# Linguistic Interval-Valued Spherical Fuzzy Soft Set and Its Application in Decision Making
## 4 Application of Linguistic Interval-Valued Spherical Fuzzy Soft Set
### Comparative Analysis  
In this subsection, we compare the MADM algorithms presented by [PERSON] et al. [23] and [PERSON] et al. [32]. for an interval-valued fuzzy soft set with our proposed algorithm. In [23], [PERSON] selected the best choice by calculating the score. The idea of our MADM algorithm is similar to [32]. We use Example 1 to demonstrate how these three MADM algorithms choose the best choice. Since each parameter is weighted equally, it does not affect the results. Orderings of the alternatives for each method are given in Table 4.  
We see that the proposed algorithm is comparable to other algorithms in terms of the selection of optimal elements, which illustrates the rationality and effectiveness of our proposed algorithm. The main difference between the proposed algorithm and previous algorithms is the use of linguistic interval-valued spherical fuzzy numbers to evaluate parameters. And we also consider the influence of parameter weight. Hence, in real life, we can deal with MADM problems described by linguistic interval-valued spherical fuzzy numbers according to our proposed algorithm. Since the goal of parameter reduction algorithms based on different fuzzy soft set models is to maintain the decision results unchanged, we compare with the parameter reduction algorithms presented by [PERSON] et al. [31]. [PERSON] proposed the Keeping optimal choice parameter reduction algorithm (KOCPR), the Keeping top three choice parameter reduction algorithm (KTTCPR) and the Standard parameter reduction algorithm (SPR) to reduce parameters. We also compare our proposed PRAKC with the above three algorithms in terms of the number of objects that keep the decision choice result unchanged (KDCRU) and the case of retaining parameters after reducing (RPR) using Example 1. The comparison results are shown in Table 5. \begin{table}
\begin{tabular}{l l} \hline \hline
**MADM Algorithms** & **Orderings of the Alternatives** \\ \hline [PERSON] et al. [23] & \(x_{1}>x_{4}>x_{3}>x_{2}\) \\ [PERSON] et al. [32] & \(x_{1}>x_{4}>x_{3}>x_{2}\) \\ Our proposed algorithm & \(x_{1}>x_{4}>x_{3}>x_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Orderings of the alternatives according to four MADM algorithms. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{U}\) & \(\mathbf{e_{2}}\) & \(\mathbf{e_{3}}\) & \(\omega\mathbf{c_{i}}\) & \(\omega\mathbf{c_{i}^{overall}}\) \\ \hline \(x_{1}\) & \(([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])\) & \(([s_{5},s_{6}],[s_{0},s_{2}],[s_{1},s_{2}])\) & [2.981,3.174] & 6.155 \\ \(x_{2}\) & \(([s_{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3}])\) & \(([s_{2},s_{4}],[s_{0},s_{3}],[s_{3},s_{4}])\) & [2.405,2.543] & 4.948 \\ \(x_{3}\) & \(([s_{3},s_{4}],[s_{1},s_{3}],[s_{2},s_{3}])\) & \(([s_{3},s_{4}],[s_{1},s_{3}],[s_{2},s_{3}])\) & [2.465,2.524] & 4.989 \\ \(x_{4}\) & \(([s_{4},s_{5}],[s_{0},s_{2}],[s_{2},s_{3}])\) & \(([s_{4},s_{5}],[s_{0},s_{2}],[s_{1},s_{3}])\) & [2.707,2.776] & 5.483 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: LVSFSS \((\tilde{F},E)\) of Example 3. \begin{table}
\begin{tabular}{l l} \hline \hline
**Algorithms** & **KDCRU** & **RPR** \\ \hline KOCPR & keep the decision result of the top choices & \(\{e_{2}\},\{e_{3}\},\{e_{1},e_{2}\},\{e_{1},e_{3}\},\{e_{2},e_{3}\}\); \\ KTTCPR & keep the decision result of the top three choices & \(\{e_{3}\},\{e_{2},e_{3}\}\); \\ SPR & keep the decision result of all & \(\{e_{2},e_{3}\}\); \\  & objects choices & \(\lambda=1\), we can obtain \(\{e_{2}\},\{e_{3}\}\), \\  & keep the decision result of the top \(\lambda\) & \(\{e_{1},e_{2}\},\{e_{1},e_{3}\},\{e_{2},e_{3}\}\); \\ PRAKC & keep the decision result of the top \(\lambda\) & \(\lambda=2\), we can obtain \(\{e_{3}\},\{e_{2},e_{3}\}\); \\  & choices, \(\lambda\in[1,4]\) & \(\lambda=3\), we can obtain \(\{e_{3}\},\{e_{2},e_{3}\}\); \\  & & \(\lambda=4\), we can obtain \(\{e_{2},e_{3}\}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of four parameter reduction algorithms.