EOQ Inventory Models for Deteriorating Item with Weibull Deterioration and Time-Varying Quadratic Holding Cost

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1 ISSN (e): Volume, 06 Issue, 0 Jnury 06 Inernionl Journl of Compuionl Engineering Reserh (IJCER) EOQ Invenory Models for eerioring Iem wih Weiull eeriorion nd ime-vrying Qudri Holding Cos Nresh Kumr Klirmn, Riu Rj, rshlini Chndr, r Hrish Chudhry Reserh Sholr, eprmen of Mhemis & Sisis, Bnshli Universiy, P.O. Bnshli Vidypih, Bnshli 00, Rjshn, Indi, Reserh Sholr, eprmen of Mhemis & Sisis, Bnshli Universiy, P.O. Bnshli Vidypih, Bnshli 00, Rjshn, Indi, Assoie Professor, eprmen of Mhemis & Sisis, Bnshli Universiy, P.O. Bnshli Vidypih,Bnshli 00, Rjshn, Indi Assisn Professor, eprmen of Mngemen Sudies, Indin Insiue of ehnology, elhi, New elhi-006, Indi, Asr his pper develops n EOQ invenory model for deerioringiemswih wo prmeers Weiull deeriorion. Shorges re permissile ndprilly klogged. In his model we onsider ime vrying qudri holding os nd rmp-ype demnd. he model is developed under wo differen replenishmen poliies: (i) Sring wih no shorges (ii) Sring wih shorges.he im of his sudy is o find he opiml soluionominimizing he ol invenory oss for oh he ove menioned sregies. o eleve he model numeril exmple hs een rried ou nd sensiiviy nlysis ourred o sudy he resul of prmeers on essenil vriles nd he enire os of his model. Keywords: Invenory, Pril Bklogging, Qudri Holding Cos, Rmp-ype emnd, Single Iem, Shorges, Weiull eeriorion I. Inroduion An eonomi order quniy model deermines he quniy ompny or reiler mus order o minimize he ol invenory os y lning he invenory holding os nd fixed ordering os. In he usul invenory sysem, i ws onsidered h he uyer pys o vendor s soon s he reeives he goods. Goods deeriore nd heir vluesdereses over ime. Eleroni equipmen my eome ouded s ehnology hnges; fshion rends depreie he vlue of lohes over ime; eries die ou s hey re old. he ouome of ime is even mosimpornfor onsumle goods suh s foodsuff nd drugs.an invenory model for fshion goods deerioring he end of he sorge period ws developed y Whiin [9]. Ghre nd Shrder [7] developed n exponenilly deerioring invenory model. A replenishmen invenory model for rdioive nulide generors presened y Emmons [6].An order- level invenory model for deying iems wih deerminisi re of deeriorion ws developed y Shh nd Jiswl []. he reiler reeives he delivery of goods nd deeriorion srs h momen in ll invenory models for deying iems. In rel life siuion, some usomers would like o wi for klogging during he shorge phse u he ohers would no. herefore, he opporuniy os due o los sles should e onsidered in he invenory models. Murdeswr[8]; Goyl e l. []; Hrig [5];Chkrri nd Chudhuri []; Hrig nd Alyn [6] developed mny invenory models ssuming h shorges re ompleely klogged. Chng nd ye [] onsidered h klogging shorges depends on led ime. An rile in he field of deerioring iems wih shorges hs provided he eonomi order quniy wih known mrke demnd re developed y Wee [8]. Sn [] nd Roy e l. [9, 0, ] developed mny invenory models onsidering pril klogging res. he lengh of he led ime for he nex replenishmen eomes min for for deermining wheher he klogging will e eped or no. A deiled review of deerioring invenory lierures ws provided y Goyl nd Giri []. hey onsidered vrile demnd re for mny invenory models. Silver nd Mel [5] onsidered ime-vrying demnd re in heir invenory models. Open Aess Journl Pge 8

2 EOQ Invenory Models for eerioring Iem wih In reen reserh, Cover nd Philip [] presened n invenory model where he ime o deeriorion is onsidered wih wo prmeer Weiull disriuion deeriorion. Ghosh nd Chudhuri [9] developed n invenory model for wo prmeers Weiull deerioring iems, wih qudri demnd re nd shorges were permissile. Hley nd Higgins [] exended he invenory poliy for wo pr rde redi, where he vendors onsider sh disoun for pying wihin speified period nd due in lrge redi period. Goyl [0] developed he EOQ model under he ondiions of permissile dely in pymens. Aggrwl nd Jggi [] onsidered deerioring iems o develop ordering poliy under he ondiions of permissile dely in pymens.srkr e l. [] developed n EOQ model for deerioring iems wih shorges nd ime vlue of money.eng e l. [7] proposed n EOQ model wih deerioring ojes, prie perepive demnd, nd differen selling nd purhsing pries under rde redi poliy nd disussed he resul oined y Goyl [0]. onldson [5] provided nnlyil soluion proedure of he si invenory poliy for se of posiive liner rend in demnd.giri e l. [8] developed n eonomi order quniy model for single deerioring iem wih rmp ype demnd nd Weiull disriuion deeriorion. Shorge is llowed nd klogged ompleely over n infinie plnning horizon. Skouri e l. [6] provided n invenory model wih ime dependen Weiull deeriorion re, pril klogging of unsisfied demnd nd generl rmp ype demnd re. he model is developed under wo differen replenishmen poliies: () sring wih no shorges nd () sring wih shorges. he klogging re is non-inresing funion of he wiing ime up o he nex replenishmen. Wu [0] presened n eonomi order quniy model wih rmp ype demnd nd Weiull disriuion deeriorion. Shorges re llowed pril klogging nd he re of klogging is dependen on wiing ime for he nex replenishmen. he min purpose of his pper is o show h here exis unique opiml yle ime o minimize he ol invenory os.a numeril illusrionnd sensiiviy nlysisis presened o show he resul of he proposed model. II. Assumpions he following ssumpions re mndory odevelop mhemil model:. he Invenory sysem onsider single iem.. he invenory level defined y I ime. i.e. I I 0, 0 I 0,, where I is reiler s sok level.. he demnd re is rmp ype funion of ime nd is defined s:,, is posiive nd oninuous for ime 0,,. he led ime is zero. 5. Shorgesre llowed nd prilly klogged. 6. Cyle horizon is onsidered s unis of ime nd replenishmen re is infinie. 7. he re of deeriorion is ime dependen, whih is wo prmeers Weiulldisriuion deeriorion, denoedy,where 0, nd 0.A vlue of defines h he filure re dereses wih ime. his hppens if imperfe iems re deerioringfirs nd he filure re dereses wih ime. A vlue of defines h he filure re is deerminisi over ime. A vlue of definesh he filure re inreses wih ime. 8. he ordering quniy level is onsidered s Q. 9. Holding os H C is ime dependen qudri funion nd is defined s H C, where 0, 0 nd 0. III. Noions he following noions re mndory o develop mhemil model: HC : he sok holding os per yer. S : he shorge os per yer. Open Aess Journl Pge 9

3 EOQ Invenory Models for eerioring Iem wih : Lengh of ime is onsidered s nnully. : ime whih invenory eomes zero. C : ol invenory os. LS : Los sle os. I :Invenory level ime. : Rmp-ype demnd funion of ime. :eeriorion os. IV. Mhemil Modelwihou Shorges: he invenory model is sring wihou shorges. A he sr of he yle, he produion srs 0 nd oninuesup o. A his ime heinvenory level rehes is mximum level nd hen produion is sopped. he invenory lessens o zero due o demnd nd deeriorionduring 0, nd flls o zero. hus, shorges our during, whih is prilly klogged.herefore, he invenoryis desried y he followingdifferenil equions: d I d, 0... I wih oundry ondiion d I d I 0,... wih oundry ondiion Where I 0,, n d, here re wo ses rise: i nd ii.. Cse (i) From Eq. (), we hve... I e Eq. (), solved y he following wo wys: d I d I d d We hve,, i,, I I ii I nd..., Open Aess Journl Pge 0

4 I B EOQ Invenory Models for eerioring Iem wih... 5 Where B 6 ol moun of deeriorion during 0, e d d 0 0 ol os of holding during 0, is... 6 H C H I d B C E F 8 0 Where... 7 B, C, E, F he shorge os during is, S I d I d I B Where d B B 6 6 Los sles os during is, L S d d B 6 Where... 9 B 6 Ordering quniy during 0, is O Q e d d d 0 O Q ol os during 0, is he sum of deeriorion os, holding os, shorge os nd los sles os is given y C H C S L S C B C 8 B E F B Open Aess Journl Pge

5 ... Soluion: C C EOQ Invenory Models for eerioring Iem wih B C... E F B C C E... F Min ojeive is o minimize he ol relevn os for he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve B C 0... E F Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. () nd he opiml vlue C of he ol relevn os is deermined y Eq. (). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C he suffiien ondiion is sisfied.... Numeril exmple: Le us onsider Fig.. Grphil represenion of C nd. $ / u n i, $ 0.5 / u n i, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r B 0. 5, B 0., B., B 0.0, C , E 0.0 0, F Open Aess Journl Pge

6 EOQ Invenory Models for eerioring Iem wih hus, he opiml vlue of is 0.. he opiml ordering quniy is OQ 7.5. he minimum relevn os isc SensiiviyAnlysis: o know, how he opiml soluion is ffeed y he vlues of onsrins, we derive he sensiiviy nlysis for some onsrins. he speifi vlues of some onsrins re inresed or deresed y 5%, 5% nd 50%, 50%.hus,we ompuehe vlues of nd C wih he help of inresed or deresed vlues of S nd.he resul of he minimum relevn os is exising in he following le. le: From he resul of ove le, we oserve h ol relevn os nd ordering quniy is muh ffeed y deeriorion os nd shorge os, nd oher prmeers re less sensiive.... When Holding Cos is Qudri Funion: We hve H C H I d e d H C B C E F F F Where B, C, E, F, F, F 5 5 ol os is he sum of qudri holding os, shorge os, deeriorion os nd los sles os during 0, Prmeers S Aul Vlues C H C S L S 50% Inresed 50% eresed 5% Inresed 5 B C 5 8 C 5 5 E F F F B B 6 5% C OQ eresed Soluion: C B C... 8 E F F 5 F 5 Open Aess Journl Pge

7 EOQ Invenory Models for eerioring Iem wih B C C E F... 9 F 5 F 5 Min ojeive is o minimize he ol relevn os of he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve B C E F 5 F F Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. (0) nd he opiml vlue C of he ol relevn os is deermined y Eq. (7). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C 0... he suffiien ondiion is sisfied. C..6. Numeril Exmple: Le us onsider Fig.. Grphil represenion of C nd. $ / u n i, $ 0.5 / u n i,, S $ 0.5 / u n i, $ 5 / u n i, 0.,, ye r, 0 0,, 0.5 ye r B 0. 5, B 0., B., B 0.0, C , E 0.0 0, F , F 0.0 0, F hus, he opiml vlue of is he opimum ordering quniy is OQ 7.5. he minimum relevn os is C SensiiviyAnlysis: o know, how he opiml soluion is ffeed y he vlues of onsrins, we derive he sensiiviy nlysis for some onsrins. he speifi vlues of some onsrins re inresed or deresed y 5%, 5% nd 50%, 50%.hus,we ompuehe vlues of nd C wih he help of inresed or deresed vlues of S nd.he resul of minimum relevn os is exising in he following le. Open Aess Journl Pge

8 Prmeers S Aul Vlues 50% Inresed EOQ Invenory Models for eerioring Iem wih le: 50% eresed 5% Inresed 5% C OQ eresed From he resul of ove le, we oserve h ol relevn os nd ordering quniy is muh ffeed y deeriorion os nd shorge os, nd oher prmeers re less sensiive... Cse (ii): he differenil Eq.() eomes d I I, 0, I I d d I d We hve I,, I 0 I e C Where C, 0... he differenil Eq. () eomes nd I e,... d I d We hve I d,... ol moun of deeriorion during 0, 0 e d e d d S Where, S ol holding os during 0, is H C H I d I d I d 0 0,, I Open Aess Journl Pge 5

9 EOQ Invenory Models for eerioring Iem wih H C C C C C C C C C C Where, C C 5 C C C, C, C C, C, C C 5 6, C, 7 9, C 8 0 he shorge os during is S, I d d... 7 Los sles os during is, L S d... 8 Ordering quniy is during 0, is O Q e d d d ol os is he sum of deeriorion os, holding os, shorge os nd los sles os during 0, C H C S L S C C C C C C 8 C C 5 C C S Open Aess Journl Pge 6

10 ...Soluion: EOQ Invenory Models for eerioring Iem wih C 0 C 9 C 6 C C 7 C C C C 5 C C C C C 7 C C... 8 Min ojeive is o minimize he ol relevn os of he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is C 0, we hve C C C C C C C 8 5 C 0... Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. () nd he opiml vlue C of he ol relevn os is deermined y Eq. (0). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C 0... he suffiien ondiion is sisfied. Fig.. Grphil represenion of C nd. Open Aess Journl Pge 7

11 ... Numeril exmple: Le us onsider EOQ Invenory Models for eerioring Iem wih $ / u n i, $ 0.5 / u n i, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r C 0., S 0., C 0.0, C 0. 8, C 0.0 0, C 0.5 6, C , C 0.0 7, C 0.0 8, C , C hus, he opiml vlue of is 0.5. he opimum ordering quniy is OQ he minimum relevn os is C When Holding Cos is Qudri Funion: ol Qudri holding os during 0, is H C H I d I d I d 0 0 C C C C 7 C 6 H C C C C C C C Where, C C C C C C C , C, 5 5 C, C, C, 5 6 C C 6 6, C, 6 7 8, C, 9 0 C ol os is he sum of qudri holding os, shorge os, deeriorion os nd los sles os during 0, C H C S L S C C C C 6 C C C C C C C C S Open Aess Journl Pge 8

12 EOQ Invenory Models for eerioring Iem wih... Soluion: C C C C 7 C C 6 C 5 C 0 C 8 C C C C C C C C... 8 C 8 C 9 C Min ojeive is o minimize he ol relevn os for he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve C C C C 7 C C C C C C Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. (9) nd he opiml vlue C of he ol relevn os is deermined y Eq. (6). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C he suffiien ondiion is sisfied...5. Numeril exmple: Le us onsider $ / u n i, $ 0.5 / u n i,, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r C 0., C 0.0, C 0.6, C 0. 9, C , C 0.0 7, C 0.0, C , C , C 0.0 5, C hus, he opiml vlue of is 0.5. he opimum ordering quniy is OQ he minimum relevn os is C 8.8. V. Mhemil Model wih Shorges: he yle srs wih shorges h our during he period 0, nd shorges re prilly klogged. Replenishmen grsps he invenory level up o Q fer ime. he invenory level deplees nd flls o zero euse of demnd nd deeriorion during, wo ses our. (i) (ii) Open Aess Journl Pge 9

13 5.. Cse (i) EOQ Invenory Models for eerioring Iem wih herefore, he invenory I is desried y he sysem of differenil equions during 0, d q d q d d, 0, q,... q q q d q q q d From Eq. (), we hve, 0... q From Eq. (), we hve... q e... 5 Where, From Eq. (), we hve q e... 6 Where, ol moun of deeriorion during e e d e d d, Where,... 7 ol holding os during,, is H C q d d H C Where, Open Aess Journl Pge 0

14 EOQ Invenory Models for eerioring Iem wih 5, 6 6 8,,,, , he shorge os during 0, is, S d d he los sles os during 0, is L S d Ordering quniy during 0, is O Q d e e d e d 0 O Q ol os during 0, is he sum of deeriorion os, holding os, shorge os nd los sles os is given y C H C S L S C Open Aess Journl Pge

15 EOQ Invenory Models for eerioring Iem wih 5... Soluion: C C Min ojeive is o minimize he ol relevn os for he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. (55) nd he opiml vlue C of he ol relevn os is deermined y Eq. (5). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C C is he suffiien ondiion is sisfied Numeril Exmple: Le us onsider $ / u n i, $ 0.5 / u n i, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r 0. 6,.0, 0., 0. 5, , 0. 7, , 0.0 5, , , , hus, he opiml vlue of is he opimum ordering quniy is OQ he minimum relevn os is C.9. Open Aess Journl Pge

16 EOQ Invenory Models for eerioring Iem wih 5...When Holding Cos is Qudri Funion: We hve H C q d H C Where, 5,,,, ,,, 5 5, ol os is he sum of qudri holding os, shorge os, deeriorion os nd los sles os during 0, C H C S L S C Soluion:, Open Aess Journl Pge

17 EOQ Invenory Models for eerioring Iem wih C C Min ojeive is o minimize he ol relevn os for he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. (6) nd he opiml vlue C of he ol relevn os is deermined y Eq. (58). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C he suffiien ondiion is sisfied Numeril Exmple: Le us onsider $ / u n i, $ 0.5 / u n i, $ / u n i, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r 0. 6,.0, 0., 0. 5, 0., 0., 0.0 6, 0.0, 0.0, , , , 0.0, hus, he opiml vlue of is he opimum ordering quniy is OQ he minimum relevn os is C Cse (ii) herefore, he invenory I is desried y he sysem of differenil equions during 0, Open Aess Journl Pge

18 d q d q d d EOQ Invenory Models for eerioring Iem wih, 0, q ,,... 6 q q d q q d q From Eq.(6), we hve,, q From Eq.(6), we hve, q 6, FromEq.(65), we hve q e, Where, ol moun of deeriorion during, e e d d ol holding os during is H C q d Where,, F F F F F F F F,, F F, F,, F F, F he shorge os during 0, is Open Aess Journl Pge 5

19 S q d q d 0 EOQ Invenory Models for eerioring Iem wih... 7 he los sles os during 0, is 0 L S d 6 d F F Where, F, F 9 0 Ordering quniy during 0, is O Q d d e e d 0 5 O Q ol os is he sum of deeriorion os, holding os, shorge os nd los sles os during 0, C H C S L S F F 6 0 F 5 6 C... 7 F F F F F F Soluion: F F C F 5 F F F F 7 6 C F F 5 F F F 6 F 7 Min ojeive is o minimize he ol relevn os for he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve Open Aess Journl Pge 6

20 EOQ Invenory Models for eerioring Iem wih F F F 6 F F F F Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. (77) nd he opiml vlue C of he ol relevn os is deermined y Eq. (7). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C he suffiien ondiion is sisfied. C 0.76, Numeril Exmple: Fig.. Grphil represenion of C nd. Le us onsider $ / u n i, $ 0.5 / u n i, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r.0, F , F 0., F , F 0.0 7, F 0.0, F 0. 0, F , F , F hus, he opiml vlue of is he opimum ordering quniy is OQ 0.9. he minimum relevn os is C SensiiviyAnlysis: o know, how he opiml soluion is ffeed y he vlues ofonsrins, we derive he sensiiviy nlysis for some onsrins. he speifi vlues of some onsrinsre inresed or deresed y 5%, 5% nd 50%, 50%.hus,we ompuehe vlues of nd C wih he help of inresed or deresed vlues of Prmeers S S nd.he resul of minimum relevn os is exising in he following le. le: Aul +50% -50% +5% -5% Vlues Inresed eresed Inresed eresed C OQ Open Aess Journl Pge 7

21 EOQ Invenory Models for eerioring Iem wih From he resul of ove le, we oserve h ol relevn os nd ordering quniy is muh ffeed y deeriorion os nd shorge os, nd oher prmeers re less sensiive When Holding Cos is Qudri Funion: We hve H C q d F F F F F F F F F Where, F F, F, F,, F 5 F, F, F, F ol os is he sum of qudri holding os, shorge os, deeriorion os nd los sles os during 0, C H C S L S C F F F F F F 6 F9 F F F F Soluion: Open Aess Journl Pge 8

22 C EOQ Invenory Models for eerioring Iem wih F F F F 6 F 5 F... 8 F 7 F 9 F 8 F F C F F F F 7 F 9 F 8 Min ojeive is o minimize he ol relevn os for he invenory model sring wihou shorges. he C essenil ondiion o minimize he ol relevn os is 0, we hve F F F F F F F F F 9 8 Using he sofwre Mhemi, we n lule he opiml vlue of y Eq. (8) nd he opiml vlue C of he ol relevn os is deermined y Eq. (80). he opiml vlue of sisfy he suffiien ondiion for minimizing ol relevn os C is C he suffiien ondiion is sisfied. C Fig.5. Grphil represenion of C nd. Open Aess Journl Pge 9

23 EOQ Invenory Models for eerioring Iem wih Numeril Exmple: Le us onsider $ / u n i, $ 0.5 / u n i, S $ 0.5 / u n i, $ 5 / u n i, 0.,, y e r, 0 0,, 0.5 y e r.0, F 0.6 8, F 0., F , F 0.0, F 0.0, F 0.0 6, F 0.0, F , F , F hus, he opiml vlue of is 0.8. he opimum ordering quniy is OQ 0.9. he minimum relevn os is C SensiiviyAnlysis: o know, how he opiml soluion is ffeed y he vlues of onsrins, we derive he sensiiviy nlysis for some onsrins. he speifi vlues of some onsrins reinresed or deresed y 5%, 5% nd 50%, 50%.hus,we ompue he vlues of nd C wih he help of inresed or deresed vlues of S nd. he resul of he minimum relevn os is exising in he following le. le: Prmeers Aul Vlues +50% Inresed -50% eresed +5% Inresed -5% eresed S C OQ From he resul of ove le, we oserve h ol relevn os nd ordering quniy is muh ffeed y deeriorion os nd shorge os, nd oher prmeers re less sensiive. VI. Conlusion We presened n order level invenory model for deerioring iems wih wo prmeers Weiull deeriorion. he model developed under wo replenishmen poliies: (i) Wih no shorges nd (ii) Wih shorges whih re prilly klogged. We onsidered rmp-ype demnd re nd ime vrying liner nd qudri holding oss, nd found h ol relevn os wih liner holding os is less hn ol relevn os wih qudri holding os. herefore, he liner ime-dependen holding os is more relisi hn qudri ime-dependen holding os. he proposed model n e exended in numerous wys like permissile dely in pymens, ime vlue of money,quniy disouns e. Aknowledgemen:We re very hnkful o he referees for heir onvenien suggesions. Referenes [] S. P. Aggrwl.nd C. K.Jggi. Ordering poliies of deerioring iems under permissile dely inpymens. Journl of Operionl Reserh Soiey, 6: , 995. []. Chkrri. nd K. S. Chudhuri. An EOQ model for deerioring iems wih liner rend in demndnd shorges in ll yles. Inernionl Journl of Produion Eonomis, 9: 05-, 997. [] H. J. Chng. nd C. Y. ye. An EOQ model for deerioring iems wih ime vrying demnd nd prilklogging. Journl of Operionl Reserh Soiey, 50: 76-8, 999. [] R. P. Cover. ndg. C. Phillip. An EOQ model for deerioring iems wih weiull disriuiondeeriorion.aiie rnsion, 5: -6, 97. [5] W. C. onldson. Invenory replenishmen poliy for liner rend in demnd-n nlyil soluion.operionl Reserh Qurerly, 8: , 977. [6] H. Emmons. A replenishmen model for rdioive nulide generors.mngemen Siene, : 6-7, 968. [7] P. M. Ghre. ndg. F. Shrder. A model for exponenilly deying invenories.journl of IndusrilEngineering, : 8-, 96. [8] B. C. Giri.A. K. Jln.nd K. S. Chudhuri.Eonomi order quniy model wih Weiull deerioriondisriuion, shorge nd rmp-ype demnd. Inernionl Journl of Sysems Siene, (): 7-, Open Aess Journl Pge 50

24 EOQ Invenory Models for eerioring Iem wih [9] S. K. Ghosh. ndk. S. Chudhuri. An order-level invenory model for deerioring iem wih weiulldisriuion deeriorion, ime-qudri demnd nd shorges, Advned Modeling nd Opimizion, 6: -5, 00. [0] S. K. Goyl. Eonomi order quniy model under ondiions of permissile dely in pymen.journl ofoperionl Reserh Soiey, 6: 5-8, 985. [] Goyl e l. he finie horizon rended invenory replenishmen prolem wih shorges.journl ofoperionl Reserh Soiey, : 7-78, 99. [] S. K. Goyl. ndb. C. Giri. Reen rends in modeling of deerioring invenory.europen Journlof Operionl Reserh, : - 6, 00. [] C. W. Hley. ndr. C. Higgin. Invenory poliy nd rde redi finning.mngemen Siene, 0: 6-7, 97. [] M. Hrig. he invenory replenishmen prolem wih oninuous liner rend in demnd.compuers ndindusril Engineering, : -50, 99. [5] M. Hrig. Opiml EOQ models for deerioring iems wih ime-vrying demnd. Journl of OperionlReserh Soiey, 7: 8-6, 996. [6] M. Hrig. nda. Alyn.A lo sizing heurisi for deerioring iems wih shorges in growing nddelining mrkes.compuers nd Operions Reserh, : , 997. [7] M. Hrig. ndl. Benkherouf. Opiml nd heurisi invenory replenishmen models for deerioringiems wih exponenil imevrying demnd.europen Journl of Operionl Reserh, 79: -7, 99. [8]. M. Murdeshwr. Invenory replenishmen poliy for linerly inresing demnd onsidering shorges-n opiml soluion.journl of Operionl Reserh Soiey, 9: , 988. [9] M. Roy., S. Sn.ndK. S. Chudhuri. An EOQ model for imperfe quliy produs wih prilklogging- omprive sudy.inernionl Journl of Servies nd Operions mngemen, 8-0, 0. [0] M. Roy., S. Sn.ndK. S. Chudhuri. An eonomi order quniy model for imperfe quliy iems wihpril klogging.inernionl Journl of Sysems Siene, : 09-9, 0. [] M. Roy., S. Sn.ndK. S. Chudhuri. An opiml shipmen sregy for imperfe iems in sok-ousiuion.mhemil nd Compuer Modeling, in press, 0. [] S. Sn. Opiml selling prie nd lo-size wih ime vrying deeriorion nd pril klogging.appliedmhemis nd Compuion, 7: 85-9, 00. [] B. R. Srkr. e l. An ordering for deerioring iems wih llowle shorge nd permissile dely inpymen.journl of Operionl Reserh Soiey, 8: 86-8, 997. [] Y. K. Shh. ndm. C. Jiswl. An order-level invenory model for sysem wih onsn re of deeriorion.opserh, : 7-8, 977. [5] E. A. Silver. ndh. C. Mel. A simple modifiion of he EOQ for he se of vrying demnd re.produion of Invenory Mngemen, 0: 5-65, 969. [6] K. Skouri, I. Konsnrs, S. Pphrisos, ndi. Gns.Invenory models wih rmp ype demnd re,pril klogging nd Weiull deeriorion re. Europen Journl of Operionl Reserh, 9(): 79-9, 009. [7] J.. eng. e l. Opiml priing nd ordering poliy under permissile dely in pymens.inernionljournl of Produion Eonomis, 97: -9, 005. [8] H. M. Wee. A deerminisi lo size invenory model for deerioring iems wih shorges nd deliningmrke.compuers nd Operions Reserh, : 5-56, 995. [9]. M. Whiin. heory of invenory mngemen.prineon Universiy Press, Prineon, NJ, 6-7, 957. [0] K. S. Wu. An EOQ invenory model for iems wih Weiull disriuion deeriorion, rmp ype demndre nd pril klogging. Produion Plnning & Conrol, (8): , Open Aess Journl Pge 5