**2. Preliminaries**

In this part, we shall briefly introduce some basic theories of intuitionistic fuzzy sets (IFSs) and review some similarity measures based on cosine functions and cotangent functions between IFSs.

**Definition 1.** *Suppose that X is a fixed set, then an intuitionistic fuzzy set (IFS) Q in X [1,2] can be denoted as*

$$Q = \left| \left\{ \mathbf{x}, a\_Q(\mathbf{x}), \beta\_Q(\mathbf{x}) \right\} \middle| \mathbf{x} \in X \right\} \tag{1}$$

*where* <sup>α</sup>*Q* : *X* → [0, 1] *means the degree of membership and* β*Q*(*x*) : *X* → [0, 1] *means the degree of non-membership which satisfies the condition of* 0 ≤ <sup>α</sup>*Q*(*x*) ≤ 1, 0 ≤ β*Q*(*x*) ≤ 1, 0 ≤ <sup>α</sup>*Q*(*x*) + β*Q*(*x*) ≤ 1, ∀ *x* ∈ *X*.

**Definition 2.** *For each intuitionistic fuzzy set (IFS) Q in X [1,2], the degree of indeterminacy membership* <sup>π</sup>*Q*(*x*) *can be expressed as*

$$
\pi\_Q(\mathbf{x}) = 1 - \alpha\_Q(\mathbf{x}) - \beta\_Q(\mathbf{x}), \forall \mathbf{x} \in X. \tag{2}
$$

The cosine similarity measures and cotangent similarity measures, which can calculate the degree of proximity between any two schemes, have been applied in many practical MADM problems. As we all know, the cosine and cotangent functions are monotone decreasing functions, thus, by considering the distance measures between any two alternatives, the bigger the distance values are, the smaller the calculating results by cosine and cotangent functions are and the lower similarity measures are. Therefore, to select best alternatives in decision-making problems, we always utilize cosine and cotangent similarity measures to obtain the similarity degree between each alternative and the ideal

*Mathematics* **2019**, *7*, 340

alternative. In what follows, we will briefly review some intuitionistic fuzzy cosine and cotangent similarity measures.

Let *M* = *xj*, <sup>α</sup>*Mxj*, <sup>β</sup>*Mxjxj* ∈ *X* and *N* = *xj*, <sup>α</sup>*Nxj*, <sup>β</sup>*Nxjxj* ∈ *X* be two intuitionistic fuzzy sets (IFSs), then the intuitionistic fuzzy cosine (*IFC*) measure between *M* and *N* proposed by Ye [13] can be shown as

$$IFC^1(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \frac{\alpha\_M(\mathbf{x}\_j)\alpha\_N(\mathbf{x}\_j) + \beta\_M(\mathbf{x}\_j)\beta\_N(\mathbf{x}\_j)}{\sqrt{\alpha\_M^2(\mathbf{x}\_j) + \beta\_M^2(\mathbf{x}\_j)}\sqrt{\alpha\_N^2(\mathbf{x}\_j) + \beta\_N^2(\mathbf{x}\_j)}}\tag{3}$$

Consider the degree of membership, non-membership and indeterminacy membership, then the intuitionistic fuzzy cosine (*IFC*) measure between *M* and *N* proposed by Shi and Ye [15] can be shown as

$$\text{IFC}^2(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \frac{a\_M(\mathbf{x}\_j)\alpha\_N(\mathbf{x}\_j) + \beta\_M(\mathbf{x}\_j)\beta\_N(\mathbf{x}\_j) + \pi\_M(\mathbf{x}\_j)\pi\_N(\mathbf{x}\_j)}{\sqrt{a\_M^2(\mathbf{x}\_j) + \beta\_M^2(\mathbf{x}\_j) + \pi\_M^2(\mathbf{x}\_j)}\sqrt{a\_N^2(\mathbf{x}\_j) + \beta\_N^2(\mathbf{x}\_j) + \pi\_N^2(\mathbf{x}\_j)}} \tag{4}$$

On account of cosine function, Ye [19] developed two intuitionistic fuzzy cosine similarity (*IFCS*) measures between two intuitionistic fuzzy sets (IFSs) *M* and *N*.

$$MFCS^1(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \cos\left[\frac{\pi}{2} \left(\max\left\{\begin{array}{c} \left|a\_M(\mathbf{x}\_j) - a\_N(\mathbf{x}\_j)\right|,\\ \beta\_M(\mathbf{x}\_j) - \beta\_N(\mathbf{x}\_j)\right|,\\ \left|\pi\_M(\mathbf{x}\_j) - \pi\_N(\mathbf{x}\_j)\right| \end{array}\right\}\right] \tag{5}$$

$$IFCS^2(M, N) = \frac{1}{n} \sum\_{i=1}^{n} \cos\left[\frac{\pi}{4} \begin{pmatrix} \left| \alpha\_M(\mathbf{x}\_j) - \alpha\_N(\mathbf{x}\_j) \right| + \\ \beta\_M(\mathbf{x}\_j) - \beta\_N(\mathbf{x}\_j) \right| + \\ \left| \pi\_M(\mathbf{x}\_j) - \pi\_N(\mathbf{x}\_j) \right| \end{pmatrix} \tag{6}$$

In addition, the intuitionistic fuzzy cotangent (*IFCot*) similarity measure between any two intuitionistic fuzzy sets (IFSs) *M* and *N* proposed by Tian [16] is shown as

$$IFcot^{1}(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \cot\left[\frac{\pi}{4} + \frac{\pi}{4} \left(\max\left(\begin{array}{c} \left| a\_M(\mathbf{x}\_j) - \alpha\_N(\mathbf{x}\_j) \right| \\ \left| \beta\_M(\mathbf{x}\_j) - \beta\_N(\mathbf{x}\_j) \right| \end{array} \right) \right) \tag{7}$$

Consider the degree of membership, non-membership, and indeterminacy membership, then the intuitionistic fuzzy cotangent (*IFCot*) similarity measure between any two intuitionistic fuzzy sets (IFSs) *M* and *N* proposed by Rajarajeswari and Uma [17] can be shown as

$$IFCot^2(M,N) = \frac{1}{n} \sum\_{j=1}^{n} \cot\left[\frac{\pi}{4} + \frac{\pi}{4} \left(\max\left(\begin{array}{c} \left| \alpha\_M(\mathbf{x}\_j) - \alpha\_N(\mathbf{x}\_j) \right| \\ \beta\_M(\mathbf{x}\_j) - \beta\_N(\mathbf{x}\_j) \right| \\ \left| \pi\_M(\mathbf{x}\_j) - \pi\_N(\mathbf{x}\_j) \right| \end{array}\right)\right] \tag{8}$$

Consider the weighting vector of the elements in IFS, the weighted intuitionistic fuzzy cosine (*WIFC*) measure, the weighted intuitionistic fuzzy cosine similarity (*WIFCS*) measure, and weighted intuitionistic fuzzy cotangent (*WIFCot*) similarity measure between any two intuitionistic fuzzy sets (IFSs), *M* and *N* can be shown as follows [13,15–17,19]

$$\text{MIFC}^{1}(M,N) = \sum\_{j=1}^{n} \alpha\_{j} \left[ \frac{\alpha\_{M}(\mathbf{x}\_{j})\alpha\_{N}(\mathbf{x}\_{j}) + \beta\_{M}(\mathbf{x}\_{j})\beta\_{N}(\mathbf{x}\_{j})}{\sqrt{\alpha\_{M}^{2}(\mathbf{x}\_{j}) + \beta\_{M}^{2}(\mathbf{x}\_{j})}\sqrt{\alpha\_{N}^{2}(\mathbf{x}\_{j}) + \beta\_{N}^{2}(\mathbf{x}\_{j})}} \right] \tag{9}$$

$$\text{MIFC}^{2}(M,N) = \sum\_{j=1}^{n} \omega\_{j} \frac{\alpha\_{M}(\mathbf{x}\_{j})\alpha\_{N}(\mathbf{x}\_{j}) + \beta\_{M}(\mathbf{x}\_{j})\beta\_{N}(\mathbf{x}\_{j}) + \pi\_{M}(\mathbf{x}\_{j})\pi\_{N}(\mathbf{x}\_{j})}{\sqrt{\alpha\_{M}^{2}(\mathbf{x}\_{j}) + \beta\_{M}^{2}(\mathbf{x}\_{j}) + \pi\_{M}^{2}(\mathbf{x}\_{j})}} \frac{\alpha\_{N}(\mathbf{x}\_{j})}{\sqrt{\alpha\_{N}^{2}(\mathbf{x}\_{j}) + \beta\_{N}^{2}(\mathbf{x}\_{j}) + \pi\_{N}^{2}(\mathbf{x}\_{j})}} \tag{10}$$

$$\mathcal{WIFCS}^{1}(M,N) = \sum\_{j=1}^{n} \alpha\_{j} \cos \left[ \frac{\pi}{2} \left( \max \left( \begin{bmatrix} \alpha\_{M}(\mathbf{x}\_{j}) - \alpha\_{N}(\mathbf{x}\_{j}) \\ \beta\_{M}(\mathbf{x}\_{j}) - \beta\_{N}(\mathbf{x}\_{j}) \\ \pi\_{M}(\mathbf{x}\_{j}) - \pi\_{N}(\mathbf{x}\_{j}) \end{bmatrix} \right) \right] \tag{11}$$

$$\text{WIFCS}^2(M, N) = \sum\_{i=1}^n \omega\_{\dot{\gamma}} \cos \left[ \frac{\pi}{4} \begin{pmatrix} \left| \alpha\_M(\mathbf{x}\_{\dot{\gamma}}) - \alpha\_N(\mathbf{x}\_{\dot{\gamma}}) \right| + \\ \beta\_M(\mathbf{x}\_{\dot{\gamma}}) - \beta\_N(\mathbf{x}\_{\dot{\gamma}}) + \\ \left| \pi\_M(\mathbf{x}\_{\dot{\gamma}}) - \pi\_N(\mathbf{x}\_{\dot{\gamma}}) \right| \end{pmatrix} \right] \tag{12}$$

$$\text{WIFCut}^{1}(M,N) = \sum\_{j=1}^{n} \omega\_{j} \cot \left[ \frac{\pi}{4} + \frac{\pi}{4} \left( \max \left( \begin{array}{c} \left| \alpha\_{M}(\mathbf{x}\_{j}) - \alpha\_{N}(\mathbf{x}\_{j}) \right| \\ \left| \beta\_{M}(\mathbf{x}\_{j}) - \beta\_{N}(\mathbf{x}\_{j}) \right| \end{array} \right) \right) \tag{13}$$

$$\text{WIFCot}^2(M, N) = \sum\_{j=1}^n \alpha\_j \cot \left[ \frac{\pi}{4} + \frac{\pi}{4} \middle| \max \left( \begin{bmatrix} \alpha\_M(\mathbf{x}\_j) - \alpha\_N(\mathbf{x}\_j) \\ \beta\_M(\mathbf{x}\_j) - \beta\_N(\mathbf{x}\_j) \\ \pi\_M(\mathbf{x}\_j) - \pi\_N(\mathbf{x}\_j) \end{bmatrix} \right) \right] \tag{14}$$

where <sup>ω</sup>*j*(*j* = 1, 2, ··· , *n*) denotes the weighting vector of elements *xj*, which satisfies the condition of <sup>ω</sup>*j* ∈ [0, 1] and *n j*=1 <sup>ω</sup>*j* = 1.

### **3. Some Similarity Measures Based on Cosine Function for q-ROFSs**

Although the intuitionistic fuzzy sets (IFSs) defined by Atanassov's [1,2] have been broadly applied in different 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 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 effective and powerful to deal with fuzzy and uncertain decision-making problems.

**Definition 3.** *Suppose P be a fix set, then a q-rung orthopair fuzzy set (q-ROFS) P in X [39,40] can be denoted as*

$$P = \{ \langle \mathbf{x}, (\alpha\_P(\mathbf{x}), \beta\_P(\mathbf{x})) \rangle | \mathbf{x} \in X \} \tag{15}$$

*where* α*P* : *X* → [0, 1] *means the degree of membership and* β*P*(*x*) : *X* → [0, 1] *means the degree of non-membership which satisfies the condition of* 0 ≤ <sup>α</sup>*P*(*x*) ≤ 1, 0 ≤ β*P*(*x*) ≤ 1, 0 ≤ (<sup>α</sup>*P*(*x*))*<sup>q</sup>* + (β*P*(*x*))*<sup>q</sup>* ≤ 1, *q* ≥ 1, ∀ *x* ∈ *X*.

**Definition 4.** *For each q-rung orthopair fuzzy set (q-ROFS) P in X [39,40], the degree of indeterminacy membership* <sup>π</sup>*P*(*x*) *can be expressed as*

$$\pi\_P(\mathbf{x}) = \sqrt[q]{(\alpha\_P(\mathbf{x}))^q + (\beta\_P(\mathbf{x}))^q - (\alpha\_P(\mathbf{x}))^q (\beta\_P(\mathbf{x}))^q}, \forall \mathbf{x} \in \mathbf{X}. \tag{16}$$

**Definition 5.** *Let p* = (<sup>α</sup>, β) *be a q-ROFN, a score function can be represented [40] as follows*

$$S(p) = \frac{1}{2}(1 + a^q - \beta^q), S(p) \in [0, 1]. \tag{17}$$

**Definition 6.** *Let rj* = α*j*, <sup>β</sup>*j*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a group of q-ROFNs with weighting vector w* = (*<sup>w</sup>*1, *w*2, ... , *wn*)*<sup>T</sup> , which satisfies wj* > 0*, i* = 1, 2, ... , *n and nj*=<sup>1</sup> *wj* = 1 *[40]. Then we can obtain the q-rung orthopair fuzzy weighted averaging (q-ROFWA) operator and the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator as follows*

$$\mathbf{q} - \text{ROFWA}(r\_1, r\_2, \dots, r\_n) = \bigoplus\_{j=1}^n w\_j r\_j = \left\langle \left( 1 - \prod\_{j=1}^n \left( 1 - \alpha\_j^q \right)^{w\_j} \right)^{1/q}, \prod\_{j=1}^n \beta\_j^{w\_j} \right\rangle \tag{18}$$

$$\text{eq}-\text{ROFWG}(r\_1, r\_2, \dots, r\_n) = \bigwedge\_{j=1}^n \left(r\_j\right)^{w\_j} = \left\langle \prod\_{j=1}^n a\_j^{w\_j} \left(1 - \prod\_{j=1}^n \left(1 - \beta\_j^q\right)^{w\_j}\right)^{1/q} \right\rangle \tag{19}$$

### *3.1. Cosine Similarity Measure for q-ROFSs*

Suppose that *P* is a q-rung orthopair fuzzy set (q-ROFS) in a universe of discourse *X* = {*x*}, the elements contained in q-ROFS can be expressed as the function of membership degree <sup>α</sup>*P*(*x*), the function of non-membership degree β*P*(*x*), and the function of indeterminacy membership degree <sup>π</sup>*P*(*x*). Thus, a cosine similarity measure and a weighted cosine similarity measure with q-rung orthopair fuzzy information are presented in an analogous manner to the cosine similarity measure based on Bhattacharya's distance and cosine similarity measure for intuitionistic fuzzy set (IFS) [13].

Let *M* = *xj*, <sup>α</sup>*Mxj*, <sup>β</sup>*Mxjxj* ∈ *X*and *N* = *xj*, <sup>α</sup>*Nxj*, <sup>β</sup>*Nxjxj* ∈ *X*be two q-rung orthopair fuzzy sets (q-ROFSs), then the q-rung orthopair fuzzy cosine (*q-ROFC*) measure between *M* and *N* can be shown as

$$q - \text{ROFC}^1(M, N) = \frac{1}{n} \sum\_{j=1}^{n} \frac{a\_M^q(\mathbf{x}\_j)a\_N^q(\mathbf{x}\_j) + \beta\_M^q(\mathbf{x}\_j)\beta\_N^q(\mathbf{x}\_j)}{\sqrt{\left(a\_M^q(\mathbf{x}\_j)\right)^2 + \left(\beta\_M^q(\mathbf{x}\_j)\right)^2}\sqrt{\left(a\_N^q(\mathbf{x}\_j)\right)^2 + \left(\beta\_N^q(\mathbf{x}\_j)\right)^2}}\tag{20}$$

Especially, when we let *n* = 1, the cosine similarity measure between q-ROFSs *M* and *N* can be depicted as *Cq*−*ROFS*(*<sup>M</sup>*, *<sup>N</sup>*), which will become the correlation coefficient between *M* and *N*, which is depicted as *Kq*−*ROFS*(*<sup>M</sup>*, *<sup>N</sup>*), i.e., *Cq*−*ROFS*(*<sup>M</sup>*, *N*) = *Kq*−*ROFS*(*<sup>M</sup>*, *<sup>N</sup>*). In addition, the cosine similarity measure between q-ROFSs *M* and *N* also satisfies some properties as follows.

