Attribute reductions in an inconsistent decision information system

Abstract

The study of attribute reductions is one of the main problems in information systems. In this paper, multi-granulation variable precision rough set (for short, MVPRS) model is introduced. Several kinds of attribute reduction in an inconsistent decision information system based on this model are proposed. Relationships among these attribute reductions given. Some examples are given to illustrate that the unmentioned relationships among these attribute reductions are often not maintained.

Keywords

Multi-granulation MVPRS inconsistent decision information system distribution reduction

1 Introduction

Rough set theory, proposed by Pawlak [13], is a new mathematical tool for data reasoning. It may be seen as an extension of classical set theory and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [14 –17].

Rough sets are based on an assumption that every object in the universe is associated with some information. Objects characterized by the same information are indiscernible with the available information about them. The indiscernibility relation generated in this way is the mathematical basis for rough set theory. Rough set theory has used successfully in the analysis of data in information systems.

In management decisions, we face with a lot of inconsistent decision information system. Researching decision problems based on this kind of information system has more practical significance. Thus, researching inconsistent decision information systems is very important.

Fuzzy decision information systems are inconsistent decision information systems under the fuzzy environment. There are some research achievements on fuzzy decision information systems [2, 5]. Besides, Ge et al. [3] presented quick general reduction algorithms for inconsistent decision information systems, Lang et al. [9] investigated characteristic matrixes-based knowledge reduction in dynamic covering decision information systems, Lang et al. [8] proposed related families-based attribute reduction of dynamic covering decision information systems, Tan et al. [18] researched matrix-based set approximations and reductions in covering decision information systems, Hu et al. [4] considered rough sets in distributed decision information systems, Dai et al. [1] studied dominance-based fuzzy rough set approach for incomplete interval-valued information systems, Li et al. [11] gave a multi-granulation decision-theoretic rough set method for distributed fc-decision information systems and Zhang et al. [20] discussed information structures and uncertainty measures in a fully fuzzy information system.

The purpose of this paper is to investigate knowledge reduction in an inconsistent decision information system. The remaining part of this paper is organized as follows. In Section 2, we recall basic concepts about inconsistent decision information systems and variable precision rough sets. In Section 3, we introduce MVPRS model. In Sections 4, we investigate mα-approximation in an inconsistent decision information system based on MVPRS model. In Section 5, we propose multi-distribution reductions in an inconsistent decision information system. In Section 6, we establish relationships among four sorts of reductions in an inconsistent decision information system. In Sections 7, we summarize this paper.

2 Preliminaries

In this section, we recall basic concepts about inconsistent decision information systems and variable precision rough sets.

Throughout this paper, “multi-granulation α" and “multi-granulation variable precision rough set" denote briefly by “mα" and “MVPRS", respectively.

2.1 Inconsistent decision information systems

A decision information system means the pair S = (U, A, F, d), where U is a finite nonempty set of objects, A is a finite nonempty set of condition attributes, d is a decision attribute, F = {f_a : f_a is a mapping from U to V_a, a ∈ A} is the relation set between U and A where V_a is domain values of the attribute a.

Let S = (U, A, F, d) be a decision information system. Denote $R_{B} = {(x, y) : \forall a \in B, f_{a} (x) = f_{a} (y)} (B \subseteq A),$ $R_{a} = R_{{a}} (a \in A);$ $R_{d} = {(x, y) : d (x) = d (y)} .$

Then R_B and R_d are two equivalence relations on U, which are called the equivalence relations induced by B and d, respectively. If R_A ⊆ R_d, then S is called a coordinate decision information system. If R_AnotsubseteqR_d, then S = (U, A, F, d) is called an inconsistent decision information system (for short, inconsistent decision information system).

Denote $U / R_{B} = {[x]_{B} : x \in U}, U / R_{d} = {[x]_{d} : x \in U},$ where $[x]_{B} = {y \in U : (x, y) \in R_{B}},$ $[x]_{d} = {y \in U : (x, y) \in R_{d}} .$ Then U/R_B and U/R_d are the partitions induced by R_B and R_d, respectively.

Example 2.1. Table 1 gives an inconsistent decision information system where U = {x₁, x₂, ⋯, x₁₀}, A = {a₁, a₂, a₃}.

U/R_A = {{x₁, x₃}, {x₂}, {x₄}, {x₅}, {x₆}, {x₇}, {x₈}, {x₉}, {x₁₀}},

U/R_d = {{x₁, x₂, x₄, x₆, x₈}, {x₃, x₅, x₇, x₉, x₁₀}}.

Table 1
An inconsistent decision information system

U	a ₁	a ₂	a₃	d
x₁	0	0	0	0
x₂	3	3	1	0
x₃	0	0	0	1
x₄	2	2	1	0
x₅	1	1	3	1
x₆	1	1	2	0
x₇	3	1	2	1
x₈	1	2	2	0
x₉	2	2	3	1
x₁₀	2	3	3	1

2.2 Variable precision rough sets

Definition 2.2. Let U be a non-empty set. For X, Y ⊆ U, D (X/Y) is called an inclusion degree if it satisfy the following conditions:

(1) 0 ⩽ D (X/Y) ⩽1;

(2) If Y ⊆ X, then D (X/Y) =1;

(3) If X ⊆ Y ⊆ Z, then D (X/Z) ⩽ D (X/Y). Then D (X/Y) is called the degree that X includes Y.

In this paper, we pick $D (Y / X) = \frac{∣ Y \cap X ∣}{∣ X ∣} .$ It is easy to check that D is an inclusion degree.

Definition 2.3. Let S = (U, A, F, d) be an inconsistent decision information system. For any B ⊆ A, α ∈ (0.5, 1] and X ∈ 2^U, define ${\underline{R}}_{B}^{α} (X) = {x : D (X / [x]_{B}) ⩾ α},$ ${\bar{R}}_{B}^{α} (X) = {x : D (X / [x]_{B}) > 1 - α} .$ Then ${\underline{R}}_{B}^{α} (X), {\bar{R}}_{B}^{α} (X)$ are called the α-lower and α-upper approximation of X on B, respectively.

For any a ∈ A, denote

${\underline{R}}_{a}^{α} (X) = {\underline{R}}_{{a}}^{α} (X),$ ${\bar{R}}_{a}^{α} (X) = {\bar{R}}_{{a}}^{α} (X) .$

Proposition 2.4. [21] Let S = (U, A, F, d) be an inconsistent decision information system. Given α ∈ (0.5, 1]. Then the following properties hold:

(1) ${\underline{R}}_{B}^{α} (U - X) = U - {\bar{R}}_{B}^{α} (X)$ . (2) ${\bar{R}}_{B}^{α} (U - X) = U - {\underline{R}}_{B}^{α} (X)$ . (3) ${\underline{R}}_{B}^{α} (X) \subseteq {\bar{R}}_{B}^{α} (X)$ . (4) ${\underline{R}}_{B}^{α} (X) = {\underline{R}}_{B}^{α} ({\underline{R}}_{B}^{α} (X))$ , ${\bar{R}}_{B}^{α} (X) = {\bar{R}}^{α} ({\bar{R}}_{B}^{α} (X))$ . (5) If X₁ ⊆ X₂, then ${\underline{R}}_{B}^{α} (X_{1}) \subseteq {\bar{R}}_{B}^{α} (X_{2})$ .

3 MVPRS model

In this section, we introduce MVPRS model.

Definition 3.1. Let S = (U, A, F, d) be an inconsistent decision information system. For any B ⊆ A, α ∈ (0.5, 1] and X ∈ 2^U, define ${\underline{R}}_{B}^{m α} (X) = {x : \forall a \in B, D (X / [x]_{a}) ⩾ α},$ ${\bar{R}}_{B}^{m α} (X) = {x : \exists a \in B, D (X / [x]_{a}) > 1 - α} .$ Then ${\underline{R}}_{B}^{m α} (X)$ and ${\bar{R}}_{B}^{m α} (X)$ are called the mα-lower approximation and mα-upper approximation of X on B, respectively.

For any a ∈ A, denote ${\underline{R}}_{a}^{m α} (X) = {\underline{R}}_{{a}}^{m α} (X), {\bar{R}}_{a}^{m α} (X) = {\bar{R}}_{{a}}^{m α} (X) .$

Obviously, ${\underline{R}}_{a}^{m α} (X) = {\underline{R}}_{a}^{α} (X), {\bar{R}}_{a}^{m α} (X) = {\bar{R}}_{a}^{α} (X) .$

Proposition 3.2. Let S = (U, A, F, d) be an inconsistent decision information system. Given α ∈ (0.5, 1]. Then the following properties hold: (1) ${\underline{R}}_{B}^{m α} (X) = \underset{a \in B}{\cap} {\underline{R}}_{a}^{m α} (X)$ ,

${\bar{R}}_{B}^{m α} (X) = \underset{a \in B}{\cup} {\bar{R}}_{a}^{m α} (X)$ .

(2) ${\underline{R}}_{B}^{m α} (X) \subseteq {\bar{R}}_{B}^{m α} (X)$ .

(3) ${\underline{R}}_{B}^{m α} (U - X) = U - {\bar{R}}_{B}^{m α} (X)$ . (4) If B₁ ⊆ B₂, then ${\underline{R}}_{B_{1}}^{m α} (X) \supseteq {\underline{R}}_{B_{2}}^{m α} (X)$ , ${\bar{R}}_{B_{1}}^{m α}$ $(X) \subseteq {\bar{R}}_{B_{2}}^{m α} (X)$ . (5) If X₁ ⊆ X₂, then ${\underline{R}}_{B}^{m α} (X_{1}) \subseteq {\underline{R}}_{B}^{m α} (X_{2})$ , ${\bar{R}}_{B}^{m α}$ $(X_{1}) \subseteq {\bar{R}}_{B}^{m α} (X_{2})$ . (6) ${\underline{R}}_{B}^{m α} (X) \subseteq {\underline{R}}_{B}^{m α} ({\underline{R}}_{B}^{m α} (X))$ . (7) ${\bar{R}}_{B}^{m α} ({\bar{R}}_{B}^{m α} (X)) \subseteq {\bar{R}}_{B}^{m α} (X)$ .

Proof.

(1) This is obvious.

(2) Since α > 0.5, we have α > 1 - α.

If $x \in {\underline{R}}_{B}^{m α} (X)$ , then ∀ a ∈ B, D (X/[x] _a) ⩾ α and so ∀ a ∈ B, D (X/[x] _a) >1 - α.

It follows from $x \in {\bar{R}}_{B}^{m α} (X)$ .

Thus, ${\underline{R}}_{B}^{m α} (X) \subseteq {\bar{R}}_{B}^{m α} (X)$ .

(3) By (1) and Proposition 2.4(1), $\begin{matrix} {\underline{R}}_{B}^{m α} (U - X) & = & \underset{a \in B}{\cap} {\underline{R}}_{a}^{m α} (U - X) = \underset{a \in B}{\cap} {\underline{R}}_{a}^{α} (U - X) \\ = & \underset{a \in B}{\cap} (U - {\bar{R}}_{a}^{α} (X)) = \underset{a \in B}{\cap} (U - {\bar{R}}_{a}^{m α} (X)) \\ = & U - \underset{a \in B}{\cup} {\bar{R}}_{a}^{m α} (X) = U - {\bar{R}}_{B}^{m α} (X) . \end{matrix}$ (4) This holds by (1).

(5) This is clear.

(6) By Proposition 2.4(4), $\forall b \in B, {\underline{R}}_{b}^{m α} (X) = {\underline{R}}_{b}^{m α} ({\underline{R}}_{b}^{m α} (X))$ .

Since ${\underline{R}}_{b}^{m α} (X) \subseteq ⋂_{a \in B} {\underline{R}}_{a}^{m α} (X)$ , by Proposition 2.4(5), we have ${\underline{R}}_{b}^{m α} (X) \subseteq {\underline{R}}_{b}^{m α} (⋂_{a \in B} {\underline{R}}_{a}^{m α} (X)) .$

So $⋂_{b \in B} {\underline{R}}_{b}^{m α} (X) \subseteq ⋂_{b \in B} {\underline{R}}_{b}^{m α} (⋂_{a \in B} {\underline{R}}_{a}^{m α} (X))$ .

Thus

${\underline{R}}_{B}^{m α} (X) \subseteq {\underline{R}}_{B}^{m α} ({\underline{R}}_{B}^{m α} (X)) .$

(7) The proof is similar to (6).

4 mα-approximations reduction in an inconsistent decision information system based on MVPRS model

In this section, we investigate mα-approximations reduction in an inconsistent decision information system based on MVPRS model.

Let S = (U, A, F, d) be an inconsistent decision information system. Denote

U = {x₁, x₂, ⋯, x_n},

U/R_d = {D₁, D₂, ⋯, D_r},

$d_{ij}^{a} = D (D_{j} / [x_{i}]_{a}) .$

Obviously,

${\underline{R}}_{B}^{m α} (D_{j}) = {x_{i} : \forall a \in B, d_{ij}^{a} ⩾ α},$

${\bar{R}}_{B}^{m α} (D_{j}) = {x_{i} : \exists a \in B, d_{ij}^{a} > 1 - α} .$

Definition 4.1. Let S = (U, A, F, d) be an inconsistent decision information system. For any B ⊆ A, denote $L_{B} = ({\underline{R}}_{B}^{m α} (D_{1}), {\underline{R}}_{B}^{m α} (D_{2}), \dots, {\underline{R}}_{B}^{m α} (D_{r})),$ $U_{B} = ({\bar{R}}_{B}^{m α} (D_{1}), {\bar{R}}_{B}^{m α} (D_{2}), \dots, {\bar{R}}_{B}^{m α} (D_{r})) .$

(1) If L_B = L_A, then B is called a mα-lower approximation coordinate set; If B is an mα-lower approximation coordinate set of S and every proper subset of B is not an mα-lower approximation coordinate set of S, then B is called a mα-lower approximation reduction.

(2) If U_B = U_A, then B is called an mα-upper approximation coordinate set of S. If B is a mα-upper approximation coordinate set of S and every proper subset of B is not an mα-upper approximation coordinate set of S, then B is called a mα-upper approximation reduction.

In this paper, the family of all mα-lower approximation (resp., all mα-upper approximation) coordinate sets of S is denoted by ${\underline{co}}^{m α} (S)$ (resp., ${\bar{co}}^{m α} (S)$ ), the family of all mα-lower approximation (resp., all mα-upper approximation) reductions of S is denoted by ${\underline{red}}^{m α} (S)$ (resp., ${\bar{red}}^{m α} (S)$ ).

Example 4.2. In Example 2.1, we have

U/R_{a
₁} = {{x₁, x₃}, {x₂, x₇}, {x₄, x₉, x₁₀}, {x₅, x₆, x₈}},

U/R_{a
₂} = {{x₁, x₃}, {x₂, x₁₀}, {x₄, x₈, x₉}, {x₅, x₆, x₇}},

U/R_{a
₃} = {{x₁, x₃}, {x₂, x₄}, {x₅, x₉, x₁₀}, {x₆, x₇, x₈}},

U/R_{a₁,a₂} = {{x₁, x₃}, {x₂}, {x₄, x₉}, {x₅, x₆}, {x₇}, {x₈}, {x₁₀}},

U/R_{a₁,a₃} = {{x₁, x₃}, {x₂}, {x₄}, {x₅}, {x₆, x₈}, {x₇}, {x₉, x₁₀}},

U/R_{a₂,a₂} = {{x₁, x₃}, {x₂}, {x₄}, {x₅}, {x₆, x₇}, {x₈}, {x₉}, {x₁₀}},

U/R_A = {{x₁, x₃}, {x₂}, {x₄}, {x₅}, {x₆}, {x₇}, {x₈}, {x₉}, {x₁₀}},

U/R_d = {{x₁, x₂, x₄, x₆, x₈}, {x₃, x₅, x₇, x₉, x₁₀}} = {D₁, D₂}.

Pick α = 0.6. Then

${\underline{R}}_{a_{1}}^{0.6} (D_{1}) = {x_{5}, x_{6}, x_{8}},$

${\underline{R}}_{a_{1}}^{0.6} (D_{2}) = {x_{4}, x_{9}, x_{10}};$

${\underline{R}}_{a_{2}}^{0.6} (D_{1}) = {x_{4}, x_{8}, x_{9}},$

${\underline{R}}_{a_{2}}^{0.6} (D_{2}) = {x_{5}, x_{6}, x_{7}};$

${\underline{R}}_{a_{3}}^{0.6} (D_{1}) = {x_{2}, x_{4}, x_{6}, x_{7}, x_{8}},$

${\underline{R}}_{a_{3}}^{0.6} (D_{2}) = {x_{5}, x_{9}, x_{10}};$

${\underline{R}}_{{a_{1}, a_{2}}}^{0.6} (D_{1}) = {x_{8}},$

${\underline{R}}_{{a_{1}, a_{2}}}^{0.6} (D_{2}) = \emptyset;$

${\underline{R}}_{{a_{1}, a_{3}}}^{0.6} (D_{1}) = {x_{6}, x_{8}},$

${\underline{R}}_{{a_{1}, a_{3}}}^{0.6} (D_{2}) = {x_{9}, x_{10}};$

${\underline{R}}_{{a_{2}, a_{3}}}^{0.6} (D_{1}) = {x_{4}, x_{8}},$

${\underline{R}}_{{a_{2}, a_{3}}}^{0.6} (D_{2}) = {x_{5}};$

${\underline{R}}_{A}^{0.6} (D_{1}) = {x_{8}},$

${\underline{R}}_{A}^{0.6} (D_{2}) = \emptyset$ .

${\bar{R}}_{a_{1}}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}},$

${\bar{R}}_{a_{1}}^{0.6} (D_{2}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{7}, x_{9}, x_{10}};$

${\bar{R}}_{a_{2}}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{8}, x_{9}, x_{10}},$

${\bar{R}}_{a_{2}}^{0.6} (D_{2}) = {x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{10}};$

${\bar{R}}_{a_{3}}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}, x_{8},},$

${\bar{R}}_{a_{3}}^{0.6} (D_{2}) = {x_{1}, x_{3}, x_{5}, x_{9}, x_{10}};$

${\bar{R}}_{{a_{1}, a_{2}}}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}},$

${\bar{R}}_{{a_{1}, a_{2}}}^{0.6} (D_{2}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{9}, x_{10}};$

${\bar{R}}_{{a_{1}, a_{3}}}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}},$

${\bar{R}}_{{a_{1}, a_{3}}}^{0.6} (D_{2}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{7}, x_{9}, x_{10}};$

${\bar{R}}_{{a_{2}, a_{3}}}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}},$

${\bar{R}}_{{a_{2}, a_{3}}}^{0.6} (D_{2}) = {x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{9}, x_{10}};$

${\bar{R}}_{A}^{0.6} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}},$

${\bar{R}}_{A}^{0.6} (D_{2}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{9}, x_{10}} .$

Theorem 4.3. Let S = (U, A, F, d) be an inconsistent decision information system. For any B ⊆ A and x ∈ U, denote

$G_{B}^{m α} (x) = {D_{j} : x \in {\underline{R}}_{B}^{m α} (D_{j})},$

$M_{B}^{m α} (x) = {D_{j} : x \in {\bar{R}}_{B}^{m α} (D_{j})} .$ Then

(1) $B \in {\underline{co}}^{m α} (S) \Leftrightarrow \forall x \in U, G_{B}^{m α} (x) = G_{A}^{m α} (x) .$

(2) $B \in {\bar{co}}^{m α} (S) \Leftrightarrow \forall x \in U, M_{B}^{m α} (x) = M_{A}^{m α} (x) .$

Proof. (1) “⇒". Suppose $B \in {\underline{co}}^{m α} (S)$ . Then L_B = L_A, that is, ∀ j, ${\underline{R}}_{B}^{m α} (D_{j}) = {\underline{R}}_{A}^{m α} (D_{j}) .$

∀ x ∈ U, $\forall D_{j} \in G_{B}^{m α} (x)$ , we have $x \in {\underline{R}}_{B}^{m α} (D_{j})$ . Then $x \in {\underline{R}}_{A}^{m α} (D_{j})$ . So $D_{j} \in G_{A}^{m α} (x)$ . This implies ∀ x ∈ U, $G_{B}^{m α} (x) \subseteq G_{A}^{m α} (x) .$

On the other hand, ∀ x ∈ U, $\forall D_{j} \in G_{A}^{m α} (x)$ , we have $x \in {\underline{R}}_{A}^{m α} (D_{j})$ . Then $x \in {\underline{R}}_{B}^{m α} (D_{j})$ . So $D_{j} \in G_{B}^{m α} (x)$ . This implies that ∀ x ∈ U, $G_{B}^{m α} (x) \supseteq G_{A}^{m α} (x) .$

Table 2
mα-lower approximation distributed functions

$G_{B}^{m α} (x_{i})$ x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈ x ₉ x ₁₀

$G_{A}^{m α} (x_{i})$ ∅ ∅ ∅ ∅ ∅ ∅ ∅ {D₁} ∅ ∅

$G_{{a_{1}}}^{m α} (x_{i})$ ∅ ∅ ∅ {D₂} {D₁} {D₁} ∅ {D₁} {D₂} {D₂}

$G_{{a_{2}}}^{m α} (x_{i})$ ∅ ∅ ∅ {D₁} {D₂} {D₂} {D₂} {D₁} {D₁} ∅

$G_{{a_{3}}}^{m α} (x_{i})$ ∅ {D₁} ∅ {D₁} {D₂} {D₁} {D₁} {D₁} {D₂} {D₂}

$G_{{a_{1}, a_{2}}}^{m α} (x_{i})$ ∅ ∅ ∅ ∅ ∅ ∅ ∅ {D₁} ∅ ∅

$G_{{a_{1}, a_{3}}}^{m α} (x_{i})$ ∅ ∅ ∅ ∅ ∅ {D₁} ∅ {D₁} {D₂} {D₂}

$G_{{a_{2}, a_{3}}}^{m α} (x_{i})$ ∅ ∅ ∅ {D₁} {D₂} ∅ ∅ {D₁} ∅ ∅

$G_{B}^{m α} (x_{i})$	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈	x ₉	x ₁₀
$G_{A}^{m α} (x_{i})$	∅	∅	∅	∅	∅	∅	∅	{D₁}	∅	∅
$G_{{a_{1}}}^{m α} (x_{i})$	∅	∅	∅	{D₂}	{D₁}	{D₁}	∅	{D₁}	{D₂}	{D₂}
$G_{{a_{2}}}^{m α} (x_{i})$	∅	∅	∅	{D₁}	{D₂}	{D₂}	{D₂}	{D₁}	{D₁}	∅
$G_{{a_{3}}}^{m α} (x_{i})$	∅	{D₁}	∅	{D₁}	{D₂}	{D₁}	{D₁}	{D₁}	{D₂}	{D₂}
$G_{{a_{1}, a_{2}}}^{m α} (x_{i})$	∅	∅	∅	∅	∅	∅	∅	{D₁}	∅	∅
$G_{{a_{1}, a_{3}}}^{m α} (x_{i})$	∅	∅	∅	∅	∅	{D₁}	∅	{D₁}	{D₂}	{D₂}
$G_{{a_{2}, a_{3}}}^{m α} (x_{i})$	∅	∅	∅	{D₁}	{D₂}	∅	∅	{D₁}	∅	∅

Thus $\forall x \in U, G_{B}^{m α} (x) = G_{A}^{m α} (x) .$

“⟸". Suppose $\forall x \in U, G_{B}^{m α} (x) = G_{A}^{m α} (x) .$ We will show that ∀ j, ${\underline{R}}_{B}^{m α} (D_{j}) = {\underline{R}}_{A}^{m α} (D_{j}) .$

∀ j, $\forall x \in {\underline{R}}_{B}^{m α} (D_{j})$ , we have $D_{j} \in G_{B}^{m α} (x)$ . Then $D_{j} \in G_{A}^{m α} (x)$ . So $x \in {\underline{R}}_{A}^{m α} (D_{j})$ . This implies that ∀ j, ${\underline{R}}_{B}^{m α} (D_{j}) \subseteq {\underline{R}}_{A}^{m α} (D_{j}) .$

On the other hand, ∀ j, $\forall x \in {\underline{R}}_{A}^{m α} (D_{j})$ , we have $D_{j} \in G_{A}^{m α} (x)$ . Then $D_{j} \in G_{B}^{m α} (x)$ . So $x \in {\underline{R}}_{B}^{m α} (D_{j})$ . This implies that ∀ j, ${\underline{R}}_{B}^{m α} (D_{j}) \supseteq {\underline{R}}_{A}^{m α} (D_{j}) .$

Thus $\forall j, {\underline{R}}_{B}^{m α} (D_{j}) = {\underline{R}}_{A}^{m α} (D_{j}) .$

Hence $B \in {\underline{co}}^{m α} (S)$ .

(2) “⇒". Suppose $B \in {\bar{co}}^{m α} (S)$ . Then U_B = U_A, that is, ∀ j, ${\bar{R}}_{B}^{m α} (D_{j}) = {\bar{R}}_{A}^{m α} (D_{j}) .$

∀ x ∈ U, $\forall D_{j} \in M_{B}^{m α} (x)$ , we have $x \in {\bar{R}}_{B}^{m α} (D_{j})$ . Then $x \in {\bar{R}}_{A}^{m α} (D_{j})$ . So $D_{j} \in M_{A}^{m α} (x)$ . This implies that ∀ x ∈ U, $M_{B}^{m α} (x) \subseteq M_{A}^{m α} (x) .$

On the other hand, ∀ x ∈ U, $\forall D_{j} \in M_{A}^{m α} (x)$ , we have $x \in {\bar{R}}_{A}^{m α} (D_{j})$ . Then $x \in {\bar{R}}_{B}^{m α} (D_{j})$ . So $D_{j} \in M_{B}^{m α} (x)$ . This implies that ∀ x ∈ U, $M_{B}^{m α} (x) \supseteq M_{A}^{m α} (x) .$

Thus $\forall x \in U, M_{B}^{m α} (x) = M_{A}^{m α} (x) .$

“⟸". Suppose $\forall x \in U, M_{B}^{m α} (x) = M_{A}^{m α} (x) .$ We will show that ∀ j, ${\bar{R}}_{B}^{m α} (D_{j}) = {\bar{R}}_{A}^{m α} (D_{j}) .$

∀ j, $\forall x \in {\bar{R}}_{B}^{m α} (D_{j})$ , we have $D_{j} \in M_{B}^{m α} (x)$ . Then $D_{j} \in M_{A}^{m α} (x)$ . So $x \in {\bar{R}}_{A}^{m α} (D_{j})$ . This implies that ∀ j, ${\bar{R}}_{B}^{m α} (D_{j}) \subseteq {\bar{R}}_{A}^{m α} (D_{j}) .$

On the other hand, ∀ j, $\forall x \in {\bar{R}}_{A}^{m α} (D_{j})$ , we have $D_{j} \in M_{A}^{m α} (x)$ . Then $D_{j} \in M_{B}^{m α} (x)$ . So $x \in {\bar{R}}_{B}^{m α} (D_{j})$ . This implies that ∀ j, ${\bar{R}}_{B}^{m α} (D_{j}) \supseteq {\bar{R}}_{A}^{m α} (D_{j}) .$

Thus $\forall j, {\bar{R}}_{B}^{m α} (D_{j}) = {\bar{R}}_{A}^{m α} (D_{j}) .$

Hence $B \in {\bar{co}}^{m α} (S)$ .

Example 4.4. In Example 4.2,

D₁ = {x₁, x₂, x₄, x₆, x₈}, D₂ = {x₃, x₅, x₇, x₉, x₁₀}.

We have Table 2 and Table 3.

Table 3

mα-upper approximation distributed functions

$M_{B}^{m α} (x_{i})$	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈	x ₉	x ₁₀
$M_{A}^{m α} (x_{i})$	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁}	{D₁, D₂}	{D₁, D₂}
$M_{{a_{1}}}^{m α} (x_{i})$	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₂}	{D₁}	{D₁}	{D₁, D₂}	{D₁, D₂}	{D₂}	{D₂}
$M_{{a_{2}}}^{m α} (x_{i})$	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁}	{D₂}	{D₂}	{D₂}	{D₁}	{D₁}	{D₁, D₂}
$M_{{a_{3}}}^{m α} (x_{i})$	{D₁, D₂}	{D₁}	{D₁, D₂}	{D₁}	{D₂}	{D₁}	{D₁}	{D₁}	{D₂}	{D₂}
$M_{{a_{1}, a_{2}}}^{m α} (x_{i})$	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁}	{D₁, D₂}	{D₁, D₂}
$M_{{a_{1}, a_{3}}}^{m α} (x_{i})$	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁}	{D₁, D₂}	{D₁}	{D₂}	{D₂}
$M_{{a_{2}, a_{3}}}^{m α} (x_{i})$	{D₁, D₂}	{D₁, D₂}	{D₁, D₂}	{D₁}	{D₂}	{D₁, D₂}	{D₁, D₂}	{D₁}	{D₁, D₂}	{D₁, D₂}

Thus, ${\underline{red}}^{m α} (S) = {\bar{red}}^{m α} (S) = {{a_{1}, a_{2}}} .$

Based on MVPRS model, each $x \in {\underline{R}}_{B}^{m α} (D_{j})$ generates a decision rule as follows: $\underset{a \in B}{\land} (a, a (x)) \to d = j .$

Example 4.5. Consider Example 4.4, we have ${\underline{red}}^{m α} (S) = {\bar{red}}^{m α} (S) = {{a_{1}, a_{2}}} .$ Then, the derived decision rules are as follows:

r₁ : (a₁, 0) ∧ (a₂, 0) → (d, 0), supported by x₁, x₃;

r₂ : (a₁, 1) ∧ (a₂, 1) → (d, 1), supported by x₄, x₅;

r₃ : (a₁, 1) ∧ (a₂, 2) → (d, 0), supported by x₈;

r₄ : (a₁, 2) ∧ (a₂, 2) → (d, 0), supported by x₄, x₉;

r₅ : (a₁, 2) ∧ (a₂, 3) → (d, 1), supported by x₁₀;

r₆ : (a₁, 3) ∧ (a₂, 1) → (d, 1), supported by x₇;

r₇ : (a₁, 3) ∧ (a₂, 3) → (d, 0), supported by x₂.

It is easy to see that r₁ is in conflict with the following decision rule derived from the original system: $(a_{1}, 0) \land (a_{2}, 0) \land (a_{3}, 0) \to (d, 0) .$

Theorem 4.6. Let S = (U, A, F, d) be an inconsistent decision information system and B ⊆ A.

(1) $B \in {\underline{co}}^{m α} (S)$ ⇒ if $G_{A}^{m α} (x) \neq G_{A}^{m α} (y)$ , then ∃ a ∈ B, s.t. [x] _a ∩ [y] _a = ∅.

(2) $B \in {\bar{co}}^{m α} (S)$ ⇒ if $M_{A}^{m α} (x) \neq M_{A}^{m α} (y)$ , then ∀ a ∈ B, s.t. [x] _a ∩ [y] _a = ∅.

Proof. (1) Let $B \in {\underline{co}}^{m α} (S)$ . Suppose that ∃ x, y ∈ U such that $G_{A}^{m α} (x) \neq G_{A}^{m α} (y)$ and ∀ a∈ B, [x] _a ∩ [y] _a ≠ ∅. Then ∀ a ∈ B, [x] _a = [y] _a. Since

$G_{B}^{m α} (x) = {D_{j} : x \in {\underline{R}}_{B}^{m α} (D_{j})} = {D_{j} : \forall a \in B s . t . D (D_{j} / [x]_{a}) \geq α};$

$G_{B}^{m α} (y) = {D_{j} : y \in {\underline{R}}_{B}^{m α} (D_{j})} = {D_{j} : \forall a \in B s . t . D (D_{j} / [y]_{a}) \geq α},$ we have $G_{B}^{m α} (x) = G_{B}^{m α} (y)$ .

Since $B \in {\underline{co}}^{m α} (S)$ , by Theorem 4.3, we have $G_{B}^{m α} (x) = G_{A}^{m α} (x)$ and $G_{B}^{m α} (y) = G_{A}^{m α} (y)$ .

Thus $G_{A}^{m α} (x) = G_{A}^{m α} (y)$ . This is a contradiction.

Therefore, if $G_{A}^{m α} (x) \neq G_{A}^{m α} (y)$ , then ∃ a ∈ B such that [x] _a∩ [y] _a = ∅.

(2) The proof is similar to (1).

Definition 4.7. Let S = (U, A, F, d) be an inconsistent decision information system, U/R_d = {D₁, D₂, ⋯, D_r}. Denote ${\underline{P}}^{m α} = {([x]_{A}, [y]_{A}) : G_{A}^{m α} (x) \neq G_{A}^{m α} (y)},$ ${\bar{P}}^{m α} = {([x]_{A}, [y]_{A}) : M_{A}^{m α} (x) \neq M_{A}^{m α} (y)} .$ The mα-lower and upper approximation discernibility sets of x_i and x_j, denoted by ${\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A})$ and ${\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A})$ , respectively, are defined as follows: ${\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A})$ $= {\begin{matrix} {a \in A : [x_{i}]_{a} \neq [x_{j}]_{a}}, ([x_{i}]_{A}, [x_{j}]_{A}) \in {\underline{P}}^{m α}, \\ \emptyset, ([x_{i}]_{A}, [x_{j}]_{A}) \notin {\underline{P}}^{m α}; \end{matrix}$ ${\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A})$ $= {\begin{matrix} {a \in A : [x_{i}]_{a} \neq [x_{j}]_{a}}, ([x_{i}]_{A}, [x_{j}]_{A}) \in {\bar{P}}^{m α}, \\ \emptyset, ([x_{i}]_{A}, [x_{j}]_{A}) \notin {\bar{P}}^{m α} . \end{matrix}$ The mα-lower and upper approximation discernibility matrixes of S, denoted by ${\underline{W}}^{m α}$ and ${\bar{W}}^{m α}$ , respectively, are defined as follows: ${\underline{W}}^{m α} = ({\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}))_{n \times n},$ ${\bar{W}}^{m α} = ({\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}))_{n \times n} .$

Theorem 4.8. Let S = (U, A, F, d) be an inconsistent decision information system and B ⊆ A.

(1) $B \in {\underline{co}}^{m α} (S)$ ⇔ if ${\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset$ , then $B ⋂ {\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset .$

(2) $B \in {\bar{co}}^{m α} (S)$ ⇔ if ${\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset$ , then $B ⋂ {\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset .$

Proof. (1) “⇒". Suppose $B \in {\underline{co}}^{m α} (S)$ .

For any ${\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset$ , ∃ x, y ∈ U such that [x_i] _A = [x] _A, [x_j] _A = [y] _A. Then $([x]_{A}, [y]_{A}) \in {\underline{P}}^{m α}$ . So $G_{A}^{m α} (x) \neq G_{A}^{m α} (y)$ .

By Theorem 4.6, [x] _B∩ [y] _B = ∅.

Then ∃ a ∈ B such that [x] _a ≠ [y] _a, i.e., [x_i] _a ≠ [x_j] _a. So, $a \in {\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A})$ .

Thus, $B ⋂ {\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset$ .

“⟸". If $\exists ([x_{i}]_{A}, [x_{j}]_{A}) \in {\underline{P}}^{m α}$ such that ${\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) \neq \emptyset$ and $B ⋂ {\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) = \emptyset .$

Then ∃ x, y ∈ U such that [x_i] _A = [x] _A and [x_j] _A = [y] _A.

So $([x]_{A}, [y]_{A}) \in {\underline{P}}^{m α}$ .

Thus $G_{A}^{m α} (x) \neq G_{A}^{m α} (y)$ .

Because $B ⋂ {\bar{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A}) = \emptyset$ , ∀ a ∈ B, $a \notin {\underline{W}}^{m α} ([x_{i}]_{A}, [x_{j}]_{A})$ . Then [x_i] _a = [x_j] _a. So [x] _B = [y] _B.

By Theorem 4.6, $B \notin {\underline{co}}^{m α} (S)$ .

Thus, we complete the proof.

(2) The proof is similar to (1).

Definition 4.9. Let S = (U, A, F, d) be an inconsistent decision information system. The mα-lower and upper approximation discernibility functions of S, denoted by ${\underline{▵}}^{m α}$ and ${\bar{▵}}^{m α}$ , respectively, are defined as follows: ${\underline{▵}}^{m α} = ⋀ ⋁ {\underline{W}}^{m α} (x_{i}, x_{j}) ((x_{i}, x_{j}) \in {\underline{P}}^{m α}),$ ${\bar{▵}}^{m α} = ⋁ ⋀ {\bar{W}}^{m α} (x_{i}, x_{j}) ((x_{i}, x_{j}) \in {\bar{P}}^{m α}) .$

Theorem 4.10. Let S = (U, A, F, d) be an inconsistent decision information system. The minuscule disjunctive normal form of the mα-lower and upper approximation discernibility functions, denoted by ${\underline{▵}}^{m α} = ⋁ ⋀ B_{1 k}, {\bar{▵}}^{m α} = ⋁ ⋀ B_{2 k} .$

Then, $B_{1 k} \in {\underline{co}}^{m α} (S)$ , $B_{2 k} \in {\bar{co}}^{m α} (S)$ .

Proof. It is similar to Theorem 4.5 in [12].

5 Biggest or least multi-distribution reductions in an inconsistent decision information system based on MVPRS model

Definition 5.1. Let S = (U, A, F, d) be an inconsistent decision information system. Suppose U = {x_i: 1 ≤ i ≤ n} and U/R_d = {D_j : 1 ≤ j ≤ r}. For any B ⊆ A and i, denote $γ_{B} (x_{i}) = \underset{a \in B}{\cup} {D_{t_{a}} : d_{i t_{a}}^{a} = \underset{j ⩽ r}{\lor} d_{i j}^{a}},$ $η_{B} (x_{i}) = \underset{a \in B}{\cap} {D_{j_{a}} : d_{i j_{a}}^{a} > 0} .$

(1) If for each i, γ_B (x_i) = γ_A (x_i), then B is called a biggest multi-distribution coordinate set; If B is a biggest multi-distribution coordinate set and any proper subset of B is not a biggest multi-distribution coordinate set, then we call B a biggest multi-distribution reduction.

(2) If for each i, η_B (x_i) = η_A (x_i), then B is called a least multi-distribution coordinate set; If B is a least multi-distribution coordinate set and any proper subset of B is not a least multi-distribution coordinate set, then we call B a least multi-distribution reduction.

In this paper, the family of all biggest (resp., least) multi-distribution coordinate sets of S is denoted by ${co}_{big}^{md} (S)$ (resp., ${co}_{le}^{md} (S)$ ), the family of all biggest (resp., least) multi-distribution reductions of S is denoted by ${red}_{big}^{md} (S)$ (resp., ${red}_{le}^{md} (S)$ ).

6 Relationships among four sorts of reductions in an inconsistent decision information system

In this section, we investigate several kinds of knowledge reduction in an inconsistent decision information system.

Theorem 6.1. Let S = (U, A, F, d) be an inconsistent decision information system. Suppose U = {x_i : 1 ≤ i ≤ n}, U/R_d = {D_j : 1 ≤ j ≤ r} and α ∈ (0.5, 1]. Denote $α^{*} = \underset{i ⩽ n, a \in A}{\land} \underset{j ⩽ r}{\lor} d_{i j}^{a} .$

If α ≤ α^*, then (1) ${\underline{co}}^{m α} (S) = {co}_{big}^{md} (S)$ ; (2) ${\underline{red}}^{m α} (S) = {red}_{big}^{md} (S)$ .

Proof. (1) Let $B \in {\underline{co}}^{m α} (S)$ . Obviously, $\forall i, γ_{B} (x_{i}) \subseteq γ_{A} (x_{i}) .$

On the other hand, ∀ i, ∀ D_j ∈ γ_A (x_i), there exists a ∈ A such that $d_{ij}^{a} = ⋁_{t ⩽ r} d_{it}^{a}$ .

Then $d_{ij}^{a} \geq α^{*} \geq α,$ which implies $x_{i} \in {\underline{R}}_{a}^{α} (D_{j})$ .

So $x_{i} \in {\underline{R}}_{A}^{m α} (D_{j})$ . Since ${\underline{R}}_{A}^{m α} (D_{j}) = {\underline{R}}_{B}^{m α} (D_{j})$ , we have $x_{i} \in {\underline{R}}_{B}^{m α} (D_{j})$ . Then $\exists b \in B, x_{i} \in {\underline{R}}_{b}^{α} (D_{j}),$ which implies that $d_{ij}^{b} ⩾ α$ .

By α > 0.5, $d_{ij}^{b} = ⋁_{t ⩽ r} d_{it}^{b}$ . Then D_j ∈ γ_B (x_i). This implies that γ_A (x_i) ⊆ γ_B (x_i).

So γ_B (x_i) = γ_A (x_i). This show $B \in {co}_{big}^{md} (S)$ .

Thus ${\underline{co}}^{m α} (S) \subseteq {co}_{big}^{md} (S) .$

b) Let $B \in {co}_{big}^{md} (S)$ . Obviously, $\forall j, {\underline{R}}_{B}^{m α} (D_{j}) \subseteq {\underline{R}}_{A}^{m α} (D_{j}) .$

On the other hand, ∀ j, $\forall x_{i} \in {\underline{R}}_{A}^{m α} (D_{j})$ , there exists a ∈ A such that $x_{i} \in {\underline{R}}_{a}^{m α} (D_{j})$ . Then $d_{ij}^{a} ⩾ α$ .

By α > 0.5, $d_{ij}^{a} = ⋁_{t ⩽ r} d_{it}^{a} .$

Then D_j ∈ γ_A (x_i).

Since γ_B (x_i) = γ_A (x_i), we have D_j ∈ γ_B (x_i). Then $\exists b \in B, d_{i j}^{b} = \underset{t ⩽ r}{\lor} d_{i t}^{b} \geq α,$ which implies that $x_{i} \in {\underline{R}}_{b}^{α} (D_{j}) \subseteq {\underline{R}}_{B}^{m α} (D_{j})$ .

Then ${\underline{R}}_{A}^{m α} (D_{j}) \subseteq {\underline{R}}_{B}^{m α} (D_{j})$ . So ${\underline{R}}_{B}^{m α} (D_{j}) = {\underline{R}}_{A}^{m α} (D_{j}) .$

Thus $B \in {\underline{co}}^{m α} (S)$ . This implies that ${co}_{big}^{md} (S) \subseteq {\underline{co}}^{m α} (S)$ .

Therefore, ${\underline{co}}^{m α} (S) = {co}_{big}^{md} (S) .$

(2) This holds by (1).

Example 6.2. $α > α^{*} {\underline{co}}^{m α} (S) \subseteq {co}_{big}^{md} (S) or {co}_{big}^{md} (S) \subseteq {\underline{co}}^{m α} (S),$ $α > α^{*} {\underline{red}}^{m α} (S) \subseteq {red}_{big}^{md} (S) or {red}_{big}^{md} (S) \subseteq {\underline{red}}^{m α} (S) .$ Given the inconsistent decision information systems S in Table 4. Then α^* = 0.5.

Pick α = 0.6. Then α^* < α.

We can observe that ${\underline{R}}_{{a_{1}}}^{m α} = ({x_{1}, x_{2}}, {, x_{3}, x_{4}, x_{5}, x_{6}}) = {\underline{R}}_{A}^{m α},$ ${\underline{R}}_{{a_{2}}}^{m α} = ({x_{2}}, {x_{3}, x_{4}, x_{6}}) \neq {\underline{R}}_{A}^{m α} .$ Then ${a_{1}} \in {\underline{co}}^{m α} (S)$ , ${a_{2}} \notin {\underline{co}}^{m α} (S)$ .

However, γ_A (x₁) = {{x₁, x₂, x₃}, {x₄, x₅, x₆}}, γ_{a₁} (x₁) = {{x₁, x₂, x₃}}. Then $γ_{A} (x_{1}) = γ_{{a_{1}}} (x_{1}) .$

So ${a_{1}} \notin {co}_{big}^{md} (S)$ .

It is easy to check that γ_{a₂} (x) = γ_A (x_i) (∀ i). Then ${a_{2}} \in {co}_{big}^{md} (S)$ .

Thus ${\underline{co}}^{m α} (S) {co}_{big}^{md} (S), {co}_{big}^{md} (S) {\underline{co}}^{m α} (S) .$

This example also illustrates that $α^{*} < α {\underline{red}}^{m α} (S) \subseteq {red}_{big}^{md} (S)$ or ${red}_{big}^{md} (S) \subseteq {\underline{red}}^{m α} (S) .$

Table 4
An inconsistent decision information systems S

U a ₁ a ₂ d

x ₁ 0 0 0

x ₂ 0 1 0

x ₃ 1 2 0

x ₄ 1 2 1

x ₅ 1 0 1

x ₆ 2 2 1

U	a ₁	a ₂	d
x ₁	0	0	0
x ₂	0	1	0
x ₃	1	2	0
x ₄	1	2	1
x ₅	1	0	1
x ₆	2	2	1

Theorem 6.3. Let S = (U, A, F, d) be an inconsistent decision information system. Suppose U = {x_i : 1 ≤ i ≤ n} and U/R_d = {D_j : 1 ≤ j ≤ r}. Denote $β^{*} = ⋀_{i ⩽ n, j ⩽ r, a \in A} {d_{ij}^{a} : d_{ij}^{a} > 0} .$

If β > 1 - β^*, then (1) ${\bar{co}}^{m β} (S) = {co}_{le}^{md} (S)$ ;

(2) ${\bar{red}}^{m β} (S) = {red}_{le}^{md} (S)$ .

Proof. (1) a) Let $B \in {\bar{co}}^{m β} (S)$ . Obviously, $\forall i, η_{A} (x_{i}) \subseteq η_{B} (x_{i}) .$

On the other hand, ∀ i, ∀ D_j ∈ η_B (x_i) and b ∈ B, D_j∩ [x_i] _b = ∅. Then $d_{ij}^{b} > 0$ .

Since $β > 1 - β^{*}, d_{ij}^{b} ⩾ β > 1 - β^{*},$ which implies that $x_{i} \in {\bar{R}}_{b}^{m β} (D_{j})$ (D_j ∈ R_d/U). Then $x_{i} \in {\bar{R}}_{B}^{m β} (D_{j})$ .

Since ${\bar{R}}_{B}^{m β} (D_{j}) = {\bar{R}}_{A}^{m β} (D_{j}), x_{i} \in {\bar{R}}_{A}^{m β} (D_{j}) .$

∀ a ∈ A, $x_{i} \in {\bar{R}}_{a}^{m β} (D_{j})$ . Then $d_{ij}^{a} > 1 - β^{*}$ . So $D_{j} \cap [x_{i}]_{a} = \emptyset (a \in A) .$

This implies D_j ∈ η_A (x_i). So η_B (x_i) ⊆ η_B (x_i). It follows that η_B (x_i) = η_B (x_i). This show $B \in {co}_{le}^{md} (S)$ .

Thus ${\bar{co}}^{m β} (S) \subseteq {co}_{le}^{md} (S)$ .

b) Let $B \in {co}_{le}^{md} (S)$ . Obviously, $\forall j, {\bar{R}}_{A}^{m β} (D_{j}) \subseteq {\bar{R}}_{B}^{m β} (D_{j}) .$

On the other hand, ∀ j, $\forall x_{i} \in {\bar{R}}_{B}^{m β} (D_{j})$ , we have $x_{i} \in {\bar{R}}_{b}^{m β} (D_{j}) (b \in B),$ which implies that $d_{ij}^{b} > 1 - β^{*}$ . Then $D_{j} \cap [x_{i}]_{b} = \emptyset .$ So D_j ∈ η_B (x_i).

Since η_A (x_i) = η_B (x_i), D_j ∈ η_A (x_i). This implies that D_j∩ [x_i] _a = ∅ (a ∈ A). By β^* > 1 - β, $x_{i} \in {\bar{R}}_{a}^{m β} (D_{j})$ . So $x_{i} \in {\bar{R}}_{A}^{m β} (D_{j})$ . Thus ${\bar{R}}_{B}^{m β} (D_{j}) \subseteq {\bar{R}}_{A}^{m β} (D_{j})$ . It follows that ${\bar{R}}_{B}^{m β} (D_{j}) = {\bar{R}}_{A}^{m β} (D_{j}) .$

This show $B \in {\bar{co}}^{m β} (S)$ .

Hence ${co}_{le}^{md} (S) \subseteq {\bar{co}}^{m β} (S)$ .

By a) and b), ${\bar{co}}^{m β} (S) = {co}_{le}^{md} (S) .$

(2) This holds by (1).

Example 6.4. $β \leq 1 - β^{*} {\bar{co}}^{m β} (S) \subseteq {co}_{le}^{md} (S) or {co}_{le}^{md} (S) \subseteq {\bar{co}}^{m β} (S),$

$β \leq 1 - β^{*} {\bar{red}}^{m β} (S) \subseteq {red}_{le}^{md} (S)$ or

${red}_{le}^{md} (S) \subseteq {\bar{red}}^{m β} (S) .$

Given a inconsistent decision information systems S in Table 5. Then β^* = 0.25.

Pick β^* = 0.6. Then 1 - β^* ≥ β.

We have

${\bar{R}}_{{a_{1}}}^{m β} = ({x_{1}, x_{2}, x_{3}, x_{4}}, {x_{5}, x_{6}}) = {\bar{R}}_{A}^{m β},$

${\bar{R}}_{{a_{2}}}^{m β} = ({x_{1}, x_{2}, x_{3}, x_{4}}, {x_{2}, x_{4}, x_{5}, x_{6}}) \neq {\bar{R}}_{A}^{m β} .$

Then ${a_{1}} \in {\bar{co}}^{m β} (S)$ , ${a_{2}} \notin {\bar{co}}^{m β} (S)$ .

However, η_A (x₃) = {{x₁, x₂, x₃}}, η_{a₁} (x₃) = {{x₁, x₂, x₃}, {x₄, x₅, x₆}}. Then $η_{A} (x_{3}) = η_{{a_{1}}} (x_{3}) .$ So ${a_{1}} \notin {co}_{le}^{md} (S)$ .

It is easy to check that ∀ i, η_{a₂} (x_i) = η_A (x_i). Then ${a_{2}} \in {co}_{le}^{md} (S)$ .

Thus ${\bar{co}}^{m β} (S) \subseteq {co}_{le}^{md} (S) or {co}_{le}^{md} (S) \subseteq {\bar{co}}^{m β} (S) .$

This example also illustrates that

$β \leq 1 - β^{*} {\bar{red}}^{m β} (S) \subseteq {red}_{le}^{md} (S)$ or

${red}_{le}^{md} (S) \subseteq {\bar{red}}^{m β} (S) .$

Table 5

An inconsistent decision information system S

U	a ₁	a ₂	d
x ₁	0	0	0
x ₁	0	0	0
x ₂	0	1	0
x ₃	0	2	0
x ₄	0	1	1
x ₅	1	3	1
x ₆	1	3	1

7 Conclusions

In this paper, we have studied MVPRS model and investigated four kinds of knowledge reduction in inconsistent decision information systems and obtained some approaches to these knowledge reductions. In future work, we will develop the proposed approaches to more complicated information systems and consider numerical experiments to illustrate the theoretical results in this paper.

Footnotes

Acknowledgements

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper. This work is supported by National Natural Science Foundation of China (11461005), Natural Science Foundation of Guangxi (2016GXNSFAA380045), National Social Science Foundation of China (16XJY015) and Research Topic of Guangxi Philosophy and Social Science Planning(15BGL003).

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Attribute reductions in an inconsistent decision information system

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Inconsistent decision information systems

Table 1 An inconsistent decision information system

3 MVPRS model

4 mα-approximations reduction in an inconsistent decision information system based on MVPRS model

6 Relationships among four sorts of reductions in an inconsistent decision information system

Table 4 An inconsistent decision information systems S U a 1 a 2 d x 1 0 0 0 x 2 0 1 0 x 3 1 2 0 x 4 1 2 1 x 5 1 0 1 x 6 2 2 1

Footnotes

Acknowledgements

References

Table 1
An inconsistent decision information system

Table 4
An inconsistent decision information systems S

U a ₁ a ₂ d

x ₁ 0 0 0

x ₂ 0 1 0

x ₃ 1 2 0

x ₄ 1 2 1

x ₅ 1 0 1

x ₆ 2 2 1