Hesitant bipolar-valued fuzzy sets and bipolar-valued hesitant fuzzy sets and their applications in multi-attribute group decision making

In this paper, we propose the concepts of hesitant bipolar-valued fuzzy sets (HBVFSs) and bipolar-valued hesitant fuzzy sets (BVHFSs). The HBVFSs encompass bipolar-valued fuzzy sets (BVFSs) and hesitant fuzzy sets (HFSs) as special cases. The BVHFS is the generalization of the concept of HFS in which the membership degrees of an element to a given set are not exactly defined, but denoted by several possible bipolar-valued fuzzy values. Then, we investigate the basic operations, properties, and comparison laws of HBVFSs and BVHFSs. Based on these basic operations, properties, and comparison laws, we also develop some aggregation operators for solving the multi-attribute group decision-making problems with numerical examples.

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