An Optimal Ordering Policy for Inventory Model with. Non-Instantaneous Deteriorating Items and. Stock-Dependent Demand

Transcription

1 Applid Mhmicl Scincs, Vol. 7, 0, no. 8, KA Ld, hp://dx.doi.org/0.988/ms.0.56 An piml rdring Policy for nvnory Modl wih Non-nsnnous rioring ms nd Sock-pndn mnd Jsvindr Kur, jndr Shrm nd A. P. Singh prmn of Mhmics, Grphic Er Univrsiy, hrdun, Urkhnd, ndi prmn of Mhmics, Grphic Er Univrsiy, hrdun, Urkhnd, ndi prmn of Mhmics, S.G... P.G. ollg, hrdun, Urkhnd, ndi opyrigh 0 Jsvindr Kur l. This is n opn ccss ricl disriud undr h riv ommons Ariuion Licns, which prmis unrsricd us, disriuion, nd rproducion in ny mdium, providd h originl work is proprly cid. Asrc n his ppr w shll discuss wih h opiml ordring policy for wo wrhous invnory modl of non-insnnous drioring ims nd sock-dpndn dmnd. r shorg is no llowd. r h holding nd drioring coss in oh wrhouss r diffrn nd ssumd h h holding cos in h Wrnd wrhous hn in h Wownd wrhous h opiml soluion r shown. Numricl xmpl is prsnd o dmonsr h dvlopd modl. Kywords: Two wrhouss, Non-insnnous driorion, Sock-dpndn dmnd. NTUTN Ls som yrs, mny rsrchrs hv givn nion o h siuion whr h dmnd r is dpndn on h lvl of h on-hnd invnory. n 976 rly discussd n invnory modl wih wo sorg fciliis. is gnrlly ssumd h h holding cos in h W is grr hn

2 4074 Jsvindr Kur, jndr Shrm nd A. P. Singh h sm in h W. nc, h ims r sord firs in h W, nd only xcss of sock is sord in h W. Furhr, h ims of h W r rlsd firs, nd hn h ims of h W. Firs Ghr nd Schrdr 96 dvlopd n Economic rdr Quniy invnory modl for h drioring ims. n 97 ovr nd Philip discussd n invnory modl wih vril driorion r nd wo-prmr Wiull disriuion. Philip 974 dvlopd h modl wih hr prmr Wiull disriuion r nd no shorgs. Goyl nd Giri 00 suggsd rviw of drioring invnory lirurs in which ll h invnory modls for drioring ims ssum h h driorion occurs s soon s h rilr rcivs h commodiis. Firs Gup nd Vr 986 discussd h invnory modls for sock-dpndn consumpion r. n 995 W prsnd drminisic lo-siz invnory modl for drioring ims wih shorgs nd dclining mrk. hng 004 dvlopd n invnory modls wih sockdpndn dmnd nd nonlinr holding coss for drioring ims. Soni nd Shh 008 discussd h opiml ordring policy for n invnory modl wih sock-dpndn dmnd undr progrssiv pymn schm. L nd su 009 dvlopd wo wrhous producion modl for drioring invnory ims wih im-dpndn dmnds. Blkhi nd Bkry 009 dls gnrl nd dynmic producion lo siz invnory modl. Bnrj nd Shrm 00 prsnd n invnory modls wih pric nd im dpndn ssonl dmnd. S., N. nd S..K. 00 discussd n invnory modl for drioring ims wih Wiull r of rplnishmn nd slling pric dpndn dmnd. hng, Tng, S. K. Goyl 00 dvlopd n opiml rplnishmn policy for non-insnnous drioring ims wih sock-dpndn dmnd. This ppr dls wih n invnory modl wih wo wrhouss o drmining h opiml rplnishmn policy for non-insnnous drioring ims nd sock-dpndn dmnd. For fiing in wih rlisic condiions, h prolm of finding h opiml rplnishmn policy for noninsnnous drioring ims wih sock-dpndn dmnd is considrd in his sudy. Th ncssry nd sufficin condiions of h xisnc nd uniqunss of h opiml soluion r givn. Finlly, numricl xmpl is prsnd o dmonsr h dvlopd modl.. ASSUMPTNS AN NTATN Th following ssumpions nd noion r usd in his ppr: Th dmnd r im is, > 0,, 0, Whr, r posiiv consns nd is h invnory lvl im. plnishmn r is infini nd h ld im is zro. is h lngh of im in which h produc hs no driorion i.., frsh produc im. nd r driorion in W nd W rspcivly. is h lngh of im in which h invnory is no shorg. dnos h invnory lvl in W im [0, ] in which h produc hs no driorion.

3 piml ordring policy for invnory modl 4075 T, is h ol rlvn invnory cos pr uni im of invnory sysm. A,,,, no h ordring cos pr ordr, invnory holding cos in W pr uni im, invnory holding cos in W pr uni im, drioring cos in W pr uni nd drioring cos in W pr uni, rspcivly. dnos h invnory lvl in W im [, ] in which h produc hs driorion. is h invnory lvl in W im [0, ] in which h produc hs no driorion. dno h invnory lvl in W im [, ] in which only driorion. dno h invnory lvl in W im [, ] in which h produc hs driorion.. MEL FMULATN uring h inrvl [0, ], h invnory lvls r posiiv W nd W nd dcrsing only owing o sock-dpndn dmnd r in W. A W, h invnory is dpld du o h comind ffcs of dmnd nd driorion. Th invnory lvl is dropping o zro du o dmnd nd driorion during h im inrvl [, ]. As dscrid ov, h invnory lvl dcrss only owing o sock-dpndn dmnd r during h im inrvl [0, ]. Th invnory lvl W nd W r govrnd y h following diffrnil quions: d [ ] d 0. d [ ] d. d 0 d 0. d 0 d.4 d [ ], d..5 wih h oundry condiions 0, 0, W, 0 rspcivly. Solving hs diffrnil quions, w g h invnory lvl s follows: { }, 0. 6 { },..7 W, 0.8 W,..9

4 4076 Jsvindr Kur, jndr Shrm nd A. P. Singh { },..0 onsidring coninuiy of, i follows from Equions 6 nd 7 h { } { } { } Susiuing quion ino 6, w g { } { } { }.., 0. According o givn condiions, { } W w log.. Now h ol rlvn invnory cos pr cycl consiss of h following lmns: rdring cos pr cycl is A. 4 nvnory holding cos pr cycl in W is givn y { d d } 0 { } { } { } { } nvnory holding cos pr cycl in W is givn y { } d d d 0 W W [ ] riorion cos pr cycl in W is givn y { } 6...5

5 piml ordring policy for invnory modl 4077 { } d { } 7 riorion cos pr cycl in W is givn y { } d d [ ] { } W W 8 Thus h ol rlvn invnory cos pr uni im is givn y T, [rdring cosnvnory holding cos pr cycl in Wnvnory holding cos pr cycl in Wriorion cos pr cycl in Wriorion cos pr cycl in W] 9 Susiuing Equion 4 8 in h ov quion 9, w g T, A { } { } { } W W. 0 Th ol rlvn invnory cos pr uni im is minimum if 0, 0, T T 0 > T T T nd. 0 > T 5. NUMEAL EXAMPLE n ordr o illusr h ov soluion procdur, considr n invnory sysm wih h following d:

6 4078 Jsvindr Kur, jndr Shrm nd A. P. Singh ; 0.; 0. A 95; 0.65; 0. 6 /0.08, W40. ; 80; 0.; 0.08;.09; hngs in T is osrvd h h ol cos T is vry snsiiv o chngs in consn dmnd. Wih incrs in h dmnd, h opimum im for invnory in W, im for invnory in W nd ol cycl im for rplnishmn dcrss. ncrsing in h r of driorion dos no produc significn chngs in h opiml soluion. Th following grphs show h rlion wn ol cos nd im priod nd. Th hr dimnsionl ol cos pr im uni grph is shown in h figur low y ploing in h rng of [0.0, 0.0] nd in h rng of [.80,.9].

7 piml ordring policy for invnory modl 4079 Th ov grph shows h h ol cos pr im uni is sricly convx. 6. NLUSNS n his ppr, w dvlopd wo wrhouss invnory modl for non-insnnous drioring ims wih sock-dpndn dmnd. Shorgs r no llowd. Furhrmor, h proposd modl cn usd in invnory conrol of crin non-insnnous drioring ims such s food ims, lcronic componns such s moil, mchins, circui, oys nd fshionl commodiis c. n h fuur sudy, i is hopd o furhr incorpor h proposd modl ino mor rlisic ssumpions, such s, vril holding cos, ckordrs, pril cklogging, xponnilly dmnd, qudric dmnd, producion dpndn dmnd. EFEENES [] Ghr, P.M., Schrdr, G.., 96. A modl for xponnilly dcying invnory sysm. nrnionl Journl of Producion srch, [] ovr,.p., Philip, G.., 97. An EQ modl for ims wih Wiull disriuion driorion. AE Trnscions 5, 6. [] Philip, G.., 974. A gnrlizd EQ modl for ims wih Wiull disriuion. AE Trnscions 6, [4] rly V. onld n h EQ modl wo lvls of sorg. psrch,,

8 4080 Jsvindr Kur, jndr Shrm nd A. P. Singh [5] Gup,., Vr, P., 986. nvnory modl wih muli-ims undr consrin sysms for sock dpndn consumpion r. prions srch 4, 4 4. [6] W,.M., 995. A drminisic lo-siz invnory modl for drioring ims wih shorgs nd dclining mrk. ompurs & prions srch, [7] Goyl, S.K., Giri, B.., 00. cn rnds in modling of drioring invnory. Europn Journl of prionl srch 4, 6. [8] hng,.-t., 004. nvnory modls wih sock-dpndn dmnd nd nonlinr holding coss for drioring ims. Asi-Pcific Journl of prionl srch, [9] Soni, Shh N 008 piml ordring policy for sock-dpndn dmnd undr progrssiv pymn schm.europn Journl of prionl srch 84:9 0. [0] L,.., & su, S.L Two Wrhous producion modl for drioring invnory ims wih im-dpndn dmnds. Europn Journl of prionl srch, 94, [] Blkhi, Z., & Bkry, A.S A gnrl nd dynmic producion lo siz invnory modl. nrnionl Journl of Mhmicl Modls nd Mhods in Applid Scincs, [] Bnrj, S. nd Shrm, A. 00. piml procurmn nd pricing policy for invnory modls wih pric nd im dpndn ssonl dmnd, Mhmicl nd ompur Modling,5, [] Sridvi G., Nirupm vi K. nd Srinivs o K. 00. nvnory modl for drioring ims wih Wiull r of rplnishmn nd slling pric dpndn dmnd. nrnionl Journl of prionl srch, Volum 9, [4] hun-to hng, Jinn-Tsir Tng, Sursh Kumr Goyl 00. piml rplnishmn policis for non-insnnous drioring ims wih sock-dpndn dmnd, nrnionl Journl of Producion Economics, Volum, civd: My 4, 0