Supply chains worldwide are grappling with the dual challenge of volatile market demand and increasingly stringent carbon emission regulations, creating pressure on companies to maintain profitability while reducing environmental impact. A new study published in Frontiers of Engineering Management presents an optimal control-based model that addresses these challenges by treating production rate as a dynamic, time-dependent variable rather than a fixed value.
The research, conducted by teams from The University of Burdwan, Jahangirnagar University and Tecnologico de Monterrey, develops a two-layer manufacturer–retailer supply chain model where market demand depends simultaneously on selling price and time. This approach integrates price- and time-sensitive demand for both retailers and consumers while linking carbon emission levels directly to production intensity. The work, available at https://doi.org/10.1007/s42524-025-4110-6, represents a significant departure from traditional supply chain models that typically assume constant production rates, overlooking real-world fluctuations and their environmental consequences.
Modern supply chains operate under volatile demand influenced by seasonality, price changes, and consumer behavior, making coordination between manufacturers and retailers increasingly difficult. Simultaneously, governments globally are enforcing carbon taxes to curb greenhouse emissions, creating additional operational pressure on production systems. The new model addresses these challenges by defining production rate as a control variable and modeling carbon emission as a linear function of production intensity, meaning higher production generates proportionally higher emissions.
To solve the non-linear variational problem, researchers applied optimal control theory and evaluated decentralized scenarios using Stackelberg game analysis. They tested six metaheuristic algorithms to obtain optimal decisions for production, pricing, inventory, and emission costs, including the Artificial Electric Field Algorithm, Firefly Algorithm, Grey Wolf Optimizer, Sparrow Search Algorithm, Whale Optimizer Algorithm, and the Equilibrium Optimizer Algorithm (EOA). Results showed that EOA outperformed other algorithms in solution accuracy, convergence, and stability.
Sensitivity analysis demonstrated how variations in tax rate, production cost, or price elasticity influence profit and emission outcomes. These findings confirm that dynamic production control can reduce environmental impact while maintaining profitability, offering a more realistic strategy than models using fixed production assumptions. The research provides a decision-support framework for industries operating under carbon regulation policies, applicable to sectors such as steel, cement, chemicals, consumer goods, and logistics where carbon output scales directly with production intensity.
"This model brings production planning closer to real industry conditions," the authors explained. "By treating production rate as a variable instead of a constant, we allow the system to react to demand and emission constraints over time. Through optimal control and algorithmic optimization, manufacturers can identify profitable operational levels without compromising environmental goals." With global emission taxes tightening, this approach may help companies develop greener strategies, lower penalties, and improve collaboration with retailers while addressing the complex interplay between economic viability and environmental responsibility in modern supply chain operations.
