Multigranulation hesitant Pythagorean fuzzy rough sets and its application in multi-attribute decision making

Abstract

Hesitant Pythagorean fuzzy set (HPFS) is a generalization of fuzzy set (FS). By integrating HPFS with rough set (RS), the notion of single granulation hesitant Pythagorean fuzzy rough sets (SGHPFRSs) is firstly put forward. Then, in view of the multigranulation framework, two types of multigranulation hesitant Pythagorean fuzzy rough sets (MGHPFRSs) are proposed, which are called the optimistic multigranulation hesitant Pythagorean fuzzy rough sets (OMGHPFRSs) and pessimistic multigranulation hesitant Pythagorean fuzzy rough sets (PMGHPFRSs). The connections between serial hesitant Pythagorean fuzzy relations and hesitant Pythagorean fuzzy approximation operators are established. The relationship between the MGHPFRSs and SGHPFRSs will be explored. Moreover, the relationship between the OMGHPFRSs and PMGHPFRSs will be established subsequently. Ultimately, an illustrative example will be given to expound the availability about the MGHPFRSs in multi-attribute decision making.

Keywords

Rough sets SGHPFRSs MGHPFRSs multi-attribute decision making

1 Introduction

RS originated by Pawlak [20, 21] is an appealing tool to handle uncertainty. However, it treats the partition of the universe as the original notion to construct the approximation operators. Aimed at this issue, many researchers exert themselves to remedy the rigidness of the condition in Pawlak’s RS, such as the equivalence relation are relaxed with fuzzy relation [7], similarity relation [26] and other generalizations of RS can be found in [2 , 59]. Furthermore, Zhang et al. [57, 60] groped the uncertainty of probabilistic rough set.

Yager originated the notion of Pythagorean fuzzy set (PFS) [39 –41], which is a loose form of intuitionistic fuzzy set (IFS) [1]. Speaking of PFS, an appealing feature is that the condition is loosened in Pythagorean fuzzy environment. Since the origination of PFS, it unlocks new avenues of study in handling uncertainty issues. In [8], Garg explored the correlation coefficient between Pythagorean fuzzy sets (PFSs). Gao et al. [9] make an exploration of Pythagorean fuzzy interaction aggregation operators. Liang and Xu [16] groped the three-way decisions using ideal TOPSIS solutions under Pythagorean fuzzy information. In [22 –24], Peng et al. studied some results of PFSs, Pythagorean fuzzy soft set, and the fundamental properties of Pythagorean fuzzy aggregation operators. Zhang [54] groped a novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making.

Hesitant fuzzy sets (HFSs) [27, 28] originated by Torra have been paid great attentions in recent years. Hesitant Pythagorean fuzzy sets (HPFSs) [15] is put forward by extending HFSs into Pythagorean fuzzy environment. Furthermore, some researches of HFSs such as the distance and correlation measures with hesitant fuzzy information [37] and hesitant fuzzy rough set [42, 55]. Further, Liang and Xu studied multiple criteria decision making with HPFSs. In [15] Wei and Lu studied dual hesitant Pythagorean fuzzy Hamacher aggregation operators in [31]. In hesitant Pythagorean fuzzy environment, it allows that the Pythagorean membership and Pythagorean nonmembership have several possible values, respectively. The notions of IFS [1], PFS [39–41 , 53], hesitant fuzzy set (HFS) [27, 28] and dual hesitant fuzzy set (DHFS) [56 , 62] are all special cases of HPFS. However, a review of existing researches exhibits there is a deficiency in integrating HPFS with RS. In present paper, by combining HPFS and RS, we propose the notion of SGHPFRSs. Some properties of this model will be groped subsequently.

Recently, the issues of fuzzy research [19], fuzzy decision making [3] and three-way decisions [38] becomes a hot topic. Due to the uncertainty and the complexity of real decision making, the viewpoint of the decision makers or experts may be inconsistency for the same object. We often need to consider each expert’s assessments to obtain the optimal solution. As multigranulation rough set [25] can handle this kind of problem mentioned above, so, in present paper, by integrating multigranulation rough set with HPFS, we propose two types of MGHPFRSs, which are called the OMGHPFRSs and PMGHPFRSs. Furthermore, some explorations of multigranulation rough sets can be found in [11 , 56]. Particularly, Zhang et al. [58] studied the fuzzy equivalence relation and its multigranulation spaces. Further, the relationship between OMGHPFRSs and PMGHPFRSs will be groped. Finally, considering the combination of research and application [29, 30], the application of MGHPFRSs in multi-attribute decision making [10 , 33] will be explored. Both a general decision making algorithm and an illustrative example will be exhibited to show the applications of MGHPFRSs.

The remainder of this paper is arranged as follows. In Section 2, we make a brief revisit of previous knowledge. After introducing the model of SGHPFRSs, some properties of this model are groped in Section 3. In Section 4, we propose two types of MGHPFRSs. Further, we probe into the properties of MGHPFRSs and the relationship between OMGHPFRSs and PMGHPFRSs. In Section 5, we give a general decision making algorithm under the context of occupation match. This paper is summarized in the last section.

2 Preliminaries

In this section, we make a brief revisit of previous knowledge include HFSs, PFSs and HPFSs.

2.1 HFSs and PFSs

Torra [27, 28] proposed the notion of HFSs, which is defined as follows.

Definition 2.1. ([27, 28]). Let U be a nonempty and finite universe. A HFS A on U can be expressed as: $A = {〈 x, h_{A} (x) 〉 | x \in U},$ where h_A (x) is a set of some different values in [0, 1], standing for the possible membership degrees of the element x ∈ U to A. As stated by Xia and Xu [34], we denote h_A (x) as a hesitant fuzzy element (HFE), and the set of all HFSs on U is denoted by HF (U).

Yager [39 , 41] proposed the notion of PFSs and defined as follows.

Definition 2.2. ([39 , 41]). Let U be a nonempty and finite universe. A PFS P on U can be expressed as: $P = {〈 x, P (μ_{P} (x), ν_{P} (x)) 〉 | x \in U},$ where the functions μ_P (x) : U → [0, 1] and ν_P (x) : U → [0, 1] are the Pythagorean membership degree and Pythagorean nonmembership degree of x ∈ U to P, respectively. ∀ x ∈ U, it needs to satisfy the condition: 0≤ (μ_P (x)) ² + (ν_P (x)) ² ≤ 1.

The set of all PFSs on U is denoted as PF (U).

Remark 2.3. If μ_P (x) + ν_P (x) ≤1 (∀ x ∈ U), then the PFS degenerates into an IFS.

2.2 HPFSs

In [15], Liang and Xu introduced the notion of HPFSs and defined as follows.

Definition 2.4. ([15]). Let U be a nonempty and finite universe. A HPFS F on U is expressed as: $F = {〈 x, F (u_{F} (x), v_{F} (x)) 〉 | x \in U},$ in which u_F (x) and v_F (x) are two sets of some different values in [0, 1], denoting the possible Pythagorean membership degrees and Pythagorean nonmembership degrees of x to F, respectively, with the conditions: $0 \leq φ, φ \leq 1, 0 \leq (φ^{+})^{2} + (φ^{+})^{2} \leq 1,$ where φ ∈ u_F (x) , φ ∈ v_F (x) , φ⁺ = max_{φ ∈_uF (x)} {φ} and φ⁺ = max_{φ ∈_vF (x)} {φ} (∀ x ∈ U).

The set of all HPFSs on U is denoted by HPF (U). The pair F (u_F (x) , v_F (x)) is called a hesitant Pythagorean fuzzy element (HPFE) and denoted by f = F (u, v).

Remark 2.5. (1) If l (u_F) (x) =1 and l (v_F) (x) =1 (∀ x ∈ U), then the HPFS degenerates into a PFS. (2) If l (u_F) (x) =1, l (v_F) (x) =1 and u_F (x) + v_F (x) ≤1 (∀ x ∈ U), then the HPFS degenerates into an IFS.

(3) If φ⁺ + φ⁺ ≤ 1 (φ⁺ = max_{φ ∈_uF (x)} {φ}, φ⁺ = max_{φ ∈_vF (x)} {φ}) (∀ x ∈ U), then the HPFS degenerates into a DHFS [61].

From Definition 2.4, the HPFS actually is composed of two functions, i.e., the Pythagorean membership hesitancy function and the Pythagorean nonmembership hesitancy function. u_F (x) and v_F (x) actually are hesitant fuzzy elements (HFEs), the numbers of u_F (x) and v_F (x) are denoted as l (u_F) (x) and l (v_F) (x), respectively. Since the number of values in hesitant Pythagorean fuzzy elements (HPFEs) may be different, similar to [6, 48], we have the following assumptions.

Assumption 1. For a HPFS F, let σ : (1, 2, …, n) → (1, 2, …, n) be a permutation satisfying $u_{F}^{σ (s)} (x) \leq u_{F}^{σ (s + 1)} (x)$ for s = 1, 2, …, n - 1, i.e. $u_{F}^{σ (s)} (x)$ be the sth largest value in u_F (x) and let δ : (1, 2, …, m) → (1, 2, …, m) be a permutation satisfying $v_{F}^{δ (t)} (x) \leq v_{F}^{δ (t + 1)} (x)$ for t = 1, 2, …, m - 1, i.e. $v_{F}^{δ (t)} (x)$ be the tth largest value in v_F (x).

Assumption 2. For two HPFSs A and B, if l (u_A) (x) ≠l (u_B) (x), l (v_A) (x) ≠ l (v_B) (x), we can make both of them have the same number of elements. In present paper, if l (u_A) (x) < l (u_B) (x), then u_A (x) should be extended by adding the maximum value until it has the same length as u_B (x); if l (v_A) (x) < l (v_B) (x), then v_A (x) should be extended by adding the maximum value until it has the same length as v_B (x).

Two special HPFSs are introduced as follows:

F is called an empty HPFS on U iff $u_{F} (x) = \tilde{0}$ and $v_{F} (x) = \tilde{1}$ (∀ x ∈ U), i.e., $u_{F}^{σ (s)} (x) = 0$ , s = 1, 2, …, n, (n = l (u_F (x))), $v_{F}^{δ (t)} (x) = 1$ , t = 1, 2, …, m, (m = l (v_F (x))) (∀ x ∈ U). For simplicity, we denote the empty HPFS as $\tilde{\emptyset}$ .

F is called a full HPFS on U iff $u_{F} (x) = \tilde{1}$ and $v_{F} (x) = \tilde{0}$ (∀ x ∈ U), i.e., $u_{F}^{σ (s)} (x) = 1$ , s = 1, 2, …, n, (n = l (u_F (x))), $v_{F}^{δ (t)} (x) = 0$ , t = 1, 2, …, m, (m = l (v_F (x))) (∀ x ∈ U). For simplicity, we denote the full HPFS as $\tilde{U}$ .

The operations of HPFEs are defined as follows.

Definition 2.6. ([15]). Let f = F (u, v), f₁ = F (u₁, v₁) and f₂ = F (u₂, v₂) be three HPFEs, then some operations among them are described as:

$f_{1} \oplus f_{2} = ⋃_{φ_{1} \in u_{1}, φ_{1} \in v_{1}, φ_{2} \in u_{2}, φ_{2} \in v_{2}} {{((φ_{1})^{2} + (φ_{2})^{2} - (φ_{1})^{2} \cdot (φ_{2})^{2})^{\frac{1}{2}}}, {φ_{1} \cdot φ_{2}}}$ ;

$f_{1} \otimes f_{2} = ⋃_{φ_{1} \in u_{1}, φ_{1} \in v_{1}, φ_{2} \in u_{2}, φ_{2} \in v_{2}} {{φ_{1} \cdot φ_{2}}, {\sqrt{(φ_{1})^{2} + (φ_{2})^{2} - (φ_{1})^{2} \cdot (φ_{2})^{2}}}}$ ;

$kf = ⋃_{φ \in u, φ \in v} {{\sqrt{1 - (1 - φ^{2})^{k}}}, {φ^{k}}}$ , where k ∈ R and k ≥ 0.

Definition 2.7. ([15]). Let f = F (u, v) be a HPFE, then the score function s (f) of f is defined as: $s (f) = (1 / l (u)) \sum_{φ \in u} φ^{2} - (1 / l (v)) \sum_{φ \in v} φ^{2} .$

Definition 2.8. Let U be a nonempty and finite universe and A, B ∈ HPF (U), then $A \tilde{\oplus} B$ is defined as: $A \tilde{\oplus} B = {〈 x, (u_{A} (x), v_{A} (x)) \oplus (u_{B} (x), v_{B} (x)) 〉 | x \in U} .$

The comparison laws of HPFEs are defined as follows.

Definition 2.9. ([13]). Let f₁ = F (u₁, v₁) , f₂ = F (u₂, v₂) be two HPFEs, then:

If s (f₁) > s (f₂), then f₁ is bigger than f₂, denoted by f₁ ≻ f₂;

If s (f₁) < s (f₂), then f₁ is smaller than f₂, denoted by f₁ ≺ f₂;

If s (f₁) = s (f₂), then f₁ is equal to f₂, denoted by f₁ ∼ f₂.

Definition 2.10. Let U be a nonempty and finite universe. ∀ A, B ∈ HPF (U), the complement of A (A^c), the union and the intersection of A and B (A⋓B, A⋒B) are defined as:

A^c = {〈x, v_A (x) , u_A (x) 〉|x ∈ U}

$= {〈 x, {v_{A}^{δ (t)} (x)}, {u_{A}^{σ (s)} (x)} 〉 | x \in U, 1 \leq s \leq k, 1 \leq t \leq l}$ ; where k = l (u_A (x)), l = l (v_A (x)).

A⋓B = {〈x, u_A⋓B (x) , v_A⋓B (x) 〉|x ∈ U}

$= {〈 x, u_{A} (x) \underline{\lor} u_{B} (x), v_{A} (x) \bar{\land} v_{B} (x) 〉 | x \in U}$

$= {〈 x, {u_{A}^{σ (s)} (x) \lor u_{B}^{σ (s)} (x)}, {v_{A}^{δ (t)} (x) \land v_{B}^{δ (t)} (x)} 〉 | x \in U, 1 \leq s \leq k, 1 \leq t \leq l}$ ;

A⋒B = {〈x, u_A⋒B (x) , v_A⋒B (x) 〉|x ∈ U}

$= {〈 x, u_{A} (x) \bar{\land} u_{B} (x), v_{A} (x) \underline{\lor} v_{B} (x) 〉 | x \in U}$

$= {〈 x, {u_{A}^{σ (s)} (x) \land u_{B}^{σ (s)} (x)}, {v_{A}^{δ (t)} (x) \lor v_{B}^{δ (t)} (x)} 〉 | x \in U, 1 \leq s \leq k, 1 \leq t \leq l}$ ; where k = max {l (u_A (x)) , l (u_B (x))}, l = max {l (v_A (x)) , l (v_B (x))}.

Definition 2.11. Let U be a nonempty and finite universe. ∀ A, B ∈ HPF (U), A is called a hesitant Pythagorean fuzzy subset of B, if u_A (x) ⪯ u_B (x) and v_A (x) ≥ v_B (x) (∀ x ∈ U), i.e.

u_{A}^{σ (s)} (x) \leq u_{B}^{σ (s)} (x), v_{A}^{δ (t)} (x) \geq v_{B}^{δ (t)} (x)

with 1 ≤ s ≤ k, 1 ≤ t ≤ l (∀ x ∈ U), where k = max {l (u_A (x)) , l (u_B (x))} and l = max {l (v_A (x)) , l (v_B (x))}. In such a circumstance, we denote it by A ⊑ B.

Based on Definitions 2.10 and 2.11, the following Theorems 2.12 and 2.13 hold.

Theorem 2.12. Let U be a nonempty and finite universe. ∀ A, B ∈ HPF (U), then: $(A ⋓ B)^{c} = A^{c} ⋒ B^{c}, (A ⋒ B)^{c} = A^{c} ⋓ B^{c} .$

Theorem 2.13. Let U be a nonempty and finite universe. ∀ A, B, A₁, B₁, C₁ ∈ HPF (U), then:

A⋓B ⊒ A, A⋓B ⊒ B;

A⋒B ⊑ A, A⋒B ⊑ B;

A₁ ⊑ B₁, A₁ ⊑ C₁ ⇒A₁ ⊑ B₁⋒C₁; A₁ ⊒ B₁, A₁ ⊒ C₁ ⇒A₁ ⊒ B₁⋓C₁.

3 SGHPFRSs and its properties

In this section, by integrating HPFS with RS, the notion of SGHPFRSs will be proposed. Some properties of this model will be examined, subsequently.

3.1 Hesitant pythagorean fuzzy relations

We first introduce the concept of hesitant Pythagorean fuzzy relations as follows.

Definition 3.1. Let U and V be two nonempty and finite universes. A hesitant Pythagorean fuzzy subset R of U × V is called a hesitant Pythagorean fuzzy relation on U × V, i.e., R is given by $R = {〈 (x, y), u_{R} (x, y), v_{R} (x, y) 〉 | (x, y) \in U \times V},$ where u_R : U × V → 2^[0,1], v_R : U × V → 2^[0,1], i.e. u_R (x, y) and v_R (x, y) are two sets of some different values in [0, 1], denoting the possible Pythagorean membership degrees and Pythagorean nonmembership degrees of the relationships between x and y respectively, with the conditions: 0 ≤ γ, η ≤ 1 and 0 ≤ (γ⁺) ² + (η⁺) ² ≤ 1, where γ ∈ u_R (x, y), η ∈ v_R (x, y), $γ^{+} = u_{R}^{+} (x, y) =$ max_{γ ∈_uR (x,y)} {γ}, $η^{+} = v_{R}^{+} (x, y) =$ max_{η ∈_vR (x,y)} {η} (∀ (x, y) ∈ U × V).

Particularly, if U = V, then we call R a hesitant Pythagorean fuzzy relation on U. The set of all hesitant Pythagorean fuzzy relations on U × V is denoted by HPFR (U × V).

Remark 3.2. R (∈HPFR (U × V)) is called serial, if ∀ x ∈ U, then ∃ y ∈ V s.t. $u_{R} (x, y) = \tilde{1}$ and $v_{R} (x, y) = \tilde{0}$ .

3.2 SGHPFRSs

The hesitant Pythagorean fuzzy lower and upper approximation operators are defined as follows.

Definition 3.3. Let U and V be two nonempty and finite universes and R be a hesitant Pythagorean fuzzy relation on U × V. The pair (U, V, R) is called a hesitant Pythagorean fuzzy approximation space (HPFAS). ∀ A ∈ HPF (V), the lower and upper approximations of A w.r.t. (U, V, R), denoted as $\underline{R} (A)$ and $\bar{R} (A)$ , respectively, are two HPFSs on U and defined as follows: $\underline{R} (A) = {〈 x, u_{\underline{R} (A)} (x), v_{\underline{R} (A)} (x) 〉 | x \in U},$ $\bar{R} (A) = {〈 x, u_{\bar{R} (A)} (x), v_{\bar{R} (A)} (x) 〉 | x \in U},$ where $u_{\underline{R} (A)} (x) = {\bar{⋀}}_{y_{j} \in V} {v_{R} (x, y_{j}) \underline{\lor} u_{A} (y_{j})}$ ;

$v_{\underline{R} (A)} (x) = {\underline{⋁}}_{y_{j} \in V} {u_{R} (x, y_{j}) \bar{\land} v_{A} (y_{j})}$ ;

$u_{\bar{R} (A)} (x) = {\underline{⋁}}_{y_{j} \in V} {u_{R} (x, y_{j}) \bar{\land} u_{A} (y_{j})}$ ;

$v_{\bar{R} (A)} (x) = {\bar{⋀}}_{y_{j} \in V} {v_{R} (x, y_{j}) \underline{\lor} v_{A} (y_{j})}$ . $\underline{R} (A)$ and $\bar{R} (A)$ are called the single granulation hesitant Pythagorean fuzzy lower and upper approximations of A w.r.t. (U, V, R), respectively. The pair $(\underline{R} (A), \bar{R} (A))$ is called a single granulation hesitant Pythagorean fuzzy rough set (SGHPFRS) of A w.r.t. (U, V, R). $\underline{R}$ , $\bar{R} : HPF (V) \to HPF (U)$ are called single granulation hesitant Pythagorean fuzzy lower and upper rough approximation operators, respectively.

Obviously, we can observe that:

$u_{\underline{R} (A)} (x) = ⋀_{y_{j} \in V} {v_{R}^{δ (s)} (x, y_{j}) \lor u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$ , where k = max_{1 ≤j ≤|V|} max (l (v_R (x, y_j)) , l (u_A (y_j)));

$v_{\underline{R} (A)} (x) = ⋁_{y_{j} \in V} {u_{R}^{σ (t)} (x, y_{j}) \land v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$ , where l = max_{1 ≤j ≤|V|} max (l (u_R (x, y_j)) , l (v_A (y_j)));

$u_{\bar{R} (A)} (x) = ⋁_{y_{j} \in V} {u_{R}^{σ (s)} (x, y_{j}) \land u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$ , where k = max_{1 ≤j ≤|V|} max (l (u_R (x, y_j)) , l (u_A (y_j)));

$v_{\bar{R} (A)} (x) = ⋀_{y_{j} \in V} {v_{R}^{δ (t)} (x, y_{j}) \lor v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$ , where l = max_{1 ≤j ≤|V|} max (l (v_R (x, y_j)) , l (v_A (y_j))).

Example 3.4. Let U = {x₁, x₂, x₃, x₄} , V = {y₁, y₂, y₃, y₄} and R₁ ∈ HPFR (U × V) (see Table 1).

Table 1
The hesitant Pythagorean fuzzy relation R₁

R ₁ y ₁ y ₂ y ₃ y ₄

x ₁ 〈{1} , {0} 〉〈{0.6, 0.8} , {0.1, 0.3} 〉〈{0.7, 0.8} , {0.2, 0.4} 〉〈{0.6, 0.7} , {0.3, 0.4} 〉

x ₂ 〈{0.3, 0.4, 0.6} , {0.5} 〉〈{1} , {0} 〉〈{0.4, 0.8} , {0.1} 〉〈{0.5, 0.7} , {0.2} 〉

x ₃ 〈{0.4, 0.6} , {0.1, 0.3} 〉〈{0.6, 0.8} , {0.2, 0.3} 〉〈{1} , {0} 〉〈{0.7, 0.8} , {0.1, 0.2} 〉

x ₄ 〈{0.4, 0.9} , {0.1, 0.2} 〉〈{0.7, 0.8} , {0.4, 0.5} 〉〈{0.5, 0.6} , {0.2, 0.3} 〉〈{1} , {0} 〉

R ₁	y ₁	y ₂	y ₃	y ₄
x ₁	〈{1} , {0} 〉	〈{0.6, 0.8} , {0.1, 0.3} 〉	〈{0.7, 0.8} , {0.2, 0.4} 〉	〈{0.6, 0.7} , {0.3, 0.4} 〉
x ₂	〈{0.3, 0.4, 0.6} , {0.5} 〉	〈{1} , {0} 〉	〈{0.4, 0.8} , {0.1} 〉	〈{0.5, 0.7} , {0.2} 〉
x ₃	〈{0.4, 0.6} , {0.1, 0.3} 〉	〈{0.6, 0.8} , {0.2, 0.3} 〉	〈{1} , {0} 〉	〈{0.7, 0.8} , {0.1, 0.2} 〉
x ₄	〈{0.4, 0.9} , {0.1, 0.2} 〉	〈{0.7, 0.8} , {0.4, 0.5} 〉	〈{0.5, 0.6} , {0.2, 0.3} 〉	〈{1} , {0} 〉

Let A = {〈y₁, {0.4, 0.8} , {0.1, 0.3} 〉, 〈y₂, {0.5, 0.7} , {0.2, 0.4} 〉, 〈y₃, {0.2, 0.6} , {0.1} 〉, 〈y₄, {0.7, 0.8} , {0.2, 0.5} 〉}.

According to Definition 3.3, we compute $\underline{R_{1}} (A)$ and $\bar{R_{1}} (A)$ as follows.

$\underline{R_{1}} (A) = {〈 x_{1}, {0.2, 0.6}, {0.2, 0.5} 〉, 〈 x_{2}, {0.2, 0.6}, {0.2, 0.5} 〉, 〈 x_{3}, {0.2, 0.6}, {0.2, 0.5} 〉, 〈 x_{4}, {0.2, 0.6}, {0.2, 0.5} 〉}$ ,

$\bar{R_{1}} (A) = {〈 x_{1}, {0.6, 0.8}, {0.1, 0.3} 〉, 〈 x_{2}, {0.5, 0.7}, {0.1} 〉, 〈 x_{3}, {0.7, 0.8}, {0.1} 〉, 〈 x_{4}, {0.7, 0.8}, {0.1, 0.3} 〉}$ .

In the following, some properties of SGHPFRSs will be groped, we only probe into the case of U = V.

Theorem 3.5. Let (U, R) be a HPFAS and R is serial. ∀ A ∈ HPF (U), we have:

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

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

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

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

Proof. (1) According to Definitions 3.3, 2.10 and Theorem 2.12, we have

$u_{\underline{R} (A^{c})} (x) = {\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} u_{A^{c}} (y_{j})}$

$= {\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} v_{A} (y_{j})}$

$= {{{\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} v_{A} (y_{j})}}^{c}}^{c}$

$= {{\underline{⋁}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} v_{A} (y_{j})}^{c}}^{c}$

$= {{\underline{⋁}}_{y_{j} \in U} {(v_{R} (x, y_{j}))^{c} \bar{⋀} (v_{A} (y_{j}))^{c}}}^{c}$

$= {{\underline{⋁}}_{y_{j} \in U} {u_{R} (x, y_{j}) \bar{⋀} u_{A} (y_{j})}}^{c}$

$= u_{(\bar{R} (A))^{c}} (x)$ , and

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

$= {\underline{⋁}}_{y_{j} \in U} {u_{R} (x, y_{j}) \bar{\land} u_{A} (y_{j})}$

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

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

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

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

$= v_{(\bar{R} (A))^{c}} (x)$ . Thus, $\underline{R} (A^{c}) = (\bar{R} (A))^{c}$ holds. (2) According to Definitions 3.3 and 2.10, we have $(\bar{R} (A^{c}))^{c} = {〈 x, u_{\bar{R} (A^{c})} (x), v_{\bar{R} (A^{c})} (x) 〉 | x \in U}^{c}$

$= {〈 x, v_{\bar{R} (A^{c})} (x), u_{\bar{R} (A^{c})} (x) 〉 | x \in U}$

$= {〈 x, v_{(\underline{R} (A))^{c}} (x), u_{(\underline{R} (A))^{c}} (x) 〉 | x \in U}$

$= {〈 x, u_{\underline{R} (A)} (x), v_{\underline{R} (A)} (x) 〉 | x \in U}$

$= \underline{R} (A)$ . Thus, $\bar{R} (A^{c}) = (\underline{R} (A))^{c}$ holds. (3) Since R is serial, then ∀ x ∈ U, ∃ y ∈ U such that $u_{R} (x, y) = \tilde{1}$ and $v_{R} (x, y) = \tilde{0}$ ;

$u_{\bar{R} (\tilde{U})} (x) = ⋁_{y_{j} \in U} {u_{R}^{σ (s)} (x, y_{j}) \land u_{\tilde{U}}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

$= ⋁_{y_{j} \in U} {u_{R}^{σ (s)} (x, y_{j}) \land 1 | 1 \leq s \leq k}$

$= u_{R} (x, y) ⊻ {{\underline{⋁}}_{y_{j} \neq y} u_{R} (x, y_{j})}$

$= \tilde{1}$

$= u_{\tilde{U}} (x)$ , where k = max_{1 ≤j ≤|U|} l (u_R (x, y_j)), and

$v_{\bar{R} (\tilde{U})} (x) = ⋀_{y_{j} \in U} {v_{R}^{δ (t)} (x, y_{j}) \lor v_{\tilde{U}}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$= ⋀_{y_{j} \in U} {v_{R}^{δ (t)} (x, y_{j}) \lor 0 | 1 \leq t \leq l}$

$= v_{R} (x, y) \bar{⋀} {{\bar{⋀}}_{y_{j} \neq y} v_{R} (x, y_{j})}$

$= \tilde{0}$

$= v_{\tilde{U}} (x)$ , where l = max_{1 ≤j ≤|U|} l (v_R (x, y_j)). Thus, $\bar{R} (\tilde{U}) = \tilde{U}$ holds. Similarly, we have $\underline{R} (\tilde{U}) = \tilde{U}$ . (4) Since R is serial, then ∀ x ∈ U, ∃ y ∈ U such that $u_{R} (x, y) = \tilde{1}$ and $v_{R} (x, y) = \tilde{0}$ ;

$u_{\bar{R} (\tilde{\emptyset})} (x) = ⋁_{y_{j} \in U} {u_{R}^{σ (s)} (x, y_{j}) \land u_{\tilde{\emptyset}}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

$= ⋁_{y_{j} \in U} {u_{R}^{σ (s)} (x, y_{j}) \land 0 | 1 \leq s \leq k}$

$= \tilde{0}$

$= u_{\tilde{\emptyset}} (x)$ , where k = max_{1 ≤j ≤|U|} l (u_R (x, y_j)), and

$v_{\bar{R} (\tilde{\emptyset})} (x) = ⋀_{y_{j} \in U} {v_{R}^{δ (t)} (x, y_{j}) \lor v_{\tilde{\emptyset}}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$= ⋀_{y_{j} \in U} {v_{R}^{δ (t)} (x, y_{j}) \lor 1 | 1 \leq t \leq l}$

$= \tilde{1}$

$= v_{\tilde{\emptyset}} (x)$ , where l = max_{1 ≤j ≤|U|} l (v_R (x, y_j)). Thus, $\bar{R} (\tilde{\emptyset}) = \tilde{\emptyset}$ holds. Similarly, we have $\underline{R} (\tilde{\emptyset}) = \tilde{\emptyset}$ .□

Theorem 3.6. Let (U, R) be a HPFAS. ∀ A, B ∈ HPF (U), we have:

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

If A ⊑ B, then $\underline{R} (A) ⊑ \underline{R} (B)$ and $\bar{R} (A) ⊑ \bar{R} (B)$ ,

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

Proof. From the above Theorem 3.5 (1) and (2), we know that $\underline{R}$ and $\bar{R}$ are dual to each other, we only explore the case of $\underline{R}$ . (1) From Definition 3.3, we have $u_{\underline{R} (A ⋒ B)} (x) = {\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} u_{A ⋒ B} (y_{j})}$

$= {\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} (u_{A} (y_{j}) \bar{⋀} u_{B} (y_{j}))}$

$= ⋀_{y_{j} \in U} {v_{R}^{δ (s)} (x, y_{j}) \lor (u_{A}^{σ (s)} (y_{j})$

$\land u_{B}^{σ (s)} (y_{j})) | 1 \leq s \leq k}$

$= ⋀_{y_{j} \in U} {(v_{R}^{δ (s)} (x, y_{j}) \lor u_{A}^{σ (s)} (y_{j}))$

$\land (v_{R}^{δ (s)} (x, y_{j}) \lor u_{B}^{σ (s)} (y_{j})) | 1 \leq s \leq k}$

$= ({\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} u_{A} (y_{j})})$

$\bar{\land} ({\bar{⋀}}_{y_{j} \in U} {v_{R} (x, y_{j}) \underline{\lor} u_{B} (y_{j})})$

$= u_{\underline{R} (A)} (x) \bar{\land} u_{\underline{R} (B)} (x)$

$= u_{\underline{R} (A) ⋓ \underline{R} (B)} (x)$ , where k = max_{1 ≤j ≤|U|} max {l (v_R (x, y_j)) , l (u_A (y_j)) , l (u_B (y_j))}, and $v_{\underline{R} (A ⋒ B)} (x) = {\underline{⋁}}_{y_{j} \in U} {u_{R} (x, y_{j}) \bar{\land} v_{A ⋒ B} (y_{j})}$

$= {\underline{⋁}}_{y_{j} \in U} {u_{R} (x, y_{j}) \bar{\land} (v_{A} (y_{j})$

$\underline{\lor} v_{B} (y_{j}))}$

$= ⋁_{y_{j} \in U} {u_{R}^{σ (t)} (x, y_{j}) \land (v_{A}^{δ (t)} (y_{j})$

$\lor v_{B}^{δ (t)} (y_{j})) | 1 \leq t \leq l}$

$= ⋁_{y_{j} \in U} {(u_{R}^{σ (t)} (x, y_{j}) \land v_{A}^{δ (t)} (y_{j}))$

$\lor (u_{R}^{σ (t)} (x, y_{j}) \land v_{B}^{δ (t)} (y_{j})) | 1 \leq t \leq l}$

$= ({\underline{⋁}}_{y_{j} \in U} {u_{R} (x, y_{j}) \bar{\land} v_{A} (y_{j})})$

$\underline{\lor} ({\underline{⋁}}_{y_{j} \in U} {u_{R} (x, y_{j}) \bar{\land} v_{B} (y_{j})})$

$= v_{\underline{R} (A)} (x) \underline{\lor} v_{\underline{R} (B)} (x)$

$= v_{\underline{R} (A) ⋓ \underline{R} (B)} (x)$ , where l = max_{1 ≤j ≤|U|} max {l (u_R (x, y_j)) , l (v_A (y_j)) , l (v_B (y_j))}. Thus, $\underline{R} (A ⋒ B) = \underline{R} (A) ⋓ \underline{R} (B)$ holds. (2) Since A ⊑ B, then ∀ y_j ∈ U, we have $u_{A}^{σ (s)} (y_{j}) \leq u_{B}^{σ (s)} (y_{j})$ and $v_{A}^{δ (t)} (y_{j}) \geq v_{B}^{δ (t)} (y_{j})$ . Consequently, ∀ x ∈ U, we have $u_{\underline{R} (A)} (x) = ⋀_{y_{j} \in U} {u_{A}^{σ (s)} (y_{j}) \lor v_{R}^{δ (s)} (x, y_{j}) | 1 \leq s \leq k}$

$⪯ ⋀_{y_{j} \in U} {u_{B}^{σ (s)} (y_{j}) \lor v_{R}^{δ (s)} (x, y_{j}) | 1 \leq s \leq k}$

$= u_{\underline{R} (B)} (x)$ , and $v_{\underline{R} (A)} (x) = ⋁_{y_{j} \in U} {v_{A}^{δ (t)} (y_{j}) \land u_{R}^{σ (t)} (x, y_{j}) | 1 \leq t \leq l}$

$\geq ⋁_{y_{j} \in U} {v_{B}^{δ (t)} (y_{j}) \land u_{R}^{σ (t)} (x, y_{j}) | 1 \leq t \leq l}$

$= v_{\underline{R} (B)} (x)$ , where k = max_{1 ≤j ≤|U|} max {l (v_R (x, y_j)) , l (u_A (y_j)) , l (u_B (y_j))},

l = max_{1 ≤j ≤|U|} max {l (u_R (x, y_j)) , l (v_A (y_j)) , l (v_B (y_j))}. Therefore, $u_{\underline{R} (A)} (x) ⪯ u_{\underline{R} (B)} (x)$ and $v_{\underline{R} (A)} (x) \geq v_{\underline{R} (B)} (x)$ (∀ x ∈ U). So, $\underline{R} (A) ⊑ \underline{R} (B)$ holds. (3) It derives immediately from the above conclusion (2) and Theorem 2.13.□

Theorem 3.7. Let (U, R) be a HPFAS. R₁, R₂ ∈ HPFR (U), ∀ A ∈ HPF (U), if R₁ ⊑ R₂, we have:

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

$\bar{R_{1}} (A) ⊑ \bar{R_{2}} (A)$ .

Proof. It derives from Definitions 2.11 and 3.3.□

Corollary 3.8. Let (U, R) be a HPFAS. R₁, R₂ ∈ HPFR (U), ∀ A ∈ HPF (U), we have:

$\underline{R_{1} ⋒ R_{2}} (A) ⊒ \underline{R_{1}} (A) ⋓ \underline{R_{2}} (A)$ , $\bar{R_{1} ⋒ R_{2}} (A) ⊑ \bar{R_{1}} (A) ⋒ \bar{R_{2}} (A)$ ,

$\underline{R_{1} ⋓ R_{2}} (A) ⊑ \underline{R_{1}} (A) ⋓ \underline{R_{2}} (A)$ , $\bar{R_{1} ⋓ R_{2}} (A) ⊒ \bar{R_{1}} (A) ⋓ \bar{R_{2}} (A)$ .

Proof. It derives from Theorems 2.13 and 3.7.□

4 MGHPFRSs and its properties

In this section, two types of MGHPFRSs will be proposed. Further, some properties of two types of MGHPFRSs will be groped. Moreover, the relationship between OMGHPFRSs and PMGHPFRSs will be examined.

4.1 The OMGHPFRSs

Definition 4.1. Let U and V be two nonempty and finite universes, and R_i ∈ HPFR (U × V) (i = 1, 2, …, m). ∀ A ∈ HPF (V), the optimistic multigranulation hesitant Pythagorean fuzzy lower and upper approximations of A w.r.t. (U, V, R_i|i = 1, 2, …, m), denoted by ${\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)$ and ${\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)$ , respectively, are two HPFSs and defined as follows: ${\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) = {〈 x, u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x), v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) 〉 | x \in U}$ , ${\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A) = {〈 x, u_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x), v_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) 〉 | x \in U}$ , where $u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = {\underline{⋁}}_{i = 1}^{m} {\bar{⋀}}_{y_{j} \in V} {v_{R} (x, y_{j}) \underline{\lor} u_{A} (y_{j})}$ ,

$v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = {\bar{⋀}}_{i = 1}^{m} {\underline{⋁}}_{y_{j} \in V} {u_{R} (x, y_{j}) \bar{\land} v_{A} (y_{j})}$ ,

$u_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = {\bar{⋀}}_{i = 1}^{m} {\underline{⋁}}_{y_{j} \in V} {u_{R} (x, y_{j}) \bar{\land} u_{A} (y_{j})}$ ,

$v_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = {\underline{⋁}}_{i = 1}^{m} {\bar{⋀}}_{y_{j} \in V} {v_{R} (x, y_{j}) \underline{\lor} v_{A} (y_{j})}$ . The pair $({\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A), {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A))$ is called an optimistic multigranulation hesitant Pythagorean fuzzy rough set (OMGHPFRS) of A.

Remark 4.2. In Definition 4.1, if m = 1, then the OMGHPFRS degenerates into the SGHPFRS given in Definition 3.3.

Remark 4.3. In Definition 4.1, if A degenerates into a PFS on V and R_i (i = 1, 2, …, m) degenerate into m Pythagorean fuzzy relations on U × V, then the OMGHPFRS degenerates into the optimistic multigranulation Pythagorean fuzzy rough set given in [52].

Remark 4.4. In Definition 4.1, if A degenerates into a FS [49] on V and R_i (i = 1, 2, …, m) degenerate into m fuzzy relations on U × V, then the OMGHPFRS degenerates into the optimistic multigranulation fuzzy rough set.

Remark 4.5. If A degenerates into an IFS on V and R_i (i = 1, 2, …, m) degenerate into m intuitionistic fuzzy relations on U × V, then the OMGHPFRS degenerates into the optimistic multigranulation intuitionistic fuzzy rough set.

Example 4.6. Let U = {x₁, x₂, x₃, x₄} , V = {y₁, y₂, y₃, y₄} and R₁, R₂, R₃ ∈ HPFR (U × V) (see Tables 1 –3, respectively).

Table 2
The hesitant Pythagorean fuzzy relation R₂

R ₂ y ₁ y ₂ y ₃ y ₄

x ₁ 〈{1} , {0} 〉〈{0.4, 0.6, 0.8} , {0.1, 0.3} 〉〈{0.6, 0.8} , {0.2, 0.4} 〉〈{0.4, 0.5} , {0.2, 0.6} 〉

x ₂ 〈{0.3, 0.5, 0.6} , {0.1, 0.2} 〉〈{1} , {0} 〉〈{0.4, 0.7} , {0.4} 〉〈{0.6, 0.7} , {0.2, 0.3} 〉

x ₃ 〈{0.6, 0.8} , {0.1, 0.4} 〉〈{0.4, 0.6} , {0.5} 〉〈{1} , {0} 〉〈{0.5, 0.6} , {0.1} 〉

x ₄ 〈{0.5, 0.7} , {0.3, 0.4} 〉〈{0.3, 0.7} , {0.2, 0.3} 〉〈{0.4.0.7} , {0.4} 〉〈{1} , {0} 〉

R ₂	y ₁	y ₂	y ₃	y ₄
x ₁	〈{1} , {0} 〉	〈{0.4, 0.6, 0.8} , {0.1, 0.3} 〉	〈{0.6, 0.8} , {0.2, 0.4} 〉	〈{0.4, 0.5} , {0.2, 0.6} 〉
x ₂	〈{0.3, 0.5, 0.6} , {0.1, 0.2} 〉	〈{1} , {0} 〉	〈{0.4, 0.7} , {0.4} 〉	〈{0.6, 0.7} , {0.2, 0.3} 〉
x ₃	〈{0.6, 0.8} , {0.1, 0.4} 〉	〈{0.4, 0.6} , {0.5} 〉	〈{1} , {0} 〉	〈{0.5, 0.6} , {0.1} 〉
x ₄	〈{0.5, 0.7} , {0.3, 0.4} 〉	〈{0.3, 0.7} , {0.2, 0.3} 〉	〈{0.4.0.7} , {0.4} 〉	〈{1} , {0} 〉

Table 3

The hesitant Pythagorean fuzzy relation R₃

R ₃	y ₁	y ₂	y ₃	y ₄
x ₁	〈{1} , {0} 〉	〈{0.2, 0.3, 0.6} , {0.4, 0.5} 〉	〈{0.3, 0.5, 0.7} , {0.4, 0.7} 〉	〈{0.7, 0.9} , {0.2, 0.3} 〉
x ₂	〈{0.3, 0.4} , {0.2, 0.7} 〉	〈{1} , {0} 〉	〈{0.2, 0.5, 0.8} , {0.6} 〉	〈{0.6, 0.8} , {0.3, 0.4} 〉
x ₃	〈{0.3, 0.5} , {0.2, 0.3} 〉	〈{0.3, 0.4, 0.5} , {0.2} 〉	〈{1} , {0} 〉	〈{0.4, 0.6} , {0.1, 0.2} 〉
x ₄	〈{0.5, 0.6} , {0.4, 0.6} 〉	〈{0.7, 0.8} , {0.4, 0.6} 〉	〈{0.8, 0.9} , {0.1, 0.2} 〉	〈{1} , {0} 〉

Let A = {〈y₁, {0.4, 0.8} , {0.1, 0.3} 〉, 〈y₂, {0.5, 0.7} , {0.2, 0.4} 〉, 〈y₃, {0.2, 0.6} , {0.1} 〉, 〈y₄, {0.7, 0.8} , {0.2, 0.5} 〉}. From Definition 3.3, the single granulation hesitant Pythagorean fuzzy lower and upper approximations of A on R₂ and R₃, respectively, can be calculated as:

$\underline{R_{2}} (A) = {〈 x_{1}, {0.2, 0.6}, {0.2, 0.5} 〉, 〈 x_{2}, {0.4, 0.6}, {0.2, 0.5} 〉, 〈 x_{3}, {0.2, 0.6}, {0.2, 0.5} 〉, 〈 x_{4}, {0.4, 0.6}, {0.2, 0.5} 〉}$ ,

$\bar{R_{2}} (A) = {〈 x_{1}, {0.4, 0.8}, {0.1, 0.3} 〉, 〈 x_{2}, {0.6, 0.7}, {0.1, 0.3} 〉, 〈 x_{3}, {0.5, 0.8}, {0.1} 〉, 〈 x_{4}, {0.7, 0.8}, {0.2, 0.4} 〉}$ ;

$\underline{R_{3}} (A) = {〈 x_{1}, {0.4, 0.7}, {0.2, 0.5} 〉, 〈 x_{2}, {0.4, 0.6}, {0.2, 0.5} 〉, 〈 x_{3}, {0.2, 0.6}, {0.2, 0.5} 〉, 〈 x_{4}, {0.2, 0.6}, {0.2, 0.5} 〉}$ ,

$\bar{R_{3}} (A) = {〈 x_{1}, {0.7, 0.8}, {0.1, 0.3} 〉, 〈 x_{2}, {0.6, 0.8}, {0.2, 0.4} 〉, 〈 x_{3}, {0.4, 0.6}, {0.1} 〉, 〈 x_{1}, {0.7, 0.8}, {0.1, 0.2} 〉}$ .

Further, according to Definition 4.1, we compute ${\underline{R_{1} + R_{2} + R_{3}}}^{O} (A)$ and ${\bar{R_{1} + R_{2} + R_{3}}}^{O} (A)$ as:

${\underline{R_{1} + R_{2} + R_{3}}}^{O} (A)$ = {〈x₁, {0.4, 0.7} , {0.2, 0.5} 〉, 〈x₂, {0.4, 0.6} , {0.2, 0.5} 〉, 〈x₃, {0.2, 0.6} , {0.2, 0.5} 〉, 〈x₄, {0.4, 0.6} , {0.2, 0.5} 〉},

${\bar{R_{1} + R_{2} + R_{3}}}^{O} (A)$ = {〈x₁, {0.4, 0.7} , {0.1, 0.3} 〉, 〈x₂, {0.5, 0.7} , {0.2, 0.4} 〉, 〈x₃, {0.4, 0.6} , {0.1} 〉, 〈x₄, {0.7, 0.8} , {0.2, 0.5} 〉}.

The following Theorem 4.7 will build the relationship between the OMGHPFRSs and SGHPFRSs.

Theorem 4.7. Let (U, R_i |i = 1, 2, …, m) be the HPFAS. ∀ A ∈ HPF (U), we have:

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

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

Proof. (1) According to Definitions 3.3 and 4.1, we have $u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = ⋁_{i = 1}^{m} ⋀_{y_{j} \in U} {v_{R_{i}}^{δ (s)} (x, y_{j}) \lor u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

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

where k = max_{1 ≤i ≤m} max_{1 ≤j ≤|U|} max {l (v_{R
_i} (x, y_j)) , l (u_A (y_j))}, and $v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = ⋀_{i = 1}^{m} ⋁_{y_{j} \in U} {u_{R_{i}}^{σ (t)} (x, y_{j}) \land v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$= ⋓_{i = 1}^{m} v_{\underline{R_{i}} (A)} (x)$ , where l = max_{1 ≤i ≤m} max_{1 ≤j ≤|U|} max {l (u_{R
_i} (x, y_j)) , l (v_A (y_j))}. Thus, ${\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) = ⋓_{i = 1}^{m} \underline{R_{i}} (A)$ holds. (2) It is analogous to the proof of (1).

In the following, some properties of OMGHPFRSs will be groped.□

Theorem 4.8. Let (U, R_i |i = 1, 2, …, m) be the HPFAS and R_i (i = 1, 2, …, m) is serial. ∀ A ∈ HPF (U), we have:

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

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

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

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

Proof. (1) It derives from Theorems 4.7, 3.5 and 2.12. (2) It derives immediately from Theorems 4.7 and 3.5.□

Theorem 4.9. Let (U, R_i |i = 1, 2, …, m) be the HPFAS. ∀ A, B ∈ HPF (U), we have:

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

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

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

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

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

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

Proof. (1) ∀ x ∈ U, according to Theorems 4.7 and 3.6, we have $u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A ⋒ B)} (x) = u_{⋓_{i = 1}^{m} \underline{R_{i}} (A ⋒ B)} (x)$

$= u_{⋓_{i = 1}^{m} (\underline{R_{i}} (A) ⋓ \underline{R_{i}} (B))} (x)$

$= ⋁_{i = 1}^{m} {u_{\underline{R_{i}} (A)}^{σ (s)} (x)$

$\land u_{\underline{R_{i}} (B)}^{σ (s)} (x) | 1 \leq s \leq k}$

$⪯ (⋁_{i = 1}^{m} {u_{\underline{R_{i}} (A)}^{σ (s)} (x) | 1 \leq s \leq k})$

$\land (⋁_{i = 1}^{m} {u_{\underline{R_{i}} (B)}^{σ (s)} (x) | 1 \leq s \leq k})$

$= u_{⋓_{i = 1}^{m} \underline{R_{i}} (A)} (x) \bar{\land} u_{⋓_{i = 1}^{m} \underline{R_{i}} (B)} (x)$

$= u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) ⋒ {\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (B)} (x)$ , where k = max max_{1 ≤i ≤m} ${l (u_{\underline{R_{i}} (A)} (x))$ , $l (u_{\underline{R_{i}} (B)} (x))}$ , and $v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A ⋒ B)} (x) = v_{⋓_{i = 1}^{m} \underline{R_{i}} (A ⋒ B)} (x)$

$= v_{⋓_{i = 1}^{m} (\underline{R_{i}} (A) ⋓ \underline{R_{i}} (B))} (x)$

$= ⋀_{i = 1}^{m} {v_{\underline{R_{i}} (A)}^{δ (t)} (x)$

$\lor v_{\underline{R_{i}} (B)}^{δ (t)} (x) | 1 \leq t \leq l}$

$\geq (⋀_{i = 1}^{m} {v_{\underline{R_{i}} (A)}^{δ (t)} (x) | 1 \leq t \leq l})$

$\lor (⋀_{i = 1}^{m} {v_{\underline{R_{i}} (B)}^{δ (t)} (x) | 1 \leq t \leq l})$

$= v_{⋓_{i = 1}^{m} \underline{R_{i}} (A)} (x) \underline{\lor} v_{⋓_{i = 1}^{m} \underline{R_{i}} (B)} (x)$

$= v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) ⋒ {\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (B)} (x)$ , where l = max max_{1 ≤i ≤m} $(l (v_{\underline{R_{i}} (A)} (x)), l (v_{\underline{R_{i}} (B)} (x))$ . Thus, we have (1) holds. (2) ∀ x ∈ U, according to Theorems 4.7 and 3.6, we have $u_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A ⋓ B)} (x) = u_{⋒_{i = 1}^{m} \bar{R_{i}} (A ⋓ B)} (x)$

$= u_{⋒_{i = 1}^{m} (\bar{R_{i}} (A) ⋓ \bar{R_{i}} (B))} (x)$

$= ⋀_{i = 1}^{m} {u_{\bar{R_{i}} (A)}^{σ (s)} (x)$

$\lor u_{\bar{R_{i}} (B)}^{σ (s)} (x) | 1 \leq s \leq k}$

$\geq (⋀_{i = 1}^{m} {u_{\bar{R_{i}} (A)}^{σ (s)} (x) | 1 \leq s \leq k})$

$\lor (⋀_{i = 1}^{m} {u_{\bar{R_{i}} (B)}^{σ (s)} (x) | 1 \leq s \leq k})$

$= u_{⋒_{i = 1}^{m} \bar{R_{i}} (A)} (x) \underline{\lor} u_{⋒_{i = 1}^{m} \bar{R_{i}} (B)} (x)$

$= u_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A) ⋓ {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (B)} (x)$ , where k = max max_{1 ≤i ≤m} $(l (u_{\bar{R_{i}} (A)} (x), l (u_{\bar{R_{i}} (B)} (x)))$ , and $v_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A ⋓ B)} (x) = v_{⋒_{i = 1}^{m} \bar{R_{i}} (A ⋓ B)} (x)$

$= v_{⋒_{i = 1}^{m} (\bar{R_{i}} (A) ⋓ \bar{R_{i}} (B))} (x)$

$= ⋁_{i = 1}^{m} {v_{\bar{R_{i}} (A)}^{δ (t)} (x)$

$\land v_{\bar{R_{i}} (B)}^{δ (t)} (x) | 1 \leq t \leq l}$

$⪯ (⋁_{i = 1}^{m} {v_{\bar{R_{i}} (A)}^{δ (t)} (x) 1 \leq t \leq l})$

$\land (⋁_{i = 1}^{m} {v_{\bar{R_{i}} (B)}^{δ (t)} (x) | 1 \leq t \leq l})$

$= v_{⋒_{i = 1}^{m} \bar{R_{i}} (A)} (x) \bar{\land} v_{⋒_{i = 1}^{m} \bar{R_{i}} (B)} (x)$

$= v_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A) ⋓ {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (B)} (x)$ , where l = max max_{1 ≤i ≤m} $(l (v_{\bar{R_{i}} (A)} (x), l (v_{\bar{R_{i}} (B)} (x)))$ . Thus, we have (2) holds.

(3) It derives from Theorems 4.7, 3.6 and 2.13.

(4) It derives from the above conclusion (3).□

Theorem 4.10. Let U be a nonempty and finite universe. $R_{i}, R'_{i} \in HPFR (U)$ (i = 1, 2, …, m). ∀ A ∈ HPF (U), if $R_{i} ⊑ R'_{i}$ (i = 1, 2, …, m), we have:

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

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

Proof. (1) Since $R_{i} ⊑ R'_{i}$ (i = 1, 2, …, m), then ∀ (x, y) ∈ U × U, $u_{R_{i}}^{σ (s)} (x, y) \leq u_{R'_{i}}^{σ (s)} (x, y)$ and $v_{R_{i}}^{δ (t)} (x, y)$ $\geq v_{R'_{i}}^{δ (t)} (x, y)$ (i = 1, 2, …, m). So, $u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = ⋁_{i = 1}^{m} ⋀_{y_{j} \in U} {v_{R_{i}}^{δ (s)} (x, y_{j}) \lor u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

$\geq ⋁_{i = 1}^{m} ⋀_{y_{j} \in U} {v_{R'_{i}}^{δ (s)} (x, y_{j})$

$\lor u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

$= u_{{\underline{\sum_{i = 1}^{m} R'_{i}}}^{O} (A)} (x)$ , and $v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = ⋀_{i = 1}^{m} ⋁_{y_{j} \in U} {u_{R_{i}}^{σ (t)} (x, y_{j}) \land v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$⪯ ⋀_{i = 1}^{m} ⋁_{y_{j} \in U} {u_{R'_{i}}^{σ (t)} (x, y_{j})$

$\land v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$= v_{{\underline{\sum_{i = 1}^{m} R'_{i}}}^{O} (A)} (x)$ , where k = max_{1 ≤i ≤m} max_{1 ≤j ≤|U|} max {l (v_{R
_i} (x, $y_{j})), l (v_{R'_{i}} (x, y_{j})), l (u_{A} (y_{j}))}$ ,

l = max_{1 ≤i ≤m} max_{1 ≤j ≤|U|} max ${l (u_{R_{i}} (x, y_{j})), l (u_{R'_{i}} (x, y_{j})), l (v_{A} (y_{j}))}$ . Thus, we have (1) holds. (2) It is similar to the proof of (1).□

4.2 The PMGHPFRSs

Definition 4.11. Let U and V be two nonempty and finite universes, and R_i ∈ HPFR (U × V) (i = 1, 2, …, m). ∀ A ∈ HPF (V), the pessimistic multigranulation hesitant Pythagorean fuzzy lower and upper approximations of A w.r.t. (U, V, R_i|i = 1, 2, …, m), denoted by ${\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)$ and ${\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)$ , respectively, are two HPFSs and defined as follows: ${\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A) = {〈 x, u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x), v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) 〉 | x \in U}$ , ${\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A) = {〈 x, u_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x), v_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) 〉 | x \in U}$ , where $u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) = {\bar{⋀}}_{i = 1}^{m} {\bar{⋀}}_{y_{j} \in V} {v_{R_{i}} (x, y_{j}) \underline{\lor} u_{A} (y_{j})}$ ,

$v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) = {\underline{⋁}}_{i = 1}^{m} {\underline{⋁}}_{y_{j} \in V} {u_{R_{i}} (x, y_{j}) \bar{\land} v_{A} (y_{j})}$ ,

$u_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) = {\underline{⋁}}_{i = 1}^{m} {\underline{⋁}}_{y_{j} \in V} {u_{R_{i}} (x, y_{j}) \bar{\land} u_{A} (y_{j})}$ ,

$v_{{\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x) = {\bar{⋀}}_{i = 1}^{m} {\bar{⋀}}_{y_{j} \in V} {v_{R_{i}} (x, y_{j}) \underline{\lor} v_{A} (y_{j})}$ . The pair $({\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A), {\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A))$ is called a pessimistic multigranulation hesitant Pythagorean fuzzy rough set (PMGHPFRS) of A w.r.t. (U, V, R_i|i = 1, 2, …, m).

Remark 4.12. In Definition 4.11, ∀ A ∈ HPF (V), if m = 1, then the PMGHPFRS degenerates into the SGHPFRS given in Definition 3.3.

Remark 4.13. In Definition 4.11, if A degenerates into a FS on V and R_i (i = 1, 2, …, m) degenerate into m fuzzy relations on U × V, then the PMGHPFRS degenerates into the pessimistic multigranulation fuzzy rough set.

Remark 4.14. If A degenerates into an IFS on V and R_i (i = 1, 2, …, m) degenerate into m intuitionistic fuzzy relations on U × V, then the PMGHPFRS degenerates into the pessimistic multigranulation intuitionistic fuzzy rough set.

Example 4.15. (Continued from Example 4.6). Based on Definition 4.11, we compute ${\underline{R_{1} + R_{2} + R_{3}}}^{P} (A)$ and ${\bar{R_{1} + R_{2} + R_{3}}}^{P} (A)$ as follows:

${\underline{R_{1} + R_{2} + R_{3}}}^{P} (A)$ = {〈x₁, {0.2, 0.6} , {0.2, 0.5} 〉, 〈x₂, {0.2, 0.6} , {0.2, 0.5} 〉, 〈x₃, {0.2, 0.6} , {0.2, 0.5} 〉, 〈x₄, {0.2, 0.6} , {0.2, 0.5} 〉},

${\bar{R_{1} + R_{2} + R_{3}}}^{P} (A)$ = {〈x₁, {0.7, 0.8} , {0.1, 0.3} 〉, 〈x₂, {0.6, 0.8} , {0.2, 0.4} 〉, 〈x₃, {0.7, 0.8} , {0.1} 〉, 〈x₄, {0.7, 0.8} , {0.2, 0.5} 〉}.

In the following, the relationship between the PMGHPFRSs and SGHPFRSs will be constructed.

Theorem 4.16. Let (U, R_i |i = 1, 2, …, m) be the HPFAS. ∀ A ∈ HPF (U), we have:

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

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

Proof. It derives immediately from Definitions 3.3 and 4.11.□

In the following, some properties of PMGHPFRSs will be groped.

Theorem 4.17. Let (U, R_i |i = 1, 2, …, m) be the HPFAS and R_i (i = 1, 2, …, m) is serial. ∀ A ∈ HPF (U), we have:

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

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

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

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

Proof. It derives from Theorems 4.16 and 3.5 and 2.12.□

Theorem 4.18. Let (U, R_i |i = 1, 2, …, m) be the HPFAS. ∀ A, B ∈ HPF (U), we have:

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

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

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

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

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

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

Proof. It is similar to the proof of Theorem 4.9.□

Theorem 4.19. Let U be a nonempty and finite universe. R_i, $R'_{i} \in HPFR (U)$ (i = 1, 2, …, m). ∀ A ∈ HPF (U), if $R_{i} ⊑ R'_{i}$ (i = 1, 2, …, m), then:

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

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

Proof. It is similar to the proof of Theorem 4.10.□

4.3 The relationship between optimistic and pessimistic multigranulation hesitant Pythagorean fuzzy rough sets

In the discussions below, we grope for the relationship between OMGHPFRSs and PMGHPFRSs.

Theorem 4.20. Let (U, R_i |i = 1, 2, …, m) be the HPFAS. ∀ A ∈ HPF (U), we have:

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

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

Proof. (1) ∀ x ∈ U, we have $u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = {\underline{⋁}}_{i = 1}^{m} {\bar{⋀}}_{y_{j} \in U} {v_{R_{i}} (x, y_{j}) \underline{\lor} u_{A} (y_{j})}$

$= ⋁_{i = 1}^{m} ⋀_{y_{j} \in U} {v_{R_{i}}^{δ (s)} (x, y_{j})$

$\lor u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

$\geq ⋀_{i = 1}^{m} ⋀_{y_{j} \in U} {v_{R_{i}}^{δ (s)} (x, y_{j})$

$\lor u_{A}^{σ (s)} (y_{j}) | 1 \leq s \leq k}$

$= u_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x)$ , where k = max_{1 ≤i ≤m}max_{1 ≤j ≤|U|} max {l (v_{R
_i} (x, y_j)) , l (u_A (y_j))}; $v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)} (x) = {\bar{⋀}}_{i = 1}^{m} {\underline{⋁}}_{y_{j} \in U} {u_{R_{i}} (x, y_{j}) \bar{\land} v_{A} (y_{j})}$

$= ⋀_{i = 1}^{m} ⋁_{y_{j} \in U} {u_{R_{i}}^{σ (t)} (x, y_{j})$

$\land v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$⪯ ⋁_{i = 1}^{m} ⋁_{y_{j} \in U} {u_{R_{i}}^{σ (t)} (x, y_{j})$

$\land v_{A}^{δ (t)} (y_{j}) | 1 \leq t \leq l}$

$= v_{{\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)} (x)$ , where l = max_{1 ≤i ≤m}max_{1 ≤j ≤|U|} max {l (u_{R
_i} (x, y_j)) , l (v_A (y_j))}. Thus, ${\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A) ⊑ {\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)$ holds. (2) Similarly, we can check it without any difficulties.□

5 Application of MGHPFRSs in multi-attribute decision making

In this section, we will explore a general approach in the process of selecting the optimal job department under the context of occupation match and give a correspondent example to illustrate it.

5.1 The application model

In this section, a general approach in the process of selecting the optimal job department will be presented. In recent years, there are more and more college students. As a college student, what he (she) concerns most lies in looking for an optimal job department, and it reflects the success rate of the job seeker at some extent. In order to obtain a better and appropriate occupation match. It is inevitable to construct a feasible assessment method for selecting the optimal job department.

Let U = {x₁, x₂, …, x_n} be a set of alternative job departments. Suppose that the job departments can be described by attributes set V = {y₁, y₂, …, y_m}. Assume that the student invites k experts to assess for him. To obtain a better occupation match, each expert needs to evaluate the memberships that the objects satisfy a certain attribute and doesn’t satisfy the same attribute simultaneously. Consider that HPFS exhibits superiorities at some extent to address this issue, so we suppose that the evaluation values of objects x_i (i = 1, 2, …, n) w.r.t. the attribute y_j (j = 1, 2, …, m) given by the expert is (u_ij, v_ij) (i = 1, 2, …, n ; j = 1, 2, …, m), where (u_ij, v_ij) is a HPFE that exhibits the nexus between job departments and attributes, i.e. u_ij and v_ij denote the possible Pythagorean membership degree and Pythagorean nonmembership degree of the job department x_i satisfies the attribute y_j, respectively, and R_k (k = 1, 2, …, s) be s hesitant Pythagorean fuzzy relations on U × V. Let A ∈ HPF (V) be a HPFS which represents the self-evaluation based on his own condition. Then we obtain several hesitant Pythagorean fuzzy decision making information systems (U, V, R_k (k = 1, 2, …, s) , A).

In what follows, an algorithm will be presented and applied to the problem for selecting the optimal job department.

5.2 Algorithm for selecting the optimal job department

Algorithm: The selection of the optimal job department by using MGHPFRSs.

Step 1. The construction of the student’s decision making information system (U, V, R_k (k = 1, 2, …, s) , A).

Step 2. Calculate the optimistic (pessimistic) multigranulation hesitant Pythagorean fuzzy lower and upper approximations of A on hesitant Pythagorean fuzzy relations R_k (k = 1, 2, …, s). So, we get the set of ${\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A)$ , ${\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)$ $({\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A)$ , ${\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A))$ ;

Step 3. Compute the ${\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)$ $({\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A))$ ;

Step 4. Compute the score values based on the conclusions obtained in Step 3, i.e.

$\tilde{s} ({\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A))$ $= {〈 x_{i}, s ({\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A)) (x_{i}) 〉 | i = 1, 2, \dots, n}$ $(\tilde{s} ({\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A)) = {〈 x_{i}, s ({\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A) (x_{i})) 〉 | i = 1, 2, \dots, n})$ ;

Step 5. Ranking the score values of MGHPFRSs. The optimal solution is the highest score value of $\tilde{s} ({\underline{\sum_{i = 1}^{m} R_{i}}}^{O} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{O} (A))$ ( $\tilde{s} ({\underline{\sum_{i = 1}^{m} R_{i}}}^{P} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{m} R_{i}}}^{P} (A))$ ). If the highest score value has more than one value, then he can choose any of them according to his preferences.

5.3 An illustrative example

Assume that a student who is majored in computer speciality wants to look for an optimal job department. There are four job departments for him to choose. Let U = {x₁, x₂, x₃, x₄} be the alternative job departments set (where x₁ stands market department, x₂ stands research and development department, x₃ stands accounting department, x₄ stands sales department), and let V = {y₁, y₂, y₃, y₄} be the attributes set (where y₁ stands software development capability, y₂ stands oral expression capability, y₃ stands organizational management ability and y₄ stands English level). The student presents the self-evaluation based on his own condition, which is denoted by a HPFS A and given as follows: A = {〈y₁, {0.4, 0.8} , {0.1, 0.3} 〉, 〈y₂, {0.5, 0.7} , {0.2, 0.4} 〉, 〈y₃, {0.2, 0.6} , {0.1} 〉, 〈y₄, {0.7, 0.8} , {0.2, 0.5} 〉}.

Simultaneously, he invites three experts to assess for him. Suppose these experts’s assessments are given by R_k (k = 1, 2, 3) (see Tables 1 –3 respectively), so we obtain the decision making information system (U, V, R_k (k = 1, 2, 3) , A) of the student.

We then compute the optimistic and pessimistic multigranulation hesitant Pythagorean fuzzy lower and upper approximations of A on hesitant Pythagorean fuzzy relations R₁, R₂ and R₃ (see Examples 4.6 and 4.15).

Further, we have:

${\underline{\sum_{i = 1}^{3} R_{i}}}^{O} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{3} R_{i}}}^{O} (A)$ = {〈x₁, {0.5426, 0.7560, 0.8602},

{0.02, 0.05, 0.06, 0.15} 〉, 〈x₂, {0.6083, 0.7211, 0.7560, 0.8207},

{0.04, 0.08, 0.1, 0.2} 〉, 〈x₃, {0.44, 0.6210, 0.6800, 0.7684} , {0.02, 0.05} 〉, 〈x₄, {0.7560, 0.8207, 0.8352, 0.8773},

{0.04, 0.1, 0.25} 〉},

${\underline{\sum_{i = 1}^{3} R_{i}}}^{P} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{3} R_{i}}}^{P} (A)$ = {〈x₁, {0.7144, 0.8089, 0.8207, 0.8773},

{0.02, 0.05, 0.06, 0.15} 〉, 〈x₂, {0.6210, 0.7684, 0.8089, 0.8773} , {0.04, 0.08, 0.1, 0.2} 〉, 〈x₃, {0.7144, 0.8089, 0.8207, 0.8773} , {0.02, 0.05} 〉, 〈x₄, {0.7144, 0.8089, 0.8207, 0.8773} , {0.04, 0.1, 0.25} 〉}.

Finally, according to Definition 2.7, we compute the score values and rank the results to obtain the optimal target.

$\tilde{s} ({\underline{\sum_{i = 1}^{3} R_{i}}}^{O} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{3} R_{i}}}^{O} (A))$ = {〈x₁, 0.5280〉, 〈x₂, 0.5193〉, 〈x₃, 0.4066〉, 〈x₄, 0.6534〉},

$\tilde{s} ({\underline{\sum_{i = 1}^{3} R_{i}}}^{P} (A) \tilde{\oplus} {\bar{\sum_{i = 1}^{3} R_{i}}}^{P} (A))$ = {〈x₁, 0.6447〉, 〈x₂, 0.5855〉, 〈x₃, 6505〉, 〈x₄, 0.6273〉}.

According to the above results, in the optimistic case, we have x₄ ≻ x₁ ≻ x₂ ≻ x₃; in the pessimistic case, we have x₃ ≻ x₁ ≻ x₄ ≻ x₂. If the student has the optimistic attitude, then he will choose to go to the sales department. If the student has the pessimistic attitude, then he will choose to go the accounting department.

6 Conclusion

In this paper, we propose a new multigranulation rough set model by integrating multigranulation rough set with HPFS. Further, we probe into some basic properties of SGHPFRSs and MGHPFRSs and the relationships among them. Furthermore, we give a decision making algorithm and a corresponding example to exhibit the application of MGHPFRSs in multi-attribute decision making. In the forthcoming research, we will study the interval-valued Pythagorean fuzzy rough sets and its application.

Footnotes

Acknowledgements

Authors would sincerely appreciate the reviewers for their highly valuable comments and precious suggestions. This work is partially supported by the National Natural Science Foundation of China (Nos. 61473181 and 11771263) and the Fundamental Research Funds for the Central Universities (GK201702008).

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