SFZN-CRITIC - Spherical Fuzzy Z-Number CRITIC Weighting

1. 1. Validate SFZN decision matrix (or per-expert tensor) and direction vector.

   $$L_o^k=[L_{o,ij}^k{]}_{m\times n},k=1,\dots ,K$$

2. 2. If per-expert matrices given, aggregate via SFŽNWA operator using expert weights Ω̆ (Niu Def II.6).

   $$SF\check{Z}NWA(L_1,\dots ,L_K)=\left(\sqrt{1-\prod _{k=1}^K(1-{ϵ}_k^2{)}^{{\Omega }_k}},\sqrt{1-\prod (1-{\tau }_{ϵ,k}^2{)}^{{\Omega }_k}},\prod {\nu }_k^{{\Omega }_k},\prod {\tau }_{\nu ,k}^{{\Omega }_k},\prod {\partial }_k^{{\Omega }_k},\prod {\tau }_{\partial ,k}^{{\Omega }_k}\right)$$

3. 3. Compute score ℑ(L_{o,ij}) for each cell of the collective SFZN matrix (Niu Def II.7).

   $$\Im (L_{o,ij})=\frac{2+({ϵ}_{ij}\cdot {\tau }_{ϵ,ij})-({\nu }_{ij}\cdot {\tau }_{\nu ,ij})-({\partial }_{ij}\cdot {\tau }_{\partial ,ij})}{3}$$

4. 4. Direction-aware linear min-max normalization of scores: benefit uses (ℑ − ℑ⁻)/(ℑ⁺ − ℑ⁻); cost uses (ℑ⁺ − ℑ)/(ℑ⁺ − ℑ⁻).

   $$\widetilde{\Im (L_{o,j})}=\{\begin{array}{ll}\frac{\Im (L_{o,j})-\Im (L_{o,j}{)}^{-}}{\Im (L_{o,j}{)}^{+}-\Im (L_{o,j}{)}^{-}},& j\in R_b\\ \frac{\Im (L_{o,j}{)}^{+}-\Im (L_{o,j})}{\Im (L_{o,j}{)}^{+}-\Im (L_{o,j}{)}^{-}},& j\in R_c\end{array}$$

5. 5. Compute standard deviation §_j of normalized scores per criterion across alternatives.

   $${\S }_j=\sqrt{\frac{{\sum }_{i=1}^m{\left(\widetilde{\Im (L_{o,ij})}-\overline{\widetilde{\Im (L_{o,j})}}\right)}^2}{m}},\overline{\widetilde{\Im (L_{o,j})}}=\frac{{\sum }_i\widetilde{\Im (L_{o,ij})}}{m}$$

6. 6. Compute Pearson correlation Υ_jl between every pair of criteria using normalized scores.

   $${{\rm Y}}_{jl}=\frac{{\sum }_{i=1}^m\left(\widetilde{\Im (L_{o,ij})}-\overline{\widetilde{\Im (L_{o,j})}}\right)\left(\widetilde{\Im (L_{o,il})}-\overline{\widetilde{\Im (L_{o,l})}}\right)}{\sqrt{{\sum }_i(\widetilde{\Im (L_{o,ij})}-\overline{\widetilde{\Im (L_{o,j})}}{)}^2{\sum }_i(\widetilde{\Im (L_{o,il})}-\overline{\widetilde{\Im (L_{o,l})}}{)}^2}}$$

7. 7. Compute information amount Γ_j per criterion by summing (1 − Υ_jl) across all other criteria, modulated by §_j.

   $${\Gamma }_j={\S }_j\sum _{l=1}^n(1-{{\rm Y}}_{jl})$$

8. 8. Normalize information amounts to sum=1 to obtain final criterion weights Ω̆_j.

   $${\breve{\Omega }}_j=\frac{{\Gamma }_j}{{\sum }_{j=1}^n{\Gamma }_j}$$