216 C. Hao et al. / Information Sciences 415–416 (2017) 213–232
Table 2
The information table
S
1
=
U,
a
1
1
, a
1
2
, a
1
3
under the first level of
scale.
U a
1
1
a
1
2
a
1
3
x
1
2 2 2
x
2
3 2 2
x
3
4 3 3
x
4
5 4 4
x
5
6 5 5
x
6
6 5 5
x
7
7 5 6
x
8
6 6 7
x
9
7 5 6
x
10
6 6 7
x
11
8 6 8
x
12
9 7 9
x
13
9 7 9
• The first level of scale is {2, 3, 4, 5, 6, 7, 8, 9} representing the money from two thousand to nine thousand;
• The second level of scale is {I, II, III, IV, V, VI} representing six preliminary assessment levels;
• The third level of scale is { A, B, C, D, E } representing “excellent”, “good”, “general”, “bad”, and “very bad”;
• The fourth level of scale is { H, S, M, L } representing “high”, “slightly high”, “medium”, and “low”;
• The last one is { Y, N } representing the trust “yes” or “no” to customers.
From Definition 3 , the data in Table 1 form a multi-scale information table. If we take k = 1 , then the information table
S
1
= (U, { a
1
1
, a
1
2
, a
1
3
} ) is generated by the first level of scale and it is shown in Table 2 . We can induce S
2
, S
3
, S
4
and S
5
in a
similar manner.
3. Sequential 3WD induced by a multi-scale information table
According to the discussion in Section 2.2 , it is known that a multi-scale information table can be decomposed into
several single-scale information tables. Moreover, each single-scale information table can lead to three-way decisions (i.e.
acceptance, rejection and uncertainty). Then, it is important to clarify the relationship between the three-way decisions in-
duced by different levels of scale. Before embarking on this issue, we need to discuss the relationship between the partitions
induced by the equivalence relations with different levels of scale.
With the attribute a
k
j
∈ AT
k
= { a
k
1
, a
k
2
, ··· , a
k
m
} under the k th level of scale, we can obtain an equivalence relation
R
a
k
j
=
(x, y ) ∈ U × U | a
k
j
(x ) = a
k
j
(y )
. (5)
Here, R
a
k
j
is regarded as the equivalence relation induced by one attribute. Furthermore, for the whole attribute set AT
k
, we
have
R
AT
k
=
a
k
j
∈ AT
k
R
a
k
j
=
(x, y ) ∈ U × U | a
k
j
(x ) = a
k
j
(y ) for any a
k
j
∈ AT
k
. (6)
Let U/R
AT
k
= { [ x ]
R
AT
k
| x ∈ U} be the partition of U induced by the equivalence relation R
AT
k
, where x
1
, x
2
∈ [ x ]
R
AT
k
means
that a
k
j
(x
1
) = a
k
j
(x
2
) for any a
k
j
∈ AT
k
. By the granular information transformation function g
k,k +1
j
, we know that for x
1
, x
2
∈
U , if a
k
j
(x
1
) = a
k
j
(x
2
) , then a
k +1
j
(x
1
) = a
k +1
j
(x
2
) . In other words, for any [ x ]
R
AT
k
∈ U/R
AT
k
, there must exist [ x
]
R
AT
k +1
∈ U/R
AT
k +1
such that [ x ]
R
AT
k
⊆ [ x
]
R
AT
k +1
. We denote this relationship by U /R
AT
k
≤ U /R
AT
k +1
. In this case, we say that the information
table under the k th level of scale is finer than the one under the (k + 1) th level of scale, or equivalently, the information
table under the (k + 1) th level of scale is coarser than the one under the k th level of scale.
Now we are ready to address the issue raised at the beginning of this section. In what follows, we respectively denote
by ACP( AT
k
, X ), REJ( AT
k
, X ) and UNC( AT
k
, X ) the three-way decisions (i.e. acceptance, rejection and uncertainty) of X in the
k th level of scale.
It deserves to be mentioned that hereinafter we reverse the order of levels of scale in a multi-scale information table (i.e.
from the coarse level to the fine level) in order to reveal the effect on three-way decisions when the levels of scale change.
Proposition 1. Let S = (U, AT = { a
k
j
| j = 1 , 2 , ··· , m, k = 1 , 2 , ··· , s } ) be a multi-scale information table, X ⊆U and AT
k
=
{ a
k
1
, a
k
2
, ··· , a
k
m
} be the attribute set AT in the kth level of scale. Then, from the kth level of scale to the (k + 1) th level of scale,
the three-way decisions of X are updated as follows: