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

Overview

B-WENSLO assigns higher weight to criteria whose linguistic expert assessments produce, after fuzzy normalisation and graded-mean defuzzification, an accumulation polyline with a longer envelope relative to its slope. Criteria on which experts disagree the most (high spread in defuzzified normalised values) gain weight; criteria where experts converge on a single label get low weight.

Output

Weight, higher is better

Data

Fuzzy (TFN), linguistic complete

Weights

Derived internally, no weight source needed

Size

0+ alternatives, 5-15 criteria works best

Used for

Expert-based subjective weighting, digital banking criteria (Demir & Ulusoy 2024), any MAGDM with linguistic criterion importance

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 ζ̃_hj = (l_hj, m_hj, u_hj) using the 9-level scale.

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

   Demir & Ulusoy 2024 Eq.(7); book Bölüm 2

2. 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.

   $${\tilde{z}}_{hj}=\left(\frac{l_{hj}}{{\sum }_{h^{\prime }}{u}_{h^{\prime }j}},\frac{m_{hj}}{{\sum }_{h^{\prime }}{m}_{h^{\prime }j}},\frac{u_{hj}}{{\sum }_{h^{\prime }}{l}_{h^{\prime }j}}\right)$$

   Demir & Ulusoy 2024 Eq.(8); book Bölüm 2 Tablo 5

3. 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.

   $$z_{hj}=\frac{l_{hj}^{(z)}+4m_{hj}^{(z)}+u_{hj}^{(z)}}{6}$$

   Demir & Ulusoy 2024 Eq.(9); book Bölüm 2

4. 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.

   $$\Delta z_j=\frac{{\max }_h{z}_{hj}-{\min }_h{z}_{hj}}{1+3.322{\log }_{10}(k)}$$

   Demir & Ulusoy 2024 Eq.(10); book Bölüm 2

5. 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.

   $$\tan {\phi }_j=\frac{{\sum }_{h=1}^k{z}_{hj}}{(k-1)\cdot \Delta z_j}$$

   Demir & Ulusoy 2024 Eq.(11); book Bölüm 2; mirror of Pamucar 2024 Eq.(7)

6. 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.

   $$E_j=\sum _{h=1}^{k-1}\sqrt{(z_{h+1,j}-z_{h,j}{)}^2+(\Delta z_j{)}^2}$$

   Demir & Ulusoy 2024 Eq.(12); book Bölüm 2; mirror of Pamucar 2024 Eq.(8)

7. 7

   Envelope/slope ratio q_j = E_j / tan(φ_j). Larger q_j ⇒ richer information variability ⇒ higher weight.

   $$q_j=\frac{E_j}{\tan {\phi }_j}$$

   Demir & Ulusoy 2024; mirror of Pamucar 2024 Eq.(9)

8. 8

   Additive normalisation of q_j yields the criterion weights.

   $$w_j=\frac{q_j}{{\sum }_{j^{\prime }=1}^n{q}_{j^{\prime }}}$$

   Demir & Ulusoy 2024 Eq.(13); book Bölüm 2 Tablo 7; mirror of Pamucar 2024 Eq.(10)

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• Expert-ordering dependence: F6 envelope sums adjacent (z_{h+1,j} - z_{h,j})^2; permuting experts changes E_j. Document expert ordering explicitly.

• Constant linguistic column (all experts pick the same label) ⇒ Δz_j = 0 ⇒ slope undefined. Input check E-3 must reject this case.

Works with

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