F-LMAW: Fuzzy Logarithm Methodology of Additive Weights (TFN)

Overview

Triangular-fuzzy linguistic expert weighting with Bonferroni aggregation; logarithmic transform around an absolute anti-ideal point. Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Triangular-fuzzy linguistic expert weighting with Bonferroni aggregation; logarithmic transform around an absolute anti-ideal point
• •Preserves triangular_fuzzy uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Assumes: Linguistic scale anchored at (1,1,1) (logarithm well-defined)
• •Assumes: γ_AIP strictly below the scale (priority TFNs ≥ 2 componentwise under (0.5,0.5,0.5) AIP)
• •Assumes: Experts assess criteria independently

Assumptions to verify

• •Linguistic scale anchored at (1,1,1) (logarithm well-defined)
• •γ_AIP strictly below the scale (priority TFNs ≥ 2 componentwise under (0.5,0.5,0.5) AIP)
• •Experts assess criteria independently

When not to use

• •Only crisp numerical importance is available → use crisp LMAW or any classical weighting
• •Interactions between criteria expected → use ANP / DEMATEL

Edge cases

• •degenerates to η̃_j = η̃_1j.
• •ties: w̃_j = ln_{∏m}(η̃_j) componentwise as ( ln(l_j)/ln(Π), ln(m_j)/ln(Π), ln(u_j)/ln(Π) ) with Π = ∏_{j=1..n} m_j. (Equivalently per Yüksel Aydın 2025 Eq. (8): w̃_j = (l_j, m_j, u_j) is transformed

Common pitfalls

• •Hatalı: 'F-LMAW bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Linguistic scale anchored at (1,1,1) (logarithm well-defined)
• •Hatalı: 'F-LMAW bu varsayımı kontrol etmeden uygulamak'. Doğrusu: γ_AIP strictly below the scale (priority TFNs ≥ 2 componentwise under (0.5,0.5,0.5) AIP)
• •Hatalı: 'F-LMAW bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Experts assess criteria independently
• •Hatalı: F-LMAW'yi 'Only crisp numerical importance is available → use crisp LMAW or any classical weighting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: F-LMAW'yi 'Interactions between criteria expected → use ANP / DEMATEL' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Each expert e_h (h=1..k) assigns a linguistic label to every criterion j; the label is mapped to its TFN γ̃_hj = (l_hj, m_hj, u_hj) using Tablo 1. Formül: \tilde{\gamma}_{hj} = \mathrm{LinguisticToTFN}(\ell_{hj}) Anchor: Yüksel Aydın 2025, Bölüm 9, Tablo 1
2. 2.Adım 2 (F2): Step 2: Compute the priority TFN η̃_hj of each cell with respect to the absolute anti-ideal point γ_AIP, using fuzzy division on each component: η̃_hj = γ̃_hj / γ_AIP = (l_hj/γ_u, m_hj/γ_m, u_hj/γ_l). With γ_AIP=(0.5,0.5,0.5) this reduces componentwise to (2·l_hj, 2·m_hj, 2·u_hj). Formül: \tilde{\eta}_{hj} = \tilde{\gamma}_{hj} \oslash \gamma_{AIP} = \big(l_{hj}/\gamma_u,\; m_{hj}/\gamma_m,\; u_{hj}/\gamma_l\big) Anchor: Yüksel Aydın 2025, Bölüm 9, Eşitlik (6)
3. 3.Adım 3 (F3): Step 3: Per criterion j, aggregate the k expert priority TFNs with the fuzzy Bonferroni mean BM^{p,q} (defaults p=q=1) into η̃_j = (l_j, m_j, u_j). With k=1 the operator degenerates to η̃_j = η̃_1j. Formül: \tilde{\eta}_{j} = \mathrm{BM}^{p,q}\big(\tilde{\eta}_{1j},\ldots,\tilde{\eta}_{kj}\big) = \left(\dfrac{1}{k(k-1)}\sum_{\substack{h_1,h_2=1\\ h_1\neq h_2}}^{k} \tilde{\eta}_{h_1 j}^{p}\otimes \tilde{\eta}_{h_2 j}^{q}\right)^{1/(p+q)} Anchor: Yüksel Aydın 2025, Bölüm 9, Eşitlik (7)
4. 4.Adım 4 (F4): Step 4: Compute the fuzzy logarithmic transform of every η̃_j against the product of m-component priorities: w̃_j = ln_{∏m}(η̃_j) componentwise as ( ln(l_j)/ln(Π), ln(m_j)/ln(Π), ln(u_j)/ln(Π) ) with Π = ∏_{j=1..n} m_j. (Equivalently per Yüksel Aydın 2025 Eq. (8): w̃_j = (l_j, m_j, u_j) is transformed through log-ratios into the final fuzzy weight before defuzzification.) Formül: \tilde{w}_{j} = \left(\dfrac{\ln l_{j}}{\ln \Pi},\; \dfrac{\ln m_{j}}{\ln \Pi},\; \dfrac{\ln u_{j}}{\ln \Pi}\right),\quad \Pi = \prod_{j=1}^{n} m_{j} Anchor: Yüksel Aydın 2025, Bölüm 9, Eşitlik (8)
5. 5.Adım 5 (F5): Step 5: Defuzzify w̃_j by the graded mean: w_j = (l_j^w + 4 m_j^w + u_j^w)/6. Formül: w_{j}^{*} = \dfrac{l_{j}^{w} + 4 m_{j}^{w} + u_{j}^{w}}{6} Anchor: Yüksel Aydın 2025, Bölüm 9, Eşitlik (9)
6. 6.Adım 6 (F6): Step 6: Final weights are renormalised on the simplex: w_j = w_j^* / Σ_j w_j^*. The output Σ_j w_j = 1. Formül: w_{j} = \dfrac{w_{j}^{*}}{\sum_{j'=1}^{n} w_{j'}^{*}} Anchor: Yüksel Aydın 2025, Bölüm 9 (simplex closure)

Commonly paired with

• •F-LMAW + FUZZY-TOPSIS (high)
• •F-LMAW + FUZZY-MARCOS (high)
• •F-LMAW + FUZZY-WASPAS (medium)

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