PROMETHEE I - Preference Ranking Organisation METHod for Enrichment Evaluations I (partial ranking)

Overview

PROMETHEE I produces a partial preorder: A_i outranks A_k (aPb) if φ⁺_i > φ⁺_k AND φ⁻_i < φ⁻_k. A_i and A_k are indifferent (aIb) if φ⁺_i = φ⁺_k AND φ⁻_i = φ⁻_k. Otherwise they are incomparable (aRb). For a complete ranking use PROMETHEE II (net flow φ = φ⁺ − φ⁻).

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 decisions

How it works

1. 1

   Compute per-criterion preference degree P_j(A_i, A_k) using selected preference function type and thresholds. For Type I (usual): P=1 if d>0, P=0 otherwise. For Type III (linear): P = min(d, p_j) / p_j.

   $$d_j(a,b)=g_j(a)-g_j(b)\text{ (benefit)},g_j(b)-g_j(a)\text{ (cost)};P_j(a,b)=H_j(d_j(a,b))\in [0,1]$$

   Brans et al. 1986, p.230 Table 1

2. 2

   Compute aggregated preference index π(A_i, A_k) = Σ_j w_j · P_j(A_i, A_k).

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

   Brans et al. 1986, p.231 Eq.(5)

3. 3

   Compute positive flow φ⁺(A_i) = Σ_k π(A_i, A_k)/(m−1) and negative flow φ⁻(A_i) = Σ_k π(A_k, A_i)/(m−1). For PROMETHEE I: partial ranking - A_i outranks A_k if φ⁺_i > φ⁺_k AND φ⁻_i < φ⁻_k.

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

   Brans et al. 1986, p.231-232 Eqs.(6)-(9)

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

• •if d>0, P=0 otherwise. For Type III (linear): P = min(d, p_j) / p_j.
• •if φ⁺_i > φ⁺_k AND φ⁻_i < φ⁻_k.

• PROMETHEE I can yield many incomparable pairs - if a total order is needed, use PROMETHEE II.

Works with

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