BF-VIKOR - Bipolar extension of VIKOR

1. 1. Construct bipolar fuzzy decision matrix B = [(μ⁺_ij, ν⁻_ij)]. Normalize cost criteria via BFN complement: (1−μ⁺, |ν⁻|−1). Convert all BFNs to crisp scores using score function Δ(B)=(1+μ⁺+ν⁻)/2 ∈ [0,1]. All subsequent steps operate on the resulting crisp score matrix x_ij = Δ(B*_ij).

   $$\begin{gathered} \Delta (B)=\frac{1+{\mu }^{+}+{\nu }^{-}}{2}\in [0,1] \\ {B}_{ij}^{\ast }=\{\begin{array}{ll}({\mu }_{ij}^{+},{\nu }_{ij}^{-})& j\in J^{+}\\ (1-{\mu }_{ij}^{+},|{\nu }_{ij}^{-}|-1)& j\in J^{-}\end{array} \\ {x}_{ij}=\Delta (B_{ij}^{\ast }) \end{gathered}$$

2. 2. Determine the best f_j* and worst f_j^- value of every criterion across alternatives, respecting benefit/cost direction.

   $$f_j^{\ast }=\{\begin{array}{ll}{\max }_i{x}_{ij}& j\in J\\ {\min }_i{x}_{ij}& j\in J^{\prime }\end{array},f_j^{-}=\{\begin{array}{ll}{\min }_i{x}_{ij}& j\in J\\ {\max }_i{x}_{ij}& j\in J^{\prime }\end{array}$$

3. 3. Compute the group utility S_i and the individual regret R_i. S aggregates weighted normalised regret (L_1-metric); R is the maximum weighted regret (L_∞-metric).

   $$S_i=\sum _{j=1}^n{w}_j\frac{f_j^{\ast }-x_{ij}}{f_j^{\ast }-f_j^{-}},R_i=\underset{j}{\max }\left[w_j\frac{f_j^{\ast }-x_{ij}}{f_j^{\ast }-f_j^{-}}\right]$$

4. 4. Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v.

   $$Q_i=v\frac{S_i-S^{\ast }}{S^{-}-S^{\ast }}+(1-v)\frac{R_i-R^{\ast }}{R^{-}-R^{\ast }},S^{\ast }=\underset{i}{\min }{S}_i,S^{-}=\underset{i}{\max }{S}_i,R^{\ast }=\underset{i}{\min }{R}_i,R^{-}=\underset{i}{\max }{R}_i$$

5. 5. Propose A^(1) (the lowest-Q alternative) as compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

   $$DQ=\frac{1}{m-1};C1:Q(A^{(2)})-Q(A^{(1)})\ge DQ;C2:A^{(1)}\text{ is best in }S\text{ or in }R$$