Perishable Inventory Model with Time Dependent Demand and Partial Backlogging

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1 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp RESEARCH ARICLE OPEN ACCESS Pershable Invenory Moel wh me Depenen Deman an Paral Backloggng Kapl Mehrora, Rajenra Sharma Research Scholar, Graphc Era Unversy, Dehraun, Ina Deparmen of Mahemacs an Sascs, Graphc Era Unversy, Dehraun, Ina Absrac An Invenory Moel For Decayng Iems Uner Inflaon Has Been Develope. Deman Rae Is aken As Lnear me Depenen. Holng Cos Is Also aken As me Depenen. We Have Develope he wo Cases: In he Frs Case Shorages Are No Allowe An In he Secon Case Parally Backlogge Shorages Are Allowe. Cos Mnmzaon echnque Is Use In hs Suy. Keywors: mpenen eman, paral backloggng, shorages, pershable nvenory moel. I. INRODUCION Plannng s he ar of specfyng meanngful nformaon abou he fuure. Long run plannng ecsons requre conseraon of many facors: general economc conons, nusry rens, probable compeor acons an so on. For proper managemen of sysem resources, here mus be an aequae knowlege base of he cusomer s eman. hs eman nees o be manage n a proper manner. he purpose of eman managemen s o coornae an conrol all he sources, so ha proucon an operaons sysems can be ulzffcenly.as nvenory represens a very mporan par of he company s fnancal asses, s very much affece by he marke s response o varous suaons, especally nflaon. Inflaon s a global phenomenon n presen ay mes. Inflaon can bfne as ha sae of sequlbrum n whch an expanson of purchasng power ens o cause or s hffec of an ncrease n he prce level. A pero of prolonge, perssen an connuous nflaon resuls n hconomc, polcal, socal an moral srupon of socey. Almos everyone hnks nflaon s evl, bu sn' necessarly so. Inflaon affecs fferen people n fferen ways. I also epens on wheher nflaon s ancpae or unancpae. If he nflaon rae correspons o wha he majory of people arxpecng (ancpae nflaon, hen we can compensae, an he cos sn' hgh. Nowaays nflaon has become a permanen feaure n he nvenory sysem. Inflaon eners n he pcure of nvenory only because may have an mpac on he presen value of he fuure nvenory cos. hus he nflaon plays a val role n he nvenory sysem an proucon managemen hough hcson makers may face ffcules n arrvng a answers relae o ecson makng. A presen, s mpossble o gnore he effecs of nflaon an s necessary o conser hffecs of nflaon on he nvenory sysem. Buzaco (975 evelope he frs EOQ moel akng nflaonary effecs no accoun. In hs moel, a unform nflaon was assume for all he assocae coss an an expresson for he EOQ was erve by mnmzng he average annual cos. Berman an homas (977 suggese he nvenory ecson polcy uner nflaonary conons.economc analyss of ynamc nvenory moels wh non-saonary coss an eman was presene by Harga (994. hffec of nflaon was also consere n hs analyss. An economc orer quany nvenory moel for eerorang ems wh nflaon was evelope by Bos al. (995. Effecs of nflaon an me-value of money on an nvenory moel was scusse by Harga (995 wh lnearly ncreasng eman rae an shorages. Harga an Ben-Daya (996 hen scusse he nvenory replenshmen problem over a fxe plannng horzon for ems wh lnearly me-varyng eman uner nflaonary conons. A generalze ynamc programmng moel for nvenory ems wh Webull srbueroraon an mpenen eman was propose by Chen (998. hffecs of nflaon an me-value of money on an economc orer quany moel have been scusse by Moon an Lee (000. Chang (004 propose an nvenory moel wh hffec of nflaon, eerorang ems an elay n paymens. Moels for amelorang/eerorang ems wh me-varyng eman paern over a fne plannng horzon were propose by Moon e al. (005. hffecs of nflaon an me value of money were also aken no accoun. Jola e al. (006 presene an opmzaon framework o erve opmal proucon over a fxe plannng horzon for ems wh a sock-epenen eman rae uner nflaonary conons. Jagg e al. (007 presene he opmal nvenory replenshmen polcy for eerorang ems uner nflaonary 40 P a g e

2 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp conons usng a scoune cash flow (DCF approach over a fne me horzon. Shorages n nvenory were allowe an compleely backlogge an eman rae was assume o be a funcon of nflaon. Chern, M. S. e al. (008 evelope an nvenory lo-sze moel for eerorang ems wh paral backloggng an me value of money. Roy, A., Pal, S. an Ma, M.K. (009 consere a proucon nvenory moel wh nflaon an me value of money an eman of he em s splaye sock epenen an lfeme of he prouc s ranom n naure an follows exponenal srbuon wh a known mean. hs paper presens an orer level nvenory sysem wh non-nsananeous eerorang ems wh mpenen eman, where supplers offer an all un quany scoun. wo cases have been scusse n hs suy: n case frs shorages are no allowe an n case secon shorages are allowe wh parally backlogge. he whole suy has been aken wh nflaonary envronmen. Specal cases are scusse n hs suy. he purpose of hs suy s o eermne an opmal orerng polcy for mnmzng hxpece oal relevan nvenory cos. Assumpons II. ASSUMPIONS AND NOAIONS. he prouc lfe me (me o eeroraon has a p..f. f ( ( e for. So ha he f( eeroraon rae s r (, for. F (. hman rae D(, where an are posve consans. 3. hffec of nflaon has been aken. 4. he replenshmen rae s fne. 5. Lea me s zero. 6. Shorages are allowe an paral backlogge. 7. here s no replacemen or repar of eerorae uns urng he pero uner conseraon. Noaons A Orerng cos per orer q he h prce breakng pon =0,,..m, where 0 q0 q... qm C he purchasng cos per un epenen on orer sze, =0,,.m, where C0 C... C m 0. ( C h Lnear holng cos C S LS Shorages cos per un backorere per un me C he cos of los sales per un e Backloggng rae Parameer of heroraon rae funcon he lengh of me n whch he prouc has no eeroraon he lengh of me wh posve sock he replenshmen cycle me, where Q Orer quany per cycle b ae Exponenal eman rae mpenen eeroraon rae b Kae Deman epenen proucon C Holng cos per un per un me C S Shorages cos per un per un me C LS Los sale cos per un per un me C ' Se up cos III. MAHEMAICAL FORMULAION 4 P a g e

3 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp hs paper s evelope he replenshmen problem of a sngle non-nsananeous eerorang em wh paral backloggng an varable holng cos. I uns of he em arrve a he nvenory sysem a he begnnng of each cycle. A, [0, ] he nvenory level ecreases only owng o sock an mpenen eman rae. he nvenory level rops o zero ue o eman an eeroraon of ems a [, ]. he shorage occurs an keeps o hn of he curren orer cycle. hnre process s repeae. We conser wo cases: frs, when shorages are compleely backlogge an secon, when shorages are parally backlogge. CASE I: When shorages are no allowe he nvenory level ecreases only ue o eman urng he nerval[, ]. Hence he fferenal equaon represenng he nvenory level s gven by: I ( ( 0 ( I (0 I. Soluon of equaon ( s: Wh he bounary conon I( I ( 0 ( I( I( ( (3 Wh he bounary conon I( 0. Soluon of equaon (3 s: ( ( ( I ( [ ( e ( e ( e ] (4 Equang hquaon ( an (4 a ( ( ( ( [ ( ( I ( e e e ] hs mples ha he mum nvenory level for each cycle s: ( ( ( I [ ( e ( ( ( e e ] (5 Subsue he value of I from equaon (5 n equaon (, we ge ( ( ( I( [ ( e ( ( ( e e ] (, 0 (6 From equaon (5, we can oban he orer quany Q as: Q I ( ( ( [ ( e ( ( ( e e ] (7 he oal relevan cos per un me consss of A Orerng cos OC (8 Invenory holng cos [ 0 ( r ( r HC C I e C I e ] [ ( ( ( { ( ( ( C e e e 3 3 r ( r ( ( ( e ( e 3 4 P a g e

4 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp r ( ( r e ( ( } r 3 8 ( ( e e { ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] (9 C r Invenory eeroraon cos DC [ I( e ] ( ( C e e [{ ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] (0 Purchase cos PC CQ ( ( ( C [ ( ( ( ( e e e ] ( he oal relevan nvenory cos per un me can be formulae as: VC(, OC HC DC PC A [ ( ( ( { ( ( ( C e e e 3 3 r ( r ( ( ( e ( e r ( ( r e ( ( } r 3 8 ( ( e e { ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] ( ( C e e [{ ( ( ( 43 P a g e

5 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] ( ( ( C [ ( ( ( ( e e e ] ( o mnmze he oal average cos per un of me, he opmal value of an can be obane by he followng equaons VC(, VC(, 0 an 0 VC(, 0 VC(, an 0 SPECIAL CASES CASE A: When 0, eman rae s consan. hen he oal relevan nvenory cos per un me can be formulae as: VC(, OC HC DC PC A [ ( { ( C e 3 r ( ( r e 6 ( ( ( e r e e { ( ( ( ( C e }] [{ ( ( ( ( r e e ( }] ( C [ ( e ( ] CASE II: When shorages are parally backlogge he nvenory level ecreases only ue o eman urng he nerval[, ]. Hence he fferenal equaon represenng he nvenory level s gven by: I ( ( 0 (3 I (0 I. Soluon of equaon (3 s: Wh he bounary conon I( I ( 0 (4 I( I( ( (5 Wh he bounary conon I( 0. Soluon of equaon (5 s: ( ( ( I ( [ ( e ( e ( e ] (6 3 } 44 P a g e

6 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp Equang hquaon (4 an (6 a ( ( ( ( [ ( ( I ( e e e ] hs mples ha he mum nvenory level for each cycle s: ( ( ( I [ ( e ( ( ( e e ] (7 Subsue he value of I from equaon (7 n equaon (4, we ge ( ( ( I( [ ( e ( ( ( e e ] ( (8 0 Durng he shorage nerval [, ], hman a any me s parally backlogge a fracon. hus he nvenory level a any me s governe by he followng equaon: I ( 3 ( e (9 3( [ ( I ( e e ( e e ] (0 Subsue = n equaon (0, we oban he mum amoun of eman backlogge per cycle as follows S ( I ( [ ( ( e e ( e e ] ( From equaon (7, we can oban he orer quany Q as: Q I S ( ( ( [ ( e ( ( ( e e ( ( ( e e ] ( he oal relevan cos per un me consss of A Orerng cos OC (3 Invenory holng cos [ ( r 0 ( r HC C I e C I e ] [ ( ( ( { ( ( ( C e e e 3 3 r ( r ( ( ( e ( e r ( ( r e ( ( } r 3 8 ( ( e e { ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( P a g e 3

7 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp ( ( r e e ( }] (4 C r Invenory eeroraon cos DC [ I( e ] ( ( C e e [{ ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] (5 Purchase cos PC CQ ( ( ( C [ ( ( ( ( e e e ( ( e e ( e e ] (6 C S r Shorages cos SC [ ( I( e ] CS e [( ( e ( ( e ( e ( e ( r ( e ( r ( e ( e e ( ( r ( e ( ] (7 C LS r Los sale cos LC [ ( e ( e ] r r ( r ( r r r CLS [ ( ( r r ( r ( r r r r r ( r ( r ( r ( r ] (8 r r ( r ( r ( r ( r he oal relevan nvenory cos per un me can be formulae as: VC(, OC HC DC PC SC LC A [ ( ( ( { ( ( ( C e e e 3 3 r ( r ( ( ( e ( e 3 46 P a g e

8 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp r ( ( r e ( ( } r 3 8 ( ( e e { ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] ( ( C e e [{ ( ( ( ( ( ( e r e e ( ( ( ( ( 3 3 r e e ( 3 3 ( ( r e e ( }] ( ( ( C [ ( ( ( ( e e e ( ( e e ( e e ] CS e [( ( e ( ( e ( e ( e ( r ( e ( r ( e ( e e ( ( r ( e ( ] r r ( r ( r r r CLS [ ( ( r r ( r ( r r r r r ( r ( r ( r ( r ] (9 r r ( r ( r ( r ( r o mnmze he oal average cos per un of me, he opmal value of an can be obane by he followng equaons VC(, 0 an VC(, 0 47 P a g e

9 Kapl Mehrora e al In. Journal of Engneerng Research an Applcaons ISSN : 48-96, Vol. 4, Issue 7( Verson 5, July 04, pp VC(, 0 an VC(, 0 IV. SPECIAL CASES CASE A: When 0, eman rae s consan. V. CONCLUSION An nvenory moel of eermnng he opmal replenshmen polcy for non-nsananeous eerorang ems wh lnear eman has been evelope. In many cases cusomers are conone o a shppng elay, an may be wllng o wa for a shor me n orer o ge her frs choce. Generally speakng, he lengh of he wang me for he nex replenshmen s he man facor for ecng wheher he backloggng wll be accepe or no. he wllngness of a cusomer o wa for backloggng urng a shorage pero eclnes wh he lengh of he wang me. hus, nvenory shorages are allowe an parally backorere n he presen chaper an he backloggng rae s consere as a ecreasng funcon of he wang me for he nex replenshmen. he man goal of hs suy s o mnmze he oal average cos. he moel evelope may furher bxene for proucon nvenory moel, nflaon an permssbllay n paymens. REFERENCES [] Berman, H.J. an homas, J. (977. Invenory ecsons uner nflaonary conons. Decson Scences, 8(, [] Bose, S., Goswam, A., Chauhur, K.S. (995. An EOQ moel for eerorang ems wh lnear me epenen eman rae an shorage uner nflaon an me scounng. J.O.R.S., 46, [3] Buzaco, J.A. (975. Economc orer quanes wh nflaon. Operaonal Research Quarerly, 6, [4] Chang, C.. (004. An EOQ moel wh eerorang ems uner nflaon when suppler cres lnke o orer quany. I.J.P.E., 88, [5] Chen. J.M. (998. An nvenory moel for eerorang ems wh me-proporonal eman an shorages uner nflaon an me scounng. I.J.P.E., 55, (, -30. [6] Chern, M. S., Yang, H. L., eng, J., an Papachrsos, S. (008. Paral backloggng nvenory losze moels for eerorang ems wh flucuang eman uner nflaon. European Journal of Operaonal Research, 9, 7 4. [7] Harga, M.A. (994. Economc analyss of ynamc nvenory moels wh non-saonary coss an eman. I.J.P.E., 36, [8] Harga. M.A. (995. Effecs of nflaon an me-value of money on an nvenory moel on an nvenory moel wh me-epenen eman rae an shorages. E.J.O.R., 8, 3, [9] Harga, M.A. an Ben-Daya, M. (996. Opmal me-varyng lo szng moels uner nflaonary conons. E.J.O.R., 89,, [0] Jagg, C.K., Aggarwal, K.K. an Goel, S.K. (007: Opmal orer polcy for eerorang ems wh nflaon nucman. I.J.P.E., 03, (, [] Jola, F., avakkol-moghaam, R., Rabban, M. an Saoughan, M.R. (006. An economc proucon lo sze moel wh eerorang ems, sock-epenen eman, nflaon, an paral backloggng. Apple Mahemacs an Compuaon (A.M.C., 8, [] Moon, I. An Lee, S. (000. hffecs of nflaon an me-value of money on an economc orer quany moel wh a ranom prouc lfe cycle. E.J.O.R., 5, [3] Moon, I., Gr, B.C. an Ko, B. (005. Economc orer quany moels for amelorang / eerorang ems uner nflaon an me scounng. E.J.O.R., 6, 3, [4] Roy, A., Pal, S. an Ma, M.K. (009. A proucon nvenory moel wh sock epenen eman ncorporang learnng an nflaonary effec n a ranom plannng horzon: A fuzzy genec algorhm wh varyng populaon sze approach. Compuers & Inusral Engneerng, 57, P a g e