WENSLO - WEight deNomination based on Slope coefficient for objective weighting

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).

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

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).

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

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).

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

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.

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

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.

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

6. 6. Additive normalisation of q_j yields the criterion weights.

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