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This paper studies the members' competition performance in a three-level multi-commodity supply chain network under vendor managed inventory(VMI) pattern over a planning period. We first analyze how demand and inventory change in a supply chain network under VMI pattern. We next establish an intertemporal equilibrium model for each level and an intertemporal network equilibrium model based on variational inequality and then conclude the equilibrium conditions for the system. In this paper we also provide economic explanations for the system, and propose a genetic algorithm to generate a relatively reasonable solution. We finally validate it with an example problem.