A Comprehensive Review of Fractional Inventory Problems: Models, Applications, and Future Directions

Abstract

The fractional inventory problem extends classical inventory models by incorporating fractional calculus to better capture memory effects and hereditary properties in real-world systems. This paper reviews the development and applications of fractional-order inventory models, highlighting their advantages over traditional models in addressing non-instantaneous dynamics and demand variability. Key approaches such as fractional-order differential equations, optimization under constraints, fuzzy logic integration, and hybrid demand functions are discussed. Additionally, the paper outlines current trends in research, identifies challenges, and suggests future directions for model improvement and practical deployment in supply chain systems. Recent studies have further expanded the applicability of fractional inventory models by integrating stochastic demand patterns, sustainability considerations, and multi-echelon supply chain structures. Researchers have demonstrated that fractional-order models provide improved flexibility and forecasting accuracy compared to integer-order formulations, particularly in environments characterized by uncertainty and long-term dependency effects. Moreover, advancements in computational techniques and numerical methods have enabled the practical implementation of these models in complex industrial scenarios. The growing intersection between fractional calculus, artificial intelligence, and data-driven optimization is expected to further enhance inventory decision-making and contribute to the development of resilient and adaptive supply chain systems.