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*_{j}_{ℓ}, 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.