CFZN-MARCOS - Complex Fuzzy Z-Number MARCOS Ranking

Overview

CFZN-MARCOS ranks alternatives based on performance scores. Higher score = better rank.

Data

Complex Fuzzy Z-Number

Weights

Needs a weight source

Edge cases and pitfalls

• Shahid 2026 paper text OMITS explicit criterion weight values W_ℛ - engine MUST require user-supplied weights and surface a paper-incompleteness warning when chained from this manifest.

• PAPER-EXTRACTED IMPLICIT WEIGHTS: From Table 13 weighted extended matrix dividing weighted U+ row by normalized U+ row (all 1's), implicit weights ≈ (0.3, 0.4, 0.1, 0.2) - not equal weights; documented as a finding, not authoritative.

• Shahid §5 Step 2 Δ (anti-ideal) and Λ (ideal) values printed in paper appear inverted relative to standard benefit-direction convention (Δ should contain mins for all-benefit case, but paper Δ contains maxes). Engine should follow paper's eq 2 definitions literally, NOT the printed numerical values.

• CFZN distance metric (eq 12) uses asymmetric weights: 1/2 on amplitude difference squared, 1/(4π²) on phase difference squared - phase contribution is ~9.87× larger than amplitude due to (4π²)⁻¹ ≈ 0.0253 vs 1/2 = 0.5. Document this in any engine implementation since naive equal-weight Euclidean will diverge from paper.

• Distance summed across criteria for Ω+_i and Ω-_i - paper presents per-cell distances in Tables 8-9 but algorithm requires summation before closeness coefficient (look at Table 10: per-cell ratios, not summed); confirm with engine whether Φ_ij is per-cell or per-row.

• MARCOS utility function (eq 21) is the canonical Stević 2020 formula - same exact form as crisp/fuzzy/SFZN MARCOS variants. CFZN extension only changes distance metric and anchor extraction, NOT the utility function.

• Ranking criterion: largest Γ(Ψ_i) wins (max utility). Engine must descending-sort, not ascending.

• Both WASPAS and MARCOS in this paper reduce CFZN to scalar via score function μ = (σϖ + τR)/2 only at the FINAL aggregation step - intermediate operations stay in CFZN space. Engine should NOT scalarize prematurely.

• Paper Tables 3-7 (WASPAS path) use scalar component-wise weighting (T'·W applied to each of σ, τ, ϖ, R independently), not CFZN multiplication operator from §3 eq 7. MARCOS Step 7 (eq 18) similarly uses scalar Q = Z·w on closeness coefficient (real-valued). This is internally consistent for MARCOS but methodologically simplified vs full CFZN operator algebra.

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