Hesitant fuzzy compatible rough set and its application in hesitant fuzzy soft set based decision making

2.1 Fuzzy soft set

In [20], Maji et al. first introduced fuzzy soft sets as follows.

Definition 2.1. ([20]) Let U be an initial universe set. Suppose that E is a universe set of parameters. If there exists a mapping F : E → F (U), where F (U) is a family of all fuzzy subsets of U, then the pair (F, E) is referred to as a fuzzy soft set on U.

Based on fuzzy soft sets above, Cagman et al. [8] introduced the concept of fuzzy soft relation.

Definition 2.2. ([8]) Suppose that (F, E) is a fuzzy soft set on U. Then a fuzzy subset of U × E is referred to as a fuzzy soft relation R from U to E, i.e.

R = {< (u, x), μ _R (u, x) > | (u, x) ∈ U × E}, where μ _R : U × E → [0, 1], μ _R (u, x) = μ_F(x) (u).

2.2 Hesitant fuzzy set

Torra [34] introduced the concept of hesitant fuzzy sets as follows:

Definition 2.3. ([34]) Let U be a fixed set, a hesitant fuzzy (HF, for short) set A on U is in terms of a function h A ( x ) that when applied to U returns a subset of [0,1].

To be easily understood, Xia and Xu [40] denoted the HF set by a mathematical symbol:

A = { < x , h A ( x ) > | x ∈ U } , where h A ( x ) is a set of some different values in [0,1], standing for the possible membership degrees of the element x ∈ U to A .

For convenience, Xia and Xu [40] called h A ( x ) an HF element, and denoted the set of all HF sets on U by HF (U). From Definition 2.3, we observe that an HF set A can be seen as a fuzzy set if there is only one element in h A ( x ) , which indicates that fuzzy sets are a special type of HF sets.

Assume that l ( h A ( x ) ) represents the number of values in h A ( x ) . Since the number of values in HF elements may be different, Xu and Xia [42] gave the following assumptions:

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

(A2) Let l = max { l ( h A ( x ) ) , l ( h B ( x ) ) } . For h A ( x ) and h B ( x ) , if l ( h A ( x ) ) ≠ l ( h B ( x ) ) , we should extend the shorter one until both of them have the same length l.

In [56], Zhang et al. introduced the following concept of the order for characterizing inclusion relation of HF sets.

Definition 2.4. ([56]) Suppose that U is a nonempty and finite universe set and A , B ∈ HF ( U ) . A is referred to as an HF subset of B , if and only if for any x ∈ U, h A ( x ) ⪯ h B ( x ) , i.e. h A ( x ) ⪯ h B ( x ) ⇔ h A σ ( k ) ( x ) ≤ h B σ ( k ) ( x ) , k = 1 , 2 , … , l . Denote it as A ⊑ B .

3.1 Hesitant fuzzy compatible relation

In [56], Zhang et al. introduced an HF relation on two different universes. In what follows we shall introduce several special HF relations based on the HF relation.

Definition 3.1. The HF relation ℝ from U to V is said to be serial if for each x ∈ U, there exists a y ∈ V such that h ℝ ( x , y ) = { 1 } ; ℝ is said to be reflexive on U if h ℝ ( x , x ) = { 1 } for all x ∈ U; ℝ is referred to as a symmetric HF relation on U if h ℝ ( x , y ) = h ℝ ( y , x ) for all x, y ∈ U; ℝ is said to be transitive on U if ⊻ y ∈ U { h ℝ ( x , y ) barwedge h ℝ ( y , z ) } ⪯ h ℝ ( x , z ) for all x, y, z ∈ U. Alternatively, ℝ is transitive if

⋁ y ∈ U ( h ℝ σ ( k ) ( x , y ) ∧ h ℝ σ ( k ) ( y , z ) ) ≤ h ℝ σ ( k ) ( x , z ) , 1 ≤ k ≤ l , where l = max { l ( h ℝ ( x , y ) ) , l ( h ℝ ( y , z ) ) , l ( h ℝ ( x , z ) ) } .

On the basis of the HF relation proposed by Zhang et al. [56], an HF compatible relation is introduced as follows.

Definition 3.2. Suppose that U and V are two nonempty and finite universes, and ℝ is a hesitant fuzzy relation from U to V. For any α ∈ (0, 1] and 1 ≤ k ≤ l ( h ℝ ( x , y ) ) , we define two HF compatible relations between the universes U and V as follows: ℝ α = { ( x , y ) ∈ U × V | h ℝ ( x , y ) Z { α } } = { ( x , y ) ∈ U × V | h ℝ σ ( k ) ( x , y ) ≥ α } , which is called the α - level cut set of ℝ ; ℝ α + = { ( x , y ) ∈ U × V | h ℝ ( x , y ) ≻ { α } } = { ( x , y ) ∈ U × V | h ℝ σ ( k ) ( x , y ) > α } , which is called the strong α - level cut set of ℝ .

And two successor neighborhoods of x ∈ U with respect to ℝ α and ℝ α + are, respectively, given as follows: ℝ α ( x ) = { y ∈ V | h ℝ ( x , y ) Z { α } } = { y ∈ V | h ℝ σ ( k ) ( x , y ) ≥ α } , ℝ α + ( x ) = { y ∈ V | h ℝ ( x , y ) ≻ { α } } = { y ∈ V | h ℝ σ ( k ) ( x , y ) > α } .

According to Definition 3.2, if ( x , y ) ∈ ℝ α , it indicates that the membership degree of the relationships of x and y with respect to the HF relation ℝ is not less than α. If ( x , y ) ∈ ℝ α + , it indicates that the membership degree of the relationships of x and y with respect to the HF relation ℝ is more than α.

In the following several special HF compatible relations are introduced. We only take the HF compatible relation ℝ α as the example and the other HF compatible relation ℝ α + is similar to ℝ α .

Definition 3.3. For a hesitant fuzzy compatible relation ℝ α from U to V, we say that ℝ α is serial if for each x ∈ U, there exists a y ∈ V such that y ∈ ℝ α ( x ) .

For an HF relation ℝ on U, ℝ α is an HF compatible relation on U.

•If for each x ∈ U, x ∈ ℝ α ( x ) , the HF compatible relation ℝ α is referred as reflexive;

•If for each x, y ∈ U, y ∈ ℝ α ( x ) implies x ∈ ℝ α ( y ) , the HF compatible relation ℝ α is referred as symmetric;

•If for each x, y, z ∈ U, y ∈ ℝ α ( x ) and z ∈ ℝ α ( y ) imply z ∈ ℝ α ( x ) , the HF compatible relation ℝ α is referred as transitive.

Theorem 3.4. Suppose that ℝ is a hesitant fuzzy relation from U to V and ℝ α is a hesitant fuzzy compatible relation from U to V.

(1) If ℝ is serial, then ℝ α and ℝ α + are serial.

Suppose that ℝ is a hesitant fuzzy relation on U and ℝ α is a hesitant fuzzy compatible relation on U, then

(2) If ℝ is reflexive, ℝ α and ℝ α + are reflexive;

(3) If ℝ is symmetric, ℝ α and ℝ α + are symmetric;

(4) If ℝ is transitive, ℝ α and ℝ α + are transitive.

Proof. The conclusions (1), (2) and (3) are straightforward. Here we are only to prove that the conclusion (4) is true.

For any y ∈ ℝ α ( x ) and z ∈ ℝ α ( y ) , by Definition 3.2, we obtain h ℝ σ ( k ) ( x , y ) ≥ α and h ℝ σ ( k ) ( y , z ) ≥ α . Thus, ⋁ y ∈ U ( h ℝ σ ( k ) ( x , y ) ∧ h ℝ σ ( k ) ( y , z ) ) ≥ α . Noting that ℝ is transitive, according to Definition 3.1, we have h ℝ σ ( k ) ( x , z ) ≥ ⋁ y ∈ U ( h ℝ σ ( k ) ( x , y ) ∧ h ℝ σ ( k ) ( y , z ) ) ≥ α . Namely, h ℝ σ ( k ) ( x , z ) ≥ α , which implies that z ∈ ℝ α ( x ) . From the above discussions, we conclude that ℝ α is transitive. Similarly, we can prove that ℝ α + is transitive.

Theorem 3.5. Suppose that ℝ α is a hesitant fuzzy compatible relation from U to V. Then

(1) ( ∼ ℝ ) α = ∼ ℝ ( 1 - α ) + , ( ∼ ℝ ) α + = ∼ ℝ 1 - α ;

(2) ℝ ⊑ S ⇒ ℝ α ⊆ S α , ℝ α + ⊆ S α + ;

(3) α ≤ β ⇒ ℝ β ⊆ ℝ α , ℝ β + ⊆ ℝ α + ;

(4) ( ℝ Cap S ) α = ℝ α ∩ S α , ( ℝ Cap S ) α + = ℝ α + ∩ S α + ;

(5) ( ℝ Cup S ) α = ℝ α ∪ S α , ( ℝ Cup S ) α + = ℝ α + ∪ S α + .

Proof. (1) Since h ( ∼ ℝ ) ( x , y ) = { 1 - h ℝ σ ( k ) ( x , y ) | k = 1 , ⋯ , l ( h ℝ ( x , y ) ) } , in terms of Definition 3.2, we have ( ∼ ℝ ) α = { ( x , y ) | 1 - h ℝ σ ( k ) ( x , y ) ≥ α } = { ( x , y ) | h ℝ σ ( k ) ( x , y ) ≤ 1 - α } . Noting that ℝ ( 1 - α ) + = { ( x , y ) | h ℝ σ ( k ) ( x , y ) > 1 - α } , we deduce that ∼ ℝ ( 1 - α ) + = { ( x , y ) | h ℝ σ ( k ) ( x , y ) ≤ 1 - α } . Thus ( ∼ ℝ ) α = ∼ ℝ ( 1 - α ) + . Similarly, we can prove that ( ∼ ℝ ) α + = ∼ ℝ 1 - α .

(2) and (3) can be proved by Definition 3.2.

(4) Since h ( ℝ Cap S ) ( x , y ) = { h A σ ( k ) ( x , y ) ∧ h B σ ( k ) ( x , y ) | k = 1, 2, …, l}, we have ( ℝ Cap S ) α = { ( x , y ) | h A σ ( k ) ( x , y ) ∧ h B σ ( k ) ( x , y ) ≥ α } = { ( x , y ) | h A σ ( k ) ( x , y ) ≥ α } ∩ { h B σ ( k ) ( x , y ) ≥ α } = ℝ α ∩ S α . Similarly, we may verify ( ℝ Cap S ) α + = ℝ α + ∩ S α + .

(5) Since h ( ℝ Cup S ) ( x , y ) = { h A σ ( k ) ( x , y ) ∨ h B σ ( k ) ( x , y ) | k = 1, 2, …, l}, we have ( ℝ Cup S ) α = { ( x , y ) | h A σ ( k ) ( x , y ) ∨ h B σ ( k ) ( x , y ) ≥ α } = { ( x , y ) | h A σ ( k ) ( x , y ) ≥ α } ∪ { h B σ ( k ) ( x , y ) ≥ α } = ℝ α ∪ S α . Similarly, we can prove that ( ℝ Cup S ) α + = ℝ α + ∪ S α + .

3.2 Hesitant fuzzy compatible rough sets

In this subsection, by employing the HF compatible relation ℝ α , an HF compatible rough set is introduced as follows.

Definition 3.6. Suppose that U, V are two nonempty and finite universes of discourse, and ℝ α is a hesitant fuzzy compatible relation from U to V. We call the triple ( U , V , ℝ α ) a hesitant fuzzy compatible approximation space. For ∀A ∈ P (V), the lower and upper approximations of A with respect to (w.r.t.) ( U , V , ℝ α ) , denoted by ℝ α _ ( A ) and ℝ α ¯ ( A ) , are defined as follows: ℝ α _ ( A ) = { x ∈ U | ℝ α ( x ) ⊆ A } ,(1) ℝ α ¯ ( A ) = { x ∈ U | ℝ α ( x ) ∩ A ≠ ∅ } .(2)

We respectively call ℝ α _ ( A ) and ℝ α ¯ ( A ) the lower and upper approximations of A w.r.t. ( U , V , ℝ α ) . The pair ( ℝ α _ ( A ) , ℝ α ¯ ( A ) ) is named as the HF compatible rough sets model of A w.r.t. ( U , V , ℝ α ) , and ℝ α _ , ℝ α ¯ : P ( V ) → P ( U ) are called lower and upper HF compatible rough approximation operators, respectively.

Furthermore, the positive region pos ℝ α ( A ) , negative region neg ℝ α ( A ) and boundary region bn ℝ α ( A ) of A about ℝ α are respectively defined as follows:

pos ℝ α ( A ) = ℝ α _ ( A ) , neg ℝ α ( A ) = U - ℝ α ¯ ( A ) , bn ℝ α ( A ) = ℝ α ¯ ( A ) - ℝ α _ ( A ) .

In Definition 3.6, the lower and upper approximations of A w.r.t. ( U , V , ℝ α ) are introduced by us. Similarly, the lower and upper approximations of A w.r.t. ( U , V , ℝ α + ) are given as: ℝ α + _ ( A ) = { x ∈ U | ℝ α + ( x ) ⊆ A } ,(3) ℝ α + ¯ ( A ) = { x ∈ U | ℝ α + ( x ) ∩ A ≠ ∅ } .(4)

In a later section, we will only consider the case of the lower and upper approximations w.r.t ( U , V , ℝ α ) .

Remark 3.7. It is worthy noting that if the HF relation ℝ degenerates to a fuzzy relation, the HF compatible rough set in Definition 3.6 converts into fuzzy rough sets on two different universes in [28].

Remark 3.8. In Definition 3.6, if α = 1 and h ℝ ( x , y ) has only one element, then the HF relation ℝ converts into a crisp binary relation. In that case, HF compatible rough sets degenerate to the generalized crisp rough set model in [49].

Remark 3.9. If there is only one element in HF elements h ℝ ( x , y ) , α = 1 and U = V, then the HF relation ℝ is a crisp binary relation on U. Furthermore, if the HF relation ℝ is also reflexive, symmetric and transitive, in terms of Theorem 3.4, HF compatible rough sets will be degenerated to the classical Pawlak rough set.

Theorem 3.10. Suppose that ( U , V , ℝ α ) is a hesitant fuzzy compatible approximation space. Then the following properties hold: ∀A, B ∈ P (V),

(HFCL1) ℝ α _ ( A ) = ∼ ℝ α ¯ ( ∼ A ) ,

(HFCU1) ℝ α ¯ ( A ) = ∼ ℝ α _ ( ∼ A ) ,

(HFCL2) A ⊆ B ⇒ ℝ α _ ( A ) ⊆ ℝ α _ ( B ) ,

(HFCU2) A ⊆ B ⇒ ℝ α ¯ ( A ) ⊆ ℝ α ¯ ( B ) ,

(HFCL3) ℝ α _ ( A ∩ B ) = ℝ α _ ( A ) ∩ ℝ α _ ( B ) ,

(HFCU3) ℝ α ¯ ( A ∪ B ) = ℝ α ¯ ( A ) ∪ ℝ α ¯ ( B ) ,

(HFCL4) ℝ α _ ( A ∪ B ) ⊇ ℝ α _ ( A ) ∪ ℝ α _ ( B ) ,

(HFCU4) ℝ α ¯ ( A ∩ B ) ⊆ ℝ α ¯ ( A ) ∩ ℝ α ¯ ( B ) ,

(HFCL5) ℝ α _ ( V ) = U ,

(HFCU5) ℝ α ¯ ( ∅ ) = ∅ .

Proof. It is straightforward by Definition 3.6.

Theorem 3.10 above shows that HF compatible rough approximation operators ℝ α ¯ and ℝ α _ are dual to each other.

Theorem 3.11. Let U, V be two nonempty and finite universes, α₁, α₂ ∈ (0, 1] and α₁ ≤ α₂. Suppose that ℝ α 1 and ℝ α 2 are two HF compatible relations. For any A ∈ P (V), then the following holds:

(1) ℝ α 1 _ ( A ) ⊆ ℝ α 2 _ ( A ) ,   (2) ℝ α 1 ¯ ( A ) ⊇ ℝ α 2 ¯ ( A ) .

Proof. It is straightforward.

Theorem 3.12. Suppose that U, V are two nonempty and finite universes, and ℝ and S are two hesitant fuzzy relations from U to V. If ℝ ⊑ S , then, for all A ∈ P (V),

(1) S α _ ( A ) ⊆ ℝ α _ ( A ) ,   (2) ℝ α ¯ ( A ) ⊆ S α ¯ ( A ) .

Proof. It is straightforward.

In what follows the connections between the properties of several special HF compatible relations and the properties of HF compatible rough approximation operators are further established.

Theorem 3.13. Assume that ℝ α is an HF compatible relation from U to V, and ℝ α _ and ℝ α ¯ are respectively the lower and upper HF compatible approximation operators; then ℝ α is serial if and only if one of the following properties holds:

( HFCL 0 ) ℝ α _ ( ∅ ) = ∅ ,

( HFCU 0 ) ℝ α ¯ ( V ) = U ,

( HFCLU 0 ) ℝ α _ ( A ) ⊆ ℝ α ¯ ( A ) , ∀ A ∈ P ( V ) .

Theorem 3.14. Assume that ℝ α is an HF compatible relation on U. Let R _ α , R ¯ α be the HF compatible approximation operators induced from ( U , ℝ α ) ; then, ∀A ∈ P (U) ℝ α i s r e f l e x i v e & ⇔ ( H F C L R ) ℝ α _ ( A ) ⊆ A , & ⇔ ( H F C U R ) A ⊆ ℝ α ¯ ( A ) . & ℝ α is symmetric & ⇔ ( HFCLS ) ℝ α ¯ ( ℝ α _ ( A ) ) ⊆ A , & ⇔ ( HFCUS ) A ⊆ ℝ α _ ( ℝ α ¯ ( A ) ) . ( 3 ) ℝ α is transitive ⇔ ( HFCLT ) ℝ α _ ( A ) ⊆ ℝ α _ ( ℝ α _ ( A ) ) , ⇔ ( HFCUT ) ℝ α ¯ ( ℝ α ¯ ( A ) ) ⊆ ℝ α ¯ ( A ) .

Proof. (1) According to the dual properties of HF compatible approximation operators, we need to prove that

ℝ is reflexive ⇔ ( HFCUR ) A ⊆ ℝ α ¯ ( A ) . If A ⊈ ℝ α ¯ ( A ) , then there exists a x₀ ∈ A such that x 0 ∉ ℝ α ¯ ( A ) , which implies that ℝ α ( x 0 ) ∩ A = ∅ . So it follows that ℝ α ( x 0 ) = ∅ , which contradicts the reflexivity of ℝ α . Hence we conclude that (HFCUR) holds.

Conversely, suppose that (HFCUR) holds. Let A = {x}, then x ∈ ℝ α ¯ ( { x } ) , from which we see that ℝ α ( x ) ∩ { x } ≠ ∅ . Namely, x ∈ ℝ α ( x ) . Consequently, ℝ α is reflexive.

(2) Based on the dual properties of HF compatible approximation operators, we need to prove that

ℝ α is symmetric ⇔ ( HFCUS ) A ⊆ ℝ α _ ( ℝ α ¯ ( A ) ) . Assume that ℝ α is symmetric. For all x ∈ A and y ∈ ℝ α ( x ) , by the assumption we have x ∈ ℝ α ( y ) . Thus ℝ α ( y ) ∩ A ≠ ∅ , which implies that y ∈ ℝ α ¯ ( A ) . It means that ℝ α ( x ) ⊆ ℝ α ¯ ( A ) . So by Equation (1), it follows that x ∈ ℝ α _ ( ℝ α ¯ ( A ) ) . Hence A ⊆ ℝ α _ ( ℝ α ¯ ( A ) ) .

Conversely, assume that A ⊆ ℝ α _ ( ℝ α ¯ ( A ) ) . Let A = {x}, then we obtain x ∈ ℝ α _ ( ℝ α ¯ ( { x } ) ) . This means that ℝ α ( x ) ⊆ ℝ α ¯ ( { x } ) . For any y ∈ ℝ α ( x ) , then y ∈ ℝ α ¯ ( { x } ) , which implies that ℝ α ( y ) ∩ { x } ≠ ∅ . Namely, x ∈ ℝ α ( y ) . Therefore, we conclude that ℝ α is symmetric.

(3) Since (HFCLT) and (HFCUT) are equivalent, we only prove that ℝ α is transitive ⇔ (HFCUT).

Suppose that ℝ α is transitive. For all x ∈ ℝ α ¯ ( ℝ α ¯ ( A ) ) , by Equation (2), we obtain ℝ α ( x ) ∩ ℝ α ¯ ( A ) ≠ ∅ . Then there exists a y ∈ ℝ α ( x ) such that y ∈ ℝ α ¯ ( A ) . It means that ℝ α ( y ) ∩ A ≠ ∅ . Namely, there exists a z ∈ ℝ α ( y ) such that z ∈ A. Since ℝ α is transitive, we see that for all x, y, z ∈ U, y ∈ ℝ α ( x ) and z ∈ ℝ α ( y ) imply that z ∈ ℝ α ( x ) . Thus ℝ α ( x ) ∩ A ≠ ∅ , which implies that x ∈ ℝ α ¯ ( A ) . Consequently, ℝ α ¯ ( ℝ α ¯ ( A ) ) ⊆ ℝ α ¯ ( A ) .

Conversely, assume that ℝ α ¯ ( ℝ α ¯ ( A ) ) ⊆ ℝ α ¯ ( A ) . Let A = {z}, then we have ℝ α ¯ ( ℝ α ¯ ( { z } ) ) ⊆ ℝ α ¯ ( { z } ) . On the other hand, let y ∈ ℝ α ( x ) and z ∈ ℝ α ( y ) . For all z ∈ ℝ α ( y ) , we can obtain y ∈ ℝ α ¯ ( { z } ) . Thus ℝ α ( x ) ∩ ℝ α ¯ ( { z } ) ≠ ∅ , which implies that x ∈ ℝ α ¯ ( ℝ α ¯ ( { z } ) ) . Namely, x ∈ ℝ α ¯ ( { z } ) . By Equation (2), we have ℝ α ( x ) ∩ { z } ≠ ∅ . Thus z ∈ ℝ α ( x ) . Consequently, we conclude that ℝ α is transitive.

By Theorem 3.14, the following conclusion is straightforward.

Corollary 3.15. Suppose that ℝ α is a reflexive and transitive HF compatible relation. Let ℝ α _ , ℝ α ¯ be the lower and upper HF compatible approximation operators; then, ∀A ∈ P (U), ( HFCLRT ) ℝ α _ ( A ) = ℝ α _ ( ℝ α _ ( A ) ) ,

( HFCURT ) ℝ α ¯ ( ℝ α ¯ ( A ) ) = ℝ α ¯ ( A ) .