Grey-TODIM - Grey extension of TODIM

Overview

grey-todim extends TODIM 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; whiten v̂_ij=½(v_ij^L+v_ij^U).

   $${\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U)$$

   Gomes-Lima 1992; Deng 1989

2. 2

   Relative weights: w̄_jr = w_j/w_r, where w_r = max_j(w_j).

   $${\overline{w}}_{jr}=w_j/w_r,w_r=\underset{j}{\max }{w}_j$$

   Gomes-Lima 1992 §relative weights

3. 3

   Dominance contributions: d_j(i,k)=|v̂_ij - v̂_kj|. φ_j(A_i,A_k)= +√(w̄_jr·d_j/Σw̄_jr) if v̂_ij>v̂_kj, 0 if equal, -(1/θ)√(Σw̄_jr·d_j/w̄_jr) if v̂_ij<v̂_kj. θ=1.

   $$\begin{gathered} d_j(i,k)=|{\hat{v}}_{ij}-{\hat{v}}_{kj}|; \\ {\varphi }_j(A_i,A_k)=\{\begin{array}{ll}+\sqrt{{\overline{w}}_{jr}{d}_j(i,k)/{\sum }_j{\overline{w}}_{jr}}& {\hat{v}}_{ij}>{\hat{v}}_{kj}\\ 0& {\hat{v}}_{ij}={\hat{v}}_{kj}\\ -(1/\theta )\sqrt{{\sum }_j{\overline{w}}_{jr}{d}_j(i,k)/{\overline{w}}_{jr}}& {\hat{v}}_{ij}<{\hat{v}}_{kj}\end{array};\theta =1 \end{gathered}$$

   Gomes-Lima 1992 §TODIM; Wang-Wang 2014 §grey TODIM

4. 4

   Overall dominance: δ(A_i,A_k) = Σ_j φ_j(A_i,A_k). Score: ξ_i = Σ_k δ(A_i,A_k). Normalize and rank descending.

   $$\begin{gathered} \delta (A_i,A_k)=\sum _j{\varphi }_j(A_i,A_k); \\ {\xi }_i=\frac{{\sum }_k\delta (A_i,A_k)-{\min }_k{\sum }_k\delta (A_k,\cdot )}{{\max }_k{\sum }_k\delta (A_k,\cdot )-{\min }_k{\sum }_k\delta (A_k,\cdot )}; \\ \text{rank descending} \end{gathered}$$

   Gomes-Lima 1992 §global dominance

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

• •if v̂_ij>v̂_kj, 0 if equal, -(1/θ)√(Σw̄_jr·d_j/w̄_jr) if v̂_ij<v̂_kj. θ=1.

• 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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