Grey-Projection - GRA yönteminin Grey uzantısı

Formül adımları

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

1. normalisation

   LaTeX

   \otimes v_{ij}=\begin{cases}[a_{ij}^L/\max_i a_{ij}^U,\;a_{ij}^U/\max_i a_{ij}^U] & j\in\Omega_b\[\min_i a_{ij}^L/a_{ij}^U,\;\min_i a_{ij}^L/a_{ij}^L] & j\in\Omega_c\end{cases},\quad\hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U)

2. ranking_computation

   $${\widehat{v}}_j^{+}={max}_i{\widehat{v}}_{ij},{\widehat{v}}_j^{-}={min}_i{\widehat{v}}_{ij}$$
   LaTeX

   \hat{v}_j^+=\max_i\hat{v}_{ij},\quad\hat{v}_j^-=\min_i\hat{v}_{ij}

3. ranking_computation

   $$P_i^{+}=\frac{{\sum }_j{w}_j{\widehat{v}}_{ij}{\widehat{v}}_j^{+}}{\sqrt{{\sum }_j{w}_j({\widehat{v}}_j^{+}{)}^2}},P_i^{-}=\frac{{\sum }_j{w}_j{\widehat{v}}_{ij}{\widehat{v}}_j^{-}}{\sqrt{{\sum }_j{w}_j({\widehat{v}}_j^{-}{)}^2}}$$
   LaTeX

   P_i^+=\frac{\sum_j w_j\hat{v}_{ij}\hat{v}_j^+}{\sqrt{\sum_j w_j(\hat{v}_j^+)^2}},\quad P_i^-=\frac{\sum_j w_j\hat{v}_{ij}\hat{v}_j^-}{\sqrt{\sum_j w_j(\hat{v}_j^-)^2}}

4. ranking_computation

   $${\mathrm{C}\mathrm{P}}_i=\frac{P_i^{+}}{P_i^{+}+P_i^{-}},\text{rank descending}$$
   LaTeX

   \mathrm{CP}_i=\frac{P_i^+}{P_i^++P_i^-},\quad\text{rank descending}

Sezgi

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

Sonucu okuma: grey-projection 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 PROJECTION 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.