Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 55

that.the.schema.theorem.cannot.be.practically.used.to.predict.the.behavior.
of.a.GA.over.multiple.generations..It.may.not.even.be.meaningful.since.only.
a.lower.bound.is.presented.
However,.in.spite.of.many.criticisms,.it.does.not.mean.that.the.schema.the-
orem.is.useless..Efforts.have.been.made.to.have.the.schema.theory.be.a.more.
quantitative. one.. By. using. the. microscopic. dynamics. of. GA. together. with.
the.appropriate.measurement.function.[1],.the.schema.theorem.is.expressed.
to.show.how.changes.in.different.macroscopic.properties.of.populations.in.
a.GA.can.be.derived..It.indeed.proves.the.intuitive.idea.about.what.makes.
a.GA.work;.that.is,.offspring.with.above-average.fitness.can.be.produced.by.
recombining.schemata.with.above-average.fitness.

4.1.3 Stephens and Waelbroeck's Model

To. further. improve. Holland's. model,. Stephens. and. Waelbroeck. derived. a.
new.schema.evolution.equation.that.gave.an.exact.formulation.(rather.than.a.
lower.bound).for.selection,.single-point.crossover,.and.mutation.[13-15]..This.
theorem.has.been.used.as.a.starting.point.for.many.other.results.explaining.
the.behavior.of.a.GA.over.multiple.generations,.with.the.assumption.of.an.
infinite.population.size.
In. Stephens. and. Waelbroeck's. model,. the. effects. of. schema. construction.
are. made. explicit.. An. exact. formulation. for. the. expected. number. of. chro-
mosomes.belonging.to.a.schema.can.then.be.obtained.by.considering.both.
schema.construction.and.destruction..In.the.process.of.the.GA,.schema.con-
struction. commonly. occurs,. and. it. is. important. for. the. search.. Offspring.
belonging. to. ξ. can. be. formed. by. mating. two. chromosomes. that. do. not.
belong.to.schema.ξ..A.similar.effect.can.be.observed.by.mutation.
It.is.possible.to.determine.the.exact.number.of.chromosomes.belonging.to.
a.schema.in.the.next.generation.by.considering.all.possible.ways.that.sche-
mata.may. survive,.be. destroyed,.or.be. created.. A.major.advantage. of.such.
an.exact.theorem.is.that.the.expected.number.of.chromosomes.matching.a.
schema.can.be.predicted.over.multiple.generations,.which.clearly.overcomes.
the.shortcomings.of.Holland's.theorem.
In.the.following,.the.exact.schema.evolution.equation.derived.in.References.
13-15. is. reviewed.. The. equation. is. derived. for. schemata. evolving. under. the.
effects.of.three.genetic.operators:.proportional.selection,.crossover,.and.muta-
tion.. For. simplicity,. single-point. crossover. is. considered. (refer. to. [12]. for. uni-
form.crossover)..Instead.of.calculating.the.expected.number.of.chromosomes.
matching.a.schema,.the.expected.relative.proportion.is.considered.as.follows:

m t
N

( , ) . ξ

(4.3)

p ( , ) ξ t =

.

where. ( , ) .is.the.number.of.chromosomes.matching.the.schema.ξ.at.gen-

eration t ,.and N .is.the.population.size.

m t