Grey-MOORA - Grey extension of MOORA

Overview

grey-moora extends MOORA 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

   Construct grey decision matrix ⊗a_ij=[a_ij^L, a_ij^U]. Normalize per column: ⊗v_ij=[a_ij^L/max_k(a_kj^U), a_ij^U/max_k(a_kj^U)]. Whiten: v̂_ij = ½(v_ij^L + v_ij^U). Vector-normalize whitenised: v̄_ij = v̂_ij / √(Σ_i v̂_ij²).

   $$\begin{gathered} \otimes v_{ij}=[\frac{a_{ij}^L}{{\max }_k{a}_{kj}^U},\frac{a_{ij}^U}{{\max }_k{a}_{kj}^U}]; \\ {\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\bar{v}}_{ij}=\frac{{\hat{v}}_{ij}}{\sqrt{{\sum }_i{\hat{v}}_{ij}^2}} \end{gathered}$$

   Stanujkic-Magdalinovic-Jovanovic-Stojanovic 2012; Deng 1989 §whitenisation

2. 2

   Ratio system score: y_i* = Σ_{j∈Ω_b} w_j·v̄_ij - Σ_{j∈Ω_c} w_j·v̄_ij. Rank descending.

   $$\begin{gathered} y_i^{\ast }=\sum _{j\in {\Omega }_b}{w}_j{\bar{v}}_{ij}-\sum _{j\in {\Omega }_c}{w}_j{\bar{v}}_{ij}; \\ \text{rank descending by }{y}_i^{\ast } \end{gathered}$$

   Stanujkic-Magdalinovic-Jovanovic-Stojanovic 2012; Brauers-Zavadskas 2006 §ratio

3. 3

   Reference point approach: r_j* = max_i(v̄_ij) benefit. y_i** = max_j(w_j·|r_j* - v̄_ij|). Rank ascending.

   $$\begin{gathered} r_j^{\ast }=\underset{i}{\max }{\bar{v}}_{ij}(\text{benefit}); \\ {y}_i^{\ast \ast }=\underset{j}{\max }(w_j|r_j^{\ast }-{\bar{v}}_{ij}|); \\ \text{rank ascending by }{y}_i^{\ast \ast } \end{gathered}$$

   Stanujkic-Magdalinovic-Jovanovic-Stojanovic 2012; Brauers-Zavadskas 2006 §ref point

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