WENSLO - WEight deNomination based on Slope coefficient for objective weighting

Overview

WENSLO assigns higher weight to criteria whose normalised values change most steeply (in absolute terms) as we move through the alternatives in rank order. A criterion with a flat profile (all alternatives similar) gets β≈0 → low weight. A criterion with a steep profile gets high weight.

Output

Weight, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Any (objective weighting)

How it works

1. 1

   Column-sum normalisation. Each column j is divided by its column sum, producing a normalised matrix Z whose columns sum to 1. The procedure is direction-agnostic - cost/benefit orientation does not enter the algorithm (Pamucar et al. 2024, p.9510 Abstract; p.9513 §III-A).

   $$z_{ij}=\frac{{\zeta }_{ij}}{{\sum }_{i=1}^m{\zeta }_{ij}}\forall j\in [1,n]$$

   Pamucar et al. 2024, p.9512 Eq.(2)

2. 2

   Criterion class interval via Sturges' rule. Δz_j scales the (max - min) range of the j-th normalised column by 1 + 3.322·log10(m).

   $$\Delta z_j=\frac{{\max }_i{z}_{ij}-{\min }_i{z}_{ij}}{1+3.322\cdot {\log }_{10}(m)}$$

   Pamucar et al. 2024, p.9513 Eq.(5)

3. 3

   Criterion slope tan(φ_j). The slope of the hypotenuse of the artificial right-angled triangle defined by ((m-1)·Δz_j, Σ_i z_ij). Since Σ_i z_ij = 1 by F1, this reduces to tan(φ_j) = 1/((m-1)·Δz_j).

   $$\tan {\phi }_j=\frac{{\sum }_{i=1}^m{z}_{ij}}{(m-1)\cdot \Delta z_j}$$

   Pamucar et al. 2024, p.9513 Eq.(7)

4. 4

   Criterion envelope E_j. Total Euclidean distance between successive normalised values along the j-th column, with the class interval Δz_j as the constant horizontal step. Captures the zig-zag length of the criterion's accumulation polyline.

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

   Pamucar et al. 2024, p.9513 Eq.(8)

5. 5

   Envelope-slope ratio q_j = E_j / tan(φ_j). Larger q_j ⇒ longer accumulation envelope relative to the average slope ⇒ richer information about criterion variability ⇒ higher weight.

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

   Pamucar et al. 2024, p.9513 Eq.(9)

6. 6

   Additive normalisation of q_j yields the criterion weights.

   $$w_j=\frac{q_j}{{\sum }_{j^{\prime }=1}^n{q}_{j^{\prime }}}\forall j\in [1,n]$$

   Pamucar et al. 2024, p.9513 Eq.(10)

Look elsewhere when

Assumptions to verify

• Decision matrix exists with measurable criteria
• Sufficient inter-alternative variation per criterion

Edge cases and pitfalls

• The slope depends on the ordering of alternatives - WENSLO weights are alternative-order dependent. A different listing order of alternatives gives different slopes.

Works with

How to cite