Geology Reference
In-Depth Information
Figure 7. Rule base construction with a) P i = 1, b) P i > 1, c) P i < 1
( )
It should be noted that it is impossible to illus-
trate the trade-off points when we consider more
than two objective functions are being considered.
To solve this difficulty, several multidimensional
visualization methods are proposed. One of these
methods which leads to comprehensive analysis of
the Pareto front is called Level Diagrams method
(Blasco et al., 2008) which is used here in to
visualize the Pareto fronts of the multi-objective
optimization.
In this method, each point of Pareto front must
be normalized to bring them between 0 and 1 based
on its minimum and maximum values (Blasco et
al., 2008) as
c
V t
J
2 =
max max
(27)
i
( )
V t
uc
i
t
i
( )
c
A t
J
3 =
max max
(28)
i
( )
uc
A t
i
t
i
, ..., indicates the number of floors
of the building; and D t
i
where i = 1
12
( ) , D t
i
( ) , V t
i
( ) ,
c
uc
c
( ) are the displacement,
velocity and acceleration of each floor of the
building in controlled and uncontrolled case,
respectively.
( ) , A t
i
( ) and A t
i
V t
i
uc
c
uc
J
M
=
max
J
,
J
m
=
min
J
,
i
=
1 2 3
,
,
i
i
i
i
J
J
m
J
=
i
i
i
M
m
J
J
i
i
(29)
Table 6. Fuzzy rule-bases
The distance of each Pareto front point from
origin can be used for comparison. Here, the
Euclidean norm of all objective functions
Input
BN
SN
Z
SP
BP
BN
BN
BN
Z
Z
Z
SN
BN
SN
Z
Z
Z
Z
SN
Z
Z
Z
SP
2
3
J
=
J i
is used for this purpose. To
SP
Z
Z
Z
SP
BP
i
=
1
2
BP
Z
Z
SP
BP
BP
represent the Pareto front, Y axis is specified for
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