A new model for deteriorating items with inflation under permissible delay in payments

Article history: Received September 22 2013 Received in Revised Format April 6 2014 Accepted April 18 2014 Available online April 29 2014 Inflation is an important factor influencing traditional economic order quality models. Marketing strategy depends on inflation due to public demand and availability of the materials. This paper presents an optimal inventory policy for deteriorating items using exponential demand rate under permissible delay in payments. Mathematical model has been derived under two cases: case I: cycle time is greater than or equal to permissible delay period, case II: cycle time is less than permissible delay period by considering holding cost as a function of time. Numerical examples and sensitivity analysis are given to reflect the numerical results. Mathematica software is used for finding optimal solutions. © 2014 Growing Science Ltd. All rights reserved


Introduction
The objective of many inventory management problems is to deal with the minimization of inventory carrying expenditures (Donalson, 1977).Thus, it is necessary to determine the optimal inventory level as well as optimal time of replenishment of inventory to meet any future demand.Among the classical inventory management models, there are many cases on solving the optimal order quantity by ignoring the type of payment.At present scenario, it is observed that supplier offers a certain fixed period to settle the account for stimulating retailers demand.During the credit period when the payment is made, some items can be sold and revenue can be accumulated to earn interest.This paper investigates inventory model for deteriorating items with exponential time dependent demand rate.In recent years, deteriorating inventory models have been widely studied.In real life situations, it is observed that demand for a particular product can be influenced by internal factor such as inflation, price and availability.The change in the demand is responsible for the change in inventory is commonly referred to as demand elasticity.Generally, inventory model considers a case in which depletion of inventory is caused by demand rate and deterioration.Most of the items deteriorate over time and this phenomena plays essential role for decision making in modern organization.During the past few decades, many researchers have developed inventory models for deteriorating items.
The analysis of deteriorating model is discussed by Ghare and Schrader (1963) with constant rate of deterioration.In the earlier period, researchers have discussed various demand patterns fitting the stage of product lifecycle.Resh et al. (1976) and Donaldson (1977) considered the situation and linearly time varying demand and established an algorithm to determine the optimal number of replenishments and timing.Henery (1976) further generalized the demand rate by considering a concave demand function increasing with time.Harris (1913) first introduced the basic economic order quantity (EOQ) model.Several interesting research papers are associated with deterioration, which are based on constant demand without any deterioration function (e.g.Sachan, 1984;Dave & Patel, 1981;Goyal & Giri, 2001;Liao & Haung, 2010;Sana, 2010;Lin et al., 2010;Sarkar, 2012;etc.).Covert and Phillip (1973) extended Ghare and Schrader's model by considering variable rate of deterioration.Data and Pal (1988) developed an EOQ model by introducing a variable deterioration rate and power demand pattern.Chung and Tang (1994) determined the replenishment schedules for deteriorating items with time proportional demand.However, in real life, the demand may increase or decrease in the course of time.Many researchers considered the varying demand (e.g.Sana, 2010;Sana & Chaudhuri, 2008;Donaldson, 1977;Goyal, 1986;Kharna & Chaudhuri, 2003;Silver & Meal, 1969;Haringa, 1996;Goyal et al., 1986).
Various researchers have developed the inflanatory effects on the inventory policy.Liao et al. (2001) developed an inventory for initial stock dependent consumption rate when a delay in payment is permissible.Hou (2006) derived an inventory model for deteriorating items with stock dependent consumption rate and shortages under inflation and time discounting over a finite planning horizon.Buzacolt (1975) developed an inventory model with inflation.Vrat and Padmanabhan (1990) developed an inventory model under a constant inflation rate and initial stock dependent consumption rate.Datta and Pal (1991) developed a model with linear time dependent demand rate and shortages to investigate the effects of inflation with time value of money on ordering policy over a finite time horizon.
Recently, Teng et al. (2012) developed an EOQ model with trade credit financing for non-decreasing demand and fundamental theoretical results obtained.Sarkar (2011) developed an EOQ model with delay in payment for time varying deterioration rate and obtained a function for maximization of profit.Pricing and lot sizing policies for deteriorating items with partial backlogging under inflation was presented by Hsieh and Dye (2010) by considering pricing and lot sizing policies for deteriorating items with partial backlogging under inflation.
In this paper, demand rate is exponential time dependent and holding cost is time dependent.The present model is discussed by using truncated Taylor's series.The conditions for convexity of optimality are obtained and numerical examples and sensitivity analysis are given.
The rest of the paper is organized as follows: In the next section assumptions and notations are given.In section 3 mathematical formulations with maximization of total inventory cost is given.In section 4 numerical examples for the cases I and II are given.In section 5 sensitivity analysis with various parameters is given to validate the inventory cost function.Finally conclusion and future research directions are given in the last section 6.

Assumptions and Notations
The mathematical model of inventory for deteriorating items is based on the following assumptions: (i) Demand rate is exponential and Inflation is constant (ii) Shortages are not allowed and lead time is zero (iii) During the permissible delay period the sales revenue generated is deposited in an interest bearing account.At the end of the trade credit period the customer pays off all units ordered and begins paying for the interest charged on the items in stock.
(iv) There is no repair or replenishment of deteriorated items during the given cycle.
(vii) Holding cost is time dependent i.e.
The following notations are used throughout the manuscript:

Mathematical Formulation
The level of inventory I(t) decreases gradually mainly to meet demands and due to deterioration.Thus, the variation of inventory with respect to time can be described by the following differential equation: The solution of equation is given by and order quantity Ordering Cost: Instantaneous ordering cost per order The number of deteriorated units = The cost of total deteriorated units Inventory Holding Cost HC I is given by Depending on the customer's choice and the length of cycle time T two possible cases are taken into account: Optimal cycle time T is greater than the permissible delay time 1 T , the interest charged IC 1 during the period [0, H] is given by Differentiating Z 1 (T) with respect to 'T' two times we get In this case, the retailer pays the procurement cost to the supplier prior to expiration of the delay period 1 T provided by the supplier.Hence, the interest charged 2 IC will be zero.Since cycle time T is less than permissible delay time T 1 , the interest earned IE 2 during [0, H] is given by Differentiating partially ( 16) w.r.t.'T' two times yields From the above tables we conclude the following results: From Table 1, we observe that: Increase in 1 T results decrease in T, Q and increase in Z, keeping , and r constant and increase in r results increase in T, Q and Z keeping T 1 constant.
From Table 2, we observe that: Increase in T 1 results decrease in T, Q and Z keeping r constant and increase in r results increase in T, Q and Z keeping T 1 constant.
From Table 3, we observe that: Increase in deterioration rate  results decrease in T, Q and increase in Z keeping  constant.An increase in  results increase in T and decrease in Q and Z keeping  and T 1 constant.
From Table 4, we observe that: Increase in  results decrease in T, Q and increase in Z keeping r constant.An increase in r results increase in T, Q and Z.

Conclusion
In this paper, an inventory model has been developed for deteriorating items under permissible delay in payments.Optimal solutions were obtained for both cases i.e. case I and II.Numerical examples and sensitivity analysis have been presented to obtain optimal cycle time and optimal total average cost per unit time.The sensitivity analysis is quite sensitive to the managerial point of view.
The proposed model can be extended in several ways for instance we may consider the demand rate as quadratic time dependent or stock dependent patterns as well as discount demand.We could extend the model for non -deteriorating demand function to stock dependent demand function.In addition, we could generate the model to allow shortages, finite capacity and others. 2 When  and r are allowed to vary by using T 1 = 225, 240, 255, 270, 285 and 300 days.We get different values of parameters as shown in Table4as follows,