BF-TOPSIS - Bipolar extension of TOPSIS

Overview

bf-topsis extends TOPSIS 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, higher 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 - Vector normalisation: scale each column to unit Euclidean norm.

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_{i=1}^m{x}_{ij}^2}},i=1,\dots ,m;j=1,\dots ,n$$

   Hwang & Yoon 1981, p.130 Eq.(4.1)

3. 3

   Step 2 - Weighted normalised matrix: multiply each column by its weight.

   $$v_{ij}=w_j\cdot r_{ij}$$

   Hwang & Yoon 1981, p.130 Eq.(4.2)

4. 4

   Step 3 - Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction.

   $$A^{+}=\{v_1^{+},\dots ,v_n^{+}\},v_j^{+}=\{\begin{array}{ll}{\max }_i{v}_{ij}& j\in J\\ {\min }_i{v}_{ij}& j\in J^{\prime }\end{array};A^{-}=\{v_1^{-},\dots ,v_n^{-}\},v_j^{-}=\{\begin{array}{ll}{\min }_i{v}_{ij}& j\in J\\ {\max }_i{v}_{ij}& j\in J^{\prime }\end{array}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.3)-(4.4)

5. 5

   Step 4 - Euclidean separation of each alternative from A⁺ and A⁻.

   $$d_i^{+}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{+}{)}^2},d_i^{-}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{-}{)}^2}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.5)-(4.6)

6. 6

   Step 5 - Relative closeness to the ideal solution; rank by C* descending.

   $$C_i^{\ast }=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},0\le C_i^{\ast }\le 1$$

   Hwang & Yoon 1981, p.131 Eq.(4.7)

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

• 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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