Grey-GRA - Grey extension of GRA

1. 1. Normalize grey matrix ⊗v_ij; whiten v̂_ij=½(v_ij^L+v_ij^U). Reference sequence: v̂_j^0 = max_i(v̂_ij) benefit.

   $$\begin{gathered} {\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\hat{v}}_j^0=\underset{i}{\max }{\hat{v}}_{ij}(\text{benefit}) \end{gathered}$$

2. 2. Deviation: Δ_ij = |v̂_j^0 - v̂_ij|. Global min/max: Δ_min = min_{i,j}(Δ_ij); Δ_max = max_{i,j}(Δ_ij).

   $$\begin{gathered} {\Delta }_{ij}=|{\hat{v}}_j^0-{\hat{v}}_{ij}|; \\ {\Delta }_{\min }=\underset{i}{\min }\underset{j}{\min }{\Delta }_{ij}; \\ {\Delta }_{\max }=\underset{i}{\max }\underset{j}{\max }{\Delta }_{ij} \end{gathered}$$

3. 3. Grey relational coefficient: ξ_ij = (Δ_min + ρ·Δ_max)/(Δ_ij + ρ·Δ_max), ρ=0.5.

   $${\xi }_{ij}=\frac{{\Delta }_{\min }+\rho {\Delta }_{\max }}{{\Delta }_{ij}+\rho {\Delta }_{\max }},\rho =0.5$$

4. 4. Grey relational grade: Γ_i = Σ_j w_j·ξ_ij. Rank descending.

   $$\begin{gathered} {\Gamma }_i=\sum _j{w}_j{\xi }_{ij}; \\ \text{rank descending by }{\Gamma }_i \end{gathered}$$