An Epq Model Having Weibull Distribution Deterioration With Exponential Demand and Production With Shortages Under Permissible Delay In Payments

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1 ISSN (Paper ISSN 5-05 (Online Vol., No., 0 An Epq Model Having Weiull Disriuion Deerioraion Wih Exponenial Demand and Producion Wih Shorages Under Permissile Delay In Paymens S.Sarkar * And T.Chakraari Universiy of Calcua Deparmen of Applied Mahemaics, 9, APC Road, Kolkaa ,India. * of he corresponding auhor sanchia77@rediffmail.com Asrac In he fundamenal producion invenory model, in order o solve he economic producion quaniy (EPQ we always fix oh he demand quaniy and he producion quaniy per day. Bu, in he real siuaion, producion is usually dependend on demand. This paper derives a producion model for he lo-size invenory sysem wih finie producion rae, aking ino consideraion he effec of decay and he condiion of permissile delay in paymens. Usually no ineres is charged if he ousanding amoun is seled wihin he permied fixed selemen period. Therefore, i makes economic sense for he cusomer o delay he selemen of he replenishmen accoun up o he las momen of he permissile period allowed y he supplier. In his model shorages are permied and fully ackordered. The purpose of his paper is o invesigae a compuing schema for he EPQ. The model is illusraed wih a numerical example. Keywords Economic producion quaniy, permissile delay, weiull disriuion, deerioraion.. Inroducion For solving he EPQ for each cycle, we always fix oh he demand quaniy and producion quaniy per day in he crisp model. Bu, in he real siuaion, oh of hem proaly will have some lile disurances per day. In recen years, many researchers have sudied invenory models for deerioraing iems such as elecronic componens, food iems, drugs and fashion goods. Deerioraion is defined as decay, change or spoilage ha preven he iems from eing used for is original purpose. There are many iems in which appreciale deerioraion can ake place during he normal sorage period of he unis and consequenly his loss mus e aken ino accoun when analyzing he model. Therefore, many auhors have considered Economic order quaniy models for deerioraing iems.acing as he driving force of he whole invenory sysem, demand is a key facor ha should e aken ino consideraion in an invenory sudy. There are mainly wo caegories demands in he presen sudies, one is deerminisic demand and he oher is sochasic demand. Some noeworhy work on deerminisic demand are : Chung and Lin [,] have considered consan demand, Giri and Chakraary[] and Teng and Chang[4] have considered ime-dependen demand where as Giri and Chaudhuri[6], Bhaacharya[7] and Wu.e.al[8] have worked on invenory leveldependen demand and Wee and Law [9] have considered price-dependen demand. Among hem, ramp ype demand is a special ype of ime-dependen demand. Hill [0] was he firs o inroduce he ramp ype demand o he invenory sudy. Then Mandal and Pal [] inroduced he ramp ype demand o he invenory sudy of he deerioraing iems. Deng.e.al [] and Shah and Jaiswal[] have exensively sudied his ype of demand. In he classical invenory model depleion of invenory is caused y a consan demand rae alone. Bu susequenly, i was noiced ha depleion of invenory may ake place due o deerioraion also. In he early sage of he sudy, mos of he deerioraing raes in he models are consan, Padmanahana and Vra [4], and Bhunia and Maii [5] worked on consan deerioraion rae. In recen research, more and more sudies have egun o consider he relaionship eween ime and deerioraing rae. Wee[8], and Mahapara[9], considered deerioraion rae as linear increasing funcion of ime. Chakraary.e.al[0] have considered hree-parameer Weiull disriuion. In his connecion, sudies of many researchers like Ghare and Schrader [], Goyel e al. [] are very imporan. Misra [] developed a wo parameer Weiull disriuion deerioraion for an invenory model. In recen research, he exensive use of rade credi as an alernaive has een addressed y Goyal[4] who developed an economic order quaniy(eoq model under he condiions of permissile delay in paymens. Chung[5] hen developed an alernaive approach o he prolem. Chand and Ward [6] analyzed Goyal s prolem under assumpions of he classical economic order quaniy model, oaining differen resuls. Nex Aggarwal and Jaggi s [7] model o shorages.there are several ineresing and relevan papers relaed o he delay of paymens such as Chu e al.[9], Chung [0], Hwang

2 ISSN (Paper ISSN 5-05 (Online Vol., No., 0 and Shinn[], Jamal e al[], Sarker e al[,4], Shah[5], Shinn[6], Khouja and Mehrez[7] and here references. However, hese sudies were developed under he assumpion ha he iems oained from an ouside supplier and he enire lo size is delivered a he same ime. In fac, when an iem can e produced in-house, he replenishmen rae is also he producion rae and is hence finie. Hence we amen Goyal s model y considering he replenishmen rae is finie, he difference eween purchasing price and selling cos and aking ino consideraion decay and shorage. In he presen paper, effors have een made o analyze an EPQ model ha have weiull disriuion deerioraion assuming demand rae o e exponenial. Here producion is demand dependen.. Assumpions and Noaions. Assumpions: The demand is aken as exponenial,r (= ae. Rae of producion varies wih demand i.e. P= k R( where k is consan. Replenishmen is insananeous. Lead-ime (i.e. he lengh eween making of a decision o replenish an iem and is acual addiion o sock is assumed o e zero. The assumpion is made so ha he period of shorage is no affeced. The rae of deerioraion a any ime >0 follow he wo parameer Weiull disriuion: where γ (0<γ< is he scale parameer and β(> 0 is he shape parameer. The implicaion of he Weiull rae (wo parameer is ha he iems in invenory sar deerioraing he insan hey are received ino invenory. Shorages are allowed and are fully acklogged.. Noaions: Replenishmen rae is finie and i is demand dependen. Lead ime is zero. T he cycle ime. I( invenory level a ime. C is he holding cos per uni ime. C is he shorage cos per uni ime. C is he uni purchase cos. C 4 is he fixed ordering cos of invenory. θ deerioraion rae of finished iems. M rade credi period. I ( he invenory level ha changes wih ime during producion period. I ( he invenory level ha changes wih ime during non-producion period. I ( he invenory level ha changes wih ime during shorage period. θ ( =γβ β,

3 ISSN (Paper ISSN 5-05 (Online Vol., No., 0 I 4 ( he invenory level ha changes wih ime during he period when shorages are fully acklogged. Mahemaical Modeling and Analysis Here we assume producion sars a =0 a he rae K and he sock aains a level Q a =. The producion sops a = and he invenory gradually deplees o zero a = mainly o mee he demands and parly for deerioraion.now shorages occur and accumulae o he level S a ime =.The producion sars again a a rae K a = and he acklog is cleared a ime =T when he sock is again zero. The cycle hen repeas iself afer ime T. The model is represened y he following diagram: Figure: Graphical represenaion of invenory siuaion Le I( e he invenory level a any ime (0 T and demand rae R( is assumed o e deerminisic and is increasing exponenially wih ime. Furher le R(= a e, 0 <, a>0. The differenial equaions descriing insananeous sae of I( in he inerval[0,t] are ( β + βγ I ( = ae (k - ( d ( β + βγ I ( = -ae d ( ( = -ae d ( 4( = -(k -ae d (4 wih he iniial condiions I ( 0 =0,I ( =Q,I ( =Q, I ( =0,I ( =0, I ( =S, I 4 ( =S, I 4 (T=0. Now solving he aove differenial equaions we ge β+ β+ β+ β+ γ γ γ I ( = a(k { γ } 0 β + β + β+ β+ β+ γ γ β+ γ I ( =-a( γ - β + β + β+ β+ β γ γ +a( β γ -γ - β + β + (5 (6

4 ISSN (Paper ISSN 5-05 (Online Vol., No., 0 a(e e I ( = (7 ( β a(e e I ( = + S T 4 (8 H o ld in g co s o ver h e p erio d [0,T ]= I ( d + I ( d 0 β + β + β + β + γ γ γ γ = C a (k [ ] β + 6 ( β + ( β + ( β + ( β + ( β + β + β + β + β + β + β + ( ( γ ( γ ( γ ( -ac [ ( β + ( β + ( β + ( β + β + β + β + β + γ ( ( γ ( ] + ac [ ( + + ( β + β + β + β + β + β + β + γ ( γ ( γ ( ] + ( β + β + ( β + S horag e C os(s.c o ver he period [0,T ] T = C I(d T = C I(d + C I(d C (k ae a C a (T = C (e e + e ( + T C (k a (e e - + C S(T ( 0 D eerioraing C os = C (P-D d-c D d 0 ac (e e (k (e = ac ( Se up Cos = C 4 ( (9 Case I:M In his siuaion since he lengh of period wih posiive sock is larger han he credi period, he uyer can use he sale revenue o earn ineres a an annual rae I e in (0,T.The ineres earned IE, is I. E = C I a e e d 0 ( C I a ( e e - = However eyond he credi period, he unsold sock is supposed o e financed wih an annual rae I c and he ineres payale I.P. is given y 4

5 ISSN (Paper ISSN 5-05 (Online Vol., No., 0 I.P = C I a e c d (4 M M = C I c a (e - e Toal average Cos (TVC= (Holding Cos+Deeoraion Cos+ Shorage Cos+Seup Cos+Ineres Ineres earned/t payale- TVC=(C 4 +H.C.+S.C.+I.P-I.E /T (5 Case II: M > T Since M > T, he uyer pays no ineres u earns ineres a an annual rae I e during he period (0,M. Ineres earned in his case, denoed y I.E, is given y M I. E = C I a e d + C I a e e e d 0 C I a ( e M e - = (6 Toal average Cos (TVC= (Holding Cos+Deeoraion Cos+ Shorage Cos+Seup Cos-Ineres earned/t TVC=(C 4 +H.C.+D.C.+S.C.-I.E /T (7 Using he iniial condiions he oal average cos ecomes funcion of and T. Hence we find he gloal opimal soluion of oal average cos y using LINGO. 4. Numerical Case (i: When permissile delay period is less han he cycle ime i.e 0<M T hen ineres is earnerd for he period [0,M] u eyond he permissile delay period [M,T] ineres I c is charged on he ousanding amoun. So oal variale cos is calculaed considering oh ineres charged and ineres earned. Le C =$ per uni; C = $.0 per uni ; C 4 = $00 per order ; =.; ea=.6; I c =%, I E =0% ; M=.year ; a= unis, gamma=.0; 0 =.7; =7. Opimal Value $ Case (ii: When permissile delay period is greaer han he cycle ime hen ineres charged I c =0. So he oal variale cos is calculaed considering only he ineres earned. Le C =$ per uni; C = $.0 per uni ; C 4 = $00 per order ; =.4; ea=.6; I E =0% ; M=.year ; a= unis, gamma=.0; 0 =.7; =7. Opimal Value $ Conclusion In he presen paper an EPQ model of deerioraing iems having weiull disriuion deerioraion has een sudied wih permissile delay in paymens. The paper, however considers only one reak in he delay period.a naural exension of he model would e o sudy he case of N reaks in he permissile delay period, i.e. o assume M=M i, if q i- q q i,i=,,..,n, Where M < M < <M N, and q 0 =0<q <.<q N =. 6.References [] Chung.K.J. and Lin.C.N.(00 Opimal invenory replenishmen models for deerioraing iems aking accoun 5

6 ISSN (Paper ISSN 5-05 (Online Vol., No., 0 of ime discouning, Compuers & Operaions Research, Vol. 8, No., pp []Benkherouf.L, Boumenir, A. and Aggoun.L.(00 A diffusion invenory model for deerioraing iems, Applied Mahemaics and Compuaion, Vol. 8, No., pp. 9. [] Giri.B.C., Chakraary.T, and Chaudhuri.K.S.(000 A noe on a lo sizing heurisic for deerioraing iems wih imevarying demands and shorages, Compuers & Operaions Research, Vol. 7, No. 6, pp [4] Teng.J.T, Chang.H.J, DyeC.Y, and Hung.C.H (00 An opimal replenishmen policy for deerioraing iems wih ime-varying demand and parial acklogging, Operaions Research Leers, Vol. 0, pp [5] Chung.K.J, Chu.P., and Lan.S.P.(000 A noe on EOQ models for deerioraing iems under sock dependen selling rae, European Journal of Operaional Research,Vol. 4, pp [6] Giri.B.C. and Chaudhuri.K.S.(998 Deerminisic models of perishale invenory wih sock-dependen demand rae and nonlinear holding cos, European Journal of Operaional Research, Vol. 05, No., pp [7] Bhaacharya.D.K.(005 On muli-iem invenory, European Journal of Operaional Research, Vol. 6, pp [8] Wu.K.S, Ouyang,L.Y. and Yang.C.T.(006 An opimal replenishmen policy for non-insananeous deerioraing iems wih sock-dependen demand and parial acklogging, Inernaional Journal of Producion Economics,Vol. 0, pp [9] Wee.H.M. and Law.S.T.(999 Economic producion lo size for deerioraing iems aking accoun of he imevalue of money, Compuers & Operaions Research, Vol. 6, pp [0] Hill.R.M.(995 Invenory model for increasing demand followed y level demand, Journal of he Operaional Research Sociey, Vol. 46, pp [] Mandal.B. and Pal.A.K.(998 Order level invenory sysem wih ramp ype demand rae for deerioraing iems, Journal of Inerdisciplinary Mahemaics, No., pp [] Deng.P.S.H, Lin.R.H.J, and Chu.P (007 A noe on he invenory models for deerioraing iems wih ramp ype demand rae, European Journal of Operaional Research, Vol. 78, No., pp. 0. [] Shah.Y.K. and Jaiswal.M.C.(977 An order-level invenory model for a sysem wih consan rae of deerioraion, Opsearch, Vol. 4, pp [4] Padmanahana.G. and Vra.P.(995 EOQ models for perishale iems under sock dependen selling rae, European Journal of Operaional Research, Vol. 86, No., pp [5] Bhunia.A.K. and Maii.M.(999 An invenory model of deerioraing iems wih lo-size dependen replenishmen cos and a linear rend in demand, Applied Mahemaical Modelling, Vol., pp [6] Bhunia.A.K. and Maii.M.(998 Deerminisic invenory model for deerioraing iems wih finie rae of replenishmen dependen on invenory level, Compuers & Operaions Research, Vol. 5, No., pp [7] Mukhopadhyay.S, Mukherjee.R.N., and Chaudhuri.K.S, (004 Join pricing and ordering policy for a deerioraing invenory, Compuers & Indusrial Engineering, Vol.47, pp [8] Wee.H.M.(999 Deerioraing invenory model wih quaniy discoun, pricing and parial ackordering, Inernaional Journal of Producion Economics, Vol. 59, pp [9] Mahapara.N.K.(005 Decision process for muliojecive,muli-iem producion-invenory sysem via ineracive fuzzy saisficing echnique, Compuers and Mahemaics wih Applicaions, Vol. 49, pp [0] Chakraary.T, Giri.B.C., and Chaudhuri.K.S.(998 An EOQ model for iems wih weiull disriuion deerioraion,shorages and rended demand: an exension of philip s model, Compuers & Operaions Research, Vol. 5, No. 7 8, pp [] Ghare, P.M., Schrader, G.F.(96, A model for exponenially decaying invenories. Journal of Indusrial Engineering. 4, 8 4. []Goel V.P, Aggarwal S.P,(98 Order level invenory sysem wih power demand paern for deerioraing iems. Proceedings AllIndia Seminar on Operaional Research and Decision Making, Universiy of Delhi, Delhi [] Misra, R.B.(975 Opimum producion lo-size model for a sysem wih deerioraing invenory. In. J. Prod. Res.,

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