Abstract: Previous work which dealing with supply chain network equilibrium is expanded in this paper to the field of closed-loop supply chain network equilibrium with dual channel and imperfect products. The closed-loop supply chain network equilibrium model consists of the supply market, manufacture market, the retail market, the consumer market, and recovery market. The suppliers provide the manufacturers raw material, the manufacturers make products and some of which includes flaw, the products satisfying consumers are sold by retailers channel and electronic channel, the used products in consumer market are returned to manufacturers by recovery centers. Firstly,bBased on finite-dimensional variational inequalities and complementary theory, the mathematic model of the closed-loop supply chain network is built and the equilibrium conditions is obtained. Then,the logarithmic-quadratic proximal prediction-correction (LQP-PC) algorithm is designed to solve the equilibrium state of the network model presented. Further,a numerical example considering different situations is used to illuminate the effects of parameters including remanufacturing rate, products flaw rate and dual channel on the equilibrium. The results indicate that when introducing the electronic channel, the electronic channel may affect the traditional channel; when the remanufacturing rate increases, the profit of manufacturers increases remarkably and the profit of the whole supply chain enhance too; when the imperfect products rate decreases, the increment rate of manufacturers' profit is less than that of the whole supply chain's profit. The conclusions obtained in this paper provide meaningful insights that lead into a number of areas for future research on network equilibrium in closed-loop supply chains with dual channel.

Key words: closed-loop supply chain network, dual channel, imperfect product, remanufacturing rate, variational inequalities