BF-WASPAS - Bipolar extension of WASPAS

Overview

bf-waspas extends WASPAS 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

   Construct the bipolar fuzzy decision matrix. Each cell B_{ij} = (μ⁺_{ij}, ν⁻_{ij}) is a BFN with μ⁺ ∈ [0,1] and ν⁻ ∈ [-1,0].

   $$X=[B_{ij}{]}_{m\times n}=[({\mu }_{ij}^{+},{\nu }_{ij}^{-}){]}_{m\times n}$$

   Wei et al. 2018b - BFN definition; Zavadskas et al. 2012 - WASPAS structure

2. 2

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

   $$B_{ij}^{\ast }=\{\begin{array}{ll}({\mu }_{ij}^{+},{\nu }_{ij}^{-})& j\in J^{+}\text{ (benefit)}\\ (1-{\mu }_{ij}^{+},|{\nu }_{ij}^{-}|-1)& j\in J^{-}\text{ (cost)}\end{array}$$

   Jana & Pal 2021, p.5 Eq.(7) - BF cost normalization

3. 3

   Weighted Sum Model (WSM) component using BFWA operator. BFWA yields a BFN for each alternative across all criteria.

   $$Q_i^{(1)}={\text{BFWA}}_{\mathbf{w}}(B_{i1}^{\ast },\dots ,B_{in}^{\ast })=(1-\prod _{j=1}^n(1-{\mu }_{ij}^{+}{)}^{w_j},-\prod _{j=1}^n|{\nu }_{ij}^{-}{|}^{w_j})$$

   Wei et al. 2018b, Eq.(4) BFWA; analogue of Zavadskas 2012 Eq.(2)

4. 4

   Weighted Product Model (WPM) component using BFWG operator. BFWG uses BFN geometric aggregation.

   $$Q_i^{(2)}={\text{BFWG}}_{\mathbf{w}}(B_{i1}^{\ast },\dots ,B_{in}^{\ast })=(\prod _{j=1}^n({\mu }_{ij}^{+}{)}^{w_j},-1+\prod _{j=1}^n(1+{\nu }_{ij}^{-}{)}^{w_j})$$

   Wei et al. 2018b, Eq.(5) BFWG; analogue of Zavadskas 2012 Eq.(3)

5. 5

   Combine WSM and WPM components via BFN arithmetic. Scale each by λ and (1−λ), then add using BFN ⊕. Convert final BFN to crisp score Δ(Q_i)=(1+μ⁺+ν⁻)/2. Rank descending.

   $$\begin{gathered} \lambda B=(1-(1-{\mu }^{+}{)}^{\lambda },-|{\nu }^{-}{|}^{\lambda }) \\ {B}_1\oplus B_2=({\mu }_1^{+}+{\mu }_2^{+}-{\mu }_1^{+}{\mu }_2^{+},-|{\nu }_1^{-}||{\nu }_2^{-}|) \\ {Q}_i=\lambda Q_i^{(1)}\oplus (1-\lambda )Q_i^{(2)} \\ \Delta (Q_i)=\frac{1+{\mu }_{Q_i}^{+}+{\nu }_{Q_i}^{-}}{2} \\ {B}_{\alpha }\succ B_{\beta }⟺\Delta (Q_{\alpha })>\Delta (Q_{\beta }) \end{gathered}$$

   Wei et al. 2018b - BFN scalar mult Eq.(9), BFN addition Eq.(7), score Eq.(2); Zavadskas 2012 - WASPAS Eq.(5) structure

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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