Overview of Multiobjective Optimization - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 15

happens,. their. niche. counts. c i . in . Equation. (2.5) . are. computed,. and. the. one.
with.a.lower.niche.count.will.be.the.winner.and.included.in.the.mating.pool.
Oei. et. al.. [55]. showed. that. the. combination. of. tournament. selection. and.
fitness.sharing.would.lead.to.chaotic.perturbations.of.the.population.compo-
sition..Therefore,.a.method.called.continuously.updated.fitness.sharing.was.
suggested..Niche.counts.are.calculated.by.using.individuals.in.the.partially.
filled.next-generation.population.rather.than.the.current-generation.popula-
tion..To.facilitate.the.calculation.of.niche.count,.the.values.of.different.objec-
tive.functions.should.be.normalized,.that.is,

(min)

f − f

m m

f ′ = . (2.6)

m

(max) (min)

f − f

.

m m

where.

f m . is. the. normalized. objective. value. of. f m ,. f (min) . and. f (max) . are. the.
minimum.and.maximum.values.of.the. m th.objective.function,.respectively.

′

2.2.3 Nondominated Sorting genetic algorithm 2

The. nondominated. sorting. genetic. algorithm. 2. (NSGA2). is. an. enhanced.
version.of.the.NSGA.[65].proposed.by.Deb.et.al..[20,21],.and.its.flowchart.
is.given.in . Figure 2.4. . It.has.four.peculiarities:.fast.nondominated.sorting,.
crowding.distance.assignment,.crowded.comparison.operator,.and.elitism.
strategy.

2.2.3.1 Fast Nondominated Sorting Approach

The.design.of.the.fast.nondominated.sorting.approach.in.NSGA2.is.aimed.
at.reducing.the.high.computational.complexity.of.the.traditional.nondomi-
nated.sorting.approach,.which.is. O ( MN 3 ).
In.the.traditional.nondominated.sorting.approach,.each.solution.has.to.be.
compared. with. every. other. solution. in. the. population. to. determine. which.
lies.on.the.irst.nondominated.front..For.a.population.of.size. N ,. O(MN) .com-
parisons.are.needed.for.each.solution,.and.the.total.complexity.is. O ( MN 2 ).for.
searching.all.solutions.of.the.irst.nondominated.front.where. M .is.the.total.
number.of.objectives.
To. obtain. the. second. nondominated. front,. the. solutions. of. the. irst. front.
are. temporarily. discounted. and. the. procedure. is. performed.. In. the. worst.
case,.locating.the.second.front.also.requires. O ( MN 2 ).computations..A.similar.
calculation.applies.to.the.other.(i.e.,.third,.fourth,.and.so.on).nondominated.
fronts..Hence,.in.the.worst.case,.overall. O ( MN 3 ).computations.are.needed.
On.the.other.hand,.the.fast.nondominated.sorting.approach.only.requires.
an. overall. computational. complexity. of. O ( MN 2 ).. Referring. to. the. lowchart.
shown. in. Figure 2.5, . its. operations. are. given. as. follows:. For. each. solution.