Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 205

G c,k G c,k

~

ξ p

ξ m

c c

~

ξ q

ξ n

k k

~

ξ n ξ q

V k V k V k V k

ξ i ξ i

p 1 p 2 p 1 p 2

Figure a.2
Visualization.of.finding.

 ξ .(logical.XOR.is.performed.in.striped.areas).

i

where

( ) ( ) ( ) ( ) ( )

ξ l ⊕ ξ l ′′ ⊕ ξ l ′′ , l V ∈ and ξ l ≠ * and ξ l ′ ≠ *

i m p k i m

. (A.11)

 ( )

( ) ( ) ( ) ( )

ξ l = ξ l ⊕ ξ l ⊕ ξ l , l V ∈ ′ and ξ l ≠ *

i i n q k i

( )

l , otherwise

ξ

i

.

( )

where. l ′′ = c + l − k ..A.way.to.understand. Equation.(A.11) . visually.is.shown.

in. Figure.A.2 .
Since. m ,. n ,. p ,.and. q .are.ixed,.we.have.

M M

( ) i ( ) j

Σ a = Σ a ,.for.any. m ,. n ,. p ,.and. q .

mn pq

i = 1 j = 1

M

( )

i

Therefore,. Σ a = C .is.a.constant..From.Lemma. 4. 2,.we.know.that

mn 1

i = 1

M M M M

∑ ∑ ∑

( )

=

i 2

a M M = .

. (A.12)

mn

.

i = 1 m = 1 n = 1 i = 1

Meanwhile,

M M M M M M M M

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

( ) ( )

i i

a

=

a

=

C = C M

2

. (A.13)

mn mn

1 1

.

i = 1 m = 1 n = 1 m = 1 n = 1 i = 1 m = 1 n = 1

M

( )

i

Therefore,.

C = ..This.is.equivalent.to.

1 Σ A = 1 .

M M ×

i = 1