The Probabilistic Approach to Preferences Measurement - Probabilistic Composition of Preferences, Theory and Applications - page 24

and

A U

ðÞ ¼sup z 2 R : l A ðÞ a f g

3.2.2 Fuzzy Logic

A way to de ne a distance between two fuzzy numbers A and B is by their α -cuts:

s

Z 1

Z 1

2 2

dA ; B ð Þ ¼ ð A L ðÞ B L ðÞ Þ da þ ð A U ðÞ B U ðÞ Þ da

0 0

Other operations involving fuzzy sets are generalizations of the crisp sets
operations of complementation, intersection and union and are called standard
fuzzy set operations. The complement of a fuzzy set A with membership function
l A is the fuzzy set A c with membership function µ A Cde ned by

l A Cx ðÞ ¼ 1 l A ð x Þ

The intersection A ∩ B between the fuzzy sets A and B with membership func-

tions

µ A and µ B has the membership function de

ned by

l A \ B ðÞ ¼ min ½ l A ðÞ ; l B ðÞ :

The union A ∪ B of the fuzzy sets A and B with membership functions

µ A and

µ B

has the membership function de ned by

l A [ B ðÞ ¼ max ½ l A ðÞ ; l B ðÞ :

cient to belong to
one of the sets and to belong to the intersection is necessary to belong to all the sets
intersected. There are other possible generalizations.

This corresponds to the idea that to belong to the union is suf

3.3 Computation of Probabilities of Preference

In the probabilistic composition, the initial point value of an attribute is seen not as
a measure of de nitive preference but as signaling the position of the location

parameter of a probability distribution of the preference that would occur if the
value of the attribute were observed under similar circumstances in a series of
evaluations of the alternative over time.
Thus, the key idea of the transformation of measurements of attributes using
natural scales into probabilities of preference is to translate each measurement of the
basic attribute into an interval of possible satisfaction evaluations that may occur if