Dominance-Based Rough Sets Using Indexed Blocks as Granules

Fundamenta Informaticae 94 (2009) 1–14 DOI 10.3233/FI-2009-121 IOS Press

Dominance-Based Rough Sets Using Indexed Blocks as Granules Chien-Chung Chan∗ Department of Computer Science, University of Akron Akron, OH, 44325-4003, USA and Department of Information Communications, Kainan University No. 1 Kainan Road, Luchu, Taoyuan County 338, Taiwan [email protected]

Gwo-Hshiung Tzeng Department of Business and Entrepreneurial Administration, Kainan University No. 1 Kainan Road, Luchu, Taoyuan County 338, Taiwan and Institute of Management of Technology, National Chiao Tung University 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan [email protected]; [email protected]

Abstract. Dominance-based rough set introduced by Greco et al. is an extension of Pawlak’s classical rough set theory by using dominance relations in place of equivalence relations for approximating sets of preference ordered decision classes satisfying upward and downward union properties. This paper introduces the concept of indexed blocks for representing dominance-based approximation spaces. Indexed blocks are sets of objects indexed by pairs of decision values. In our study, inconsistent information is represented by exclusive neighborhoods of indexed blocks. They are used to define approximations of decision classes. It turns out that a set of indexed blocks with exclusive neighborhoods forms a partition on the universe of objects. Sequential rules for updating indexed blocks incrementally are considered and illustrated with examples.

Keywords: Rough sets, Dominance-based rough sets, Multiple criteria decision analysis (MCDA), Classification, Sorting, Indexed blocks, Granular computing ∗

Address for correspondence: Department of Computer Science, University of Akron, Akron, OH, 44325-4003, USA

1

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C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

1. Introduction Dominance-based Rough Set Approach (DRSA) introduced by Greco, Matarazzo and Slowinski [1, 2, 3] extend Pawlak’s classical rough sets (CRS) [8, 9, 10] by considering attributes, called criteria, with preference-ordered domains and by substituting the indiscernibility relation in CRS with a dominance relation that is reflexive and transitive. The DRSA approach was motivated by representing preference models for multiple criteria decision analysis (MCDA) problems, where preference orderings on domains of attributes are quite typical in exemplary based decision-making. It is also assumed that decision classes are ordered by some preference ordering. More precisely, let Cl = {Clt |t ∈ T }, T = {1, 2, ..., n}, be a set of decision classes such that for each x in the universe U , x belongs to one and only one Clt ∈ Cl and for all r, s ∈ T , if r > s, the decision from Clr are preferred to the decision from Cls . A consistent preference model is taken to be one that respects the dominance principle when assigning actions (objects) to the preference ordered decision classes. Action x is said to dominate action y if x is at least as good as y under all considered criteria. The dominance principle requires that if action x dominates action y, then x should be assigned to a class not worse than y. Given a total ordering on decision classes, in DRSA, the sets to be approximated are the upward union and downward union of decision classes [5]. The DRSA approach has been shown to be an effective tool for MCDA [12] and has been applied to solve multiple-criteria sorting problems [4, 5]. Algorithms for inducing decision rules consistent with dominance principle were introduced in [6, 7]. Two extensions of DRSA are the variable consistency model [13] and the stochastic model which allows varible precision of inconsistencies [14]. The indexed blocks representation proposed in this work provides a general representation of dominanced-based approximation space for any subsets of decision classes. It is related to granular computing approaches based on neighborhood systems [15, 16]. This work is motivated by trying to study the relationship between the structures of approximation spaces based on dominance relations and the structures of preference ordered decision classes satisfying upward and downward union property. The basic idea is to consider relations on decision classes and the representation of objects related by pairs of decisions in a multiple-criteria decision table. In addition, we are interested in computing the reduction of inconsistency when criteria are aggregated one by one incrementally. Here inconsistency is as a result of violating the dominance principle. In this study, we consider only decision tables with multiple criteria which are quantitative and totally ordered. The remainder of this paper is organized as follows. In Section 2, after reviewing related concepts, the concept of indexed blocks is defined. In Section 3, we consider the combination of criteria and how to update indexed blocks. The concept of exclusive neighborhoods is introduced, and three sequential rules for combining criteria are presented. Examples are given to illustrate the concepts and rules. In Section 4, we formulate approximations of sets of decision classes in terms of indexed blocks and exclusive neighborhoods. We show that a set of indexed blocks forms a partition on the universe of objects when the neighborhoods of its blocks are exclusive. Section 5 presents a brief analysis of time complexities for computing indexed blocks from a multiple-criteria decision table. Finally, conclusions are given in Section 6.

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

3

2. Related Concepts 2.1. Information Systems, Rough Sets, and Dominance-Based Rough Sets In rough set theory [8, 9, 10], information of objects in a domain is represented by an information system IS = (U, A, V, f ), where U is a finite set of objects, A is a finite set of attributes, V = ∪q∈A Vq and Vq is the domain of attribute q, and f : U × A → V is a total information function such that f (x, q) ∈ Vq for every q ∈ A and x ∈ U. In many applications, we use a special case of information systems called decision tables to represent data sets. In a decision table (U, C ∪D = {d}), there is a designated attribute {d} called decision attribute, and attributes in C are called condition attributes. Each attribute q in C ∪D is associated with an equivalence relation Rq on the set U of objects such that for each x and y ∈ U , xRq y means f (x, q) = f (y, q). For each x and y ∈ U , we say that x and y are indiscernible on attributes P ⊆ C if and only if xRq y for all q ∈ P. In dominance-based rough sets, attributes with totally ordered domains are called criteria. More precisely, each criterion q in C is associated with an outranking relation [11] Sq on U such that for each x and y ∈ U, xSq y means f (x, q) ≥ f (y, q). For each x and y ∈ U, we say that x dominates y on criteria P ⊆ C if and only if xSq y for all q ∈ P. The dominance relations are taken to be total pre-ordered, i.e., strongly complete and transitive binary relations [5]. Dominance-based rough set approach is capable of dealing with inconsistencies in MCDA problems based on the principle of dominance, namely: given two objects x and y, if x dominates y, then x should be assigned to a class not worse than y. Assignments of objects to decision classes are inconsistent if the dominance principle is violated. The sets of decision classes to be approximated are considered to have upward union and downward union properties. More precisely, let Cl = {Clt |t ∈ T }, T = {1, 2, ..., n}, be a set of decision classes such that for each x ∈ U , x belongs to one and only one Clt ∈Cl and for all r, s in T , if r > s, the decision from Clr is preferred to the decision from Cls . Based on this total ordering of decision classes, the upward union and downward union of decision classes are defined respectively as: Clt≥ = ∪s≥t Cls , Clt≤ = ∪s≤t Cls , t = 1, 2, ..., n. An object x is in Clt≥ means that x at least belongs to class Clt , and x is in Cl≤ t means that x at most belongs to class Clt .

2.2. Indexed Blocks To study the relationships between approximation spaces based on dominance relations and the sets of decision classes to be approximated, we introduce a new concept called indexed blocks, which are sets of objects indexed by pairs of decision values. Let (U, C ∪ D = {d}) be a multiple-criteria decision table where condition attributes in C are criteria and decision attribute d is associated with a total preference ordering. For each condition criterion q and a decision value di of d, let minq (di ) = min{f (x, q)|f (x, d) = di } and maxq (di ) = max{f (x, q)|f (x, d) = di }. That is, minq (di ) denotes the minimum value of q among objects with decision value di , and maxq (di ) denotes the maximum value. For each condition criterion q, the mapping Iq (i, j) : D × D →℘(Vq ) is defined as Iq (i, j) = {f (x, q) = v|v ≥ minq (dj ) and v ≤ maxq (di ), f or i < j; i, j = 1, ..., VD },

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C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

Iq (i, j) = Iq (j, i), if i > j, and Iq (i, i) = {f (x, q)|f (x, d) = i and f (x, q) ∈ / ∪i6=j Iq (i, j)}, where ℘(Vq ) denotes the power set of Vq . Intuitively, Iq (i, j) denotes the set of values of criterion q shared by objects of decision values i and j. We will denote the set of values as [minq (j), maxq (i)] or simply as [minj , maxi ] for a decision value pair i and j with i < j. The set Iq (i, i) denotes the values of criterion q where objects can be consistently labeled with decision value i. For i < j, values in Iq (i, j) are conflicting or inconsistent in the sense that objects with higher values of criterion q are assigned to a lower decision class or vice versa, namely, the dominance principle is violated. For each Iq (i, j), the corresponding set of ordered pairs [Iq (i, j)]: D × D → ℘(U × U ) is defined as [Iq (i, j)] = {(x, y) ∈ U ×U |f (x, d) = i, f (y, d) = j such that f (x, q) ≥ f (y, q) f or f (x, q), f (y, q) ∈ Iq (i, j)}. For simplicity, we will take the set [Iq (i, i)] to be reflexive. For each [Iq (i, j)], the restrictions of [Iq (i, j)] to i and j are defined as: [Iq (i, j)]i = {x ∈ U | there exists y ∈ U such that (x, y) ∈ [Iq (i, j)]} and [Iq (i, j)]j = {y ∈ U | there exists x ∈ U such that (x, y) ∈ [Iq (i, j)]}. The corresponding indexed block Bq (i, j) ⊆ U of [Iq (i, j)] is defined as Bq (i, j) = [Iq (i, j)]i ∪ [Iq (i, j)]j . For each criterion q, the union of its indexed blocks is a covering of U generally. Example 2.1. The above concepts are illustrated using the multiple-criteria decision table shown in Table 1, where U is the universe of objects, q1 and q2 are condition criteria and d is the decision with preference ordering 3 > 2 > 1. Table 1. Example of a multiple-criteria decision table.

U

q1

q2

d

1

1

2

2

2

1.5

1

1

3

2

2

1

4

1

1.5

1

5

2.5

3

2

6

3

2.5

3

7

2

2

3

8

3

3

3

In order to find out the minimum and maximum values for each decision class, we can apply sorting on the decision d first, followed by sorting on the criterion q1 . The result is shown in Table 2. From Table 2, we can derive inconsistent intervals Iq1 (i, j) as shown in Table 3.

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

Table 2. Result after sorting in terms of {d} followed by {q1 }.

U

q1

d

4

1

1

2

1.5

1

3

2

1

1

1

2

5

2.5

2

7

2

3

6

3

3

8

3

3

Table 3. Inconsistent intervals of values.

D×D

1

2

3

1

[]

[1, 2]

[2, 2]

[]

[2, 2.5]

2 3

[3, 3]

The sets of ordered pairs derived from Table 3 are: [Iq1 (1, 1)] = [Iq1 (2, 2)] = ∅, [Iq1 (1, 2)] = {(4, 1), (2, 1), (3, 1)}, [Iq1 (1, 3)] = {(3, 7)}, [Iq1 (2, 3)] = {(5, 7)}, [Iq1 (3, 3)] = {(6, 6), (8, 8)}. The indexed blocks are shown in Table 4. Table 4.

Index blocks derived from [Iq1 (i, j)].

D×D

1

2

3

1

Ø

{1, 2, 3, 4}

{3, 7}

Ø

{5, 7}

2 3

{6, 8}

5

6

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

3. Combination of Criteria In this section, we consider the combination of two criteria using indexed blocks and the underlying sets of ordered pairs. For a criterion q and for each decision value i ∈ Vd , and for each indexed block Bq (i, i) of [Iq (i, i)], the neighborhood of Bq (i, i) is defined as N B(Bq (i, i)) = {Bq (k, i)|k ≥ 1 and k < i} ∪ {Bq (i, k)|k ≥ 1 and k > i}. Note that Bq (i, i) is not part of its neighborhood, and the “exclusive” neighborhood of Bq (i, i) corresponds to sets of objects which have inconsistent decision class assignments associated with decision i. Objects in Bq (i, i) are assigned to decision i consistently. Blocks in the neighborhood of Bq (i, i) are inconsistent blocks. Alternatively, we may take the neighborhood of Bq (i, i) as a set of objects, i.e., ∪N B(Bq (i, i)). For x in U , we say x does not belong to the neighborhood of Bq (i, i) iff x ∈ / B, for all B in NB(Bq (i, i)) iff x ∈ / ∪NB(Bq (i, i)). Let BP (i, j) − N B(BP (i, i)) denote the set of objects in block BP (i, j) but not in N B(BP (i, i)) − {BP (i, j)}, and let BP (i, j) − N B(BP (j, j)) denote the set of objects in BP (i, j) but not in N B(BP (j, j)) − {BP (i, j)}. The neighborhood of BP (i, j), for i 6= j, is defined as N B(BP (i, j)) = [N B(BP (i, i)) ∪ N B(BP (j, j)) − {BP (i, j)}]

∪

{BP (i, j) − N B(BP (i, i))} ∪ {BP (i, j) − N B(BP (j, j))}. In general, neighborhood of BP (i, j) is not exclusive. It is partially inclusive, i.e., some objects of BP (i, j) may be associated with some blocks other than BP (i, j) in the neighborhood. When combining two criteria q1 and q2 , we use the following three rules to update the sets of ordered pairs [I{q1 ,q2 } (i, j)] and indexed blocks B{q1 ,q2 } (i, j) : Rule 1: For decision pairs (i, i): [I{q1 ,q2} (i, i)] = [Iq1 (i, i)] ∪ [Iq2 (i, i)]. Rule 2: For decision pairs (i, j) and i < j: [I{q1 ,q2 } (i, j)] = [Iq1 (i, j)] ∩ [Iq2 (i, j)]. Rule 3: For pairs (x, y) in [Iq1 (i, j)] − [I{q1 ,q2 } (i, j)] or [Iq2 (i, j)] − [I{q1 ,q2 } (i, j)]: If {x} does not belong to the neighborhood of B{q1 ,q2 } (i, i), then add (x, x) to [I{q1 ,q2 } (i, i)], which is the same as adding x to B{q1 ,q2} (i, i). If {y} does not belong to the neighborhood of B{q1 ,q2 } (j, j), then add (y, y) to [I{q1 ,q2 } (j, j)], which is the same as adding y to B{q1 ,q2} (j, j). After applying the above three rules in sequence, we can obtain the updated indexed blocks accordingly. The working of the rules is illustrated in the following example. Example 3.1. Consider the criterion q2 in the multiple-criteria decision table given in Table 1. The result of sorting the table by {d} followed by {q2 } is given in Table 5. Table 6 shows the set of inconsistent intervals Iq2 (i, j). The corresponding sets of ordered pairs are given as follows:

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

Table 5. Result after sorting in terms of {d} followed by {q2 }.

Table 6.

U

q2

d

2

1

1

4

1.5

1

3

2

1

1

2

2

5

3

2

7

2

3

6

2.5

3

8

3

3

Inconsistent intervals Iq2 (i, j).

D×D

1

2

3

1

[1, 1.5]

[2, 2]

[2, 2]

[]

[2, 3]

2 3

[]

[Iq2 (1, 1)] = {(2, 2), (4, 4)}, [Iq2 (1, 2)] = {(3, 1)}, [Iq2 (1, 3)] = {(3, 7)}, [Iq2 (2, 2)] = Ø, [Iq2 (2, 3)] = {(1, 7), (5, 7), (5, 6), (5, 8)}, [Iq2 (3, 3)] = Ø. The indexed blocks Bq2 (i, j) are shown in Table 7. Table 7. Indexed blocks Bq2 (i, j).

D×D

1

2

3

1

{2, 4}

{1, 3}

{3, 7}

Ø

{1, 5, 6, 7, 8}

2 3

Ø

Applying the three rules of combining q1 and q2 we have [I{q1,q2} (1, 1)] = {(2, 2), (4, 4)}, [I{q1,q2} (1, 2)] = {(3, 1)},

7

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C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

[I{q1,q2} (1, 3)] = {(3, 7)}, [I{q1,q2} (2, 2)] = Ø, [I{q1,q2} (2, 3)] = {(5, 7)}, [I{q1,q2} (3, 3)] = {(6, 6), (8, 8)}. The combined indexed blocks are shown in Table 8. Table 8. Indexed blocks B{q1,q2} (i, j).

D×D

1

2

3

1

{2, 4}

{1, 3}

{3, 7}

Ø

{5, 7}

2 3

{6, 8}

From Table 8, we have the following exclusive neighborhoods: NB(B{q1,q2} (1, 1)) = {B{q1,q2} (1, 2), B{q1,q2} (1, 3)}, NB(B{q1,q2} (2, 2)) = {B{q1,q2} (1, 2), B{q1,q2} (2, 3)}, NB(B{q1,q2} (3, 3)) = {B{q1,q2} (1, 3), B{q1,q2} (2, 3)}.

4. Approximating Sets of Decision Classes Let (U, C ∪ D = {d}) be a multiple-criteria decision table, P ⊆ C, and {BP (i, j)| (i, j) ∈ Vd × Vd } be the indexed blocks derived from P for (i, j) ∈ Vd × Vd . For a decision class Cli = {x ∈ U |f (x, d) = di , di ∈ Vd } with decision value di , the lower approximation of Cli by P is the indexed block BP (i, i), the boundary set of Cli is ∪N B(BP (i, i)) the union of blocks in the neighborhood of BP (i, i), and the upper approximation of Cli is the union of BP (i, i) and the boundary set. For a set of two decision values D2 = {i, j} ⊆ Vd , the neighborhoods of indexed blocks BP (i, i) and BP (j, j) are reduced by removing objects associated with decision values {i, j} only. This can be done by checking objects in the indexed block BP (i, j). For x in BP (i, j), if f (x, d) = i and x does not belong to N B(BP (i, i)) – {BP (i, j)}, then x is removed from BP (i, j) and added to BP (i, i). Similarly, if f (x, d) = j and x does not belong to N B(BP (j, j)) – {BP (i, j)}, then x is removed from BP (i, j) and added to BP (j, j). More precisely, we define the updated indexed blocks as BP (i, i)D2 = BP (i, i) ∪ {x ∈ BP (i, j)|f (x, d) = i and x ∈ / N B(BP (i, i)) − {BP (i, j)}} and BP (j, j)D2 = BP (j, j) ∪ {x ∈ BP (i, j)|f (x, d) = j and x ∈ / N B(BP (j, j)) − {BP (i, j)}}. The lower approximation of ClD2 = Cli ∪ Clj is defined as PClD2 = BP (i, i) ∪ BP (j, j) ∪ (BP (i, j) − ∪NB(BP (i, j))) = BP (i, i)D2 ∪ BP (j, j)D2 , the boundary set of Cli ∪ Clj is defined as BN P (ClD2 ) = ∪NB(BP (i,j)), and the upper approximation of Cli ∪Clj is defined as P¯ ClD2 = P ClD2 ∪ BNP (ClD2 ).

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

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Now, for a set of k decision values Dk = {di1 , . . . , dik } ⊆ Vd , we need to compute the neighborhoods NB(BP (i, j)) for each pair D2 = (i, j) ∈ Dk × Dk and update the diagonal indexed blocks BP (i, i)D2 , for all i ∈ Dk using the above definitions. The lower approximation of the set ClDk = ∪{Cli |i ∈ Dk } of decision classes by the set P of criteria is defined as P ClDk = ∪{P ClD2 |D2 = (i, j) ∈ Dk × Dk and i < j}, the boundary set of ClDk is defined as BNP (ClDk ) = ∪{ ∪ NB(BP (i, j))|(i, j) ∈ Dk × Dk }, and the upper approximation of ClDk is defined as P¯ ClDk = P ClDk ∪ BN P (ClDk ). Based on the indexed blocks representation of dominance-based approximation spaces, one useful property is that the family of indexed blocks form a partition on the set U of objects when BP (i, j) are exclusive for all (i, j) ∈ T × T . This is formulated and shown in the following fact and example. Fact 4.1. Let (U , C ∪ D={d}) be a multiple-criteria decision table with total preference orderings on domains of all criteria q ∈ C and the decision classes Cl = {Clt |t ∈ T }, T = {1, . . . , n}. Then, the neighborhoods of indexed blocks BC (i, j) are exclusive for all (i, j) ∈ T × T iff the set of indexed blocks BC (i, j) is a partition on U . Example 4.1. For convenience, we will use the multiple-criteria decision table taken from [6], which is shown in Table 9. The inconsistent intervals for each criteria q1 , q2 , and q3 are shown in Tables 10, 11, and 12. From the tables, we have the sets of ordered pairs for criterion q1 as: [Iq1 (1, 1)] = [Iq1 (2, 2)] = Ø, [Iq1 (1, 2)] = {(4, 12), (3, 12), (7, 12), (7, 6), (9, 12), (9, 6), (14, 12), (14, 6), (14, 13), (14, 1), (14, 2), (14, 10), (14, 15)}, [Iq1 (1, 3)] = Ø, [Iq1 (2, 3)] = {(11, 8), (11, 16)}, [Iq1 (3, 3)] = {(17, 17), (5, 5)}. The sets of ordered pairs for criterion q2 are: [Iq2 (1, 1)] = {(4, 4)}, [Iq2 (2, 2)] = Ø, [Iq2 (1, 2)] = {(7, 13), (3, 13), (3, 16), (14, 13), (14, 6), (9, 13), (9, 6), (9, 10), (9, 1), (9, 12)}, [Iq2 (1, 3)] = Ø, [Iq2 (2, 3)] = {(11, 8), (11, 16), (15, 8), (15, 16), (15, 5), (2, 8), (2, 16), (2, 5)}, [Iq2 (3, 3)] = {(17, 17)}. And, the sets of ordered pairs for criterion q3 are: [Iq3 (1, 1)] = {(4, 4), (3, 3)}, [Iq3 (2, 2)] = Ø, [Iq3 (1, 2)] = {(9, 1), (9, 6), (14, 10), (14, 6), (14, 12), (7, 10), (7, 6), (7, 12), (7, 13)}, [Iq3 (1, 3)] = Ø, [Iq3 (2, 3)] = {(2, 5), (2, 8), (11, 5), (11, 8), (1, 5), (1, 8), (15, 5), (15, 8), (15, 16)}, [Iq3 (3, 3)] = {(17, 17)}. The indexed blocks for q1 , q2 , and q3 are shown in Tables 13, 14, and 15. The result of combining criteria q1 and q2 is shown in Table 16.

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C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

Table 9. A multiple-criteria decision table adapted from [6].

U

q1

q2

q3

d

1

1.5

3

12

2

2

1.7

5

9.5

2

3

0.5

2

2.5

1

4

0.7

0.5

1.5

1

5

3

4.3

9

3

6

1

2

4.5

2

7

1

1.2

8

1

8

2.3

3.3

9

3

9

1

3

5

1

10

1.7

2.8

3.5

2

11

2.5

4

11

2

12

0.5

3

6

2

13

1.2

1

7

2

14

2

2.4

6

1

15

1.9

4.3

14

2

16

2.3

4

13

3

17

2.7

5.5

15

3

Table 10. Inconsistent intervals Iq1 (i, j).

D×D

1

2

3

1

[]

[0.5, 2]

[]

[]

[2.3, 2.5]

2 3

[2.7, 3.0]

Table 11. Inconsistent intervals Iq2 (i, j).

D×D

1

2

3

1

[0.5, 0.5]

[1, 3]

[]

[]

[3.3, 5]

2 3

[5.5, 5.5]

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

Table 12. Inconsistent intervals Iq3 (i, j).

D×D

1

2

3

1

[1.5, 2.5]

[3.5, 8]

[]

[]

[9, 14]

2 3

[15, 15]

Table 13. Indexed blocks Bq1 (i, j).

D×D

1

2

3

1

Ø

{1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15}

Ø

Ø

{8, 11, 16}

2

{5, 17}

3

Table 14. Indexed blocks Bq2 (i, j).

D×D

1

2

3

1

{4}

{1, 3, 6, 7, 9, 10, 12, 13, 14, }

Ø

Ø

{2, 5, 8, 11, 15, 16}

2

{17}

3

Table 15. Indexed blocks Bq3 (i, j).

D×D

1

2

3

1

{3, 4}

{6, 7, 9, 10, 12, 13,14}

Ø

Ø

{1, 2, 5, 8, 11, 15, 16}

2

{17}

3

Table 16. Indexed blocks B{q1 ,q2 } {i, j).

D×D

1

2

3

1

{3, 4, 7}

{6, 9, 12, 13, 14}

Ø

{1, 2, 10, 15}

{8, 11, 16}

2 3

{5, 17}

11

12

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

We have the sets of ordered pairs after combining q1 and q2 (for simplicity, we leave out the sets for indices (i, i)): [I{q1 ,q2} (1, 2)] = {(14, 13), (14, 6), (9, 6), (9, 12)} and [I{q1 ,q2} (2, 3)] = {(11, 8), (11, 16)}. After combined with q3 , we have: [I{q1 ,q2,q3 } (1, 2)] = {(14, 6), (9, 6)} and [I{q1 ,q2,q3 } (2, 3)] = {(11, 8)}. The final indexed blocks after combining q1 , q2 , and q3 are shown in Table 17. It is clear that the set of indexed blocks in Table 17 forms a partition on the universe. Table 17. Indexed blocks B{q1,q2,q3} (i, j).

D×D

1

2

3

1

{3, 4, 7}

{6, 9, 14}

Ø

{1, 2, 10, 12, 13, 15}

{8, 11}

2 3

{5, 16, 17}

From Table 17, we have the following neighborhoods for single decision classes: ∪NB(B{q1,q2,q3} (1, 1)) = {B{q1,q2,q3} (1, 2), B{q1,q2,q3} (1, 3)} = {6, 9, 14}, ∪NB(B{q1,q2,q3} (2, 2)) = {B{q1,q2,q3} (1, 2), B{q1,q2,q3} (2, 3)} = {6, 9, 14} ∪ {8, 11}, ∪NB(B{q1,q2,q3} (3, 3)) = {B{q1,q2,q3} (1, 3), B{q1,q2,q3} (2, 3)} = {8, 11}. The neighborhoods of sets of decision classes {1, 2} and {2, 3} are: ∪NB(B{q1,q2,q3} (1, 2)) = ∪(NB(B{q1,q2,q3} (1, 1)) ∪ NB(B{q1,q2,q3} (2, 2)) − B{q1,q2,q3} (1, 2)) = {8, 11}, ∪NB(B{q1,q2,q3} (2, 3)) = ∪(NB(B{q1,q2,q3} (2, 2)) ∪ NB(B{q1,q2,q3} (3, 3)) − B{q1,q2,q3} (2, 3)) = {6, 9, 14}.

5. Analysis of Time Complexities In the following, we consider the time it takes to compute the indexed blocks from a multiple-criteria decision table with N objects, M criteria, and K decision classes. For each criterion, the operations include sorting the objects by decision values, computing the inconsistent intervals and sets of ordered pairs, and deriving indexed blocks from the ordered pairs. Sorting N objects can be accomplished in O(N logN ). To compute one inconsistent interval for a pair (i, j) of decision values, we need to compare the maximum value of objects in decision class i with objects in decision class j. For simplicity, let each class have N/K objects, then it takes N ∗ (K − 1)/K comparisons for the first row. For the second row, it takes N ∗ (K − 2)/K comparisons, and so on. Therefore, the total number of comparisons is N ∗ (K − 1)/2, which is in the order of O(N K). To determine one indexed block from an inconsistent interval, we need to examine 2 ∗ N/K objects from two decision values. Since there are K(K + 1)/2 indexed blocks in a table, the number of look-up operations is in O(N (K + 1)).

C.-C. Chan and G.-H. Tzeng / Dominance-Based Rough Sets Using Indexed Blocks as Granules

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The cost of combining two criteria is determined by combining two indexed block tables as outlined in the three rules given in Section 3. The total number of intersection and union operations is K(K + 1)/2 according to Rule 1 and 2. From Rule 3, the number of look-up operations for block BP (i, j) is K + j − i − 3 which includes (i − 1) vertical look-ups, (K − i − 1) horizontal look-ups, and (j − i − 1) horizontal look-ups in the next row. Therefore, the number of look-ups is in O(K 3 ) for combining two criteria with K(K − 1)/2 pairs of decision values, and the total number of look-up opertations for combining M criteria is in O(M K 3 ).

6. Conclusions In this paper we introduced the concept of indexed blocks to represent dominance based approximation spaces derived from multiple-criteria decision tables. Inconsistencies caused by violating the dominance principle are represented as sets of objects indexed by pairs of decision values. Therefore, blocks indexed with decision values i and j are inconsistent when i 6= j. Approximations of decision classes are defined in terms of neighborhoods of indexed blocks, which represents inconsistent information related to the indexing decision values. It turns out that a family of indexed blocks form a partition on the universe of objects when all its blocks’ neighborhoods are exclusive, namely, all blocks are not part of their neighborhoods. We also introduced sequential rules for updating indexed blocks by combining criteria incrementally. The concept of indexed blocks provides a new way for understanding and studying dominance-based rough sets. It can be used to study approximations of any sets of decision classes including those satisfying upward and downward union property. The time complexity of generating indexed block tables from a multiple-criteria decision table with N objects, M criteria, and K decision classes is in the order of O(N M K), and the look-up operations for combining indexed block tables is in the order of O(M K 3 ). It can facilitate the development of efficient algorithms for generating rules from multiple-criteria decision tables.

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