Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 78

where. C 0 . and. C 1 . are. constants.. Using. Equation. (4.53). and. assuming. that.
,0 ϕ ∈

S ,.we.have

ϕ

M ' M '

∑ ∑

( ) i

P ( ϕ ) = a P ( ϕ ) ( P ϕ )

i mn m n

m = 1 n = 1

∑ ∑ ∑ ∑

( ) i ( ) i

C = a P ( ϕ ) ( P ϕ ) + a P ( ϕ ) ( P ϕ )

0 mn m n mn m n

ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z

m even n even m even n odd

∑ ∑ ∑ ∑

( ) i ( ) i

+ a P ( ϕ ) ( P ϕ ) + a P ( ϕ ) ( P ϕ )

mn m n mn m n

ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z

m odd n even m odd n odd

∑ ∑ ∑ ∑

a C

( ) i 2

a C C

( ) i

= +

mn 0 mn 0 1

ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z

m even n even m even n odd

. (4.68)

∑ ∑ ∑ ∑

( ) i ( ) i 2

+ a C C + a C

mn 1 0 mn 1

ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z ϕ ∈ Z

m odd n even m odd n odd

2

2

2

= Q C + ( Q Q C C Q P C + ) +

1 3 4 0 1 2

2

1 1

+

2

= Q C + ( Q Q C + ) − C Q − C

1 3 4 0 0 2 0

M M

C
M M Q

1

2 0

= ( Q Q Q Q C + − − ) + ( Q Q + − 2 Q ) +

1 2 3 4 0 3 4 2 2

2

1

= ( C + C C ) + (1 − C − C )

row col 0 row col

2 M

1

=

2 M

.

M

′

( )

Since. C =

1

(1 − C M × ) =

1

= C . .Therefore,. P i ϕ =

1

, ∀ϕ ∈.

Σ P ( ϕ ) = 1,

1 0 0 i

i M 2 M 2 M

i = 1

S ϕ .By.mathematical.induction,.the.proof.is.completed. .

4.3.5 Proof of Theorem 4.2

By.using.Lemmas.4.8.to.4.13,.the.proof.of.Theorem.4.2.can.be.obtained.by.
following.the.same.steps.given.in.the.proof.of.Theorem.4.1.