TELKOM
NIKA
, Vol. 11, No. 4, April 2013, pp. 1781
~
1
786
ISSN: 2302-4
046
1781
Re
cei
v
ed
Jan
uary 10, 201
3
;
Revi
sed Fe
brua
ry 6, 201
3; Acce
pted
February 18,
2013
An Approach to Determining the Optimal Cell Number
of Manufacturing Cell Formation
Jian
w
e
i Wan
g
Coll
eg
e of Mechan
ical En
gi
ne
erin
g, Dali
an U
n
iversit
y
, Da
lia
n, Chin
a
Corresp
on
din
g
author, e-mai
l
:
w
a
ng
j
w
7
2
@1
63.com
A
b
st
r
a
ct
An a
ppro
a
ch t
o
det
ermin
i
ng
the o
p
ti
ma
l ce
ll
nu
mb
er of
ma
n
u
facturin
g ce
ll f
o
rmatio
n
is
pre
s
ented
.
F
i
rstly, the differenc
e of w
e
ig
hting ex
po
ne
nt, cluste
r center
and
metrics h
o
w
to have an
imp
a
ct up
on th
e
clusteri
ng res
u
lts and
me
mbe
r
ship functi
on
are studi
ed.
S
e
con
d
ly, a
met
hod to d
e
ter
m
i
ne the o
p
ti
mal
m
valu
e is
giv
e
n
.
T
w
o-order p
a
rtial
der
ivativ
e of th
e
obj
e
c
tive functi
on
for F
C
M is c
a
lculat
ed, a
n
d
th
e
variati
ona
l w
e
ightin
g exp
one
nt m is
obta
i
ne
d that can prev
ent the par
ameter from
b
e
in
g the uni
qu
e va
l
u
e
and
play a
n
i
m
portant rol
e
in
the pr
oc
ess of fu
zz
y
cl
usterin
g
. Moreover, i
n
order to av
oid
a sing
le va
lid
it
y
i
n
de
x ca
n
no
t a
sse
ss co
rrectl
y
, p
a
r
ti
ti
o
n
co
e
ffi
ci
en
t (PC
)
, cl
a
ssi
fi
ca
ti
on
e
n
t
ro
p
y
(C
E), Fu
ku
ya
ma
and
Suge
no (F
S)
a
nd Xi
e
and
Be
ni (XB)
are c
o
nsid
ered
as
multi-p
e
rformanc
e in
dex
es to
e
v
alu
a
te the
clu
s
ter
valid
ity, and th
en an o
p
ti
ma
l nu
mb
er c is chosen b
a
se
d
on
these vali
dity me
asur
es. F
i
nally, test exa
m
pl
s
are giv
en to ill
u
s
trate the valid
i
t
y of the propo
sed ap
pro
a
ch.
Ke
y
w
ords
: cel
l
formati
on, cel
l
number, fu
zz
y
c-mea
n
, eval
u
a
tion
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Cellula
r ma
n
u
facturi
ng i
s
a useful
way
to
improve o
v
erall man
u
fa
cturin
g pe
rformance.
Grou
p techn
o
logy is u
s
ed
to incre
a
se the produ
ctivity for manufa
c
turin
g
hig
h
quality pro
d
u
c
ts
and im
provin
g the flexibilit
y of manufa
c
turing
syst
e
m
s. Cell form
a
t
ion is
an im
portant
step
i
n
grou
p tech
nol
ogy. It is use
d
in desi
gnin
g
good
cellul
a
r man
u
factu
r
ing
system
s. The key ste
p
in
desi
gning
an
y cellula
r ma
nufactu
ring
system is th
e
identificatio
n
of part famili
es a
nd ma
ch
ine
grou
ps fo
r the creation
of cells that
use
s
the si
milaritie
s
bet
wee
n
part
s
i
n
relation to
the
machi
n
e
s
in
their ma
nufa
c
ture
[1]. Clu
s
ter
analy
s
is
is a m
e
thod
for cl
uste
ring
a data
set i
n
to
grou
ps of si
m
ilar individu
al
s, its prin
ciple
ac
cord
s with
the requi
rem
ent of cell formation.
In c
l
us
ter
analys
is
, the fuzzy c
-
means
(
F
C
M
)
c
l
us
ter
i
ng algor
i
thm is the bes
t
k
n
ow
n and
use
d
meth
od
for
cell fo
rmation p
r
o
b
lem. Du
ri
n
g
t
he la
st two
decade
s
of rese
arch, a
l
a
rge
numbe
r
of ce
ll formatio
n
method
s b
a
sed o
n
F
C
M
h
a
ve be
en
de
veloped. Xu
and
Wa
ng [2]
first
applie
d the fu
zzy
clu
s
teri
ng
to cell
form
ation.
Ch
u a
nd
Hayya [3] the
n
improved it
s u
s
ag
e. Gin
d
y
et al. [4]
considered optim
al
numbers of part families and mach
i
n
e groups usi
ng
som
e
vali
dity
indexe
s
. Ven
ugop
al [5] ga
ve a state
-
of
-the-art revie
w
on th
e u
s
e
of soft com
puting in
clu
d
ing
fuzzy
clu
s
teri
ng. Moreove
r
, Gün
gör
a
nd Ari
k
an [6
] applied fu
zzy de
cisi
on
makin
g
in
CF.
Ho
wever, it is ne
ce
ssary to pre
-
a
s
sum
e
the cell nu
mber
c in tho
s
e F
C
M clu
s
t
e
ring al
go
rithms,
the cell number c i
s
generally unknown. If cell numbe
r c i
s
assigned an inaccurate value, it will
cau
s
e i
n
valid
or
wo
rse
clu
s
ter. T
herefore, it is
worthy
how to d
e
termine the
opti
m
al cell nu
m
ber
c of manufa
c
t
u
ring
cell formation.
In this
pap
er,
an
app
roa
c
h
to dete
r
mini
ng the
optim
al cell n
u
mbe
r
of m
anufa
c
t
u
ring
cell
formation is
pre
s
ente
d
. Firstly, the influen
ce
facto
r
s of FCM algorithm and cl
u
s
ter validity are
analysed. Se
con
d
ly, ba
se
d on
the
relat
i
onship fu
zzy
obje
c
tive
fun
c
tion with wei
ghting expon
ent
m, a novel method of choo
sing m i
n
FCM i
s
pr
opo
sed. Ta
ki
ng into a
c
co
unt the effect of
clu
s
terin
g
ce
nter
subj
ect t
o
FCM, th
e
obje
c
tive
fun
c
tion i
s
mo
dified by revisin
g
the
con
s
tra
i
nt
term b
a
sed
o
n
si
mulated
a
nneali
ng to
a
v
oid the
acco
rdant
cl
uste
r
cente
r
s h
app
ening.
The
n
e
w
measure style of fuzzy cl
uster i
s
ado
pted to dec
rease the defect of Euclid
distan
ce in cell
formation. Ai
ming at n
one
of uniform p
e
rform
a
n
c
e i
ndex for
eval
uating the
cl
uster validity, a
synthetic p
e
rf
orma
nce inde
xes ar
e ado
p
t
ed to asse
ss the clu
s
ter
v
a
lidity and se
lect the optim
al
cell n
u
mbe
r
.
Finally, set o
f
test exampl
es a
r
e
given,
the sim
u
lati
on results d
e
m
onst
r
ate th
e
prop
osed ap
p
r
oa
ch is b
o
th effective and
feasibl
e
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No. 4, April 2013 : 1781 – 1
786
1782
The re
st of the pap
er is
orga
nized a
s
follo
ws: Sections 2 intro
d
u
ce the d
e
tai
l
s of the
prop
osed al
g
o
rithm. The t
e
st exampl
es are give
n a
nd sim
u
lation
results a
nd
discu
ssi
on
s
are
pre
s
ente
d
in Section 3. Fin
a
lly, concl
u
si
ons a
r
e given
.
2. An Appr
o
ach to
Deter
m
ining the O
p
timal Cell Number of Ma
nufa
c
turing
Cell Formati
on
2.1. FCM Clu
s
tering
Algo
rithm
The FCM i
s
an iterative algorithm u
s
ing
the
nece
s
sary condition
s for a minimu
m of the
FCM obj
ectiv
e
function
J
m
(
μ
,v) [7].
It can be descri
bed
as follows:
Nc
m
mi
k
k
i
k1
i
1
J(
,
v
)
d
z
,
v
(1)
whe
r
e
μ
={
μ
1
,
μ
2
,…,
μ
c
}. v={v
1
, v
2
,…,v
c
} is
t
he set
of
c
clu
s
t
e
r
cent
e
r
s.
μ
ik
is the membe
r
ship
of
the kth
sam
p
l
e
to the ith
cl
uster center,
it
s value i
s
a
ssi
gne
d in th
e interval [0,1
] and
c
ik
i1
1
.
m is the weig
hting expone
nt,
m1
,
. d(z
k
,v
i
) is the Euclidea
n distan
ce be
tween the
sa
mple
z
k
an
d the
cl
u
s
ter ce
nter v
i
,
1/
2
s
2
ki
k
j
i
j
j1
d(
z
,
v
)
z
v
.
s
12
N
Zz
,
z
,
,
z
R
is the
dat
a set, N
is the sa
mple
numbe
r of data set.
The
ne
ce
ssa
r
y conditio
ns for
a mi
nim
um (
μ
,v) of
J
m
(
μ
,v)
are
t
he follo
w
in
g
update
equatio
ns:
2
(m
1
)
1
c
(l
)
ik
ik
j1
jk
d
d
i,j=1,2,…,c, k=1,2,…,
N.
(2)
ii
k
i
k
NN
mm
(l)
(
l)
(
l
)
k
k1
k1
vz
i=1,2,…,c
(3)
2.1.1. The
w
e
ighting Exp
onent
m
In
fluence
s
on F
C
M Algori
t
h
m
The wei
ghtin
g expone
nt m is calle
d
the fuzzifier
which
can h
a
ve an influen
ce on the
clu
s
terin
g
pe
rforma
nce of
FCM. Th
e b
e
st choi
ce fo
r m i
s
p
r
oba
bly in the int
e
rval [1.5, 2.
5],
who
s
e m
ean
and midp
oint
m=2, have o
f
ten been the
preferre
d ch
oice fo
r man
y
users of F
C
M
[8]. It is important to choo
se corre
c
tly m according to the different p
r
oble
m
s.
There is the i
m
plicit rel
a
tio
n
shi
p
betw
e
e
n
J
m
(
μ
,v) and
m, then
cn
cn
m2
2
m
m1
ik
ik
ik
ik
ik
ik
ik
i1
k
1
i1
k
1
J,
v
lg
d
l
g
d
0
m
(4)
From the e
q
u
a
tion, it can b
e
found that J
m
(
μ
,v) will monotoni
cally
decrea
s
e wit
h
the increa
si
ng
m. However,
the de
cre
asi
n
g rate of
J
m
(
μ
,v) can b
e
div
ided into t
w
o
parts: a
sh
arp dro
p
an
d sl
ow
drop, th
en th
ere i
s
a i
n
flection point b
e
twee
n
the tw
o
part
s
. The
o
p
timal weighti
ng expo
nent
is
the value co
rresp
ondi
ng to the inflection
point. m*
can
be cal
c
ul
ated
by the following equ
ation:
*
m
J(,
v
)
mm
0
mm
(5)
2.1.2. The Fu
zzy
Distanc
e
d(z
k
,v
i
)
Influ
e
nces o
n
FCM Algorithm
Durin
g
the cell formation,
the Euclid di
stan
ce is m
o
stly adopted
to determin
e
d(z
k
,v
i
).
Ho
wever, th
e same o
r
d
i
fferent elem
ent numbe
r are taken int
o
accou
n
t firstly, the Euclid
distan
ce
sho
uld not reflect the cha
r
a
c
teristi
c
of
cell
formation p
r
oblem. Th
e d
i
stan
ce fun
c
ti
on
d(z
k
,v
i
) bet
we
en part
z
k
an
d clu
s
ter
cent
er v
i
can be d
e
scrib
ed a
s
follow
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Approach
to Determ
ining the Optim
a
l Cell
Num
b
er of Manufacturing Cell... (Jianwei
Wang)
1783
ss
i
k
k
i
kj
i
j
kj
i
j
j1
j1
dz
,
v
0
.
5
z
v
0
.
5
z
v
(6)
whe
r
e
s
kj
i
j
j1
zv
is th
e
numbe
r
of u
s
ed different
machi
ne
bet
wee
n
p
a
rt
z
k
and clu
s
ter
center
v
i
.
s
kj
i
j
j1
zv
is the numbe
r of use
d
sam
e
machi
ne be
tween p
a
rt z
k
and cl
uste
r center v
i
.
2.1.3. The Cluster
Cen
t
er
v
i
Influence
s
on FCM Al
gorithm
In FCM al
go
rithm, the
clu
s
ter
ce
nter v
i
sho
uld b
e
keep the
differentiation d
egrees
and
avoid the
con
s
iste
ncy. T
h
e
philo
so
phy o
f
simul
a
ted
a
nneali
ng i
s
re
feren
c
ed
a
nd
the value
of
γ
is
revise
d con
s
tantly, then the influen
ce d
egre
e
of
clu
s
ter ce
nter
ca
n be imp
r
ove
d
. In the initial
stage, the val
ue of
γ
is l
a
rge to en
su
re
the se
pa
ratio
n
between
cl
usters. In the
final stag
e, the
value of
γ
de
cre
ases to 0 t
o
ensure the
comp
actn
ess betwee
n
clu
s
ters.
2.1.4. Sub Bab 2
Nc
c
m
mi
k
k
i
i
t
k1
i
1
t
1
J
(
,v
)
d
z
,
v
d
(
v
,v
)
c
(7)
whe
r
e
"
i,k,
ik
0,
1
,
c
ik
i1
1
, i=1,2,…,c, k=1,2,…,N. T
he wei
ghting
exponent m
*
, fuzzy
distan
ce d
(
z
k
,v
i
), cluste
r ce
nter v
i
and m
e
mbe
r
ship fu
nction
μ
ik
are sho
w
n a
s
foll
ows:
**
*
m
JU
,
V
ma
r
g
m
i
n
m
(8)
ss
k
i
kj
i
j
kj
i
j
j1
j1
dz
,
v
0
.
5
z
v
0
.
5
z
v
(9)
NN
mm
ii
k
k
i
k
k1
k1
vz
，
i=1,2,…,c
(10)
2
(m
1
)
1
c
ik
ik
j1
jk
d
d
(11)
2.2. Cluste
r Validit
y
for Fuzz
y
Clustering
Wheth
e
r d
o
e
s
the F
C
M al
gorithm
accu
rately
re
pre
s
ent the struct
ure of the
da
ta set?
There are fou
r
most cite
d v
alidity indexes sh
own as fo
llows:
(1)
Partition co
efficient (P
C) [9]
:
ma
x
m
a
x
cN
2
ik
2c
c
2
c
c
i1
k
1
1
ma
x
P
C
(
c
)
ma
x
(
)
N
(12)
whe
r
e
1c
P
C
(
c
)
1
.
(2)
Cla
ssifi
cation
entropy
(
C
E
)
[7]:
ma
x
m
a
x
cN
ik
2
i
k
2c
c
2
c
c
i1
k
1
1
mi
n
C
E
(
c
)
mi
n
l
o
g
(
)
N
(13)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No. 4, April 2013 : 1781 – 1
786
1784
whe
r
e
2
0C
E
(
c
)
l
o
g
c
.
(3)
Fukuya
ma an
d Sugeno
(F
S) [10]:
ma
x
m
a
x
cN
cN
22
mm
ik
k
i
ik
i
2c
c
2
c
c
i1
k
1
i1
k
1
mi
n
F
S
(
c
)
m
i
n
z
v
v
v
(14)
whe
r
e
c
i
i1
1
vv
c
.
(4)
Xie and Beni
(XB) [11]:
ma
x
m
a
x
cN
2
m
ik
k
i
i
,
j
i
j
2c
c
2
c
c
i1
k
1
mi
n
X
B
(
c
)
mi
n
z
v
N
m
i
n
v
v
(15)
Note that sin
c
e no
single
validity index
is the be
st, a better way of usin
g validity
indexe
s
to solve th
e
clu
s
ter vali
dity probl
em i
s
to co
n
s
ide
r
all inform
atio
n propo
se
d
by all sele
cted
indexe
s
, an
d
then ma
ke
an
optimal
de
ci
sion.
The
fou
r
validity ind
e
x
es a
r
e
loo
ke
d a
s
a
synth
e
t
ic
perfo
rman
ce i
ndexe
s
to asse
ss the
clu
s
ter validity and cho
o
se the optimal c.
3. Case Stud
y
To demon
strate the performance of the propo
se
d method, four in
stan
ce
s of Referen
c
e
[8] is adopte
d
and u
s
ed t
he sa
me dat
a set. The in
itial machin
e-part matrix o
f
four instan
c
es
own
s
diffe
r
e
n
t data
scal
es
5×7, 10
×15, 24
×40
a
nd 40
×1
00
respe
c
tively. The
pro
p
o
s
ed
approa
ch a
n
d
four p
e
rfo
r
mance ind
exes a
r
e em
plo
yed for sim
u
l
a
ting test. Th
e paramete
r
s for
test
cas
e
s
w
e
re
s
e
t a
s
fo
llow
s:
ε
=
0
.001
, c
min
=2. Th
e pe
rforman
c
e ind
e
xes for test
ca
se
a
r
e
sho
w
n i
n
Tab
les 1
-
4. Th
e
optimal cell n
u
mbe
r
c i
s
si
gned
with g
r
a
y
for every te
st ca
se
sh
ow
n in
Table
s
1
-
4.
The validity indexe
s
with
different cl
ust
e
r nu
mbe
rs
c for te
st ca
se a
r
e
sho
w
n in
Figures 1
-
4. Table 5 sho
w
s the soluti
ons of optim
al cell num
b
e
r c obtai
ned
by the prop
ose
d
method in thi
s
pap
er an
d Referen
c
e [8].
Figure 1. Validity Indexes with Differe
nt Clu
s
ter
Num
b
er
s c (
5
×7)
Figure 2. Validity Indexes with Differe
n
t
Clu
s
ter N
u
m
b
ers
c (1
0×
15)
Table 1. Synthetic Perfo
r
m
ance Indexe
s
for Test Ca
se (5×7)
c
S
y
nthetic perfo
r
m
ance indexes
PC(c
)
CE(c
)
FS(c
)
XB(c
)
2
0.9188
0.2323
-5.0955
0.1635
3 0.8954
0.3060
-3.2208
0.2454
4 0.9155
0.2588
-0.4343
0.4016
2
3
4
5
6
7
8
0
0.2
0.4
0.6
0.8
1
1.2
PC
CE
XB
c
Performance ind
e
xes
PC
XB
CE
1
2
3
4
-5
-4
-3
-2
-1
0
1
PC
CE
FS
XB
c
Performance ind
e
xes
PC
XB CE
FS
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Approach
to Determ
ining the Optim
a
l Cell
Num
b
er of Manufacturing Cell... (Jianwei
Wang)
1785
Figure 3. Validity Indexes with Differe
nt
Clu
s
ter N
u
m
b
ers
c (2
4×
40)
Figure 4. Validity Indexes with Differe
nt
Clu
s
ter N
u
m
b
ers
c (4
0×
10
0)
Table 2. Synthetic Perfo
r
m
ance Indexe
s
for Test Ca
se (10
×
15
)
c
S
y
nthetic perfo
r
m
ance indexes
PC(c
)
CE(c
)
FS(c
)
XB(c
)
2 0.9809
0.1752
2.7661
0.6984
3
0.9997
0.0130
-25.9917
0.1023
4 0.9996
0.0759
-28.0633
0.2249
5 0.9954
0.0758
-29.6039
0.1042
6 0.9954
0.0563
-31.6841
0.0498
7 NaN
NaN
NaN
NaN
8 NaN
NaN
NaN
NaN
Table 3. Synthetic Perfo
r
m
ance Indexe
s
for Test Ca
se (24
×
40
)
c
S
y
nthetic perfo
r
m
ance indexes
PC(c
)
CE(c
)
FS(c
)
XB(c
)
2 0.5000
1.0000
51.2776
1.82e+2
3 0.3333
1.5850
32.8266
3.73e+3
4 0.5676
1.2215
-12.9215
0.8657
5 0.3057
2.0124
5.5703
3.03e+2
6 0.2756
2.2390
1.3559
1.16e+2
7 0.6895
1.0437
-62.0475
0.2796
8 0.1251
2.9995
11.1527
3.17e+2
9 0.7480
0.9104
-74.8708
0.1732
10 0.7814
0.8124
-80.7597
0.1316
11 0.5134
1.8330
-42.3003
4.15e+1
12 0.3171
2.6495
-23.1775
2.53e+2
13
0.8875
0.4481
-92.8612
0.0560
14 0.5010
2.0237
-44.5946
1.10e+2
15 0.5362
1.9192
-48.4852
1.23e+2
Table 4. Synthetic Perfo
r
m
ance Indexe
s
for Test Ca
se (40
×
10
0)
c
S
y
nthetic perfo
r
m
ance indexes
PC(c
)
CE(c
)
FS(c
)
XB(c
)
2
0.9879
0.0504
1.48e+02
1.9185
3 0.9799
0.0843
6.87
e+01
1.7285
4 0.9222
0.2403
3.14
e+01
2.2233
5
0.9085
0.3001
-3.7551
2.1446
6 0.9692
0.1237
-9.81
e+01
1.2114
7 0.9716
0.1097
-1.40
e+02
1.2571
8 0.9683
0.1231
-1.49
e+02
1.8824
9 0.9966
0.0182
-2.28
e+02
0.6216
10
0.9994
0.0041
-2.61
e+02
0.4600
11 0.9932
0.0336
-2.27
e+02
0.8729
12 0.9993
0.0044
-2.80
e+02
0.7497
13 0.9938
0.0228
-2.78
e+02
0.8178
14 0.9993
0.0045
-2.87
e+02
1.2127
15 0.9950
0.0196
-2.92 e+02
0.6904
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-3
-2
-1
0
1
2
3
PC
CE
FS
XB
Performance ind
e
xes
FS
CE
XB
PC
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-100
-80
-60
-40
-20
0
20
40
55
PC
CE
FS
XB
Performance ind
e
xes
FS
XB
CE
PC
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No. 4, April 2013 : 1781 – 1
786
1786
Table 5. Simulated Data a
nd Clu
s
ter
Nu
mber
Example
5×7 10×15 24×40
40×100
c
Reference [8]
2
3
7
10
Proposed metho
d
2
3
13
10
As can b
e
se
en in Table
s
1-5 an
d Figu
res 1-4,
the optimal cell nu
mber o
b
taine
d by the
prop
osed
ap
proa
ch
is i
n
a
c
cord
with th
at given in
R
e
feren
c
e
[8]. More
over, th
ere
are
still t
h
ree
or four extre
m
ums of pe
rforma
nce ind
exes corr
e
s
p
ondin
g with the optimal ce
ll numbe
r, thus it
can be p
r
ove
d
that the presente
d
metho
d
is effe
ctive and feasi
b
le. Ho
wever, the
fact is existing
that som
e
si
mulation
re
su
lts of the p
r
o
posed al
gorit
hm are a
c
cord with the
oth
e
rs, it
reflect
s
the
fact that the
com
plexity of clu
s
ter problem
s
a
n
d
the differen
tia of pe
rformance in
dex
es.
Therefore, it can be
con
c
l
uded that the approa
ch
for dete
r
minin
g the optimal
cell numbe
r of
manufa
c
turi
n
g
cell form
ation is availa
ble and ro
bu
st.
4. Conclusio
n
Takin
g
into
accou
n
t the cha
r
a
c
teri
stics
of cell formation probl
em and an
al
ysing the
differen
c
e of
weig
hting ex
pone
nt, clust
e
r center
and
metrics ho
w
to have an i
m
pact u
pon t
h
e
clu
s
terin
g
re
sults a
nd m
e
m
bership
fun
c
tion, an
ap
pro
a
ch
to d
e
termining th
e o
p
t
imal cell
nu
m
ber
of manufact
u
ring
cell fo
rmation i
s
p
r
opo
se
d.
FC
M algorithm
is adopte
d
to calcul
ate
th
e
membe
r
ship
and cl
uste
r center of pa
rt
s in the de
si
gnated
rang
e
of cell num
ber. In order to
asse
ss
corre
c
tly the clust
e
ring p
e
rfo
r
m
ance, f
our sy
nthetic pe
rformanc
e index
es are emplo
yed
to sel
e
ct
the
optimal
cell
n
u
mbe
r
. Fin
all
y, a set
of te
st example
s
are given,
the
simulation
re
sults
demon
strate the pro
p
o
s
ed
approa
ch is b
o
th effective and fea
s
ible.
Ackn
o
w
l
e
dg
ment
The p
r
oje
ct
is p
a
rtly
sup
porte
d b
y
National
Nature a
n
d
Scien
c
e
F
ound
ation
(No.5
127
506
1,5097
503
3),
Science & Tech
nolo
gy
Re
sea
r
ch Fo
undatio
n of the Educatio
nal
Dep
a
rtme
nt of Liaonin
g Province
(No.LS20
1
0006,LT
201
0
006), Scie
n
c
e & Tech
nology
Re
sea
r
ch Fo
undatio
n of Liaonin
g
Provin
cial (No.20
08
2190
13) a
nd
Do
ctoral Start
ing Foun
datio
n
of Dalian
Uni
v
ersity.
Referen
ces
[1]
Hun
g
W
L
, Yang MS, Lee ES. Cell formati
on
usin
g fuzz
y
re
l
a
tion
al cluster
i
ng al
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Mathe
m
atic
al
and C
o
mput
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ng
. 20
1
1
; 53: 177
6-17
87.
[2]
X
u
H, Wang HP. Part fa
mil
y
f
o
rmatio
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GT
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ons
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u
zz
y mathem
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I
n
ternati
o
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a
l
Journ
a
l of Prod
uction R
e
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CH,
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yya
JC. A
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y c
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usteri
ng
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a
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pi
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z
y
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usterin
g
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oach
w
i
th
valid
it
y
m
eas
ure.
Internati
o
nal Jo
urn
a
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e
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go
pa V.
Soft-computin
g
-
base
d
a
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gr
oup
tech
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og
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st
ate-of-the-art
revie
w
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Intern
a
t
iona
l Journ
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e
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h
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[6]
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Ari
k
an F
.
Appl
ica
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e
cisio
n
mak
i
ng
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ou
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.
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l of Prod
uction Eco
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n
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zz
y
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gor
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w
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r
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z
z
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a
l of Cyb
e
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u
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l
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zz
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s
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edi
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