An Analysis of Manufacturer Benefits under Vendor Managed Systems

Transcription

1 An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY Nesim Ekip 1 Depatment of Industial Engineeing, Bilkent Univesity, 06800, Bilkent, Ankaa, TURKEY [email protected] Vendo Managed Inventoy (VMI) has attacted a lot of attention due to its benefits such as fewe stockouts, highe sales and lowe inventoy levels at the etailes. Vendo Managed Availability (VMA) is an impovement that exploits the advantages beyond VMI. We complement VMA by analyzing the benefits beyond infomation shaing and by clealy assessing the motivation fo the manufactue (vendo) behind joining to such a pogam. We show that such vendo managed systems povide inceased flexibility in manufactue s opeations and may bing additional benefits. We analyze how the system paametes affect the pofitability and detemine the conditions that make the vendo managed system a viable stategy fo the manufactue. Keywods: Vetical Collaboation; Vendo Managed Inventoy; Capacity Management; Opeational Flexibility; Consignment Stock 1 Intoduction Vendo Managed Inventoy (VMI) is a collaboative pocess between a supplie/manufactue and a manufactue/etaile/distibuto, whee the manufactue gains access to the demand and inventoy infomation at the etaile and uses this infomation to bette manage the etaile s inventoy. VMI stated as a pilot pogam in etail industy between Pocte&Gamble and WalMat in 80 s and esulted in significant benefits, such as lowe inventoy levels, fewe stock-outs and inceased sales, and has been adopted by many othe supply chains such as Dell s, Bailla s o Nestle s. In many eseach and business aticles, the benefits of VMI ae attibuted to infomation shaing between the manufactue and the etaile (see Cachon and Fishe 1997; Schenck and McIneney 1998). Howeve, thee is moe to VMI than just the infomation availability; thee ae benefits hidden in the inceased flexibility of the manufactue s poduction opeations. Thee exist limited analytical wok in liteatue on how the manufactue can tanslate this flexibility into benefit, and why the paties join to a VMI pogam. We believe that it is impotant to emphasize the benefits of VMI additional to infomation shaing, so that the motivation behind joining to a VMI pogam is bette compehended. In a vendo managed setting, although the manufactue takes contol of inventoy, it is the etaile that 1 Coesponding autho. 1

2 usually benefits fom manufactue managing the inventoy (Dong and Xu 2002). The eason is, the etaile can always set the tems of the ageement such that the pefomance measues (such as numbe of stock-outs, aveage inventoy level, etc.) will impove. Whethe the manufactue benefits fom the vendo managed system on the othe hand, depends on how well the manufactue can take advantage of the inceased flexibility. In the ageement, the etaile may eflect a equied poduct availability on the shelf, o sevice level by imposing a lowe bound on the inventoy level. Similaly, due to shelf space constaints o to avoid high inventoy levels, the etaile may limit the amount of eplenishment fom the manufactue. Theefoe a contact may consist of an uppe and a lowe bound on inventoy level, whee oveshooting o undeshooting by the manufactue is penalized. While penalties compel the manufactue to confom with the inventoy limits, it is definitely a challenging task fo the etaile to detemine the penalties as well as to set the bounds on the inventoy level that will esult in the desied sevice level o inventoy holding cost. Ou modeling of VMI is close to vendo managed availability (Hausman 2003), whee the vendo is moe flexible in tems of eplenishment opeations than VMI, since in VMI, eplenishments ae moe esticted due to the bounds on etaile s inventoy level. Vendo managed availability has been pacticed by seveal majo etailes such as J.C.Penney, o Costco. J.C.Penney supplies shits fom a Hong-Kong based shit-make whee the supplie completely contols the inventoy by monitoing J.C.Penney s stock levels and making eplenishments diectly to the stoe, if necessay. To ensue availability, at times the supplie expedites the delivey by shipping though ai (Kahn 2002; Hausman 2003). Similaly, Kimbely-Clak, a supplie of poducts such as diapes, tissues o pape towels fo Costco in U.S., is vey flexible in its eplenishment opeations. The company simply keeps each [Costco] stoe s inventoy as low as possible without isking empty shelves (Nelson and Zimmeman 2000). These examples descibe moe flexible ageement tems between the manufactue and the etaile. To eflect this pactice, in ou vendo managed model we assume that the sevice level is the only constaint fo the manufactue, which esults in an inceased flexibility even compaed to VMI. Fo instance, at times the manufactue may not pefe to eplenish etaile s stock if the capacity can be used fo a moe pofitable ode. At othe times when thee is excess capacity, i.e., when the capacity is less valuable, seveal eplenishments may enable an inceased sevice level at the etaile. The etaile ends up with the same sevice level wheeas the manufactue effectively manages its poduction, capacity allocation and eplenishment opeations. In this pape, we conside the notions intoduced by VMA, an enhanced vesion of VMI. In the est of the text, we use the tems VMA o vendo managed system to epesent this enhanced vesion of VMI. In this study, we model a supply chain consisting of a single manufactue and a etaile. We fist define the taditional system unde which the manufactue and the etaile opeate, and then intoduce the vendo managed system and compae the two systems. We assume that the etaile sets the tems of the contact 2

3 such that she is neve wose off unde the new (vendo managed) system. We make the analysis fom the pespective of the manufactue who caies most of the collaboation buden. The etaile faces stochastic demand and in the taditional system peiodically places odes to the manufactue. Manufactue has limited capacity to meet the odes fom the etaile, and a moe expensive outsoucing option. To analyze benefits due to vendo managed system alone, ou poposed model fo the taditional system consides a manufactue that has full infomation on end-demand distibution, demand ealization and inventoy levels at the etaile and hence evisits capacity planning aspects of opeating a taditional manufactuing system. We assume that the paties do not shae cost infomation. Futhemoe, infomation on available capacity o end-of-peiod inventoy level at the manufactue is not shaed with the etaile. Ou focus is on the vetical collaboation pocess in the supply chain unde this asymmetic and patially shaed infomation setting. In vendo managed systems the issue of who owns the inventoy depends on the elationship between the manufactue (supplie) and the etaile (manufactue). If the manufactue is vey poweful (such as Dell) it may foce the supplies to own the inventoy at the manufactue s site o at a supply hub neaby. On the othe hand, if supplie is poweful then inventoy may not be consigned. Intel, fo instance, although has an ageement with Dell, does not opeate though supply hub like othe supplies (Banes et al. 2000). We conside two types of vendo managed ageements, consignment stock and no-consignment stock, and fo each type analyze how the manufactue may benefit fom managing the etaile s inventoy. In ou model thee does not exist an uppe and lowe bound estiction at the etaile s inventoy level; howeve etaile explicitly specifies sevice level and aveage inventoy level equiements. Given this setting we addess the following questions: (i) Ae thee any benefits fo the manufactue in managing the etaile s inventoy apat fom what is aleady achieved by shaing demand and inventoy infomation? (ii) What ae the conditions that make the manufactue bette off unde the vendo managed system consideed? (iii) Unde the vendo managed system should the manufactue consign the stock o not? Ou wok contibutes to the liteatue in seveal ways. Ou wok is one of the few studies that analyzes benefits due to vendo managed systems fom the manufactue s pespective and that identifies the conditions to make the manufactue willing to join such an ageement. Ealie studies eithe ignoe the motivation behind vendo managed systems, o focus only on total supply chain benefits athe than the individuals. Futhemoe, we make a compaison of benefits unde consignment stock and no-consignment stock models, to detemine the type of ageement the manufactue will benefit, wheeas pevious liteatue mostly assume centalized, consignment stock models. The emainde of the pape is oganized as follows. In 2 we eview the pevious wok on vendo managed inventoy systems. In 3 and 4 the model chaacteistics and stuctual popeties ae pesented. In 3

4 5 we make an expeimental analysis and discuss the esults, and based on these discussions we povide manageial insights. We pesent ou conclusions in 6. 2 Liteatue Review Majoity of existing studies analyze the vendo managed system in a manufactuing-etaile setting, while a few conside a supplie-manufactue setting (Choi, Dai and Song 2004). Inventoy owneship is modeled eithe by totally consigned stock, o by the tansfe of the title at the time of aival. In most of the pevious studies, the focus of the analysis is limited to designing an optimal opeating policy fo the vendo in a vendo managed system, and the motivation of the vendo in managing the inventoy is not unde consideation. In the analysis of the vendo managed systems unde a single manufactue and multiple etailes, the focus is mainly on the savings in tanspotation due to bette ode consolidation o savings due to coodination of etaile eplenishments. To analyze the benefit of VMI, Cetinkaya and Lee (2000) compae a VMI system with a taditional system. In the taditional system the manufactue sends a shipment immediately when the demand aives, while in VMI system shipments ae consolidated. Authos detemine the optimal dispatch quantity unde VMI consideing the inventoy cost and the tanspotation cost incued by the manufactue, and conclude that when inventoy holding cost and dispatching cost ae low, VMI esults in significant savings fo the manufactue. Kleywegt, Noi and Savelsbeg (2002) study an inventoy outing poblem of a manufactue who owns the inventoy at the etailes. An appoximation method is developed to find the minimum cost outing policy, howeve, thee does not exist a discussion on whethe the manufactue is bette off unde the vendo managed system. Walle, Johnson and Davis (1999) also conside a multiple etaile setting and though a simulation analysis demonstate the effects of VMI on the inventoy levels at the etailes and on the capacity utilization at the manufactue. VMI esults in savings due a decease in the inventoy levels, which is a consequence of the inceased fequency of etaile eplenishments. Aviv and Fedeguen (1998) conside a capacitated supplie with multiple etailes and analyze how coodination of etaile odes unde VMI decease the system-wide cost of opeation. They explicitly model a taditional system with no infomation shaing and with full infomation shaing to assess the benefits of VMI beyond infomation shaing. Fy, Kapuscinski and Olsen (2001) compae a VMI system with a taditional system in a single manufactue, single etaile setting unde full infomation shaing. The authos identify the optimal opeating policies of both the manufactue and the etaile in a stochastic setting. Unde VMI the etaile detemines the maximum inventoy level and the vendo incus a penalty if the inventoy level is outside the limits. Authos find that VMI pefoms close to a centalized model in the pesence of high demand 4

5 vaiance and high cost of outsoucing. Seveal othe papes study the optimal decisions of the manufactue unde VMI in a deteministic envionment. Valentini and Zavanella (2003) and Shah and Goh (2006) conside a consignment stock system whee the demand is deteministic with a constant ate. Jauphongsa, Cetinkaya and Lee (2004) study a poblem with delivey time windows and ealy shipment penalties unde dynamic demand. The authos popose a dynamic pogamming algoithm to obtain the minimum cost unde VMI. Depending on the fom of ageement between the etailes and the manufactue, the system unde vendo managed egime can be vey close to a centalized system. A numbe of papes analyze the ole of VMI as a channel coodinato. Benstein, Chen and Fedeguen (2006) study the constant wholesale pice and quantity discount contacts that lead to pefect coodination in a supply chain with multiple competing etailes, and show how VMI helps achieve the coodination. Nagaajan and Rajagopalan (2008) show that simple contacts in VMI can impove the pefomance of the oveall system unde cetain conditions. Dong and Xu (2002) analyze the benefits of VMI both in tems of total channel cost and vendo s cost. In thei model the etailes set the puchasing pice in the contact and the supplie in tun detemines the selling quantity. Authos detemine the conditions unde which the supplie benefits fom VMI and conclude that VMI can always decease the cost of channel as a whole. Fy, Kapuscinski and Olsen (2001) also discuss centalization of the supply chain. Thee has been few wok on the sevice level consideations in a VMI system. In most of the papes the sevice level is assumed implicit in the lowe inventoy level set by the lowe-echelon. Choi, Dai and Song (2004) study the sevice level elationship between a supplie and a manufactue in a VMI famewok and show that high sevice levels at the supplie does not guaantee the desied sevice level at the manufactue and that expected backodes should also be taken into account. Ou study is most closely elated to Fy, Kapuscinski and Olsen (2001). We study a single manufactue single etaile system and compae the vendo managed system with the taditional system to quantify the benefits beyond infomation shaing. Howeve, we focus on the benefits to the manufactue to detemine the motivation to make an ageement. We futhemoe conside capacity management as an impotant facto in detemining the benefits of vendo managed system. Additionally, we study both consignment and no-consignment models to identify the conditions that make eithe model beneficial fo the manufactue. In ou model, we do not necessaily egad the vendo managed system as a coodinated system. We popose a moe ealistic setting with asymmetic and patial infomation shaing and focus on the collaboation pocess. Since usually it is the manufactue that is eluctant in these ageements, we analyze the poblem fom manufactue s pespective. Finally, we take sevice level consideations explicitly into account. In summay, ou model diffes fom the existing studies in the following aspects: (i) We look at manufac- 5

6 tue benefits in joining to the vendo managed system. (ii) We identify the benefits beyond infomation shaing to clealy assess the manufactue s motivation. (iii) We explicitly model the consignment and no-consignment systems and povide a compaison of these systems to detemine which type of ageement is moe beneficial to the manufactue. In pactice, if the lowe echelon is moe poweful, the stock is usually consigned by the manufactue. Othewise, if the manufactue is poweful, the stock is not necessaily consigned. Theefoe it is not appaent whethe the manufactue should consign the stock o not. (iv) Finally, we analyze how benefits unde vendo managed system change with system paametes. Specifically, we measue the effect of capacity management and povide a detailed analysis of the benefits fom poduction and tanspotation flexibility. 3 A Modeling Famewok fo the Manufactue We compae two settings; a taditional system whee the etaile manages and owns the inventoy, and a vendo managed system. In the vendo managed system we model two cases based on the owneship of stock. Unde no-consignment stock model (VM-NC), the stock is managed by the manufactue while owned by the etaile. Unde consignment stock model (VM-C), the inventoy is both managed and owned by the manufactue. We assume the etaile accepts the ageement only if the pefomance measues ae as good compaed to the taditional case. We conside a peiodic-eview model whee the manufactue has limited and non-stationay capacity, which is known by the manufactue in advance. The non-stationaity in the capacity eflects an envionment whee the manufactue has seveal customes and allocates some potion of the capacity to the etaile and the emaining to the othe odes. We assume that the capacity allocated to the etaile may be 0 in some peiods, i.e., the manufactue poduces fo the etaile in evey T p peiods, and without loss of geneality we assume non-negative capacity in the fist peiod of T p. We call the time span between two positive capacity levels as the poduction cycle. Note that cyclic poduction concept is a well-known and utilized idea in the liteatue. Maxwell and Muckstadt (1985) intoduced the idea of consistent and ealistic eode intevals. Li and Wang (2007) mention cyclic stuctues within the supply chain as a coodination mechanism. Fy, Kapuscinski and Olsen (2001) conside a simila cyclic stuctue in thei study. We futhe assume that the level of capacity may be non-stationay fo the peiods in which the manufactue poduces fo the etaile. We assume this non-stationaity also shows a cyclic behaviou. In othe wods, in evey T m peiods the level of the capacity is the same and T m may consist of seveal T p cycles, each cycle with possibly a diffeent capacity level (see Figue 1). We call this lage cycle as the capacity cycle. Similaly, due to scheduling pactices the etaile places a eplenishment ode to the manufactue in evey T peiods. We call the etaile s cycle as the eplenishment cycle. 6

7 Capacity allocated to the etaile Replenishment cycle Peiods Poduction cycle Capacity cycle Figue 1: Manufactue s capacity cycle is 12 peiods, poduction cycle is 6 peiods, etaile s eplenishment cycle is 4 peiods. We assume the eplenishment odes ae quantized, whee unit eplenishment size Q eflects economies of scale in manufactuing and tanspotation and is an ageed-upon quantity between the manufactue and the etaile. Note that, this assumption implies that the manufactue is expected to opeate with this bucket size Q with all of the customes. Hence, we can assume that the capacity at the manufactue is a non-negative intege multiple of Q. This type of envionment can be obseved in pactice. Fo example, DMC, a Fench thead company, loweed its shipment size fom 24-unit cases to 12-unit cases afte an ageement made with WalMat. Since switching to 12-unit case equied significant investment now the company is shipping in 12-unit cases to all of its customes (Fishman 2006). The end-item demand is stochastic and stationay. Holding cost is incued based on end-of-peiod inventoy level, and the etaile opeates based on a sevice level constaint. Excess demand at the etaile can be backlogged (thee is no cost associated), howeve the manufactue (always) meets the etaile ode eithe though egula stock o by subcontacting (fo a simila usage of subcontacting option, see Gavineni, Kapuscinski and Tayu 1999). Hee, the tem subcontacting actually coesponds to a vaiety of altenatives to meet the unsatisfied demand. The manufactue can use an additional set up fom the capacity of othe poducts/customes, ovetime poduction, expedite the supply, o let the etaile to take cae of unmet demand but pay a(n implied) penalty. We assume that tanspotation time is negligible and hence the poduced amount is deliveed at the same peiod (ovenight). Note that this is consistent with the JIT delivey concept. We model the etaile s and the manufactue s poblem unde the taditional system, and the manufactue s poblem unde the vendo managed system as a Makov Decision Pocess (MDP). We detemine the optimal opeating policy unde each system. Model paametes, decision vaiables and state vaiables ae pesented in Table 1. One of the objectives of this study is to quantify the benefits of the vendo managed system fo the 7

8 manufactue when demand and inventoy infomation of the etaile is available. Specifically, we make the following assumptions on infomation shaing: 1. The infomation of peiodic demand ealization, end-of-peiod inventoy level at the etaile, and etaile s demand distibution is povided by the etaile to the manufactue. 2. Infomation of unit inventoy holding cost o any othe cost infomation at the etaile is not shaed with the manufactue. Similaly, cost infomation of the manufactue is not shaed with the etaile. Cost infomation is mutually unavailable. 3. Infomation on capacity level and end-of-inventoy level at the manufactue is not shaed with the etaile. Theefoe infomation shaing is asymmetic and patial. Table 1: Notation fo Taditional and Vendo Managed System Models Paametes T p : length of the poduction cycle fo the manufactue T m : length of the capacity cycle fo the manufactue T : length of the eplenishment cycle fo the etaile unde taditional system D i : andom vaiable denoting demand ove i peiods, i {1,, T } P i : pobability mass function fo D i Q: batch ode (dispatch) quantity c: unit poduction cost w: unit outsoucing cost h: (manufactue s) unit holding cost 1 β: sevice level at the etaile z: the numbe of poduction cycles in a capacity cycle, zt p = T m Decision Vaiables R: eode level at the etaile p n : numbe of lots of Q poduced in peiod n d n : numbe of lots of Q dispatched in peiod n State Vaiables Im: n numbe of lots on-hand at the manufactue at the end of peiod n 1, Im n {0, 1,, }. I n : net inventoy at the etaile at the end of peiod n 1, I n {,, }. t n m : the elative position of peiod n in capacity cycle, t n m {1,, T p,, 2T p,, zt p = T m } t n : the elative position of peiod n in eplenishment cycle, t n {1,, T } K n : the capacity level in peiod n (implied by t n m), K n {0, K 1,, K z }. S T : state unde taditional system, S T = (I m, I, t m, t ) S NC : state unde no-consignment vendo managed system, S NC = (I m, I, t m ) S C : state unde consignment vendo managed system, S C = (I, t m ) 8

9 3.1 Taditional System In the taditional model, at the beginning of each peiod the manufactue decides on how much to poduce and/o to outsouce. The manufactue poduces fo the etaile in evey T p peiods, while the etaile places an ode in evey T peiods. T is known by the manufactue. We assume that fixed cost of tanspotation is zeo unde taditional and unde vendo managed systems. We assume that the etaile places odes based on an (R, nq) type policy, whee R is the eode point that guaantees a specified sevice level (Zheng and Chen 1992). Note that due to quantized shipments the analysis would not change unde a fixed cost of tanspotation pe batch. The sequence of events unde taditional system is as follows: 1. At the beginning of a peiod, the manufactue gives the decision of how many units to poduce and/o to outsouce, consideing the allocated capacity (if allocated capacity is zeo, thee is no poduction). If an ode is placed by the etaile in the last peiod of the eplenishment cycle, a dispatch is made to the etaile in the fist peiod of the following eplenishment cycle. Poduction, outsoucing and dispatch lead times ae negligible. Theefoe the dispatched quantity is immediately eady at the etaile at the beginning of the eplenishment cycle, befoe any demand is ealized at the etaile. 2. Demand is ealized at the etaile. If thee is enough inventoy in stock, the etaile fulfills the demand. If the etaile can not meet the demand completely, the unmet amount is backodeed (at no explicit penalty). If it is the last peiod of the eplenishment cycle, the etaile places an ode at the manufactue (if any), which is a non-negative intege multiple of Q. Othewise, if it is not the last peiod, the etaile only passes the demand infomation to the manufactue, and updates the inventoy level Retaile s poblem unde the taditional system The poblem of the etaile is to minimize the expected inventoy level unde a sevice level equiement (thee is no explicit backode cost fo the etaile). We only conside the opeating policies with (R, nq) stuctue. In the last peiod of the eplenishment cycle, afte the demand is ealized, etaile places an ode if the inventoy level is equal to o less than the eode point, R. The eode point, R, is the decision vaiable and Q is assumed to be a paamete. Fist, conside the two measues fo a given R and Q: (i) Expected aveage inventoy level (Ī), and (ii) Aveage sevice level (1 β). 9

10 The expected aveage inventoy level is expessed as follows: Ī = 1 Q R+Q i i=r+1 j=0 (i j) P 1(j) + P 2 (j) + + P T (j) T (1) In (1), P 1 is the pobability mass function of single peiod demand and P k, k {1,, T }, is the k- convoluted pobability mass function (i.e., pobability mass function of k-peiod demand). Conside the eplenishment cycle T. Unde the quantized odeing policy, (R, nq), at the beginning of each cycle the inventoy level at the etaile is i with pobability 1 Q, whee i {R + 1,, R + Q}. In the long-un, R+Q fo the fist peiod of the cycle, expected end-of-peiod inventoy level is 1 i Q i=r+1 j=0 (i j)p 1(j), Similaly, fo the second peiod, expected end-of-peiod inventoy level is 1 R+Q i Q i=r+1 j=0 (i j)p 2(j), and so on. Since in the long-un, pobability of being in any peiod in the eplenishment cycle is equal to 1 T, the time-aveaged expected inventoy level is expessed as in (1). We define the aveage sevice level as 1 β, whee β is the expected aveage faction of backodeed demand pe peiod. Let β i, i = 1, 2,, T, denote the expected faction of backodeed demand in the i th peiod of the eplenishment cycle. Then β i would be expessed as follows: β i = I i P(I i ) E[(D 1 I i ) + ] E[D 1 ] whee I i is the beginning inventoy level of the i th peiod, P(I i ) is the pobability that the beginning inventoy level is I i, and D 1 is the andom vaiable denoting one-peiod demand. Then expected aveage faction of backodeed demand, β, is expessed as: β = β 1 + β β T T (2) Equivalently, β is expessed as follows: β = 1 Q R+Q i=r+1 j=i+1 (j i) P T (j) T E[D 1 ]. (3) We limit the opeating policy of the etaile to the (R, nq) policy. Unde this policy, to minimize the expected aveage inventoy level in (1), the etaile simply chooses the minimum eode point that guaantees the desied sevice level. Howeve, as we analyze below, unde quantized odeing (R, nq) type policy is not necessaily the optimal policy fo the etaile. In othe wods, even if the optimal eode point is chosen, expected inventoy level may not be minimized. In Poposition 1 below, we identify the conditions unde which the optimal policy is indeed an (R, nq) type policy fo T = 1. We pesent the poofs in Appendix. Each eode point implies a sevice level (1 β), and an expected inventoy level (Ī). Let S be the set of the β values implied by all (intege and non-negative) eode points (note that the elements of set S 10

11 vay with Q). Fo β S, let R(β) denote the eode point that esults in the sevice level of 1 β. (We assume thee exists a unique R(β) fo each β S. Unde T = 1 this is possible if R(β) + 1 max(d 1 )). Poposition 1 Suppose T = 1. (i) Fo β S, (R(β), nq) policy is the unique inventoy level minimizing policy fo the etaile. (ii)fo β S, thee may exist moe than one optimal odeing policy fo the etaile, none of which is an (R, nq) policy. Poposition 1 implies that fo β S the only policy that achieves the minimum inventoy level is (R(β), nq) policy. We use this esult late in Section 4 when analyzing the manufactue s policy Manufactue s poblem unde the taditional system We detemine the optimal opeating policy of the manufactue unde the taditional system. We model the manufactue s poblem as a Makov Decision Pocess unde aveage cost citeia as follows. g(s) = min δ lim N 1 N Eδ s [ N n=1 ] (s n, a n ) (4) whee g(s) indicates the optimal aveage cost given that initial state is s, δ is any Makovian policy (note, the undelying chain is weakly communicating and unde aveage cost citeia an optimal policy exists), s n indicates the state in peiod n, a n indicates the action in peiod n, and (s n, a n ) is the (immediate) cost of taking action a n in state s n. We define the states unde taditional model as, S T = (I m, I, t m, t ) whee, I m is the numbe of lots on-hand at the manufactue at the end of the pevious peiod, o at the beginning of the cuent peiod. Since capacity in evey peiod is a non-negative intege multiple of Q, without loss of optimality, I m indicates a non-negative intege multiple of Q, I m = {0, 1, }. If I m = 2 fo instance, thee exists 2Q units in inventoy (see the discussion on action space). I is the net inventoy at the etaile at the end of the pevious peiod, I = {,, }. t m denotes the elative position of a peiod in the capacity cycle, t m {1,, T m }. We assume T m implies the following capacity stuctue, (K 1, 0, 0,, 0, K 2, 0, 0,, 0, K z, 0, 0,, 0) t denotes the elative position of a peiod in the eplenishment cycle, t {1,, T }. In the taditional model, the action is defined only by the poduction quantity in peiod n, p n. The quantity to be outsouced can aleady be infeed fom the etaile s ode quantity at the end of the eplenishment cycle. If the ode quantity exceeds the amount in stock and the poduction capacity of 11

12 the manufactue, then the emaining quantity should be outsouced. This implies outsoucing is not an independent decision. Note that outsoucing takes place only at the beginning of the eplenishment cycle, since othewise it will esult in additional holding cost. The etaile odes in multiples of Q, and capacity available is a multiple of Q, theefoe without loss of optimality, we limit the poduction quantity in evey peiod to multiples of Q (this implies I m is a multiple of Q). The action space in a peiod is denoted by p n {0, 1,, K n }, whee each value coesponds to the multiple of Q. We assume that single peiod demand is chaacteized by a discete pobability distibution. Next, we define the components of Equation (4) unde the taditional model. We define (s, a), whee s = (Im, n I n, t n m, t n ) and a = (p n ), as follows: ) (cp n + h(im n + p n L) + + w( Im n p n + L) + Q if t n = 1 (s, a) = ( ) cp n + h(im n + p n ) Q if t n 1 (5) whee L denotes the numbe of lots equested by the etaile at the end of eplenishment cycle, i.e., at the end of peiod T. The amount equested is dispatched by the manufactue in the fist peiod of the eplenishment cycle, and is eady at the etaile befoe the demand is ealized. Note that, the quantity L is deteministic and can be infeed fom I. Tansition pobability P(j s, a) denotes the pobability that next state is j given cuent state is s and action taken is a, whee j = (Im n+1, I n+1, t n+1 m, t n+1 ). We categoize all possible tansitions unde the taditional system as follows: Fo t n 1, I P P(j s, a) = 1 (I n I n+1 m n+1 = Im n + p n, ) if t n+1 m = t n m(1 1 {tm=t m}) + 1, t n+1 = t n (1 1 {t=t }) + 1 0, othewise whee 1 {tm=t m} takes value of 1 fo the last peiod of the capacity cycle. P 1 (I n I n+1 ) is the pobability that single peiod demand is I n I n+1. Fo t n = 1 thee ae two possibilities. The etaile s ode quantity does not exceed the available stock and poduction quantity, and theefoe outsoucing is not necessay. When this is the case, I n m+p n I n+1 m = L. Othewise, if outsoucing is necessay, then I n+1 m = 0. We pesent the tansition pobability as follows. ) I P P(j s, a) = 1 (I n I n+1 m n+1 = (Im n + p n L) + + LQ, if t n+1 m = t n m(1 1 {tm=t m}) + 1, t n+1 = t n + 1 0, othewise 12

13 3.2 Vendo Managed System Unde the vendo managed system, we focus only on the manufactue s poblem since the etaile does not make any decisions. Retaile only equies he pefomance measues to be as good as those unde the taditional system. At the beginning of the poduction cycle the manufactue decides on how much to poduce, and in evey peiod how much to outsouce and to dispatch. The dispatched quantity immediately aives at the etaile, i.e., lead time of tanspotation is zeo. Note that due to the ageement thee does not exist a eplenishment cycle. The sequence of events is as follows: 1. At the beginning of a peiod, the manufactue gives the decision of how many units to poduce (if possible), to outsouce and to dispatch. Inventoy status of the manufactue and the etaile ae updated based on the dispatch quantity. 2. The demand is ealized at the etaile s site. Inventoy status of the etaile is updated and endof-peiod holding costs at the manufactue and at the etaile ae incued. We analyze the vendo managed setting unde two cases; no-consignment stock and consignment stock No-consignment stock Unde no-consignment stock model (VM-NC) the owneship of the stock is tansfeed to the etaile once the dispatch aives at the etaile. To be compatible with the taditional system, we assume that unde the vendo managed system the etaile equies the aveage inventoy investment to be as low as, and aveage sevice level to be as high as those levels unde the taditional system. In othe wods, the etaile is indiffeent between the taditional and the vendo managed system. We detemine the manufactue s optimal opeating policy unde the no-consignment system. We model the manufactue s poblem as a Makov Decision Pocess unde the aveage-cost citeia as follows: g(s) = min δ lim N 1 N Eδ s [ N n=1 ] (s n, a n ) s.t. Aveage Inventoy Level at Retaile Ī (7) Sevice Level at Retaile 1 β (8) (6) whee β is defined as in (3), and Ī as in (1). The constaint on inventoy level in (7) eflects the case whee the etaile is not willing to pay fo inventoy investment moe than what it pays unde the taditional system. In pactice, the etaile may equie that at least one of the pefomance measues descibed by Equation (7) o (8) ae impoved as a esult of the ageement. Hence the sevice level specified by (1 β) 13

14 can be egaded as a lowe bound, and similaly the limit specified by Ī on the aveage inventoy level unde vendo managed system can be egaded as an uppe bound. In all ou analysis, ight-hand-side (RHS) of Equation (7) o (8) is used as is, so that we have compaable cases. Note that it is though these constaints that the availability is ensued at the etaile at the ight level of inventoy. If instead, the etaile wee to opeate with min-max bounds on inventoy, compaed to the taditional system the etaile s eithe the sevice level would be lowe o aveage inventoy level would be highe o both. Futhemoe, the manufactue s benefits would decease due to deceased opeational flexibility. The state is S NC = (I m, I, t m ) whee I m, I, and t m ae as defined in Table 1. At the beginning of the poduction cycle the manufactue decides on how much to poduce, p n, and in evey peiod how much to outsouce, y n, and to dispatch, d n. The manufactue poduces, outsouces and dispatches in multiples of Q, and capacity available at the beginning of the poduction cycle is a multiple of Q. The action space is denoted as, p n {0, 1,, K n } and d n {0, 1,, }. Without loss of optimality, we limit the action space to multiples of Q. Note that the outsouced quantity at peiod n, y n, is defined by (d n I n m p n ) +, and is not a(n independent) decision vaiable. Next, we define the components of Equations (6) though (8). In (6) we define (s, a), whee s = (I n m, I n, t n m) and a = (p n, d n ), as follows: (s, a) = ( c.p n + h(i n m + p n d n ) + + w( I n m p n + d n ) +) Q (9) Note that L in Equation (5) is now a decision vaiable and is denoted with d n. We define tansition pobabilities, P(j s, a) whee j = (Im n+1, I n+1, t n+1 ), as follows: P 1 (I n I n+1 + d n Im Q), if n+1 = (Im n + p n d n ) + t P(j s, a) = n+1 m = t n m(1 1 {tm=t m}) + 1, 0, othewise m Left-hand-side (LHS) of (7) eflects the expected inventoy level pe peiod at the etaile unde the manufactue s optimal opeating policy and is expessed as: I m,t m i>0 iπnc(i M m, I = i, t m ). whee π M NC (I m, I = i, t m ) is the faction of time spent (o the steady-state pobability) in state (I m, I = i, t m ) unde the manufactue s optimal opeating policy unde no-consignment system. LHS of (8) eflects the aveage sevice level at the etaile unde the manufactue s optimal opeating policy: 1 I m,i,t m d(i):i+d(i)>0 πnc(i M m, I = i, t m ) E[(D 1 i d(i)) + ] E[D 1 ] I m,i,t m d(i):i+d(i) 0 π M NC(I m, I = i, t m ) E[D 1] E[D 1 ]. 14

15 (10) In (10) d(i) denotes the set of possible dispatch actions that can be taken at state i. In the expession note that expected backodeed demand is calculated diffeently if i + d(i) 0. When i + d(i) > 0, the amount of available stock at the etaile befoe the demand is ealized is positive. Then the expected backodeed demand is E[(D 1 i d(i)) + ]. On the othe hand, if i + d(i) 0, then all demand occued in that peiod should be backodeed and expected backodeed demand is E[D 1 ]. Fo those peiods sevice level is 1 E[D 1] E[D 1 ] = 0. Aveaging ove all peiods gives the expession in (10). Finally, we note that if unde optimal dispatch policy the available stock at the etaile befoe the demand is ealized is always positive, then sevice level is always positive in all of the peiods. When this is the case, the sevice level expession in (10) can be eplaced with the following expession: I m,t m i<0 i E[D 1 ] πm NC(I m, I = i, t m ). (11) Consignment Stock In the consignment stock system (VM-C) the sequence of events is the same with no-consignment system except that the manufactue owns and manages the inventoy at the etaile s site. We detemine the manufactue s optimal opeating policy unde the consignment system. We model the manufactue s poblem as a Makov Decision Pocess unde aveage cost citeia as follows: g(s) = min δ lim N 1 N Eδ s [ N n=1 ] (s n, a n ) (12) s.t. Sevice Level at Retaile 1 β (13) Note that since the stocking cost is incued by the manufactue thee does not exist any constaint on the aveage inventoy level. Futhemoe, as we descibe below, the ewad function, (s, a), now includes the holding cost at both the manufactue and the etaile. Obseve that Equation (13) is same as Equation (8). In the consignment stock model, we assume that unit holding cost is the same at the manufactue s and the etaile s site. The caying chage of the inventoy at a site is detemined by the oppotunity cost and isk level at the site. Since stocks at both echelons belong to the same fim (manufactue), the oppotunity costs of the tied up capital that could be used in some othe investment is the same at both sites. Futhemoe, the isk levels at both sites ae the same, since the manufactue has a single etaile. If thee wee multiple etailes, the manufactue would pefe to keep stock at the uppe echelon to minimize the isks and send the items to lowe echelon only when necessay. Due to inceased isks, the implied unit holding cost at the lowe echelon would be highe. Howeve, in this single etaile setting 15

16 keeping the items at the lowe echelon athe than at both echelons does not affect the inventoy holding cost while impoving the sevice level. Since unit holding cost is the same at manufactue s and etaile s site, the manufactue keeps inventoy only at the etaile s site and as a esult immediately dispatches whateve it poduces and outsouces to the etaile s site. Unde consignment stock the state is defined as S C = (I, t m ) whee I and t m ae defined as befoe, and the actions ae only how much to dispatch at the beginning of peiod n, d n {0, 1,, }, whee each d n value coesponds to the multiple of Q. We define (s, a) whee s = (I n, t n m) and a = (d n ), as follows: (s, a) = ( ) c.min{d n, K n } + w. max{d n K n, 0} Q + he[(i n + d n Q D 1 ) + ]. (14) Note that in (s, a) the holding cost at the manufactue s site is not expessed, since stock is kept only at the etaile s site. The tansition pobabilities ae expessed as follows: { P1 (I P(j s, a) = n I n+1 + d n Q), if t n+1 m = t n m(1 1 {tm=t m}) + 1 0, othewise The consignment and no-consignment models ae diffeent but elated. Note that in the Makov Decision Pocess the ewad functions and the constaints ae diffeent (see Equations (6)-(9) and (12)-(14)). Howeve, the two systems ae elated in that thee ae paamete settings unde which the actions taken unde both systems ae the same. Note that Equation (7) in the no-consignment model implies a unit holding cost. If the implied holding cost is equal to the manufactue s holding cost, h, then consignment and no-consignment system can be egaded as equivalent in tems of the actions taken. Fo tighte o moe elaxed inventoy estictions consignment and no-consignment systems ae expected to esult in diffeent opeating policies. 4 Analysis and compaison of taditional and vendo managed systems In this pat, we fist povide an analysis on the stuctual popeties of the optimal policy unde taditional and no-consignment systems. Then we compae the cost unde no-consignment system with the cost unde taditional system and the cost unde consignment system. In the emainde of the text, we denote taditional system with TRAD, no-consignment vendo managed system with VM-NC, and consignment system with VM-C. 4.1 Stuctual popeties of the optimal policy We analyze the stuctual popeties of the optimal policy unde taditional and unde no-consignment systems. We show in Popety 1 that the optimal policy unde the taditional system is a modified 16

17 base-stock policy. Fo the no-consignment system, we discuss how the etaile s optimal policy and the esultant inventoy level and sevice level constaints affect the manufactue s optimal policy. In Popety 2 we show that unde cetain conditions, the optimal policy unde no-consignment system is also a modified base-stock policy. Popety 1 Optimal policy of the manufactue unde taditional system is a modified base-stock policy. Next, we discuss optimal opeating policy of the manufactue unde VM-NC. Unde VM-NC the manufactue decides on how much to poduce, outsouce and dispatch to the etaile s site. The dispatch policy is subject to the following two constaints: (i) Expected inventoy level at the etaile can not exceed a cetain level (as expessed in (7)), and (ii) Sevice level at the etaile should satisfy a minimum level (as expessed in (8)). These constaints make it difficult to chaacteize the optimal opeating policy of the manufactue. Howeve, as we show in Popety 2 unde cetain conditions the optimal policy of the manufactue has the athe simple base-stock stuctue. To define the manufactue s policy unde VM-NC, we should focus on the etaile s opeating policy unde the taditional system. In Poposition 1 we show that when T = 1, fo β S thee exists a unique optimal policy which is (R(β), nq). This esult leads to the following obsevation. Obsevation 1 Fo T = 1 and β S: (i) Unde VM-NC the manufactue s optimal opeating policy is defined by a unique dispatch policy. This unique dispatch policy is the same policy as the etaile s ode policy unde the taditional system, which is (R(β), nq). (ii) Manufactue s optimal opeating policy unde VM-NC is independent of β. Obsevation 1 states that fo β S, unde optimality the only possible dispatch policy of the manufactue that satisfies constaints (7) and (8) is the etaile s (R(β), nq) policy. In othe wods, in evey peiod the manufactue dispatches the minimum amount (in multiples of Q) to bing the etaile s stock level above R(β). Futhemoe, the dispatch policy is the same fo all β S. The eason is, discete eode points define β and the value of the eode point does not have an impact on the dispatch policy of the manufactue (This stuctue esembles the one in a base-stock system whee the ode-up-to point does not affect the quantity odeed evey peiod). Fo β S, multiple dispatch policies may satisfy the constaints (7) and (8). Unde optimality the manufactue may select one of the eligible dispatch policies. Using Obsevation 1, in Popety 2 we povide a chaacteization of the optimal policy of the manufactue. Popety 2 Fo T = 1 and β S, unde VM-NC the manufactue s optimal policy is a modified base-stock policy. 17

18 4.2 Compaison of taditional and vendo managed systems In this section we make two compaisons. Fist, we compae the no-consignment system with the taditional system. Using the stuctual esults of the pevious subsection, we show that the cost unde the no-consignment system is always lowe than o equal to the cost unde the taditional system (Popety 3). We then compae the no-consignment system with the consignment system. In Poposition 4 we show that unde cetain settings and unde cetain sufficient conditions the cost of the consignment system is lowe than the cost of the no-consignment system Compaison of no-consignment and taditional systems Popety 3 The cost of the manufactue unde VM-NC is always lowe than o equal to the cost unde TRAD. Popety 3 states that if unde vendo managed system stock is not consigned, then vendo managed system esults in lowe cost than the taditional system, i.e., VM-NC is a no-isk case fo the manufactue Compaison of no-consignment and consignment systems Although VM-NC is a no-isk case, the cost unde VM-NC is not always lowe than the cost unde VM-C. As we show in the analysis below, unde vendo managed system consigning the stock may be less costly than not consigning it. In the following, we intoduce a specific instance. Fo this instance, we fist obtain a lowe bound on the optimal cost unde VM-NC (Poposition 2), and an uppe bound on the optimal cost unde VM-C (Poposition 3). We then identify a set of sufficient conditions that make VM-C less costly than VM-NC (Poposition 4). Assume that Q = 1 and that the (single-peiod) demand, D, has the following pobability distibution: { 1/2, if D = µ 1 P(D) = 1/2, if D = µ + 1 Assume that capacity pe peiod is E[D] = µ, and T = 1, i.e., unde the taditional system etaile places odes in evey peiod. In the analysis below, we focus on the cases whee β S. In this setting since Q = 1, unde the taditional system the etaile opeates unde the base-stock policy. Poposition 2 Fo the instance defined above, LB(VM-NC) = (w c)h h 2 on the manufactue s optimal cost unde VM-NC. + cµ is a lowe bound In the following, we detemine an uppe bound on the optimal cost of the manufactue unde VM-C (Poposition 3). Unde VM-C the manufactue dispatches whateve he poduces and outsouces, and 18

19 the poblem unde consideation is how much to poduce and outsouce evey peiod whee the decisions ae subject to the sevice level constaint. Below we popose two uppe bounds on the optimal cost unde VM-C. Poposition 3 Fo the instance unde consideation, w c (i) Suppose w 3h+c, and w is such that h + 1 Z+. Then UB(VM-C) = h( is an uppe bound on the optimal aveage cost unde VM-C. k 2 k+1 w c h )+cµ (ii) Suppose fo k 1 and k Z +, 1 β 1 q 2µ w c, w (5k + 3)h + c and w is such that h +(k2 k+1) ( ) w c h + (k2 k + 1) Z + w c. Then UB(VM-C) = h h + (k2 k + 1) (k ) + cµ is an uppe bound on the optimal aveage cost unde VM-C. Poposition 3 suggests two uppe bounds fo the optimal cost unde VM-C. Pat(i) implies moe elaxed sufficient conditions fo the uppe bound, and does not equie any condition on the sevice level. When the sevice level is as high as 100%, the uppe bound in pat(i) is applicable. The uppe bound in pat(ii) equies tighte sufficient conditions, and in etun gives a tighte uppe bound. Note that U B(VM-C) in pat (ii) is deceasing in the paamete k. Paamete k denotes how low inventoy level can be set at the etaile. As the sevice level equiement is lowe (i.e., as β gets highe) k inceases, and U B(VM-C) deceases. Using Poposition 2 and Poposition 3, in Poposition 4 we pesent the main esult of this subsection. Poposition 4 Suppose the following conditions ae satisfied: (i) β q k 2 k+1, 2µ w c h +(k2 k+1) (ii) w > (5k + 3)h + c, whee k 2 and optimal cost unde VM-NC. w c h + (k2 k + 1), k Z +. Then the optimal cost unde VM-C is lowe than the Poposition 4 compaes the no-consignment and consignment models unde a deteministic eode point at the etaile. In pactice, fims pefe a fixed opeating policy athe than a andomized one due to opeational difficulties, even if a andomized policy may yield lowe costs. The fist condition in the poposition states that if sevice level equiement at the etaile is not high, then consignment stock is pefeed. This esult is in line with ou expeimental study whee we obseved that unde 99% sevice level consignment stock is neve pefeed (see Section 5.1). The intuition behind this esult is as follows. If the sevice level equiement of the etaile is low then this implies the expected inventoy level equiement at the etaile is also low (i.e., RHS of the constaint in (7)). This coesponds to a high implied unit holding cost fo the stock at the etaile s site. If the implied cost is vey high (i.e., if 19

20 expected inventoy level is vey low) then the manufactue simply pefes owning the stock athe than tying to meet the equiement unde no-consignment. In pactice, fo items with low implicit stock-out costs, the etaile may allow low sevice levels. Examples ae the items fo which the etaile also cay the substitutes, o poducts that ae not competitive. Fo these items inventoy equiement imposed by the etaile to the manufactue would be low, and the manufactue might pefe consigning the stock to no-consignment. The second condition states that if outsoucing cost is high, then consigning the stock is pefeed. This esult also suppots ou obsevations fom the computational study. Unde high outsoucing cost the manufactue would pefe to keep high levels of inventoy, which is allowed unde the consignment stock model but not unde no-consignment model. Note that ou constuction assumes T = 1 and β S. Unde these assumptions the cost and opeating policies unde TRAD and unde VM-NC ae the same. Theefoe the intuition obtained fom Poposition 4 could be extended to the compaison of the consignment system with the taditional system. We conclude that the manufactue pefes VM-C to TRAD when the inventoy level constaint is tight, i.e., when the opeating policy of the etaile imposes an inflexible dispatch policy fo the manufactue. 5 Computational Analysis We conduct expeiments to analyze how the system paametes affect the manufactue s savings unde the vendo managed system and identify the conditions unde which manufactue is willing to make an ageement. In designing the expeiments we keep unit holding cost, unit poduction cost, expected demand pe peiod as constant at h = 1, c = 10, and E[D 1 ] = 20. We assume the lot size is, Q = 5. We assume that the capacity cycle is two peiods T m = 2, poduction cycle is one peiod, T p = 1, and eplenishment cycle, T, can be one o two peiods. Capacity levels in the capacity cycle ae indicated with K 1 and K 2. We conside the effect of the following paametes on aveage cost pe peiod: 1. Total capacity. We assumed the capacity levels ae tight, medium, o excessive. Unde tight capacity K = K 1+K 2 2 = E[D 1 ] = 20, unde medium capacity K = 25, and unde excessive capacity K = Outsoucing cost. w = 11, 15, 20, and Capacity non-stationaity. (K 1, K 2 ) =(40,0), (30,10), (20,20), (10,30), and (0,40). 4. Replenishment cycle, T = 1, 2. When T = 2, unde the taditional system the etaile places odes in evey two peiods, wheeas shaes the demand and inventoy level infomation in evey peiod. Compaing the taditional system unde T = 2 with the vendo managed system, 20

21 the manufactue has a gain in tems of both dispatch quantity and dispatch time (i.e., dispatch fequency) flexibility. In we quantify the benefit of flexibility. 5. Demand coefficient of vaiation. The demand faced by the etaile is modeled via a discete distibution. The distibutions consideed and the coesponding values of the coefficient of vaiation, cv, ae as follows: Unifom [11, 29] (cv = 0.28), Tuncated Nomal (µ = 20, σ = 30) (cv = 0.57), Beta (0.3, 0.3) (cv = 0.80). 6. Sevice level at the etaile. We assumed sevice levels ae 90%, 95%, 99%. When detemining the sevice level at the etaile, we only conside discete eode points, and we set the eode point such that the sevice level is highe than 90% (o 95%, o 99%). Fo example, fo unifom distibuted demand, when T = 1 the etaile s eode point that gives a sevice level of at least 90% is R = 18 and the sevice level implied by this eode point is 90.26%. We pesent the sevice levels in Table 2. Table 2: Sevice Levels T=1 T=2 Unifom Nomal Beta Unifom Nomal Beta 90% 90.26% 91.06% 90.41% 90.66% 90.72% 90.40% 95% 95.79% 95.90% 95.17% 95.18% 95.49% 95.42% 99% 99.47% 99.25% 99.41% 99.11% 99.12% 99.22% In this section we obseve the effect of system paametes on the benefits of vendo managed system. We have aleady shown that no-consignment system is a no-isk case fo the manufactue, and fo T = 1, cost unde TRAD and VM-NC ae the same, so when making the obsevations we only compae TRAD and VM-C, unless othewise stated. Howeve, fo cetain cases, when we believe that compaison with VM-NC povides futhe insights we explicitly state this in the discussions. In the following subsections, we pesent ou esults unde two main titles: Analysis unde Stationay Capacity and Analysis unde Non-Stationay Capacity. We use linea pogamming (LP) model to solve (4) and (12) and use aveage cost pe peiod citeia fo the analysis. When LP is used to solve the coesponding MDP poblem unde VM-C we obtain at most a single andomized action as we have one additional constaint (coesponding to sevice level). We use the esults as is. Numbe of vaiables of the LP model (catesian space of states and actions of the MDP) ae 55,000 fo the taditional system, and 4,060 fo the VM-C system. Total numbe of expeiments caied out fo the stationay capacity case is 216, and fo the non-stationay case 288. Due to computational buden that Equation (10) bings, we use the expession in (11) as a suogate of the sevice level at the etaile. Note that (10) and (11) 21

22 ae equivalent if etaile s beginning inventoy level in evey peiod is non-negative. In ou expeimental setting unde vendo managed system we expect this to be the case, and believe that using (11) instead of (10) has a negligible effect in the esults. 5.1 Analysis unde Stationay Capacity We fist make a compaison of the vendo managed system with the taditional system unde stationay capacity, when capacity pe peiod is tight ( K = 20), medium ( K = 25), and excessive ( K = 30). Unde stationay capacity, the capacity level ove the peiods is constant: (K 1, K 2 ) = (20, 20), (25, 25), and (30, 30). Based on the insights obtained in this section, we extend the analysis to the non-stationay capacity Effect of unit outsoucing cost on savings We analyze the effect of outsoucing cost (w) on % savings unde the vendo managed system (= TRADcost V MScost TRADCost.100) fo sevice levels 90%, 95%, and 99% (We efe to the sevice levels in Table 2, while we indicate the levels with 90%, 95%, and 99%). As unit outsoucing cost inceases, unde both taditional and vendo managed system, the aveage inventoy level at the manufactue inceases while numbe of units outsouced deceases. Howeve, although the numbe of outsouced units deceases, the total outsoucing cost inceases unde both systems. We obseve that the incease in inventoy level and decease in numbe of outsouced units is moe dastic unde the taditional system compaed to the vendo managed system. As a esult of this, the numbe of units poduced in-house inceases significantly unde taditional system. We conclude that, the taditional system is less obust to the changes in unit outsoucing cost compaed to the vendo managed system. As a esult of the changes the poduction cost, total outcoucing cost and inventoy holding cost, the cost unde TRAD inceases at a steepe ate than the cost unde VM-C. We obseve that the savings unde VM-C incease with unit ousoucing cost. Expeimental esults fo T = 1 suppot the conclusions deived in Poposition 4: we obseve that when sevice level is 99% and demand coefficient of vaiance is high, cost unde VM-C is always highe than the cost unde TRAD o VM-NC. Othewise, cost unde VM-C can be lowe especially if the outsoucing cost is high. Fo T = 2, the manufactue keeps highe inventoy compaed to T = 1 unde taditional system, and theefoe VM-C can be moe beneficial. Finally, as the demand coefficient of vaiance inceases we obseve that manufactue s savings unde VM-NC incease while savings unde VM-C deceases. We conclude that although highe coefficient of vaiation helps manufactue to manage the opeations moe effectively, incuing the etaile s inventoy holding cost outweighs the savings. In the oveall setting, savings unde VM-C can be as high as 5.37% (when outsoucing cost is high, 22

23 demand coefficient of vaiance is low, and sevice level equiement is low) and as low as 9.80% Effect of the vendo managed system on capacity utilization We analyze how the capacity utilization change as system moves fom taditional to vendo managed system. Capacity utilization is a measue of manufactue s ability to meet the demand though in-house poduction. The unmet demand is outsouced and in this espect outsoucing cost functions as a lost sales penalty, and capacity utilization eflects the sevice level povided by the manufactue. Analysis indicates that capacity utilization is always highe unde vendo managed system than the taditional system (see Figue 2). Also as unit outsoucing cost incease the capacity utilization incease. The capacity utilization inceases at a highe ate unde the taditional system as the unit outsoucing cost incease. Pecentage utilization 100% 90% 80% TRAD VMC 70% 60% Outsoucing cost Figue 2: Effect of outsoucing cost on utilization when total capacity is tight. We also analyze the effect of capacity level on the cost unde TRAD and VM-C. An incease in capacity level fom tight to medium o medium to excessive, deceases the cost unde both taditional and vendo managed systems. We obseve that how the two systems eact to an incease in capacity level slightly diffes with espect to the coefficient of vaiation in demand: 1. When the coefficient of vaiation of demand is low, unde vendo managed system inventoy buden on the manufactue is low. The manufactue aleady uses the capacity effectively, and theefoe the benefit of additional capacity is elatively low. Beyond a cetain theshold, the incease in capacity does not decease the cost fo vendo managed system. On the othe hand, unde taditional system, the additional capacity is moe beneficial, since additional capacity will help the manufactue to meet the etaile s odes moe effectively. Fo sufficiently high capacity, we expect that unde taditional system in-house poduction will be equal to the demand, and no outsoucing cost o 23

24 holding cost will be incued. This implies unde sufficiently high capacity, cost unde taditional system will be lowe than the cost unde vendo managed system, since unde VM-C thee always be the buden of inventoy holding due to the etaile. Theefoe when coefficient of vaiation is low, as the capacity level inceases, benefit of vendo managed inventoy deceases. Figue 3(a) shows the costs unde two systems when coefficient of vaiation is low. 2. When coefficient of vaiation of demand is high, we obseve that both vendo managed system and taditional system benefit an incease in the capacity level. When capacity level is excessively high, both systems each to a stability in tems of cost and the cost does not decease futhe with an incease in capacity level. We obseve that unde sufficiently high capacity the aveage cost unde taditional system can be as low as the total in-house poduction cost (which is expessed as p.e[d] and which is the lowest level fo the cost), wheeas unde VM-C the cost consists of the in-house poduction cost and the inventoy cost at the etaile. We conclude that, unde sufficiently high capacity, vendo managed system is not beneficial. In Figue 3(b) we show how the costs change with espect to the capacity level when coefficient of vaiation is high. The amount of pe peiod capacity necessay to attain 100% in-house poduction is highe unde high coefficient of vaiation of demand compaed to the case whee coefficient of vaiation is low.!"#$#% *+,+-./0,12 &'()"#$#%,12.34,12.34 *+,+- Figue 3: Effect of capacity incease costs unde taditional and vendo managed system Effect of vendo managed system on inventoy levels We compae the expected total inventoy level in the system (at the manufactue and the etaile) and the expected inventoy level at the etaile s site unde taditional system and the vendo managed system. Expeimental esults show that the expected total inventoy level in the system is lowe unde VM-C. 24

25 Popety 4 Expected inventoy level at the etaile s site: (i) is highe unde VM-C compaed to the taditional system when T = 1, (ii) may o may not be highe unde VM-C compaed to the taditional system when T = 2. The inventoy level at the etaile s site may o may not be lowe unde the vendo managed system depending on how the etaile opeates unde the taditional system. If etaile aleady equies small and fequent eplenishments unde the taditional system (i.e., if T = 1), then inventoy at etaile s site inceases unde the vendo managed system. The eason is, unde taditional system the etaile opeates with the minimum inventoy level fo a given sevice level, since (R, nq) is the optimal opeating policy (see Poposition 1). When the manufactue manages the etaile s inventoy, due to capacity estictions at the manufactue, the vendo managed system coesponds to a constained system compaed to the taditional system. Theefoe expected inventoy level at the etaile is highe. If unde taditional system, the etaile places infequent and lumpy odes (i.e., when T = 2), then unde high outsoucing cost and low sevice levels the inventoy level at etaile s site is highe unde VM-C. Unde high sevice levels the inventoy level at etaile s site is lowe unde VM-C. We would like to note that the esults may vay if we elax the assumption on same unit holding cost at both echelons unde VM-C. In that case unde both consignment and no-consignment stock systems, inventoy would be held at both sites and the inventoy kept at the etaile s site would be lowe. Howeve, thee will still be instances unde low sevice levels and high outsoucing cost whee the inventoy level at etaile s site is highe unde VM-C. Unde low outsoucing cost, we obseve that inventoy level at the etaile is always lowe unde VM-C compaed to the inventoy level unde the taditional system Quantifying dispatch time and dispatch quantity flexibility Unde the vendo managed system the manufactue decides on how much to dispatch in evey peiod depending on the capacity level and demand. In some peiods the manufactue may not make any dispatches, othe times may pefe moe fequent dispatches. The vendo managed system implies a gain in dispatch time and dispatch quantity flexibility compaed to the taditional system. In the following we quantify the benefits due to dispatch quantity flexibility and due to both dispatch time and dispatch quantity flexibility: Measuing dispatch quantity flexibility. To analyze benefits due to dispatch quantity flexibility only, we compae the following two settings: T = 2 unde taditional system, and T = 2 unde no-consignment vendo managed system. We estict the dispatch time unde vendo managed system to one dispatch in two peiods. This implies compaed to the taditional system thee does not exist an incease in dispatch time flexibility unde vendo managed system, but only an incease in dispatch quantity flexibility. (To obtain optimal cost of manufactue unde VM-NC with T = 2 estiction we use the constaint in (10)). 25

26 Measuing dispatch time and quantity flexibility. To analyze benefits unde joint dispatch time and quantity flexibility, we elax the estiction on the dispatch time unde vendo managed system, we simply compae the taditional system (with T = 2) with VM-NC (To obtain optimal cost of manufactue unde VM-NC with no dispatch time estiction we use the constaint in (10)). Note that when quantifying the benefits of flexibility, we assume inventoy is not consigned, to detemine the benefits of flexibility. Pecentage savings due to quantity and time flexibility 12% 10% 8% 6% 4% 2% 0% QTY TIMEQTY Outsoucing cost Figue 4: Effect of dispatch time and dispatch quantity flexibility on savings Figue 4 shows that savings due to dispatch quantity flexibility ae high especially when unit outsoucing cost is high (fo this analysis we assumed capacity is tight, K = 20, and demand has a simplified stuctue such that demand pe peiod is eithe 15 o 25 each with pobability 1 2 ). Analysis show that, while dispatch quantity flexibility may o may not decease the inventoy level at the manufactue, the additional flexibility due to dispatch fequency deceases the inventoy level in the system and significantly inceases the savings. Finally, we note that unde vendo managed system dispatch time flexibility not necessaily implies moe fequent shipments. Analysis show that depending on capacity estictions, unde VM-C the manufactue may pefe less fequent shipments compaed to the taditional system which may esult in lowe inventoy cost and lowe total cost compaed to taditional system. 5.2 Analysis unde Non-Stationay Capacity We now analyze how the costs diffe unde taditional and the vendo managed system as the capacity levels change thoughout the peiods. When eplenishment cycle is one peiod, unde both taditional and vendo managed system lowest cost is incued when capacity is stationay at (K 1, K 2 ) = (20, 20) (Figue 5.a). This is expected since the end-demand is stationay. In the two-peiod eplenishment cycle, unde taditional system as moe capacity is allocated close to the eplenishment point (which is the fist peiod of the eplenishment cycle), total cost deceases. This is because the manufactue can use the end-demand infomation available in the pevious peiod and eact accodingly in the eplenishment 26

27 597 :;<= Cost pe peiod BCDEDFBGDEHDFBIDEIDFBHDEGDFBDECDF peiod (Figue 5.b) Cost pe peiod BCDEDFBGDEHDFBIDEIDFBHDEGDFBDECDF (a) One-peiod eplenishment cycle (b) Two-peiod eplenishment cycle Figue 5: Effect of capacity non-stationaity on savings The analysis unde non-stationay capacity povides insights on how a manufactue should allocate the capacity unde taditional system vesus unde the vendo managed system (of couse if possible; i.e., if the manufactue has the flexibility in shifting its capacity fom one peiod to anothe). Unde taditional system the manufactue would schedule the odes fom diffeent customes so that bulk of poduction fo a cetain custome can be ealized as the eplenishment time fo that custome appoaches. This may esult in eatic odes placed by the manufactue to the uppe echelons. On the othe hand, unde the vendo managed system the manufactue pefes smoothing out the poduction and dispatch pocess by allocating the same capacity in evey peiod. Allocation capacity unifomly would esult in much less fluctuation in the odes placed by the manufactue, theefoe the vendo managed system would also benefit the playes in the uppe-echelons of the supply chain. Note that Lee, Padmanabhan and Whang (1997) specifies a simila manufactuing situation to show the bullwhip effect. In ou case, we show that capacity management is a useful tool to educe the bullwhip effect. The analysis povides futhe insights on the type of settings that a manufactue should o should not pefe the vendo managed system. If the odes fom etaile ae staggeed, fequent odes with small lot sizes (in ou case, when eplenishment cycle is one peiod), then managing the etaile s inventoy would not bing much benefit to the manufactue, and the cost of consigning the stock may outweigh the benefits (Figue 5.a). On the othe hand, if the etaile place the odes infequently and in lage lot sizes (in ou case, when eplenishment cycle is two peiods) then the manufactue may o may not benefit managing the etaile s inventoy depending on the flexibility in its opeations, explained as follows. When the manufactue does not have the flexibility in shifting the capacity, (if, fo instance, the manufactue has customes with stict delivey time equiements) expeimental esults indicate that savings ae high (aound 1.9% unde consignment system and 11.6% unde no-consignment system). The inflexible system is modeled with (K 1, K 2 ) = (0, 40) unde the vendo managed system and unde the taditional system. 27

28 These figues imply that when the manufactue does not have much contol ove allocating the capacity to espond to the odes effectively, dispatch quantity and dispatch time flexibility ae most useful. Howeve, if the manufactue can shift the capacity to eact to the demand pattens, then savings unde the vendo managed system ae athe limited. This is because it is aleady possible to opeate the taditional system effectively. Analysis yield that on the aveage savings unde consignment system ae 5.1%, and savings unde no-consignment system ae 4.8% (hee best pefomances ae compaed, i.e., aveaged cost unde (40, 0) fo the taditional system is compaed with the cost unde (20, 20) fo the vendo managed systems). Fully consigning inventoy may not be a pefeable option if manufactue has sufficient flexibility in shifting its capacity. 5.3 Manageial Insights In this section we pesent the highlights of ou analysis. Seveal of the insights we obtained fom the study suppot and build up on the pevious findings in the liteatue, while othes eveal new infomation. Insight 1 The benefits that the vendo managed system will bing to the manufactue depends on the type of the vendo managed system elation. Thee may be benefits beyond shaing demand and inventoy infomation. Howeve, thee ae a numbe of cases whee VM-C does not yield any additional savings ove infomation shaing, as well. Hence, infomation shaing should be consideed as a fist step in the elation with a etaile befoe going into isky the vendo managed elations. Howeve, the next level of elationship should not necessaily follow; equies caeful evaluation of tade-offs. Insight 1 complements the findings in the liteatue by assessing the benefit of vendo-managed system fom the manufactue s pespective. Fy, Kapuscinski and Olsen (2001) quantify the effect of VMI on the system-wide cost given that infomation is aleady shaed, and conclude that if cetain paametes (such as dispatch quantity o penalty of violating inventoy bounds) ae not chosen popely, vendo managed setting can be moe costly fo the chain than the taditional setting. We show that if conditions o the tems of the ageement ae tight fo the manufactue, then vendo-managed system does not bing additional benefit to infomation shaing. Insight 2 VM-NC constitutes a no-isk case fo the manufactue. Howeve, thee can be cases whee VM-C is moe pofitable than the VM-NC fo the manufactue. We explicitly compae the no-consignment stock system with the consignment stock system, and conclude that if the sevice level equiement at the etaile and the coefficient of vaiance fo demand ae low, and as a esult inventoy level has to be tightly kept at a low level, then consigning the stock might be 28

29 less costly fo the manufactue. In the liteatue eithe totally consigned stock o no-consignment stock models ae studied and the question of whethe the stock should be consigned is unaddessed. Insight 3 If capacity allocated is sufficiently high, then the manufactue is less likely to benefit fom the vendo managed system. Insight 4 Manufactues usually have the pactice of allocating capacity fo a poduct o custome. It tuns out that the way to allocate this level optimally is not vey staightfowad; whethe the system opeates with full infomation only, o unde a cetain type of the vendo managed system may lead to diffeent capacity allocation schemes, significantly affecting the pefomance. If the manufactue can not easily change the capacity allocation, i.e., if opeating in a igid system, then it is most likely to benefit the vendo managed system. We analyze the inteaction of capacity management and the vendo managed system. Gavineni, Kapuscinski and Tayu (1999) quantify the benefit of infomation shaing in a supply chain and show that as capacity level inceases, the cost savings at manufactue inceases (with diminishing etuns). We obseve that the savings of the manufactue due to vendo managed system deceases with the incease in capacity and thee exist capacity levels whee taditional system is less costly. We also analyze the impact of the opeating stategy on the capacity allocation decisions. Unde vendo-managed system, the manufactue pefes smoothing the poduction decisions by allocating equal capacity in each peiod, wheeas unde taditional system if odes placed by the etaile ae lumpy and intemittent, then capacity allocation is unbalanced and poduction amounts may fluctuate. Lee, Padmanabhan and Whang (1997) specify a simila manufactuing situation to show that lumpy odes incease the bullwhip effect. We complement the study by showing that vendo-managed system benefits the uppe echelon though smooth poduction pattens. Ou finding is also in line with Disney and Towill (2003). Ou analysis on capacity management contibutes to the liteatue by connecting the benefit of vendo managed system to capacity management decisions. Insight 5 The vendo managed system may o may not decease the inventoy level at the etaile. Insight 6 Main benefits of the vendo managed system can be descibed with dispatch time flexibility and dispatch quantity flexibility, with the fome potentially leading to eduction in inventoy investment in the system. Reduction in inventoy investment may occu even if dispatches ae less fequent. Finally, we quantify the benefit of vendo managed system in tems of dispatch quantity flexibility and dispatch time flexibility. Cetinkaya and Lee (2000) contasts a taditional system with fequent shipments 29

30 with a vendo-managed system whee shipments ae consolidated at the expense of inceased inventoy levels. Walle, Johnson and Davis (1999) shows though a simulation study that vendo-managed inventoy may incease the fequency of dispatches to etailes which helps decease the inventoy level. We show that dispatch time flexibility may contibute significantly to the eduction of inventoy in the system, and this is not necessaily achieved though inceased fequency of the shipments. 6 Conclusions In this study we analyze a vendo managed system fo a supply chain consisting of a single manufactue and a single etaile. We model the manufactue effectively, so that benefits going unde a vendo managed ageement can be analyzed. We assume that etaile demand infomation is fully available to the manufactue and hence only study the benefits beyond infomation shaing. We study both the consignment stock and no-consignment stock systems unde the vendo managed system. We conside the capacity limitation of the manufactue in ou analysis, which tuned out to be a vey impotant facto fo the manufactue and analyze the poblem unde diffeent capacity allocation schemes to identify the effect of capacity management on benefits. Ou main findings ae the vendo managed system indeed bings benefit to the manufactue beyond infomation shaing. The benefits ae high especially unde modeate o tight poduction capacity athe than excessive capacity, o unde low sevice level equiements. Analyses indicate that if the inventoy and sevice level equiements ae too tight, athe than confoming with the specifications, owning the inventoy might be less costly fo the manufactue. Unde the vendo managed system manufactue can take a poactive appoach in esponding to etaile s demand and thus can incease the capacity utilization. The vendo managed system povides the manufactue with both dispatch time and dispatch quantity flexibility, and this flexibility is most valuable when unde taditional system etaile equests iegula/lage shipments athe than small and fequent shipments. We also compae the inventoy levels unde taditional and vendo managed systems. Total inventoy level in the system is lowe, howeve inventoy level at the etaile may not be lowe. We analyze the effect of end demand vaiability on the savings. Highe vaiability esults in highe savings, but the savings ae outweighed by the inventoy holding costs unde consignment stock model. Finally, we obseve the savings unde vaying capacity allocation schemes. Unde the vendo managed system the manufactue pefes unifomly allocated capacity, thus helps educing the bullwhip effect in the total chain, wheeas unde taditional system allocates most of the capacity towads the time of the eplenishment. We conclude that consigning inventoy unde the vendo managed system may not be a pefeable option if manufactue has sufficient flexibility in allocating the esouces. If manufactue has limited o no flexibility, then the vendo managed system povides the highest benefit. 30

31 In the vendo managed system, we limit the analysis to the two exteme cases, totally consigned stock vesus no-consignment stock. Between the two exteme cases, in geneal eithe the cost of stock at the etaile might be shaed by the manufactue and the etaile, o the owneship of inventoy can be tansfeed fom the manufactue to the etaile afte some time peiod between 0 and sales time. Futue wok includes analysis of a moe geneal owneship model. Acknowledgements. Secil Savasaneil acknowledges the eseach gant povided by Middle East Technical Univesity, BAP The authos thank the depatment edito, the anonymous associate edito and two efeees fo thei valuable comments, which helped to impove the pape. 7 Appendix 7.1 Poof of Poposition 1 Befoe stating the poof, we fomally state the etaile s poblem of minimizing aveage inventoy level as a Makov Decision Pocess unde aveage-cost citeia and pesent the coesponding linea pogamming fomulation as follows. (P) min iπi,a R (15) i>0 s.t. πj,a R πi,ap R j i,a = 0 j (16) a i,a πi,a R = 1 (17) i,a i<0 i πr i,a E[D 1 ] πi,a R R + β (18) In (P), i is the end-of-peiod inventoy level at the etaile πi,a R is steady-state pobability that inventoy level is i at the etaile a actions ae (quantized) ode quantities 1 β is the equied sevice level P j i,a is the tansition pobability of being in state j in the next peiod given that cuent state is i and action is a. Note that P j i,a = P 1 (i + aq j), whee P 1 is defined as in Table 1. The objective function denotes the expected aveage end-of-peiod inventoy level in steady-state. Equation (16) peseves the flow balance and Equation (17) ensues that sum of the steady-state pobabilities do not exceed 1. Finally equation (18) ensues that the equied sevice level is satisfied. 31

32 We now pesent the poof. (i) We show that fo β S thee is a unique solution to (P) which is an (R, nq) policy with eode paamete R(β). Conside the lagange elaxation of (P): L(λ) = min iπ i,a + λ i πi,a R λβe[d 1 ] i>0 i<0 s.t. πj,a R πi,ap R j i,a = 0 j a i,a πi,a R = 1 i,a π R i,a R + whee λ > 0 is the lagange multiplie. Note that L(λ) is equivalent to the peiodic-eview stochastic dynamic inventoy poblem with batch odeing whee oveage cost is linea with ate h = 1 and undeage cost is linea with ate λ. Fo any λ (0, ) optimal solution fo L(λ) is an (R, nq) policy (Veinott 1965). It is easy to see that λ, namely λ β, fo which (R(β), nq) is the optimal inventoy policy. Note that by definition, the value of the lagangean elaxation poblem (fo any λ) is a lowe bound on the optimal value of the main poblem (P). Since the optimal solution of L(λ β ) makes (18) an equality and is a feasible solution fo (P), we conclude that λ β is the optimal lagange multiplie fo L(λ) and (R(β), nq) is the optimal solution fo (P). (Note that fo L(λ), λ β may not be unique.) Futhemoe, (R(β), nq) is unique, since any othe policy (i.e., if eode point is R(β) + 1 o R(β) 1) would eithe violate the sevice level constaint in (18) o incease the objective function value in (15). (ii) Though an example, we show that fo β S, thee may be moe than one solution to (P). Example 1. Let Q = 1. Unde this assumption (R, nq) policy is a base-stock policy with odeup-to level R + 1. Fo some β S, conside the optimal odeing policy. Suppose this policy could be a andomized o a deteministic policy. A policy simply states a set of ode-up-to levels and the pecentage of the time each ode-up-to level is eached. Theefoe any policy can be expessed as a convex combination of seveal ode-up-to levels. Conside the following instance. Let (single-peiod) demand take values 1, 2, o 3 with pobability 1/3. The optimal policy unde β = 0.1 is a andomized policy with the following ode-up-to levels: at states (i.e., beginning inventoy level) -1 and 1 ode up to 2; at state 0, 1/5 of the time ode up to 2, 4/5 of the time ode up to 3; at state 2, ode up to 3. Steady-state pobabilities unde optimal policy ae π 1 = 6 30, π 0 = π 1 = 10 30, π 2 = Theefoe the optimal policy implies the following: 3/5 of the time ode up to 2, 2/5 of the time ode up to 3. Thee may be othe policies that coesponds to the same scheme. Fo example, conside the policy: At all states 3/5 of the time ode up to 2, and 2/5 of the time ode up to 3. This policy also yields β = 0.1 and minimizes the inventoy level. We conclude 32

33 that when β S, thee may be multiple optimal policies which ae andomized. Note that none of these optimal policies can be a non-andomized (R, nq) policy since in that case β S. 7.2 Poof of Popety 1 Unde the taditional system, the etaile places odes to the manufactue at the end of each eplenishment cycle. At the beginning of evey peiod (befoe dispatch) the manufactue obseves the etaile s inventoy level, I. If it is the fist peiod of the eplenishment cycle, the manufactue dispatches the quantity equied by the etaile, that is, dispatches the minimum amount that will bing the inventoy level at the etaile above R. The dispatch quantity is simply the demand of the manufactue fo the fist peiod of the eplenishment cycle, which is known due to I. Fo the othe peiods demand is zeo. Since the dispatch quantity is aleady implied by I and thee does not exist any uncetainty. This system is equivalent to a poduction-inventoy system whee the manufactue s demand is known with cetainty in the cuent peiod. If total poduction quantity plus the available stock is not sufficient to meet the dispatch, the emaining amount is outsouced. Outsoucing is simply equivalent to a lost sales stuctue whee pe unit lost sales cost is w. Theefoe manufactue s system is a peiodic eview singleechelon capacitated poduction-inventoy system with makov-modulated demand, peiodically changing capacity levels, and linea oveage and lost sales costs. The optimal policy of the manufactue is a modified base-stock policy (see Aviv and Fedeguen 1997; Kapuscinski and Tayu 1998; Gavineni, Kapuscinski and Tayu 1999). 7.3 Poof of Popety 2 To show why Popety 2 holds, we ague that the manufactue s poblem unde VM-NC is a single-echelon capacitated poduction-inventoy model with makov-modulated demand and peiodically changing capacities. Unde VM-NC the manufactue detemines the optimal poduction, outsoucing and dispatch policy. Based on Obsevation 1 the manufactue s dispatch policy is aleady detemined as (R(β), nq). Specifically, the manufactue obseves the etaile s end-of-peiod inventoy level and dispatches exactly the minimum amount that bings the inventoy level above R(β) befoe demand in the cuent peiod is ealized at the etaile. The dispatch quantity is simply the demand of the manufactue. The dispatch quantity is infeed fom etaile s inventoy level, I, and theefoe does not involve any uncetainty. Simila to the taditional system, this system is equivalent to a poduction-inventoy system whee the manufactue s demand is known with cetainty in the cuent peiod. Theefoe manufactue s system is a peiodic eview single-echelon capacitated poduction-inventoy system with makov-modulated demand, peiodically changing capacity levels, and linea oveage and lost sales costs. The optimal policy of the manufactue is a modified base-stock policy. 33

34 Finally note that, fo β S the manufactue s policy is not necessaily a modified base-stock policy since thee may exist moe than one dispatch policy that esult in the same β and Ī(β). Manufactue may pefe any of the dispatch policies to minimize the cost and the opeating policy can be andomized. 7.4 Poof of Popety 3 We make the compaison unde two cases: (i) Fo β S and T = 1, and (ii) Fo β S o T 1. Fo β S and T = 1, Popety 2 shows that the dispatch policy of the manufactue can be chaacteized as (R(β), nq). Note that the dispatch policy of the manufactue unde TRAD is also (R(β), nq) as imposed by the etaile. Fom Popety 1 and Popety 2 both systems opeate unde modified base-stock policies and the policies ae identical. Theefoe the costs ae equal. Fo β S o T 1, the cost unde no-consignment is lowe than the cost unde taditional system. The eason is, when β S, the manufactue can now conside andomized policies and theefoe is moe flexible in tems of dispatch policies. Fo T 1, unde the taditional system dispatches ae not allowed in any peiod except the fist peiod of the eplenishment cycle. In no-consignment system on the othe hand, thee is no estiction on the dispatch quantity in any peiod. This coesponds to a moe flexible system and theefoe the cost unde no-consignment system is lowe compaed to the taditional system. 7.5 Poof of Poposition 2 We make the poof in two steps. In Step 1 fo the simple scenaio defined, we povide an exact chaacteization of the optimal makov-modulated modified base-stock policy unde VM-NC (in Lemma 1). Using this chaacteization, in Step 2 we povide a lowe bound on the optimal cost unde VM-NC. Step 1. We assumed β S and Q = 1. This implies unde optimality thee exists a unique deteministic dispatch policy fo the manufactue, which is simply dispatch the last peiod s ealized demand. The states ae (I m, I ) whee I m and I ae the inventoy levels at the beginning of the peiod at the manufactue and at the etaile, espectively. If etaile s demand in last peiod is µ 1, I = etaile s base-stock level (µ 1). Retaile s base-stock level is diectly implied by the sevice level equiement and is ielevant to the manufactue s optimal poduction and dispatch policy (see Obsevation 1). Fo notational simplicity, we ignoe etaile s base-stock level and simply indicate I with (µ 1) o (µ + 1). Lemma 1 chaacteizes optimal poduction policy of the manufactue. Lemma 1 Unde VM-NC the optimal poduction policy of the manufactue is as follows (since in ou model outsoucing and lost sales ae equivalent, we use them intechangeably): 34

35 (i) If I m = 0 and etaile s demand in the last peiod is µ + 1 (i.e., if I = µ 1), poduce µ and outsouce 1 (lose the sale of 1 unit), (ii) If I m = I max and demand in the last peiod is µ 1 (i.e., if I = µ + 1), poduce µ 1, (iii) Else poduce µ (i.e., fo 0 < I m I max and I = µ 1, o fo 0 I m < I max and I = µ + 1 poduce µ), whee I max indicates the maximum inventoy level at the manufactue (see Figue 6). 1/2 (0, µ+1) 1/2 (1, µ+1) (Imax,-µ+1) 1/2 1/2 1/2 1/2 1/2 (0, µ 1) 1/2 (1, µ 1) (Imax,-µ-1) Figue 6: The tansition diagam of the undelying Makov chain of the optimal policy unde VM-NC. Poof (Lemma 1). Popety 2 states that unde VM-NC the optimal poduction policy is a modified base-stock policy. The modified base-stock policy states the following opeating policy stuctue: Fo states I m = 0, 1,, k poduce µ, fo I m = k + 1 poduce µ 1 and decease the poduction quantity in unit incements as beginning stock level, I m, inceases. Fo I m = 0 poducing less than µ might as well be optimal. Note that k might be diffeent fo diffeent I levels. In the following, fo each state we detemine the optimal poduction quantity (i.e., we detemine k fo the two I levels, I = µ + 1 and I = µ 1). We analyze the states (I m, µ+1) and (I m, µ 1) sepaately. The eason is, the poblem has a makov-modulated stuctue and the base-stock levels might be diffeent unde I = µ + 1 and I = µ 1. Note that I m 0. The optimal poduction policy unde VM-NC is as follows: (i) Fo states (I m, µ + 1) dispatch quantity is µ 1. We stat with the analysis of I m = 0. Fo (0, µ + 1) the manufactue may pefe to poduce µ o µ 1. Note that poducing µ 2 and outsoucing one unit, o outsoucing seveal units to accumulate stock ae moe costly actions. Theefoe fo (0, µ + 1) only two actions, poduce µ o µ 1, ae unde consideation. Note futhemoe that, outsoucing is not an optimal action in any state (I m, µ + 1). Fo any (I m, µ + 1) if the poduction quantity is µ, then the next state is (I m + 1, µ + 1) o (I m + 1, µ 1) (each with pobability 1 2 ). If, on the othe hand, poduction quantity is µ 1, then next state is (I m, µ + 1) o (I m, µ 1). In othe wods, the peiod whee the manufactue 35

36 decides to poduce µ 1 unde (I m, µ + 1) detemines a candidate fo the maximum stock level unde optimal policy, say I max. In (ii) below we show that I max is indeed the maximum stock level unde optimal policy, and in Step 2 we detemine the I max value in tems of poblem paametes. Unde the optimal policy, fo 0 I m < I max the manufactue poduces µ and fo I m = I max the manufactue poduces µ 1 in state (I m, µ + 1) (note, I max can be zeo). (ii) Fo states (I m, µ 1) dispatch quantity is µ+1. We stat with the analysis of I m = 0. Fo I m = 0 optimal poduction quantity is µ and the emaining one unit is outsouced. Note that poducing less than µ and/o outsoucing moe than one unit ae moe costly actions. Also fo I m > 0, outsoucing to accumulate stock is a moe costly action. This implies that when the cuent state is (I m, µ 1), in the next state the I m level deceases (fo I m > 0). In othe wods the candidate I max in (i) is indeed the maximum stock level. Modified base-stock policy states that poduction quantity is µ fo states 0 I m k and then deceases in unit incements. We would like to detemine the optimal value of k (which might as well be zeo). Fist obseve that unde (I m, µ 1) if poduction quantity is µ, then the next state is (I m 1, µ + 1) o (I m 1, µ + 1) (each with pobability 1 2 ), if poduction quantity is µ 1 then next state is (I m 1, µ + 1) o (I m 1, µ + 1), and so on. In the following, we show that fo 0 I m I max optimal poduction quantity is µ. Define i, as the state whee optimal poduction quantity is µ fo 0 I m i (whee i can be anything in {0, 1,, I max 1}). At state (i+1, µ 1) possible actions ae to poduce µ o µ 1. The optimality equation (unde aveage cost citeia) at state (i + 1, µ 1) is as follows: v(i + 1, µ 1) = g + min{poduce µ,poduce µ 1}, v(i + 1, µ 1) = g + min{cµ + (i)h v(i, µ + 1) + 1 v(i, µ 1), 2 c(µ 1)+(i 1)h v(i 1, µ + 1) + 1 v(i 1, µ 1)}, (19) 2 whee v(i, j) is the optimal bias value of stating in state (i, j), and g is the optimal aveage cost in the Makov Decision Pocess (Puteman 1994). Now conside the optimality equation at (i 1, µ + 1). Possible actions ae poduce µ o µ 1: v(i 1, µ + 1) = g + min{poduce µ,poduce µ 1}, v(i 1, µ + 1) = g + min{cµ + (i)h v(i, µ + 1) + 1 v(i, µ 1), 2 c(µ 1)+(i 1)h v(i 1, µ + 1) + 1 v(i 1, µ 1)}, (20) 2 36

37 Note that ight hand side of Equation (19) and Equation (20) ae the same, theefoe the optimal actions taken at states (i + 1, µ 1) and (i 1, µ + 1) must be the same. We have shown in (i) that fo 0 I m < I max in state (I m, µ + 1) optimal action is to poduce µ. We conclude that fo 0 i I max, optimal action at state (i, µ 1) is to poduce µ. This equivalently implies that fo I = µ 1, optimal value of k is I max. Analysis in (i) and (ii) completes the poof of Lemma 1. Step 2. Given the manufactue s optimal policy stuctue, it is possible to detemine the I max value that minimizes the cost. Unde the descibed optimal policy stuctue, the coesponding Makov chain implies that the steady-state pobability of a state (I m, I ) is: πnc M (I 1 m, I ) = 2.(I max+1). The one-step ewad at state (I m, I ) is expessed as follows: cµ + w(1) if I m = 0, I = µ 1 c(µ 1) + h(i (I m, I ) = max ) if I m = I max, I = µ + 1 cµ + h(i m ) if 0 < I m I max and I = µ 1, o, 0 I m I max and I = µ + 1 Based on the steady-state pobabilities and the ewad function, we expess the aveage-cost unde VM- NC as: cost(vm-nc) = w(1) + h(i max)(i max + 1) c(1) 2(I max + 1) + cµ, (21) whee I max is intege. Note that cost(vm-nc) is convex in I max. The I max value that minimizes (21) is: w c Imax = 1 (22) h In (22) Imax can be a eal numbe o an intege numbe. Replacing I max in (21) with Imax gives a lowe bound on the optimal aveage cost unde VM-NC: LB(vm-nc) = (w c)h h 2 + cµ (23) We wite LB(vm-nc) cost(vm-nc), since I max in (22) is not necessaily an intege numbe. 7.6 Poof of Poposition 3 We conside the following opeating policy. Policy-VMC: (i) Fo I = I min, poduce (and dispatch) µ + 1 (ii) Fo I = I max, poduce (and dispatch) µ 1, 37

38 (iii) Othewise, fo I min < I < I max, poduce (and dispatch) µ, whee we assume I max I min + 3 (i.e., unde Policy-VMC thee exist at least fou states). The poposed policy, Policy-VMC, is not necessaily the optimal policy unde VM-C, theefoe the cost implied by this policy will be an uppe bound on the optimal cost unde VM-C. Note that the tansition pobabilities fo all I ae 1 2. The tansition diagam of the undelying Makov chain is pesented in Figue 7. 1/2 1/2 1/2 1/2 1/2 1/2 I min I min +1 I min I max I max I max /2 1/2 1/2 Figue 7: The tansition diagam of the undelying Makov chain unde Policy-VMC unde VM-C. The steady-state pobabilities, πc M(I ), implied by Policy-VMC ae as follows: 1 πc M 2 fo I = I min, I min + 1, I max 1, I max 1 (I ) = fo I min + 1 < I < I max 1 0 o/w whee = I max (I ) = I min 1. The ewad in state I is expessed as follows: cµ + w(1) if I = I min c(µ 1) + h(i max ) if I = I max cµ + h(i ) + I min < I < I max (24) Ou aim is to detemine an uppe bound on the optimal aveage cost unde VM-C. Note that in (24), I max is always positive, since othewise β 1. Futhemoe, I min possible to attain 100% when I min = 0). When detemining I min is non-positive (since it is aleady and we make sue that the sevice level constaint in (13), whee β S, is satisfied. In othe wods the following constaint is satisfied: i=0 i=i min i πc M(I = i) β (25) E[D] Based on (I ) and π M C (I ) we wite an uppe bound on optimal aveage cost, UB(vm-c), as follows: UB(vm-c) = (I )πc M (I ) = w h( Imax 2 + Imax Imax 2 ) c cµ. The cost, UB(vm-c), consists of the cost of poduction and outsoucing in all states {I min,, I max } and the cost of holding inventoy in states {1, 2,, I max }. Replacing I max with I min + 1: UB(vm-c) = w h( Imin 1 + Imin 2 which is equivalent to, h( I min +1/2) 2 +w c+ 3 4 h 2 + cµ if I max 3 UB(vm-c) = w h( ) c cµ if Imax = 2 w h(1 2 ) c cµ if Imax = 1 To make the analysis simple we focus on the case whee I max In summay we make the following assumptions when defining Policy-VMC Imin ) c cµ, (26) 3. This is equivalent to I min 2.

39 A1. I min 0, A2. I min 2 Fom second ode condition, fo I max condition, the minimizing is obtained as a function of I min w c = + ( I h min 2 I min + 1) 3 the expession in (26) is convex in. Fom the fist ode as follows: In ode to obtain a easonable uppe bound (i.e., an uppe bound which is not vey elaxed), we conside the that minimizes (26). To detemine the values I min could take, we make the following obsevation. Obsevation. Poduction and outsoucing cost of the manufactue is independent of I min, and the inventoy holding cost deceases as I min The obsevation implies that fo a given, as I min deceases. level constaint, to lowe the cost one should lowe the I min value I min consideing I max 3: deceases total cost deceases. If thee wee no sevice value. Howeve, thee is a limit on the lowest can take when minimizing U B(vm-c), detemined by the sevice level. Based on (25) and I min 2 + Imin Imin βµ. In the expession above, fo β = 0 (i.e, fo 100% sevice level) I min set I min as high as k, β should satisfy: should be 0. Othewise, to be able to β k2 k + 1 2µ fo k = 1, (27) Fo a equied sevice level, if the I min level is set to its highest possible value, then this would coespond to the lowest possible inventoy holding cost unde that sevice level. A limit on how high I min can be set is obtained fom Equation (27). Fo a given I min, minimizes the UB(vm-c) value. We place in equation (27) and we suggest two possible uppe bound values fo optimal aveage cost unde VM-C: UB1 fo I min Uppe bound 1 (UB1). We set I min = 0, and a genealized uppe bound fo I min > 0. to the highest possible value, I min UB(VM-C) deceases, uppe bound obtained unde I min bound. Fo I min = 0, we obtain = = 0. Since as I min deceases, = 0 can be egaded as a elaxed uppe w c h + 1. Fom Assumption A2, must be geate than o equal to 2. This is equivalent to, w 3h + c. Futhemoe to guaantee that the expession in (26) is an uppe bound, must be an intege. Equivalently, we say, w should be such that is intege-valued. Placing I min = 0 and in (26) we obtain the following. 39

40 Suppose that w W, whee W = {w w 3h+c, 1 2 ) + cµ is an uppe bound on the optimal aveage cost unde VM-C. w c h + 1 Z+ w c }. Then UB(VM-C) = h( h + 1+ A genealized uppe bound. We constuct a moe geneal set of conditions fo detemining an uppe bound. Let I min = k. Fo I min = k, we obtain = A2, must be geate than o equal to I min must be an intege. w c h + ( Imin 2 I min + 1). Fom Assumption + 2. This is equivalent to, w (5 I min + 3)h + c. Also, Fo a equied sevice level 1 β, highest I min level is obtained as follows: 1 β 1 w c 2µ h I min 2 I min Imin 2 I min + 1 (28) Placing I min and in (26) we obtain the following. w c Suppose that w W, whee W = {w w (5k + 3)h + c, h + (k2 k + 1) Z + } and SL k 1 2 k+1. Then UB(VM-C) = h( + k2 k + 1 (k 1 2 )) + cµ, whee k Z+, is an q 2µ w c h +k2 k+1 w c h uppe bound on the optimal aveage cost unde VM-C. 7.7 Poof of Poposition 4 We compae LB(VM-NC)and U B(VM-C) and obtain the (sufficient) conditions unde which VM-NC esults in highe cost than VM-C. In Poposition 3 we obtain two uppe bounds fo the optimal aveage cost of VM-C. Fist we compae UB1 with LB(VM-NC). UB1 may be a easonable uppe bound when the sevice level equiement is vey high. When I min w c = 0, uppe bound is expessed as: U B(VM-C) = h( h )+cµ. Compaing this with LB(VM-NC) = (w c)h h 2 +cµ, we obtain that UB(VM-C) is always highe than the LB(VM-NC). Similaly, when I min = 1, we again obtain that UB(VM-C) is highe than the LB(VM-NC). Theefoe we conclude that when sevice level equiement at the etaile is high, it is less likely that VM-C esults in lowe cost. Fo I min 2, unde the uppe bound in (26) is expessed as: w c UB(VM-C) = h( h + Imin 2 I min + 1) ( I min 1 2 )) + cµ. Compaing this with LB(VM-NC), we obtain the following: U B(VM-C) < LB(VMNC) w c h( h + ( Imin 2 I min + 1) ( I min 1 2 )) < h( w c h 1 2 ) I min w c < 2 h ( Imin 1) 40

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