An Inventory Model of Repairable Items with Exponential Deterioration and Linear Demand Rate

Transcription

1 IOSR Journal of Mahemaics (IOSR-JM) e-issn: , p-issn: X. Volume 1, Issue Ver. IV (May - June 017), PP An Invenory Model of Repairable Iems wih Exponenial Deerioraion and Linear Demand Rae U. B. Gohi 1, Malav Joshi, Kiran Parmar 1 Head & Associae Prof.,Dep. of Saisics, S. Xavier s College (Auonomous), Ahmedabad, Gujara, India. Research Scholar, Dep. of Saisics, S. Xavier s College (Auonomous), Ahmedabad, Gujara, India. Assian Prof., Dep. of Saisics, S. Xavier s College (Auonomous), Ahmedabad, Gujara, India. Absrac: In his paper an invenory model for deerioraing and repairable iemsis developed wih linear demand. An Exponenial disribuion is used o represen he disribuion of ime for deerioraion.in he model considered here, shorages are allowed o occur and defecive iems can be repaired. The model is solved analyically o obain he opimal soluion of he problem. The derived model is illusraed by a numerical example and is sensiiviy analysis is carried ou. I. Inroducion Deerioraion means damage, spoilage, dryness, vaporizaion, ec. I is defined as decay or damage such ha he iem canno be used for is original purposes. The effec of deerioraion is very imporan in many invenory sysems. The pioneering work of Harris [7] invenory models are being reaed by mahemaical echniques. He developed he simples invenory model, he Economic Order Quaniy (EOQ) model which was laer popularized by Wilson [4]. Relaxaion of some assumpions in he formulaion of he EOQ model led o he developmen of oher invenory models ha effecively ackles several oher invenory problems occurring in day-o-day life. Invenory of deerioraing iems was firs sudied by Whiin [5] where he considered he deerioraion of fashion goods a he end of prescribed sorage period. Ghare and Schrader [5] exended he classical EOQ formula o include exponenial decay, wherein a consan fracion of on hand invenory is assumed o be los due o deerioraion. Daa, T. K. and Pal, A. K. [4]gave order level invenory sysem wih power demand paern for iems wih variable rae of deerioraion. Mandal, B. N. and Ghosh, A. K. [14]wroe a noe on an invenory model wih differen demand raes during sock in and sock ou period. Mandal, B. N. and Pal A. K. [15]have given order level invenory sysem for perishable iems wih power demand paern. Teng, J. T. [1]has worked on model wih linear rend in demand. Manna and Chiang [1] developed an EPQ model for deerioraing iems wih ramp ype demand. Teng and Chang [] considered he economic producion quaniy model for deerioraing iems wih sock level and selling price dependen demand. Jain e al. [8] developed an economic producion quaniy model wih shorages by incorporaing he deerioraion effec and sock dependen demand rae. Roy and Chaudhary [17] developed wo producion raes invenory model for deerioraing iems when he demand rae was assumed o be sock dependen. In he research of Sana e al. [18] shorages are allowed o occur a he end of a cycle. Wih he consideraion of ime varying demand and consan deerioraing rae, he opimal producion invenory policy was sudied. Raman Pael [16] developed a producion invenory model for deerioraing iems following Weibull disribuion wih price and quaniy dependen demand and varying holding cos wih shorages. Boh Skouri and Papachrisos [0] and Chen e al. [] developed a producion invenory model in which he shorages are allowed a he beginning of a cycle. In conras,manna and Chaudhari [1] have allowed shorages o occur a he end of each cycle. Goyal[6] deals wih producion invenory problem of a produc wih ime varying demand, producion and deerioraion raes in which he shorages occur a he beginning of he cycle. There are four synonyms of reuse according o Thierry e al. []. They are: Direc Reuse, Repair, Recycling and Remanufacuring. Schrady[19] was he firs o consider reuse in a deerminisic model.recenly, Mabini e al. [11] exended Schrady s model o consider sock-ou service level consrains and muli-iem sysem sharing he same repair faciliy. In he policy, expressions for he opimal conrol parameer values were derived. Koh[10] developed a join EOQ and EPQ model in which he saionary demand can be saisfied by recycled producs and newly purchased producs. The model assumes a fixed proporion of he used producs ha are colleced from he cusomers. Recenly, Bhojak and Gohi [1] have developed invenory models for amelioraing and deerioraing iems wih ime dependen demand and IHC. Kiran Parmar and Gohi [9] have developed an EPQ model of deerioraing iems using hree parameer Weibull disribuion wih consan producion rae and ime varying DOI: / Page

2 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand holding cos.devyani Chaerji and Gohi [] have developed hree-parameric Weibully deerioraed EOQ model wih price dependen demand and shorages under fully backlogged condiion. R. K. Yadav and Rajeev Kumar [6] developed an invenory model for deerioraing and repairable iems wih linear demand.here, we have ried o redevelop he same model and correced mos of he resuls. II. Noaions The mahemaical model in his paper is developed using he following noaions: 1. k : Producion rae (unis/uni ime).. D : Demand rae.. θ() : The deerioraion rae (unis/uni ime). 4. b :The decreasing rae of he demand (unis/uni ime). 5. c : Fracion of defecive produc. 6. r :Fracion of sock-ou demand sales los due o some sock-ou demands.(0 < r < 1) 7. Q() :The insananeous sae of he invenory level a any ime (0 T). 8.Q 1 : The maximum invenory level of he produc. 9. Q : The maximum invenory level during shorage period. 10. A : Ordering cos per order. 10. C h :Invenory holding cos per uni per uni ime. 11. C d : Deerioraion cos per uni per uni ime. 1. C s :Shorage cosper uni. 1. C p : The penaly cos of a los sale including los profi (uni ime). 14. Pc : Purchase cos per uni. 15. TC : The average oal cos for he ime period [0, T]. III. Basic Assumpions The model is derived under he following assumpions 1. The invenory sysem deals wih single iem.. The producion rae is finie and consan, which is larger han he demand rae and is unaffeced by he lo size.. Once a uni of he produc is produced, i is available o mee he demand. 4. Once he producion is sared he produc sars being deerioraed. 5. The annual demand rae is a linear funcion of ime and i is D = α + β. (α, β > 0). 6.Shorages are allowed and compleely backlogged. 7. Replenishmen rae is infinie and insananeous. 8. All defecive producs can be repaired and reused. 9. The second and higher powers of θ, b and care negleced in he analysis of he derived model. 10. Toal invenory cos is a real, coninuous funcion which is convex o he origin. IV. Mahemaical Model And Analysis Here, we consider a single commodiy deerminisic producion invenory model wih a ime dependenlinear demand rae. The disribuion of he ime o deeriorae is random variable following he exponenial disribuion. The probabiliy densiy funcion for exponenial disribuion is given by f () e ; ( > 0 and 0 < θ < 1) The insananeous rae of deerioraion θ() of he non-deerioraed invenory a ime can be obained from f () (), where F() = 1 F() 1 e is he cumulaive disribuion funcion for he exponenial disribuion.thus, he insananeous rae of deerioraion of he on-hand invenory is (). The probabiliy densiy funcion represens he disribuion of he ime o deeriorae which may have a decreasing, consan or increasing rae of deerioraion. Iniially, invenory level is zero. A ime = 0, he producion sars and simulaneously supply also beginsandhe producion sops a = 1 when he maximum invenory level Q 1 is reached. In he inerval [0, 1 ], before he producion sops, he invenory is buil up a a rae k D and is depleed a he rae (θ b + c). In he inerval[ 1, ] he invenory is depleed a he rae D and rae (θ b). The invenory is finiely decreasing in he ime inerval [ 1, ] unil invenory level reaches zero. I is decided o backlog he demands up o Q level which occurs during sock-ou ime. Thereafer, shorages can occur during he ime inerval [, ], and all of he demand during he period [, ] is compleely backlogged. Thereafer, producion is sared a a rae (1 r)(k D)so as o clear he backlog, and he invenory level reaches o 0 (i.e. he backlog is cleared) a =T. The picorial presenaion is shown in he Figure 1. DOI: / Page

3 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand Figure 1: Graphical presenaion of he invenory sysem The differenial equaions which govern he insananeous sae of Q() over he ime inervals [0, 1 ], [ 1, ], [, ] and [, T] are given by dq() ( b c) Q( ) k ( ), 0 1 d (1) dq() ( b) Q( ) ( ), 1 d () dq() (1 r)( ), d () dq() (1 r) c Q( ) (1 r) k ( ), T d (4) Under he boundary condiions Q(0) = 0, Q( 1 ) = Q 1, Q( ) = 0, Q( ) = Q &Q(T) = 0, he soluions of equaions(1) o (4)are given by Q( ) ( k ) 0 (5) Q( ) (1 ) ) (6) 1 Q( ) (1 r) ( ) ( ) DOI: / Page 1 1 (7) T (8) k 1 k T 1 Q( ) (1 r) (1 T )(1 ) [Where δ = θ b + c, ξ = θ b and μ = (1 r). c] From (5), Q( 1 ) Q ( k ) 1 1 (9) Q ( ) Q (1 ) ) (10) and from (6), Eliminaing Q( 1 ) from equaions (9) and (10), we ge ( ) 1 k (11)

4 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand Thus, 1 can be wrien in erms of and so 1 is no a decision variable. 1 From (7), Q ( ) Q (1 r) ( ) ( ) and from (8), Q( ) Q (1 r) Eliminaing Q( ) from equaions (1) and (1), we ge (1) k 1 k T 1 (1 T)(1 ) 1 k 1 T T k T (14) Thus, can be wrien in erms of and so is alsono a decision variable. Cos Componens: The oal cos per replenishmen cycle consiss of he following cos componens: 1) Ordering Cos (OC) The ordering cos OC over he period [0, T] is OC = A (Fixed) (15) ) Deerioraion Cos (DC) The deerioraion cos DC over he period [0, 1 ] and [, T] is T 1 DC C d c Q ( ) d (1 r ) Q ( ) d 0 DC C c d ) Invenory Holding Cos (IHC) 1 (1 r) 1 k T k 1 (1 )(1 ) 1 T r k T ( ) T DOI: / Page (1) (1 T )(1 r) ( T ) 1 ( k ) (1 r ) (1 r ) ( T ) The invenory holding cos IHC over he period [0, ] is 1 IHC C h Q( ) d Q( ) d IHC Ch ( k ) 1 ( )( 1 ) ( 1) ( 1) (17) 4) Shorage Cos (SC) Demand during he ime [, T] is saisfied a a ime as he producion hasalready sared a ime = and so shorage cos during his inerval is no aken ino accoun. The shorage cos SC over he period [, ] is (16)

5 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand SC C Q () d s SC Cs 1 r ( ) ( ) 6 (18) 5) Los Sale Cos (LSC) The los sale cos LSC over he period [, ] is LSC C r ( ) d p 1 ( ) LSC C r p (19) 6) Purchase Cos (PC) The purchase cos PC over he period [0, T] is PC P k( ) c 1 ( ) 1 PC Pc k k Hence, he oal cos per uni ime for he ime period [0, T] is given by k T T k T 1 1 TC OC DC IHC SC LSC PC (1) T Now, our objecive is o deermine opimum values * and T* of and T respecively o minimize he oal cos TC TC. Using mahemaical sofware, he opimal values * and T* can be obained by solving 0 and TC 0 T which can saisfy he following sufficien condiions: (0) TC TC TC 0 T T * *,TT TC 0 * *,TT () DOI: / Page

6 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand V. Numerical Example To illusrae he proposed model, an invenory sysem wih he following hypoheical values is considered. By aking α = 40, β = 0.001, C h = 0.8, C d =, C p = 5, Cs =, θ = 0.01, b = 0.005,c = 0.06, r = 0.09, k = 60 and A = 00(wih appropriae unis), opimal values of and T are * = ,T* = unis and he opimal oal cos per uni ime TC = unis. VI. Sensiiviy Analysis Sensiiviy analysis depics he exen o which he opimal soluion of he model is affeced by he changes in is inpu parameer values. Here, we sudy he sensiiviy for he cycle lengh T and oal cos per ime uni TC wih respec o he changes in he values of he parameers α, β, C h, C d, Cp, Cs, θ, b, c, r, k and A. The sensiiviy analysis is performed by considering differen values in each one of he above parameers keeping all oher parameers as fixed. The resuls are presened in he Table. Table:Parial Sensiiviy Analysis Parameer % T TC % change in TC α β C h C d C p Cs θ b c r k A DOI: / Page

7 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand VII. Graphical Presenaion Figure Figure VIII. Conclusions From he Tablewe can conclude ha TC is highly sensiive o he change in C p and k, moderaely sensiive o he change in α and C d and less sensiive o change in β, C h, Cs, θ, b, c, A and r. I is observed from Figure ha he effec of increase or decrease in he values of α does no affeche oal cos TC much. I is observed fromfigure ha when he values of A, C h, C d, Cp Cs and k increase simulaneously he average oal cos TC also increases. I is observed from Figure ha when he values of b increase hen TC also increases and when he values of β, θ, c and r increase, TC decreases. References [1] Bhojak, A., Gohi, U.B. A noe on an order-level model for a sysem wih consan rae of deerioraion. Inernaional Journal of Managemen and Social Science Research Review, 1(), 016, 85 98p. [] Chen S. J., Chen S. P., and Liu X. B., A producion invenory model for deerioraing iems, Journal of Zhaoqing Universiy, Vol., No., 00, pp [] Chaerji, D. A. and Gohi, U. B., Three-parameric Weibully deerioraed EOQ model wih price dependen demand and shorages under fully backlogged condiion, Inernaional Journal of Innovaive Research in Science, Engineering and Technology, Vol. 4, Issue 1, 015, [4] Daa, T. K. and Pal, A. K. Order level invenory sysem wih power demand paern for iems wih variable rae of deerioraion. Ind Journal of Pure & Applied. Mahs, 19 (11),1988, DOI: / Page

8 An Invenory Model Of Repairable Iems Wih Exponenial Deerioraion And Linear Demand [5] Ghare, P. M., and Schrader, G.F., A model for an exponenially decaying invenory, Journal of Indusrial Engineering, 14, pp. 196, 8 4. [6] Goyal and Giri, The producion-invenory problem of a produc wih ime varying demand, producion anddeerioraion raes, European Journal of Operaional Research, Vol. 147, 00, pp [7] Harris F.W., Operaions and cos, 1915, Shaw Company, Chicago: A. W. [8] Jain, M. and Sharma, G. C., Rahore, S., Economic order quaniy model wih shorages and sock dependen demand for deerioraing iems, IJE Transacions A: Basics, Vol. 0(), 007, pp [9] Kiran Parmar and Gohi U. B.,An EPQ model of deerioraing iems using hree parameer Weibull disribuion wih consan producion rae and ime varying holding cos, Inernaional Journal of Science, Engineering and Technology Research, Vol. 4, Issue, 015, pp [10] Koh, S.G., Hwang, H., Sohn, K.I., & Ko, C.S. An opimal ordering and recovery policy for reusable iems, Compuers & Indusrial Engineering, 4, 00, [11] Mabini, M.C., Pinelon, L.M., & Gelders, L.F. EOQ ype formulaion for conrolling repairable invenories, Inernaional Journal of Producion Economics, 8, 199, 1-. [1] Manna and Chaudhuri, An EOQ model wih ramp ype demand rae, ime dependen deerioraion rae, Uni producion cos and shorages, European Journal of Operaional Research, Vol.171, 006, pp [1] Manna, S. K. and Chiang, C., Economic producion quaniy models for deerioraing iem wih ramp ype demand, Inernaional Journal Operaion Research, Vol. 7, 010, pp [14] Mandal, B. N. and Ghosh, A. K., A noe on an invenory model wih differen demand raes during sock in and sock and period. In. Journal of Mg. & Sys, 7 (1), 1991, -6. [15] Mandal, B. N. and Pal A. K., Order level invenory sysem for perishable iems wih power demand paern In. Journal of Mg.& Sys. 16 (), 000, [16] Raman Pael, Producion invenory model for Weibull deerioraing iems wih price and quaniy dependen demand, ime varying holding cos and shorages, Indian Journal of Applied Research, Vol. 4, Issue 1, 014. [17] Roy, T. and Chaudhary, K. S., A producion invenory model under sock dependen demand, Weibull disribuion deerioraion and shorages, Inernaional Transacions in Operaions Research, Vol. 16, 009, pp [18] Sana, S. Goyal, S. K. and Chaudhuri, K. S., A producion - invenory model for a deerioraing iem wih rended demand and shorages, European Journal of Operaional Research, Vol. 157, 004, [19] Schrady, D.A. A Deerminisic invenory model for repairable iems, Naval Research Logisics Quarerly, 14, 1967, [0] Skouri, K and Papachrisos, S., Opimal sopping and resaring producion imes for an EOQ model wih deerioraing iems and ime-dependen parial backlogging, Inernaional Journal of Producion Economics, Vol. 81 8, 00, pp [1] Teng, J. T., A deerminisic replenishmen model wih linear rend in demand. Opns. Res. Le. 19,1996,-41. [] Teng, J.T. and Chang, H.T., Economic producion quaniy model for deerioraing iems wih price and sock dependen demand; Compuers and Oper. Res., Vol., 005, pp [] Thierry, M.C., Salomon, M., Van Numen, J.,& Van Wassenhove, L. Sraegy issues in produc recovery managemen, California Managemen Review, 7(), 1995, [4] Wilson R.H., A scienific rouine for sock conrol, Hary Bus Rev 1, 194, pp [5] Whiin, T.M., The heory of Invenory Managemen, nd ediion, Princeon Universiy Press, Princeon, New Jersey, 1957, pp [6] Yadav, R. and Rajeev Kumar, An invenory model for deerioraing and repairable iems wih linear demand, vol. 10 (6), 014, DOI: / Page