Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 214

By. adding. columns. 1. and. 4. and. then. subtracting. columns. 2. and. 3,. we.
can.get

0.78
0.64
0.64
0.54

0.1
0.04
0.04
0.02

0.24
0.14
0.14
0.08

0.24
0.14
0.14
0.08

0.4
0.4
0.4
0.4

+ − + = .

. .

as. there. are. two. positions. that. contain. don't-care. bits. only. (i.e.,. *00 ** *. and.
*00* ** )..Thus

N
L L

2
5

0

N = 2, and C = = = 0.4. .(Using.Lemma.4.4)

0 row

− + 1

.

g

By.adding.rows.1.and.4.and.then.subtracting.rows.2.and.3,.we.can.get

([0.78 0.24 0.24 0.1] + [0.54 0.08 0.08 0.02])

− ([0.64 0.14 0.14 0.04] + [0.64 0.14 0.14 0.04])

= [0.04 0.04 0.04 0.04]

. .

as.there.is.only.a.single.position.that.contains.all.the.actual.bits.(i.e.,.* 00 ***)..Thus

N
L L

1
25

all

N = 1, and C = = = 0.04. (Using.Lemma.4.5)

all col

( )

2

− + 1

g

.

ProofofLemma4.6
If.

ξ ∈ ,. C row . and. C col . are. both. nonnegative.. Otherwise,. they. are. both.
nonpositive.
Apparently,

Z

≤ N N + ≤ L L − +

0 1

0 all g

N
L L

N
L L

0 all

g

≤ + ≤

0 1

− + 1 − + 1

.

g

.