Interval-Valued q-Spherical Fuzzy Rough Sets and TOPSIS for Multi-Criteria Decision-Making: Application to Sustainable Smart City Development
This study develops an interval-valued q-spherical fuzzy rough set TOPSIS framework (IVq-SFRS-TOPSIS) for multi-criteria group decision-making when expert judgments contain interval uncertainty, neutrality, and granular indiscernibility. The revised framework clarifies the relationship between interval-valued q-spherical and interval-valued T-spherical fuzzy models, defines admissible approximation operators over compatible domains, and introduces a radial projection step that guarantees closure under the IVq-SFN constraint whenever component-wise extrema would otherwise violate it. The proposed framework provides a mathematically balanced representation of interval-valued q-spherical fuzzy information, reflecting the concept of symmetry and supporting reliable group decision-making under uncertainty. The TOPSIS procedure is then formulated through expert aggregation, benefit–cost normalization, entropy-based criteria weighting, ideal-solution distance calculation, and closeness-coefficient ranking. The method is illustrated through a sustainable smart city development case using four AI-based alternatives and six criteria. Rather than claiming unconditional superiority, the revised comparative and sensitivity analyses examine how the ranking changes under alternative fuzzy decision models, different q values, perturbations to criteria weights, and perturbations to the decision matrix. The results indicate that the proposed framework provides an interpretable rough-boundary representation and a reproducible ranking mechanism for complex MCDM problems under interval-valued q-spherical uncertainty.