In this paper, the generalized multi-rate multicast weight-based fair rate allocation problem is studied. First, a multi-rate multicast network model is formalized, and the minimum rate requirement and weight of each virtual session is considered in the model. Based on the model, the concept of generalized multi-rate multicast weight-based max-min fairness is defined, and the theory of generalized multi-rate multicast weight-based max-min fair rate allocation is established. At last, a centralized algorithm to solve the generalized multi-rate multicast weight-based fair rate allocation problem is presented, and its correctness is also proved.