AN INVENTORY MODEL FOR DETERIORATING ITEMS WITH EXPONENTIAL DECLINING DEMAND AND PARTIAL BACKLOGGING

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1 Yugoslav Journal of Operaions Researh 5 (005) Number AN INVENTORY MODEL FOR DETERIORATING ITEMS WITH EXPONENTIAL DECLINING DEMAND AND PARTIAL BACKLOGGING Liang-Yuh OUYANG Deparmen of Managemen Sienes and Deision Making Tamkang Universiy Tamsui Taipei 5 Taiwan liangyuh@mail.ku.edu.w Kun-Shan WU Deparmen of Business Adminisraion Tamkang Universiy Tamsui Taipei 5 Taiwan Mei-Chuan CHENG Graduae Insiue of Managemen Sienes Tamkang Universiy Tamsui Taipei 5 Taiwan. Reeived: Marh 003 / Aeped: Augus 00 Absra: This sudy proposes an EOQ invenory mahemaial model for deerioraing iems wih exponenially dereasing demand. In he model he shorages are allowed and parially bakordered. The baklogging rae is variable and dependen on he waiing ime for he nex replenishmen. Furher we show ha he minimized objeive os funion is joinly onvex and derive he opimal soluion. A numerial example is presened o illusrae he model and he sensiiviy analysis is also sudied. Keywords: Invenory deerioraing iems exponenial delining demand parial baklogging.. INTRODUCTION In daily life he deerioraing of goods is a ommon phenomenon. Pharmaeuials foods vegeables and frui are a few examples of suh iems. Therefore he loss due o deerioraion anno be negleed. Deerioraing invenory models have been widely sudied in reen years. Ghare and Shrader [7] were he wo earlies researhers o onsider oninuously deaying invenory for a onsan demand. Laer

2 78 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems Shah and Jaiswal [3] presened an order-level invenory model for deerioraing iems wih a onsan rae of deerioraion. Aggarwal [] developed an order-level invenory model by orreing and modifying he error in Shah and Jaiswal s analysis [3] in alulaing he average invenory holding os. Cover and Philip [5] used a variable deerioraion rae of wo-parameer Weibull disribuion o formulae he model wih assumpions of a onsan demand rae and no shorages. Then Philip [] exended he model by onsidering a variable deerioraion rae of hree-parameer Weibull disribuion. However all he above models are limied o he onsan demand. Reenly Goyal and Giri [8] provides a deailed review of deerioraing invenory lieraures. They indiaed: The assumpion of onsan demand rae is no always appliable o many invenory iems (for example eleroni goods fashionable lohes e.) as hey experiene fluuaions in he demand rae. Many produs experiene a period of rising demand during he growh phase of heir produ life yle. On he oher hand he demand of some produs may deline due o he inroduion of more araive produs influening usomers preferene. Moreover he age of he invenory has a negaive impa on demand due o loss of onsumer onfidene on he qualiy of suh produs and physial loss of maerials. This phenomenon promped many researhers o develop deerioraing invenory models wih ime varying demand paern. In developing invenory models wo kinds of ime varying demands have been onsidered so far: (a) oninuous-ime and (b) disree-ime. Mos of he oninuous-ime invenory models have been developed onsidering eiher linearly inreasing/dereasing demand or exponenially inreasing/dereasing demand paerns. Dave and Pael [6] developed an invenory model for deerioraing iems wih ime proporional demand insananeous replenishmen and no-shorage. The onsideraion of exponenially dereasing demand for an invenory model was firs proposed by Hollier and Mak [0] who obained opimal replenishmen poliies under boh onsan and variable replenishmen inervals. Hariga and Benkherouf [9] generalized Hollier and Mak s model [0] by aking ino aoun boh exponenially growing and delining markes. Wee [5 6] developed a deerminisi lo size model for deerioraing iems where demand delines exponenially over a fixed ime horizon. Laer Benkherouf [] showed ha he opimal proedure suggesed by Wee [5] is independen of he demand rae. Chung and Tsai [] demonsraed ha he Newon s mehod by Wee [5] is no suiable for he firs order ondiion of he oal os funion. They deomposed i o drop he nonzero par and hen applied he Newon s mehod. Su e al. [] proposed a produion invenory model for deerioraing produs wih an exponenially delining demand over a fix ime horizon. In he menion above mos researhers assumed ha shorages are ompleely baklogged. In praie some usomers would like o wai for baklogging during he shorage period bu he ohers would no. Consequenly he opporuniy os due o los sales should be onsidered in he modeling. Wee [6] presened a deerioraing invenory model where demand dereases exponenially wih ime and os of iems. In his paper he baklogging rae was assumed o be a fixed fraion of demand rae during he shorage period. Many researhers suh as Park [] and Hollier and Mak [0] also onsidered onsan baklogging raes in heir invenory models. In some invenory sysems however suh as fashionable ommodiies he lengh of he waiing ime for he nex replenishmen is he main faor in deermining wheher he baklogging will be aeped or no. The longer he waiing ime is he smaller he baklogging rae would be

3 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems 79 and vie versa. Therefore he baklogging rae is variable and dependen on he waiing ime for he nex replenishmen. In a reen paper Chang and Dye [3] invesigaed an EOQ model allowing for shorage. During he shorage period he baklogging rae is variable and dependen on he lengh of he waiing ime for he nex replenishmen. In his paper an EOQ invenory model wih deerioraing iems is developed in whih we assume ha he demand funion is exponenially dereasing and he baklogging rae is inversely proporional o he waiing ime for he nex replenishmen. The primary problem is o minimize he oal relevan os by simulaneously opimizing he shorage poin and he lengh of yle. We also show ha he minimized objeive os funion is joinly onvex and obain he opimal soluion. A numerial example is proposed o illusrae he model and he soluion proedure. The sensiiviy analysis of he major parameers is performed.. NOTATION AND ASSUMPTIONS The mahemaial model in his paper is developed on he basis of he following noaion and assumpions. Noaion: : holding os $/per uni/per uni ime : os of he invenory iem $/per uni 3 : ordering os of invenory $/per order : shorage os $/per uni/per uni ime 5 : opporuniy os due o los sales $/per uni : ime a whih shorages sar T : lengh of eah ordering yle W : he maximum invenory level for eah ordering yle S : he maximum amoun of demand baklogged for eah ordering yle Q : he order quaniy for eah ordering yle I(): he invenory level a ime Assumpions:. The invenory sysem involves only one iem and he planning horizon is infinie.. The replenishmen ours insananeously a an infinie rae. 3. The deerioraing rae θ (0 < θ < ) is onsan and here is no replaemen or repair of deerioraed unis during he period under onsideraion.. The demand rae R () is known and dereases exponenially. λ R () = Ae I () > 0 = D I( ) 0 where A (>0) is iniial demand and λ (0< λ < θ ) is a onsan governing he dereasing rae of he demand.

4 80 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems 5. During he shorage period he baklogging rae is variable and is dependen on he lengh of he waiing ime for he nex replenishmen. The longer he waiing ime is he smaller he baklogging rae would be. Hene he proporion of usomers who would like o aep baklogging a ime is dereasing wih he waiing ime ( T ) waiing for he nex replenishmen. To ake are of his siuaion we have defined he baklogging rae o be when invenory is negaive. The + ( T ) baklogging parameer is a posiive onsan T. 3. MODEL FORMULATION Here he replenishmen poliy of a deerioraing iem wih parial baklogging is onsidered. The objeive of he invenory problem is o deermine he opimal order quaniy and he lengh of ordering yle so as o keep he oal relevan os as low as possible. The behavior of invenory sysem a any ime is depied in Figure. I ( ) W 0 T Time I ( ) vs ime Figure : Invenory level I() vs. ime Replenishmen is made a ime = 0 and he invenory level is a is maximum W. Due o boh he marke demand and deerioraion of he iem he invenory level dereases during he period [0 ] and ulimaely falls o zero a =. Thereafer shorages are allowed o our during he ime inerval [ T ] and all of he demand during he period [ T ] is parially baklogged.

5 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems 8 As desribed above he invenory level dereases owing o demand rae as well as deerioraion during invenory inerval [0 ]. Hene he differenial equaion represening he invenory saus is given by di() λ + θ I( ) = A e 0 () d wih he boundary ondiion I(0) = W. The soluion of equaion () is λ Ae ( θ λ)( ) I() = [ e ] 0. () θ λ So he maximum invenory level for eah yle an be obained as A ( ) W I(0) [ e θ = = λ ]. (3) θ λ During he shorage inerval [ T ] he demand a ime is parly baklogged a he fraion. Thus he differenial equaion governing he amoun of demand + ( T ) baklogged is as below. di() D = < T () d + ( T ) wih he boundary ondiion I ( ) = 0. The soluion of equaion () an be given by D I( ) = {ln [ + ( T )] ln [ + ( T )]} T. (5) Le = T in (5) we obain he maximum amoun of demand baklogged per yle as follows: D S = I( T ) = ln[ + ( T )]. (6) Hene he order quaniy per yle is given by A Q = W + S = e θ λ θ λ The invenory holding os per yle is D ln[ + ( )]. (7) ( ) [ ] + T A λ θ θ λ HC = 0 I() d = e [ e ( e )]. (8) θ ( θ λ) λ The deerioraion os per yle is DC = [ W R( ) d ] 0 λ [ ] 0 = W A e d ( θ λ) λ = A ( e ) ( e ) θ λ λ. (9)

6 8 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems The shorage os per yle is T T SC = [ I( ) d] = D ln[ + ( T )]. (0) The opporuniy os due o los sales per yle is T BC = 5 [ ] Dd = 5 D ( T ) ln[ + ( T )]. () + ( T ) Therefore he average oal os per uni ime per yle is TVC TVC( T ) = (holding os + deerioraion os + ordering os + shorage os + opporuniy os due o los sales)/ lengh of ordering yle ( θ λ) λ A θ e e e e e + A + 3 T θθ ( λ) λ θ λ λ λ θ λ = ( ) [ ] ln + ( T ) + D + 5 T A ( - ) = ( + θ ) θ λ A ( ) ( + θ ) λ e θ λ λ e + T θ( θ λ) θλ 3 D ( + ln ( 5) + T ) + T. () The objeive of he model is o deermine he opimal values of and T in order o minimize he average oal os per uni ime TVC. The opimal soluions T need o saisfy he following equaions: and TVC A ( + θ ) ( ) ( D 5) e θ λ e λ + = = 0 (3) T θ + ( T ) and TVC D ( ( + 5) T )( ) = + ln [ + ( T )] T T + ( T ) A ( ) A ( ) + = 0. () θ ( θ λ) θλ + θ ( θ λ) λ e + θ e 3

7 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems 83 A( + θ ) D ( + 5) For onveniene we le M = and N = and hen from (3) θ and () we ge M ( θ λ) λ M ( θ λ) λ T = + e e e e N N (5) and ( T )( ) M ( θ λ) ln[ ( N + + T )] e + ( T ) θ λ M λ + ( e ) 3 = 0 respeively. (6) λ Subsiuing (5) ino (6) we obain M (θ ) (θ ) e λ N M e λ ( ) ln e λ e λ N ( ) M ( θ λ) M λ e + e 3 = 0. (7) θ λ λ λ N N θ λ If we le P = ( + ) ( + ) hen we have he following resuls. M M Theorem. MP N N M MP( λ ) θ If θ > λ and ln[ + ] ln[ P] N 3 > 0 hen θ λ M N λ λ( θ λ) he soluion o (3) and () no only exiss bu also is unique (i.e. he opimal value ( T ) is uniquely deermined). Proof: By assumpion 5 we have T > and hene from (5) we obain M (θ λ) λ e e > 0 (8) N whih implies ˆ N < ln[ ] θ λ +. M Nex from (7) we le M (θ ) N M (θ ) F ( ) e λ e λ λ λ ( ) ln e e = N ( ) M + θ λ λ M (θ λ ) λ e e 3.

8 8 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems Taking he firs derivaive of F( ) wih respe o (0 ˆ ) we ge df( ) ( θ λ) ( ) ( ) ( ) λ MM θ λ λ M θ λ e e M e e e λ θ λ λ e = d + + N N > 0. ( by equaion (8) ) Hene F( ) is a srily inreasing funion in [0 ˆ ). Furhermore we have F (0) = 3 < 0 and M ( θ λ) λ N M ( θ λ) λ lim F ( ) = lim e e ( ) ln e e ˆ ˆ N M ( θ λ) M λ e + ( e ) 3 θ λ λ MP N N MP MP( λ ) Nθ = ln[ + ] ln[ ] 3 θ λ M N λ λ( θ λ) λ N N θ λ where P = ( + ) ( + ). M M MP N N M MP( λ ) θ Thus if θ > λ and ln[ + ] ln[ P] N 3 > 0 θ λ M N λ λ( θ λ) we obain lim F ( ) > 0. Therefore we an find an unique (0 ˆ ) suh ha ˆ F ( ) = 0. One we obain he value hen he opimal value T an be uniquely deermined by equaion (5). This omplees he proof. Now we an obain he following main resul. Theorem. MP N N M MP( λ ) θ If θ > λ and ln[ + ] ln[ P] N 3 > 0 θ λ M N λ λ( θ λ) he oal os per uni ime TVC( T ) is onvex and reahes is global minimum a poin ( T ). Proof: From equaions (3) and () we have and ( ) N TVC M[( ) e θ λ e λ ] = 0 θ λ + λ + > T [ + ( T ( )] T ) N TVC = T T [ + ( T ( ) )] T

9 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems 85 N TVC = > 0. T T [ + ( T )] ( T ) Then TVC TVC TVC ( ) ( ) ( T T T ) T T ( ) N M[( ) e θ θ λ λ λe λ ] = 0. + > T [ + ( T )] This omplees he proof. Nex by using and T we an obain he opimal maximum invenory level and he minimum average oal os per uni ime from equaions (3) and () respeively (we denoe hese values by he opimal order quaniy (we denoe i by TVC ). Furhermore we an also obain Q ) from equaion (7).. NUMERICAL EXAMPLE AND ITS SENSITIVITY ANALYSIS Aording o he resuls of Seion 3 we will provide an example o explain how he soluion proedure works. Suppose ha here is a produ wih an exponenially dereasing funion of demand f () = Ae λ where A and λ are arbirary onsans saisfying A > 0 and λ > 0. The remaining parameers of he invenory sysem are A = θ = 0.08 = λ = 0.03 = 0.5 =.5 3 = 0 =.5 5 = and D = 8. Under he above-given parameer values we hek he ondiion MP N N M ln[ + ] ln[ P ] θ λ M N MP( λ ) θ N 3 = 77. > 0 λ λ( θ λ) and hen obain he opimal shorage poin =.775 uni ime and he opimal lengh of ordering yle T =.8536 uni ime. Thereafer we ge he opimal maximum invenory level W = 8.0 unis he opimal order quaniy Q = 0.83 unis and he minimum average oal os per uni ime TVC = $.65. Nex we sudy he effes of hanges in he model parameers suh as D θ and on he opimal shorage poin he opimal lengh of A λ 3 5 ordering yle he opimal order quaniy he opimal maximum invenory level and he minimum average oal os per uni ime. The sensiiviy analysis is performed by hanging eah of he parameers by -50 % -5 % +5 % and +50 % aking one parameer a a ime while keeping remaining unhanged. The resuls are presened in Table.

10 86 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems Table : Sensiiviy Analysis Parameer A λ 3 5 D θ % hange T % hange in Q W TVC +50% -.76% -.077% % 5.78% 6.335% +5% -.660% % % 8.657% 8.953% -5% 8.536% 0.87% -.6% % -.7% -50% 7.799% % % % -5.56% +50%.5905%.06% % 0.503% -0.59% +5% 0.808% 0.59% 0.03% 0.750% -0.97% -5% % % -0.9% -0.67% 0.90% -50% % % % % % +50% % % -5.30% -9.3% % +5% % % % % % -5%.003% 8.558%.7973%.60% % -50% 3.6%.505% 9.083% 35.96% -9.5% +50% % -.98% -.0% % % +5% -.780% % -.907% %.8795% -5%.9753%.699%.75% 3.097% -.959% -50% 6.739% 3.53% 5.500% 6.% -.005% +50%.03% % 3.03%.99%.68% +5%.36% 3.877%.0705%.606%.337% -5% % -.659% -3.6% -3.73% % -50% % % % -9.85% % +50%.88% % %.905%.880% +5%.093% -.87% -0.63%.059%.037% -5% -.866%.903% 0.05% -.335% % -50% % 6.075% 0.58% % -.990% +50%.6579% -.9% -0.37%.768%.7037% +5%.5357% -.680% -0.80%.596%.56% -5% -.9%.697% 0.368% -.309% -.60% -50% -5.77% % 0.867% % % +50% 3.693% %.799% 3.838% % +5%.78% %.79%.3635%.30% -5% -.05% % % % -.63% -50% infeasible soluion +50% -7.30% -.875% % -.965%.35% +5% % -.09% -.933% -.539%.656% -5%.6%.8%.066%.7009% % -50% 8.693% 5.703%.768% 5.575% -.830% +50%.97% -.7% %.599%.576% +5% 0.89% % % 0.85% 0.878% -5% -.000% 0.658% 0.05% -.036% -.050% -50% -.660%.393%.050% -.37% -.997%

11 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems 87 On he basis of he resuls shown in Table he following observaions an be made.. and T derease while Q TVC inrease wih inrease in he value of he model parameer A. The obained resuls show ha T Q TVC are moderaely sensiive o hanges in he value of A. Moreover is highly sensiive o hanges in A.. TVC dereases while T Q and W inrease wih inrease in he value of he 3. model parameer λ. I is seen ha T hanges in he value of he parameer λ. T Q and W derease while Q model parameers or. Moreover TVC sensiive o hanges in he value of he parameer hanges in he value of.. As he value of 3 inreases T T Q T and TVC are insensiive o TVC inreases wih inrease in he value of he Q T Q and W are highly and moderaely sensiive o TVC inrease. I is seen ha TVC are highly sensiive o hanges in he value of 3. Q derease while he model parameers or 5. I is seen ha W sensiive o hanges in he values of and 5. However T dereases while Q model parameer D. T Q and model parameer θ. T and W derease while Q derease while he model parameer. In addiion W o hanges in he value of. TVC inrease wih inrease in he value of TVC and T are lowly Q is almos insensiive. TVC inrease wih inrease in he value of he TVC inreases wih inrease in he value of he TVC inrease wih inrease in he value of TVC 5. CONCLUSIONS T and Q are lowly sensiive The lassial eonomi order quaniy (EOQ) model assumes a predeermined onsan demand rae and no effes on shorages. In realiy however no only demand varies wih ime bu also oss are affeed by shorages. In he proposed model we presen an EOQ invenory model for deerioraing iems wih exponenial delining demand and parial baklogging. The rae of deerioraion is assumed o be onsan and he baklogging rae is inversely proporional o he waiing ime for he nex replenishmen. We also show ha he minimized objeive os funion is joinly onvex and derive he opimal soluion. Furhermore a numerial example and is sensiiviy analysis for parameers are provided o assess he soluion proedure.

12 88 L.-Y. Ouyang K.-S. Wu M.-C. Cheng / An Invenory Model for Deerioraing Iems The proposed model an be exended in several ways. For insane i ould be of ineres o relax he resriion of onsan deerioraion rae. Also we may exend he deerminisi demand funion o sohasi fluuaing demand paerns. Finally we ould generalize he model o he eonomi produion lo size model. Aknowledgemen: The auhors grealy appreiae he anonymous referees for heir value and helpful suggesions on an earlier version of he paper. REFERENCES [] Aggarwal S.P. A noe on an order-level model for a sysem wih onsan rae of deerioraion Opsearh 5 (978) [] Benkherouf L. Noe on a deerminisi lo size invenory model for deerioraing iems wih shorages and a delining marke Compuers and Operaions Researh 5 (998) [3] Chang H.J. and Dye C.Y. An EOQ model for deerioraing iems wih ime varying demand and parial baklogging Journal of he Operaional Researh Soiey 50 (999) [] Chung K.J. and Tsai S.F. A soluion proedure o deermine invenory replenishmen poliies for deerioraing iems in a delining marke Journal of Informaion & Opimizaion Sienes 0 (999) -5. [5] Cover R.B. and Philip G.S. An EOQ model wih Weibull disribuion deerioraion AIIE Transaions 5 (973) [6] Dave U. and Pael L.K. (T S i ) Poliy invenory model for deerioraing iems wih ime proporional demand Journal of he Operaional Researh Soiey 3 (98) 37-. [7] Ghare P.M. and Shrader G.H. A model for exponenially deaying invenory sysem Inernaional Journal of Produion Researh (963) [8] Goyal S.K. and Giri B.C. Reen rends in modeling of deerioraing invenory European Journal of Operaional Researh 3 (00) -6. [9] Hariga M. and Benkherouf L. Opimal and heurisi replenishmen models for deerioraing iems wih exponenial ime varying demand European Journal of Operaional Researh 79 (99) [0] Hollier R.H. and Mak K.L. Invenory replenishmen poliies for deerioraing iems in a delining marke Inernaional Journal of Produion Researh (983) [] Park K.S. Invenory models wih parial bakorders Inernaional Journal of Sysems Siene 3 (98) [] Philip G.C. A generalized EOQ model for iems wih Weibull disribuion AIIE Transaions 6 (97) [3] Shah Y.K. and Jaiswal M.C. An order-level invenory model for a sysem wih onsan rae of deerioraion Opsearh (977) 7-8. [] Su C.T. Lin C.W. and Tsai C.H. A deerminisi produion invenory model for deerioraing iems wih an exponenial delining demand Operaional Researh Soiey of India 36 () (999) [5] Wee H.M. A deerminisi lo-size invenory model for deerioraing iems wih shorages and a delining marke Compuers and Operaions Researh (995a) [6] Wee H.M. Join priing and replenishmen poliy for deerioraing invenory wih delining marke Inernaional Journal of Produion Eonomis 0 (995b) 63-7.