Optimal Policies for Perishable Items when Demand Depends on Freshness of Displayed Stock and Selling Price

Display of stock plays an effective role to boost up the demand as it encourages customer to buy more. For perishable good freshness is also an important factor affecting its demand, as consumers look for fresh items. Selling price is also a major factor affecting the demand. This paper proposes an inventory model where demand depends on selling price, freshness of item and displayed stock. The traditional assumption of zero ending inventories is relaxed to a non-zero ending inventory. As it may be profitable to have a closeout sale at a markdown price, and always keep on-hand fresh displayed stocks if the demand is freshnessand-stock dependent. The objective is to maximize the total profit with respect to three decision variables (i.e., unit price, cycle time and ending-inventory level). Numerical examples are presented to validate the model and sensitivity analysis of inventory parameters is done to understand their effect in determining optimal policies.


Introduction
Deterministic inventory models usually consider the demand rate to be either constant or time-varying, but independent of level of inventory i.e. level of stock. However, it has been noted, especially in the retail market, that these inventory models are not suitable for representations of the reality of inventory control situation in the retail field. Levin et al. quoted that "large piles of goods attract more customers" this is termed as stock-dependent demand [1]. To incorporate this scenario into inventory models, a variety of stock-dependent demand models have been proposed. Baker and Urban established an Economic Order Quantity (EOQ) inventory model by specifying the stock-dependent demand as a power function with diminishing return of displayed stock level [2]. Urban extended the classical EOQ inventory model from zero ending inventories to non-zero ending inventory when the demand is dependent on the amount of displayed stocks [3]. Urban  in recent years. In today's market, determining price and order quantity jointly for retailers to better manage perishable products is recognized as an important way to increase profitability and maintain competitiveness in a supply chain. The traditional EOQ models assume that products can be stored indefinitely to meet future demand. However, many perishable products such as fruits, vegetables, medicines, and volatile liquids degrade or deteriorate continuously due to evaporation, spoilage, obsolescence or some other reasons. Most researchers modeled the effect of perishability in the past only from the retailer's perspective as the on-hand stocks shrink due to damage, spoilage, dryness, vaporization, etc. Researchers rarely developed the effect of perishability into their models from the costumer's perspective. In fact, freshness is one of the most critical criteria affecting customers purchasing decisions. Fujiwara and Perera took the effect of product freshness on consumer demand into consideration [13]. Subsequently, Sarker et al. studied the inventory policies for perishable products when the demand is negatively impacted by the age of the on-hand stocks [14]. Bai [18][19][20].
It has been noted that the demand of an item is affected by its price, consumer may opt for an alternative where he gets better deal. So in today's competitive markets, determining price and order quantity jointly for retailers is recognized as an important way to increase profitability and maintain competitiveness in a supply chain. The demand rate is usually influenced by price of an item. Urban and Baker further generalized the demand rate as a multivariate function of price and displayed stocks level [21]. Teng and Chang expanded the EOQ inventory model with price and stock-dependent demand to an Economic Production Quantity (EPQ) inventory model for deteriorating items [22]. Feng et al. presented an EOQ model with demand as a multivariate function of price, stock and freshness of the product [23]. So the research in area of price and stock dependent demand is extensive.
This paper develops an EOQ model in which the demand of product is dependent on the freshness of item its selling price and displayed stock. Selling price crucially affects not only the demand rate but also the total profit. As a result, the selling price is considered as a decision variable in this paper. The objective of this study is to determine jointly the optimal unit price, cycle time and nonzero ending inventory that maximize the total profit of the retailer. The remainder of this paper includes notations and assumptions used in this paper, then the mathematical modelling is done by incorporating the demand as a multivariate function of unit selling price, displayed stocks and freshness of the item. Thereafter, numerical example is presented to validate the model. Sensitivity analysis for key parameters is done to observe their effect on decision variables. Lastly, we have described the conclusion of this paper with future research directions.

Notation and Assumptions
The proposed model uses the following notation and assumptions.

2.
Demand of the product is dependent on the freshness of the stock displayed and its unit selling price. Here the standard assumption of zero ending inventory is given up as It might be beneficial to close out the sale at a nonzero ending inventory and have fresh stock for sale, because fresher items would attract more. 3. The nonzero ending inventory is sold out with attractive discount rate. 4. Replenishment is infinite, lead time is zero and shortages are not allowed.

Mathematical Model
In the proposed model, we consider a retailer dealing with a single item. The demand is dependent on freshness of the displayed stock and its selling price. It might be beneficial to close out sale at a lower rate, and keep on-hand fresh displayed stocks, because the demand is freshness and stock dependent. At the end of cycle time inventory level depletes to non-negative ending inventory i.e.
0, E ≥ which is sold for salvage value. The following differential equation represents the status of inventory during interval [0, ] T : with (0) and ( ) .
The solution of using ( ) Using the other condition (0)

I Q
= and, we have The inventory holding cost is given by, Therefor the total profit per unit time is given by,

S C E S T > ≥
Next, we follow the following steps listed below to have an optimal solution.

Algorithm
Step 1: Assign numerical values to all the inventory parameters.
Step 2: Set and solve simultaneously for as nonzero ending inventory. The concavity of the given profit function is also shown in Figures 1-3.

Sensitivity analysis
With the values of inventory parameters taken in example 1, sensitivity analysis of the optimal solution with respect to inventory parameters is done by changing one parameter at a time. The results obtained are shown in Table 1.
The observations made from these calculations are: Increase in scale demand increases the order quantity, total profit and ending inventory; whereas the selling price and cycle time decreases (which implies higher demand shortens cycle time and increases profit). For increase in stock dependent parameter Concavity of profit function for S=$10.37. Figure 1 Concavity of profit function for T=0.0956 year. Figure 2 International Journal of Applied Science -Research and Review ISSN 2394-9988 Concavity of profit function for E=3.64 units. Figure 3 Parameter per unit increases the ending inventory, selling price and cycle time; which in result decreases order quantity and total profit. Increase in ordering cost hikes up all decision variables lowering the total profit. Lastly, increase in holding cost increases the ending inventory, selling price; whereas cycle time, order quantity and total profit decreases.
Numerical results in Table 1 demonstrates that the total profit is more sensitive to scale demand , λ price efficiency of demand , λ maximum lifetime of item m , discount rate and the purchase cost of item. Hence, the retailer needs to pay more attention to these parameters than others (Table 1).

Conclusion
In this paper, the objective is to determine optimal policies for retailer which maximizes his profit. The demand of an item is considered to be dependent on stock displayed, its selling price and the freshness of the items. The standard assumption of inventory level depleting to zero is relaxed here, instead it might be beneficial for the retailer to close out sale and stock up fresh items which boosts his demand. The retailer profit is maximized by considering cycle time, selling price and nonzero ending inventory decision variables Proposed model can be extended by allowing shortages, trade credit and investment in preservation technology. Model can be reviewed with stochastic demand also.