Geology Reference
In-Depth Information
L
U
ρ
≤ ≤
ρ
ρ
(
i
=
1 2
,
, ...,
N
)
(39)
3.4 Multi-Objective
Optimization Algorithm
i
i
i
i
By varying the upper bound value,
ε
2
, for the
objective
f
The multi-objective optimization problem given in
Eqs. (33)-(35) can be solved by a
Pareto optimal
set
- formed by a large number (infinite number)
of Pareto optimal solutions. The Pareto optimal
set can provide an overview of all the tradeoffs
to the designer. To establish a Pareto optimum
point, multi-objective optimization algorithms
should be constructed, one of which transforms
the multi-objective problem into a single-objective
optimization. The key to the transformation is
that the solution of the single-objective problem
should be a point in the Pareto set with respect to
a feasible region and a set of objective functions.
Among transformation methods, the ε-constraint
method is one of the commonly used approaches in
practical problems (Kaisa 1999; Marler and Arora
2004). The ε-constraint method is a technique
that transforms a multi-criteria objective function
into a single criterion by retaining one selected
objective function as the primary criterion to be
optimized and treating the remaining criteria as
constraints.
Herein, the construction cost,
f
2
(
ρ
and minimizing the objective
i
1
(
ρ
, all Pareto optimal points are, in principle,
attainable. Theoretically, there is no limitation on
the range of
ε
2
(
−∞ ≤ ≤ +∞
f
i
ε
2
) regardless of
the convexity conditions. However, this may
result in extensive computational time to find a
Pareto set. In fact, the main difficulty of the ε-
constraint method lies in finding the range of
reasonable values for the upper bound,
ε
2
. Details
can be found in Zou (2002) and Zou et al (2007).
3.5 Multi-Objective Design
Optimization Procedure
The multi-objective inelastic optimal design
procedure is outlined as follows:
1. The optimal member sizes
B
i
and
D
i
are
first found based on the elastic design opti-
mization, which are fixed in the inelastic
design optimization. The initial reinforce-
ment ratios for each member are taken as
the minimum values of the design variable,
ρ
i
, obtained from the elastic optimization
results.
2. Establish the explicit optimization problem
given in Eqs. (33)-(35).
3. Transfer multi-objective design problem Eqs.
(33)-(35) into a single objective optimization
Eqs.(36)-(39).
4. Change
ε
2
in Eq.(40) and generate one Pareto
optimum solution by employing the OC
method. The global convergence is checked
based on the change in the objective function
and the violation of the constraints.
5. Repeat Steps 4 and 5 until the upper bounds
of the active design variables are achieved.
1
(
ρ
, is taken
as a primary objective, while the damage loss,
f
i
2
(
ρ
, is transformed into a design constraint.
The reason for this consideration is that both the
damage loss
f
i
2
(
ρ
in Eq. (32) and the inter-story
drift constraints in Eq. (34) are expressed explic-
itly in terms of the inter-story drifts. The similar-
ity may bring advantages in numerical calcula-
tions. Thus, the multi-objective optimization
problem given in Eqs. (33)-(35) can be transformed
as
i
Minimize:
f
1
(
ρ
(36)
i
Subject to:
f
(
ρ
)
≤
(37)
2
2
i
g
(
ρ
≤
)
1
(
j
=
1 2
,
,
...,
N
)
(38)
j
i
j
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