Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 206

example a.1

Let. S =

{

*00 * **, * 01* **, * 10 * **, * 11* * *

}

.and. L g = ;.we.can.obtain.the. 2

ξ

following.distribution.matrices:

0.78 0.24 0.24 0.1 0.1 0.64 0.04 0.18

0.64 0.14 0.14 0.04 0.16 0.66 0.06 0.16 ( ) ( )

1 2

A = , A = ,

0.64 0.14 0.14 0.04 0.16 0.66 0.06 0.16

0.54 0.08 0.08 0.02 0.18 0.64 0.04 0.1

. .

0.1 0.04 0.64 0.18 0.02 0.08 0.08 0.54

0.16 0.06 0.66 0.16 0.04 0.14 0.14 0.64

( ) ( )

3 4

A = , A = .

0.16 0.06 0.66 0.16 0.04 0.14 0.14 0.64

0.18 0.04 0.64 0.1 0.1 0.24 0.24 0.78

. .

It.can.be.easily.verified.that

4 4 4 4 4 4 4 4

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

(1) (2) (3)

.and.

(4)

.(Using.Lemma.4.2) a mn = 4, a mn = 4, a mn = 4, a mn = 4

. .

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

and

1 1 1 1

4

∑

1 1 1 1

( )

A i

= . .(Using.Lemma.4.3)

1 1 1 1

i = 1

1 1 1 1

.

remark a.1
Let.us.consider

m − 1

∏

∆ ξ ( , V ; ξ , G ) = d v j v j ( ( ), ( ))

k m c k , 1 2

i

.

j = 0

where.

v

= f ( ξ , V ) .and. v = f ( ξ , G ) ..Based.on.Definition.4.4,.we.assume.

T k 2 T m c

i

that. there. are. u + v . actual. bits. in.

v j ,. where.

( ) u c k . bits. are. compared.

c k , c k , ,

2 v j ,.as.

( )

with.the.actual.bits.in.

2 v j ,.and.

( ) v c k .bits.are.compared.with.*.in.

,

shown.in . Figure.A.3 .. If.all. u c k .actual.bits.in.

v j .are.the.same.as.those.in.

( )

,

2 v j ,.we.have

( )

− v

∆ ξ ( , V ; ξ , G ) 2 = ,

c k ,

.

k m c k ,

i