Interval-valued dual hesitant fuzzy rough set over two universes and its application

Abstract

In many real decision-making problems, since the assessment information is usually inaccurate and incomplete, decision makers may be hard to get enough information and precisely quantify their opinions. To solve this problem, we aim to introduce interval-valued dual hesitant fuzzy rough set over two universes model. By integrating interval-valued dual hesitant fuzzy set and rough set models, this paper provides a systematic framework for the study of interval-valued dual hesitant fuzzy rough set over two universes, which can better handle imprecise and uncertain information. Firstly, in constructive approach, we define the lower and upper approximation operators under an interval-valued dual hesitant fuzzy relation, and some relevant properties are discussed. Then, we establish a general decision-making approach based on the proposed model. Finally, two practical examples are provided to elaborate the created method and illustrate its effectiveness.

Keywords

Interval-valued dual hesitant fuzzy rough set interval-valued dual hesitant fuzzy set dual hesitant fuzzy relation interval-valued fuzzy rough approximation operators rough set over two universes

1 Introduction

In many practical decision-making (DM) problems, it’s difficult to accurately depict decision information. Since fuzzy set [1] can effectively describe vague and imprecise information, it has attracted great attention. In light of limitations of fuzzy set in some practical applications, scholars have proposed various improved models of fuzzy set [2 –4]. The detailed overview of fuzzy research can be found in [5]. Considering that it’s irresolute for experts sometimes to make a choice between several possible membership degrees of an element to a set, Torra [3] developed hesitant fuzzy (HF) set theory, which could retain all these values. Lately, many scholars have researched HF set model and its properties from different angles to better handle DM problems [6 –22]. Dual hesitant fuzzy (DHF) set, proposed by Zhu et al. [4], is a novel extension of HF set. By providing both membership degrees and non-membership degrees with some possible values between 0 and 1 respectively, DHF set contains more useful information to evaluate DM problems. It can be seen that DHF set is a generalization of fuzzy set, intuitionistic fuzzy (IF) set and HF set. Many scholars have noticed this novel extended model and carried on related research on it [23 –29]. Although many meaningful achievements have been obtained, it’s noted that one of important properties of DM problems is vagueness and inaccuracy. More specially, in many practical DM problems, since the DM environment is usually complicated and uncertain, which leads to the insufficiency of available information, it may be difficult for experts to precisely quantify their opinions with crisp numbers. In this case, interval values are more reasonable to reflect experts’ judgements and can solve this situation well. Therefore, numerous interval-valued fuzzy structures have been studied and fruitful results have been obtained [21 –23]. Especially, Chen et al. [21] introduced interval-valued hesitant fuzzy (IVHF) set. Ju et al. [23] further investigated interval-valued dual hesitant fuzzy (IVDHF) set and introduced some aggregation operators of IVDHF set. It’s noted that when the upper and lower bounds of interval values are identical, IVDHF set turns into DHF set. In addition, if there are only membership degrees or non-membership degrees, IVDHF set degenerates into IVHF set. To sum up, we could see that IVDHF set encompasses both DHF set and IVHF set, which indicates that IVDHF set is more objective than other fuzzy set models to reflect human’s thinking mode.

As another important mathematical tool dealing with imprecise and uncertain information, Rough set theory, introduced by Pawlak [30], has been successfully applied to many areas, especially in knowledge discovery and decision making etc. [31 –34] However, there are some limitations for traditional rough set. Therefore, scholars have made many improvements, which are mainly in the extension of universe and relation [35 –49]. In terms of relation, the equivalence relation of classical rough set is too strict, which cannot directly handle real data. Fuzzy rough set theory, proposed by Dubois et al. [35], offered an effective solution for this problem by replacing indiscernible relation with fuzzy relation. On that basis, many new extended models are researched in detail [36 –42]. In particular, Yang et al. [39] investigated both constructive and axiomatic methods of HF rough set. Zhang et al. [42] further introduced DHF rough set. In terms of universe, traditional rough set mainly focuses on single universe which are limited in some practical problems. Thus, Pei et al. [50] introduced rough set models over two universes and discussed the relevant properties. Sun et al. [51] further researched fuzzy rough set model over two universes. Moreover, many studies of rough set over two universes were made [42 –48].

In fuzzy environment, we should further analyze the decision information to obtain the final result. namely, it’s important to utilize a useful method to analyze the fuzzy information. As mentioned above, we could see that rough set is closely related and complementary to fuzzy set model. Moreover, rough set has advantages to explain decisions and acquire knowledge. To better analyze and process the IVDHF information in DM problems, we intend to introduce a new model called IVDHF rough set over two universes. We summary the intentions of this novel combination model of rough set and IVDHF set as follows:

The combined rough set model is an effective mathematical tool to characterize the relationships between fuzzy elements and classes. Until now, most current studies about IVDHF set mainly focus on IVDHF set itself and there is little discussion about the fusion of IVDHF set and rough set. Since the combination of rough set and IVDHF set is an important and useful way in DM problems, which could effectively deal with the incomplete information and find potential decision rules, it’s necessary to develop a systematic framework for the study of IVDHF rough set over two universes.

Although Zhang et al. [42] developed DHF rough set model, however, one of important properties of DM problems is inaccuracy and incompleteness. The evaluation information is usually insufficient for experts to precisely quantify their opinions with crisp numbers, which means that DHF rough set may not effectively deal with this situation. While the interval values can be used to overcome this problem. Moreover, in many actual DM processes, experts usually need to consider different aspects of decision information, which are related to each other. That is to say, two or more correlative universes are required to describe the information. For instance, in medical diagnosis, a symptom can imply several diseases. Conversely, the patient with one disease may show kinds of symptoms at the same time. In this case, we should consider the DM problem from point of view of two universes, which are sets of symptoms and diseases respectively, rather than single universe. In light of the need of applications in practice, as well as a supplement for both rough set and IVDHF set theory, it’s necessary to introduce the concept of IVDHF rough set over two universes. Therefore, it’s essential to investigate a novel combination model of IVDHF set and rough set.

The organization of this paper is as follows. Some basic knowledge of DHF set and IVDHF set are introduced in the next section. Moreover, some novel properties of IVDHF set are discussed and developed. In Section 3, the upper and lower IVDHF rough approximation operators are defined. Some necessary properties of IVDHF rough approximation operators are also explored. In Section 4, we develop a DM approach based on the IVDHF rough set model over two universes. Section 5 provides two numerical examples to illustrate the DM approach raised in previous section. Moreover, a contrastive analysis is made. Finally, we conclude this paper in Section 6.

2 Preliminaries

2.1 Hesitant fuzzy set

By returning a set of possible membership degrees, Torra [3] presented hesitant fuzzy set.

Definition 1. [3] Let U be a reference set, then a HF set $\tilde{A}$ on U is defined in terms of a function returning a set of several possible membership degrees in [0,1], which is denoted by: $\tilde{A} = {x, h_{\tilde{A}} (x) | x \in U}$ (1) where $h_{\tilde{A}} (x)$ is a set of several possible membership degrees of x ∈ U to $\tilde{A}$ in [0,1], and $h_{\tilde{A}} (x)$ is called a hesitant fuzzy element (HFE).

To compare the magnitude of different HFEs, various kinds of score functions were proposed [8 –10]. Xia and Xu [8] present the concept of arithmetic-mean score function for HFEs. Furthermore, Farhadinia discussed basic properties of score functions and introduced several score functions for HFE [9]. For convenience, we only introduce the arithmetic-mean score function here and more details can be seen in [9].

Definition 2. [8] For a HFE $h_{\tilde{A}} (x)$ , the score function of $h_{\tilde{A}} (x)$ is defined as follows: $s (h_{\tilde{A}} (x)) = \frac{1}{l_{h_{\tilde{A}} (x)}} \sum_{γ \in h} γ$ (2) where $l_{h_{\tilde{A}} (x)}$ is the number of membership degrees in $h_{\tilde{A}} (x)$ . $s (h_{\tilde{A}} (x))$ a value between 0 and 1. For two HFEs $h_{\tilde{A}} (x)$ and $h_{\tilde{B}} (x)$ , $h_{\tilde{A}} (x) \geq h_{\tilde{B}} (x)$ if $s (h_{\tilde{A}} (x)) \geq s (h_{\tilde{B}} (x))$ .

2.2 Interval-valued dual hesitant fuzzy set

By providing sets of possible membership degrees and non-membership degrees respectively, Zhu et al. [4] presented concept of dual hesitant fuzzy set as follows.

Definition 3. [4] Let U be a reference set, then a DHF set $𝔻$ on U is defined as follows: $𝔻 = {x, h_{𝔻} (x), g_{𝔻} (x) | x \in U}$ (3) where $h_{𝔻} (x)$ and $g_{𝔻} (x)$ are two sets of several possible membership degrees and non-membership degrees in [0,1], respectively, with the conditions:0 ≤ γ, η ≤ 1, 0 ≤ γ⁺ + η⁺ ≤ 1, in which $γ \in h_{𝔻} (x), η \in g_{𝔻} (x), γ^{+} = ⋃_{γ \in h_{𝔻} (x)} max {γ}, η^{+} = ⋃_{η \in g_{𝔻} (x)} \max {η}$ for all x ∈ U. The pair $d_{𝔻} (x) = (h_{𝔻} (x), g_{𝔻} (x))$ is said to be a dual hesitant fuzzy element (DHFE).

Definition 4. [21] Let $γ_{1} = [γ_{1}^{l}, γ_{1}^{u}]$ and $γ_{2} = [γ_{2}^{l}, γ_{2}^{u}]$ be two interval values, λ ≥ 0, then

$γ_{1} = γ_{2} \Leftrightarrow γ_{1}^{l} = γ_{2}^{l}$ and $γ_{1}^{u} = γ_{2}^{u}$

$γ_{1} + γ_{2} = [γ_{1}^{l} + γ_{2}^{l}, γ_{1}^{u} + γ_{2}^{u}]$

$λ γ_{1} = [λ γ_{1}^{l}, λ γ_{1}^{u}]$

$p (γ_{1} \geq γ_{2}) = \max {1 - \max (\frac{γ_{2}^{u} - γ_{1}^{l}}{l_{γ_{1}} + l_{γ_{2}}}, 0), 0}$

where l_{γ
_i} (i = 1, 2) represents length of interval value γ_i. p (γ₁ ≥ γ₂) is the possibility degree of γ₁ ≥ γ₂. For convenience, hereinafter, we denote an interval value [γ^l, γ^u] by γ.

Definition 5. [23] Let U be a reference set, then an IVDHF set $D$ on U is defined as follows: $D = {x, h_{D} (x), g_{D} (x) | x \in U}$ (4) where $h_{D} (x)$ and $g_{D} (x)$ are two sets of membership degrees and non-membership degrees, respectively, denoted by several possible interval values in [0,1], with the conditions: [γ^l, γ^u] , [η^l, η^u] ⊆ [0, 1] , 0 ≤ (γ^u) ⁺ + (η^u) ⁺ ≤ 1, in which $[γ^{l}, γ^{u}] \in h_{D} (x), [η^{l}, η^{u}] \in g_{D} (x), (γ^{u})^{+} = ⋃_{[γ^{l}, γ^{u}] \in h_{D} (x)} max {γ^{u}}$ and $(η^{u})^{+} = ⋃_{[η^{l}, η^{u}] \in g_{D} (x)} \max {η^{u}}$ for all x ∈ U. The pair $d_{D} (x) = (h_{D} (x), g_{D} (x))$ is said to be an interval-valued dual hesitant fuzzy element (IVDHFE). It is noted that if there are γ^l = γ^u and η^l = η^u, then IVDHF set degenerates into DHF set.

Suppose that U is a reference set, then we denote all IVDHF sets on U by IVDHF(U). Moreover, several special IVDHF sets are defined as follows: $\forall D \in IVDHF (U)$ :

$D$ is called an empty IVDHF set, denoted by ∅, if and only if $h_{D} (x) = {[0, 0]}$ and $g_{D} (x) = {[1, 1]}$ are satisfied for all x ∈ U.

$D$ is called a full IVDHF set, denoted by $U$ , if and only if $h_{D} (x) = {[1, 1]}$ and $g_{D} (x) = {[0, 0]}$ are satisfied for all x ∈ U.

$D$ is called a constant IVDHF set if and only if $h_{D} (x) = \hat{γ_{1, \dots, m}} = {[γ_{1}^{l}, γ_{1}^{u}], [γ_{2}^{l}, γ_{2}^{u}], \dots, [γ_{m}^{l}, γ_{m}^{u}]}$ and $g_{D} (x) = \hat{η_{1, \dots, n}} = {[η_{1}^{l}, η_{1}^{u}], [η_{2}^{l}, η_{2}^{u}], \dots, [η_{n}^{l}, η_{n}^{u}]}$ are satisfied for all x ∈ U. In this paper, we denote a constant IVDHF set by $\hat{γ_{1, \dots, m}, η_{1, \dots, n}}$ .

Further, we define some other special IVDHF sets including [1, 1] _y, [1, 1] _U-y and [1, 1] _M as follows: $\begin{matrix} (1) & h_{{[1, 1]}_{y}} (x) = {\begin{matrix} {[1, 1]}, x = y \\ {[0, 0]}, x \neq y \end{matrix}, \\ g_{{[1, 1]}_{y}} (x) = {\begin{matrix} {[0, 0]}, x = y \\ {[1, 1]}, x \neq y \end{matrix} . \\ (2) & h_{{[1, 1]}_{U - y}} (x) = {\begin{matrix} {[0, 0]}, x = y \\ {[1, 1]}, x \neq y \end{matrix}, \\ g_{{[1, 1]}_{U - y}} (x) = {\begin{matrix} {[1, 1]}, x = y \\ {[0, 0]}, x \neq y \end{matrix} . \\ (3) & h_{{[1, 1]}_{M}} (x) = {\begin{matrix} {[1, 1]}, x \in M, M \subseteq U \\ {[0, 0]}, x \notin M, M \subseteq U \end{matrix}, \\ g_{{[1, 1]}_{M}} (x) = {\begin{matrix} {[0, 0]}, x \in M, M \subseteq U \\ {[1, 1]}, x \notin M, M \subseteq U \end{matrix} . \end{matrix}$

Definition 6. Let U be a reference set, $\forall A, B \in IVDHF (U)$ , then for any x ∈ U,

The complement of $A$ is defined as follows: $\begin{matrix} A^{c} & = & {x, h_{A^{c}} (x), g_{A^{c}} (x) | x \in U} \\ = & {x, g_{A} (x), h_{A} (x) | x \in U} . \end{matrix}$

The union of $A$ and $B$ is denoted by $A ⋓ B$ so that $A ⋓ B = {x, h_{A ⋓ B} (x), g_{A ⋓ B} (x) | x \in U} .$ where $\begin{matrix} h_{A ⋓ B} (x) = h_{A} (x) \underline{\lor} h_{B} (x) \\ = {[max (γ_{1}^{l}, γ_{2}^{l}), max (γ_{1}^{u}, γ_{2}^{u})] \\ | γ_{1} \in h_{A} (x), γ_{2} \in h_{B} (x)} \\ g_{A ⋓ B} (x) = g_{A} (x) \bar{\land} g_{B} (x) \\ = {[min (η_{1}^{l}, η_{2}^{l}), min (η_{1}^{u}, η_{2}^{u})] \\ | η_{1} \in g_{A} (x), η_{2} \in g_{B} (x)} . \end{matrix}$

The intersection of $A$ and $B$ is denoted by $A ⋒ B$ so that $A ⋒ B = {x, h_{A ⋒ B} (x), g_{A ⋒ B} (x) | x \in U} .$ where $\begin{matrix} h_{A ⋒ B} (x) = h_{A} (x) \bar{\land} h_{B} (x) \\ = {[min (γ_{1}^{l}, γ_{2}^{l}), min (γ_{1}^{u}, γ_{2}^{u})] \\ | γ_{1} \in h_{A} (x), γ_{2} \in h_{B} (x)} . \\ g_{A ⋒ B} (x) = g_{A} (x) \underline{\lor} g_{B} (x) \\ = {[max (η_{1}^{l}, η_{2}^{l}), max (η_{1}^{u}, η_{2}^{u})] \\ | η_{1} \in g_{A} (x), η_{2} \in g_{B} (x)} . \end{matrix}$

The ⊞-union of $A$ and $B$ is denoted by $A ⊞ B$ so that $A ⊞ B = {x, h_{A ⊞ B} (x), g_{A ⊞ B} (x) | x \in U} .$ where $\begin{matrix} h_{A ⊞ B} (x) = h_{A} (x) \underline{\lor} h_{B} (x) \\ = {[γ_{1}^{l} + γ_{2}^{l} - γ_{1}^{l} γ_{2}^{l}, γ_{1}^{u} + γ_{2}^{u} - γ_{1}^{u} γ_{2}^{u}] \\ | γ_{1} \in h_{A} (x), γ_{2} \in h_{B} (x)} . \\ g_{A ⊞ B} (x) = g_{A} (x) \otimes g_{B} (x) \\ = {[η_{1}^{l} η_{2}^{l}, η_{1}^{u} η_{2}^{u}] \\ | η_{1} \in g_{A} (x), η_{2} \in g_{B} (x)} . \end{matrix}$

The ⊞-intersection of $A$ and $B$ is denoted by $A ⊠ B$ so that $A ⊠ B = {x, h_{A ⊠ B} (x), g_{A ⊠ B} (x) | x \in U} .$ where $\begin{matrix} h_{A ⊠ B} (x) = h_{A} (x) \otimes h_{B} (x) \\ = {[γ_{1}^{l} γ_{2}^{l}, γ_{1}^{u} γ_{2}^{u}] | γ_{1} \in h_{A} (x), γ_{2} \in h_{B} (x)} . \\ g_{A ⊠ B} (x) = g_{A} (x) \oplus g_{B} (x) \\ = {[η_{1}^{l} + η_{2}^{l} - η_{1}^{l} η_{2}^{l}, η_{1}^{u} + η_{2}^{u} - η_{1}^{u} η_{2}^{u}] \\ | η_{1} \in g_{A} (x), η_{2} \in g_{B} (x)} . \end{matrix}$

Remark 1. In Definition 6, the operations ⋓, ⋒, ⊞, ⊠ are defined on IVDHF sets, respectively, and the operations $\underline{\lor}$ , $\bar{\land}$ Z, ⊕, ⊗ are defined on IVDHFEs respectively.

Theorem 1. Let U be a reference set, for any $A, B \in IVDHF (U)$ , we have

${(A ⋓ B)}^{c} = A^{c} ⋒ B^{c}$

${(A ⋒ B)}^{c} = A^{c} ⋓ B^{c}$

Proof. It follows directly from Definition 6.

IVDHFE is the basic unit of an IVDHF set. To compare the magnitude of different IVDHFEs, we introduce the concepts of score function and accuracy function for IVDHFEs.

Definition 7. Let d_i = (h_{d
_i}, g_{d
_i}) (i = 1, 2) be two IVDHFEs, then the score function sf (d_i) and the accuracy function pf (d_i) are defined as follows: $\begin{matrix} sf (d_{i}) & = & s (h_{d_{i}}) - s (g_{d_{i}}) \\ = & \frac{1}{l_{h_{d_{i}}}} \sum_{γ_{i} \in h_{d_{i}}} γ_{i} - \frac{1}{l_{g_{d_{i}}}} \sum_{η_{i} \in g_{d_{i}}} η_{i} \end{matrix}$ (5) $\begin{matrix} pf (d_{i}) & = & s (h_{d_{i}}) + s (g_{d_{i}}) \\ = & \frac{1}{l_{h_{d_{i}}}} \sum_{γ_{i} \in h_{d_{i}}} γ_{i} + \frac{1}{l_{g_{d_{i}}}} \sum_{η_{i} \in g_{d_{i}}} η_{i} \end{matrix}$ (6) where $γ_{i} = [γ_{i}^{l}, γ_{i}^{u}]$ , $η = [η_{i}^{l}, η_{i}^{u}]$ are possible interval values in h_{d
_i} and g_{d
_i} respectively. l_{h
_{d
_i}} and l_{g
_{d
_i}} are numbers of interval values in h_{d
_i} and g_{d
_i} respectively. Then,

if sf (d₁) > sf (d₂), then d₁ is said to be superior to d₂,denoted by d₁ > d₂.

if sf (d₁) = sf (d₂), then,

if pf (d₁) > pf (d₂), then d₁ > d₂.

if pf (d₁) = pf (d₂), then d₁ is equivalent to d₂,denoted by d₁ = d₂.

Theorem 2. Let d_i = (h_{d
_i}, g_{d
_i}) (i = 1, 2) be two IVDHFEs, then the following properties hold. $\begin{matrix} (1) & sf (d_{1} \oplus d_{2}) = s (h_{d_{1}}) s \oplus (h_{d_{2}}) \\ - s (g_{d_{1}}) \otimes s (g_{d_{2}}) \\ pf (d_{1} \oplus d_{2}) = s (h_{d_{1}}) s \oplus (h_{d_{2}}) \\ + s (g_{d_{1}}) \otimes s (g_{d_{2}}) \\ (2) & sf (d_{1} \otimes d_{2}) = s (h_{d_{1}}) \otimes s (h_{d_{2}}) \\ - s (g_{d_{1}}) \oplus (s g_{d_{2}}) \\ pf (d_{1} \otimes d_{2}) = s (h_{d_{1}}) \otimes s (h_{d_{2}}) \\ + s (g_{d_{1}}) s (g_{d_{2}}) \end{matrix}$

Proof. (1) First of all, We are going to prove s (h_{d
₁}) ⊕ (h_{d
₂}) = s (h_{d
₁}h_{d
₂}) . Suppose that $h_{d_{1}} = {[γ_{1}^{1 l}, γ_{1}^{1 u}], [γ_{1}^{2 l}, γ_{1}^{2 u}], \dots, [γ_{1}^{nl}, γ_{1}^{nu}]}$ and $h_{d_{2}} = {[γ_{2}^{1 l}, γ_{2}^{1 u}], [γ_{2}^{2 l}, γ_{2}^{2 u}], \dots, [γ_{2}^{ml}, γ_{2}^{mu}]}$ . Then, we have

$\begin{matrix} s (h_{d_{1}}) \oplus s (h_{d_{2}}) \\ = [\frac{\sum_{i = 1}^{n} γ_{1}^{il}}{n} \oplus \frac{\sum_{j = 1}^{m} γ_{2}^{jl}}{m}, \frac{\sum_{i = 1}^{n} γ_{1}^{iu}}{n} \oplus \frac{\sum_{j = 1}^{m} γ_{2}^{ju}}{m}] \\ = [\frac{\sum_{i = 1}^{n} γ_{1}^{il}}{n} + \frac{\sum_{j = 1}^{m} γ_{2}^{jl}}{m} - \frac{\sum_{i = 1}^{n} γ_{1}^{il}}{n} \frac{\sum_{j = 1}^{m} γ_{2}^{jl}}{m}, \\ \frac{\sum_{i = 1}^{n} γ_{1}^{iu}}{n} + \frac{\sum_{j = 1}^{m} γ_{2}^{ju}}{m} - \frac{\sum_{i = 1}^{n} γ_{1}^{iu}}{n} \frac{\sum_{j = 1}^{m} γ_{2}^{ju}}{m}] \\ = [\frac{m \sum_{i = 1}^{n} γ_{1}^{il} + n \sum_{j = 1}^{m} γ_{2}^{jl} - \sum_{i = 1}^{n} γ_{1}^{il} \sum_{j = 1}^{m} γ_{2}^{jl}}{mn}, \\ \frac{m \sum_{i = 1}^{n} γ_{1}^{iu} + n \sum_{j = 1}^{m} γ_{2}^{ju} - \sum_{i = 1}^{n} γ_{1}^{iu} \sum_{j = 1}^{m} γ_{2}^{ju}}{mn}] \\ = [\frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} (γ_{1}^{il} + γ_{2}^{jl} - γ_{1}^{il} γ_{2}^{jl})}{mn}, \\ \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} (γ_{1}^{iu} + γ_{2}^{ju} - γ_{1}^{iu} γ_{2}^{ju})}{mn}] \\ = [\frac{\sum_{γ_{3} \in (h_{d_{1}} \oplus h_{d_{2}})} γ_{3}^{l}}{mn}, \frac{\sum_{γ_{3} \in (h_{d_{1}} \oplus h_{d_{2}})} γ_{3}^{u}}{mn}] \\ = s (h_{d_{1}} \oplus h_{d_{2}}) \end{matrix}$

On the other hand, suppose that $g_{d_{1}} = {[η_{1}^{1 l}, η_{1}^{1 u}], [η_{1}^{2 l}, η_{1}^{2 u}], \dots, [η_{1}^{nl}, η_{1}^{nu}]}$ and $g_{d_{2}} = {[η_{2}^{1 l}, η_{2}^{1 u}], [η_{2}^{2 l}, η_{2}^{2 u}], \dots, [η_{2}^{ml}, η_{2}^{mu}]}$ , then $\begin{matrix} s (g_{d_{1}}) \otimes s (g_{d_{2}}) \\ = [\frac{\sum_{i = 1}^{n} η_{1}^{il}}{n} \otimes \frac{\sum_{j = 1}^{m} η_{2}^{jl}}{m}, \frac{\sum_{i = 1}^{n} η_{1}^{iu}}{n} \otimes \frac{\sum_{j = 1}^{m} η_{2}^{ju}}{m}] \\ = [\frac{\sum_{i = 1}^{n} η_{1}^{il} \sum_{j = 1}^{m} η_{2}^{jl}}{mn}, \frac{\sum_{i = 1}^{n} η_{1}^{iu} \sum_{j = 1}^{m} η_{2}^{ju}}{mn}] \\ = [\frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} η_{1}^{il} η_{2}^{jl}}{mn}, \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{m} η_{1}^{iu} η_{2}^{ju}}{mn}] \\ = s (g_{d_{1}} \otimes g_{d_{2}}) \end{matrix}$

therefore, we can get that property (1) holds.

(2) Similarly, it is easy to prove the next property.

Assume that the number of membership and non-membership degrees in d_i is m_i and n_i, respectively. Then, the computation complexity of computing sf (d₁ ⊕ d₂) and pf (d₁ ⊕ d₂) by Definition 7 is O (m₁m₂ + n₁n₂), which will increase quickly as m_i and n_i increase. However, if we computing it by Theorem 2, the computation complexity is O (m₁ + m₂ + n₁ + n₂), which can effectively decrease the complexity of computation.

The score function and the accuracy function provide a way to compare the magnitude of different IVDHFEs. However, the comparisons between different IVDHF sets are also significant. Hereinafter, the concept of IVDHF subset is introduced to make a comparison between two IVDHF sets. For convenience, $\forall D \in IVDHF (U)$ we denote the maximum and minimum upper bounds of interval values in an IVDHFE by $d_{D}^{u +} = (h_{D}^{u +} (x), g_{D}^{u +} (x))$ and $d_{D}^{u -} = (h_{D}^{u -} (x), g_{D}^{u -} (x))$ respectively. Similarly, the maximum and minimum lower bounds of interval values in an IVDHFE are denoted by $d_{D}^{l +} = (h_{D}^{l +} (x), g_{D}^{l +} (x))$ and $d_{D}^{l -} = (h_{D}^{l -} (x), g_{D}^{l -} (x))$ respectively.

Definition 8. Let U be a reference set, $\forall A, B \in IVDHF (U)$ , if $d_{A} (x) ≼ d_{B} (x) \Leftrightarrow h_{A} (x) ≼ h_{B} (x)$ and $g_{A} (x) ≽ g_{B} (x)$ holds for any x ∈ U so that $\begin{matrix} h_{A} (x) ≼ h_{B} (x) \Leftrightarrow h_{A}^{l -} (x) \leq h_{B}^{l -} (x), \\ h_{A}^{l +} (x) \leq h_{B}^{l +} (x), h_{A}^{u -} (x) \leq h_{B}^{u -} (x), \\ h_{A}^{u +} (x) \leq h_{B}^{u +} (x), \\ g_{A} (x) ≽ g_{B} (x) \Leftrightarrow g_{A}^{l -} (x) \geq g_{B}^{l -} (x), \\ g_{A}^{l +} (x) \geq g_{B}^{l +} (x), \\ g_{A}^{u -} (x) \geq g_{B}^{u -} (x), g_{A}^{u +} (x) \geq g_{B}^{u +} (x) \end{matrix}$ then $A$ is called an IVDHF subset of $B$ , denoted by $A ⊑ B$ or $B ⊒ A$ .

Remark 2. (1) It’s noted that if there are only membership degrees or non-membership degrees with the condition that the upper and lower bounds of interval values are identical, then Definition 8 degenerates into the concept of HF subset. (2) It’s evident that ⊑ is reflexive and transitive, while the anti-symmetry doesn’t hold.

Theorem 3. Let U be a reference set, $\forall A_{i}, B, C \in IVDHF (U)$ , (i = 1, 2, …, n), the following properties are true:

$A_{i} ⊑ B \Rightarrow ⋓_{i = 1}^{n} A_{i} ⊑ B; A_{i} ⊒ B \Rightarrow ⋒_{i = 1}^{n} A_{i} ⊒ B .$

$B ⋒ C ⊑ B, C; B ⋓ C ⊒ B, C$ .

$\emptyset ⊑ A_{i}, B, C; U ⊒ A_{i}, B, C$ ;

Proof. Since the proving processes are similar, we only prove $A_{i} ⊑ B \Rightarrow ⋓ A_{i} ⊑ B$ , and the others are proved in the same way. By Definition 8, ∀x ∈ U, we can get that $h_{A_{i}}^{l -} (x) \leq h_{B}^{l -} (x), h_{A_{i}}^{l +} (x) \leq h_{B}^{l +} (x), h_{A_{i}}^{u -} (x) \leq h_{B}^{u -} (x), h_{A_{i}}^{u +} (x) \leq h_{B}^{u +} (x)$ . Then, we can deduce that $h_{⋓ A_{i}}^{l -} (x) = ⋃_{i = 1}^{n} \max {h_{A_{i}}^{l -} (x)} \leq h_{B}^{l -} (x), h_{⋓ A_{i}}^{l +} (x) = ⋃_{i = 1}^{n} max {h_{A_{i}}^{l +} (x)} \leq h_{B}^{l +} (x), h_{⋓ A_{i}}^{u -} (x) = ⋃_{i = 1}^{n} \max {h_{A_{i}}^{u -} (x)} \leq h_{B}^{u -} (x), h_{⋓ A_{i}}^{u +} (x) = ⋃_{i = 1}^{n} max {h_{A_{i}}^{u +} (x)} \leq h_{B}^{u +} (x),$ which means that $h_{⋓ A_{i}} (x) ≼ h_{B} (x)$ . On the other hand, there are $g_{A_{i}}^{l -} (x) \geq g_{B}^{l -} (x), g_{A_{i}}^{l +} (x) \geq g_{B}^{l +} (x), g_{A_{i}}^{u -} (x) \geq g_{B}^{u -} (x)$ and $g_{A_{i}}^{u +} (x) \geq g_{B}^{u +} (x) .$ Hence, we can conclude that $g_{⋓ A_{i}}^{l -} (x) = ⋃_{i = 1}^{n} min {g_{A_{i}}^{l -} (x)} \geq g_{B}^{l -} (x), g_{⋓ A_{i}}^{l +} (x) = ⋃_{i = 1}^{n} \min {g_{A_{i}}^{l +} (x)} \geq g_{B}^{l +} (x), g_{⋓ A_{i}}^{u -} (x) = ⋃ ⋃_{i = 1}^{n} \min {g_{A_{i}}^{u -} (x)} \geq g_{B}^{u -} (x), g_{⋓ A_{i}}^{u +} (x) = ⋃_{i = 1}^{n} \min {g_{A_{i}}^{u +} (x)} \geq g_{B}^{u +} (x),$ which means that $g_{⋓ A_{i}} (x) ≽ g_{B} (x)$ . From the above discussion, we have $⋓ A_{i} ⊑ B$ .

Corollary 1. Let U be a reference set, $\forall A_{i}, B, C \in IVDHF (U)$ , (i = 1, 2, …, n), the following properties are true:

$d_{A_{i}} (x) ≼ d_{B} (x) \Rightarrow {\underline{\lor}}_{i = 1}^{n} d_{A_{i}} (x) ≼ d_{B} (x)$ , nam- ely, ${\underline{\lor}}_{i = 1}^{n} h_{A_{i}} (x) ≼ h_{B} (x)$ and ${\bar{\land}}_{i = 1}^{n} g_{A_{i}} (x) ≽ g_{B} (x)$ .

$d_{A_{i}} (x) ≽ d_{B} (x) \Rightarrow {\bar{\land}}_{i = 1}^{n} d_{A_{i}} (x) ≽ d_{B} (x)$ , namely, ${\bar{\land}}_{i = 1}^{n} h_{A_{i}} (x) ≽ h_{B} (x)$ and ${\underline{\lor}}_{i = 1}^{n} g_{A_{i}} (x) ≼ g_{B} (x)$ .

$d_{B} (x) \bar{\land} d_{C} (x) ≼ d_{B} (x), d_{C} (x)$ , namely, $h_{B} (x) \bar{\land} h_{C} (x) ≼ h_{B} (x), h_{C} (x)$ and $g_{B} (x) \underline{\lor} g_{C} (x) ≽ g_{B} (x), g_{C} (x) .$

$d_{B} (x) \underline{\lor} d_{C} (x) ≽ d_{B} (x), d_{C} (x)$ , namely, $h_{B} (x) \underline{\lor} h_{C} (x) ≽ h_{B} (x), h_{C} (x)$ and $g_{B} (x) \bar{\land} g_{C} (x) ≼ g_{B} (x), g_{C} (x)$ .

Proof. It follows directly from Theorem 3.

3 Interval-valued dual hesitant fuzzy rough set over two universes

3.1 Definition of IVDHF rough set

Differing from classical rough set, an indiscernible relation is replaced by a fuzzy relation in fuzzy rough set such that a fuzzy target can be approximated. Hence, before introducing IVDHF rough set over two universes, we first define an IVDHF relation.

Definition 9. Let U, V be two reference sets, then an IVDHF relation $R$ from U to V is an IVDHF subset so that $R \in IVDHF (U \times V)$ where $R = {(x, y), h_{R} (x, y), g_{R} (x, y) | (x, y) \in (U \times V)} . \forall (x, y) \in (U \times V),$ $h_{R} (x, y)$ and $g_{R} (x, y)$ are two sets of membership degrees and non-membership degrees of the relationships between x and y, respectively, denoted by some possible interval values in [0,1]. Specifically, $R$ is called an IVDHF relation on U while U = V.

Definition 10. Let $R$ be an IVDHF relation from U to V, then

$R$ is said to be serial, if for any x ∈ U, there is a y ∈ V satisfying $h_{R} (x, y) = {[1, 1]}$ and $g_{R} (x, y) = {[0, 0]}$ .

$R$ is said to be reflexive on U, if for any x ∈ U, there exist $h_{R} (x, x) = {[1, 1]}$ and $g_{R} (x, x) = {[0, 0]}$ .

$R$ is said to be symmetric on U, if $h_{R} (x, y) = h_{R} (y, x)$ and $g_{R} (x, y) = g_{R} (y, x)$ hold for any x, y ∈ U.

$R$ is said to be transitive on U, if $h_{R} (x, y) \bar{\land} h_{R} (y, z) ≼ h_{R} (x, z)$ and $g_{R} (x, y) \underline{\lor} g_{R} (y, z) ≽ g_{R} (x, z)$ hold for any x, y, z ∈ U.

Definition 11. Let U, V be two reference sets. $R$ is an IVDHF relation from U to V, for any $A \in IVDHF (V)$ , the lower and upper approximations of $A$ in regard to $R$ , denoted by $\underline{R} (A)$ and $\bar{R} (A)$ , respectively, are two IVDHF sets and defined as follows: $\underline{R} (A) = {x, h_{\underline{R} (A)} (x), g_{\underline{R} (A)} (x) | x \in U},$ (7) $\bar{R} (A) = {x, h_{\bar{R} (A)} (x), g_{\bar{R} (A)} (x) | x \in U} .$ (8) where $\begin{matrix} h_{\underline{R} (A)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{A} (y)}, \\ g_{\underline{R} (A)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} g_{A} (y)}, \\ h_{\bar{R} (A)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} h_{A} (y)}, \\ g_{\bar{R} (A)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} g_{A} (y)} . \end{matrix}$

The pair $(\underline{R} (A), \bar{R} (A))$ is called an IVDHF rough set of $A$ with respect to $R$ . The notations $\underline{R}$ and $\bar{R}$ are lower and upper IVDHF rough approximation operators respectively.

Remark 3. (1) It’s noted that if there are only membership degrees or non-membership degrees, then $(\underline{R} (A), \bar{R} (A))$ becomes an IVHF rough set. (2) when the upper and lower bounds of interval values are identical, $(\underline{R} (A), \bar{R} (A))$ turns into a DHF rough set. In this case, if there are only membership degrees or non-membership degrees, then $(\underline{R} (A), \bar{R} (A))$ becomes a HF rough set. From what have been discussed above, we know that IVDHF rough set is a generalization of HF rough set, IVHF rough set and DHF rough set.

3.2 Properties of IVDHF rough set

In this subsection, we discuss the basic properties of IVDHF rough set.

Theorem 4. Let U, V be two reference sets. $R$ is an IVDHF relation from U to V, $\forall A, B \in IVDHF (V)$ , the following properties are true:

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

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

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

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

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

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

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

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

Proof. (1) By Definition 11, we have $h_{\underline{R} (V)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{V} (y)} = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} {[1, 1]}} = {[1, 1]}, g_{\underline{R} (V)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} g_{V} (y)} = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} {[0, 0]}} = {[0, 0]} .$ Hence, $\underline{R} (V) = U$ holds, and it’s similar to prove that $\bar{R} (\emptyset) = \emptyset .$

(2) By Definitions 6 and 11, we can obtain that $h_{\underline{R} (A^{c})} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{A^{c}} (y)} = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} g_{A} (y)} = g_{\bar{R} (A)} (x) = h_{(\bar{R} (A))^{c}} (x)$ and $g_{\underline{R} (A^{c})} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} g_{A^{c}} (y)} = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} h_{A} (y)} = h_{\bar{R} (A)} (x) = g_{(\bar{R} (A))^{c}} (x) .$ Hence, $\underline{R} (A^{c}) = (\bar{R} (A))^{c}$ holds, and it’s similar to prove that $\bar{R} (A^{c}) = (\underline{R} (A))^{c} .$

(3) By Definitions 8 and 11, we can easily obtain that, for any $D \in IVDHF (U)$ , $\begin{matrix} h_{\underline{R} (D)}^{l -} (x) \\ = min {max {g_{R}^{l -} (x, y), h_{D}^{l -} (y)} | y \in V}, \\ h_{\underline{R} (D)}^{l +} (x) \\ = min {max {g_{R}^{l +} (x, y), h_{D}^{l +} (y)} | y \in V}, \\ h_{\underline{R} (D)}^{u -} (x) \\ = min {max {g_{R}^{u -} (x, y), h_{D}^{u -} (y)} | y \in V}, \\ h_{\underline{R} (D)}^{u +} (x) \\ = min {max {g_{R}^{u +} (x, y), h_{D}^{u +} (y)} | y \in V}, \\ g_{\underline{R} (D)}^{l -} (x) \\ = max {min {h_{R}^{l -} (x, y), g_{D}^{l -} (y)} | y \in V}, \\ g_{\underline{R} (D)}^{l +} (x) \\ = max {min {h_{R}^{l +} (x, y), g_{D}^{l +} (y)} | y \in V}, \\ g_{\underline{R} (D)}^{u -} (x) \\ = max {min {h_{R}^{u -} (x, y), g_{D}^{u -} (y)} | y \in V}, \\ g_{\underline{R} (D)}^{u +} (x) \\ = max {min {h_{R}^{u +} (x, y), g_{D}^{u +} (y)} | y \in V} . \end{matrix}$

Since $A ⊑ B$ , we can obtain that $h_{A}^{l -} (x) \leq h_{B}^{l -} (x), h_{A}^{l +} (x) \leq h_{B}^{l +} (x), h_{A}^{u -} (x) \leq h_{B}^{u -} (x), h_{A}^{u +} (x) \leq h_{B}^{u +} (x)$ and $g_{A}^{l -} (x) \geq g_{B}^{l -} (x), g_{A}^{l +} (x) \geq g_{B}^{l +} (x),$ $g_{A}^{u -} (x) \geq g_{B}^{u -} (x), g_{A}^{u +} (x) \geq g_{B}^{u +} (x) .$ Therefore, we can deduce that $h_{\underline{R} (A)}^{l -} (x) \leq h_{\underline{R} (B)}^{l -} (x), h_{\underline{R} (A)}^{l +} (x) \leq h_{\underline{R} (B)}^{l +} (x), h_{\underline{R} (A)}^{u -} (x) \leq h_{\underline{R} (B)}^{u -} (x), h_{\underline{R} (A)}^{u +} (x) \leq h_{\underline{R} (B)}^{u +} (x)$ and $g_{\underline{R} (A)}^{l -} (x) \geq g_{\underline{R} (B)}^{l -} (x), g_{\underline{R} (A)}^{l +} (x) \geq g_{\underline{R} (B)}^{l +} (x),$ $g_{\underline{R} (A)}^{u -} (x) \geq g_{\underline{R} (B)}^{u -} (x), g_{\underline{R} (A)}^{u +} (x) \geq g_{\underline{R} (B)}^{u +} (x)$ , which means that $h_{\underline{R} (A)} ≼ h_{\underline{R} (B)}$ and $g_{\underline{R} (B)} ≽ g_{\underline{R} (B)}$ , namely, $\underline{R} (A) ⊑ \underline{R} (B)$ . Similarly, it’s not difficult to prove that $\bar{R} (A) ⊑ \bar{R} (B)$ .

(4) By Theorem 3, we know that $A ⋓ B ⊒ A, B$ , then $\underline{R} (A ⋓ B) ⊒ \underline{R} (A), \underline{R} (B)$ and $\bar{R} (A ⋒ B) ⊑ \bar{R} (A), \bar{R} (B)$ . Therefore, $\underline{R} (A ⋓ B) ⊒ \underline{R} (A) ⋓ \underline{R} (B)$ and $\bar{R} (A ⋒ B) ⊑ \bar{R} (A) ⋒ \bar{R} (B)$ hold.

(5) By Definitions 6 and 11, we can obtain that $h_{\underline{R} (A ⋒ B)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{A ⋒ B} (y)} = {[{\bar{\land}}_{y \in V} (η_{g}^{l} \underline{\lor} γ_{A ⋒ B}^{l}), {\bar{\land}}_{y \in V} (η_{g}^{u} \underline{\lor} γ_{A ⋒ B}^{u})] | η_{g} \in g_{R} (x, y), γ_{A ⋒ B} \in h_{A ⋒ B} (y)} = {[{\bar{\land}}_{y \in V} (η_{g}^{l} \underline{\lor} (γ_{A}^{l} \bar{\land} γ_{B}^{l})), {\bar{\land}}_{y \in V} (η_{g}^{u} \underline{\lor} (γ_{A}^{u} \bar{\land} γ_{B}^{u}))] | η_{g} \in g_{R} (x, y), γ_{A} \in h_{A} (y), γ_{B} \in h_{B} (y)} = {[({\bar{\land}}_{y \in V} (η_{g}^{l} \underline{\lor} γ_{A}^{l})) \bar{\land} ({\bar{\land}}_{y \in V} (η_{g}^{l} \underline{\lor} γ_{B}^{l})), ({\bar{\land}}_{y \in V} (η_{g}^{u} \underline{\lor} γ_{A}^{u})) \bar{\land} ({\bar{\land}}_{y \in V} (η_{g}^{u} \underline{\lor} γ_{B}^{u}))] | η_{g} \in g_{R} (x, y), γ_{A} \in h_{A} (y), γ_{B} \in h_{B} (y)} = h_{\underline{R} (A)} (x) \bar{\land} h_{\underline{R} (B)} (x) = h_{\underline{R} (A) ⋒ \underline{R} (B)} (x) .$ and $g_{\underline{R} (A ⋒ B)} (x) = {\underline{\lor}}_{y \in V}$ ${h_{R} (x, y) \bar{\land} g_{A ⋒ B} (y)} = {[{\underline{\lor}}_{y \in V} (γ_{g}^{l} \bar{\land} η_{A ⋒ B}^{l}), {\underline{\lor}}_{y \in V} (γ_{g}^{u} \bar{\land} η_{A ⋒ B}^{u})] | γ_{g} \in h_{R} (x, y), η_{A ⋒ B} \in g_{A ⋒ B} (y)} = {[{\underline{\lor}}_{y \in V} (γ_{g}^{l} \bar{\land} (η_{A}^{l} \underline{\lor} η_{B}^{l})), {\underline{\lor}}_{y \in V} (γ_{g}^{u} \bar{\land} (η_{A}^{u} \underline{\lor} η_{B}^{u}))] | γ_{g} \in h_{R} (x, y), η_{A} \in g_{A} (y), η_{B} \in g_{B} (y)} = {[({\underline{\lor}}_{y \in V}$ $(γ_{g}^{l} \bar{\land} η_{A}^{l})) \underline{\lor} ({\underline{\lor}}_{y \in V} (γ_{g}^{l} \bar{\land} η_{B}^{l})), ({\underline{\lor}}_{y \in V} (γ_{g}^{u} \bar{\land} η_{A}^{u})) \underline{\lor}$ $({\underline{\lor}}_{y \in V} γ_{g}^{u} \bar{\land} η_{B}^{u}) | γ_{g} \in h_{R} (x, y), η_{A} \in g_{A} (y), η_{B} \in g_{B} (y)} = g_{\underline{R} (A)} (x) \underline{\lor} g_{\underline{R} (B)} (x) = g_{\underline{R} (A) ⋒ \underline{R} (B)} (x)$ . Hence, $\underline{R} (A ⋒ B) = \underline{R} (A) ⋒ \underline{R} (B)$ . And it’s not difficult to prove that $\bar{R} (A ⋓ B) = \bar{R} (A) ⋓ \bar{R} (B) .$

Theorem 5. Let U, V be two reference sets. $R_{1}, R_{2}$ are two IVDHF relations from U to V, respectively. $R_{1} ⊑ R_{2}$ , $\forall A \in IVDHF (V)$ , the following properties are true:

$\underline{R_{1}} (A) ⊒ \underline{R_{2}} (A)$ .

$\bar{R_{1}} (A) ⊑ \bar{R_{2}} (A) .$

Proof. By Definitions 8 and 11, we can easily obtain that, for any x ∈ U, there exist $\begin{matrix} h_{\underline{R_{i}} (A)}^{l -} (x) \\ = min {max {g_{R_{i}}^{l -} (x, y), h_{A}^{l -} (y)} | y \in V}, \\ h_{\underline{R_{i}} (A)}^{l +} (x) \\ = min {max {g_{R_{i}}^{l +} (x, y), h_{A}^{l +} (y)} | y \in V}, \\ h_{\underline{R_{i}} (A)}^{u -} (x) \\ = min {max {g_{R_{i}}^{u -} (x, y), h_{A}^{u -} (y)} | y \in V}, \\ h_{\underline{R_{i}} (A)}^{u +} (x) \\ = min {max {g_{R_{i}}^{u +} (x, y), h_{A}^{u +} (y)} | y \in V} . \\ g_{\underline{R_{i}} (A)}^{l -} (x) \\ = max {min {h_{R_{i}}^{l -} (x, y), g_{A}^{l -} (y)} | y \in V}, \\ g_{\underline{R_{i}} (A)}^{l +} (x) \\ = max {min {h_{R_{i}}^{l +} (x, y), g_{A}^{l +} (y)} | y \in V}, \\ g_{\underline{R_{i}} (A)}^{u -} (x) \\ = max {min {h_{R_{i}}^{u -} (x, y), g_{A}^{u -} (y)} | y \in V}, \\ g_{\underline{R_{i}} (A)}^{u +} (x) \\ = max {min {h_{R_{i}}^{u +} (x, y), g_{A}^{u +} (y)} | y \in V} . \end{matrix}$

Since $R_{1} ⊑ R_{2}$ , by Definition 8, we can obtain that $h_{R_{1}}^{l -} (x, y) \leq h_{R_{2}}^{l -} (x, y), h_{R_{1}}^{l +} (x, y) \leq h_{R_{2}}^{l +} (x, y), h_{R_{1}}^{u -} (x, y) \leq h_{R_{2}}^{u -} (x, y), h_{R_{1}}^{u +} (x, y) \leq h_{R_{2}}^{u +} (x, y)$ and $g_{R_{1}}^{l -} (x, y) \geq g_{R_{2}}^{l -} (x, y), g_{R_{1}}^{l +} (x, y) \geq g_{R_{2}}^{l +} (x, y), g_{R_{1}}^{u -} (x, y) \geq g_{R_{2}}^{u -} (x, y), g_{R_{1}}^{u +} (x, y) \geq g_{R_{2}}^{u +} (x, y) .$ Therefore, we can deduce that $h_{\underline{R_{1}} (A)}^{l -} (x) \geq h_{\underline{R_{2}} (A)}^{l -} (x), h_{\underline{R_{1}} (A)}^{l +} (x) \geq h_{\underline{R_{2}} (A)}^{l +} (x), h_{\underline{R_{1}} (A)}^{u -} (x) \geq h_{\underline{R_{2}} (A)}^{u -} (x), h_{\underline{R_{1}} (A)}^{u +} (x) \geq h_{\underline{R_{2}} (A)}^{u +} (x),$ and $g_{\underline{R_{1}} (A)}^{l -} (x) \leq g_{\underline{R_{2}} (A)}^{l -} (x), g_{\underline{R_{1}} (A)}^{l +} (x) \leq g_{\underline{R_{2}} (A)}^{l +} (x), g_{\underline{R_{1}} (A)}^{u -} (x) \leq g_{\underline{R_{2}} (A)}^{u -} (x), g_{\underline{R_{1}} (A)}^{u +} (x) \leq g_{\underline{R_{2}} (A)}^{u +} (x)$ , which means that $h_{\underline{R_{1}} (A)} ≽ h_{\underline{R_{2}} (A)}$ and $g_{\underline{R_{1}} (A)} ≼ g_{\underline{R_{2}} (A)}$ , namely, $\underline{R_{1}} (A) ⊒ \underline{R_{2}} (A)$ . Similarly, it’s not difficult to prove that $\bar{R_{1}} (A) ⊑ \bar{R_{2}} (A) .$

3.3 Associations between IVDHF rough sets and IVDHF relations

From the previous subsection, we know that the upper and lower IVDHF rough approximations are based on an IVDHF relation. Furthermore, some IVDHF rough approximations can reflect special properties of an IVDHF relation, in this subsection, we will discuss the associations between them.

Theorem 6. Let U be a reference set. $R$ is an IVDHF relation on U, M ⊆ U, then for any x ∈ U and (x, y) ∈ (U × U), the following properties hold.

$h_{\bar{R} ({[1, 1]}_{y})} (x) = h_{R} (x, y),$

$g_{\bar{R} ({[1, 1]}_{y})} (x) = g_{R} (x, y) .$

$h_{\underline{R} ({[1, 1]}_{y})} (x) = {\bar{\land}}_{z \neq y} g_{R} (x, z),$

$g_{\underline{R} ({[1, 1]}_{y})} (x) = {\underline{\lor}}_{z \neq y} h_{R} (x, z) .$

$h_{\bar{R} ({[1, 1]}_{U - y})} (x) = {\underline{\lor}}_{z \neq y} h_{R} (x, z),$

$g_{\bar{R} ({[1, 1]}_{U - y})} (x) = {\bar{\land}}_{z \neq y} g_{R} (x, z)$

$h_{\underline{R} ({[1, 1]}_{U - y})} (x) = g_{R} (x, y),$

$g_{\underline{R} ({[1, 1]}_{U - y})} (x) = h_{R} (x, y) .$

Proof. (1) By Definition 11, we can obtain that $h_{\bar{R} ({[1, 1]}_{y})} (x) = {\underline{\lor}}_{z \in U} {h_{R} (x, z) \bar{\land} h_{{[1, 1]}_{y}} (z)} = ({\underline{\lor}}_{z \neq y} {h_{R} (x, z) \bar{\land} {[0, 0]}}) \underline{\lor} (h_{R} (x, y) \bar{\land} {[1, 1]}) = {[0, 0]} \underline{\lor} h_{R} (x, y) = h_{R} (x, y)$ and $g_{\bar{R} ({[1, 1]}_{y})} (x) = {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} g_{{[1, 1]}_{y}} (z)} = ({\bar{\land}}_{z \neq y} {g_{R} (x, z) \underline{\lor} {[1, 1]}}) \bar{\land} (g_{R} (x, y) \underline{\lor} {[0, 0]}) = {[1, 1]} \bar{\land} g_{R} (x, y) = g_{R} (x, y)$ .

(2) Similarly, it is not difficult to prove that (2) holds.

(3) By Definition 11, we can obtain that $h_{\bar{R} ({[1, 1]}_{U - y})} (x) = {\underline{\lor}}_{z \in U} {h_{R} (x, z) \bar{\land} h_{{[1, 1]}_{U - y}} (z)} = ({\underline{\lor}}_{z \neq y} {h_{R} (x, z) \bar{\land} {[1, 1]}}) \underline{\lor} (h_{R} (x, y) \bar{\land} {[0, 0]}) = ({\underline{\lor}}_{z \neq y} h_{R} (x, z)) \underline{\lor} {[0, 0]} = {\underline{\lor}}_{z \neq y} h_{R} (x, z),$ and $g_{\bar{R} ({[1, 1]}_{U - y})} (x) = {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} g_{{[1, 1]}_{U - y}} (z)} = ({\bar{\land}}_{z \neq y} {g_{R} (x, z) \underline{\lor} {[0, 0]}}) \bar{\land} (g_{R} (x, y) \underline{\lor} {[1, 1]}) = ({\bar{\land}}_{z \neq y} g_{R} (x, z)) \bar{\land} {[1, 1]} = {\bar{\land}}_{z \neq y} g_{R} (x, z)$ .

(4) Similarly, it is not difficult to prove that (4) holds.

Theorem 7. Let U, V be two reference sets. $R$ is an IVDHF relation from U to V. $\underline{R}$ and $\bar{R}$ are the lower and upper IVDHF rough approximation operators respectively, then we have

$\begin{matrix} R is serial & \Leftrightarrow & (IVDHFL 0) \underline{R} (\emptyset) = \emptyset; \\ \Leftrightarrow & (IVDHFU 0) \bar{R} (V) = U; \\ \Leftrightarrow & (IVDHFLU 0) \underline{R} (A) ⊑ \bar{R} (A), \\ \forall A \in IVDHF (V); \end{matrix}$ (9)

Proof. Firstly, we are going to prove that $R$ is serial ⇔ (IVDHFL0).

⇒ : Suppose that $R$ is serial, then for any x ∈ U, there exists one z ∈ V satisfying $h_{R} (x, z) = {[1, 1]}$ and $g_{R} (x, z) = {[0, 0]}$ . By Definition 11, we have $h_{\underline{R} (\emptyset)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{\emptyset} (y)} = ({\bar{\land}}_{y \neq z} {g_{R} (x, y) \underline{\lor} {[0, 0]}}) \bar{\land} (g_{R} (x, z) \underline{\lor} {[0, 0]}) = ({\bar{\land}}_{y \neq z} g_{R} (x, y)) \bar{\land} {[0, 0]} = {[0, 0]}$ and $g_{\underline{R} (\emptyset)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} g_{\emptyset} (y)} = ({\underline{\lor}}_{y \neq z} {h_{R} (x, y) \bar{\land} {[1, 1]}}) \underline{\lor} (h_{R} (x, z) \bar{\land} {[1, 1]}) = ({\underline{\lor}}_{y \neq z} h_{R} (x, y)) \underline{\lor} {[1, 1]} = {[1, 1]}$ . Therefore, we obtain that $\underline{R} (\emptyset) = \emptyset .$

⇐ : Suppose that $\underline{R} (\emptyset) = \emptyset$ , then for any x ∈ U, there exist $h_{\underline{R} (\emptyset)} (x) = {[0, 0]}$ and $g_{\underline{R} (\emptyset)} (x) = {[1, 1]}$ . If $R$ is not serial, then for any y ∈ V, $h_{R} (x, y) = {[1, 1]}$ and $g_{R} (x, y) = {[0, 0]}$ don’t hold, namely, $h_{\underline{R} (\emptyset)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{\emptyset} (y)} = {\bar{\land}}_{y \in V} g_{R} (x, y) \neq {[0, 0]}, g_{\underline{R} (\emptyset)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} g_{\emptyset} (y)} = {\underline{\lor}}_{y \in V} h_{R} (x, y) \neq {[1, 1]}$ . The result contradicts the assumption, hence, $R$ is serial.

Similarly, it is easy to prove that $R$ is serial⇒ (IVDHFU0). Next, we are going to prove that $R$ is serial ⇔ (IVDHFLU0).

⇒ : Suppose that $R$ is serial, then for any x ∈ U, there exists one z ∈ V satisfying $h_{R} (x, z) = {[1, 1]}$ and $g_{R} (x, z) = {[0, 0]}$ . By Definition 11, $\forall A \in IVDHF (V)$ , we have $h_{\underline{R} (A)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} h_{A} (y)} = ({\bar{\land}}_{y \neq z} {g_{R} (x, y) \underline{\lor} h_{A} (y)}) \bar{} \land (g_{R} (x, z) \underline{\lor} h_{A} (z)) = ({\bar{\land}}_{y \neq z} {g_{R} (x, y) \underline{\lor} h_{A} (y)}) \bar{\land} h_{A} (z)$ and $h_{\bar{R} (A)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} h_{A} (y)} = ({\underline{\lor}}_{y \neq z} {h_{R} (x, y) \bar{\land} h_{A} (y)}) \underline{\lor} (h_{R} (x, z) \bar{\land} h_{A} (z)) = ({\underline{\lor}}_{y \neq z} {h_{R} (x, y) \bar{\land} h_{A} (y)}) \underline{\lor} h_{A} (z)$ . By Corollary 1, we can obtain that $h_{\underline{R} (A)} (x) ≼ h_{A} (z) ≼ h_{\bar{R} (A)} (x)$ . On the other hand, we have $g_{\underline{R} (A)} (x) = {\underline{\lor}}_{y \in V} {h_{R} (x, y) \bar{\land} g_{A} (y)} = ({\underline{\lor}}_{y \neq z} {h_{R} (x, y) \bar{\land} g_{A} (y)}) \underline{\lor} (h_{R} (x, z) \bar{\land} g_{A} (z)) = ({\underline{\lor}}_{y \neq z} {h_{R} (x, y) \bar{\land} g_{A} (y)}) \underline{\lor} g_{A} (z)$ and $g_{\bar{R} (A)} (x) = {\bar{\land}}_{y \in V} {g_{R} (x, y) \underline{\lor} g_{A} (y)} = ({\bar{\land}}_{y \neq z} {g_{R} (x, y) \underline{\lor} g_{A} (y)}) \bar{\land} (g_{R} (x, z) \underline{\lor} g_{A} (z)) = ({\bar{\land}}_{y \neq z} {g_{R} (x, y) \underline{\lor} g_{A} (y)}) \bar{\land} g_{A} (z)$ . By Corollary 1, we can obtain that $g_{\underline{R} (A)} (x) ≽ g_{A} (z) ≽ g_{\bar{R} (A)} (x)$ . From the above discussions, we can conclude that $\underline{R} (A) ⊑ \bar{R} (A)$ .

⇐ : Suppose that for any $A \in IVDHF (V)$ , $\underline{R} (A) ⊑ \bar{R} (A)$ . Hence, there are $h_{\underline{R} (A)} (x) ≼ h_{\bar{R} (A)} (x)$ and $g_{\underline{R} (A)} (x) ≽ g_{\bar{R} (A)} (x)$ . It follows that $h_{\underline{R} (\emptyset)} (x) ≼ h_{\bar{R} (\emptyset)} (x)$ and $g_{\underline{R} (\emptyset)} (x) ≽ g_{\bar{R} (\emptyset)} (x)$ . Since $h_{\bar{R} (\emptyset)} (x) = {[0, 0]}$ and $g_{\bar{R} (\emptyset)} (x) = {[1, 1]}$ , by Definition 8, we can deduce that $h_{\underline{R} (\emptyset)} (x) = {[0, 0]}$ and $g_{\underline{R} (\emptyset)} (x) = {[1, 1]}$ . That is to say, $h_{\underline{R} (\emptyset)} (x) = {\bar{\land}}_{y \in V} g_{R} (x, y) = {[0, 0]}$ and $g_{\underline{R} (\emptyset)} (x) = \underline{\lor} ¯_{y \in V} h_{R} (x, y) = {[1, 1]}$ . Therefore, for any x ∈ U, there is a y ∈ V such that $h_{R} (x, y) = {[1, 1]}$ and $g_{R} (x, y) = {[0, 0]}$ , which means that $R$ is serial.

Theorem 8. Let U be a reference set. $R$ is an IVDHF relation over U. $\underline{R}$ and $\bar{R}$ are the lower and upper IVDHF rough approximation operators respectively, then we have

$R$ is reflexive $\Leftrightarrow (IVDHFLR) \underline{R} (A) ⊑ A, \forall A \in IVDHF (U); \Leftrightarrow (IVDHFUR) A ⊑ \bar{R} (A), \forall A \in IVDHF (U);$

$R$ is symmetric $\Leftrightarrow (IVDHFLS) h_{\underline{R} ({[1, 1]}_{U - y})} (x) = h_{\underline{R} ({[1, 1]}_{U - x})} (y), g_{\underline{R} ({[1, 1]}_{U - y})} (x) = g_{\underline{R} ({[1, 1]}_{U - x})} (y) . \Leftrightarrow (IVDHFUS) h_{\bar{R} ({[1, 1]}_{y})} (x) = h_{\bar{R} ({[1, 1]}_{x})} (y), g_{\bar{R} ({[1, 1]}_{y})} (x) = g_{\bar{R} ({[1, 1]}_{x})} (y) .$

$R$ is transitive $\Leftrightarrow (IVDHFLT) \underline{R} (A) ⊑ \underline{R} (\underline{R} (A)), \forall A \in IVDHF (U); \Leftrightarrow (IVDHFUT) \bar{R} (\bar{R} (A)) ⊑ \bar{R} (A), \forall A \in IVDHF (U);$

Proof. (1) We are going to prove that $R$ is reflexive ⇔ (IVDHFLR) first.

⇒ : Suppose that $R$ is reflexive, then for any x ∈ U, there are $h_{R} (x, x) = {[1, 1]}$ and $g_{R} (x, x) = {[0, 0]}$ , hence, we have $h_{\underline{R} (A)} (x) = {\bar{\land}}_{y \in U} {g_{R} (x, y) \underline{\lor} h_{A} (y)} = ({\bar{\land}}_{y \neq x} {g_{R} (x, y) \underline{\lor} h_{A} (y)}) \bar{\land} (g_{R} (x, x) \underline{\lor} h_{A} (x)) = ({\bar{\land}}_{y \neq x} {g_{R} (x, y) \underline{\lor} h_{A} (y)}) \bar{\land} h_{A} (x)$ and $g_{\underline{R} (A)} (x) = {\underline{\lor}}_{y \in U} {h_{R} (x, y) \bar{\land} g_{A} (y)} = ({\underline{\lor}}_{y \neq x} {h_{R} (x, y) \bar{\land} g_{A} (y)}) \underline{\lor} (h_{R} (x, x) \bar{\land} g_{A} (x)) = ({\underline{\lor}}_{y \neq x} {h_{R} (x, y) \bar{\land} g_{A} (y)}) \underline{\lor} g_{A} (x)$ . By Corollary 1, we can obtain that $h_{\underline{R} (A)} (x) ≼ h_{A} (x)$ and $g_{\underline{R} (A)} (x) ≽ g_{A} (x)$ , namely, $\underline{R} (A) ⊑ A$ .

⇐ : Conversely, if $\underline{R} (A) ⊑ A$ holds for any $A \in IVDHF (U)$ , then $h_{\underline{R} (A)} (x) ≼ h_{A} (x)$ and $g_{\underline{R} (A)} (x) ≽ g_{A} (x)$ hold. It follows that $h_{\underline{R} ({[1, 1]}_{U - x})} (x) ≼ h_{{[1, 1]}_{U - x}} (x), g_{\underline{R} ({[1, 1]}_{U - x})} (x) ≽ g_{[1, 1]_{U - x}} (x)$ . By Theorem 6, we obtain that $g_{R} (x, x) ≼ {[0, 0]}, h_{R} (x, x) ≽ {[1, 1]}$ . By Definition 8, we can deduce that $g_{R} (x, x) = {[0, 0]}$ and $h_{R} (x, x) = {[1, 1]}$ . Hence, $R$ is reflexive.

Similarly, it is easy to prove that $R$ is reflexive⇔ (IVDHFUR) .

(2) We can get (2) directly by Theorem 6.

(3) Next, we first prove that $R$ is transitive ⇔ (IVDHFLT).

⇒ : Suppose that $R$ is transitive, then for any x, y, z ∈ U, there are $h_{R} (x, y) \bar{\land} h_{R} (y, z) ≼ h_{R} (x, z)$ and $g_{R} (x, y) \underline{\lor} g_{R} (y, z) ≽ g_{R} (x, z)$ . By Corollary 1, we can obtain that ${\underline{\lor}}_{y \in U} (h_{R} (x, y) \bar{\land} h_{R} (y, z)) ≼ h_{R} (x, z)$ and ${\bar{\land}}_{y \in U} (g_{R} (x, y) \underline{\lor} g_{R} (y, z)) ≽ g_{R} (x, z)$ . Then, $h_{\underline{R} (\underline{R} (A))} (x) = {\bar{\land}}_{y \in U} {g_{R} (x, y) \underline{\lor} h_{\underline{R} (A)} (y)} = {\bar{\land}}_{y \in U} {g_{R} (x, y) \underline{\lor} ({\bar{\land}}_{z \in U} {g_{R} (y, z) \underline{\lor} h_{A} (z)})} = {\bar{\land}}_{y \in U} {\bar{\land}}_{z \in U} {g_{R} (x, y) \underline{\lor} g_{R} (y, z) \underline{\lor} h_{A} (z)} = {\bar{\land}}_{z \in U} {{\bar{\land}}_{y \in U} (g_{R} (x, y) \underline{\lor} g_{R} (y, z)) \underline{\lor} h_{A} (z)} ≽ {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} h_{A} (z)}$ and $g_{\underline{R} (\underline{R} (A))} (x) = {\underline{\lor}}_{y \in U} {h_{R} (x, y) \bar{\land} g_{\underline{R} (A)} (y)} = {\underline{\lor}}_{y \in U} {h_{R} (x, y) \bar{\land} ({\underline{\lor}}_{z \in U} {h_{R} (y, z) \bar{\land} g_{A} (z)})} = {\underline{\lor}}_{y \in U} {\underline{\lor}}_{z \in U} {h_{R} (x, y) \bar{\land} h_{R} (y, z) \bar{\land} g_{A} (z)} = {\underline{\lor}}_{z \in U} {{\underline{\lor}}_{y \in U} (h_{R} (x, y) \bar{\land} h_{R} (y, z)) \bar{\land} g_{A} (z)} ≼ {\underline{\lor}}_{z \in U} {h_{R} (x, z) \bar{\land} g_{A} (z)} .$ Therefore, $\underline{R} (A) ⊑ \underline{R} (\underline{R} (A))$ .

⇐ : Conversely, if $\underline{R} (A) ⊑ \underline{R} (\underline{R} (A))$ holds for any $A \in IVDHF (U)$ , then ∀x, y ∈ U. By Definition 11, we have $h_{\underline{R} ({[1, 1]}_{U - y})} (x) = {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} h_{{[1, 1]}_{U - y}} (z)} = ({\bar{\land}}_{z \neq y} {g_{R} (x, z) \underline{\lor} {[1, 1]}}) \bar{\land} (g_{R} (x, y) \underline{\lor} {[0, 0]}) = g_{R} (x, y)$ and $h_{\underline{R} (\underline{R} ({[1, 1]}_{U - y}))} (x) = {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} h_{\underline{R} ([1, 1]_{U - y})} (z)} = {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} g_{R} (z, y)}$ . Similarly, we can obtain that $g_{\underline{R} ({[1, 1]}_{U - y})} (x) = h_{R} (x, y)$ and $g_{\underline{R} (\underline{R} ({[1, 1]}_{U - y}))} (x) = {\underline{\lor}}_{z \in U} {h_{R} (x, z) \bar{\land} h_{R} (z, y)}$ . By Corollary 1, we can conclude that $g_{R} (x, y) ≼ {\bar{\land}}_{z \in U} {g_{R} (x, z) \underline{\lor} g_{R} (z, y)} ≼ g_{R} (x, z) \underline{\lor} g_{R} (z, y)$ and $h_{R} (x, y) ≽ {\underline{\lor}}_{z \in U} {h_{R} (x, z) \bar{\land} h_{R} (z, y)} ≽ h_{R} (x, z) \bar{\land} h_{R} (z, y)$ , which means that $R$ is transitive.

It is similar to prove that $R$ is transitive ⇔ (IVDHFUT).

4 A decision approach based on interval-valued dual hesitant fuzzy rough set over two universes

In this section, a general DM approach based on IVDHF rough set over two universes model is proposed. Firstly, we introduce a brief background description for DM problem in practice.

Suppose that U ={ x₁, x₂, …, x_m }, representing the set of alternatives, and V ={ y₁, y₂, …, y_n }, representing the set of attributes, are two universes of discourse respectively. $R = {(x_{i}, y_{j}), h_{R} (x_{i}, y_{j}), g_{R} (x_{i}, y_{j}) | (x_{i}, y_{j}) \in (U \times V)}$ is an IVDHF relation from U to V representing the relational degrees of alternatives and characteristics. ∀ (x_i, y_j), $h_{R} (x_{i}, y_{j})$ is a set of possible interval-valued membership degrees of relations between x_i and y_j. $g_{R} (x_{i}, y_{j})$ is a set of possible interval-valued non-membership degrees of relations between x_i and y_j. Suppose that the evaluation set $A = {y_{j}, h_{A} (y_{j}), g_{A} (y_{j}) | y_{j} \in V}$ is also denoted by an IVDHF rough set, which reflects the attributive character of $A$ . Then, the problem is how to properly evaluate $A$ with one of the alternatives in U. To handle this problem, we present an approach based on IVDHF rough set over two universes as follows, which mainly includes four steps.

Step 1. First, according to Definition 11, calculate $\underline{R} (A)$ and $\bar{R} (A)$ .

Step 2. According to equations (5) and (6), compute $sf (d_{\underline{R} (A)} (x)), pf (d_{\underline{R} (A)} (x)), sf (d_{\bar{R} (A)} (x))$ and $pf (d_{\bar{R} (A)} (x))$ . Further, according to Theorem 2, we can obtain that $\forall x \in U, sf (d_{\underline{R} (A) ⊞ \bar{R} (A)} (x)) = sf (d_{\underline{R} (A)} (x) d_{\bar{R} (A)} (x)) = s (h_{\underline{R} (A)} (x)) s (h_{\bar{R} (A)} (x)) - s (g_{\underline{R} (A)} (x)) s (g_{\bar{R} (A)} (x))$ and $pf (d_{\underline{R} (A) ⊞ \bar{R} (A)} (x)) = pf (d_{\underline{R} (A)} (x) d_{\bar{R} (A)} (x)) = s (h_{\underline{R} (A)} (x)) s (h_{\bar{R} (A)} (x)) + s (g_{\underline{R} (A)} (x)) s (g_{\bar{R} (A)} (x))$ .

Step 3. Based on the DM strategy presented in [51], the following three indicators are computed by the score functions and accuracy functions obtained in Step2: $T_{1} (A) = {i | \max_{x_{i} \in U} {\underline{R} (A) (x_{i})}}$ (10) $T_{2} (A) = {j | \max_{x_{j} \in U} {\bar{R} (A) (x_{j})}}$ (11) $T_{3} (A) = {k | \max_{x_{k} \in U} {\underline{R} (A) ⊞ \bar{R} (A) (x_{k})}}$ (12)

It is noted that Equations (9) to (11) are based on the risk DM strategy in classical operational research. Since the lower approximation reflects the degree of x_i necessarily matches $A$ , $T_{1} (A)$ is the DM way with the minimum uncertainties. The upper approximation reflects the degree of x_i possibly matches $A$ , which means that $T_{2} (A)$ is the DM way with maximum uncertainties. More specifically, $T_{1} (A)$ represents the conservative DM strategy which avoids risk. $T_{2} (A)$ represents the positive DM strategy which seek risk and $T_{3} (A)$ is the weighted DM strategy of $T_{1} (A)$ and $T_{2} (A)$ . By making full use of the three indicators, it’s conducive for experts to reach a more comprehensive DM result. Therefore, we introduce the following decision rules.

Step 4. Calculate $T_{1} (A) ⋂ T_{2} (A) ⋂ T_{3} (A)$ and $T_{1} (A) ⋂ T_{2} (A)$ .

If $T_{1} (A) ⋂ T_{2} (A) ⋂ T_{3} (A) \neq \emptyset$ , then $A$ matches $x_{l} (l \in T_{1} (A) ⋂ T_{2} (A) ⋂ T_{3} (A))$ .

Otherwise, if $T_{1} (A) ⋃ T_{2} (A) \neq \emptyset$ , then $A$ matches $x_{p} (p \in T_{1} (A) ⋂ T_{2} (A))$ . If $T_{1} (A) ⋂ T_{2} (A) = \emptyset$ , then $A$ matches $x_{k} (k \in T_{3} (A))$ .

The situation (1) in step 4 reflects that the DM results of all indicators are largely consistent, hence $x_{l} (l \in T_{1} (A) ⋂ T_{2} (A) ⋂ T_{3} (A))$ is the optimal DM result. As for situation (2), there exist some difference between the results of three indicators. In such case, if the results of conservative DM strategy ( $T_{1} (A)$ ) and positive strategy ( $T_{2} (A)$ ) are largely consistent, $x_{p} (p \in T_{1} (A) ⋂ T_{2} (A))$ is the optimal DM result, otherwise, the result of weighted DM strategy ( $T_{3} (A)$ ) is optimal. It can be seen from the above analysis that the proposed DM approach is multifaceted and is more comprehensive to obtain a reliable DM result.

5 An illustrative example

In this section, we will first utilize a medical diagnosis problem adapted from Zhang [42] and Sun [51] to illustrate the main steps of the decision method presented in section 4.

Example 1. [42, 51]. Let U ={ x₁, x₂, x₃, x₄, x₅ } be a set of diseases, representing fever, malaria, typhoid, stomach problem and chest problem respectively, and V ={ y₁, y₂, y₃, y₄, y₅ } be a set of symptoms, representing temperature, headache, cough, stomach pain and chest pain respectively. In reality, a concrete disease usually shows several different symptoms at the same time. Since the relationships between diseases and symptoms are complicated and uncertain which means that crisp numbers are limited in describing them, we use both interval-valued membership degrees and interval-valued non-membership degrees to depict the relation. Suppose that $R = {(x, y), h_{R} (x, y), g_{R} (x, y) | (x, y) \in (U \times V)}$ is an IVDHF relation denoting the relationships between diseases and symptoms, which is presented in Table 1. $A_{i} = {A_{i}, h_{A_{i}} (y_{j}), g_{A_{i}} (y_{j}) | y_{j} \in V, i = 1, 2, 3}$ are three IVDHF sets, which are shown in Table 2, representing three patients with the above symptoms.

Table 1
Symptoms characteristic for the considered diagnoses

$R$ y₁ y₂ y₃ y₄ y₅

x ₁ {{, [0.5, 0.6]}, {[0.1, 0.3], [0.2, 0.4]}} {{[0.3, 0.4]}, {[0.2, 0.3]}} {{[0.2, 0.3], [0.1, 0.4]}, {[0.3, 0.5], [0.4, 0.6]}} {{[0.2, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}} {{[0.1, 0.2], [0.3, 0.4]}, {[0.3, 0.5], [0.4, 0.6]}}

x ₂ {{[0.6, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}} {{[0.3, 0.4], [0.5, 0.6]}, {[0.1, 0.3], [0.2, 0.4]}} {{[0.2, 0.3], [0.5, 0.6]}, {[0.2, 0.3]}} {{[0.5, 0.6]}, {[0.2, 0.4], [0.3, 0.4]}} {{[0.3, 0.4]}, {[0.3, 0.4], [0.3, 0.5]}}

x ₃ {{[0.1, 0.3], [0.2, 0.3]}, {[0.4, 0.5], [0.5, 0.7]}} {{[0.6, 0.7]}, {[0.1, 0.3]}} {{[0.3, 0.4], [0.5, 0.6]}, {[0.2, 0.4]}} {{[0.3, 0.4]}, {[0.3, 0.4]}} {{[0.2, 0.3]}, {[0.4, 0.6], [0.5, 0.7]}}

x ₄ {{[0.2, 0.4]}, {[0.2, 0.4], [0.3, 0.5]}} {{[0.2, 0.3], [0.4, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}} {{[0.5, 0.6]}, {[0.2, 0.3], [0.1, 0.4]}} {{[0.7, 0.8]}, {[0.1, 0.2]}} {{[0.1, 0.2], [0.3, 0.4]}, {[0.2, 0.4], [0.3, 0.4]}}

x ₅ {{[0.1, 0.2], [0.2, 0.3]}, {[0.4, 0.7], [0.5, 0.6]}} {{[0.1, 0.2], [0.3, 0.3]}, {[0.3, 0.6], [0.4, 0.5]}} {{[0.2, 0.3]}, {[0.4, 0.6]}} {{[0.4, 0.5]}, {[0.3, 0.5], [0.4, 0.5]}} {{[0.7, 0.8]}, {[0.1, 0.2]}}

$R$	y₁	y₂	y₃	y₄	y₅
x ₁	{{, [0.5, 0.6]}, {[0.1, 0.3], [0.2, 0.4]}}	{{[0.3, 0.4]}, {[0.2, 0.3]}}	{{[0.2, 0.3], [0.1, 0.4]}, {[0.3, 0.5], [0.4, 0.6]}}	{{[0.2, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}}	{{[0.1, 0.2], [0.3, 0.4]}, {[0.3, 0.5], [0.4, 0.6]}}
x ₂	{{[0.6, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}}	{{[0.3, 0.4], [0.5, 0.6]}, {[0.1, 0.3], [0.2, 0.4]}}	{{[0.2, 0.3], [0.5, 0.6]}, {[0.2, 0.3]}}	{{[0.5, 0.6]}, {[0.2, 0.4], [0.3, 0.4]}}	{{[0.3, 0.4]}, {[0.3, 0.4], [0.3, 0.5]}}
x ₃	{{[0.1, 0.3], [0.2, 0.3]}, {[0.4, 0.5], [0.5, 0.7]}}	{{[0.6, 0.7]}, {[0.1, 0.3]}}	{{[0.3, 0.4], [0.5, 0.6]}, {[0.2, 0.4]}}	{{[0.3, 0.4]}, {[0.3, 0.4]}}	{{[0.2, 0.3]}, {[0.4, 0.6], [0.5, 0.7]}}
x ₄	{{[0.2, 0.4]}, {[0.2, 0.4], [0.3, 0.5]}}	{{[0.2, 0.3], [0.4, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}}	{{[0.5, 0.6]}, {[0.2, 0.3], [0.1, 0.4]}}	{{[0.7, 0.8]}, {[0.1, 0.2]}}	{{[0.1, 0.2], [0.3, 0.4]}, {[0.2, 0.4], [0.3, 0.4]}}
x ₅	{{[0.1, 0.2], [0.2, 0.3]}, {[0.4, 0.7], [0.5, 0.6]}}	{{[0.1, 0.2], [0.3, 0.3]}, {[0.3, 0.6], [0.4, 0.5]}}	{{[0.2, 0.3]}, {[0.4, 0.6]}}	{{[0.4, 0.5]}, {[0.3, 0.5], [0.4, 0.5]}}	{{[0.7, 0.8]}, {[0.1, 0.2]}}

Table 2

Symptoms characteristic for the considered patients

	y ₁	y ₂	y ₃	y ₄	y ₅
$A_{1}$	{{[0.2, 0.4]}, {[0.3, 0.4]}}	{{[0.5, 0.6], [0.2, 0.3]}}	{{[0.4, 0.6], [0.2, 0.3]}}	{{[0.7, 0.8], [0.1, 0.2]}}	{{[0.5, 0.7], [0.1, 0.2]}}
$A_{2}$	{{[0.5, 0.7]}, {[0.1, 0.2]}}	{{[0.4, 0.6], [0.2, 0.3]}}	{{[0.1, 0.3], [0.5, 0.6]}}	{{[0.4, 0.5], [0.3, 0.5]}}	{{[0.3, 0.6], [0.2, 0.3]}}
$A_{3}$	{{[0.3, 0.5]}, {[0.3, 0.4]}}	{{[0.3, 0.4], [0.3, 0.4]}}	{{[0.7, 0.8], [0.1, 0.2]}}	{{[0.2, 0.5], [0.2, 0.4]}}	{{[0.6, 0.8], [0.1, 0.2]}}

Next, we will utilize the DM approach in section 4 to judge which disease the patient is suffering.

Step 1. According to equations (7) and (8), calculate $\underline{R} (A_{i})$ and $\bar{R} (A_{i})$ , which are shown in Table 3.

Table 3

The lower and upper approximations of $A_{i}$

	x ₁	x ₂	x ₃	x ₄	x ₅
$\underline{R} (A_{1})$	{{[0.2, 0.4]}, {[0.3, 0.4]}}	{{[0.2, 0.4]}, {[0.3, 0.4]}}	{{[0.4, 0.5], [0.4, 0.6]}, {[0.2, 0.3]}}	{{[0.2, 0.4], [0.3, 0.5]}, {[0.2, 0.4]}}	{{[0.4, 0.6]}, {[0.2, 0.3]}}
$\bar{R} (A_{1})$	{{[0.3, 0.4]}, {[0.2, 0.3]}}	{{[0.5, 0.6]}, {[0.2, 0.3]}}	{{[0.5, 0.6]}, {[0.2, 0.3]}}	{{[0.7, 0.8]}, {[0.1, 0.2]}}	{{[0.5, 0.7]}, {[0.1, 0.2]}}
$\underline{R} (A_{2})$	{{[0.3, 0.5], [0.4, 0.5]}, {[0.2, 04]}}	{{[0.2, 0.3]}, {[0.3, 0.5], [0.5, 0.6]}}	{{[0.2, 0.4]}, {[0.3, 0.4], [0.5, 0.6]}}	{{[0.2, 0.3], [0.1, 0.4]}, {[0.5, 0.6]}}	{{[0.3, 0.5]}, {[0.3, 0.5]}}
$\bar{R} (A_{2})$	{{[0.4, 0.5], [0.5, 0.6]}, {[0.1, 0.3], [0.2, 0.3]}}	{{[0.5, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}}	{{[0.4, 0.6]}, {[0.2, 0.3]}}	{{[0.4, 0.5]}, {[0.2, 0.4], [0.3, 0.4]}}	{{[0.4, 0.6]}, {[0.2, 0.3]}}
$\underline{R} (A_{3})$	{{[0.3, 0.4]}, {[0.3, 0.4]}}	{{[0.2, 0.4], [0.3, 0.4]}, {[0.3, 0.4]}}	{{[0.3, 0.4]}, {[0.3, 0.4]}}	{{[0.2, 0.4], [0.2, 0.5]}, {[0.2, 0.4], [0.3, 0.4]}}	{{[0.3, 0.5], [0.4, 0.5]}, {[0.2, 0.4], [0.3, 0.4]}}
$\bar{R} (A_{3})$	{{[0.3, 0.5]}, {[0.3, 0.4]}}	{{[0.3, 0.5], [0.5, 0.6]}, {[0.2, 0.3]}}	{{[0.3, 0.4], [0.5, 0.6]}, {[0.2, 0.4]}}	{{[0.5, 0.6]}, {[0.1, 0.4], [0.2, 0.3]}}	{{[0.6, 0.8]}, {[0.1, 0.2]}}

Step 2. Next, we calculate the score function and accuracy function of $\underline{R} (A_{i})$ and $\bar{R} (A_{i})$ respectively. Then, by Theorem 2, we can obtain the score function and the accuracy function of $\underline{R} (A_{i}) ⊞ \bar{R} (A_{i})$ . The results are presented in Tables 4 and 5.

Table 4

Results of score functions

sf	x ₁	x ₂	x ₃	x ₄	x ₅
$\underline{R} (A_{1})$	[– 0.1, 0]	[– 0.1, 0]	[0.2, 0.25]	[0.05, 0.05]	[0.2, 0.3]
$\underline{R} (A_{2})$	[0.1, 0.15]	[– 0.2, – 0.15]	[– 0.2, – 0.1]	[– 0.35, – 0.25]	[0, 0]
$\underline{R} (A_{3})$	[0, 0]	[– 0.05, 0]	[0, 0]	[– 0.05, 0.05]	[0.1, 0.1]
$\bar{R} (A_{1})$	[0.1, 0.1]	[0.3, 0.3]	[0.3, 0.3]	[0.6, 0.6]	[0.4, 0.5]
$\bar{R} (A_{2})$	[0.25, 0.3]	[0.35, 0.45]	[0.2, 0.3]	[0.1, 0.15]	[0.2, 0.3]
$\bar{R} (A_{3})$	[0, 0.1]	[0.2, 0.25]	[0.1, 0.2]	[0.25.0.35]	[0.5, 0.6]
$\underline{R} (A_{1}) ⊞ \bar{R} (A_{1})$	[0.38, 0.52]	[0.54, 0.64]	[0.66, 0.73]	[0.76, 0.81]	[0.68, 0.82]
$\underline{R} (A_{2}) ⊞ \bar{R} (A_{2})$	[0.61, 0.65]	[0.54, 0.65]	[0.44, 0.61]	[0.36, 0.43]	[0.52, 0.65]
$\underline{R} (A_{3}) ⊞ \bar{R} (A_{3})$	[0.42, 0.54]	[0.49, 0.61]	[0.52, 0.54]	[0.56, 0.64]	[0.72, 0.82]

Table 5

Results of accuracy functions

pf	x ₁	x ₂	x ₃	x ₄	x ₅
$\underline{R} (A_{1})$	[0.5, 0.8]	[0.5, 0.8]	[0.6, 0.85]	[0.45, 0.85]	[0.6, 0.9]
$\underline{R} (A_{2})$	[0.55, 0.9]	[0.6, 0.85]	[0.6, 0.9]	[0.65, 0.95]	[0.6, 1]
$\underline{R} (A_{3})$	[0.6, 0.8]	[0.55, 0.8]	[0.6, 0.8]	[0.45, 0.85]	[0.6, 0.9]
$\bar{R} (A_{1})$	[0.5, 0.7]	[0.7, 0.9]	[0.7, 0.9]	[0.8, 1]	[0.6, 0.9]
$\bar{R} (A_{2})$	[0.6, 0.85]	[0.65, 0.95]	[0.6, 0.9]	[0.65, 0.9]	[0.6, 0.9]
$\bar{R} (A_{3})$	[0.6, 0.9]	[0.6, 0.85]	[0.6, 0.9]	[0.65.0.95]	[0.7, 1]
$\underline{R} (A_{1}) ⊞ \bar{R} (A_{1})$	[0.5, 0.76]	[0.66, 0.88]	[0.74, 0.91]	[0.795, 0.97]	[0.72, 0.94]
$\underline{R} (A_{2}) ⊞ \bar{R} (A_{2})$	[0.6725, 0.895]	[0.66, 0.9275]	[0.6, 0.91]	[0.615, 0.915]	[0.64, 0.95]
$\underline{R} (A_{3}) ⊞ \bar{R} (A_{3})$	[0.6, 0.86]	[0.61, 0.85]	[0.64, 0.86]	[0.6375, 0.92]	[0.765, 0.98]

Step 3. Since the score functions and the accuracy functions have been obtained, then we can compute the three indicators as follows: $\begin{matrix} T_{1} (A_{1}) = {i | \max_{x_{i} \in U} {\underline{R} (A_{1}) (x_{i})}} = 5 . \\ T_{2} (A_{1}) = {j | \max_{x_{j} \in U} {\bar{R} (A_{1}) (x_{j})}} = 4 . \\ T_{3} (A_{1}) = {k | \max_{x_{k} \in U} {\underline{R} (A_{1}) ⊞ \bar{R} (A_{1}) (x_{k})}} \\ = 4 . \end{matrix}$

Similarly, we can obtain that $T_{1} (A_{2}) = 1$ , $T_{2} (A_{2}) = 2$ , $T_{3} (A_{2}) = 1$ and $T_{1} (A_{3}) = 5$ , $T_{2} (A_{3}) = 5$ , $T_{3} (A_{3}) = 5$ .

Step 4. Since $T_{1} (A_{1}) ⋂ T_{2} (A_{1}) ⋃ T_{3} (A_{1}) = \emptyset$ and $T_{1} (A_{1}) ⋂ T_{2} (A_{1}) = \emptyset$ , then we conclude that $A_{1}$ matches x₄, namely, $A_{1}$ has stomach problem. It is similar to conclude that $A_{2}$ has fever (x₁), and $A_{3}$ has chest problem (x₅).

In what follows, to illustrate the effectiveness of the DM approach proposed in this paper, a comparison method is provided next. Since distance measures are fundamentally important and widely used in DM problems [22 , 52–54], we utilize the distance measure of IVDHF sets based on [22 , 52–54] to conduct a comparative analysis.

It’s noted that from the standpoint of distance measures, both the diseases and patients should be described by IVDHF sets. By the IVDHF relation between diseases and symptoms, we can represent the disease as $X_{j} = {y_{litsc}, h_{R} (x_{j}, y_{litsc}), g_{R} (x_{j}, y_{litsc}) | y_{litsc} \in V}$ . In addition, the lengths of different IVDHFEs may be different and the interval values in IVDHFEs are usually out of order. According to [21], we can rearrange the interval values in an ascending order and repeat the largest interval value in the short-length IVDHFE until the lengths are identical. For an IVDHF set $A$ , we denote the ith smallest value in $h_{A} (x)$ and $g_{A} (x)$ by $h_{A}^{σ (i)} (x)$ and $g_{A}^{σ (i)} (x)$ respectively. Then, the distance measure of IVDHF sets is introduced as follows:

For any $A, B \in IVDHF (U)$ , the interval-valued dual hesitant normalized Euclidean distance is defined as follows: $d (A, B) = \sum_{i = 1}^{n} (\frac{1}{n} (\frac{1}{2 l_{x i}} (\sum_{j = 1}^{# h_{x i}} ({| h_{A}^{σ (j) L} (x_{i}) - h_{B}^{σ (j) L} (x_{i}) |}^{2} + {| h_{A}^{σ (j) U} (x_{i}) - h_{B}^{σ (j) U} (x_{i}) |}^{2}) + \sum_{k = 1}^{# g x_{i}} ({| g_{A}^{σ (j) L} (x_{i}) - g_{B}^{σ (j) L} (x_{i}) |}^{2} + {| g_{A}^{σ (j) U} (x_{i}) - g_{B}^{σ (j) U} (x_{i}) |}^{2}))))$ where λ > 0, l_{x
_i} = (# h_{x
_i}) + (# g_{x
_i}), # h_{x
_i} and # g_{x
_i} are the numbers of the interval values in h and g respectively.

In distance measures, the basic idea is that if $d (A, X_{j}) = \min_{1 \leq i \leq n} d (A, X_{i})$ , then $A$ matches with $X_{j}$ . Based on this principle, we can match the patients described by IVDHF sets with the corresponding diseases. The distance between $A_{i}$ and $X_{j}$ are calculated respectively. Table 6 shows the decision outcomes. From Table 6, it’s not hard to conclude that $A_{1}, A_{2}$ and $A_{3}$ match x₄ (stomach problem), x₁ (fever) and x₅ (chest problem), respectively.

Table 6

Decision results of distance measures

	x ₁	x ₂	x ₃	x ₄	x ₅
$A_{1}$	0.5467	0.4164	0.4697	0.2777	0.4948
$A_{2}$	0.3015	0.3028	0.5111	0.5484	0.4574
$A_{3}$	0.4979	0.4725	0.5410	0.4371	0.4327

Next, a practical example in steam turbine fault diagnosis is employed here to further justify the applicability of the proposed method.

Example 2. [44 , 57–58]. Let U = {x₁, x₂, …, x₁₀} be a set of failures, representing unbalance, pneumatic force couple, offset center, oil-membrane oscillation, radial impact friction of rotor, symbiosis loose fault, damage of antithrust bearing, surge, looseness of bearing block and non-uniform bearing stiffness respectively, and V = {y₁, y₂, …, y₉} be a set of different frequency ranges, representing the maximum amplitude ratio of vibration signal in different frequency ranges. In generator sets, the failures are closely related to the maximum amplitude ratio of vibration in different frequency ranges [57]. However, in practice, the amplitude ratio is usually uncertain and inaccuracy due to the measurement error and environmental disturbance. Thus, we use an IVDHF relation $R = {(x, y), h_{R} (x, y), g_{R} (x, y) | (x, y) \in (U \times V)}$ to describe the relationships between failures and failure features. $B = {B, h_{B} (y_{i}), g_{B} (y_{i}) | y_{i} \in V}$ is a failure sample with IVDHF information to be evaluated. The fault knowledge $R$ and evaluated sample $B$ derived from [44 , 57–58] are presented in Table 7.

Table 7

Fault knowledge of steam turbine system

$R$	y₁ (0.01 - 0.39f)	y₂ (0.4 - 0.49f)	y₃ (0.5f)	y₄ (0.51 - 0.99f)	y₅ (f)
x ₁	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.85, 0.88],[0.85, 0.93]}, {[0.04, 0.06],[0.05, 0.07]}}
x ₂	{{[0, 0]}, {[1, 1]}}	{{[0.27, 0.32],[0.28, 0.33]}, {[0.58, 0.61],[0.6, 0.62]}}	{{[0.08, 0.1],[0.09, 0.13]}, {[0.84, 0.86]}}	{{[0.54, 0.6],[0.56, 0.7]}, {[0.25, 0.28]}}	{{[0, 0]}, {[1, 1]}}
x ₃	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.3, 0.58],[0.4, 0.6]}, {[0.15, 0.38]}}
x ₄	{{[0.09, 0.1],[0.1, 0.12]}, {[0.83, 0.86]}}	{{[0.78, 0.8],[0.8, 0.82]}, {[0.13, 0.16],[0.15, 0.17]}}	{{[0, 0]}, {[1, 1]}}	{{[0.08, 0.1],[0.1, 0.11]}, {[0.85, 0.87]}}	{{[0, 0]}, {[1, 1]}}
x ₅	{{[0.09, 0.11],[0.11, 0.12]}, {[0.84, 0.85]}}	{{[0.09, 0.1],[0.1, 0.11]}, {[0.86, 0.87]}}	{{[0.08, 0.1],[0.1, 0.12]}, {[0.84, 0.86],[0.85, 0.87]}}	{{[0.09, 0.11],[0.11, 0.12]}, {[0.85, 0.87]}}	{{[0.18, 0.2],[0.19, 0.21]}, {[0.65, 0.68],[0.66, 0.67]}}
x ₆	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.18, 0.2],[0.2, 0.22]}, {[0.75, 0.77]}}
x ₇	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.08, 0.11],[0.1, 0.12]}, {[0.84, 0.86]}}	{{[0.86, 0.89],[0.88, 0.93]}, {[0.02, 0.05]}}	{{[0, 0]}, {[1, 1]}}
x ₈	{{[0, 0]}, {[1, 1]}}	{{[0.27, 0.3],[0.29, 0.32]}, {[0.58, 0.6],[0.59, 0.62]}}	{{[0.08, 0.1],[0.09, 0.12]}, {[0.82, 0.85]}}	{{[0.54, 0.6],[0.57, 0.62]}, {[0.27, 0.3],[0.28, 0.33]}}	{{[0, 0]}, {[1, 1]}}
x ₉	{[0.85, 0.9],[0.89, 0.93]}, {[0.01, 0.06]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
x ₁₀	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
$B$	{{[0.39, 0.4],[0.39, 0.41]}, {[0.32, 0.37]}}	{{[0.07, 0.09],[0.08, 0.1]}, {[0.85, 0.88]}}	{{[0, 0]}, {[1, 1]}}	{{[0.06, 0.07],[0.06, 0.08]}, {[0.88, 0.9]}}	{{[0, 0]}, {[1, 1]}}
$R$	y₆ (2f)	y₇ (3 -5f)	y₈ (oddtimesoff)	y₉ (>5f)
x ₁	{{[0.03, 0.05],[0.04, 0.06]}, {[0.91, 0.93]}}	{{[0.04, 0.06],[0.05, 0.07]}, {[0.89, 0.91]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
x ₂	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.07, 0.12],[0.08, 0.14]}, {[0.79, 0.83],[0.82, 0.85]}}
x ₃	{{[0.4, 0.6],[0.45, 0.62]}, {[0.22, 0.31],[0.2, 0.35]}}	{{[0.08, 0.12],[0.1, 0.13]}, {[0.82, 0.86]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
x ₄	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
x ₅	{{[0.08, 0.12],[0.1, 0.13]}, {[0.82, 0.85]}}	{{[0.08, 0.12],[0.1, 0.13]}, {[0.83, 0.85]}}	{{[0.08, 0.1],[0.1, 0.12]}, {[0.85, 0.87]}}	{{[0.08, 0.1],[0.1, 0.12]}, {[0.85, 0.86]}}
x ₆	{{[0.12, 0.15],[0.15, 0.17]}, {[0.78, 0.81],[0.8, 0.82]}}	{{[0.37, 0.4],[0.4, 0.45]}, {[0.46, 0.49],[0.45, 0.51]}}	{{[0, 0]}, {[1, 1]}}	{{[0.22, 0.25],[0.24, 0.28]}, {[0.6, 0.63],[0.62, 0.66]}}
x ₇	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
x ₈	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
x ₉	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.08, 0.1],[0.09, 0.12]}, {[0.74, 0.79]}}	{{[0, 0]}, {[1, 1]}}
x ₁₀	{{[0.77, 0.8],[0.8, 0.83]}, {[0.07, 0.1],[0.08, 0.11]}}	{{[0.19, 0.2],[0.2, 0.23]}, {[0.66, 0.69]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}
$B$	{{[0.12, 0.14],[0.13, 0.15]}, {[0.77, 0.8]}}	{{[0, 0]}, {[1, 1]}}	{{[0, 0]}, {[1, 1]}}	{{[0.33, 0.36],[0.34, 0.37]}, {[0.51, 0.54]}}

Analogously, by using the proposed DM approach, we can obtain that $T_{1} (B) ⋂ T_{2} (B) ⋂ T_{3} (B) = \emptyset$ and $T_{1} (B) ⋂ T_{3} (B) = 5 .$ Thus, we can conclude that the vibration fault is most likely resulted from radial impact friction of rotor (x₅). Moreover, it’s not hard to find that the same result can be obtained by using distance measures.

It’s noted that by using the DM method proposed in this paper, both results of the above two examples are consistent with the method of distance measures. This shows that the proposed method is effective. However, as mentioned above, the method of distance measure is on the premise of assumption that different IVDHFEs have the same length, otherwise extra elements should be added to the shorter IVDHFE until they have equal lengths. To some degree, this operation may influence and change the information contained in IVDHFEs, which may be less well justified and objective.

In contrast, the proposed DM model takes advantage of rough set, which provides us another feasible method. More specially, by developing approximation space into IVDHF environment, decision makers can obtain two different kinds of decision results representing different risk preferences, which are the lower IVDHF rough approximation with the minimum uncertainties and the upper IVDHF rough approximation with the maximum uncertainties respectively. By utilizing the pair of approximations, we could make full use of original decision information without any changes to it. Moreover, multifaced decision rules are designed to reach a final DM result based on the approximation information. Therefore, the interval-valued dual hesitant fuzzy rough set over two universes model is more reliable and justified.

6 Conclusion

In this paper, we provide a systematic framework for the study of IVDHF rough set over two universes by constructive approach, which develops a new study view to handle uncertain decision information. We first extend and explore some basic operations and properties of IVDHF sets. The comparisons for IVDHFEs and IVDHF sets are introduced next. Then, we define the upper and lower IVDHF rough approximation operators and discuss properties of IVDHF rough approximation operators. Finally, we present a DM method based on IVDHF rough set over two universes model. The outcomes of two numerical examples show its effectiveness. Comparing to the method of distance measures, which needs process original decision information first, the proposed approach in this paper provides another effective way to solve decision problems. By using IVDHF rough set over two universes, it’s beneficial for experts to better handle uncertain information in DM problems and get a reliable result.

It’s noted that there are some recent studies on information loss problem of HF set theory, such as extended HF set [55] and expanded HF set [56], etc. These researches have tried to provide a more general and comprehensive model for DM problems from different angles, which can also identify the information of different decision makers. Thus, in the future, our group will mainly investigate the potential correlations and applications between IVDHF rough set and these novel extended models.

Footnotes

Acknowledgments

The authors are very grateful to the anonymous reviewers for their invaluable and insightful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 61573240 and 61503241).

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