Spherical Fuzzy Multisets: Symmetric Algebraic Structures and Applications in Multi-Criteria Decision-Making

Uncertainty remains a central challenge in multi-stage decision-making, yet existing spherical fuzzy models cannot explicitly preserve repeated assessments of the same element. In this paper, we introduce Spherical Fuzzy Multisets (SFMSs), a framework that combines the expressive power of Spherical Fuzzy Sets (SFSs) with multiset theory to retain multiple evaluations individually, rather than collapsing them into a single value. SFMSs preserve repeated membership, abstention, and non-membership information under the spherical constraint μ² + η² + ν² ≤ 1, thereby capturing uncertainty more faithfully in repeated or sequential assessments. We establish the theoretical foundation of SFMSs by defining their basic structure, set operations, algebraic properties, level subsets, (α, β, γ)-cuts, score and accuracy functions, and modal operators, and prove the principal properties of each construction. A composite relation on SFMSs is further developed to support decision analysis and MCDM. For validation, the framework is applied to a higher education course placement problem in which student performance is evaluated across multiple assessment rounds. The results show that the proposed method distinguishes placement outcomes while preserving nuanced patterns of uncertainty, hesitation, and repeated evaluations that would otherwise be aggregated away. This richer representation improves the discrimination of alternatives and yields more informative decision outputs than existing spherical fuzzy models. Overall, SFMSs extend spherical fuzzy modelling in a mathematically rigorous way and provide a strong basis for MCDM applications in education, healthcare, management, and engineering—particularly when repeated or staged evaluations must be retained explicitly.