Applied Mathematical Modelling

Transcription

1 Appled Matematcal Modellng 7 (01) ontents lsts avalable at cverse cencerect Appled Matematcal Modellng journal omepage: A mult-product mult-ecelon nventory control model wt jont replensment strategy We-Q Zou, Long en, Hu-Mng Ge cool of Automoble and Traffc Engneerng, Jangsu Unversty, Zenjang 101, na artcle nfo abstract Artcle story: Receved January 011 Receved n revsed form 11 Aprl 01 Accepted 1 Aprl 01 Avalable onlne 1 May 01 Keywords: Inventory Mult-product Mult-ecelon Genetc Algortm (GA) Jont replensment strategy On te bass of analyzng te sortages of present studes on mult-ecelon nventory control, and consderng some restrctons, ts paper apples te jont replensment strategy nto te nventory system and bulds a mult-product mult-ecelon nventory control model. Ten, an algortm desgned by Genetc Algortm (GA) s used for solvng te model. Fnally, we respectvely smulate te model under tree dfferent orderng strateges. Te smulaton result sows tat te establsed model and te algortm desgned by GA ave obvous superorty on reducng te total cost of te mult-product multecelon nventory system. Moreover, t llustrates te feasblty and te effectveness of te model and te GA metod. rown opyrgt Ó 01 Publsed by Elsever Inc. All rgts reserved. 1. Introducton A supply can s a networ of nodes cooperatng to satsfy customers demands, and te nodes are arranged n ecelons. In te networ, eac node s poston s correspondng to ts relatve poston n realty. Te nodes are nterconnected troug supply demand relatonsps. Tese nodes serve external demand wc generates orders to te downstream ecelon, and tey are served by external supply wc responds to te orders of te upstream ecelon. Te problem of mult-ecelon nventory control as been nvestgated as early as te 1950s by researcers suc as Arrow et al. [1] and Love []. Te man callenge n tese problems s to control te nventory levels by determnng te sze of te orders for eac ecelon durng eac perod so as to optmze a gven objectve functon. Many researcers ave studed ow to reduce te nventory cost of eter supplers or dstrbutors, or ave consdered eter te dstrbuton system or te producton system. Burns and vazlan [] nvestgated te dynamc response of a mult-ecelon supply can to varous demands placed upon te system by a fnal consumer. Van Bee [] carred out a model n order to compare several alternatves for te way n wc goods are forwarded from factory, va stores to te customers. Zjm [5] presented a framewor for te plannng and control of te materals flow n a mult-tem producton system. Te prme objectve was to meet a presanctfed customer servce level at mnmum overall costs. Van der Hejden [6] determned a smple nventory control rule for mult-ecelon dstrbuton systems under perodc revew wtout lot szng. Yoo et al. [7] proposed an mproved RP metod to scedule mult-ecelon dstrbuton networ. s and Ko [8] consdered a dvergent mult-ecelon nventory system, suc as a dstrbuton system or a producton system. Andersson and Melcors [9] consdered a one wareouse several retalers nventory system, assumng lost sales at te retalers. Huang et al. [10] orrespondng autor. Tel.: ; fax: E-mal address: zwqsy@16.com (W.-Q. Zou) X/$ - see front matter rown opyrgt Ó 01 Publsed by Elsever Inc. All rgts reserved. ttp://dx.do.org/ /j.apm

2 00 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) consdered a one-wareouse mult-retaler system under constant and determnstc demand, wc s subjected to transportaton capacty for every delvery perod. Lagodmos and Kououmalos [11] developed closed-form customer servce models. And many researcers ave modeled an nventory system of only two-ecelon or two-layer. Gupta and Albrgt [1] modeled a two-ecelon mult-ndentured reparable-tem nventory system. Axsäter and Zang [1] consdered a two-level nventory system wt a central wareouse and a number of dentcal retalers. Axsäter [1] consdered a two-ecelon dstrbuton nventory system wt stocastc demand. en et al. [15] consdered a two-level nventory system n wc tere are one suppler and multple retalers. Tee and Rossett [16] developed a smulaton model to explore te model s ablty to predct system performance for a two-ecelon one-wareouse, multple retaler system. eferls and Gannelos [17] developed a new two-layered optmzaton-based control approac for mult-product, mult-ecelon supply can networs. Hll et al. [18] consdered a sngle-tem, two-ecelon, contnuous-revew nventory model. Al-Rfa and Rossett [19] presented a two-ecelon non-reparable spare parts nventory system. Mtra [0] consdered a two ecelon system wt returns under more generalzed condtons, and developed a determnstc model as well as a stocastc model under contnuous revew for te system. Tere are also many researces on mult-ecelon nventory control, consderng eter te dstrbuton system or te supply system. o et al. [1] evaluated conventonal lot-szng rules n a mult-ecelon coalescence MRP system. án and Vastag [] descrbed a mult-ecelon producton nventory system and developed a eurstc suggeston. Bregman et al. [] ntroduced a eurstc algortm for managng nventory n a mult-ecelon envronment. Van der Vorst et al. [] presented a metod for modelng te dynamc beavor of mult-ecelon food supply cans and evaluatng alternatve desgns of te supply can by applyng dscrete-event smulaton. Te model consdered a producer, a dstrbuton center and retaler outlets. Ida [5] studed a dynamc mult-ecelon nventory problem wt nonstatonary demands. Lau and Lau [6] appled dfferent demand-curve functons to a smple nventory/prcng model. Routroy and Kodal [7] developed a treeecelon nventory model for sngle product, wc conssts of sngle manufacturer, sngle wareouse and sngle retaler. ong and Lee [8] consdered a mult-ecelon seral perodc revew nventory system and ecelons for numercal example. Te system extended te approxmaton to te tme correlated demand process of lar and carf [9], and studed n partcular for an auto-regressve demand model te mpact of leadtmes and auto-correlaton on te performance of te seral nventory system. Gumus and Guner [0] structured an nventory management framewor and determnstc/stocastc-neurofuzzy cost models wtn te context of ts framewor for effectve mult-ecelon supply cans under stocastc and fuzzy envronments. aggano et al. [1] descrbed and valdated a practcal metod for computng cannel fll rates n a mult-tem, mult-ecelon servce parts dstrbuton system. Yang and Ln [] provded a seral mult-ecelon ntegrated just-n-tme (JIT) model based on uncertan delvery lead tme and qualty unrelablty consderatons. Gumus et al. [] structured an nventory management framewor and determnstc/ stocastc-neuro-fuzzy cost models wtn te context of te framewor. Ten, a numercal applcaton n a tree-ecelon tree-structure can s presented to sow te applcablty and performance of proposed framewor. Te model only andled one product type. Only one oter paper we are aware of addresses a problem smlar to ours and consderes nventory optmzaton n a mult-ecelon system, consderng bot te dstrbuton system and te supply system. Rau et al. [] developed a multecelon nventory model for a deteroratng tem and to derve an optmal jont total cost from an ntegrated perspectve among te suppler, te producer, and te buyer. Te model consdered te sngle suppler, sngle producer and sngle buyer. Te basc dfference between our model and Rau et al. [] s tat our model consders multple supplers, one producer, and multple dstrbutors and buyers. Addtonally, an algortm desgned by Genetc Algortm (GA) s used for solvng te model, and we apply te jont replensment strategy nto te model. Te remander of ts paper s organzed as follows: In ecton, te varous assumptons are made and te mult-product mult-ecelon nventory control model s developed. In ecton, GA s used for solvng te model and te algortm based on GA s desgned. Ten, we smulate te model under tree dfferent orderng strateges, respectvely. In ecton, conclusons and lmtatons n ts researc are presented. o f g j e Layer 1 Layer - Layer -1 Layer Layer +1 Layer + Layer N upply Networ ore Enterprse strbuton Networ Fg. 1. Te mult-product mult-ecelon nventory control model.

3 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) Matematcal model.1. Te mult-product mult-ecelon nventory control model descrpton In ts model, te raw materals, accessores or products can be suppled from te nodal enterprse of layer to te nodal enterprse of layer + 1, but tere s no logstcs between nodal enterprses of te same layer or te non-adjacent layers. And also tere s no reverse logstcs from te nodal enterprse of g-layer to te nodal enterprse of low-layer. Te mult-product mult-ecelon nventory system s dvded nto tree subsystems (supply networ, core enterprse and dstrbuton networ) by te core enterprse as a dvdng lne (Fg. 1). Te ey ssue to te mult-product mult-ecelon nventory system s to determne te optmal order quantty and te optmal order cycle for eac nodal enterprse n order to mnmze te nventory cost of te wole system. In ts paper, te (T, ) nventory control strategy based on mult-product jont replensment s used. Te mult-product jont replensment strategy s an orderng strategy tat to order varetes of products n one order cycle. Eac nodal enterprse determnes a mnmum order cycle as te basc order cycle, and te order cycle of te same enterprse to order eac product s an ntegral multple of te basc order cycle... Assumptons (1) In ts supply can, tere s only one core enterprse. () Allow a varety of products, but te prce of eac product s fxed. And also allow a varety of raw materals or accessores, but one suppler only provdes one raw materal or accessory. () Te demand of eac nodal enterprse per day s random, but t obeys Posson dstrbuton. () Lead tme of eac nodal enterprse s fxed. (5) torage cost per product per unt tme s constant. And te storage cost of dfferent nodal enterprses s allowed to be dfferent... Notatons P w prce of product w (tere are W nd of products, and w =1,,...,W) P g prce of raw materal or accessory provded by te nodal enterprse g of layer l(g =1,,...,m l l ; l =1,,..., 1; m l s te number of nodal enterprses of layer l) T basc order cycle of te nodal enterprse of layer to order products from te nodal enterprses of layer 1 T ðg;þ order cycle of te nodal enterprse of layer to order products from te nodal enterprse g of layer 1 Z ðg;þ rato of T ðg;þ and T, wc s a postve nteger, so Tðg;Þ ¼ Z ðg;þ T A publc orderng cost of te nodal enterprse of layer to order products from te nodal enterprses of layer 1n eac order cycle, wc s ndependent of te order quantty and te order varetes A ðg;þ ndvdual orderng cost of te nodal enterprse of layer to order products from te nodal enterprse g of layer 1 A ð;;wþ þl T þl T ð;wþ þl Z ð;wþ þl ðg;þ E ðg;þ L ðg;þ Y ðg;þ n eac order cycle, wc s dependent of te order quantty and te order varetes ndvdual orderng cost of te nodal enterprse of layer + l to order te product w from te nodal enterprse of layer + l 1 n eac order cycle, n te dstrbuton networ basc order cycle of te nodal enterprse of layer + l to order products from te nodal enterprses of layer + l 1, n te dstrbuton networ order cycle of te nodal enterprse of layer + l to order product w from te nodal enterprses of layer 1, n te dstrbuton networ rato of T ð;wþ þl and T þl, wc s a postve nteger, so Tð;wÞ þl ¼ Z ð;wþ þl T þl maxmum nventory level of te nodal enterprse of layer to order products from te nodal enterprse g of layer 1 average demand of te nodal enterprse of layer to order products from te nodal enterprse g of layer 1 per day lead tme of te nodal enterprse of layer to order products from te nodal enterprse g of layer 1 L ð;wþ te average lead tme of te nodal enterprse of layer + l to order te product w from te nodal enterprse of layer þl + l 1 H ðg;þ storage cost of te nodal enterprse of layer per product per year quantty demand of te nodal enterprse of layer to order products from te nodal enterprse g of layer 1 per year, so Y ðg;þ ¼ 65E ðg;þ te number of trps from te nodal enterprse g of layer 1 to te nodal enterprse of layer per year, wc s n ðg;þ nversely proportonal to order cycle, so n ðg;þ ¼ Z ðg;þ 1 T

4 0 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) f ðg;þ t ðg;þ X ð;;wþ 1 ðg;;wþ fxed transportaton cost from te nodal enterprse g of layer 1 to te nodal enterprse of layer n eac transportaton (suc as drver s wage) varable transportaton cost to transport te unt product from te nodal enterprse g of layer 1 to te nodal enterprse of layer (suc as cost of fuels), wc s te functon of transport effcency and order quantty n te case of fxed transportaton dstance te expected value of te produce w of te nodal enterprse of layer relatve to order quantty of te nodal enterprse of layer +1 converson rate of product w produced by te nodal enterprse of layer relatve to raw materals or accessores suppled by te nodal enterprse g of layer 1 g ð;;wþ supply coeffcent of product w suppled from te nodal enterprse of layer to te nodal enterprse of layer +1, and P m þ1 ¼1 gð;;wþ ¼ 1 b ðg;;wþ B ð;;wþ proportonalty coeffcent of raw materals or accessores used to produce product w, wc are suppled from te nodal enterprse g of layer 1 to te nodal enterprse of layer, and P W w¼1 bðg;;wþ ¼ 1 sortage penalty per produce w per order cycle from te nodal enterprse of layer + 1 to te nodal enterprse of layer.. Mult-product mult-ecelon nventory control model We dvde te nventory cost nto orderng cost, oldng cost, transportaton cost and sortage cost. (1) Orderng cost Te total orderng cost of te core enterprse per year s defned as follows: Order ¼ Xm A þ Xm 1X m ¼1 T g¼1 ¼1 A ðg;þ Z ðg;þ T : ð1þ Te total orderng cost of te supply networ per year s defned as follows: Order ¼ X X m l A g l T g þ X mx l1 X m l l¼1 g¼1 l l¼1 f ¼1 g¼1 Z Al l T g l : ðþ Te total orderng cost of te dstrbuton networ per year s defned as follows: Order ¼ XN X m þl A þl þ XN mx þl1 X m þlx W l¼1 ¼1 T þl l¼1 ¼1 ¼1 w¼1 A ð;;wþ þl Z ð;wþ þl T þl : ðþ Terefore, te total orderng cost of te mult-product mult-ecelon nventory system per year s defned as follows: T Order ¼ Order þ Order þ Order : ðþ () Holdng cost Te nventory level of te nodal enterprse of layer wen t as receved te order quantty from te nodal enterprse of layer 1 s: ðg;þ E ðg;þ L ðg;þ ; ð5þ and te nventory level of te nodal enterprse of layer before t receves te order quantty next order cycle s: ðg;þ E ðg;þ L ðg;þ E ðg;þ Z ðg;þ T : ð6þ Terefore, te average nventory level n one order cycle s: 1 E ðg;þ L ðg;þ þ ðg;þ E ðg;þ L ðg;þ E ðg;þ ðg;þ Z ðg;þ T ¼ ðg;þ E ðg;þ L ðg;þ E ðg;þ Z ðg;þ T : ð7þ Te total oldng cost of te core enterprse per year s defned as follows: X Hold ¼ Xm1 m E ðg;þ ðg;þ E ðg;þ L ðg;þ Z ðg;þ T 5H ðg;þ : ð8þ g¼1 ¼1

5 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) As a practcal matter, we must ensure tat te average nventory level s greater tan zero, as sown n Eq. (9): E ðg;þ ðg;þ E ðg;þ L ðg;þ Z ðg;þ T > 0: ð9þ Te total oldng cost of te supply networ per year s defned as follows: Hold ¼ X mx l1 X m l l¼1 f ¼1 g¼1 l under te followng constrant: E E L l l l E l l Z L l l T g l E l Z l T g l 5Hl ; ð10þ > 0: ð11þ Te total oldng cost of te dstrbuton networ per year s defned as follows: Hold ¼ XN X m þlx W E ð;wþ ð;wþ þl E ð;wþ þl L ð;wþ þl Z ð;wþ þl T þl þl 5H ð;wþ þl ; l¼1 ¼1 w¼1 under te followng constrant: ð;wþ þl E ð;wþ þl L ð;wþ þl E ð;wþ þl Z ð;wþ þl T þl > 0: ð1þ Terefore, te total oldng cost of te mult-product mult-ecelon nventory system per year s defned as follows: T Hold ¼ Hold g¼1 ¼1 þ Hold þ Hold : ð1þ () Transportaton cost Te total transportaton cost of te core enterprse per year s defned as follows: X Trans ¼ Xm1 m n ðg;þ f ðg;þ þ t ðg;þ Y ðg;þ : ð15þ Te total transportaton cost of te supply networ per year s defned as follows: Trans ¼ X mx l1 X m l l¼1 f ¼1 g¼1 n l fl þ t l Y l ð1þ ; ð16þ 1; were nl ¼ Zl T g l Y l ¼ 65E l Te total transportaton cost of te dstrbuton networ per year s defned as follows: Trans ¼ XW X N mx þl1 X m þl n ð;wþ þl f ð;þ þl w¼1 l¼1 ¼1 ¼1 1; were n ð;wþ þl ¼ Z ð;wþ þl T ð;;wþ þl Y þl þ t ð;þ þl Yð;;wÞ þl ¼ 65E ð;;wþ þl ; ð17þ Terefore, te total transportaton cost of te mult-product mult-ecelon nventory system per year s defned as follows: T Trans ¼ Trans þ Trans þ Trans : ð18þ () ortage cost Assumng X ð;;wþ obeys Posson dstrbuton p ð;;wþ X ð;;wþ ¼ X1 u¼a u g ð;;wþ 1 ðg;;wþ b ðg;;wþ ðg;þ p ð;;wþ Z ðg;þ Z ðg;þ T þ Lðg;Þ durng te perod Z ðg;þ T þ Lðg;Þ, so: T þ Lðg;Þ : ð19þ Te total sortage cost of te core enterprse per year s defned as follows: ortage ¼ Xm Xm þ1 X W ¼1 ¼1 w¼1 B ð;;wþ Z ðg;þ X ð;;wþ T : ð0þ

6 0 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) Te total sortage cost of te supply networ per year s defned as follows: ortage ¼ X X m l mx lþ1 B ðg;þ l X ðg;þ l l¼1 g¼1 ¼1 Zl T g l were X ðg;þ l ¼ P 1 u¼a u gðg;þ l 1l l ; ð1þ p ðg;þ l Z l T g l þ Lðf ;gþ l ; g ðg;þ l ¼ E ðg;þ lþ1 P mlþ1 E ðg;þ ¼1 lþ1, and P m lþ1 ¼1 g ðg;þ l ¼ 1. Te total sortage cost of te dstrbuton networ per year s defned as follows: were ortage X ð;j;wþ þl ¼ XN X m þl mx þlþ1 X W ¼ X1 l¼1 ¼1 j¼1 w¼1 u¼a u g ð;j;wþ þl ð;wþ þl B ð;j;wþ þl X ð;j;wþ þl Z ð;wþ þl T þl p ð;j;wþ þl ; ðþ Z ð;wþ þl T þl þ Lð;wÞ þl ; g ð;j;wþ þl ¼ E ð;j;wþ þl X ; and X E ð;j;wþ j¼1 þl m þlþ1 j¼1 m þlþ1 g ð;j;wþ ¼ 1: Terefore, te total sortage cost of te mult-product mult-ecelon nventory system per year s defned as follows: T ortage ¼ ortage þ ortage þ ortage : ðþ In concluson, we develop te mult-product mult-ecelon nventory control model as follows: mn T ¼ T Order þ T Hold þ T Trans þ T ortage ; ðþ s:t: E ðg;þ E ðg;þ L ðg;þ Z ðg;þ T þ ðg;þ < 0; ð5þ E l Zl T g l E l Ll þ l < 0; l ¼ 1; ;...; ; f ¼ 1; ;...; m l1 ; g ¼ 1; ;...; m l ; ð6þ E ð;wþ E ð;wþ þl L ð;wþ þl Z ð;wþ þl T þl þl þ ð;wþ þl < 0; l ¼ 1; ;...; N ; ¼ 1; ;...; m þl ; w ¼ 1; ;...; W; ð7þ mn Z ; Z ;...; Z 1 ;Þ ¼ 1; ð8þ ðf ¼1;gÞ ðf ¼;gÞ mn Z ; Z ;...; Z ðf ¼m l1 ;gþ ¼ 1; l ¼ 1; ; ;...; ; ð9þ l mn Z ð;w¼1þ þl l ; Z ð;w¼þ þl l ;...; Z ð;w¼wþ þl ¼ 1; l ¼ 1; ; ;...; N : (8) (0) can ensure tat at least one product s order cycle s te basc order cycle. Te decson varables n te model are all ntegers greater tan or equal to zero.. mulaton and analyss.1. mulaton model based on GA Te objectve functon of ts optmzaton model s mnmzaton, and te objectve functon of GA s maxmzaton, so te objectve functon of ts optmzaton model cannot be taen as te ftness functon of GA. We must convert te objectve functon to te ftness functon of GA as follows: FðXÞ ¼ T max T; T < T max ; ð1þ 0; T P T max ; were F(X) s te ndvdual ftness. T max s a relatvely large number, and n ts smulaton model, we may put T max as te largest objectve functon value durng evoluton. Te mult-product mult-ecelon nventory control model can be reduced to a nonlnear programmng problem as follows: ð0þ

7 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) mn f ðxþ; ðþ s:t: g ðxþ 6 0 ð ¼ 1; ; ;...; mþ: In ts paper, penalty functon s used as constrant. o, we construct te penalty functon as follows: /ðx; c Þ¼ Xm ¼1 c mn ðg ðxþ; 0Þ ; ðþ were s teraton tmes of GA. c s penalty factor, wc s a monotone ncreasng sequence and postve value. And c þ1 ¼ e c. Te experence n computaton sows tat f c ¼ 1 and e =5 10, we can aceve satsfactory results. o, we cange (1) to te functon as follows: ( FðXÞ ¼ T max T /ðx; c Þ; T < T max ; ðþ 0; T P T max : Moreover, we use te floatng pont number codng (te cromosome s lengt equals te number of decson varables), te roulette weels selecton mecansm as te selecton operator, te artmetc cross tecnque as te crossover operator, te Gauss mutaton operator as te mutaton operator, and algebra (ts values range from 100 to 500) as te termnaton crtera... mulaton As an llustraton, we develop a mult-product mult-ecelon nventory control model wc as four supplers (te four supplers are dvded nto two levels and eac level as two supplers), one core enterprse and two dstrbutors, and as two products (Fg. ). Te average demand of te customers to order product 1 and product from te dstrbutor 1 of layer per day s 6 unts and unts. Te average demand of te customers to order product 1 and product from te dstrbutor of layer per day s unts and 7 unts. Te values of oter parameters are sown n Tables 1. uppler 1 uppler 1 strbutor 1 ore Enterprse uppler uppler strbutor Layer 1 Layer Layer Layer Fg.. Te example model. Table 1 Te values of te parameters of te core enterprse. Parameters A 1 A ð1;1þ A ð;1þ L ð1;1þ L ð;1þ H ð1;1þ H ð;1þ f ð1;1þ f ð;1þ Values $100 $0 $0 5 5 $5 $0 $00 $50 Parameters t ð1;1þ t ð;1þ g ð1;1;1þ g ð1;;1þ g ð1;1;þ g ð1;;þ 1 ð1;1;1þ 1 ð;1;1þ 1 ð1;1;þ Values $15 $ Parameters 1 ð;1;þ b ð1;1;1þ b ð1;1;þ b ð;1;1þ b ð;1;þ B ð1;;1þ B ð1;1;þ B ð1;;þ B ð1;1;1þ Values $150 $10 $180 $00 Table Te values of te parameters of te supply networ. Parameters A 1 A ð1;1þ A ð;1þ A A ð1;þ A ð;þ L ð1;1þ L ð;1þ L ð1;þ Values $70 $00 $180 $60 $50 $ Parameters L ð;þ H ð1;1þ H ð;1þ H ð1;þ H ð;þ f ð1;1þ f ð;1þ f ð1;þ f ð;þ Values 6 $ $6 $ $15 $00 $10 $50 $150 Parameters t ð1;1þ t ð;1þ t ð1;þ t ð;þ g ð1;1þ g ð;1þ g ð1;þ g ð;þ 1 ð1;1þ Values $ $6 $5 $ Parameters 1 ð;1þ 1 ð1;þ 1 ð;þ B ð1;1þ B ð;1þ B ð1;þ B ð;þ Values $150 $10 $160 $10

8 06 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) Table Te values of te parameters of te dstrbuton networ. Parameters A ð1;1þ A ð1;þ A ð1;1;1þ A ð1;1;þ A ð1;;1þ A ð1;;þ L ð1;1þ L ð1;þ L ð;1þ Values $10 $10 $00 $50 $0 $0 8 8 Parameters L ð;þ H ð1;1þ H ð1;þ H ð;1þ H ð;þ f ð1;1þ f ð1;þ t ð1;1þ t ð1;þ Values $60 $0 $60 $50 $00 $00 $0 $5 Parameters g ð1;1;1þ g ð;1;1þ g ð;1;þ g ð1;1;þ B ð1;1;1þ B ð1;1;þ B ð;1;1þ B ð;1;þ Values $50 $00 $60 $180 Table Te decson varables of te mult-ecelon nventory control model. Nodal enterprses ecson varables ore enterprse T 1 Z ð1;1þ Z ð;1þ ð1;1þ ð;1þ upply networ T 1 T Z ð1;1þ Z ð;1þ Z ð1;þ Z ð;þ ð1;1þ ð;1þ ð1;þ ð;þ strbuton networ T 1 T Z ð1;1þ Z ð;1þ Z ð1;þ Z ð;þ ð1;1þ ð1;þ ð;1þ ð;þ Table 5 Te optmum values of decson varables under te jont replensment strategy. ore enterprse ecson varables T 1 Z ð1;1þ Z ð;1þ ð1;1þ ð;1þ Optmum values upply networ ecson varables T 1 T Z ð1;1þ Z ð;1þ Z ð1;þ Optmum values ecson varables Z ð;þ ð1;1þ ð;1þ ð1;þ ð;þ Optmum values strbuton networ ecson varables T 1 T Z ð1;1þ Z ð;1þ Z ð1;þ Optmum values ecson varables Z ð;þ ð1;1þ ð1;þ ð;1þ ð;þ Optmum values Fg.. Te optmzaton results and evolutonary process of GA under te jont replensment strategy.

9 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) Table 6 Te optmum values of decson varables under te separate replensment strategy. ore enterprse ecson varables T ð1;1þ T ð;1þ ð1;1þ ð;1þ Optmum values upply networ ecson varables T ð1;1þ T ð;1þ T ð1;þ T ð;þ Optmum values ecson varables ð1;1þ ð;1þ ð1;þ ð;þ Optmum values strbuton networ ecson varables T ð1;1þ T ð;1þ T ð1;þ T ð;þ Optmum values 0 0 ecson varables ð1;1þ ð1;þ ð;1þ ð;þ Optmum values Fg.. Te optmzaton results and evolutonary process of GA under te separate replensment strategy. We use te Genetc Algortm and rect earc Toolbox (GAT) desgned by MATLAB. And te decson varables to be optmzed are sown n Table. Based on lots of testng of dfferent combnatons of te parameters, we determne te optmal parameters as follows: Te populaton sze s 100. Te crossover probablty s 0.8. Te mutaton probablty s Te maxmum evoluton generaton s 150. Te elte reserved quantty s. Besdes te jont replensment strategy, tere are also separate replensment strategy and unfed replensment strategy. o we calculate and optmze te nventory cost under te tree replensment strateges usng te GA, respectvely, and ten compare te results. (1) Te jont replensment strategy. Te optmzaton results are sown n Table 5. Te evolutonary process of GA s convergent and te total nventory cost s declnng as te evolutonary process gong (Fg. ). Te total cost s $ () Te separate replensment strategy. Under te separate replensment strategy, te order processes of eac product are ndependent of eac oter, and eac nodal enterprse bears ts own cost of te respectve order. o, we compute te orderng cost repettvely, wc leads to an ncrease n te total nventory cost. Te optmzaton results are sown n Table 6. Te total cost s $ (Fg. ).

10 08 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) Table 7 Te optmum values of decson varables under te unfed replensment strategy. ore enterprse ecson varables T 1 ð1;1þ ð;1þ Optmum values upply networ ecson varables T 1 T ð1;1þ ð;1þ ð1;þ ð;þ Optmum values strbuton networ ecson varables T 1 T ð1;1þ ð1;þ ð;1þ ð;þ Optmum values Fg. 5. Te optmzaton results and evolutonary process of GA under te unfed replensment strategy. Fg. 6. Te comparson on te nventory cost under te tree replensment strateges.

11 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) () Te unfed replensment strategy. Under te unfed replensment strategy, eac nodal enterprse orders products wt te basc order cycle (Z ð1;1þ ; Z ð;1þ ; Z ð1;1þ ; Z ð;1þ ; Z ð1;þ ; Z ð;þ ; Z ð1;1þ ; Z ð;1þ ; Z ð1;þ and Z ð;þ are equal to 1). Te optmzaton results are sown n Table 7. Te total cost s $ (Fg. 5)... Analyss (1) Te total cost under te jont replensment strategy s te lowest (Fg. 6). It s 1% lower tan te total cost under te unfed replensment strategy, and 5% lower tan te total cost under te separate replensment strategy. It sows tat te mult-product mult-ecelon nventory control model and te algortm desgned by GA ave a clear advantage on decreasng te nventory cost of te mult-product mult-ecelon nventory system. () Almost all te order cycles n Table 5 are sorter tan te order cycles n Table 6 or Table 7. Ts s because t allows multple products sare te same orderng cost under te jont replensment strategy. Terefore, t can reduce orderng cost of eac order, mprove order frequency and sorten order cycle. In general, t can reduce te total nventory cost. In ts llustraton, we assume tat t only ave two products n te system. If t as more products n te system, eac product would sare less orderng cost, and te total nventory cost would be lower.. oncluson Te problem of mult-ecelon nventory control s becomng more mportant. Amng at ts problem, we apply te jont replensment strategy nto te mult-ecelon nventory system and buld a mult-product mult-ecelon nventory control model. onsderng bot supplers and dstrbutors, or bot te dstrbuton system and te producton system, ts model can ntegrally express te actons and relatons between every entty n te system. And an algortm desgned by GA s used for solvng te model. Ten we respectvely smulate te model under tree dfferent orderng strateges. Te smulaton results sow tat te establsed model and te algortm desgned by GA ave a clear advantage on decreasng te nventory cost of te mult-ecelon nventory system. Tere are some lmtatons n ts researc. Frst, we assume te lead tme of eac nodal enterprse s fxed. In practce, owever, t s often random varable. econd, ts model only consders te nventory cost. In real lfe, t also needs to consder te tme cost. If consder bot te nventory cost and te tme cost, t s te nventory control model based on agle supply can. Ts wll be done n our future researc. References [1] K.J. Arrow,. Karln, H. carf, tudes n te Matematcal Teory of Inventory and Producton, tanford Unversty Press, tanford, alforna, [].F. Love, Inventory ontrol, McGraw-Hll, [] J.F. Burns, B.. vazlan, ynamc analyss of mult-ecelon supply systems, omput. Ind. Eng. (1978) [] P. Van Bee, Modelng and analyss of a mult-ecelon nventory system: a case study, Eur. J. Oper. Res. 6 (1981) [5] W.H.M. Zjm, Herarccal producton plannng and mult-ecelon nventory management, Int. J. Prod. Econ. 6 (199) [6] M.. Van der Hejden, upply ratonng n mult-ecelon dvergent systems, Eur. J. Oper. Res. 101 (1997) [7] Y.J. Yoo, W.. Km, J.T. Ree, Effcent nventory management n mult-ecelon dstrbuton systems, omput. Ind. Eng. (1997) [8] E.B. s, A.G. Ko, Optmal control of a dvergent mult-ecelon nventory system, Eur. J. Oper. Res. 111 (1998) [9] J. Andersson, P. Melcors, A two-ecelon nventory model wt lost sales, Int. J. Prod. Econ. 69 (001) [10] H.. Huang, E.P. ew, K.H. Go, A two-ecelon nventory system wt transportaton capacty constrant, Eur. J. Oper. Res. 167 (005) 9 1. [11] A.G. Lagodmos,. Kououmalos, ervce performance of two-ecelon supply cans under lnear ratonng, Int. J. Prod. Econ. 11 (008) [1] A. Gupta,.. Albrgt, teady-state approxmatons for a mult-ecelon mult-ndentured reparable-tem nventory system, Eur. J. Oper. Res. (199) 0 5. [1]. Axsäter, W.F. Zang, A jont replensment polcy for mult-ecelon nventory control, Int. J. Prod. Econ. 59 (1999) 50. [1]. Axäter, calng down mult-ecelon nventory problems, Int. J. Prod. Econ. 71 (001) [15] F.Y. en, Y.Y. Feng,.. Lev, Unform dstrbuton of nventory postons n two-ecelon perodc revew systems wt batc-orderng polces and nterdependent demands, Eur. J. Oper. Res. 10 (00) [16] Y.. Tee, M.. Rossett, A robustness study of a mult-ecelon nventory model va smulaton, Int. J. Prod. Econ. 80 (00) [17] P. eferls, N.F. Gannelos, A two-layered optmsaton-based control strategy for mult-ecelon supply can networs, omput. Aded em. Eng. 1 (00) [18] R.M. Hll, M. efbargy,.k. mt, A two-ecelon nventory model wt lost sales, Eur. J. Oper. Res. 181 (007) [19] M.H. Al-Rfa, M.. Rossett, An effcent eurstc optmzaton algortm for a two-ecelon (R,Q) nventory system, Int. J. Prod. Econ. 109 (007) [0]. Mtra, Analyss of a two-ecelon nventory system wt returns, Omega 7 (1) (009) [1] H.G. o, E.M. Malstrom, R.J. lassen, omputer smulaton of lot-szng algortms n tree-stage mult-ecelon nventory systems, J. Oper. Manage. (198) [] A. án, G. Vastag, Analyss of a mult-ecelon producton nventory system wt random supply, Eng. ost Prod. Econ. 1 (1987) [] R.L. Bregman, L.P. Rtzman, L.J. Krajews, A eurstc algortm for managng nventory n a mult-ecelon envronment, J. Oper. Manage. 8 (1989) [] J.G.A.J. van der Vorst, A.J.M. Beulens, P. van Bee, Modellng and smulatng mult-ecelon food systems, Eur. J. Oper. Res. 1 (000) [5] T. Ida, Te nfnte orzon non-statonary stocastc mult-ecelon nventory problem and near-myopc polces, Eur. J. Oper. Res. 1 (001) [6] A.H.L. Lau, H.. Lau, Effects of a demand-curve s sape on te optmal solutons of a mult-ecelon nventory/prcng model, Eur. J. Oper. Res. 17 (00) [7]. Routroy, R. Kodal, fferental evoluton algortm for supply can nventory plannng, J. Manuf. Tecnol. Manage. 16 (005) 7 17.

12 050 W.-Q. Zou et al. / Appled Matematcal Modellng 7 (01) [8] L. ong, H.L. Lee, Optmal polces and approxmatons for a seral multecelon nventory system wt tme-correlated demand, Oper. Res. 51 (00) [9] A. lar, H. carf, Optmal polces for a mult-ecelon nventory problem, Manage. c. 6 (1960) [0] A.T. Gumus, A.F. Guner, A mult-ecelon nventory management framewor for stocastc and fuzzy supply cans, Exp. yst. Appl. 6 (009) [1] K.E. aggano, P.L. Jacson, J.A. Mucstadt, J.A. Rappold, Effcent computaton of tme-based customer servce levels n a mult-tem, multecelon supply can: a practcal approac for nventory optmzaton, Eur. J. Oper. Res. 199 (009) [] M.F. Yang, Y. Ln, Applyng te lnear partcle swarm optmzaton to a seral mult-ecelon nventory model, Exp. yst. Appl. 7 (010) [] A.T. Gumus, A.F. Guner, F. Ulengn, A new metodology for mult-ecelon nventory management n stocastc and neuro-fuzzy envronments, Int. J. Prod. Econ. 18 (010) [] H. Rau, M.Y. Wu, H.M. Wee, Integrated nventory model for deteroratng tems under a mult-ecelon supply can envronment, Int. J. Prod. Econ. 86 (00)