Grey-MARCOS - MARCOS yönteminin Grey uzantısı

Formül adımları

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

1. Adım 1 — Gri normalleştirme, beyazlatma, AI ve AAI.

   $${\widehat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U);(A{I}_j,AA{I}_j)=\{\begin{array}{cc}({max}_i{\widehat{v}}_{ij},{min}_i{\widehat{v}}_{ij})\hfill & j\in J^{+}\hfill \\ ({min}_i{\widehat{v}}_{ij},{max}_i{\widehat{v}}_{ij})\hfill & j\in J^{-}\hfill \end{array}$$
   LaTeX

   \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U);\quad (AI_j,AAI_j)=\begin{cases}(\max_i\hat v_{ij},\min_i\hat v_{ij})&j\in J^+\\(\min_i\hat v_{ij},\max_i\hat v_{ij})&j\in J^-\end{cases}

2. Adım 2 — AI'ya göre normalleştir.

   $$n_{ij}=\{\begin{array}{cc}{\widehat{v}}_{ij}/A{I}_j\hfill & j\in J^{+}\hfill \\ A{I}_j/{\widehat{v}}_{ij}\hfill & j\in J^{-}\hfill \end{array}$$
   LaTeX

   n_{ij}=\begin{cases}\hat v_{ij}/AI_j&j\in J^+\\AI_j/\hat v_{ij}&j\in J^-\end{cases}

3. Adım 3 — Ağırlıklı toplam S_i, S_AI ve S_AAI.

   $$S_i=\sum _j{w}_j{n}_{ij};S_{AI}=1;S_{AAI}=\sum _j{w}_j{n}_{AAI,j}$$
   LaTeX

   S_i=\sum_j w_j n_{ij};\quad S_{AI}=1;\quad S_{AAI}=\sum_j w_j\,n_{AAI,j}

4. Adım 4 — Fayda dereceleri K_i+, K_i- ve kompozit f(K_i). Azalan sıra.

   $$\begin{gathered} K_i^{+}=S_i; \\ {K}_i^{-}=S_i/S_{\mathrm{A}\mathrm{A}\mathrm{I}}; \\ f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+(1-f(K_i^{+}))/f(K_i^{+})+(1-f(K_i^{-}))/f(K_i^{-})} \\ \text{where }f(K_i^{+})=K_i^{+}/(K_i^{+}+K_i^{-}); \\ \text{rank descending} \end{gathered}$$
   LaTeX

   K_i^+=S_i;\\ K_i^-=S_i/S_{\mathrm{AAI}};\\ f(K_i)=\frac{K_i^++K_i^-}{1+(1-f(K_i^+))/f(K_i^+)+(1-f(K_i^-))/f(K_i^-)}\\ \text{where }f(K_i^+)=K_i^+/(K_i^++K_i^-);\\ \text{rank descending}

Sezgi

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

Sonucu okuma: grey-marcos extends MARCOS 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 MARCOS 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.