BF-EDAS: Bipolar extension of EDAS

Overview

Bipolar outranking/ranking: Bipolar Fuzzy Set (BFS: positive membership μ⁺ ∈ [0,1], negative μ⁻ ∈ [-1,0]). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Bipolar outranking/ranking: Bipolar Fuzzy Set (BFS: positive membership μ⁺ ∈ [0,1], negative μ⁻ ∈ [-1,0])
• •Preserves bipolar_fuzzy uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Bipolar Fuzzy numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

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

When not to use

• •Crisp data sufficient: use base EDAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for BF-EDAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'BF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Bipolar Fuzzy numbers/tuples
• •Hatalı: 'BF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'BF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: BF-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BF-EDAS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: M = [B_{\rho\tau}^\xi]_{\sigma\times\eta} = \bigl(\mu^{+\xi}_{\rho\tau},\,\nu^{-\xi}_{\rho\tau}\bigr)_{\sigma\times\eta},\quad \mu^+ \in [0,1],\; \nu^- \in [-1,0] Anchor: Jana & Pal 2021, p.4 Eq.(6)
2. 2.Adım 2 (F2): Step 2: Normalize the matrix for criterion type. Benefit criteria are kept unchanged; cost criteria are complemented: (1 − μ⁺, |ν⁻| − 1). Formül: M_{\rho\tau} = \begin{cases} B_{\rho\tau} = \langle\mu^+_{\rho\tau},\,\nu^-_{\rho\tau}\rangle & \text{if } G_\tau \text{ benefit} \\ 1 - B_{\rho\tau} = \langle 1 - \mu^+_{\rho\tau},\;|\nu^-_{\rho\tau}| - 1\rangle & \text{if } G_\tau \text{ cost} \end{cases} Anchor: Jana & Pal 2021, p.5 Eq.(7)
3. 3.Adım 3 (F3): Step 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. Formül: \text{BFWA}_\psi(B_1,\ldots,B_\eta) = \Bigl(1 - \prod_{\xi=1}^{k}(1-\mu^+_\xi)^{\theta_\xi},\;-\prod_{\xi=1}^{k}|\nu^-_\xi|^{\theta_\xi}\Bigr) Anchor: Jana & Pal 2021, p.3 Eq.(4)
4. 4.Adım 4 (F4): Step 4: Compute Average Solution (AV) per criterion using BFWA geometric mean across alternatives. AV_τ is the geometric-mean BFN of column τ. Formül: AV_\tau = \Bigl(1 - \prod_{\rho=1}^{\sigma}(1-\mu^{\prime+}_{\rho\tau})^{1/\sigma},\;-\prod_{\rho=1}^{\sigma}|\nu^{\prime-}_{\rho\tau}|^{1/\sigma}\Bigr) Anchor: Jana & Pal 2021, p.5 Eq.(11)
5. 5.Adım 5 (F5): Step 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. Formül: \Delta(B) = \dfrac{1 + \mu^+ + \nu^-}{2} \in [0,1] \\[6pt] PDA_{\rho\tau} = \dfrac{\max\bigl(0,\,\Delta(B^\prime_{\rho\tau}) - \Delta(AV_\tau)\bigr)}{\Delta(AV_\tau)} \\[6pt] NDA_{\rho\tau} = \dfrac{\max\bigl(0,\,\Delta(AV_\tau) - \Delta(B^\prime_{\rho\tau})\bigr)}{\Delta(AV_\tau)} Anchor: Jana & Pal 2021, p.5 Eqs.(2)(14)(15)
6. 6.Adım 6 (F6): Step 6: Compute weighted sums SP_ρ and SN_ρ using criterion weights ψ_τ. Formül: SP_\rho = \sum_{\tau=1}^{\eta} \psi_\tau\, PDA_{\rho\tau}, \qquad SN_\rho = \sum_{\tau=1}^{\eta} \psi_\tau\, NDA_{\rho\tau} Anchor: Jana & Pal 2021, p.5 Eq.(16)
7. 7.Adım 7 (F7): Step 7: Normalize SP and SN by their respective maxima. Formül: NSP_\rho = \dfrac{SP_\rho}{\max_k SP_k}, \qquad NSN_\rho = 1 - \dfrac{SN_\rho}{\max_k SN_k} Anchor: Jana & Pal 2021, p.6 Eq.(17)
8. 8.Adım 8 (F8): Step 8: Compute Appraisal Score AS_ρ as average of NSP and NSN. Formül: AS_\rho = \dfrac{1}{2}\bigl(NSP_\rho + NSN_\rho\bigr), \quad 0 \le AS_\rho \le 1 Anchor: Jana & Pal 2021, p.6 Eq.(18)
9. 9.Adım 9 (F9): Step 9: Rank alternatives in descending order of AS_ρ. Higher AS is better. Formül: B_\alpha \succ B_\beta \iff AS_\alpha > AS_\beta Anchor: Jana & Pal 2021, p.6 Step 9

Commonly paired with

• •n_a + BF-EDAS (common)

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