Periodic Resource Reallocation in Two-Echelon Repairable Item Inventory Systems

Transcription

1 -ASE Perodc Resource Reallocao wo-echelo Reparable Iem Iveory Sysems Hoog Chu LAU, Je PAN, Huawe SON Absrac ve a exsg sock allocao a veory sysem, s ofe ecessary o perform reallocao over mulple me pos o address veory mbalace ad maxmze avalably. I hs paper, we focus o he suao where here are wo opporues o perform reallocao wh a repleshme cycle. We derve a mahemacal model o deerme whe ad how o perform reallocao. Furhermore, we cosder he exeso of hs model o he suao allowg a arbrary umber of reallocaos. Expermeal resuls show ha he wo-reallocao approach acheves beer performace compared wh he sgle-reallocao approach foud he leraure. We also llusrae how o apply he proposed model o desg cos-opmal perodc resupply polces. Idex erms wo-echelo Iveory, Perodc Resupply, Reallocao, Reparable Iem, Mlary Logscs I. INRODUCION We cosder a arboreal veory sysem whch a ceral depo serves bases. Mlary sysems such as arcrafs ad aks are deployed a he bases. hese sysems break dow because he uderlyg compoes, whch are called LRU le replaceable us, are eher wor ou over me ad/or damaged durg usage. Socks are allocaed a he depo ad bases o sure couy of operaos. Whe a LRU fals a a base, a spare replaces f oe s avalable; oherwse a backorder s curred. I mlary pracce, due o he lmed space a he bases, he faled LRUs are usually se back o he depo for repar. I he mea me, a order s placed by he base o he depo o sed a spare. A spare wll be se o he base f oe s avalable; oherwse here s a backorder a he depo. Afer he falure s repared, wll be se o he depo veory o fulfll fuure demads. As demads are sochasc, veory mbalace wll occur ad eds o grow wh me. hs mbalace ulmaely Mauscrp receved Ocober 0, 007. hs work was suppored par by he Sgapore Msry of Defese uder he JAUAR research gra. Hoog Chu Lau s wh School of Iformao Sysems, Sgapore Maageme Uversy, 80 Samford Road, Sgapore 7890 phoe: ; e-mal: hclau@smu.edu.sg. Je Pa s wh School of Iformao Sysems, Sgapore Maageme Uversy, 80 Samford Road, Sgapore 7890 e-mal: jepa@smu.edu.sg. Huawe Sog s wh Decso Scece ad Egeerg Sysems, Resselaer Polyechc Isue, 0 8 h Sree, roy, NY 80 USA e-mal: sogh@rp.edu. reaches a suao where some bases hold excess veores, whle ohers face crcal shorage. o correc he mbalace, socks eed o be reallocaed. o crease he effcecy ad reduce uavalably, we also allow excess veores a some bases o be reallocaed laerally o ohers wh shorage. I hs paper, we are cocered wh a wo-sa reallocao scheme wh a sysem repleshme cycle. hs work s respose o he ope challege pos by Cao ad Slver [] o cosder wo or more possble reallocaos wh a cycle. I pracce, hs problem s also faced by plaers who eed deerme he me bewee perodc reallocaos. Alhough he S-,S repleshme polcy s geerally assumed he leraure.e. depo wll sed a spare o he base oce a falure occurs, hs s a sylzed suao sce s almos mpossble o supply couously pracce, especally a aval evrome. he depo has o sed spares o offshore bases ad brg he falures back for repar perodcally. Hece, s mpora o deerme whe ad how o dsrbue socks o he bases perodcally he cycle. ve ha we have wo reallocao sas, s mpora o deerme whe each reallocao should occur. If we perform he frs reallocao oo early, may preve early backorders bu could lead o growh of backorders before he ex reallocao or durg he remag me he cycle. Coversely, performg reallocao laer preses wo problems: Frs, may cause hgh levels of early backorders. Secod, lae reallocao may leave o me o perform he secod reallocao. Furhermore, he me erval bewee he frs ad he secod reallocaos s also mpora: f s oo shor, falures brough back o depo for repar may o have bee compleed ad cosequely he depo has oo few spares o perform he secod reallocao; f he me erval s oo log, causes hgher levels of backorders a bases bewee reallocaos. herefore, he key ssue s how bes o sychroze he wo reallocaos. May papers have aalyzed resource reallocao, perodc resupply ad rsk poolg effec. Oe example s he classc paper by Eppe ad Schrage [4] whch aalyzed a mulechelo veory sysem cosderg exeral lead mes ad radom demads, where he opmal allocao of socks amog mulple ses may o be feasble due o sock mbalace. I her model, he depo wll order eough socks from a ousde suppler o esure a cera level of sysemwde veory poso, ad he perform complee allocao of he receved sock o he ses accordg o he demads durg he exeral lead me. Jösso ad Slver [7] saw ha

2 -ASE hs scheme has o reallocao possbly, ad proposed a scheme ha performs complee rasshpme of all se veores a a fxed sa, whch s oe perod before he ed of he order cycle sce her raoale s ha sockous prmarly occur durg he las perods of a order cycle. Jackso ad Mucksad [6] also cosdered a sgle, predeermed reallocao me ad derve boh exac ad approxmae opmaly codos ha do o gore he possbly of mbalace a he me of reallocao. Oe lmao of hs work however s ha, sce hey do o perm laeral resupply bewee ses, hey ecouered dffcules ryg o ascera how much sock o allocae o each se. Aoher aveue of exeso of he Eppe ad Schrage [4] work, whch also mproves he suao gve Jackso ad Mucksad [6], s foud Jackso [5], whch allows he ceral warehouse o hold sock ad make allocaos o he realers every perod of he cycle. he proposed allocao polcy s a "shp-up-o-s" polcy: he warehouse makes shpmes o resore he veory poso of each realer o some predeermed value, S, every perod so log as he warehouse has suffce sock. he cocep of pooled-rsk perod was roduced, whch refers o he laes perod of allocao. sao ad Ekawa [3] proposed a wo-phase push corol polcy for cosderg he opmal reallocao sa a wo-echelo veory sysem. her mehod predeermes a fxed reallocao sa all repleshme cycles, depede of he dyamc behavor of he veores a he realers. hs suao was mproved by Cao ad Slver [] recely, who proposed a heursc mehod o dyamcally deerme he opmal reallocao sa each repleshme cycle ad perform reallocao a ha sa. he abovemeoed papers deal mosly wh cosumable ems. I a mlary coex, s mpora o perform reallocaos of spare pars perodcally because hey are ofe very expesve ad affec sysem avalably grealy [3], [9]. Sysem avalably s usually measured erms of Expeced Backorders EBO e.g. [], [8], []. o our kowledge, here are few works o redsrbuo of spare pars mulechelo sysems, excep a bref meo of he problem [] ad a feaure wh a propreary commercal ool OPUS [0]. hs paper makes echcal corbuos he followg ways. Frs ad foremos, we cosder how o perform more ha oe reallocao wh a repleshme cycle, ad sead of fxg reallocao sas o cera me pos, we propose how o deerme he me po sas for reallocao. hs s a respose o he challege pos by Cao ad Slver [], whch ssued a ope queso for he problem of mulple reallocaos wh a cycle. Furhermore, we relax he classcal assumpos he followg maer. he repleshme of socks from he depo o bases s perodc,.e. socks ad falures ca be raspored oly a cera me pos a bach. he eral lead or raspor me bewee he depo ad bases s ozero. rasshpmes amog bases are allowed. he remader of hs paper s orgazed as follows. Seco gves he problem defo, assumpos ad oaos. Our mahemacal model ad approach are he preseed Seco 3. Seco 4 preses exesve expermeal resuls. Seco 5 shows how our model ca be exeded o desg cos-opmal perodc resupply polcy. Fally, coclusos ad fuure work are provded Seco 6. II. PROBLEM DEFINIION Our resource reallocao problem s based o wo mpora ad smplfyg assumpos. Frs we assume he eral lead or raspor mes for movg ems from he ceral depo o each base s o eglgble, bu he lead me laerally bewee bases s eglgble. I pracce, he raspor me bewee echelos s more mpora o mlary plaers such ha usually cao be gored whle he assumpo of eglgble laeral lead me s cosse wh [], [5], [3] whch assume reallocao ca be acheved saly. hus he spares should be raspored from he depo ahead of eral lead me for he desao so ha ca arrve o me for reallocao. Secod, o smplfy he problem, we assume all falures are oly reparable a he depo whch has fe repar capaces. he repar me s expoeally dsrbued wh mea. Because of raspor me, repar a he depo ca oly ake place lead me afer reallocao. he sysem repleshme cycle s H base perods,.e. every H base perods, he ceral depo places orders o a ousder suppler. herefore our reallocao decso me horzo s wh hs repleshme cycle. Demads over me a he bases are assumed o be depede, Posso varables wh mea a base durg each perod. he ma oaos used hs paper are as follows. : dex of se 0 for he depo : umber of bases S : al sock level of LRU a se H: sysem repleshme cycle, base perods L: eral lead me,.e. raspor me bewee depo ad base : mea repar me of LRU y : Posso radom varable demads a sgle perod a base : mea value of y τ: dex of he perod a he ed of whch reallocao akes place I τ: sock level a se saly before reallocao a he ed of perod τ U τ: sock level a se saly afer reallocao a he ed of perod τ : me po a whch he frs reallocao s performed : me po a whch he secod reallocao s performed EBO : expeced backorder a he ed of perod a base If reallocao akes place a, deoes he EBO saly before reallocao EBO: expeced sum of backorders over all bases a he ed of perod

3 -ASE E, : oal.e. aggregae EBO a base a me pos, ad H E, : oal EBO across all bases a me pos, ad H. ve he al umber of socks a each se, we eed o decde he varables, a whch reallocao akes place, as well as he umber of socks a each se afer reallocao so as mmze he oal EBO over all bases a hree me pos a he ed of perod ad jus before he reallocaos respecvely ad a he ed of he cycle. We lke o clarfy a hs sage ha we use he erm reallocao o refer o hree separae reallocao acves of dffere ypes of veores a dffere me pos wh a cycle: a Reallocao of spare ems from he depo o bases a me pos L ad L; b Reallocao of spare ems amog bases a me pos ad ; ad c Sedg faled ems from bases o he depo a me po. Noe ha he value of he objecve fuco depeds o he mes a whch he reallocaos are carred ou ad how he reallocaos are doe. Noaoally herefore, our am s o fd,, U, U such ha he fuco E, E, s mmzed, where E, EBO + EBO + EBO H III. MAHEMAICAL MODEL A. Before he Frs Reallocao ve he al sock allocaos a all ses, he expeced backorders over all bases a me, EBO, saly before reallocao ca be calculaed accordg o he sadard defo of EBO as follows: EBO x S f x S EBO dx where x s a realzao of radom varable of D. D y deoes he demads of LRUs a base durg me erval [0, ], whch s a Posso radom varable wh mea ad f s he probably desy fuco of D. I hs paper, we approxmae D by a ormally dsrbued radom varable wh mea E D, ad varace Var D ad hece he probably desy fuco s x f x exp. We jusfy hs π approxmao as follows. Sadard sascs have show ha hs approxmao s good whe he mea value of he Posso radom varable s o smaller ha 0. Furhermore, urs ou from our dealed umercal vesgao ha he approxmao s sll sasfacory for our purpose eve wh mea values o smaller ha 4 see Appedx A. I he mlary coex, we wess a prologed repleshme cycle, where he sum of demads arsg he erval bewee he sar ad he frs allocao or bewee he wo allocaos exhb relavely large mea values. hs pheomeo s also see a varey of commercal segs revewed [4], such as copyg maches ad rasporao equpme, whch have relavely log produc lfecycle, or elecrocs, whch requre a relavely large umber of reparable ems. Hece, afer sadardzao ad compuao, EBO ca be expressed by: S EBO 3 where k z k φ z dz s he u ormal loss k fuco ad φz s he probably desy fuco of he sadard ormal dsrbuo. B. he Frs Reallocao A me, we wll perform he frs reallocao. he spares wll be dsrbued from he depo o bases ad amog dffere bases whle falures a all bases wll be se back o depo for repar. From our assumpo, he spares a he depo wll commece rasporao for he bases a me L ad arrve a me for reallocao, whle rasshpme amog bases wll occur saly. O he oher had, repar for faled ems wll sar a he depo a + L because of he raspor me. ve he veory levels a all ses before reallocao I 0,,,, our goal s o fd he veory levels a all ses afer reallocao U 0,,, such ha he EBO over all bases by jus before he secod reallocao wll be mmzed. ha s, he reallocaed spares wll be used o las ul he ex reallocao. Mahemacally, EBO over all bases by ca be expressed as: EBO m EBO U ' s U m U ' s Ad we have he cosras U + U 0 U 0, 0,..., I + I 0 Whou rasshpmes, he equaly cosras 5 should be chaged o U I. We kow ha hose spares he raspor ppeles from he depo have o effec o he EBO a bases ul hey arrve a bases saly before he reallocao. herefore, we cosra I 0 τ I 0 τ-l ad U 0 τ U 0 τ-l where reallocao akes place a he ed of perod τ. So we have I 0 I 0 L S 0 ad we kow I S D for,, where D y ad > L. Hece, Equao 5 ca be chaged o: + U 0 S0 + S D U 6 Usg a Lagrage mulpler, he opmzao ca be represeed as 4 5

4 -ASE U m U ' s + U + U 0 I I0 Dffereag wh respec o U,, ad seg he resul o zero, we oba U Ψ + 0,, 8 where Ψ k φ z dz s he rgh-had al area of he k sadard ormal dsrbuo. So accordg o he propery of sadard ormal dsrbuo, U c,, 9 where c s a cosa, depede of. Usg 9 o sum over all bases ad usg 6 leads o U + j j [ S + S D U ] 0 0 j j 7 0 C. he Secod Reallocao Usg he same mehod as above, for he secod reallocao, our purpose s o fd U 0,,, such ha he EBO over all bases a he ed of he cycle wll be mmzed. However, due o he raspor me L bewee he depo ad bases, he repar cao sar ul falures arrve a he depo a me + L ad smlarly spares mus be se ou o bases a L. I order o have more spares for he secod reallocao, we cosra > + L. hose falures comg ou of he repar ppele afer L ca oly be used for reallocao ex me f here are subseque reallocao sas, bu sce here s o more opporuy o perform a hrd reallocao, we wll compleely redsrbue all socks o had a he depo o bases by L. Assumg complee redsrbuo a he secod reallocao sa, we have U 0 U 0 L 0. Our objecve fuco s: EBO H m EBO H U ' s U H m H U ' s H subjec o he cosras U I + I 0 We kow I U D for,, where D + y. he umber of avalable spares a he depo jus before he secod reallocao equals o he umber of spares lef a he depo afer he frs reallocao U 0, plus hose faled ems se o depo durg he frs reallocao whch have fshed repar ad se o depo veory by me L. Le R be he radom varable ha represes he umber of such ems. Hece, we have I 0 I 0 L U 0 + R. Assumg he repar me a he depo follows a expoeal dsrbuo wh mea, he probably ha a faled em has fshed repar ad se o veory by L s gve by exp[- L/]. Sce we kow E D ad Var, hs mples / E R L e ad L/ Var R e. herefore, Equao ca be rewre as: 0 + U U D + U R 3 Usg 6, Equao 3 ca be furher chaged o: + R 4 D U S0 + S D As he above mehod, usg a Lagrage mulpler ad dffereag wh respec o U,, ad seg he resul o zero, we oba: U H + j 0+ j [ S S Y H ] j j 0 + j j D y j j 5 where Y Y + Y R, Y D, Y D. D. Compue Opmal EBO Usg 0 ad 5, we ca compue he opmal spare allocaos of LRU afer each reallocao. However, we have o specfed how o compue EBO by 4 ad EBOH by. I addo, U 0 s sll cluded 0,.e. he veory level a each base afer he frs reallocao depeds o dffere veory levels lef a he depo. From 5, we kow ha o maer how may spares are lef a he depo afer he frs reallocao, uder he complee redsrbuo assumpo for he secod reallocao, he erm U 0 wll dsappear,.e. he veory level s depede of U 0. herefore, order o reduce EBO jus before he secod reallocao a, U 0 should be se o zero accordg o 0. ha s, we also adop complee redsrbuo for he frs reallocao. hus, U + 6 [ S S Y ] where Y. Subsug U 4 by 6, we ca compue he

5 -ASE EBO jus before he secod reallocao for a gve value of Y. However, Y s a ormally dsrbued radom varable wh mea EY ad varace Var Y. hus, weghg EBO for a gve value of Y by he desy of Y ad egrag over Y, we have U EBO f y dy 7 By subsug Y ξ ad usg 6, Equao 7 ca be rewre as EBO aξ + b ϕ ξ dξ 8 where a ad S0 + S b Usg he resul [] ad [] where b ax + b φ x dx + a, we oba + a + EBO 9 S0 S + + Respecvely, usg ad 5, we ca also cosruc he formula of EBOH for a gve value of Y, whch Y s a ormally dsrbued radom varable. YY +Y Y D y, ad y + R, Y D EY EY + EY ER. hus, has mea L/ [ ] L/ + e e ad varace Var Y Var Y + Var Y + Var R L/ L/ + + e + e Weghg EBOH for a gve value of Y by he desy of Y ad egrag over Y, we have U H EBO H H f y dy 0 H ad usg he same mehod as EBO, we oba L/ EBO H H + + e L/ S0 S H e + + L/ H + + e E. wo-allocao Approach Usg 3, 9 ad, we ca calculae our objecve fuco E, he oal expeced backorders over all bases a hree me pos for a gve reallocao sa par ad : E, EBO + EBO + EBOH Our purpose s o fd such ad 0 < < < H ha E s mmzed. Due o he raspor me L from he depo o bases, he frs reallocao cao ake place earler ha he perod L. I addo, he secod reallocao has o ake place before he perod H. he wo reallocao mes are cosraed by > + L as meoed Seco 3.3. herefore, he frs reallocao ca ake place a he erval [L, H-L-] whereas he secod reallocao ca ake place a he erval [ +L+, H-]. Hece, we ca compue E for possble pars of, o deerme he mmal E ad he correspodg,. However, f he secod reallocao sa s he ed of he cycle, we do o make good use of wo reallocao opporues. E wll be he same as ha [] wh oly oe reallocao because our objecve fuco s he expeced backorders jus before reallocao. [] proved ha he E value wll decrease frs ad he crease as creases for a gve. Hece our heursc sraegy ca be saed as a smple search procedure as follows: for L; < H L ; ++ for + L + ; < H; ++ { // perform d reallocao laer f E, > E, + coue; // perform d reallocao a hs else sore hs value of E, ad break; } Compare he sored E, values for dffere values of ad choose he mmum wh correspodg ad. Compuaoally speakg, he wors case umber of eraos s bouded by H-3L*H-3L-/, ad hece he compuaoal me complexy so H, sce each f saeme requres O compuao. We prese he compuao performace for varous repleshme horzos H Appedx B. hese resuls show ha he compuao me for our approach s reasoably accepable. F. Exeso o Mulple Reallocaos Based o he resul of wo-reallocao for he reparable em veory sysem, he exeso o M-reallocao s ow preseed. he EBO over all bases a M + me pos for a gve se of reallocao sas {,,, M } are gve as E,,, M EBO + EBO EBO M + EBOH 3

6 -ASE Wh he assumpo abou complee dsrbuo of ems a ceral depo a each reallocao sa, he order-up-o level a each reallocao sa ad he correspodg expeced backorder from hs reallocao o he ex reallocao ca be smlarly calculaed as wo-reallocao problem saed before. Here, we oly prese he formula o he order-up-level a fal reallocao sa M ad he correspodg expeced backorder from M ll he ed of repleshme cycle as follows: U M H M + 4 [ S + S Y H ] j 0 M j j j ad R, 3,, M M where Y y R M deoes he arrvg repared ems a reallocao sa from he depo o bases. We use X k o deoe he umber of falures geeraed durg perod [ k-, k ] ad we kow E Xk k k ad Var X k k k for all k k,, M f we assume 0 0. hus, we ca calculae R as follows: [ k L]/ / k k L/ R X e e + X [ e ] [ ]/ / k L k k k j j L/ [ ] j j e [ k L]/ / k k j e k j L/ [ ] j j e ER e e + Var R e + 5 Furhermore, he mea ad varace of Y ca be calculaed as follows: M M M M E R EY Ey [ ] ER ad M M + M M. Var R Var Y Var y Var R + Weghg EBOH for a gve value of Y by he desy of Y ad egrag over Y, we have U M H M EBO H H M f y dy 6 H M ad usg he same mehod as he wo-reallocao problem, we oba M M + M + EBO H H Var R 7 M S0 S H E R + + M H M + M + Var R Aga, our purpose s o fd such se of reallocao sas {,,, M } 0 < < < M < H ha E s mmzed. Due o he raspor me L from he depo o bases, he frs reallocao cao ake place earler ha he perod L. I addo, fal reallocao M has o ake place before he perod H. wo cosecuve reallocao sas are cosraed by > - + L M for he same reaso as he wo-reallocao problem saed Seco 3.3. Due o hs cosra, s also easy o see ha Normal dsrbuo approxmaes Posso dsrbuo well. herefore, he frs reallocao ca ake place a he erval [L, H-M-*L+-] whereas he reallocao sa M ca ake place a he erval [ - +L+, H-M*L+-]. he wors-case oal eraos s hus H LM + L, ad hece he compuaoal me M M complexy ca be measured as H M O. M! IV. EXPERIMENAL RESUL AND SENSIIVIY ANALYSIS Our expermeal resuls are preseed hs seco. I Seco 4., we use es cases o compare he resuls uder wo reallocao sas wh hose havg sgle sa []. I Seco 4., we perform exesve sesvy aalyss o show he effecs of each depede parameer o he values of oal EBO ad reallocao sas. A. Comparso Frs, we use es case o deerme whe ad how o reallocae for wo reallocao sas, ad compare he resuls wh hose allowg sgle reallocao, as see []. I our experme, we have oe depo suppors 5 bases 5. he legh of he repleshme cycle s 30 perods H30. I order o make meagful comparso bewee he woreallocao ad he sgle-reallocao scheme proposed [], we se boh he eral lead me ad he mea repar me o be zero L0, 0. Frs we focus o he decal depede demad dsrbuos a all bases. We se 4 for all,,. he sock level a each base s se o S H* 0. We deerme he sock level a he depo o be 0 S.33 H 57.07, where.33 represes he probably of % probably ha he oal sysem demads a cycle H exceeds he oal socks a he depo ad all bases. We mpleme boh our mehod ad ha of [] so ha we ca compare hem o he effec of wo reallocaos durg a cycle. he resuls are show Fg., ogeher wh Fg. whch provdes a elarged vew o he comparso amog wo-reallocaos over dffere frs-reallocao me sas. From Fg., we ca see frsly ha gve he frs reallocao sa, he oal EBO E decreases ad he creases as he secod reallocao sa creases. hs s cosse wh wha we meoed he above seco. herefore, he me erval bewee reallocaos ca be eher oo shor because of fewer repared falures a he depo or oo log because of more falures a bases.

7 -ASE Fg.. Comparso of EBO vs. me bewee sgle- ad wo-reallocao Fg.. Comparso of EBO vs. me amog wo-reallocaos Secodly, we ca see from Fg. ha E decreases ad he creases as he frs reallocao sa creases. hs s dcaed by comparg E0,, E4, ad E7,. he curve of E4, s below ha of E0,, dcag beer o perform he frs reallocao a 4 ha 0 whle he curve of E7, s above ha of E4,, dcag worse o perform he frs reallocao a 7 ha 4. hrdly, Fg. shows ha he curve of E7, ersecs wh ha of E0,. hs dcaes ha should o always delay he frs reallocao ad perform he secod oe a hurry. I fac, here s also a erseco bewee he curve of E4, ad he curve of E7, alhough o obvous. Comparg all combaos of,, we fd he opmal reallocao sas par s 4, 0 wh E 3.7e-4. Fourhly, we ca see from Fg. ha mulple reallocaos ca reduce he oal EBO compared wh sgle reallocao. From Fg., he opmal reallocao sa [] s a 4 wh E We ca also see ha our frs reallocao sa should be before 4. hs s because f we reallocae a laer ha 4, here wll be a large umber of backorders a bases. More eresgly, we compare [] s resul wh hose whose frs reallocao akes place a 4. he resuls are show able I. From able I, we ca see ha whe he frs reallocao sa s 4, f we perform he secod reallocao mmedaely afer he frs oe, E ca also be mproved because more falures ca be repared as more falures are brough back he depo uder he assumpo of fe repar capaces. However, f we perform he secod reallocao a he ed of he cycle, s equvale o reallocae oly oce recall our objecve fuco s ha jus before reallocao. Hece, E should be he same as ha of ha preseed [], whch s proved o be able I. ABLE I OAL EBO WHEN HE FIRS REALLOCAION INSAN IS 4 H30, E 4, , , , , , Nex, we se lower sock levels a bases o vesgae wha wll happe uder a hgher level of EBO. he resuls are show Fg. 3, where we mulply he sock level a each base he prevous case by 0.8. Fg. 4 s o hghlgh he comparso wh wo-reallocaos whe he frs reallocao akes place a dffere me sa. Cao ad Slver [] clam ha because geerally s more cosly o delay allocao beyod he bes me ha o perform somewha early, eds o hedge agas he hgher peales by commg o a earler allocao me. Fg. 3 shows frsly ha he opmal reallocao sa for [] s 8 wh E From Fg. 4, our opmal reallocao sa par s 4, 9 wh E3.68e-5 correspodgly, he secod oe beg brough forward. hs s cosse wh he clam []. Furhermore, Fg. 4 shows secodly ha uder he same reallocao sa for he frs me, he secod reallocao sa s also brough forward. I he prevous case, whe 0, he opmal s 6 whle he curre case he opmal s. Smlarly whe 4, 9 he prevous case whle he curre oe. Nex, we ru a es case by relaxg he assumpo of decal demad dsrbuos a bases whle demad s sll assumed o be depede. For o-decal demad dsrbuos a bases, we use he same mehod as [], seg he mea demad of each base by */ +,,, 4. he resuls are show able II. From able II, we observe ha hs chage does o brg abou cosse effec o he resuls a hgh sock level ad low sock level wh he mehod []. However, usg our mehod, eher a hgh sock level or a low sock level, he o-decal demad wll curs hgher expeced backorder ha decal demad. I s probably due o he hgher CV a some bases wh o-decal demad. ABLE II E FOR IDENICAL DEMAND DISRIBUIONS VS. NON-IDENICAL DEMAND DISRIBUIONS Case []- Ours- []-0.8 Ours-0.8 Idecal e e-5 No-decal e e-3

8 -ASE show he chages of E wh dffere depede parameers ad he chages of reallocao sas wh repar me ad eral lead me. ables IV o VI llusrae ha E decreases as k he safey facor creases, ha E decrease as he average demad level creases ad ha E decreases wh H he sysem cycle legh, all of whch are cosse wh correspodg oes [] for sgle reallocao cases. ABLE IV: EFFECS OF k ON E k Idecal.4e e-5 No-decal 3.3e-3.63e-3 Fg. 3. Comparso of EBO vs. me bewee sgle- ad wo-reallocao wh fewer socks ABLE V: EFFECS OF ON E Idecal 3.68e-5 3.3e e- No-decal.63e e e-7 Fg. 4. Comparso of EBO vs. me amog wo-reallocaos wh fewer socks B. Sesvy Aalyss As preseed [], hs subseco we wll show he effecs of each depede parameer o he values of oal EBO.e. E ad reallocao sas for wo reallocao sas cases. We use he parameers Seco 4., plus repar me ad eral lead me se by us see able III ad focus o he suao of low sock level as saed Seco 4.. We ru he es cases for boh decal ad o-decal demad dsrbuos whle demad s sll assumed o be depede. For o-decal demad dsrbuos a bases, we use he same mehod as [], seg he mea demad of each base as * / +,,. ABLE III: PARAMEERS FOR SENSIIVIY ANALYSIS Parameer Values k.645,.33 4, 6, 0 H 30, 50, 00 0, 0, 0, 30, 00 L 0,, 5, 8 Isead of usg graphcs as [], whch seems o gve a vsual lluso ha he relaoshp s lear, we use ables o ABLE VI: EFFECS OF H ON E H Idecal 3.68e-5 3.e-8 5.e-6 No-decal.63e-3.37e-4.43e-8 More eresgly, we show he effecs of repar me ad eral lead me o he oal EBO E ad reallocao sas,, whch s o he model of []. I s o surprsg ha E creases as repar me creases able VII sce akes more me o repar a falure so ha fewer ems come ou gve a cera me erval. From able VII, s eresg for us o fd ha as creases, he me erval bewee wo reallocaos creases 8 whe 0, 9 whe 0, 0 whe 30 for decal case ad 8 whe 0, 0 whe 0, whe 30 for odecal case. However, whe 00, recommeds ha he frs reallocao should be shfed earler 3 ad he creases he me erval bewee wo reallocaos for decal case. I able IX, s also o surprsg ha E creases as L eral lead me creases sce akes more me o raspor ems back ad forh excep ha he effec s o obvous whe L s small E remas same for L 0 ad L.. I able X, we show he eresg effec of L o reallocao sas especally. From able X, we kow ha he frs opmal reallocao sa s 4 f raspor me L s zero ad repar wll ake place a he depo saly afer he frs sa. Bu f L, repar ca oly ake place me perods afer he frs sa. Whe mea repar me s assumed o be zero, hs chage o raspor me wll o affec reallocao sas. However, whe raspor me becomes larger lke L5, he frs sa has o be pu earler so as o le he secod reallocao o so hurry. I addo, raspor mus ake place from he depo o he bases a he begg of he cycle 0 for reallocao such a way ha he frs reallocao wll o be doe earler ha 8 as show able X.

9 -ASE ABLE VII: EFFECS OF ON E Idecal 3.68e No-decal.63e ABLE VIII: EFFECS OF ON REALLOCAION INSANS, Idecal 4,9 4, 5,4 4,4 3,4 No-decal 3,9 4, 4,4 3,4 4,5 ABLE IX: EFFECS OF L ON E L Idecal 3.68e e-5.05e No-decal.63e-3.63e e ABLE X: EFFEC OF L ON REALLOCAION INSANS, L Idecal 4, 9 4, 9 0, 8, 5 No-decal 3, 9 3, 9 0, 8, 5 V. APPLICAION: PERIODIC RESUPPLY POLICY I mos sudes o perodc resupply polces, oe usually deermes he opmal allocao of veores uder a gve fxed me erval bewee perodc allocaos, for example, every h me us. he queso we eed o ask s wheher hs h me u s cos-opmal. Noe ha whou cosderg coss, would be opmal o reallocae as frequely as possble,.e. le he me erval bewee reallocaos ed o be fely small so ha approxmaes couous resupply as [], [8]-[]. ve ha reallocao s cosly, freque reallocao wll cur hgh operag cos, whle shorage of socks wll also cur pealy coss. I s herefore mpora o fd he rgh value for h so as o balace bewee he coss of reallocaos ad shorages. I hs seco, we show how our reallocao model ca be exeded ad appled o derve a perod resupply polcy. o acheve hs purpose, we frs exed our proposed model o mulple more ha reallocaos, where he challege s o deerme he me erval bewee reallocaos assumg he ervals are he same for a gve me horzo. hs would pave he way for he desg of cos-opmal perodc resupply polces by sudyg he balace bewee he coss of reallocaos ad shorages. We roduce more oaos based o hose Seco. Isead of radom reallocao sas, here reallocaos are assumed o ake place every h perod so ha he oal umber of reallocaos s m [H /h] s uecessary o reallocae a he ed of he cycle, where [x] s he maxmal eger umber o greaer ha x. We also assume h s greaer ha L, he lead me o raspor back ad forh bewee he depo ad bases. C r s he u cos of a reallocao ad C s s he u cos of a shorage. hus, our objecve s o mmze he oal cos; m C Cr m + Cs E 8 Exeded from 7, E here s he expeced backorders over all bases a a seres of reallocao sas ad a he ed of he cycle,.e. m p m + EBO p p E EBO ph + EBO H 9 where EBO p EBO ph p,,m ad EBO m+ EBO H Accordg o 3, he expeced backorders over all bases a me h, jus before he frs reallocao s S h EBO h 30 h Respecvely accordg o 9, he expeced backorders over all bases a me h, jus before he secod reallocao s EBO h + h 3 S0 S h + h + h From he secod reallocao oward, we mus cosder hose falures ha have fshed repar ad are se o depo veory me o be raspored o bases for he k h reallocao, R k. We use X o deoe he umber of falures geeraed durg perod [ h, h] ad we kow EX h ad Var X h for all,, m Hece, for he secod reallocao, we have R X [ e -h- L/ ], ER h [ e-h-l/ ] ad VarR h [ e -h-l/ ]. For he hrd reallocao, we have R 3 X e -h- L/ e -h/ +X [ e -h-l/ ], ER 3 h [ e-h-l/ ] ad VarR 3 h {e-h-4l/ e -h/ +[ e -h-l/ ] }. Ad geeral, for he k h k,, m reallocao, we have k [ k h L]/ h / k h L/ Xk [ e ] [ k h L]/ k [ ] k h k [ k h L]/ h/ h L/ e e + e R X e e + ER h e Var R { [ ] } Hece, accordg o 6, p p + + EBO h p h Var R p S0 S ph E R + + p h + p h + Var R p 3,, m ad 3 33

10 -ASE m m+ + + EBO H mh mh Var R 34 m S0 S H E R + + m H mh + mh + Var R Usg 8 34, we ca compue he oal cos cossg of reallocao cos ad shorage pealy cos for a gve me erval bewee perodc reallocaos. Mahemacally, hs oal cos s a fuco of h. hus uvely, we ca compue he frs dervave ad se he resul o be zero.e. dc/dh 0 o oba he opmal perodc polcy erms of he rgh h value. However, due o he complexy he equao for EBO as defed 33 ad 34 especally he u ormal loss fuco, s compuaoally esve o compue he opmal h by usg he frs dervave. o overcome compuaoal effcecy, he followg heursc approach may be appled o compue EBO: for h L+; h < H ; h++ { Compue EBO,,EBO m+ by Compue oal EBO E by 9. Compue Ch+ by 8. f Ch Ch+ oupu he opmal perodc polcy h ad correspodg sock reallocaos } VI. CONCLUSION AND FURHER RESEARCH I hs paper, we were eresed aalyzg he performace of reallocao wh a mul-echelo veory sysem. Durg he repleshme cycle, we have wo opporues o reallocae he spares by redsrbug he depo socks o he bases ad by laeral rasshpme. We developed a mahemacal model ad use a Lagrage mulpler o deerme how o reallocae he spares o acheve a mmzed oal expeced backorders uder a gve reallocao sa par. he we derve a dyamc reallocao mehod o deerme whe o perform he frs ad secod reallocao respecvely. Expermeal resuls show ha wo-reallocao s beer ha sgle-reallocao. he logc of our approach s easy o mpleme ad effcely compued. Several possble aveues of exeso of our work are worh cosderg: Echelo Srucure. We have cosdered he wo-echelo ree srucure. A aural exeso s o hadle more ha wo echelos where reallocao sas a dffere echelos ca be dffere. Moreover, oe ca exed he supply cha srucure from a ree srucure o a ework graph. Demad dsrbuo. I s also eresg o cosder osaoary demad dsrbuos,.e. demads a each perod are o decal wh me-varyg mea ad sadard devao. 3 Cos cosderao. he objecve of hs paper s o cosder he mg of reallocao ha seeks o mprove effcecy ad reduce uavalably. I s obvous ha more freque reallocao yelds beer performace compare o sgle reallocao f he model does o corporae he cos of reallocao. 4 Perodc Resupply. Fally, from he pracce sadpo, s eresg o experme o he dea of compug opmal perodc resupply proposed Seco 5. APPENDIX A. Expermes o Approxmao of Posso dsrbuo by Normal dsrbuo I he followg, we vesgae he effec of approxmao of Posso dsrbuo by a Normal dsrbuo whe he mea of Posso radom varable our case, demad s smaller ha 0. More precsely for he purpose of our paper, we are cocered wh he approxmao error o he value of he expeced backorder. I s clear ha he sum of Posso demads wh mea durg he erval [0, ] sll follows a Posso dsrbuo wh mea *. ve al veory S, he expeced backorder based o Posso demad dsrbuo deoed EBO_Posso s calculaed as k e k S. If we approxmae he Posso k S k! dsrbuo wh a Normal dsrbuo, he he expeced backorder based o Normal demad dsrbuo deoed EBO_Normal s calculaed as S. hus, he relave approxmao error ca be calculaed as EBO_Normal - EBO_Posso/EBO_Posso. I able A., we compare he expeced backorder values as well as he relave approxmao error compued uder he wo demad dsrbuos usg dffere values of * ad assumg S o be equal o *. We observe ha he approxmao error s very small % or less whe he mea of Posso dsrbuo s o smaller ha 4. We coclude ha he Normal dsrbuo ca effecvely approxmae Posso dsrbuo, for he purpose of hs work. ABLE A: RELAIVE APPROXIMAION ERROR ON EXPECED BACKORDER * EBO_Normal EBO_Posso Relave approxmao error

11 -ASE B. Expermes o Compuaoal performace of our woreallocao algorhm Usg he same expermeal seup as Seco 4., we measure he compuaoal me requred o execue he woreallocao algorhm for dffere repleshme horzos H o a mache wh CPU.66Hz ad RAM B. he resul s show as follows: ABLE A COMPUAION PERFORMANCE FOR DIFFEREN REPLENISHMEN HORIZONS Repleshme horzo H perod Compuao me mllsecod REFERENCES [] P. Alfredsso, Opmzao of mul-echelo reparable em veory sysems wh smulaeous locao of repar facles, Europea Joural of Operaoal Research, v99, , 997. [] D. B. Cao ad E. A. Slver, Dyamc Allocao Heursc for Ceralzed Safey Sock, Naval Research Logscs, v5:6, 53-56, 005. [3] A. Díaz ad M. C. Fu, Models for mul-echelo reparable em veory sysems wh lmed repar capacy, Europea Joural of Operaoal Research, v97, , 997. [4]. D. Eppe ad L. Schrage, Ceralzed orderg polces a mulwarehouse sysem wh lead mes ad radom demad, I L. B. Schwarz ed IMS Sudes he Maageme Sceces, v6, 5-67, 98. [5] P. L. Jackso, Sock allocao a wo-echelo dsrbuo sysem or wha o do ul your shp comes, Maageme Scece, v34, , 988. [6] P. L. Jackso ad J. A. Mucksad, Rsk poolg a wo-perod, woechelo veory sockg ad allocao problem, Naval Research Logscs, v36, -6, 989. [7] H. Jösso ad E. A. Slver, Aalyss of a wo-echelo veory corol sysem wh complee redsrbuo, Maageme Scece, v33, 5-7, 987. [8] H. C. Lau, H. Sog, C.. See ad S.Y. Cheg, Evaluao of me- Varyg Avalably Mul-Echelo Spare Pars Sysems wh Passvao, Europea Joural of Operaoal Research, v70:, 9-05, 006. [9] H. C. Lau ad H. Sog, Mul-Echelo Reparable Iem Iveory Sysem wh Lmed Repar Capacy uder Nosaoary Demads, Ieraoal Joural of Iveory Research,, 67-9, 008. [0] OPUS0 User s Referece Logscs Suppor ad Spares Opmzao, Syseco AB, May 998. [] C. C. Sherbrooke, Opmal Iveory Modelg of Sysem: Mul- Echelo echques, Joh Wley & Sos, New York, 99. [] E. A. Slver, D. F. Pyke ad R. Peerso, Iveory maageme ad produco plag ad schedulg, 3 rd edo, 998. [3] D. B. sao ad. Ekawa, Opmal secod repleshme polcy wo-phased push corol sysem, Joural of Operaoal Research Socey Japa, v35, 73-89, 99. [4] V. D. R. ude Jr. ad R. Srvasava, Reparable veory heory: Models ad applcaos, Europea Joural of Operaoal Research, v0, -0, 997. Hoog Chu LAU receved he B.S. ad M.S. degrees from he Uversy of Mesoa 987 ad 988 respecvely, ad he D.Eg. degree from okyo Isue of echology, Japa 996. He s currely Assocae Professor of Iformao Sysems a he Sgapore Maageme Uversy, ad holds a cocurre appome as Drecor of Defese Logscs a he Logscs Isue Asa Pacfc, Naoal Uversy of Sgapore. Hs research eress are suaed a he erseco of arfcal ellgece ad operaos research, wh applcao o plag ad schedulg large-scale rasporao, logscs ad supply cha maageme. o dae, he has publshed more ha 90 papers jourals ad eraoal cofereces. Par of hs research has resuled ovave ools ad sysems for dusry, ad oe of such ools wo he Sgapore Naoal Iovao ad Qualy Crcles Sar Award 006. He serves acvely program commees of eraoal cofereces, cludg he IEEE/WIC/ACM Ieraoal Coferece o Iellge Age echology, IEEE Coferece o Auomao Scece ad Egeerg, ad Ieraoal Coferece o Auomaed Plag ad Schedulg ICAPS. Je PAN receved hs B.Eg. ad M.Eg. degrees from Behag Uversy, Bejg, Cha, 999 ad 00 respecvely. He worked as research egeer he School of Iformao Sysems a Sgapore Maageme Uversy, 009. He s currely Ph.D. caddae of Idusral ad Sysems Egeerg a Naoal Uversy of Sgapore. Hs research eress are suaed a operaos research, compuaoal game heory wh applcao o logscs ad supply cha maageme. Huawe Sog receved hs B.S. from Fuda Uversy, Shagha, Cha, 000, ad M.S. from Naoal Uversy of Sgapore 00. He worked as a research egeer he Logscs Isue - Asa Pacfc , where he parcpaed he JAUAR defese logscs projecs. He s currely a Ph.D. caddae of Decso Scece ad Egeerg Sysems a Resselaer Polyechc Isue. Hs research eress are sued a sascs, sochasc process, Moe Carlo smulao, ad opmzao wh applcao o sysem relably, rsk maageme, logscs ad supply cha maageme.