Group Decision Making With Probabilistic Hesitant Multiplicative Preference Relations Based on Consistency and Consensus

The probabilistic hesitant multiplicative preference relations (PHMPRs) can model the opinions of the decision makers for pairwise comparisons over alternatives using some possible values from Saaty's 1-9 scale with the probability information. The existing study normalized the PHMPRs by adding additional preference values with the probability information into the shorter element before modifying the consistency and consensus. As a supplement, in this paper, we focus on dealing with the consistency and consensus for PHMPRs by means of multiplicative preference relations. We first present the definition of geometric consistency index for PHMPRs and develop an automatic iterative algorithm for checking and improving the geometric consistency index for PHMPRs. We also put forward the concept of geometric consensus degree for PHMPRs and then devise an automatic consensus reaching algorithm to modify the geometric consensus degree of PHMPRs. After that, a novel complete group decision making model with PHMPRs is put forward. Finally, an illustrative example is shown to verify the proposed model and we compare it with the existing study to show its superiority.