Information Technology Reference
In-Depth Information
process of (nonlinear) diffusion and an individual mechanism of (nonlinear) inertia.
The overall effect of the dynamics is to outline and enhance the natural segmentation
of the decision makers group into homogeneous preference subgroups. For extended
review see [12].
In the meantime a number of different fuzzy approaches have been proposed.
The linguistic approach of Zadeh [28-30] is applicable when the information in-
volved either at individual level or at group level present qualitative aspects that
cannot be effectively represented by means of precise numerical values. Innovative
approaches to the modelling of consensus in fuzzy environments were developed
under linguistic assessments and the interested reader is referred, among others, to
Ben Arieh et al. [2], Cabrerizo et al. [4], and Herrera-Viedma et al. [16, 17]. The
typical problem addressed is that in which decision makers have different levels of
knowledge about the alternatives and use linguistic term sets with different cardinal-
ity to assess their preferences. This is the so-called group decision making problem
in a multigranular fuzzy linguistic context.
21.4
Fuzzy Preference-Based Consensus in Diagnosis
By following the approach adopted in [10] now we introduce the dynamical consen-
sus model aiming at finding the consensual diagnosis alternative.
If
A
=
{
a
1
,...,
a
m
}
is
a
set
of
alternative
diagnoses
considered
and
E
is a set of experts, then the fuzzy preference relation
R
i
of expert
e
i
is given by its membership function
R
i
:
A
=
{
e
1
,...,
e
n
}
×
A
−→
[
0
,
1
]
, such that
⎧
⎨
1
if
a
k
is definitely preferred over
a
l
ξ
1
∈
(
0
.
5
,
1
)
if
a
k
is preferred over
a
l
R
i
(
a
k
,
a
l
)=
0
.
5
if there is indifference between
a
k
and
a
l
⎩
ξ
2
∈
(
0
,
0
.
5
)
if
a
l
is preferred over
a
k
0
if
a
l
is definitely preferred over
a
k
m
. Moreover, with
r
kl
where
i
=
1
,...,
n
and
k
,
l
=
1
,...,
:
=
R
i
(
a
k
,
a
l
)
, we impose
r
kl
+
r
lk
=
1. This implies that
r
kk
=
m
. Here,
for the sake of simplicity, we assume that the alternative diagnoses available are
only two (
m
0
.
5forall
i
=
1
,...,
n
and
k
=
1
,...,
2), which means that each individual preference relation
R
i
has only
one degree of freedom, denoted by
x
i
=
=
r
i
12
.
In the dynamical consensus model each expert
e
i
for
i
n
, is represented by
a pair of connected nodes, a primary node (dynamic) and a secondary node (static).
The
n
primary nodes, denoted
r
i
∈
[
=
1
,...,
, form a fully connected subnetwork and
each of them encodes the individual opinion of a single expert. The
n
secondary
nodes, denoted
s
i
∈
[
0
,
1
]
, on the other hand, encode the individual opinions origi-
nally declared by the experts, and each of them is connected only with the associated
primary node.
The iterative process of opinion transformation corresponds to the gradient dy-
namics of a cost function
W
, depending on both the present and the original network
configurations.
W
combines a measure
V
of the overall agreement in the present
0
,
1
]
