Inventory management is crucial in commerce. The primary aim of inventory management is to minimize the overall system cost via informed decision-making based on various mathemati-cal methodologies such as optimal control analysis, dynamic programming, and network optimization. Inventory decisions determine the optimal timing for replenishment and the quantity to be added. The core objective of inventory theory is to identify managerial guidelines for reducing inventory expenses while fulfilling market demands. To achieve the optimal total cost using an inventory model, most industrial entities emphasize inventory control and strive to navigate challenges such as managing and sustaining stocks of perishable items. The concept of deteriora-tion, which encompasses various forms of damage, spoilage, dryness, and vaporization [1], is crucial in the management of inventory models. Deterioration is important in inventory analy-sis, as it affects cost reduction and profit maximization. Failure to remove degraded items from a system can significantly affect inventory operations. Without relevant monitoring, deterioration will cause excessive waste and losses, which may reach up to 30% in some countries [2]-[3]. In recent decades, researchers have focused primarily on controlling and sustaining inventories. In practical scenarios, item decay or deterioration is natural and evident in various products such as vegetables, fruits, foods, perfumes, chemicals, pharmaceuticals, radioactive substances, and electronic equipment.

Many businesses prioritize inventory management and trou-bleshooting inventory issues to achieve the economic order quan-tity (EOQ), which minimizes the overall average inventory costs. Generally, for deteriorating products, the inventory decays over time, thus causing profit losses because the degraded items can-not be sold and must be removed from the stock. Unsold items in the inventory increase the holding costs over time until all items are depleted, whereas the ordering cost decreases over time. The aggregated cost features an optimum point at which replenish-ment minimizes the overall cost, as illustrated in Figure 1.

Figure 1: 
General concept of EOQ in supply chain

The foundational inventory model for deteriorating items as-serts that inventory depletion primarily stems from a consistent demand rate. Whitin [4] examined the effect of deterioration on fashion items once they surpassed the recommended date. Wag-ner and Whitin [5] introduced a dynamic version of the classical EOQ model developed for deteriorating items. Ghare and Schrader [6] investigated an inventory model for deteriorating items, which was characterized by constant deterioration and demand rates. Shah and Jaiswal [7] investigated an order-level inventory model developed for deteriorating items at a constant deterioration rate. Aggarwal [8] expanded Shah and Jaiswal’s work by calculating the average holding cost. Dave and Patel [9] developed an inventory model for deteriorating items character-ized by linearly increasing demand and deterioration rates, which remained a constant fraction of the on-hand inventory. All the aforementioned models were developed based on constant deteri-oration rates, constant demand rates, infinite replenishment, and no shortages. Heng et al. [10] introduced an exponentially decay-ing inventory model for deteriorating items by assuming a finite replenishment rate and constant demand rate. Reviews summariz-ing advancements in deteriorating inventory are available in the literature [11]-[15].

Pricing policy is widely recognized as one of the most preva-lent and effective methods in businesses for influencing customer demand. As mentioned in [16], realizing optimal pricing policies can offer competitive advantages to firms. Many studies have approached pricing policies as an optimization problem to max-imize the expected cumulative profit, where the pricing strategy was prioritized in the objective function. The Hamilton–Jacobi–Bellman (HJB) equation has been proposed in numerous studies, either in a continuous or discrete domain, as a robust optimization tool to yield an analytical solution [17]-[20].

The HJB equation has been widely applied in many previous studies, particularly in continuous models using the analytical approach. In the case of deterministic problems, for instance, Dye’s model [21] formulates customer demand as a multiplicative function comprising price, the advertising goodwill, and a fresh-ness index. Yang and Zhang [22] used the HJB equation for the EOQ model, where stationary demand was expressed in a mul-tinominal logit (MNL) model, along with inventory-cost and inventory penalty functions. Furthermore, the MNL model was used to derive a deterministic relaxation formulation for station-ary demand [23]. Recently, stochastic problems have been inves-tigated extensively as they can describe issues more realistically. In such problems, customer demand is typically expressed as a stochastic process, such as the Poisson process [24], Brownian motion [25], or Bernoulli random variable [23]. However, most studies focused on stochastic events instead of variations in sys-tematic parameters as a source of uncertainty. Most real-world systems are difficult to model owing to the change in parameters over time. To describe the behavior of these systems precisely, one should consider the effect of parametric variation.

Hence, this study is performed to address the research gap mentioned above by introducing a discrete stochastic deteriorat-ing inventory model based on the EOQ concept, whose deteriora-tion rate is uniformly distributed in a specified range. Customer demand is assumed to depend on a product’s price as a linear function with varying coefficients over time. Within this uncer-tain environment, the final objective is to craft an optimal policy that maximizes the total cumulative profit, as well as to realize the replenishment quantity for every cycle. The main contributions of this study are as follows:

• ① A stochastic model for an inventory system is constructed, which includes uncertainties in the deterioration rate and the coefficients of the demand function.
• ② An analytical solution for achieving the optimal pricing using the HJB equation is derived comprehensively.
• ③ The solution convergence and the optimal replenishment quantity are analyzed.
• ④ Numerical simulations are conducted to reveal the perfor-mance of the proposed policy and compare the outputs in ideal and uncertain scenarios.

The remainder of this paper is organized as follows: Section 2 describes the proposed inventory system using a mathematical model. Section 3 provides a detailed analysis of the convergence condition of the solution, including the closed forms of the coef-ficients and the compact form of the optimal pricing policy, as well as the best quantity to be replenished per cycle. Simulations performed to validate the analytical results are presented in Sec-tion 4, and Section 5 concludes the paper.

In this study, an optimal control technique for a discrete EOQ-decaying inventory system was applied using the HJB equation to maximize the cumulative profit. The proposed pricing policy is robust for cases involving oscillating parameters and complex environments, as the difference in profit between the output in ideal and realistic scenarios remains under 5%. For a specified replenishment cycle length, the proposed method provided an optimal pricing that maximized the profit, even under imperfect conditions. The pricing policy significantly affects an enterprise’s profit. Therefore, the proposed technique is promising for deci-sion makers as it can yield the best policy under predefined oper-ating conditions and markets, even for volatile ones.

In this study, all unknown and variation sources of the supply chain were reflected in the model as oscillations of systematic parameters. In addition, customer demand was modeled as a linear function of pricing, which may not be suitable for practical cases as more factors affect customer behavior in reality. In the future, this study can be expanded by adding realistic constraints. For instance, the selection of customers, which is typically un-predictable and randomized, can be modeled stochastically. This renders the study more practical and highlights the robustness of optimal control methods.