This paper considers the Balanced Multi-Weighted Attribute Set Partitioning (BMWASP) problem which requires finding a partition of a given set of objects with multiple weighted attributes into a certain number of groups so that each attribute is evenly distributed amongst the groups. Our approach is to define an appropriate criterion allowing to compare the degree of deviation from the " perfect balance " for different partitions and then produce the partition that minimizes this criterion. We have proposed a mathematical model for the BMWASP and its mixed-integer linear reformulation. We evaluated its efficiency through a set of computational experiments. To solve instances of larger problem dimensions, we have developed a heuristic method based on a Variable Neighborhood Search (VNS). A local search procedure with efficient fast swap-based local search is implemented in the proposed VNS-based approach. Presented computational results show that the proposed VNS is computationally efficient and quickly reaches all optimal solutions for smaller dimension instances obtained by exact solver and provide high-quality solutions on large-scale problem instances in short CPU times.