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.

Entropy is an important tool of information measurement in the fuzzy set and its inference. The research on information measurement based on interval-valued Pythagorean fuzzy sets mostly involves the distance formula for interval-valued Pythagorean fuzzy numbers, but seldom involves the measurement of fuzziness. In view of this situation, we have aimed to propose new interval-valued Pythagorean fuzzy entropy and weighted exponential entropy schemes. Based on the interval-valued Pythagorean fuzzy weighted averaging operator, a strategy based on weighted exponential entropy is proposed to solve the multi-criteria group decision-making (MCGDM) problem in the interval-valued Pythagorean environment. Two examples illustrate that this paper provides a feasible new method to solve the MCGDM problem in an interval-valued Pythagorean fuzzy (IVPF) environment. Finally, by comparing with the existing methods, it is concluded that the entropy measure of IVPF schemes and the corresponding MCGDM method can select the optimal solution of the practical problem more precisely and accurately. Therefore, the comparative analysis shows that the proposed measurement method has the characteristics of flexibility and universality.

This paper mainly studies and proves the roughness bound of disturbation fuzzy sets. Firstly, based on the theory of determining self-increment and uncertain self-decrement operators, the problem that the execution subsets are not equal sets is effectively solved, which hinders the quantitative study of disturbed fuzzy sets and lays a foundation for the quantitative study of the related properties of disturbed fuzzy sets in the future. The boundary of roughness measure of disturbing fuzzy set is further studied and proved. The new territories proposed in this paper can effectively avoid the unnecessary calculation space outside the boundary in the calculation process, so as to improve the work efficiency.