On variable-precision-based rough set approach to incomplete interval-valued fuzzy information systems and its applications

1 Introduction

As an important tool to deal with the uncertain and incomplete information, the basic framework of classical rough set theory [1] is an approximate space composed of the universes and equivalent binary relations. And based on this approximate space, the lower and upper approximation operators (connotation and epitaxy) are defined to describe the approximate objects. However, the classification basis of classical rough set theory is the equivalence classes, which limits the universality and effectiveness of its applications in a certain extent. As an extension of rough set theory, the variable precision rough set model [2] based on the majority inclusion relation has the characteristics and advantages of being able to deal with "the inclusion or belong in a way", so that the classical rough set theory has been widely applied and popularized. At the same time, in order to deal with the problems of information systems with fuzzy attributes, such as knowledge discovery, the fuzzy rough set model [3] has been extensively studied and used in the fuzzy rough relational database [4], the data set with real value attributes [5] and other fields. It is worth mentioning that, zhang et al. [6] proposed a fast feature selection algorithm based on the information entropy of fuzzy rough set. The conditional entropy was computed by iterated reduction examples, which significantly shortened the computing time. Considering the characteristics of fuzzy rough set and variable precision rough set, variable precision fuzzy rough set theory [7] has been discussed as a more effective tool to deal with the uncertain theoretical problems. As the applications, in recent years, many scholars have proposed various models to deal with uncertainty multi-attribute decision making problems [8–11], such as, Zhang et al. [11] by means of a fuzzy co-implication operator J and a triangular conorm S, put forward the two pairs of (J, S)-fuzzy rough set models and analyzed the relationship between them and some existing rough set models. It is worth mentioning that Zhan et al. [12] proposed a covering based Pythagorean fuzzy rough set model and presented two Pythagorean fuzzy TOPSIS methodologies. In 2020, Zhang and Zhan et al. [13] combined CVPIFRS model with TOPSIS idea, proposed an effective method to solve the complex and changeable bone graft selection in multi-attribute decision making problem.

As a generalization of fuzzy sets, Turken and Gorzafczary discussed the properties of interval-valued fuzzy sets [14, 15], respectively. In 2008, Gong and Sun et al. [16, 17] defined rough set theory of interval-valued fuzzy information system on the basis of studying interval-valued fuzzy set and classical Pawlak rough set theory. In 2010, Zhang [18] combined the classical Pawlak rough set theory with the interval-valued intuitionistic fuzzy set theory, proposed the interval-valued rough intuitionistic fuzzy set model, and extended the classical fuzzy rough set model. However, as we all know, the incompleteness of information is inevitable due to information loss, omission, measurement error, data noise and transmission medium fault, so it is necessary to study the incomplete information system. Subsequently, Gong and Tao [19] gave the rough approximate set, decision table and decision rules of interval-valued fuzzy set based on incomplete information systems, and improved the fuzzy rough set model. Meng [20] and Gehrke [21] further gave some basic theory of interval-valued fuzzy sets and Zeng [22] introduced the decomposition theorem of interval-valued fuzzy sets. However, it is relatively difficult to deal with the massive data containing missing information in incomplete information systems. To solve this problem, literature [23, 24] gave the dominance-based rough set approach to incomplete ordered information systems and incomplete interval-valued information systems, respectively, and proved its basic properties. In 2013, Wang et al. [25] defined h-similar class on the basis of similarity relation and proposed a new uncertainty measurement method, named h-rough degree. Dai [26] proposed an uncertainty measurement method based on weak similarity degree over the incomplete interval-valued information systems and defined a fuzzy dominance relation, and based on this relation they proposed some fuzzy approximation operators [27]. In 2017, in order to solve the incompleteness of the multi-source information system caused by the corresponding lack of data, Li and Zhang [28] proposed an incomplete multi-source system fusion method based on information entropy. In 2020, with the neighborhood relation classes, Sun et al. [29] establish the variable precision diversified attribute multi-granulation fuzzy rough set model over diversified attribute fuzzy decision making space.

From the perspective of the decision model, although they have been proven to be effective, there are still some problems worthy of consideration, i.e., the upper and lower approximation operators of incomplete interval-valued rough fuzzy set discussed at present have two shortcomings:

The upper and lower approximations have limiting properties. A relatively small misclassification rate may lead to the exclusion of some indistinguishable classification from the lower approximation, and may also lead to the unnecessary increase of the upper approximation.

Using the idea of variable precision into the incomplete interval-valued fuzzy information systems theory, we discuss the above problems in the form of variable precision, which not only overcomes the above two problems, but also helps to discover the relevant data from the data that is "considered" irrelevant by rough set theory.

It is well known that the multiple attribute decision making problems are an important research topic in decision theory. Because the objects are fuzzy and uncertain, the attributes involved in decision problems are not always expressed as real numbers under uncomplete information systems. In this paper, in order to analyze the incomplete information more precisely, we consider the data classification problems of uncertain relation on incomplete interval-valued fuzzy information systems, and discuss the variable precision problems of incomplete interval-valued fuzzy information systems with variable precision rough set model. We unified and extended the existing work, and finally give the decision model and its’ corresponding specific calculation steps.

The structure of this paper is as follows. In Section 2, we introduce some basic concepts of interval-valued rough fuzzy set theory. In Section 3, we define the variable precision interval-valued rough fuzzy set model over incomplete information system. Meanwhile,the basic properties of the variable precision interval-valued rough fuzzy set model in the incomplete information system are investigated. In Section 4, we give the decision table and decision rules based on incomplete interval-valued fuzzy information system. In Section 5, we propose some attribute reduction methods of the incomplete interval-valued fuzzy information system. Finally, in Section 6, we give some concluding remarks and future research.

2 Preliminary

3 Variable precision interval-valued rough fuzzy set theory based on incomplete information system

One limitation of the interval-valued rough fuzzy set model on incomplete information systems is that the classification it deals with must be completely correct or affirmative. Since it is strictly classified according to similar classes, its classification is accurate, that is, “inclusive” or “non-inclusive”, without a certain degree of “inclusive” or “belong”. Based on this limitation of the interval-valued rough fuzzy set model on incomplete information systems, an extension of the model is given below. We introduce parameter β (0.5 < β ≤ 1) on the basis of the incomplete interval-valued rough fuzzy set model and propose a variable precision interval-valued rough fuzzy set on incomplete information systems. This model allows a certain degree of error classification rate. This is not only beneficial to use the incomplete information rough fuzzy set theory to find relevant data, but also improves rough set theory.

Definition 3.1. Let U = {x₁, x₂, …, x _n } be a non-empty finite universe, IS = 〈U, AT, V, f〉 be an incomplete interval-valued fuzzy information system, 0.5 < β ≤ 1, δ = [δ₁, δ₂] ∈ [I] (δ ≠ [0, 1]). μ ∈ F ⁱ  (U) is an interval-valued fuzzy set, and its membership degree is μ (x), A ⊆ AT. Then the β-lower approximation and β-upper approximation of μ with respect to A are defined as follows, respectively, A _ μ β = { ⋀ y ∈ S α 1 ( x ) μ ( y ) , ∃ α 1 , x ∈ U [ 0 , 0 ] , other and A ¯ μ β = { ⋁ y ∈ S α 2 ( x ) μ ( y ) , ∃ α 2 , x ∈ U [ 0 , 0 ] , other where α₁ = sup {δ ∈ (0, 1] × (0, 1] |P _δ  (S _A  (x) , μ)≥β}, S _{α 1}  (x) = S _A  (x) ∩ μ _{α 1} , α₂ = sup {δ ∈ (0, 1] × (0, 1] |P _δ  (S _A  (x) , μ) >1 - β}, S _{α 2}  (x) = S _A  (x) ∩ μ _{α 2} , P δ ( S A ( x ) , μ ) = | { y ∈ S A ( x ) | μ ( y ) ≥ δ } | | S A ( x ) | .

It is well known that P _δ  (S _A  (x) , μ) represents the ratio of the number of elements in set S _A  (x) whose membership μ (y) (y ∈ S _A  (x)) is not less than δ to the number of elements in set S _A  (x), that is, the correct classification rate.

In addition, we call the operators A _ μ β : ( U → F i ( U ) ) and A ¯ μ β : ( U → F i ( U ) ) are β-lower approximation and β-upper approximation operator of variable precision interval-valued fuzzy rough set, respectively. The ordered pair ( A _ μ β , A ¯ μ β ) is called the variable precision interval-valued rough fuzzy set.

In the β-lower approximation of interval-valued fuzzy set μ with respect to A, if α₁ exists, then set S _{α 1} contains elements that its membership degree μ (y) (y ∈ S _A  (x)) in S _A  (x) is at least α₁. A _ μ β is determined by “better” elements in S _A  (x _i ) (i = 1, 2, …, n) (i.e., elements with corresponding membership degree μ (y) (y ∈ S _A  (x)) not less than δ), rather than determined by all S _A  (x _i ). Obviously, this explanation still holds for the β-upper approximation.

It is not difficult to see that the β-lower approximation and the β-upper approximation of interval-valued fuzzy set μ with respect to A can be calculated by the following formula: ∀x ∈ U, A _ μ β ( x ) = [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] A ¯ μ β ( x ) = [ ∨ { μ - ( y ) : y ∈ S α 2 ( x ) } , ∨ { μ + ( y ) : y ∈ S α 2 ( x ) } ] In fact, A _ μ β ( x ) = ∧ { μ ( y ) : y ∈ S α 1 ( x ) } = ∧ { [ μ - ( y ) , μ + ( y ) ] : y ∈ S α 1 ( x ) } = { [ ∧ μ - ( y ) , ∧ μ + ( y ) ] : y ∈ S α 1 ( x ) } = [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] A ¯ μ β ( x ) = ∨ { μ ( y ) : y ∈ S α 2 ( x ) } = ∨ { [ μ - ( y ) , μ + ( y ) ] : y ∈ S α 2 ( x ) } = { [ ∨ μ - ( y ) , ∨ μ + ( y ) ] : y ∈ S α 2 ( x ) } = [ ∨ { μ - ( y ) : y ∈ S α 2 ( x ) } , ∨ { μ + ( y ) : y ∈ S α 2 ( x ) } ]

Theorem 3.1. Let μ be an interval-valued fuzzy set on U. If β = 1, then A _ μ 1 = A _ μ , A ¯ μ 1 = A ¯ μ .

Proof. If β = 1, then P _δ  (S _A  (x) , μ) =1, that is P δ ( S A ( x ) , μ ) = | { y ∈ S A ( x ) | μ ( y ) ≥ δ } | | S A ( x ) | = 1 , At this time, there are no abandoned elements in S _A  (x), so { y ∈ S A ( x ) | μ ( y ) ≥ δ } = S A ( x ) . If α₁ = sup {δ|P _δ  (S _A  (x) , μ) =1} , then S A ( x ) = S A ( x ) ∩ μ α 1 = S A ( x ) . So A _ μ 1 = A _ μ . Similarly, we can prove that A ¯ μ 1 = A ¯ μ . This completes the proof.

Therefore, interval-valued fuzzy rough set is a special form of variable precision interval-valued rough fuzzy set.

Definition 3.2. Let U be a non-empty finite universe, 〈U, AT, V, f〉 is an incomplete interval-valued information system, A ⊆ AT, μ ∈ F ⁱ  (U), δ = [δ₁, δ₂] ∈ [I] (δ ≠ [0, 1]), The lower and upper approximations of level set μ_[δ1, δ2] of μ with respect to A are defined as follows, respectively, A _ μ [ δ 1 , δ 2 ] β = { x ∈ U : P ( S A ( x ) , μ [ δ 1 , δ 2 ] ) ≥ β } , A ¯ μ [ δ 1 , δ 2 ] β = { x ∈ U : P ( S A ( x ) , μ [ δ 1 , δ 2 ] ) > 1 - β } , where P ( S A ( x ) , μ [ δ 1 , δ 2 ] ) = | { S A ( x ) , μ [ δ 1 , δ 2 ] } | | S A ( x ) | .

Theorem 3.2. Let μ be an interval-valued fuzzy set on U. Then

A _ μ β ( x ) = ∨ { [ δ 1 , δ 2 ] : x ∈ A _ μ [ δ 1 , δ 2 ] β } = ∨ { [ δ 1 , δ 2 ] : P ( S A ( x ) , μ [ δ 1 , δ 2 ] ) ≥ β } ,

Proof. Let A _ μ ′ β ( x ) = ∨ { [ δ 1 , δ 2 ] : x ∈ A _ μ [ δ 1 , δ 2 ] β } = ∨ { [ δ 1 , δ 2 ] : P ( S A ( x ) , μ [ δ 1 , δ 2 ] ) ≥ β } .

We just have to prove that A _ μ β ( x ) = A _ μ ′ β ( x ) . In fact, for any x ∈ X, let [r₁, r₂] satisfy P (S _A  (x) , μ_[r1,r2]) ≥ β. If y ∈ S _{α 1}  (x), then μ (y) ≥ [r₁, r₂] and ⋀_{y∈S α 1 (x)}μ (y) ≥ [r₁, r₂], so A _ μ β ( x ) ≥ [ r 1 , r 2 ] , we have A _ μ β ( x ) ≥ A _ μ ′ β ( x ) .

Conversely, since [ r 1 , r 2 ] ≤ A _ μ ′ β ( x ) , by the definition of A _ μ ′ β ( x ) , there is y ∈ S _{α 1}  (x), so that y ∉ S _A  (x) ∩ μ_[r1,r2]. That is μ ( y ) ≤ [ r 1 , r 2 ] , and μ ( y ) ≥ [ α 1 , α 2 ] , so [ r 1 , r 2 ] ≥ A _ μ β ( x ) , therefore A _ μ ′ β ( x ) ≥ A _ μ β ( x ) , that is A _ μ ′ β ( x ) = A _ μ β ( x ) . Similarly, we can prove that (2) is true. This completes the proof.

Theorem 3.3. Let 〈U, C, D, V, f〉 be an incomplete information system. For any A ⊆ C, 0.5 < α < β ≤ 1, δ = [δ₁, δ₂] ∈ [I], the β-lower approximation and β-upper approximation of interval-valued fuzzy set μ, ω ∈ D have the following properties:

A _ μ β ⊆ μ ⊆ A ¯ μ β ,

Proof. (1) For any x ∈ U, if A _ μ β ( x ) = [ 0 , 0 ] , the conclusion clearly holds. If P δ ( S A ( x ) , μ ) = | { y ∈ S A ( x ) | μ ( y ) ≥ δ } | | S A ( x ) | ≥ β then A _ μ β ( x ) = [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] , we have ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } ≤ μ - ( x ) , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ≤ μ + ( x ) . Thus [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] ≤ [ μ - ( x ) , μ + ( x ) ] . So, we have A _ μ β ⊆ μ . Similarly, it can be proved that μ ⊆ A ¯ μ β .

(2) For any x ∈ U, since A _ μ β ( x ) = [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] A _ ω β ( x ) = [ ∧ { ω - ( y ) : y ∈ S α 1 ( x ) } , ∧ { ω + ( y ) : y ∈ S α 1 ( x ) } ] then A _ μ β ( x ) ∩ A _ ω β ( x ) = [ [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } ] ∧ [ ∧ { ω - ( y ) : y ∈ S α 1 ( x ) } ] , [ ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] ∧ [ ∧ { ω + ( y ) : y ∈ S α 1 ( x ) } ] ] = [ ∧ { μ - ( y ) ∧ ω - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) ∧ ω + ( y ) : y ∈ S α 1 ( x ) } ] = A _ μ ∩ ω β ( x ) . Similarly, it can be proved that A ¯ μ ∪ ω β = A ¯ μ β ∪ A ¯ ω β .

(3) Since A _ μ β ( x ) ∪ A _ ω β ( x ) = [ [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } ] ∨ [ ∧ { ω - ( y ) : y ∈ S α 1 ( x ) } ] , [ ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] ∨ [ ∧ { ω + ( y ) : y ∈ S α 1 ( x ) } ] ] ≤ [ ∧ { μ - ( y ) ∨ ω - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) ∨ ω + ( y ) : y ∈ S α 1 ( x ) } ] = A _ μ ∪ ω β ( x ) . Then A _ μ ∪ ω β ⊇ A _ μ β ∪ A _ ω β . Similarly, it can be proved that A ¯ μ ∩ ω β ⊆ A ¯ μ β ∩ A ¯ ω β .

(4) Since μ ⊆ ω, then μ^- (y) ≤ ω^- (y), μ⁺ (y) ≤ ω⁺ (y), so we have [ ∧ { μ - ( y ) : y ∈ S α 1 ( x ) } , ∧ { μ + ( y ) : y ∈ S α 1 ( x ) } ] ≤ [ ∧ { ω - ( y ) : y ∈ S α 1 ( x ) } , ∧ { ω + ( y ) : y ∈ S α 1 ( x ) } ] . So A _ μ β ⊆ A _ ω β . Similarly, it can be proved that if μ ⊆ ω, then A ¯ μ β ⊆ A ¯ ω β .

(5) Since ∼μ (y) =1 - μ (y), that is ∼μ (y) = [1 - μ⁺ (y) , 1 - μ^- (y)], so we have A _ ∼ μ β ( x ) = [ ∧ { 1 - μ + ( y ) : y ∈ S α 1 ( x ) } , ∧ { 1 - μ - ( y ) : y ∈ S α 1 ( x ) } ] = [ 1 - ∨ { μ + ( y ) : y ∈ S α 1 ( x ) } , 1 - ∨ { μ - ( y ) : y ∈ S α 1 ( x ) } ] = [ ∨ { μ - ( y ) : y ∈ S α 1 ( x ) } , ∨ { μ + ( y ) : y ∈ S α 1 ( x ) } ] c = ( ∼ A ¯ μ β ) ( x ) . Similarly, it can be proved that A ¯ ∼ μ β = ∼ A _ μ β .

(6) If α < β, let γ 1 = sup { δ ∈ ( 0 , 1 ] × ( 0 , 1 ] | P δ ( S A ( x ) , μ ) ≥ α } , γ 2 = sup { δ ∈ ( 0 , 1 ] × ( 0 , 1 ] | P δ ( S A ( x ) , μ ) ≥ β } , then γ₁ > γ₂, so S γ 1 = S A ( x ) ∩ μ γ 1 ⊆ S γ 2 = S A ( x ) ∩ μ γ 2 , then

A _ μ α = ⋀ y ∈ S γ 1 μ ( y ) ≥ A _ μ β = ⋀ y ∈ S γ 2 μ ( y ) , thus A _ μ β ⊆ A _ μ α . Similarly, it can be proved that A ¯ μ α ⊆ A ¯ μ β . This completes the proof.

Definition 3.3. Let 〈U, AT, V, f〉 be an incomplete interval-valued fuzzy information system, and μ be an interval-valued fuzzy set on U. For any [δ₁, δ₂] , [γ₁, γ₂] ∈ [I], the [δ₁, δ₂]-level set of μ of the lower approximation with respect to A and the [γ₁, γ₂]-level set of μ of the upper approximation with respect to A are defined as follows, respectively, ( A _ μ β ) [ δ 1 , δ 2 ] = { μ : A _ μ β ≥ [ δ 1 , δ 2 ] } = { μ : A _ μ - β ≥ δ 1 , A _ μ + β ≥ δ 2 } ( A ¯ μ β ) [ γ 1 , γ 2 ] = { μ : A ¯ μ β ≥ [ γ 1 , γ 2 ] } = { μ : A ¯ μ - β ≥ γ 1 , A ¯ μ + β ≥ γ 2 }

According to Definition (3), we have the following Theorem (3).

Theorem 3.4. Let 〈U, AT, V, f〉 be an incomplete inte rval-valued fuzzy information system, and μ be an interval-valued fuzzy set on U. For any [δ₁, δ₂] , [γ₁, γ₂] ∈ [I], the [δ₁, δ₂]-level set of μ of the lower approximation with respect to A and the [γ₁, γ₂]-level set of μ of the upper approximation with respect to A have the following properties:

( A _ μ β ) [ δ 1 , δ 2 ] ⊇ A _ μ [ δ 1 , δ 2 ] β ,

Proof. (1) By the Definition (3), we have ( A _ μ β ) [ δ 1 , δ 2 ] = { x ∈ U : A _ μ β ( x ) ≥ [ δ 1 , δ 2 ] } . We know that A _ μ [ δ 1 , δ 2 ] β = { x ∈ U : P ( S A ( x ) , μ [ δ 1 , δ 2 ] ) ≥ β } , so for any x ∈ A _ μ [ δ 1 , δ 2 ] β , μ (x) ≥ [δ₁, δ₂]. So min μ (x) ≥ [δ₁, δ₂], thus A _ μ β ( x ) ≥ [ δ 1 , δ 2 ] , we have x ∈ ( A _ μ β ) [ δ 1 , δ 2 ] . This proves that ( A _ μ β ) [ δ 1 , δ 2 ] ⊇ A _ μ [ δ 1 , δ 2 ] β .

(2) It can be proved by using a similar method as in (1).

(3) Since [δ₁, δ₂] ≥ [γ₁, γ₂], we have x ∈ ( A _ μ β ) [ δ 1 , δ 2 ] ⇒ ( A _ μ β ) ( x ) ≥ [ δ 1 , δ 2 ] ≥ [ γ 1 , γ 2 ] , so for any x ∈ U, x ∈ ( A _ μ β ) [ δ 1 , δ 2 ] ⊆ ( A ¯ μ β ) [ γ 1 , γ 2 ] . Therefore ( A _ μ β ) [ δ 1 , δ 2 ] ⊆ ( A ¯ μ β ) [ γ 1 , γ 2 ] .

(4) For any x ∈ ( A _ μ β ) [ δ 1 , δ 2 ] , we have [ δ 1 , δ 2 ] ≤ A _ μ β ( x ) . Since A _ μ β ( x ) = ∧ { μ ( y ) : y ∈ S α 1 ( x ) } ≤ ∨ { μ ( y ) : y ∈ S α 1 ( x ) } = A ¯ μ β ( x ) , so we have A ¯ μ β ( x ) ≥ [ δ 1 , δ 2 ] , that is x ∈ ( A ¯ μ β ) [ δ 1 , δ 2 ] . Therefore A _ μ [ δ 1 , δ 2 ] β ⊆ ( A _ μ β ) [ δ 1 , δ 2 ] ⊆ μ [ δ 1 , δ 2 ] ⊆ ( A ¯ μ β ) [ δ 1 , δ 2 ] ⊆ A ¯ μ [ δ 1 , δ 2 ] β . This completes the proof.

4 Decision table and decision rules based on incomplete information system

Incomplete decision table (DT) is a typical incomplete information system, DT = {U, AT ∪ {d}}, where d (d ∉ ATand ∗ ∉ V _d ) is called decision attribute, and the elements in AT are called conditional attribute. Define function δ _A  : U → P (V _d ) , A ⊆ AT as follows, δ A ( x ) = { i : i = d ( y ) , y ∈ S A ( x ) } , x ∈ U δ _A is called the generalized decision function in DT, where P (V _d ) represents the power set of V _d .

Obviously, we known that the DT is coordinated if |δ _A  (x) |=1; otherwise it is uncoordinated.

Let (U, A, D, G) be an information system or decision table, (U, A, D) is an interval-valued fuzzy information system, A is the conditional attribute set, D is the decision attribute set, D = {d₁, d₂, …, d _m }, G = { g j , j ≤ p } , ( g j : U → V j ′ , V j ′ is the range of d _j ). In information system (U, A, D, G), if U/R _A  ⊆ U/R _D , it is called coordinated information system.

Let (U, AT, D, G) be an incomplete interval-valued fuzzy information system, where ( U , AT , D ˜ ) is an incomplete interval-valued fuzzy information system, D ˜ = { D ˜ 1 , D ˜ 2 , … , D ˜ n } , D ˜ k ( k = 1 , 2 , … , n ) is an interval-valued fuzzy set, and G is a relational set on U → D.

Theorem 4.1. Let ( U , AT , D ˜ ) be an incomplete variable precision interval-valued fuzzy information system, and B ⊆ AT, if r i ( x ) = B _ D ˜ i β ( x ) > B _ D ˜ j β ( x ) , then S α 1 ( x ) = ( D ˜ i ) [ r i - ( x ) , r i + ( x ) ] , and S α 1 ( x ) ∩ ( ∼ D ˜ j ) [ r i - ( x ) , r i + ( x ) ] ≠ ∅ , where r i ( x ) = [ r i - ( x ) , r i + ( x ) ] , and r _i  (x) ≠ [0, 1], D ˜ j ∈ U / R D .

Proof. We know that r i ( x ) = [ r i - ( x ) , r i + ( x ) ] = B _ D ˜ i β ( x ) = ∧ { D ˜ i ( y ) : y ∈ S α 1 ( x ) } = [ ⋀ y ∈ S α 1 ( x ) D ˜ i - ( y ) , ⋀ y ∈ S α 1 ( x ) D ˜ i + ( y ) ] , for any x ∈ U, y ∈ S _{α 1}  (x) we have S α 1 ( x ) ⊆ ( D ˜ i ) [ r i - ( x ) , r i + ( x ) ] .

Since r i ( x ) = B _ D ˜ i β ( x ) > B _ D ˜ j β ( x ) , then ∃y ∈ S _{α 1}  (x), we have D ˜ j ( y ) < r i ( x ) . So S α 1 ( x ) ∩ ( ∼ D ˜ j ) [ r i - ( x ) , r i + ( x ) ] ≠ ∅ . This completes the proof.

If let k _x  ≤ r, r k x ( x ) = B _ D ˜ k x β ( x ) > B _ D ˜ j β ( x ) , then the decision rules of information system ( U , AT , D ˜ ) are defined as: if y ∈ S _{α 1}  (x), then y ∈ ( D ˜ k x ) [ r k x - ( x ) , r k x + ( x ) ] , the accuracy of decision-making is as follows: r k x ( x ) = B _ D ˜ k x β ( x ) .

The specific calculation steps are as follows,

Step 1. According to the conditional attribute set AT and the similarity relation SIM (A on the incomplete interval-valued information system, S _AT  (x _i ) , x _i  ∈ U is calculated, respectively.

Step 2. According to Definition (3), the lower approximation operators of the element x _i in U are calculated, respectively.

Step 3. According to the above calculation results, the corresponding decision rules are given.

Step 4. End.

Example 4.1. Table 1 presents an incomplete interval-valued fuzzy information system, in which U = {x₁, x₂, x₃, x₄, x₅, x₆} is the object set, AT = {a, b, c, d} is the conditional attribute set, D ˜ = { D ˜ 1 , D ˜ 2 , D ˜ 3 , D ˜ 4 } is the decision attribute set, where D ˜ i ∈ F i ( U ) , i = 1 , 2 , 3 , 4 . Let β = 0.65.

Using the method proposed in Definition (3), the following results are obtained: U / SIM ( AT ) = { S AT ( x 1 ) , S AT ( x 2 ) , S AT ( x 3 ) , S AT ( x 4 ) , S AT ( x 5 ) , S AT ( x 6 ) } where S AT ( x 1 ) = { x 1 } , S AT ( x 2 ) = { x 2 , x 6 } , S AT ( x 3 ) = { x 3 } , S AT ( x 4 ) = { x 4 , x 5 } , S AT ( x 5 ) = { x 4 , x 5 , x 6 } , S AT ( x 6 ) = { x 2 , x 5 , x 6 } .

Thus, we have

for S _AT  (x₁) = {x₁}, there are AT _ D ˜ 1 0.65 ( x 1 ) = [ 0.50 , 0.80 ] , AT _ D ˜ 2 0.65 ( x 1 ) = [ 0.45 , 0.70 ] , AT _ D ˜ 3 0.65 ( x 1 ) = [ 0.20 , 0.60 ] , AT _ D ˜ 4 0.65 ( x 1 ) = [ 0.30 , 0.50 ] .