Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 63

The.probability.of.a.particular.pair.of.ξ m .and.ξ n .is

E t ( ) = P ( ξ , ) ( t P ξ , ) . t (4.27)

.

D m n

and.the.expected.construction.of.ξ.in.this.case.is.computed.as

∑ ∑

P ( , ) ξ t = PC × E t ( )

D D D

ξ ∈ S ξ ∈ S

m ξ n ξ

p
L L

copy

. = . (4.28)

2

( − + 1)

g

L L − L L −

g g

∑ ∑ ∑ ∑

[ ]

× ∆ ξ ( , V ; ξ , G ) ( , ∆ ξ V ′ ξ ; , V P ′ ) ( ξ , ) ( t P ξ , ) t

k m c k , k n k m n

S S c 0 k 0

ξ ∈ ξ ∈ = =

m n

. .

ξ ξ

Summarizing.the.construction.and.the.destruction.of.ξ.stated.in.all.four.
cases,. the. expected. proportion. of. ξ at. time. t +. 1. after. the. copy-and-paste.
transposition.becomes

( ) ( ) ( ) ( ) ( ) ( )

P ξ , t + 1 = P ξ , t − P ξ , t + P ξ , t − P ξ , t + P ξ , t

A B C D

p
L L

copy

( )

= P ξ , t + ×

2

( − + 1)

g

L L − L L −

g g

∑ ∑

− [1 − ∆ ξ ( , V ; ξ , G )] P ( , ) ( , ) ξ t P ξ t k c k ,

c = 0 k = 0

L L − L L −

g g

∑ ∑ ∑

[ ] + ∆ ξ ( , V ; ξ , G . ) ( , ∆ ξ V ′ ξ ; , V P ′ ) ( , ) ( ξ t P ξ , ) t (4.29)

.

k c k , k n k n

ξ ∈ S c = 0 k = 0

n

ξ

L L − L L −

g g

∑ ∑ ∑

− [1 − ∆ ξ ( , V ; ξ , G )] P ( ξ , ) ( , ) t P ξ t

k m c k , m

ξ ∈ S c = 0 k = 0

m ξ

L L − L L −

g g

∑ ∑ ∑ ∑

[ ]

+ ∆ ξ ( , V ; ξ , G ) ( , ∆ ξ V ′ ξ ; , V P ′ ) ( ξ , ) ( t P ξ , ) t

k m c k , k n k m n

ξ ∈ S ξ ∈ S c = 0 k = 0

m

ξ n ξ