Ranking for Objects and Attribute Reductions in Intuitionistic Fuzzy Ordered Information Systems

Abstract

We aim to investigate intuitionistic fuzzy ordered information systems. The concept of intuitionistic fuzzy ordered information systems is proposed firstly by introducing an intuitionistic fuzzy relation to ordered information systems. And a ranking approach for all objects is constructed in this system. In order to simplify knowledge representation, it is necessary to reduce some dispensable attributes in the system. Theories of rough set are investigated in intuitionistic fuzzy ordered information systems by defining two approximation operators. Moreover, judgement theorems and methods of attribute reduction are discussed based on discernibility matrix in the systems, and an illustrative example is employed to show its validity. These results will be helpful for decisionmaking analysis in intuitionistic fuzzy ordered information systems.

1. Introduction

Rough set theory was first proposed by Pawlak in the early 1980s [1]. The theory is an extension of the classical set theory for modeling uncertainty or imprecision information. The research has recently roused great interest in the theoretical and application fronts, such as machine learning, pattern recognition, and data analysis. It is a new mathematical approach to uncertain and vague data analysis and plays an important role in many fields of data mining and knowledge discovery.

Partition or equivalence (indiscernibility relation) is an important and primitive concept in Pawlak's original rough set theory. However, partition or equivalence relation is still restrictive for many applications. To overcome this limitation, classical rough sets have been extended to several interesting and meaningful general models in recent years by proposing other binary relations, such as tolerance relations [2], neighborhood operators [3], and others [4–11]. However, the original rough set theory does not consider attributes with preference ordered domain, that is criteria. Particularly, in many real situations, we are often faced with the problems in which the ordering of properties of the considered attributes plays a crucial role. One such type of problem is the ordering of objects. For this reason, Greco et al. [12–17] proposed an extension rough set theory, called the dominance-based rough set approach (DRSA) to take into account the ordering properties of criteria. This innovation is mainly based on substitution of the indiscernibility relation by a dominance relation. In DRSA, condition attributes are criteria and classes that are preference ordered; the knowledge approximated is a collection of upward and downward unions of classes and the dominance classed are sets of objects defined by using a dominance relation. In recent years, several studies have been made about properties and algorithmic implementations of DRSA [18–22].

Another important mathematical structure to cope with imperfect and/or imprecise information is called the “intuitionistic fuzzy (IF, for short) set” initiated by Atanassov [23, 24] on the basis of orthopairs of fuzzy sets. (Though the term of intuitionistic fuzzy set has been the argument of a large debate [25–27], we still use this notion due to its underlying mathematical structure, and because it is becoming increasing popular topic of investigation in the fuzzy set community.) An IF set is naturally considered as an extension of Zadeh's fuzzy sets [28] defined by a pair of membership functions: while a fuzzy set gives a degree to which an element belongs to a set, an IF set gives both a membership degree and a nonmembership degree. The membership and nonmembership values induce an indeterminacy index, which models the hesitancy of deciding the degree to which an object satisfies a particular property. Recently, IF set theory has been successfully applied in decision analysis and pattern recognition (see, e.g., [27, 29–31]).

Combining IF set theory and rough set theory may result in a new hybrid mathematical structure for the requirement of knowledge-handling systems. Research on this topic has been investigated by a number of authors. Çoker [32] first revealed the relationship between IF set theory and rough set theory and showed that a fuzzy rough set was in fact an intuitionistic fuzzy set. Various tentative definitions of IF rough sets were explored to extend rough set theory to the IF environment [33–38]. For example, according to fuzzy rough sets in the sense of Nanda and Majumdar [39], Jena et al. [35] and Chakrabarty et al. [33] independently proposed the concept of an IF rough set in which the lower and upper approximations are both IF sets.

In this paper, the intuitionistic fuzzy relation is introduced to DRSA. Actually, in real life, the intuitionistic fuzzy relation is an important type of data tables in ordered information systems. We aim to introduce dominance relation to ordered information systems with intuitionistic fuzzy relation and establish a rough set approach and evidence theory in this system.

The rest of this paper is organized as follows. Some preliminary concepts of rough sets and ordered information systems are briefly recalled in Section 2. In Section 3, the intuitionistic fuzzy ordered information system is introduced and some important properties are discussed. In Section 4, a rank approach with dominance class is considered by proposing the concept of dominance degree in intuitionistic fuzzy ordered information system. In Section 5, a rough set approach is investigated by establishing the upper and lower approximation operators in intuitionistic fuzzy ordered information system. In Section 6, attribute reductions are discussed in this system. Finally, we conclude the paper with a summary and outlook for further research in Section 7.

2. Preliminaries

The following recalls necessary concepts and preliminaries required in the sequel of our work. Detailed description of the theory can be found in the source papers [12–17]. A description has also been made in [40].

The notion of information system (sometimes called data tables, attribute-value systems, knowledge representation systems, etc.) provides a convenient tool for the representation of objects in terms of their attribute values.

An information system is an ordered quadruple ℐ=(𝑈,𝐴𝑇,𝑉,𝑓), where 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛} is a nonempty finite set of objects, and 𝐴𝑇={𝑎₁,𝑎₂,…,𝑎_𝑝} is a nonempty finite set of attributes, 𝑉=⋃_{𝑎∈𝐴𝑇}𝑉_𝑎 and 𝑉_𝑎 is a domain of attribute 𝑎, 𝑓∶𝑈×𝐴𝑇→𝑉 is a function such that 𝑓(𝑥,𝑎)∈𝑉_𝑎, for every 𝑎∈𝐴𝑇,𝑥∈𝑈, called an information function. A decision table is a special case of information systems in which, among the attributes, we distinguish one called a decision attribute. The other attributes are called condition attributes. Therefore, ℐ=(𝑈,𝐶∪{𝑑},𝑉,𝑓) and 𝐶∩{𝑑}=𝜙, where set 𝐶 and {𝑑} are condition attributes and the decision attribute, respectively.

In information systems, if the domain of an attribute is ordered according to a decreasing or increasing preference, then the attribute is a criterion.

Definition 2.1 (see [12–17]). An information system is called an ordered information system (OIS) if all condition attributes are criteria.
Assuming that the domain of a criterion 𝑎∈𝐴𝑇 is completely preordered by an outranking relation ≽_𝑎, then 𝑥≽_𝑎𝑦 means that 𝑥 is at least as good as 𝑦 with respect to criterion 𝑎. And we can say that 𝑥 dominates 𝑦. In the following, without any loss of generality, we consider criteria having a numerical domain, that is, 𝑉_𝑎⊆ℛ (ℛ denotes the set of real numbers). Being of type gain, that is, 𝑥≽_𝑎𝑦⇔𝑓(𝑥,𝑎)≥𝑓(𝑦,𝑎) (according to increasing preference) or 𝑥≽_𝑎𝑦⇔𝑓(𝑥,𝑎)≤𝑓(𝑦,𝑎) (according to decreasing preference), where 𝑎∈𝐴𝑇,𝑥,𝑦∈𝑈. Without any loss of generality and for simplicity, in the following we only consider condition attributes with increasing preference.
For a subset of attributes 𝐵⊆𝐴𝑇, we define 𝑥≽_𝐵𝑦⇔forall𝑎∈𝐵,𝑓(𝑥,𝑎)≥𝑓(𝑦,𝑎), and that is to say 𝑥 dominates 𝑦 with respect to all attributes in 𝐵. In general, we denote an ordered information system by ℐ^≽=(𝑈,𝐴𝑇,𝑉,𝑓).
For a given OIS, we say that 𝑥 dominates 𝑦 with respect to 𝐵⊆𝐴𝑇 if 𝑥≽_𝐵𝑦 and denote by 𝑥𝑅^≽_𝐵𝑦. That is 𝑅^⪰_𝐵=𝑥,𝑦∈𝑈×𝑈∣𝑥⪰_𝐵𝑦=𝑥,𝑦∈𝑈×𝑈∣𝑓(𝑥,𝑎)≥𝑓𝑦,𝑎,∀𝑎∈𝐵. (2.1)𝑅^≽_𝐵 are called dominance relations of ordered information system ℐ^≽.
Let 𝑥_𝑖^⪰_𝐵=𝑥_𝑗∈𝑈∣𝑥_𝑗,𝑥_𝑖∈𝑅^⪰_𝐵=𝑥_𝑗∈𝑈∣𝑓𝑥_𝑗,𝑎≥𝑓𝑥_𝑖,𝑎,∀𝑎_𝑙,𝑈∈𝐵𝑅^⪰_𝐵=𝑥_𝑖^⪰_𝐵∣𝑥_𝑖∈𝑈, (2.2) where 𝑖∈{1,2,…,|𝑈|}, then [𝑥_𝑖]^≽_𝐵 will be called a dominance class or the granularity of information, and 𝑈/𝑅^≽_𝐵 be called a classification of 𝑈 about attribute set 𝐵.

The following properties of a dominance relation are trivial by the above definition.

Proposition 2.2 (see [12–17]). Let 𝑅^≽_𝐴 be a dominance relation.(1)𝑅^≽_𝐴 is reflexive, transitive, but not symmetric, so it is not an equivalence relation.(2)If 𝐵⊆𝐴, then 𝑅^≽_𝐴⊆𝑅^≽_𝐵.(3)If 𝐵⊆𝐴, then [𝑥_𝑖]^≽_𝐴⊆[𝑥_𝑖]^≽_𝐵.(4)If 𝑥_𝑗∈[𝑥_𝑖]^≽_𝐴, then [𝑥_𝑗]^≽_𝐴⊆[𝑥_𝑖]^≽_𝐴 and [𝑥_𝑖]^≽_𝐴=∪{[𝑥_𝑗]^≽_𝐴∣𝑥_𝑗∈[𝑥_𝑖]^≽_𝐴}.(5)[𝑥_𝑗]^≽_𝐴=[𝑥_𝑖]^≽_𝐴 if and only if 𝑓(𝑥_𝑖,𝑎)=𝑓(𝑥_𝑗,𝑎) for all 𝑎∈𝐴.(6)|[𝑥_𝑖]^≽_𝐵|≥1 for any 𝑥_𝑖∈𝑈.(7)𝑈/𝑅^≽_𝐵 constitute a covering of 𝑈, that is, for every 𝑥∈𝑈 we have that [𝑥]^≽_𝐵≠𝜙 and ⋃_𝑥∈𝑈[𝑥]^≽_𝐵=𝑈,where |⋅| denotes cardinality of the set.

For any subset 𝑋⊆𝑈 and 𝐴⊆𝐴𝑇 in ℐ^≽, the lower and upper approximation of 𝑋 with respect to a dominance relation 𝑅^≽_𝐴 could be defined as follows (see [12–17]): 𝑅^⪰_𝐴(𝑋)=𝑥∈𝑈∣[𝑥]^⪰_𝐴⊆𝑋;𝑅^⪰_𝐴(𝑋)=𝑥∈𝑈∣[𝑥]^⪰_𝐴∩𝑋≠𝜙. (2.3)

Unlike classical rough set theory, one can find easily that 𝑅^≽_𝐴(𝑋)=⋃{[𝑥]^≽_𝐴[𝑥]^≽_𝐴⊆𝑋} and 𝑅^≽_𝐴(𝑋)=⋃{[𝑥]^≽_𝐴[𝑥]^≽_𝐴∩𝑋≠𝜙} do not hold.

Example 2.3. An OIS is presented in Table 1, where 𝑈={𝑥₁,𝑥₂,…,𝑥₆}, 𝐴𝑇={𝑎₁,𝑎₂,𝑎₃}.
From the table we can have𝑥₁^⪰_𝐴𝑇=𝑥₁,𝑥₂,𝑥₅,𝑥₆,𝑥₂^⪰_𝐴𝑇=𝑥₂,𝑥₅,𝑥₆,𝑥₃^⪰_𝐴𝑇=𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₄^⪰_𝐴𝑇=𝑥₄,𝑥₆,𝑥₅^⪰_𝐴𝑇=𝑥₅,𝑥₆^⪰_𝐴𝑇=𝑥₆. (2.4) If 𝑋={𝑥₂,𝑥₃,𝑥₅}, then 𝑅^⪰_𝐴𝑇(𝑋)=𝑥₅,𝑅^⪰_𝐴𝑇(𝑋)=𝑥₁,𝑥₂,𝑥₃,𝑥₅. (2.5)

Table 1 

An ordered information system.

3. Intuitionistic Fuzzy Ordered Information Systems

An intuitionistic fuzzy information system is an ordered quadruple ℐ^≀=(𝑈,𝐴𝑇,𝑉,𝑓), where 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛} is a nonempty finite set of objects, and 𝐴𝑇={𝑎₁,𝑎₂,…,𝑎_𝑝} is a nonempty finite set of attributes, 𝑉=⋃_{𝑎∈𝐴𝑇}𝑉_𝑎 and 𝑉_𝑎 is a domain of attribute 𝑎, 𝑓∶𝑈×𝐴𝑇→𝑉 is a function such that 𝑓(𝑥,𝑎)∈𝑉_𝑎, for every 𝑎∈𝐴𝑇,𝑥∈𝑈, called an information function, where 𝑉_𝑎 is a intuitionistic fuzzy set of universe 𝑈. That is 𝑓(𝑥,𝑎)=𝜇_𝑎(𝑥),𝜈_𝑎(𝑥),∀𝑎∈𝐴𝑇, (3.1)

where 𝜇_𝑎∶𝑈→[0,1] and 𝜈_𝑎∶𝑈→[0,1] satisfy 0≤𝜇_𝑎(𝑥)+𝜈_𝑎(𝑥)≤1, for all 𝑥∈𝑈. And 𝜇_𝑎(𝑥) and 𝜈_𝑎(𝑥) are, respectively, called the degree of membership and the degree of nonmembership of the element 𝑥∈𝑈 to attribute 𝑎. We denote 𝑎(𝑥)=(𝜇_𝑎(𝑥),𝜈_𝑎(𝑥)), then it is clear that 𝑎 is an intuitionistic fuzzy set of 𝑈.

In other words, an intuitionistic fuzzy information system is an information system in which the relation between universe 𝑈 and attributes set 𝐴𝑇 is an intuitionistic fuzzy relation.

Example 3.1. An intuitionistic fuzzy information system is presented in Table 2, where 𝑈={𝑥₁,𝑥₂,…,𝑥₁₀}, 𝐴𝑇={𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₅}.
In practical decision-making analysis, we always consider a binary dominance relation between objects that are possibly dominant in terms of values of attributes set in information systems based on intuitionistic fuzzy relation, in general, an increasing preference and a decreasing preference, then the attribute is a criterion.

Table 2 

An IS based on intuitionistic fuzzy relation.

Definition 3.2. An intuitionistic fuzzy information system is called an intuitionistic fuzzy ordered information system (IFOIS) if all condition attributes are criteria.
In general, we denote an intuitionistic fuzzy ordered information system by ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓).
Assuming that the domain of a criterion 𝑎∈𝐴𝑇 is completly preordered by an outranking relation ≽_𝑎, then 𝑥≽_𝑎𝑦 means that 𝑥 is at least as good as 𝑦 with respect to criterion 𝑎. And we can say that 𝑥 dominates 𝑦. For a subset of attributes 𝐴⊆𝐴𝑇, we define 𝑥≽_𝐴𝑦⇔forall𝑎∈𝐴,𝑥≽_𝑎𝑦. In other words, 𝑥 is at least as good as 𝑦 with respect to all attributes in 𝐴.
In the following, we introduce a dominance relation that identifies dominance classes to an intuitionistic fuzzy ordered information system. In a given IFOIS, we say that 𝑥 dominates 𝑦 with respect to 𝐴⊆𝐴𝑇 if 𝑥≽_𝐴𝑦, and is denoted by 𝑥𝑅_𝐴^≀≽𝑦. That is 𝑅_𝐴^≀⪰=𝑥,𝑦∈𝑈×𝑈∣𝑥⪰_𝐴𝑦. (3.2) Obviously, if (𝑥,𝑦)∈𝑅_𝐴^≀≽, then 𝑥 dominates 𝑦 with respect to 𝐴. 𝑅_𝐴^≀≽ are called a dominant relations of IFOIS.
Similarly, the relation 𝑅_𝐴^≀⪯, which is called a dominated relation, can be defined as follows: 𝑅_𝐴^≀⪯=𝑥,𝑦∈𝑈×𝑈∣𝑦⪰_𝐴𝑥. (3.3)
For simplicity and without any loss of generality, in the following we only consider condition attributes with increasing preference. Let us define this dominant relation in intuitionistic fuzzy ordered information systems as follows: 𝑅_𝐴^≀⪰=𝑥,𝑦∈𝑈×𝑈∣𝜇_𝑎(𝑥)≥𝜇_𝑎𝑦,𝜈_𝑎(𝑥)≤𝜈_𝑎𝑦,∀𝑎∈𝐴. (3.4)
That is to say that 𝑅_𝐴^≀≽ are called dominance relations of IFOIS ℐ^≀≽.
Let 𝑥_𝑖𝐴^≀⪰=𝑥_𝑗∈𝑈∣𝑥_𝑗,𝑥_𝑖∈𝑅_𝐴^≀⪰=𝑥_𝑗∈𝑈∣𝜇_𝑎𝑥_𝑗≥𝜇_𝑎𝑥_𝑖,𝜈_𝑎𝑥_𝑗≤𝜈_𝑎𝑥_𝑖,𝑈,∀𝑎∈𝐴𝑅_𝐴^≀⪰=𝑥_𝑖𝐴^≀⪰∣𝑥_𝑖∈𝑈, (3.5) where 𝑖∈{1,2,…,|𝑈|}, then [𝑥_𝑖]_𝐴^≀≽ describe the set of objects that may dominate 𝑥_𝑖 in terms of 𝐴 in IFOIS ℐ^≀≽ and will be called a dominance class of IFOIS ℐ^≀≽, and 𝑈/𝑅_𝐴^≀≽ be called a classification of 𝑈 about attribute set 𝐴 in IFOIS ℐ^≀≽.

Definition 3.3. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝐵,𝐴⊆𝐴𝑇.(1)If [𝑥]^≽_𝐵=[𝑥]^≽_𝐴 for any 𝑥∈𝑈, then we call that classification 𝑈/𝑅^≽_𝐵 is equal to 𝑅/𝑅^≽_𝐴, denoted by 𝑈/𝑅^≽_𝐵=𝑈/𝑅^≽_𝐴.(2)If [𝑥]^≽_𝐵⊆[𝑥]^≽_𝐴 for any 𝑥∈𝑈, then we call that classification 𝑈/𝑅^≽_𝐵 is finer than 𝑅/𝑅^≽_𝐴, denoted by 𝑈/𝑅^≽_𝐵⊆𝑈/𝑅^≽_𝐴.(3)If [𝑥]^≽_𝐵⊆[𝑥]^≽_𝐴 for any 𝑥∈𝑈 and [𝑥]^≽_𝐵≠[𝑥]^≽_𝐴 for some 𝑥∈𝑈, then we call that classification 𝑈/𝑅^≽_𝐵 is properly finer then 𝑅/𝑅^≽_𝐴, denoted by 𝑈/𝑅^≽_𝐵⊂𝑈/𝑅^≽_𝐴.

From the definitions of 𝑅_𝐴^≀≽ and [𝑥]_𝐴^≀≽, the following properties can easily be obtained.

Proposition 3.4. Letting ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, and 𝐴⊆𝐴𝑇, one can have 𝑅_𝐴^≀⪰=_𝑎∈𝐴𝑅_{𝑎}^≀⪰. (3.6)

Proposition 3.5. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, and 𝐴⊆𝐴𝑇. Then(1)𝑅_𝐴^≀≽ is reflexive,(2)𝑅_𝐴^≀≽ is unsymmetric, (3)𝑅_𝐴^≀≽ is transitive.

Proposition 3.6. Letting ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, and 𝐴,𝐵⊆𝐴𝑇, one has the following results.(1)If 𝐵⊆𝐴, then 𝑅_𝐴^≀≽⊆𝑅_𝐵^≀≽. (2)If 𝐵⊆𝐴, then[𝑥_𝑖]_𝐴^≀≽⊆[𝑥_𝑖]_𝐵^≀≽. (3)If 𝑥_𝑗∈[𝑥_𝑖]_𝐴^≀≽, then [𝑥_𝑗]_𝐴^≀≽⊆[𝑥_𝑖]_𝐴^≀≽ and [𝑥_𝑖]_𝐴^≀≽=∪{[𝑥_𝑗]_𝐴^≀≽∣𝑥_𝑗∈[𝑥_𝑖]_𝐴^≀≽}. (4)[𝑥_𝑗]_𝐴^≀≽=[𝑥_𝑖]_𝐴^≀≽ iff 𝜇_𝑎(𝑥_𝑖)=𝜇_𝑎(𝑥_𝑗) and 𝜈_𝑎(𝑥_𝑖)=𝜈_𝑎(𝑥_𝑗) for all 𝑎∈𝐴.

These properties mentioned above can be understood through the following example.

Example 3.7 (continued from Example 3.1). Computing the classification induced by the dominance relation 𝑅^≀≽_𝐴𝑇 in Table 2.
From the table, one can obtain that 𝑥₁^≀⪰_𝐴𝑇=𝑥₁,𝑥₅,𝑥₇,𝑥₈,𝑥₂^≀⪰_𝐴𝑇=𝑥₁,𝑥₂,𝑥₃,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥₃^≀⪰_𝐴𝑇=𝑥₁,𝑥₃,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₄^≀⪰_𝐴𝑇=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥₅^≀⪰_𝐴𝑇=𝑥₅,𝑥₆^≀⪰_𝐴𝑇=𝑥₅,𝑥₆,𝑥₈,𝑥₇^≀⪰_𝐴𝑇=𝑥₅,𝑥₇,𝑥₈,𝑥₈^≀⪰_𝐴𝑇=𝑥₈,𝑥₉^≀⪰_𝐴𝑇=𝑥₉,𝑥₁₀^≀⪰_𝐴𝑇=𝑥₁₀. (3.7) If 𝐴={𝑎₁,𝑎₂,𝑎₃,𝑎₅}⊆𝐴𝑇, we can get that 𝑥_1𝐴^≀⪰=𝑥₁,𝑥₅,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥_2𝐴^≀⪰=𝑥₁,𝑥₂,𝑥₃,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥_3𝐴^≀⪰=𝑥₁,𝑥₂,𝑥₃,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥_4𝐴^≀⪰=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥_5𝐴^≀⪰=𝑥₅,𝑥₁₀,𝑥_6𝐴^≀⪰=𝑥₅,𝑥₆,𝑥₈,𝑥₉,𝑥₁₀,𝑥_7𝐴^≀⪰=𝑥₅,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀,𝑥_8𝐴^≀⪰=𝑥₅,𝑥₈,𝑥₉,𝑥₁₀,𝑥_9𝐴^≀⪰=𝑥₉,𝑥₁₀,𝑥_10𝐴^≀⪰=𝑥₁₀. (3.8)
Obviously, [𝑥_𝑖]^≀≽_𝐴𝑇⊆[𝑥_𝑖]_𝐴^≀≽ and 𝑈𝑅^≀⪰_𝐴𝑇=𝑈𝑅_𝐴^≀⪰=𝑥_𝑖𝐴^≀⪰∣𝑖=1,2,…,10. (3.9)

From this example, we can easily verify above propositions of information systems based on intuitionistic fuzzy relation.

4. Ranking for Objects in IFOIS

In general, there are two classes of problems in intelligent decision making. One is to find satisfactory results through ranking with information aggregation. And the other is to find dominance rules through relations.

In this section, we mainly investigate how to rank all objects by the dominance relation in an intuitionistic fuzzy ordered information system.

Definition 4.1. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝐴⊆𝐴𝑇. Dominance degree between two objects 𝑥_𝑖,𝑥_𝑗∈𝑈 with respect to the dominance relation 𝑅_𝐴^≀≽ is defined as 𝑑_𝐴𝑥_𝑖,𝑥_𝑗𝑥=1−^{𝑖𝐴≀⪰}∩∼𝑥_𝑗𝐴^≀⪰|𝑈|. (4.1) We say that dominance degree of 𝑥_𝑖 to 𝑥_𝑗 is 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗).
From the definition, the dominance degree 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗) depict the proportion of some objects which are at least as good as 𝑥_𝑗 in dominance class [𝑥_𝑖]_𝐴^≀≽. Moreover, we can obtain the following properties.

Proposition 4.2. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, 𝐴⊆𝐴𝑇 and dominance degree between two objects 𝑥_𝑗 and 𝑥_𝑖 be 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗) with respect to the dominance relation 𝑅_𝐴^≀≽, then the following holds.(1)0≤𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)≤1 and 𝑑_𝐴(𝑥_𝑖,𝑥_𝑖)=1.(2)If 𝑥_𝑖∈[𝑥_𝑗]_𝐴^≀≽, then 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)=1.(3)If 𝑥_𝑗∈[𝑥_𝑘]_𝐴^≀≽, then 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)≤𝑑_𝐴(𝑥_𝑖,𝑥_𝑘).(4)If 𝑥_𝑗∈[𝑥_𝑘]_𝐴^≀≽ and 𝑥_𝑘∈[𝑥_𝑖]_𝐴^≀≽, then 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)≤𝑑_𝐴(𝑥_𝑘,𝑥_𝑗) and 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)≤𝑑_𝐴(𝑥_𝑖,𝑥_𝑘).

Proof. (1) is directly obtained by the definition.
(2) Since 𝑥_𝑖∈[𝑥_𝑗]_𝐴^≀≽, one can have [𝑥_𝑖]_𝐴^≀≽⊆[𝑥_𝑗]_𝐴^≀≽ by Proposition 3.6. So, we have [𝑥_𝑖]_𝐴^≀≽∩(∼[𝑥_𝑗]_𝐴^≀≽)=𝜙. That is to say 𝑑_𝐴𝑥_𝑖,𝑥_𝑗𝑥=1−^{𝑖𝐴≀⪰}∩∼𝑥_𝑗𝐴^≀⪰|𝑈|=1. (4.2)
(3) If 𝑥_𝑗∈[𝑥_𝑘]_𝐴^≀≽, then we can obtain [𝑥_𝑗]_𝐴^≀≽⊆[𝑥_𝑘]_𝐴^≀≽. So we have (∼[𝑥_𝑗]_𝐴^≀≽)⊇(∼[𝑥_𝑘]_𝐴^≀≽). Thus 𝑥_𝑖𝐴^≀⪰∩∼𝑥_𝑗𝐴^≀⪰|𝑈|≥𝑥_𝑖𝐴^≀⪰∩∼𝑥_𝑘𝐴^≀⪰|𝑈|. (4.3) Thus 𝑑_𝐴𝑥_𝑖,𝑥_𝑗≤𝑑_𝐴𝑥_𝑖,𝑥_𝑘. (4.4)
(4) If 𝑥_𝑗∈[𝑥_𝑘]_𝐴^≀≽ and 𝑥_𝑘∈[𝑥_𝑖]_𝐴^≀≽, then we can obtain [𝑥_𝑗]_𝐴^≀≽⊆[𝑥_𝑘]_𝐴^≀≽⊆[𝑥_𝑖]_𝐴^≀≽. That is (∼[𝑥_𝑗]_𝐴^≀≽)⊇(∼[𝑥_𝑘]_𝐴^≀≽)⊇(∼[𝑥_𝑖]_𝐴^≀≽) hold. So we have 𝑥_𝑖𝐴^≀⪰∩∼𝑥_𝑗𝐴^≀⪰|𝑈|≥𝑥_𝑘𝐴^≀⪰∩∼𝑥_𝑗𝐴^≀⪰|𝑈|,𝑥_𝑖𝐴^≀⪰∩∼𝑥_𝑗𝐴^≀⪰|𝑈|≥𝑥_𝑖𝐴^≀⪰∩∼𝑥_𝑘𝐴^≀⪰|𝑈|. (4.5) Thus 𝑑_𝐴𝑥_𝑖,𝑥_𝑗≤𝑑_𝐴𝑥_𝑘,𝑥_𝑗,𝑑_𝐴𝑥_𝑖,𝑥_𝑗≤𝑑_𝐴𝑥_𝑖,𝑥_𝑘. (4.6)
The proposition was proved.

Definition 4.3. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝐴⊆𝐴𝑇. Denote 𝑀_𝐴^≀⪰=𝑟_{𝑖𝑗|𝑈|×|𝑈|},where𝑟_𝑖𝑗=𝑑_𝐴𝑥_𝑖,𝑥_𝑗. (4.7) Then, we call the matrix 𝑀_𝐴^≀≽ to be a dominance matrix with respect to 𝐴 induced by the intuitionistic fuzzy dominance relation 𝑅_𝐴^≀≽.
Moreover, if 𝑑_𝐴𝑥_𝑖=1|𝑈|^{𝑥𝑗∈𝑈}𝑑_𝐴𝑥_𝑖,𝑥_𝑗, (4.8) then we call 𝑑_𝐴(𝑥_𝑖) to be dominance degree of 𝑥_𝑖 with respect to relation 𝑅_𝐴^≀≽, for every 𝑥_𝑖∈𝑈.
By definition of dominance matrix and dominance degree of the object with respect to relation 𝑅_𝐴^≀≽, we can directly receive the following properties. For all 𝑥_𝑖∈𝑈, the degree can be calculated according to the following formula: 𝑑_𝐴𝑥_𝑖=1|𝑈|^|𝑈|𝑗=1𝑟_𝑖𝑗. (4.9)

As a result of the above discussions, we come to the following two corollaries.

Corollary 4.4. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝐴⊆𝐴𝑇. If 𝑅_𝐴^≀≽=𝑅^≀≽_𝐴𝑇, then 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)=𝑑_𝐴𝑇(𝑥_𝑖,𝑥_𝑗), 𝑑_𝐴(𝑥_𝑖)=𝑑_𝐴𝑇(𝑥_𝑖) and 𝑀_𝐴^≀≽=𝑀^≀≽_𝐴𝑇, for 𝑥_𝑖,𝑥_𝑗∈𝑈.

From the dominance degree of each object on the universe, we can rank all objects according to the number of 𝑑_𝐴. A larger number implies a better object. This idea can be understood by the following example.

Example 4.5 (continued from Example 3.1). Rank all objects in 𝑈 according to the dominance relation 𝑅^≀≽_𝐴𝑇 in the system of Example 3.1.
By Example 3.7, we can easily obtain the dominance degree of two objects and dominance matrix in the system as follows: 𝑀^≀⪰_𝐴𝑇=711118109107106106105101710121041041021021021081051011171071051041041041091061011103103101101101109101111111910910910810111191081071071081011119101910710710910111191011191091019101910910910910191091019101910910910910910110. (4.10) So, we can have 𝑑_𝐴𝑇𝑥₁=0.83,𝑑_𝐴𝑇𝑥₂=0.48,𝑑_𝐴𝑇𝑥₃𝑑=0.7,_𝐴𝑇𝑥₄=0.39,𝑑_𝐴𝑇𝑥₅=0.97,𝑑_𝐴𝑇𝑥₆𝑑=0.88,_𝐴𝑇𝑥₇=0.90,𝑑_𝐴𝑇𝑥₈=0.97,𝑑_𝐴𝑇𝑥₉𝑑=0.93,_𝐴𝑇𝑥₁₀=0.93. (4.11) Therefore, according the above we rank all objects in the following: 𝑥₅=𝑥₈⪰𝑥₉=𝑥₁₀⪰𝑥₇⪰𝑥₆⪰𝑥₁⪰𝑥₃⪰𝑥₂⪰𝑥₄. (4.12)

5. Rough Set Approach to IFOIS

In this section, we investigate the problem of set approximation with respect to a dominance relation 𝑅_𝐴^≀≽ in intuitionistic fuzzy ordered information systems.

Similar to ordered information systems, we can define the upper and lower approximation sets in IFOIS.

Definition 5.1. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system. For any 𝑋⊆𝑈 and 𝐴⊆𝐴𝑇, the upper and lower approximations of 𝑋 with respect to the dominance relation 𝑅_𝐴^≀≽ are defined as follows: 𝑅_𝐴^≀⪰(𝑋)=𝑥∈𝑈∣[𝑥]_𝐴^≀⪰,𝑅∩𝑋≠𝜙_𝐴≀⪰(𝑋)=𝑥∈𝑈∣[𝑥]_𝐴^≀⪰⊆𝑋. (5.1)

From above definition, one can briefly notice that 𝑅_𝐴^≀≽(𝑋) is a set of objects that belong to 𝑋 with certainty and 𝑅_𝐴^≀≽(𝑋) is a set of objects that probably belong to 𝑋. If 𝑅_𝐴^≀≽(𝑋)≠𝑅_𝐴^≀≽(𝑋), we say the subset 𝑋 of 𝑈 is rough, otherwise 𝑋 is precise. 𝐵𝑛_𝐴(𝑋)=𝑅_𝐴^≀≽(𝑋)−𝑅_𝐴^≀≽(𝑋) is called to a boundary of the rough set.

Moreover, we can directly obtain the following results.

Proposition 5.2. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝐴⊆𝐴𝑇. For any 𝑋⊆𝑈, the following always holds:(1)𝑅_𝐴^≀≽(𝑋)⊆𝑅^≀≽_𝐴𝑇(𝑋) and 𝑅_𝐴^≀≽(𝑋)⊇𝑅^≀≽_𝐴𝑇(𝑋).(2)If 𝑅_𝐴^≀≽=𝑅^≀≽_𝐴𝑇, then 𝑅_𝐴^≀≽(𝑋)=𝑅^≀≽_𝐴𝑇(𝑋) and 𝑅_𝐴^≀≽(𝑋)=𝑅^≀≽_𝐴𝑇(𝑋).

Proposition 5.3. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system. For any 𝑋,𝑌⊆𝑈 and 𝐴⊆𝐴𝑇, then the properties in Table 3 hold.

Table 3 

Proof. The proof are similar to the case of Properties in [41].

Another topic is uncertainty of a rough set in rough set theory. Uncertainty of a rough set is due to the existence of a borderline region. The greater the borderline region of a rough set, the lower the accuracy of the rough set. In order to measure the imprecision of a rough set induced by intuitionistic dominance relation in ordered information systems, definition of accuracy measure is introduced in the following.

Definition 5.4. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, 𝑋⊆𝑈 and 𝐴⊆𝐴𝑇. Roughness measure of 𝑋 with respect to the dominance relation 𝑅_𝐴^≀≽ is defined as 𝜌𝑅_𝐴^≀⪰𝑅,𝑋=1−^𝐴≀⪰(𝑋)𝑅_𝐴^≀⪰(𝑋). (5.2)

By the definition, we can easily find that the roughness measure expresses the rough degree of the knowledge about 𝑋, given the knowledge 𝑈/𝑅_𝐴^≀≽. It is clear that 0≤𝜌≤1. We can directly get that 𝑋 is definable in IFOIS when 𝜌=0, in which 𝑅_𝐴^≀≽(𝑋)=𝑅_𝐴^≀≽(𝑋).

We have the following properties about roughness 𝜌(𝑅_𝐴^≀≽,𝑋).

Proposition 5.5. Letting ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, 𝑋⊆𝑈 and 𝐴⊆𝐴𝑇, the following results hold: (1)𝜌(𝑅_𝐴^≀≽,𝑋)=1−|𝑅_𝐴^≀≽(𝑋)|/|𝑅_𝐴^≀≽(𝑋)|=1−|𝑅_𝐴^≀≽(𝑋)|/(|𝑈|−|𝑅_𝐴^≀≽(∼𝑋)|); (2)if 𝑅_𝐴^≀≽=𝑅^≀≽_𝐴𝑇, then 𝜌(𝑅_𝐴^≀≽,𝑋)=𝜌(𝑅^≀≽_𝐴𝑇,𝑋);(3)if 𝐵⊆𝐴⊆𝐴𝑇, then 𝜌(𝑅^≀≽_𝐴𝑇,𝑋)≤𝜌(𝑅_𝐴^≀≽,𝑋)≤𝜌(𝑅_𝐵^≀≽,𝑋).

Proof. (1) and (2) can be directly obtained by Definitions 5.1 and 5.4 and Proposition 5.3.
(3) Since 𝐵⊆𝐴, we can get that 𝑅_𝐵^≀≽(𝑋)⊆𝑅_𝐴^≀≽(𝑋) and 𝑅_𝐵^≀≽(𝑋)⊇𝑅_𝐴^≀≽(𝑋). So we have 𝑅_𝐵^≀⪰(𝑋)𝑅_𝐵^≀⪰(𝑋≤𝑅)^𝐴≀⪰(𝑋)𝑅_𝐴^≀⪰(𝑋). (5.3) That is 𝜌𝑅_𝐴^≀⪰,𝑋≤𝜌𝑅_𝐵^≀⪰,𝑋. (5.4) Similarly, we can obtain 𝜌(𝑅^≀≽_𝐴𝑇,𝑋)≤𝜌(𝑅_𝐴^≀≽,𝑋).
The proof is completed.

Example 5.6 (continued from Examples 3.1 and 3.7). Consider the ordered information system based on intuitionistic fuzzy relation in Example 3.1.
Letting 𝐴={𝑎₁,𝑎₂,𝑎₃,𝑎₅}⊆𝐴𝑇 and 𝑋={𝑥₁,𝑥₅,𝑥₆,𝑥₈}, compute approximation operators of set 𝑋 approximated by intuitionistic fuzzy relation 𝑅_𝐴^≀≽ and 𝑅^≀≽_𝐴𝑇, respectively.
According to Definition 5.1 and Example 3.7, the lower and upper approximations of 𝑋 with respect to relation 𝑅^≀≽_𝐴𝑇(𝑋) can be received as follows: 𝑅^≀⪰_𝐴𝑇(𝑋)=𝑥₅,𝑥₆,𝑥₈,𝑅^≀⪰_𝐴𝑇(𝑋)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇,𝑥₈. (5.5) Similarly, the lower and upper approximations of 𝑋 with respect to relation 𝑅_𝐴^≀≽(𝑋) can be received as follows: 𝑅_𝐴^≀⪰(𝑋)=𝜙,𝑅_𝐴^≀⪰(𝑋)=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇,𝑥₈. (5.6) Thus, we have that 𝜌𝑅_𝐴^≀⪰𝑅,𝑋=1−^𝐴≀⪰(𝑋)𝑅_𝐴^≀⪰(𝑋𝜌𝑅)=1,_{≀⪰𝐴𝑇}𝑅,𝑋=1−^{≀⪰𝐴𝑇}(𝑋)𝑅^≀⪰_𝐴𝑇(𝑋=5)8. (5.7) Hence, one can get 𝜌𝑅^≀⪰_𝐴𝑇,𝑋≤𝜌𝑅_𝐴^≀⪰,𝑋. (5.8)
From the above, we can easily find that the subset 𝑋⊆𝑈 is more rough in system based on intuitionistic fuzzy relation 𝑅_𝐴^≀≽ than relation 𝑅^≀≽_𝐴𝑇, which is consistent with the fact.

6. Attribute Reduction Based on Discernibility Matrix in IFOIS

In order to simplify knowledge representation, it is necessary to reduce some dispensable attributes in a given intuitionistic fuzzy ordered information system. In this section, an approach to attribute reduction in ordered information systems based intuitionistic fuzzy relation will be established and an illustrative example is employed to show its validity.

Definition 6.1. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝑋⊆𝑈. An attribute subset 𝐴⊆𝐴𝑇 is referred to as a classical consistent set of ℐ^≀≽ if 𝑅_𝐴^≀≽=𝑅^≀≽_𝐴𝑇. Moreover, if 𝐴 is a classical consistent set of ℐ^≀≽ and no proper subset of 𝐴 is a classical consistent set of ℐ^≀≽, then 𝐴 is referred to as a classical reduction of ℐ^≀≽.

From the definition, we can directly obtain the following property.

Proposition 6.2. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝐴⊆𝐴𝑇. If 𝐴 is a classical reduction, then 𝑑_𝐴(𝑥_𝑖,𝑥_𝑗)=𝑑_𝐴𝑇(𝑥_𝑖,𝑥_𝑗) for any 𝑥_𝑖,𝑥_𝑗∈𝑈.

It is obvious that a classical reduction of an IFOIS is a minimal attribute subset satisfying 𝑅_𝐴^≀≽=𝑅^≀≽_𝐴𝑇. An attribute 𝑎∈𝐴𝑇 is called dispensable with respect to 𝑅^≀≽_𝐴𝑇 if 𝑅^≀≽_𝐴𝑇=𝑅^≀≽_{𝐴𝑇−{𝑎}}; otherwise 𝑎 is called indispensable. The set of all indispensable attributes is called a core with respect to the dominance relation 𝑅^≀≽_𝐴𝑇 and is denoted by 𝑐𝑜𝑟𝑒(𝐴𝑇). An attribute in the core must be in every attribute reduction. In other words, 𝑐𝑜𝑟𝑒(𝐴𝑇) is the intersection of all classical reductions of the system. Thus the core may be empty set.

Definition 6.3. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system. For any 𝑥_𝑖,𝑥_𝑗∈𝑈, if we denote Dis𝑥_𝑖,𝑥_𝑗=𝑎∈𝐴𝑇∣𝜇_𝑎𝑥_𝑗<𝜇_𝑎𝑥_𝑖,or𝜈_𝑎𝑥_𝑗>𝜈_𝑎𝑥_𝑖,ℳ_Dis=𝑙_{𝑖𝑗|𝑈|×|𝑈|},where𝑙_𝑖𝑗=Dis𝑥_𝑖,𝑥_𝑗, (6.1) then we call Dis(𝑥_𝑖,𝑥_𝑗) to be a discernibility attribute set between objects 𝑥_𝑖 and 𝑥_𝑗, and matrix ℳ_Dis to be a discernibility matrix of an IFOIS.

One can easily get the following property by the above definition.

Proposition 6.4. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system and 𝑥_𝑖,𝑥_𝑗∈𝑈, then Dis(𝑥_𝑖,𝑥_𝑖)=𝜙 and Dis(𝑥_𝑖,𝑥_𝑗)∩Dis(𝑥_𝑗,𝑥_𝑖)=𝜙.

The following property provides a judgement method of a classical reduction of an IFOIS.

Proposition 6.5. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, 𝐴⊆𝐴𝑇 and Dis(𝑥_𝑖,𝑥_𝑗) the discernibility attributes set of ℐ^≀≽ with respect to 𝑅^≀≽_𝐴𝑇. Then the following two proposition are equivalent. (1)𝐴 is a classical consistent set of ℐ^≀≽.(2)If Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙, then 𝐴∩Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙 for 𝑥_𝑖,𝑥_𝑗∈𝑈.

Proof. “(1)⇒(2)” If 𝐴 is a classical consistent set of ℐ^≀≽, then we have 𝑅_𝐴^≀≽=𝑅^≀≽_𝐴𝑇. By the definition of the dominance relation, one can know [𝑥]_𝐴^≀≽=[𝑥]^≀≽_𝐴𝑇 for any 𝑥∈𝑈. On the other hand, since Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙, then 𝑥_𝑗∉[𝑥_𝑖]^≀≽_𝐴𝑇. So 𝑥_𝑗∉[𝑥_𝑖]_𝐴^≀≽. That is to say that there exists 𝑎∈𝐴 such that 𝑓(𝑎,𝑥_𝑗)<𝑓(𝑎,𝑥_𝑖). Therefore 𝑎∈ Dis(𝑥_𝑖,𝑥_𝑗). Thus 𝐴∩Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙.
“(2)⇒(1)” Because we have known 𝑅^≀≽_𝐴𝑇⊆𝑅_𝐴^≀≽ by the Proposition 3.6, we need only prove 𝑅_𝐴^≀≽⊆𝑅^≀≽_𝐴𝑇.
For 𝑥_𝑖,𝑥_𝑗∈𝑈, if Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙, then 𝑥_𝑗∉[𝑥_𝑖]^≀≽_𝐴𝑇. Moreover, we know that 𝐴∩Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙 when Dis(𝑥_𝑖,𝑥_𝑗)≠𝜙. So there exists 𝑎∈𝐴 such that 𝜇_𝑎(𝑥_𝑗)<𝜇_𝑎(𝑥_𝑖)or𝜈_𝑎(𝑥_𝑗)>𝜈_𝑎(𝑥_𝑖), that is to say 𝑥_𝑗∉[𝑥_𝑖]_𝐴^≀≽. Hence, we can find that if 𝑥_𝑗∉[𝑥_𝑖]^≀≽_𝐴𝑇 then 𝑥_𝑗∉[𝑥_𝑖]_𝐴^≀≽. In other words, if 𝑥_𝑗∈[𝑥_𝑖]_𝐴^≀≽ then 𝑥_𝑗∈[𝑥_𝑖]^≀≽_𝐴𝑇. That is [𝑥_𝑖]_𝐴^≀≽⊆[𝑥_𝑖]^≀≽_𝐴𝑇, that is,𝑅_𝐴^≀≽⊆𝑅^≀≽_𝐴𝑇.
This completes the proof.

Definition 6.6. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system, 𝐴⊆𝐴𝑇 and ℳ_Dis be discernibility matrix. Denoted by 𝐹^≀⪰=𝑎_𝑘∣𝑎_𝑘∈Dis𝑥_𝑖,𝑥_𝑗,𝑥_𝑖,𝑥_𝑗∈𝑈, (6.2) then 𝐹^≀≽ is called discernibility formula.

Based on the discernibility formula, we can design a practical approach to classical reduction in an IFOIS as follows.

Proposition 6.7. Let ℐ^≀≽=(𝑈,𝐴𝑇,𝑉,𝑓) be an intuitionistic fuzzy ordered information system. The minimal disjunctive normal form of discernibility formula of distribution is 𝐹^≀⪰=^𝑝_𝑘=1^𝑞𝑘_𝑠=1𝑎_𝑠. (6.3) Denote 𝐵_𝑘^≀≽={𝑎_𝑠|𝑠=1,2,…,𝑞_𝑘}, then {𝐵_𝑘^≀≽∣𝑘=1,2,…,𝑝} are just sets of all distribution reductions of ℐ^≽.

Proof. It follows directly from Proposition 6.5 and the definition of minimal disjunctive normal of the discernibility formula.

In the following, we analyze how to obtain classical reductions from all attributes in an IFOIS by an illustrative example.

Example 6.8 (continued from Example 3.1). Calculate all classical reductions of the IFOIS in Example 3.1.
By the definition of discernibility matrix, one can obtain the discernibility of the system in Table 4.
Hence, we can have that 𝐹^≀⪰=𝑎₁∨𝑎₂∨𝑎₃∧𝑎₁∨𝑎₂∨𝑎₃∨𝑎₄∨𝑎₅∧𝑎₂∨𝑎₅∧𝑎₁∨𝑎₂∨𝑎₃∨𝑎₅∧𝑎₄∧𝑎₁∨𝑎₃∧𝑎₁∨𝑎₃∨𝑎₄=𝑎₁∨∨𝑎₃∧∨𝑎₂∨𝑎₅∧𝑎₄=𝑎₁∧𝑎₂∧𝑎₄∨𝑎₁∧𝑎₄∧𝑎₅∨𝑎₂∧𝑎₃∧𝑎₄∨𝑎₃∧𝑎₄∧𝑎₅. (6.4)
So, there are four classical reductions for the system, which are {𝑎₁,𝑎₂,𝑎₄}, {𝑎₁,𝑎₄,𝑎₅}, {𝑎₂,𝑎₃,𝑎₄}, and {𝑎₃,𝑎₄,𝑎₅}. And it is clear that the core of the system is {𝑎₄}.

Table 4 

The discernibility matrix of the system in Example 3.1.

7. Conclusions

Rough set theory is a new mathematical tool to deal with vagueness and uncertainty. Development of a rough computational method is one of the most important research tasks. While, in practise, intuitionistic fuzzy ordered information system confines the applications of classical rough set theory. In this paper, we mainly considered some important concepts and properties in this system. We defined two approximation operators and established the rough set approach to intuitionistic fuzzy ordered information systems. Moreover, we also investigated the problem of attribute reductions based on rough sets and presented method of the reduction. However, extracted dominance rules from the system is another important class of problem in decision-making analysis. So, our further works discuss ordered decision table based on intuitionistic fuzzy relation and dominance rules extracted from this type of decision tables.

Acknowledgments

This work is supported by National Natural Science Foundation of China (nos. 61105041, 71071124, and 11001227), Postdoctoral Science Foundation of China (no. 20100481331), and Natural Science Foundation Project of CQ CSTC (no. cstc2011jjA40037).