TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 10, Octobe
r 20
14, pp. 7533
~ 754
2
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.598
9
7533
Re
cei
v
ed Ma
rch 1
7
, 2014;
Re
vised Aug
u
st
2, 2014;
Acce
pted Au
gust 25, 20
14
Recent Study on Distance Formula an
d Similarity
Measur
es between Two Vague Sets
Zhang Kun*,
Wang Hong
-xu, Wa
ng Hai-fe
ng, Li Zhuang
Coll
eg
e of Elec
tronics an
d Informatio
n
Engi
n
eeri
ng, Qion
gz
hou U
n
ivers
i
t
y
,
San
y
a Ha
in
an
5720
22, Ch
ina
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: zk0588
@16
3
.
com
A
b
st
r
a
ct
F
i
rstly, tw
o new
distance
for
m
u
l
as w
e
r
e
p
u
t
forw
ard an
d p
r
oved
to satisfy
w
i
th know
n
ax
io
ms. A
n
appr
opri
a
te di
stance for
m
ul
a
may giv
e
Va
gue set reg
u
l
a
tions a
more r
easo
n
a
b
le cl
u
s
ter, and red
u
c
e
search sc
ale
of rule b
a
se a
nd
compl
e
xity of calcul
atio
n. T
h
e
n
, by studyi
ng
t
he defic
ienc
ie
s and re
aso
n
s
of
similar
i
ty
meas
ures b
e
tw
een
Vagu
e sets (v
alu
e
s), a
new
defin
ition, w
h
ic
h is
a certa
i
n
nu
mb
er i
n
o
p
e
n
interva
l
(0, 1), w
a
s put forw
ard. Moreov
er, a new
form
u
l
a
satisfying th
is defin
ition w
a
s
built acc
o
rd
ing
t
o
this defin
itio
n.
Ke
y
w
ords
: axi
o
ms of the d
i
stance for
m
u
l
a, new
distanc
e formula, cl
usteri
ng of the Vag
u
e
set rules
Co
p
y
rig
h
t
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Since 1
993, t
he Vagu
e set theory [1] propo
sed
by G
au an
d Bueh
rer
ha
s be
en
widely
use
d
in intell
igent system
s. It has be
en studi
ed i
n
a variety of vague set
reasonin
g
[2-9].
Gene
rally
sp
eaki
ng, the
system ne
ed
s
to sea
r
ch the
entire
rule
b
a
se i
n
o
r
de
r t
o
find a
suita
b
le
matchin
g
rule
,; howeve
r
,
when th
e
rule
base i
s
la
rg
e, the
se
archin
g scal
e al
so
result
s in
a
large
one, which g
r
eatly increa
ses the
compu
t
ational co
mp
lexity. A concept of cl
uste
ri
ng vagu
e rul
e
s
is pre
s
e
n
ted i
n
Referen
c
e [2] in order to
achi
eve red
u
c
tion in the si
ze of the sea
r
ch a
nd re
du
ce
the computati
onal
com
p
lexi
ty reasonin
g
.
The a
pproa
ch
is: A
s
sumin
g
the
a
c
cura
cy of cl
uste
rin
g
is
ε
before
s
e
lec
t
ing the
rules
.
If the firs
t rule is
s
e
lec
t
ed, it is
c
l
ass
i
fied in the firs
t
c
a
tegory; if the
sele
cted
rul
e
is i
(i
>1
), m
easure
the
e
a
ch
di
st
an
ce
between
the
prere
qui
site
in the
rul
e
i
and
o
t
h
e
r
p
r
er
eq
uis
i
te
s in
th
e
k
n
ow
n
r
u
le
s
s
e
pa
r
a
te
ly
(
i
nd
ic
a
t
ed
b
y
d). If d
≤ε
alwa
ys exist
s
in
a
n
y
rule
within
a
cla
ss, the
sel
e
cted
rul
e
ca
n be i
n
cl
uded
in the
cla
ss,
and the
cl
ust
e
ring
calculation
of the n
e
xt rule
can
be
continue
d. On
the oth
e
r
ha
nd, if the
rul
e
can
not be
cla
s
sified in
the
clu
s
ter
cal
c
ul
ation of a
n
y existing
cla
s
s, it s
houl
d be
inclu
ded i
n
a ne
w
cla
ss.
After the ab
o
v
e
pro
c
e
ss i
s
re
done, a
clu
s
t
e
r
will be o
b
tained
with th
e accu
ra
cy of
ε
. By the s
a
me method, s
u
b
requi
site
clu
s
tering
of the
sam
e
rule
b
a
se
A can
b
e
con
s
tru
c
te
d, and th
en
the ap
proxim
ate
rea
s
oni
ng on
clu
s
terin
g
Va
gue can be
concl
ude
d.
It should
be
empha
si
zed
that a di
stan
ce formul
a i
s
need
ed to
ca
lculate th
e di
stan
ce
betwe
en Va
g
ue sets i
n
thi
s
ap
proximate re
asonin
g
of vague
set.
Relying
on t
h
is fo
rmula, t
h
e
clu
s
terin
g
of
the vagu
e ru
les
ca
n b
e
d
one.
Referen
c
e [2]
sh
ows a fo
rmula
for the
di
stan
ce
betwe
en vag
ue set
s
, whil
e Refe
ren
c
e
[10] gives si
x distance fo
rmula
e
with t
he axiom
s
th
at
these
six formulae shoul
d
be followe
d. In this
study, two ne
w dista
n
ce fo
rmula
e
betwe
en vag
ue
sets
are presented. The
ai
m is to
p
r
ove
that they are
suitabl
e wi
th t
he kno
w
n axi
o
ms, the
r
efo
r
e,
there
will be
more
choi
ce
s about sele
cting dista
n
ce
formul
ae in o
r
der to facilitat
e the se
arch
and
redu
ce
the co
mputational complexity
purposes
in the
pro
c
e
ss of
clusteri
ng Vag
ue rule
s.
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 753
3
– 7542
7534
2. Preliminary
Kno
w
ledge
2.1.The First Distan
ce Fo
rmula for the
Vague Sets
(Values
)
Assu
me
the di
screte dom
ain :
1
,1
/
n
Ai
A
i
i
i
A
tu
f
u
u
,
1
,1
/
n
B
iB
i
i
i
B
tu
f
u
u
,mak
e
(
(
)
)
()
()
,
(
{
1
,
2
,
,
}
)
iA
i
A
i
SA
u
t
u
f
u
i
n
.
Referen
c
e [2] sho
w
s the fo
rmula for Va
g
ue set
s
(valu
e
s) a
s
follo
ws: definiteany:
(
(
)
,
()
)
(
()
)
(
()
)
4
()
()
()
()
4
i
i
i
i
A
i
Bi
A
i
Bi
dA
u
B
u
S
A
u
S
B
u
t
u
t
u
f
u
f
u
11
11
(
,
)
(
(
)
,
(
))
(
(
))
(
(
))
4
|
(
)
(
)|
|
(
)
(
)|
4
nn
ii
i
i
ii
Ai
B
i
A
i
B
i
d
A
B
d
A
u
Bu
S
A
u
S
Bu
nn
tu
tu
f
u
f
u
The above t
w
o form
ulae
referred to
the first dist
ance formul
a
for the Vag
ue set
s
(value
s).
2.2. The Axi
o
ms tha
t
the
Distan
ce Fo
rmula for the
Vague Set (Value) to be
Obe
y
ed
Presented by
Referen
c
e [10], the axioms t
hat thedi
stan
ce form
u
l
a for theVag
ue set
(value
) to be obeyed a
r
e a
s
followi
ng:
Based
on th
e
domain
X, D
∈∈
V (U), any
x, y, z
X, then t
he dista
n
ce d
(D (x), D
(y)),
d
(D (x),
D
(z)), d (D
(y),
D (z)) b
e
twe
e
n
Vague
Sets D
(x),
D
(y) and
D (z)
should
follo
w
th
e
following four axioms
.
Axiom 1: Bounded
ne
ss
0(
(
)
,
(
)
)
1
dD
x
D
y
;
Axiom 2: Boundary conditi
ons
((
)
,
(
)
)
0
dD
x
D
x
,whe
n
()
(
)
D
xD
y
;
Axiom 3: Symmetry
(
(
)
,
()
)
(
()
,
(
)
)
dD
x
D
y
d
D
y
D
x
;
Axiom 4: Triangle ine
qualit
y
(
(
)
,
()
)
(
(
)
(
)
)
(
()
,
(
)
)
d
D
x
D
y
d
Dx
Dz
d
D
y
D
z
.
As
s
u
me the dis
c
rete domain is
U, A
、、
∈
B
C
V(U),
and
12
,,
,
n
Uu
u
u
,
1
,1
/
n
Ai
A
i
i
i
A
tu
f
u
u
,
1
,1
/
n
B
iB
i
i
i
Bt
u
f
u
u
,
1
,1
/
n
Ci
Ci
i
i
Ct
u
f
u
u
,
the dist
ance f
o
rmul
a for th
e Vagu
e
sets A and B
is
d
e
fined a
s
1
1
(,
)
(
(
)
,
(
)
)
n
ii
i
dA
B
d
A
u
B
u
n
,
then the dista
n
ce d
(
A,B), d(A,C) an
d d(B
,
C) am
o
ng th
e Vague sets A, B and C sho
u
ld obey the
four axioms
as
follows
:
Axiom 5: Bounded
ne
ss
0(
,
)
1
dA
B
Axiom 6: Boundary conditi
ons
(,
)
0
dA
A
,whe
n
A
B
Axiom 7: Symmetry
(,
)
(
,
)
dA
B
d
B
A
Axiom 8: Triangle ine
qualit
y
(,
)
(
,
)
(,
)
dA
B
d
A
C
d
B
C
Theo
rem 1: T
he first di
stan
ce form
ula for
the Vague set (value) a
ccord
s to axiom
s
1-8.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
cent Study on Dista
n
ce Form
ula and
Sim
ilarity Measu
r
e
s
betwe
en Two
…
(Z
h
ang Kun
)
7535
3. The Ne
w
Dista
n
ce Fo
r
m
ulae for th
e Vague Set
(Value)
3.1. The Sec
ond Distanc
e
Formula fo
r the Vague
Set (Value
)
The
discrete
domain
U a
n
d
the
vague
sets A a
nd B
app
ea
red
in
the pa
rt of
prelimina
r
y
kno
w
le
dge, a
nd the se
co
n
d
distan
ce formula for the
Vague set (value) i
s
defin
ed as:
For any
{1
,
2
,
,
}
in
,
(
(
)
,
()
)
(
()
)
(
()
)
2
ii
i
i
dA
u
B
u
S
A
u
S
B
u
(1)
11
11
(
,
)
(
()
,
(
)
)
{
(
()
)
(
()
)
2
}
nn
ii
i
i
ii
dA
B
d
A
u
B
u
S
A
u
S
B
u
nn
(2)
Theo
rem 2: The se
con
d
distan
ce
fo
rm
ula
fo
r the V
ague
set
(val
ue)
acco
rd
s t
o
axiom
s
1-8.
3.2. The Third Dista
n
ce F
o
rmula for the Vague Se
t (Value
)
The di
screte
domain
U
an
d the Vag
ue
sets A an
d B
appe
are
d
in t
he pa
rt of p
r
e
liminary
kno
w
le
dge, a
nd the thi
r
d
distan
ce
formula fo
r the
Vague
set (value) i
s
defi
ned
as: fo
r
any
{1
,
2
,
,
}
in
,
(
(
)
,
()
)
(
()
)
(
()
)
ii
i
i
dA
u
B
u
S
A
u
S
B
u
,
11
11
(
,
)
(
(
)
,(
)
)
|(
(
)
,(
)
)
|
nn
ii
ii
ii
dA
B
d
A
u
B
u
dA
u
B
u
nn
In the above formul
ae, parameter
sati
sfies the conditi
on:
01
2
.
Theo
rem 3: Whe
n
12
, the third di
stan
ce f
o
rmul
a for th
e Vagu
e set
(value
) eq
ual
s
to the second
distan
ce formula for the
Vague set (value).
Theo
rem 4: T
he third di
sta
n
ce formula f
o
r t
he Vagu
e set (value
) a
c
cords to axio
ms 1-8.
4. Analy
s
is o
f
Examples
Example 4.1:
Assume
12
3
,,
,
(
)
Uu
u
u
A
B
C
V
U
、、
an
d
12
3
[
0
.5
,
0
.8
]
[
0
.
6
,
0
.
9
]
[
0
.
7
,
0
.
9
]
A
uu
u
,
12
3
[
0
.
4
,
0
.
7
]
[
0.
5
,
0.
8
]
[
0
.
6
,
0
.
9
]
B
uu
u
,
12
3
[
0
.3
,
0
.6
]
[
0
.
4,
0
.
7
]
[
0
.5
,
0
.9
]
Cu
u
u
.
Then, the
calcul
ation b
a
s
ed
on th
e
first di
stan
ce fo
rmula
i
s
(
,
)
0
.
083
dA
B
,
(,
)
0
.
1
7
dA
C
Cal
c
ulation b
a
se
d on the seco
nd di
stan
ce form
ula is
(
,
)
0
.
083
dA
B
,
(,
)
0
.
1
7
dA
C
Cal
c
ulation
based on th
e third dista
n
ce formula
(
0.
4
) is
(
,
)
0
.
067
dA
B
,
(,
)
0
.
1
3
dA
C
.
Example 4.2: assume
12
3
,,
,
(
)
Xx
x
x
A
B
C
V
X
、、
, and
12
3
[
0
.4
,
0
.6
]
[
0
.
5
,
0
.
7
]
[
0
.7
,
0
.9
]
A
xx
x
,
12
3
[
0
.3
,
0
.4
]
[
0
.
5
,
0
.
5
]
[
0
.
7
,
0
.8
]
B
xx
x
,
12
3
[
0
.3
,
0
.5
]
[
0
.
4
,
0
.
7
]
[
0
.6
,
0
.7
]
Cx
x
x
,
Then, the
cal
c
ulatio
n based on t
he first di
stance formul
a is
(,
)
0
.
1
0
dA
B
,
(
,
)
0
.
083
dA
C
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 753
3
– 7542
7536
The
cal
c
ula
t
ion ba
sed
on the
se
con
d
di
stance formul
a is
(,
)
0
.
1
0
dA
B
;
(,
)
0
.
1
0
dA
C
.
The calculati
on ba
sed
on
the third di
stance fo
rmul
a is (
0.
4
) is
(,
)
0
.
0
8
dA
B
;
(,
)
0
.
0
8
dA
C
.
From the ab
o
v
e two examples, it indica
tes
that different distan
ce formul
ae are applie
d
sep
a
rately for the different vague sets. Und
e
r this
way, generally,
the calculati
ng re
sults
will
not
be the same.
Select a suitable di
stan
ce
formula,
an
d
a much more rea
s
o
nabl
e
clu
s
terin
g
of the
vague set co
uld be obtai
n
ed.
In this
study,
two n
e
w
dist
ance formula
e
fo
r th
evagu
e set are p
r
o
posed. By proving the
accordan
ce
with the
kno
w
n
axioms,
there
a
r
e
mo
re choi
ce
s of t
he di
stan
ce
formul
ae
wh
e
n
the
clu
s
terin
g
of t
he vagu
e set is p
r
o
c
e
ssi
n
g
. Then,
in th
e re
asoning
of vague
rule
s, it is b
enefi
c
ial
for de
cre
a
si
n
g
the sea
r
chi
ng scop
e an
d red
u
cin
g
th
e com
p
lexity in the pro
c
e
s
s of rea
s
o
n
ing
and
cal
c
ulati
ng. Similarity mea
s
ures
b
e
twee
n V
agu
e set
(
value
)
play an im
po
rtant rol
e
in t
he
Vague
set of appli
c
ation
s
and ha
s be
co
me an impo
rt
ant part of the Vague
set theory. Howe
ver,
it has be
en
n
o
ted that th
e
simila
rity me
asu
r
e
s
b
e
twe
en vag
ue
set
s
a
r
e i
nad
eq
uate for a l
o
ng
time. For exa
m
ple, the
ref
e
ren
c
e
s
[2-4]
have
pointe
d
out
the Va
gue X
=
Y= [0-1]. X and
Y t
h
e
similarity measure
M(X,Y)=1 i
s
un
scientific. This
study aims to
re
sea
r
ch a
nd
a
nalyse t
h
is i
s
sue
and try to put forwa
r
d the m
e
thod of co
rrection in
ade
q
uaci
e
s.
5. Inadequac
i
es of the Ex
isting Sim
ila
rit
y
bet
w
een Vague Value Metrics
Assu
me is Va
gue (valu
e
):
1
(,
)
1
2
x
yx
y
tt
f
f
Mx
y
(
3
)
2
()
(
)
(,
)
1
2
xy
x
y
tt
f
f
Mx
y
(
4
)
3
()
(
)
(,
)
1
44
x
yx
y
x
y
x
y
t
t
ff
t
t
ff
Mx
y
(5)
4
(,
)
1
22
x
yx
y
SS
K
K
Mx
y
(6)
Among
,,
,
x
xx
y
y
y
x
x
x
St
f
S
t
f
K
t
f
yy
y
Kt
f
.
5
(,
)
1
2
x
yx
y
x
y
tt
f
f
Mx
y
(
7
)
6
(,
)
1
11
xy
x
y
x
yx
y
tt
f
f
Mx
y
tt
f
f
(
8
)
Example 5.1.
Use
s
formu
l
a (3)-(8
) to cal
c
ulate
Ta
ble 1 Vague
set date, got vague
simila
rity betwee
n
vague
values.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
2302-4
046
Re
cent Study on Dista
n
ce Form
ula and
Sim
ilarity Measu
r
e
s
betwe
en Two
…
(Z
h
ang Kun
)
7537
Dis
c
u
s
sion
：
i
n
the Tabl
e 1,
(
,
)
1
,
(
1
,
2
,
3
,
4
,
5,
6;
1
,
2
)
ji
i
Mx
y
j
i
. Bec
a
us
e
11
x
y
and
22
x
y
are ordi
nary coll
ectio
n
. Their un
ce
rtainty is
0(
1
,
2)
ii
xy
i
. Therefo
r
e, they
measure the
simila
rity max is 1, whi
c
h i
s
con
s
iste
nt with pe
ople’
s intuition. But
33
(,
)
1
j
Mx
y
,
(1
,
2
,
3
,
4
,
5
,
6
)
j
. Be
c
a
us
e
33
x
y
is
fuzz
y s
e
t.Set
33
0
xy
. Thus, they
mea
s
ure th
e
simila
rity max is 1
,
which i
s
co
nsi
s
te
nt with p
eople’
s int
u
ition. Because
44
(
,
)
1
,
(
1
,
2,
3
,
4,
5
,
6
)
.
j
Mx
y
j
It is count
erintuitive with
people.It is because [2-4]
44
x
y
[0,1] is particula
r Vague,
which i
s
char
a
c
te
rized
by its uncertainty, the max
44
1
xy
. People
kn
o
w
not
hing
ab
out. this el
e
m
ent, Calcul
ate the
simil
a
rity mea
s
u
r
e is
equal to 1. It mean
s the t
w
o Vag
ue a
n
s
wer i
s
44
x
y
[0,1]
is
the mos
t
similarity, which is
contrary to
intuition. But for the no
rmal
V
ague,
su
ch a
s
55
6
6
[0
.4
,
0
.8
]
,
[
0
.7
,
0
.9
]
xy
x
y
.
Their ch
ara
c
t
e
risti
cs are
u
n
ce
rtainty
0:
55
6
6
0.4
,
0.2
xy
x
y
.For the
re
ason, wo
rk
out the
an
swer i
s
1,
whi
c
h mea
n
s tha
t
they
are
m
o
st
simila
ry and
un
certai
n. Thu
s
,.a m
o
re
rea
s
on
able
n
u
mbe
r
of si
mi
larity metri
cs
sho
u
ld b
e
in t
he op
en inte
rval (0-1). Th
e
uncertainty in
the simila
rity of vague valu
e plays a pivo
tal role.
Table 1. Co
m
pari
s
on
of the Similarity Measu
r
e
s
betwe
en Vague Val
ues
i
x
1
x
=[1,1]
2
x
=[0,0]
3
x
=[0.2,0.2]
4
x
=[0,1]
5
x
=[0.4,0.8]
6
x
=[0.7,0.9]
i
y
1
y
=[1,1]
2
y
=[0,0]
3
y
=[0.2,0.2]
4
y
=[0,1]
5
y
=[0.4,0.8]
6
y
=[0.7,0.9]
1
(,
)
ii
M
xy
1 1
1
1
1
1
2
(,
)
ii
M
xy
1 1
1
1
1
1
3
(,
)
ii
M
xy
1 1
1
1
1
1
4
(,
)
ii
M
xy
1 1
1
1
1
1
5
(,
)
ii
M
xy
1 1
1
1
1
1
6
(,
)
ii
M
xy
1 1
1
1
1
1
(2
)
(,
)
ii
M
xy
1 1
1
0
0.64
0.008
Example 5.2.
Use
s
formu
l
a (3)-(8
) to cal
c
ulate
Ta
ble 2 Vague
set date, got vague
simila
rity betwee
n
vague
values.
Observed f
r
o
m
the Table
2, although t
he re
sol
u
ti
on
of formula
(3)-(7
) is
not hig
h
, it can
be intuitivelydetermin
ed tha
tformula (8
) is of highe
r re
solutio
n
than
formula (3)-(7
) .
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
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KA
Vol. 12, No. 10, Octobe
r 2014: 753
3
– 7542
7538
Table 2. Co
m
pari
s
on of Accura
cy of t
he Similarity Measure
s
bet
ween Vagu
e Value
i
x
1
x
=[0.4,0.8]
2
x
=[0.4,0.8]
3
x
=[0.4,0.8]
4
x
=[0.4,0.8]
i
y
1
y
=[0.3,0.8]
2
y
=[0.4,0.9]
3
y
=[0.5,0.8]
4
y
=[0.4,0.7]
1
(,
)
ii
M
xy
0.950
0.950
0.950
0.950
2
(,
)
ii
M
xy
0.950
0.950
0.950
0.950
3
(,
)
ii
M
xy
0.950
0.950
0.950
0.950
4
(,
)
ii
M
xy
0.900
0.900
0.900
0.900
5
(,
)
ii
M
xy
0.900
0.900
0.900
0.900
6
(,
)
ii
M
xy
0.941
0.923
0.947
0.933
(2
)
(,
)
ii
M
xy
0.837
0.799
0.902
0.868
6. The Ne
w
Vague Value
of the Similarit
y
and New
F
o
rmula
The data is gi
ven to the new formul
a.
Definition
1 [1
1]. For Vague
[,
1
]
,
x
x
x
tf
：
define
(0)
(
0
)
,
x
xx
x
tt
f
f
,
(0
)
1
x
xx
x
tf
,and
()
2
(1
)
mm
x
xx
x
x
tt
,
()
2
(1
)
mm
x
xx
x
x
ff
,
()
1
,
mm
xx
(
0,
1
,
2,
.
m
).
Lemma 1[1
1
].
(
)
()
()
[,
1
]
mm
m
xx
x
x
tf
(
0,
1
,
2,
.
m
) is the vag
ue value.
Definition
2. Assume V
a
gue
[,
1
]
,
x
x
x
tf
[,
1
]
yy
yt
f
, if the formul
a suit
the
(,
)
M
xy
:
1) No
rmative
0(
,
)
1
Mx
y
;
2) Symmetry
(,
)
(
,
)
M
xy
M
y
x
;
3) Mix when
the
[0
,
0
]
,
[
1
,1
]
xy
or
[1,
1
]
,
[
0
,
0
]
xy
,
(,
)
0
Mx
y
;
4) Max
(,
)
1
Mx
y
0
xy
xy
and
;
5) Parti
c
ula
r
ityFor
[0
,1
]
xy
,
(,
)
0
.
Mx
y
[,
1
]
zz
zt
f
;
6) Re
sol
u
tio
n
when
,
x
y
and
[,
1
]
zz
zt
f
is of arbitra
r
y Vagueval
ue ,
(,
)
(
,
)
M
xz
M
y
z
.
So
(,
)
M
xy
is Vague
value X and Y the similarit
y
.
Explanation:
most formul
a
has the lem
m
a, normativ
e
, particul
a
rit
y
and so on. Max and
particula
rity is trying to re
solve the ex
ampl
e 5.1
ca
se
s of an exi
s
ti
ng di
scussi
on vague val
ue
establi
s
h
ed b
y
the inade
q
uaci
e
s
simila
rity measur
e
s
. Resolution
is e
s
tabli
s
he
d wh
en trying
to
overcome th
e
dra
w
b
a
cks o
f
existing
simi
larity me
a
s
u
r
es in
the exa
m
ple 5.2
simi
larity ,based
on
pape
r [12].
Theo
rem 5 it’s the Vague v
a
lue
[,
1
]
x
x
x
tf
and
[,
1
]
yy
yt
f
: s
i
milarity measures
,
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
cent Study on Dista
n
ce Form
ula and
Sim
ilarity Measu
r
e
s
betwe
en Two
…
(Z
h
ang Kun
)
7539
()
()
()
()
()
()
()
(,
)
1
2
mm
m
m
m
m
xy
x
y
x
y
m
tt
f
f
Mx
y
(
9
)
Among
0,
1
,
2,
m
.
For exampl
e 5.1 Vague in the value data,the app
lication of the formula (9
) ( pa
ramet
e
r
m=2 )
cal
c
ula
t
ion cal
c
ulate
d
the simila
rity measu
r
e
s
b
e
twee
n the o
b
tained vag
u
e
value is al
so
sho
w
n in
Ta
ble 1, in thel
astro
w
. We l
ook fo
rw
ard t
o
these re
sul
t
s exac
tly, se
en Tabl
e 1, the
formula (9) h
a
s be
en overcome vag
ue i
n
example 5.
2.
Definition 3. If
12
{,
,
,
}
n
X
xx
x
, which h
a
s V
ague:
11
2
2
([
(
)
,
1
(
)
],
[
(
),
1
(
)],
,
[
(
)
,
1
(
)])
GG
G
G
G
n
G
n
Gt
x
f
x
t
x
f
x
t
x
f
x
,
11
2
2
([
(
)
,
1
(
)
],
[
(
),
1
(
)],
,
[
(
),
1
(
)])
S
S
S
S
Sn
Sn
St
x
f
x
t
x
f
x
t
x
f
x
Label
ed
11
2
2
1
1
([
,
1
],
[
,
1
]
,
,
[
,
1
]),
([
,
1
],
,
[
,
1
]).
G
G
x
G
Gn
Gn
S
S
S
n
S
n
Gt
f
t
f
t
f
B
t
f
t
f
If the formula
(,
)
M
GS
suit
s it
:
1) Lemm
a
0(
,
)
1
MG
S
;
2) Parti
c
ula
r
ity
(,
)
(
,
)
M
GS
M
G
S
;
3) Mix formul
a
([
0
,
0],
[
0
,
0],
,
[
0
,
0
])
G
,
(
[
1
,
1]
,
[
1
,
1]
,
,
[
1
,
1
]
)
S
or
([
1
,
1
]
,
[
1
,
1
]
,
,
[
1
,
1
])
,
(
[
0
,
0
],
[
0
,
0
],
,
[
0
,
0
]
)
GS
时
,
(,
)
0
MG
S
;
4) Max
(,
)
1
MG
S
A
B
,and
0,
1
,
2,
Gk
S
k
kn
;
5) Parti
c
ula
r
ity when
([
0,
1
]
,
[
0
,
1
]
,
,
[
0
,
1
]
)
GS
,
(,
)
0
MG
S
;
6)
Re
solutio
n
when
,
GS
but
R
is
anyone
value
,
so
(,
)
(
,
)
M
GR
M
S
R
.
(,
)
M
GS
is
the simila
rity
Vague bet
we
en G and M.
Theo
rem 6. T
h
is is
(,
)
M
GS
vague similarity between G an
d M.
()
()
()
()
()
()
()
1
1
(,
)
1
2
n
mm
m
m
m
m
mG
i
S
i
G
i
S
i
G
i
S
i
i
MG
S
t
t
f
f
n
(10
)
0,
1
,
2,
m
.
Theo
rem 7. T
h
is is
(,
)
WM
G
S
Vague
betwee
n
G a
nd M:
(
)
()
()
()
(
)
()
()
1
1
(,
)
1
2
n
mm
m
m
m
m
m
i
G
i
Si
G
i
Si
G
i
S
i
i
WM
G
S
w
t
t
f
f
(11)
0,
1
,
2,
m
, and
(0
1
)
ii
ww
is the weight of
i
x
,and suffice
1
1
n
i
i
w
.
Example 6.
1.
Let the
domain
12
3
{,
,
}
X
xx
x
，
in
it’s stan
da
rd mod
e
Vagu
e set
12
3
,,
GG
G
and prepa
rati
on re
cog
n
itio
n mode
S
:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 753
3
– 7542
7540
12
3
(
[
0.2
,
0.
4]
,
[
0.3
,
0.6]
,
[
0.6
,
0.7
]),
([
0.5
,
0.6],
[
0.7,
0.
9]
,
[
0.6
,
0.8
]
),
([
0.
4
,
0.7
]
,
[
0.3
,
0.8
]
,
[
0.5
,
0.9])
;
(
[
0.3
,
0.5
]
,
[
0.4
,
0.7
]
,
[
0.7,
0.9])
.
GG
GS
Applicatio
n of the formula (10)
，
tak
e
parameters
2
m
calculate
，：
get
(2
)
1
(2
)
2
(2
)
3
(
,
)0
.
8
3
,
(
,
)0
.
7
9
,
(
,
)0
.
8
6
MG
S
M
G
S
MG
S
Applicatio
n V
ague
crite
r
ion
of pattern re
cog
n
ition
，
ge
tthe pre
paration recognitio
n
mod
e
S
itis vested in the stan
dard mode
3
G
.
Applicatio
n of the formula (8) cal
c
ul
ate
，：
get
61
6
2
6
3
(
,
)
0
.8
6
,
(
,
)
0
.8
1
,
(
,
)
0
.8
7
MG
S
M
G
S
MG
S
The a
ppli
c
ati
on of
pattern
re
cog
n
ition
crite
r
ia Va
gu
e
，
getthe pre
paratio
n re
co
gnition
mode
S
Shoul
d be veste
d
i
n
the stan
da
rd mode
3
G
. And appli
c
ation
of the formul
a (10
)
the
results obtai
n
ed is the sam
e
as.
If the two
re
sults a
r
e
different, th
en t
he a
ppli
c
atio
n of th
e fo
rmula
(10
)
th
e results
obtaine
d are
more
credibl
e. Becau
s
e f
o
rmul
a (1
0)
satisfie
s the
definition of
2,
it
s st
ru
ct
ur
e is
more rea
s
on
able. Thu
s
, the con
c
lu
sio
n
t
hat it is more
efficient inference.
7. Practical a
pplication
Example 7.1.
A
ssume
12
3
{,
,
}
X
xx
x
, among it. Vague G1, G2, G
3
and S:
12
3
(
[
0.2
,
0.
4]
,
[
0.3
,
0.6]
,
[
0.6
,
0.7
]),
([
0.5
,
0.6],
[
0.7,
0.
9]
,
[
0.6
,
0.8
]
),
([
0.
4
,
0.7
]
,
[
0.3
,
0.8
]
,
[
0.5
,
0.9])
,
([
0.3
,
0.5
]
,
[
0.4
,
0.7
]
,
[
0.
7,
0.9])
.
GG
GS
So (8), and
2
m
got :
(2
)
1
(2
)
2
(2
)
3
(
,
)0
.
8
3
,
(
,
)0
.
7
9
,
(
,
)0
.
8
6
MG
S
M
G
S
MG
S
Based o
n
the Vague patt
e
rn reco
gniti
on crit
e
r
ion,
the resultant
pattern S should be
attributed to the stan
dard p
a
ttern G3, wh
ich is
id
entica
l
to that obtained by formul
a (10
)
.
61
6
2
6
3
(
,
)
0
.8
6
,
(
,
)
0
.8
1
,
(
,
)
0
.8
7
MG
S
M
G
S
MG
S
Note that
re
sults yiel
ded
by formula
(10
)
is
more
reliabl
e, for formula
(1
0) satisfie
s
definition 2 a
nd feature
s
a
better structu
r
e.
8. Conclusio
n
Dista
n
ce bet
wee
n
Vague
sets i
s
alway
s
the study focu
s
，
ap
plications al
so mo
re widely
in re
ce
nt
y
e
a
r
s,
su
ch a
s
r
e
f
e
ren
c
e
s
[
1
4-21]
.
A
c
co
rd
ing to a
pplica
t
ion ca
se
s, di
fferent di
stan
ce
formula
s
sho
u
ld be a
pplie
d for differe
n
t
Vague sets,
while the
ca
lculate
d
dista
n
ce valu
es
a
r
e
also
differe
nt. With a
ppro
p
riate
dista
n
ce form
ula, th
e mo
re
rea
s
onabl
e
cluste
r for Vagu
e
set
rule
s, whi
c
h
may be con
d
u
ctive to red
u
ce
sea
r
ch scale an
d cal
c
ulatio
n com
p
lexity in Va
gue
rule
-ba
s
e
d
reasonin
g
, ca
n be o
b
tain
ed. In additi
on, ne
w defi
n
ition of si
m
ilarity mea
s
u
r
es
betwe
en Va
g
ue
sets reserves the
ba
si
c natu
r
e
of
si
milarity mea
s
ure
s
betwe
en
Vague
sets
and
effectively exclud
es some
deficie
nci
e
s o
f
simila
rity m
easure
s
between Va
gue
sets. In
parti
cu
lar,
this d
e
finition
point
s out th
at the value
should
be
a
ce
rtain n
u
mbe
r
in ope
n inte
rval (0,
1),
rath
er
than 1, which is more a
c
curate and in
novative.
The sustai
nable i
m
provem
ent of definition for
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
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046
Re
cent Study on Dista
n
ce Form
ula and
Sim
ilarity Measu
r
e
s
betwe
en Two
…
(Z
h
ang Kun
)
7541
simila
rity measu
r
e
s
bet
we
en Vagu
e sets is
not t
he n
eed for Vagu
e set a
ppli
c
at
ion, but al
so t
h
e
new tre
nd of Vague set theory.
Ackn
o
w
l
e
dg
ements
The work wa
s supp
orte
d by
the
20
13 Hain
an e
d
u
c
ation scie
nce
study topi
cs of the
"12th Five
-Ye
a
r Pla
n
(No.
QJY12
511
9),
the
Hain
an
Planning
p
r
oj
ects of
philo
sophy a
nd S
o
cial
Scien
c
e
s
(No. HNSK1
4-50), the
hain
an key sci
e
n
t
ific and te
chnolo
g
ical
pl
an p
r
oje
c
ts
(No.
zdxm20
140
8
7
), the
20
13
innovatio
n
and
entre
pr
e
neurshi
p
trai
ning
coll
ege
nation
a
l
coll
ege
proje
c
ts (No.
2013
1110
006
1).
Referen
ces
[1]
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an Qin-W
en, Yin Gua
ng-zh
i, He Yo
u-fang
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ation
o
f
GIS and F
u
zz
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a
tio
n
T
heor
y
t
o
Locati
on of T
a
ilings D
a
m.
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a
l
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0
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14.
[2]
Z
hang
Kun, W
ang
Ho
ng-
xu,
W
ang H
a
i-F
e
n
g
, Li Z
h
ua
ng.
Ne
w
E
x
pl
orati
on
on
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i
tio
n
of Sim
ilar
i
t
y
Measur
es bet
w
e
e
n
Vag
ue S
e
ts.
Journa
l of
Natura
lScie
n
c
e
of He
ilo
ng
jia
ng U
n
ivers
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ty
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2; 29(
03)
:
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[3]
W
ang H
o
n
g
-
x
u. S
y
nthes
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ue S
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ts and
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p
p
licatio
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me Optimu
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ang H
ong-
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r
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rmu
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a
lue
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ang Ho
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hang F
u
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e
rgro
un
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he Internation
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ang H
ong-
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lu
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ang H
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a
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ang Ho
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r
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u
zz
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gue Va
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a
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ua-
w
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ang F
e
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i
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e S
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ang H
o
n
g
-
x
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.
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i
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e
t
w
e
e
n
Vag
ue S
e
ts a
nd Its Ap
plic
ati
on.
C
o
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r
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h
a
ng-
ya
n. Measu
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e
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en Va
g
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l of Softw
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. 200
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926.
[12]
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n
SM. Similarit
y
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w
e
e
n
Va
gue Sets an
d bet
w
e
en Elem
ents.
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m CA. A Note Simil
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e S
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eme
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a
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lin
g and
Contro
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y
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d
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a
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u
zz
y
Log
ic Co
ntrol
T
e
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T
E
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g
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[15]
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i
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T
y
peI
fuzz
y
bo
un
dar
y
an
d R
e
se
arc
h
o
n
Bo
un
dar
y Defi
nitio
n
of
High
Order
Fu
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y
Re
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n
.
T
E
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nesi
a
n
Jo
u
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of El
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al E
ngi
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ao
L
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w
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Nun
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u
zz
y
Mu
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Crit
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Decisi
on
Maki
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ode
l
w
i
t
h
F
u
zz
y T
i
me
W
e
ight Schem
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T
E
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1
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Ni
gh
at
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Ya
pin
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, Amir M
a
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,
Xu
Xia
ng Y
ang. Id
entific
ati
on
of No
nl
ine
a
r
S
y
stem Base
d on F
u
zz
y
Mo
d
e
l
w
i
th Enh
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E
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ngi
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. 2
014;
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[18]
Brahim F
e
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Samira Di
b, Brahim Ber
bao
ui
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h
id D
ehi
ni. Desi
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and Sim
u
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on
of D
y
n
a
mi
c
Voltag
e R
e
sto
r
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a
sed
o
n
F
u
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y
Co
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ANF
I
S.
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l J
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u
r
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r
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d Drive Syste
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[19]
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a
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Manso
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r Bazr
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i Akbari, Mojd
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age
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e
Based
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u
zz
y T
e
c
hni
que
w
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th
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a
rall
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l
i
d
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Mode O
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Inter
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l Jo
urna
l o
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Advanc
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. 2013; 2(4): 1
71-1
84.
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Vol. 12, No. 10, Octobe
r 2014: 753
3
– 7542
7542
[20]
Krishn
an M
ani
ckavasa
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F
u
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c co
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Sing
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u
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r P
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ngi
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: 1-8.
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p
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ontr
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for S
a
tel
l
i
t
e Attitude
Co
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b
y
T
w
o
State actuator
to reduc
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i
mit C
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c
l
e b
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a
kagi
Suge
no M
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th
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urna
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