A game-theoretic framework for dual-channel scrap metal collection with quality-dependent production in a closed-loop supply chain

This paper develops a game-theoretic optimisation framework for a multi-echelon closed-loop supply chain with dual-channel scrap metal collection and quality-dependent production. The system consists of a supplier, manufacturer, retailer, and third-party recycler, where scrap metals are collected through both retailer-led and recycler-driven channels. The manufacturer carries the production using a combination of virgin and recycled materials, with product quality endogenously determined by input quality. The study aims to determine equilibrium decisions, including prices, collection and quality improvement effort, greening level, advertising effort and quality, under different supply chain structures. The model is analysed under centralised coordination, Manufacturer Stackelberg, and Vertical Nash frameworks. Although first-order conditions and concavity properties are derived symbolically, the resulting system of nonlinear equations is not analytically tractable. Therefore, equilibrium solutions are obtained numerically using constrained nonlinear optimisation. The centralised solution is computed using the trust-constr algorithm with concavity verification. The Stackelberg equilibrium is obtained via Differential Evolution combined with a fixed-point best-response procedure, followed by local refinement, while the Nash equilibrium is computed by solving the system of first-order conditions. Results show that centralised coordination significantly improves system performance, increasing total profit by 73% over Stackelberg and 57% over Nash. It also enhances recycling efforts and greening investment, whereas decentralised structures lead to weak recycling incentives, higher prices, and reduced demand. Sensitivity analysis confirms the robustness of outcomes.