# Machine Selection for Inventory Tracking with a Continuous Intuitionistic Fuzzy Approach
## 3 Preliminaries
### Intuitionistic Fuzzy Sets (IFSs)  
Fuzzy set theory, first introduced by [PERSON] [13], is a key concept for representing uncertainty in decision-making. [PERSON] [14] introduced single-valued intuitionistic fuzzy sets and offered a better representation of uncertainty. These sets include both the membership degree and non-membership degree of an element in a fuzzy set, with the sum of these degrees being less than or equal to one. The difference between one and this sum yields the hesitancy associated with an element's membership.  
**Definition 1**.: _An IFS 1 is represented in a universal set \(X\) as given by Equation (1):_  
\[I=\{(x,m_{I}(x),\vartheta_{I}(x))|x\in X\} \tag{1}\]  
_where \(m_{I}(x)\) and \(\vartheta_{I}(x)\), both in \([0,1]\) interval, represent the membership and non-membership degree of \(x\) to the fuzzy set \(I\), respectively, with the rule: \(0\leq m_{I}(x)+\vartheta_{I}(x)\leq 1\), for every \(x\in X\). Then, the degree of hesitancy is defined by: \(\pi_{I}(x)=1-m_{I}(x)-\vartheta_{I}(x)\)._  
**Definition 2**.: _Suppose that \(\tilde{A}=(m_{\tilde{A}},\vartheta_{\tilde{A}})\) and \(\tilde{B}=(m_{\tilde{B}},\vartheta_{\tilde{B}})\) are two single-valued intuitionistic fuzzy numbers (SVIFNs). The mathematical operations in single-valued intuitionistic fuzzy sets (SVIFSs) are defined by Equations (2)-(5):_  
\[\tilde{A}\oplus\tilde{B}=(m_{\tilde{A}}+m_{\tilde{B}}-m_{\tilde{A}}m_{\tilde{B }},\vartheta_{\tilde{A}}\vartheta_{\tilde{B}}) \tag{2}\]  
\[\tilde{A}\otimes\tilde{B}=(m_{\tilde{A}}m_{\tilde{B}},\vartheta_{\tilde{A}}+ \vartheta_{\tilde{B}}-\vartheta_{\tilde{A}}\vartheta_{\tilde{B}}) \tag{3}\]  
\[\lambda\tilde{A}=(1-(1-m_{\tilde{A}})^{\lambda},\vartheta_{\tilde{A}}^{\lambda}) \tag{4}\]  
\[\tilde{A}^{\lambda}=(m_{\tilde{A}}^{\lambda},1-(1-\vartheta_{\tilde{A}})^{ \lambda}) \tag{5}\]_where \(m_{\tilde{A}}\) and \(\vartheta_{\tilde{A}}\) refer to the membership and non-membership degrees of \(\tilde{A}\) in the fuzzy set, indicating the degree to which \(\tilde{A}\) satisfies and does not satisfy the criterion, respectively._  
**Definition 3**.: _Suppose that \(\tilde{A}i=(m_{\tilde{A}_{i}},\vartheta_{\tilde{A}_{i}}),(i=1,2,\cdots,n)\) is a set of SVIFNs and \(\rho_{i}=(p_{1},p_{2},\cdots,p_{n})^{T}\) is the weight vector of \(\tilde{A}_{i}\). Then, the intuitionistic fuzzy weighted arithmetic (IFWA) operator, which maps \(\tilde{A}^{n}\rightarrow\tilde{A}\), is defined by Equation (6):_  
\[IFWA(\tilde{A}_{1},\tilde{A}_{2},\ldots,\tilde{A}_{n})=\left(1-\prod_{i=1}^{n} \Bigl{(}1-\mu_{\tilde{A}_{i}}\Bigr{)}^{\rho_{i}}.\prod_{i=1}^{n}\vartheta_{ \tilde{A}_{i}}^{\rho_{i}}\right) \tag{6}\]  
**Definition 4**.: _Suppose that \(\tilde{A}i=(m\tilde{A}i,\vartheta\tilde{A}_{i}),i=1,2,...,n\) is a set of SVIFNs and \(\rho_{i},i=1,2,...,n\) is the weight vector of \(\tilde{A}_{i}\). Then, the intuitionistic fuzzy weighted geometric (IFWG) operator, which maps \(\tilde{A}^{n}\rightarrow\tilde{A}\), is defined by Equation (7)._  
\[IFWG\bigl{(}\tilde{A}_{1},\tilde{A}_{2},\ldots,\tilde{A}_{n}\bigr{)}=\left( \prod_{i=1}^{n}\mu_{\tilde{A}_{i}}^{\rho_{i}},1-\prod_{i=1}^{n}\Bigl{(}1- \vartheta_{\tilde{A}_{i}}\Bigr{)}^{\rho_{i}}\right) \tag{7}\]  
**Definition 5**.: _The score function \(S(\tilde{A})\) and the accuracy function \(H(\tilde{A})\) of an SVIFN \(\tilde{A}=(m_{\tilde{A}},\vartheta_{\tilde{A}})\) are defined by Equations (8) and (9)._  
\[S(\tilde{A})=m_{\tilde{A}}-\vartheta_{\tilde{A}} \tag{8}\]  
\[H(\tilde{A})=m_{\tilde{A}}+\vartheta_{\tilde{A}} \tag{9}\]  
**Definition 6**.: _Suppose that \(\tilde{A}_{i}=(m_{\tilde{A}_{i}},\vartheta_{\tilde{A}_{i}})\) and \(\tilde{B}_{i}=(m_{\tilde{B}_{i}},\vartheta_{\tilde{B}_{i}})\) are two sets of SVIFNs. Then, the Euclidean distance between any two sets of SVIFNs is obtained by Equation (10)._  
\[d(\tilde{A}_{i},\tilde{B}_{i})=\sqrt{\frac{1}{2n}\sum_{i=1}^{n}[(m_{\tilde{A} i}-m_{\tilde{B}i})^{2}+(\vartheta_{\tilde{A}_{i}}-\vartheta_{\tilde{B}_{i}})^{2}+( \tau_{\tilde{A}_{i}}-\pi_{\tilde{B}_{i}})^{2}]} \tag{10}\]