DNMA - Double Normalization-Based Multiple Aggregation

Overview

U_i ∈ [0,1] approximately. Higher U means better. DNMA combines two normalisation paradigms - linear (min-max) and vector (Euclidean) - to reduce the influence of any single normalisation method on the ranking. λ=0.5 gives equal weight to both. λ=1 reduces to a min-max SAW; λ=0 gives a vector-normalised SAW.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Three normalisations: linear-max, vector, and target-based.

   $$r_{ij}^{(1)}=\frac{x_{ij}}{\max x_{ij}};r_{ij}^{(2)}=\frac{x_{ij}}{\sqrt{\sum x_{kj}^2}};r_{ij}^{(3)}\text{ target-based}$$

   Liao-Wu 2020, p.6 Eqs.(1)-(3)

2. 2

   Three sub-utilities: complete (WSM), incomplete (WPM), and TOPSIS-like.

   $$u_i^{(1)}=\sum w_j{r}_{ij}^{(1)},u_i^{(2)}=\prod r_{ij}^{(2)w_j},u_i^{(3)}=\text{TOPSIS closeness}$$

   Liao-Wu 2020, p.7 Eqs.(4)-(6)

3. 3

   Compute three ranks R^{(k)}_i; deviation measure considers rank spread.

   $$R_i^{(k)}=\text{rank}(u_i^{(k)});S_i^{(k)}=u_i^{(k)}-\underset{l}{\min }{u}_l^{(k)}$$

   Liao-Wu 2020, p.8 Eq.(7)

4. 4

   Combined DNMA score D_i = Σ α_k (u^{(k)}_i)^2 − Σ β_k (R^{(k)}_i)^2; descending ranking.

   $$D_i=\sum _{k=1}^3{\alpha }_k(u_i^{(k)}{)}^2-\sum _{k=1}^3{\beta }_k(R_i^{(k)}{)}^2$$

   Liao-Wu 2020, p.8 Eq.(8)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• •ties: complete (WSM), incomplete (WPM), and TOPSIS-like.

• Constant criterion column (max = min): linear normalisation denominator is zero.

Works with

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