## 2 Preliminaries
### Intuitionistic Fuzzy Archimedean Copula Weighted Arithmetic Aggregation Operator  
**Definition 7**. : _Let \(\alpha_{i}=\left\langle\mu_{\alpha_{i}},\
u_{\alpha_{i}}\right\rangle\),\(i=1,\ldots,n\), be a collection of IFVs and \(w=\left\{w_{1},\ldots,w_{n}\right\}^{T}\) 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 arithmetic aggregation (IFACWAA\({}_{w}\)) operator of dimension \(n\) is a mapping, IFACWAA\({}_{w}:V^{n}\to V\), according to: IFACWAA\({}_{w}(\alpha_{1},\ldots,\alpha_{n})=\oplus_{C}{}_{i=1}^{n}w_{i}\alpha_{i}\)._  
Based on the proposed addition and scalar multiplication operations of IFVs, the following theorem is expressed. **Theorem 3**. : _Let \(\alpha_{i}=\left\langle\mu_{\alpha_{i}},\
u_{\alpha_{i}}\right\rangle\),\(i=1,\ldots,n\), be a collection of IFVs and \(w=\left\{w_{1},\ldots,w_{n}\right\}^{T}\) 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 IFACWAA\({}_{w}\) operator, can be expressed as:_  
\[IFACWAA_{w}(\alpha_{1},\ldots,\alpha_{n})=\left\langle n_{\lambda}(\varphi^{-1 }[\sum_{i=1}^{n}w_{i}\varphi(n_{\lambda}(\mu_{\alpha_{i}}))]),\varphi^{-1}[ \sum_{i=1}^{n}w_{i}\varphi(\
u_{\alpha_{i}})]\right\rangle\]  
_The value of IFACWAA\({}_{w}\) operator is also an IFV._  
Proof of Theorem 3.: For \(n=1\), we get:  
\[IFACWAA_{w}(\alpha_{1})=\left\langle n_{\lambda}(\varphi^{-1}[w_{1}\varphi(n_{ \lambda}(\mu_{\alpha_{1}}))])\,\varphi^{-1}[w_{1}\varphi(\
u_{\alpha_{1}})]\right\rangle\]  
But we have that \(w_{1}=1\), so we get:  
\[IFACWAA_{w}(\alpha_{1})=\left\langle n_{\lambda}(\varphi^{-1}[\varphi(n_{ \lambda}(\mu_{\alpha_{1}}))]),\varphi^{-1}[\varphi(\
u_{\alpha_{1}})]\right\rangle\Leftrightarrow\]  
\[IFACWAA_{w}(\alpha_{1})=\left\langle\mu_{\alpha_{1}},\
u_{\alpha_{1}}\right\rangle \Leftrightarrow IFACWAA_{w}(\alpha_{1})=\alpha_{1},\]  
as \(\alpha_{1}\in V\),\(IFACWAA_{w}(\alpha_{1})\in V\). Next, we suppose that for \(n=k\) it holds, so we calculate:  
\[IFACWAA_{w}(\alpha_{1},\ldots,\alpha_{k})=\left\langle n_{\lambda}(\varphi^{-1 }[\sum_{i=1}^{k}w_{i}\varphi(n_{\lambda}(\mu_{\alpha_{i}}))]),\varphi^{-1}[ \sum_{i=1}^{k}w_{i}\varphi(\
u_{\alpha_{i}})]\right\rangle\]  
For \(n=k+1\), we get:  
\[IFACWAA_{w}(\alpha_{1},\ldots,\alpha_{k},\alpha_{k+1})=\oplus_{c =  
\[\varphi^{-1}[\varphi(\varphi^{-1}[\sum\limits_{i=1}^{k}w_{i}\varphi(v_{\alpha_{i}})] )+\varphi(\varphi^{-1}[w_{k+1}\varphi(v_{\alpha_{k+1}})])]\right)=\left\{n_{ \lambda}(\varphi^{-1}[\sum\limits_{i=1}^{k}w_{i}\varphi(n_{\lambda}(\mu_{\alpha_{ i}}))+w_{k+1}\varphi(n_{\lambda}(\mu_{\alpha_{i+1}}))]),\]  
\[\varphi^{-1}[\sum\limits_{i=1}^{k}w_{i}\varphi(v_{\alpha_{i}})])+w_{k+1}\varphi( v_{\alpha_{k+1}})]\right)=\left\{n_{\lambda}(\varphi^{-1}[\sum\limits_{i=1}^{k+1}w_{i} \varphi(n_{\lambda}(\mu_{\alpha_{i}}))]),\varphi^{-1}[\sum\limits_{i=1}^{k+1}w_ {i}\varphi(v_{\alpha_{i}})]\right\}\]  
It holds, so as a result, by mathematical induction the proof is complete. **Proposition 4**. : _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 IFACWAA\({}_{w}\) operator is:_  
1. _Idempotent, i.e., if_\(\left\langle\mu_{\alpha_{i}},\
u_{\alpha_{i}}\right\rangle=\left\langle\mu_{ \alpha},\
u_{\alpha}\right\rangle\)_, then: for_ \(i=1,\dots,n\)_, IFACWAA\({}_{w}(\alpha_{1},\dots,\alpha_{n})=\alpha\)_._
2. _Bounded, i.e., for_ \(i=1,\dots,n\)_,_ \(\left\langle 0,1\right\rangle\leq\)_IFACWAA\({}_{w}(\alpha_{1},\dots,\alpha_{n})\leq\left\langle 1,0\right\rangle\)_._
3.