Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 936340, 8 pages

http://dx.doi.org/10.1155/2015/936340

## Decision Rules Acquisition for Inconsistent Disjunctive Set-Valued Ordered Decision Information Systems

School of Mathematical Sciences, University of Jinan, Jinan 250022, China

Received 4 January 2015; Accepted 11 March 2015

Academic Editor: Wanquan Liu

Copyright © 2015 Hongkai Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Set-valued information system is an important formal framework for the development of decision support systems. We focus on the decision rules acquisition for the inconsistent disjunctive set-valued ordered decision information system in this paper. In order to derive optimal decision rules for an inconsistent disjunctive set-valued ordered decision information system, we define the concept of reduct of an object. By constructing the dominance discernibility function for an object, we compute reducts of the object via utilizing Boolean reasoning techniques, and then the corresponding optimal decision rules are induced. Finally, we discuss the certain reduct of the inconsistent disjunctive set-valued ordered decision information system, which can be used to simplify all certain decision rules as much as possible.

#### 1. Introduction

Rough set theory, proposed by Pawlak [1–3], has been regarded as a useful tool to conceptualize and analyze various types of data. It is applied to many fields such as machine learning, knowledge discovery, and pattern recognition. With replacement of the equivalence relation by other relations such as the tolerance relation [4–9] and the dominance relation [10–13], Pawlak rough set model has been extended to numerous generalized rough set models.

Pawlak rough set model is based on an assumption that every object in the universe of discourse is associated with some information, and objects characterized by the same information are indiscernible. It is successfully used in decision rules acquisition and attribute reduction for nominal information systems. However, Pawlak rough set model is not able to discover inconsistency for attributes in preference-ordered domains. Ordered information systems are firstly proposed and studied by Iwinski [14]. Yao and Sai [15, 16] transformed an ordered information table into a binary information table and then applied classical machine learning and data mining algorithms to derive ordering rules. Greco et al. [10–12] proposed the dominance-based rough set approach (DRSA), which is mainly based on substitution of the indiscernibility relation by a dominance relation, to deal with such kind of inconsistency.

The DRSA proposed by Greco et al. [10–12] mainly focuses on discussing sorting problem and extracting the dominance decision rules for a complete ordered decision information system (ODIS), in which all attribute values are exactly known. Many researchers have investigated various types of ODISs by the dominance-based rough set approach and its extended models [17–29].

Set-valued information systems are generalized models of single-valued information systems. There are many ways to give a semantic interpretation of the set-valued information systems. Guan and Wang [5] summarized them in two types: disjunctive and conjunctive systems. For set-valued ordered information systems (OISs), Qian et al. [27] defined two dominance relations in a conjunctive set-valued ODIS and a disjunctive set-valued ODIS, respectively, and used them to define lower and upper approximations. From the lower and upper approximations, certain decision rules and possible decision rules can be derived from these two types of set-valued ODISs. Furthermore, Qian et al. [28] introduced four types of dominance relations and discussed criterion reductions of disjunctive set-valued ordered information systems. However, for the reduction of the system, their discussions are restricted on set-valued OISs and consistent set-valued ODISs. Nevertheless, inconsistent set-valued ODISs are common in practice, so, in this paper, we will investigate the reduction of the inconsistent set-valued ODISs based on the discussion of decision rules optimization. Our discussion mainly focuses on disjunctive set-valued ODISs, and this approach is also applicable to the conjunctive set-valued ODISs.

The rest of this paper is organized as follows. In Section 2, some notations and basic concepts for the ODIS and the DRSA are introduced. In Section 3, dominance-based rough set model in disjunctive set-valued ODISs is reviewed. In Section 4, the optimization problem of decision rules in inconsistent disjunctive set-valued ODIS is discussed. The concept of reduct of an object is proposed. By constructing the dominance discernibility function for an object, reducts of an object are computed via utilizing Boolean reasoning techniques, and then the corresponding optimal decision rules are induced. In Section 5, the certain reduct of the inconsistent disjunctive set-valued ODIS, which can be used to simplify all certain decision rules as much as possible, is also discussed. Finally, we conclude our work in Section 6.

#### 2. The Dominance-Based Rough Set Approach

##### 2.1. The Ordered Decision Information System (ODIS)

An information system is represented by a quadruple , where is a finite nonempty set of objects, called the universe of discourse; is a finite set of attributes; , where is the domain of attribute ; is an information function satisfying , , . We denote for simplicity. In general, if attributes in an information system are classified into condition attributes and decision attributes, then is called a decision information system (DIS) or called a decision table (DT). In this case, the set of condition attributes is denoted by , and the set of decision attributes is denoted by ; that is, with . Without loss of generality, we assume that and . The partition of determined by is usually denoted as , where , , is called a decision class.

Further, we denote for any . Then one can conclude that .

If the domain of a condition attribute is ordered according to a decreasing or increasing preference, then the attribute is called a criterion [10–13]. Next we will introduce the concept of the ordered decision information system.

*Definition 1 (see [10, 12]). *For a DIS , if the condition attributes are criteria and the value of decision attribute represents an overall preference of objects, that is, for any and with , the objects in are more preferred than the objects in , then one calls the information system an ordered decision information system (ODIS) or ordered decision table (ODT).

In an ODIS, the domain of a criterion is completely preordered by an outranking relation , where implies that is at least as good as with respect to the criterion . We denote such and as or .

For , if and the domain of every attribute in is ordered according to a descending preference while the domain of the attribute in is ordered according to an ascending preference, then the dominance relation determined by can be defined as

From mathematical point of view, the ascending and descending order relations are dual, and so they can be handled similarly. Therefore, without loss of generality, we make an assumption for simplicity that values in the domain of every condition attribute are only descendingly ordered. With this assumption, we obtain

Since is equivalent to , is also denoted as or . From (2), we obtain that and for .

For , one can see that is transitive, and it is also reflexive under the assumption that , , and . Furthermore, for , we denote if , , and if such that . Then, and can imply . Therefore, is antisymmetric. Hence, is a partial ordered set [30] and so is .

##### 2.2. The Dominance-Based Rough Set Model

In an ODIS, for , if , one says that dominates or is dominated by with respect to . Let us denoteThen is called a dominating set and a dominated set of with respect to .

Further, we denoteThen, both and are coverings of .

In the DRSA [10, 12, 17], and are regarded as basic knowledge granules and they are used to define the lower and upper approximations as shown below.

*Definition 2 (see [10, 12, 17]). *In an ODIS , for and , letThen is called lower approximation and is called upper approximation of with respect to .

Analogously, one can define lower approximation and upper approximation , respectively, as

In the DRSA, the set to be approximated is an upward union or a downward union of decision classes; that is, or with , where implies that belongs to at least decision class and indicates that belongs to at most decision class . By the lower and upper approximations of and , Greco et al. [12] proposed five types of dominance decision rules. For example, from , one can get an “at least” decision rule with the form “if , then ”; from , one can get an “at most” decision rule with the form “if , then .”

#### 3. Dominance-Based Rough Set Model in the Disjunctive Set-Valued ODIS

##### 3.1. Disjunctive Set-Valued Ordered Decision Information Systems

A set-valued ordered decision table (ODT) is a set-valued decision information system , where is a nonempty finite set of objects; is a finite set of condition attributes; is a decision attribute with ; , where is the set of conditional attribute values and is the set of decisional attribute values; is a mapping from to such that is a set-valued mapping and is a single-valued mapping, where is the power set of .

Table 1 shows a set-valued ordered decision table.