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members. The former satisfy all the constraint functions while the latter do not. Thus,
the ranking of the HM members consists of two corresponding stages: ranking of the
feasible HM members and ranking of the infeasible ones. The ranking of the feasible
HM members is straightforward: they can be sorted using their objective function val-
ues. However, for the infeasible ones, the ranking is based on the Pareto-dominance of
the HM members [12]. An infeasible HM member dominates another member if none
of its constraint function values is larger and at least one is smaller. Formally, the
Pareto-dominance is defined as follows. Suppose there are two infeasible HM members,
1
x
G
and
x
G
. If
{
2
}
G
G
{
}
G
G
1
2
1
2
∀
i
∈
1
2
"
,
I
:
g
(
x
)
≤
g
(
x
)
∧
∃
i
∈
1
2
"
,
I
:
g
(
x
)
<
g
(
x
)
,
i
i
i
i
x
G
. For each infeasible HM member, we can calculate
the number of others that dominate it, which implies its relative degree of violation of
the constraint functions. That is, the rank of an infeasible HM member is determined
by the number of other infeasible HM members by which it is dominated.
After the HM is ranked, the worst HM member
x
G
dominates
we conclude that
1
2
G
x
#
can be selected and compared
G
G
G
with a new solution candidate
x
*
. Note,
x
*
does not need to be feasible. When
x
#
is
G
will replace
G
,
G
only in one of the following three cases:
compared with
x
*
x
*
x
#
G
is feasible, and
G
is infeasible
1)
x
x
*
#
G
and
G
are feasible, and
G
G
2) both
x
*
x
#
f(
x
*
)
<
f(
x
#
)
G
and
G
are infeasible, and
G
dominates
G
3) both
x
x
x
x
*
#
*
#
x
G
replaces
x
G
on condition that
More precisely,
*
#
G
{
}
*
⎧
∀
i
∈
1
2
"
,
I
:
g
(
x
)
≤
0
i
⎪
⎪
⎨
G
{
}
⎧
*
∧
∀
i
∈
1
2
"
,
I
:
g
(
x
)
≤
0
i
⎪
⎨
G
⎪
{
}
#
∧
∨
∀
i
∈
1
2
"
,
I
:
g
(
x
)
≤
0
i
G
⎪
⎩
⎪
⎪
⎩
{
}
#
∧
∃
i
∈
1
2
"
,
I
:
g
(
x
)
>
0
i
G
G
⎪
*
#
f(
x
)
<
f(
x
)
G
{
}
⎧
*
∃
i
∈
1
2
"
,
I
:
g
(
x
)
>
0
i
(6)
⎪
⎪
∧
G
⎪
⎪
⎪
{
}
#
∃
i
∈
1
2
"
,
I
:
g
(
x
)
>
0
i
∨
∧
⎨
G
G
⎪
⎪
{
}
*
#
∀
i
∈
1
2
"
,
I
:
g
(
x
)
≤
g
(
x
)
i
i
⎪
⎪
∧
G
G
{
}
*
#
⎪
∃
i
∈
1
2
"
,
I
:
g
(
x
)
<
g
(
x
)
⎩
i
i
x
G
and
x
G
and replace-
Figure 4 illustrates this procedure of comparison between
#
*
x
G
with
x
G
.
ment of
#
*
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