Fuzzy AROMAN: Fuzzy extension of AROMAN

Overview

Fuzzy outranking/ranking: Triangular Fuzzy Number (TFN: l, m, u). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Fuzzy outranking/ranking: Triangular Fuzzy Number (TFN: l, m, u)
• •Preserves fuzzy_TFN uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

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

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

When not to use

• •Crisp data sufficient: use base AROMAN directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for FUZZY-AROMAN-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'FUZZY-AROMAN bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• •Hatalı: 'FUZZY-AROMAN bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'FUZZY-AROMAN bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: FUZZY-AROMAN'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-AROMAN'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Extract TFN decision matrix X̃=[x̃ij] and weights w̃j=(wjα,wjβ,wjγ). Formül: X̃=[x̃_{ij}]=(x_{ij}^{\alpha},x_{ij}^{\beta},x_{ij}^{\gamma});\quad\tilde{w}_j=(w_j^{\alpha},w_j^{\beta},w_j^{\gamma})
2. 2.Adım 2 (F2): Vector (Euclidean) normalisation per column using COA defuzzified values: d̃ij = x̃ij / sqrt(Σi COA(x̃ij)²). Formül: \text{COA}(\tilde{x}_{ij})=\frac{x_{ij}^{\alpha}+x_{ij}^{\beta}+x_{ij}^{\gamma}}{3};\quad d_{ij}^{\text{norm}}=\frac{\text{COA}(\tilde{x}_{ij})}{\sqrt{\sum_{i=1}^{m}\text{COA}(\tilde{x}_{ij})^2}};\quad \tilde{d}_{ij}=\tilde{x}_{ij}\,/\,\sqrt{\sum_{i=1}^{m}\text{COA}(\tilde{x}_{ij})^2}
3. 3.Adım 3 (F3): Min-max normalisation on vector-normalised TFNs: benefit r̃ij = d̃ij / max_i COA(d̃ij); cost r̃ij = min_i COA(d̃ij) / d̃ij. Formül: \tilde{r}_{ij}=\begin{cases}\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{cases}
4. 4.Adım 4 (F4): Weighted TFN product: ṽij = r̃ij ⊗ w̃j = (rijα·wjα, rijβ·wjβ, rijγ·wjγ). Formül: \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.Adım 5 (F5): Compute AROMAN score Qi = Σj COA(ṽij) and rank alternatives in descending order; highest Qi is best. Formül: 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};\quad\text{rank descending by }Q_i

Commonly paired with

• •FUZZY-AHP + FUZZY-AROMAN (common)

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