Key words: EOQ, Deterioration, Stock dependent demand pattern

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2 An Invenory Model Wih Sock Dependen Demand, Weibull Disribuion Deerioraion R. Babu Krishnaraj Research Scholar, Kongunadu Ars & Science ollege, oimbaore amilnadu, INDIA. & K. Ramasamy Deparmen of Mahemaics Kongunadu Ars & Science ollege, oimbaore amilnadu, INDIA. Absrac An invenory problem can be solved by using several mehods saring from rial and error mehods o mahemaical and simulaion mehods. Mahemaical mehods help in deriving cerain rule and which may sugges how o minimize he oal invenory cos in case of deerminisic demand. Here an aemp has been made for obaining a deerminisic invenory model for sock dependen demand paern incorporaing woparameer Weibull disribuion deerioraion and wih reserve invenory. Key words: EOQ, Deerioraion, Sock dependen demand paern 79

3 Inroducion A number of researchers have worked on invenory wih consan demand rae, ime varying demand paerns. A few of he researchers have considered he demand of he iems as sock dependen demand paern. Daa and Pal [3] have developed an order level invenory sysem wih power demand paern, assuming he deerioraion of iems governed by a special form of Weibull densiy funcion ( ) ;,. hey used special form of Weibull densiy funcion o siderack he mahemaical complicaions in deriving a compac EOQ model. Gupa and Jauhari [4] has developed an EOQ model for deerioraing iems wih power demand paern wih an addiional feaure of permissible delay in paymens. hey also used special form of Weibull densiy funcion for deerioraion of iems. A sep forward o special form of Weibull densiy funcion, here we shall develop an EOQ model wih sock dependen demand paern and using acual form of Weibull densiy funcion Z(), where ( ), for deerioraion of iems. Despie of all mahemaical inricacies, expressions for various invenory parameers are obained. Here we shall develop he same problem wih sock dependen demand paern wih reserve invenory. 8

4 Assumpions and Noaions noaions. Invenory model is developed under he following assumpions and Assumpions Replenishmen rae is infinie. he leadime is zero. Shorages in invenory are no allowed. he demand is given by he sock dependen demand paern for which, demand upo ime is assumed o be, D() abi(), Where D is he demand size during he fixed cycle ime, and is he demand rae a ime he rae of deerioraion a any ime follows he woparameer Weibull disribuion Z(), where ( ), is he scale parameer and ( ) is he shape parameer. Noaions: he fixed lengh of each ordering/producion cycle. he holding cos, per uni ime. 3 he cos of each deerioraed uni. he reserve invenory cos per uni ime. 8

5 Developmen of he Model Le Q be he number of iems produced or purchased a he beginning of he cycle and (QS) iems be delivered ino he reserve invenory sock. Balance of S iems as he iniial invenory of he cycle. I will be he iniial invenory a ime and d be he demand during period. Now, he invenory level S gradually falls during ime period (, ), due o demand and deerioraion. A ime invenory level becomes zero. Shorages are fulfilled from he reserve sock (QS), afer he period. Le I() be he onhand invenory, hen he various saes of he sysem are governed by he following differenial equaions: di() d I( ) ( a bi( )) Where Z() di() d () () a (3) Using () in (), he Soluion of I() is, I() e ( b ) e e ( b) d ( b ) ( b ) ae d ( b ) ( b ) ( b ) I() Se ae e d (4) ( b) Solving furher, on expanding e ( b), as ( ) gives I() b b b ( ) ( Se ae ) [ ] (5) Soluion of (3) using I ( ) gives, 8

6 I() a( ) (6) Using I( ), in (5) gives, S b a [ ] he oal amoun of deerioraed unis, (7) S ( a bi ( )) d Using (7) in he above, he oal amoun of deerioraed unis in [, ] b a [ ] b [ ] a b a [ ] b [ a ] Average oal cos per uni is given by (S, ) 3 I () d (8) I () d (9) Subsiuing he values of I() from (5) and (6), eliminaing S using (7) and inegraing yields, (S,) 3 ( ( b) ( b b Se ae ) [ ]) d ( a( ) )d 83

7 3 ( b ( b) ( b) b a[ ] e ae [ ] )d a( ) d 3 b a[ ][ ( b)] b a[ ( b)][ ] d a( ) 3 d b a b b b [ ] 3 b b b b [ ] d a [ ] 3 a b ( ) b) b b b b b ( )( ) 6 ( 3) ( )( 3)

8 a [ ] a b b ( b) 3 8 ( )( ) a ( 3) ( )( 3) 3 b 3 (neglecing higher powers of,... ) () Furher, for he minimizaion of he cos, we se d( ) d a b (( b) )( ) b 3 3 ( ( ) )( 3) ( ) b a [ ] On solving i, we obain value of and le his value of be he opimum value of Solving he opimum value of in (7), opimum value of S is, S * onclusion * * * b a [ ] From he above work, An invenory model is sudied using Weibull disribuion. Wih Sock dependen demand rae, we have obained Opimal value of S, wih minimum cos. 85

9 REFERENES. over, R.P. and Philip, G..: An EOQ model for iems wih Weibull disribuion deerioraion, AIEE ransacion, 5 (973), pp hakrabory,., Giri, B.. and haudhuri, K.S.: An EOQ model for iems wih Weibull disribuion deerioraion, shorage and rended demand, ompuers and Operaions Research, 5 (998), pp Daa,.K. and Pal, A.K.: Order level invenory sysem wih power demand paern for iems wih variable rae of deerioraion, Indian J. Pure Appl. Mah, 9(), (998), pp Gupa, P.N. and Jauhari, R. : An EOQ model for deerioraing iems wih power demand paern subjec o permissible delay in paymens, Gania Sandesh, 9(), (995), pp Jalan, A.K., Giri, R.R. and haudhuri, K.S.: EOQ model for iems wih Weibull disribuion deerioraion, shorage and rended demand, Inyernaional J. of Sysem Sciences, 7 (996). 6. Jalan, A.K., Giri, R.R. and haudhuri, K.S. : EOQ model for iems wih Weibull disribuion deerioraion, shorage and ramp ype demand, Recen developmen in O.R., Narosa Pub. House, New Delhi (). 7. Sanjay Jain and Mukesh Kumar: An Invenory model wih power demand paern, weibull disribuion Deerioraion and shorages, Journal of Indian Acad., Mah, Vol.3. No. (8), pp Babu Krishnaraj. R and Ramasamy. K: An Invenory model wih power demand paern, weibull disribuion deerioraion and wihou shorages, Bullein of Sociey of Mahemaical Services and Sd., Vol., Issue.. (), pp