Civil Engineering Reference
In-Depth Information
other particular subset of
X
l
(Klir 1995). Essentially, this property shows
that, for example, {
x
2
,
x
3
} is a subset of
X
l
but does not belong to
{
x
1
,
x
2
,
x
3
}
.
For a given evidence, every subset
X
l
for which
p
(
X
l
)
≠
0 is called a
focal
∑
()
=
element
, where
p
(
X
l
)
→
[0, 1];
p
(
φ
)
=
0 and
pX
l
1
.
Xl
X
The lower bound, belief (
bel
), for a set
X
l
is defi ned as the sum of all
basic probability assignments
(
bpa
) of the proper subsets
X
r
of the set of
interest
X
l
, i.e.,
X
r
⊆
X
l
. The general relation between
bpa
and belief can be
written as
∑
()
=
()
bel X
p X
[6.4a]
l
r
Xr
Xl
where it can be shown that
()
=
()
=
bel
0
;
bel x
1
[6.4b]
The upper bound,
plausibility
(
pl
), is the summation of
basic probability
assignment
of the sets
X
r
that intersect with the set of interest
X
l
, i.e.,
X
r
∩
X
l
≠
φ
, and therefore it can be written as
∩
∑
φ
()
=
()
pl X
p X
[6.5]
l
r
Xr
Xl
The belief interval is an interval representing range [
plausibility
-
belief
],
where 'true' probability may lie. A narrow belief interval represents more
precise probabilities. It can be shown that the probability is determined
uniquely if
bel
(
X
l
)
pl
(
X
l
); in other words, probability theory is applicable
only where all probabilities are unique and disjoint (Yager 1987). The belief
interval of 'one' implies that no information is available for
X
l
and 'zero'
implies that it has been completely confi rmed by
p
(
X
l
).
If the decision making process requires information from multiple
sources, i.e., more than one decision maker provides the assessment of
p
ij
,
where
i
and
j
are indices for alternatives and critera, respectively) aggrega-
tion (fusion) of evidence is required to estimate a valuation function
V
(
A
i
).
The DS rule of combination strictly emphasizes the agreement among
multiple sources and ignores all confl icting evidence through
normalization
.
A strict
conjunctive logic
through an AND-type operator (product) is
employed in the aggregation of evidence. The DS theory also assumes that
the sources of information are independent. The DS rule of combination
determines the joint
p
DS
(
X
l
) from the aggregation of basic probability
assignments
p
1
(
X
p
) and
p
2
(
X
q
) as follows:
=
∑
(
)
(
)
pX pX
1
p
2
q
Xp
∩=
Xq
Xl
()
=
()
=
pX
;
when
X
≠
φ
,
p
φ
0
[6.6]
DS
l
l
DS
1
−
K
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