Civil Engineering Reference
In-Depth Information
1
MCS
Possibility
Necessity
Limit state
0.8
0.6
0.4
0.2
0
5
000
25
000
45
000
65
000
85
000
Force,
F
(kN)
6.2
Bounds for the probability measure based on the possibility
function.
=
supremum, the largest value over the universe of discourse of
u
; min
=
minimization operator;
Inf
u
=
infi mum, the smallest value over the universe
of discourse of
u
; and max
the maximization operator. Thus, from Fig. 6.2
and Equations 6.2 and 6.3, the possibility
=
Π
and necessity
N
can be com-
puted to be 1 and 0.222. Indeed, the MCS result is bounded between the
two results, i.e.
N
<
MCS
<
Π
.
6.2.3 Evidence theory (Dempster-Shafer, DS, theory)
Evidential reasoning or Dempster-Shafer (DS) theory (Dempster 1967;
Shafer 1976) provides a generalization for Bayesian theory. The DS theory
is suited to handle non-specifi c, confl icting, and ambiguous information. The
DS theory combines the rigor of probability theory with the fl exibility of
rule-based systems.
Let
x
{
x
1
,
x
2
, . . . ,
x
n
} be a mutually exclusive and exhaustive set of
payoffs (for
n
criteria), which is referred to as the
frame of discernment
. The
power set
c
has a total of 2
|
x
|
subsets in the domain, where
=
|
x
|
is the cardinal-
ity of the
frame of discernment
. For example, if
|
x
| =
n
=
3, the power set
c
contains 8 subsets
X
l
(
l
=
1, 2, . . . , 8), i.e.,
{
{}{}{}{
}{
}{
}{
}
}
c
=
φ
,
x
,
x
,
x
,
xx
,
,
xx
,
xx
,
,
xxx
,
,
.
1
2
3
1
2
2
3
1
3
1
2
3
Three important concepts in DS theory are
basic probability assignment
,
belief
, and
plausibility
. The
basic probability assignment
(
p
) expresses the
proportion of all available relevant evidence that supports the claim that a
particular element of the power set
c
belongs to the (sub)set
X
l
but to no
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