This is a method from the family of outranking (i.e. based on pair-wise comparison) MCDA methods. The result is a partial ordering of alternatives (see Partial Order Preference Structure)
Here:
ELECTRE I chooses alternatives that are preferred by the majority of the criteria and which don;t cause an unacceptable level of discontentment on other criteria.
The ELECTRE methods are based on two concepts:
Or quantification of positive arguments
For ELECTRE 1, when comparing $a$ and $b$
Note that weight should sum up to 1
There are two extreme cases:
Or quantification of negative arguments
We want to find a strong argument against $a \ P \ b$
For ELECTRE I, define $d(a, b)$ as
Another way
For ELECTRE I we define $S \equiv P \lor I$ as: (the "as good as relations", also see Voting Theory Relations)
Or:
Note that this relation gives us partial order:
Graph Kernels are used to identify good alternatives.
It is possible to express the outranking relation $S$ in form of a directed graph
Example:
A kernel of a graph $K \subset A$ is
For the example above:
Another example
Remarks:
The kernel $K$ of a set $A$ forms a set of preferred alternatives.
price | comfort | speed | aesthetic | ||
---|---|---|---|---|---|
1 | 300 | ex | fast | good | |
2 | 250 | ex | med | good | |
3 | 250 | med | fast | good | |
4 | 200 | med | fast | med | |
5 | 200 | med | med | good | |
7 | 100 | poor | med | med | |
$w$ | 5 | 4 | 3 | 3 | $W = \sum w = 15$ |
Perform pair-wise comparisons for all elements $A$
This way we compare each with each:
1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|
1 | - | 10 | 10 | 10 | 10 | 10 | 10 |
2 | 12 | - | 12 | 7 | 10 | 7 | 10 |
3 | 11 | 11 | - | 10 | 10 | 10 | 10 |
4 | 8 | 8 | 12 | - | 12 | 12 | 10 |
5 | 8 | 11 | 12 | 12 | - | 12 | 10 |
6 | 11 | 11 | 11 | 11 | 11 | - | 10 |
7 | 5 | 8 | 5 | 8 | 8 | 9 | - |
(note that this is not normalized - need also to divide on 15)
In this case we define discordance by enumerating cases that are forbidden
The following tables shows what veto we define
price | comfort | |
---|---|---|
$a$ | 250 | poor |
$b$ | 100 | excellent |
So the establish the following relation $S$ and the following graph:
There are two kernels in this case:
Kernel:
The following evaluation table is obtained
$A$ | $B$ | $C$ | $D$ | $E$ | Weight | |
---|---|---|---|---|---|---|
$D$ | 7 | 11 | 15 | 11 | 16 | 15 |
$K$ | 12 | 18 | 6 | 8 | 10 | 15 |
$P$ | 13 | 13 | 14 | 19 | 10 | 15 |
$L$ | 18 | 16 | 19 | 13 | 19 | 25 |
$S$ | 10 | 20 | 16 | 14 | 20 | 20 |
Total weight is $W = 15 + 15 + 15 + 25 + 20 = 100$
Then we construct the concordance index by comparing alternatives pair-wise
For example, consider alternatives $A$ and $B$
$A$ vs $B$ | $B$ vs $A$ | |
---|---|---|
$D$ | 15 | |
$K$ | 15 | |
$P$ | 25 | 25 |
$L$ | 25 | |
$S$ | 20 | |
Total | 50 | 100 |
Normalized | 0.5 | 0.75 |
Repeating this for all pairs, construct the following table:
$A$ | $B$ | $C$ | $D$ | $E$ | |
---|---|---|---|---|---|
$A$ | - | 0.75 | 0.15 | 0.4 | 0.4 |
$B$ | 0.5 | - | 0.35 | 0.75 | 0.6 |
$C$ | 0.85 | 0.65 | - | 0.6 | 0.5 |
$D$ | 0.6 | 0.25 | 0.4 | - | 0.25 |
$E$ | 0.6 | 0.6 | 0.75 | 0.75 | - |
Bold are alternatives that should be preferred provided that there's no discordance
The discordance index is $\tilde d = 6$
Let's compare $A$ with $B$ and $A$ with $C$:
$A$ vs $B$ | $B$ vs $A$ | $A$ vs $C$ | $C$ vs $A$ | |
---|---|---|---|---|
$D$ | 7 | - | 8 | - |
$K$ | 6 | - | - | 6 |
$P$ | 0 | 0 | 1 | - |
$L$ | - | 2 | 1 | - |
$S$ | 10 | - | 6 | - |
Max | 10 | 2 | 8 | 6 |
Rel | $A \ J \ B$ | $A \ J \ C$ | $C \ J \ A$ |
Based on the Concordance and Discordance we define the outranking relation $S$