Cubic-EDAS - Kübik Pisagor Bulanık EDAS (CuP-EDAS)

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

Adım 1 — K uzman karar matrisini CuPyFWG operatörü ile birleştir (Tanım 5.3): uzman ağırlıkları ω_k (varsayılan 1/K). Birleşik karar matrisi A = (a_sj) üretir.

$C u P y F W G ({\bar{h}}_{1}, \dots, {\bar{h}}_{K}) = (\langle [\prod_{k} (Y_{k}^{-})^{ω_{k}}, \prod_{k} (Y_{k}^{+})^{ω_{k}}], [\sqrt{1 - \prod_{k} (1 - (F_{k}^{-})^{2})^{ω_{k}}}, \sqrt{1 - \prod_{k} (1 - (F_{k}^{+})^{2})^{ω_{k}}}] \rangle, \langle \prod_{k} Y_{k}^{ω_{k}}, \sqrt{1 - \prod_{k} (1 - F_{k}^{2})^{ω_{k}}} \rangle)$

LaTeX
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)
Adım 2 — Doğrusal olmayan optimizasyon ile kriter ağırlıkları ω_j hesaplanır. Amaç: toplam mesafe maksimize edilir. Kısıt sistemi W uzmanlar tarafından belirlenir. ω_j = (ω*_j)².

$max F (ω^{*}) = \sum_{j = 1}^{n} ω_{j}^{* β} \sum_{1 \leq s < t \leq m} D^{q} (a_{s j}, a_{t j}), s.t. ω_{j}^{*} \in W, \sum_{j} (ω_{j}^{*})^{2} = 1$

LaTeX
\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
Adım 3 — Her kriter j için ortalama çözüm AV_j hesaplanır. Skor fonksiyonu: S(h̄) = (1/2)[(1/2)((Y⁻)²+(Y⁺)²−(F⁻)²−(F⁺)²) + Y²−F²].

$A V_{j} = \frac{1}{m} ⨁_{s = 1}^{m} a_{s j}, S (\bar{h}) = \frac{1}{2} [\frac{1}{2} ((Y^{-})^{2} + (Y^{+})^{2} - (F^{-})^{2} - (F^{+})^{2}) + Y^{2} - F^{2}]$

LaTeX
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]
Adım 4 — PDA matrisi Z = (ζ_sj) ve NDA matrisi E = (η_sj) hesaplanır. Skor fonksiyonu S(h̄) kullanılır: fayda kriteri için ζ_sj = max{0, S(a_sj)−S(AV_j)} / S(AV_j).

$ζ_{s j} = {\begin{matrix} \frac{max (0, S (a_{s j}) - S (A V_{j}))}{S (A V_{j})} & c_{j} \in B \\ \frac{max (0, S (A V_{j}) - S (a_{s j}))}{S (A V_{j})} & c_{j} \in L \end{matrix}, η_{s j} = {\begin{matrix} \frac{max (0, S (A V_{j}) - S (a_{s j}))}{S (A V_{j})} & c_{j} \in B \\ \frac{max (0, S (a_{s j}) - S (A V_{j}))}{S (A V_{j})} & c_{j} \in L \end{matrix}$

LaTeX
\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}
Adım 5 — PDA ve NDA'nın ağırlıklı toplamları: S⁺_s = Σ_j ω_j · ζ_sj; S⁻_s = Σ_j ω_j · η_sj.

$S_{s}^{+} = \sum_{j = 1}^{n} ω_{j} ζ_{s j}, S_{s}^{-} = \sum_{j = 1}^{n} ω_{j} η_{s j}$

LaTeX
S^+_s = \sum_{j=1}^n \omega_j \zeta_{sj}, \quad S^-_s = \sum_{j=1}^n \omega_j \eta_{sj}
Adım 6 — S⁺ ve S⁻'yi maksimumlarına göre normalize et: NS⁺_s = S⁺_s / max; NS⁻_s = 1 − S⁻_s / max.

$N S_{s}^{+} = \frac{S_{s}^{+}}{{max}_{s} {S_{s}^{+}}}, N S_{s}^{-} = 1 - \frac{S_{s}^{-}}{{max}_{s} {S_{s}^{-}}}$

LaTeX
NS^+_s = \frac{S^+_s}{\max_s\{S^+_s\}}, \quad NS^-_s = 1 - \frac{S^-_s}{\max_s\{S^-_s\}}
Adım 7 — Değerlendirme skoru AS_s = (1/2)(NS⁺_s + NS⁻_s) hesaplanır, azalan sırada sırala.

$A S_{s} = \frac{1}{2} (N S_{s}^{+} + N S_{s}^{-}), 0 \leq A S_{s} \leq 1; S_{i} ≻ S_{j} ⟺ A S_{i} > A S_{j}$

LaTeX
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

Sezgi

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).

Sonucu okuma: CuP-EDAS extends EDAS into the Cubic Pythagorean Fuzzy environment. Each decision-maker assessment is a CuPyFN = (⟨[Y⁻,Y⁺],[F⁻,F⁺]⟩,⟨Y,F⟩) with Pythagorean constraint (Y⁺)²+(F⁺)² ≤ 1 (stronger than intuitionistic). Expert opinions are aggregated via CuPyFWG. Distance-based optimization determines criteria weights. PDA/NDA use score function comparison vs average solution (avoids ideal solution bias). Higher appraisal score AS = (NS⁺+NS⁻)/2 is better.

Varsayımlar

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)

Ne zaman kullanılmaz

Classical 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

Sınırlılıklar

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)

Sık yapılan hatalar

Değer uzayı ihlali: her CuPyFN girişinin (Y⁺)²+(F⁺)² ≤ 1 Pisagor kısıtını sağladığından emin olun.
Ağırlık optimizasyonu doğrusal değildir: LINGO veya eşdeğeri gerekir; ağırlıklar doğrudan ωj = (ω*j)² olarak da verilebilir.
S(AV_j) sıfır veya negatif olabilir: PDA/NDA'da sıfıra bölme için epsilon koruması kullanın.

Hesap adımları ve dayanakları

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}.

Dayanak: Paul-Jana-Pal 2023, Def.5.3 (CuPyFWG), Eq.(Step1)
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.

Dayanak: Paul-Jana-Pal 2023, Eq.(1)-(2), Def.6.1, Box I (Eq.11-13)
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].

Dayanak: Paul-Jana-Pal 2023, Eq.(3), Def.5.5
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).

Dayanak: Paul-Jana-Pal 2023, Eqs.(4)-(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.

Dayanak: Paul-Jana-Pal 2023, Eqs.(6)-(7)
Normalize S⁺ and S⁻ by their maxima: NS⁺_s = S⁺_s / max_s{S⁺_s}; NS⁻_s = 1 − S⁻_s / max_s{S⁻_s}.

Dayanak: Paul-Jana-Pal 2023, Eqs.(8)-(9)
Compute appraisal score AS_s = (1/2)(NS⁺_s + NS⁻_s), rank alternatives in descending order. Best alternative has highest AS.

Dayanak: Paul-Jana-Pal 2023, Eq.(10)