BF-EDAS - Bipolar extension of EDAS

Overview

BF-EDAS extends EDAS to Bipolar Fuzzy Numbers (BFN: μ⁺ ∈ [0,1], ν⁻ ∈ [-1,0]). Input each cell as a pair (μ⁺, ν⁻). Cost criteria are automatically complemented in Step 2. The algorithm computes a BFN average solution via geometric-mean BFWA (Step 4), converts BFNs to crisp scores via Δ(B)=(1+μ⁺+ν⁻)/2 (Step 5), then applies standard EDAS PDA/NDA logic on those scores (Steps 5-9). Higher Appraisal Score (AS) is better. Supports multiple experts via BFWA aggregation (Step 3).

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

   Construct the bipolar fuzzy evaluation matrix. Each cell B_{ρτ}^ξ = (μ⁺_{ρτ}^ξ, ν⁻_{ρτ}^ξ) is a BFN provided by expert γ_ξ for alternative B_ρ under criterion G_τ.

   $$M=[B_{\rho \tau }^{\xi }{]}_{\sigma \times \eta }=({\mu }_{\rho \tau }^{+\xi },{\nu }_{\rho \tau }^{-\xi }{)}_{\sigma \times \eta },{\mu }^{+}\in [0,1],{\nu }^{-}\in [-1,0]$$

   Jana & Pal 2021, p.4 Eq.(6)

2. 2

   Normalize the matrix for criterion type. Benefit criteria are kept unchanged; cost criteria are complemented: (1 − μ⁺, |ν⁻| − 1).

   $$M_{\rho \tau }=\{\begin{array}{ll}{B}_{\rho \tau }=⟨{\mu }_{\rho \tau }^{+},{\nu }_{\rho \tau }^{-}⟩& \text{if }{G}_{\tau }\text{ benefit}\\ 1-B_{\rho \tau }=⟨1-{\mu }_{\rho \tau }^{+},|{\nu }_{\rho \tau }^{-}|-1⟩& \text{if }{G}_{\tau }\text{ cost}\end{array}$$

   Jana & Pal 2021, p.5 Eq.(7)

3. 3

   Aggregate expert opinions using the BFWA operator with expert weights θ_ξ (Σθ_ξ=1). For a single expert (ξ=1) this step is trivial: B_{ρτ} = B_{ρτ}^1.

   $${\text{BFWA}}_{\psi }(B_1,\dots ,B_{\eta })=(1-\prod _{\xi =1}^k(1-{\mu }_{\xi }^{+}{)}^{{\theta }_{\xi }},-\prod _{\xi =1}^k|{\nu }_{\xi }^{-}{|}^{{\theta }_{\xi }})$$

   Jana & Pal 2021, p.3 Eq.(4)

4. 4

   Compute Average Solution (AV) per criterion using BFWA geometric mean across alternatives. AV_τ is the geometric-mean BFN of column τ.

   $$A{V}_{\tau }=(1-\prod _{\rho =1}^{\sigma }(1-{\mu }_{\rho \tau }^{\prime +}{)}^{1/\sigma },-\prod _{\rho =1}^{\sigma }|{\nu }_{\rho \tau }^{\prime -}{|}^{1/\sigma })$$

   Jana & Pal 2021, p.5 Eq.(11)

5. 5

   Compute PDA and NDA using the score function Δ(B) = (1 + μ⁺ + ν⁻)/2. Criterion direction is already encoded in the normalized matrix from F2.

   $$\begin{gathered} \Delta (B)=\frac{1+{\mu }^{+}+{\nu }^{-}}{2}\in [0,1] \\ PD{A}_{\rho \tau }=\frac{\max (0,\Delta (B_{\rho \tau }^{\prime })-\Delta (A{V}_{\tau }))}{\Delta (A{V}_{\tau })} \\ ND{A}_{\rho \tau }=\frac{\max (0,\Delta (A{V}_{\tau })-\Delta (B_{\rho \tau }^{\prime }))}{\Delta (A{V}_{\tau })} \end{gathered}$$

   Jana & Pal 2021, p.5 Eqs.(2)(14)(15)

6. 6

   Compute weighted sums SP_ρ and SN_ρ using criterion weights ψ_τ.

   $$S{P}_{\rho }=\sum _{\tau =1}^{\eta }{\psi }_{\tau }PD{A}_{\rho \tau },S{N}_{\rho }=\sum _{\tau =1}^{\eta }{\psi }_{\tau }ND{A}_{\rho \tau }$$

   Jana & Pal 2021, p.5 Eq.(16)

7. 7

   Normalize SP and SN by their respective maxima.

   $$NS{P}_{\rho }=\frac{S{P}_{\rho }}{{\max }_{k}S{P}_k},NS{N}_{\rho }=1-\frac{S{N}_{\rho }}{{\max }_{k}S{N}_k}$$

   Jana & Pal 2021, p.6 Eq.(17)

8. 8

   Compute Appraisal Score AS_ρ as average of NSP and NSN.

   $$A{S}_{\rho }=\frac{1}{2}(NS{P}_{\rho }+NS{N}_{\rho }),0\le A{S}_{\rho }\le 1$$

   Jana & Pal 2021, p.6 Eq.(18)

9. 9

   Rank alternatives in descending order of AS_ρ. Higher AS is better.

   $$B_{\alpha }\succ B_{\beta }⟺A{S}_{\alpha }>A{S}_{\beta }$$

   Jana & Pal 2021, p.6 Step 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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