<p>This paper investigates how firms in a green supply chain can plan production and environmental investment when both output levels and carbon emissions vary over time. We develop a mathematical framework that integrates variational calculus with metaheuristic optimization to determine production and inventory strategies that balance cost efficiency with environmental responsibility. The variational approach yields analytical expressions showing how production rates should adapt dynamically under different carbon-tax levels and greening costs. The retailer’s demand is modeled as a nonlinear function of selling price, green index, and time, capturing consumer preferences for both affordability and sustainability. Two decision structures are analyzed: a centralized system maximizing joint profit and a decentralized system modeled through a Stackelberg game with the retailer as leader. The resulting nonlinear optimization problems are solved using the Artificial Hummingbird Algorithm (AHA) and benchmarked against several established metaheuristics. Comparative results show that while all algorithms achieve similar solution quality, AHA attains the same accuracy with substantially lower computational effort, highlighting its efficiency for complex supply chain problems. Numerical results further demonstrate that flexible, time-varying production policies outperform constant-rate strategies by improving both profitability and emission reduction, especially under centralized coordination. Managerially, the analysis underscores how synchronized decision-making and well-designed policy measures, such as balanced carbon-tax rates and incentives for green technology, can promote sustainable production without compromising competitiveness. Overall, the proposed approach provides both theoretical insights and practical guidance for designing adaptive, low-carbon supply chain strategies in evolving market environments.</p>