Multistage supply chain inventory model for controllable deterioration and imperfect production with carbon emissions regulations under Stackelberg game approach

In today's highly competitive and environmentally conscientious business environment, integrating environmental considerations, such as carbon emissions reduction strategies is receiving significant attention for its social, environmental, and economic benefits. Several inventory stages within a supply chain are considered in this paper that address controllable deterioration, imperfect production, and carbon emissions under a Stackelberg game approach. In this game, the manufacturer occupies a dominant position, with the retailer following his/her decision. Imperfect production processes occur throughout the production run time which reflect a real-world scenario. Moreover, time-reliant deterioration is envisaged and the deterioration rate can be reduced through appropriate strategy. The primary goal of this investigation is to determine the optimal replenishment policies and pricing strategies that maximizing supply chain profit while meeting carbon emission targets. Two different situations are addressed based on the carbon emission regulations. The analytical derivations and numerical experiments provide a significant insight into the optimal decisions for inventory management. The findings revealed that carbon cap-and-trade regulation is more sustainability advantages for its environmental, and economic benefits. Under this policy, the total profits of the retailer and the manufacturer are increased by 0.041% and 0.049% respectively. For theoretical validation, the convexity/concavity criteria are established utilizing Hessian matrix optimality and eigenvalue characterization theorem. Finally, the validity, reliability, and robustness of decision support of the study are ensured by providing several numerical examples, comprehensive graphical illustrations, and in-depth sensitivity analysis is performed.