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[9]-[15] show their outperform abilities. The paper is organized as follows. Section 1
is the introduction. In section 2, 4 multi-objective strategies are presented. The 3LM-
CDE algorithm is in section 3. MO-3LM-CDE algorithms are tested in 6 benchmark
functions in Section 4. The application of MO-3LM-CDE algorithms used in
optimizing 3 control parameters of Texaco coal-water slurry gasifier is in section 5.
The last section is the conclusion.
2
Multi-objective Optimization Strategies
The multi-objective problem considered in this paper has the following form:
Minimize
: (
F
X
)
=
[
f
(
X
), (
f
X
), ..., (
f
X
)]
(
r
≥
2)
1
2
r
n
XR
∈
.
(1)
Subject to
: (
g
X
X
)
≥=
=
0 (
i
1, 2, ..., )
k
i
h
(
)=0 (
i
1, 2, ..., )
l
i
where the decision vector
X
= (
x
1
,
x
2
, …,
x
n
) belong to the feasible n-dimensional
space
R
n
. The purpose is to search
X
*
= (
x
1
*
,
x
2
*
, …,
x
n
*
), which can satisfy all
constraints and at the same time make the objectives achieve optimum.
Some basic definitions [15] are given as follows.
Definition 1
(Pareto dominance): A vector
u
= (
u
1
,
u
2
, …,
u
r
) is called to dominate
another vector
v
= (
v
1
,
v
2
, …,
v
r
) iff
u
is partially less than
v
, i.e.
{1, 2, ..., }, (
∀
i
r
∈
u
v
)
(
j
{1, 2, ..., }:
≤ ∧ ∃ ∈
r
u
<
v
)
.
i
i
i
i
Definition 2
(Pareto optimal solution): A solution
X
1
is said to be Pareto optimal
solution with respect to feasible regions iff there is no
X
2
for
f
(
X
2
) dominating
f
(
X
1
).
Definition 3
(Pareto optimal set): The Pareto optimal set is defined as the set of all
Pareto optimal solution.
In this paper, 4 strategies of transform multi-objective into single objective are
used. They are listed below.
1.
Main objective method: According to the general technical requirements of
optimization design, the most important objective function is chosen from each
branch objective function as the main objective function. At the same time, other
branch objective functions are transformed into appropriate constraints.
2.
Linear weighted method: According to the importance in the entire optimization
design of each branch objective function, we can get a set of corresponding
weighting factors ω
1
, ω
2
, …, ω
r
. The new unified objective function is made up of
the linear combinations of
f
j
(
X
) and ω
j
as equation (2).
r
f
() min[
X
=
ω
f
( ]
X
(2)
j
j
j
=1
3.
Max-min method: The value of evaluation function is the maximum value of each
branch objective function shown in formula (3).
f
(
X
)
=
min{max [
f
(
X
)]}
(3)
j
1
≤≤
jr
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