Grey-MOORA - MOORA yönteminin Grey uzantısı

Formül adımları

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

1. Adım 1 — Gri karar matrisi ⊗a_ij=[a_ij^L, a_ij^U] oluştur. Sütun normalleştirme, beyazlatma, vektör normalleştirme.

   $$\begin{gathered} \otimes v_{ij}=[\frac{a_{ij}^L}{{max}_k{a}_{kj}^U},\frac{a_{ij}^U}{{max}_k{a}_{kj}^U}]; \\ {\widehat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\overline{v}}_{ij}=\frac{{\widehat{v}}_{ij}}{\sqrt{{\sum }_i{\widehat{v}}_{ij}^2}} \end{gathered}$$
   LaTeX

   \otimes v_{ij}=\Bigl[\frac{a_{ij}^L}{\max_k a_{kj}^U},\frac{a_{ij}^U}{\max_k a_{kj}^U}\Bigr];\\ \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U);\\ \bar{v}_{ij}=\frac{\hat{v}_{ij}}{\sqrt{\sum_i \hat{v}_{ij}^2}}

2. Adım 2 — Oran sistemi skoru: y_i* = Σ_{j∈Ω_b} w_j·v̄_ij - Σ_{j∈Ω_c} w_j·v̄_ij. Azalan sırayla sırala.

   $$\begin{gathered} y_i^{*}=\sum _{j\in {\Omega }_b}{w}_j{\overline{v}}_{ij}-\sum _{j\in {\Omega }_c}{w}_j{\overline{v}}_{ij}; \\ \text{rank descending by }{y}_i^{*} \end{gathered}$$
   LaTeX

   y_i^*=\sum_{j\in\Omega_b}w_j\bar{v}_{ij}-\sum_{j\in\Omega_c}w_j\bar{v}_{ij};\\ \text{rank descending by }y_i^*

3. Adım 3 (İsteğe bağlı) — Referans nokta yaklaşımı: r_j* = max_i(v̄_ij). y_i** = max_j(w_j·|r_j* - v̄_ij|). Artan sıra.

   $$\begin{gathered} r_j^{*}={max}_i{\overline{v}}_{ij}(\text{benefit}); \\ {y}_i^{**}={max}_j(w_j|r_j^{*}-{\overline{v}}_{ij}|); \\ \text{rank ascending by }{y}_i^{**} \end{gathered}$$
   LaTeX

   r_j^*=\max_i\bar{v}_{ij}\;(\text{benefit});\\ y_i^{**}=\max_j\bigl(w_j|r_j^*-\bar{v}_{ij}|\bigr);\\ \text{rank ascending by }y_i^{**}

Sezgi

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

Sonucu okuma: grey-moora extends MOORA 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 MOORA directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• 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.