Fuzzy AROMAN - AROMAN yönteminin Fuzzy uzantısı

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

1. TFN karar matrisi X̃=[x̃ij] ve ağırlık vektörü w̃j=(wjα,wjβ,wjγ)'yi çıkar.

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

   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. İki-adım normalizasyon: lineer (Eq.2) + vektör (Eq.3), sonra agrege ortalama (Eq.4) t^norm=(β·lin+(1-β)·vec)/2, β=0.5. TFN bileşen-bazlı.

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

   \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. Ağırlıklı normalize matris (Eq.5): t̂ij = w̃j ⊗ t^norm_ij.

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

   \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. Yön ayrımı (Eq.6/7): Li=Σ min-tip, Ai=Σ max-tip (COA); λ-güç (Eq.8/9): L^=Li^λ, A^=Ai^(1-λ), λ=0.5.

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

   \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. Final sıralama (Eq.10): Ri = exp(A^ − L^); azalan sıra.

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

   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

Sezgi

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

Sonucu okuma: 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.

Varsayımlar

• 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

Ne zaman kullanılmaz

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

Sınırlılıklar

• 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

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin TFN: l ≤ m ≤ u ve tümü ≥ 0 koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: centroid (l+m+u)/3 kanonik seçimdir.