# Machine Selection for Inventory Tracking with a Continuous Intuitionistic Fuzzy Approach
## 3 Preliminaries
### Continuous Intuitionistic Fuzzy Sets (CINFUSs)  
CINFUSs are introduced as a way to represent and manage continuous and uncertain information more accurately in real-world decision problems.  
**Definition 10**.: _Consider the fixed set \(X\). In \(X\), a CINFUS \(A\) is an object of Equation (14):_  
\[A=\{(x,F(\mu_{A}(x),\zeta),F(\vartheta_{A}(x),\zeta))|x\in X\} \tag{14}\]  
_where the continuous membership and non-membership degrees of \(x\) to the set \(A\) are indicated, respectively, by the functions \(F(\mu_{A}(x),\zeta):X\rightarrow[0,1]\) and \(F(\vartheta_{A}(x),\zeta):X\rightarrow[0,1]\), where \(F(\mu_{A}(x),\zeta)=\gamma\zeta-\gamma\zeta^{2}\) and \(F(\vartheta_{A}(x),\zeta)=(4-\gamma)\zeta-(4-\gamma)\zeta^{2}\)._  
The hesitancy degree of the judgment can be defined by Equation (15).  
\[F(\pi,\zeta)=1-(\gamma\zeta-\gamma\zeta^{2})-((4-\gamma)\zeta-(4-\gamma)\zeta ^{2})=1-4(\zeta-\zeta^{2}) \tag{15}\]  
**Definition 11**.: _Let \(CINFUS_{A}=\left\langle(\gamma_{A}\zeta-\gamma_{A}\zeta^{2})-((4-\gamma_{A}) \zeta-(4-\gamma_{A})\zeta^{2})\right\rangle\) and \(CINFUS_{B}=\left\langle(\gamma_{B}\zeta-\gamma_{B}\zeta^{2})-((4-\gamma_{B}) \zeta-(4-\gamma_{B})\zeta^{2})\right\rangle\) be two CINFUISs. The mathematical operations on two continuous intuitionistic fuzzy numbers (CINFUS) are defined by Equations (16)-(19)._  
\[CINFUS_{A}\oplus CINFUS_{B}=\left\langle\begin{matrix}\big{(}(\gamma_{A}+ \gamma_{B})\zeta-(\gamma_{A}+\gamma_{B})\zeta^{2}-\big{(}\gamma_{A}\tau-\gamma _{A}\zeta^{2}\big{)}\big{(}\gamma_{B}\zeta-\gamma_{B}\zeta^{2}\big{)}\big{)},\\ \big{(}(4-\gamma_{A})\zeta-(4-\gamma_{A})\zeta^{2}\big{(}(4-\gamma_{B})\zeta-(4 -\gamma_{B})\zeta^{2}\big{)}\big{)}\end{matrix}\right\rangle \tag{16}\]  
\[CINFUS_{A}\otimes CINFUS_{B}=\left\langle\begin{matrix}\big{(}(\gamma_{A}+ \gamma_{B})\big{(}\zeta-\zeta^{2}\big{)}\big{)},\\ \big{(}(8-\gamma_{A}-\gamma_{B})\big{(}\zeta-\zeta^{2}\big{)}-(4-\gamma_{A})(4 -\gamma_{B})\big{(}\zeta-\zeta^{2}\big{)}^{2}\big{)}\end{matrix}\right\rangle \tag{17}\]  
\[k\times CINFUS_{A}=\left\langle\bigg{(}1-\Big{(}1-\gamma_{A}\zeta+\gamma_{A} \zeta^{2}\Big{)}^{k}\bigg{)},\bigg{(}(4-\gamma_{A})^{k}\Big{(}\zeta-\zeta^{2} \Big{)}^{k}\bigg{)}\right\rangle \tag{18}\]  
\[CINFUS_{A}^{k}=\left\langle\Big{(}\gamma_{A}\zeta-\gamma_{A}\zeta^{2}\Big{)}^ {k},\Big{(}1-(1-(4-\gamma_{A})(\zeta-\zeta))^{k}\Big{)}\right\rangle \tag{19}\]  
**Definition 12**.: _Let \(\tilde{A}_{i}=\left\langle(\gamma\zeta-\gamma_{A}\zeta^{2})_{i},((4-\gamma) \zeta-(4-\gamma)\zeta^{2})_{i}\right\rangle\), \((i=1,2,\cdots,n)\) is a set of CINFUSs and \(\rho=(\rho_{1},\rho_{2},\cdots,\rho_{n})^{T}\) is the weight vector of \(\tilde{A}_{i}\) where \(\sum_{i=1}^{n}p_{i}=1\). Then, a continuous intuitionistic fuzzy weighted geometric (CINFUS) operator is described as in Equation (20)._  
\[CINFUS(\tilde{A}_{1},\tilde{A}_{2},\ldots,\tilde{A}_{n})=\left(\prod_{i=1}^{ n}\Bigl{(}\gamma\zeta-\gamma\zeta^{2}\Big{)}_{i}^{p_{i}},1-\prod_{i=1}^{n} \Bigl{(}1-\Bigl{(}(4-\gamma)\zeta-(4-\gamma)\zeta^{2}\Bigr{)}_{i}\Bigr{)}^{p_{ i}}\right) \tag{20}\]