Overview of Multiobjective Optimization - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 19

. 5.. Calculate. I [ ] distance . for. each. of. the. remaining. nondominated. solu-

tions. i = 2,3,⋯,( l −1).by

I i [ + 1] − I i [ − 1]

m m

I i [ ] = I i [ ] +

. (2.8) distance distance

f

(max)

f

(min)

− m m

.

. where l .is.the.total.number.of.solutions.in.the.nondominated.set. I ,.

I [ i ] m .is.the. m th.objective.function.value.of.the. i th.solution.in I ,.and.
f (min) .and. (max) f m .are.the.minimum.and.maximum.values.of.the. m th.

objective.function,.respectively.

. 6.. Update. m .by.setting. m = m + 1..If. m ≤ M .where. M .is.the.total.number.

of.objective.functions,.go.to.step.3.

A.solution.with.a.smaller.crowding.distance.means.that.it.is.more.crowded..
In.other.words,.this.solution.and.its.other.surrounding.solutions.are.close.
in.distance.

2.2.3.2.2 Crowded-ComparisonOperator
The. crowded-comparison. operator,. implemented. in. the. selection. process,.
aims.to.obtain.more.uniformly.distributed.nondominated.solutions..If.any.two.
solutions. picked. have. different. ranks,. the. solution. with. the. lower. rank. will.
be.chosen..However,.if.their.ranks.are.the.same,.the.solution.with.the.larger.
crowding.distance.(i.e.,.located.in.a.less-crowded.region).is.preferred.

2.2.3.2 Elitism Strategy

The.procedure.for.executing.elitism.is.outlined.as.follows:

. 1.. Assuming.that.the. t th.generation.is.considered,.combine.the.original.

population. P t .and.the.offspring.population. Q t .to.form.a.population. R t .

. 2.. Sort. R t .by.the.fast.nondominated.sorting.approach.and.then.identify.

different.nondominated.fronts.

F , F  . ,

1 2

. 3.. Include.these.fronts.in.the.new.population. P t 1 .one.by.one.until. P t 1 .

+ +

with.size. N .is.full.

If.there.is.insufficient.room.to.include.all.members.of.a.particular.front,.say.
F i .into. P t 1 ,.the.irst j .members.of. F i .with.larger.crowding.distances.will.be.

selected.to.fill.

+

P t 1 ..The.lowchart.of.the.elitism.strategy.is.given.in . Figure 2.8 .

+

2.2.4 Strength Pareto evolutionary algorithm 2

The. strength. Pareto. evolutionary. algorithm. 2. (SPEA2). [69]. is. an. improved.
version.of.SPEA.[70].that.includes.three.distinct.features:.strength.value.and.