Fuzzy AROMAN - Fuzzy extension of AROMAN

Overview

fuzzy-aroman extends AROMAN to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Output

utility, higher is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Extract TFN decision matrix X̃=[x̃ij] and weights w̃j=(wjα,wjβ,wjγ).

   $$\tilde{X}=[{\tilde{x}}_{ij}]=(x_{ij}^{\alpha },x_{ij}^{\beta },x_{ij}^{\gamma });{\tilde{w}}_j=(w_j^{\alpha },w_j^{\beta },w_j^{\gamma })$$
2. 2

   Vector (Euclidean) normalisation per column using COA defuzzified values: d̃ij = x̃ij / sqrt(Σi COA(x̃ij)²).

   $$\text{COA}({\tilde{x}}_{ij})=\frac{x_{ij}^{\alpha }+x_{ij}^{\beta }+x_{ij}^{\gamma }}{3};d_{ij}^{\text{norm}}=\frac{\text{COA}({\tilde{x}}_{ij})}{\sqrt{{\sum }_{i=1}^m\text{COA}({\tilde{x}}_{ij}{)}^2}};{\tilde{d}}_{ij}={\tilde{x}}_{ij}/\sqrt{\sum _{i=1}^m\text{COA}({\tilde{x}}_{ij}{)}^2}$$
3. 3

   Min-max normalisation on vector-normalised TFNs: benefit r̃ij = d̃ij / max_i COA(d̃ij); cost r̃ij = min_i COA(d̃ij) / d̃ij.

   $${\tilde{r}}_{ij}=\{\begin{array}{ll}{\tilde{d}}_{ij}/{\max }_i\text{COA}({\tilde{d}}_{ij})& \text{benefit}\\ {\min }_i\text{COA}({\tilde{d}}_{ij})/{\tilde{d}}_{ij}& \text{cost}\end{array}$$
4. 4

   Weighted TFN product: ṽij = r̃ij ⊗ w̃j = (rijα·wjα, rijβ·wjβ, rijγ·wjγ).

   $${\tilde{v}}_{ij}={\tilde{r}}_{ij}\otimes {\tilde{w}}_j=(r_{ij}^{\alpha }{w}_j^{\alpha },r_{ij}^{\beta }{w}_j^{\beta },r_{ij}^{\gamma }{w}_j^{\gamma })$$
5. 5

   Compute AROMAN score Qi = Σj COA(ṽij) and rank alternatives in descending order; highest Qi is best.

   $$Q_i=\sum _{j=1}^n\text{COA}({\tilde{v}}_{ij})=\sum _{j=1}^n\frac{v_{ij}^{\alpha }+v_{ij}^{\beta }+v_{ij}^{\gamma }}{3};\text{rank descending by }{Q}_i$$

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy TFN: l ≤ m ≤ u, all ≥ 0 before computation.

• Defuzzification method affects ranking: centroid (l+m+u)/3 is the canonical choice but alternatives exist.

Works with

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