Cubic-EDAS: Cubic Pythagorean Fuzzy EDAS (CuP-EDAS)

Overview

Cubic Pythagorean Fuzzy ranking: CuPyFN = ⟨IvPyFN, PyFN⟩ = (⟨[Y⁻,Y⁺],[F⁻,F⁺]⟩,⟨Y,F⟩); Pythagorean constraint (Y⁺)²+(F⁺)² ≤ 1; average-solution EDAS with score-function PDA/NDA. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Cubic Pythagorean Fuzzy ranking: CuPyFN = ⟨IvPyFN, PyFN⟩ = (⟨[Y⁻,Y⁺],[F⁻,F⁺]⟩,⟨Y,F⟩); Pythagorean constraint (Y⁺)²+(F⁺)² ≤ 1; average-solution EDAS with score-function PDA/NDA
• •Preserves pythagorean 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 EDAS; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid CuPyFNs: (Y⁺)²+(F⁺)² ≤ 1 and Y²+F² ≤ 1
• •Assumes: Expert weight vector sums to 1
• •Assumes: S(AV_j) > 0 for all criteria (required for PDA/NDA denominator)

Assumptions to verify

• •Decision matrix entries are valid CuPyFNs: (Y⁺)²+(F⁺)² ≤ 1 and Y²+F² ≤ 1
• •Expert weight vector sums to 1
• •S(AV_j) > 0 for all criteria (required for PDA/NDA denominator)

When not to use

• •Crisp data sufficient: use base EDAS directly
• •Pythagorean uncertainty not justified: consider CUBIC-TOPSIS (CIFN, intuitionistic) or IF-EDAS
• •Weight information completely unknown without expert-defined constraint structure

Edge cases

• •default 1/K): CuPyFWG(h̄_1,...,h̄_K) = (⟨[∏(Y⁻_k)^ωk, ∏(Y⁺_k)^ωk], [√(1−∏(1−(F⁻_k)²)^ωk), √(1−∏(1−(F⁺_k)²)^ωk)]⟩, ⟨∏Y_k^ωk, √(1−∏(1−F_k²)^ωk)⟩). Produces aggregated decision matrix A = (a_sj)_{m×n}.
• •default β=0.5; q ≥ 1, default q=3.

Common pitfalls

• •Hatalı: 'CUBIC-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid CuPyFNs: (Y⁺)²+(F⁺)² ≤ 1 and Y²+F² ≤ 1
• •Hatalı: 'CUBIC-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Expert weight vector sums to 1
• •Hatalı: 'CUBIC-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: S(AV_j) > 0 for all criteria (required for PDA/NDA denominator)
• •Hatalı: CUBIC-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: CUBIC-EDAS'yi 'Pythagorean uncertainty not justified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: CUBIC-EDAS'yi 'Weight information completely unknown without expert-defined constraint structure' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Aggregate K expert decision matrices A^(k) using CuPyFWG operator (Def.5.3) with expert weights ω_k (default 1/K): CuPyFWG(h̄_1,...,h̄_K) = (⟨[∏(Y⁻_k)^ωk, ∏(Y⁺_k)^ωk], [√(1−∏(1−(F⁻_k)²)^ωk), √(1−∏(1−(F⁺_k)²)^ωk)]⟩, ⟨∏Y_k^ωk, √(1−∏(1−F_k²)^ωk)⟩). Produces aggregated decision matrix A = (a_sj)_{m×n}. Formül: CuPyFWG(\bar{h}_1,\ldots,\bar{h}_K) = \Bigl(\bigl\langle\bigl[\prod_{k}(Y^-_k)^{\omega_k},\prod_{k}(Y^+_k)^{\omega_k}\bigr],\bigl[\sqrt{1-\prod_k(1-(F^-_k)^2)^{\omega_k}},\sqrt{1-\prod_k(1-(F^+_k)^2)^{\omega_k}}\bigr]\bigr\rangle, \bigl\langle\prod_k Y_k^{\omega_k},\sqrt{1-\prod_k(1-F_k^2)^{\omega_k}}\bigr\rangle\Bigr) Anchor: Paul-Jana-Pal 2023, Def.5.3 (CuPyFWG), Eq.(Step1)
2. 2.Adım 2 (F2): Step 2: Determine criteria weights ω_j via non-linear optimization. Objective: maximize total generalized distance F(ω*) = Σ_j ω*_j · Σ_{s<t} D^q_stj, where D^q_stj is the generalized distance between alternatives s and t on criterion j. Subject to partial weight constraint system W (expert-specified bounds). Convert ω*_j → ω_j = (ω*_j)² / Σ_j(ω*_j)² if needed, or directly ω_j = (ω*_j)² (when Σ(ω*_j)²=1 by constraint). Parameter β ∈ (0,1], default β=0.5; q ≥ 1, default q=3. Formül: \max F(\omega^*) = \sum_{j=1}^n \omega^{*\beta}_j \sum_{1 \le s < t \le m} D^q(a_{sj}, a_{tj}), \quad \text{s.t. } \omega^*_j \in W,\; \sum_j (\omega^*_j)^2 = 1 Anchor: Paul-Jana-Pal 2023, Eq.(1)-(2), Def.6.1, Box I (Eq.11-13)
3. 3.Adım 3 (F3): Step 3: Compute average solution AV_j for each criterion j using Pythagorean fuzzy arithmetic mean via ⊕ operator: AV_j = (⊕_{s=1}^{m} a_sj) / m. Score function S(h̄) = (1/2)[(1/2)((Y⁻)²+(Y⁺)²−(F⁻)²−(F⁺)²) + Y²−F²], S ∈ [−1,1]. Formül: AV_j = \frac{1}{m}\bigoplus_{s=1}^{m} a_{sj}, \quad S(\bar{h}) = \frac{1}{2}\!\left[\frac{1}{2}\bigl((Y^-)^2+(Y^+)^2-(F^-)^2-(F^+)^2\bigr) + Y^2 - F^2\right] Anchor: Paul-Jana-Pal 2023, Eq.(3), Def.5.5
4. 4.Adım 4 (F4): Step 4: Compute PDA matrix Z = (ζ_sj) and NDA matrix E = (η_sj) using score function S(h̄): For benefit criterion j ∈ B: ζ_sj = max{0, S(a_sj)−S(AV_j)} / S(AV_j); η_sj = max{0, S(AV_j)−S(a_sj)} / S(AV_j). For cost criterion j ∈ L: ζ_sj = max{0, S(AV_j)−S(a_sj)} / S(AV_j); η_sj = max{0, S(a_sj)−S(AV_j)} / S(AV_j). Formül: \zeta_{sj} = \begin{cases}\dfrac{\max(0,\,S(a_{sj})-S(AV_j))}{S(AV_j)} & c_j \in B \\ \dfrac{\max(0,\,S(AV_j)-S(a_{sj}))}{S(AV_j)} & c_j \in L\end{cases}, \quad \eta_{sj} = \begin{cases}\dfrac{\max(0,\,S(AV_j)-S(a_{sj}))}{S(AV_j)} & c_j \in B \\ \dfrac{\max(0,\,S(a_{sj})-S(AV_j))}{S(AV_j)} & c_j \in L\end{cases} Anchor: Paul-Jana-Pal 2023, Eqs.(4)-(5)
5. 5.Adım 5 (F5): Step 5: Compute weighted sums of PDA and NDA across criteria: S⁺_s = Σ_j ω_j · ζ_sj; S⁻_s = Σ_j ω_j · η_sj. Formül: S^+_s = \sum_{j=1}^n \omega_j \zeta_{sj}, \quad S^-_s = \sum_{j=1}^n \omega_j \eta_{sj} Anchor: Paul-Jana-Pal 2023, Eqs.(6)-(7)
6. 6.Adım 6 (F6): Step 6: Normalize S⁺ and S⁻ by their maxima: NS⁺_s = S⁺_s / max_s{S⁺_s}; NS⁻_s = 1 − S⁻_s / max_s{S⁻_s}. Formül: NS^+_s = \frac{S^+_s}{\max_s\{S^+_s\}}, \quad NS^-_s = 1 - \frac{S^-_s}{\max_s\{S^-_s\}} Anchor: Paul-Jana-Pal 2023, Eqs.(8)-(9)
7. 7.Adım 7 (F7): Step 7: Compute appraisal score AS_s = (1/2)(NS⁺_s + NS⁻_s), rank alternatives in descending order. Best alternative has highest AS. Formül: AS_s = \tfrac{1}{2}(NS^+_s + NS^-_s), \quad 0 \le AS_s \le 1; \quad S_i \succ S_j \iff AS_i > AS_j Anchor: Paul-Jana-Pal 2023, Eq.(10)

Commonly paired with

• •internal_optimization + CUBIC-EDAS (canonical)

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