An approximation reduction approach in multi-granulation hesitant fuzzy decision information system

Abstract

Through a combination of hesitant fuzzy sets with rough sets, this study develops a single-granulation hesitant fuzzy rough set model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough set model, called the optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets. In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the single-granulation hesitant fuzzy rough sets, optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets are also investigated. Finally, we develop an approximation reduction approach of multi-granulation hesitant fuzzy rough sets to eliminate redundant hesitant fuzzy granulations with a detailed example.

Keywords

Single-granulation hesitant fuzzy rough set Multi-granulation hesitant fuzzy rough set Hesitant fuzzy granulations Approximation reduction

1 Introduction

As a valid means of granular computing, rough set theory introduced by Pawlak [23, 24] is a mathematical approach to handle uncertainty in data analysis in which the equivalence relation is a key tool and can represent information systems or decision tables. However, the equivalence relation seems to be a very stringent condition that may limit the application of rough sets in practical problems. Therefore many researchers have generalized the notion of classical rough set model by using non-equivalence relations rather than the equivalence relation. It may be a covering, dominance, neighborhood, fuzzy, intuitionistic fuzzy, interval-valued fuzzy, hesitant fuzzy or other indiscernibility one within the generalized rough sets [1 , 61].

In existing rough set models mentioned above, such as fuzzy rough set [1 , 33], intuitionistic fuzzy rough set [8 , 57], hesitant fuzzy rough set [4 , 47], and so on, set approximations are depicted only by a single granulation on the universe from the perspective of granular computing. However, in many cases a target concept should be described through multiple binary relations on the universe according to the different requirements of a user or targets of problem solving [25]. Hence, in order to more widely apply rough set theory in practical applications, Qian et al. [25] extended classical Pawlak’s rough set theory to multi-granulation environment, and introduced multi-granulation rough set theory. In Qian’s model, two basic models are introduced: one is the optimistic multi-granulation rough set; the other is the pessimistic multi-granulation rough set. Up to now, rapid developments have been achieved in multi-granulation rough set field [6 , 54].

In order to deal with the indecisiveness in decision-making issues, Torra and Narukawa [31] and Torra [32] initiated hesitant fuzzy set theory in which several different values are used to characterize the degree of membership of an element belonging to a set rather than a value. As one of the extensions of fuzzy set [59], the biggest advantage of hesitant fuzzy set is that it can express the hesitant information more comprehensively than other extensions of fuzzy set. Ever since its appearance, hesitant fuzzy set theory has been applied in dealing with lots of decision making problems successfully [2 , 60].

As two different tools to handle uncertainty, although the relationships between rough set theory and hesitant fuzzy set theory were not still explicit, Yang et al. [44] took a giant stride forward in understanding hesitant fuzzy sets and rough sets, and proposed a hesitant fuzzy rough set by combing the two theories. However, the disadvantage of this model is that the order on the hesitant fuzzy power set for representing inclusion relation of two hesitant fuzzy sets is not necessarily antisymmetric. These situations do not occur in classical set theory. To overcome the difficulty, Zhang et al. [47] introduced the order for characterizing inclusion relation of hesitant fuzzy sets which is antisymmetric. Based on this order relation, we shall attempt to build a novel hesitant fuzzy rough set called the single-granulation hesitant fuzzy rough set in the point view of granular computing.

On the one hand, it is well known that the existing multi-granulation rough sets, including multi-granulation fuzzy rough sets, can handle some group decision-making problems to quantify the ideas of decision makers by using a crisp number. However, when facing the problem that the basic features of decision-making activities are described by several numbers within [0,1] to depict the hesitancy situation of decision makers, we cannot offer a comprehensive, accurate and flexible solution by using the existing multi-granulation rough sets. Therefore, on the basis of the above-mentioned analysis, it is natural for us to explore multi-granulation rough sets under hesitant fuzzy environment. In this context, multi-granulation rough set is generalized into hesitant fuzzy environment and a multi-granulation hesitant fuzzy rough set is initiated, in which two types of the multi-granulation hesitant fuzzy rough sets model, called the optimistic multi-granulation hesitant fuzzy rough set and the pessimistic multi-granulation hesitant fuzzy rough set, are constructed. On the other hand, although there exist many rough set models under hesitant fuzzy environment, approximate reduction methods in hesitant fuzzy information systems are scarce and blank. As a result, in this paper we mainly focus on the study of reduction approaches in multi-granulation hesitant fuzzy decision information system to eliminate redundant hesitant fuzzy granulations.

The rest of the paper is organized as follows. The next section reviews some basic concepts related to hesitant fuzzy sets. Section 3 develops single-granulation hesitant fuzzy rough set theory and two types of multi-granulation hesitant fuzzy rough sets model where set approximations are described by multiple hesitant fuzzy tolerance relations. Then we further establish the relationships among single-granulation hesitant fuzzy rough set theory, optimistic multi-granulation hesitant fuzzy rough set and pessimistic multi-granulation hesitant fuzzy rough set. In Section 4, we propose a reduction approach of the multi-granulation hesitant fuzzy rough set to eliminate redundant hesitant fuzzy granulations by a numerical example. Finally, we conclude the paper in Section 5.

2 Hesitant fuzzy sets

In [31, 32], Torra and Narukawa introduced the concept of hesitant fuzzy sets as follows.

Definition 2.1. ([31, 32]) 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 [35] denoted the hesitant fuzzy set as 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 [35] called $h_{A} (x)$ a hesitant fuzzy element, and denoted the set of all hesitant fuzzy sets on U by HF (U).

Assume that $l (h_{A} (x))$ represents the number of values in $h_{A} (x)$ . Since the number of values in hesitant fuzzy elements may be different, Xu and Xia [40] gave the following assumptions:

(A1) All the elements in $h_{A} (x)$ are arranged in increasing order, and then $h_{A}^{σ (k)} (x)$ is said to be the kth largest value in $h_{A} (x)$ .

(A2) For $h_{A} (x)$ and $h_{B} (x)$ , if $l (h_{A} (x)) \neq l (h_{B} (x))$ , we should extend the shorter one until both of them have the same length l with $l = \max {l (h_{A} (x)),$ $l (h_{B} (x))}$ . In terms of the pessimistic principle, the smallest element will be added while in the opposite case, the optimistic principle may be adopted. In this paper, we adopt the optimistic principle. Specifically, if $l (h_{A} (x)) < l (h_{B} (x))$ , then $h_{A} (x)$ should be extended by adding its maximum value in it until it has the same length with $h_{B} (x)$ .

In [16], Liao et al. developed some new methods when operating the hesitant fuzzy elements, which are slightly different from the ones introduced by Torra [32] and Xia and Xu [35] (see [16] in detail). In this paper, without special statement, we shall adopt adjusted operational laws proposed by Liao et al. [16] and the assumptions given by Xu and Xia [40].

3 Multi-granulation hesitant fuzzy rough sets in the generalized hesitant fuzzy tolerance approximation space

In this section, we shall propose a novel hesitant fuzzy rough set which is called the single-granulation hesitant fuzzy rough sets. First, we introduce the hesitant fuzzy relation proposed by Yang et al. [44].

Definition 3.1. ([44]) Suppose that U is a nonempty and finite universe. A hesitant fuzzy subset $ℝ$ of the universe U is referred to as a hesitant fuzzy relation on U, namely, $ℝ$ is given by

$ℝ = {< (x, y), h_{ℝ} (x, y) > | (x, y) \in U \times U},$ where $h_{ℝ} : U \times U \to [0, 1]$ is a set of some different values in [0, 1].

Definition 3.2. ([44]) Suppose that $ℝ$ is a hesitant fuzzy relation on U.

(1) $ℝ$ is serial, if for any x ∈ U, there exists a y ∈ U such that $h_{ℝ} (x, y) = {1}$ ;

(2) $ℝ$ is reflexive, if $h_{ℝ} (x, x) = {1}$ for all x ∈ U;

(3) $ℝ$ is symmetric, if for all (x, y) ∈ U × U, $h_{ℝ} (x, y) = h_{ℝ} (y, x)$ .

If a hesitant fuzzy relation $ℝ$ on U is reflexive and symmetric, it is referred to as a hesitant fuzzy tolerance relation on U.

By using the hesitant fuzzy relation, Yang et al. [44] introduced the hesitant fuzzy rough sets. However, hesitant fuzzy subset based on the hesitant fuzzy rough sets proposed by them is not necessarily antisymmetric. For example, suppose that $F = {0.2, 0.3, 0.8}$ and $G = {0.2, 0.4, 0.6, 0.8} .$ According to the definition of hesitant fuzzy subset in [44], we have $F ⊑ G$ and $G ⊑ F$ . However, it is clear that $F \neq G$ . These situations do not occur in classical set theory. To remedy this defect, Zhang et al. [47] introduced the order for characterizing inclusion relation of hesitant fuzzy sets which is antisymmetric. Based on this order relation, a single-granulation hesitant fuzzy rough set in the hesitant fuzzy tolerance approximation space is introduced as per the adjusted operational laws proposed by Liao et al. [16] and the assumptions given by Xu and Xia [40].

Definition 3.3. Suppose that U is a nonempty and finite universe of discourse, and $ℝ$ is a hesitant fuzzy tolerance relation on U, then the pair $(U, ℝ)$ is called a hesitant fuzzy tolerance approximation space. For any $A \in HF (U),$ the lower and upper approximations of $A$ with respect to $(U, ℝ)$ , denoted by $\underline{ℝ} (A)$ and $\bar{ℝ} (A)$ , are two hesitant fuzzy sets and are, respectively, defined as follows:

$\begin{matrix} \underline{ℝ} (A) = {< x, h_{\underline{ℝ} (A)} (x) > | x \in U}, \\ \bar{ℝ} (A) = {< x, h_{\bar{ℝ} (A)} (x) > | x \in U}, \end{matrix}$ where

$h_{\underline{ℝ} (A)} (x) = {\bar{\land}}_{y_{j} \in U} {h_{ℝ^{c}} (x, y_{j}) ⊻ h_{A} (y_{j})},$

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

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

In [25], multi-granulation rough sets involve in two models: one is called the optimistic multi-granulation rough set; the other is called the pessimistic multi-granulation rough set. In this section, we generalize the multi-granulation rough set model into hesitant fuzzy environment, and propose multi-granulation hesitant fuzzy rough sets in the generalized hesitant fuzzy tolerance approximation space by integrating multi-granulation rough set with hesitant fuzzy set theory. Because this new hybrid model called multi-granulation hesitant fuzzy rough sets includes both ingredients of multi-granulation rough set and hesitant fuzzy set, it is more flexible to handle imprecise information than both multi-granulation rough set and hesitant fuzzy set in a decision making process. Along the lines of Qian’s multi-granulation rough sets model, we will also present two types of multi-granulation hesitant fuzzy rough sets induced by multiple hesitant fuzzy tolerance relations.

Definition 3.4. Suppose that U is a nonempty and finite universe of discourse, and $ℝ_{i} (1 \leq i \leq m)$ are m hesitant fuzzy tolerance relations on U; the pair $(U, {ℝ_{i} | 1 \leq i \leq m})$ is called a generalized hesitant fuzzy tolerance approximation space. For any $A \in HF (U),$ the optimistic multi-granulation 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 hesitant fuzzy sets and are, respectively, defined as follows: $\begin{matrix} {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A) = {< x, h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A)} (x) > | x \in U}, \\ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A) = {< x, h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A)} (x) > | x \in U}, \end{matrix}$ where

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

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

The pair $({\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A), {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A))$ is called an optimistic multi-granulation hesitant fuzzy rough set 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 $(U, {ℝ_{i} | 1 \leq i \leq m})$ .

The following Theorem establishes the relationship between optimistic multi-granulation hesitant fuzzy rough sets and single-granulation hesitant fuzzy rough sets.

Theorem 3.5. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be a generalized hesitant fuzzy tolerance approximation space. For any $A \in HF (U),$ ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A)$ and ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A)$ are the optimistic multi-granulation hesitant fuzzy lower and upper approximations of $A$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ , respectively. Then,

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

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

Proof. It can be directly derived from Definitions 3.3 and 3.4. □

Definition 3.6. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be a generalized hesitant fuzzy tolerance approximation space. For any $A \in HF (U),$ the pessimistic multi-granulation 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 hesitant fuzzy sets and are, respectively, defined as follows: $\begin{matrix} {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A) = {< x, h_{{\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A)} (x) > | x \in U}, \\ {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A) = {< x, h_{{\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A)} (x) > | x \in U}, \end{matrix}$ where

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

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

The pair $({\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A), {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A))$ is called a pessimistic multi-granulation hesitant fuzzy rough set 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 $(U, {ℝ_{i} | 1 \leq i \leq m})$ .

In what follows we establish the relationship between pessimistic multi-granulation hesitant fuzzy rough sets and single-granulation hesitant fuzzy rough sets.

Theorem 3.7. Let $(U, {ℝ_{i} | 1 \leq i \leq m})$ be a generalized hesitant fuzzy tolerance approximation space. For any $A \in HF (U),$ ${\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A)$ and ${\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A)$ are the pessimistic multi-granulation hesitant fuzzy lower and upper approximations of $A$ with respect to $(U, {ℝ_{i} | 1 \leq i \leq m})$ , respectively. Then,

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

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

Proof. It can be directly derived from Definitions 3.3 and 3.6. □

4 Approximation reduction approach in multi-granulation hesitant fuzzy decision information system

In this section, we establish a practical reduction approach in multi-granulation hesitant fuzzy decision information system based on the multi-granulation hesitant fuzzy rough set model. The objective of reduction is to obtain a smallest subset of hesitant fuzzy relations that preserves the consistence of multi-granulation hesitant fuzzy decision information system.

Definition 4.1. Suppose that apr= $(U, ℝ = {ℝ_{i} | 1 \leq i \leq m})$ is a generalized hesitant fuzzy tolerance approximation space, $A$ is the hesitant fuzzy set and ${\underline{ℝ}}^{O}, {\bar{ℝ}}^{O}, {\underline{ℝ}}^{P}, {\bar{ℝ}}^{P} \subseteq ℝ$ .

(1) If ${\underline{\sum_{ℝ_{i} \in {\underline{ℝ}}^{O}} ℝ_{i}}}^{O} (A) = {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A),$ then ${\underline{ℝ}}^{O}$ is referred to as a consistent optimistic lower approximation of apr. If ${\underline{ℝ}}^{O}$ is a consistent optimistic lower approximation, and no proper subset of ${\underline{ℝ}}^{O}$ is a consistent optimistic lower approximation, then ${\underline{ℝ}}^{O}$ is called an optimistic lower approximation reduct of apr.

(2) If ${\underline{\sum_{ℝ_{i} \in {\underline{ℝ}}^{P}} ℝ_{i}}}^{P} (A) = {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A),$ then ${\underline{ℝ}}^{P}$ is referred to as a consistent pessimistic lower approximation of apr. If ${\underline{ℝ}}^{P}$ is a consistent pessimistic lower approximation, and no proper subset of ${\underline{ℝ}}^{P}$ is a consistent pessimistic lower approximation, then ${\underline{ℝ}}^{P}$ is called a pessimistic lower approximation reduct of apr.

(3) If ${\bar{\sum_{ℝ_{i} \in {\bar{ℝ}}^{O}} ℝ_{i}}}^{O} (A) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (A),$ then ${\bar{ℝ}}^{O}$ is referred to as a consistent optimistic upper approximation of apr. If ${\bar{ℝ}}^{O}$ is a consistent optimistic upper approximation set, and no proper subset of ${\bar{ℝ}}^{O}$ is a consistent optimistic upper approximation, then ${\bar{ℝ}}^{O}$ is called an optimistic upper approximation reduct of apr.

(4) If ${\bar{\sum_{ℝ_{i} \in {\bar{ℝ}}^{P}} ℝ_{i}}}^{P} (A) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (A),$ then ${\bar{ℝ}}^{P}$ is referred to as a consistent pessimistic upper approximation of apr. If ${\bar{ℝ}}^{P}$ is a consistent pessimistic upper approximation, and no proper subset of ${\bar{ℝ}}^{P}$ is a consistent pessimistic upper approximation, then ${\bar{ℝ}}^{P}$ is called a pessimistic upper approximation reduct of apr.

From Definition 4.1, we know that a lower approximation reduct is the smallest subset of $ℝ = {ℝ_{i} | 1 \leq i \leq m}$ which preserves the lower approximations of all hesitant fuzzy sets in U. And so is for the upper approximation reduct.

Definition 4.2.A multi-granulation hesitant fuzzy decision information system is a quads MGHFDIS $= (U, {ℝ_{i} | 1 \leq i \leq m}, D, V),$ where U is a nonempty and finite universe; ${ℝ_{i} | 1 \leq i \leq m}$ is a set of m hesitant fuzzy relations on U; D is a nonempty and finite set of decision attributes; V = {g (x, d) |x ∈ U, d ∈ D} is a set of the relationships between U and D, and g (x, d) is a hesitant fuzzy element. We call g (x, d) a decision hesitant fuzzy value of x under decision attribute d.

Example 4.3. A multi-granulation hesitant fuzzy decision information system MGHFDIS $= (U, {ℝ_{i} | 1 \leq i \leq 5}, D, V),$ can be described as follows: U = {x₁, x₂, ⋯ , x₈}; $ℝ_{i} (1 \leq i \leq 5)$ are five hesitant fuzzy relations on U shown as Tables 1-5; D = {d₁, d₂}; V = {g (x_i, d_j) |x_i ∈ U, d_j ∈ D} , where g (x₁, d₁) = {0.3, 0.6} , g (x₂, d₁) = {0.3, 0.5} , g (x₃, d₁) = {0.2, 0.7} , g (x₄, d₁) = {0.4, 0.6} , g (x₅, d₁) = {0.6, 0.8} , g (x₆, d₁) = {0.5, 0.7} , g (x₇, d₁) = {0.6, 0.7} , g (x₈, d₁) = {0.4, 0.6} , and g (x₁, d₂) = {0.4, 0.7} , g (x₂, d₂) = {0.5, 0.6} , g (x₃, d₂) = {0.4, 0.7} , g (x₄, d₂) = {0.8, 0.9} , g (x₅, d₂) = {0.3, 0.6} , g (x₆, d₂) = {0.2, 0.4} , g (x₇, d₂) = {0.6, 0.7} , g (x₈, d₂) = {0.5, 0.8} .

Table 1
Hesitant fuzzy relation $ℝ_{1}$ in Example 4.3

U × U x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈

x ₁ {1,1} {0.3,0.5} {0.2,0.8} {0.5,0.9} {0.3,0.7} {0.2,0.4} {0.6,0.7} {0.2,0.3}

x ₂ {0.3,0.5} {1,1} {0.4,0.5} {0.2,0.7} {0.4,0.5} {0.3,0.6} {0.5,0.8} {0.7,0.9}

x ₃ {0.2,0.8} {0.4,0.5} {1,1} {0.4,0.6} {0.2,0.8} {0.7,0.9} {0.4,0.5} {0.6,0.6}

x ₄ {0.5,0.9} {0.2,0.7} {0.4,0.6} {1,1} {0.3,0.8} {0.2,0.4} {0.1,0.2} {0.3,0.4}

x ₅ {0.3,0.7} {0.4,0.5} {0.2,0.8} {0.3,0.8} {1,1} {0.5,0.8} {0.4,0.6} {0.2,0.2}

x ₆ {0.2,0.4} {0.3,0.6} {0.7,0.9} {0.2,0.4} {0.5,0.8} {1,1} {0.5,0.8} {0.4,0.6}

x ₇ {0.6,0.7} {0.5,0.8} {0.4,0.5} {0.1,0.2} {0.4,0.6} {0.5,0.8} {1,1} {0.6,0.8}

x ₈ {0.2,0.3} {0.7,0.9} {0.6,0.6} {0.3,0.4} {0.2,0.2} {0.4,0.6} {0.6,0.8} {1,1}

Table 2

Hesitant fuzzy relation $ℝ_{2}$ in Example 4.3

U × U	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
x ₁	{1,1}	{0.4,0.5}	{0.3,0.6}	{0.1,0.4}	{0.4,0.8}	{0.5,0.7}	{0.6,0.7}	{0.3,0.4}
x ₂	{0.4,0.5}	{1,1}	{0.5,0.7}	{0.4,0.6}	{0.2,0.4}	{0.2,0.6}	{0.2,0.4}	{0.7,0.8}
x ₃	{0.3,0.6}	{0.5,0.7}	{1,1}	{0.5,0.7}	{0.4,0.6}	{0.3,0.4}	{0.2,0.3}	{0.7,0.9}
x ₄	{0.1,0.4}	{0.4,0.6}	{0.5,0.7}	{1,1}	{0.7,0.8}	{0.6,0.9}	{0.3,0.5}	{0.6,0.7}
x ₅	{0.4,0.8}	{0.2,0.4}	{0.4,0.6}	{0.7,0.8}	{1,1}	{0.5,0.8}	{0.1,0.2}	{0.4,0.6}
x ₆	{0.5,0.7}	{0.2,0.6}	{0.3,0.4}	{0.6,0.9}	{0.5,0.8}	{1,1}	{0.6,0.7}	{0.2,0.3}
x ₇	{0.6,0.7}	{0.2,0.4}	{0.2,0.3}	{0.3,0.5}	{0.1,0.2}	{0.6,0.7}	{1,1}	{0.5,0.6}
x ₈	{0.3,0.4}	{0.7,0.8}	{0.7,0.9}	{0.6,0.7}	{0.4,0.6}	{0.2,0.3}	{0.5,0.6}	{1,1}

Table 3

Hesitant fuzzy relation $ℝ_{3}$ in Example 4.3

U × U	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
x ₁	{1,1}	{0.3,0.5}	{0.2,0.6}	{0.4,0.7}	{0.5,0.8}	{0.3,0.4}	{0.6,0.7}	{0.1,0.3}
x ₂	{0.3,0.5}	{1,1}	{0.4,0.5}	{0.5,0.7}	{0.2,0.5}	{0.5,0.7}	{0.5,0.8}	{0.3,0.4}
x ₃	{0.2,0.6}	{0.4,0.5}	{1,1}	{0.3,0.5}	{0.6,0.9}	{0.7,0.8}	{0.4,0.5}	{0.7,0.9}
x ₄	{0.4,0.7}	{0.5,0.7}	{0.3,0.5}	{1,1}	{0.3,0.4}	{0.2,0.3}	{0.4,0.7}	{0.8,0.8}
x ₅	{0.5,0.8}	{0.2,0.5}	{0.6,0.9}	{0.3,0.4}	{1,1}	{0.6,0.9}	{0.2,0.4}	{0.6,0.7}
x ₆	{0.3,0.4}	{0.5,0.7}	{0.7,0.8}	{0.2,0.3}	{0.6,0.9}	{1,1}	{0.7,0.8}	{0.3,0.5}
x ₇	{0.6,0.7}	{0.5,0.8}	{0.4,0.5}	{0.4,0.7}	{0.2,0.4}	{0.7,0.8}	{1,1}	{0.6,0.7}
x ₈	{0.1,0.3}	{0.3,0.4}	{0.7,0.9}	{0.8,0.8}	{0.6,0.7}	{0.3,0.5}	{0.6,0.7}	{1,1}

Table 4

Hesitant fuzzy relation $ℝ_{4}$ in Example 4.3

U × U	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
x ₁	{1,1}	{0.3,0.4}	{0.7,0.8}	{0.5,0.9}	{0.2,0.5}	{0.4,0.7}	{0.6,0.7}	{0.3,0.4}
x ₂	{0.3,0.4}	{1,1}	{0.2,0.4}	{0.6,0.8}	{0.4,0.6}	{0.6,0.7}	{0.3,0.5}	{0.6,0.8}
x ₃	{0.7,0.8}	{0.2,0.4}	{1,1}	{0.3,0.4}	{0.6,0.8}	{0.5,0.6}	{0.4,0.5}	{0.2,0.3}
x ₄	{0.5,0.9}	{0.6,0.8}	{0.3,0.4}	{1,1}	{0.7,0.9}	{0.2,0.5}	{0.5,0.6}	{0.1,0.2}
x ₅	{0.2,0.5}	{0.4,0.6}	{0.6,0.8}	{0.7,0.9}	{1,1}	{0.5,0.9}	{0.7,0.8}	{0.6,0.7}
x ₆	{0.4,0.7}	{0.6,0.7}	{0.5,0.6}	{0.2,0.5}	{0.5,0.9}	{1,1}	{0.2,0.3}	{0.3,0.5}
x ₇	{0.6,0.7}	{0.3,0.5}	{0.4,0.5}	{0.5,0.6}	{0.7,0.8}	{0.2,0.3}	{1,1}	{0.4,0.6}
x ₈	{0.3,0.4}	{0.6,0.8}	{0.2,0.3}	{0.1,0.2}	{0.6,0.7}	{0.3,0.5}	{0.4,0.6}	{1,1}

Table 5

Hesitant fuzzy relation $ℝ_{5}$ in Example 4.3

U × U	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
x ₁	{1,1}	{0.3,0.6}	{0.4,0.8}	{0.5,0.5}	{0.6,0.7}	{0.3,0.6}	{0.7,0.8}	{0.1,0.2}
x ₂	{0.3,0.6}	{1,1}	{0.3,0.6}	{0.2,0.3}	{0.5,0.8}	{0.4,0.6}	{0.2,0.3}	{0.5,0.6}
x ₃	{0.4,0.8}	{0.3,0.6}	{1,1}	{0.6,0.8}	{0.4,0.6}	{0.7,0.9}	{0.4,0.5}	{0.6,0.7}
x ₄	{0.5,0.5}	{0.2,0.3}	{0.6,0.8}	{1,1}	{0.8,0.9}	{0.5,0.7}	{0.1,0.2}	{0.6,0.7}
x ₅	{0.6,0.7}	{0.5,0.8}	{0.4,0.6}	{0.8,0.9}	{1,1}	{0.4,0.5}	{0.3,0.4}	{0.2,0.5}
x ₆	{0.3,0.6}	{0.4,0.6}	{0.7,0.9}	{0.5,0.7}	{0.4,0.5}	{1,1}	{0.4,0.6}	{0.2,0.4}
x ₇	{0.7,0.8}	{0.2,0.3}	{0.4,0.5}	{0.1,0.2}	{0.3,0.4}	{0.4,0.6}	{1,1}	{0.4,0.6}
x ₈	{0.1,0.2}	{0.5,0.6}	{0.6,0.7}	{0.6,0.7}	{0.2,0.5}	{0.2,0.4}	{0.4,0.6}	{1,1}

Based on Definition 4.1, approximation reducts in multi-granulation hesitant fuzzy decision information system based on the multi-granulation hesitant fuzzy rough set model are defined as follows:

Definition 4.4. Let MGHFDIS $= (U, {ℝ_{i} | 1 \leq i \leq m}, D, V)$ be a multi-granulation hesitant fuzzy decision information system, where U = {x₁, x₂, ⋯ , x_n}, D = {d₁, d₂, ⋯ , d_l} and D_j = {< x_i, g (x_i, d_j) > |x_i ∈ U, d_j ∈ D} ∈ HF (U).

(1) If ${\underline{\sum_{ℝ_{i} \in {\underline{ℝ}}^{O}} ℝ_{i}}}^{O} (D_{j}) = {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (D_{j}),$ for all j (1 ≤ j ≤ l), then ${\underline{ℝ}}^{O}$ is called a consistent optimistic lower approximation of MGHFDIS. If ${\underline{ℝ}}^{O}$ is a consistent optimistic lower approximation, and no proper subset of ${\underline{ℝ}}^{O}$ is a consistent optimistic lower approximation, then ${\underline{ℝ}}^{O}$ is referred to as an optimistic lower approximation reduct of MGHFDIS.

(2) If ${\underline{\sum_{ℝ_{i} \in {\underline{ℝ}}^{P}} ℝ_{i}}}^{P} (D_{j}) = {\underline{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (D_{j}),$ for all j (1 ≤ j ≤ l), then ${\underline{ℝ}}^{P}$ is called a consistent pessimistic lower approximation of MGHFDIS. If ${\underline{ℝ}}^{P}$ is a consistent pessimistic lower approximation, and no proper subset of ${\underline{ℝ}}^{P}$ is a consistent pessimistic lower approximation, then ${\underline{ℝ}}^{P}$ is referred to as a pessimistic lower approximation reduct of MGHFDIS.

(3) If ${\bar{\sum_{ℝ_{i} \in {\bar{ℝ}}^{O}} ℝ_{i}}}^{O} (D_{j}) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{O} (D_{j}),$ for all j (1 ≤ j ≤ l), then ${\bar{ℝ}}^{O}$ is called a consistent optimistic upper approximation of MGHFDIS. If ${\bar{ℝ}}^{O}$ is a consistent optimistic upper approximation, and no proper subset of ${\bar{ℝ}}^{O}$ is a consistent optimistic upper approximation, then ${\bar{ℝ}}^{O}$ is referred to as an optimistic upper approximation reduct of MGHFDIS.

(4) If ${\bar{\sum_{ℝ_{i} \in {\bar{ℝ}}^{P}} ℝ_{i}}}^{P} (D_{j}) = {\bar{\sum_{i = 1}^{m} ℝ_{i}}}^{P} (D_{j}),$ for all j (1 ≤ j ≤ l), then ${\bar{ℝ}}^{P}$ is called a consistent pessimistic upper approximation of MGHFDIS. If ${\bar{ℝ}}^{P}$ is a consistent pessimistic upper approximation, and no proper subset of ${\bar{ℝ}}^{P}$ is a consistent pessimistic upper approximation, then ${\bar{ℝ}}^{P}$ is referred to as a pessimistic upper approximation reduct of MGHFDIS.

In the text that follows, without loss of generality, suppose that all the hesitant fuzzy elements have the same length k on the basis of the assumptions given by Xu and Xia [40]. In what follows, in order to acquire the optimistic and pessimistic approximation reducts of multi-granulation hesitant fuzzy decision information system, we introduce the concepts of hesitant fuzzy vectors and hesitant fuzzy matrices proposed by Liao and Xu [17].

Definition 4.5. ([17]) Let n-dimensional vector $\vec{α} = (α_{1}, α_{2},$ ⋯, α_n) ^T, where $α_{i} = {α_{i}^{σ (s)} | 1 \leq s \leq k}$ are n hesitant fuzzy elements. Then we call $\vec{α}$ a n-dimensional hesitant fuzzy vector. If $M_{nm} = ({\vec{α}}_{1}, {\vec{α}}_{2}, \dots, {\vec{α}}_{m}),$ where ${\vec{α}}_{j} (1 \leq j \leq m)$ are m n-dimensional hesitant fuzzy vectors, then we call M_nm a n × m hesitant fuzzy matrix. Specially, a n-dimensional hesitant fuzzy vector can be viewed as a n × 1 hesitant fuzzy matrix.

Based on Definition 4.5, multi-granulation hesitant fuzzy decision information system can be described as multiple hesitant fuzzy matrices (hesitant fuzzy relation matrices) and vectors (called decision hesitant fuzzy vectors). For example, by using hesitant fuzzy relation matrices and decision hesitant fuzzy vectors, the multi-granulation hesitant fuzzy decision information system in Example 4.3 can be described by the following $M_{ℝ_{1}},$ $M_{ℝ_{2}}$ , $M_{ℝ_{3}},$ $M_{ℝ_{4}},$ and $M_{ℝ_{5}}$ (see the next page) and decision hesitant fuzzy vectors: $\begin{matrix} D_{1} = ({0.3, 0.6}, {0.3, 0.5}, {0.2, 0.7}, {0.4, 0.6}, \\ {0.6, 0.8}, {0.5, 0.7}, {0.6, 0.7}, {0.4, 0.6})^{T}, \\ D_{2} = ({0.4, 0.7}, {0.5, 0.6}, {0.4, 0.7}, {0.8, 0.9}, \\ {0.3, 0.6}, {0.2, 0.4}, {0.6, 0.7}, {0.5, 0.8})^{T} . \end{matrix}$

Now, the union, intersection and complement of two hesitant fuzzy vectors and matrices can be defined as follows:

Definition 4.6. Let ${\vec{α}}_{1} = (α_{11}, α_{12}, \dots, α_{1 n})^{T}$ and ${\vec{α}}_{2} = (α_{21}, α_{22}, \dots, α_{2 n})^{T}$ be two n-dimensional hesitant fuzzy vectors, where $α_{1 i} = {α_{1 i}^{σ (s)} | 1 \leq s \leq k}$ and $α_{2 i} = {α_{2 i}^{σ (s)} | 1 \leq s \leq k} (1 \leq i \leq n)$ are 2n hesitant fuzzy elements. Assume that $M_{1} = ({\vec{α}}_{11}, {\vec{α}}_{12}, \dots, {\vec{α}}_{1 m})$ and $M_{2} = ({\vec{α}}_{21}, {\vec{α}}_{22}, \dots, {\vec{α}}_{2 m})$ are two n × m hesitant fuzzy matrices, where ${\vec{α}}_{1 j}$ and ${\vec{α}}_{2 j} (1 \leq j \leq m)$ are n-dimensional hesitant fuzzy vectors. Then,

(1) ${\vec{α}}_{1} ⋓ {\vec{α}}_{2} = (α_{11} ⊻ α_{21}, α_{12} ⊻ α_{22}, \dots, α_{1 n} ⊻ α_{2 n})^{T},$ where $α_{1 i} ⊻ α_{2 i} = {α_{1 i}^{σ (s)} \lor α_{2 i}^{σ (s)} | 1 \leq s \leq k} (1 \leq i \leq n);$

(2) ${\vec{α}}_{1} ⋒ {\vec{α}}_{2} = (α_{11} \bar{\land} α_{21}, α_{12} \bar{\land} α_{22}, \dots, α_{1 n} \bar{\land} α_{2 n})^{T},$ where $α_{1 i} \bar{\land} α_{2 i} = {α_{1 i}^{σ (s)} \land α_{2 i}^{σ (s)} | 1 \leq s \leq k} (1 \leq i \leq n);$

(3) The complementary vector of ${\vec{α}}_{1}$ , $({\vec{α}}_{1})^{c} = (\sim α_{11}, \sim α_{12}, \dots, \sim α_{1 n})^{T},$ where $\sim α_{1 i} = {1 - α_{1 i}^{σ (s)} | 1 \leq s \leq k} (1 \leq i \leq n);$

$M_{ℝ_{1}} = (\begin{matrix} {1, 1} & {0.3, 0.5} & {0.2, 0.8} & {0.5, 0.9} & {0.3, 0.7} & {0.2, 0.4} & {0.6, 0.7} & {0.2, 0.3} \\ {0.3, 0.5} & {1, 1} & {0.4, 0.5} & {0.2, 0.7} & {0.4, 0.5} & {0.3, 0.6} & {0.5, 0.8} & {0.7, 0.9} \\ {0.2, 0.8} & {0.4, 0.5} & {1, 1} & {0.4, 0.6} & {0.2, 0.8} & {0.7, 0.9} & {0.4, 0.5} & {0.6, 0.6} \\ {0.5, 0.9} & {0.2, 0.7} & {0.4, 0.6} & {1, 1} & {0.3, 0.8} & {0.2, 0.4} & {0.1, 0.2} & {0.3, 0.4} \\ {0.3, 0.7} & {0.4, 0.5} & {0.2, 0.8} & {0.3, 0.8} & {1, 1} & {0.5, 0.8} & {0.4, 0.6} & {0.2, 0.2} \\ {0.2, 0.4} & {0.3, 0.6} & {0.7, 0.9} & {0.2, 0.4} & {0.5, 0.8} & {1, 1} & {0.5, 0.8} & {0.4, 0.6} \\ {0.6, 0.7} & {0.5, 0.8} & {0.4, 0.5} & {0.1, 0.2} & {0.4, 0.6} & {0.5, 0.8} & {1, 1} & {0.6, 0.8} \\ {0.2, 0.3} & {0.7, 0.9} & {0.6, 0.6} & {0.3, 0.4} & {0.2, 0.2} & {0.4, 0.6} & {0.6, 0.8} & {1, 1} \end{matrix})$

$M_{ℝ_{2}} = (\begin{matrix} { 1,1} & { 0.4,0.5} & { 0.3,0.6} & { 0.1,0.4} & { 0.4,0.8} & { 0.5,0.7} & { 0.6,0.7} & { 0.3,0.4} \\ { 0.4,0.5} & { 1,1} & { 0.5,0.7} & { 0.4,0.6} & { 0.2,0.4} & { 0.2,0.6} & { 0.2,0.4} & { 0.7,0.8} \\ { 0.3,0.6} & { 0.5,0.7} & { 1,1} & { 0.5,0.7} & { 0.4,0.6} & { 0.3,0.4} & { 0.2,0.3} & { 0.7,0.9} \\ { 0.1,0.4} & { 0.4,0.6} & { 0.5,0.7} & { 1,1} & { 0.7,0.8} & { 0.6,0.9} & { 0.3,0.5} & { 0.6,0.7} \\ { 0.4,0.8} & { 0.2,0.4} & { 0.4,0.6} & { 0.7,0.8} & { 1,1} & { 0.5,0.8} & { 0.1,0.2} & { 0.4,0.6} \\ { 0.5,0.7} & { 0.2,0.6} & { 0.3,0.4} & { 0.6,0.9} & { 0.5,0.8} & { 1,1} & { 0.6,0.7} & { 0.2,0.3} \\ { 0.6,0.7} & { 0.2,0.4} & { 0.2,0.3} & { 0.3,0.5} & { 0.1,0.2} & { 0.6,0.7} & { 1,1} & { 0.5,0.6} \\ { 0.3,0.4} & { 0.7,0.8} & { 0.7,0.9} & { 0.6,0.7} & { 0.4,0.6} & { 0.2,0.3} & { 0.5,0.6} & { 1,1} \end{matrix})$

$M_{ℝ_{3}} = (\begin{matrix} { 1,1} & { 0.3,0.5} & { 0.2,0.6} & { 0.4,0.7} & { 0.5,0.8} & { 0.3,0.4} & { 0.6,0.7} & { 0.1,0.3} \\ { 0.3,0.5} & { 1,1} & { 0.4,0.5} & { 0.5,0.7} & { 0.2,0.5} & { 0.5,0.7} & { 0.5,0.8} & { 0.3,0.4} \\ { 0.2,0.6} & { 0.4,0.5} & { 1,1} & { 0.3,0.5} & { 0.6,0.9} & { 0.7,0.8} & { 0.4,0.5} & { 0.7,0.9} \\ { 0.4,0.7} & { 0.5,0.7} & { 0.3,0.5} & { 1,1} & { 0.3,0.4} & { 0.2,0.3} & { 0.4,0.7} & { 0.8,0.8} \\ { 0.5,0.8} & { 0.2,0.5} & { 0.6,0.9} & { 0.3,0.4} & { 1,1} & { 0.6,0.9} & { 0.2,0.4} & { 0.6,0.7} \\ { 0.3,0.4} & { 0.5,0.7} & { 0.7,0.8} & { 0.2,0.3} & { 0.6,0.9} & { 1,1} & { 0.7,0.8} & { 0.3,0.5} \\ { 0.6,0.7} & { 0.5,0.8} & { 0.4,0.5} & { 0.4,0.7} & { 0.2,0.4} & { 0.7,0.8} & { 1,1} & { 0.6,0.7} \\ { 0.1,0.3} & { 0.3,0.4} & { 0.7,0.9} & { 0.8,0.8} & { 0.6,0.7} & { 0.3,0.5} & { 0.6,0.7} & { 1,1} \end{matrix})$

$M_{ℝ_{4}} = (\begin{matrix} { 1,1} & { 0.3,0.4} & { 0.7,0.8} & { 0.5,0.9} & { 0.2,0.5} & { 0.4,0.7} & { 0.6,0.7} & { 0.3,0.4} \\ { 0.3,0.4} & { 1,1} & { 0.2,0.4} & { 0.6,0.8} & { 0.4,0.6} & { 0.6,0.7} & { 0.3,0.5} & { 0.6,0.8} \\ { 0.7,0.8} & { 0.2,0.4} & { 1,1} & { 0.3,0.4} & { 0.6,0.8} & { 0.5,0.6} & { 0.4,0.5} & { 0.2,0.3} \\ { 0.5,0.9} & { 0.6,0.8} & { 0.3,0.4} & { 1,1} & { 0.7,0.9} & { 0.2,0.5} & { 0.5,0.6} & { 0.1,0.2} \\ { 0.2,0.5} & { 0.4,0.6} & { 0.6,0.8} & { 0.7,0.9} & { 1,1} & { 0.5,0.9} & { 0.7,0.8} & { 0.6,0.7} \\ { 0.4,0.7} & { 0.6,0.7} & { 0.5,0.6} & { 0.2,0.5} & { 0.5,0.9} & { 1,1} & { 0.2,0.3} & { 0.3,0.5} \\ { 0.6,0.7} & { 0.3,0.5} & { 0.4,0.5} & { 0.5,0.6} & { 0.7,0.8} & { 0.2,0.3} & { 1,1} & { 0.4,0.6} \\ { 0.3,0.4} & { 0.6,0.8} & { 0.2,0.3} & { 0.1,0.2} & { 0.6,0.7} & { 0.3,0.5} & { 0.4,0.6} & { 1,1} \end{matrix})$

$M_{ℝ_{5}} = (\begin{matrix} { 1,1} & { 0.3,0.6} & { 0.4,0.8} & { 0.5,0.5} & { 0.6,0.7} & { 0.3,0.6} & { 0.7,0.8} & { 0.1,0.2} \\ { 0.3,0.6} & { 1,1} & { 0.3,0.6} & { 0.2,0.3} & { 0.5,0.8} & { 0.4,0.6} & { 0.2,0.3} & { 0.5,0.6} \\ { 0.4,0.8} & { 0.3,0.6} & { 1,1} & { 0.6,0.8} & { 0.4,0.6} & { 0.7,0.9} & { 0.4,0.5} & { 0.6,0.7} \\ { 0.5,0.5} & { 0.2,0.3} & { 0.6,0.8} & { 1,1} & { 0.8,0.9} & { 0.5,0.7} & { 0.1,0.2} & { 0.6,0.7} \\ { 0.6,0.7} & { 0.5,0.8} & { 0.4,0.6} & { 0.8,0.9} & { 1,1} & { 0.4,0.5} & { 0.3,0.4} & { 0.2,0.5} \\ { 0.3,0.6} & { 0.4,0.6} & { 0.7,0.9} & { 0.5,0.7} & { 0.4,0.5} & { 1,1} & { 0.4,0.6} & { 0.2,0.4} \\ { 0.7,0.8} & { 0.2,0.3} & { 0.4,0.5} & { 0.1,0.2} & { 0.3,0.4} & { 0.4,0.6} & { 1,1} & { 0.4,0.6} \\ { 0.1,0.2} & { 0.5,0.6} & { 0.6,0.7} & { 0.6,0.7} & { 0.2,0.5} & { 0.2,0.4} & { 0.4,0.6} & { 1,1} \end{matrix})$

(4) $M_{1} ⋓ M_{2} = ({\vec{α}}_{11} ⋓ {\vec{α}}_{21}, {\vec{α}}_{12} ⋓ {\vec{α}}_{22}, \dots, {\vec{α}}_{1 m} ⋓ {\vec{α}}_{2 m});$

(5) $M_{1} ⋒ M_{2} = ({\vec{α}}_{11} ⋒ {\vec{α}}_{21}, {\vec{α}}_{12} ⋒ {\vec{α}}_{22}, \dots, {\vec{α}}_{1 m} ⋒ {\vec{α}}_{2 m});$

(6) The complementary matrix of M₁, ${M_{1}}^{c} = (({\vec{α}}_{11})^{c}, ({\vec{α}}_{12})^{c}, \dots, ({\vec{α}}_{1 m})^{c})^{T} .$

In the following we define the product operation of hesitant fuzzy matrices as follows:

Definition 4.7. Let P and Q be two hesitant fuzzy matrices, and $P = (\begin{matrix} p_{11} & p_{12} & \dots & p_{1 l} \\ p_{21} & p_{22} & \dots & p_{2 l} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{m 1} & p_{m 2} & \dots & p_{ml} \end{matrix}), Q = (\begin{matrix} q_{11} & q_{12} & \dots & q_{1 n} \\ q_{21} & q_{22} & \dots & q_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ q_{l 1} & q_{l 2} & \dots & q_{\ln} \end{matrix})$ where $p_{it} = {p_{it}^{σ (s)} | 1 \leq s \leq k} (1 \leq i \leq m, 1 \leq t \leq l)$ and $q_{tj} = {q_{tj}^{σ (s)} | 1 \leq s \leq k} (1 \leq j \leq n, 1 \leq t \leq l)$ . Then, the product of P and Q is a m × n hesitant fuzzy matrix, denoted as follows:

M = P ∘ Q = (r_ij) _{1≤i≤m,1≤j≤n}, where $r_{ij} = ⊻_{1 \leq t \leq l} {p_{it} \bar{\land} q_{tj}} = {⋁_{1 \leq t \leq l} (p_{it}^{σ (s)} \land q_{tj}^{σ (s)}) | 1 \leq s \leq k} .$

For convenience, we don’t distinguish between hesitant fuzzy vectors and hesitant fuzzy sets on U.

Theorem 4.8. Suppose that $ℝ$ is the hesitant fuzzy relation on U, and $M_{ℝ}$ is the hesitant fuzzy matrix of $ℝ$ and $A \in HF (U)$ . Then

(1) $\underline{ℝ} (A) = (M_{ℝ} \circ A^{c})^{c},$

(2) $\bar{ℝ} (A) = M_{ℝ} \circ A,$ where $\underline{ℝ} (A)$ and $\bar{ℝ} (A)$ are respectively the single-granulation lower and upper approximations defined in Definition 3.3.

Proof. It can be easily derived from Definitions 4.7 and 3.3.

According to Theorems 4.8, 3.5 and 3.7, we conclude that the following theorem holds:

Theorem 4.9. Suppose that $ℝ_{i} (1 \leq i \leq m)$ are m hesitant fuzzy relations on U, $M_{ℝ_{i}}$ are m hesitant fuzzy relation matrices of $ℝ_{i} (1 \leq i \leq m)$ and $A \in HF (U)$ . Then

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

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

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

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

Example 4.10. (Continued from Example 4.3) According to Theorem 4.8(2), we have $\begin{matrix} \begin{matrix} \bar{ℝ_{1}} (D_{1}) = M_{ℝ_{1}} \circ D_{1} \\ = ({0.6, 0.7}, {0.5, 0.7}, {0.5, 0.8}, {0.4, 0.8}, \\ {0.6, 0.8}, {0.5, 0.8}, {0.6, 0.7}, {0.6, 0.7})^{T}, \\ \bar{ℝ_{2}} (D_{1}) = M_{ℝ_{2}} \circ D_{1} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} = ({0.6, 0.8}, {0.4, 0.7}, {0.4, 0.7}, {0.6, 0.8}, \\ {0.6, 0.8}, {0.6, 0.8}, {0.6, 0.7}, {0.5, 0.7})^{T}, \\ \bar{ℝ_{3}} (D_{1}) = M_{ℝ_{3}} \circ D_{1} \\ = ({0.6, 0.8}, {0.5, 0.7}, {0.6, 0.8}, {0.4, 0.7}, \\ {0.6, 0.8}, {0.6, 0.8}, {0.6, 0.7}, {0.6, 0.7})^{T}, \\ \bar{ℝ_{4}} (D_{1}) = M_{ℝ_{4}} \circ D_{1} \\ = ({0.6, 0.7}, {0.5, 0.7}, {0.6, 0.8}, {0.6, 0.8}, \\ {0.6, 0.8}, {0.5, 0.8}, {0.6, 0.8}, {0.6, 0.7})^{T}, \\ \bar{ℝ_{5}} (D_{1}) = M_{ℝ_{5}} \circ D_{1} \\ = ({0.6, 0.7}, {0.5, 0.8}, {0.5, 0.7}, {0.6, 0.8}, \\ {0.6, 0.8}, {0.5, 0.7}, {0.6, 0.7}, {0.4, 0.7})^{T} . \end{matrix} \end{matrix}$

Then by Theorem 4.9(2) and (4), we obtain $\begin{matrix} \begin{matrix} {\bar{\sum_{i = 1}^{5} ℝ_{i}}}^{O} (D_{1}) \\ = ({0.6, 0.7}, {0.4, 0.7}, {0.4, 0.7}, {0.4, 0.7}, \\ {0.6, 0.8}, {0.5, 0.7}, {0.6, 0.7}, {0.4, 0.7})^{T}, \\ {\bar{\sum_{i = 1}^{5} ℝ_{i}}}^{P} (D_{1}) \\ = ({0.6, 0.8}, {0.5, 0.8}, {0.6, 0.8}, {0.6, 0.8}, \\ {0.6, 0.8}, {0.6, 0.8}, {0.6, 0.8}, {0.6, 0.7})^{T} . \end{matrix} \end{matrix}$ Similarly, we have $\begin{matrix} \begin{matrix} \bar{ℝ_{1}} (D_{2}) = M_{ℝ_{1}} \circ D_{2} \\ = ({0.6, 0.9}, {0.5, 0.8}, {0.5, 0.7}, {0.8, 0.9}, \\ {0.4, 0.8}, {0.5, 0.7}, {0.6, 0.8}, {0.6, 0.8})^{T}, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} \bar{ℝ_{2}} (D_{2}) = M_{ℝ_{2}} \circ D_{2} \\ = ({0.6, 0.7}, {0.5, 0.8}, {0.5, 0.8}, {0.8, 0.9}, \\ {0.7, 0.8}, {0.6, 0.9}, {0.6, 0.7}, {0.6, 0.8})^{T}, \\ \bar{ℝ_{3}} (D_{2}) = M_{ℝ_{3}} \circ D_{2} \\ = ({0.6, 0.7}, {0.5, 0.7}, {0.5, 0.8}, {0.8, 0.9}, \\ {0.5, 0.7}, {0.6, 0.7}, {0.6, 0.7}, {0.8, 0.8})^{T}, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} \bar{ℝ_{4}} (D_{2}) = M_{ℝ_{4}} \circ D_{2} \\ = ({0.6, 0.9}, {0.6, 0.8}, {0.4, 0.7}, {0.8, 0.9}, \\ {0.7, 0.9}, {0.5, 0.7}, {0.6, 0.7}, {0.5, 0.8})^{T}, \\ \bar{ℝ_{5}} (D_{2}) = M_{ℝ_{5}} \circ D_{2} \\ = ({0.6, 0.7}, {0.5, 0.6}, {0.6, 0.8}, {0.8, 0.9}, \\ {0.8, 0.9}, {0.5, 0.7}, {0.6, 0.7}, {0.6, 0.8})^{T} . \end{matrix} \end{matrix}$ Then $\begin{matrix} \begin{matrix} {\bar{\sum_{i = 1}^{5} ℝ_{i}}}^{O} (D_{2}) \\ = ({0.6, 0.7}, {0.5, 0.6}, {0.4, 0.7}, {0.8, 0.9}, \\ {0.4, 0.7}, {0.5, 0.7}, {0.6, 0.7}, {0.5, 0.8})^{T}, \\ {\bar{\sum_{i = 1}^{5} ℝ_{i}}}^{P} (D_{2}) \\ = ({0.6, 0.9}, {0.6, 0.8}, {0.6, 0.8}, {0.8, 0.9}, \\ {0.8, 0.9}, {0.6, 0.9}, {0.6, 0.8}, {0.8, 0.8})^{T} . \end{matrix} \end{matrix}$ By virtue of Theorem 4.8(1), we have $\begin{matrix} \begin{matrix} \underline{ℝ_{1}} (D_{1}) = (M_{ℝ_{1}} \circ {D_{1}}^{c})^{c} \\ = ({0.2, 0.6}, {0.3, 0.5}, {0.2, 0.6}, {0.3, 0.6}, \\ {0.2, 0.6}, {0.2, 0.6}, {0.3, 0.5}, {0.3, 0.5})^{T}, \\ \underline{ℝ_{2}} (D_{1}) = (M_{ℝ_{2}} \circ {D_{1}}^{c})^{c} \\ = ({0.3, 0.6}, {0.3, 0.5}, {0.2, 0.5}, {0.3, 0.6}, \\ {0.3, 0.6}, {0.3, 0.6}, {0.3, 0.6}, {0.2, 0.5})^{T}, \\ \underline{ℝ_{3}} (D_{1}) = (M_{ℝ_{3}} \circ {D_{1}}^{c})^{c} \\ = ({0.3, 0.6}, {0.3, 0.5}, {0.2, 0.6}, {0.3, 0.5}, \\ {0.2, 0.6}, {0.2, 0.5}, {0.3, 0.5}, {0.2, 0.6})^{T}, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} \underline{ℝ_{4}} (D_{1}) = (M_{ℝ_{4}} \circ {D_{1}}^{c})^{c} \\ = ({0.2, 0.6}, {0.3, 0.5}, {0.2, 0.6}, {0.3, 0.5}, \\ {0.2, 0.6}, {0.3, 0.5}, {0.3, 0.6}, {0.3, 0.5})^{T}, \\ \underline{ℝ_{5}} (D_{1}) = (M_{ℝ_{5}} \circ {D_{1}}^{c})^{c} \\ = ({0.2, 0.6}, {0.3, 0.5}, {0.2, 0.6}, {0.2, 0.6}, \\ {0.3, 0.5}, {0.2, 0.6}, {0.3, 0.6}, {0.3, 0.5})^{T} . \end{matrix} \end{matrix}$ Then according to Theorem 4.9(1) and (3), we obtain $\begin{matrix} \begin{matrix} {\underline{\sum_{i = 1}^{5} ℝ_{i}}}^{O} (D_{1}) \\ = ({0.3, 0.6}, {0.3, 0.5}, {0.2, 0.6}, {0.3, 0.6}, \\ {0.3, 0.6}, {0.3, 0.6}, {0.3, 0.6}, {0.3, 0.6})^{T}, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} {\underline{\sum_{i = 1}^{5} ℝ_{i}}}^{P} (D_{1}) \\ = ({0.2, 0.6}, {0.3, 0.5}, {0.2, 0.5}, {0.2, 0.5}, \\ {0.2, 0.5}, {0.2, 0.5}, {0.3, 0.5}, {0.2, 0.5})^{T} . \end{matrix} \end{matrix}$ Similarly, we have $\begin{matrix} \begin{matrix} \underline{ℝ_{1}} (D_{2}) = (M_{ℝ_{1}} \circ {D_{2}}^{c})^{c} \\ = ({0.3, 0.7}, {0.4, 0.6}, {0.2, 0.4}, {0.3, 0.7}, \\ {0.2, 0.5}, {0.2, 0.4}, {0.2, 0.5}, {0.4, 0.6})^{T}, \\ \underline{ℝ_{2}} (D_{2}) = (M_{ℝ_{2}} \circ {D_{2}}^{c})^{c} \\ = ({0.3, 0.5}, {0.4, 0.6}, {0.4, 0.6}, {0.2, 0.4}, \\ {0.2, 0.5}, {0.2, 0.4}, {0.3, 0.4}, {0.4, 0.6})^{T}, \\ \underline{ℝ_{3}} (D_{2}) = (M_{ℝ_{3}} \circ {D_{2}}^{c})^{c} \\ = ({0.3, 0.6}, {0.3, 0.5}, {0.2, 0.4}, {0.4, 0.6}, \\ {0.2, 0.4}, {0.2, 0.4}, {0.2, 0.4}, {0.3, 0.6})^{T}, \\ \underline{ℝ_{4}} (D_{2}) = (M_{ℝ_{4}} \circ {D_{2}}^{c})^{c} \\ = ({0.3, 0.6}, {0.3, 0.4}, {0.3, 0.5}, {0.3, 0.6}, \\ {0.2, 0.5}, {0.2, 0.4}, {0.3, 0.6}, {0.3, 0.6})^{T}, \\ \underline{ℝ_{5}} (D_{2}) = (M_{ℝ_{5}} \circ {D_{2}}^{c})^{c} \\ = ({0.3, 0.6}, {0.3, 0.6}, {0.2, 0.4}, {0.3, 0.4}, \\ {0.3, 0.6}, {0.2, 0.4}, {0.4, 0.6}, {0.4, 0.6})^{T} . \end{matrix} \end{matrix}$ Then $\begin{matrix} \begin{matrix} {\underline{\sum_{i = 1}^{5} ℝ_{i}}}^{O} (D_{2}) \\ = ({0.3, 0.7}, {0.4, 0.6}, {0.4, 0.6}, {0.4, 0.7}, \\ {0.3, 0.6}, {0.2, 0.4}, {0.4, 0.6}, {0.4, 0.6})^{T}, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} {\underline{\sum_{i = 1}^{5} ℝ_{i}}}^{P} (D_{2}) \\ = ({0.3, 0.5}, {0.3, 0.4}, {0.2, 0.4}, {0.2, 0.4}, \\ {0.2, 0.4}, {0.2, 0.4}, {0.2, 0.4}, {0.3, 0.6})^{T} . \end{matrix} \end{matrix}$

It is well known that in rough set theory the discernibility function is a key notion to various reduction algorithms. Therefore, we develop a practical method to determining the optimistic and pessimistic approximation reducts of multi-granulation hesitant fuzzy decision information system by constructing the discernibility functions.

Definition 4.11. Let MGHFDIS $= (U, {ℝ_{j} | 1 \leq j \leq m}, D = {d_{i} | 1 \leq i \leq l}, V),$ |U| = n and D_i (1 ≤ i ≤ l) be decision vectors. Denote ${\underline{\sum_{j = 1}^{m} ℝ_{j}}}^{O} (D_{i}) = ({\underline{o}}_{i 1}, {\underline{o}}_{i 2}, \dots, {\underline{o}}_{in}) (1 \leq i \leq l),$ where ${\underline{o}}_{it} = {{\underline{o}}_{it}^{σ (s)} | 1 \leq s \leq k} (1 \leq t \leq n);$ ${\bar{\sum_{j = 1}^{m} ℝ_{j}}}^{O} (D_{i}) = ({\bar{o}}_{i 1}, {\bar{o}}_{i 2}, \dots, {\bar{o}}_{in}) (1 \leq i \leq l),$ where ${\bar{o}}_{it} = {{\bar{o}}_{it}^{σ (s)} | 1 \leq s \leq k} (1 \leq t \leq n);$ ${\underline{\sum_{j = 1}^{m} ℝ_{j}}}^{P} (D_{i}) = ({\underline{p}}_{i 1}, {\underline{p}}_{i 2}, \dots, {\underline{p}}_{in}) (1 \leq i \leq l),$ where ${\underline{p}}_{it} = {{\underline{p}}_{it}^{σ (s)} | 1 \leq s \leq k} (1 \leq t \leq n);$ ${\bar{\sum_{j = 1}^{m} ℝ_{j}}}^{P} (D_{i}) = ({\bar{p}}_{i 1}, {\bar{p}}_{i 2}, \dots, {\bar{p}}_{in}) (1 \leq i \leq l),$ where ${\bar{p}}_{it} = {{\bar{p}}_{it}^{σ (s)} | 1 \leq s \leq k} (1 \leq t \leq n);$ $\underline{ℝ_{j}} (D_{i}) = ({\underline{r}}_{ij 1}, {\underline{r}}_{ij 2}, \dots, {\underline{r}}_{ijn}) (1 \leq i \leq l, 1 \leq j \leq m),$ where ${\underline{r}}_{ijt} = {{\underline{r}}_{ijt}^{σ (s)} | 1 \leq s \leq k} (1 \leq t \leq n);$ $\bar{ℝ_{j}} (D_{i}) = ({\bar{r}}_{ij 1}, {\bar{r}}_{ij 2}, \dots, {\bar{r}}_{ijn}) (1 \leq i \leq l, 1 \leq j \leq m),$ where ${\bar{r}}_{ijt} = {{\bar{r}}_{ijt}^{σ (s)} | 1 \leq s \leq k} (1 \leq t \leq n) .$

Then, the optimistic lower approximation discernibility function of MGHFDIS is

${\underline{f}}^{O} = ⋀_{i = 1}^{l} ⋀_{t = 1}^{n} ⋀_{s = 1}^{k} (⋁_{{\underline{r}}_{ijt}^{σ (s)} = {\underline{o}}_{it}^{σ (s)}, 1 \leq j \leq m} ℝ_{j});$ the optimistic upper approximation discernibility function of MGHFDIS is

${\bar{f}}^{O} = ⋀_{i = 1}^{l} ⋀_{t = 1}^{n} ⋀_{s = 1}^{k} (⋁_{{\bar{r}}_{ijt}^{σ (s)} = {\bar{o}}_{it}^{σ (s)}, 1 \leq j \leq m} ℝ_{j});$ the pessimistic lower approximation discernibility function of MGHFDIS is

${\underline{f}}^{P} = ⋀_{i = 1}^{l} ⋀_{t = 1}^{n} ⋀_{s = 1}^{k} (⋁_{{\underline{r}}_{ijt}^{σ (s)} = {\underline{p}}_{it}^{σ (s)}, 1 \leq j \leq m} ℝ_{j});$ the pessimistic upper approximation discernibility function of MGHFDIS is

${\bar{f}}^{P} = ⋀_{i = 1}^{l} ⋀_{t = 1}^{n} ⋀_{s = 1}^{k} (⋁_{{\bar{r}}_{ijt}^{σ (s)} = {\bar{p}}_{it}^{σ (s)}, 1 \leq j \leq m} ℝ_{j}) .$

By Definitions 4.11 and 4.4, we can easily obtain the following theorem.

Theorem 4.12. Let MGHFDIS $= (U, {ℝ_{j} | 1 \leq j \leq m}, D = {d_{i} | 1 \leq i \leq l}, V),$ |U| = n be a multi-granulation hesitant fuzzy decision information system. We can convert the approximation discernibility functions ${\underline{f}}^{O}$ , ${\bar{f}}^{O}$ , ${\underline{f}}^{P}$ and ${\bar{f}}^{P}$ of MGHFDIS into their disjunction forms ${\underline{f}}^{O} = ⋁_{u = 1}^{u_{1}} (⋀_{v = 1}^{v_{1}} ℝ_{uv 1}),$ ${\bar{f}}^{O} = ⋁_{u = 1}^{u_{2}} (⋀_{v = 1}^{v_{2}} ℝ_{uv 2}),$ ${\underline{f}}^{P} = ⋁_{u = 1}^{u_{3}} (⋀_{v = 1}^{v_{3}} ℝ_{uv 3}),$ and ${\bar{f}}^{P} = ⋁_{u = 1}^{u_{4}} (⋀_{v = 1}^{v_{4}} ℝ_{uv 4}),$ respectively. Then, ${\underline{B}}_{u}^{O} = {ℝ_{uv 1} | v = 1, 2, \dots, v_{1}}$ (u = 1, 2, ⋯ , u₁) , ${\bar{B}}_{u}^{O} = {ℝ_{uv 2} | v = 1, 2, \dots, v_{2}}$ (u = 1, 2, ⋯ , u₂) , ${\underline{B}}_{u}^{P} = {ℝ_{uv 3} | v = 1, 2, \dots, v_{3}}$ (u = 1, 2, ⋯ , u₃) , and ${\bar{B}}_{u}^{P} = {ℝ_{uv 4} | v = 1, 2, \dots, v_{4}}$ (u = 1, 2, ⋯ , u₄) are the optimistic lower and upper, and pessimistic lower and upper approximation reducts of MGHFDIS, respectively.

From Theorem 4.12, we see that all the approximation reducts of multi-granulation hesitant fuzzy decision information system can be obtained through using the discernibility functions defined as in Definition 4.11.

Example 4.13. (Continued from Example 4.10) From Definition 4.11, we obtain

${\underline{f}}^{O} = (ℝ_{2} \land ℝ_{3} \land (ℝ_{1} \lor ℝ_{4})) \land (ℝ_{1} \land ℝ_{2} \land ℝ_{3} \land ℝ_{5}) = ℝ_{1} \land ℝ_{2} \land ℝ_{3} \land ℝ_{5},$

${\bar{f}}^{O} = (ℝ_{2} \land ℝ_{3} \land ℝ_{5}) \land (ℝ_{1} \land ℝ_{3} \land ℝ_{4} \land ℝ_{5}) = ℝ_{1} \land ℝ_{2} \land ℝ_{3} \land ℝ_{4} \land ℝ_{5},$

${\underline{f}}^{P} = (ℝ_{2} \land ℝ_{5} \land (ℝ_{3} \lor ℝ_{4}) \land (ℝ_{1} \lor ℝ_{3})) \land (ℝ_{2} \land ℝ_{3} \land ℝ_{4}) = ℝ_{2} \land ℝ_{3} \land ℝ_{4} \land ℝ_{5},$ and

${\bar{f}}^{P} = (ℝ_{4} \land ℝ_{5} \land (ℝ_{2} \lor ℝ_{3})) \land (ℝ_{1} \land ℝ_{2} \land ℝ_{3} \land ℝ_{4} \land ℝ_{5}) = ℝ_{1} \land ℝ_{2} \land ℝ_{3} \land ℝ_{4} \land ℝ_{5} .$ Hence, we conclude that

the optimistic lower approximation reduct of MGHFDIS is ${ℝ_{1}, ℝ_{2}, ℝ_{3}, ℝ_{5}};$

the optimistic upper approximation reduct of MGHFDIS is ${ℝ_{1}, ℝ_{2}, ℝ_{3}, ℝ_{4}, ℝ_{5}};$

the pessimistic lower approximation reduct of MGHFDIS is ${ℝ_{2}, ℝ_{3}, ℝ_{4}, ℝ_{5}};$

the pessimistic upper approximation reduct of MGHFDIS is ${ℝ_{1}, ℝ_{2}, ℝ_{3}, ℝ_{4}, ℝ_{5}} .$

5 Conclusion

Since Qian et al. [25] proposed multi-granulation rough set model, rough set theory has been extended to other uncertainty environment from the perspective of granular computing. In this study, a novel hesitant fuzzy rough set model has been developed by us to improve Yang et al.’s one in [44]. Then we have proposed two types of multi-granulation hesitant fuzzy rough sets in the generalized hesitant fuzzy tolerance approximation space based on multiple hesitant fuzzy tolerance relations, and established an approximation reduction approach in multi-granulation hesitant fuzzy decision information system. In the future, we mainly focus on investigating the multi-granulation rough set model under interval-valued hesitant fuzzy environment. Moreover, it’s interesting to further explore the practical application of this model.

Footnotes

Acknowledgement

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the Natural Science Foundation of Gansu Province (Nos. 17JR5RA284), the Research Project Funds for Higher Education Institutions of Gansu Province (No. 2016B-005), the Fundamental Research Funds for the Central Universities of Northwest MinZu University (No. 31920190055) and Gansu Provincial first-class discipline program of Northwest Minzu University.

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