## 2 Preliminaries
### Intuitionistic Fuzzy Archimedean Copula Weighted Geometric Aggregation Operator  
**Definition 8**. : _Let \(\alpha_{i}=\left\langle\mu_{\alpha_{i}},\
u_{\alpha_{i}}\right\rangle\), \(i=1,\ \dots,\ n\), be a collection of IFVs and \(w=\{w_{1},\ \dots,\ w_{n}\}^{T}\)be the weighting vector of \(\alpha_{i}\), with \(0\leq w_{i}\leq 1\) and \(\sum_{i=1}^{n}w_{i}=1\). An intuitionistic fuzzy Archimedean copula weighted geometric aggregation (IFACWGA\({}_{w}\)) operator of dimension \(n\) is a mapping, IFACWGA\({}_{w}:V^{n}\to V\), according to: IFACWGA\({}_{w}(\alpha_{1},\dots,\alpha_{n})=\infty_{i=1}^{n}\alpha_{i}^{w_{i}}\)._  
Based on the proposed multiplication and power operations of IFVs, the following theorem is expressed. **Theorem 5**. : _Let \(\alpha_{i}=\left\langle\mu_{\alpha_{i}},\
u_{\alpha_{i}}\right\rangle\), \(i=1,\dots,n\), be a collection of IFVs and \(w=\{w_{1},\dots,w_{n}\}^{T}\) be the associated weighting vector, with \(0\leq w_{i}\leq 1\) and \(\sum_{i=1}^{n}w_{i}=1\). Then, the aggregated value, based on the IFACWGA\({}_{w}\) operator, can be expressed as: IFACWGA\({}_{w}(\alpha_{1},\dots,\alpha_{n})=\left\langle\varphi^{-1}[\sum\limits_{i=1}^{n}w_{i} \varphi(\mu_{\alpha_{i}})],\)\(n_{\lambda}(\varphi^{-1}[\sum\limits_{i=1}^{n}w_{i}\varphi(n_{\lambda}( \
u_{\alpha_{i}}))])\right\rangle\)._  
_The value of IFACWGA\({}_{w}\) operator is also an IFV._  
Proof of Theorem 5.: For \(n=1\), we get: \(IFACWGA_{w}(\alpha_{1})=\left\langle\varphi^{-1}[w_{1}\varphi(\mu_{\alpha_{1}})], \ n_{\lambda}(\varphi^{-1}[w_{1}\varphi(n_{\lambda}(\
u_{\alpha_{1}}))])\right\rangle\), but we have that \(w_{1}=1\) so we get: \(IFACWGA_{w}(\alpha_{1})=\left\langle\varphi^{-1}[\varphi(\mu_{\alpha_{1}})], \ n_{\lambda}(\varphi^{-1}[\varphi(n_{\lambda}(\
u_{\alpha_{1}}))])\right\rangle\). As a result, we have: \(IFACWGA_{w}(\alpha_{1})=\left\langle\mu_{\alpha_{1}},\
u_{\alpha_{1}}\right\rangle=\alpha_ {1}\). As \(\alpha_{1}\in V\), \(IFACWGA_{w}(\alpha_{1})\in V\). Next, suppose that for \(n=k\) it holds. For \(n=k+1\) we calculate:  
\[IFACWGA_{w}(\alpha_{1},\dots,\alpha_{k+1})=\infty_{i=1}^{k+1}\alpha_{i}^{w_{i}}=( \infty_{i=1}^{k}\alpha_{i}^{w_{i}})\otimes_{\mathbb{C}}\left(\alpha_{k+1}^{w_{k+ 1}}\right)\]  
\[=\left\langle\varphi^{-1}[\sum\limits_{i=1}^{k}w_{i}\varphi(\mu_{\alpha_{i}})], \ n_{\lambda}(\varphi^{-1}[\sum\limits_{i=1}^{k}w_{i}\varphi(n_{\lambda}( \
u_{\alpha_{i}}))])\right\rangle\otimes_{\mathbb{C}}\left\langle\varphi^{-1}[w_{k+ 1}\varphi(\mu_{\alpha_{1}})],\ n_{\lambda}(\varphi^{-1}[w_{k+1}\varphi(n_{ \lambda_{i+1}})])\right\rangle\]\[=\left\langle\varphi^{-1}[\varphi(\varphi^{-1}[\sum\limits_{i=1}^{k}w_{ i}\varphi(\mu_{a_{i}})])+\varphi(\varphi^{-1}[w_{k+1}\varphi(\mu_{a_{k+1}})])],n_{ \lambda}(\varphi^{-1}[\varphi(n_{\lambda}(n_{\lambda}(\varphi^{-1}[\sum\limits_ {i=1}^{k}w_{i}\varphi(n_{\lambda}(\
u_{a_{i}}))]))))\]  
It holds, so by mathematical induction the proof is complete. \(\square\)  
**Remark 1**. : _IFACWGA\({}_{w}\) operator possesses the same properties as IFACWAA\({}_{w}\), such as idempotency, boundary property and commutativity._  
In order to depict some of the produced _IFACWAA\({}_{w}\)_ and _IFACWGA\({}_{w}\)_ operators we use the proposed operators for IFVs as described in Table 3, which are expressed in the Tables 4 and 5, respectively. **4. ** _IFACWAA\({}_{w}\)_**and** _IFACWGA\({}_{w}\)_**Operators in MADM**  
In this section an algorithm for Multiple-Attribute Decision Making (MADM) with IFVs is introduced. Let \(U=\{u_{1},\ldots,u_{m}\}\) be the set of alternatives and \(C=\{C_{1},\ldots,C_{n}\}\) be the set of attributes, with a weighting vector \(w=\{w_{1},\ldots,w_{n}\}^{T}\), with \(0\leq w_{i}\leq 1\) and \(\sum\limits_{i=1}^{n}w_{i}=1\). The intuitionistic fuzzy decision matrix \(R=(<\mu_{ij},\
u_{ij}>)_{m\times n}\) where \(\alpha_{ij}=<\mu_{ij},\
u_{ij}>\), with \(i=1,\ldots,m\) and \(j=1,\ldots,n\) represent the degree that \(u_{i}\) satisfies \(C_{j}\), such that \(\mu_{ij},\
u_{ij}\in[0,1]\), with \(\mu_{ij}+\
u_{ij}\leq 1\), i.e.,  
\[R=\left[