Inventory Control in a Multi-Supplier System

Transcription

1 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 Inventory Control n a Mult-Suppler Syste Yasen Arda and Jean-Claude Hennet LAAS-CRS, 7 Avenue du Colonel Roche, 3077 Toulouse Cedex 4, FRACE Abstract An enterprse network s analyzed fro the vewpont of an end-product anufacturer who receves custoer orders and organses hs producton and supply polcy so as to nze the su of hs average holdng cost and average stock-out cost. For each an coponent to be ordered, the producer has several possble supplers. The arrvals of custoers orders are rando and delvery tes fro supplers are also supposed rando. Ths supply syste s represented as a queung network and the producer uses a base-stock nventory control polcy that keeps constant the poston nventory level (current nventory level pendng replenshent orders). The decson varables are the reference poston nventory level and the percentages of orders sent to the dfferent supplers. In the queung network odel, the percentages of orders are pleented as Bernoull branchng paraeters. A close-for expresson of the expected cost crteron s obtaned as a coplex non-lnear functon of decson varables. A decoposed approach s proposed for solvng the optzaton proble n an approxate anner. The qualty of the approxate soluton s evaluated by coparson to the exact soluton, whch can be coputed nuercally n soe sple cases, n partcular n the two-suppler case. uercal applcatons show the portant econoc advantage for the producer of sendng orders to several supplers rather than to a sngle one. Keywords: Inventory Control, Supply Chan, Stochastc Models Correspondng author : Jean-Claude Hennet, LAAS-CRS, 7 Avenue du Colonel Roche, 3077 Toulouse Cedex 4, France, Tel : , Fax : , e-al: hennet@laas.fr. - -

2 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 Inventory Control n a Mult-Suppler Syste Yasen Arda and Jean-Claude Hennet LAAS-CRS, 7 Avenue du Colonel Roche, 3077 Toulouse Cedex 4, FRACE Tel : , Fax : e-al: hennet (yarda)@laas.fr). Introducton A aor dffculty n supply chan organsaton and anageent s to conclate global effcency wth local autonoy. When consderng a network of cooperatng enterprses, a basc obectve s to organze ntegraton n a non-copulsory anner, so as to antan the autonoy of partners. A possble approach for cobnng ntegraton and autonoy s through partly autoated negotaton processes (Jennngs et al, 00, Besebel et al., 00). Once the negotaton has started, paraeters can be updated and other crtera can enter nto play, such as costs (fxed and varable orderng costs), qualty and non-foralzed preference. In such a way, the syste coplexty nherently attached to supply chan organzaton, can be anaged through negotaton between the an actors, each one of the basng hs decson upon a local optzaton process. Such a schee sees appealng by preservng the autonoy and decson optzaton aong the partners of an Enterprse etwork. However, t has been shown to be globally sub-optal (see e.g. Cachon and Zypkn, 999), by drvng the syste to a ash equlbru whch can globally perfor very poorly wth respect to the nal total cost crteron. Several correctve actons have been proposed to copensate for ths bas. They anly consst n sharng rsks and costs aong partners and ths can be pleented through contracts odfyng local crtera n a globally ore effcent anner (Cachon and Larvere, 00, Chen et al., 00). Another well-known factor of neffcency n supply chans s the so-called bullwhp effect, whch tends to propagate and aplfy dsturbances upward along the supply chan. A supply chan generally nvolves several sources of dsturbances, and coordnaton of product flows s fragle snce varatons n external supply and deand ay be aplfed through nterconnectons between partners. Soe typcal causes for such aplfcaton are capacty ltatons and the use of dfferent batch szes between partners (Lee and Bllngton, 99). - -

3 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 The study analyzes one of the basc eleents of a supply chan: the operatonal relatonshps between an end-producer and hs drect supplers. A sple queung odel s constructed, based on the assuptons of a Posson external deand for end-products, nstantaneous delvery to the custoer fro the producer s stock and an exponentally dstrbuted servce te for each suppler. Only one basc coponent of the end-product s consdered and supplers are supposed equvalent n ters of qualty and cost. They only dffer by ther average servce te. In spte of ts splcty, such a odel grasps the an ssues for the producer: should he use only one suppler, the best one n ters of a relevant perforance ndex, or should he dspatch hs orders between dfferent supplers? In the latter case, what supplers should be selected and for what percentage of the deand? The potental usefulness of the odel for the suppler s n the a-pror deternaton of hs optal nventory level and of the volues (or frequency) of hs orders to supplers, based on a-pror evaluaton of ther average delvery te. In practce, ths a-pror knowledge can be consdered as a startng pont n the negotaton process that wll be undertaken wth the supplers. By assung untary deands and orders and eoryless arrval, dspatchng and servce processes, we get rd of the bullwhp effect n the odel, to concentrate on the ean perforance analyss. Fro the lterature on the bullwhp effect, t s assued that t can be treated separately, through a ore detaled odel, ether through synchronous schedulng (Cachon, 999) or/and through an adequate choce of batch szes (Rddalls and Bennett, 00). To optze the producer s nventory level and the order dspatchng proportons, t s essental to cobne the effects of rando fluctuatons on deand flows, and delays of delveres fro supplers. Rando deands have often been consdered n the exstng odels of nventory control, specally the ones based on the newsvendor paradg (Arrow et al., 95, Porteus, 990). On the contrary, rando delays n part delveres have not often been ntegrated n odels explctly. An excepton s the work of Dolgu and Louly (00), n whch several supplers wth rando delvery delays are consdered. In ther work, the dfferent supplers provde dfferent parts to be assebled by the producers. Then, there s nterdependency between the nventory postons of the end product and of ts coponents. However, the nventory postons of the dfferent coponents can be ndependently controlled through the nforaton and orderng syste. On the contrary, the case of a centralzed nventory of a sngle coponent does not offer the sae possblty of decoposton. In ths paper, a centralzed nventory control odel s constructed to cobne supply and deand randoness. The obectve of the producer s to nze hs average cost by - 3 -

4 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 constructng an orderng polcy, defned by an optal reference nventory poston and a rule for selectng the suppler of each nventory replenshent order. Secton forulates the optal nventory and orderng proble for one producer and several supplers. Then secton 3 solves optally the order dspatchng proble n the partcular ake to order case, and an approxate resoluton technque s presented n secton 4 for the ake to stock case. The perforance of the approxate soluton s coparatvely evaluated on sple exaples n secton 5. Fnally, soe conclusons and perspectves are presented.. The optal nventory and orderng polcy The current nventory level of the product consdered at te t s denoted I(t). It s defned as the dfference between the on-hand nventory and the aount of backorders. In general, an orderng decson should not be based only on the nventory level. One should also consder the nuber of replenshent orders whch have been placed earler and not yet been delvered, denoted u(t). The global state of the syste can then be characterzed by the nventory poston, denoted P(t) defned by: P(t) I(t) u(t) () Dependng on the nforaton syste avalable, an nventory poston ay be controlled at any te through a contnuous revew polcy, or at perodc tes through a perodc revew polcy. Then, the control polcy deternes when and how uch to order. Dfferent control polces ay be appled, wthn the lts of the legal agreeents between producer and suppler. One of the ost popular contnuous revew polcy s the (s,s) polcy, n whch s stands for the nventory poston order pont and S for the nventory poston replenshent level. The basestock polcy can be seen as a varant of the (s,s) polcy, for whch an order s placed whenever a deand coes, so as to peranently antan the nventory poston S. Ths type of a polcy has been shown to be optal under constant average deand rates or untary deands wth ndependent dentcally dstrbuted (..d.) arrval dates, whenever the cost crteron only depends on the nventory poston (Axsater, 000). Moreover, under untary deands wth (..d.) arrval dates, the optal base stock polcy reduces to the polcy (s,s) wth s S-. Ths polcy s denoted the reference nventory polcy. It wll be studed n the sequel, n the ult-suppler case

5 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 In the ake-to stock context, when an order coes to the producer, t s edately satsfed f ts aount s avalable n the stock. If not, t has to wat untl the nventory has been suffcently replenshed by the arrval of products fro supplers. In both cases, an order s placed fro the producer to the suppler whenever a deand coes and has the sae aount ( n the untary case). As shown n (Bollon et al., 000), such a base stock control polcy can also be nterpreted as a Kanban echans After an ntal nventory replenshent stage, the reference nventory polcy antans constant the nventory poston of the producer: P( t) S t t () 0 If the rando processes of deand arrvals and supply delveres are statonary, then, under a statonary (S-,S) base stock polcy, the syste (producer supplers) reaches statonary condtons characterzed by statonary probabltes of the nuber of orders placed by the producer and not yet delvered by the supplers. In the sequel, these probabltes wll be coputed n the case of exponental dstrbutons of deand arrvals and supply lead tes. Fro the producer vewpont, the cost functon to be nzed s the su of the average holdng cost and the average stock-out cost. Consder the followng notatons: I s the rando varable representng the producer nventory level n statonary condtons u s the rando varable representng the nuber of uncopleted orders fro the producer to the suppler n ergodc condtons h s the unt holdng cost, b s the unt stock-out cost. In statonary condtons under the (S-,S) base stock polcy, rando varables I and w are related by the followng equalty, derved fro () and (): IS-u (3) Usng notaton (x) for ax(x,0) and (x) - for ax(-x,0), the average cost crteron takes the for: C(S) E [ h (I) b (I) ]. (4) Deands are assued untary. They enter the syste as a Posson process wth rate > 0. When a untary deand arrves at te t, the producer serves t edately f I(t) 0. He wats f I(t)<0. In both cases, he apples the (S-,S) orderng polcy by sendng a correspondng order to suppler wth probablty α satsfyng 0 α and α. Such - 5 -

6 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 untary orders can be seen as a lt case that axzes the nforaton effcency of order transfer fro the producer to the supplers and nzes the bullwhp effect by avodng transsson dstortons due to dfferences n batch polces. Each suppler s supposed to have an exponental servce rate and treats the requests n the FIFO (Frst In Frst Out) order. Let denote the ean servce rate of suppler,,, wth for (,,-,,,). Such an order dspatchng polcy fro the producer to the suppler s known as a Bernoull splttng process. A well known property of ths process s { (t), t 0},, are Posson processes wth rates p. Moreover, these processes are utually ndependent.the proof of ths property can be found, n partcular, n (Ross, 000). As a consequence, each suppler can be odelled as an M/M/ queue wth arrval rate α and servce rate. The set of supplers s represented as a network of ndependent M/M/ queues n parallel. The probablty of havng k orders n queue s gven by: k α α P( k ) Pr{ k orders n } (5) The necessary and suffcent condton for stablty of queue s α ρ <, wth ρ /. The probablty for the network of queues to be n state { k,k } product-for expresson (Baskett et al, 975): K,...,k,...,k s gven by the k P ( k,..., k ) ( α ρ ) ( α ρ ) (6) In the order dspatchng proble, Bernoull paraeters α, α α are decson varables. Ther optal values express the optal assgnent probabltes n steady state. The consdered optzaton crteron s the su of the ean holdng cost and the ean stock-out cost per te unt. The obectve of the study s to copute the optal Bernoull paraeters and the optal base stock value, S nzng the average cost crteron. In the case of supplers, the nuber of orders not yet delvered to the producer s equal to the total nuber of orders n the open queung network coposed by the queues of orders cong fro the producer. Let K denote the nuber of orders sttng n the th suppler queue. P(k ) Pr{K k } s defned by equaton (5). Then, the probablty of havng w orders watng n the supplers queue s gven by P { K K... K w } w Pr Probablty P w can be obtaned by coposton of the probabltes related to the queues. Such a coposton can be coputed fro the probablty generatng functon. Assung - 6 -

7 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 α ρ α ρ, the product for () can be transfored nto a su as n (Klenrock, 975). Then one obtans (Arda and Hennet, 003) w- w- ( ) ρ Pw H b α wth H() ( α ρ ), b k α k ρ k α ρ.(7) The ean value of the nuber of pendng orders s denoted Z, wth w 0 Z E[ u] wp w. (8) Crteron (4) can be re-wrtten: C(S) (h b) (S-w)P S w0 and the followng expresson s obtaned: C(S, α, α,..., α ) (h b) H( ) w b ( Z S ) (9) - - S S Sα ρ α ρ (-α ρ ) αρ (0) b b( S ). ρ ( ρ ) α α ρ α Convexty of crteron (0) wth respect to varables α,, α, S s not guaranteed n general. Therefore, nzaton of crteron (0) subect to constrants 0 α for,, and α appears to be a hard optzaton proble. 3. The optal soluton n the MTO case In the Make-to Order case, the base stock level s supposed equal to zero. Then, the cost functon reduces to the backorder cost : α C( α, α,..., α ) bz b () α Mnzng the backorder cost () s equvalent to nzng the nuber of unsatsfed orders or equvalently, fro Lttle forula, nzng the expected watng te E[T] Z /. Wthout loss of generalty, the supplers can be ranked n the decreasng order of ther servce rate : > >... > 0. The proble constrants are based on the followng > condtons: - Bernoull paraeters should be feasble. Ths condton requres the followng constrants: - 7 -

8 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp α,..., () α (3) - Stablty of the queung network requres the followng condtons : α <,..., (4) - Moreover, a necessary and suffcent condton for the exstence of a set of Bernoull paraeters, (α ;,,) satsfyng constrants (), (3), and (4) s : < (5) Suppose that condton (5) s satsfed by the proble data. Then, the Make-to Order optzaton proble, denoted proble (P) takes the followng for : α nze (6) α,..., α α under constrant (3), and constrant (7) whch replaces () and (4) : 0 α n(, ),..., (7) All the constrants are lnear and n the feasble doan, crteron E[T] s convex: d E[ T ] d dα dα ( α ) ( α ) 4 ( α ) ( α ) 3. on negatvty of d E[ T ] dα s always guaranteed under constrant (4). Therefore, proble (P) s convex and has a unque nu defned by the frst order optalty condtons: d E[ T ] 0 for,, under constrants (3), and (7). dα 3. Resoluton of a relaxed proble Consder now the case when the optal soluton of the proble defned by (6) and (3) naturally satsfes condton (7). Then, ths soluton s optal for proble (P). The Lagrangean of the relaxed proble can be wrtten : - 8 -

9 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 α L p α α wth p the Lagrange paraeter assocated wth the equalty constrant : α. Let α be the optal value of the Bernoull paraeter α for,,. Then, the optal soluton of the relaxed proble satsfes the followng set of condtons: d L p 0 dα ( α ),..., α (9) For any par ( α, α ), condton (8) can be re-wrtten : ( α ) α (0) By sung over both ters of equaton (0), one obtans: ( α ) ( α ) (8) α Under constrant α, ths equaton becoes : the followng result s obtaned. Property : and thus, The optal values of Bernoull paraeters wth respect to the relaxed proble, are defned by the followng expressons: α ( ),..., () where s defned by : () If the optal values (α ) satsfy constrants (7), then the nu of the relaxed proble s feasble and therefore optal for proble (P). The feasblty condton s re-wrtten: 0 ( ) n(, ),..., (3) - 9 -

10 - 0 - Fro condton (3), > 0. Therefore, nequaltes (3) can be replaced by :,..., 0 (4) eanng 0 α. The left-sde nequalty can be re wrtten :,..., (5) And, usng α, the rght-sde nequalty becoes satsfed. 3. The restrcted choce proble Fro the rankng of servce rates n the decreasng order 0... > > > >, f. nequalty (5) s not satsfed for 0, wth 0, then, t s also volated for 0,,. In ths case, the restrctve choce proble s obtaned by posng 0 α for 0,, To show the relevance of the restrcted choce proble, the followng paraeter s defned for,,, under the conventon 0: (6) The evoluton of satsfes the followng propertes. Property For postve values of paraeters et ( -), the evoluton of satsfes the followng rules : () < < < () (3) > > > Proof Fro (6), one obtans (7) (8) ths ples : 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6

11 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 ) ( - (9) ( - ) (30) The rules of property drectly derve fro these two equaltes. Property 3 : Paraeter ncreases wth for. Then, paraeter onotonously decreases wth for <. The axal value of paraeter s obtaned for ( ), whch s the unque ndex satsfyng: > and < (3) Proof : The proof s presented n two parts. ) Exstence of the ndex : For any set of paraeters (,,, ), condton (6) ples > 0. Let n be the sallest ndex satsfyng n >. Replacng by n n equaton (3), one obtans: n- n n n ( n - n- ) (3) Fro n > 0 and n- 0, equaton (33) ples n > n. If n n, then n. If not, relaton n > n ples n n the process s terated for n,,. > and n > n by the thrd rule of Property. And so, Then, startng fro 0, we obtan. Then, f >,. If not, relaton ples - et - fro the frst rule of Property. Thus, there exsts a unque ndex, wth, that satsfes relatons (3). ote that n the case n, equaton (3) ples > and thus. ) The evoluton of paraeter follows property

12 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 For ndces,,, relaton > > ples > 0.Then, fro, we can derve fro Property, and Therefore, relaton < ples <. Applyng the sae reasonng for 3,,, shows that paraeter onotonously decreases wth for <. Accordng to equatons (9) and (30), paraeter ncreases wth for n-, snce > 0 for,,. Then, knowng that > 0 for n,, and that >, one can wrte > - and > fro Property. Consequently, relaton > > - ples - > -. The sae reasonng can then be appled to - n,, -3. And thus, the value of paraeter ncreases wth for and fnally, the axal value of paraeter s obtaned for. If condton (5) s not satsfed, the constraned proble can be solved usng the followng property : Property 4 : Suppose that condton (3) s satsfed and consder the ndex ( ) whch satsfes relatons (3). Then, the optal values of Bernoull paraeters are gven by: ( ) for,..., α (33) 0 for,..., Proof : () Feasblty of polcy α ( ) defned by property 4 : The set of Bernoull paraeters ( α ;,,) defned by (33) satsfes constrant (3) : α ( ) ( ). If, the set ( α, α 0 for,,) s feasble. If >, snce > for, - and >, then > > whch ples > for,. > Fro expresson (33), t followsα 0 for,. Moreover, the rght sde nequalty of constrant (5) s satsfed snce > 0. Thus, property 4 defnes a feasble polcy. () Optalty of polcy α ( ) : - -

13 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 Fro property 4, the polcy α ( ) s optal f. For <, fro the convexty of proble P wth respect to paraeters α, for,...,,, t suffces to show that the set of values ( α, α,..., α,0,..., 0 ) s locally optal. So, consder an ncrease α > 0 wth <. fro polcy α ( ). Then, constrant (3) ples a decrease of α for soe,, ( α < 0) under the followng feasblty constrant: α 0 (34) α The crteron varaton then takes the followng for : E[ T ] or, equvalently, E[ T ] ( α α α ( α ) α α α ( α α ))( α ) α α α α Fro relatons α > 0, α < 0, and expresson (33), the followng nequalty s obtaned. (35) (36) E[ T ] > ( α α ) α α α α α (37) Then, usng equaton (34), α can be replaced by - α n nequalty (37). And fro for -,,, we obtan : E[ T ] > α > 0 (38) Therefore, the polcy defned by paraeters (33) s optal. 4. An approxate soluton n the ake-to-stock case The ake-to-stock case corresponds to the general case, ncludng the ake-to-order case, whch can be characterzed by a null base-stock level (S0). Due to the coplexty of the cost functon (0), t s proposed to decopose the proble nto two parts. In the frst part, Bernoull paraeters are the decson varables whle the base stock level s supposed to take the zero value. These assuptons are the sae as for Proble (P). They correspond to the MTO (Make to Order) case solved at the precedng secton. In the second part of the proble; - 3 -

14 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 denoted (P), the values of Bernoull paraeters are supposed gven and the only decson varable to be deterned s the base stock level. In ths second part, the Bernoull paraeters values obtaned n (P) are used as nput data for proble (P) and the optal value of the nventory capacty, S, s coputed usng the dscrete verson of the newsvendor proble. 4. Coputaton of the dspatchng paraeters Proble (P) s solved as descrbed n secton 3. The value of s calculated by Property 3. Then, the reference values of Bernoull paraeters are drectly coputed by explct expressons (33). The error derved fro the applcaton of ths approxaton schee wll be evaluated n secton Deternaton of the base stock level Fro crteron expresson (9), consder the ncreental functon G( S) C( S ) C( S). One obtans: G(S) (h b) Prob (u S) -b. The PDF F(S) Prob{ u S } beng a onotonous ncreasng functon, so s G(S). Then, the value S for whch C(S ) s optal satsfes: C( S ) C( S ) G( S ) 0 C( S ) < C( S ) G( S ) > 0. Therefore, a necessary and suffcent condton for optalty s gven by the condton: (39) S w 0 S b P w < Pw. (40) h b w 0 For the order dspatchng polcy α ( ), expresson (7) of P w leads to evaluaton the followng quantty, fro whch the soluton S of Proble (P) can be coputed fro (40): S S S α ρ ( α ρ ) ( ) 0 ρ Pw H b α w. 5. Evaluaton of the Approxate Method The approxaton schee descrbed n secton 4 reles on two splfcatons. The frst one conssts n replacng the global optsaton proble, wth varables α,...,α and S by an ndependent proble (proble P) n α,...,α, followed by a proble n S (proble P). The second splfcaton conssts n solvng proble (P) for a value of S (S0) whch s not, n general, the optal one. It can be noted that wth the value of S posed n proble - 4 -

15 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 (P), t s not possble to terate the approxaton schee by updatng the value of S. As a consequence, the qualty of the approxate soluton s not guaranteed and there s a possblty to dentfy soe bas n the ethod. The approxate schee has been evaluated n the partcular case of one producer and two supplers. In ths case, the global optal soluton can be easly coputed by exploraton of the feasble doan (Arda and Hennet, 003). uercal evaluatons reported on table, show an econoc advantage for the producer of sendng orders to several supplers rather than to a sngle one, even when the second one s clearly less effcent than the frst one. They also show that the approxaton ethod s satsfactory wth an average devaton of less than 8% fro the optu, but a strong tendency (ore than %) to over-evaluate the dspatchng paraeters assocated wth the ost effcent supplers

16 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 Exaple Exaple Exaple Exaple Exaple Exaple 6.5 Exaple Exaple Exaple Exaple Exaple 0.9 Exaple α α S Crteron Optal soluton wth suppler,000 0,000 30,000 30,957 Optal soluton wth supplers 0,740 0,60 5,000 4,494 Sub-optal soluton wth supplers 0,79 0,09 6,000 5,50 Optal soluton wth suppler,000 0,000 30,000 30,957 Optal soluton wth supplers 0,698 0,30 4,000 3,75 Sub-optal soluton wth supplers 0,748 0,5 4,000 3,969 Optal soluton wth suppler,000 0,000 30,000 30,957 Optal soluton wth supplers 0,660 0,340 3,000,90 Sub-optal soluton wth supplers 0,707 0,93 3,000,77 Optal soluton wth suppler,000 0,000 30,000 30,957 Optal soluton wth supplers 0,66 0,374,000,449 Sub-optal soluton wth supplers 0,667 0,333,000,76 Optal soluton wth suppler,000 0,000 30,000 30,957 Optal soluton wth supplers 0,596 0,404,000 0,738 Sub-optal soluton wth supplers 0,68 0,37,000 0,908 Optal soluton wth suppler,000 0,000 30,000 30,957 Optal soluton wth supplers 0,63 0,368 8,000 7,906 Sub-optal soluton wth supplers 0,667 0,333 8,000 8,049 Optal soluton wth suppler,000 0,000 9,000 9,955 Optal soluton wth supplers 0,845 0,55 9,000 8,64 Sub-optal soluton wth supplers,000 0,000 9,000 9,955 Optal soluton wth suppler,000 0,000 9,000 9,955 Optal soluton wth supplers 0,85 0,75 8,000 8,80 Sub-optal soluton wth supplers 0,966 0,034 9,000 9,438 Optal soluton wth suppler,000 0,000 9,000 9,955 Optal soluton wth supplers 0,790 0,0 8,000 7,99 Sub-optal soluton wth supplers 0,93 0,068 9,000 9,05 Optal soluton wth suppler,000 0,000 9,000 9,955 Optal soluton wth supplers 0,760 0,40 8,000 7,780 Sub-optal soluton wth supplers 0,897 0,03 8,000 8,673 Optal soluton wth suppler,000 0,000 9,000 9,955 Optal soluton wth supplers 0,730 0,70 8,000 7,67 Sub-optal soluton wth supplers 0,863 0,37 8,000 8,56 Optal soluton wth suppler,000 0,000 9,000 9,955 Optal soluton wth supplers 0,70 0,90 7,000 7,356 Sub-optal soluton wth supplers 0,88 0,7 8,000 7,953 Table Coparatve Results 6. Conclusons Cooperaton between the actors of a supply chan s a dffcult proble due to the dstrbuted nature of the syste and the assocated degrees of decsonal autonoy of the actors

17 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 egotaton can be seen as a basc tool to cobne autonoy and ntegraton. However, at the present te, there s a lack of decson support tools for negotaton. In the partcular case of a negotaton between one producer and supplers, the producer needs to have a clear vson of hs own nterest n ters of costs and delay. The study has shown that n the case of a rando deand fro custoers and rando delvery delays fro supplers, t s generally proftable to dspatch the orders between several supplers rather than to drect all the replenshent orders toward a sngle one. More specfcally, the addressed proble was to deterne the percentages of orders to be drected toward each suppler and the base stock level. An approxate technque has been proposed to solve ths proble. Even f the qualty of ths technque s satsfactory, an on-gong research s devoted to ts proveent. References Arda Y. and J.C. Hennet, (003) Optzng the orderng polcy n a supply chan, LAAS Report Arrow, K., T. Harrs and J. Marschak (95). Optal Inventory Polcy. Econoetrca, 9, Axsater, S. (000). Inventory control. Ed. Kluwer Acadec Publshers. Baskett, F., K. M. Chandy, R. R. Muntz, F. G. Palacos (975). Open, closed and xed networks of queues wth dfferent classes of custoers. Journal of the Assocaton for Coputng Machnery,, o., Besebel, I., J.C. Hennet, E. Chacon (00) Coordnaton by herarchcal negotaton wthn an enterprse network, Proc. ICE 00, Roa (Itala), Bollon J.M., M. D Mascolo, Y. Fren, (000) Unfed forulaton of pull control polces usng (n,plus) algebra, Proc. 5th IAR Annual Meetng, ancy. Cachon, G.P.(999) Managng supply chan deand varablty wth scheduled order polces, Manageent Scence 45 (6) Cachon, G.P. and P.H. Zypkn, (999) Copettve and cooperatve nventory polces n a two-stage supply chan, Manageent Scence 45 (7) Cachon, G.P. and M. Larvere, (00), Contracts to assure supply: how to share deand forecasts n a supply chan?, Manageent Scence 47 (5) Chen, F., A. Federgruen and Y.S. Zheng, (00), Coordnaton echanss for a dstrbuton syste wth ons suppler and ultple retalers, Manageent Scence 47 (5)

18 3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.5-6 Dolgu A., M. A. Louly (00), A odel for supply plannng under lead te uncertanty Int. J. Producton Econocs 78 () Jennngs,.R., P. Faratn, A. R. Lousco, S. Parsons, C. Serra and M. Wooldrdge, (00) Autoated negotaton: prospects, ethods and challenges, Int. J. of Group Decson and egotaton 0 () Klenrock, L. (975). Queueng systes. Volue : Theory. Ed. John Wley & Sons. Lee, H.L., C. Bllngton, (99) Managng the supply chan nventory: ptfalls and opportuntes, Sloan Manageent Revew 33 (3) Porteus, E.L. (990) Stochastc Inventory Theory, n Handbooks n operatons research and anageent scence. Volue : Stochastc Models, Ed. orth-holland. Rddalls, C.E., S. Bennett (00) The optal control of batched producton and ts effect on deand aplfcaton, Int. J. Producton Econocs 7, Ross, S.M. (000) Introducton to probablty odels. Ed. A Harcourt Scence and Technology Copany, Acadec Press