#### 3.5.2 Fuzzy method –1 (Non–dominated layers)

Ranking procedures based on fuzzy logic assume a fuzzy preference relation m : A´ AÕ Š 0,1‹ ‹ on a given set A of alternatives. The membership function is represented by the matrix . The values rij = m (ai, aj) are understood as the degrees to which the preference expressed by the statements "ai is preferred to aj" are true.

It may be pointed out that non–fuzzy weak preferences are usually modeled by quasi–order relations, i.e., they are assumed to possess both reflexivity and transitivity properties. In the case of strict preferences, anti–reflexivity and transitivity are assumed. But, in real life situations, transitivity may not be an inherent property of the relation r; hence, when modeling this relation, it seems reasonable to consider a more general class of reflexive or anti–reflexive fuzzy relations. It is therefore assumed:

In the case of weak preference, m is reflexive

m (ai, ai) = rii = 1 for all aiÎ A.

In the case of strict preference, m is anti–reflexive

m (ai, ai) = rii = 0 for all ai Î A

In this method, we are concerned with the determination of non-dominated (hereafter denoted as ND) alternatives, and we consider a set of all non-dominated alternatives as a solution to the problem of identifying the highest- level core of alternatives. The reason is that ND alternatives are either equivalent to one another, or not comparable to one another on the basis of the preference relations considered, and they are not dominated in a strict sense by others. Therefore, not being in a position to prefer any one of them, we should consider them as potentially rational choices.

To determine the fuzzy set of ND alternatives, we define two fuzzy relations corresponding to the given preference relation R: Fuzzy quasi-equivalence relation and strict preference relation. Formally, they are defined as follows:

Fuzzy quasi-equivalence relation Re:

Re = R Ç R-1

Fuzzy strict preference relation Rs:

Rs = R \ Re = R \ (R Ç R-1) = R \ R-1

where R-1 is a relation opposite to the relation R..

Furthermore, the following membership functions are defined respectively for Re and Rs:

m e(ai, aj) = min(rij, rji)

For any fixed alternative ajÎ A the function m s(aj, ai) describes a fuzzy set of alternatives which are strictly dominated by aj.. The complement of this fuzzy set, described by the membership function 1 - m s(aj, ai), is for any fixed aj the fuzzy set of all the alternatives which are not strictly dominated by aj. Then the intersection of all such complement fuzzy sets (over all ajÎ A) represents the fuzzy set of those alternatives aiÎ A, which are not strictly dominated by any of the alternatives from the set A. Thus, according to the definition of intersection

The value of represents the degree to which the alternative ai is not strictly dominated by any of the alternatives from the set A.

The highest level core of alternatives contains those alternatives ai,, which have the greatest degree of non-dominance or, in other words, which gives a value for that is equal to the value:

The value of MND is called the certainty level corresponding to the core defined by:

The subsequent cores are constructed by a repeated application of the procedure described above.

For adapting this method to our original problem, let us consider now the sequence in the following table:

 Sequence No. of cores Set of alternatives Core Certainty level 1 A1 = A 2 . . . . . . . . q q+1 Ø

These sequences can be constructed by a repeated application of the above procedure. We take the sequence of cores (q £ n) for the solution of rank-ordering the elements of A.

Indeed, these cores represent a successive sequence of non-dominated layers in the set of alternatives, the highest level core being removed at each set from the relational structure.