CFZN-MARCOS - Complex Fuzzy Z-Number MARCOS Ranking

1. 1. Validate CFZN decision matrix, criterion weights, and direction vector.

   $$T=[T_{ij}{]}_{m\times n},T_{ij}=\{{\sigma }_{ij}{e}^{2\pi i{\tau }_{ij}},{\varpi }_{ij}{e}^{2\pi i{\Re }_{ij}}\};w=(w_1,\dots ,w_n),\sum w_j=1$$

2. 2. Augment with anti-ideal Δ (least favorable per criterion type) and ideal Λ (most favorable per criterion type) per Shahid §5 Step 2.

   $$\text{Benefit: }{\Lambda }_j=\{\max {\sigma }_{ij}{e}^{2\pi i\max {\tau }_{ij}},\max {\varpi }_{ij}{e}^{2\pi i\max {\Re }_{ij}}\},{\Delta }_j=\{\min {\sigma }_{ij}{e}^{2\pi i\min {\tau }_{ij}},\min {\varpi }_{ij}{e}^{2\pi i\min {\Re }_{ij}}\};\text{Cost: swap min}\leftrightarrow \max$$

3. 3. Compute CFZN distance from each cell to ideal Λ (Ω+) and to anti-ideal Δ (Ω-) per Shahid §5 Step 3 eq 12-13.

   $${\Omega }_{ij}^{+}=\frac{1}{2}(({\sigma }_{ij}{\varpi }_{ij})-({\sigma }_{{\Lambda }_j}{\varpi }_{{\Lambda }_j}){)}^2+\frac{1}{4{\pi }^2}(({\tau }_{ij}{\Re }_{ij})-({\tau }_{{\Lambda }_j}{\Re }_{{\Lambda }_j}){)}^2;{\Omega }_{ij}^{-}\text{analogous vs}{\Delta }_j$$

4. 4. Compute closeness coefficient Φ_ij = Ω-_ij / (Ω+_ij + Ω-_ij) per Shahid §5 Step 4 eq 14.

   $${\Phi }_{ij}=\frac{{\Omega }_{ij}^{-}}{{\Omega }_{ij}^{+}+{\Omega }_{ij}^{-}}$$

5. 5. Build (m+2)×n extended matrix A by prepending anti-ideal row (U- = Φ-_ℛ values) and appending ideal row (U+ = Φ+_ℛ values) per Shahid §5 Step 5 eq 15.

   $$A={\left(\begin{array}{c}{\Phi }_j^{-}\\ {\Phi }_{ij}\\ {\Phi }_j^{+}\end{array}\right)}_{(m+2)\times n};{\Phi }_j^{-}=\{\begin{array}{ll}{\min }_i{\Phi }_{ij},& j\in B\\ {\max }_i{\Phi }_{ij},& j\in C\end{array};{\Phi }_j^{+}\text{symmetric}$$

6. 6. Normalize extended matrix A → A' direction-aware per Shahid §5 Step 6 eq 16-17.

   $${\vartheta }_{ij}=\{\begin{array}{ll}\frac{{\Phi }_{ij}}{{\Phi }_j^{+}},& j\in B\\ \frac{{\Phi }_j^{+}}{{\Phi }_{ij}},& j\in C\end{array}$$

7. 7. Compute weighted normalized extended decision matrix Q_ij = ϑ_ij · w_j per Shahid §5 Step 7 eq 18.

   $$Q_{ij}={\vartheta }_{ij}\cdot w_j$$

8. 8. Compute utility degrees Ψ-_i = ß_i / ß- and Ψ+_i = ß_i / ß+ where ß_i = Σ_j Q_ij, ß- = Σ_j Q_{U-,j}, ß+ = Σ_j Q_{U+,j} per Shahid §5 Step 8 eq 19-20.

   $${\beta }_i=\sum _{j=1}^n{Q}_{ij};{\beta }^{-}=\sum _j{Q}_{U^{-},j};{\beta }^{+}=\sum _j{Q}_{U^{+},j};{\Psi }_i^{-}={\beta }_i/{\beta }^{-};{\Psi }_i^{+}={\beta }_i/{\beta }^{+}$$

9. 9. Compute utility function Γ(Ψ_i) per canonical Stević 2020 formula (Shahid §5 Step 9 eq 21-23).

   $$\Gamma ({\Psi }_i^{+})=\frac{{\Psi }_i^{-}}{{\Psi }_i^{+}+{\Psi }_i^{-}};\Gamma ({\Psi }_i^{-})=\frac{{\Psi }_i^{+}}{{\Psi }_i^{+}+{\Psi }_i^{-}};\Gamma ({\Psi }_i)=\frac{{\Psi }_i^{+}+{\Psi }_i^{-}}{1+\frac{1-\Gamma ({\Psi }_i^{+})}{\Gamma ({\Psi }_i^{+})}+\frac{1-\Gamma ({\Psi }_i^{-})}{\Gamma ({\Psi }_i^{-})}}$$

10. 10. Rank alternatives descending by utility function Γ(Ψ_i); largest is best per Shahid §5 Step 10.

    $${\text{rank}}_i={\text{argsort}}_{\text{desc}}(\Gamma ({\Psi }_i))$$