CFZN-MARCOS - Karmaşık Bulanık Z-Sayıları MARCOS Sıralama

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

CFZN karar matrisi, kriter ağırlıkları ve yön vektörünü doğrula.

$T = [T_{i j}]_{m \times n}, T_{i j} = {σ_{i j} e^{2 π i τ_{i j}}, ϖ_{i j} e^{2 π i ℜ_{i j}}}; w = (w_{1}, \dots, w_{n}), \sum w_{j} = 1$

LaTeX
T = [T_{ij}]_{m \times n},\ T_{ij} = \{\sigma_{ij} e^{2\pi i \tau_{ij}},\ \varpi_{ij} e^{2\pi i \Re_{ij}}\};\ w = (w_1, \ldots, w_n),\ \sum w_j = 1
Anti-ideal Δ (kriter tipine göre en az tercih edilir) ve ideal Λ (kriter tipine göre en çok tercih edilir) ile Shahid §5 Adım 2'ye göre genişlet.

$Benefit: Λ_{j} = {max σ_{i j} e^{2 π i max τ_{i j}}, max ϖ_{i j} e^{2 π i max ℜ_{i j}}}, Δ_{j} = {min σ_{i j} e^{2 π i min τ_{i j}}, min ϖ_{i j} e^{2 π i min ℜ_{i j}}}; Cost: swap min \leftrightarrow max$

LaTeX
\text{Benefit: } \Lambda_j = \{\max \sigma_{ij} e^{2\pi i \max \tau_{ij}},\ \max \varpi_{ij} e^{2\pi i \max \Re_{ij}}\},\ \Delta_j = \{\min \sigma_{ij} e^{2\pi i \min \tau_{ij}},\ \min \varpi_{ij} e^{2\pi i \min \Re_{ij}}\};\ \text{Cost: swap min} \leftrightarrow \max
Her hücreden ideal Λ'ya (Ω+) ve anti-ideal Δ'ya (Ω-) CFZN uzaklığını Shahid §5 Adım 3 denk 12-13'e göre hesapla.

$Ω_{i j}^{+} = \frac{1}{2} ((σ_{i j} ϖ_{i j}) - (σ_{Λ_{j}} ϖ_{Λ_{j}}))^{2} + \frac{1}{4 π^{2}} ((τ_{i j} ℜ_{i j}) - (τ_{Λ_{j}} ℜ_{Λ_{j}}))^{2}; Ω_{i j}^{-} analogous vs Δ_{j}$

LaTeX
\Omega^+_{ij} = \frac{1}{2}((\sigma_{ij} \varpi_{ij}) - (\sigma_{\Lambda_j} \varpi_{\Lambda_j}))^2 + \frac{1}{4\pi^2}((\tau_{ij} \Re_{ij}) - (\tau_{\Lambda_j} \Re_{\Lambda_j}))^2;\ \Omega^-_{ij}\ \text{analogous vs}\ \Delta_j
Yakınlık katsayısı Φ_ij = Ω-_ij / (Ω+_ij + Ω-_ij) hesapla Shahid §5 Adım 4 denk 14'e göre.

$Φ_{i j} = \frac{Ω_{i j}^{-}}{Ω_{i j}^{+} + Ω_{i j}^{-}}$

LaTeX
\Phi_{ij} = \frac{\Omega^-_{ij}}{\Omega^+_{ij} + \Omega^-_{ij}}
Anti-ideal satırı (U- = Φ-_ℛ değerleri) öne ve ideal satırı (U+ = Φ+_ℛ değerleri) sona ekleyerek (m+2)×n genişletilmiş matris A oluştur, Shahid §5 Adım 5 denk 15.

$A = (_{\begin{matrix} Φ_{j}^{-} \\ Φ_{i j} \\ Φ_{j}^{+} \end{matrix}) (m + 2) \times n}; Φ_{j}^{-} = {\begin{matrix} {min}_{i} Φ_{i j}, & j \in B \\ {max}_{i} Φ_{i j}, & j \in C \end{matrix}; Φ_{j}^{+} symmetric$

LaTeX
A = \begin{pmatrix} \Phi^-_{j} \\ \Phi_{ij} \\ \Phi^+_{j} \end{pmatrix}_{(m+2) \times n};\ \Phi^-_j = \begin{cases} \min_i \Phi_{ij}, & j \in B \\ \max_i \Phi_{ij}, & j \in C \end{cases};\ \Phi^+_j\ \text{symmetric}
Genişletilmiş matris A → A' yön-farkındalıklı normalize et Shahid §5 Adım 6 denk 16-17'ye göre.

$ϑ_{i j} = {\begin{matrix} \frac{Φ_{i j}}{Φ_{j}^{+}}, & j \in B \\ \frac{Φ_{j}^{+}}{Φ_{i j}}, & j \in C \end{matrix}$

LaTeX
\vartheta_{ij} = \begin{cases} \frac{\Phi_{ij}}{\Phi^+_j}, & j \in B \\ \frac{\Phi^+_j}{\Phi_{ij}}, & j \in C \end{cases}
Ağırlıklı normalize genişletilmiş karar matrisi Q_ij = ϑ_ij · w_j hesapla Shahid §5 Adım 7 denk 18'e göre.

$Q_{i j} = ϑ_{i j} \cdot w_{j}$

LaTeX
Q_{ij} = \vartheta_{ij} \cdot w_j
Fayda dereceleri Ψ-_i = ß_i / ß- ve Ψ+_i = ß_i / ß+ hesapla, ß_i = Σ_j Q_ij, ß- = Σ_j Q_{U-,j}, ß+ = Σ_j Q_{U+,j}, Shahid §5 Adım 8 denk 19-20'ye göre.

$β_{i} = \sum_{j = 1}^{n} Q_{i j}; β^{-} = \sum_{j} Q_{U^{-}, j}; β^{+} = \sum_{j} Q_{U^{+}, j}; Ψ_{i}^{-} = β_{i} / β^{-}; Ψ_{i}^{+} = β_{i} / β^{+}$

LaTeX
\beta_i = \sum_{j=1}^{n} Q_{ij};\ \beta^- = \sum_j Q_{U^-, j};\ \beta^+ = \sum_j Q_{U^+, j};\ \Psi^-_i = \beta_i / \beta^-;\ \Psi^+_i = \beta_i / \beta^+
Fayda fonksiyonu Γ(Ψ_i) hesapla kanonik Stević 2020 formülüne göre (Shahid §5 Adım 9 denk 21-23).

$Γ (Ψ_{i}^{+}) = \frac{Ψ_{i}^{-}}{Ψ_{i}^{+} + Ψ_{i}^{-}}; Γ (Ψ_{i}^{-}) = \frac{Ψ_{i}^{+}}{Ψ_{i}^{+} + Ψ_{i}^{-}}; Γ (Ψ_{i}) = \frac{Ψ_{i}^{+} + Ψ_{i}^{-}}{1 + \frac{1 - Γ (Ψ_{i}^{+})}{Γ (Ψ_{i}^{+})} + \frac{1 - Γ (Ψ_{i}^{-})}{Γ (Ψ_{i}^{-})}}$

LaTeX
\Gamma(\Psi^+_i) = \frac{\Psi^-_i}{\Psi^+_i + \Psi^-_i};\ \Gamma(\Psi^-_i) = \frac{\Psi^+_i}{\Psi^+_i + \Psi^-_i};\ \Gamma(\Psi_i) = \frac{\Psi^+_i + \Psi^-_i}{1 + \frac{1 - \Gamma(\Psi^+_i)}{\Gamma(\Psi^+_i)} + \frac{1 - \Gamma(\Psi^-_i)}{\Gamma(\Psi^-_i)}}
Alternatifleri Γ(Ψ_i) fayda fonksiyonuna göre azalan sırala; en büyük en iyidir, Shahid §5 Adım 10.

${rank}_{i} = {argsort}_{desc} (Γ (Ψ_{i}))$

LaTeX
\text{rank}_i = \text{argsort}_\text{desc}(\Gamma(\Psi_i))

Sezgi

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

Sık yapılan hatalar

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.