An Economic Production Quantity model with inventory dependent demand and deterioration

EPQ models are used to control the inventory effectively. The present work proposes EPQ model exclusively for deteriorating items. In the present work the EPQ model has been developed by assuming that deterioration of the items start after some constant time as it enters into the inventory. For that during production up time, demand rate is considered as inventory dependent and kept constant during down time. An efficient procedure is developed to find the optimal production run length. The influence of inventory dependent consumption rate on the production up time is also discussed in this model. A numerical example is provided to illustrate the theoretical results. The result of sensitivity analysis indicates that production up time and total cost per unit time, both are highly sensitive to change in production and demand rate. Demand consumption parameter reduces the total inventory and hence holding cost.

Most of the inventory models considered various demand patterns like stock dependent demand, power form demand, ramp type of demand, time dependent demand, selling price dependent demand and exponential demand. It is well known that, the demand rate vary with change in inventory level. From the literature review, it is observed that the different demand pattern has not been discussed by any researcher so far on different time period. The present study may be significant in filling this gap since it aims to develop EPQ model by assuming demand as inventory dependent criteria during inventory buildup time and constant during inventory depletion period. This paper has several sections. Research motivation and literature is narrated in introduction. Next section contains notations and assumption. The following sections formulate the model and derives the optimal solutions. The aspect of numerical and sensitivity analysis is discussed in some details followed by some concluding remarks.
II. ASSUMTIONS AND NOTATIONS Following assumption and notations are used to develop the model.

A. Assumptions
The following assumptions are made in development of the model.
a) The production rate is known and is kept constant. b) The production rate is greater than the demand. c) The demand is inventory level dependent in up time and kept constant in down time. d) Deterioration of the items is at constant rate. e) Inventory holding cost is known and termed as constant. f) Deterioration of the items start after some constant time as it enters into inventory. g) Shortages are not allowed. h) Every produced items needs inspection B. Notations I 1 -Inventory level during production up time. I 2 -Inventory level during production down time. III. MODEL FORMULATION Present work deals with an EPQ model for deteriorating items. Inventories are buildup gradually during production up time, new production run will start after complete consumption of the buildup inventory. At the beginning it is assumed that the inventory is zero. As the production rate is greater than the demand rate, inventory is gradually buildup at a rate of (P-D). During production up time the demand rate is inventory dependent. At time T 2 the inventory will be maximum. At this stage production is terminated and on hand inventory will be used to meet the demand and to offset the loss due to deterioration. During production down time demand remains constant. As shown in fig 1,the production will start at t = 0, During the time period (0, T 1 ) the inventory will gradually build up with no deterioration. For the time period (0,T 2 ), inventory will build up under the action of inventory dependent demand and deterioration. Maximum inventory will be at time t = T 2 . Later, production stops and buildup inventory is consumed to fulfill the demand. Production system can be described by the following differential equations. According to the assumptions, over time span [0, T 1 ], demand rate is inventory dependent and no deterioration, which makes variation of the inventory level with respect to time for the reference time ,be governed by.
In the time interval (0, T 2 ), the system is affected by the combined effect of inventory dependent demand and deterioration. Hence, the change in inventory level is governed by the following differential equation.
In the time interval (0, T 3 ), the system is affected by the combined effect of constant demand and deterioration. Hence, the change in inventory level is governed by the following differential equation.
Initial boundary conditions associated with this equations are, at t = 0, I 1 (t) = 0 , at t = T 2 , I 2 (T 2 ) = Q 1 and at t = T 3 , I 3 (T 3 )=0 the solution to above equations is as follows. These three equations are used in the derivation of our model.
By using initial boundary conditions I 3 By using boundary conditions I 2 (0) = 0 and from Eq.8 Total inventory is given by Inventory holding cost is given by 12. All produced items are inspected , inspection cost is given by,  T  T  T  T  T  3  1  2  1 2 TC = A + h I t dt + I t dt + I t dt + C I t dt + I t dt Production cycle time = T = T + T + T 1 2 3  Where, The optimum production up time can be derived by satisfying the equation 17. dTCT = 0 dT 2 1 7 . Ignoring the higher power terms of T 2 , following quadratic equation can be obtained.
The closed form solution can be obtained by considering positive root of the equation. IV. NUMERICAL AND SENSITIVITY ANALYSIS Numerical example and sensitivity analysis has been carried out to validate the theoretical aspects. The numerical data is adopted from (Jie,) [17]  The optimum value of T 2 can be found, as the total cost function is convex (Fig. 2). The optimum value of T 2 is 0.221 The optimum total cost per unit time is TCT = Rs.763.26. Sensitivity analysis is carried out by changing each parameter by -40% to +40%, taking one parameter at a time and keeping others unchanged.   As shown in Fig.3, Production uptime is highly sensitive to production rate, demand rate. It is moderately sensitive to holding cost and inspection cost while it is slightly sensitive to deterioration rate and inventory consumption rate. Production up time decreases due to increase in the production rate. For the first 40% increase in the production rate ,it is observed that rate of decrease in the production up time is more as compared to last 40% increase in production rate. When rate of production is higher, inventory will build at higher rate. As the demand is inventory dependent in the production up time, net demand of the product increases. This indicates that at higher rate of production demand is higher so inventory will build up at slower rate. Production up time increases due to increase in the demand rate. From fig.4, It is observed that total cost per unit time is highly sensitive to production and demand rate. It is moderately sensitive to holding and inspection cost while slightly sensitive to inventory consumption parameter and deterioration rate.   5 shows that non production time is highly sensitive to production and demand rate. It is slightly sensitive to holding cost, inspection cost, inventory consumption parameter and deterioration rate.  Fig.6 and 7 shows that holding cost and total inventory is sensitive to change in parameter α. It is observed from Fig 3 that, TCT increases with increase in the production rate. This happens because increase in production rate increases the inventory and hence the holding cost. But if the demand pattern is inventory level dependent, then increase in inventory increases the demand and hence decrease in total inventory and decrease in inventory holding cost. Subsequently it can be commented referring Fig.6 that, increase in the inventory dependent consumption rate parameter decreases the holding cost. Again from Fig.7 increase in α decrease the total inventory. This attracts the attention of inventory managers to estimate the accurate value of the inventory dependent consumption rate parameter.
V. CONCLUSION In this study, theoretical EPQ model has been developed over an infinite horizon. During inventory buildup time, demand used is inventory level dependent, and during inventory depletion it is constant. No deterioration for time period (0, T 1 ) and there after constant deterioration rate. Increase in demand parameter α decreases the holding cost and total inventory. Sensitivity analysis shows that TCT as well as production up and down time are highly sensitive to production and demand rate. Increase in inventory dependent consumption parameter decrease the total inventory and hence holding cost. This indicates that higher inventory stock can attract the customer to buy more. This model can be useful for the inventory managers in decision making especially for the perishable items. The model can be further developed by considering different deterioration rate, production rate.
VI. REFERENCES