esearch Journal of ecen Sciences ISSN - Vol. -9 July es.j.ecen Sci. EO odel wih ime Dependen Holding os under rade redis bsrac ohan.* Pericia Leema oseline Dep of ahemaics ollege of iliary Engineering Pune H INDI Deparmen of ompuer Science hris ollege Pune H INDI vailable online a: www.isca.in www.isca.me eceived h January revised h pril acceped rd July In his paper we have developed a deerminisic invenory model wih consan dem wih variable holding cos under rade redi. Here we have considered shorages are allowed he deerioraion is assumed as funcion of ime. Suiable numerical example sensiiviy analysis is also carried ou a he end. Keywords: onsan dem deerioraion rae shorages ime dependen deerioraion. Inroducion esearchers sudied deerminisic invenory models under uadraic linear exponenially increasing/decreasing dem rae for he producs like elecronic goods food suffs fashionable clohes ec. some invenory models sudied under price dependen dem considering variable deerioraion rae. In business buying invenory wih full capial amoun is no viable. In developing differen ypes of invenory models considering permissible under rade credi is also a worh aemp. esearchers developed rade credis in differen ways. Here rade credi means ha supplier offers he cusomer ase a permissible delay in paymens i.e. rade credis o arac cusomers o increase dem in ase II o moivae quick paymen reduce credi expenses. hough many EO models proposed under permissible delay in paymens we have developed an invenory model wih consan dem considering variable holding cos wih shorages. In his model deerioraion rae is funcion of ime. Hariga developed an invenory model of deerioraing iems for ime-varying dem also considering shorages. hakrabori houdhuri proposed an EO model for deerioraing producs of linear rend in dem wih shorages in all cycles. Giri haudhuri developed an economic order quaniy model for deerioraing iems of ime varying dem coss considering shorages. Goyal Giri sudied complee survey of recen rend in deerioraing invenory models. ondal e. al developed an invenory model of amelioraing producs for price dependen dem rae. You sudied he invenory sysem for he producs wih price ime dependen dems. jana oy proposed an invenory model for deerioraing iems of ime varying holding cos wih wihou shorages in he price dependen dem. ishra Singh sudied an invenory model for deerioraing iems wih ime dependen dem parial backlogging. ishra developed an invenory model wih Weibull rae of deerioraion rae of dem is consan wih variable holding cos considering shorages salvage value 9. ukesh e al proposed a deerminisic invenory model for deerioraing producs wih price dependen dem under rade credis. Vikas Sharma ekha developed an invenory model for ime dependan dem for deerioraing iems wih Weibull rae of deerioraion shorages. Venkaeswarlu ohan sudied an EO model wih - parameer Weibull deerioraion ime dependen quadraic dem salvage value. Venkaeswarlu ohan proposed an EO model for ime varying deerioraion price dependen quadraic dem wih salvage value. ohan Venkaeswarlu developed an EO models wih variable holding cos salvage value. ohan Venkaeswarlu proposed an invenory model for ime Dependen quadraic dem wih salvage considering deerioraion rae is ime dependen. ecenly ohan Venkaeswarlu developed an invenory model wih uadraic Dem Variable Holding os wih Salvage value. In his paper we developed a model wih ime-dependen deerioraion when he dem rae is consan under rade credis. Shorages are allowed. ime horizon is infinie. he opimal oal cos is obained. he sensiiviy analysis is done wih numerical example. he robusness of his model by increasing /deceasing of parameers which gives he objecives from he sensiiviy analysis carried ou a he end. ssumpions noaions he mahemaical model is developed on he following assumpions noaions: he dem rae D a ime is assumed o be D Lead ime is zero eplenishmen rae is infinie is he deerioraion rae < <. he ordering cos h h> > he holding cos per uni he cos per uni I is he invenory level a ime. he order quaniy in one cycle is q. is he shorage cos per uni per order NDU he number of deerioraing unis per order wih one cycle ime. Inernaional Science ongress ssociaion
esearch Journal of ecen Sciences ISSN - Vol. -9 July es.j.ecen Sci. Inernaional Science ongress ssociaion ahemaical odel he differenial equaion which governs describes he insananeous invenory level a ime is given by Figure- ime - Invenory graph I d d d d Wih he iniial condiion a Using he iniial condiion soluion of equaion is given by e K e K Using he condiion we ge he value of K. Subsiuing he value K in he above equaion we ge { } Using we obain he soluion of equaion using he boundary condiion is Invenory odel wih Shorages he oal cos of he sysem consiss of he following coss: Ordering cos O Invenory holding cos per cycle IH d h 9 h IH he number of unis ha deerioraed during his cycle ime is D where d D NDU NDU os due o deerioraion *D D Shorage cos S d S 9 Since we are considering he invenory sysem under rade credis he following wo cases arises: ase i: Here he paymen on or before when invenory reaches o zero In case-i he credi period ended on or before when he invenory level reaches o zero. fer due dae he ime horizon for he ineres payable is for he invenory which are no sold afer he due dae. Figure- ime invenory graph for case i Hence
esearch Journal of ecen Sciences ISSN - Vol. -9 July es.j.ecen Sci. Inernaional Science ongress ssociaion 9 E d E IE c d c I oal verage os is given by I I S O D IH 9 a a a a h E c he necessary condiion for minimizing he oal cos is ie. *. Ear a h. a a a h Provided > where a h. h. h
esearch Journal of ecen Sciences ISSN - Vol. -9 July es.j.ecen Sci. Inernaional Science ongress ssociaion 9 ase ii: > In his case Ineres payable per cycle is zero when Figure- ime invenory graph for case ii oal verage os is given by IE S O D IH 9 h he necessary condiion for minimizing he oal cos is i.e. * a h. h Provided > Numerical Example for case I: By puing proper unis of parameric values for a.9. Ie.9. h.9... Ic. Using HD Sofware he opimal values he invenory sysem are:..9. Numerical Example for case II: a. Ie.9. h.9... Ic. Sensiiviy nalysis We will sudy he effec of changes made in he values of he parameers a γ h on he opimal cycle ime oal cos EO of hese models. We have he following inferences can be made from able- able using case I II Decrease in he parameers a decreases he oal opimal sysem cos. Similarly he increase in hese parameers increases he oal opimal cos. he effec of hese parameers is more pronounced on he oal cos of he sysem. he effec of he parameer on he oal opimal cos is very significan. he oal opimal cos of he sysem is very significan by considering all variables ogeher decreasing or increasing. onclusion We have developed invenory managemen model for deerioraing iems when he dem rae is assumed o be consan. I is assumed ha he deerioraion rae is proporional o ime. We have solved he model considering shorages.he objecives are discussed in he sensiiviy analysis for he various parameers he sensiiviy analysis is calculaed in percenile variaion raher han acual values from he ables. his model can be exended furher incorporaing inflaion using fuzzy conceps.
esearch Journal of ecen Sciences ISSN - Vol. -9 July es.j.ecen Sci. able- ase : Parameer % change Θ -.9. -. -.9.9 -. -.9 -.9. -.9 -.. - -. -. -. - -. -. -.9...99... -..9 -.9 -.9. -.9 -. -.. -9.9 -.9. Α -.9.9 -. -.9. -. -.9 -..9 -. -..9 -.9. -. -.9. -. -.9 -.. -.9 -.9. Π - -.9. -. - -.9.999 -.9 -..9 -. -.. -. H -.. -.99 -.. -.9 -. -.9. -. -..9 ll parameers eferences - -..9 -. -..9 -. 9. -..9 -. -. 9.9. Hariga. n EO model for deerioraing iems wih shorages ime-varying dem Jour. of Operaional es Socy 9-99. hakrabori. haudhuri K.S. n EO model for iems wih linear rend in dem shorages in all cycles In. Jour of Producion Eco 9-99. Giri B.. haudhuri K.S. Heurisic model for deerioraing iems wih shorages In. Jour of Sysem Science -9 99. Goyal S.K Giri B.. ecen rends in modeling of deerioraing invenory European Jour of Ops resh -. ondal B. Bhunia.K aii. n invenory able- ase : > Parameer % change -.9. -. -.9. -.9 -.. -. -..9 - -. -. -. - -. -.9 -.999.9..9... a -.99. -9.9 -.9. -. -.9 -.99. -. -.. -.9.9 -.9 -.. -. -. -.. -.9 -..9 - -.9.9.99 - -...99.99 -. -.9. -. -. - -..9 -.9 - -.. -..9 -... -9. 9.999 h -. -. -. -. -.99 -. -..99. -9... ll parameers -.9. -. - -. -.9 -. -. -.99. -.9 -.9. sysem of amelioraing iems for price dependen dem rae omps Inds Engg -. You S.P. Invenory policy for producs wih price ime dependen dems Jour of Oper es Soc -. jana oy n Invenory model for deerioraing iems wih price dependan dem ime-varying holding cos O-dv modeling opim. ishra V.K. Singh L.S. Deerioraing invenory model wih ime dependen dem parial backlogging pp ah Sci. -9 9. Vinod Kumar ishra Invenory model for ime dependen holding cos deerioraion wih salvage value shorages he Jour of ah omp Sci. -. ukesh Kumar e al. Deerminisic invenory model Inernaional Science ongress ssociaion 9
esearch Journal of ecen Sciences ISSN - Vol. -9 July es.j.ecen Sci. for Deerioraing Iems wih price Dependen Dem ime Varying Holding os under rade redis Inernaional Journal of Sof ompuing EngineeringIJSE. Vikas Sharma ekha ani haudhuri n invenory odel for deerioraing iems wih Weibull Deerioraion wih ime Dependen Dem shorages es Jour of anagemen Sci. -. Venkaeswarlu ohan. n Invenory odel wih Weibull Deerioraion ime Dependen uadraic Dem Salvage Value IS - Proceedings Bangalore a. Venkaeswarlu ohan. n Invenory odel for ime Varying Deerioraion Price Dependen uadraic Dem wih salvage value Ind. J. of ompuaional ppld ah. -. ohan Venkaeswarlu. Invenory anagemen odels wih Variable Holding os Salvage Value IOS J. of Busi. gm IOS-JB - a. ohan Venkaeswarlu. Invenory odel for ime Dependen Deerioraion ime Dependen uadraic Dem Salvage Value J of In. ah. Socy In press b. ohan Venkaeswarlu. Invenory anagemen odel wih uadraic Dem Variable Holding os wih Salvage value es. Journal of anagemen Sci - Inernaional Science ongress ssociaion 9