Fuzzy MULTIMOORA - Fuzzy extension of MULTIMOORA

1. 1. Extract TFN decision matrix x̃ij=(xijα,xijβ,xijγ) and scalar weights wj (defuzzify fuzzy weights immediately via centroid if provided as TFNs).

   $$\tilde{X}=[{\tilde{x}}_{ij}];w_j=\frac{w_j^{\alpha }+w_j^{\beta }+w_j^{\gamma }}{3}\text{ (defuzzified weight for sub-method scalars)}$$

2. 2. Normalize each TFN coordinate by its own Euclidean column-coordinate norm, as in Eq. (13).

   $$D_{jk}=\sqrt{\sum _i(x_{ij}^k{)}^2};{\tilde{r}}_{ij}=\left(\frac{x_{ij}^l}{D_{jl}},\frac{x_{ij}^m}{D_{jm}},\frac{x_{ij}^u}{D_{ju}}\right)$$

3. 3. Ratio System uses TFN addition/subtraction then COA. Reference Point uses component-wise fuzzy maxima/minima and weighted vertex distance; lower z is better.

   $${\tilde{y}}_i=\underset{j\in {\Omega }_{+}}{⨁}{w}_j{\tilde{r}}_{ij}\ominus \underset{j\in {\Omega }_{-}}{⨁}{w}_j{\tilde{r}}_{ij};z_i=\underset{j}{\max }{w}_j{d}_v({\tilde{r}}_j^{\ast },{\tilde{r}}_{ij}),d_v=\sqrt{\frac{1}{3}\sum _{k\in \{l,m,u\}}(r_{jk}^{\ast }-r_{ijk}{)}^2}$$

4. 4. Weighted fuzzy Full Multiplicative Form on raw positive TFNs; fuzzy division reverses denominator endpoints, then COA produces crisp U. Rank descending.

   $${\tilde{A}}_i=\underset{j\in {\Omega }_{+}}{⨂}{\tilde{x}}_{ij}^{w_j},{\tilde{B}}_i=\underset{j\in {\Omega }_{-}}{⨂}{\tilde{x}}_{ij}^{w_j};{\tilde{U}}_i=(A_i^l/B_i^u,A_i^m/B_i^m,A_i^u/B_i^l),U_i=\text{COA}({\tilde{U}}_i)$$

5. 5. Aggregate the three equally important ordinal rankings by majority dominance, transitivity, equability, and equal-rank treatment of cycles. Rank sums are not used.

   $$a{\succ }_{D}b⟺\#\{q\in \{RS,RP,FMF\}:r_q(a)<r_q(b)\}\ge 2;\text{cycles form one equivalence class}$$