Grey-PROMETHEE - Grey extension of PROMETHEE

Overview

grey-promethee extends PROMETHEE 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

preference flow, 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

   Choose preference function P_j and parameters (q,p) per criterion (d_j direction-aware).

   $$d_j(a,b)=\{\begin{array}{ll}{x}_{aj}-x_{bj}& j\in J^{+}\\ x_{bj}-x_{aj}& j\in J^{-}\end{array};P_j(d):\mathbb{R}\to [0,1]$$

   Brans-Vincke 1985, p.649 Eq.(2)

2. 2

   Pairwise preference π(a,b) = Σ w_j P_j(d_j(a,b)).

   $$\pi (a,b)=\sum _{j=1}^n{w}_j{P}_j(d_j(a,b))$$

   Brans-Mareschal 2002, p.41 Eq.(5)

3. 3

   Positive flow φ^+(a) = (1/(m-1)) Σ_b π(a,b).

   $${\varphi }^{+}(a)=\frac{1}{m-1}\sum _{b\ne a}\pi (a,b)$$

   Brans-Mareschal 2002, p.41 Eq.(6)

4. 4

   Negative flow φ^−(a) = (1/(m-1)) Σ_b π(b,a).

   $${\varphi }^{-}(a)=\frac{1}{m-1}\sum _{b\ne a}\pi (b,a)$$

   Brans-Mareschal 2002, p.41 Eq.(7)

5. 5

   Net flow φ(a) = φ^+(a) − φ^−(a); descending complete ranking (PROMETHEE II).

   $$\varphi (a)={\varphi }^{+}(a)-{\varphi }^{-}(a)$$

   Brans-Mareschal 2002, p.41 Eq.(8)

Fits when / Look elsewhere when

Assumptions to verify

• Thresholds (q, p, v) can be expressed in Grey scale
• Veto + concordance semantics adapted to fuzzy arithmetic

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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