A Deterministic Inventory Model for Perishable Items with Lead Time and Price Dependent Demand

-In the present paper, an attempt has been made to develop a deterministic inventory model for perishable items with lead time and price dependent demand. Shortages are allowed and completely backlogged. The problem of perishability or deterioration plays an important role in the field of inventory control and management. The purpose of our study is to minimize the total variable inventory cost during a given period of time. A numerical example is given to demonstrate the developed model


INTRODUCTION
Academicians as well as industrialists have great interest in the development of inventory control and their uses. There are many goods which either deteriorate or become obsolete with passage of time. For such perishable products different modeling techniques are applied. Perishable inventory forms a small part of total inventory and includes fashionable garments, electronic items, digital products and periodicals. The perishable products can be classified based on two categories: (1) deterioration (2) obsolescence. Deterioration is defined as damage, decay or spoilage of the items that are stored for future use and which always loose part of their value with passage of time. Obsolescence occurs due to the arrival of new and better products in the market.
In the existing literature, some inventory models which were developed by contemporary researchers considering some or all of the parameters related to constant demand rate, increasing/ decreasing function of time, price and stock dependent have been quoted. The demand of newly arrived products in market is influenced by their prices, because the attractive prices or offers on the products motivate the customers to buy more. This situation increases the order quantity of the retailers or customers. In recent years some researchers also gave their attention towards a time dependent rate, because the demand of newly launched products such as fashionable garments, electronic items, motor vehicles, mobiles etc. increases with time and later it becomes constant.
But in the real life there are many situations in which these assumptions are not valid such as seasonal products, bakery products, electronic items and medicines. Some researches in the area are worth mentioning. Goswami and Chaudhuri [1] developed an EOQ model for deteriorating items with linear trend in demand and shortages. Padmanabhan and Vrat [2] considered an EOQ model for perishable items with stock dependent selling rate. Giri et al. [3] proposed an inventory model for deteriorating items with stock dependent demand rate. Hargia [4] presented an EOQ model for deteriorating items with time varying demand. Giri and Chaudhuri [5] developed a deterministic inventory model for deteriorating items with non-linear holding cost and stock dependent demand rate. Chang and Dye proposed two inventory models [6] and [11]. The model [6] is an EOQ model for deteriorating items with time varying demand and partial backlogging. And the model [11] is an inventory model for perishable items with permissible delay in payments and shortages. Chung et al. [7] presented a note on EOQ models for deteriorating items with stock dependent selling rate. Lin et al. [8] proposed an EOQ model for deteriorating items with time varying demand and allowing shortages. Papachristos and Skouri developed two inventory models [9] and [12]. In model [9] they presented an optimal replenishment policy for deteriorating items with exponential type backlogging rate and time varying demand. The model [12] is a continuous review inventory model for deteriorating items with time dependent demand and allowing shortages. Goyal and Giri developed two inventory models [10] and [15]. In model [10] they considered recent trends in modeling of deteriorating inventory. And the model [15] is a production inventory model with time varying demand, production and deterioration rate. Wu [13] proposed an EOQ model for Weibull deteriorating items with time varying demand and allowing shortages. Wang [16] presented a note on EOQ model for perishable items with exponential distribution, deterioration and time dependent demand rate. They also considered shortages in their inventory model. Dye and Ouyang [17] developed an EOQ model for perishable products with stock dependent selling rate and allowing shortages. Shah [18] proposed an inventory model for deteriorating items with time value of money and permissible delay in payments. She considered a finite planning horizon in her inventory model. Hou and Lin [19] developed an EOQ model for deteriorating items with price and stock dependent selling rate. They considered the effect of inflation and time value of money in their inventory model. Dye [20] presented a joint pricing and ordering policy for deteriorating items with partial backlogging. Roy et al. [21] presented an inventory model for deteriorating items with stock dependent demand rate and fuzzy type inflation. They also considered time discounting over a random planning horizon. Min and Zhou [22] developed an inventory model for deteriorating items with stock dependent selling rate and allowing shortages. Jain et al. [23] proposed an inventory model for deteriorating items with fuzzy type inflation and cash discounting over random planning horizon. Panda et al. [24] developed a two warehouse inventory model for deteriorating items with fuzzy type demand rate and lead time. Roy [25] proposed a fuzzy inventory model for deteriorating items with price dependent demand rate. Chaudhary and Sharma [26] presented an inventory model for Weibull deteriorating items with price dependent demand rate under inflation. Maragatham and Palani [27] developed an inventory model for perishable items with lead time, price dependent demand and allowing shortages.

II. ASSUMPTIONS NOTATIONS
We consider the following assumptions and notations Here p is the selling price.

3.
C o is the ordering cost per order.

4.
c h is the holding cost per unit time.

5.
The solutions of the equations (1) and (2) are given by the equations (3) and (4) respectively. By considering the first degree terms in , we have The maximum inventory level is obtained by putting L t  in equation (3) The ordering cost per cycle is The holding cost per cycle is The deterioration cost per cycle is The shortage cost per cycle is The purchase cost per cycle is The total variable inventory cost per cycle is Putting the values of    (   T   T  T  L  TC  T  T   T  T  L  TC  L  T   T  T  L  TC   T  T   T  T  L  TC  T   T  T  L  TC  L  T   T  T  L  TC   T  L   T  T  L  TC  T  L   T  T  L  TC  L   T  T  L  TC H Partially differentiating equation (13), we have

V. CONCLUSION
The results of the proposed model show that the total variable inventory cost is deeply impacted by the parameters a and b in comparison with the parameter  . This is due to the reason that the newly arrived goods/products in the super market increase the demand. The cycle length and lead time are main components for optimizing the cost/profit of an organization. The products such as vegetables, milk, bakery products and news papers are necessarily to be sold in the market as the cycle length decreases.