ELECTRE I - ELimination Et Choix Traduisant la REalité I (kernel / choice)

Overview

ELECTRE I produces a choice recommendation (kernel), NOT a complete ranking. The kernel K contains alternatives not dominated by any other. Increase c̄ or decrease d̄ to obtain a smaller, more selective kernel.

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

   Vector normalisation of decision matrix.

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_k{x}_{kj}^2}}$$

   Roy 1968, p.61 Eq.(1)

2. 2

   Weighted normalised matrix v_ij = w_j r_ij.

   $$v_{ij}=w_j{r}_{ij}$$

   Roy 1968, p.61 Eq.(2)

3. 3

   Concordance index C(a,b) = Σ_{j: Δ_j(a,b)≥0} w_j.

   $${\Delta }_j(a,b)=\{\begin{array}{ll}{v}_{aj}-v_{bj}& j\in J^{+}\\ v_{bj}-v_{aj}& j\in J^{-}\end{array};C(a,b)=\sum _{\{j:{\Delta }_j(a,b)\ge 0\}}{w}_j$$

   Roy 1968, p.62 Eq.(3)

4. 4

   Discordance index D(a,b) = max(-Δ_j(a,b))/max|v_kj−v_lj|.

   $$D(a,b)=\frac{{\max }_{\{j:{\Delta }_j(a,b)<0\}}(-{\Delta }_j(a,b))}{{\max }_{k,l,j}|v_{kj}-v_{lj}|},{\Delta }_j(a,b)=\{\begin{array}{ll}{v}_{aj}-v_{bj}& j\in J^{+}\\ v_{bj}-v_{aj}& j\in J^{-}\end{array}$$

   Roy 1968, p.62 Eq.(4)

5. 5

   Concordance/discordance thresholds c̄ and d̄.

   $$\bar{c}=\frac{1}{m(m-1)}\sum _{a\ne b}C(a,b),\bar{d}=\frac{1}{m(m-1)}\sum _{a\ne b}D(a,b)$$

   Roy 1968, p.63 Eqs.(5)-(6)

6. 6

   Outranking relation a S b ⟺ C(a,b)≥c̄ AND D(a,b)≤d̄; kernel via graph.

   $$aSb⟺C(a,b)\ge \bar{c}\text{AND}D(a,b)\le \bar{d}$$

   Roy 1968, p.63 Eq.(7)

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

• Empty kernel: thresholds too strict - lower c̄ or raise d̄.

• Full kernel (no outranking relations): thresholds too lenient - raise c̄ or lower d̄.

Works with

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