Weibull Deterioration, Quadratic Demand Under Inflation

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1 IOS Journal of Mahemaics IOS-JM e-issn: 78-78, p-issn: 9 7X. Volume 0, Issue Ver. V May-Jun. 0, PP 09-7 Weibull Deerioraion, Quadraic Demand Under Inflaion. Mohan *,.Venkaeswarlu Dep of Mahemaics, F-ivil, ollege of Miliary Engineering Pune - 0 INDIA GIAM School of Inernaional Business GIAM Universiy Visakhapanam INDIA Absrac: Deerminisic invenory model is developed for deerioraing producs when he demand rae is considered as quadraic funcion of ime, furher incorporaing wo parameer Weibull rae of deerioraion wih inflaion. he model is solved for when shorages are no allowed. A numerical example and sensiiviy of he models is also sudied a he end. Key words: Weibull deerioraion, Quadraic demand, salvage value, Inflaion I. Inroducion he assumpion of consan, linear, exponenial demand rae is no always applicable o many invenory producs like vegeables, food suffs, fashionable clohes, elecronic goods, ec., since unsable siuaions in demand rae. Inroducing new iems will facinae more in demand during he growh phase of heir life cycle. I is eviden ha, some iems may decline due o he inroducion of new producs due o he choice or influencing cusomers. So o develop deerioraing invenory models wih quadraic demand paern is worh aemp. Buzaco 97 developed an EOQ model wih inflaion.ghare and Schrader 9 sudied an invenory model for exponenially decaying. Su..,ong L.I., sai S.,99 sudied an invenory model under inflaion for he sock dependen demand consumpion rae wih exponenial decay. over and Philip 97 developed an invenory model for ime dependen rae of deerioraion. Aggarwal 978 proposed an order level invenory model for a sysem wih consan rae of deerioraion. Dave and Pael 98 sudied a lo size model for ime proporional demand wih consan deerioraion. Deb and houdhuri 987 sudied a heurisic approach for replenishmen of rended invenories wih shorages. Hariga 99 developed an invenory model of deerioraing iems for ime-varying demand wih shorages. Daa and Pal 99 proposed an EOQ model in he effecs of inflaion and ime value of money wih linear ime dependen demand rae incorporaing shorages. hakrabori and houdhuri 99 proposed an EOQ model for deerioraing iems of linear rend in demand wih shorages in all cycles. Giri and haudhuri 997 developed a heurisic model for deerioraing iems of ime varying demand and coss considering shorages. Goyal and Giri 00 sudied survey of recen rend in deerioraing invenory model. Mondal e. al 00 developed an invenory model of amelioraing iems for price dependen demand rae. You 00 sudied he invenory sysem for he producs wih price and ime dependen demands. Ajana oy 008 developed an invenory model for deerioraing iems wih price dependen demand and ime varying holding cos wih shorages and wihou shorages. Mishra and Singh 00 sudied an invenory model for deerioraing iems wih ime dependen demand and parial backlogging. Mishra 0 proposed an invenory model wih Weibull rae of deerioraion and consan demand. He incorporaed variable holding cos wih shorages. salvage value also considered for deerioraing iems. Vikas Sharma and ekha 0 proposed an invenory model for ime dependan demand for deerioraing iems wih Weibull rae of deerioraion.in his model shorages are aken consideraion. Mohan and Venkaeswarlu 0a developed an invenory model wih variable holding cos and salvage value. Mohan and Venkaeswarlu 0b proposed an invenory model for quadraic demand wih respec o ime wih salvage considering deerioraion iems. Mohan and Venkaeswarlu0c developed an invenory model wih Quadraic Demand, Weibull disribuion deerioraion rae wih Variable Holding os and Salvage value. Shial S. Pael,aman Pael sudied an invenory model for deerioraing iems wih linear demand under permissible delay in paymens and inflaion in corporaing shorages. In his paper, we consider an invenory model wih weibull deerioraion rae and demand rae is quadraic funcion of ime. Shorages are no allowed in his case and he ime horizon is infinie. he opimal oal cos is obained considering he salvage value for deerioraed iems. he numerical example and sensiiviy analysis is done a he end. 09 Page

2 Weibull Deerioraion, Quadraic Demand Under Inflaion II. Mahemaical Assumpions and Noaions his model is developed using he following assumpions and noaions: i he demand rae D a ime is assumed o be D = a + b + c, a 0, b 0, c 0. Here c is he iniial rae of demand, b is he rae wih which he demand rae increases and a is he rae wih which he change in he rae demand rae iself increases. ii he deerioraion rae follows wo parameer Weibull disribuion and is given by ; 0 α, β. iii D, he number of deerioraed unis iv eplenishmen rae is infinie v Lead ime is zero. vi, he cos per uni vii Q is he invenory level a ime. viii A is he order cos per uni order. ix he salvage value γ*, 0 γ < is associaed wih deerioraed unis during a cycle ime. x, inflaion rae III. Formulaion and soluion of he model I is assumed ha he invenory level deplees as he ime passes due o demand rae and deerioraion. he differenial equaion which describes he invenory level a ime can be wrien as Fig. d Q d where Q D ;, D a b c, wih Q 0 a 0 a b a b c Q e c k where k is an inegral consan. Here we have expanded order erms as α is small. e and ignored higher he soluion of he above differenial equaion using he boundary condiions is given by 0 Page

3 Weibull Deerioraion, Quadraic Demand Under Inflaion Page e c b a c b a Q Since Q Q 0, we ge Q IV. Invenory Model wih Salvage value: he following coss are calculaed o find he oal cos of he sysem when shorages are no allowed: Ordering cos, O = A Invenory Holding cos, IH = d e Q h h he number of unis ha deerioraed during his cycle ime is D = Q - d e D 0 where D= c b a is he rae of Demand. os due o deerioraion D = hus he oal cos is obained as oal os = Ordering cos + Holding cos + os due o deerioraion-salvage value

4 Weibull Deerioraion, Quadraic Demand Under Inflaion Page h he necessary condiion for a minimize oal cos per uni ime is

5 Weibull Deerioraion, Quadraic Demand Under Inflaion Page h h and he second derivaive is > 0. i.e., 0 7 he opimum order quaniy is given by Q 8 he opimum value of is obained by solving Equaion using MAHAD and classified ino four models. he following ables show he acceleraed growh /decline models or rearded growh/decline models depending on he sign of a and b: Depending on he signs of a' and b', one may have he following differen ypes of relaive demand paerns: a > 0 and b > 0 gives acceleraed growh in demand

6 Weibull Deerioraion, Quadraic Demand Under Inflaion a > 0 and b < 0 gives rearded growh in demand a < 0 and b > 0, gives rearded decline in demand a < 0 and b < 0 gives acceleraed decline in demand. he above four ypes of demand curves are given below: Acceleraed Growh Demand urve Fig earded Growh Demand urve Fig earded Decline Demand urve Fig. Page

7 Weibull Deerioraion, Quadraic Demand Under Inflaion Acceleraed Decline Demand urve Fig.. Numerical Example: o illusrae he models developed, we assume he following daa: a = b = 0 c = 00 A = 00 = h = = 0.0 able : MODEL-I: a >0, b > 0 and c > 0 Model ype Q Quadraic Demand Linear Demand able : MODEL-II: a < 0, b > 0 and c > 0 Model ype Q Quadraic Demand Linear Demand able : MODEL-III: a > 0, b < 0 and c > 0 Model ype Q Quadraic Demand Linear Demand able : MODEL-IV: a < 0, b < 0 and c > 0 Model ype Q Quadraic Demand Linear Demand I is observed ha he Models II has shown marginal improvemen in oal cos of he invenory sysem when comparing he quadraic ime dependen demand models wih linear dependen demand models when he deerioraion rae follows Weibull deerioraion rae. Hence we presen he sensiiviy analysis for he model II which is significan for furher analysis. Page

8 Weibull Deerioraion, Quadraic Demand Under Inflaion. Sensiiviy analysis MODEL-II: a < 0, b > 0 and c > 0 able : Sensiiviy of he scale parameer α Q α = α = α = α = α = able : Sensiiviy of he shape parameer β Q β = β = β = β = β = able 7: Sensiiviy of he salvage parameer γ Q γ = γ = γ = γ = γ = From ables- o 7, he following poins are noiced: he oal cos is more sensiive o he changes made in scale parameer and shape parameer bu i does saisfy when he salvage value greaer han 0. V. Discussions: Neglecing α and higher powers of α and puing = 0 he oal cos funcion of will be reduced o Venkaeswarlu and Mohan, An Invenory Model for ime dependen Quadraic Demand Weibull ae of Deerioraion ae and Salvage value enh AIMS Inernaional onference on Managemen Proceedings 8-89'. When we compared his model wih inflaion and as menioned above model he following variaions are sudied. enh AIMS Inernaional onference on ManagemenProceedings 8-89'wihou inflaion ycle ime is less he number of order quaniyq is less oal cos is less wo models are exising Invenory model wih inflaion of his paper In his model cycle ime is more he number of order quaniyq is significanly more oal cos is significanly more One model is exising VI. onclusions: We have developed invenory managemen models for deerioraing iems when he demand rae is assumed o be quadraic funcion of ime. I is assumed ha he deerioraion rae is wo parameer Weibull disribuion. We have solved he model wihou shorages under inflaion. eferences [] Buzaco J.A.,97 Economic Order Quaniy wih Inflaion, Operaions esearch Sociey, Vol,-8 [] Ghrae P.M. and Schrader G.F.,9, An Invenory model for exponenially deerioraing iems, Journal of Indusrial Engineering,,8- [] Su. e al, 99 An Invenory model under inflaion for sock dependen demand consumpion rae and exponenial decay, Opsearch Vol 0, 99-0 [] over.p and Philip G..,97 An EOQ model for iems wih Weibull disribuion deerioraion, AIIE ransacions,, - [] Aggarwal S.P 978, A noe on an order-level invenory model for a sysem wih consan rae of deerioraion, Opsearch,, 8-87 [] Dave U. and Pael L.K., 98,S i- policy invenory model for deerioraing iems wih ime proporional demand. Journal of Operaional esearch Sociey,,7- Page

9 Weibull Deerioraion, Quadraic Demand Under Inflaion [7] Deb M. and haudhuri K.,987 A noe on he heurisic for replenishmen of rended invenories considering shorages. Journal of Operaional esearch Sociey,8,9- [8] Hariga M. 99 An EOQ model for deerioraing iems wih shorages and ime-varying demand. Journal of Operaional esearch Sociey,,98-0 [9] Daa.K. and Pal A.K.,99 Effecs of inflaion and ime value of money wih linear ime dependen demand rae and shorages European Journal of Operaions esearch Vol, -8 [0] hakrabori. and haudhuri K.S. 99 An EOQ model for iems wih linear rend in demand and shorages in all cycles Inernaional Journal of Producion Economics,9,0- [] Giri B. and haudhuri K.S. 997 Heurisic model for deerioraing iems wih shorages Inernaional Journal of Sysem Science,8,-9 [] Goyal S.K and Giri. B.., 00, ecen rends in modeling of deerioraing invenory European Journal of Operaions research, Vol., pp.-. [] Mondal B., Bhunia A.K and Maii M., 00 An invenory sysem of amelioraing iems for price dependen demand rae, ompuers and Indusrial Engineering,,- [] You S.P., 00 Invenory policy for producs wih price and ime dependen demands, Journal of Operaional esearch Sociey,, [] Ajana oy 008, An Invenory model for deerioraing iems wih price dependan demand and ime-varying holding cos, AMO- Advanced modeling and opimizaion, volume 0, number, 008 [] Mishra V.K and Singh L.S 00, Deerioraing invenory model wih ime dependen demand and parial backlogging, Applied Mahemaical sciences 7, -9 [7] Vinod Kumar Mishra 0, Invenory model for ime dependen holding cos and deerioraion wih salvage value and shorages. he Journal of Mahemaics and ompuer Science Vol. No [8] Vikas Sharma and ekha ani haudhuri 0, An invenory Model for deerioraing iems wih Weibull Deerioraion wih ime Dependen Demand and shorages, esearch Journal of Managemen Sciences Vol., 8-0 [9] Mohan. and Venkaeswarlu.,0a, Invenory Managemen Models wih Variable Holding os and Salvage Value, IOS J. of Busi. and Mgm IOS-JBM, Vol., pp. 7- [0] Mohan. and Venkaeswarlu., 0b Invenory Model for ime Dependen Deerioraion, ime Dependen Quadraic Demand and Salvage Value, J of In. Mah.Socy, vol 8 Nos - - [] Mohan. and Venkaeswarlu., 0c Invenory Managemen Model wihquadraic Demand, Variable Holding os wih Salvage value,es J of Managemen Sci. Vol [] Shial S. Pail, aman Pael 0, An Invenory Model for Weibull Deerioraing Iems wih Linear demand, Shorages under permissible delay in Paymens and inflaion, Inernaional Journal of Mahemaics and Sasisics Invenion Vol Issue Aug 0 PP -0 7 Page