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

Overview

F-LMAW returns crisp criterion weights on the simplex (Σ w_j = 1, w_j ≥ 0) from linguistic expert assessments. Use it when (i) only verbal expert judgements are available, (ii) you want a logarithmic transform that compresses extreme priorities, and (iii) you have at least one expert and at least two criteria. With multiple experts, the Bonferroni mean (p=q=1 default) aggregates expert priorities before the logarithmic transform.

Output

Weight, higher is better

Data

Fuzzy (TFN), linguistic expert assessments

Weights

Derived internally, no weight source needed

Size

0+ alternatives, 3-12 criteria works best

Used for

Criterion-importance elicitation, expert-driven group decision making, linguistic evaluation under uncertainty

How it works

1. 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.

   $${\tilde{\gamma }}_{hj}=\mathrm{LinguisticToTFN}({\ell }_{hj})$$

   Yüksel Aydın 2025, Bölüm 9, Tablo 1

2. 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).

   $${\tilde{\eta }}_{hj}={\tilde{\gamma }}_{hj}\oslash {\gamma }_{AIP}=(l_{hj}/{\gamma }_u,m_{hj}/{\gamma }_m,u_{hj}/{\gamma }_l)$$

   Yüksel Aydın 2025, Bölüm 9, Eşitlik (6)

3. 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.

   $${\tilde{\eta }}_j={\mathrm{BM}}^{p,q}({\tilde{\eta }}_{1j},\dots ,{\tilde{\eta }}_{kj})={\left(\frac{1}{k(k-1)}\sum _{\begin{array}{c}{h}_1,h_2=1\\ h_1\ne h_2\end{array}}^k{\tilde{\eta }}_{h_{1}j}^p\otimes {\tilde{\eta }}_{h_{2}j}^q\right)}^{1/(p+q)}$$

   Yüksel Aydın 2025, Bölüm 9, Eşitlik (7)

4. 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.)

   $${\tilde{w}}_j=\left(\frac{\ln l_j}{\ln \Pi },\frac{\ln m_j}{\ln \Pi },\frac{\ln u_j}{\ln \Pi }\right),\Pi =\prod _{j=1}^n{m}_j$$

   Yüksel Aydın 2025, Bölüm 9, Eşitlik (8)

5. 5

   Defuzzify w̃_j by the graded mean: w_j = (l_j^w + 4 m_j^w + u_j^w)/6.

   $$w_j^{\ast }=\frac{l_j^w+4m_j^w+u_j^w}{6}$$

   Yüksel Aydın 2025, Bölüm 9, Eşitlik (9)

6. 6

   Final weights are renormalised on the simplex: w_j = w_j^* / Σ_j w_j^*. The output Σ_j w_j = 1.

   $$w_j=\frac{w_j^{\ast }}{{\sum }_{j^{\prime }=1}^n{w}_{j^{\prime }}^{\ast }}$$

   Yüksel Aydın 2025, Bölüm 9 (simplex closure)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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

• γ_AIP lower bound ≥ any l_ij of the linguistic scale → log argument ≤ 0 or fuzzy division ill-defined. Fix γ_AIP=(0.5,0.5,0.5) (book convention) and a scale starting at (1,1,1).

• Single expert (k=1): Bonferroni operator with p=q=1 is undefined as written (1/(k(k-1))). By convention F-LMAW collapses to η̃_j = η̃_1j (identity). The verifier applies this convention.

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