Effective inventory management is essential for operational efficiency, particularly in industries facing highly uncertain and stock-dependent demand patterns. Traditional deterministic models often fail to capture the complexities of real-world systems, such as demand variability and stochastic fluctuations, leading to suboptimal decisions. To address these challenges, this study proposes a novel inventory model based on stochastic differential equations (SDEs) that integrates Brownian motion and white noise to represent fluctuating demand in a continuous-time framework. The model avoids shortages, assumes a fixed setup cost, and employs fuzzy set theory—including \(\alpha \) -cut techniques—to handle imprecise parameters and quantify uncertainty. The production rate is defined as a linear function of inventory, and Itô’s calculus is applied to derive analytical solutions for inventory dynamics. A numerical algorithm is developed to compute the Average Total Cost (ATC), and sensitivity analyses are performed to assess the effects of key parameters. Results reveal that unit holding cost, demand variability, and responsiveness parameters significantly influence ATC, while fuzzy analysis confirms a stable optimal replenishment interval. The model consistently identifies an optimal switching point in the production-depletion cycle that minimizes cost. This research contributes a robust, adaptable framework for managing inventory under uncertainty and provides actionable insights for practitioners in sectors such as pharmaceuticals, perishable goods, and fast-moving consumer goods. Future work will expand the model to include multi-echelon supply chains, sustainability considerations, and scenarios allowing for shortages and variable lead times.
