Some New Operations on Picture Fuzzy Multisets

Abstract

A multiset is an extension of a crisp set in which elements are allowed to occur more than once, enabling the representation of data where frequency or multiplicity is essential. Building upon this, fuzzy multisets introduce a degree of membership to each element, capturing both quantity and uncertainty. Further developments have led to the formulation of picture fuzzy multisets a robust framework that extends picture fuzzy sets. In this paper, we propose some new operations on picture fuzzy multisets by analyzing existing operations on fuzzy sets, and fuzzy multisets, extending these foundations using picture fuzzy logic principles to define operations that maintain the consistency of positive membership, neutral membership, and negative membership and refusal degrees, establishing a set of axioms and properties (such as commutativity, and distributivity) that the operations must satisfy, proving theorems that verify these properties and constructing a detailed numerical example to illustrate and validate the behavior and correctness of the proposed operations. It was shown that the proposed operations are well-defined, internally consistent, and closed under the structure of PFMs. The example demonstrates that the operations handle ambiguity, contradiction, and repetition effectively, making them suitable for applications in multi-criteria decision-making, knowledge representation, and information systems. The new operations significantly broaden the mathematical toolkit available for handling picture fuzzy multisets. They lay a foundation for future research into more complex structures such as picture fuzzy multirelations, aggregation operators, and soft computing models based on picture fuzzy multisets.