Grey-SAW - SAW yönteminin Grey uzantısı

Formül adımları

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

1. Adım 1 — Gri matris, normalleştirme, beyazlatma.

   $$\begin{gathered} \otimes v_{ij}=[v_{ij}^L,v_{ij}^U]; \\ \text{benefit: }{\widehat{v}}_{ij}=\frac{1}{2}(\frac{v_{ij}^L}{{max}_k{v}_{kj}^U}+\frac{v_{ij}^U}{{max}_k{v}_{kj}^U}); \\ \text{cost: }{\widehat{v}}_{ij}=\frac{1}{2}(\frac{{min}_k{v}_{kj}^L}{v_{ij}^U}+\frac{{min}_k{v}_{kj}^L}{v_{ij}^L}) \end{gathered}$$
   LaTeX

   \otimes v_{ij}=[v_{ij}^L,v_{ij}^U];\\ \text{benefit: }\hat{v}_{ij}=\tfrac{1}{2}\Bigl(\frac{v_{ij}^L}{\max_k v_{kj}^U}+\frac{v_{ij}^U}{\max_k v_{kj}^U}\Bigr);\\ \text{cost: }\hat{v}_{ij}=\tfrac{1}{2}\Bigl(\frac{\min_k v_{kj}^L}{v_{ij}^U}+\frac{\min_k v_{kj}^L}{v_{ij}^L}\Bigr)

2. Adım 2 — Ağırlıklı toplam skoru.

   $$S_i=\sum _{j=1}^n{w}_j{\widehat{v}}_{ij}$$
   LaTeX

   S_i=\sum_{j=1}^n w_j\hat{v}_{ij}

3. Adım 3 — Azalan sıra.

   $$\text{rank descending by }{S}_i$$
   LaTeX

   \text{rank descending by }S_i

Sezgi

Grey outranking/ranking - Grey Interval Number (GIN: [x̲, x̄]). Output typically utility (higher value = preferred).

Sonucu okuma: grey-saw extends SAW to handle Grey uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Grey Interval Number (GIN: [x̲, x̄]) algebra. The final scores are defuzzified via whitenisation: (x̲ + x̄)/2 before ranking.

Varsayımlar

• Decision matrix entries are valid Grey 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 SAW directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• Assumes: Decision matrix entries are valid Grey 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 GIN: x̲ ≤ x̄ (lower and upper bounds of interval) koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: whitenisation: (x̲ + x̄)/2 kanonik seçimdir.