Grey-Projection - Grey extension of GRA

Overview

grey-projection 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
   $$\otimes v_{ij}=\{\begin{array}{llc}[a_{ij}^L/{\max }_i{a}_{ij}^U,a_{ij}^U/{\max }_i{a}_{ij}^U]& j\in {\Omega }_b\text{[}{\min }_i{a}_{ij}^L/a_{ij}^U,{\min }_i{a}_{ij}^L/a_{ij}^L]& j\in {\Omega }_c\end{array},{\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U)$$

   GREY-PROJECTION-base

2. 2
   $${\hat{v}}_j^{+}=\underset{i}{\max }{\hat{v}}_{ij},{\hat{v}}_j^{-}=\underset{i}{\min }{\hat{v}}_{ij}$$

   GREY-PROJECTION-base

3. 3
   $$P_i^{+}=\frac{{\sum }_j{w}_j{\hat{v}}_{ij}{\hat{v}}_j^{+}}{\sqrt{{\sum }_j{w}_j({\hat{v}}_j^{+}{)}^2}},P_i^{-}=\frac{{\sum }_j{w}_j{\hat{v}}_{ij}{\hat{v}}_j^{-}}{\sqrt{{\sum }_j{w}_j({\hat{v}}_j^{-}{)}^2}}$$

   GREY-PROJECTION-base

4. 4
   $${\mathrm{CP}}_i=\frac{P_i^{+}}{P_i^{+}+P_i^{-}},\text{rank descending}$$

   GREY-PROJECTION-base

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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