Interval-Valued Spherical Fuzzy Soft Rough Sets: A Hybrid Framework for Multi-Criterion Group Decision-Making

Decision-making problems often involve simultaneous sources of uncertainty: interval-valued hesitation in expert judgments, positive–neutral–negative assessments, parameter-dependent evaluations, and boundary-region indiscernibility. To address these combined forms of uncertainty, this paper develops an interval-valued spherical fuzzy soft rough set (IVSFSRS) framework. The model integrates interval-valued spherical fuzzy information, soft parameterization, and rough lower–upper approximations in a unified approximation space. In response to the need for a more explicit theoretical foundation, an IVSF soft relation is formally defined and its role in constructing the approximation operators is clarified. The paper also presents basic operations, Hamacher-type extensions for the lower and upper approximations, and proof sketches showing the validity of the proposed operators. Furthermore, an IVSFSRS-based multi-criterion group decision-making procedure is developed, and a TOPSIS-oriented formulation is outlined to improve practical applicability. A demonstrative location-selection example, comparative analysis with related models, and sensitivity discussion illustrate the expressive power, robustness, and limitations of the proposed approach.