LPF-CRITIC-EDAS - CRITIC ağırlıklı Dilsel Pisagor Bulanık EDAS (Akram-Ramzan-Deveci 2023)

Formül adımları

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

1. Adım 1 — Her karar verici t = 1,…,T için d^(t)_pq = (I_ψ_pq^t, I_ζ_pq^t), ψ²+ζ² ≤ τ² olan dilsel Pisagor bulanık karar matrisleri D^(t) = [d^(t)_pq] oluşturulur.

   $$D^{(t)}=[d_{pq}^{(t)}{]}_{m\times n},d_{pq}^{(t)}=(I_{{\psi }_{pq}^t},I_{{\zeta }_{pq}^t}),({\psi }_{pq}^t{)}^2+({\zeta }_{pq}^t{)}^2\le {\tau }^2$$
   LaTeX

   D^{(t)} = [d^{(t)}_{pq}]_{m \times n}, \; d^{(t)}_{pq} = (I_{\psi^t_{pq}}, I_{\zeta^t_{pq}}), \; (\psi^t_{pq})^2 + (\zeta^t_{pq})^2 \le \tau^2

2. Adım 2 — DM ağırlıkları Γ = (γ_1,…,γ_T) ile LPFHWA (Hamacher ağırlıklı ortalama, Akram 2023 Teorem 3.1) üzerinden karar vericiler arası birleştirme: α̃_pq = LPFHWA_Γ(d^(1)_pq,…,d^(T)_pq). κ=1'de (cebirsel) sadeleşir.

   $${\tilde{\alpha }}_{pq}=LPFHW{A}_{\Gamma }(d_{pq}^{(1)},\dots ,d_{pq}^{(T)});\text{(κ=1)}{\tilde{\psi }}_{pq}=\tau \sqrt{1-{\prod }_{t=1}^T(1-({\psi }_{pq}^t/\tau {)}^2{)}^{{\gamma }_t}},{\tilde{\zeta }}_{pq}=\tau \prod _{t=1}^T({\zeta }_{pq}^t/\tau {)}^{{\gamma }_t}$$
   LaTeX

   \tilde{\alpha}_{pq} = LPFHWA_\Gamma\big(d^{(1)}_{pq},\ldots,d^{(T)}_{pq}\big); \;\;\text{(κ=1)}\;\; \tilde{\psi}_{pq} = \tau\sqrt{1 - \prod_{t=1}^{T}\big(1 - (\psi^t_{pq}/\tau)^2\big)^{\gamma_t}}, \;\; \tilde{\zeta}_{pq} = \tau \prod_{t=1}^{T}(\zeta^t_{pq}/\tau)^{\gamma_t}

3. Adım 3 — Birleştirilmiş LPF matrisinden CRITIC ile kriter ağırlıkları ϖ_r. (i) Skor S(α̃_pq) = √((τ²+ψ̃²−ζ̃²)/2) (Eq.4.3). (ii) Yön-farkındalı standartlaştırma E_pr ∈ [0,1] (Eq.4.4): fayda (BA) E_pr = (S_pr − S^−_r)/(S^+_r − S^−_r); maliyet (CA) E_pr = (S^+_r − S_pr)/(S^+_r − S^−_r). (iii) Korelasyon λ_{rk} (Eq.4.5). (iv) σ_r std-sapma (Eq.4.6). (v) Bilgi içeriği γ_r = σ_r · Σ_k(1 − λ_{rk}) (Eq.4.7). (vi) Normalize ϖ_r = γ_r / Σ_k γ_k (Eq.4.8); Σ ϖ_r = 1.

   $$S(\tilde{\alpha })=\sqrt{({\tau }^2+{\tilde{\psi }}^2-{\tilde{\zeta }}^2)/2};E_{pr}=\{\begin{array}{cc}(S_{pr}-S_r^{-})/(S_r^{+}-S_r^{-})\hfill & r\in BA\hfill \\ (S_r^{+}-S_{pr})/(S_r^{+}-S_r^{-})\hfill & r\in CA\hfill \end{array};{\lambda }_{rk}=\frac{{\sum }_p(E_{pr}-{\overline{E}}_r)(E_{pk}-{\overline{E}}_k)}{\sqrt{{\sum }_p(E_{pr}-{\overline{E}}_r{)}^2{\sum }_p(E_{pk}-{\overline{E}}_k{)}^2}};{\sigma }_r=\sqrt{\frac{1}{m}{\sum }_p(E_{pr}-{\overline{E}}_r{)}^2};{\gamma }_r={\sigma }_r\sum _{k=1}^n(1-{\lambda }_{rk});{\varpi }_r={\gamma }_r/\sum _{k=1}^n{\gamma }_k$$
   LaTeX

   S(\tilde\alpha) = \sqrt{(\tau^2 + \tilde\psi^2 - \tilde\zeta^2)/2}; \;\; E_{pr} = \begin{cases} (S_{pr} - S^-_r)/(S^+_r - S^-_r) & r \in BA \\ (S^+_r - S_{pr})/(S^+_r - S^-_r) & r \in CA \end{cases}; \;\; \lambda_{rk} = \frac{\sum_p (E_{pr}-\bar E_r)(E_{pk}-\bar E_k)}{\sqrt{\sum_p(E_{pr}-\bar E_r)^2 \sum_p(E_{pk}-\bar E_k)^2}}; \;\; \sigma_r = \sqrt{\tfrac{1}{m}\sum_p(E_{pr}-\bar E_r)^2}; \;\; \gamma_r = \sigma_r \sum_{k=1}^{n}(1-\lambda_{rk}); \;\; \varpi_r = \gamma_r / \sum_{k=1}^{n}\gamma_k

4. Adım 4 — Kriter bazında eşit alternatif ağırlığı 1/m ile LPFAWA kullanılarak LPF Ortalama Çözüm δ̃ = (δ̃_1,…,δ̃_n) hesaplanır: δ̃_q = LPFAWA_{1/m}(α̃_{1q},…,α̃_{mq}).

   $${\tilde{\delta }}_q=(\tau \sqrt{1-{\prod }_{p=1}^m(1-({\tilde{\psi }}_{pq}/\tau {)}^2{)}^{1/m}},\tau \prod _{p=1}^m({\tilde{\zeta }}_{pq}/\tau {)}^{1/m})$$
   LaTeX

   \tilde\delta_q = \Big(\tau\sqrt{1 - \prod_{p=1}^{m}\big(1-(\tilde\psi_{pq}/\tau)^2\big)^{1/m}},\; \tau\prod_{p=1}^{m}(\tilde\zeta_{pq}/\tau)^{1/m}\Big)

5. Adım 5 — LPFN çıkarma (Eq.2.8) ve skaler çarpma (Eq.2.11) ile AVS'den pozitif/negatif uzaklık. Fayda kriterleri için PDA_pq = (α̃_pq ⊖ δ̃_q)/S(δ̃_q) eğer α̃_pq ≥ δ̃_q aksi halde 0̃; NDA simetrik. Maliyet kriterlerinde roller değişir.

   $${\text{PDA}}_{pq}=\{\begin{array}{cc}({\tilde{\alpha }}_{pq}\ominus {\tilde{\delta }}_q)/S({\tilde{\delta }}_q)\hfill & \text{if }{\tilde{\alpha }}_{pq}⪰{\tilde{\delta }}_q\text{ (benefit)}\hfill \\ ({\tilde{\delta }}_q\ominus {\tilde{\alpha }}_{pq})/S({\tilde{\delta }}_q)\hfill & \text{if }{\tilde{\alpha }}_{pq}⪯{\tilde{\delta }}_q\text{ (cost)}\hfill \\ \tilde{0}\hfill & \text{otherwise}\hfill \end{array};{\text{NDA}}_{pq}\text{ symmetric with reversed inequality}$$
   LaTeX

   \text{PDA}_{pq} = \begin{cases} (\tilde\alpha_{pq} \ominus \tilde\delta_q) / S(\tilde\delta_q) & \text{if } \tilde\alpha_{pq} \succeq \tilde\delta_q \text{ (benefit)} \\ (\tilde\delta_q \ominus \tilde\alpha_{pq}) / S(\tilde\delta_q) & \text{if } \tilde\alpha_{pq} \preceq \tilde\delta_q \text{ (cost)} \\ \tilde 0 & \text{otherwise} \end{cases}; \;\; \text{NDA}_{pq} \text{ symmetric with reversed inequality}

6. Adım 6 — CRITIC ağırlıkları ϖ_r ile alternatif bazında ağırlıklı pozitif/negatif LPF uzaklıklar: WPDA_p = ⊕_q (ϖ_q · PDA_pq); WNDA_p = ⊕_q (ϖ_q · NDA_pq). κ=1'de cebirsel LPFAWA toplamlarına denk gelir.

   $${\text{WPDA}}_p={⨁}_{q=1}^n({\varpi }_q·{\text{PDA}}_{pq});{\text{WNDA}}_p={⨁}_{q=1}^n({\varpi }_q·{\text{NDA}}_{pq})$$
   LaTeX

   \text{WPDA}_p = \bigoplus_{q=1}^{n} (\varpi_q \cdot \text{PDA}_{pq}); \;\; \text{WNDA}_p = \bigoplus_{q=1}^{n} (\varpi_q \cdot \text{NDA}_{pq})

7. Adım 7 — WPDA/WNDA skor (Eq.4.3) ile defuzzify: wsp_p = S(WPDA_p), wsn_p = S(WNDA_p). Normalize: NSP_p = wsp_p / max_p wsp_p; NSN_p = 1 − wsn_p / max_p wsn_p.

   $${\text{wsp}}_p=S({\text{WPDA}}_p);{\text{wsn}}_p=S({\text{WNDA}}_p);{\text{NSP}}_p=\frac{{\text{wsp}}_p}{{max}_p{\text{wsp}}_p};{\text{NSN}}_p=1-\frac{{\text{wsn}}_p}{{max}_p{\text{wsn}}_p}$$
   LaTeX

   \text{wsp}_p = S(\text{WPDA}_p); \;\; \text{wsn}_p = S(\text{WNDA}_p); \;\; \text{NSP}_p = \frac{\text{wsp}_p}{\max_p \text{wsp}_p}; \;\; \text{NSN}_p = 1 - \frac{\text{wsn}_p}{\max_p \text{wsn}_p}

8. Adım 8 — Değerlendirme Skoru S_q^A = (NSP_p + NSN_p)/2 ∈ [0,1].

   $$S_p^A=\frac{1}{2}({\text{NSP}}_p+{\text{NSN}}_p)$$
   LaTeX

   S^A_p = \tfrac{1}{2}(\text{NSP}_p + \text{NSN}_p)

9. Adım 9 — Alternatifler S^A_p azalan sıraya göre sıralanır (en büyük = en iyi).

   $$\text{rank desc by }{S}_p^A$$
   LaTeX

   \text{rank desc by } S^A_p

Sezgi

Linguistic Pythagorean fuzzy ranking - LPFN (I_ψ, I_ζ) with ψ²+ζ² ≤ τ². Output typically utility (higher value = preferred).

Sonucu okuma: LPF-EDAS uses linguistic Pythagorean fuzzy numbers (LPFN: (I_ψ, I_ζ) with ψ²+ζ² ≤ τ²) to encode expert linguistic judgments at granularity τ. Group judgments are aggregated via LPFHWA (Hamacher weighted average; algebraic at κ=1). CRITIC derives objective criterion weights from inter-criterion correlation and standard deviation. EDAS structure (PDA/NDA distances from average solution) produces the final appraisal score. The score function S(α) = √((τ²+ψ²−ζ²)/2) defuzzifies LPFNs throughout.

Varsayımlar

• All cells satisfy LPFN constraint ψ²+ζ² ≤ τ²
• All decision-makers use the same linguistic term set granularity τ
• DM weights γ_t form a simplex (Σγ_t = 1, γ_t ≥ 0)
• Hamacher parameter κ is chosen explicitly (default κ=1 = algebraic)

Ne zaman kullanılmaz

• Classical data sufficient - use base EDAS directly (avoid unnecessary uncertainty layer)
• Decision-makers disagree on linguistic granularity τ - convert to common τ first
• Number of criteria n < 2 - CRITIC weight derivation degenerates

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp EDAS base; cf. Keshavarz Ghorabaee 2015)
• Assumes: All cells satisfy LPFN constraint ψ²+ζ² ≤ τ²
• Assumes: All decision-makers use the same linguistic term set granularity τ
• Assumes: DM weights γ_t form a simplex (Σγ_t = 1, γ_t ≥ 0)
• Assumes: Hamacher parameter κ is chosen explicitly (default κ=1 = algebraic)

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce her LPFN hücresi için ψ²+ζ² ≤ τ² doğrulayın.
• Granülerlik kayması: tüm DM'ler aynı τ'yu kullanmalıdır; karışık-τ matrisler LPFN uyumlu değildir.
• Hamacher κ seçimi: κ=1 (cebirsel, varsayılan) en yaygın; κ=2 ve üstü duyarlılık varyantları - raporlarda κ belirtilmeli.
• CRITIC dejenerasyonu: skor matrisinin herhangi bir sütunu sabitse σ_r = 0 ve ϖ_r = 0 - bu kriter bilgi üretmez.