Fermatean Vague Soft Set and Its Application in Decision Making

Abstract

In this paper, we introduce the concept of Fermatean vague soft set and its operations, which are union, intersection, complement, subset, AND, and OR. An application of Fermatean vague soft set in decision-making according to the degree of preference is demonstrated. Fermatean vague soft set is profound, as it bridges gaps in traditional vague and soft set theories. By offering enhanced precision and flexibility, the Fermatean vague soft set provides decision-makers with a powerful tool to tackle challenges in fields like artificial intelligence, operational research, and engineering design, ultimately driving more accurate and reliable outcomes

1. Introduction

In several real-world scenarios, data frequently exhibit incompleteness, imprecision, or inconsistency, owing to variables such as randomness, measurement mistakes, and human subjectivity. This uncertainty presents difficulties in areas such as engineering, economics, medical research, and environmental studies, where decision-making necessitates the management of unclear or ambiguous information. Several mathematical frameworks have been developed throughout the years to address these issues, each enhancing knowledge and modeling of uncertainty.

Beginning with the introduction of fuzzy set theory by Zadeh [1], which offered a mathematical framework to handle information that is unclear, there have been considerable advancements in the management of uncertainty and imprecision in decision-making processes. When compared with classical sets, fuzzy sets provide a more complex view of reality, since they allow partial membership. By adding a hesitation margin, membership and non-membership values, and a further degree of uncertainty, Atanassov’s intuitionistic fuzzy sets expanded this even further. Adding to this earlier work, Atanassov [2,3] extended the ability to deal with more complicated data uncertainty by developing operators for interval-valued intuitionistic fuzzy sets.

Neutrosophic theory, introduced to address indeterminacy in decision-making and mathematical modeling, has found diverse applications in areas such as earthquake response, smart grid systems, and internet streaming services. Recent studies highlight significant contributions to the field. For instance, Al-Omeri et al. [4,5,6,7,8,9,10,11,12] explored the role of neutrosophic graphs in earthquake response centers in Japan, showcasing their potential in disaster management by handling ambiguous and uncertain data effectively. This application underscores the relevance of neutrosophic approaches in addressing complex real-world challenges. In the domain of fuzzy set theory, examined fuzzy totally continuous mappings based on fuzzy αm-Open sets within Šostak’s framework. This work expands the theoretical foundation of fuzzy set theory, emphasizing its utility in computational and applied mathematics. Similarly, Al-Omeri’s research on fuzzy irresolute mappings demonstrated their significance in smart decision-making for electric vehicle systems, highlighting the integration of fuzzy logic in enhancing energy-efficient transportation systems. Neutrosophic algebra also witnessed advancements through the study of INK-algebras, which investigated for translation and application in complex systems. This exploration broadens the scope of neutrosophic sets in algebraic structures, aiding in the development of novel computational methods. Moreover, the study on identifying internet streaming services using the max product of complements in neutrosophic graphs illustrates the application of these graphs in optimizing decision-making processes in digital environments. Topological spaces, another key area of mathematical research, benefited from investigations into nano soft ideal topological spaces, shedding light on novel topological properties. Al-Omeri’s contributions to neutrosophic topology, particularly through -closed sets and their properties, provided new insights into the behavior of neutrosophic topological spaces, enriching the theoretical landscape, Furthermore, studies on bipolar neutrosophic nano-open sets expanded the application of neutrosophic systems in complex data structures.

Simultaneously, Gau and Buehrer [13] introduced vague set theory, which is identical to intuitionistic fuzzy sets but distinguishes itself in that it uses lower and upper boundaries for membership degrees. As Bustince and Burillo [14] pointed out, intuitionistic fuzzy sets are the derivatives of vague sets, adding to this relation. In system reliability analysis, for example, Chen [15,16] modeled and evaluated the reliability of systems in the presence of fuzzy conditions, providing a contribution by defining the criteria for comparing and combining fuzzy environments and similarity measurements between vague sets.

Molodtsov [17] proposed the soft set theory as a typical mathematical method in order to deal with unidentified information that typical mathematical methods fail to handle. He demonstrated several applications of this theory in addressing various practical issues in engineering, economics, medical sciences, and social sciences. Afterwards, researchers, such as Maji et al. [18], conducted more research on the notion of soft sets and applied this theory to solve other decision-making problems. They developed the idea of the fuzzy soft set, a more generalized notion that combines the fuzzy set and soft set, and explored its properties. In 2007, Aktas and Cagman [19] defined the concept of soft groups. Majumdar and Samanta [20] studied the problem of similarity measurements among soft sets.

The significance of vague sets in decision-making was heightened by Hong and Choi [21], who applied vague set theory to MADM problems. Gorzalzany’s [22] work on interval-valued fuzzy sets provided a method for inference in approximate reasoning, establishing a foundation for using interval-valued sets in decision-making. In practical applications, Kumar et al. [23,24] used interval-valued vague sets to analyze the reliability of marine power plants and extended this work with arithmetic operations on interval-valued vague sets for system reliability analysis.

Vague soft set theory, introduced by Xu et al. [25], is a flexible and advanced mathematical tool for handling situations where the information is uncertain and vague. Recent research has built upon these foundations, incorporating more sophisticated methods such as the transitive closure of vague soft set relations and its operators, possibility interval-valued vague soft set, generalized interval-valued vague soft set, and cubic vague sets, which have been applied in MADM, introduced by Alhazaymeh et al. [26,27,28,29,30,31]. Additionally, these operations on vague soft sets have further expanded the theoretical and practical scope of soft set theory, particularly in uncertain environments.

A more recent study expanded upon these foundations by integrating advanced methodologies, including the fuzzy emergency model and robust emergency strategy for supply chain systems under random supply disruptions, as defined by Zhang et al. [32,33]. This study emphasized effective emergency techniques that utilize fuzzy decision-making to reduce risks and ensure supply chain continuity during unforeseen catastrophes.

Recent advancements in fuzzy decision-making have significantly improved operational strategies across various domains. Zhang et al. [25] proposed a fuzzy control model for nonlinear supply chain systems with lead times, demonstrating its effectiveness in enhancing stability and responsiveness under dynamic conditions.

Fermatean membership functions (characterized by a pair of values representing degrees of membership and non-membership) allow for a more detailed modeling of uncertainty, with a condition ensuring that the sum of membership and non-membership degrees, squared, is less than or equal to one. The motivation for creating the Fermatean vague soft set (FVSS) comes from the necessity to overcome deficiencies in current models for handling uncertainty, imprecision, and vagueness in nuanced decision-making situations. Primary motives encompass FVSS combining the strengths of Fermatean fuzzy sets, allowing higher flexibility in defining membership, non-membership, and hesitation, with the vague soft set’s capacity to manage imprecise parameterization. By integrating vague soft sets and Fermatean fuzzy logic, FVSS can handle overlapping uncertainties in data, improving analysis accuracy.

Uncertainty modeling is made easier by the Fermatean vague soft set (FVSS), which takes the best parts of fuzzy, vague, and soft set theories and fixes their worst parts. Because of its dynamic interaction with truth, falsity, and degrees of uncertainty, it can accurately handle the complexities of real-world problems. To capture the richness of human reasoning, FVSS uses Fermatean parameters to depict deeper, non-linear associations, as opposed to conventional methods that are constrained by interval-based or linear memberships. Advanced decision-making in the face of disagreements is now possible because of the recently established framework, which successfully handles conflicting, overlapping, and nebulous data. Redefining our approach to uncertainty, FVSS decreases computational needs and promotes flexibility, offering a simple and adaptable framework that quickly adapts to varied issues. Whether dealing with complicated datasets or ambiguous scenarios, FVSS is a one-of-a-kind method that enhances confidence and clarity in decision analysis.

The objective of introducing the Fermatean vague soft set is to establish a comprehensive and robust framework for addressing complex decision-making challenges by enhancing existing methodologies. This hybrid framework aims to manage multi-criteria and multi-parameter scenarios while developing novel operations and algorithms for information aggregation and analysis within this framework.

The paper’s structure is as follows: In Section 2, we recall the basic concept of both the soft set and the vague soft set, like union, intersection, and complement. In Section 3, we introduce the new concept of the Fermatean vague soft set and explain its properties. In Section 4, we discuss some properties of union, intersection, and complementation. In Section 5, we offer an application in decision making that is useful for handling uncertainty and vagueness. In Section 6, Fermatean vague soft sets (FVSS) are compared with other conventional models. Section 7 contains our last remarks.

2. Preliminary

In this section, we offer some definitions that are required for this paper.

2.1. Soft Set

Let $U$ be the universal set and $E$ be the set of attributes with respect to $U$. Let $P(U)$ be the power set of $U$ and $A\subseteq E.$ The pair $(F,A)$ is called a soft set over $U$, and its mapping is given as $F:A\to P(U)$. It is also defined as $(F,A)=\left\{F(e)\in P(U):e\in E,F(e)=\varnothing ife\notin A\right\}$.

Definition 1.

If the union of two soft sets of $(F,A)$ and $(G,B)$ over the universe $U$ is a soft set $(H,Q)$, then $Q=A\cup B$.

$$H(e)=\left\{\begin{array}{c}F\left(e\right) \mathrm{If} e\in A-B\\ G\left(e\right) \mathrm{If} e\in B-A\\ F(e)\cup G\left(e\right) \mathrm{If} e\in B\cap A\end{array}\right.$$

Then, we denote that $\left(F,A\right)\cup \left(G,B\right)=(H,Q)$.

Definition 2.

If the intersection of two soft sets of $(F,A)$ and $(G,B)$ over the universe $U$ is a soft set $(H,Q)$, then $Q=A\cap B$.

$$H(e)=\left\{\begin{array}{c}F\left(e\right) \mathrm{If} e\in A-B\\ G\left(e\right) \mathrm{If} e\in B-A\\ F(e)\cap G\left(e\right) \mathrm{If} e\in B\cap A\end{array}\right.$$

Then, we denote that $\left(F,A\right)\cap \left(G,B\right)=(H,Q)$.

Definition 3.

The complement of a soft set $(F,A)$ is denoted by ${(F,A)}^c=(F^c,\neg A)$, where $F^c$ : $\neg A\to P(U)$ is the mapping given by $F^c\left(\alpha \right)=U-F\left(\neg \alpha \right), for all \alpha \in \neg A$.

2.2. Vague Soft Sets

Definition 4.

The pair $(\widehat{F},A)$ is called a vague soft set over $U$, where $\widehat{F}$ is mapped as follows:

$$\widehat{F}:A\to V(U)$$

A vague soft set over $U$ is the parameters family of the vague set of the universe $U.$ For $\epsilon \in A$, ${\mu }_{\widehat{F}(\epsilon )}: U\to {[0,1]}^2$ is considered the collection set of $\epsilon$-approximate elements of the vague soft set $\left(\widehat{F},A\right).$

Definition 5.

For a vague soft set, the complement of $\left(\widehat{F},A\right)$ is denoted by ${\left(\widehat{F},A\right)}^c$, defined by ${\left(\widehat{F},A\right)}^c=\left({\widehat{F}}^c,\neg A\right)$, where ${\widehat{F}}^c :\neg A\to V(U)$ is the mapping given by:

i.

$t_{{\widehat{F}\left(\alpha \right)}^c}(x)=f_{\widehat{F}\left(\neg \alpha \right)}(x)$,

ii.

$1-f_{{\widehat{F}\left(\alpha \right)}^c}\left(x\right)=f_{\widehat{F}\left(\neg \alpha \right)}\left(x\right)$.

Example 1.

Let a vague soft set be $\left(\widehat{F},E\right)$, where $U$ is a set of six houses under the consideration of a decision maker to purchase, which is denoted by $U=\left\{u_1,u_2,\dots ,u_6\right\}$, and $E$ is a parameter set, where $E=\left\{e_1,e_2,e_3,e_4,e_5\right\}=\left\{cheap,expensive,wooden,beautiful,green\right\}$. The vague soft set $\left(\widehat{F},A\right)$ describes the ‘‘attractiveness of the houses’’ to this decision maker.

Suppose that:

$$\widehat{F}\left(e_1\right)=\left(\left[0.1,0.2\right]/u_1,\left[0.9,1.0\right]/u_2,\left[0.3,0.5\right]/u_3,\left[0.8,0.9\right]/u_4,\left[0.2,0.4\right]/u_5,\left[0.4,0.6\right]/u_6\right),$$

$$\widehat{F}\left(e_2\right)=\left(\left[0.9,1.0\right]/u_1,\left[0.2,0.7\right]/u_2,\left[0.6,0.9\right]/u_3,\left[0.2,0.4\right]/u_4,\left[0.3,0.4\right]/u_5,\left[0.1,0.6\right]/u_6\right),$$

$$\widehat{F}\left(e_3\right)=\left(\left[0.0,0.0\right]/u_1,\left[0.0,0.0\right]/u_2,\left[1.0,1.0\right]/u_3,\left[1.0,1.0\right]/u_4,\left[1.0,1.0\right]/u_5,\left[0.0,0.0\right]/u_6\right),$$

$$\widehat{F}\left(e_4\right)=\left(\left[0.8,0.9\right]/u_1,\left[0.0,0.1\right]/u_2,\left[0.5,0.7\right]/u_3,\left[0.1,0.2\right]/u_4,\left[0.6,0.8\right]/u_5,\left[0.4,0.6\right]/u_6\right),$$

$$\widehat{F}\left(e_5\right)=\left(\left[0.9,1.0\right]/u_1,\left[0.2,0.3\right]/u_2,\left[0.1,0.4\right]/u_3,\left[0.1,0.2\right]/u_4,\left[0.2,0.4\right]/u_5,\left[0.7,0.9\right]/u_6\right).$$

Then, the complement of a vague soft set $\left(\widehat{F},E\right)$ is given below:

$${\widehat{F}\left(e_1\right)}^c=\left(\left[0.8,0.9\right]/u_1,\left[0.0,0.1\right]/u_2,\left[0.5,0.7\right]/u_3,\left[0.1,0.2\right]/u_4,\left[0.6,0.8\right]/u_5,\left[0.4,0.6\right]/u_6\right),$$

$${\widehat{F}\left(e_2\right)}^c=\left(\left[0.0,0.1\right]/u_1,\left[0.3,0.8\right]/u_2,\left[0.1,0.4\right]/u_3,\left[0.6,0.8\right]/u_4,\left[0.6,0.7\right]/u_5,\left[0.4,0.9\right]/u_6\right),$$

$${\widehat{F}\left(e_3\right)}^c=\left(\left[1.0,1.0\right]/u_1,\left[1.0,1.0\right]/u_2,\left[0.0,0.0\right]/u_3,\left[0.0,0.0\right]/u_4,\left[0.0,0.0\right]/u_5,\left[1.0,1.0\right]/u_6\right),$$

$${\widehat{F}\left(e_4\right)}^c=\left(\left[0.1,0.2\right]/u_1,\left[0.9,1.0\right]/u_2,\left[0.3,0.5\right]/u_3,\left[0.8,0.9\right]/u_4,\left[0.2,0.4\right]/u_5,\left[0.4,0.6\right]/u_6\right),$$

$${\widehat{F}\left(e_5\right)}^c=\left(\left[0.0,0.1\right]/u_1,\left[0.7,0.8\right]/u_2,\left[0.6,0.9\right]/u_3,\left[0.8,0.9\right]/u_4,\left[0.6,0.8\right]/u_5,\left[0.1,0.3\right]/u_6\right).$$

Definition 6.

A null, vague soft set $\left(\widehat{F},A\right)$ over $U$ is denoted by $\phi$; if $\epsilon \in A$, then:

i.

$t_{\widehat{F}\left(\epsilon \right)}(x)=0$,

ii.

$1-f_{\widehat{F}\left(\epsilon \right)}\left(x\right)=0.$

Example 2.

Consider $U=\left\{u_1,u_2,u_3\right\}$ to be a vague soft set of the universe defined on $U$. Then, the null vague soft set is defined as follows:

$$\phi =\left\{\left({\displaystyle \frac{u_1}{[0,0]}},{\displaystyle \frac{u_2}{\left[0,0\right]}},{\displaystyle \frac{u_3}{\left[0,0\right]}}\right)\right\}$$

Definition 7.

An absolute of the vague soft set $\left(\widehat{F},A\right)$ over $U$ is denoted by $\widehat{A}$; if $\epsilon \in A$ then:

i.

$t_{\widehat{F}\left(\epsilon \right)}(x)=1$,

ii.

$1-f_{\widehat{F}\left(\epsilon \right)}\left(x\right)=1.$

Example 3.

Let $U=\left\{u_1,u_2,u_3\right\}$ be a vague soft set of the universe defined on $U$. Then, the absolute vague soft set is defined as follows:

$$\widehat{A}=\left\{\left({\displaystyle \frac{u_1}{[1,1]}},{\displaystyle \frac{u_2}{\left[1,1\right]}},{\displaystyle \frac{u_3}{\left[1,1\right]}}\right)\right\}$$

Definition 8.

If the union of two vague soft sets of $\left(\widehat{F},A\right)$ and $(\widehat{G},B)$ over the universe $U$ is the vague soft set $(\widehat{H},Q)$, then $Q=A\cup B$.

$$t_{H\left(e\right)}(x)=\left\{\begin{array}{c}{t}_{\widehat{F}\left(e\right)}(x), \mathrm{If} e\in A-B\\ t_{\widehat{G}\left(e\right)}(x), \mathrm{If} e\in B-A\\ \max \left(t_{\widehat{F}\left(e\right)}\left(x\right),t_{\widehat{G}\left(e\right)}\left(x\right)\right), \mathrm{If} e\in B\cap A\end{array}\right.$$

$$1-f_{H\left(e\right)}(x)=\left\{\begin{array}{c}1-f_{\widehat{F}\left(e\right)}(x), \mathrm{If} e\in A-B\\ 1-f_{\widehat{G}\left(e\right)}(x), \mathrm{If} e\in B-A\\ \max \left(1-f_{\widehat{F}\left(e\right)}(x),1-f_{\widehat{G}\left(e\right)}(x)\right), \mathrm{If} e\in B\cap A\end{array}\right.$$

Then, we denote that $\left(\widehat{F},A\right)\cup \left(\widehat{G},B\right)=(\widehat{H},Q)$.

Example 4.

Let $X$ be a universe set. Suppose $\left(\widehat{F},A\right)$ and $\left(\widehat{G},B\right)$ are two vague soft sets of the universe $U$, defined by:

$$\left(\widehat{F},A\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.5,0.5\right]}},{\displaystyle \frac{u_2}{\left[0.4,0.8\right]}},{\displaystyle \frac{u_3}{\left[0.2,0.3\right]}}\right)\right\},$$

$$\left(\widehat{G},B\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.4,0.4\right]}},{\displaystyle \frac{u_2}{\left[0.1,0.7\right]}},{\displaystyle \frac{u_3}{\left[0.6,0.9\right]}}\right)\right\}.$$

Then, by Definition 8, we have:

$$(\widehat{H},Q)=\left\{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{\widehat{F}\left(e\right)}\left(u\right),t_{\widehat{G}\left(e\right)}\left(u\right)\right),\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{\widehat{F}\left(e\right)}\left(u\right),{1-f}_{\widehat{G}\left(e\right)}\left(u\right)\right)\right)\right\}$$

$$(\widehat{H},Q)=\left\{{\displaystyle \frac{u_1}{\left[0.5,0.5\right]}},{\displaystyle \frac{u_2}{\left[0.4,0.8\right]}},{\displaystyle \frac{u_3}{\left[0.6,0.9\right]}}\right\}.$$

Definition 9.

If the intersection of two vague soft sets of $(\widehat{F},A)$ and $(\widehat{G},B)$ over the universe $U$ is the vague soft set $(\widehat{H},Q)$, then $Q=A\cap B$.

$$t_{H\left(e\right)}(x)=\left\{\begin{array}{c}{t}_{\widehat{F}\left(e\right)}(x), \mathrm{If} e\in A-B\\ t_{\widehat{G}\left(e\right)}(x), \mathrm{If} e\in B-A\\ \min \left(t_{\widehat{F}\left(e\right)}\left(x\right),t_{\widehat{G}\left(e\right)}\left(x\right)\right), \mathrm{If} e\in B\cap A\end{array}\right.$$

$$1-f_{H\left(e\right)}(x)=\left\{\begin{array}{c}1-f_{\widehat{F}\left(e\right)}(x), \mathrm{If} e\in A-B\\ 1-f_{\widehat{G}\left(e\right)}(x), \mathrm{If} e\in B-A\\ \min \left(1-f_{\widehat{F}\left(e\right)}(x),1-f_{\widehat{G}\left(e\right)}(x)\right), \mathrm{If} e\in B\cap A\end{array}\right.$$

Then, we denote that $\left(\widehat{F},A\right)\cap \left(\widehat{G},B\right)=(\widehat{H},Q)$.

3. Fermatean Vague Soft Set

In this section, we introduce the notation of the Fermatean vague soft set (FVSS).

Definition 10.

Let $U$ represent a universal set and $E$ be a set of parameters. Suppose $A\subseteq E$ and $F:A\to FV(U)$, and $\alpha$ represents a vague subset of $A$, $\alpha :A\to [0,1]$, where $FV(U)$ indicates the collection of all Fermatean vague soft subsets of $U$. Let $\tilde{F}:A\to FV(U)$ be a function defined as follows:

$$\tilde{F}\left(a\right)=\left\{{u,t}_{F(a)},{1-f}_{F(a)}\right\}$$

Then, $\tilde{F}\left(a\right)$ is called the Fermatean vague soft set over $(U,E)$.

The values $t_{F(a)}\left(u\right)$ and $f_{F(a)}\left(u\right)$ for each $u\in U$ represent the degrees of truth membership and falsity membership, respectively, satisfying:

$${t_{F(a)}\left(u\right)}^3+{f_{F(a)}\left(u\right)}^3\le 1.$$

Example 5.

Consider a Fermatean vague soft set $(\tilde{F},E), where$ $U$ is a set of three Ph.D. students nominated as smart, and $E$ is a set of parameters. Let $A\subseteq E$ and $A=\left\{r=“results” , c=“conduct” , g=“game and sport performance”\right\}$. Then, we define $\tilde{F}$ as follows:

$$\tilde{F}\left(r\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.6,0.6\right]}},{\displaystyle \frac{u_2}{\left[0.6,0.8\right]}},{\displaystyle \frac{u_3}{\left[0.2,0.5\right]}}\right)\right\}$$

$$\tilde{F}\left(c\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.6,0.7\right]}},{\displaystyle \frac{u_2}{\left[0.7,0.8\right]}},{\displaystyle \frac{u_3}{\left[0.9,0.9\right]}}\right)\right\}$$

$$\tilde{F}\left(g\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.1,0.6\right]}},{\displaystyle \frac{u_2}{\left[0.3,0.5\right]}},{\displaystyle \frac{u_3}{\left[0.5,0.7\right]}}\right)\right\}$$

Then, $\tilde{F}$ is a Fermatean vague soft set over $(U,E)$.

Definition 11.

Consider $\tilde{\theta }$ to be a Fermatean vague soft set of the universe $U$, where $\forall { u}_i\in U$, $t_{F(a)}\left(u\right)=0$, and $f_{F(a)}\left(u\right)=1;$ then, $\tilde{\theta }$ is called the null Fermatean vague set, where $1\le i\le n$.

Example 6.

Consider $U=\left\{u_1,u_2,u_3\right\}$ to be a Fermatean vague soft set of the universe defined on $U$. Then, the absolute Fermatean vague soft set is defined as follows:

By Definition 11: $f_{F(a)}\left(u\right)=1$ , then $1-f_{F(a)}=0$

$$\tilde{\theta }=\left\{\left({\displaystyle \frac{u_1}{[0,0]}},{\displaystyle \frac{u_2}{\left[0,0\right]}},{\displaystyle \frac{u_3}{\left[0,0\right]}}\right)\right\}$$

Definition 12.

Let $\tilde{\vartheta }$ be a Fermatean vague soft set of the universe $U$, where $\forall { u}_i\in U$, $t_{F(a)}\left(u\right)=1$, and $f_{F(a)}\left(u\right)=0$; then, $\tilde{\vartheta }$ is called a unit Fermatean vague soft set, where $1\le i\le n$.

Example 7.

Let $U=\left\{u_1,u_2,u_3\right\}$ be a Fermatean vague soft set of the universe defined on $U$. Then, the unit Fermatean vague soft set is defined as follows:

By Definition 12: $f_{F(a)}\left(u\right)=0$, then $1-f_{F(a)}\left(u\right)=1$

$$\tilde{\vartheta }=\left\{\left({\displaystyle \frac{u_1}{[0,1]}},{\displaystyle \frac{u_2}{\left[0,1\right]}},{\displaystyle \frac{u_3}{\left[0,1\right]}}\right)\right\}$$

Definition 13.

Let $U$ be a universe set. Suppose ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague sets of the universe $U$. If $\forall { u}_i\in U$, then:

$$t_{AP}\left(u_i\right)\le t_{BP}(u_i)$$

$${1-f}_{AP}\left(u_i\right)\le {1-f}_{BP}(u_i)$$

Thus, the Fermatean vague set ${\tilde{F}}_A$ is subset by ${\tilde{F}}_B$, denoted by ${\tilde{F}}_A\subseteq {\tilde{F}}_B$, where $1\le i\le n$. It may not be evident why we chose this definition of containment instead of the more obvious “subinterval” relation ($t_{AP}\left({ u}_i\right)\le t_{BP}\left({ u}_i\right), 1-f_{AP}(u_i)\ge 1-f_{BP}({ u}_i)$). The definition makes more sense when viewed in terms of the lower bounds for the truth/false membership functions. The statement $A\subseteq B$ implies that the lower bound on the truth of $A$ also serves as a lower bound on the truth of $B (t_{AP}\left({ u}_i\right)\le t_{BP}\left({ u}_i\right))$, and the lower bound on the falsity of $B$serves as a lower bound on the falsity of $A (f_{BP}({ u}_i)\le f_{AP}({ u}_i)\iff 1-f_{AP}({ u}_i)\le 1-f_{BP})(u_i)$. When the truth/falsity values are restricted to 0 or 1, the relation $A\subseteq B$ reverts to the traditional definition where $x\in A\to x\in B,$and $x\notin B\to x\notin A.$

Example 8.

Let $U$be a universe set. Suppose ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague soft sets of the universe $U$, defined by:

$${\tilde{F}}_A=\left\{\left({\displaystyle \frac{u_1}{\left[0.5,0.5\right]}},{\displaystyle \frac{u_2}{\left[0.6,0.7\right]}},{\displaystyle \frac{u_3}{\left[0.5,0.6\right]}}\right)\right\},$$

$${\tilde{F}}_B=\left\{\left({\displaystyle \frac{u_1}{\left[0.7,0.9\right]}},{\displaystyle \frac{u_2}{\left[0.7,0.7\right]}},{\displaystyle \frac{u_3}{\left[0.6,0.7\right]}}\right)\right\}.$$

It is clear that ${\tilde{F}}_A$ is a Fermatean vague soft subset of ${\tilde{F}}_B$.

4. Some Properties of Union, Intersection, and Complementation

In this section, we will define some operations of a Fermatean vague soft set (FVSS).

Definition 14.

The complement of a Fermatean vague soft set $\tilde{F}$ is denoted by ${\left(\tilde{F}\right)}^c$, where:

i.

${\left(t_{F(a)}\left(u\right)\right)}^c=f_{F(a)}\left(u\right)$,

ii.

$1-{f_{F(a)}\left(u\right)}^c={1-t}_{F(a)}\left(u\right)$,

Example 9.

Consider Example 5. Then:

$$\tilde{F}\left(r\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.6,0.6\right]}},{\displaystyle \frac{u_2}{\left[0.6,0.8\right]}},{\displaystyle \frac{u_3}{\left[0.2,0.5\right]}}\right)\right\},$$

$$\tilde{F}\left(c\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.6,0.7\right]}},{\displaystyle \frac{u_2}{\left[0.7,0.8\right]}},{\displaystyle \frac{u_3}{\left[0.9,0.9\right]}}\right)\right\},$$

$$\tilde{F}\left(g\right)=\left\{\left({\displaystyle \frac{u_1}{\left[0.1,0.6\right]}},{\displaystyle \frac{u_2}{\left[0.3,0.5\right]}},{\displaystyle \frac{u_3}{\left[0.5,0.7\right]}}\right)\right\}.$$

Definition 15.

Let $U$be a universe set. Suppose ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague sets of the universe $U$. The union of two Fermatean vague sets is defined by:

$${\tilde{F}}_A\cup {\tilde{F}}_B={\tilde{F}}_C, \forall u\in U.$$

where:

i.

$t_{F_C}\left(u\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),$

ii.

${1-f}_{F_C}\left(u\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)=1-\mathrm{m}\mathrm{i}\mathrm{n}\left(f_{F_A}\left(u\right),f_{F_B}\left(u\right)\right).$

Definition 16.

Let $U$be a universe set. Suppose ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague soft sets of the universe $U$. Then, the intersection of two Fermatean vague soft sets is defined by:

$${\tilde{F}}_A\cap {\tilde{F}}_B={\tilde{F}}_C, \forall u\in U.$$

where:

i.

$t_{F_C}\left(u\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),$

ii.

${1-f}_{F_C}\left(u\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)=1-\mathrm{m}\mathrm{a}\mathrm{x}\left(f_{F_A}\left(u\right),f_{F_B}\left(u\right)\right).$

Example 10.

Consider Example 8; to find the union and intersection of two Fermatean vague soft sets, we used the vague set method:

i.

To find the union of two Fermatean vague soft sets:

$${\tilde{F}}_A\cup {\tilde{F}}_B=\left\{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)\right\}$$

$${\tilde{F}}_A\cup {\tilde{F}}_B=\left\{{\displaystyle \frac{u_1}{\left[0.7,0.9\right]}},{\displaystyle \frac{u_2}{\left[0.7,0.7\right]}},{\displaystyle \frac{u_3}{\left[0.6,0.7\right]}}\right\}.$$

ii.

To find the intersection of two Fermatean vague soft sets:

$${\tilde{F}}_A\cap {\tilde{F}}_B=\left\{\left(\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)\right\}$$

$${\tilde{F}}_A\cap {\tilde{F}}_B=\left\{{\displaystyle \frac{u_1}{\left[0.5,0.5\right]}},{\displaystyle \frac{u_2}{\left[0.6,0.7\right]}},{\displaystyle \frac{u_3}{\left[0.5,0.6\right]}}\right\}.$$

Definition 17.

Let $U$be a universe set. If ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague soft sets of the universe $U$, “ ${\tilde{F}}_A$ AND ${\tilde{F}}_B$”, denoted by “${\tilde{F}}_A$ $\wedge$ ${\tilde{F}}_B$”, is defined as:

$${\tilde{F}}_A\wedge {\tilde{F}}_B={\tilde{F}}_C, \forall u\in U.$$

where:

i.

$t_{F_C}\left(u\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),$

ii.

${1-f}_{F_C}\left(u\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)=1-\mathrm{m}\mathrm{a}\mathrm{x}\left(f_{F_A}\left(u\right),f_{F_B}\left(u\right)\right).$

Definition 18.

Let $U$be a universe set. If ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague soft sets of the universe $U$, “ ${\tilde{F}}_A$ OR ${\tilde{F}}_B$”, denoted by “${\tilde{F}}_A$ $\vee$ ${\tilde{F}}_B$”, is defined as:

$${\tilde{F}}_A\vee {\tilde{F}}_B={\tilde{F}}_C, \forall u\in U.$$

where:

i.

$t_{F_C}\left(u\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),$

ii.

${1-f}_{F_C}\left(u\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)=1-\mathrm{m}\mathrm{i}\mathrm{n}\left(f_{F_A}\left(u\right),f_{F_B}\left(u\right)\right).$

Proposition 1.

Consider ${\tilde{F}}_A$ and ${\tilde{F}}_B$ as two Fermatean vague soft sets of the universe $U$. Then, the following holds:

i.

${\tilde{F}}_A\cup {\tilde{F}}_B={\tilde{F}}_B\cup {\tilde{F}}_A$,

ii.

${\tilde{F}}_A\cap {\tilde{F}}_B={\tilde{F}}_B\cap {\tilde{F}}_A$.

Proof of Proposition 1.

(i)

$$\begin{array}{c}=\left\{\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right\},\hfill \\ =\left\{\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),1-\mathrm{m}\mathrm{i}\mathrm{n}\left(f_{F_A}\left(u\right),f_{F_B}\left(u\right)\right)\right\},\hfill \\ =\left\{\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_B}\left(u\right),t_{F_A}\left(u\right)\right),1-\mathrm{m}\mathrm{i}\mathrm{n}\left(f_{F_B}\left(u\right),f_{F_A}\left(u\right)\right)\right\},\hfill \\ ={\tilde{F}}_B\cup {\tilde{F}}_A.\hfill \end{array}$$

(ii)

$$\begin{array}{c}=\left\{\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right\},\hfill \\ =\left\{\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right),1-\mathrm{m}\mathrm{a}\mathrm{x}\left(f_{F_A}\left(u\right),f_{F_B}\left(u\right)\right)\right\},\hfill \\ =\left\{\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_B}\left(u\right),t_{F_A}\left(u\right)\right),1-\mathrm{m}\mathrm{a}\mathrm{x}\left(f_{F_B}\left(u\right),f_{F_A}\left(u\right)\right)\right\},\hfill \\ ={\tilde{F}}_B\cap {\tilde{F}}_A.\hfill \end{array}$$

□

Proposition 2.

If ${\tilde{F}}_A$ and ${\tilde{F}}_B$ are two Fermatean vague sets of the universe $X$, then the following holds:

i.

${\left({\tilde{F}}_A\cup {\tilde{F}}_B\right)}^c={\left({\tilde{F}}_A\right)}^c\cap {\left({\tilde{F}}_B\right)}^c$

ii.

${\left({\tilde{F}}_A\cap {\tilde{F}}_B\right)}^c={\left({\tilde{F}}_A\right)}^c\cup {\left({\tilde{F}}_B\right)}^c$

Proof of Proposition 2.

(i)

$$\begin{array}{c}={\left\{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right),\left(\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)\right\}}^c,\hfill \\ =\left\{{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right)}^c,{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)}^c\right\},\hfill \\ =\left\{\left(\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right),\left(\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)\right\},\hfill \\ =\left\{\left(\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right),\left(1-\mathrm{m}\mathrm{a}\mathrm{x}\left(f_{F_B}\left(u\right),f_{F_A}\left(u\right)\right)\right)\right\},\hfill \\ ={\left({\tilde{F}}_A\right)}^c\cap {\left({\tilde{F}}_B\right)}^c.\hfill \end{array}$$

(ii)

$$\begin{array}{c}={\left\{\left(\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right),\left(\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)\right\}}^c,\hfill \\ =\left\{{\left(\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right)}^c,{\left(\mathrm{m}\mathrm{i}\mathrm{n}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)}^c\right\},\hfill \\ =\left\{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right),\left(\mathrm{m}\mathrm{a}\mathrm{x}\left({1-f}_{F_A}\left(u\right),{1-f}_{F_B}\left(u\right)\right)\right)\right\},\hfill \\ =\left\{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{F_A}\left(u\right),t_{F_B}\left(u\right)\right)\right),\left(1-\mathrm{m}\mathrm{i}\mathrm{n}\left(f_{F_B}\left(u\right),f_{F_A}\left(u\right)\right)\right)\right\},\hfill \\ ={\left({\tilde{F}}_A\right)}^c\cup {\left({\tilde{F}}_B\right)}^c.\hfill \end{array}$$

□

5. An Application Based on the Fermatean Vague Soft Set

An application of the Fermatean vague soft set in decision-making is presented here.

Example 11.

Assume that the universe consists of three solar panel systems, that is, $U=\left\{\left.u_1,u_2,u_3\right\}\right.$, and there are three parameters, $E=\left\{e_1,e_2,e_3\}\right.$, where:

• $e_1 : Off-grid system suitability,$
• $e_2 : Hybrid system adaptability,$
• $e_3 : Zero-export system performance.$

which describe their performances according to certain specific tasks? Suppose Mr. Z wants to buy one such system, depending on either of the parameters only. Evaluations are illustrated using FVSS, where:

$$F_A(e_i)=\left\{\left(u_1,\left(m_{u_1},n_{u_1},h_{u_1}\right)\right),\left(u_2,\left(m_{u_2},n_{u_2},h_{u_2}\right)\right),\left(u_3,\left(m_{u_3},n_{u_3},h_{u_3}\right)\right)\right\},$$

where:

• $m_{u_j}$: Membership degree
• $n_{u_j}$: Non-membership degree
• $h_{u_j}$: Hesitation degree ($h_{u_j}=1-{m_{u_j}}^3-{n_{u_j}}^3$).

Improved definitions:

A Fermatean vague soft set is defined by the subsequent conditions:

• $m_{u_j},n_{u_j}\in [0,1]$
• ${m_{u_j}}^3+{n_{u_j}}^3\le 1$, ensure valid $h_{u_j}\ge 0.$
• It offers greater flexibility than intuitionistic and Pythagorean vague sets.

Let there be two observations, ${\tilde{F}}_A$ and ${\tilde{F}}_B$, by two experts with access to the system:

$${\tilde{F}}_A(e_1)=\left\{\left({\displaystyle \frac{u_1}{⟨0.2,0.6⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.4,0.6⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.3,0.6⟩}}\right)\right\};$$

$${\tilde{F}}_A(e_2)=\left\{\left({\displaystyle \frac{u_1}{⟨0.3,0.4⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.2,0.9⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.7,0.8⟩}}\right)\right\};$$

$${\tilde{F}}_A(e_3)=\left\{\left({\displaystyle \frac{u_1}{⟨0.6,0.6⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.3,0.9⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.3⟩}}\right)\right\};$$

$${\tilde{F}}_B(e_1)=\left\{\left({\displaystyle \frac{u_1}{⟨0.3,0.5⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.2,0.8⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.9⟩}}\right)\right\};$$

$${\tilde{F}}_B(e_2)=\left\{\left({\displaystyle \frac{u_1}{⟨0.2,0.4⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.7,0.7⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.3⟩}}\right)\right\};$$

$${\tilde{F}}_B(e_3)=\left\{\left({\displaystyle \frac{u_1}{⟨0.4,0.5⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.1,0.4⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.3⟩}}\right)\right\}.$$

The evaluations by ${\tilde{F}}_A$ and ${\tilde{F}}_B$ for each parameter:

Expert ${\tilde{F}}_A$:

$${\tilde{F}}_A(e_1)=\left\{\left({\displaystyle \frac{u_1}{⟨0.2,0.6,0.776⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.4,0.6,0.72⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.3,0.6,0.757⟩}}\right)\right\};$$

$${\tilde{F}}_A(e_2)=\left\{\left({\displaystyle \frac{u_1}{⟨0.3,0.4,0.909⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.2,0.9,0.263⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.7,0.8,0.145⟩}}\right)\right\};$$

$${\tilde{F}}_A(e_3)=\left\{\left({\displaystyle \frac{u_1}{⟨0.6,0.6,0.568⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.3,0.9,0.244⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.3,0.972⟩}}\right)\right\};$$

Expert ${\tilde{F}}_B$:

$${\tilde{F}}_B(e_1)=\left\{\left({\displaystyle \frac{u_1}{⟨0.3,0.5,0.848⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.2,0.8,0.48⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.9,0.27⟩}}\right)\right\};$$

$${\tilde{F}}_B(e_2)=\left\{\left({\displaystyle \frac{u_1}{⟨0.2,0.4,0.928⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.7,0.7,0.314⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.3,0.972⟩}}\right)\right\};$$

$${\tilde{F}}_B(e_3)=\left\{\left({\displaystyle \frac{u_1}{⟨0.4,0.5,0.811⟩}}\right),\left({\displaystyle \frac{u_2}{⟨0.1,0.4,0.935⟩}}\right),\left({\displaystyle \frac{u_3}{⟨0.1,0.3,0.972⟩}}\right)\right\}.$$

Application of decision-making:

To determine which solar panel system to purchase, we may adhere to the following steps:

Compute aggregation: evaluating ${\tilde{F}}_A$ and ${\tilde{F}}_B$ for each $e_i$ using the weighted average:

$$F(e_i)=\alpha F_A(e_i)+\beta F_B(e_i),\alpha +\beta =1.$$

• Alternative rank: calculate the score for each $u_j$ using the score function:

  $$S(u_j)=m_{u_j}-n_{u_j}.$$
• Select the best system: Identify the system with the highest aggregate score across all $e_i$:

  $$u^{\mathrm{*}}=\arg \underset{{\mathrm{u}}_{\mathrm{j}}}{\mathrm{m}\mathrm{a}\mathrm{x}}{\sum }_{i=1}^{3}S(u_j)$$

Example calculation:

Assume an aggregation score with $\alpha =0.5$ and $\beta =0.5$.

For parameter $e_1$:

$$u_1:m=0.5\left(0.2\right)+0.5\left(0.3\right)=0.25$$

$$u_1:n=0.5\left(0.6\right)+0.5\left(0.5\right)=0.55$$

$$u_1:h=1-{0.25}^3-{0.55}^3=0.818$$

$$u_2:m=0.5\left(0.4\right)+0.5\left(0.2\right)=0.3$$

$$u_2:n=0.5\left(0.6\right)+0.5\left(0.8\right)=0.7$$

$$u_2:h=1-{0.3}^3-{0.7}^3=0.63$$

$$u_3:m=0.5\left(0.3\right)+0.5\left(0.1\right)=0.2$$

$$u_3:n=0.5\left(0.6\right)+0.5\left(0.9\right)=0.75$$

$$u_3:h=1-{0.2}^3-{0.75}^3=0.5701$$

After calculation, we obtain the following:

For parameter $e_2$:

$$u_1:m=0.25,n=0.4,h=0.920$$

$$u_2:m=0.45,n=0.8,h=0.396$$

$$u_3:m=0.4,n=0.55,h=0.769$$

For parameter $e_3$:

$$u_1:m=0.5,n=0.55,h=0.708$$

$$u_2:m=0.2,n=0.65,h=0.717$$

$$u_3:m=0.1,n=0.3,h=0.972$$

Calculate the score of each solar panel system.

For parameter $e_1$:

$$m_{u_1}^{e_1}-n_{u_1}^{e_1}=0.25-0.55=-0.3$$

$$m_{u_1}^{e_2}-n_{u_1}^{e_2}=0.25-0.4=-0.15$$

$$m_{u_1}^{e_3}-n_{u_1}^{e_3}=0.5-0.55=-0.05$$

$$S\left(u_1\right)=\left(-0.3\right)+\left(-0.15\right)+\left(-0.05\right)=-0.50$$

For parameter $e_2$:

$$m_{u_2}^{e_1}-n_{u_2}^{e_1}=0.3-0.7=-0.4$$

$$m_{u_2}^{e_2}-n_{u_2}^{e_2}=0.45-0.8=-0.35$$

$$m_{u_2}^{e_3}-n_{u_2}^{e_3}=0.2-0.65=-0.45$$

$$S\left(u_2\right)=\left(-0.4\right)+\left(-0.35\right)+\left(-0.45\right)=-1.20$$

For parameter $e_3$:

$$m_{u_3}^{e_1}-n_{u_3}^{e_1}=0.2-0.75=-0.55$$

$$m_{u_3}^{e_2}-n_{u_3}^{e_2}=0.4-0.55=-0.15$$

$$m_{u_3}^{e_3}-n_{u_3}^{e_3}=0.1-0.3=-0.2$$

$$S\left(u_3\right)=\left(-0.4\right)+\left(-0.35\right)+\left(-0.15\right)=-0.90$$

The optimal solar panel system to select, according to our analysis, is $u_1$, since it possesses the greatest aggregate score.

6. Results and Comparisons of Fermatean Vague Soft Sets (FVSS) with Other Conventional Models

The realm of uncertainty representation is expanded by FVSS by the use of a constraint, based on the Fermatean logic ${t_{F(a)}\left(u\right)}^3+{f_{F(a)}\left(u\right)}^3\le 1$. This, in comparison to intuitionistic or Pythagorean frameworks, provides a greater level of freedom. Because of this, FVSS can describe complex indeterminacy in addition to exact membership and non-membership degrees; this is crucial for real-world applications, such as data categorization, risk analysis, and medical diagnostics.

This table compares Fermatean vague soft sets (FVSS) with traditional models, including fuzzy sets (FS), intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and vague sets (VS), according to key parameters (Figure 1).

Figure 1. Comparison of Fermatean vague soft sets (FVSS) with traditional models, including fuzzy sets (FS), intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and vague sets (VS), according to key parameters.

7. Conclusions

This paper reviewed the fundamental concepts of vague soft sets, redefines the intersection and union of two vague soft sets, and revises several propositions presented. The concept of a Fermatean vague soft set was presented as an extension of the vague soft set. The fundamental properties of Fermatean vague soft sets were presented and examined. This extension significantly enhanced existing theories for managing uncertainties and suggested potential areas for further research and relevant applications. Further research should be conducted to investigate the relationship among soft sets, rough sets, and vague soft sets and to integrate these theories for addressing uncertainties. Further exploration of the applications of the Fermatean vague soft set approach for addressing real-world decision-making problems is warranted. Furthermore, we introduced decision-making for the first time into the Fermatean vague soft set and where it was not defined into the vague set and vague soft set. Additionally, the FVSS algorithm’s introduction into the decision-making procedure encouraged a more organized and robust method to managing the difficulties associated with uncertain environments. This progression not only thrusts the field of vague science forward but also opens up new openings for application in other restraints seeking to process their decision-making developments under uncertainty. The future of decision-making may perceive the prevalent adoption of FVSS models, theoretically leading to more informed, capable, and accurate results across various fields. This research sets the podium for the constant exploration and alteration of Fermatean vague soft sets, paving the way for advanced keys to some of the most ambitious problems encountered in theoretical and functional research. Expanding the study of Fermatean vague soft sets into advanced control systems research is an innovative and promising avenue. The incorporation of these mathematical instruments might mitigate uncertainty, vagueness, and complexity in diverse control systems. It examines potential extensions of this work in the designated regions through observer-based sliding mode control for fuzzy stochastic switching systems with deception attacks, introduced by Wu et.al. [34]. Sliding mode control of discrete-time interval type-2 fuzzy Markov jump systems with the preview target signal, introduced by Sun et.al. [35], might be solved by constructing a comprehensive framework utilizing Fermatean vague soft sets to represent uncertainty and deceptive effects and integrating Fermatean vague soft sets to enhance the modeling of uncertainty in Markov jump characteristics.