Biomedical Engineering Reference
In-Depth Information
2
ε
=
min
ε
∀
f
∈
B
.
(5.17)
2
1
2
f
f
,
f
1
f A
∈
.
(
)
I A B
,
=
max
2
ε
.
(5.18)
ε
2
f
f
∈
B
.
or.equivalently
1
f
f
(
)
i
I A B
,
=
maxminmax
1
.
(5.19)
ε
2
2
1
f
∈
B f A
∈
≤ ≤
i M
i
.
While. comparing. two. solution. sets.
A
. and.
B
,. three. different. cases. can.
result:
Case.I:.
I A B
(
,
) 1
≤
.and.
I B A
( ,
)
>
1
.⇒.Set.
A
.is.better.than.Set.
B
.
ε
ε
Case.II:.
I A B
(
,
)
>
1
.and.
I B A
( ,
) 1
≤
.⇒.Set.
B
.is.better.than.Set.
A
.
ε
ε
Case.III:.
I A B
(
,
)
>
1
.and.
I B A
( ,
)
>
1
.⇒.Sets.
A
.and.
B
.are.incomparable.
ε
ε
This. gives. rise. to. three. scenarios. when. two. multiobjective. optimization.
algorithms,.say.
MO
A
.and.
MO
B
,.are.compared:
.
1.. Favorable.to.the.
MO
A
:.
N
Γ
> Γ
.and.
N
Γ
> Γ
N
N
I
II
I
III
.
2.. Unfavorable.to.the.
MO
A
:.
N
Γ
> Γ
.and.
N
Γ
> Γ
N
N
II
I
II
III
.
3.. Inconclusive:.
N
Γ
> Γ
.and.
N
Γ
> Γ
N
N
III
I
III
II
where.
N
Γ ,.
N
I
Γ ,.and.
N
II
Γ .are.the.number.of.occurrences.of.Cases.I,.II,.and.
III,.with.the.solution.sets.
A
.and.
B
.obtained.by.
MO
A
.and.
MO
B
,.respectively.
5.6 StatisticalTestUsingPerformanceMetrics
Judging.the.convergence.and.diversity.performance.of.a.multiobjective.evo-
lutionary.algorithm.(MOEA).through.performance.metrics.is.an.important.
issue. in. multiobjective. optimization. [6,7].. Since. MOEAs. are. probabilistic.
search.techniques,.their.offered.solution.quality.is.not.guaranteed.to.be.good.
for.all.simulation.runs.
It.turns.out.that.if.only.one.or.several.sets.of.nondominated.solutions.pro-
duced. by. a. particular. MOEA. are. examined,. a. conclusion. about. the. perfor-
mance.of.the.MOEA.is.probably.wrong..The.reason.is.that.the.acquired.results.
can.be.biased..Hence,.a.statistical.test.is.necessary.to.assess.the.performance.of.
MOEAs.[6]..To.carry.out.the.statistical.test,.a.sufficiently.large.number.of.sim-
ulation.runs,.say.
K
=.50.runs,.is.needed..Given.that.a.nondominated.solution.
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