A Production Inventory Model for Different Classes of Demands with Constant Production Rate Considering the Product s Shelf-Life Finite

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1 nrnionl Confrnc on Mchnicl nusril n Mrils Enginring 5 CMME5 - Dcmbr 5 RUE Rjshhi Bnglsh. Ppr D: E-6 A Proucion nvnory Mol for Diffrn Clsss of Dmns wih Consn Proucion R Consiring h Prouc s Shlf-Lif Fini Mohmm Ekrmol slm Shirjul slm Ukil n M Shrif Uin Dprmn of Businss Aminisrion Norhrn Univrsiy Bnglsh Dhk Bnglsh n Dprmn of Mhmics Jhngirngr Univrsiy Svr Bnglsh E-mil: shirjukil@yhoo.com Phon: Absrc his ppr unfols how mol is vlop on h bsis of mrk mns n compny s proucion prn. vncs in qus of opimum cos consiring h prouc s shlf-lif fini. h ppr iscusss bou proucion invnory mol whr compny proucs ims wih consn r bu mns vry u o h cusomrs ns. Wihou hving ny sor of bcklogs proucion srs. Rching h sir lvl of invnoris i sops proucion. Afr h u o mns long wih h riorion of h ims i iniis is plion n fr crin prios h invnory gs zro. is ssum h h cy of h proucs is lvl pnn. h objciv of his ppr is o fin ou h opimum cos n im. Ky wors:proucion invnory Shlf-lif im Dmn clss Proucion r.. nroucion n minimizing invnory cos his ppr vlops n nvnory Mol of Drminisic Dmn of mrils which hv h fini shlf-lif. h Mol vncs by consiring h consn proucion r smll moun of cy vrying mn prn whil rching in crin moun of invnory lvl h proucion sops. Hrris [] prsn h fmous conomic orr quniy EO formul. Whiin [] ws h firs rsrchr who vlops h invnory mol wih cy for fshion goos. Ghr n Schrr [] firs poin ou h ffc of cy n iscovr EO mol. Rosnbl n L [4] ssum h im s imporn fcor for sock. Jml Srkr n Monl [5] us singl n Ekrmol [67] us vrious proucion sgs. Abullh n Chuhuri [8] n Ji-zr n Li-Frn [9] inclu h fciv ims wih imprfc procss n bckorrs in h mol. ng Chrn n Yng [] consir flucuing mn n Skouri n Ppchrisos [] iscuss coninuous rviw mol. Vino [] scrib im pnn rioring ims n Brojswr Shib n Chuhuri [] iscuss wih rwork bl ims n supply isribuion. h imporn formul from Nor [4] n Whiin [5] hs bn inrouc o vlop his ppr. Subsqunly h mol is formul by proving h h ol vribl is convx which shows h h opimum invnory cos is miniml.. Assumpions. Proucion r is consn which srs whn invnory is zro n cy is vry smll. b. nvnory lvl is highs. From his poin h ol ims mus b livr rly o voi is cy. Sinc h proucion sops whil invnory is highs invnory pls quickly u o mn.

2 nrnionl Confrnc on Mchnicl nusril n Mrils Enginring 5 CMME5 - Dcmbr 5 RUE Rjshhi Bnglsh.. Noions = Proucion r n i oix whr i= o. = Vry smll moun of consn cy r for uni invnory. Afr h proucion sops is zro. = nvnory lvl insn n n = nvnory im x = Dmn siz uring im i = mn r im x 4 = vrg invnory whil no proucion occurs in h lmm. which pics h invnory lvl rspcivly n. n = invnory consiring h mn prn inx n n m = An ingr. n = mn prn inx q = nvnory lvl whil proucion sops n S = Smll porion of C = ol cos in rms of 4. Dvlopmn of h mol K = S up cos n h = vrg holing cos. = invnory fr proucion sops h bginning. W mn whil no proucion occurs n n = opimum orr quniy. V smll im sgmn. A h bginning whil im h proucion srs wih zro invnory wih h r consn for nir proucion cycl. Bu mns will vry im o im which is shown in h figur. which rmins λ - λ λ = Fig.. nvnory lvl vrious sgs From h bov figur w g h vlu of hrfor w g in iffrn im sgmns n hos r During o invnory incrss h r of n w g h iffrnil quion s:

3 nrnionl Confrnc on Mchnicl nusril n Mrils Enginring 5 CMME5 - Dcmbr 5 RUE Rjshhi Bnglsh. Applying h bounry coniion n w g h soluion s Consiring up o scon gr of n using quion ol un-cy invnory uring o 4 During o pplying h bounry coniion w consir.hn w g 5 Bing s smll quniy nglcing is highr powr h ol un-cy invnory uring o is 6 During o similr pproch n bounry coniion sy bing us n bing vry smll nglcing is highr powr w g h ol un-cy invnory s } }{ - { Afr rching h sir lvl of invnoris h proucion sops n in his sg invnory rchs zro u o consn mn n ngligibl moun of cy. Lmm: f h mximum invnory lvl is n mn occurs in uniform wy h moun of invnory will b s 8 4 Proof: W know from Nor [4] h h mn prn cn b gnrlly rprsn s 9 n x S An on h bsis of bov quion from Nor [9] w cn xprss h vrg moun of invnory by

4 nrnionl Confrnc on Mchnicl nusril n Mrils Enginring 5 CMME5 - Dcmbr 5 RUE Rjshhi Bnglsh. whr q n. V V q 4 m n n n W W which cn b compr wih quion no 9. n our cs w us h ls im sgmn h mn occurs uniformly n s w nglc cy i.. V W m i.. m n compring s 4 n mn prn inx q=mv =m +μ V or θ W -=mθ =mv Fig.. nvnory lvl whil no proucion occurs Puing hs vlus in h quion no w g h following rsul 4 Hnc from h quion numbr 8 n w g h proof. ol im cycl Now ol im cycl cn b xprss s m - ol cos funcion K h 4 W us h quion no 6 8 9n o g h ol cos C K h C {

5 ol Cos C nrnionl Confrnc on Mchnicl nusril n Mrils Enginring 5 CMME5 - Dcmbr 5 RUE Rjshhi Bnglsh. 5 7 K o rmin h opimum orr quniy h 9 4 h h firs n scon riviv of h quion wih rspc o n o vrify h h quion no is convx in } w mus show is zro n posiiv rspcivly. Hnc h convx propris imply h h firs riviv C n h scon riviv C K K { } which is lwys posiiv s h quniy r posiiv. hrfor ol cos is convx in K. Hnc for opimum vlu of by Hly n Whiin [5] w g h opimum orr quniy h ol cos funcion will b minimum n s blow h9 4 K 4 5. Numricl illusrion wih snsiiviy nlysis h 6 L h prmrs K.. hn from n 4 w g h opimum orr quniy C = 8.7 unis n ol opimum cos = 4.58 unis. ol cos crss if mn n λ incrss; ol cos incrss if h mn n incrss. Orr quniy vrss ol cos C Fig.. uniy Vrss ol Cos Orr uniy

6 nrnionl Confrnc on Mchnicl nusril n Mrils Enginring 5 CMME5 - Dcmbr 5 RUE Rjshhi Bnglsh. 6. Conclusion ol cos crss in h firs sg s i hs sufficin invnory wih rspc o is mn incrss. his cos incrss uring h scon n hir sgs u o is ruc invnory lvl s h proucion sops h n of his sgs hving incrsing mn. ol cos incrss in h fourh sg whil mn incrss fr no proucion. h Mol coul sblish h wih priculr orr lvl i.. = 8.7 unis ol cos is minimum i.. C = 4.58 unis. Bfor n fr his poin ol Cos incrss shrply. his ppr iscuss h in vrying mn prn how mol is vlop by rciving ppropri moun of orr consiring h mrk mns prouc s shlf-lif n compny s proucion r which hs ulimly nsur h opimum invnory cos. 7. Rfrncs [] F. W. Hrris How Mny Prs o Mk Onc h Mgzin of Mngmn Vol. No. pp []. Whiin hory of nvnory Mngmn Princon Univrsiy Prss Princon NJ pp [] P. M. Ghr n G. P. Schrr A Mol for n Exponnilly Dcying nvnory Journl of nusril Enginring Vol. 4 No [4] M. J. Rosnbl n H. L. L Economic Proucion Cycls wih mprfc Proucion Procsss E rnscions Vol. 8 No. pp [5] A. M. M. Jml B. R. Srkr n S. Monl Opiml Mnufcuring Bch Siz wih Rwork Procss Singl-Sg Proucion Sysm Compurs n nusril Enginring Vol. 47 pp [6] M. Ekrmol slm A Proucion nvnory wih hr Proucion Rs n Consn Dmns Bnglsh slmi Univrsiy Journl Vol.. ssu. pp. 4-. [7] M. Ekrmol slm A Proucion nvnory Mol for Dioring ms wih Vrious Proucion Rs n Consn Dmn Proc. of h Annul Confrnc of KMA n Nionl Sminr on Fuzzy Mhmics n Applicions Pyynnur Krl pp [8] E. Abullh n O. Gulkin An Economic Orr uniy Mol wih Dfciv ms n Shorgs nrnionl Journl of Proucion Economics Vol. 6 8 pp [9] H. Ji-zr n H. Li-Frn ngr Vnor-Buyr Coopriv Mol in n mprfc Proucion Procss wih Shorg Bckorring nrnionl Journl of Avnc Mnufcuring chnology Vol. 65 ssu. -4 pp [] J.. ng M. S. Chrn n H. L. Yng Drminisic Lo-Siz nvnory Mols wih Shorgs n Drioring for Flucuing Dmn Oprion Rsrch Lrs Vol. 4 8 pp [] K. Skouri n S. Ppchrisos A Coninuous Rviw nvnory Mol wih Drioring ms im-vrying Dmn Linr Rplnishmn Cos Prilly im-vrying Bcklogging Appli Mhmicl Moling Vol. 6 pp [] K. M. Vino n S. S. Ll Proucion nvnory Mol for im Dpnn Drioring ms wih Proucion Disrupions nrnionl Journl of Mngmn Scinc n Enginring Mngmn Vol. 6 No. 4 pp [] P. Brojswr S. S. Shib n K. Chuhuri A Muli-Echlon Supply Chin Mol for Rwork bl ms in Mulipl- Mrks wih Supply Disrupion Economic Moling Vol. 9 pp [4] E. Nor nvnory Conrol Sysm pp [5] G. Hly n. Whiin Anlysis of nvnory Sysms Prnic Hll Engl-woo Cliffs pp