Profit–cost strategy for fuzzy fractional inventory of deteriorating items with reliability and green efforts

This study proposes a fractional-fuzzy sustainable inventory model for retailers, integrating green investment strategies with demand shaped by price sensitivity, consumer green awareness, and product reliability. Using Caputo–Fabrizio fractional calculus, the framework captures memory-dependent demand dynamics, promotional efforts, and stock visibility, while triangular fuzzy numbers address uncertainties in ordering costs, deterioration rates, and demand parameters. The graded mean integration method (GMIM) optimizes replenishment cycles and greening levels, outperforming traditional approaches like the Signed Distance Method. Results demonstrate that long-term memory effects significantly enhance profitability, and strategic green investments improve sustainability without compromising margins. Sensitivity analysis identifies deterioration rates and promotional efficiency as critical cost drivers, with excessive eco-initiatives risking profit erosion. The model provides actionable insights for balancing profitability-sustainability trade-offs, emphasizing adaptive ordering policies and targeted green campaigns. Challenges include data limitations, necessitating future validation under dynamic market conditions. This work advances sustainable supply chain research by merging fractional calculus, fuzzy logic, and eco-conscious decision-making for resilient retail operations. The idea is also illustrated in graphical abstract form in Fig. 1 .