Joint Pricing and Inventory Policies for Perishable Items with Price Discount based on Freshness Index

Article History: Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published online: 10 May 2021 Abstract It is generally observed that the products losses its freshness with the course of time that stimulates depression in demand of the product. In these circumstances, price discounts are necessary to raise the market. This is why, when the product's the index of freshness reaches a certain level, we created an inventory model wherein price reductions are provided at a sale price. The main goal is to figure out what the best selling price and cycle time are in order to maximise profit. The meaning and uniqueness of an ideal model solution are incorporated into the circumstances. The next move is to use a simple algorithm to find an optimal solution. Finally, a numerical example is presented, followed by a sensitivity analysis.


Introduction
The term "Perishable products" can be considered for those commodities which lose their value with the course of time, such as fruits, vegetables, flowers, medicines, etc. The freshness of the product and price are the two most substantial elements for customers, hence reducing the prices for less fresh products can be taken into consideration as the most convenient approach to enhance selling of products.
In the literature several models for a peregrinable inventory were suggested. Stock sample tests for deteriorating components have been provided in excellent detail Goyal and Giri (2001), Nahmias (1982), Raafat (1991). Janssen et al. (2016). Bakker et al.(2012) most of the perishable item stock models built follow the random lifetime of items, but their freshness will remain the same and catch the same demand until their expiry date. The concept that freshness of the product effect demand was taken into consideration first by Fujiwara and Parera (1993). Sarker et al. (1997) then considered that the age of stock products had a negative effect on demand. The conduct of consumers regarding the expiry dates for perishable goods was studied by Tsiros and Heilman (2005). Bai and Kendall (2008) studied an inventory model linked with fresh produce where demand is dependent on freshness and displayed inventory. Wang and Li (2012) developed an inventory model with fixed expiry date for perishable goods in which quality decay is one of the key factor. Zhou and Piramuthu (2013) have developed a declining product inventory model where the demand for each item is based on its instantaneous quality and the allocation of the shelter area. Wu et al. (2016) created a fresh produce stock model in which product freshness, inventory level, and expiration date all influence time-diversity. In 2016, Chen et al. looked into an EOQ for fresh produce, in which demand determines the freshness expiration date and stock levels. Demirage et al. (2017) formulated the inventory ordering problem using a deterministic demand function that is concave in the product age. Dobson et al. (2017) developed an EOQ model based on the assumption that commodity freshness reduces demand rate linearly. One of the most important factors in deciding whether or not a product will be popular is its price. A variety of studies have looked into the impact of pricing on demand. The inventory models for products that deteriorate because of price-based demand were created Wee (1997) created and assumed a demand model for inventory management that is focused on price, freshness, and inventory sale. Li and Teng (2018) developed a perishable commodity inventory model in which market price, reference price, product freshness, and the stock level shown all influence demand. A deterministic model for age, stock, and price-based demand goods was submitted by Agi and Soni(2019).
Request price discounts are used to increase profits because fresh products are in higher demand than those with a stale appearance. For perishable goods, Rabbani et al. (2016) established an inventory strategy in which the demand rate is dictated by inventory quality and price fluctuations over time. (Agrawal and Banerjee (2017) presented an inventory model in which demand for a perishable good is initially determined by the purchase price and then by the freshness state, as well as optimum discounting and ordering strategies to optimise net profit. Bahula et al. (2019) created an ideal inventory model for perishable items with reasonable latency, recommending optimal discounting and ordering policies to optimise net profit in the initial selling price and later on freshness, where the conditions for perishable goods were based on optimal discounting and ordering policies to optimise net profit under subsequent price discounts. Kamaruzaman and M. Omar (2020) proposed an EOQ model in which the expiration date, price, and inventory level all influenced demand.
Researchers (Rabbani et al. (2016), Banerjee and Agrawal (2017), Kamaruzaman and M. Omar (2020)) believe the product's freshness tends to deteriorate after a period of storage. Certain items, such as fruits and vegetables, are not suitable because their freshness begins to deteriorate shortly after receiving inventory. This proposed study would then produce a model for inventory in which the freshness of the product degrades as it comes into the inventory and a discount on the sales price will be offered when the product freshness exceeds a certain level. The primary objective is to identify the best sale prices and cycle times to maximise benefit. The rest of the article is structured in the same way. The notations and assumptions of the model are discussed in section 2. Section 3 shows both the generated mathematical model and the theoretical results. The numerical findings and sensitivity analyses are presented in Section4. Section 6 concludes the paper and recommends prospective research.

Notations and Assumptions
"The mathematical model in this article was built using the notations and assumptions mentioned below".   The cost of keeping a unit for a given amount of time.


Level of freshness index from which discount is offered, 01   . 1 t Time at which the freshness index reaches to .
p Until the discount on the purchase price is given, the selling price per product. 1 p After the discount on the sale price is applied, the selling price per unit is calculated.

It
The inventory level "at time t".
( ) dp "The price portion of the product's production. It may be non-negative continuous, convex", lower sale price function.

Assumptions
1. The inventory in question suffers from "two types of degradation over time: a gradual physical deterioration" of the current stock and a deterioration of the freshness of the commodity. 2. Several factors that include high temperatures, humidity, chilling, etc.Can decrease product freshness. A clear index of freshness for a commodity does not seem possible. It is obviously understood nevertheless that, over time, the freshness of every product has constantly degenerated and eventually expires. In consequence, we can deduce that the freshness index begins at 1, at time 0, and ultimately decreases to 0. As Chen et al. (2016) did, we infer that the freshness index at times t decreases linearly from 1 at the start of life to 0 at the close of life: 3. A Discount is available when a product's freshness index falls to a certain level  ( ) The Demand for the fresh product is definitive. It is both based on the freshness index and the price: 5. Deficiencies are not permissible. 6. The pace of renewal is unlimited, and there is no lead time.. 7. "The time horizon is infinite".

Theoretical Results and Model Formulation
At the start of each period, Q units of goods in good condition arrive at the inventory system. The inventory deteriorates [0, T] at a steady rate over the inventory period. Furthermore, it loses its freshness over time in response to this physical decay. The price and the freshness of the commodity are expected to decrease by the demand rate. We believe the freshness index falls to a certain level  at time 1 tt = ,  %Discount is provided to increase demand for sale price. Our project aims to investigate the resulting stock management process and determine the most cost-effective solution. * p , the best cycle time * T , and the best order amount * Q . Since the request is deterministic, the significance of theQ if the values of p and Tare known. "Hence, p, and T are judgement variables".

3.1Model Formulation
The inventory amount is calculated based on the above conventions.

( )
ItDuring the span of time, at time t   1 0, t the following differential equation governs the situation: Solving the differential equation in (3), we express the inventory level as follows:" tT"the inventory level

( )
It at time tis governed by the following differential equation: "Solving the differential equations in (5) (12) Therefore the profit per cycle is c SR OC HC PC And the benefit per unit time may be as follows: Where v and ware given by equation (8) 3

.2 Solution and Results
The following is the optimization topic discussed in this article: Subject to 1 , 0 c p t T n     This topic can be divided into two different optimization problems. The first is a problem of optimization "with respect to T, and the second is a problem of optimization with respect top.

Optimization with respect to T
Here we study the function ( ) b. The answer * T "satisfies the second order criterion for a" maximal value in (a).

"Proof.
By taking first order derivative of ( ) Then there is a one-of-a-kind attribute of T, say * T such that ( )     , pT  at * TT = is a maximum."

p -related optimization
In this section, we'll look at situations where the best price is available and is one-of-a-kind. Regardless of the circumstances * T , the first and most important requirement for ( )  Step 2. Find the maximum values of ( ) , j pT  in (14) by calculus method.
Step 3. Use the result in step 2 to determine the optimal price

 =
, n = 0.5 years. The following optimal prices, with this model parameter value and the implementation of the above algorithm, the inventory cycle, the quantity of "the order, and the corresponding optimal profit are obtained. If the product's refreshes reduces the overall period time, the optimum sale price and the optimum order amount decreases and benefit decreases first and subsequently increases as product freshness decreases. It is visible from table 2 that higher the percentage discount offered on selling price results in short cycle length, lower the profit, but optimal selling price and order quantity first increases and then decreases.
For the various values of the request parameters a and b the optimum solution of our model is shown in table 3, while other model parameters remain unchanged respectively. From table 3 one can observed that higher values of a, results in higher values of optimal selling price, order quantity and profit but short cycle length. Higher value of b results in higher value of order quantity, longer the cycle length but lower the optimal selling price and profit.

Conclusions and Research Directions in the Future
This paper created a model for the inventory of perishable products that is based on the sales price and freshness index. When the object is saved, the lack of freshness is taken into account. The percentage discount on the sales price will be calculated by using freshness. In terms of cycle time, the benefit function appears concave. For the presence of the specific optimal solution, conditions are listed. Numerical examples and sensitivity analysis were presented to the key parameters of the model. The proposed model can be generalized by allowing shortages. In addition, a decision variable on the percentage reduction in selling prices will generalize the model.