Covering-based generalized variable precision fuzzy rough set

Abstract

Zhan and Jiang defined covering-based compact and loose variable precision fuzzy rough set models. Soon after, they proposed a reflexive fuzzy β-neighborhood operator and defined a covering-based generalized variable precision fuzzy rough set. Based on them, in this paper, we use the reflexive fuzzy β-neighborhood operator to establish two covering-based generalized variable precision fuzzy rough set models, which are called covering-based generalized compact and loose variable precision fuzzy rough set models, respectively. Then, we investigate the important properties of the two rough set models and their relationship to the original models. Finally, we apply the covering-based generalized compact variable precision fuzzy rough set model to decision-making problems. A simple example is given to verify the validity of the model and compare the results with other models.

Keywords

Fuzzy rough set compact loose fuzzy logical operator

1 Introduction

Pawlak [24] proposed rough set theory in 1982, which can approximately describe any subset of a universe by using two definable subsets: the upper and lower approximations [5] of sets. Rough set theory can deal with uncertainty and incompleteness in information system and data analysis. One of the basic assumptions of rough set theory is the indiscernibility relation between objects. It is defined by a partition of equivalence relations. Therefore, Pawlak rough sets have some limitations [24]. To overcome this shortcoming, researchers began various generalizations. Many researchers replace equivalence relations or partitions with general binary relations [29, 30] or concepts such as neighborhood systems and Boolean algebras [32]. Since covering is a generalization of partition, many researchers naturally extend it to covering-based rough sets. Among them, Zakowski [40] firstly extended the equivalence class to covering and proposed the covering-based rough set model. However, the upper and lower approximation operators of this model do not satisfy the duality property. In order to solve this problem, Pomykala [25] studied approximation operators in approximation space and he defined two pairs of dual approximation operators: covering-based compact rough set model and covering-based loose rough set model. The results have been widely recognized and popularized by researchers. Yao and Yao [37] proposed a covering-based rough set approximation method on the basis of neighborhood system. Zhu and Wang [45] introduced three types of covering rough set models. Xu and Zhang [33] introduced roughness measurement of covering-based generalized rough set model. In particular, in 1998, Yao [36] proposed a covering rough set model based on neighborhood system on the basis of the concept of neighborhood and granularity. Then some covering rough set models based on neighborhood and topological properties of upper and lower approximation operators in combination with topology was studied by Ma [18]. After that, he also discussed twin approximation operators in covering approximation spaces [19]. In 2016, D^′eer et al. [10] classified the neighborhood operators in the covering approximation space in detail and constructed a partial order among these neighborhood operators. These studies have greatly advanced the field. However, covering rough set also has limitations, and it can only be used for symbol data processing, so fuzzy subset theory was introduced, and then researchers put forward a new model.

Fuzzy sets are different from rough sets. Fuzzy sets are used to deal with qualitative data. They can deal with fuzzy concepts and some indiscernibility relation effectively. But fuzzy subset faces great limitations when dealing with real valued data sets [12]. As we know, attribute values in database may be symbol values or real values, so both rough set theory and fuzzy subset theory are needed when dealing with uncertain and incomplete information in information system. In 1990, Dubois and Prade [7] first proposed the concept of fuzzy rough set which can effectively deal with real values data. Radzikowska and Kerre [26] defined the upper and lower approximation operators of fuzzy subsets using the general boundary implicators in 2002. After that, fuzzy rough set model based on fuzzy logic operator has been studing by many researchers. Hu and Wong [21] constructed generalized interval valued fuzzy rough set and generalized interval valued fuzzy variable precision rough sets on the basis of fuzzy logic operators. In addition, L-fuzzy rough set is studied on the basis of constructional method and axiomatic method. Similar to the thought of studying classical rough sets, people began to replace fuzzy relation with fuzzy covering. Similarly, covering-based fuzzy rough set model was defined. For example, Li et al. [16] studied two pairs of covering-based fuzzy rough approximation operators, which are called covering-based compact fuzzy rough approximation and covering-based loose fuzzy rough approximation respectively. Subsequently, D^′eer et al. [8] defined four types of fuzzy neighborhood operators and studied their related properties. In addition, Ma [20] introduced the concepts of fuzzy β-covering and fuzzy β-neighborhood, and he came up with two covering-based fuzzy rough set models. Many researchers have also generalized [23, 38] this finding.

However, these rough set models have a common problem: it is sensitive to the classification of data, that is, each object is required to obtain an evaluation value of 1 on at least one criterion. As a result, they are limited in the process of use. Then, researchers began to explore the stability of rough set models. Ziarko [41] proposed a variable precision rough set model based on the classical rough set. Subsequently, many people use the idea of variable precision [3 , 31] to study fuzzy rough sets. Zhao et al. [44] proposed a variable precision fuzzy rough set model. On the basis of fuzzy logic operator and the thought of variable precision, Jiang et al. [13] defined a covering-based variable precision fuzzy rough set on fuzzy β-covering, and studied its application in multi-attribute decision making. After that, Jiang and Zhan [14] defined the covering-based compact variable precision fuzzy rough set model and the covering-based loose variable precision fuzzy rough set model, and their related properties were studied. Recently, they defined a kind of reflexive fuzzy β-neighborhood [43] on fuzzy β-covering approximation space, and defined a covering-based generalized variable precision fuzzy rough set model [43] on fuzzy logic operator and the thought of variable precision, then a new multi-attribute decision making method was established.

Fuzzy set [11 , 27] and rough set [1 , 35] have made irreplaceable achievements in describing uncertain problems. So in this paper, we will combine the advantages of covering-based fuzzy rough set and the thought of compact and loose, to introduce the thought of compact and loose into the covering-based variable precision fuzzy rough set model. Then we study two kinds of covering-based generalized variable precision fuzzy rough set models and discuss their properties. This model can reduce the requirement of attribute values, that is, the object doesn’t have to obtain an evaluation value of 1 to satisfy some conclusions. Moreover, this model is a generalization of [14], and it can be used in a wider range, for example in decision-making problems.

The rest of this paper is organized as follows. Section 2 reviews the main results from existing studies about fuzzy logic operator, fuzzy neighborhood, and fuzzy covering. The major contributions of this paper are covered in Sections 3, 4 and Sections 5. In Section 3, the covering-based generalized compact and loose variable precision fuzzy rough set models are defined. And some of their detail properties are studied in Subsections 3.1 and 3.2. Section 4 presents the relationship of lower and upper approximations for two kinds of newly defined fuzzy rough sets, and their relationship to the original model. Section 5 presents an example using covering-based generalized compact variable precision fuzzy rough set model to illustrate the model’s effectiveness and variety. It can provide decision makers with more options. Section 6 is a summary of this paper.

2 Preliminaries

In this section, we present some concepts such as t-norm, R-implicator, fuzzy β-neighborhood, fuzzy topology and so on.

2.1 Fuzzy logic operator

Definition 2.1. [22 , 30] A binary operator $T : [0, 1]^{2} \to [0, 1]$ is called a triangle norm (or t-norm in short), if for all u, v, w, z ∈ [0, 1],

(1) $T (u, 1) = u$ ;

(2) $T (u, v) = T (v, u)$ ;

(3) $T (T (u, v), w) = T (u, T (v, w))$ ;

(4) $u \leq z, v \leq w \Rightarrow T (u, v) \leq T (z, w)$ .

Further, if for an arbitrary index set Λ and ∀i ∈ Λ u, v_i ∈ [0, 1], $T (u, \lor_{i \in I} v_{iitsc}) = \lor_{i \in I} T (u, v_{iitsc})$ , then $T$ is called a left continuous t-norm.

The most popular t-norms are:

•the standard min operator:

$T (u, v) = \min {u, v}$ (the largest t-norm);

•the algebraic product: $T (u, v) = u \cdot v$ ;

•the Łukasiewicz t-norm:

$T (u, v) = \max {0, u + v - 1}$ .

Definition 2.2. [16, 26] Let $N$ be a decreasing unary operation on interval [0, 1]İf $N$ satisfies $N (1) = 0$ and $N (0) = 1$ , then $N$ is called a negator.

Further, if $N (N (u)) = u (\forall u \in [0, 1])$ , then $N$ is called an involutive negator. In particular, if $N (u) = 1 - u (\forall u \in [0, 1])$ , then $N$ is called a standard negator.

Definition 2.3. [16 , 28] A binary operator $I : [0, 1]^{2} \to [0, 1]$ is called a fuzzy implicator, if for all u, v, w ∈ [0, 1],

(1) $u \leq v \Rightarrow I (u, w) \geq I (v, w)$ ;

(2) $v \leq w \Rightarrow I (u, v) \leq I (u, w)$ ;

(3) $I (0, 0) = 1, I (1, 1) = 1, I (1, 0) = 0$ .

Definition 2.4. [16 , 28] Given a left continuous triangular norm $T$ , the following binary operation on [0, 1], $I (u, v) = \lor {w \in [0, 1] | T (u, w) \leq v} (\forall u, v \in [0, 1])$ is called the residuation implicator of $T$ , or the R-implicator.

The most popular R-implicators are:

•the Łukasiewicz implicator:

$I (u, v) = \min {1, 1 - u + v}$ ;

•the Kleene-Dienes implicator:

$I (u, v) = \max {1 - u, v}$ ;

•the Godel implicator: for all u ≤ v,

$I (u, v) = 1$ , otherwise, $I (u, v) = v$ ;

•the Gaines implicator: for all u ≤ v,

$I (u, v) = 1$ , otherwise, $I (u, v) = \frac{v}{u}$ .

Remark 2.5. In this article, we always assume that X is a finite universe, Λ is a nonempty index set, $T$ is a left continuous t-norm, $I$ is a R-implicator, $N$ is the negator induced by $I ($ which means that $N (a) = I (a, 0))$ , and $N$ satisfies $N (N (a)) = a$ .

Lemma 2.6. [16] The $T$ , $I$ and $N$ defined as above have the following properties:

(1) $T (u, 0) = 0, T (u, v) \leq u,$

$u \leq v \Leftrightarrow I (u, v) = 1$ .

(2) $T (u I (u v)) \leq v,$

$v \leq I (u T (u, v))$

$u \leq I (I (u, v), v)$ .

(3) $I (T (u, v), w) = I (u, I (v, w)) .$

(4) $I (u, \land_{i \in Λ} v_{iitsc}) = \land_{i \in Λ} I (u, v_{iitsc}),$

$I (\lor_{i \in λ} u_{i}, v) = \land_{i \in λ} I (u_{i}, v) .$

(5) $I (\land_{i \in λ} u_{iitsc}, v) \geq \lor_{i \in λ} I (u_{iitsc}, v),$

$I (u, \lor_{i \in λ} v_{i}) \geq \lor_{i \in λ} I (u, v_{i}) .$

(6) $T (b, \lor_{i \in Λ} a_{iitsc}) = \lor_{i \in Λ} T (b, a_{iitsc})$ ,

$T (b, \land_{i \in Λ} a_{iitsc}) \leq \land_{i \in Λ} T (b, a_{iitsc})$ .

(7) $I (a N (b)) = N (T (a, b))$ ,

$T (a, N (b)) \leq N (I (a, b))$ .

2.2 Fuzzy covering, fuzzy neighborhood and fuzzy topology

Definition 2.7. [39] Let X be a finite nonempty universe. A fuzzy subset F is a function of X to the interval [0, 1]. The set family of all fuzzy subsets on X is denoted as $𝔽 (X)$ .

Fuzzy subsets $A, B \in 𝔽 (X)$ , if for all a ∈ X, A (a) ≤ B (a), then we denote A ⊆ B. A fuzzy subset F satisfying F (x) = γ for all x ∈ X denoted as γ_X. In addition, for all fuzzy subset $A \in 𝔽 (X)$ , the symbol co_N (A) represents the fuzzy complement of the fuzzy subset A determined by the negator $N$ , i.e., ${co}_{N} (A) (a) = N (A (a))$ . A fuzzy subset $R \in 𝔽 (X \times X)$ is called a fuzzy binary relation on X.

Definition 2.8. [8, 9] Let $ℂ = {C_{i} \in 𝔽 (X) | C_{i} \neq \emptyset, i \in Λ}$ . If for all a ∈ X, there exists a $C \in ℂ$ such that C (a) =1, then the set $ℂ$ is called a fuzzy covering on X, and the pair $(X, ℂ)$ is called a fuzzy covering approximation space.

Definition 2.9. [20] Let $ℂ = {C_{i} \in 𝔽 (X) | C_{i} \neq \emptyset, i \in Λ}$ and β ∈ (0, 1]. If for all a ∈ X, there are (⋃ _i∈ΛC_i) (a) ≥ β, then the set $ℂ$ is called a fuzzy β-covering. The pair $(X, ℂ)$ is called a fuzzy β-covering approximation space.

Definition 2.10. [34]Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all a ∈ X, the fuzzy β-minimum description of a is defined by: ${\tilde{Md}}_{a}^{β} = {C \in ℂ | (C (a) \geq β) \land ((\forall D \in ℂ) \land (D (a) \geq β) \land (D \subseteq C) \Rightarrow C = D)}$ .

Definition 2.11. [43] Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all a, b ∈ X, the fuzzy β-neighborhood of a is defined by: $ℕ_{β} (a) (b) = ⋀_{C \in {\tilde{Md}}_{a}^{β}} I (C (a), C (b)) .$

Proposition 2.12. [43] Let $(X, ℂ)$ be a fuzzy β-covering approximation space, then the operator $ℕ_{β}$ has the following properties:

(1) $ℕ_{β}$ is reflexive, that is, for all a ∈ X,

$ℕ_{β} (a) (a) = 1$ .

(2) $ℕ_{β}$ is $T$ -transitive, that is, for all a, b, c ∈ X,

$T (N_{β} (a) (b), N_{β} (b) (c)) \leq N_{β} (a) (c)$ .

Definition 2.13. [43] Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all $F \in 𝔽 (X)$ and parameter γ ∈ [0, 1], the covering-based generalized variable precision fuzzy rough upper and lower approximation operator are defined as follows: for all a ∈ X, $\begin{array}{l} {\bar{C}}_{ℕ_{β}}^{γ} (F) (a) = \underset{b \in X}{v} T (ℕ_{β} (a) (b),^F (b)), \\ {\underline{C}}_{ℕ_{β}}^{γ} (F) (a) = \underset{b \in X}{^} T (ℕ_{β} (a) (b), v F (b)), \end{array}$

The pair $({\bar{C}}_{ℕ_{β}}^{γ} (F), {\underline{C}}_{ℕ_{β}}^{γ} (F))$ is called the covering-based generalized variable precision fuzzy rough set of fuzzy subset F on X.

Proposition 2.14. [43] Let $(X, ℂ)$ be a fuzzy β-covering approximation space, then for all fuzzy subsets $F, H, F_{i} \in 𝔽 (X)$ , γ, λ ∈ [0, 1), the following properties hold:

(1) ${\underline{C}}_{ℕ_{β}}^{γ} (X) = X,$

${\bar{C}}_{ℕ_{β}}^{γ} (\emptyset) = \emptyset .$

(2) ${\underline{C}}_{ℕ_{β}}^{γ} (F) \subseteq γ_{X} \cup F,$

$N (γ)_{X} \cap F \subseteq {\bar{C}}_{ℕ_{β}}^{γ} (F) .$

(3) ${\underline{C}}_{ℕ_{β}}^{γ} (u_{X}) = (γ \lor u)_{X}$

${\bar{C}}_{ℕ_{β}}^{γ} (u_{X}) = (N (γ) \land u)_{X} .$

(4) if γ ≤ λ, then ${\underline{C}}_{ℕ_{β}}^{γ} (F) \subseteq {\underline{C}}_{ℕ_{β}}^{λ} (F),$

${\bar{C}}_{ℕ_{β}}^{λ} (F) \subseteq {\bar{C}}_{ℕ_{β}}^{γ} (F) .$

(5) if F ⊆ H, then ${\underline{C}}_{ℕ_{β}}^{γ} (F) \subseteq {\underline{C}}_{ℕ_{β}}^{γ} (H),$

${\bar{C}}_{ℕ_{β}}^{γ} (F) \subseteq {\bar{C}}_{ℕ_{β}}^{γ} (H) .$

(6) ${\underline{C}}_{ℕ_{β}}^{γ} (\cap_{i \in Λ} F_{i}) = \cap_{i \in Λ} {\underline{C}}_{ℕ_{β}}^{γ} (F_{i})$

${\bar{C}}_{ℕ_{β}}^{γ} (\cup_{i \in Λ} F_{i}) = \cup_{i \in Λ} {\bar{C}}_{ℕ_{β}}^{γ} (F_{i}) .$

(7) ${\underline{C}}_{ℕ_{β}}^{γ} (\cup_{i \in Λ} F_{i}) \supseteq \cup_{i \in Λ} {\underline{C}}_{ℕ_{β}}^{γ} (F_{i})$

(8) ${\underline{C}}_{ℕ_{β}}^{γ} (γ_{X} \cup F) = {\underline{C}}_{ℕ_{β}}^{γ} (F)$

(9) ${\bar{C}}_{ℕ_{β}}^{γ} ({\bar{C}}_{ℕ_{β}, γ}^{'} (F)) = γ_{X} \cup {\underline{C}}_{ℕ_{β}}^{γ} (F)$

${\bar{C}}_{ℕ_{β}}^{γ} ({\bar{C}}_{ℕ_{β}, γ}^{'} (F)) = N (γ)_{X} \cap {\bar{C}}_{ℕ_{β}}^{γ} (F) .$

(10) ${\underline{C}}_{ℕ_{β}}^{γ} (F) \subseteq {co}_{N} ({\bar{C}}_{ℕ_{β}}^{γ} ({co}_{N} (F))),$

${\bar{C}}_{ℕ_{β}}^{γ}^{'} (F) \subseteq {co}_{N} ({\underline{C}}_{ℕ_{β}}^{γ} ({co}_{N} (F))) .$

Definition 2.15. [17] A fuzzy subset family $T$ is called a fuzzy topology on X if the following conditions hold:

(1) For all constant α ∈ [0, 1], $α \in T$ ,

(2) For all $u, v \in T$ , $u \land v \in T$ ,

(3) For all ${u_{i} | i \in Λ} \subseteq T$ , $⋃_{i \in Λ} u_{i} \in T$ .

3 Covering-based generalized compact and loose variable precision fuzzy rough set models

In this section, on the basis of Definition 2.13, two pairs of fuzzy rough sets are defined, which called covering-based generalized compact variable precision fuzzy rough set and covering-based generalized loose variable precision fuzzy rough set. Moreover, their detail properties are also studied.

Definition 3.1. Let $(X, ℂ)$ be a finite fuzzy β-covering approximation space. For all $F \in 𝔽 (X)$ and parameter γ ∈ [0, 1) (γ → 0), the covering-based generalized compact and loose variable precision fuzzy rough upper and lower approximation operator are defined as follows: for all a ∈ X,

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋁_{b \in X} T (ℕ_{β} (b) (a),$

$⋀_{c \in X} I (ℕ_{β} (b) (c), γ \lor F (c)))$ ,

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋀_{b \in X} I (ℕ_{β} (b) (a),$

$⋁_{c \in X} T (ℕ_{β} (b) (c), N (γ) \land F (c)))$ ,

$\begin{matrix} {\underline{C}}_{ℕ_{β}, γ}'' (F) (a) = \underset{b \in X}{⋁} ℐ (ℕ_{β} (b) (a), \end{matrix} \begin{matrix} \underset{c \in X}{⋁} ℐ (ℕ_{β} (b) (c), γ \lor F (c))) \end{matrix}$

$⋀_{c \in X} I (ℕ_{β} (b) (c), γ \lor F (c)))$ ,

$\begin{matrix} {\underline{C}}_{ℕ_{β}, γ}'' (F) (a) = \underset{b \in X}{⋁} ℐ (ℕ_{β} (b) (a), \\ \underset{c \in X}{⋁} ℐ (ℕ_{β} (b) (c), γ \lor F (c))) \end{matrix}$

$⋁_{c \in X} T (ℕ_{β} (b) (c), N (γ) \land F (c)))$ .

The pair $({\underline{C}}_{ℕ_{β}, γ}^{'} (F), {\bar{C}}_{ℕ_{β}, γ}^{'} (F))$ is called covering-based generalized compact variable precision fuzzy rough set. Similarly, the pair $({\underline{C}}_{ℕ_{β}, γ}'' (F), {\bar{C}}_{ℕ_{β}, γ}'' (F))$ is called the covering-based generalized loose variable precision fuzzy rough set.

Remark 3.2. (1) Definition 3.1 is two pairs of upper and lower approximations generalized from a pair of upper and lower approximations of definition 3.6 in [43]. And Definition 3.2 in [14] is a special case of Definition 3.1 with β=1.

(2) When γ = 0, the covering-based generalized compact and loose variable precision fuzzy rough set models in Definition 3.1 becomes the following forms: for all a ∈ X,

${\underline{C}}_{ℕ_{β}, 0}^{'} (F) (a) = ⋁_{b \in X} T (ℕ_{β} (b) (a),$

$⋀_{c \in X} I (ℕ_{β} (b) (c), F (c)))$ ,

${\bar{C}}_{ℕ_{β}, 0}^{'} (F) (a) = ⋀_{b \in X} I (ℕ_{β} (b) (a),$

$⋁_{c \in X} T (ℕ_{β} (b) (c), F (c)))$ ,

$\begin{matrix} {\underline{C}}_{ℕ_{β}, 0}'' (F) (a) = \underset{b \in X}{⋁} ℐ (ℕ_{β} (b) (a), \\ \underset{c \in X}{⋁} ℐ (ℕ_{β} (b) (c), γ \lor F (c))) \end{matrix}$

$⋀_{c \in X} I (ℕ_{β} (b) (c), F (c)))$ ,

$\begin{matrix} {\underline{C}}_{ℕ_{β}, 0}'' (F) (a) = \underset{b \in X}{⋁} ℐ (ℕ_{β} (b) (a), \\ \underset{c \in X}{⋁} ℐ (ℕ_{β} (b) (c), γ \lor F (c))) \end{matrix}$

$⋁_{c \in X} T (ℕ_{β} (b) (c), F (c)))$ .

The pair $({\underline{C}}_{ℕ_{β}}^{0}^{'} (F), {\bar{C}}_{ℕ_{β}}^{0}^{'} (F))$ is called covering-based generalized compact fuzzy rough set. Similarly, the pair $({\underline{C}}_{N}^{o}^{' '} (F), {\bar{C}}_{N_{β}}^{o} (F))$ is called the covering-based generalized loose fuzzy rough set.

(3) Fuzzy neighborhood operator $ℕ_{β}$ can be regarded as a special kind of fuzzy binary relation. Therefore, if fuzzy binary relation R is used to replace fuzzy neighborhood operator $ℕ_{β}$ , then Definition 3.1 becomes the following forms: for all a ∈ X,

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋁_{b \in X} T (R (b, a)$

$⋀_{c \in X} I (ℕ_{β} (b) (c), γ \lor F (c)))$ ,

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋀_{b \in X} I (R (b, a)$

$⋁_{c \in X} T (ℕ_{β} (b) (c), N (γ) \land F (c)))$ ,

${\underline{C}}_{ℕ_{β, γ}}^{' '} (F) (a) =^_{b \in X} I (R (b, a),^_{c \in X} I (ℕ_{β} (b) (c), γ ⋁ F (c))),$ ,

${\underline{C}}_{ℕ_{β, γ}}^{' '} (F) (a) =^_{b \in X} I (R (b, a),^_{c \in X} I (ℕ_{β} (b) (c), γ \lor F (c))),$ .

(4) If F is a classical set, then Definition 3.1 becomes the following forms: for all a ∈ X,

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋁_{b \in X} T (ℕ_{β} (b) (a)$

$⋀_{F (c) = 0} I (ℕ_{β} (b) (c), γ))$ ,

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋀_{b \in X} I (ℕ_{β} (b) (a)$

$⋁_{F (c) = 1} T (ℕ_{β} (b) (c), N (γ)))$ ,

${\underline{C}}_{ℕ_{β, γ}}^{' '} (F) (a) =^_{b \in X} I (ℕ_{β} (b) (a), V_{F_{(c) = 0}} I (ℕ_{β} (b) (c), γ)),$ ,

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) (a) = ⋀_{b \in X} I (ℕ_{β} (b) (a)$

$⋁_{F (c) = 1} T (ℕ_{β} (b) (c), N (γ)))$ .

3.1 Properties of covering-based generalized compact variable precision fuzzy rough set model

The properties of covering-based generalized compact variable precision fuzzy rough set model are studied in this section. For the unsatisfied property, we give the necessary and sufficient conditions to make it hold.

Proposition 3.3. Let $(X, ℂ)$ be a fuzzy β-covering approximation space, for any $F, H, F_{i} \in 𝔽 (X)$ , and γ ∈ [0, 1), the following properties hold:

(1) ${\underline{C}}_{ℕ_{β}, γ}^{'} (X) = X,$

(2) ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \subseteq γ_{X} \cup F,$

${N (γ)}_{X} \cap F \subseteq {\bar{C}}_{ℕ_{β}, γ}^{'} (F) .$

(3) ${\underline{C}}_{ℕ_{β}, γ}^{'} (μ_{X}) = (γ \lor μ)_{X},$

(4) If γ ≤ λ, then ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \subseteq {\underline{C}}_{ℕ_{β}, λ}^{'} (F)$

(5) If F ⊆ H, then ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{'} (H)$

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) \subseteq {\bar{C}}_{ℕ_{β}, γ}^{'} (H) .$

(6) ${\underline{C}}_{ℕ_{β}, γ}^{'} (γ_{X} \cup F) = {\underline{C}}_{ℕ_{β}, γ}^{'} (F),$

(7) ${\underline{C}}_{ℕ_{β}, γ}^{'} (⋃_{i \in Λ} F_{i}) \supseteq ⋃_{i \in Λ} {\underline{C}}_{ℕ_{β}, γ}^{'} (F_{i})$

(8) ${\underline{C}}_{ℕ_{β}, γ}^{'} (⋂_{i \in Λ} F_{i}) \subseteq ⋂_{i \in Λ} {\underline{C}}_{ℕ_{β}, γ}^{'} (F_{i})$

(9) ${\underline{C}}_{ℕ_{β}, γ}^{'} ({\underline{C}}_{ℕ_{β}, γ}^{'} (F)) = γ_{X} \cup {\underline{C}}_{ℕ_{β}, γ}^{'} (F),$

(10) ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \subseteq {co}_{N} ({\bar{C}}_{ℕ_{β}, γ}^{'} ({co}_{N} (F))),$

Proof. Propositions (1) to (8) can be directly proved by Definition 3.1, now we will give a proof of propositions (9) and (10). For all a ∈ X,

(9) ${\underline{C}}_{ℕ_{β}, γ}^{'} ({\underline{C}}_{ℕ_{β}, γ}^{'} (F)) (a)$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a) ⋀_{c \in X} I (ℕ_{β} (b) (c)$

$γ \lor {\underline{C}}_{ℕ_{β}, γ}^{'} (F) (c)))$

$\geq ⋁_{b \in X} T (ℕ_{β} (b) (a), (⋀_{c \in X} I (ℕ_{β} (b) (c), γ) \lor$

$⋀_{c \in X} I (ℕ_{β} (b) (c), {\underline{C}}_{ℕ_{β}, γ}^{'} (F) (c))))$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} I (ℕ_{β} (b) (c), γ)) \lor$

$⋁_{b \in X} T (ℕ_{β} (b) (a) ⋀_{c \in X} I (ℕ_{β} (b) (c)$

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) (c)))$

$= {\underline{C}}_{ℕ_{β}, γ}^{'} (γ_{X}) (a) \lor ⋁_{b \in X} T (ℕ_{β} (b) (a),$

$⋀_{c \in X} I (ℕ_{β} (b) (c), {\underline{C}}_{ℕ_{β}, γ}^{'} (F) (c)))$

$= γ_{X} (a) \lor ⋁_{b \in X} T (ℕ_{β} (b) (a),$

$⋀_{c \in X} I (ℕ_{β} (b) (c), {\underline{C}}_{ℕ_{β}, γ}^{'} (F) (c)))$ .

Moreover,

$⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} I (ℕ_{β} (b) (c),$

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) (c)))$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} I (ℕ_{β} (b) (c)$

$⋁_{b^{'} \in X} T (ℕ_{β} (b^{'}) (c), ⋀_{c^{'} \in X} I (ℕ_{β} (b^{'}) (c^{'}),$

γ ∨ F (c′)))))

$\geq ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} I (ℕ_{β} (b) (c),$

$T (N_{β} (b) (c), ⋀_{c^{'} \in X} I (N_{β} (b) (c^{'}),$

γ ∨ F (c′)))))

$\geq ⋁_{b \in X} T (ℕ_{β} (b) (a)$

$⋀_{c \in X} (⋀_{c^{'} \in X} I (ℕ_{β} (b) (c^{'}), γ \lor F (c^{'}))))$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a) ⋀_{c^{'} \in X} I (ℕ_{β} (b) (c^{'})$

$γ \lor F (c^{'}))) = {\underline{C}}_{ℕ_{β}, γ}^{'} (F) (a)$ .

So, we get ${\underline{C}}_{ℕ_{β}, γ}^{'} ({\underline{C}}_{ℕ_{β}, γ}^{'} (F)) \supseteq γ_{X} \cup {\underline{C}}_{ℕ_{β}, γ}^{'} (F) .$

And then from Proposition 3.3 (2) we can get ${\underline{C}}_{ℕ_{β}, γ}^{'} ({\underline{C}}_{ℕ_{β}, γ}^{'} (F)) \subseteq γ_{X} \cup {\underline{C}}_{ℕ_{β}, γ}^{'} (F) .$

In conclusion, ${\underline{C}}_{ℕ_{β}, γ}^{'} ({\underline{C}}_{ℕ_{β}, γ}^{'} (F)) = γ_{X} \cup {\underline{C}}_{ℕ_{β}, γ}^{'} (F) .$

Similarly, we have ${\bar{C}}_{ℕ_{β}, γ}^{'} ({\bar{C}}_{ℕ_{β}, γ}^{'} (F)) = {N (γ)}_{X} \cap {\bar{C}}_{ℕ_{β}, γ}^{'} (F) .$

(10) ${co}_{N} ({\bar{C}}_{ℕ_{β}, γ}^{'} ({co}_{N} (F))) (a)$

$= N ({\bar{C}}_{ℕ_{β}, γ}^{'} ({co}_{N} (F)) (a))$

$= N (⋀_{b \in X} I (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$N (γ) \land {co}_{N} (F) (c))))$

$= ⋁_{b \in X} N (I (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$N (γ) \land N (F (c)))))$

$\geq ⋁_{b \in X} T (ℕ_{β} (b) (a), N (⋁_{c \in X} T (ℕ_{β} (b) (c),$

$N (γ) \land N (F (c)))))$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} N (T (ℕ_{β} (b) (c),$

$N (γ) \land N (F (c)))))$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} I (ℕ_{β} (b) (c), γ \lor F (c)))$

$= {\bar{C}}_{ℕ_{β}, γ}^{'} (F) (a)$ .

That is to say,

Similarly, we have

□ In general, the following conclusions does not hold for approximation operators ${\underline{C}}_{ℕ_{β}, γ}^{'}$ and ${\bar{C}}_{ℕ_{β}, γ}^{'}$ : ${\bar{C}}_{ℕ_{β, γ}}^{'} (F \cap H) = {\underline{C}}_{ℕ_{β, γ}}^{'} (F) \cap {\underline{C}}_{ℕ_{β, γ}}^{'} (H) .$ (1) ${\underline{C}}_{ℕ_{β, γ}}^{'} (F \cap H) = {\bar{C}}_{ℕ_{β, γ}}^{'} (F) \cup {\bar{C}}_{ℕ_{β, γ}}^{'} (H) .$ (2)

Example 3.4. Let X = {a₁, a₂, a₃, a₄, a₅}, a family of fuzzy subsets $ℂ = {C_{1}, C_{2}, C_{3}, C_{4}}$ on X are given by: $C_{1} = \frac{0.26}{a_{1}} + \frac{0.37}{a_{2}} + \frac{1.00}{a_{3}} + \frac{0.57}{a_{4}} + \frac{0.45}{a_{5}}$ $C_{2} = \frac{0.60}{a_{1}} + \frac{0.55}{a_{2}} + \frac{0.41}{a_{3}} + \frac{0.76}{a_{4}} + \frac{0.23}{a_{5}}$ $C_{3} = \frac{0.19}{a_{1}} + \frac{0.70}{a_{2}} + \frac{0.32}{a_{3}} + \frac{0.68}{a_{4}} + \frac{0.71}{a_{5}}$ $C_{4} = \frac{0.72}{a_{1}} + \frac{0.52}{a_{2}} + \frac{0.74}{a_{3}} + \frac{0.16}{a_{4}} + \frac{0.64}{a_{5}}$ Clearly, $ℂ$ is a fuzzy β-covering (β ∈ (0, 0.70]) on X. If β=0.60, then according to Definition 2.10, the fuzzy β-minimum description of a_i (i = 1, 2, …, 5) are:

${\tilde{Md}}_{a_{1}}^{0.60} = {C_{2}, C_{4}}$ , ${\tilde{Md}}_{a_{2}}^{0.60} = {C_{3}}$ , ${\tilde{Md}}_{a_{3}}^{0.60} = {C_{1}, C_{4}}$ , ${\tilde{Md}}_{a_{4}}^{0.60} = {C_{2}, C_{3}}$ , ${\tilde{Md}}_{a_{5}}^{0.60} = {C_{3}, C_{4}}$ .

Let $I$ be the Łukasiewicz implicator, then according to Definition 2.11 the neighborhood of object $ℕ_{β} (a_{i})$ of a_i (i=1,2,3,4,5) are given in Table 1.

Table 1

Fuzzy β-neighborhood (β=0.60) of object a_i

$ℕ_{0.60} (a_{i}) / X$	a ₁	a ₂	a ₃	a ₄	a ₅
$ℕ_{0.60} (a_{1})$	1.00	0.80	0.81	0.44	0.63
$ℕ_{0.60} (a_{2})$	0.49	1.00	0.62	0.98	1.00
$ℕ_{0.60} (a_{3})$	0.26	0.37	1.00	0.42	0.45
$ℕ_{0.60} (a_{4})$	0.51	0.79	0.64	1.00	0.47
$ℕ_{0.60} (a_{5})$	0.48	0.88	0.61	0.52	1.00

Suppose that two fuzzy subsets on X are defined as follows: $F = \frac{0.26}{a_{1}} + \frac{0.71}{a_{2}} + \frac{0.15}{a_{3}} + \frac{0.43}{a_{4}} + \frac{0.84}{a_{5}}$ Then, $F \cap H = \frac{0.26}{a_{1}} + \frac{0.61}{a_{2}} + \frac{0.15}{a_{3}} + \frac{0.17}{a_{4}} + \frac{0.76}{a_{5}},$ $F \cup H = \frac{0.92}{a_{1}} + \frac{0.71}{a_{2}} + \frac{0.44}{a_{3}} + \frac{0.43}{a_{4}} + \frac{0.84}{a_{5}} .$ Take $T$ as the Łukasiewicz t-norm, γ=0.2, then by the Definition 2.13, we can get ${\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (F) = \frac{0.26}{a_{1}} + \frac{0.47}{a_{2}} + \frac{0.20}{a_{3}} + \frac{0.43}{a_{4}} + \frac{0.59}{a_{5}},$ ${\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (H) = \frac{0.63}{a_{1}} + \frac{0.56}{a_{2}} + \frac{0.44}{a_{3}} + \frac{0.20}{a_{4}} + \frac{0.68}{a_{5}},$ ${\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (F \cap H) = \frac{0.11}{a_{1}} + \frac{0.47}{a_{2}} + \frac{0.20}{a_{3}} + \frac{0.20}{a_{4}} + \frac{0.23}{a_{5}} .$ And ${\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (F) = \frac{0.51}{a_{1}} + \frac{0.71}{a_{2}} + \frac{0.25}{a_{3}} + \frac{0.50}{a_{4}} + \frac{0.80}{a_{5}},$ ${\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (H) = \frac{0.80}{a_{1}} + \frac{0.61}{a_{2}} + \frac{0.44}{a_{3}} + \frac{0.40}{a_{4}} + \frac{0.76}{a_{5}},$ ${\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (F \cup H) = \frac{0.80}{a_{1}} + \frac{0.71}{a_{2}} + \frac{0.62}{a_{3}} + \frac{0.56}{a_{4}} + \frac{0.80}{a_{5}} .$ We can see, ${\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (F) \cap {\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (H) = \frac{0.26}{a_{1}} + \frac{0.47}{a_{2}} + \frac{0.20}{a_{3}} +$ $\frac{0.20}{a_{4}} + \frac{0.59}{a_{5}} \supset {\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (F \cap H),$

${\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (F) \cup {\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (H) = \frac{0.80}{a_{1}} + \frac{0.71}{a_{2}} + \frac{0.44}{a_{3}} +$ $\frac{0.50}{a_{4}} + \frac{0.80}{a_{5}} \subset {\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (F \cup H) .$

So, we will discuss the conditions that approximation operators ${\underline{C}}_{ℕ_{β}, γ}^{'}$ and ${\bar{C}}_{ℕ_{β}, γ}^{'}$ satisfy formulae (1) (2).

Definition 3.5. Let $(X, ℂ)$ be a fuzzy β-covering approximation space, then for all $F \in 𝔽 (X)$ and parameter γ ∈ [0, 1), we define

(1) ${\underline{F}}_{γ} = {F \in 𝔽 (X) | {\underline{C}}_{ℕ_{β}, γ}^{'} (F) = γ_{X} \cup F}$ .

(2) ${\bar{F}}_{γ} = {F \in 𝔽 (X) | {\bar{C}}_{ℕ_{β}, γ}^{'} (F) = N (γ)_{X} \cap F}$ .

Theorem 3.6. Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all $F \in {\underline{F}}_{γ}$ and parameter γ ∈ [0, 1), we have the following conclusions:

(1) For all constant α ∈ [0, 1], $α \in {\underline{F}}_{γ}$ .

(2) For all i ∈ Λ, if $F_{i} \in {\underline{F}}_{γ}$ , then $⋃_{i \in Λ} F_{i} \in {\underline{F}}_{γ}$ .

(3) If $F \in {\underline{F}}_{γ}$ , then ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \in {\underline{F}}_{γ} .$

Proof. (1) It can be obtained from Proposition 3.3 (3).

(2) For any $F_{i} \in {\underline{F}}_{γ}$ , by the Proposition 3.3 (2) and (7), we have $γ_{X} \cup (⋃_{i \in Λ} F_{i}) = ⋃_{i \in Λ} (γ_{X} \cup F_{i}) = ⋃_{i \in Λ} {\underline{C}}_{ℕ_{β}, γ}^{'} (F_{i}) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{'} (⋃_{i \in Λ} F_{i}) \subseteq γ_{X} \cup (⋃_{i \in Λ} F_{i}),$ then ${\underline{C}}_{ℕ_{β}, γ}^{'} (⋃_{i \in Λ} F_{i}) = γ_{X} \cup (⋃_{i \in Λ} F_{i})$ . Thus there are $⋃_{i \in Λ} F_{i} \in {\underline{F}}_{γ}$ .

(3) That’ Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all $F \in {\bar{F}}_{γ}$ and parameter γ ∈ [0, 1), we have the following conclusions:

(1) For all constant α ∈ [0, 1], $α \in {\bar{F}}_{γ}$ .

(2) For all i ∈ Λ, if $F_{i} \in {\bar{F}}_{γ}$ , then $⋃_{i \in Λ} F_{i} \in {\bar{F}}_{γ}$ .

(3) If $F \in {\bar{F}}_{γ}$ , then ${\bar{C}}_{ℕ_{β}, γ}^{'} (F) \in {\bar{F}}_{γ} .$

Proof. This proof is similar to that of Theorem 3.6.

□ Theorem 3.8. Let $(X, ℂ)$ be a fuzzy β-covering approximation space, and parameter γ ∈ [0, 1), then the following conclusions hold:

(1) For all $F, H \in {\underline{F}}_{γ}$ , ${\underline{C}}_{ℕ_{β}, γ}^{'} (F \cap H) = {\underline{C}}_{ℕ_{β}, γ}^{'} (F) \cap {\underline{C}}_{ℕ_{β}, γ}^{'} (H)$ holds if and only if ${\underline{F}}_{γ}$ is a fuzzy topology on X.

(2) For all $F, H \in {\bar{F}}_{γ}$ , ${\bar{C}}_{ℕ_{β}, γ}^{'} (F \cup H) = {\bar{C}}_{ℕ_{β}, γ}^{'} (F) \cup {\bar{C}}_{ℕ_{β}, γ}^{'} (H)$ holds if and only if ${\bar{F}}_{γ}$ is a fuzzy topology on X.

Proof. (1) (⇒) For all $F, H \in {\underline{F}}_{γ}$ , ${\underline{C}}_{ℕ_{β}, γ}^{'} (F \cap H) = {\underline{C}}_{ℕ_{β}, γ}^{'} (F) ⋂ {\underline{C}}_{ℕ_{β}, γ}^{'} (H) = (γ_{X} \cup F) ⋂ (γ_{X} \cup H) = γ_{X} \cup (F \cap H)$ , thus $F \cap H \in {\underline{F}}_{γ}$ . Furthermore, it is known from Theorem 3.6 that formula eq1 is a fuzzy topology on X.

(⇐) Let ${\underline{F}}_{γ}$ be a fuzzy topology on X. For all $F, H \in {\underline{F}}_{γ}$ , According to the definition of ${\underline{F}}_{γ}$ , we can get ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) = γ_{X} \cup F$ , ${\underline{C}}_{ℕ_{β}, γ}^{'} (H) = γ_{X} \cup H$ . Then, ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \cap {\underline{C}}_{ℕ_{β}, γ}^{'} (H) = (γ_{X} \cup F) \cap (γ_{X} \cup H)$ . Because ${\underline{F}}_{γ}$ is a fuzzy topology, there is $F \cap H \in {\underline{F}}_{γ}$ . That is, ${\underline{C}}_{ℕ_{β}, γ}^{'} (F \cap H) = γ_{X} \cup (F \cap H)$ . As we all know (γ_X ∪ F) ∩ (γ_X ∪ H) = γ_X ∪ (F ∩ H), so ${\underline{C}}_{ℕ_{β}, γ}^{'} (F) \cap {\underline{C}}_{ℕ_{β}, γ}^{'} (H) = {\underline{C}}_{ℕ_{β}, γ}^{'} (F \cap H)$ .

(2) This proof is similar to that of (1). □

3.2 Properties of covering-based generalized loose variable precision fuzzy rough set model

The properties of covering-based generalized loose variable precision fuzzy rough set model are studied in this section. And we give an example to show that the two new models are different.

Proposition 3.9. Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all $F, H, F_{i} \in 𝔽 (X)$ , and γ ∈ [0, 1), the following properties hold:

(1) ${\underline{C}}_{ℕ_{β}, γ}^{″} (X) = X$

(2) ${\underline{C}}_{ℕ_{β}, γ}^{″} (F) \subseteq γ_{X} \cup F,$

${N (γ)}_{X} \cap F \subseteq {\bar{C}}_{ℕ_{β}, γ}^{″} (F) .$

(3) ${\underline{C}}_{ℕ_{β}, γ}^{″} (μ_{X}) = (γ \lor μ)_{X}$

${\bar{C}}_{ℕ_{β}, γ}^{″} (μ_{X}) = (N (γ) \land μ)_{X} .$

(4) If γ ≤ λ, then ${\underline{C}}_{ℕ_{β}, γ}^{″} (F) \subseteq {\underline{C}}_{ℕ_{β}, λ}^{″} (F)$

${\bar{C}}_{ℕ_{β}, λ}^{″} (F) \subseteq {\bar{C}}_{ℕ_{β}, γ}^{″} (F) .$

(5) If F ⊆ H, then ${\underline{C}}_{ℕ_{β}, γ}^{″} (F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{″} (H)$

${\bar{C}}_{ℕ_{β}, γ}^{″} (F) \subseteq {\bar{C}}_{ℕ_{β}, γ}^{″} (H) .$

(6) ${\underline{C}}_{ℕ_{β}, γ}^{″} (γ_{X} \cup F) = {\underline{C}}_{ℕ_{β}, γ}^{″} (F)$

(7) ${\underline{C}}_{ℕ_{β}, γ}^{″} (⋃_{i \in Λ} F_{i}) \supseteq ⋃_{i \in Λ} {\underline{C}}_{ℕ_{β}, γ}^{″} (F_{i})$

(8) ${\underline{C}}_{ℕ_{β}, γ}^{″} (⋂_{i \in Λ} F_{i}) = ⋂_{i \in Λ} {\underline{C}}_{ℕ_{β}, γ}^{″} (F_{i})$

(9) ${\bar{C}}_{ℕ_{β}, γ}^{″} ({\underline{C}}_{ℕ_{β}, γ}^{″} (F)) \subseteq N (γ)_{X} \cap (γ_{X} \cup F)$

$γ_{X} \cup (N (γ)_{X} \cap F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{″} ({\bar{C}}_{ℕ_{β}, γ}^{″} (F)) .$

Proof. Propositions (1) to (8) can be directly proved by Definition 3.1, now we will give a proof of (9). For all a ∈ X,

${\bar{C}}_{ℕ_{β}, γ}^{″} ({\underline{C}}_{ℕ_{β}, γ}^{″} (F)) (a)$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$N (γ) \land {\underline{C}}_{N_{β}, γ}^{″} (F) (c)))$

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$N (γ) \land ⋀_{b^{'} \in X} I (N_{β} (b^{'}) (c), ⋀_{c^{'} \in X} I (N_{β} (b^{'}) (c^{'}),$

γ ∨ F (c′)))))

$= ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$⋀_{b^{'} \in X} (N (γ) \land I (ℕ_{β} (b^{'}) (c), ⋀_{c^{'} \in X} I (ℕ_{β} (b^{'}) (c^{'}),$

γ ∨ F (c′))))))

$\leq ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$⋀_{b^{'} \in X} (I (ℕ_{β} (b^{'}) (c), N (γ) \land ⋀_{c^{'} \in X} I (ℕ_{β} (b^{'}) (c^{'}),$

γ ∨ F (c′))))))

$\leq ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋁_{c \in X} T (ℕ_{β} (b) (c),$

$I (ℕ_{β} (b) (c), I (ℕ_{β} (b) (c) N (γ) \land$

( γ ∨ F (c))))))

$\leq ⋁_{b \in X} T (ℕ_{β} (b) (a), I (ℕ_{β} (b) (a), N (γ) \land$

$(γ \lor F (a)))) \leq N (γ) \land (γ \lor F (a))$ .

So, we get

Similarly, we have

$γ_{X} \cup (N (γ)_{X} \cap F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{″} ({\bar{C}}_{ℕ_{β}, γ}^{″} (F)) .$ □ Here is an example to show that covering-based generalized compact and loose variable precision fuzzy rough set models are different.

Example 3.10. (Continue to Example 3.1) From Definition 3.1 we can get ${\underline{C}}_{ℕ_{0.60}, 0.2}^{″} (F) = \frac{0.26}{a_{1}} + \frac{0.45}{a_{2}} + \frac{0.20}{a_{3}} + \frac{0.43}{a_{4}} + \frac{0.45}{a_{5}},$ ${\bar{C}}_{ℕ_{0.60}, 0.2}^{″} (F) = \frac{0.51}{a_{1}} + \frac{0.84}{a_{2}} + \frac{0.46}{a_{3}} + \frac{0.82}{a_{4}} + \frac{0.84}{a_{5}}$ ${\underline{C}}_{ℕ_{0.60}, 0.2}^{″} (H) = \frac{0.63}{a_{1}} + \frac{0.22}{a_{2}} + \frac{0.44}{a_{3}} + \frac{0.20}{a_{4}} + \frac{0.22}{a_{5}}$ ${\bar{C}}_{ℕ_{0.60}, 0.2}^{″} (H) = \frac{0.92}{a_{1}} + \frac{0.76}{a_{2}} + \frac{0.73}{a_{3}} + \frac{0.74}{a_{4}} + \frac{0.76}{a_{5}}$ So obviously, ${\underline{C}}_{ℕ_{0.60}, 0.2}^{″} (F) \subset {\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (F),$ ${\underline{C}}_{ℕ_{0.60}, 0.2}^{″} (H) \subset {\underline{C}}_{ℕ_{0.60}, 0.2}^{'} (H) .$

And, ${\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (F) \subset {\bar{C}}_{ℕ_{0.60}, 0.2}^{″} (F),$ ${\bar{C}}_{ℕ_{0.60}, 0.2}^{'} (H) \subset {\bar{C}}_{ℕ_{0.60}, 0.2}^{″} (H) .$

4 The relationship between the two models

The relationship between covering-based generalized compact and loose variable precision fuzzy rough set models is studied in this section.

Theorem 4.1. Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all $F \in 𝔽 (X)$ and γ ∈ [0, 1), the following properties hold:

(1) ${\underline{C}}_{ℕ_{β}, 0}^{'} (F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{'} (F)$ ,

(2) ${\underline{C}}_{ℕ_{β}, 0}^{″} (F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{″} (F)$ ,

Proof. They can be obtained from Proposition 3.3 (4) and Proposition 3.3 (4).□ Theorem 4.2. Let $(X, ℂ)$ be a fuzzy β-covering approximation space. For all $F \in 𝔽 (X)$ and γ ∈ [0, 1), the following properties hold:

(1) ${\underline{C}}_{ℕ_{β}, γ}^{″} (F) \subseteq {\underline{C}}_{ℕ_{β}}^{γ} (F) \subseteq {\underline{C}}_{ℕ_{β}, γ}^{'} (F) \subseteq γ_{X} \cup F,$

(2)

Proof. (1) For all a ∈ X,

$\begin{matrix} = & ⋁_{b \in X} T (ℕ_{β} (b) (a), ⋀_{c \in X} I (ℕ_{β} (b) (c), γ \lor F (c))) \\ \geq & T (ℕ_{β} (a) (a), ⋀_{c \in X} I (ℕ_{β} (a) (c), γ \lor F (c))) \\ = & T (1, ⋀_{c \in X} I (ℕ_{β} (a) (c), γ \lor F (c))) \\ = & ⋀_{c \in X} I (ℕ_{β} (a) (c), γ \lor F (c)) \\ = & {\underline{C}}_{ℕ_{β}}^{γ} (F) (a), \end{matrix}$

So, ${\underline{C}}_{ℕ_{β, γ}}^{' '} (F) (a) \leq {\underline{C}}_{ℕ_{β}}^{γ} (F) (a) \leq {\underline{C}}_{ℕ_{β, γ}}^{'} π (F) (a) .$

Therefore,

And then from Proposition 3.3 (2) we can get

(2) This proof is similar to that of (1). □ Corollary 4.3. If γ = 0, then for all $F \in 𝔽 (X)$ , the following conclusion hold:

${\underline{C}}_{ℕ_{β}, 0}^{″} (F) \subseteq {\underline{C}}_{ℕ_{β}}^{0} (F) \subseteq {\underline{C}}_{ℕ_{β}, 0}^{'} (F) \subseteq F \subseteq {\bar{C}}_{ℕ_{β}, 0}^{'} (F) \subseteq {\bar{C}}_{ℕ_{β}}^{0} (F) \subseteq {\bar{C}}_{ℕ_{β}, 0}^{″} (F)$ .

Remark 4.4. Though the above conclusions, we can explain the above relationships by drawing a partial diagram (see Fig. 1). From Fig. 1, we can easily get that the covering-based generalized compact variable precision fuzzy rough set model is tighter than the other models. Therefore, the covering-based generalized compact variable precision fuzzy rough set is more accurate than the other in approximating a set fuzzy subset F.

In Fig. 1,→ represents the partial order relation ⊆ of these models. For example, ${\underline{C}}_{ℕ_{β}, γ}^{″} (F) \to {\underline{C}}_{ℕ_{β}}^{γ} (F)$ means ${\underline{C}}_{ℕ_{β}, γ}^{″} (F) \subseteq {\underline{C}}_{ℕ_{β}}^{γ} (F)$ .

Fig.1

Partial diagram of the relationships among covering-based generalized compact and loose variable precision fuzzy rough set models and the other models.

Remark 4.5. In Definition 3.1, we defined the rough set models by using R-implicator. Indeed, if replacing R- with S-, we can defined two new similar models, and we can obtain similar properties. Here, we omit the specific definition and properties.

5 Application of covering-based generalized compact variable precision fuzzy rough set models

In this section, we will give an application example of the new model. For the convenience of comparison, we will take the example of Reference [43] and design our algorithm by combining our rough set model with the algorithm of Reference [43].

Input: A real-valued decision-making matrix and the values of p and γ.

Output: A ranking result of all alternatives.

Step 1: Normalize the MADM matrix.

Step 2: Compute the deviation of evaluation values among alternatives.

Step 3: Calculate the preference values among alternatives.

Step 4: Calculate the overall preference indices among alternatives.

Step 5: Calculate the leaving flow and entering flow of every alternative, respectively.

Step 6: Determine the average leaving flow and the average entering flow, respectively.

Step 7: Calculate the distance between the leaving flow of every alternative and the average leaving flow, and the distance between entering flow of every alternative and the average leaving flow, respectively.

Step 8: Calculate the appraisal score of every alternative and determine the evaluation fuzzy set F.

Step 9: The upper and lower approximations $({\underline{C}}_{ℕ_{β}, γ}^{'} (F)$ , ${\bar{C}}_{ℕ_{β}, γ}^{'} (F))$ of the fuzzy set F are calculated using the coverage-based generalized compact variable precision fuzzy rough set model in Definition 3.1.

Step 10: Compute the overall evaluation value V_i of every alternative and arrange in descending order to get the complete ranking of alternatives.

Remark 5.1. The general methods of decision-making algorithms are listed here. For specific decision-making algorithms, please refer to Reference [43]. Note that the Step 9 in this paper is different from that in Reference [43]. For the convenience of comparison, this paper adopts the case in Reference [43] for analysis.

A family plans to buy a private car. After preliminary evaluation, there are now six car models to choose from, and X = {a₁, a₂, a₃, a₄, a₅, a₆} represents the set of six car models. $ℂ = {C_{1}, C_{2}, C_{3}, C_{4}, C_{5},}$ represents the set of five attributes, where C₁ stands for power, C₂ stands for design, C₃ stands for safety coefficients, C₄ stands for fuel economy and C₅ stands for price. In addition, C₁, C₂, C₃ and C₄ are benefit attribute and C₅ is a cost attribute. The weight vector of five attributes W = {0.25, 0.25, 0.2, 0.15, 0.15}. The evaluation results of the six car models on the five attributes are shown in a real-valued decision-making matrix (see Table 2), where the unit of attribute C₁ is kw, the unit of attribute C₅ is 10000 RMB and the value of car ai with respect to the attribute C₂, C₃ and C₄ are scored by experts (ranging from 0 to 100).

Table 2
A real-valued MADM matrix of six cars

$X / ℂ$ C ₁ C ₂ C ₃ C ₄ C ₅

a ₁ 1.8 74 65 77 18

a ₂ 1.4 64 67 44 20

a ₃ 2.4 67 72 52 12

a ₄ 1.2 42 58 37 14

a ₅ 2.5 78 56 44 8

a ₆ 1.7 54 46 42 11

$X / ℂ$	C ₁	C ₂	C ₃	C ₄	C ₅
a ₁	1.8	74	65	77	18
a ₂	1.4	64	67	44	20
a ₃	2.4	67	72	52	12
a ₄	1.2	42	58	37	14
a ₅	2.5	78	56	44	8
a ₆	1.7	54	46	42	11

Normalize the real-valued MADM matrix of six cars. The normalized decision-making matrix of six cars is listed as in Table 3.

The preference index among six car models is calculated according to the algorithm formula, where the preference threshold p = 0.5.

Calculate the leaving flow and entering flow of each car model according to the algorithm formula, and then obtain the average leaving flow and average entering flow. Then calculate the distance between the leaving flow and the average leaving flow for each car model and the distance between the entering flow and the average entering flow for each car model.

Finally, calculate the appraisal score for each car model, and a evaluation fuzzy set F is constructed according to the evaluation scores, which $F = \frac{0.8022}{a_{1}} + \frac{0.3236}{a_{2}} + \frac{0.8749}{a_{3}} + \frac{0}{a_{4}} + \frac{1}{a_{5}} + \frac{0.2191}{a_{6}} .$

Next, use the coverage-based generalized compact variable precision fuzzy rough set model defined in Definition 3.1 to calculate the upper and lower approximation $({\underline{C}}_{ℕ_{β}, γ}^{'} (F), {\bar{C}}_{ℕ_{β}, γ}^{'} (F))$ of the fuzzy set F.

Besides, from Table 3, we can observe that $ℂ$ is a fuzzy β-covering on X (β ∈ (0, 0.7273]). In this section, we set β = 0.7. Of course, other values of β can also be given according to the requirements. Suppose that $I$ is replaced by the Łukasiewicz implicator in the fuzzy β-neighborhood operator. Then, according to Definition 2.11 the neighborhood of object $ℕ_{β} (a_{i})$ of a_i (i = 1, 2, 3, 4, 5, 6) are given in Table 4.

Table 3

Normalized decision-making matrix of six cars

$X / ℂ$	C ₁	C ₂	C ₃	C ₄	C ₅
a ₁	0.7200	0.9487	0.9028	1.0000	0.4444
a ₂	0.5600	0.8205	0.9306	0.5714	0.4000
a ₃	0.9600	0.8500	1.0000	0.6753	0.6667
a ₄	0.4800	0.6588	0.8056	0.4805	0.5714
a ₅	1.0000	1.0000	0.7778	0.5714	1.0000
a ₆	0.6800	0.6923	0.6389	0.5455	0.7273

Table 4

Fuzzy β-neighborhoods (β=0.70) of six automobile types

$ℕ_{0.70} (a_{i}) / X$	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
$ℕ_{0.70} (a_{1})$	1	0.5714	0.6753	0.4805	0.5714	0.5455
$ℕ_{0.70} (a_{2})$	0.9722	1	1	0.7179	0.8472	0.7083
$ℕ_{0.70} (a_{3})$	0.76	0.6	1	0.52	0.7778	0.6389
$ℕ_{0.70} (a_{4})$	1	1	1	1	0.9722	0.8333
$ℕ_{0.70} (a_{5})$	0.4444	0.4	0.6667	0.48	1	0.6800
$ℕ_{0.70} (a_{6})$	0.7172	0.6727	0.9394	0.8	1	1

From the definition of coverage-based generalized compact variable precision fuzzy rough set model in Definition 3.1, we know that the upper and lower approximation operators are defined by an implicator $I$ and a triangular norm $T$ . In the following, we divide the decision-making process into two cases. Among them, Case I is selected with two models defined by R-implicator, and Case II is selected with two models defined by S-implicator.

Case I: Suppose that $I$ is replaced by the Łukasiewicz implicator, $T$ is replaced by the Łukasiewicz t-norm and precision parameter γ = 0.1, by calculation, we have

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) = \frac{0.6195}{a_{1}} + \frac{0.3236}{a_{2}} + \frac{0.58}{a_{3}} + \frac{0.1}{a_{4}}$

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) = \frac{0.8022}{a_{1}} + \frac{0.8749}{a_{2}} + \frac{0.8749}{a_{3}} + \frac{0.8749}{a_{4}}$

Then, we compute the overall evaluation value V_i of every car, which are shown as follows: V₁ = 0.9247, V₂ = 0.9154, V₃ = 0.9475, V₄ = 0.8874, V₅ = 0.9539, V₆ = 0.9219. Therefore, the ranking result of six cars is a₅ ≻ a₃ ≻ a₁ ≻ a₆ ≻ a₂ ≻ a₄. In this case, we find that the car a₅ is the optimal car.

Case II: Suppose that $I$ is replaced by the Kleene-Dienes implicator, $T$ is replaced by the algebraic product and precision parameter γ = 0.1, by calculation, we have

${\underline{C}}_{ℕ_{β}, γ}^{'} (F) = \frac{0.4286}{a_{1}} + \frac{0.2821}{a_{2}} + \frac{0.3611}{a_{3}} + \frac{0.2059}{a_{4}}$

$+ \frac{0.32}{a_{5}} + \frac{0.2338}{a_{6}} .$

${\bar{C}}_{ℕ_{β}, γ}^{'} (F) = \frac{0.8022}{a_{1}} + \frac{0.8022}{a_{2}} + \frac{0.8022}{a_{3}} + \frac{0.8022}{a_{4}}$

$+ \frac{0.8022}{a_{5}} + \frac{0.8022}{a_{6}} .$

Then, we compute the overall evaluation value V_i of every car, which are shown as follows: V₁ = 0.8870, V₂ = 0.8580, V₃ = 0.8736, V₄ = 0.8429, V₅ = 0.8655, V₆ = 0.8484. Therefore, the ranking result of six cars is a₁ ≻ a₃ ≻ a₅ ≻ a₂ ≻ a₆ ≻ a₄. In this case, we find that the car a₁ is the optimal car.

Remark 5.2. By the above calculations, the ranking results of six cars in Case I is a₅ ≻ a₃ ≻ a₁ ≻ a₆ ≻ a₂ ≻ a₄, in Case II is a₁ ≻ a₃ ≻ a₅ ≻ a₂ ≻ a₆ ≻ a₄. The ranking results of the six car models are not exactly the same in both cases, but the worst car model is the same, i.e., a₄. That is, different fuzzy logic operators have little effect on the worst alternative.

We can see that the best model in Reference [14] is the same in both cases. But the best car model for the six car models is different in both cases in this article. The best car model obtained by the decision generated by the R-implicator is a₅, and the best car model obtained by the decision generated by the S-implicator is a₁. That means decision makers have more options. Decision makers can choose different decision algorithms according to their own needs and personal preferences.

6 Conclusions

In this paper, we defined two pairs of rough approximations on the basis of fuzzy rough set and variable precision rough set: covering-based generalized compact variable precision fuzzy rough set and covering-based generalized loose variable precision fuzzy rough set. Furthermore, we studied their properties. We also give their necessary and sufficient conditions for some unsatisfied properties. Furthermore, we discussed the relationship between two pairs of fuzzy rough set models, and we also discuss the relationship between these two new models and the original models. At the end of this article, we give an example to verify the validity of the new model and compare the results with other models.

Funding

This work is supported by National Natural Science Foundation of China (No. 12171220) and Natural Science Foundation of Shandong Province (No. ZR2020MA042).

Compliance with ethical standards

Conflicts of interest

The authors declare that there is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Covering-based generalized variable precision fuzzy rough set

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy logic operator

2.2 Fuzzy covering, fuzzy neighborhood and fuzzy topology

3 Covering-based generalized compact and loose variable precision fuzzy rough set models

3.1 Properties of covering-based generalized compact variable precision fuzzy rough set model

4 The relationship between the two models

Table 2 A real-valued MADM matrix of six cars X / ℂ C 1 C 2 C 3 C 4 C 5 a 1 1.8 74 65 77 18 a 2 1.4 64 67 44 20 a 3 2.4 67 72 52 12 a 4 1.2 42 58 37 14 a 5 2.5 78 56 44 8 a 6 1.7 54 46 42 11

Funding

Compliance with ethical standards

Conflicts of interest

Ethical approval

References

Table 2
A real-valued MADM matrix of six cars

$X / ℂ$ C ₁ C ₂ C ₃ C ₄ C ₅

a ₁ 1.8 74 65 77 18

a ₂ 1.4 64 67 44 20

a ₃ 2.4 67 72 52 12

a ₄ 1.2 42 58 37 14

a ₅ 2.5 78 56 44 8

a ₆ 1.7 54 46 42 11