More on Dombi operations and Dombi aggregation operators for q -rung orthopair fuzzy values

Abstract

Dombi operations which include the Dombi product and Dombi sum are special cases of t-norms and t-conorms besides the algebraic operations. Recently, operations and aggregation operators for q-rung orthopair fuzzy values (q-ROFVs) based on Dombi operations were proposed. In this paper, we further discuss some additional issues relating to Dombi operations and Dombi aggregation operators of q-ROFVs. First, we give a reasonable explanation for the definition of the Dombi scalar multiplication and Dombi exponentiation which are constructed respectively by the Dombi sum and Dombi product over q-ROFVs, and then investigate the fundamental properties of these operations. Subsequently, the shift-invariance and homogeneity properties of the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators are analyzed. And the boundedness of aforementioned aggregation operators are precisely characterized with respect to the parameter in Dombi operations. Finally, a method for multiattribute decision making is proposed by utilizing the developed operators under the q-rung orthopair fuzzy environment and an example of the selection of investment companies is given to illustrate the detailed decision making process.

Keywords

aggregation operator Dombi operation multiattribute decision making

1 Introduction

The concept of fuzzy set was initiated by Zadeh [1] to deal with imprecision and uncertainty, especially in multiattribute decision making (MADM) problems. In fuzzy set theory, the degree of membership lying in the unit interval is given and the degree of nonmembership is taken as one minus the membership degree naturally. To tackle more complex problems with vague information in the real world, several extensions of fuzzy sets were developed, such as intuitionistic fuzzy sets [2], Pythagorean fuzzy sets [3], Fermatean fuzzy sets [4], generalized orthopair fuzzy sets [5], picture fuzzy sets [6], and T-spherical fuzzy sets [7]. Orthopair fuzzy sets are fuzzy sets in which each membership grade is described by a pair of values both from [0, 1], the former indicating the degree of support for membership and the latter support against membership. Yager [5] proposed the q-rung orthopair fuzzy sets (q-ROFSs, q ≥ 1) in which the sum of the qth powers of the support for and the support against is equal to or less than one. It is proved that with the increase of the rung q, the space of acceptable orthopairs increases. Thus it gives the decision makers more freedom in expressing their beliefs about the membership grade.

Due to its wide scope of applications in practice, the q-rung orthopair fuzzy set theory has attracted the attention of many scholars since its inception. Du [8, 9] presented the Minkowski distances and correlation coefficients between q-ROFSs and discussed their applications in MADM and cluster analysis, respectively. Liu and Wang [10] proposed the addition and multiplication operations for q-rung orthopair fuzzy values (q-ROFVs) via the algebraic t-(co)norm, while Du [11] defined the subtraction and division operations over q-ROFVs in two different ways. Gao et al. discussed continuities, derivatives and differentials of q-rung orthopair fuzzy functions based on arithmetic operations over q-ROFVs [12] and further studied the q-rung orthopair fuzzy integrals under additive operations [13]. Shu et al. [14] developed the q-rung orthopair fuzzy definite integrals to aggregate q-rung orthopair fuzzy continuous information. In the seminal paper [5], Yager provided a general framework of constructing aggregation operators of q-ROFVs. Specially, Yager et al. [5, 15] developed the Sugeno and the Choquet aggregation operators on q-ROFVs based on a pair of dual aggregation operators. Along this line of research, Liu and his colleagues introduced a large number of operators on q-ROFVs, such as the weighted averaging/geometric operators [10], (weighted) Archimedean Bonferroni mean operators [16], power (weighted) Maclaurin symmetric mean operators [17], and weighted generalized (geometric) Maclaurin symmetic mean operators [18]. Wei et al. [19] put forward the generalized (weighted) Heronian mean operators for q-ROFVs. Wang et al. [20] presented the q-rung orthopair fuzzy (weighted, dual) Muirhead mean operators for fusing q-ROFVs. Du [21] developed the (ordered) weighted power means of q-ROFVs and presented their applications in MADM. To arrange a pleasant decision, Garg and Chen [22] established some neutrality aggregation operators based on neutral operational laws for q-ROFVs.

It deserves mentioning that in the most existing literature, operations on q-ROFVs are established with the help of the algebraic t-norm and algebraic t-conorm which are employed to model intersection and union of q-ROFVs, respectively. Besides the algebraic operations, there are many other operational laws that are often used in the fuzzy set theory, Dombi operations, for example. Dombi operations are special cases of t-(co)norms which have the advantage of good flexibility with a general parameter, the sign of which determines the type of the operations [23, 24]. According to Dombi, there are two main reasons for a great interest in such parametrical families from the practical viewpoint: in applications, by changing a single parameter, a different logic can be modeled; through a learning algorithm, the appropriate parameter and the appropriate operator can be found. Due to these advantages, Liu et al. [25] proposed some Dombi Bonferroni mean operators to deal with the aggregation of intuitionistic fuzzy values. He [26] developed some Dombi aggregation operators for hesitant fuzzy elements by utilizing Dombi operations under hesitant fuzzy environment and further extended them to interval-valued intuitionistic fuzzy cases [27]. Jana et al. presented some Dombi operations based aggregation operators to fuse picture fuzzy information [28], bipolar fuzzy elements [29], and Pythagorean fuzzy values [30]. Ashraf et al. [31] defined some aggregation operators on the basis of Dombi operations in the spherical fuzzy context. Recently, Jana et al. [32] introduced some aggregation operators for aggregating q-ROFVs based on Dombi operations, and demonstrated that these aggregation operators are idempotent, monotonic and bounded. In Ref. [32], a realistic instance for implement emerging software system was provided to show the applicability and the effectiveness of the proposed operations in MADM. However, there are still some additional issues that need to be addressed. Why do we define the Dombi scalar multiplication and Dombi exponentiation of q-ROFVs in such a way? Are there any other properties of the Dombi aggregation operators on q-ROFVs? Could the boundedness of the aggregation results be further characterized? The initial motivation of this paper is to solve these problems and the main contributions are listed as follows:

It is shown that how to construct the Dombi scalar multiplication and Dombi exponentiation by the Dombi sum and Dombi product, respectively. The fundamental properties of four Dombi operations over q-ROFVs are investigated as well.

Two more properties, the shift-invariance property and the homogeneity property, of the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators are proposed.

As for the boundedness property, we present a more accurate characterization of these two aggregation operators in terms of the parameter in Dombi operations. The results are finer than those determined by the max and min operators.

Moreover, we apply the proposed operators to MADM problems in the q-rung orthopair fuzzy setting. The results show that the developed operators can be used to aggregate any fuzzy information by choosing suitable q; the aggregations obtained by the operators are monotonic with respect to the parameter therein; and the best selection obtained by the proposed method are consistent with most of the other existing approaches.

The other parts of this paper are organized as follows. In Section 2, we recall some basic concepts of Dombi operations and q-rung orthopair fuzzy values. In Section 3, we investigate the operational rules of Dombi operations over q-ROFVs. In Section 4, we examine some essential properties of aggregation operators for q-ROFVs based on Dombi operations. In Section 5, we provide a new method for handling MADM problems based on the proposed operators under q-rung orthopair fuzzy environment and give a numerical example to illustrate the detailed decision making procedure. In Section 6, we conclude the present paper with a summary and suggestions for further work.

2 Preliminaries

In this section, we briefly review basic concepts concerning Dombi operations, q-rung orthopair fuzzy values and some operations defined on them.

2.1 Dombi operations

Dombi [23] proposed the Dombi sum and Dombi product which are special cases of t-norms and t-conorms, respectively. Definition 2.1. [23] Let x and y be any two real numbers in the unit interval [0, 1]. Then the Dombi sum and Dombi product of x and y are defined as follows: $\begin{matrix} x +_{D}^{κ} y & = 1 - \frac{1}{1 + ((\frac{x}{1 - x})^{κ} + (\frac{y}{1 - y})^{κ})^{\frac{1}{κ}}}, \\ x \times_{D}^{κ} y & = \frac{1}{1 + ((\frac{1 - x}{x})^{κ} + (\frac{1 - y}{y})^{κ})^{\frac{1}{κ}}}, \end{matrix}$ where κ > 0.

The Dombi t-norm and Dombi t-conorm satisfy the following properties: [33]

For each κ, the Dombi t-norm and Dombi t-conorm are dual to each other.

$lim_{κ \to 0^{+}} x +_{D}^{κ} y = {\begin{matrix} x \land y, & if x \lor y = 1, \\ 0, & otherwise . \end{matrix}$ $lim_{κ \to 0^{+}} x \times_{D}^{κ} y = {\begin{matrix} x \lor y, & if x \land y = 0, \\ 1, & otherwise . \end{matrix}$

$lim_{κ \to \infty} x +_{D}^{κ} y = x \lor y$ , $lim_{κ \to \infty} x \times_{D}^{κ} y = x \land y$ .

The Dombi t-(co)norms are continuous Archimedean and strict.

The family of Dombi t-norms is strictly increasing and the family of Dombi t-conorms is strictly decreasing with respect to the parameter κ.

Additive generators of the Dombi t-norms and Dombi t-conorms are $g (t) = (\frac{1 - t}{t})^{κ}$ and $h (t) = (\frac{t}{1 - t})^{κ}$ , respectively.

2.2 q-Rung orthopair fuzzy values

Definition 2.2. [5] An orthopair 〈a, b〉 is said to be a q-rung orthopair fuzzy value if a, b ∈ [0, 1] and $a^{q} + b^{q} \leq 1,$ (1) where q ≥ 1.

Note that if q = 1, 〈a, b〉 is an intuitionistic fuzzy value [34]; if q = 2, 〈a, b〉 is a Pythagorean fuzzy value [35]; if q = 3, 〈a, b〉 is a Fermatean fuzzy value [4]. In this way, we can see that q-rung orthopair fuzzy values are generalizations of these three types of fuzzy values.

Denote S_q the space of all q-rung orthopair fuzzy values, namely, $S_{q} = {〈 a, b 〉 : a, b \in [0, 1], a^{q} + b^{q} \leq 1},$ (2) in which the positive ideal orthopair 〈1, 0〉, denoted by e, can be seen as true or full membership; the negative ideal orthopair 〈0, 1〉, denoted by θ, can be seen as false or full nonmembership. Thus we can represent the classical binary case as either 〈1, 0〉 or 〈0, 1〉. For two q-ROFVs x = 〈a₁, b₁〉 and y = 〈a₂, b₂〉, the basic operations and order relation ⪯ between them are given by [5] $\begin{matrix} x \land y & = 〈 a_{1} \land a_{2}, b_{1} \lor b_{2} 〉, \\ x \lor y & = 〈 a_{1} \lor a_{2}, b_{1} \land b_{2} 〉, \\ x ⪯ y & iff a_{1} \leq a_{2} and b_{1} \geq b_{2} . \end{matrix}$

3 Dombi operations on q-rung orthopair fuzzy values

In this section, some primary properties of operational laws of q-ROFVs based on Dombi operations are examined.

Definition 3.1. [32] Let x = 〈a₁, b₁〉 and y = 〈a₂, b₂〉 be two q-ROFVs. The addition and multiplication of x and y based on Dombi operations are defined as follows: κ > 0,

$\begin{matrix} x & \oplus_{D}^{κ} y = 〈 {(1 - \frac{1}{1 + ((\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ} + (\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ} + (\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (3) $\begin{matrix} x & \otimes_{D}^{κ} y = 〈 {(\frac{1}{1 + ((\frac{1 - a_{1}^{q}}{a_{1}^{q}})^{κ} + (\frac{1 - a_{2}^{q}}{a_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + ((\frac{b_{1}^{q}}{1 - b_{1}^{q}})^{κ} + (\frac{b_{2}^{q}}{1 - b_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ (4)

Because function f (κ) = (a^κ + b^κ) ^1/κ is monotonic decreasing (by f′ (κ) ≤0), we have $x \oplus_{D}^{κ} y ≼ x \oplus_{D}^{κ^{*}} y$ and $x \otimes_{D}^{κ} y ⪯ x \otimes_{D}^{κ^{*}} y$ if κ ≤ κ^*. Moreover, the parameter κ is always omitted in the expression when no confusion arises, namely, for notational convenience, $x \oplus_{D}^{κ} y$ and $x \otimes_{D}^{κ} y$ are shortly denoted by x ⊕ _Dy and x ⊗ _Dy, respectively. Note that for q-ROFVs x and y, their Dombi sum x ⊕ _Dy and Dombi product x ⊗ _Dy are still q-ROFVs. From Definition 3.1, it follows easily that x ⊕ _Dθ = x, x ⊕ _De = e, x ⊗ _Dθ = θ, x ⊗ _De = x, ∀x ∈ S_q.

Example 3.2. Take x = 〈0.7, 0.8〉 and y = 〈0.4, 0.9〉. They are both q-ROFVs, here q = 3. If we set κ = 2, then

$\begin{matrix} x \oplus_{D} y & = 〈 {(1 - \frac{1}{1 + ((\frac{0 . 7^{3}}{1 - 0 . 7^{3}})^{2} + (\frac{0 . 4^{3}}{1 - 0 . 4^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3}, \\ {(\frac{1}{1 + ((\frac{1 - 0 . 8^{3}}{0 . 8^{3}})^{2} + (\frac{1 - 0 . 9^{3}}{0 . 9^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3} 〉 \\ = 〈 0.7013, 0.7907 〉, \\ x \otimes_{D} y & = 〈 {(\frac{1}{1 + ((\frac{1 - 0 . 7^{3}}{0 . 7^{3}})^{2} + (\frac{1 - 0 . 4^{3}}{0 . 4^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3}, \\ {(1 - \frac{1}{1 + ((\frac{0 . 8^{3}}{1 - 0 . 8^{3}})^{2} + (\frac{0 . 9^{3}}{1 - 0 . 9^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3} 〉 \\ = 〈 0.3989, 0.9056 〉 . \end{matrix}$

Theorem 3.3. Let x, y, z be q-ROFVs. Then we have the following:

x ⊕ _Dy = y ⊕ _Dx,

x ⊗ _Dy = y ⊗ _Dx,

x ⊕ _Dy = (x^c ⊗ _Dy^c) ^c,

x ⊗ _Dy = (x^c ⊕ _Dy^c) ^c,

x ⊕ _D (y ⊕ _Dz) = (x ⊕ _Dy) ⊕ _Dz,

x ⊗ _D (y ⊗ _Dz) = (x ⊗ _Dy) ⊗ _Dz.

Proof. We only prove (5). For x = 〈a₁, b₁〉, y = 〈a₂, b₂〉, z = 〈a₃, b₃〉 ∈ S_q, by Definition 3.1, we have $\begin{matrix} x \oplus_{D} (y \oplus_{D} z) \\ = 〈 a_{1}, b_{1} 〉 \oplus_{D} 〈 {(1 - \frac{1}{1 + ((\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ} + (\frac{a_{3}^{q}}{1 - a_{3}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ} + (\frac{1 - b_{3}^{q}}{b_{3}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + ((\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ} + (\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ} + (\frac{a_{3}^{q}}{1 - a_{3}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ} + (\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ} + (\frac{1 - b_{3}^{q}}{b_{3}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + ((\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ} + (\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ} + (\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \oplus_{D} 〈 a_{3}, b_{3} 〉 \\ = (x \oplus_{D} y) \oplus_{D} z . \end{matrix}$ □

Theorem 3.3 reveals that operations ⊕_D and ⊗_D are dual to each other and that they are both commutative and associative, i.e., they preserve the most desired properties of classical addition and multiplication operations. In this way, for x_i ∈ S_q, x₁ ⊕ _Dx₂ ⊕ _D ⋯ ⊕ _Dx_n and x₁ ⊗ _Dx₂ ⊗ _D ⋯ ⊗ _Dx_n can be shortly denoted without any confusion by ${\oplus_{D}}_{i = 1}^{n} x_{i}$ and ${⨂_{D}}_{i = 1}^{n} x_{i}$ , respectively.

Theorem 3.4. Let ${x_{i} = 〈 a_{i}, b_{i} 〉}_{i = 1}^{n}$ be a collection of q-ROFVs. Then the followings hold. $\begin{matrix} {\oplus_{D}}_{i = 1}^{n} x_{i} & = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (5) $\begin{matrix} {⨂_{D}}_{i = 1}^{n} x_{i} & = 〈 {(\frac{1}{1 + (\sum_{i = 1}^{n} (\frac{1 - a_{i}^{q}}{a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} (\frac{b_{i}^{q}}{1 - b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ (6)

Proof. Eq. (5) can be proved by mathematical induction. For n = 1, $\begin{matrix} 〈 {(1 - \frac{1}{1 + ((\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 = 〈 a_{1}, b_{1} 〉 . \end{matrix}$ Thus the result is true for n = 1. Assume now that Eq. (5) holds for n = k, that is, $\begin{matrix} {\oplus_{D}}_{i = 1}^{k} x_{i} & = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{k} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{k} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$

Then, when n = k + 1, we have $\begin{matrix} {\oplus_{D}}_{i = 1}^{k + 1} x_{i} = {\oplus_{D}}_{i = 1}^{k} x_{i} \oplus_{D} x_{k + 1} \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{k} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{k} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \oplus_{D} 〈 a_{k + 1}, b_{k + 1} 〉 \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{k} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ} + (\frac{a_{k + 1}^{q}}{1 - a_{k + 1}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{k} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ} + (\frac{1 - b_{k + 1}^{q}}{b_{k + 1}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{k + 1} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{k + 1} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ i.e., Eq. (5) holds for n = k + 1. Therefore, Eq. (5) holds for all positive integer n.

Eq. (6) holds similarly.□

For a positive integer n, denote $n \cdot_{D} x = {\overset{︷}{x \oplus_{D} x \oplus_{D} \dots \oplus_{D} x}}^{n times}$ and $x^{D n} = {\overset{︷}{x \oplus_{D} x \oplus_{D} \dots \oplus_{D} x}}^{n times} .$ Then from Theorem 3.4, it follows immediately that, for x = 〈a, b〉, $\begin{matrix} n \cdot_{D} x & = 〈 {(1 - \frac{1}{1 + (n (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (n (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (7) $\begin{matrix} x^{D^{n}} & = 〈 {(\frac{1}{1 + (n (\frac{1 - a^{q}}{a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + (n (\frac{b^{q}}{1 - b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (8)

Denote $y = \frac{1}{n} \cdot_{D} x$ if x = n · _Dy. Set $\frac{1}{n} \cdot_{D} x = 〈 a_{*}, b_{*} 〉$ , by $n \cdot_{D} (\frac{1}{n} \cdot_{D} x) = x$ , we have $\begin{matrix} 〈 {(1 - \frac{1}{1 + (n (\frac{a_{*}^{q}}{1 - a_{*}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (n (\frac{1 - b_{*}^{q}}{b_{*}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 = 〈 a, b 〉 . \end{matrix}$ It is easy to solve this equation and we can get $\begin{matrix} a_{*} & = {(1 - \frac{1}{1 + (\frac{1}{n} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ b_{*} & = {(\frac{1}{1 + (\frac{1}{n} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} . \end{matrix}$ That is, $\begin{matrix} \frac{1}{n} \cdot_{D} x = & 〈 {(1 - \frac{1}{1 + (\frac{1}{n} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\frac{1}{n} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$

Furthermore, it is well known that for rational number λ, there exist integers m, n such that $λ = \frac{m}{n}$ . Then, $\begin{matrix} λ \cdot_{D} x & = m \cdot_{D} (\frac{1}{n} \cdot_{D} x) \\ = m \cdot_{D} 〈 {(1 - \frac{1}{1 + (\frac{1}{n} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\frac{1}{n} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (\frac{m}{n} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\frac{m}{n} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (λ (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ Similarly, for rational number λ, we have $x^{D λ} = 〈 {(\frac{1}{1 + (λ (\frac{1 - a^{q}}{a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, {(1 - \frac{1}{1 + (λ (\frac{b^{q}}{1 - b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 .$

Extending the above parameter λ to any positive real number comes the following definition.

Definition 3.5. [32] Let x = 〈a, b〉 be a q-ROFV and λ be a positive number. The λ times and λth power of x based on Dombi operations are given by $\begin{matrix} λ \cdot_{D} x & = 〈 {(1 - \frac{1}{1 + (λ (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (9) $\begin{matrix} x^{D^{λ}} & = 〈 {(\frac{1}{1 + (λ (\frac{1 - a^{q}}{a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + (λ (\frac{b^{q}}{1 - b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (10)

It deserves mentioning that for λ > 0, if x is a q-ROFV, then its Dombi scalar multiplication λ · _Dx and Dombi exponentiation $x^{D^{λ}}$ are still q-ROFVs.

Example 3.6. (Continued from Example 3.2). Take q-ROFVs x = 〈0.7, 0.8〉 and y = 〈0.4, 0.9〉, here q = 3. If we set κ = 2, then $\begin{matrix} 0.5 \cdot_{D} x & = 〈 {(1 - \frac{1}{1 + (0.5 \times (\frac{0 . 7^{3}}{1 - 0 . 7^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3}, \\ {(\frac{1}{1 + (0.5 \times (\frac{1 - 0 . 8^{3}}{0 . 8^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3} 〉 \\ = 〈 0.6460, 0.8422 〉, \\ x^{D 0.5} & = 〈 {(\frac{1}{1 + (0.5 \times (\frac{1 - 0 . 7^{3}}{0 . 7^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3}, \\ {(1 - \frac{1}{1 + (0.5 \times (\frac{0 . 8^{3}}{1 - 0 . 8^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3} 〉 \\ = 〈 0.7517, 0.7524 〉 . \end{matrix}$ Similarly, we have $\begin{matrix} 0.5 \cdot_{D} y & = 〈 0.3586, 0.9252 〉, \\ y^{D 0.5} & = 〈 0.4451, 0.8686 〉 . \end{matrix}$

Theorem 3.7. Let x, y be two q-ROFVs and λ, λ₁, λ₂ > 0. Then the followings hold.

λ · _D (x ⊕ _Dy) = (λ · _Dx) ⊕ _D (λ · _Dy),

(x ⊗ _Dy)^Dλ=x^Dλ ⊗ _Dy^Dλ,

(λ₁ + λ₂) · _Dx = (λ₁ · _Dx) ⊕ _D (λ₂ · _Dx),

λ₁ · _D (λ₂ · _Dx) = (λ₁λ₂) · _Dx,

Proof. (1) For x = 〈a₁, b₁〉, y = 〈a₂, b₂〉 ∈ S_q,

$\begin{matrix} λ & \cdot_{D} (x \oplus_{D} y) \\ = λ \cdot_{D} 〈 {(1 - \frac{1}{1 + ((\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ} + (\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ} + (\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (λ (\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ} + λ (\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ (\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ} + λ (\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (λ (\frac{a_{1}^{q}}{1 - a_{1}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ (\frac{1 - b_{1}^{q}}{b_{1}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ \oplus_{D} 〈 {(1 - \frac{1}{1 + (λ (\frac{a_{2}^{q}}{1 - a_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ (\frac{1 - b_{2}^{q}}{b_{2}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = (λ \cdot_{D} x) \oplus_{D} (λ \cdot_{D} y) . \end{matrix}$

(3) For x = 〈a, b〉 ∈ S_q, $\begin{matrix} ( & λ_{1} \cdot_{D} x) \oplus_{D} (λ_{2} \cdot_{D} x) \\ = 〈 {(1 - \frac{1}{1 + (λ_{1} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ_{1} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ \oplus_{D} 〈 {(1 - \frac{1}{1 + (λ_{2} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ_{2} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + ((λ_{1} + λ_{2}) (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((λ_{1} + λ_{2}) (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = (λ_{1} + λ_{2}) \cdot_{D} x . \end{matrix}$ (5) For x = 〈a, b〉 ∈ S_q, $\begin{matrix} λ_{1} & \cdot_{D} (λ_{2} \cdot_{D} x) \\ = λ_{1} \cdot_{D} 〈 {(1 - \frac{1}{1 + (λ_{2} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ_{2} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (λ_{1} λ_{2} (\frac{a^{q}}{1 - a^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ_{1} λ_{2} (\frac{1 - b^{q}}{b^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = (λ_{1} λ_{2}) \cdot_{D} x . \end{matrix}$

The others can be proved by the duality.□

4 q-Rung orthopair fuzzy Dombi aggregation operators

In this section, a brief proof of the expression of the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators are provided and some of their fundamental properties are investigated.

Definition 4.1. [32] Let ${x_{i} = 〈 a_{i}, b_{i} 〉}_{i = 1}^{n}$ be a collection of q-ROFVs, and w=(w₁, w₂, …, w_n)’ be the associated weighting vector such that all the weights are positive and their sum is one. The q-rung orthopair fuzzy Dombi weighted averaging (q-ROFDWA) operator and the q-rung orthopair fuzzy Dombi weighted geometric (q-ROFDWG) operator are mappings from $S_{q}^{n}$ to S_q given by $q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) = {\oplus_{D}}_{i = 1}^{n} w_{i} \cdot_{D} x_{i},$ (11) $q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) = {⨂_{D}}_{i = 1}^{n} x_{i}^{D w_{i}},$ (12) respectively.

Based on results presented in Section 3, the aggregations obtained by the q-ROFDWA and q-ROFDWG operators are still q-ROFVs.

Theorem 4.2. [32] Let $q {- ROFDWA}_{w}^{κ}$ and $q {- ROFDWG}_{w}^{κ}$ be the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators. Then we have $\begin{matrix} q & {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (13) $\begin{matrix} q & {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - a_{i}^{q}}{a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{b_{i}^{q}}{1 - b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ (14)

In Ref. [32], Jana et al. gave a routine but a little tedious proof of Theorem 4.2. The following we give a briefer and more effective proof of this theorem.

Proof. Based on the operational rules of q-ROFVs, we have $\begin{matrix} q & {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) = {\oplus_{D}}_{i = 1}^{n} w_{i} \cdot_{D} x_{i} \\ = {\oplus_{D}}_{i = 1}^{n} 〈 {(1 - \frac{1}{1 + (w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ The other equation follows analogously.□

If q = 1, Eqs. (28) and (29) reduce to the intuitionistic fuzzy Dombi weighted averaging/geometric (IFDWA/IFDWG) operators as follows: for x_i = 〈μ_i, ν_i〉, [36] $\begin{matrix} {IFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{μ_{i}}{1 - μ_{i}})^{κ})^{\frac{1}{κ}}}, \\ \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - ν_{i}}{ν_{i}})^{κ})^{\frac{1}{κ}}} 〉, \end{matrix}$ (15) $\begin{matrix} {IFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - μ_{i}}{μ_{i}})^{κ})^{\frac{1}{κ}}}, \\ 1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{ν_{i}}{1 - ν_{i}})^{κ})^{\frac{1}{κ}}} 〉 . \end{matrix}$ (16)

If q = 2, these two formulae reduce to the Pythagorean fuzzy Dombi weighted averaging/geometric (PFDWA/PFDWG) operators as follows: for x_i = 〈μ_i, ν_i〉, [30, 37] $\begin{matrix} {PFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 \sqrt{1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{μ_{i}^{2}}{1 - μ_{i}^{2}})^{κ})^{\frac{1}{κ}}}}, \\ \sqrt{\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - ν_{i}^{2}}{ν_{i}^{2}})^{κ})^{\frac{1}{κ}}}} 〉, \end{matrix}$ (17) $\begin{matrix} {PFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 \sqrt{\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - μ_{i}^{2}}{μ_{i}^{2}})^{κ})^{\frac{1}{κ}}}}, \\ \sqrt{1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{ν_{i}^{2}}{1 - ν_{i}^{2}})^{κ})^{\frac{1}{κ}}}} 〉 . \end{matrix}$ (18)

Example 4.3. (Continued from Examples 3.2 and 3). Take q-ROFVs x = 〈0.7, 0.8〉 and y = 〈0.4, 0.9〉, here q = 3. If we set κ = 2 and w=(0.5,0.5)’, then

$\begin{matrix} q & {- ROFDWA}_{w}^{κ} (x, y) = 0.5 \cdot_{D} x \oplus_{D} 0.5 \cdot_{D} y \\ = 〈 {(1 - \frac{1}{1 + (0.5 \times (\frac{0 . 7^{3}}{1 - 0 . 7^{3}})^{2} + 0.5 \times (\frac{0 . 4^{3}}{1 - 0 . 4^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3}, \\ {(\frac{1}{1 + (0.5 \times (\frac{1 - 0 . 8^{3}}{0 . 8^{3}})^{2} + 0.5 \times (\frac{1 - 0 . 9^{3}}{0 . 9^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3} 〉 \\ = 〈 0.6474, 0.8341 〉, \\ q & {- ROFDWG}_{w}^{κ} (x, y) = x^{D 0.5} \otimes D y^{D 0.5} \\ = 〈 {(\frac{1}{1 + (0.5 \times (\frac{1 - 0 . 7^{3}}{0 . 7^{3}})^{2} + 0.5 \times (\frac{1 - 0 . 4^{3}}{0 . 4^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3}, \\ {(1 - \frac{1}{1 + (0.5 \times (\frac{0 . 8^{3}}{1 - 0 . 8^{3}})^{2} + 0.5 \times (\frac{0 . 9^{3}}{1 - 0 . 9^{3}})^{2})^{\frac{1}{2}}})}^{1 / 3} 〉 \\ = 〈 0.4439, 0.8756 〉 . \end{matrix}$

Theorem 4.4. Let $q {- ROFDWA}_{w}^{κ}$ and $q {- ROFDWG}_{w}^{κ}$ be the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators. Then they satisfy the following properties.

(Shift-invariance). Let x, x_i ∈ S_q, i = 1, 2, …, n. Then $\begin{matrix} q & {- ROFDWA}_{w}^{κ} (x \oplus_{D} x_{1}, x \oplus_{D} x_{2}, \dots, x \oplus_{D} x_{n}) \\ = x \oplus_{D} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}), \\ q & {- ROFDWG}_{w}^{κ} (x \otimes_{D} x_{1}, x \otimes_{D} x_{2}, \dots, x \otimes_{D} x_{n}) \\ = x \otimes_{D} q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) . \end{matrix}$

(Homogeneity). Let λ > 0 and x_i ∈ S_q, i = 1, 2, …, n. Then $\begin{matrix} q & {- ROFDWA}_{w}^{κ} (λ \cdot_{D} x_{1}, λ \cdot_{D} x_{2}, \dots, λ \cdot_{D} x_{n}) \\ = λ \cdot_{D} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}), \\ q & {- ROFDWG}_{w}^{κ} (x_{1}^{D λ}, x_{2}^{D λ}, \dots, x_{n}^{D λ}) \\ = (q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}))^{D λ} . \end{matrix}$

Proof. (1) For x = 〈a, b〉, x_i = 〈a_i, b_i〉 ∈ S_q,

$\begin{matrix} q {- ROFDWA}_{w}^{κ} (x \oplus_{D} x_{1}, x \oplus_{D} x_{2}, \dots, x \oplus_{D} x_{n}) \\ = q {- ROFDWA}_{w}^{κ} \\ (〈 {(1 - \frac{1}{1 + ((\frac{a^{q}}{1 - a^{q}})^{κ} + (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + ((\frac{1 - b^{q}}{b^{q}})^{κ} + (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉) \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} ((\frac{a^{q}}{1 - a^{q}})^{κ} + (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ}))^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} ((\frac{a^{q}}{1 - a^{q}})^{κ} + (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ}))^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 a, b 〉 \oplus_{D} 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = x \oplus_{D} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) . \end{matrix}$

(2) For x_i = 〈a_i, b_i〉 ∈ S_q,

$\begin{matrix} q & {- ROFDWA}_{w}^{κ} (λ \cdot_{D} x_{1}, λ \cdot_{D} x_{2}, \dots, λ \cdot_{D} x_{n}) \\ = q {- ROFDWA}_{w}^{κ} (〈 {(1 - \frac{1}{1 + (λ (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉) \\ = 〈 {(1 - \frac{1}{1 + (λ \sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (λ \sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = λ \cdot_{D} 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = λ \cdot_{D} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) . \end{matrix}$

The other parts of the theorem follow similarly.□

It has been proven the boundedness property of the q-ROFDWA/q-ROFDWG operators as follows. Denote $\underline{x}$ and $\bar{x}$ the min and max of ${x_{i}}_{i = 1}^{n}$ , respectively. That is, $\underline{x} = ⋀_{i = 1}^{n} x_{i} = 〈 ⋀_{i = 1}^{n} a_{i}, ⋁_{i = 1}^{n} b_{i} 〉$ , $\bar{x} = ⋁_{i = 1}^{n} x_{i} = 〈 ⋁_{i = 1}^{n} a_{i}, ⋀_{i = 1}^{n} b_{i} 〉$ . Then [32] $\underline{x} ⪯ q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) ⪯ \bar{x},$ (19) $\underline{x} ⪯ q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) ⪯ \bar{x} .$ (20)

By the monotonicity of the weighed power mean function $f (κ) = (\sum_{i = 1}^{n} w_{i} x_{i}^{κ})^{1 / κ}$ , i.e., the weighed power mean inequality [38] $(\sum_{i = 1}^{n} w_{i} x_{i}^{κ})^{1 / κ} \leq (\sum_{i = 1}^{n} w_{i} x_{i}^{κ^{*}})^{1 / κ^{*}}$ if κ ≤ κ^*, it follows that the aggregations by the q-ROFDWA operator are monotonic increasing with respect to κ and are monotonic decreasing with respect to κ by the q-ROFDWG operator. More precisely, we have, ∀κ > 0, $\begin{matrix} lim_{κ \to 0^{+}} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ lim_{κ \to \infty} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}), \\ lim_{κ \to \infty} q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ lim_{κ \to 0^{+}} q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) . \end{matrix}$

Theorem 4.5. Let $q {- ROFDWA}_{w}^{κ}$ and $q {- ROFDWG}_{w}^{κ}$ be the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators. Then the following equalities hold. $\begin{matrix} lim_{κ \to 0^{+}} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉, \end{matrix}$ (21) $\begin{matrix} lim_{κ \to 0^{+}} q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - a_{i}^{q}}{a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{b_{i}^{q}}{1 - b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉, \end{matrix}$ (22) $\begin{matrix} lim_{κ \to \infty} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 ⋁_{i = 1}^{n} a_{i}, ⋀_{i = 1}^{n} b_{i} 〉, \end{matrix}$ (23) $\begin{matrix} lim_{κ \to \infty} q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 ⋀_{i = 1}^{n} a_{i}, ⋁_{i = 1}^{n} b_{i} 〉 . \end{matrix}$ (24)

Proof. Consider now the membership and nonmembership degrees of the aggregation results. $\begin{matrix} lim_{κ \to 0^{+}} (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}} \\ = exp (lim_{κ \to 0^{+}} \frac{1}{κ} ln (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})) \\ = exp (lim_{κ \to 0^{+}} \frac{\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ} ln (\frac{a_{i}^{q}}{1 - a_{i}^{q}})}{\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ}}) \\ = exp (\sum_{i = 1}^{n} w_{i} ln (\frac{a_{i}^{q}}{1 - a_{i}^{q}})) \\ = \prod_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{w_{i}} . \end{matrix}$ $\begin{matrix} lim_{κ \to 0^{+}} (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}} \\ = exp (lim_{κ \to 0^{+}} \frac{1}{κ} ln (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})) \\ = exp (lim_{κ \to 0^{+}} \frac{\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ} ln (\frac{1 - b_{i}^{q}}{b_{i}^{q}})}{\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ}}) \\ = exp (\sum_{i = 1}^{n} w_{i} ln (\frac{1 - b_{i}^{q}}{b_{i}^{q}})) \\ = \prod_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{w_{i}} . \end{matrix}$ Thus, it follows that $\begin{matrix} lim_{κ \to 0^{+}} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = lim_{κ \to 0^{+}} 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + {lim}_{κ \to 0^{+}} (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + {lim}_{κ \to 0^{+}} (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉 . \end{matrix}$ Moreover, by the fact that $lim_{κ \to \infty} (a^{κ} + b^{κ})^{1 / κ} = max {a, b}$ for a, b ≥ 0, we have $\begin{matrix} lim_{κ \to \infty} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(1 - \frac{1}{1 + {lim}_{κ \to \infty} (\sum_{i = 1}^{n} w_{i} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + {lim}_{κ \to \infty} (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 \\ = 〈 {(1 - \frac{1}{1 + {max}_{1 \leq i \leq n} {\frac{a_{i}^{q}}{1 - a_{i}^{q}}}})}^{1 / q}, \\ {(\frac{1}{1 + {max}_{1 \leq i \leq n} {\frac{1 - b_{i}^{q}}{b_{i}^{q}}}})}^{1 / q} 〉 \\ = 〈 arg max_{1 \leq i \leq n} {\frac{a_{i}^{q}}{1 - a_{i}^{q}}}, arg max_{1 \leq i \leq n} {\frac{1 - b_{i}^{q}}{b_{i}^{q}}} 〉 \\ = 〈 ⋁_{i = 1}^{n} a_{i}, ⋀_{i = 1}^{n} b_{i} 〉 . \end{matrix}$ The others follow similarly.□

From the above analysis, the following conclusions hold: ∀κ > 0, $\begin{matrix} 〈 {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉 \\ ⪯ q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ 〈 ⋁_{i = 1}^{n} a_{i}, ⋀_{i = 1}^{n} b_{i} 〉, \end{matrix}$ (25) $\begin{matrix} 〈 ⋀_{i = 1}^{n} a_{i}, ⋁_{i = 1}^{n} b_{i} 〉 ⪯ q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ 〈 {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - a_{i}^{q}}{a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{b_{i}^{q}}{1 - b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉 . \end{matrix}$ (26) Furthermore, due to

$\begin{matrix} 〈 {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉 ≼ 〈 ⋀_{i = 1}^{n} a_{i}, ⋁_{i = 1}^{n} b_{i} 〉, \\ 〈 {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - a_{i}^{q}}{a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{b_{i}^{q}}{1 - b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉 ⪯ 〈 ⋁_{i = 1}^{n} a_{i}, ⋀_{i = 1}^{n} b_{i} 〉, \end{matrix}$

we then conclude that the boundedness property characterized by Eqs. 25 and 26 are more refined than those determined by Eqs. 19 and 20.

Moreover, by $lim_{κ \to 0^{+}} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) = lim_{κ \to 0^{+}} q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n})$ , more precisely, $\begin{matrix} 〈 {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{a_{i}^{q}}{1 - a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - b_{i}^{q}}{b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉 \\ = 〈 (\frac{\prod_{i = 1}^{n} a_{i}^{q w_{i}}}{\prod_{i = 1}^{n} a_{i}^{q w_{i}} + \prod_{i = 1}^{n} (1 - a_{i}^{q})^{w_{i}}})^{1 / q}, \\ (\frac{\prod_{i = 1}^{n} b_{i}^{q w_{i}}}{\prod_{i = 1}^{n} b_{i}^{q w_{i}} + \prod_{i = 1}^{n} (1 - b_{i}^{q})^{w_{i}}})^{1 / q} 〉 \\ = 〈 {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - a_{i}^{q}}{a_{i}^{q}})^{w_{i}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{b_{i}^{q}}{1 - b_{i}^{q}})^{w_{i}}})}^{1 / q} 〉, \end{matrix}$ we have, ∀κ > 0,

$\begin{matrix} q {- ROFDWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ q {- ROFDWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) . \end{matrix}$ (27)

Similar conclusions hold for the q-rung orthopair fuzzy Dombi ordered weighted averaging/geometric (q-ROFDOWA/q-ROFDOWG) operators, which are defined as follows: [32] $\begin{matrix} q {- ROFDOWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) & = {\oplus_{D}}_{i = 1}^{n} w_{i} \cdot_{D} x_{τ (i)}, \\ q {- ROFDOWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) & = {⨂_{D}}_{i = 1}^{n} x_{τ (i)}^{D w_{i}}, \end{matrix}$ where x_τ(i) = 〈a_τ(i), b_τ(i)〉 is the ith largest of ${x_{i}}_{i = 1}^{n}$ .

These two aggregation operators are computed by $\begin{matrix} q & {- ROFDOWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{a_{τ (i)}^{q}}{1 - a_{τ (i)}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - b_{τ (i)}^{q}}{b_{τ (i)}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉, \end{matrix}$ (28) $\begin{matrix} q & {- ROFDOWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ = 〈 {(\frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{1 - a_{τ (i)}^{q}}{a_{τ (i)}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + (\sum_{i = 1}^{n} w_{i} (\frac{b_{τ (i)}^{q}}{1 - b_{τ (i)}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ (29)

The boundedness of the q-ROFDOWA/q-ROFDOWG operators can be expressed by $\begin{matrix} 〈 {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{a_{τ (i)}^{q}}{1 - a_{τ (i)}^{q}})^{w_{i}}})}^{1 / q}, \\ {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - b_{τ (i)}^{q}}{b_{τ (i)}^{q}})^{w_{i}}})}^{1 / q} 〉 \\ ⪯ q {- ROFDOWA}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ 〈 ⋁_{i = 1}^{n} a_{i}, ⋀_{i = 1}^{n} b_{i} 〉, \\ 〈 ⋀_{i = 1}^{n} a_{i}, ⋁_{i = 1}^{n} b_{i} 〉 ⪯ q {- ROFDOWG}_{w}^{κ} (x_{1}, x_{2}, \dots, x_{n}) \\ ⪯ 〈 {(\frac{1}{1 + \prod_{i = 1}^{n} (\frac{1 - a_{τ (i)}^{q}}{a_{τ (i)}^{q}})^{w_{i}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + \prod_{i = 1}^{n} (\frac{b_{τ (i)}^{q}}{1 - b_{τ (i)}^{q}})^{w_{i}}})}^{1 / q} 〉 . \end{matrix}$

5 Multiattribute decision making based on the q-ROFDWA/q-ROFDWG operators

In this section, we apply the developed operators to multiattribute decision making problems in the q-rung orthopair fuzzy setting.

In an MADM problem within the q-rung orthopair fuzzy context, there are m (m ≥ 2) feasible alternatives A₁, A₂, …, A_m, which are evaluated in terms of n attributes U = {u₁, u₂, …, u_n} whose weighting vector being w=(w₁, w₂, …, w_n)’. Let R=(γ_ij)_m ×n be the decision matrix given by the decision makers, where γ_ij = 〈a_ij, b_ij〉 is a q-ROFV indicating the evaluation value of alternative A_i with respect to attribute u_j, which means that the degree to which alternative A_i satisfies attribute u_j is a_ij and the degree to which alternative A_i dissatisfies attribute u_j is b_ij. Thus each alternative is described by a q-ROFS on U, and in the following we use A_i to denote the alternative and its corresponding q-ROFS interchangeably. Then we can rank all the alternatives based on the q-ROFDWA and q-ROFDWG operators by the steps as follows.

[Step 1.] Normalize the decision matrix.

Since there are two types of attributes, that is, the benefit type and cost type, in order to relieve the effect of the different attribute types, we need convert the cost type to benefit one by the following formula (For convenience of expression, the converted value is still denoted by γ_ij): $γ_{ij} = {\begin{matrix} 〈 a_{ij}, b_{ij} 〉, & if u_{j} is a benefit attribute, \\ 〈 b_{ij}, a_{ij} 〉, & if u_{j} is a cost attribute . \end{matrix}$ (30) Note that if there is no cost type attribute, this step can be omitted.

[Step 2.] Aggregate attribute values.

For each alternative A_i, calculate the aggregation of γ_i1, γ_i2, …, γ_in, denoted by η_i = 〈a_i, b_i〉, by the $q {- ROFDWA}_{w}^{κ}$ operator or the $q {- ROFDWG}_{w}^{κ}$ operator as follows: $\begin{matrix} q & {- ROFDWA}_{w}^{κ} (γ_{i 1}, γ_{i 2}, \dots, γ_{in}) = {\oplus_{D}}_{j = 1}^{n} w_{j} \cdot_{D} γ_{ij} \\ = 〈 {(1 - \frac{1}{1 + (\sum_{j = 1}^{n} w_{j} (\frac{a_{ij}^{q}}{1 - a_{ij}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(\frac{1}{1 + (\sum_{j = 1}^{n} w_{j} (\frac{1 - b_{ij}^{q}}{b_{ij}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ (31) $\begin{matrix} q & {- ROFDWG}_{w}^{κ} (γ_{i 1}, γ_{i 2}, \dots, γ_{in}) = {\oplus_{D}}_{j = 1}^{n} γ_{ij}^{D w_{j}} \\ = 〈 {(\frac{1}{1 + (\sum_{j = 1}^{n} w_{j} (\frac{1 - a_{ij}^{q}}{a_{ij}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q}, \\ {(1 - \frac{1}{1 + (\sum_{j = 1}^{n} w_{j} (\frac{b_{ij}^{q}}{1 - b_{ij}^{q}})^{κ})^{\frac{1}{κ}}})}^{1 / q} 〉 . \end{matrix}$ (32)

[Step 3.] Rank all the alternatives.

Rank all the alternatives by virtue of the score and accuracy of η_i for each A_i given by the following formulae: $s (η_{i}) = a_{i}^{q} - b_{i}^{q},$ (33) $t (η_{i}) = a_{i}^{q} + b_{i}^{q},$ (34) and then select the best one(s).

[Step 4.] End.

Example 5.1. An investor wants to invest in a company chosen from five potential alternatives A₁, A₂, A₃, A₄, and A₅. There are six attributes that are used to evaluate the ability of these companies, including the technical ability (u₁), the expected benefit (u₂), the competitive power on the market (u₃), the ability to bear risk (u₄), the management capability (u₅), and the organizational culture (u₆). The associated weighting vector is w = (0.2, 0.1, 0.3, 0.15, 0.15, 0.1)’. Assume that the assessment value of each alternative with respect to each attribute is represented by a q-ROFV as shown in the q-rung orthopair fuzzy decision matrix R given in Table 1. Then choose the most suitable company to invest.

Table 1

vq-Rung orthopair fuzzy decision matrix

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
A ₁	〈0.5, 0.2〉	〈0.8, 0.3〉	〈0.8, 0.3〉	〈0.7, 0.3〉	〈0.4, 0.2〉	〈0.4, 0.8〉
A ₂	〈0.6, 0.3〉	〈0.5, 0.8〉	〈0.6, 0.5〉	〈0.6, 0.5〉	〈0.7, 0.4〉	〈0.5, 0.6〉
A ₃	〈0.3, 0.4〉	〈0.8, 0.5〉	〈0.7, 0.6〉	〈0.6, 0.4〉	〈0.6, 0.2〉	〈0.4, 0.7〉
A ₄	〈0.7, 0.4〉	〈0.5, 0.6〉	〈0.7, 0.4〉	〈0.5, 0.5〉	〈0.7, 0.6〉	〈0.6, 0.5〉
A ₅	〈0.7, 0.6〉	〈0.6, 0.4〉	〈0.4, 0.7〉	〈0.4, 0.3〉	〈0.7, 0.7〉	〈0.5, 0.4〉

First, because all six attributes are the benefit type, there is no need to normalize the decision matrix.

Second, take q = 3 and κ = 0.5. Then the aggregations of these five alternatives by the $q {- ROFDWA}_{w}^{κ}$ operator are $\begin{matrix} η_{1} = 〈 0.6715, 0.2642 〉, & η_{2} = 〈 0.6002, 0.4420 〉, \\ η_{3} = 〈 0.6066, 0.3899 〉, & η_{4} = 〈 0.6507, 0.4596 〉, \\ η_{5} = 〈 0.5595, 0.4974 〉 . \end{matrix}$

Then, the scores of η_i are $\begin{matrix} s (η_{1}) = 0.2844, & s (η_{2}) = 0.1298, \\ s (η_{3}) = 0.1640, & s (η_{4}) = 0.1784, \\ s (η_{5}) = 0.0521 . \end{matrix}$

Finally, we can rank all the alternatives in descending order as follows: $A_{1} ≻ A_{4} ≻ A_{3} ≻ A_{2} ≻ A_{5} .$ And the best one is A₁ among these alternatives.

If one employs the $q {- ROFDWG}_{w}^{κ}$ operator to fuse q-ROFVs, the aggregations of these five alternatives are $\begin{matrix} η_{1} = 〈 0.5721, 0.3644 〉, & η_{2} = 〈 0.5887, 0.5149 〉, \\ η_{3} = 〈 0.4976, 0.4958 〉, & η_{4} = 〈 0.6264, 0.4831 〉, \\ η_{5} = 〈 0.4984, 0.5894 〉 . \end{matrix}$

Then, the scores of η_i are $\begin{matrix} s (η_{1}) = 0.1389, & s (η_{2}) = 0.0676, \\ s (η_{3}) = 0.0014, & s (η_{4}) = 0.1331, \\ s (η_{5}) = - 0.0810 . \end{matrix}$

Finally, we can rank all the alternatives in descending order as follows: $A_{1} ≻ A_{4} ≻ A_{2} ≻ A_{3} ≻ A_{5} .$ And the best one is A₁ among these alternatives.

In order to analyze the sensitivity of parameter κ in the developed operators, we set different κs to rank the alternatives while keeping other conditions unchanged. Then, based on the proposed operators we can obtain the aggregations and ranking of alternatives. The detailed results are shown in Tables 2 and 3.

Table 2

Aggregations and ranking of alternatives based on the $q {- ROFDWA}_{w}^{κ}$ operator (q = 3)

κ	η ₁	η ₂	η ₃	η ₄	η ₅	Ranking of alternatives
κ → 0	〈0.6237, 0.2933〉	〈0.5944, 0.4751〉	〈0.5576, 0.4489〉	〈0.6395, 0.4707〉	〈0.5278, 0.5480〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
κ = 0.1	〈0.6341, 0.2853〉	〈0.5956, 0.4680〉	〈0.5686, 0.4377〉	〈0.6419, 0.4684〉	〈0.5341, 0.5383〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
κ = 0.2	〈0.6441, 0.2786〉	〈0.5967, 0.4611〉	〈0.5791, 0.4262〉	〈0.6442, 0.4661〉	〈0.5405, 0.5282〉	A₁ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₅
κ = 1	〈0.7062, 0.2496〉	〈0.6058, 0.4148〉	〈0.6406, 0.3346〉	〈0.6597, 0.4501〉	〈0.5881, 0.4508〉	A₁ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₅
κ = 5	〈0.7757, 0.2143〉	〈0.6461, 0.3327〉	〈0.7384, 0.2267〉	〈0.6869, 0.4173〉	〈0.6678, 0.3387〉	A₁ ≻ A₃ ≻ A₅ ≻ A₄ ≻ A₂
κ = 10	〈0.7879, 0.2071〉	〈0.6707, 0.3160〉	〈0.7689, 0.2129〉	〈0.6934, 0.4087〉	〈0.6838, 0.3190〉	A₁ ≻ A₃ ≻ A₅ ≻ A₂ ≻ A₄
κ = 20	〈0.7940, 0.2035〉	〈0.6854, 0.3079〉	〈0.7847, 0.2064〉	〈0.6967, 0.4043〉	〈0.6919, 0.3094〉	A₁ ≻ A₃ ≻ A₅ ≻ A₂ ≻ A₄
κ→ ∞	〈0.8, 0.2〉	〈0.7, 0.3〉	〈0.8, 0.2〉	〈0.7, 0.4〉	〈0.7, 0.3〉	A₁ ≈ A₃ ≻ A₂ ≈ A₅ ≻ A₄

Table 3

Aggregations and ranking of alternatives based on the $q {- ROFDWG}_{w}^{κ}$ operator (q = 3)

κ	η ₁	η ₂	η ₃	η ₄	η ₅	Ranking of alternatives
κ → 0	〈0.6237, 0.2933〉	〈0.5944, 0.4751〉	〈0.5576, 0.4489〉	〈0.6395, 0.4707〉	〈0.5278, 0.5480〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
κ = 0.1	〈0.6132, 0.3030〉	〈0.5933, 0.4825〉	〈0.5460, 0.4594〉	〈0.6370, 0.4731〉	〈0.5215, 0.5573〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
κ = 0.2	〈0.6027, 0.3148〉	〈0.5922, 0.4902〉	〈0.5341, 0.4694〉	〈0.6345, 0.4756〉	〈0.5154, 0.5662〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
κ = 1	〈0.5298, 0.4785〉	〈0.5830, 0.5597〉	〈0.4439, 0.5295〉	〈0.6122, 0.4958〉	〈0.4753, 0.6174〉	A₄ ≻ A₁ ≻ A₂ ≻ A₃ ≻ A₅
κ = 5	〈0.4352, 0.7358〉	〈0.5446, 0.7358〉	〈0.3327, 0.6318〉	〈0.5407, 0.5581〉	〈0.4202, 0.6759〉	A₄ ≻ A₃ ≻ A₅ ≻ A₂ ≻ A₁
κ = 10	〈0.4176, 0.7689〉	〈0.5237, 0.7689〉	〈0.3160, 0.6644〉	〈0.5204, 0.5784〉	〈0.4101, 0.6877〉	A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁
κ = 20	〈0.4087, 0.7847〉	〈0.5118, 0.7847〉	〈0.3079, 0.6822〉	〈0.5102, 0.5892〉	〈0.4050, 0.6939〉	A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁
κ→ ∞	〈0.4, 0.8〉	〈0.5, 0.8〉	〈0.3, 0.7〉	〈0.5, 0.6〉	〈0.4, 0.7〉	A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁

As can be seen from Table 2, the ranking results are slightly different with respect to different parameter κ. But the best selection is always the company A₁, which demonstrates the robustness of the proposed approach from one facet. And with the increase of κ, the aggregation of each alternative based on the $q {- ROFDWA}_{w}^{κ}$ operator becomes larger and larger. More precisely, as κ increases, the degree of support for is larger and larger, whereas the degree of support against is smaller and smaller. While for the $q {- ROFDWA}_{w}^{κ}$ operator, the situation is just the opposite as seen from Table 3. Consequently, the score function of each aggregation is monotonic based on both operators, which is shown in Fig. 1. Additionally, the parameter κ indicates the attitude of decision makers. Especially when κ→ ∞ the decision makers are the most optimistic by the $q {- ROFDWA}_{w}^{κ}$ operator, i.e., the aggregation of q-ROFVs is their maximum; while the decision makers are the most pessimistic by the $q {- ROFDWG}_{w}^{κ}$ operator, i.e., the aggregation of q-ROFVs is their minimum.

Fig. 1

Scores of aggregations of alternatives based on the q - ROFDWA and q - ROFDWG operators with different parameters κ.

Now turn our attention to the rung q, since γ₁₂ = 〈0.8, 0.3〉 is not an intuitionistic fuzzy value, aggregation operators on intuitionistic fuzzy values do not work in this problem. If γ₁₂ is changed to 〈0.8, 0.7〉, by the fact that 〈0.8, 0.7〉 is not a Pythagorean fuzzy value, aggregation operators on Pythagorean fuzzy values will not work neither. But aggregation operators on q-rung orthopair fuzzy values can always work by choosing suitable q. The following we set different qs to rank the alternatives, and the results based on the $q {- ROFDWA}_{w}^{κ}$ and $q {- ROFDWG}_{w}^{κ}$ operators are shown in Tables 4 and 5, respectively.

Table 4

Aggregations and ranking of alternatives based on the $q {- ROFDWA}_{w}^{κ}$ operator (κ = 0.5)

q	η ₁	η ₂	η ₃	η ₄	η ₅	Ranking of alternatives
q = 2	〈0.6710, 0.2720〉	〈0.6000, 0.4531〉	〈0.6045, 0.4119〉	〈0.6505, 0.4635〉	〈0.5579, 0.5158〉	A₁ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₅
q = 5	〈0.6775, 0.2536〉	〈0.6013, 0.4233〉	〈0.6159, 0.3513〉	〈0.6524, 0.4531〉	〈0.5675, 0.4653〉	A₁ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₅
q = 10	〈0.7021, 0.2376〉	〈0.6068, 0.3885〉	〈0.6432, 0.2873〉	〈0.6602, 0.4409〉	〈0.5957, 0.4099〉	A₁ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₅
q = 20	〈0.7375, 0.2215〉	〈0.6198, 0.3505〉	〈0.6799, 0.2417〉	〈0.6735, 0.4261〉	〈0.6349, 0.3600〉	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂
q = 30	〈0.7551, 0.2144〉	〈0.6320, 0.3337〉	〈0.7032, 0.2270〉	〈0.6810, 0.4184〉	〈0.6540, 0.3400〉	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂
q = 40	〈0.7652, 0.2108〉	〈0.6425, 0.3251〉	〈0.7200, 0.2199〉	〈0.6853, 0.4140〉	〈0.6646, 0.3298〉	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂
q = 50	〈0.7716, 0.2086〉	〈0.6511, 0.3199〉	〈0.7326, 0.2158〉	〈0.6881, 0.4112〉	〈0.6714, 0.3236〉	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂

Table 5

Aggregations and ranking of alternatives based on the $q {- ROFDWG}_{w}^{κ}$ operator (κ = 0.5)

q	η ₁	η ₂	η ₃	η ₄	η ₅	Ranking of alternatives
q = 2	〈0.5889, 0.3534〉	〈0.5907, 0.5123〉	〈0.5192, 0.4913〉	〈0.6310, 0.4820〉	〈0.5082, 0.5873〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
q = 5	〈0.5437, 0.4023〉	〈0.5851, 0.5260〉	〈0.4604, 0.5092〉	〈0.6176, 0.4876〉	〈0.4825, 0.5982〉	A₄ ≻ A₁ ≻ A₂ ≻ A₃ ≻ A₅
q = 10	〈0.4974, 0.5147〉	〈0.5770, 0.5668〉	〈0.3997, 0.5429〉	〈0.5975, 0.5037〉	〈0.4564, 0.6236〉	A₄ ≻ A₂ ≻ A₁ ≻ A₃ ≻ A₅
q = 20	〈0.4556, 0.6358〉	〈0.5621, 0.6418〉	〈0.3513, 0.5860〉	〈0.5662, 0.5318〉	〈0.4319, 0.6523〉	A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅
q = 30	〈0.4379, 0.6862〉	〈0.5494, 0.6870〉	〈0.3338, 0.6111〉	〈0.5469, 0.5495〉	〈0.4216, 0.6656〉	A₄ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₂
q = 40	〈0.4285, 0.7130〉	〈0.5397, 0.7131〉	〈0.3251, 0.6279〉	〈0.5355, 0.5606〉	〈0.4162, 0.6733〉	A₄ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₂
q = 50	〈0.4228, 0.7296〉	〈0.5325, 0.7296〉	〈0.3199, 0.6400〉	〈0.5284, 0.5679〉	〈0.4130, 0.6782〉	A₄ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₂

From Tables 4 and 5, one can see that for any q ≥ 2 we can aggregate each of the alternatives by the q - ROFDWA and q - ROFDWG operators. And with the increase of q, the aggregation of each alternative becomes larger and larger based on the q - ROFDWA operator while smaller and smaller based on the q - ROFDWG operator. The scores of aggregations of all five alternatives by the q - ROFDWA and q - ROFDWG operators are presented in Fig. 2. On the other hand, for all the listed cases, although the ranking results of these alternatives are slightly different, the best selection is always the company A₁ based on the q - ROFDWA operator while the company A₄ based on the q - ROFDWG operator when q > 3, which shows the robustness of this approach from another facet.

Fig. 2

Scores of aggregations of alternatives based on the q - ROFDWA and q - ROFDWG operators with different rungs q.

The following we compare the q - ROFDWA and q - ROFDWG [32] operators with other aggregation operators in MADM, including the IFWDBM (intuitionistic fuzzy weighted Dombi Bonferroni mean) [25], IFDGBM (intuitionistic fuzzy Dombi geometric Bonferroni mean) [25], q-ROFWA [10], q-ROFWG [10], q-ROFEWA (q-rung orthopair fuzzy Einstein weighted averaging), and q-ROFEWG (q-rung orthopair fuzzy Einstein weighted geometric) operators. The aggregated results and ranking of alternatives based on these operators are summarized in Table 6.

Table 6

Aggregations and ranking of alternatives based on the existing aggregation operators (q = 3)

Operator	η ₁	η ₂	η ₃	η ₄	η ₅	Ranking of alternatives
IFWDBM	–	–	–	–	–	Cannot be calculated
IFDGBM	–	–	–	–	–	Cannot be calculated
q-ROFWA	〈0.6876, 0.2871〉	〈0.6033, 0.4660〉	〈0.6269, 0.4403〉	〈0.6555, 0.4681〉	〈0.5779, 0.5344〉	A₁ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₅
q-ROFWG	〈0.6002, 0.4409〉	〈0.5920, 0.5402〉	〈0.5406, 0.5213〉	〈0.6337, 0.4926〉	〈0.5181, 0.6083〉	A₄ ≻ A₁ ≻ A₂ ≻ A₃ ≻ A₅
q-ROFEWA	〈0.6771, 0.2897〉	〈0.6017, 0.4698〉	〈0.6187, 0.4442〉	〈0.6528, 0.4693〉	〈0.5711, 0.5405〉	A₁ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₅
q-ROFEWG	〈0.6097, 0.4199〉	〈0.5931, 0.5301〉	〈0.5480, 0.5154〉	〈0.6362, 0.4902〉	〈0.5223, 0.6019〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
q-ROFDWA	〈0.6715, 0.2642〉	〈0.6002, 0.4420〉	〈0.6066, 0.3899〉	〈0.6507, 0.4596〉	〈0.5595, 0.4974〉	A₁ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₅
q-ROFDWG	〈0.5721, 0.3644〉	〈0.5887, 0.5149〉	〈0.4976, 0.4958〉	〈0.6264, 0.4831〉	〈0.4984, 0.5894〉	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅

As seen from Table 6, these operators can be applied to aggregate the fuzzy information in this example except the IFWDBM and IFDGBM operators. Moreover, the ranking results corresponding to the q-ROFDWA operator are the same as those corresponding to the q-ROFEWA operator; the ranking results corresponding to the q-ROFDWG operator are the same as those corresponding to the q-ROFEWG operator. And the best selection is always the company A₁ based on these aggregation operators except the q-ROFWG operator, which shows our methods may serve as an alternative approach in practical MADM problems. Additionally, for the operators designed for handling q-rung orthopair fuzzy information, the fact that the q-ROFDWA and q-ROFDWG operators having a dedicated parameter in their operational rules makes them more flexible than the others.

6 Conclusions

Dombi operations with a general parameter are a special class of operations that are usually used in the fuzzy set theory. Jana et al. introduced some Dombi operations and Dombi aggregation operators of q-ROFVs which are extensions of (intuitionistic/Pythagorean/Fermatean) fuzzy values. In this paper, we show how to construct the Dombi scalar multiplication and Dombi exponentiation over q-ROFVs by their Dombi sum and Dombi product, respectively. These operations satisfy the same laws of arithmetic as those for real numbers, which confirms that they are indeed well-defined. For aggregation operators of q-ROFVs based on Dombi operations, we discuss some of their desirable properties including the shift-invariance and the homogeneity. Moreover, the boundedness of aggregations obtained by the q-rung orthopair fuzzy Dombi aggregation operators are narrowed down by the parameter in Dombi operations. Finally, we propose a novel method for solving multiattribute decision making problems based on developed operators under q-rung orthopair fuzzy circumstance. A numerical example of investment selection is given to illustrate the decision making procedure in detail. The experimental results show that the parameter κ in the model reflects the attitude of decision makers. More precisely, as κ trends to infinity, the decision makers are the most optimistic for the q-ROFDWA operator based model, and the most pessimistic for the q-ROFDWG operator based model.

Considering that there are many other types of aggregation manners on q-ROFVs based on algebraic operations, such as Bonferroni mean, Heronian mean, Maclaurin symmetric mean and Muirhead mean. Our future work will combine them with Dombi operations and develop some new aggregation operators on q-ROFVs. Moreover, we can extend this work to the environment of picture fuzzy sets and T-spherical fuzzy sets.

Footnotes

Acknowledgments

The author gratefully appreciates the anonymous reviewers for their constructive comments and suggestions that greatly helped to improve the quality of this paper. This research was supported by the National Natural Science Foundation of China (Grant no. 61806182), the Scientific Research Fund for Young Teachers of Zhengzhou University (Grant no. 32220326), the Research Base Program of New Disciplines in Economics and Management of Zhengzhou University (Grant no. 101/32610168) and the Training Project for Young Backbone Teachers of Colleges and Universities of Henan Province.

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