#### 3.5.1 Ranking by Classical Logic (ELECTRE)

In this method, the matrix P (n, m) is used as the initial data for e analysis. It may be pointed out that the character of preference relation (weak or strong) plays a role only in the steps leading to the construction of the preference matrix. In the subsequent steps of the analysis, the procedure is controlled by other parameters, such as rank difference for concordance and rank difference for discordance.

The procedure ELECTRE consists of two major steps:

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

Concordance relation: Two parameters are used in creating a relation, which reflects the collective opinion that " ai is preferred to aj"

Dc = the rank difference for concordance ( 0 £ Dc £ m–1)

Pc = the minimum proportion for concordance ( 0 £ Pc £ 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 RCk(dc), which measure the dominance of ai over aj according to the evaluation k, are defined as follows:

The aggregation of these matrices measures the average dominance of ai over aj and has the form of a fuzzy relation described by the matrix.

where

Note that higher dc values lead to more rigorous construction rules, since implies

and

Minimum proportion for concordance makes it possible to transform the fuzzy relation RC (dc) into a non–fuzzy one, called the concordance relation, described by the matrix.

the elements of which are defined as follows:

The condition rcij(dc, r c) = 1 means that the collective opinion is in concordance with the statement "ai is preferred to aj" at the level (dc, r c).

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

1. 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:

dd = the rank difference for concordance (0 £ dd £ 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 RDk (dd), which measure the dominance of aj over ai according to the evaluation k, are defined as follows:

The aggregation of these matrices measures the average dominance of aj over ai 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(dd) into a non–fuzzy one, called the discordance relation, described by the matrix

the elements of which are defined as follows:

The condition rdij(dd, r d) = 1 means that the collective opinion is in disconcordance with the statement "ai is preferred to aj", i.e., supports the opposite statement "aj is preferred to ai", at the level (dd, r d). This can be interpreted as a "collective veto" against the statement "ai is preferred to aj"

Note that higher values of dd 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 "ai is preferred to aj" can be accepted if the collective opinion

• is in concordance with it, i.e. rcij(dc, r c) = 1, and
• is not in discordance with it, i.e. rdij(dd, 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 rij 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 dc, r c and /or decreasing dd, 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 aj is non–dominated, if and only if

rij = 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.