Phase Wise Supply Chain Model of EOQ with Normal Life Time for Queued Customers: A Computational Approach

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1 Amrca Joural of Opraos Rsarch, 0,, hp://dx.do.org/0.436/ajor Publshd Ol Spmbr 0 (hp:// Phas Ws Supply Cha Modl of EOQ wh Normal Lf Tm for Quud Cusomrs: A Compuaoal Approach Sa Shara Mshra, Prm Praash Mshra Dparm of Mahmacs & Sascs (Cr of Excllc o Advacd Compug), Dr. Ram Maohar Loha Avadh Uvrsy, Fazabad, Ida Emal: sa_x003@yahoo.co., lov_lgh59@yahoo. Rcvd May, 0; rvsd Ju 3, 0; accpd July 4, 0 ABSTRACT I hs papr, w maly am o compu h opmal vory h phas ws supply cha for quud cusomrs h rval of lowr ad uppr bouds wh parcular lf of h ms. Impora prformac masurs such as oal opmal cos of h sysm ad oal xpcd dlvry hav also b compud by applyg h dyamc programmg ad Drchl horm. Fally, umrcal dmosrao ad ssvy aalyss hav also b prsd o ga h br prspcv of h modl. Kywords: Supply Cha; Corrlao; Normal Lf Tm; Dyamc Programmg. Iroduco Wh a m s producd a ay maufacurg cr, hs ms hav o pass hrough dffr locaos from h supply bfor rachg h ms a fal sllg po. Ths locaos ar ow as varous phass of supply (hs phass ar o accssbl by h d cosumrs or cusomrs). For xampl, xplorao of crud ols whch s o radly avalabl for h cosumrs ad hs has o pass hough dffr phass of supply bcaus of gographc accssbly du o rra faurs such as mouas or las c.; ably o purchas las rghof-ways, ad du o rsrcd wldlf aras. Smlarly, ohr vory ms whos maufacurg ca b asly accomplshd bcaus of asly avalabl raw marals ha rgo bu has o go hough dffr supply phass bcaus of rasos as sad abov bfor rachg h d cosumrs, for xampl, produco of a Ida. Sc hs ms hav o udr go varous supply phass, hr lf ms as wll as hr avalably a parcular m ad a parcular phas ar subjc o chag du o m ad clmac facors c. I hs suao aalyss of opmum vory rsulg avalably of vory ad hr lf ms a ay m s a rsg phomo o vsga by h rsarchrs gagd hs fld. Ch ad L [] mployd h rplshm problm for drorag ms wh ormally dsrbud shlf lf, couous m-varyg dmad, ad shorags udr h flaoary ad m dscou vrom. Abdl [] laborad hr mchasms, amly food avalably, food affordably ad food accssbly. Accordg o hm may ohr sa-capals, avalably of food s o major cosra h dyamcs of h food sysm supply cha. Shuj [3] llusrad brfly h cocp of lf m ad h rlably of ay opraor whch s worg varous flds ad h cocp of avalably s also corrlad wh h cocp of rlably ad s lf m. Ma al. [4] drvd saoary dsrbuo of jo quu lgh ad vory procss xplc produc form for varous M/M/ sysms wh vory udr couous rvw ad dffr vors maagm polcs, ad wh los sals. Dmad s Posso, srvc ms ad lad ms ar xpoally dsrbud. Ths dsrbuos usd o calcula prformac masurs of h rspcv sysms cas of f wag room h y rsul s ha h lmg dsrbuos of h quu lgh procss ar h sam as h classcal M/M//. Grard al. [5] dscussd a quug modl whch wo sragc srvrs basd o hr prformac; h fasr a srvr wors, h mor dmad h srvr s allocad. Th buyr s objcv s o mmz h avrag lad m rcvd from srvrs. Thy foud h cosdrabl varao h prformac of allocao polcs ad cocludd a ffcv procurm sragy for a buyr as log as h buyr xplcly accous for h srvr s sragc bhavour. Mshra ad Yadav [6] dal h prof opmzao of a loss quug sysm wh h f capacy ad compud oal xpcd cos (TEC), oal xpcd rvu (TER) ad oal opmal prof (TOP) of h sysm. Mshra [7] dscussd h cos aalyss of G/M/C/K/N modl cludg srvc ra ad hypr gomrc fucos of ohr paramrs ordr o g opmal cos. Copyrgh 0 ScRs.

2 S. S. MISHRA, P. P. MISHRA 97 Ravdra ad Phlps [8] solvd a problm ol raspor chology whch h Blac Gold Prolum Compay had foud larg dposs of ol o h Norh Slop of Alasa by cosdrg h smpl rasporao worg ad solvg h problm for h opmum cos. Vswaaha ad Mahur [9] cosdrd a dsrbuo sysm wh a cral warhous ad may ralrs ha soc a umbr of dffr producs. Drmsc dmad occurs a h ralrs for ach produc. Th warhous acs as a bra-bul cr ad dos o p ay vory. Th producs ar dlvrd from h warhous o h ralrs by vhcl rous usd for h dlvry, so as o mmz h log-ru avrag vory ad rasporao coss. Irava ad To [0] cosdrd h procssg of M jobs a flow shop wh N saos whch oly a sgl srvr s charg of all saos. Thr objcv was o mmz h oal sup ad holdg ar cos, a class of asly mplm abl schduls s asympocally opmal. Bllma [] laborad h hory of dyamc programmg o solv h dffr yp mul-chlo dcso mag problm of h maagm. W g far movao from Amraj [], Joh [3], Mshra ad Sgh [4,5] ad Sudhr al. [6] hs fld of vsgao. Lod ad Lus [7] addrssd ha problm from a dyamc opmzao of local dcsos po of vw, o sur a global opmum for h supply cha prformac. Ths s do udr h framwors of Collcv Illgc (COIN) hory ad Mul-Ag Sysms (MAS). By COIN, a larg MAS whr hr s o cralzd corol ad commucao, bu also, whr hr s a global as o compl h global supply cha wor opmzao. Th proposd modl focuss o h racos a local ad global lvls bw ags ordr o mprov h ovrall supply cha busss procss bhavor. Bsds, collcv larg cosss of adapg h local bhavor of ach ag (mcro-larg) o h opmzao of h bhavor globally (macro-larg). Carryg cos whch rprss a larg chu of a compay s oal supply cha coss so has prom rol h supply cha maagm. Ivory carryg coss ar xprssd as a prcag of h avrag dollar valu of vory ovr a fxd prod usually a yar. As a rul of humb, vory carryg cos s 5% of a compay s avrag vory vsm, bu wh you ally up all h rlva carryg coss, ca ru as hgh as 40% or mor. A whl bac a cl asd m jus how much was cosg hm o carry vory. I g hs quso all h m, ad my aswr s always h sam: a compay s vory carryg cos s, o avrag, 5% of s aual o had vory vsm. From a facal po of vw, udrsadg ad maagg vory carryg cos wll hav a mpac o your compay s oprag com. I wll also hlp you balac your oprag x- ps wh vory lvls. I opraos, wh purchasg ad vory corol saff rplsh a m, hy as hmslvs wo basc qusos: how much should w ordr, ad wh hs ar rval mars. Ordr mor frquly, ad your ordr cos crass whl carryg coss dcras; lss frquly, ad you rad off lowr ordr coss wh a largr avrag vory. Th mos ffc way o fgur ou how much s o us h coomc ordr quay modl. Ths modl mmzs h oal varabl coss rqurd o ordr ad hold vory. Ivory ordrg cos, also ow as purchasg cos or s-up cos, cluds h clrcal wor rqurd o prpar, rlas, moor, ad rcv ordrs. I maufacurg, vory ordrg cos cluds produco schdulg m, mach s-up m, ad spco. Shaul al. [8] llusrad a EOQ-yp vory problm whr h dmad ra s a fuco of h vory lvl. I has b od by marg rsarchrs ad pracors ha a cras a produc s shlf spac usually has a posv mpac o h sals of h produc. I such a cas, h dmad ra s o logr a cosa, bu dpds o h amou of o-had vory. Erl [9] xuld h ssvy of h basc coomc ordr quay (EOQ) modl o couous purchas prc. Th phomo of couous prc chags xss svral cours ad s o lly o mprov. Wu ad Low [0] xuld ha h currly avalabl coomc ordr quay-jus--m (EOQ-JIT) cos dffrc po modls suggs ha h JIT purchasg approach s always prfrrd o h EOQ approach wh h JIT purchasg approach ca capa- lz o physcal pla spac rduco. I was foud ha hs modls dd o mprcally sudy h capably of a vory facly o hold h EOQ-JIT cos dffrc po s amou of vory. I addo, som mpora cos compos udr h vory maagm sysms wr gord by h modls, for xampl, h ou-of-soc coss ad h mpac of vory polcy o produc qualy, produco flxbly. By dvlopg h JIT purchasg hrshold valu (JPTV) modls, suggss ha h advaags of JIT purchasg may hav b ovrsad hory. Th JPTV modls of hs sudy ovrcom h wo lmaos of h xsg EOQ-JIT cos dffrc po modls. Ds [] usd h formao maagral chqus o solv h logsc problms whch cludd h supply cha. Tmo ad Ma [] xam h curr sa of vory maagm Flad by rvwg ff cas suds mad a h Lappraa Uvrsy of Tchology ad svral aspcs of vory maagm ar cosdrd by hm. I grar dal, w hav rd o aswr qusos such as: wha s h rol of vory maagm corpora plag, ad how ar vory dcsos mad maufacurg orgazaos? I addo parcular a- Copyrgh 0 ScRs.

3 98 S. S. MISHRA, P. P. MISHRA o s drcd a h goal sg ad prformac masurm vory maagm. W hav foud ou ha mpora dcsos cocrg vors ar usually mad a a low lvl h orgazaoal hrarchy wh ou ay gudls from op or mddl maagm. Furhrmor, compas lac accura ral m ad suably aggrga formao of maral flow ad soc lvls. Ths mas sg prcs quaav goals for vory maagm dffcul. Som compas hav had facal prssur o dcras vors ad bcaus of h lac of propr vory corol, hs had ld o boh xral ad ral soc ous. I h pracc of phas ws supply cha modl whr cusomrs ar quud v srous ao of profssoals o compu h ovrall ffcvss of h modl havg ormal lf m of h vory. Prvously, hs d of modl was o ampd for modl had soluo complxy bcaus of us of mulpl soluo chqus. I hs papr, prformac masurs rlad o h sysm has b aalyzd ad compud umrcally. Th oal opmal cos of h sysm cludg vory ad quug boh sysms, oal opmal rvu ad prof, opmal vory lowr ad uppr bouds, ad oal xpcd wag m of h cusomrs o m h vory (oal xpcd dlvry m), hav b dscussd ad fally compud. A s of mulpl chqus such as dyamc programmg, rasporao worg ad o-lar quadrac quaos cludg a mpora Drchl horm hav b usd o aalyz ad compu prformac masurs of h sysm. Compug algorhm has b dvlopd ordr o compu prformac masurs. Obsrvaos hav b draw o dscuss h ssvy aalyss rlad o varous paramrs of varaos. Th whol papr s orgazd varous scos such as roduco, dscrpo of h modl, mahmacal aalyss, compug algorhm, umrcal dmosrao, obsrvaos ad cocluso. Carryg coss ar xprssd as a prcag of h avrag dollar valu of vory ovr a fxd prod usually a yar. As a rul of humb, vory carryg cos s 5% of a compay s avrag vory vsm, bu wh you ally up all h rlva carryg coss, ca ru as hgh as 40% or mor. A whl bac a cl asd m jus how much was cosg hm o carry vory. I g hs quso all h m, ad my aswr s always h sam: a compay s vory carryg cos s, o avrag, 5% of s aual o had vory vsm. From a facal po of vw, udrsadg ad maagg vory carryg cos wll hav a mpac o your compay s oprag com. I wll also hlp you balac your oprag xps wh vory lvls. I opraos, wh purchasg ad vory corol saff rplsh a m, hy as hmslvs wo basc qusos: how much should w ordr, ad wh hs ar rval mars. Ordr mor frquly, ad your ordr cos crass whl carryg coss dcras; lss frquly, ad you rad off lowr ordr coss wh a largr avrag vory. Th mos ffc way o fgur ou how much s o us h coomc ordr quay modl. Ths modl mmzs h oal varabl coss rqurd o ordr ad hold vory. Ivory ordrg cos, also ow as purchasg cos or s-up cos, cluds h clrcal wor rqurd o prpar, rlas, moor, ad rcv ordrs. I maufacurg, vory ordrg cos cluds produco schdulg m, mach s-up m, ad spco. Shaul al. [8] llusrad a EOQ-yp vory problm whr h dmad ra s a fuco of h vory lvl. I has b od by marg rsarchrs ad pracors ha a cras a produc s shlf spac usually has a posv mpac o h sals of h produc. I such a cas, h dmad ra s o logr a cosa, bu dpds o h amou of o-had vory. Erl [9] xuld h ssvy of h basc coomc ordr quay (EOQ) modl o couous purchas prc. Th phomo of couous prc chags xss svral cours ad s o lly o mprov. Wu ad Low [0] xuld ha h currly avalabl coomc ordr quay-jus--m (EOQ-JIT) cos dffrc po modls suggs ha h JIT purchasg approach s always prfrrd o h EOQ approach wh h JIT purchasg approach ca capalz o physcal pla spac rduco. I was foud ha hs modls dd o mprcally sudy h capably of a vory facly o hold h EOQ-JIT cos dffrc po s amou of vory. I addo, som mpora cos compos udr h vory maagm sysms wr gord by h modls, for xampl, h ou-of-soc coss ad h mpac of vory polcy o produc qualy, produco flxbly. By dvlopg h JIT purchasg hrshold valu (JPTV) modls, suggss ha h advaags of JIT purchasg may hav b ovrsad hory. Th JPTV modls of hs sudy ovrcom h wo lmaos of h xsg EOQ-JIT cos dffrc po modls. Ds [] usd h formao maagral chqus o solv h logsc problms whch cludd h supply cha. Tmo ad Ma [] xam h curr sa of vory maagm Flad by rvwg ff cas suds mad a h Lappraa Uvrsy of Tchology ad svral aspcs of vory maagm ar cosdrd by hm. I grar dal, w hav rd o aswr qusos such as: wha s h rol of vory maagm corpora plag, ad how ar vory dcsos mad maufacurg orgazaos? I addo parcular ao s drcd a h goal sg ad prformac masurm vory maagm. W hav foud ou ha mpora dcsos cocrg vors. Copyrgh 0 ScRs.

4 S. S. MISHRA, P. P. MISHRA 99. Dscrpo of h Modl Hr, s assumd ha vory ms dmadd h mar follow ormal avalably as wll as ormal lfm.th rasos of choosg ormal ar wofold: s o of h mos mpora probably phoma h ral world du o h classcal cral lm horm, ad s also o of h mos commoly usd lfm dsrbuos rlably coxs ad Morovr, ucras hr cusomr dmads ma dffcul for supply chas o achv jus--m vory rplshm, rsulg loosg sals opporus or pg xcssv cha-wd vors. Hr, supply cha cosss of o supplr ad mulpl ralrs. Th vory-corol paramrs of h supplr ad ralrs ar lad m ad socs. Cusomr-dmad par s supposd o b proporoal o h arrval of h cusomrs a sllg po whch follows Posso dsrbuo. Also supply cha of ms s cosdrd from frm o h sllg po of vory ms ad supply m of h vory ms s xpoal dsrbuo o fulfll dmad ad rarrval m of dmad follows xpoal dsrbuo a dffr phass. Ths shows ha supply of vory ally passs hrough dffr chals, ad from ach chal passs hrough dffr phass o rach h sllg pos. Furhr, s assumd ha l s h umbrs of chals ad m s h paral gral quay of h vory avalabl a o chal, s rqurd a h sllg po. Hc, paral amou of avalably of ms, a a sllg po s f = lm. Th paral quay whch s o rach sllg po (las phas) of h avrag avalabl quay a h frm s q gv by f. Ths avrag avalably of vory q f rolls hrough all h locaos (phass) o cha of supply. Ths wll b h avrag avalably a h sllg po (las phas) ad cusomrs (cosumrs) com a h sllg po o g vory wh ra λ ad follow Posso q Probably Dsrbuo. A las amou s f suppld by o phas o aohr phas ll las phas (sllg po). Sc arrval of cosumrs a sllg po, wh ra λ follows Posso dsrbuo ad srvc ra μ a sllg po follow xpoal dsrbuo. Assumg ha supply ra of vory s R, r supply m of vory s also follow xpoal dsrbuo. Du o aforsad assumpos cosumrs hav o wa quu fro of h sllg po o g vory ms. Th followg oaos hav b usd hroughou h papr q = Avalabl quay a m such ha q 0,, τ = Lfm of vory m such ha 0,, μ = Ma of avalabl quay, μ = Ma of lfm, σ = Sadard dvao of avalabl quay, σ = Sadard dvao of lfm Hr, q = Avrag avalabl quay of gv lfm τ =, β = Corrlao coffc bw avalably ad lfm of vory ms, ( q / ) = Paral amou of avalably of ms, f a a sllg po, whr f shows a posv gral quay whch s chos by a sllg po as pr s rqurm ad hs amou passs hrough phass, h = Holdg cos pr u m pr u quay a ach phas, PC 0 = Cos of raw maral vsd by Producr, PC = Purchasg cos pr u quay a h h phas, r% = Prof ard by ach sllr a vry phas, L q = Wag lgh of cusomr o fd srvc a sllg po (a las phas), ω = Wag cos pr cusomr quu, o = Ordrg cos pr ordr a h phas, O = Toal ordrg cos pr cycl of whol sysm, R = Quay rcvd pr ordr.. suppld quay pr ordr, λ = Arrval ra of cusomr a las phas, μ = Srvc ra a las phas o srv cusomrs, Q = Avrag quay dmadd by pr cusomr, s = Sup cos of h phas, Tp = Trasporao cos from ( )h phas o h phas pr shpm, T = Toal lad m of vory from producr o las sllr. I s also assumd o cycl m. Th -phass ar ascrad afr solvg supply cha worg; vd a sampl worg graph by usg h cocp of dyamc programmg as dscussd blow brf. Hr, cusomrs ar arrvg wh Posso ra λ a h phas (a sllg po for h cusomrs) ad rcvg ms wh srvc ra μ. W hr oly dscuss abou h quay whch wll rach a h las po of sllg avalabl for h cosumrs. Th problm s solvd by usg dyamc programmg f μ = 3σ, μ = 3σ ad a , b ad f μ =.5σ, μ =.5σ, a , b I pracc, f μ.5σ, μ.5σ w assum a, b. For 0 < q < ad 0 < τ <, whr σ > 0, σ > 0 ad ρ <. To sudy hs jo probably, l us frs show ha paramr μ ad μ, σ ad σ ar h mas ad sadard dvaos of h wo varabls q, τ. To bg wh, w gra o y from Copyrgh 0 ScRs.

5 300 S. S. MISHRA, P. P. MISHRA 0 o gg pos so, appars l wor of supply of vory. I hs wor w hav o sarch h shors rou of supply of ms. To fd ou mmum rasporao cos rou hs wor by usg Dyamc Programmg. W prs som sampl worg o llusra h modl ad whch ca furhr b gralzd as pr rqurm of h supply cha w cosdr sv rgos ad sv sags o compl h cha from sllg pos o h po of org (maufacurg po). Rgo ad sags rfr o varous aras of oprao ad dsac from h org rspcvly. For xampl sag-i rfrs o las po of sllg of cusomrs lvl ad rgo VII sads for a parcular ara f oprao/ dsrbuo rlad o d cosumrs. Furhr, rgo VII ad sag I cosu h cocp of phas (hr las phas). Whl solvg, o fd ou shors rou (shors cha of supply), w assum all h sllg pos udr rgo-vii, ad sag-, bcaus w sar from h sllg po h sarch of shors rou. Afr hs w rach udr rgo-vi, ad sag-, h procss of sorg s cou ul w rach a h frm whch s udr rgo I ad sag-7. I ach rgo, w assum ha a las, hr ar hr rous hr drcos a ach phas (locao).ths drcos ar Forward, Lf ad Rgh dod by F, L, ad R rspcvly. Tag dcso a ach Phas o choos h mmum valu amog F, L & R wghs whch s dod by d Afr choosg mmum wghd pah, w hav o mov ha drco. Ths procss s coud ul w hav o rach Rgo-I. Ths yp of chqu s also usd Graph Thory o fd ou h mmal wghd spag r. I s calld Prm s Algorhm. Accordg o hs algorhm w sar from ay vrx of graph ad a ha vrx w hav o choos mmum lgh dg amog h cdc dgs. W ravrs o ad rach a ohr vrx, aga w mployd h sam procdurs as afor sad ul w ravrs a all vrcs of graph. Nwor of Supply Cha Mmum Trasporao Rou Sag (REGION VII) S /d R L F f s A 8-5 F 5 C R 4 E 0 5 F 5 F F 8 G F 6 H R 9 I L 6 J R 8 Sag (REGION VI) d S /d R L F f s A 6-7 F 7 B R 8 C F 5 D 0 4 L E R 8 F F 5 Sag 3 (REGION V) d S /d R L F f s A F 5 B L 0 C 4 4 F D 3 0 L 0 E L 6 F F 9 Sag 4 (REGION IV) d S /d R L F f s A 7-37 R 7 B 0 0 F 0 C F 09 D 3 6 L E L 7 F L 0 Sag5 (REGION III) d S /d R L F f s A R 35 B 0-0 F 0 C R 5 D F 9 E 4 5 L 5 F L 5 d Sag 6 (REGION II) S /d R L F f s A R 5 B R 30 C F 7 D L 8 d Copyrgh 0 ScRs.

6 S. S. MISHRA, P. P. MISHRA 30 g q q q 0 d For margal dsy of q, h mporarly mag q h subsuo u o smplfy h oao ad chagg h varabl of grao by lg v, w obad as g q q 0 v uv dv Afr complg h squar by lg v βuv = (v βu) β u ad collcg rms, hs bcoms (afr dfyg h quay parhss as h ormal dsy gral from 0 o + ), w ca qualz o o. I cosquly gvs us g q g q u u 0 v u dv q Accordac wh dfo ad lg u q ad o smplfy h oaos, w g u uvv v u vu Th, xprssg h rm orgal varabls, w oba w q q q I hs sco, w hav o ma a smpl modl of rou of rasporao from frm o h sllg pos whch ar suad a dffr locaos. Sc hr ar umbr of rous from frm o sll. Shors Supply Chas (dffr phas ws supply rous) Aalogously, sa ca r oly from sa sc hr s o sa +. W ow solv Equaos ()-(4) rms of h saoary probably ha h maagr s dl,.. rms of P 0. W g P P0, P P0,, P P0 (5) Copyrgh 0 ScRs.

7 30 S. S. MISHRA, P. P. MISHRA W ca ow us h fac ha P o solv for P 0 xplcly. W g 0 P 0 (6) L us assum ha h s holdg cos pr u m a h phas. Tp s h rasporao cos from o h phas. Pc s h purchasg cos of pr u m for h phas. S s h srvc cos pr u m o dlvr srvc a h Phas. I s assum ha capacy of sysm s K. Thus h sa spac of h quu ca b dxd by, whr =,, 3,, K. Tha s wh hr ar K cusomr h sysm, Maagr wll o provd vory o ay mor cusomrs cra m prod. So, P P () 0 whr λ ad μ ar h paramrs of h wo xpoal dsrbuos, w g P P P (3) Fally for sa h ra qualy prcpl gvs m h followg balac quao Usg Equaos (5) ad (6), w g P ; =,, 3, (7) 0 0 Ls P L q 0 0 d d, d 0 d whr 0 L q, W cosruc h oal cos as d d hq r Toal Cos T Pc0 f 00 q/ Tp. (8) fr q/ O S Lq fr Hr, w assum ha dmad of m s lar fuc- o of arrval.. D =Qλ whr Q s aggrga quay Q dmadd by pr cusomr, ad d Q ad vory m s suppld from frm o h frs whol sllr of ach chal ad hrafr hs wll pass hrough umbr of -phass a ach phas posssss q q f quay bu h las sllr posssss approxma avalabl quay q lm whr l s h umbr of chals ad m s paral gral quay of avalably a o chal rachd a las sllr. Sc boh l, m ar cosa valus so w cosdr lm = f. Hc q q ; lm f Sc hs quay wll also roll hrough varous phass of. I cas of vory sysm, hr xs maly wo yps of corary coss, o s carryg cos ad aohr s ordrg cos. I cas of quug sysm, also has maly wo corary coss. Ths ar wag cos ad srvc cos. Wh w cosdr vory sysm wh quug sysm h w g a supply cha sysm of vory. I hs supply cha sysm hr ar maly four yps of coss ou of hs coss, carryg cos ad srvc coss ar of h sam aur, ordrg cos ad wag cos, boh ar also of sam. Bu sum of hs pars ar corary o ach ohr. Th rsco po of h corary coss, whch s foud afr sum, gvs h opmal po for vory as wll as for opmal cos, s gv as blow. Carryg cos of ms + Srvc cos (o srv ms) = Ordrg cos o vory+ Wag cos of cusomr quu hq r r T Pc0 f QT O f L us cosdr q Ths mpls ha qht Pc whr 0 L r 00 q q q, f r OQT 00 q ( ), λ s h arrval ra of cusomr ad μ Copyrgh 0 ScRs.

8 S. S. MISHRA, P. P. MISHRA 303 s srvc ra of vory. I fally urs ou o yld q Pc q q OQT ht r r r r r r ht Pc0 Pc0 4 OQ T q ht From abov s obvous ha r r ht Pc0 4 OQT I furhr gvs us r r ht Pc0 4 OQT ad fally. r r ht Pc0 OQT Accordg o h abov qualy, w hav o sudy abou q followg wo cass. Cas I Wh r r Pc0 T hoq I mpls ha as q max q m whr d E d d d d d ; Proof: ST, R, R T C T! 0 RT RT R C,0 T,.! PTTTTT TT 3 PT PT PT PT dtdt dt RT [Sc PT R 3 ]. Now applyg Drchl s horm of mulpl grals, RT R T ; 0. r r r r ht Pc0 Pc0 4 OQT ht r r r r ht Pc0 Pc0 4 OQT ht Cas II q OQ r r Pc0 T hoq f h ; OQ h q op ad so, f q f h OQ Subsug h valu of q op Equao (), w g oal opmum cos L O s ordrg cos vsd by h phas ad ω s wag cos pr cusomr. ; Copyrgh 0 ScRs.

9 304 S. S. MISHRA, P. P. MISHRA hq r Toal Cos T Pc0 f 00 q q Tp O f R f R S L I fally gvs oal opmal cos as h OQ r TOC T Pc0 h 00 OQ OQ + Tpm O R h R h S W ca also compu h oal opmal prof as Toal opmal prof for o sllr = Toal opmal Rvu Toal opmal cos; TOP Pc q Tprf Pc TC OQ h q h OQ r T Pc0 h 00 f OQ OQ Tp O R h R h S T Tp O OQ h TOP Pc T h R R r S Pc0 ( ) 00 Thus w hav dlvry ms T, T, T 3,, T phass whch ar xpoally dsrbud varabls wh a commo ma R, ad h T = T + T + T T hav h (Gamma) dsrbuo wh phass ad param r R. Hc xpcd lad m o dlvr m from frs o las phas s gv as E T R RT T! Hc (Toal xpcd m (for wag vory) R ET ETE R R 3. Compug Algorhm Sp : Bg, Sp : Ipu all h cd dgs a o od, Sp3: Ipu all h paramrs whch ar cssary o compu EOQ, TOC, TOP, ad TEDT, Sp 4: Compu shors dg from h cd dgs, Sp 5: Compu h mmum rasporao cos from frm o sllg po, Sp 6: Compu h avrag avalably, Sp 7: Compu avrag lf-m of vory, Sp 8: Compu h opmal paral amou, Sp 9: Compu h maxmum quay whch may avalabl a sllg po, Sp 0: Compu h mmum quay whch may b dmadd a sllg po, Sp : Compu h TOC, Sp: Compu TOP, Sp 3: Compu TEDT, Sp 4: Sop. Varous umrcal dmosraos ar gv o h bass of hypohcal daa bas sysm o sudy h varao of o paramr o h ohrs. 4. Obsrvaos ad Coclusos Th followg mpora obsrvaos ar draw from h coducd rsarch: From Tabl, s obvous ha probably of quay avalabl h sor s dpd of avrag lf m, oal opmal cos ad also oal xpcd dlvry m. Tabl, shows ha wh lf-m of h vory s crasd h opmal paral amous also cras. From Tabl 3, w obsrv, ha wh corrlao b- w lf m ad avalably dcrass h avrag avalably, avrag lf-m ad opmal paral amou wll also dcras. Tabl. Probably of quay avalabl rms of q Vrss TOC, TOP ad TEDT Varabls. q Avrag avalably Avr. lf-m... q ma x q m Opmal quay Toal opmal cos Opmal paral amou Toal opmal prc Toal xpcd dlvry m ; Copyrgh 0 ScRs.

10 S. S. MISHRA, P. P. MISHRA 305 Tabl. Lfm τ vrss TOC, TOP ad TEDT q = 0.8, 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, C h = 00, C p = q 00, r = 30, = 0, ω = 90,000, = 3, μ = 50, λ = 40, R = 0, 0 ρ = 0.7, Q = 40, S 00, C0 = 50. τ Avr. Aval Avr. lfm... q max q m q op f TOC TOP TEDT Tabl 3. Corrlao coffc r vrss oal opmal prof, oal xpcd dlvry m τ = 0.7, q 0.5, μ τ = 0.3, σ = 0., σ = 0., C h = 00, C p = 00, r = 30, = 0, μ = 50, λ = 40, R = 0, ω = 90,000, = 3, Q = 40, = 50. Corrlao bw l f m ad avalably S 00, C 0 Avrag avalably Avrag lf Opmal paral amou Toal opmal cos Toal xpcd dlvry m I s also vd from Tabl 4 h a wh rasporao cos s crasd h oal opmal co s as wll as oal opmal prof wll also cras ad avrag avalably, opmal quay ad opmal paral amou wll rma uaffcd. I s obvous from Tabl 5, ha arrval ra of cusomrs a sllg po dcass h opmal quay, oal opmal cos ad Toal opmal prof wll dcras bu opmal paral amou wll cr as ad ohr hgs rma ualrd. Tabl 6 shows ha wh srvc ra a sllg dcass h oly oal xpcd m wll also dcras ad aohr paramrs rma uchagd. W ca coclud from Tabl 7, ha wh oal dmadd quay a sllg po crass h oal opmal cos wll cras bu oal opmal prof of h sysm rmas cosa. I vw of Tabl 8, w ca say ha wh wag m crass h maxmum quay as wll as mmum Tabl 4. Toal cos vrss oal opmal cos, oal opmal prof q = 0.8, τ = 0. 7, q 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, C h = 00, C p = 00, r = 30, = 0, μ = 50, λ = 40, R = 0 0, ω = 90000, ρ = 0.7, Q = 40, S 00, C 0 = 50. M. rasporao cos Avr. avalably Opmal quay Opmal paral amou Toal opmal cos TOP Tabl 5. Toal arrval ra λ vrss oal opmal co s, oal opmal prof, TEDT q = 0.8, τ = 0.7, q 0.5, μτ = 0.3, σ = 0., σ = 0., ρ = 0.7, C h = 00, C p = 00, r = 30, = 0, μ 0 = 50, R = 0, ω = 90,000, ρ = 0.7, Q = 40, S 00, C 0 = 50. Arrval ra of cusomr a sllg po Avr. avalably Avr. lf... Opmal quay Opmal paral amou Toal opmal cos TOP Toal xpcd dlvry Tm Tabl 6. Srvc ra μ vrss oal opmal cos, oal opmal prof ad TEDT q = 0.8, τ = 0.7, q 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, Ch = 00, C p = 00, r = 30, = 0, λ 0 = 40, R = 0, ω = 90000, ρ = 0.7, Q = 40, S 00, C 0 = 50. Srvc ra of cusomr a sllg po Avr. avalably Avr. lf Opmal paral amou Toal opmal cos TOP Toal xpcd dlvry m quay also crass. I mas ha rag of quay dd a sllg po wll cras. Tabl 9, jusfs ha wh avalably as wll as opmal paral amou rmas uchagd for sgfca chags sadard Copyrgh 0 ScRs.

11 306 S. S. MISHRA, P. P. MISHRA Tabl 7. Quay Q vrss oal opmal cos, oal opmal prof ad TEDT q = 0.8, τ = 0.7, q 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, C = 00, C = 00, r = 30, = 0, ω = h p 90,000, λ= 40, R = 0, ω = 90,000, Q = 40, S 00, C 0 = Toal dmadd quay Q a sllg po Avr. avalably TOP qmax q m TEDT (housads) Tabl 8. Wag cos ω vrss oal opmal cos, oal opmal prof ad TEDT, q max, q m q = 0.8, τ = 0.7, 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, C h = 00, C p = q 00, 0 S 00, r = 30, = 0, λ = 40, R = 0, ρ = 0.7, Q = 40, C 0 = 50. Wag cos ω ( housads) Avr. avalably Op. quay Op. paral amou TOC TOP q max q m TEDT (housads) Tabl 9. Purchasg cos C p vrss oal opmal cos, oal opmal prof ad TEDT, q max, q m q = 0.8, τ = 0.7, q 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, C h = 00, r = 30, = 0, ω = 90,000, λ = 40, R = 0, ρ = 0.7, Q = 40, 0 S 00, C 0 = 50. C p purchasg cos Avr. av q max q m Opmal paral amou Toal opmal cos TOP dvaos cras. I Tabl 0, a s sadard dvao crass h opmum paral amou dcr ass ad op- mal quay. From Tabl, s obvous ha wh ordrd quay Tabl 0. Sadard dvao of lf σ, Sadard dvao of avalably σ vrss opmal quay, Opmal paral amou f q = 0.8, τ = 0.7, q 0.5, σ = 0., μ τ = 0.3, σ = 0., ρ = 0.7, C h = 00, C p = 00, r = 30, = 0, ω = 90,000, λ = 0 40, R = 0, ρ = 0.7, Q = 40, S 00, C 0 = 5. Sadard dvao of lf m Sadard dvao of avalably Avr. lf m Avr. avalably Opmal paral amou Opmal quay Tabl. Ordrd qamou R, m, C0 brss q ma x, q m ad TOC, TOP, TEDT q = 0.8, τ = 0.7, q 0.5, μ τ = 0.3, σ = 0., σ = 0., ρ = 0.7, C h = 00, C p = 00, r = 30, ω = 90,000, λ 0 = 40, R = 0, ρ = 0.7, Q = 40, S 00, C 0 = 5. Ordrd quay ach phas Numbr of phass Ordrg cos Max. quay M. quay TOC TOP q max q mm TEDT (Thousads) s crasd h oal opmal prof as wll as oal xpcd dlvry m also crass bu wh umbr of phas crass h maxmum quay dcrass as wll as mmum quay also dcrass bu oal opmal cos as wll as oal opmal prof ad oal xpcd dlvry m cras. I hs papr a frsh amp has b mad o compu h uppr ad lowr bouds of h vory ms whch ca asly corbu o h maagm lvl of h orgazao. Dcso o avalably as wll as lf m of vory ca ffcly hac h maagral dcso mags, whch ar par of our papr dals wh wo sysms smulaously o s h supply cha of v- ors ad aohr s h quug sys m of cos umrs. Ths modl s bfcal h fl d of md c supply cha as wll a s ohr xpry ms whos lf m fol- lows h characrscs of h ormal dsrbuo. I pracc for ay orgazao s vry crcal o fd ou h mmum rasporao ro u for h o pmum cos of h rasporao. Ths par of problm s wll-dwl Copyrgh 0 ScRs.

12 S. S. MISHRA, P. P. MISHRA 307 upo by us. Expcd dlvry m of vory closly affc h dcso mags rlad o supply cha maagm ad sysm ca b calbrad for br srvc o h cusomrs mor-plad way. REFERENCES [] J. M. Ch ad C. S. L, A Opmal Rplshm Modl for Ivor y Ims wh Normally Dsrbud Drorao, Produco Plag ad Corol, Vol. 3, No. 5, 00, pp do:0.080 / [] Abl, Avalably, Affordably ad Accssbly of Food Kharoum, GoJoural, Vol. 34, No. 3, 004, pp [3] S. Osa, Sochasc Sysm Rl ably Modlg, Word Scfc Publshg, Sgap or Cy, 985. [4] M. Schwarz, C. Saur, H. Dadu a, R. Kul ad R. Szl, M/M/ Quug Sysms wh Ivory, Quug Sysm, Vol. 54, No., 006, pp do:0.007/s [5] G. P. Cacho ad F. Q. Zhag, Obag Fas Srvc a Quug Sysm va Prformac-Basd Allocao of Dmad, Maagm Scc, Vol. 53, No. 3, 007, pp do:0.87/msc [6] S. S. Mshra ad D. K. Yadav, Cos ad Prof Aalyss of M/E / Quug Sysm wh Rmovabl Srvc Sao, Bulgara Joural of Appld Mahmacal Sccs, Vol., 008, pp [7] S. S. Mshra, Compuaoal Approach o h Cos Aalyss of Dual Mach Irfrc Modl wh Gral Arrval Dsrbuo, PAMM, Vol. 7, No., 007, pp do:0.00/pamm [8] Ravdra ad Phlps, O.R. Prcpl ad Pracc, Solbrg, Nw Yor, 00. [9] S. Vswaaha ad K. Mahur, Igrag Roug ad Ivory Dcsos O-Warhous Mulralr Mulproduc Dsrbuo Sysms, Maagm Scc, Vol. 40, 997, pp [0] S. M. R. Irava ad C. P. To, Asympocally Opmal Schduls for Sgl-Srvr Flow Shop Problms wh Sup Coss ad Tms, Opraos Rsarch Lrs, Vol. 33, No. 4, 005, pp do:0.06/j.orl [] R. Bllma, Som Problm h Thory of Dyamc Programmg, Ecoomrca, Vol., No., 954, pp do:0.307/ [] A. A. Baabyal, Th Quug Thorc Approach o Groudwar Maagm, Ecologcal Modlg, Vol. 85, 996, pp [3] J. E. Frud, Mahmacal Sascs, Prc Hall, Ic., Nw Jrsy, 99. [4] S. S. Mshra ad P. K. Sgh, Compuaoal Approach o a Ivory Modl wh Ramp-Typ Dmad ad Lar Drorao, Iraoal Joural of Opraos Rsarch, Vol. 5, No. 3, 0, pp [5] S. S. Mshra ad P. K. Sgh, Compuaoal Approach o EOQ Modl wh Powr Form Soc-Dpd Dmad ad Cubc Drorao, Amrca Joural of Opraos Rsarch, Vol., 0, pp [6] S. J. Rya Dal, Chadrashar ad Rajdra, Iraoal Trasacos Oprao Rsarch, Wly Ir Scc, Vol., 005, pp [7] L. Shrmov ad L. Rocha-Mr, Supply Cha Nwor Opmzao Basd o Collcv-Illgc ad Ag Tchologs, Huma Sysms Maagm, Vol. 7, 008, pp [8] K. Shaul, Bar-Lv, P. Mahmu ad D. Prry, O h EOQ Modl wh Ivory-Lvl-Dpd Dmad Ra ad Radom Yld, Oprao Rsarch, Vol. 6, 994, pp [9] E. Erl, Th Effc of Couous Prc Chag h EOQ, Omga, Vol. 0, 00, pp do:0.06/ (9) [0] M. Wu ad L. S. Phg, Modlg Jus--Tm Purchasg h Rady Mxd Cocr Idusry, Iraoal Joural of Produco Ecoomc, Vol. 07, 007, pp [] D. R. Towll, Logscs Iformao Maagm, MCB UP Ld., Vol. 9, 996, pp [] T. Prlä ad V.-M. Vrola, A Ovrvw of h Sa ad Problms of Ivory Maagm Flad, Iraoal Joural of Produco Ecoomcs, Vol. 6, 99, pp Copyrgh 0 ScRs.