Geology Reference
In-Depth Information
Single and Multi-Objective
Optimization Problems
in real-life design problems that have to be dealt
with simultaneously. This situation forces the
designer to look for a good compromise among
the conflicting requirements. Problems of this kind
constitute multi-objective optimization problems.
In general, a multi-objective optimization problem
can be stated as follows:
In the following paragraphs, the single and the
two-criteria design optimization problems and the
optimum designs obtained are presented.
Problem Formulations
T
min
[
C
( ),
s
C
(t, )]
s
s
F
IN
LS
The mathematical formulation of the optimization
problem for the single-objective formulation, as it
was presented in (Lagaros et al., 2004), is defined
as follows:
where
C
( )
s
=
C
( ) +
s
C
( )
s
+
C
( )
s
+
C
( )
s
IN
b
sl
cl
ns
Capacity
subject to
g
( )
s
0 =1,...,
j
k
j
g
PBD s
( )
0 =1,...,
j
k
j
(2)
min
C
C
( )
s
s
where s represents the design vector, F is the
feasible region where all the constraint functions
g Capacity and g PBD are satisfied for the PBD formula-
tion. The objective functions considered are the
initial construction cost C IN and the life-cycle
cost C LS . Several methods have been proposed for
treating structural multi-objective optimization
problems (Coello, 2000, Marler & Arora, 2004).
In this work, the Nondominated Sorting Evolu-
tion Strategies II (NSES-II) algorithm, proposed
by Lagaros and Papadrakakis (2007), is used in
order to handle the two-objective optimization
problem at hand. This algorithm is denoted as
NSES-II( μ+λ ) or NSES-II( μ,λ ), depending on the
selection operator.
Various sources of uncertainty are considered:
on the ground motion excitation which influences
the level of seismic demand and on the model-
ing and the material properties which affects the
structural capacity. The structural stiffness is
directly connected to the modulus of elasticity
E s and E c of the longitudinal steel reinforcement
and concrete respectively, while the strength is
influenced by the yield stress f y of the steel and
the cylindrical strength for the concrete f c and the
hardening b of the steel. In addition to the mate-
rial properties, the cross-sectional dimensions are
considered as random variables. Thus, both for
beams and columns four random variables are
s
F
IN
where
( )
=
C
( )
s
+
C
( )
s
+
C
( ss
)
+
C
( )
s
IN
b
sl
cl
ns
subject to
g
SERV
( )
s
0 =1,...,
j
m
j
ULT s
g
( )
0 =1,...,
j
k
j
(1)
where s represents the design vector, F is the
feasible region where all the serviceability and
ultimate constraint functions (g SERV and g ULT )
are satisfied. In this formulation the boundaries
of the feasible region are defined according to
the recommendations of the EC8. The single
objective function considered is the initial con-
struction cost C IN , while C b ( s ), C sl ( s ), C cl ( s ) and
C ns ( s ) correspond to the total initial construction
cost of beams, slabs, columns and non-structural
elements, respectively. The term “initial cost” of
a new structure corresponds to the cost just after
construction. The initial cost is related to material,
which includes concrete, steel reinforcement, and
labor costs for the construction of the building. The
solution of the resulting optimization problem is
performed by means of Evolutionary Algorithms
(EA) (Mitropoulou, 2010).
In practical applications of sizing optimiza-
tion problems, the initial cost rarely gives a
representative measure of the performance of
the structure. In fact, several conflicting and
usually incommensurable criteria usually exist
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