A differential game model for closed-loop supply chain participants under carbon emission permits

In the low carbon economy background, supply chain participants need to make their investment and production decisions under the carbon emission consideration. In this study, we propose a closed-loop supply chain (CLSC) network including multiple manufacturing/remanufacturing plants and multiple distribution/collection centers. From a long-term horizon and dynamic perspective, we analyze the optimality conditions of CLSC participants, who compete in a non-cooperative manner under carbon emission permits. We develop a differential game model for the CLSC network based on differential variational inequality, and illustrate that the differential variational inequality is equivalent to the nonlinear complementary problem. A successive linearization algorithm is proposed. In addition, we analyze the impact of subsidy and remanufacturing ratio on the equilibrium results. The results indicate that, under carbon permits, receiving subsidy can motivate plants to input more investment into carbon emission abatement technologies to reduce the carbon emission, which results in the reduction of the total carbon stocks and the increase of the total profits for the plants and the whole CLSC network. In addition, plants should enhance their remanufacturing capability, which brings more profits and a higher sustainability level for the whole supply chain. Our study provides implications for companies in the CLSC to enhance their profits through reducing carbon emissions by increasing technology investment and enhancing remanufacturing capabilities.