Deteriorating Items Inventory Model with Different Deterioration Rates and Shortages

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1 Volume IV, Issue IX, September 5 IJLEMAS ISSN 78-5 Deteriorating Items Inventory Model with Different Deterioration Rates and Shortages Raman Patel, S.R. Sheikh Department of Statistics, Veer Narmad South Gujarat University, Surat, INDIA Department of Statistics, V.B. Shah Institute of Commerce and Management, Amroli, Surat, INDIA Abstract: An inventory model with linear trend in demand and time varying holding cost is developed. Different deterioration rates are considered in a cycle. Shortages are allowed and completely backlogged. Numerical example is provided to illustrate the model and sensitivity analysis is also carried out for parameters. Key Words: Inventory model, Varying Deterioration, Linear demand, ime varying holding cost, Shortages D I. INRODUCION eterioration of items is a general phenomenon in real life and effect of deterioration cannot be ignored in many inventory systems. EOQ models for deteriorating items are studied by many researchers. Ghare and Schrader [] considered no-shortage inventory model under constant deterioration. Shah and Jaiswal [] considered an order level inventory model for items deteriorating at a constant rate. Aggarwal [] discussed an order level inventory model with constant rate of deterioration. Dave and Patel [] developed the deteriorating items inventory model with linear trend in demand. hey considered demand as linear function of time. Hill [5] considered inventory model with ramp type demand rate. Mandal and Pal [7] developed inventory model with ramp type demand with shortages. Other research work related to deteriorating items can be found in, for instance (Raafat [9], Goyal and Giri [], Ruxian et al. []). Hung [6] considered inventory model with arbitrary demand and arbitrary deterioration rate. Mathew [8] developed an inventory model for deteriorating items with mixture of Weibull rate of decay and demand as function of both selling price and 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 different deterioration rates for the cycle time. Demand is considered as linear functions of time. Shortages are allowed and completely backlogged. o illustrate the model, numerical example is taken and sensitivity analysis for major parameters on the optimal solutions is also carried out. II. ASSUMPIONS AND NOAIONS NOAIONS he following notations are used for the development of the model: D(t) : Demand rate is a linear function of time t (a+bt, a>, <b<) A : Replenishment cost per order c : Purchasing cost per unit p : Selling price per unit : Length of inventory cycle I(t) : Inventory level at any instant of time t, t Q : Order quantity initially Q : Quantity of shortages Q : Order quantity HC : Holding cost per unit time is a linear function of time t (x+yt, x>, <y<) c : Shortage cost per unit θ : Deterioration rate during t, < θ < θt : Deterioration rate during, t, < θ < π : otal relevant profit per unit time. ASSUMPIONS: he following assumptions are considered for the development of two warehouse 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 allowed and completely backlogged. Deteriorated units neither be repaired nor replaced during the cycle time. III. HE MAHEMAICAL MODEL AND ANALYSIS Let I(t) be the inventory at time t ( t ) as shown in figure. Figure Page 6

2 Volume IV, Issue IX, September 5 IJLEMAS ISSN 78-5 he differential equations which describes the instantaneous states of I(t) over the period (, ) are given by di(t) = - (a ), t () di(t) + θi(t) = - (a ), di(t) + θti(t) = - (a ), di(t) = - (a ), with initial conditions I() = Q, I( ) = S, I(t ) = and I()=-Q. Solutions of these equations are given by: I(t) = Q - (at ), a - t + b - t + aθ - t I(t) = + bθ - t - aθt - t - bθt - t t () t t () t t () + S + θ - t a t - t - t - t 6 I(t) =. (7) + bθt -t - aθt t-t - bθt t -t 8 I(t) = - a - b + at. (8) (by neglecting higher powers of θ) From equation (5), putting t =, we have Q = S + a + b. (9) From equations (6) and (7), putting t =, we have a + b + aθ I( ) = () + bθ - aθ - bθ + S + θ a t 6 I( )=. () + bθt - - aθ t- - bθ t - 8 So from equations () and (), we get (5) (6) S = + θ a t + aθ t 6 + bθt - - aθ t- - bθ t - () 8. - a - b - aθ - bθ - +aθ - + bθ - Putting value of S from equation () into equation (9), we have Q = + θ a t + aθ t 6 + bθt - - aθ t - - bθ t - 8 () - a - b - aθ - bθ - +aθ - + bθ - + a + b. Putting t = in equation (8), we have Q = a - b - at - bt. () Using () in (5), we have I(t) = + θ a t + aθ t 6 + bθt - - aθ t- - bθ t - 8 (5) - a - b - aθ - bθ - +aθ - + bθ - + a - t b - t. Based on the assumptions and descriptions of the model, the total annual relevant profit (π), include the following elements: (i) Ordering cost (OC) = A (6) (ii) Holding cost (HC) is given by t HC = (x+yt)i(t) Page 7

3 Volume IV, Issue IX, September 5 IJLEMAS ISSN 78-5 = (x+yt)i(t) + (x+yt)i(t) + (x+yt)i(t) t = xbθ + yaθ t +x at + bθt t x - b - aθt - bθt -ya t 5 + -xa + yat + bθt t ybθ + ybθ - ybθ 8 - xaθ + y- b - aθt - bθt - x - b - aθt - bθt - ya - -xa + yat + bθt ybθ - x at + bθt xbθ + yaθ 5 8 a + b + aθ + bθ + + θ + x a t + aθ t 6 + bθ t - aθ t - bθ t 8 - a - b - aθ - bθ + aθ + bθ + θ a t + aθ t 6 + bθ t - aθ t - bθ t + x 8 - a - b - aθ - bθ + aθ + bθ +a + b +θ - xbθ + y- b + aθ 6 x - b + aθ - a - + θ a t + aθ t bθt - - aθ t- - bθ t y θ - a - b - aθ - bθ - +aθ - + bθ - - bθ - aθ - a - + θ a t + aθ t 6 + bθt - - aθ t - - bθ t - 8 x θ - a - b - aθ - bθ - +aθ - + bθ - - bθ - aθ - a + b + aθ + bθ + + θ a t + aθ t + y 6 + bθt- - aθ t - - bθ t a - b - aθ - bθ - +aθ - + bθ - (+θ ) Page 8

4 Volume IV, Issue IX, September 5 IJLEMAS ISSN θ a t + aθ t 6 + bθ t - - aθ t - - bθ t xa + y 8 - a - b - aθ - bθ - +aθ - + bθ - +a b a b aθ b + + θ a t + b t + aθ t - yb - x bθt - - aθ t - - bθ t a - b - aθ - bθ - +aθ - + bθ - +θ (7) (iii) Deterioration cost (DC) is given by DC = c θi(t) + θti(t) (by neglecting higher powers of θ) a + b + aθ + bθ -aθ - bθ + + θ( - ) at- =cθ 6 + bθt - aθ t- - bθ t- 8 -a(- )- b - - aθ - - bθ- +aθ (- )+ bθ - +θ + b + aθ + bθ θ( - ) a t - + aθ t 6 + bθt - aθ t- - cθ 8 - bθ t - a( - ) - b - aθ - bθ +aθ (- ) + bθ + θ bθt + - b - aθt - bθt t 8 5 +cθ - at at + bθt t bθ + aθ + - b - aθt - bθt cθ - a at + bθt (iv) Shortage cost (SC) is given by (8) SC = - c I(t) = - c (- at - bt + at ) t t - a - t - b - t 6 = - c (9) + at - t - t (v) SR = p (a+bt) = pa + b () he total profit during a cycle, π(t,) consisted of the following: π(t, ) = SR - OC - HC - DC - SC () Substituting values from equations (6) to () in equation (), we get total profit per unit. he optimal value of t = t * and = * (say), which maximizes profit π(t,) can be obtained by differentiating it with respect to t and and equate it to zero Page 9

5 Volume IV, Issue IX, September 5 IJLEMAS ISSN 78-5 π(t, ) π(t, ) i.e. =, = t provided it satisfies the condition π(t, ) π(t, ) t t t >. π(t, ) d π(t, ) IV. NUMERICAL EXAMPLES () () Considering A= Rs., a = 5, b=.5, c=rs. 5, p= Rs., θ=.5, x = Rs. 5, y=.5, c = Rs. 8, in appropriate units. he optimal value of t * =.6, * =.7, Profit*= Rs and optimum order quantity Q* = he second order conditions given in equation () are also satisfied. he graphical representation of the concavity of the profit function is also given. t and Profit Parameter a x θ A c able Sensitivity Analysis % t Profit Q +% % % % % % % % % % % % % % % % % % % % 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 θ, x and c, there is corresponding decrease/ increase in total profit and optimum order quantity. From the table we observe that as parameter A increases/ decreases average total profit decreases/ increases and optimum order quantity also decreases/ increases. VI. CONCLUSION Graph and Profit In this paper, we have developed an inventory model for deteriorating items with linear demand under different deterioration rates and shortages. Sensitivity with respect to parameters have been carried out. he results show that with the increase/ decrease in the parameter values there is corresponding increase/ decrease in the value of profit. REFERENCES Graph V. SENSIIVIY 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. [] Aggarwal, S.P. (978): A note on an order level inventory model for a system with constant rate of deterioration; Opsearch, Vol. 5, pp [] Dave, U. and Patel, L.K. (98): (, si)-policy inventory model for deteriorating items with time proportional demand; J. Oper. Res. Soc., Vol., pp. 7-. [] Ghare, P.N. and Schrader, G.F. (96): A model for exponentially decaying inventories; J. Indus. Engg., Vol. 5, pp. 8-. [] Goyal, S.K. and Giri, B. (): Recent trends in modeling of deteriorating inventory; Euro. J. Oper. Res., Vol., pp. -6. [5] Hill, R.M. (995): Inventory models for increasing demand followed by level demand; J. Oper. Res. Soc., Vol. 6, No., pp [6] Hung, K.C. (): An inventory model with generalized type demand, deterioration and backorder rates; Euro. J. Oper. Res, Vol. 8, pp. 9-. [7] Mandal, B. and Pal, A.K. (998): Order level inventory system with ramp type demand rate for deteriorating items; J. Interdisciplinary Mathematics, Vol., No., pp Page 5

6 Volume IV, Issue IX, September 5 IJLEMAS ISSN 78-5 [8] Mathew, R.J. (): Perisable inventory model having mixture of Weibull lifetime and demand as function of both selling price and time; International J. [9] Raafat, F. (99): Survey of literature on continuous deteriorating inventory model, J. of O.R. Soc., Vol., pp [] Ruxian, L., Hongjie, L. and Mawhinney, J.R. (): A review on deteriorating inventory study; J. Service Sci. and management; Vol., pp [] Shah, Y.K. and Jaiswal, M.C. (977): An order level inventory model for a system with constant rate of deterioration; Opsearch; Vol., pp Page 5