#### 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 cjp, of statements "aj is exactly at the pth place in the ordered sequence of the alternatives in A", denoted Tjp. The Cjp 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 Cjp’s they must be decomposed into already known credibility levels rij, and thus the statements Tjp must be decomposed into elementary statements with known credibility levels rij. For that, further notations are introduced. Note that for an alternative aj being exactly at the pth 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 aj is fixed, then

= the subset of alternatives to which aj is preferred,

= the subset of alternatives which are preferred to aj,

= the subset

Obviously,

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

Furthermore, the statement "aj is preferred to all the elements of " is a conjunction of the already known statements "aj is preferred to a", with the credibility level equal to rj, for all the elements ai of .

Similarly, the statement "all the elements of are preferred to aj" is a conjunction of the already known statements "aj is preferred to aj", with the credibility level equal to rij, for all the elements ai of .

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

The computation of the cjp 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 aj, a fuzzy membership function values show the credibility of having this alternative at the pth 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 aj(j = 1, 2, …, m) to be at this place. Also the most credible alternatives, candidates for the place, are listed.