A hybrid FFHS-TOPSIS framework based on trigonometric similarity measures for multi-attribute decision-making with application to bridge design optimization

The Fermatean Fuzzy Hypersoft Set (FFHS) 0 ≤ (μA(x))3 + (νA(x))3 ≤ 1, ∀x ∈ X is a novel model that addresses the critical limitations of traditional fuzzy theories in multi-parameter systems, with sub-parametrized uncertainty
modeling. The Fermatean fuzzy structure offers more reliability over existing fuzzy theories by accommodating a more flexible form of uncertainty. It is intended to establish a comprehensive structure of FFHSS, its
axiomatic properties, and rigorous proofs of the fundamental theorems, such as identity, range, monotonicity, and symmetry, for trigonometric similarity measures (TSM). It aims to develop a novel FFHS-TOPSIS algorithm
that incorporates hypersoft structures to preserve attribute and multiple attribute sets. The proposed algorithm will be validated through an extensive on-bridge design optimization problem, evaluating different alternatives
across attribute sets. The proposed FFHS-Cosine method achieves superior discrimination (ΔCC = 0.0094) and perfect Spearman correlation (ρ = 1.0) against five benchmarks, FFHS-TOPSIS, IFS-Cosine, PFS-Cosine, VIKOR,
and EDAS. The ranking stability across expert weight variations was confirmed through the sensitivity analysis. Results recommend a construction timeline, balancing cost, structural integrity, efficiency, and environmental
impact. The proposed framework elaborates O(n) hypersoft capability, computational efficiency, and practical applicability for sustainable projects. The findings demonstrate that the proposed method enhances ranking,
improves discrimination among alternatives, and yields more consistent and interpretable findings compared to existing approaches.