Hila, "On fuzzy relation
and fuzzy quotient [GAMMA]-groups," Quasigroups and Related Systems, vol.
C1 C2 C3 C1 -- -- SC IM SC IM C2 IM MC -- -- SC EL C3 IM MC EU MC -- -- C4 MC MC IM ML ML EL C5 IM MC IM ML MC EL C4 C5 C1 SC SC SC IM C2 VLC IM VLC IM C3 EU VLC EU SC C4 -- -- VLC IM C5 IM ML -- -- Table 6: Collective evaluation with fuzzy relation
A fuzzy preference relation on the attributes is a binary fuzzy relation
P on C, where P is a mapping C x C [right arrow] [0, 1] and [p.sub.jr] denotes the preference degree of attribute [c.sub.j] over [c.sub.r].
Step 2 (establishment of the fuzzy relation
matrix R : C [right arrow] V).
where I denotes implication, P denotes premise, and C denotes conclusion, while "[omicron]" is a max-min composition, defined on the sets X, Y, and Z, whose result for fuzzy relations
A [subset] X x Y and B [subset] Y x Z is a fuzzy relation
A [omicron] B [subset] x Z with a membership function:
Since fuzzy relation
equations with max-min composition were firstly introduced by Sanchez [1-3], they have attracted much research attention.
Since the membership function is applicable to all evaluation objects, an evaluation matrix can be obtained as a fuzzy relation
, that is, R: Q x U [right arrow] [0,1], which is defined as
In the formula, [u.sub.i] as the characteristic value, [r.sub.ik] is the form the membership degree of the fuzzy relation
matrix, the fuzzy relation
matrix is obtained, which is not enough for the evaluation of the things.
A fuzzy graph is a pair of functions G = ([sigma], [mu]) where [sigma] is a fuzzy subset of a nonempty set V and [mu] is a symmetric fuzzy relation
on [sigma], i.e.
Apart from sets, another basic element used by the model is the fuzzy relation
. Just as the fuzzy set is a generalization of a non-fuzzy set, so is the fuzzy relation
a generalization of a non-fuzzy relation.
Similarly, assume that experts evaluate scientific research conditions guarantee, the scientific research achievement and the project management situation, then the evaluation results constitute the fuzzy relation
matrix of project guarantee index valuation results: