A Globally Optimal Local Inventory Control Policy for Multistage Supply Chains

Transcription

1 Inernaonal Journal of Producon Research, Vol. X, o. X, Monh 2X, xxx xxx A Globally Opmal Local Invenory Conrol Polcy for Mulsage Supply Chans J.-C. HEE CRS-LSIS, Unversé Paul Cézanne, Faculé de San Jérôme Avenue Escadrlle ormande émen,3397 Marselle Cedex 2 Absrac In he consdered mulsage supply chan, he produc srucure deermnes he organzaon of he nework of enerprses ha cooperae o manufacure end-producs from raw maerals. hs paper proposes o descrbe such a mulsage supply chan by a sequence of lnear dynamcal models wh dsrbued delays. he assumed auonomy and nformaon prvacy of producon uns mposes srucural consrans on producon and nvenory polces. Each local nvenory polcy s supposed o be a base-sock polcy appled o he poson nvenory. A each sage, demand s supposed random bu saonary wh a known mean value. hs paper shows ha he collecon of local nvenory polces s equvalen o he global polcy whch mnmzes he long erm average cos of he whole supply chan. Keywords: Supply Chan Managemen, Vrual Enerprse, Invenory Conrol, Srucural consrans AMS Subjec Classfcaon: 9B3, 9B5, 9B5, 93E2, 9B, 5A48 Correspondng auhor. E-mal: jean-claude.henne@lss.org

2 J.-C. Henne Inroducon In conras o a mulsage sysem owned by a sngle producer, a nework of auonomous enerprses s characerzed by s heerogeney and dsrbued decsonal srucure. hese srucural consrans pose obsacles o he mplemenaon of negraed plannng sofware such as ERP and APS. eworks of enerprses are ypcally characerzed by possble conflcs of neres, randomness and uncerany of lead-mes and ransporaon mes, and conrol acons may have lmed mpacs. However, hey also have advanages regardng flexbly, agly and quck adapaon o he marke, wh respec o qualy and quany. Mulsage producon ofen movaes he desgn of a supply chan as a vrual enerprse, gaherng n an enerprse nework he mos effecve companes for each producon sage. In a mulsage manufacurng nework, he produc srucure dcaes he organzaon. Producers of prmary producs play he role of supplers. Producers of nermedae producs play boh roles of supplers and producers, and producers of endproducs are boh producers and realers. Supply chan desgn should be frs opmzed hrough selecon of he bes coalon. hen, nformaon and produc flows should be organzed hrough negoaon on ask assgnmen and busness conracs. In convenonal mulsage MRP, all he calculaons are performed on a sngle compuer; Quanes and release daes of all he producs are compued o sasfy n me and quany he scheduled shpmens of fnal producs, deermned by he Maser Producon Schedule (MPS). Several auhors have proposed o mplemen MRP n a decenralzed manner. A ypcal applcaon conex s cellular manufacurng (Love and Bareka 989). Among he advanages of dsrbued decenralzed MRP, Buzaco (997) emphaszes he possbly of adjusng lo-szes and changng mngs o compensae for local dsurbances such as scrap and nvenory losses. In he case of a nework of enerprses workng as a mulsage sysem, decsons are naurally dsrbued. However, an addonal dffculy s he combnaon of a hgh level of neracon beween produc flows and a low level of decsonal coordnaon. Demand varably ends o ncrease up he supply chan. Small changes n cusomer demand can resul n large varaons n orders placed upsream. Snce he work of Lee e al. (997), such a phenomenon has been referred o as he Bullwhp Effec. I has been recognzed as a nonrval conrol problem. In parcular, classcal orderng polces such as he (s,s) polcy are unable o aenuae he perurbaon and damp he oscllaons (Dejonckheere e al. 23). Many sudes have shown ha local effecveness s far from auomacally nducng global effcency of he whole producon process. wo facors concur o dsadvanage dsrbued decsons n erms of global effcency: lmed nformaon a local levels and based opmaly crera. Several auhors have proposed conracs o mprove global effcency whle respecng he decsonal auonomy of he parners ( Cachon and Zpkn 999, Caldeney and Wen 23). One of he leadng mehods for Supply Chan Managemen s based on he SCOR (Supply Chan Operaons Reference) Model. hs model provdes a framework for assessng and evaluang a Supply Chan n erms of process models. hree levels of process models are dsngushed (Supply Chan Councl 28). he op level conans 5 core managemen processes called: Plan, Source, Make, Delver, Reurn. he second level s he confguraon level wh 3 processes: Plannng, Execuon, Enable. he hrd level deals level 2 processes, ofen n he form of workflows. Supply chans may have comparave advanages over negraed sysems wh respec o flexbly, mproved delvery servce and nvenory reducon (Fawce e al. 28). However, nformaon sharng appears as a key success facor, parcularly a he sraegc level.

3 Local Invenory Conrol Polcy he pcure s no as clear a he operaonal level; some auhors such as Lau e al. (24) have shown by expermenal smulaon ha nformaon does no provde sgnfcan advanage wh respec o performance ndcaors such as operang cos, nvenory level and backlog level. he scenaro suded n hs paper s n agreemen wh hs observaon. A hgh level of sraegc negraon s assumed hrough he use of he same ype of base sock nvenory polces by all he acors of he supply chan. In he mulsage case consdered, he nformaon avalable a each sage s manly local. Parners share he knowledge of he produc srucure, whch only conans sac daa. hey also know he mean demand for endproducs hrough he nduced mean demand for he producs ha hey manufacure. he man queson addressed n hs paper s "can local orderng and producon polces perform as well as an opmal negraed producon polcy n a mulsage manufacurng nework?" Raher surprsngly, he answer wll be "yes" when he supply chan s n saonary condons wh random bu lmed varaons of demand for end-producs. he local orderng and producon polces ha have been seleced for he medum-erm and long-erm horzons are of he base sock ype, appled o he nvenory posons raher han o he nvenory levels. In he ermnology of (Axsaer 23), he consdered polces are nsallaon sock polces raher han echelon sock polces. An nsallaon sock poson s classcally defned as he sum of he sock level and he expeced orders, from whch we subrac he backordered fnal demands n he case of he socks of fnal producs. An echelon sock s defned as he number of ems ordered plus he oal number of ems held locally and n all he downsream nsallaons, mnus he number of ems embedded n backordered demands for fnal producs. he class of echelon sock polces clearly conans he class of nsallaon sock polces and herefore may yeld beer performance (Axsaer 23). In he decenralzed approach of hs paper, nsallaon socks are consdered snce only local nformaon on socks s assumed avalable. As a consequence, hese polces may no be opmal n a larger class of polces, bu hey are known o possess good properes n erms of economcal performance and dsurbance aenuaon (Poreus 99, Henne 28). hs paper shows ha under some classcal assumpons such as fxed lead mes and common orderng perodcy, he local base sock polces are equvalen o an negraed polcy. I has been shown n Henne (23), ha hs polcy s opmal wh respec o a mean value creron, robus o unceranes, random varaons and dsurbances and able o guaranee sasfacon of producon and sorage capacy consrans. However, such properes are only vald n a parcular se of saes, whch has o be reached as a frs sep n he mplemenaon of he seady-sae local orderng and producon polces. he second par of he paper presens he model of manufacurng sages and formally unfes hem n a global model. he hrd par descrbes he proposed local nvenory replenshmen polces and shows he equvalence of local polces o he negraed opmal polcy. he fourh par llusraes he resuls on an example. Fnally, some conclusons and possble exensons of hs work are presened. 2 he mulsage model 2. Mulsage manufacurng processes A large varey of manufacurng sysems can be modelled as mulsage manufacurng processes. Mulsage processes conss of several sages conneced by hree ypes of branchng: lnear, assembly and dsrbuon. Each producon acvy has several npu producs, and producs of he same ype may be used by several acves. he oal number of producs s noed. I s assumed ha here s a one-o-one correspondence beween

4 J.-C. Henne acves and oupu producs. In mahemacal erms, he oupu marx s he deny marx of dmenson, noed I. A ls of noaons used n hs paper s gven n appendx. Graphcally, he produc srucure ha descrbes he Bll of Maerals (BOM) s a dreced graph wh no loop. Such a graph can be decomposed by levels. By convenon (see e.g. (Baker 993)), level s for end-producs, level l for producs ha are componens of producs of levels srcly less han l and of a leas one level l- produc. For {,n e }, produc s an end-produc; for =n e +,...,, produc s a componen. Accordng o a gven manufacurng recpe, producon of one un of produc j consumes componens =,..., n quanesπ j, for j=,...,. Producs beng ordered n agreemen wh he ncreasng order of her level, he npu marx Π = π )) s nonnegave, lower-rangular wh zeros on he man dagonal: L L π 2 O O Π = π L π, (( j M. () Le I denoe he deny marx of he approprae dmenson. he ne producon marx, I Π has all s off-dagonal elemens nonposve. I may be called an essenally nonposve marx (Berman and Plemmons 979). Furhermore, s lower-rangular, and s dagonal elemens are equal o. herefore, de ( I Π )=, and more generally, all s prncpal mnors are equal o. I s a well-known characersc propery of an M-marx o be an essenally nonposve marx wh all s prncpal mnors posve (Berman and Plemmons 979). herefore, marx ( I Π ) s an M-marx. hs mples, n parcular, ha s nverse ( I Π ) s a nonnegave marx. he vecor v of produc requremens o sasfy an exernal demand represened by vecor d R+ s gven by he Leoneff formula: v = ( I Π) d. (2) on-negavy of vecor v clearly derves from non-negavy of ( I Π ) and d. A lead me s classcally defned as he allowed me nerval beween he release me of a producon lo and s delvery. As s classcal n he MRP leraure, planned lead-mes are supposed consan and ndependen of lo szes, provded ha capacy consrans are sasfed. In he consdered mulsage manufacurng process, lead me θ corresponds o produc,. { } 2.2 Local producon and nvenory models n a manufacurng nework Each produc manufacurer s supposed o receve orders from hs cusomers and delver producs from hs sock. In he ermnology of Baker (993), demand for fnal producs s ndependen, whereas demand for componens s dependen. However, from he local vewpon of a parcular manufacurer producng a parcular produc, orders receved from downsream parners n he supply chan can be consdered ndependen (or exogeneous). In addon o havng a perfec knowledge of shor-erm demand n he form of frm orders, each manufacurer s supposed o know he mean value of he demand for hs produc. hs knowledge s eher obaned from a local esmaor (ndependen vew) or from esmaon of demands for fnal producs and use of he Leoneff formula (2) (dependen vew).

5 Local Invenory Conrol Polcy he oupu o of produc a perod s delvered o cusomers f s a fnal produc ( {,n e} ). If s a componen ( { ne +, } ), o s he quany of produc delvered o downsream producon sages a perod. he exernal demand d for produc a perod s such ha d = f s no a fnal produc ( ne +, ). { } Backorders are allowed for end-producs ( { },n e ). Le β denoe he amoun of backorders for end-produc a he end of perod. Assumng ha demand for end-produc s sasfed as much and as soon as possble, he delvery varable s relaed o he backorder hrough equaon (3) : o = d + β, β for neger, =, {, ne}. β (3) Backorders are no allowed for componens, snce componen avalably s a necessary condon for producon. he sock level on hand a he end of perod s denoed s. hs s a nonnegave varable. An nvenory varable, denoed r (possbly negave for fnal producs) can be defned as follows: r r = s = s β for {, n } for e { n +, }. (4) e he decson o produce a quany u of produc s aken a he begnnng of perod. he correspondng manufacurng acvy s supposed o sar a he begnnng of perod, for he producon lo o be delvered a he begnnng of perod + θ, he neger θ beng he producon lead-me for produc. Under convenon u = for <, he nvenory balance equaon for produc { ne +, } akes he followng form: r, o for, s gven, { n, }. (5) = r + u, θ e + and for fnal producs, r = r, + u, θ d for, r = s gven, {, n }. (6) e Producon varables u are consraned o be nonnegave: u >, {, }. (7) I s mporan o noe ha non-negavy of nvenory varables for componens s an essenal logcal condon ha deermnes feasbly of producon varables u : r >, { n + }. (8) e,

6 J.-C. Henne Le p n denoe he amoun of space p needed o sore one un of produc. For each of he p sorage zones, locaed n he dfferen enerprses, nvenory capacy consrans ake he form: p n r = p. (9) r Le R be he oal number of resources n he manufacurng nework. Le m denoe he amoun of resource r, wh r {,..., R}, needed per perod o produce one un of produc. I s supposed ha he same amoun of resource s requred durng he lead me. Producon capacy consrans hen ake he followng form: θ = l= m r u, l M r for r=,...r, =,... () Model (6)-() descrbes he sysem evoluon over he whole plannng horzon (wh possble rescalng f he sze of he me buckes vares). Only he assumpons on demand and orders change from he shor erm o he medum erm horzon. A he local level of he producon un for produc, he only global nformaon are he sac daa menoned below. In addon, he local varables avalable a he begnnng of perod are: - curren nvenory level a he end of perod -, r, - pas conrols, u,l u,,, θ - curren oupu, o, - nvenory poson a he end of perod -,, whch s he sum of he nvenory level p, and expeced orders (manufacurng orders launched bu no compleed ye). he lead me θ beng assumed consan and precsely known, hen compued by: p, can be p, = r, + θ u k k=,. () he local model wll hen be defned by he balance equaon (6) and by he expresson of he conrol npu a perod, u ha wll be consruced n secon 3.2. he dsrbued naure of he sysem appears as a srucural consran n consrucng he conrol vecor u = [ u, L, u ]. 2.3 he global mulsage model I s assumed ha all he acors of he supply chan use he same orderng perodcy, whch corresponds o he un me bucke of he plannng horzon. In order o oban a quany u j of produc j avalable a he begnnng of perod + θ j, componens { ne +, } are requred n quanes π ju j a he begnnng of perod.

7 Local Invenory Conrol Polcy Under he base sock polcy, componens are delvered from sock and he producon order for componen a he begnnng of each me perod exacly maches he consumpon of hs componen durng hs perod. Delvery lead mes are no explcly consdered bu could be easly negraed n he model hrough an ancpaon of componens consumpon. As a consequence, he oal amoun of componen delvered a perod s gven by: π for neger, u = for <, { n,}. o u = j= j j (2) j e + oe ha for he produc srucures consdered n hs paper, he upper lm,, of he summaon n (2) could be equvalenly replaced by - (wh 2 ) snce marx Π n () verfes π = j; j. j Equaons (5), (6) can hen be re-wren n a unfed manner: r = r, + u, θ ju j d π for, r = s gven, {,}. j = o smplfy noaon, he delay operaor, noed } (3) z, wll now be used. By defnon, noaon z h for me seres{ h represens h : z h h. In a smlar manner, θ u z u., θ = he global manufacurng sysem can be represened by a dscree-me, lnear, mulvarable model wh dsrbued delays ( θ, Lθ ), an exernal npu vecor, d = [ d, L, d ], capacy consrans on nvenory vecor, r = [ r, L, r ] and producon vecor, u = [ u, L, u ]. oe ha he erm delay s used n reference o he delay operaor. In he model, delays represen normal lead mes assocaed wh producon processes. Usng he sysem npu marx, Π, see equaon (), he sysem dynamc oupu marx also has dmenson x and s defned by: θ z θ Dag( z ) = M θ z 2 O L L O O M θ z. (4) he maxmal lead-me s denoed as follows: θ = max θ {, }. Equaons (3) for all he producs can be gahered n he vecor balance equaon below (5). θ r = r Dag z + [ ( ) Π] u d (5) Marx ( z) = [ Dag( z ) Π] θ can be decomposed as follows: ( z) = + z + L + z. (6) θ θ

8 J.-C. Henne Due o he srucure and delay erms of marx (z), ( z) s purely ancpave. Le ρ be he maxmal degree n z of polynomal marx ( z). One can wre: ( z) = z Q( z) where marx Q(z), whch s polynomal n z, s decomposed as follows: ρ ρ Q ρ Q( z) = Q + z Q + Lz. (7) Marces, {, ρ} Q are lower-rangular wh nonnegave coeffcens. he sysem delay ρ hen corresponds o he longes me pah n he produc srucure graph from raw maerals o end-producs. 2.4 A predcve model for medum erm demand Afer he shor erm me horzon, denoed S, exernal demands for end-producs (,n e ) are decomposed no a predced componen, supposed consan, and a dsurbance: { } d = d ˆ + e for > S (8) e has mean value and sandard varaon σ. I s assumed ha here s no exernal demand for prmary and nermedae producs: = ne +,. In vecor form, d for { } equaon (8) s wren: d = d ˆ + e for > S, wh d = [ d Ldn L], e d ˆ dˆ ˆ = [ L ], e = [ e Le L]. d ne L ne Demand predcon for fnal produc a any perod > S s denoed dˆ. he only global nformaon avalable o each supply chan parner s he sac daa gven by vecor dˆ, marx Π and he vecor of lead-mes θ ). ( =,..., In seady-sae condons, nvenory levels are supposed consan and he vecor of reference producon levels are obaned from equaon (5) n he form: uˆ = ( I Π) dˆ. (9) oe ha equaon (9) reduces o he Bll Of Maerals (BOM) equaon (2). herefore, as prevously dscussed, vecor u ˆ s unquely defned and nonnegave. A nomnal polcy for he supply chan can be assocaed wh predced demands. I can be decomposed by producs and s defned by he followng predcons: predced oupu per perod for end-producs: oˆ = ˆ for,n }, (2) d { e predced producon levels per perod: uˆ = [( I Π) ] dˆ where [( I Π) ] denoes he h row of marx ( I Π), (2) predced oupu per perod for prmary and nermedae producs:

9 Local Invenory Conrol Polcy oˆ π uˆ. (22) = j= j j Addonally, s naural o assume ha sysem capacy s suffcen o sasfy exernal demand, a leas around nomnal condons. A range where producon capacy condons (9) apply can hen be defned around he nomnal producon vecor, defned for nsance by a posve vecor δ = [ δ, L, δ ] such ha θ = l= m r u, l M r for r=,...r, u ˆ ˆ, l ; u δ u, l u + δ (23) Due o non-negavy of demand sequences ( d ), varaons of orders for end-producs can be assumed bounded from below. I s also realsc o assume hem bounded from above. hen, f s an end-produc ( {,n e }), { n } ω e ω, wh ω dˆ, ω,, e for > S. (24) If he probably dsrbuon of e s unform on ( ω, ω ), wh ω ˆ d, s ω 3 sandard devaon s σ = ; and he demand for end-produc durng perod, > S, 3 supposed saonary, s unformly dsrbued on dˆ ω, dˆ + ω ]. 3 Producon and nvenory conrol polces 3. Base-sock polcy [ he proposed producon conrol polcy for each produc follows he base sock approach. Due o he exsence of capacy consrans and lead mes a each sage, orders may no be served mmedaely. o smooh he varaons, nvenory levels are replaced by nvenory posons. he seleced base sock polcy consss of launchng producon and supply orders a he begnnng of each lead-me perod o manan consan he local nvenory poson (sock level - backorders + orders no ye delvered). hs ype of a polcy has been shown o be opmal wh respec o he oal average cos under saonary condons n he absence of se-up coss (Federgruen and Zpkn 984). However, he dffculy s n he compuaon of he opmal order-up-o levels, especally n he dsrbued conex where coss and benefs are locally compued. In he saonary case, each base sock level can be chosen for each produc usng he classc approach whch guaranees some qualy bound on he sockou probably, denoed by: ε. Prob(demand durng θ s ) ε. * * s

10 J.-C. Henne,n e durng lead-me θ can be decomposed no he predced par and a random par. Accordngly, he base sock level can be decomposed as follows: * he cumulaed demand for produc { } s s ~ = ˆ + s wh sˆ = θ [( I Π) ] dˆ and ~ s = Φ ( ε ) (25) where Φ s he CDF (Cumulave Dsrbuon Funcon) of supplemenal demand for produc durng lead me θ. oe ha he par of he base sock level correspondng o he predced par of demand s: sˆ = θuˆ. he oher par of he base sock level, s~, can be consdered as safey sock. Under he proposed base-sock polcy, he curren nvenory level for produc s never * larger han s. As a consequence, he sysem can be desgned beforehand so ha nvenory consrans (9) are never volaed by mposng, for each sorage zone p, p * n s = p. (26) Under saonary condons, he sockou lm probably for end-producs can be h + c compued by he classcal newsboy formula (see e.g. (Poreus 99)): ε =, where: h + b c s he un producon cos for produc, h s he un holdng cos per me un for produc, b s he un backorder cos per me un for produc. For example, under unformly dsrbued demand perurbaon for end-producs, q * 2b Φ ( q) = + wh a = θ[( I Π) ] ω h 2c, and s = θ [( I Π) ] dˆ + a. 2 2a h + b 3.2 Opmaly of local base sock conrol In he decenralzed framework, he producon un assocaed wh produc ( ne +, ) s supposed o have nformaon abou local nvenory and producon varables ( r u, k ), and demand predcons for end-producs, dˆ. Addonally, he,,, k producon un receves oupu orders from downsream sages, polcy for produc s hen defned by u * + o. = s p, (27) o o. he local base-sock { } Oupu s he oal replenshmen supply for produc a he begnnng of perod, ordered by downsream producon uns and/or cusomers: o = π ju j + d j=. In hs

11 Local Invenory Conrol Polcy manner, nformaon abou producon levels propagaes upward n he supply chan a each perod. As menoned n secon 3., base sock polces naurally sasfy nvenory consrans (9), provded ha consrans (26) are sasfed. On he conrary, manufacurng consrans () are no auomacally sasfed by he producon levels gven by (27). However, f a nonempy range of he nomnal producon vecor s feasble, as gven by condons (23), producon levels (27) are supposed o sasfy consrans (9) and he doman n whch hs assumpon holds rue wll be specfed as he arge se n secon 3.3. On he bass of mean expeced demands for end-producs, he producon order for produc sared a perod can be decomposed as follows: uˆ = [( I Π) ] dˆ and by defnon, v = u uˆ (28) hen, usng (22), (25) and (28), he nvenory poson of produc a he begnnng of perod can be decomposed as follows: p, = r, + θ[( I Π) ˆ θ ] d + v k k=,. (29) Usng (28), (29), local polces (27) can be re-wren as: v = s * θ r I Π dˆ v [( I Π) ] dˆ, θ [( ) ] + o. (3) k k=, hen, wh (25) and nong ha dˆ + Πuˆ = uˆ v θ = ~ s r, v, k + π j k= j= v j + e Consderng for =,...,, each local nomnal polcy defned by he uple ( ~ s, uˆ ), he sysem polcy can be descrbed around hs nomnal pon by he ncremenal varables y = r ~ s, v = u uˆ ). ( Globally, vecors y = [ y, L, y ] and v = [ v, L, v ] are defned and he ncremenal sysem can be represened by sae vecor (Henne 23): (3) x θ = [ y v L v ]. (32) Under he local nformaon assumpon nvoked n hs sudy, he enre sae vecor s no observed locally. Bu s used o formally defne he nfne horzon average cos creron around he nomnal pon ( ~ s, uˆ ): V ( x, v) = lm E + = lm E + = c( x, v c u ) + h max( r,)) + n = = = e b max( r,). (33)

12 J.-C. Henne he elemenary cos funcon c( x, v ) s he sum of ncremenal producon coss, backorder coss and nvenory coss. I s supposed o reach s mnmum value for x, v =, wh c(,) =. he followng resul can now be saed. = Proposon he local base-sock polcy defned by (27) s globally opmal wh respec o creron (33). Proof For k =,..., θ, defne marces Γ by: Γ =. Equvalenly, k k k θ k j j= k Γ k = Dag( Γ ) wh Γ = f and only f k θ, else, Γ =. (34) he ncremenal conrol vecor derved from expressons (3) can be re-wren: = Fy + θ Gkv k e wh F = ( I Π) ; Gk = I Π) Γk, k= v + k ( k =,... θ. (35) hs expresson s almos dencal o he globally opmal conrol law consruced n (Henne 23). he only dfference s n he addonal erm, whch appears n (35). he law proposed n (Henne 23) was smply he sae feedback par: wh respec o creron (33) s obaned f he followng condons apply: E[ x ] = for S + θ +. E[ v ] = for S + e Fy + θ Gk v k. Opmaly k= Snce E[ e ] = S +, he same opmaly condons apply, and conrol law (35), whch s equvalen o local polces (27), s globally opmal wh respec o creron (33). e oe ha he addonal erm n expresson (3) derves from he synchronzaon assumpon ha producon levels a he begnnng of each perod are decded sage by sage, upward from fnal demands o prmary componens. If such a synchronzaon mechansm were no used as an operang phlosophy for he supply chan, he erm as well as π j= j v j would be deleed from (3); and, as a consequence, Proposon would no apply. In fac, s no dffcul o show ha hs synchronzaon assumpon s necessary for opmaly. 3.3 Defnon and accessbly of he arge se Opmaly of local polces gven by (27) has been shown under he assumpon ha such polces are feasble wh respec o consrans (9). Anoher propery of he opmal polcy proposed n (Henne 23) s ha naurally sasfes consrans (9) and () f he nal sae of he sysem belongs o a gven se, whch s posvely nvaran and ncluded n he e

13 Local Invenory Conrol Polcy doman of consrans. he same scheme can be used here, wh he mnor dfference of he addonal e erm n expresson (3), whch may be seen as an addve dsurbance on he sae feedback conrol. hus, provded ha he maxmal nvaran se conaned n he doman of consrans (9) s no empy, hs se can be seleced as a doman of admssble nal saes for ncremenal polces (3). hs se can also be seleced as a arge se for he shor-erm polcy. In hs sudy, no exernal demand wll be served durng he ramp up shor-erm horzon. In oher words, s assumed ha d =, {, }, {, S}. hen, snce shor-erm daa are deermnsc, he shor-erm par of he conrol can be smplfed by mposng o reach, n mnmal me, S, he nomnal sae of he sysem defned by local ermnal condons: r, S =, u, u u. (36) S = L =, = S θ s ~ ˆ In a global plannng scheme, s naural o solve hs deermnsc problem by Lnear Programm ng, o easly negrae consrans (9) and (), se he arge sae by (36), and mnmze S. Acually, such a echnque can also be appled n a oally decenralzed manner, snce no real-me daa s needed. Only sac daa and nal condons are needed o run such a program and can seem reasonable o assume ha a mnmal level of cooperaon can be esablshed among he supply chan parners o jonly opmze he sar-up sage. he mnmal value of S s he smalles value of for whch he followng se of lnear consrans s sasfed: { } { } u,,,, u = r {, }, { ne +, }, r gven p n r p p {, P}, {, }, {, } = θ r mu, l M r r, R,,,, = l= r = r, + u, θ π ju j {, }, {, }, r j= r = ~, s, u, = L = u, θ = uˆ. { } { } { } gven, (37) A global lnear cos funcon can be added o consrans (37) n order o solve he Lnear Program for each enave value of, so ha nal producon flows can be synchronzed n he mos economcal manner. Usng he cos parameers defned n secon 3., he cos funcon o be mnmzed akes he followng form: J = c u + h r. = = (38) Oher echnques, such as he ones based on MRP and MRPII, can be used o buld nal nvenores. hey can be mplemened n a decenralzed, dsrbued manner accordng o he mulsage producon srucure, as proposed n Buzaco (997). However, such

14 J.-C. Henne echnques do no make an opmal use of nvenory and producon capaces. I s, herefore, advsable o organze he sar-up of a supply chan nework n a collaborave manner, explong he nal nvenores of he parners o reduce he duraon of sar-up and globally mnmze sar-up coss. 4. An example he example s aken from Henne (23). Arc valuaons n he gozno graph of Fg. correspond o he bll of maerals wh npu marx Π = and oupu marx I. Producon lead mes are θ = me uns for =, 3, 5, θ = 2 me uns for =2, 4. Fgure abou here z he npu-oupu marx s ( z) = 2 z 2 z = 2 3 ( z) 2z 2z z z 6z 3z z z + 4z 2z + 4z 2z z 2 z 2 z and hus 2 3 z 2 2 z and ( I Π) = In hs example, he maxmal degree n z of polynomal marx ( z ) s ρ = 5. I s he mnmal duraon of he ransen phase. Demands per perod for producs and 2 have been randomly genera ed accordng o unform dsrbuons around he r mean values dˆ = 2, d ˆ 2 = 5, on nervals [ 3] and [ 5 25], respecvely. he arge sae for he shor-erm ransen polcy s gven by: r ~, S = s for {, L,5}, u, S = uˆ for {, L,5}, u, S 2 = uˆ for = { 2,4}. From expresson (9), predced producon levels per perod durng saonary runnng condons are:

15 Local Invenory Conrol Polcy dˆ = 2 dˆ 2 = 5 u ˆ = 2dˆ + 2 ˆ = d 7. he seleced base sock levels are gven by: 2 6d ˆ + 6 ˆ d2 = 2 5d ˆ + 6 ˆ d2 = 9 ~ s 6 45 = ransen rajecores are compued by solvng he Lnear Program defned by creron (38) and consrans (37), wh he followng parameers: c =, c 2 = 9, c 3 = 5, c4 = c5 = 3, h = 5, h 2 = 4, h 3 = h4 = h5 =. Manufacurng and sorage capacy per perod are supposed dsrbued by producs, wh he followng upper bounds, respecvely: M = 8, M 2 = 6, M 3 = 28, M 4 = 84, M 5 = 76 and = 8, 2 = 75, 3 = 27, 4 = 3, 5 = 82. For he value =5, he LP happens o be feasble. If he problem s nfeasble, mus be ncreased unl feasbly s aaned. he base sock polcy s mplemened from S = 5. Smulaon resuls are presened by fgure 2. hey show he sablzng propery of he proposed base sock polcy; demand varance s no amplfed upward hrough he supply chan. I s clear from he curves ha as soon as he arge sae s reached a he end of he sar-up sage, saonary dynamcs are reached wh very moderae oscllaons due o perurbaons on end-produc demand. 5. Conclusons Fg. 2 abou here radonally, producon sysems were generally organzed under he phlosophy of Compuer Inegraed Manufacurng, amng a full coordnaon and global opmzaon of he sysem. owadays, he new prevalng paradgms are Supply Chan Managemen and Enerprse eworks, focusng on organzaonal adapably and local decsonal auonomy. Such dsrbued srucures are ceranly more realsc and manageable, bu hey ofen preclude mplemenaon of globally opmal managemen polces. hs sudy has shown ha, n spe of he lack of nformaon a he level of producon uns n mulsage sysems, s possble o buld manufacurng polces ha perform as well as he opmal negraed polcy. hs resul, somewha surprsng and very neresng n pracce, akes advanage of he nerconneced srucure of he sysem. More generally, he sudy rases he ssue of deermnng he mnmal nformaon necessary o locally mplemen a polcy ha s globally opmal. Acknowledgemens he auhor wans o hank he gues Edor, Alexandre Dolgu, and hree anonymous referees for her helpful commens and suggesons.

16 J.-C. Henne References Axsaer, S., Supply chan operaons: Seral and dsrbuon nvenory sysems. Handbooks n Operaons Research and Managemen Scence, : Supply Chan Managemen: Desgn, Coordnaon and Operaon, eded by de Kok, A.G. e S.C. Graves, pp , 23 (orh Holland Amserdam). Baba, M.Z. and Dallery, Y., An analyss of forecas based reorder pon polces: he benef of usng forecass, n Informaon Conrol Problems n Manufacurng 26, A Proceedngs volume from he 2h IFAC Inernaonal Symposum, eded by Dolgu A., Morel G. and Perera C.E., 26, 3, pp (Elsever Ld, Oxford, UK). Baker, K.R., Requremens plannng, Handbooks n Operaons Research and Managemen Scence, 4, eded by Graves S.C., Rnnooy Kan A.H.G., Zpkn P.H., pp , 993 (orh-holland, Amserdam). Berman, A. and Plemmons, R.J., on egave Marces n he Mahemacal Scences, 979 (Academc Press, ew York, London). Buzaco, J.A., Connuous me dsrbued decenralzed MRP, Producon Plannng and Conrol, 997, 8(), Cachon, G. P., and Zpkn, P. H., Compeve and Co-operave Invenory Polces n a wo- Sage Supply Chan, Managemen Scence, 999, 45, Caldeney, R., Wen, L., Analyss of a Decenralzed Producon-Invenory Sysem Manufacurng & Servce Operaons Managemen, 23, 5, 7. Dejonckheere, J., Dsney, S.M., Lambrech, M.R. and owll, D.R., Measurng and avodng he bullwhp effec: A conrol heorec approach, European J. Op. Res., 23, 47(3), Fawce, S.E., Magnan, G.M. and McCarer, M.W., Benefs, barrers, and brdges o effecve supply chan managemen, Supply Chan Managemen: An Inernaonal Journal, 28, 3(), Federgruen, A., & Zpkn, P., Compuaonal ssues n an nfne-horzon, mul-echelon nvenory model. Operaons Research, 984, 32, Graves, S.C., A sngle-em nvenory model for a nonsaonary demand process, Manufacurng & Servce Operaons Managemen, 999,, 5-6. Hadley, G., Whn,., Analyss of Invenory Sysems,963 (Prence-Hall, Englewood Clffs, J). Henne, J.-C., A bmodal scheme for mul-sage producon and nvenory conrol, Auomaca, 23, 39,

17 Local Invenory Conrol Polcy Henne, J.-C., Load and nvenory flucuaons n supply chans, n Smulaon-Based Case- Sudes n Logscs: Educaon and Appled Research, eded by Merkuryev Y., Pera M.A., Guash., Merkuryeva G., 28 (Sprnger, o appear). Lau, J.S.K., Huang, G.Q., and Mak KL., Impac of nformaon sharng on nvenory replenshmen n dvergen supply chans, Inernaonal Journal of Producon Research, 24, 42(5): Lee, L.H., Padmanabhan, P. and Whang, S., Informaon Dsoron In A Supply Chan: he Bullwhp Effec, Managemen Scence, 997, 43(4), Love, D. and Bareka, M.M., Decenralzed, dsrbued MRP: solvng conrol problems n cellular manufacurng, Producon and Invenory Managemen Journal, 989, 3, Poreus, E.L., Sochasc nvenory heory, n Handbooks n Operaons Research and Managemen Scence, 2 Sochasc Models, 99 (orh-holland, Amserdam). Supply Chan Councl, SCOR bookle, verson Appendx Ls of noaons n e number of end-producs oal number of producs Π = (( π j )) npu marx of he gozno graph I deny marx d exernal demand for produc a perod (quany known a he begnnng of perod or even a he perod before) o quany of produc delvered o downsream producon sages a perod β amoun of backorders for end-produc a he end of perod s sock level of produc on hand a he end of perod r arhmec nvenory level of produc on hand a he end of perod. I s posve represen a sock level, negave o represen a level of backorders u quany of produc whose producon s sared a he begnnng of perod θ producon lead-me for produc p n amoun of space p needed o sore one un of produc Capacy of space p p r m amoun of resource r, needed per perod o produce one un of produc M r capacy of resource r per me un p nvenory poson a he end of perod z advance operaor z delay operaor θ = max θ {, } maxmal lead-me

18 J.-C. Henne ρ S dˆ e maxmal degree n z of polynomal marx [ Dag( z shor erm me horzon θ ) Π] predced componen of exernal demand for produc a perod dsurbance of exernal demand for produc a perod û predced quany of produc whose producon s sared a he begnnng of each perod ô predced quany of produc delvered o downsream producon sages per perod δ maxmal devaon from nomnal quany per perod ( ω, ω ) maxmal negave/posve devaons from predced exernal demand per ε * s ŝ perod sockou probably for produc base sock nvenory poson for produc runnng sock par of he base sock nvenory poson for produc s~ safey sock level for produc Φ (.) CDF (cumulave dsrbuon funcon) of supplemenal demand for produc durng lead me θ c un producon cos for produc, h un holdng cos per me un for produc, b un backorder cos per me un for produc v = u uˆ ncremenal producon of produc sared a he begnnng of perod y = r ~ s ncremenal nvenory level a he end of perod û dˆ

19 Local Invenory Conrol Polcy Ls of fgures Average demand rae d Average demand rae d 2 Fgure A 5-produc mulsage example Invenory for produc Producon of produc Invenory for produc 2 Producon of produc 2

20 J.-C. Henne Invenory for produc 3 Producon of produc Invenory for produc 4 Producon of produc Invenory for produc 5 Producon of produc 5 Fgure 2 Invenory and producon curves