GRA - Gri İlişkisel Analiz

Formül adımları

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

1. Adım 1 — Kriter yönüne göre [0,1] normalleştirme x*_ij.

   $$x_{ij}^{*}=\{\begin{array}{cc}{\displaystyle \frac{x_{ij}-min{x}_{ij}}{max{x}_{ij}-min{x}_{ij}}}\hfill & j\in J^{+}\hfill \\ {\displaystyle \frac{max{x}_{ij}-x_{ij}}{max{x}_{ij}-min{x}_{ij}}}\hfill & j\in J^{-}\hfill \end{array}$$
   LaTeX

   x^{*}_{ij} = \begin{cases}\dfrac{x_{ij}-\min x_{ij}}{\max x_{ij}-\min x_{ij}} & j\in J^{+}\\\dfrac{\max x_{ij}-x_{ij}}{\max x_{ij}-\min x_{ij}} & j\in J^{-}\end{cases}

2. Adım 2 — Referans dizisi x_0 = (1,1,...,1) (normalize sonrası ideal).

   $$x_{0,j}=1,j=1,\dots ,n$$
   LaTeX

   x_{0,j} = 1,\quad j=1,\ldots,n

3. Adım 3 — Mutlak farklar Δ_ij = |x_0,j − x*_ij|.

   $${\Delta }_{ij}=|x_{0,j}-x_{ij}^{*}|$$
   LaTeX

   \Delta_{ij} = \lvert x_{0,j} - x^{*}_{ij}\rvert

4. Adım 4 — Gri ilişki katsayısı ξ_ij = (min_min + ρ·max_max)/(Δ_ij + ρ·max_max); ρ ∈ (0,1] genelde 0.5.

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

   \xi_{ij} = \dfrac{\min_{i,j}\Delta_{ij} + \rho\,\max_{i,j}\Delta_{ij}}{\Delta_{ij} + \rho\,\max_{i,j}\Delta_{ij}}

5. Adım 5 — Gri İlişki Derecesi Γ_i = Σ w_j · ξ_ij ∈ [0,1]; azalan sıralama.

   $${\Gamma }_i=\sum _{j=1}^n{w}_j{\xi }_{ij},0\le {\Gamma }_i\le 1$$
   LaTeX

   \Gamma_{i} = \sum_{j=1}^{n} w_{j}\,\xi_{ij},\quad 0\le \Gamma_{i}\le 1

Sezgi

Grey relational grade (reference sequence comparison). Output typically utility (higher value = preferred).

Sonucu okuma: Γ_i ∈ (0,1]. Higher Γ means greater similarity to the ideal reference sequence. ζ=0.5 is Deng's recommended default - it balances the contribution of the minimum and maximum absolute differences. Smaller ζ amplifies the discriminating power (differences between alternatives become larger); larger ζ reduces discrimination.

Varsayımlar

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Ne zaman kullanılmaz

• Criteria strongly correlated → consider DEMATEL/ANP for interdependence
• Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)

Sınırlılıklar

• Assumes: Criteria preferences are independent (no synergistic interactions)
• Assumes: Compensation is acceptable: high score on one criterion can offset low on another
• Assumes: Decision matrix is complete (no missing values)

Sık yapılan hatalar

• Sabit kriter sütunu (max = min): normalleştirme paydası sıfırdır - E-5'i kontrol edin.
• ζ duyarlılığı: ζ değiştikçe sıralamalar değişebilir; ζ=0.5 geleneksel olmakla birlikte doğrulanmalıdır.