Ma
h
w
a e&So
C
om
pu i
ng 3(
19
94)23
9-25
1
TheU
s
eo F
uzz
yC
onnec
i
es
oD
esign
Real-CodedGene icA
lgo i hm
s
3
F.He
e a,M.Lozanoa
nd J.L.V
e dega
y
Dep . o Com
pu e Sci
enceandA
ici
alIn
e
ll
igence
Uni
e si
y o G
anada,18071-G anada,S
pa in
e-mail:he
e
a,loz
ano, e de
gay@
obinson.ug
.es
Ab
s ac
Gen e icalgo i hmsa eadap i e me hods ha usep incip lesi
nspi edb
y
na u alpopu
la
iongene ics oe
ol
esol
u
ions osea c
handop imiza i on
p oble
ms.Gene icalgo i hmsp o cessapopula iono sea c
hspac esolu ions
wi h h eeope a ions:sele
c ion,c osso
e andm
u a
io
n.
Ag ea p o
blemin heuseo gene icalgo i hmsisp ema u econ
e gence;
h esea c
hbecomes appedinal
ocalop im
umbe
o e hegloba
lop im
um
is ound.F
uzzylogic ec
hniquesma
ybeused
o sol in
g hisp oblem.This
pape p esen soneo hem: hedesign o c osso
e ope a
o s o eal-coded
gene icalgo i hmsusing
uz
zyconnec i
esandi sex ens
ionbasedon he
useo pa ame e ized
uzzyconnec i
esa
s
ools o ac
kling hep ema u e
con
e gencep oblem.
Keyw
o ds:
Gene icAlgo i
hms,RealC
oding,F
uzzyConne
c i es.
1In
oduc
i
on
G
ene i
cal
go i hm
s(
GAs)a esea
c
halgo
i
hms ha useope a io
ns oundi
n
na
u algen
e ics
og
ui
de he ek h oughase
a ch space.G
Asa e heo e ical
ly
a
ndem
pi ical
ly p o
en op o i
de obus sea c
hi
nco
m
plexs
paces,g
i i
nga
al
id
app oa
ch o p o
blem
s equi
i
ng ecien
and e ec i
esea
c
h([6]
).
A GA s a s wi
hapopula i
on o ando
mly ge ne a e d sol
u i
ons, c
h o
mosom
es,
and ad
ance o
wa ds be e so
lu io
ns b
yapply
ing g
e ne ic op e a o
s m
odel
ed o
n he
g
e ne i
c p o ce sse s o cc u ing i
n na u e. In hes e al
go i hm
swemain
ai
n a p opula
ion
o
sol
u ions
o a gi
e n p oblem; his p opul
a io
n unde go es e
olu i
on in a
o m o
na
u al s elec i
on. In eac
h ge ne a ion, ela i
el
y go o d sol
u i
ons ep o duc e o gi
e
o sp ing ha
eplace he ela i
el
ybadso
lu ions whic
hdi
e. An e
al
ua io
no
 nes s unc i
on pla
ys he ol
eo heen
i
o
nmen odi
s i
nguish b e
wee n good and
badsolu ions.
3
This esea chhasbeen suppo ed by DGICY T PB92- 0933.
239
240
F.He e a,M
.L
oz ano& J.
L. V
e dega
y
Al
h o
ugh he ea em
a
n
ypossi
ble
a
i
an
so hebasi
cGA,
he undam
en
al
un de lyi
ngm
echa
nismope a es on apo
pul
a i
ono c
h o
mosome s(
ep esen
i
ngpo
s-
si
blesol
u ions o hep oblem) andco
nsi
s so he o
llo
wing op e a io
nswhic
ha e
appli
eddu ingea
chg
ene a
i
on
:
1.e
alua
ion o in di
idua
l ness,
2. o m
a iono ag
enepoo
lb
yc
hoo
si
ngi
ndi i
dualsi
np
opo
ion o hei ela i
e
 ness,
3.
ec
om
bi
na io
nb
ym
eanso heg
ene i
cope a o c o
sso
e andm
u a ion.
Thep ocessisi e a
e
du
n
il hesys emceases oim
p o
eo
ag
i
engene a ion
T
is ea
ched.
Fi
xe d-l
eng
handbi
na yencodeds
ings
o ep esen
a ion so
lu io
nha
edom
-
ina edG
A esea c
h,since he e exis heo e ical esul s ha sho
w hem obe he
m
os a
pp op i
a e,a
nd hey a eam
ena
bl
e osim
ple im
plem
en
a io
n.Bu heGA's
goodp ope iesdono s em om heuseo
bi s i
ngs(
[
1]
).F
o his eason,
hepa hha
sbeenlai
n
ow
a ds heuseo al
phabe swi
h ahi
ghe ca dinal
,
ol-
lo
w
edb
y hede
el
opme n o newgene icope a o
s(c osso
e andm
u a
ion)on
hesealpha
be s.Nonbina yencodingsi
ncl
ude ealnum
be ep esen
a i
ons,whi
ch
w
ouldseempa i
cul
a lyna u alwhenw
ea
e
a
ckl
ingop i
m
iza i
onp oble m
so
pa am
e e swi
h
a ia
blesi
ncon
i
n
uousdo
mai
ns.Thenac
h o
m
osom
ei
sa
ec o
o oa ingpoi
n
n
um
be swhosesi
zei
sk
ep hes
am
eas heleng ho
he
ec o
whic
hi
s hesol
u ion o hep obl
em
.G
Aswi h his
ypeo encodi
ngwil
lbecall
ed
eal
-codedGAs(R
CGAs).
Theuseo
ealpa am
e e sm
a
k
esi possi
bl
e ou
se la
gedom
a
ins(e
enunknown
dom
ai
ns )
o he
a ia
bles,whi
c
hi
sd
icul oac
hie
ei
nbi
na yim
plem
en
a ions,
whe e inc
easing hedo
m
ai
nw
oul
dm
ea
ns
ac icingp eci
si
on,assumi
ng axed
leng h
o hec
h om
osom
es.Ano he a
d
an
ag
ewhenusi
ng ealpa
a
me e
sis hei
capa
ci
y
oe
xplo
i heg aduali
yo he
unc ionswi hcon
i
n
uous
a iabl
es(whe e
heco
nc ep o
g
adual
i
y e
e s o h
e
ac
ha
sl
igh
c
hangesi
n
he a i
ables
co e spond osl
igh
c
hangesi
n
he unc i
on).Las l
y
, h ey al
lo
w he oo
ls ha
handlen
on- i
i
al es ic ions
ob
ed
esi
gnedm
o eeasi
ly
,a
nd he
ep esen
a ion
o heso
lu io
nsi
s
e yc
lo
se o hena u al
o m
ul
a io
no m
an
yp obl
em
s.
The m
u a io
no
pe
a o a
bi
a
i
ly al
e s one o m
o e c o
mp onen
s (c a
lled ge nes)
o asel
ec ed s uc u e s o as o i
nc e a
se he s uc u al
a i
abil
i
y o he p opula i
on,
e.
g., i
explo es he se a c
hspac e. Unde bi
na y c od i
ng, gi enag
enewi h
al
ue 0
",
i is e pl
ace d b
y
1", and i
ce
e sa. Unde e a
l co ding di
e e n
e sions o hi
s
op e a o w
e e p ese n
ed ([
9]
). Eac
h gene o e ac
hc
h om
osom
ein he p opul
a ion
un de go es a andom chang
eacco
di
ng o a p obabil
i
y dene d by he m
u a ion
a
e, he m
u a
ion p oba
bi
li
y
,
p
m
.
The c osso
e o
p e a o e xpl
oi
s he a
ai
la
ble i
n o m
a ion
om he p opul
a ion
ab ou he se a chspace. I combines he ea u e s o wopa en s uc u es o o m
wosimila o sp ing. The classical c osso e op e a o unde bina y co ding builds
TheUseo F
uzzyC
onne c i
es o Desi
gn.
..
2
41
an osp i
ngb
yl
inki
ng oge he w
ogenesegm
en
s,eac
ho
ne be
longi
ng oadi
e en
pa
e
n
.Thi
sope a
o i
sappl
iedw
i
hap obabi
li
yo
pe
o ma
nce, hec osso
e
p obabi
li
y
,
p
c
,
ha de e m
i
nes hen
um
be o
c
h om
o
som
esin hepopula ion o
be c
ossed.[
7,9
] epo
c osso
e o
pe a o m
odelsunde b
ina yand ealcodi
ng
.
Thec osso
e op e a o pla
ysacen
a
l ole in h eGA'
spe o m
ance.I could
beconside ed o be o
neo healg
o i hm
's deningc
ha ac e is ics,andi
i
so
ne
o
hecom
po
ne n s
obea inm
ind o im
p o
i
ng heG
A'sbeha
io
u ([
12]
). The
m
u a ion op e a
o
isacom
plem
en
o hec
o
sso
e o
pe a o .I isneed
ed oa
oid
helo
sso
u
se
ulin o
m
a io
np oduce db
y
hec osso
e .
Ani
m
po an
p o
blemin heu
seo G
Asis
p
ema
u
ec
on e genc
e
; hesea ch
becom
es appedi
naloc
al op i
m
umbe o
e hegl
obalo
p im
um i
s
ound.Thisi
s
p oducedb
y hel
ac
ko
di e si
y
in hepopul
a ionandadi
sp opo io
na e
exploi a-
ion
/
explo
a ion
ela
i
onship;anadequa eba
lancebe
w
ee nab oa
dsea c
handa
sucien e nem
en
isno es a
bl
ished .
F
uzzy logi
c
ec
hniquesm
a
ybeused
os
ol e hesep o
blem
s.Ana em
p con-
sis s o heu
seo
uzzyl
ogi
cbased sys em
s o
hedy
nam
i
ccon
ol pa am
e e s
o
R
CGA(
p
m
,
p
c
,popula
i
onsize,e c)insuc
haw
a
y h
a heco
ec exploi
a-
i
on/explo a i
on el
a ionsh
i
pandsui abl
edi
e si
yl
e
elsbecam
ees abli
shed([8
,
1
1]).Ano he o
neuses
uzzy co
nnec i
es odesig
nc osso
e o
pe a o s([8]
).
In hispape w
ep esen
c osso
e ope a
o
s o R
CGAsba
sedo
n heus eo
uzzyco
nnec i
es,and hei
e
x ensi
onbasedo
n heuseo pa am
e e i
zed uzzy
connec i
es o designi
ngdynam
i
cc osso
e ope a o swi h hem
ai
nobj
ec i
eo
in
oducingpopula i
ondi
e si
yi
n heGAsea c
h.
2 D
esi
gno C
osso
e O
pe a o s o R
CGAus-
i
ngF
uzz
yConne
c i
es
Ash
as a
l ea
dybeenpoi
n
edo
u ,a ch om
o
som
eisa
ec o o e al nu m
be s,and
i
s p ec i
si
onwi
llbem
a k
edby h
a o hecom
pu e unde whi
ch h
ea
l
go i
hmi
s
ca iedou .Thesi
ze o hec
h om
osom
ei
sk
ep hes
am
ea
s
hel
eng ho
he
ec o
ha is heso
lu io
n
o
he p oblem
;in hisw
a
y
,e
ac
hg
ene
ep
esen
sa
a i
abl
e
o
hep oblem
.The
a
lueso hec
h om
o
som
egenesa e o
ced o em
ai
nin he
i
n e
al e s abli
s hed b
y he
a i
able ha he c
h om
oso
me e
p e sen
s , so he gene i
c
op e a o s m
us p ese
e his e
qui
em
en
.
In[3]i
w
as p oi
n ed ou ha he c osso
e op e a
o
i
sak
ey poi
n o sol
ing he
p em
a u e con
e genc e p oblem
.Th
us , s ol
u ions o hi
sp ob
lem m
aybe
ound b
y
desi
gni
ng new al e na
i
es o hi
sope a
o . He e, he de
el
opm
en o
suc
h c oss o
e
op e a o s is a em
p e d. W
e p e sen
c osso
e op e a o s o
R
CG
A base d o
n he
use o
uzz y connec i
es: -
no
m
s , -c o
no m
s, a
e age unc ions and gene al
ize d
com
p ens a
ion op e a
o s ([14
,1
5]) whic
h induce di
 e e n
di
e si
y le
els in he
p opula ion , and he e o e he p ema u e c on e genc e p oblem maybe con olled.
Le usassume ha he ch omos omes
C
1
=(
c
1
1
;c
1
2
;:::; c
1
n
)and
C
2
=(
c
2
1
;c
2
2
;:::; c
2
n
)
242
F.He e a,M
.L
oz ano& J.
L. V
e dega
y
a es
elec ed oapply hec
osso
e ope a o o hem
,a
nd w
og
en es
c
1
i
and
c
2
i
o
be c
ossedo
e ,
c
1
i
;c
2
i
2
[
m
i
;M
i
], bei
ng
x
i
=m
in(
c
1
i
;c
2
i
)a
nd
y
i
=m
ax(
c
1
i
;c
2
i
). I
se
em
s easo
nable oim
agine
he po
ssibi
li
yo ob ai
ningg
oodd
escenden
sou si
de
h
is in
e
al.Insho
, hei
n e
alo ac io
no heg
en
e
i
,[
m
i
;M
i
], ma
ybedi
ided
in
o h ee egio
ns[
m
i
;x
i
],[
x
i
;y
i
]
,a
nd[
y
i
;M
i
],whe egooddescende n sm
ay be
ob ained;e
enconside
inga egi
on[
x
0
i
;y
0
i
]wi
h
x
0
i

x
i
and
y
0
i

y
i
w
ouldseem
easonabl
e.G
a
phi
ca
ll
y
Fi
gu e1
:Ac
io
nin
e
al o ag
ene
W
esh
al
lno
wg
oon opu o w
a dase o c osso
e ope a
o s ha al
lo
w
desce
nden
s obeob ai
nedin hep e i
ousin
e
al
s.Ino de o do ha ,w
euse
ou unc ions
F
,
S
,
M
a
nd
L
dened
om[
a;b
]
2
[
a;b
]i
n[
a; b
]
,
a;b
2<
,whi
ch
ull
l:
(P1
)
8
x;y
2
[
a; b
]
F
(
x;y
)

m
in(
x;y
),
(P2
)
8
x;y
2
[
a
;b
]
S
(
x
;y
)

m
ax(
x
;y
),
(P3
)
8
x; y
2
[
a
;b
]m
in(
x;y
)

M
(
x;y
)

ma
x(
x;y
),
(P4
)
8
x;y
2
[
a
;b
]
F
(
x;y
)

L
(
x;y
)

S
(
x
;y
),
(P5
)
F
,
S
,
M
,and
L
a e mono onenon-dec easing.
Le usassum
e ha
Q
2
F;S;M;L
g
,and
C
1
=(
c
1
1
:: :c
1
n
)a
nd
C
2
=(
c
2
1
:::c
2
n
)
a e
wo ch om
osom
es ha ha
e b ee n s elec ed o apply he c osso
e o
p e a o o
hem
.W
em
ay gene a e he c
h om
osom
e
H
=(
h
1
:::h
n
)as
H
=
Q
(
C
1
;C
2
)
;h
i
=
Q
(
c
1
i
;c
2
i
)
; i
=1
;:
::
;n:
W
i h he -no m o
p e a o s , -cono m
s, a
e a
ging
unc io
ns and gene al
ize d
co
mp e nsa io
nope a
o
s used as uz zy c onnec i
es , w
e shall a
s so cia e
F
wi
h a
-
no m
,
S
wi h a -cono m
,
M
wi
h ana
e a
ging o
p e a o and
L
wi
h a ge ne al-
ize d comp ens a ion op e a o . Fi s , weneedase o linea ans o ma ions o be
able o apply he se op e a o s unde he gen e deni ionin e als.
TheUseo F
uzzyC
onne c i
es o Desi
gn.
..
2
43
Le
O
beano
pe a o belonging o hese o
m
edb
y he -no
m
s, -co
no m
s,
a
e a
ging
unc io
nsandgene
ali
zedco
m
pensa io
no
pe a o s. F
o eac
hpo
si
ion
i
2
1
;:
::
n
g
, he
oll
o
wi
ng op e a i
onswil
lbeca
i
edou :
1.T
ans o m
c
1
i
and
c
2
i
in o
he
alues
s
1
i
;s
2
i
2
[0
;
1]s
uc
h ha
s
k
i
=
c
k
i
0
m
i
M
i
0
m
i
;k
=1
;
2
:
Usin g
hi
ss
ep,w
e ans
o m he
alueso
hegenesso ha heo
pe a o
m
a
ybea
ppl
ied o hem
.
2.Appl
y heope a o
O
(
s
1
i
;s
2
i
)andca
lcula e he alu e
h
i
:
h
i
=
m
i
+(
M
i
0
m
i
)
1
O
(
s
1
i
;s
2
i
)
;
so ha
hegene
h
i
i
si
n el
a ion oi
s o ig
inalli
m
i
s,
h
i
2
[
m
i
;M
i
]
.
3.
h
i
wi
llbe he
alue o
heg
enea posi
i
on
i
o
hec
h o
m
osom
e esul
ing
om hec o
sso
e o hec
h o
m
osom
es
C
1
and
C
2
.
Com
plyi
ngwi
hase o uzzyconnec i
es,(
T
j
;G
j
;P
j
;
^
C
j
),
j
=1
;::
:;k
,ase o
unc io
ns
F
j
,
S
j
,
M
j
and
L
j
j
=1
;::
:;k
isbuil
a
sw
edesc i
bebelo
w:
F
j
(
c
1
i
;c
2
i
)=
m
i
+(
M
i
0
m
i
)
1
T
j
(
s
1
i
;s
2
i
)
S
j
(
c
1
i
;c
2
i
)=
m
i
+(
M
i
0
m
i
)
1
G
j
(
s
1
i
;s
2
i
)
M
j
(
c
1
i
;c
2
i
)=
m
i
+(
M
i
0
m
i
)
1
P
j
(
s
1
i
;s
2
i
)
L
j
(
c
1
i
;c
2
i
)=
m
i
+(
M
i
0
m
i
)
1
^
C
j
(
s
1
i
;s
2
i
)
Thesec oss o
e op e
a o sha
edi
e e n ea u es: he
F
-and
S
-c os so
e s show
explo
a
i
on, he
M
-c osso
e ope a o ssho
wexpl
oi a i
onand he
L
-c osso
e sho
w
elaxed exploi
a io
n.
3Exa m
pl
e
W
eha
e ca i
e d ou die en
ex
pe im
en
s h a
help o com
pa e hebeha
iou o
ab
ina y c o ded GA, and som
eR
CG
As wi
h c os so
e op e a o s h a
ha
ebeen
p op ose d i
n o he publi
ca
ions, wi h a s e o
alg
o i hm
s bas ed on he c oss o
e
op e a o s p op ose d, which us e he uzzy connec i es (
T
j
;G
j
;P
j
;
^
C
j
),
j
=1
;:::;
5
showed inTable1.

244
F.He e a,M
.L
oz ano& J.
L. V
e dega
y
-no m -cono mA
e agingO
pe a o
LogicalP
o duc LogicalS
um
T
1
(
x;y
)=min(
x;y
)
G
1
(
x;y
)=max(
x;y
)
P
1
(
x
;y
)=(1
0
p
)
x
+
py
Hamac
he
P
o duc Hamac
he
S
um
(
x
)=
1
0
x
x
T
2
(
x;y
)=
xy
x
+
y
0
xy
G
2
(
x;y
)=
x
+
y
0
2
xy
1
0
xy
P
2
(
x;y
)=
1
y
0
yp
0
xy
+
xp
xy
+1
Algeb aicP oduc Algeb aicSum
(
x
)=
0
log
x
T
3
(
x
;y
)=
xyG
3
(
x
;y
)=
x
+
y
0
xyP
3
(
x;y
)=
x
1
0
p
y
p
Eins einP oduc Eins einSu m
(
x
)=log
2
0
x
x
T
4
(
x;y
)=
xy
1+(1
0
x
)(1
0
y
)
G
4
(
x;y
)=
x
+
y
1+
xy
P
4
(
x;y
)=
2
1+(
2
0
x
x
)
p
(
2
0
y
y
)
1
0
p
BoundedP oduc BoundedSum
(
x
)=1
0
x
T
5
(
x;y
)=
0
_
(
x
+
y
0
1)
G
5
(
x;y
)=1
^
(
x
+
y
)
P
5
(
x
;y
)= (1
0
p
)
x
+
py
T
a
ble1:
Se o Op
e
a o s
Eac
h -cono
mi
sdual
o
he -no
msho
wn oi sl
e
. Thea
e a
ging unc ion
iscal
cula
ed om
he o m
ulao hequasi-a
i
hme i
ca
e ages,usingas he
unc i
on hea
ddi
i e gene a o
unc io
no he
-no mp
la
cedin hesa
m
eline,
excep
o he s one wh i
c
h,no bei
ngA c
him
edean,d
oesno
ha
eag
ene a
o
unc i
on,andw
eshall us e
(
x
)=
x
.Thi
s
unc io
ni
ss
ho
wni
n heuppe pa o
hecel
lswhe e hesea
e agi
ngope a
o
sa especi
ed.F
o ea
ch
am
i
lyo o
pe a o s
inT
abl
e1
,agene al
izedcom
pensa ionope a o
^
C
j
wil
lbeconside ed,denedas
ol
lo
ws:
^
C
j
=
P
j
(
T
j
;G
j
)
Fo he
am
i
lyo Lo
gicalope
a o s,w
eshal
lconside
^
C
1
=
T
1
0
p
1
:S
p
1
.
TheGA am
il
iesa edi
e en
i
a edacco
di
ng oho
w heyca y ou he ol
lo
wing
w
os eps:
1.Gene a i
on o osp
ingusi
ng hedie en
c
osso
e ope a o s.
2.Sel
ec iono osp i
ng esu
l
ing om hec osso
e whic
hwil
l o mpa o he
po
pula i
on.
Ap
opo
sali
s he o
llo
wing
:F
o ea
ch pai o c
h om
oso
m
es om a
o al o
1
2
1
p
c
1
N
(
p
c
c osso
e p obabi
li
y
,
N
po
pul
a ionsize) , ou osp inga eg
ene a
ed,
he es ul
o applyi
ng sp e cic
u nc io
ns
F
,
S
,
M
,a
nd
L
o hem
.The
wom
os
p o
misi
n g osp i
ng o he
ou epla
ce hei
pa en
s in he p opula i
on.
Thi
ss
el
e c ion s a egy in
oduce
sahi
gh explo
i a i
on l
e
el wi
h a
n unde l
ying
expl
o a ion c ause d b
y he use o he die en
Q
-c oss o
e o
p e a o s .
Nex , he esul
s o
nR
ose n
b o c
k's Gene al
ize d
un c io
n([
4]) a e sho
wn. The
anal
y i
ca
l and g aphi
ca
l o m
ul
a io
n oge he wi
h he elem
en
h
a ep es en
s he
glo
bal o
p i
mum(m
inim
um
) a e:
(
~x
)=
n
0
1
X
i
=1
(100
1
(
x
i
+1
0
x
2
i
)
2
+(
x
i
0
1)
2
)
TheUseo F
uzzyC
onne c i
es o Desi
gn.
..
2
45
0
5
:
12

x
i

5
:
12
min(
)=
(1
;:::;
1)=0
Fig
u e2:
whe e
n
=5.
Ase o nineR
CGAbasedo
nc osso
e ope a o sp es
en
edi
n heli
e a u e
w
e econside ed o
he expe i
m
en
s(R
GA1-R
GA9
). Belo
w he ei
sa a
bleindica -
i
ng he
y
peo c osso
e a
ndm
u a io
nusedb
yeac
ho
hem
, og
e he wi
h hei
na
m
es.
Algo i hmsMu a ionC
os
so
e
R
GA
1
RandomSimpl
e[13,17]
R
GA
2
Non-
Uni o mSimple[13,17]
R
GA
3
RandomUni
o mA i hme i cal[13]
a
=0
:
35
R
GA
4
Non-
Uni o mUni
o mA i hme ical [13]
a
=0
:
35
R
GA5-

Non-
Uni o mBLX-

[5]
(

=0
;:
15
;:
3
;:
5)
R
GA
6
Non-
Uni
o mDisc e e [16]
R
GA
7
Non-
Uni o mLinea [17]
R
GA
8
Non-
Uni o m Ex endedI
n e medi
a
e[16]
R
GA
9
Non-
Uni o mEx endedLine[16]
Tabl
e2
:
Real Co
ded Gen e i
cA
lgo
i hms
By BGA w
edeno
eabi
na y co ded GA wi h 30 genes pe
a i
able,m
ul ipl
e
c oss o
e wi
h
wopoi
n s and p o
po i
onal se l
ec i
on p obabi
li y.
By
NRG
A
1
; :::
; N RGA
5w
edeno
e a a
mil
yo
G
A based on he
uzzy conne c-
i e s c oss o
e and he gene a i
on a
nd sel
ec i
on o os p ing p op osals, using he
amilie s o uz zy conne c i
es: Log
ical, Ham
am
che , Al
ge b aic , Eins e i
n and Bound,
es pec i
el
y.Inal
l case s w
euse he s o cha
s i
c uni
e sa
lsam
pl
ing ([
2]) se l
ec ion
p o ce du e and he e li is model. Weca ied ou ou exp e imen s u sing h e ol-
lowing pa ame e s: h e p opula ion size is 61 indi iduals, he c osso e p obabili y
246
F.He e a,M
.L
oz ano& J.
L. V
e dega
y
p
c
=0
:
6,and hep obabi
li
yo
c
h om
osom
eupda e
p
u
=
p
m
1
5=0
:
6,and hepa-
a
me e
b
usedb
y henon-un
i o mm
u a i
onis5.W
eex
ecu ed al
l
heal
go i hm
s3
i
mes,ea
chonewi h10
0,500a
nd 5000gene a ions,a
ndp es en
hea
e age
al
ue
o hemi
nT
a
bl
e3.
Al go i hms10 050050 00
BGA
1.1262e+013.6069e+
001.9045e+00
R
GA1
3.0446e+012.0070e
+0 16.0669e+00
R
GA2
6.9230e+002.1448e+
004.7343e-01
R
GA3
4.1941e+00 8.6031e+
00 6.3745e+00
RGA4
3.7915e+002.9624e+
008.9244e-01
R
GA5-0.0
4.5504e+002.3454e+
009.1602e-01
R
GA5-0.15
3.215 7e+002.9556e+
007.092 9e-01
R
GA5-0.3
3.7477e+001.5844e+
004.8854e-01
R
GA5-0.5
8.2653e+002.1099e+
001.7329e+00
R
GA6
5.5393
e+002.8379 e+
003.5106
e-01
R
GA7
4.7115e+001.8487e+
005.1499e-01
R
GA8
4.6861e+003.5337e+
005.3325e-01
R
GA9
4.2196e+002.9374e+
003.8014e-02
NR
GA1
4.1431e+001.9687e+
004.9364e-03
NR
GA2
1.2378e+017.
38 68e-014.0848e-02
NR
GA3
1.003 1e+013.5037e+
001.0099e+00
NR
GA4
1.2425e+014.3985e
+
001.8347e+00
NR
GA5
6.0121e+002.6930e+
001.2150e+00
T
a
ble3:
Resul s
Thebes beha
i
ou co espo
nds o heLog
icalc osso
e o
pe a o .Thi
sop-
e a o oge he wi h heosp i
ngsel
ec i
onm
ec
hanismoe asui a
bleexplo
i a-
i
on/explo
a
i
onbal
ance, al
houghw
em
us poi
n
ou ha
hi
sm
ec
hani
smi
sm
o e
im
eexpensi
ebecauseneed mo e e
a
lua io
ns.O he osp i
ng sel
ec ionm
ec
ha-
nism
sa ep oposedin[
10]
.
4 D
esi
gno Dynam
i
cC oss
o
e Op e
a o sUsi
ng
Pa a me
e i
ze
dF
uzz
yC
on ne
c i
es
A n idea o a
oiding he p e m
a u e c on
e genc e consis s in al
lo
wi
ng he e xplo a ion
in he b e gi
nning o he sea ch p o c es s and he e xpl
oi a i
on a he end o i . Wi h
he explo a i
on he di
e si
ybeca
mes g ea e , i
nc easing he p o
babil
i
yo
nding
z ones whic
ha
e clo
sed o op i
mal s o
lu io
ns . Then, supp os i
ng ha
he p o
pula-
ion ha
ei
n
o m
a i
on ab ou hese zones, he co
n e genc e o
wa ds he op i
mumi
s
p o duce d h ough e xpl
oi a i
on.
Am
u a i
on op e a
o o
R
CG
Acal
le d non-uni
o m m
u a io
n ([1
3]) is base d
on he a o em
en
ioned p i
nc i
ple.T
h e p o
po
i
on in whic
ha ea
lgene is m
u a e d
de c ease s as he GA's e xec u ion ad ance. Thu s, he changes p o duc ed on he ge nes
a e smalle in he las ge ne a ions p o duc ing a local uning.
TheUseo F
uzzyC
onne c i
es o Desi
gn.
..
2
47
We ma
yex end heuseo hec
o
ss
o
e o
pe a o sp esen
e
di
no
de o
oll
o
w
hea
o em
en
i
onedi
deas. Die en
uzzyconnec i
esc
ould be useddu ing he
GA's un.Fi s l
y
,w
eshall us e
uz zyconnec i
es ha p od
uceh
ighdi
e si
yle
els,
andl
a
e o he onesp oducingasu

ci
en
di
e si
yle el
oa
llo
w heco
n
e g
ence
obe ea
ched.
W
ep esen
ase o d
ynam
ic c osso
e ope a o sb
ased on heuseo pa am
e e -
i
zed -no m
s, -co
no
m
sa
nda
e ag
ing unc i
ons(
[14, 15
]).In he s s ag
es, we
shal
luse -no
m
sand -
co
no
m
sdis an
om hemi
nimu m -no
ma
ndm
a
xim
um
-cono m espec i
ely
,sohi
ghdi
e si
yisi
nduced.La e , -
no
m
scl
ose o he
m
ini
m
umand -cono m
sclo
se o hem
a
xim
uma
e conside ed.Theco
n
e g
ence
i
scausedand hego od beha
io
u o
helog
ical uzz
yconnec i
eswil
lbek
ep . As
w
assho
wninT
able3, hes
eope a o sa eingene al he m
os p o
 able.
T
odo his,w
ep op osease o c o
sso
e ope a o sbasedon he
unc io
ns
am
ilie s:
F
p
g
p
=1
;:::;G
,
S
p
g
p
=1
;:::;
G
,and
M
p
g
p
=1
;:
::;G
,
G
2
N
de
ned om[
a;b
]
2
[
a;b
]in [
a;b
]
,
a;b
2<
,whic
h ul
ll hec
o espondingP1
-P5p op e
ies and:
(P6)
8
x;y
2
[
a;b
]
;
l
im
p
!
G
F
p
(
x; y
)

=
m
in(
x;y
)
(P7)
8
x;y
2
[
a;b
]
;
l
im
p
!
G
S
p
(
x;y
)

=
m
ax(
x
;y
)
(P8)
8
x;y
2
[
a;b
]
;
m
i
n(
x; y
)

M
p
(
x
;y
)

x
+
y
2
o
x
+
y
2

M
p
(
x;y
)

m
ax(
x
;y
)
and
8
x;y
2
[
a; b
]
;
l
im
p
!
G
M
p
(
x
;y
)

=
x
+
y
2
Le us conside aGAwi
ham
axim
um num
be o gene a io
ns

a
nd
C
1
=
(
c
1
1
;::
:c
1
n
)and
C
2
=(
c
2
1
;:::c
2
n
) w
oc
h om
o
som
es ha w
e es
el
ec
e din hegene -
a
i
on
oa
ppl
y hec osso
e o
pe
a o
o hem
.I
Q
p
2
F
p
;S
p
;M
p
g
p
=1
;:::; 
w
em
a
yg
ene a
e hech om
osome
H
=(
h
1
;::
:;h
n
)a
s
H
=
Q
(
C
1
;C
2
)
; h
i
=
Q
(
c
1
i
;c
2
i
)
; i
=1
;:
::;n:
W
esha
ll buil
d unc i
ons a
mil
ies wi h he (P6) and (P7) p op e i
e s using he
pa
a
me e ize d -no m
sa
nd -
co
no
ms desc ib ed in T
able 4. Ta
ble 5 sho
ws he
p ope i
e s o he pa am
e e i
zed -no m
si
n hi
s a
ble. The p op e ies o he pa-
ame e iz ed -cono ms a e analogous. Wemus p oin ou ha
T
6
is he d as ic
-no m ( he smalles -no m).