3.5.3 Fuzzy method –2 (Ranks)

The input relation to this method is the same as to Method-1 i.e., the matrix R which has to be reflexive or anti-reflexive. However, the question to be answered here is quite different.

Fuzzy method-2 looks for the level of credibility, denoted by c_jp, of statements "a_j is exactly at the p^th place in the ordered sequence of the alternatives in A", denoted T_jp. The C_jp values form a matrix M of m ´ m dimensions representing a fuzzy membership function, in which the rows correspond to the alternative and the columns to the possible positions in the sequence 1, 2, . . ., m.

In order to make possible the calculation of C_jp’s they must be decomposed into already known credibility levels r_ij, and thus the statements T_jp must be decomposed into elementary statements with known credibility levels r_ij. For that, further notations are introduced. Note that for an alternative a_j being exactly at the p^th place means that it is preferred to m – p alternatives and is preceded by the remaining p – 1 alternatives. When the subset of alternatives after a_j is fixed, then

= the subset of alternatives to which a_j is preferred,

= the subset of alternatives which are preferred to a_j,

= the subset

Obviously,

and the statement T_jp is equivalent to a sequence of statements "a_j is preferred to all the elements of and all the elements of are preferred to a_j", connected by the disjunctive (logic) operator.

Furthermore, the statement "a_j is preferred to all the elements of " is a conjunction of the already known statements "a_j is preferred to a_ℓ", with the credibility level equal to r_jℓ, for all the elements a_i of .

Similarly, the statement "all the elements of are preferred to a_j" is a conjunction of the already known statements "a_j is preferred to a_j", with the credibility level equal to r_ij, for all the elements a_i of .

Applying the corresponding fuzzy operators, the elements of the matrix M can be obtained as follows:

The computation of the c_jp values is performed using an optimization procedure, which produces a series of subsets (while keeping j and p fixed) with strictly monotonically increasing values of the function to be maximized in successive steps.

The program provides two ways of interpretation of the matrix M.

FUZZY SUBSETS OF ALTERNATIVES BY RANKS

For each alternative a_j, a fuzzy membership function values show the credibility of having this alternative at the p^th place (p = 1,2, …, m). Also, the most credible ranks (places) for each alternative are listed.

Fuzzy subsets of alternatives by ranks

For each rank (place) p, a fuzzy membership function value shows the credibility of the alternative a_j(j = 1, 2, …, m) to be at this place. Also the most credible alternatives, candidates for the place, are listed.