**Proof.**

<sup>(1)</sup> It is clear that the proposition is true based on the cosine result.


Therefore, we have finished the proofs. -

In what follows, we shall study the distance measure of the angle as *d*(*<sup>M</sup>*, *N*) = arccos*<sup>C</sup>*1*q*−*ROFS*(*<sup>M</sup>*, *N*). It satisfies some properties as follows.


**Proof.** Clearly the distance measure *d*(*<sup>M</sup>*, *N*) satisfies properties (1)–(3). In what follows we shall prove that the distance measure *d*(*<sup>M</sup>*, *N*) satisfies property (4).

For any q-rung orthopair fuzzy set (q-ROFS) *T* = *xj*,<sup>α</sup>*Txj*, <sup>β</sup>*Txjxj* ∈ *x*, *M* ⊆ *N* ⊆ *T*, let us investigate the distance measures of the angle between the vectors:

$$\begin{array}{l} d\_{j}(\mathcal{M}(\mathbf{x}\_{j}),\mathcal{N}(\mathbf{x}\_{j})) = \arccos(q - ROFC\_{i}^{1}(\mathcal{M}(\mathbf{x}\_{i}),\mathcal{N}(\mathbf{x}\_{i})))(j = 1,2,\cdots,n) \\ d\_{j}(\mathcal{M}(\mathbf{x}\_{j}),\mathcal{T}(\mathbf{x}\_{j})) = \arccos(q - ROFC\_{i}^{1}(\mathcal{M}(\mathbf{x}\_{i}),\mathcal{T}(\mathbf{x}\_{i})))(j = 1,2,\cdots,n) \\ d\_{j}(\mathcal{N}(\mathbf{x}\_{j}),\mathcal{T}(\mathbf{x}\_{j})) = \arccos(q - ROFC\_{i}^{1}(\mathcal{N}(\mathbf{x}\_{i}),\mathcal{T}(\mathbf{x}\_{i})))(j = 1,2,\cdots,n) \end{array}$$

where

$$\begin{split} &q - ROFC\_{j}^{1}\{\mathcal{M}\{\mathbf{x}\_{j}\},\mathcal{N}\{\mathbf{x}\_{j}\}\}=\frac{a\_{M}^{q}\left(\mathbf{x}\_{j}\right)a\_{N}^{q}\left(\mathbf{x}\_{j}\right) + \beta\_{M}^{q}\left(\mathbf{x}\_{j}\right)\beta\_{N}^{q}\left(\mathbf{x}\_{j}\right)}{\sqrt{\left(a\_{M}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2} + \left(\beta\_{M}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2}}\sqrt{\left(a\_{N}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2} + \left(\beta\_{N}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2}}} \\ &q - ROFC\_{j}^{1}\{\mathcal{M}\{\mathbf{x}\_{j}\},\mathcal{T}\{\mathbf{x}\_{j}\}\}=\frac{a\_{M}^{q}\left(\mathbf{x}\_{j}\right)a\_{T}^{q}\left(\mathbf{x}\_{j}\right) + \beta\_{M}^{q}\left(\mathbf{x}\_{j}\right)\beta\_{T}^{q}\left(\mathbf{x}\_{j}\right)}{\sqrt{\left(a\_{M}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2} + \left(\beta\_{M}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2}}\sqrt{\left(a\_{T}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2} + \left(\beta\_{T}^{q}\left(\mathbf{x}\_{j}\right)\right)^{2}}} \\ &q - ROFC\_{j}^{1}\{\mathcal{N}\{\mathbf{x}\_{j}\},\mathcal{T}\{\mathbf{x}\_{j}\}\}=\frac{a\_{N}^{q}\left(\mathbf{x}\_{j}\right)a\_{T}^{q}\left(\mathbf{x}\_{j}\right) + \beta\_{N}^{q}$$

*<sup>M</sup>xj* = <sup>α</sup>*Mxj*, <sup>β</sup>*Mxj*, *<sup>N</sup>xj* = <sup>α</sup>*Nxj*, <sup>β</sup>*Nxj*, *<sup>T</sup>xj* = <sup>α</sup>*Txj*, <sup>β</sup>*Txj* are three vectors in one plane, if *<sup>M</sup>xj* ⊆ *<sup>N</sup>xj* ⊆ *<sup>T</sup>xj*, *j* = 1, 2, ··· , *n*. Therefore, it is clear that *djMxj*, *<sup>T</sup>xj* ≤ *djMxj*, *<sup>N</sup>xj* + *djNxj*, *<sup>T</sup>xj* based on the triangle inequality. Combining the inequality 0 ≤ <sup>α</sup>*Pxj<sup>q</sup>* + <sup>β</sup>*Pxj<sup>q</sup>* ≤ 1, we can ge<sup>t</sup> *d*(*<sup>M</sup>*, *T*) ≤ *d*(*<sup>M</sup>*, *N*) + *d*(*<sup>N</sup>*, *<sup>T</sup>*). Therefore *d*(*<sup>M</sup>*, *N*) meets the property (4). So we completed the process of proof. -

If we consider three terms—membership degree, non-membership degree, and indeterminacy membership—which are contained in q-ROFSs, assume that there are two q-rung orthopair fuzzy sets, *M* = *xj*, <sup>α</sup>*Mxj*, <sup>β</sup>*Mxj*, <sup>π</sup>*Mxjxj* ∈ *X*(*j* = 1, 2, ... , *n*) and *N* = *xj*, <sup>α</sup>*Nxj*, <sup>β</sup>*Nxj*, <sup>π</sup>*Nxjxj* ∈ *X*(*j* = 1, 2, ... , *<sup>n</sup>*), then the q-rung orthopair fuzzy cosine (*q-ROFC*) measures between q-ROFSs can be expressed as

$$q-ROFC^{2}(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \left[ \frac{\left(\begin{array}{c} \alpha\_{M}^{q}\big(\mathbf{x}\_{j}\big)a\_{N}^{q}\big(\mathbf{x}\_{j}\big) + \beta\_{M}^{q}\big(\mathbf{x}\_{j}\big)\beta\_{N}^{q}\big(\mathbf{x}\_{j}\big) \\ + \pi\_{M}^{q}\big(\mathbf{x}\_{j}\big)\pi\_{N}^{q}\big(\mathbf{x}\_{j}\big) \end{array}\right)}{\left[\left(\begin{array}{c} \left(\alpha\_{M}^{q}\big(\mathbf{x}\_{j}\big)\right)^{2} + \left(\beta\_{M}^{q}\big(\mathbf{x}\_{j}\big)\right)^{2} + \left(\pi\_{M}^{q}\big(\mathbf{x}\_{j}\big)\right)^{2} \\ \times \sqrt{\left(\alpha\_{N}^{q}\big(\mathbf{x}\_{j}\big)\right)^{2} + \left(\beta\_{N}^{q}\big(\mathbf{x}\_{j}\big)\right)^{2} + \left(\pi\_{N}^{q}\big(\mathbf{x}\_{j}\big)\right)^{2}}\end{array}}\right] \tag{21}$$

Especially when we let *n* = 1, the cosine similarity measure between q-ROFSs *M* and *N* will become the correlation coefficient between q-rung orthopair fuzzy sets (q-ROFSs) *M* and *N*. Of course, the cosine similarity measure *<sup>q</sup>*−*ROFC*<sup>2</sup>(*<sup>M</sup>*, *N*) also satisfies some properties which are listed as follows.


Consider the weighting vector of the elements in q-ROFS, the q-rung orthopair fuzzy weighted cosine (q-ROFWC) measure between two q-rung orthopair fuzzy sets (q-ROFSs) *M* and *N* can be shown as follows.

$$q-\text{ROFWC}^{1}(M,N) = \sum\_{j=1}^{n} \alpha\_{j} \frac{a\_{M}^{q}(\mathbf{x}\_{j})a\_{N}^{q}(\mathbf{x}\_{j}) + \beta\_{M}^{q}(\mathbf{x}\_{j})\beta\_{N}^{q}(\mathbf{x}\_{j})}{\sqrt{\left(a\_{M}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\beta\_{M}^{q}(\mathbf{x}\_{j})\right)^{2}}\sqrt{\left(a\_{N}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\beta\_{N}^{q}(\mathbf{x}\_{j})\right)^{2}}}\tag{22}$$
 
$$\left[\quad \left(\quad a\_{M}^{q}(\mathbf{x}\_{j})a\_{N}^{q}(\mathbf{x}\_{j}) + \beta\_{M}^{q}(\mathbf{x}\_{j})\beta\_{N}^{q}(\mathbf{x}\_{j})\right)\right]$$

$$q-\text{ROFWC}^{2}(M,N) = \sum\_{j=1}^{n} \omega\_{j} \left| \frac{\begin{pmatrix} \alpha\_{M}^{q}(\mathbf{x}\_{j})\alpha\_{N}^{q}(\mathbf{x}\_{j}) + \beta\_{M}^{q}(\mathbf{x}\_{j})\beta\_{N}^{q}(\mathbf{x}\_{j})\\+\pi\_{M}^{q}(\mathbf{x}\_{j})\pi\_{N}^{q}(\mathbf{x}\_{j}) \end{pmatrix}}{\left| \begin{pmatrix} \sqrt{\left(\alpha\_{M}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\beta\_{M}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\pi\_{M}^{q}(\mathbf{x}\_{j})\right)^{2}}\\\times\sqrt{\left(\pi\_{N}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\beta\_{N}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\pi\_{N}^{q}(\mathbf{x}\_{j})\right)^{2}} \end{pmatrix}} \right| \tag{23}$$

where ω = (<sup>ω</sup>1, ω2, ··· , <sup>ω</sup>*n*)*<sup>T</sup>* indicates the weighting vector of the elements *xj*(*j* = 1, 2, ··· , *n*) contained in q-ROFS and the weighting vector satisfies <sup>ω</sup>*j* ∈ [0, 1], *j* = 1, 2, ··· , *n*, *nj*=<sup>1</sup> <sup>ω</sup>*j* = 1. Especially, when we let weighting vector be ω = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, ··· , 1/*n*)*<sup>T</sup>*, then the weighted cosine similarity measure will reduce to cosine similarity measure. In other words, when <sup>ω</sup>*j* = 1*n* , *j* = 1, 2 ··· , *n*, the *q* − *ROFWC*<sup>1</sup>(*<sup>M</sup>*, *N*) = *q* − *ROFC*<sup>1</sup>(*<sup>M</sup>*, *<sup>N</sup>*).

**Example 1.** *Suppose there are two q-ROFSs M* = (*<sup>x</sup>*1, 0.7, 0.4),(*<sup>x</sup>*2, 0.5, 0.6),(*<sup>x</sup>*3, 0.3, 0.8) *and N* = (*<sup>x</sup>*1, 0.9, 0.2),(*<sup>x</sup>*2, 0.4, 0.3),(*<sup>x</sup>*3, 0.7, 0.6)*, assume q* = 3, <sup>ω</sup>*j* = (0.2, 0.3, 0.5) *then according to Equation (19), the weighted cosine similarity measure between M and N can be calculated as*

$$\begin{split} q-\text{ROFWC}^{1}(M,N) &= \sum\_{j=1}^{n} \omega\_{j} \frac{\alpha\_{\text{M}}^{q}(\mathbf{x}\_{j})\alpha\_{\text{N}}^{q}(\mathbf{x}\_{j}) + \beta\_{\text{M}}^{q}(\mathbf{x}\_{j})\beta\_{\text{N}}^{q}(\mathbf{x}\_{j})}{\sqrt{\left(\alpha\_{\text{M}}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\beta\_{\text{M}}^{q}(\mathbf{x}\_{j})\right)^{2}} \sqrt{\left(\alpha\_{\text{N}}^{q}(\mathbf{x}\_{j})\right)^{2} + \left(\beta\_{\text{N}}^{q}(\mathbf{x}\_{j})\right)^{2}}} \\ &= \left(\frac{0.2 \times \left(0.7^{3} \times 0.9^{3} + 0.4^{3} \times 0.2^{3}\right)}{\sqrt{\left(0.7^{3}\right)^{2} + \left(0.4^{3}\right)^{2} + \left(0.2^{3}\right)^{2}} + \frac{0.3 \times \left(0.5^{3} \times 0.4^{3} + 0.6^{3} \times 0.3^{3}\right)}{\sqrt{\left(0.5^{3}\right)^{2} + \left(0.6^{3}\right)^{2} \times \sqrt{\left(0.4^{3}\right)^{2} + \left(0.3^{3}\right)^{2}}}}\right) \\ &\qquad + \frac{0.5 \times \left(0.3^{3} \times 0.7^{3} + 0.8^{3} \times 0.6^{3}\right)}{\sqrt{\left(0.3^{3}\right)^{2} + \left(0.8^{3}\right)^{2} \times \sqrt{\left(0.7^{3}\right)^{2} + \left(0.6^{3}\right)^{2}}}} \\ &= 0.7247 \end{split}$$

**Example 2.** *Suppose there are two q-ROFSs M* = (*<sup>x</sup>*1, 0.7, 0.4),(*<sup>x</sup>*2, 0.5, 0.6),(*<sup>x</sup>*3, 0.3, 0.8) *and N* = (*<sup>x</sup>*1, 0.9, 0.2),(*<sup>x</sup>*2, 0.4, 0.3),(*<sup>x</sup>*3, 0.7, 0.6)*, assume q* = 3, <sup>ω</sup>*j* = (0.2, 0.3, 0.5) *then according to Equation (3) and Equation (20), the weighted cosine similarity measure between M and N can be calculated as*

*q* − *ROFWC*<sup>2</sup>(*<sup>M</sup>*, *N*) = *n j*=1 <sup>ω</sup>*j* ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ α*qMxj*α*qNxj* + <sup>β</sup>*qMxj*β*qNxj* <sup>+</sup>π*qMxj*π*qNxj* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ α*qMxj*<sup>2</sup> + β*qMxj*<sup>2</sup> + π*qMxj*<sup>2</sup> × α*qNxj*<sup>2</sup> + β*qNxj*<sup>2</sup> + π*qNxj*<sup>2</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0.2 × ⎛⎜⎜⎜⎜⎜⎝ 0.7<sup>3</sup> ×0.93+0.4<sup>3</sup> ×0.23+0.7<sup>3</sup> ×0.9<sup>3</sup> (0.7<sup>3</sup>)<sup>2</sup> + (0.4<sup>3</sup>)<sup>2</sup> + (0.7<sup>3</sup>)<sup>2</sup> × (0.9<sup>3</sup>)<sup>2</sup> + (0.2<sup>3</sup>)<sup>2</sup> + (0.9<sup>3</sup>)<sup>2</sup> ⎞⎟⎟⎟⎟⎟⎠ +0.3 × ⎛⎜⎜⎜⎜⎜⎝ 0.5<sup>3</sup> ×0.43+0.6<sup>3</sup> ×0.33+0.7<sup>3</sup> ×0.4<sup>3</sup> (0.5<sup>3</sup>)<sup>2</sup> + (0.6<sup>3</sup>)<sup>2</sup> +(0.7<sup>3</sup>)<sup>2</sup> × (0.4<sup>3</sup>)<sup>2</sup> + (0.3<sup>3</sup>)<sup>2</sup>+(0.4<sup>3</sup>)<sup>2</sup> ⎞⎟⎟⎟⎟⎟⎠ +0.5 × ⎛⎜⎜⎜⎜⎜⎝ 0.3<sup>3</sup> ×0.73+0.8<sup>3</sup> ×0.63+0.8<sup>3</sup> ×0.8<sup>3</sup> (0.3<sup>3</sup>)<sup>2</sup> + (0.8<sup>3</sup>)<sup>2</sup> + (0.8<sup>3</sup>)<sup>2</sup> × (0.7<sup>3</sup>)<sup>2</sup> + (0.6<sup>3</sup>)<sup>2</sup> + (0.8<sup>3</sup>)<sup>2</sup> ⎞⎟⎟⎟⎟⎟⎠ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = 0.8789

Evidently, similar to cosine similarity measure *q* − *ROFC*<sup>1</sup>(*<sup>M</sup>*, *<sup>N</sup>*), the weighted cosine similarity measure *q* − *ROFWC*<sup>1</sup>(*<sup>M</sup>*, *N*) also meets three properties as follows.


### *3.2. Similarity Measures of q-ROFSs Based on Cosine Function*

In this section, according to the cosine function, we will present some q-rung orthopair fuzzy cosine similarity measures (q-ROFCS) between q-ROFSs and discuss their properties.

**Definition 7.** *Assume that there are any two q-rung orthopair fuzzy sets (q-ROFSs) M* = *xj*,<sup>α</sup>*Mxj*, <sup>β</sup>*Mxjxj* ∈ *x* and *N* = *xj*,<sup>α</sup>*Nxj*, <sup>β</sup>*Nxjxj* ∈ *x. Then, we shall propose two q-rung orthopair fuzzy cosine similarity (q-ROFCS) measures between q-ROFSs M and N as follows*

$$q-ROFCS^1(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \cos\left[\frac{\pi}{2} \left(\max\left(\begin{array}{c} \left| \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \right|\_\prime \\ \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \right|\_\prime \end{array}\right) \right) \tag{24}$$

$$q - ROFCS^2(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \cos\left[\frac{\pi}{4} \left( \begin{vmatrix} \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \end{vmatrix} \right) \right] \tag{25}$$

**Proposition 1.** *Assume that there are any two q-rung orthopair fuzzy sets (q-ROFSs) M and N in X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}*, the q-rung orthopair fuzzy cosine similarity measures q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *N*)(*k* = 1, 2) *should satisfy the properties (1)–(4):*


*(4) Let M*, *N*, *T be three q-ROFSs in X and M* ⊆ *N* ⊆ *T, then q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *T*) ≤ *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *<sup>N</sup>*), *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *T*) ≤ *q* − *ROFCS<sup>k</sup>*(*<sup>N</sup>*, *<sup>T</sup>*).

**Proof.** (1) Since the calculated results based on the cosine function are within [0, 1], the q-rung orthopair fuzzy cosine similarity measures based on the cosine function are also within [0, 1]. Thus 0 ≤ *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *N*) ≤ 1, *k* = 1, 2.

(2) For two q-rung orthopair fuzzy sets (q-ROFSs) *M* and *N* in *X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}, if *M* = *N*, then α*qMxj* = <sup>α</sup>*qNxj*, <sup>β</sup>*qMxj* = <sup>β</sup>*qNxj*, *j* = 1, 2, ··· , *n*. Thus, α*qMxj* − α*qNxj* = 0, β*qMxj* − <sup>β</sup>*qNxj* = 0. So, *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *N*) = 1, *k* = 1, 2. If *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *N*) = 1, *k* = 1, 2, it implies α*qMxj* − α*qNxj* = 0, *j* = 1, 2, ··· , *n*, β*qMxj* − <sup>β</sup>*qNxj* = 0, *j* = 1, 2, ··· , *n*. Since cos(0) = 1. Then, there are α*qMxj* = <sup>α</sup>*qNxj*, <sup>β</sup>*qMxj* = <sup>β</sup>*qNxj*, *j* = 1, 2, ··· , *n*. Hence *M* = *N*.

(3) Proof is straightforward.

(4) If *M* ⊆ *N* ⊆ *T*, that means <sup>α</sup>*Mxj*≤ <sup>α</sup>*Nxj*≤ <sup>α</sup>*Txj*,β*Mxj*≥ <sup>β</sup>*Nxj*≥ <sup>β</sup>*Txj*, for *j* = 1, 2, ··· , *n*. Then α*qMxj* ≤ α*qNxj* ≤ <sup>α</sup>*qTxj*, <sup>β</sup>*qMxj* ≥ <sup>β</sup>*qNxj* ≥ <sup>β</sup>*qTxj*. Thus, we have

$$\begin{vmatrix} \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \end{vmatrix} \le \left| \alpha\_M^q(\mathbf{x}\_j) - \alpha\_T^q(\mathbf{x}\_j) \right| \cdot \left| \alpha\_N^q(\mathbf{x}\_j) - \alpha\_T^q(\mathbf{x}\_j) \right| \le \left| \alpha\_M^q(\mathbf{x}\_j) - \alpha\_T^q(\mathbf{x}\_j) \right|.$$

$$\left| \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \right| \le \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_T^q(\mathbf{x}\_j) \right| \cdot \left| \beta\_N^q(\mathbf{x}\_j) - \beta\_C^q(\mathbf{x}\_j) \right| \le \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_T^q(\mathbf{x}\_j) \right|.$$

Thus *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *T*) ≤ *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *<sup>N</sup>*), *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *T*) ≤ *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *<sup>N</sup>*), as the cosine function is a decreasing function with the interval [0, <sup>π</sup>/2]. Then, we finished the process of proofs. -

If we consider three terms including membership degree, non-membership degree, and indeterminacy membership, which are contained in q-ROFSs, assume that there are two q-rung orthopair fuzzy sets *M* = *xj*, <sup>α</sup>*Mxj*, <sup>β</sup>*Mxj*, <sup>π</sup>*Mxjxj* ∈ *X*(*j* = 1, 2, ... , *n*) and *N* = *xj*, <sup>α</sup>*Nxj*, <sup>β</sup>*Nxj*, <sup>π</sup>*Nxjxj* ∈ *X*(*j* = 1, 2, ... , *<sup>n</sup>*), then the q-rung orthopair fuzzy cosine similarity (q-ROFCS) measures between *M* and *N* can be expressed as

$$q-ROFCS^3(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \cos\left[\frac{\pi}{2} \left| \max\left\{ \begin{vmatrix} \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \\ \pi\_M^q(\mathbf{x}\_j) - \pi\_N^q(\mathbf{x}\_j) \end{vmatrix} \right\} \right|\right] \tag{26}$$

where *q* − *ROFCS*<sup>3</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy cosine similarity measures between *M* and *N*, which consider the maximum distance based on the membership, indeterminacy membership, and non-membership degree.

$$q - ROFCS^4(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \cos\left[\frac{\pi}{4} \begin{pmatrix} \left| \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \right| + \\ \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \right| + \\ \left| \pi\_M^q(\mathbf{x}\_j) - \pi\_N^q(\mathbf{x}\_j) \right| \end{pmatrix} \tag{27}$$

where *q* − *ROFCS*<sup>4</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy cosine similarity measures between *M* and *N*, which consider the sum of distance based on the membership, indeterminacy membership, and non-membership degree.

Consider the weighting vector of the elements in q-ROFS, the q-rung orthopair fuzzy weighted cosine similarity (q-ROFWCS) measure between two q-rung orthopair fuzzy sets (q-ROFSs) *M* and *N* can be shown as follows.

$$q-ROF\mathsf{WCS}^{1}(M,N) = \sum\_{j=1}^{n} \alpha\_{j} \cos\left[\frac{\pi}{2} \left(\max\left(\begin{array}{c} \left| a\_{M}^{q}(\mathbf{x}\_{j}) - a\_{N}^{q}(\mathbf{x}\_{j}) \right| \\ \left| \beta\_{M}^{q}(\mathbf{x}\_{j}) - \beta\_{N}^{q}(\mathbf{x}\_{j}) \right| \end{array}\right) \right) \right] \tag{28}$$

where *q* − *ROFWCS*<sup>1</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cosine similarity measures between *M* and *N*, which consider the maximum distance based on the membership and non-membership degree.

$$q-ROFWCS^2(M,N) = \sum\_{j=1}^{n} \omega\_j \cos\left[\frac{\pi}{4} \begin{pmatrix} \left| a\_M^q(\mathbf{x}\_j) - a\_N^q(\mathbf{x}\_j) \right| + \\ \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \right| \end{pmatrix} \right] \tag{29}$$

where *q* − *ROFWCS*<sup>2</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cosine similarity measures between *M* and *N*, which consider the sum of distance based on the membership and non-membership degree.

$$q - \text{ROFWCS}^3(M, N) = \sum\_{j=1}^{n} \omega\_j \cos \left[ \frac{\pi}{2} \left( \max \left( \begin{bmatrix} \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \\ \pi\_A^2(\mathbf{x}\_j) - \pi\_B^2(\mathbf{x}\_j) \end{bmatrix} \right) \right) \right] \tag{30}$$

where *q* − *ROFWCS*<sup>3</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cosine similarity measures between *M* and *N*, which consider the maximum distance based on the membership, indeterminacy membership, and non-membership degree.

$$q - \text{ROFWCS}^4(M, N) = \sum\_{j=1}^n \alpha\_j \cos \left[ \frac{\pi}{4} \begin{pmatrix} \left| \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \right| + \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \left| + \\ \left| \pi\_A^2(\mathbf{x}\_j) - \pi\_B^2(\mathbf{x}\_j) \right| \end{pmatrix} \right] \tag{31}$$

where *q* − *ROFWCS*<sup>4</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cosine similarity measures between *M* and *N*, which consider the sum of distance based on the membership, indeterminacy membership, and non-membership degree.

where ω = (<sup>ω</sup>1, ω2, ··· , <sup>ω</sup>*n*)*<sup>T</sup>* indicates the weighting vector of the elements *xj*(*j* = 1, 2, ··· , *n*) contained in q-ROFS, and the weighting vector satisfies <sup>ω</sup>*j* ∈ [0, 1], *j* = 1, 2, ··· , *n*, *nj*=<sup>1</sup> <sup>ω</sup>*j* = 1. Especially, when we let weighting vector be ω = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, ··· , 1/*n*)*<sup>T</sup>*, then the weighted cosine similarity measure will reduce to cosine similarity measure. In other words, when <sup>ω</sup>*j* = 1*n* , *j* = 1, 2 ··· , *n*, the *q* − *ROFWCS<sup>k</sup>*(*<sup>M</sup>*, *N*) = *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *N*)(*k* = 1, 2, 3, <sup>4</sup>).

**Example 3.** *Suppose there are two q-ROFSs, M* = (*<sup>x</sup>*1, 0.7, 0.4),(*<sup>x</sup>*2, 0.5, 0.6),(*<sup>x</sup>*3, 0.3, 0.8) *and N* = (*<sup>x</sup>*1, 0.9, 0.2),(*<sup>x</sup>*2, 0.4, 0.3),(*<sup>x</sup>*3, 0.7, 0.6)*, assume q* = 3, <sup>ω</sup>*j* = (0.2, 0.3, 0.5)*, then according to Equation (25), the weighted cosine similarity measure between M and N can be calculated as*

$$\begin{split} \boldsymbol{q} - \boldsymbol{R} \boldsymbol{\theta} \boldsymbol{F} \boldsymbol{W} \boldsymbol{\zeta} \boldsymbol{S}^{1} (\boldsymbol{M}, \boldsymbol{N}) &= \sum\_{j=1}^{n} \boldsymbol{\omega}\_{j} \cos \left[ \frac{\pi}{2} \left( \max \left| \boldsymbol{\left[ \boldsymbol{\alpha}\_{M}^{q} \left( \boldsymbol{x}\_{j} \right) - \boldsymbol{\alpha}\_{N}^{q} \left( \boldsymbol{x}\_{j} \right) \right] \left| \boldsymbol{\beta}\_{M}^{q} \left( \boldsymbol{x}\_{j} \right) - \boldsymbol{\beta}\_{N}^{q} \left( \boldsymbol{x}\_{j} \right) \right| \right) \right) \right] \\ &= \left( \boldsymbol{0}.2 \times \cos \left[ \frac{\pi}{2} \max \left| \left| \boldsymbol{0}.7^{3} - \boldsymbol{0}.9^{3} \right| \boldsymbol{\beta}\_{r} \left| \boldsymbol{0}.4^{3} - \boldsymbol{0}.2^{3} \right| \right) + \boldsymbol{0}.3 \times \cos \left[ \frac{\pi}{2} \max \left( \begin{array}{c} \left| \boldsymbol{0}.5^{3} - \boldsymbol{0}.4^{3} \right| \\ \boldsymbol{0}.6^{3} - \boldsymbol{0}.3^{3} \right| \end{array} \right) \right) \\ & \quad + 0.5 \times \cos \left[ \frac{\pi}{2} \left( \max \left| \left| \boldsymbol{0}.3^{3} - \boldsymbol{0}.7^{3} \right| \boldsymbol{\beta}\_{r} \left| \boldsymbol{0}.8^{3} - \boldsymbol{0}.6^{3} \right| \right) \right) \right] \\ &= 0.8909 \end{split} \right) $$

Evidently, similar to cosine similarity measure *q* − *ROFCS<sup>k</sup>*(*<sup>M</sup>*, *N*)(*k* = 1, 2, 3, <sup>4</sup>), the weighted cosine similarity measure *q* − *ROFWCS<sup>k</sup>*(*<sup>M</sup>*, *N*)(*k* = 1, 2, 3, 4) also meets some properties as follows.

**Proposition 2.** *Assume that there are any two q-rung orthopair fuzzy sets (q-ROFSs) M and N in X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}*, the q-rung orthopair fuzzy weighted cosine similarity measures q* − *ROFWCS<sup>k</sup>*(*<sup>M</sup>*, *N*)(*k* = 1, 2, 3, 4) *should satisfy the properties (1)–(4):*


The proof is similar to Proposition 1, so it is omitted here.

*3.3. Similarity Measures of q-ROFSs Based on Cotangent Function*

In this section, according to the cotangent function, we will present some q-rung orthopair fuzzy cotangent similarity measures (q-ROFCot) between q-ROFSs and discuss their properties.

**Definition 8.** *Assume that there are any two q-rung orthopair fuzzy sets (q-ROFSs) M* = - *xj*, α*M xj* , β*M xj xj* ∈ *x and N* = - *xj*, α*N xj* , β*N xj xj* ∈ *x . Then, we shall propose two q-rung orthopair fuzzy cotangent (q-ROFCot) measures between q-ROFSs M and N* as follows

$$q-ROFcot^{1}(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \cot\left[\frac{\pi}{4} + \frac{\pi}{4} \left(\max\left(\begin{array}{c} \left| a\_M^q(\mathbf{x}\_j) - a\_N^q(\mathbf{x}\_j) \right| \\ \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \right| \end{array} \right) \right] \tag{32}$$

*where q* − *ROFCot*<sup>1</sup>(*<sup>M</sup>*, *N*) *means the q-rung orthopair fuzzy cotangent similarity measures between M and N, which consider the maximum distance based on the membership and non-membership degree.*

$$q - \text{ROFCol}^2(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \text{cot}\left[\frac{\pi}{4} + \frac{\pi}{8} \left( \begin{vmatrix} a\_M^q(\mathbf{x}\_j) - a\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \end{vmatrix} \right) \right] \tag{33}$$

*where q* − *ROFCot*<sup>2</sup>(*<sup>M</sup>*, *N*) *means the q-rung orthopair fuzzy cotangent similarity measures between M and N, which consider the sum of distance based on the membership and non-membership degree.*

If we consider three terms—membership degree, non-membership degree and indeterminacy membership—which are contained in q-ROFSs, assume that there are two q-rung orthopair fuzzy sets *M* = - *xj*, α*M xj* , β*M xj* , π *M xj xj* ∈ *X* (*j* = 1, 2, ... , *n*) and *N* = - *xj*, α*N xj* , β*N xj* , π *N xj xj* ∈ *X* (*j* = 1, 2, ... , *<sup>n</sup>*), then the q-rung orthopair fuzzy cotangent (q-ROFCot) similarity measures between *M* and *N* can be expressed as

$$q - \text{ROF} \text{cot}^3(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \text{cot} \left[ \frac{\pi}{4} + \frac{\pi}{4} \left| \max \left( \begin{vmatrix} \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \\ \pi\_A^2(\mathbf{x}\_j) - \pi\_B^2(\mathbf{x}\_j) \end{vmatrix} \right) \right| \right] \tag{34}$$

where *q* − *ROFCot*<sup>3</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy cotangent similarity measures between *M* and *N*, which consider the maximum distance based on the membership, indeterminacy membership, and non-membership degree.

$$q - ROFCat^4(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \cot \left[ \frac{\pi}{4} + \frac{\pi i}{8} \left( \begin{vmatrix} \alpha\_M^q(\mathbf{x}\_j) - \alpha\_N^q(\mathbf{x}\_j) \\ \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \\ \left| \pi\_A^2(\mathbf{x}\_j) - \pi\_B^2(\mathbf{x}\_j) \right| \end{vmatrix} \right) \tag{35}$$

where *q* − *ROFCot*<sup>4</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy cotangent similarity measures between *M* and *N*, which consider the sum of distance based on the membership, indeterminacy membership, and non-membership degree.

Consider the weighting vector of the elements in q-ROFS, the q-rung orthopair fuzzy weighted cotangent (q-ROFWCot) similarity measure between two q-rung orthopair fuzzy sets (q-ROFSs) *M* and *N* can be shown as follows.

$$q - \text{ROFWC}^1(M, N) = \sum\_{j=1}^n \omega\_j \cot \left[ \frac{\pi}{4} + \frac{\pi}{4} \left( \max \left( \begin{vmatrix} a\_M^q(\mathbf{x}\_j) - a\_N^q(\mathbf{x}\_j) \\ \end{vmatrix} \Big| \rho\_M^q(\mathbf{x}\_j) - \rho\_N^q(\mathbf{x}\_j) \Big| \right) \right) \right] \tag{36}$$

where *q* − *ROFWCot*<sup>1</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cotangent similarity measures between *M* and *N*, which consider the maximum distance based on the membership and non-membership degree.

$$q - \text{ROFWC}^2(M, N) = \sum\_{j=1}^{n} \omega\_j \cot \left[ \frac{\pi}{4} + \frac{\pi}{8} \begin{pmatrix} \left| a\_M^q(\mathbf{x}\_j) - a\_N^q(\mathbf{x}\_j) \right| + \\ \left| \beta\_M^q(\mathbf{x}\_j) - \beta\_N^q(\mathbf{x}\_j) \right| \end{pmatrix} \right] \tag{37}$$

where *q* − *ROFWCot*<sup>2</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cotangent similarity measures between *M* and *N*, which consider the sum of distance based on the membership and non-membership degree.

$$q - \text{ROFWC}^{3}(M, N) = \sum\_{j=1}^{n} \omega\_{j} \cot \left[ \frac{\pi}{4} + \frac{\pi}{4} \left| \max \left( \begin{vmatrix} \alpha\_{M}^{q}(\mathbf{x}\_{j}) - \alpha\_{N}^{q}(\mathbf{x}\_{j}) \\ \beta\_{M}^{q}(\mathbf{x}\_{j}) - \beta\_{N}^{q}(\mathbf{x}\_{j}) \\ \pi\_{A}^{2}(\mathbf{x}\_{j}) - \pi\_{B}^{2}(\mathbf{x}\_{j}) \end{vmatrix} \right) \right| \tag{38}$$

where *q* − *ROFWCot*<sup>3</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cotangent similarity measures between *M* and *N*, which consider the maximum distance based on the membership, indeterminacy membership and non-membership degree.

$$q - \text{ROFWcot}^{4}(M, N) = \sum\_{j=1}^{n} \omega\_{j} \cot \left[ \frac{\pi}{4} + \frac{\pi i}{8} \left| \begin{array}{c} \left| a\_{M}^{q}(\mathbf{x}\_{j}) - a\_{N}^{q}(\mathbf{x}\_{j}) \right| + \\ \left| \beta\_{M}^{q}(\mathbf{x}\_{j}) - \beta\_{N}^{q}(\mathbf{x}\_{j}) \right| + \\ \left| \pi\_{A}^{2}(\mathbf{x}\_{j}) - \pi\_{B}^{2}(\mathbf{x}\_{j}) \right| \end{array} \right] \tag{39}$$

where *q* − *ROFWCot*<sup>4</sup>(*<sup>M</sup>*, *N*) means the q-rung orthopair fuzzy weighted cotangent similarity measures between *M* and *N*, which consider the sum of distance based on the membership, indeterminacy membership and non-membership degree.

Where ω = (<sup>ω</sup>1, ω2, ··· , <sup>ω</sup>*n*) *T* indicates the weighting vector of the elements *xj*(*j* = 1, 2, ··· , *n*) contained in q-ROFS and the weighting vector satisfies <sup>ω</sup>*j* ∈ [0, 1], *j* = 1, 2, ··· , *n*, *n j*=1 <sup>ω</sup>*j* = 1. Especially, when we let weighting vector be ω = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, ··· , 1/*n*) *T*, then the weighted cotangent similarity measure will reduce to cotangent similarity measure. In other words, when <sup>ω</sup>*j* = 1 *n* , *j* = 1, 2 ··· , *n*, the *q* − *ROFWCot<sup>k</sup>*(*<sup>M</sup>*, *N*) = *q* − *ROFWCot<sup>k</sup>*(*<sup>M</sup>*, *N*)(*k* = 1, 2, 3, <sup>4</sup>).

**Example 4.** *Suppose there are two q-ROFSs M* = (*<sup>x</sup>*1, 0.7, 0.4),(*<sup>x</sup>*2, 0.5, 0.6),(*<sup>x</sup>*3, 0.3, 0.8) *and N* = (*<sup>x</sup>*1, 0.9, 0.2),(*<sup>x</sup>*2, 0.4, 0.3),(*<sup>x</sup>*3, 0.7, 0.6) *, assume q* = 3, <sup>ω</sup>*j* = (0.2, 0.3, 0.5)*, then according to Equation (33), the weighted cotangent similarity measure between M and N can be calculated as*

$$\begin{split} &q - \text{ROFW} \text{Cost}^{1}(M, N) = \sum\_{j=1}^{n} \omega\_{j} \text{cost} \left[ \frac{\pi}{4} + \frac{\pi}{4} \Big( \max \Big( \left| a\_{M}^{q} \{ \mathbf{x}\_{j} \} - a\_{N}^{q} \{ \mathbf{x}\_{j} \} \right|, \left| \mathcal{C}\_{M}^{q} \{ \mathbf{x}\_{j} \} - \mathcal{C}\_{N}^{q} \{ \mathbf{x}\_{j} \} \right| \Big) \Big) \right] \\ &= \left( \begin{array}{ll} 0.2 \times \text{cot} \Big[ \frac{\pi}{4} + \frac{\pi}{4} \max \Big( \left| 0.7^{3} - 0.9^{3} \right|, \left| 0.4^{3} - 0.2^{3} \right| \Big) \Big] + 0.3 \times \text{cot} \Big[ \frac{\pi}{4} + \frac{\pi}{4} \max \Big( \begin{array}{ll} \left| 0.5^{3} - 0.0.4^{3} \right|, \\ \left| 0.6^{3} - 0.3^{3} \right| \end{array} \Big) \right) \\ &+ 0.5 \times \text{cot} \Big[ \frac{\pi}{4} + \frac{\pi}{4} \Big( \max \Big( \left| 0.3^{3} - 0.7^{3} \right|, \left| 0.8^{3} - 0.6^{3} \right| \Big) \Big) \Big] \right) \\ &= 0.6245 \end{split} \right)$$
