A hybrid decision making method based on q -rung orthopair fuzzy soft information

Abstract

Soft set (S_fS) theory is a basic tool to handle vague information with parameterized study during the process as compared to fuzzy as well as q-rung orthopair fuzzy theory. This research article is devoted to establish some general aggregation operators (AOs), based on Yager’s norm operations, to cumulate the q-rung orthopair fuzzy soft data in decision making environments. In this article, the valuable properties of q-rung orthopair fuzzy soft set (q - ROFS_fS) are merged with the Yager operator to propose four new operators, namely, q-rung orthopair fuzzy soft Yager weighted average (q - ROFS_fYWA), q-rung orthopair fuzzy soft Yager ordered weighted average (q - ROFS_fYOWA), q-rung orthopair fuzzy soft Yager weighted geometric (q - ROFS_fYWG) and q-rung orthopair fuzzy soft Yager ordered weighted geometric (q - ROFS_fYOWG) operators. The dominant properties of proposed operators are elaborated. To emphasize the importance of proposed operators, a multi-attribute group decision making (MAGDM) strategy is presented along with an application in medical diagnosis. The comparative study shows superiorities of the proposed operators and limitations of the existing operators. The comparison with Pythagorean fuzzy TOPSIS (PF-TOSIS) method shows that PF-TOPSIS method cannot deal with data involving parametric study but developed operators have the ability to deal with decision making problems using parameterized information.

Keywords

Yager operators aggregation operators TOPSIS method

1 Introduction

In our daily life problem, we face uncertainties in making right decisions. Several decision making methods have been investigated in medical field. Decision making is a technique of nominating the best option among various objects. That’s why decision making performs an important role in the human life problems. The life style can be changed by a good decision. First of all, a decision maker tries to know about the restrictions, advantages and qualities of each alternative and then he/she finds out the final decision. To handle with such situations, Zadeh [45] proposed the prominent model of fuzzy set (FS). FS brought the revolution in different fields of science. In FS, a membership degree (MD) within the interval [0, 1] is assigned to each object. Atanassov [9] proposed the intuitionistic fuzzy set (IFS) which is characterized by both the MD μ and the nonmembership degree (NMD) ν within the interval [0, 1], respectively such that the summation of these two values is less than or equal to 1. From last few years, different researchers developed various operators on the existing ideas in which the best one idea is the AOs. Yager [40] introduced the dominant concept of ordered weighted (OW) AOs. The concepts of novel geometric AOs for IFS in multi attribute decision making (MADM) were presented by Xu and Yager [39]. The study of intuitionistic fuzzy (IF) AOs was given by Xu [37]. Wang and Liu [36] discussed the IF Einstein weighted geometric (IFEWG) operators. Li [21] studied the generalized OW averaging operators for IF data. Garg and Rani [13] initiated the idea of complex IF AOs and applied this concept in MADM problems. IFS is a robust concept and many researchers have done work on it. However, there are some drawbacks in the model of IFS, due to which this theory fails to handle the problems.

To overcome such difficulties, Yager [42] discussed the model of Pythagorean fuzzy set (PFS). This model has the quality to replace the constraint of IFS with the condition μ² + ν² ≤ 1. Yager [41] proposed PF arithmetic and geometric operators. Peng and Yang [28] developed interval-valued PF AOs. A hybrid structure for PF MADM was proposed by Zeng et al. [46]. Akram et al. [1] worked for the development of Pythagorean Dombi fuzzy AOs with applications. Shahzadi et al. [31] developed the theory of Yager AOs under PF data. However, the PFS has also some limitations, if MD of an element is 0.9 and NMD is 0.6, then sum of square of these values is greater than 1 and PFS cannot deal such situations. That’s why researchers needed some new extensive model for such type of shortcomings.

Yager [43] investigated new notion of q-rung orthopair fuzzy set (q-ROFS) in which μ^q + ν^q ≤ 1. In decision making problems, q-ROFS is more capable than IFS and PFS, as shown in Fig. 1. All IF numbers (IFNs) and PF numbers (PFNs) are q-rung orthopair fuzzy numbers (q-ROFNs) but converse is not true. Jana et al. [17] studied q-rung orthopair fuzzy Dombi AOs. Joshi and Gegov [20] developed the confidence levels q-rung orthopair fuzzy AOs. Akram and Shahzadi [4] introduced Yager AOs for q-ROFS. Garg [10] defined some trigonometric operations laws for q-ROFS. The concept of neutrality AOs of q-ROFS was studied by Garg and Chen [12]. As FS, IFS, PFS, q-ROFS 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.

Fig. 1

Comparison of q-ROFN, PFN and IFN spaces.

In 1999, a pioneer model of S_fS was proposed by Molodstov [27] which maintains the parametrization tool to handle with vague data. Many ideas are combined with S_fS and established new theories. Maji [25, 26] discussed the models of fuzzy S_fS (FS_fS) and intuitionistic fuzzy S_fS (IFS_fS). The intuitionistic fuzzy soft (IFS_f) AOs were studied by Arora and Garg [8]. Garg and Arora [11] discussed the generalized IFS_f power AOs and their application in MADM. q-rung orthopair fuzzy soft (q - ROFS_f) average operators were initiated by Hussain et al. [16]. Zhang et al. [47] developed the theory of consensus reaching for social network group decision making. Zhang et al. [48] discussed the concept of consistency improvement for fuzzy preference relations with self-confidence. For other terminologies not discussed in the paper, the readers are suggested to [3 , 44].

The motivations of this article are described as:

Advantages of q - ROFS_fS: q - ROFS_fS, a remarkable extension of IFS_fS and Pythagorean fuzzy S_fS (PFS_fS), permits modeling of the situations with higher more generality than IFS_fS and PFS_fS, because it still applies in cases where μ + ν ≤ 1 and μ² + ν² ≤ 1 but it satisfies μ^q + ν^q ≤ 1 corresponding to different parameters.

Importance of S_fS: S_fS theory handles the uncertainty problems with parameterized information.

Logics behind the use of Yager AOs: Yager AOs are the simplest and quite creative approach for dealing with decision making affairs. Basically, this article is devoted to Yager AOs in q - ROFS_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 MADM problems in q - ROFS_f data.

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

The contributions of this article are outlined as:

The idea of Yager AOs is extended to q - ROFS_f numbers (q - ROFS_fNs) and fundamental properties are discussed.

To handle complex realistic problems with q - ROFS_f data, an algorithm is developed. The proposed algorithm is supported by an illustrative example for the diagnosis of the illness of patients.

The comparison analysis with existing operators shows the superiority of proposed model.

The remaining paper is composed as follows: In Section 2, some basic definitions are discussed. In Section 3, we study Yager operations for q - ROFS_fNs, q - ROFS_fYWA operator and discuss some results in the light of these operations. Section 4 presents q - ROFS_fYOWA operator and some properties related to them. In Sections 5 and 6, we discuss q - ROFS_fYWG and q - ROFS_fYOWG operators to aggregate the q - ROFS_fNs. Section 7 gives an algorithm for MAGDM and discuss a numerical example in the field of medical diagnosis based on q - ROFS_fNs. Section 8 gives the comparison analysis with existing operators to show the superiority and validity of proposed theory. In Section 9, we have concluded results about proposed theory.

2 Preliminaries

In this section, we recall some basic definitions.

Definition 2.1. [29] Let $U$ be a universe of discourse and $\tilde{W}$ be the set of all parameters, $\tilde{E} \subseteq \tilde{W}$ . $P (U)$ represents the set of all PFSs over $U$ . $(\tilde{J}, \tilde{E})$ is called a PFS_fS over $U$ , where $\tilde{J}$ is a function given by $\tilde{J} : \tilde{E} \to P (U)$ , which is $\tilde{J} (e_{p}) = {≺ x_{t}, f_{p} (x_{t}), g_{p} (x_{t}) ≻} | x_{t} \in U},$ with condition $0 \leq (f_{p} (x_{t}))^{2} + (g_{p} (x_{t}))^{2} \leq 1$ .

Definition 2.2. [16] Let $U$ be a universe of discourse and $\tilde{W}$ be the set of all parameters, $\tilde{E} \subseteq \tilde{W}$ . $P (U)$ represents the set of all q-ROFSs over $U$ . $(τ, \tilde{E})$ is called a q - ROFS_fS over $U$ , where τ is a function given by $τ : \tilde{E} \to P (U)$ , which is $τ (e_{p}) = {≺ x_{t}, f_{p} (x_{t}), g_{p} (x_{t}) ≻} | x_{t} \in U},$ with condition $0 \leq (f_{p} (x_{t}))^{q} + (g_{p} (x_{t}))^{q} \leq 1$ . $π_{τ_{e_{t p}}} = \sqrt[q]{1 - ((f_{t p})^{q} + (g_{t p})^{q})}$ is the hesitancy degree, where $f_{t p}$ denotes $f_{p} (x_{t})$ and $g_{t p}$ denotes $g_{p} (x_{t})$ .

Definition 2.3. [41] For any two real numbers x and y, Yager’s t-norms and Yager’s t-conorms are given as: $T (x, y) = 1 - min (1, ((1 - x)^{ϑ} + (1 - y)^{ϑ})^{\frac{1}{ϑ}}),$ (1) $T^{'} (x, y) = min (1, (x^{ϑ} + y^{ϑ})^{\frac{1}{ϑ}}), ϑ \in (0, \infty) .$ (2)

3 q-rung orthopair fuzzy soft Yager weighted average operators

In this section, we develop Yager t-norm and t-conorm for q - ROFS_fNs and Yager weighted average operators for q - ROFS_fNs.

Definition 3.1. Let $τ_{e_{1 p}} = (f_{1 p}, g_{1 p}) (p = 1, 2),$ and τ = (f, g) be three q - ROFS_fNs and λ > 0, ϑ ∈ (0, ∞). Then, Yager norm operations for q - ROFS_fNs are

$\begin{matrix} 1 . & τ_{e_{11}} \oplus τ_{e_{12}} = (\sqrt[q]{min (1, (f_{11}^{q ϑ} + f_{12}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, ((1 - g_{11}^{q})^{ϑ} + (1 - g_{12}^{q})^{ϑ})^{\frac{1}{ϑ}})}), \\ 2 . & τ_{e_{11}} \otimes τ_{e_{12}} = (\sqrt[q]{1 - min (1, ((1 - f_{11}^{q})^{ϑ} + (1 - f_{12}^{q})^{ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{min (1, (g_{11}^{q ϑ} + g_{12}^{q ϑ})^{\frac{1}{ϑ}})}), \end{matrix}$ $\begin{matrix} 3 . & λ τ = (\sqrt[q]{min (1, (λ f^{q ϑ})^{\frac{1}{ϑ}})}, \sqrt[q]{1 - min (1, (λ (1 - g^{q})^{ϑ})^{\frac{1}{ϑ}})}), \\ 4 . & τ^{λ} = (\sqrt[q]{1 - min (1, (λ (1 - f^{q})^{ϑ})^{\frac{1}{ϑ}})}, \sqrt[q]{min (1, (λ g^{q ϑ})^{\frac{1}{ϑ}})}) . \end{matrix}$

Definition 3.2. The score function and accuracy function for q - ROFS_fNs $τ_{e_{t p}} = (f_{t p}, g_{t p})$ are represented by $S (τ_{e_{t p}}) = f_{t p}^{q} - g_{t p}^{q}, S (τ_{e_{t p}}) \in [- 1, 1]$ and $H (τ_{e_{t p}}) = f_{t p}^{q} + g_{t p}^{q}, H (τ_{e_{t p}}) \in [0, 1]$ respectively.

Definition 3.3. Let τ_{e
₁₁} = (f₁₁, g₁₁) and τ_{e
₁₂} = (f₁₂, g₁₂) be two q - ROFS_fNs,

if S (τ_e
₁₁) > S (τ_e
₁₂), then τ_e
₁₁≻ τ_e
₁₂ ;

if S (τ_e
₁₁) < S (τ_e
₁₂), then τ_e
₁₁≺ τ_e
₁₂ ;

if S (τ_e
₁₁) = S (τ_e
₁₂), then

if H (τ_{e
₁₁}) > H (τ_{e
₁₂}), then τ_{e
₁₁}≻ τ_{e
₁₂} ;

if H (τ_{e
₁₁}) < H (τ_{e
₁₂}), then τ_{e
₁₁}≺ τ_{e
₁₂} ;

if H (τ_{e
₁₁}) = H (τ_{e
₁₂}), then τ_{e
₁₁} ∼ τ_{e
₁₂} .

Definition 3.4. Let $τ_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a collection of q - ROFS_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 q - ROFS_fYWA operator is a function q - ROFS_fYWA: $M^{n} \to M$ , where $M$ is the set of q - ROFS_fNs. $\begin{matrix} q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \\ = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} (w_{t} τ_{e_{t p}})) . \end{matrix}$

Theorem 3.1. The aggregated value by utilizing q - ROFS_fYWA operator is given by $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$ $\begin{matrix} = (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ (3)

Proof. Applying mathematical induction to prove it.

(i) When $n = 2, m = 2$ , q - ROFS_fYWA (τ_{e
₁₁}, τ_{e
₁₂}) $\begin{matrix} = & \oplus_{p = 1}^{2} v_{p} (\oplus_{t = 1}^{2} w_{t} τ_{e_{t p}}) \\ = & v_{1} (\oplus_{t = 1}^{2} w_{t} τ_{e_{t 1}}) \oplus v_{2} (\oplus_{t = 1}^{2} w_{t} τ_{e_{t 2}}) \\ = & v_{1} (w_{1} τ_{e_{11}} \oplus w_{2} τ_{e_{21}}) \oplus v_{2} (w_{1} τ_{e_{12}} \oplus w_{2} τ_{e_{22}}) \\ = & v_{1} {(\sqrt[q]{min (1, (w_{1} f_{11}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (w_{1} (1 - g_{11}^{q})^{ϑ})^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[q]{min (1, (w_{2} f_{21}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (w_{2} (1 - g_{21}^{q})^{ϑ})^{\frac{1}{ϑ}})})} \oplus \\ v_{2} {(\sqrt[q]{min (1, (w_{1} f_{12}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (w_{1} (1 - g_{12}^{q})^{ϑ})^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[q]{min (1, (w_{2} f_{22}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (w_{2} (1 - g_{22}^{q})^{ϑ})^{\frac{1}{ϑ}})})} \\ = & v_{1} (\sqrt[q]{min (1, (\sum_{t = 1}^{2} w_{t} f_{t 1}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{t = 1}^{2} w_{t} (1 - g_{t 1}^{q})^{ϑ})^{\frac{1}{ϑ}})}) \oplus \\ v_{2} (\sqrt[q]{min (1, (\sum_{t = 1}^{2} w_{t} f_{t 2}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{t = 1}^{2} w_{t} (1 - g_{t 2}^{q})^{ϑ})^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, (v_{1} \sum_{t = 1}^{2} w_{t} f_{t 1}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (v_{1} \sum_{t = 1}^{2} w_{t} (1 - g_{t 1}^{q})^{ϑ})^{\frac{1}{ϑ}})}) \oplus \\ (\sqrt[q]{min (1, (v_{2} \sum_{t = 1}^{2} w_{t} f_{t 2}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (v_{2} \sum_{t = 1}^{2} w_{t} (1 - g_{t 2}^{q})^{ϑ})^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{2} v_{p} (\sum_{t = 1}^{2} w_{t} f_{t p}^{q ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{2} v_{p} (\sum_{t = 1}^{2} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ Hence, result is true for $n = 2, m = 2$ .

Suppose Equation 3 is true for $n = k_{1}, m = k_{2}$ , $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{k_{1} k_{2}}})$ $\begin{matrix} = & (\sqrt[q]{min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} f_{t p}^{q ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ Now for $n = k_{1} + 1, m = k_{2} + 1$ , $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{k_{1} k_{2}}})$

$\begin{matrix} = & {\oplus_{p = 1}^{k_{2}} v_{p} (\oplus_{t = 1}^{k_{1}} (w_{t} τ_{e_{t p}}))} \oplus v_{(k_{2} + 1)} (w_{(k_{1} + 1)} τ_{e_{(k_{1} + 1) (k_{2} + 1)}}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} f_{t p}^{q ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{k_{2}} v_{p} (\sum_{t = 1}^{k_{1}} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus v_{(k_{2} + 1)} (w_{(k_{1} + 1)} τ_{e_{(k_{1} + 1) (k_{2} + 1)}}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{k_{2} + 1} v_{p} (\sum_{t = 1}^{k_{1} + 1} w_{t} f_{t p}^{q ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{k_{2} + 1} v_{p} (\sum_{t = 1}^{k_{1} + 1} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$

Hence, Equation 3 is true for $n = k_{1} + 1, m = k_{2} + 1$ . Therefore, by mathematical induction the Equation 3 is true, $\forall m, n \geq 1$ .□

Remark 3.1. (a) For q = 1, q - ROFS_fYWA operator becomes intuitionistic fuzzy soft Yager weighted average (IFS_fYWA) operator.

(b) For q = 2, q - ROFS_fYWA operator becomes Pythagorean fuzzy soft Yager weighted average (PFS_fYWA) operator.

(c) For only one parameter, i.e., e₁, q - ROFS_fYWA operator becomes q-rung orthopair fuzzy Yager weighted average (q-ROFYWA) operator.

From Remark 3.1, we conclude that q - ROFS_fYWA operator is generalized case of IFS_fYWA, PFS_fYWA and q-ROFYWA operators.

Example 3.1. Suppose Mr. $X$ wants to purchase a watch from a set of four watches in the domain set $U = {x_{1}, x_{2}, x_{3}, x_{4}}$ and let $\tilde{E} = {e_{1}, e_{2}, e_{3}, e_{4}}$ be the set of parameters for the selection of a watch, i.e., $e_{p} (p = 1, 2, 3, 4)$ where e₁ =expensive, e₂ =beautiful, e₃ =synthetic sapphire crystal and e₄ =water resistance. Suppose the WVs for experts $x_{t}$ and parameters $e_{p}$ are w = (0.34, 0.24, 0.20, 0.22) ^T and v = (0.25, 0.20, 0.30, 0.25) ^T, respectively. By applying Equation 3, q - ROFS_fYWA (τ_{e
₁₁}, τ_{e
₁₂}, ⋯ , τ_{e
₄₄}) $\begin{matrix} = & (\sqrt[3]{min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = & (0.82, 0.41) . \end{matrix}$

Example 3.2. Suppose Mr. $X$ wants to purchase a car from a set of four cars in the domain set $U = {x_{1}, x_{2}, x_{3}, x_{4}}$ 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 3, q - ROFS_fYWA (τ_{e
₁₁}, τ_{e
₁₂}, ⋯ , τ_{e
₄₄}) $\begin{matrix} = & (\sqrt[3]{min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = & (0.77, 0.45) . \end{matrix}$

Property 3.1. (Idempotency) If all $τ_{e_{t p}} = L_{e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical, then $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) = L_{e} .$

Proof. As $τ_{e_{t p}} = L_{e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ , then $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$

$\begin{matrix} = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - c^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, (b^{q ϑ})^{\frac{1}{ϑ}})}, \sqrt[q]{1 - min (1, ((1 - c^{q})^{ϑ})^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, b^{q})}, \sqrt[q]{1 - min (1, (1 - c^{q}))}) \\ = & (b, c) . \end{matrix}$

□

Property 3.2. (Boundedness) Let $τ_{e_{t p}} = (f_{t p}, g_{t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a family of q - ROFS_fNs and $τ_{e_{t p}}^{-} = (min_{p} min_{t} {f_{t p}}, max_{p} max_{t} {g_{t p}})$ and $τ_{e_{t p}}^{+} = (max_{p} max_{t} {f_{t p}}, min_{p} min_{t} {g_{t p}})$ , $τ_{e_{t p}}^{-} \leq q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq τ_{e_{t p}}^{+} .$

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

Property 3.3. (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 q - ROFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ , then $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq q - ROF S_{f} YWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Proof. Consider $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) = (I, J)$ and $q - {ROFS}_{f} YWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) = (I^{'}, J^{'}) .$ First, we show that I ≤ I′. As $f_{t p} \leq b_{t p}, f_{t p}^{q} \leq b_{t p}^{q} .$ Moreover, $\begin{matrix} \sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ}) & \leq & \sum_{t = 1}^{n} (w_{t} b_{t p}^{q ϑ}) \\ \sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})) & \leq & \sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{q ϑ})) \\ (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}} & \leq & (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{q ϑ})))^{\frac{1}{ϑ}} \\ min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}}) & \leq & min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{q ϑ})))^{\frac{1}{ϑ}}) \end{matrix}$ This implies $\begin{matrix} \sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})} \\ \leq \sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b_{t p}^{q ϑ})))^{\frac{1}{ϑ}})} . \end{matrix}$ Therefore, I ≤ I′. Similarly we can show that J ≥ J′. Hence, $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq q - {ROFS}_{f} YWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$ □ Property 3.4. (Shift Invariance) If $L_{e} = (b, c)$ is another q - ROFS_fN, then $q - {ROFS}_{f} YWA (τ_{e_{11}} \oplus L_{e}, τ_{e_{12}} \oplus L_{e}, \dots, τ_{e_{n m}} \oplus L_{e}) = q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \oplus L_{e} .$

Proof. Since $L_{e} = (b, c)$ and $τ_{e_{t p}} = (f_{t p}, g_{t p})$ are q - ROFS_fNs, so

$\begin{matrix} τ_{e_{11}} \oplus L_{e} & = & (\sqrt[q]{min (1, (f_{11}^{q ϑ} + b^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, ((1 - g_{11}^{q})^{ϑ} + (1 - c^{q})^{ϑ})^{\frac{1}{ϑ}})}) . \end{matrix}$

Therefore, $q - {ROFS}_{f} YWA (τ_{e_{11}} \oplus L_{e}, τ_{e_{12}} \oplus L_{e}, \dots, τ_{e_{n m}} \oplus L_{e})$ $\begin{matrix} = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} w_{t} (τ_{e_{t p}} \oplus L_{e})) \\ = (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (f_{t p}^{q ϑ} + p^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} ((1 - g_{t p}^{q})^{ϑ} + (1 - c^{q})^{ϑ})))^{\frac{1}{ϑ}}})) . \end{matrix}$ (4) $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \oplus L_{e}$ $\begin{matrix} = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \oplus (b, c) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[q]{min (1, b^{q})}, \sqrt[q]{1 - min (1, (1 - c^{q}))}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[q]{min (1, (b^{q ϑ})^{\frac{1}{ϑ}})}, \sqrt[q]{1 - min (1, ((1 - c^{q})^{ϑ})^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ \oplus (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} b^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - c^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (f_{t p}^{q ϑ} + p^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} ((1 - g_{t p}^{q})^{ϑ} + (1 - c^{q})^{ϑ})))^{\frac{1}{ϑ}}})) . \end{matrix}$ (5)

From Equations 30 and 31, $q - {ROFS}_{f} YWA (τ_{e_{11}} \oplus L_{e}, τ_{e_{12}} \oplus L_{e}, \dots, τ_{e_{n m}} \oplus L_{e}) = q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \oplus L_{e} .$ □ Property 3.5. (Homogeneity) Let λ ≥ 0 then $q - {ROFS}_{f} YWA (λ τ_{e_{11}}, λ τ_{e_{12}}, \dots, λ τ_{e_{n m}}) = λ q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) .$

Proof. As $\begin{matrix} λ τ_{e_{t p}} & = & (\sqrt[q]{min (1, (λ f_{t p}^{q ϑ})^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (λ (1 - g_{t p}^{q})^{ϑ})^{\frac{1}{ϑ}})}) . \end{matrix}$ Now $q - {ROFS}_{f} YWA (λ τ_{e_{11}}, λ τ_{e_{12}}, \dots, λ τ_{e_{n m}})$ $\begin{matrix} = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (λ f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (λ (1 - g_{t p}^{q})^{ϑ})))^{\frac{1}{ϑ}})}) . \end{matrix}$ (6) and $q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$ $\begin{matrix} = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ $λ q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$ $\begin{matrix} = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (λ f_{t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (λ (1 - g_{t p}^{q})^{ϑ})))^{\frac{1}{ϑ}})}) . \end{matrix}$ (7)

From Equations 32 and 33, $q - {ROFS}_{f} YWA (λ τ_{e_{11}}, λ τ_{e_{12}}, \dots, λ τ_{e_{n m}}) = λ q - {ROFS}_{f} YWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) .$ □

4 q-rung orthopair fuzzy soft Yager ordered weighted average operators

q - ROFS_fYWA operator only weighted the q - ROFS_fN′s values. But q - ROFS_fYOWA operator weights the ordered positions via scoring the q - ROFS_f values rather than weighting the q - ROFS_f values themselves. In this section, we develop q - ROFS_fYOWA operator and related properties.

Definition 4.1. Let $τ_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a family of q - ROFS_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 q - ROFS_fYOWA operator is a function q - ROFS_fYOWA: $M^{n} \to M$ $\begin{matrix} q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \\ = \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} (w_{t} τ_{σ e_{t p}})) . \end{matrix}$

Theorem 4.1. The aggregated value by utilizing q - ROFS_fYOWA operator is given by $q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$ $\begin{matrix} = & \oplus_{p = 1}^{m} v_{p} (\oplus_{t = 1}^{n} (w_{t} τ_{σ e_{t p}})) \\ = & (\sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{σ t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{σ t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) . \end{matrix}$ (8) where $τ_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$ , is the permutation of $t$ th and $p$ th largest object of the collection of $t \times p$ q - ROFS_fNs $τ_{e_{t p}} = (f_{t p}, g_{t p}) .$

Proof. It is similar to Theorem 3.1.□

Remark 4.1. (a) For q = 1, q - ROFS_fYOWA operator becomes intuitionistic fuzzy soft Yager ordered weighted average (IFS_fYOWA) operator.

(b) For q = 2, q - ROFS_fYOWA operator becomes Pythagorean fuzzy soft Yager ordered weighted average (PFS_fYOWA) operator.

(c) For only one parameter, i.e., e₁, q - ROFS_fYOWA operator becomes q-rung orthopair fuzzy Yager ordered weighted average (q-ROFYOWA) operator.

Therefore, IFS_fYOWA, PFS_fYOWA and q-ROFYOWA operators are special cases of q - ROFS_fYOWA operator.

Example 4.1. Consider the collections q - ROFS_fNs of Example 3.2 as given in Table 2, by using definition of score function, the tabular representation of $τ_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$ is given in Table 3. By using Equation 8, q - ROFS_fYOWA (τ_{e
₁₁}, τ_{e
₁₂}, ⋯ , τ_{e
₄₄}) $\begin{matrix} = & (\sqrt[3]{min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} (w_{t} f_{σ t p}^{q ϑ})))^{\frac{1}{ϑ}})}, \\ \sqrt[3]{1 - min (1, (\sum_{p = 1}^{4} v_{p} (\sum_{t = 1}^{4} w_{t} (1 - g_{σ t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}) \\ = & (0.82, 0.40) . \end{matrix}$ We give some properties without their proofs.

Table 1

Tabular representation of q - ROFS_fNs

$U$	e ₁	e ₂	e ₃	e ₄
x ₁	(0.7, 0.40)	(0.96, 0.44)	(0.67, 0.40)	(0.91, 0.43)
x ₂	(0.91, 0.50)	(0.62, 0.11)	(0.44, 0.20)	(0.89, 0.11)
x ₃	(0.81, 0.62)	(0.5, 0.35)	(0.69, 0.51)	(0.65, 0.34)
x ₄	(0.7, 0.32)	(0.81, 0.40)	(0.8, 0.61)	(0.6, 0.24)

Table 2

Tabular representation of q - ROFS_fNs

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

Table 3

Tabular representation of q - ROFS_fNs $τ_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$

$U$	e ₁	e ₂	e ₃	e ₄
x ₁	(0.91, 0.50)	(0.96, 0.44)	(0.8, 0.61)	(0.89, 0.11)
x ₂	(0.70, 0.32)	(0.81, 0.40)	(0.67, 0.40)	(0.91, 0.43)
x ₃	(0.81, 0.62)	(0.62, 0.11)	(0.69, 0.51)	(0.65, 0.34)
x ₄	(0.70, 0.40)	(0.50, 0.35)	(0.44, 0.20)	(0.60, 0.24)

Property 4.1. (Idempotency) If all $τ_{e_{t p}} = L_{σ e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical, then $q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) = L_{σ e} .$

Proof. It is similar to Property 3.1.□

Property 4.2. (Boundedness) Let $τ_{σ e_{t p}} = (f_{σ t p}, g_{σ t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a collection of q - ROFS_fNs and $τ_{σ e_{t p}}^{-} = (min_{p} min_{t} {f_{σ t p}}, max_{p} max_{t} {g_{σ t p}})$ and $τ_{σ e_{t p}}^{+} = (max_{p} max_{t} {f_{σ t p}}, min_{p} min_{t} {g_{σ t p}})$ ,

$τ_{σ e_{t p}}^{-} \leq q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq τ_{σ e_{t p}}^{+} .$

Proof. It is similar to Property 3.2.□

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 q - ROFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ , then $q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq q - {ROFS}_{f} YOWA (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Proof. It is similar to Property 3.3.□

Property 4.4. (Shift Invariance) If $L_{e} = (b, c)$ is another q - ROFS_fN, then $q - {ROFS}_{f} YOWA (τ_{e_{11}} \oplus L_{e}, τ_{e_{12}} \oplus L_{e}, \dots, τ_{e_{n m}} \oplus L_{e}) = q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \oplus L_{e} .$

Proof. It is similar to Property 3.4.□

Property 4.5. (Homogeneity) Let λ ≥ 0 then $q - {ROFS}_{f} YOWA (λ τ_{e_{11}}, λ τ_{e_{12}}, \dots, λ τ_{e_{n m}}) = λ q - {ROFS}_{f} YOWA (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) .$

Proof. It is similar to Property 3.5.□

5 q-rung orthopair fuzzy soft Yager weighted geometric operators

In this section, we develop Yager weighted geometric operators for q - ROFS_fNs.

Definition 5.1. Let $τ_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a family of q - ROFS_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 q - ROFS_fYWG operator is a function q - ROFS_fYWG: $M^{n} \to M$ $\begin{matrix} q - {ROFS}_{f} YWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \\ = ⨂_{p = 1}^{m} (⨂_{t = 1}^{n} τ_{e_{t p}}^{w_{t}})^{v_{p}} . \end{matrix}$

Theorem 5.1. The aggregated value by utilizing q - ROFS_fYWG operator is given by $q - {ROFS}_{f} YWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$ $\begin{matrix} = & (\sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - f_{t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} g_{t p}^{q ϑ})))^{\frac{1}{ϑ}})},) . \end{matrix}$ (9)

Proof. It is similar to Theorem 3.1.□

Remark 5.1. (a) For q = 1, q - ROFS_fYWG operator reduces to intuitionistic fuzzy soft Yager weighted geometric (IFS_fYWG) operator.

(b) For q = 2, q - ROFS_fYWG operator reduces to Pythagorean fuzzy soft Yager weighted geometric (PFS_fYWG) operator.

(c) If parameter is only one, i.e., e₁, then q - ROFS_fYWG operator reduces to q-rung orthopair fuzzy Yager weighted geometric (q-ROFYWG) operator.

It is clear from Remark 5.1 that q - ROFS_fYWG operator is generalized case of IFS_fYWG, PFS_fYWG and q-ROFYWG operators.

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

Proof. It is similar to Property 3.1.□

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

Proof. It is similar to Property 3.2.□

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 q - ROFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ , then $q - {ROFS}_{f} YWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq q - ROF S_{f} YWG (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Proof. It is similar to Property 3.3.□

Property 5.4. (Shift Invariance) If $L_{e} = (b, c)$ is another q - ROFS_fN, then $q - {ROFS}_{f} YWG (τ_{e_{11}} \otimes L_{e}, τ_{e_{12}} \otimes L_{e}, \dots, τ_{e_{n m}} \otimes L_{e}) = q - {ROFS}_{f} YWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \otimes L_{e} .$

Proof. It is similar to Property 3.4.□

Property 5.5. (Homogeneity) Let λ ≥ 0 then $q - {ROFS}_{f} YWG (τ_{e_{11}}^{λ}, τ_{e_{12}}^{λ}, \dots, τ_{e_{n m}}^{λ}) = (q - {ROFS}_{f} YWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}))^{λ} .$

Proof. It is similar to Property 3.5.□

6 q-rung orthopair fuzzy soft Yager ordered weighted geometric operators

q - ROFS_fYWG operator only weighted the q - ROFS_fN′s values. But q - ROFS_fYOWG operator weights the ordered positions via scoring the q - ROFS_f values rather than weighting the q - ROFS_f values themselves. Here, we will develop q - ROFS_fYOWG operator and related properties.

Definition 6.1. Let $τ_{e_{t p}} = (f_{t p}, g_{t p}) (t = 1, 2, \dots, n; p = 1, 2, \dots, m)$ be a family of q - ROFS_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 q - ROFS_fYOWG operator is a function q - ROFS_fYOWG: $M^{n} \to M$ $\begin{matrix} q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \\ = ⨂_{p = 1}^{m} (⨂_{t = 1}^{n} τ_{σ e_{t p}}^{w_{t}})^{v_{p}} . \end{matrix}$

Theorem 6.1. The aggregated value by utilizing q - ROFS_fYOWG operator is given by $q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}})$ $\begin{matrix} = & ⨂_{p = 1}^{m} (⨂_{t = 1}^{n} τ_{σ e_{t p}}^{w_{t}})^{v_{p}} \\ = & (\sqrt[q]{1 - min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} w_{t} (1 - g_{σ t p}^{q})^{ϑ}))^{\frac{1}{ϑ}})}, \\ \sqrt[q]{min (1, (\sum_{p = 1}^{m} v_{p} (\sum_{t = 1}^{n} (w_{t} f_{σ t p}^{q ϑ})))^{\frac{1}{ϑ}})}) . \end{matrix}$ (10) where $τ_{σ e_{t p}} = (f_{σ t p}, g_{σ t p})$ , is the permutation of $t$ th and $p$ th largest object of the collection of $t \times p$ q - ROFS_fNs $τ_{e_{t p}} = (f_{t p}, g_{t p}) .$

Proof. It is similar to Theorem 3.1.□

Remark 6.1. (a) For q = 1, q - ROFS_fYOWG operator reduces to intuitionistic fuzzy soft Yager ordered weighted geometric (IFS_fYOWG) operator.

(b) For q = 2, q - ROFS_fYOWG operator reduces to Pythagorean fuzzy soft Yager ordered weighted geometric (PFS_fYOWG) operator.

(c) If parameter is only one, i.e., e₁ then q - ROFS_fYOWG operator reduces to q-rung orthopair fuzzy ordered weighted geometric (q-ROFYOWG) operator.

Therefore, IFS_fYOWG, PFS_fYOWG and q-ROFYOWG operators are special cases of q - ROFS_fYOWG operator.

We give some properties without their proofs.

Property 6.1. (Idempotency) If all $τ_{e_{t p}} = L_{σ e} = (b, c) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ are identical, then $q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) = L_{σ e} .$

Proof. It is similar to Property 3.1.□

Property 6.2. (Boundedness) Let $τ_{σ e_{t p}} = (f_{σ t p}, g_{σ t p}) (\forall t = 1, 2, \dots, m; p = 1, 2, \dots, n)$ be a collection of q - ROFS_fNs and $τ_{σ e_{t p}}^{-} = (min_{p} min_{t} {f_{σ t p}}, max_{p} max_{t} {g_{σ t p}})$ and $τ_{σ e_{t p}}^{+} = (max_{p} max_{t} {f_{σ t p}}, min_{p} min_{t} {g_{σ t p}})$ ,

$τ_{σ e_{t p}}^{-} \leq q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq τ_{σ e_{t p}}^{+} .$

Proof. It is similar to Property 3.2.□

Property 6.3. (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 q - ROFS_fNs such that $f_{t p} \leq b_{t p}, g_{t p} \geq c_{t p}$ , then $q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \leq q - {ROFS}_{f} YOWG (L_{e_{11}}, L_{e_{12}}, \dots, L_{e_{n m}}) .$

Proof. It is similar to Property 3.3.□

Property 6.4. (Shift Invariance) If $L_{e} = (b, c)$ is another q - ROFS_fN, then $q - {ROFS}_{f} YOWG (τ_{e_{11}} \otimes L_{e}, τ_{e_{12}} \otimes L_{e}, \dots, τ_{e_{n m}} \otimes L_{e}) = q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}) \otimes L_{e} .$

Proof. It is similar to Property 3.4.□

Property 6.5. (Homogeneity) Let λ ≥ 0 then $q - {ROFS}_{f} YOWG (τ_{e_{11}}^{λ}, τ_{e_{12}}^{λ}, \dots, τ_{e_{n m}}^{λ}) = (q - {ROFS}_{f} YOWG (τ_{e_{11}}, τ_{e_{12}}, \dots, τ_{e_{n m}}))^{λ} .$

Proof. It is similar to Property 3.5.□

7 Model for MAGDM under q-rung orthopair fuzzy soft information

In this section, we give a mathematical description of proposed theory for MAGDM under q - ROFS_f data. The basic idea and steps of an algorithm for given theory are as follows:

Let $U = {x_{1}, x_{2}, \dots, x_{l}}$ be the set of alternatives, assessed by $n$ senior experts ${\tilde{D}}_{1}, {\tilde{D}}_{2}, \dots, {\tilde{D}}_{n}$ 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 q - ROFS_fNs $τ_{e_{t p}} = (f_{t p}, g_{t p})$ 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) $M = [τ_{e_{t p}}]_{n \times m}$ . By using proposed AOs, the aggregated q - ROFS_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.

The algorithm consists on the following steps:

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

Step 2. 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., $P_{t p} = {\begin{matrix} τ_{e_{t p}}^{c}; & for CP, \\ τ_{e_{t p}}; & for BP . \end{matrix}$

Step 3. For each alternative x_s (s = 1, 2, ⋯ , l), aggregate the q - ROFS_fNs by using proposed operators in the collective DMx Ω_s.

Step 4. Find the score values for Ω_s of all alternatives by utilizing the score function.

Step 5. Arrange the alternatives in the decreasing order of score value and choose the best result.

7.1 Numerical example

This section gives presentation of an illustrating example to indicate the importance and validity of proposed model under q - ROFS_f data.

The team of experts involve five senior doctors ${\tilde{D}}_{1}, {\tilde{D}}_{2}, {\tilde{D}}_{3}, {\tilde{D}}_{4}$ and ${\tilde{D}}_{5}$ whose WVs w = (0.16, 0.20, 0.21, 0.22, 0.21) ^T present their judgment for different patients x₁, x₂, x₃, x₄ under the parameters $\tilde{E} = {Fever, Headache, Diarrhea, Dry cough}$ having WV v = (0.26, 0.24, 0.27, 0.23) ^T. Doctors give their interpretation for alternatives to their corresponding symptom in the form of q - ROFS_fNs. Now, we utilize the proposed algorithm of our model to interpret the illness of desirable patients.

By applying q - ROFS_fYWA operator:

Step 1. The evaluation of doctors for the illness of patients in terms of q - ROFS_fNs is given in Tables 7, respectively.

Table 4
q - ROFS_f matrix for x₁

Experts e ₁ e ₂ e ₃ e ₄

${\tilde{D}}_{1}$ (0.8,0.4) (0.91,0.1) (0.5,0.1) (0.55,0.44)

${\tilde{D}}_{2}$ (0.7,0.5) (0.83,0.45) (0.7,0.5) (0.61,0.3)

${\tilde{D}}_{3}$ (0.4,0.3) (0.7,0.5) (0.8,0.55) (0.7,0.1)

${\tilde{D}}_{4}$ (0.8,0.1) (0.33,0.11) (0.9,0.33) (0.8,0.4)

${\tilde{D}}_{5}$ (0.7,0.3) (0.77,0.4) (0.8,0.11) (0.9,0.3)

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{D}}_{1}$	(0.8,0.4)	(0.91,0.1)	(0.5,0.1)	(0.55,0.44)
${\tilde{D}}_{2}$	(0.7,0.5)	(0.83,0.45)	(0.7,0.5)	(0.61,0.3)
${\tilde{D}}_{3}$	(0.4,0.3)	(0.7,0.5)	(0.8,0.55)	(0.7,0.1)
${\tilde{D}}_{4}$	(0.8,0.1)	(0.33,0.11)	(0.9,0.33)	(0.8,0.4)
${\tilde{D}}_{5}$	(0.7,0.3)	(0.77,0.4)	(0.8,0.11)	(0.9,0.3)

Table 5

q - ROFS_f matrix for x₂

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{D}}_{1}$	(0.6,0.5)	(0.71,0.21)	(0.74,0.2)	(0.8,0.18)
${\tilde{D}}_{2}$	(0.66,0.4)	(0.8,0.14)	(0.8,0.17)	(0.5,0.24)
${\tilde{D}}_{3}$	(0.78,0.56)	(0.6,0.1)	(0.7,0.24)	(0.75,0.2)
${\tilde{D}}_{4}$	(0.91,0.4)	(0.64,0.33)	(0.77,0.14)	(0.86,0.15)
${\tilde{D}}_{5}$	(0.7,0.33)	(0.54,0.2)	(0.6,0.4)	(0.6,0.23)

Table 6

q - ROFS_f matrix for x₃

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{D}}_{1}$	(0.7,0.24)	(0.77,0.11)	(0.87,0.11)	(0.8,0.17)
${\tilde{D}}_{2}$	(0.8,0.14)	(0.84,0.12)	(0.9,0.1)	(0.64,0.24)
${\tilde{D}}_{3}$	(0.77,0.1)	(0.87,0.13)	(0.83,0.11)	(0.85,0.1)
${\tilde{D}}_{4}$	(0.79,0.21)	(0.74,0.24)	(0.74,0.28)	(0.74,0.24)
${\tilde{D}}_{5}$	(0.6,0.23)	(0.7,0.18)	(0.73,0.17)	(0.6,0.1)

Table 7

q - ROFS_f matrix for x₄

Experts	e ₁	e ₂	e ₃	e ₄
${\tilde{D}}_{1}$	(0.75,0.21)	(0.74,0.21)	(0.84,0.14)	(0.77,0.2)
${\tilde{D}}_{2}$	(0.71,0.11)	(0.78,0.17)	(0.65,0.11)	(0.72,0.14)
${\tilde{D}}_{3}$	(0.81,0.15)	(0.82,0.12)	(0.84,0.13)	(0.81,0.11)
${\tilde{D}}_{4}$	(0.59,0.26)	(0.69,0.31)	(0.71,0.21)	(0.82,0.21)
${\tilde{D}}_{5}$	(0.54,0.11)	(0.8,0.11)	(0.8,0.14)	(0.71,0.16)

Step 2. The normalize DMx is not required because all parameters are of same type.

Step 3. The evaluation of doctors for each patient is aggregated by using Equation 3, given by $Ω_{1} = (0.79, 0.37), Ω_{2} = (0.76, 0.31),$ $Ω_{3} = (0.79, 0.19), Ω_{4} = (0.77, 0.18) .$

Step 4. The score values of aggregated value Ω_s are $\begin{matrix} S (x_{1}) = 0.44, S (x_{2}) = 0.41, S (x_{3}) = 0.49, S (x_{4}) = 0.45 . \end{matrix}$

Step 5. Rank the patients in the decreasing order of score values. The ranking order (RO) is $S (x_{3}) > S (x_{4}) > S (x_{1}) > S (x_{2}) .$ Hence, patient x₃ has more severe illness as compared to other patients.

By applying q - ROFS_fYWG operator:

Step 1. Same as above.

Step 2. Same as above.

Step 3. The evaluation of doctors for each patient is aggregated by using Equation 3, given by $Ω_{1} = (0.70, 0.44), Ω_{2} = (0.69, 0.42),$ $Ω_{3} = (0.75, 0.22), Ω_{4} = (0.74, 0.23) .$

Step 4. The score values of aggregated value Ω_s are $\begin{matrix} S (x_{1}) = 0.26, S (x_{2}) = 0.25, S (x_{3}) = 0.41, S (x_{4}) = 0.39 . \end{matrix}$

Step 5. Rank the patients in the decreasing order of score values. The RO is $S (x_{3}) > S (x_{4}) > S (x_{1}) > S (x_{2}) .$ Hence, patient x₃ has more severe illness as compared to other patients.

8 Comparison analysis

This section gives comparison analysis of proposed operators with existing operators to show the effectiveness and superiority of proposed model. It is clear from Tables 4-7, there are some values for which sum of MD and NMD is greater than 1. Therefore, methods given in [8, 11] are fail to handle the situations.

8.1 Comparison with q-rung orthopair fuzzy soft aggregation operators

Here, we discuss the comparison of proposed operators with q - ROFS_f weighted averaging (q - ROFS_fWA) [16] and q - ROFS_f ordered weighted averaging (q - ROFS_fOWA) [16] operators. The computed results by applying these operators are summarized in Table 8 and shown in Fig. 4. It is clear from Table 8 and Fig. 4 that best alternative obtained by using q - ROFS_fWA and q - ROFS_fOWA operators remain same as obtained from using proposed operators. This implies our proposed methods are authentic and can be applied in MAGDM problems.

Table 8
Comparison with q - ROFS_fWA and q - ROFS_fOWA operators

Methods S (x₁) S (x₂) S (x₃) S (x₄) RO

q - ROFS_fYWA 0.44 0.41 0.49 0.45 x₃ > x₄ > x₁ > x₂

q - ROFS_fYWG 0.26 0.25 0.41 0.39 x₃ > x₄ > x₁ > x₂

q - ROFS_fWA 0.42 0.41 0.49 0.45 x₃ > x₄ > x₁ > x₂

q - ROFS_fOWA 0.40 0.36 0.47 0.44 x₃ > x₄ > x₁ > x₂

Methods	S (x₁)	S (x₂)	S (x₃)	S (x₄)	RO
q - ROFS_fYWA	0.44	0.41	0.49	0.45	x₃ > x₄ > x₁ > x₂
q - ROFS_fYWG	0.26	0.25	0.41	0.39	x₃ > x₄ > x₁ > x₂
q - ROFS_fWA	0.42	0.41	0.49	0.45	x₃ > x₄ > x₁ > x₂
q - ROFS_fOWA	0.40	0.36	0.47	0.44	x₃ > x₄ > x₁ > x₂

Fig. 2

Framework for the selection of best alternative.

Fig. 3

RO of alternatives.

Fig. 4

Comparison with existing operators.

8.2 Comparison with Pythagorean fuzzy yager aggregation operators

This subsection discusses the comparison with PF Yager weighted averaging (PFYWA) [31] and PF Yager weighted geometric (PFYWG) [31] operators. If we consider the data given in Tables 4-7, then the methods presented in [31] are fail to handle the problems but proposed theory cover all such situations. For that reason distinct parameters of q - ROFS_fNs are aggregated by utilizing weighted averaging operator with WV w = (0.16, 0.20, 0.21, 0.22, 0.21) ^T and aggregated q - ROFS_f matrix for different patients is given in Table 9. Using this evaluated matrix, comparison with PFYWA and PFYWG operators is given in Table 10. From above table. it is clear that x₃ is more serious patient of given diseases as compared to others.

Table 9
Aggregated q - ROFS_f matrix

x ₁ x ₂ x ₃ x ₄

e ₁ (0.675,0.312) (0.739,0.4349) (0.7335,0.1819) (0.6753,0.1674)

e ₂ (0.6929,0.3192) (0.4349,0.1972) (0.7837,0.1595) (0.7664,0.1841)

e ₃ (0.754,0.3272) (0.7208,0.2312) (0.8096,0.158) (0.765,0.1473)

e ₄ (0.6976,0.3024) (0.7007,0.2001) (0.7233,0.17) (0.7668,0.1629)

x ₁	x ₂	x ₃	x ₄
e ₁	(0.675,0.312)	(0.739,0.4349)	(0.7335,0.1819)	(0.6753,0.1674)
e ₂	(0.6929,0.3192)	(0.4349,0.1972)	(0.7837,0.1595)	(0.7664,0.1841)
e ₃	(0.754,0.3272)	(0.7208,0.2312)	(0.8096,0.158)	(0.765,0.1473)
e ₄	(0.6976,0.3024)	(0.7007,0.2001)	(0.7233,0.17)	(0.7668,0.1629)

Table 10

Comparison with PFYWA and PFYWG operators

Methods	S (x₁)	S (x₂)	S (x₃)	S (x₄)	RO
q - ROFS_fYWA	0.44	0.41	0.49	0.45	x₃ > x₄ > x₁ > x₂
q - ROFS_fYWG	0.26	0.25	0.41	0.39	x₃ > x₄ > x₁ > x₂
PFYWA	0.58	0.52	0.66	0.63	x₃ > x₄ > x₁ > x₂
PFYWG	-0.02	-0.03	0.15	0.13	x₃ > x₄ > x₁ > x₂

8.3 Comparison with Pythagorean fuzzy TOPSIS method

Here, we discuss the comparison of proposed operators with PF-TOPSIS [2] method. The steps to find out the best alternative by PF-TOPSIS method are:

Construction of a PF DMx in which each entry corresponds to a PFN.

Construction of a weighted PF DMx, which is given in Table 11.

The PF positive ideal solution (PFPIS) $S^{+}$ and PF negative ideal solution (PFNIS) $S^{-}$ are given as $\begin{matrix} S^{+} & = & {(0.2036, 0.0883), (0.1946, 0.064), \\ (0.2186, 0.0462), (0.2065, 0.0397)}, \\ S^{-} & = & {(0.1534, 0.0665), (0.1043, 0.047), \\ (0.1591, 0.0374), (0.1756, 0.0435)} . \end{matrix}$

The distance between the alternative $x_{t}$ and PFPIS $S^{+}$ together with the PFNIS $S^{-}$ are given in the Table 12.

The revised closeness degree of each alternative is given as ξ (x₁) = -0.4770, ξ (x₂) = -0.6096, ξ (x₃) = -0.2980, ξ (x₄) =0.4570 .

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

x₃ has more serious diseases.

Table 11
Weighted PF DMx

x ₁ x ₂ x ₃ x ₄

e ₁ (0.1755,0.0811) (0.1921,0.1132) (0.1907,0.0473) (0.1756,0.0435)

e ₂ (0.1663,0.0766) (0.1043,0.047) (0.1881,0.0383) (0.1839,0.0442)

e ₃ (0.2036,0.0883) (0.1946,0.0624) (0.2186,0.0462) (0.2065,0.0397)

e ₄ (0.1534,0.0665) (0.1542,0.0440) (0.1591,0.0374) (0.1687,0.0358)

x ₁	x ₂	x ₃	x ₄
e ₁	(0.1755,0.0811)	(0.1921,0.1132)	(0.1907,0.0473)	(0.1756,0.0435)
e ₂	(0.1663,0.0766)	(0.1043,0.047)	(0.1881,0.0383)	(0.1839,0.0442)
e ₃	(0.2036,0.0883)	(0.1946,0.0624)	(0.2186,0.0462)	(0.2065,0.0397)
e ₄	(0.1534,0.0665)	(0.1542,0.0440)	(0.1591,0.0374)	(0.1687,0.0358)

Table 12

Distance of alternatives from PFPIS and PFNIS

$D (x_{t}, S^{+}$ )	$D (x_{t}, S^{-}$ )
0.0197	0.0147
0.0301	0.0225
0.0187	0.0179
0.0194	0.0148

We observe from Table 13 that the methods given in [2, 31] have no knowledge about parameterized study. The superiority of proposed theory is that it has the ability to handle real life problems by utilizing parametrization information. Therefore, developed approach can be applied for handling decision making problems involving parameters.

Table 13

Characteristic analysis with different operators

Methods	Fuzzy Information	Parametric Information
PFYWA [31]	Yes	No
PFYWG [31]	Yes	No
PF-TOPSIS [2]	Yes	No
q - ROFS_fWA [16]	Yes	Yes
Proposed operators	Yes	Yes

9 Conclusions

In our daily life problem, we face uncertainties in making right decisions. In this study, we have proposed two different decision making methods in medical diagnose. S_fS is a parametrization tool to deal with uncertainty and vague information. q - ROFS_fS as an extension of IFS_fS and PFS_fS is more flexible tool to solve decision making problems involving vagueness corresponding to different parameters. Moreover, Yager’s t-norm and t-conorm have more generalized structure that operate efficiently to integrate the complex information. By combining these two structures, we have developed a new hybrid model to solve MAGDM problems. We have proposed four families of AOs, namely, q - ROFS_fYWA, q - ROFS_fYOWA, q - ROFS_fYWG and q - ROFS_fYOWG operators. Some properties of proposed operators including idempotency, boundedness, monotonicity, shift invariance and homogeneity have been established. The main objective of this article is to develop a method based on the proposed operators to handle decision making problem under q - ROFS_f data. A practical example for the diagnosis of illness of patients is given to demonstrate the application of the proposed strategy. To show the validity and authenticity of our model, we have performed the comparison analysis of proposed theory with existing operators. In short, this article builds up a tool that has the rich properties of Yager AOs and flexibility of q - ROFS_f model. In future, we will extend our models to q-rung picture fuzzy soft set data.

Conflict of iterest: The authors declare no conflict of interest.

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