Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Shem Ben Jouda Saouen Krchen Wald Klb November 014 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Shem Ben Jouda 1, Saouen Krchen 1, Wald Klb,* 1 LARODEC, Inttut Supéreur de Geton, Unverté de Tun, 41, rue de la Lberté, 000 Le Bardo, Tune Interunverty Reearch Centre on Enterpre Network, Logtc and Tranportaton (CIRRELT) and Kedge Bune School, 680 cour de la Lbératon, 33405 Talence Cedex, France Abtract. Th paper tude a coalton formaton problem for cooperatve replenhment wth a ngle uppler and multple frm. We conder that a vertcal cooperaton between the uppler and the et of frm already et ung perodc contract, and nvetgate the proftablty of a horzontal cooperaton between frm. Three bune tuaton are npected, focung ether on the collaboraton opportunte n the orderng, the nventory holdng or the tranportaton proce. In each collaboratve tuaton, a decon-makng approach appled to ae the expected proftablty of each frm and to determne f frm hould form coalton n order to maxmze ther proft. An exact oluton method baed on a game-theoretc approach developed, named cooperatve replenhment algorthm (CRA), whch generate core-table coalton for cot avng regardng all partner tandpont. An analytcal tudy of tablty condton regardng the orderng, holdng and tranportaton cot tructure wa performed to enhance the CRA reoluton procedure. Extenve experment baed on realtc ntance are provded to valdate the performance of the CRA propoed n term of oluton tablty and the convergence n term of runnng tme. The computatonal reult confrmed alo the potental beneft of horzontal collaboraton between frm n term of proft maxmzaton compared to the tand alone tuaton. They provded encouragng fndng on the proft enhancement that harng nventore and tranportaton poolng practce could produce wthn the upply chan. Keyword: Supply network, game theory, coalton formaton, quantty dcount, contract, jont replenhment, collaboratve tranportaton. Reult and vew expreed n th publcaton are the ole reponblty of the author and do not necearly reflect thoe of CIRRELT. Le réultat et opnon contenu dan cette publcaton ne reflètent pa nécearement la poton du CIRRELT et n'engagent pa a reponablté. * Correpondng author: Wald.Klb@crrelt.ca Dépôt légal Bblothèque et Archve natonale du uébec Bblothèque et Archve Canada, 014 Ben Jouda, Krchen, Klb and CIRRELT, 014
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network I. Introducton Supply chan (SC) have been characterzed a organzatonal network that are lnked through uptream and downtream procee and actvte that produce value n the form of product and ervce delvered to the hand of the ultmate cutomer (Chrtopher, 1998). A Supply Network (SN) a confguraton of faclte geographcally deployed n order to erve a predetermned cutomer bae. To repond to cutomer' demand wth the adequate ervce level, product are procured, tored and dtrbuted to demand zone ung a SN nvolvng everal faclte owned or rented by the frm. In uch network, day-to-day procurement, warehoung, tranportaton and demand management decon generate product flow n the network, wth aocated cot and revenue. For a gven product-market, where a et of frm are operatng, ther SN can be vewed a a complex network encompang a large number of decon makng unt. Each unt may have t own objectve and make t decon o a to maxmze elf-proft. Unfortunately, by actng olely, thee frm could dm collaboraton opportunte enhancng ther proft. At the trategc level, each frm decde on the number, the locaton and the mon of the faclte to operate and elect the portfolo of uppler to employ n order to degn a value-creatng SN (Klb et al., 010a). When the SN degned by the frm become operatonal, t managed to repond on a daly ba to cutomer demand through plannng and control procee. At the tactcal level, frm have to antcpate future demand and decde on nventory management, orderng and replenhment polce, whch organze the concrete relatonhp between the frm and ther uppler. Th generally done by electng approprate forecatng method, quantte to order, tranportaton opton, and coordnaton/collaboraton mechanm uch a contract and nformaton harng (Tay et al., 1999; Cachon and Zpkn, 1999). Fgure 1 llutrate the bune context condered, where a two-tage SN depcted: the frm' faclte tage where product are tored and delvered to cutomer when order are receved; and the uppler' tage dedcated to the replenhment of the frm' faclte from pre-agned uppler' te ung dfferent tranportaton mean. Suppler 1 Suppler j Vertcal collaboraton/ External collaboraton Frm 1 Frm Frm n Internal collaboraton Horzontal collaboraton/ External collaboraton Poble coalton Cutomer 1 Cutomer Cutomer k Fgure 1- The Bune Context under Collaboraton Opportunte Nowaday, major preoccupaton of frm are to deal wth the growng volatlty n demand (Chrtopher and Holweg, 011), the nherent carcty of raw materal and non-renewable energe and ther ncreang cot (Shell, 013), and the uncertanty and drupton n global ourcng (Klb and Martel, 013). In thee crcumtance, more effcency, utanablty and flexblty n the SN are neceary for frm n order to reman compettve. In th lne of thnkng, collaboraton foreeen a one of mechanm to help mtgatng thee preoccupaton CIRRELT-014-61 1
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network and to contrbute to the achevement of contemporary bune objectve. A llutrated n Fgure 1, two man categore of collaboraton ext: vertcal collaboraton (Chen et al., 001) and horzontal collaboraton (Krchen et al., 011). In counterpart of ther concentraton on core competence to be effectve, current frm mut rely on actvte externalzaton, ubcontractng and on vertcal collaboraton n the SC. The vblty offered by the latter n the SN nowaday wdely accepted, but eem not uffcent to overcome the ue underlned above. New acton uch a dual ourcng, aet harng, reource poolng, mutualaton or jont replenhment have been recently propoed to offer more flexblty and utanablty to SN (Chrtopher and Holweg, 011). Thee practce are uually aocated to horzontal cooperaton n the SC and ther mplementaton necetate effcent coordnaton tratege, n order to fnd the bet comprome between procurement, nventory holdng and tranportaton operaton optmzaton wthn the SC. The horzontal collaboraton matche compettor or non-compettor SC entte that have bult relatonhp and ntegrated procee by jonng ther order/goal (Mangan et al., 008) or harng ther reource (Smatupang and Srdharan, 00). Horzontal collaboraton among the contemporary practce a one could ee n recent large cale project uch a CO3 (www.co3-project.eu/), Modulucha (www.moduluhca.eu/) and the Phycal Internet Manfeto (www.phycalnternetntatve.org/). The nteret of uch horzontal cooperatve practce can be attrbuted to the ncreaed appeal of allance/coalton n dfferent ndutre where a group of frm ntated collaboratve replenhment, warehoung, orderng and/or tranportaton actvte wth a potve mpact on ther SN performance. However, there tll a lack of model that can addre thee operaton jontly n a collaboratve proce. In practce, frm could be ntereted to collaborate n the orderng, the nventory holdng, and/or the tranportaton actvty n a upply context. Accordngly, n the context of SN, a coalton/allance characterze the group of frm that decde to work jontly by groupng ther order for a gven product-market and ynchronzng ther replenhment polce. The am of the approach to fnd the mot proftable coalton tructure for each frm under collaboraton opportunte. To addre thee queton, we conder n th paper a coalton formaton mechanm that may allow frm accordng to the ncentve of varou collaboratve tuaton to maxmze ther own proft. Th tactcal decon-makng proce tend to fnd the mot proftable coalton/allance for all frm whle degnng the bet ourcng contract. In order to tudy extenvely the beneft of horzontal collaboraton between a et of frm, th paper propoe a collaboraton-baed modelng approach of the coalton formaton problem for the degn of ourcng contract. The contrbuton of th paper threefold. Frt, whle mot paper n the lterature on horzontal collaboraton addre the jont replenhment problem aumng the coalton already formed, we propoe here to tackle the coalton formaton problem, when the ourcng contract mut be degned, n order to better antcpate the expected proft from collaboraton under orderng, nventory holdng and tranportaton cot. Second, to account for the partcularte of realtc upply network, three collaboratve context are tuded n order to nvetgate the collaboraton opportunte offered alternatvely by the orderng, the nventory and the tranportaton actvte. Accordngly, three typcal cooperatve replenhment tuaton are nvetgated, namely: (a) Cooperatve Orderng wth Drect Shpment (CODS), (b) Jont Replenhment wth Shared Warehoung (JRSW) and (c) Jont Replenhment wth Synchronzed Shpment (JRSS). Thrd, to buld the mot proftable coalton, we propoe a game theoretc modelng approach baed on core-tablty concept. Game theory vewed a a powerful tool to CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network cope wth uch decon nce t reache to generate poble coalton and to dentfy an equlbrum tuaton that atfy all player n the game (Shenoy, 1979). We develop n th paper a cooperatve replenhment algorthm (CRA) that manage the coalton formaton proce ung the core-tablty condton. Several problem ntance are addreng to ae the effcency of the CRA n generatng approprate coalton tructure that fulfll all frm requrement. We how through an emprcal nvetgaton the avng n computatonal tme and convergence of our approach. Furthermore, an expermental tudy, of everal cooperatve replenhment tuaton, underlne the mportance of the collaboraton n the cot avng and llutrate how frm tend to further collaborate when jont orderng, nventory holdng and tranportaton cot are propoed. The ret of the paper organzed a follow. Secton cont n an overvew on vertcal and horzontal collaboraton n the context of SN. Secton 3 tate the decrpton and the formulaton of the coalton formaton problem. Secton 4 preent the three replenhment context nvetgated. Secton 5 dcue the tablty condton of the coalton tructure and propoe a cooperatve replenhment algorthm to olve the coalton formaton problem. Numercal reult are preented and dcued n Secton 6 n the objectve to how the performance of the propoed algorthm n generatng table coalton. Fnally, Secton 7 ummarze our fndng and propoe future reearch avenue. II. Vertcal and Horzontal Collaboraton n the Supply Chan Th ecton attempt to brefly overvew the extng work n the vertcal coordnaton and collaboraton n the upply chan, baed on contractng mechanm. In addton, t revew current tate of the art on the horzontal collaboraton n the upply network. It preent alo typcal collaboraton cae between frm reported from practce n the recent year. Comprehenve revew on the coordnaton mechanm n the upply chan are found n Arhnder et al., (011) and Cachon, (003). On the one hand, vertcal collaboraton between uppler and frm generally governed by a contract or an nformaton harng mechanm that mprove the performance of the SC. Such protocol offer the potental to prevent and mulate the level of the effcency when actng n collaboratve SC (Chen et al., 001; Cachon and Fher, 000). The focu of th paper dedcated to the contractng coordnaton mechanm baed on t potve mpact for the better management of entte relatonhp, for cot avng and for rk reducton. A hown n Fgure 1, the vertcal coordnaton take place on the uptream of the SC between uppler and frm that agree wth a pecfc contract tructure a wth the vendor managed nventory (VMI) approach; and on downtream de managng relatonhp between frm and cutomer a wth the cutomer relatonhp management (CRM). State of the art contractng coordnaton characterzed by the object of the contract, the number of perod and the nvolved entte. The object of the contract can be ether the quantty dcount (Weng, 004), the buy back (We et al., 013), the quantty flexblty (Tay, 1999), the revenue harng (Yao et al., 008, Zhou and Wang, 009) and the ubde/penalte (Cachon, 003). Note that regardng the number of perod n contract, mot of the extng paper condered ngle perod model. Generally, uch contract are degned a two-echelon n the SC ncludng one uppler and one frm (Jema and Karaemen, 007; Cachon, 003; Cachon, 005); or multple uppler and one frm (Hu et al., 013). Zaho et al, (010) addreed a coordnaton ue for manufacturer-retaler upply chan ung opton contract. They demontrated that the contract opton can coordnate the upply CIRRELT-014-61 3
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network chan wth rk harng. One hould menton that thee former work and alo Hu et al., (013) tuded the contractng problem under the aumpton of a determntc demand proce. Fnally, Cachon and Larvere (005) modeled a tochatc veron of the demand proce when conderng the contract-baed coordnaton mechanm. Among the manageral nght underlned n the lterature cted above that the quantty and prcng decon n the SC are the mot nfluent component that can encourage uppler and frm to cooperate. Regardng the quantty dcount contract, the man ncentve to encourage uppler and frm to produce and purchae proftable quantte for the whole SC. Comprehenve lterature revew on quantty dcount are gven by Dolan and Frey (1987) and Vwanthan and Wang (003). The latter clafed the extng paper accordng to whether there are one or multple frm nvolved n the coordnaton mechanm. Corbett and De Groote, (000) and Corbett et al., (004) demontrated that the optmal contract for the uppler turn out to be quantty dcount contract. Chen et al. (001) focued on the tranacton and channel effcency ung the dcount quantty contract for a SC ncludng one uppler and one frm. Alternatvely, many reearcher tuded the role of quantty dcount a an effcent channel coordnaton mechanm (Cachon, 003; Jaber et al., 006). On the other hand, the horzontal collaboraton make frm cooperate n order to hare prvate nformaton and reource, wth the ultmate goal of ncreang ther ndvdual proft. Such opton are generally announced n a uppler contract (Cachon, 00; Krchen et al., 011). A a reult, there ext a trong ncentve to decreae frm cot by coordnatng actvte, uch a orderng, warehoung and tranportaton. In the recent year, we oberved a hgh nteret to the tudy of the man tructure of frm allance for rk and cot reducton (Mollenkopf et al., 010). Elomr et al., (01) focued on the tranportaton actvty a a horzontal collaboraton mechanm between multple frm wth one uppler. They ued a coalton formaton model for cot allocaton n whch frm nteract on the tranportaton level. They formulated the problem a a cooperatve game that am to determne effcent coalton and proved numercally that coretable coalton tructure are proftable for the whole ytem. Moreover, horzontal coordnaton mechanm on the orderng actvty between multple frm and one uppler wa addreed n Meca et al., (005). The objectve wa to determne the Economc Order uantty (EO) conderng the orderng and holdng cot of the collaboratng frm. In th cae, when gatherng ther order, frm hare orderng cot baed on ther mean number of order. The model aume that frm hare only the nformaton of ther order frequence and how that the problem converge to a non-empty core of the game. In order to mnmze frm cot for a ngle uppler and multple frm, Krchen et al., (011) modeled an EO wth quantty dcount and delay n payment. They propoed a game theoretc approach that generate coalton tructure for frm wth delay n payment and quantty dcount. Smlarly, Ozen et al. (008) condered a SC nvolvng multple frm, multple warehoue and a ngle uppler. They propoed a cooperatve game for the torage of tem n common warehoue where order are uppled va warehoue avalable wthn a predefned lead tme. A two-level centralzed model propoed wth permble delay n payment and proft harng cenaro to encourage frm gather ther order and jontly upply ther order ung common warehoue. The lterature revew reveal that depte the ncreaed nteret n horzontal collaboraton, to the bet of our knowledge all extng model don't ntegrate multaneouly, orderng, and warehoung and tranportaton cooperaton. Ther ncluon at the ourcng contract degn level crucal to adequately calbrate the revenue and operatonal cot of coalton formaton. Th paper amed at overcomng th drawback. 4 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Although SC contract are ued to be an effcent tool for makng entte coordnate ndependently, game theoretc approach (Ozen et al., 008, Nagarajan and Soc, 008) play an mportant role to help entte ether to agree or not wth the propoed contract and to anwer how the proft/cot harng mechanm employed. Indeed the game theory provde a mathematcal background for modelng the SC decon-makng ytem and generatng oluton n compettve or conflctng tuaton. One hould notce that the extng lterature on game theory clafed nto cooperatve and non-cooperatve game. In the cooperatve game, condered n th paper, partcpant are called to deal wth a pecfc contract n uch way all member are atfed wth the decded contract, and thu there no room for conflct or competton when each entty called to tempt h optmal coalton of partner. The core concept n game theory approach can be an effcent way to form proftable coalton and ha been uccefully appled n Meca et al, (005). Varou cooperatve tuaton are preented n Table 1 baed on typcal cae collected from the lterature. Thee cae concern a well the orderng, the warehoung and the tranportaton actvte n the SN. Table 1 promote for everal large frm ther recent collaboraton ntatve and whch actvte they tend to cooperate n. It how the portfolo of actvte nvolved n the cae of horzontal collaboraton and underlne ome nnovatve collaboratve practce uch a poolng and harng reource. Actvte Cae (Sector, Reference) Acton and beneft Orderng Warehoung Tranportaton Mar and Netle (Food, Logtc Shared truck to delver combned load (over 60 load hared durng 11 week) Manager, 009) Kellogg and Kmberly Clark (Food, Logtc Manager, 008) Netle Water and Danone (Food, Netle, 010) Henkel Colgate, Palmolve and Sara-lee (Peronal Care, Henkel, 010) Spar Belgum (Retal, CO3 project, 014) Elmnaton of 7.5 mle of duplcate trp Shared warehoung to conoldate mall order Reducton of empty runnng vehcle and of replenhment cycle Shared warehoung and dtrbuton of good Full utlzaton of warehoue nvolved and mproved truck utlzaton Reducton of warehoung and tranport cot Manufacturer tore ther product on the ame warehoue Synchronzaton of flow ung common VMI and dtrbuton Le nventory on retaler warehoue whle ncreang the ervce level Better optmzaton of the truck, reducton of 41 ton of CO Savng generated 50% le delvere, 9.300 Km, 97 lter of fuel n 008 Conderaton of a neutral Trutee company to manage the collaboraton Degnate 3PL to provde jont replenhment from to 4 uppler Reducton of the tranport klometer wth 66% and the number of truck of /3 Table - Practcal Example of Horzontal Collaboraton ntatve Decrpton and formulaton of the coalton formaton Problem III. In th ecton, we tart wth a decrpton of the bune context of the replenhment problem and the defnton and the modellng of the tand-alone tuaton experenced by a gven frm. Then, the concept of coalton formaton ntroduced and the problem modellng for a et of frm propoed. Th ecton end wth the preentaton of varou cooperatve replenhment tuaton, baed on the coalton formaton, and ther modellng feature. III.1. The Bune Context The bune context tuded n th paper the followng. A et of frm (retaler or wholealer) operatng n the ame geographcal area purchae the ame product (t could refer to a CIRRELT-014-61 5
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network famly of mlar product) from a unque upply ource (uppler or manufacturer) that potoned locally or abroad (.e. global ource). Th product old by each frm to t own cutomer by hppng the daly order receved to a number of demand zone ung contract carrer. In order to provde a good ervce, each frm operate t own dtrbuton center (DC) located n the product-market coverage area o that t able to provde next day delvery to the predetermned et of demand zone. In addton, to atfy the varyng quantty of product requeted by t cutomer on a daly ba, each frm have to keep t own nventory level wthn the located DC (aumed to be large enough to meet the requrement of the nventory polcy of the frm). For each frm, durng t bune perod, the nventory level mut be flled perodcally, baed on t replenhment plan, and ung ndvdual hppng. Th context decrbe the tandalone cae ncurred by each frm when actng olely. When lookng to the entre productmarket zone, the replcaton of uch bune operaton gve re to the cluter of ndependent SN llutrated n Fgure 1 where a et of frm operate n a tandalone ettng. We aume that trategc decon on the SN tructure, nvolvng the electon of the upply ource and the locaton of the DC are already fxed for each frm and wll not be revoked for a long term (Klb et al., 010a). In order to maxmze t return on nvetment durng t lfepan, each frm mut nure that the cot and revenue generated when ung the SN degned are optmzed perodcally (.e. on a tactcal rollng horzon). More pecfcally, each frm have to mnmze for the tactcal horzon (typcally a year) the replenhment (orderng and tranportaton) cot, the nventory holdng cot and the cutomer' delvery cot. In addton each frm mut maxmze t revenue by atfyng the requeted cutomer' demand wth the adequate ervce and by decreang the purchang prce contracted wth the uppler for that perod. Fgure llutrate a realzaton of the tactcal plan over the plannng horzon T for a et of DC, each one repreentng a dtnct frm that replenhng olely. The plannng horzon T parttoned nto a et of dcrete perod tt repreentng day or week dependng on the granularty of the demand proce. Let I denote the et of frm, I correpond to the ndex of each frm n th et, and j denote the unque upply ourcng (uppler) of the SN. For each frm I, along the plannng horzon, t cutomer order varyng quantte of the product on a perodc ba. Let d t denote the um of order receved by the DC of frm from t cutomer at perod t. A depcted by the Fgure, we aume that the cumulatve demand receved for perod t at a gven DC from a ubet of t agned cutomer charactered by a Normal proce 1. When conderng that d t a random varable fttng a Normal dtrbuton, th provde explctly that the total demand of frm (.e. d ) alo a random varable t T t characterzed by a Normal proce. However, n the begnnng of the plannng horzon, when the tactcal decon have to be fxed, the realaton of the demand proce d t not known wth certanty. Thu, only an etmate of the future demand could be antcpated n order to optmze the tactcal plan. Th antcpaton uually baed on the htorcal path of the frm and ued to provde t future replenhment plan. Accordngly, at the decon moment, let Dˆ denote the etmated total demand of frm n t DC for the plannng horzon T (the hat "" ndcate that th value repreent an approxmaton of the future demand). 1 Th aumpton reaonable nce the number of cutomer agned to each depot uffcently large to beneft from the Central Lmt Theorem, even f the cutomer demand proce modeled a a compound Poon proce. 6 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Frm 1 Antcpaton D ˆ, ˆ (0) 1 1 Plannng Horzon Order receved at DC for perod t d t Normal Proce Frm Dˆ, ˆ (0) Frm I Dˆ, I ˆ (0) I Contract Selecton and Coalton Formaton Decon t 0 t T Fgure - Frm Demand Proce at DC for the Plannng Horzon When future demand antcpated, each frm can optmze olely t own revenue and cot for the plannng horzon condered by fndng the bet comprome between all the cot component and the purchang prce,.e. by optmzng the orderng quantty. Th latter crucal nce t depend on the replenhment cot value ncurred by the frm, on the demand behavor from t cutomer, and on the purchang cot tructure offered by the contracted uppler. For ntance, t well known that when demand charactered by a Normal proce the optmal orderng quantty can be determned by the o-called economc orderng quantty (EO) formula. Let denote the ndvdual orderng quantty defned n the replenhment plan of frm I ˆ baed on the demand Dˆ, I. For a gven frm, an orderng equence of ˆ mut be defned for a ubet of perod tt n order to atfy t demand Dˆ durng the plannng horzon. Under normalty aumpton on Dˆ, the replenhment become a equence of equal and ndvdual order ze ˆ for regular perod along the plannng horzon. Let N be the number of replenhment (.e. order frequency) along the plannng horzon, correpondng mplctly to the repartton of the demand Dˆ nto lot ˆ for frm I. When the frm operatng olely, the antcpated value correpond to ˆ (0) D and underlnng the tandalone ettng (referred by ˆ ubcrpt 0) of the frm' decon a hown n Fgure. Once the orderng quantty and frequency are determned by the frm, an agreement on the contract term for the tactcal horzon wth the upply entty pnponted. The pecfcaton of the contract feature and the charactertc of the relatonhp between the two SC entte, determne the degree of the vertcal collaboraton between them. Flexble contract, VMI or CPFR approache dcued n Secton pertan to th type of collaboraton. At the operatonal level, when the replenhment plan executed and delvere are made, the cutomer order are fulflled from the agned DC. It through th order fulflment proce that ale revenue and operatonal cot are generated. Let Pˆ., Rˆ. and Cˆ. denote, the proft, the ale revenue and the total operatonal cot antcpated for frm I, repectvely. One hould menton that, at t replenhment perod, each frm (baed on t order ze ˆ(0) ) requet from t contracted carrer the truck requred to upply the product from the ource CIRRELT-014-61 7
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network locaton to the DC ung Sngle Truckload (STL) tranportaton. The STL refer to the cae where a not full traler delvered to a ngle detnaton (llutrated later on n Fgure 4). The context decrbed n th ubecton correpond to the tand-alone replenhment tuaton, where each frm operate ndvdually. Note that the cutomer' delvery cot are aumed to be unk cot n what follow nce the modelng approach employed not optmzng the DC to demand zone tranportaton tage. The next ubecton provde the modelng feature of th problem. Next, the tandalone replenhment plannng problem decrbed here wll be contrated wth varou collaboratve bune tuaton. III.. The Standalone Replenhment Problem Modellng Intally, the uppler j, mut develop the purchang cot tructure n term of the quantty dcount to propoe on the product-market, ndependently from the number of frm and ther requrement. More pecfcally, the uppler mut fx for each orderng level the aocated untary purchang cot. In practce, the dcount offer propoed encourage the maxmzaton of the ale volume for the uppler that attempt to beneft from econome of cale n producton, n nbound tranportaton and warehoung avng. Let c j denote the untary purchang cot ncurred by the frm for each unt of product ourced from uppler j. A mentoned, the c j value depend on the ordered quantty requred, and thu the purchang cot ncurred by the frm are functon of ˆ. Let Gˆ ˆ denote the purchang cot functon appled by the uppler baed on quantty ˆ of a gven frm. A the dcount a potental opton to encourage orderng larger quantte, we propoe a dgreve purchang cot functon that controlled by a mnmal prefxed threhold mn and a maxmal threhold max. The purchang cot functon Gˆ ˆ rele on a maxmal value c that defne the untary cot for purchang product and on a quantty dcount rate e reflectng the purchang cot decreae when the ordered quantte ncreae. Thee value are defned o that for any orderng level beyond the maxmal quantty we have c c e max and when the quantty exceed the maxmal level max we have cmn c emax, where c mn the fxed mnmal cot. Accordngly, the untary purchang cot for a gven frm I could be antcpated ung the etmated quantty to order and the uppler cot tructure ˆ gven by equaton (1), adapted from Fazel et al. (1998): c f mn c j c ( e) f mn cmn f max max The two threhold max and mn are obtaned a follow: c cmn max max, mn e Afterward, the antcpated cot and revenue of each frm I are derved baed on the antcpated demand ˆ 0 D and the decon on quantte to order ˆ. Let R, 0 C and ( ) 0 G ( ) denote the ale revenue, the total operatonal cot, and the purchang cot of the frm I n a tandalone replenhment context, repectvely. In th cae, frm am to maxmze t proft, denoted by 0 ˆ, by mnmzng t operatonal cot 0 C ( ), and ncreang t net revenue P (1) In the cae of a global uppler, the local locaton could be aocated to an entry port or to an nland cargo hub. 8 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network 0 (.e. ale revenue R mnu purchang cot G 0 ) ung the followng orderng quanttybaed equaton : ( ) 0 0 0 0 () P ( ) R C ( ) G ( ) where the purchang cot are computed by G ( ) c jd ung relaton (1). By ntegratng the purchang cot preented n (1), the proft functon could be wrtten a follow: 0 ˆ 0 0 ˆ R C ( ) cd f 0 mn 0 0 0 P ( ) ( ) ( ( )) ˆ R C c e D f mn 0 0 R ( ) ˆ C cmnd f max max (3) 0 Moreover, antcpated ale revenue R, for a gven frm, are obtaned by multplyng the etmated total demand D by the untary ale prce, denoted by p, whch gve the followng 0 relaton: Rˆ ˆ pd. Regardng the operatonal cot 0 C, recall that t take nto account the ( ) entre replenhment proce by computng the orderng cot, the nventory holdng cot, and the nbound tranportaton cot. Accordngly, we propoe n th paper a typcal modelng of the orderng and holdng cot functon, mlarly to Elomr et al. (01), but a more elaborated tranportaton cot functon provded here. Th latter propoe to capture varable and fxed part of the tranportaton fee ung, a dtance and weght-baed, lnear functon that etmate the unt arc-flow cot. The varable cot part related to the hpment weght that depend on the antcpated quantty to order by hpment and t gven by cˆ v, where cv the untary varable cot by product unt aocated to the hpment. The fxed cot part aocated to the traveled dtance to hp from the ource locaton (local te or an entry port/hub) to the frm DC. Snce the dtance to travel known n advance, the fxed cot computed a pror by ca cdd j where ca a fxed charge for the truck uage, cd the untary dtance-rate to hp one unt of product, and d j the travelng dtance from locaton j to. Hence the antcpated tranportaton cot ncurred by a gven frm could be computed a follow: TC 0 ˆ c c d c ˆ, I a d j v 0 The orderng cot for frm, denoted by OC, depend manly on a fxed untary orderng cot A and the number of placed order N (recall that Nˆ ˆ ˆ ). In addton, the nventory holdng D 0 cot, denoted by HC, nvolve a lnear relaton between the untary cot of captal to hold one unt of product (denoted by h ) and the average ze of nventore kept at DC between two replenhment, calbrated to be ˆ. The orderng and the holdng cot functon are gven by (5) and (6), repectvely. OC 0 ( ) AN, I 0 HC ( ) h, I (4) (5) (6) CIRRELT-014-61 9
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network The antcpated total operatonal cot ncurred by a frm are thu computed ung the followng ummaton: 0 0 0 0 C ( ) OC ( ) HC ( ) TC ( ) Baed on the et of operatonal cot functon and revenue formula ntroduced above, the proft tructure for a gven frm preented n (3) can be rewrtten a follow: ˆ ˆ ˆ D p ˆ ˆ 0 ˆ D A h ca cdd j cv mn ˆ cd f ˆ ˆ ˆ 0 ˆ ˆ D P ˆ ˆ ˆ ˆ pd A h ca cdd j cv c e D f mn ˆ ˆ ˆ ˆ D p ˆ ˆ ˆ D A h ca cdd j cv cmnd f max ˆ 0,* Accordng to the formulaton gven n (7), the optmal ordered quantty that maxmze the 0 proft for frm obtaned by olvng d P ( ) whch can be wrtten accordng to the purchang cot tructure n (1) a follow: d D( A ca cd D) f 0 mn h 0,* D( A ca cd D) f mn h e D D ( ) A ca cd D f max h In what follow, gven the tandalone ettng decrbed above, the opportunty for a gven frm to form a coalton dcued. III.3. Coalton Formaton Problem Decrpton and Modellng max When lookng to the entre tructure of the SN pertanng to the cluter of operatng frm I n the product-market condered here, the collecton of ndvdual frm-baed decon correpond to a et of a decentralzed decon makng procee. In th tuaton everal queton may are: I there any beneft for thee frm to mprove ther ndvdual economc performance when they collaborate throughout ther SN? If o, what the bet coalton formaton for each frm n order to maxmze t proft? How wll the frm react toward collaboraton accordng to the proft harng mechanm propoed? Fnally, when varou jont replenhment tuaton are nvetgated, one expect the ame coalton to be formed? More pecfcally, the man ue to npect f the frm Icould ncreae ther ndvdual proft P., when they collaborate? Dependng on the replenhment tuaton, whch the bet comprome for each frm between the purchang cot G. and the operatonal cot C.? and thu n each replenhment tuaton how the bet orderng quantty could be determned? To anwer to thee queton we mut defne the coalton formaton problem for a jont replenhment proce. Aaou et al. (007) reume the purchang decon proce nto x major component: make/buy, uppler electon, contract negotaton, degn collaboraton, procurement, and ourcng analy. Here, we are clearly focung to the contract negotaton and the degn max (7) (8) 10 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network collaboraton component by decdng for each frm on the coalton to form n order to degn the ourcng contract takng nto account the decon to collaborate or not wth other frm. A mentoned, the ourcng contract wa uually degned by the frm olely or jontly baed only on the quantty dcount ncentve, but not by antcpatng the beneft of jont replenhment n tranportaton and/or n warehoung. Accordngly, the ourcng contract degn can be vewed a a dtrbuted decon-makng proce (Schneewe, 003) amng to dentfy proftable coalton to form by antcpatng the revenue and cot generated by the frm collaboraton. Fgure 3 decrbe the four tep encountered n the collaboratve contract degn deconal proce when one uppler and a et of ndependent frm are nvolved. In tep (1), the uppler provde for the et of frm requrng t product a purchang cot tructure encompang the quantty-baed untary cot feature a gven n equaton (1). In uch contract propoton, the uppler ha to fx the man contract parameter: the dcount value, the untary cot ncurred and the mnmum and maxmum quantte. In tep (), gven the uppler offer, frm are tempted to evaluate the opportunty to operate ndvdually or n cooperaton wth other frm n order to maxmze ther own proft. In the cooperatve tuaton, frm wll try to form proftable coalton regardng all the partner wthn. The antcpaton of a proftable coalton eentally mply to hare among the partner a number of nformaton about ther etmated annual demand Dˆ, a well a ther etmated order quantte ˆ. In ome cae th could be ploted by an external logtc ervce provder (.e. bddng 3PL, currently contracted carrer) to guarantee the neutralty and confdentalty of the proce. For that reaon the degn proce llutrated here ncorporate the nterventon of an external entty n tep (). Recall that we aume here that n the tandalone tuaton each frm rely olely on a contracted carrer to replenh t product. Such entty could be reponble for the whole coalton formaton project or mply be mandated for the executon of orderng, holdng, and/or tranportaton actvte for the formed coalton. Typcally, n tep () the external entty condered provded the neceary nformaton n term of bd and charge, and n term of tranportaton capacte. See for ntance the cae of Netle Water and Danone n (010) reported n Table 1. The dea of Trutee entty wa alo ntroduced n the cae of retal nbound horzontal collaboraton n Belgum a a neutral actor that facltate the collaboraton among a large number of uppler (www.co3-project.eu/). Step (3) reflect the decon made by the frm partcpatng n the collaboraton opportunty to jon or not the coalton baed on the antcpated proftablty. Remnd that a coalton/allance characterze the group of frm that decde to work jontly whle groupng ther order and ynchronzng ther polce. All jonng frm wll operate a a ngle decon makng unt a howed n Fgure 3 to decde about the contract tructure propoed by the uppler j. Fnally, n tep (4) when coalton formaton decon acted, a jont replenhment plan optmed and ubmtted by the cooperatng frm to the uppler. More pecfcally, the coalton decde on the jont orderng level and orderng frequency contracted wth the uppler. CIRRELT-014-61 11
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Frm 1 D, 1 1 (1) Contract propoton Predpoed to cooperate Suppler Contract propoal Maxmum quantty Rebate tructure Mnmum cot cj ( ) Frm D, (4) Decon on the orderng level and frequency, N Predpoed to cooperate External entty Tranportaton cot, bd and capacte Warehoung cot and capacte Frm I D, I I Informaton exchange () (3) Decon of the coalton formaton S Fgure 3- Decon Proce of the Contract Degn Problem wth collaboraton Baed on the decon-makng proce decrbed above, the followng modelng approach for the coalton formaton problem provded. Let S denote a gven coalton compoed by a collecton of frm I uch a S I. For a gven coalton S, let I S be the ubet of frm agned to S, uch a IS arg I S and I I. Converely, let S refer to the coalton of frm S. Note that the et of all poble coalton denoted by o that each coalton ndexed by S. When a coalton S formed, the et of frm I S pertanng to the allance mut pool ther ordered quantty and mut be algned on the number of order. Let S denote the orderng quantty for coalton S and let N denote the order frequency S for coalton S along the plannng horzon. We aume alo the extence of a collectve orderng cot AS aocated to the charactertc of a gven coalton S. Frt, one hould determne a unque number of order N S, for all frm S o that they are nvted to adopt the ame replenhment cycle N S a propoed n Meca et al., (005) and n Fetra-Janero et al. (011). Conequently, an agreement hould be adopted relatvely to the ndvdual N for all S, whch are generally not necearly algned when frm are beforehand actng olely. A the proft functon nverely proportonal to cooperatve orderng number, N S can be expreed a follow (Krchen et al., 011): N mn( N ), I Baed on the algnment of the orderng cycle, the ordered quantty of each frm n the coalton S mut be adjuted. Let, denote the quantty ordered by the frm S whch defned n accordance to the ngle frequency N and baed on the new cooperatve N a follow: accordng to the cooperatve order number a: (10) D S f N N S, N S f N N S Fnally, by formng a coalton, frm pertanng to the allance mut coordnate ther order and thu hare a unque orderng cot A S. Hence, a communcaton cot wll be needed to manage bune operaton on day to day ba between coalton member. Furthermore, that cot can be (9) 1 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network alo ued to upport the degn of a dtrbuted databae among all coalton member. Such mechanm called n practce a Electronc Data Interchange (EDI) whch a good example of tandard developed n order to hare document between organzaton n a tandardzed electronc form and n an automated manner (Bhatt, 001; Audy et al., 01). In a generc way, one hould expect that the coalton orderng cot be functon of the frm' ndvdual orderng cot and the ze of the coalton: AS f ( A, S ). Conequently, we propoe to nclude a communcaton cot to be ncurred by frm belongng to the coalton. Th cot ncreae proportonally to the number of frm n the coalton. Gven a coalton S, t upplementary orderng cot equal to cc S where cc the unt communcaton cot. Then, the coalton orderng cot to be placed wth any order can be tated a: A Ac S c In addton, when frm collaborate, they hare nformaton n a tmely manner and ue dfferent approache to ynchronze ther actvte effcently. In uch cae, a cumulatve holdng cot agned once a coalton formed. Elomr et al., 01 propoed that cot where an EO ued a a reordered polcy. In th way, frm formng a coalton are poolng ther order to maxmze proft by reducng the unt purchang cot (ee equaton (1)). Furthermore, once a coalton formed, one hould ak how much proft each frm could gan? Th depend on the harng opportunte offered by the replenhment tuaton. To th end, the next ecton decrbe a number of replenhment tuaton that offer varou opportunte for allance to hare not only order but alo tranportaton and/or torage actvte. By formng coalton, frm group ther order and decde about the mot proftable coalton to be adopted. The decon of jonng coalton taken accordng to the realzed proft of each frm n a gven coalton. Let Pˆ S (.), denote the cooperatve proft evaluated for a coalton S. We defne by Pˆ S, (.) the proft allocated to a frm S that uppoe a harng mechanm that the coalton proft wll be dtrbuted among all the member proportonally to the quantty S,. Th volume-baed gan harng mechanm ha been uccefully appled recently n a retalng context (www.co3- project.eu ). The ndvdual proft P S, (.) calculated by: ˆ ˆ, S PS (.) PS, (.), S (1) S For each frm, varou coalton n can be proftable wth regard to the cot avng realzed by the cooperaton. Let ' the et of all proftable coalton belongng to, '. Equaton (1) aume alo that each frm avng on purchang, tranportaton and torage cot would be proportonal to the ordered quantty wthn the coalton. Cooperatve game theory offer a natural paradgm to deal wth cot avng problem (Meca el al., 005). To defne a cooperatve game, two ngredent are needed: a et of player (here the frm) and a charactertc functon whch agn to each poble coalton of player a numercal value (here the proft value) to be l l l l nterpreted a a meaure of t payoff. Let P { P1,, P,,..., P, } be the et of proft realzed by frm n a gven coalton S. Indeed, the member of a coalton S may be ntereted n formng a l 0 coalton S, f P P. The man objectve to fnd among all proftable coalton the table I * * one. We denote by the et of table coalton; '. Coalton tructure are table n the ene that no frm ha an ncentve to devate (e.g tablty condton dcued n detal n (11) CIRRELT-014-61 13
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Secton V). Stable coalton n the game theory n the ene of the Coalton Structure Core (Aumann and Drze, 1974). Core concept defned a follow: P (.) P (.) I, I,, ' ',, ' * * Once table coalton are obtaned, let P, (.) and, be repectvely the table proft and ordered quantty for a frm S. Conequently, utable replenhment decon to jon table coalton are to be taken from frm to procure product at the lowet poble prce. Three replenhment tuaton are to be defned and dcued n the next ecton. IV. The Invetgaton of Three Cooperatve Replenhment Stuaton A decrbed n the bune context and llutrated by decon-makng proce n Fgure 3, n repone to the uppler offer, each frm can contract ndvdually or collaborate wth other frm. Typcally, frm are ntereted by new purchang contract baed on dcount tructure offered by the uppler whch could be reached only when they decde to jon ther orderng proce. Moreover, a reported n Table 1, everal practcal example howed that frm could alo beneft from collaboraton n the tranportaton and warehoung operaton by poolng ther order and the entre replenhment proce. In th paper, we propoe to tudy three alternatve replenhment tuaton that could favor horzontal collaboraton on thee operaton. Thee three tuaton are namely: Collaboratve Orderng wth Drect Shpment (CODS), Jont Replenhment wth Shared Warehoung (JRSW), and Jont Replenhment wth Shared Shpment (JRSS). Thee tuaton are explaned hereafter and are llutrated n Fgure 4 n order to mmc the replenhment proce n each cae. To npect eparately the outcome of thee tuaton, let ˆ l P S (.) be the cooperatve proft where an ndex l=1,, 3, ued to denote CODS, JRSW and JRSS tuaton, repectvely (recall that l=0 correpond to the tandalone tuaton). Recall that the objectve of the tudy of thee three alternatve tuaton frt to npect n each cae the mpact of the replenhment ncentve on the coalton to form, and econdly to analye the nght toward horzontal collaboraton baed on the drver offered n the orderng, tranportaton and warehoung operaton. A llutrated n Fgure 4, thee replenhment tuaton rely on two tranportaton opton: Sngle truckload (STL) or Mult-drop truckload route (MTL) (Klb et al., 010b). Baed on tranportaton cot functon (4), the tranportaton fee charged wth each of thee two mode were computed dfferently dependng on the replenhment tuaton (.e. ubcrpt 1 for STL and for MTL were added to all the parameter of the tranportaton cot functon (4)). a) CODS b) JRSW c) JRSS ' (13) Fgure 4- The Replenhment Proce of Each of the Three Stuaton Invetgated 14 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network The frt cooperatve tuaton cont n collaboratng olely n the orderng proce. In th cae, frm group ther quantte to order for each cheduled tme perod. Baed on the orderng polcy n (9), jont order are computed wth (10) for each tme perod. Once order are placed, each frm could plan wth t own tranportaton ervce provder (3PL) for the delvery of t ndvdual part of the jont order ung a ngle truckload hpment (STL). CODS tuaton llutrated n Fgure 4.a), and correpond to clacal jont orderng proce a dcued n (Krchen et al., 011) and n everal practcal cae n Table 1. For a frm I, the optmaton of the tranportaton cot not alway poble becaue the replenhment level S, depend on the coalton S parameter S, N S and thu, are not alway deally calbrated to have a full truckload a n the ndvdual orderng cae. For th frt replenhment tuaton, the coalton 1 proft Pˆ S (.) gven by expreon (14). It nclude the four cot component: orderng, holdng, tranportaton and purchang, and conder the jont orderng cot expreed n (11). For the et of the collaboratng frm S, ung S, the um of ndvdual hpment tranportaton cot n STL 1 computed baed on expreon (4) and the jont purchang cot ( Gˆ ˆ S( S) ) are gven by equaton (3). The collaboratve proft ˆ 1 (.) accordng to the CODS replenhment tuaton a follow: P S D 1 1 1 D p D ( A ( ) ) 0 h ca cd dj cv c D mn I I I P p D A h c c d c c e D 1 D 1 1 1 D ( ( ) ) ( ) a d j v mn max I I I D 1 1 1 D p D ( A ( ) ) h ca cddj cv c mn D max I I I In the econd cooperatve tuaton JRSW, frm gather ther ndvdual order to be delvered n jont hpment to a common warehoue or hub. The ncentve behnd propong uch bune cae are from the conderaton of long dtance between geographcal locaton. In fact, when uppler are located n foregn countre, ntermedate warehoue are requred for temporary torage (ee Fgure 4.b). A reported n (Ozen et al., 008), th ue wa oberved n everal manufacturng faclte n Aa to market n Europe and North Amerca, whch uffer from long uppler lead tme. The author propoed that frm collaborate through nventory poolng and make allocaton after demand realzaton n order to enhance ther proft. In the ame way, we condered n th paper that n the JRSW tuaton, a et of collaboratve warehoue, denoted S W, would be avalable for coalton S. Thee latter are aumed to be part of the prvate network of hub managed by the 3PL and thu n-between connecton tranportaton fee are not condered n the frm charge. Subequently, the tranportaton lane take nto account of the S nbound tranportaton (uppler - 3PL orgn warehoue ww ) and outbound tranportaton S (3PL detnaton warehoue ww DC, S ). The nbound tranportaton lane accomplhed jontly n STL whch could be n full truckload f correpond to the 3PL truck capacty. A fnal hpment tep cont n delverng to frm va ngle STL hpment. Here, the total travelled dtance from the uppler to the S hp-to pont gven by djw d S w' be the travelled dtance to the S detnaton, where S the number of frm n the coalton S. Hence, for a gven coalton S, the total cot n the JRSW tuaton baed on hared orderng and nventory holdng cot component, and thu, the total proft ˆ (.) wrtten a follow: P S (14) CIRRELT-014-61 15
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network 1 1 1 D D p D ( A h ca S 1 cddj cv ) c D 0 mn I I P p D A h c S c d c c e D D 1 1 1 D ( a 1 d j v ) ( ) mn max I I D 1 1 1 D p D ( A h ca S 1 cddj cv ) c mn D I I max (15) Fnally, for the thrd replenhment tuaton JRSS, the delvery proce from the uppler' warehoue to the frm' local warehoue ynchronzed n a mult-drop truck load (MTL) through poolng product tranportaton. A MTL refer to the cae where the truck delvery route nvolve more than one detnaton (Klb et al., (010b)) and thu erve multple DC wthn one trp a preented n Fgure 4.c). The dea npred from the tranportaton poolng and t beneft a dcued n Pan et al., (01). The author propoed a poolng trategy at the tranportaton level by harng the freght network at a natonal level n order to reduce CO emon. Th opportunty offer lower tranportaton cot nce the travelled dtance hared by the nvolved frm. One way to evaluate the tranportaton cot here by olvng the jont nventory-routng problem (Coelho et al., 014). Another approach, ued n th paper, to buld an evaluaton functon that baed on a lnear regreon route length etmator (ntroduced by Daganzo (1984) and extended n Klb et al., (010b)) to account for STL and MTL tranportaton. In th cae, let be the total dtance travelled by the frm of the coalton from the uppler to ther repectve DC. Th dtance approxmated by the followng expreon:. S ( 0.57 ) S 1 Hence, n the JRSS tuaton frm formng a coalton are harng orderng, nventory holdng and tranportaton cot and ther cooperatve proft can be wrtten a follow: D D p D ( A ) 0 h ca S cd cv c D mn I I P p D A h c S c c c e D 3 D D ( ) ( ) a d v mn max I I D D p D ( A h ca S cd cv ) c mn D I I l The hared proft of frm I regardng tuaton l {1,,3} denoted by P,. So, the harng mechanm reported n equaton (1) become: Pˆ ˆ Pˆ ˆ S l l l, S, S S S Smlarly, the proft ncurred by the table coalton for each frm, greater than the proft generated by any other coalton. Conequently, utable replenhment decon to jon table coalton are to be taken from frm to procure product at the lowet poble prce. The three cooperatve tuaton preented above wll be evaluated n varou ndutral context and compared to the tandalone tuaton n Secton V. max (16) (17) 16 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network V. Soluton Approach V.1. Stablty Condton of the Coalton Structure Th ecton propoe a game theoretcal oluton approach for generatng proftable coalton tructure and electng table one baed on decon rule relyng on the core (Aumann and Drze, 1974). The core tablty concept ued n cooperatve game take nto account the ndvdual and coaltonal ratonalte needed for reachng bet comprome between all frm n the upply chan (Krchen et al, 011). The core tablty concept determned wth the followng two feature: Coaltonal ratonalty condton: A proft atfe: P l P, l 1,,3 0 l P ad to be coaltonally ratonal f t Stablty condton: A coalton ad to be table f t atfe: P P, I, I,, ' '.,, ' ' For the coalton formaton problem, enumeratng all the poble coalton and tetng ther tablty too tme conumng, f not mpoble. For that reaon, we frt developed n th ecton nequalty condton that elmnate beforehand ome non-proftable coalton, and thu enhance conderably the oluton approach effcency. Next to that, the cooperatve replenhment Algorthm (CRA) tep wll be preented n detal. To tart wth, only proftable coalton are nteretng to accept wth repect to proft equaton (3) and (14)-(17), and thu mut be computed and compared effcently. To th end, a et of nequalte wa developed and ther valdty wa proved analytcally. Dependng on the replenhment tuaton, thee latter are baed on the trade-off produced on the orderng cot, the nventory holdng cot and the tranportaton cot. Accordngly, for each replenhment tuaton l 1, or 3, two coaltonbaed expreon, denoted by l l, and,, mut be calculated and compared to check f t benefcal or not for a gven frm to jon a coalton S. Conderng a gven replenhment tuaton l, the decon to accept or to dcard the coalton S for a gven frm baed on the followng decon rule: l l Accept the coalton f Refue t otherwe,, Conderng the CODS replenhment tuaton, each frm ha nteret to accept to jon coalton 1 1 S f the nequalty above hold where and are gven by expreon (18) :,, 1 1 D ( ) 1 ( ), cc S,, A( 1), ; f mn max D, D, For the JRSW replenhment tuaton (l=) we prove that the nventory holdng cot of the common local warehoue h may affect the tablty of the core tructure. Then, to mprove the tablty condton for the JRSW, the followng condton (19) requred: (18), h, h,, ; f mn max, (19) CIRRELT-014-61 17
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Fnally, for the JRSS replenhment tuaton (l=3), we proved that the 3PL travelled dtance can perturb the tranportaton cot and thu mpact on the coalton. Conequently, a gven coalton expected to be table f: 3 d 3 3 j D,,,, ; f mn max k, cd, The mathematcal proof of condton (18)-(0) gven n Appendx A. Thee three condton are encompaed n the reoluton algorthm hereafter to enhance t effcency.,, V.. The Cooperatve Replenhment Algorthm We preent now the chema of the CRA whch compoed by three man tep. The frt determne the ndvdual proft for all frm n the tandalone context. The econd tep a flterng proce that keep only cot avng coalton baed on the proft tructure gven by (3) and (14)-(17). It proceed by enumeratng for each frm the ubet of frm that can be benefcal to cooperate wth. Subequently, all non-proftable coalton are dcarded ntally when performng the prelmnary calculaton of threhold l l and n order to compare coaltonbaed and tandalone proft. The thrd tep of the proce cont n applyng the core-tablty tet n order to fnd, among the current et of proftable coalton, the table one. Wth thee tep, the CRA produce the mot proftable repartton of the et of frm nto a et of table coalton that are mutually excluve and collectvely exhautve. Subequently, the CRA teted and valdated on extenve numercal ntance. (0) 18 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network l ' ' *, l *, l * CRA (Input:,,, I ;,, P,, ',, '' ; Output:,, P,,, * * ) * * ) 1. For all I do Compute the SA proft P 0 (.) ung equaton (3) End for ', * S. For all do Stop=true.1 Whle I and NotStop 1 1 1 1 Compute and baed on and wth equaton 3,, P,, (10), (14) and (18) 1 1 If (, >, ) then Stop= true End f End whle. If NotStop then ' S S ' ' ' { S } End f End for 3. For all S ' ' do Stable=true 3.1 Whle S '' ' and I and Stable=True 1 1 If ( P, " P, ' ) then Stable = fale End f End whle 3. If Stable=true do * ' S S { S } * * * *,1 1 P, P,, I *,1 1,,, I End f End for Fgure 5- The Cooperatve Replenhment Algorthm VI. Numercal Reult and Dcuon VI.1. Plan of Experment In order to tet the modelng and oluton approache propoed to olve the coalton formaton problem, everal problem ntance were generated baed on the followng dmenon: market ze, network confguraton and cot tructure (hgh / low cot). All thee ntance wll be teted for the et of replenhment tuaton propoed (CODS, JRSW or JRSS) and compared to the SA. The horzon length fxed to one year and one product famly 3 (18) hould be replaced by (19) f l = (JRSW) and by (0) f l = 3 (JRSS) CIRRELT-014-61 19
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network condered n th tudy. We condered that the number of frm hould not exceed 30, whch congruent wth the ndutral example reported n Table 1 and uffcent to valdate the oluton approach effcency. Thu, the market ze n n our tet range between 5 and 30 frm. The frm' market-ze for the product-market condered and total demand receved are calbrated baed a North Amercan dtrbuton context tuded n Klb et al. (010b). Ung the cutomer' characterzaton (large, medum, or mall) and ther aocated demand ze, a repartton wthn the et of frm provded to calbrate ther market hare. Accordngly, an aggregated yearly demand for each frm derved and two man clae of frm are generated:. Small-zed Frm / product-demand (SF) wth a yearly total demand level wthn the nterval of [10, 0] pallet.. Large-zed Frm / product-demand (LF) wth a yearly total demand level wthn the nterval of [480, 580] pallet. Wth the frm clae defned above, three confguraton of frm network wll be nvetgated: (1) a network contanng only large zed frm (LN), () network ncludng only mall zed frm (SN) and network comprng both large and mall zed frm (HN). Furthermore, n order to capture dfferent cot tructure, two level of fxed and varable cot were condered. A provded n Table, for each level, orderng, nventory holdng and purchang cot are vared. Note that the value for low cot attrbute are fne-tuned baed on the work of Elomr et al. (01) for the nventory holdng component, and npred from the work of Krchen et al. (011) for the purchang tructure. Thee value are augmented by factor to obtan the correpondng value for hgh cot attrbute. Note alo that the orderng cot component wa calbrated baed on an average truck fulflment n pallet uch a the hpment frequency on a yearly ba about N [6, 10[ for SF and N [1, 0[ for LF. Fnally, the product famly average prce fxed to 60$ for all frm. Regardng the tranportaton cot, the fxed and varable charge are etmated n Table 3 for the STL and MTL tranportaton mode. Thee are etmated by regreon wth htorcal data n Klb et al. (010b) under the aumpton that a 400-mle dtance lmt defned between uppler and frm depot. Low cot Orderng cot A = 10, ; c c = 10 A = 100 ; c c = 0 Hgh cot Holdng cot h [15, 5]; h = 35 h [30, 50]; h = 70 Purchang cot e = 0.01 ;c = 15;c mn =1 e = 0.0 ;c = 30;c mn =4 Table - Input Problem Cot Structure k c a k c d k c v k=1 (STL) 60 0.15 0.375 k= (MTL) 60 0.3 0.75 Table 3- Parameter for the Tranportaton Structure Each problem ntance denoted by the trplet (n, x, y) where n refer to the market ze, x to the network confguraton and y to the cot tructure. It defned a follow: ( n, x, y): n{5,10,15,0,5,30}; x{ SN, LN, HN}; y{ LC, HC} 0 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Recall that each of thee 36 ntance (n, x, y) mut be olved for the SA tuaton and threefold for the cooperatve tuaton (CODS, JRSW, JRSS). Th combnaton yeld 144 problem to olve wth the CRA. All the computatonal experment are performed n Java language on a 3 bt computer wth Pentum (R) CPU,.13 GHz, and 4GB of RAM. The next ecton preent the numercal reult of the teted ntance and dcued ther manageral nght. VI.. Numercal Reult Gven the 36 problem ntance pecfed prevouly, th ecton npect for each cooperatve tuaton f the collaboraton a promng upport to further ncreae the proft of frm nvolved. In addton, baed on the proft realed by the frm n the coalton formed, we compare the ndvdual proft before and after cooperatng n order to dcu the effcency of the ncentve offered by a gven cooperatve tuaton. Note that th ecton focung on the analye of all the ntance wth mlar compane' network ether wth large zed frm (LN) or wth mall zed frm (SN). The ntance conderng hybrd network attrbute are dcued n the ubequent ecton. Analy of the proft allocaton Frt, to meaure the degree of cooperaton between frm, we compare the gap n term of proft realzed between the tand-alone cae and the cooperatve one for each replenhment tuaton l = 1,, 3 for hgh. Th latter baed on the um of the proft of all the coalton l formed n that market. For a gven replenhment tuaton l, Gap computed a follow: l l 0 0 Gap P (.) P (.) P (.) I I Fgure 6 llutrate uch gap for the two followng ntance wth oppote attrbute: (., SN, LC) and (., LN, HC) where hgh and low orderng cot value are compared on each market ze. It how eparately n graphc a), b) and c) the dfference between the SA and CODS, JRSW and JRSS, repectvely. One can oberve from Fgure 6 that all gap value are potve and that n general they are hgher when the orderng fxed cot hgh wth the three cooperatve tuaton (except wth ntance of ze 0 for the JRSW). Th underlne that when the fxed orderng cot are hgh, frm are more encouraged to collaborate n order to reduce th cot component. Epecally under JRSW and JRSS tuaton, the gap wth the SA much more marked. We clearly deduce that the degree of collaboraton ncreangly proportonal to the fxed orderng cot A. The detaled gap value are gven n Appendx B for varou ntance wth orderng cot A value rangng between 10 and 100. (a) SA v. CODS (b) SA v. JRSW (c) SA v. JRSS Fgure 6- Gap Comparon between SA and Cooperatve Stuaton CIRRELT-014-61 1
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network Analy of the Cardnalty of the table coalton tructure The mpact of collaboraton alo meaured n term of the number of coalton formed wthn the network: the more th number decreae the more the degree of collaboraton hgh. Ideally the grand coalton formed wth all the frm preent n the network. Table 4 report the * number of table coalton wthn the tructure belongng to et when the market ze ncreae, and ung ntance wth attrbute (. SN, LC) and (., LN, HC). Th table expree that the cardnalty of the coalton tructure for the three tuaton, for all ntance, le than the number of frm. Th well explan the cooperatve tendency of the propoed cenaro a they reflect varou behavoral cooperatve protocol. Indeed, t notceable that for 34 out of 36 of the ntance the number of frm wthn the coalton ncreae wth the problem ze (ee Table 4, Fgure 1 and Fgure n Appendx B). However, the cardnalty of the coalton tructure ncreae proportonally to the market ze, whch probably underlne the lmt of collaboraton opportunte n th context. Th behavor appear to be qute mlar n all the replenhment tuaton. In th table, are hghlghted the lowet number of entte oberved n the table coalton tructure. For ntance, the lat raw of Table 4 provde the average proporton of dtnct entte n the table coalton tructure for the CODS, JRSW and JRSS tuaton n both ntance npected. Thee reult confrm the good ncentve toward collaboraton, regardng nventory poolng, propoed by JRSW tuaton. Hence, we notce that the proporton of the table coalton ze better wth the hgh cot attrbute whch explan the effcency of collaboraton n term of cot avng. Bede, the obtaned reult how that collaboraton ha a gnfcant performance for the three cooperatve tuaton. Cardnalty of the coalton tructure (., SN, LC) (., LN, HC) n CODS JRSW JRSS CODS JRSW JRSS 5 1 1 1 1 1 1 10 3 1 5 3 1 3 15 9 6 6 8 5 6 0 14 10 11 1 9 9 5 18 14 15 16 11 11 30 0 18 19 18 15 17 Average coalton 53.1% 39.3% 48.1% 47.9% 33.7% 39.3% formaton (n %) Table 4- Stable Coalton Sze n SA and n Cooperatve Stuaton In complement, Table 5 report the number of frm nvolved n each coalton accordng to each replenhment tuaton. For ntance, when n = 10, the CODS tuaton generate a coalton tructure compoed of three coalton: the frt coalton nvolve 8 frm, and the remanng frm operate ndvdually. Th gve the followng tructure: Coalton 1: 8 frm /Coalton : 1 frm /Coalton 3: 1 frm. CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network n CODS JRSW JRSS 5 (5) (5) (5) 10 (8,1,1) (10) (6,1,1,1,1) 15 (7,1,1,1,1,1,1,1,1) (10,1,1,1,1,1) (10,1,1,1,1,1) 0 (7,1,1,1,1 1) (11,1,1,1,1 1) (10,1,1,1,1 1) 5 (8,1,1,1,.1) (1,1,1,1,.1) (11,1,1,1,.1) 30 (11, 1,1,,1) (13, 1,1,,1,1) (1, 1,1,,1) Table 5- Number of Frm n SA and Cooperatve Stuaton for (., SN, LC) Intance A reported n Table 5, for each ntance, the CRA alway generate a table coalton tructure that plt the whole et of frm nto: one coalton that encompae all cooperatve frm and a et of ngleton ndcatng the tand-alone behavor of the remanng one. We alo notce that, n pte of the extence of collaboraton, the tand-alone poton domnant n the generated coalton wth repect to the number of tand-alone frm. The grand coalton obtaned only when n = 5 for all replenhment tuaton whch probably due to the cot of coalton mpacted. Th congruent wth real world cae reported n Table 1 that pont out the manageral and cot dffculte to go beyond larger number of frm wthn a coalton. A unque cae of grand coalton wa oberved when n = 10 wth the JRSW tuaton whch could be explaned by the mportance of the nventory poolng ncentve when mall frm are nvolved. Under the CODS tuaton, the number of frm beng n tand-alone more mportant n comparon to JRSW and JRSS, nce the former offer only cooperaton on the orderng cot. Thee reult underlne the mportance of ncentve to cooperate n warehoung and tranportaton wthn the replenhment proce. Performance Analy n term of degree of collaboraton All collaboratve tuaton ncur an orderng cot A expreed n term of the fxed orderng cot A and the communcaton cot cc, a reported n equaton (11). Hence, for a fxed value of A, the coalton formaton proportonal to cc. We conducted a entvty analy on the effect of the cc by olvng the CRA wth varou cc value, n order to pont out key threhold value Th( c c ) that enhance frm cooperaton. Fgure 7 llutrate uch analy for the problem ntance wth attrbute (10, LN, HC) and regardng all replenhment tuaton. In th cae, tartng wth a c c =0 and olvng teratvely the CRA, a coalton formaton oberved untl Th( c c ) reache the value of 30A, 35A and 43A for CODS, JRSS and JRSW tuaton, repectvely. Thee value are very ueful to determne whether a coalton formaton poble or not, and could be of key mportance for frm n order to etablh the adequate communcaton cot c c wthout alterng the cooperaton opportunty. In addton, th fgure underlne alo that the threhold value for the CODS lower than the other tuaton. Th explaned by the dependency of th collaboratve tuaton to the orderng cot wherea the two other benefcate from addton ncentve n the tranportaton and the warehoung. Smlar analy for problem ntance wth low orderng cot attrbute (.e. (10, LN, LC)) provde Th( c c ) value of 15A, 17A and 1A, for CODS, JRSS and JRSW tuaton, repectvely. Th congruent wth the prevou reult due to the huge dfference between low cot and hgh cot tructure ued n thee ntance. CIRRELT-014-61 3
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network CODS 0 30A cc c JRSS 0 35A cc c JRSW 0 43A cc c Collaboraton Stand-alone Fgure 7- Extreme Value for Cooperaton for ntance (10,LN,HC) Performance analy n term of the cot avng A mentoned, the antcpated total operatonal cot ncurred by a frm are compoed by the orderng cot (eq. 5), the nventory holdng cot (eq. 6) and the tranportaton cot (eq. 4). In the ubequent analy we computed the proporton of each of thee cot component n the total operatonal cot n order to npect ther relevance n the coalton formaton decon-makng proce. A CODS tuaton commtted wth the degn of a jont orderng between frm, we drve an analy on the mprovement of the orderng cot n the tand alone tuaton and th collaboratve one. We report n Table 6, the proporton of the orderng cot regardng the total operatonal cot baed on the reult of (., SN,, LC) ntance attrbute. The emprcal reult how that the orderng cot proporton when frm collaborate range acendngly between 6.3% to 75.43% for n=5 to n=30, repectvely. Th gve re to an average gap of 11% mprovement compared to the tand-alone tuaton. Th reult well explan the ncentve of frm to collaborate under orderng cot avng. Jont orderng computaton wth a hgh market cot gven n Table 5 n Appendx B. n Stand alone Collaboraton 5 68.94% 6.3% 10 85.39% 7.13% 15 85.73% 73.1% 0 86.4% 73.58% 5 87.1% 74.47% 30 87.61% 75.43% Average 83. 50% 71. 85% Table 6- Proporton of the Orderng Cot n the Total Cot Functon Regardng the purchang cot, the electon of the untary purchang landng (ee eq. (1)), for each frm a key decon n the cot avng purut. One can note from Fgure 8 the beneft from collaboraton where all the frm move from the frt landng n the SA to the econd or the thrd landng n CODS. Wth (., SN,, LC) ntance attrbute, all frm n collaboraton benefcate from the econd landng n 4 out of 6 ntance and from the thrd landng for the remanng one. Alternatvely, wth (., LN,, HC) ntance attrbute, the untary purchang cot for the collaboratve frm move n all the cae to the thrd landng. 4 CIRRELT-014-61
Coalton Formaton for Sourcng Contract Degn wth Cooperatve Replenhment n Supply Network From the JRSW tandpont, the nventory holdng cot the mot relevant component of the total operatonal cot. We explore n what follow t mpact on the collaboraton degree. Table 7 how that the proporton of the nventory holdng cot when each frm operate ndvdually (econd column of Table 7), for all problem ntance, greater than t equvalent n the collaboratve framework (thrd column of Table 7). In average, uch proporton move from 15.39% for tand-alone poton to 11.33% for the collaboratve tuaton. Such decreae due to the harng of warehoue once the cooperaton launched whch well explan the effcency of ung collaboratve warehoue n the cot avng. In addton, we notce that the proporton of the holdng cot ncreae wth the market ze whch ndcate that the number of collaboratve frm ncreae. Fnally, we hould menton that the ame behavor oberved for ntance (., LN, HC) (ee Table 6 n Appendx B). n Stand alone Collaboraton 5 14.1% 10.11% 10 15.4% 10.4% 15 15.7% 11.34% 0 15.07% 11.6% 5 15.87% 1.33% 30 16.13% 1.39% Average 15. 39% 11. 33% Fgure 8- Optmal Purchang Cot Landng n SA and CODS Table 7- Proporton of the Holdng Cot n the Total Cot Functon Snce for the JRSS tuaton the tranportaton cot the mot domnant component, we compute, n Table 8, t nfluence on the frm collaboraton. We alo report the proporton of the tranportaton cot when each frm take the tand-alone poton veru the cooperaton under a low market aumpton. We can underlne that the tranportaton cot proportonal to the number of frm. It alo notceable that the tranportaton cot decreae from 19.3% to 11.03% n average. Th cot drop due to the reducton of the number of ued STL, a collaboratve frm group ther order n common MTL. Hence, the collaboraton concern frm dtrbuton actvte by harng logtc reource. Bede, the traveled dtance can be mnmzed regardng the hp to pont n the tand-alone tuaton. The mpact of the jont replenhment n the reducton of the tranportaton cot alo oberved wth the hgh market cot (ee Table 6 n Appendx B). n Stand alone Collaboraton 5 16.85% 9.4% 10 19.7% 10.8% 15 19.74% 10.99% 0 19.01% 11.56% 5 19.96% 11.76% 30 0.15% 11.83% Average 19. 3% 11. 03% Table 8- Proporton of the Tranportaton Cot n the Total Cot Functon CIRRELT-014-61 5