An Optimal Inventory Policy for Items having Linear Demand and Variable Deterioration Rate with Trade Credit Singh Sarbjit and Singh Shivraj

Problem statement: Demand considered in most of the classical inventor y models is constant, while in most of the practical cases the demand changes with time. In this study model has been framed to study the items whose demand changes with time and deterioration rate increases with time. The effect of permissible delay is also incor porated in this study. The objective of this resear ch is to develop an inventory model for perishable items whose perish-ability rate as well as demand increases with time Approach: Firstly, problem is framed in the form of linear d ifferential equation model and this model had been solved using general solution techniques of linear differential equations. The solution obtained gives the inventor y level at any particular time of the cycle period. With the help of this inventory level, total as wel l as average inventory cost has been obtained. Results: This study developed a model to determine an optima l order quantity by using calculus technique of maxima and minima. Thus it helps retai l r to decide its optimal ordering quantity under the constraints of variable deterioration rate and li ear pattern of demand. Conclusion: Numerical solution of the suggested model had also been propo sed, the above model can be converted into constant demand model, or for items having no deter ioration. This study can further be extended for items having some other demand pattern, also time v alue of money and inflation can be incorporated in this model to make it more realistic and present bu siness environment suited.


INTRODUCTION
The concept of permissible delay is not new, it has been there from time immortal, even when currency was not in circulation and barter system was prevalent, than also permissible delay was provided by suppliers to buyers. In general practice, suppliers are known to offer their customers a fixed period of time and do not charge any interest for this period. However, a higher interest is charged if the payment is not settled by the end of credit period. The permissible delay in payment produces three advantages to the supplier Firstly it helps to attract new customer as it can be considered some sort of loan. Secondly it helps in the bulk sale of goods. The existence of credit period serves to reduce the cost of holding stock to the user, because it reduces the amount of capital invested in stock for the duration of the credit period. Thirdly it may be applied as an alternative to price discount because it does not provoke competitors to reduce their prices and thus introduce lasting price reductions. Goyal [7] was the first to introduce the concept of permissible delay. Aggarwal and Jaggi [1] then extended Goyal's model for deteriorating items. Jamal et al. [10] further generalized the model to allow for shortages and deterioration. Teng [11] considered the EOQ under condition of permissible delay in payment which is further extended by Ken Chung Kun and Yun-Fu-Huang [4] for limited storage capacity. Goyal [12] developed the optimal pricing and ordering policies for items under permissible delay. Ghare and Schrader [6] were the first to use the concept of deterioration followed by Covert and Philip [5] who formulated a model with variable rate of deterioration with two parameter Weibull distributions, which was further extended by Shah [8] . Buzacott [3] , Bierman and Thomas [2] investigated the inventory models with inflation followed by Misra [11] . Mishra also developed a discount model in which the effect of both inflation and time value of money was considered. Chandra and Bahner [4] had also developed model under inflation and time value of money. Chang [3] presented model for the situation where the demand rate is a time-continuous function and items deteriorate at a constant rate with partial backlogging. Su et al. [14] also developed model under inflation for stock dependent consumption rate and exponential decay. Sarbjit and Shivraj [10] developed an inventory model having linear demand rate under permissible delay in payments with constant rate of deterioration.
This study is related to products having linear demand and variable rate of deterioration, which is the extension of author's earlier paper having linear demand pattern with constant rate of deterioration. Generally in trade credit, if cycle period is more than permissible delay, the buyer settles its account only after completion of cycle period. Plus most of the models on permissible delay consider cost price and selling price equal which is not feasible. But it is a fact that it is more beneficial to settle the account in between the permissible delay and cycle period. This time is known as payment time. With the help of this model retailer can easily extract its cycle period as well as payment time to reduce the total inventory cost.

Mathematical model and analysis:
In this model variable rate of demand is considered with variable rate of deterioration. Depletion of the inventory occurs due to demand (supply) as well as due to deterioration which occurs only when there is inventory i.e., during the period [0, T]. For this period the inventory at any time t is given by: where, at t = 0, I (t) = I 0 (initial inventory level). Putting this value in the above equation, we get: This gives: It is obvious that at t = T i.e., at the end of cycle period, I (T) = 0. So Eq. 2 yields: Substituting the value of I o in Eq. 2 we get: The total demand during cycle period T is ( ) Evaluating the cost functions: • For most of the inventory systems ordering cost for items is fixed at A Rupees/order • The cost of deterioration: Interest earned per cycle, I T is the interest earned during the positive inventory is given by: The variable cost is aggregately comprised of ordering cost, carrying cost, cost due to deterioration of materials and the interest payable minus the interest earned. Thus, the total variable cost per cycle, C VT is defined as: The variable cost per unit time is given by: VT T C (P,T) C (P,T) T = The optimal values of P and T will provide the optimal solution which can be obtained by variation techniques

RESULTS
One of the objectives of this study is to develop an inventory model for perishable items having linear demand pattern. Then with the help of this model total inventory cost is obtained. The practical aspects of inventory management like opportunity cost and effect of permissible delay in payment is also considered.
The total cost obtained then can be used to obtain an average inventory cost, which can be optimized using calculus techniques.
Numerical illustrations proves the applicability of the suggested model

DISCUSSION
The model considered above is suited for items having variable deterioration rate, earlier models have considered items having constant rate of deterioration. This model can be used for items like fruits and vegetables whose deterioration rate increases with time. Demand pattern considered here is linear demand patterns, which can also be converted into constant demand pattern.
The suggested model can further be extended for fixed credit period as well as allowable shortages. This study will help retailers in deciding their optimal ordering quantity to have minimum inventory cost.

CONCLUSION
This study can help substantially retailers or buyers in deciding their payment time, considering the benefit of permissible delay in payments. In this study items having linear demand are considered with variable rate of deterioration. The cost price and selling price are different which increases the practicality of this model as most of the earlier models considered cost price and selling price of the articles same. This model can be further extended for items having quadratic demand or power demand. As well as effect of inflation and time value of money can also be considered. This study will act as a catalyst for the study of permissible delay in payments.