# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 3 Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Basic Concept of LIVSFSS  
**Definition 6**. : _Let \(U\) be an initial universe, \(E\) be a set of parameters,\(LIVSF(U)\) be the set of all LIVSFS of \(U\). A pair \(<\tilde{F},E>\) is called a linguistic interval-valued spherical fuzzy soft set on \(U\), where \(\tilde{F}\) is the mapping given by_  
\[\tilde{F}:\ E\to LIVSF(U) \tag{10}\]  
An LIVSFSS is a parameterized family of linguistic interval-valued spherical fuzzy subsets of \(U\), thus, its universe is the set of all LIVSFS on \(U\). In other words, because LIVSFSS is still a mapping from parameters to \(LIVSF(U)\), it is regarded as a powerful extention of soft set. For \(\forall\in E\), \(\tilde{F}(\varepsilon)\) is referred to as the LIVSFS with parameter \(\varepsilon\), and it is actually an LIVSFS on \(U\). For \(x\in U\) and \(\varepsilon\in E\), it can be written as follows. \[\tilde{F}(\varepsilon)=\Big{\{}\Big{\langle}x_{r}\Big{(}\tilde{s}_{u\tilde{F} (\varepsilon)}(x),\tilde{s}_{u\tilde{F}(\varepsilon)}(x),\tilde{s}_{v\tilde{F} (\varepsilon)}(x)\Big{)}\Big{\rangle}\Big{|}x\in U\Big{\}} \tag{11}\]  
where \(\tilde{F}_{x_{i}}(\varepsilon)=([s^{L}_{u\tilde{F}(\varepsilon)}(x_{i}),s^{ L}_{u\tilde{F}(\varepsilon)}(x_{i})],[s^{L}_{u\tilde{F}(\varepsilon)}(x_{i}),s^{ U}_{u\tilde{F}(\varepsilon)}(x_{i})],[s^{L}_{v\tilde{F}(\varepsilon)}(x_{i}),s^{ U}_{v\tilde{F}(\varepsilon)}(x_{i})])\) are the membership degree, the hesitancy degree and the non-membership degree of object \(x_{i}\) respectively. **Example 1**. : _Under uncertain environments, how do we deal with the situation where the sum of membership degree, hesitancy and non-membership degree exceeds 1 after the experts evaluate each parameter? And experts prefer to evaluate each parameter in natural language. LIVSFSS can facilitate the handing of the above situations. Let \(U=\{x_{1},x_{2},x_{3},x_{4}\}\) be a set of teachers and \(E=\{e_{1},e_{2},e_{3}\}=\{\)teaching quality, blackboard writing, research ability\(\}\) be a set of parameters for evaluation indicators of teachers. Experts assign a value to each parameter according to a continuous LTS \(\overline{S}\) defined as \(\overline{S}=\{s_{0}=``extremely poor\",\)\(s_{1}=``very poor\",\)\(s_{2}=``poor\",\)\(s_{3}=``slightly poor\",\)\(s_{4}=``medium\",\)\(s_{5}=``slightly good\",s_{6}=``good\",\)\(s_{7}=``very good\",\)\(s_{8}=``extremely good\"\}\). The result described by the LIVSFSS \((\tilde{F},E)\) is presented in Table \(1\) and \((\tilde{F},E)\) is defined as follows._\[ \begin{array}{l}\tilde{F}(e_{1})=\{<x_{1},([s_{3},s_{5}],[s_{0},s_{3}],[s_{2},s_{ 3}])>,<x_{2},([s_{3},s_{5}],[s_{0},s_{3}],[s_{2},s_{3}])>,<x_{3},([s_{5},s_{6}], \\ [s_{0},s_{2}],[s_{1},s_{2}])>,<x_{4},([s_{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3} ])>\};\\ \tilde{F}(e_{2})=\{<x_{1},([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])>,<x_{2},([s _{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3}])>,<x_{3},([s_{3},s_{4}],\\ [s_{1},s_{3}],[s_{2},s_{3}])>,<x_{4},([s_{4},s_{5}],[s_{0},s_{2}],[s_{3} ])>\};\\ \tilde{F}(e_{3})=\{<x_{1},([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])>,<x_{2},([s _{2},s_{4}],[s_{0},s_{3}],[s_{3},s_{4}])>,<x_{3},([s_{3},s_{4}],\\ [s_{1},s_{3}],[s_{2},s_{3}])>,<x_{4},([s_{4},s_{5}],[s_{0},s_{2}],[s_{1},s_{3} ])>\}.\\ \end{array} \]  
**Definition 7**. : _Let \(U\) be an initial universe set, \(E\) be a set of parameters and suppose that \(A\), \(B\subset\,E\), \((\tilde{F},A)\) and \((\tilde{G},B)\) are two linguistic interval-valued spherical fuzzy soft sets; \((\tilde{F},A)\) is a linguistic interval-valued spherical fuzzy soft subset of \((\tilde{G},B)\), which can be denoted by \((\tilde{F},A)\!\subset\! (\tilde{G},B)\), if and only if_  
1. \(A\subset B\)_;_
2. \(\forall\varepsilon\in A,\!\tilde{F}(\varepsilon)\) _is a linguistic interval-valued spherical fuzzy subset of_ \(\tilde{G}(\varepsilon)\)_._  
**Definition 8**.