AN EOQ INVENTORY MODEL FOR ITEMS WITH RAMP TYPE DEMAND, THREE-PARAMETER WEIBULL DISTRIBUTION DETERIORATION AND STARTING WITH SHORTAGE

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1 Yugoslav Journal of Opraions Rsarc Volum0 00, Numr, DOI:0.98/YJOR0049J N EOQ INVENORY MODEL FOR IEMS WIH RMP YPE DEMND, HREE-PRMEER WEIBULL DISRIBUION DEERIORION ND SRING WIH SHORGE Sanjay JIN Dparmn of Mamaical Scincs Govrnmn ollg jmr, jmr , Inia rjainsanjay@gmail.com Muks KUMR Dparmn of Mamaics Govrnmn ollg Kisangar, Kisangar , Inia Rciv: Ocor 006 / ccp: Novmr 00 srac: In is prsn papr an invnory mol is vlop wi ramp yp man, saring wi sorag an r paramr Wiull isriuion rioraion. rif analysis of cos involv is carri ou y an xampl. Kywors: EOQ, Wiull isriuion rioraion, Sorag, ramp yp man. MS Sujc lassificaion: 90B05. INRODUION numr of invnory mols wr vlop y rsarcrs assuming man of ims o consan, linarly incrasing or crasing man or xponnially incrasing or crasing wi im. Lar i was xprinc a aov man parns o no prcisly pic man of crain ims suc as nwly launc fasion goos an cosmics, garmns, auomoils c; for wic man incrass wi im as y ar launc ino mark an afr som im i coms consan. In orr o consir man of suc ims, concp of ramp yp man is inrouc. Ramp yp man funcion pics a man, wic incrass up o a crain im afr wic i sailizs an coms consan.

2 50 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims In rcn yars, r is a spa of inrs in suying invnory mols for rioraing ims. Gar an Scrar [5] wr arlis rsarcrs wo inrouc aspc of rioraion in invnory mols; y vlop an invnory mol for xponnially cay in wic invnory is no only pl y man alon u also y irc spoilag, pysical plion or rioraion. fr Gar an Scrar s work, a numr of rsarcrs work on invnory mol for rioraing ims, assuming ra of rioraion o consan an pnn on im. mong s rsarcrs ovr an Pilip [], Misra [3], Elsay an rsi [4], Jalan al [8] us woparamr Wiull isriuion an Pilip [5], akraary al [] us rparamr Wiull isriuion o rprsn im o rioraion. work of rsarcrs wo us ramp-yp man as man funcion an various form of rioraion, for vloping conomic orr quaniy mols is summariz low: Rfrnc Ojcivs onsrains onriuions Limiaions Manal & Pal [], 998 Kun-San & Ouyang [0], 000 Jalan, Giri & auuri [8], 00 Kun-San [], 00 Fining EOQ Fining EOQ Fining EOQ Fining EOQ Ramp yp man, ons. ra of rioraion, Sorag no allow Ramp yp man, ons. ra of rioraion, Sorag allow Ramp yp man, Wiull rioraion, Sorag allow Ramp yp man, Wiull rioraion, Sorag allow, Parial acklogging n approxima Sol n for EOQ is oain n xac Sol n for EOQ is oain EOQ givn y Numrical cniqu EOQ oain for 3 numrical xampls. pproxima Sol n, onsan ra of rioraion onsan ra of rioraion EOQ can oain analyically Mo xplain y numrical xampls. aov al sows a only a fw rsarcrs vlop EOQ mols y aking ramp yp man, rioraion consan / Wiull isriuion an sorag allow / no allow. mong s rsarcrs only [0] oain an xac soluion for EOQ. I is commonly osrv a r ar som ims, wic o no sar rioraing as soon as y ar rciv; insa, rioraion sars afr som im, as y ar acually inclu in sock. For suc ims r-paramr Wiull isriuion can us o rprsn im o rioraion. moivaion in vloping an invnory mol in prsn aricl is o prpar a mor gnral invnory mol, wic inclus r-paramr Wiull isriuion rioraion, incorporaing ramp yp man an saring wi sorag. n xac soluion of vlop mol is oain. Numrical xampl is prsn o illusra ffcivnss of mol.

3 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims 5. SSUMPIONS ND NOIONS mol is vlop unr following assumpions an noaions.. ssumpions invnory sysm is consir ovr an infini im orizon. Sorags in invnory ar allow an ar complly acklogg. Ra of rplnismn is assum o infini. La-im is pracically assum o zro. insananous ra funcion Z for wo-paramr Wiull isriuion is givn y Z wr 0 < << is scal paramr, > 0 is sap paramr; >0 is im of rioraion. From an figur. i is clar a wo paramr Wiull isriuion is appropria for an im wi crasing ra of rioraion only if iniial ra of rioraion is xrmly ig an wi incrasing ra of rioraion only if iniial ra of rioraion is approximaly zro. Ra of < cras > > incrasing Drioraion ra ra consan ra im Figur. Ra of rioraion-im rlaionsip for a wo-paramr Wiull isriuion Howvr, s limiaions can rmov y using r-paramr Wiull isriuion o rprsn im o rioraion. nsiy funcion f for is isriuion is givn y f wr,, ar fin as arlir an is locaion paramr. insananous ra of rioraion of non-riora invnory a im, Z can oain y using rlaion

4 5 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims f Z 3 F Wr F is cumulaiv isriuion funcion for r-paramr Wiull isriuion an is givn y F 4 susiuing valus of f an F from an 4 in 3 an simplifying, w oain Z 5 < 0 0 Incrasing ra >> > 0 Dcrasing ra < 0, < im Figur. Ra of rioraion-im rlaionsip for r-paramr Wiull isriuion From figur. i is clar a r-paramr Wiull isriuion is suial for ims wi any iniial valu of rioraion an also for ims, wic sar rioraing only afr a crain prio of im. I is assum r a ra of rioraion a any im > 0 follows r-paramr Wiull isriuion Z wr,, ar fin as arlir an 0< < is locaion paramr. rason in imposing coniion 0 < < on locaion paramr lis in fac a w ar vloping invnory mol for wic ims sar rioraing afr a sor prio of im, as y ar inclu ino sock.

5 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims 53. Noaions man funcion R is akn o a ramp yp funcion of im: [ H R ] Wr H is wll known Havisi s funcion fin as :, H 0, < iniial man ra, a consan govrning xponnial man ra Invnory mol is vlop o sar wi sorags an only for <. fix lng of ac orring cycl S maximum invnory lvl for ac orring cycl invnory oling cos pr uni pr uni im s sorag cos pr uni pr uni im cos of rioraion for singl uni I on an invnory a im ovr [0, ] Procurmn im 3. DEVELOPMEN OF HE MODEL invnory sysm vlop is pic y following figur Invnory Lvl S im Figur 3. Invnory lvl-im rlaionsip invnory sysm sars wi zro invnory a 0. Sorags ar allow o accumula up o im. im invnory is rplnis. quaniy rciv a is parly us o m sorags wic accumula from im 0 o, laving a alanc of S ims a im. s im passs, invnory lvl S clins only u o man uring prio [, ], an mainly u o man an parly u o

6 54 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims rioraion of ims uring prio [, ]. im invnory lvl graually falls o zro. invnory lvl of sysm a any im ovr prio [0, ] can scri following iffrnial quaions: or I, 0 I, I, I Z I I I, soluions of iffrnial quaions 6, 7, 8 an 9 wi ounary coniions I0 0, I 0 an I S ar implis a I ; 0 0 I ; S ; I I { } ; In quaions an 3 valus of I a S { } { } Now oal numr of ims riora uring [, ] is S Dman uring [, ] soul coinci, wic 4

7 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims 55 or, { } { } By using 4 an R [ H ], w av { } { } { } 5 sorag wic accumula uring [ 0, ] is 0 s 6 invnory ol ovr prio [, ] is { } 7 Nglcing igr orr of oal rlvan cos of sysm uring im inrval [ 0, ] y using 5, 6 an 7 is s s 8 rfor avrag oal cos pr uni im is

8 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims 56 s 9 In orr o minimiz avrag oal cos pr uni of im, opimal valu of no y can oain y solving [ ] 0 0 wic also saisfis coniion [ ] 0 > Equaion 9 ogr wi givs { } { } 0 s Equaion is a nonlinar quaion in, using iraiv numrical mo on can solv i for iffrn valus of various paramrs. suprioriy of Nwon- Rapson mo ovr iscion mo is salis y Wan an u [7]. valiiy for using Nwon-Rapson mo lis in slcing a propr iniial roo so a an xclln convrgn squnc is oain. Hnc Nwon-Rapson mo wi propr slcion of iniial roo is suial o loca xac roo of. Using opimal valu oain y Nwon-Rapson mo minimum avrag oal cos pr uni of im can oain from 9. Furr oal man ackorr R B can oain as:

9 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims 57 R B 0 3 lso invnory lvl a from 4 { } { } S 4 rfor opimal orr quaniy Q using 3 an 4 is Q 5 4. NUMERIL EXMPLE For numrical illusraion of vlop mol, valus of various paramrs can akn as follows: 00 unis, Rs. 5 pr uni, Rs. 3 pr uni pr yar, s Rs.5 pr uni pr yar, yar,.08,.00, Ramp in man. an.5 assuming ra of rioraion o incrasing wi im, locaion paramr {.08,.,.} man funcion R [ H ] will as follows < R, 00, Now using aov aa w fin following rsul for iffrn valus of Q vrag os vrag oal os Drioraion Holing Sorag

10 58 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims I can osrv from al a as im of sar of rioraion of ims afr rplnismn incrass, prio of sorag opimal orr quaniy crass. cos of rioraion also crass as incrass. Sinc valu of crass wi incras in valu of, sorag cos crass; an ims ar ol in invnory for a comparaivly longr prio; r y incrasing oling cos. avrag oal cos of invnory sysm crass wi incras in valu of. 5. ONLUSION invnory mol vlop aov is concrn wi ramp yp man, saring wi sorag an r-paramr Wiull isriuion rioraion. Ramp yp man prcisly rprsns man of a numr of consumr ims of prsn ra; also mol saring wi sorag is a salin faur of vlop mol. W av provi an xac soluion procur for mol; a numrical xampl is also givn in suppor of ory. possil ircion for furr rsarc may o consir siuaion wn >. cknowlgmns scon auor is graful o Univrsiy Grans ommission UG/RO, Inia for awaring acr Rsarc Fllowsip. REFERENES [] akraary,., Giri, B.., an auuri, K.S., n EOQ mol for ims wi Wiull isriuion rioraion, sorags an rn man: n xnsion of Pilip s mol, ompurs an Opraions Rsarc, [] ovr, R.P., an Pilip, G.., n EOQ mol for ims wi Wiull isriuion rioraion, IEE ransacions, [3] Dav, U., an Pal, L.K.,, S i policy invnory mol for rioraing ims wi im proporional man, Journal of Opraion Rsarc Sociy, [4] Elsay, E.., an rsi,., nalysis of invnory sysms wi rioraing ims, Inrnaional Journal of Proucion Rsarc, [5] Gar, P.M., an Scrar, G.P., mol for xponnially caying invnory, Journal of Inusrial Enginring, [6] Goyal, S.K., Economic orring policy for rioraing ims ovr an infini im orizon, Europan Journal of Opraions Rsarc, [7] Jain, S., an Kumar, M., n EOQ invnory mol wi ramp yp man, Wiull isriuion rioraion an saring wi sorag, Opsarc [8] Jalan,.K., Giri, B.., an auuri, K.S., n EOQ mol for ims wi Wiull isriuion rioraion, sorags an ramp yp man, Rcn Dvlopmn in Opraions Rsarc, Narosa Pulising Hous, Nw Dli, Inia, 00, [9] Jalan,.K., Giri, R.R., an auuri, K.S., EOQ mol for ims wi Wiull isriuion rioraion, sorags an rn man, Inrnaional Journal of Sysm Scinc, [0] Wu, K.S., an Ouyang L.Y., rplnismn policy for rioraing ims wi ramp yp man ra, Proc. Na. Sci. ounc. RO,

11 S., Jain, M., Kumar / n EOQ Invnory Mol for Ims 59 [] Wu, K.S., n EOQ invnory mol for ims wi Wiull isriuion rioraion, ramp yp man ra an parial acklogging, Proucion Planning an onrol, [] Manal, B., an Pal,.K., Orr lvl invnory sysm wi ramp yp man ra for rioraing ims, Journal of Inrisciplinary Mamaics, [3] Misra, R.B., Opimum proucion lo siz mol for a sysms wi rioraing invnory, Inrnaional Journal of Proucion Rsarc, [4] Naamias, S., Prisal invnory ory: rviw, Opsarc, [5] Pilip, G.., Gnraliz EOQ mol for ims wi Wiull isriuion rioraion, IIE ransacion, [6] Raafa, F., Survy of liraur on coninuously rioraing invnory mols, Journal of Opraion Rsarc Sociy, [7] Wan, W.J. an u, P., Nwon-Rapson mo for xpc prsn valu of oal invnory coss, Journal of Informaion an Opimizaion Scincs,