## 2 Preliminaries  
In this section, some basic definitions and their examples are presented such as fuzzy set (FS), interval-valued fuzzy set (IVFS), complex fuzzy set (CFS), interval-valued complex fuzzy set (IVCFS), Cartesian product (CP) of two IVCIFSs, interval-valued complex fuzzy relation (IVCFR), intuitionistic fuzzy set (IFS), complex intuitionistic fuzzy set (CIFS), and interval-valued complex intuitionistic fuzzy set (IVCIFS). **Definition 1**. : _Ref. [1] A fuzzy set (FS) \(\tilde{\mathbb{E}}\) in a universe \(T\) is of the following form_  
\[\tilde{\mathbb{E}}=\left\{(x,\hat{\mathbb{m}}(x)):x\in T\right\}\]  
_where \(\hat{\mathbb{m}}:\tilde{\mathbb{E}}\rightarrow[0,\,1]\) is a mapping called degree of membership._  
**Example 1**. : \(\tilde{\mathbb{E}}\ =\ \left\{(s,0),(t,0.63),(u,0.97),(v,0.26),(w,0.18),(x,1), \ (y,0.85),(z,0.35)\right\}\) _is an FS._  
**Definition 2**. : _Ref. [4] An interval-valued fuzzy set (IVFS) \(\tilde{\mathbb{E}}\) in a universe \(T\) is of the following form_  
\[\tilde{\mathbb{E}}=\left\{\left(x,\hat{\mathbb{m}}^{-}(x),\hat{\mathbb{m}}^{+ }(x)\right]\right):x\in T\right\}\]  
_where \(\hat{\mathbb{m}}^{-}:\tilde{\mathbb{E}}\rightarrow[0,\,1]\) and \(\hat{\mathbb{m}}^{+}:\tilde{\mathbb{E}}\rightarrow[0,\,1]\) are the mappings called lower and upper degrees of membership, respectively._  
**Example 2**. : \(\tilde{\mathbb{E}}=\left\{ \begin{array}{c}(u,[0.32,0.68]),(v,[0.16,0.26]),(w,[0.58,0.63]),\\ (x,[0.55,0.66]),(y,[0.23,0.49]),(z,[0.87,1])\end{array} \right\}\) _is an IVFS._  
**Definition 3**. : _Ref. [13] A complex fuzzy set (CFS) \(\tilde{\mathbb{E}}\) in a universe \(T\) is of the following form_  
\[\tilde{\mathbb{E}}=\left\{(x,\hat{\mathbb{m}}_{\mathbb{C}}(x)):x\in T\right\}\]  
_where \(\hat{\mathbb{m}}_{\mathbb{C}}:\tilde{\mathbb{E}}\rightarrow\mathcal{Z}\
i 0 \leq|\mathcal{Z}|\leq 1\) and \(\mathcal{Z}\) is a complex number. The mapping \(\hat{\mathbb{m}}_{\mathbb{C}}\) is called degree of membership. Equivalently, the CFS can also be represented in the following form_  
\[\tilde{\mathbb{E}}=\left\{\left(x,\alpha(x)e^{i2\rho(x)\pi}\right):x\in T\right\}\]  
_where \(\alpha:\tilde{\mathbb{E}}\rightarrow[0,\,1]\) and \(\rho:\tilde{\mathbb{E}}\rightarrow[0,\,1]\) are mappings called amplitude term and phase term of degree of membership and \(i=\sqrt{-1}\)._  
**Example 3**. : \(\tilde{\mathbb{E}}=\left\{ \begin{array}{c}\left(u,0.15e^{i2(0.58)\pi} \right),\left(v,0.75e^{i2(0.23)\pi}\right),\left(w,0.65e^{i2(0.42)\pi}\right), \\ \left(x,0e^{i2(1)\pi}\right),\left(y,1e^{i2(0)\pi}\right),\left(z,0.93e^{i2(0.3 7)\pi}\right)\end{array} \right\}\) _is a CFS._  
**Definition 4**. : _Ref. [15] An interval-valued complex fuzzy set (IVCFS) \(\tilde{\mathbb{E}}\) in a universe \(T\) is of the following form_  
\[\tilde{\mathbb{E}}=\left\{\left(x,\hat{\mathbb{m}}_{\mathbb{C}}^{-}(x),\hat{ \mathbb{m}}_{\mathbb{C}}^{+}(x)\right]\right):x\in T\right\}\]_where \(\hat{\mathbb{m}}_{C}^{-}:\hat{\mathbb{E}}\rightarrow\mathcal{Z}\) and \(\hat{\mathbb{m}}_{C}^{+}:\hat{\mathbb{E}}\rightarrow\mathcal{Z}\
i 0\leq| \mathcal{Z}|\leq 1\) and \(\mathcal{Z}\) is a complex number. The mappings \(\hat{\mathbb{m}}_{C}^{-}\) and \(\hat{\mathbb{m}}_{C}^{+}\) are called the lower and upper degrees of membership, respectively. Equivalently, the IVCFS can also be represented in the following form_  
\[\hat{\mathbb{E}}=\Big{\{}\Big{(}x,\big{[}\alpha^{-}(x),\alpha^{+}(x)\big{]}e^{ i2[\rho^{-}(x),\rho^{+}(x)]\pi}\Big{)}:x\in T\Big{\}}\]  
_where \(\alpha^{-}:\hat{\mathbb{E}}\rightarrow[0,\,1]\), \(\alpha^{+}:\hat{\mathbb{E}}\rightarrow[0,\,1]\), \(\rho^{-}:\hat{\mathbb{E}}\rightarrow[0,\,1]\) and \(\rho^{+}:\hat{\mathbb{E}}\rightarrow[0,\,1]\) are mappings called the lower amplitude term, upper amplitude term, lower phase term and upper phase term of degree of membership and \(i=\sqrt{-1}\)._  
**Example 4. **: \(\hat{\mathbb{E}}=\left\{