Interval-Valued Probabilistic Dual Hesitant Fuzzy Sets for Multi-Criteria Group Decision-Making

As a powerful extension to hesitant fuzzy sets (HFSs), dual hesitant fuzzy sets (DHFSs) have been closely watched by many scholars. The DHFSs can reflect the disagreement and hesitancy of decision-makers (DMs) flexibly and conveniently. However, all the evaluation values under the same membership degree are endowed with similar importance. And DHFSs are not able to express DMs' preference degrees on different variables. To overcome this drawback, in this paper, we propose the concept of interval-valued probabilistic dual hesitant fuzzy sets (IVPDHFSs) by providing each element with an interval-valued probability value, which can describe DMs' preferences, hesitancy and disapproval simultaneously. Then we define the basic operation laws, score function and deviation function for interval-valued probabilistic dual hesitant fuzzy elements (IVPDHFEs). Besides, the ordered distance and similarity measures are proposed to calculate the deviation of any two IVPDHFSs and to derive the weight vector for DMs objectively, respectively. To aggregate decision-making information, we present interval-valued probabilistic dual hesitant fuzzy ordered weighted averaging (IVPDHFOWA) operator. Moreover, the water-filling theory is first introduced into IVPDHFSs environment and utilized to obtain unified criteria weights mathematically. Furthermore, a three-phased multi-criteria group decision-making (MCGDM) framework is constructed to address IVPDHFSs information. Finally, a case study concerning Arctic risk evaluation is provided to verify the effectiveness and superiority of the proposed three-phased framework.