In this method, the matrix ** P** (

The procedure ELECTRE consists of two major steps:

- construction of the relations,
- identification of the cores.

: In this step, two working relations (concordance and discordance) are constructed, which are subsequently used to construct the final dominance relation.*Construction of the relations*

*Concordance relation: *Two parameters are
used in creating a relation, which reflects the collective opinion that " a_{i} is preferred
to a_{j}"

D_{c} = the rank difference for concordance
( 0 £ D_{c }_{£ }m–1)

P_{c} = the minimum proportion for
concordance ( 0 £
P_{c }_{£ }1)

Rank difference for concordance enables the user to influence the evaluation of the data, when constructing the individual preference matrices

where *i*,
*j* = 1,2, . . . , *m*.

The elements of *RC*^{k}(*d*_{c}),
which measure the dominance of *a*_{i}
over *a*_{j} according to the evaluation
*k*, are defined as follows:

The aggregation of these matrices measures the
average dominance of *a*_{i} over *a*_{j} and has the form of a fuzzy relation
described by the matrix.

where

Note that higher *d*_{c} values lead
to more rigorous construction rules, since implies

and

Minimum proportion for concordance makes it
possible to transform the fuzzy relation *RC *(*d*_{c}) into
a non–fuzzy one, called the concordance relation, described by the
matrix.

the elements of which are defined as follows:

The condition *rc*_{ij}(*d*_{c}, r
_{c}) = 1 means that the collective opinion is in concordance with the
statement "*a*_{i} is preferred to *a*_{j}" at the level (*d*_{c},
r _{c}).

It is clear again that increasing the r _{c} value one obtains stricter
conditions for concordance.

*Discordance relation:*The construction of the discordance relation follows the same way as was explained for the concordance. The two parameters controlling the construction are:

*d*_{d}
= the rank difference for concordance (0 £
*d*_{d}_{ }_{£ }m–1)

*r *_{d}
= the minimum proportion for concordance (0 £
r _{d }_{£ }1)

The individual discordance relations are determined first in the matrices

where *i*, *j* = 1,2, . . . , *m*.

The elements of *RD*^{k}^{
}(*d*_{d}), which measure the
dominance of *a*_{j} over *a*_{i} according to the evaluation *k*,
are defined as follows:

The aggregation of these matrices measures the
average dominance of *a*_{j} over *a*_{i} and has the form of a fuzzy relation
described by the matrix.

where

As for the concordance, the second parameter
(maximum proportion for discordance), enables the user to transform the fuzzy
relation *RD*(*d*_{d})
into a non–fuzzy one, called the discordance relation, described by the
matrix

the elements of which are defined as follows:

The condition *rd*_{ij}(*d*_{d}, r _{d}) = 1 means that the collective
opinion is in disconcordance with the statement
"*a _{i}* is preferred to

Note that higher values of *d*_{d} and r
_{d} lead to less rigorous construction rules and thus to weaker
conditions for discordance.

The* Dominance Relation *is composed of the concordance and discordance
relations. The basic idea is that the statement "*a*_{i}
is preferred to *a*_{j}" can be
accepted if the collective opinion

- is in concordance with it, i.e.
*rc*_{ij}(*d*_{c}, r_{c}) = 1, and - is not in discordance with it, i.e.
*rd*_{ij}(*d*_{d}, r_{d}) = 0;

otherwise, this statement has to be rejected. So
the dominance relation, being a function of four parameters, is described by
the matrix *R* of *m* ´ *m*
dimensions

where the elements are obtained according to the expression

The *r*_{ij} is a monotonically
decreasing function of the first two parameters, and a monotonically increasing
function of the last two parameters. This implies that:

- By increasing the
*d*_{c}, r_{c}and /or decreasing*d*_{d}, r_{d}one can diminish the number of connections in the dominance relation, and - By changing the parameters in the opposite direction, one can
create more connections.

*Identification of Cores* The *cores* are subsets of *A* (set
of alternatives) consisting of non–dominated alternatives. An alternative
*a _{j}* is non–dominated, if and
only if

*r*_{ij}
= 0 for all* I* = 1, 2, . . . , *m*.

(i) According
to this criterion the core of the set *A *(the highest level core) is the
subset

- if
*C*(*A*) = ø then all the alternatives are dominated. - if
*C*(*A*) =*A*then all the alternatives are non–dominated.

(ii) In order to find the subsequent core, the elements of the previous core are first removed from the dominance relation. This means that the corresponding rows and columns are removed from the relational matrix. Then the search for a new core is repeated in the reduced structure.

The successive application of (i) and (ii) gives a series of cores . These cores represent consecutive layers of alternatives with decreasing ranks in the preference structure, while the alternatives belonging to the same core are assumed to be of the same rank.