Applied Mathematics, 2011, 2, 918-921
doi:10.4236/am.2011.27125 Published Online July 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Lower Approximation Reduction in Ordered Information
System with Fuzzy Decision
Xiaoyan Zhang, Weihua Xu
School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China
E-mail: zhangxyms@gmail.com , chxuwh@gmail.com
Received March 28 , 20 1 1; Revised June 5, 2011; accepted J un e 8, 2011
Abstract
Attribute reduction is one of the most important problems in rough set theory. This paper introduces the
concept of lower approximation reduction in ordered information systems with fuzzy decision. Moreover, the
judgment theorem and discernable matrix are obtained, in which case an approach to attribute reduction in
ordered information system with fuzzy decision is constructed. As an application of lower approximation
reduction, some examples are applied to examine the validity of works obtained in our works.
Keywords: Fuzzy Decision, Lower Approximation Reduction, Ordered Information Systems, Rough Set
1. Introduction
Rough set theory, proposed by Pawlak Z. in the early
1980s [1], is an extension of classical set theory and can
be regarded as a soft computing tool to deal with uncer-
tainty or imprecision information. It was well known that
this theory is based upon the classification mechanism,
from which the classification can be viewed as an
equivalence relation and knowledge blocks induced by it
be a partition on universe. For this reason, it has been
applied widely and successfully in pattern recognition [2],
medical diagnosis [3], granular computing [4] and so on.
Attributes reduction, as one important portion of rough
set researching, its main idea is not only to delete re-
dundant attributes but also to pr eserve th e invariability o f
classification ability. In fact, for the reason of noise or
information losing, there have many information systems
that the relation is not equivalence relation. Therefore,
how to deal with this type of information systems has be-
came a very hot topic in rough set theory. Meantime,
many experts have studied attribute reduction by extend
ing the equivalence relation to consistent relation, similar
relation, dominance relation [5-7] and so on. Also, some
useful works, take dominance relation based information
systems, [8-14] for example, have been done in detail.
At the same time, for decision makers, there may exist
one case that the decision objects are not certain or
precision but fuzzy. So the fuzzy decision must be taken
into consideration. Our work in this paper is to consider
the attribute reduction in ordered information systems
with fuzzy decision. Firstly, concept of lower approxi-
mation consistent set is proposed by comparing decision
attribute values. What is more, lower approximation re-
duction and judgment theo rem is introduced, from which
one can define the discernable matrix and find that it is
an useful approach to attribute reduction in ordered in-
formation system with fuzzy decision. As an applica-
tion of lower approximation reduction, examples are
considered to illustrate the validity of some results ob-
tained in our works.
The rest of this paper is organized as follows. Some
preliminary concepts required in our work are briefly
recalled in Section 2. In Section 3 the lower approxi-
mation reduction in ordered information system with
fuzzy decision is investigated. The discernable matrix and
approach to attribute reduction are defined in Section 4.
Section 5 concludes this paper.
2. Information System with Fuzzy Decision
In this section, we shall begin our work with some nec-
essary concepts required. For more detailed description
of the theory, please refer to references [15,16].
Definition 2.1. An information system is a quaternion
(, ,,)
I
UA DFG
, where
12
,,,
n
Uuu u is nonempty finite set of objects;
12
,,,
p
A
aa a is nonempty finite set of condi-
tional attributes;
{}Dd
is set of decision attribute.
,
kk
F
fUVk p is relation set ofand U
A
,
X. Y. ZHANG ET AL.919
and

ikk i
VfuuU
is the domain of k
aA
. Table 1. An ordered IS with fuzzy decision.
Attributes
d
dUV is th,
and
Ge relation set of d U anD


U
is the domain of dD.
Inith fuzzy decisio
d
dii
Vuu
formation system w that the
de
n is
cision value of objects is expressed as fuzzy form, i.e.,
for any i
uU,

0, 1
i
du .
Defin2.ition 2. Let

,,,
I
UADFG
A, if denote be an infor-
mation system and B



,,Ruu fuf 
kBij kij
UUu B ,
then
k
a
B
R
to
means a dominance relation on re- U with
io
spect B, in which case the informatn system

,,,
I
UA FG is called an ordered information
y
D
systnoted bem and de
I
.
If an information system is based on domrela-
tio inance
n and the decision value of objects is fuzzy, then the
information system is called ordered information system
with fuzzy decision. For convenience, the notation
f
I
is used to express ordered information systems w
fuzzy decision, from which the dominance class of any
i
uU is denoted as

ith

,
ij
uuuuR
.
i
B
Pro t
jB
position 2.1. Le
f
I
be an informm
an
ation syste
dBA.
1)
B
R
is reflexive, transitive, but noric, so it
is
th.
t symmet
B
.
21
i
B
.
stem
not an equivalence relation.
2) If 12
BBA, thenA
R
 
21
B
R R

uu

3) If , then
 
u
12
BBA ii
AB
4) If en uu

ji
A
uu,
ition 2.3.

iA
A
j

n information syDefin For a
f
I
re, the
lower and lower approximation sets of D withspect
to
A
, which are fuzzy sets, are denoteby d
D
A
and
D
A
respectively. And their membership functs are
ned by
,
defi ion



min
Dij ji
A
Auduu u
,




.
ation sy
max
Dij ji
A
Auduu u
Example 2.1. Consider an ordered informstem
wi
and
th fuzzy dec ision in Table 1.
By comput i ng we hav e that

0.5, du du



 
12
34
56
0.3,
0.7, 0.9,
0.1, 0.6,
du du
du du






256
46
6
, ,,
, ,
.
uuu
uu
u






1 12562
3 234564
55 6
,,,,
,,,,,
,
AA
A
A
AA
u uuuuu
uuuuuuu
uu u





U
1
a 2
a 3
a d
1
u 1 2 1 0.5
2
u 3 2 2 0.3
1 1 0.7
2 1 3 0.9
3
u 2
4
u
5
u 3 3 2 0.1
6
u 3 2 3 0.6
So we have tht a
1
u2 4 6
23456
.1.60 0.6
,
0.6 0.6 0.9 0.9 0.10.6.
D
Au uu
uuuuu


3. Lower Approximation Reduction of
Ordered Information System with Fuzzy
Decision
n,
a-
tio
wiision. Therefore, the lower approximation
consistent set is constructed and lower approximation
3
u5
u
1
D
Au

0.10 0.10.1
Since dominance relation is not the equivalence relatio
the approach to attribute reduction in original inform
n systems is not fitted for ordered information system
th fuzzy dec
reduction is defined, from which the judgment theorem is
introduced.
Definition 3.1. Let
f
I
be an information system and
BA. If
D
iDi
A
uBu

for any i
uU, then B
is called lower approximation consistent set of this in-
formation system. Moreover, if any proper subset of B
is not a lower approximation consistenen
on n
system
t set, thB is
e lower approximatio reduction of this information
.
Example 3.1. (Continued from Example 2.1) If take
23
,Baa, we will have that
 
ii
AB
uu

holds for
any i
uU
. Hence

D
iDi
A
uBu

, that is,
23
,aa
is one lower approximation consistent set of this infor-
mation system. Moreover, if take

13
,Caa, by com-
inputg we ha ve




1 6
2256
,
,, ,
,,,,
C
C
u u
uuuu
uuuuu






12 5
3 23456
446
5256
66
34
,,,,
,
,,
,, ,
,
,
C
C
C
C
uuuuu
u
uuu
uuuu
uu
Copyright © 2011 SciRes. AM
X. Y. ZHANG ET AL.
920
and
123456
0.10.10.1 0.60.1 0.6.
D
D
CA
uuuuuu


Hence,
31,aa
set. Mois also another lower approximation
consistentreover, we can obtain that
3
a
concl
n redu
is
lower appron consistent sets. Hence weude
that d only lower approximatioc-
tion ation system.
the next, the judgment theorem in ordered informa-
tion system with fuzzy decision is proposed.
Theorem 3.1. Let
oximati
is one an
is inform

3
a
of th
In
f
I
wer be an information syst and
. is a loapproximation consistent set if
em
BA
an B
d only if for every i
u, j
uU
, if

D
iDj
A
uAu

,
then there exist k
aB such that


j
kki
f
ufu
Proof:
: Suppose that
.

j
kki
f
ufu for every
k
aB when() ()
D
iDj
A
uAu

. So we can obtain
ij
B
uu


, that is,

ij
B
B
uu


. By




min
Di i
B
Buduu u
and



min
Dj j
B
Buduuu



,
we ht ave tha

D
jD
Bu B

Since er approximaonsistent set, we
have th
i
u.
B
at is lowtion c
 
D
jDj
A
u
dBu
an

D
iDi
A
uBu
. So
we can
obtain


D
jDi
A
uAu

. It is a contction.
”: Supposeis not lower approximation con-
et, then thust exist one such that
radi
sisten B
ere mt s0
i
uU
 
Di
00
Di
A
uBu

. According
have
to Proposition 2.1, we
that
 
Di
uu
k00
00
Di
BA

If we tae j
B
i
uu


such that




000
jiDi
B
duuB u

 .
min duu
We have known that



00
min A
jj
du uduu

. From above, we have that because of00
jA
j
uu

 
00
j
Di D
A
u
Au
, thus there exist such that
j
k
aB

00
kki
f
ufcontradiction
u
. It is a with
00
j
B
i
uu


.
The theorem is proved.
4. Approach to Lower Approximation
on
In theorem 3.1, we proposed an equivalent description of
lower approximatch it can be
used to estimate whether the attribute set is an lower ap-
proximation consistent set or not. In next, discernable
mnd reduction approach is con-
structed im
Reducti
ion consistent set, from whi
atrix is introduced first a
mediately.
Definition 4.1. Let
f
I
be an information system and

iDj
D uAu
. If denote ,
fijD
uuA






,
,,
kijijf
fij
ij
kk
f
auuuuD
Duu uu D
Af f


,
then
f
D is called the lower approximation discernable
attribute set between i
u and
j
u, and
,MDuu
,
ij
ffij
uu U
is the discernable matrix of this information system.
Theorem 4.1. Let
f
I
is lower approxim
be an information system and
ation consistent set if an
only if
BA, B d
,
fij
BDuu
 for all
of m 3.1 wehere exist

,
ij f
uu D.
Pro :
”: By theore have that t
k
aB
s.t.
j
kki
f
ufu for, then

,
ij f
uu D
u, hence BD,u
kf
aDij fij

,uu
.
”: For

,
ij f
uuD , if ,
fij
BDuu
,
erexist k
aB
thee n th
s.t.

,
k
aDuu, that is
fij
j
kki
f
um 3.1 wB
fu
lowe
r a
. By Theoree have that is a
pproximation consistent set of
A
.
Ttheorem is proved.
ition 4ormat
he
Defin .2. For an infion system
f
I
and
er approle f
BA, the lowximation discernaboula is
denoted as rm



,,
,,,
fkf i
kkfi jijf
aa D uuuuD
 

Theorem 4.2. Let

,
.
kj ij
FaaD uuuuU 
 
f
I
be an information system and
th normal form is defined as e minimum alternative

1
pk
f
f
B
 . If take

,1,2,,
k
k
E
f
sk
Basq then
,,p
the lower approxim
,1,2
k
f
Bkis the set with all its elements have
ation reduction form.
Proof: For any
,
ij f
uu D
normal form
, by the definitin of
minimum alternative and theorem 4.1, we
have that
o
k
f
B is consistent set. If
on
lower approximation
e element of k
f
B in

1
pk
f
f
k
F
B
 is deleted and
k
f
B becomes then the re must exis t

`
k
f
B,from which
Copyright © 2011 SciRes. AM
X. Y. ZHANG ET AL.
Copyright © 2011 SciRes. AM
921
Table 2. Discernable matrix of the IS in example 2.1.
Objects
U
1
u 2
u 3
u 4
u 5
u 6
u
1
u 13
aa 13
aa
2
u  3
a 3
a
3
u 123
aaa
13
aa
4
u
5 3
a 3
a
u
6
u
such that

00
,
ijf
uu D


,
k
ffij
BDuu
00
. So
is not a lower approximation consistent set.

`
k
f
B
Hen eck
f
B r approxiation reduction. Sinceis a lowem
f
M
includes all

,
f
ij
Duu
redutions
, there don’t exist other
atit at have different forms.
Eple 4.1. (Continued fro Example 2.1 and
ample 3.1)y comutinghave Table .
Hence, a, that
to say, ation
eduction sion ac
quired in this consistent with the resul
Examp
.
systems are based on dominance rel
f va
ation consistent set is constructed and
te reduction in ordered information
on is proposed. Moreover, the
lower approximon ch
m
xam Ex-
Bp we 2

3 1233faa a aaa 
3
{}a is the one and only lower approxim
from which we have that the conclu
1 3
E 
is
r,-
Example is ts in
le 3.1.
5Conclusions
Attribute reduction, as one hot research topic, has played
an important role in rough set theory. However, most of
informationations
because orious factors. To acquire brief decision
rules from inconsistent ordered information systems with
fuzzy decision, attribute reductions are needed. The main
aim of this paper is to study the problem. In this paper,
the lower approxim
approach to attribu
system with fuzzy decisi
judgment theorem of lower approximation consistent set
is obtained and some useful works are done in detail.
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