A differential game model based on the government subsidy strategy considering the green pharmaceutical problem and the goodwill of the company

China has emerged as a major global player in the pharmaceutical industry and a significant consumer of drugs. However, the rapid growth of the raw material pharmaceutical manufacturing sector has resulted in a substantial amount of waste generation, leading to environmental pollution issues. In order to mitigate the pollution problems associated with the pharmaceutical manufacturing industry, this research proposes a tripartite game model with two dynamic parameters and develops optimization models for green supply chains under three decision-making scenarios, aiming to contribute to the efforts for achieving low-carbon development. Additionally, the study considers the government's green subsidy system and incorporates the optimization of green pharmaceutical practices and goodwill of the enterprises as dynamic parameters. By employing differential game methods, equilibrium solutions among the three parties are determined. The research findings reveal that: firstly, the green pharmaceutical levels are equivalent among the three models; secondly, in scenarios where retailers receive subsidies and make decentralized decisions, API manufacturers achieve the highest level of goodwill and the supply chain profit is maximized; thirdly, when manufacturers have sufficiently high marginal profits, adopting a decentralized decision-making approach encourages retailers and enhances the overall goodwill level of the supply chain. This study provides valuable insights for the formulation of effective strategies and policies to promote sustainable development within the pharmaceutical industry.