Group decision-making for the selection of an antivirus mask under fermatean fuzzy soft information

Abstract

With the rapid increase of COVID-19, mostly people are facing antivirus mask shortages. It is necessary to select a good antivirus mask and make it useful for everyone. For maximize the efficacy of the antivirus masks, we propose a decision support algorithm based on the concept of Fermatean fuzzy soft set (FFS_fS). The basic purpose of this article is to introduce the notion of FFS_fS to deal with problems involving uncertainty and complexity corresponding to various parameters. Here, the valuable properties of FFS_fS are merged with the Yager operator to propose four new operators, namely, Fermatean fuzzy soft Yager weighted average (FFS_fYWA), Fermatean fuzzy soft Yager ordered weighted average (FFS_fYOWA), Fermatean fuzzy soft Yager weighted geometric (FFS_fYWG) and Fermatean fuzzy soft Yager ordered weighted geometric (FFS_fYOWG) operators. The fundamental properties of proposed operators are discussed. For the importance of proposed operators, a multi-attribute group decision-making (MAGDM) strategy is presented along with an application for the selection of an antivirus mask over the COVID-19 pandemic. The comparison with existing operators shows that existing operators cannot deal with data involving parametric study but developed operators have the ability to deal decision-making problems using parameterized information.

Keywords

Fermatean fuzzy soft numbers Yager operators Aggregation operators Antivirus mask selection TOPSIS method

1 Introduction

COVID-19 is a pandemic disease. Most of the infected people with the COVID-19 virus suffer respiratory illness and some recover without requiring special treatment. Older people and people having diseases like cardiovascular disease, diabetes, chronic respiratory disease and cancer are more likely to develop serious illness. Mostly COVID-19 spreads through the sneezing or coughing of an infected person. So, it is necessary to follow respiratory etiquette, for example, by coughing into a flexed elbow. This pandemic is moving like a waveone that infected rapidly many people in a few months. COVID-19 is affecting adversely the economic, social and politically condition of every country all over the world. Under the COVID-19 pandemic, masks have become essential items. Hence, optimizing the use of antivirus masks according to disparate people is the efficacious basic measure to deal with the mask shortage and COVID-19 diffusion.

Decision-making (DM) is a system of nominating the best option among various objects. That’s why DM performs an important role in the human life problems. A decision maker firstly tries to know about the restrictions, advantages and qualities of each alternative and then he finds out the final decision. To handle with such situations, Zadeh [44] proposed the prominent model of fuzzy set (FS). In this theory, each object is assigned by a membership degree (MD) belongs to [0, 1]. As an extension of FS, Atanassov [8] proposed the intuitionistic fuzzy set (IFS) which has both the MD μ and the nonmembership degree (NMD) ν in [0, 1], respectively such that μ + ν ≤ 1. From last few years, different researchers proposed various operators on the existing ideas in which the best one idea is the aggregation operators (AOs). The applications of novel geometric AOs for IFS in multi-attribute decision-making (MADM) were presented by Xu and Yager [39]. Xu [37] studied the intuitionistic fuzzy (IF) AOs. Wang and Liu [33] discussed the IF Einstein weighted geometric (IFEWG) operators. The notion of complex IF AOs with application in MADM was initiated by Garg and Rani [13]. IFS is a robust concept and many researchers have done work on it. However, there are some difficulties in the model of IFS, due to which this theory fails to handle the problems. Yager [40] proposed the theory of Pythagorean fuzzy set (PFS) to reduce the limitations of IFS. PFS replaces the constraint of IFS with the condition μ² + ν² ≤ 1. Akram et al. [1] worked for the development of Pythagorean Dombi fuzzy AOs. Shahzadi et al. [31] proposed the model of Yager AOs under Pythagorean fuzzy (PF) data. Liang et al. [20] introduced the idea of linear assignment method for MAGDM based on interval-valued PF Bonferroni mean operators. Wei and Lu [34] proposed PF Maclaurin symmetric mean operators. The idea of PF power AOs was initiated by Wei and Lu [35]. Wang and Li [32] used the idea of PF interaction power Bonferroni mean AOs in MADM. However, PFS has also some limitations.

Yager [41] defined q-rung orthopair fuzzy set (q-ROFS) in which μ^q + ν^q ≤ 1. Jana et al. [16] discussed the q-rung orthopair fuzzy (q-ROF) Dombi AOs. Joshi and Gegov [19] developed the confidence levels q-ROF AOs. The idea of Yager AOs for q-ROFS was introduced by Akram and Shahzadi [4]. The trigonometric operations laws for q-ROFS were studied by Garg [9]. Garg and Chen [11] examined the concept of neutrality AOs for q-ROFS. Recently, Senapati and Yager [30] gave the concept of Fermatean fuzzy set (FFS) as a generalization of IFS and PFS. The theory of Fermatean fuzzy (FF) weighted averaging/geometric operators was also given by Senapati and Yager [28]. As FS, IFS, PFS, FFS and so forth are generally applied by researchers to deal with vagueness and uncertainty during analysis but all these concepts have no information about parameterized study.

Molodstov [36] developed the model of S_fS for parametrization tool to handle with vague data. Maji [24, 25] introduced the theories of fuzzy S_fS (FS_fS) and intuitionistic fuzzy S_fS (IFS_fS). Arora and Garg [7] studied the IFS_f aggregation operators. Garg and Arora [10] discussed the generalized IFS_f power AOs and their application in MADM. q-rung orthopair fuzzy soft (q - ROFS_f) AOs were initiated by Hussain et al. [15]. For other terminologies not discussed in the paper, the readers are suggested to [2 , 46].

The motivations of this article are described as:

What are the advantages and restrictions of FFS_fS? FFS_fS is a peculiar extension of IFS_fS and PFS_fS which works with more generality than IFS_fS and PFS_fS. It allows the sum of the MD and NMD to be larger than one and the square sum of the MD and NMD to be larger than but restricts that their cubic sum is bounded by one corresponding to different parameters. However, this model fails when cubic sum of MD and NMD is not bounded by one.

Why we need S_fS? S_fS theory handles the uncertainty problems with parameterized information.

What are the logics behind the use of Yager AOs? Yager AOs are the simplest and quite creative approach for dealing with DM affairs. Basically this article directs Yager AOs in FFS_f surroundings to face complex issues corresponding to different parameters. The outcomes based on conclusion are quite accurate under Yager AOs when it is put on to the reality based MAGDM problems in FFS_f data.

The limitations of previously existing operators can be overcome by proposed operators as these operators are more general that work excellently not only for FFS_f information but also for IFS_f and PFS_f data.

The contributions of this article are described as:

The concept of Yager AOs is extended to FFS_f numbers (FFS_fNs) and fundamental properties including idempotency, boundedness, monotonicity, shift invariance and homogeneity are discussed.

An algorithm is developed to handle complex realistic problems with FFS_f data.

The proposed algorithm is supported by an illustrative example for the selection of an antivirus mask over the COVID-19 pandemic.

The superiority and importance of proposed model is shown through comparison analysis.

The rest of this article is composed as follows: In Section 2, we recall some basic definitions. In Section 3, we introduce the concept of FFS_fS, Yager operations for FFS_fNs and related score functions. Section 6 defines FFS_fYWA operator, FFS_fYOWA operator and some properties related to them. In Section 5, we analyze FFS_fYWG and FFS_fYOWG operators to aggregate the FFS_fNs. In Section 6, we propose an algorithm for MAGDM along with a numerical example in the field of an antivirus mask selection based on FFS_fNs and discuss the comparison analysis with existing operators to show the superiority and validity of proposed theory. In Section 7, results are concluded about proposed theory.

2 Preliminaries

Definition 2.1. [24] Let $V$ be a universe of discourse and $\tilde{W}$ be the set of all parameters, $\tilde{E} \subseteq \tilde{W}$ . $P (V)$ represents the set of all FSs over $V$ . $(\tilde{I}, \tilde{E})$ is called a FS_fS over $V$ , where $\tilde{I}$ is a function given by $\tilde{I} : \tilde{E} \to P (V)$ , which is ${\tilde{I}}_{e_{p}} (x_{t}) = {≺ x_{t}, f_{p} (x_{t}) ≻ | x_{t} \in V} .$

Definition 2.2. [27] Let $V$ be a universe of discourse and $\tilde{W}$ be the set of all parameters, $\tilde{E} \subseteq \tilde{W}$ . $P (V)$ represents the set of all PFSs over $V$ . $(\tilde{J}, \tilde{E})$ is called a PFS_fS over $V$ , where $\tilde{J}$ is a function given by $\tilde{J} : \tilde{E} \to P (V)$ , which is ${\tilde{J}}_{e_{p}} (x_{t}) = {≺ x_{t}, f_{p} (x_{t}), g_{p} (x_{t}) ≻ | x_{t} \in V},$ with condition $0 \leq (f_{p} (x_{t}))^{2} + (g_{p} (x_{t}))^{2} \leq 1$ .

Definition 2.3. [30] A Fermatean fuzzy set $F$ over the domain $V$ is defined as $F = {〈 x, f_{F} (x), g_{F} (x) 〉},$ where $f_{F} : V \to [0, 1]$ , $g_{F} : V \to [0, 1]$ and $ϖ_{F} (x) = \sqrt[3]{1 - (f_{F} (x))^{3} - (g_{F} (x))^{3}}$ specify MD, NMD and Indeterminacy degree (InD), respectively.

Definition 2.4. [15] Let $V$ be a universe of discourse and $\tilde{W}$ be the set of all parameters, $\tilde{E} \subseteq \tilde{W}$ . $P (V)$ represents the set of all q-ROFSs over $V$ . $(T^{'}, \tilde{E})$ is called a q - ROFS_fS over $V$ , where $T^{'}$ is a function given by $T^{'} : \tilde{E} \to P (V)$ , which is $T_{e_{p}}^{'} (x_{t}) = {≺ x_{t}, f_{p} (x_{t}), g_{p} (x_{t}) ≻ | x_{t} \in V},$ with condition $0 \leq (f_{p} (x_{t}))^{q} + (g_{p} (x_{t}))^{q} \leq 1$ . $ϖ_{T_{e_{t p}}^{'}} = \sqrt[q]{1 - ((f_{t p})^{q} + (g_{t p})^{q})}$ is the InD, where $f_{t p}$ denotes $f_{p} (x_{t})$ and $g_{t p}$ denotes $g_{p} (x_{t})$ .

3 Fermatean fuzzy soft set

Definition 3.1. Let $V$ be a universe of discourse and $\tilde{W}$ be the set of all parameters, $\tilde{E} \subseteq \tilde{W}$ . $P (V)$ represents the set of all FFSs over $V$ . $(T, \tilde{E})$ is called a FFS_fS over $V$ , where $T$ is a function given by $T : \tilde{E} \to P (V)$ , which is $T_{e_{p}} (x_{t}) = {≺ x_{t}, f_{p} (x_{t}), g_{p} (x_{t}) ≻ | x_{t} \in V},$ with condition $0 \leq (f_{p} (x_{t}))^{3} + (g_{p} (x_{t}))^{3} \leq 1$ . $π_{T_{e_{t p}}} = \sqrt[3]{1 - ((f_{t p})^{3} + (g_{t p})^{3})}$ is the hesitancy degree. For simplicity $T_{e_{p}} (x_{t}) = 〈 x_{t}, f_{p} (x_{t}), g_{p} (x_{t}) 〉$ is denoted by $T_{e_{t p}} = (f_{t p}, g_{t p})$ represents a FFS_fN.

Example 3.1. Suppose a person wants to purchase a washing machine from a set of four alternatives, i.e., $V = {x_{1}, x_{2}, x_{3}, x_{4}}$ . Let $\tilde{E} = {e_{1}, e_{2}, e_{3}, e_{4}}$ be the corresponding set of parameters where e₁=spin cycle, e₂=load capacity, e₃=efficiency, e₄=wash settings.

On the basis of above criteria a decision maker evaluates the alternatives with rating values and described the results in the form of FFS_fNs as given in Table 1.

Table 1
Tabular representation (TR) of FFS_fS

$V$ e ₁ e ₂ e ₃ e ₄

x ₁ (0.81,0.69) (0.90,0.60) (0.70,0.56) (0.67,0.45)

x ₂ (0.78,0.51) (0.66,0.23) (0.55,0.11) (0.77,0.34)

x ₃ (0.88,0.66) (0.91, 0.61) (0.68,0.21) (0.57,0.31)

x ₄ (0.71,0.55) (0.67,0.41) (0.87,0.56) (0.78,0.21)

$V$	e ₁	e ₂	e ₃	e ₄
x ₁	(0.81,0.69)	(0.90,0.60)	(0.70,0.56)	(0.67,0.45)
x ₂	(0.78,0.51)	(0.66,0.23)	(0.55,0.11)	(0.77,0.34)
x ₃	(0.88,0.66)	(0.91, 0.61)	(0.68,0.21)	(0.57,0.31)
x ₄	(0.71,0.55)	(0.67,0.41)	(0.87,0.56)	(0.78,0.21)

Definition 3.2. Let $T_{e_{1 p}} = (f_{1 p}, g_{1 p}) (p = 1, 2)$ and $T = (f, g)$ be three FFS_fNs and ζ > 0. Then some operations for FFS_fNs are

$T_{e_{11}} \cup T_{e_{12}} = (max (f_{11}, f_{12}), min (g_{11}, g_{12}))$ ,

$T_{e_{11}} \cap T_{e_{12}} = (min (f_{11}, f_{12}), max (g_{11}, g_{12}))$ ,

$T^{c} = (g, f)$ , $T^{c}$ is the complement of $T$ ,

$T_{e_{11}} \oplus T_{e_{12}} = (\sqrt[3]{(f_{11})^{3} + (f_{12})^{3} - (f_{11})^{3} (f_{12})^{3}}, g_{11} g_{12})$ ,

$T_{e_{11}} \otimes T_{e_{12}} = (f_{11} f_{12}, \sqrt[3]{(g_{11})^{3} + (g_{12})^{3} - (g_{11})^{3} (g_{12})^{3}})$ ,

$ζ T = (\sqrt[3]{1 - (1 - f^{3})^{ζ}}, g^{ζ}),$

$T^{ζ} = (f^{ζ}, \sqrt[3]{1 - (1 - g^{3})^{ζ}}) .$

Definition 3.3 Let $T_{e_{1 p}} = (f_{1 p}, g_{1 p}) (p = 1, 2)$ and $T = (f, g)$ be three FFS_fNs and ζ > 0. Then, Yager norm operations for FFS_fNs are

$T_{e_{11}} \oplus T_{e_{12}} = (\sqrt[3]{min (1, (f_{11}^{3 ϑ} + f_{12}^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, ((1 - g_{11}^{3})^{ϑ} + (1 - g_{12}^{3})^{ϑ})^{\frac{1}{ϑ}})})$ ,

$T_{e_{11}} \otimes T_{e_{12}} = (\sqrt[3]{1 - min (1, ((1 - f_{11}^{3})^{ϑ} + (1 - f_{12}^{3})^{ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{min (1, (g_{11}^{3 ϑ} + g_{12}^{3 ϑ})^{\frac{1}{ϑ}})})$ ,

$ζ T = (\sqrt[3]{min (1, (ζ f^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, (ζ (1 - g^{3})^{ϑ})^{\frac{1}{ϑ}})}),$

$T^{ζ} = (\sqrt[3]{1 - min (1, (ζ (1 - f^{3})^{ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{min (1, (ζ g^{3 ϑ})^{\frac{1}{ϑ}})}) .$

Why the name “Yager operations”? These operations are derived using the theoretical foundations of Yager t-norm and Yager t-conorm for the FFS_f environment. These operations carry the accuracy feature and aggregation skills of the Yager norm for the flexible model of FFS_fS. Therefore, these operations are named as “Yager operations”.

Definition 3.4. The score function and accuracy function for FFS_fNs $T_{e_{t p}} = (f_{t p}, g_{t p})$ are represented by $S (T_{e_{t p}}) = f_{t p}^{3} - g_{t p}^{3}, S (T_{e_{t p}}) \in [- 1, 1]$ and $H (T_{e_{t p}}) = f_{t p}^{3} + g_{t p}^{3}, H (T_{e_{t p}}) \in [0, 1]$ respectively.

Definition 3.5. Let $T_{e_{11}} = (f_{11}, g_{11})$ and $T_{e_{12}} = (f_{12}, g_{12})$ be two FFS_fNs,

if $S (T_{e_{11}}) > S (T_{e_{12}})$ , then $T_{e_{11}} ≻ T_{e_{12}};$

if $S (T_{e_{11}}) < S (T_{e_{12}})$ , then $T_{e_{11}} ≺ T_{e_{12}};$

if $S (T_{e_{11}}) = S (T_{e_{12}})$ , then

if $H (T_{e_{11}}) > H (T_{e_{12}})$ , then $T_{e_{11}} ≻ T_{e_{12}};$

if $H (T_{e_{11}}) < H (T_{e_{12}})$ , then $T_{e_{11}} ≺ T_{e_{12}};$

if $H (T_{e_{11}}) = H (T_{e_{12}})$ , then $T_{e_{11}} \sim T_{e_{12}} .$

4 Fermatean fuzzy soft Yager average operators

4.1 Fermatean fuzzy soft Yager weighted average operators

In this section, we develop Yager weighted average operators for FFS_fNs.

Definition 4.1. Let $T_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a collection of FFS_fNs and $w_{t}$ , $v_{p}$ are the weight vectors (WVs) for the experts $x_{t}^{'} s$ and parameters $e_{p}^{'} s$ , respectively, satisfying $w_{t} > 0, v_{p} > 0$ with $\sum_{t = 1}^{n} w_{t} = 1$ and $\sum_{p = 1}^{m} v_{t} = 1$ . Then FFS_fYWA operator is a function FFS_fYWA: $M^{n} \to M$ , where $M$ is the set of FFS_fNs. ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} (w_{t} T_{e_{t p}})) .$

Theorem 4.1. The aggregated value by utilizing FFS_fYWA operator is given by ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$ $\begin{matrix} = & (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ (1)

Proof. Applying mathematical induction to prove it.

(i) When $n = 2, m = 2$ , ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}})$ $\begin{matrix} = \oplus_{p = 1}^{2} v_{p} (\oplus_{t = 1}^{2} w_{t} T_{e_{t p}}) \\ = v_{1} (\oplus_{t = 1}^{2} w_{t} T_{e_{t 1}}) \oplus v_{2} (\oplus_{t = 1}^{2} w_{t} T_{e_{t 2}}) \\ = v_{1} (w_{1} T_{e_{11}} \oplus w_{2} T_{e_{21}}) \oplus v_{2} (w_{1} T_{e_{12}} \oplus w_{2} T_{e_{22}}) \\ = v_{1} {(\sqrt[3]{min (1, (w_{1} f_{11}^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, (w_{1} (1 - g_{11}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[3]{min (1, (w_{2} f_{21}^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, (w_{2} (1 - g_{21}^{3})^{ϑ})^{\frac{1}{ϑ}})})} \oplus \\ v_{2} {(\sqrt[3]{min (1, (w_{1} f_{12}^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, (w_{1} (1 - g_{12}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[3]{min (1, (w_{2} f_{22}^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, (w_{2} (1 - g_{22}^{3})^{ϑ})^{\frac{1}{ϑ}})})} \\ = v_{1} (\sqrt[3]{min (1, (\sum_{t = 1}^{2} w_{t} f_{t 1}^{3 ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{t = 1}^{2} w_{t} (1 - g_{t 1}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \oplus \\ v_{2} (\sqrt[3]{min (1, (\sum_{t = 1}^{2} w_{t} f_{t 2}^{3 ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{t = 1}^{2} w_{t} (1 - g_{t 2}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, (v_{1} \sum_{t = 1}^{2} w_{t} f_{t 1}^{3 ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (v_{1} \sum_{t = 1}^{2} w_{t} (1 - g_{t 1}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \oplus \\ (\sqrt[3]{min (1, (v_{2} \sum_{t = 1}^{2} w_{t} f_{t 2}^{3 ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (v_{2} \sum_{t = 1}^{2} w_{t} (1 - g_{t 2}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{2} v_{p} (\sum_{t = 1}^{2} w_{t} f_{t p}^{3 ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{2} v_{p} (\sum_{t = 1}^{2} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ Hence, result is true for $n = 2, m = 2$ . Suppose Equation 1 is true for $n = k_{1}, m = k_{2}$ , ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{k_{1} k_{2}}})$ $\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} f_{t p}^{3 ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$

Now for $n = k_{1} + 1, m = k_{2} + 1$ , ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{k_{1} k_{2}}})$

$\begin{matrix} = {\oplus_{p = 1}^{k_{2}} v_{p} (\oplus_{t = 1}^{k_{1}} (w_{t} T_{e_{t p}}))} \oplus \\ v_{(k_{2} + 1)} (w_{(k_{1} + 1)} T_{e_{(k_{1} + 1) (k_{2} + 1)}}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} f_{t p}^{3 ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus v_{(k_{2} + 1)} (w_{(k_{1} + 1)} T_{e_{(k_{1} + 1) (k_{2} + 1)}}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{k_{2} + 1} v_{p} (\sum_{t = 1}^{k_{1} + 1} w_{t} f_{t p}^{3 ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{k_{2} + 1} v_{p} (\sum_{t = 1}^{k_{1} + 1} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ Hence, Equation 1 is true for $n = k_{1} + 1, m = k_{2} + 1$ . Therefore, by mathematical induction the Equation 1 is true, $\forall m, n \geq 1$ . □

Example 4.1. Suppose Mr. $B$ wants to purchase a car from a set of four cars in the domain set $V = {x_{1}, x_{2}, x_{3}, x_{4}}$ and let $\tilde{E} = {e_{1}, e_{2}, e_{3}, e_{4}}$ be the set of parameters for the selection of a car, i.e., $e_{p} (p = 1, 2, 3, 4)$ where e₁ =fuel efficiency, e₂ =good quality, e₃ =safety and e₄ =good warranty. Suppose the WVs for experts $x_{t}$ and parameters $e_{p}$ are w = (0.35, 0.25, 0.15, 0.25) ^T and v = (0.30, 0.20, 0.20, 0.30) ^T, respectively.

By applying Equation 1, ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{44}})$ $\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = (0.77, 0.45) . \end{matrix}$

Property 4.1. (Idempotency) If all $T_{e_{t p}} = L_{e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical. Then ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = L_{e}$

Proof. As $T_{e_{t p}} = L_{e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ , then ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$ $\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - c^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, (b^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, ((1 - c^{3})^{ϑ})^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, b^{3})}, \sqrt[3]{1 - min (1, (1 - c^{3}))}) \\ = (b, c) . \end{matrix}$ □

Property 4.2. (Boundedness) Let $T_{e_{t p}} = (f_{t p}, g_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a family of FFS_fNs, and $T_{e_{t p}}^{-} = (min_{p} min_{t} {f_{t p}}, max_{p} max_{t} {g_{t p}})$ and $T_{e_{t p}}^{+} = (max_{p} max_{t} {f_{t p}}, min_{p} min_{t} {g_{t p}})$ . Then $T_{e_{t p}}^{-} \leq {FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq T_{e_{t p}}^{+} .$

Proof. Let $T_{e_{t p}}^{-} = (min_{p} min_{t} {f_{t p}}, max_{p} max_{t} {g_{t p}})$ and $T_{e_{t p}}^{+} = (max_{p} max_{t} {f_{t p}}, min_{p} min_{t} {g_{t p}})$ . The inequalities for membership value are $\begin{matrix} \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{- 3 ϑ})))^{\frac{1}{ϑ}})} \\ \leq \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})} \\ \leq \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{+ 3 ϑ})))^{\frac{1}{ϑ}})} . \end{matrix}$ Similarly, for nonmembership value $\begin{matrix} \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{+ 3})^{ϑ}))^{\frac{1}{ϑ}})} \\ \leq \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})} \\ \leq \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{- 3})^{ϑ}))^{\frac{1}{ϑ}})} . \end{matrix}$ Therefore, $T_{e_{t p}}^{-} \leq {FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq T_{e_{t p}}^{+} .$ □

Property 4.3. (Monotonicity) Let $L_{e_{t p}} = (b_{t p}, c_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be another collection of FFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ . Then ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq {FFS}_{f} YWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Proof. Consider ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = (I, J)$ and ${FFS}_{f} YWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) = (I^{'}, J^{'}) .$ First, we show that I ≤ I′. As $f_{t p} \leq b_{t p}, f_{t p}^{3} \leq b_{t p}^{3} .$ Moreover, $\begin{matrix} \sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ}) \leq \sum_{t = 1}^{n} (w_{t} b_{t p}^{3 ϑ}) \\ \sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})) \leq \sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{3 ϑ})) \\ (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}} \leq (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{3 ϑ})))^{\frac{1}{ϑ}} \\ min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}}) \\ \leq min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{3 ϑ})))^{\frac{1}{ϑ}}) \\ \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})} \\ \leq \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})} . \end{matrix}$

Therefore, I ≤ I′. Similarly we can show that J ≥ J′. Hence, ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq {FFS}_{f} YWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$ □

Property 4.4. (Shift Invariance) If $L_{e} = (b, c)$ is another FFS_fN. Then ${FFS}_{f} YWA (T_{e_{11}} \oplus L_{e}, T_{e_{12}} \oplus L_{e}, \dots, T_{e_{n m}} \oplus L_{e}) = {FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \oplus L_{e} .$

Proof. Since $L_{e} = (b, c)$ and $T_{e_{t p}} = (f_{t p}, g_{t p})$ are FFS_fNs, so $T_{e_{11}} \oplus L_{e} = (\sqrt[3]{min (1, (f_{11}^{3 ϑ} + b^{3 ϑ})^{\frac{1}{ϑ}})}$ , $\sqrt[3]{1 - min (1, ((1 - g_{11}^{3})^{ϑ} + (1 - c^{3})^{ϑ})^{\frac{1}{ϑ}})})$ . Therefore, ${FFS}_{f} YWA (T_{e_{11}} \oplus L_{e}, T_{e_{12}} \oplus L_{e}, \dots, T_{e_{n m}} \oplus L_{e})$

$\begin{matrix} = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} w_{t} (T_{e_{t p}} \oplus L_{e})) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (f_{t p}^{3 ϑ} + b^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} ((1 - g_{t p}^{3})^{ϑ} + (1 - c^{3})^{ϑ})))^{\frac{1}{ϑ}}})) . \end{matrix}$ (2) ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \oplus L_{e}$ $\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \oplus (b, c) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[3]{min (1, b^{3})}, \sqrt[3]{1 - min (1, (1 - c^{3}))}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[3]{min (1, (b^{3 ϑ})^{\frac{1}{ϑ}})}, \sqrt[3]{1 - min (1, ((1 - c^{3})^{ϑ})^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - c^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (f_{t p}^{3 ϑ} + b^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} ((1 - g_{t p}^{3})^{ϑ} + (1 - c^{3})^{ϑ})))^{\frac{1}{ϑ}}})) . \end{matrix}$ (3)

From Equations 2 and 3, ${FFS}_{f} YWA (T_{e_{11}} \oplus L_{e}, T_{e_{12}} \oplus L_{e}, \dots, T_{e_{n m}} \oplus L_{e}) = {FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \oplus L_{e} .$ □

Property 4.5. (Homogeneity) Let ζ ≥ 0. Then ${FFS}_{f} YWA (ζ T_{e_{11}}, ζ T_{e_{12}}, \dots, ζ T_{e_{n m}}) = ζ {FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) .$

Proof. As $\begin{matrix} ζ T_{e_{t p}} \\ = (\sqrt[3]{min (1, (ζ f_{t p}^{3 ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (ζ (1 - g_{t p}^{3})^{ϑ})^{\frac{1}{ϑ}})}) \end{matrix}$ Now ${FFS}_{f} YWA (ζ T_{e_{11}}, ζ T_{e_{12}}, \dots, ζ T_{e_{n m}})$

$\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (ζ f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (ζ (1 - g_{t p}^{3})^{ϑ})))^{\frac{1}{ϑ}})}) . \end{matrix}$ (4) and ${FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$ $\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ $ζ {FFS}_{f} YWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$

4.2 Fermatean fuzzy soft yager ordered weighted average operators

Why we need FFS_fYOWA operator? FFS_fYWA operator only weighted the FFS_fN′s values but FFS_fYOWA operator weights the ordered positions via scoring the FFS_f values rather than weighting the FFS_f values themselves. In this section, we develop FFS_fYOWA operator and related properties.

Definition 4.2. Let $T_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a family of FFS_fNs and $w_{t}$ , $v_{p}$ are the WVs for the experts $x_{t}^{'} s$ and parameters $e_{p}^{'} s$ , respectively, satisfying $w_{t} > 0, v_{p} > 0$ with $\sum_{t = 1}^{n} w_{t} = 1$ and $\sum_{p = 1}^{m} v_{t} = 1$ . Then FFS_fYOWA operator is a function FFS_fYOWA: $M^{n} \to M$ ${FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} (w_{t} T_{σ e_{t p}})) .$

Theorem 4.2. The aggregated value by utilizing FFS_fYOWA operator is given by ${FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$ $\begin{matrix} = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} (w_{t} T_{σ e_{t p}})) \\ = (\sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{σ t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{σ t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ (6) where $T_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$ , is the permutation of $t$ th and $p$ th largest object of the collection of $t \times p$ FFS_fNs $T_{e_{t p}} = (f_{t p}, g_{t p}) .$

Proof. It is similar to Theorem 4.1. □

Example 4.2. Consider the collections FFS_fNs of Example 4.1 as given in Table 2, by using definition of score function, the TR of $T_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$ is given in Table 3.

Table 2

TR of FFS_fNs

$V$	e ₁	e ₂	e ₃	e ₄
x₁	(0.72, 0.40)	(0.76, 0.44)	(0.47, 0.30)	(0.81, 0.43)
x₂	(0.81, 0.67)	(0.69, 0.11)	(0.46, 0.20)	(0.89, 0.31)
x₃	(0.84, 0.62)	(0.78, 0.55)	(0.89, 0.51)	(0.67, 0.34)
x₄	(0.75, 0.42)	(0.87, 0.47)	(0.67, 0.61)	(0.91, 0.20)

Table 3

TR of FFS_fNs $T_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$

$V$	e ₁	e ₂	e ₃	e ₄
x₁	(0.84, 0.62)	(0.87, 0.47)	(0.89, 0.51)	(0.91, 0.20)
x₂	(0.75, 0.42)	(0.76, 0.44)	(0.46, 0.20)	(0.89, 0.31)
x₃	(0.72, 0.40)	(0.69, 0.11)	(0.47, 0.30)	(0.81, 0.43)
x₄	(0.81, 0.67)	(0.78, 0.55)	(0.67, 0.61)	(0.67, 0.34)

By using Equation 6, ${FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{44}})$ $\begin{matrix} = (\sqrt[3]{min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} (w_{t} f_{σ t p}^{3 ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} w_{t} (1 - g_{σ t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = (0.80, 0.48) . \end{matrix}$

We give some properties without their proofs.

Property 4.6. (Idempotency) If all $T_{e_{t p}} = L_{σ e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical. Then ${FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = L_{σ e} .$

Property 4.7. (Boundedness) Let $T_{σ e_{t p}} = (f_{σ t p}, g_{σ t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a collection of FFS_fNs and $T_{σ e_{t p}}^{-} = (min_{p} min_{t} {f_{σ t p}}, max_{p} max_{t} {g_{σ t p}})$ and $T_{σ e_{t p}}^{+} = (max_{p} max_{t} {f_{σ t p}}, min_{p} min_{t} {g_{σ t p}})$ . Then $T_{σ e_{t p}}^{-} \leq {FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq T_{σ e_{t p}}^{+} .$

Property 4.8. (Monotonicity) Let $L_{e_{t p}} = (b_{t p}, c_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be another collection of FFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ . Then ${FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq {FFS}_{f} YOWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Property 4.9. (Shift Invariance) If $L_{e} = (b, c)$ is another FFS_fN. Then ${FFS}_{f} YOWA (T_{e_{11}} \oplus L_{e}, T_{e_{12}} \oplus L_{e}, \dots, T_{e_{n m}} \oplus L_{e}) = {FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \oplus L_{e} .$

Property 4.10. property(Homogeneity) Let ζ ≥ 0. Then ${FFS}_{f} YOWA (ζ T_{e_{11}}, ζ T_{e_{12}}, \dots, ζ T_{e_{n m}}) = ζ {FFS}_{f} YOWA (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) .$

5 Fermatean fuzzy soft Yager geometric operators

5.1 Fermatean fuzzy soft Yager weighted geometric operators

In this section, we develop Yager weighted geometric operators for FFS_fNs.

Definition 5.1. Let $T_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a family of FFS_fNs and $w_{t}$ , $v_{p}$ are the WVs for the experts $x_{t}^{'} s$ and parameters $e_{p}^{'} s$ , respectively, satisfying $w_{t} > 0, v_{p} > 0$ with $\sum_{t = 1}^{n} w_{t} = 1$ and $\sum_{p = 1}^{m} v_{t} = 1$ . Then FFS_fYWG operator is a function FFS_fYWG: $M^{n} \to M$ ${FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = ⨂_{p = 1}^{m} (⨂_{t = 1}^{n} T_{e_{t p}}^{w_{t}})^{v_{p}} .$

Theorem 5.1. The aggregated value by utilizing FFS_fYWG operator is given by ${FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$ $\begin{matrix} = (\sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - f_{t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} g_{t p}^{3 ϑ})))^{\frac{1}{ϑ}})},) . \end{matrix}$ (7)

Proof. It is similar to Theorem 4.1. □

Property 5.1. (Idempotency) If all $T_{e_{t p}} = L_{e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical. Then ${FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = L_{e}$

Property 5.2. (Boundedness) Let $T_{e_{t p}} = (f_{t p}, g_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a collection of FFS_fNs and $T_{e_{t p}}^{-} = (min_{p} min_{t} {f_{t p}}, max_{p} max_{t} {g_{t p}})$ and $T_{e_{t p}}^{+} = (max_{p} max_{t} {f_{t p}}, min_{p} min_{t} {g_{t p}})$ . Then $T_{e_{t p}}^{-} \leq {FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq T_{e_{t p}}^{+} .$

Property 5.3. (Monotonicity) Let $L_{e_{t p}} = (b_{t p}, c_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be another collection of FFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ . Then ${FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq {FFS}_{f} YWG (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Property 5.4. (Shift Invariance) If $L_{e} = (b, c)$ is another FFS_fN. Then ${FFS}_{f} YWG (T_{e_{11}} \otimes L_{e}, T_{e_{12}} \otimes L_{e}, \dots, T_{e_{n m}} \otimes L_{e}) = {FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \otimes L_{e} .$

Property 5.5. (Homogeneity) Let ζ ≥ 0. Then ${FFS}_{f} YWG (T_{e_{11}}^{ζ}, T_{e_{12}}^{ζ}, \dots, T_{e_{n m}}^{ζ}) = ({FFS}_{f} YWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}))^{ζ} .$

5.2 Fermatean fuzzy soft yager ordered weighted geometric operators

Why we need FFS_fYOWG operator? FFS_fYWG operator only weighted the FFS_fN′s values but FFS_fYOWG operator weights the ordered positions via scoring the FFS_f values rather than weighting the FFS_f values themselves. Here, we will develop FFS_fYOWG operator and related properties.

Definition 5.2. Let $T_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a family of FFS_fNs and $w_{t}$ , $v_{p}$ are the WVs for the experts $x_{t}^{'} s$ and parameters $e_{p}^{'} s$ , respectively, satisfying $w_{t} > 0, v_{p} > 0$ with $\sum_{t = 1}^{n} w_{t} = 1$ and $\sum_{p = 1}^{m} v_{t} = 1$ . Then FFS_fYOWG operator is a function FFS_fYOWG: $M^{n} \to M$ ${FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = ⨂_{p = 1}^{m} (⨂_{t = 1}^{n} T_{σ e_{t p}}^{w_{t}})^{v_{p}} .$

Theorem 5.2. The aggregated value by utilizing FFS_fYOWG operator is given by ${FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}})$ $\begin{matrix} = & ⨂_{p = 1}^{m} (⨂_{t = 1}^{n} T_{σ e_{t p}}^{w_{t}})^{v_{p}} \\ = & (\sqrt[3]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{σ t p}^{3})^{ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{σ t p}^{3 ϑ})))^{\frac{1}{ϑ}})}) . \end{matrix}$ (8) where $T_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$ , is the permutation of $t$ th and $p$ th largest object of the collection of $t \times p$ FFS_fNs $T_{e_{t p}} = (f_{t p}, g_{t p}) .$

Proof. It is similar to Theorem 4.1.□

We give some properties without their proofs.

Property 5.6. (Idempotency) If all $T_{e_{t p}} = L_{σ e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical. Then ${FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) = L_{σ e} .$

Property 5.7. (Boundedness) Let $T_{σ e_{t p}} = (f_{σ t p}, g_{σ t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a collection of FFS_fNs and $T_{σ e_{t p}}^{-} = (min_{p} min_{t} {f_{σ t p}}, max_{p} max_{t} {g_{σ t p}})$ and $T_{σ e_{t p}}^{+} = (max_{p} max_{t} {f_{σ t p}}, min_{p} min_{t} {g_{σ t p}})$ . Then $T_{σ e_{t p}}^{-} \leq {FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq T_{σ e_{t p}}^{+} .$

Property 5.8. (Monotonicity) Let $L_{e_{t p}} = (b_{t p}, c_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be another collection of FFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ . Then ${FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \leq {FFS}_{f} YOWG (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Property 5.9. (Shift Invariance) If $L_{e} = (b, c)$ is another FFS_fN. Then ${FFS}_{f} YOWG (T_{e_{11}} \otimes L_{e}, T_{e_{12}} \otimes L_{e}, \dots, T_{e_{n m}} \otimes L_{e}) = {FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}) \otimes L_{e} .$

Property 5.10. (Homogeneity) Let ζ ≥ 0. Then ${FFS}_{f} YOWG (T_{e_{11}}^{ζ}, T_{e_{12}}^{ζ}, \dots, T_{e_{n m}}^{ζ}) = ({FFS}_{f} YOWG (T_{e_{11}}, T_{e_{12}}, \dots, T_{e_{n m}}))^{ζ} .$

6 Model for MAGDM using fermatean fuzzy soft information

This section discusses a mathematical description of proposed approach for MAGDM under FFS_f information. The basic idea and steps of an algorithm for given approach are described as:

Let $V = {x_{1}, x_{2}, \dots, x_{l}}$ be the set of alternatives, assessed by $n$ senior experts ${\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{n}$ under the constraints of $m$ parameters $\tilde{E} = {e_{1}, e_{2}, \dots, e_{m}}$ . These $n$ experts give their preferences for the l alternatives in terms of FFS_fNs $T_{e_{t p}} = (f_{t p}, g_{t p})$ with WVs $w = (w_{1}, w_{2}, \dots, w_{n})^{T}$ and $v = (v_{1}, v_{2}, \dots, v_{m})^{T}$ for experts and parameters, respectively such that $\sum_{t = 1}^{n} w_{t} = 1$ and $\sum_{p = 1}^{m} v_{p} = 1$ . The collective information is given in the form of decision matrix (DMx) $M = [T_{e_{t p}}]_{n \times m}$ . By using proposed AOs, the aggregated FFS_fN Ω_s (s = 1, 2, ⋯ , l) for alternative x_s is given by Ω_s = (f_s, g_s) . For getting the optimal solution, determine the score values for the alternatives and rank them.

For solving a MAGDM problem, the steps of Algorithm 1 are given as:

Algorithm 1: Steps to deal MAGDM problem by proposed operator

In the first step, collect the assessment information of expert’s corresponding to their parameters for each alternative and compose a DMx $M = [T_{e_{t p}}]_{n \times m}$ in the following way: $M = [\begin{matrix} (f_{11}, g_{11}) & (f_{12}, g_{12}) & \dots & (f_{1 m}, g_{1 m}) \\ (f_{21}, g_{21}) & (f_{22}, g_{22}) & \dots & (f_{2 m}, g_{2 m}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (f_{n 1}, g_{n 1}) & (f_{n 2}, g_{n 2}) & \dots & (f_{n m}, g_{n m}) \end{matrix}]$

Find the normalize DMx by interchanging the assessment value of cost parameter (CP) into benefit parameter (BP) by using the expression taken from [38], i.e., $P_{t p} = {\begin{matrix} T_{e_{t p}}^{c}; & for CP, \\ T_{e_{t p}}; & for BP . \end{matrix}$

For each alternative x_s (s = 1, 2, ⋯ , l), aggregate the FFS_fNs by using proposed operators in the collective DMx Ω_s.

By utilizing the score function, find the score values for Ω_s of all alternatives.

Arrange the alternatives in the decreasing order of score values and choose the best alternative.

6.1 Selection of an effective antivirus mask

For the importance and validity of proposed model under FFS_f data, we discuss a numerical example for the selection of an antivirus mask in the critical situation of COVID-19 pandemic. Everyone in the society is facing many difficulties in getting a good and effective antivirus mask to prevent himself/herself from COVID-19. Demand of an antivirus mask is increased in such critical situation of COVID-19. Due to increasing demand, it is difficult to get good and effective antivirus masks in local markets. Increasing demand has also led to low quality antivirus masks entering the market. Yang et al. [43] found out the effective antivirus mask over COVID-19 pandemic based on spherical normal fuzzy set and the main motive of this application is to select an effective antivirus mask to mitigate transmission of coronavirus by applying FFS_fYWA (or FFS_fYWG) operator based on FFS_fS.

The team of experts involve four members in a group ${\tilde{Y}}_{1}, {\tilde{Y}}_{2}, {\tilde{Y}}_{3}$ and ${\tilde{Y}}_{4}$ whose weight vectors w = (0.28, 0.22, 0.25, 0.25) ^T present their judgment for different antivirus masks x₁, x₂, x₃, x₄, where $\begin{matrix} x_{1} & = & N 95 mask, \\ x_{2} & = & FFP 1 mask, \\ x_{3} & = & Surgical mask, \\ x_{4} & = & Cloth mask, \end{matrix}$ under the parameters $\begin{matrix} \tilde{E} = {Leakage rate, Reusability, Quality of \\ raw material, Filtration efficiency} \end{matrix}$ having WV v = (0.25, 0.20, 0.25, 0.30) ^T. They give their interpretation for alternatives to their corresponding parameters in the form of FFS_fNs. Now, we utilize the proposed algorithm of our model to evaluate the best antivirus mask.

The evaluation of experts for the evaluation of antivirus masks in terms of FFS_fNs is given in Tables 4 –7, respectively. The evaluation of antivirus masks is being evaluated by four experts to give their grades in terms of FFS_fNs, given in Tables 4 –7, respectively.

The normalize DMx is not required because all parameters are of same type.

The evaluation of group members for each mask is aggregated by using Equation 1, given by $\begin{matrix} Ω_{1} = (0.8170, 0.4448), \\ Ω_{2} = (0.7724, 0.4841), \\ Ω_{3} = (0.7872, 0.4674), \\ Ω_{4} = (0.6264, 0.5038) . \end{matrix}$

The score values of aggregated value Ω_s are $S (x_{1}) = 0.46, S (x_{2}) = 0.35, S (x_{3}) = 0.39, S (x_{4}) = 0.12 .$

Rank the masks in the decreasing order of score values. The ranking order (RO) is $S (x_{1}) > S (x_{3}) > S (x_{2}) > S (x_{4}) .$ Hence, the most suitable antivirus mask is x₁ as compared to other masks.

By applying {FFS}_{f} WG operator :

Same as above.

The evaluation of group members for each mask is aggregated by using Equation 7, given by $\begin{matrix} Ω_{1} = (0.8170, 0.4108), \\ Ω_{2} = (0.7724, 0.4566), \\ Ω_{3} = (0.7872, 0.4393), \\ Ω_{4} = (0.6264, 0.4702) . \end{matrix}$

The score values of aggregated value Ω_s are $S (x_{1}) = 0.48, S (x_{2}) = 0.37, S (x_{3}) = 0.40, S (x_{4}) = 0.14 .$

Rank the masks in the decreasing order of score values. The RO is $S (x_{1}) > S (x_{3}) > S (x_{2}) > S (x_{4}) .$ Hence, the most suitable antivirus mask is x₁ as compared to other masks.

The RO of masks from both methods is shown in the Fig. 1.

Fig. 1

RO of alternatives.

The framework for the selection of most effective antivirus mask is shown in Fig. 2.

Fig. 2

Framework for the selection of an effective antivirus mask.

6.2 Comparison analysis

Here, we discuss the comparison analysis of proposed operators with existing operators to elaborate the importance of proposed model. We can see that from Tables 4-7, there are some values for which sum of MD and NMD is greater than 1 and square sum of MD and NMD is also greater than 1. Therefore, methods given in [7, 10] are fail to handle the situations.

Table 4
FFS_f matrix for x₁

Experts e ₁ e ₂ e ₃ e ₄

${\tilde{Y}}_{1}$ (0.85,0.40) (0.71,0.30) (0.65,0.22) (0.70,0.42)

${\tilde{Y}}_{2}$ (0.63,0.31) (0.80,0.52) (0.75,0.6) (0.83,0.33)

${\tilde{Y}}_{3}$ (0.9,0.50) (0.60,0.22) (0.93,0.46) (0.94,0.41)

${\tilde{Y}}_{4}$ (0.91,0.60) (0.81,0.52) (0.98,0.30) (0.78,0.53)

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{Y}}_{1}$	(0.85,0.40)	(0.71,0.30)	(0.65,0.22)	(0.70,0.42)
${\tilde{Y}}_{2}$	(0.63,0.31)	(0.80,0.52)	(0.75,0.6)	(0.83,0.33)
${\tilde{Y}}_{3}$	(0.9,0.50)	(0.60,0.22)	(0.93,0.46)	(0.94,0.41)
${\tilde{Y}}_{4}$	(0.91,0.60)	(0.81,0.52)	(0.98,0.30)	(0.78,0.53)

Table 5

FFS_f matrix for x₂

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{Y}}_{1}$	(0.80,0.40)	(0.65,0.45)	(0.51,0.23)	(0.68,0.45)
${\tilde{Y}}_{2}$	(0.64,0.42)	(0.78,0.54)	(0.71,0.65)	(0.82,0.36)
${\tilde{Y}}_{3}$	(0.85,0.58)	(0.60,0.26)	(0.86,0.46)	(0.91,0.33)
${\tilde{Y}}_{4}$	(0.86,0.60)	(0.78,0.65)	(0.90,0.42)	(0.72,0.55)

Table 6

FFS_f matrix for x₃

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{Y}}_{1}$	(0.81,0.39)	(0.65,0.43)	(0.55,0.22)	(0.69,0.44)
${\tilde{Y}}_{2}$	(0.65,0.39)	(0.78,0.53)	(0.70,0.61)	(0.82,0.35)
${\tilde{Y}}_{3}$	(0.87,0.57)	(0.61,0.25)	(0.89,0.45)	(0.93,0.30)
${\tilde{Y}}_{4}$	(0.89,0.59)	(0.80,0.60)	(0.93,0.41)	(0.72,0.55)

Table 7

FFS_f matrix for x₄

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{Y}}_{1}$	(0.70,0.51)	(0.61,0.46)	(0.49,0.25)	(0.45,0.55)
${\tilde{Y}}_{2}$	(0.62,0.47)	(0.72,0.55)	(0.69,0.60)	(0.40,0.56)
${\tilde{Y}}_{3}$	(0.75,0.51)	(0.58,0.28)	(0.71,0.50)	(0.39,0.43)
${\tilde{Y}}_{4}$	(0.69,0.57)	(0.75,0.66)	(0.76,0.46)	(0.50,0.44)

6.3 Comparison with fermatean fuzzy yager aggregation operators

This subsection discusses the comparison with FF Yager weighted average (FFYWA) [12] and FF Yager weighted geometric [12] operators. If we take the data given in Tables 4-7, then the existing model presented in [12] is fail to handle the problems but proposed theory cover all such situations. For that reason distinct parameters of FFS_fNs are aggregated by utilizing Fermatean weighted averaging operator with WV w = (0.28, 0.22, 0.25, 0.25) ^T and aggregated FFS_f matrix for different masks is given in Table 8. Using this evaluated matrix, comparison with FFYWA and FFYWG operators is given in Table 9.

Table 8
Aggregated FFS_f matrix

e ₁ e ₂ e ₃ e ₄

x₁ (0.8423,0.4787) (0.7366,0.4242) (0.8463,0.4337) (0.8184,0.4364)

x₂ (0.8008,0.5155) (0.7074,0.5107) (0.7703,0.4743) (0.7889,0.4420)

x₃ (0.8194,0.5031) (0.7153,0.4833) (0.7934,0.4529) (0.7977,0.4338)

x₄ (0.6953,0.5186) (0.6694,0.5208) (0.6727,0.4755) (0.4408,0.5019)

	e ₁	e ₂	e ₃	e ₄
x₁	(0.8423,0.4787)	(0.7366,0.4242)	(0.8463,0.4337)	(0.8184,0.4364)
x₂	(0.8008,0.5155)	(0.7074,0.5107)	(0.7703,0.4743)	(0.7889,0.4420)
x₃	(0.8194,0.5031)	(0.7153,0.4833)	(0.7934,0.4529)	(0.7977,0.4338)
x₄	(0.6953,0.5186)	(0.6694,0.5208)	(0.6727,0.4755)	(0.4408,0.5019)

Table 9

Comparison with FFYWA and FFYWG operators

Methods	S (x₁)	S (x₂)	S (x₃)	S (x₄)	RO
FFS _f YWA	0.46	0.35	0.39	0.12	x₁ > x₃ > x₂ > x₄
FFS _f YWG	0.48	0.37	0.40	0.14	x₁ > x₃ > x₂ > x₄
FFYWA	0.46	0.35	0.39	0.13	x₁ > x₃ > x₂ > x₄
FFYWG	0.47	0.35	0.39	0.13	x₁ > x₃ > x₂ > x₄

From above table, it is clear that x₁ is more effective antivirus mask as compared to others.

6.4 Comparison with fermatean fuzzy TOPSIS method

Here, we discuss the comparison with FF TOPSIS [30] method. The steps to find out the best alternative by FF TOPSIS method are:

Table 8 represents the FF decision matrix in which each entry corresponds to a FFN.

The FF positive ideal solution (FFPIS) $S^{+}$ and FF negative ideal solution (FFNIS) $S^{-}$ are given as $\begin{matrix} S^{+} & = & {(0.8423, 0.4787), (0.7366, \\ 0.4242), (0.8463, 0.4337), (0.8184, \\ 0.4364)}, \\ S^{-} & = & {(0.6953, 0.5186), (0.6694, \\ 0.5208), (0.6727, 0.4755), \\ (0.4408, 0.5019)} . \end{matrix}$

The Euclidean distance between the alternative $x_{l}$ and FFPIS $S^{+}$ together with the FFNIS $S^{-}$ are given in the Table 10.

The revised closeness degree of each alternative is given as $\begin{matrix} ς (x_{1}) = 1, ς (x_{2}) = - 0.83, \\ ς (x_{3}) = - 0.19, ς (x_{4}) = - 5.08 . \end{matrix}$

We get the following ranking list by arranging the alternatives in decreasing order with respect to $ς (L_{l})$ : $x_{1} > x_{3} > x_{2} > x_{4} .$

x₁ is the best alternative.

The results obtained from proposed operators and existing methods are shown in Fig. 3.

Table 10
Euclidean distance of alternatives from FFPIS and FFNIS

$D (x_{l}, S^{+}$ ) $D (x_{l}, S^{-}$ )

0 0.2717

0.0823 0.1932

0.0535 0.2212

0.2716 0

$D (x_{l}, S^{+}$ )	$D (x_{l}, S^{-}$ )
0	0.2717
0.0823	0.1932
0.0535	0.2212
0.2716	0

Fig. 3

Comparison with existing methods.

It is clear from Fig. 3 that the results obtained from proposed operators, FFYWA operator, FFYWG operator and FF TOPSIS method are same but FF TOPSIS method, FFYWA and FFYWG operators has no knowledge about parameterized study. The superiority of proposed theory is that it has the ability to solve real life problems by utilizing parameterized information. Therefore, developed approach can be applied for handling DM problems involving parameters.

7 Conclusions

As an extension of IFS_fS and PFS_fS, FFS_fS are more flexible to solve DM problems involving vagueness and complexity corresponding to different parameters. To solve MAGDM problems, we have combined Yager norm operators with FFS_fS. By combining these two structures, we have developed a new hybrid model to solve MAGDM problems. We have proposed four families of AOs, namely, FFS_fYWA, FFS_fYOWA, FFS_fYWG and FFS_fYOWG operators. Some basic properties of proposed operators including idempotency, boundedness, monotonicity, shift invariance and homogeneity have been elaborated. This article develops a method based on the proposed operators to handle MAGDM problem under FFS_f data. The proposed approach helps in the the selection of an effective antivirus mask in the critical situations of COVID-19. The comparison analysis with FF TOPSIS method and other operators show that the developed approach has the ability to deal with DM problems involving parameterized study which shows the superiority of proposed approach over existing study. In short, this article builds up a tool that has the rich properties of Yager AOs and flexibility of FFS_f model. However, there are some restrictions of proposed theory, it cannot applied on those DM problems where (i) cubic sum of MD and NMD is greater than 1, (ii) multi-polarity of an alternative cannot discussed in this model. Therefore, in future, we will extend our models to q-rung orthopair fuzzy set, m-polar fuzzy soft set, bipolar fuzzy soft set and neutrosophic set.

Conflict of interest

The authors declare no conflict of interest.

References

Akram

, Dudek

W.A.

and Dar

J.M.

, Pythagorean Dombi fuzzy aggregation operators with application in multi-criteria decision-making, International Journal of Intelligent Systems 34(11) (2019), 3000–3019.

Akram

, Dudek

W.A.

and Illyas

, Group decision-making based on Pythagorean fuzzy TOPSIS method, International Journal of Intelligent Systems 34 (2019), 455–1475.

Akram

, Garg

and Zahid

, Extensions of ELECTRE-I and TOPSIS methods for group decision-making under complex Pythagorean fuzzy environment, Iranian Journal of Fuzzy Systems 17(5) (2020), 147–164.

Akram

and Shahzadi

, A hybrid decision-making model under q-rung orthopair fuzzyYager aggregation operators, Granular Computing (2020), 1–15.

Akram

, Shahzadi

and Peng

, Extension of Einstein geometric operators to multi-attribute decision-making under q-rung orthopair fuzzy information, Granular Computing (2020), https://doi.org/10.1007/s41066-020-00233-3.

Ali

M.I.

, Another view on-rung orthopair fuzzy sets, International Journal of Intelligent Systems 33(11) (2018), 2139–2153.

Arora

and Garg

, Robust aggregation operators for multicriteria decision-making with intuitionistic fuzzy soft set environment, Scientia Iranica Transaction E Industrial Engineering 25(2) (2018), 931–942.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

Garg

, A novel trigonometric operation based-rung orthopair fuzzy aggregation operator and its fundamental properties, Neural Computing and Applications 32 (2020), 15077–15099.

10.

Garg

and Arora

, Generalized intuitionistic fuzzy soft power aggregation operator based on t-norm and their application in multicriteria decision-making, International Journal of Intelligent Systems 34(2) (2019), 215–246.

11.

Garg

and Chen

S.M.

, Multi-attribute group decisionmaking based on neutrality aggregation operators of-rung orthopair fuzzy sets, Information Sciences 517 (2020), 427–447.

12.

Garg

, Shahzadi

and Akram

, Decision-making analysis based on Fermatean fuzzy Yager aggregation operators with application in COVID-19 testing facility, Mathematical Problems in Engineering Article ID 7279027, (2020), 16.

13.

Garg

and Rani

, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision making process, Arabian Journal for Science and Engineering 44(3) (2019), 2679–2698.

14.

, Wang

, Zhang

and Li

, Some-rung picture fuzzy Dombi Hamy Mean operators with their application to project assessment, Mathematics 7(5) (2019), 468–.

15.

Hussain

, Ali

M.T.

, Mahmood

and Munir

, q-rung orthopair fuzzy soft average aggregation operators and their application in multi-criteria decision-making, International Journal of Intelligent Systems 35(4) (2020), 571–599.

16.

Jana

, Muhiuddin

and Pal

, Some Dombi aggregation of-rung orthopair fuzzy numbers in multiple-attribute decision-making, International Journal of Intelligent Systems 34(12) (2019), 3220–3240.

17.

Jana

and Pal

, A robust single-valued neutrosophic soft aggregation operators in multi-criteria decision-making, Symmetry 11(1) (2019), 110–.

18.

Jana

, Pal

and Wang

, A robust aggregation operator for multi-criteria decision-making method with bipolar fuzzy soft environment, Iranian Journal of Fuzzy Systems 16(6) (2019), 1–16.

19.

Joshi

B.P.

and Gegov

, Confidence levels-rung orthopair fuzzy aggregation operators and its applications to MCDM problems, International Journal of Intelligent Systems 35(1) (2020), 125–149.

20.

Liang

, Darko

A.P.

, Xu

and Quan

, The linear assignment method for multicriteria group decision-making based on interval-valued Pythagorean fuzzy Bonferroni mean, International Journal of Intelligent Systems 33(11) (2018), 2101–2138.

21.

Liu

, Akram

and Sattar

, Extensions of prioritized weighted aggregation operators for decision-making under complex q-rung orthopair fuzzy information, Journal of Intelligent and Fuzzy System (2020), 1–25. DOI: 10.3233/JIFS-200789

22.

Liu

, Shahzadi

and Akram

, Specific types of-rung picture fuzzy Yager aggregation operators for decisionmaking, International Journal of Computational Intelligence Systems 13(1) (2020), 1072–1091.

23.

Liu

and Wang

, Some-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision-making, International Journal of Intelligent Systems 33(2) (2018), 259–280.

24.

Maji

P.K.

, Biswas

R.K.

and Roy

, Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 589–602.

25.

Maji

P.K.

, Biswas

R.K.

and Roy

, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 677–692.

26.

Molodtsov

, Soft set theory Ufirst results, Computers and Mathematics with Applications 37(4-5) (1999), 19–31.

27.

Peng

X.D.

, Yang

, Song

J.P.

and Jiang

, Pythagorean fuzzy soft set and its application, Computer Engineering 41(7) (2015), 224–229.

28.

Senapati

and Yager

R.R.

, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision-making methods, Engineering Applications of Artificial Intelligence 85 (2019), 112–121.

29.

Senapati

and Yager

R.R.

, Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision-making, Informatica 30(2) (2019), 391–412.

30.

Senapati

and Yager

R.R.

, Fermatean fuzzy sets, Journal of Ambient Intelligence and Humanized Computing 11(2) (2020), 663–674.

31.

Shahzadi

, Akram

and Al-Kenani

A.N.

, Decision-making approach under Pythagorean fuzzy Yager weighted operators, Mathematics 8(1) (2020), 70–.

32.

Wang

and Li

, Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision-making, International Journal of Intelligent Systems 35(1) (2020), 150–183.

33.

Wang

and Liu

, Intuitionistic fuzzy geometric aggregation operators based on Einstein operations, International Journal of Intelligent Systems 26(11) (2011), 1049–1075.

34.

Wei

and Lu

, Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision-making, International Journal of Intelligent Systems 33(5) (2018), 1043–1070.

35.

Wei

and Lu

, Pythagorean fuzzy power aggregation operators in multiple attribute decision-making, International Journal of Intelligent Systems 33(1) (2018), 169–186.

36.

, Dong

, Qin

and Pedrycz

, Flexible linguistic expressions and consensus reaching with accurate constraints in group decision-making, IEEE Transactions on Cybernetics 50(6) (2019), 2488–2501.

37.

, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15(6) (2007), 1179–1187.

38.

and Hu

, Projection models for intuitionistic fuzzy multiple attribute decision-making, International Journal of Information Technology and Decision-Making 9(02) (2010), 267–280.

39.

and Yager

R.R.

, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems 35(4) (2006), 417–433.

40.

Yager

R.R.

, Pythagorean fuzzy subsets, In 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS). IEEE (2013), 57–61.

41.

Yager

R.R.

, Generalized orthopair fuzzy sets, IEEE Transactions on Fuzzy Systems 25(5) (2016), 1222–1230.

42.

Yager

R.R.

, Alajlan

and Bazi

, Aspects of generalized orthopair fuzzy sets, International Journal of Intelligent Systems 33(11) (2018), 2154–2174.

43.

Yang

, Li

, Garg

and Qi

, Decision support algorithm for selecting an antivirus mask over COVID-19 pandemic under spherical normal fuzzy environment, International Journal of Environmental Research and Public Health 17(10) (2020), 3407–.

44.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

45.

Zhang

, Li

and Liang

, Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes, Information Sciences 507 (2020), 665–683.

46.

Zeng

, Chen

and Li

, A hybrid method for Pythagorean fuzzy multiple criteria decision making, International Journal of Information Technology and Decision-Making 15(02) (2016), 403–422.

Group decision-making for the selection of an antivirus mask under fermatean fuzzy soft information

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Fermatean fuzzy soft set

Table 1 Tabular representation (TR) of FFS f S V e 1 e 2 e 3 e 4 x 1 (0.81,0.69) (0.90,0.60) (0.70,0.56) (0.67,0.45) x 2 (0.78,0.51) (0.66,0.23) (0.55,0.11) (0.77,0.34) x 3 (0.88,0.66) (0.91, 0.61) (0.68,0.21) (0.57,0.31) x 4 (0.71,0.55) (0.67,0.41) (0.87,0.56) (0.78,0.21)

4.1 Fermatean fuzzy soft Yager weighted average operators

5.1 Fermatean fuzzy soft Yager weighted geometric operators

6.1 Selection of an effective antivirus mask

Table 4 FFS f matrix for x1 Experts e 1 e 2 e 3 e 4 Y ˜ 1 (0.85,0.40) (0.71,0.30) (0.65,0.22) (0.70,0.42) Y ˜ 2 (0.63,0.31) (0.80,0.52) (0.75,0.6) (0.83,0.33) Y ˜ 3 (0.9,0.50) (0.60,0.22) (0.93,0.46) (0.94,0.41) Y ˜ 4 (0.91,0.60) (0.81,0.52) (0.98,0.30) (0.78,0.53)

Table 10 Euclidean distance of alternatives from FFPIS and FFNIS D ( x l , S + ) D ( x l , S - ) 0 0.2717 0.0823 0.1932 0.0535 0.2212 0.2716 0

Conflict of interest

References

Table 1
Tabular representation (TR) of FFS_fS

$V$ e ₁ e ₂ e ₃ e ₄

x ₁ (0.81,0.69) (0.90,0.60) (0.70,0.56) (0.67,0.45)

x ₂ (0.78,0.51) (0.66,0.23) (0.55,0.11) (0.77,0.34)

x ₃ (0.88,0.66) (0.91, 0.61) (0.68,0.21) (0.57,0.31)

x ₄ (0.71,0.55) (0.67,0.41) (0.87,0.56) (0.78,0.21)

Table 10
Euclidean distance of alternatives from FFPIS and FFNIS

$D (x_{l}, S^{+}$ ) $D (x_{l}, S^{-}$ )

0 0.2717

0.0823 0.1932

0.0535 0.2212

0.2716 0