ELECTRE III - ELimination Et Choix Traduisant la REalité III (pseudo-criteria, fuzzy outranking)

Overview

ELECTRE III uses pseudo-criteria (with q, p, v thresholds per criterion) instead of binary true/false concordance, producing a fuzzy outranking relation. The credibility matrix σ is then used in a two-pass distillation to produce a complete ranking. It is more discriminating than ELECTRE II but requires more parameter elicitation.

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

   Three thresholds per criterion: indifference q_j, preference p_j, veto v_j.

   $$0\le q_j\le p_j\le v_j$$

   Roy 1978, p.244 Sec.3

2. 2

   Per-criterion concordance c_j(a,b) using fuzzy thresholds q,p (direction-aware Δ_j).

   $${\Delta }_j(a,b)=\{\begin{array}{ll}{x}_{bj}-x_{aj}& j\in J^{+}\\ x_{aj}-x_{bj}& j\in J^{-}\end{array};c_j(a,b)=\{\begin{array}{ll}1& {\Delta }_j(a,b)\le q_j\\ 0& {\Delta }_j(a,b)\ge p_j\\ \frac{p_j-{\Delta }_j(a,b)}{p_j-q_j}& \text{else}\end{array}$$

   Roy 1978, p.245 Eq.(1)

3. 3

   Comprehensive concordance C(a,b) = Σ w_j c_j(a,b).

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

   Roy 1978, p.245 Eq.(2)

4. 4

   Per-criterion discordance d_j(a,b) using veto v_j (direction-aware Δ_j).

   $$d_j(a,b)=\{\begin{array}{ll}0& {\Delta }_j(a,b)\le p_j\\ 1& {\Delta }_j(a,b)\ge v_j\\ \frac{{\Delta }_j(a,b)-p_j}{v_j-p_j}& \text{else}\end{array}$$

   Roy 1978, p.246 Eq.(3)

5. 5

   Credibility σ(a,b) = C · Π (1−d_j)/(1−C) over criteria with d_j > C.

   $$\sigma (a,b)=C(a,b)\prod _{\{j:d_j(a,b)>C(a,b)\}}\frac{1-d_j(a,b)}{1-C(a,b)}$$

   Roy 1978, p.246 Eq.(4)

6. 6

   Distillation procedures (descending + ascending) build two complete preorders.

   $$\text{Descending: }{\sigma }_{\max }-\delta ({\sigma }_{\max });\text{Ascending: reverse direction}$$

   Roy 1978, p.247 Sec.4

7. 7

   Final partial preorder = intersection of descending and ascending.

   $${⪰}^{\text{final}}={⪰}^d\cap {⪰}^a$$

   Roy 1978, p.248

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

• Threshold elicitation is the most difficult part - q, p, v must be set per criterion and reflect the DM's discrimination capability.

Works with

How to cite