PROMETHEE II - Preference Ranking Organisation METHod for Enrichment of Evaluations

Overview

φ(a) ∈ [−1, 1]. Higher net flow φ means stronger overall preference. φ⁺ measures how much a is preferred over all others (leaving flow); φ⁻ measures how much others are preferred over a (entering flow). PROMETHEE I (partial ranking) uses (φ⁺, φ⁻) and allows incomparability. PROMETHEE II (complete ranking) uses net flow φ alone. Choice of preference function type and thresholds strongly affects results - select them with care based on criterion nature.

Output

preference flow, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Complex multi-stakeholder problems, Environmental assessment

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)

Look elsewhere when

Assumptions to verify

• Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
• Non-compensatory preference structure

Edge cases and pitfalls

• Using Type I (usual) preference function for criteria with continuous scales ignores indifference zones - prefer Type III or V.

• Threshold values p and q require domain-expert elicitation - arbitrary values distort the preference structure.

Works with

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