Novel Fermatean Fuzzy Aczel–Alsina Model for Investment Strategy Selection

Abstract

For choosing the optimal option for multiple issues, the multiattribute decision-making (MADM) technique within a Fermatean fuzzy environment is a well-known and significant method. This paper presents a novel superiority inferiority ranking (SIR) approach for Fermatean fuzzy sets in group decision-making using multicriteria to reduce investment risk. This approach aims to evaluate the strategies for selecting the optimal investment company. The SIR method is depicted, and its effectiveness in decision-making is explored. In this manuscript, we develop new types of A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ operations on the Fermatean fuzzy environment and Fermatean Fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ (FF-AA) average aggregation operators, including their properties such as idempotency, monotonicity, and boundedness. Further, we introduce a Fermatean fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ weighted average closeness coefficient (FF-AA-WA-$\mathcal{C}\mathcal{C}$) aggregation operator (AO) based on the closeness coefficient for MAGDM issues. By utilizing the proposed technique, we solve a numerical example of an MAGDM problem. The results show that this approach is accurate and practical, and consistent with a realistic investment circumstance. A demonstration was created to emphasize the significance and credibility of this approach and assess its validity by comparing its outcomes with the established methods.

1. Introduction

There are numerous risks involved in selecting an investment strategy, including economic, policy, social, credit, and technology hazards. Shyng et al. [1] proposed that individuals’ past experiences influence their investment decision-making beliefs. Many studies have been aimed at assessing an investment strategy and its risk [2,3,4]. Now, investment management is an active subject of study [5,6,7,8,9,10,11]. Through the lens of the investor, the decision-making procedure can be categorized into four stages: identification of the problem, seeking relevant information, appraisal of options, and investment decision [12]. The evaluation of possibilities holds the utmost significance, which can produce the optimal investment strategy for investors’ demands. For the preference of the best or worst option [13], the SIR model is helpful for decision-makers. The primary elements of this paper are the SIR approach, Fermatean fuzzy information, triangular norms, A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ aggregation operators, and the MAGDM problem.

The SIR technique for MAGDM plays a vital role in addressing decision-making (DM) issues. It is a significant mathematical technique that is used to deal with daily life problems, such as correlation and similarity measures [14,15]. It determines the most desired and appealing option from a group of alternatives. Xu [16] was the first to propose the SIR technique. This approach ranks alternatives based on superiority flow and inferiority flow. This approach has the advantage of merging the features of various MCDM issues such as PROMETHEE, SAW, and TOPSIS [17]. Tam et al. [18] used the superiority and inferiority ranking method for the selection of concrete pumps. Tam et al. [19] applied this strategy with a grey aggregation approach for locating large-scale harbour-front project developments. In 2010, Liu [20] used the concept of multicriteria group decision-making for IFSs with the SIR model. Later on, Ma et al. [21] extended the idea of group decision-making for the environment of HFSs and IVHFSs. Peng et al. [22] established this same approach for the generalization of fuzzy sets which were PFSs and demonstrated some outcomes. Rouhani [23] developed a fuzzy superiority and inferiority (SIR) ranking method and applied it to the area of software selection. Chen [24] extended the decision areas and developed a PROMETHEE-based multicriteria group decision-making outranking method for PFNs. Ali et al. [25] established the multicriteria decision-making model for intuitionistic hesitant fuzzy connection numbers. Geetha et al. [26] developed an extension of this approach of SIR techniques for HPFSs. Riaz et al. [27] employed this technique for the healthcare tourism supply chain under a bipolar fuzzy environment.

Atanassov [28] proposed the notion of IFS, an efficient generalization of Zadeh’s fuzzy set theory [29], that handles data uncertainty and ambiguity. Each element within the IFS is denoted by an ordered pair of values, namely, the membership status (MS) and the non-membership status (NMS); the sum of these values varies from zero to one. Notably, in some instances, the sum may be greater than the one provided by the decision-makers, but the sum of their squares varies from zero to one. Thus, Yager [30,31,32] presented PFSs that satisfy the required condition that the sum of squares varies from zero to one. Moreover, Yager [30,31,32] introduced various types of aggregation operators for DM challenges, especially addressing situations where ambiguity arises from alternatives. Senapati and Yager [33] introduced FFSs, another extension of IFSs. FFS meets the condition that the cube sum of its MS and NMS ranges from zero to one.

Triangular norms are crucial in DM situations. Menger [34] was the pioneer in introducing triangular norms. Deschrijver et al. [35] analyzed t-norm ($\mathrm{T}\mathrm{N}$) and t-conorm ($\mathrm{T}\mathrm{C}\mathrm{N}$) for IFSs. There are several $\mathrm{T}\mathrm{N}$s and $\mathrm{T}\mathrm{C}\mathrm{N}$s that are employed for data aggregation, including Lukasiewicz $\mathrm{T}\mathrm{N}$ and $\mathrm{T}\mathrm{C}\mathrm{N}$ [36], product$\mathrm{T}\mathrm{N}$ and sum $\mathrm{T}\mathrm{C}\mathrm{N}$ [37], Archimedian $\mathrm{T}\mathrm{N}$ and $\mathrm{T}\mathrm{C}\mathrm{N}$ [38], and drastic $\mathrm{T}\mathrm{N}$ and $\mathrm{T}\mathrm{C}\mathrm{N}$ [39]. These standards play an important role in the creation of AOs [40,41]. A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$ and A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ (AA) [42] suggested a new concept of $\mathrm{T}\mathrm{N}$ and $\mathrm{T}\mathrm{C}\mathrm{N}$ known as A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ $\mathrm{T}\mathrm{N}$ and $\mathrm{T}\mathrm{C}\mathrm{N}$. Many researchers employed AA $\mathrm{T}\mathrm{N}$ and$\mathrm{T}\mathrm{C}\mathrm{N}$ and came to the conclusion that the parameters of these norms and co-norms provide superior outcomes. Senapati et al. [43,44,45,46] accepted the IF, IVIF, HF, and PF operators of aggregation based on the AA-$\mathrm{T}\mathrm{N}$ and AA-$\mathrm{T}\mathrm{C}\mathrm{N}$ by utilizing MADM problems. Further, Gokasar et al. [47] integrated alternatives for fuzzy decision-making models. Deveci et al. [48] analysed parking operation in urban areas using a fuzzy Dombi-based RAFSI model. Senapati et al. [49] and Mishra et al. [50,51] discussed aggregation operators for FF data. Deveci et al. [52] discussed a FF score function-based SWARA method. Pamučar et al. [53] used an A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ function for selection of healthcare waste management treatment. Hussain et al. [54,55] utilized AA AOs on PFSs and TSFSs for MADM challenges. Recently, the multicriteria decision-making methods and fuzzy sets have been employed widely to solve different complex engineering problems [56,57,58,59,60,61,62,63,64].

Research Gap and Motivation of the Study

During the literature review, it was observed that the SIR model is a very important tool to rank the multiple alternatives with respect to their attributes and given criteria. It was found that the above-mentioned model can be extended to the data of fermentation fuzzy situations, which is more compatible than IFSs and PFSs. Moreover, it was observed that AA AOs played a very crucial role in aggregating fuzzy numbers. Inspired by this role, we investigated these operators for FFNs. To fill the gap, we developed a novel approach in this paper and explored its superiority using this model in the business field.

The major contributions of the study are the following:

• A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ operations on Fermatean fuzzy numbers;
• Fermatean Fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ (FF-AA) average AOs;
• Aggregation operator properties such as idempotency, monotonicity, and boundedness;
• Fermatean fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ weighted average closeness coefficient (FF-AA-WA-$\mathcal{C}\mathcal{C}$) aggregation operator for MAGDM challenges based on the closeness coefficient;
• A numerical example of MAGDM problems;
• Comparison analysis with results by existing [65] FFEWA operators.

The main contribution of this research article is the application of the Fermatean fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ weighted average closeness coefficient (FF-AA-WA-$\mathcal{C}\mathcal{C}$) aggregation operators to the Fermatean environment and assessment of the options’ superiority and inferiority. In particular, we assessed the group decision-making process of a multicriteria model referred to as MAGDM and rated the most effective and ineffective options’ outcomes. We applied the SIR criteria to the Fermatean data. Moreover, we established the reliability and authenticity of the suggested strategy by solving a mathematical illustration for picking the best alternative for a valuable investment.

The remaining part of the article is structured as follows. Section 2 briefly investigates the fundamental notions of PFS, FFS,$\mathrm{T}\mathrm{N}, \mathrm{T}\mathrm{C}\mathrm{N}$, AA-$\mathrm{T}\mathrm{N}$, and AA-$\mathrm{T}\mathrm{C}\mathrm{N}$. Section 3 proposes AA operations on Fermatean Fuzzy Numbers (FFNs). Section 4 introduces Fermatean Fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ (FF-AA), average and geometric aggregation operators and their features. Section 5 presents a novel FF-AA-WA-$\mathcal{C}\mathcal{C}$ aggregation operator for overcoming MAGDM issues with the SIR method, as well as its features. In Section 6, we present an SIR approach algorithm to cope with MAGDM issues for the FF-AA-WA-$\mathcal{C}\mathcal{C}$ operator. In Section 7, a numerical example depicts the proposed approach. Section 8 gives the comparison analysis. Finally, Section 9 concludes this article.

2. Preliminaries

This section summarizes the required information connected with IFSs, PFSs, FFSs, and basic notions of TN, TCN, AA-$\mathrm{T}\mathrm{N}$, and AA-$\mathrm{T}\mathrm{C}\mathrm{N}$ for the creation of this article. We also review more common concepts that are useful in sequential analysis.

2.1. IFSs, PFSs, FFSs

Atanassov [28] proposed IFS as an extension of Zadeh’s fuzzy set theory [29].

Definition 1

([28]). Let $X$ be universe of discourse and$\mathrm{ẋ}\in X$. Then, IFS ‘$ꝉ$’ is defined on $X$ as

$$ꝉ=\left\{\left(\mathrm{ẋ},{\mu }_{ꝉ}\left(\mathrm{ẋ}\right){, \nu }_{ꝉ}\left(\mathrm{ẋ}\right)\right):\mathrm{ẋ}\in X\right\},$$

(1)

where ${\mu }_{ꝉ}\left(\mathrm{ẋ}\right)$ $\in$[0, 1] represents the membership status and ${ \nu }_{ꝉ}\left(\mathrm{ẋ}\right)$ ∈ [0, 1] represents the non-membership status of $\mathrm{ẋ}$ to $ꝉ$, respectively, with $0\le {\mu }_{ꝉ}\left(\mathrm{ẋ}\right){+\nu }_{ꝉ}\left(\mathrm{ẋ}\right)\le 1$, $\forall \mathrm{ẋ}\in X$.

Yager et al. [30,31,32] presented a concept for PFSs—the sum of the MS and NMS is greater than 1—while IFSs fail in such cases.

Definition 2

([30,31,32]). Let $X$ be universe of discourse and $\mathrm{ẋ}\in X$. A PFS ‘$Ꝓ$’ is defined on $X$ as

$$Ꝓ=\left\{\left(\mathrm{ẋ},{\mu }_{Ꝓ}\left(\mathrm{ẋ}\right){, \nu }_{Ꝓ}\left(\mathrm{ẋ}\right)\right):\mathrm{ẋ}\in X\right\},$$

(2)

where ${\mu }_{Ꝓ}$ ∈ [0, 1] represents the membership status and ${\nu }_{Ꝓ}$ ∈ [0, 1] represents the non-membership status of $\mathrm{ẋ}$ to $Ꝓ$, respectively, with $0\le {\left({\mu }_{Ꝓ}(\mathrm{ẋ})\right)}^2+{({\nu }_{Ꝓ}\left(\mathrm{ẋ}\right))}^2\le 1$. The indeterminacy is ${\pi }_{Ꝓ}\left(\mathrm{ẋ}\right)$ = $\sqrt{{1-\left({\mu }_{Ꝓ}(\mathrm{ẋ})\right)}^2-{({\nu }_{Ꝓ}\left(\mathrm{ẋ}\right))}^2}$.

Senapati and Yager [33] proposed an idea for FFSs—the sum of the squares of MS and NMS is greater than 1—while PFSs fail in such cases.

Definition 3

([33]). Let $X$ be a universe of discourse and $\mathrm{ẋ}\in X$. A Fermatean fuzzy set ‘$Ӻ$’ is defined on $X$ as

$$Ӻ=\left\{\left(\mathrm{ẋ},{\mu }_{Ӻ}(\mathrm{ẋ}){, \nu }_{Ӻ}\left(\mathrm{ẋ}\right)\right):\mathrm{ẋ}\in X\right\},$$

(3)

where ${\mu }_{Ӻ}$ $\in$ [0, 1] represents the membership status and ${\nu }_{Ӻ}\in$ [0, 1] represents the non-membership status of $\mathrm{ẋ}$ to $Ӻ$, respectively, with $0\le {\left({\mu }_{Ӻ}(\mathrm{ẋ})\right)}^3+{({\nu }_{Ӻ}\left(\mathrm{ẋ}\right))}^3\le 1$. The indeterminacy is ${\pi }_{Ӻ}\left(\mathrm{ẋ}\right)$ = $\sqrt[3]{{1-\left({\mu }_{Ӻ}(\mathrm{ẋ})\right)}^3-{({\nu }_{Ӻ}\left(\mathrm{ẋ}\right))}^3}$. To make things easier, a Fermatean fuzzy number (FFN) $\left({\mu }_{Ӻ}(\mathrm{ẋ}){, \nu }_{Ӻ}\left(\mathrm{ẋ}\right)\right)$ is represented by $({\mu }_{Ӻ}{, \nu }_{Ӻ}$).

Definition 4

([33]). The distance between two FFNs ${Ӻ}_1=({\mu }_{{Ӻ}_1},{\nu }_{{Ӻ}_1}) and {Ӻ}_2=({\mu }_{{Ӻ}_2},{\nu }_{{Ӻ}_2})$ is defined as

$$\begin{array}{l}d ({Ӻ}_1, {Ӻ}_2)=\\ \sqrt{\frac{1}{2}\left({\left({\left({\mu }_{{Ӻ}_1}\right)}^3-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)}^2+{\left({\left({\nu }_{{Ӻ}_1}\right)}^3-{\left({\nu }_{{Ӻ}_2}\right)}^3\right)}^2+{\left({\left({\pi }_{{Ӻ}_1}\right)}^3-{\left({\pi }_{{Ӻ}_2}\right)}^3\right)}^2\right)}.\end{array}$$

(4)

Definition 5

([33]). The score function of FFN $Ӻ=\left({\mu }_{Ӻ}{, \nu }_{Ӻ}\right)$ is defined as

$$Șc\left(Ӻ\right)={\left({\mu }_{Ӻ}\right)}^3-{\left({\nu }_{Ӻ}\right)}^3,$$

(5)

where $Șc(Ӻ)$ ∈ [−1, 1].

For any two FFNs ${Ӻ}_1$, ${Ӻ}_2$:

• If $Șc$ (${Ӻ}_1$) < $Șc$ (${Ӻ}_2$), then ${Ӻ}_1$ ≺ ${Ӻ}_2$;
• If $Șc$ (${Ӻ}_1$) > $Șc$ (${Ӻ}_2$), then ${Ӻ}_1 \succ {Ӻ}_2$;
• If $Șc$ (${Ӻ}_1$) = $Șc$ (${Ӻ}_2$), then ${Ӻ}_1$ ∼ ${Ӻ}_2$.

For comparison of FFNs, the efficiency of the score function is no doubt accepted. However, in some cases, it cannot be applied to decide the selection of better FFNs. For example, if ${Ӻ}_1$ = (0.6, 0.6) and ${Ӻ}_2$ = (0.7, 0.7), then $Șc$(${Ӻ}_1$) = $Șc$ (${Ӻ}_2$) = 0. That is, for any FFNs with ${\mu }_{Ӻ}{=\nu }_{Ӻ}$, the score function is always equal to zero. However, these FFNs may be different from each other. This can be rectified by the accuracy function for FFNs.

Definition 6

([33]). The accuracy function of FFN $Ӻ=\left({\mu }_{Ӻ}{, \nu }_{Ӻ}\right)$ is defined as

$$Ȃc\left(Ӻ\right)={\left({\mu }_{Ӻ}\right)}^3+{\left({\nu }_{Ӻ}\right)}^3,$$

(6)

where, $Ȃc(Ӻ)$ ∈ [0, 1].

• If $Ș$c(${Ӻ}_1$) < $Ș$c(${Ӻ}_2$), then ${Ӻ}_1$ ≺ ${Ӻ}_2$;
• If $Ș$c(${Ӻ}_1$) = $Ș$c(${Ӻ}_2$), then

  (a) 

  If $Ȃ$c(${Ӻ}_1$) $>$ $Ȃ$c(${Ӻ}_2$), then ${Ӻ}_1$ ≻ ${Ӻ}_2$;

  (b) 

  If $Ȃ$c(${Ӻ}_1$) $<$ $Ȃ$c(${Ӻ}_2$), then ${Ӻ}_1\prec$ ${Ӻ}_2$;

  (c) 

  If $Ȃ$c(${Ӻ}_1$) = $Ȃ$c(${Ӻ}_2$), then ${Ӻ}_1$ = ${Ӻ}_2$.

Definition 7.

Let $Ӻ=\left({\mu }_{Ӻ}{, \nu }_{Ӻ}\right)$, ${Ӻ}_1=\left({\mu }_{{Ӻ}_1},{\nu }_{{Ӻ}_1}\right), and {Ӻ}_2=({\mu }_{{Ӻ}_2},{\nu }_{{Ӻ}_2})$ be three FFNs; then, operations of FFNs are defined as follows:

• ${Ӻ}_1\cap {Ӻ}_2=\left(min\left\{{\mu }_{{Ӻ}_1},{\mu }_{{Ӻ}_2}\right\},ma\mathrm{ẋ}\left\{{\nu }_{{Ӻ}_1},{\nu }_{{Ӻ}_2}\right\}\right);$
• ${Ӻ}_1\cup {Ӻ}_2=\left(ma\mathrm{ẋ}\left\{{\mu }_{{Ӻ}_1},{\mu }_{{Ӻ}_2}\right\},min\left\{{\nu }_{{Ӻ}_1},{\nu }_{{Ӻ}_2}\right\}\right);$
• ${Ӻ}^c=({\nu }_{Ӻ},{\mu }_{Ӻ})$.

Definition 8.

Let $Ӻ=\left({\mu }_{Ӻ}{, \nu }_{Ӻ}\right)$, ${Ӻ}_1=\left({\mu }_{{Ӻ}_1},{\nu }_{{Ӻ}_1}\right), and {Ӻ}_2=({\mu }_{{Ӻ}_2},{\nu }_{{Ӻ}_2})$ be three different FFNs, and $\Omega$ > 0. Then, operations of FFNs are defined as follows:

• ${Ӻ}_1⊞{Ӻ}_2$ = $\left(\sqrt[3]{{\left({\mu }_{{Ӻ}_1}\right)}^3+{\left({\mu }_{{Ӻ}_2}\right)}^3-{\left({\mu }_{{Ӻ}_1}\right)}^3{\left({\mu }_{{Ӻ}_2}\right)}^3},{\nu }_{{Ӻ}_1}{\nu }_{{Ӻ}_2}\right);$
• ${Ӻ}_1⊠{Ӻ}_2$ = $\left({\mu }_{{Ӻ}_1}{\mu }_{{Ӻ}_2},\sqrt[3]{{\left({\nu }_{{Ӻ}_1}\right)}^3+{\left({\nu }_{{Ӻ}_2}\right)}^3-{\left({\nu }_{{Ӻ}_1}\right)}^3{\left({\nu }_{{Ӻ}_2}\right)}^3}\right);$
• $\mathsf{\Omega }Ӻ$ = $\left(\sqrt[3]{{1-\left(1-{\left({\mu }_{Ӻ}\right)}^3\right)}^{\mathsf{\Omega }}}, {\left({\nu }_{Ӻ}\right)}^{\mathsf{\Omega }}\right)$;
• ${Ӻ}^{\mathsf{\Omega }}$ = $\left({\left({\mu }_{Ӻ}\right)}^{\mathsf{\Omega }},\sqrt[3]{{1-\left(1-{\left({\nu }_{Ӻ}\right)}^3\right)}^{\mathsf{\Omega }}},\right)$.

2.2. Boundary Conditions for Fermatean Fuzzy Sets

In Fermatean fuzzy sets, there are three types of boundary conditions, namely, lower membership boundary, upper membership boundary, and non-membership boundary:

• The lower membership boundary condition identifies the minimal degree of membership at which an element is deemed to be a component of the Fermatean fuzzy set. The element is said to be a part of the set if its membership degree is equal to or higher than the lower membership limit;
• The upper membership boundary condition establishes the highest degree of membership at which an element may be regarded as a component of the Fermatean fuzzy set. If an element has a membership degree that is lower than the upper membership limit, it does not fully belong to the set;
• The non-membership boundary condition establishes the threshold for the non-membership degree of an element in the Fermatean fuzzy set. It reflects the highest level of non-membership that an element may be given. The element does not belong to the set if its non-membership degree is equal to or greater than the non-membership border.

In order to specify the extent to which an element is a part of the Fermatean fuzzy set or is not, boundary conditions are constructed based on triples: $\left({{\mu }_{Ӻ}}^3+{{\nu }_{Ӻ}}^3+{{\pi }_{Ӻ}}^3=1\right)$.

2.3. T Norm, T Conorm, A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ T Norm, A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ T Conorm

Triangular norms $\mathrm{T}\mathrm{N}\mathrm{s}$ are a type of function that is used to understand the combination of fuzzy logic and the fuzzy set intersection. Menger [34] is credited with being the first to establish triangle norms for statistical metric spaces. They have several applications, including DM and acting as aggregation operators. We look at various concepts that are important for the creation of this paper.

Definition 9

([66,67,68]). A t-norm is a mapping$Ṫ :\left[0, 1\right]\times \left[0, 1\right]\to \left[0, 1\right]$, $\forall \dot{\alpha },\acute{ \beta }, \stackrel{`}{\gamma }\in \left[0, 1\right]$ for which the following axioms are satisfied:

• Symmetry: $i.e.$, $Ṫ\left(\dot{\alpha } , \acute{\beta }\right)$ = $Ṫ\left(\acute{\beta } ,\dot{\alpha }\right)$;
• Associativity: $i.e.$, Ṫ($\dot{\alpha }$,Ṫ($\acute{\beta }$, $\stackrel{`}{\gamma }$))$\dot{ \alpha }$ = Ṫ(Ṫ($\dot{\alpha }$, $\acute{\beta }$), $\stackrel{`}{\gamma }$);
• Monotonicity:$i.e.$, Ṫ($\dot{\alpha }$, $\acute{\beta }$) $\le$ Ṫ($\dot{\alpha }$, $\stackrel{`}{\gamma }$) if $\acute{\beta }$ $\le \stackrel{`}{\gamma }$;
• One Identity: $i.e.$, Ṫ(1, $\dot{\alpha }$) = $\dot{\alpha }.$

Examples:

$\forall \dot{\alpha },\acute{ \beta }, \stackrel{`}{\gamma }\in \left[0, 1\right]$:

• Product triangular norm: ${Ṫ}_{pro}$ ($\dot{\alpha }$, $\acute{\beta }$) = $\dot{\alpha }$. $\acute{\beta }$;
• Minimum triangular norm: ${Ṫ}_{min}$ ($\dot{\alpha }$, $\acute{\beta }$) = min ($\dot{\alpha }$, $\acute{\beta }$);
• Lukasiewicz triangular norm: ${Ṫ}_{luk}$ ($\dot{\alpha }$, $\acute{\beta }$) = max ($\dot{\alpha }$ + $\acute{\beta }-1 , 0$);
• Drastic triangular norm:

  $${Ṫ}_{dra} (\dot{\alpha }, \acute{\beta })=\left\{\begin{array}{c}\dot{\alpha }, if \acute{\beta }=1\\ \acute{\beta }, if \dot{\alpha }=1\\ 0, \mathrm{otherwise}. \end{array}\right.$$

Definition 10

([66,67,68]). A T-Conorm is a mapping $\mathrm{Ṡ}:\left[0, 1\right]\times \left[0, 1\right]\to \left[0, 1\right],$ $\forall \dot{\alpha },\acute{ \beta }, \stackrel{`}{\gamma }\in \left[0, 1\right]$ for which the following axioms are satisfied:

• Symmetry: $i.e.$, $\mathrm{Ṡ}\left(\dot{\alpha } , \acute{\beta }\right)$ = $\mathrm{Ṡ}\left(\acute{\beta } ,\dot{\alpha }\right)$;
• Associativity: $i.e.$, Ṡ($\dot{\alpha }$, Ṡ($\acute{\beta }$, $\gamma$)) = Ṡ(Ṡ($\dot{\alpha }$, $\acute{\beta }$), $\gamma$);
• Monotonicity: $i.e.$, Ṡ($\dot{\alpha }$, $\acute{\beta }$) $\le$ Ṡ($\dot{\alpha }$, $\stackrel{`}{\gamma }$) if $\acute{\beta }$ $\le \stackrel{`}{\gamma }$;
• Zero Identity: $i.e.$, Ṡ(0, $\dot{\alpha }$) = $\dot{\alpha }$.

Examples

$\forall \dot{\alpha },\acute{ \beta }, \stackrel{`}{\gamma }\in \left[0, 1\right]$:

• Probabilistic sum triangular conorm: ${\mathrm{Ṡ}}_{PS}$ ($\dot{\alpha }$, $\acute{\beta }$) = $\dot{\alpha }$ + $\acute{\beta }-\dot{\alpha }, \acute{\beta }$;
• Maximum triangular conorm: ${\mathrm{Ṡ}}_{ma\mathrm{ẋ}}$ ($\dot{\alpha }$, $\acute{\beta }$) = max ($\dot{\alpha }$, $\acute{\beta }$);
• Lukasiewicz triangular conorm: ${\mathrm{Ṡ}}_{luk}$ ($\dot{\alpha }$, $\acute{\beta }$) = min {$\dot{\alpha }$ + $\acute{\beta }, 1$};
• Drastic triangular conorm:

  $${\mathrm{Ṡ}}_{dra} (\dot{\alpha }, \acute{\beta })=\left\{\begin{array}{c}\dot{\alpha }, if \acute{\beta }=0\\ \acute{\beta }, if \dot{\alpha }=0\\ 1, \mathrm{otherwise}. \end{array}\right.$$

Definition 11.

Aczel et al. [42] proposed the t-norm in the context of functional equations, which is known as the A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ t-norm (AA-$TN)$. For all $\phi \in \left[0, \infty \right]$, it is defined as

$${Ṫ}_{\mathrm{Ǎ}}^{\phi }\left(\dot{\alpha }, \beta \right)=\left\{\begin{array}{c}{Ṫ}_{dra}\left(\dot{\alpha }, \acute{\beta }\right), if \phi =0\\ min\left(\dot{\alpha }, \acute{\beta }\right), if \phi =\infty \\ e^{-{({\left(-\mathit{log}\dot{\alpha }\right)}^{\phi }+{(-log \acute{\beta })}^{\phi })}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\phi $}\right.}}, \mathit{otherwise}.\end{array}\right.$$

AA-$TCN$ is defined as

$${\mathrm{Ṡ}}_{\mathrm{Ǎ}}^{\phi }\left(\dot{\alpha }, \acute{\beta }\right)=\left\{\begin{array}{c}{\mathrm{Ṡ}}_{dra}\left(\dot{\alpha }, \acute{\beta }\right), if \phi =0\\ max\left(\dot{\alpha }, \acute{\beta }\right) if \phi =\infty \\ 1-e^{-{({\left(-\mathit{log}(1-\dot{\alpha })\right)}^{\phi }+{(-log (1-\acute{\beta }\left)\right)}^{\phi })}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\phi $}\right.}}, \mathrm{otherwise}.\end{array}\right.$$

Definition 12.

If $F$ and $H$ are two FFSs, then the generalization of their intersection and union are

$$F{\cap }_{Ṫ,\mathrm{Ṡ}}H=\left\{<\mathrm{ẋ},{Ṫ\{\mu }_F\left(\mathrm{ẋ}\right){, \mu }_H\left(\mathrm{ẋ}\right)\},\mathrm{Ṡ}\{{\nu }_F\left(\mathrm{ẋ}\right){, \nu }_H\left(\mathrm{ẋ}\right)\}>:\mathrm{ẋ}\in X\right\};$$

$$F{\cup }_{Ṫ,\mathrm{Ṡ}}H=\left\{<\mathrm{ẋ},{\mathrm{Ṡ}\{\mu }_F\left(\mathrm{ẋ}\right){, \mu }_H\left(\mathrm{ẋ}\right)\},Ṫ\{{\nu }_F\left(\mathrm{ẋ}\right){, \nu }_H\left(\mathrm{ẋ}\right)\}>:\mathrm{ẋ}\in X\right\};$$

Here, $Ṫ$ and $\mathrm{Ṡ}$ are $TN and TCN$, respectively.

3. Proposed A𝒸𝓏ℯℓ–Aℓ𝓈𝒾𝓃𝒶 Operations on Fermatean Fuzzy Numbers

For $\mathrm{T}\mathrm{N}$(Ṫ) and $\mathrm{T}\mathrm{C}\mathrm{N}$(Ṡ), if the AA product is denoted by ${Ṫ}_{\mathrm{Ǎ}}$ and AA sum is denoted by${\mathrm{Ṡ}}_{\mathrm{Ǎ}}$, then the extension of intersection and union across two FFSs ‘$\mathrm{F}$’ and ‘$\mathrm{H}$’ turns out to be the AA product (denoted by $\mathrm{F}⊠\mathrm{H}$) and AA sum (denoted by $\mathrm{F}⊞\mathrm{H}$) across FFSs ‘$\mathrm{F}$’ and ‘$\mathrm{H}$’; these are symbolically defined as

$$\mathrm{F}⊠\mathrm{H}=\left\{<\mathrm{ẋ},{{Ṫ}_{\mathrm{Ǎ}}\{\mu }_{\mathrm{F}}\left(\mathrm{ẋ}\right){, \mu }_{\mathrm{H}}\left(\mathrm{ẋ}\right)\},{\mathrm{Ṡ}}_{\mathrm{Ǎ}}\{{\nu }_{\mathrm{F}}\left(\mathrm{ẋ}\right){, \nu }_{\mathrm{H}}\left(\mathrm{ẋ}\right)\}>:\mathrm{ẋ}\in X\right\};$$

$$\mathrm{F}⊞\mathrm{H}=\left\{<\mathrm{ẋ},{{\mathrm{Ṡ}}_{\mathrm{Ǎ}}\{\mu }_{\mathrm{F}}\left(\mathrm{ẋ}\right){, \mu }_{\mathrm{H}}\left(\mathrm{ẋ}\right)\},{Ṫ}_{\mathrm{Ǎ}}\{{\nu }_{\mathrm{F}}\left(\mathrm{ẋ}\right){, \nu }_{\mathrm{H}}\left(\mathrm{ẋ}\right)\}>:\mathrm{ẋ}\in X\right\}.$$

Definition 13.

Let $Ӻ=\left({\mu }_{Ӻ}{, \nu }_{Ӻ}\right)$, ${Ӻ}_1=\left({\mu }_{{Ӻ}_1},{\nu }_{{Ӻ}_1}\right), and {Ӻ}_2=({\mu }_{{Ӻ}_2},{\nu }_{{Ӻ}_2})$ be three FFNs, $\beth$ ≥ 1 and $\Omega$ > 0. Then, by using Definition 11, the following operations are defined:

$${Ӻ}_1⊞{Ӻ}_2=\left(\sqrt[3]{1-e^{-{\left({\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+{\left(-log\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}} , e^{-{\left({\left(-log\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }+{\left(-log\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right);$$

$${Ӻ}_1⊠{Ӻ}_2=\left(e^{-{\left({\left(-log\left({\mu }_{{Ӻ}_1}\right)\right)}^{\beth }+{\left(-log\left({\mu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}, \sqrt[3]{1-e^{-{\left({\left(-log\left(1-{\left({\nu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+{\left(-log\left(1-{\left({\nu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}\right);$$

$$\Omega Ӻ=\left(\sqrt[3]{1-e^{-{\left(\Omega { \left(-log\left(1-{\left({\mu }_{Ӻ}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}, e^{-{\left(\Omega {\left(-log\left({\nu }_{Ӻ}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right);$$

$${Ӻ}^{\Omega }=\left(e^{-{\left(\Omega {\left(-log\left({\mu }_{Ӻ}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}, \sqrt[3]{1-e^{-{\left(\Omega { \left(-log\left(1-{\left({\nu }_{Ӻ}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}\right).$$

Theorem 1.

Let$Ӻ=\left({\mu }_{Ӻ}{, \nu }_{Ӻ}\right)$, ${Ӻ}_1=\left({\mu }_{{Ӻ}_1},{\nu }_{{Ӻ}_1}\right), and {Ӻ}_2=({\mu }_{{Ӻ}_2},{\nu }_{{Ӻ}_2})$ be three FFNs with $\Omega$, ${\Omega }_1, {\Omega }_2$ > 0. Then, the following hold:

• ${Ӻ}_1⊞{Ӻ}_2={Ӻ}_2⊞{Ӻ}_1$;
• ${Ӻ}_1⊠{Ӻ}_2={Ӻ}_2⊠ {Ӻ}_1;$
• $\Omega \left({Ӻ}_1⊞{Ӻ}_2\right)=\Omega { Ӻ}_1⊞\Omega {Ӻ}_1$;
• $\left({\Omega }_1+{\Omega }_2\right)Ӻ={\Omega }_1Ӻ⊞{\Omega }_2 Ӻ$;
• ${({Ӻ}_1⊠{Ӻ}_2)}^{\Omega }={{Ӻ}_1}^{\Omega }⊠{{Ӻ}_2}^{\Omega }$;
• ${(Ӻ)}^{{\Omega }_1}⊠{(Ӻ)}^{{\Omega }_2}={(Ӻ)}^{\left({\Omega }_1+{\Omega }_2\right)}$.

Proof. 

We shall prove (1), (3), and (5); the others, namely, (2), (4), and (6), can be proved analogously.

$$\left(1\right) {Ӻ}_1⊞{Ӻ}_2=\left(\sqrt[3]{1-e^{-{\left({\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+{\left(-log\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}} , e^{-{\left({\left(-log\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }+{\left(-log\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right)$$

$$=\left(\sqrt[3]{1-e^{-{\left({\left(-log\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }+{\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}} , e^{-{\left({\left(-log\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }+{\left(-log\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right)={Ӻ}_2⊞{Ӻ}_1.$$

$$\left(3\right) \mathsf{\Omega }\left({Ӻ}_1⊞{Ӻ}_2\right)=$$

$$\mathsf{\Omega }\left(\sqrt[3]{1-e^{-{\left({\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+{\left(-log\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}, e^{-{\left({\left(-log\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }+{\left(-log\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right)$$

$$=\left(\sqrt[3]{1-e^{-{\left(\mathsf{\Omega }{\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+\mathsf{\Omega }{\left(-log\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}, e^{-{\left(\mathsf{\Omega }{\left(-log\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }+{\mathsf{\Omega }\left(-log\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right)$$

$$=\left(\sqrt[3]{1-e^{-{\left(\mathsf{\Omega }{\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}, e^{-{\left(\mathsf{\Omega }{\left(-log\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right)$$

$$⊞\left(\sqrt[3]{1-e^{-{\left(\mathsf{\Omega }{\left(-log\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}, e^{-{\left(\mathsf{\Omega }{\left(-log\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}\right) =\mathsf{\Omega }{Ӻ}_1⊞\mathsf{\Omega }{Ӻ}_2.$$

$${\left(5\right) ({Ӻ}_1⊠{Ӻ}_2)}^{\mathsf{\Omega }}=$$

$$\left(e^{-{\left(\mathsf{\Omega }{\left(-log\left({\mu }_{{Ӻ}_1}\right)\right)}^{\beth }+{\mathsf{\Omega }\left(-log\left({\mu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}},\sqrt[3]{1-e^{-{\left(\mathsf{\Omega }{\left(-log\left(1-{\left({\nu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+\mathsf{\Omega }{\left(-log\left(1-{\left({\nu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}\right)$$

$$=\left(e^{-{\left(\mathsf{\Omega }{\left(-log\left({\mu }_{{Ӻ}_1}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}},\sqrt[3]{1-e^{-{\left(\mathsf{\Omega }{\left(-log\left(1-{\left({\nu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}\right)$$

$$⊠\left(e^{-{\left(\mathsf{\Omega }{\left(-log\left({\mu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}},\sqrt[3]{1-e^{-{\left(\mathsf{\Omega }{\left(-log\left(1-{\left({\nu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\frac{1}{\beth }}}}\right)={{Ӻ}_1}^{\mathsf{\Omega }}⊠{{Ӻ}_2}^{\mathsf{\Omega }}.$$

□

4. Proposed Fermatean Fuzzy A𝒸𝓏ℯℓ–Aℓ𝓈𝒾𝓃𝒶 (FF-AA) Aggregation Operators

In this section, inspired by the PF-AA-AOs, we propose FF-AA-AOs, with some properties and numerical examples under FFSs related to uncertain information.

4.1. Proposed Fermatean Fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ (FF-AA) Averaging Aggregation Operators

Definition 14.

Let ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right), (i=1, 2,\dots ,n)$ be a collection of FFNs. Suppose the weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i, (i=1, 2,\dots ,n)$ follows ${\varpi }_i$ $>$ 0, ${\varpi }_i\in \left[\mathrm{0,1}\right],$ and ${\sum }_{i=1}^n{\varpi }_i=1.$ Then, FF-AA-WA operator is a function: (FF-AA-WA): ${\left(Ӻ\right)}^n\to Ӻ$, denoted by

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-} \mathrm{WA}\left({Ӻ}_1,{Ӻ}_2,\dots , {Ӻ}_n\right)={⊞}_{i=1}^n\left({\varpi }_i{Ӻ}_i\right)={\varpi }_1{Ӻ}_1⊞{\varpi }_2{Ӻ}_2⊞,\dots , ⊞{\varpi }_n{Ӻ}_n.$$

(7)

Theorem 2.

Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right), (i=1, 2,\dots ,n)$, is a collection of FFNs. Suppose the weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i, (i=1, 2,\dots ,n)$ follows ${\varpi }_i$ $>$ 0, ${\varpi }_i\in \left[\mathrm{0,1}\right]$, and ${\sum }_{i=1}^n{\varpi }_i=1.$ Then, the accumulated value of FFNs by FF-AA-WA operator is defined as

$$\begin{array}{l}\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WA}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\\ =\left(\sqrt[3]{1-e^{-{\left({\sum }_{i=1}^n{\varpi }_i{\left(-log\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({{\sum }_{i=1}^n{\varpi }_i\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right).\end{array}$$

(8)

Proof. 

We use a mathematical induction technique to prove this theorem. For $n=2$,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2\right)={⊞}_{i=1}^2\left({\varpi }_i{Ӻ}_i\right)={\varpi }_1{Ӻ}_1⊞{\varpi }_2{Ӻ}_2$$

$$=\left(\sqrt[3]{{1-e}^{-{\left({\varpi }_1{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({\varpi }_1{\left(-\mathit{log}\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$⊞\left(\sqrt[3]{{1-e}^{-{\left({\varpi }_2{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({\varpi }_2{\left(-\mathit{log}\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$=\left(\sqrt[3]{1-e^{-{\left({\varpi }_1{\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+{\varpi }_2{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({\varpi }_1{\left(-\mathit{log}({\nu }_{{Ӻ}_1})\right)}^{\beth }+{\varpi }_2{\left(-\mathit{log}\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$=\left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i=1}^2{\varpi }_i{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, {\mathrm{e}}^{-{\left({{\sum }_{i=1}^2{\varpi }_i\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right); \mathrm{thus}, \mathrm{this} \mathrm{is} \mathrm{true} \mathrm{for} n=2$$

Let this be true for $n$, $\mathrm{i}.\mathrm{e}$.,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)={⊞}_{i=1}^n\left({\varpi }_i{Ӻ}_i\right)={\varpi }_1{Ӻ}_1⊞{\varpi }_2{Ӻ}_2⊞,\dots ,⊞{\varpi }_n{Ӻ}_n.$$

We prove, for $n+1$,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_{n+1}\right)={⊞}_{i=1}^{n+1}\left({\varpi }_i{Ӻ}_i\right)={⊞}_{i=1}^n\left({\varpi }_i{Ӻ}_i\right)⊞{\varpi }_{n+1}{Ӻ}_{n+1}$$

$$=\left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i=1}^n{\varpi }_i{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, {\mathrm{e}}^{-{\left({{\sum }_{i=1}^n{\varpi }_i\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$⊞\left(\sqrt[3]{{1-e}^{-{\left({\varpi }_{n+1}{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_{n+1}}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({\varpi }_{n+1}{\left(-\mathit{log}\left({\nu }_{{Ӻ}_{n+1}}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$=\left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i=1}^{n+1}{\varpi }_i{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, {\mathrm{e}}^{-{\left({{\sum }_{i=1}^{n+1}{\varpi }_i\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right).$$

Hence, it is true $\forall n$.  □

Theorem 3.

Let ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),(i=1, 2,\dots ,n)$, be a collection of FFNs, and weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i,(i=1, 2,\dots ,n)$ follows ${\varpi }_i$ $>$ 0, ${\varpi }_i\in \left[\mathrm{0,1}\right],$ and ${\sum }_{i=1}^n{\varpi }_i=1.$ Then, the following hold:

(P1)

(Idempotency): For equal FFNs, ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),(i=1, 2,\dots ,n)$, that is, ${Ӻ}_i=Ӻ$ $\forall$ $i$, then

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)=Ӻ.$$

(P2)

(Boundedness): Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2, \dots ,n\right),$ is a collection of FFNs, with

$${Ӻ}^{-}=min\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right) \mathrm{and} {Ӻ}^{+}=ma\mathrm{ẋ}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right). \mathrm{Then},$$

$${Ӻ}^{-}\le \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\le {Ӻ}^{+}.$$

(P3)

(Monotonicity): Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right)$ and $\acute{{Ӻ}_i}=\left(\acute{{\mu }_{{Ӻ}_i}},\acute{{\nu }_{{Ӻ}_i}}\right),(i=1, 2,\dots ,n)$ are any two collections of FFNs. If ${Ӻ}_i$ $\le$ $\acute{{Ӻ}_i}$, $\forall (i=1, 2,\dots ,n)$, with ${Ӻ}^{-}$ = $min\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$ and ${Ӻ}^{+}$ = $ma\mathrm{ẋ}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$, then

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\le \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left(\acute{{Ӻ}_1},\acute{{Ӻ}_2},\dots ,\acute{{Ӻ}_n}\right).$$

Example 1.

Let ${Ӻ}_1=\left(0.9, 0.5\right), {Ӻ}_2=\left(0.7, 0.8\right), {Ӻ}_3=\left(0.6, 0.9\right), {Ӻ}_4=\left(0.8, 0.6\right)$ be four FFNs with weight vector $\varpi ={\left(0.26, 0.20, 0.24, 0.30\right)}^T$; then,

$$\begin{array}{l}\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\\ =\left(\sqrt[3]{1-e^{-{\left({\sum }_{i=1}^n{\varpi }_i{\left(-log\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({{\sum }_{i=1}^n{\varpi }_i\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right);\end{array}$$

$$\mathrm{FF}-\mathrm{AA}\text{-}\mathrm{WA}\left({Ӻ}_1,{Ӻ}_2,{Ӻ}_3,{Ӻ}_4\right)=$$

$$\left(\begin{array}{c}\sqrt[3]{1-e^{-{\left(0.26{\left(-log\left(1-{\left(0.9\right)}^3\right)\right)}^4+0.20{\left(-log\left(1-{\left(0.7\right)}^3\right)\right)}^4+0.24{\left(-log\left(1-{\left(0.6\right)}^3\right)\right)}^4+0.30{\left(-log\left(1-{\left(0.8\right)}^3\right)\right)}^4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}}, \\ e^{-{\left({0.26\left(-\mathit{log}(0.5)\right)}^4+{0.20\left(-\mathit{log}(0.8)\right)}^4+{0.24\left(-\mathit{log}(0.9)\right)}^4+{0.30\left(-\mathit{log}(0.6)\right)}^4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}\end{array}\right);$$

$$=\left(0.85101,0.58658\right).$$

4.2. Proposed Fermatean Fuzzy A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ (FF-AA) geometric Aggregation Operators

Definition 15.

Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ is a collection of FFNs, and weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i, (i=1, 2,\dots ,n)$ follows ${\varpi }_i$ $>$ 0, ${\varpi }_i\in \left[0, 1\right],$ and ${\sum }_{i=1}^n{\varpi }_i=1.$ Then, FF-AA-WG operator is a function: (FF-AA-WG): ${\left(Ӻ\right)}^n\to Ӻ,$ denoted by

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WG}\left({Ӻ}_1,{Ӻ}_2,\dots , {Ӻ}_n\right)={⊠}_{i=1}^n\left({{Ӻ}_i}^{{\varpi }_i}\right)={{Ӻ}_1}^{{\varpi }_1}⊠{{Ӻ}_2}^{{\varpi }_2}⊠,\dots , ⊠{{Ӻ}_n}^{{\varpi }_n}.$$

(9)

Theorem 4.

Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ is a collection of FFNs, and weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i, (i=1, 2,\dots ,n)$ follows ${\varpi }_{i }>$ 0, ${\varpi }_i\in \left[\mathrm{0,1}\right],$ and ${\sum }_{i=1}^n{\varpi }_i=1.$ Then, the aggregated value of FFNs by FF-AA-WG operator is defined as

$$\begin{array}{l}\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WG}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)=\\ \left(e^{-{\left({\sum }_{i=1}^n{\left(-log{\left({\mu }_{{Ӻ}_i}\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}, \sqrt[3]{1-e^{-{\left({\sum }_{i=1}^n{\left(-log{\left(1-{\left({\nu }_{{Ӻ}_i}\right)}^3\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right).\end{array}$$

(10)

Proof. 

We use the mathematical induction technique to prove this theorem. For $n=2$,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left({Ӻ}_1,{Ӻ}_2\right)={⊠}_{i=1}^2\left({{Ӻ}_1}^{{\varpi }_1}\right)={{Ӻ}_1}^{{\varpi }_1}⊠{{Ӻ}_2}^{{\varpi }_2}$$

$$=\left({\mathrm{e}}^{-{\left({\left(-log{\left({\mu }_{{Ӻ}_1}\right)}^{{\varpi }_1}\right)}^{\beth }+{\left(-log{\left({\mu }_{{Ӻ}_2}\right)}^{{\varpi }_2}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}},\sqrt[3]{1-{\mathrm{e}}^{-{\left({\left(-log{\left(1-{\left({\nu }_{{Ӻ}_1}\right)}^3\right)}^{{\varpi }_1}\right)}^{\beth }+{\left(-log{\left(1-{\left({\nu }_{{Ӻ}_2}\right)}^3\right)}^{{\varpi }_2}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right)$$

$$=\left({\mathrm{e}}^{-{\left({\sum }_{i=1}^2{\left(-log{\left({\mu }_{{Ӻ}_i}\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}},\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i=1}^2{\left(-log{\left(1-{\left({\nu }_{{Ӻ}_i}\right)}^3\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right); \mathrm{this}, \mathrm{this} \mathrm{is} \mathrm{true} \mathrm{for} n=2.$$

Let it be true for $n,$ $\mathrm{i}.\mathrm{e}$.,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)={⊠}_{i=1}^n\left({{Ӻ}_i}^{{\varpi }_i}\right)={{Ӻ}_1}^{{\varpi }_1}⊠{{Ӻ}_2}^{{\varpi }_2}⊠,\dots ,⊠{{Ӻ}_n}^{{\varpi }_n}.$$

We now have to show that it is true for $n+1$, that is,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_{n+1}\right)={⊠}_{i=1}^{n+1}\left({{Ӻ}_i}^{{\varpi }_i}\right)={⊠}_{i=1}^n\left({{Ӻ}_i}^{{\varpi }_i}\right)⊠{{Ӻ}_{n+1}}^{{\varpi }_{n+1}}=$$

$$\begin{gathered} \left({\mathrm{e}}^{-{\left({\sum }_{i=1}^n{\left(-log{\left({\mu }_{{Ӻ}_i}\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}},\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i=1}^n{\left(-log{\left(1-{\left({\nu }_{{Ӻ}_i}\right)}^3\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right) \\ ⊠\left({\mathrm{e}}^{-{\left({\left(-log{\left({\mu }_{{Ӻ}_{n+1}}\right)}^{{\varpi }_{n+1}}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}},\sqrt[3]{1-{\mathrm{e}}^{-{\left({\left(-log{\left(1-{\left({\nu }_{{Ӻ}_{n+1}}\right)}^3\right)}^{{\varpi }_{n+1}}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right) \end{gathered}$$

$$=\left({\mathrm{e}}^{-{\left({\sum }_{i=1}^{n+1}{\left(-log{\left({\mu }_{{Ӻ}_i}\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}},\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i=1}^{n+1}{\left(-log{\left(1-{\left({\nu }_{{Ӻ}_i}\right)}^3\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right).$$

Hence, it is true $\forall n$.  □

Theorem 5.

Let ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ be a collection of FFNs, and weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i,(i=1, 2,\dots ,n)$ follows ${\varpi }_i$ $>$ 0, ${\varpi }_i\in \left[\mathrm{0,1}\right],$ and ${\sum }_{i=1}^n{\varpi }_i=1$. Then, the following hold:

(P1)

(Idempotency): For equal FFNs ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),(i=1, 2,\dots ,n)$, that is, ${Ӻ}_i=Ӻ$ $\forall$ $i$, then

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)=Ӻ.$$

(P2)

(Boundedness): Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ is an accumulation of FFNs, with ${Ӻ}^{-}$ = $min\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$ and ${Ӻ}^{+}$ = $ma\mathrm{ẋ}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$. Then,

$${Ӻ}^{-}\le \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\le {Ӻ}^{+}.$$

(P3)

(Monotonicity): Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right)$ and $\acute{{Ӻ}_i}=\left(\acute{{\mu }_{{Ӻ}_i}},\acute{{\nu }_{{Ӻ}_i}}\right),\left(i=1, 2,\dots ,n\right),$ are any two collections of FFNs. If ${Ӻ}_i$ $\le$ $\acute{{Ӻ}_i}$, $\forall (i=1, 2,\dots ,n)$, with ${Ӻ}^{-}$ = $min\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$ and ${Ӻ}^{+}$ = $ma\mathrm{ẋ}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$, then

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\le \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{G}\left(\acute{{Ӻ}_1},\acute{{Ӻ}_2},\dots ,\acute{{Ӻ}_n}\right).$$

Example 2.

Let ${Ӻ}_1=\left(0.9, 0.5\right), {Ӻ}_2=\left(0.7, 0.8\right), {Ӻ}_3=\left(0.6, 0.9\right), {Ӻ}_4=\left(0.8, 0.6\right)$ be four FFNs with weight vector $\varpi ={\left(0.26, 0.20, 0.24, 0.30\right)}^T$; then,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WG}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)=\left(e^{-{\left({\sum }_{i=1}^n{\left(-log{\left({\mu }_{{Ӻ}_i}\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}, \sqrt[3]{1-e^{-{\left({\sum }_{i=1}^n{\left(-log{\left(1-{\left({\nu }_{{Ӻ}_i}\right)}^3\right)}^{{\varpi }_i}\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}\right);$$

$$\begin{array}{c}\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WG}\left({Ӻ}_1,{Ӻ}_2,{Ӻ}_3, {Ӻ}_4\right)=\hfill \\ \left(\begin{array}{c}{e}^{-{\left({\left(-log{\left(0.9\right)}^{0.26}\right)}^4+{\left(-log{\left(0.7\right)}^{0.20}\right)}^4+{\left(-log{\left(0.6\right)}^{0.24}\right)}^4+{\left(-log{\left(0.8\right)}^{0.30}\right)}^4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}},\\ \sqrt[3]{1-e^{-{\left({\left(-log{\left(1-{\left(0.5\right)}^3\right)}^{0.26}\right)}^4+{\left(-log{\left(1-{\left(0.8\right)}^3\right)}^{0.20}\right)}^4+{\left(-log{\left(1-{\left(0.9\right)}^3\right)}^{0.24}\right)}^4+{\left(-log{\left(1-{\left(0.6\right)}^3\right)}^{0.30}\right)}^4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}}\end{array}\right)\end{array}$$

$$=\left(0.87943,0.64764\right).$$

5. Proposed FF-AA-WA-$\mathcal{C}\mathcal{C}$ Operator for Solving MAGDM Problems

In this part, we proposed FF-AA-WA-$\mathcal{C}\mathcal{C}$ operators, with some characteristics of FFSs related to MAGDM challenges for DM.

Definition 16.

Let ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ be a collection of FFNs. Suppose ${\xi }_k(k=1, 2,\dots ,l)$ are the individual weighted measures of expert status$\varpi ={\varpi }_1,{\varpi }_2,\dots {,\varpi }_l$. Let ${\mathsf{ϛ}}_k={{(\mathsf{ϛ}}_1,{\mathsf{ϛ}}_2,\dots ,{\mathsf{ϛ}}_l)}^T$ be the normalized vector of individual degree measurements. Then, FF-AA-WA-$\mathcal{C}\mathcal{C}$ operator is a mapping

$$\left(\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WA}\text{-}\mathcal{C}\mathcal{C}\right): {\left(Ӻ\right)}^n\to Ӻ\mathit{denoted}\mathit{by}$$

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WA}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)={⊞}_{i,k=1}^n\left({\mathsf{ϛ}}_k{Ӻ}_i\right)={\mathsf{ϛ}}_1{Ӻ}_1⊞{\mathsf{ϛ}}_2{Ӻ}_2⊞,\dots ,⊞{\mathsf{ϛ}}_n{Ӻ}_n.$$

(11)

Theorem 6.

Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ is a collection of FFNs. Suppose ${\xi }_k(k=1, 2,\dots ,l)$ are the individual weighted measures of expert status $\varpi ={\varpi }_1,{\varpi }_2,\dots {,\varpi }_l$. Let ${\mathsf{ϛ}}_k={{(\mathsf{ϛ}}_1,{\mathsf{ϛ}}_2,\dots ,{\mathsf{ϛ}}_l)}^T$ be the normalized vector of individual degree measurements. Then, the aggregated value of FFNs by FF-AA-WA-$\mathcal{C}\mathcal{C}$ operator is defined as

$$\begin{array}{l}\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{WA}\text{-}\mathcal{C}\mathcal{C}\left({{Ӻ}_i}^{\left(1\right)}, {{Ӻ}_i}^{\left(2\right)},\dots , {{Ӻ}_i}^{\left(l\right)}\right)=\\ \left(\sqrt[3]{1-e^{-{\left({\sum }_{k=1}^l{\mathsf{ϛ}}_k{\left(-\mathit{log}(1-{({{\mu }_i}^{\left(k\right)} )}^3)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}} , e^{-{\left({{\sum }_{k=1}^l{\mathsf{ϛ}}_k\left(-\mathit{log}({{\nu }_i}^{\left(k\right)})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right).\end{array}$$

(12)

Proof. 

We use the mathematical induction technique to prove this theorem. For $n=2$,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1, {Ӻ}_2\right)={⊞}_{i,k=1}^2\left({\mathsf{ϛ}}_k{Ӻ}_i\right)={\mathsf{ϛ}}_1{Ӻ}_1⊞{\mathsf{ϛ}}_2{Ӻ}_2$$

$$=\left(\sqrt[3]{{1-e}^{-{\left({\mathsf{ϛ}}_1 {\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}},{ e}^{-{\left({\mathsf{ϛ}}_1 {\left(-\mathit{log}\left({\nu }_{{Ӻ}_1}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$⊞\left(\sqrt[3]{{1-e}^{-{\left({\mathsf{ϛ}}_2 {\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}},{ e}^{-{\left({\mathsf{ϛ}}_2 {\left(-\mathit{log}\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$=\left(\sqrt[3]{1-e^{-{\left({\mathsf{ϛ}}_1{\left(-log\left(1-{\left({\mu }_{{Ӻ}_1}\right)}^3\right)\right)}^{\beth }+{\mathsf{ϛ}}_2{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_2}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}} , e^{-{\left({\mathsf{ϛ}}_1{\left(-\mathit{log}({\nu }_{{Ӻ}_1})\right)}^{\beth }+{\mathsf{ϛ}}_2{\left(-\mathit{log}\left({\nu }_{{Ӻ}_2}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$=\left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i,k=1}^2{\mathsf{ϛ}}_k{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, {\mathrm{e}}^{-{\left({{\sum }_{i,k=1}^2{\mathsf{ϛ}}_k\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right), \mathrm{which} \mathrm{is} \mathrm{true} \mathrm{for} n=2.$$

Let it be true for $n$, $\mathrm{i}.\mathrm{e}$.,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)={⊞}_{i,k=1}^n\left({\mathsf{ϛ}}_k{Ӻ}_i\right)={\mathsf{ϛ}}_1{Ӻ}_1⊞{\mathsf{ϛ}}_2{Ӻ}_2⊞,\dots ,⊞{\mathsf{ϛ}}_n{Ӻ}_n$$

We prove, for$n+1$,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_{n+1}\right)={⊞}_{i,k=1}^{n+1}\left({\mathsf{ϛ}}_k{Ӻ}_i\right)={⊞}_{i,k=1}^n\left({\mathsf{ϛ}}_k{Ӻ}_i\right)⊞{\mathsf{ϛ}}_{n+1}{Ӻ}_{n+1}$$

$$=\left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i,k=1}^n{\mathsf{ϛ}}_k{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}},{\mathrm{e}}^{-{\left({{\sum }_{i,k=1}^n{\mathsf{ϛ}}_k\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$⊞\left(\sqrt[3]{{1-e}^{-{\left({\mathsf{ϛ}}_{n+1} {\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_{n+1}}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}},{ e}^{-{\left({\mathsf{ϛ}}_{n+1} {\left(-\mathit{log}\left({\nu }_{{Ӻ}_{n+1}}\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

$$=\left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{i,k=1}^{n+1}{\mathsf{ϛ}}_k{\left(-\mathit{log}\left(1-{\left({\mu }_{{Ӻ}_i}\right)}^3\right)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}},{\mathrm{e}}^{-{\left({{\sum }_{i,k=1}^{n+1}{\mathsf{ϛ}}_k\left(-\mathit{log}({\nu }_{{Ӻ}_i})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right)$$

Hence, it is true$\forall n$.  □

Theorem 7.

Let ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),\left(i=1, 2,\dots ,n\right),$ be a collection of FFNs, and weight vector $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_n\right)}^T$ of the ${Ӻ}_i, (i=1, 2,\dots ,n)$ follows ${\varpi }_i$ > 0, ${\varpi }_i\in \left[\mathrm{0,1}\right],$ and ${\sum }_{i=1}^n{\varpi }_i=1.$ Then, the following hold:

(P1)

(Idempotency): For equal FFNs${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),(i=1, 2,\dots ,n)$, that is, ${Ӻ}_i=Ӻ$ $\forall$ $i$,

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)=Ӻ.$$

(P2)

(Boundedness): Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right),(i=1, 2,\dots ,n)$ is an accumulation of FFNs, with

$${Ӻ}^{-}=min\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right) \mathrm{and} {Ӻ}^{+}=ma\mathrm{ẋ}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right).\mathrm{Then},$$

$${Ӻ}^{-}\le \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\le {Ӻ}^{+}.$$

(P3)

(Monotonicity): Suppose ${Ӻ}_i=\left({\mu }_{{Ӻ}_i},{\nu }_{{Ӻ}_i}\right)$ and $\acute{{Ӻ}_i}=\left(\acute{{\mu }_{{Ӻ}_i}}, \acute{{\nu }_{{Ӻ}_i}}\right),(i=1, 2,\dots ,n)$, are any two collections of FFNs. If ${Ӻ}_i$ $\le$ $\acute{{Ӻ}_i}$, $\forall (i=1, 2,\dots ,n)$, with ${Ӻ}^{-}$ = $min\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$ and ${Ӻ}^{+}$ = $ma\mathrm{ẋ}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)$, then

$$\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({Ӻ}_1,{Ӻ}_2,\dots ,{Ӻ}_n\right)\le \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left(\acute{{Ӻ}_1},\acute{{Ӻ}_2},\dots ,\acute{{Ӻ}_n}\right).$$

6. Application of the SIR Approach for FFSs with Novel AOs to Closeness Coefficient-Based MAGDM Problems

Next, we applied the FF-AA-$\mathrm{W}\mathrm{A}$-$\mathcal{C}\mathcal{C}$ operator to deal with MAGDM problems using FF information. Suppose $\mathrm{ẋ}$ = {${\mathrm{ẋ}}_1,{\mathrm{ẋ}}_2,\dots {\mathrm{ẋ}}_m$} is a group of alternatives and $c$ = {$c_1,c_2,\dots c_n$} is a set of attributes. Let $e$ = {$e_1,e_2,\dots e_l$} be a group of experts with $\varpi ={\left({\varpi }_1,{\varpi }_2,\dots {,\varpi }_l\right)}^T$ as weight vector. Let $Ӻ\left(k\right)=$ ${\left({{Ӻ}_{ij}}^{\left(k\right)}\right)}_{m\times n},\left(i=1, 2,\dots ,m\right),$($j=1, 2,\dots ,n)$, ($k=1, 2,\dots ,l)$, be the FF decision matrix. ${{Ӻ}_{ij}}^{\left(k\right)}$ indicates the attribute value, and the alternative ${\mathrm{ẋ}}_i$ fulfils the attribute $c_j$ specified by the expert $e_k$. $Ⱳ$ = ${\left({{\varpi }_j}^{\left(k\right)}\right)}_{l\times n}$ is the attribute weight decision matrix, and ${{\varpi }_j}^{\left(k\right)}$ represents the weight of attribute $c_j$ recommended by expert $e_k$.

Step 1. Find the degree of individual measurement ${\xi }_k(k=1, 2,\dots ,l)$ via weights of experts. The relative closeness coefficient is determined as

$${\xi }_k=\frac{d({\varpi }_k, {\varpi }^{-})}{d\left({\varpi }_k, {\varpi }^{-}\right)+d({\varpi }_k, { \varpi }^{+})}$$

(13)

$${\varpi }^{-}=\left(min\left\{{\mu }_{{Ӻ}_i}\right\},ma\mathrm{ẋ}\left\{{\nu }_{{Ӻ}_i}\right\}\right), {\varpi }^{+}=\left(ma\mathrm{ẋ}\left\{{\mu }_{{Ӻ}_i}\right\},min\left\{{\nu }_{{Ӻ}_i}\right\}\right) \mathrm{with} 0\le {\varpi }_k\le 1.$$

If ${\xi }_k\to 1$, then ${\varpi }_k\to {\varpi }^{+}$; if ${\xi }_k\to 0$, then ${ \varpi }_k\to {\varpi }^{-}$.

Step 2. Normalize ${\xi }_k(k=1, 2,\dots ,l)$:

$${\mathsf{ϛ}}_k=\frac{{\xi }_k}{{\sum }_{k=1}^l{\xi }_k}$$

(14)

The normalized vector ${\mathsf{ϛ}}_k={{(\mathsf{ϛ}}_1,{\mathsf{ϛ}}_2,\dots ,{\mathsf{ϛ}}_l)}^T$ is used as degree of individual measurement.

Step 3. Use FF-AA-$\mathrm{W}\mathrm{A}$-$\mathcal{C}\mathcal{C}$ operator for aggregating group perspectives:

(a)

Weights integration of individual attributes:

$$\begin{gathered} {\overline{\varpi }}_j=\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({{\varpi }_j}^{\left(1\right)}, {{\varpi }_j}^{\left(2\right)}, \dots , {{\varpi }_j}^{\left(l\right)}\right)= \\ \left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{k=1}^l{\mathsf{ϛ}}_k{\left(-\mathit{log}(1-{({{\mu }_j}^{\left(k\right)} )}^3)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, {\mathrm{e}}^{-{\left({{\sum }_{k=1}^l{\mathsf{ϛ}}_k\left(-\mathit{log}({{\nu }_j}^{\left(k\right)})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right); \end{gathered}$$

(15)

(b)

Integration of individual decision matrix:

$$\begin{gathered} {\overline{Ӻ}}_{ij}=\mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({{Ӻ}_{ij}}^{\left(1\right)}, {{Ӻ}_{ij}}^{\left(2\right)}, \dots , {{Ӻ}_{ij}}^{\left(l\right)}\right)= \\ \left(\sqrt[3]{1-{\mathrm{e}}^{-{\left({\sum }_{k=1}^l{\mathsf{ϛ}}_k{\left(-\mathit{log}(1-{({{\mu }_{ij}}^{\left(k\right)} )}^3)\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}}, e^{-{\left({{\sum }_{k=1}^l{\mathsf{ϛ}}_k\left(-\mathit{log}({{\nu }_{ij}}^{\left(k\right)})\right)}^{\beth }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beth $}\right.}}\right). \end{gathered}$$

(16)

The attribute weights vector $\overline{\varpi }=({\overline{\varpi }}_1,{ \overline{\varpi }}_2, \dots , {\overline{\varpi }}_n$), as well as group integrated decision matrix ${\left({\overline{Ӻ}}_{ij}\right)}_{m\times n}$, are accomplished.

Step 4. Find FF superiority/inferiority matrix:

(a)

Performance function:

$${ƒ}_{ij}=ƒ ({\overline{Ӻ}}_{ij})=\frac{d({\overline{Ӻ}}_{ij}, { \overline{Ӻ}}^{-})}{d({\overline{Ӻ}}_{ij}, { \overline{Ӻ}}^{-})+d({\overline{Ӻ}}_{ij}, { \overline{Ӻ}}^{+})}$$

(17)

$${\overline{Ӻ}}^{-}=\left\{c_j, \left(min\left\{{\mu }_{{Ӻ}_{ij}}\right\}, ma\mathrm{ẋ}\left\{{\nu }_{{Ӻ}_{ij}}\right\}\right)\right\}, {\overline{Ӻ}}^{+}=\left\{c_j, \left(ma\mathrm{ẋ}\left\{{\mu }_{{Ӻ}_{ij}}\right\}, min\left\{{\nu }_{{Ӻ}_{ij}}\right\}\right)\right\},$$

With $0\le {ƒ}_{ij}\le 1$. If ${ƒ}_{ij}\to 1$, then ${\overline{Ӻ}}_{ij}\to {\overline{Ӻ}}^{+};$ if ${ƒ}_{ij}\to 0$, then ${\overline{Ӻ}}_{ij}\to {\overline{Ӻ}}^{-}$;

(b)

Preference intensity ${PI}_j$(${\mathrm{ẋ}}_i,{\mathrm{ẋ}}_t$):

Define ${PI}_j$(${\mathrm{ẋ}}_i,{\mathrm{ẋ}}_t$), $\left(t,i=1, 2,\dots ,m,t\ne i;j=1,2,\dots ,n\right)$, as $PI$ of alternative ${\mathrm{ẋ}}_i$ with alternative ${\mathrm{ẋ}}_t$ to the corresponding attribute $c_j$, $i.e$.,

$${PI}_j({\mathrm{ẋ}}_i, {\mathrm{ẋ}}_t)={ꝕ}_j\left({ƒ}_{ij}-{ƒ}_{tj}\right)={ꝕ}_j\left(d\right).$$

(18)

Here, ${ꝕ}_j\left(d\right)$ is the generalized threshold function;

(c)

Evaluate superiority index (S-I): S = ${\left(S_{ij}\right)}_{m\times n}$.

$$S_{ij}={\displaystyle {\sum }_{t=1}^n}{PI}_j({\mathrm{ẋ}}_i, {\mathrm{ẋ}}_t)={\displaystyle {\sum }_{t=1}^n}{ꝕ}_j\left({ƒ}_{ij}-{ƒ}_{tj}\right).$$

(19)

(d)

Evaluate inferiority index (I-I): I = ${\left(I_{ij}\right)}_{m\times n}.$

$$I_{ij}={\displaystyle {\sum }_{t=1}^n}{PI}_j({\mathrm{ẋ}}_t, {\mathrm{ẋ}}_i)={\displaystyle {\sum }_{t=1}^n}{ꝕ}_j\left({ƒ}_{tj}-{ƒ}_{ij}\right).$$

(20)

Step 5. Determine the following:

Superiority-Flow:

$$\begin{gathered} {ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)= \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({\overline{\varpi }}_1,{ \overline{\varpi }}_2, \dots , {\overline{\varpi }}_n\right)= \\ \left(\sqrt[3]{1-{\mathrm{e}}^{-{\displaystyle {\sum }_{j=1}^n}{S}_{ij} {\overline{\varpi }}_j}}, {\mathrm{e}}^{-{\displaystyle {\sum }_{j=1}^n}{S}_{ij}{ \overline{\varpi }}_j}\right); \end{gathered}$$

(21)

Inferiority-Flow:

$$\begin{gathered} {ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)= \mathrm{FF}\text{-}\mathrm{AA}\text{-}\mathrm{W}\mathrm{A}\text{-}\mathcal{C}\mathcal{C}\left({\overline{\varpi }}_1,{\overline{\varpi }}_2, \dots , {\overline{\varpi }}_n\right)= \\ \left(\sqrt[3]{1-{\mathrm{e}}^{-{\displaystyle {\sum }_{j=1}^n}{I}_{ij} {\overline{\varpi }}_j}}, {\mathrm{e}}^{-{\displaystyle {\sum }_{j=1}^n}{I}_{ij}{ \overline{\varpi }}_j}\right). \end{gathered}$$

(22)

Determine the score function of ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)$ that provides S-flow and I-flow of alternatives ${\mathrm{ẋ}}_i$ as ${\mathrm{ẋ}}_{i }\left({ꝕ}^{>}\left({\mathrm{ẋ}}_i\right), {ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)\right)$. For greater ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)$ and lower ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)$, alternative ${\mathrm{ẋ}}_i$ is superior.

Step 6. Superiority/inferiority ranking:

(a)

Superiority ranking rules:

If ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)>{ꝕ}^{>}\left({\mathrm{ẋ}}_t\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)<{ꝕ}^{<}\left({\mathrm{ẋ}}_t\right)$, then ${\mathrm{ẋ}}_i \succ {\mathrm{ẋ}}_t;$

If ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)>{ꝕ}^{>}\left({\mathrm{ẋ}}_t\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)={ꝕ}^{<}\left({\mathrm{ẋ}}_t\right)$, then ${\mathrm{ẋ}}_i \succ {\mathrm{ẋ}}_t;$

If ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)={ꝕ}^{>}\left({\mathrm{ẋ}}_t\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)<{ꝕ}^{<}\left({\mathrm{ẋ}}_t\right)$, then ${\mathrm{ẋ}}_i \succ {\mathrm{ẋ}}_t;$

(b)

Inferiority ranking rules:

If ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)<{ꝕ}^{>}\left({\mathrm{ẋ}}_t\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)>{ꝕ}^{<}\left({\mathrm{ẋ}}_t\right)$, then ${\mathrm{ẋ}}_i\prec {\mathrm{ẋ}}_t;$

If ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)<{ꝕ}^{>}\left({\mathrm{ẋ}}_t\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)={ꝕ}^{<}\left({\mathrm{ẋ}}_t\right)$, then ${\mathrm{ẋ}}_i\prec {\mathrm{ẋ}}_t;$

If ${ꝕ}^{>}\left({\mathrm{ẋ}}_i\right)={ꝕ}^{>}\left({\mathrm{ẋ}}_t\right)$ and ${ꝕ}^{<}\left({\mathrm{ẋ}}_i\right)>{ꝕ}^{<}\left({\mathrm{ẋ}}_t\right)$, then ${\mathrm{ẋ}}_i\prec {\mathrm{ẋ}}_t$.

Step 7. Combine the rules of superiority and inferiority ranking in order to achieve the finest alternative (${\mathrm{ẋ}}_i$).

7. Numerical Example

To demonstrate the suggested strategy, we offer a realistic example of a software development project with FF information in this part.

A venture capital firm is interested in investing in a software development project. As a result, the organization employs three experts: project manager ($e_1$), CEO${(e}_2)$, and chairman$\left(e_3\right)$. First, they choose four possible alternatives:

• ${\mathrm{ẋ}}_1$: mail development project (MDP);
• ${\mathrm{ẋ}}_2$: game development project (GDP);
• ${\mathrm{ẋ}}_3$: browser development project (BDP);
• ${\mathrm{ẋ}}_4$: music player development project (MPDP).

The firm chooses three attributes in order to evaluate the four probable software development projects: economic feasibility ($c_1$); technological feasibility$(c_2$); and staff feasibility ($c_3)$. Next, the experts $e_k (k=1, 2, 3)$ assess the software development ${\mathrm{ẋ}}_i (i=1, 2, \mathrm{3,4})$ relating to attributes $c_j (j=1, 2, 3)$ and form the following three FF decision matrices $Ӻ\left(k\right)=$ ${\left({{Ӻ}_{ij}}^{\left(k\right)}\right)}_{4\times 3}$ in

Table 1. Fermatean Fuzzy (FF) decision matrices.

${\mathit{e}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$
${\mathbf{ẋ}}_{\mathbf{1}}$	(0.9, 0.6)	(0.8, 0.7)	(0.6, 0.7)
${\mathbf{ẋ}}_{\mathbf{2}}$	(0.8, 0.3)	(0.6, 0.4)	(0.7, 0.4)
${\mathbf{ẋ}}_{\mathbf{3}}$	(0.7, 0.5)	(0.5, 0.6)	(0.8, 0.5)
${\mathbf{ẋ}}_{\mathbf{4}}$	(0.6, 0.4)	(0.7, 0.5)	(0.6, 0.6)
${\mathit{e}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$
${\mathbf{ẋ}}_{\mathbf{1}}$	(0.7, 0.5)	(0.8, 0.6)	(0.9, 0.6)
${\mathbf{ẋ}}_{\mathbf{2}}$	(0.9, 0.3)	(0.7, 0.4)	(0.8, 0.5)
${\mathbf{ẋ}}_{\mathbf{3}}$	(0.8, 0.4)	(0.5, 0.7)	(0.7, 0.2)
${\mathbf{ẋ}}_{\mathbf{4}}$	(0.8, 0.5)	(0.6, 0.3)	(0.6, 0.4)
${\mathit{e}}_{\mathbf{3}}$	${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$
${\mathbf{ẋ}}_{\mathbf{1}}$	(0.8, 0.2)	(0.8, 0.7)	(0.7, 0.5)
${\mathbf{ẋ}}_{\mathbf{2}}$	(0.7, 0.5)	(0.9, 0.6)	(0.6, 0.4)
${\mathbf{ẋ}}_{\mathbf{3}}$	(0.6, 0.3)	(0.7, 0.4)	(0.8, 0.2)
${\mathbf{ẋ}}_{\mathbf{4}}$	(0.9, 0.4)	(0.6, 0.5)	(0.5, 0.3)

, weights of experts in

Table 2. Weights of experts.

Experts	PFEs
${\mathit{e}}_{\mathbf{1}}$	(0.9, 0.6)
${\mathit{e}}_{\mathbf{2}}$	(0.7, 0.5)
${\mathit{e}}_{\mathbf{3}}$	(0.8, 0.7)

, and attribute weights in

Table 3. Weights of attributes.

	${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$
${\mathit{e}}_{\mathbf{1}}$	(0.8, 0.7)	(0.7, 0.5)	(0.9, 0.3)
${\mathit{e}}_{\mathbf{2}}$	(0.7, 0.6)	(0.8, 0.2)	(0.7, 0.4)
${\mathit{e}}_{\mathbf{3}}$	(0.8, 0.3)	(0.9, 0.6)	(0.8, 0.6)

.

The FFSs, a generalization of IFSs and PFSs, are more flexible and have a broader area to solve decision-making problems for ranking alternatives, Figure 1 represents maximum superiority and minimum inferiority for best alternative.

The steps below find the most appealing alternative(s):

Step 1. Find the degree of individual measure ${\xi }_k(k=1, 2, 3)$ utilizing expert weights by Equation (13):

$${\xi }_k={(0.7892, 0.4613, 0.4373)}^T.$$

Step 2. Using Equation (14), calculate the normalized vector ${\mathsf{ϛ}}_k(k=1, 2, 3)$:

$${\mathsf{ϛ}}_k={(0.4676, 0.2733, 0.2591)}^T.$$

Step 3. (a) Using Equation (15), calculate the attributes weights integration ${\overline{\varpi }}_j(j=1, 2, 3)$:

$${\overline{\varpi }}_1=(0.62953, 0.70684), {\overline{\varpi }}_2=\left(0.68504, 0.62029\right),{\overline{\varpi }}_3=(0.71538, 0.64215).$$

(b) Using Equation (16), compute the aggregated decision matrix integration:

$${\left({Ӻ}_{ij}\right)}_{4\times 3}=\left(\begin{array}{c}\left(0.7154, 0.6277\right) \left(0.6445, 0.8365\right) \left(0.6778, 0.7964\right)\\ \left(0.6923, 0.6167\right) \left(0.6748, 0.6927\right) \left(0.5375, 0.6865\right)\\ \left(0.5864, 0.6662\right) \left(0.4833, 0.7542\right) \left(0.6445, 0.5602\right)\\ \left(0.6826, 0.6865\right) \left(0.5198, 0.6777\right) \left(0.4528, 0.6770\right)\end{array}\right).$$

Step 4. (a) Using Equation (17), calculate the performance function ${ƒ}_{ij}$:

$${\left({ƒ}_{ij}\right)}_{4\times 3}=\left(\begin{array}{c}0.9181 0.3762 0.3989\\ 0.7706 0.9157 0.5125\\ 0.1619 0.4797 0.8672\\ 0.6015 0.6106 0.5051\end{array}\right).$$

(b) Using Equation (18), calculate the preference intensity ${PI}_j$(${\mathrm{ẋ}}_i,{\mathrm{ẋ}}_t$):

Set attribute threshold function:

$${ꝕ}_k\left(d\right)=\left\{\begin{array}{c}0.01 if d>0\\ 0.00 if d\le 0.\end{array}\right.$$

(c) Using Equation (19), calculate the Superiority matrix (S. Matrix):

$$\mathrm{S}=\left(\begin{array}{c}0.03 0.00 0.00\\ 0.02 0.03 0.02\\ 0.00 0.01 0.03\\ 0.01 0.02 0.01\end{array}\right).$$

(d) Using Equation (20), calculate the inferiority matrix (I. Matrix):

$$\mathrm{I}=\left(\begin{array}{c}0.00 0.03 0.03\\ 0.01 0.00 0.01\\ 0.03 0.02 0.00\\ 0.02 0.01 0.02\end{array}\right).$$

Step 5. Equations (21) and (22) are used to calculate the S-Flow and I-Flow (Figure 1), as illustrated in

Table 4. The FF-SIR Flow.

Software Development Projects	${\mathbf{ꝕ}}^{>}\left({\mathbf{ẋ}}_{\mathit{i}}\right)$	$\mathbf{Ș}\mathbf{c}\left({\mathbf{ꝕ}}^{>}\left({\mathbf{ẋ}}_{\mathit{i}}\right)\right)$	${\mathbf{ꝕ}}^{<}\left({\mathbf{ẋ}}_{\mathit{i}}\right)$	$\mathbf{Ș}\mathbf{c}\left({\mathbf{ꝕ}}^{<}\left({\mathbf{ẋ}}_{\mathit{i}}\right)\right)$
${\mathbf{ẋ}}_{\mathbf{1}}$ (MDP)	(0.2655, 0.9790)	$-$0.919599	(0.3452, 0.9628)	$-$0.851365
${\mathbf{ẋ}}_{\mathbf{2}}$ (GDP)	(0.3592, 0.9554)	$-$0.825733	(0.2373, 0.9866)	$-$0.946974
${\mathbf{ẋ}}_{\mathbf{3}}$ (BDP)	(0.3034, 0.9749)	$-$0.898646	(0.3177, 0.9669)	$-$0.871884
${\mathbf{ẋ}}_{\mathbf{4}}$ (MPDP)	(0.2992, 0.9744)	$-$0.898365	(0.3214, 0.9674)	$-$0.872154

.

Step 6. (a) When Superiority Ranking Rules are combined with Table 4, the result is

$$Ș\mathrm{c}\left({ꝕ}^{>}\left({\mathrm{ẋ}}_2\right)\right)>Ș\mathrm{c}\left({ꝕ}^{>}\left({\mathrm{ẋ}}_4\right)\right)>Ș\mathrm{c}\left({ꝕ}^{>}\left({\mathrm{ẋ}}_3\right)\right)>Ș\mathrm{c}\left({ꝕ}^{>}\left({\mathrm{ẋ}}_1\right)\right)$$

$${\mathrm{ẋ}}_2 \succ {\mathrm{ẋ}}_4 \succ {\mathrm{ẋ}}_3 \succ {\mathrm{ẋ}}_{1 }.$$

(b) When Inferiority ranking rules are combined with Table 4, the result is

$$Ș\mathrm{c}\left({ꝕ}^{<}\left({\mathrm{ẋ}}_2\right)\right)<Ș\mathrm{c}\left({ꝕ}^{<}\left({\mathrm{ẋ}}_4\right)\right)<Ș\mathrm{c}\left({ꝕ}^{<}\left({\mathrm{ẋ}}_3\right)\right)<Ș\mathrm{c}\left({ꝕ}^{<}\left({\mathrm{ẋ}}_1\right)\right)$$

$${\mathrm{ẋ}}_2 \succ {\mathrm{ẋ}}_4 \succ {\mathrm{ẋ}}_3 \succ {\mathrm{ẋ}}_{1 }.$$

Step 7. By the superiority and inferiority ranking rules’ outcomes, the best alternative is ${\mathrm{ẋ}}_2$(GDP) in the software development project.

Finally, our proposed approach provides the best alternative, ${\mathrm{ẋ}}_2$(GDP), compared to all the others for the software development project investment, using the direction of the chosen experts. In this way, the established approach plays a vital role for a company or investor in making the best investment strategy by choosing the most efficient option from multiple participants with the guidance of experts.

8. Comparison Analysis

To ensure the authenticity and the feasibility of our suggested technique, we aggregated the same information (with weight attribute vector $\overline{\varpi }={\left(0.25, 0.35, 0.40\right)}^T)$ and applied different operators, namely, Fermatean Fuzzy Einstein Weighted Average (FFEWA) and Fermatean Fuzzy Hamacher Weighted Average (FFHWA) operators (Figure 2).

Definition 17

([65]). The FFEWA operator is defined as

$$\begin{array}{l}FFEWA\left({\mathrm{ẋ}}_i,i=1, 2, \dots m\right) = \\ \left(\sqrt[3]{\frac{{\prod }_{j=1}^n{\left(1+{{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}-{\prod }_{j=1}^n{\left(1-{{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}}{{\prod }_{j=1}^n{\left(1+{{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}+{\prod }_{j=1}^n{\left(1-{{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}}},\frac{\sqrt[3]{2} {\prod }_{j=1}^n{\left({\nu }_{{Ӻ}_{ij}}\right)}^{{\overline{\varpi }}_j}}{\sqrt[3]{{\prod }_{j=1}^n{\left(2-{{\nu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}+{\prod }_{j=1}^n{\left({{\nu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}}}\right).\end{array}$$

(23)

$$FFEWA\left({\mathrm{ẋ}}_1\right) =\left(\begin{array}{c}\sqrt[3]{\frac{{\left(1+{0.7154}^3\right)}^{0.25} {\left(1+{0.6445}^3\right)}^{0.35 }{\left(1+{0.6778}^3\right)}^{0.40 }-{\left(1-{0.7154}^3\right)}^{0.25 }{\left(1-{0.6445}^3\right)}^{0.35 }{\left(1-{0.6778}^3\right)}^{0.40 }}{{\left(1+{0.7154}^3\right)}^{0.25 }{\left(1+{0.6445}^3\right)}^{0.35 }{\left(1+{0.6778}^3\right)}^{0.40 }+{\left(1-{0.7154}^3\right)}^{0.25 }{\left(1-{0.6445}^3\right)}^{0.35 }{\left(1-{0.6778}^3\right)}^{0.40}}} , \\ \frac{\sqrt[3]{2} {\left(0.6277\right)}^{0.25 }{\left(0.8365\right)}^{0.35 }{\left(0.7964\right)}^{0.40 }}{\sqrt[3]{{\left(2-{0.6277}^3\right)}^{0.25 }{\left(2-{0.8365}^3\right)}^{0.35 }{\left(2-{0.7964}^3\right)}^{0.40 }+{\left({0.6277}^3\right)}^{0.25 }{\left({0.8365}^3\right)}^{0.35 }{\left({0.7964}^3\right)}^{0.40 }}}\end{array}\right)$$

$$=\left(0.6770, 0.7672\right).$$

$$\begin{gathered} FFEWA\left({\mathrm{ẋ}}_2\right) =\left(\begin{array}{c}\sqrt[3]{\frac{{\left(1+{0.6923}^3\right)}^{0.25} {\left(1+{0.6748}^3\right)}^{0.35 }{\left(1+{0.5375}^3\right)}^{0.40 }-{\left(1-{0.6923}^3\right)}^{0.25 }{\left(1-{0.6748}^3\right)}^{0.35 }{\left(1-{0.5375}^3\right)}^{0.40 }}{{\left(1+{0.6923}^3\right)}^{0.25 }{\left(1+{0.6748}^3\right)}^{0.35 }{\left(1+{0.5375}^3\right)}^{0.40 }+{\left(1-{0.6923}^3\right)}^{0.25 }{\left(1-{0.6748}^3\right)}^{0.35 }{\left(1-{0.5375}^3\right)}^{0.40}}} , \\ \frac{\sqrt[3]{2} {\left(0.6167\right)}^{0.25 }{\left(0.6927\right)}^{0.35 }{\left(0.6865\right)}^{0.40 }}{\sqrt[3]{{\left(2-{0.6167}^3\right)}^{0.25 }{\left(2-{0.6927}^3\right)}^{0.35 }{\left(2-{0.6865}^3\right)}^{0.40 }+{\left({0.6167}^3\right)}^{0.25 }{\left({0.6927}^3\right)}^{0.35 }{\left({0.6865}^3\right)}^{0.40 }}}\end{array}\right) \\ =\left(0.6335, 0.6708\right). \end{gathered}$$

$$\begin{gathered} FFEWA\left({\mathrm{ẋ}}_3\right) =\left(\begin{array}{c}\sqrt[3]{\frac{{\left(1+{0.5864}^3\right)}^{0.25} {\left(1+{0.4833}^3\right)}^{0.35 }{\left(1+{0.6445}^3\right)}^{0.40 }-{\left(1-{0.5864}^3\right)}^{0.25 }{\left(1-{0.4833}^3\right)}^{0.35 }{\left(1-{0.6445}^3\right)}^{0.40 }}{{\left(1+{0.5864}^3\right)}^{0.25 }{\left(1+{0.4833}^3\right)}^{0.35 }{\left(1+{0.6445}^3\right)}^{0.40 }+{\left(1-{0.5864}^3\right)}^{0.25 }{\left(1-{0.4833}^3\right)}^{0.35 }{\left(1-{0.6445}^3\right)}^{0.40}}} , \\ \frac{\sqrt[3]{2} {\left(0.6662\right)}^{0.25 }{\left(0.7542\right)}^{0.35 }{\left(0.5602\right)}^{0.40 }}{\sqrt[3]{{\left(2-{0.6662}^3\right)}^{0.25 }{\left(2-{0.7542}^3\right)}^{0.35 }{\left(2-{0.5602}^3\right)}^{0.40 }+{\left({0.6662}^3\right)}^{0.25 }{\left({0.7542}^3\right)}^{0.35 }{\left({0.5602}^3\right)}^{0.40 }}}\end{array}\right) \\ =\left(0.5827, 0.6518\right). \end{gathered}$$

$$\begin{gathered} FFEWA\left({\mathrm{ẋ}}_4\right) =\left(\begin{array}{c}\sqrt[3]{\frac{{\left(1+{0.6826}^3\right)}^{0.25} {\left(1+{0.5198}^3\right)}^{0.35 }{\left(1+{0.4528}^3\right)}^{0.40 }-{\left(1-{0.6826}^3\right)}^{0.25 }{\left(1-{0.5198}^3\right)}^{0.35 }{\left(1-{0.4528}^3\right)}^{0.40 }}{{\left(1+{0.6826}^3\right)}^{0.25 }{\left(1+{0.5198}^3\right)}^{0.35 }{\left(1+{0.4528}^3\right)}^{0.40 }+{\left(1-{0.6826}^3\right)}^{0.25 }{\left(1-{0.5198}^3\right)}^{0.35 }{\left(1-{0.4528}^3\right)}^{0.40}} }, \\ \frac{\sqrt[3]{2} {\left(0.6865\right)}^{0.25 }{\left(0.6777\right)}^{0.35 }{\left(0.6770\right)}^{0.40 }}{\sqrt[3]{{\left(2-{0.6865}^3\right)}^{0.25 }{\left(2-{0.6777}^3\right)}^{0.35 }{\left(2-{0.6770}^3\right)}^{0.40 }+{\left({0.6865}^3\right)}^{0.25 }{\left({0.6777}^3\right)}^{0.35 }{\left({0.6770}^3\right)}^{0.40 }}}\end{array}\right) \\ =\left(0.5512, 0.6796\right). \end{gathered}$$

$$Șc\left({\mathrm{ẋ}}_1\right)=-0.141282, Șc\left({\mathrm{ẋ}}_2\right)=-0.047604, Șc\left({\mathrm{ẋ}}_3\right)=-0.079063, Șc\left({\mathrm{ẋ}}_4\right)=-0.146411$$

$$Șc\left({\mathrm{ẋ}}_2\right)>Șc\left({\mathrm{ẋ}}_3\right)>Șc\left({\mathrm{ẋ}}_1\right)>Șc\left({\mathrm{ẋ}}_4\right)$$

$${\mathrm{ẋ}}_2 \succ {\mathrm{ẋ}}_3 \succ {\mathrm{ẋ}}_1 \succ {\mathrm{ẋ}}_{4 }.$$

Definition 18

([69]). The FFHWA operator is defined as

$$\begin{array}{l}FFHWA \left({\mathrm{ẋ}}_i,i=1, 2, \dots m\right) = \\ \left(\sqrt[3]{\frac{{\prod }_{j=1}^n{\left(1+{(\lambda -1){\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}-{\prod }_{j=1}^n{\left(1-{{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}}{{\prod }_{j=1}^n{\left(1+(\lambda -1){{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}+(\lambda -1){\prod }_{j=1}^n{\left(1-{{\mu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}}},\frac{\sqrt[3]{\lambda } {\prod }_{j=1}^n{\left({\nu }_{{Ӻ}_{ij}}\right)}^{{\overline{\varpi }}_j}}{\sqrt[3]{{\prod }_{j=1}^n{\left(1+(\lambda -1)(1-{{\nu }_{{Ӻ}_{ij}}}^3)\right)}^{{\overline{\varpi }}_j}+(\lambda -1){\prod }_{j=1}^n{\left({{\nu }_{{Ӻ}_{ij}}}^3\right)}^{{\overline{\varpi }}_j}}}\right)\end{array}$$

(24)

$$FFHWA\left({\mathrm{ẋ}}_1\right)=\left(0.6582, 0.7689\right), FFHWA\left({\mathrm{ẋ}}_2\right)=\left(0.6121, 0.6710\right), FFHWA\left({\mathrm{ẋ}}_3\right)=\left(0.5666, 0.6529\right),$$

$$FFHWA\left({\mathrm{ẋ}}_4\right)=\left(0.5325, 0.7070\right).$$

$$Șc\left({\mathrm{ẋ}}_1\right)=-0.169429, Șc\left({\mathrm{ẋ}}_2\right)=-0.072778, Șc\left({\mathrm{ẋ}}_3\right)=-0.096418, Șc\left({\mathrm{ẋ}}_4\right)=-0.202399$$

$$Șc\left({\mathrm{ẋ}}_2\right)>Șc\left({\mathrm{ẋ}}_3\right)>Șc\left({\mathrm{ẋ}}_1\right)>Șc\left({\mathrm{ẋ}}_4\right)$$

$${\mathrm{ẋ}}_{2 }\succ {\mathrm{ẋ}}_{3 }\succ {\mathrm{ẋ}}_{1 }\succ {\mathrm{ẋ}}_{1 }.$$

Table 5. Comparison analysis with FFEWA and FFHWA operators.

Methods	$\mathbf{Ș}\mathbf{c}\left({\mathbf{ẋ}}_{\mathbf{1}}\right)$	$\mathbf{Ș}\mathbf{c}\left({\mathbf{ẋ}}_{\mathbf{2}}\right)$	$\mathbf{Ș}\mathbf{c}\left({\mathbf{ẋ}}_{\mathbf{3}}\right)$	$\mathbf{Ș}\mathbf{c}\left({\mathbf{ẋ}}_{\mathbf{4}}\right)$	Ranking
FFEWA operator	$-0.141282$	$-0.047604$	$-0.079063$	$-0.146411$	${\mathrm{ẋ}}_2$ ≻ ${\mathrm{ẋ}}_3$ ≻ ${\mathrm{ẋ}}_1$ ≻ ${\mathrm{ẋ}}_{4 }$
FFHWA operator	$-0.169429$	$-0.072778$	$-0.096418$	$-0.202399$	${\mathrm{ẋ}}_{2 }$ ≻ ${\mathrm{ẋ}}_3$ ≻ ${\mathrm{ẋ}}_1$ ≻ ${\mathrm{ẋ}}_{4 }$
$\mathbf{F}\mathbf{F}$-AA-$\mathbf{W}\mathbf{A}$-𝓒𝓒 Operator (proposed)	$-$0.919599	$-$0.825733	$-$0.898646	$-$0.898365	${\mathrm{ẋ}}_{2 }$ ≻ ${\mathrm{ẋ}}_4$ ≻ ${ \mathrm{ẋ}}_3$ ≻ ${\mathrm{ẋ}}_1$

shows that FFEWA, FFHWA found the best alternative and $\mathrm{F}\mathrm{F}$-AA-$\mathrm{W}\mathrm{A}$-$\mathcal{C}\mathcal{C}$ operator had the same, that is, ${\mathrm{ẋ}}_2$ (GDP) from among the software development projects. The attribute weight vector for the FFEWA and FFHWA operators caused a slight change in the other alternatives, i.e., ${\mathrm{ẋ}}_1$ (MDP), ${\mathrm{ẋ}}_3$ (BDP), and ${\mathrm{ẋ}}_4$ (MPDP). Thus, our proposed approach is correct and can be used in DM challenges.

The advantage of the novel approach is that, even if the non-membership of any one alternative is zero, grades of non-membership will have an influence on the aggregated value. As a result, there is a relationship between membership and non-membership grades. This is not so for the FFEWA operator, which is the drawback of the FFEWA operator. For example, for FFNs ${Ӻ}_1=\left(0.90, 0\right)$, ${Ӻ}_2=\left(0.77, 0.45\right)$, ${Ӻ}_3=\left(0.80, 063\right)$, ${Ӻ}_4=\left(0.58, 0.67\right)$ with weights $\overline{\varpi }={\left(0.3, 0.3, 0.2, 0.2\right)}^T$, FFEWA $\left({Ӻ}_1, {Ӻ}_2, {Ӻ}_3, {Ӻ}_4\right)$ = $\left(0.88, 0\right)$. This shows that non-membership is independent of the non-membership (non-zero) of other FFNs.

Additionally, the proposed approach is well worked for the Fermatean fuzzy environment, and this approach helps to cope with the uncertain environment, based on the principles of Pythagorean fuzzy and Intuitionistic fuzzy, by reducing the exponents of membership status and non-membership status. This approach will fail for q-rung orthopairs, but, in the future, this issue can be resolved easily. Therefore, the limitation of the developed approach is it is only applicable to Fermatean fuzzy and all the above-mentioned generalized forms of fuzzy sets.

9. Conclusions and Future Work

In daily life problems, the SIR model has a significant role in resolving uncertainty for decision-making. For this consideration, the SIR techniques with Fermatean information are best for MAGDM problems. This research article’s foremost contributions are as follows.

Based on FFSs, we developed aggregation operators, namely, $\mathrm{F}\mathrm{F}$-AA-$\mathrm{W}\mathrm{A}$ operator and $\mathrm{F}\mathrm{F}$-AA-$\mathrm{W}\mathrm{A}$-$\mathcal{C}\mathcal{C}$ operator, with SIR methods. Some A$\mathcal{c}\mathcal{z}\mathcal{e}\mathcal{l}$–A$\mathcal{l}\mathcal{s}\mathcal{i}\mathcal{n}\mathcal{a}$ operational laws over FFSs have been discussed in detail. Meanwhile, we investigated features such as idempotency, boundedness, and monotonicity based on FF aggregation operators. This more generalized structure of AA AOs using SIR techniques and t-norm and t-conorm efficiently combines the difficult challenges. We mainly employed Fermatean data to construct the MAGDM technique for decision-makers’ convenience. To illustrate the legitimacy of the proposed work and assess its validity, a MAGDM issue for the selection of software development projects was addressed, and its results were compared to approaches already in use. Our advanced technique is also feasible for bipolar, Pythagorean, intuitionistic, cubic, and fuzzy information; thus, this study makes a valuable addition to the literature.

In the future, we intend to extend our study to q-rung orthogonal pair information and other fuzzy environments [70,71,72,73]. Further, we will make them available to other AOs, such as power mean aggregation operators, Bonferroni mean operators, Hamacher aggregation operates, Hamy mean operators, Dombi aggregation operators, and so on. We may apply this technique in marketing management, engineering, information technology, human resources management, energy management, etc.