Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 76

i
12

( )

Based.on.the.deinition.of. Q i .in.Lemma.4.7,.one.can.have. a

( ) i

11

= Q ,. a = Q ,

1 3

( ) i

21

( ) i

22

a = Q ,.and. a = Q ..Hence,

4 2

P ( , t 1) Q P

2

( , ) t ( Q Q P ) ( , ) ( t P , ) t Q P

2

( , ) t

ξ + = ξ + + ξ ξ + ξ

1 1 1 3 4 1 2 2 2

2 2

= Q P ( ξ , ) t + ( Q Q P + ) ( ξ , )(1 t − P ( ξ , )) t + Q (1 − P ( ξ , )) t

1 1 3 4 1 1 2 1

. (4.64)

2

= ( Q Q Q Q P + − − ) ( ξ , ) t + ( Q Q + − 2 Q P ) ( ξ , ) t Q +

1 2 3 4 1 3 4 2 1 2

1
2 (1

= ( C + C P ) ( ξ , ) t + − C − C )

row col 1 row col

.

By.iterating.backward,.one.has

t

1
2 (1

1 ( − C + C )

t row col

P ( ξ , ) t = ( C + C ) P ( ξ , 0) + − C − C ) ×

1 row col 1 row col

(1 ( − C + C )

row col

. (4.65)

1
2

1
2

t

= + ( C + C ) P ( ξ , 0) −

row col 1

.

t

Based.on.Lemma.4.6,.one.has. C ( + C ) → 0 .when. t → ∞ ..From. Equation.

row col

(4.65) ,. ( .and. ( ..Therefore,.the.theorem.is.true.

P ξ , ∞ = )

1
2

P ξ , ∞ = ) 1 − P ( ξ , ∞ = )

1
2

1 2 1

for.all. S ξ .with. o (ξ) = 1.
Now,.it.is.assumed.that.the.theorem.is.true.for.any. S ξ .with. o (ξ).= n .and. M = 2

n

,.

that.is,. Y

× .or.

P ( ) S

ξ ,.where. P ( ξ .and. Y .represent. P i

( ξ , ∞ . )

=

1

1 ξ =

1

∀ξ ∈

M M

1 i i

M

and. Y ( ∞ ,.respectively,.for.simplicity.
Consider. any. primary. schemata. competition. set. S ϕ . where. ( . It.

o ϕ ) = n + 1.

i

is.easily.proved.that.there.exists. n + 1.subsets.of. S ξ .with. o (ξ) = n .and. M = .

such.that.the.locations.of.the.actual.bits.in. S ξ .are.also.the.locations.of.actual.
bits.in. S ϕ ..Mathematically,.

n

ξ ⊂ ϕ .
S ϕ .can.be.divided.into.two.subgroups.

f f

( ) ( )

L L

S ϕ .and. S ϕ ,.which.are.formed.by.

,0 ,1

the.following.rules:

The.actual.bits.of.the.first.element.in. S ϕ .are.all.zeros,.while.the.other.

elements.are.found.by.flipping.the.last.actual.bit.and.changing.the.
other.bits.according.to.Gray.coding.
The. actual. bits. of. the. irst. element. in.

,0

S ϕ . are. all. zeros. except. the. last.
actual. bit,. while. the. other. elements. are. found. by. lipping. the. last.
actual.bit.and.changing.the.other.bits.according.to.Gray.coding.

,1

Remark4.7

Obviously,.it.can.be.proved.that.

ϕ .and.

S S  = ∅ ..Moreover,.

S  = S S ϕ ,0 ϕ ,1

ϕ ,0 ϕ ,1

size S ( ) = size S ( ) = M .

ϕ ,0 ϕ ,1