Production Inventory Model with Weibull Deterioration Rate, Time Dependent Quadratic Demand and Variable Holding Cost

Transcription

1 roduion nvenory Model wih Weiull Deeriorion Re Time Dependen Qudri Demnd nd Vrile Holding Cos BN: R Venkeswrlu GTAM Universiy rngvjhlv@yhoooin M Reddy BVR Engineering College nveensrinu@gmilom This pper presens produion lo size invenory models for deerioring iems wih ime dependen qudri demnd re is ssumed h he deeriorion re follows Weiull disriuion is furher ssumed h he holding os is liner funion of ime nvenory models re developed wihou onsidering shorges The slvge vlue is onsidered while luling he opiml poliies h mximize he revenue of he sysem Numeril exmple is given nd disussed he sensiiviy of hese models Keywords: roduion Qudri Demnd Weiull Holding Cos nvenory Model nroduion Operions Reserh OR ddresses he proess of deision mking in usiness enerprises nd indusries is known h he invenory mngemen sysem is one of he imporn field of sudy in OR The sudy of deerioring iems in invenory sysem hs gined he enion of mny reserhers in his re of reserh The sudy of he invenory of deerioring iems ws opened up y wihin [] n his sudy he disussed he deeriorion of fshion goods he end of presried sorge period Ghre nd hrder [] exended he lssil EOQ formul wih exponenil dey of invenory due o deeriorion nd gve mhemil model of invenory of deerioring iems The lierure is replee wih invenory models for deerioring iems on he sis of demnd vriions nd vrious oher ondiions or onsrins One imporn prolem fed in supply hin mngemen in ody s onex is o onrol he invenory for deerioring iems Usully deeriorion is defined s he dmge spoilge pilferge dryness vporizion e h resul in derese of usefulness of he originl one is elieved h goods deeriore over ime The re of deeriorion depends on he ype of good Eleroni produs my eome solue s ehnology hnges Fshion goods end o depreie he vlue of lohing over ime The effe of ime is even more riil for perishle goods suh s food suffs nd igrees The derese or loss of uiliy due o dey is usully funion of he on-hnd invenory n relisi erms he produ my e undersood o hve life ime whih ends when uiliy rehes zero Hiping nd Wng [7] developed n eonomi poliy model for deerioring iems wih ime proporionl demnd Donldson [8] derived n nlyil soluion o he prolems of oining he opiml numer of replenishmens nd he opiml replenishmen imes of n EOQ model wih linerly ime dependen demnd pern over finie ime horizon Zngwill [9] developed disree-in-ime dynmi progrmming lgorihm o solve n invenory model y llowing he invenory levels o e negive where he demnd pern is ime dependen Following he pproh of Donldson [8] Murdeshwr [6] hs ried o derive n ex soluion for finie horizon invenory model o oin he opiml numer of replenishmens opiml replenishmens imes nd he opiml imes whih he invenory level flls o Zero ssuming he demnd re o e linerly ime dependen nd shorges Hmid [] Kun-hn Wu el [5] presened heurisi model for deermining he ordering shedule when invenory iems re sujes o deeriorion nd demnd hnges linerly over ime nd oined n opiml replenishmen yle lengh Goswmi nd Chudhuri [] presened n EOQ model deerioring iems wih shorge nd liner rend in demnd Brd hw nd Erol [] pulished pper in whih hey derived unounded onrol poliies for lss of liner ime invrin produion invenory sysems All hese works were sed on he ssumpion h he demnd re is eiher liner or exponenil funion of ime everl reserhers rgued h in relisi erms he demnd need no follow eiher liner or exponenil rend is well known h he demnd for spre prs of new ero plnes ompuer hips of dvned ompuer mhines e inrese very rpidly while he demnds for spres of he osolee ero plnes ompuers e derese very rpidly wih ime This ype of phenomen n well e ddressed y invenory models wih qudri demnd re [ie D ; ] The funionl form of ime-dependen qudri demnd explins he elered or rerded growh or deline in he demnd perns whih my rise due o sesonl demnd re Khnr nd Chudhuri [4] One n explin differen ypes of relisi demnd perns depending on he signs of nd Bhndri nd hrm [5] hve sudied single period invenory prolem wih qudri demnd disriuion under he influene of mrkeing poliies Khnr nd Chudhuri [4] hve disussed n order-level invenory prolem wih he demnd re represened y oninuous qudri funion of ime n nd Chudhuri [6] hve developed sok-review invenory model for perishle iems wih uniform replenishmen re nd sok-dependen demnd Ghosh nd Chudhuri [7] hve developed n invenory model for deerioring iem hving n insnneous supply qudri ime-vrying demnd nd shorges in invenory They hve used wo-prmeer Weiull disriuion o represen he ime o deeriorion Venkeswrlu nd Mohn [8] hve developed invenory models for deerioring iems wih ime dependen qudri demnd nd slvge vlue Venkeswrlu

2 6 Foureenh AM nernionl Conferene on Mngemen nd Mohn [9] sudied invenory model for ime vrying deeriorion nd prie dependen qudri demnd wih slvge vlue Venkeswrlu nd Reddy [] developed ime dependen qudri demnd invenory model under inflion Venkeswrlu nd Reddy [] sudied invenory models when he demnd is ime dependen qudri demnd nd he dely in pymens is eple Begum e l [5] developed n EOQ model wih shorges for deerioring iems wih Weiull disriuion nd uni produion os wih qudri demnd in ime They hve furher ssumed h he produion os is inversely proporionl o he demnd re Klm e l [6] lso developed lo-size invenory model for deerioring iems wih Weiull disriuion qudri demnd nd shorges Thus in his pper i is proposed o develop invenory models for deerioring iems whih follow Weiull disriuion vrile holding os nd ime dependen qudri demnd re is furher ssumed h he slvge vlue o opimize he ol revenue of he sysem Numeril exmple is given o es he rousness of he model ensiiviy nlysis is rried ou o deermine he mos sensiive prmeers in he model Assumpions nd Noions The demnd re is ssumed o e D where nd eing onsns is he nvenory level ime The led-ime is Zero nd shorges re llowed 4 lnning horizon is finie 5 The produion re sy K D where > 6 The frion of he on-hnd invenory deeriores per uni ime where < < > nd 7 The produion domines demnd nd deeriorion during he ime o nd he nvenory level umules 8 There is no produion during he ime o nd demnd nd deeriorion domine nd so he invenory level grdully deplees o zero 9 Holding os is liner funion of ime h is he Deerioring os per uni ime is he slvge vlue ssoied wih deeriored unis during yle ime r is he selling prie per uni T is he presried ime period Formulion nd oluion of he Mhemil Model The ojeive of he model is o deermine he opimum profi for iems hving ime dependen qudri demnd nd he re of deeriorion follows Weiull disriuion is ssumed h he produion domines demnd nd deeriorion during he ime o Furher i is ssumed h here is no produion during he ime o nd demnd nd deeriorion domine so h he invenory level grdully deplees o zero he end f e he invenory level ime he differenil equions whih desries he invenory level ime re given y d θ K R d θ d θ R d Where K R R nd θ Our ssumpions imply h nd The soluion of equion wih he ondiion is 4

3 Foureenh AM nernionl Conferene on Mngemen 6 Here he higher powers of is negleed s he vlue is so smll Now he soluion of equion is e 5 where is onsn of inegrion ine when nd from equion 5 we hve 6 One gin he higher powers of is negleed The ol os TC is given y TC OC HC DC V 7 where OC-ordering os HC- holding os DC-deeriorion os nd V-slvge os Now The Ordering os A nvenory holding os per uni is given y d h HC d h d h Where d d And d d The deeriorion os is given y d d DC

4 64 Foureenh AM nernionl Conferene on Mngemen 4 lvge vlue is given y V CD The les Revenue of he sysem is given y r d d R r Thus he Tol rofi of he sysem is TC R T d d A T r d d A T r 8 The opimum vlue of nd re oined y solving 9 The following ondiions re neessry nd suffiien o mximize he Tol rofi per uni ime < < And >

5 Foureenh AM nernionl Conferene on Mngemen 65 Using MATHCAD he ove equions 9 re solved for opimliy 4 Numeril Exmple To es he vlidiy of he model he following vlues in suile unis re ssumed for vrious prmeers in he model: A 5 T 5 5 r The opimliy ondiions given y equions nd re sisfied in ll ypes of demnd perns ie elered growh rerded growh elered deline nd rerded deline in demnd models Tle- shows he resuls of vrious models is oserved h he ehvior of hese models is similr Tle Model Aelered Growh Model Rerded Growh Model Aelered Deline Model Rerded Deline Model ensiiviy Anlysis We now sudy sensiiviy of he models developed o exmine he impliions of underesiming nd overesiming he prmeers individully nd ll ogeher on opiml vlue of ol profi The ensiive nlysis is performed y hnging eh of he prmeer y -5% -% % nd 5% king one prmeer ime nd keeping he remining prmeers re unhnged nd finlly ll prmeers re onsidered ine ll models show similr resuls we will presen only he sensiiviy for elered growh model The resuls re shown in Tle- The following oservions re mde from his le: The profi funion of he sysem inreses dereses wih n inrese derese in he vlues of he prmeers nd r while i dereses inreses wih inrese derese in he vlues of he prmeers nd However he profi is highly sensiive o he hnges in he vlues of he prmeers nd r moderely sensiive o he hnges in nd nd slighly sensiive o he hnges in he vlues of he prmeers nd As expeed he inrese derese in he vrile holding os dereses inreses in he vlue of he profi funion of he sysem 4 imilrly he inrese derese in he slvge vlue inreses derese he profi of he sysem 5 The profi funion of he sysem is highly sensiive o he hnges in he vlues of ll prmeers ken ogeher in he model The vriions of wih respe o he vlues of some imporn prmeers is shown in Figure he vriions of wih respe o nd re shown in Figure- nd Figure- respeively % -5% % 5% % - -4 r -6-8 Figure

6 66 Foureenh AM nernionl Conferene on Mngemen 5 5 Figure prmeer Figure Tle f > > > % Chnge hnge% in rmeer % % %

7 Foureenh AM nernionl Conferene on Mngemen 67 r All Referene A Brdshw Y Erol Conrol oliies for roduion nvenory ysems wih Bounded npu nernionl Journl of ysem iene 98: Whiin TM 957 Theory of nvenory Mngemen nd ed rineon NJ: rineon Universiy ress Ghre M hrder GF 96 A model for n exponenilly deying invenory Journl of ndusril Engineering Vol B Hmid Replenishmen hedule for Deerioring ems wih Time roporionl Demnd Journl of Operionl Reserh oiey 4989: Khnr nd K Chudhuri A noe on order-level invenory model for deerioring iem wih imedependen qudri demnd Compuers nd Operions reserh Vol pp RM Bhndri nd K hrm A single period invenory prolem wih qudri demnd disriuion under he influene of Mrke poliies Eng iene Vol No pp hishnkr n nd KChudhry 4 A ok-review EOQ Model wih ok-dependen Demnd Qudri Deeriorion Re Advned Modeling nd Opimizion vol6 No pp 5-8 K Ghosh nd K Chudhuri 4 An order- level invenory model for deerioring iem wih Weiull Deeriorion Time-qudri demnd nd shorges Advned Modeling nd OpimizionVol6 No pp-5 9 RVenkeswrlu nd R Mohn nvenory Models for Deerioring ems wih Time Dependen Qudri Demnd nd lvge Vlue nernionl Journl of Applied Mhemil ienesvol5no- pp-8 RVenkeswrlu nd R Mohn An nvenory Model for Time Vrying Deeriorion nd rie Dependen Qudri Demnd wih lvge Vlue ndin Journl of Compuionl nd Applied mhemis Vol No pp- 7 RVenkeswrlu nd M Reddy 4 Time Dependen Qudri Demnd nvenory Models Under nflion Glol Journl of ure nd Applied Mhemis Volume Numer 4 pp RVenkeswrlu nd M Reddy 4 Time Dependen Qudri Demnd nvenory Models when Dely in ymens is Aeple nernionl Journl of Modern Engineering Reserh Vol 4 pp 6-7

8 68 Foureenh AM nernionl Conferene on Mngemen K Wu L Y Ouyng CT Yng An Opiml Replenishmen oliy for Non-insnneous Deerioring ems wih ok-dependen Demnd nd ril klogging nernionl Journl of roduion Eonomis 6 : TM Murdeshwr nvenory replenishmen poliy for linerly inresing demnd onsidering shorges n opiml soluion Journl of operionl reserh soiey 9988: U Hiping H wng An Eonomi ordering poliy model for deerioring iems wih ime proporionl demnd Europen Journl of operionl reserh46 99:-7 6 WA Donldson nvenory replenishmen poliy for liner rend in demnd n nlyil soluion Operionl reserh Querly8 977: W Zngwill A deerminisi muli period produion sheduling model wih klogging Mngemen siene 966:5-9 8 R Begum K hu nd RRhoo An EOQ Model for Deerioring ems wih Weiull Disriuion Deeriorion Uni roduion Cos wih Qudri Demnd nd horges Applied Mhemil ienes Vol4 No A Klm D ml K hu nd M Mishr A roduion Lo-size nvenory Model for Weiull Deerioring em wih Qudri Demnd Qudri roduion nd horges Vol No Jnury-June pp 59-6