Applied Mahemaical Sciences, Vol. 4, 00, no. 7, 36-369 Deerioraing Invenory Model wih Time Dependen Demand and Parial Backlogging Vinod Kumar Mishra Deparmen of Compuer Science & Engineering Kumaon Engineering College Dwaraha, Almora, - 63653, (Uarakhand), India vkmishra005@gmail.com Lal Sahab Singh Deparmen of Mahemaics & Saisics Dr.Ram Manohar Lohia Avadh Universiy Faizabad-400, (Uar Pradesh), India singhdrlalsahab7@gmail.com Absrac In his paper, a deerminisic invenory model is developed for deerioraing iems in which shorages are allowed and parially backlogged. Deerioraion rae is consan, Demand rae is linear funcion of ime, backlogging rae is variable and is dependen on he lengh of he nex replenishmen. The model is solved analyically by minimizing he oal invenory cos. Mahemaics Subjec Classificaion: 90B05 Keywords: Invenory, deerioraing iems, shorages, ime dependen demand, parial backlogging..0 Inroducion Deerioraion is defined as decay, damage, spoilage evaporaion and loss of uiliy of he produc. Deerioraion in invenory is a realisic feaure and need o consideraion i. Ofen we encouner producs such as fruis, milk, drug, vegeables, and phoographic films ec ha have a defined period of life ime. Such iems are referred as deerioraing iems. The loss due o deerioraion canno be avoided. Due o deerioraion, invenory sysem faces he problem of shorages Elecronic copy available a: hp://ssrn.com/absrac=804874
36 V. K. Mishra and L. S. Singh and loss of good will or loss of profi. Shorage is a fracion of hose cusomers whose demand is no saisfied in he curren period reacs o his by no reurning he nex period. Invenory in deerioraing iems firs considered by Wihin [957], he considered he deerioraion of fashion goods a he end of prescribed sorage period. In 963 Ghare and Schrader exended he classical EOQ formula wih exponenial decay of invenory due o deerioraion and gave a mahemaical modeling of invenory in deerioraing iems. Dave and Pael [98] developed he firs deerioraing invenory model wih linear rend in demand. He considers demand as a linear funcion of ime. Nahmias [98] gave a review on perishable invenory heory. He reviewed he relevan lieraure on he problem of deermining suiable ordering policies for boh fixed life perishable invenory, and invenory subjeced o coninuous exponenial decay. Rafaa [99] gave a survey of lieraure on coninuously deerioraing invenory models and he considered he effec of deerioraion as a funcion of he on hand level of invenory. He focused o presen an up-o-dae and complee review of he lieraure for he coninuously deerioraing mahemaical invenory models. Bu all researchers assume ha during shorage period all demand eiher backlogged or los. In realiy i is observed ha some cusomers are willing o wai for he nex replenishmen. Abad [996] considered his phenomenon in his model, opimal pricing and lo sizing under he condiions of perishable and parial backordering. He assumed ha he backlogging rae depends upon he waiing ime for he nex replenishmen. Bu he does no include he sock ou cos (back order cos and los sale cos). Chang and Dye [999] developed an invenory model wih ime varying demand and parial backlogging. He considered ha if longer he waiing ime smaller he backlogging rae would be. So he proporion of he cusomer who would like o accep backlogging a ime is decreasing wih he waiing ime for he nex replenishmen. So o ake care for his siuaion he defined a backlogging rae s.. B() = + α ( i ) Where i is he ime a which he i h replenishmen is making and α is backlogging parameer. Goyal and Giri [00] gave recen rends of modeling in deerioraing iems invenory. They classified invenory models on he basis of demand variaions and various oher condiions or consrains. Ouyang, Wu and Cheng [005] developed an invenory model for deerioraing iems wih exponenial declining demand and parial backlogging. Dye, Hsieh and Ouyang [007] find an opimal selling price and lo size wih a varying rae of deerioraion and exponenial parial backlogging. They assume ha a fracion of cusomers who backlog heir orders increases exponenially as he waiing ime for he nex replenishmen decreases. Shah and Shukla [009] developed a deerioraing invenory model for waiing ime parial backlogging when demand is consan and deerioraion rae is consan. They made Abad [996, 00] more realisic and applicable in pracice. In his paper, an invenory model for deerioraing iems is developed wih Elecronic copy available a: hp://ssrn.com/absrac=804874
Deerioraing invenory model 363 ime dependen demand and consan rae of deerioraion. Shorages are allowed and parially backlogged; backlogging rae is variable and is dependen on he lengh of he nex replenishmen.. Assumpion and Noaions The mahemaical model is based on he following noaions and assumpions.. Noaions A he ordering cos per order. C he purchase cos per uni. θ he deerioraion rae. h he invenory carrying cos per uni per ime uni. π he backordered cos per uni shor per ime uni. b π L he cos of los sales per uni. he ime a which he invenory level reaches zero, 0 he lengh of period during which shorages are allowed, 0 T (= + ) he lengh of cycle ime IM he maximum invenory level during [0, T]. IB he maximum invenory level during shorage period. Q (= IM + IB) he order quaniy during a cycle of lengh T. I () he level of posiive invenory a ime, 0 I () he level of negaive invenory a ime, + TC (, ) he oal cos per ime uni.. Assumpions The demand rae is ime dependen ha is if a is fix fracion of demand and b is ha fracion of demand which is vary wih ime hen demand funcion is f() = a + b, where a>0,b>0. Shorages are allowed and parially backlogged. The lead ime is zero. The replenishmen rae is infinie. The planning horizon is infinie. The deerioraion rae is consan. During sock ou period, he backlogging rae is variable and is dependen on he lengh of he waiing ime for nex replenishmen. So ha he backlogging rae for negaive invenory is, B () = + ( T ) is backlogging parameer and (T-) is waiing ime ( T ). where
364 V. K. Mishra and L. S. Singh 3.0 Mahemaical Model The rae of change of invenory during posiive sock period [0, ] and shorage period [,T] is governed by he differenial equaions di () + θ I d () = ( a+ b ) 0 () di () ( a+ b) = d + ( T ) () Wih boundary condiion I () =I () =0 a = and I () =IM a =0 4.0 Analyical Soluion Case I: Invenory level wihou shorage During he period [0, ], he invenory deplees due o he deerioraion and demand. Hence, he invenory level a any ime during [0, ] is described by differenial equaion di() + θ() = a 0 (3) d Wih he boundary condiion I ( ) = 0 a = The soluion of equaion (3) is ( ) a b θ a b I () = ( ) + e ( ) θ θ θ θ θ θ ; 0 (4) Case II: Invenory level wih shorage During he inerval [,T] he invenory level depends on demand and a fracion of demand is backlogged. The sae of invenory during [,T] can be represened by he differenial equaion di () ( a+ b) = ; + (5) d + ( + ) Wih he boundary condiion I ( ) = 0 a = The Soluion of equaion (5) is a + ( + ) b[ + ( + )] [ + ( + )] b( ) I () = log + log (6) + + Therefore he oal cos per replenishmen cycle consiss of he following componens: Invenory holding cos per cycle;
Deerioraing invenory model 365 IHC = h I () d 0 θ θ θ IHC = ( ( h( e aθ e b θ+ e b + aθ + bθ + aθ b))) (7) 3 θ Backordered cos per cycle; + BC = π ( I () d) b ( (a + b + b + b + b log( ) 3 + BC = π b + blog( ) + alog( ) + b log( ))) + + + (8) Los sales cos per cycle; + LS = π ( ) ( ) ( a b ) d l + + + LS= ( ( a + b + b a log( + ) π l (9) blog( + ) blog( + ) blog( + ) + b)) Purchase cos per cycle = (purchase cos per uni) X (Order quaniy in one cycle) PC = C.Q When = 0 he level of invenory is maximum and i is denoed by IM (= I (0)) hen from he equaion (4) a b θ a b IM = + + e ( ) θ θ θ θ... (0) θ The maximum backordered invenory is obained a = + hen from he equaion (6) IB = -I ( + ) a b[ + ( + )] b IB = [ log + log + ]... () + + Thus he order size during oal ime inerval [0,T] Q = IM + IB Now from equaions (0) and ()
366 V. K. Mishra and L. S. Singh Q Thus a b θ a b a + + e ( ) lo g θ θ θ θ θ + = b[ + ( + ) ] b log( + ) PC = C.Q a b θ a b a + + e ( ) log θ θ θ θ θ + C = b[ + ( + )] b log( + )... () Therefore he oal cos per ime uni is given by, = [Ordering cos + carrying cos + backordering cos + los sale cos + ( + ) purchase Cos] TC (, ) = [OC+ IHC + BC + LS + PC] ( + ) Now puing he values in his equaion of OC, IHC, BC, LS and PC hen, TC (, ) = θ θ θ ( ( A h e aθ e b θ + e b+ aθ + bθ + aθ b)) 3 θ + ( π (a + b + b a log( + ) b log( + ) l b log( + ) b log( + ) + b )) + ( π (a + b + b + b + 3 b ( + ) b log( ) + b log( ) + a log( ) + b log( ) )) + + + + a b θ a b a + + e ( ) log θ θ θ θ θ + c + b[ + ( + )] b log( + ) (3) The necessary condiion for he oal cos per ime uni, o be minimize is
Deerioraing invenory model 367 TC TC = 0 and = 0 Provided TC TC TC > 0 (4) 5.0 Sensiiviy Analysis Consider an invenory sysem wih he following parameer in proper uni A=500, h=.5, C=4, pib=, pil=5, dela=8, a=5, b=0.0, hea=.005.the compuer oupu by using maple mahemaical sofware is =5.4 =0.4 and TC= 95.30. i.e. he value of a which he invenory level become zero is 5.40 uni and shorage period is 0.04 uni.the variaion in he parameer is as follows Table 5. Variaion in parameer Table-5. Variaion in parameer θ TC 6.4 5.40.04 95.07 8.0 5.40.04 95.30 8.8 5.40.03 95.39 9. 5.4.03 95.44 Table- 5.3 Variaion in parameer b b TC.0 5.30.04 93.5 0.0 5.40.04 95.30 8.0 5.6.04 88.44 6.0 5.87.04 847.7 θ TC.0040 5.4.04 93.99.0045 5.4.04 94.65.0050 5.40.04 95.30.0055 5.40.04 95.96 Table- 5.4 Variaion in parameer a a TC 6.0 5.40.04 888. 5.0 5.40.04 888..5 5.4.04 888.0 0.0 5.44.04 888.8 From able 5., 5., 5.3 and 5.4 we observed ha he oal cos increases if we increases he parameer a,b, θ and.i s also observed ha he parameer a and b is more sensiive han he parameer θ and If we plo he oal cos funcion (3) wih some values of and s.., = 5.0 o 5.80 wih equal inerval = 0.0 o.09.then we ge a hree dimensional convex
368 V. K. Mishra and L. S. Singh Graph of TC given by he figure (5.) Fig-5. 6.0 Concluding Remarks In his paper, we developed a model for deerioraing iem wih ime dependen demand and parial backlogging and give analyical soluion of he model ha minimize he oal invenory cos. The deerioraion facor aken ino consideraion in he presen model, as almos all iems undergo eiher direc spoilage (like fruis, vegeable ec) or physical decay (in case of radioacive subsance ec.) in he course ime, deerioraion is naural feaure in he invenory sysem. The model is very useful in he siuaion in which he demand rae is depending upon he ime. References [] Deb, M., and Chaudhary, K., A noe on he heurisic for replenishmen of rended invenories considering shorages, Journal of he Operaional Research Sociey, 38(987), 459-463. [] Ghare, P. M. and Schrader, G. F., A model for an exponenially decaying invenory, Journal of Indusrial Engineering, 4(963), 38-43. [3] Goyal, S. K. and B. C. Giri, Recen rends in modeling of deerioraing invenory, European Journal of Operaional Research, 34(00), -6. [4] Hadley, G., Whiin, T., Analysis of Invenory Sysems. Prenice Hall, Englewood Cliffs (963). [5] Hariga, M. A., An EOQ model for deerioraing iems wih shorages and ime-varying demand, Journal of he Operaional Research Sociey, 46(995), 398-404. [6] Hariga, M. A., Economic analysis of dynamic invenory models wih non-saionary coss and demand, Inernaional Journal of Producion Economics, 35(994), 55-66. [7] Harris, F.W., Operaions and cos. Chicago (95).
Deerioraing invenory model 369 [8] Jalan, A.K., Giri, R. R., and Chaudhary, K.S., EOQ model for iems wih weibull disribuion deerioraion shorages and rended demand. Inernaional Journal of Sysem Science. 7(996), 85-855. [9] Naddor, E., Invenory Sysem. Willey New York, (966). [0] Nahmias, S., Perishable invenory heory: A review. Operaions Research, 30(978), 680-708. [] Pareek,S.,Mishra,V.K. and Rani,S., An Invenory Model for ime dependen deerioraing iem wih salvage value and shorages, Mahemaics Today, 5(009), 3-39. [] Roy, Ajana, An invenory model for deerioraing iems wih price dependen demand and ime varying holding cos. Advanced Modeling and Opimizaion, 0(008), 5-37. [3] Shah, Nia H. and Acharya, Anki S, A ime dependen deerioraion order level invenory model for declining demand, Applied Mahemaical Sciences, (008), 795-80. [4] Sarr, M.K., and D.W. Miller., Invenory Conrol-Theory and Pracice /e.prenice hall, Englewood Cliffs, N.J. (974). [5] Wee, H. M., A deerminisic lo-size invenory model for deerioraion iems wih shorages and a declining marke, Compuers and Indusrial Engineering, (995), 345-356. [6] You, S.P, Invenory Policy for producs wih price and ime dependen demands, Journal of Operaional Research Sociey, 56(005), 870-873. Received: June, 00