**4. Applications**

In this section, we shall give two applications about the cosine similarity measures and cotangent similarity measures under q-rung orthopair fuzzy environment. The methods proposed in this paper are applied to pattern recognition and scheme selection to demonstrate the e ffectiveness of these methods.

### *4.1. Numerical Example 1—Pattern Recognition*

There is no doubt that the quantity of construction mainly depends on the quality of building materials. Therefore, building material inspection is the premise of good engineering quality. In the selection of materials must be strictly controlled. Inspection can not only enable the builders to accurately identify qualified materials, but also ensure and improve the quality of the project. Let us consider the pattern recognition problems about classification of building materials, suppose there are five known building materials *Ai*(*i* = 1, 2, 3, 4, <sup>5</sup>), which are depicted by the q-ROFSs *Ai*(*i* = 1, 2, 3, 4, 5) in the feature space *X* = {*<sup>x</sup>*1, *x*2, *x*3, *x*4, *<sup>x</sup>*5} as

$$\begin{array}{l} A\_{1} = \{ (\mathbf{x}\_{1}, 0.5, 0.8), (\mathbf{x}\_{2}, 0.6, 0.4), (\mathbf{x}\_{3}, 0.8, 0.3), (\mathbf{x}\_{4}, 0.6, 0.9), (\mathbf{x}\_{5}, 0.1, 0.4) \} \\ A\_{2} = \{ (\mathbf{x}\_{1}, 0.6, 0.7), (\mathbf{x}\_{2}, 0.7, 0.3), (\mathbf{x}\_{3}, 0.6, 0.2), (\mathbf{x}\_{4}, 0.8, 0.6), (\mathbf{x}\_{5}, 0.3, 0.5) \} \\ A\_{3} = \{ (\mathbf{x}\_{1}, 0.3, 0.4), (\mathbf{x}\_{2}, 0.7, 0.5), (\mathbf{x}\_{3}, 0.9, 0.3), (\mathbf{x}\_{4}, 0.4, 0.8), (\mathbf{x}\_{5}, 0.2, 0.3) \} \\ A\_{4} = \{ (\mathbf{x}\_{1}, 0.5, 0.3), (\mathbf{x}\_{2}, 0.4, 0.4), (\mathbf{x}\_{3}, 0.6, 0.2), (\mathbf{x}\_{4}, 0.4, 0.7), (\mathbf{x}\_{5}, 0.2, 0.6) \} \\ A\_{5} = \{ (\mathbf{x}\_{1}, 0.4, 0.7), (\mathbf{x}\_{2}, 0.2, 0.6), (\mathbf{x}\_{3}, 0.5, 0.4), (\mathbf{x}\_{4}, 0.5, 0.3), (\mathbf{x}\_{5}, 0.4, 0.2) \} \end{array}$$

Consider an unknown building material *A* ∈ *q* − *ROFSs*(*X*) that will be recognized, which is depicted as

$$A = \{ (\mathbf{x}\_1, 0.7, 0.6), (\mathbf{x}\_2, 0.8, 0.2), (\mathbf{x}\_3, 0.4, 0.3), (\mathbf{x}\_4, 0.7, 0.8), (\mathbf{x}\_5, 0.4, 0.2) \}$$

The purpose of this problem is classify the pattern *A* in one of the following classes, *A*1, *A*2, *A*3, *A*4, or *A*5. For it, the cosine similarity measures and cotangent similarity measures proposed in this paper have been utilized to compute the similarity from *A* to *Ai*(*i* = 1, 2, 3, 4, 5) and the results are listed as follows. (Suppose *q* = 3)

For q-rung orthopair fuzzy cosine (*q-ROFC*1) measures, we can obtain

$$=\frac{1}{8}\begin{pmatrix}q-ROFC^{1}\left(A\_{1},A\right)\\ \hline \sqrt{\left(0.5^{3}\times0.7^{3}+0.8^{3}\times0.6^{3}\right)}\\ \cline{2-4} \\ \hline \frac{1}{8}\left(\begin{array}{c}\sqrt{\left(0.5^{3}\right)^{2}+\left(0.8^{3}\right)^{2}}\times\sqrt{\left(0.7^{3}\right)^{2}+\left(0.6^{3}\right)^{2}}\\ +\frac{\left(0.8^{3}\times0.4^{3}+0.3^{3}\times0.3^{3}\right)}{\sqrt{\left(0.8^{3}\right)^{2}+\left(0.3^{3}\right)^{2}}\times\sqrt{\left(0.4^{3}\right)^{2}+\left(0.3^{3}\right)^{2}}\end{array}+\frac{\left(0.6^{3}\times0.7^{3}+0.9^{3}\times0.8^{3}\right)}{\sqrt{\left(0.6^{3}\right)^{2}+\left(0.9^{3}\right)^{2}}\times\sqrt{\left(0.7^{3}\right)^{2}+\left(0.8^{3}\right)^{2}}}\\ +\frac{\left(0.1^{3}\times0.4^{3}+0.4^{3}\times0.2^{3}\right)}{\sqrt{\left(0.1^{3}\right)^{2}+\left(0.4^{3}\right)^{2}}\times\sqrt{\left(0.4^{3}\right)^{2}+\left(0.2^{3}\right)^{2}}}\\ = 0.7443\end{cases}$$

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

 Similarly, we can ge<sup>t</sup>

$$q-ROFC^1(A\_2, A) = 0.8033, q-ROFC^1(A\_3, A) = 0.7988, \\ q-ROFC^1(A\_4, A) = 0.7345, q-ROFC^1(A\_5, A) = 0.6897.$$

(1) For q-rung orthopair fuzzy cosine (*q-ROFC*2) measures we can obtain

$$q-ROFC^{2}(A\_{1},A)=0.8795, q-ROFC^{2}(A\_{2},A)=0.9116,\\\ q-ROFC^{2}(A\_{3},A)=0.9124, q-ROFC^{2}(A\_{4},A)=0.8766,\\\ q-ROFC^{2}(A\_{5},A)=0.8543.$$

(2) For q-rung orthopair fuzzy cosine similarity (*q-ROFCS*1) measures we can obtain

> *q* − *ROFCS*<sup>1</sup>(*<sup>A</sup>*1, *A*) = 0.8975, *q* − *ROFCS*<sup>1</sup>(*<sup>A</sup>*2, *A*) = 0.9588, *q* − *ROFCS*<sup>1</sup>(*<sup>A</sup>*3, *A*) = 0.8496, *q* − *ROFCS*<sup>1</sup>(*<sup>A</sup>*4, *A*) = 0.9057, *q* − *ROFCS*<sup>1</sup>(*<sup>A</sup>*5, *A*) = 0.8654.

(3) For q-rung orthopair fuzzy cosine similarity (*q-ROFCS*2) measures we can obtain

> *q* − *ROFCS*<sup>2</sup>(*<sup>A</sup>*1, *A*) = 0.9559, *q* − *ROFCS*<sup>2</sup>(*<sup>A</sup>*2, *A*) = 0.9774, *q* − *ROFCS*<sup>2</sup>(*<sup>A</sup>*3, *A*) = 0.9498, *q* − *ROFCS*<sup>2</sup>(*<sup>A</sup>*4, *A*) = 0.9561, *q* − *ROFCS*<sup>2</sup>(*<sup>A</sup>*5, *A*) = 0.9291.

(4) For q-rung orthopair fuzzy cosine similarity (*q-ROFCS*3) measures we can obtain

> *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*1, *A*) = 0.8975, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*2, *A*) = 0.9588, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*3, *A*) = 0.8364, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*4, *A*) = 0.8880, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*5, *A*) = 0.8540.

(5) For q-rung orthopair fuzzy cosine similarity (*q-ROFCS*4) measures we can obtain

> *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*1, *A*) = 0.8964, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*2, *A*) = 0.9630, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*3, *A*) = 0.8386, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*4, *A*) = 0.8701, *q* − *ROFCS*<sup>3</sup>(*<sup>A</sup>*5, *A*) = 0.8362.

(6) For q-rung orthopair fuzzy cotangent similarity (*q-ROFCot*1) measures we can obtain

$$q-ROFCat^1(A\_1, A) = 0.6618, q-ROFCat^1(A\_2, A) = 0.7633, 

 \eta - ROFCat^1(A\_3, A) = 0.6362, q-ROFCat^1(A\_4, A) = 0.6613, 

 \eta - ROFCat^1(A\_5, A) = 0.6766.$$

(7) For q-rung orthopair fuzzy cotangent similarity (*q-ROFCot*2) measures we can obtain

$$q-ROFcot^2(A\_1, A) = 0.7571, q-ROFcot^2(A\_2, A) = 0.8257, 
\quad q-ROFcot^2(A\_3, A) = 0.7613, q-ROFcot^2(A\_4, A) = 0.7544, 
\quad q-ROFcot^2(A\_5, A) = 0.7522.$$

(8) For q-rung orthopair fuzzy cotangent similarity (*q-ROFCot*3) measures we can obtain

> *q* − *ROFCot*<sup>3</sup>(*<sup>A</sup>*1, *A*) = 0.6618, *q* − *ROFCot*<sup>3</sup>(*<sup>A</sup>*2, *A*) = 0.7633, *q* − *ROFCot*<sup>3</sup>(*<sup>A</sup>*3, *A*) = 0.6198, *q* − *ROFCot*<sup>3</sup>(*<sup>A</sup>*4, *A*) = 0.6318, *q* − *ROFCot*<sup>3</sup>(*<sup>A</sup>*5, *A*) = 0.6596.

(9) For q-rung orthopair fuzzy cotangent similarity (*q-ROFCot*4) measures we can obtain

$$q-ROFCat^4(A\_1, A) = 0.6588, q-ROFCat^4(A\_2, A) = 0.7702, q-ROFCat^4(A\_3, A) = 0.6259, q-ROFCat^4(A\_4, A) = 0.6085, q-ROFCat^4(A\_5, A) = 0.6496.$$

Thereafter, the above computing results are concluded to list in Table 1 as follows.


**Table 1.** The similarity measures between *Ai*(*i* = 1, 2, 3, 4, 5) and *A*.

According to the above calculated results listed in Table 1, except for *q* − *ROFC*<sup>2</sup>(*Ai*, *<sup>A</sup>*), we can easily find that the degree of similarity between *A*2 and *A* is the largest as derived by the other nine similarity measures. This indicates the nine similarity measures allocate the unknown building material *A* to the known building material *A*2 based on the principle of maximum similarity between q-rung orthopair fuzzy sets (q-ROFSs).

In practical decision-making problems, it is important to take the weights of elements into account, if we let the weights of elements *xi*(*<sup>i</sup>* = 1, 2, 3, 4, 5) be ω*i* = (0.15, 0.20, 0.25, 0.10, 0.30), respectively. Then the weighted cosine similarity measures and weighted cotangent similarity measures proposed in this paper have been utilized to compute the similarity from *A* to *Ai*(*i* = 1, 2, 3, 4, 5) and the results are listed in Table 2 (suppose *q* = 3). (The calculation process is similar to not weighted situation, so it is omitted here.)

According to the above calculated results listed in Table 2, except for *q* − *ROFC*<sup>2</sup>(*Ai*, *A*) and *q* − *ROFWC*<sup>2</sup>(*Ai*, *<sup>A</sup>*), we can easily find that the degree of similarity between *A*2 and *A* is the largest one as derived by the other eight similarity measures. This indicates the eight similarity measures allocate the unknown building material *A* to the known building material *A*2 based on the principle of maximum similarity between q-rung orthopair fuzzy sets (q-ROFSs).


**Table 2.** The weighted similarity measures between *Ai*(*i* = 1, 2, 3, 4, 5) and *A*.

In order to illustrate the effective and scientific of our proposed methods, we shall compare with other decision-making methods, such as the q-rung orthopair fuzzy weighted averaging (q-ROFWA) operator and the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator proposed by Liu and Wang [40], we obtained the following results in Table 3.

**Table 3.** The fused values of *Ai*(*i* = 1, 2, 3, 4, 5) and *A*.


Then, based on distance measure between q-rung orthopair fuzzy numbers (q-ROFNs), we can allocate the unknown building material *A* to the known building material *Ai*, the distance measure *d*(*<sup>M</sup>*, *N*) between q-ROFNs *M* = (<sup>α</sup>1, β1) and *N* = (<sup>α</sup>2, β2) can be depicted as

$$d(M,N) = \frac{\left| (\alpha\_1)^q - (\alpha\_2)^q \right| + \left| (\beta\_1)^q - (\beta\_2)^q \right|}{2} \tag{40}$$

For q-ROFWA operator, we can obtain the distance results *d*(*Ai*, *A*) as

$$d(A\_1, A) = 0.0323 
 \mu(A\_2, A) = 0.0165 
 \mu(A\_3, A) = 0.0275 
 \mu(A\_4, A) = 0.0222 
 \mu(A\_5, A) = 0.0197$$

For q-ROFWG operator, we can obtain the distance results *d*(*Ai*, *A*) as

$$d(A\_1, A) = 0.0556 
 \mu(A\_2, A) = 0.0049 
 \mu(A\_3, A) = 0.0352 
 \mu(A\_4, A) = 0.0484 
 \mu(A\_5, A) = 0.0474$$

From above analysis, for q-ROFWA and q-ROFWG operators, the distance measure between *A*2 and *A* is the minimum one. This indicates that q-ROFWA and q-ROFWG operators allocate the unknown building material *A* to the known building material *A*2. Although, based on the two operators and our developed methods, we can derive the same results, however, the q-ROFWA and q-ROFWG operators have the limitation of considering the interrelationship between attributes; our developed methods can overcome this disadvantage and derive more accuracy and scientific decision-making results.

### *4.2. Numerical Example 2—Scheme Selection*

In this section, we shall present a numerical example to show scheme selection of construction project with q-rung orthopair fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible construction projects. *Yi*(*i* = 1, 2, 3, 4, 5) to select. Experts selected five attributes to evaluate from the five possible construction projects: -1 G1 is the capital and technical factors; -2 G2 is the Hoisting construction operation factors; -3 G3 is the PC component installation factor; -4 G4 is the internal and external environmental risk factors; and -5 G5 is the professional managemen<sup>t</sup> level factors. The five possible construction projects *Yi*(*i* = 1, 2, 3, 4, 5) are to be evaluated using the q-rung orthopair fuzzy information by the decision maker under the above five attributes which listed as follows.

$$\begin{array}{l} Y\_1 = \{ (G\_1, 0.6, 0.7), (G\_2, 0.5, 0.8), (G\_3, 0.6, 0.3), (G\_4, 0.7, 0.3), (G\_5, 0.4, 0.6) \} \\ Y\_2 = \{ (G\_1, 0.7, 0.4), (G\_2, 0.8, 0.3), (G\_3, 0.5, 0.6), (G\_4, 0.2, 0.5), (G\_5, 0.6, 0.3) \} \\ Y\_3 = \{ (G\_1, 0.6, 0.3), (G\_2, 0.4, 0.2), (G\_3, 0.7, 0.4), (G\_4, 0.5, 0.2), (G\_5, 0.9, 0.4) \} \\ Y\_4 = \{ (G\_1, 0.8, 0.7), (G\_2, 0.5, 0.6), (G\_3, 0.4, 0.6), (G\_4, 0.6, 0.3), (G\_5, 0.4, 0.2) \} \\ Y\_5 = \{ (G\_1, 0.7, 0.2), (G\_2, 0.4, 0.3), (G\_3, 0.5, 0.6), (G\_4, 0.3, 0.5), (G\_5, 0.7, 0.2) \} \end{array}$$

Let

$$\mathcal{Y}^{+} = \left\{ \begin{pmatrix} \mathcal{G}\_{1\prime} \max\_{i}(\boldsymbol{\alpha}\_{i1}), \min\_{i}(\boldsymbol{\beta}\_{i1}) \\ \mathcal{G}\_{3\prime} \max\_{i}(\boldsymbol{\alpha}\_{i3}), \min\_{i}(\boldsymbol{\beta}\_{i3}) \\ \vdots \\ \mathcal{G}\_{5\prime} \max\_{i}(\boldsymbol{\alpha}\_{i5}), \min\_{i}(\boldsymbol{\beta}\_{i3}) \end{pmatrix} \Big| \begin{pmatrix} \mathcal{G}\_{2\prime} \max\_{i}(\boldsymbol{\alpha}\_{i2}), \min\_{i}(\boldsymbol{\beta}\_{i2}) \\ \mathcal{G}\_{4\prime} \max\_{i}(\boldsymbol{\alpha}\_{i4}), \min\_{i}(\boldsymbol{\beta}\_{i4}) \\ \vdots \end{pmatrix} \right\}$$

According to the evaluation results given in *Y*1,*Y*2,*Y*3,*Y*4 *and Y*5, we can easily obtain

$$Y^+ = \{ (G\_1, 0.8, 0.2), (G\_2, 0.8, 0.2), (G\_3, 0.7, 0.3), (G\_4, 0.7, 0.2), (G\_5, 0.9, 0.2) \}$$

Then the weighted cosine similarity measures and weighted cotangent similarity measures proposed in this paper have been utilized to compute the similarity from *Y*<sup>+</sup> to *Yi*(*i* = 1, 2, 3, 4, 5) and the results are listed in Table 4 (suppose *q* = 3).


**Table 4.** The similarity measures between *Yi*(*i* = 1, 2, 3, 4, 5) and *Y*+.

According to the above calculated results listed in Table 4, we can easily find that the degree of similarity between *Y*3 and *Y* is the largest one as derived by all ten similarity measures. This indicates all ten similarity measures think the alternative *Y*3 is closest to be best alternative *Y*<sup>+</sup> based on the principle of maximum similarity between q-rung orthopair fuzzy sets (q-ROFSs). In other words, *Y*3 is the best scheme selection for the construction project.

In practical decision-making problems, it is important to take the weights of elements into account, if we let the weights of elements *xi*(*<sup>i</sup>* = 1, 2, 3, 4, 5) be ω*i* = (0.15, 0.20, 0.25, 0.10, 0.30), respectively. Then the weighted cosine similarity measures and weighted cotangent similarity measures proposed

.

in this paper have been utilized to compute the similarity from *Y* to *Yi*(*i* = 1, 2, 3, 4, 5) and the results are listed in Table 5 (suppose *q* = 3).


**Table 5.** The weighted similarity measures between *Yi*(*i* = 1, 2, 3, 4, 5) and *Y*+.

According to the above calculated results listed in Table 5, we can easily find that the degree of similarity between *Y*3 and *Y* is the largest one as derived by other nine similarity measures. This indicates all ten similarity measures; the alternative *Y*3 is closest to be best alternative *Y*<sup>+</sup> based on the principle of maximum similarity between q-rung orthopair fuzzy sets (q-ROFSs). In other words, *Y*3 is the best scheme selection for the construction project.

In order to illustrate the effective and scientific of our proposed methods, we shall compare with other decision-making methods such as the q-rung orthopair fuzzy weighted averaging (q-ROFWA) operator and the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator proposed by Liu and Wang [40], we can obtain the result which is listed in Table 6.


**Table 6.** The fused results of *Yi*(*i* = 1, 2, 3, 4, <sup>5</sup>).

Then according to the score functions of q-rung orthopair fuzzy numbers (q-ROFNs), we can obtain the score values of *Yi*(*i* = 1, 2, 3, 4, 5) which is listed in Table 7.

**Table 7.** The score values of *Yi*(*i* = 1, 2, 3, 4, <sup>5</sup>).


Then based on score values, the ordering of *Yi*(*i* = 1, 2, 3, 4, 5) can be determined in Table 8.

From above analysis, based on the two operators and our developed methods, we can obtain that the ordering of alternatives are slightly different and the best results are same, however, the q-ROFWA and q-ROFWG operators have the limitation of considering the interrelationship between attributes, our developed methods can overcome this disadvantage and derive more accuracy and scientific decision-making results.



### *4.3. Advantages of the Proposed Similarity Measures*

Although, the intuitionistic fuzzy sets (IFSs), defined by Atanassov's [1,2], have been broadly applied in di fferent areas, for some special cases, such as when membership degree and non-membership degree are given as 0.7 and 0.8, it is clear that IFSs theory cannot satisfy this situation. The q-rung orthopair fuzzy set (q-ROFS) is also denoted by the degree of membership and non-membership, whose *q-th* power sum of them is restricted to 1; obviously, the q-ROFS is more general than the q-ROFS and can express more fuzzy information. In other words, the q-ROFS can deal with the MADM problems which IFS cannot and it is clear that IFS is a part of the q-ROFS, which indicates q-ROFS can be more e ffective and powerful to deal with fuzzy and uncertain decision-making problems. Thus, the MADM problem with q-rung orthopair fuzzy information is more e ffective and suitable for practical scientific and engineering applications.

To date, we can ge<sup>t</sup> that the cosine similarity measures and cotangent similarity measures [13,15–17,19] with IFSs have been investigated by a large amount of scholars; however, as mentioned above, there are some special cases that cannot be described by IFS. Therefore, the algorithms based on similarity measures with IFS can't deal with such problems. The cosine similarity measures and cotangent similarity measures with intuitionistic fuzzy information are special case of our proposed similarity measures with q-rung orthopair fuzzy information in this paper. Thus, our defined similarity measures are more suitable and generalized to deal with the real-life problem more accurately than the existing ones.
