Multi-hesitant fuzzy sets (MHFSs) can deal effectively with cases where some values are repeated more than once in hesitant fuzzy sets (HFSs). In this paper, the novel convex combination of multi-hesitant fuzzy elements (MHFEs) is introduced. The generalized multi-hesitant fuzzy weighted average (GMHFWA) operator based on convex operation is developed, and corresponding property is discussed. Then, based on the proposed aggregation operator, a novel approach for multi-criteria decision-making (MCDM) problems is proposed for ranking alternatives. Finally, an example is provided in order to verify the developed approach and demonstrate its validity and feasibility.