An EOQ Model for Deteriorating Items Quadratic Demand and Shortages

Transcription

1 Ieriol Jourl of Iveory Corol d Mgeme Speil Issue o Ieriol Coferee o Applied Mhemis & Sisis De ISSN , AACS. ( All righ reserved. A EOQ Model for Deeriorig Iems Qudri Demd d Shorges Volume Numer July-De: pp ARICLE INFO Arile Hisory Reeived : 8.5. Aeped : 8.. R. Begum*, S. K. Shu Deprme of Mhemis, C.V.Rm College of Egieerig, Bhueswr- 755, Odish, Idi, e-mil: egum_rehe@rediffmil.om P.G. Deprme of Sisis, Smlpur Uiversiy, Jyoi Vihr, Burl,Smlpur- 7688, Idi, e-mil: drsudhir97@gmil.om * Correspodig uhor Sope d Purpose: he sope of his model iludes he ppliio of iveory mgeme. I hs ee empirilly oserved h he life expey of my iems e expressed i erms of Weiull disriuio. his empiril oservio hs promped reserhers o represe he ime o deeriorio of produ y Weiull disriuio. I is lso oserved h demd of osumer produ usully vries wih ime d hee, he demd re should e ke s ime-depede. Our purpose is o fous o mhemil model o iveory produio sysem osiderig ll hese fors. herefore, we develop eoomi iveory model for deeriorig iems wih qudri demd d shorges i iveory. Asr his pper preses iveory model for deeriorig iems wih qudri demd, iseous supply d shorges i iveory. A woprmeer Weiull disriuio is ke o represe he ime o deeriorio. he heory for fidig he opiml soluio of he prolem is developed. A umeril exmple is ke o illusre he soluio proedure. Keywords: EOQ, ime-qudri demd, shorges, deeriorio.. Iroduio: I iveory prolems, deeriorio is defied s dmge, dey, spoilge, evporio, osolesee, loss of uiliy or loss of mrgil vlue of goods h resuls i derese he usefuless of he origil oe. Deeriorio should o e egleed i iveory prolems for he iems like foodsuff, hemils,

2 A EOQ Model for Deeriorig Iems Qudri Demd d Shorges 58 phrmeuils, eleroi goods, rdioive suses, e. Emmos s (968) models wih wo-prmeer Weiull disriuio deeriorio were disussed y Cover d Philip (97), Philip (97), Giri e l.(), Ghosh d Chudhuri () e. wheres Chkrry, Giri d Chudhuri (998) d oher reserhers used hree-prmeer Weiull disriuio deeriorio i heir iveory models. Giri e l. (999), S e l. (), S d Chudhuri (), e., developed iveory models i his direio. Misr (975) developed EOQ model wih Weiull deeriorio re for perishle produ where korderig is o llowed. hese ivesigios were followed y severl reserhers like De d Chudhuri (986), Goswmi d Chudhuri (99), Giri e l. (996) e. where imeproporiol deeriorio re is osidered. I hs ee empirilly oserved h he filure d life expey of my iems e expressed i erms of Weiull disriuio. his empiril oservio hs eourged reserhers o represe he ime of deeriorio of produ y Weiull disriuio. Ghre d Shrder s (96) model ws exeded y Cover d Philip (97) d oied EOQ model wih vrile re of deeriorio y ssumig wo-prmeer Weiull disriuio. Ler, my reserhers like dikmll (978), Chkrry e l. (998), Mukhopdhyy e l. (, 5) developed eoomi order quiy models. herefore, he re of deeriorio is reed s ime vryig fuio i relisi models. Begum e l. () develop EOQ model for vryig deeriorig iems wih Weiull disriuio deeriorio d priedepede demd. hey ssume h he demd d deeriorio res re oiuous d differeile fuio of prie d ime. Demd plys key role i modelig of deeriorig iveory, reserhers hve reogized d sudied he vriios (or heir omiios) of demd from he viewpoi of rel life siuios. Demd my e os, ime-vryig, sokdepede, prie-depede, e. he os demd is vlid, oly whe he phse of he produ life yle is mured d lso for fiie periods of ime. Wger d Whii (958) disussed he disree se of he dymi versio of EOQ. he lssil o-shorge iveory poliy for lier red i demd ws disussed y Doldso (977). EOQ models for deeriorig iems wih reded demd were osidered y Bhri-Kshi (989), Goswmi d Chudhuri (99, 99), Xu d Wg (99), Kim (995), Jl e l. (996), Jl d Chudhuri (999), Li e l. (), e. My reserh riles y Silver (979), Heery (979), MDold (979),

3 59 R. Begum, S. K. Shu Dve d Pel (98), Sh (98), De d Chudhuri (986), Murdeshwr (988), Hrig (99), e. lyzed lier ime-vryig demd. Ler, Ghosh d Chudhuri (,6), Khr d Chudhuri (), e. eslished heir models wih qudri ime-vryig demd. I he prese pper,we reosider he model of Cover d Philip(97) d exed i o ilude ime-qudri demd re d shorges i iveory. A wo-prmeer Weiull disriuio is osidered o represe he ime o deeriorio. he heory for fidig he opiml soluio of he prolem is developed. A umeril exmple is ke o illusre he soluio proedure. Sesiiviy of he opiml soluio wih respe o hges i differe prmeer vlues is exmied.. Assumpios d Noios of he Model he model is developed usig he followig ssumpios: () he deermiisi demd re D () vries i qudri wih ime. i.e. D where, d re oss. Here ' ' is iiil re of demd, ' ' is he re whih he demd re ireses. () Led ime is zero. () he repleishme is iseous. (d) Shorges re llowed. (e) he holdig os, orderig os, shorge os d ui os remi os over ime. (f) he disriuio of he ime o deeriorio follows wo prmeer Weiull disriuio d he deeriored uis re o repled durig give yle. o develop he mhemil model of he iveory repleishme, he oios doped i his pper is lised elow: K os vlue rryig os per ui per ui ime shorge os per ui per ui ime. orderig os per order os of ui

4 A EOQ Model for Deeriorig Iems Qudri Demd d Shorges 6 legh of he iveory yle. D demd re y is. i.e. where,, re posiive oss. iseous re of deeriorio of he iveory is followed y woprmeer Weiull disriuio. i.e. sle prmeer, shpe prmeer,. Formulio of he Model Le e he iseous iveory level y ime. he iseous se of y ime is desried y he differeil equio () kig () Equio () eomes () he soluio of equio () yields () (5) Usig he odiio i equio (5), we ge (6) he soluio of equio () eomes (7) Expdig equio (6) i ifiie series d iegrig erm y erm, we hve

5 6 R. Begum, S. K. Shu!!! q (8) Usig equio (6) i equio(5), we hve,.. e d e d e Q (9) d, () he iveory level he egiig of he yle mus e suffiie for meeig he ol demd is d. d he ol deeriored iems is q d q. he verge iveory holdig os i, is q. he verge shorge os i, is d 6 herefore, he ol vrile os per ui ime is q q VC 6, () As he legh of he shorge iervl is pr of yle ime, herefore we my ssume, K K ; where K is os o e deermied i opiml mer. Usig equio (7) i equio(); we hve,

6 A EOQ Model for Deeriorig Iems Qudri Demd d Shorges 6 K K K K K K K K K d e K d e K d e K VC K K K 6. herefore, i e wrie s, () Cosiderig s deisio vrile, he eessry odiios for he miimizio of verge sysem os re d () he equio() eomes () d

7 6 R. Begum, S. K. Shu = +! +6 = he opiml vlues of d of re oied y solvig equio () (5) d (5). he suffiie odiios h hese vlues miimize re (6) d Equios () d (5) oly e solved wih he help of ompuer for give se of prmeer vlues y ruig he ifiie series if. Numeril Alysis Equios () d (5) re solved wih he help of ompuer sed ehique usig he followig prmeer vlues: (7) per ui per dy, per ui per dy, per order, per ui, he he opiml yle ime re, opiml vlue eoomi order quiy he vlue of suffiie odiios give i equio (6) d (7). ol verge os I is heked h his soluio sisfies he 5. Sesiiviy Alysis We ow sudy he effes of hges i he vlue of sysem prmeers o he opiml yle ime, he opiml legh of

8 A EOQ Model for Deeriorig Iems Qudri Demd d Shorges 6 iveory d he miimum ol relev os per ui ime. he sesiiviy lysis is performed y hgig eh of he prmeer y 5%,%, % d - 5%, kig oe prmeer ime d keepig he remiig prmeers uhged. he resuls re show i le. O he sis of he resuls of le-, he followig oservios e mde: ) d re ll isesiive o hges i he prmeer. ) d re ifesile owrds he hges i d. ) ; d ireses, u dereses wih irese i d d ireses wih irese i. ) is ifesile owrds he soluio. Wih irese i ; d re isesiive o hges i he prmeer. Chgig Prmeer le- Sesiiviy Alysis % hge % % hge % i he hge hge i sysem prmeer i i

9 65 R. Begum, S. K. Shu - 5 () represes ifesile soluio 6. Colusio A iveory repleishme poliy is developed for deeriorig iems wih imequdri demd. he re of deeriorio is ime-proporiol d he ime o deeriorio is followed y wo-prmeer Weiull disriuio. I he prese pper, we reosider he model of Cover d Philip (97) d exeded i o imedepede demd re d shorges i iveory d lso usig Weiull disriuio. A umeril exmple is ke o illusre he heory. he sesiiviy of he opiml soluio o hges i he prmeer vlues is exmied. From he ove lysis, i is see h d re he riil prmeer i he sese h y error i he esimio of d resulig errors i he opiml resuls. herefore, proper re mus e ke o esime. Agi he ove lysis shows h gre re should e ke o esime he vlue of he prmeer d. Referees []. Bhri-Kshi, H. (989), Repleishme shedule for deeriorig iems wih ime-proporiol demd, Jourl of he Operiol Reserh Soiey,, []. Begum, R, Shoo, R. R., Shu, S. K. d Mishr, M. (), A EOQ model for vryig iems wih Weiull disriuio deeriorio d prie-depede demd, Jourl of Sieifi Reserh, :, -6. []. Chkrry,., Giri, B.C. d Chudhuri, K.S. (998), A EOQ model for iems wih Weiull disriuio deeriorio, shorges d red demd: A exesio of Philip s model, Compuers & Operios Reserh, 5, []. Chug, K.J. d ig, P.S. (99), A heurisi for repleishme of deeriorig iems wih lier red i demd, Jourl of Operiol Reserh Soiey,, 5-. [5]. Cover, R.P. d Philip, G.C. (97), A EO model for iems wih Weiull disriuio deeriorio, AIIE rsios, 5, -6.

10 A EOQ Model for Deeriorig Iems Qudri Demd d Shorges 66 [6]. Dve, U. d Pel, L. K. (98), (, S i ) poliy iveory model for deeriorig iems wih ime-proporiol demd, Jourl of he Operiol Reserh Soiey,, 7-. [7]. De, M. d Chudhuri, K.S. (986), A EOQ model for iems wih fiie re of produio d vrile re of deeriorio, Opserh,, [8]. Doldso, W. A. (977), Iveory repleishme poliy for lier red i demd: lyil soluio, Operiol Reserh Qurerly, 8, [9]. Emmos, H. (968), A repleishme model for rdioive ulide geerors, Mgeme Siee,, 6-7. []. Ghre, P.M. d Shrder, G.F. (96), A model for expoeilly deyig iveories, Jourl of Idusril Egieerig,, 8-. []. Ghosh, S.K. d Chudhuri, K.S. (), A order-level iveory model for deeriorig iem wih Weiull disriuio deeriorio, ime-qudri demd d shorges, Ieriol Jourl of Adved Modelig d Opimizio, 6 (), -5. []. Ghosh, S.K. d Chudhuri, K.S. (6), A EOQ model for wih qudri demd, ime-proporiol deeriorio d shorges i ll yles, Ieriol Jourl of Sysems Siee, 7 (), []. Giri, B. C., Goswmi, A. d Chudhuri, K. S. (996), A EOQ model for deeriorig iems wih ime vryig demd d oss, Jourl of he Operiol Reserh Soiey, 7, []. Giri, B.C., Chkrry,. d Chudhuri, K.S. (999), Reiler s opiml poliy for perishle produ wih shorges whe supplier offers ll-ui quiy d freigh os disous, Proeedigs of Niol Ademy of Siees, 69 (A), III, 5-6. [5]. Goswmi, A. d Chudhuri, K. S. (99), A EOQ model for deeriorig iems wih shorges d lier red i demd, Jourl of he Operiol Reserh Soiey,, 5-. [6]. Goswmi, A. d Chudhuri, K.S. (99), Vriio of order-level iveory models for deeriorig iems, Ieriol Jourl of Produio Eoomis, 7, -7. [7]. Hrgi, M.A. (99), he iveory repleishme prolem wih oiuous lier red i demd, Compuers d Idusril Egieers,, -5.

11 67 R. Begum, S. K. Shu [8]. Heery, R.J. (979), Iveory repleishme poliy for iresig demd, Jourl of Operiol Reserh Soiey, 6, [9]. Jl, A.K. d Chudhuri, K.S. (999), Sruurl properies of iveory sysem wih deeriorio d reded demd, Ieriol Jourl of Sysems Siee, (6), []. Jl, A.K., Giri, R.R. d Chudhuri, K.S. (996), EOQ model for iems wih Weiull disriuio deeriorio, shorges d reded demd, Ieriol Jourl of Sysem Siee, 7(9), []. Khr, S. d Chudhuri, K.S. (), A oe o order level iveory model for deeriorig iem wih ime-depede qudri demd, Compuers Operios Reserh,, []. Kim, K.H. (995), A heurisi for repleishme of deeriorig iems wih lier red i demd, Ieriol Jourl of Produio Eoomis, 9, []. Li, C.,, B. d Lee, W.C. (), A EOQ model for deeriorig iems wih ime-vryig demd d shorges, Ieriol Jourl of Sysems Siee, (), 9-. []. MDold, J.J. (979), Iveory repleishme poliies- ompuiol soluios, Jourl of Operiol Reserh Soiey,, [5]. Misr, R. B. (975), Opimum produio lo size model for sysem wih deeriorig iveory, Ieriol Jourl of Produio Reserh, [6]. Mukhopdhyy, S., Mukherjee, R.N. d Chudhuri, K.S. (), Joi priig d orderig poliy for deeriorig iveory, Compuers d Idusril Egieerig, 7, 9-9. [7]. Mukhopdhyy, S., Mukherjee, R.N. d Chudhuri, K.S. (5), A EOQ model wih wo-prmeer Weiull disriuio deeriorio d priedepede demd, Ieriol Jourl of Mhemil Eduio I Siee d ehology, 6, 5-. [8]. Murdeshwr,.M. (988), Iveory repleishme poliy for lierly iresig demd osiderig shorges- lyi opiml soluio, Jourl of Operiol Reserh Soiey, 9,

12 A EOQ Model for Deeriorig Iems Qudri Demd d Shorges 68 [9]. Philip, G.C. (97), A geerlized EOQ model for iems wih Weiull disriuio deeriorio, AIIE rsio, 6, []. Sh, R.S. (98), O (, S i ) poliy iveory model for deeriorig iems wih ime proporiol demd, Jourl of Operiol Reserh Soiey, 5, -9. []. S, S. d Chudhuri, K.S. (), O volume flexile produio poliy for deeriorig iem wih sok-depede demd re, Nolier Pheome i Complex sysems, 7:, []. S, S., Goyl, S.K., d Chudhuri, K.S. (), A produio-iveory model for deeriorig iem wih reded demd d shorges Europe Jourl of Operio Reserh, 57, []. Silver, E. A. (979), A simple iveory deisio rule of lier red i demd, Jourl of Operiol Reserh Soiey,, []. dikmll, P.R. (978), A EOQ iveory model for iems wih gmm disriuio, AIIE rsios,, 8-. [5]. Wger, H. M. d Whii,. M. (958), Dymi versio of he eoomi lo size model, Mgeme Siee, 5 (), [6]. Xu, H. d Wg, H.P. (99), A eoomi orderig poliy for deeriorig iems wih ime proporiol demd, Europe Jourl of Operio Reserh, 6, -7. BIOGRAPHICAL NOES: Dr. Rehe Begum is urrely Assis Professor i Assis Professor i Mhemis i C.V.Rm College of Egieerig, Bhueswr, Odish, Idi. She go her M.S degree i Mhemis d M.Phil degree i Operiol Reserh from Smlpur Uiversiy, Odiss. She oied her PhD degree from Berhmpur Uiversiy. Her reserh ieress re i he field of lysis of deeriorig iveory produio sysem, produio iveory orol, opimizio d osmology. She hs pulished riles i repued jourls like Ieriol Jourl of Sysems Siee, Briish Jourl of Applied Siee d ehology, Jourl of Sieifi Reserh d Applied Mhemil Siees. Dr. Sudhir Kumr Shu is Leurer i he Deprme of Sisis, Smlpur Uiversiy, Oriss, Idi. He did his MS i Sisis, Meh i Compuer Siee d PhD i Operiol Reserh from Ukl Uiversiy, Oriss. He hs so fr guided PhD heses i he res of Compuer Siee d Iveory Corol.He hs pulished severl ppers o iveory modellig i iol d ieriol jourls of repue.