Q-rung orthopair triangular fuzzy linguistic Frank aggregation operators and their application

Abstract

The aim of this paper is to introduce a Frank operator in the q-rung orthopair triangular fuzzy linguistic environment on the basis of the notion of the Frank operator and the q-rung orthopair fuzzy set. Firstly, the concept of a q-rung orthopair triangular fuzzy linguistic set (q-ROTrFLS) is proposed, then several basic operations, score, and accuracy functions to compare the q-ROTrFLS values are defined. Secondly, a series of q-rung orthopair triangular fuzzy linguistic Frank aggregation operators are developed, such as q-rung orthopair triangular fuzzy linguistic Frank weighted average (q-ROTrFLWA)operator,q-rung orthopair triangular fuzzy linguistic Frank weighted geometric (q-ROTrFLWG) operator, and we introduce several relevant properties of these operators and prove their validity, and show the relevant relationship between some operators. Thirdly, two different decision-making approaches are constructed in the q-rung orthopair triangular fuzzy linguistic environment. Furthermore, a practical example is given to explain the developed methods. Finally, a comparative study is conducted, and the relevant sensitivity analysis is also discussed, and the outcome shows the prominence and the effectiveness of the developed methods compared to previous studies.

Keywords

q-rung orthopair triangular fuzzy linguistic set Frank operator multi-attribute decision making (MADM)q-rung orthopair set

1 Introduction

In practical application, due to the variability and uncertainty of social environments, many factors cannot be described with accurate values. In 1965, Zadeh introduced fuzzy theory [1] to deal with multi-attribute group decision-making (MADM). Afterward, Atanassov [2]introduced intuitionistic fuzzy sets, which require the sum of the membership and non-membership degrees to be less than or equal to 1; However, in real life, there will be situations where the sum of the membership and non-membership degrees is greater than 1. Therefore, in 2017, Yager [3 –5] proposed the Pythagorean fuzzy set (PFS), which allows the sum of membership and non-membership to be greater than 1, and the square of membership and non-membership is less than or equal to 1. The PFS is more general than the intuitionistic fuzzy set (IFS), and many studies [6 –15] have been conducted on this type of fuzzy set.

As the increasing complexity of society and theory develops, a new concept, the q-rung orthopair fuzzy set (q-ROFS), was proposed by Yager [16]in 2017, which allows the sum of the membership and non-membership degrees to be greater than 1, and the sum of the membership to the power q and the non-membership to the power q is restricted to one. This concept expanded the research scope of IFS and PFS, and thus q-ROFS is more flexible and more general than IFS and PFS due to the presence of the parameter q in dealing with uncertain information. Since q-ROFS was first proposed, a series of studies on it have been carried out by scholars. Liu and Liu [17] proposed several q-rung orthopair fuzzy Bonferroni mean operators. Liu and Wang [18] defined the q-rung orthopair fuzzy weighted average (q-ROFWA) operator. Liu [19] defined the q-rung orthopair fuzzy power Maclaurin symmetric mean operator. Wei et al. [20, 21] extended the Heronian average operator and the Maclaurin symmetric mean operator to q-ROFS. Ju et al. [22 –24] defined the q-rung orthopair fuzzy two-tuple linguistic Muirhead aggregation operator, interval q-rung orthopair fuzzy weighted average operator, and q-rung orthopair fuzzy power weighted aggregation operator. Wang et al. [25] proposed several q-rung orthopair fuzzy Hamy mean operators. Liu et al. [26] defined the concept of complex q-ROFSs. Peng et al. [27] developed several exponential operations for q-ROFSs. Garg et al. [28] developed the connection number of q-ROFSs. Du et al. [29] developed q-rung trapezoidal fuzzy linguistic Hamacher operators. Since it was initiated, many researchers have utilized it in various areas [30 –40].

Frank operational laws [41] are an extended form of Archimedes’ T mode and Archimedes’ S mode. Frank operational laws are generalizations of operational laws such as algebraic, Einstein, and Hamacher operational laws, and are a flexible family of continuous triangular norms. So, Frank operational laws are more widely used than many other operators such as algebraic operator, Einstein operator, and Dombi operator. Because Frank operational laws have e a certain parameter which makes them more flexible and more adequate in the process of information fusion, and Frank operator has less calculation quantity than many operators such as Bonferroni mean (BM) operator and Heronian Mean(HM) operator, and Frank operations are only one type of t-norm that can satisfy the compatibility rule. Some scholars have made a series of extensions of the Frank operator. For example, Qin et al. [42, 43] applied the Frank operator to solve the MADM problem under triangular interval type-2 fuzzy information, and they proposed the hesitant fuzzy Frank weighted average (HFFWA) operator. Zhang [44] defined the interval intuitionistic fuzzy Frank weighted arithmetic operators. Ji et al. [45] proposed a simplified neutral Frank standardized priority Bonferroni average operator. Liu [46] developed Normal neutrosophic frank aggregation operators. Mahmood et al. [47] examined several interval-valued picture fuzzy frank aggregation operators. Xing et al. [48] proposed the Pythagorean fuzzy Choquet Frank weighted arithmetic operator. Du et al. [49] defined interval intuitionistic fuzzy linguistic Frank aggregation operators.

In numerous decision-making issues, decision-makers are often unable to accurately express their opinions by only a linguistic variable from a predefined linguistic term set or a q-rung orthopair fuzzy number. Thus, some novel notions combining a linguistic term set and a q-rung orthopair fuzzy set have been introduced,such as Linguistic q-rung orthopair fuzzy set [50], q-rung orthopair uncertain linguistic fuzzy set [51], probabilistic linguistic q-rung orthopair fuzzy set [52], and complex q-rung orthopair linguistic fuzzy set [53], Since Triangular fuzzy linguistic numbers [54] can effectively represent information that is difficult to describe with precise numerical values, it is a method to convert the fuzzy and uncertain linguistic variables into definite values. The triangular fuzzy linguistic number can be used to solve the problem that the evaluated object cannot be accurately measured and can only be defined in linguistic variables. However, the existing fuzzy sets cannot solve MADM problems under a q-rung orthopair triangular linguistic fuzzy environment. Based on these limitations, in this paper, we introduce a q-rung orthopair triangular fuzzy linguistic variable < [s_α, s_β, s_r], (u, v)>to present the capability by giving the membership degree u and non-membership degree v to [s_α, s_β, s_r], which is combined with triangular fuzzy linguistic variables and a q-rung orthopair fuzzy set. This fuzzy set has a more flexible range of applications than the q-rung fuzzy set. The main advantages of (q-ROTrFLS) include (1) q rung orthopair triangular fuzzy linguistic term which can deal with the uncertainty more precisely than q rung orthopair fuzzy set and q-rung orthopair uncertain linguistic fuzzy set and Linguistic q-rung orthopair fuzzy set in qualitative; (2) membership and non-membership degree are complements of the triangular linguistic terms, which can show us how much degree that an attribute value belongs to and not belong to a q rung triangular linguistic term in quantitative; (3) Many fuzzy set can not deal with MADM under the linguistic environment such as the rough set and neutrosophic set, But the q rung orthopair triangular fuzzy linguistic number can be used to solve such multi-attribute decision-making problems effectively.

Moreover, since the Frank t-norm and t-conorm are interesting generalizations of algebraic operational laws, it is more general and more adequate to deal with MCDM problems. Based on the Frank operator and the-q-rung orthopair triangular fuzzy linguistic set, some q-rung orthopair triangular fuzzy linguistic Frank operators are given in this paper.

The remainder of this article is arranged as follows: In Section 2, we review fundamental conceptions related to q-ROFSs. In Section 3, we present the q-rung orthopair triangular fuzzy linguistic Frank weighted (q-ROFTRLDHM) operator and the q-rung orthopair triangular fuzzy linguistic Frank weighted (q-ROFDHM) operator, and then several desirable properties are proved along with some of their special cases. In Section 4, we introduce two novel methods to solve the MADM problems based on the q-ROFWHM (q-ROFWDHM) operator. In Section 5, we give a numerical example to illustrate the superiority and effectiveness of the proposed methods. Section 6 concludes the paper.

2 Preliminarie

2.1 q-rung orthopair fuzzy set

Definition 1. [16]. Let A ={ 〈 x, u (x) , v (x) 〉 |x ∈ X } be the q-rung orthopair fuzzy set on X, where X is a non-empty set of real numbers, u (x) : X → [0, 1], v (x) : X → [0, 1], u (x) indicates the degree of membership of x in A, v (x) indicates the degree of non-membership of x in A, there is 0 ⩽ (u (x)) ^q + (v (x)) ^q ⩽ 1, and let $π (x) = \sqrt[q]{1 - {(u (x))}^{q} - {(v (x))}^{q}}$ be the degree of hesitation.

2.2 Triangular fuzzy linguistic variables

A linguistic set is proposed as a finite and completely ordered discrete term set, S = (s₀, s₁, …, s_l-1), where l is the odd value. If l = 5, then the linguistic term set S can be developed as follows:

S = (s₀ : extremelypoor, s₁ : verypoor, s₂ : poor, s₃ : fair, s₄ : good,).

Definition 2. [54]. Let S = {s_θ|s₀ ⩽ s_θ ⩽ s_l-1, θ ∈ [0, l - 1]}, which is the continuous form of the linguistic set S. s_α(x), s_β(x) and s_γ(x) are three linguistic terms in S, and s₀ ⩽ s_α(x) ⩽ s_β(x) ⩽ s_γ(x) ⩽ s_l-1 if 0 ⩽ α (x) ⩽ β (x) ⩽ γ (x) ⩽ l - 1. Then, the triangular fuzzy linguistic variable is introduced as $\tilde{s} = [s_{α (x)}, s_{β (x)}, s_{γ (x)}]$ , and S denotes a set of triangular fuzzy linguistic variables.

3 The q-rung orthopair triangular fuzzy linguistic set (q-ROTrFLS)

According to the concept of q-rung orthopair set and Triangular fuzzy linguistic set, we define a new fuzzy set called q-rung orthopair triangular fuzzy linguistic set (q-ROTrFLS).

Definition 3. Let $\bar{S} = {s_{i} | s_{0} ⩽ s_{i} ⩽ s_{g}, i \in [0, g]}$ be the continuous form of S, X be a non-empty set of the universe, a q rung orthopair triangular fuzzy linguistic set A in X can be represented as follows: $\begin{matrix} A = {〈 x [s_{α (x)}, s_{β (x)}, s_{γ (x)}], (u (x), v (x)) 〉 | x \in X}, \\ \forall x \in X \end{matrix}$ where $[s_{α (x)}, s_{β (x)}, s_{γ (x)}] \in \bar{S}$ , u (x) and v (x) are two sets of some values in [0,1], denoting the possible membership degrees and non-membership degrees of the element xîX to the linguistic variable [s_α(x), s_β(x), s_γ(x)], respectively, with the conditions: 0 ⩽ (u (x)) ^q + (v (x)) ^q ⩽ 1, and its hesitation degree is π (x) = (1 - (u (x)) ^q - (v (x)) ^q) ^1/q.

For convenience, we call a = 〈 [s_α(x), s_β(x), s_γ(x)] , (u (x) , v (x)) 〉 a q-rung orthopair triangular fuzzy linguistic variable (q-ROTRFLV) and A the set of all q-rung orthopair triangular fuzzy linguistic variables (q-ROTFLVs). Afterwards, a q-rung orthopair triangular fuzzy linguistic number (q-ROTRFLN) is usually abbreviated as a =〈 [s_α, s_β, s_γ] , (u, v) 〉.

Theorem 1. Let a_i and a_j be two q-ROTRFLVs, then we can obtain the following rules.

$a_{i} \oplus a_{j} = 〈 [s_{α_{1} + α_{2}}, s_{β_{1} + β_{2}}, s_{γ_{1} + γ_{2}} s_{η_{1} + η_{2}}], (\sqrt[q]{u_{i}^{q} + u_{j}^{q} - u_{i}^{q} u_{j}^{q}}, v_{i} v_{j}) 〉;$

$a_{i} \otimes a_{j} = 〈 [s_{α_{1} α_{2}}, s_{β_{1} β_{2}}, s_{γ_{1} γ_{2}}], (u_{i} u_{j}, \sqrt[q]{v_{i}^{q} + v_{j}^{q} - v_{i}^{q} v_{j}^{q}}) 〉;$

${ka}_{i} = 〈 [s_{k \times α_{i}}, s_{k \times β_{i}}, s_{k \times r_{i}}], (\sqrt[q]{1 - {(1 - u_{i}^{q})}^{k}}, v_{i}^{k}) 〉, k > 0;$

${(a_{i})}^{k} = 〈 [s_{{(α_{i})}^{k}}, s_{{(β_{i})}^{k}}, s_{(r_{i}))}^{k}],$ $(u_{i}^{k}, \sqrt[q]{1 - {(1 - v_{i}^{q})}^{k}}) 〉, k > 0$ .

Definition 4. Let a_i =〈 [s_{α
_i}, s_{β
_i}, s_{γ
_i}] , (u_i, v_i 〉 and a_j =〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}] , (u_j, v_j) 〉 be two q-ROTrFLVs, the expected function E (a_i) and the accuracy function H (a_i) of the q-ROTrFLVs can be respectively defined as follows: $E (a_{i}) = (u_{i}^{q} - v_{i}^{q}) (α_{i} + β_{i} + γ_{i}) / 3$ (1) $H (a_{i}) = (u_{i}^{q} + v_{i}^{q}) (α_{i} + β_{i} + γ_{i}) / 3$ (2)

Definition 5. Let a_i =〈 [s_{α
_i}, s_{β
_i}, s_{γ
_i}] , (u_i, v_i 〉 and a_j =〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}] , (u_j, v_j) 〉 be two q-ROTrFLVs, they can be compared by the following rules:

If E (a_i) > E (a_j), we get a_i > a_j;

If E (a_i) = E (a_j), we get

If H (a_i) > H (a_j), we get a_i > a_j;

If H (a_i) = H (a_j), we get a_i = a_j.

Definition 6. Let a_j = 〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}] , (u_j, v_j) 〉 (j = 1, 2, . . . , n) be a set of the q-ROTrFLVs. Then, the aggregated value of q-ROTrFLVs, by using the q-ROTrFLWA operator, is also a q-ROTRFLV: $\begin{matrix} q - ROTrFLWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} \\ = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], \\ (\sqrt[q]{1 - \prod_{j = 1}^{n} ({(1 - u_{j}^{q})}^{w_{j}})}, \prod_{j = 1}^{n} {(v_{j})}^{w_{j}}) 〉 \end{matrix}$ (3) where w = (w₁, w₂, …, w_n) ^T is the corresponding weight of a_j (j = 1, 2, . . . , n), such that 0 ⩽ w_j ⩽ 1 and $\sum_{j = 1}^{n} w_{j} = 1$ .

Definition 7. Let a_j = 〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}] , (u_j, v_j) 〉 (j = 1, 2, . . . , n) be a set of the q-ROTrFLVs. Then, the aggregated value of q-ROTrFLVs, by using the q-ROTrFLWG operator, is also a q-ROTrFLFV: $\begin{matrix} q - ROTrFLWG (a_{1}, a_{2}, \dots, a_{n}) = \otimes_{j = 1}^{n} a_{j}^{w_{j}} \\ = 〈 [s_{\prod_{i = 1}^{n} {(α_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(β_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(γ_{j})}^{w_{j}}}], \\ (\prod_{j = 1}^{n} {(u_{j})}^{w_{j}}, \sqrt[\begin{matrix} q \end{matrix}]{1 - \prod_{j = 1}^{n} ({(1 - v_{j}^{q})}^{w_{j}})}) 〉 \end{matrix}$ (4) where w = (w₁, w₂, …, w_n) ^T is the corresponding weight of a_j (j = 1, 2, . . . , n), such that 0 ⩽ w_j ⩽ 1, and $\sum_{j = 1}^{n} w_{j} = 1$ .

4 q-rung orthopair triangular fuzzy linguistic set based on Frank aggregation operators

Definition 8 [55]. Frank T-norm and S-norm under the q-rung orthopair fuzzy environment are developed as follows: $\begin{matrix} T (x, y) = \sqrt[q]{{log}_{τ} (1 + \frac{(τ^{x^{q}} - 1) (τ^{y^{q}} - 1)}{τ - 1}),} \\ S (x, y) = \\ \sqrt[q]{1 - {log}_{τ} (1 + \frac{(τ^{1 - x^{q}} - 1) (τ^{1 - y^{q}} - 1)}{τ - 1})}, \\ (τ > 1) \end{matrix}$

Based on the Definition 8, we discuss several special forms of Frank T-norm and S-norm under the q-rung orthopair fuzzy environment as below:

If τ → 1, the q-rung orthopair Frank T-norm is simplified to the q-rung orthopair algebraic T module, so, we have $T (x, y) = x \otimes y = xy$ and,the q-rung orthopair Frank S-norm is simplified to the q-rung orthopair algebraic S module, that is $S (x, y) = x \oplus y = \sqrt[q]{x^{q} + y^{q} - x^{q} y^{q}}$

If τ→ + ∞, the q-rung orthopair Frank T-norm is reduced to the q-rung orthopair Einstein T-norm, so, we have $T (x, y) = x \otimes y = \frac{xy}{\sqrt[q]{1 - (1 - x^{q}) (1 - y^{q})}}$ and, the q-rung orthopair Frank S-norm is reduced to the q-rung orthopair Einstein S-norm, that is $S (x, y) = x \oplus y = \sqrt[q]{\frac{x^{q} + y^{q}}{1 + x^{q} y^{q}}}$

According to the concept of q-rung orthopair triangular fuzzy linguistic numbers and the Frank T-norm and S-norm, the Frank information aggregation operators in the q-rung orthopair triangular fuzzy linguistic environment can be defined.

Definition 9. Let a_i =〈 [s_{α
_i}, s_{β
_i}, s_{γ
_i}] , (u_i, v_i 〉 and a_j =〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}] , (u_j, v_j) 〉 be two q-ROTrFLVs, λ > 0, and τ > 1, then we have the following rules:

$a_{1} \oplus a_{2} = 〈 [s_{α_{1} + α_{2}}, s_{β_{1} + β_{2}}, s_{γ_{1} + γ_{2}}], (\sqrt[q]{1 - {log}_{τ} (1 + \frac{(τ^{1 - u_{1}^{q}} - 1) (τ^{1 - u_{2}^{q}} - 1)}{(τ - 1)})}, \sqrt[q]{{log}_{τ} (1 + \frac{(τ^{v_{1}^{q}} - 1) (τ^{v_{2}^{q}} - 1)}{τ - 1})}) 〉$ ;

$a_{1} \otimes a_{2} = 〈 [s_{α_{1} \times α_{2}}, s_{β_{1} \times β_{2}}, s_{r_{1} \times r_{2}}], (\sqrt[q]{{log}_{τ} (1 + \frac{(τ^{u_{1}^{q}} - 1) (τ^{u_{2}^{q}} - 1)}{τ - 1})}, \sqrt[q]{1 - {log}_{τ} (1 + \frac{(τ^{1 - v_{1}^{q}} - 1) (τ^{1 - v_{2}^{q}} - 1)}{(τ - 1)})}) 〉$ ;

$λ a_{1} = 〈 [s_{λ \times α_{1}}, s_{λ \times β_{1}}, s_{λ \times r_{1}}], (\sqrt[q]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u^{q}} - 1)}^{λ}}{{(τ - 1)}^{λ - 1}})}, \sqrt[q]{{log}_{τ} (1 + \frac{{(τ^{v^{q}} - 1)}^{λ}}{{(τ - 1)}^{λ - 1}})}) 〉;$

$a_{1}^{λ} = 〈 [s_{{(α_{1})}^{λ}}, s_{{(β_{1})}^{λ}}, s_{{(r_{1})}^{λ}}], (\sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{{(τ^{u^{q}} - 1)}^{λ}}{{(τ - 1)}^{λ - 1}})}, \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - v^{q}} - 1)}^{λ}}{{(τ - 1)}^{λ - 1}})}) 〉$ .

Furthermore, several properties of the operational law can be obtained easily.

Theorem 2. Let a₁, a₂, a be any three q-ROTrFLVs, and τ > 1, λ > 0, k, k₁, k₂ > 0, then

a₁ ⊕ _Fa₂ = a₂ ⊕ _Fa₁

k · (a₁ ⊕ _Fa₂) = ka₁ ⊕ _Fka₂

(k₁k₂) · a = k₁ · (k₂ · a)

k₁a ⊕ _Fk₂a = (k₁ + k₂) a

$a_{1}^{k} \otimes_{F} a_{2}^{k} = {(a_{2} \otimes_{F} a_{1})}^{k}, k ⩾ 0$

a^k
₁ ⊗ _Fa^k
₂ = (a) ^k₁+k₂.

4.1 The q-rung orthopair triangular fuzzy linguistic Frank weighted operators

Based on the operation rules of the Frank operator under the q-rung orthopair fuzzy linguistic fuzzy environment, several q-rung orthopair triangular fuzzy linguistic Frank aggregation operators are defined as follows.

Definition 10. Let a_j (j = 1, 2, . . . , n) is a set of q-ROTRFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , with τ > 1. Then, the q-rung orthopair triangular fuzzy linguistic Frank weighted average (q-ROTrFLFWA) operator is a mapping q-ROTrFLFWA: Ωⁿ → Ω, if $\begin{matrix} q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} \\ = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], (\sqrt[q]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})}, \sqrt[q]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})}) 〉 \end{matrix}$ (5) where Ω is the set of all q-rung orthopair triangular fuzzy linguistic variables. Then the function q - ROTrFLFWA is known as the q - ROTrFLFWA operator.

Proof By using the mathematical induction, mathematical induction is used as follows:

(1) For n = 1, Theorem 2 is established. (2) For n = 2, we have $w_{1} a_{1} = 〈 [s_{w_{1} \times α_{1}}, s_{w_{1} \times β_{1}}, s_{w_{1} \times r_{1}}], (\sqrt[q]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{1}^{q}} - 1)}^{w_{1}}}{{(τ - 1)}^{w_{1} - 1}})}, \sqrt[q]{{log}_{τ} (1 + \frac{{(τ^{v_{1}^{q}} - 1)}^{w_{1}}}{{(τ - 1)}^{w_{1} - 1}})}) 〉$ $w_{2} a_{2} = 〈 [s_{w_{2} \times α_{2}}, s_{w_{2} \times β_{2}}, s_{w_{2} \times r_{2}}] \sqrt[q]{1 - \log_{τ} (1 + \frac{{(τ^{1 - u_{2}^{q}} - 1)}^{w_{2}}}{{(τ - 1)}^{w_{2} - 1}})}, \sqrt[]{\log_{τ} (1 + \frac{{(τ^{v_{2}^{q}} - 1)}^{w_{2}}}{{(τ - 1)}^{w_{2} - 1}})} 〉$

Then, we get $q - RFFWA (a_{1}, a_{2}) = w_{1} a_{1} + w_{2} a_{2} = 〈 [s_{w_{1} \times α_{1} + w_{2} \times α_{1}}, s_{w_{1} \times β_{1} + w_{2} \times β_{2}}, s_{w_{1} \times r_{1} + w_{2} \times r_{2}}],$ $(\sqrt[q]{1 - {log}_{τ} (1 + \frac{(\frac{{(τ^{1 - u_{1}^{q}} - 1)}^{w_{1}}}{{(τ - 1)}^{w_{1} - 1}}) (\frac{{(τ^{1 - u_{2}^{q}} - 1)}^{w_{2}}}{{(τ - 1)}^{w_{2} - 1}})}{(τ - 1)})}, \sqrt[q]{{log}_{τ} (1 + \frac{(\frac{{(τ^{v_{1}^{q}} - 1)}^{w_{1}}}{{(τ - 1)}^{w_{1} - 1}}) (\frac{{(τ^{v_{2}^{q}} - 1)}^{w_{2}}}{{(τ - 1)}^{w_{2} - 1}})}{τ - 1})}) 〉$ $= 〈 [s_{w_{1} \times α_{1} + w_{2} \times α_{1}}, s_{w_{1} \times β_{1} + w_{2} \times β_{2}}, s_{w_{1} \times r_{1} + w_{2} \times r_{2}}],$ $(\sqrt[q]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{1}^{q}} - 1)}^{w_{1}} {(τ^{1 - u_{2}^{q}} - 1)}^{w_{2}}}{{(τ - 1)}^{w_{1} + w_{2} - 1}})}, \sqrt[q]{{log}_{τ} (1 + \frac{{(τ^{v_{1}^{q}} - 1)}^{w_{1}} {(τ^{v_{2}^{q}} - 1)}^{w_{2}}}{{(τ - 1)}^{w_{1} + w_{2} - 1}})}) 〉$

Hence, that is true for n = 2.

(3) Suppose that Formula (5) is true when n = m. Next, verify that Formula (5) is true when n = m + 1. $q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{k + 1}) = q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{k}) \oplus w_{k + 1} a_{k + 1}$ $= 〈 [s_{\oplus_{j = 1}^{k} w_{j} α_{j}}, s_{\oplus_{j = 1}^{k} w_{j} β_{j}}, s_{\oplus_{j = 1}^{k} w_{j} γ_{j}}], (\sqrt[q]{1 - {log}_{τ} (1 + \prod_{j = 1}^{k} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})}, \sqrt[q]{{log}_{τ} (1 + \prod_{j = 1}^{k} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}) 〉$ $\oplus 〈 [s_{w_{k + 1} α_{k + 1}}, s_{w_{k + 1} β_{k + 1}}, s_{w_{k + 1} γ_{k + 1}}], \sqrt[q]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{k + 1}^{q}} - 1)}^{w_{k + 1}}}{{(τ - 1)}^{w_{k + 1} - 1}})}, \sqrt[q]{{log}_{τ} (1 + \frac{{(τ^{v_{k + 1}^{q}} - 1)}^{w_{k + 1}}}{{(τ - 1)}^{w_{k + 1} - 1}})} 〉$ $= 〈 [s_{\oplus_{j = 1}^{k + 1} w_{j} α_{j}}, s_{\oplus_{j = 1}^{k + 1} w_{j} β_{j}}, s_{\oplus_{j = 1}^{k + 1} w_{j} γ_{j}}], (\sqrt[q]{1 - {log}_{τ} (1 + \frac{(\prod_{j = 1}^{k} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) (\frac{{(τ^{1 - u_{k + 1}^{q}} - 1)}^{w_{k + 1}}}{{(τ - 1)}^{w_{k + 1} - 1}})}{(τ - 1)})},$ $\sqrt[q]{{log}_{τ} (1 + \frac{(\prod_{j = 1}^{k} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}) (\frac{{(τ^{v_{k + 1}^{q}} - 1)}^{w_{k + 1}}}{{(τ - 1)}^{w_{k + 1} - 1}})}{τ - 1})}) 〉$ $= 〈 [s_{\oplus_{j = 1}^{k + 1} w_{j} α_{j}}, s_{\oplus_{j = 1}^{k + 1} w_{j} β_{j}}, s_{\oplus_{j = 1}^{k + 1} w_{j} γ_{j}}]$ $(\sqrt[q]{1 - {log}_{τ} (1 + \frac{(\prod_{j = 1}^{k} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) {(τ^{1 - u_{k + 1}^{q}} - 1)}^{w_{k + 1}}}{{(τ - 1)}^{\sum_{j = 1}^{k + 1} w_{j} - 1}})}, \sqrt[q]{{log}_{τ} (1 + \frac{(\prod_{j = 1}^{k} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}) {(τ^{v_{k + 1}^{q}} - 1)}^{w_{k + 1}}}{{(τ - 1)}^{\sum_{j = 1}^{k + 1} w_{j} - 1}})}) 〉$ $= 〈 [s_{\oplus_{j = 1}^{k + 1} w_{j} α_{j}}, s_{\oplus_{j = 1}^{k + 1} w_{j} β_{j}}, s_{\oplus_{j = 1}^{k + 1} w_{j} γ_{j}}] (\sqrt[q]{1 - {log}_{τ} (1 + \prod_{j = 1}^{k + 1} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})}, \sqrt[q]{{log}_{τ} (1 + \prod_{j = 1}^{k + 1} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}) 〉$

This shows that is true for all values of n.

In the following, we will explore some characteristics of q - ROTrFLFWA operator.

Theorem 3. Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, with 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ . When τ → 1, we have $\begin{matrix} lim_{τ \to 1} q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) = \\ 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], \sqrt[q]{1 - \prod_{j = 1}^{n} ({(1 - u_{j}^{q})}^{w_{j}})}, \prod_{j = 1}^{n} {(v_{j})}^{w_{j}} 〉 \end{matrix}$

Proof: Because τ → 1, $\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}} \to 0$ , therefore, we can obtain that $\sqrt[q]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})} = \sqrt[q]{1 - \frac{ln (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})}{ln τ}} = \sqrt[q]{1 - \frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}}{ln τ}}$

According to Taylor’s expansion, we can get $τ^{1 - u_{j}^{q}} = 1 + (1 - u_{j}^{q}) ln τ + \frac{(1 - u_{j}^{q})}{2} {(ln τ)}^{2} + . . .$

Since τ > 1, we have $τ^{1 - u_{j}^{q}} - 1 = (1 - u_{j}^{q}) ln τ + O (ln τ)$ $\begin{matrix} \sqrt[q]{1 - \frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}}{ln τ}} = \\ \sqrt[q]{1 - \frac{\prod_{j = 1}^{n} {((1 - u_{j}^{2}) ln τ)}^{w_{j}}}{ln τ}} \end{matrix}$ $\begin{matrix} = \sqrt[\begin{matrix} q \end{matrix}]{1 - \frac{\prod_{j = 1}^{n} ({(1 - u_{j}^{q})}^{w_{j}}) ln τ}{ln τ}} \\ = \sqrt[\begin{matrix} q \end{matrix}]{1 - \prod_{j = 1}^{n} ({(1 - u_{j}^{q})}^{w_{j}})} \end{matrix}$

Similarly, if τ → 1, $\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}} \to 0$ then we get $\begin{matrix} \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})} \\ = \sqrt[\begin{matrix} q \end{matrix}]{\frac{ln (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}{ln τ}} \\ = \sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}}{ln τ}} \end{matrix}$

According to Taylor’s expansion, we can get $τ^{v_{j}^{q}} = 1 + v_{j}^{q} ln τ + \frac{v_{j}^{2 q}}{2} {(ln τ)}^{2} + . . .$ since τ > 1, we have $τ^{v_{j}^{q}} - 1 = v_{j}^{q} ln τ + O (ln τ)$ $\begin{matrix} \sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}}{ln τ}} = \sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{n} {(v_{j}^{q} ln τ)}^{w_{j}}}{ln τ}} \\ = \prod_{j = 1}^{n} {(v_{j})}^{w_{j}} \end{matrix}$

Therefore, we can obtain $lim_{τ \to 1} q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})$ $\begin{matrix} = lim_{τ \to 1} 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], \\ (\sqrt[q]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})}, \\ \sqrt[q]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}) 〉 \end{matrix}$ $\begin{matrix} = lim_{τ \to 1} 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{1 - \frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}}{ln τ}}, \\ \sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}}{ln τ}}) 〉 \end{matrix}$ $\begin{matrix} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{1 - \prod_{j = 1}^{n} ({(1 - u_{j}^{q})}^{w_{j}})}, \prod_{j = 1}^{n} {(v_{j})}^{w_{j}}) 〉 . \end{matrix}$

Theorem 4. Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ . When τ→ + ∞, the q - ROTrFLFWA operator is reduced to the arithmetic weighted average operator, that is: $\begin{matrix} lim_{τ \to + \infty} q - ROTrFLFWA = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{\sum_{j = 1}^{n} w_{j} {(u_{j})}^{q}}, \sqrt[\begin{matrix} q \end{matrix}]{\sum_{j = 1}^{n} w_{j} {(v_{j})}^{q}}) 〉 \end{matrix}$ (6)

Proof. In order to prove (4), we only need to prove that $lim_{τ \to + \infty} \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})} = \sum_{j = 1}^{n} w_{j} u_{j},$ $lim_{τ \to + \infty} \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})} = \sqrt[\begin{matrix} q \end{matrix}]{\sum_{j = 1}^{n} w_{j} {(v_{j})}^{q}}$

1) We first prove that $lim_{τ \to + \infty} \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})} = \sqrt[\begin{matrix} q \end{matrix}]{\sum_{j = 1}^{n} w_{j} {(u_{j})}^{q}}$

Firstly, we have ${(\prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})}^{'} = {({(τ^{1 - {(u_{1})}^{q}} - 1)}^{w_{1}} {(τ^{1 - {(u_{2})}^{q}} - 1)}^{w_{2}} . . . . \times {(τ^{1 - {(u_{n})}^{q}} - 1)}^{w_{n}})}^{'}$ $= \frac{(w_{1} (1 - {(u_{1})}^{q}) τ^{- {(u_{1})}^{q}} {(τ^{1 - {(u_{1})}^{q}} - 1)}^{w_{1}} {(τ^{1 - {(u_{2})}^{q}} - 1)}^{w_{2}} . . . . \times {(τ^{1 - {(u_{n})}^{q}} - 1)}^{w_{n}})}{(τ^{1 - {(u_{1})}^{q}} - 1)} + . . .$ $+ \frac{(w_{n} {(τ^{1 - {(u_{n})}^{q}} - 1)}^{w_{n}} (1 - {(u_{1})}^{q}) τ^{- {(u_{1})}^{q}} {(τ^{1 - {(u_{2})}^{q}} - 1)}^{w_{2}} . . . . \times {(τ^{1 - {(u_{n - 1})}^{q}} - 1)}^{w_{n - 1}})}{(τ^{1 - {(u_{n})}^{q}} - 1)}$ $= \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}} (\sum_{j = 1}^{n} \frac{w_{j} (1 - {(u_{j})}^{q}) τ^{- {(u_{j})}^{q}}}{(τ^{1 - {(u_{j})}^{q}} - 1)})$

Since $1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})$ is continuous, then we get $lim_{τ \to + \infty} \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})} = \sqrt[\begin{matrix} q \end{matrix}]{lim_{τ \to + \infty} (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}}))}$

Then, based on logarithmic transform, we can have $= \sqrt[\begin{matrix} q \end{matrix}]{1 - lim_{τ \to + \infty} {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})}$ $= \sqrt[\begin{matrix} q \end{matrix}]{1 - lim_{τ \to + \infty} \frac{ln (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}})}{ln τ}}$

Based on the L’Hospital’s rule, we have $= \sqrt[\begin{matrix} \begin{matrix} \begin{matrix} q \end{matrix} \end{matrix} \end{matrix}]{1 - lim_{τ \to + \infty} \frac{τ \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}} (\sum_{j = 1}^{n} \frac{w_{j} (1 - {(u_{j})}^{q}) τ^{- {(u_{j})}^{q}}}{(τ^{1 - {(u_{j})}^{q}} - 1)})}{1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}}}}$ $= \sqrt[\begin{matrix} \begin{matrix} \begin{matrix} q \end{matrix} \end{matrix} \end{matrix}]{1 - lim_{τ \to + \infty} \frac{\prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}}}{1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j})}^{q}} - 1)}^{w_{j}}} (\sum_{j = 1}^{n} w_{j} (1 - {(u_{j})}^{q}) \frac{τ^{1 - {(u_{j})}^{q}}}{(τ^{1 - {(u_{j})}^{q}} - 1)})}$ $= \sqrt[\begin{matrix} q \end{matrix}]{1 - \sum_{j = 1}^{n} w_{j} (1 - {(u_{j})}^{q})}$ $= \sqrt[\begin{matrix} q \end{matrix}]{\sum_{j = 1}^{n} w_{j} {(u_{j})}^{q}}$

Similarly, we have $lim_{τ \to + \infty} \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})} = \sqrt[\begin{matrix} q \end{matrix}]{(\sum_{j = 1}^{n} w_{j} {(v_{j})}^{q})}$

We prove that $lim_{τ \to + \infty} \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})}$ $= \sqrt[\begin{matrix} q \end{matrix}]{lim_{τ \to + \infty} \frac{ln (1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})}{ln τ}}$ $= \sqrt[\begin{matrix} q \end{matrix}]{lim_{τ \to + \infty} \frac{\frac{\prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}}}{(1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})} (\sum_{j = 1}^{n} \frac{w_{j} ({(v_{j})}^{q}) τ^{{(v_{j})}^{q} - 1}}{(τ^{{(v_{j})}^{q}} - 1)})}{\frac{1}{τ}}}$ $= \sqrt[\begin{matrix} q \end{matrix}]{lim_{τ \to + \infty} \frac{\prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}}}{(1 + \prod_{j = 1}^{n} {(τ^{{(v_{j})}^{q}} - 1)}^{w_{j}})} (\sum_{j = 1}^{n} \frac{w_{j} ({(v_{j})}^{q}) τ^{{(v_{j})}^{q}}}{(τ^{{(v_{j})}^{q}} - 1)})} = \sqrt[\begin{matrix} q \end{matrix}]{\sum_{j = 1}^{n} w_{j} {(v_{j})}^{q}} .$

Which completes the proof of Theorem 14.

Theorem 5. (boundedness). Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, $s_{α^{-}} = min_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}$ , $s_{α^{+}} = max_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}$ $s_{β^{-}} = min_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}$ , $s_{β^{+}} = max_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}$ ; $s_{γ^{-}} = min_{1 ⩽ j ⩽ n} {s_{γ (a_{j})}}$ , , $u_{+} = max_{1 ⩽ j ⩽ n} {u_{j}}$ , $v_{-} = min_{1 ⩽ j ⩽ n} {v_{j}}$ , $v_{+} = max_{1 ⩽ j ⩽ n} {v_{j}}$ , then $< [s_{α^{-}}, s_{β^{-}}, s_{γ^{-}}], (u_{-}, v_{+}) > ⩽ q - ROTrFLFWA ⩽ < [s_{α^{+}}, s_{β^{+}}, s_{γ^{+}}], (u_{+}, v_{-}) >$

Proof. Because $u_{-} = min_{1 ⩽ j ⩽ n} {u_{j}}$ , $u_{+} = max_{1 ⩽ j ⩽ n} {u_{j}}$ , $v_{-} = min_{1 ⩽ j ⩽ n} {v_{j}}$ , $v_{+} = max_{1 ⩽ j ⩽ n} {v_{j}}$ $s_{α^{-}} = min_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}$ $s_{α^{+}} = max_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}$ ; $s_{β^{-}} = min_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}$ , $s_{β^{+}} = max_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}$ therefore, we have

u_- ⩽ u_j ⩽ u₊, v_- ⩽ v_j ⩽ v₊, s_{α
^-} ⩽ s_{α
_j} ⩽ s_{α
⁺}, s_{β
^-} ⩽ s_{β
_j} ⩽ s_{β
⁺}, s_{γ
^-} ⩽ s_{γ
_j} ⩽ s_{γ
⁺}

Then, we can obtain $u_{-} = \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{-}^{q}} - 1)}^{w_{j}})} ⩽ \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})}$ $⩽ \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{+}^{q}} - 1)}^{w_{j}})} = u_{+}$ $v_{-} = \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{-}^{q}} - 1)}^{w_{j}})} ⩽ \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})} ⩽ \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{+}^{q}} - 1)}^{w_{j}})} = v_{+}$

According to the definition 3, we can get $S (a_{j}) = 1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})$ $⩽ 1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{+}^{q}} - 1)}^{w_{j}}) - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{-}^{q}} - 1)}^{w_{j}})$

Based on the definition 4, we have $q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) ⩽ < u_{+}, v_{-} >$

Similarly, we can obtain $< u_{-}, v_{+} > ⩽ q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})$

and $< [s_{α^{-}}, s_{β^{-}}, s_{γ^{-}}], (u_{-}, v_{+}) > ⩽ q - ROTrFLFWA ⩽ < [s_{α^{+}}, s_{β^{+}}, s_{γ^{+}}], (u_{+}, v_{-}) > .$

Theorem 6. (Idempotency) Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, if all a_j = a₀ (j = 1, 2, . . . , n), then, $q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) = a_{0}$

Theorem 7. (Monotonicity) Let a_j (j = 1, 2, . . . , n) and $a_{j}^{'} (j = 1, 2, . . ., n)$ are two collections of q-ROTrFLVs, if $a_{j}^{'} ⩽ a_{j} (j = 1, 2, . . ., n)$ , then $q - ROTrFLFWA (a_{1}^{'}, a_{2}^{'}, \dots, a_{n}^{'}) ⩽ q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})$

Theorem 8. Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ and k > 0, then $q - ROTrFLFWA ({ka}_{1}, {ka}_{2}, \dots, {ka}_{n}) = k q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})$

Proof. Since k > 0, we have ${ka}_{1} = 〈 [s_{w_{1} \times α_{1}}, s_{w_{1} \times β_{1}}, s_{w_{1} \times r_{1}}], (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{1}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{{(τ^{v_{1}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}) 〉$ $w_{1} ({ka}_{1}) = 〈 [s_{w_{1} \times k α_{1}}, s_{w_{1} \times k β_{1}}, s_{w_{1} \times {kr}_{1}}], \sqrt[\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} q \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(\frac{{(τ^{1 - u_{1}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}^{w_{1}}}{{(τ - 1)}^{w_{1} - 1}})}, \sqrt[\begin{matrix} \begin{matrix} \begin{matrix} q \\ \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]{{log}_{τ} (1 + \frac{{(\frac{{(τ^{v_{1}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}^{w_{1}}}{{(τ - 1)}^{w_{1} - 1}})} 〉$ $w_{2} ({ka}_{2}) = 〈 [s_{w_{2} \times k α_{2}}, s_{w_{2} \times k β_{1}}, s_{w_{2} \times {kr}_{2}}] \sqrt[\begin{matrix} \begin{matrix} q \\ \begin{matrix} \end{matrix} \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(\frac{{(τ^{1 - u_{2}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}^{w_{2}}}{{(τ - 1)}^{w_{2} - 1}})}, \sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{{log}_{τ} (1 + \frac{{(\frac{{(τ^{v_{2}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}^{w_{2}}}{{(τ - 1)}^{w_{2} - 1}})} 〉$

Therefore, we can get $w_{1} ({ka}_{1}) \oplus w_{2} ({ka}_{2}) =$ $= 〈 \begin{matrix} [s_{w_{1} \times k α_{1} + w_{2} \times k α_{2}}, s_{w_{1} \times k β_{1} + w_{2} \times k β_{2}}, s_{w_{1} \times {kr}_{1} + w_{2} \times {kr}_{2}}] \\ \sqrt[q]{1 - {log}_{τ} (1 + \prod_{j = 1}^{2} {(\frac{{(τ^{1 - u_{j}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}^{w_{j}})}, \sqrt[q]{{log}_{τ} (1 + \prod_{j = 1}^{2} {(\frac{{(τ^{v_{j}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}^{w_{j}})} \end{matrix} 〉$

According to mathematical induction, we have $q - ROTrFLFWA ({ka}_{1}, {ka}_{2}, \dots, {ka}_{n})$ $= 〈 [s_{k \oplus_{j = 1}^{n} w_{j} α_{j}}, s_{k \oplus_{j = 1}^{n} w_{j} β_{j}}, s_{{k \oplus}_{j = 1}^{n} w_{j} γ_{j}}],$ $(\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} \frac{{(τ^{1 - u_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} \frac{{(τ^{v_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}})}) 〉$

Since $q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}]$ $(\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}) 〉$

Then we have $k q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) =$ $〈 [s_{k \oplus_{j = 1}^{n} w_{j} α_{j}}, s_{k \oplus_{j = 1}^{n} w_{j} β_{j}}, s_{{k \oplus}_{j = 1}^{n} w_{j} γ_{j}}]$ $(\sqrt[\begin{matrix} \begin{matrix} \begin{matrix} q \end{matrix} \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}))} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}, \sqrt[\begin{matrix} \begin{matrix} \begin{matrix} q \end{matrix} \end{matrix} \end{matrix}]{{log}_{τ} (1 + \frac{{(τ^{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}) 〉$ $= 〈 [s_{k \oplus_{j = 1}^{n} w_{j} α_{j}}, s_{k \oplus_{j = 1}^{n} w_{j} β_{j}}, s_{{k \oplus}_{j = 1}^{n} w_{j} γ_{j}}]$ $(\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}})}) 〉$

Then we can get q - ROTrFLFWA (ka₁, ka₂, …, ka_n) = kq - ROTrFLFWA (a₁, a₂, …, a_n).

Theorem 9. Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , then $q - ROTrFLFWA (a_{1} \oplus a, a_{2} \oplus a, \dots, a_{n} \oplus a) = q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) \oplus a$

Proof (1) Since $a_{j} \oplus a = 〈 [s_{α_{j} + α}, s_{β_{j} + β}, s_{γ_{j} + γ}], \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{(τ^{1 - u_{j}^{q}} - 1) (τ^{1 - u^{q}} - 1)}{(τ - 1)})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{(τ^{v_{j}^{q}} - 1) (τ^{v^{q}} - 1)}{τ - 1})} 〉,$

Then we have $q - ROTrFLFWA (a_{1} \oplus a, a_{2} \oplus a, \dots, a_{n} \oplus a) = 〈 [s_{\oplus_{j = 1}^{n} w_{j} (α_{j} + α)}, s_{\oplus_{j = 1}^{n} w_{j} (β_{j} + β)}, s_{\oplus_{j = 1}^{n} w_{j} (γ_{j} + γ)}],$ $(\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{log}_{τ} (1 + \frac{(τ^{1 - u_{j}^{q}} - 1) (τ^{1 - u^{q}} - 1)}{(τ - 1)})} - 1)}^{w_{j}})} \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(1 + \frac{(τ^{v_{j}^{q}} - 1) (τ^{v^{q}} - 1)}{τ - 1} - 1)}^{w_{j}})}) 〉$ $= 〈 [s_{\oplus_{j = 1}^{n} w_{j} (α_{j} + α)}, s_{\oplus_{j = 1}^{n} w_{j} (β_{j} + β)}, s_{\oplus_{j = 1}^{n} w_{j} (γ_{j} + γ)}],$ $(\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} {(1 + \prod_{j = 1}^{n} (\frac{(τ^{1 - u_{j}^{q}} - 1) (τ^{1 - u^{q}} - 1)}{(τ - 1)}))}^{w_{j}}}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(\frac{(τ^{v_{j}^{q}} - 1) (τ^{v^{q}} - 1)}{τ - 1})}^{w_{j}})}) 〉$

(2) q - ROTrFLFWA (a₁, a₂, …, a_n) ⊕ a $= 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j} + α}, s_{\oplus_{j = 1}^{n} w_{j} β_{j} + β}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j} + γ}] 〈 \sqrt[\begin{matrix} \begin{matrix} \begin{matrix} q \end{matrix} \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \frac{(τ^{({log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}))} - 1) (τ^{1 - u^{q}} - 1)}{(τ - 1)})},$ $\sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{(\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}) (τ^{v^{q}} - 1)}{τ - 1})} 〉$ $= 〈 \begin{matrix} [s_{\oplus_{j = 1}^{n} w_{j} α_{j} + α}, s_{\oplus_{j = 1}^{n} w_{j} β_{j} + β}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j} + γ}] \\ (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{(\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) (τ^{1 - u^{q}} - 1)}{(τ - 1)})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{(\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}) (τ^{v^{q}} - 1)}{τ - 1})}) \end{matrix} 〉$

So, we have $q - ROTrFLFWA (a_{1} \oplus a, a_{2} \oplus a, \dots, a_{n} \oplus a) = q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) \oplus a .$

Theorem 10. Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , k > 0, then $q - ROTrFLFWA (k \cdot a_{1} \oplus a, k \cdot a_{2} \oplus a, \dots, k \cdot a_{n} \oplus a) = k \cdot q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) \oplus a$

Proof. (1) Since ${ka}_{j} = 〈 [s_{k α_{j}}, s_{k β_{j}}, s_{{kr}_{j}}], (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{j}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{{(τ^{v_{j}^{q}} - 1)}^{k}}{{(τ - 1)}^{k - 1}})}) 〉$

Then we have ka_j ⊕ a $= 〈 \begin{matrix} [s_{k α_{j} + α}, s_{k β_{j} + β}, s_{{kr}_{j} + γ}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{j}^{q}} - 1)}^{k} (τ^{1 - u^{q}} - 1)}{{(τ - 1)}^{k}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{{(τ^{v_{j}^{q}} - 1)}^{k} (τ^{v^{q}} - 1)}{{(τ - 1)}^{k}})}) \end{matrix} 〉$

Therefore, we have $q - ROTrFLFWA (k \cdot a_{1} \oplus a, k \cdot a_{2} \oplus a, \dots, k \cdot a_{n} \oplus a)$ $= 〈 [s_{\sum_{j = 1}^{n} {kw}_{j} α_{j} + n α}, s_{\sum_{j = 1}^{n} {kw}_{j} β_{j} + n β}, s_{\sum_{j = 1}^{n} {kw}_{j} r_{j} + n γ}], (\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - (1 - {log}_{τ} (1 + \frac{{(τ^{1 - u_{j}^{q}} - 1)}^{k} (τ^{1 - u^{q}} - 1)}{{(τ - 1)}^{k}}))} - 1)}^{w_{j}})},$ $\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{log}_{τ} (1 + \frac{{(τ^{v_{j}^{q}} - 1)}^{k} (τ^{v^{q}} - 1)}{{(τ - 1)}^{k}})} - 1)}^{w_{j}})}) 〉$ $= 〈 \begin{matrix} [s_{\sum_{j = 1}^{n} {kw}_{j} α_{j} + n α}, s_{\sum_{j = 1}^{n} {kw}_{j} β_{j} + n β}, s_{\sum_{j = 1}^{n} {kw}_{j} r_{j} + n γ}] \\ (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} \frac{{(τ^{1 - u_{j}^{q}} - 1)}^{{kw}_{j}} (τ^{1 - u^{q}} - 1)}{{(τ - 1)}^{k}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} \frac{{(τ^{v_{j}^{q}} - 1)}^{{kw}_{j}} (τ^{v^{q}} - 1)}{{(τ - 1)}^{k}})}) \end{matrix} 〉$

(2) k · q - ROTrFLFWA (a₁, a₂, …, a_n) $= 〈 \begin{matrix} [s_{\oplus_{j = 1}^{n} {kw}_{j} α_{j}}, s_{\oplus_{j = 1}^{n} {kw}_{j} β_{j}}, s_{\oplus_{j = 1}^{n} {kw}_{j} γ_{j}}], \\ \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}})} \end{matrix} 〉$

Then we have $k \cdot q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) \oplus a$ $= 〈 \begin{matrix} [s_{\oplus_{j = 1}^{n} ({kw}_{j} α_{j} + α)}, s_{\oplus_{j = 1}^{n} ({kw}_{j} β_{j} + β)}, s_{\oplus_{j = 1}^{n} ({kw}_{j} γ_{j} + γ)}] \\ \sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \frac{(\frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}}) (τ^{1 - u^{q}} - 1)}{(τ - 1)})}, \sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{{log}_{τ} (1 + \frac{(\frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{{kw}_{j}}}{{(τ - 1)}^{k - 1}}) (τ^{v^{q}} - 1)}{τ - 1})} \end{matrix} 〉$ $= 〈 \begin{matrix} [s_{\oplus_{j = 1}^{n} {kw}_{j} α_{j} + n α}, s_{\oplus_{j = 1}^{n} {kw}_{j} β_{j} + n β}, s_{\oplus_{j = 1}^{n} {kw}_{j} γ_{j} + n γ}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{{kw}_{j}} (τ^{1 - u^{q}} - 1)}{{(τ - 1)}^{k}})}, \sqrt[\begin{matrix} q \\ \begin{matrix} \end{matrix} \end{matrix}]{{log}_{τ} (1 + \frac{\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{{kw}_{j}} (τ^{v^{q}} - 1)}{{(τ - 1)}^{k}})}) \end{matrix} 〉$ Then we get $q - ROTrFLFWA (k \cdot a_{1} \oplus a, k \cdot a_{2} \oplus a, \dots, k \cdot a_{n} \oplus a) = k \cdot q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) \oplus a$

Theorem 11. Let a_j (j = 1, 2, . . . , n) and $a_{j}^{'} (j = 1, 2, . . ., n)$ are two collections of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , then $\begin{matrix} q - ROTrFLFWA (a_{1} \oplus a_{1}^{'}, a_{2} \oplus a_{2}^{'}, \dots, a_{n} \oplus a_{n}^{'}) \\ = q - ROTrFLFWA (a_{1}, a_{2}, . . ., a_{n}) \oplus q - ROTrFLFWA (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'}) \end{matrix}$

Proof (1) Since $a_{j} \oplus a_{j}^{'} = 〈 \begin{matrix} [s_{α_{j} + α_{j}^{'}}, s_{β_{j} + β_{j}^{'}}, s_{γ_{j} + γ_{j}^{'}}], \\ \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{(τ^{1 - u_{j}^{q}} - 1) (τ^{1 - {(u_{j}^{'})}^{q}} - 1)}{(τ - 1)})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{(τ^{v_{j}^{q}} - 1) (τ^{{(v_{j}^{'})}^{q}} - 1)}{τ - 1})} \end{matrix} 〉$ Then $q - ROTrFLFWA (a_{1} \oplus a_{1}^{'}, a_{2} \oplus a_{2}^{'}, \dots, a_{n} \oplus a_{n}^{'})$ $\begin{matrix} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j} + \oplus_{j = 1}^{n} w_{j} α_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j} + \oplus_{j = 1}^{n} w_{j} β_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j} + \oplus_{j = 1}^{n} w_{j} γ_{j}^{'}}], \\ (\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - (1 - {log}_{τ} (1 + \frac{(τ^{1 - u_{j}^{q}} - 1) (τ^{1 - {(u_{j}^{'})}^{q}} - 1)}{(τ - 1)}))} - 1)}^{w_{j}})} \end{matrix}$ $\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{({log}_{τ} (1 + \frac{(τ^{v_{j}^{q}} - 1) (τ^{{(v_{j}^{'})}^{q}} - 1)}{τ - 1}))} - 1)}^{w_{j}})}) 〉$ $= 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j} + \oplus_{j = 1}^{n} w_{j} α_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j} + \oplus_{j = 1}^{n} w_{j} β_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j} + \oplus_{j = 1}^{n} w_{j} γ_{j}^{'}}], (\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(\frac{(τ^{1 - u_{j}^{q}} - 1) (τ^{1 - {(u_{j}^{'})}^{q}} - 1)}{(τ - 1)})}^{w_{j}})}$ $\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(\frac{(τ^{v_{j}^{q}} - 1) (τ^{{(v_{j}^{'})}^{q}} - 1)}{τ - 1})}^{w_{j}})}) 〉$

(2) For the right-hand side of Equation (7), we can obtain $q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}) \oplus q - ROTrFLFWA (a_{1}^{'}, a_{2}^{'}, \dots, a_{n}^{'})$ $= 〈 \begin{matrix} [s_{\oplus_{j = 1}^{n} w_{j} α_{j} + \oplus_{j = 1}^{n} w_{j} α_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j} + \oplus_{j = 1}^{n} w_{j} β_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j} + \oplus_{j = 1}^{n} w_{j} γ_{j}^{'}}], \\ (\begin{matrix} \sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{1 - {log}_{τ} (1 + \frac{(τ^{1 - (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}))} - 1) (τ^{1 - (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{j}^{'})}^{q}} - 1)}^{w_{j}}))} - 1)}{(τ - 1)})}, \\ \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{(\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}) (\prod_{j = 1}^{n} {(τ^{{(v_{j}^{'})}^{q}} - 1)}^{w_{j}})}{τ - 1})} \end{matrix}) \end{matrix} 〉$ $= 〈 \begin{matrix} [s_{\oplus_{j = 1}^{n} w_{j} α_{j} + \oplus_{j = 1}^{n} w_{j} α_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j} + \oplus_{j = 1}^{n} w_{j} β_{j}^{'}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j} + \oplus_{j = 1}^{n} w_{j} γ_{j}^{'}}], \\ (\begin{matrix} \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \frac{(\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) (\prod_{j = 1}^{n} {(τ^{1 - {(u_{j}^{'})}^{q}} - 1)}^{w_{j}})}{(τ - 1)})}, \\ \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \frac{(\prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}}) (\prod_{j = 1}^{n} {(τ^{{(v_{j}^{'})}^{q}} - 1)}^{w_{j}})}{τ - 1})} \end{matrix}) \end{matrix} 〉$

Then, we have $\begin{matrix} q - ROTrFLFWA (a_{1} \oplus a_{1}^{'}, a_{2} \oplus a_{2}^{'}, \dots, a_{n} \oplus a_{n}^{'}) \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} = q - ROTrFLFWA (a_{1}, a_{2}, . . ., a_{n}) \oplus q - ROTrFLFWA (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'}) \end{matrix} .$

Definition 11. Let a_j (j = 1, 2, . . . , n) is a set of of q-ROTrFLVs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , with τ > 1. Then, the q-rung orthopair triangular fuzzy linguistic Frank weighted geometric (q-ROTrFLWG) operator is a mapping q-ROTrFLFWG: Ωⁿ → Ω, if $q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n}) = \otimes_{j = 1}^{n} a_{j}^{w_{j}}$ $= 〈 [s_{\prod_{i = 1}^{n} {(α_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(β_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(γ_{j})}^{w_{j}}}] (\sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})}, \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{w_{j}})}) 〉$ (7)

Where Ω is the set of all q-rung orthopair triangular fuzzy linguistic variables, then the function q-ROTrFLFWG is known as the q-ROTrFLFWG operator.

In particular, if $w = (\frac{1}{n}, \frac{1}{n}, . . ., \frac{1}{n})^{T}$ , the q - ROTrFLFWG operator will be simplified to q - ROTrFLFG operator, that is $q - ROTrFLFG (a_{1}, a_{2}, \dots, a_{n}) = \otimes_{j = 1}^{n} a_{j}^{\frac{1}{n}}$ $= 〈 [s_{\prod_{i = 1}^{n} {(α_{j})}^{\frac{1}{n}}}, s_{\prod_{i = 1}^{n} {(β_{j})}^{\frac{1}{n}}}, s_{\prod_{i = 1}^{n} {(γ_{j})}^{\frac{1}{n}}}] \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{\frac{1}{n}})}, \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{\frac{1}{n}})} 〉$

Similar to the q-ROTrFLFWA operator, the q-ROTrFLFWG operator also has bounded, monotonic and idempotent properties.

Lemma 1. [56] Let x_j > 0, λ_j > 0, j = 1, 2, . . . , n, and $\sum_{j = 1}^{n} λ_{j} = 1$ , then $\prod_{j = 1}^{n} x_{j}^{λ_{j}} ⩽ \sum_{j = 1}^{n} λ_{j} x_{j}$ (8) with equality if and only if x₁ = x₂ = ⋯ = x_n.

Theorem 12. Let a_j (j = 1, 2, . . . , n) is a set of of q-ROTrFLVs, w = (w₁, w₂, . . . , w_n) ^T be the weight vector of them, and 0 ⩽ w_j ⩽ 1. At the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , then we have $q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n}) ⩽ q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})$

Proof. (1) We first prove $\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})} ⩾ \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})}$

According to Lemma 1,

Then we have $\prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}} ⩽ \sum_{j = 1}^{n} w_{j} (τ^{1 - u_{j}^{q}} - 1)$ $\prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}} ⩽ \sum_{j = 1}^{n} w_{j} (τ^{u_{j}^{q}} - 1)$

Since $1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})$ $= {log}_{τ}^{τ} - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})$ $= {log}_{τ} \frac{τ}{(1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})}$ $⩾ {log}_{τ} \frac{τ}{(1 + \sum_{j = 1}^{n} w_{j} (τ^{1 - u_{j}^{q}} - 1)) (1 + \sum_{j = 1}^{n} w_{j} (τ^{u_{j}^{q}} - 1))}$ $⩾ {log}_{τ} \frac{τ}{(\sum_{j = 1}^{n} w_{j} τ^{1 - u_{j}^{q}}) (\sum_{j = 1}^{n} w_{j} τ^{u_{j}^{q}})}$ $⩾ {log}_{τ} \frac{1}{(\sum_{j = 1}^{n} w_{j} τ^{- u_{j}^{q}}) (\sum_{j = 1}^{n} w_{j} τ^{u_{j}^{q} - 1})} ⩾ 0$

Therefore $\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})} ⩾ \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})}$

(2) If u₁ = u₂ = . . . u_n, v₁ = v₂ = . . . v_n, according to Lemma 1, $\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})} = \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})}$ $\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{w_{j}})} = \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}$ $\begin{matrix} S (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})) \\ = (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})) (α_{j} + β_{j} + γ_{j}) / 3 \end{matrix}$

Then we get $\begin{matrix} S (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n})) \\ = ({log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}}) - (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{w_{j}}))) (α_{j} + β_{j} + γ_{j}) / 3 \end{matrix}$ $S (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})) = S (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n}))$

Furtherefore, $\begin{matrix} H (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})) \\ = (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) + {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})) (α_{j} + β_{j} + γ_{j}) / 3 \end{matrix}$ $\begin{matrix} H (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n})) \\ = ({log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}}) + (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{w_{j}}))) (α_{j} + β_{j} + γ_{j}) / 3 \end{matrix}$

Then, we have $H (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})) = H (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n}))$

(3) If u₁ = u₂ = . . . u_n, v₁ = v₂ = . . . v_n cannot be established at the same time, according to lemma 1 we can obtain $\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}})} > \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}})}$ $\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{w_{j}})} > \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})}$

Then $\begin{matrix} S (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})) \\ = (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{j}^{q}} - 1)}^{w_{j}}) - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{j}^{q}} - 1)}^{w_{j}})) (α_{j} + β_{j} + γ_{j}) / 3 \end{matrix}$

Furthermore, $\begin{matrix} S (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n})) \\ = {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{u_{j}^{q}} - 1)}^{w_{j}}) - (1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - v_{j}^{q}} - 1)}^{w_{j}})) (α_{j} + β_{j} + γ_{j}) / 3 \end{matrix}$

Therefore, we can get $S (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n})) < S (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n}))$

According to (1), (2) and (3), we have $S (q - ROTrFLFWG (a_{1}, a_{2}, \dots, a_{n})) ⩽ S (q - ROTrFLFWA (a_{1}, a_{2}, \dots, a_{n})) .$

4.2 The q-rung orthopair trapezoid fuzzy linguistic Frank ordered weighted aggregation operators

Definition 12. Let a_j (j = 1, 2, . . . , n) is a set of of q-ROTrFLVs, and ω = (ω₁, ω₂, . . . , ω_n) ^T is the position weight vector of a_j (j = 1, 2, …, n), and ω_j ∈ [0, 1] , j = 1, 2, . . . , n, at the same time $\sum_{i = 1}^{n} ω_{i} = 1$ . Then, the q-rung orthopair triangular fuzzy linguistic Frank ordered weighted average (q - ROTrFLFOWA) operator is defined as follows: $q - ROTrFLFOWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} ω_{j} a_{σ (j)}$ $= 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} β_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{σ (j)}}], (\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - u_{σ (j)}^{q}} - 1)}^{ω_{j}})}, \sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{v_{σ (j)}^{q}} - 1)}^{ω_{j}})}) 〉$

a_σ(j) is the jth largest element in a_j (j = 1, 2, …, n), and a_σ(j-1) ⩾ a_σ(j), then their aggregated value by the q-ROTrFLFOWA operator is still a q-ROTrFLVs.

Similar to the q-ROTrFLFWA operator, the q-ROTrFLFOWA operator also has the properties of boundedness, idempotency and monotonicity. In addition, the q-ROTrFLFOWA operator also has the property of commutativity.

Theorem 13. (Commutativity) Let a_j (j = 1, 2, . . . , n) is a set of q-ROTrFLVs, and ω = (ω₁, ω₂, . . . , ω_n) ^T be the aggregation-associated vector of a_j, and ω_i ∈ [0, 1] (i = 1, 2, . . . , n), at the same time $\sum_{i = 1}^{n} ω_{i} = 1$ . If $(a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'})$ is any permutation of (a₁, a₂, . . . , a_n), then $\begin{matrix} q - ROTrFLFOWA (a_{1}, a_{2}, . . ., a_{n}) = \\ q - ROTrFLFOWA (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'}) \end{matrix}$

5 The methods based on q - ROTrFLFWA and q - ROTrFLFWG operators

For a MADM problem, let A = {A₁, A₂, . . . , A_m} be the set of decision-makers, C = {C₁, C₂, . . . , C_n} be the collection of n attributes. S = {s₀, s₁, …, s_l} be a finite set of linguistic terms, w = (w₁, w₂, …, w_n) ^T represents an attribute weight vector, w_i ⩾ 0, (i = 1, 2, …, n), and $\sum_{i = 1}^{n} w_{i} = 1$ . Assume that the expert’s evaluation of decision schem A_i under attribute C_j is expressed as a_ij = 〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}] , (u_j, v_j) 〉 (j = 1, 2, . . . , n) by q-rung orthopair triangular fuzzy linguistic information, and A = [a_ij] _m×n is the q-rung orthopair triangular fuzzy linguistic decision matrix.

Then, to determine the most desirable alternative(s),the q-ROTrFLFWA operator and q - ROTrFLFWG are utilized to propose an approach to MADM under q-rung orthopair triangular fuzzy linguistic fuzzy environment, which involves the following steps:

Step 1: Use the q-ROTrFLFWA operator or q - ROTrFLFWG operator to integrate the ith row of matrix A = [a_ij] _m×n as $\begin{matrix} a_{i} = q - ROTrFLFWA (a_{i 1}, a_{i 2}, \dots, a_{in}) = \\ \oplus_{j = 1}^{n} w_{j} a_{ij} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{ij}}, s_{\oplus_{j = 1}^{n} w_{j} β_{ij}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{ij}}], \end{matrix}$ $(\sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(u_{ij})}^{q}} - 1)}^{w_{j}})},$ $\sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{(v_{ij})}^{q}} - 1)}^{w_{j}})}) 〉$ (9) or $\begin{matrix} a_{i} = q - ROTrFLFWG (a_{i 1}, a_{i 2}, \dots, a_{in}) = \otimes_{j = 1}^{n} a_{ij}^{w_{j}} \\ = 〈 [s_{\prod_{i = 1}^{n} {(α_{ij})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(β_{ij})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(γ_{ij})}^{w_{j}}}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{{log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{{(u_{ij})}^{q}} - 1)}^{w_{j}})}, \\ \sqrt[\begin{matrix} q \end{matrix}]{1 - {log}_{τ} (1 + \prod_{j = 1}^{n} {(τ^{1 - {(v_{ij})}^{q}} - 1)}^{w_{j}})}) 〉 . \end{matrix}$ (10)

Step 2: Calculate the score function S (a_i) (i = 1, 2, . . . , m), the exact functions H (a_i) (i = 1, 2, . . . , m).

Step 3: Rank all feasible alternatives according to Definition 5, and obtain the best desirable alternative(s).

6 Numerical example

An investment company has a sum of money (adapted from [57, 58]), the top administrator wants to choose a company to invest. There are four companies A₁, A₂, A₃, A₄, and the top administrator investigates these companies from four factors: (1) C₁, the growth factor; (2) C₂, the risk factor; (3) C₃, the social impact; (4) C₄, the environmental impact, whose weight vector is given as w = (0.26, 0.36, 0.16, 0.22) ^T. By using the set S = {s₀: neither, s₁: very low, s₂: low, s₃: fair, s₄: high, s₅: very high, s₆: absolute}, the top administrator must make a decision according to the following the q-rung orthopair triangular fuzzy linguistic decision matrix R = (a_ij) _m×n which is constructed as shown in Table 1.

Table 1
q-rung orthopair triangular fuzzy linguistic decision matrix R = (a_ij) _m×n

C ₁ C ₂ C ₃ C ₄

A ₁ < [s₂, s₃, s₄], (0.7, 0.4)> < [s₄, s₅, s₆], (0.6,0.4)> < [s₁, s₂, s₃], (0.3,0.4)> < [s₂, s₃, s₅], (0.5,0.4)>

A ₂ < [s₁, s₂, s₃], (0.8,0.3)> < [s₁, s₂, s₃], (0.7,0.3)> < [s₃, s₄, s₆], (0.6,0.3)> < [s₁, s₂, s₃], (0.8,0.2)>

A ₃ < [s₃, s₄, s₅], (0.8,0.5)> < [s₂, s₃, s₄], (0.1,0.2)> < [s₂, s₃, s₄], (0.3,0.2)> < [s₄, s₅, s₆], (0.4,0.3)>

A ₄ < [s₁, s₂, s₄], (0.5,0.2)> < [s₃, s₄, s₆], (0.5,0.6)> < [s₁, s₂, s₃], (0.6,0.5)> < [s₂, s₄, s₅], (0.5,0.4)>

A ₅ < [s₃, s₄, s₅], (0.3,0.2)> < [s₁, s₂, s₆], (0.6,0.2)> < [s₂, s₄, s₅], (0.1,0.4)> < [s₁, s₃, s₅], (0.4,0.1)>

C ₁	C ₂	C ₃	C ₄
A ₁	< [s₂, s₃, s₄], (0.7, 0.4)>	< [s₄, s₅, s₆], (0.6,0.4)>	< [s₁, s₂, s₃], (0.3,0.4)>	< [s₂, s₃, s₅], (0.5,0.4)>
A ₂	< [s₁, s₂, s₃], (0.8,0.3)>	< [s₁, s₂, s₃], (0.7,0.3)>	< [s₃, s₄, s₆], (0.6,0.3)>	< [s₁, s₂, s₃], (0.8,0.2)>
A ₃	< [s₃, s₄, s₅], (0.8,0.5)>	< [s₂, s₃, s₄], (0.1,0.2)>	< [s₂, s₃, s₄], (0.3,0.2)>	< [s₄, s₅, s₆], (0.4,0.3)>
A ₄	< [s₁, s₂, s₄], (0.5,0.2)>	< [s₃, s₄, s₆], (0.5,0.6)>	< [s₁, s₂, s₃], (0.6,0.5)>	< [s₂, s₄, s₅], (0.5,0.4)>
A ₅	< [s₃, s₄, s₅], (0.3,0.2)>	< [s₁, s₂, s₆], (0.6,0.2)>	< [s₂, s₄, s₅], (0.1,0.4)>	< [s₁, s₃, s₅], (0.4,0.1)>

Table 2

If τ = 4, ranking results based on q - ROTrFLFWA operator by using the different q

q	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
1.1	0.6305	1.1303	0.5969	0.3607	0.7347	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
1.5	0.6577	1.1699	0.6446	0.3638	0.6608	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
2	0.6326	1.1247	0.6436	0.3354	0.5354	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
5	0.2532	0.5510	0.3427	0.0931	0.1041	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
10	0.0351	0.1499	0.1055	0.0055	0.0074	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
20	0.0008	0.0138	0.0111	0.00002	0.00004	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
30	0.00002	0.0014	0.0011	0.0000001	0.0000003	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
50	0.00000002	0.00002	0.00001	0.000000000004	0.00000000001	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

6.1 Rank the alternatives by the proposed methods

Approach: The decision making steps based on q-ROTrFLFWA operator

The q-ROTrFLFWA operator is used to integrate all the q-rung orthopair triangular fuzzy linguistic information a_ij (i = 1, 2, . . . , 4) (j = 1, 2, . . . , 4) to the q-rung orthopair triangular fuzzy linguistic value, and then the score function is calculated, and sorted. Taking q = 1.5 and τ = 4 as examples.

Step 1. Aggregate all assessment values a_ij =〈 [s_{α
_ij}, s_{β
_ij}, s_{γ
_ij}, s_{η
_ij}] , (u_ij, v_ij) 〉 of the alternative A_i (i = 1, 2, 3, 4, 5) on attributes C_j (j = 1,2,3,4) into the overall assessment values a_i = 〈 [s_{α
_i}, s_{β
_i}, s_{γ
_i}] , (u_i, v_i) 〉 by Equation (9).

a₁ = {, (0 . 573, 0 . 4)}; a₂ ={[s_1.32, s_2.32, s_3.48] (0 . 7404, 0 . 2748)}; a₃ ={[s_2.7, s_3.7, s_4.7] , (0 . 4709, 0 . 2810)}; a₄ ={[s_1.94, s_3.16, s_4.78] , (0 . 5174, 0 . 4092)}; a₅ ={[s_1.68, s_3.06, s_5.36] , (0 . 4294, 0 . 1935)}.

Step 2. Calculate the score values E (a_i) of the overall assessment values a_i (i = 1,2,3,4). According to Equation (1), we have

E (a₁) = 0.6577; E (a₂) = 1.1699; E (a₃) = 0.6445; E (a₄) = 0.3638; E (a₅) = 0.6607.

Step 3. Rank these alternatives A_i (i = 1,2,3,4) and select the best one by the the score values E (a_i). $A_{2} ≻ A_{5} ≻ A_{1} ≻ A_{3} ≻ A_{4} .$

Therefore, the best alternative is A₂.

The integration process using the q - ROTrFLFWG operator and the solution process are similar to the q - ROTrFLFWA operator, so omitted here.

Table 3
If τ = 4, ranking results based on q - ROTrFLFWG operator by using the different q

q S (a₁) S (a₂) S (a₃) S (a₄) S (a₅) Ranking

1.1 0.5030 1.0434 0.0435 0.2014 0.4443 A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃

1.5 0.5090 1.0753 –0.0665 0.1809 0.3473 A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃

2 0.4660 1.0274 –0.0826 0.1389 0.2309 A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃

5 0.1225 0.4660 –0.0257 0.0019 0.0099 A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃

10 0.0065 0.09603 –0.0009 –0.0032 0.00002 A₂ ≻ A₁ ≻ A₅ ≻ A₃ ≻ A₄

20 0.000013 0.00397 –0.0000008 –0.00003 –0.000000006 A₂ ≻ A₁ ≻ A₅ ≻ A₃ ≻ A₄

30 0.000000026 0.00017 –0.0000000008 –0.0000002 –0.0000000000006 A₂ ≻ A₁ ≻ A₅ ≻ A₃ ≻ A₄

40 0.0000000000000003 0.000007 0.0000000000009 0.000000001 0.0 A₂ ≻ A₁ ≻ A₄ ≻ A₁ ≻ A₅

q	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
1.1	0.5030	1.0434	0.0435	0.2014	0.4443	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
1.5	0.5090	1.0753	–0.0665	0.1809	0.3473	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
2	0.4660	1.0274	–0.0826	0.1389	0.2309	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
5	0.1225	0.4660	–0.0257	0.0019	0.0099	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
10	0.0065	0.09603	–0.0009	–0.0032	0.00002	A₂ ≻ A₁ ≻ A₅ ≻ A₃ ≻ A₄
20	0.000013	0.00397	–0.0000008	–0.00003	–0.000000006	A₂ ≻ A₁ ≻ A₅ ≻ A₃ ≻ A₄
30	0.000000026	0.00017	–0.0000000008	–0.0000002	–0.0000000000006	A₂ ≻ A₁ ≻ A₅ ≻ A₃ ≻ A₄
40	0.0000000000000003	0.000007	0.0000000000009	0.000000001	0.0	A₂ ≻ A₁ ≻ A₄ ≻ A₁ ≻ A₅

6.2 Influence of the parameter on the final result

In order to further analyze the flexibility and sensitivity of the parameter q, when τ = 4, we select different q to study the ranking of enterprises. The ranking results are shown in the following Tables 2 and 3.

Table 4
If q = 4, ranking results based on the q - ROTrFLFWA operator by using the different q

τ S (a₁) S (a₂) S (a₃) S (a₄) S (a₅) Ranking

1.1 0.3711 0.7264 0.4713 0.1573 0.1842 A₂ ≻ A₁ ≻ A₃ ≻ A₅ ≻ A₄

2 0.3680 0.7228 0.4538 0.1561 0.1828 A₂ ≻ A₁ ≻ A₃ ≻ A₅ ≻ A₄

5 0.3643 0.7181 0.4335 0.1543 0.1811 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

10 0.3623 0.7154 0.4225 0.1530 0.1803 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

30 0.3602 0.7123 0.4109 0.1510 0.1794 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

50 0.3596 0.7112 0.4074 0.1501 0.1792 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

70 0.3593 0.7107 0.4056 0.1495 0.1790 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

100 0.3590 0.7101 0.4039 0.1489 0.1789 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

1000 0.3582 0.7083 0.3985 0.1449 0.1786 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

10000 0.3580 0.7077 0.3969 0.1408 0.1785 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

1000000 0.3580 0.7074 0.3957 0.1324 0.1785 A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

τ	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
1.1	0.3711	0.7264	0.4713	0.1573	0.1842	A₂ ≻ A₁ ≻ A₃ ≻ A₅ ≻ A₄
2	0.3680	0.7228	0.4538	0.1561	0.1828	A₂ ≻ A₁ ≻ A₃ ≻ A₅ ≻ A₄
5	0.3643	0.7181	0.4335	0.1543	0.1811	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
10	0.3623	0.7154	0.4225	0.1530	0.1803	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
30	0.3602	0.7123	0.4109	0.1510	0.1794	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
50	0.3596	0.7112	0.4074	0.1501	0.1792	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
70	0.3593	0.7107	0.4056	0.1495	0.1790	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
100	0.3590	0.7101	0.4039	0.1489	0.1789	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
1000	0.3582	0.7083	0.3985	0.1449	0.1786	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
10000	0.3580	0.7077	0.3969	0.1408	0.1785	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
1000000	0.3580	0.7074	0.3957	0.1324	0.1785	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄

Table 5

If q = 4, ranking results based on q - ROTrFLFWG operator by using the different τ

τ	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
1.1	0.1985	0.6208	–0.0478	0.0188	0.0289	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
2	0.2020	0.6249	–0.0467	0.02028	0.02949	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
5	0.2074	0.6309	–0.0452	0.0219	0.0304	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
10	0.2114	0.6349	–0.0439	0.0229	0.0311	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
30	0.2177	0.6408	–0.0420	0.0239	0.0322	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
50	0.2206	0.6433	–0.0410	0.0243	0.0328	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
70	0.2225	0.6449	–0.0405	0.0245	0.0331	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
90	0.2239	0.6459	–0.0399	0.0247	0.0334	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
100	0.2245	0.6464	–0.0398	0.0247	0.0334	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
1000	0.2369	0.6546	–0.0347	0.0256	0.0359	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
10000	0.2485	0.6603	–0.0286	0.0263	0.0386	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
1000000	0.2686	0.6667	–0.0120	0.0275	0.0444	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃

Table 6

When k = 3, ranking results based on the q - ROTrFLHWA operator by using the different q

q	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
1.1	0.75618	1.18833	0.6037	0.4739	0.76	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
1.5	1.0283	1.3116	0.77772	0.7128	0.7484	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
2	0.9989	1.2395	0.7407	0.6903	0.5909	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
3	0.6929	0.9608	0.5457	0.4329	0.3237	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
10	0.0352	0.1471	0.1002	0.0059	0.0074	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
20	0.0008	0.0138	0.0110	0.000021	0.000044	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃

Further, when q = 4, we select different τ to study the ranking of enterprises, as shown in the following Tables 4 and 5.

From Tables 2 and 3, we can know the aggregation results are slightly different with parameter q increasing in the q - ROTrFLFWA operator and the q - ROTrFLFWG operator, and A₂ is always the optimal company. From Tables 4 and 5, we can get the aggregation results are different with parameter τ increasing in the q - ROTrFLFWA operator and the q - ROTrFLFWG operator, but A₂ is always the optimal company.

Further, when the value of the parameter q becomes larger and larger (from 1.1 to 40), the fused results by q - ROTrFLFWA operator and q - ROTrFLFWG operator ROFHWA are smaller, and the attitude of the decision -maker is more pessimistic from Tables 2, 3. Meanwhile, the fused results become more and more steady. Afterward, when the value of the parameter τ is larger (from 1.1 to 1000000), the attitude of the decision- maker is more optimistic from Tables 4 and 5.

6.3 Comparative analysis

6.3.1 q - ROTrFLHWA operator and q - ROTrFLHWG operator

To facilitate the comparative analysis, we give the q-rung orthopair triangular fuzzy linguistic Hamacher weighted average (q - ROTrFLHWA) and q - ROTrFLHWG operators on the base of [32], which are given in the follows:

Table 7
When k = 3, ranking results based on q - ROTrFLHWG operator by using the different q

q S (a₁) S (a₂) S (a₃) S (a₄) S (a₅) Ranking

1.1 0.3538 0.9279 –0.0899 0.0703 0.3752 A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

1.5 –0.0607 0.5114 –0.2687 –0.2981 0.0827 A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

2 –0.2099 0.2417 –0.2632 –0.4123 –0.0266 A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

3 –0.1636 0.0526 –0.1422 –0.3189 –0.0339 A₂ ≻ A₅ ≻ A₃ ≻ A₁ ≻ A₄

8 –0.00038 –0.0000052 –0.0009441 –0.007746 –0.000056 A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

10 –0.000000039 –0.00000000006 –0.0000009 –0.00004 –0.0000000059 A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

q	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
1.1	0.3538	0.9279	–0.0899	0.0703	0.3752	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
1.5	–0.0607	0.5114	–0.2687	–0.2981	0.0827	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
2	–0.2099	0.2417	–0.2632	–0.4123	–0.0266	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
3	–0.1636	0.0526	–0.1422	–0.3189	–0.0339	A₂ ≻ A₅ ≻ A₃ ≻ A₁ ≻ A₄
8	–0.00038	–0.0000052	–0.0009441	–0.007746	–0.000056	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
10	–0.000000039	–0.00000000006	–0.0000009	–0.00004	–0.0000000059	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

Table 8

Ranking results based on four operators

q	S (a₁)	S (a₂)	S (a₃)	S (a₄)	S (a₅)	Ranking
PTrFLWA	0.6506	1.1327	0.7356	0.3568	0.5548	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
PTrFLWG	0.4664	1.0703	–0.1224	0.1298	0.2171	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
PTrFLHWA (k = 3)	0.9989	1.2395	0.7407	0.6903	0.5909	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
PTrFLHWG (k = 3)	–0.2099	0.2417	–0.2632	–0.4123	–0.0266	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄

Definition 13 Let a_j (j = 1, 2, . . . , n) be a collection of q-ROTRFLEs, with weighting vector be w = (w₁, w₂, . . . , w_n) ^T, which satisfies 0 ⩽ w_j ⩽ 1 (j = 1, 2, . . . , n), at the same time, $\sum_{j = 1}^{n} w_{j} = 1$ , then we have $q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}],$ $(\sqrt[q]{\frac{\prod_{j = 1}^{n} (1 + (k - 1) u_{j}^{q})^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} (1 + (k - 1) u_{j}^{q})^{w_{j}} + (k - 1) \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}},} \frac{\sqrt[q]{k} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{j}^{{qw}_{j}}}}) 〉;$ $q - ROTrFLHWG (a_{1}, a_{2}, \dots, a_{n}) = \otimes_{j = 1}^{n} a_{j}^{w_{j}} = 〈 [s_{\prod_{i = 1}^{n} {(α_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(β_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(γ_{j})}^{w_{j}}}],$ $(\frac{{(k)}^{1 / q} \prod_{j = 1}^{n} u_{j}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - u_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} u_{j}^{{qw}_{j}})}^{1 / q}}, {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) v_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - v_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) v_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - v_{j}^{q})}^{w_{j}}})}^{1 / q}) 〉$

Let k = 3, we can utilize overall a_ij by q - ROTrFLHWA and q - ROTrFLHWG operators to solve this MADM problem, then the ranking results are given in the following Tables 6 and 7.

6.3.2 PTrFLWA operator, PTrFLWG operator, PTrFLHWA operator and PTrFLHWG operator

We use the PTrFLWA and PTrFLWG operators [23], PTrFLHWA operator and PTrFLHWG operator to solve the Numerical example, the results are given in the following Table 8.

6.3.3 q-ROFWHM operator

To use the Hamy operator to deal with this example, we remove the language variables from Table 1, and the data after removal are shown in Table 9.

Table 9
The data after removal the language variables from Table 1

C ₁ C ₂ C ₃ C ₄

A ₁ < (0.7, 0.4)> < (0.6,0.4)> < (0.3,0.4)> < (0.5,0.4)>

A ₂ < (0.8,0.3)> < (0.7,0.3)> < (0.6,0.3)> < (0.8,0.2)>

A ₃ <, (0.8,0.5)> < (0.1,0.2)> < (0.3,0.2)> < (0.4,0.3)>

A ₄ < (0.5,0.2)> < (0.5,0.6)> < (0.6,0.5)> < (0.5,0.4)>

A ₅ < [(0.3,0.2)> < (0.6,0.2)> < (0.1,0.4)> < (0.4,0.1)>

C ₁	C ₂	C ₃	C ₄
A ₁	< (0.7, 0.4)>	< (0.6,0.4)>	< (0.3,0.4)>	< (0.5,0.4)>
A ₂	< (0.8,0.3)>	< (0.7,0.3)>	< (0.6,0.3)>	< (0.8,0.2)>
A ₃	<, (0.8,0.5)>	< (0.1,0.2)>	< (0.3,0.2)>	< (0.4,0.3)>
A ₄	< (0.5,0.2)>	< (0.5,0.6)>	< (0.6,0.5)>	< (0.5,0.4)>
A ₅	< [(0.3,0.2)>	< (0.6,0.2)>	< (0.1,0.4)>	< (0.4,0.1)>

Table 10

The aggregation operator processes the result

q	Ranking
PTrFLWA	A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄
PTrFLWG	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
PTrFLHWAk = 3)	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃
PTrFLHWGk = 3)	A₂ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₄
q-ROFWHM	A₂ ≻ A₁ ≻ A₄ ≻ A₃ ≻ A₅

Table 11

The Comparison of the different aggregation operators

Aggregation	Whether a	Whether the
operators	parameter	operator
vector exists	can deal with
to manipulate	MADM in triangular
the ranking	fuzzy linguistic
results	environment between
the q-ROFNs
TrFLWA [61]	No	No
TrFLOWA [61]	No	No
ITrFLWA [62]	No	No
ITrFLOWA [62]	No	No
ITrFLHWA [62]	No	No
PTRFLBM [63]	No	No
PTRFLHWA [63]	No	No
TPFLIOWA [64]	No	No
TPFLIOWG [64]	No	No
q-ROFBM [65]	Yes	No
q-ROFGBM [65]	Yes	No
Our proposed methods
based on given operators	Yes	Yes

Combined with Table 9,we use the q-rung orthopair fuzzy Hamy mean operator [60] to solve the given example, and it is concluded that the final ranking of investment enterprises is A₂ ≻ A₁ ≻ A₄ ≻ A₃ ≻ A₅. As we can see, depending on the aggregation operators used, the ordering of the emerging technology enterprises is the same and the best enterprise is A₂.

From Table 10, we proved the feasibility and correctness of the proposed method. Firstly, we compared the proposed methods in this paper with each other. Secondly, we compared the proposed method with other existing methods in detail. Although we use different operators for aggregation from Tables 2–10, the results are still the same. The best desirable alternative is A₁. By comparing the proposed method with PTrFLWA, PTrFLWG, PTrFLHWA, PTrFLHWG, q-ROFWHM, q - ROTrFLHWA and q - ROTrFLHWG operators, the optimal enterprise is also the same. The deficiencies of Jing’s [59]and Du’s [23] based on the PTrFLWA operator, PTrFLWG operator, PTrFLHWA operator, PTrFLHWG operator cannot deal with q-rung orthopair triangular fuzzy values, and the computational complexity of Du’s method [32] is higher than that of the methods proposed in this paper. Moreover, Wang’s method [60] cannot solve the MAGDM under triangular fuzzy values. In comparison with other methods, the advantage of the introduced methodology is that it can solve decision problems according to different parameters of and q values.

The prominent characteristic of the q-ROTrFLHWA operator and q-ROTrFLHWG operator is that the decision-makers can choose the appropriate parameter value q and τ by their preferences. We investigated relevant comparative analyses to show the advantages of the developed operators. Table 11 shows further details.

Compared with TrFLWA, TrFLOWA, ITrFLWA, ITrFLOWA operators, these operators are only several special cases of our proposed operators. So the operators and methods proposed in this paper are more general and more flexible. Furthermore, based on Frank aggregation operators, the proposed operators are very robust and can capture the relationship between the arguments.

Compared with PTRFLBM, PTRFLHWA, TPFLIOWA, TPFLIOWG, q-ROFBM and q-ROFGBM operators proposed in Ref.63,64,65,66, the computational complexity of the defined methods are simple than PTRFLBM, PTRFLHWA, TPFLIOWA, TPFLIOWG, q-ROFGBM operator. Moreover, the developed methods introduced in Ref. 65, 66 can deal with MADM under the q rung triangular fuzzy linguistic environment.

The method we defined has an ideal property about the parameter value q and τ which provides people to choose appropriate values on the basis of risk preferences. We can make the parameter τ as large as possible if the decision-maker is risk-averse, and as small as possible if the decision-maker is risk-loving. So, decision-makers can use appropriate parameter values based on their risk appetite and actual needs. Since Frank’s algorithm is a generalization of algorithms such as Algebra, Einstein, and Hamacher’s algorithm, it is more flexible and general in the process of information fusion.

7 Conclusion

The q-rung orthopair fuzzy aggregation operators are suitable for aggregating information taking the form of numerical values, but they cannot deal with q-rung orthopair triangular fuzzy linguistic information. To reflect the uncertainty of the experts, in this paper, a new class of fuzzy sets named q-rung orthopair triangular fuzzy linguistic set is introduced. Some operational laws and the score function for q-rung orthopair triangular fuzzy linguistic elements are then introduced. Afterward, we develop some aggregation operators based on the q-rung orthopair triangular fuzzy linguistic information and the Frank operator. Moreover, the prominent characteristics of these proposed operators and the relationships between them are discussed. Furthermore, we utilize q - ROTrFLFWA and q - ROTrFLFWG to develop some approaches to solving q-rung orthopair triangular linguistic fuzzy MADM problems. Finally, a practical example is presented to illustrate the application of the proposed procedure.

The key contributions of the paper can be summarized as follows: (1)We develop a q-rung orthopair triangular fuzzy linguistic set. (2) We give a ranking method for q-ROTrFLNs. (3) We introduce the q- ROTrFLWA, q- ROTrFLWG,q-ROTrFLFWA, q- ROTrFLFWG, q-ROTrFLFOWA, and q-ROTrFLFHA operators. (4) Frank operators are degenerated and can be induced on fuzzy aggregation operators, such as the q-ROTrFLHWA operator being induced to a q-ROTrFLWA operator. (5) We prove several desired properties, such as idempotency, commutativity, and boundedness. (6) We present two methods to MADM problems in q-rung orthopair triangular linguistic fuzzy environments. (7) We provide a systematic analysis of the proposed methods with other existing methods.

The Frank aggregation operators for the q-rung orthopair triangular fuzzy linguistic set proposed in this paper are an important complement to relevant research. This article makes some contribution to the development of the IFS theory, the defined methods in this paper may add a new direction in the context of solving MCDM problems. Since the defined methods in this article can not deal with MADM under complex q-rung orthopair fuzzy environments, in future work, we will develope the Frank operator under complex q-rung orthopair fuzzy environments, complex fuzzy N-soft environments [67] or T-spherical fuzzy environments [68]. Furthermore, we shall further generalize these operators by the use of q-rung orthopair trapezoidal fuzzy linguistic information or extend the applications of the aggregation operators to other domains, such as investment decisions, pattern recognition, fuzzy control, cluster analysis, uncertainty optimization, and supply chain management.

Footnotes

Acknowledgments

This research is supported by National Natural Science Foundation of China (71771025, 71571019), Social Science Foundation of Beijing (19YJB013), and School level Excellence Program of University of Chinese Academy of Social Sciences (20220026).

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