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Int. J. Production Economics 81–82 (2003) 215–224The consignment stock of inventories: industrial case andperformance analysisGiovanni Valentini a , Lucio Zavanella b, *a IESE Business School, Barcelona, Spainb Universit "a degli Studi di Brescia, Dp.to Ing. Meccanica, Facolt"a di Ingegneria, Via Branze, 38, I-25123 Brescia, ItalyAbstractThe ‘‘Consignment Stock’’ technique is a novel approach to the management of inventories in supply chains. It isbased on an improved collaboration between the company and its suppliers, one that is acquiring growing importancein industrial environments, as the authors have found in Italy. The main aim of the present work is to describe thetechnique itself, thus underlining its potential benefits and pitfalls. The case proposed refers to a companymanufacturing components for the automotive industry. Essentially, the company offered its suppliers the opportunityof stocking part of the items in its own warehouses, with the agreement that they would guarantee over time aninventory level between a set minimum and a maximum value.r 2002 Elsevier Science B.V. All rights reserved.Keywords: Supply chain; Inventory management; Consignment stock1. Introduction and consignment stock practiceThere is no question that over the past decadesan increasing amount of research has concernedthe importance of establishing a profitable verticalrelationship between companies and suppliers,especially in management literature. In particular,strong interaction and reliable collaboration betweenthese two actors have emerged as strategicissues and powerful instruments for maintainingor acquiring competitive advantages in a dynamicand selective market. Indisputably, this issue alsoplays a pivotal role in inventory policies and*Corresponding author. Tel.: +39-0303715579; fax: +39-0303702448.E-mail address: zavanell@ing.unibs.it (L. Zavanella).management. Yet, despite the growing number ofstudies and theoretical models developed, theoperational research still seems to be frequentlydivorced from industrial reality. Meanwhile, severalpractices that are not dealt with in theliterature show up, acquire importance, and proveto be successful.This is the case of consignment stock (CS)management of provisioning, which, regardless ofsome similarities with the common (s, S) policy,reveals significant innovative contributions.Nowadays, this practice has been widely adoptedin Italy, and is consistently gaining consensusamong both small and large firms. Both businesspress and private sources have also confirmedrecent CS applications between Italy and othercountries. Under a CS policy, the relationship0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.PII: S 0925-5273(02)00300-6

216G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224between a company and a supplier is based on thefollowing simple rules:1. The supplier will guarantee the company thecontinuity of an available stock between aminimum level s and a maximum level S: thestock will be stored in the company’s rawmaterial depots, close to the production lines.2. The company may draw on raw materials daily,according to its needs. The supplier is paid forthese materials according to their agreement,hypothetically up to a daily frequency, so thatthe information concerning the consumptiontrend is also constantly refreshed and immediatelytransferred to the supplier.In such a way, the continuous replenishmentfrom the supplier protects the company againstdemand fluctuations and costs determined byeventual stockout may also be debited to thesupplier, by means of contract penalties. On theother hand, the supplier has a better perception ofhis customer’s requirements: lower stocking costsare incurred and the continuous evolution ofmarket demand is directly perceived thanks to anelectronic data interchange (EDI) interface. Severalbenefits are immediately evident:1. The company always has raw material available.2. The company pays for raw material consumptiononly when the items are drawn on for use.3. The supplier saves holding costs and mayorganise his production in different ways, alsowith respect to eventual third parties.4. A renewed and reinforced link is set up betweenthe company and its supplier.However, CS requires the accurate definition ofvarious parameters, i.e. s and S; and a constantattention to the information flow, that is, theelectronic transmission to the supplier of itemconsumption. On this basis, the supplier canforesee the consequences of a better managementof his own production, being freed from thebounds implicit in the strict EOQ practice (e.g.handling large but infrequent orders).This paper seeks to provide a coherent frameworkwithin which to understand the success ofCS. We ask why these changes are taking place. Isit a mere coincidence that various firms areadopting CS policies or is there a rationale behindtheir choice? If this is the case, how should theirchoice be implemented?So as to grasp both the why and the how of CS,the rest of the paper is organised as follows.Section 2 briefly presents the main results of theliterature, while Section 3, using the results of acomputational simulation, shows how CS policymay outperform previous models. In other words,we provide a tentative proof that CS is a rationalchoice. Section 4 addresses the main tacticalquestions that follow the decision to adopt CS.This problem is tackled with the benefit of insightsoffered by a case study. Concluding remarksfollow.2. Literature reviewOne of the major outcomes of the inventorytheory has been to show that (s, S) policies areoptimal for a class of dynamic inventory modelswith random periodic demands and fixed orderingcosts. Under a common (s, S) policy, if theinventory level at the beginning of a period islower than the reorder point s; then a sufficientquantity must be ordered to achieve an inventorylevel S: Literature on the subject is ample andcovers a wide time span. E.g., Iglehart (1963)obtained the stationary distribution of the inventory/backlogand developed an explicit formulafor evaluating the stationary average cost, with theappropriate assumptions. Zheng (1991) has provideda rigorous proof of the optimality of an (s, S)policy for the model of Veinott and Wagner(1965). Sethi and Cheng (1997) broadened somepreviously rigid assumptions in favour of morerealism, and still demonstrated the optimality of(s, S) policies dealing with a given distribution ofdemand.In general, many practical inventory replenishmentproblems satisfy reasonably well the mathematicalconditions under which this type of policyis convenient; however, complex analytical methodsfor computing the best (or even ‘‘a good’’)policy are rarely used, because, according to somepractitioners, they are prohibitively expensive.

G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224 217Several researches tried to show that this objectiondoes not respond to reality. Generally speaking,inventory models have been discussed extensivelyin literature, whereas the treatment of interactionsbetween buyers and suppliers has developed morerecently (Goyal and Gupta, 1989). In 1977, Goyal(Goyal, 1977) suggested a joint economic lot sizemodel where the objective is to minimise the totalrelevant costs for both the vendor and the buyer.The model was generalised by Banerjee (1986) andGoyal (1988) himself. While these later modelsassume that there’s a perfect balance of powerbetween the vendor and the buyer, enforced bycontractual agreement, there are also modelsdeveloped so as to minimise the vendor’s totalannual cost subject to the maximum cost that thebuyer may be prepared to incur (e.g. Lu, 1995).However, it is clear how a partnership may bestrategically useful to both sides, and how it canimprove the inventory management policies (i.e.reduce the costs) by loosening some constraints.This is described in Hill (1997), who introduced areference model in an integrated perspective. Heshowed how the system as a whole can enjoysignificant improvements without any relevant lossfor the parties.To summarise, the literature highlights thetheoretical optimality of (s, S) policies, but it isinconclusive on a lot of issues regarding the levelsof inventory, variables to which the results of themodel are very sensitive. Moreover, the benefits ofa collaboration between companies and suppliersare not taken into account: in other words,suppliers’ costs are not considered a key variable.On the other hand, the models that allow forprofitable interaction between the parties oftenlack the flexibility and the practicality of general(s, S) policies, and provide complex formulas oforder quantities.As we shall see, CS seeks to use the best of bothapproaches.3. Considerations on CS fundamentalsThe basic idea of CS consists in the fact that thephysical inventory of raw material resides withinthe company, ranging freely in quantity between aminimum required level s and a maximumpermitted level S: In addition, under the agreement,the material is formally purchased only atthe moment of its consumption. In other words,even if the raw material is stored in the company’swarehouse, it is still ‘‘owned’’ by the supplier. Letus see what changes in the cost structure this policyentails.We may think of the per unit inventory cost h asdriven by two main components: a financial one(h fin ) and a storage one (h stock ). The financial partregards the opportunity costs a firm incurs whileinvesting financial resources in producing a good.The operational component has to do with thepure storage and movement costs, insurance costs,etc. Under a typical supply agreement, these costsare borne as indicated in Table 1. That is, eachfirm, the supplier and the company in turn,sustains the entire inventory cost (both thefinancial and the storage components) as determinedby the firms’ endogenous characteristicswhile the material is stored in their warehouse. Inthe simplest terms, no inventory cost is sustainedby the supplier after delivering the goods, and nocost is borne by the company before the delivery.Finally, it should be noted that h c;fin þ h c;stock isgenerally greater than h s;fin þ h s;stock ; mainly becauseof the financial component, which increasesas it goes down the supply chain. The differentsituation brought about by CS is outlined inTable 2.As can clearly be seen, the main difference is tobe found in the case where the material has alreadybeen delivered to the company. In fact, thecompany incurs the storage cost, given that thematerial is located in its warehouse, but it doesnot yet sustain the financial cost. In fact, given thata good is formally purchased only after itsTable 1Relevant inventory costs in traditional agreementsPosition of raw materialSupplierCompanyRelevant costs Supplier h s;fin þ h s;stock 0Company 0 h c;fin þ h c;stock

218G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224Table 2Relevant inventory costs under CS policyPosition of raw materialSupplierCompanyRelevant costs Supplier h s;fin þ h s;stock h s;finCompany 0 h c;stockconsumption, the supplier is still bearing thefinancial opportunity cost. 1 Thus, while calculatingthe total cost for the system, we may reasonablyassume that the storage component h stock ofthe total inventory cost may be considered as moreor less identical for the supplier and the company,i.e. h c;stock ¼ h s;stock : As a consequence, referring tothe same average stock level, the total storage costof the system is lower in the CS case (as we assumeh c;fin > h s;fin ), even if a part of the cost is ‘‘shifted’’onto the supplier. However, the supplier perceivessome advantages as counterpart: (i) the averagequantity of the material stored in his owninventories decreases and, consequently, (ii) spaceis available to allocate other items; finally, (iii) thesupplier may manage his production plan moreflexibly as it is not constrained by closed-orders.On the other hand, the company ‘‘sees’’ a lowerper unit inventory cost, that is, only h c;stock insteadof the entire (h c;fin þ h c;stock ). 2Furthermore, it should be noted that there is nolonger any administrative cost for placing anorder, as, in fact, there is no longer any order. 3It should be added that, in some applications ofCS strategy, the item carrier may also be involvedin the partnership, as it may delivery goods fromdifferent suppliers and update the information oninventory levels.To give substance to our arguments, we will tryto show more formally how adopting the CSpolicy improves the performance of the company/1 Obviously, we are not considering the delay between thepurchase and the payment by the company. Still, our argumentholds ceteris paribus.2 Admittedly, the average quantity stored in its owninventories may be larger.3 Because of the EDI interface between the company and thesupplier.supplier system; more specifically we will compareit to Hill’s model (Hill, 1997) while dealing withGoyal’s (1988) classic example. The followingnumerical values and parameters are taken intoaccount: 4A 1 =supplier’s set-up cost=400$,A 2 =cost of placing one order=25$h 1 =supplier’s inventory cost/unit=4$,h 2 =company’s inventory cost/unit=5$,D=demand rate=1000 units/year,P=production rate=3200 units/yearn=number of deliveries per lot.The ‘delivery lead-time’ is set equal to zero.According to Hill (1997), the order quantity q that minimises Cðn; qÞ; the average joint cost perunit of time, is given byq ¼s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA 1 þ nA 2 Þ D Dh 1n P þ ðP DÞn þ h 2 h 11:2P 2Applying the classic EOQ solution (Buffa andSarin, 1987), we would get an optimal EOQ of 100items. Hill’s solution identifies a minimum cost of1903$ in correspondence to a modified EOQ ¼ 110and an EPQ ¼ 550; thus providing the conditionof minimum cost for the whole buyer–vendorsystem.In order to compare Hill’s model to CS, a seriesof tests has been run.If the cost function is modified according to theCS policy as described in Table 2, (i.e. removingcompany order costs and splitting the inventorycosts into a financial component and in a storageone of equal amount), 5 Fig. 1 is obtained: itrepresents the total average joint costs—on thebasis of a 20-year period 6 —depending on the Slevel (s is set to zero, because of the null lead-timeassumption).4 In the model there is no transportation cost. This may berealistic, for instance, when geographical proximity is assumed.Goyal assumes linear costs to be usual.5 Results depend on this assumption, but a basic sensitivityrevealed how simulation main outcomes hold for a wide enoughrange of h stock and h fin :6 This assumption is made in order to obtain better qualitygraphs, but it does not influence the indications offered by theanalysis carried out.

G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224 219Fig. 1. Total joint costs in the CS policy depending on S:Fig. 2. Hill’s model costs depending on S in the case of a nondeterministicdemand.At the present stage, it may even be misleadingto take into account all the variables at stakewhich may determine the supplier deliveries inbetween s and S: Thus, we assumed that when s isreached, an (S2s) quantity is immediately deliveredby the supplier. The minimum cost obtainedis 1793$, in correspondence to S ¼ 426: AdoptingCS policy and the cost savings implicit in it, it istherefore possible to achieve a 6% reduction in thetotal cost. Moreover, neither the supplier’s allowedflexibility and the company’s reliability in guaranteeingservice levels were quantified in monetaryterms, in spite of their indisputable value.Even if it is quite obvious to obtain a reducedcost with respect to Hill’s model (as some costs arelower), the test results show that this CS performancedepends on the S level fixed. Thus, a firstinsight is possible: when the company and itssupplier bargain the CS agreement, they must takeinto account that the convenient S level is to beidentified for the entire system.Now, let us remove the hypothesis of deterministicdemand and divide the year into 100 periods,each with a demand profile sampled from thenormal distribution dBNð10; 1Þ: In such a way,and still considering Hill’s model, the possibility ofincurring a stockout is introduced and a morerealistic market situation is taken into account.Simulating this policy over 50 years, we obtain thecost curve of Fig. 2. The average cost is slightlylower (1897$) than Hill’s model performance fordeterministic demand, but what matters is that inthis point an average demand of nine items peryear remains unsatisfied. The costs determined bythe shortage event were not considered in thecalculation but, especially when the companyconsiders the service level as a strategic key forgaining competitive advantages, they may be ofgreat relevance. As far as stockout events areconcerned, it is also interesting to analyse theaverage number of items undelivered by thecompany per year, while varying the S value(Fig. 3). Of course, the magnitude of the phenomenonmay be decreased while increasing the S level(thus incurring additional storage costs), but thegreater the level, the lower the possibility offurther service improvement.In conclusion, it may argued that, with the sametotal cost, the CS policy allows the company tokeep an s level of even more than 20 items, thusreducing stockout risk. The results proposedsuggest a second insight. For a fluctuatingdemand, the CS policy allows a higher servicelevel (i.e. less shortage events) at acceptable costs.Once again, this performance depends on the (s, S)levels adopted. The same positive results describedabove may be extended to the common case of avariable lead time in the supplier deliveries. In fact,generally speaking, s level may help the companyin facing both demand variations and lead time

220G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224Fig. 3. Common supplier–vendor model and non-deterministicdemand: undelivered items because of stockout as a function ofS:uncertainties. Furthermore, CS policy, thanks toEDI interface, allows the supplier to perceivedirectly the trend of the demand, thus reducing orat least stabilising the lead time.These considerations reinforce our initial beliefthat the importance that CS is acquiring inpractical terms, although it has often been ignoredin literature, rests on a rational basis. However,several practical issues remain unresolved. Addressingthem is the subject of the following section,starting from the case observed.4. The industrial caseHaving shown that CS could be the best choicedoes not solve the implementation problems. Is CSthe best solution for every item? What are theforces that determine the levels of S and s? Once Sand s have been fixed, how does the suppliermanage his degrees of freedom acquired throughCS? To provide an answer to these questions, wewill consider general insights and the case studytogether.The case analysed refers to an Italian company,which manufactures components for the automotiveindustry (braking systems). The company sawa fast growth in the recent past, roughly doublingboth gross income and personnel from 1995 to1999. The most recent evolution gave it anindustrial presence in Japan and a consolidatedposition as a supplier to the most importantEuropean manufacturers of cars, industrial vehiclesand motorbikes. The industrial activity alsoexpanded into the racing sector, strategicallyidentified in dedicated areas of the productiveunits. The production covers both the OriginalEquipment and the After Market demand. Therelevant performances obtained are also to berelated to the effective introduction of techniquessuch as TQM, Just in Time and Kaizen, flexibilityin assembling and production, personnel motivation.As far as supplies are concerned, thecompany recently introduced the CS techniqueand some interesting questions arise from itsexperience.The first insights we get refer to the choice of theitems and suppliers to be involved in the CSprogram. The suppliers involved in the projectmust be among the most active and critical,starting from considerations on the type and/orquantity of pieces supplied together with theeconomic relevance of their supplies. Thus, aninitial step consists in identifying:* The most attractive items for CS management(e.g. criticality for the assembling tasks, strategicvalue of the system to which the componentpertains etc.): such an analysis also leads to theidentification of the suppliers to be contacted.* According to the resulting list, the mostqualified suppliers are selected (e.g. on the basisof their turnover with respect to the company,on their performance in quality assurance andstability in delivery time) for the first implementationof the CS supply.Afterwards, a technical comparison between thecompany and the supplier is necessary to identifythe salient parameters for the items to be providedunder CS management (e.g. safety stocks, leadtimes, packaging conditions, transport quantity).Consequently, the minimum and maximum itemlevels (i.e. s and S) to be guaranteed over timeare jointly fixed. The forces that constitute thevariables at stake, while suppliers and companynegotiate the level of s and S; are summed up inTable 3.

G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224 221Table 3Desired (s; S) levels and related motivation for the supplier and the companys levelS levelSupplier kKeep as low as possible mKeep as high as possibleIt represents a capital ‘‘frozen’’ in the company’swarehouse, which determines an opportunity costIt consists in a space offered by the company to stockproducts, thus freeing space in own warehouses.Consequently, production flexibility may be increased.Company mKeep as high as possible kKeep as close as possible to s levelIt is a sort of safety stock which enables a higherservice level, whereas its economic burden isborne by the supplierThere is a need for limiting the space occupied by theproducts and for reducing the problems linked to theirmanagement and handlingFig. 4. Inventory level of an item through time (dashed lines indicate s and S level)On the basis of the industrial case, one field tobe explored is related to the supplier’s behaviourwhile maintaining the stock within the (s; S)range. In fact, two limiting conditions may behypothesised.In the first case, the supplier keeps his stockclose to the upper level S or, alternatively, he mayfill the company warehouse up to the S level andconsequently wait for the stock erosion, due todemand requirements, down to level s (Fig. 4). Insuch a situation, the profile of the companyinventory is similar to the classic behaviour of acontinuous-review model, with fixed order pointand order quantity. E.g., this situation may occurwhen ðS sÞEEPQ: According to Table 2 motivations,the supplier may produce according to hisEPQ, but his own warehouses are freed, as space isprovided by the company. It is also possible tounderstand that, in these cases, the company hasnot fully used its bargaining power (or the supplierhas somey), since S is comparatively high even ifthe item does not seem to have any strategicrelevance despite its high rate of consumption.In the second limiting condition, the suppliermay decide to keep the company’s inventory closeto the s level, thus incurring the risk of penaltypayment due to the lower level violation, butreducing his economic exposure (e.g., if EPQ5S).Of course, an intermediate behaviour is representedby the suppliers delivering quantities so thatthe inventory level fluctuates between the two s2Slimits. This situation, graphically described inFig. 5, refers to the interesting case of a componentsubjected to frequent delivery and irregular,though intensive, consumption: the supplier continuouslyswitches within the s and S levels, thanksto a flexibility in production volume and a ‘‘flat’’behaviour of its characteristic EPQ curve. Also in

222G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224Fig. 5. Inventory level of an item through time (dashed lines indicate s and S level).this case, the risk of shortage could arise in somesituations. The curve shows the stock level of a realitem of the industrial case observed. It is a verystrategic one (this justifies a reasonably high slevel), and is produced by a very close supplier,extremely flexible, and with a number of importantcustomers. In fact, the erratic profile of the supplycurve not only points to several factors relative tothe specific item, but is also an index of theincreased supplier flexibility in deciding his productionplan while trying to cope with customers’needs. In the absence of precise and imperativeproduction order and time deadline from a givencompany, the supplier has a greater degree offreedom to organise and adapt his manufacturingresources. This item also highlights anotherimportant variable in CS policy: item dimension.The high number of s and S is also an index of thesmall dimensions of the item involved. It may seemtrivial, but this is another important factor thathas to be taken into account while decidinginventory levels.The problem of transported quantities is alsorelevant, this being related in turn to the geographicalposition of the supplier with respect to thecompany location. Generally speaking, distantsuppliers aim to fill the stocks up to S level (largequantities transported, also by ship and/or train),whereas the closest suppliers (which deliver smallquantities by local truck transportation) may keepa stock close to s level and/or show the irregularpattern of deliveries already described.According to the industrial experience analysed,some other issues emerge:* Items to be included in the CS programme arethose characterised by a constant consumption(‘‘open order’’ provisioning).* Items of potential interest for CS pertain tostandard production, but they are subjected to a‘‘close order’’ provisioning system. Comingunder the CS management, they will pass toan ‘‘open order’’ provisioning.* Items to be excluded from CS management arenon-standard products and prototypes (e.g. inthe industrial case under examination, items forracing models are excluded, together withproduction for prototypes and first-samplesupply).* The minimum level s may be roughly estimatedas the safety stock which enables the companyto cover the production of a period the lengthof which depends on the lead time of thesupplier.The agreement between the company and thesupplier may include further obligations, such as:* The agreed lead time in case of sudden demandpeaks for the company.* The level of the safety stock the supplier shouldmaintain in his own depots, taking into accountthe provisioning time of the item considered.This parameter may also influence s and Svalues.

G. Valentini, L. Zavanella / Int. J. Production Economics 81–82 (2003) 215–224 223* The type and capacity of the pallets for delivery,as s and S values are an integer multiple of it.This parameter is also to be fixed to interfaceCS standards with the kanban system.* The company may agree to pay for the goodsstored in its warehouse, even if it has notconsumed them yet, after a given amount oftime.Some of the above topics are prudentiallyintroduced when starting the CS management ofitems: they can be removed once the systemparameters have been conveniently arranged.Nevertheless, safety stocks and planned lead timesmay be also providential measures to meetunpredictable or turbulent market demand, especiallyin the case of suppliers that lack productionflexibility. It must be highlighted how the finalagreement generally involves further parameterswhich are linked more closely to fiscal and legalregulation. Thus, it may be difficult to discussthem exhaustively, as they may differ in aninternational context. Though, it is interesting tocomment on some of them:* The eventual dispatching of items exceeding Slevel allows the company to send them back tothe supplier without incurring transport costs.* Shortage costs may be included in the agreementas a penalty for the supplier, if the eventoccurs because of failure to respect the s level.Such a penalty may be large enough to coverthe entire economic damage caused to thecompany.* Every single item pertaining to a stock is theproperty of the supplier until the companydraws on it for production.5. ConclusionsThis paper gives the account of an innovativeinventory policy called consignment stock. Itsrapid diffusion and the absence of any importantreference to it in the scientific literature led us toask ourselves whether CS was just a fad or whetherit had some value.We showed how, by combining the practicalityof (s; S) policies with the ‘‘system approach’’ ofjoint-profit maximising models, CS is able tooutperform the usual inventory models. Not onlydoes CS allow savings in inventory costs, but italso entails several complementary intangibleadvantages, such as a higher degree of flexibility,an increased service level in turbulent environments,a reinforced and reliable relationshipbetween companies and suppliers. Even Authorsthat argue that CS may be harmful for the supplierfrom a purely economic point of view, recognisethe strategic importance of the relational rentsstemming from CS (Ferrozzi and Shapiro, 2000).Hence, the diffusion of CS has a clear rationalebehind it, which at present we have shown bysimulation experiments.Moreover, thanks to a case study, we haveprovided some insights into tactical issues thatfirms have to address once they decide to adopt theCS policy. We believe we have provided reasonsthat show why CS is catching on as well as how it isand should be implemented.AcknowledgementsThe Authors gratefully acknowledge BremboSpa management and technical staff.ReferencesBanerjee, A., 1986. A joint economic lot size model forpurchase and vendor. Decision Sciences 17, 292–311.Buffa, E.S., Sarin, R.K., 1987. Modern Production. OperationsManagement, 8th Edition.. Wiley & Sons, New York.Ferrozzi, C., Shapiro, R., 2000. Dalla Logistica al Supply ChainManagement. ISEDI, Turin, Italy.Goyal, S.K., 1977. An integrated inventory model for a singlesupplier–single customer problem. International Journal ofProduction Research 15, 107–111.Goyal, S.K., 1988. A joint economic lot size model forpurchase and vendor: A comment. Decision Sciences 19,236–241.Goyal, S.K., Gupta, Y.P., 1989. Integrated inventory models:The buyer–vendor coordination. European Journal ofOperational Research 41, 261–269.Hill, R.M., 1997. The single-vendor single-buyer integratedproduction–inventory model with a generalisedpolicy. European Journal of Operational Research 97,493–499.

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