Grey-TODIM - TODIM yönteminin Grey uzantısı

Formül adımları

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

Adım 1 — Gri normalleştirme ve beyazlatma.

${\hat{v}}_{i j} = \frac{1}{2} (v_{i j}^{L} + v_{i j}^{U})$

LaTeX
\hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U)
Adım 2 — Göreli ağırlıklar.

${\overset{―}{w}}_{j r} = w_{j} / w_{r}, w_{r} = {max}_{j} w_{j}$

LaTeX
\overline{w}_{jr}=w_j/w_r,\quad w_r=\max_j w_j
Adım 3 — Hâkimiyet katkıları φ_j.

$d_{j} (i, k) = | {\hat{v}}_{i j} - {\hat{v}}_{k j} |; ϕ_{j} (A_{i}, A_{k}) = {\begin{matrix} + \sqrt{{\overset{―}{w}}_{j r} d_{j} (i, k) / \sum_{j} {\overset{―}{w}}_{j r}} & {\hat{v}}_{i j} > {\hat{v}}_{k j} \\ 0 & {\hat{v}}_{i j} = {\hat{v}}_{k j} \\ - (1 / θ) \sqrt{\sum_{j} {\overset{―}{w}}_{j r} d_{j} (i, k) / {\overset{―}{w}}_{j r}} & {\hat{v}}_{i j} < {\hat{v}}_{k j} \end{matrix}; θ = 1$

LaTeX
d_j(i,k)=|\hat{v}_{ij}-\hat{v}_{kj}|;\\ \phi_j(A_i,A_k)=\begin{cases}+\sqrt{\overline{w}_{jr}d_j(i,k)/\textstyle\sum_j\overline{w}_{jr}}&\hat{v}_{ij}>\hat{v}_{kj}\\0&\hat{v}_{ij}=\hat{v}_{kj}\\-(1/\theta)\sqrt{\textstyle\sum_j\overline{w}_{jr}d_j(i,k)/\overline{w}_{jr}}&\hat{v}_{ij}<\hat{v}_{kj}\end{cases};\quad\theta=1
Adım 4 — Genel hâkimiyet ve normalize edilmiş skor. Azalan sıra.

$δ (A_{i}, A_{k}) = \sum_{j} ϕ_{j} (A_{i}, A_{k}); ξ_{i} = \frac{\sum_{k} δ (A_{i}, A_{k}) - {min}_{k} \sum_{k} δ (A_{k}, \cdot)}{{max}_{k} \sum_{k} δ (A_{k}, \cdot) - {min}_{k} \sum_{k} δ (A_{k}, \cdot)}; rank descending$

LaTeX
\delta(A_i,A_k)=\sum_j\phi_j(A_i,A_k);\\ \xi_i=\frac{\sum_k\delta(A_i,A_k)-\min_k\sum_k\delta(A_k,\cdot)}{\max_k\sum_k\delta(A_k,\cdot)-\min_k\sum_k\delta(A_k,\cdot)};\\ \text{rank descending}

Sezgi

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

Sonucu okuma: grey-todim extends TODIM 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 TODIM 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.

Hesap adımları ve dayanakları

Normalize grey matrix; whiten v̂_ij=½(v_ij^L+v_ij^U).

Dayanak: Gomes-Lima 1992; Deng 1989
Relative weights: w̄_jr = w_j/w_r, where w_r = max_j(w_j).

Dayanak: Gomes-Lima 1992 §relative weights
Dominance contributions: d_j(i,k)=|v̂_ij - v̂_kj|. φ_j(A_i,A_k)= +√(w̄_jr·d_j/Σw̄_jr) if v̂_ij>v̂_kj, 0 if equal, -(1/θ)√(Σw̄_jr·d_j/w̄_jr) if v̂_ij<v̂_kj. θ=1.

Dayanak: Gomes-Lima 1992 §TODIM; Wang-Wang 2014 §grey TODIM
Overall dominance: δ(A_i,A_k) = Σ_j φ_j(A_i,A_k). Score: ξ_i = Σ_k δ(A_i,A_k). Normalize and rank descending.

Dayanak: Gomes-Lima 1992 §global dominance