A robust correlation coefficient for probabilistic dual hesitant fuzzy sets and its applications

As a generalization of the hesitant fuzzy sets (HFSs) and dual HFSs (DHFSs), probabilistic dual hesitant fuzzy sets (PDHFSs) are a strong and valuable tool to represent the imprecise information by embedding both the features of HFSs and probabilistic information instantaneously. Meanwhile, a correlation coefficient is a prominent measure to measure the relationship between two sets. Motivated by these primary characteristics, it is interesting to present some information measures to the PDHFSs and hence decision-making approach based on the correlation coefficient. In this paper, we develop a method to solve the multi-criteria decision-making (MCDM) problem under PDHFS environment. For it, firstly, we define the informational energy and the covariance between the two PDHFSs and study their properties. Secondly, we develop correlation coefficients and the weighted correlation coefficients for PDHFSs. In the formulation, DHFSs are able to represent the information in terms of their respective degrees, while the assigned probabilities give more details about the level of agreeness or disagreeness. Also, some properties of the proposed measures are also studied. Thirdly, a novel algorithm is developed based on the proposed operators to solve MCDM problems. A practical example is provided to verify the developed approach and to demonstrate its practicality and feasibility. Also, a comparative analysis with several existing studies reveals the proposed method is better during solving the decision-making problems.