**2. Preliminaries**

In this section, we recall some basic notions and definitions which will be used in this paper.

**Definition 1** ([1])**.** *Let U be a non-empty finite universe and R be an equivalence relation on U. We use U*/*R to denote the family of all equivalence classes of R (or classifications of U), and* [*x*]*R to denote an equivalence class of R containing the element x* ∈ *U. The pair* (*<sup>U</sup>*, *R*) *is called an approximation space. For any X* ⊆ *U, we can define the lower and upper approximations of X as follows:*

$$\underline{R}(X) = \{ \mathbf{x} \in \mathcal{U} : [\mathbf{x}]\_{\mathcal{R}} \subseteq X \},$$

$$\overline{R}(X) = \{ \mathbf{x} \in \mathcal{U} : [\mathbf{x}]\_{\mathcal{R}} \cap X \neq \emptyset \}.$$

*The pair* (*R*(*X*), *R*(*X*)) *is referred to as the rough set of X. The rough set* (*R*(*X*), *R*(*X*)) *gives rise to a description of X under the present knowledge, i.e., the classification of U.*

*Furthermore, the positive region, negative region, and boundary region of X about the approximation space* (*<sup>U</sup>*, *R*) *are defined as follows, respectively:*

$$pro(X) = \underline{R}(X), \neg \underline{\operatorname{reg}}(X) = \sim \overline{\mathcal{R}}(X), \operatorname{bm}(X) = \overline{\mathcal{R}}(X) - \underline{\mathcal{R}}(X),$$

*where* ∼ *X stands for complementation of the set X.*

**Definition 2** ([7])**.** *Let E be the set of parameters with the connection to the objects in U. A pair* (*<sup>F</sup>*, *E*) *is called a soft set over U, where F is a mapping given by F* : *E* −→ *<sup>P</sup>*(*U*)*, P*(*U*) *is a set of all subsets of U.*

*This definition shows that a soft set over U is a parameterized family of subsets of the universe U. For e* ∈ *E*, *<sup>F</sup>*(*e*) *is regarded as the set of e-approximate elements of the soft set* (*<sup>F</sup>*, *<sup>E</sup>*)*.*

**Definition 3** ([5])**.** *Given a non-empty subset A of X, a hesitant fuzzy set HX* = {(*<sup>x</sup>*, *hX*(*x*) : *x* ∈ *X*)} *on X satisfying the following condition:*

> *hX*(*x*) = *φ* for all *x* ∈/ *A*

*is called a hesitant fuzzy set related to A (briefly, A-hesitant fuzzy set) on X and is represented by HA* = {(*<sup>x</sup>*, *hA*(*x*) : *x* ∈ *<sup>X</sup>*)}*, where hA is a mapping from X to p*([0, 1]) *with hA*(*x*) = *φ for all x* ∈/ *A.*

**Definition 4** ([34])**.** *Let H* ˜ (*U*) *be the set of all hesitant fuzzy sets in U. A pair* (*F*˜, *A*˜) *is called a hesitant fuzzy soft set over U, where F is a mapping given by* ˜

$$
\mathcal{F}: A \longrightarrow \mathcal{R}(\mathcal{U}).
$$

*A hesitant fuzzy soft set is a mapping from parameters to H* ˜ (*U*)*. It is a parameterized family of hesitant fuzzy subsets of U. For e* ∈ *A, F* ˜(*e*) *may be considered as the set of e-approximate elements of the hesitant fuzzy soft set* (*F*˜, *<sup>A</sup>*)*.*

#### **3. Multi** *Q***-Hesitant Fuzzy Soft Sets**

We first introduce the notion of *Q*-hesitant fuzzy soft sets as a generalization of *Q*-fuzzy soft sets.

**Definition 5.** *Let U be a universal set and Q be non-empty set. A Q-hesitant fuzzy set AQ is a set given by*

$$A\_Q = \{ \langle (uq), h\_{A\_Q}(uq) \rangle : u \in \mathcal{U}, q \in Q \},$$

*where hAQ* : *U* × *Q* −→ [0, 1]*. The function hAQ* (*uq*) *is called the membership function of Q-hesitant fuzzy set, and the set of all Q-hesitant fuzzy sets over U* × *Q will be denoted by QHF*(*U* × *Q*)*.*

**Definition 6.** *Let U be a non-empty finite universe and Q be a non-empty set. For any AQ*, *BQ* ∈ *QHF*(*U* × *Q*)*, then, for all u* ∈ *U*, *q* ∈ *Q, we have*


3. *AQ* ∩ *BQ* = {(*uq*), *hAQ* (*uq*) ∧ *hBQ* (*uq*), *u* ∈ *U*, *q* ∈ *Q*}.


**Definition 7.** *Let U be a universal set and Q be non-empty set, I be a unit interval* [0, 1] *and k be a positive integer. A multi Q-hesitant fuzzy set H* ˜ *Q in U* × *Q is a set defining by*

$$\tilde{H}\_Q = \{ \langle (uq), h\_{\tilde{H}\_Q}^i(uq) \rangle : u \in \mathcal{U}, q \in Q \text{ for all } i = 1, 2, \dots, k \}, \ell$$

*where hiH*˜ *Q* : *U* × *Q* −→ *Ik for all i* = 1, 2, ··· , *k*. *The function h*1*H*˜ *Q* (*uq*), *h*2*H*˜ *Q* (*uq*), ··· , *hkH*˜ *Q* (*uq*) *is called the membership function of multi Q-hesitant fuzzy set and k is called the dimension of hiH*˜ *Q . The set of all multi Q-hesitant fuzzy set of dimension k in U* × *Q is denoted by MkQHFS*(*U* × *Q*)*.*

**Definition 8.** *Let AQ, BQ be a multi Q-hesitant fuzzy sets over U* × *Q. Then, AQ is said to be a multi Q-hesitant fuzzy subset of BQ if*

$$h\_{A\_Q}^i(uq) \le h\_{B\_Q}^i(uq)$$

*holds for any u* ∈ *U*, *q* ∈ *Q*, *i* = *i*, 2, ··· , *k and it is denoted by AQ* ⊆ *BQ.*

**Definition 9.** *Let U be a universal set and be non-empty set, E be the set of parameters and MkQHF*(*U* × *Q*) *be the set of all multi Q-hesitant fuzzy sets on U* × *Q with the dimension k. Let A* ⊆ *E the pair* (*HQ*, *A*) *is called a multi Q-hesitant fuzzy soft set* (*MkQHFSS*) *over U, where* (*HQ*, *A*) *is given by the form*

$$\{(H\_{\mathbb{Q}'}A) = \{ (\varepsilon, h\_{\mathbb{Q}}^i(\varepsilon)) : \varepsilon \in A, h\_{\mathbb{Q}}^i(\varepsilon) \in M^k \mathbb{Q}HFS(\mathcal{U} \times \mathbb{Q}) \},$$

*where hiQ* : *A* −→ *MkQHF*(*U* × *Q*) *such that <sup>h</sup>iQ*(*e*) = *φ if e* ∈/ *A. The set of all multi Q-hesitant fuzzy soft sets over U* × *Q will be denoted by MkQHFSS*(*U* × *Q*)*.*

**Example 1.** *Suppose that a company wants to buy three types of products from two brands and wants to take the opinion of two specialists about these products (k= 2). Let U* = {*<sup>u</sup>*1, *u*2, *<sup>u</sup>*3} *be a set of products, Q* = {*p , q*} *be a set of brands, and E* = {*<sup>e</sup>*1 = *easy to use*,*e*2 = *quality*,*e*<sup>3</sup> = *price*} *is the set of decision parameters. Then we can define the multi Q-hesitant fuzzy soft sets*(*HQ*, *A*) *as follows:*

$$(H\_{Q\_{\prime}}A) = \{ \langle e\_1, \left( \frac{u\_1 p}{(0.2, 0.3)(0.1)} \right), \left( \frac{u\_1 q}{(0.1, 0.3)(0.4, 0.8)} \right), \left( \frac{u \imath p}{(0.6, 0.5)(0.2, 0.2)} \right), \left( \frac{u \imath q}{(0.2, 0.4)(0.1)} \right) \},$$

$$\langle e\_{2\prime}, \left( \frac{u\_2 p}{(0.3, 0.1)(0.2, 0.3, 0.6)} \right), \left( \frac{u\_2 q}{(0.5, 0.3)(0.5, 0.5, 0.2)} \right), \left( \frac{u \imath p}{(0.2, 0.2)(0.4)} \right), \left( \frac{u \imath q}{(0.7, 0.3)(0.2, 0.9)} \right) \rangle,$$

$$\langle e\_{3\prime}, \left( \frac{u\_1 p}{(0.1, 0.1)(0.4, 0.4)} \right), \left( \frac{u\_1 q}{(0.1, 0.3)(0.7, 0.6)} \right), \left( \frac{u\_2 p}{(0.4, 0.3)(0.4, 0.1)} \right), \left( \frac{u\_2 q}{(0.2, 0.6)(0.7, 0.3)} \right) \rangle.$$

**Definition 10.** *Let* (*HQ*, *A*) *and* (*FQ*, *B*) *be two multi Q-hesitant fuzzy soft sets,* (*HQ*, *A*) *is said to be multi Q-hesitant fuzzy soft subset of* (*FQ*, *B*) *if A* ⊆ *B and HQ*(*e*) ⊆ *FQ*(*e*) *for all e* ∈ *E and denoted by* (*HQ*, *A*) ⊆ (*FQ*, *<sup>B</sup>*)*.*

**Proposition 1.** *Let* (*HQ*, *<sup>A</sup>*)*,* (*FQ*, *B*) *and* (*GQ*, *C*) *be three multi Q-hesitant fuzzy soft sets. Then,*


**Definition 11.** *A multi Q-hesitant fuzzy soft set* (*HQ*, *A*) *of dimension k over U* × *Q is called the null multi Q-hesitant fuzzy soft set if HQ*(*e*) = *φk for all e* ∈ *A and it is denoted by φkA.*

**Definition 12.** *A multi Q-hesitant fuzzy soft set* (*HQ*, *A*) *of dimension k over U* × *Q is called the absolute multi Q-hesitant fuzzy soft set if HQ*(*e*) = 1*k for all e* ∈ *A and it is denoted by UkA.*

**Definition 13.** *Let* (*HQ*, *A*) *be a multi Q-hesitant fuzzy soft set of dimension k over U* × *Q. Then, the complement of* (*HQ*, *A*) *is denoted by* (*HQ*, *A*)*c and defined by* (*HQ*, *<sup>A</sup>*)*c=*(*HcQ*, *<sup>A</sup>*), *where HcQ* : *A* −→ *MkQHFS*(*U* × *Q*) *is mapping given by HcQ*(*e*)=(*HQ*(*e*))*<sup>c</sup> for all e* ∈ *A.*

**Remark 1.** *Clearly,* ((*HQ*, *A*)*c*)*c=* (*HQ*, *A*) *and* (*φkA*)*c* = *UkA,* (*UkA*)*c* = *φkA.*

**Definition 14.** *The union of two multi Q-hesitant fuzzy soft sets of dimension k over U,* (*HQ*, *A*) *and* (*FQ*, *B*) *is the multi Q-hesitant fuzzy soft set* (*GQ*, *<sup>C</sup>*)*, where C* = *A* ∪ *B, and for all e* ∈ *C, GQ*(*e*) = *HQ*(*e*) ∪ *FQ*(*e*). *We write* (*HQ*, *A*) ∪ (*FQ*, *<sup>B</sup>*)=(*GQ*, *<sup>C</sup>*).

**Definition 15.** *The intersection of of two multi Q-hesitant fuzzy soft sets of dimension k over U,* (*HQ*, *A*) *and* (*FQ*, *B*) *with A* ∩ *B* = *φ is the multi Q-hesitant fuzzy soft set* (*GQ*, *<sup>C</sup>*)*, where C* = *A* ∩ *B, and for all e* ∈ *C,*

$$G\_Q(e) = \begin{cases} \begin{array}{c} H\_Q(e) \quad \text{for} \quad e \in A - B, \\ F\_Q(e) \quad \text{for} \quad e \in B - A, \\ H\_Q(e) \cup F\_Q(e) \quad \text{for} \quad e \in A \cap B. \end{array} \end{cases}$$

*In this case, we write* (*HQ*, *A*) ∩ (*FQ*, *<sup>B</sup>*)=(*GQ*, *<sup>C</sup>*).

**Theorem 1.** *Let* (*HQ*, *A*) *and* (*FQ*, *B*) *be two multi Q-hesitant fuzzy soft sets of dimension k over U* × *Q. Then,*



#### **4. Multi** *Q***-Hesitant Fuzzy Soft Rough Set**

**Definition 16.** *Let* (*HQ*, *A*) *be a multi Q-hesitant fuzzy soft set over U* × *Q. A multi Q-hesitant fuzzy subset of* (*U* × *Q*) × (*E* × *Q*) *is called a multi Q-hesitant fuzzy soft relation (MkQHFSR*) *from* (*U* × *Q*) *to* (*E* × *Q*) *given by*

$$\mathcal{R}\_Q = \{ \langle (uq, eq), h\_{\mathcal{R}\_Q}^i (uq, eq) \rangle, \mathfrak{u}q \in \mathcal{U} \times \mathcal{Q}, \mathfrak{e}q \in \mathcal{E} \times \mathcal{Q}, i = 1, 2, \dots, k \},$$

*where hiRQ* : (*U* × *Q*) × (*E* × *Q*) −→ [0, 1]*k.*

**Definition 17.** *Let U be nonempty universe, Q be a nonempty set and E be the set of parameters. RQ is a multi Q-hesitant fuzzy soft relation RQ* ∈ *MkQHFSR*((*U* × *Q*) × (*E* × *Q*)) *and the triple* ((*<sup>U</sup>*, *Q*),(*<sup>E</sup>*, *Q*), *RQ*) *is multi Q-hesitant fuzzy soft approximation space. For any AQ* ∈ *<sup>M</sup>kQHFS*(*E*), *the lower and upper approximations of AQ with respect to* (*<sup>U</sup>*, *E*, *Q*, *RQ*) *denoted by RQ*(*AQ*) *and RQ*(*AQ*)*, are two multi Q-hesitant fuzzy soft sets, respectively, defined as follows:*

$$\begin{aligned} \underline{R\_{\underline{Q}}}(A\_{\underline{Q}}) &= \{ \langle (\iota \underline{\iota}q), h\_{\underline{R\_{\underline{Q}}}(A\_{\underline{Q}})}(\iota \underline{\iota}q) \rangle : (\iota \underline{\iota}q) \in \mathcal{U} \times \mathcal{Q} \}, \\ \overline{R\_{\underline{Q}}}(A\_{\underline{Q}}) &= \{ \langle (\iota \underline{\iota}q), h\_{\overline{\underline{R\_{Q}}}(A\_{\underline{Q}})}(\iota \underline{\iota}q) \rangle : (\iota \underline{\iota}q) \in \mathcal{U} \times \mathcal{Q} \}. \end{aligned}$$

*where*

$$\begin{split} h\_{\underline{R\_{Q}(A\_{Q})}}(\iota\eta) &= \{ \langle \bigwedge\_{\epsilon \in E} \{ (1 - h\_{R\_{Q}}^{i}(\iota\eta\_{\epsilon}\epsilon q)) \vee h\_{A\_{Q}}^{i}(\epsilon q) \} \rangle : (\iota\eta) \in \mathsf{U} \times \mathsf{Q}, i = 1, 2, \ldots, k \}, \\ h\_{\overline{R\_{Q}(A\_{Q})}}(\iota\eta) &= \{ \langle \bigvee\_{\epsilon \in E} \{ h\_{R\_{Q}}^{i}(\iota\eta\_{\epsilon}\epsilon q) \wedge h\_{A\_{Q}}^{i}(\epsilon q) \} \rangle : (\iota\eta) \in \mathsf{U} \times \mathsf{Q}, i = 1, 2, \ldots, k \}. \end{split}$$

*RQ*(*AQ*) *and RQ*(*AQ*) *are, respectively, called the lower and upper Q-hesitant fuzzy soft rough approximations' operators. The pair* (*RQ*(*AQ*)*, RQ*(*AQ*)) *is called the multi Q-hesitant fuzzy soft rough set of AQ with respect to* (*<sup>U</sup>*, *E*, *Q*, *RQ*)*. Moreover, if RQ*(*AQ*) *= RQ*(*AQ*)*, then AQ is called definable.*

**Example 2.** *Suppose that U* = {*<sup>u</sup>*1, *u*2, *<sup>u</sup>*3} *is the set of cars that Mr X wants to buy and Q* = {*q*1, *q*2} *represents the companies of the different cars. They form the universe (U,Q) and let E* = {*<sup>e</sup>*1 = *size*,*e*<sup>2</sup> = *price*,*e*<sup>3</sup> = *colour*} *be the set of parameters. Consider a multi Q-hesitant fuzzy soft relation RQ* : *U* × *Q* −→ *E* × *Q with dimension k = 2 is given by Table 1.*


**Table 1.** Multi *Q*-hesitant fuzzysoft relation *RQ*.

*Now, if Mr X gives the optimum decision object AQ* ∈ *<sup>M</sup>kQHF*(*E*), *which is a Q-hesitant fuzzy subset defined as follows:*

*AQ* = {((*<sup>e</sup>*1*q*1), {(0.1, 0.3)(0.4, 0.5)}),((*<sup>e</sup>*1*q*2), {(0.2, 0.4)(0.5, 0.6)}),((*<sup>e</sup>*2*q*1), {(0.3, 0.6)(0.6, 0.7)}), ((*<sup>e</sup>*2*q*2), {(0.2, 0.5),(0.2, 0.8)})}.

*Then, by Definition 17, we have*

*hRQ* (*<sup>u</sup>*1*q*1) = D*e*∈*<sup>E</sup>*{(<sup>1</sup> − *<sup>h</sup>*<sup>2</sup>*RQ* )(*<sup>u</sup>*1*q*1,*eq*) ∨ *<sup>h</sup>*<sup>2</sup>*AQ* (*eq*)} = ({(0.8),(0.4, 0.6)}∨{(0.1, 0.3)(0.4, 0.5)}) ∧ ({(0.7, 0.3),(0.4)}∨{(0.2, 0.4)(0.5, 0.6)}) ∧ ({(0.5, 0.6, 0.4),(0.4, 0.5)}∨{(0.3, 0.6)(0.6, 0.7)}) ∧ ({(0.6, 0.8),(0.9, 0.7)}∨{(0.2, 0.5),(0.2, 0.8)}) = {(0.8, 0.8),(0.4, 0.6)}∧{(0.7, 0.4),(0.5, 0.6)}∧{(0.5, 0.6, 0.6),(0.6, 0.7)}∧{(0.6, 0.8),(0.9, 0.8)} = {(0.5, 0.4, 0.4),(0.4, 0.6)}*.*

*Similarly, we have*

$$\begin{aligned} h\_{\underline{R\_Q}}(u\_1 q\_2) &= \{ (0.2, 0.4, 0.4)(0.8, 0.6) \}, \\ h\_{\underline{R\_Q}}(u\_2 q\_1) &= \{ (0.5, 0.6)(0.4, 0.5, 0.7) \}, \\ h\_{\underline{R\_Q}}(u\_2 q\_2) &= \{ (0.2, 0.5)(0.4, 0.5, 0.5) \}, \\ h\_{\underline{R\_Q}}(u\_1 q\_1) &= \{ (0.3, 0.4, 0.6)(0.6, 0.6) \}, \\ h\_{\overline{R\_Q}}(u\_1 q\_2) &= \{ (0.3, 0.4, 0.4)(0.2, 0.7) \}, \\ h\_{\overline{R\_Q}}(u\_2 q\_1) &= \{ (0.3, 0.4)(0.5, 0.8, 0.8) \}, \\ h\_{\overline{R\_Q}}(u\_2 q\_2) &= \{ (0.3, 0.6)(0.5, 0.6, 0.5) \}. \end{aligned}$$

*Thus, we conclude that:*

> *RQ*(*AQ*) = {(*<sup>u</sup>*1*q*1), {(0.5, 0.4, 0.4),(0.4, 0.6)},(*<sup>u</sup>*1*q*2), {(0.2, 0.4, 0.4)(0.8, 0.6)}, (*<sup>u</sup>*2*q*1), {(0.5, 0.6)(0.4, 0.5, 0.7)},(*<sup>u</sup>*2*q*2), {(0.2, 0.5)(0.4, 0.5, 0.5)}}, *RQ*(*AQ*) = {(*<sup>u</sup>*1*q*1), {(0.3, 0.4, 0.6)(0.6, 0.6),(*<sup>u</sup>*1*q*2), {(0.3, 0.4, 0.4)(0.2, 0.7)}, (*<sup>u</sup>*2*q*1), {(0.3, 0.4)(0.5, 0.8, 0.8)},(*<sup>u</sup>*2*q*2), {0.3, 0.6)(0.5, 0.6, 0.5)}}.

*The pair* (*RQ*(*AQ*), *RQ*(*AQ*)) *is called a multi Q-hesitant fuzzy soft rough set with dimension 2.*

**Theorem 2.** *Let* (*<sup>U</sup>*, *E*, *Q*, *RQ*) *be multi Q-hesitant fuzzy soft approximation space. The lower and upper Q-hesitant fuzzy soft rough approximations operators RQ*(*AQ*) *and RQ*(*AQ*)*, respectively, for any AQ*, *BQ* ∈ *MkQHF*(*E*) *satisfy the following properties:*

*1. RQ*(*AcQ*)=(*RQ*(*AQ*))*c, RQ*(*AcQ*)=(*RQ*(*AQ*))*c, 2. AQ* ⊆ *BQ* ⇒ *RQ*(*AQ*) ⊆ (*RQ*(*BQ*)) *, AQ* ⊆ *BQ* ⇒ *RQ*(*AQ*) ⊆ (*RQ*(*AQ*))*, 3. RQ*(*AQ* ∩ *BQ*) = *RQ*(*AQ*) ∩ (*RQ*(*BQ*))*, RQ*(*AQ* ∪ *BQ*) = *RQ*(*AQ*) ∪ (*RQ*(*BQ*))*,4.RQ*(*AQ*∪*BQ*)⊇*RQ*(*AQ*)∪(*RQ*(*BQ*))*,RQ*(*AQ*∩*BQ*)⊆*RQ*(*AQ*)∩(*RQ*(*BQ*)).
