Abstract: It has already run into a common view among industry, academia, government and consumers that reducing carbon emissions in production and consumption because greenhouse gases have caused environmental disasters such as rising temperatures and climate change. Thus, carbon emission reduction has become an important issue to be solved at present all over the world. In the supply chain, the production is one of the main process of carbon emission. In practice, the supply chain is a network structure for the complex trading relationships. At the same time, in China, after several years of pilot, carbon trading market has begun to conduct carbon trading. Herein, the enterprises' production and carbon reduction strategy under the Cap-and-Trade system in a closed-loop supply chain network consisting of two types of manufacturers and various demand markets. The first type of manufacturers can be called high-emission manufacturers, because they emit more carbon when producing unit product; and the second type of manufacturers can be called low-emission manufacturers, because they emit less carbon when producing unit product. Manufacturers of the same type produce homogeneous products, while manufacturers of different types produce heterogeneous products, and there is no competition between them. No loss of generality, we assume that the high-emission manufacturers buy carbon emission rights from low-emission manufacturers, and they trade carbon emission right in carbon trading center based on carbon quotas from government, which forms a carbon trading subnetwork; meanwhile, the manufacturers and demand markets trade products, which forms a product trading subnetwork. Combining with variational inequality and complementarity theory, a profit maximization model for two types of manufacturers and carbon trade center is established, and the nonlinear programming of maximizing profit with constraints is transtformed into a variational inequality problem. In turn, the optimal conditions for the trading price, production and carbon emission reduction strategy in the closed-loop supply chain network are obtained further the conditions that how to reach equilibrium between various decision-makers in the supply chains acquired. The modified projection method is employed for solving this variational inequality. In the numerical examples, the impacts of carbon quotas of the two types of manufacturers respectively, and the recovery rate of EOL products on the equilibrium states and the profits of enterprises are paid attention to. Thus, several conclusions with certain reference value are deduced by analyzing the changing of relevant parameters. The results show that there is consistency between enterprise profit target and government emission reduction target under some conditions. The carbon trading volume of the two types of manufacturers is the same in most cases, but when the carbon quota increases to a certain extent, there may be inconsistencies; it is not feasible to set the same recovery rate for different types of manufacturers, the manufacturer's actual recovery rate may be higher than the government's minimum recovery rate. The research in this paper analyzes the production and carbon trading strategy of enterprises considering the competition of the closed-loop supply chain network, which enriches the carbon trading theory and expands the research scope of supply chain network. The background and models of carbon trading in this paper have enabled us to deepen our understanding of Cap-and-Trade system. In the background of green development, this study provides certain instructions and reference for government and enterprises to achieve the goal of carbon reduction.

Key words: cap-and-trade, carbon trading subnet, product transaction subnet