Dominance-based Rough Set Approach (DRSA), as the extension of Pawlak's Rough Set theory, is effective and fundamentally important in Multiple Criteria Decision Analysis (MCDA). In previous DRSA models, the definitions of the upper and lower approximations are preserving the class unions rather than the singleton class. In this paper, we propose a new Class-based Rough Approximation with respect to a series of previous DRSA models, including Classical DRSA model, VC-DRSA model and VP-DRSA model. In addition, the new class-based reducts are investigated.
Multiple Criteria Decision Analysis (MCDA) aims at providing the decision maker (DM) a knowledge recommendation while considering the finite objects evaluated from multiple viewpoints (known as criteria). Roy [9] considered four problems in MCDA, including criteria analysis, choice, ranking, sorting. The first one is the essential procedure for optimization of decision information and the latter three ones can produce specific decision outcomes.
Apart from several valid and classical MCDA approaches (see the state-of-the-art survey in [3]), the non-classical methods and techniques (like [1] [2]) are significant since it attempts to address the risk and uncertainty of MCDA catering to the real world. Classical Rough Set Approach (CRSA for short) initially proposed by Pawlak (see [8]) is an effective mathematical tool for decision analysis. But, it fails to deal with the preference-ordered data in MCDA. In this reason, Dominance-based Rough Set Approach (DRSA for short) was generated by Greco and his colleagues [5] [10]. Unlike the CRSA which makes use of the indiscernibility relations for construction of knowledge granular, DRSA considers the dominance relations of these preference-ordered data in given decision table.
The target by using DRSA is to induce the decision rule as classifier for providing the suitable assignment of both learning objects (from given decision table) and new objects. Recently, the classical DRSA model had been extended to VC-DRSA [4], VP-DRSA [6], etc.
In all previous DRSA models, the upper and lower approximations are defined in consideration of the union of decision class (i.e. upward union t Cl ≥ and downward union t Cl ≤ ). We call them as union-based rough approximation. In this paper, we attempt to investigate the issue: whether one singleton decision class can be used to define the upper and lower approximation in a series of DRSA models. To this end, we firstly analyze the partition of objects preserving one particular decision class, and provide a new Three Region Model (TRM). Then, we develop the so-called class-based rough approximation in a series of previous DRSA models, including the classical DRSA model, VC-DRSA model and VP-DRSA model. Finally, inspired by Inuiguchi’s initial works [6][7], the class-based criteria reduction is also studied. This paper is organized as follows: The next section briefly reviews the basic principles of DRSA theory. Section 3 studies the class-based rough approximation in a series of DRSA models. Section 4 investigates the class-based criteria reduction. Finally, we draw the conclusion in section 5.
In this section, we concisely revisit the basic theory of DRSA. Despite the various problem domains regarding MCDA, three elementary factors are usually involved, including objects, criteria and DM(s). These factors can generally be organized as decision table with columns of criteria and rows of objects. Formally, a decision table is the 4-tuple ), and { } 1 ,…, n q Q q q ∈ = ; (3) the domain of criterion q denoted by q V , where
; (4) information function denoted by ( ) :
. In addition, the following properties are valid: ( ) ( )
⊆ ⊆ , we have the following properties:
( ) ( )
. The definitions of the classical DRSA model are based on the strict dominance principle (as shown in above). Inspired by the Variable Precision Rough Set [11], which is the extension of CRSA via relaxation of strict indiscernibility relation, Greco et al. [10] provided the VC-DRSA model. This model accepts a limited number of inconsistency objects controlled by a predefined threshold called consistency level.
The lower approximations of VC-DRSA model can be represented as follows. For any P C ⊆ , we have:
Classical DRSA model can be regarded as a special case of VC-DRSA model with the consistency level fulfilling 1 2 1 l l = = (the strict dominance principle), while,
Cl .
Classical DRSA model with 1 2 1 Cl + , the constraint conditions are given in Table 1. Each consistency level 1 l (or 2 l ) divides the entire objects into two regions: Low region and High region. These regions are constrained by different conditions. For the class t Cl , the constraint conditions are (A), (B), (C), (D). For its adjacent classes And also, we can obtain following properties, which can be easily proved: and its adjacent classes
Bn Cl P Cl P Cl Bn Cl . The concept of lower approximations at some consistency levels 1 l and 2 l are formally presented as:
Then, the TRM preserving the predefined class t Cl ( 2,…, 1 t l = -) can be presented as: Low boundary region 2 ( ) .
As such, the predefined levels 1 l and 2 l are used to control the precision degrees 1 β and 2 β in definitions of lower approximation, respectively.
objects: t x Cl ∈
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