Grey-GRA - GRA yönteminin Grey uzantısı

Formül adımları

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

1. Adım 1 — Gri normalleştirme, beyazlatma, referans dizisi.

   $$\begin{gathered} {\widehat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\widehat{v}}_j^0={max}_i{\widehat{v}}_{ij}(\text{benefit}) \end{gathered}$$
   LaTeX

   \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U);\\ \hat{v}_j^0=\max_i\hat{v}_{ij}\;(\text{benefit})

2. Adım 2 — Sapma Δ_ij ve global min/max.

   $$\begin{gathered} {\Delta }_{ij}=|{\widehat{v}}_j^0-{\widehat{v}}_{ij}|; \\ {\Delta }_{min}={min}_i{min}_j{\Delta }_{ij}; \\ {\Delta }_{max}={max}_i{max}_j{\Delta }_{ij} \end{gathered}$$
   LaTeX

   \Delta_{ij}=|\hat{v}_j^0-\hat{v}_{ij}|;\\ \Delta_{\min}=\min_i\min_j\Delta_{ij};\\ \Delta_{\max}=\max_i\max_j\Delta_{ij}

3. Adım 3 — Gri ilişkisel katsayı ξ_ij, ρ=0.5.

   $${\xi }_{ij}=\frac{{\Delta }_{min}+\rho {\Delta }_{max}}{{\Delta }_{ij}+\rho {\Delta }_{max}},\rho =0.5$$
   LaTeX

   \xi_{ij}=\frac{\Delta_{\min}+\rho\Delta_{\max}}{\Delta_{ij}+\rho\Delta_{\max}},\quad\rho=0.5

4. Adım 4 — Gri ilişkisel derece Γ_i ve azalan sıra.

   $$\begin{gathered} {\Gamma }_i=\sum _j{w}_j{\xi }_{ij}; \\ \text{rank descending by }{\Gamma }_i \end{gathered}$$
   LaTeX

   \Gamma_i=\sum_j w_j\xi_{ij};\\ \text{rank descending by }\Gamma_i

Sezgi

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

Sonucu okuma: grey-gra extends GRA 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 GRA 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.