# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 4 Application of Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Parameter Reduction  
In the process of decision making, some redundant parameters are not necessary. Thus, the parameter reduction becomes important. According to Algorithm 1, we can obtain a descending queue based on the linguistic weighted overall choice value. In real life, not all decision-making problems require only one choice, sometimes they require several choices. Therefore, it is necessary to consider multiple scenarios in the context of parameter reduction.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{U}\) & \(\mathbf{e_{1}}\) & \(\mathbf{e_{2}}\) & \(\mathbf{e_{3}}\) & \(\mathbf{\omega c_{i}}\) & \(\mathbf{\omega c_{i}^{overall}}\) \\ \hline \(x_{1}\) & \(([s_{3},s_{5}],[s_{0},s_{3}],[s_{2},s_{3}])\) & \([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])\) & \(([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])\) & \([4.214,4.562]\) & \(8.776\) \\ \(x_{2}\) & \(([s_{3},s_{5}],[s_{0},s_{3}],[s_{2},s_{3}])\) & \([[s_{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3}])\) & \(([s_{2},s_{4}],[s_{0},s_{3}],[s_{3},s_{4}])\) & \([3.637,3.931]\) & \(7.568\) \\ \(x_{3}\) & \(([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])\) & \(([s_{3},s_{4}],[s_{1},s_{3}],[s_{2},s_{3}])\) & \(([s_{3},s_{4}],[s_{1},s_{3}],[s_{2},s_{3}])\) & \([3.955,4.111]\) & \(8.066\) \\ \(x_{4}\) & \(([s_{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3}])\) & \(([s_{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3}])\) & \(([s_{4},s_{5}],[s_{0},s_{2}],[s_{1},s_{3}])\) & \([4.040,4.164]\) & \(8.204\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: LVSSPSS \((\tilde{F},E)\) of Example 2.  
Figure 2: Flow chart of Algorithm 1.  
**Definition 20**.: _Let \((\bar{F},E)\) be a linguistic interval-valued spherical fuzzy soft set over \(U\), \(U=\{x_{1},x_{2},\ldots,x_{n}\}\), \(E=\{e_{1},e_{2},\ldots,e_{m}\}\), \(\bar{S}=\{s_{\alpha}|s_{0}\leq s_{\alpha}\leq s_{h},\alpha\in[0,h]\}\), \(D_{E}=\{x_{i},....x_{j}|\)\(i,j\in[1,n]\}\) be the set of all objects sorted in descending order by linguistic weighted overall choice values, \(\lambda\in[1,n]\) be called decision-making number, \(D_{E}^{\lambda}\) be the set of \(\lambda\) objects sorted in descending order by linguistic weighted overall choice values. If a subset \(A=\{e_{1}^{\prime},e_{2}^{\prime},\ldots,e_{n}^{\prime}\}\subset E\) satisfies \(D_{E-A}^{\lambda}=D_{E}^{\lambda}\), \(A\) is called the unnecessary parameter set in \(E\) with \(\lambda\) candidates, otherwise \(A\) is called the necessary parameter set in \(E\) with \(\lambda\) candidates._  
In other words, if A is the unnecessary parameter set with \(\lambda\) candidates, it means that A can be reduced, otherwise A cannot be reduced. Based on the above definition, the parameter reduction algorithm of keeping \(\lambda\) candidates (PRKAC) (Algorithm 2) is proposed as follows.  
The flow chart of the PRAKC algorithm is shown in Figure 3.  
**Example 3**.: _We use \((\bar{F},E)\), \(U=\{h_{1},h_{2},h_{3},h_{4}\}\), \(E=\{e_{1},e_{2},e_{3}\}\), \(\omega=\{\omega_{1},\omega_{2},\omega_{3}\}=\{1/3,\ 1/3,1/3\}\) shown in Table 2 and set \(\lambda=4\) to demonstrate how PRAKC reduces unnecessary parameters whilst maintaining the four candidates as invariable. We obtain that \(x_{1},x_{4},x_{3},x_{2}\) are the four choice candidates from Example 2. So, \(D_{E}^{\pm}=\{x_{1},x_{4},x_{3},x_{2}\}\). We can find \(A=\{e_{1}\}\) satisfies \(D_{E-A}^{2}=D_{E}^{2}\). Therefore, \(E-A=\{e_{2},e_{3}\}\) is the parameter reduction with two candidates, which is given in Table 3._  
Figure 3: Flow chart of Algorithm 2.