Two level production inventory model with exponential demand and time dependent deterioration rate

Transcription

1 Mly Journl of Mtemtik, Vol. S, No., 0-4, 08 wo level production inventory model with exponentil demnd nd time dependent deteriortion rte Nrendr Kumr *, Dhrmendr Ydv nd Rchn Kumri Astrct In this pper, production inventory model is developed y considering two different rtes of production with exponentil demnd rte. It is ssumed tht rte of deteriortion is liner function of time nd shortges re not llowed. Production rtes re different in two different phses of production. Likewise demnd rte lso chnged. Optiml production inventory model is developed to determine the optiml production quntity nd minimized the totl cost, i.e. minimized the sum of production cost, setup cost, holding cost nd deteriortion cost. A numericl exmple for the verifiction of the prolem is lso given. Sensitivity nlysis is lso crried out for the different prmeters. Keywords EPQ, time dependent deteriortion rte, exponentilly incresing demnd, nd different production. Government Girls Degree College, Khrkhud, Meerut, Uttr Prdesh, Indi. College, Bijnore, Uttr Prdesh, Indi. Meerut College, Meerut, Uttr Prdesh, Indi. *Corresponding uthor: nrendr4ysh@gmil.com; dhrmendrydv580@gmil.com; kumrn.inde@gmil.com; Article History: Received 4 Decemer 07 ; Accepted Jnury 08 Vrdhmn Contents Introduction Assumptions nd Nottions Mthemticl Formultion nd it s Solutions Numericl Exmple Sensitivity nlysis Conclusions References Introduction Inventory level decrese with the demnd rtes nd lso, one of the fctor decrese inventory level s well s profit which clled deteriortion nd we cnnot e ignored it. In generl, deteriortion is defined s spoilge, decy, dmge, loss of utility, osolescence etc. like fruits, fertilizer, food items, medicine. Some items hve low rte of deteriortion such s hrdwre, toys, glsswre etc. herefore, rte of deteriortion is lso different for different items. Shh nd Jiswl [] discussed n inventory model y ssuming constnt rte of deteriortion. Misr [] investigted deterministic model c 08 MJM. in which rte of replenishment is finite, shortge not llowed nd rte of deteriortion considered oth vrile nd constnt. Chudhri nd Chudhri [] developed n inventory model. hey considered finite rte of replenishment nd constnt rte of deteriortion. Blki nd Benkherouf [4] developed production lot size inventory model in their studied they considered ritrry production rte nd time dependent demnd. Bhuni nd Miti [5] investigted deterministic inventory model for vriles production. hey lso ssumed production rte depends on either demnd or on hnd inventory. Cho nd Chng [6] developed production inventory model, they ssumed tht vriles demnd rte, production is proportionl to demnd without shortge nd no deteriorting items. Wee nd Wng [7] introduced production inventory model y ssuming finite rte of production, time dependent demnd, deteriorting items nd shortges. Sn et. l. [8] presented production inventory model for deteriorting items over finite plnning horizon with finite rte of production, time vrying liner demnd nd constnt deteriortion rte. hey lso considered completely cklogged shortges. Yng nd Wee [9] investigted n integrted multi-lot-size production inventory model in which rte of production nd demnd re constnt with deteriorting items. Avikr et l. [0] presented

2 wo level production inventory model with exponentil demnd nd time dependent deteriortion rte /4 production inventory model for deteriorting items with exponentilly incresing demnd over fixed time horizon, rte of production nd deteriortion oth constnt. In the present study, we consider two different rtes of production. Sivshnkri nd Pnyppn [] developed production inventory model for two level production with deteriortive items nd shortges. hey considered rte of production, deteriortion nd demnd ws constnt over time horizon. Islm et l. [] presented production model with constnt production rte nd ssumed tht mrket demnds re exponentilly decresing. Ptr nd Mity [] discussed production inventory model in which vrile production rte, deteriortion nd two type of demnd rtes for defective nd non-defective items re considered. In this pper, presented two level production oth rte of production is constnt, demnd is the function of time which is incresing exponentilly nd rte of deteriortion is lso liner function of time. suppose tht I is mximum inventory level. During the time intervl [ ], the production rte is λ P nd demnd rte is λ et where λ is constnt nd λ > 0 nd hence, inventory level increses in this intervl t the rte λ [P et ]. At time production stopped nd suppose tht I is mximum inventory t tht time. Due to demnd nd deteriortion inventory level strts to decrese nd fter time inventory ecomes zero. Let I(t e the inventory level t ny time t where 0 t. hen the model is governing y the following differentil equtions ( ti = P et 0 t t ( ti = λ (P e ( ti = e t t t (. (. (. he oundry conditions of ove Differentil equtions re given y. Assumptions nd Nottions I(0 = 0, I( = I, I( = I, I( = 0 (.4 Assumptions ( he demnd rte is exp(t, where nd re oth constnt such tht > 0 nd > 0. ( wo rtes of production re considered. (c Production run only single product. (d he production rte is lwys greter thn or equl to sum of the demnd rte nd deteriortion rte. (e Rte of deteriortion is liner function of time. (f Shortges re not llowed. Nottions ( P is initil production rte in units per unit time. ( I : On-hnd inventory level t time. ( I : On-hnd inventory level t time. (4 I Optiml inventory. (5 C p production cost per unit time. (6 Ch Holding cost per unit per yer. (7 C0 Setup cost per set up. (8 t Rte of deteriortion where 0 <,. (9 cycle time. (0 i : unit time in period, i =,,... ( c : otl cost. Figure Using the oundry condition (.4 nd neglecting second nd higher power of nd, eing nd re very smll then, the solution of eqution (., (. nd (. re given y respectively I= t t et P t ( t et t t [ ( et ] 0 t λ t t t ( e λ P t I=. Mthemticl Formultion nd it s Solutions λ et t ( t λ e t ( e P ( λ ( e λ e ( e P( λ In mthemticl formultion of production inventory model for two level production with ove descried ssumptions nd nottions is depicted in Figure. Let the cycle strts t time t = 0, nd during the time intervl [0 ], the production rte is P nd the demnd rte is et such tht P > et nd the inventory level increses t the rte P et. At time (.5

3 wo level production inventory model with exponentil demnd nd time dependent deteriortion rte /4 e ( e P ( λ λ e t (.6 t e t et (t I= e [ ( ] θ θ e θ e ( (e (e Cp eθ ( λ Pθ C p ( θ ( θ {( C p ( } [{ ( }e t { ( e ] he totl cost function c involved two vriles nd. he vlues of nd for which the cost function c minimize is otined y solving the simultneous equtions (.7 otl cost: he totl cost is the sum of the production cost, setup cost, holding cost nd deteriortion cost. Let us consider = θ, where θ > 0. otl Cost Function c = (.8 c =0 c =0 Provided tht c > 0, c > 0 nd C p C0 [( λ eθ ( λ e e ] Ch [ (θ nd (.9 c c c >0 (.0 θ Pθ (6 θ θ 4 {θ (6 θ } 6 4 { (θ }eθ eθ {λ P( ( λ Pθ 6 Equtions (.9 nd (.0 re solved numericlly with the help of Mthemtic. 4. Numericl Exmple Let us consider n inventory model with the following : P = 4000, = 600, = 0., = 0.0, = 0., θ = 0.4, λ =, C0 = 80, C p = 40, Ch =. On the sis of ove cost prmeters, we otin the optiml solution s: = 5.8, I = 09, =.46, =.654, I = 48.8, I = 77.8, c = 9. Production cost = 958.4, Setup cost =.7, Holding cost = 6706., Deteriortion cost = 440.8, }( θ eθ {P ( λ Pθ } 6 ( θ P ( λ ( θ 4 4 (e θ eθ 4 4 { ( λ λ ( }(e eθ { }( { e ( e } 5. Sensitivity nlysis ( Incresing of deteriortion, cycle time, optiml inventory I, production time ( nd, mximum inventory I nd totl cost c decreses ut mximum inventory I increses. ( Incresing of setup cost, cycle time, optiml inventory I, production time ( nd, mximum inventory I nd I nd totl cost c remin unchnged. e {( ( } ( Incresing of holding cost, cycle time, optiml in6 ventory I, production time ( nd, nd mximum Pθ C p ( (e e ] ( θ inventory I decreses ut totl cost c nd mximum 6 inventory I increses. C p [(θ ( θ (4 Incresing of production cost, cycle time, optiml inventory I, production time ( nd, mximum { ( θ }eθ ] inventory I nd totl cost c increses ut mximum P λc p P inventory I decreses. ( θ ( θ

4 wo level production inventory model with exponentil demnd nd time dependent deteriortion rte /4 le. Vrition in rte of deteriorting items with inventory nd totl cost I Production cost Setup cost Holding cost Deteriorting cost otl cost le. Effect of demnd nd cost prmeters on optiml vlues I I I c C Ch Cp θ λ

5 wo level production inventory model with exponentil demnd nd time dependent deteriortion rte 4/4 6. Conclusions In this pper, we proposed two level production inventory models with exponentil demnd nd time dependent deteriortion rte. he ojective is to determine optiml production policy nd minimize the totl inventory cost. he vrition in production rte provides wy resulting consumer stisfction nd erning potentil profit. he present model differs from the existing models, s two-level production nd exponentilly incresing demnd re considered here. A numericl exmple is provided to demonstrte mens prcticl usge. his model cn e extended y tking more relistic ssumptions such s shortge, other types of demnd ptterns etc. []????????? ISSN(P:9-786 Mly Journl of Mtemtik ISSN(O:-5666????????? References [] [] [] [4] [5] [6] [7] [8] [9] [0] [] [] stnt production where shelf-life of the product is finite, Open Journl of Applied Sciences, 06(06, 0. K. Ptr nd R. Mity, A single item inventory model with vrile production rte nd defective items, Interntionl Journl of Applied nd Computtionl Mthemtics, 0(07,. Y.K. Shh nd M.C. Jiswl, Order level inventory model for system of constnt rte of deteriortion, Opserch, 4(977, R.B. Misr, Optimum Production Lot-size Model for System with deteriting inventory, Interntionl Journl of Production Reserch, 5(975, M. Roy Choudhri nd K. S. Chudhri, An order level inventory model for deteriorting items with finite rte of replenishment, Opserch, 0(98, Z.. Blki nd L. Benkherouf, A production lot size inventory model for deteriorting items nd ritrry production nd demnd rtes, Europen Journl of Opertionl Reserch, 9(996, A.K. Bhuni nd M. Miti, Deterministic inventory models for vriles production, Journl of Opertionl Reserch Society, 48(997, 4. S.U. Cho-on nd L, Chng, A production inventory model for vrile demnd nd production, Yugoslv Journl of Opertions Reserch, (999, H.M. Wee nd W.. Wng, A vrile production scheduling policy for deteriorting items with time vrying demnd, Computers & Opertions Reserch, 6(999, S. Sn, S.K. Goyl nd K.S. Chudhri, A production inventory model for deteriorting item with trended demnd nd shortges, Europen Journl of Opertionl Reserch, 57(004, P.C. Yng nd H.M. Wee, Computers nd Opertions Reserch, 0(00, Avikr, R.K. uli nd A. Shrm, Optiml inventory mngement for exponentilly increse demnd with finite rte of production nd deteriortion, IJCS, ((0, C.K. Sivshnkri nd S. Pnyppn (05, Production inventory model for two-level production with deteriortive items nd shortges, Int. Adv. Mnuf. echonl., 76(05, M.E. Islm, S.I. Ukil nd M.S. Uddin, A time dependent inventory model for exponentil demnd rte with con4