Hesitant fuzzy β covering rough sets and applications in multi-attribute decision making

Abstract

In present paper, we put forward four types of hesitant fuzzy β covering rough sets (HFβCRSs) by uniting covering based rough sets (CBRSs) and hesitant fuzzy sets (HFSs). We firstly originate hesitant fuzzy β covering of the universe, which can induce two types of neighborhood to produce four types of HFβCRSs. We then make further efforts to probe into the properties of each type of HFβCRSs. Particularly, the relationships of each type of rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped. Moreover, the relationships between our proposed models and some other existing related models are established. Finally, we give an application model, an algorithm, and an illustrative example to elaborate the applications of HFβCRSs in multi-attribute decision making (MADM) problems. By making comparative analysis, the HFβCRSs models proposed by us are more general than the existing models of Ma and Yang and are more applicable than the existing models of Ma and Yang when handling hesitant fuzzy information.

Keywords

MADM

1 Introduction

Rough set (RS) originated by Pawlak [15, 16] is a significant tool to handle uncertainty. However, it concentrates on the partition of the universe to construct the approximation operators. Aimed at this issue, substantial number of researchers devote themselves to remedy the rigidness of the condition in Pawlak’s RS, such as loosened the equivalence relation to fuzzy relation, similarity relation, coverings of the universe and so on. Massive literatures relevant to the extension of Pawlak’s RS appeared, such as fuzzy rough sets (FRSs) and rough fuzzy sets (RFSs) originated by Dubois and Prade [2] and others, such as [4 , 44].

As is known to all, Pawlak’s RS model is aimed at processing qualitative data, it doesn’t work when dealing with real-valued data sets. Fuzzy set (FS) [33] theory is very useful to overcome these limitations. However, in real life decision making process, the decision makers may have different evaluations for the same object w.r.t. the same attribute. We usually need to consider each decision maker’s evaluation to obtain more reasonable and scientific decision making results, i.e., several possible values should be adopted to depict each object. Therefore, hesitant fuzzy set (HFS) [19, 20] originated by Torra and Narukawa which is a generalized form of FS, is very appropriate to cope with the issue. Since the origination of HFS, it unlocks new avenues of study in handling uncertainties, which enormously promoted the forward development of research work about uncertainty problems. Further, Zhang et al. groped the hesitant fuzzy rough sets (HFRSs) in [38]. In [25, 26], Xu and Xia originated and groped some measures of HFSs. Further, Xia and Xu probed into the hesitant fuzzy information aggregation in [24]. In [21], Tang et al. developed novel distance and similarity measures for HFSs. Yang et al. [27] explored the constructive and axiomatic approaches to hesitant fuzzy rough approximation operators. Zhang et al. [41] discussed uncertainty and equivalence relation analysis for HFRSs. Liao and Xu studied decision making under hesitant fuzzy environment with incomplete weights in [6]. Liang and Liu [7] made an exploration of three-way decision making model by employing hesitant fuzzy TOPSIS method and so on.

Since the origination of CBRSs [34], a great many of literatures related with CBRSs and its generalized models appeared in succession, among which there are [1 , 42–46]. It loosens the equivalence relation of Pawlak’s RS with covering, fuzzy covering, intuitionistic fuzzy covering and so on. Further, Ma [13] explored classification of coverings under the setting of the finite approximation spaces. In [14], Ma pointed out there exist some limitations of the definition of fuzzy covering in practical applications. Consequently, Ma generalized the fuzzy covering to fuzzy $\tilde{β}$ covering [14]. Moreover, Ma [14] defined two new types of fuzzy coverings rough set models. Further, Yang and Hu [30] proposed three new types of fuzzy coverings rough set models under fuzzy $\tilde{β}$ covering approximation space. Huang et al. [3] define intuitionistic fuzzy β-minimal description of intuitionistic fuzzy β-covering, and they explored four intuitionistic fuzzy β-covering-based rough sets. However, as the models in [3 , 30] can’t deal with the hesitancy fuzzy information, we generalize the fuzzy $\tilde{β}$ covering to hesitant fuzzy β covering. Our contributions are summarized as follows.

(1) By using the two types of neighborhood that the hesitant fuzzy β covering induced, we put forward four types of HFβCRSs which build a bridge among CBRSs and HFSs.

(2) We establish the relationships between our proposed models and some other existing related models.

(3) The relationships of each type of lower and upper approximation operators w.r.t. two different hesitant fuzzy β coverings are groped.

(4) We elaborate the applications of the new proposed HFβCRSs in MADM problems.

The remainder of this paper is arranged as follows. In Section 2, we make a revisit of the concepts and results of HFSs. In Section 3, four types of HFβCRSs will be introduced, some properties of HFβCRSs are examined, the relationships between the proposed HFβCRSs and some other existing related models are established. Particularly, the relationships of each type of rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped. In Section 4, we present an application model, an algorithm and an example to validate the feasibility of HFβCRSs in MADM problems. Moreover, the comparative analysis are made between our models and other existing related models. This paper is summarized in the last section with some prospects.

2 Preliminaries

In this section, we make a brief revisit of HFSs. Throughout the paper, we denote U as a finite and nonempty domain of discourse.

2.1 HFSs

Torra [19, 20] proposed the notion of HFSs, which is defined as follows.

Definition 2.1. ([19, 20]). Let U be a domain of discourse. A HFS A on U can be expressed as: $A = {〈 x, h_{A} (x) 〉 | x \in U},$ where h_A (x) is a set of some different values in [0, 1], standing for the possible membership degrees of x ∈ U to A. As stated by Xia and Xu [24], we call β = h_A (x) a hesitant fuzzy element (HFE), the set of all HFSs on U is denoted by HF (U).

From Definition 2.1, the number of values in hesitant fuzzy elements (HFEs) may be different. To operate HFEs, Xu and Xia [25, 26] give the following assumptions. Let l (h_A (x)) be the number of values in h_A (x),

Assumption 1. For HFS A = {〈x, h_A (x) 〉|x ∈ U}, let σ : (1, 2, . . . , n) → (1, 2, . . . , n) be a permutation satisfying $h_{A}^{σ (s)} (x) \leq h_{A}^{σ (s + 1)} (x)$ (s = 1, 2, . . . , n - 1), i.e. $h_{A}^{σ (s)} (x)$ be the sth largest value in h_A (x).

Assumption 2. For two HFSs A and B, if l (h_A (x)) ≠ l (h_B (x)), we can make both of them have the same number of elements by adding some elements to the HFE with less number of elements. In present paper, if l (h_A (x)) < l (h_B (x)), then we add the maximum element to h_A (x) until it has the same length as h_B (x).

To facilitate the discussions below, empty HFS and full HFS [20] are defined as follows.

∀ A ∈ HF (U), A is called an empty HFS on U iff $h_{A} (x) = \tilde{0}$ (∀ x ∈ U), i.e., ∀ x ∈ U, $h_{A}^{σ (s)} (x) = 0, s = 1, 2, . . ., l (h_{A} (x))$ , and we denote it as $\tilde{\emptyset}$ .

∀ A ∈ HF (U), A is called a full HFS on U iff $h_{A} (x) = \tilde{1}$ (∀ x ∈ U), i.e., ∀ x ∈ U, $h_{A}^{σ (s)} (x) = 1, s = 1, 2, . . ., l (h_{A} (x))$ , and we denote it as $\tilde{U}$ .

In the following, we make a revisit of the operations among HFEs.

Definition 2.2. ([24, 38]). Let h, h₁ and h₂ be three HFEs, some operations among them are defined as follows.

(1) $h_{1} \oplus h_{2} = ⋃_{φ_{1} \in h_{1}, φ_{2} \in h_{2}} {φ_{1} + φ_{2} - φ_{1} \cdot φ_{2}}$ ;

(2) $h_{1} \otimes h_{2} = ⋃_{φ_{1} \in h_{1}, φ_{2} \in h_{2}} {φ_{1} \cdot φ_{2}};$

(3) h^c = {1 - h^σ(s)|s = 1, 2, . . . , k}, where k = l (h);

(4) $h_{1} ⊻ h_{2} = {h_{1}^{σ (s)} \lor h_{2}^{σ (s)} | s = 1, 2, . . ., k}$ , where k=max(l (h₁) , l (h₂));

(5) $h_{1} ⌅ h_{2} = {h_{1}^{σ (s)} \land h_{2}^{σ (s)} | s = 1, 2, . . ., k}$ , where k=max(l (h₁) , l (h₂)).

In [24], Xia and Xu introduced the grade function and the comparison laws of HFEs, both of them are defined as follows.

Definition 2.3. ([24]) Let h be a HFE, then the grade value s (h) of h is defined as follows.

$s (h) = \frac{1}{l (h)} \sum_{s = 1}^{l (h)} h^{σ (s)}$ .

Definition 2.4. ([24]) Let h₁, h₂ be two HFEs, then

(1) If s (h₁) > s (h₂) , then h₁ is bigger than h₂, denoted by h₁≻ h₂ ;

(2) If s (h₁) = s (h₂) , then h₁ is equal to h₂, denoted by h₁ ∼ h₂ .

Zhang et al. [38] initiated the complement, union and intersection of HFSs, which are defined as follows.

Definition 2.5([38]). Let U be a domain of discourse. ∀ A, B ∈ HF (U), then

(1) $A^{c} = {〈 x, h_{A}^{c} (x) 〉 | x \in U}$ ;

(2) A ⋓ B = {〈x, h_{A ⋓ B} (x) 〉|x ∈ U}

= {〈x, h_A ⊻ h_B (x) 〉|x ∈ U};

(3) A ⋒ B = {〈x, h_{A ⋒ B} (x) (x) 〉|x ∈ U}

= {〈x, h_A (x) ⌅ h_B (x) 〉|x ∈ U}.

In [39], Zhang et al. originated the notion of hesitant fuzzy subsets, we make a revisit of it.

Definition 2.6([39]). Let U be a domain of discourse. ∀ A, B ∈ HF (U) , A is called a hesitant fuzzy subset of B if h_A (x) ⪯ h_B (x) (h_B (x) ≽ h_A (x)) (∀ x ∈ U), i.e. $h_{A}^{σ (s)} (x) \leq h_{B}^{σ (s)} (x)$ , s = 1, 2, . . . , k (∀ x ∈ U), where k = max {l (h_A (x)) , l (h_B (x))}. Under this circumstance, we denote it by A ⊑ B (B ⊒ A).

Theorem 2.1. Let U be a domain of discourse. ∀ A, B ∈ HF (U), we have

(1) (A^c) ^c = A;

(2) (A ⋓ B) ^c = A^c ⋒ B^c;

(3) (A ⋒ B) ^c = A^c ⋓ B^c.

Proof. It derives from Definition 2.5.

Theorem 2.2. Let U be a domain of discourse. ∀ A, B, A₁, B₁, C₁ ∈ HF (U), we have

(1) A ⋓ B ⊒ A, A ⋓ B ⊒ B;

(2) A ⋒ B ⊑ A, A ⋒ B ⊑ B;

(3) A₁ ⊑ B₁, A₁ ⊑ C₁ ⇒A₁ ⊑ B₁ ⋒ C₁;

A₁ ⊒ B₁, A₁ ⊒ C₁ ⇒A₁ ⊒ B₁ ⋓ C₁.

Proof. It derives from Definitions 2.5 and 2.6.

3 HFβCRSs and their properties

In this section, we firstly define the hesitant fuzzy β covering of U, it can induce two types of neighborhood, which are regarded as building blocks to induce HFβCRSs. Some properties of HFβCRSs will be explored. We then construct the relationships among the proposed HFβCRSs and other existing related models. Particularly, the relationships of the rough approximation operators w.r.t. two different hesitant fuzzy β coverings are established.

3.1 Four types of HFβCRSs

We first introduce the hesitant fuzzy β covering of U and hesitant fuzzy β neighborhood of x (∈ U) as follows.

Definition 3.1. (1) Let U be a domain of discourse and $C$ ={ C₁, C₂, . . . , C_m}, where C_i ∈ HF (U), i = 1, 2, . . . , m . ∀ HFE β, $C$ is called a hesitant fuzzy β covering of U if $h_{⋓_{i = 1}^{m} C_{i}} (x) ≽ β$ (∀ x ∈ U). (U, $C$ ) is called a hesitant fuzzy covering approximation space (HFCAS).

(2) Let (U, $C$ ) be a HFCAS for some HFEs β = {β^σ(s)|s = 1, 2, . . . , l (β)} and $C$ ={C₁, C₂, . . . , C_m}. Then, we call ${\hat{N}}_{x}^{β} = ⋒ {C_{j} \in C$ |h_{C
_j} (x) ≽ β, j = 1, 2, . . . , m} a hesitant fuzzy β neighborhood of x ∈ U.

Remark 3.1. Set l (β) =1, then a hesitant fuzzy β covering of U degenerates into a fuzzy $\tilde{β}$ covering [14] if C_i (i = 1, . . . , m) degenerates into a FS.

In addition, we denote $N_{C}^{β} = {{\hat{N}}_{x}^{β} | x \in U}$ as the hesitant fuzzy β neighborhood system induced by hesitant fuzzy β covering $C$ , where ${\hat{N}}_{x}^{β} = {〈 y, h_{{\hat{N}}_{x}^{β}} (y) 〉 | y \in U}$ .

Definition 3.2. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A ∈ HF (U), the lower and upper approximations of A w.r.t. $N_{C}^{β}$ , denoted as $\underline{N_{C}^{β}} (A)$ and $\bar{N_{C}^{β}} (A)$ , respectively, are two HFSs on U, defined by: $\underline{N_{C}^{β}} (A) = {〈 x, h_{\underline{N_{C}^{β}} (A)} (x) 〉 | x \in U},$ $\bar{N_{C}^{β}} (A) = {〈 x, h_{\bar{N_{C}^{β}} (A)} (x) 〉 | x \in U},$ where

$h_{\underline{N_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)}$ ;

$h_{\bar{N_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A} (y)}$ .

$\underline{N_{C}^{β}} (A)$ , $\bar{N_{C}^{β}} (A) : HF (U) \to HF (U)$ are called the first type of lower and upper rough approximation operators of A w.r.t. $N_{C}^{β}$ . The pair $(\underline{N_{C}^{β}} (A), \bar{N_{C}^{β}} (A))$ is referred to as the first type of HFβCRSs (1-HFβCRSs).

Analogous to [30], we put forward the concept of hesitant fuzzy complement β neighborhood of x ∈ U.

Definition 3.3. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ x ∈ U, set $h_{{\hat{M}}_{x}^{β}} (y) = h_{{\hat{N}}_{y}^{β}} (x)$ (∀ y ∈ U), ${\hat{M}}_{x}^{β}$ is called a hesitant fuzzy complement β neighborhood of x ∈ U.

We denote $M_{C}^{β} = {{\hat{M}}_{x}^{β} | x \in U}$ as the hesitant fuzzy complement β neighborhood system induced by hesitant fuzzy β covering $C$ , where ${\hat{M}}_{x}^{β} = {〈 y, h_{{\hat{M}}_{x}^{β}} (y) 〉 | y \in U}$ .

Definition 3.4. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A ∈ HF (U), the lower and upper approximations of A w.r.t. $M_{C}^{β}$ , denoted as $\underline{M_{C}^{β}} (A)$ and $\bar{M_{C}^{β}} (A)$ , respectively, are two HFSs on U, defined by: $\underline{M_{C}^{β}} (A) = {〈 x, h_{\underline{M_{C}^{β}} (A)} (x) 〉 | x \in U},$ $\bar{M_{C}^{β}} (A) = {〈 x, h_{\bar{M_{C}^{β}} (A)} (x) 〉 | x \in U},$ where

$h_{\underline{M_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)}$ ;

$h_{\bar{M_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{A} (y)}$ .

$\underline{M_{C}^{β}} (A)$ , $\bar{M_{C}^{β}} (A) : HF (U) \to HF (U)$ are called the second type of lower and upper rough approximation operators of A w.r.t. $M_{C}^{β}$ . The pair $(\underline{M_{C}^{β}} (A), \bar{M_{C}^{β}} (A))$ is called the second type of HFβCRSs (2-HFβCRSs).

Example 3.1. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. Take β = {0.2, 0.4, 0.8} (see Table 1).

Table 1
The hesitant fuzzy β covering $C$

U/ $C$ C ₁ C ₂ C ₃ C ₄ C ₅

x ₁ {0.2, 0.3, 0.6} {0.4, 0.8} {0.2, 0.3, 0.4} {0.4, 0.6} {0.3, 0.4, 0.8}

x ₂ {0.3, 0.4} {0.3, 0.8} {0.2, 0.8} {0.3, 0.4} {0.1, 0.2}

x ₃ {0.5, 0.6} {0.4, 0.6} {0.3, 0.8} {0.4, 0.8} {0.2, 0.3}

x ₄ {0.2, 0.8} {0.1, 0.2, 0.4}) {0.2, 0.4} {0.6, 0.9} {0.3, 0.4}

x ₅ {0.3, 0.8} {0.2, 0.3} {0.4, 0.6} {0.1, 0.2} {0.3, 0.9}

U/ $C$	C ₁	C ₂	C ₃	C ₄	C ₅
x ₁	{0.2, 0.3, 0.6}	{0.4, 0.8}	{0.2, 0.3, 0.4}	{0.4, 0.6}	{0.3, 0.4, 0.8}
x ₂	{0.3, 0.4}	{0.3, 0.8}	{0.2, 0.8}	{0.3, 0.4}	{0.1, 0.2}
x ₃	{0.5, 0.6}	{0.4, 0.6}	{0.3, 0.8}	{0.4, 0.8}	{0.2, 0.3}
x ₄	{0.2, 0.8}	{0.1, 0.2, 0.4})	{0.2, 0.4}	{0.6, 0.9}	{0.3, 0.4}
x ₅	{0.3, 0.8}	{0.2, 0.3}	{0.4, 0.6}	{0.1, 0.2}	{0.3, 0.9}

Let A = {〈x₁, {0.3, 0.4} 〉, 〈x₂, {0.2, 0.6} 〉, 〈x₃, {0.4, 0.6, 0.8} 〉, 〈x₄, {0.3, 0.5, 0.6} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉}.

According to Definition 3.1, we have ${\hat{N}}_{x_{1}}^{β} = C_{2} ⋒ C_{5}$ , ${\hat{N}}_{x_{2}}^{β} = C_{2} ⋒ C_{3}$ , ${\hat{N}}_{x_{3}}^{β} = C_{3} ⋒ C_{4}$ , ${\hat{N}}_{x_{4}}^{β} = C_{1} ⋒ C_{4}$ , ${\hat{N}}_{x_{5}}^{β} = C_{1} ⋒ C_{5}$ .

According to Definition 3.2, we compute $\underline{N_{C}^{β}} (A)$ and $\bar{N_{C}^{β}} (A)$ as:

$\underline{N_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.7,$ 0.8} 〉, 〈x₃, {0.4, 0.7, 0.8} 〉, 〈x₄, {0.3, 0.6, 0.7} 〉, 〈 x₅, {0.3, 0.7, 0.8} 〉},

$\bar{N_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4} 〉, 〈 x_{2}, {0.3, 0.6} 〉, 〈 x_{3}$ , {0.3, 0.6, 0.8} 〉, 〈x₄, {0.4, 0.6} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉}.

According to Definition 3.4, $\underline{M_{C}^{β}} (A)$ and $\bar{M_{C}^{β}} (A)$ are calculated as:

$\underline{M_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.7} 〉,$ 〈x₃, {0.4, 0.6} 〉, 〈x₄, {0.3, 0.8} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉},

$\bar{M_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4, 0.6} 〉, 〈 x_{2}, {0.3, 0.6} 〉,$ 〈x₃, {0.3, 0.6, 0.8} 〉, 〈x₄, {0.2, 0.5, 0.6} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉}.

From Example 3.1, we find that the arguments $\underline{N_{C}^{β}} (A) ⊑ A ⊑ \bar{N_{C}^{β}} (A)$ and $\underline{M_{C}^{β}} (A) ⊑ A ⊑ \bar{M_{C}^{β}} (A)$ don’t hold. Moreover, in Table 1, take β = {0.8}, if we only retain the largest membership of x_i w.r.t. C_j by deleting other possible membership of x_i w.r.t. C_j (i, j = 1, 2, . . . , 5), then the hesitant fuzzy β covering of U degenerates into a fuzzy $\tilde{β}$ covering of U.

Remark 3.2. There is no direct correlation between 1-HFβCRSs and 2-HFβCRSs.

In the following, the other two types of HFβCRSs will be explored, respectively.

Definition 3.5. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ = { C₁, C₂, . . . , C_m}. ∀ A ∈ HF (U), the lower and upper approximations of A w.r.t. $N_{C}^{β}$ and $M_{C}^{β}$ , denoted as $\underline{P_{C}^{β}} (A)$ and $\bar{P_{C}^{β}} (A)$ , respectively, are two HFSs on U, defined by: $\underline{P_{C}^{β}} (A) = {〈 x, h_{\underline{P_{C}^{β}} (A)} (x) 〉 | x \in U},$ $\bar{P_{C}^{β}} (A) = {〈 x, h_{\bar{P_{C}^{β}} (A)} (x) 〉 | x \in U};$ where

$h_{\underline{P_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A} (y)}$ ;

$h_{\bar{P_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)}$ .

$\underline{P_{C}^{β}} (A)$ , $\bar{P_{C}^{β}} (A) : HF (U) \to HF (U)$ are referred to as the third type of lower and upper rough approximation operators of A w.r.t. $N_{C}^{β}$ and $M_{C}^{β}$ . The pair $(\underline{P_{C}^{β}} (A), \bar{P_{C}^{β}} (A))$ is referred to as the third type of HFβCRSs (3-HFβCRSs).

Example 3.2. (Continued from Example 3.1) According to Definition 3.5, $\underline{P_{C}^{β}} (A), \bar{P_{C}^{β}} (A)$ are calculated as follows.

$\underline{P_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.7,$ 0.8} 〉, 〈x₃, {0.4, 0.7, 0.8} 〉, 〈x₄, {0.3, 0.8} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉},

$\bar{P_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4} 〉, 〈 x_{2}, {0.3, 0.6} 〉, 〈 x_{3},$ {0.3, 0.6, 0.8} 〉, 〈x₄, {0.2, 0.5, 0.6} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉}.

Definition 3.6. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A ∈ HF (U), the lower and upper approximations of A w.r.t. $N_{C}^{β}$ and $M_{C}^{β}$ , denoted as $\underline{Q_{C}^{β}} (A)$ and $\bar{Q_{C}^{β}} (A)$ , respectively, are two HFSs on U, defined by: $\underline{Q_{C}^{β}} (A) = {〈 x, h_{\underline{Q_{C}^{β}} (A)} (x) 〉 | x \in U},$ $\bar{Q_{C}^{β}} (A) = {〈 x, h_{\bar{Q_{C}^{β}} (A)} (x) 〉 | x \in U};$ where

$h_{\underline{Q_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A} (y)}$ ;

$h_{\bar{Q_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)}$ .

$\underline{Q_{C}^{β}} (A)$ , $\bar{Q_{C}^{β}} (A) : HF (U) \to HF (U)$ are referred to as the fourth type of lower and upper rough approximation operators of A w.r.t. $N_{C}^{β}$ and $M_{C}^{β}$ . The pair $(\underline{Q_{C}^{β}} (A), \bar{Q_{C}^{β}} (A))$ is called the fourth type of HFβCRSs (4-HFβCRSs).

Example 3.3. (Continued from Example 3.1) According to Definition 3.6, we compute $\underline{Q_{C}^{β}} (A)$ and $\bar{Q_{C}^{β}} (A)$ as follows.

$\underline{Q_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.7} 〉,$ 〈x₃, {0.4, 0.6} 〉, 〈x₄, {0.3, 0.6, 0.7} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉},

$\bar{Q_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4, 0.6} 〉, 〈 x_{2}, {0.3, 0.6} 〉,$ 〈x₃, {0.3, 0.6, 0.8} 〉, 〈x₄, {0.4, 0.6} 〉, 〈x₅,

{0.3, 0.7, 0.8} 〉}.

From Examples 3.2 and 3.3, we find that the arguments $\underline{P_{C}^{β}} (A) ⊑ A ⊑ \bar{P_{C}^{β}} (A)$ and $\underline{Q_{C}^{β}} (A) ⊑ A ⊑ \bar{Q_{C}^{β}} (A)$ don’t hold. Furthermore, we find that $\underline{P_{C}^{β}} (A) = \underline{N_{C}^{β}} (A) ⋓ \underline{M_{C}^{β}} (A)$ , $\bar{P_{C}^{β}} (A) = \bar{N_{C}^{β}} (A) ⋒ \bar{M_{C}^{β}} (A)$ , $\underline{Q_{C}^{β}} (A) = \underline{N_{C}^{β}} (A) ⋒ \underline{M_{C}^{β}} (A)$ and $\bar{Q_{C}^{β}} (A) = \bar{N_{C}^{β}} (A) ⋓ \bar{M_{C}^{β}} (A)$ .

In the following discussions of Subsection 3.2, we can prove that these arguments hold.

Remark 3.3. There is no direct correlation between 3-HFβCRSs and 4-HFβCRSs.

3.2 The properties of HFβCRSs

In the following, some properties of 1-HFβCRSs will be groped.

Theorem 3.1. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A, B ∈ HF (U), then

(1) $\underline{N_{C}^{β}} (A^{c}) = (\bar{N_{C}^{β}} (A))^{c}$ ,

(2) $\bar{N_{C}^{β}} (A^{c}) = (\underline{N_{C}^{β}} (A))^{c}$ ,

(3) $\underline{N_{C}^{β}} (\tilde{U}) = \tilde{U}$ ,

(4) $\bar{N_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(5) $\underline{N_{C}^{β}} (A ⋒ B) = \underline{N_{C}^{β}} (A) ⋒ \underline{N_{C}^{β}} (B)$ ,

(6) $\bar{N_{C}^{β}} (A ⋓ B) = \bar{N_{C}^{β}} (A) ⋓ \bar{N_{C}^{β}} (B)$ ,

(7) If A ⊑ B, then $\underline{N_{C}^{β}} (A) ⊑ \underline{N_{C}^{β}} (B)$ and $\bar{N_{C}^{β}} (A) ⊑ \bar{N_{C}^{β}} (B)$ ,

(8) $\underline{N_{C}^{β}} (A) ⋓ \underline{N_{C}^{β}} (B) ⊑ \underline{N_{C}^{β}} (A ⋓ B)$ ,

(9) $\bar{N_{C}^{β}} (A ⋒ B) ⊑ \bar{N_{C}^{β}} (A) ⋒ \bar{N_{C}^{β}} (B)$ .

Proof. (1) According to Definition 3.2, we have

$h_{\underline{N_{C}^{β}} (A^{c})} (x) = \underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A^{c}} (y)}$

$= \underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A^{c}}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= {{\underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A^{c}}^{σ (s)} (y) | s = 1, 2, . . ., k}}^{c}}^{c}$

$= {\underset{y \in U}{⋁} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A^{c}}^{σ (s)} (y) | s = 1, 2, . . ., k}^{c}}^{c}$

$= {\underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}}^{c}$

$= h_{(\bar{N_{C}^{β}} (A))^{c}}$ where k =max_y ∈Umax ${l (h_{{\hat{N}}_{x}^{β}} (y)), l (h_{A} (y))}$ .

Therefore, we have $\underline{N_{C}^{β}} (A^{c}) = (\bar{N_{C}^{β}} (A))^{c}$ .

(2) Since $\underline{N_{C}^{β}} (A^{c}) = (\bar{N_{C}^{β}} (A))^{c}$ , thus, we have

$(\bar{N_{C}^{β}} (A^{c}))^{c} = {〈 x, h_{\bar{N_{C}^{β}} (A^{c})} (x) 〉 | x \in U}^{c}$

$= {〈 x, h_{\underline{(N_{C}^{β}} (A))^{c}} (x) 〉 | x \in U}^{c}$

$= {〈 x, h_{\underline{N_{C}^{β}} (A)} (x) 〉 | x \in U}$ .

Therefore, we have $\bar{N_{C}^{β}} (A^{c}) = (\underline{N_{C}^{β}} (A))^{c}$ .

(3) ∀ x ∈ U, we have $h_{\tilde{U}}^{σ (s)} (x) = 1$ (s = 1, 2, . . . , k).

$h_{\underline{N_{C}^{β}} (\tilde{U})} (x) = \underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{\tilde{U}} (y)}$

$= \underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor u_{\tilde{U}}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor 1 | s = 1, 2, . . ., k}$

$= \tilde{1}$

$= h_{\tilde{U}} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{N}}_{x}^{β}} (y)), l (h_{\tilde{U}} (y))}$ .

Therefore, we have $\underline{N_{C}^{β}} (\tilde{U}) = \tilde{U}$ .

(4) ∀ x ∈ U, we have $h_{\tilde{\emptyset}}^{σ (s)} (x) = 0$ (s = 1, 2, . . . , k).

$h_{\bar{N_{C}^{β}} (\tilde{\emptyset})} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{\tilde{\emptyset}} (y)}$

$= \underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{\tilde{\emptyset}}^{σ (s)} | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land 0 | s = 1, 2, . . ., k}$

$= \tilde{0}$

$= h_{\tilde{\emptyset}} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{N}}_{x}^{β}} (y)), l (h_{\tilde{\emptyset}} (y))}$ .

Therefore, we have $\bar{N_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ .

(5) According to Definitions 3.2 and 2.5, we have

$h_{\underline{N_{C}^{β}} (A ⋒ B)} (x) = \underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A ⋒ B} (y)}$

$= \underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ (h_{A} (y) ⌅ h_{B} (y))}$

$= \underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor (h_{A}^{σ (s)} (y) \land h_{B}^{σ (s)} (y)) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋀} {(h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A}^{σ (s)} (y))$

$\land (h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{B}^{σ (s)} (y)) | s = 1, 2, . . ., k}$

$= (\underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}} (y) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k})$

$⌅ (\underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k})$

$= h_{\underline{N_{C}^{β}} (A)} (x) ⌅ h_{\underline{{\hat{N}}_{C}^{β}} (B)} (x)$

$= h_{\underline{N_{C}^{β}} (A) ⋒ \underline{N_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{N}}_{x}^{β}} (y)), l (h_{A} (y)), l (h_{B} (y))}$ .

Therefore, we have $\underline{N_{C}^{β}} (A ⋒ B) = \underline{N_{C}^{β}} (A) ⋒ \underline{N_{C}^{β}} (B)$ .

(6) According to Definitions 3.2 and 2.5, we have

$h_{\bar{N_{C}^{β}} (A ⋓ B)} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A ⋓ B} (y)}$

$= \underset{y \in U}{\underline{⋁}} {h_{{\hat{N}}_{x}^{β}} (y) ⌅ (h_{A} (y) ⊻ h_{B} (y))}$

$= \underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land (h_{A}^{σ (s)} (y) \lor h_{B}^{σ (s)} (y)) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋁} {(h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{A}^{σ (s)} (y)) \lor (h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{B}^{σ (s)} (y)) | s = 1, 2, . . ., k}$

$= (\underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k})$

$⊻ (\underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k})$

$= h_{\bar{N_{C}^{β}} (A)} (x) ⊻ h_{\bar{N_{C}^{β}} (B)} (x)$

$= h_{\bar{N_{C}^{β}} (A) ⋓ \bar{N_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{N_{x}^{β}} (y)), l (h_{A} (y)), l (h_{B} (y))}$ .

Therefore, we have $\bar{N_{C}^{β}} (A ⋓ B) = \bar{N_{C}^{β}} (A) ⋓ \bar{N_{C}^{β}} (B)$ .

(7) Since A ⊑ B, thus, ∀ x ∈ U, we have h_A (x) ⪯ h_B (x).

$h_{\underline{N_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)}$

$= \underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋀} {h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\underline{N_{C}^{β}} (B)} (x)$ ;

$h_{\bar{N_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A} (y)}$

$= \underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\bar{N_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{N}}_{x}^{β}} (y)), l (h_{A} (x)), l (h_{B} (y))}$ .

Therefore, we have $\underline{N_{C}^{β}} (A) ⊑ \underline{N_{C}^{β}} (B)$ and $\bar{N_{C}^{β}} (A) ⊑ \bar{N_{C}^{β}} (B)$ .

(8-9) It follows immediately from (7) and Theorem 2.2.

In the following, the relationships of the first type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped.

Theorem 3.2. Let U be a domain of discourse, $C$ and $T$ are two hesitant fuzzy β coverings of U for HFE β, where $C$ ={C₁, C₂, . . . , C_m}, $T$ = ${C_{1}^{'}, C_{2}^{'}, . . ., C_{n}^{'}}$ . Suppose that ${({\hat{N}}_{1})_{x}^{β} | x \in U}$ and ${({\hat{N}}_{2})_{x}^{β} | x \in U}$ are two hesitant fuzzy β neighborhood systems induced by $C$ and $T$ , respectively. ∀ A ∈ HF (U), if $({\hat{N}}_{1})_{x}^{β} ⊑ ({\hat{N}}_{2})_{x}^{β}$ (∀ x ∈ U), then

(1) $\underline{N_{T}^{β}} (A) ⊑ \underline{N_{C}^{β}} (A)$ ,

(2) $\bar{N_{T}^{β}} (A) ⊒ \bar{N_{C}^{β}} (A)$ .

Proof. (1) According to Definitions 3.2 and 2.6, we have

$h_{\underline{N_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {h_{(({\hat{N}}_{1})_{x}^{β})^{c}} (y) ⊻ h_{A} (y)}$

$= \underset{y \in U}{⋀} {h_{(({\hat{N}}_{1})_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$≽ \underset{y \in U}{⋀} {h_{(({\hat{N}}_{2})_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\underline{N_{T}^{β}} (A)} (x)$ ; where k =max_y ∈Umax ${l (h_{({\hat{N}}_{1})_{x}^{β}} (y)), l (h_{({\hat{N}}_{2})_{x}^{β}} (y)),$ l (h_A (x))}.

Thus, we have (1) holds.

(2) According to Definitions 3.2 and 2.6, we have

$h_{\bar{N_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {h_{({\hat{N}}_{1})_{x}^{β}} (y) ⌅ h_{A} (y)}$

$= \underset{y \in U}{⋁} {h_{({\hat{N}}_{1})_{x}^{β}}^{σ (s)} (y) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋁} {h_{({\hat{N}}_{2})_{x}^{β}}^{σ (s)} (y) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\bar{N_{T}^{β}} (A)} (x)$ ; where k =max_y ∈Umax ${l (h_{({\hat{N}}_{1})_{x}^{β}} (y)), l (h_{({\hat{N}}_{2})_{x}^{β}} (y)),$ l (h_A (x))}.

Thus, we have (2) holds.

In the following, some properties of 2-HFβCRSs will be groped.

Theorem 3.3. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A, B ∈ HF (U), then

(1) $\underline{M_{C}^{β}} (A^{c}) = (\bar{M_{C}^{β}} (A))^{c}$ ,

(2) $\bar{M_{C}^{β}} (A^{c}) = (\underline{M_{C}^{β}} (A))^{c}$ ,

(3) $\underline{M_{C}^{β}} (\tilde{U}) = \tilde{U}$ ,

(4) $\bar{M_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(5) $\underline{M_{C}^{β}} (A ⋒ B) = \underline{M_{C}^{β}} (A) ⋒ \underline{M_{C}^{β}} (B)$ ,

(6) $\bar{M_{C}^{β}} (A ⋓ B) = \bar{M_{C}^{β}} (A) ⋓ \bar{M_{C}^{β}} (B)$ ,

(7) If A ⊑ B, then $\underline{M_{C}^{β}} (A) ⊑ \underline{M_{C}^{β}} (B)$ and $\bar{M_{C}^{β}} (A) ⊑ \bar{M_{C}^{β}} (B)$ ,

(8) $\underline{M_{C}^{β}} (A) ⋓ \underline{M_{C}^{β}} (B) ⊑ \underline{M_{C}^{β}} (A ⋓ B)$ ,

(9) $\bar{M_{C}^{β}} (A ⋒ B) ⊑ \bar{M_{C}^{β}} (A) ⋒ \bar{M_{C}^{β}} (B)$ .

Proof. It is similar to the proof of Theorem 3.1.

In the following, the relationships of the second type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped.

Theorem 3.4. Let U be a domain of discourse, $C$ and $T$ are two hesitant fuzzy β coverings of U for HFE β, where $C$ ={C₁, C₂, . . . , C_m}, $T$ = ${C_{1}^{'}, C_{2}^{'}, . . ., C_{n}^{'}}$ . Suppose that ${({\hat{N}}_{1})_{x}^{β} | x \in U}$ and ${({\hat{N}}_{2})_{x}^{β} | x \in U}$ are two hesitant fuzzy β neighborhood systems induced by $C$ and $T$ , respectively. ∀ A ∈ HF (U), if $({\hat{N}}_{1})_{x}^{β} ⊑ ({\hat{N}}_{2})_{x}^{β}$ (∀ x ∈ U), then

(1) $\underline{M_{T}^{β}} (A) ⊑ \underline{M_{C}^{β}} (A)$ ,

(2) $\bar{M_{T}^{β}} (A) ⊒ \bar{M_{C}^{β}} (A)$ .

Proof. It is similar to the proof of Theorem 3.2.

In the following, the relationships of the proposed four types of HFβCRSs will be established.

Theorem 3.5. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A ∈ HF (U), then

(1) $\underline{Q_{C}^{β}} (A) ⊑ \underline{N_{C}^{β}} (A) ⊑ \underline{P_{C}^{β}} (A)$ ,

(2) $\underline{Q_{C}^{β}} (A) ⊑ \underline{M_{C}^{β}} (A) ⊑ \underline{P_{C}^{β}} (A)$ ,

(3) $\bar{P_{C}^{β}} (A) ⊑ \bar{N_{C}^{β}} (A) ⊑ \bar{Q_{C}^{β}} (A)$ ,

(4) $\bar{P_{C}^{β}} (A) ⊑ \bar{M_{C}^{β}} (A) ⊑ \bar{Q_{C}^{β}} (A)$ ,

(5) $\underline{Q_{C}^{β}} (A) = \underline{N_{C}^{β}} (A) ⋒ \underline{M_{C}^{β}} (A)$ ,

(6) $\bar{Q_{C}^{β}} (A) = \bar{N_{C}^{β}} (A) ⋓ \bar{M_{C}^{β}} (A)$ ,

(7) $\underline{P_{C}^{β}} (A) = \underline{N_{C}^{β}} (A) ⋓ \underline{M_{C}^{β}} (A)$ ,

(8) $\bar{P_{C}^{β}} (A) = \bar{N_{C}^{β}} (A) ⋒ \bar{M_{C}^{β}} (A)$ .

Proof. (1)-(4) It follows immediately from Definitions 3.2, 3.4, 3.5, 3.6 and 2.6.

(5) According to Definitions 3.2, 3.4 and 3.6, we have

$h_{\underline{Q_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A} (y)}$

$= \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)) ⌅ (h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y))}$

$= (\underset{y \in U}{\bar{⋀}} {h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)}) ⌅ (\underset{y \in U}{\bar{⋀}} {h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)})$

$= h_{\underline{M_{C}^{β}} (A)} (x) ⌅ h_{\underline{N_{C}^{β}} (A)} (x)$

$= h_{\underline{M_{C}^{β}} (A) ⋒ \underline{N_{C}^{β}} (A)} (x)$ .

(6) According to Definitions 3.2, 3.4 and 3.6, we have

$h_{\bar{Q_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)}$

$= \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{A} (y)) ⊻ (h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A} (y))}$

$= (\underset{y \in U}{\underline{⋁}} {h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{A} (y)}) ⊻ (\underset{y \in U}{⋁} {h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A} (y)})$

$= h_{\bar{M_{C}^{β}} (A)} (x) ⊻ h_{\bar{N_{C}^{β}} (A)} (x)$

$= h_{\bar{M_{C}^{β}} (A) ⋓ \bar{N_{C}^{β}} (A)} (x)$ .

(7-8) It can be easily checked without any difficulties.

In the following, some properties of 3-HFβCRSs will be groped.

Theorem 3.6. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A, B ∈ HF (U), then

(1) $\underline{P_{C}^{β}} (A^{c}) = (\bar{P_{C}^{β}} (A))^{c}$ ,

(2) $\bar{P_{C}^{β}} (A^{c}) = (\underline{P_{C}^{β}} (A))^{c}$ ,

(3) $\underline{P_{C}^{β}} (\tilde{U}) = \tilde{U}$ ,

(4) $\bar{P_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(5) $\underline{P_{C}^{β}} (A ⋒ B) = \underline{P_{C}^{β}} (A) ⋒ \underline{P_{C}^{β}} (B)$ ,

(6) $\bar{P_{C}^{β}} (A ⋓ B) = \bar{P_{C}^{β}} (A) ⋓ \bar{P_{C}^{β}} (B)$ ,

(7) If A ⊑ B, then $\underline{P_{C}^{β}} (A) ⊑ \underline{P_{C}^{β}} (B)$ and $\bar{P_{C}^{β}} (A) ⊑ \bar{P_{C}^{β}} (B)$ ,

(8) $\underline{P_{C}^{β}} (A) ⋓ \underline{P_{C}^{β}} (B) ⊑ \underline{P_{C}^{β}} (A ⋓ B)$ ,

(9) $\bar{P_{C}^{β}} (A ⋒ B) ⊑ \bar{P_{C}^{β}} (A) ⋒ \bar{P_{C}^{β}} (B)$ .

Proof. (1) According to Theorems 2.1, 3.1, 3.3 and 3.5, we have

$\underline{P_{C}^{β}} (A^{c}) = \underline{N_{C}^{β}} (A^{c}) ⋓ \underline{M_{C}^{β}} (A^{c})$

$= (\bar{N_{C}^{β}} (A))^{c} ⋓ (\bar{M_{C}^{β}} (A))^{c}$

$= (\bar{N_{C}^{β}} (A) ⋒ \bar{M_{C}^{β}} (A))^{c}$

$= (\bar{P_{C}^{β}} (A))^{c}$ .

Therefore, we have $\underline{P_{C}^{β}} (A^{c}) = (\bar{P_{C}^{β}} (A))^{c}$ .

(2) Based on the results of (1), we have

$\bar{P_{C}^{β}} (A^{c}) = {〈 x, h_{\bar{P_{C}^{β}} (A^{c})} (x) 〉 | x \in U}$

$= {〈 x, h_{(\underline{P_{C}^{β}} (A))^{c}} (x) 〉 | x \in U}$

$= {〈 x, h_{\underline{P_{C}^{β}} (A)} (x) 〉 | x \in U}^{c}$ .

Therefore, we have $\bar{P_{C}^{β}} (A^{c}) = (\underline{P_{C}^{β}} (A))^{c}$ .

(3) ∀ x ∈ U, we have $h_{\tilde{U}}^{σ (s)} (x) = 1$ (s = 1, 2, . . . , k).

$h_{\underline{P_{C}^{β}} (\tilde{U})} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{\tilde{U}} (y)}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{\tilde{U}}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor 1 | s = 1, 2, . . ., k}$

$= \tilde{1}$

$= h_{\tilde{U}} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y))$ , $l (h_{\tilde{U}} (y))}$ .

Therefore, we have $\underline{P_{C}^{β}} (\tilde{U}) = \tilde{U}$ .

(4) ∀ x ∈ U, we have $h_{\tilde{\emptyset}}^{σ (s)} (x) = 0$ (s = 1, 2, . . . , k).

$h_{\bar{P_{C}^{β}} (\tilde{\emptyset})} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{\tilde{\emptyset}} (y)}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \land h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land h_{\tilde{\emptyset}}^{σ (s)} | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \land h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land 0 | s = 1, 2, . . ., k}$

$= \tilde{0}$

$= h_{\tilde{\emptyset}} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ $l (h_{\tilde{\emptyset}} (y))}$ .

Therefore, we have $\bar{P_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ .

(5) According to Definitions 2.5 and 3.5, we have

$h_{\underline{P_{C}^{β}} (A ⋒ B)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A ⋒ B} (y)}$

$= \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ (h_{A} (y) ⌅ h_{B} (y))}$

$= \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y))$

$⌅ (h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{B} (y))}$

$= (\underset{y \in U}{\bar{⋀}} {h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{A} (y)})$

$⌅ (\underset{y \in U}{\bar{⋀}} {h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y) ⊻ h_{B} (y)})$

$= h_{\underline{P_{C}^{β}} (A)} (x) ⌅ h_{\underline{P_{C}^{β}} (B)} (x)$

$= h_{\underline{P_{C}^{β}} (A) ⋒ \underline{P_{C}^{β}} (B)} (x)$ .

Therefore, we have $\underline{P_{C}^{β}} (A ⋒ B) = \underline{P_{C}^{β}} (A) ⋒ \underline{P_{C}^{β}} (B)$ .

(6) According to Definitions 2.5 and 3.5, we have

$h_{\bar{P_{C}^{β}} (A ⋓ B)} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A ⋓ B} (y)}$

$= \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ (h_{A} (y) ⊻ h_{B} (y))}$

$= \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)$

$⊻ (h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{B} (y))}$

$= (\underset{y \in U}{\underline{⋁}} {h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A} (y)})$

$⊻ (\underset{y \in U}{\underline{⋁}} {h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{B} (y)})$

$= h_{\bar{P_{C}^{β}} (A)} (x) ⊻ h_{\bar{P_{C}^{β}} (B)} (x)$

$= h_{\bar{P_{C}^{β}} (A) ⋓ \bar{P_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ l (h_A (y)) , l (h_B (y))}.

Therefore, we have $\bar{P_{C}^{β}} (A ⋓ B) = \bar{P_{C}^{β}} (A) ⋓ \bar{P_{C}^{β}} (B)$ .

(7) Since A ⊑ B, thus, ∀ x ∈ U, we have h_A (x) ⪯ h_B (x).

$h_{\underline{P_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⊻ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A} (y)}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \lor h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\underline{P_{C}^{β}} (B)} (x)$ ;

$h_{\bar{P_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⌅ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \land h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \land h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\bar{P_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ l (h_A (y)) , l (h_B (y))}.

Therefore, we have $\underline{P_{C}^{β}} (A) ⊑ \underline{P_{C}^{β}} (B)$ and $\bar{P_{C}^{β}} (A) ⊑ \bar{P_{C}^{β}} (B)$ .

(8-9) It derives from (7) and Theorem 2.2.

In the following, the relationships of the third type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped.

Theorem 3.7. Let U be a domain of discourse, $C$ and $T$ are two hesitant fuzzy β coverings of U for HFE β, where $C$ ={C₁, C₂, . . . , C_m}, $T$ = ${C_{1}^{'}, C_{2}^{'}, . . ., C_{n}^{'}}$ . Suppose that ${({\hat{N}}_{1})_{x}^{β} | x \in U}$ and ${({\hat{N}}_{2})_{x}^{β} | x \in U}$ are two hesitant fuzzy β neighborhood systems induced by $C$ and $T$ , respectively. ∀ A ∈ HF (U), if $({\hat{N}}_{1})_{x}^{β} ⊑ ({\hat{N}}_{2})_{x}^{β}$ (∀ x ∈ U), then

(1) $\underline{P_{T}^{β}} (A) ⊑ \underline{P_{C}^{β}} (A)$ ,

(2) $\bar{P_{T}^{β}} (A) ⊒ \bar{P_{C}^{β}} (A)$ .

Proof. It is similar to the proof of Theorem 3.2.

In the following, some properties of 4-HFβCRSs will be groped.

Theorem 3.8. Let (U, $C$ ) be a HFCAS for some HFEs β and $C$ ={C₁, C₂, . . . , C_m}. ∀ A, B ∈ HF (U), then

(1) $\underline{Q_{C}^{β}} (A^{c}) = (\bar{Q_{C}^{β}} (A))^{c}$ ,

(2) $\bar{Q_{C}^{β}} (A^{c}) = (\underline{Q_{C}^{β}} (A))^{c}$ ,

(3) $\underline{Q_{C}^{β}} (\tilde{U}) = \tilde{U}$ ,

(4) $\bar{Q_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(5) $\underline{Q_{C}^{β}} (A ⋒ B) = \underline{Q_{C}^{β}} (A) ⋒ \underline{Q_{C}^{β}} (B)$ ,

(6) $\bar{Q_{C}^{β}} (A ⋓ B) = \bar{Q_{C}^{β}} (A) ⋓ \bar{Q_{C}^{β}} (B)$ ,

(7) If A ⊑ B, then $\underline{Q_{C}^{β}} (A) ⊑ \underline{Q_{C}^{β}} (B)$ and $\bar{Q_{C}^{β}} (A) ⊑ \bar{Q_{C}^{β}} (B)$ ,

(8) $\underline{Q_{C}^{β}} (A) ⋓ \underline{Q_{C}^{β}} (B) ⊑ \underline{Q_{C}^{β}} (A ⋓ B)$ ,

(9) $\bar{Q_{C}^{β}} (A ⋒ B) ⊑ \bar{Q_{C}^{β}} (A) ⋒ \bar{Q_{C}^{β}} (B)$ .

Proof. (1) According to Theorems 2.1, 3.1, 3.3 and 3.5, we have

$\underline{Q_{C}^{β}} (A^{c}) = \underline{N_{C}^{β}} (A^{c}) ⋒ \underline{M_{C}^{β}} (A^{c})$

$= (\bar{N_{C}^{β}} (A))^{c} ⋒ (\bar{M_{C}^{β}} (A))^{c}$

$= (\bar{N_{C}^{β}} (A) ⋓ \bar{M_{C}^{β}} (A))^{c}$

$= (\bar{Q_{C}^{β}} (A))^{c}$ .

Therefore, we have $\underline{Q_{C}^{β}} (A^{c}) = (\bar{Q_{C}^{β}} (A))^{c}$ .

(2) Since $\underline{Q_{C}^{β}} (A^{c}) = (\bar{Q_{C}^{β}} (A))^{c}$ , thus, we have

$\bar{Q_{C}^{β}} (A^{c}) = {〈 x, h_{\bar{Q_{C}^{β}} (A^{c})} (x) 〉 | x \in U}$

$= {〈 x, h_{(\underline{Q_{C}^{β}} (A))^{c}} (x) 〉 | x \in U}$

$= {〈 x, h_{\underline{Q_{C}^{β}} (A)} (x) 〉 | x \in U}^{c}$ .

Therefore, we have $\bar{Q_{C}^{β}} (A^{c}) = (\underline{Q_{C}^{β}} (A))^{c}$ .

(3) ∀ x ∈ U, we have $h_{\tilde{U}}^{σ (s)} (x) = 1$ (s = 1, 2, . . . , k).

$h_{\underline{Q_{C}^{β}} (\tilde{U})} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{\tilde{U}} (y)}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \land h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor u_{\tilde{U}}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \lor u_{\tilde{U}}^{σ (s)} (y))$

$\land (h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor u_{\tilde{U}}^{σ (s)} (y)) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \lor 1) \land (h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y) \lor 1) | s = 1, 2, . . ., k}$

$= \tilde{1}$

$= h_{\tilde{U}} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ $l (h_{\tilde{U}} (y))}$ .

Therefore, we have $\underline{Q_{C}^{β}} (\tilde{U}) = \tilde{U}$ .

(4) ∀ x ∈ U, we have $h_{\tilde{\emptyset}}^{σ (s)} (x) = 0$ (s = 1, 2, . . . , k).

$h_{\bar{Q_{C}^{β}} (\tilde{\emptyset})} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{\tilde{\emptyset}} (y)}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \lor h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land h_{\tilde{\emptyset}}^{σ (s)} | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \land h_{\tilde{\emptyset}}^{σ (s)}) \lor (h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land h_{\tilde{\emptyset}}^{σ (s)}) | s = 1, 2, . . ., k}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \land 0) \lor (h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y) \land 0) | s = 1, 2, . . ., k}$

$= \tilde{0}$

$= h_{\tilde{\emptyset}} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ $l (h_{\tilde{\emptyset}} (y))}$ .

Therefore, we have $\bar{Q_{C}^{β}} (\tilde{\emptyset}) = \tilde{\emptyset}$ .

(5) According to Definitions 2.5 and 3.6, we have

$h_{\underline{Q_{C}^{β}} (A ⋒ B)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A ⋒ B} (y)}$

$= \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ (h_{A} (y) ⌅ h_{B} (y))}$

$= \underset{y \in U}{\bar{⋀}} {((h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A} (y))$

$⌅ ((h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{B} (y))}$

$= (\underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \land h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k})$

$⌅ (\underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \land h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k})$

$= h_{\underline{Q_{C}^{β}} (A)} (x) ⌅ h_{\underline{Q_{C}^{β}} (B)} (x)$

$= h_{\underline{Q_{C}^{β}} (A) ⋒ \underline{Q_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ l (h_A (y)) , l (h_B (y))}.

Therefore, we have $\underline{Q_{C}^{β}} (A ⋒ B) = \underline{Q_{C}^{β}} (A) ⋒ \underline{Q_{C}^{β}} (B)$ .

(6) According to Definitions 2.5 and 3.6, we have

$h_{\bar{Q_{C}^{β}} (A ⋓ B)} (x) = \underset{y \in U}{\underline{⋁}} {h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y) ⌅ h_{A ⋓ B} (y)}$

$= \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ (h_{A} (y) ⊻ h_{B} (y))}$

$= \underset{y \in U}{\underline{⋁}} {((h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y))$

$⊻ ((h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{B} (y))}$

$= (\underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)})$

$⊻ (\underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{B} (y)})$

$= h_{\bar{Q_{C}^{β}} (A)} (x) ⊻ h_{\bar{Q_{C}^{β}} (B)} (x)$

$= h_{\bar{Q_{C}^{β}} (A) ⋓ \bar{Q_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ l (h_A (y)) , l (h_B (y))}.

Therefore, we have $\bar{Q_{C}^{β}} (A ⋓ B) = \bar{Q_{C}^{β}} (A) ⋓ \bar{Q_{C}^{β}} (B)$ .

(7) Since A ⊑ B, thus, ∀ x ∈ U, we have h_A (x) ⪯ h_B (x).

$h_{\underline{Q_{C}^{β}} (A)} (x) = \underset{y \in U}{\bar{⋀}} {(h_{({\hat{M}}_{x}^{β})^{c}} (y) ⌅ h_{({\hat{N}}_{x}^{β})^{c}} (y)) ⊻ h_{A} (y)}$

$= \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \land h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋀} {(h_{({\hat{M}}_{x}^{β})^{c}}^{σ (s)} (y) \land h_{({\hat{N}}_{x}^{β})^{c}}^{σ (s)} (y)) \lor h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\underline{Q_{C}^{β}} (B)} (x)$ ;

$h_{\bar{Q_{C}^{β}} (A)} (x) = \underset{y \in U}{\underline{⋁}} {(h_{{\hat{M}}_{x}^{β}} (y) ⊻ h_{{\hat{N}}_{x}^{β}} (y)) ⌅ h_{A} (y)}$

$= \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \lor h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land h_{A}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$⪯ \underset{y \in U}{⋁} {(h_{{\hat{M}}_{x}^{β}}^{σ (s)} (y) \lor h_{{\hat{N}}_{x}^{β}}^{σ (s)} (y)) \land h_{B}^{σ (s)} (y) | s = 1, 2, . . ., k}$

$= h_{\bar{Q_{C}^{β}} (B)} (x)$ ; where k =max_y ∈Umax ${l (h_{{\hat{M}}_{x}^{β}} (y)), l (h_{{\hat{N}}_{x}^{β}} (y)),$ l (h_A (y)) , l (h_B (y))}.

Thus, we have $\underline{Q_{C}^{β}} (A) ⊑ \underline{Q_{C}^{β}} (B)$ and $\bar{Q_{C}^{β}} (A) ⊑ \bar{Q_{C}^{β}} (B)$ .

(8-9) It follows immediately from (7) and Theorem 2.2.

In the following, the relationships of the fourth type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped.

Theorem 3.9. Let U be a domain of discourse, $C$ and $T$ are two hesitant fuzzy β coverings of U for HFE β, where $C$ ={C₁, C₂, . . . , C_m}, $T$ = ${C_{1}^{'}, C_{2}^{'}, . . ., C_{n}^{'}}$ . Suppose that ${({\hat{N}}_{1})_{x}^{β} | x \in U}$ and ${({\hat{N}}_{2})_{x}^{β} | x \in U}$ are two hesitant fuzzy β neighborhood systems induced by $C$ and $T$ , respectively. ∀ A ∈ HF (U), if $({\hat{N}}_{1})_{x}^{β} ⊑ ({\hat{N}}_{2})_{x}^{β}$ (∀ x ∈ U), then

(1) $\underline{Q_{T}^{β}} (A) ⊑ \underline{Q_{C}^{β}} (A)$ ,

(2) $\bar{Q_{T}^{β}} (A) ⊒ \bar{Q_{C}^{β}} (A)$ .

Proof. It is similar to the proof of Theorem 3.2.

3.3 Relationships between the proposed HFβCRSs and some other existing related models

In Theorems 3.2, 3.4, 3.7 and 3.9, we established the relationships of each type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings. In Theorem 3.5, we constructed the relationships of the proposed four types of HFβCRSs. In this subsection, the relationships between the proposed HFβCRSs and some other existing related models will be explored.

Theorem 3.10. If we set hesitant fuzzy relation $\hat{R}$ as $h_{\hat{R}} (x, y) = h_{{\hat{N}}_{x}^{β}} (y)$ (∀ x, y ∈ U), then the 1-HFβCRSs degenerate into HFRSs [39].

Proof. It derives from Definition 3.2.

Theorem 3.11. If we set hesitant fuzzy relation $\hat{R}$ as $h_{\hat{R}} (x, y) = h_{{\hat{M}}_{x}^{β}} (y)$ (∀ x, y ∈ U), then the 2-HFβCRSs degenerate into HFRSs [39].

Proof. It derives from Definition 3.4.

Theorem 3.12. If the hesitant fuzzy β covering of U degenerates into a fuzzy $\tilde{β}$ covering of U and A ∈ HF (U) degenerates into a FS on U, then the 1-HFβCRSs degenerate into Ma’s fuzzy covering rough set model for FSs [14].

Proof. It derives from Definition 3.2.

Theorem 3.13. If the hesitant fuzzy β covering of U degenerates into a fuzzy $\tilde{β}$ covering and A ∈ HF (U) degenerates into a FS on U, then the 2-HFβCRSs degenerate into Yang’s first type of fuzzy covering-based rough set [30].

Proof. It derives from Definition 3.4.

Theorem 3.14. If the hesitant fuzzy β covering of U degenerates into a fuzzy $\tilde{β}$ covering of U and A ∈ HF (U) degenerates into a FS on U, then the 3-HFβCRSs degenerate into Yang’s second type of fuzzy covering-based rough set [30].

Proof. It derives from Definition 3.5.

Theorem 3.15. If the hesitant fuzzy β covering of U degenerates into a fuzzy $\tilde{β}$ covering of U and A ∈ HF (U) degenerates into a FS on U, then the 4-HFβCRSs degenerate into Yang’s third type of fuzzy covering-based rough set [30].

Proof. It derives from Definition 3.6.

4 Applications of HFβCRSs in MADM

In this section, we give an application model, an algorithm and an illustrative example to elaborate the applications of the proposed HFβCRSs in MADM problems.

4.1 The application model

In recent years, there are more and more college graduates. What they concern most lies in looking for an optimal job department, and it reflects the success rate of the job seeker at some extent. In order to obtain a better and appropriate occupation match. It is inevitable for us to construct a feasible assessment method of selecting the optimal objects.

In what follows, an application model will be presented and applied to the problem of selecting the optimal job department.

Let U = {x₁, x₂, . . . , x_n} be a set of alternative job departments that is called the object set. Suppose that the job departments can be described by attribute set $C$ = { C₁, C₂, . . . , C_m}. We suppose that the evaluation values of objects x_i (i = 1, 2, . . . , n) w.r.t. the attribute C_j (j = 1, 2, . . . , m) is h_{C
_j} (x_i) = h_ij (i = 1, 2, . . . , n ; j = 1, 2, . . . , m), where h_{C
_j} (x_i) is a HFE, which denotes the possible membership degrees of the object x_i satisfies C_j is h_ij, i.e. C_j (j = 1, 2, . . . , m) is a HFS, which presents the relationships between objects and attributes, i.e. $C$ = { C₁, C₂, . . . , C_m} is a hesitant fuzzy β covering of U for some HFEs β. Then, to determine the optimal object by utilizing HFβCRSs. In what follows, we give an algorithm which will be applied into the MADM problems.

4.2 Algorithm for the MADM by employing HFβCRSs

Algorithm: The selection of the optimal object by using HFβCRSs.

Step 1. Construct the multi-attribute hesitant fuzzy decision making information system (U, $C$ , $N_{C}^{β}$ ( $M_{C}^{β}$ ), A).

Step 2. Compute the lower and upper approximations of A = {〈x, h_A (x) 〉|x ∈ U} w.r.t. $N_{C}^{β}$ ( $M_{C}^{β}$ ).

Step 3. Compute the grade value of $s ((\underline{▵_{C}^{β}} (A) \tilde{\oplus} \bar{▵_{C}^{β}} (A)) (x))$ (∀x ∈ U), where ▵ = N, M, P, Q, $\underline{▵_{C}^{β}} (A) \tilde{\oplus} \bar{▵_{C}^{β}} (A) = {〈 x, h_{\underline{▵_{C}^{β}} (A)} (x) \oplus h_{\bar{▵_{C}^{β}} (A)} (x) 〉 | x \in U}$ ;

Step 4. Ranking the grade values obtained in Step 3, the optimal solution is the highest grade value of $s ((\underline{▵_{C}^{β}} (A) \tilde{\oplus} \bar{▵_{C}^{β}} (A)) (x))$ (∀x ∈ U) (▵ = N, M, P, Q). If the highest grade value has more than one, we can choose any of them according to our preferences.

4.3 An illustrative example

Assume that a college student who is majored in computer speciality wants to seek a job department which is suitable for him. There are five departments for him to choose. Suppose that the decision maker represents the evaluation with HFE for each object w.r.t. every attribute (see Table 1), where U = {x₁, x₂, x₃, x₄, x₅} be the alternative job departments set, where x₁ stands market department, x₂ stands research and development department, x₃ stands accounting department, x₄ stands sales department, x₅ stands administration department and let $C$ ={y₁, y₂, y₃, y₄, y₅} be the attribute set, where y₁ stands software development ability, y₂ stands oral expression and communication ability, y₃ stands organizational management ability, y₄ stands English level and y₅ stands sales ability. Simultaneously, the job seeker presents the evaluation with HFEs based on his own condition, which is denoted by A.

A = {〈x₁, {0.3, 0.4} 〉, 〈x₂, {0.2, 0.6} 〉, 〈x₃, {0.4, 0.6, 0.8} 〉, 〈x₄, {0.3, 0.5, 0.6} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉}.

Take β = {0.2, 0.4, 0.8}, the representations of ${\hat{N}}_{x}^{β}$ and ${\hat{M}}_{x}^{β} (x \in U)$ are given as Tables 2 and 3. Thus, we construct a multi-attribute hesitant fuzzy decision making information system (U, $C$ , $N_{C}^{β}$ ( $M_{C}^{β}$ ), A). Further, we calculate the lower and upper approximations of A = {〈x, h_A (x) 〉|x ∈ U} w.r.t $N_{C}^{β}$ according to Definition 3.2 as follows.

Table 2
The representation of ${\hat{N}}_{x}^{β}$

x ₁ x ₂ x ₃ x ₄ x ₅

${\hat{N}}_{x_{1}}^{β}$ {0.3, 0.4, 0.8} {0.1, 0.2} {0.2, 0.3} {0.1, 0.2, 0.4} {0.2, 0.3}

${\hat{N}}_{x_{2}}^{β}$ {0.2, 0.3, 0.4} {0.2, 0.8} {0.3, 0.6} {0.1, 0.2, 0.4} {0.2, 0.3}

${\hat{N}}_{x_{3}}^{β}$ {0.2, 0.3, 0.4} {0.2, 0.4} {0.3, 0.8} {0.2, 0.4} {0.1, 0.2}

${\hat{N}}_{x_{4}}^{β}$ {0.2, 0.3, 0.6} {0.3, 0.4} {0.4, 0.6} {0.2, 0.8} {0.1, 0.2}

${\hat{N}}_{x_{5}}^{β}$ {0.2, 0.3, 0.6} {0.1, 0.2} {0.2, 0.3} {0.2, 0.4} {0.3, 0.8}

	x ₁	x ₂	x ₃	x ₄	x ₅
${\hat{N}}_{x_{1}}^{β}$	{0.3, 0.4, 0.8}	{0.1, 0.2}	{0.2, 0.3}	{0.1, 0.2, 0.4}	{0.2, 0.3}
${\hat{N}}_{x_{2}}^{β}$	{0.2, 0.3, 0.4}	{0.2, 0.8}	{0.3, 0.6}	{0.1, 0.2, 0.4}	{0.2, 0.3}
${\hat{N}}_{x_{3}}^{β}$	{0.2, 0.3, 0.4}	{0.2, 0.4}	{0.3, 0.8}	{0.2, 0.4}	{0.1, 0.2}
${\hat{N}}_{x_{4}}^{β}$	{0.2, 0.3, 0.6}	{0.3, 0.4}	{0.4, 0.6}	{0.2, 0.8}	{0.1, 0.2}
${\hat{N}}_{x_{5}}^{β}$	{0.2, 0.3, 0.6}	{0.1, 0.2}	{0.2, 0.3}	{0.2, 0.4}	{0.3, 0.8}

Table 3

The representation of ${\hat{M}}_{x}^{β}$

	x ₁	x ₂	x ₃	x ₄	x ₅
${\hat{M}}_{x_{1}}^{β}$	{0.3, 0.4, 0.8}	{0.2, 0.3, 0.4}	{0.2, 0.3, 0.4}	{0.2, 0.3, 0.6}	{0.2, 0.3, 0.6}
${\hat{M}}_{x_{2}}^{β}$	{0.1, 0.2}	{0.2, 0.8}	{0.2, 0.4}	{0.3, 0.4}	{0.1, 0.2}
${\hat{M}}_{x_{3}}^{β}$	{0.2, 0.3}	{0.3, 0.6}	{0.3, 0.8}	{0.4, 0.6}	{0.2, 0.3}
${\hat{M}}_{x_{4}}^{β}$	{0.1, 0.2, 0.4}	{0.1, 0.2, 0.4}	{0.2, 0.4}	{0.2, 0.8}	{0.2, 0.4}
${\hat{M}}_{x_{5}}^{β}$	{0.2, 0.3}	{0.2, 0.3}	{0.1, 0.2}	{0.1, 0.2}	{0.3, 0.8}

$\underline{N_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.7,$ 0.8} 〉, 〈x₃, {0.4, 0.7, 0.8} 〉, 〈x₄, {0.3, 0.6, 0.7} 〉, 〈x₅, {0.2, 0.7, 0.8} 〉},

$\bar{N_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4} 〉, 〈 x_{2}, {0.3, 0.6} 〉,$ 〈x₃, {0.3, 0.6, 0.8} 〉, 〈x₄, {0.4, 0.6} 〉, 〈x₅, {0.3, 0.7, 0.8} 〉}.

In Step 2 of the algorithm given in Subsection 4.2, if we compute the second type of the lower and upper approximations of A = {〈x, h_A (x) 〉} w.r.t $M_{C}^{β}$ , then

$\underline{M_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.8} 〉, 〈 x_{3}, {0.4, 0.6} 〉, 〈 x_{4}, {0.3, 0.8} 〉, 〈 x_{5}, {0.3,$ 0.7, 0.8} 〉},

$\bar{M_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4, 0.6} 〉, 〈 x_{2}, {0.3, 0.6} 〉, 〈 x_{3}, {0.3, 0.6, 0.8} 〉, 〈 x_{4}, {0.2, 0.5, 0.6} 〉,$ 〈x₅, {0.3, 0.7, 0.8} 〉}.

In Step 2 of the algorithm given in Subsection 4.2, if we compute the third type of the lower and upper approximations of A = {〈x, h_A (x) 〉|x ∈ U} w.r.t $N_{C}^{β}$ and $M_{C}^{β}$ , then

$\underline{P_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.8} 〉, 〈 x_{3}, {0.4, 0.7, 0.8} 〉, 〈 x_{4}, {0.3, 0.8} 〉, 〈 x_{5},$ {0.3, 0.7, 0.8} 〉},

$\bar{P_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4} 〉, 〈 x_{2}, {0.3, 0.6} 〉, 〈 x_{3}, {0.3, 0.6, 0.8} 〉, 〈 x_{4}, {0.2, 0.5, 0.6} 〉, 〈 x_{5},$ {0.3, 0.7, 0.8} 〉}.

In Step 2 of the algorithm given in Subsection 4.2, if we compute the fourth type of the lower and upper approximations of A = {〈x, h_A (x) 〉|x ∈ U} w.r.t $N_{C}^{β}$ and $M_{C}^{β}$ , then

$\underline{Q_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.6, 0.7} 〉, 〈 x_{2}, {0.2, 0.7, 0.8} 〉, 〈 x_{3}, {0.4, 0.6} 〉, 〈 x_{4}, {0.3, 0.6, 0.7} 〉,$ 〈x₅, {0.2, 0.7, 0.8} 〉},

$\bar{Q_{C}^{β}} (A) = {〈 x_{1}, {0.3, 0.4, 0.6} 〉, 〈 x_{2}, {0.3, 0.6} 〉, 〈 x_{3}, {0.3, 0.6, 0.8} 〉, 〈 x_{4}, {0.4, 0.6} 〉, 〈 x_{5},$ {0.3, 0.7, 0.8} 〉}.

Furthermore, we calculate $\underline{N_{C}^{β}} (A) \tilde{\oplus} \bar{N_{C}^{β}} (A)$ , $\underline{M_{C}^{β}} (A) \tilde{\oplus} \bar{M_{C}^{β}} (A)$ , $\underline{P_{C}^{β}} (A) \tilde{\oplus} \bar{P_{C}^{β}} (A)$ , and $\underline{Q_{C}^{β}}$ $(A) \tilde{\oplus} \bar{Q_{C}^{β}} (A)$ as follows.

$\underline{N_{C}^{β}} (A) \tilde{\oplus} \bar{N_{C}^{β}} (A) = {〈 x_{1}, (0.5100, 0.5800, 0.7200, 0.7600, 0.7900, 0.8200) 〉, 〈 x_{2}, (0.4400$ , 0.6800, 0.7900, 0.8600, 0.8800, 0.9200) 〉, 〈x₃, (0.5800, 0.7600, 0.7900, 0.8600, 0.8800, 0.8800, 0.9200, 0.9400, 0.9600) 〉, 〈x₄, (0.5800, 0.7200, 0.7600, 0.8200, 0.8400, 0.8800) 〉, 〈x₅, (0.5100, 0.7900, 0.7900, 0.8600, 0.8600, 0.9100, 0.9400, 0.9400, 0.9600) 〉}.

$\underline{M_{C}^{β}} (A) \tilde{\oplus} \bar{M_{C}^{β}} (A) = {〈 x_{1}, (0.5100, 0.5800, 0.7200, 0.7200, 0.7600, 0.7900, 0.8200, 0.8400$ , 0.8800) 〉, 〈x₂, (0.4400, 0.6800, 0.7900, 0.8800) 〉, 〈x₃, (0.5800, 0.7200, 0.7600, 0.8400, 0.8800, 0.9200) 〉, 〈 x₄, (0.4400, 0.6500, 0.7200, 0.8400, 0.9000, 0.9200) 〉, 〈x₅, (0.5100, 0.7900, 0.7900, 0.8600, 0.8600, 0.9100, 0.9400, 0.9400, 0.9600) 〉}.

$\underline{P_{C}^{β}} (A) \tilde{\oplus} \bar{P_{C}^{β}} (A) = {〈 x_{1}, (0.5100, 0.5800, 0.7200, 0.7600, 0.7900, 0.8200) 〉, 〈 x_{2}, (0.4400, 0.6800, 0.7900, 0.8600, 0.8800, 0.9200) 〉, 〈 x_{3}, (0.5800, 0.7600, 0.7900, 0.8600, 0.8800, 0.8800, 0.9200, 0.9400, 0.9600) 〉,$ 〈x₄, (0.4400, 0.6500, 0.7200, 0.8400, 0.9000, 0.9200) 〉, 〈x₅, (0.5100, 0.7900, 0.7900, 0.8600, 0.8600, 0.9100, 0.9400, 0.9400, 0.9600) 〉}.

$\underline{Q_{C}^{β}} (A) \tilde{\oplus} \bar{Q_{C}^{β}} (A) = {〈 x_{1}, (0.5100, 0.5800, 0.7200, 0.7200, 0.7600, 0.7900, 0.8200, 0.8400, 0.8800) 〉$ , 〈x₂, (0.4400, 0.6800, 0.7900, 0.8800) 〉, 〈x₃, (0.5800, 0.7200, 0.7600, 0.8400, 0.8800, 0.9200) 〉, 〈x₄, (0.5800, 0.7200, 0.7600, 0.8200, 0.8400, 0.8800) 〉, 〈x₅, (0.5100, 0.7900, 0.7900, 0.8600, 0.8600, 0.9100, 0.9400, 0.9400, 0.9600) 〉}.

Finally, we compute the grade values by using Definition 2.3.

$s (\underline{N_{C}^{β}} (A) \tilde{\oplus} \bar{N_{C}^{β}} (A)) = {〈 x_{1}, 0.6967 〉, 〈 x_{2}, 0.7617 〉, 〈 x_{3}, 0.8411 〉, 〈 x_{4}, 0.7667 〉, 〈 x_{5}, 0.8400 〉}$ .

$s (\underline{M_{C}^{β}} (A) \tilde{\oplus} \bar{M_{C}^{β}} (A)) = {〈 x_{1}, 0.7356 〉, 〈 x_{2}, 0.6975 〉, 〈 x_{3}, 0.7833 〉, 〈 x_{4}, 0.7450 〉, 〈 x_{5}, 0.8400 〉}$ .

$s (\underline{P_{C}^{β}} (A) \tilde{\oplus} \bar{P_{C}^{β}} (A)) = {〈 x_{1}, 0.6967 〉, 〈 x_{2}, 0.7617 〉, 〈 x_{3}, 0.8411 〉, 〈 x_{4}, 0.7450 〉, 〈 x_{5}, 0.8400 〉}$ .

$s (\underline{Q_{C}^{β}} (A) \tilde{\oplus} \bar{Q_{C}^{β}} (A)) = {〈 x_{1}, 0.7356 〉, 〈 x_{2}, 0.6975 〉, 〈 x_{3}, 0.7833 〉, 〈 x_{4}, 0.7667 〉, 〈 x_{5}, 0.8400 〉}$ .

According to the above results, we rank the objects (see Table 4). From Table 4, we find that the highest grade value is x₃ in the case of 1-HFβCRSs, so, the most suitable department for him (her) is accounting department. We can make similar analysis for other situations.

Table 4

Grade values obtained by the HFβCRSs and the rankings of objects

	x ₁	x ₂	x ₃	x ₄	x ₅	Ranking
1-HFβCRSs	0.6967	0.7617	0.8411	0.7667	0.8400	x₃ ≻ x₅ ≻ x₄ ≻ x₂ ≻ x₁
2-HFβCRSs	0.7356	0.6975	0.7833	0.7450	0.8400	x₅ ≻ x₃ ≻ x₄ ≻ x₁ ≻ x₂
3-HFβCRSs	0.6967	0.7617	0.8411	0.7450	0.8400	x₃ ≻ x₅ ≻ x₂ ≻ x₄ ≻ x₁
4-HFβCRSs	0.7356	0.6975	0.7833	0.7667	0.8400	x₅ ≻ x₃ ≻ x₄ ≻ x₁ ≻ x₂

4.4 The comparative analysis of our models and other existing related models

The example given in Subsection 4.3 illustrates that each type of HFβCRSs model can be used for MADM under the evaluation of hesitant fuzzy information. The decision making method given in Subsection 4.2 helps a student to obtain a better and appropriate occupation match by using the proposed HFβCRSs models. In the case of 1-HFβCRSs, the most suitable department for him (her) is accounting department. In the case of 2-HFβCRSs, the most suitable department for him (her) is administration department. In the case of 3-HFβCRSs, the most suitable department for him (her) is accounting department. In the case of 4-HFβCRSs, the most suitable department for him (her) is administration department. We obtain the same decision making results by using 1-HFβCRSs and 3-HFβCRSs. Meanwhile, we obtain the same decision making results by using 2-HFβCRSs and 4-HFβCRSs.

Both Ma’s fuzzy covering rough set model [14] and Yang’s fuzzy covering-based rough set model [14] enrich the fuzzy rough set theory. However, their models have limitations in the practical applications, in which their models are not applicable when dealing with hesitant fuzzy information, the models proposed by us can cope with the issue. Moreover, four types of HFβCRSs models proposed by us are generalizations of their models. Thus, the HFβCRSs models proposed by us are more general than their models and are more applicable than their models when handling hesitant fuzzy information. However, as we only explored the relationships of each type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings, we will explore more general properties of the lower and upper rough approximation operators under different hesitant fuzzy β coverings and the reduct of hesitant fuzzy β coverings of a universe U in the following research.

5 Conclusion

In this paper, we put forward four types of HFβCRSs which can be viewed as bridges between CBRSs theory and HFS theory. We then discussed their properties and constructed the relationships of each type of lower and upper rough approximation operators w.r.t. two different hesitant fuzzy β coverings. Further, we not only established the relationships of our proposed models, but also constructed the relationships among our proposed models, Ma’s model and Yang’s model. In fact, our proposed models are generalizations of Ma’s model and Yang’s model. Finally, we presented an application model, an algorithm and an illustrative example to show the feasibility of HFβCRSs in MADM. The new four types of HFβCRSs proposed by us enrich the fuzzy rough set theory and its applications.

Footnotes

Acknowledgments

Authors would like to thank Editors and the anonymous reviewers for their constructive comments in improving this paper.

References

Couso

and Dubois

, Rough Sets, coverings and incomplete information, Fundamenta Informaticae 108(3-4) (2011), 223–247.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990), 191–209.

Huang

, Li

H.X.

, Feng

G.F.

and Guo

C.X.

, Intuitionistic fuzzy β-covering-based rough sets, Artificial Intelligence Review 53 (2020), 2841–2873.

Jena

S.P.

and Ghosh

S.K.

, Intuitionistic fuzzy rough sets, Notes on Intuitionistic Fuzzy Sets 8 (2002), 1–18.

T.J.

and Zhang

W.X.

, Rough fuzzy approximation on two universes of discourse, Information Sciences 178 (2008), 892–906.

Liao

H.C.

and Xu

Z.S.

, Decision making under hesitant fuzzy environment with incomplete weights, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 22(4) (2014), 553–572.

Liang

D.C.

and Liu

, A novel risk decision making based on decision-theoretic rough sets under hesitant fuzzy Information, IEEE Transactions on Fuzzy Systems 23(2) (2015), 237–247.

Lin

G.P.

, Qian

Y.H.

and Li

J.J.

, NMGRS: Neighborhood-based multigranulation rough sets, International Journal of Approximate Reasoning 53(7) (2012), 1080–1093.

Lin

G.P.

, Liang

J.Y.

and Qian

Y.H.

, Multigranulation rough sets: From partition to covering, Information Sciences 241 (2013), 101–118.

10.

Liu

C.H.

and Wang

M.Z.

, Covering fuzzy rough set based on multi-granulations, International Conference on Uncertainty Reasoning and Knowledge Engineering Indonesia, (2011), 146–149.

11.

Liu

C.H.

, Miao

D.Q.

and Qian

, On multi-granulation covering rough sets, International Journal of Approximate Reasoning 55(6) (2014), 1404–1418.

12.

L.W.

, On some types of neighborhood-related covering rough sets, International Journal of Approximate Reasoning 53(6) (2012), 901–911.

13.

L.W.

, Classification of coverings in the finite approximation spaces, Information Sciences 276(20) (2014), 31–41.

14.

L.W.

, Two fuzzy coverings rough set models and their generalizations over fuzzy lattices, Fuzzy Sets and Systems 294 (2016), 1–17.

15.

Pawlak

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

16.

Pawlak

, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, Boston, (1991).

17.

Restrepo

, Cornelis

and Gómez

, Partial order relation for approximation operators in covering based rough sets, Information Sciences 284(10) (2014), 44–59.

18.

Slowinski

and Vanderpooten

, A generalized definition of rough approximations based on similarity, IEEE Transaction on Knowledge and Data Engineering 12 (2000), 331–336.

19.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Koera, (2009), 1378–1382.

20.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25(6) (2010), 529–539.

21.

Tang

X.A.

, Peng

Z.L.

, Ding

H.N.

, Cheng

M.L.

and Yang

S.L.

, Novel distance and similarity measures for hesitant fuzzy sets and their applications to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 34 (2018), 3903–3916.

22.

Tong

J.L.

, Leung

and Zhang

W.X.

, Generalized fuzzy rough approximation operators based on fuzzy coverings, International Journal of Approximate Reasoning 48(3) (2008), 836–856.

23.

M.F.

, Han

H.H.

and Si

Y.F.

, Properties and axiomatization of fuzzy rough sets based on fuzzy coverings, Proceedings of the 2012 International Conference on Machine Learning and Cybernetics (2012), 184–189.

24.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52(3) (2011), 395–407.

25.

Z.S.

and Xia

M.M.

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(11) (2011), 2128–2138.

26.

Z.S.

and Xia

M.M.

, On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligent Systems 26(5) (2011), 410–425.

27.

Yang

X.B.

, Song

X.N.

, Qi

Y.S.

and Yang

J.Y.

, Constructive and axiomatic approaches to hesitant fuzzy rough sets, Soft Computing 18(6) (2014), 1067–1077.

28.

Yao

Y.Y.

, On generalizing pawlak approximation operators, LNAI 1424 (1998), 298–307.

29.

Yao

Y.Y.

and Yao

B.X.

, Covering based rough set approximations, Information Sciences 200 (2012), 91–107.

30.

Yang

and Hu

B.Q.

, On some types of fuzzy covering-based on rough sets, Fuzzy Sets and Systems 312 (2017), 36–65.

31.

Yang

H.L.

, Zhang

C.L.

, Guo

Z.L.

, Liu

Y.L.

and Liao

X.W.

, A hybrid model of single valued neutrosophic sets and rough sets: single valued neutrosophic rough set model, Soft Computing 21(21) (2017), 6253–6267.

32.

Yang

H.L.

and Zhou

J.J.

, Interval-valued pythagorean fuzzy rough approximation operators and its application, Journal of Intelligent and Fuzzy Systems 39(3) (2020), 3067–3084.

33.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

34.

Zakowski

, Approximations in the space (U, II), Demonstratio Mathematica IXV (1983), 761–769.

35.

Zhou

J.J.

and Yang

H.L.

, Multigranulation hesitant Pythagorean fuzzy rough sets and its application in multi-attribute decision making, Journal of Intelligent and Fuzzy Systems 36 (2019), 5631–5644.

36.

Zhang

and Li

D.Y.

, Pythagorean fuzzy rough sets and its applications in multi-attribute decision making, Journal of Chinese Computer Systems 37(7) (2016), 1531–1535.

37.

Zhang

, Li

D.Y.

and Ren

, Pythagorean fuzzy multigranulation rough set over two universes and its applications in merger and acquisition, International Journal of Intelligent Systems 31(9) (2016), 921–943.

38.

Zhang

H.D.

, Shu

and Liao

S.L.

, Hesitant fuzzy rough set over two universes and its application in decision making, Soft Computing 21 (2017), 1803–1816.

39.

Zhang

H.D.

, Shu

and Xiong

L.L.

, On novel hesitant fuzzy rough sets, Soft Computing (2019), https://doi.org/10.1007/s00500-019-04037-9.

40.

Zhang

Z.M.

, Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings, Information Sciences 198 (2012), 186–206.

41.

Zhang

H.Q.

, Li

D.W.

, Wang

, Li

T.R.

and Bouras

, Uncertainty and equivalence relation analysis for hesitant fuzzy-rough sets and their applications in classification, Computing in Science and Engineering (2018), DOI: 10.1109/MCSE.2018.110150747

42.

Zhang

, Zhan

J.M.

and Xu

Z.X.

, Covering-based generalized IF rough sets with applications to multi-attribute decision-making, Information Sciences 478 (2019), 275–302.

43.

Zhu

and Wang

F.Y.

, Reduction and axiomization of covering generalized rough sets, Information Sciences 152 (2003), 217–230.

44.

Zhu

and Wang

F.Y.

, The fourth type of covering-based rough sets, Information Sciences 201 (2012), 80–92.

45.

Zhan

J.M.

, Sun

B.Z.

and Alcantud

J.C.R.

, Covering based multigranulation (I,T)-fuzzy rough set models and applications in multi-attribute group decision-making, Information Sciences 476 (2019), 290–318.

46.

Zhan

J.M.

and Xu

W.H.

, Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artificial Intelligence Review 53(1) (2020), 167–198.