Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 68

CaseC
ξ

m n , .
The.last.case.is.that.both.schemata.are.not.ξ..The.condition.to.have. ′

ξ ∈ S

.is

ξ = ξ

m

( ) ( ) ( ) ∆ ξ , I ; ξ ′ , I ∆ ξ , M ; ξ ′ , M ∆ ξ , R ; ξ ′ , R ≠ 0

c , k m c , k c , k m c , k c , k m c , k

m m m m m m m m m m m m

( )

⇒ ∆ ξ ( , I ; ξ , G ) ∆ ξ , M ; ξ , M ′ ∆ ξ ( , R ; ξ , R ) ≡ ∆ ≠ 0

c , k n c , c , k c , k m c , k c , k m c , k IMR

.

m m n m m m m m m m m m m m

. . (4.48)

Similarly,.to.have. ′ ξ = ξ ,.the.condition.is

n

( ) ( ) ( )

∆ ξ , I ; ξ ′ , I ∆ ξ , M ; ξ ′ , M ∆ ξ , R ; ξ ′ , R ≠ 0

c , k n c , k c , k n c , k c , k n c , k

n n n n n n n n n n n n

( )

⇒ ∆ ξ ( , I ; ξ , G ) ∆ ξ , M ; ξ , M ′ ∆ ξ ( , R ; ξ , R ) ≡ ∆ ≠ 0

c , k m c , c , k c , k n c , k c , k n c , k IMR

. ..

n n m n n n n n n n n n n n

. . (4.49)

Therefore,.the.constructive.rate.ξ.for.a.particular.pair.of. ξ m .and. ξ n .is

L L − L L −

g g

p
L L

1
2

∑ ∑ ∑ ∑

cut

PC = ∆ + ∆

C IMR IMR

3

m n

( − + 1)

g

c = 0 c = 0 k ∉κ ( c ) k ∉κ ( c )

. . (4.50)

n m m m n n

Together.with.the.occurrence.probability.of. ξ ,.the.construction.of.

, ξ ∈

S

m n

ξ.in.case.C.can.be.computed.as

L L − L L −

g g

p
L L

∑ ∑ ∑ ∑

cut

P ( , ) ξ t = ×

C

3

( − + 1)

g

ξ ∈ S ξ ∈ S c = 0 c = 0

m n n m

ξ ξ

. (4.51)

1
2

∑ ∑

∆ + ∆ P ( ξ , ) ( t P ξ , ) t

IMR IMR m n

m n

k ∉κ ( c ) k ∉κ ( c )

.

m m n n

By summing.all.three.cases.and.regrouping.the.terms,.the.schema.evolu-
tion.equation.of.ξ.at.time.( t +.1).under.the.cut-and-paste.operation.is

P ( , t 1) P ( , ) t P ( , ) t P ( , ) t P ( , ) t P ( , ) t

ξ + = ξ − ξ − ξ + ξ + ξ

A B 1 B 2 C

= (1 − p ) ( , ) P ξ t

cut

.(4.52)

L L − L L −

g g

p
L L

∑ ∑ ∑ ∑ ∑

cut

+ ∆ P ( ξ , ) ( t P ξ , ) t

IMR m n

3

( − + 1)

n

g

S S c 0 c 0 k ( c )

ξ ∈ ξ ∈ = = ∉κ

.

m n m n n n

ξ ξ

where. κ ( c ) = ( c , c + L ) .

n n n g