*6.2. Example*

For illustrating the efficiency of the proposed algorithm, we use a photovoltaic system fault diagnose problem with multi *Q*-hesitant fuzzy soft decision information.

Suppose that *U* = {*<sup>u</sup>*1, *u*2, *<sup>u</sup>*3} be the set of fault type where *uv* stands for, partial shading, delamination, cracks in cells, respectively. *Q* = {*q*1 = *low*, *q*2 = *high*} represent the set of status levels and *E* = {*<sup>e</sup>*1,*e*2,*e*3} be the set of power measurement where *es* stands for current, voltage, and series resistance, respectively. The photovoltaic system fault detection knowledge base with *MkQHFS* information with dimension k = 1 is presented in Tables 2–4.

In photovoltaic system fault detection, assume that we take a fault testing sample, which is presented by the following multi *Q*-hesitant fuzzy soft information:

*AQ* = {((*<sup>e</sup>*1, *q*1), 0.9, 0.4),((*<sup>e</sup>*1, *q*2), 0.6, 0.8, 0.4),((*<sup>e</sup>*2, *q*1), 0.1, 0.9),((*<sup>e</sup>*2, *q*2), 0.2, 0.5), ((*<sup>e</sup>*3, *q*1), 0.2, 0.4, 0.1),((*<sup>e</sup>*3, *q*2), 0.3, 0.7)}.


**Table 2.** Knowledge given by expert 1.

**Table 3.** Knowledge given by expert 2.


**Table 4.** Knowledge given by expert 3.


Now, by applying the steps of algorithm that we mentioned above, we first calculate the lower and upper approximation of optimistic and pessimistic multi *Q*-hesitant fuzzy soft multi-granulation rough sets of *AQ* with respect to (*<sup>U</sup>*, *E*, *Q*, *RQj*), respectively:

$$\begin{aligned} \frac{\sum\_{j=1}^{3} R\_{Q\_j}}{\langle (u\_2 q\_1), \{0.2, 0.5, 0.5\} \rangle}, \langle (u\_2 q\_2), \{0.5, 0.5, 0.5\} \rangle, \\ \langle (u\_2 q\_1), \{0.2, 0.5, 0.5\} \rangle, \langle (u\_2 q\_2), \{0.5, 0.5, 0.5\} \rangle, \langle (u\_3 q\_1), \{0.3, 0.7, 0.6\} \rangle, \langle (u\_3 q\_2), \{0.6, 0.7, 0.5\} \rangle \rangle, \\ \overline{\sum\_{j=1}^{3} R\_{Q\_j}}^{\overline{\mathcal{S}}}(A\_Q) = \{ \langle (u\_1 q\_1), \{0.6, 0.5, 0.5\} \rangle, \langle (u\_1 q\_2), \{0.3, 0.6, 0.4\} \rangle, \\ \langle (u\_2 q\_1), \{0.4, 0.5, 0.3\} \rangle, \langle (u\_2 q\_2), \{0.6, 0.5, 0.6\} \rangle, \langle (u\_3 q\_1), \{0.4, 0.7, 0.5\} \rangle, \langle (u\_3 q\_2), \{0.6, 0.5, 0.6\} \rangle \rangle, \\ \text{and} \end{aligned}$$

$$\begin{aligned} \Sigma\_{j=1}^3 R\_{Q\_j}^{-p}(A\_Q) &= \{ \langle (u\_1 q\_1), \{0.2, 0.4, 0.4\} \rangle, \langle (u\_1 q\_2), \{0.2, 0.4, 0.1\} \rangle, \\ \langle (u\_2 q\_1), \{0.2, 0.4, 0.1\} \rangle, \langle (u\_2 q\_2), \{0.2, 0.4, 0.1\} \rangle, \langle (u\_3 q\_1), \{0.2, 0.4, 0.4\} \rangle, \langle (u\_3 q\_2), \{0.1, 0.4, 0.2\} \rangle \}, \end{aligned}$$

<sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) = {(*<sup>u</sup>*1*q*1), {0.6, 0.8, 0.9},(*<sup>u</sup>*1*q*2), {0.5, 0.7, 0.5}, (*<sup>u</sup>*2*q*1), {0.8, 0.8, 0.7},(*<sup>u</sup>*2*q*2), {0.9, 0.9, 0.9},(*<sup>u</sup>*3*q*1), {0.6, 0.9, 0.6},(*<sup>u</sup>*3*q*2), {0.8, 0.8, 0.8}}.

#### Then, by Definition 21, we have:

<sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) ⊕ <sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) = {(*<sup>u</sup>*1*q*1), {0.8, 0.75, 0.7},(*<sup>u</sup>*1*q*2), {0.72, 0.84, 0.7}, (*<sup>u</sup>*2*q*1), {0.52, 0.75, 0.65},(*<sup>u</sup>*2*q*2), {0.8, 0.75, 0.8}, (*<sup>u</sup>*3*q*1), {0.58, 0.91, 0.8},(*<sup>u</sup>*3*q*2), {0.84, 0.85, 0.8}},

<sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) ⊕ <sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) = {(*<sup>u</sup>*1*q*1), {0.68, 0.88, 0.94},(*<sup>u</sup>*1*q*2), {0.6, 0.82, 0.55}, (*<sup>u</sup>*2*q*1), {0.84, 0.88, 0.73},(*<sup>u</sup>*2*q*2), {0.92, 0.94, 0.91}, (*<sup>u</sup>*3*q*1), {0.86, 0.94, 0.76},(*<sup>u</sup>*3*q*2), {0.82, 0.88, 0.84}},

$$=\{\langle\langle\underbrace{\iota\_{j=1}}\_{j=1}R\_{Q\_{j}}\overset{\circ}{\ }(A\_{Q})\oplus\overline{\langle\rangle^{3}\_{j=1}R\_{Q\_{j}}\overset{\circ}{\ }(A\_{Q})\rangle}{\langle\!\langle\!u\_{1}q\_{1}\rangle,\!\{0.936,0.97,0.982\}\rangle}\rangle\oplus\langle\underbrace{\left\{\sum\_{j=1}^{3}R\_{Q\_{j}}\overset{\circ}{\ }(A\_{Q})\oplus\overline{\langle\rangle^{3}\_{j=1}R\_{Q\_{j}}\overset{\circ}{\ }(A\_{Q})\right\}\rangle}{\langle\!\langle\!u\_{1}q\_{1}\rangle,\!\{0.936,0.982\}\rangle}\rangle,$$

$$\langle\langle\!u\_{2}q\_{1}\rangle,\!\{0.9232,0.97,0.905\}\rangle,\langle\langle\!u\_{2}q\_{2}\rangle,\!\{0.984,0.985,0.982\}\rangle,$$

$$\langle\langle\!u\_{3}q\_{1}\rangle,\!\{0.8656,0.9946,0.952\}\rangle,\langle\langle\!u\_{3}q\_{2}\rangle,\!\{0.9712,0.982,0.968\}\rangle.$$

In what follows, according to Definition 20, we calculate the score function values of multi *Q*-hesitant fuzzy soft elements

$$\begin{array}{l} \mathrm{S}\left(\sum\_{j=1}^{3} R\_{Q\_{j}} \right) \stackrel{\scriptstyle{o}}{\longleftrightarrow} \overline{\sum\_{j=1}^{3} R\_{Q\_{j}}} \, ^{o}(A\_{Q}) \, \overline{\,} = \{ \langle (u\_{1}q\_{1}), \{0.75\} \rangle, \langle (u\_{1}q\_{2}), \{0.753\} \rangle \}. \\ \langle (u\_{2}q\_{1}), \overline{\{0.64\} \rangle, \langle (u\_{2}q\_{2}), \{0.78\} \rangle, \langle (u\_{3}q\_{1}), \{0.76\} \rangle, \langle (u\_{3}q\_{2}), \{0.83\} \rangle \}. \end{array}$$

$$\begin{array}{c} \mathcal{S}\left(\sum\_{j=1}^{3} R\_{Q\_j}^{-p}(A\_Q) \oplus \overline{\sum\_{j=1}^{3} R\_{Q\_j}}^{3}(A\_Q)\right) = \{\langle(u\_1q\_1), \{0.83\}\rangle, \langle(u\_1q\_2), \{0.65\}\rangle, -1\}, \\ \langle(u\_2q\_1), \{(0.81)\}, \langle(u\_2q\_2), \{0.92\}\rangle, \langle(u\_3q\_1), \{0.79\}\rangle, \langle(u\_3q\_2), \{0.84\}\rangle\}. \end{array}$$

*S* ∑<sup>3</sup>*j*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) ⊕ <sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) ⊕ ∑<sup>3</sup>*j*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) ⊕ <sup>∑</sup><sup>3</sup>*j*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) = {(*<sup>u</sup>*1*q*1), {0.96},(*<sup>u</sup>*1*q*2), {0.90},(*<sup>u</sup>*2*q*1), {0.93},(*<sup>u</sup>*2*q*2), {0.98}, (*<sup>u</sup>*3*q*1), {0.94},(*<sup>u</sup>*3*q*2), {0.97}}.

Then, we obtain that

$$T\_1 = \left\{ (S, T) \mid \max\_{\mathbf{u}, q\_t} S \left( \underbrace{\sum\_{j=1}^m R\_{Q\_j}}\_{\mathbf{u} \in \mathbf{g}} (A\_Q)(u\_S q\_t) \oplus \overbrace{\sum\_{j=1}^m R\_{Q\_j}}\_{\mathbf{u} \in \mathbf{g}} (A\_Q)(u\_S q\_t) \right) \right\} = (3, 2),$$

$$T\_2 = \left\{ (X, Y) \mid \max\_{\mathbf{u}, q\_t} S \left( \underbrace{\sum\_{j=1}^m R\_{Q\_j}}\_{\mathbf{u} \in \mathbf{g}} (A\_Q)(u\_A q\_t) \oplus \overbrace{\sum\_{j=1}^m R\_{Q\_j}}\_{\mathbf{u} \in \mathbf{g}} (A\_Q)(u\_A q\_y) \right) \right\} = (2, 2),$$

$$T\_3 = \left\{ (V, N) \mid \max\_{\mathbf{u}, q\_t} S \left( \left( \sum\_{j=1}^m R\_{Q\_j} \left(A\_Q \right) \oplus \overbrace{\sum\_{j=1}^m R\_{Q\_j} \left(A\_Q \right)}^{\mathbf{u}} (A\_Q) \right) \oplus \left( \sum\_{j=1}^m R\_{Q\_j} \left(A\_Q \right) \oplus \overbrace{\sum\_{j=1}^m R\_{Q\_j} \left(A\_Q \right)}^{\mathbf{u}} \right) \right) \right\} = (2, 2).$$

According to the above results, the decision maker will choose the type of fault *u*2 and condition degree *q*2. Thus, we find that the photovoltaic systems fault is initiated by a high degree of delamination.

#### *6.3. Comparative Analysis and Discussion*

To explore the effectiveness of the proposed model based on multi-Q hesitant fuzzy soft multi-granulation rough sets, we compare it with the method proposed in [27]. The method given in [27] deals with the decision-making problems of one-dimensional universal sets U and V with hesitant fuzzy information, while the model proposed in the present paper can handle the decision-making problems of two-dimensional universal sets *U* × *Q* and *E* × *Q* with multi hesitant fuzzy soft information that contains much more information to deal with uncertainties in data related to an object with parameter value and the information expressed more precisely and objectively during the decision-making process. Thus, the proposed method is more general and its application domain is wider than that of the method in [27]. Reference [27] proposed a decision-making method based on the TODIM approach, and the basic parts of the previous method compute the dominance degree *ζ*(*pi*, *pk*) = ∑*nj*=<sup>1</sup> <sup>Φ</sup>*j*(*pi*, *pk*) of each alternative *pi* over each alternative *pk* and the overall prospect values *ζ*(*pi*) for alternative *pi* according to the following expression, respectively:

$$\Phi\_{j}(p\_{i},p\_{k}) = \begin{cases} \sqrt{w\_{jr}(h\_{ij}-h\_{kj})/(\sum\_{j=1}^{n}w\_{jr})} & \text{if } \quad h\_{ij}-h\_{kj} > 0, \\\ 0 & \text{if } \quad h\_{ij}-h\_{kj} = 0, \\\ -\frac{\sqrt{(\sum\_{j=1}^{n}w\_{jr})(h\_{ij}-h\_{kj})/w\_{jr}}}{\theta} & \text{if } \quad h\_{ij}-h\_{kj} < 0, \end{cases}$$

and

$$\zeta(p\_i) = \frac{\sum\_{j=1}^{n} \Phi\_j(p\_{i\prime}, p\_k) - \min\_i \left\{ \sum\_{j=1}^{n} \Phi\_j(p\_{i\prime}, p\_k) \right\}}{\max\_i \left\{ \sum\_{j=1}^{n} \Phi\_j(p\_{i\prime}, p\_k) \right\} - \min\_i \left\{ \sum\_{j=1}^{n} \Phi\_j(p\_{i\prime}, p\_k) \right\}}$$

.

As presented in [27], the optimistic decision criterion ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*) ⊕ ∑*mj*=<sup>1</sup> *RQj <sup>o</sup>*(*AQ*), pessimistic decision criterion ∑*mj*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) ⊕ ∑*mj*=<sup>1</sup> *RQj <sup>p</sup>*(*AQ*) and the weighted decision criterion

$$\frac{1}{2} \left( \underbrace{\sum\_{j=1}^{m} R\_{Q\_j}}\_{j=1} (A\_Q) \oplus \overline{\sum\_{j=1}^{m} R\_{Q\_j}}^{\overline{m}} (A\_Q) \right) \oplus \frac{1}{2} \left( \underbrace{\sum\_{j=1}^{m} R\_{Q\_j}}\_{j=1} (A\_Q) \oplus \overline{\sum\_{j=1}^{m} R\_{Q\_j}}^{\overline{m}} (A\_Q) \right)$$

are three alternatives, the fault types with condition degrees are the criteria, and the obtained evaluation values of the alternative with respect to the criterion are the elements in the decision matrix. The alternative with the largest overall prospect value is the optimal alternative. Then, in the optimal alternative, the fault type and condition degree with the largest score value are the determined fault type with its degree. Through utilizing the above procedure, we could obtain that *ζ*(*p*1) = 0.22, *ζ*(*p*2) = 0.35 and *ζ*(*p*3) = 0.36. Since the greater *ζ*(*pi*) is, the better alternative *pi* will be, the weighted decision criterion can be considered as the best alternative.

Then, we compute the score value of the fault types with condition degrees in the weighted decision criterion, which means the type of fault *u*2 and condition degree *q*2. Thus, we find that the photovoltaic systems fault is initiated by a high degree of delamination.

**Discussion**: Based on the above analysis, the results obtained by the proposed method in this paper are consistent with the one obtained using the compared method in [27], which further demonstrate the effectiveness and feasibility of the proposed model. There are two advantages of a multi *Q*-hesitant fuzzy soft multi-granulation rough set model in photovoltaic systems fault detection procedure. One advantage is that the hesitancy membership function in multi *Q*-hesitant fuzzy soft sets provides the electrical engineers with much more access to convey their understanding about the electrical knowledge base and another advantage is that the decision makers can control the size of the loss of information by adding another dimension to the universal sets. In light of the above, the greatness of the multi *Q*-hesitant fuzzy soft multi-granulation rough set model could decline the uncertainty to a grea<sup>t</sup> extent and enhance the accuracy and reliability of electrical detection effectively.
