A Programming Model for Consistency and Consensus in Group Decision Making with Probabilistic Hesitant Fuzzy Preference Relations

Individual consistency and group consensus are important topics in group decision-making process with preference relations. This study introduces a programming model to meet the individual consistency level and group consensus threshold value simultaneously. First, the distance measure between two probabilistic hesitant fuzzy preferences (PHFPRs) is proposed, and a consistency index for PHFPR is defined based on the proposed distance measure. Then, a mathematical programming model is constructed to improve its consistency when the PHFPR does not meet the consistency level. Second, considering that some values in PHFPRs provided by the decision makers may be missing in the decision-making process, a mathematical programming model is constructed to derive the missing values. Third, the proximity degree between any two decision makers is proposed, and a consensus index among the decision makers is defined based on the proposed proximity degree. Then, a mathematical programming model is constructed to obtain the expected consistency level and consensus threshold value simultaneously. Finally, an example is provided to illustrate the effectiveness of the proposed method. Comparative studies with several existing methods are also provided.