Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 56

If. mutation. is. carried. out. after. crossover,. the. expected. proportion. of.
schema.ξ.in.the.population.is

∑

. (4.4)

p ( , ξ t + 1) = P ( ξ → ξ ) p ( , ) ξ t + P ( ξ → ξ ) p ( ξ , ) t

c i c i

ξ

.

i

where

( )

H H

p m o

ξ

d

( , ξ ξ )

m o

( ) ξ − d ( , ξ ξ )

P ( ξ → ξ = ) (1 − ) . and. P ( ξ → ξ = ) p (1 − p ) . (4.5)

i i

.

i

The.former.is.the.probability.that.no.mutation.is .c arried.out.on.schema.ξ;.
the.latte r .is.the.probability.that.a.different.schema. ξ i .is.mutate d .into.schema.
ξ.. d H

( , ) ).is.the.Hamming.distance.between.schemata.ξ.and.ξi. i .
The.quantity.

ξ ξ

i

c ( , ) .in. Equation.(4.4) . is.the.expected.proportion.of.chro-
mosomes.belonging.to.schema.ξ.after.selection.and.single-point.crossover..
Explicitly,

p ξ t

L − 1

p
L

∑

( )

x

p ( , ) ξ t = p ′ ξ ( , ) t − ( p ′ ξ ( , ) t − p ′ ξ ( k , ) t p ′ ξ ( ( ), )) k t . (4.6)

c L R

− 1

.

k = 1

where. k .is.the.crossover.point,.ξ k

L ( ) .is.the.schema.obtained.by.replacing.all.
the.elements.of.ξ.from.position. k +1.to.position. L .with.don't-care.bits.(*),.and.
ξ k

R ( ) .is.the.schema.obtained.by.replacing.all.the.elements.of.ξ.from.position.
1.to.position. k .with.don't-care.bits.
Finally,. ( , ) . in . Equation. (4.6) . is. the. expected. proportion. of. schema. ξ.

after.the.proportional.selection.process,.which.is.calculated.as.follows:

p ′ ξ t

f ( , )

( )

t

ξ

p ′ ξ ( , ) t = p ( , ) ξ t . (4.7)

f t

.

Equations. (4.4) . and . (4.6) . take. into. account. the. effects. of. both. destruction.
and. construction. of. schema. during. mutation. and. crossover,. respectively..
They.thus.yield.an.exact.expression.for.the.expectation.values.of. m ξ ,.that.

is,.the.number.of.strings.matching.ξ..In.the.limit.of. N → ∞ ,. Equations.(4.4)-
(4.7) . present.the.correct.probability.distribution.governing.GA.evolution.
It.should.also.be.noted.that.the.development.of.an.exact.schema.theorem.is.
useful.for.providing.insights.on.the.effect.of.different.operations..For.exam-
ple,.as.shown.in.Reference.[12],.mutation.can.achieve.higher.levels.of.disrup-
tion.than.uniform.crossover,.while.uniform.crossover.is.more.powerful.from.
a.construction.point.of.view.

( , )