Multi-granulation dual hesitant fuzzy rough sets

Abstract

In this paper, a single-granulation dual hesitant fuzzy rough set model based on dual hesitant fuzzy tolerance relations is first introduced. By the combination of multi-granulation rough sets and the single-granulation dual hesitant fuzzy rough sets, we then develop a new multi-granulation rough set model, called a multi-granulation dual hesitant fuzzy rough set. In the multi-granulation framework, two types of multi-granulation dual hesitant fuzzy rough sets are proposed, which are named the optimistic multi-granulation dual hesitant fuzzy rough sets and pessimistic multi-granulation dual hesitant fuzzy rough sets. Finally, the relationships among the single-granulation dual hesitant fuzzy rough sets, optimistic multi-granulation dual hesitant fuzzy rough sets and pessimistic multi-granulation dual hesitant fuzzy rough sets are investigated.

Keywords

Dual hesitant fuzzy tolerance relation Single-granulation dual hesitant fuzzy rough set Multi-granulation dual hesitant fuzzy rough set

1 Introduction

Rough set theory, introduced by Pawlak [25, 26], is a new mathematical approach to cope with imprecision and uncertainty in data analysis, and can be regarded as a valid means of granular computing [27]. In Pawlak’s rough set model, a key and primitive notion is equivalence relation. However, the equivalence relation is a very stringent condition which limits the application of rough sets. Therefore, by replacing the equivalence relation with other binary relations, such as dominance, fuzzy, interval-valued fuzzy, intuitionistic fuzzy, hesitant fuzzy and interval-valued hesitant fuzzy, lots of researchers have proposed many new rough sets model [4 , 61–63]. In recent years, from the perspective of granular computing, the generalization of Pawlak’s rough set model has become a new research focus. Since Qian and Liang [28] introduced multi-granulation rough set (MGRS) theory, lots of fruitful results about MGRS theory have been achieved [7 , 52].

As one of the extensions of Zadeh’s fuzzy set [60], hesitant fuzzy (HF) set theory, initiated by Torra [36, 37], permits the membership degree of an element to a set having several possible values, and can express the hesitant information more comprehensively than other extensions of fuzzy set. Since the appearance of hesitant fuzzy set, it has attracted more and more scholars’ attention [2 , 61]. Dual hesitant fuzzy (DHF) set, introduced by Zhu et al. [53], is a comprehensive set encompassing fuzzy sets, intuitionistic fuzzy sets [1], hesitant fuzzy sets, and fuzzy multisets [23] as special cases. Unlike a hesitant fuzzy set, a dual hesitant fuzzy set gives both membership degrees and nonmembership degrees which are respectively represented by two sets of possible values. By several possible values for the membership and nonmembership degrees, dual hesitant fuzzy sets are more objective than hesitant fuzzy sets to describe the vagueness of data or information. More recently, many authors have developed multiple attribute decision-making theories and methods under dual hesitant fuzzy environment. For example, Wang et al. [38] investigated the multiple attribute decision making problem based on the aggregation operators with dual hesitant fuzzy information. Ye [49] and Chen [3] proposed some correlation coefficient formulas for dual hesitant fuzzy sets and applied them to multiple attribute decision making under dual hesitant fuzzy environment. Meantime, Farhadinia [6] also presented an approach for deriving the correlation coefficient of dual hesitant fuzzy sets, and gave a practical example to illustrate the application of correlation coefficient for dual hesitant fuzzy sets in medical diagnosis.

It is well known that the combination of rough sets and other uncertain theories has always been a research hotspot. By integrating HF sets with rough sets, Yang et al. [50] first proposed the concept of hesitant fuzzy rough sets, and investigated an axiomatic approach of this model. The purpose of this paper is first to extend hesitant fuzzy rough sets into dual hesitant fuzzy environment and propose a dual hesitant fuzzy rough set model, called the single-granulation dual hesitant fuzzy rough set (SGDHFRS) from the perspective of granular computing. Furthermore, inspired by Qian et al.’s work, we then generalize the SGDHFRS to multi-granulation environment by the combination of DHF set theory and MGRS, and construct two types of multi-granulation dual hesitant fuzzy rough set (MGDHFRS) models in which set approximations are described by multiple DHF tolerance relations.

The rest of the paper is organized as follows. The next section reviews the basic concepts considered in the study. In Section 3, SGDHFRS theory is proposed and some properties of this model are examined.Section 4 introduces two types of MGDHFRS models in which set approximations are described by multiple DHF tolerance relations. And some of their properties are discussed. In Section 5, we establish the relationships among SGDHFRS, the optimistic MGDHFRS and pessimistic MGDHFRS. Finally, we conclude the paper in Section 6.

2 Preliminaries

In this section, we briefly recall the concepts of HF sets and DHF sets.

2.1 Hesitant fuzzy sets

Torra and Narukawa [36] and Torra [37] firstly proposed the concept of hesitant fuzzy sets defined as follows:

Definition 2.1. [36, 37] Let U be a fixed set, a hesitant fuzzy set $𝔸$ on U is in terms of a function $h_{𝔸} (x)$ that when applied to U returns a subset of [0,1].

To be easily understood, Xia and Xu [41] denoted the HF set by a mathematical symbol: $𝔸 = {< x, h_{𝔸} (x) > | x \in U},$ where $h_{𝔸} (x)$ is a set of some different values in [0,1], standing for the possible membership degrees of the element x ∈ U to $𝔸$ .

For convenience, Xia and Xu [41] called $h_{𝔸} (x)$ a HF element, and denoted the set of all HF sets on U by HF (U).

2.2 Dual hesitant fuzzy sets

As an extension of hesitant fuzzy sets, dual hesitant fuzzy sets are defined by Zhu et al. [53] as follows.

Definition 2.2. [53] Let U be a fixed set, a dual hesitant fuzzy set $𝔻$ on U is described as: $𝔻 = {< x, h_{𝔻} (x), g_{𝔻} (x) > | x \in U},$ in which $h_{𝔻} (x)$ and $g_{𝔻} (x)$ are two sets of some values in [0,1], denoting the possible membership degrees and non-membership degrees of the element x ∈ U to the set $𝔻$ respectively, with the conditions: 0 ≤ γ, η ≤ 1 and 0 ≤ γ ⁺ + η ⁺ ≤ 1, where for all x ∈ U, $γ \in h_{𝔻} (x), η \in g_{𝔻} (x),$ $γ^{+} \in h_{𝔻}^{+} (x) = \cup_{γ \in h_{𝔻} (x)} \max {γ}, η^{+} \in g_{𝔻}^{+} (x) = \cup_{η \in g_{𝔻} (x)} \max {η}$ .

For convenience, the pair $d (x) = (h_{𝔻} (x), g_{𝔻} (x))$ is called a DHF element denoted by d = (h, g). The set of all DHF sets on U is denoted by DHF (U).

Remark 2.3. From Definition 2.2, we can see that the existing sets, such as fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets, can be regarded as special cases of DHF sets.

Example 2.4. Let U = {x ₁, x ₂} be a reference set, $d (x_{1}) = (h_{𝔻} (x_{1}), g_{𝔻} (x_{1})) = ({0.2, 0.4}, {0.6})$ and $d (x_{2}) = (h_{𝔻} (x_{2}), g_{𝔻} (x_{2})) = ({0.3, 0.4}, {0.1, 0.5})$ be the DHF elements of x _i (i = 1, 2) to a set $𝔻$ , respectively. Then $𝔻$ can be considered as a DHF set, that is, $\begin{matrix} 𝔻 & = & {< x_{1}, {0.2, 0.4}, {0.6} >, \\ < x_{2}, {0.3, 0.4}, {0.1, 0.5} >} . \end{matrix}$

Here, we introduce several special DHF sets as follows: $\forall 𝔻 \in DHF (U)$ ,

1. $𝔻$ is referred to as an empty DHF set if and only if $h_{𝔻} (x) = {0}$ and $g_{𝔻} (x) = {1}$ for all x ∈ U. In this paper, the empty DHF set is denoted by ∅.

2. $𝔻$ is referred to as a full DHF set if and only if $h_{𝔻} (x) = {1}$ and $g_{𝔻} (x) = {0}$ for all x ∈ U. In this paper, the full DHF set is denoted by $𝕌$ .

It’s worth noting that the number of values in different DHF elements may be different and the values of DHF elements are usually given in a disorder. Suppose that $l (h_{𝔸} (x))$ and $l (g_{𝔸} (x))$ stand for the number of values in $h_{𝔸} (x)$ and $g_{𝔸} (x)$ , respectively. To operate correctly, Ye [49] and Chen [3] gave the following assumptions:

(A1) For a DHF set $𝔸 = {< x, h_{𝔸} (x), g_{𝔸} (x) > | x \in U}$ , let σ : (1, 2, ⋯ , n) ⟶ (1, 2, ⋯ , n) be a permutation satisfying $h_{𝔸}^{σ (s)} (x) \leq h_{𝔸}^{σ (s + 1)} (x)$ for s = 1, 2, …, n - 1, and $h_{𝔸}^{σ (s)} (x)$ be the sth largest value in $h_{𝔸} (x)$ ; let σ : (1, 2, ⋯ , m) ⟶ (1, 2, ⋯ , m) be a permutation satisfying $g_{𝔸}^{σ (t)} (x) \leq g_{𝔸}^{σ (t + 1)} (x)$ for t = 1, 2, …, m - 1, and $g_{𝔸}^{σ (t)} (x)$ be the tth largest value in $g_{𝔸} (x)$ .

(A2) For two DHF sets $𝔸$ and $𝔹$ , when $l (h_{𝔸} (x)) \neq l (h_{𝔹} (x))$ , $l (g_{𝔸} (x)) \neq l (g_{𝔹} (x))$ , one can make them have the same number of elements by adding some elements to the DHF element which has less number of elements. In terms of the pessimistic principle, the smallest element will be added while in the opposite case, the optimistic principle may be adopted. In the present work, we use the latter case. Therefore, if $l (h_{𝔸} (x)) < l (h_{𝔹} (x))$ , then $h_{𝔸} (x)$ should be extended by adding the maximum value in it until it has the same length as $h_{𝔹} (x)$ ; if $l (g_{𝔸} (x)) < l (g_{𝔹} (x))$ , then $g_{𝔸} (x)$ should be extended by adding the maximum value in it until it has the same length as $g_{𝔹} (x)$ .

On the basis of the assumptions given by Ye [49] and Chen [3], we develop some new methods when operating the DHF elements, which are slightly different from the ones introduced by Farhadinia [6]. The adjusted operational laws are defined as follows.

Definition 2.5. Let U be a nonempty and finite universe of discourse. For any $𝔸$ and $𝔹 \in DHF (U)$ , then

(1) the complement of $𝔸$ , denoted by $𝔸^{c}$ , is given by $\begin{matrix} 𝔸^{c} & = & {< x, g_{𝔸} (x), h_{𝔸} (x) > | x \in U} \\ = & {< x, {g_{𝔸}^{σ (t)} (x)}, {h_{𝔸}^{σ (s)} (x)} > | x \in U}; \end{matrix}$

(2) the union of $𝔸$ and $𝔹$ , denoted by $𝔸 ⋓ 𝔹$ , is given by $\begin{matrix} 𝔸 ⋓ 𝔹 & = & {< x, h_{𝔸} (x) ⊻ h_{𝔹} (x), g_{𝔸} (x) ⌆ g_{𝔹} (x) > | x \in U} \\ = & {< x, {h_{𝔸}^{σ (s)} (x) \lor h_{𝔹}^{σ (s)} (x)}, \end{matrix}$ $\begin{matrix} {g_{𝔸}^{σ (t)} (x) \land g_{𝔹}^{σ (t)} (x)} > | x \in U, 1 \\ \leq s \leq k, 1 \leq t \leq l}; \end{matrix}$

(3) the intersection of $𝔸$ and $𝔹$ , denoted by $𝔸 ⋒ 𝔹$ , is given by $\begin{matrix} 𝔸 ⋒ 𝔹 & = & {< x, h_{𝔸} (x) ⌆ h_{𝔹} (x), g_{𝔸} (x) ⊻ g_{𝔹} (x) > | x \in U} \\ = & {< x, {h_{𝔸}^{σ (s)} (x) \land h_{𝔹}^{σ (s)} (x)}, \end{matrix}$ $\begin{matrix} {g_{𝔸}^{σ (t)} (x) \lor g_{𝔹}^{σ (t)} (x)} > | x \in U, 1 \\ \leq s \leq k, 1 \leq t \leq l}, \end{matrix}$ where $k = \max {l (h_{𝔸} (x)), l (h_{𝔹} (x))},$ and $l = \max {l (g_{𝔸} (x)), l (g_{𝔹} (x))} .$

From Definition 2.5, we can easily verify that the following theorem holds.

Theorem 2.6. Let U be a reference set. For any $𝔻_{1},$ $𝔻_{2} \in DHF (U)$ , then the followings are valid:

$(𝔻_{1} ⋓ 𝔻_{2})^{c} = 𝔻_{1}^{c} ⋒ 𝔻_{2}^{c},$

$(𝔻_{1} ⋒ 𝔻_{2})^{c} = 𝔻_{1}^{c} ⋓ 𝔻_{2}^{c} .$

In what follows, we first introduce the concept of the DHF subset as follows.

Definition 2.7. Let U be a nonempty and finite universe of discourse. Denote $k = \max {l (h_{𝔸} (x)), l (h_{𝔹} (x))},$ and $l = \max {l (g_{𝔸} (x)), l (g_{𝔹} (x))} .$ For all $𝔸, 𝔹 \in DHF (U)$ , $𝔸$ is said to be a DHF subset of $𝔹$ , if $h_{𝔸} (x) ⪯ h_{𝔹} (x)$ and $g_{𝔸} (x) ⪰ g_{𝔹} (x)$ holds for any x ∈ U, i.e.

$h_{𝔸} (x) ⪯ h_{𝔹} (x),$ $g_{𝔸} (x) ⪰ g_{𝔹} (x) \Leftrightarrow h_{𝔸}^{σ (s)} (x) \leq h_{𝔹}^{σ (s)} (x), g_{𝔸}^{σ (t)} (x) \geq g_{𝔹}^{σ (t)} (x),$ with 1 ≤ s ≤ k and 1 ≤ t ≤ l . We denote it by $𝔸 ⊑ 𝔹$ .

Obviously, we conclude that the following results hold: $\forall 𝔸, 𝔹, ℂ \in DHF (U)$ ,

$𝔸 ⊑ 𝔸,$

$𝔸 ⊑ 𝔹, 𝔹 ⊑ ℂ \Rightarrow 𝔸 ⊑ ℂ,$

$𝔸 ⊑ 𝔹, 𝔹 ⊑ 𝔸 \Leftrightarrow 𝔸 = 𝔹 .$

That is, the notation ⊑ is reflexive, transitive and antisymmetric on DHF(U).

3 Single-granulation dual hesitant rough sets in the HF tolerance approximation space

Inspired by the concept of the HF relation in [50], we will further extend the HF relation into DHF environment and first introduce the concept of a DHF relation as follows.

Definition 3.1. Let U, V be two nonempty and finite universes. A DHF subset $ℝ$ of the universe U × V is called a DHF relation from U to V, namely, $ℝ$ is given by

$ℝ = {< (x, y), h_{ℝ} (x, y), g_{ℝ} (x, y) > | (x, y) \in U \times V},$ where $h_{ℝ}, g_{ℝ} : U \times V \to [0, 1]$ are two sets of some values in [0,1], denoting the possible membership degrees and non-membership degrees of the relationships between x and y respectively, with the conditions: 0 ≤ γ, η ≤ 1 and 0 ≤ γ ⁺ + η ⁺ ≤ 1, where for all (x, y) ∈ U × V, $γ \in h_{ℝ} (x, y), η \in g_{ℝ} (x, y),$ $γ^{+} \in h_{ℝ}^{+} (x, y) = \cup_{γ \in h_{ℝ} (x, y)} \max {γ}, η^{+} \in g_{ℝ}^{+} (x, y) = \cup_{η \in g_{ℝ} (x, y)} \max {η}$ .

In particular, if U = V, we call $ℝ$ a DHF relation on U. In what follows several special DHF relations are introduced as follows.

Definition 3.2. The DHF relation $ℝ$ from U to V is said to be serial if for each x ∈ U, there exists a y ∈ V such that $h_{ℝ} (x, y) = {1}$ and $g_{ℝ} (x, y) = {0}$ ; $ℝ$ is said to be reflexive on U if $h_{ℝ} (x, x) = {1}$ and $g_{ℝ} (x, x) = {0}$ for all x ∈ U; $ℝ$ is referred to as a symmetric DHF relation on U if $h_{ℝ} (x, y) = h_{ℝ} (y, x)$ and $g_{ℝ} (x, y) = g_{ℝ} (y, x)$ for all x, y ∈ U.

If a DHF relation $ℝ$ on U is reflexive and symmetric, it is called a DHF tolerance relation on U.

Based on the above DHF relation, lower and upper DHF approximation operators are defined asfollows.

Definition 3.3. Let U be a nonempty and finite universes and $ℝ$ be a DHF tolerance relation on U. The pair $(U, ℝ)$ is called a DHF tolerance approximation space. For any $𝔸 \in DHF (U),$ the lower and upper approximations of $𝔸$ with respect to $(U, ℝ)$ , denoted by $\underline{ℝ} (𝔸)$ and $\bar{ℝ} (𝔸)$ , are two DHF sets of U and are, respectively, defined as follows: $\underline{ℝ} (𝔸) = {< x, h_{\underline{ℝ} (𝔸)} (x), g_{\underline{ℝ} (𝔸)} (x) > | x \in U},$ (1) $\bar{ℝ} (𝔸) = {< x, h_{\bar{ℝ} (𝔸)} (x), g_{\bar{ℝ} (𝔸)} (x) > | x \in U},$ (2) where

$h_{\underline{ℝ} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ h_{𝔸} (y_{j})},$

$g_{\underline{ℝ} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ g_{𝔸} (y_{j})};$

$h_{\bar{ℝ} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ h_{𝔸} (y_{j})},$

$g_{\bar{ℝ} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ g_{𝔸} (y_{j})} .$

$\underline{ℝ} (𝔸)$ and $\bar{ℝ} (𝔸)$ are, respectively, called the single-granulation lower and upper approximations of $𝔸$ with respect to $(U, ℝ)$ . The pair $(\underline{ℝ} (𝔸), \bar{ℝ} (𝔸))$ is called a SGDHFRS of $𝔸$ with respect to $(U, ℝ)$ , and $\underline{ℝ}, \bar{ℝ} : DHF (U) \to DHF (U)$ are referred to as single granulation lower and upper DHF rough approximation operators, respectively.

Clearly, the above definition implies the following equivalent forms:

$h_{\underline{ℝ} (𝔸)} (x) = {⋀_{y_{j} \in U} g_{ℝ}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j}) |$ $1 \leq s \leq max_{1 \leq j \leq | U |} \max {l (g_{ℝ} (x, y_{j})), l (h_{𝔸} (y_{j}))}},$

$g_{\underline{ℝ} (𝔸)} (x) = {⋁_{y_{j} \in U} h_{ℝ}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j}) |$ $1 \leq t \leq max_{1 \leq j \leq | U |} \max {l (h_{ℝ} (x, y_{j})), l (g_{𝔸} (y_{j}))}},$

$h_{\bar{ℝ} (𝔸)} (x) = {⋁_{y_{j} \in U} h_{ℝ}^{σ (s)} (x, y_{j}) \land h_{𝔸}^{σ (s)} (y_{j}) |$ $1 \leq s \leq max_{1 \leq j \leq | U |} \max {l (h_{ℝ} (x, y_{j})), l (h_{𝔸} (y_{j}))}},$

$g_{\bar{ℝ} (𝔸)} (x) = {⋀_{y_{j} \in U} g_{ℝ}^{σ (t)} (x, y_{j}) \lor g_{𝔸}^{σ (t)} (y_{j}) |$ $1 \leq t \leq max_{1 \leq j \leq | U |} \max {l (g_{ℝ} (x, y_{j})), l (g_{𝔸} (y_{j}))}} .$

Theorem 3.4. Let $(U, ℝ)$ be a DHF tolerance approximation space. For the DHF set $𝔸 \in DHF (U),$ the single-granulation lower and upper approximations of $𝔸$ with respect to $(U, ℝ)$ satisfy the following properties:

$\underline{ℝ} (𝔸) ⊑ 𝔸,$

$\bar{ℝ} (𝔸) ⊒ 𝔸,$

$\underline{ℝ} (𝔸^{c}) = (\bar{ℝ} (𝔸))^{c},$

$\bar{ℝ} (𝔸^{c}) = (\underline{ℝ} (𝔸))^{c},$

$\underline{ℝ} (𝕌) = \bar{ℝ} (𝕌) = 𝕌,$

$\bar{ℝ} (\emptyset) = \underline{ℝ} (\emptyset) = \emptyset .$

Proof. We just prove the odd terms, and even terms can be proved in the similar way.

(1) Since $ℝ$ is a DHF tolerance relation, then ∀x ∈ U, there are $h_{ℝ} (x, x) = {1}$ and $g_{ℝ} (x, x) = {0}$ . From Equation (1), we have $\begin{matrix} h_{\underline{ℝ} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ h_{𝔸} (y_{j})} \\ = {⋀_{y_{j} \in U} (g_{ℝ}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j})) | 1 \leq s \leq k} \\ = {(g_{ℝ}^{σ (s)} (x, x) \lor h_{𝔸}^{σ (s)} (x)) \land \\ (⋀_{y_{j} \neq x} (g_{ℝ}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j}))) | 1 \leq s \leq k} \\ = {h_{𝔸}^{σ (s)} (x) \land (⋀_{y_{j} \neq x} (g_{ℝ}^{σ (s)} (x, y_{j}) \\ \lor h_{𝔸}^{σ (s)} (y_{j}))) | 1 \leq s \leq k} \\ ⪯ {h_{𝔸}^{σ (s)} (x) | 1 \leq s \leq k} = h_{𝔸} (x), \end{matrix}$ where $k = max_{1 \leq j \leq | U |} \max {l (g_{ℝ} (x, y_{j})), l (h_{𝔸} (y_{j})) .$

On the other hand, $\begin{matrix} g_{\underline{ℝ} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ g_{𝔸} (y_{j})} \\ = {⋁_{y_{j} \in U} (h_{ℝ}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j})) | 1 \leq t \leq l} \\ = {(h_{ℝ}^{σ (t)} (x, x) \land g_{𝔸}^{σ (t)} (x)) \lor \\ (⋁_{y_{j} \neq x} (h_{ℝ}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j}))) | 1 \leq t \leq l} \\ = {g_{𝔸}^{σ (t)} (x) \lor (⋁_{y_{j} \neq x} (h_{ℝ}^{σ (t)} (x, y_{j}) \\ \land g_{𝔸}^{σ (t)} (y_{j}))) | 1 \leq t \leq l} \\ ⪰ {g_{𝔸}^{σ (t)} (x) | 1 \leq t \leq l} = g_{𝔸} (x), \end{matrix}$ where $l = max_{1 \leq j \leq | U |} \max {l (h_{ℝ} (x, y_{j})), l (g_{𝔸} (y_{j})) .$

From the above discussions, we conclude that $\underline{ℝ} (𝔸) ⊑ 𝔸 .$

(3) According to Definitions 3.3, we obtain $\begin{matrix} h_{\underline{ℝ} (𝔸^{c})} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ h_{𝔸^{c}} (y_{j})} \\ = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ g_{𝔸} (y_{j})} \\ = g_{\bar{ℝ} (𝔸)} (x) = h_{(\bar{ℝ} (𝔸))^{c}} (x) \end{matrix}$ and $\begin{matrix} g_{\underline{ℝ} (𝔸^{c})} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ g_{𝔸^{c}} (y_{j})} \\ = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ h_{𝔸} (y_{j})} \\ = h_{\bar{ℝ} (𝔸)} (x) = g_{(\bar{ℝ} (𝔸))^{c}} (x) . \end{matrix}$ Hence it follows that $\underline{ℝ} (𝔸^{c}) = (\bar{ℝ} (𝔸))^{c}$ .

(5) The conclusion $\underline{ℝ} (𝕌) = 𝕌$ is straightforward by Equation (1). On the other hand, since $ℝ$ is a DHF tolerance relation, then ∀x ∈ U, there are $h_{ℝ} (x, x) = {1}$ and $g_{ℝ} (x, x) = {0}$ . By Equation (2), we obtain $\begin{matrix} h_{\bar{ℝ} (𝕌)} (x) = {⋁_{y_{j} \in U} (h_{ℝ}^{σ (s)} (x, y_{j}) \land h_{𝕌}^{σ (s)} (y_{j})) | 1 \leq s \leq k} \\ = {⋁_{y_{j} \in U} (h_{ℝ}^{σ (s)} (x, y_{j}) \land 1) | 1 \leq s \leq k} \\ = {h_{ℝ}^{σ (s)} (x, x) \lor (⋁_{y_{j} \neq x} h_{ℝ}^{σ (s)} (x, y_{j})) | 1 \leq s \leq k} \\ = {1} = h_{𝕌} (x), \end{matrix}$ where $k = max_{1 \leq j \leq | U |} l (h_{ℝ} (x, y_{j})) .$

On the other hand, we have $\begin{matrix} g_{\bar{ℝ} (𝕌)} (x) = {⋀_{y_{j} \in U} (g_{ℝ}^{σ (t)} (x, y_{j}) \lor g_{𝕌}^{σ (t)} (y_{j})) | 1 \leq t \leq l} \\ = {⋀_{y_{j} \in U} (g_{ℝ}^{σ (t)} (x, y_{j}) \lor 0) | 1 \leq t \leq l} \\ = {g_{ℝ}^{σ (t)} (x, x) \land (⋀_{y_{j} \neq x} g_{ℝ}^{σ (t)} (x, y_{j})) | 1 \leq t \leq l} \\ = {0} = g_{𝕌} (x), \end{matrix}$ where $l = max_{1 \leq j \leq | U |} l (g_{ℝ} (x, y_{j})) .$

From the above discussions, we conclude that $\bar{ℝ} (𝕌) = 𝕌 .$ □

Theorem 3.5. Let $(U, ℝ)$ be a DHF tolerance approximation space. Then the lower and upper DHF rough approximation operators induced from $(U, ℝ)$ satisfy the following properties: $\forall 𝔸, 𝔹 \in DHF (U),$

$\underline{ℝ} (𝔸 ⋒ 𝔹) = \underline{ℝ} (𝔸) ⋒ \underline{ℝ} (𝔹),$

$\bar{ℝ} (𝔸 ⋓ 𝔹) = \bar{ℝ} (𝔸) ⋓ \bar{ℝ} (𝔹),$

$𝔸 ⊑ 𝔹 \Rightarrow \underline{ℝ} (𝔸) ⊑ \underline{ℝ} (𝔹),$

$𝔸 ⊑ 𝔹 \Rightarrow \bar{ℝ} (𝔸) ⊑ \bar{ℝ} (𝔹),$

$\underline{ℝ} (𝔸 ⋓ 𝔹) ⊒ \underline{ℝ} (𝔸) ⋓ \underline{ℝ} (𝔹),$

$\bar{ℝ} (𝔸 ⋒ 𝔹) ⊑ \bar{ℝ} (𝔸) ⋒ \bar{ℝ} (𝔹),$

Proof. Due to the duality of $\underline{ℝ}$ and $\bar{ℝ}$ , we only prove the odd terms.

(1) By virtue of Equation (1), we have $\begin{matrix} h_{\underline{ℝ} (𝔸 ⋒ 𝔹)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ h_{𝔸 ⋒ 𝔹} (y_{j})} \\ = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ (h_{𝔸} (y_{j}) ⌆ h_{𝔹} (y_{j}))} \\ = {⋀_{y_{j} \in U} (g_{ℝ}^{σ (s)} (x, y_{j}) \lor (h_{𝔸}^{σ (s)} (y_{j}) \\ \land h_{𝔹}^{σ (s)} (y_{j}))) | 1 \leq s \leq k} \\ = {⋀_{y_{j} \in U} (g_{ℝ}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j})) \land \\ ⋀_{y_{j} \in U} (g_{ℝ}^{σ (s)} (x, y_{j}) \lor h_{𝔹}^{σ (s)} (y_{j})) | 1 \leq s \leq k} \\ = h_{\underline{ℝ} (𝔸)} (x) ⌆ h_{\underline{ℝ} (𝔹)} (x) = h_{\underline{ℝ} (𝔸) ⋒ \underline{ℝ} (𝔹)} (x), \end{matrix}$ where $l = max_{1 \leq j \leq | U |} \max {l (g_{ℝ} (x, y_{j})), l (h_{𝔸} (y_{j})), l (h_{𝔹} (y_{j}))} .$

On the other hand, $\begin{matrix} g_{\underline{ℝ} (𝔸 ⋒ 𝔹)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ g_{𝔸 ⋒ 𝔹} (y_{j})} \\ = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ (g_{𝔸} (y_{j}) ⊻ g_{𝔹} (y_{j}))} \\ = {⋁_{y_{j} \in U} (h_{ℝ}^{σ (t)} (x, y_{j}) \land (g_{𝔸}^{σ (t)} (y_{j}) \\ \lor g_{𝔹}^{σ (t)} (y_{j}))) | 1 \leq t \leq l} \\ = {⋁_{y_{j} \in U} (h_{ℝ}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j})) \lor \\ ⋁_{y_{j} \in U} (h_{ℝ}^{σ (t)} (x, y_{j}) \land g_{𝔹}^{σ (t)} (y_{j})) | 1 \leq t \leq l} \\ = g_{\underline{ℝ} (𝔸)} (x) ⊻ g_{\underline{ℝ} (𝔹)} (x) = g_{\underline{ℝ} (𝔸) ⋒ \underline{ℝ} (𝔹)} (x), \end{matrix}$ where $l = max_{1 \leq j \leq | U |} \max {l (h_{ℝ} (x, y_{j})), l (g_{𝔸} (y_{j})), l (g_{𝔹} (y_{j}))} .$

From the above discussions, we conclude that the result (1) holds.

(3) Since $𝔸 ⊑ 𝔹$ , for all y _j ∈ U, then there are $h_{𝔸}^{σ (s)} (y_{j}) \leq h_{𝔹}^{σ (s)} (y_{j})$ and $g_{𝔸}^{σ (t)} (y_{j}) \geq g_{𝔹}^{σ (t)} (y_{j})$ . So it follows that $\begin{matrix} ⋀_{y_{j} \in U} (h_{𝔸}^{σ (s)} (y_{j}) \lor g_{ℝ}^{σ (s)} (x, y_{j})) \\ \leq ⋀_{y_{j} \in U} (h_{𝔹}^{σ (s)} (y_{j}) \lor g_{ℝ}^{σ (s)} (x, y_{j})), 1 \leq s \leq k, \end{matrix}$ and $\begin{matrix} ⋁_{y_{j} \in U} (g_{𝔸}^{σ (t)} (y_{j}) \land h_{ℝ}^{σ (t)} (x, y_{j})) \\ \geq ⋁_{y_{j} \in U} (g_{𝔹}^{σ (t)} (y_{j}) \land h_{ℝ}^{σ (t)} (x, y_{j})), 1 \leq t \leq l . \end{matrix}$ Hence, there are $h_{\underline{ℝ} (𝔸)} (x) ⪯ h_{\underline{ℝ} (𝔹)} (x)$ and $g_{\underline{ℝ} (𝔸)} (x) ⪰ g_{\underline{ℝ} (𝔹)} (x)$ , which imply that $\underline{ℝ} (𝔸) ⊑ \underline{ℝ} (𝔹) .$

(5) It can be directly followed from the above result (3). □

4 Multi-granulation dual hesitant fuzzy rough sets in the DHF tolerance approximation space

In this section, we extend Qian et al.’s MGRS and Huang et al.’s IFMGRS theory into dual hesitant fuzzy environment. Meanwhile, we also generalize SGDHFRSs proposed in Section 3 to multi-granulation environment and propose MGDHFRSs in a DHF tolerance approximation space. Following the lines of Qian et al.’s MGRS model, we shall also present two types of MGDHFRSs induced by multiple DHF tolerance relations.

4.1 The optimistic multi-granulation dual hesitant fuzzy rough set in the DHF tolerance approximation space

Definition 4.1. Let U be a nonempty and finite universe of discourse and $ℝ_{i} (1 \leq i \leq m)$ be m DHF tolerance relations on U; the pair $(U, {ℝ_{i} | 1 \leq i \leq m})$ is called the DHF tolerance approximation space. For any $𝔸 \in DHF (U),$ the optimistic multi-granulation dual hesitant fuzzy lower and upper approximations of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ , denoted by ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ and ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ , are two DHF sets and are, respectively, defined as follows:

$\begin{matrix} {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) & = & {< x, h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x), \\ g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) > | x \in U}, \end{matrix}$ (3)

$\begin{matrix} {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) & = & {< x, h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x), \\ g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) > | x \in U}, \end{matrix}$ (4) where

$h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⊻_{i = 1}^{m} ⌆_{y_{j} \in U} {g_{ℝ_{i}} (x, y_{j}) ⊻ h_{𝔸} (y_{j})},$

$g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⌆_{i = 1}^{m} ⊻_{y_{j} \in U} {h_{ℝ_{i}} (x, y_{j}) ⌆ g_{𝔸} (y_{j})},$

$h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⌆_{i = 1}^{m} ⊻_{y_{j} \in U} {h_{ℝ_{i}} (x, y_{j}) ⌆ h_{𝔸} (y_{j})},$

$g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⊻_{i = 1}^{m} ⌆_{y_{j} \in U} {g_{ℝ_{i}} (x, y_{j}) ⊻ g_{𝔸} (y_{j})} .$

The pair $({\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸), {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸))$ is called an optimistic MGDHFRS of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ . If ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ , then $𝔸$ is referred to as optimistic-definable in $(U, {ℝ_{i} | 1 \leq i \leq m})$ ; otherwise, $𝔸$ is referred to as optimistic-undefinable in $(U, {ℝ_{i} | 1 \leq i \leq m})$ .

Remark 4.2. In Definition 4.1, for all $𝔸 \in DHF (U),$ if m = 1, we observe that

$h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ h_{𝔸} (y_{j})} = h_{\underline{ℝ} (𝔸)} (x),$

$g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ g_{𝔸} (y_{j})} = g_{\underline{ℝ} (𝔸)} (x),$

$h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ h_{𝔸} (y_{j})} = h_{\bar{ℝ} (𝔸)} (x),$

$g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ g_{𝔸} (y_{j})} = g_{\bar{ℝ} (𝔸)} (x) .$

In that case, the optimistic MGDHFRSs become the SGDHFRSs in Definition 3.3, which indicates that SGDHFRSs are a special type of optimistic MGDHFRSs.

Remark 4.3. In Definition 4.1, if there only exists one element in all the DHF elements $h_{𝔸} (y)$ , $g_{𝔸} (y)$ , $h_{ℝ_{i}} (x, y)$ and $g_{ℝ_{i}} (x, y)$ , that is, $𝔸$ degenerates to an intuitionistic fuzzy set A and $ℝ_{i} (1 \leq i \leq m)$ degenerate to m intuitionistic fuzzy relations R _i on U, then

$h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⋁_{i = 1}^{m} ⋀_{y_{j} \in U} [γ_{R_{i}} (x, y_{j}) \lor μ_{A} (y_{j})]$ $= μ_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x),$

$g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⋀_{i = 1}^{m} ⋁_{y_{j} \in U} [μ_{R_{i}} (x, y_{j}) \land γ_{A} (y_{j})]$ $= γ_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x),$

$h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⋀_{i = 1}^{m} ⋁_{y_{j} \in U} [μ_{R_{i}} (x, y_{j}) \land μ_{A} (y_{j})]$ $= μ_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x),$

$g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x) = ⋁_{i = 1}^{m} ⋀_{y_{j} \in U} [γ_{R_{i}} (x, y_{j}) \lor γ_{A} (y_{j})]$ $= γ_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) .$

In that case, the optimistic MGDHFRSs convert into the optimistic IFMGRSs proposed by Huang et al. [7], which implies that the optimistic IFMGRSs are a special case of optimistic MGDHFRSs.

Remark 4.4. From Remark 2.3, we know that fuzzy sets are a special case of DHF sets. Therefore, if $𝔸$ degenerates to a fuzzy set A and $ℝ_{i} (1 \leq i \leq m)$ are m fuzzy tolerance relations R _i on U, then the optimistic MGDHFRSs become the optimistic MGFRSs proposed by Xu et al. [42]. That is, the optimistic MGDHFRSs are an extension of the optimistic MGFRSs.

Example 4.5. Let U = {x ₁, x ₂, x ₃}. Assume that $ℝ$ and $𝕊$ are two DHF tolerance relations on U defined by two matrices as follows, respectively: $ℝ = \begin{matrix} x_{1} & x_{2} & x_{3} \\ \begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix} ( & \begin{matrix} {1}, {0} \\ {0.3, 0.4, 0.6}, {0.2, 0.3} \\ {0.3, 0.4}, {0.2, 0.5} \end{matrix} & \begin{matrix} {0.3, 0.4, 0.6}, {0.2, 0.3} \\ {1}, {0} \\ {0.2, 0.4, 0.8}, {0.1} \end{matrix} & \begin{matrix} {0.3, 0.4}, {0.2, 0.5} \\ {0.2, 0.4, 0.8}, {0.1} \\ {1}, {0} \end{matrix} & ) \begin{matrix} \end{matrix} \end{matrix}$ $𝕊 = \begin{matrix} x_{1} & x_{2} & x_{3} \\ \begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix} ( & \begin{matrix} {1}, {0} \\ {0.3, 0.5, 0.9}, {0, 0.1} \\ {0.2, 0.6, 0.7}, {0.2, 0.3} \end{matrix} & \begin{matrix} {0.3, 0.5, 0.9}, {0, 0.1} \\ {1}, {0} \\ {0.3, 0.5, 0.6}, {0.2, 0.4} \end{matrix} & \begin{matrix} {0.2, 0.6, 0.7}, {0.2, 0.3} \\ {0.3, 0.5, 0.6}, {0.2, 0.4} \\ {1}, {0} \end{matrix} & ) \begin{matrix} \end{matrix} \end{matrix}$

Let $\begin{matrix} 𝔸 & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4, 0.8}, {0.1} >} . \end{matrix}$

Then by Definition 3.3, the single-granulation hesitant fuzzy lower and upper approximations of $𝔸$ on fuzzy tolerance relations $ℝ$ and $𝕊$ can be calculated as follows, respectively: $\begin{matrix} \underline{ℝ} (𝔸) & = & {< x_{1}, {0.2, 0.4}, {0.3, 0.4, 0.6} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4}, {0.2, 0.4, 0.6} >}, \end{matrix}$ $\begin{matrix} \bar{ℝ} (𝔸) & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.3, 0.4, 0.8}, {0.1} >, \\ < x_{3}, {0.3, 0.4, 0.8}, {0.1} >}, \end{matrix}$ $\begin{matrix} \underline{𝕊} (𝔸) & = & {< x_{1}, {0.2, 0.4}, {0.3, 0.5, 0.6} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4}, {0.3, 0.5, 0.6} >}, \end{matrix}$ $\begin{matrix} \bar{𝕊} (𝔸) & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.3, 0.5, 0.7}, {0.2, 0.3} >, \\ < x_{3}, {0.2, 0.6, 0.8}, {0.1} >}, \end{matrix}$ $\begin{matrix} \underline{ℝ ⋓ 𝕊} (𝔸) & = & {< x_{1}, {0.2, 0.4}, {0.3, 0.5, 0.6} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4}, {0.3, 0.5, 0.6} >}, \end{matrix}$ $\begin{matrix} \bar{ℝ ⋓ 𝕊} (𝔸) & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.3, 0.5, 0.8}, {0.1} >, \\ < x_{3}, {0.3, 0.6, 0.8}, {0.1} >}, \end{matrix}$ $\begin{matrix} \underline{ℝ ⋒ 𝕊} (𝔸) & = & {< x_{1}, {0.2, 0.4}, {0.3, 0.4, 0.6} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4}, {0.2, 0.4, 0.6} >}, \end{matrix}$ $\begin{matrix} \bar{ℝ ⋒ 𝕊} (𝔸) & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.3, 0.4, 0.6}, {0.2, 0.3} >, \\ < x_{3}, {0.2, 0.4, 0.8}, {0.1} >} . \end{matrix}$

On the other hand, according to Definition 4.1, we compute the optimistic multi-granulation dual hesitant fuzzy lower and upper approximations of $𝔸$ on dual hesitant fuzzy tolerance relations $ℝ$ and $𝕊$ as follows: $\begin{matrix} {\underline{ℝ + 𝕊}}^{O} (𝔸) & = & {< x_{1}, {0.2, 0.4}, {0.3, 0.4, 0.6} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4}, {0.2, 0.4, 0.6} >}, \end{matrix}$ $\begin{matrix} {\bar{ℝ + 𝕊}}^{O} (𝔸) & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.3, 0.4, 0.7}, {0.2, 0.3} >, \\ < x_{3}, {0.2, 0.4, 0.8}, {0.1} >}, \end{matrix}$

Consequently, we can draw the conclusion that

${\underline{ℝ + 𝕊}}^{O} (𝔸) = \underline{ℝ} (𝔸) ⋓ \underline{𝕊} (𝔸),$ ${\bar{ℝ + 𝕊}}^{O} (𝔸) = \bar{ℝ} (𝔸) ⋒ \bar{𝕊} (𝔸),$

$\underline{ℝ ⋓ 𝕊} (𝔸) ⊑ {\underline{ℝ + 𝕊}}^{O} (𝔸) ⊑ 𝔸 ⊑ {\bar{ℝ + 𝕊}}^{O} (𝔸) ⊑ \bar{ℝ ⋓ 𝕊} (𝔸) .$

In what follows we will establish the relationship between the optimistic MGDHFRSs and SGDHFRSs.

Theorem 4.6. Let U be a nonempty and finite universe of discourse and $ℝ_{i} (1 \leq i \leq m)$ be m DHF tolerance relations on U. For any $𝔸 \in DHF (U),$ ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ and ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ are the optimistic multi-granulation dual hesitant fuzzy lower and upper approximations of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ , respectively. Then,

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) = ⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) = ⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔸) .$

Proof. It can be easily verified from Definitions 3.3 and 4.1. □

Theorem 4.7. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space. For the DHF set $𝔸 \in DHF (U),$ the optimistic multi-granulation DHF lower and upper approximations of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ satisfy the following properties:

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊑ 𝔸,$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊒ 𝔸;$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸^{c}) = ({\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸))^{c},$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸^{c}) = ({\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸))^{c};$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝕌) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝕌) = 𝕌,$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (\emptyset) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (\emptyset) = \emptyset .$

Proof. It can be directly obtained from Theorems 4.6, 3.4 and 2.7. □

Theorem 4.8. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space. For the DHF sets $𝔸$ and $𝔹,$ then the following properties hold.

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋒ 𝔹) ⊑ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋓ 𝔹) ⊒ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹);$

$𝔸 ⊑ 𝔹 \Rightarrow {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊑ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹),$

$𝔸 ⊑ 𝔹 \Rightarrow {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊑ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹);$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋓ 𝔹) ⊒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋓ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋒ 𝔹) ⊑ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋒ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹) .$

Proof. (1) For all x ∈ U, by virtue of Theorems 4.6, 3.5 and Definition 2.5, we have $\begin{matrix} h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋒ 𝔹)} (x) = h_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸 ⋒ 𝔹)} (x) \\ = h_{⋓_{i = 1}^{m} (\underline{ℝ_{i}} (𝔸) ⋒ \underline{ℝ_{i}} (𝔹))} (x) \\ = {⋁_{i = 1}^{m} (h_{\underline{ℝ_{i}} (A)}^{σ (s)} (x) \land h_{\underline{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ ⪯ {(⋁_{i = 1}^{m} h_{\underline{ℝ_{i}} (A)}^{σ (s)} (x)) \land (⋁_{i = 1}^{m} h_{\underline{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ = h_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) ⌆ h_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔹)} (x) \\ = h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹)} (x), \end{matrix}$ where $k = max_{1 \leq i \leq m} \max (l (h_{\underline{ℝ_{i}} (A)} (x)), l (h_{\underline{ℝ_{i}} (B)} (x))) .$

Meanwhile, we obtain $\begin{matrix} g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋒ 𝔹)} (x) = g_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸 ⋒ 𝔹)} (x) \\ = g_{⋓_{i = 1}^{m} (\underline{ℝ_{i}} (𝔸) ⋒ \underline{ℝ_{i}} (𝔹))} (x) \end{matrix}$

$\begin{matrix} = {⋀_{i = 1}^{m} (g_{\underline{ℝ_{i}} (A)}^{σ (t)} (x) \lor g_{\underline{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ ⪰ {(⋀_{i = 1}^{m} g_{\underline{ℝ_{i}} (A)}^{σ (t)} (x)) \lor (⋀_{i = 1}^{m} g_{\underline{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ = g_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) ⊻ g_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔹)} (x) \\ = g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹)} (x), \end{matrix}$ where $l = max_{1 \leq i \leq m} \max (l (g_{\underline{ℝ_{i}} (A)} (x)), l (g_{\underline{ℝ_{i}} (B)} (x))) .$

From the above discussions, we conclude that ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋒ 𝔹) ⊑ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹) .$

(2) For all x ∈ U, according to Theorems 4.6, 3.5 and Definition 2.5, we have $\begin{matrix} h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋓ 𝔹)} (x) = h_{⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔸 ⋓ 𝔹)} (x) \\ = h_{⋒_{i = 1}^{m} (\bar{ℝ_{i}} (𝔸) ⋓ \bar{ℝ_{i}} (𝔹))} (x) \\ = {⋀_{i = 1}^{m} (h_{\bar{ℝ_{i}} (A)}^{σ (s)} (x) \lor h_{\bar{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ ⪰ {(⋀_{i = 1}^{m} h_{\bar{ℝ_{i}} (A)}^{σ (s)} (x)) \lor (⋀_{i = 1}^{m} h_{\bar{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ = h_{⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔸)} (x) ⊻ h_{⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔹)} (x) \\ = h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹)} (x), \end{matrix}$ where $k = max_{1 \leq i \leq m} \max (l (h_{\bar{ℝ_{i}} (A)} (x)), l (h_{\bar{ℝ_{i}} (B)} (x))) .$

On the other hand, $\begin{matrix} g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋓ 𝔹)} (x) = g_{⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔸 ⋓ 𝔹)} (x) \\ = g_{⋒_{i = 1}^{m} (\bar{ℝ_{i}} (𝔸) ⋓ \bar{ℝ_{i}} (𝔹))} (x) \\ = {⋁_{i = 1}^{m} (g_{\bar{ℝ_{i}} (A)}^{σ (t)} (x) \land g_{\bar{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ ⪯ {(⋁_{i = 1}^{m} g_{\bar{ℝ_{i}} (A)}^{σ (t)} (x)) \land (⋁_{i = 1}^{m} g_{\bar{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ = g_{⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔸)} (x) ⌆ g_{⋒_{i = 1}^{m} \bar{ℝ_{i}} (𝔹)} (x) \\ = g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹)} (x), \end{matrix}$ where $l = max_{1 \leq i \leq m} \max (l (g_{\bar{ℝ_{i}} (A)} (x)), l (g_{\bar{ℝ_{i}} (B)} (x))) .$

From the above discussions, we conclude that ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸 ⋓ 𝔹) ⊒ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹) .$

(3) Since $𝔸 ⊑ 𝔹$ , by Theorem 3.5, then $\underline{ℝ_{i}} (𝔸) ⊑ \underline{ℝ_{i}} (𝔹)$ , which implies that $h_{\underline{ℝ_{i}} (𝔸)}^{σ (s)} (x) \leq h_{\underline{ℝ_{i}} (𝔹)}^{σ (s)} (x)$ and $g_{\underline{ℝ_{i}} (𝔸)}^{σ (t)} (x) \geq g_{\underline{ℝ_{i}} (𝔹)}^{σ (t)} (x)$ for x ∈ U. Then $⋁_{1 \leq i \leq m} h_{\underline{ℝ_{i}} (𝔸)}^{σ (s)} (x) \leq ⋁_{1 \leq i \leq m} h_{\underline{ℝ_{i}} (𝔹)}^{σ (s)} (x)$ and $⋀_{1 \leq i \leq m} g_{\underline{ℝ_{i}} (𝔸)}^{σ (t)} (x) \geq ⋀_{1 \leq i \leq m} g_{\underline{ℝ_{i}} (𝔹)}^{σ (t)} (x)$ . Hence, for all x ∈ U, there are $h_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) ⪯ h_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔹)} (x)$ and $g_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) ⪰ g_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔹)} (x)$ . According to Theorem 4.6, 3.5, we conclude that ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ $⊑ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹) .$

(4) It is similar to the proof of the result (3).

(5) and (6) can be obtained from the conclusions (3) and (4), respectively. □

Theorem 4.8 states the basic properties about the optimistic multi-granulation DHF lower and upper approximations of two different concepts. Properties (1) and (2) say that the optimistic multi-granulation DHF lower approximation is included into the intersection of the two optimistic multi-granulation DHF lower approximations, while the optimistic multi-granulation DHF upper approximation includes the union of the two optimistic multi-granulation DHF upper approximations. Properties (3) and (4) show the monotone of the optimistic multi-granulation DHF lower and upper approximations with respect to the variety of the target concept. Meanwhile, it’s worthy noting that properties (1) and (2) in Theorem 4.8 are different from the ones in Theorem 3.5 which satisfy the multiplicativity and additivity.

4.2 The pessimistic multi-granulation dual hesitant fuzzy rough set in the DHF tolerance approximation space

Definition 4.9. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space. For any $𝔸 \in DHF (U),$ the pessimistic multi-granulation dual hesitant fuzzy lower and upper approximations of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ , denoted by ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)$ and ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)$ , are two DHF sets and are, respectively, defined as follows:

$\begin{matrix} {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) & = & {< x, h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x), \\ g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) > | x \in U}, \end{matrix}$ (5)

$\begin{matrix} {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) & = & {< x, h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x), \\ g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) > | x \in U}, \end{matrix}$ (6) where

$h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⌆_{i = 1}^{m} ⌆_{y_{j} \in U} {g_{ℝ_{i}} (x, y_{j}) ⊻ h_{𝔸} (y_{j})},$

$g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⊻_{i = 1}^{m} ⊻_{y_{j} \in U} {h_{ℝ_{i}} (x, y_{j}) ⌆ g_{𝔸} (y_{j})},$

$h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⊻_{i = 1}^{m} ⊻_{y_{j} \in U} {h_{ℝ_{i}} (x, y_{j}) ⌆ h_{𝔸} (y_{j})},$

$g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⌆_{i = 1}^{m} ⌆_{y_{j} \in U} {g_{ℝ_{i}} (x, y_{j}) ⊻ g_{𝔸} (y_{j})} .$

The pair $({\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸), {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸))$ is called a pessimistic MGDHFRS of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ . If ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)$ , then $𝔸$ is referred to as pessimistic-definable in $(U, {ℝ_{i} | 1 \leq i \leq m})$ ; otherwise, $𝔸$ is referred to as pessimistic-undefinable in $(U, {ℝ_{i} | 1 \leq i \leq m})$ .

Remark 4.10. In Definition 4.9, for all $𝔸 \in DHF (U),$ if m = 1, we observe that

$h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ h_{𝔸} (y_{j})} = h_{\underline{ℝ} (𝔸)} (x),$

$g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ g_{𝔸} (y_{j})} = g_{\underline{ℝ} (𝔸)} (x),$

$h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{ℝ} (x, y_{j}) ⌆ h_{𝔸} (y_{j})} = h_{\bar{ℝ} (𝔸)} (x),$

$g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{ℝ} (x, y_{j}) ⊻ g_{𝔸} (y_{j})} = g_{\bar{ℝ} (𝔸)} (x) .$

In that case, the pessimistic MGDHFRSs become the SGDHFRSs in Definition 3.3, which indicates that SGDHFRSs are a special type of pessimistic MGDHFRSs.

Remark 4.11. In Definition 4.9, if there only exists one element in all the DHF elements $h_{𝔸} (y)$ , $g_{𝔸} (y)$ , $h_{ℝ_{i}} (x, y)$ and $g_{ℝ_{i}} (x, y)$ , that is, $𝔸$ degenerates to an intuitionistic fuzzy set A and $ℝ_{i} (1 \leq i \leq m)$ degenerate to m intuitionistic fuzzy relations R _i on U, then

$h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⋀_{i = 1}^{m} ⋀_{y_{j} \in U} [γ_{R_{i}} (x, y_{j}) \lor μ_{A} (y_{j})] = μ_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x),$

$g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⋁_{i = 1}^{m} ⋁_{y_{j} \in U} [μ_{R_{i}} (x, y_{j}) \land γ_{A} (y_{j})] = γ_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x),$

$h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⋁_{i = 1}^{m} ⋁_{y_{j} \in U} [μ_{R_{i}} (x, y_{j}) \land μ_{A} (y_{j})] = μ_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x),$

$g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x) = ⋀_{i = 1}^{m} ⋀_{y_{j} \in U} [γ_{R_{i}} (x, y_{j}) \lor γ_{A} (y_{j})] = γ_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) .$

In that case, the pessimistic MGDHFRSs convert into the pessimistic IFMGRSs proposed by Huang et al. [7], which implies that the pessimistic IFMGRSs are a special case of pessimistic MGDHFRSs.

Remark 4.12. From Remark 2.3, we know that fuzzy sets are a special case of DHF sets. Therefore, if $𝔸$ degenerates to a fuzzy set A and $ℝ_{i} (1 \leq i \leq m)$ are m fuzzy tolerance relations R _i on U, then the pessimistic MGDHFRSs become the pessimistic MGFRSs proposed by Xu et al. [42]. That is, the pessimistic MGDHFRSs are an extension of the pessimistic MGFRSs.

Example 4.13. (Continued from Example 4.5) According to Definition 4.9, we can calculate the pessimistic multi-granulation DHF lower and upper approximations of $𝔸$ on DHF tolerance relations $ℝ$ and $𝕊$ as follows: $\begin{matrix} {\underline{ℝ + 𝕊}}^{P} (𝔸) & = & {< x_{1}, {0.2, 0.4}, {0.3, 0.5, 0.6} >, \\ < x_{2}, {0.2, 0.4}, {0.3, 0.6} >, \\ < x_{3}, {0.2, 0.4}, {0.3, 0.5, 0.6} >}, \end{matrix}$

$\begin{matrix} {\bar{ℝ + 𝕊}}^{P} (𝔸) & = & {< x_{1}, {0.5, 0.7}, {0.2, 0.3} >, \\ < x_{2}, {0.3, 0.5, 0.8}, {0.1} >, \\ < x_{3}, {0.3, 0.6, 0.8}, {0.1} >}, \end{matrix}$

Hence, we see that

${\underline{ℝ + 𝕊}}^{P} (𝔸) = \underline{ℝ} (𝔸) ⋒ \underline{𝕊} (𝔸),$

${\bar{ℝ + 𝕊}}^{P} (𝔸) = \bar{ℝ} (𝔸) ⋓ \bar{𝕊} (𝔸),$

$\underline{ℝ ⋓ 𝕊} (𝔸) = {\underline{ℝ + 𝕊}}^{P} (𝔸) ⊑ \underline{ℝ ⋒ 𝕊} (𝔸) ⊑ 𝔸 ⊑ \bar{ℝ ⋒ 𝕊} (𝔸) ⊑ {\bar{ℝ + 𝕊}}^{P} (𝔸) = \bar{ℝ ⋓ 𝕊} (𝔸) .$

In what follows we establish the relationship between the pessimistic MGDHFRSs and SGDHFRSs in Section 3, and investigate the properties of the pessimistic MGDHFRSs.

Theorem 4.14. Let U be a nonempty and finite universe of discourse and $ℝ_{i} (1 \leq i \leq m)$ be m DHF tolerance relations on U. For any $𝔸 \in DHF (U),$ ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)$ and ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)$ are the pessimistic multi-granulation DHF lower and upper approximations of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ , respectively. Then,

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) = ⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) = ⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔸) .$

Proof. It is straightforward from Definitions 3.3and 4.9.□

Theorem 4.15. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space. For the DHF set $𝔸 \in DHF (U),$ the pessimistic multi-granulation DHF lower and upper approximations of $𝔸$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ satisfy the following properties:

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊑ 𝔸,$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊒ 𝔸;$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸^{c}) = ({\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸))^{c},$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸^{c}) = ({\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸))^{c};$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝕌) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝕌) = 𝕌,$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (\emptyset) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (\emptyset) = \emptyset .$

Proof. It can be easily verified from Theorems 4.14, 3.4 and 2.6.□

Theorem 4.16. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space. For the DHF sets $𝔸$ and $𝔹 \in DHF (U),$ then the following properties hold.

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋒ 𝔹) = {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋓ 𝔹) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹);$

$𝔸 ⊑ 𝔹 \Rightarrow {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊑ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹),$

$𝔸 ⊑ 𝔹 \Rightarrow {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊑ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹);$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋓ 𝔹) ⊒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋓ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋒ 𝔹) ⊑ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋒ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔹) .$

Proof. We only prove the items (1) and (2), the others are similar to the proof of the ones in Theorem 4.8.

(1) For all x ∈ U, by virtue of Theorems 4.14, 3.5 and Definition 2.5, we have $\begin{matrix} h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋒ 𝔹)} (x) = h_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸 ⋒ 𝔹)} (x) \\ = h_{⋒_{i = 1}^{m} (\underline{ℝ_{i}} (𝔸) ⋒ \underline{ℝ_{i}} (𝔹))} (x) \\ = {⋀_{i = 1}^{m} (h_{\underline{ℝ_{i}} (A)}^{σ (s)} (x) \land h_{\underline{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ = {(⋀_{i = 1}^{m} h_{\underline{ℝ_{i}} (A)}^{σ (s)} (x)) \land (⋀_{i = 1}^{m} h_{\underline{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ = h_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) ⌆ h_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔹)} (x) \\ = h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹)} (x), \end{matrix}$ and $\begin{matrix} g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋒ 𝔹)} (x) = g_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸 ⋒ 𝔹)} (x) \\ = g_{⋒_{i = 1}^{m} (\underline{ℝ_{i}} (𝔸) ⋒ \underline{ℝ_{i}} (𝔹))} (x) \\ = {⋁_{i = 1}^{m} (g_{\underline{ℝ_{i}} (A)}^{σ (t)} (x) \lor g_{\underline{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ = {(⋁_{i = 1}^{m} g_{\underline{ℝ_{i}} (A)}^{σ (t)} (x)) \lor (⋁_{i = 1}^{m} g_{\underline{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ = g_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) ⊻ h_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔹)} (x) \\ = g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹)} (x) . \end{matrix}$

From the above discussions, we conclude that

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋒ 𝔹) = {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋒ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹) .$

(2) For all x ∈ U, according to Theorems 4.2, 3 and Definition 2.2, we obtain $\begin{matrix} h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋓ 𝔹)} (x) = h_{⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔸 ⋓ 𝔹)} (x) \\ = h_{⋓_{i = 1}^{m} (\bar{ℝ_{i}} (𝔸) ⋓ \bar{ℝ_{i}} (𝔹))} (x) \\ = {⋁_{i = 1}^{m} (h_{\bar{ℝ_{i}} (A)}^{σ (s)} (x) \lor h_{\bar{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ = {(⋁_{i = 1}^{m} h_{\bar{ℝ_{i}} (A)}^{σ (s)} (x)) \lor (⋁_{i = 1}^{m} h_{\bar{ℝ_{i}} (B)}^{σ (s)} (x)) | 1 \leq s \leq k} \\ = h_{⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔸)} (x) ⊻ h_{⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔹)} (x) \\ = h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹)} (x), \end{matrix}$ and $\begin{matrix} g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋓ 𝔹)} (x) = g_{⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔸 ⋓ 𝔹)} (x) \\ = g_{⋓_{i = 1}^{m} (\bar{ℝ_{i}} (𝔸) ⋓ \bar{ℝ_{i}} (𝔹))} (x) \\ = {⋀_{i = 1}^{m} (g_{\bar{ℝ_{i}} (A)}^{σ (t)} (x) \land g_{\bar{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ = {(⋀_{i = 1}^{m} g_{\bar{ℝ_{i}} (A)}^{σ (t)} (x)) \land (⋀_{i = 1}^{m} g_{\bar{ℝ_{i}} (B)}^{σ (t)} (x)) | 1 \leq t \leq l} \\ = g_{⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔸)} (x) ⌆ g_{⋓_{i = 1}^{m} \bar{ℝ_{i}} (𝔹)} (x) \\ = g_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹)} (x), \end{matrix}$

From the above discussions, we deduce that ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸 ⋓ 𝔹) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⋓ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔹) .$ □

Theorem 4.16 states the basic properties about the pessimistic multi-granulation DHF lower and upper approximations of two different concepts. Properties (1) and (2) explain that the pessimistic multi-granulation DHF lower approximation is equal to the intersection of the two pessimistic multi-granulation DHF lower approximations, and the pessimistic multi-granulation DHF upper approximation is equivalent to the union of the two pessimistic multi-granulation DHF upper approximations. Properties (3) and (4) show the monotone of pessimistic multi-granulation DHF lower and upper approximations with respect to the variety of the target concept. Meanwhile, it should be noted that properties (1) and (2) in Theorem 4.16 are different from the ones in Theorem 4.8 which don’t satisfy the multiplicativity and additivity.

5 The relationship among SGDHFRS, the optimistic MGDHFRS and pessimistic MGDHFRS

In the above section, we mainly discuss the properties of the optimistic and pessimistic MGDHFRSs. In this section, the relationships among SGDHFRS, the optimistic MGDHFRS and pessimistic MGDHFRS will be further established.

Theorem 5.1. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space and the DHF set $𝔸 \in DHF (U) .$ Then the following properties hold.

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊑ \underline{⋒_{i = 1}^{m} ℝ_{i}} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊒ \bar{⋒_{i = 1}^{m} ℝ_{i}} (𝔸);$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) = \underline{⋓_{i = 1}^{m} ℝ_{i}} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) = \bar{⋓_{i = 1}^{m} ℝ_{i}} (𝔸) .$

Proof. (1) For all x ∈ U, according to Equation (1), we have $\begin{matrix} h_{\underline{⋒_{i = 1}^{m} ℝ_{i}} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{(⋒_{i = 1}^{m} ℝ_{i})} (x, y_{j}) ⊻ h_{𝔸} (y_{j})} \\ = {⋀_{y_{j} \in U} (⋁_{i = 1}^{m} g_{ℝ_{i}}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j}) | 1 \leq s \leq k} \\ ⪰ {⋁_{i = 1}^{m} ⋀_{y_{j} \in U} (g_{ℝ_{i}}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j})) | 1 \leq s \leq k} \\ = h_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) = h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x), \end{matrix}$ where $k = max_{1 \leq i \leq m} max_{1 \leq j \leq | U |} \max {l (g_{ℝ_{i}} (x, y_{j})), l (h_{𝔸} (y_{j}))} .$

On the other hand, $\begin{matrix} g_{\underline{⋒_{i = 1}^{m} ℝ_{i}} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{(⋒_{i = 1}^{m} ℝ_{i})} (x, y_{j}) ⌆ g_{𝔸} (y_{j})} \\ = {⋁_{y_{j} \in U} (⋀_{i = 1}^{m} h_{ℝ_{i}}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j}) | 1 \leq t \leq l} \\ ⪯ {⋀_{i = 1}^{m} ⋁_{y_{j} \in U} (h_{ℝ_{i}}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j})) | 1 \leq t \leq l} \\ = g_{⋓_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) = g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)} (x), \end{matrix}$ where $l = max_{1 \leq i \leq m} max_{1 \leq j \leq | U |} \max {l (h_{ℝ_{i}} (x, y_{j})), l (g_{𝔸} (y_{j}))} .$

From the above discussions, it follows that ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸)$ $⊑ \underline{⋒_{i = 1}^{m} ℝ_{i}} (𝔸) .$

(2) It is similar to the proof of the conclusion (1).

(3) For all x ∈ U, from Equation (1), we obtain $\begin{matrix} h_{\underline{⋓_{i = 1}^{m} ℝ_{i}} (𝔸)} (x) = ⌆_{y_{j} \in U} {g_{(⋓_{i = 1}^{m} ℝ_{i})} (x, y_{j}) ⊻ h_{𝔸} (y_{j})} \\ = {⋀_{y_{j} \in U} (⋀_{i = 1}^{m} g_{ℝ_{i}}^{σ (s)} (x, y_{j})) \lor h_{𝔸}^{σ (s)} (y_{j}) | 1 \leq s \leq k} \\ = {⋀_{i = 1}^{m} ⋀_{y_{j} \in U} (g_{ℝ_{i}}^{σ (s)} (x, y_{j}) \lor h_{𝔸}^{σ (s)} (y_{j})) | 1 \leq s \leq k} \\ = h_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) = h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x), \end{matrix}$ where $k = max_{1 \leq i \leq m} max_{1 \leq j \leq | U |} \max {l (g_{ℝ_{i}} (x, y_{j})), l (h_{𝔸} (y_{j}))} .$

On the other hand, $\begin{matrix} g_{\underline{⋓_{i = 1}^{m} ℝ_{i}} (𝔸)} (x) = ⊻_{y_{j} \in U} {h_{(⋓_{i = 1}^{m} ℝ_{i})} (x, y_{j}) ⌆ g_{𝔸} (y_{j})} \\ = {⋁_{y_{j} \in U} (⋁_{i = 1}^{m} h_{ℝ_{i}}^{σ (t)} (x, y_{j})) \land g_{𝔸}^{σ (t)} (y_{j}) | 1 \leq t \leq l} \\ = {⋁_{i = 1}^{m} ⋁_{y_{j} \in U} (h_{ℝ_{i}}^{σ (t)} (x, y_{j}) \land g_{𝔸}^{σ (t)} (y_{j})) | 1 \leq t \leq l} \\ = g_{⋒_{i = 1}^{m} \underline{ℝ_{i}} (𝔸)} (x) = g_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸)} (x), \end{matrix}$ where $l = max_{1 \leq i \leq m} max_{1 \leq j \leq | U |} \max {l (h_{ℝ_{i}} (x, y_{j})), l (g_{𝔸} (y_{j}))} .$

Therefore, we conclude that ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) = \underline{⋓_{i = 1}^{m} ℝ_{i}} (𝔸) .$

(4) It is similar to the proof of the conclusion (3). □

Theorem 5.1 states the relationship between the optimistic MGDHFRS and the SGDHFRS, and also reflects the connection between the pessimistic MGDHFRS and the SGDHFRS.

In what follows we will establish the relationship among SGDHFRS, optimistic MGDHFRS and pessimistic MGDHFRS.

Theorem 5.2. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space and the DHF set $𝔸 \in DHF (U) .$ Then the pessimistic MGDHFRS, the SGDHFRS and the optimistic MGDHFRS have relations below.

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊑ \underline{ℝ_{i}} (𝔸) ⊑ {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊒ \bar{ℝ_{i}} (𝔸) ⊒ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) .$

Proof. It can be directly obtained from Theorems 4.6 and 4.14. □

Theorem 5.3. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be the DHF tolerance approximation space and the DHF set $𝔸 \in DHF (U) .$ Then

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊒ \underline{⋓_{i = 1}^{m} ℝ_{i}} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (𝔸) ⊑ \bar{⋓_{i = 1}^{m} ℝ_{i}} (𝔸);$

${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊑ \underline{⋒_{i = 1}^{m} ℝ_{i}} (𝔸),$

${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (𝔸) ⊒ \bar{⋒_{i = 1}^{m} ℝ_{i}} (𝔸) .$

Proof. It is easy to be proved from Theorems 5.1 and 5.2. □

6 Conclusion

DHF sets and rough sets are two new mathematical approaches to cope with imprecision and uncertainty in data analysis. Although lots of fruitful results about DHF sets and rough sets have been achieved, little attention has been paid to the research of combining DHF sets and rough sets. The novelty of the current paper lies in two aspects. On one hand, by combining DHF set theory and rough sets we first introduce a single-granulation dual hesitant fuzzy rough set model. On the other hand, the single-granulation dual hesitant fuzzy rough set is extended to multi-granulation environment, and a new multi-granulation rough set model, called a MGDHFRS, is proposed. By investigating the properties of MGDHFRSs, we conclude that many existing MGRS models are special types of MGDHFRS models, such as MGRSs [28], MGFRSs in a fuzzy tolerance approximation space [42] and IFMGRSs [7]. Since the presented MGDHFRS includes both ingredients of DHF sets and rough sets, it is more effective and flexible than both DHF sets and rough sets to handle imprecise and imperfect information. Therefore, the research about the applications of the presented MGDHFRS is very important and necessary to us.

In the future, the application of the presented MGDHFRS is the main research direction considered by our group. We also believe that the presented MGDHFRS is of great practical value in many applications.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 11461082) and the Research Project Funds for Higher Education Institutions of Gansu Province (No. 2015B-006).

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