AROMAN - Alternative Ranking Order Method Accounting for Two-Step Normalisation

Overview

Q_i > 0. Higher Q means better. AROMAN applies two-step normalisation (linear-max then vector) to reduce normalisation-method sensitivity, then aggregates with a power function. α=0.5 is the default; α→1 approaches weighted sum; α→0 approaches weighted product (geometric mean behaviour).

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

   Dual normalisation: linear (T¹) + vector (T²).

   $$T_{ij}^1=\frac{x_{ij}-{\min }_i{x}_{ij}}{{\max }_i{x}_{ij}-{\min }_i{x}_{ij}};T_{ij}^2=\frac{x_{ij}}{\sqrt{{\sum }_k{x}_{kj}^2}}$$

   Bošković 2023, p.4 Eqs.(1)-(2)

2. 2

   Combined normalisation Z_ij = (β T¹_ij + (1−β) T²_ij)/2 with β ∈ [0,1].

   $$Z_{ij}=\frac{\beta T_{ij}^1+(1-\beta )T_{ij}^2}{2}$$

   Bošković 2023, p.4 Eq.(3)

3. 3

   Weighted matrix v_ij = w_j Z_ij.

   $$v_{ij}=w_j{Z}_{ij}$$

   Bošković 2023, p.5 Eq.(4)

4. 4

   Sum of benefit Σ^+_i and cost Σ^−_i criteria.

   $${\Sigma }_i^{+}=\sum _{j\in J^{+}}{v}_{ij},{\Sigma }_i^{-}=\sum _{j\in J^{-}}{v}_{ij}$$

   Bošković 2023, p.5 Eqs.(5)-(6)

5. 5

   AROMAN score R_i = (Σ^−_i)^λ + (Σ^+_i)^(1−λ) and descending ranking. λ=0.5 default (paper Eq. 8). Note: normalization in Steps 1-2 has NO direction inversion; L_i (cost sum) and A_i (benefit sum) are raw minmax/vector values. Bošković et al. 2023, IEEE Access DOI:10.1109/ACCESS.2023.3265818.

   $$R_i=({\Sigma }_i^{-}{)}^{\lambda }+({\Sigma }_i^{+}{)}^{1-\lambda },\lambda \in [0,1],\text{rank descending}$$

   Bošković 2023, p.6 Eq.(7)

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

• •default (paper Eq. 8). Note: normalization in Steps 1-2 has NO direction inversion; L_i (cost sum) and A_i (benefit sum) are raw minmax/vector values. Bošković et al. 2023, IEEE Access DOI:10.1109/ACC

• α sensitivity: different α values can change rankings - always run sensitivity analysis.

Works with

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