A New Multi-objective Inventory Model under Stochastic Conditions with Considering Perishable Costs

Transcription

1 Ausrala Joural of Bas a Appl Ss, : -, SSN A Nw Mul-obv vory Mol ur Sohas Coos wh Cosrg rshabl Coss Abolfazl Mrzazah Dparm of usral Egrg, slam Aza Uvrsy, Kara Brah, Kara, ra Absra: hs papr prss a w mulpl obvs mol for h opmal prouo for a vory orol sysm. h so ms may b roras a h sysms oss wll b hag ovr h m. h ral suao, som bu o all usomrs wll wa for balogg ms urg a shorag pro a hrfor, h mol orporas paral baloggg. h ma ra a b a fuo of flao a m valu of moy whr h flao a m horzo.., pro of busss, boh ar raom aur. h obvs of h problm ar: Mmzao of h oal xp prs valu of oss ovr m horzo osss of h rorao os, prouo os, vory holg os, baorrg os, los sal os a orrg os a Drasg h oal quay of goos h warhous ovr m horzo. W propos h al po approah o formula h mol. h umral xampl has b prov for valuao a valao of h horal rsuls. Ky wors: supply ha maagm; Mul-obv; vory; Sohas; Opmzao. NODUCON h praal xprs rval ha h supply ha maagm SCM s ur ura a varabl oos. O of h mos mpora pars of SCM s vory sysm maagm whh s hrly o-rms suao. h may parms of orgazao suh as warhous, marg, sal, purhasg, faal, plag, prouo, maa a. ar rlva o h vory problm. h pas as, h rplshm shulg problms wr ypally aa by vlopg propr mahmaal mols ha osr praal faors ral worl suaos, suh as ura oos, physal hararss of vor goos, ffs of flao a m valu of moy, paral baloggg of usasf ma,. vor goos a b broaly lassf o four ma-agors bas o. Obsols rfrs o ms ha los hr valu hrough m baus of rap hags of hology or h rouo of a w prou by a ompor. For xampl, spar pars for mlary arraf ar syl goos, a hy bom obsol wh a rplam mol s rou.. Drorao rfrs o h amag, spolag, ryss, vaporzao,. of h prous. For xampl, h ommoly us goos l frus, vgabls, ma, foosuffs, prfums, alohol, gasol, raoav subsas, phoograph flms, lro ompos,. whr rorao s usually obsrv urg hr ormal sorag pro.. Amlorao rfrs o ms whos valu or uly or quay ras wh m. s a praal xpr h valu of rsa arp rass by ag. Ohr xampls a b w maufaurg usry a fas growg amals l brolr, shp, pg,. farmg yar. V. h las o rfrs o obsols, rorao a amlorao. h shlf-lf of som prous a b f a h hy woul fall ur h o obsols/rorao/amlorao agory. S 97, a srs of rla paprs appar ha osr h ffs of m valu of moy a flao o h vory sysm. hr ar a fw problms h flaoary vory sysms o obsols a amlorao ms whh hav b arss by h rsarhrs, baus, w wll o us obsols ms h fuur a h amlorao prous ar lm h ral worl. For xampl, Moo al. osr amlorag/rorag ms wh a m-varyg ma par. Aohr rsarh for amlorag ms has b o by Saa. h o obsolsg, rorag a amlorag ms hav b osr som rsarhs o h flaoary vory sysm. Msra 979 vlop a sou os mol a lu ral ompay a xral gral oomy flao ras for varous oss assoa wh a vory sysm. Sarr a a 99 survy h ffs of flao a h m valu of moy o orr quay wh f rplshm ra. Som ffors wr x h prvous wors o osr mor omplx a rals assumpo, suh as Uhayaumar a Gha 9, May 8, Vra a amaabha 99, Daa a al 99, Harga 99, Harga a B-Daya 99, Chug a Cha H.H.,. Corrspog Auhor: Abolfazl Mrzazah, Dparm of usral Egrg, slam Aza Uvrsy, Kara Brah, Kara, ra l: Fax: E-mal: a.mrzazah@au.a.r.

2 Aus. J. Bas & Appl. S., : -, h rorag vory sysms hav b su osrably h r yars. For xampl, Chug a sa prs a vory mol for rorag ms wh h ma of lar r osrg h m-valu of moy. W a Law rv a rorag vory mol ur flaoary oos wh h ma ra s a lar rasg fuo of h sllg pr. Ch a L suss a vory mol for rorag ms wh a ormally srbu shlf lf, ouous m-varyg ma, a shorags ur a flaoary a m soug vrom. Yag suss h wo-warhous vory problm for rorag ms wh a osa ma ra a shorags. Chag sablsh a rorag EOQ mol wh h supplr offrs a prmssbl lay o h purhasr f h orr quay s grar ha or qual o a prrm quay. Ma al. propos a vory mol wh so-p ma ra a wo sorag fals ur flao a m valu of moy. Lo al. 7 vlop a gra prouo-vory mol wh assumpos of varyg ra of rorao, paral baorrg, flao, mprf prouo prosss a mulpl lvrs. A wo sorag vory problm wh yam ma a rval valu la-m ovr a f m horzo ur flao a m-valu of moy osr by Dy al. 8. Ohr ffors o flaoary vory sysms for rorag ms hav b ma by Hsh a Dy, Su al. 99, Ch 998, W a Law 999, Sarr al., Yag al.,, Lao a h, Balh a, b, Hou a L, Hou, Jagg al., Chr al. 8 a Sarar a Moo. S.K. Ghosh, S. Khara, K.S. Chauhur.,, Hr, M.S., Yag, H.L., g, J.., apahrsos, S., 8 a b s ha h mo rsarhs, ra of flao has b assum omplly ow a ra. Y, flao rs h vory pur oly baus may hav a mpa o h fuur vory oss, a h fuur ra of flao s hrly ura a usabl. Bu, hr ar a fw wors h flaoary vory rsarhs ur sohas oos, spally wh mulpl sohas paramrs. Mrzazah a Sarfaraz 997 prs mulpl-ms vory sysm wh a bug osra a h uform srbuo fuo for h xral flao ra for o obsols, rorao a amlorao ms a Horowz suss a EOQ mol wh a ormal srbuo for h flao ra. Mrzazah 7 ompar h avrag aual os a h sou os mhos h vory sysm's molg wh osrg sohas flao. h rsuls show ha hr s a glgbl ffr bw wo prours for w rag valus of h paramrs. Furhrmor, Mrzazah 8 aohr wor, propos a vory mol ur m-varyg flaoary oos for rorag ms. Mrzazah 9 vlop A aral Baloggg Mahmaal Mol ur Varabl flao a Dma. h obvs of h problm ar: Mmzao of h oal xp prs valu of oss a Drasg h oal quay of goos h warhous ovr h raom m horzo. h so obv has slom osr h prvous rsarh of h vory sysms. Bu, rasg h vory lvl s mpora for ompay, baus: rasg vory lvl auss rasg ompay flxbly agas hags h mar oos, usomr s a so o, h quay of h rora goos s rla o vory lvl so ha rasg vory auss rasg sroy goo, low vory sysm auss fasr ompay aapao wh hology hags, rasg vory auss br ash flow a ra of rur. Furhrmor, h ma s a fuo of h flao ra, hs papr. h xsg lraur, flaoary vory mols ar usually vlop ur h assumpo of osa a wll ow m horzo. Howvr, hr ar may ral lf suaos whr hs assumpos ar o val,.g., for a sasoal prou, hough m horzo s ormally assum as f a rsp aur, bu, vry yar fluuas pg upo h vromal ffs a s br o sma hs horzo as a sohas paramr, whh has b osr hs papr. Aoally, h rplshm ra s f a rorag ms ar survy wh osrg rorao os. may ral suaos, urg a shorag pro, h logr h wag m s, h smallr h baloggg ra woul b. For sa, for fashoabl ommos a hgh-h prous wh h shor prou lf yl, h wllgss for a usomr o wa for baloggg s mshg wh h lgh of h wag m. hrfor, h paral baloggg has b osr hs papr. Ur h mo suaos, a w mahmaal mol for h opmal prouo for a vory orol sysm s formula ur sohas vrom, a h papr has b orgaz as follow. Frs, h assumpos a oaos a h, h mul-obv mol formulao ar rv. h, h soluo prour has b prpar wh usg h al po approah. h umral xampl has b prov o larfy how h propos mol s appl. h fal so has vo o h susso.. Assumpos, Noaos a Dsrpo of h Mol: h vlop mol has b ma bas o h followg assumpos:. h sysm brgs abou for a raom m-horzo;

3 Aus. J. Bas & Appl. S., : -,. A osa frao of h o-ha vory roras pr u m, as soo as h m s rv o vory;. Shorags ar allow a paral balogg, xp for h fal yl;. All of h sysm oss wll b ras ovr m horzo va sohas flao ra;. h ma ra hr s a lar fuo of h flao ra;. h osa aual prouo plshm ra s f. h plshm ra s hghr ha h sum of osumpo a rorao ras; 7. La m s glgbl. Also, h al a fal vory lvl s zro. h followg oaos ar us: h sohas flao ra f h pf of flao ra h rs ra h sou ra of flao: = r- D h ma ra pr u m s a fuo of flao ra a b a, b a a b ar h osa ral umbr. h osa rorao ra pr u m. h orrg os pr orr a m zro h purhas os a m zro h vory arryg os pr u pr u m a m zro h baloggg os pr u pr u m, f h shorag s balogg h u opporuy os u o los sal, f h shorag s los h rorao os pr u of h rora m a m zro H h sohas f m horzo fh h pf of H h osa aual prouo plshm ra h frao of shorags baorr ha s a ffrabl a rasg fuo of m, whr s h wag m up o h x rplshm, wh = a =. No ha f = or for all, h shorags ar omplly balogg or los. W assum = -α whr α. h rval of m bw rplshm h proporo of m ay gv vory yl whh orrs a b fll from h xsg so h umbr of rplshms urg m horzo EVC, h oal prs valu of oss ovr h m horzo, h oal quay of goos warhous ovr m horzo Aoal oaos wll b rou lar. h graphal rprsao of h vory sysm s show Fgur. h ral m horzo H has b v o qual pars ah of lgh so ha =H/. al a fal vory lvls ar boh zro. Eah vory yl xp h las yl a b v o four pars. h prouo sars a m zro a h vory lvl s graually rasg u o prouo, ma a rorao ras. hs fa ous ll h prouo sops a m α. h h vory lvl graually rasg maly u o osumpo a parly u o rorao a rahs zro a m a shorags our a ar aumula ul m λ. Durg h m rval [,], w o o hav ay rorao a hrfor, shorags lvl larly hag. A m λ h prouo sars aga a shorags lvl larly rass ul h mom of. h parally baorr quay s suppl o usomrs urg h m rval [λ,]. A m, h so yl sars a hs bhavor ou ll h of h --h yl. h las yl shorags ar o allow a h vory yl a b v o wo pars. h prouo sops a m -+λ a h h vory lvl rass o la zro a h of h m horzo.. h Mahmaal Molg a Aalyss: h obvs of h problm a b xpla as follows:. Mmzao of h xp prs valu of oss ovr m horzo M Z EVC,. Drasg of h oal quay of goos warhous ovr m horzo

4 Aus. J. Bas & Appl. S., : -, M Z, Fg. : Graphal rprsao of h vory sysm. vory rasg s mpora, baus: rasg h vory lvl auss rasg ompay flxbly agas hags h mar oos, usomr s a so o, h quay of h rora goos s rla o h vory lvl so ha rasg h vory rass h sroy goo, h low vory sysm auss fasr ompay aapao wh h hology hags, a rasg h vory auss br ash flow a ra of rur. h mulpl obv fuo of h vory sysm a b osr as follows M Z MEVC,, M, h vory yls s v o four ffr pars. L o h vory lvl a ay m h h par of h frs o --h yls =,,,. h ffral quaos srbg h vory lvl a ay m h yl ar gv as,,, 7, 8 h las yl shorags ar o allow a h vory lvl s govr by h followg ffral quaos o h vory lvl a ay m h -h par of h las yl ha =,, 9, h soluo of h abov ffral quaos alog wh h bouary oos =, -λ =, =, -λ =, = a -λ =, ar

5 Aus. J. Bas & Appl. S., : -,,,,,,, h valus of λ, λ a λ a b alula wh rsp o a, usg h abov quaos. Solvg λ = for λ w hav L 7 λ a b alula by solvg λ -= [ ] 8 Fally, solvg λ = for λ w hav L 9.. h Exp rs Valu of Coss: L EC as h xp prs valu EV of rplshm oss, EC as h EV of purhasg oss, ECH as h EV of arryg oss, ECS as h EV of shorags oss baorrg a los sal a ECD as h EV of rorao oss, rspvly. h al aalyss s gv as follows.... h Exp rs Valu of Orrg Cos EC: Assum C as h orrg os C By rplag quao quao 7 a ag h xp valu w hav EC E b arb ra p p... h Exp rs Valu of urhasg Cos EC: L EC a EC as h EV of h purhas os h frs o --h yls a h las yl, rspvly. h frs purhas os ha s orr a m zro quals o: λ. h, h x purhas wll our a m λ a hrfor, h frs yl purhas os s h purhas os for -h yl, =,,, - s smlar o h abov quao wh osrg h sou faor, hrfor, h EV of h purhas os h frs --h yls s 7

6 Aus. J. Bas & Appl. S., : -, 8 L E EC - -- h prouo quay h las yl wll our a m - a quals o λ. hrfor, h EV of h purhas os h las yl wll b L E EC h oal xp purhas os ovr h m horzo woul b EC EC EC... Exp rs Valu of Holg Cos ECH: Cosr ECH as h EV of h holg os urg h frs o --h yls. h EV of h holg os urg h las yl a b f wh ECH. h frs pro, h holg oss for -h yl s,,...,, CH Afr som omplx alulaos a ag h xp valu w hav E ECH For h las yl, holg os wll b CH 8 Afr som omplx alulaos a ag h xp valu w hav E ECH 9 So, h oal EV of h holg oss ovr h m horzo s ECH ECH ECH... h Exp rs Valu of Shorags Cos ECS: ECS shows h EV of h shorags os, lug baorr a los sals, urg h frs o --h yls. Shorags ar o allow h las yl. hrfor E ECS Or

7 Aus. J. Bas & Appl. S., : -, ECS E h Exp rs Valu of Drorag Cos ECD: Do D h quay of vory ms whh hav b rora pr yl h frs o h --h yls D a b a b Now, assum ECD as h EV of h rorao os urg h frs o h --h yls. Also, ECD s f h EV of h rorao os urg h las yl. ECD afr ag h xp valu wll b ECD E a b a b -- - E a b a b For h las yl, rorao os wll b ECD E E a b a b hrfor, h oal EV of h rorao os ovr h m horzo s ECD ECD ECD Cosrg h abov mo aalyss, h EV of h oal sysm oss ovr h m horzo for a gv valu of H, s as follow EC, EC EC ECH ECS ECD 7 No ha h m horzo H has a p..f. fh. So, h prs valu of xp oal os from ompl yls, EVC,, s gv by h EVC, EC, f h 8 hrfor 9

8 Aus. J. Bas & Appl. S., : -, M b a b a L E M b a b a L E M E EVC H p p ra rb a b -, 9 Whh M H - s h mom grag fuo of H... h oal Quay of vory, : L a as h oal quay of h goos hl as vory h warhous urg h frs - yls a h las yl rspvly. quals o quay of h goos hl as vory urg ah yl mulpl by -. s alula as follow wh osrg quaos a

9 Aus. J. Bas & Appl. S., : -, No s a rasg fuo wh rsp o. frrg Fgur, a b obsrv ha Smlarly, s qual o So, h oal vory ovr m horzo s, h abov mo quao lus sohas paramrs, h flao ra, a =H., h umbr of rplshms urg raom m horzo. hrfor, h xp valu approah a b us. L E, as h xp oal vory ovr m horzo. E, wll b as follow E, E f hh. h Soluo rour: h problm s o rm h opmal valus of, h umbr of rplshms o b ma urg pro H, a, h proporo of m ay gv vory yl whh orrs a b fll from h xsg so. h al po approah wll b us o solv h mol. Cosr h followg mulobv programmg problm M for =,,..., s..: g x< for =,,, m Whr x s a -msoal so vor. For ay, f h al po as whh x * mmzs. s all h al po. h masur s losss a L-mr s us. Lmr fs h sa bw wo pos -msoal spa as * f x f x whr 7 Whr for =... s rlav mpora wghs of h obv fuo h omproms soluo for a gv valu of wll b mmzs h -mr 7. h masurm u of h mol obvs s o qual o ah ohr a hrfor, w o ormalz h sa famly of 7 by usg h rfr po as follow

10 Aus. J. Bas & Appl. S., : -, * x f x * f x f whr 8 hrfor M, EVC E * *, EVC, * * EVC, * *, E, * * E, whr 9 f =, h oal vory ovr m horzo, E,, wll b mmz for a gv valu of. hs oo, h vory lvl s zro ovr m horzo, xp h las yl, baus, h shorags ar o allowabl. Wh rass, h m rval bw rplshms, spally h las yl, wll b rass. So, E, h absolu zro vory lvl s mpossbl for ay ompay a hrfor, h vory sysm maagr has o osr a mmum valu o zro vory up o ral ompay a xral mar suaos. L E*,* as h rm mmum vory. S EVC, s a fuo of a sr varabl a a ouous varabl < <, hrfor, for ay gv, h ssary oo for h mmum of EVC, s EVC, For a gv valu of, rv * from Equao. EVC,* rvs by subsug, * o quao 9. h, ras by h rm of o oually a EVC,* alula aga. h abov sags rpa ul h mmum EVC,* b fou. h *,* a EVC*,* valus osu h opmal soluo a sasfy h followg oos EVC, EVC, Whr EVC, EVC, EVC,. Numral Exampl: h followg umral xampl s prov o larfy how h propos mol s appl. L h orrg, prouo, holg, baorrg, los sals a rorao oss a h bgg of h m horzo as follow =$/orr; =$8/u; =$/u/yar; =$/u/yar; =$/u a =$/u. h flao ras a h m horzo ar sohas wh h followg p..f.s: ~ U $.8/$/yar,$./$/yar H ~ N,. h ompay rs ra s pr a h rorao ra of h o ha vory pr u m s fv pr. h osa aual prouo ra s us. r=$./$/yar, τ=./u/yar, =us/yar h baloggg ra s δ= -., a h ma paramr valus ar a=us/yar a b=-. h problm s h opmum orrg poly for mmzg h xp prs valu of h oal vory sysm oss, EVC,, a h xp oal quay of h goos warhous ovr m horzo, E,. As sa h al po mho, w hav o frs opmz obvs, sparaly. h al po of h frs obv wh osrg h abov mo paramrs valus a usg h umral mhos, s alula a h rsuls ar llusra abl. a b s ha h mmum xp os s 7.8$ for *=7 a *=.8 h shorags our afr lapsg.8% of h yl m.

11 Aus. J. Bas & Appl. S., : -, abl : h opmal soluo of EVC, for h umral xampl. EVC, EVC, *.8 * 7.8 * L E*,*=. as h rm mmum vory up o h ral ompay a h xral mar suaos. Aorg o = a h ffr ombaos of a wh osrg EVC *, * =7.8 a E*,*=., h problm s valua a h rsuls ar show abl. h maagr a rm h opmum valu of a wh osrg ompay poly abou h mpora of h goals. abl : Soluo of h problm. * * * EVC* Dsusso: h vory sysms usually hav b survy o mmz h oal os. h r as, h ompas ry o maa survval a ras hr orbuos h mar wh osrg aoal obvs. Drasg h vory lvl s o of h mos mpora obv for h ompay, baus: rasg h vory lvl auss rasg ompay flxbly agas hags h mar oos, usomr s a so o, h quay of h rora goos s rla o h vory lvl so ha rasg h vory rass h sroy goo, low vory sysm auss fasr ompay aapao wh h hology hags, a rasg h vory auss br ash flow a ra of rur. hrfor, a b-obv vory mol has b vlop hs papr. raly, h valu or uly of goos rass ovr m for rorag ms, whh ur suggss smallr yl lgh, whras prs of flao os a s mpa o ma suggss largr yl lgh. hs arl, vory mol has b vlop osrg boh h oppos hararss rorao a flao of h ms, wh shorags ovr a sohas m horzo. Shorags ar parally balogg a ma s a fuo of h flao ra. h umral xampl has b gv o llusra h horal rsuls. h suy has b ou ur h Dsou Cash Flow DCF approah. EFEENCES Balh, Z..,. O h Opmaly of vory Mols wh Drorag ms for Dma a O- Ha vory Dp rouo a. MA Joural Maagm Mahmas, : 7-8. Balh, Z..,. A Opmal Soluo of a Gral Lo Sz vory Mol wh Drora a mprf rous, ag o Aou flao a m Valu of Moy. raoal Joural of Sysms S, : Buzao, J.A., 97. Eoom Orr Quas wh flao. Opraoal sarh Quarrly, : - 8. Chag, C..,. A EOQ Mol wh Drorag ms ur flao wh Supplr Crs L o Orr Quay. raoal Joural of rouo Eooms, 88: 7-. Ch, J.M., 998. A vory Mol for Drorag ms wh m-roporoal Dma a Shorags ur flao a m Dsoug. raoal Joural of rouo Eooms, : -. Ch, J.M., C.S. L,. A Opmal plshm Mol for vory ms wh Normally Dsrbu Drorao. rouo lag a Corol, : 7-8. Chr, M.S., H.L. Yag a J.. g, 8. aral baloggg vory lo-sz mols for rorag ms wh fluuag ma ur flao. Europa Joural of Opraoal sarh, 9: 7-. Chug, K.J.. A Algorhm for a vory Mol wh vory-lvl-dp Dma a. Compurs a Opraos sarh, : -7. Chug, K.J. a S.F. sa,. vory Sysms for Drorag ms wh Shorag a a Lar r Dma-ag Aou of m Valu. Compurs a Opraos sarh, 8: 9-9.

12 Aus. J. Bas & Appl. S., : -, Daa,.K. a A.K. al, 99. Effs of flao a m-valu of Moy o a vory Mol wh Lar m-dp Dma a a Shorags. Europa Joural of opraoal sarh, : -. Dy, J.K., S.K. Moal a M. Ma, 8. wo sorag vory problm wh yam ma a rval valu la-m ovr f m horzo ur flao a m-valu of moy. Europa Joural of Opraoal sarh, 8: 7-9. Goyal, S.K. a B.C. Gr,. rs Molg of Drorag vory. Europa Joural of Opraoal sarh, : -. Haly, G.A., 9. Comparso of Orr Quas Compu Usg h Avrag Aual Cos a h Dsou os. Maagm S, : 7-7. Harga, M.A., 99. Effs of flao a m-valu of Moy o a vory Mol wh m- Dp Dma a a Shorags. Europa Joural of opraoal sarh, 8: -. Harga, M.A. a M. B-Daya, 99. Opmal m Varyg Lo-Szg Mols ur flaoary Coos. Europa Joural of opraoal sarh, 89: -. Hsh,.. a C.Y. Dy,. rg a lo-szg pols for rorag ms wh paral baloggg ur flao. Expr Sysms wh Applaos, rss. Horowz,.,. EOQ a flao Uray. raoal Joural of rouo Eooms, : 7-. Hou, K.L.,. A vory Mol for Drorag ms wh So-Dp Cosumpo a a Shorags ur flao a m Dsoug. Europa Joural of Opraoal sarh, 8: -7. Hou, K.L. a L.C. L,. Opmal vory Mol wh So-Dp Sllg a ur Maxmal oal rs Valu of rofs. ro Fourh ASED raoal Cofr o Molg, Smulao, a Opmzao, 7-. Jagg, C.K., K.K. Aggarwal a S.K. Gol,. Opmal orr poly for rorag ms wh flao u ma. raoal Joural of rouo Eooms, : Lao, H.C. a Y.K. Ch,. Opmal aym m for alr's vory Sysm. raoal Joural of Sysms S, : -. Lo, S.., H.M. W a W.C. Huag, 7. A gra prouo-vory mol wh mprf prouo prosss a Wbull srbuo rorao ur flao. raoal Joural of rouo Eooms, : 8-. May, A.K.,. O Mah Mulpl-rou roblm wh rouo-vory Sysm ur Fuzzy qualy Cosra. Appl Sof Compug, rss. Ma, A.K., M.K. Ma, M. Ma,. wo sorag vory mol wh raom plag horzo, Appl Mahmas a Compuao, 8: Mrzazah, A., A.. Sarfaraz, 997. Cosra Mulpl ms Opmal Orr oly ur Sohas flaoary Coos, ro so Aual raoal Cofr o usral Egrg Applao a ra, USA, Sa Dgo, 997: 7-7. Mrzazah, A., 8. Eoom orr rval ur varabl flaoary oos. Hamburg raoal Cofr of Logss HCL8, Hamburg, Grmay. Mrzazah, A., 7. Effs of ura flaoary oos o vory mols usg h avrag aual os a h sou os. Eghh raoal Cofr o Opraos & Quaav Maagm COQM-8, Bago, 7-. Mrzazah, A., 9. A aral Baloggg Mahmaal Mol ur Varabl flao a Dma wh Cosrg Drorao Cos. Worl Appl Ss Joural 7 Spal ssu for Appl Mah: 9-9. Msra,.B., 979. A No o Opmal vory Maagm ur flao. Naval sarh Logss Quarrly, : -. Moo,., B.C. Gr a B. Ko,. Eoom Orr Quay Mols for Amlorag/Drorag ms ur flao a m Dsoug. Europa Joural of Opraoal sarh, : Saa, S.S.,. Dma flu by rprss avs - A mul-m EOQ mol of rorag a amlorag ms. Mahmaal a Compur Mollg, : 8-. Sarar, B. a. Moo,. A EQ mol wh flao a mprf prouo sysm. Appl Mahmas a Compuao, 7: 9-. Sarr, B.., A.M.M. Jamal a S. Wag,. Supply Cha Mols for rshabl rous ur flao a rmssbl Dlay ayms. Compurs a Opraos sarh, 7: 9-7. Sarr, B.. a H. a, 99. Effs of flao a h m Valu of Moy o Orr Quay a Allowabl Shorag. raoal Joural of rouo Eooms, : -7. Su, C.., L.. og a H.C. Lao, 99. A vory Mol ur flao for So Dp Dma a a Expoal Day. Opraos sarh, : 7-8. Vra,. a G. amaabha, 99. A vory Mol ur flao for So Dp Cosumpo a ms. Egrg Coss a rouo Eooms, 9: 79-8.

13 Aus. J. Bas & Appl. S., : -, W, H.M. a S.. Law, 999. Eoom rouo Lo Sz for Drorag ms ag Aou of h m Valu of Moy. Compurs a Opraos sarh, : -8. W, H.M. a S.. Law,. plshm a rg oly for Drorag ms ag o Aou h m-valu of Moy. raoal Joural of rouo Eooms, 7: -. Yag, H.L.,. wo-warhous vory Mols for Drorag ms wh Shorags ur flao. Europa Joural of Opraoal sarh, 7: -. Yag H.L.,. wo-warhous paral baloggg vory mols for rorag ms ur flao. raoal Joural of rouo Eooms, : -7. Yag, H.L., J.. g a M.S. Chr,. A vory mol ur flao for rorag ms wh so-p osumpo ra a paral baloggg shorags. raoal Joural of rouo Eooms, : 8-9. Yag, H.L., J.. g a M.S. Chr,. Drms vory Lo-Sz Mols ur flao wh Shorags a Drorao for Fluuag Dma. Naval sarh Logss, 8: -8. Uhayaumar,. a K.V. Gha, 9. plshm poly for sgl m vory mol wh moy flao. Opsarh, : -7.