Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 74

∑ ∑

M

( ) i

Q = a =

4 (1

− C + C );

3 mn row col

.

ξ ∈ Z ξ ∈ Z ′

m n

∑ ∑

M

( ) i

Q = a =

4 (1

+ C − C )

4 mn row col

.

ξ ∈ Z ′ ξ ∈ Z

m n

where. Z .is. Z odd .or. Z even .

Lemma4.8

M
∑

M

( )

i

{ }

b = M , ∀ ∈ i 1, 2,  , M .

mn

.

.

m = 1 n = 1

Lemma4.9

M
∑

( )

i

b mn = 1,

{ }

∀ m n , ∈ 1, 2,  , M .

.

.

i = 1

Lemma4.10

∑ ∑

N
L L

′

( ) i ( ) i 0

b − b = ± ≡ C ′ ,

mn mn row

2

( − + 1)

g

.

ξ ∈ Z ξ ∈ Z ′

n n

where. Z . is. Z odd . or. Z even ,. and. N ′. is. the. number. of. possible. combinations. of.
cutting.and.pasting.positions.such.that.the.transposon.and.moving.regions.
in. ξ .contain.don't-care.bits.(*).only.

Lemma4.11

∑ ∑

N
L L

( ) i ( ) i all

g

b − b = ± ≡ C ′

mn mn col

2

( − + 1)

.

ξ ∈ Z ξ ∈ ′ Z

m m

where. Z .is. Z odd .or. Z even ,.and. N all .is.the.number.of.possible.cutting.positions.
such.that.the.transposon.region.in.ξ i .includes.all.the.actual.bits.