Grey-GRA - Grey extension of GRA

Overview

grey-gra extends GRA 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.

Output

utility, higher is better

Data

Grey Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Grey MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

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.

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

   Deng 1989 §GRA

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).

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

   Deng 1989 §grey relational coefficient

3. 3

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

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

   Deng 1989 §Eq.(3); Liu-Lin 2006

4. 4

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

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

   Deng 1989 §grey relational grade

Fits when / Look elsewhere when

Assumptions to verify

• 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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy GIN: x̲ ≤ x̄ (lower and upper bounds of interval) before computation.

• Defuzzification method affects ranking: whitenisation: (x̲ + x̄)/2 is the canonical choice but alternatives exist.

Works with

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