BF-MARCOS - Bipolar extension of MARCOS

Overview

bf-marcos extends MARCOS 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 - Form extended decision matrix with AAI (anti-ideal) and AI (ideal) rows.

   $$X^{\ast }=[X^T;AAI;AI{]}^T,AA{I}_j=\min /\max ,A{I}_j=\max /\min$$

   Stević 2020, p.6 Eq.(1)

3. 3

   Step 2 - Normalise vs ideal AI per criterion direction.

   $$n_{ij}=\{\begin{array}{ll}{x}_{ij}/x_{AI,j}& j\in J^{+}\\ x_{AI,j}/x_{ij}& j\in J^{-}\end{array}$$

   Stević 2020, p.6 Eq.(2)

4. 4

   Step 3 - Weighted normalised matrix v_ij = w_j · n_ij.

   $$v_{ij}=w_j{n}_{ij}$$

   Stević 2020, p.6 Eq.(3)

5. 5

   Step 4 - Sum each alternative's weighted matrix row: S_i = Σ v_ij.

   $$S_i=\sum _{j=1}^n{v}_{ij}$$

   Stević 2020, p.7 Eq.(4)

6. 6

   Step 5 - Utility ratios vs ideal K+ and anti-ideal K− rows.

   $$K_i^{+}=\frac{S_i}{S_{AI}},K_i^{-}=\frac{S_i}{S_{AAI}}$$

   Stević 2020, p.7 Eqs.(5)-(6)

7. 7

   Step 6 - Utility functions f(K+_i) and f(K−_i) of the utility ratios.

   $$f(K_i^{+})=\frac{K_i^{-}}{K_i^{+}+K_i^{-}},f(K_i^{-})=\frac{K_i^{+}}{K_i^{+}+K_i^{-}}$$

   Stević 2020, p.7 Eqs.(7)-(8)

8. 8

   Step 7 - Final utility f(K_i) and descending ranking.

   $$f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+\frac{1-f(K_i^{+})}{f(K_i^{+})}+\frac{1-f(K_i^{-})}{f(K_i^{-})}}$$

   Stević 2020, p.7 Eq.(9)

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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