Biomedical Engineering Reference
In-Depth Information
.
and
M
∑
(
)
2
.
(5.14)
d
=
f
(
I
)
−
f
(
e
)
2
m i
m
2
.
m
=
1
.
5.. If. either. of. the. following. two. conditions. is. satisfied,. (a).
d
≤ ,. (b).
r
1
d
≤ ,.
C
=
C
.+.1.
r
2
.
6.. Increment.
i
by.1,.and
if.
i
≤.
N
.where.
N
.is.the.total.number.of.solutions.
in.the.
j
th.nondominated.set,.go.to.step.4.
.
7.. Increment.
j
by.1,.and
if.
j
≤.
S
.where.
S
.is.the.total.number.of.nondomi-
nated.sets,.go.to.step.3.
.
8.. Output.the.value.of.
C
.
A. larger. metric. value. indicates. that. the. nondominated. set. offers. more.
extreme.nondominated. solutions. to.decision. makers.. Note. that. this. metric.
cannot. be. used. alone. to. assess. the. diversity. of. nondominated. sets.. This. is.
because.a.large.metric.value.does.not.guarantee.that.the.obtained.nondomi-
nated.solutions.are.uniformly.distributed..Thus,.it.must.be.adopted.together.
with.other.diversity.metrics.at.the.same.time.
5.5 Binary
ε
-Indicator
It.was.criticized.that.the.unary.metrics.described.in.Sections.5.1-5.4.are.inca-
pable. of. indicating. whether. one. nondominated. set. is. better. than. another.
[22,23]..Therefore,.a.binary.ε-indicator.has.been.proposed.[23].and.used.to.
identify.the.better.performer.when.a.pair.of.nondominated.solution.sets.is.
examined.
Consider. a. minimization. problem. with.
M
. positive. objectives;. an. objec-
tive.vector.
f
1
(
f
1
,
f
1
,
,
f
M
1
)
.is.said.to.ε-dominate.another.objective.vector.
=
1
2
2
2
2
2
,.written.as.
f
1
2
,.if.and.only.if
f
=
(
f
,
f
,
,
f
M
)
ε
f
1
2
1
2
∀
1
≤
i M f
≤
:
≤ ε ⋅
f
.
(5.15)
.
i
i
for.a.given.ε > 0.
The.binary.ε-indicator.
I
ε
.is.deined.as.
2
1
1
2
I A B
(
,
)
=
inf{
∀
f
∈
B f A f
,
∃
∈
:
f
}
ε
ε
ε∈ℜ
for.any.two.nondominated.solution.sets.
A
and.
B
..It.can.be.calculated.as
f
f
1
i
1
2
ε
=
max
∀
f
∈
A f
,
∈
B
.
(5.16)
1
2
f
,
f
2
1
≤ ≤
i M
.
i
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