Overview of Multiobjective Optimization - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 21

START

Strength, raw fitness,
and density assignment

Set the required
parameters and
generation counter
g = 0

Yes

g = MAXIMUM
GENERATION?

Fitness assignment

No

Population initialization

Environmental selection

Binary tournament
selection

Objective value
evaluation

g = g + 1

Crossover

Strength, raw fitness,
and density assignment

Mutation

Output Pareto-optimal
solutions

Fitness assignment

Objective value
evaluation

Environmental selection

END

Figure 2.9
Flowchart.of.SPEA2.

dominate.each.other..As.a.result,.density.estimation.is.employed.to.discrimi-
nate.between.individuals.having.identical.raw.fitness.values.
The.density.estimation.technique.used.in.the.SPEA2.is.called.the. k th.near-
est-neighbor.method.[63];.the.density.at.any.point.is.a.decreasing.function.of.
the.distance.to.the. k th.nearest.data.point,.which.can.be.expressed.as

1

D i ( ) =

2 .

(2.11)

d i k +

.

where. D ( i ).and. d i k .are.the.density.and.the.found.distance.value.of.solution i ,.
respectively.
A.value.of.two.is.added.in.the.denominator.in . Equation.(2.11) .to.avoid.divi-
sion.by.zero.and.make.the.density.value.always.less.than.one..The.value.of.
d i k .is.obtained.by.the.following.procedures:

. 1.. The.distances.in.objective.space.of.all.individuals j in. A .and. P .are.

calculated.and.stored.in.a.list.

. 2.. Sort.the.list.in.ascending.order.

3.. The. k th.element.gives.the.value.of. d i k ,.where. .

| | | | .

k = P + A