EFFECT OF ADVANCE PAYMENT ON AN INVENTORY MODEL FOR A DETERIORATING ITEM CONSIDERING EXPIRATION TIME OF PRODUCT, RESTRICTED SHELF SPACE AND AGE DEPENDENT SELLING PRICE WITH DISCOUNT FACILITY

The life-time of a deteriorating product is a major issue in inventory management. In practise the quality as well as quantity of deteriorating product deteriorates over time and the quality (freshness) of the product influences the customer’s demand. With the consideration of product life time, we develop an inventory model with price and freshness dependent demand under advance payment system with discount facility. The selling price is freshness sensitive. Goal of our model are twofold. The first one is customer’s demand is dependent upon the newness/freshness state of the product as well as selling price of the product. In the last one stock level at the end of cycle is relaxed. The solution process of projected optimization model is illustrated theoretically. Couple of numerical examples and sensitivity analysis are provided to demonstrate the feature of the profit function. Concavity of the average profit function is shown by plotting graphs. This study shows that all parameters in the proposed maximization model significantly influence the optimal solution.


1.INTRODUCTION
To maintain proper balance between production supply and customer's demand, we observe that inventory or stock management is essential in any kind of business sector or industries.
In the classical economic order quantity (EOQ) model, it was often assumed that purchaser 2 LAKSHMI NARAYAN DE pays the cost of items or products at the time of release of the product from the seller. Goyal [1] was the first person who introduced the concept of delay-in-payments in his EOQ model to encounter inventory backlog problem due to the retailer's financial constraints for the time being. There is no interest charge if the cost of items is paid with in recognized delay period.
On the other hand, if the payment is not paid in full by termination of the permissible delay period, interest is charged on the unpaid amount. There have been extensive researches concerning the EOQ with permissible delay in payments. Aggarwal & Jaggi [2] extended Goyal's [1] model by considering perishable products. Chang et al. [3] developed an economic order quantity model under the assumption that delay period is linked with ordering quantity. De and Goswami [4] provided the probabilistic EOQ model for declining items under tolerable delay in payments.
Offering delay period in payment sometimes leads to several disadvantages, one of being that the supplier does not receive money instantly and this may lead to a supply crisis. In order to address this issue a different tactic practised in the marketplace is advanced payment. The reverse of delay-in-payment is advance payment strategy. Sometimes it becomes much difficult for business person or seller to carry on himself/herself in the competition when the stock level of the particular product is volatized in the market due to higher demand of the product and its insufficient supply. To catch the attention of the consumers in addition to make the potential consumers in to the regular buyers, the business person use different types of price cut scheme, for example, price concession, seasonal price cut, discount due to prepayment or advance payment, etc. At the present time, the price cut due to the advance payment provision becomes a new inclination in the market dynamics. When a retailer places an order for the product, the supplier demands money in advance. Next, supplier provides some price discounts for advance payment. Some suppliers also tolerate the retailer to disburse a part of the purchased cost by instalment. Thus, from the last few decades, researchers or academicians are truly engrossed to study the EOQ model with price cut under advance payment provision. Gupta et al. [5] firstly proposed the advance payment scheme in their EOQ model. Maiti et al. [6] also introduced the concept of advance payment in an inventory model by considering selling price dependent demand and stochastic lead time.
Thangam [7] discussed advance payment policy to find out optimal lot-sizing strategy for the perishable goods. Taleizadeh [8] well thought-out a number of prepayments for declining items with shortages. Teng et al. [9] developed an EOQ model with expiration time dependent deterioration rate and advance payment strategy. Diabat et al. [10] introduced partial downstream delayed payment, partial up-stream advance payment in their model for deteriorating items with partial back ordering. Recently Mashud et al. [11] established joint pricing inventory model of deteriorating products with expiration time dependent 3 EFFECT OF ADVANCE PAYMENT ON AN INVENTORY MODEL deterioration rate under the effect of advance payment with discount facility. Rahman et al. [12] also developed a hybrid price and stock dependent EOQ model for deteriorating items with advance payment related to price cut facility under preservation technology. Duary et al. [13] invented a price-discount inventory model for deteriorating product with partially backlogged shortage under the joint effect of advance and delay in payment. In this projected work we incorporate advance payment policy with price discount to promote sales and decrease supply crunch simultaneously.
Deterioration is a common occurrence of the most of the goods, so researchers can not overlook it to make strong inventory model. It is well known that most of the products of grocery shop, vegetable and fruit shop, dairy farm, medicine shop, alcohol shop etc., will spoil or damage or expire over time. The life time of the deteriorating item is limited and these have an expiration date. Specifically, the rate of deterioration accelerates over time and the item will completely spoils at the expiration date and has no utility for consumers.
Therefore time dependent deterioration rate should be well thought-out to develop an inventory policy. Researchers developed several extensive models for declining items with an expiration date or utmost life time. Sarkar and Sarkar [14] established a model for deteriorating item with stock dependent demand and time dependent deterioration rate. Teng et al. [9] studied another inventory model of deteriorating products in which the deterioration rate is a function of time and life time of the product. Wu et al [15] considered an inventory model of deteriorating items in which deterioration rate function approaches the full item value near the expiration date. Tiwari et al. [16] invented a supply chain EOQ model with an expiration date to find out the optimum cost and ordering cycle. Recently, Iqbal and Sarkar [17] studied deteriorating items with life-time-reliant demand rate and incorporated the consequence of preservation technology. Mashud et al. [11] also established an inventory model of deteriorating products with expiration time dependent deterioration rate. In the projected work, we formulate and look into an inventory model where the deterioration rate is a function of time and life time of the product.
Consumers are generally aware to quality changes of fresh produce and foods. In view of supermarket consumers, they will like better to purchase fresh products instead of old ones.
When price is the same, they will have a first choice to the newer ones. In our present effort, we think about a demand which decreases with the age of the product. To best of our information, there are just a small number of papers in the unpreserved inventory invented story that take into consideration the declining effectiveness of perishable products all through their life time. Fujiwara and Perera [18] was the pioneer who well thought-out declining effectiveness of perishable products related with lifetime. Though, they make use of a constant demand rate. Bai and Kendall [19] established an inventory model where demand 4 LAKSHMI NARAYAN DE rate is dependent on the displayed stock and the freshness of the product. Amorim et al. [20] considered multi-item production based inventory model for deteriorating products where demand is age dependent. Chen et al. [21] established EOQ model with positive stock level at the end of cycle with stock level dependent and linearly declining demand function with age of the product. Dobson et al. [22] formulated an inventory model using an only age reliant demand function that decreases linearly with age of the product until stock of item vanishes.
Price is one of the key factors in a consumer's purchasing decision. Buyers like to purchase from a shop which has minimum selling price. It is evident that the less selling price makes the demand high. If the seller raises the selling price of the manufactured goods, the clients would shift other shopping places to fulfil their demand. As a consequence, demand for perishable goods is dependent upon the joint effect of selling price and product newness.
There are several investigations have been made on the consequence of price variations.
Kotler [23] included business policies into inventory decisions and debated the connection among economic order quantity and pricing decision. Wee [24] discussed a replenished plan for perishable products where demand is price sensitive and deterioration follows Weibull distribution. Papers related to this field are Shah et al. [25], Ranganayki et al. [26], etc.
Another most important issue of demand is changeable selling price. Alturki and Alfares [27] established a storehouse assortment model where selling price is dependent on time. In general, demand of a product declines with raise in selling-price and vice versa. Also, life span of such goods affects the selling-price. To endorse to sell products of little life seller uses low selling price. Although not too many research papers have been published in this area. Iqbal and Sarkar [28] established a supply chain model where selling price is dependent upon the life time of the product. Recently, De [29] developed an EOQ model where selling price has a reverse association with the newness of the manufactured goods to increase the demand of the product left over in store.
This article presents an inventory model of deteriorating products having maximum lifetime with following considerations: i) the deterioration rate increases over time and the goods are entirely deteriorated at the date of expiration. ii) demand of the product is dependent on selling price and freshness of the product, after exceeding life time there is no demand i.e., cycle time must be shorter than product's life time iii) freshness sensitive selling price of the product iii) the retailer prepays a portion of his purchase cost as an advance through equal multiple instalments to the supplier before receiving the product iv) the inventory level at the end of each cycle may be positive or zero v) the shelf space for holding the product of the retailer is limited. The purpose of this paper is to find out the utmost earnings of this model.

ASSUMPTIONS AND NOTATIONS
To build up the present model, the succeeding assumptions and notations are used all through this paper.

Assumptions:
i) Infinite replenishment rate and zero lead time are considered.
ii) There is no shortage allowed iii) No chance for substitute or renovate in this model for the single perishable item is considered.
iv) Product's selling price is depended upon age of the product i.e., = 0 (1 − ), ix) To attract advance payment policy, the supplier also offers a discount or concession on the cost of the purchasing products. To receive the discount the retailer accepts the condition and pays × ℎ on equal instalments during years before delivery. : inventory level at the end of cycle ( ≥ 0).

MATHEMATICAL MODEL FORMULATION
In this paper, two models with advance payments have been discussed. In the first case, we have supposed that the prepayment is completed in a number of equal instalments.
In the 2 nd case, the retailer will pay in advance the entire cost at the specified time to enjoy a cash reduction on that cost by borrowing some funds for his purchasing cost from some financial organization like bank.

Case 1(Advance payment is done in instalments)
In the EOQ model developed in case 1, fraction of the entire purchased cost of the product is been prepaid by equal instalments within years before the delivery and the rest amount inventory level ( ) starting with units decreases gradually due to both demand and deterioration and reaches to (≥ 0).

Fig 1. Graphical representation of the inventory system when prepayment is done in instalments
The governing differential equation is as follows: To receive the maximum (0≤ ≤ 100) percent discount on purchase cost, the retailer must pay fraction of the entire purchased cost in uniform instalment before delivery. As is the discount rate for a single instalment of the prepayment, so the discount rate = .
The total discount (DC) for the advance payment =

Fig 2. Graphical representation of the inventory system when prepayment is done at a time
From Fig 2, it is clear that supplier offers percent discount rate to the retailer for complete advance payment before delivery. When the retailer does not enough money in hand during the time M year, he may take loan from a bank or any financial sector at an interest of %.
So here total discount (DC) for the advance payment= (12) Total interest paid ( 2 ) to the bank on the purchased cost per cycle due to advance Now our objective is to obtain optimal cycle time * and inventory remaining * at end of cycle in order to maximize the average profit per unit time.

Case1. (Advance payment is done in instalments)
Taking 1st and 2nd derivatives of 1 ( , )in (11) with respect to , we find and 2 ( 1 ( , )) 2 = 0 Here we see that 1 ( , ) is linear function of . So 1 ( , ) is either rising or declining function in . Therefore, two cases may arise.
As Hessian matrix for 1 ( , ) is less than zero, so there exists unique * at which the profit function 12 ( ) attains a maximum.

Subcase 2.
( 2 ( , )) ≤ 0 For this case, putting = 0 in equation (14), we obtain the following expression of the profit function as a function of only: Here also there exists unique * at which the profit function 22 ( ) attains a maximum.

ALGORITHM
Now we sketch the algorithm to achieve optimal solution for both cases of the projected model.

SENSITIVITY ANALYSIS
To inspect the sensitivity of the model, we study the impact of changes in different inventory parameters against optimal solutions ( , ), optimal order quantities and average profit for the example1(for case 1) and example 4 (for case 2) by varying the value of one parameter at a time and fixing other left over parameters, the analysis has been completed.
The results of sensitivity analysis are shown in Table 1 Table 2, the following analysis can be made.
(i) It is observed from the Table 2 that, superior value in prepayment time ( ) gives lesser profit of the inventory system especially for case 2. This is happening because due to bigger prepayment the retailer needs to wait for a product for longer time and have to pay more banking interest in taking that time.
(ii) Another noticeable observation is that, large number of instalments ( ) reduces the total profit of the system as the retailer will be given a smaller discount rate.
(iii) If we notice the sensitivity of one of the most important parameter , life time of the product, we will see that, higher profit will be obtained for a product having bigger life time as the product deteriorates slowly.
(iv) From Table 2, it is readily observed that, the increment of shelf space ( ) increases the total profit as the retailer may purchase higher amount of quantity.
(v) There is positive impact on the profit ( ) with respect to the value of the parameter , 0 , , that is increases when the values of , 0 , , increase, while for the parameters , , ℎ , , , , there is negative impact on .
(vi) The optimal cycle time * is dependent on parameters , , , , , , in a positive way but it depends on parameters 0, , ℎ, , , , , in negative way.
(vii) The impact on the optimal ordering quantity * and end inventory level * at end in both cases with respect to the value of all parameters is not easy to say. Sometimes it may have positive impact, sometimes negative. Sometimes optimal solution will occur by taking zero end inventory model sometimes positive end inventory models.
But it can be definitely concluded that relaxation of end inventory level produces improved result.

CONCLUSIONS
In this article, we discussed an EOQ model for deteriorating item with expiration date, time dependent deterioration rate and freshness and price sensitive demand. We have assumed that the selling price of the product is freshness sensitive. We have also assumed that, the retailer pays the purchase cost of the product in advance either in equal instalments or in a single payment. In the first case, the retailer will receive a price discount when he prepays the some portion of purchase cost before the delivery time, while in the second case he prepays the same at a time before delivery of the product to get price discount. The present article extends the earlier existing work by incorporating these two concepts such as freshness and price There are some limitations in the present study. One limitation is that shortages are not allowed, which would not be true in fact. In addition, we also assume that the lead time is zero. In reality, the lead time is not zero. So more research is need to be carried out in the future.