An Inventory Model for Time Dependent Weibull Deterioration with Partial Backlogging

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1 American Journal of Operaional Research 0, (): -5 OI: 0.593/j.ajor An Invenory Model for Time ependen Weibull eerioraion wih Parial Backlogging Umakana Mishra,, Chaianya Kumar Tripahy eparmen of Mahemaics, P.K.A.College of Engineering, Chakarkend, Bargarh, 76808, India eparmen of Saisics, Sambalpur Universiy, Jyoi Vihar, Sambalpur, 76809, India Absrac This paper deals wih developing an invenory model for Weibull deerioraion wih wo parameers. Here shorages are allowed and are parially backlogged. We have derived he opimal order quaniy by minimizing he oal invenory cos. To illusrae he model a numerical example has been provided and sensiiviy analysis has also been carried ou o sudy he effec of parameers on decision variables and he oal cos of an invenory of his model. Keywords Invenory Model, eerioraion, eerminisic emand, Parial Backlogging. Inroducion The effec of deerioraion is very imporan for mos of he iems which canno be negleced in invenory model. eerioraion may be defined as decay, change or spoilage ha prevens he iem from being used for is original purpose. Food iems, drugs, pharmaceuicals, phoographic film, elecronic componens and radioacive subsances are a few examples of iems in which appreciable deerioraion can ake place during he normal sorage period of he unis and consequenly his loss mus be aken ino accoun when analysing he model. Ghare and Schrader[7 were he pioneer in sudy of invenory model considering he effec of deerioraion. In heir paper hey have considered consan rae of deerioraion in an invenory model wih no shorages. There afer a lo of research work has been done in his direcion. The lieraure survey by Raafa[9, Shah and Shah[ and Goyal and Giri[8 give review on deerioraing invenory models. Abad[, has derived pricing and ordering policy for a variable rae of deerioraion and parial backlogging. In his papers he parial backlogging was assumed o be exponenial funcion of waiing ime ill he nex replenishmen. The backorder cos and los sale have been considered by ye e al.[4 in heir invenory model. Shah and Shukla[ have developed deerminisic invenory model for deerioraing iems wih shorages. In his case hey considered parial backlogging for unsaisfied demand. Tripahy and Mishra[4 have improved upon heir model by considering deerioraion o be a linear Corresponding auhor: umakana.mah@gmail.com (Umakana Mishra) Published online a hp://journal.sapub.org/ajor Copyrigh 0 Scienific & Academic Publishing. All Righs Reserved funcion of ime insead of being a consan. Some of he imporan research works in his direcion are due o Roy e al.[0, eng e al.[3, ye e al.[6, ye[5 and Teng e al.[3, ec. In his paper we have developed a deerminisic invenory model for wo parameer Weibull deerioraion. Shorages are allowed and are parially backlogged for his model. The unsaisfied demand is backlogged and is a funcion of ime. The opimal order quaniy has derived by minimizing he oal invenory cos. A numerical example has been provided o illusrae he model. Sensiiviy analysis has also been carried ou o sudy he effec of parameers on decision variables and he oal cos of an invenory of his model.. Noaions We need he following noaions for developing mahemaical model. i. A : ordering cos per order ii. c : purchase cos per uni. iii. h : invenory holding cos per uni per ime uni. iv. πb : backorder cos per uni shor per ime uni. v. πl : cos of los sales per uni. vi. : ime a which he invenory level reaches zero, 0 vii. : lengh of period during which shorages are allowed, 0 viii. T : Toal lengh of cycle ime i.e. T = ( + ). ix. Im : Maximum invenory level during [ 0,T. x. Ib : Maximum backordered unis during sock ou period. xi. : Toal order quaniy during a cycle of lengh T i.e. = Im + Ib xii. I ( ): level of posiive invenory a ime, 0 xiii. I ( ): level of negaive invenory a ime, T

2 Umakana Mishra e al.: An Invenory Model for Time ependen Weibull eerioraion wih Parial Backlogging xiv. : oal average cos per ime uni. 3. Assumpions The following are he assumpions needed for developing he model: a. The model developed for single iem invenory. b. The rae of demand is known and consan. c. The rae of deerioraion, θ = αβ β, follows a wo parameer Weibull disribuion, where α(0 < α << ) is he scale parameer, β > is he shape parameer. I is assumed ha he deerioraion increases wih ime > 0. d. Infinie replenishmen rae. e. Lead - ime is zero or negligible. f. Planning horizon is infinie. g. The rae of backlogging during he shorage period is variable and depends on he lengh of he waiing ime ill he nex replenishmen. The negaive invenory backlogging rae will be B () = ; where > + ( T ) 0 denoes he backlogging parameer and T 4. Mahemaical Modeling Considering he effec of demand and deerioraion during [0, he invenory level a any insan of ime during [0, can be described by he following differenial equaion di() + θ I() = 0 d Now puing he value of θ in above equaion we ge, di() β + αβ I() = 0 () d Furher he invenory level reaches zero a ime =. Using his boundary condiion we have I ( ). Again since α is very small, using series expansion and ignoring second and higher powers of α, from () we ge, β + β + α α β β+ I( ) = + α + α,0 () β + β + Since invenory level reaches o zero a ime and here afer shorages occur. uring he inerval [, T, he invenory level depends on demand and a fracion of he demand is backlogged. The sae of invenory during [, T can be represened by he differenial equaion, di () = T (3) d + ( T ) Since invenory level reaches o zero a ime, he boundary condiion I ( ) and he soluion of he differenial equaion (3) is I( ) = [ ln ( + ( + )) ln ( + ), T (4) Thus he maximum invenory will be β + α Im = I(0) = +, (5) β + and he maximum backordered unis will be Ib = I( + ) = ln ( + ) (6) So, he order size during [ 0,T will be = Im + Ib β + α i.e. = + + ln ( + ) (7) β + The following cos componens need o be considered o calculae oal cos per cycle: i. Ordering cos per cycle is A ii. Holding cos per cycle is β+ β+ α α h I() d = h + ( β + )( β + ) β + 0 iii. Backorder cos per cycle + πb = πb I( ) d = [ ln( + ) iv. Cos of los sales per cycle + = πl d ( + + ) = π l [ ln( + ) v. Purchase cos per cycle β + α = c = c + + ln ( + ) β + Therefore, he oal average cos per uni ime is ordering cos + holding cos = + backorder cos + cos of los sales + + purchase cos β+ β+ α α = A + h + + ( β + )( β + ) β + πb πl + [ ln( + ) + [ ln( + ) β + (8) α + c + + ln ( + ) β + The necessary condiions for minimizing he oal average cos per uni will be (9) and (0) The pair of (, ) calculaed from equaions (9) and (0), subjec o he condiion > 0 will minimize. Now we have

3 American Journal of Operaional Research 0, (): = A + h ( + ) β+ β+ α αβ ( + ) ( + ) ( β + )( β + ) β+ β α + αβ ( + ) ( + ) + ( β + ) ( πb + π l ) ( ln( + ) ) β β+ αβ ( + ) ( + ) α + ( β + ) + c ln ( + ) β + α ( β + )( β + ) = A h ( + ) β + αβ + β + ( πb + πl) + ( ) ( + ) ln( + ) + + β + α + β + + c ( + ) ln + + ( )( ) ( ) () () + The above se of parameric equaions () and () can be solved for and using mahemaica-5. sofware. Now we will consider a numerical example o illusrae and validae he proposed model in he following secion. 5. Numerical Example Example-: Le an invenory sysem have he following parameric values in proper unis. [ A, c, h, π b π l,,, α, β =[00, 0, 0.7,, 0, 0,, 0.3, 0.. Using hese values for he above model, we ge opimum value of = , =.367 puing he opimum values of and in equaion (7) and (8) we ge he opimum values of = and minimum oal average cos per uni ime = respecively. 6. Sensiiviy Analysis Now we will perform sensiiviy analysis by increasing parameers α, β, and, one a a ime and keeping he remaining parameers a heir original values. We will be using he values of he numerical example given in he previous secion for performing he sensiiviy analysis. The resuls are given in abulaed form in Table- o Table-4. In able-, i is observed ha increase in demand parameer resuls in increase in of an invenory sysem and also increase ordering quaniy. Parameer Table. Variaion in demand Table. Variaion in deerioraion rae α Parameer α In able-, i is observed ha increase in deerioraion rae α resuls in increase of an invenory sysem and also increase ordering quaniy. Table 3. Variaion in deerioraion rae β Parameer β Change in Change in In able-3, i is observed ha increase in deerioraion rae β resuls in increase of an invenory sysem and also increase ordering quaniy. Table 4. Variaion in backlogging parameer Parameer In able-4, i is observed ha increase in backlogging parameer resuls in decrease in of an invenory sysem and decrease ordering quaniy. The resuls observed above can also be represened graphically in form of he following Figures Figure. Toal cos versus and for able-.

4 4 Umakana Mishra e al.: An Invenory Model for Time ependen Weibull eerioraion wih Parial Backlogging Figure. Toal cos versus for able- demand is backlogged and is a funcion of ime. The opimal order quaniy has been derived by minimizing he oal invenory cos. This will help in invenory decision making under similar condiion. A numerical example and sensiiviy analysis are presened o illusrae he proposed model. I is observed ha increase in demand parameer or scale or shape parameers of deerioraion funcion resuls in increase in oal cos of an invenory sysem and also increase in ordering quaniy. Furher sudies in his direcion can be carried ou incorporaing more realisic assumpion such as sochasic demand and finie replenishmen rae. ACKNOWLEGEMENTS We are highly indebed o he anonymous referees for heir valuable suggesion and commens o improve upon he earlier version of his aricle o he presen form Figure 3. Toal cos versus and for able REFERENCES [ Abad, P.L., 996, Opimal pricing and lo-sizing under condiions of perish abiliy and parial backordering, Managemen Science, 4, [ Abad, P.L., 00, Opimal price and order-size for a reseller under parial backlogging, Compuers and Operaion Research, 8, [3 eng, P. S., Lin, Rober, 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, [4 ye, C.Y., Hsieh, T.P., and Ouyang, L.Y., 007, eermining opimal selling price and lo size wih a varying rae of deerioraion and exponenial parial backlogging, European Journal of Operaional Research, 8, Figure 4. Toal cos versus and for able-4 7. Conclusions In he presen scenario of invenory sysem, large amoun of capial is invesed for purchase of invenory. I is imporan o consider deerioraion in invenory decision making. In his paper a deerminisic invenory model has been developed for Weibull deerioraion. Shorages are allowed and are parially backlogged for his model. The unsaisfied. [5 ye, C.Y., 007, Join pricing and ordering policy for a deerioraing invenory wih parial backlogging, Omega, [6 ye, C.Y., Ouyang, L.Y., and Hsieh, T.P., 007, eerminisic invenory model for deerioraing iems wih capaciy consrain and ime-proporional backlogging rae, European Journal of Operaional Research, 78, [7 Ghare, P.M., and Schrader, G.H., 963, A model for exponenially decaying invenory sysem, Journal of indusrial Engg.4, [8 Goyal, S.K., and Giri, B.C., 00, Recen rends in modeling of deerioraing invenory, European Journal of Operaional Research, 34, -6. [9 Raafa, F., 99, Survey of lieraure on coninuously deerioraing invenory models, Journal of he Operaional Research Sociey, 40, [0 Roy, T., and Chaudhuri, K.S., 009, A producion-invenory model under sock-dependen demand, Weibull disribuion deerioraion and shorage, Inl. Trans. in Op. Res. 6,

5 American Journal of Operaional Research 0, (): -5 5 [ Shah, N.H., and Shah, Y. K., 000, Lieraure survey on invenory model for deerioraing iems, Economic annals, 44, -37. [ Shah, N.H., and Shukla, K.T., 009, eerioraing invenory model for waiing ime parial backlogging, Applied mahemaical Sciences, vol.3, no.9, [3 Teng, J.T, Ouyang, L.Y., and Chenb, L.H., 007, A comparison beween wo pricing and lo-sizing models wih parial backlogging and deerioraed iems, In. J. Producion Economics, 05, [4 Tripahy, C. K., and Mishra, U., 00, An invenory Model wih Time ependen Linear eerioraing Iems wih Parial Backlogging, In. Journal of Mah. Analysis, Vol. 4, no. 38,