Abstract
Picture Fuzzy Sets (PFS) extend fuzzy sets and intuitionistic fuzzy sets, offering a more robust framework for handling ambiguous, uncertain, and imprecise data. Interval-valued Picture Fuzzy Set represent membership, neutral, and non-membership degrees as intervals, providing a flexible solution for situations where precise values are difficult to determine. This paper introduces novel Hamacher operations for interval-valued picture fuzzy matrices, building upon existing Hamacher operations for Picture Fuzzy matrices. We develop and examine scalar multiplication and exponentiation operations using these new Hamacher operations. This research advances the development of sophisticated medical diagnosis systems, leading to improved patient outcomes and more accurate decision-making.
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