Analysis of Two-Echelon Perishable Inventory System with Direct and Retrial demands

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1 O Joural of Mathematc (O-JM) e-: p-: 9-765X. Volume 0 ue 5 Ver. (ep-oct. 04) aly of Two-chelo erhable vetory ytem wth rect ad etral demad M. amehpad C.eryaamy K. Krha epartmet of Mathematc. M. Thevar College Ulampatt Madura t.taml adu da. epartmet of Mathematc Madura Kamara Uverty College attur Vrudhuagar t.taml adu da. eearch epartmet of Mathematc C.. College odayaaur The t. Taml adu da. btract: th paper we coder a cotuou revew perhable vetory ytem wth Marova demad. The operatg polce are ( ) ad (0M). The lfe tme of a tem ha a expoetal dtrbuto. The ordered tem are receved after a radom tme whch aumed to be expoetal dtrbuto. The demad that occur drectly to the dtrbuto cetre are called drect demad. The arrval proce for the drect demad follow oo proce. The demad proce to the retaler ode depedet to the drect demad proce ad follow oo proce. The demad that occur durg toc out perod are eter to the orbt of fte ze. Thee orbtg demad retry for ther demad after a radom tme whch aumed to be expoetal dtrbuto. The ot probablty dtrbuto of the vetory level at lower echelo hgher echelo ad the umber of cutomer the orbt obtaed the teady tate cae. Varou ytem performace meaure the teady tate are derved ad the log ru total expected cot rate calculated. Keyword: Two-echelo vetory ytem erhable vetory etral demad rect demad.. troducto upply cha may be defed a a tegrated proce where a umber of varou bue ette (uppler dtrbutor ad retaler) wor together a effort to () acqure raw materal () proce them ad the produce valuable product ad () traport thee fal product to retaler. The proce ad delvery of good through th etwor eed effcet commucato ad traportato ytem. The upply cha tradtoally characterzed by a forward flow of materal ad product ad bacward flow of formato. Over the lat two decade reearcher ad practtoer have prmarly vetgated the varou proce of upply cha dvdually. Complete revew wa provded by eta M. eamo (998)[0]. ecetly however there ha bee creag atteto placed o performace deg ad aaly of the upply cha a a whole. H'(Hawlett acard) trategc lag ad Modelg(aM) group tated th d of reearch 977. From practcal tad pot the upply cha cocept aroe from a umber of chage the maufacturg evromet cludg the rg cot of maufacturg the hrg reource of maufacturg bae horteed product lfe cycle the levellg of plag feld wth maufacturg vetory drve cot (C) volved dtrbuto ad the globalzato of maret ecoomc. Wth maufacturg reearch the upply cha cocept grow largely out of two-tage mult-echelo vetory model ad t mportat to ote that coderable reearch th area baed o the clac wor of Clar ad carf (960)[]. complete revew o th developmet were recorded by Federgrue (99)[9]. ecet developmet two-echelo model may be foud. M. He ad. M. Jewe (000)[] xater. (99)[7] ad ahma (98)[4]. Cotuou revew erhable vetory wth tataeou replehmet wa codered by Kalpaam ad rvarga G(988)[5]. cotuou revew ( ) polcy wth potve lead tme two echelo upply Cha wa codered by Krha.K ad lago.c. 005 [5][6]. Th paper deal wth a mple upply cha that modelled a a gle warehoue ad multple retaler ytem hadlg a gle product. order to avod the complexty at the ame tme wthout lo of geeralty we aumed detcal demad patter at each ode. Th retrct our tudy to deg ad aalyze a tadem etwor of vetory whch a buldg bloc for the whole upply cha ytem. The ret of the paper orgazed a follow. The model formulato decrbed ecto alog wth ome mportat otato ued the paper. ecto teady tate aaly are doe. ecto 4 deal wth the operatg charactertc of the ytem ad ecto 5 deal wth the cot aaly for the operato. The ecto 6 deal wth ome umercal llutrato ad the lat ecto 7 coclude the paper. 5 age

2 aly of Two-chelo erhable vetory ytem wth rect ad etral demad. Model ecrpto th paper we coder a two level upply cha vetory ytem. t cot of oe warehoug faclty ad oe retaler. We aumed that the demad to the trbuto Cetre follow oo proce wth parameter λ (> 0). The drect demad get ut at a tme. The demad tated at retaler ode follow oo proce wth parameter λ (> 0). The demad to the retaler ode requre gle tem at a tme. The lead tme are expoetally dtrbuted wth parameter µ(> 0). The retaler follow ( ) polcy to mata vetory ad the dtrbutor follow (0 ) polcy for matag vetory. The tem are perhable ature ad t aumed that the tem are perhe oly at the retaler ode. The lfe tme of a tem expoetally dtrbuted wth parameter γ (> 0).The uatfed cutomer are treated a retral cutomer ad they are watg the orbt wth fte capacty. The repeated cutomer from the orbt (wth capacty ) are etered to the ytem wth rate λ (> 0). The arrvg demad fd the empty toc ad the orbt full are codered to be lot. otato: [] : ( ) th elemet/bloc of the matrx : detty matrx of order e : colum vector of oe wth approprate dmeo = The maxmum vetory level at retaler ode = eorder level at retaler ode = = { = 0 = 0 = }. aly Let { T (t) T (t) T (t) : t 0 } repectvely deote the umber of demad the orbt the o had vetory level the retaler ode ad the umber of tem the trbuto cetre at tme t. From the aumpto made o the put ad output procee X(t) = { ( T (t) T (t) T ( t) ) : t 0} a Marov proce wth tate pace. The ftemal geerator of th proce = ( a( m : l )) ( m) ( l ) ε ca be obtaed from the followg argumet. The prmary arrval of demad to the retaler ode mae a trato the Marov proce from ( m) to ( m) wth tety of trato λ. The arrval of a demad at retaler ode from orbt mae a trato the Marov roce from ( m) to ( m) wth tety of trato λ. The tem perhe mae the trato from ( m) to ( - m) wth tety of trato γ. ad The replehmet of a vetory at retaler ode mae a trato from ( m) to ( + m ) wth rate of trato µ. The arrval of demad to the dtrbuto cetre mae a trato the Marov proce from ( m) to ( m ) wth tety of trato λ. The the ftemal geerator ha the followg fte tructure: 0 Where C 0 otherwe C C C C 0. otherwe age

3 aly of Two-chelo erhable vetory ytem wth rect ad etral demad 5 age otherwe F ) ( where Where ) ( ad 0 00 Where ) ( F Where ) ( Here the matrce ad C are the quare matrce of order ( + )* ad ote that the matrx the quare matrx of order.

4 aly of Two-chelo erhable vetory ytem wth rect ad etral demad. teady tate aly ce the tate pace fte ad rreducble the tatoary probablty vector П for the geerator alway ext ad atfe П = 0 & Пe =. The vector П ca be repreeted by 0 П = ( ) Where 0 П <> 0. = ( ) П <<>> = ( ) 0... ow the tructure of how the model uder tudy a fte brth death model the Marova evromet. Hece we ue the Gaver algorthm for computg the lmtg probablty vector. For the ae of completee we provde the algorthm here. lgorthm:. eterme recurvely the matrx 0 by ug 0 = 0. olve the ytem. Compute recurvely the vector = ( ) C ug П <> = ( ) e-ormalze the vector П ug e..k V. erformace Meaure th ecto we computed the followg ytem performace meaure teady for propoed vetory ytem 4. Mea vetory Level Let deote the expected vetory level the teady tate at retaler ode ad deote the expected vetory level at dtrbuto cetre. = 0 e = xpected erhable ate Let deote the expected perhable rate ad whch gve by = 0 e 4. xpected umber Of emad The Orbt Let (o) deote the expected umber of cutomer the orbt whch gve by (o) = 4.4 Mea eorder ate The mea reorder rate at retaler ode gve by r = 0 e ( ( ) ) 4.5 hortage ate hortage occur oly at retaler ode ad the hortage rate for the retaler deoted by α ad whch gve by e 54 age

5 aly of Two-chelo erhable vetory ytem wth rect ad etral demad = V. Cot aly th ecto we aalye the cot tructure for the propoed model by coderg the mmzato of the teady tate total expected cot per tme. The log ru expected cot rate for the model defed to be TC( ) h h r g pc c0 ( o ) Where h - deote the vetory holdg cot at retaler ode per ut tem h - deote the vetory carryg cot at dtrbuto cetre per ut tem - deote the etup cot/ order at retaler ode g - deote the hortage cot/ ut hortage at retaler ode c o - deote the bac orderg of a demad the orbt / ut tme. pc deote cot per ut falure V. umercal llutrato We fx the followg put to calculate the ytem performace meaure ad the total vetory cot curred all the three ode We aalyed the followg the umercal ecto.. From Fgure we have how that TC decreae whe µ γ creae.. From Table to table 4 we oberve the followg a) to be expected ad creae TC creae b) to be expected creae TC decreae. c) to be expected the total cot fucto creae whe the cot et-up cot holdg cot hortage cot cot for falure ad the bac orderg cot creae. Fgure: µ v γ o TC (4 4 6) = =4 =6 =8 =0 = = = = = Table : v o TC =80 =85 =90 =95 =00 = = = = = Table : v o TC 55 age

6 aly of Two-chelo erhable vetory ytem wth rect ad etral demad h =0.04 h =0.08 h =0. h =0.6 h =0.0 h = h = h = h = h = Table : h v h o TC(4 4 6) g =0. g =0.4 g =0.6 g =0.8 g =.0 = = = = = Table 4: v g o TC(4 4 6) pc=0. pc=0.4 pc=0.6 pc=0.8 pc=.0 c 0= c 0= c 0= c 0= c 0= Table 5: pc v c 0 o TC(4 4 6) V. Cocludg emar th paper we aalyed a cotuou revew perhable vetory ytem wth upply cha. The tructure of the cha allow vertcal movemet of good from dtrbuto ceter to retaler. ( ) type vetory ytem wth oo demad ad expoetal dtrbuted lead tme for tem are aumed at C (mddle echelo). emad occurrg durg the toc out perod are aumed to be lot. The C replehe ther toc wth expoetal dtrbuted lead tme from warehoue (upper echelo) ha abudat upply ource. The tem are uppled to the C pac of (= -) tem from the warehoue. The model deal wth lot ale at C ad the upply from maufacturer term of pocet. t would be teretg to aalye the problem dcued th artcle where the lfe tme of tem are cotat. aturally wth the cluo of cotat lfe tme of each tem the problem wll be more challegg. other mportat exteo could be made by relaxg the aumpto of expoetally dtrbuted lead tme to a cla of arbtrarly dtrbuted lead tme ug techque from reewal theory ad em-regeeratve procee. Oce th doe the geeral model ca be ued to geerate varou pecal cae. For example three dfferet lead tme dtrbuto oe wth coeffcet of varato greater tha oe oe wth coeffcet of varato le tha oe ad aother wth coeffcet of varato equal to oe ca be compared. Cot aaly ca the be carred out for ( ) ( -) ad lot-for-lot model ug each of the three dfferet lead tme dtrbuto to determe whch polcy optmal for ay gve lead tme dtrbuto. The author curretly worg o the above exteo ad thee wll be reported future publcato. eferece [] Clar. J. ad H. carf 960. Optmal olce for a Mult- chelo vetory roblem. Maagemet cece 6(4): [] Hadley G ad Wht T. M. (96) aly of vetory ytem retce- Hall glewood Clff ew Jerey. [] Clar. troducto to tochatc rocee. retce Hall glewood Clff J 975. [4] ahma. 98. erhable vetory theory. revew. Operato eearch [5] Kalpaam ad rvarga G Cotuou revew erhable vetory Model tattc [6] voroo toy ad aul Zp 99. valuato of Oe-for-Oe eplehmet olce for Multechelo vetory ytem Maagemet cece 7(): [7] xater. 99a. xact ad approxmate evaluato of batch orderg polce for two level vetory ytem. Oper. e [8] uzacott J.. ad J. G. hathumar. 99. tochatc Model of Maufacturg ytem. retce Hall J. [9] Federgrue Cetralzed plag model for mult echelo vetory ytem uder ucertaty.c.grave et al.ed. Hadboo OMvol4orth-Hollad mterdam The etherlad -7. [0] eta M. eamo upply Cha eg ad aly: Model ad Method. teratoal Joural of roducto coomc.vol.55 o. pp []. Mg He ad. M. Jewe erformace meaure of a mae-to- order vetory- producto ytem. Traacto [] lago C. 00 cotuou revew perhable vetory ytem at ervce faclte upublhed h.. The Madura Kamara Uverty Madura [] rvarga G. lago C. ad rumugam. (00) cotuou revew perhable vetory cotrol ytem at ervce faclte dvace tochatc Modellg otable ublcato c [4] Gapaolo Callo Xaver de Motgro et. al vetory rve Cot Harvard ue evew age

7 aly of Two-chelo erhable vetory ytem wth rect ad etral demad [5] Krha. K ad lago.c cotuou revew ( ) polcy wth potve lead tme upply Cha roceedg of atoal Coferece o tellget Optmzato Modelg. lled ublher vt. ltd. ew elh [6] Krha. K 007 tochatc Modelg upply Cha Maagemet ytem upublhed h.. The Madura Kamara Uverty Madura age