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MCDM Methods

Step-by-Step Algorithms

ELECTRE Method

ELimination Et Choix Traduisant la REalité

1968 Bernard Roy 11 Steps

ELECTRE (ELimination Et Choix Traduisant la REalité) is a family of outranking MCDM methods used to compare and rank alternatives under multiple conflicting criteria. ELECTRE determines whether one alternative outranks another by considering two concepts: Concordance, the degree to which the criteria support the statement that one alternative is at least as good as another and Discordance, the degree to which some criterion strongly contradicts that statement.

Video Demonstration

Step-by-Step Algorithm

Calculated cell
Source cells in formula
1

Form the decision matrix

Create the decision matrix with $m$ alternatives and $n$ criteria :

\[ X=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n}\\ x_{21} & x_{22} & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} \]

\(x_{ij}\) presents the performance value of ith alternative on jth criterion, m and n are the numbers of alternatives and criteria respectively.

AlternativeCriterion 1 ($C_1$)Criterion 2 ($C_2$)Criterion 3 ($C_3$)
$A_1$102030
$A_2$152535
$A_3$251535
Criterion type: C₁ Benefit C₂ Benefit C₃ Benefit
Weights: $w_1=0.7279$ · $w_2=0.2009$ · $w_3=0.0712$ (Entropy method)
2

Normalize the decision matrix

Formula

The normalized value is calculated as :

$$ r_{ij}=\frac{x_{ij}}{\sqrt{\displaystyle\sum_{i=1}^{m}x_{ij}^{2}}} $$

\(r_{ij}\) indicates the normalized value of the decision matrix of ith alternative in jth criteria.

AlternativeCriterion 1 ($C_1$)Criterion 2 ($C_2$)Criterion 3 ($C_3$)
$A_1$0.32440.56570.5183
$A_2$0.48670.70710.6047
$A_3$0.81110.42430.6047
3

Construct the weighted normalized decision matrix

Formula

Multiply each normalized value by the corresponding criterion weight :

$$ v_{ij}=w_j \cdot r_{ij} $$

The weighted normalized matrix is :

$$ V=[v_{ij}]_{m\times n} $$

where \(w_j\) is the weight of criterion \(j\).

Using weights from the Entropy method: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$

AlternativeCriterion 1 ($C_1$)Criterion 2 ($C_2$)Criterion 3 ($C_3$)
$A_1$0.23610.11360.0369
$A_2$0.35430.14210.0431
$A_3$0.59040.08520.0431
4

Construct the concordance sets

Formula

For each pair of alternatives \(A_k\) and \(A_l\), the concordance set identifies the criteria where \(A_k\) performs at least as well as \(A_l\).

For a benefit criterion \(j\in J^{+}\) :

$$ C_{kl}=\left\{ j \mid v_{kj} \geq v_{lj} \right\} $$

For a non-benefit criterion \(j\in J^{-}\) :

$$ C_{kl}=\left\{ j \mid v_{kj} \leq v_{lj} \right\} $$
$C_{kl}$$A_1$$A_2$$A_3$
$A_1$—$\emptyset$$\{C_2\}$
$A_2$$\{C_1, C_2, C_3\}$—$\{C_2, C_3\}$
$A_3$$\{C_1, C_3\}$$\{C_1, C_3\}$—
5

Construct the discordance sets

Formula

For each pair of alternatives \(A_k\) and \(A_l\), the discordance set identifies the criteria where \(A_k\) performs worse than \(A_l\).

For a benefit criterion \(j\in J^{+}\) :

$$ D_{kl}=\left\{ j \mid v_{kj} < v_{lj} \right\} $$

For a non-benefit criterion \(j\in J^{-}\) :

$$ D_{kl}=\left\{ j \mid v_{kj} >v_{lj} \right\} $$
$D_{kl}$$A_1$$A_2$$A_3$
$A_1$—$\{C_1, C_2, C_3\}$$\{C_1, C_3\}$
$A_2$$\emptyset$—$\{C_1\}$
$A_3$$\{C_2\}$$\{C_2\}$—
6

Calculate the concordance matrix

Formula

The concordance index is calculated by summing the weights of the criteria in the concordance set :

$$ c_{kl}=\sum_{j\in C_{kl}}w_j $$

The concordance matrix is :

$$ C=[c_{kl}] $$
$c_{kl}$$A_1$$A_2$$A_3$
$A_1$—0.00000.2009
$A_2$1.0000—0.2721
$A_3$0.79910.7991—
7

Calculate the discordance matrix

Formula

The discordance index can be calculated as :

$$ d_{kl}=\frac{\displaystyle\max_{j\in D_{kl}}|v_{kj}-v_{lj}|}{\displaystyle\max_j|v_{kj}-v_{lj}|} $$

The discordance matrix is :

$$ D=[d_{kl}] $$
$d_{kl}$$A_1$$A_2$$A_3$
$A_1$—1.00001.0000
$A_2$0.0000—1.0000
$A_3$0.08020.2410—
8

Determine the concordance threshold and the discordance threshold

Formula

The average concordance index can be used as the concordance threshold :

$$ \bar{c}=\frac{1}{m(m-1)} \sum_{k=1}^{m}\sum_{\substack{l=1\\l\neq k}}^{m}c_{kl} $$

The average discordance index can be used as the discordance threshold :

$$ \bar{d}=\frac{1}{m(m-1)} \sum_{k=1}^{m}\sum_{\substack{l=1\\l\neq k}}^{m}d_{kl} $$
ThresholdValue
Concordance threshold ($\bar{c}$)0.5119
Discordance threshold ($\bar{d}$)0.5535
9

Construct the dominant concordance matrix

Formula
$$ f_{kl}=\begin{cases} 1, & c_{kl}\geq\bar{c}\\ 0, & c_{kl}<\bar{c} \end{cases} $$
$f_{kl}$$A_1$$A_2$$A_3$
$A_1$—00
$A_2$1—0
$A_3$11—
10

Construct the dominant discordance matrix

Formula
$$ g_{kl}=\begin{cases} 1, & d_{kl}\leq\bar{d}\\ 0, & d_{kl}>\bar{d} \end{cases} $$
$g_{kl}$$A_1$$A_2$$A_3$
$A_1$—00
$A_2$1—0
$A_3$11—
11

Construct the aggregate dominance matrix

The aggregate dominance matrix is obtained by multiplying the corresponding elements of the dominant concordance and dominant discordance matrices.

Formula
$$ e_{kl}=f_{kl} \times g_{kl} $$

\(A_k\) outranks \(A_l\) if and only if \(e_{kl}=1\).

Both concordance and discordance conditions must be satisfied simultaneously for an outranking relationship to hold.

$e_{kl}$$A_1$$A_2$$A_3$
$A_1$—00
$A_2$1—0
$A_3$11—
12

Determine the ranking

Using the aggregate dominance matrix, identify which alternatives are dominated and determine the final ranking.

Interpretation

An alternative \(A_k\) outranks \(A_l\) when \(e_{kl}=1\).

An alternative \(A_l\) is dominated if there exists at least one \(A_k\) such that \(e_{kl}=1\) and no \(e_{lk}=1\).

Non-dominated alternatives form the kernel (best set).

AlternativeOutranksIs outranked byStatus
$A_1$None$A_2, A_3$Dominated
$A_2$$A_1$$A_3$Dominated
$A_3$$A_1, A_2$NoneNon-dominated (Kernel)
Outranking relationships: $A_3 \to A_1$, $A_3 \to A_2$, $A_2 \to A_1$
Non-dominated alternative (Kernel): $A_3$
Final ranking: $A_3 \succ A_2 \succ A_1$

ELECTRE Algorithm Flowchart