Three-way decisions for composed set-valued decision tables

Abstract

In this paper, composition of set-valued decision tables are discussed based on the quasi-order relation on set-valued decision tables. Based on the theory of the three-way decisions, decision method of the set-valued decision tables are discussed, the three-way decisions of the object-composed set-valued decision tables and the three-way decisions of the attribute-composed set-valued decision tables are also discussed. Furthermore, the three-way decisions of composed set-valued decision tables are discussed.

Keywords

Composed set-valued decision tables object-composed set-valued decision tables attribute-composed set-valued decision tables three-way decisions

1 Introduction

In 1982, a tool for data analysis and data processing was proposed by Polish mathematician Pawlak— rough set theory [1], which can be used to process incomplete and imprecise information. It also can make decisions on information systems or get the classification rules under the premise of keeping date classification unchanged. Since then, the study on rough set theory has received extensive attention for the artificial intelligence researchers. At present, rough set theory has been successfully applied to various fields, such as knowledge discovery, data mining, decision support and analysis, machine learning, pattern recognition, etc [2, 3].

The knowledge discovery of information system is an important content of rough set theory. By researching the attributes of information systems to obtain classification rules is the basic method in the rough set theory. But attributes in information systems are not equally important. It has important theoretical values and practical significance that researching on classifying information systems, rule extraction [4, 6], knowledge acquisition [5] and the importance of attributes in information system [7]. Pawlak’s rough set theory is used to deal with complete information systems with discrete attribute values. When dealing with the data under the condition of lack of information, we have to study incomplete information systems or set-valued information systems [8 –12]. In [13], Song defined a quasi-relation in set-valued information systems, and discussed the knowledge reduction and attribute characteristics of set-valued information systems in this kind of relation.

When we process information, we should consolidate all kinds of relevant information to analyze and manipulate. Thus, it is a worth discussing problem that how to better incorporating different information systems. On the basis of incorporation, researches about the composed information system and the original information systems on the knowledge base, knowledge acquisition, classification and mining rules, etc. have a certain value. In [14], C.C. Chan’s recursive algorithm of rough set approximation [15] was promoted, and the relationship between the upper and lower approximation operators of synthetic information systems, information subsystems and the original information systems are discussed. In [16, 17], authors discussed synthetic information systems, as well as further discussed the attributes reduction of composed information systems. In [18], the random composed object-composed information systems and the attribute-composed information systems are studied.

Making decision is an important subject in the study about information systems and knowledge discovery [19 –21], then the study of decision tables is the further development of the study of information systems, which has a certain theoretical significance and a certain practical significance. The theory of three-way decision is a kind of decision model based on human cognition. It was proposed by Yao et al. [22, 23] in 1990s. It is developed on the basis of the two-way decision. In the traditional two-way decision, a set is divided into two disjoint regions on the basis of one threshold value, there are only two decision results, that is acceptance or refusal. However, in practical application, the price of forced decision of acceptance or refusal will be very big, but three-way decision divided a set into three disjoint collections by introducing a pair of thresholds. In addition to accept and reject, three-way decision offered the third decision, that is, delay selection. The three-way decision model is practical and effective. It can avoid the unnecessary cost which is induced by two-way decision, and this method is easy to understand. Any question on two-way decision can be extended to the three-way decision problem. The introduction of three-way decision theory unified the decision-making method in many disciplines. The three-way decision thought and method are simple and effective. At present, it has become a focus for many researchers [24 –31], and it has been widely used in many fields [32 –35].

Based on the former researches of set-valued information systems and composed set-valued information systems, this paper studies the composition of set-valued decision tables, and joins the three-way decision thought, as well as further discussed the three-way decisions of set-valued decision tables and the three-way decisions of composed set-valued decision tables.

2 Set-valued decision tables

In this section, we will define set-valued decision table and introduce its properties.

Definition 1. (U, A, f, D, g) is called a set-valued decision table, where U = {x₁, x₂, …, x_n} is a finite non-empty set of objects called the universe of discourse. A = {a₁, a₂, …, a_m} is a non-empty finite set of condition attributes, D = {d₁, d₂, …, d_l} is a non-empty finite set of decision attributes. f : U × A → P₀ (V_A) is an information function, where $V_{A} = \underset{a \in A}{\cup} V_{a}$ , a non-empty set V_a is the values set of a, and P₀ (V_A) is the non-empty power set of V_A, and it is the value set of A. g : U × D → P₀ (V_D) is an information function that maps an object in U to exactly one value in P₀ (V_D), where $V_{D} = \underset{d \in D}{\cup} V_{d}$ , a non-empty set V_d is the value set of d, and P₀ (V_D) is the non-empty power set of V_D and it is the value set of D.

Definition 2. [13] Let (U, A, f, D, g) be a set-valued decision table. For any subset B ⊆ A, a binary relation on U is defined as: $R_{B}^{\subseteq} = {(x, y) \in U \times U : f (x, a) \subseteq f (y, a), \forall a \in B}$ , Similarly, $R_{D}^{\subseteq} = {(x, y) \in U \times U : g (x, d) \subseteq g (y, d), \forall d \in D}$ .

The relations are called quasi-order relation on U. It is easy to prove that $R_{B}^{\subseteq}$ and $R_{D}^{\subseteq}$ satisfy the property of reflexive, transitive, they are not equivalence relations.

Denote $R_{B}^{\subseteq} (x) = {y \in U : (x, y) \in R_{B}^{\subseteq}}$ , $R_{D}^{\subseteq} (x) = {y \in U : (x, y) \in R_{D}^{\subseteq}}$ , they are called the information granule of x about the condition attribute set B and about the decision attribute set D, respectively.

All the information granules $U / R_{B}^{\subseteq} = {R_{B}^{\subseteq} (x) : x \in U}$ or $U / R_{D}^{\subseteq} = {R_{D}^{\subseteq} (x) : x \in U}$ can form a covering of U.

Property 1. Let (U, A, f, D, g) be a set-valued decision table, $R_{B}^{\subseteq}$ , $R_{D}^{\subseteq}$ quasi-order relations on U, B ⊆ A, E ⊆ D. For any x, y ∈ U, the following conclusions hold:

If B ⊆ A, then $R_{A}^{\subseteq} (x) \subseteq R_{B}^{\subseteq} (x)$ and $R_{B}^{\subseteq} = \underset{a \in B}{\cap} R_{a}^{\subseteq}$ , If E ⊆ D, then $R_{D}^{\subseteq} \subseteq R_{E}^{\subseteq}$ and $R_{E}^{\subseteq} = \underset{d \in E}{\cap} R_{d}^{\subseteq}$ .

If B ⊆ A, then $R_{A}^{\subseteq} (x) \subseteq R_{B}^{\subseteq} (x)$ and $R_{B}^{\subseteq} (x) = \underset{a \in B}{\cap} R_{a}^{\subseteq} (x)$ , If E ⊆ D, then $R_{D}^{\subseteq} (x) \subseteq R_{E}^{\subseteq} (x)$ and $R_{E}^{\subseteq} (x) = \underset{d \in E}{\cap} R_{d}^{\subseteq} (x)$ .

If $y \in R_{B}^{\subseteq} (x)$ , then $R_{B}^{\subseteq} (y) \subseteq R_{B}^{\subseteq} (x)$ and $R_{B}^{\subseteq} (x) = \cup {R_{B}^{\subseteq} (y) : y \in R_{B}^{\subseteq} (x)}$ .

Definition 3. [10] Let U be a non-empty and finite set of objects, if there are C₁, C₂, …, C_k ⊆ U satisfying:

C_i ≠ ∅ (i ≤ k) ,

C_i ∩ C_j = ∅ (i ≠ j) ,

$⋃_{i = 1}^{k} C_{i} = U$ .

Then, $ℛ = {C_{1}, C_{2}, \dots, C_{k}}$ is called a partition of the universe U.

Theorem 1. A partition of U can be generated by a quasi-order relation.

Proof. Let {C₁, C₂, ⋯ C_k} be a partition of U, define R^⊆ = {(x_i, x_j) : there exist p ≤ k, satisfying x_i, x_j ∈ C_p}, then R^⊆ is a quasi-order relation on U . And when x_i ∈ C_p, there must be [x_i] _{R
^⊆} = C_p.

For any B ⊆ A, denote $R (B) = {[x]_{R_{B}^{\subseteq}} : x \in U}$ as a set of all quasi-order classes about attribute set B under the quasi-order relation. For any $[x]_{R_{A}^{\subseteq}} \in ℛ (A)$ , there always exists $[x]_{R_{B}^{\subseteq}} \in ℛ (B)$ satisfying $[x]_{R_{A}^{\subseteq}} \subseteq [x]_{R_{B}^{\subseteq}}$ . Then, $ℛ (A)$ is thinner than $ℛ (B)$ , which is denoted as $ℛ (A) \leq ℛ (B)$ .

Definition 4. Let (U, A, f, D, g) be a set-valued decision table. If $R_{A}^{\subseteq} \subseteq R_{D}^{\subseteq}$ , that is, $ℛ (A) \leq ℛ (D)$ , then the set-valued decision table is a consistent decision table. Otherwise, it is said to be inconsistent.

3 Composed set-valued decision tables

Definition 5. Let (U₁, A, f₁, D, g₁) and (U₂, A, f₂, D, g₂) be two set-valued decision tables with the same condition attribute set and the same decision attribute set, and U₁ ∩ U₂ = ∅ , (U₁ ∪ U₂, A, f_U, D, g_U) is called an object-composed set-valued decision table, in which:

$\begin{matrix} f_{1} : U_{1} \times A \to P_{0} (V_{A}^{1}), f_{2} : U_{2} \times A \to P_{0} (V_{A}^{2}), \\ f_{U} : (U_{1} \cup U_{2}) \times A \to P_{0} (V_{A}^{1, 2}), V_{A}^{1, 2} = V_{A}^{1} \cup V_{A}^{2}, \\ f_{U} (x, a) = {\begin{matrix} f_{1} (x, a), x \in U_{1}, \\ f_{2} (x, a), x \in U_{2} . \end{matrix} \\ g_{1} : U_{1} \times D \to P_{0} (V_{D}^{1}), g_{2} : U_{2} \times D \to P_{0} (V_{D}^{2}), \\ g_{U} : (U_{1} \cup U_{2}) \times D \to P_{0} (V_{D}^{1, 2}), V_{D}^{1, 2} = V_{D}^{1} \cup V_{D}^{2}, \\ g_{U} (x, d) = {\begin{matrix} g_{1} (x, d), x \in U_{1}, \\ g_{2} (x, d), x \in U_{2} . \end{matrix} . \end{matrix}$

Let (U₁ ∪ U₂, A, f_U, D, g_U) be an object-composed set-valued decision table, for any B ⊆ A, E ⊆ D, denote $\begin{matrix} R_{B}^{\subseteq_{i}} & = & {(x, y) : x, y \in U_{i}, f_{i} (x, a) \\ \subseteq f_{i} (y, a), (\forall a \in B)}, i = 1, 2, \\ R_{B}^{\subseteq_{1, 2}} & = & {(x, y) : x, y \in U_{1} \cup U_{2}, f_{U} (x, a) \\ \subseteq f_{U} (y, a), (\forall a \in B)}, \\ R_{E}^{\subseteq_{i}} & = & {(x, y) : x, y \in U_{i}, g_{i} (x, d) \\ \subseteq g_{i} (y, d), (\forall d \in E)}, i = 1, 2, \\ R_{E}^{\subseteq 1, 2} & = & {(x, y) : x, y \in U_{1} \cup U_{2}, g_{U} (x, d) \\ \subseteq g_{U} (y, d), (\forall d \in E)} . \end{matrix}$

They are all quasi-order relations on (U₁ ∪ U₂, A, f_U, D, g_U), then their corresponding information granules are respectively denoted as: $\begin{matrix} R_{B}^{\subseteq_{i}} (x) & = & {y : x, y \in U_{i}, (x, y) \in R_{B}^{\subseteq_{i}}}, i = 1, 2, \\ R_{B}^{\subseteq_{1, 2}} (x) & = & {y : x, y \in U_{1} \cup U_{2}, (x, y) \in R_{B}^{\subseteq_{1, 2}}}, \\ R_{E}^{\subseteq_{i}} (x) & = & {y : x, y \in U_{i}, (x, y) \in R_{E}^{\subseteq_{i}}}, i = 1, 2, \\ R_{E}^{\subseteq_{1, 2}} (x) & = & {y : x, y \in U_{1} \cup U_{2}, (x, y) \in R_{E}^{\subseteq_{1, 2}}} . \end{matrix}$

Definition 6. Let (U₁, A, f₁, D, g₁) and (U₂, A, f₂, D, g₂) be two set-valued decision tables with the same condition attribute set and the same decision attribute set. When U₁∩ U₂ ≠ ∅, there exists x ∈ U₁ ∩ U₂, then we have two kinds of methods to compound:

The first one, in order to keep information in an integral state, is defined as follows: $\begin{matrix} f_{U}^{\cup} (x, a) & = & f_{1} (x, a) \cup f_{2} (x, a), x \in U_{1} \cap U_{2}, \\ g_{U}^{\cup} (x, d) & = & g_{1} (x, d) \cup g_{2} (x, d), x \in U_{1} \cap U_{2} . \end{matrix}$

The second one, in order to reflect the core of the information, is defined as follows: $\begin{matrix} f_{U}^{\cap} (x, a) & = & f_{1} (x, a) \cap f_{2} (x, a), x \in U_{1} \cap U_{2}, \\ g_{U}^{\cap} (x, d) & = & g_{1} (x, d) \cap g_{2} (x, d), x \in U_{1} \cap U_{2} . \end{matrix}$

Example 1. Tables 1 and 2 are two set-valued decision tables, which have the same condition attribute set and the same decision attribute set but different object sets, let condition attributes a₁ a₂ a₃ represent three attributes, d₁ represent a property. Tables 1 and 2 are two statistical tables in different periods.

Table 1

a ₁ a ₂ a ₃ d ₁

x ₁ {1, 2} {1, 3} {3, 4} {1, 2}

x ₂ {1, 2, 4} {2, 3} {1} {1, 3}

x ₃ {1, 3} {2, 3} {2, 4} {2, 3}

	a ₁	a ₂	a ₃	d ₁
x ₁	{1, 2}	{1, 3}	{3, 4}	{1, 2}
x ₂	{1, 2, 4}	{2, 3}	{1}	{1, 3}
x ₃	{1, 3}	{2, 3}	{2, 4}	{2, 3}

Table 2

	a ₁	a ₂	a ₃	d ₁
x ₁	{2, 3}	{2, 4}	{3, 4}	{2, 3}
x ₂	{1, 3, 4}	{1, 4}	{3, 4}	{1, 2, 3}
x ₄	{2, 3}	{1, 2, 4}	{3, 4}	{2}

We can get Table 3 by compound Tables 1 and 2 in the first method of Definition 8. In Table 3, we can get all the values of these attributes of the object set in two different periods. We get Table 4 by compound Tables 1 and 2 in the second method of Definition 8. From Table 4 we obtain the same values of these attributes of objects in different periods. Similarly, for decision attributes, these two different compound methods can show the change of values of decision attributes in different periods. Such a model can be used to study the formation of a certain type of people through the analysis of the behavior in different periods.

Table 3

	a ₁	a ₂	a ₃	d ₁
x ₁	{1, 2, 3}	{1, 2, 3, 4}	{3, 4}	{1, 2, 3}
x ₂	{1, 2, 3, 4}	{1, 2, 3, 4}	{1, 3, 4}	{1, 2, 3}
x ₃	{1, 3}	{2, 3}	{2, 4}	{2, 3}
x ₄	{1, 2, 3}	{1, 2, 3, 4}	{2, 3, 4}	{2, 3}

Table 4

	a ₁	a ₂	a ₃	d ₁
x ₁	{2}	∅	{3, 4}	{2}
x ₂	{1, 4}	∅	∅	{1, 3}
x ₃	{1, 3}	{2, 3}	{2, 4}	{2, 3}
x ₄	{3}	{2}	{4}	{2}

In the following, we only discuss the first case.

Definition 7. Let (U, A₁, f₁, D₁, g₁) and (U, A₂, f₂, D₂, g₂) be two set-valued decision tables with the same object set, A₁∩ A₂ = ∅. (U, A₁ ∪ A₂, f_A, D₁ ∪ D₂, g_D) is called a attribute-composed set-valued decision table, in which: $\begin{matrix} f_{1} : U \times A_{1} \to P_{0} (V_{A}^{1}), f_{2} : U \times A_{2} \to P_{0} (V_{A}^{2}), \\ f_{A} : U \times (A_{1} \cup A_{2}) \to P_{0} (V_{A}^{1, 2}), V_{A}^{1, 2} = V_{A}^{1} \cup V_{A}^{2}, \\ f_{A} (x, a) = {\begin{matrix} f_{1} (x, a), a \in A_{1}, \\ f_{2} (x, a), a \in A_{2} . \end{matrix} \\ g_{1} : U \times D_{1} \to P_{0} (V_{D}^{1}), g_{2} : U \times D_{2} \to P_{0} (V_{D}^{2}), \\ g_{D} : U \times (D_{1} \cup D_{2}) \to P_{0} (V_{D}^{1, 2}), V_{D}^{1, 2} = V_{D}^{1} \cup V_{D}^{2}, \\ g_{D} (x, d) = {\begin{matrix} g_{1} (x, d), d \in D_{1}, \\ g_{2} (x, d), d \in D_{2} . \end{matrix} \end{matrix}$

Definition 8. Let (U, A₁, f₁, D₁, g₁) and (U, A₂, f₂, D₂, g₂) be two set-valued decision tables with the same object set. When A₁∩ A₂ ≠ ∅, there exists a ∈ A₁ ∩ A₂. Then, we have two kinds of methods to compound:

The first one, in order to keep information in an integral state, is defined as follows: $\begin{matrix} f_{A}^{\cup} (x, a) & = & f_{1} (x, a) \cup f_{2} (x, a), a \in A_{1} \cap A_{2}; \\ g_{D}^{\cup} (x, d) & = & g_{1} (x, d) \cup g_{2} (x, d), d \in D_{1} \cap D_{2}; \end{matrix}$

The second one, in order to reflect the core of the information, is defined as follows: $\begin{matrix} f_{A}^{\cap} (x, a) & = & f_{1} (x, a) \cap f_{2} (x, a), a \in A_{1} \cap A_{2}; \\ g_{D}^{\cap} (x, d) & = & g_{1} (x, d) \cap g_{2} (x, d), d \in D_{1} \cap D_{2}; \end{matrix}$

Example 2. Tables 5 and 6 are two set-valued decision tables, they represent the features in the two different periods of the object set. We can get Table 7 by compound Tables 5 and 6 in the first method, and Table 7 shows all the attribute values of the object set in the two periods. And we can get Table 8 by compound Tables 5 and 6 in the second method. Table 8 shows the unchanged attribute values of the objects in the two periods.

Table 5

	a ₁	a ₂	a ₃	d ₁	d ₂
x ₁	{1, 2}	{2, 3}	{1, 3, 4}	{2, 3}	{1, 3}
x ₂	{2, 4}	{2, 3, 4}	{3}	{3}	{1, 2}
x ₃	{3, 4}	{1, 2}	{1, 4}	{1, 2}	{2, 3}

Table 6

a ₂	a ₃	a ₄	d ₂	d ₃
x ₁	{2, 4}	{1, 4}	{2, 4}	{2, 3}	{3}
x ₂	{1, 3}	{2, 3}	{1, 2, 3}	{1, 3}	{2}
x ₃	{1, 4}	{1, 2, 4}	{2, 3, 4}	{1, 3}	{2}

Table 7

	a ₁	a ₂	a ₃	a ₄	d ₁	d ₂	d ₃
x ₁	{1, 2}	{2, 3, 4}	{1, 3, 4}	{2, 4}	{2, 3}	{1, 2, 3}	{3}
x ₂	{2, 4}	{1, 2, 3, 4}	{2, 3}	{1, 2, 3}	{3}	{1, 2, 3}	{2}
x ₃	{3, 4}	{1, 2, 4}	{1, 2, 4}	{2, 3, 4}	{1, 2}	{1, 2, 3}	{2}

Table 8

	a ₁	a ₂	a ₃	a ₄	d ₁	d ₂	d ₃
x ₁	{1, 2}	{2}	{1, 4}	{2, 4}	{1, 3}	{3}	{3}
x ₂	{2, 4}	{3}	{3}	{1, 2, 3}	{1, 2}	{1}	{2}
x ₃	{3, 4}	{1}	{1, 4}	{2, 3, 4}	{2, 3}	{3}	{2}

In what follows, we also take the first method as an example to discuss.

Definition 9. Let (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂) be two set-valued decision tables with different object sets and condition attribute sets. When A₁∩ A₂ ≠ ∅, U₁∩ U₂ ≠ ∅. Then, we can compound the two tables step by step.

First, expand (U_i, A_i, f_i, D_i, g_i) to (U_i ∪ U_j, A_i, f_i, D_i, g_i), i ≠ j, i, j ∈ {1, 2}. For x ∈ U_j, x ∉ U_i, ∀a ∈ A_i, d ∈ D_i, denote f_i (x, a) = ∅ , g_i (x, d) =∅.

Then, (U₁ ∪ U₂, A₁, f₁, D₁, g₁) and (U₁ ∪ U₂, A₂, f₂, D₂, g₂) are two set-valued decision tables with the same object set, we can get a new attribute-composed set-valued decision table (U₁ ∪ U₂, A₁ ∪ A₂, f₃, D₁ ∪ D₂, g₃). $\begin{matrix} For \forall x \in U_{1} \cup U_{2}, a \in A_{1} \cup A_{2}, f_{3} (x, a) \\ = {\begin{matrix} f_{i} (x, a); a \in A_{i}, a \notin A_{j}, \\ f_{i} (x, a) \cup f_{j} (x, a); a \in A_{i} \cap A_{j}, \end{matrix} i \neq j . \\ For \forall x \in U_{1} \cup U_{2}, d \in D_{1} \cup D_{2}, g_{3} (x, d) \\ = {\begin{matrix} g_{i} (x, d); d \in D_{i}, d \notin D_{j}, \\ g_{i} (x, d) \cup g_{j} (x, d); d \in D_{i} \cap D_{j}, \end{matrix} i \neq j . \end{matrix}$

4 Three-way decisions of composed set-valued decision tables

Three-way decision theory is a new decision theory which is based on the two-way decision theory, it was proposed by Professor Yao [22, 23]. Compared with two-way decision theory, three-way decisions are more perfect in the theory than two-way decisions, and it has more extensive application areas. This paper will give a brief introduction to the three-way decision theory and we will introduce three-way decision theory to composed set-valued decision tables.

4.1 Three-way decisions

Definition 10. Let U = {x₁, x₂, …, x_n} be a finite and non-empty universe of discourse, C a finite non-empty condition set, and V_C a non-empty value set of C. The core work of three-way decisions is to divide the object set into three disjoint parts based on the condition set. We denote the three parts as domain L, domain M, and domain R. They can be interpreted as the left domain, the middle domain and the right domain, the meaning of them are accept, delay selection and reject, and they satisfy the following conditions: $\begin{matrix} (1) L \cup M \cup R & = & U, (2) L \cap M = \emptyset, \\ L \cap R & = & \emptyset, R \cap M = \emptyset . \end{matrix}$

Generally speaking, these three domains may include empty set, so the set {L, M, R} may not be a partition of U. We call it a partition of U for convenience.

In order to structure these three domains, we need to introduce the definition of decision function. Let t_C : U → V_C be a information function on U. If for any x ∈ U, t_C (x) ∈ V_C, then t_C is called a decision function of condition set C.

4.2 Three-way decisions of set-valued decision tables

Let U be a set of objects, C be a finite condition set of three-way decisions, P₀ (V_C) is the value set of C. The set of objects are divided into three domains (which are domain L_C, domain M_C and domain R_C) by the information function t_C : U → P₀ (V_C). In a decision table, The finite condition set C can be the condition attribute set A or decision attribute set D.

Definition 11. Let (U, A, f, D, g) be a set-valued decision table, D a finite condition set of three-way decisions. If t_D : U → P₀ (V_D) is the information function, and P₀ (V_D) is the value set of D, then t_D is called the decision function of the three-way decisions in a set-valued decision table. Suppose α_D, β_D, γ_D are three nonempty subsets of V_D, and they satisfy the following properties:

α_D ∪ β_D ∪ γ_D = V_D,

α_D ∩ β_D = ∅ , β_D ∩ γ_D = ∅ , γ_D ∩ α_D = ∅ .

Based on the condition set D, the decision function of the three-way decisions can divide the object set into three domains as follows:

If $\frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |}$ and $\frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |}$ , then x ∈ L_D;

If $\frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |} \geq \frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |}$ and $\frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |} \geq \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |}$ , then x ∈ M_D;

If $\frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |}$ and $\frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |}$ , then x ∈ R_D.

In which, $\frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |}, \frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |}, \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |} \in [0, 1] .$

Let A be the finite condition set of three-way decisions, we can obtain the three-way decisions in a similar way.

Definition 12. Denote $^{*} L_{D} = {x \in L_{D} | \frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap (V_{D} - α_{D}) |}{| t_{D} (x) |}}$ , we call it a absolute advantage L–domain. Denote $^{*} R_{D} = {x \in R_{D} | \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap (V_{D} - γ_{D}) |}{| t_{D} (x) |}}$ , we call it a absolute advantage R–domain.

Definition 13. If t_D (x) ⊆ t_D (y) implying $(x, y) \in R_{t_{D}}^{\subseteq}$ , then $R_{t_{D}}^{\subseteq}$ is called a quasi-order relation about t_D on (U, A, f, D, g). Similarly, $R_{t_{A}}^{\subseteq}$ is called a quasi-order relation about t_A on (U, A, f, D, g).

Denote $R_{t_{D}}^{\subseteq} (x) = {y | (x, y) \in R_{t_{D}}^{\subseteq}} = [x]_{R_{t_{D}}^{\subseteq}} .$ It represents the information granule of x under the quasi-order relation $R_{t_{D}}^{\subseteq}$ , which is also known as quasi-order class. Similarly, $R_{t_{A}}^{\subseteq} (x)$ represents the information granule of x under the quasi-order relation $R_{t_{A}}^{\subseteq}$ .

By Theorem 1, $ℛ_{t_{D}}$ is the partition of U generated by the quasi-order relation $R_{t_{D}}^{\subseteq}$ , and $ℛ_{t_{A}}$ is the partition of U generated by the quasi-orderrelation $R_{t_{A}}^{\subseteq}$ .

Definition 14. The set-valued decision table (U, A, f, D, g) is consistent about the three-way decision function t, if $\forall [x]_{R_{t_{A}}^{\subseteq}} \in ℛ_{t_{A}}$ , there exists $[x]_{R_{t_{D}}^{\subseteq}} \in ℛ_{t_{D}}$ satisfying $[x]_{R_{t_{A}}^{\subseteq}} \subseteq [x]_{R_{t_{D}}^{\subseteq}}$ , then $R_{t_{A}}^{\subseteq} \subseteq R_{t_{D}}^{\subseteq} .$

For the convenience of discussion, we denote U = U¹ ∪ U², in which, $\begin{matrix} U^{1} & = & {x \in U | \frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} = 1 \lor \frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |} = 1 \lor \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |} = 1}, \\ U^{2} & = & {x \in U | \frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} \neq 1 \land \frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |} \neq 1 \land \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |} \neq 1} . \end{matrix}$

Theorem 2. Let D be the finite condition set of three-way decisions, t_D the decision function, for x ∈ U¹, the following conclusions hold:

If x ∈ L_D, $(y, x) \in R_{t_{D}}^{\subseteq}$ then y ∈ L_D,

If x ∈ R_D, $(y, x) \in R_{t_{D}}^{\subseteq}$ then y ∈ R_D,

If x ∈ M_D, $(y, x) \in R_{t_{D}}^{\subseteq}$ then y ∈ M_D.

Let A be the finite condition set of three- way decisions, t_A the decision function. For x ∈ U¹, the following conclusions hold:

If x ∈ L_A, $(y, x) \in R_{t_{A}}^{\subseteq}$ then y ∈ L_A,

If x ∈ R_A, $(y, x) \in R_{t_{A}}^{\subseteq}$ then y ∈ R_A,

If x ∈ M_A, $(y, x) \in R_{t_{A}}^{\subseteq}$ then y ∈ M_A.

Proof. (1) If x ∈ L_D, $(y, x) \in R_{t_{D}}^{\subseteq}$ , then t_D (y) ⊆ t_D (x) ⊆ α_D, thus y ∈ L_D. The others can be proved in the same way.

For any x ∈ U², Theorem 2 does not necessarily hold, for example:

If x ∈ L_D, $\frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap β_{D} |}{| t_{D} (x) |}$ and $\frac{| t_{D} (x) \cap α_{D} |}{| t_{D} (x) |} > \frac{| t_{D} (x) \cap γ_{D} |}{| t_{D} (x) |}$ are satisfied, and it holds $(y, x) \in R_{t_{D}}^{\subseteq}$ , thus, t_D (y) ⊆ t_D (x), but it may be t_D (y) ⊆ t_D (x) ∩ β_D, so y ∈ M_D, Theorem 2 no longer holds.

Theorem 3. Let (U, A, f, D, g) be a set-valued decision table. It is consistent about the three-way decision function t. When α_A = α_D, for any x ∈ U¹, if all the information granules of x under the decision function t_D are included in domain L_D, then x belongs to domain L_A.

Proof. Since the set-valued decision table is consistent about t, so $R_{t_{A}}^{\subseteq} (x) \subseteq R_{t_{D}}^{\subseteq} (x)$ . If $R_{t_{D}}^{\subseteq} (x) \subseteq L_{D}$ , then t_A (x) ⊆ t_A (y) ⊆ t_D (y) ⊆ α_D, and $y \in R_{t_{D}}^{\subseteq} (x)$ α_A = α_D, so x ∈ L_A.

When β_A = β_D, γ_A = γ_D, similar results can be got.

4.3 Three-way decisions of object-composed set-valued decision tables

Let (U₁, A, f₁, D, g₁) and (U₂, A, f₂, D, g₂) be two set-valued decision tables with the same condition attribute set and the same decision attribute set D is the finite condition set, their value sets are $P_{0} (V_{D}^{1})$ and $P_{0} (V_{D}^{2}) .$ (U₁ ∪ U₂, A, f_U, D, g_U) is the object-composed set-valued decision table, the value set of D is $P_{0} (V_{D}^{1, 2})$ , $V_{D}^{1, 2} = V_{D}^{1} \cup V_{D}^{2}$ , we can see which was changed in the composed set-valued decision table, so threshold values of composed set-valued decision table should be changed.

(U_i, A, f_i, D, g_i) are two set-valued decision tables, i = 1, 2 . D is the finite condition set, $P_{0} (V_{D}^{i})$ is the value set of D. The three-way decision function is $t_{D}^{i} : U_{i} \to P_{0} (V_{D}^{i})$ . Let $α_{D}^{i}$ $β_{D}^{i}$ $γ_{D}^{i}$ be three nonempty subsets of $V_{D}^{i}$ , satisfying the following conditions:

$α_{D}^{i} \cup β_{D}^{i} \cup γ_{D}^{i} = V_{D}^{i},$

$α_{D}^{i} \cap β_{D}^{i} = \emptyset, β_{D}^{i} \cap γ_{D}^{i} = \emptyset, γ_{D}^{i} \cap α_{D}^{i} = \emptyset .$

(U₁ ∪ U₂, A, f_U, D, g_U) is the object-composed set-valued decision table of (U₁, A, f₁, D, g₁) and (U₂, A, f₂, D, g₂), D is the finite condition set of three-way decisions, $P_{0} (V_{D}^{1, 2})$ is the value set of D. The three-way decision function is $t_{D}^{U} : U_{1} \cup U_{2} \to P_{0} (V_{D}^{1, 2})$ , $V_{D}^{1, 2} = V_{D}^{1} \cup V_{D}^{2}$ . Let $α_{D}^{1, 2}$ $β_{D}^{1, 2}$ $γ_{D}^{1, 2}$ be three nonempty subsets of $V_{D}^{1, 2}$ , and they satisfy the following conditions:

$α_{D}^{1, 2} \cup β_{D}^{1, 2} \cup γ_{D}^{1, 2} = V_{D}^{1, 2},$

$α_{D}^{1, 2} \cap β_{D}^{1, 2} = \emptyset . β_{D}^{1, 2} \cap γ_{D}^{1, 2} = \emptyset, γ_{D}^{1, 2} \cap α_{D}^{1, 2} = \emptyset .$

In which, $α_{D}^{1, 2} = α_{D}^{1} \cup α_{D}^{2}$ , $γ_{D}^{1, 2} = γ_{D}^{1} \cup γ_{D}^{2}$ , $β_{D}^{1, 2} = V_{D}^{1, 2} - α_{D}^{1, 2} - γ_{D}^{1, 2}$ .

The composed three-way decision function is: $t_{D}^{U} (x) = {\begin{matrix} t_{D}^{1} (x), x \in U_{1} \land x \notin U_{2}, \\ t_{D}^{2} (x), x \in U_{2} \land x \notin U_{1}, \\ t_{D}^{1} (x) \cup t_{D}^{2} (x), x \in U_{1} \cap U_{2} . \end{matrix}$

Based on the condition set and new threshold values, the object set can be divided into three domains (that are $L_{D}^{U}, M_{D}^{U}, R_{D}^{U}$ ) by the three-way decision function.

Let A be the finite condition set, the three-way decisions of object-composed set-valued decision tables can be got in a similar way.

Note: In this paper, we only discuss the case $α_{D}^{i} \cap γ_{D}^{j} = \emptyset, i \neq j$ , for the case of $α_{D}^{i} \cap γ_{D}^{j} \neq \emptyset, i \neq j$ , we think it is contradictory.

4.4 Three-way decisions of attribute-composed set-valued decision tables

Similar to the three-way decisions of object-composed set-valued decision tables, we can get the three-way decisions of attribute-composed set-valued decision tables in the same way.

Let (U, A₁, f₁, D₁, g₁) be a set-valued decision table, in which D₁ is the finite condition set. The three-way decision function is t_{D
₁} : U → P₀ (V_{D
₁}), P₀ (V_{D
₁}) is the value set of D₁. Let α_{D
₁}, β_{D
₁}, γ_{D
₁} be three nonempty subsets of V_{D
₁}, and they satisfy the following conditions:

α_{D
₁} ∪ β_{D
₁} ∪ γ_{D
₁} = V_{D
₁},

α_{D
₁} ∩ β_{D
₁} = ∅ . β_{D
₁} ∩ γ_{D
₁} = ∅ , γ_{D
₁} ∩ α_{D
₁} = ∅ .

Let (U, A₂, f₂, D₂, g₂) be a set-valued decision table, in which D₂ is finite condition set. The three-way decision function is t_{D
₂} : U → P₀ (V_{D
₂}), P₀ (V_{D
₂}) is the value set of D₂. Let α_{D
₂}, β_{D
₂}, γ_{D
₂} be three nonempty subsets of V_{D
₂}, and they satisfy the following conditions:

α_{D
₂} ∪ β_{D
₂} ∪ γ_{D
₂} = V_{D
₂},

α_{D
₂} ∩ β_{D
₂} = ∅ , β_{D
₂} ∩ γ_{D
₂} = ∅ , γ_{D
₂} ∩ α_{D
₂} = ∅ .

(U, A₁ ∪ A₂, f_A, D₁ ∪ D₂, g_D) is the attribute-composed set-valued decision table of (U, A₁, f₁, D₁, g₁) and (U, A₂, f₂, D₂, g₂), D₁ ∪ D₂ is the finite condition set, V_{D
_1,2} = V_{D
₁} ∪ V_{D
₂}, P₀ (V_{D
_1,2}) is the value set of D₁ ∪ D₂. The three-way decision function is t_{D
_1,2} : U → P₀ (V_{D
_1,2}). Let α_{D
_1,2}, β_{D
_1,2}, γ_{D
_1,2} be threenonempty subsets of V_{D
_1,2}, and they satisfy the following properties:

α_{D
_1,2} ∪ β_{D
_1,2} ∪ γ_{D
_1,2} = V_{D
_1,2},

α_{D
_1,2} ∩ β_{D
_1,2} = ∅ , β_{D
_1,2} ∩ γ_{D
_1,2} = ∅ , γ_{D
_1,2} ∩ α_{D
_1,2} = ∅ .

In which, α_{D
_1,2} = α_{D
₁} ∪ α_{D
₂}, γ_{D
_1,2} = γ_{D
₁} ∪ γ_{D
₂}, β_{D
_1,2} = V_{D
_1,2} - α_{D
_1,2} - γ_{D
_1,2}.

The three-way decision function of composed decision table is: $\begin{matrix} t_{D_{1, 2}} (x) \\ = {\begin{matrix} t_{D_{1}} (x), when t_{D_{1}} (x) \neq \emptyset and t_{D_{2}} (x) = \emptyset, \\ t_{D_{2}} (x), when t_{D_{1}} (x) = \emptyset and t_{D_{2}} (x) \neq \emptyset, \\ t_{D_{1}} (x) \cup t_{D_{2}} (x), when t_{D_{1}} (x) \neq \emptyset and t_{D_{2}} (x) \neq \emptyset . \end{matrix} \end{matrix}$

Let A be the finite condition set, three-way decision rules of attribute-composed set-valued decision tables can be got in a similar way.

Note: In this paper, we only discuss the case of α_{D_i} ∩ γ_{D_j} = ∅ , i ≠ j, for the case of α_{D_i} ∩ γ_{D_j} ≠ ∅ , i ≠ j, we think it is contradictory.

Let (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂) be two set-valued decision tables, U₁∩ U₂ ≠ ∅, A₁ ∩ A₂ ≠ ∅ , D_i two finite attribute condition sets, t_{D_i} two three-way decision functions, i = 1, 2 . First, expand (U_i, A_i, f_i, D_i, g_i) to (U_i ∪ U_j, A_i, f_i, D_i, g_i) for x ∈ U_i and x ∉ U_j, denote t_{D_i} (x) = ∅ , i ≠ j. Then, we can get (U₁ ∪ U₂, A₁ ∪ A₂, f₃, D₁ ∪ D₂, g₃) by compound (U₁ ∪ U₂, A₁, f₁, D₁, g₁) and (U₁ ∪ U₂, A₂, f₂, D₂, g₂).

According to the definition of three-way decisions of attribute-composed set-valued decision tables we can get the three-way decisions of (U₁ ∪ U₂, A₁ ∪ A₂, f₃, D₁ ∪ D₂, g₃).

Theorem 4. Let (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2) be the set-valued decision table composed by (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂), A_i finite attribute condition sets of three-way decisions, t_i two three-way decision functions, i = 1, 2. For x ∈ U₁ ∩ U₂, if x ∈ ^*L₁ and x ∈ ^*L₂, then x ∈ L_1,2.

In which, ^*L_i is the absolute advantage L–domain of (U_i, A_i, f_i, D_i, g_i), i = 1, 2, and L_1,2 is the domain L of (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2).

Proof. By Definition 10, first, we prove $\frac{| t_{12} (x) \cap α_{12} |}{| t_{12} (x) |} > \frac{| t_{12} (x) \cap γ_{12} |}{| t_{12} (x) |}$ as (1):

(1) Since (t_i (x) ∩ α_i) ⊃ (t_i (x) ∩ (V_{D_i} - α_i)), and α_i∩ γ_j = ∅, then (γ_j ∩ (V_{D_i} - α_i)) ⊂ (V_{D_i} - α_i). Thus, t₁ (x)∩ (α₁ ∪ α₂) ⊇ (t₁ (x) ∩ α₁) ⊃ (t₁ (x) ∩ γ₁₂), t₂ (x) ∩ (α₁ ∪ α₂) ⊇ (t₂ (x) ∩ α₂) ⊃ (t₂ (x) ∩ γ₁₂). So, we have (t₁ (x) ∪ t₂ (x)) ∩ (α₁ ∪ α₂) ⊃ (t₁ (x) ∩ γ₁₂), t₁₂ (x) ∩ α₁₂ ⊃ (t₂ (x) ∩ γ₁₂). That is t₁₂ (x) ∩ α₁₂ ⊃ t₁₂ (x) ∩ γ₁₂, $\frac{| t_{12} (x) \cap α_{12} |}{| t_{12} (x) |} > \frac{| t_{12} (x) \cap γ_{12} |}{| t_{12} (x) |}$ .

Then we prove $\frac{| t_{12} (x) \cap α_{12} |}{| t_{12} (x) |} > \frac{| t_{12} (x) \cap β_{12} |}{| t_{12} (x) |}$ as (2):

(2) By t₁ (x) ∪ t₂ (x) ∩ (α₁ ∪ α₂) ⊇ t₁ (x) ∩ α₁ ∪ (t₂ (x) ∩ α₂) ⊃ (t₁ (x) ∩ β₁) ∪ (t₂ (x) ∩ β₂) ⊃ (t₁ (x) ∩ β_1,2) ∪ (t₂ (x) ∩ β_1,2) =(t₁ (x) ∪ t₂ (x)) ∩ β₁₂ = t₁₂ (x) ∩ β₁₂, and t₁ (x) ∪ t₂ (x) ∩ (α₁ ∪ α₂) = t₁₂ (x) ∩ α₁₂, then $\frac{| t_{12} (x) \cap α_{12} |}{| t_{12} (x) |} > \frac{| t_{12} (x) \cap β_{12} |}{| t_{12} (x) |}$ .

In summary, x ∈ L₁₂.

If D is the finite condition set, similar results can be got.

Theorem 5. Let (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2) be the set-valued decision table composed by (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂) , A_i finite condition sets of three-way decisions, t_i two three-way decision functions, i = 1, 2. For x ∈ U₁ ∩ U₂, if x ∈ ^*R₁ and x ∈ ^*R₂, then x ∈ R_1,2.

In which, ^*R_i is the absolute advantage R–domain of (U_i, A_i, f_i, D_i, g_i) , i = 1, 2, R₁₂ is the domain R of (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2).

Proof. It is similar to Theorem 4.

Theorem 6. Let (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2) be the set-valued decision table composed by (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂), A the finite attribute condition set, t_i, i = 1, 2 two three-way decision functions. For x ∈ U_1,2, if x ∈ M_1,2, then x ∈ M₁ or x ∈ M₂.

In which, M_i is the domain M of (U_i, A_i, f_i, D_i, g_i), i = 1, 2, M_1,2 is the domain M of (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2).

Proof. Proof by contradiction, suppose x ∉ M₁ and x ∉ M₂, we might as well let x ∈ L₁ and x ∈ L₂. Since t₁ (x) ∩ β₁ ⊆ t₁ (x) ∩ α₁ ⊆ t₁ (x) ∩ α_1,2, t₂ (x) ∩ β₂ ⊆ t₂ (x) ∩ α₂ ⊆ t₂ (x) ∩ α_1,2, then t_1,2 (x) ∩ β_1,2 ⊆ t_1,2 (x) ∩ α_1,2. It is contrary to x ∈ M_1,2.

If D is the finite condition set, similar results can be got.

Theorem 7. Let (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2) be the set-valued decision table composed by (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂), A the finite attribute condition set, t_i, i = 1, 2, two three-way decision functions. For x ∈ U_1,2, if $\frac{| t_{1} (x) \cap α_{1} |}{| t_{1} (x) |} = 1, x \in L_{1}$ and $\frac{| t_{2} (x) \cap α_{2} |}{| t_{2} (x) |} = 1, x \in L_{2}$ , then $\frac{| t_{1, 2} (x) \cap α_{1, 2} |}{| t_{1, 2} (x) |} = 1, x \in L_{1, 2} .$

In which, L_i is the domain L of (U_i, A_i, f_i, D_i, g_i), i = 1, 2,L_1,2 is the domain L of (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2).

Proof. For x ∈ U₁, since $\frac{| t_{1} (x) \cap α_{1} |}{| t_{1} (x) |} = 1$ , that is, t₁ (x) ⊆ α₁. If x ∉ U₂, then t₂ (x) =∅. If x ∈ U₂ and $\frac{| t_{2} (x) \cap α_{2} |}{| t_{2} (x) |} = 1$ , that is, t₂ (x) ⊆ α₂. And as well t_1,2 (x) = t₁ (x) ∪ t₂ (x) , α_1,2 = α₁ ∪ α₂, thus, t_1,2 (x) ⊆ α_1,2.

Let D be the finite condition set, similar results can be got.

Theorem 8. Let (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2) be the set-valued decision table composed by (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂), A the finite attribute condition set, t_i, i = 1, 2, two three-way decision functions. For x ∈ U_1,2, if $\frac{| t_{1} (x) \cap γ_{1} |}{| t_{1} (x) |} = 1, x \in R_{1},$ and $\frac{| t_{2} (x) \cap γ_{2} |}{| t_{2} (x) |} = 1, x \in R_{2},$ then $\frac{| t_{1, 2} (x) \cap γ_{1, 2} |}{| t_{1, 2} (x) |} = 1, x \in R_{1, 2} .$

In which, R_i is the domain R of (U_i, A_i, f_i, D_i, g_i), i = 1, 2, R_1,2 is the domain R of (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2).

Proof. It is similar to Theorem 7. If D is the finite condition set, similar results can be got.

Theorem 9. Let (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2) be the set-valued decision table composed by (U₁, A₁, f₁, D₁, g₁) and (U₂, A₂, f₂, D₂, g₂), A the finite attribute condition set, t_i, i = 1, 2, two three-way decision functions. For x ∈ U_1,2, if $\frac{| t_{1, 2} (x) \cap β_{1, 2} |}{| t_{1, 2} (x) |} = 1, x \in M_{1, 2},$ then $\frac{| t_{1} (x) \cap β_{1} |}{| t_{1} (x) |} = 1, x \in M_{1},$ and $\frac{| t_{2} (x) \cap β_{2} |}{| t_{2} (x) |} = 1, x \in M_{2} .$

In which, M_i is the domain M of (U_i, A_i, f_i, D_i, g_i), i = 1, 2, M_1,2 is the domain M of (U_1,2, A_1,2, f_1,2, D_1,2, g_1,2).

Proof. Since $\frac{| t_{1, 2} (x) \cap β_{1, 2} |}{| t_{1, 2} (x) |} = 1$ , x ∈ M_1,2, we have t_1,2 (x) ⊆ β_1,2 ⊆ β₁ and t_1,2 (x) ⊆ β_1,2 ⊆ β₂ . And as well t_1,2 (x) = t₁ (x) ∪ t₂ (x), so t₁ (x) ⊆ β₁ and t₂ (x) ⊆ β₂.

If D is the finite condition set, similar results can be got.

Definition 15. If L_A ⊆ L_D, the set-valued decision table (U, A, f, D, g) is said to be L–consistent, if M_A ⊆ M_D, the set-valued decision table (U, A, f, D, g) is said to be M–consistent, if R_A ⊆ R_D, the set-valued decision table (U, A, f, D, g) is said to be R–consistent.

Theorem 10. Let $(U_{1}^{1}, A_{1}, f_{1}, D_{1}, g_{1})$ and $(U_{2}^{1}, A_{2}, f_{2}, D_{2}, g_{2})$ be two set-valued decision tables, $(U_{1, 2}^{1}, A_{1, 2}, f_{1, 2}, D_{1, 2}, g_{1, 2})$ the composed set-valued decision table. If $(U_{1}^{1}, A_{1}, f_{1}, D_{1}, g_{1})$ is L–consistent and $(U_{2}^{1}, A_{2}, f_{2}, D_{2}, g_{2})$ is L–consistent, then $(U_{1, 2}^{1}, A_{1, 2}, f_{1, 2}, D_{1, 2}, g_{1, 2})$ is L–consistent.

Proof. It can be proved by Theorem 7.

Theorem 11. Let $(U_{1}^{1}, A_{1}, f_{1}, D_{1}, g_{1})$ and $(U_{2}^{1}, A_{2}, f_{2}, D_{2}, g_{2})$ be two set-valued decision tables, $(U_{1, 2}^{1}, A_{1, 2}, f_{1, 2}, D_{1, 2}, g_{1, 2})$ the composed set-valued decision table. If $(U_{1}^{1}, A_{1}, f_{1}, D_{1}, g_{1})$ is R–consistent and $(U_{2}^{1}, A_{2}, f_{2}, D_{2}, g_{2})$ is R–consistent, then $(U_{1, 2}^{1}, A_{1, 2}, f_{1, 2}, D_{1, 2}, g_{1, 2})$ is R–consistent.

Proof. It can be proved by Theorem 8.

Theorem 12. Let $(U_{1}^{1}, A_{1}, f_{1}, D_{1}, g_{1})$ and $(U_{2}^{1}, A_{2}, f_{2}, D_{2}, g_{2})$ be two set-valued decision tables, $(U_{1, 2}^{1}, A_{1, 2}, f_{1, 2}, D_{1, 2}, g_{1, 2})$ the composed set-valued decision table. If $(U_{1, 2}^{1}, A_{1, 2}, f_{1, 2}, D_{1, 2}, g_{1, 2})$ is M–consistent, then $(U_{1}^{1}, A_{1}, f_{1}, D_{1}, g_{1})$ is M–consistent and $(U_{2}^{1}, A_{2}, f_{2}, D_{2}, g_{2})$ is M–consistent.

Proof. It can be proved by Theorem 9.

5 Conclusion

In this paper, we studied the composed set-valued decision tables and we defined three-way decisions of set-valued decision tables. Further more, we discussed the properties of three-way decisions of set-valued decision tables. Threshold values of three-way decisions are changed after compounded the decision tables, the new threshold values expand the scope of domain L and domain R, reduced the domain M, this makes the original object which is not able to make clear decision of accept or refuse gets a more clear decision.

Footnotes

Acknowledgments

This work is supported by grants from National Natural Science Foundation of China under Grant (61573127, 61502144, 61300121), Natural Science Foundation of Hebei Province under Grant (A2014205157), Training Program for Leading Talents of Innovation Teams in the Universities of of Hebei Province under Grant (LIRC022), Natural Science Foundation of Higher Education Institutions of Hebei under Grant (QN2016133), Doctor Science Foundation of Hebei Normal University under Grant (L2015B01), Graduate student innovation fund project of Hebei Province Office of Education under Grant (sj2015001), Graduate student innovation fund project of College of Mathematics and Information Science of Hebei Normal University.

References

Pawlak

, Rough sets, International Journal of Computer and Information Science 11 (1982), 341–356.

Jackson

A.G.

, Pawlak

and LeClair

S.R.

, Rough sets applied to the discovery of materials knowledge, Journal of Alloys and Compounds 279 (1998), 14–21.

Skowron

and Rauszer

, The discernibility matrices and functions in information systems. Boston:Kluwer Academic Publishers, (1992), 331–362.

Wong

S.K.M.

and Ziarko

, On optional decision rules in decision tables, Bulletin of Polish Academy of Science 33 (1985), 693–696.

X.H.

and Cercone

, Learning in relational database:a rough set approach, Computation Intelligence 11(2) (1995), 323–338.

Chang

L.Y.

, Wang

G.Y.

and Wu

, A method of attribute reduction and rule extraction based on rough set theory, Journal of Software 10(11) (1999), 1206–1211.

Zhang

W.X.

, Qiu

G.F.

and Wu

W.Z.

, General Theory of attribute reduction in rough set, Information Science 35(12) (2005), 1304–1313.

Wang

G.Y.

, The extension of rough set theory in incomplete information system rough, Journal of Computer Research and Development 39(10) (2002), 1238–1243.

Song

X.X.

, Xie

Z.G.

and Zhang

W.X.

, Knowledge reduction and rule extraction in set-valued decision information systems, Computer Science 34(4) (2007), 182–184, 191.

10.

Zhang

W.X.

, Liang

and Wu

W.Z.

, Information systems and knowledge discovery. Beijing: Science Press, 2003.

11.

X.K.

, Knowledge reduction and attribute characteristics of set valued information system, Microcomputer and Its Applications 24(32) (2015), 14–15.

12.

Tao

and Yue

M.X.

, Variable precision rough set model of dominance relation set-valued systems, Journal of Civil Aviation University of China 33(1) (2015), 55–58.

13.

Song

X.X.

, Li

H.R.

and Zhang

W.X.

, Knowledge reduction and attribute characteristics of set-valued information systems, Computer Engineering 32(22) (2006), 26–27, 36.

14.

Shao

M.W.

, Zhang

W.X.

and Mi

J.S.

, Combined information systems and subinformation systems, Computer Science 31(3) (2004), 137–140.

15.

Chan

C.C.

, A rough set approach to attribution generalization in data mining, Information Science 107 (1998), 169–176.

16.

Shao

M.W.

and Long

W.J.

, Attribute characteristics of combined information systems and subinformation systems, Computer Engineering 33(17) (2007), 7–9.

17.

J.M.

, Pan

X.C.

and Zhang

W.X.

, Attribute characteristics of composed set-valued information systems, Fuzzy Systems and Mathematics 29(3) (2015), 174–180.

18.

Qiu

W.G.

and Hu

Z.B.

, Rough set model of combined information system based on random sets, Computer Engineering 37(9) (2011), 210–212.

19.

Cheng

H.H.

, Zhang

X.Q.

, Li

F.J.

and Qian

Y.H.

, Decision table attribute reduction algorithm based on correspondence constraints, Computer Science 42(6) (2015), 50–53.

20.

Cheng

Z.C.

, Liu

P.H.

and Guo

C.F.

, Attribute reduction in interval ordered decision table based on fuzzy dominance relation, Fuzzy Systems and Mathematics 27(6) (2013), 167–175.

21.

, Wang

J.H.

, Liang

J.Y.

, Mi

and Dang

C.Y.

, Compacted decision tables based attribute reduction, Knowledge-Based Systems 86 (2015), 261–277.

22.

Yao

Y.Y.

, Wong

S.K.M.

, Lingras

, A decision-theoretic rough set model, in: Ras

Z.W.

, Zemankova

, Emrich

M.L.

(Eds.), Methodologies for Intelligent Systems, North-Holland, New York, (1990) 17–24.

23.

Yao

Y.Y.

and Wong

S.K.M.

, A decision theoretic framework for approximating concepts, International Journal of Man-Machine Studies 37(6) (1992), 793–809.

24.

Yao

Y.Y.

, Probabilistic rough set approximation, International Journal of Approximate Reasoning (49) (2008), 255–271.

25.

Yao

Y.Y.

, Three-way decisions with probabilistic rough sets, Information Science, (180) (2010), 341–353.

26.

Yao

Y.Y.

and Zhou

, Naive bayesian rough sets, Lecture Notes in Computer Science, Springer, Berlin, 6401(4) (2010), 719–726.

27.

Yao

Y.Y.

, The superiority of three-way decision in probabilistic rough set models, Information Science 181(6) (2011), 1080–1096.

28.

Yao

Y.Y.

, Three-way decisions using rough sets. Rough Sets: Selected Methods and Applications in Management and Engineering, Advanced Information and Knowledge Processing, (2012), 79–93.

29.

Yao

Y.Y.

, An outline of a theory of three-way decisions. Rough Sets and Current Trends in Computing, Leture Notes in Computer Science. Springer-Verlag, Berlin, (2012), 7413:1–7417.

30.

Xiao

Y.C.

, Hu

B.Q.

and Zhao

X.R.

, Three-way decisions based on type-2 fuzzy sets and interval-valued type-2 fuzzy sets, Journal of Intelligent & Fuzzy Systems 31 (2016), 1385–1395.

31.

Dai

J.H.

, Zheng

G.J.

, Hu

Q.H.

, Liu

M.F.

and Su

H.S.

, Decision-theoretic rough set approach for fuzzy decisions based on fuzzy probability measure and decision making, Journal of Intelligent & Fuzzy Systems 31 (2016), 1341–1353.

32.

Jia

X.Y.

, Zheng

, Li

W.W.

, Liu

T.T.

and Shang

, Three-way decisions solution to spam filter email: An empirical study, Rough Sets and Current Trends in Computing, Leture Notes in Computer Science. Springer-Verlag. Berlin 7413 (2012), 287–296.

33.

, Zhang

and Swan

J.R.

, An information filtering model on the web and its application in JobAgent, Knowledge-Based Systems 13 (2000), 285–296.

34.

, Miao

D.Q.

, Wang

W.L.

, Zhang

, Hierarchical rough decision theoretic framework for text classification, in: Proceedings of the 9th IEEE International Conference on Cognitive Informatics 2010, 484–489.

35.

Liu

, Li

T.R.

and Liang

D.C.

, Three-way government decision analysis with decision-theoretic rough sets, International Journal of Uncertainty, Fuzziness Knowledge-Based Systems 20(1) (2012), 119–132.

Three-way decisions for composed set-valued decision tables

Abstract

Keywords

1 Introduction

2 Set-valued decision tables

3 Composed set-valued decision tables

Table 1 a 1 a 2 a 3 d 1 x 1 {1, 2} {1, 3} {3, 4} {1, 2} x 2 {1, 2, 4} {2, 3} {1} {1, 3} x 3 {1, 3} {2, 3} {2, 4} {2, 3}

4.1 Three-way decisions

4.2 Three-way decisions of set-valued decision tables

4.3 Three-way decisions of object-composed set-valued decision tables

4.4 Three-way decisions of attribute-composed set-valued decision tables

5 Conclusion

Footnotes

Acknowledgments

References

Table 1

a ₁ a ₂ a ₃ d ₁

x ₁ {1, 2} {1, 3} {3, 4} {1, 2}

x ₂ {1, 2, 4} {2, 3} {1} {1, 3}

x ₃ {1, 3} {2, 3} {2, 4} {2, 3}