# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 2 Preliminaries
### Linguistic Interval-Valued Spherical Fuzzy Set  
**Definition 1**. : _[_13_]_ _Let \(S=\{s_{t}|t=0,1,2,\ldots,g\}\) be a finite linguistic term set (LTS) with an odd number of linguistic terms, where \(g\) is a positive integer. The LTS \(S\) satisfies the following characteristics._  
1. _order relation:_ \(s_{i}\geq s_{k}\)_, if_ \(i\geq k\)_;_
2. _negation operator:_ \(Neg(s_{i})=s_{k}\)_, where_ \(k=g-i\)_;_
3. _maximization operator:_ \(max\{s_{i},s_{k}\}=s_{i}\)_, if_ \(i\geq k\)_;_
4. _minimization operator:_ \(max\{s_{i},s_{k}\}=s_{k}\)_, if_ \(i\geq k\)_._  
For example, an LTS with five linguistic terms can be represented as \(S=\{s_{0}=\textit{none},\)\(s_{1}=\textit{low},s_{2}=\textit{medium},s_{3}=\textit{high},s_{4}=\textit{ perfect}\}\). **Definition 2**. : _[_14_]_ _Let \(S=\{s_{t}|t=0,1,2,\ldots,g\}\) be an LTS. The discrete LTS \(S\) is extended to a continuous linguistic term set \(\overline{S}=\{s_{a}|s_{0}\leq s_{a}\leq s_{h},\alpha\in[0,h]\}\), where \(h\) (\(h>g\)) is a sufficiently large positive integer. Consider any linguistic terms \(s_{i}\), \(s_{k}\in\overline{S}\), and \(\lambda,\lambda_{1},\lambda_{2}\in[0,1]\), some operation rules are as follows._  
\[s_{i}\oplus s_{k}=s_{k}\oplus s_{i}=s_{i+k} \tag{1}\] \[s_{i}\otimes s_{k}=s_{k}\otimes s_{i}=s_{i*k}\] (2) \[\lambda s_{i}=s_{\lambda i}\] (3) \[(\lambda_{1}+\lambda_{2})s_{i}=\lambda_{1}s_{i}\oplus\lambda_{2}s_ {i}\] (4) \[\lambda(s_{i}\oplus s_{k})=\lambda s_{i}\oplus\lambda s_{k} \tag{5}\]  
**Definition 3**. : _[_16_]_ _Let \(U\
eq\varnothing\) be an initial universe and \(\overline{S}=\{s_{a}|s_{0}\leq s_{a}\leq s_{h},\alpha\in[0,h]\}\) be a continuous linguistic term set. An LIVS \(A\) in \(U\) is defined as_  
\[A=\{\langle x,(s_{uA}(x),s_{\pi A}(x),s_{\pi A}(x))\rangle|x\in U\} \tag{6}\]  
_where \(\bar{s}_{uA}(x)= \begin{bmatrix}s_{uA}^{L}(x),s_{uA}^{U}(x)\end{bmatrix} \subseteq [s_{0},s_{h}],\bar{s}_{\pi A}(x)= \begin{bmatrix}s_{\pi A}^{L}(x),s_{\pi A}^{U }(x)\end{bmatrix} \subseteq[s_{0},s_{h}]\), \(\bar{s}_{\
u A}(x)= \begin{bmatrix}s_{\
u A}^{L}(x),\bar{s}_{\
u A}^{U}(x) \end{bmatrix} \subseteq[s_{0},s_{h}]\) are the membership degree, the hesitatory degree and the non-membership degree of \(x\) to \(A\) such that, for all \(x\in U\), respectively,_  
\[0\leq(\bar{s}_{uA}^{U}(x))^{2}+\left(\bar{s}_{\pi A}^{U}(x)\right)^{2}+(\bar{ s}_{\
u A}^{U}(x))^{2}\leq h^{2} \tag{7}\]  
_then \(\bar{s}_{rA}(x)= \begin{bmatrix}\bar{s}_{rA}^{L}(x),\bar{s}_{rA}^{U}(x)\end{bmatrix} \) is the waiter degree of \(x\) to \(U\), where \(\bar{s}_{rA}^{L}(x)=\begin{smallmatrix}s\\ \sqrt{h^{2-(uA^{L}(x))^{2}+(\pi A^{L}(x))^{2}+(\pi A^{L}(x))^{2}}},\ \bar{s}_{rA}^{U}(x)=s_{\sqrt{h^{2}-(uA^{U}(x))^{2}+(\pi A^{U}(x))^{2}}}.\) The set of all LIVS \(A\) in \(U\) is denoted by \(LIVS(U)\). For notational simplicity, the linguistic interval-valued spherical fuzzy number (LIVSFN) is denoted by \(\alpha= \begin{pmatrix}[s_{a},s_{b}],[s_{c},s_{d}],[s_{e},s_{o}]\end{pmatrix} \), where \([s_{a},s_{b}]\subseteq[s_{0},s_{h}],[s_{b},s_{c}]\subseteq[s_{0},s_{h}],\)\([s_{e},s_{o}]\subseteq[s_{0},s_{h}]\), \(0\leq b^{2}+d^{2}+o^{2}\leq h^{2}\) and \(s_{a},s_{b},s_{c},s_{d},s_{e},s_{o}\in\overline{S}\). The basic operations of LIVSFs are defined as follows._  
_Let \(A=\{\langle x,(\bar{s}_{uA}(x),\bar{s}_{\pi A}(x),\bar{s}_{\pi A}(x))\rangle|x \in U\}\), \(B=\{\langle x,(\bar{s}_{uB}(x),\bar{s}_{\pi B}(x),\bar{s}_{\pi B}(x))\rangle|x\in U\}\)\(\in\)\(LIVS(U)\), then_  
1. \(A\subseteq B\Leftrightarrow\bar{s}_{\mu A}^{L}(x)\leq\bar{s}_{uB}^{L}(x),\bar{s }_{\mu A}^{U}(x)\leq\bar{s}_{uB}^{U}(x),\bar{s}_{\pi A}^{L}(x)\leq\bar{s}_{\pi B }^{L}(x),\bar{s}_{\
u A}^{U}(x)\leq\bar{s}_{\
u B}^{U}(x),\)__ \(\bar{s}_{\
u A}^{L}(x)\geq\bar{s}_{\
u B}^{U}(x),\bar{s}_{\
u A}^{U}(x)\geq \bar{s}_{\
u B}^{U}(x);\)__
2. \(A=B\Leftrightarrow A\subseteq B,B\subseteq A\)_;_
3.