Grey-VIKOR - Grey extension of VIKOR

Overview

grey-vikor extends VIKOR 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, lower 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 decision matrix; whiten: v̂_ij = ½(v_ij^L + v_ij^U). Compute best/worst whitenised values: v̂_j* = max_i(v̂_ij) benefit; v̂_j- = min_i(v̂_ij).

   $$\begin{gathered} \otimes v_{ij}=[v_{ij}^L,v_{ij}^U]; \\ {\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\hat{v}}_j^{\ast }=\underset{i}{\max }{\hat{v}}_{ij}(\text{benefit}); \\ {\hat{v}}_j^{-}=\underset{i}{\min }{\hat{v}}_{ij} \end{gathered}$$

   Deng 1989 §whitenisation; Opricovic-Tzeng 2004 §VIKOR normalization

2. 2

   3 - Normalised gap: f_ij = (v̂_j* - v̂_ij)/(v̂_j* - v̂_j-). Utility S_i = Σ_j w_j·f_ij; regret R_i = max_j(w_j·f_ij).

   $$\begin{gathered} f_{ij}=\frac{{\hat{v}}_j^{\ast }-{\hat{v}}_{ij}}{{\hat{v}}_j^{\ast }-{\hat{v}}_j^{-}}; \\ {S}_i=\sum _{j=1}^n{w}_j{f}_{ij}; \\ {R}_i=\underset{j}{\max }(w_j{f}_{ij}) \end{gathered}$$

   Opricovic 1998; Opricovic-Tzeng 2004 §2 Eqs.(3)-(4)

3. 3

   Step 4-5 - Compromise index Q_i = v·(S_i-S*)/(S⁻-S*) + (1-v)·(R_i-R*)/(R⁻-R*) with v=0.5. Rank ascending; check conditions C1 and C2.

   $$\begin{gathered} S^{\ast }=\underset{i}{\min }{S}_i,S^{-}=\underset{i}{\max }{S}_i;R^{\ast }=\underset{i}{\min }{R}_i,R^{-}=\underset{i}{\max }{R}_i; \\ {Q}_i=v\frac{S_i-S^{\ast }}{S^{-}-S^{\ast }}+(1-v)\frac{R_i-R^{\ast }}{R^{-}-R^{\ast }}; \\ \text{rank ascending by }{Q}_i \end{gathered}$$

   Opricovic 1998; Opricovic-Tzeng 2004 §2 Eq.(6)

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