DETERMINATION OF RETAILER’S OPTIMAL POLICY IN A PRICE- SENSITIVE INVENTORY MODEL FACILIATING TWO TYPES OF PAYMENT SCHEME FOR THE CUSTOMERS

How to cite this article (APA): Adhikari, S. (2022). Determination of Retailer’s Optimal Policy in A Price-Sensitive Inventory Model Faciliating Two Types of Payment Scheme for the Customers. International Journal of Engineering Technologies and Management Research, 9(3), 1-11. doi: 10.29121/ijetmr.v9.i3.2022.1129 1 DETERMINATION OF RETAILER’S OPTIMAL POLICY IN A PRICESENSITIVE INVENTORY MODEL FACILIATING TWO TYPES OF PAYMENT SCHEME FOR THE CUSTOMERS


INTRODUCTION
Mathematical modeling in inventory become very attractive day by day to the researchers and practitioners as problems associated with inventory has a close relation with business, economy, and finance. Harris (1913) was first to introduce inventory in mathematical modeling. There after many developments has been made. Determination of price of a commodity is very common problem in inventory. General perception is that decrease in selling price increases demand rate. Abad (1996) considered dynamic pricing together with lot-sizing in an inventory model. In another work Abad (2001). he assumed selling price dependent demand rate for items which decays instantaneously. After that many inventory model has been developed taking demand rate as a monotonic decreasing function of selling price. It should be noted that demand rate may be linear or non-linear function of selling price. As limitation of capital is a major challenge in today's business world, different types of credit policies have a great impact in that world. Studies based on the effect of inclusion of trade credit policy in an inventory system have become much attractive to the modelers and interpreters This work was started by Goyal (1985). He was the pioneer to include the effect of 2 introduction of single stage trade credit policy in the inventory system. Huang (2003) extended the concept between supplier and retailer. He assumed that retailer's offered credit period to the customer is less than the credit period offered by the supplier to the retailer. Jaggi et al. (2008) introduced credit linked demand rate in their model. Chung (2013) implemented the concept of two-level trade credit periods in economic production quantity model with storage capacity constraints. Chung et al. (2014) considered the facility of permissible delay in payment in economic production quantity model for deteriorating items. Giri and Sharma (2016) incorporated the concept of permissible delay in payment in a model where demand rate increases linearly with time. Pal (2018) examined the effect of trade credit policy with partial backlogging of shortages. Giri and Sharma (2019) developed inventory model with partial trade credit policy. Tiwary et al. (2018) introduced two level partial trade credit policy in a three-echelon supply chain associated with perishable items. Mandal et al. (2020) introduces reliability in a production inventory model with two tier credit policy. An inventory model with two stage deterioration under the atmosphere of permissible delay option was formulated by Pal et al. (2021). ƞ A fraction portion of the total customer make immediate payment on purchase of goods ( ) Retailer's inventory level at time

BASIC ASSUMPTIONS
• Retailer has some capital to arrange set-up cost. But he has to go for bank loan for ordering quantity . Banks provide a credit period. After that period bank will charge interest for remaining unpaid amount. Retailer can enjoy interest from sells revenue up to the time .
• Obtained interest rate is less than paid interest rate.
• Retailer faces two types of customers. One category wants to pay immediately after purchase of a thing while another category wants to avail some time delay for making payment.
• Retailer offers a credit period to the customer and sells all items by the time .
From the point of category of customer enjoying permissible delay in payment scheme, if buys item at time (0 < ≤ ) arranges payment at time + .

3
• Retailer's offered credit period is less than his cycle length( < ) • Demand rate is a monotonic function which decreases linearly with selling price . Exact form of demand rate is = ( ) = − with > 0, > 0.

Figure 1 Retailer's time weighted inventory
Retailer ordering quantity is . This amount depletes at the rate -.
The differential equation which governs inventory level of the retailer is given below.
= − subject to initial condition (0) = (1) Solution is given by Terminal condition ( ) = 0 gives, = Total holding cost is given as (2) and (3)] Retailer goes for bank loan for purchasing amount . So, bank loan amount is = . Two cases come up.  Either Case1. ≤ or Case2. < . Now, Case 2 can be subdivided into three Sub Cases on the basis of the fact that retailer collects his final payment from the customer at time + . Possible three Sub Cases are given below.
Case 1. ≤ < < + Average profit function of the retailer for different situations is given in the following table.

SOLUTION METHODOLOGY
We first solve the following partial differential equations = 0, = 0 = 1,2,3,4 Then we get solutions for cycle length and selling price as = * , = * . These solutions are optimal if the eigen values of the following Hessian matrix at ( * , * ) are negative.

NUMERICAL EXAMPLE
Values of parameters for different situations are given below. Now we show three-dimensional plotting of retailer's average profit function with respect to two decision variables selling price and cycle length for Case1, Sub Case 2.1, Sub Case 2.2, Sub Case 2.3 respectively.

CONCLUSION
In this work we have presented a mathematical model which maximizes retailer average profit depending on decision variables namely cycle time and selling price. Different situations associated with two stage trade credit have been considered. It has been observed from numerical results that retailer profit increases as length of the credit period offered by the bank increases. Here we take demand pattern based on selling price linearly. This work may be extended by taking non-linear demand pattern. Also credit linked as well as selling price dependent demand pattern may be a possible extension of this modelling frame work.