Three-layer probabilistic hesitant fuzzy knowledge measures and their applications in multi-attribute decision-making

Purpose The purpose of this study is to overcome the limitations of traditional entropy-based methods in processing uncertain information within hesitant and probabilistic hesitant fuzzy environments. It aims to systematically develop the theoretical framework of knowledge measures to more effectively characterize a system's cognitive certainty for complex decision-making. Design/methodology/approach An innovative two-parameter knowledge measure for hesitant fuzzy sets is proposed, with its parameter adjustment mechanism verified through bidirectional projection. The framework of knowledge measures is extended to probabilistic hesitant fuzzy sets, establishing a comprehensive system that includes general, one-parameter, and two-parameter types. The research thoroughly investigates the monotonicity of two-parameter knowledge measures, alongside fuzziness quantification methods and attribute weight calculation techniques. To demonstrate applicability, the approach is integrated with the TOPSIS method and validated through comparative experiments on three case studies: volunteer team selection, CCUS site selection, and PhD interview evaluation. Findings The results demonstrate that the proposed approach not only maintains decision stability but also significantly enhances ranking discrimination. This offers a powerful and effective tool for supporting decision-making in probabilistic hesitant fuzzy environments. Originality/value The study introduces a novel two-parameter knowledge measure for hesitant fuzzy sets and extends the knowledge measure framework to probabilistic hesitant fuzzy sets, establishing a comprehensive system with multiple parameterized forms. The work systematically investigates key properties, including monotonicity and fuzziness quantification, and demonstrates practical value through integration with TOPSIS and validation across diverse case studies.