Dr.Pravat Kumar Sukla P.S College, Koksara,Kalahandi,Odisa,INDIA

Transcription

1 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September - 0 An nventory Ordering Poliy Using Constant Deteriorating tems With Constant Demand. Abstrat Dr.Pravat Kumar Sukla P.S College, Koksara,Kalahandi,Odisa,NDA n this paper, a deterministi inventory model for onstant deteriorating items with onstant demand is developed. optimal solution with respet to the parameter of the system is arried out with the help of numerial example using the software MAHEMACA 4. version. Keywords: nventory, Ordering poliy, Constant demand, Shortages, Deterioration ntrodution: he analysis of deteriorating inventory began with Ghare and Shrader (963), who established the lassial no shortage inventory with a onstant rate of deay. However it has been empirially observed that failure and life expetany of many items an be expressed in items of weibull distribution. his empirial observation has prompted researhers to represent the time to deterioration of a produt by a weibull distribution. Covert and Philip (973) extended Ghare and Shrader (963) model and obtain an eonomi order quantity model for a variable rate of deterioration buy assuming a twoparameter weibull distribution. Researhers inluding Philip (974), Misra (975) adikamalla (978), Wee (997, 93), Chakrabarty etal. (998) and Mukhopadhaya etal (005), developed eonomi order quantity models that foused on this type of produts. herefore, a realisti model is the ones that treat the deterioration rate as a time varying funtion. n general deterioration is defined as deay damage, spoilage, evaporation, obsolese, pilferage, loss of utility or loss of marginal value of a ommodity that results in derease of usefulness from the original one. he derease of loss of utility due to deay is usually a funtion of the on hand inventory. t is reasonable to note that a produt may be understood to have life time whih ends when utility reahes zero. he ontinuously deaying/deterioration of items is lassified as age dependent ongoing

2 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September - 0 deterioration and age independent ongoing deterioration. Blood, fish, strawberry are some of the examples of the former while alohol, gasoline and radioative hemial and grain produts are examples of the latter. ime varying demand patterns are ommonly used to reflet sales in different phases of the produt life yle in the market. For example, the demand for inventory items inreases over time in the growth phase and dereases in the deline phase. Donaldson (997) initially developed an inventory model with a linear trend in demand. After that, many researh works for example Dave and Patel (98), Mak (98), Holier and Mak (983), Sahan (984), Goyal (987), Bahari- Kashani (989), Aggrawal and Bahari-Kashani (99), Goswami and Chaudhuri (99), Xu and Wang (99), Chung and ing (994), Hargia and Benkherouf (994), Jain and Silver (994), Hariga (994,995) Sahu,Dash and Sukla(007), have been devoted to inorporating a time-varying demand rate into their models for deteriorating items with or with out shortages under a variety of irumstane. n this paper, a deterministi inventory model for onstant deteriorating items with onstant demand is developed. optimal solution with respet to the parameter of the system is arried out with the help of numerial example using the software MAHEMACA 4. version. Assumptions: (i) (ii) (iii) (iv) Lead time is negligible, Replenishment is instantaneous, A deteriorating item is neither repaired nor replaed during a given yle, Shortages are allowed and fully bak-logged, Notations: Length of one yle, q Order quantity, A Order Cost per order, r( Constant demand rate,

3 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September - 0 C Unit ost of item, nventory arrying, Shortage ost per unit time, Constant rate of deterioration, 0, ( nventory level at time t, Mathematial Formulation and Solutions : Let ( be the inventory level at any time t ( 0 t ). and demand rate is depleting due to the simultaneously ourrene of demand and the deterioration. he differential equations for the instantaneous state of ( over ( 0, ) are given by d( ( R dt di( R dt 0 t t () t t () he solution of () (boundary ondition at t=0, ( (0) ) is t R t ( (0) e ( e ) he solution of (3) boundary onditions at t t ( t ) 0, is (3) ( R( t t t (4) at t t, ( t ) 0, hene (3) beomes R t t ( e ) (0) e t (0) log R 0 Stok loss dueto deterioration in the interval ( 0, t ) is given by (5) t D (= (0)- Rdt ( (6) 0 3

4 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September - 0 Again (3), gives R t R ( 0) ( t e (7) t R t Using (7) in (6),we get D( ( e ) ( ( e ) Rt Loss of stok due to deterioration within yle is till time t. D t R Rt t t ( ) ( e ) ( ( e ) Sine ( t ) 0, D ( t ) 0, (8) R t D( t ) ( e ) Rt (9) B t ) ( t Rdt R( Now q D( 0 Rdt R t q ( e ) Rt R R ( t q ) R ( t ) (0) and ( 0) R t ( e ) () R t R t t hus (= ( e ) ( e ) e () he total ost per unit time C, t ), is the sum of ost of the units, ordering ost, ( inventory holding ost and shortage ost C(, t ) A q t 0 ( dt t t ( dt (3) 4

5 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September - 0 t e R( A C R C(, R t R t + e t + Our objetive is to minimize the ost funtion C, t ), whih is a differentiable funtion. ( Let t Z, 0 Z (5) By substituting (5) in(4), it an be written as C( ) A C R ( e Z R ) R( Z) ( e Z R ( Z ) Z) (4) A C R ( e Z ) R( Z) R Z R ( e Z) ( Z Z) dc( ) minimize the ost funtion C( ), the neessary ondition is 0. d dc( ) A C R Z ( ) ( ) e R Z d C R Z R ( ) ( ) e Z Z C RZe Z R( Z) (7) d C d A C R ( e 3 3 Z C ) R( Z) ( RZe Z ( R( Z) C R Rz Z Z Z ( RZ e ) ( e Z) ( e ) 3 he optimal value of say RZ e Z (8) * an be obtained from equation (8) and optimal order quanity d C d * q an be obtained from (0) 0 5

6 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September - 0 Numerial Example: As an illustration to the above-developed model we onsider a system with the following parameters: A Rs.00/ Per order. C Rs.0/ Per unit, Rs.5 / Per unit time, Rs.5 / Per unit per unit time, 0.05, R=700.,we gets the optimal length of ordering yle * is 3.04, the optimal order quantity q * is and the optimal total variable ost C * is he optimal solution with respet to the above parameters of the system is arried out with the help of the software MAHEMACA 4.version. CONCLUSON: nventory ordering poliy using onstant deteriorating items with onstant demand is presented in this paper, method is effiient for getting aurate results, and no doubt it is a bit lengthy and ompliated. While the method is faster and good for mathematial simpliation. Referenes:. Aggarwal,V. and H.Bahari-Kashani(99) Synhronized prodution poliies for deteriorating items iln a delining market.e ransations,3(), Bahari-Kashani,H.(989) Replenishment shedule for deterioratin with time proportional demand.journal of the Operational Researh Soiety,40(), Chung,K.J. and P.S. ing (994) On replenishment shedule for deteriorating items with time proportional demand.prodution Planning and Control,5(4), Chakrabarty, B.C.Giri, K.S.Chaudhuri, An EOQ model for items with Weibull distribution deterioration shortages and trended demand: An extension of Philips model, Computers and Operations Researh 5(998) R.B.Covert, G.S.Philip, An EOQ model with Weibull distribution, AEE ransations 5(973)

7 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September Dave, U. and Patel, L.K. (98)(, S) poliy inventory model for deteriorating items with time proportional demand, Journal of the Operational Researh soiety, 3, Donaldson, W.A. (977) inventory replenishment poliy for a linear trend in demands an analyti solution Operational Researh Quarterly, 8, Goswami,A. and K.S.Chaudhuri (99) Variations of order-level inventory models for deteriorating items.nternational Journal of Prodution Eonomis,7, Goyal,S.K.(987) Eonomis ordering poliy for deterilorating tiems ovesr an infinite time horizon.european Journal of Operationsl Researh,8, P.M.Ghare, G.H.Shrader, A model for exponentially deaying inventory systems, nternational Journal of Prodution Researh (),(963) Hariga, M.A. and Benkherouf, L. Optimal and heuristi inventory replenishment model for deteriorating items with exponential time-varying demand. European Journal of the Operational Researh, 79,(994) Hollier, R.H. and Mak, K.L. nventory replenishment poliies for deteriorating items in a delining market. nternational Journal of Prodution Researh, (6),(983) S.Mukhopadhaya, R.N.Mukherjee, K.S.Chaudhuri.An EOQ model with two parameter weibull distribution deterioration and prie dependent demand, nternational Journal of Mathematial eduation in Siene and ehnology 36(005) R.B.Misra, Optimum prodution lot size model for a system with deteriorating 5. inventory, nternational Journal Of Prodution researh, 3(975) Mak, K.L. (98) A prodution lot size inventory model for deteriorating items. Computers and ndustrial Engineering, 6, G.C.Philip, A generalized EOQ model for items with weibull distribution deterioration. AEE ransations 6(974) Sahan, R.S. (984) On (, Si) poliy inventory model for deteriorating items with time proportional demand. Journal of the Operational Researh soiety, 35(),(984)

8 nternational Journal of Engineering Researh & ehnology (JER) SSN: Vol. ssue 7, September P.R.adikamalla, An EOQ inventory model for items with gamma distribution, AEE ransations 0(978) H.M.Wee, A replenishment poliy for items with a prie dependent demand and a varying rate of deterioration, Prodution Planning &Control 8(997) H.Wee, Eonomi prodution lot size model for deteriorating items with partial blakordering, Computer &ndustrial Engineering 4(993) Xu, H. and Wang, H.P. Optimal inventory poliy for perishable items with time proportional demand. E ransation.4 (5),(99) Sahu, S.K, Dash, D., R.and Sukla, P., K An inventory model for weibull deteriorating items with time dependent demand publiation in the journal of ACCS RESEARCH JOURNAL, Vol. V.No.,(007)