Grey-SAW - Grey extension of SAW

Overview

grey-saw extends SAW 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 matrix ⊗v_ij=[v_ij^L, v_ij^U]; normalize benefit/cost; whiten: v̂_ij=½(v_ij^L+v_ij^U).

   $$\begin{gathered} \otimes v_{ij}=[v_{ij}^L,v_{ij}^U]; \\ \text{benefit: }{\hat{v}}_{ij}=\frac{1}{2}(\frac{v_{ij}^L}{{\max }_k{v}_{kj}^U}+\frac{v_{ij}^U}{{\max }_k{v}_{kj}^U}); \\ \text{cost: }{\hat{v}}_{ij}=\frac{1}{2}(\frac{{\min }_k{v}_{kj}^L}{v_{ij}^U}+\frac{{\min }_k{v}_{kj}^L}{v_{ij}^L}) \end{gathered}$$

   Fishburn 1967; Deng 1989

2. 2

   Weighted sum score S_i = Σ_j w_j·v̂_ij.

   $$S_i=\sum _{j=1}^n{w}_j{\hat{v}}_{ij}$$

   Fishburn 1967 §SAW

3. 3

   Rank descending by S_i. Best: max S_i.

   $$\text{rank descending by }{S}_i$$

   Fishburn 1967 §SAW ranking

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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