Fuzzy CoCoSo - Fuzzy extension of COCOSO

1. 1. Extract TFN decision matrix x̃ij=(xijα,xijβ,xijγ) and fuzzy weight vector w̃j=(wjα,wjβ,wjγ).

   $$\tilde{X}=[{\tilde{x}}_{ij}]=(x_{ij}^{\alpha },x_{ij}^{\beta },x_{ij}^{\gamma });{\tilde{w}}_j=(w_j^{\alpha },w_j^{\beta },w_j^{\gamma })$$

2. 2. Min-max normalization per component. Benefit: r̃ijk=(xijk-min_i(xijα))/(max_i(xijγ)-min_i(xijα)). Cost: r̃ijα=(max_i(xijγ)-xijγ)/denom, r̃ijβ=(max_i(xijγ)-xijβ)/denom, r̃ijγ=(max_i(xijγ)-xijα)/denom.

   $${\tilde{r}}_{ij}^{+}=\frac{x_{ij}^k-{\min }_i{x}_{ij}^{\alpha }}{{\max }_i{x}_{ij}^{\gamma }-{\min }_i{x}_{ij}^{\alpha }},{\tilde{r}}_{ij}^{-}=\frac{{\max }_i{x}_{ij}^{\gamma }-x_{ij}^k}{{\max }_i{x}_{ij}^{\gamma }-{\min }_i{x}_{ij}^{\alpha }},k\in \{\alpha ,\beta ,\gamma \}$$

3. 3. Weighted sum score S̃i=Σj w̃j⊗r̃ij (TFN component-wise multiplication then sum). Power score Pi=Σj(COA(r̃ij))^(COA(w̃j)) using centroid scalar approximation.

   $${\tilde{S}}_i=\sum _j{\tilde{w}}_j\otimes {\tilde{r}}_{ij};P_i=\sum _j{\left(\frac{r_{ij}^{\alpha }+r_{ij}^{\beta }+r_{ij}^{\gamma }}{3}\right)}^{\frac{w_j^{\alpha }+w_j^{\beta }+w_j^{\gamma }}{3}}$$

4. 4. Three compromise appraisal scores using centroid Si=COA(S̃i). Ka_i=(Si+Pi)/Σ(Si+Pi); Kb_i=Si/min(Si)+Pi/min(Pi); Kc_i=(λSi+(1-λ)Pi)/(λmax(Si)+(1-λ)max(Pi)), λ=0.5.

   $$K_{a,i}=\frac{S_i+P_i}{{\sum }_i(S_i+P_i)};K_{b,i}=\frac{S_i}{\min S}+\frac{P_i}{\min P};K_{c,i}=\frac{\lambda S_i+(1-\lambda )P_i}{\lambda \max S+(1-\lambda )\max P}$$

5. 5. Final compromise score Ki combining geometric mean and arithmetic mean of (Ka, Kb, Kc). Rank descending.

   $$K_i=(K_{a,i}\cdot K_{b,i}\cdot K_{c,i}{)}^{1/3}+\frac{1}{3}(K_{a,i}+K_{b,i}+K_{c,i});\text{rank descending}$$