Hamming distance-based knowledge measure and entropy for interval-valued Pythagorean fuzzy sets

The development of knowledge measures and uncertainty measures for constructing interval-valued Pythagorean fuzzy sets (IVPFS) have garnered significant attention in recent years. Nevertheless, existing uncertainty measures predominantly depend on entropy-based approaches, which exhibit limitations in effectively characterizing the knowledge inherent in interval intuitionistic fuzzy sets. This study extends the axiomatic framework of knowledge measures for fuzzy sets by introducing a novel distance-based knowledge measure function. The proposed measure is rigorously validated through comprehensive mathematical analysis and supported by extensive numerical examples. Furthermore, this research extends the entropy properties from interval-valued intuitionistic fuzzy sets to their Pythagorean counterparts while providing rigorous proofs of their compliance with axiomatic definitions. To demonstrate practical applicability, the proposed entropy measure is implemented in multi-attribute group decision-making scenarios involving unknown interval-valued Pythagorean fuzzy information. Experimental results substantiate both the validity and practical utility of the proposed measures.