Grey-SAW - Grey extension of SAW

1. 1. Construct grey matrix ⊗v_ij=[v_ij^L, v_ij^U]; normalize benefit/cost; whiten: v̂_ij=½(v_ij^L+v_ij^U).

   $$\begin{gathered} \otimes v_{ij}=[v_{ij}^L,v_{ij}^U]; \\ \text{benefit: }{\hat{v}}_{ij}=\frac{1}{2}(\frac{v_{ij}^L}{{\max }_k{v}_{kj}^U}+\frac{v_{ij}^U}{{\max }_k{v}_{kj}^U}); \\ \text{cost: }{\hat{v}}_{ij}=\frac{1}{2}(\frac{{\min }_k{v}_{kj}^L}{v_{ij}^U}+\frac{{\min }_k{v}_{kj}^L}{v_{ij}^L}) \end{gathered}$$

2. 2. Weighted sum score S_i = Σ_j w_j·v̂_ij.

   $$S_i=\sum _{j=1}^n{w}_j{\hat{v}}_{ij}$$

3. 3. Rank descending by S_i. Best: max S_i.

   $$\text{rank descending by }{S}_i$$