Inventory Model with Different Deterioration Rates with Shortages, Time and Price Dependent Demand under Inflation and Permissible Delay in Payments

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1 Global Journal of Pure and Applied athematics. ISSN Volume 3, Number 6 (07), pp Research India Publications Inventory odel with Different Deterioration Rates with Shortages, Time and Price Dependent Demand under Inflation and Permissible Delay in Payments Raman Patel Department of Statistics Veer Narmad South Gujarat University Surat, INDIA D.. Patel Department of Commerce Narmada College of Science and Commerce Zadeshwar, Bharuch, INDIA Abstract An inventory model for deteriorating items with stock and price dependent demand is developed. Holding cost is considered as function of time. Shortages are allowed and completely backlogged. Numerical example is provided to illustrate the model and sensitivity analysis is also carried out for parameters. Keywords: Inventory model, Deterioration, Price dependent demand, time dependent demand, Time varying holding cost, Shortages. INTRODUCTION: In real life, deterioration of items is a general phenomenon for many inventory systems and therefore deterioration effect cannot be ignored. any researchers have studied EOQ models for deteriorating items in past. Ghare and Schrader [] considered no-shortage inventory model with constant rate of deterioration. The model was extended by Covert and Philip [] by considering variable rate of deterioration. By considering shortages, the model was further extended by Shah and

2 500 Raman Patel and D.. Patel Jaiswal [4]. The related work are found in (Nahmias [9], Raffat [], Goyal and Giri [3], Ouyang et al. [0], Wu et al. [6]). Hill [4] considered inventory model with ramp type demand rate. andal and Pal [6] developed inventory model with ramp type demand with shortages. Hung [5] considered inventory model with arbitrary demand and arbitrary deterioration rate. Salameh and Jaber [3] developed a model to determine the total profit per unit of time and the economic order quantity for a product purchased from the supplier. ukhopadhyay et al. [8] developed an inventory model for deteriorating items with a price-dependent demand rate. The rate of deterioration was taken to be timeproportional and a power law form of the price-dependence of demand was considered. Teng and Chang [5] considered the economic production quantity model for deteriorating items with stock level and selling price dependent demand. athew [7] developed an inventory model for deteriorating items with mixture of Weibull rate of decay and demand as function of both selling price and time. Patel and Parekh [] developed an inventory model with stock dependent demand under shortages and variable selling price. Inventory models for non-instantaneous deteriorating items have been an object of study for a long time. Generally the products are such that there is no deterioration initially. After certain time deterioration starts and again after certain time the rate of deterioration increases with time. Here we have used such a concept and developed the deteriorating items inventory models. In this paper we have developed an inventory model with stock and price dependent demand with different deterioration rates for the cycle time. Shortages are allowed and completely backlogged. To illustrate the model, numerical example is taken and sensitivity analysis for major parameters on the optimal solutions is also carried out.. ASSUPTIONS AND NOTATIONS: NOTATIONS: The following notations are used for the development of the model: D(t) : Demand rate is a linear function of price and inventory level (a + bt - ρp, a>0, 0<b<, ρ>0) A : Replenishment cost per order c : Purchasing cost per unit p : Selling price per unit h(t) : x+yt, Inventory variable holding cost per unit excluding interest charges c : Shortage cost per unit : Permissible period of delay in settling the accounts with the supplier T : Length of inventory cycle Ie : Interest earned per year Ip : Interest paid in stocks per year R : Inflation rate

3 Inventory odel with Different Deterioration Rates with Shortages, Time and Price 50 I(t) : Inventory level at any instant of time t, 0 t T Q : Order quantity initially Q : Shortages of quantity Q : Order quantity θ : Deterioration rate during μ t μ, 0< θ < θt : Deterioration rate during, μ t t0, 0< θ < π : Total relevant profit per unit time. ASSUPTIONS: The following assumptions are considered for the development of the model. The demand of the product is declining as a function of price and time. Replenishment rate is infinite and instantaneous. Lead time is zero. Shortages are permitted and completely backlogged. Deteriorated units 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. 3. THE ATHEATICAL ODEL AND ANALYSIS: Let I(t) be the inventory at time t (0 t T) as shown in figure. Figure The differential equations which describes the instantaneous states of I(t) over the period (0, T) is given by di(t) = - (a + bt - ρp), dt 0t μ () di(t) + θi(t) = - (a + bt - ρp), dt μ t μ ()

4 50 Raman Patel and D.. Patel di(t) + θti(t) = - (a + bt - ρp), dt μ t t0 (3) di(t) = - (a + bt - ρp), dt t0 t T (4) With initial conditions I(0) = Q, I(μ) = S,I(t0) = 0, and I(T) = -Q. Solutions of these equations are given by I(t) = Q - (at - ρpt + bt ), (5) a μ - t - ρpμ - t + aθ μ - t - ρpθ μ - t + bμ - t I(t) = bθμ - t - aθt μ - t + ρpθt μ - t - bθt μ - t (6) 3 I(t) = + S + θ+b μ - t a t - t - ρp t - t + b t - t + aθ t - t - ρpθ t - t bθ t - t - aθt t - t + ρpθt t - t - bθt t - t I(t) = a t 0 - t - ρpt 0 - t + bt 0 - t. (by neglecting higher powers of θ) (7) (8) From equation (5), putting t = μ, we have Q = S + aμ - ρpμ + bμ. From equations (6) and (7), putting t = μ, we have a μ - ρp μ + aθ μ - ρpθ μ + b μ I(μ ) = bθμ - aθμ μ + ρpθμ μ - bθt μ 3 + S + θ+b μ (9) (0) a t 0 - ρpt 0 + bt 0 + aθ t 0 - ρpθt I(μ ) =. () bθt 0 - aθμ t 0 + ρpθμ t 0 - bθμ t 0 8 4

5 Inventory odel with Different Deterioration Rates with Shortages, Time and Price 503 So from equations (0) and (), we get S = + θ+bμ a t 0 - ρpt 0 + bt 0 + aθ t 0 - ρθpt bθt 0 - aθμ t 0 + ρθpμ t 0 - bθμ t a μ + ρpμ - aθ μ + ρpθ μ - b μ bθμ + aθμ μ - ρpθμ μ bθμ μ 3. () Putting value of S from equation () into equation (6), we have I(t) = + + θ+b μ - t + θ+b μ a t 0 - ρpt 0 + bt 0 + aθ t 0 - ρθp t bθt 0 - aθμ t 0 + ρθpμ t 0 - bθμ t a μ + ρp μ - aθ μ + ρpθ μ - b μ bθμ + aθμ μ - ρpθμ μ bθμ μ 3 a μ - t - ρpμ - t + aθ μ - t - ρpθ μ - t + b μ - t bθμ - t - aθt μ - t + ρpθt μ - t - bθt μ - t 3 (3) Similarly, putting value of S from equation () into equation (9), we have Q = + θ+bμ a t 0 - ρpt 0 + bt 0 + aθ t 0 - ρθpt bθt 0 - aθμ t 0 + ρθpμ t 0 - bθμ t a μ + ρpμ - aθ μ + ρpθ μ - b μ bθμ + aθμ μ - ρpθμ μ bθμ μ 3 + aμ - ρpμ + bμ. (4)

6 504 Raman Patel and D.. Patel Putting value of Q in equation (5), we get I(t) = + θ+bμ a t 0 - ρpt 0 + bt 0 + aθ t 0 - ρθpt bθt 0 - aθμ t 0 + ρθpμ t 0 - bθμ t a μ + ρpμ - aθ μ + ρpθ μ - b μ bθμ + aθμ μ - ρpθμ μ bθμ μ 3 + a μ - t - ρpμ - t + bμ - t. Putting t = T in equation (8), we have (5) Q = a T - t 0 - ρpt - t 0 + bt - t 0.. (6) Based on the assumptions and descriptions of the model, the total annual relevant profit (π), include the following elements: (i) Ordering cost (OC) = A (7) (ii) Holding cost (HC) is given by t 0 HC = (x+yt)i(t)e 0 dt μ μ t0 (8) = (x+yt)i(t)e dt + (x+yt)i(t)e dt + (x+yt)i(t)e dt 0 μ μ (iii) Deterioration cost (DC) is given by μ μ (iv) Shortage cost (SC) is given by μ T DC = c θi(t)e dt + θti(t)e dt (9) T SC = - c I(t)e dt (0) t0 (v) Sales revenue (SR) is given by T SR = p (a+bt - ρp)e dt () 0 (by neglecting higher powers of θ) To determine the interest earned, there will be two cases i.e. Case I: (0 t0) and Case II: (0 t0 ).

7 Inventory odel with Different Deterioration Rates with Shortages, Time and Price 505 Case I: (0 t0): In this case the retailer can earn interest on revenue generated from the sales up to. Although, he has to settle the accounts at, for that he has to arrange money at some specified rate of interest in order to get his remaining stocks financed for the period to t0. (vi) Interest earned per cycle: IE = pie a + bt - ρp te dt 0 Case II: (0 t0 ): () In this case, the retailer earns interest on the sales revenue up to the permissible delay period. So (vii) Interest earned up to the permissible delay period is: t 0 IE = p Ie a + bt - ρp t e dt + a + bt 0 - ρp t0 - t0 (3) 0 To determine the interest payable, there will be four cases i.e. (viii) Interest payable per cycle for the inventory not sold after the due period is Case I: (0 μ): (viii) IP t 0 = cip I(t)e dt μ μ t 0 = cip I(t)e dt + I(t)e dt + I(t)e dt μ μ (4) Case II: (μ μ): (ix) IP t 0 = cip I(t)e dt μ t0 = cip I(t)e dt + I(t)e dt μ Case III: (μ t0): (5) (x) IP3 t 0 = cip I(t)e dt (6)

8 506 Raman Patel and D.. Patel Case IV: (t0 T): (xi) IP4 = 0 (7) (by neglecting higher powers of b and R) The total profit (πi), i=,,3 and 4 during a cycle consisted of the following: π = SR - OC - HC - DC - SC - IP + IE T i i i Substituting values from equations (7) to (7) in equation (8), we get total profit per unit. Putting µ = vt0, µ = vt0 in equation (8), we get profit in terms of t0 and T for the four cases will be as under: π = SR - OC - HC - DC - SC - IP + IE T π = SR - OC - HC - DC - SC - IP + IE T π = SR - OC - HC - DC - SC - IP + IE T 3 3 π = SR - OC - HC - DC - SC - IP + IE T 4 4 (8) (9) (30) (3) (3) The optimal value of t0*, T and p* (say), which maximizes πi can be obtained by solving equation (9), (30), (3) and (3) by differentiating it with respect to t0 and T and equate it to zero, we have π (t,t,p) π (t,t,p) π (t,t,p) = 0, = 0, = 0, i=,,3,4 t T p i.e. i 0 i 0 i 0 provided it satisfies the condition 0 (33) π (t,t,p) π (t,t,p) π (t,t,p) t t T t p i 0 i 0 i π (t,t,p) π (t,t,p) π (t,t,p) Tt T Tp i 0 i 0 i 0 0 π (t,t,p) π (t,t,p) π (t,t,p) i 0 i 0 i 0 pt 0 pt p > 0 i=,,3,4. (34)

9 Inventory odel with Different Deterioration Rates with Shortages, Time and Price NUERICAL EXAPLE: Case I: Considering A= Rs.00, a = 500, b=0.05, c=rs. 5, ρ= 5, θ=0.05, x = Rs. 5, y=0.05, c=rs. 8, v=0.30, v=0.50, R = 0.06, Ie = 0., Ip = 0.5, = 0.04 in appropriate units. The optimal value of t0* = 0.639, T* =0.3396, p* = , Profit*= Rs and Q*= Case II: Considering A= Rs.00, a = 500, b=0.05, c=rs. 5, ρ= 5, θ=0.05, x = Rs. 5, y=0.05, c=rs. 8, v=0.30, v=0.50, R = 0.06, Ie = 0., Ip = 0.5, = 0.07 in appropriate units. The optimal value of t0* = 0.695, T* =0.333, p* = , Profit*= Rs and Q*= Case III: Considering A= Rs.00, a = 500, b=0.05, c=rs. 5, ρ= 5, θ=0.05, x = Rs. 5, y=0.05, c=rs. 8, v=0.30, v=0.50, R = 0.06, Ie = 0., Ip = 0.5, = 0.5 in appropriate units. The optimal value of t0* = 0.85, T* =0.394, p* = , Profit*= Rs and Q*= Case IV: Considering A= Rs.00, a = 500, b=0.05, c=rs. 5, ρ= 5, θ=0.05, x = Rs. 5, y=0.05, c=rs. 8, v=0.30, v=0.50, R = 0.06, Ie = 0., Ip = 0.5, = 0.5 in appropriate units. The optimal value of t0* = 0.999, T* =0.990, p* = 50.47, Profit*= Rs and Q*= The second order conditions given in equation (34) are also satisfied. The graphical representation of the concavity of the profit function is also given. Case I t 0 and Profit T and Profit p and Profit Graph Graph Graph 3

10 508 Raman Patel and D.. Patel Case II t 0 and Profit T and Profit p and Profit Graph 4 Graph 5 Graph 6 Case III t 0 and Profit T and Profit p and Profit Graph 8 Graph 9 Graph 7 Case IV t 0 and Profit T and Profit p and Profit Graph 0 Graph Graph

11 Inventory odel with Different Deterioration Rates with Shortages, Time and Price SENSITIVITY ANALYSIS: On the basis of the data given in example above we have studied the sensitivity analysis by changing the following parameters one at a time and keeping the rest fixed. Table : Case I (0 μ) Sensitivity Analysis Parameter a θ x A R ρ c % t 0 T p Profit Q +0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

12 50 Raman Patel and D.. Patel Parameter a θ x A R ρ c Table Case II (μ μ) Sensitivity Analysis % t 0 T p Profit Q +0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

13 Inventory odel with Different Deterioration Rates with Shortages, Time and Price 5 Parameter a θ x A R ρ c Table 3 Case III (μ t0) Sensitivity Analysis % t 0 T p Profit Q +0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

14 5 Raman Patel and D.. Patel Parameter a θ x A R ρ c Table 4 Case I (t0 T) Sensitivity Analysis % t 0 T p Profit Q +0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

15 Inventory odel with Different Deterioration Rates with Shortages, Time and Price 53 From the table we observe that as parameter a increases/ decreases average total profit and optimum order quantity also increases/ decreases. Also, we observe that with increase and decrease in the value of x, R and c, there is corresponding decrease/ increase in total profit and optimum order quantity. From the table we observe that as parameter A and ρ increases/ decreases average total profit decreases/ increases and optimum order quantity increases/ decreases. Also, we observe that with increase and decrease in the value of there is corresponding increase/ decrease in total profit and decrease/ increase in optimum order quantity. From the table we observe that as parameter θ increases/ decreases there is almost very little change in average total profit and optimum order quantity. CONCLUSION: In this paper, we have developed an inventory model for deteriorating items with price and inventory dependent demand with different deterioration rates. Sensitivity with respect to parameters have been carried out. The results show that with the increase/ decrease in the parameter values there is corresponding increase/ decrease in the value of profit. REFERENCES [] Covert, R.P. and Philip, G.C. (973): An EOQ model for items with Weibull distribution deterioration; American Institute of Industrial Engineering Transactions, Vol. 5, pp [] Ghare, P.N. and Schrader, G.F. (963): A model for exponentially decaying inventories; J. Indus. Engg., Vol. 5, pp [3] Goyal, S.K. and Giri, B. (00): Recent trends in modeling of deteriorating inventory; Euro. J. Oper. Res., Vol. 34, pp. -6. [4] Hill, R.. (995): Inventory models for increasing demand followed by level demand; J. Oper. Res. Soc., Vol. 46, No. 0, pp [5] Hung, K.C. (0): An inventory model with generalized type demand, deterioration and backorder rates; Euro. J. Oper. Res, Vol. 08, pp [6] andal, B. and Pal, A.K. (998): Order level inventory system with ramp type demand rate for deteriorating items; J. Interdisciplinary athematics, Vol., No., pp [7] athew, R.J. (03): Perishable inventory model having mixture of Weibull lifetime and demand as function of both selling price and time; International J. of Scientific and Research Publication, Vol. 3(7), pp. -8.

16 54 Raman Patel and D.. Patel [8] ukhopadhyay, R.N., ukherjee, R.N. and Chaudhary, K.S. (004): Joint pricing and ordering policy for deteriorating inventory; Computers and Industrial Engineering, Vol. 47, pp [9] Nahmias, S. (98): Perishable inventory theory: a review; Operations Research, Vol. 30, pp [0] Ouyang, L. Y., Wu, K.S. and Yang, C.T. (006): A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments; Computers and Industrial Engineering, Vol. 5, pp [] Patel, R. and Parekh, R. (04): Deteriorating items inventory model with stock dependent demand under shortages and variable selling price, International J. Latest Technology in Engg. gt. Applied Sci., Vol. 3, No. 9, pp [] Raafat, F. (99): Survey of literature on continuously deteriorating inventory model, Euro. J. of O.R. Soc., Vol. 4, pp [3] Salameh,.K. and Jaber,.Y. (000): Economic production quantity model for items with imperfect quality; J. Production Eco., Vol. 64, pp [4] Shah, Y.K. and Jaiswal,.C. (977): An order level inventory model for a system with constant rate of deterioration; Opsearch; Vol. 4, pp [5] Teng, J.T. and Chang, H.T. (005): Economic production quantity model for deteriorating items with price and stock dependent demand; Computers and Oper. Res., Vol. 3, pp [6] Wu, K.S., Ouyang, L. Y. and Yang, C.T. (006): An optimal replenishment policy for non-instantaneous deteriorating items with stock dependent demand and partial backlogging; International J. of Production Economics, Vol. 0, pp