Thompson and Srickland 200 and Barnes e al. 2000). Under VMI in he supplier-manufacurer seing, he supplier is responsible for all decisions regarding he componen invenory a he manufacurer. Despie he success and poenial benefis of MTO and VMI, companies ha are adoping hese business models ofen face complex execuion challenges. For example, o ensure saisfacory order fulfillmen performance, MTOcalls for more criical componen invenory managemen han ever, which in urn requires closer and improved relaionships wih suppliers. A common issue is how o monior he supplier s performance so as o guaranee a saisfacory service level a he cusomer end. Wihou excepion, he manufacurer would like o have componens available whenever he needs hem, bu his would cos he supplier oo much o be feasible. A soluion o his dilemma is ofen a conrac beween he manufacurer and he supplier, in which boh paries agree on cerain requiremens regarding he supplier s performance in componen delivery. The requiremens can be expressed in various ways. To simplify he monioring process, which is ofen done by hird-pary companies, measurable quaniies are preferred. As widely adoped performance measures in supply chain pracice, service levels such as fill rae and sockou rae, which quanify how much and/or how soon downsream orders are delivered, easily come o managers minds as desirable candidaes. Such measures are indeed being used under VMI conracs in various companies; see, e.g., Fry e al. (200). However if he manufacurer ries o conrol his manufacuring process o achieve cerain cusomer service level, how can he accuraely quanify he service level o specify in a supply conrac, and how does ha specified service level affec he end cusomer service level? These are he quesions asked by he managers.

3 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 55 We address hese quesions by considering a decenralized wo-pary supply chain consising of one manufacurer and one supplier. The manufacurer has a finie capaciy and makes a produc o serve marke demand following an MTOpolicy. The supplier provides componens for he produc and manages he componen invenory a a warehouse near or a he manufacurer s sie under a VMI program. In he general model, we do no impose any assumpions abou he supplier s operaional characerisics, such as her capaciy and invenory policy. The wo paries are conneced hrough componen requiremen and fulfillmen. The manufacurer can influence he supplier only hrough requiremens on he supplier s componen delivery performance. We examine several commonly used service level measures, and demonsrae he irrelevance of supplier s service level o cusomer service level. For insance, in 4 we consider a deailed supplier model in which he supplier can manipulae hree parameers in order o achieve her service level, say a sockou rae no more han 5%, specified by he manufacurer. The hree parameers are excess capaciy, producion reliabiliy (yield), and base-sock level. We show ha many differen choices of hese parameers can resul in he desired 5% service level. However, he resuling sockou rae a he end cusomer level demonsraes a wide range and someimes can be as high as 34 46%. In oher words, he cusomer service level depends no only on he quaniaive measure of he service level from he supplier, bu also on how he supplier achieves ha service level. A major conribuor o he flucuaing cusomer service is he manufacurer s capaciy. These findings imply he necessiy of oher supplier performance measures ha can secure he manufacurer s cusomer service. By developing bounds on he cusomer service level, we idenify plausible supplier performance measures ha allow he manufacurer o conrol he end cusomer service regardless of how he supplier manages he invenory. The resuls sugges ha in addiion o he service level, we also need o measure he average componen shorage or he componen backlogs. We propose a new conrac form ha consiss of a menu of combinaions of hese wo quaniies. The design of he menu is hrough a linear funcion of he wo quaniies ha guaranee he manufacurer s cusomer service level. The parameers of he linear funcion depend on he manufacurer s capaciy and demand disribuion only. Therefore, he conrac design is independen of he supplier s operaing characerisics and cos informaion. The supplier can choose any combinaion of he wo quaniies lised on he menu, which leaves he supplier complee flexibiliy in opimizing her own cos. Thus, his ype of conrac is deail free (Wilson 987), so i is expeced o be easily implemenable. Moreover, he developmen of he bound (he linear funcion) assumes no informaion on he supplier s operaing characerisics, and hence he resul is robus. Our resuls have direc implicaions o supply chain performance merics in general. Tha is, he local (or inernal) service levels in a supply chain are insufficien measures o guaranee he enire supply chain s service level o is end cusomer. (I is worh menioning ha his observaion is rue even for uncapaciaed supply chains bu wih posiive ransporaion leadimes beween sages.) Thus, alernaive local measures and/or coordinaion mechanisms should be employed in order o achieve desired sysem performance. Alhough individual researchers may have noiced or suspeced he ineffeciveness of local service levels in supply chain coordinaion, o our knowledge, our paper provides he firs documened sudy o bring he awareness of his issue. Our sudy also goes a sep furher by demonsraing ha how he supplier achieves he prespecified service level also affecs he performance of he downsream pary, and hus provides deeper insighs. Developing performance measures solely from wha can be observed from supplier s oupu process, wihou knowing he supplier s operaing characerisics, presens a remendous echnical challenge. As such, he analysis should be, in general, sysem specific. We hope ha he approach we ake here using bounds on he downsream service level provides some encouragemen and inspiraion for fuure research endeavor along his line. Finally, we noe ha while VMI has been discussed in numerous papers, our paper appears o be he firs o examine he environmen in which he downsream pary is a capaciaed manufacurer. The majoriy of

4 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 56 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS he VMI lieraure o be reviewed in he nex secion focuses on a supplier-reailer seing, so here is no downsream capaciy issue. In such a seing, he supplier s performance can be easily aligned wih he cusomer service level: because cusomers are served direcly from he supplier managed invenory, he supplier s service level is precisely he cusomer service level. Because his is no longer rue in a capaciaed supply chain as demonsraed in our paper, i is reasonable o conjecure ha he resuls obained in he supplier-reailer seing need o be reexamined in he supplier-manufacurer seing. The remainder of he paper is organized as follows. Secion 2 reviews he lieraure. Secion 3 describes a general model and provides some preliminary discussions on he relaionship beween he supplier s and he manufacurer s service levels. Secion 4 analyzes several special cases of he general model and demonsraes he ineffeciveness of using service levels o measure he supplier s performance. Secion 5, focusing on he general model again, derives alernaive supplier performance measures. Secion 6 provides an example on how o design a supply conrac using he alernaive measures. We end he paper wih a few concluding remarks, including some discussions on oher possible supplier coordinaion mechanisms. 2. Lieraure Review We firs review he lieraure on VMI. Plambeck and Zenios (2003) sudy a make-o-sock model, in which he manufacurer bears he invenory-holding and backorder coss of he finished good bu delegaes he producion of he finished good o a supplier. The supplier dynamically conrols he producion rae and incurs a convex producion cos. The manufacurer canno monior he producion rae, bu can draw inference from increases in he invenory level. By making paymens coningen on he invenory level, he manufacurer moivaes he supplier o conrol he producion rae in a manner ha will minimize he manufacurer s oal expeced discouned cos. They show ha he opimal incenive paymen scheme consiss of piece raes and invenory penalies ha vary dynamically wih he invenory level. This scheme coordinaes he sysem if he supplier is risk neural. Oherwise operaional performance is degraded by he conflic in incenives beween manufacurer and supplier. Their model seing is differen from ours in several ways. Firs, in our model he manufacurer conrols he finished-good invenory, while he supplier conrols he componen invenory, so here are wo socking posiions in he supply chain insead of. Second, because he manufacurer, in heir model, delegaes he producion o he supplier, here is no capaciy issue a he manufacurer; whereas in our model, he manufacurer has a finie producion capaciy. Third, hey assume he manufacurer ries o minimize oal discouned coss, which includes backorder coss, while we do no consider backorder cos explicily bu assume ha he manufacurer ries o achieve cerain service level. Fry e al. (200) consider he z Z -ype VMI conrac in a one supplier, one reailer supply chain: The reailer ses a minimum invenory level z and a maximum invenory level Z, and he supplier is agreed o pay a penaly o he reailer for every uni of reailer s invenory ha is ouside his band afer cusomer demand. Boh paries know he reailer s demand disribuion. The supplier produces every T periods wih no capaciy limi. I also has he opion of ousourcing in order o mainain he desired reailer s invenory level. The supplier s decisions are hus how much o produce in each producion cycle, how much o ousource, and how much o send o he reailer in each period. Wih he ousourcing opion (so ha he supplier can always supply wha is needed a he reailer), he reailer s problem becomes a single-locaion invenory problem, whose backorder coss influence he supplier s coss. Recall ha our model reas a wo-locaion invenory problem. In heir paper, here is also no capaciy issue. As indicaed by hese auhors, in all he VMI agreemens hey observed in pracice, he penalies are no incurred immediaely (i.e., on a daily basis), bu are based on long-erm (approximaely yearly) performance, ofen as par of balanced scorecard evaluaion. The service level considered in our paper measures long-erm performance. Cachon (200) sudies how o achieve channel coordinaion in a one-supplier, muli-reailer compeiive supply chain using VMI. Boh he supplier and he reailers incur invenory and backorder coss. Cachon shows ha VMI is no guaraneed o coordinae he chain unless all members are willing o accep or pay

5 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 57 fixed ransfer paymens. A numerical sudy shows ha VMI provides no improvemen in supply chain coss when fixed ransfer paymens are forbidden. Narayanan and Raman (2002) examine a reailer and a supplier under a newsvendor seing. The reailer carries a privae label produc ha is a subsiue o he produc he carries from he supplier. Thus, he cos associaed wih a sockou is differen for he supplier and he reailer, and consequenly heir arge fill raes are differen. They derive condiions under which socking decisions should be ransferred from reailer o supplier (VMI). Clark and Hammond (997) and Cachon and Fisher (997) sudy he issue of wheher VMI coupled wih informaion sharing provides greaer benefis han informaion sharing alone. Bernsein and Federgruen (2003) analyze he consandemand-rae case and consider a model of VMI where he replenishmen decision is ransferred o he supplier, bu he reailer is able o make his own pricing decisions. Oher papers on VMI sudy logisics issues; Fry e al. (200) provide an excellen review. Our sudy here has a differen focus from hese works. We now review he lieraure on supply chains wih paricular concerns on service levels. Much of his lieraure has deal wih he problem of achieving a arge service level a he mos downsream sage in a cenralized supply chain (see van Houum e al. 996 and Diks e al. 996 for lieraure reviews). Mos auhors focus on how o coordinae he elemens in he supply chain o achieve a sysem-wide arge. Operaional decisions on producion, disribuion, and/or invenory conrol a each sage are coordinaed by a cenral planner. Under a VMI program, however, he manufacurer and he supplier are separae organizaions, so each pary in he supply chain makes is own operaional decisions. Usually, he supply chain paries can affec each oher s operaions only by specifying requiremens on observable measures in a conrac; canno dicae how he oher accomplishes hose requiremens. The manufacurer, for insance, may include in a supply conrac service performance requiremen regarding componen supply bu no deailed invenory policies. Several auhors adop a decomposiion approach (Bollapragada e al. 2000, Cohen and Lee 988, Lee and Billingon 993, Paschalidis and Liu 2003). Insead of cenralizing he enire operaion in he supply chain o achieve a arge sysem service level, a local service level arge is se for each sage of he supply chain. These local arges work as links beween sages so ha he enire sysem accomplishes he arge service level. The decomposiion mehod is relevan o he VMI seing we consider, because defining local service arges can be viewed as a way of specifying service requiremens in he supply conrac. When i comes o how o define local arges, however, he exising lieraure provides no noable resuls. Cohen and Lee (988) simply assume ha hey are given while Lee and Billingon (993) use a simple search heurisic o find he bes local arges, assuming each sage follows a base-sock policy. In Bollapragada e al. (2000) and Paschalidis and Liu (2003), he service level of he supplier is se o be greaer han or equal o he arge cusomer service level. This rule-of-humb is based on a common belief ha he downsream service level is guaraneed regardless of how he upsream service level is achieved, as long as he upsream service level is high enough. In his paper, we show ha his belief is no necessarily valid. Several researchers have sudied fill raes in cenralized capaciaed serial sysems under modified basesock policies; see, for example, Glasserman (997) and Sobel (2002). However, hese works focus on he sysem fill rae and do no discuss local fill raes and heir relaionship wih he sysem fill rae. 3. Model and Preliminaries We consider he following model: There is a single manufacurer who makes a produc o order (we relax his assumpion laer), and here is a single supplier who provides componens for he produc. The demands for he produc in differen periods are independen and idenically disribued random variables. One uni of componen is used o produce one uni of he produc. The supplier manages he componen invenory a a warehouse near or a he manufacurer s sie under a VMI program. The manufacurer has no knowledge abou he supplier s capaciy or invenory policy. Periodic-review sysems are used o conrol invenory and producion a boh he manufacurer and he supplier.

6 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 58 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS A he beginning of each period, afer observing cusomer demand for he period, he manufacurer decides producion quaniy considering demand and producion capaciy. Then i rerieves componens from he componen invenory. If here are no enough componens in invenory, he manufacurer reduces producion quaniy o he number of available componens. A he end of he period, he produc is delivered o he cusomer and he supplier resocks he componen invenory according o is invenory policy. When he manufacurer is no able o deliver he enire cusomer demand due o capaciy limi or componen shorage, he unsaisfied porion of he demand is backordered. The following noaion is used hroughou he paper. D : demand for period, B : backorders of he produc a he beginning of period, B = 0, I : componen invenory level a he beginning of before rerieval by he manufacurer, c: producion capaciy of he manufacurer in a period, P : number of he produc manufacured in period, R : number of he componen requesed in period, Q : number of he componen requesed bu no available in period. For any sochasic process X = 2 wih a saionary disribuion, denoe X o be a random variable ha follows he saionary disribuion. There are several definiions of service levels, common in boh indusry pracice and he academic lieraure; see, for example, Schneider (98). One of he mos popular measures is he -ype service level, which measures he likelihood of sockou. In paricular, le be he long-run fracion of periods ha has sockou. Then is called he -ype service level. The -ype service level of he manufacurer in our model is m = long-run fracion of periods wih demand backorders = lim B >0 Similarly, he -ype service level of he supplier is s = long-run fracion of periods wih componen sockou = lim Q >0 Defined using upper limis, m and s ake ino accoun even he case where /T T B >0 or /T T Q >0 does no have a limi. (When B and Q have a saionary disribuion and saisfy a srong law of large numbers, m = P B > 0 and s = P Q > 0 wih probabiliy, where B and Q are he corresponding saionary random variables. See 4 for an example.) Le be he unfill rae, which is he long-run proporion of demands ha canno be fulfilled immediaely. I can be shown ha equals he raio of he expeced backorders o he expeced demand. Then is he -ype service level, and is corresponding expression for he manufacurer is m expeced unsaisfied demand per uni ime = expeced demand per uni ime = E U E D where U = min B + D is he unsaisfied demand ou of D. While m uses he new backorders incurred in he curren period, m uses he cumulaive backorders up o a period. expeced cumulaive unsaisfied demand per uni ime m = expeced demand per uni ime = E B E D The - and -ype service levels of he supplier in our model are idenical because unsaisfied componen requiremen is no backlogged. s = s = E Q E R For simpliciy in exposiion, mos of his paper focuses on he -ype service level, bu we remark

7 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 59 ha he major findings hold for he oher wo ypes of service levels as well and provide more deailed discussions whenever possible. We assume he manufacurer follows a modified base-sock policy o conrol is producion. Tha is, in each period he manufacurer produces as much as possible, wihin he manufacuring capaciy, so as o keep he invenory level as close as possible o a arge base-sock level. This ype of policy has been shown o be opimal for he manufacurer o minimize he long-run average invenory-backorder cos, provided he demand is saionary and here is infinie componen supply (i.e., he only resricion of he producion level is he manufacurer s own capaciy limi). See Federgruen and Zipkin (986). More recenly, Parker and Kapuscinski (200) show ha his kind of policy is also opimal for boh finie-horizon and infiniehorizon problems if he manufacurer s capaciy does no exceed he supplier s capaciy. Because we consider an MTOmanufacurer here, here is no finishedgoods invenory; he base-sock level is se a zero. Le R be he planned producion quaniy for period, hen his policy implies R = min B + D c () Recall ha P is he acual producion quaniy in period. So, Q, he difference beween he required and available componen quaniies, can be expressed as Q = R P The number of produc backorders a he end of period is hen B + = B + D P = B + D R + Q = Q + B + D c + (2) where x + = max x 0. Equaion (2) provides some preliminary insighs ino how he supplier s performance is relaed o he cusomer service level m = P B = 0. There are wo condiions for B + o be 0. Firs, Q should be 0. Componen sockou immediaely causes backorders of cusomer demand and hus m is a leas as much as s. The second condiion for B + = 0 from (2) is B + D c. A recursive expansion of B changes he condiion o Q + B + D c + + D c. From his inequaliy, i is clear ha no only he frequency (he firs condiion, wheher Q > 0 or no) bu also he amoun of componen shorage (Q in he second condiion) has influence on m. If his quaniy is large, due o he capaciy limi c, i can ake many periods for he manufacurer o clear a large amoun of backorders, which would resul in a high value of m. Of course, Q depends on he supplier s operaional characerisics. To see his, we assume for he momen ha he supplier produces componens in a manufacuring faciliy whose capaciy is V. There is ample upsream supply, so he supplier s producion quaniy is solely consrained by her own producion capaciy. Similar o he manufacurer, he supplier follows a revised base-sock policy o conrol he componen invenory. Le s be he arge componen base-sock level. Then he acual componen invenory level I is updaed as follows: Because Q can be expressed as I = min s I P + V (3) Q = R I + P = min R I (4) = max 0 R s R I + P V Thus, he supplier s operaional characerisics, such as V and s, direcly influence he size of Q, which in urn influences m. In oher words, he supplier s producion/invenory operaions evenually have an effec on he manufacurer s cusomer service level. This raises he quesion of wheher he supplier s service level s provides sufficien informaion abou he supplier s performance for he manufacurer o predic he cusomer service level. In he nex secion, we sudy a more deailed model o shed ligh on his issue. Quesions under invesigaion include: Are here definiive relaionships beween he supplier s service level and he manufacurer s service level? Does i maer how he supplier achieves he arge service level specified by he manufacurer? Wha is he role of he supplier s operaing characerisics such as invenory level and capaciy?

8 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 60 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 4. Relevance of Service Levels 4.. The -Type Service Level For simpliciy and o highligh he impac of he supplier s operaional characerisics, in his secion we assume ha boh he cusomer demand and he manufacurer s capaciy are consan. Le demand D = d > 0 for all periods and he manufacurer s capaciy c = d + b. The supplier has a finie and variable producion capaciy. Is capaciy in period is { d + e wih probabiliy p V = (5) 0 wih probabiliy p where 0 <p<. We assume c>d or b>0 o preven backorders from exploding. For a similar reason, he supplier s maximum capaciy is assumed o be greaer han d, in oher words, e>0. The manufacurer s capaciy is always b unis greaer han demand. On he oher hand, he supplier s capaciy depends on wo parameers: e and p. Parameer e is inerpreed as he maximum exra capaciy of he supplier, while parameer p represens he reliabiliy of he supplier s resource. A higher value of p means less variance in he supplier s capaciy. The expeced capaciy, p d + e is assumed o be greaer han d. Oherwise, he supplier could no deliver enough componens o saisfy cusomer demand. We se he iniial componen invenory level I equal o he base-sock level s. Ifs d, hen here will always be backorders afer he firs backorder occurs, because he base-sock level is he maximum number of componens available. (If B > 0, hen R >d, which implies Q = R P >d s 0 and B + > 0.) For his reason, we assume s o be greaer han d. We will show shorly ha he process I B = 2 is a discree-ime Markov chain. Using is saionary disribuion, we can compue m and s. This allows us o make several key observaions as illusraed in Table, which liss several combinaions of e, p, and s generaing s close o 5% when d = 0. As we can see, alhough s is kep roughly a he same level (5%), m varies significanly (7% o 34%) wih differen combinaions of he parameers. Thus, he impac of he supplier on he cusomer service level canno be prediced simply by he supplier s service level. How he supplier achieves is service level maers. The resuls also help us o see which of he supplier s characerisics has he greaes impac on he cusomer service level. We observe ha he gap beween m and s becomes smaller when he manufacurer has more exra capaciy b. Bu even wih he same b, he cusomer service level sill depends heavily on he supplier s characerisic parameers such as exra capaciy e and reliabiliy facor p. I appears ha having reliable resources, e.g., p = 0 95, has he greaes effec on improving he cusomer service level ( m ). This is perhaps he mos expensive mehod, however, for he supplier o improve is operaions. Now we show how -ype service levels have been compued for Table. I follows from () and (4) ha P = min B + d d + b I (6) and { min s I P I + = + d + e wih probabiliy p I P wih probabiliy p min s max I + e B I + e b d+ e wih probabiliy p = (7) max I B d I d b 0 wih probabiliy p Backorders are updaed by B + = B + d P = max 0 B b B + d I (8) Table Supplier s Characerisic Parameer Ses Yielding s 5% b e p s s (%) m (%)

9 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 6 From (7) and (8), i is clear ha I B = 2 is a discree-ime Markov chain. Is sae space is = i j i j s 0 0 i s j 0 where i j s 0 means i j communicaes wih s 0. Because I B = s 0, only he saes ha communicae wih s 0 are included in. No all invenory posiions can be reached from he iniial sae. For example, if d = b = e = 2 and s = 2k for a posiive ineger k, any sae in 2 k i 2j + i = 0 k j = 0 does no communicae wih s 0. I can be easily shown ha I B = 2 is posiive recurren if p d + e >d. Because he Markov chain is irreducible and posiive recurren, here is a unique saionary disribuion i j i j. Furher, he srong law of large numbers holds for I B. In paricular, wih probabiliy, m = lim B >0 = P B > 0 Thus, using he saionary probabiliy, m expressed as m = P B > 0 = can be s i j (9) i=0 j= Similarly, he expression for s is s = P Q > 0 j+d b = i j + j=0 i=0 j=b+ b+d i=0 i j (0) The following resuls reveal ha under cerain condiions here is a definiive relaionship beween m and s. The proofs can be found in he appendix. Proposiion 4.. If b e, i j is independen of b. Proposiion 4. implies ha adding capaciy does no help he manufacurer improve he cusomer service level when is capaciy is greaer han or equal o he supplier s peak capaciy. Proposiion 4.2. If b>e, m = s. Proposiion 4.2 indicaes ha m is equal o s if he manufacurer always has excessive capaciy b> e. In his case, he manufacurer knows exacly wha kind of service level i can offer o he cusomer once i makes an agreemen wih he supplier on he expeced sockou rae s. Similarly, once he manufacurer ses a arge cusomer service level, i should require a supplier s service level no less han he arge level when signing a supply conrac. Proposiion 4.2 illusraes a special case where he cusomer service level is predicable from he supplier s service level. Unforunaely, such a direc relaionship beween he cusomer and he supplier s service levels does no always exis. As demonsraed in Table, if b e, here is no general resul he manufacurer can use o esimae he cusomer service level from he supplier s service level Oher Types of Service Levels We now show ha similar conclusions can be reached for - and -ype service levels as well. We use he same example as in 4. for which saionary disribuion of B and Q exis. Recall ha demand and he manufacurer s capaciy are assumed o be consan. The supplier can change is performance by hree parameers: e, p and s. Table 2 liss several combinaions of he supplier s characerisic parameers which yield s = s 0 5% when d = 0 and b = 5. While s = s remains close o 5%, m ranges from 2% o 20%. I urns ou ha m and m vary significanly as well (7% o % and 9% o 9%, respecively). Thus, no maer wha ype of definiion is used, he cusomer service level is no predicable jus by s (or s ). Noe ha s involves he average amoun of componen sockou, which is one of he measures recommended in 5. This example demonsraes again ha a single quaniy is no adequae o serve as he supplier performance measure. The manufacurer mus use wo supplier performance measures: he frequency and he average amoun of componen sockou, joinly. Table 2 Supplier s Characerisic Parameer Ses Yielding s = s 5% e p s s = s % m % m % m %

10 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 62 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS The examples in his secion show ha he cusomer service level can vary even when he supplier s service level is consan. The degree of cusomer service improvemen depends no only on he supplier s service level, bu also on how he supplier has achieved ha level. Thus, in general, he manufacurer needs more informaion on he supplier s operaion in order o guaranee he cusomer service level. 5. New Supplier Performance Measures To overcome he shorcomings of he radiional service level as a measure of supplier performance, in his secion we develop alernaive supplier performance measures ha can allow he manufacurer o secure is cusomer service level regardless of how he supplier delivers he performance. Noe ha an upper bound on m gives a lower bound on m, which in urn can serve as he minimum cusomer service level he manufacurer can guaranee. Our goal is hus o consruc an upper bound on m, say u m, in erms of he supplier s delivery saisics of Q = 2. This way, if we require he supplier o deliver a performance measured by a specified value u m, hen he manufacurer is assured o deliver a cusomer service level u m. To obain a robus performance measure, i is imporan ha boh he bound and he derivaion of he bound depend only on he observable quaniies Q. Thus, in our analysis here we do no make any specific assumpion abou he supplier s operaional characerisics. Equaion (2) plays a key role in his analysis. Our firs main resul, an upper bound on m,is saed in he heorem below. All he proofs can be found in he appendix. Theorem 5.. Assume demands are i.i.d. and have he same disribuion as D. Le c be he manufacurer s producion capaciy in each period. Le Q u m = + c s + + E D c + c + E D + E D c + where Q = lim T /T T Q and = P D > c. Then, wih probabiliy, m u m () The resul of Theorem 5. and is derivaion do no make use of any informaion on he supplier s side excep Q = 2. In paricular, he bound is valid whichever policy he supplier adops o manage he componen invenory. The following heorem provides a lower bound on m. Q defined as in Theo- Theorem 5.2. Le D, c, and rem 5., and le l m = Q + E D c + c E D + E D c + If lim T B T /T = 0, hen, wih probabiliy, m l m (2) The smaller he gap beween u m and m is, he beer he upper bound u m is. The following lemma gives a bound on he gap. Lemma 5.. If lim T B T /T = 0, u m m c s + c + E D + E D c + (3) The nex heorem shows he convergence of he upper bound u m. Is proof makes use of Lemma 5. and can be found in he appendix. In he heorem, superscrip r is used as an index of sequences. Theorem 5.3. Assume ha here exis consans C and r such ha c r c r E D r C for r> r If r m 0 as r, hen ur m converges o 0 as well. Tigher upper bounds are available when demand is deerminisic, as in he previous secion. Corollary 5.. If demand is a consan, i.e., D = d<c for all, Q m c d + s (4) Furhermore, if d and c are inegers and Q = 2 akes only inegral values, Q m ( c d + ) c d s (5)

11 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 63 Theorem 5. and is corollary give a hin as o how he supplier s performance should be measured o guaranee he cusomer service level. The manufacurer can conrol he supplier s influence on he cusomer service level hrough service level s and average sockou Q ogeher by including an agreemen on hese wo measures when signing he supply conac. We can also learn from he expression of he bounds ha he requiremens on he supplier mus be adjused depending on he manufacurer s capaciy. To demonsrae he performance of he bounds, we firs consider he deerminisic demand case, as in 4. The examples in Table saisfy he assumpions for (5) of Corollary 5.. Table 3 shows he corresponding lower bound l m and upper bound (in he righ-hand side of (5), denoed by u d m ) wih he service levels. Recall ha b is he difference of capaciy and demand b = c d. As menioned in he previous secion, he cusomer service level m varies significanly from 65% o 94% depending on he supplier s operaing parameers, even hough he supplier s service level s remains almos he same as 95% in all insances. In Table 3, we can see ha u d m, which akes ino accoun boh s and Q, reflecs he variaion of he cusomer service level effecively. The gap beween he upper bound and he cusomer service level is less han 3%. Especially when b =, m = u d m = Q ( m Q from (5) and m Q from (2)). The gap beween l m and m is also no significan less han 3% in mos cases. The lower bound as well as he upper bound migh be used o esimae he cusomer service level. However, for he purpose of Table 3 Upper Bound on m (Deerminisic Demand) b e p s s % l m % m % u d m % conrolling he supplier s performance o guaranee he cusomer service level m, upper bounds on m have a beer use han lower bounds. Thus, we focus on he upper bound. To illusrae he performance of he bounds for he general i.i.d. demand case (Theorem 5.), we consider hree seings of he supplier s operaional characerisics. The firs seing is similar o he example in 4. The supplier follows a revised base-sock policy wih base-sock level s o conrol producion of componens. Insead of aking one of wo possible values (see (5)), he supplier s capaciy V is now assumed o have a more general disribuion. Anoher difference is ha he demand is an i.i.d. process insead of a consan. In he second seing, he supplier uses an s S policy o conrol he componen invenory. In each period, if he invenory posiion is below or equal o s afer he manufacurer rerieves componens, he supplier sars o produce or places an order from an upsream source o bring he invenory posiion o S. The replenishmen akes place afer a fixed lead ime, LT. The las seing is similar o he second one excep ha he invenory policy is an R S policy. The parameer R represens a review period and he oher parameer S has he same meaning as in he s S policy. Afer checking he componen invenory posiion every R periods, he supplier iniiaes a replenishmen o bring he invenory posiion back o S. Tables 4, 5, and 6 show he simulaion resuls when D = 2 and V = 2 follow normal, Poisson, and gamma disribuion, respecively. E D is se o 20 in all examples. X 2 and c2 X denoe he variance and squared coefficien of variaion of a random variable X, respecively. All simulaion resuls are averaged values from 0 replicaions. Each replicaion runs for T = 0 5 periods. In simulaions wih normal disribuions, negaive random numbers have been rese o 0. For each invenory policy, he simulaion is done wih wo differen levels of he variaions of demand and/or he supplier s capaciy (hree wih gamma disribuion). For each se of demand and supplier parameers, we illusrae he effec of manufacurer s capaciy c on he performance of boh he supplier and he manufacurer. While increasing c improves manufacurer s service level m, i ends o have he

12 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 64 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS Table 4Supplier Performance Measures (wih Normal Disribuions) Supplier s Invenory Policy c s (%) Q (%) lm (%) m (%) u m (%) m (%) Revised 2 D = Base-Sock 2 V = Policy E V = s = D 2 V = = s S Policy s = S = LT = D = s = S = LT = D = R S Policy R = S = LT = D = R = S = LT = D = opposie effec on supplier s performance measured by s and Q. We also observe ha m improves (ges smaller) wih smaller variaions in demand and supplier capaciy. The upper bound u m also decreases as m decreases o 0, which is expeced from Theorem 5.3. When m is less han %, u m is no more han 3%. To decide wha kind of performance o ask of he supplier, he manufacurer needs an esimae of m ha is expressed in erms of he supplier performance measures. An ideal candidae would be a igh upper bound on m so ha i can be used o derive supplier s performance requiremens ha guaranee a arge cusomer service level. In he case of deerminisic demand, he upper bounds in (4) and (5) serve exacly his purpose. In general, Theorem 5.3 shows ha u m ges smaller when m decreases. Bu, unlike he deerminisic demand case, he numerical examples in Tables 4 6 reveal ha in mos cases he difference beween u m and m is no small enough for u m iself o be used o approximae he cusomer service level. To resolve his discrepancy, we develop he following closer esimae of m in erms of s and Q, which is anoher main resul in his secion: Q m = + c E D s + + E D c + c E D + + E D c +

13 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 65 Table 5 Supplier Performance Measures (wih Poisson Disribuions) Supplier s Invenory Policy c s (%) Q (%) lm (%) m (%) u m (%) m (%) Revised E V = Base-Sock Policy s = E V = s S Policy s = S = LT = s = S = LT = R S Policy R = S = LT = R = S = LT = Tables 4 6 lis m in he las column. In all cases, m is closer o m han u m. In general, m is no an upper bound on m. I would be an upper bound if he following inequaliies held (see (26) and (28)). All he new noaions have been used for he proof of Theorem 5. and heir definiions can be found in he appendix. lim K G c R c E D lim K G and lim N L lim N lim L Due o he dependency of B + and Q on D, i canno be guaraneed ha he wo inequaliies hold in general. Noneheless, as shown in Tables 4 6, m is greaer han m excep for 2 ou of 08 insances. Figure plos he numerical resuls for he revised base-sock policy sysems repored in Tables 4 6. As expeced, he variaion of demand has subsanial influence on he cusomer service level because

14 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 66 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS Table 6 Supplier Performance Measures (wih Gamma Disribuions) Supplier s Invenory Policy c s (%) Q (%) lm (%) m (%) u m (%) m (%) Revised c 2 D = Base-Sock Policy c 2 V = E V = s = c 2 D c 2 V c 2 D c 2 V = = = = s S Policy s = S = LT = c 2 D = s = S = LT = c 2 D = s = S = LT = c 2 D = R S Policy R = S = LT = c 2 D = R = S = LT = c 2 D = R = S = LT = c 2 D = demand exceeding manufacurer s capaciy resuls in backorders. I is also obvious ha he variaion of he supplier s performance affecs m. For example, in he case wih a revised base-sock policy, flucuaions of he supplier s capaciy cause more frequen componen sockou. The cusomer service level m ges lower wih higher variaions (larger values of cd 2 and/or c2 V ) han wih lower variaions (smaller values cd 2 and/or c2 V ). Figure shows ha m is an accurae esimae of m when m is small <0%. We have obained he same resuls wih he oher invenory policies. Thus, when he arge cusomer service level is high, i.e., greaer han 90%, he manufacurer can derive from m he supplier s performance requiremens, which will resul in a cusomer service level close o he arge. 6. Proposed Conrac Form As shown in he previous secion, we can use m o approximae he manufacurer s cusomer service level m wih confidence. We now show how o use his resul o design a supply conrac.

15 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 67 Figure Effecs of he Variaion of Demand and Supplier s Capaciy % 60 u m α m θ m 20 0 l m c % u m θ m 5 α m 0 l m c (a-) Normal (c D 2 = 0.25, c V 2 = 0.08) (a-2) Normal (c D 2 = 0.05, c V 2 = 0.032) % u m θ m α m 0 l m c % u m θ m α m 0 l m c (b-) Poisson (c D 2 = 0.05, c V ) (b-2) Poisson (c D 2 = 0.05, c V 2 = 0.04) % 60 % 0 50 u m 40 θ m α m 0 l m c u m θ m 5 α m l m c (c-) Gamma (c D 2 = 2, c V 2 = 2) (c-2) Gamma (c D 2 = 0.5, c V 2 = 0.5) Noe ha m is a linear funcion of s and Q; m = s + 2 Q + where = c E D / c E D + + E D c + 2 = / c E D + + E D c +, = c E D + E D c + / c E D + + E D c +, and = P D > c. If we se m o a arge level m, hen we ge a linear relaionship beween s and Q: s + 2 Q = m (6) Frequency and average of shorage are posiive quaniies unless boh are zero. Thus he range of s saisfying (6) is 0 < s < m (7) Now for any fixed s in his range, we can solve (6) o obain he corresponding Q. If he supplier can achieve he service level no greaer han s and a he same ime can conrol he average backorders no o exceed Q, hen he manufacurer s cusomer service level (sockou rae) is bounded above by m, and herefore he arge cusomer service level will be

16 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 68 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS guaraneed. This pair of values s Q can hen be specified in a supply conrac. In fac, he supply conrac can be composed of a lis of such pairs. Once a base pair is specified, we can use ( ) Q = s (8) o choose he oher pairs o be included in he menu. Noe ha (8), derived from (6), provides he (approximae) rade-off beween s and Q ha yields he same cusomer service level. For example, if he cusomer demand has a Poisson disribuion wih mean E D = 20 and he manufacurer s capaciy is c = 30, we have From (7) he range of s adequae for 95% arge service level (i.e., m = 0 05) is < s < 3 5% and he slope of he linear rade-off in (8) is These lead o he following sample menu for he 95% arge cusomer service level: s (%) Q Concluding Remarks In his paper we examine he role of he convenional service levels such as fill rae and sockou rae in decenralized supply chains. We specifically sudy he relevance of local service levels in a supply chain o is end cusomer service level. The research was moivaed by our ineracions wih managers who are ineresed o know how o specify supplier s service level in a VMI conrac so o allow he manufacurer o conrol his manufacuring process o achieve cerain desired cusomer service level. Using a wo-pary capaciaed supply chain model, we demonsrae ha even if he supplier provides seady performance in erms of hese service measures, he evenual cusomer service level a he end of he supply chain can vary considerably. Thus, he rule-of-humb of specifying a higher upsream service level indicaed in he lieraure proves o be invalid. Alhough our model assumes an MTOmanufacurer, he resuls can be easily exended o a make-o-sock (MTS) manufacurer, wih a posiive base-sock level. This finding also indicaes ha he commonly used VMI conrac ha requires a minimum invenory level ha he supplier mus mainain, while working for a supplier-reailer seing, fails o work for a suppliermanufacurer seing when he manufacurer has a finie producion capaciy. Noe ha he probabiliy ha he invenory level is below a lower limi is similar in naure o he sockou rae (in he laer, he lower limi is zero). So, mainaining a minimum invenory level wih high probabiliy is equivalen o a -ype service level. In a supplier-reailer seing, cusomers wihdraw direcly from he invenory supplier mainains, so supplier s service level is precisely he cusomer service level. Under he suppliermanufacurer seing, even if he componen supply is available, he manufacurer s capaciy may limi his abiliy o fulfill he cusomer demand in ime. Indeed, he irrelevance of supplier s service level holds even if we ask he supplier o mainain a leas c unis (he manufacurer s capaciy) componen invenory. By esablishing a bound on he manufacurer s cusomer service level, we show ha, in addiion o he supplier s sockou rae, he average componen backorders is anoher imporan measure ha should be specified in a supply conrac for he manufacurer o guaranee his desired cusomer service level. The supply conrac can be designed as a menu of differen combinaions of sockou rae and average componen backorders along a linear funcion. The virue of his kind of conrac is ha i requires minimum informaion sharing and is easy o monior. Also, is implemenaion is independen of he supplier s cos. I is worh menioning ha our findings on he irrelevance of supplier s service level hold even if demand informaion is shared wih he supplier. Assume he manufacurer rerieves a componen whenever a demand occurs, even hough i migh no be required for producion in he curren period. This way he acual demand informaion is passed o

17 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 69 he componen supplier, which should help preven he manufacurer s capaciy from being wased due o componen sockou. However, using D insead of R o reques componens in he example in 4, wih unavailable componens backlogged a he supplier, we have observed he same lack of connecion beween cusomer and supplier s service levels (for all -, -, and -ype service levels; he daa are no repored here). Insead of local performance measures, ransfer paymens from he manufacurer o he supplier based on he supply chain performance is anoher way for he manufacurer o conrol he supplier s impac on he final cusomer service level. Examples include he linear ransfer paymen agreemen discussed in Cachon (999), Cachon and Zipkin (999), Caldeney and Wein (2003), or he fixed paymen agreemen proposed in Cachon (200). See a review by Cachon (2003). To carry ou hese ypes of conracs, besides demand informaion sharing, several addiional requiremens are in need. Firs, o figure ou he paymen schemes, he manufacurer needs o assess he penaly associaed wih cusomer backorders. So he supply chain performance is no only measured by he service level bu also by he backorder level. Second, because he supplier is measured by he end cusomer service, i is necessary for he manufacurer o share his demand, capaciy, and holding cos informaion wih he supplier, so ha he supplier can be in effec a cenral planner. Or, boh he supplier and he manufacurer share heir capaciy and cos informaion wih a cenral planner, who can hen deermine he paymen schemes. These requiremens may render hese ypes of conracs difficul o implemen. Acknowledgmens The auhors hank Gerard Cachon for valuable commens on an earlier version of his paper. The paper has also benefied from simulaing discussions wih Derek Akins, Marin Puerman, Paul Zipkin, and oher seminar paricipans a Duke Universiy, he Universiy of Briish Columbia, he Universiy of Minnesoa a Twin Ciies, and he Universiy of Texas a Dallas. The research was suppored in par by Naional Science Foundaion Grans DMI , DMI , DMI , and DMI , and by The Logisics Insiue Asia Pacific, a parnership beween The Naional Universiy of Singapore and The Georgia Insiue of Technology. Appendix Proof of Proposiion 4.. We show ha b has no influence on P = 2. I implies ha B = 2 and I = 2 are updaed independenly of b. From (6), if B b or I d +b = 2 (9) P is no dependen on b. For =, i is obvious ha (9) holds. Assuming ha (9) holds for = k, we consider hree exclusive and exhausive condiions on B k and I k : (i) if B k b and I k d, P k I k d implies B k+ = B k P k + d B k b. (ii) if B k b and I k <d, P k = I k implies I k+ I k P k + d + e d + b. (iii) if B k >b and I k d + b, P k = I k as in (ii). This leads o I k+ d + b. Therefore, (9) also holds for = k+. By inducion, i holds for = 2 Proof of Proposiion 4.2. We will show ha if b>e, B + > 0 is equivalen o I <R. This implies m = s. (i) if B b, R = B + d. Therefore, B + > 0iffI <R. (ii) if B >b, R = d + b and B + B + d R > 0 We need o show ha I <R. We can modify (9) in he proof of Proposiion 4. as follows. If b<e, B b or I <d+b = 2 I implies I <d+ b if B >b. Therefore, I <d+ b = R. Proof of Theorem 5.. Firs, i is convenien for he proof o define he following subses of ime horizon 2. L = D > c M = B > 0 N = Q > 0 F = M c B + > 0 G = M B + = 0 where A c denoes he complemen se of A. L is he se of he periods when manufacurer s capaciy is less han cusomer demand. M and N are he se of he periods when here is sockou of finished produc and componen, respecively. The m and s defined in 3 can be expressed as m = lim s = lim B >0 = lim Q >0 = lim M N F is he se of he periods when he manufacurer does no have backorders a he beginning bu does a he end. G is for he opposie case. Noe ha cardinaliy of G is eiher equal o or less han ha of F by.

18 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 70 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS From he second equaliy in (2) ha, for any T, B T + = B + D P (20) Because B = 0, B T + 0 implies R D Q (2) By definiion, R = min D c for M c. Thus, R = D for M c L c. Using his, he righ-hand side in (2) is reduced o R D = R D + R D M c L c M c L c c = R D M L = R D + R D (22) M From (2) and (22), R D Q + M M c L M c L D R Afer adding M c R on boh sides, we ge c D Q + c R + M M M c L D R (23) For L M\G N c, R = c. I is because (i) L is he se of he periods when cusomer demand exceeds manufacurer s capaciy. Because he manufacurer needs componens no less han demand and no more han capaciy, R = c for L. (ii) If M\G N c, he manufacurer has rerieved all he componens i requires ( N c and i means P = R ). Bu, here are backorders a he end of ( G which means B + > 0). This implies ha R = min D + B c = c. Oherwise R = D + B >c, here would be no backorder a he end B + = B +P D = B +R D = 0, which conradics M\G. Thus, (23) becomes c D Q + c R + M M L M\G N c c M c L D c (24) To simplify he index se of he second summaion on he righ-hand side, we define K = M L c N. Then, The hird equaliy is from he fac ha G is a subse of M by definiion and also of L c because L D >c implies B + > 0. Because M c L D c = M c D c + = we ge he following from (24) M D c + D c + c D + D c + Q + c R + D c + which is equal o M K G M D c Q + c R + D c + (25) K G where x = x +. We can rewrie (25) using indicaor funcions. The range of summaions is suppressed for simple exposiion. M D c Q + K G c R + D c + Using nonnegaiviy of R,wege M D c Q + c K G + D c + (26) Because K G = K N c G N ) and T G T F, he second summaion on he righ-hand side of (26) is changed o K G = K + G K + F (27) Now we show ha F = M c N L. For M c, B + = D P = D R + Q = D c + + Q This implies ha B + > 0iffD >c L or Q > 0 N. M c B + > 0 = M c N L The lef-lend side is F by definiion. Wih his equaliy abou F, (27) is changed o K G M L c N M c N L = N L \ M L = N L M L M L M\G N c c = M L c M c G N = M L c G M L c N = G M L c N = G K = N + N + L N L M L L M L (28)

19 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS 7 Using (28), (26) is reduced o M D c + c [ Q + c N + M L ] L + D c + Dividing by T and aking upper limis as T lead o he inequaliy below. lim ( Q+c M D c +c lim s + lim M L L )+ lim D c + (29) Because D = 2 is i.i.d., by he srong law of large numbers, he wo summaions on he righ-hand side converge o and E D c + almos surely as T, respecively. Once he wo upper limis on he lef-hand side are shown o be equivalen o m E D c and m respecively, he proof is finished because (29) ges reduced o he following, which equals (). m E D c + c m Q + c s + + E D c + We firs prove ha lim T /T T M D c is equal o m E D c wih probabiliy. lim T T M D c = lim T T T T M M D c T T M The firs par of he righ-hand side is he definiion of m. If D M, in urn D c M, is i.i.d, hen he srong law of large numbers can apply o he second par and is limi is E D c. Le n = inf > n B > 0 for n = 2 and 0 = 0. Then, D M can be represened by D n n= 2. We use f X for he p.d.f. (or p.m.f.) of random variable X. Since D is independen of B and has he same disribuions as D, f D n = f D for an arbirary n. Thus, D n n= 2 is idenically disribued. I is also independen because f D 2 n x x 2 x n = E f D 2 n x x 2 x n where = 2 n. = E f D x f D 2 x 2 f D n x n = E f D x f D 2 x 2 f D n x n = E f D x f D x 2 f D x n = f D x f D x 2 f D x n = f D x f D 2 x 2 f D n x n Because D M is i.i.d., so is D c M.If m > 0 (his is assumed here because () holds obviously when m = 0), T M goes o infiniy as T goes o infiniy. Now we can apply he srong law of large numbers and ge T M D c T M a.s. E D c as T I can be proven similarly ha he second upper limi in (29) is equal o m wih probabiliy. Proof of Theorem 5.2. From (20) (25), we can see he following equaion holds. B T + + D c = Q + c R + D c + M M K G Because R c for all, B T + + D c Q + D c + Using he similar argumens as in he proof of Theorem 5., we ge B lim T + + m E D c Q + E D c + (30) which equals (2) if lim T B T /T = 0. Because B 0 for all, he condiion lim T B T /T = 0 is equivalen o lim T B T /T = 0. Proof of Lemma 5.. Because u m m u m l m, Q u m m + c s + + E D c + c + E D + E D c + Q + E D c + c E D + E D c + c s + c + E D + E D c + Proof of Theorem 5.3. Le = lim T B T /T. From (30) and a similar argumen in he proof of Lemma 5., u r m r m + c r r s + r + r c r + r E D r + E D r c r + r m + cr r s + r + r /c r c r E D r By he assumpion, for r> r, u r m r m + C ( r s + r + r We show ha r s, r, and r /c r converge o 0 as r m 0. Demand exceeding producion capaciy and componen shorage give rise o backorders of finished produc, which implies r m r and r m r s Thus, as r m converges o 0, r and r s also converge o 0. c r )

20 Choi, Dai, and Song: Supplier Performance Under Vendor-Managed-Invenory Programs 72 Manufacuring & Service Operaions Managemen 6(), pp , 2004 INFORMS Now we show he convergence of r /c r o 0. Assume ha r = lim T BT r /T > 0. Then, for an arbirary 0 < < r, BT r is greaer han r T infiniely ofen. Because he backorders canno be reduced more han c r in one period, r m lim r T /c r + r T /c r = r c r + r Thus, when r m <, r r m cr r m Because he inequaliy holds for an arbirary >0, r r m cr 0 r m which equals r c r m r r m Hence, r /c r converges o 0 as r m 0. Proof of Corollary 5.. If D = d, (25) is reduced o he following inequaliy and we use i insead of (26) o ge an upper bound: c d M Q + c d M N G Also, if d c, and Q = 2 are all inegers, so is R = 2. Then R d + for M, and he above inequaliy can be replaced wih a igher, c d M Q + c d M N G The remaining procedure is similar, as in Theorem 5.. References Barnes, E., J. G. Dai, S. Deng, D. Down, M. Goh, H. C. Lau, M. Sharafali Elecronics manufacuring service indusry. TLI-AP Repor, Georgia Insiue of Technology. Bernsein, F., A. Federgruen Pricing and replenishmen sraegies in a disribuion sysem wih compeing reailers. Oper.Res.5(3) Bollapragada, R., U. S. Rao, J. Zhang A decomposiion approach for managing assembly sysems wih demand and supply uncerainy. Working paper, Carnegie Mellon Universiy, Pisburgh, PA. Buzzell, R. D., G. Ormeyer Channel parnership sreamline disribuion. Sloan Managemen Rev. 36(3) Cachon, G Compeiive and cooperaive invenory managemen in a wo-echelon supply chain wih los sales. Working paper, Duke Universiy, Chapel Hill, NC. Cachon, G Sock wars: Invenory compeiion in a wo echelon supply chain. Oper.Res.49(5) Cachon, G Supply chain coordinaion wih conracs. A. de Kok, S. Graves, eds. Handbooks in Operaions Research and Managemen Science: Supply Chain Managemen. Norh-Holland, Amserdam, The Neherlands. Cachon, G., M. Fisher Campbell Soup s coninuous produc replenishmen program: Evaluaion and enhanced decision rules. Producion Oper.Managemen 6(3) Cachon, G., P. Zipkin Compeiive and cooperaive invenory policies in a wo-sage supply chain. Oper.Res.45(7) Caldeney, R., L. Wein Analysis of a decenralized producion-invenory sysem. Manufacuring Service Oper.Managemen 5 7. Cohen, M. A., H. L. Lee Sraegic analysis of inegraed producion-disribuion sysems: Models and mehods. Oper. Res. 36(2) Diks, E. B., A. G. de Kok, A. G. Lagodimos Muli-echelon sysems: A service measure perspecive. Eur.J.Oper.Res.95(2) Federgruen, A., P. Zipkin An invenory model wih limied producion capaciy and uncerain demands I. The averagecos crierion. Mah.Oper.Res.(2) Fisher, M. L Wha is he righ supply chain for your produc? Harvard Business Rev. (March April) Fry, M., R. Kapuscinski, T. Olsen Coordinaing producion and delivery under a z Z -ype vendor-managed invenory conrac. Manufacuring Service Oper.Managemen 3(2) Glasserman, P Bounds and asympoics for planning criical safey socks. Oper.Res Hammond, J., T. Clark Reengineering channel reordering processes o improve oal supply chain performance. Producion Oper.Managemen 6(3) Kleijnen, J., M. Smis Performance merics in supply chain managemen. Producion Oper.Managemen 6(3) Lee, H. L., C. Billingon Maerial managemen in decenralized supply chains. Oper.Res.4(5) Narayanan, V., A. Raman, S. Kraiselburd Conracing for invenory in a supply chain wih sochasic demand and subsiue producs. Producion Oper.Managemen. Forhcoming. Parker, R., R. Kapuscinski Opimal policies for a capaciaed wo-echelon invenory sysem. Oper.Res.Forhcoming. Paschalidis, I. C., Y. Liu Large deviaions-based asympoics for invenory conrol in supply chains. Oper.Res.5(3) Plambeck, E., S. Zenios Incenive efficien conrol of a makeo-sock producion sysem. Oper.Res Schneider, H. 98. Effec of service-levels on order-poins or orderlevels in invenory models. In.J.Prod.Res.9(6) Simchi-Levi, D., P. Kaminsky, E. Simchi-Levi Designing and Managing he Supply Chain: Conceps, Sraegies, and Case Sudies. McGraw-Hill, New York. Sobel, M Fill raes of single-sage and muli-sage supply sysems. Manufacuring Service Oper.Managemen.Forhcoming. Thompson, A. A., A. J. Srickland Sraegic Managemen: Conceps and Cases, 2h ed. McGraw-Hill, New York. van Houum, G. J., K. Inderfurh, W. H. M. Zijm Maerials coordinaion in sochasic muli-echelon sysems. Eur.J.Oper. Res. 95() 23. Wilson, R Game heoreic analyses of rading processes. T. F. Bewley, ed. Advances in Economic Theory: Fifh World Congress. Cambridge Universiy Press, Cambridge, U.K., Chaper 2,