Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

A ⊗
−γ γ
γ B = (A ∩ B)
1
⋅(A
∪ B)
(X.4)
where γ is between 0 and 1 and controls the amount of “mixing” of the union and intersection
components, i.e., if γ is close to 0, the hybrids acts like an intersection, near 1 produces a union-like
response, and for γ around 0.5, the hybrid takes on the characteristics of a generalized mean.
Zimmermann and Zysno [1980] proposed a hybrid operator for multicriteria aggregation that was
modeled after the compensatory nature of human aggregation. This hybrid operator (γ model) is an
example of Equation (X.4) and is given by:
where,
n
and∑ δ i = n
i=
1
aggregated.
n
n
δ −γ
δ
Y = (
γ
∏(a
) i )
1
(1 −∏(1
− a ) i
i
i ) (X.5)
i=
1
i=
1
a i ∈[0,1] are the criteria satisfactions to be aggregated, 0 ≤ γ ≤ 1 is the mixing coefficient,
. Here, δ i are weights associated with each criterion a i and n is the number of criteria being
Y
δ 1
γ
………
Final Node
δ n
Node 1
γ
…….…..……
γ
Node n
δ 1
….
δ n
δ 1
…..
δ n
….
……
a 1 a n a 1 a n
Figure X.4 A fuzzy aggregation network of multiplicative hybrids used in [Parekh and Keller, 2007].
Krishnapuram and Lee [1992a,b] also developed a back-propagation algorithm to learn the parameters of
operators of this type in a network-based decision application. While the algorithm converged, the