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

Overview

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

Strengths

• •Method-specific: Linguistic Pythagorean fuzzy ranking: LPFN (I_ψ, I_ζ) with ψ²+ζ² ≤ τ²
• •Preserves linguistic_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 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)

Assumptions to verify

• •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)

When not to use

• •Crisp 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

Edge cases

• •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).

Common pitfalls

• •Hatalı: 'LPF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All cells satisfy LPFN constraint ψ²+ζ² ≤ τ²
• •Hatalı: 'LPF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic term set granularity τ
• •Hatalı: 'LPF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: DM weights γ_t form a simplex (Σγ_t = 1, γ_t ≥ 0)
• •Hatalı: 'LPF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Hamacher parameter κ is chosen explicitly (default κ=1 = algebraic)
• •Hatalı: LPF-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: LPF-EDAS'yi 'Decision-makers disagree on linguistic granularity τ' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: LPF-EDAS'yi 'Number of criteria n < 2' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: 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 Anchor: Akram 2023 Eq.(4.1); Garg 2018 Definition 2.4 (LPFN definition)
2. 2.Adım 2 (F2): Step 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}. Formül: \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} Anchor: Akram 2023 Theorem 3.1, Eq.(3.2); Garg 2018 (algebraic limit at κ=1); Hamacher 1978 (t-norm family)
3. 3.Adım 3 (F3): Step 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. Formül: 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 Anchor: Akram 2023 Eqs.(4.3)-(4.8); Diakoulaki et al. 1995 (CRITIC original)
4. 4.Adım 4 (F4): Step 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}). Formül: \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) Anchor: Akram 2023 Eq.(4.9); Garg 2018 LPFAWA (κ=1)
5. 5.Adım 5 (F5): Step 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). Formül: \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} Anchor: Akram 2023 Eqs.(4.10)-(4.13); Eqs.(2.8)+(2.11) for LPFN ⊖ and scalar mult
6. 6.Adım 6 (F6): Step 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. Formül: \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}) Anchor: Akram 2023 Eqs.(4.14)-(4.15); Eqs.(2.9)+(2.11) for Hamacher ⊕ and scalar mult
7. 7.Adım 7 (F7): Step 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. Formül: \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} Anchor: Akram 2023 Eqs.(4.16)-(4.17); Keshavarz Ghorabaee 2015 Eqs.(8)-(9)
8. 8.Adım 8 (F8): Step 8: Appraisal Score S_q^A = (NSP_p + NSN_p)/2 ∈ [0,1]. Formül: S^A_p = \tfrac{1}{2}(\text{NSP}_p + \text{NSN}_p) Anchor: Akram 2023 Eq.(4.18); Keshavarz Ghorabaee 2015 Eq.(10)
9. 9.Adım 9 (F9): Step 9: Rank alternatives in descending order of S^A_p (largest = best). Formül: \text{rank desc by } S^A_p Anchor: Akram 2023 §4 Step 9; Keshavarz Ghorabaee 2015 Step 10

Commonly paired with

• •CRITIC (endogenous) + LPF-EDAS (canonical (Akram 2023))

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