SFZN-MARCOS - Spherical Fuzzy Z-Number MARCOS Ranking

Overview

Output ranks alternatives by F(U_i) ∈ [0, 1] (MARCOS utility function combining utility to ideal and anti-ideal). Best alternative has highest F(U). MARCOS differs from TOPSIS by constructing an EXTENDED decision matrix that explicitly includes anti-ideal and ideal rows, then computing utility degrees relative to these extended-matrix anchors. The canonical Stević 2020 utility function F(U) = (U+ + U-)/(1 + (1-F(U+))/F(U+) + (1-F(U-))/F(U-)) combines both utilities. SFZN reliability (τ_ε, τ_ν, τ_∂) modulates each component via the (component × reliability) Euclidean distance in Step 6, then collapses to a crisp closeness coefficient in Step 7.

Data

Spherical Fuzzy Z-Number

Weights

Needs a weight source

Edge cases and pitfalls

• PAPER-LEVEL F(U) BUG: Niu 2024 Table XXV reports F(U_T1)=0.69628 and F(U_T3)=0.70045 which appear swapped relative to canonical Stević 2020 formula applied to Niu's own Table XXIV U values (canonical produces T1=0.70046, T3=0.69630). Niu's reported ranking T3≥T1 (Table XXVI) is therefore inconsistent with the documented algorithm. Manifest uses canonical formula; engine reproduction will produce T1≥T3≥T2≥T4, not T3≥T1≥T2≥T4.

• Forgetting to extend the decision matrix in Step 8 - MARCOS REQUIRES adding anti-ideal and ideal rows to the closeness coefficient matrix BEFORE normalization in Step 9; skipping the extension produces wrong utility degrees.

• Confusing closeness coefficient C_ij = d-/(d++d-) (Niu §V.A Step 7 - collapses SFZN to crisp) with the closeness coefficient form in CRADIS (no such collapse step; uses distances directly).

• Using a different MARCOS utility function variant (e.g., F(U) = U+ + U- average) instead of canonical Stević 2020 nested form - only the canonical formula is documented in Niu §V.A Step 12.

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