Economic order Quantity (EOQ) for Deteriorating Items with Non-instantaneous Receipt under Trade Credits

 This paper presents an inventory model with non-instantaneous receipt under the condition of permissible delay in payments. Certain items like volatiles deteriorate during production process. In this paper, we consider deterioration into two phases i.e. phase 1 and phase 2. The purpose of this paper is to determine the optimal replenishment policies under conditions of non-instantaneous receipt and permissible delay in payments. The replenishment rate is assumed to be greater than demand rate. Second order approximation have been used for in exponential terms for finding closed form solution of optimal order quantity, order cycle and order receipt period, so that the total profit per unit time is maximized. Some results have also been obtained. Numerical examples are presented to validate the proposed model. The sensitivity analysis of the solution with the variation of the parameters associated with the model is also discussed.


INTRODUCTION
The deterioration of goods is a realistic phenomenon in much inventory system. Maximum items or commodities undergo deterioration over time. The controlling of deteriorating items is a measure problem in any inventory system. Fruits, vegetables and food products suffer from depletion by direct spoilage while stored. Highly deteriorating items like volatile liquids such as alcohol, gasoline and turpentine undergo physical depletion over time through the evaporating process. Radioactive substances, electronic goods, photographic film, grains etc deteriorate through continuous loss of utility or potential with time. Hence, while developing an optimal inventory policy for such products, the loss of inventory due to deterioration cannot be ignored. The two researchers Ghare and Schrader (1963) were considered decaying inventory for a constant demand. Covert and Philip (1973) extended Ghare and Schrader's model for variable deterioration rate by assuming two parameters Weibull distribution function. Later, Shah and Jaiswal (1977) presented an order level inventory model for deteriorating items with a constant rate of deterioration. Aggarwal (1978) corrected and modified the error in Shah and Jaiswal's model (1977) by considering an order level inventory model and calculated the average inventory holding cost. Goyal and Giri (2001) developed a detailed review of deteriorating inventory. The models for these type products have been developed by Mishra (1975) , Chakrabarty et al. (1998), Hariga (1996) , Wee (1995) , Jalan et al. (1996) , Su et al. (2007) , Giri and Chaudhuri ((1997 In today's business competition, it can be observed that suppliers offer a certain fixed credit period to settle the account for stimulating retailer's demand. The classical inventory management is almost concentrated on solving the optimal order quantity and reorder point but neglecting the type of payment. In the above models, authors/ researchers assumed that an entire order is received into inventory at one time (instantaneously). In real world the order quantity is frequently received gradually over time and the inventory level is depleted at the same time it is being replenished. Hence the more realistic assumption is the non-instantaneous receipt. Ouyang et al.(2004) developed an inventory system with non-instantaneous receipt under condition of permissible delay in payments. Sugapriya and Jeyaraman (2008) considered the economic production quantity for non-instantaneous deteriorating items allowing price discount with constant production and demand rate extending the facility of permissible delay in payments. Choi and Hwang (1986) developed a model determining the production rate for deteriorating items to minimize the total cost function over a finite planning horizon. Raafat (1985) extended Choi and Hwang (1986) model, given in Park (1983) to deal with a case in which the finished product is also subject to a constant rate of deterioration. Yang and Wee (2003) presented a multi-lot-size production inventory system for decaying items with constant demand and production rates. Ghiami et al. (2013) investigated a two-echelon supply chain model for deteriorating inventory in which the retailer's warehouse has a limited capacity. Ouyang and Cheng (2008) presented the optimal ordering with decaying items under permissible delay in payments, and considered two possible ways for retailer to pay off the loan. Li et al. (2014) developed a model for inventory game with permissible delay in payments. Ouyang and Chang (2013) explored the effects of the reworking imperfect quality item and trade credit on the EPQ model with imperfect production process and complete backlogging. Bhunia et al.(2014) developed an inventory model for single deteriorating item with two separate warehouses having different preserving facilities.
In this paper, we consider the order cycle [0, T] into two parts (i) inventory replenishment period and (ii) inventory depleted period. The deterioration is taken in both parts. In this study we develop an EOQ model with non-instantaneous receipt under trade credits. We then establish numerical solution for finding the optimal order quantity, order cycle and order receipt period so that the total relevant profit per unit time is maximized. Truncated Taylor's series expansion is used for finding closed form optimal solution. Numerical examples and sensitivity analysis is given to validate the proposed model. Three results have also been obtained from the optimal solution.
The rest of this paper is organized as follows. Section 2 is notation and assumption we adopt through this paper. In section 3 we develop mathematical models for the two different situations. Section 4 presents the determination of the optimal replenishment time with some results. In section 5 numerical examples are given to illustrate the proposed model followed by sensitivity analysis in section 6. Finally, we provide conclusion and future research direction in the last section 7.

Notations
The following notations are used throughout this manuscript to develop the proposed model: 3. The inventory system under consideration deals with single item. 4. The replenishment rate λ, is finite and greater than demand rate D, i.e. λ > D. 5. Supplier offers a certain fixed period, m to settle the account. 6. Retailer would not consider paying the payment until receiving all items. 7. The order cycle period [0, T] is divided into two phases (i) inventory replenished period (phase 1) (ii). Inventory depleted period (phase 2). 8. There is no replenishment or repair for a deteriorated item.

MATHEMATICAL FORMULATION
According to the assumption the order cycle [0, T] is divided into two parts (i) inventory replenished period (ii) inventory depleted period. The two different cases are shown in the following fig 1. The change of inventory in the above two phases can be described as follows: Phase 1. In this phase replenishment rate is greater than the demand rate, the inventory go up to maximum level (called order quantity). The rate of change of inventory at time 't', 1 ( ) dI t dt is given by With the boundary condition I 1 ( 0 ) = 0 Phase 2. Replenishment is stopped and the inventory decreases due to demand and deterioration. The rate of change of inventory at time 't', 2 ( ) dI t dt can be described by With the boundary condition 2 ( ) 0 I T = . The solution of (1) and (2) are respectively given by But the order quantity (3) and (4) we obtain ( ) We can obtain the total profit per unit time for the following two cases: This case is shown in Fig 1 (a). In this case, the total profit per cycle consists of sales revenue, ordering cost, holding cost, interest payable and interest earned. The components are calculated as follows: (a). Sales Revenue The ordering cost per order s = (c) The holding cost during the interval [0, T] is given by ( ) (d). The interest payable per cycle is given by (e) The interest earned per cycle is given by Therefore, the total profit per unit time is given by

Case 2. T ≤ m
This case is shown in fig.1 (b). In this case, the total profit per cycle consists of the sales revenue, ordering cost, holding cost and interest earned. Since cycle time is less than credit period, the retailer pays no interest and earns the interest during the period [0, m]. The interest earned in this case is given by Total profit per unit time is given by

DETERMINATION OF OPTIMAL REPLENISHMENT TIME
Since it is difficult to handle above equations for finding the exact value of T, therefore, we make use of the second order approximation for the exponential and logarithm in equations (10), (12) and (5) , which follows as Also from (5) we obtain . Using (16) in (14) and (15), we obtain Note that the purpose of this approximation is to obtain the unique closed form solution for the optimal value of T. By taking first and second order derivatives of Z1(T) and Z2(T) from (17) and (18), with respect to T, we obtain From (21) and (22) it is clear that Z 1 (T) and Z 2 (T) both are concave function of T. It can be seen from the following graph: The objective is to determine the optimal value of * T = T for case I which maximize the total profit per unit time 1 1 Z (T *) . The necessary condition for Z 1 (T) to be maximum at point The optimal value of ** T = T is obtain by solving, 2 ( ) From the above discussion we observe the following properties: Result 1.Substituting (23)  The optimal economic order quantity for each case given by In classical EOQ model with non-instantaneous receipt, the retailer must pay the payment at the beginning of each cycle. Hence the classical optimal economic order From the above discussion we obtain the following theorem. The change in the values of parameters may happen due to variation or uncertainties in any decision -making situation. The sensitivity analysis will be very useful in decision making in order to examine the effect and variation of these changes. Using the above data, the sensitivity analysis of various parameters has been done. The results of sensitivity analysis are given in the following tables.        Table 8: Effect of unit holding cost 'h' on optimal replenishment policy. The following inferences can be made from the result obtained. (a). When ordering cost per order 's' increases, the optimal receipt T 1 , optimal cycle time T , and optimal order quantity Q increases while total profit per cycle decreases. That is , the change in 's' will cause the positive change in optimal