Fuzzy PROMETHEE - Fuzzy extension of PROMETHEE

Overview

fuzzy-promethee extends PROMETHEE to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Output

preference flow, higher is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) 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).

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

   Goumas-Lygerou 2000, EJOR 123:606-613 (preference function definition)

2. 2

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

   $$\pi (a,b)=\sum _{j=1}^n{w}_j\cdot P_j({\tilde{x}}_{aj}\ominus {\tilde{x}}_{bj})\text{ (Fuzzy Strict Preference)}$$

   Goumas-Lygerou 2000, EJOR 123:606-613 (pairwise preference index)

3. 3

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

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

   Goumas-Lygerou 2000, EJOR 123:606-613 (positive flow)

4. 4

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

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

   Goumas-Lygerou 2000, EJOR 123:606-613 (negative flow)

5. 5

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

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

   Goumas-Lygerou 2000, EJOR 123:606-613 (net flow & ranking)

Fits when / Look elsewhere when

Assumptions to verify

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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy TFN: l ≤ m ≤ u, all ≥ 0 before computation.

• Defuzzification method affects ranking: centroid (l+m+u)/3 is the canonical choice but alternatives exist.

Works with

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