Applications of a novel probabilistic hesitant fuzzy correlation coefficient in medical diagnosis and students’ performance assessment

Probabilistic hesitant fuzzy sets (PHFSs) are effective for modeling uncertainty by combining hesitant assessments with probability information. However, existing correlation coefficients (CCs) for PHFSs are limited in scope and typically require that the compared probabilistic hesitant fuzzy elements (PHFEs) have equal lengths. This requirement is usually met by padding shorter PHFEs with artificial elements, which may introduce bias, increase computational burden, and distort the correlation analysis. To overcome these drawbacks, this paper develops a new class of CCs for PHFSs based on a probability-splitting strategy that unifies the lengths of PHFEs without introducing fictitious elements. The proposed approach reduces the number of operations, preserves fuzzy information more faithfully, and provides a more reliable and accurate assessment of the association between PHFSs. Furthermore, motivated by the central role of entropy in uncertainty modeling, we propose a mixed exponential probabilistic hesitant fuzzy entropy for PHFEs, which jointly characterizes their individual and overall uncertainties and addresses several deficiencies of existing entropy measures in terms of coherence and interpretability. The effectiveness and advantages of the proposed CCs and entropy measure are demonstrated through a medical diagnosis study. In addition, their general applicability is further illustrated through a students' performance assessment problem formulated as a multi-criteria decision-making (MCDM) example, together with comparative analyses against alternative methods.