**Proof.** 1. By Definition 17, we have

*RQ*(*AcQ*) = {(*uq*), *<sup>h</sup>iRQ*(∼*AQ*)(*uq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = {(*uq*), <sup>D</sup>*e*∈*<sup>E</sup>*{*hi*∼*RQ* (*uq*,*eq*) ∨ *hi*∼*AQ* (*eq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = {(*uq*), ∼ (E*e*∈*<sup>E</sup>*{*hiRQ* (*uq*,*eq*) ∧ *hiAQ* (*eq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = {(*uq*), ∼ *<sup>h</sup>iRQ*(*AQ*)(*uq*) : (*uq*) ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = (*RQ*(*AQ*))*<sup>c</sup>*.

Similarly, we can obtain that *RQ*(*AcQ*)=(*RQ*(*AQ*))*<sup>c</sup>*.


Hence, *RQ*(*AQ* ∩ *BQ*) = *RQ*(*AQ*) ∩ *RQ*(*BQ*).

Similarly, we can prove that *RQ*(*AQ* ∩ *BQ*) = *RQ*(*AQ*) ∩ *RQ*(*BQ*).

$$\begin{split} 4. \quad \underline{R\_{Q}(A\_{Q}\cup B\_{Q})} &= \{ \langle (uq), h\_{\overline{R}\_{Q}(A\_{Q}\cup B\_{Q})}^{\dagger}(uq) \rangle : uq \in \mathcal{U}\times Q, i = 1, 2, \ldots, k \} \\ &= \{ \langle (uq), \wedge\_{\alpha\in E}(1 - h\_{\overline{R}\_{Q}}^{\dagger})(uq, eq) \vee h\_{A\_{Q}\cup B\_{Q}}^{\dagger}(eq) \rangle : uq \in \mathcal{U}\times Q, i = 1, 2, \ldots, k \} \\ &= \{ \langle (uq), \wedge\_{\alpha\in E}(1 - h\_{\overline{R}\_{Q}}^{\dagger})(uq, eq) \vee (h\_{A\_{Q}}^{\dagger}(eq) \vee h\_{\overline{R}\_{Q}}^{\dagger}(eq)) \rangle : uq \in \mathcal{U}\times Q, i = 1, 2, \ldots, k \} \\ &= \{ \langle (uq), \wedge\_{\mathbb{R}\in E}((1 - h\_{\overline{R}\_{Q}}^{\dagger})(uq, eq) \vee h\_{A\_{Q}}^{\dagger}(eq)) \rangle \vee \left( \bigwedge\_{u\in E}((1 - h\_{\overline{R}\_{Q}}^{\dagger})(uq, eq) \vee h\_{\overline{R}\_{Q}}^{\dagger}(eq)) \right) \rangle : uq \in \mathcal{U}\times Q, i = 1, 2, \ldots, k \} \\ &= \{ (\langle uq \rangle, h\_{\overline{R}\_{Q}(A\_{Q})}^{\dagger}(uq) \vee h\_{\overline{R}\_{Q}(B\_{Q})}^{\dagger}(uq) \rangle : uq \in \mathcal{U}\times Q, i = 1, 2, \ldots, k \} \\ &= \underline{R\_{Q}(A\_{Q})} \overline{\cup (\underline{R\_{Q}(B\_{Q})})}. \end{split}$$

Hence, *RQ*(*AQ* ∪ *BQ*) = *RQ*(*AQ*) ∪ *RQ*(*BQ*).

Similarly, we can prove that *RQ*(*AQ* ∪ *BQ*) = *RQ*(*AQ*) ∪ *RQ*(*BQ*).

**Theorem 3.** *Let RQ, SQ be multi Q-hesitant fuzzy soft relations from* (*U* × *Q*) *to* (*E* × *Q*) *, if RQ* ⊆ *SQ, for any A* ∈ *QHF*(*E*), *then:*

*1. RQ*(*AQ*) ⊇ *SQ*(*AQ*), *2. RQ*(*AQ*) ⊆ *SQ*(*AQ*).

**Proof.** 1. If *RQ* ⊆ *SQ* , then, by Definition 8, we have *hiRQ* (*uq*,*eq*) ≤ *hiSQ* (*uq*,*eq*) for all *uq* ∈ *U* × *Q*, *eq* ∈ *E* × *Q*, then *RQ*(*AQ*) = {(*uq*), *<sup>h</sup>iRQ*(*AQ*)(*uq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = {(*uq*), D*e*∈*<sup>E</sup>*{(<sup>1</sup> − *hiRQ* )(*uq*,*eq*) ∨ *hiAQ* (*eq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} ≥ {(*uq*), D*e*∈*<sup>E</sup>*{(<sup>1</sup> − *hiSQ* )(*uq*,*eq*) ∨ *hiAQ* (*eq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = {(*uq*), *<sup>h</sup>iSQ*(*AQ*)(*uq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = *SQ*(*AQ*).

2. Similarly, it can be proved.

#### **5. Multi** *Q***-Hesitant Fuzzy Soft Multi-Granulation Rough Set**

**Definition 18.** *Let U be a universal set and Q be non-empty set, and E be the set of parameters and RQj ,(j=1,2,. . . ,m) be multi Qm-hesitant fuzzy soft relations over* (*U* × *Q*) × (*E* × *Q*)*, and* (*<sup>U</sup>*, *E*, *Q*, *RQj*) *be called multi Q-hesitant fuzzy soft multi-granulation approximation space, for any AQ* ∈ *<sup>M</sup>kQHF*(*E*)*, the optimistic lower and upper approximation of AQ with respect to* (*<sup>U</sup>*, *E*, *Q*, *RQj*) *are defined as follows:*

$$\begin{aligned} \sum\_{j=1}^{m} R\_{Q\_j} \stackrel{\circ}{\ }(A\_Q) &= \{ \langle (uq) \rangle \prime \underline{h\_{\sum\_{j=1}^{m} R\_{Q\_j}} ^\circ (A\_Q)} (u, q) \rangle : uq \in \mathcal{U} \times Q \},\\ \overline{\sum\limits\_{j=1}^{m} R\_{Q\_j} \ (A\_Q)} &= \{ \langle (uq) \rangle \prime \underline{h\_{\sum\_{j=1}^{m} R\_{Q\_j}} ^\circ (A\_Q)} (u, q) \rangle : uq \in \mathcal{U} \times Q \}. \end{aligned}$$

*where*

$$\begin{aligned} \frac{h\_{\sum\_{j=1}^m R\_{Q\_j}} ^\circ (A\_Q)}{} &= \{ (\bigvee\_{j=1}^m \bigwedge\_{i=1}^k \{ (1 - h\_{R\_{Q\_j}}^i)(uq, eq) \lor h\_{A\_Q}^i(eq) \} ) : uq \in \mathcal{U} \times Q \}, \\ h\_{\overline{\sum\_{j=1}^m R\_{Q\_j}} ^\circ (A\_Q)}{} &= \{ (\bigwedge\_{j=1}^m \bigvee\_{i=1}^k \{ h\_{R\_{Q\_i}}^i(uq, eq) \land h\_{A\_Q}^i(eq) \} ) : uq \in \mathcal{U} \times Q \}. \end{aligned}$$

*The pair* (∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*), ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*)) *is called an optimistic multi Q-hesitant fuzzy soft multi-granulation rough set of AQ with respect to* (*<sup>U</sup>*, *E*, *Q*, *RQj*)*.*

**Theorem 4.** *Let* (*<sup>U</sup>*, *E*, *Q*, *RQj*) *be multi Q-hesitant fuzzy soft multi-granulation approximation space and RQj* ∈ *MkQHFSR*((*U* × *Q*) × (*E* × *Q*))*,(j =1,2,...,m) be multi Qm hesitant fuzzy soft relations over* (*U* × *Q*) × (*E* × *Q*)*, for any AQ*, *BQ* ∈ *<sup>M</sup>kQHF*(*E*)*, the optimistic lower and upper approximation satisfy the following properties:*

$$\begin{array}{ll} 1. & \frac{\sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{\prime} (A\_{Q}^{c})}{\overbrace{\sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{\prime}} (A\_{Q}^{c})} = (\sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{\prime} (A\_{Q}))^{c}, \\ & \frac{\sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{\prime}}{\sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{\prime}} (A\_{Q}^{c}) = (\sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{\prime} (A\_{Q}))^{c}. \end{array}$$

$$\begin{array}{ll} 2. & A\_Q \subseteq B\_Q \Rightarrow \sum\_{j=1}^m R\_{Qj} \, ^\circ(A\_Q) \subseteq \underbrace{\sum\_{j=1}^m R\_{Qj} \, ^\circ(B\_Q)}\_{\sum\_{j=1}^m R\_{Qj} \, ^\circ(A\_Q)} \; ^\circ(B\_Q) \\ & A\_Q \subseteq B\_Q \Rightarrow \overbrace{\sum\_{j=1}^m R\_{Qj} \, ^\circ(A\_Q)}\_{\sum\_{j=1}^m R\_{Qj} \, ^\circ(B\_Q)} \; ^\circ(B\_Q). \end{array}$$

$$\frac{\begin{array}{c} \sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{(A\_{Q} \cap B\_{Q})} = \sum\_{j=1}^{m} R\_{Q\_{j}} \prescript{o}{}{(A\_{Q})} \cap \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}}}\_{\sum\_{j=1}^{m} R\_{Q\_{j}}} (A\_{Q}) \cap \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}}}\_{\sum\_{j=1}^{m} R\_{Q\_{j}}} (B\_{Q}), \end{array}$$

$$\begin{array}{llll} 4. & \frac{\sum\_{j=1}^{m} R\_{Q\_{j}} \, ^{o}(A\_{Q} \cup B\_{Q})}{\sum\_{j=1}^{m} R\_{Q\_{j}} \, ^{o}(A\_{Q} \cap B\_{Q})} \stackrel{\supset}{\in} \frac{\sum\_{j=1}^{m} R\_{Q\_{j}} \, ^{o}(A\_{Q}) \cup \sum\_{j=1}^{m} R\_{Q\_{j}} \, ^{o}(B\_{Q})}{\sum\_{j=1}^{m} R\_{Q\_{j}} \, ^{o}(A\_{Q}) \cap \overline{\sum\_{j=1}^{m} R\_{Q\_{j}}} ^{o}(B\_{Q})}. \\ \end{array}$$

#### **Proof.** 1. By Definition 18, we have,

$$\begin{split} & \underline{\sum}\_{j=1}^{m} \mathcal{R}\_{Q\_{j}} ^{o} (A\_{Q}^{c}) = \{ \langle (uq), h\_{\sum\_{j=1}^{m} \mathcal{R}\_{Q\_{j}} ^{o} (\sim \mathcal{A}\_{Q})} (uq) \rangle : uq \in \mathcal{U} \times \mathcal{Q}, i = 1, 2, \ldots, k \} \\ & = \{ \langle (uq), \mathcal{V}\_{j=1}^{m} \mathcal{N}\_{i=1}^{k} \{ \sim h\_{\mathcal{R}\_{Q\_{j}}}^{i} (uq, eq) \vee h\_{\sim \mathcal{A}\_{Q}}^{i} (eq) \rangle : uq \in \mathcal{U} \times \mathcal{Q} \} \\ & = \{ \langle (uq), \sim \langle \bigwedge\_{j=1}^{m} \mathcal{V}\_{i=1}^{k} \{ h\_{\mathcal{R}\_{Q\_{j}}}^{i} (uq, eq) \wedge h\_{\mathcal{A}\_{Q}}^{i} (eq) \rangle \} : uq \in \mathcal{U} \times \mathcal{Q} \} \\ & = \{ \langle (uq), \sim \mathcal{h} \xrightarrow{\mathbf{m}}\_{\sum\_{j=1}^{m} \mathcal{R}\_{Q\_{j}} ^{o}} (uq) \rangle : uq \in \mathcal{U} \times \mathcal{Q}, i = 1, 2, \ldots, k \} \\ & = (\overline{\sum\_{j=1}^{m} \mathcal{R}\_{Q\_{j}}}^{\text{un}} (A\_{Q}))^{c}. \end{split}$$

Similarly, we can obtain that ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AcQ*)=(∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*))*<sup>c</sup>*.

2. If *AQ* ⊆ *BQ*, by Definition 8, *hiAQ* (*<sup>u</sup>*, *q*) ≤ *hiBQ* (*uq*) for all *u* ∈ *U*, *q* ∈ *Q*, therefore, E*mj*=<sup>1</sup> D*ki*=<sup>1</sup>{(<sup>1</sup> − *<sup>h</sup>iRQj*)(*uq*,*eq*) ∨ *hAQ* (*e*, *q*)} ≤ E*mi*=<sup>1</sup> D*ki*=<sup>1</sup>{(<sup>1</sup> − *<sup>h</sup>iRQj*)(*uq*,*eq*) ∨ *hiBQ* (*eq*)}, thus *<sup>h</sup>i*∑*mj*=<sup>1</sup> *RQj o* (*AQ*)(*uq*) ≤ *<sup>h</sup>i*∑*mj*=<sup>1</sup> *RQj o* (*BQ*)(*uq*) it follows that ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) ⊆ ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*BQ*).

$$\begin{split} 3. \quad \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}} ^{o} (A\_{Q} \cap B\_{Q})}\_{=\begin{subarray}{c} \{ (\!\!u q) , \!\!u\_{i}^{j} \} \_{\sum\_{j=1}^{m} R\_{Q\_{j}} ^{o} (A\_{Q} \cap B\_{Q})} (\!\!u q) \/ \!\!u \/ \!u q \/ \!\!u \/ \!u \/ \!u \/ \!u \/ \!u \/ \!u \/ \!\!u \/ \!\!u \/ \!\!u \/ \!\!Q \/ \!u \/ \!\!Q \/ \!\!u \/ \!\!Q \/ \!\!u \/ \!\!Q \/ \!\!u \/ \!\!Q \/ \!\!u \/ \!\!Q \/ \!\!Q \/ \!\!u \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!Q \/ \!\!\!\!Q \/ \!\!\!Q \/ \!\!\!\!Q \/ \!\!\!\!\!\!Q \/ \!\!\!\!\!\!\!\!\!\!\!\!\/$$

Hence, ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ* ∩ *BQ*) = ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) ∩ ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*BQ*). Similarly,wecanprovethat∑*mj*=<sup>1</sup>*RQj<sup>o</sup>*(*AQ*∩ *BQ*)= ∑*mj*=<sup>1</sup>*RQj<sup>o</sup>*(*AQ*)∩ ∑*mj*=<sup>1</sup>*RQj*

 *<sup>o</sup>*(*BQ*).

$$\begin{split} 4. \quad \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}}{}^{o} (A\_{Q} \cup B\_{Q})} &= \{ \langle (uq), h\_{\sum\_{j=1}^{m} R\_{Q\_{j}})}^{i} (A\_{Q} \cup B\_{Q}) \, | \, uq \rangle : \! uq \in \! II \times Q, i = 1, 2, \ldots, k \} \\ &= \{ \langle (uq), \bigvee\_{j=1}^{m} \bigwedge\_{k=1}^{k} (1 - h\_{R\_{Q\_{j}}}^{i}) (\overline{uq\_{\epsilon}}, \overline{eq}) \, \bigvee h\_{A\_{Q} \cup B\_{Q}}^{i} (eq) \rangle : \! uq \in \! II \times Q \} \\ &= \{ \langle (uq), \bigvee\_{j=1}^{m} \bigwedge\_{k=1}^{k} (1 - h\_{R\_{Q\_{j}}}^{i}) (uq, eq) \vee (h\_{A\_{Q}}^{i} (eq) \vee h\_{B\_{Q}}^{i} (eq)) \rangle : \! uq \in \! II \times Q \} \\ &= \{ \langle (uq), \left( \bigvee\_{j=1}^{m} \bigwedge\_{k=1}^{k} (1 - h\_{R\_{Q\_{j}}}^{i}) (uq, eq) \vee h\_{A\_{Q}}^{i} (eq) \right) \rangle : \! (uq) \in ! \! I \times Q \} \\ &= \{ \langle (uq), h\_{\sum\_{j=1}^{m} R\_{Q\_{j}}}^{i} (A\_{Q\_{j}})^{o} (uq) \vee h\_{\sum\_{j=1}^{m} R\_{Q\_{j}}}^{i} (eq) \rangle : \! uq \in ! I \times Q, i = 1, 2, \ldots, k \} \\ &= \sum\_{j=1}^{m} \, \, \mathcal{R}\_$$

$$\text{Hence, } \underbrace{\sum\_{j=1}^{m} R\_{Q\_j}}\_{}^{o} (A\_Q \cup B\_Q) = \underbrace{\sum\_{j=1}^{m} R\_{Q\_j}}\_{}^{o} (A\_Q) \cup \underbrace{R\_Q}\_{}^{o} (B\_Q).$$
 
$$\text{Similarly, we can prove that } \overbrace{\sum\_{j=1}^{m} R\_{Q\_j}}^{}^{o} (A\_Q \cup B\_Q) = \overbrace{\sum\_{j=1}^{m} R\_{Q\_j}}^{m} (A\_Q) \cup \overbrace{\sum\_{j=1}^{m} R\_{Q\_j}}^{}^{o} (B\_Q).$$

**Theorem 5.** *Let RQj* , *SQj* ∈ *MkQHFSR*((*U* × *Q*) × (*E* × *Q*)) (*j* = 1, 2, ..., *m*) *be multi Qm hesitant fuzzy soft relations over* (*U* × *Q*) × (*E* × *Q*)*, if RQj* ⊆ *SQj , for any AQ* ∈ *<sup>M</sup>kQHF*(*E*), *the following properties are true:*

$$1.\qquad \sum\_{j=1}^{m} R\_{Q\_j} \prescript{\bullet}{}{(A}\_Q) \supseteq \sum\_{j=1}^{m} S\_{Q\_j} \prescript{\bullet}{}{(A}\_Q)\_{\prime}$$

$$2. \qquad \overline{\sum\_{j=1}^{m} R\_{Q\_j}}^{\overline{\text{m}}}(\mathcal{A}\_Q) \subseteq \overline{\sum\_{j=1}^{m} S\_{Q\_j}}^{\overline{\text{m}}}(\mathcal{A}\_Q).$$

**Proof.** 1. If *RQj* ⊆ *SQj* , then, by Definition 8, we have *<sup>h</sup>iRQj*(*uq*,*eq*) ≤ *<sup>h</sup>iSQj*(*uq*,*eq*) for all(*<sup>u</sup>*, *q*) ∈ *U* × *Q*, *eq* ∈ *E* × *Q*, then ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) = {(*uq*), *<sup>h</sup>i*∑*mj*=<sup>1</sup> *RQj o* (*AQ*)(*uq*) : *uq* ∈ *U* × *Q*, *i* = 1, 2, ..., *k*} = {(*uq*), E*mj*=<sup>1</sup> D*ki*=<sup>1</sup>{(<sup>1</sup> − *<sup>h</sup>iRQj*)(*uq*,*eq*) ∨ *hiAQ* (*eq*) : *uq* ∈ *U* × *Q*} ≥ {(*uq*), E*mj*=<sup>1</sup> D*ki*=<sup>1</sup>{(<sup>1</sup> − *<sup>h</sup>iSQj*)(*uq*,*eq*) ∨ *hiAQ* (*eq*) : *uq* ∈ *U* × *Q*} = {(*uq*), *<sup>h</sup>i*∑*mj*=<sup>1</sup> *SQj o* (*AQ*)(*uq*) : *uq* ∈ *U* × *Q*} = ∑*mj*=<sup>1</sup> *SQj <sup>o</sup>*(*AQ*).

2. It can be proved similarly to 1.

**Definition 19.** *Let U be a universal set and Q be a non-empty set, and E be the set of parameters and RQj ,(j=1,2,...,m) are multi Qm-hesitant fuzzy soft relations over* (*U* × *Q*) × (*E* × *Q*)*, the triple*(*<sup>U</sup>*, *E*, *Q*, *RQj*) *is called multi Q-hesitant fuzzy soft multi-granulation approximation space, for any AQ* ∈ *<sup>M</sup>kQHF*(*E*)*, and the pessimistic lower and upper approximation of AQ with respect to* (*<sup>U</sup>*, *E*, *Q*, *RQj*) *are defined as follows:*

$$\begin{aligned} \underbrace{\sum\_{j=1}^{m} R\_{Q\_j}^p(A\_Q)}\_{j=1} &= \{ \langle (\iota \eta), h\_{\sum\_{j=1}^{m} R\_{Q\_j}^p(A\_Q)}^i(\iota, q) \rangle : \iota \eta \in \mathcal{U} \times Q \}, \\ \underbrace{\sum\_{m=1}^{m} R\_{Q\_j}^p(A\_Q)}\_{j=1} (A\_Q) &= \{ \langle (\iota \eta), h\_{\sum\_{j=1}^{m} R\_{Q\_j}^p(A\_Q)}^i(\iota, q) \rangle : \iota \eta \in \mathcal{U} \times Q \}, \end{aligned}$$

*where*

$$\begin{aligned} \frac{h\_{\sum\_{j=1}^m R\_{Q\_j}}(A\_Q)}{} (\!(\bigwedge\_{j=1}^m \bigwedge\_{i=1}^k \{(1 - h\_{R\_{Q\_j}}^i)(uq, eq) \lor h\_{A\_Q}^i(eq)\} ) : uq \in \mathcal{U} \times Q \}, \\\ h\_{\sum\_{j=1}^m R\_{Q\_j}}(uq) = \{ (\bigvee\_{j=1}^m \bigvee\_{i=1}^k \{h\_{R\_{Q\_i}}^i(uq, eq) \land h\_{A\_Q}^i(eq)\} ) : uq \in \mathcal{U} \times Q \}. \end{aligned}$$

*The pair* (∑*mj*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*), ∑*mj*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*)) *is called an pessimistic multi Q-hesitant fuzzy soft multi-granulation rough set of AQ with respect to* (*<sup>U</sup>*, *E*, *Q*, *RQj*)*.*

**Theorem 6.** *Let* (*<sup>U</sup>*, *E*, *Q*, *RQj*) *be multi Q-hesitant fuzzy soft multi-granulation approximation space and RQj* ∈ *MkQHFSR*((*U* × *Q*) × (*E* × *Q*)*,(i=1,2,...,m) be multi Qm hesitant fuzzy soft relations over* (*U* × *Q*) × (*E* × *Q*)*, for any AQ*, *BQ* ∈ *<sup>M</sup>kQHF*(*E*)*, the pessimistic lower and upper approximation satisfy the following properties:*

$$1. \quad \underline{\sum\_{j=1}^{m} R\_{\mathbb{Q}\_{j}^{\times}}}(A\_{\mathbb{Q}}^{c}) = \overline{(\sum\_{j=1}^{m} R\_{\mathbb{Q}\_{j}^{\times}})}(A\_{\mathbb{Q}}))^{c}, \\ \overline{\sum\_{j=1}^{m} R\_{\mathbb{Q}\_{j}^{\times}}}(A\_{\mathbb{Q}}^{c}) = (\underline{\sum\_{j=1}^{m} R\_{\mathbb{Q}\_{j}^{\times}}}(A\_{\mathbb{Q}}))^{c}.$$

$$\begin{array}{ll} 2. & A\_Q \subseteq B\_Q \Rightarrow \sum\_{j=1}^m R\_{Qj} \, ^p(A\_Q) \subseteq \underbrace{\sum\_{j=1}^m R\_{Qj} \, ^p(B\_Q)}\_{\sum\_{j=1}^m R\_{Qj} \, ^p(A\_Q)} \; ^p(B\_Q) \; . \\ & A\_Q \subseteq B\_Q \Rightarrow \overline{\sum\_{j=1}^m R\_{Qj} \, ^p(A\_Q)} \; ^p(A\_Q) \subseteq \overline{\sum\_{j=1}^m R\_{Qj} \, ^p(B\_Q)} \; . \end{array}$$

$$\underbrace{\begin{array}{c} \sum\_{j=1}^{m} R\_{Q\_{j}} ^{\mathcal{O}}(A\_{Q} \cap B\_{Q}) \ = \sum\_{j=1}^{m} R\_{Q\_{j}} ^{\mathcal{P}}(A\_{Q}) \cap \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}}}\_{\text{---}-} (A\_{Q}) \end{array} \underbrace{\begin{array}{c} \sum\_{j=1}^{m} R\_{Q\_{j}} ^{\mathcal{P}}(B\_{Q}) \end{array}}\_{\text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-$$

$$\begin{array}{llll} & \sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}(A\_{Q} \cup B\_{Q}) = \sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}(A\_{Q}) \cup \sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}(B\_{Q}).\\ 4. & \frac{\sum\_{i=1}^{m} R\_{Q\_{i}}}{\sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}}(A\_{Q} \cup B\_{Q}) \supseteq \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}(A\_{Q})}\_{\subseteq} \cup \underbrace{\sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}(A\_{Q})}\_{\subseteq} \cap \overbrace{\overline{\sum\_{j=1}^{m} R\_{Q\_{j}}^{-p}}(A\_{Q})}\_{\subseteq}. \end{array}$$

 **Proof.** It can easily be proved by using Theorem 4 and Definition 19.

**Theorem 7.** *Let* (*<sup>U</sup>*, *E*, *Q*, *RQj*) *be multi Q-hesitant fuzzy soft multi-granulation approximation space and RQj* , *SQj* ∈ *MkQHFSR*((*U* × *Q*) × (*E* × *Q*)*,(i=1,2,...,m) be multi Qm hesitant fuzzy soft relations over* (*U* × *Q*) × (*E* × *Q*)*, if RQj*⊆ *SQj, for any AQ* ∈ *<sup>M</sup>kQHF*(*E*), *the following properties are true :*

$$\begin{array}{ll} 1. & \underline{\sum\limits\_{j=1}^{m} R\_{Q\_{j}}}^{m}(A\_{Q}) \supseteq \underline{\sum\limits\_{j=1}^{m} S\_{Q\_{j}}}^{m}(A\_{Q}). \end{array}$$

$$\text{2. } \quad \overline{\sum\_{j=1}^{m} R\_{Q\_j}}^{m}(A\_Q) \subseteq \overline{\sum\_{j=1}^{m} S\_{Q\_j}}^{m}(A\_Q).$$

**Proof.** It can be easily proved by Theorem 5 and Definition 19.

#### **6. Photovoltaic Systems Fault Detection Approach**

Fuzzy sets and rough sets are both mathematical tools to handle uncertainties, they have a wide applications in many practical problems, especially in the area of decision-making. In many instances, we can not successfully utilize these classical methods to deal with decision-making problems since various types of uncertainties involved in these problems which require that second dimension must be added to the expression of the membership value.

Inspired by this, we construct a new model to the decision-making problem of photovoltaic system fault detection depending on the notion of *MkQHFS* multi-granulation rough set.

#### *6.1. The Application Model*

Photovoltaic systems (solar panel) can be explained as a piece of equipment converting sunlight (photons) to electric energy. Loss of power in photovoltaic systems can occur suddenly any time. Therefore, it is necessary to detect faults as early as possible. Unexpected power loss is usually detected by comparing the output to a reference figure.

By employing the model of multi *Q*-hesitant fuzzy soft multi-granulation rough sets, we can indicatethelossofpowerinphotovoltaicsystemsexpressedasmulti*Q*-hesitantfuzzysoftelements.

 Let *U* = {*<sup>u</sup>*1, *u*2, ...., *uv*} be the fault type set, *Q* = {*q*1, *q*2} represents the set of condition degrees and *E* = {*<sup>e</sup>*1,*e*2, ....,*es*} be the set of power measurement. Let *RQj* ∈ *MkQHFSR*((*U* × *Q*) × (*E* × *Q*)) (*j* = 1, 2, ..., *<sup>m</sup>*), which was employed to indicate the electrical information given by m experts via the membership degrees between the fault detected with condition degrees and the power measurement with condition degrees. In addition, *AQ* ∈ *MkQHF*(*E*) represents the power measurements with the condition degree of each measurement. Then, we construct a multi *Q*-hesitant fuzzy soft decision information system (*<sup>U</sup>*, *E*, *Q*, *RQj*) of the electrical detection procedure.

First, based on the the score function definition given by Xia and Xu [29], we define the score function of *MkQHFS* element as follows:

**Definition 20.** *Let <sup>h</sup>iQ*(*uq*) *be MkQHFS element, then the score function can be fined as follows:*

$$S(h\_Q^i(\mu q)) = \{ \frac{1}{l(h\_Q^i)} \sum\_{\gamma \in h\_Q^i} \gamma, i = 1, 2, \dots, k \},$$

*where l*(*hiQ*) *is the number of values in* (*hiQ*(*uq*))*.*

By Definition 20, we can define the sum of *AQ* and *BQ* as follows:

**Definition 21.** *Letting AQ and BQ be two MkQHFSS in U* × *Q, we define the sum of hiAQ* (*uq*) *and hiBQ* (*uq*) *such that i* = 1, 2, ..., *k by*

$$\{h^i\_{A\_Q}(uq) \oplus h^i\_{B\_Q}(uq) = \{ (h^1\_{A\_Q}(uq) + h^1\_{B\_Q}(uq) - h^1\_{A\_Q}(uq)h^1\_{B\_Q}(uq), h^2\_{A\_Q}(uq)h^2\_{B\_Q}(uq), \dots, h^k\_{A\_Q}(uq)h^k\_{B\_Q}(uq) \} \} $$

Based on the decision-making strategy developed in [14], we introduce the following three measurement indices which are denoted by:

$$\begin{array}{l} T\_{1} = \left\{ (S,T) \,|\,\mathrm{max}\_{\mathsf{u},q\_{t}} \, \mathrm{S} \left( \sum\_{j=1}^{m} \mathrm{R}\_{Q\_{j}}^{\,\,o}(A\_{Q})(u\_{s}q\_{t}) \oplus \overline{\sum\_{j=1}^{m} \mathrm{R}\_{Q\_{j}}}^{\mathrm{m}}(A\_{Q})(u\_{s}q\_{t}) \right) \right\}, \\ T\_{2} = \left\{ (X,Y) \,|\,\mathrm{max}\_{\mathsf{u},q\_{t}} \, \mathrm{S} \left( \sum\_{j=1}^{m} \mathrm{R}\_{Q\_{j}}^{\,p}(A\_{Q})(u\_{s}q\_{t}) \oplus \overline{\sum\_{j=1}^{m} \mathrm{R}\_{Q\_{j}}^{\,p}}^{\mathrm{m}}(A\_{Q})(u\_{s}q\_{t}) \right) \right\}, \\ T\_{3} = \left\{ (V,N) \,|\,\mathrm{max}\_{\mathsf{u},q\_{t}} \, \mathrm{S} \left( \sum\_{j=1}^{m} \mathrm{R}\_{Q\_{j}}^{\,o}(A\_{Q}) \oplus \overline{\sum\_{i=1}^{m} \mathrm{R}\_{Q\_{i}}^{\,o}}^{\mathrm{m}}(A\_{Q}) \right) \oplus \left( \overline{\sum\_{i=1}^{m} \mathrm{R}\_{Q\_{i}}^{\,o}}^{\mathrm{m}}(A\_{Q}) \right) \oplus \overline{\sum\_{j=1}^{m} \mathrm{R}\_{Q\_{j}}^{\,p}}^{\mathrm{m}}(A\_{Q}) \right\} \,\mathrm{J} \,\, \mathrm{s} \,\, \mathrm{d} \,\, \mathrm{s}^{-1} \end{array}$$

Now, the decision rules for photovoltaic systems fault detection by using a multi *Q*-hesitant fuzzy soft multi-granulation rough set are given as follows :


In the following, we present our method in an Algorithm 1 for the photovoltaic systems fault detection model by using a multi *Q*-hesitant fuzzy soft multi-granulation rough set.

**Algorithm 1.** Photovoltaic systems fault detection

