Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 72

4.3.2 Distribution Matrix for Cut-and-Paste

Similarly,.let.us.consider.the.cut-and-paste.operation..Assuming.that. p cut = . 1,

Equation.(4.52). becomes

∑ ∑

( ) i

P ( ξ , t + 1) = b P ( ξ , ) ( t P ξ , ) t .for. ξ ∈ S i , = 1, 2,  , M . (4.59)

i mn m n i ξ

ξ ∈ S ξ ∈ S

.

m ξ n ξ

where

L L − L L −

g g

1

∑ ∑ ∑

( ) i

b = ∆ . (4.60)

mn IMR

3

n

( L L − + 1)

g

( )

c = 0 c = 0 k ∉κ c

.

n m n n

and.the.distribution.matrix.can.be.defined.as.follows:

( ) i ( ) i ( ) i

b b b

11 12 1 M

( ) i ( ) i ( ) i

b b b

( ) i

B =

21 22 2 M

. (4.61)

( ) i ( ) i ( ) i

b b b

M 1 M 2 MM

.

such.that

( )

i

y t ( + 1) = Y t B Y t ′ ( ) ( ) . for. i = 1,.2, …,M . (4.62)

.

i

where. Y ( t ).is.deined.as.given.in . Equation.(4.57) .

4.3.3 lemmas

Before.presenting.the.proof.of.Theorems.4.1.and.4.2,.some.lemmas.are.given..
The.proofs.of.these.lemmas.are.given.in.Appendix.A.

Lemma4.1

Considering.a.quadruple. q = ξ ( , ξ , c k , ) ,.a.set.of q is.deined.as

m n

Q q c = { : ∈ [0, L L − ]; k ∈ [0, L L − ]; ξ ∈ S ; ξ ∈ S },

g g m ξ n ξ

.

and.a.function.is.specified.by

( )

i

f ( ) q = δ ( f ( ξ , V f ), ( ξ , G )) × δ ( f ( ξ , V f ′ ), ( ξ , V ′ )),

.

T k T m c k , T i k T n k

i

 =

there exists. a. bijective. mapping g: Q → Q such. that q g ( ) . and.

( ) j

 =

( ) i

f ( ) q f ( ). q