Dynamic Optimization in Pharmaceutical Supply Chains: Balancing Inventory and Environmental Sustainability

Background: Pharmaceutical supply chains are highly sensitive systems due to medicine deterioration, strict storage requirements, recycling challenges, and environmental concerns. Traditional inventory models mainly focus on cost minimization and demand satisfaction while often neglecting sustainability, recycling mechanisms, and long-term inventory stability. Therefore, there is a growing need for mathematical frameworks capable of integrating pharmaceutical inventory management with environmentally sustainable practices. Objective: This study aimed to develop and analyze a nonlinear mathematical inventory model for sustainable pharmaceutical supply chains incorporating production, distribution, deterioration, recycling, and disposal processes within a Manufacturer–Vendor–Hospital (MVH) framework. Methods: A system of nonlinear ordinary differential equations was formulated to describe pharmaceutical inventory dynamics among manufacturers, vendors, and hospitals. The model incorporated recycling and green supply chain concepts through a sustainability-oriented framework. Positivity and boundedness of solutions were established mathematically. Equilibrium points and an Inventory Sustainability Index were derived to evaluate long-term inventory persistence. Local and global stability analyses were performed using Jacobian matrix analysis, Routh–Hurwitz criteria, Lyapunov functions, and LaSalle’s invariance principle. An optimal control problem was further formulated using Pontryagin’s Maximum Principle to determine cost-effective inventory management strategies. Numerical simulations and sensitivity analyses were conducted to validate analytical findings. Results: The model demonstrated the existence of two equilibrium states corresponding to inventory shortage and sustainable inventory persistence. The Inventory Sustainability Index was identified as a critical threshold determining whether stable vendor inventory levels could be maintained. Stability analyses confirmed that the sufficient-inventory equilibrium remained locally and globally asymptotically stable when the sustainability threshold exceeded unity. Numerical simulations supported the theoretical findings and showed that reducing deterioration rates improved inventory persistence and accelerated convergence toward equilibrium. Sensitivity analysis revealed that production and delivery rates positively influenced inventory sufficiency, whereas deterioration, recycling losses, and redistribution rates negatively affected system stability. The optimal control framework provided strategies for minimizing operational costs while maintaining sustainable inventory levels. Conclusion: The proposed mathematical framework offers a rigorous approach for analyzing sustainable pharmaceutical inventory systems with recycling and deterioration effects. The findings highlight the importance of balancing production, recycling, and distribution processes to maintain stable pharmaceutical supply chains while reducing environmental and operational burdens. This model may support policymakers and healthcare supply chain managers in designing resilient and environmentally sustainable pharmaceutical inventory systems.