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Research Article | Open Access
Adrián Macías-López, Leopoldo Eduardo Cárdenas-Barrón, Rodrigo E. Peimbert-García, Buddhadev Mandal, "An Inventory Model for Perishable Items with Price-, Stock-, and Time-Dependent Demand Rate considering Shelf-Life and Nonlinear Holding Costs", Mathematical Problems in Engineering, vol. 2021, Article ID 6630938, 36 pages, 2021. https://doi.org/10.1155/2021/6630938
An Inventory Model for Perishable Items with Price-, Stock-, and Time-Dependent Demand Rate considering Shelf-Life and Nonlinear Holding Costs
Nowadays, consumers are more health conscious than before, and their demand of fresh items has intensely increased. In this context, an effective and efficient inventory management of the perishable items is needed in order to avoid the relevant losses due to their deterioration. Furthermore, the demand of products is influenced by several factors such as price, stock, and freshness state, among others. Hence, this research work develops an inventory model for perishable items, constrained by both physical and freshness condition degradations. The demand for perishable items is a multivariate function of price, current stock quantity, and freshness condition. Specific to price, six different price-dependent demand functions are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial. By working with perishable items that eventually deteriorate, this inventory model also takes into consideration the expiration date, a salvage value, and the cost of deterioration. In addition, the holding cost is modelled as a quadratic function of time. The proposed inventory model jointly determines the optimal price, the replenishment cycle time, and the order quantity, which together result in maximum total profit per unit of time. The inventory model has a wide application since it can be implemented in several fields such as food goods (milk, vegetables, and meat), organisms, and ornamental flowers, among others. Some numerical examples are presented to illustrate the use of the inventory model. The results show that increasing the value of the shelf-life results in an increment in price, inventory cycle time, quantity ordered, and profits that are generated for all price demand functions. Finally, a sensitivity analysis is performed, and several managerial insights are provided.
Inventory management is a valuable function for companies around the world. It seeks to control the materials from acquisition up to sales-related decision-making (how much and when to buy items) in order to avoid overstock and/or stockout. According to Yavari et al. , one of the challenges when managing inventories is the inherent perishability of many items, which means their freshness and quality decrease over time, and these cannot be sold after their expiration date. Tirkolaee et al.  noted that the inherent perishability widely occurs in food goods (e.g., milk, vegetables, and meat), organisms, and ornamental flowers. These authors also stated that the time window between preparation and sales of perishable items is very significant for producers and purchasers.
Given that the inherent perishability can occur immediately, Pal et al.  addressed a production-inventory model for deteriorating products when the production cost depends on both production order quantity and production rate. Later, meanwhile, Mashud et al.  determined the optimal replenishment policy of deteriorating goods for the classical newsboy inventory problem by considering multiple just-in-time deliveries. Additionally, Mashud et al.  derived an inventory model for deteriorating products that calculates the optimal vales for replenishment time, price, and green investment cost.
There are also some items in which the inherent perishability is noninstantaneous. Mashud et al.  considered this by developing an inventory model for noninstantaneous deteriorating items, which jointly optimizes the cycle time, price, the spending in preservation technology, and credit financing. Alongside, Mashud et al.  proposed another inventory model for noninstantaneous deteriorating goods which determines the cycle length, price, and preservation cost. On the contrary, Hasan et al.  introduced a noninstantaneous inventory model for agricultural goods taking into account the effects of the inherent perishability. This model states the optimal pricing and timing inventory policies.
Similarly, there is currently an increasing demand for fresh items since consumers are more concerned about their health habits. In this context, consumers want to buy fresh goods far from their expiry date, so these can be stored during longer periods of time. Thus, companies should carefully manage and control their inventories of fresh items in the warehouse. One of the most common approaches to tracking the freshness and quality of perishable items in the warehouse is the radio frequency identification (RFID) smart tags, which allows to decrease the risk of selling items after their expiration date. Some examples have been exemplified by Herbon et al. , Herbon et al. , and Herbon and Ceder , who investigated the effect of implementing the time-temperature-indicator (TTI) in order to make information about expiration dates (for different items) available online.
Once items are available on shelves, exhibiting large quantities is one of the factors that often influence consumers to purchase more products. Then, retailers might become more profitable by increasing their goods’ availability. However, this is different for cases where time plays a key role, such as perishable items, whose degradation tends to make them less attractive and unsalable at the end of shelf-life. Another factor is price, which inversely stimulates demand; lower prices result in higher demand, and vice versa. Overall, price, stock availability, time, and shelf-life are the critical factors that affect the demand of perishable products and should, consequently, be considered when developing inventory models.
The literature on inventory models with time-, price-, and stock-dependent demand is also abundant. For instance, Mashud et al.  constructed a price-sensitive inventory model by considering that the demand follows an exponential or isoelastic price-dependent demand.
Avinadav et al.  proposed two inventory models under the assumption that the demand function is dependent on both price and stock age and determined the optimal pricing and inventory policies. While the first inventory model assumes a multiplicative demand function, the second assumes an additive demand function. During the same year, Qin et al.  formulated an inventory model that determines the pricing and inventory policies for a perishable item considering stock-dependent demand and the effects of item’s quality degradation. Later, Chen et al.  derived the optimal inventory policy and shelf-space size for fresh products considering the expiration time and a demand rate that depends on freshness and stock. The demand function treated by Chen et al.  grows in the on-hand stock level and diminishes with respect to the age of the item. Afterwards, Feng et al.  resolved the inventory model of Chen et al.  considering the demand rate jointly dependent of price, stock, and age. Meanwhile, Dobson et al.  built an economic order quantity (EOQ) inventory model for a perishable item with fixed shelf-life when the demand rate decreases linearly with respect to the age of the stock.
Other authors that work on a similar problem are Herbon and Khmelnitsky , who developed an inventory model that determines the optimal replenishment time and price when the demand rate actually depends on both time and price. Alongside, Banerjee and Agrawal  defined the optimal discounting and ordering policies for deteriorating products including a demand rate that depends on price and freshness. Hsieh and Dye  also provided the optimal pricing policy for deteriorating goods by considering the impact of reference prices under the assumption that stocks stimulate demand. Finally, Li and Teng  obtained the lot sizing and pricing strategies for a perishable item when the demand rate is dependent on reference price, exhibited inventory, and item freshness.
Recently, Agi and Soni  developed an inventory model for joint optimal pricing and inventory management for a perishable item under stock-, age-, and price-dependent demand, allowing surplus inventory at the end of cycle. These authors stated that finishing the inventory cycle period with a positive inventory level results in a benefit because the demand increases when large quantities are ordered and exhibited in the shelf space. Conversely, having inventory of perishable products at the end of their cycle is not desirable, as these cannot be stored for the next inventory cycle. Hence, these must be sold at a salvage price.
Although most inventory models consider a constant holding cost, different factors that affect inventories on the storage place make the holding cost intrinsically variable. There are common scenarios in the real world where the holding cost increases over time because longer storage periods require more sophisticated and costly warehouse facilities. For instance, a longer storage of fresh products needs refrigeration and some specific conditions in place to prevent damages. Alfares and Ghaithan  provided an excellent state-of-the-art review on the economic order quantity (EOQ) and economic production quantity (EPQ) inventory models with variable holding costs. Specifically, Alfares and Ghaithan  classified the variable holding cost into three categories: time-dependent holding cost, stock-dependent holding cost, and multiple dependence cost variability. The types of holding cost functions are constant, linear, nonlinear, step, or general. For modelling the case of variable time-dependent holding cost, there exist several functions. One is the quadratic holding cost function that is exemplified by Valliathal and Uthayakumar , who constructed two EPQ inventory models for deteriorating items by including the holding cost as nonlinear time dependent. Years later, Pal et al.  investigated the single-period newsvendor model and obtained the optimal lot size when the customers’ balking happens by taking into consideration a nonlinear holding cost that depends on both lot size and the stock level. Tripathi and Mishra  studied two inventory models, with time-varying holding cost, that obtain both the optimal order quantity and the optimal replenishment cycle time. While the first inventory model presents a linearly time-dependent holding cost, the second one has a quadratic time dependence, carrying inventory cost. In the same year, Sivashankari  also presented a comparative study of three EPQ inventory models under the assumption that the carrying inventory cost is either constant, a linear function of time, or a quadratic function of time.
Since closing the stock cycle time with zero inventory is the most appropriate strategy when dealing with perishable items, this research work builds an inventory model for perishable items with zero inventory at the end of stock cycle. On the one hand, these perishable items are subject to physical deterioration and freshness degradation over time, and on the other hand, the demand is a multivariate function of price, on-hand stock quantity, and freshness state. For the demand that is related to price, six distinct price-dependent demand functions are considered: linear, isoelastic, exponential, logit, logarithmic, and polynomial. The inventory model also considers the expiration date, a salvage value, and a deterioration cost of the perishable item. Moreover, the holding cost is considered as nonlinear with a quadratic function of time. The proposed inventory model conjointly derives the optimal policy for the price, the replenishment cycle time, and the order quantity, which together maximize the total profit per unit of time.
The rest of this research work is comprised of several sections as follows. Section 2 introduces the notation and assumptions. Section 3 develops the inventory model with price-, stock-, and time-dependent demand with nonlinear holding cost. Section 4 develops the solution procedure to determine the optimal solution. Section 5 presents the solution to six different price-dependent demand functions and solves some numerical examples. Section 6 performs a sensitivity analysis and proposes some managerial insights. Finally, Section 7 provides the conclusions and outlines several areas for further research.
2. Notation and Assumptions
The notation used to develop the inventory model with price-, stock-, and age-dependent demand, with zero inventory at the end of cycle, is given as follows. The symbols of Agi and Soni  are used, and some more additional symbols are defined here, in order to have a standard notation (Table 1).
The inventory model is based on the following assumptions:(1)The item in the storage is subject to two types of degradations over time: physical degradation at a constant rate and freshness degradation.(2)The item has a definite and limited shelf-life after which it is not salable.(3)The demand of the item depends on price, the current stock amount, and its freshness. For the demand that is price-dependent, six different functions of the price-dependent demand are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial.(4)The number of remaining items at the end of the cycle is zero.(5)The holding cost is nonlinear and modelled with a quadratic function that depends on time (i.e., ).(6)The salvage value and deterioration cost are taken into account for the items deteriorated throughout the inventory cycle.(7)The time horizon planning is infinite. The lead time is zero, and therefore, the replenishment rate is instantaneous.(8)At the beginning of the stock period = 0, the item is fresh and has no age effect on demand. Thus, the item loses its freshness over time, and its demand consequently decreases.(9)The length of the stock period must not surpass the item’s shelf-life since the item is unsalable after its expiration date is over .
3. Inventory Model with Price-, Stock-, and Age-Dependent Demand, with Zero Inventory at the End of the Cycle
The problem under study is described as follows. A company manages a product that has an inherent perishability, and this item is subject to both physical and freshness degradation. Moreover, it is known that the item has a limited shelf-life, after which it is not marketable. Thus, the length of the stock cycle must not be greater than the item’s shelf-life. Due to the nature of the item, the demand depends on price, the current stock, and its freshness. On the contrary, the target is to have zero inventory at the end of the inventory cycle. The salvage value and deterioration cost are also considered for the deteriorated items throughout the inventory period. Figure 1 shows the behavior of the inventory level over time. At the beginning of the stock cycle = 0, the retailer receives a lot size of units, and the inventory level immediately starts to decrease due to both demand and deterioration, until the stock level reaches zero units at .
For the duration of the stock cycle , the on-hand stock deteriorates at a constant rate , while it also loses its freshness over time. The demand depends on price, stock, and age, according to the following function:where the part of demand that is dependent on price takes one of the following expressions:
The six demand functions used in this study, which depend on price, adequately model the scenario where the demand increases as the price decreases, and vice versa.
By considering the aforementioned assumptions, the behavior of the on-hand inventory level is modelled by the following differential equation:with the boundary condition:
By solving the differential equation (3), the inventory level is expressed as follows:
The order quantity , which occurs when , is expressed as follows:
The profit is calculated as the difference between the sales revenue of those perfect and deteriorated items and the total costs resulting from sum of the ordering cost, holding cost, purchase cost, and deterioration cost. A brief explanation about the calculation of sales revenue and costs follows. The sales revenue is the product of selling price and the total demand occurred within the replenishment time , where the total demand is obtained with the definite integral from zero to of the demand function . The salvage value is computed as the product of salvage cost, salvage coefficient, and the number of deteriorated units per cycle. The ordering cost represents the cost of placing an order. The holding cost is calculated by the definite integral from zero to , of the product of the quadratic holding cost function , and the stock level function . The purchase cost is computed by multiplying the unit purchase cost by the order quantity . Finally, the deterioration cost is the product of unit deterioration cost by the number of deteriorated units per cycle.
The detailed calculation of all components of the profit function is mathematically presented below.(1)Sales revenue per cycle:(2)Salvage value of deteriorated items per cycle:(3)Ordering cost per cycle:(4)The holding cost per cycle:(5)The purchase cost per cycle:(6)The deterioration cost per cycle: Therefore, the total profit per cycle is And, it is expressed as Hence, the total profit per unit of time is expressed as follows: In general, the optimization problem is formulated as follows: The total profit per unit of time function given in equation (15) shows that this is highly nonlinear in both selling price and replenishment cycle time . Therefore, it is not possible to determine a close form for these decision variables. Nonetheless, the optimal solution to and are found through an optimization procedure that is based on the traditional conditions for optimality. The solution procedure is explained in the following section.
4. Solution Procedure to Obtain the Optimal Solution
4.1. Theoretical Results
The goal is to determine the optimal selling price and optimal replenishment cycle time that maximize the total profit. Since the total profit per unit of time function is continuous and twice differentiable with respect to both variables on the interval [0, ], there exists, then, a global maximum on that interval.
The necessary conditions for the total profit per unit of time function to be maximized are as follows:
Moreover, for the expected total profit per unit of time function to be concave, the sufficient conditions are given as follows:
The optimal solution is determined by simultaneously solving the first partial derivatives of the total profit per unit of time function given in equation (15), with respect to and equalizing to zero.
If the solution satisfies the conditions given by equations (19)–(22), it demonstrates that the function is strictly concave in both decision variables with a negative-definite Hessian matrix. If so, the solution is optimal.
In the optimization problem given in equation (16), it is stated that the selling price has an upper bound of and the replenishment cycle time has an upper bound of . The component of the demand that corresponds to price is modelled with six different functions. For isoelastic, exponential, and logit functions, the interval for the price is [0, ]. On the contrary, for linear, logarithmic, and polynomial, there exists a maximum permissible value . The upper bound is and for the linear, logarithmic, and polynomial, respectively. For these demand functions, the solution for selling price is when the solution for the selling price is greater than . This allows avoiding a negative value for the demand that is dependent on price. This is mathematically right, but it does not make sense in any business of the real world.
Notice that the replenishment cycle time is on the interval [0, ]. When the solution for the replenishment cycle time is greater than , the solution for the replenishment cycle time, then, is due to the shelf-life constrain.
4.2. Algorithm for Finding the Optimal Solution
Considering the theoretical results presented in the previous section, the following algorithm is proposed (Algorithm 1).
5. Optimal Inventory Policy for Six Different Price-Dependent Demands
5.1. Optimal Inventory Policy Using the Price-Dependent Linear Demand Function
The first partial derivative of with respect to is
The first partial derivative of with respect to is
Example 1. In order to represent a real scenario, let us consider a place that sells a fresh item. Assume that the price-dependent component of the demand for the fresh item is linear as follows: (i.e., and ). The shelf-life of the fresh item is week. There is a maximum shelf space of units. The cost for placing an order to the supplier is euros per order, the purchase cost is euros per unit, and the salvage value of the deteriorated item is euros per unit. The sensitivity coefficient to the level of stock is , and the stock deterioration rate is . The aforementioned data were taken from Agi and Soni . Additional data are still needed to solve the numerical examples. The values of the holding cost are euros per unit per week, euros per unit per week2, and euros per unit per week3. The deterioration cost of the item is euros per unit, and the salvage coefficient is . Since week, the replenishment cycle time must be week. For the price linear demand, the upper bound for price is . This means that the price must satisfy .
By applying the proposed algorithm, the following optimal solution for the inventory system is calculated: euros per unit, weeks, units, and euros per week. This solution satisfies all conditions for the optimality: , , , and . As the Hessian determinant is greater than zero, the total profit function then is strictly concave with a negative-definite Hessian matrix. Consequently, the solution is optimal.
By plotting the total profit per unit of time function with distinct values of (taking values between 7 and 27) given and distinct values of (taking values between 0 and 1) given , it is observed that is strictly concave with respect to when is fixed (see Figure 2(a)) and with respect to when is given (see Figure 2(b)). Additionally, is also strictly concave with respect to both and (see Figure 2(c)). Thus, it is ensured that the solution corresponds a global maximum.
5.2. Optimal Inventory Policy Using the Price-Dependent Isoelastic Demand Function
The first partial derivative of with respect to is
The first partial derivative of with respect to is