GRA - Grey Relational Analysis

Overview

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.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Normalise sequences per criterion type to [0,1].

   $$x_{ij}^{\ast }=\{\begin{array}{ll}\frac{x_{ij}-\min x_{ij}}{\max x_{ij}-\min x_{ij}}& j\in J^{+}\\ \frac{\max x_{ij}-x_{ij}}{\max x_{ij}-\min x_{ij}}& j\in J^{-}\end{array}$$

   Deng 1989, p.4 Eq.(2)

2. 2

   Reference sequence x_0 = (1, 1, ..., 1) (post-normalisation ideal).

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

   Deng 1989, p.5

3. 3

   Absolute differences Δ_ij = |x_0,j − x*_ij|.

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

   Deng 1989, p.5 Eq.(3)

4. 4

   Grey relational coefficient ξ_ij with distinguishing coefficient ρ (typically 0.5).

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

   Deng 1989, p.5 Eq.(4)

5. 5

   Grey Relational Grade Γ_i = Σ w_j · ξ_ij and descending ranking.

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

   Deng 1989, p.5 Eq.(5)

Look elsewhere when

Assumptions to verify

• 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)

Edge cases and pitfalls

• Constant criterion column (max = min): normalisation denominator is zero - check E-5.

• ζ sensitivity: rankings can change as ζ varies; ζ=0.5 is conventional but should be validated.

Works with

How to cite