Abstract: A decision matrix is typically used to express hesitant information when solving multi-attribute decision problems in an uncertain environment. To further investigate the decision problem, this paper takes the property of matrix rank as the starting point, introduces it into the hesitant fuzzy theory, presents the concept of the rank of the hesitant fuzzy decision matrix and discusses the related properties, and then studies the hesitant fuzzy linear relation, obtaining the attribute reduction method based on the hesitant fuzzy linear relation and applying it to the multi-attribute decision making. It adds to the theoretical understanding of the hesitant fuzzy decision matrix. The aggregation operator first transforms the hesitant fuzzy information into a comprehensive decision matrix, and the row echelon transformation determines the rank of the matrix. Second, the hesitant fuzzy linear relationship is obtained using the rank property. A new hesitant fuzzy matrix is obtained after attribute reduction based on the hesitant fuzzy linear relationship, and the alternatives are sorted using the TOPSIS method. Finally, the effectiveness and superiority of the proposed method are demonstrated through a comparison of actual case analysis and existing methods, and the expected research purpose is met.
