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where
x
i
is the design variable;
g
j
,
h
k
, and
w
l
represent the state variables;
N
is the number of
design variables and
m
1
+
m
2
+
m
3
is the number of
state variables. The bar above/below variables
represents the lower/upper bound.
optimization method and the differential equation
method can all solve constrained optimization
problems. In this chapter, the first-order optimiza-
tion method and the penalty concept are utilized.
With regard to this optimization method, Equation
(1) for the constrained case is transformed into
an unconstrained one through penalty functions,
expressed as:
Analytical Hierarchy Process (AHP)
The AHP is a decision-aiding method proposed
by Saaty (1980). It aims at quantifying relative
priorities for a given set of alternatives on a ratio
scale, based on the judgment of the decision-
maker, and stresses the importance of the intui-
tive judgments of a decision-maker as well as the
consistency of the comparison of alternatives in
the decision-making process. The AHP method has
been widely used in evaluation system of many
fields (Skibniewski & Chao, 1992; Mikhailov,
2004; Zhang et al., 2005) and improved or com-
bined with other analysis methods in evaluation
(Klocke et al., 1997; Bao et al., 2004).
Procedures of the AHP solution are generally
as follows (Skibniewski, 1988): (1) A complex
problem is structured by decomposing it into a
hierarchy with enough levels to include all at-
tribute elements to reflect the goals and concerns
of decision-makers; (2) Elements are compared
in a systematic manner using the same scale to
measure their relative importance, and the overall
priorities among the elements within the hierarchy
are established; (3) The relative standing of each
alternative with respect to each criterion element
in the hierarchy is determined using the same
scale; (4) The overall score for each alternative
can then be aggregated, and the sensitivity analysis
can be performed to see the effect of change in
the initial priority setting, while the consistency
of comparison can be measured using Saaty's
consistency ration.
m
m
m
J
J
N
1
2
3
(
)
=
∑
∑
∑
∑
Q x q
,
+
P x
(
)
+
q
P g
(
)
+
P h
(
)
+
P
(
w
l
)
x
i
g
j
h
k
w
i
=
1
j
=
1
k
=
1
l
=
1
0
(2)
where
Q(x,q)
is the unconstrained objective func-
tion;
q
is the bound control parameter;
J
0
is the
reference objective function value that is selected
from the current group of design sets;
P
x
,
P
g
,
P
h
and
P
w
represent penalties of the constrained design
and state variables respectively. The optimization
iteration formula is:
(
j
+
1
)
( )
j
( )
j
x
=
x
+
s d
(3)
j
where
s
j
is the line search parameter and
d
(j)
is
the search direction vector which leads to the
minimum value of
Q(x,q)
. Various slopes and
direction searches are performed for the iteration
until convergence is obtained.
( )
j
(
j
−
1
)
( )
j
( )
b
J
−
J
≤
τ
and
J
−
J
≤
τ
(4)
where
J
(j)
,
J
(j-1)
and
J
(b)
refer to the current, previ-
ous and best objective function values, respec-
tively (Jaishi & Ren, 2005).
τ
is the objective
function tolerance.
The zero-order optimization method is simi-
lar to the first-order optimization method with
the main difference being in the disposition of
variables. Derivatives of the variables are used
in the first-order optimization method whereas
variables themselves are used in the zero-order
optimization method.
First-Order Optimization Method
The first-order optimization method (Wang, Li,
& Mao, 2005; Jaishi & Ren, 2005), the zero order
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