PROMETHEE II - Preference Ranking Organisation Method for Enrichment of Evaluations II

Overview

Φ(a) ∈ [−1,+1]. Higher net flow means the alternative outranks more alternatives than it is outranked by. Φ = Φ⁺ − Φ⁻ gives a complete ranking (PROMETHEE II), unlike PROMETHEE I which yields a partial preorder. The 'usual' preference function is equivalent to BORDA count; richer functions (vshape, level) allow modelling indifference and preference thresholds.

Output

net flow, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Supplier selection, Project ranking, Environmental assessment, Energy planning

How it works

1. 1

   For each criterion j, compute pairwise difference matrix d_j(a,b): benefit criteria → x_aj − x_bj; cost criteria → x_bj − x_aj.

   $$d_j(a,b)=\{\begin{array}{ll}{x}_{aj}-x_{bj}& j\in J^{+}\\ x_{bj}-x_{aj}& j\in J^{-}\end{array}$$

   Brans & Vincke 1985, p.649 Eq.(1)

2. 2

   Apply preference function F_j to each pairwise difference to get unicriterion preference degree P_j(a,b) ∈ [0,1]. Common types: usual (0/1), vshape (linear ramp), ushape (step at q), level (combines p and q), vshape_2 (linear with q cutoff).

   $$P_j(a,b)=F_j(d_j(a,b))\in [0,1]$$

   Brans & Vincke 1985, p.649-650 Table 1 (6 preference functions)

3. 3

   Aggregate unicriterion preferences into global pairwise preference index π(a,b) = Σ_j w_j · P_j(a,b).

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

   Brans & Vincke 1985, p.650 Eq.(2)

4. 4

   Compute positive outranking flow Φ⁺(a) and negative outranking flow Φ⁻(a).

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

   Brans et al. 1986, p.230 Eqs.(2)-(3)

5. 5

   Net flow Φ(a) = Φ⁺(a) − Φ⁻(a). Rank descending (higher net flow = better). Complete ranking (PROMETHEE II, as opposed to partial PROMETHEE I).

   $$\Phi (a)={\Phi }^{+}(a)-{\Phi }^{-}(a),\text{rank descending}$$

   Brans et al. 1986, p.231 Eq.(4)

Look elsewhere when

Assumptions to verify

• Decision-maker can specify appropriate preference function type per criterion
• Criteria are measured on commensurate or comparable scales
• Rank reversal is acceptable (or use SPOTIS/RAFSI if it is not)

Edge cases and pitfalls

• All alternatives have equal Φ: occurs when π(a,b) = π(b,a) for all pairs - very symmetric data.

• Rank reversal is known for PROMETHEE II (Mareschal 2008) - adding/removing alternatives can change rankings.

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