General Bounds for the Optimal Value of Retailers Reorder Point in a Two-level Inventory Control System with and without Information Sharing

Transcription

1 Genera Bon for te Optia Vae of etaier eorer oint in a Two-eve Inventory Contro Syte wit an witot Inforation Saring Nia Yazan Sena.D. Caniate Departent of Intria Engineering Sarif Univerity of Tecnoogy one: ax: E-ai: nia_yazanena@er.arif.e Aboai Erag Jaroi.D. Aociate rofeor Departent of Intria Engineering Sarif Univerity of Tecnoogy one: ax: E-ai: erag@sarif.e Seye Tagi Akavan Niaki.D. rofeor Departent of Intria Engineering Sarif Univerity of Tecnoogy.O. Box Azai Ave. Teran Iran one: ax: E-ai: niaki@sarif.e Abtract In ti ty an inventory yte coniting of a inge proct one ppier an tipe ientica retaier i coniere. Eac retaier repenie inventory fro te ppier accoring to te we known poicy. Tranit tie are contant an retaier face inepenent oion ean. Te ppier tiizing te retaier' inforation in eciion aking for repenient poicy wit a given orer ize tart wit initia batce of ize an pace an orer in a batc of ize to an otie orce wen a new orer i pace. In ti inventory yte exce ean i backorere eaye orer are atifie on a firt-coe firterve bai an no partia ipent i aowe. By partitioning te cot fnction of ti yte genera pper an ower bon for te optia vae of are eterine. Bae on evera nerica exape it i own tat tee bon Correponing Ator

2 epeciay te ower bon aow te optia reorer point to be fon ore effectivey wit a orter oving tie. Keywor: Two-eve inventory yte Inforation aring Contino review Bon oion ean. Introction an iteratre eview Sppy cain anageent SCM a been tie for ore tan for ecae. Since orreter [] icovere te fctation an apification of ean fro owntrea to ptrea of te ppy cain tere a been conierabe aont of iteratre anayzing te SCM. Nero copanie c a grocery tore [2] itribtion center [3] beer anfactrer [4] etc. in vario intrie ave nertaken ajor initiative c a re-engineering effort an invetent in inforation tecnoogy to anage teir canne an rece inefficiencie in teir ppy yte. An iportant ie in SCM i ow to contro ti-eve inventory yte [5]. Meto for controing repenient in c yte can be bae on centraize eceon tock poicy or ecentraize tock an ean inforation intaation tock poicy. In intaation tock poicy orering eciion are ae bae on te inventory poition of te inivia intaation i.e. on te tock on an an on orer in te backog. Te eceon tock inventory poition of a certain intaation i obtaine by aing te intaation inventory poition at te coniere intaation an te intaation inventory poition at a it owntrea intaation. It i obvio tat intaation tock poicie o not reqire inforation abot te inventory itation at oter intaation. However tee poicie o i 2

3 te avantage of ing inforation to anage teir canne an rece inefficiencie in teir ppy yte. Severa reearcer ave anayze te intaation tock poicy an it inor variation ing te oion oe. oion oe wit one-for-one orering poicie in ppy cain can be ove very efficienty. Axäter [6] eveope a ipe recrive eto to eterine te oing an tock-ot cot of a yte coniting of one centra wareoe an tipe non-ientica retaier. He erive exact eto to evaate te appicabiity of oion oe in one-for-one orering poicie. Axäter [7] expree tota inventory cot a a weigte ean of cot for one-for-one orering poice of a two-eve inventory yte wit one ppier an evera ientica retaier in wic a faciitie epoy poicy. In ti poicy wen te inventory poition of a retaier ecine to an orer of ize i pace fro te ppier. Uing [6] orberg [8] cacate te cot of a two-eve inventory yte wit one wareoe an evera retaier wo face copon oion ean procee. orberg [9] erive an exact fora for te tota cot of a two-eve inventory yte wit one wareoe an evera retaier in wic te retaier face ifferent oion ean procee. Wen tere are two non-ientica retaier Axäter [] propoe an exact eto to evaate te inventory yte. However for te cae of ore tan two retaier e eveope an approxiate procere. Seifbargi an Akbari [] invetigate te tota cot of a two-eve inventory yte were te nfie ean are ot an ence te ean i approxiatey a oion proce. Axäter [2] coniere a two-eve inventory yte wit one centra wareoe an tipe non-ientica retaier facing ifferent inepenent copon oion ean procee in wic a faciitie appy contino review intaation tock poicie wit ifferent 3

4 reorer point an batc qantitie. He preente a new approxiate eto to evaate oing an ortage cot to eect optia poicie. A oe of ecentraize inventory contro in a two-eve itribtion yte wit one centra wareoe an N non-ientica retaier wa coniere by Aneron an Markn [3] in wic a intaation e contino review intaation tock -poicie for repeniing teir inventorie. In ti reearc an approxiate cot evaation tecniqe an a oifie cot-trctre at te wareoe wa introce to ecopoe te ti-eve inventory contro probe into N+ inge eve b-probe. Ten te b-probe were ove in an iterative anner by a ipe coorination procere. Tey owe tat teir propoe eto converge to a near optia otion wen te ean foow nora itribtion. Moreover to ae te qaity of te invove approxiation an pper bon for te reative cot increae of ing te obtaine otion wa erive. rterore for pecia cae of poicie vario approxiate an exact eto ave been preente in te iteratre exape of c eto are [4-9]. Wie inforation tecnoogy aow fir to tranfer ata qicky an econoicay it a graay ifte reearcer conieration fro intaation to eceon tock poicie. Te eceon tock concept wa firt introce by Cark an Scarf [2]. Migro an obert [2] ientifie inforation a a btitte for inventory on econoic ter. A brief overview of te iteratre revea tat tere are evera recent paper concerne wit ow te eceon tock poicy iprove inventory contro of itribtion yte. ee an Wang [22] ice te e of inforation aring in ppy cain in practice reate it to acaeic reearc an otine te caenge facing 4

5 te area. Cacon an ier [23] anayze a two-eve itribtion yte wit ientica retaier batc orering an perioic review. In ti yte a faciitie foow an n orering poicy wit te ppier batc ize being an integer tipe of retaier' batc ize. Tey erive a iation-bae ower bon over a feaibe poicie an copare te ipact of ing an aocation poicy bae on are inforation to te effect of rece ea-tie an ceaper orer proceing. Tey owe ow te ppier can e c inforation to better aocate tock to retaier. Hiao an Sie [24] coniere a two-eceon ppy cain tat containe one ppier an one retaier. Tey tie te qantification of te bwip effect an te vae of inforation aring between te ppier an te retaier. Tey owe wen tere exit inforation aring te vae of te bwip effect i greater tan wen it i witot inforation aring. Moinzae [25] coniere a tanar ppy cain oe coniting of a inge proct one ppier an M ientica retaier. He ae tat te ean at eac retaier foowe a oion proce an eac retaier pace orer to te ppier accoring to te poicy. He ten tie te benefit of inforation aring in te ppy cain caracterize a te avaiabiity of on-ine inforation of retaier' inventory poition to te ppier an provie an exact anayi for te operating eare of c yte. At te en e owe tat inforation aring i ot beneficia in yte wit pecific caracteritic. Sajaifar an Haji [26] invetigate an inventory yte wit one ppier an one retaier. Tey epoye a of te aption of Moinzae reearc [25] an erive te exact cot fnction of te inventory yte ner conieration. rterore Axäter an Markn [27] coniere a two-eceon inventory yte wit one centra wareoe an tipe non-ientica retaier. Tey erive a new poicy for te 5

6 wareoe orering wic wa optia in te broa ca of poition-bae poicie reying on copete inforation on te retaier inventory poition tranportation tie cot trctre an ean itribtion in a faciitie. A ti-eve inventory contro yte can be eigne in ifferent way. Often te contro i carrie ot eiter by an intaation tock or by an eceon tock reorer point poicy [28]. Axäter an Zang [29] entione tat it i ot coon to e an intaation tock reorer point poicy to contro c yte. Tey ecare tat oter contro poicie are bae on te eceon tock reorer point intea of te intaation tock poicy. Te eterination of te reorer point in ti-eve inventory contro yte a a contro paraeter i a coon practice [ ]. Terefore if a genera bon i erive for te reorer point of c yte it wi be ot beneficia to bot practitioner an reearcer. In ti reearc we e a of te aption of Moinzae' reearc [25] an conier an inventory yte coniting of a proct one ppier an M ientica retaier. Eac retaier repenie inventory fro te ppier accoring to a poicy. Te ppier can take avantage of te onine inforation on te inventory tat an ean activitie of a retaier to repeni i tock fro an otie orce in batce of. A fixe orer qantity i ae an effective bon epeciay a ower bon for te optia vae of reorer point in te retaier eve are fon. Tee bon enabe te practitioner an reearcer to fin te optia reorer point ore effectivey in a orter oving tie. In wat foow firt te expecte tota cot of te inventory yte i fon. Next by new foration it i own tat te average tota cot rate of te yte coprie of oe part an i bae on te pecification of eac part. 6

7 Ten a genera pper bon an a genera ower bon for te optia vae of are erive. Tee bon can be e a genera bon for retaier' reorer point in any probe eiter wit inforation aring te ppier e te are inforation to repeni it tock or witot inforation aring te ppier oe not ave acce to te retaier' inforation. inay te perforance of tee bon i exaine trog etiating te copter CU-tie ave to ove te probe. 2. robe Definition an oration Conier an inventory/itribtion yte coniting of a inge ite a ppier an M ientica retaier were ean at eac retaier foow a oion proce. Eac retaier carrie inventory an repenie tock fro te ppier accoring to a poicy. Wen te inventory poition ecine to a batc of ize i orere. Exce ean i backorere in eac retaier an no partia ipent i aowe. Deaye retaier-orer are atifie on a firt-coe firt-erve bai. Te ppier tart wit initia batce of ize an pace an orer to an otie orce wen a new orer i pace by a retaier. To eiinate te inferior poicy we ony conier non-negative vae of in or anayi. In aition to te poibe rano eay at te ppier te tranit tie fro te ppier to eac retaier i contant. rterore we ae tat te repenient ea-tie fro te otie orce to te ppier i ao contant i.e. te otie orce a ape capacity. 2. Notation Te paraeter an te variabe of te oe are: 7

8 : Te ean qantity at eac retaier p : Te oion probabiity a fnction of te ean wit ean : Te oion copientary cative itribtion fnction of te ean wit ean : : : : : : : : : Orer qantity of eac retaier an te ppier eorer point at eac retaier Nber of batce of ize initiay aocate to te ppier Tranit tie fro te ppier to eac retaier Tranit tie fro te otie orce to te ppier Hoing cot per nit per tie nit at eac retaier Hoing cot per nit per tie nit at te ppier Sortage cot per nit per tie nit at eac retaier Eape tie between te paceent of a ppier' orer nti it i reqete to fi a retaier' orer f. : robabiity enity fnction of. : Copientary cative itribtion fnction of E. : Expecte vae of K : Te retaier average tota cot rate were ea tie i an te retaier carrie inventory accoring to te poicy. B : A part of K tat i e to cacate te average ortage cot 2.2 Cacating te Cot Te average tota cot in te yte i coprie of te cot at eac retaier an te oing cot at te ppier. Te retaier an te ppier cot can be 8

9 cacate by focing on a ppier orer an evaating it eary an tary tie. Wie eary tie i te tie an orer pen in ppier inventory nti it i e to atify a retaier orer tary tie i te aont of tie a eaye retaier orer t wait nti it i fie by an incoing ppier orer. However in orer to copte tee cot one nee to copte te itribtion of. Note tat eac retaier face a oion itribtion wit a rate of an exce ean i backorere in te retaier. Terefore te ppier face a oion itribtion wit a rate of M an eape tie between te paceent of a ppier' orer nti it i reqete to fi a retaier' orer foow an Erang M itribtion. A entione before we ae tat i given an fixe ignoring te ppier an retaier orer cot. Bae on Haey an Witin [3] if ea tie i eqa to an a retaier carrie inventory accoring to a poicy te average tota cot rate at te retaier can be expree a: K K B 2 were: B 2 Aing te expecte vae of te tary tie to te repenient ea-tie fro te ppier to a retaier an repacing ti vae in we can fin te average tota cot rate at a retaier. Moreover aving te itribtion of an ing te expecte vae of te eary tie it i eay to fin te oing cot at te ppier. Te e of eiter te intaation or te eceon tock reorer point poicie i coon practice [29]. Terefore in orer to generaize te fining an epoy te 9

10 oe for bot inforation aring an witot inforation aring yte we e te Moinzae' [25] cot fnction. Moinzae [25] coniere a two-eve inventory yte wit te ae aption e in ti reearc. However in i yte te ppier a on-ine inforation abot te ean a we a inventory activitie of te proct at eac retaier. Te ppier tart wit initia batce of ize an pace an orer to an otie orce ieiatey after a retaier inventory poition reace were. Wen te ppier orer ieiatey after receiving an orer fro a retaier iiar to a part of ti reearc i.e. te ppier oe not e te are inforation. Wen te ppier epoy te inforation are by te retaier to ake orer/repenient eciion. Terefore ing Moinzae' cot fnction we can are not ony te yte of i reearc an intaation tock poicy bt ao te yte wit te are inforation an eceon tock poicy a we. In orer to e Moinzae' [25] cot fnction et introce te foowing aitiona notation: G. : Copientary cative fnction of ean at a retaier ring orer ea-tie TC : Expecte tota cot of te yte per tie nit wen te ppier tart wit initia batce of ize an pace an orer to an otie orce ieiatey after te retaier inventory poition reace. B : A part of TC tat i e to cacate te average ortage cot. : A part of B in te firt partitioning eto

11 O : A part of B in te econ partitioning eto Moinzae [25] erive te expecte tota cot of te yte a: 2 TC M B E 3 were B G G 4 in wic G f p f. 5 He ao cacate te copientary cative itribtion fnction of ee Appenix. In te next ection we firt partition te cot fnction in 3 an ten we obtain pper an ower bon for reorer point of retaier' poicy. 3. Te Cot artitioning Ti ection preent two ifferent partitioning of te expecte tota cot of te yte by wic a partition wi be e to erive a ower bon an te oter wi be epoye to fin an pper bon on te bet vae of. epacing 5 in 4 an ing 2 we get: B G G

12 2 p p B 6 In oter wor B i coprie of two part. Te firt part tat i te firt bracket of 6 i enote by B an te econ part tat i te econ bracket of 6 i. Bae on 6 an 3 TC can ten be written a: 2 E M M MK E M M B M TC 7 Ti partitioning wi be e to erive te pper bon. Aternativey rewriting 5. G can be own to be: f G 8 epacing 8 in 4 an ing 2 we ave:.. O B f f G G B 9

13 3 In oter wor B conit of two part te firt part tat i te firt bracket of 9 i enote by B an te econ part tat i te econ bracket of 9 i cae O. Uing 9 an 3 it i poibe to verify tat: 2 E M M O M MK E M M O M B M TC Ti partitioning wi be appie to fin te ower bon. 4. Deriving te Bon To rive te bon firt a pecification of K o be aree. Differentiating K in wit repect to an 2 we ave: B K B K Coniering 2 it can be own tat: B B 2 Hence bae on an 2 for a given vae of an K i convex in.

14 4.. Te Upper Bon Bae on 7 TC i coprie of tree part. Te firt part i K were an in ection 4 it a been own to be convex in. Wie te econ part i an increaing fnction on ee Appenix 2 te tir part i inepenent of an can be coniere a a contant vae. It ean tat te optia reorer point for K i an pper bon for te bet vae of te retaier' reorer point in te yte Te ower Bon Bae on one can ow tat TC i coprie of for part. Te firt part i K were wic i convex in. Te econ part i a ecreaing fnction on ee Appenix 3. Te tir an fort part are inepenent of an teir can be coniere a a contant vae. It ean tat te optia reorer point for i a ower bon for te optia vae of te K retaier' reorer point in te yte Deterination of te Bon To eterine te bon te vae of tat iniize an K o be fon. Tee vae are te ower an te pper bon repectivey. A proven before for a given vae of an K i convex in. Terefore te optia vae of tat iniize K wi be te ini vae of tat i vai for: K K 3 4 K

15 Uing an 2 we can rewrite 3 a: 2 4 Terefore te ini vae of tat i vai in 4 wi be te pper ower bon of te retaier reorer point in te efine yte were i given for te pper bon an for te ower bon. Meanwie it o be note tat te bet vae of for K i greater tan or eqa to [7]. In ary te fowcart of te propoe eto in eriving te ower bon i given in igre. epacing wit te pper bon can be obtaine by a iiar fowcart. Inert igre abot ere 5. aity of te Bon In orer to invetigate te efficiency of te propoe bon evera tet probe are ove in ti ection. Te tet probe invove tree ifferent ppy cain coprie of one two an for retaier for eac of wic a paraeter except an are fixe. Tee probe are contrcte by taking a poibe cobination of te foowing vae of te paraeter: =2 5 an = an 2 an = 5 an i.e. 8 ifferent cobination. Te vae of te paraeter an are contant an for exape are ae to be = = =. an =. Witot epoying te bon Moinzae [25] ove te probe an obtaine te optia vae of te eciion variabe an. A entione before in cae were te probe wit inforation aring are aree 5

16 an wen probe witot inforation aring are exaine. or te fixe vae of an te average tota cot rate in te yte i convex in [25-26] an te bet vae of i greater tan or eqa to [7]. An iportant criterion for evaating te efficiency of te bon i te nber of iteration reqire by te earc eto. Bae on te convexity pecification of te cot fnction on one an witot ing te bon te optia vae of tat iniize TC i fon by exaining vae of tarting wit to. On te oter an by ing te ower bon te optia vae of tat iniize TC i fon by exaining vae of tarting wit to iiting te nber of iteration. et efine te efficiency a te pee rate of te earc agorit in fining te otion. Coniering te nber of eeent ina b wic ib a for eac probe we expre te coptationa efficiency of te bon a te nber of eeent in In oter wor ivie by te nber of eeent in 2 efficiency 5 2. Ti efficiency inex ow ow any tie one can ove te probe fater by ing te ower bon. Tabe ow te efficiency ret of 8 ifferent probe for te cain wit one retaier. Te ret of ifferent probe for te two oter cain are own in Tabe 2 an 3. In tee tabe te opti vae of te yte paraeter ave been own by an te pper an te bon ave been cacate.. In eac probe for te bet vae of an 6

17 Te ret in Tabe 2 an 3 ow tat in a probe te optia reorer point i very coe to te eterine ower bon an tat on te average te probe can be ove 6.27 tie fater tan te one obtaine witot eterining te ower bon. Inert Tabe abot ere Inert Tabe 2 abot ere Inert Tabe 3 abot ere 6. Concion an ecoenation for tre eearc In ti paper an inventory contro yte coniting of one proct a ppier an M ientica retaier wa coniere. It wa own tat tere are two genera bon for te optia vae of te retaier reorer point. Te pper bon i te optia vae of a retaier reorer point in a inge tage yte were te ea tie i eqa to axi poibe ea tie for te retaier in te yte. Te ower bon i te optia vae of a retaier reorer point in a inge tage yte were te ea tie i eqa to ini poibe ea tie for te retaier in te yte. Te average tota cot rate at a retaier in a inge tage yte i convex in te retaier reorer point an terefore te genera bon can be eaiy cacate. It o be note tee genera bon are vai for te yte eiter wit or witot inforation aring. or reveaing te coptationa efficiency of te above ggete pper an ower bon 54 nerica exape ave been ove. Bae on te efficiency inex efine in te paper on te average te probe can be ove 6.27 tie fater tan wen we o not e te ower bon. 7

18 A ajor iitation of te oe ice in ti paper i te aption of te ientica retaier. Atog it i c ore iffict to ea wit non-ientica retaier it i poibe to exten oe of te iea preente in ti paper to ore genera cae. or frter reearc we gget eveoping an anaytica approac bae on te trctre given in ti paper to fin or to iit te vae of an TC. 7. Acknowegeent Te ator are tankf for te contrctive coent of te reviewer tat iprove te preentation of te paper. 8. eference. orreter JW 96 Intria Dynaic. MIT re Cabrige MA. 2. Krt Saon Aociate Inc. 993 Efficient coner repone: Enancing coner vae in te grocery intry. oo Marketing Intitte Waington D.C. USA. 3. Stak G Evan San E 992 Copeting on capabiitie: Te new re of corporate trategy. HAVAD BUS EV Marc-Apri: orbe 997 Heineken on te Internet. Marc 24: Markn J 22 Centraize inventory contro in a two-eve itribtion yte wit oion ean. NAV ES OG 49: Axäter S 99 Sipe otion procere for a ca of two-eceon inventory probe. OE ES 38:

19 7. Axäter S 993 Exact an approxiate evaation of batc-orering poicie for two-eve inventory yte. OE ES 4: orberg 995 Optiization of orer-p-to- poicie for two-eve inventory yte wit copon oion ean. EU J OE ES 8: orberg 996 Exact evaation of -poicie for two-eve inventory yte wit oion ean. EU J OE ES 96: Axäter S 998 Evaation of intaation tock-bae poicie for two-eve inventory yte. OE ES 46:S35-S45.. Seifbargi M Akbari M 26 Cot evaation of a two-eceon inventory yte wit ot ae an approxiatey oion ean. INT J OD ECON 2: Axäter S 995 Approxiate evaation of batc-orering poicie for a one-wareoe N-non-ientica retaier yte ner copon oion ean. NAV ES OG 42: Aneron J Markn j 2 Decentraize inventory contro in a twoeve itribtion yte. EU J OE ES 27: Moinzae K ee H 986 Batc ize an tocking eve in tieceon repairabe yte. MANAGE SCI 32: ee H Moinzae K 987a Two-paraeter approxiation for tieceon repairabe inventory oe wit batc orering poicy. IIE TANS 9:

20 6. ee H an Moinzae K 987b Operating caracteritic of a twoeceon inventory yte for repairabe an conabe ite ner batc orering an ipent poicy. NAV ES OGIST 34: Svorono A Zipkin 988 Etiating te perforance of ti-eve inventory yte. OE ES 36: Axäter S orberg Zang W 994 Approxiating genera tieceon inventory yte by oion oe. INT J OD ECON 35: Aneron J Mecior 2 A two-eceon inventory oe wit ot ae. INT J OD ECON 69: Cark AJ Scarf HE 96 Optia poicie for a ti-eceon inventory probe. MANAGE SCI 6: Migro obert J 99 Te econoic of oern anfactring Tecnoogy trategy an organization. AM ECON EV 8: ee H Wang S 2 Inforation aring in a ppy cain. INT J TECHNO MANAGE 2: Cacon G ier M 2 Sppy cain inventory anageent an te vae of are inforation. MANAGE SCI 46: Hiao JM Sie CJ 26 Evaating te vae of inforation aring in a ppy cain ing an AIMA oe. INT J ADV MANU TECH 27: Moinzae K 22 A ti-eceon inventory yte wit inforation excange. MANAGE SCI 48:

21 26. Sajaifar SM Haji 27 Optia otion for a two-eve inventory yte wit inforation excange eaing to a ore coptationay efficient earc. A MATH COMUT 89: Axäter S Markn J 28 Optia poition-bae wareoe orering in ivergent two-eceon inventory yte. OE ES 56: Axäter S Jntti 997 Coparion of eceon tock an intaation tock poicie wit poicy ajte orer qantitie. INT J OD ECON 48: Axäter S Zang W 999 A joint repenient poicy for ti-eceon inventory contro. INT J OD ECON 59: Haey G Witin T 963 Anayi of Inventory Syte. rentice a Eagewoo Ciff NJ USA. 2

22 22 Appenix : Te cative itribtion fnction of Te cative itribtion fnction of i M k k t k k M t In wic k i i k j k M k k t j i k t j t i t k t i t i t i an t i t i t i

23 23 Appenix 2: or fixe vae of an i an increaing fnction on. p p p p p p p p p p p

24 24 Appenix 3: or fixe vae of an O i a ecreaing fnction on f f f f f f f O G f f f O O O

25 it of figre caption - igre : Te fowcart of te propoe eto to fin te ower bon it of tabe caption - Tabe : Te bon an te efficiency of te ower bon for te cain wit one retaier 2- Tabe 2: Te bon an te efficiency of te ower bon for te cain wit two retaier 3- Tabe 3: Te bon an te efficiency of te ower bon for te cain wit for retaier 25

26 igre Start 2 Ye En No igre : Te fowcart of te propoe eto to fin te ower bon 26

27 Tabe Tabe : Te bon an te efficiency of te ower bon for te cain wit one retaier efficiency

28 Tabe 2: Te bon an te efficiency of te ower bon for te cain wit two retaier efficiency

29 Tabe 3: Te bon an te efficiency of te ower bon for te cain wit for retaier efficiency