An Analysis of Manufacturer Benefits under Vendor Managed Systems


 Priscilla Ward
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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 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 stockouts 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 stockouts, 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 HongKong based shitmake 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, KimbelyClak, 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 enddemand 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 endofpeiod 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 noconsignment 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 noconsignment 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 manufactuingetaile setting, while a few conside a suppliemanufactue 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 systemwide 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 loweechelon. 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 noconsignment 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 noconsignment 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 noconsignment stock model (VMNC), the stock is managed by the manufactue while owned by the etaile. Unde consignment stock model (VMC), 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 peiodiceview model whee the manufactue has limited and nonstationay capacity, which is known by the manufactue in advance. The nonstationaity 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 nonnegative 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 wellknown 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 nonstationay fo the peiods in which the manufactue poduces fo the etaile. We assume this nonstationaity 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 ageedupon 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 nonnegative 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 24unit cases to 12unit cases afte an ageement made with WalMat. Since switching to 12unit case equied significant investment now the company is shipping in 12unit cases to all of its customes (Fishman 2006). The enditem demand is stochastic and stationay. Holding cost is incued based on endofpeiod 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, endofpeiod 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 endofinventoy 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 onhand 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 noconsignment 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 nonnegative 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 kpeiod 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 longun, R+Q fo the fist peiod of the cycle, expected endofpeiod inventoy level is 1 i Q i=r+1 j=0 (i j)p 1(j), Similaly, fo the second peiod, expected endofpeiod inventoy level is 1 R+Q i Q i=r+1 j=0 (i j)p 2(j), and so on. Since in the longun, pobability of being in any peiod in the eplenishment cycle is equal to 1 T, the timeaveaged 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 onepeiod 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 nonnegative) 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 onhand 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 nonnegative intege multiple of Q, without loss of optimality, I m indicates a nonnegative 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 endofpeiod holding costs at the manufactue and at the etaile ae incued. We analyze the vendo managed setting unde two cases; noconsignment stock and consignment stock Noconsignment stock Unde noconsignment stock model (VMNC) 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 noconsignment system. We model the manufactue s poblem as a Makov Decision Pocess unde the aveagecost 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, ighthandside (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 minmax 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 Lefthandside (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 steadystate pobability) in state (I m, I = i, t m ) unde the manufactue s optimal opeating policy unde noconsignment 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 (VMC) the sequence of events is the same with noconsignment 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 noconsignment 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 noconsignment model implies a unit holding cost. If the implied holding cost is equal to the manufactue s holding cost, h, then consignment and noconsignment system can be egaded as equivalent in tems of the actions taken. Fo tighte o moe elaxed inventoy estictions consignment and noconsignment 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 noconsignment systems. Then we compae the cost unde noconsignment 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, noconsignment vendo managed system with VMNC, and consignment system with VMC. 4.1 Stuctual popeties of the optimal policy We analyze the stuctual popeties of the optimal policy unde taditional and unde noconsignment systems. We show in Popety 1 that the optimal policy unde the taditional system is a modified 16
17 basestock policy. Fo the noconsignment 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 noconsignment system is also a modified basestock policy. Popety 1 Optimal policy of the manufactue unde taditional system is a modified basestock policy. Next, we discuss optimal opeating policy of the manufactue unde VMNC. Unde VMNC 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 basestock stuctue. To define the manufactue s policy unde VMNC, 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 VMNC 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 VMNC 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 basestock system whee the odeupto 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 VMNC the manufactue s optimal policy is a modified basestock policy. 17
18 4.2 Compaison of taditional and vendo managed systems In this section we make two compaisons. Fist, we compae the noconsignment system with the taditional system. Using the stuctual esults of the pevious subsection, we show that the cost unde the noconsignment system is always lowe than o equal to the cost unde the taditional system (Popety 3). We then compae the noconsignment 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 noconsignment system Compaison of noconsignment and taditional systems Popety 3 The cost of the manufactue unde VMNC 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., VMNC is a noisk case fo the manufactue Compaison of noconsignment and consignment systems Although VMNC is a noisk case, the cost unde VMNC is not always lowe than the cost unde VMC. 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 VMNC (Poposition 2), and an uppe bound on the optimal cost unde VMC (Poposition 3). We then identify a set of sufficient conditions that make VMC less costly than VMNC (Poposition 4). Assume that Q = 1 and that the (singlepeiod) 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 basestock policy. Poposition 2 Fo the instance defined above, LB(VMNC) = (w c)h h 2 on the manufactue s optimal cost unde VMNC. + cµ is a lowe bound In the following, we detemine an uppe bound on the optimal cost of the manufactue unde VMC (Poposition 3). Unde VMC 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 VMC. Poposition 3 Fo the instance unde consideation, w c (i) Suppose w 3h+c, and w is such that h + 1 Z+. Then UB(VMC) = h( is an uppe bound on the optimal aveage cost unde VMC. 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(VMC) = h h + (k2 k + 1) (k ) + cµ is an uppe bound on the optimal aveage cost unde VMC. Poposition 3 suggests two uppe bounds fo the optimal cost unde VMC. 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(VMC) 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(VMC) 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 VMNC. w c h + (k2 k + 1), k Z +. Then the optimal cost unde VMC is lowe than the Poposition 4 compaes the noconsignment 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 noconsignment. In pactice, fo items with low implicit stockout 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 noconsignment. 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 noconsignment model. Note that ou constuction assumes T = 1 and β S. Unde these assumptions the cost and opeating policies unde TRAD and unde VMNC 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 VMC 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 nonstationaity. (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
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