WEIBULL DETERIORATING ITEMS OF PRICE DEPENDENT DEMAND OF QUADRATIC HOLDING FOR INVENTORY MODEL

Transcription

1 WEIBULL DEERIORAING IEM OF PRIE DEPENDEN DEMAND OF QUADRAI OLDING FOR INVENORY MODEL. Mohn Prhu Reserh nd Develomen enre, Bhrhir Universiy, oimore-6 6. Leurer, Muhymml ollege of Ars nd iene, Rsiurm, Nmkkl-67 Dr. P. Puhiynygm Assoie Professor, ulkshmi Lkshmihi ollege of iene, Mduri-65. ABRA In his er disuss wih develomen of n invenory model when he deeriorion re follows Weiull disriuions. ere i is ssumed h demnd re is funion of selling rie nd holding os. Wih shorge nd wihou shorge oh ses hve een ken re of in develoing in he invenory models. horges re omleely klogged whenever hey re llowed. he onlusions re illusred wih hel of he numeril exmles. he sensiiviy nlysis for he model hs een erformed o sudy he effe hnges of he vlues of he Invenory model. Keywords: Deerioring iems; EOQ model; Prie deenden demnd; horge; Weiull disriuion. Volume:, Numer:, De- : RJEB Pge

2 . Inroduion he onrol nd minenne invenories for deerioring iems wih shorges hve reeived muh enion of severl reserhers in he reen yers euse mos of he hysil goods deeriore over ime. In reliy, some of he iems re eiher dmged or deyed or ffeed y some oher fors nd is no in erfe ondiion o sisfy he demnd. Food iems, drugs, hrmeuils, rdioive susnes re exmles of suh iems where deeriorion n ke le during he norml sorge eriod of he ommodiy nd onsequenly his loss mus e ken ino oun when nlyzing he sysem. o dey or deeriorion of hysil goods in sok is very relisi feure nd reserhers fel he neessiy o use his for ino onsiderion in develoing invenory models. hkrory nd hudhuri [], Invenory models wih ime deenden re of deeriorion were onsidered y over nd Phili [6], develoed model for n exonenilly deying invenory. An order level invenory model for iems deerioring onsn re ws roosed y Ghre nd hrder [], Burwell [] develoed eonomi lo size model for rie-deenden demnd under quniy nd freigh disouns, hh nd Jiswl, [], Aggrwl [], ome of he signifin reen work in his field hve een done y hung nd ing [5], Dve nd Pel [7], De nd hudhuri [8], Giri nd hudhuri [], hkrory e l., [], Fujiwr [9], Jln nd hudhuri [5], Goh [], rig nd Benkherouf [], Jln e l., [6], Muhlemnn nd Vlis nooulos [9], Mishr [7], rig [], A he eginning, demnd re were ssumed o e onsn whih is in generl likely o e ime deenden nd sok deenden. Invenory model for melioring iems for rie deenden demnd re ws roosed y Mondl, Bhuni nd Mii [8], Weiss [], rihy nd Mishr [], Invenory model wih rie nd ime deenden demnd ws develoed y You,.P.,[5], Wee [], In generl holding os is ssumed o e known nd onsn. Bu in relisi ondiion holding os my no lwys e onsn. o severl reserhers like Vn der Veen [], nd onsidered vrious funions o desrie holding os nd e. In his er, we hve develoed generlized EOQ model for deerioring iems where deeriorion re follows wo-rmeer Weiull disriuion nd holding os re exressed s linerly inresing funions of ime nd demnd re is onsidered o e funion of selling rie. For he model where shorges re llowed hey re omleely klogged. If he holding os is non-liner, similr nlysis is resened for he rend funions. hey hve nlyzed wih shorge nd wihou shorge invenory model for liner rend h h, where, h in holding os nd i is exended y king he holding os h,,,. ere we hve onsidered oh he se of wih shorge nd wihou shorge in develoing he model.. Assumions nd Noions he fundmenl ssumions of he model re s follows: he demnd re is funion of selling rie. horges, whenever llowed, re omleely klogged. he deeriorion re is roorionl of fixed ime. d olding os h er iem er ime-uni is ime ssumed o e h =, where >, >, > onsidered he se of wih shorge nd wihou shorge. e Relenishmen is insnneous nd led ime is zero. f is he lengh of he yle. Volume:, Numer:, De- : RJEB Pge

3 A g he order quniy in one yle is q. h A is he os of ling n order. i he selling rie er uni iem is. j is he uni os of n iem. k he invenory holding os er uni er uni ime is h. l is he shorge os er uni er uni ime. m he deeriorion of unis follows he wo rmeer Weiull disriuion sy where < < is he sle rmeer nd > is he she rmeer. n During ime, invenory is deleed due o deeriorion nd demnd of he iem. ime he invenory eomes zero nd shorges sr ourring. o elling rie follows n inresing rend, nd demnd re ossess he negive derivive hrough ou is domin where demnd re is f = >. Mhemil formulion nd soluion Le Q e he invenory level ime. he differenil equions o desrie insnneous se over, re given y, dq d Q dq, d Wih Q = = olving equion nd equion nd negleing higher owers of α, we ge In equion. dq { } Q d Q e e Q e d d d e d Q e Qe Qe α α α Qe d ] α Volume:, Numer:, De- : RJEB Pge

4 α α Q e ] α Q α ] Q α Volume:, Numer:, De- : RJEB Pge ; α nd in equion dq, d king inegrion on oh sides, dq d d Q Q [ Q Q d ],. Now sok loss due o deeriorion D e d d d α [] α [ ] α D α q D d α d α [] d

5 Volume:, Numer:, De- : RJEB Pge 5 [] α α q olding os is, d du e e u Negleing higher owers of α, we ge d du u. α d u u α d αα αα d αα αα α αα α αα α αα α αα αα αα α αα α αα αα α αα α α α α αα α

6 Volume:, Numer:, De- : RJEB Pge 6 α α α α α α α α α 6. Now shorge os during he yle, d d

7 Volume:, Numer:, De- : RJEB Pge From, nd 5 ol rofi er uni ime is given y,, q A P A, P, 6 6 Le =, < < ene we ge he rofi funion, A P, 6 y 7 Our ojeive is o Mximize he rofi funion P,. he neessry ondiions for mximizing he rofi re, P P We ge,

8 Volume:, Numer:, De- : RJEB Pge ] [ ] [ ] [ 6 ] [ 8 From Equion 8, We n lule he oimum vlues of ] [ ] [ 6 And he oiml vlue P, of he verge ne rofi is deermined y 7 rovided hey sisfy he suffiieny ondiions for mximizing P, is, P,

9 If he soluions oined from equion 8 do no sisfy he suffiieny ondiion. We onlude h no fesile soluion will e oiml for he se of rmeer vlues ken o solve equion 8. uh siuion will imly h he rmeer vlues re inonsisen nd here is some error in heir esimion.. Numeril Exmle. se-i Wih horges Exmle- Le A=, =, =, =., =., =.6, =., α=., =., γ=.5, =.65. Bsed on hese inu d, he omued ouus re s follows: = = =.5785 q = se-ii Wihou horges Exmle- Le A=, =, =, =., =., =.6, α=., =., γ=.5, nd =.998. Bsed on hese inu d, he omued ouus re s follows: = = =.9699 q = le- hnging Prmeer % hnge in ysem hnge in hnge in q hnge in Volume:, Numer:, De- : RJEB Pge 9

10 We sudy from ove le- revels he following i Inrese in he vlues of eiher of he rmeer, will resul in inrese of P, ndq. ii Derese in he vlues of eiher of he rmeer, will resul in derese of P, ndq. iii Inrese in he vlues of eiher of he rmeer α, will resul in derese of P u inrese nd q. iv Derese in he vlues of eiher of he rmeer α, will resul in inrese of P u derese nd q. v Inrese in he vlues of eiher of he rmeer, will resul in inrese of P u derese nd q. vi Derese in he vlues of eiher of he rmeer, will resul in derese of P u inrese nd q. hnging Prmeer % hnge in ysem le- hnge in hnge in hnge in q We sudy from ove le- revels he following i Inrese in he vlues of rmeer, will resul in inrese of P, ndq. ii Derese in he vlues of rmeer will resul in derese of P, ndq. iii Inrese in he vlues of rmeer, will resul in derese of P u inrese nd q. iv Derese in he vlues of rmeer, will resul in inrese of P u derese nd q. v Inrese in he vlues of rmeer, will resul in inrese of P u derese nd q. vi Derese in he vlues rmeer, will resul in derese of P u inrese nd q. 5. ensiiviy Anlysis nd onlusion we hve develoed deerminsi invenory model for deerioring iems for wih shorge nd wihou shorge ses. he deerminsi demnd re is ssumed o e Volume:, Numer:, De- : RJEB Pge

11 funion of selling rie. Whenever shorge re llowed nd hey re omleely klogged nd holding os is ssumed here o e ime vrying. We n mke good omrive sudy eween he resuls of he wih-shorge se nd wihou shorge se. In he numeril exmles, i is found h he oimum verge rofi in wihoushorge se is more hn h of he shorge se. o sudy he effes of hnges of he rmeers on he oiml rofi derived y roosed mehod, sensiiviy nlysis is erformed onsidering he numeril exmle given ove. ensiiviy nlysis is erformed y hnging inresing or deresing he rmeers y % nd 5% nd king one rmeer ime, keeing he remining rmeers heir originl vlues. he resuls re shown in le- nd le- for wih shorge se nd wihou shorge se reseively. REFERENE. Aggrwl,.P., A noe on n order-level invenory model for sysem wih onsn re of deeriorion, Oserh, 5978, Burwell.., Dve D.., Fizrik K.E., Roy M.R., Eonomi lo size model for rie-deenden demnd under quniy nd freigh disouns, Inernionl Journl of Produion Eonomis, 8 997, 55.. hkrri, e l., An EOQ model for iems Weiull disriuion deeriorion shorges nd rended demnd nd exension of Phili s model. omuers nd Oerions Reserh, 5 997, hkrori,., nd hudhuri, K.., An EOQ model for deerioring iems wih liner rend in demnd nd shorges in ll yles, Inernionl Journl of Produion Eonomis, 9 997, hung, K., nd ing, P., A heurisi for relenishmen of deerioring iems wih liner rend in demnd, Journl of he Oerionl Reserh oiey, 99, over, R.P., nd Phili, G.., An EOQ model for iems wih Weiull disriuion deeriorion. AIIE rnsions, 5 97, Dve, U., nd Pel. L.K.,, oliy invenory model for deerioring iems wih ime roorionl demnd, Journl of he Oerionl Reserh oiey. 98, De, M., nd hudhuri, K.., An EOQ model for iems wih finie re of roduion nd vrile re of deeriorion, Oserh, 986, Fujiwr, O., EOQ models for oninuously deerioring rodus using liner nd exonenil enly oss, Euroen Journl of Oerionl Reserh, 7 99, -.. Ghre, P.M., nd hrder, G.F., An invenory model for exonenilly deerioring iems, Journl of Indusril Engineering, 96, 8-.. Giri, B.., nd hudhuri, K.., eurisi models for deerioring iems wih shorges nd ime-vrying demnd nd oss, Inernionl Journl of ysems iene, 8 997, Goh, M., EOQ models wih generl demnd nd holding os funions, Euroen Journl of Oerionl Reserh, 799, rig, M., Oiml EOQ models for deerioring iems wih ime-vrying demnd, Journl of Oerionl Reserh oiey, 7 996, 8-6. Volume:, Numer:, De- : RJEB Pge

12 . rig, M.A., nd Benkherouf, L., Oiml nd heurisi invenory relenishmen models for deerioring iems wih exonenil ime-vrying demnd, Euroen Journl of Oerionl Reserh. 7999, Jln, A.K., nd hudhuri, K.., ruurl roeries of n invenory sysem wih deeriorion nd rended demnd, Inernionl Journl of sysems iene, 999, Jln, A.K., Giri, R.R., nd udhuri, K.., EOQ model for iems wih Weiull disriuion deeriorion shorges nd rended demnd. Inernionl Journl of ysems iene , Mishr, R.B., Oimum roduion lo-size model for sysem wih deerioring invenory, Inernionl Journl of Produion Reserh, 975, Mondl, B., Bhuni, A.K., Mii, M., An invenory sysem of melioring iems for rie deenden demnd re, omuers nd Indusril Engineering, 5, Muhlemnn, A.P. nd Vlis nooulos, N.P., A vrile holding os re EOQ model, Euroen Journl of Oerionl Reserh, 98, -5.. hh, Y.K. nd Jiswl, M.., An order-level invenory model for sysem wih onsn re of deeriorion, Oserh, 997, rihy,.k nd Mishr, U., An invenory model for weiull deerioring iems wih rie deenden demnd nd ime vrying holding os, Alied Mhemil ienes, Vol.,, Vn Der Veen, B., Inroduion o he heory of Oerionl Reserh, Phili ehnil Lirry, ringer-verlg, New York, Wee,.M., A deerminisi lo-size invenory model for deerioring iems wih shorges nd delining mrke, omuers nd Oerions, 995, Weiss,.J., Eonomi Order Quniy models wih nonliner holding os, Euroen Journl of Oerionl Reserh, 7 98, You,.P., Invenory oliy for rodus wih rie nd ime-develomen demnds, Journl of he Oerionl Reserh oiey, 565, Volume:, Numer:, De- : RJEB Pge