Development of novel distance measures for picture hesitant fuzzy sets and their application in medical diagnosis

The picture hesitant fuzzy set (PHFS) integrates elements of picture fuzzy sets and hesitant fuzzy sets, incorporating membership, abstinence, and non-membership degrees to provide a robust framework for addressing uncertainties and complex data in real-world scenarios. In this study, we introduce key characteristics of picture hesitant fuzzy elements, including average functions, variance functions, and hesitancy degrees, to enhance its descriptive capability. Based on these characteristics, we proposed novel distance measures for PHFS. Further, we investigated their properties and proved the triangle inequality of distance measure. These measures were systematically applied in a medical diagnostic context, where they demonstrated significant improvements in diagnostic accuracy by effectively distinguishing patient conditions. Sensitivity analyses and comparative evaluations further validated the practicality and robustness of the proposed methods, highlighting their potential for broader applications in decision-making under uncertainty.