Coordinating the pharmaceutical supply chain with a two-invoice mechanism: A differential game approach

Introduction：It is well known that the pharmaceutical industry is crucial to human health as well as social well-being. This paper uses a differential game approach to investigate coordination strategies in China’s pharmaceutical supply chain under a two-invoice mechanism.

Methods：Using a differential game approach, this paper modeled a three-tier system to analyze dynamic interactions between R&D effort, logistics effort, market demand, and quality evolution. A state equation described drug quality evolution, influenced by R&D effort, logistics service effort, and natural decay rate. A market demand function (positively related to quality, negatively to retail price) and long-term profit functions (with quadratic convex costs for R&D and logistics) were established. Three scenarios were examined: centralized optimization, decentralized Stackelberg game, and a novel combination contract incorporating government subsidies and cost-sharing. For each scenario, optimal strategies, drug quality trajectories, and profits were derived using backward induction and solving differential equations. Finally, numerical experiments were conducted to verify the models by comparing key indicators.

Results：Drug quality converges to a steady state independent of initial levels. Outcomes vary significantly: centralized decision-making yields the highest quality and profit, followed by the combination contract, with decentralized decision-making the lowest.

Conclusions：This paper integrates pharmaceutical supply chain performance and drug quality with time by using differential games. The convergence of drug quality to a steady state indicates that long-term quality depends on decision behaviors, not initial conditions. The combination contract effectively coordinates decentralized decision-making, improving quality, efforts, and profits, making it more feasible.