Chromosome Representation - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 228

Appendix C: Chromosome
Representa tion

Two. types. of. chromosome. representations,. real. and. binary,. were. adopted.
for. the. evaluation. of. test. functions.. Regarding. the. real. representation,. the.
encoding.and.decoding.of.genes.are.straightforward..One.decision.variable.
can.be.directly.encoded.to.a.gene,.and.this.gene.can.be.decoded.back.to.the.
original.decision.variable.with.no.extra.effort..But.a.method.of.determining.
the.total.length.of.a.chromosome.and.a.decoding.function.are.required.for.
binary.representation.[2]..Their.details.are.given.next.
First,. let. us. consider. the. former. method. and. assume. that. there. are. two.
decision. variables. in. an. optimization. problem,. and. both. of. their. bounds.
are. 4, 4 − ..If.the.required.precision.is.six.decimal.places,.the.range. 4, [ − .

should. be. divided. into. at. least. 8. ×. 1000000. + 1. equal. size. ranges.. Since.
2

[ ]

22 23
< < ,. it. turns. out. that. 23. bits. are. needed. to. represent. one.
decision. variable,. and. the. total. length. of. a. chromosome. for. encoding. two.
decision.variables.is.46.bits.
On.the.other.hand,.the.decoding.function.starts.with.converting.a.binary.
string. (

8, 000, 001 2

b b ...... b ) .to.the.corresponding.decimal.value. a .using.the.formula

n 1 n 2 0

− −

n − 1

∑

i

. (C.1)

a = b ⋅ 2 , ∀ b ∈ {0, 1}

i i

.

i = 0

where. n is.the.total.number.of.bits.in.the.string.
Then,.the.value.of.a.decision.variable. x .can.be.obtained.by

a

x B = + ⋅ ( B − B ) . (C.2)

L U L

n

2 − 1

.

where. B L .and. B U .are.the.lower.and.upper.bounds.of.the.decision.variable,.
respectively.
Consider.an.example.of.a.binary.string.(10100001000100001001001).with. n .=.
23.bits;.the.value.of. a .is.5277769..Assuming.that. B L = − .and. 4 B U = ,.the.value. 4

of.the.decision.variable.is

5277769
2

x = − 4 + ⋅ (4 ( 4)) 1.0333 − − ≈

23

− 1

. .

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