Analysis of Constant Deteriorating Inventory Management with Quadratic Demand Rate

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1 Amerin Journl of Operionl Reserh, (): 98- DOI:.9/j.jor.. Anlysis of onsn Deerioring Invenory Mngemen wih Qudri Demnd Re Goind hndr Pnd,*, Syji Shoo, Prv Kumr Sukl Dep of Mhemis,Mhvir Insiue of Engineering nd Tehnlogy,Odish, Indi Pnhy Smii ollege, Koksr, Klhndi, Odish, Indi Asr This pper invesiges invenory-produion sysems where iems follow onsn deeriorion. The ojeive is o develop n opiml poliy h minimies ol verge os. The qudri demnd ehnique is pplied o onrol he prolem in order o deermine he opiml produion poliy, holding os nd os of deeriorion. Sensiiviy nlysis is ondued o sudy he effe of he os prmeers on he ojeive funion. Keywords Produion, Invenory, Deeriorion, Shorge, Qudri demnd. Inroduion The purpose of he presen pper is o give new dimension o he invenory lierure on ime vrying demnd perns. Reserhers hve exensively disussed vrious ypes of invenory models wih liner rend (posiive or negive) in demnd. The min Limiion in liner ime-vrying demnd re is h i implies uniform hnge in he demnd per uni ime. This rrely hppens in he se of ny ommodiy in he mrke. In reen yers, some models hve een developed wih demnd re h hnges exponenilly wih ime. Demnds for spre prs of new eroplnes, ompuer hips of dvned ompuer mhines, e. derese very rpidly wih ime. Some modellers sugges h his ype of rpid hnge in demnd n e represened y n exponenil funion of ime. The presen uhors feel h n exponenil re of hnge in demnd is exrordinrily high nd he demnd fluuion of ny ommodiy in he rel mrke nno e so high.a relisi pproh is o hink of elered growh (or deline) in he demnd re in he siuions ied ove nd i n e es represened y qudri funion of ime. Thus, his pper hs he sope of dire ppliion in he very pril siuions noed ove. Goods deeriore nd heir vlue redues wih ime. Eleroni produs my eome osolee s ehnology hnges. Fshion ends o depreie he vlue of lohing over ime. Beries die ou s hey ge. The effe of ime is even more riil for perishle goods suh s foodsuff nd igrees. The effe of deeriorion nd ime/ge is h he lssil invenory model hs o e redjused K. Heng, J. * orresponding uhor: prvsukl@gmil.om (Prv Kumr Sukl) Pulished online hp://journl.spu.org/jor opyrigh Sienifi & Ademi Pulishing. All Righs Reserved Ln, R. Linn () In generl, deeriorion is defined s dey, dmge, spoilge, evporion, osolese, pilferge, loss of uiliy or loss of mrginl vlue of ommodiy h resuls in derese of usefulness from he originl one. The derese or loss of uiliy due o dey is usully funion of he on-hnd invenory. I is resonle noe h produ my e undersood o hve lifeime, whih ends when uiliy rehes ero. The oninuously deying/deeriorion of iems is lssified s ge-dependen ongoing deeriorion, nd ge-independen ongoing deeriorion. Blood, fish, srwerry re some of he exmples of he former while lohol, gsoline nd rdioive hemil nd grin produs re exmples of he ler H. Wee (). 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 horion. 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 The pproh of Donldson (8), Murdeshwr (,)Shu nd Sukl () hs ried o derive n ex soluion for fin ie horion invenory model o oin he opiml numer of replenishmens, opiml replenishmen imes nd he opiml imes whih he invenory level flls o ero, ssuming he demnd re o e linerly ime dependen nd shorges. Hmid () presened heurisi model for deermining he ordering shedule when invenory iems re suje o deeriorion nd demnd hnges linerly over ime nd oined n opiml replenishmen yle lengh. Goswmi nd hudhuri () presened n EOQ model for deerioring

2 99 Amerin Journl of Operionl Reserh, (): 98- iems wih shorge nd liner rend in demnd. Brdshw nd Errol (), pulished pper in whih hey derived unounded onrol poliies for lss of liner ime invrin produion-invenory sysems. This pper invesiges invenory-produion sysems where iems follow onsn deeriorion. The ojeive is o develop n opiml poliy h minimies he os ssoied wih invenory nd produion re. The qudri demnd ehnique is pplied o onrol he prolem in order o deermine he opiml produion poliy. Sensiiviy nlysis is ondued o sudy he effe of he os prmeers on he ojeive funion.. Assumpions nd Noions The following ssumpions nd noions hve een used in developing he model. (i) The demnd re is ssumed o e R,, nd re onsns. Suh h >, >, >. Here snds for he iniil demnd re nd for he posiive rend in demnd. (ii) The produion re Sy k R, where >. A frion, < < of he on-hnd invenory deeriores per uni ime. (iii) The led-ime is ero nd shorges re no llowed. per uni ime nd uni (i v) Uni holding os deeriorion os per uni ime re known nd onsns. (v) is he ol verge os for he produion yle. Mhemil Formulion nd Soluion Le q e he invenory level ny ime ) (. The differenil equions governing he sysem in he inervl, ) re ( q k R, d q R(), d () () The sok level iniilly is ero. Produion egins jus fer, oninues up o nd sops s soon s he sok level eomes S. Then he invenory level dereses due o demnd nd deeriorion oh ill i eomes ero. The yle hen repes iself. Our ojeive is o deermine he opimum vlues of S,, nd. The inensiy of deeriorion is very low iniilly u i inreses wih ime. However, i remins ounded for >> Using he vlue of R, he wo equions () nd () ke he form q d q, d ( ), () nd () The soluion of equion () wih iniil ondiions is qe ( ) ( )( )d nd S is he sok level rehed in he yle. or (vi) The se up os is no onsidered in his model euse i is ken o e fixed for he whole yle ime. (vii) Plnning horion is finie. q () Negleing he powers of greer hn. Similrly, he soluion of equion () lso is (negleing he powers of greer hn ) q exp ( ) exp d or q () For, q S S (7) From () nd (7) we ge he relion q S

3 Goind hndr Pnd e l.: Anlysis of onsn Deerioring Invenory Mngemen wih Qudri Demnd Re (8) Using he ondiion q for in equion (8), we ge S (9) Now he verge holding os eomes d q d q H d ( S Now susiuing he vlue of S from (9) nd simplifying we ge H 8 ϑ () The verge os due o deeriorion in he ol yle ime is d d d () From () nd () he ol verge os of he invenory I 8 8 )

4 Amerin Journl of Operionl Reserh, (): 98- By puing (where <<) in equion (), we ge ( ) ( ) ( ) () ( ) 8 8 ( ( ) ( ) ( ) () For luling he opimum vlue of we differenie i prilly wih respe o nd eque hem o ero. Thus we ge he following equion:- d d ( ) ( ) ( ) ) () )) This equion gives us he opimum vlue of whih, when susiued equion (), give he ol verge os, d provided >. Equion () is highly non-liner in d nd nno e solved nlyilly. This equion, herefore, n e solved y some suile numeril mehod like Newon-Rphson, nd opiml vlue of n e oined. This opiml vlue of gives he minimum os of he sysem in quesion. We hve solved his equion on ompuer for se of vlues of he prmeers wih he help of Newon-Rphson mehod. A numeril exmple is given elow s n illusrion... Exmple- Le., ɣ.,.7,.,,,, in suile unis. The soluion for opiml vlues of nd is.,., whih gives minimum verge os * 9. Following re numer of les represening he opiml vlues of, nd s lso he no-produion inervl.. Sensiiviy Anlysis We hve disussed he effes of he differen prmeers. (i) Inrese in he vlue of dereses he vlue of, nd. (ii) Inrese in he vlue of he prmeer, derese he vlues of, nd. (iii) Inrese in he vlue of holding os inreses he vlue of he os, nd dereses he vlue of. (iv) Inrese in he vlue of deeriorion os inreses he vlue of he os nd. However he vlues of derese. Inrese in he vlue of, dereses he vlue of,,. Inrese in he vlue of, inreses he vlues of, nd. (v) Inrese in he vlue of, dereses he vlue of,,. (vi) Keeping hese vriions in mind of he deision mker of he invenory sysem n onrol he prmeers so s o opimie he ojeive funion. The deision mker my onrol priulrly he holding os nd he os of deeriorion for minimiing he ol verge os.

5 Goind hndr Pnd e l.: Anlysis of onsn Deerioring Invenory Mngemen wih Qudri Demnd Re Tle. Vri ions in prmeers Prme er ɣ onlusions In his rile, deerminisi invenory model hs een proposed for deerioring iem wih qudri demnd re, where shorges re no llowed. The gol of he pper is o inorpore he deeriorion phenomenon ogeher ino n invenory model over finie plnning horion. This pper invesiges invenory-produion sysems where iems follow onsn deeriorion. The ojeive is o develop n opiml poliy h minimies ol verge os. The qudri demnd ehnique is pplied o onrol he prolem in order o deermine he opiml produion poliy, holding os nd os of deeriorion. REFERENES [] Goswmi, K. hudhuri, n EOQ model for deerioring iems wih shorges nd liner rend in demnd, J. Opl.

6 Amerin Journl of Operionl Reserh, (): 98- Res. So. (99) -. [] A. Brdshw, Y. Erol, onrol poliies for produion invenory sysems wih ounded inpu, In. J. Sys. Si. (98) [] B. Hmid, Replenishmen shedule for deerioring iems wih ime proporionl demnd. J. Oper. Res. So. (989) 7-8. [] H. Wee, Eonomi produion lo sie model for deerioring iems wih pril k ordering, ompu. Ind. Eng. (99) 9-8. [] K. Heng, J. Ln, R. Linn. An order level for deerioring iems wih pril k ordering, ompu. Ind. Eng. (99) [] T.M. Murdeshwr Invenory replenishmen poliy for linerly inresing demnd onsidering shorges- n opiml soluion. J.OplRes.So.9, (988) 87 9 [7] U. Hiping, H. Wng, An eonomi ordering poliy model for deerioring iems wih ime proporionl demnd, Eur. J. Oper. Res. (99) -7. [8] W.A. Donldson Invenory replenishmen poliy for liner rend in demnd- n nlyil soluion. Opl Res. Q. 8,(977) -7. [9] W.I Zngwill (9) A deerminisi muli-period produion sheduling model wih klogging. Mgm. Si., (9) -9. [] S.K.Shu,P.K.Sukl,A noe for weiul deerioring model wih ime vrying demnd nd pril k-ordering.a ieni Indi,vol,xxxiv M,No,,(8) 7-7.