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Ranking

ELECTRE II - ELimination Et Choix Traduisant la REalité II (complete ranking)

Outranking with strong/weak concordance thresholds (complete preorder)

Roy, B., Bertier, P.1973

Overview

ELECTRE II produces a complete preorder (total ranking with possible ties) unlike ELECTRE I which produces a kernel. The distillation procedure (forward + backward passes) resolves the outranking graphs into a linear order. Thresholds p (strong concordance), q (weak concordance), and v (discordance) control the sensitivity of the outranking relation.

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
Concordance C(a,b) and discordance D(a,b) indices (same as ELECTRE I).
$Δ_{j} (a, b) = {v_{aj} - v_{bj} v_{bj} - v_{aj} j \in J^{+} j \in J^{-}; C (a, b) = {j : Δ_{j} (a, b) \geq 0} \sum w_{j}; D (a, b) = j max (- Δ_{j} (a, b)) /Δ$
Roy-Bertier 1973, p.4 Eqs.(1)-(2)
2
Strong outranking thresholds c⁺, d⁺ and weak c⁻, d⁻ define a S^F b and a S^f b.
$a S^{F} b ⟺ C \geq c^{+}, D \leq d^{+}; a S^{f} b ⟺ C \geq c^{-}, D \leq d^{-}$
Roy-Bertier 1973, p.6 Eqs.(3)-(4)
3
Construct strong graph G^F and weak graph G^f from outranking relations.
$G^{F} = (A, S^{F}), G^{f} = (A, S^{f})$
Roy-Bertier 1973, p.7
4
Direct (descending) preorder by removing strongly-dominated alternatives iteratively.
$V_{1}^{d} = {a : no b S^{F} a}, V_{2}^{d} = V_{1}^{d} ∖ {newly dominated}, \dots$
Roy-Bertier 1973, p.8 Procedure A1
5
Inverse (ascending) preorder by reversing relations.
$V_{k}^{a} built on reverse outranking$
Roy-Bertier 1973, p.8 Procedure A2
6
Median preorder is the average of direct and inverse preorders.
$V_{i}^{m} = \frac{V _{i}^{d} + V _{i}^{a}}{2}$
Roy-Bertier 1973, p.8 Procedure A3
7
Final ranking from median preorder.
$rank (A_{i}) = ⌈ V_{i}^{m} ⌉$
Roy-Bertier 1973, p.9

Look elsewhere when

•Small dataset (m<3). Outranking benefit minimal.
•Compensatory preferences acceptable. Simpler ranking method.

Assumptions to verify

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

Limitations

•Rank reversal known on alternative-set changes (ref: general MCDM literature)

Edge cases and pitfalls

Threshold selection heavily influences results - perform sensitivity analysis on p, q, v.

Works with

Commonly takes its weights from

AHP BWM ENTROPY CRITIC SWARA

How to cite

Roy, B.; Bertier, P. (1973). La méthode ELECTRE II: une application au media-planning. Operational Research '72 (Proceedings IFORS), North-Holland.

BibTeX RIS APA (.txt)

System ID, as it appears in reports and the API

ELECTRE-II

Overview

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

Concordance C(a,b) and discordance D(a,b) indices (same as ELECTRE I).

Δ_{j} (a, b) = {v_{aj} - v_{bj} v_{bj} - v_{aj} j \in J^{+} j \in J^{-}; C (a, b) = {j : Δ_{j} (a, b) \geq 0} \sum w_{j}; D (a, b) = j max (- Δ_{j} (a, b)) /Δ

Roy-Bertier 1973, p.4 Eqs.(1)-(2)

Strong outranking thresholds c⁺, d⁺ and weak c⁻, d⁻ define a S^F b and a S^f b.

a S^{F} b ⟺ C \geq c^{+}, D \leq d^{+}; a S^{f} b ⟺ C \geq c^{-}, D \leq d^{-}

Roy-Bertier 1973, p.6 Eqs.(3)-(4)

Construct strong graph G^F and weak graph G^f from outranking relations.

G^{F} = (A, S^{F}), G^{f} = (A, S^{f})

Roy-Bertier 1973, p.7

Direct (descending) preorder by removing strongly-dominated alternatives iteratively.

V_{1}^{d} = {a : no b S^{F} a}, V_{2}^{d} = V_{1}^{d} ∖ {newly dominated}, \dots

Roy-Bertier 1973, p.8 Procedure A1

Inverse (ascending) preorder by reversing relations.

V_{k}^{a} built on reverse outranking

Roy-Bertier 1973, p.8 Procedure A2

Median preorder is the average of direct and inverse preorders.

V_{i}^{m} = \frac{V _{i}^{d} + V _{i}^{a}}{2}

Roy-Bertier 1973, p.8 Procedure A3

Final ranking from median preorder.

rank (A_{i}) = ⌈ V_{i}^{m} ⌉

Roy-Bertier 1973, p.9