1 Introduction
2 Preliminaries
Definition 1 (See Atanassov, 1986, 1989).
(1)
\[ A=\big\{\big\langle x,{\mu _{A}}(x),{\nu _{A}}(x)\big\rangle \hspace{0.2778em}\big\hspace{0.2778em}x\in X\big\},\]Definition 2 (See Atanassov, 1989).
(3)
\[ {\mathit{IFC}^{1}}(A,B)=\frac{1}{n}{\sum \limits_{j=1}^{n}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})}}.\](4)
\[\begin{array}{l}\displaystyle {\mathit{IFC}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\frac{1}{n}{\sum \limits_{j=1}^{n}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})+{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})+{\pi _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})+{\pi _{B}^{2}}({x_{j}})}}.\end{array}\](5)
\[\begin{array}{l}\displaystyle {\mathit{IFCS}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle \frac{1}{n}{\sum \limits_{j=1}^{n}}\cos \bigg\{\frac{\pi }{2}\big[\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\vee \big{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})\big\big]\bigg\},\end{array}\](6)
\[\begin{array}{l}\displaystyle {\mathit{IFCS}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle \frac{1}{n}{\sum \limits_{j=1}^{n}}\cos \bigg\{\frac{\pi }{4}\big[\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big+\big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big+\big{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})\big\big]\bigg\}.\end{array}\](7)
\[ {\mathit{IFCT}^{1}}(A,B)=\frac{1}{n}{\sum \limits_{j=1}^{n}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\big(\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\big)\bigg],\](8)
\[\begin{array}{l}\displaystyle {\mathit{IFCT}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle \frac{1}{n}{\sum \limits_{j=1}^{n}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\big(\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\vee \big{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})\big\big)\bigg].\end{array}\](9)
\[ {\mathit{IFC}^{1}}(A,B)={\sum \limits_{j=1}^{n}}{\omega _{j}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})}},\](10)
\[\begin{array}{l}\displaystyle {\mathit{IFC}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}={\sum \limits_{j=1}^{n}}{\omega _{j}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})+{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})+{\pi _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})+{\pi _{B}^{2}}({x_{j}})}},\end{array}\](11)
\[\begin{array}{l}\displaystyle {\mathit{WIFCS}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cos \bigg\{\frac{\pi }{2}\big[\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\vee \big{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})\big\big]\bigg\},\end{array}\](12)
\[\begin{array}{l}\displaystyle {\mathit{WIFCS}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cos \bigg\{\frac{\pi }{4}\big[\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big+\big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big+\big{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})\big\big]\bigg\},\end{array}\](13)
\[\begin{array}{l}\displaystyle {\mathit{WIFCT}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}={\sum \limits_{j=1}^{n}}{\omega _{j}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\big(\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\bigg)\bigg],\end{array}\](14)
\[\begin{array}{l}\displaystyle {\mathit{WIFCT}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\big(\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\vee \big{\pi _{A}}({x_{j}}){\pi _{B}}({x_{j}})\big\big)\bigg],\end{array}\]3 Some Similarity Measure Based on Cosine Function for Picture Fuzzy Sets
Definition 3 (See Cuong, 2014).
(15)
\[ A=\big\{\big\langle x,{\mu _{A}}(x),{\eta _{A}}(x),{\nu _{A}}(x)\big\rangle \hspace{0.2778em}\big\hspace{0.2778em}x\in X\big\},\]3.1 Cosine Similarity Measure for Picture Fuzzy Sets
(16)
\[ {\mathit{PFC}^{1}}(A,B)=\displaystyle \frac{1}{n}{\sum \limits_{j=1}^{n}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\eta _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\eta _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})}}.\]Proof.

(1) It is obvious that the proposition is true according to the cosine value.

(2) It is obvious that the proposition is true.

(3) When $A=B$, there are ${\mu _{A}}({x_{j}})={\mu _{B}}({x_{j}})$, ${\eta _{A}}({x_{j}})={\eta _{B}}({x_{j}})$ and ${\nu _{A}}({x_{j}})={\nu _{B}}({x_{j}})$ for $j=1,2,\dots ,n$. So ${C_{\mathit{PFS}}^{1}}(A,B)=1$. Therefore, we have finished the proofs.
Proof.
(17)
\[ {\mathit{WPFC}^{1}}(A,B)=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\eta _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\eta _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})}},\](18)
\[\begin{array}{l}\displaystyle {\mathit{PFC}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle \frac{1}{n}{\sum \limits_{j=1}^{n}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})+{\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\eta _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})+{\rho _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\eta _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})+{\rho _{B}^{2}}({x_{j}})}},\\ {} \displaystyle {\mathit{WPFC}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\frac{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})+{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})+{\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})}{\sqrt{{\mu _{A}^{2}}({x_{j}})+{\eta _{A}^{2}}({x_{j}})+{\nu _{A}^{2}}({x_{j}})+{\rho _{A}^{2}}({x_{j}})}\sqrt{{\mu _{B}^{2}}({x_{j}})+{\eta _{B}^{2}}({x_{j}})+{\nu _{B}^{2}}({x_{j}})+{\rho _{B}^{2}}({x_{j}})}},\end{array}\]3.2 Similarity Measures of Picture Fuzzy Sets Based on Cosine Function
Definition 4.
(19)
\[ {\mathit{PFCS}^{3}}(A,B)=\frac{1}{n}\hspace{0.1667em}{\sum \limits_{j=1}^{n}}\hspace{0.1667em}\cos \bigg\{\hspace{0.1667em}\frac{\pi }{2}\hspace{0.1667em}\left(\substack{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\vee {\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\vee \\ {} {\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\vee {\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})}\right)\bigg\},\](20)
\[ {\mathit{PFCS}^{4}}(A,B)=\frac{1}{n}\hspace{0.1667em}{\sum \limits_{j=1}^{n}}\hspace{0.1667em}\cos \bigg\{\hspace{0.1667em}\frac{\pi }{4}\hspace{0.1667em}\left(\substack{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})+\\ {} {\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})+{\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})}\right)\bigg\}.\]Proposition 1.
Proof.
(21)
\[\begin{array}{l}\displaystyle {\mathit{WPFCS}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cos \bigg\{\frac{\pi }{2}\big[\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\big]\bigg\},\end{array}\](22)
\[\begin{array}{l}\displaystyle {\mathit{WPFCS}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\hspace{0.1667em}\cos \hspace{0.1667em}\bigg\{\frac{\pi }{4}\big[\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big+\big{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\big+\big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\big]\bigg\},\end{array}\](23)
\[ {\mathit{WPFCS}^{3}}(A,B)=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cos \bigg\{\frac{\pi }{2}\left(\substack{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\vee {\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\vee \\ {} {\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\vee {\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})}\right)\bigg\},\](24)
\[ {\mathit{WPFCS}^{4}}(A,B)=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cos \bigg\{\frac{\pi }{4}\left(\substack{{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})+{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})+\\ {} {\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})+{\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})}\right)\bigg\},\]Proposition 2.
3.3 Similarity Measures of Picture Fuzzy Sets Based on Cotangent Function
(25)
\[\begin{array}{l}\displaystyle {\mathit{PFCT}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle \frac{1}{n}{\sum \limits_{j=1}^{n}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\big(\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\big)\bigg],\end{array}\](26)
\[\begin{array}{l}\displaystyle {\mathit{PFCT}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}=\frac{1}{n}{\sum \limits_{j=1}^{n}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\left(\substack{\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\big\vee \\ {} \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\vee \big{\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})\big}\right)\bigg].\end{array}\](27)
\[\begin{array}{l}\displaystyle {\mathit{WPFCT}^{1}}(A,B)\\ {} \displaystyle \hspace{1em}=\displaystyle {\sum \limits_{j=1}^{n}}{\omega _{j}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\big(\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\big\vee \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\big)\bigg],\end{array}\](28)
\[\begin{array}{l}\displaystyle {\mathit{WPFCT}^{2}}(A,B)\\ {} \displaystyle \hspace{1em}={\sum \limits_{j=1}^{n}}{\omega _{j}}\cot \bigg[\frac{\pi }{4}+\frac{\pi }{4}\left(\substack{\big{\mu _{A}}({x_{j}}){\mu _{B}}({x_{j}})\big\vee \big{\eta _{A}}({x_{j}}){\eta _{B}}({x_{j}})\big\vee \\ {} \big{\nu _{A}}({x_{j}}){\nu _{B}}({x_{j}})\big\vee \big{\rho _{A}}({x_{j}}){\rho _{B}}({x_{j}})\big}\right)\bigg],\end{array}\]4 Numerical Example
Table 1
${A_{1}}$  ${A_{2}}$  ${A_{3}}$  ${A_{4}}$  A  
${S_{1}}$  (0.53,0.33,0.09)  (1.00,0.00,0.00)  (0.91,0.03,0.02)  (0.85,0.09,0.05)  (0.90,0.05,0.02) 
${S_{2}}$  (0.89,0.08,0.03)  (0.13,0.64,0.21)  (0.07,0.09,0.05)  (0.74,0.16,0.10)  (0.68,0.08,0.21) 
${S_{3}}$  (0.42,0.35,0.18)  (0.03,0.82,0.13)  (0.04,0.85,0.10)  (0.02,0.89,0.05)  (0.05,0.87,0.06) 
${S_{4}}$  (0.08,0.89,0.02)  (0.73,0.15,0.08)  (0.68,0.26,0.06)  (0.08,0.84,0.06)  (0.13,0.75,0.09) 
${S_{5}}$  (0.33,0.51,0.12)  (0.52,0.31,0.16)  (0.15,0.76,0.07)  (0.16,0.71,0.05)  (0.15,0.73,0.08) 
${S_{6}}$  (0.17,0.53,0.13)  (0.51,0.24,0.21)  (0.31,0.39,0.25)  (1.00,0.00,0.00)  (0.91,0.03,0.05) 
Table 2
Similarity measures  $({A_{1}},A)$  $({A_{2}},A)$  $({A_{3}},A)$  $({A_{4}},A)$ 
${\mathit{WPFC}^{1}}({A_{i}},A)$  0.813  0.656  0.787  0.994 
${\mathit{WPFC}^{2}}({A_{i}},A)$  0.810  0.656  0.638  0.993 
${\mathit{WPFCS}^{1}}({A_{i}},A)$  0.813  0.765  0.762  0.992 
${\mathit{WPFCS}^{2}}({A_{i}},A)$  0.840  0.765  0.831  0.991 
${\mathit{WPFCS}^{3}}({A_{i}},A)$  0.813  0.765  0.709  0.992 
${\mathit{WPFCS}^{4}}({A_{i}},A)$  0.813  0.757  0.707  0.989 
${\mathit{WPFCT}^{2}}({A_{i}},A)$  0.486  0.442  0.469  0.666 
${\mathit{WPFCT}^{2}}({A_{i}},A)$  0.486  0.442  0.440  0.665 