B-WENSLO: Fuzzy WEight deNomination based on Slope coefficient (triangular fuzzy extension)

Overview

Weight_Subjective (linguistic-TFN expert weighting; envelope/slope ratio on TFN accumulation polyline). Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Weight_Subjective (linguistic-TFN expert weighting; envelope/slope ratio on TFN accumulation polyline)
• •Preserves triangular_fuzzy uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Assumes: Experts use the same 9-level linguistic scale consistently
• •Assumes: Sufficient inter-expert disagreement per criterion (no constant columns)
• •Assumes: Expert ordering documented (algorithm is order-dependent)

Assumptions to verify

• •Experts use the same 9-level linguistic scale consistently
• •Sufficient inter-expert disagreement per criterion (no constant columns)
• •Expert ordering documented (algorithm is order-dependent)

When not to use

• •All experts agree on identical labels for a criterion → Δz_j = 0 (algorithm undefined)
• •Crisp numeric data available → use crisp WENSLO instead
• •Pairwise comparisons preferred → use BWM/AHP/FUCOM

Edge cases

• •See F.steps and D.parameters for B-WENSLO-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'B-WENSLO bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Experts use the same 9-level linguistic scale consistently
• •Hatalı: 'B-WENSLO bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Sufficient inter-expert disagreement per criterion (no constant columns)
• •Hatalı: 'B-WENSLO bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Expert ordering documented (algorithm is order-dependent)
• •Hatalı: B-WENSLO'yi 'All experts agree on identical labels for a criterion → Δz_j = 0 (algorithm undefined)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: B-WENSLO'yi 'Crisp numeric data available → use crisp WENSLO instead' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: B-WENSLO'yi 'Pairwise comparisons preferred → use BWM/AHP/FUCOM' 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 ζ̃_hj = (l_hj, m_hj, u_hj) using the 9-level scale. Formül: \tilde{\zeta}_{hj} = \mathrm{LinguisticToTFN}(\ell_{hj}) Anchor: Demir & Ulusoy 2024 Eq.(7); book Bölüm 2
2. 2.Adım 2 (F2): Step 2: Fuzzy column-sum normalisation. The column TFN sum is S̃_j = (Σ l_hj, Σ m_hj, Σ u_hj); each cell is divided by S̃_j using TFN division with the reverse order: z̃_hj = (l_hj / Σu_·j, m_hj / Σm_·j, u_hj / Σl_·j). Direction-agnostic. Formül: \tilde{z}_{hj} = \left(\dfrac{l_{hj}}{\sum_{h'} u_{h'j}},\;\dfrac{m_{hj}}{\sum_{h'} m_{h'j}},\;\dfrac{u_{hj}}{\sum_{h'} l_{h'j}}\right) Anchor: Demir & Ulusoy 2024 Eq.(8); book Bölüm 2 Tablo 5
3. 3.Adım 3 (F3): Step 3: Graded-mean defuzzification of normalised TFN cells. z_hj = (l + 4m + u) / 6 produces a crisp normalised matrix on which the envelope/slope ratio is computed. Formül: z_{hj} = \dfrac{l_{hj}^{(z)} + 4\, m_{hj}^{(z)} + u_{hj}^{(z)}}{6} Anchor: Demir & Ulusoy 2024 Eq.(9); book Bölüm 2
4. 4.Adım 4 (F4): Step 4: Sturges class interval Δz_j on the defuzzified normalised column. Δz_j = (max_h z_hj − min_h z_hj) / (1 + 3.322·log10(k)) where k is the number of experts. Formül: \Delta z_j = \dfrac{\max_{h} z_{hj} - \min_{h} z_{hj}}{1 + 3.322\,\log_{10}(k)} Anchor: Demir & Ulusoy 2024 Eq.(10); book Bölüm 2
5. 5.Adım 5 (F5): Step 5: Criterion slope tan(φ_j) = Σ_h z_hj / ((k-1)·Δz_j). Identical structure to crisp WENSLO Eq.(7) with experts replacing alternatives. Formül: \tan\varphi_j = \dfrac{\sum_{h=1}^{k} z_{hj}}{(k-1)\cdot\Delta z_j} Anchor: Demir & Ulusoy 2024 Eq.(11); book Bölüm 2; mirror of Pamucar 2024 Eq.(7)
6. 6.Adım 6 (F6): Step 6: Criterion envelope E_j as the sum of partial Euclidean distances between successive defuzzified normalised values along the criterion column, with constant horizontal step Δz_j. Formül: E_j = \sum_{h=1}^{k-1} \sqrt{(z_{h+1,j} - z_{h,j})^{2} + (\Delta z_j)^{2}} Anchor: Demir & Ulusoy 2024 Eq.(12); book Bölüm 2; mirror of Pamucar 2024 Eq.(8)
7. 7.Adım 7 (F7): Step 7: Envelope/slope ratio q_j = E_j / tan(φ_j). Larger q_j ⇒ richer information variability ⇒ higher weight. Formül: q_j = \dfrac{E_j}{\tan\varphi_j} Anchor: Demir & Ulusoy 2024; mirror of Pamucar 2024 Eq.(9)
8. 8.Adım 8 (F8): Step 8: Additive normalisation of q_j yields the criterion weights. Formül: w_j = \dfrac{q_j}{\sum_{j'=1}^{n} q_{j'}} Anchor: Demir & Ulusoy 2024 Eq.(13); book Bölüm 2 Tablo 7; mirror of Pamucar 2024 Eq.(10)

Commonly paired with

• •B-WENSLO + FF-TOPSIS (medium)
• •B-WENSLO + FF-MARCOS (medium)
• •B-WENSLO + FF-EDAS (medium)

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