An EOQ Inventory Model for Deteriorating Items with Linear Demand, Salvage Value and Partial Backlogging

Transcription

1 Inernaional Journal of Applied Engineering Research ISSN 97-6 Volume Number 9 (6) pp Research India Publicaions hp://wwwripublicaioncom An EOQ Inenory Model for Deerioraing Iems wih Linear Dem Salage Value Parial Backlogging Pi Jagaana Mishra Research Scholar Deparmen of Mahemaics Raenshaw Uniersiy Cuack-7 Odisha India railokyanah Singh Deparmen of Mahemaics V Raman College of Engineering Bhubaneswar-7 Odisha India Hadibhu Paanayak Deparmen of Mahemaics Insiues of Mahemaics Applicaions Andharua Bhubaneswar-7 Odisha India Absrac In his paper an EOQ (economic order quaniy) model is deeloped for a deerioraing iem when he dem paern is considered as a linear funcion of ime Shorages are allowed parially backlogged he salage alue is included ino deerioraed unis he backlogging rae is ariable dependen on he waiing ime for he nex replenishmen he objecie of he model is o deermine he alue of he shorage poin cycle lengh order quaniy in order o minimize he aerage oal cos he resuls obained in his paper are illusraed wih he help of a numerical example sensiiiy analysis Keywords: EOQ Inenory Linear Dem Paern Parial Backlogging Salage Value Mahemaics Subjec Classificaion No: 9B Inroducion In he recen hree decades arious researches hae been done on inenory problems for deerioraing iems such as fashion goods elecronic componens hi-fi equipmens medicines drugs blood banks chemicals olaile liquids radioacie subsances Deerioraion is a phenomenon by which iems may decay damage spoilage eaporae loss of uiliy or loss of marginal alue ha resuls in decreasing usefulness from he original one he deerioraion rae of inenory in sock during he sorage period consiues an imporan facor which has araced he researchers he research on inenory was begun wih Whiin [] who sudied he deerioraion of fashion iems afer he prescribed ime of sorage Laer Ghare Schrader [] firs sudied he consumpion of deerioraing iems which was closely relaed o he negaie exponenial funcion of ime hey saed he deerioraion of inenory wih he help of a linear differenial equaion where inenory leel dem rae as funcion of ime Shah Jaiswal [] presened an order leel inenory model for deerioraing iems considering he dem as funcion of ime Aggarwal [] re-esablished heir model by recifying he errors of heir work In all hese models he deerioraion rae as well as he dem paern was aken as consans replenishmen was infinie no shorage in inenory was allowed Earlier in he classical EOQ (Economic Order Quaniy) model he dem rae of an iem was consan Many models on inenory were deeloped in he inenory lieraure considering he dem rae as consan Bu in he real life siuaion dem rae of any produc is always in a dynamic sae herefore many researchers are moiaed for deeloping he models wih ime dependen dem Siler Meal [] he earlies researchers who deeloped he modified EOQ wih ime-arying dem Howeer Donaldson [6] he firs researcher who gae a fully analyical reamen o he problem of inenory replenishmen wih a linearly ime-dependen dem Many researchers like Siler [7] Richie [8] Mira e al [9] Singh Panayak [] Singh Paanayak [] made he aluable conribuion in his direcion he possibiliies of shorage deerioraion in inenory were no considered in all models Dae Pael [] deeloped an inenory model for deerioraing iems wih ime-proporional dem Sachan [] exended heir model by considering he backlogging opion he assumpion of consan deerioraion rae was relaxed by Coer Philip [] who used a wih wo-parameer Weibull disribuion o represen he disribuion of ime o deerioraion Furher Philip [] generalized heir model by considering a hree-parameer Weibull disribuion Goyal Giri [6] sudied recen rends in modeling deerioraing inenory Li e al [7] also sudied he reiew lieraure on deerioraing inenory In real life siuaion shorages may occur in inenory model Iems like fashionable goods hi-fi equipmens clohes cusomers may prefer o wai for back orders when shorage occurs his ype of shorage is called he exernal shorage In his case he cusomer s order is no fulfilled due o he limied resources he inernal shorage occurs when he dem is uncerain he shorage can resul in back order coss profi loss delay cos ec During a shorage period he 679

2 Inernaional Journal of Applied Engineering Research ISSN 97-6 Volume Number 9 (6) pp Research India Publicaions hp://wwwripublicaioncom backlogging rae is ariable dependen on he waiing ime for he nex replenishmen Abad [8] sudied a lo-sizing inenory model for deerioraing iems wih ariable rae of deerioraion allowing shorages backlogging Chang Dye [9] deeloped an EOQ model wih arying dem parial backlogging by considering sock ou cos Consequenly he opporuniy cos due o los sales was considered in he model Researches in his direcion came from Papachrisos Skouri [] Ouyang e al [] Singh e al [] ec Mos of he aricles addressed assume ha he deerioraion of a uni is a complee loss o he inenory sysem ha hese deerioraed unis hae no sale alue Howeer in pracice he endor can offer a reduced uni cos o his buyer for he deerioraed sock Jaggi Aggarwal [] deeloped an EOQ model for deerioraing iems wih salage alues Mishra Shah [] formulaed a mahemaical model for iems which deeriorae wih respec o ime In heir paper he salage alue is incorporaed ino he deerioraed unis Annadurai [] sudied an opimal replenishmen policy considering shorages salage alue Recenly Singh Panayak [6] sudied he EOQ model for deerioraing iems considering linear dem parial backlogging In his sudy an EOQ inenory model for deerioraing iems has been deeloped cosidering he linear dem paern ariable deerioraion rae Shorages are alowed parially backlogged Iems such as paddy food suffs fruis egeables ec whose deerioraion rae increase wih ime is considered as linear dem paern he backlogging rae is inersely proporional o he waiing ime for he nex replenishmen he salage alue is incorporaed ino hese deerioraing iems I is obsered ha inclusion of he salage alue resuls in a reduced oal cos of he inenory sysem Furhermore we hae used he numerical example by minimizing he oal cos by simulaneously opimizing he shorage period he lengh of cycle Finally we hae sudied he sensiiiy analysis of he arious parameers on he effec of he opimal soluion Assumpions he following assumpions are made in deeloping he model (i) he inenory sysem inoles only one iem he planning horizon is infinie (ii) he replenishmen occurs insananeously a an infinie rae (iii) he deerioraing rae θ() = θ < θ << is a ariable deerioraion here is no replacemen or repair of deerioraed unis during he period under consideraion a+ b I( ) > (i) he dem rae D() = D I() where a > b > a is iniial dem () During he shorage period he backlogging rae is ariable is dependen on he lengh of he waiing ime for he nex replenishmen he longer he waiing ime is he smaller he backlogging rae would (i) be Hence he proporion of cusomers who would like o accep backlogging a ime is decreasing wih he waiing ime ( ) waiing for he nex replenishmen o ake care of his siuaion we hae defined he backlogging rae o be + δ ( ) when inenory is negaie he backlogging parameer δ is a posiie consan Salage alue associaed wih deerioraed unis during he cycle ime Noaions he following noaions hae been used in deeloping he model (i) C : holding cos $/per uni/per uni ime (ii) C : cos of he inenory iem $/per uni (iii) C : ordering cos of inenory $/per order (i) C : shorage cos $/per uni/per uni ime () C : opporuniy cos due o los sales $/per uni (i) C : salage alue parameer where C < associaed wih deerioraed unis during he cycle ime (ii) : ime a which shorages sar (iii) : lengh of each ordering cycle (ix) W : he maximum inenory leel for each ordering cycle (x) S : he maximum amoun of dem backlogged for each ordering cycle (xi) : (xii) ( ) : (xiii) (xi) (x) (xi) (xii) Q he economic order quaniy for each ordering cycle I he inenory leel a ime : he opimal soluion of : he opimal soluion of Q : he opimal economic order quaniy W : he opimal maximum inenory leel C : he minimum aerage oal cos per uni ime Mahemaical Model We consider he deerioraing inenory model wih linear dem Replenishmen occurs a ime = when he inenory leel aains is maximum W From = o he inenory leel reduces due o dem deerioraion A ime he inenory leel achiees zero hen shorage is all of allowed o occur during he ime ineral [ ] 68

3 Inernaional Journal of Applied Engineering Research ISSN 97-6 Volume Number 9 (6) pp Research India Publicaions hp://wwwripublicaioncom he dem during shorage period [ ] parially backlogged As he inenory leel reduces due o dem rae as well as deerioraion during he inenory ineral [ ] he differenial equaion represening he inenory saus is goerned by di () + θ () I() = D() () d θ θ D = a+ b where ( ) = ( ) he inegraing facor is θ IF = e I = is he soluion of Equaion () using he condiion ( ) θ θ I() = a + + b a θ b θ e θ + + () 6 8 (by neglecing he higher power of θ as < θ << ) Maximum inenory leel for each cycle is obained by puing he boundary condiion I ( ) = W in Equaion () herefore θ θ W = I( ) = a + + b + () 6 8 he dem a ime is During he shorage ineral [ ] ( ) parially backlogged a he fracion + δ herefore he differenial equaion goerning he amoun of dem backlogged is di () D = () d + δ ( ) wih he boundary condiion I( ) = he soluion of Equaion () is D I () = ln ( + δ ( ) ) δ ( δ ( )) ln + () Maximum amoun of dem backlogged per cycle is obained by puing = in Equaion () herefore D S = I( ) = ln + δ ( ) (6) δ Hence he economic order quaniy per cycle is θ Q= W + S = a + 6 θ D + b + = ln + δ ( ) 8 δ (7) he inenory holding cos per cycle is θ HC = C I() d = C a θ θ θ θ b + a + b + e d θ θ = C a + + b + (8) (by neglecing he higher power of θ as < θ << ) he deerioraion cos per cycle is DC = θc W D() d a b = θc W ( a+ b) d = θc + (9) 6 8 he ordering cos is OC = C () he shorage cos per cycle is SC = C I() d CD = ln ( + δ ( ) ) ln ( + δ ( )) d δ SC = CD ln ( + δ ( ) ) δ δ () he opporuniy cos due o los sales per cycle is OCLS = C D d ( ) + δ = CD ln ( + δ ( ) ) δ () he salage alue of he deerioraed iems is () () SV = C θ I d θ θ = C θ a + + b

4 Inernaional Journal of Applied Engineering Research ISSN 97-6 Volume Number 9 (6) pp Research India Publicaions hp://wwwripublicaioncom θ θ θ a + b + e d 6 8 a b = θc () (by neglecing he higher power of θ as < θ << ) herefore he aerage oal cos per uni ime per cycle = (holding cos + deerioraion cos + ordering cos + shorage cos + opporuniy cos due o los sales - salage alue of he deerioraed iems)/lengh of he ordering cycle ie ( ) C = C = HC DC OC SC OCLS SV [ ] θ θ a b () C a + + b + + θc + + C 6 8 = + CD ln ( + δ ( ) ) + CD ln ( + δ ( ) ) δ δ δ a b θc Our aim is o deermine he opimal alues of in order o minimize he aerage oal cos per uni ime C Using calculus we now minimize C he opimum alues of for he minimum aerage cos he soluions of he equaions = = () proided ha hey saisfy he sufficien condiions > > > Equaion () can be wrien as θ = ( a+ b) C + θ ( δ )( ) δ ( ) D C + C + + ( C C ) = (6) D( C+ δc)( ) = ( C) = (7) + δ ( ) Now are obained from he Equaions (6) (7) respeciely Nex by using we can obained he opimal economic order quaniy he minimum aerage oal cos per uni ime from Equaions (7) () respeciely Numerical Example In his secion we proide a numerical example o illusrae he aboe heory Example : Le us ake he parameer alues of he inenory sysem as follows: a = b = = = = = C = D = 8 = θ = δ = Soling Equaions (6) (7) we hae he opimal shorage period = 879 uni ime he opimal lengh of ordering cycle = 996 uni ime hereafer we ge he opimal order quaniy Q = 99 unis he minimum aerage oal cos per uni ime C = 88 Sensiiiy Analyasis We sudy now sudy he effecs of changes in he alues of he sysem parameers a b D θ δ on he opimal oal cos number of reorder he sensiiiy analysis is performed by changing each of parameers by +% +% % % aking one parameer a a ime keeping he remaining parameers unchanged he analysis is based on he Example he resuls are shown in able he following poins are obsered ) decrease while C increases wih he ) ) ) increase in he alue of he parameer a Here all highly sensiiiy o change in a decrease while C increases wih he increase in he alue of he parameer b Here all lowly sensiiiy o change in b decrease while C increases wih he increase in he alue of he parameer C Here all in C moderaely sensiiiy o change decrease while C increases wih he increase in he alue of he parameer C Here all in C moderaely sensiiiy o change 68

5 Inernaional Journal of Applied Engineering Research ISSN 97-6 Volume Number 9 (6) pp Research India Publicaions hp://wwwripublicaioncom ) 6) 7) 8) 9) ) ) Para meer C increase wih he increase in he alue of he parameer C Here all highly sensiiiy o change in C C increase while decreases in he alue of he parameer C Here all lowly sensiiiy o change in C C increase while decreases in he alue of he parameer C Here all lowly sensiiiy o change in C C increase while decreases in he alue of he parameer D Here all moderaely sensiiiy o change in D increase while of he parameer insensiiiy o change in decrease while C decreases in he alue C Here all C C increases in he alue of he parameer θ Here all lowly sensiiiy o change in θ C increase while decreases in he alue of he parameer δ Here all lowly sensiiiy o change in δ % Change in parameer a b C C C able Sensiiiy analysis C % Change in C C C D C θ δ Conclusions In his sudy an EOQ inenory model for deerioraing iems has been deeloped cosidering he linear dem paern ariable deerioraion rae Shorages are alowed parially backlogged Iems such as paddy food suffs fruis egeables ec whose deerioraion rae increase wih ime is considered as linear dem paern he backlogging rae is inersely proporional o he waiing ime for he nex replenishmen he salage alue is incorporaed ino hese deerioraing iems I is obsered ha inclusion of he salage alue resuls in a reduced oal cos of he inenory sysem Furhermore we hae used he numerical example by minimizing he oal cos by simulaneously opimizing he shorage period he lengh of cycle Finally we hae sudied he sensiiiy analysis of he arious parameers on he effec of he opimal soluion here are a numerous direcions in which his research can be exended One possible exension sems from he dem For insance we may exend he linear dem ino a more realisic ime-arying dem ha is funcion of ime selling price ohers Also we could consider he effecs of he ariable deerioraions (wo-parameer Weibull hreeparameer Weibull Gamma disribuion) Finally we could generalize he model o sochasic flucuaing dem paerns he economic producion lo size model allowing quaniy discouns inflaion rae References [] M Whiin "heory of inenory managemen Prineon Uni ersiy Press Pri nceon NJ 97 pp 6-7 [] P M Ghare G P Schrader A model for an exponenially decaying inenory Journal of 68

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