An Inventory Model for Constant Deteriorating Items with Price Dependent Demand and Time-varying Holding Cost

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1 Inernaional Journal of Compuer Science & Communicaion An Invenory Model for Consan Deerioraing Iems wih Price Dependen Demand and ime-varying Holding Cos N.K.Sahoo, C.K.Sahoo & S.K.Sahoo 3 Maharaja Insiue of echnology (M.I.), Khurda, Bhubaneswar Xavier Insiue of echnology (X.I.), Gangapaana, Bhubaneswar, Insiues of Mahemaics & Applicaions, Andharua, Bhubaneswar narenmahs@yahoo.co.in, sahoock@yahoo.com, 3 sahoosk@rediffmail.com Vol., No., January-June 00, pp ABSRAC In his paper a deerminisic invenory model is developed when he deerioraion is consan. Demand rae is a funcion of selling price and holding cos is aken as ime dependen. he model is solved allowing shorage in invenory. he resuls are solved wih he help of numerical example by using Mahemaica 5.. he sensiiviy analysis of he soluion has done wih he changes of he values of he parameers associaed wih he model is discussed. Keywords: EOQ Model; Deerioraing Iems; Invenory Model; Shorage; Price Dependen Demand; Holding Cos.. INRODUCION A lo of work has been done for deermining he invenory level of deerioraing iems which allows and does no allow shorage by differen researchers over las hree decades Maximum physical goods undergo decay or deerioraion over ime. Fruis, vegeables and food iems suffer from depleion by direc spoilage while sored. Highly volaile liquids such as gasoline, alcohol and urpenine undergo physical depleion over ime hrough he process of evaporaion. Elecronic goods, radioacive subsances, phoographic film, grain, ec. deeriorae hrough a gradual loss of poenial or uiliy wih he passage of ime. So decay or deerioraion of physical goods in sock is a very realisic feaure and invenory researchers fel he necessiy o use his facor ino consideraion. Whiin [957] considered he deerioraion of he fashion goods a he end of a prescribed shorage period. Ghare and Schrader [963] developed a model for an exponenially decaying invenory. An order level invenory model for iems deerioraing a a consan rae was presened by Shah and Jaiswal [997], Aggarwal [978], Dave and pael [98]. Invenory models wih a ime dependen rae of deerioraion were considered by Cover and Philip [973], Philip [974], Mishra [975] and Deb and chaudhuri [986]. Some of he recen work in his field has been done by Chung and ing [993], Fujiwara [993], Hariga [996], Hariga and Benkherouf [994], Wee [995], Jalan, e al [996], Su, e al [996], Chakrabory and Chaudhuri [997], Giri and Chaudhuri [997], Chakrabory, e al [998] and Jalan and Chaudhuri [999]. In classical invenory models he demand rae is assumed o be a consan. In realiy demand for physical goods may be ime dependen, sock dependen and price dependen. Selling price plays an imporan role in invenory sysem. Burwell [997] developed economic lo size model for price-dependen demand under quaniy and freigh discouns. An invenory sysem of amelioraing iems for price dependen demand rae was considered by Mondal, e al [003]. You [005] developed an invenory model wih price and ime dependen demand. In mos models, holding cos is known and consan. Bu holding cos may no always be consan. In generalizaion of EOQ models, various funcions describing holding cos were considered by several researchers like Naddor [966], Van der Venn [967], Muhlemann and Valis Spanopoulos [980], Weiss [98] and Goh [994]. In his presen paper, we have developed a generalized EOQ model for deerioraing iems where deerioraion rae and holding cos are expressed as linearly increasing funcions of ime and demand rae is a funcion of selling price. Shorages are allowed here and are compleely backlogged.. ASSUMPIONS AND NOAIONS he presen model is developed under following assumpions and noaions: (i) he demand rae is a funcion of selling price. (ii) Shorages are allowed and are fully backlogged.

2 68 Inernaional Journal of Compuer Science & Communicaion (IJCSC) (iii) he deerioraion rae is ime proporional. (iv) Holding cos h() per iem per ime-uni is ime dependen and is assumed as h() = h + α where α > 0, h > 0. (v) Replenishmen is insananeous and lead ime is zero. (vi) is he lengh of he cycle. (vii) he order quaniy in one cycle is q. (viii) A is he cos of placing an order. (ix) he selling price per uni iem is p. (x) C is he uni cos of an iem. (xi) he invenory holding cos per uni per uni ime is h(). (xii) C is he shorage cos per uni per uni ime. (xiii) θ () = β is he consan deerioraion, 0 < β <<. (xiv) During ime, invenory is depleed due o consan deerioraion and demand of he iem. A ime he invenory becomes zero and shorages sar occurring. (xv) Selling price p follows an increasing rend and demand rae posses he negaive derivaive hrough ou is domain where demand rae is f ()() p a p >. 3. MAHEMAICAL FORMULAION AND SOLUION Le Q() be he invenory level a ime (0). he deferenial equaions for he insananeous sae over (0, ) are given by + θ Q = f p, 0 = f p, Wih he condiion Q() a =. () () Using θ () = β and f ()() p = a p, hen equaion () and () becomes + β Q = a p, 0 = a p, (3) (4) Wih he condiion Q () a =. Solving (3) and (4) and neglecing higher powers of β we ge ( ) +β + () a p,0 θ () = 3 3 +β () a p(, ) Now sock loss due o deerioraion Holding cos is β D = a p e a p 0 0 β β = () a p q = D + a p 3 3 β β = () a p (5) (neglecing higher powers of β ) β β u H = ( h + α ) e ( a p) e du β β = h() a p β β a p (6) Now shorage during he cycle = Le ( )( ) S a p = ( )( ) a p (7)

3 Anomaly Inrusion Deecion Sysem using Hamming Nework Approach 69 oal profi per uni ime is given by P p p a p A Cq H CS (,, ) = ( ) ( ) = p () a p [From (5), (6) and (7)] 3 β β A + C () a p C + () a p ( ) 3 4 β β + h() a p β β + + (8) a p Le = ν, 0 < ν < Hence we have he profi funcion P (,() p) = p a p 3 3 β ν β ν A + C() a p C + () a p ( ν) ν βν β ν + h() a p ν βν β ν + + (9) a p Our objecive is o maximize he profi funcion (, ) P profi are p. he necessary condiions for maximizing he (, p) P his implies and (, p) P p ( ) ν α ( ) ν 3 3 A h a p a p C ( ) ( ν ) a p β C ( a p) ν + h ( a p) ν + α ( a p) ν β C ( a p) ν + h ( a p) ν + α ( a p) ν (0) And + + ν + αν 3 a p C h + C ( ν) β Cν + hν + αν β Cν + hν + αν () he soluions of (0) and () will give * and p *. he values * and p *, so obained, he opimal value p * (, p) of he average ne profi is deermined by (9) provided hey saisfy he sufficien condiions for maximizing (, ) P p are And (, ) (, ) P p P p < 0, < 0 p (, ) (, ) (, ) P p P p P p > 0 a p p p = p () and = (3) If he soluions obained from equaion (0) and () do no saisfy he sufficien condiions () and (3) we may conclude ha no feasible soluion will be opional for he se of parameer values aken o solve equaions (0) and (). Such a siuaion will imply ha he parameer values are inconsisen and here is some error in heir esimaion. Numerical Example: Suppose A = 50, a=00, C, β.95, α., c =.3, h.4, ν =0.0 he in appropriae unis. Based on hese inpu daa, he compuer oupus are as follows: Profi = , =. and p.97 able Changing % change Parameer in sysem p q Profi α β Cond...

4 70 Inernaional Journal of Compuer Science & Communicaion (IJCSC) Cond... ν h c c A SENSIIVIY ANALYSIS o sudy he effecs of changes in he sysem parameers, c, h, c, α, β, A and on he opimal cos derived by he proposed mehod and a is consan. Sensiiviy analysis is performed by changing (increasing or decreasing) he parameers by 0% and 50% and aking one parameer a a ime, keeping he remaining parameers a heir original values. On he basis of he resuls of able, he following observaion can be made. (i) Decrease in he value of he parameers hen p, q, and P (, p) is increased. (ii) Decrease in he values of eiher of he parameers β, A hen p, q, is decreased. (iii) Decrease in he value of he parameers c hen p, q, is increased and P(, p) is decreased. (iv) Decrease in he value of he parameers hen p, q, is decreased and P(, p) is increased. (v) Decrease in he value of he parameers h hen p, q is decreased and, P(, p) is increased. (vi) Decrease in he value of he parameer C hen p, q, is decreased and P(, p) is increased.. 5. CONCLUSION In he presen paper a deerminisic invenory model is developed for deerioraing iems. Shorages are allowed and are compleely backlogged in he presen model. In many pracical siuaions, sock ou is unavoidable due o various uncerainies. here are many siuaions in which he profi of he sored iem is high han is backorder cos Which is mos applicable o he physical goods under delay or deerioraion over ime. Commodiies such as fruis, vegeables, food suff, ec. Suffer from depleion by direc spoilage while kep in sore. REFERENCE [] Aggarwal, S.P., 978. A Noe on an Order-level Invenory Model for a Sysem wih Consan Rae of Deerioraion, Opsearch, 5, [] Burwell,.H., Dave, D.S., Fizparick, K.E., Roy, M.R., 997. Economic Lo Size Model for Price-dependen Demand under Quaniy and Freigh Discouns. Inernaional Journal of Producion Economics, 48(), [3] Chakrabari, e al, 997. An EOQ Model for Iems Weibull Disribuion Deerioraion Shorages and rended Demand an Exension of Philip s Model. Compuers and Operaions Research, 5, [4] Chakrabori,., and Chaudhuri, K.S., 997. An EOQ Model for Deerioraing Iems wih a Linear rend in Demand and Shorages in all Cycles. Inernaional Journal of Producion Economics, 49, [5] Chung, K., and ing, P., 993. An Heurisic for Replenishmen of Deerioraing Iems wih a Linear rend in Demand. Journal of he Operaional Research Sociey, 44, [6] Cover, R.P., and Philip, G.C., 973. An EOQ Model for Iems wih Weibull Disribuion Deerioraion. AIIE ransacions, 5, [7] Dave, U., and Pael, L.K., 98. Policy Invenory Model for Deerioraing Iems wih ime Proporional Demand. Journal of he Operaional Research Sociey, 3, [8] Deb, M., and Chaudhuri. K.S., 986. An EOQ Model for Iems wih Finie Rae of Producion and Variable Rae of Deerioraion. Opsearch, 3, [9] Fujiwara, O., 993. EOQ Models for Coninuously Deerioraing Producs using Linear and Exponenial Penaly Coss. European Journal of Operaional Research, 70, [0] Ghare, P.M., and Schrader, G.F., 963. An Invenory Model for Exponenially Deerioraing Iems. Journal of Indusrial Engineering, 4, [] Giri, B.C., and Chaudhuri, K.S., 997. Heurisic Models for Deerioraing Iems wih Shorages and ime-varying Demand and Coss. Inernaional Journal of Sysems Science, 8, [] Goh, M., 994. EOQ Models wih General Demand and Holding Cos Funcions. European Journal of Operaional Research, 73, [3] Hariga, M., 996. Opimal EOQ Models for Deerioraing Iems wih ime-varying Demand. Journal of Operaional Research Sociey, 47, [4] Hariga, M.A., and Benkherouf, L., 994. Opimal and Heurisic Invenory Replenishmen Models for

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