Grey-COPRAS - Grey extension of COPRAS

Overview

grey-copras extends COPRAS 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

   Whiten grey matrix. Column-sum normalization: v̄_ij=v̂_ij/Σ_k(v̂_kj). Weighted: v_ij=w_j·v̄_ij.

   $$\begin{gathered} {\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\bar{v}}_{ij}=\frac{{\hat{v}}_{ij}}{{\sum }_k{\hat{v}}_{kj}}; \\ {v}_{ij}=w_j{\bar{v}}_{ij} \end{gathered}$$

   Zavadskas-Kaklauskas 1996; Deng 1989

2. 2

   Benefit sum S_i+ = Σ_{j∈Ω_b} v_ij; cost sum S_i- = Σ_{j∈Ω_c} v_ij.

   $$\begin{gathered} S_i^{+}=\sum _{j\in {\Omega }_b}{v}_{ij}; \\ {S}_i^{-}=\sum _{j\in {\Omega }_c}{v}_{ij} \end{gathered}$$

   Zavadskas-Kaklauskas 1996 §COPRAS

3. 3

   Relative significance Q_i and utility degree N_i. Rank descending.

   $$\begin{gathered} Q_i=S_i^{+}+\frac{{\sum }_k{S}_k^{-}}{S_i^{-}{\sum }_k(1/S_k^{-})}; \\ {N}_i=\frac{Q_i}{Q_{\max }}\times 100\%; \\ \text{rank descending by }{Q}_i \end{gathered}$$

   Zavadskas-Kaklauskas 1996 §relative significance

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

How to cite