Abstract: This article was designed to develop an equilibrium model of a uncertain multi-period closed-loop supply chain network which was comprised by noncooperative manufacturers, retailers, demand markets and recovery centers. The decision-makers in the network had sufficient information about the future and sought to determine their optimal plans that maximize their profits over the multi-period planning horizon. With the help of equilibrium theory and variational inequality theory, the network structure of the multi-period closed-loop supply chain based on uncertainty and the resulting demand satisfaction was identified and the decision-makers’ equilibrium conditions were derived. An algorithm for solving the variational inequality was designed via logarithmic-quadratic proximal prediction-correction algorithm to compute the equilibrium solution. Finally, numerical examples are solved to test the efficiency of the model and the proposed algorithm for finding the above equilibrium, and sensitivity analysis are also be conducted to illustrate the effect of the resulting demand satisfaction on the decision-makers’ strategies, together with their respective profits.

Key words: variational inequality, closed-loop supply chain network, multi-period, Nash equilibrium