Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 80

based.on . Equations.(4.29) .and . (4.51) ,.respectively..The.initial.proportion.of.
each.schema.is.randomly.assigned,.and.it.is.clearly.observed.that.the.pro-
portion.goes.to.the.equilibrium.of.0.125. (1/2 3 ,.despite.the.initial.states..It.is.
true.even.though.some.schemata.in.set. S ξ .may.not.exist.initially.

4.4.2 Simulation 4.2: Primary Schemata Competition
Sets with Different Orders

In.Simulation.4.2,.the.growth.of.different.order.primary.schemata.competi-
tion. sets. in. the. population. is. recorded.. For. the. order. 1,. 2,. 3,. 4. competition.
sets,. the. representatives. are. taken. as. ***#************ , ***#*****#****** ,
***#*#***#****** , and. ***#*#**##****** ,. respectively,. where. #. is. the. actual.
bit..The.initial.proportion.of.each.schema.of.order.4.is.randomly.assigned,.
while.the.initial.portions.of.the.other.three.sets.can.be.calculated..For.exam-
ple,.if.the.proportions.of.schemata.***0*0**00******.and.***0*0**10******.are.0.1.
and.0.05,.respectively,.the.proportion.of.schema.***0*0***0******.is.then.deter-
mined.by.0.1.+ 0.05.= 0.15.
Figures 4.5a . and. 4.5b . show. that,. using. either. the. copy-and-paste. or. the.
cut-and-paste.operation,.the.primary.schemata.competition.sets.of.different.
orders. ultimately. reach. their. own. equilibria.. It. is. also. noticed. that. it. takes.
longer.for.a.higher-order.primary.schemata.competition.set.to.reach.equilib-
rium.than.for.its.corresponding.lower-order.set.

4.5 Discussion
4.5.1 assumptions

It. should. be. remarked. that. the. proof. of. the. theorem. of. equilibrium. is.
based. on. the. assumption. that. the. don't-care. bit. (*). has. an. equal. chance.
to.be.0.or.1..This.assumption.is.not.common.in.other.schema.theories..A.
don't-care. bit. is. not. to. be. compared. with. any. deinite. bits. in. traditional.
genetic. operations;. therefore,. its. chance. to. be. 0. or. 1. is. not. important..
However,. this. assumption. is. required. in. our. proofs;. fortunately,. if. the.
initial.population.is.large.enough.and.contains.sufficient.randomness,.it.
is.more.or.less.true.

4.5.2 implications

As.demonstrated.by.many.previous.reports,.techniques.such.as.fitness.shar-
ing.[4,5,11].and.niche.crowding.[2,18].adopted.in.an.MOEA.are.essential.to.
maintain.the.diversity.in.the.population.and.prevent.a.crowded.population..
From.this.point.of.view,.there.are.at.least.two.implications.for.the.theorems.of.
equilibrium..First,.any.possible.individual.would.have.the.same.occurrence.