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Now we are ready to define a non-domination matrix, instead of the discernibility
matrix of RSM. The non-domination matrix
M
=
(
m
ij
)
i
,
j
=
1
,
2
,...,
n
in DRSM is
defined as follows:
m
ij
={
c
∈
C
|
(
u
j
,
u
i
)
∈
D
}
}
{
c
Based on
M
, we define four non-domination functions.
Definition 19
Non-domination functions
F
≥
,
F
≤
and
F
L
are defined as follows.
F
≥
(
˜
c
1
,...,
˜
c
m
)
=
m
ij
˜
c
,
c
∈
i
,
j
|
l
C
(
u
j
)<
l
C
(
u
i
)
F
≤
(
˜
c
1
,...,
˜
c
m
)
=
m
ji
˜
c
,
c
∈
i
,
j
|
u
C
(
u
j
)>
u
C
(
u
i
)
⊛
⊞
F
L
⊝
⊠
,
(
˜
c
1
,...,
˜
c
m
)
=
m
ij
˜
c
∧
m
ji
˜
c
c
∈
c
∈
i
:
l
C
(
x
i
)
=
u
C
(
x
i
)
j
|
l
C
(
u
j
)<
l
C
(
u
i
)
j
|
u
C
(
u
j
)>
u
C
(
u
i
)
where
c
i
is a Boolean variable corresponding to
i
th condition attribute
c
i
.
From Lemma
6
, we have the following theorem. Let
A
˜
c
A
ↆ
C
. Remember that
˜
c
i
is a Boolean vector such that
i
th element
˜
is true iff
c
i
∈
A
, and
ˆ
A
is the term
{˜
c
|
c
∈
A
}
.
Theorem 10
([
31
,
49
,
52
])
Let A be a subset of C . We have the following equiva-
lences:
•
A satisfies (DlG), i.e., (
DL
≥
) if and only if F
≥
(
˜
c
A
)
=
1
. Moreover, A is an
L
≥
-reduct in DRSM if and only if
ˆ
A
is a prime implicant of F
≥
,
A satisfies (DuG), i.e., (
DL
≤
) if and only if F
≤
(
˜
c
A
•
)
=
1
. Moreover, A is an
L
≤
-reduct in DRSM if and only if
ˆ
A
is a prime implicant of F
≤
,
A satisfies (DLG), i.e., (DL) if and only if F
L
c
A
•
(
˜
)
=
1
. Moreover, A is an L-reduct
ˆ
A
is a prime implicant of F
L
.
From Theorem
10
,allL
≥
-, L
≤
- and L-reducts can be obtained as all prime impli-
cants of Boolean functions
F
≥
,
F
≤
and
F
L
, respectively.
The proposed non-domination matrices have an advantage when compared with
the previous ones. We need to calculate neither lower, upper approximations nor
boundary regions of unions but only the lower bounds
l
C
and the upper bounds
u
C
of all objects. Namely, the computation of the proposed approach is free from the
number of decision classes.
in DRSM if and only if
Example 13
Remember the decision table given
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.7
.
In Table
7.8
,weshowagain
with the lower bounds
l
C
and the upper bounds
u
C
of
the generalized decisions of the objects which appear in the rightmost two columns
of the table. To obtain
l
C
(
D
, we search the minimum class in
D
C
(
u
i
)
and
u
C
(
u
i
)
u
i
)
and the maximum class in
D
C
(
u
i
)
, respectively.
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