THE CONSIGNMENT STOCK OF INVENTORIES IN PRESENCE OF OBSOLESCENCE

Transcription

1 HE CONSIGNMEN SOCK OF INVENORIES IN RESENCE OF OBSOLESCENCE Marco CAENA a, Adrea GRASSI b, Alessadro ERSONA a, a ipartimeto di ecica di Gestioe dei Sistemi Idustriali, Uiversità di adova, Stradella S. Nicola 3, 3600 Viceza, Italy b ipartimeto di Scieze e Metodi dell Igegeria, Uiversità degli Studi di Modea e Reggio Emilia, Via Fogliai, 400 Reggio Emilia, Italy Abstract he Cosigmet Stoc (CS) ivetory policy is becomig a importat strategy that compaies adopt to face ew maufacturig ad supply chai maagemet challeges. he CS policy implies great collaboratio betwee the buyer ad his supplier, pushig them toward a complete exchage of iformatio ad a cosistet sharig i maagemet riss. I such a cotext, the effects of the product obsolescece have to be carefully evaluated sice they fall oto both actors, causig a icrease i total supply chai costs. his paper proposes a aalytical model able to tae ito accout the effects of obsolescece i a supply chai maaged with a CS policy. he determiistic sigle-vedor sigle-buyer CS model is herei used as a base to develop the proposed model. A compariso with a o-obsolescece optimal solutio available i literature is preseted. Moreover, the stochastic behavior of the product lifetime estimatio is also tae ito cosideratio. Results demostrate that effects of obsolescece ca cosistetly ifluece the global optimum coditio. Keywords: Cosigmet stoc, Ivetory maagemet, Obsolescece, Supply Chai. Correspodig author: rof. Alessadro ersoa ipartimeto di ecica di Gestioe dei Sistemi Idustriali, Uiversità di adova, Stradella S. Nicola 3, 3600 Viceza, Italy. el Fax persoa@gest.uipd.it (A. ersoa)

2 HE CONSIGNMEN SOCK OF INVENORIES IN RESENCE OF OBSOLESCENCE Abstract he Cosigmet Stoc (CS) ivetory policy is becomig a importat strategy that compaies adopt to face ew maufacturig ad supply chai maagemet challeges. he CS policy implies great collaboratio betwee the buyer ad his supplier, pushig them toward a complete exchage of iformatio ad a cosistet sharig i maagemet riss. I such a cotext, the effects of the product obsolescece have to be carefully evaluated sice they fall oto both actors, causig a icrease i total supply chai costs. his paper proposes a aalytical model able to tae ito accout the effects of obsolescece i a supply chai maaged with a CS policy. he determiistic sigle-vedor sigle-buyer CS model is herei used as a base to develop the proposed model. A compariso with a o-obsolescece optimal solutio available i literature is preseted. Moreover, the stochastic behavior of the product lifetime estimatio is also tae ito cosideratio. Results demostrate that effects of obsolescece ca cosistetly ifluece the global optimum coditio. Keywords: Cosigmet stoc, Ivetory maagemet, Obsolescece, Supply Chai.. Itroductio he ivetory maagemet ad cotrol is a problem widely discussed i iteratioal literature. While the first studies were orieted to a idepedet optimisatio of ivetory levels for the supplier ad the retailer ad to a discout approach to maximize the vedor profit, recet approaches aalyse the iteractios betwee the two actors. I fact, it is acowledged that the Ecoomic Order Quatity (EOQ) model gives a optimal solutio for the buyer, but it is always uacceptable to the vedor. he Ecoomic roductio Quatity (EQ) of the vedor usually differs greatly from the EOQ, triggerig off egotiatios o the price per item ad o the size of the batch

3 supplied. Moaha (984) developed a optimal uatity discout pricig schedule for the vedor to obtai larger orders from customers, ad coseuetly fewer maufacturig set-ups per year ad fewer trasportatio discouts, allowig the buyer to have moey i use earlier i the year. Lee ad Roseblatt (986) improved Moaha s method imposig a ecoomic costrait o the maximum discout ad relaxig the order-for-order policy for the supplier. Lal ad Staeli (984) proposed a pricig discout schedule for multiple buyer groups. Kim ad Hwag (988) developed a model for a optimal discout schedule assumig a sigle icremetal discout system. Models, whose objectives were to icrease supplier profit through a discout policy, were proposed by Goyal (987a, 987b), rezer ad Wesolowsy (989). Weg (995) studied supplier ad buyer coordiatio, aalyzig the effects of uatity discout o chael coordiatio i case of price sesitive demad ad order uatities fuctio of trasactio costs. Corbett ad e Groote (000) dropped symmetric iformatio assumptio ad derived the supplier s optimal uatity discout scheme where the buyer holds private iformatio about his costs structure. o overcome the local miimizatio of costs both for the vedor ad the buyer, ad to move toward the global miimizatio of the costs for the two parties, it is ecessary to view the system as a itegrated whole ad to exchage iformatio about productio, demad ad shipmets data. hat is possible, if collaborative arragemet ca be eforced by some cotractual agreemet betwee the buyer ad the supplier (Lee ad Roseblatt, 986). Goyal (977) suggested a Joit Ecoomic Lot Size (JELS) model where the objective was to miimize the total relevat costs for both the vedor ad the buyer. Baerjee (986), geeralizig Goyal model, examied the ecoomic beefits of a joit optimal orderig policy i which the vedor maufactured each buyer shipmet as a separate batch. Goyal (988) relaxed the Baerjee assumptio for the vedor to produce the purchaser s orders o a lot-for-lot basis ad demostrated that producig i multiple lots of purchasers orders was a better solutio. A wide review of models which provided a coordiate mechaism betwee buyer ad vedor up to 989 was proposed by Goyal ad Gupta (989). Hill (997) showed that sedig shipmets with sizes icreased by a geeral fixed factor was a better

4 solutio tha the previous oes. Hill (999), combiig a eual shipmet size policy ad Goyal s (995) policy, where the shipmets were made as soo as the buyer was about to ru out of stoc, determied the form of the globally-optimal batchig ad shippig policy for a sigle-vedor sigle-buyer cotext, assumig a determiistic eviromet. his policy gave lower total costs tha the previous oes. Houe ad Goyal (000) developed a optimal policy assumig that a lot was trasferred to the buyer i a fiite umber of eual ad ueual batch sizes, icreased by a fixed factor, imposig a trasport euipmet fiite capacity costrait. Recetly, a ew ivetory policy, defied as Cosigmet Stoc (CS) policy (Braglia ad Zavaella 003, Valetii ad Zavaella 003) or Supplier-Owed Ivetory (SOI) strategy (Yap 999, iplai ad Viswaatha 003) was studied. he CS is based o the cooperatio betwee the buyer ad the vedor ad it has bee more ad more followed as idustrial practice. With a CS policy, the vedor removes his ivetory ad maitais this stoc at the buyer s premises. he buyer ca draw o the material whe eed arises ad oly pays for the uatity draw, while the vedor maitais the ivetory maagemet ad repleishmet resposibility. Valetii ad Zaella (003) described the ew policy ad the mai fudametals, uderliig potetial beefits ad pitfalls, ad showed that a CS policy allows for both vedor ad buyer to reduce total joit costs ad stoc-out riss while assurig a higher service level i case of fluctuatig demad, whe compared to the Hill model. iplai ad Viswaatha (003), assumig a oe-buyer multi-vedor approach, proposed a umerical study that cofirmed the lower total joit costs of a CS policy i compariso with a cotiuous review ivetory policy. his reductio icreased with the icrease i share of total demad from the buyer that employed a CS policy. Fially Braglia ad Zavaella (003), startig from the Hill model (999), proposed a model to evaluate the performace of the policy ad of the situatios where it could be adopted successfully, showig that a CS policy was a useful approach to ivetory maagemet i case of delivery lead times or maret demad which were variable over time. 3

5 A essetial assumptio i the etire previous model is the ifiite lifetime of the products. I fact, buyer ad vedor determie productio, ivetory ad shipmets which miimize the average total cost per uit time eglectig the obsolescece costs. his obsolescece is caused by the strog competitio i the marets, the products differetiatio, the reductio i the products life cycle ad the rapid techical iovatios. hese treds imply that the fuctio fulfilled by a compoet is o loger reuired, or that there is a suitable item which performs similar fuctios (Brow et al., 964). he obsolescece causes a partial or a total loss of value of the ivetory o had, forcig buyer ad vedor to deviate from the optimal lot sizig, ivetory level ad shipmet uatity calculated by meas of the previous models. May wors study the optimal lot size i order to miimize holdig costs i presece of obsolescece (Jogleart ad Lee 993, Karua 994, ohi ad Osai 995, Sog ad Zipi 996, Va elft ad Vial 996), but igore the total cost of itegrated supply chais. he obsolescece presets difficulties i ivetory maagemet, ad these problems are amplified i a CS policy sice a basic assumptio of this strategy is the freedom o the vedor part to maage the buyer ivetory stoc betwee the miimum level s ad the maximum S, shiftig the greater part of the obsolescece ris to the buyer. his paper proposes a model able to maage the effects of the product obsolescece o a itegrated sigle-vedor ad sigle-buyer system worig with a Cosigmet Stoc ivetory policy. Such a model is a extesio of the Braglia ad Zavaella s (003) approach to the case of limited product lifetime, allowig the aalyst to determie the globally-optimal batchig ad shippig uatity. I the ext sectio a Cosigmet Stoc policy will be described, the a aalytical model for maagig the Cosigmet Stoc i presece of obsolescece will be itroduced. Later, the optimizatio approach for the buyer stoc value S ad the compariso with Braglia ad Zavaella s model will be preseted. Fially the stochastic item life period case is aalyzed. 4

6 . Cosigmet stoc policy he Cosigmet Stoc policy is a recet maagig ivetory policy that is comig i use as idustrial practice. A CS agreemet betwee a vedor ad a buyer implies a great collaboratio ad itegratio betwee the two parties. I fact, the supplier places the products stoc i the buyer s warehouse without charge, ad all items belog to the supplier util the buyer draws o them accordig to its productio plaig. his policy allows the vedor to elimiate its ivetory by movig it to the buyer s premises, but the supplier should assure the buyer a available stoc level betwee a miimum s ad a maximum S values, located o the buyer s premises. he cotiuous exchage of ivetory data level betwee the two parties, implicit i a CS agreemet, allows the buyer to achieve a higher service level through the elimiatio of the demad perturbatio perceived by the vedor ad caused by the buyer s order policy. Literature highlights several beefits for the buyer: A suppressio of the ivetory holdig costs, because the buyer oly pays whe drawig o the raw material; A higher service level compared to a traditioal policy because the vedor guaratees a miimum stoc level s i the buyer s warehouse; A reductio i admiistrative costs because the purchase order processig is lower tha that of a traditioal policy; A applicatio of just-i-time procuremet without high repleishmet costs. he CS strategy allows the vedor to orgaize the productio i a more cost savig maer (Valetii ad Zavaella 003). he beefits for the vedor are: Access to the fial demad without the filter of the buyer s order policy; A reductio i the warehouse dimesios; A reductio i productio costs for the icreased batch productio size ad a reductio i set-up umbers per period; A reductio i shippig costs compared to a just-i-time procuremet. 5

7 he CS strategy allows the vedor to produce a batch of material of a higher size tha the Ecoomic Joit Order (Corbett 00) ad to ship this to the buyer i a uiue shipmet, i the evet of a high vedor productio rate compared to buyer s demad, or i multiple shipmets i other cases. he size of the vedor s productio uatity is related to a ey decisio i a CS agreemet, that is the determiatio of the maximum level S ad the miimum s of the ivetory. revious researches (Valetii ad Zavaella, 003; Braglia ad Zavaella, 003) showed that the vedor ad the buyer had two cotrastig objectives i this choice. he supplier wats to eep the s level as low as possible to reduce holdig costs, while maitaiig the S level as high as possible to icrease productio flexibility ad reduce shippig costs. he productio uatity for the vedor is lower or eual to S s. O the cotrary, the buyer wats to eep the s level as high as possible for assurig a high service level ad the S level close to s for limitig the space occupied by the products. 3. Cosigmet stoc model with obsolescece 3. Aalytical model he model proposed i this paper is a extesio of Braglia ad Zavaella s (BZ) approach i order to tae ito accout the effects of obsolescece. he authors preseted a model able to determie the optimal shipmet uatity, ad coseuetly the maximum ivetory level S o the buyer s premises for a CS policy, adaptig Hill s (999) model. he model proposed here, as the BZ oe, assumes a sigle-vedor sigle-buyer system, where the vedor produces i batches ad icurs i set-up costs, ad the buyer cosumes with a fixed rate. Each batch is dispatched to the buyer i a umber of shipmets, some of which occur while productio is still taig place. he lead time of each shipmet is eual to zero ad the basic model assumes that shipmet batches are set as soo as the vedor s ivetory achieves the shipmet uatity. he buyer is subject to a fixed order emissio cost ad the vedor to a fixed trasport cost, both idepedet from the uatity shipped. 6

8 Both parties icur i time-proportioal material holdig costs, at differet rates. he s ivetory level has bee set eual to zero. As i previous researches (Goyal 988, Hill 999, Braglia ad Zavaella 003), the demad ad the vedor s productio rates are cosidered both costat ad cotiuous. Moreover, the vedor s productio rate is assumed higher tha the demad rate ( > ). he item holdig costs per time uit is greater for the buyer tha for the vedor, sice the item value icreases as it desceds alog the supply chai as a coseuece of trasport cost ad vedor profit. Figure shows the tred of the ivetory level i the case of a CS policy as proposed by BZ. hey assume a batch size Q = 550 (items), a umber of trasports per batch = 4 ad a trasport lot size = 8 (items). he supplier ships a batch with multiple travels, reducig his stoc level at his best. Accordig to the hypothesis of determiistic demad, it is assumed that the vedor starts the productio whe the buyer s ivetory level is eual to the total demad durig the productio time of the trasport lot size. ae i Figure Ulie BZ model behaviour, the product obsolescece causes a fiite umber of cycles, with a legth eual to Q /. Moreover, the last cycle ca be icomplete. he cosigmet stoc agreemet lets the vedor maage his productio i a flexible way, with the sole costrait of assurig the buyer s ivetory level withi the rage s ad S. Whe the obsolescece occurs, the item demad falls to zero ad a step ca be see i the demad tred. his step ca occur durig a vedor productio period or i a productio iterval. I the first case, the vedor will cotiue the productio of the batch ad the shipmet of the remaiig lots, accordig to the CS freedom, util the buyer s ivetory level is less or eual to S. Figure shows a example of this situatio, where the obsolescece occurs after 4 periods from the vedor s productio start. he vedor ships the first shipmet lots while the last are stoced i the 7

9 vedor s warehouse because the level of the buyer s ivetory ca ot exceed S. Both parties icur i obsolescece costs. he obsolescece ca occur durig the vedor s productio period without costs arisig. I fact, if the uatity i the remaiig shipmets is less or eual to the residual stoc available i the buyer s warehouse, the vedor ca sed all of his productio ad coseuetly he does ot icur i obsolescece costs. ae i Figure I the secod case the vedor has completed the productio ad the shipmet of the batch before the icurrig of obsolescece ad does ot have ay additioal cost. O the cotrary, the buyer should pay the obsolescece costs for the residual items. Figure 3 shows the ivetory tred i a situatio where obsolescece occurs after the batch productio eds. ae i Figure 3 I order to develop the model the followig otatios are used: A A C c p h h batch set-up cost for the vedor, e.g. 400 ($/set-up) order emissio cost for the buyer, e.g. 5 ($/order); average total costs of the system per time uit, e.g. ($/year); item s productio cost for the vedor, e.g. 8 ($/item); cotiuous demad rate experieced by the buyer, e.g. 000 (uits/year); vedor holdig cost per item ad per time period, e.g. 4 ($/item year); buyer holdig cost per item ad per time period, e.g. 5 ($/item year); i capital cost rate, e.g. 0%; umber of shipmets per productio batch; 8

10 * umber of ot set shipmets durig the last productio batch from the vedor; p cotiuous vedor productio rate, e.g. 000 (uits/ year); price per item paid by the buyer to the vedor, e.g..5 ($/item); uatity trasported per delivery, e.g. (uits/ delivery), for which the productio batch size is defied as Q= ; S maximum level of buyer s stoc (uit); item life period, e.g. (year); x is eual to the smaller iteger if x 0, to 0 if x < 0. Accordig to Valetii ad Zavaella (003) the per uit ivetory cost h is composed by two mai compoets: a fiacial oe h fi ad a storage oe h stoc. Assumig i as the capital cost rate, the item price ad item productio costs ca be calculated as follows: h,fi Item price : p = () i h,fi Item productio cost : c p = () i he relatio betwee the maximum level of buyer s stoc S ad the uatity trasported per delivery is eual to (Braglia ad Zavaella, 003): Maximum buyer's stoc level : S = (3) ( ) o simplify the model formulatio we defie t * as the time betwee the last productio batch start ad the obsolescece occurrece: t * = (4) where / is the CS cycle time. he umber of shipmets * that are set durig the last productio batch by the vedor are calculated as followig: 9

11 * * S t = (5) where / is the product level i the buyer s stoc whe the vedor starts the productio batch. Hece, vedor s set-up, holdig ad obsolescece average costs per year ca be calculated as: A Set up cost : C v s = (6) * v Holdig cost : C = m h (7) c * ( ) v p Obsolesce ce cost : Co = (8) I (7) the cotributio is the product of the average uatity i stoc, /, the productio batch time, /, ad the umber of whole CS cycles durig the item life period,. he average ivetory level i the vedor warehouse durig the last cycle is calculated cosiderig the umber * of shipmets set to the buyer. I (8) the obsolescece cost for the vedor is obtaied as the product betwee the umber of * shipmet ot delivered to the buyer, ( ) he buyer s costs are defied as followig:, ad the ecoomic value of a shipmet, c p. b A * Order emissio cost : Ce = (9) b h S Holdig cost : C = ( ) * * ( ) t t m (0) p Obsolesce ce cost :C b o = * t* () 0

12 where is the umber of shipmets set by the vedor durig the last cycle ad before the obsolescece occurrece, defied as: * t = mi ; () he Order emissio cost (9) cosiders the costs of the orders made durig the CS iteger cycles ad the umber of orders * completed i the last CS cycle. he holdig cost (0) is determied by the sum of the average ivetory level durig the CS iteger cycles, S, ad the average ivetory level i the last cycle cosiderig the time iterval betwee the cycle start ad the occurrece of the obsolescece. his last term is the sum of two factors. he former represets the average stoc level util the -th shipmet, while the latter is the average stoc level startig from the istat i which the last shipmet is set to the istat t * represetig the obsolescece occurrece. he total costs for the system are determied by the sum of the buyer ad vedor s costs: C() = C C C C C C (3) v s v m v o b e b m b o Give the umber of shipmets per cycle, the optimal maximum level S of the buyer s stoc ca be computed by meas of (3) oce the value of that miimized the total costs is determied. 3. Optimizatio of value Ulie the CS model proposed by Braglia ad Zavaella (003), the cost fuctio of the Obsolescece CS (OCS) model is characterized by several discotiuity poits, each of them represetig a local miimum of the fuctio. Hece, the value of that miimizes total costs ca ot be evaluated by derivig (3). Nevertheless, it ca be observed that all the local miimum poits for the cost fuctio are idetified by a uiue miimum coditio. I particular, the values of miimizig total costs are

13 those that imply a value of * t tedig to a CS cycle time. With respect to the umber of CS cycles i the obsolescece iterval, the uiue miimum coditio ca be writte as follows:, lim * = t (4) meaig that miimum poits are those for which teds to a whole umber, that is:.,,, with = (5) Lemma : whe (4) holds, the *. roof: = t * * < < = 0 * verified. Lemma : whe (4) holds, the. roof:. beig ; mi ; mi * t > = = By imposig the coditio expressed by (4) to the total costs fuctio (3), a discrete costs fuctio idetifyig all the miima ca be obtaied. I particular the euatio (4) ca be rewritte as:, = (6) ad substituted i (3) gives:

14 ( ) ( ) ( ), mi p h h A A C = (7) subject to the costrait (5), meaig that such a euatio is valid if ad oly if teds to a whole umber. his is the reaso why (7) represets a discrete series of the miimum poits. he euatio (7) ca be further rewritte by substitutig =, i which the sole variable is sice,, ad are defied a priori. ( ) ( ). mi = p h h A A C (8) he euatio (8) represets the series of the costs local miima as a fuctio of the iteger series =,,,. o compute the iteger value of which ivolves the absolute miimum cost a search amog the series represeted by (8) should be carried out. Nevertheless, the search ca be sped up by facig the problem from a cotiuous poit of view. I particular, sice (8) represets all the miimum costs poits, if a cotiuous variable is cosidered, istead of the iteger variable, such a euatio becomes a cotiuous fuctio iterpolatig all the miimum costs poits. Hece, by fidig the absolute miimum, amed R, of the cotiuous fuctio we ca restrict the search of the valid (iteger) absolute miimum to the two values, if ad sup, represetig the iteger immediately lower ad higher tha R respectively. By computig the costs i correspodece of if ad sup, the oe that ivolves miimum costs ca be easily idetified. R represets the value of that wipes out the first derivative, holdig the coditio of positiveess of the secod derivative. 3

15 ( ) ( ) ( ), mi = = p h h A A C R R (9) ( ) ( ) ( ). 3 mi = = p h h C R R (0) It ca be observed that the secod derivative is always greater tha zero, sice the term is implicitly positive. I fact: > > 0 verified. he value of R is the draw as follow: ( ) ( ) A A p h h R = () Oce R is determied, we ca obtai the suitable whole values of as: R = if () sup = R (3) By substitutig if ad sup i the euatio (5) we ca determie the two values, if ad sup, which ivolve miimum total costs. if if (4) sup sup (5) he euatios (4) ad (5) restrict the search of the optimal amog two cadidate values. o idetify which of the if ad sup actually ivolves global miimum costs, attetio to two mai aspects has to be paid. Firstly, values of applicable i the practice have to be iteger, ad secodly, the miimum coditio is a limit. articularly, this latter aspect implies that the values 4

16 if ad sup which ivolve miimum costs are ot exactly eual to the ratios if ad sup respectively, but are a ifiitesimal higher to those ratios. Cocludig, the real applicable which ivolves global miimum costs is either the first iteger greater tha if or the first iteger greater tha sup that ivolves the miimum value of the euatio (3) Model compariso i the determiistic eviromet I Figure 4 the tred of the total costs for the two models are reported, assumig, as the iput data, the followig values: =, =5, A =400, A =5, =000, =000, h =4, h =5, p=.5, c p =8. As ca be see, the total costs are obviously lower for the BZ model, which cosiders ifiite lifetime, while the total cost i presece of obsolescece is always higher ad presets a irregular tred. he miimum of the total costs proposed by the BZ model, assumig ifiite item lifetime, is eual to 890 ($/year) with a optimal value of (uits/delivery) ad a optimal S value of 333 (uits). he miimum of the total costs usig the OCS model, assumig a fiite items lifetime, is eual to 707 ($/year) with a optimal value of 0 (uits/delivery) ad a optimal S value of 303 (uits). However, applyig the value calculated with the BZ model to the case of obsolescece, the total costs would become 506 ($/year), 89% greater tha the solutio proposed by the OCS model. ae i Figure 4 he above reported example demostrates how the total costs ca cosistetly icrease eve if the differece betwee the two calculated values of is ot large. Such a differece is maily iflueced by the values of,, ad. I Figures 5 ad 6 the depedece of the optimal value, 5

17 ad the coseuet total costs, from the demad ad the ratio / is show for differet values of. ae i Figure 5 ae i Figure 6 I fact the OCS model shows how the optimal value of has a low depedece from the ratio /, eve if differet values of are tae ito accout. O the opposite, a positive correlatio betwee ad the demad rate is emphasized. By comparig the results proposed by the OCS model to the oes carried out by the BZ model, it ca be see that the effects of the obsolescece imply optimal values always lower tha those related to the o-obsolescece case. I other words, the aalyst is drive to adopt values of higher tha the optimal oes if he does ot tae ito accout the presece of obsolescece, icurrig i cosistet icreases i total costs. I particular, the differece betwee the optimal values computed i the two cases, obsolescece ad oobsolescece, is the higher the value of / teds to. Hece, obsolescece has to be particularly tae ito accout whe, i a CS maaged supply chai, the vedor s productivity rate is close to the buyer s demad rate. he aual total costs i presece of items obsolescece is decreasig for greater values, as showed i figure 7, where the compariso betwee the BZ model ad the OCS model is preseted for differet values. he ew approach implies lower or eual costs compared to the other model. he irregular tred of the BZ model is caused by the aual obsolescece costs. he tred of the fuctio is decreasig with because the value of the obsolete items is spread over a greater time period. 6

18 ae i Figure 7 Figure 6 shows the compariso betwee the optimal maximum level of the buyer s stoc for ifiite life time items model ad for the OCS model. he aalysis highlights that the itroductio of a fiite life time for the compoets implies a reductio of the S level for the buyer, allowig to recover free space i the warehouse ad to dedicate smaller space to material stocig. his reductio is o average greater whe the value of is lower. For a value lower tha years the S level should be reduced by about 30%. ae i Figure 8 he reductio of the buyer level S compared to the stoc level determied by the BZ model ivolves lower total costs, but the cost reductio is i fuctio of the item price. Figure 9 shows the total costs agaist the item price cosiderig the other variables costat. he applicatio of the OCS model implies a strog reductio i the total costs for compoets at a higher price. ae i Figure Stochastic case for the item life period A determiistic value for the item life period is rare i realistic situatio. his ca happe, for example, i food idustry, where usually raw materials have a expiry date, or i the pharmaceutical idustry, where products have a strict ad moitored life time period. Usually, 7

19 buyers ad vedors do ot ow the life time of the items, because this value is geerally determied by the maret demad for the fial products sold by the buyer ad for which these items are assembled or used, or by ay evetual redesig of the compoets carried out by oe of the two actors. he variability of causes some difficulties i the estimatio of the optimal uatity trasported per delivery ad coseuetly the maximum level for the buyer stoc S. o aalyze the impact of this ucertaity a simulatio study has bee performed. he parameter has bee assumed ormally distributed with mea μ ad stadard deviatio σ. he stadard deviatio σ is i fuctio of the ucertaity about the life time period ad a low value of the idicator has bee hypothesized as beig eual to /00, while a high value eual to /. For a sigle couple of value of μ ad σ, 000 differet values of have bee geerated ad the mea total costs value for differet values of the uatity trasported per delivery were calculated. Figure 0 shows some results of this study assumig a umber of shipmets per productio batch eual to 5. his figures shows that for value eual to the total costs tred is strogly irregular ad a little variatio of the value ear the optimum ivolves a strog rise i the total costs. his variatio is higher for high stadard deviatio values, while for low stadard deviatios the total costs tred is ear to the total costs tred for determiistic value. For items of higher life periods the values assumed by the total costs i fuctio of the uatity trasported per delivery is more idepedet from the stadard deviatio of. ae i Figure 0 his aalysis shows how for stochastic item life time period, the optimal value is lower compared to the oe calculated for the determiistic case, while for greater values the optimal shipmet uatity arises toward the optimal value calculated for the determiistic eviromet. 8

20 Moreover, the simulatio highlights how very critical is, for a low value of the item life period, the determiatio of the optimal uatity trasported per delivery, ad coseuetly the maximum buyer stoc level S, sice a wrog estimatio implies substatial higher costs for both vedor ad buyer. For higher values of, the ad S estimatio is less critical, because the total costs tred is smoother ad less irregular. ae i able able shows the optimal value ad the total costs for a OCS model with determiistic value, for a OCS model with stochastic value of the item life period ad with low, medium ad high values of the stadard deviatio. he optimal value is lower for a OCS model with stochastic compared to the OCS model with determiistic item lifetime, ad is lower for greater stadard deviatio of the item life period. his percetage reductio is smaller for higher value of because the obsolescece costs, caused by the ucertaity of, are distributed o a loger period. Figure plots the impact of the umber of shipmets o the total costs for small value of, showig that there is ot a direct relatio betwee the optimal umber of deliveries ad the total costs i a stochastic item life time eviromet. ae i Figure 4. Coclusios he Cosigmet Stoc (CS) ivetory maagemet policy has bee proved to be particularly suitable for facig ew maufacturig ad supply chai maagemet challeges. he CS policy 9

21 implies a complete exchage of iformatio betwee the buyer ad his supplier, ad a cosistet sharig i maagemet riss. I such a cotext, the cosideratio of the product obsolescece effects o the total costs of the supply chai is of great iterest. his wor proposes a aalytical model, based o the Braglia ad Zavaella s approach, which cocers the determiistic sigle-vedor sigle-buyer productive situatio, allowig the aalyst to idetify the optimal ivetory level ad shipmet policy for optimizig total costs whe products are characterized by a fiite lifetime. Results show that the presece of obsolescece reduces the optimal ivetory level, particularly i case of a short period of life. he ratio betwee the vedor s productio rate ad the buyer s demad rate is aother importat parameter to tae ito accout. I particular, the effects of obsolescece o the correct estimatio of the optimal shipmet dimesio are higher whe the productio rate is close to the demad rate. Moreover, simulatios have bee carried out to assess the impact of the stochastic estimatio of the item lifetime period. Results show how the optimal shipmet dimesio for the stochastic life time case is always smaller tha that cocerig the determiistic case. he higher the ucertaity i product life time estimatio the lower is the dimesio of the shipmet, with respect to the determiistic case. Moreover, results emphasize that there is o relatio betwee the umber of deliveries ad the total costs. I coclusio, this wor provides to the aalyst a robust methodology ad some importat isights i how to maage the effects of obsolescece i such a way as to reduce total costs i a supply chai maaged with a CS policy. Future wors Further wor might evaluate the multi-buyer or multi-vedor eviromets, the case of obsolescece ad stochastic demad, ad the evaluatio of the ivetory level for buyer ad vedor i case of trasfer of owership, from vedor to buyer, after a fixed period from the delivery of the products. 0

22 Refereces BANERJEE, A., 986, A joit ecoomic lot size model for purchaser ad vedor. ecisio Scieces, 7, 9-3. BRAGLIA, M., ad ZAVANELLA, L., 003, Modellig a idustrial strategy for ivetory maagemet i supply chais: the Cosigmet Stoc case. Iteratioal Joural of roductio Research, 4, BROWN, G.W., LU, J.Y., ad WOLFSON, R.J., 964, yamic modellig of ivetories subject to obsolescece. Maagemet Sciece,, CORBE, C.J., 00, Stochastic ivetory systems i a supply chai with asymmetric iformatio: cycle stocs, safety stocs, ad cosigmet stoc. Operatios Research, 49, CORBE, C. J., ad E GROOE, X., 000, A supplier s optimal uatity discout policy uder asymmetric iformatio. Maagemet Sciece, 46, OHI,., ad OSAKI, S., 995, Optimal ivetory policies uder product obsolescet circumstace. Computers & Mathematics with Applicatios, 9, REZNER, Z., ad WESOLOWSKY, G.O., 989, Multi-buyer discout pricig. Europea Joural of Operatioal Research, 40, GOYAL, S.K., 977, etermiatio of optimum uatity for a two-stage productio system. Operatioal Research Quarterly, 8, GOYAL, S.K., 987a, etermiatio of a supplier s ecoomic orderig policy. Joural of Operatioal Research Society, 38, GOYAL, S.K., 987b, Commet o: A geeralized uatity discout pricig model to icrease supplier s profits. Maagemet Sciece, 33, GOYAL, S. K., 988, A joit ecoomic lot size model for purchaser ad vedor: A commet. ecisio Sciece, 9, GOYAL, S.K., ad GUA,., 989, Itegrated ivetory models: he buyer-vedor coordiatio. Europea Joural of Operatioal Research, 4, GOYAL, S.K., 995, A oe-vedor multi-buyer itegrated ivetory model: A commet. Europea Joural of Operatioal Research, 8, HILL, R.M., 997, he sigle-vedor sigle-buyer itegrated productio-ivetory model with a geeralised policy. Europea Joural of Operatioal Research, 97, HILL, R.M., 999, he optimal productio ad shipmet policy for the sigle-vedor sigle-buyer itegrated productio-ivetory problem. Iteratioal Joural of roductio Research, 37, HOQUE, M.A., ad GOYAL, S.K., 000, A optimal policy for a sigle-vedor sigle-buyer itegrated productio-ivetory system with capacity costrait of the trasport euipmet. Iteratioal Joural of roductio Ecoomics, 65, JOGLEKAR,., ad LEE,., 993, Exact formulatio of ivetory costs ad optimal lot size i face of sudde obsolescece. Operatios Research Letters, 4, KARUNA, J., 994, Lot sizig for a product subject to obsolescece or perishability. Europea Joural of Operatioal Research, 75, KIM, K.H., ad HWANG, H., 988, A icremetal discout pricig schedule with multiple customer ad sigle price brea. Europea Joural of Operatioal Research, 35, LAL, R., ad SAELIN, R., 984, A approach for developig a optimal discout pricig policy. Maagemet Sciece, 30,

23 LU, L., 995, A oe-vedor multi-buyer itegrated ivetory model. Europea Joural of Operatioal Research, 8, LEE, H. L., ad ROSENBLA, M. J., 986, A geeralized uatity discout pricig model to icrease supplier s profits. Maagemet Sciece, 3, MONAHAN, J.., 984, A uatity discout pricig model to icrease vedor profits. Maagemet sciece, 30, ILANI, R., ad VISWANAHAN, S., 003, A model for evaluatig supplier-owed ivetory strategy. Iteratioal Joural of roductio Ecoomics, 8-8, 3-4. SONG, J.S., ad ZIKIN,.H., 996, Maagig ivetory with the prospect of obsolescece. Operatios Research, 44, 5. VALENINI, G., ad ZAVANELLA, L., 003, he cosigmet stoc of ivetories: idustrial case ad performace aalysis. Iteratioal Joural of roductio Ecoomics, 8-8, 3-4. VAN ELF, C., ad VIAL, J.., 996, iscouted costs, obsolescece ad plaed stocouts with the EOQ formula. Iteratioal Joural of roductio Ecoomics, 44, WENG, Z.K., 995, Chael coordiatio ad uatity discouts. Maagemet Sciece, 4, YA, R., 999, Holdig the Aces i Maufacturig Logistics. roceedigs of Maufacturig Logistics for the st Cetury, Sigapore,

24 = =3 =5 =7 OCS EERMINISIC OCS SOCHASIC, σ LOW OCS SOCHASIC, σ MEIUM OCS SOCHASIC, σ HIGH C tot C tot C tot C tot able. Optimal uatity trasported per delivery ad total costs for obsolescece CS model with determiistic ad stochastic values, for eual to 5. 3

25 Figure. Braglia ad Zavaella s model: ivetory level at the buyer ad vedor premises. 4

26 Figure. Ivetory level whe obsolescece occurs durig the vedor productio period. 5

27 Figure 3. Ivetory level whe obsolescece occurs after the completio of the vedor productio batch. 6

28 Figure 4. otal cost for CS policy without obsolescece (Braglia ad Zavaella) ad with obsolescece assumig = ad =5. 7

29 OCS =000 BZ =000 OCS =000 BZ =000 OCS =3000 BZ = OCS =000 BZ =000 OCS =000 BZ =000 OCS =3000 BZ = mi otal costs / / OCS /=. OCS /=. OCS /=3.3 BZ /=. BZ /=. BZ /= mi otal costs OCS /=. OCS /=. OCS /=3.3 BZ /=. BZ /=. BZ /= Figure 5: reds of mi ad total costs as a fuctio of / ad with =. 8

30 mi OCS =000 BZ =000 OCS =000 BZ =000 OCS =3000 BZ =3000 otal costs / OCS =000 BZ =000 OCS =000 BZ =000 OCS =3000 BZ = / mi OCS /=. OCS /=. OCS /=3.3 BZ /=. BZ /=. BZ /= otal costs OCS /=. OCS /=. OCS /=3.3 BZ /=. BZ /=. BZ /= Figure 6: reds of mi ad total costs as a fuctio of / ad with =4. 9

31 Figure 7. otal costs for differet values ad assumig eual to 5. 30

32 Figure 8. Optimal S value i fuctio of for Braglia ad Zavaella ad obsolescece CS model assumig eual to 5. 3

33 Figure 9. Compariso of total costs betwee BZ ad obsolescece CS models for differet values of item price, assumig = ad =5. 3

34 Figure 0. otal costs agaist the uatity trasported per delivery for differet values of the stochastic item life period, assumig eual to 5. 33

35 Figure. otal costs agaist the uatity trasported per delivery for differet values of the umber of shipmets. 34