BF-VIKOR - Bipolar extension of VIKOR

Overview

bf-vikor extends VIKOR to handle Bipolar uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Bipolar Fuzzy Set (BFS: positive membership μ⁺ ∈ [0,1], negative μ⁻ ∈ [-1,0]) algebra. The final scores are defuzzified via score (μ⁺ + μ⁻)/2 before ranking.

Output

utility, lower is better

Data

Bipolar Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Bipolar Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Step 0 - 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).

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

   Wei et al. 2018b - BFN score function Eq.(2); Jana & Pal 2021 - cost normalization Eq.(7)

2. 2

   Step 1 - Determine the best f_j* and worst f_j^- value of every criterion across alternatives, respecting benefit/cost direction.

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

   Opricovic & Tzeng 2004, §2 Eq.(3)

3. 3

   Step 2 - 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).

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

   Opricovic & Tzeng 2004, §2 Eq.(4)

4. 4

   Step 3 - Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v.

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

   Opricovic & Tzeng 2004, §2 Eq.(5)

5. 5

   Step 4 - 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)}.

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

   Opricovic & Tzeng 2004, §2 Eqs.(6)-(7)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Bipolar Fuzzy numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• •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)}.

• Value-space violation: ensure all entries satisfy BFS: μ⁺ ∈ [0,1], μ⁻ ∈ [-1,0] before computation.

• Defuzzification method affects ranking: score (μ⁺ + μ⁻)/2 is the canonical choice but alternatives exist.

• Bipolar Fuzzy Set (BFS) ≠ 2-polar Fuzzy Set. BFS encodes a single attribute as a counter-property pair ⟨μ⁺ ∈ [0,1], μ⁻ ∈ [-1,0]⟩ (positive evidence vs negative evidence). A 2-polar Fuzzy Set encodes TWO INDEPENDENT attributes as a pair (μ₁, μ₂) ∈ [0,1]² - both non-negative, no counter-property semantics. Despite the superficial 'two component' similarity, the value spaces, arithmetic, and aggregation are distinct. For genuine independent-attribute m-polar problems with m=2, use the MPF-* method family (see MPF-ELECTRE-I.json) - not this Bipolar manifest. Reference: Chen et al. 2014 (m-polar foundations) explicitly contrast their construction with Zhang's 1994 bipolar fuzzy sets.

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