LPF-CRITIC-EDAS - Linguistic Pythagorean Fuzzy EDAS with CRITIC weighting (Akram-Ramzan-Deveci 2023)

1. 1. Form per-DM linguistic Pythagorean fuzzy decision matrices D^(t) = [d^(t)_pq], where d^(t)_pq = (I_ψ_pq^t, I_ζ_pq^t), ψ²+ζ² ≤ τ², for each decision-maker t = 1,…,T.

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

2. 2. Aggregate across DMs via LPFHWA (Hamacher weighted average, Akram 2023 Theorem 3.1) with DM weights Γ = (γ_1,…,γ_T): α̃_pq = LPFHWA_Γ(d^(1)_pq,…,d^(T)_pq). At κ=1 (algebraic) this simplifies to ψ̃² = τ²·(1 − ∏_t(1 − (ψ^t/τ)²)^{γ_t}) and ζ̃ = τ·∏_t(ζ^t/τ)^{γ_t}.

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

3. 3. CRITIC weights ϖ_r for criteria from the aggregated LPF matrix. (i) Score S(α̃_pq) = √((τ²+ψ̃²−ζ̃²)/2) (Eq.4.3). (ii) Direction-aware standardization E_pr ∈ [0,1] (Eq.4.4): benefit (BA) E_pr = (S_pr − S^−_r)/(S^+_r − S^−_r); cost (CA) E_pr = (S^+_r − S_pr)/(S^+_r − S^−_r). (iii) Correlation λ_{rk} between columns r and k (Eq.4.5). (iv) σ_r = std-dev of column r (Eq.4.6). (v) Information content 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.

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

4. 4. Compute the LPF Average Solution δ̃ = (δ̃_1,…,δ̃_n) by per-criterion LPFAWA with equal alternative weights 1/m: δ̃_q = LPFAWA_{1/m}(α̃_{1q},…,α̃_{mq}).

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

5. 5. Positive/Negative Distance from AVS using LPFN subtraction (Eq.2.8) and scalar multiplication (Eq.2.11) with direction-aware sign. For benefit criteria: PDA_pq = (α̃_pq ⊖ δ̃_q)/S(δ̃_q) if α̃_pq ≥ δ̃_q else 0̃; NDA_pq symmetric. For cost criteria the roles swap. Division by S(δ̃_q) is the LPFN scalar mult by 1/S(δ̃_q) (Eq.2.11).

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

6. 6. Weighted positive/negative LPF distances per alternative via CRITIC weights ϖ_r: WPDA_p = ⊕_q (ϖ_q · PDA_pq); WNDA_p = ⊕_q (ϖ_q · NDA_pq), where ⊕ is LPFN Hamacher addition (Eq.2.9) and ϖ·α is LPFN scalar multiplication (Eq.2.11). At κ=1 these are the algebraic LPFAWA aggregates.

   $${\text{WPDA}}_p=\underset{q=1}{\overset{n}{⨁}}({\varpi }_q\cdot {\text{PDA}}_{pq});{\text{WNDA}}_p=\underset{q=1}{\overset{n}{⨁}}({\varpi }_q\cdot {\text{NDA}}_{pq})$$

7. 7. Defuzzify WPDA/WNDA via score (Eq.4.3): 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.

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

8. 8. Appraisal Score S_q^A = (NSP_p + NSN_p)/2 ∈ [0,1].

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

9. 9. Rank alternatives in descending order of S^A_p (largest = best).

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