EOQ Model for Weibull Deteriorating Items with Linear Demand under Permissable Delay in Payments

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1 International Journal of Computational Science and Mathematics. ISSN Volume 4, Number 3 (0), pp International Research Publication House EOQ Model for Weibull Deteriorating Items with Linear Demand under Permissable Delay in Payments Shital S. Patel and R.D. Patel Department of Statistics Veer Narmad South Gujarat University, Surat, India patelramanb@yahoo.co.in Abstract A deterministic inventory model is developed when the deterioration rate is Weibull distribution. Here it is assumed that the demand rate is linear function of time. In this study, model has been framed to study the items whose demand changes with time and deterioration rate increase with time under permissible delay in payments. Numerical example is taken to support the model. Keywords: Economic order quantity, Deteriorating item, Weibull deterioration, rade credit, Linear demand Introduction Most of the existing inventory models in the literature assume that items can be stored indefinitely to meet the future demand. However, certain types of commodities either deteriorate or become obsolete in the course of time and hence are unstable. For example, the commonly used goods like fruits, vegetables, electronic components, etc. where deterioration is usually observed during their normal storage period. herefore, if the rate of deterioration is not sufficiently low, its impact on modeling of such an inventory system cannot be ignored. Inventory problems for deterioration items have been studied extensively by many researchers from time to time. Research in this area started with the work of Whitin [], who considered fashion goods deteriorating at the end of prescribed storage period. Ghar and Schrader [] developed an inventory model with a constant rate of deterioration. Shah and Jaiswal [9] considered an order level inventory model for items deteriorating at a constant rate. Raafat [8] provided a comprehensive survey on

2 76 Shital S. Patel and R.D. Patel continuous deteriorating inventory models where the deterioration is considered as a function of the on-hand inventory. Many times the supplier offers a permissible credit period to the retailor if the outstanding amount is paid within the allowable fixed period and the order quantity is large. he credit period is treated as a promotional tool to attract more customers. It is a type of price discount because paying later indirectly reduces the purchase cost which motivates retailers to increase their order quantity. Goyal [3] developed an EOQ model under the conditions of permissible delay in payments. Mandal and Phaujdar [6] extended this issue by considering the interest earned from the sales revenue. eng [] amended Goyal s [3] model by identifying the difference between unit price and unit cost. Chang et al. [] established EOQ model with deteriorating items under supplier trade credits linked to order quantity. Shah [0] derived an inventory model by assuming constant rate of deterioration of units in an inventory, time value of money under the conditions of permissible delay in payments. Recently, Huang [4] developed EOQ model in which the supplier offers a partially permissible delay in payments when the order quantity is smaller than the predetermined quantity. Ouyang et al. [7] have considered trade credit linked to order quantity for deteriorating items. Mahata [6] developed an EOQ model for deteriorating items with instantaneous replenishment, exponential decay rate and a time varying linear demand without shortages under permissible delay in payments. In this paper we have developed EOQ model when demand rate is linear function of time, deterioration is two parameter Weibull distribution and inventory holding costy is linear function of time. Numerical example is taken and sensistivity analysis is also done. Notations and Assumptions he following notations and assumptions are used in this paper: Notations R(t) = a + bt, Annual demand is linear function of time, where a > 0, b>0 A= Ordering cost per orderc c = Unit purchasing cost per item p = the unit selling price with (p>c) h(t) = x+yt, inventory variable holding cost per unit per year excluding interest charges I e = Interest earned per year I c = Interest charged in stocks per year M = Permissible period of delay in settling the accounts with the supplier = ime interval (in years) between two successive orders. I(t) = Inventory level at any instant of time t, 0 t

3 EOQ Model for Weibull Deteriorating Items with Linear Demand 77 Q = Order quantity. αβt β- is the two parameter Weibull deterioration rate, where 0 α, β>0. Assumptions he following assumptions are used in the development of the model: he demand of the product is declining as a linear function of time. Replenishment rate is infinite and instantaneous. Lead time is zero. Shortages are not allowed. he deteriorated units can neither be repaired nor replaced during the cycle time. During the time, the account is not settled, generated sales revenue is deposited in an interest bearing account. At the end of the credit period, the account is settled as well as the buyer pays off all units sold and starts paying for the interest charges on the items in stocks. he Mathematical Model and Analysis he inventory level of the product at time t over the period [0, ] can be represented by the differential equation di(t) β + αβt I(t) = (a+ bt), 0 t, () dt with the boundary conditions Q(0) = Q, Q() = 0. () he solution of equation () using () is: α αt = β+ β+ β+ β+ β I(t) a -t+ - +aα (t-) -t α αt β+ β+ β+ β+ β +b + - +bα (t-) (3) he initial order quantity at t =0 is: β+ β+ b aα bα Q = I(0) = a+ + + β+ β+ he associated costs are: A (i) Ordering cost: OC= (4) (5)

4 78 Shital S. Patel and R.D. Patel (ii) Cost due to deterioration per unit time; β+ β+ c aα bα DC = + β+ β+ (6) (iii) Inventory variable holding cost per unit time is HC = h(t)q(t)dt 0 3 β+ 3 β+ β+ 3 a b bα aα bα + + x 3 (β + ) ( β + )( β + ) ( β + )( β + 3 ) = β 3 β β bα aα bα + + ( β + ) ( β + ) ( β + 3) 3 4 β+ 3 β+ 3 β+ 4 a b aα aα bα + + y 6 8 ( β + ) ( β + )( β + 3 ) (β + )(β + 4) + β+ 3 β+ 3 β+ 4 aα aα bα + + ( β + ) ( β + 3) ( β + 4) (7) Now we consider the two cases based on the length of and M, using the fact that interest charged or earned. Case (I): M < : From the assumption, the retailer sells R(M)M units by the end of the permissible tread credit M and has CR(M)M to pay the supplier. he supplier charges an interest rate I C from time M onwards for the unsold items in the stock. Hence the interest charged IC(I) per time unit is (iv) Interest charged per time unit is cic IC(I) = Q(t)dt M 3 β+3 β+ β+3 β+3 a b bα aα bα bα β+ ( β+)( β+) ( β+)( β+3) ( β+) β+ β+3 β+ β+ aα bα bm amα bmα + + -am- - - ci c β+ β+3 β+ β+ = 3 β+ β+3 β+ am bm aαm bαm aαm ( β+)( β+) ( β+)( β+3) ( β+) β+ β+ β+3 bαm aαm bαm ( β+) ( β+) ( β+3) (8)

5 EOQ Model for Weibull Deteriorating Items with Linear Demand 79 During [0, M] the retailer sells the product and deposits the revenue into an interest earning account at the rate Ic per monetary unit per year, we get the interest earned per unit time as: M pie (v) IE(I) = ( a+bt) tdt 3 pi e am bm = + 3 (9) 0 Hence the total cost C ( ) per cycle = Ordering cost + Cost of deteriorated units + Inventory Holding cost + Interest payable beyond permissible period interest earned during the cycle i.e. C() = OC + DC + HC + IC(I) - IE(I) A c aα bα C () = + + β+ β+ β+ β+ 3 β+3 β+ β+3 a b bα aα bα (β+) ( β+)( β+) ( β+)( β+3 x ) + β+3 β+ β+3 bα aα bα ( β+) ( β+) ( β+3) β+ 3 β+ 3 β+ 4 a b aα aα bα ( β ) ( β )( β 3 ) (β )(β 4) y β 3 β 3 β aα aα bα + + ( β + ) ( β + 3) ( β + 4) 3 β+3 β+ β+3 a b bα aα bα β+ ( β+)( β+) ( β+)( β+3) β+3 β+ β+3 β+ bα aα bα bm amα am- - ( β+) β+ ( β+3) β+ 3 ci c PIe am bm + β+ 3 β+ β bmα am bm aαm bαm β+ 6 ( β+)( β+) ( β+)( β+3) β+ β+ β+ β+3 aαm bαm aαm bαm β+ ( β+) ( β+) ( β+3 ) (0) Now we have to minimize C() for a given value of M. he necessary and sufficient conditions to minimize C() for a given value of M are respectively dc () 0 d = ()

6 80 Shital S. Patel and R.D. Patel and dc() d > 0. () Equation () yields the following equation after simplification: β β- A aαβ bα β+ - + c + β+ β+ β β+ a b β+ aα bα + + bα 3 ( β + ) ( β + 3) bα( β+ ) aαβ ( + ) bαβ ( + ) + + ( β + ) ( β + ) ( β + 3) + x β+ β β+ β+ β+ β+ a 3b aα β+ aαβ+ bαβ β + β + β + 3 (β + )(β + 4) + y β+ β+ β+ aα β+ bαβ+ 3 aα + + β + 3 β + 4 β β+ a b β+ aα bα + +bα ( β+) ( β+3) bα( β+) aα( β+) bα( β+) bm amαβ ( β+) β+ ( β+3) β+ + cic β 3 β+ β+3 bmα( β+) am bm aαm bαm bαm aαm bαm ( β+) ( β+) ( β+3) 3 pie am bm + + =0 3 β+ β β+ β- β+ 6 β+ β+ β+ β+3 β+ β+ β+3 (3) By solving equation (3) for, we obtain the optimal cycle length it satisfies equation (). he EOQ Q( ) for this case is given by, = provided b aα bα Q( ) = a β+ β+ β+ β+ (4) he minimum annual variable cost C ( ) is then obtain from equation (0) for =.

7 EOQ Model for Weibull Deteriorating Items with Linear Demand 8 Case(II): M : In this case, the retailor earns interest on the sales revenue up to the permissible delay period and no interest is payable during this period for the items kept in stock. So Interest charges IC(II)= 0. (5) And the interest earned during ( M-) i.e. up to the permissible delay period is: PI a b 3 3 e IE(II) = am + bm - - (6) + Hence the total cost C () per cycle = Ordering cost + Cost of deteriorated units Inventory Holding cost + Interest charged cost interest earned during the cycle. C () = OC + DC + HC + IC(II) - IE(II) A c aα bα C() = + + β+ β+ β+ β+ 3 β+3 β+ β+3 a b bα aα bα (β+) ( β+)( β+) ( β+)( β+3 x ) + β+3 β+ β+3 bα aα bα ( β+) ( β+) ( β+3) 3 4 β+3 β+3 β+4 a b aα aα bα ( β+) ( β+)( β+3 ) (β+)(β+4) y + β+3 β+3 β+4 aα aα bα ( β+) ( β+3) ( β+4) 3 PI e a b - am + bm (7) Now we have to minimize C () for a given value of M. he necessary and sufficient conditions to minimize C () for a given value of M are respectively dc () = 0 (8) d and d C () > 0. (9) d

8 8 Shital S. Patel and R.D. Patel Equation (8) yields the following equation after simplification: β β- A aαβ bα β+ - + c + β+ β+ β β+ a b β+ aα bα + + bα 3 ( β + ) ( β + 3) bα( β+ ) aαβ ( + ) bαβ ( + ) + + ( β + ) ( β + ) ( β + 3) + x β+ β β+ β+ β+ β+ a 3b aα β+ aα β+ bα β β+ β+ β+3 (β+)(β+4) + y β+ β+ β+ aα β+ bα β+3 -aα + + β+3 β+4 a 4b - PIe bm- - = 0 3 (0) By solving equation (0) for, we obtain the optimal cycle length it satisfies equation (9). he EOQ, Q( ) for this case is given by, = provided b aα bα Q( ) = a β+ β+ β+ β+ () he minimum annual variable cost =. C() is then obtain from equation (7) for Numerical Example Example : aking a = 50 units per year, b= units per year, A= Rs 00 units per year, c = Rs. 30 per unit, p = Rs 40 per unit, I c = Rs 0.5 per year, I e =0.09 per year, M = years, α =0.04, β =, x=5.5, y=0.5 in appropriate unit. Solving equation (3), we obtained the optimal value of =0.694 and from equations (0) and (4), the optimal total cost C () = Rs and the optimum order quantity Q= Example : aking a = 50 units per year, b= units per year, A= Rs 00 units per year, c = Rs. 30 per unit, p = Rs 40 per unit, I c = Rs 0.5 per year, I e =0.09 per year, M = years, α =0.04, β =, x=5.5, y=0.5 in appropriate unit. Solving equation (0), we obtained the optimal value of =0.765 and from equations (7) and (0), the optimal total cost C () = Rs and the optimum order quantity Q=

9 EOQ Model for Weibull Deteriorating Items with Linear Demand 83 Sensitivity Analysis We have performed sensitivity analysis for the examples given in section 4, by changing parameters a, b, x, y, α and β as 50%, 0%, -0% and -50% and keeping the remaining parameters at their original values. he corresponding changes in the cycle time, total cost and purchase quantities are shown in table able Parameter Changes % C() Q C() Q Remarks a M > M > M < M < b M > M > M > M > x M > M > M < M < y M > M > M > M > α M > M > M > M > β M > M > M > M >

10 84 Shital S. Patel and R.D. Patel From the table we observe that as parameter a increases/ decreases, order quantity and average total cost increases/ decreases in both case I and case II. Similarly for change in parameter b, there is slight change in total cost and quantity in both cases. Also we observe that with increase and decrease in parameters x and y, there is corresponding increase/ decrease in total cost and quantity in case I, but in case II, for increase/decrease in x and y values there is increase/decrease in total cost but decrease/increase in quantity. We observe that with increase and decrease in parameter α, there is corresponding increase/ decrease in total cost but there is decrease/increase in quantity for both the cases. We observe that with increase and decrease in parameter β, there is corresponding increase/decrease in quantity but there is decrease/increase in total cost for both the cases. able Case I ( M < ) Case II ( M > ) M C () Q M C () Q From the above table it can be observed that for case I, increasing the delay period, there is decrease in cost and in case II, increasing the delay period there is increase in cost but there is very slight change in the order quantity for both the cases. Conclusion he model developed in this paper assumes thet demand of the product is declining as a linear function of time. Shortages are not allowed and replenishment rate is infinite and instantaneous. he effect of delay period offered by the supplier to retailer is analyzed when the demand of the product is linear and also the linear variable holding cost taken to derive the model. he units in the inventory are assumed to be time dependent two parameter Weibull deterioration. It is observed that increase in delay period results in increase in total inventory cost in period I but decrease in period II that s the main point of this paper. References [] Chang, C.., Ouyang, L.Y. and eng, J.. (003): An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Appl. Math. Model, Vol. 7, pp

11 EOQ Model for Weibull Deteriorating Items with Linear Demand 85 [] Ghare, P.N. and Schrader, G.F. (963): A model for exponentially decaying inventories, J. Indus. Engg., Vol. 5, pp [3] Goyal, S.K. (985): Economic order quantity under conditions of permissible delay in payments, J. O.R. Soc., Vol. 36, pp [4] Huang, Y.F. (007): Economic order quantity under conditionally permissible delay in payments, Euro. J. O.R., Vol. 76, pp [5] Mahata, G.C. (0): EOQ model for items with exponential distribution deterioration and linear trend demand under permissible delay in payments, Int. J. Soft Comput., Vol. 6(3), pp [6] Mandal, B.N. and Phaujdar, S. (989): Some EOQ models under permissible delay in payments, Int. J. Mag. Sci., Vol. 5(), pp [7] Ouyang, L.Y., eng, J.., Goyal S.K. and Yang, C.. (009): An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, Euro. J. O.R., Vol. 94, pp [8] Raafat, F. (99): Survey of literature on continuous deteriorating inventory model, J. Oper. Res. Soc., Vol. 4, pp [9] Shah, Y.K. and Jaiswal, M.C. (977): An order level inventory model for a system with constant rate of deterioration, Opsearch, Vol. 4, pp [0] Shah, N.H. (006): Inventory models for deteriorating items and time value of money for a finite horizon under the permissible delay in payments, Int. J. Syst. Sci., Vol. 37, pp [] eng, J.. (00): On the economic order quantity under conditions of permissible delay in payments; J. O.R. Soc., Vol. 53, pp [] Whitin,.M. (957): heory of inventory management, Princeton Univ. Press, Princeton, NJ.