A consensus-based approach for multi-criteria decision making with probabilistic hesitant fuzzy information

As a generalized fuzzy number, probabilistic hesitant fuzzy element (PHFE) improves the flexibility for decision makers in expressing hesitant information, and it has been receiving increased attention. This study develops a multi-criteria decision-making (MCDM) approach that considers consensus reaching among decision makers with probabilistic hesitant fuzzy information. To obtain this aim, first, a new approach to derive normalized PHFE (NPHFE) is proposed to overcome the shortcomings in previous studies. Subsequently, a new Euclidean distance and some operations related to PHFEs are developed based on the new proposed NPHFEs. At the same time, the effectiveness and rationality of the new proposed approaches are discussed. Second, a consensus index of group with PHFEs is presented, which based on the proposed Euclidean distance of decision-makers' evaluation information on all the criteria. Third, if the consensus level of the group does not reach the expect threshold value, an iteration algorithm is designed to improve its consensus level. Moreover, the proof of the convergence of the proposed algorithm is provided to verify its effectiveness, and a MCDM approach based on group consensus is proposed. Finally, the most comprehensive candidate selection problems are provided to demonstrate the effectiveness of the proposed MCDM approach. And a comparative study with other methods is conducted with the same illustrative example.