Some q-rung orthopair trapezoidal fuzzy linguistic hamacher aggregation operators and their applications

Abstract

A Hamacher operator in a q-rung orthopair trapezoidal fuzzy linguistic environment is studied based on the definition of the q-rung orthopair fuzzy set and the Hamacher aggregation operator. First, we define a new fuzzy variable called q-rung orthopair trapezoidal fuzzy linguistic sets, and the operational laws, score function, accuracy function, comparison rules, and distance measures of the IVPFLVS are defined. Second, based on the Hamacher operator and the q-rung orthopair trapezoidal fuzzy linguistic sets, we propose several q-rung trapezoidal fuzzy linguistic Hamacher operator information aggregation operators, such as the generalized q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted averaging (q-GROTrFLHWA) operator, and the generalized q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted geometric (q-GROTrFLHWG) operator. Third, some desirable properties of the correlation operators, such as idempotency, boundedness, and monotonicity are discussed. Finally, there are two group decision schemes based on q-rung orthopair trapezoidal fuzzy information with known attribute weights. The decision-making scheme is applied to the evaluation of school teaching quality, and the practicability and effectiveness of the scheme are demonstrated by different methods.

Keywords

The q-rung trapezoidal fuzzy linguistic set multi-attribute decision making Hamacher operator application

1 Introduction

The complexity and uncertainty of objective things cause many practical problems in a fuzzy environment. Zadeh [1] introduced fuzzy theory to manage decision making problems in 1965. Atanassov [2] introduced intuitionistic fuzzy sets that consider membership degree, non-membership degree, and hesitation degree, which require that the sum of membership degree and non-membership degree should be less than or equal to 1. However, in practical decision making, the sum of the membership degree and the non-membership degree is sometimes greater than 1, which cannot be described using intuitionistic fuzzy sets. Thus, Yager [3] defined the Pythagorean fuzzy set in 2013, and many studies [4 –6] have been conducted.

To study the decision-making problem more widely in 2016, Yager [7] proposed the q-rung orthopair fuzzy set based on an intuitive fuzzy set and Pythagorean fuzzy set, where the sum of the membership to the power q and non-membership to the power q is less than or equal to 1, namely, 0 ⩽ u^q + v^q ⩽ 1. The q-rung fuzzy set is an extension of the intuitionistic fuzzy set and Pythagorean fuzzy set, thus it has a wider range of applications, greater flexibility, and it can better deal with decision-making problem. Experts have performed some extended studies on this fuzzy set and applied some operators to the q-rung orthopair fuzzy environment. Wang et al. [8] introduced the q-rung orthopair fuzzy HM (q-ROFHM) operator and q-rung orthopair fuzzy WHM (q-ROFWHM) operator. Abosuliman et al. [9] introduced fractional orthotriple fuzzy β-covering (FOF β-covering), fractional orthotriple fuzzy β-neighbor (FOF β-neighbor) and proposed a new FOF-covering decision theoretical rough sets model (FOFCDTRSs) or developed related properties. Xing et al. [10] proposed some point operators under q-rung orthopair fuzzy environment to redistribute the membership and non-membership. Liu et al. [11] developed a concept of q-rung orthopair hesitant fuzzy set and introduced the operational laws between two q-ROHFSs, then, they developed the distance measures between q-ROHFSs. Liu et al. [12] proposed a MAGDM method based on the novel aggregation operator under the linguistic intuitionistic cubic fuzzy set (LICFS) to solve the selection problem of the online education platform in the period of the COVID-19. Gao et al. [13] proposed derivatives and differentials of the q-rung orthopair fuzzy functions (q-ROFFs), introduced a accurate concept of definite integrals of q-ROFFs, then discussed basic properties. Naeem et al. [14] proposed a new similarity measures (SMs) based on the fractional orthotriple fuzzy set (FOFS), and applied these SMs and WSMs to the pattern recognition problem. Gong et al. [15] proposed a new UTAE approach based on q-rung orthopair fuzzy sets and the multi-attribute border area comparison method to evaluation. Garg et al. [16] proposed a concept of Complex q-rung orthopair fuzzy set (Cq-ROFS), further to aggregate the information.

The Hamacher operator [17 , 21] is an extended form of Archimedes’ T and Archimedes’ S modes, the Hamacher product and sum can realize the flexibility of calculation by its own parameters, which has been studied by experts and scholars recently. For example, Huang [19] proposed the intuitionistic fuzzy Hamacher operator. Zhou et al. [18] extended the Hamacher operator to hesitant fuzzy set and defined the hesitant fuzzy Hamacher operator. Son et al. [20] introduced a new hesitant fuzzy Hamacher aggregation operators based on Hamacher t-norm or t-conorm. Ayaz et al. [22] applied Hamacher operators such as spherical cubic fuzzy Hamacher weighted average (SCFHWA) operator, spherical cubic fuzzy Hamacher ordered weighted average (SCFHOWA) operator for the appraisal of the optimal choice. Xu [23] introduced a concept of trapezoidal fuzzy linguistic variable or some operational laws, then developed a similarity measure between two trapezoidal fuzzy linguistic numbers. Akram et al. [24] introduced q-rung orthopair fuzzy Hamacher graphs (q-ROFHGs) to solve decision making numerical example related to the selection of optimal term. Aydemir et al. [25] developed Fermatean aggregation operators based on Dombi, Hamacher or Einstein algebraic operators. Then, to understand the impact of the proposed operators, Fermatean fuzzy TOPSIS was established.

Although the q-rung orthopair fuzzy set is superior to the intuitionistic fuzzy set and the Pythagorean fuzzy set, its application scope is further expanded, however, it can not solve the trapezoidal fuzzy multi-attribute decision making problem and the operator has a high complexity. The trapezoidal fuzzy model can better describe the critical state evaluation, and the model and the appropriate parameters is an important means to reduce uncertainty. Thus, we combines the q-rung orthopair fuzzy linguistic set and the trapezoidal fuzzy set, a new q-rung orthopair trapezoidal fuzzy linguistic set is proposed. It has wider and more flexible than q-rung fuzzy set for multi-attribute problems. Since the advantage of Hamacher aggregation operators, we are utilized it to develop new operator in fuzzy linguistic environment.

The rest of this paper is organized as follows. In Section 2, some basic definitions are briefly reviewed, such as q-rung orthopair fuzzy sets, trapezoidal fuzzy linguistic numbers and q-rung orthopair trapezoidal fuzzy linguistic set, moreover, the operational laws, score and accuracy functions are defined, the standard hamming distance for the q-ROTrFLNs is given. In Section 3, q-rung orthopair trapezoidal fuzzy linguistic Hamacher operators are proposed, some desirable properties are investigated. In Section 4 and 5, two approaches for MAGDM with q-rung orthopair trapezoidal fuzzy linguistic are developed. In Section 6, an example is presented to illustrate effectiveness of the method. The paper is concluded in Section 7.

2 Preliminarie

2.1 q-rung orthopair fuzzy set

Definition 1 [7, 8]. Let A ={〈 x, u (x) , v (x) 〉 |x ∈ X} be a 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, 0 ⩽ (u (x)) ^q + (v (x)) ^q ⩽ 1, $π (x) = \sqrt[q]{1 - {(u (x))}^{q} - {(v (x))}^{q}}$ be the hesitation degree.

2.2 trapezoidal fuzzy linguistic set

The linguistic set is proposed as a finite and ordered discrete term set, S = (s₀, s₁, …, s_l-1), where l is the odd value. If l = 5, then $S = (s_{0} : extreme poor, s_{1} : very poor, s_{2} : poor, s_{3} : fair, s_{4} : good) .$

Definition 2 [11]. Let $\tilde{S} = {s_{θ} | s_{0} ⩽ s_{θ} ⩽ s_{l - 1}, θ \in [0, l - 1]}$ is the continuous form of linguistic set S. s_α(x), s_β(x), s_γ(x), and s_η(x) are four linguistic terms, s₀ ⩽ s_α(x) ⩽ s_β(x) ⩽ s_γ(x) ⩽ s_η(x) ⩽ s_l-1 and the trapezoidal fuzzy linguistic information variable is defined as $\tilde{s} = [s_{α (x)}, s_{β (x)}, s_{γ (x)}, s_{η (x)}]$ .

2.3 q-rung orthopair trapezoidal fuzzy linguistic set

Since the q-rung orthopair fuzzy set is an extension of the Pythagorean fuzzy set, a new q-rung orthopair trapezoidal fuzzy linguistic set will be defined.

Definition 3. Let X be a non-empty set of real numbers, and the q-rung orthopair trapezoidal fuzzy linguistic set can be expressed as: $A = (〈 x [s_{α (x)}, s_{β (x)}, s_{γ (x)}, s_{η (x)}], (u (x), v (x)) 〉 | x \in X)$ where $u (x) : X \in [0, 1], \begin{matrix} \end{matrix} v (x) : X \in [0, 1]$ , u (x) indicates the degree of membership of x in A, v (x) indicates the degree of non-membership of x in A, and 0 ⩽ (u (x)) ^q + (v (x)) ^q ⩽ 1, q > 1, and its hesitation degree is $π (x) = \sqrt[q]{1 - {(u (x))}^{q} - {(v (x))}^{q}}$ . We note the q-rung orthopair trapezoidal fuzzy linguistic number (q-ROTrFLN) as a =〈 [s_α(x), s_β(x), s_γ(x), s_η(x)] , (u, v) 〉.

Theorem 1. Let a_i =〈 [s_{α(a_i)}, s_{β(a_i)}, s_{γ(a_i)}, s_{η(a_i)}] , (u (a_i) , v (a_i)) 〉 and a_j = 〈 [s_{α(a_j)}, s_{β(a_j)}, s_{γ(a_j)}, s_{η(a_j)}] , (u (a_j) , v (a_j)) 〉 are two q-ROTrFLNs, the operational laws as follows.

Definition 4. Let a_i and a_j are two q-ROTrFLNs, k, k_i, k_j ⩾ 0, and then

a_i ⊕ a_j = a_j ⊕ a_i;

a_i ⊗ a_j = a_j ⊗ a_i

λ (a_i ⊕ a_j) = λa_j ⊕ λa_i;

(λ_i ⊕ λ_j) a_i = λ_ia_i ⊕ λ_ja_i

$a_{i}^{λ_{1}} \otimes a_{i}^{λ_{2}} = a_{i}^{λ_{1} + λ_{2}}$ ; (6) $a_{i}^{λ} \otimes a_{j}^{λ} = (a_{i} \otimes a_{j})^{λ}$ .

The following defines the score function S (a_i) and accuracy function H (a_i) for the q-rung orthopair trapezoidal fuzzy linguistic set.

Definition 5. Let a_i =〈 [s_{α(a_i)}, s_{β(a_i)}, s_{γ(a_i)}, s_{η(a_i)}] , (u (a_i) , v (a_i)) 〉 be a q-ROTrFLN. $E (a_{i}) = (u^{q} (a_{i}) - v^{q} (a_{i})) (α (a_{i}) + β (a_{i}) + γ (a_{i}) + η (a_{i})) / 4$ (1) $H (a_{i}) = (u^{q} (a_{i}) + v^{q} (a_{i})) (α (a_{i}) + β (a_{i}) + γ (a_{i}) + η (a_{i})) / 4$ (2)

Definition 6. Let a_i, a_j be two q-ROTrFLNs, then

if E (a_i) > E (a_j), then a_i > a_j;

if E (a_i) < E (a_j), then a_i < a_j;

if E (a_i) = E (a_j), then:

if H (a_i) > H (a_j), then a_i > a_j;

if H (a_i) < H (a_j), then a_i < a_j.

Definition 7. Let a_i and a_j be two q-ROTrFLNs, where S = {s₀, s₁, ... , s_l} is a limited set of linguistic terms, and the standard hamming distance can be defined [16]. $\begin{matrix} d (a_{i}, a_{j}) = \frac{1}{2 (l - 1)} (| (1 + u^{q} (a_{i}) - v^{q} (a_{i})) \times \frac{α (a_{i}) + β (a_{i}) + r (a_{i}) + η (a_{i})}{4} - \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} (1 + u^{q} (a_{j}) - v^{q} (a_{j})) \times \frac{α (a_{j}) + β (a_{j}) + γ (a_{j}) + η (a_{j})}{4} |) \end{matrix} \end{matrix}$ (3)

3 q-rung orthopair trapezoidal fuzzy linguistic set based on Hamacher aggregation operators

Definition 8 [16]. Hamacher T-norm or S-norm in the q-rung orthopair fuzzy set can be defined $T (a, b) = a \otimes b = \frac{ab}{(k + (1 - k) (a + b - ab))}, (k > 0)$ (4) $S (a, b) = a \oplus b = \frac{a + b - a b - (1 - k) ab}{1 - (1 - k) ab}, (k > 0)$ (5)

According to the concept of q-ROTrFLNs and the Hamacher T-norm and S-norm, the Hamacher aggregation operators in the q-rung orthopair trapezoidal fuzzy linguistic set can be defined.

Theorem 2. Let a₁ =〈 [s_{α
₁}, s_{β
₁}, s_{γ
₁}, s_{η
₁}] , (u₁, v₁) 〉, a₂ =〈 [s_{α
₂}, s_{β
₂}, s_{γ
₂}, s_{η
₂}] , (u₂, v₂) 〉 be any two q-rung orthopair trapezoidal fuzzy linguistic numbers, and λ > 0. Then, the operational laws as follows: $(1) a_{1} \oplus a_{2} = 〈 [s_{α_{1} + α_{2}}, s_{β_{1} + β_{2}}, s_{γ_{1} + γ_{2}}, s_{η_{1} + η_{2}}], (\sqrt[\begin{matrix} q \end{matrix}]{\frac{u_{1}^{q} + u_{2}^{q} - u_{1}^{q} u_{2}^{q} - (1 - k) u_{1}^{q} u_{2}^{q}}{1 - (1 - k) u_{1}^{q} u_{2}^{q}}} \frac{v_{1} v_{2}}{\sqrt[\begin{matrix} q \end{matrix}]{k + (1 - k) (v_{1}^{q} + v_{2}^{q} - v_{1}^{q} v_{2}^{q})}}) 〉;$ $(2) a_{1} \otimes a_{2} = 〈 [s_{α_{1} \times α_{2}}, s_{β_{1} \times β_{2}}, s_{r_{1} \times r_{2}}, s_{η_{1} \times η_{2}}], (\frac{u_{1} u_{2}}{\sqrt[q]{k + (1 - k) (u_{1}^{q} + u_{2}^{q} - u_{1}^{q} u_{2}^{q})}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{v_{1}^{q} + v_{2}^{q} - v_{1}^{q} v_{2}^{q} - (1 - k) v_{1}^{q} v_{2}^{q}}{1 - (1 - k) v_{1}^{q} v_{2}^{q}}}) 〉;$ $(3) λ a_{1} = [s_{λ \times α_{1}}, s_{λ \times β_{1}}, s_{λ \times r_{1}}, s_{λ \times η_{1}}], (\sqrt[\begin{matrix} q \end{matrix}]{\frac{{(1 + (k - 1) u_{1}^{q})}^{λ} - {(1 - u_{1}^{q})}^{λ}}{{(1 + (k - 1) u_{1}^{q})}^{λ} + (k - 1) {(1 - u_{1}^{q})}^{λ}}}, 〉〈 \frac{\sqrt[q]{k} v_{1}^{λ}}{\sqrt[\begin{matrix} q \end{matrix}]{{(1 + (k - 1) (1 - v_{1}^{q}))}^{λ} + (k - 1) v_{1}^{q λ}}}) ;$ $(4) a_{1}^{λ} = 〈 [s_{{(α_{1})}^{λ}}, s_{{(β_{1})}^{λ}}, s_{{(r_{1})}^{λ}}, s_{{(η_{1})}^{λ}}], (\frac{\sqrt[q]{k} u_{1}^{λ}}{\sqrt[\begin{matrix} q \end{matrix}]{{(1 + (k - 1) (1 - u_{1}^{q}))}^{λ} + (k - 1) u_{1}^{q λ}}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{{(1 + (k - 1) v_{1}^{q})}^{λ} - {(1 - v_{1}^{q})}^{λ}}{{(1 + (k - 1) v_{1}^{q})}^{λ} + (k - 1) {(1 - v_{1}^{q})}^{λ}}}) 〉 .$

3.1 The q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted operators

Definition 9. Let a_j =〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}, s_{η
_j}] , (u_j, v_j) 〉 be a set of q-ROTrFLNs, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, 0 ⩽ w_j ⩽ 1. Then, the q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted average can be defined, Hⁿ → H. $\begin{matrix} q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}} s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], \\ (\sqrt[q]{\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix}$ (6) where Hⁿ → H expressed as an q-ROTrFLHWA operator is a mapping, H is the set of all q-ROTrFLNs.

Proof. Mathematical induction is used as follows:

When n = 1, the q-ROTrFLHWA operator clearly holds.

Then, verification that the above equation is true when n = 2. $\begin{matrix} q - ROTrFLHWA (a_{1}, a_{2}) = w_{1} a_{1} \oplus w_{2} a_{2} \\ = 〈 [s_{w_{1} α_{1} + w_{2} α_{2}}, s_{w_{1} β_{1} + w_{2} β_{2}}, s_{w_{1} γ_{1} + w_{2} γ_{2}}, s_{w_{1} η_{1} + w_{2} η_{2}}], (\sqrt[q]{\frac{k {(1 + (k - 1) u_{2}^{q})}^{w_{2}} {(1 + (k - 1) u_{1}^{q})}^{w_{1}} - k (1 - u_{2}^{q})^{w_{2}} (1 - u_{1}^{q})^{w_{1}}}{k {(1 + (k - 1) (1 - v_{2}^{q}))}^{w_{2}} {(1 + (k - 1) (1 - v_{1}^{q}))}^{w_{1}} + k (k - 1) (1 - v_{2}^{q})^{w_{2}} (1 - v_{1}^{q})^{w_{1}}}}, \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \frac{\sqrt[q]{k} v_{1}^{w_{1}} v_{2}^{w_{2}}}{\sqrt[q]{{(1 + (k - 1) (1 - v_{1}^{q}))}^{w_{1}} {(1 + (k - 1) (1 - v_{2}^{q}))}^{w_{2}} + (k - 1) v_{1}^{{qw}_{1}} v_{2}^{{qw}_{2}}}}) \end{matrix} 〉 \\ = 〈 [s_{w_{1} α_{1} + w_{2} α_{2}}, s_{w_{1} β_{1} + w_{2} β_{2}}, s_{w_{1} γ_{1} + w_{2} γ_{2}}, s_{w_{1} η_{1} + w_{2} η_{2}}], \\ (\sqrt[q]{\frac{\prod_{j = 1}^{2} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{2} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{2} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{2} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{2} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{2} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{2} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix}$

Suppose the q-ROTrFLHWA operator is true when n = m, next, verify that it is true when n = m + 1. $\begin{matrix} q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{m + 1}) = \oplus_{j = 1}^{m} w_{j} a_{j} \oplus w_{m + 1} a_{m + 1} = 〈 [s_{\oplus_{j = 1}^{m} w_{j} α_{j}}, s_{\oplus_{j = 1}^{m} w_{j} β_{j}}, s_{\oplus_{j = 1}^{m} w_{j} γ_{j}} s_{\oplus_{j = 1}^{m} w_{j} η_{j}}], \\ (\sqrt[q]{\frac{\prod_{j = 1}^{m} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{m} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{m} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{m} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{m} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} v_{j}^{{qw}_{j}}}}) 〉 \oplus \\ 〈 [s_{w_{m + 1} \times α_{m + 1}}, s_{w_{m + 1} \times β_{m + 1}}, s_{w_{m + 1} \times r_{m + 1}}, s_{w_{m + 1} \times η_{m + 1}}], \\ (\sqrt[q]{\frac{{(1 + (k - 1) u_{m + 1}^{q})}^{w_{m + 1}} - {(1 - u_{m + 1}^{q})}^{w_{m + 1}}}{{(1 + (k - 1) u_{m + 1}^{q})}^{v_{m + 1}} + (k - 1) {(1 - u_{m + 1}^{q})}^{w_{m + 1}}}, \frac{\sqrt[q]{k} v_{m + 1}^{w_{m + 1}}}{\sqrt[q]{{(1 + (k - 1) (1 - v_{m + 1}^{q}))}^{w_{m + 1}} + (k - 1) v_{m + 1}^{{qv}_{m + 1}}}}}) 〉 \end{matrix}$ $\begin{matrix} = 〈 [s_{\oplus_{j = 1}^{m + 1} w_{j} α_{j}}, s_{\oplus_{j = 1}^{m + 1} w_{j} β_{j}}, s_{\oplus_{j = 1}^{m + 1} w_{j} γ_{j}} s_{\oplus_{j = 1}^{m + 1} w_{j} η_{j}}], \\ (\sqrt[q]{\frac{\prod_{j = 1}^{m + 1} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{m + 1} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{m + 1} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{m + 1} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{m + 1} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{m + 1} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{m + 1} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix}$

We derive the q-ROTrFLHWA operator by mathematical induction and verify its correctness. By adjusting the parameter k, some special forms of the operator are discussed, as shown below.

When k = 1, the q-ROTrFLHWA operator is induced to a q-rung orthopair trapezoidal fuzzy linguistic weighted average (q-ROTrFLWA) operator, and $q - ROTrFLWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}}, s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], (\sqrt[\begin{matrix} q \end{matrix}]{1 - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}, \prod_{j = 1}^{n} v_{j}^{w_{j}}) 〉 .$

When k = 2, the q-ROTrFLHWA operator is induced to a q-rung orthopair trapezoidal fuzzy linguistic Einstein weighted average (q-ROTrFLEWA) operator, and

\begin{matrix} q - ROTrFLEWA (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}} s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], (\sqrt[q]{\frac{\prod_{j = 1}^{n} {(1 + u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + u_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{2} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{n} {(2 - v_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix}

The q-ROTrFLHWA operator also have some desirable properties such as idempotency, boundedness and monotonicity. The desirable properties are discussed below.

Theorem 3. (idempotency): Let a_j (j = 1, 2, . . . , n) be a set of q-ROTrFLNs, a_j = a₀ =〈 [s_{α
₀}, s_{β
₀}, s_{γ
₀}, s_{η
₀}] , (u₀, v₀) 〉. $q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} = a_{0}, (j = 1, 2, . . ., n)$

Proof. $\begin{matrix} q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}} s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], \\ \begin{matrix} (\sqrt[q]{\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix} \\ = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{0}}, s_{\oplus_{j = 1}^{n} w_{j} β_{0}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{0}} s_{\oplus_{j = 1}^{n} w_{j} η_{0}}], (\sqrt[q]{\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{0}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{0}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{0}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} {(1 - u_{0}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{n} v_{0}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{0}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{0}^{{qw}_{j}}}}) 〉 \\ = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{0}}, s_{\oplus_{j = 1}^{n} w_{j} β_{0}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{0}} s_{\oplus_{j = 1}^{n} w_{j} η_{0}}], (\sqrt[q]{\frac{{(1 + (k - 1) u_{0}^{q})}^{\sum_{j = 1}^{n} w_{j}}}{{(1 + (k - 1) u_{0}^{q})}^{\sum_{j = 1}^{n} w_{j}} + (k - 1) (1 - u_{0}^{q})^{\sum_{j = 1}^{n} w_{j}}}}, \frac{\sqrt[q]{k} v_{0}^{\sum_{j = 1}^{n} w_{j}}}{\sqrt[q]{{(1 + (k - 1) (1 - v_{0}^{q}))}^{\sum_{j = 1}^{n} w_{j}} + (k - 1) v_{0}^{q \sum_{j = 1}^{n} w_{j}}}}) 〉 \\ = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{0}}, s_{\oplus_{j = 1}^{n} w_{j} β_{0}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{0}} s_{\oplus_{j = 1}^{n} w_{j} η_{0}}], (u_{0}, v_{0}) 〉 = a_{0} \end{matrix}$

Theorem 4. (boundedness): Let a_j (j = 1, 2, . . . , n) be a set of q-rung orthopair trapezoidal fuzzy linguistic numbers, a_j = a₀ =〈 [s_{α
₀}, s_{β
₀}, s_{γ
₀}, s_{η
₀}] , (u₀, v₀) 〉.

If $s_{α^{-}} = min_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}, s_{α^{+}} = max_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}, s_{β^{-}} = min_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}$ , $s_{β^{+}} = max_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}, s_{γ^{-}} = min_{1 ⩽ j ⩽ n} {s_{γ (a_{j})}}, s_{γ^{+}} = max_{1 ⩽ j ⩽ n} {s_{γ (a_{j})}}$ , $s_{η^{-}} = min_{1 ⩽ j ⩽ n} {s_{η (a_{j})}}, s_{η^{+}} = max_{1 ⩽ j ⩽ n} {s_{η (a_{j})}}, u_{min} = min_{1 ⩽ j ⩽ n} {u (a_{j})}$ , $u_{max} = max_{1 ⩽ j ⩽ n} {u (a_{j})}, v_{min} = min_{1 ⩽ j ⩽ n} {v (a_{j})}, v_{max} = max_{1 ⩽ j ⩽ n} {v (a_{j})}$ , then $〈 [s_{α^{-}}, s_{β^{-}}, s_{γ^{-}}, s_{η^{-}}], (u_{min}, v_{max}) 〉 ⩽ q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) ⩽ 〈 [s_{α^{+}}, s_{β^{+}}, s_{γ^{+}}, s_{η^{+}}], (u_{max}, v_{min}) 〉$

Proof. (1) Let $u_{q - ROTrFLHWA} = {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - u_{j}^{q})}^{w_{j}}})}^{1 / q} = {(1 - \frac{k}{\prod_{j = 1}^{n} {(1 + \frac{{ku}_{j}^{q}}{(1 - u_{j}^{q})})}^{w_{j}} + (k - 1)})}^{1 / q},$

let $H (x) = \frac{{kx}^{q}}{(1 - x^{q})}, 0 ⩽ x ⩽ 1, k > 0, \begin{matrix} {H^{'} (x) = \frac{{qkx}^{q - 1}}{(1 - x^{q})^{2}} > 0, \end{matrix}$ and it is known that H (x) is a monotonically increasing function. Since u_min ⩽ u_j ⩽ u_max, it is known that, H (u_j) _min ⩽ H (u_j) ⩽ H (u_j) _max and its expression is as follows: $u_{min} = {(1 - \frac{k}{\prod_{j = 1}^{n} {(1 + \frac{k (u_{min})^{q}}{(1 - (u_{min})^{q})})}^{w_{j}} + (k - 1)})}^{1 / q} ⩽ {(1 - \frac{k}{\prod_{j = 1}^{n} {(1 + \frac{{ku}_{j}^{q}}{1 - u_{j}^{q}})}^{w_{j}} + (k - 1)})}^{1 / q} ⩽ (1 - \frac{k}{\prod_{j = 1}^{n} {(1 + \frac{k (u_{max})^{q}}{1 - (u_{max})^{q}})}^{w_{j}} + (k - 1)}) = u_{max}$

It is known that $min u_{j} ⩽ u_{q - ROTrFLHWA} ⩽ max u_{j}$ .

(2) Since $v_{q - ROTrFLHWA} = \frac{{(k)}^{1 / q} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{j}^{{qw}_{j}})}^{1 / q}} = {(\frac{k}{\prod_{j = 1}^{n} {(\frac{k}{v_{j}^{q}} - k + 1)}^{w_{j}} + (k - 1)})}^{1 / q},$

let $G (x) = \frac{k}{x^{q}}, 0 < x ⩽ 1, k > 0; G^{'} (x) = \frac{- {kqx}^{q - 1}}{x^{2 q}} < 0$ ; therefore, G (x) is a monotonically decreasing function. Since v_min ⩽ v_j ⩽ v_max, it is known that $min v_{j} = {(\frac{k}{\prod_{j = 1}^{n} {(\frac{k}{{(min v_{j})}^{q}} - k + 1)}^{w_{j}} + (k - 1)})}^{1 / q} ⩽ {(\frac{k}{\prod_{j = 1}^{n} {(\frac{k}{v_{j}^{q}} - k + 1)}^{w_{j}} + (k - 1)})}^{1 / q} ⩽ {(\frac{k}{\prod_{j = 1}^{n} {(\frac{k}{{(max v_{j})}^{q}} - k + 1)}^{w_{j}} + (k - 1)})}^{1 / q} = max v_{j}$

Then $min v_{j} ⩽ v_{q - ROTrFLHWA} ⩽ max v_{j}$ , since s_{α
^-} ⩽ s_{α(a_j)} ⩽ s_{α
⁺}, s_{β
^-} ⩽ s_{β(a_j)} ⩽ s_{β
⁺}, s_{γ
^-} ⩽ s_{γ(a_j)} ⩽ s_{γ
⁺}, and s_{ι
^-} ⩽ s_{γ(a_j)} ⩽ s_{η
⁺}. From Definition 6, it is known that $〈 [s_{α^{-}}, s_{β^{-}}, s_{γ^{-}}, s_{η^{-}}], (u_{min}, v_{max}) 〉 ⩽ q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) ⩽ 〈 [s_{α^{+}}, s_{β^{+}}, s_{γ^{+}}, s_{η^{+}}], (u_{max}, v_{min}) 〉$

Theorem 5. (monotonicity): If $a_{j} ⩽ a_{j}^{'} (j = 1, 2, . . ., n)$ , then $q - ROTrFLHWA (a_{1}, \dots, a_{n}) ⩽ q - ROTrFLHWA (a_{1}^{'}, \dots, a_{n}^{'})$ .

The proof is similar to Theorem 4.

Definition 10. Let a_j (j = 1, 2, . . . , n) be a q-ROTrFLNs, and then the q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted geometric operator can be defined as given below: Hⁿ → H. $\begin{matrix} q - ROTrFLHWG (a_{1}, a_{2}, \dots, a_{n}) = \otimes_{j = 1}^{n} a_{j}^{w_{j}} = 〈 [s_{\prod_{i = 1}^{n} {(α_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(β_{j})}^{w_{j}}}, s_{\prod_{i = 1}^{n} {(γ_{j})}^{w_{j}}} s_{\prod_{i = 1}^{n} {(η_{j})}^{w_{j}}}], \\ (\frac{{(k)}^{1 / q} \prod_{j = 1}^{n} u_{j}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - u_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} u_{j}^{{qw}_{j}})}^{1 / q}}, {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) v_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - v_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) v_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - v_{j}^{q})}^{w_{j}}})}^{1 / q}) 〉 \end{matrix}$ (7)

Similar to the q-ROTrFLHWA operator, the q-ROTrFLHWG operator also have idempotency, monotonicity and boundedness, and the proof is also similar to Theorem 3 and Theorem 4.

Theorem 6. (idempotency):

If a_j = a₀ =〈 [s_{α
₀}, s_{β
₀}, s_{γ
₀}] , (u₀, v₀) 〉, then $q - ROTrFLHWG (a_{1}, a_{2}, \dots, a_{n}) = \otimes_{j = 1}^{n} a_{j}^{w_{j}} = a_{0}$

Theorem 7. (boundedness):

If $s_{α^{-}} = min_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}, s_{α^{+}} = max_{1 ⩽ j ⩽ n} {s_{α (a_{j})}}, s_{β^{-}} = min_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}$ , $s_{β^{+}} = max_{1 ⩽ j ⩽ n} {s_{β (a_{j})}}, s_{γ^{-}} = min_{1 ⩽ j ⩽ n} {s_{γ (a_{j})}}, s_{γ^{+}} = max_{1 ⩽ j ⩽ n} {s_{γ (a_{j})}}$ , $s_{η^{-}} = min_{1 ⩽ j ⩽ n} {s_{η (a_{j})}}, s_{η^{+}} = max_{1 ⩽ j ⩽ n} {s_{η (a_{j})}}, u_{min} = min_{1 ⩽ j ⩽ n} {u (a_{j})}$ , $u_{max} = max_{1 ⩽ j ⩽ n} {u (a_{j})}, v_{min} = min_{1 ⩽ j ⩽ n} {v (a_{j})}, v_{max} = max_{1 ⩽ j ⩽ n} {v (a_{j})}$ then 〈 [s_{α
^-}, s_{β
^-}, s_{γ
^-}, s_{η
^-}] , (u_min, v_max)〉 ⩽ q - ROTrFLHWG (a₁, a₂, …, a_n) ⩽ 〈 [s_{α
⁺}, s_{β
⁺}, s_{γ
⁺}, s_{η
⁺}] , (u_max, v_min) 〉.

Theorem 8. (monotonicity): If $a_{i} ⩽ a_{i}^{'} (i = 1, 2, . . ., n)$ , then $q - ROTrFLHWG (a_{1}, \dots, a_{n}) ⩽ q - ROTrFLHWG (a_{1}^{'}, \dots, a_{n}^{'})$ . The operator mentioned above is extended£¬a ordered weighted integration operatorn is defined.

3.2 The q-rung orthopair trapezoidal fuzzy linguistic Hamacher ordered weighted operator

Definition 11. Let a_j (j = 1, 2, . . . , n) be a q-ROTrFLNs, w = (w₁, w₂, …, w_n) ^T is the position weight vector of a_j (j = 1, 2, . . . , n), 0 ⩽ w_j ⩽ 1, $\sum_{j = 1}^{n} = 1$ . Then the q-rung orthopair trapezoidal fuzzy linguistic Hamacher ordered weighted average operator is defined: Hⁿ → H. $\begin{matrix} q - ROTrFLHOWA = \oplus_{j = 1}^{n} w_{j} a_{σ (j)} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} β_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{σ (j)}}], \\ 〈 {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{σ (j)}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{σ (j)}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{σ (j)}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - u_{σ (j)}^{q})}^{w_{j}}})}^{1 / q}, \frac{{(k)}^{1 / q} \prod_{j = 1}^{n} v_{σ (j)}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{σ (j)}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{σ (j)}^{{qw}_{j}})}^{1 / q}} 〉 \end{matrix}$ (8) where a_σ(j) is the j^th largest value in a_j.

The q-ROTrFLHOWA operator have idempotency, boundedness and monotonicity similar to q-ROTrFLHWA operator, moreover, the operator is also commutative.

Theorem 9. (commutativity): Let a_j (j = 1, 2, . . . , n) be a set of q-ROTrFLNs, and if $a_{j}^{'} (j = 1, 2, . . ., n)$ is any substitute for a_j (j = 1, 2, . . . , n), then $q - ROTrFLHOWA (a_{1}, \dots, a_{n}) = q - ROTrFLHOWA (a_{1}^{'}, \dots, a_{n}^{'})$

Proof. Because if $a_{j}^{'} (j = 1, 2, . . ., n)$ is any substitute for a_j (j = 1, 2, . . . , n), then $\begin{array}{l} q - ROTrFLHOWA = \oplus_{j = 1}^{n} w_{j} a_{σ (j)}^{'} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} β_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{σ (j)}}], \\ {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u^{'}_{σ (j)}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u^{'}_{σ (j)}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u^{'}_{σ (j)}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - u^{'}_{σ (j)}^{q})}^{w_{j}}})}^{1 / q}, \frac{{(k)}^{1 / q} \prod_{j = 1}^{n} v^{'}_{σ (j)}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{σ (j)}^{^{'} q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{σ (j)}^{^{'} q w_{j}})}^{1 / q}} 〉 \end{array}$ $\begin{array}{l} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} β_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{σ (j)}}], \\ {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{σ (j)}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{σ (j)}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{σ (j)}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - u_{σ (j)}^{q})}^{w_{j}}})}^{1 / q}, \frac{{(k)}^{1 / q} \prod_{j = 1}^{n} v_{σ (j)}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{σ (j)}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{σ (j)}^{q w_{j}})}^{1 / q}} 〉 \\ = \oplus_{j = 1}^{n} w_{j} a_{σ (j)} \end{array}$

3.3 The q-rung orthopair trapezoidal fuzzy linguistic Hamacher hybrid average operator

Since the q-ROTrFLHWA operator only considers importance of each q-rung orthopair trapezoidal fuzzy linguistic set, q-ROTrFLHOWA operator only gives a weight to the position of the q-rung orthopair trapezoidal fuzzy linguistic variable. In order to overcome the shortcomings of these two kinds of operators, the Hamacher mixed-weighted average operator is given below.

Definition 12. Let a_j (j = 1, 2, . . . , n) be a set of q-rung orthopair fuzzy linguistic numbers, n is the balancing coefficient, w_j is the weight vector corresponding to a_j (j = 1, 2, . . . , n), 0 ⩽ w_j ⩽ 1, Then the q-rung orthopair trapezoidal fuzzy linguistic Hamacher hybrid average (q-ROTrFLHHA) operator can be defined as given below: Hⁿ → H. $\begin{matrix} q - ROTrFLHHA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} {\dot{a}}_{σ (j)} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} {\dot{α}}_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} {\dot{β}}_{σ (j)}}, s_{\oplus_{j = 1}^{n} w_{j} {\dot{γ}}_{σ (j)}} s_{\oplus_{j = 1}^{n} w_{j} {\dot{η}}_{σ (j)}}], \\ = 〈 {(\frac{\prod_{j = 1}^{n} {(1 + (k - 1) {\dot{u}}_{σ (j)}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{u}}_{σ (j)}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) {\dot{u}}_{σ (j)}^{q})}^{w_{j}} + \prod_{j = 1}^{n} (k - 1) {(1 - {\dot{u}}_{σ (j)}^{q})}^{w_{j}}})}^{1 / q}, \frac{{(k)}^{1 / q} \prod_{j = 1}^{n} {\dot{v}}_{σ (j)}^{w_{j}}}{{(\prod_{j = 1}^{n} {(1 + (k - 1) (1 - {\dot{v}}_{σ (j)}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} {\dot{v}}_{σ (j)}^{{qw}_{j}})}^{1 / q}} 〉 \end{matrix}$ (9) where ${\dot{a}}_{σ (j)}$ is the j^th largest element in ${\dot{a}}_{j}$ , ${\dot{a}}_{j} = {nw}_{j} a_{j} (j = 1, 2, . . ., n)$ .

If $w = (\frac{1}{n}, \frac{1}{n}, . . ., \frac{1}{n})^{T}$ , then the q-ROTrFLHHA operator is induced to the q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted average (q-ROTrFLHWA) operator.

3.4 Generalized q-rung orthopair trapezoidal fuzzy linguistic Hamacher aggregation operators

Definition 13. Let a_j (j = 1, 2, . . . , n) be a q-ROTRFLN, w = (w₁, w₂, …, w_n) ^T is the weight vector of a_j, 0 ⩽ w_j ⩽ 1 (j = 1, 2, . . . , n). Then, the generalized q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted averaging (Gq-ROTrFLHWA) operator is defined. $\begin{matrix} Gq - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = {(\oplus_{j = 1}^{n} w_{j} a_{j}^{λ})}^{1 / λ} = 〈 [s_{{(\sum_{j = 1}^{m + 1} w_{j} {(α_{j})}^{λ})}^{1 / λ}}, s_{{(\sum_{j = 1}^{m + 1} w_{j} {(β_{j})}^{λ})}^{1 / λ}}, s_{{(\sum_{j = 1}^{m + 1} w_{j} {(γ_{j})}^{λ})}^{1 / λ}}, s_{{(\sum_{j = 1}^{m + 1} w_{j} {(η_{j})}^{λ})}^{1 / λ}}] \\ (\frac{\sqrt[q]{k} (\sqrt[\begin{matrix} q \end{matrix}]{\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}}}) 1 / λ}{\sqrt[\begin{matrix} q \end{matrix}]{{(\prod_{j = 1}^{n} A_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ} + (k - 1) {(\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ}}} \sqrt[\begin{matrix} q \end{matrix}]{\frac{{(\prod_{j = 1}^{n} C_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ} - {(\prod_{j = 1}^{n} C_{j}^{w_{j}} - \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ}}{{(\prod_{j = 1}^{n} C_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ} + (k - 1) {(\prod_{j = 1}^{n} C_{j}^{w_{j}} - \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ}}}) 〉 \end{matrix}$ (10) where H is the set of all q-ROTRFLNs, λ > 0. $\begin{matrix} A_{j} = {(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} + (k^{2} - 1) u_{j}^{q λ}, \\ B_{j} = {(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} - u_{j}^{q λ}, \\ C_{j} = {(1 + (k - 1) v_{j}^{q})}^{λ} + (k^{2} - 1) {(1 - v_{j}^{q})}^{λ}, \\ D_{j} = {(1 + (k - 1) v_{j}^{q})}^{λ} - {(1 - v_{j}^{q})}^{λ} . \end{matrix}$

Proof. $\begin{matrix} a_{j}^{λ} = 〈 [s_{{(α_{j})}^{λ}}, s_{{(β_{j})}^{λ}}, s_{{(r_{j})}^{λ}}, s_{{(η_{j})}^{λ}}], (\frac{\sqrt[q]{k} u_{j}^{λ}}{\sqrt[\begin{matrix} q \end{matrix}]{{(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} + (k - 1) u_{j}^{q λ}}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{{(1 + (k - 1) v_{j}^{q})}^{λ} - {(1 - v_{j}^{q})}^{λ}}{{(1 + (k - 1) v_{j}^{q})}^{λ} + (k - 1) {(1 - v_{j}^{q})}^{λ}}}) 〉 . \end{matrix}$

$w_{j} a_{j}^{λ} = 〈 \begin{matrix} [s_{w_{j} {(α_{j})}^{λ}}, s_{w_{j} {(β_{j})}^{λ}}, s_{w_{j} {(r_{j})}^{λ}}, s_{w_{j} {(η_{j})}^{λ}}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{\frac{{(({(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} + (k^{2} - 1) u_{j}^{q λ}))}^{w_{j}} - {({(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} - u_{j}^{q λ})}^{w_{j}}}{{({(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} + (k^{2} - 1) u_{j}^{q λ})}^{w_{j}} + (k - 1) {({(1 + (k - 1) (1 - u_{j}^{q}))}^{λ} - u_{j}^{q λ})}^{w_{j}}}}, \frac{\sqrt[q]{k} (\sqrt[\begin{matrix} q \end{matrix}]{{(1 + (k - 1) v_{j}^{q})}^{λ} - {(1 - v_{j}^{q})}^{λ}}) w_{j}}{\sqrt[\begin{matrix} q \end{matrix}]{{({(1 + (k - 1) v_{j}^{q})}^{λ} + (k^{2} - 1) {(1 - v_{j}^{q})}^{λ})}^{w_{j}} + (k - 1) ({(1 + (k - 1) v_{j}^{q})}^{λ} - {(1 - v_{j}^{q})}^{λ}) w_{j}}}) \end{matrix} 〉$

Then, calculate $\oplus_{j = 1}^{n} {(w_{j} a_{j}^{λ})}^{1 / λ}$ , and for convenience, let $\begin{matrix} w_{j} a_{j}^{λ} = 〈 [s_{w_{j} {(α_{j})}^{λ}}, s_{w_{j} {(β_{j})}^{λ}}, s_{w_{j} {(r_{j})}^{λ}}, s_{w_{j} {(η_{j})}^{λ}}], (\sqrt[\begin{matrix} q \end{matrix}]{\frac{A_{j}^{w_{j}} - B_{j}^{w_{j}}}{A_{j}^{w_{j}} + (k - 1) B_{j}^{w_{j}}}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{{kD}_{j}^{w_{j}}}{C_{j}^{w_{j}} + (k - 1) D_{j}^{w_{j}}}}) 〉, λ > 0; \\ \oplus_{j = 1}^{n} w_{j} a_{j}^{λ} = 〈 \begin{matrix} [s_{\sum_{j = 1}^{n} w_{j} {(α_{j})}^{λ}}, s_{\sum_{j = 1}^{n} w_{j} {(β_{j})}^{λ}}, s_{\sum_{j = 1}^{n} w_{j} {(γ_{j})}^{λ}}, s_{\sum_{j = 1}^{n} w_{j} {(η_{j})}^{λ}}], \\ (\sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}}}{\prod_{j = 1}^{n} A_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} B_{j}^{w_{j}}}}, \sqrt[\begin{matrix} q \\ \begin{matrix} \end{matrix} \end{matrix}]{\frac{k \prod_{j = 1}^{n} D_{j}^{w_{j}}}{\prod_{j = 1}^{n} C_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} D_{j}^{w_{j}}}}) \end{matrix} 〉 \end{matrix}$ (11)

When n = 1, Equation (11) is true obviously.

Assume that when n = m, Equation (11) is true. Then, n = m + 1 is discussed.

\begin{matrix} \oplus_{j = 1}^{m + 1} w_{j} a_{j}^{λ} = \oplus_{j = 1}^{m} w_{j} a_{j}^{λ} + w_{m + 1} a_{m + 1}^{λ} \\ = 〈 [s_{\sum_{j = 1}^{m} w_{j} {(α_{j})}^{λ}}, s_{\sum_{j = 1}^{m} w_{j} {(β_{j})}^{λ}}, s_{\sum_{j = 1}^{m} w_{j} {(γ_{j})}^{λ}}, s_{\sum_{j = 1}^{m} w_{j} {(η_{j})}^{λ}}], (\sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{m} A_{j}^{w_{j}} - \prod_{j = 1}^{m} B_{j}^{w_{j}}}{\prod_{j = 1}^{m} A_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} B_{j}^{w_{j}}}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{k \prod_{j = 1}^{m} D_{j}^{w_{j}}}{\prod_{j = 1}^{m} C_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} D_{j}^{w_{j}}}}) 〉 \\ \oplus 〈 [s_{w_{m + 1} {(α_{m + 1})}^{λ}}, s_{w_{m + 1} {(β_{m + 1})}^{λ}}, s_{w_{m + 1} {(r_{m + 1})}^{λ}}, s_{w_{m + 1} {(η_{m + 1})}^{λ}}], (\sqrt[\begin{matrix} q \end{matrix}]{\frac{A_{m + 1}^{w_{m + 1}} - B_{m + 1}^{w_{m + 1}}}{A_{m + 1}^{w_{m + 1}} + (k - 1) B_{m + 1}^{w_{m + 1}}}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{{kD}_{m + 1}^{w_{m + 1}}}{C_{m + 1}^{w_{m + 1}} + (k - 1) D_{m + 1}^{w_{m + 1}}}}) 〉 \end{matrix}

\begin{matrix} = 〈 [s_{\sum_{j = 1}^{m + 1} w_{j} {(α_{j})}^{λ}}, s_{\sum_{j = 1}^{m + 1} w_{j} {(β_{j})}^{λ}}, s_{\sum_{j = 1}^{m + 1} w_{j} {(γ_{j})}^{λ}}, s_{\sum_{j = 1}^{m + 1} w_{j} {(η_{j})}^{λ}}], \\ (\sqrt[\begin{matrix} \begin{matrix} q \end{matrix} \end{matrix}]{\frac{\frac{\prod_{j = 1}^{m} A_{j}^{w_{j}} - \prod_{j = 1}^{m} B_{j}^{w_{j}}}{\prod_{j = 1}^{m} A_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} B_{j}^{w_{j}}} + \frac{A_{m + 1}^{w_{m + 1}} - B_{m + 1}^{w_{m + 1}}}{A_{m + 1}^{w_{m + 1}} + (k - 1) B_{m + 1}^{w_{m + 1}}} - \frac{\prod_{j = 1}^{m} A_{j}^{w_{j}} - \prod_{j = 1}^{m} B_{j}^{w_{j}}}{\prod_{j = 1}^{m} A_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} B_{j}^{w_{j}}} \frac{A_{m + 1}^{w_{m + 1}} - B_{m + 1}^{w_{m + 1}}}{A_{m + 1}^{w_{m + 1}} + (k - 1) B_{m + 1}^{w_{m + 1}}} (2 - k)}{1 - (1 - k) \frac{\prod_{j = 1}^{m} A_{j}^{w_{j}} - \prod_{j = 1}^{m} B_{j}^{w_{j}}}{\prod_{j = 1}^{m} A_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} B_{j}^{w_{j}}} \frac{A_{m + 1}^{w_{m + 1}} - B_{m + 1}^{w_{m + 1}}}{A_{m + 1}^{w_{m + 1}} + (k - 1) B_{m + 1}^{w_{m + 1}}}}}, \\ \frac{\sqrt[\begin{matrix} q \end{matrix}]{\frac{k \prod_{j = 1}^{m} D_{j}^{w_{j}}}{\prod_{j = 1}^{m} C_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} D_{j}^{w_{j}}} (\frac{{kD}_{m + 1}^{w_{m + 1}}}{C_{m + 1}^{w_{m + 1}} + (k - 1) D_{m + 1}^{w_{m + 1}}})}}{\sqrt[\begin{matrix} q \end{matrix}]{k + (1 - k) (\frac{k \prod_{j = 1}^{m} D_{j}^{w_{j}}}{\prod_{j = 1}^{m} C_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} D_{j}^{w_{j}}} + \frac{{kD}_{m + 1}^{w_{m + 1}}}{C_{m + 1}^{w_{m + 1}} + (k - 1) D_{m + 1}^{w_{m + 1}}} - \frac{k \prod_{j = 1}^{m} D_{j}^{w_{j}}}{\prod_{j = 1}^{m} C_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m} D_{j}^{w_{j}}} \frac{{kD}_{m + 1}^{w_{m + 1}}}{C_{m + 1}^{w_{m + 1}} + (k - 1) D_{m + 1}^{w_{m + 1}}})}}) 〉 \end{matrix}

Since the formula is too complicated, it is divided into the left and right parts for discussion, as shown below. $\begin{matrix} Left side = \sqrt[\begin{matrix} q \end{matrix}]{\frac{k \prod_{j = 1}^{m + 1} A_{j}^{w_{j}} - k \prod_{j = 1}^{m + 1} B_{j}^{w_{j}}}{k \prod_{j = 1}^{m + 1} A^{w_{j}} + k (k - 1) \prod_{j = 1}^{m + 1} B_{j}^{w_{j}}}} = \sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{m + 1} A_{j}^{w_{j}} - \prod_{j = 1}^{m + 1} B_{j}^{w_{j}}}{\prod_{j = 1}^{m + 1} A_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m + 1} B_{j}^{w_{j}}}} \\ Right side = \frac{\sqrt[\begin{matrix} q \end{matrix}]{k^{2} \prod_{j = 1}^{m + 1} D_{j}^{w_{j}}}}{\sqrt[\begin{matrix} q \end{matrix}]{k \prod_{j = 1}^{m + 1} C_{j}^{w_{j}} + k (k - 1) \prod_{j = 1}^{m + 1} D_{j}^{w_{j}}}} = \sqrt[\begin{matrix} q \\ \begin{matrix} \end{matrix} \end{matrix}]{\frac{k \prod_{j = 1}^{m + 1} D_{j}^{w_{j}}}{\prod_{j = 1}^{m + 1} C_{j}^{w_{j}} + (k - 1) \prod_{j = 1}^{m + 1} D_{j}^{w_{j}}}} \end{matrix}$

So when n = m + 1, Equation (11) is right, combining steps 1) and 2), we can obtain it is right.

Next, calculate ${(\oplus_{j = 1}^{n} w_{j} a_{j}^{λ})}^{1 / λ}$ . $\begin{matrix} {(\oplus_{j = 1}^{n} w_{j} a_{j}^{λ})}^{1 / λ} = 〈 [s_{{(\sum_{j = 1}^{n} w_{j} {(α_{j})}^{λ})}^{1 / λ}}, s_{{(\sum_{j = 1}^{n} w_{j} {(β_{j})}^{λ})}^{1 / λ}}, s_{{(\sum_{j = 1}^{n} w_{j} {(γ_{j})}^{λ})}^{1 / λ}}, s_{{(\sum_{j = 1}^{n} w_{j} {(η_{j})}^{λ})}^{1 / λ}}], \\ (\frac{\sqrt[q]{k} (\sqrt[\begin{matrix} q \end{matrix}]{\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}}}) 1 / λ}{\sqrt[\begin{matrix} q \end{matrix}]{{(\prod_{j = 1}^{n} A_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ} + (k - 1) {(\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ}}}, \sqrt[\begin{matrix} q \end{matrix}]{\frac{(\prod_{j = 1}^{n} C_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} D_{j}^{w_{j}}) 1 / λ - {(\prod_{j = 1}^{n} C_{j}^{w_{j}} - \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ}}{{(\prod_{j = 1}^{n} C_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ} + (k - 1) {(\prod_{j = 1}^{n} C_{j}^{w_{j}} - \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ}}}) 〉 \end{matrix}$

Therefore, the Equation (10) is true for all n, and the Gq-ROTrFLHWA operator has desirable properties.

Theorem 11. (idempotency):If all a_j =〈 [s_α, s_β, s_γ, s_η] , (u_j, v_j) 〉 are equal, and a_j = a, then $Gq - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} {(w_{j} a_{j}^{λ})}^{1 / λ} = a$

Theorem 12. (boundedness): Let a^- = (min(r_j) , max(η_j)), a⁺ = (max(r_j) , min(η_j)), and then $a^{-} ⩽ G q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) ⩽ a^{+} .$ where u_min ⩽ u_j ⩽ u_max, v_min ⩽ v_j ⩽ v_max, s_{α
_min} ⩽ s_{α
_j} ⩽ s_{α
_max} s_{β
_min} ⩽ s_{β
_j} ⩽ s_{β
_max}, s_{γ
_min} ⩽ s_{γ
_j} ⩽ s_{γ
_max}, s_{η
_min} ⩽ s_{η
_j} ⩽ s_{η
_max}. Since s (a^-) < s (a_j) < s (a⁺), then according to the order relation between two q-ROTRFLNs, a^- ⩽ Gq - ROTrFLHWA (a₁, a₂, …, a_n) ⩽ a⁺.

Theorem 13. (monotonicity) If $a_{j} ⩽ a_{j}^{'} (j = 1, . . ., n)$ , then $Gq - ROTrFLHWA (a_{1}, \dots, a_{n}) ⩽ G q - ROTrFLHWA (a_{1}^{'}, \dots, a_{n}^{'})$

The proof is similar to that of Theorem 4, and therefore it is omitted here. But some special cases about the parameters are discussed.

If λ = 1, then the Gq-ROTrFLHWA operator is induced to the q-ROTrFLHWA operator as follows: $\begin{matrix} q - ROTrFLHWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} \\ = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}} s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], \\ (\sqrt[q]{\frac{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (k - 1) u_{j}^{q})}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{k} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{\sqrt[q]{\prod_{j = 1}^{n} {(1 + (k - 1) (1 - v_{j}^{q}))}^{w_{j}} + (k - 1) \prod_{j = 1}^{n} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix}$ (12)

if k = 1, then Formula (12) will be induced as the q-rung orthopair trapezoidal fuzzy linguistic weighted averaging (q-ROTrFLWA) operator, as follows: $q - ROTrFLWA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} w_{j} a_{j} = 〈 [s_{\oplus_{j = 1}^{n} w_{j} α_{j}}, s_{\oplus_{j = 1}^{n} w_{j} β_{j}}, s_{\oplus_{j = 1}^{n} w_{j} γ_{j}} s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], (\sqrt[\begin{matrix} q \end{matrix}]{1 - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}, \prod_{j = 1}^{n} v_{j}^{w_{j}}) .$

If k = 2, then Formula (12) will be induced as the q-rung orthopair trapezoidal fuzzy linguistic Einstein weighted averaging (q-ROTrFLEWA) operator, as follows:

\begin{matrix} q - ROTrFLEWA (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}} s_{\oplus_{j = 1}^{n} w_{j} η_{j}}], (\sqrt[\begin{matrix} q \end{matrix}]{\frac{\prod_{j = 1}^{n} {(1 + u_{j}^{q})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + u_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - u_{j}^{q})}^{w_{j}}}}, \frac{\sqrt[q]{2} \prod_{j = 1}^{n} v_{j}^{w_{j}}}{\sqrt[\begin{matrix} q \end{matrix}]{\prod_{j = 1}^{n} {(2 - v_{j}^{q})}^{w_{j}} + \prod_{j = 1}^{n} v_{j}^{{qw}_{j}}}}) 〉 \end{matrix}

Definition 14. Let a_j (j = 1, 2, . . . , n) be a q-ROTrFL Ns, the generalized q-rung orthopair trapezoidal fuzzy linguistic Hamacher weighted geometric (Gq-ROTrFLHWG) operator is defined: Hⁿ → H $Gq - ROTrFLHWG (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{λ} (\otimes_{j = 1}^{n} (λ a_{j})^{w_{j}})$ where H is a q-rung orthopair trapezoidal fuzzy linguistic set, w = (w₁, w₂, . . . , w_n) ^T is the weight vector of them, 0 ⩽ w_j ⩽ 1 (j = 1,2, ... ,n), $\sum_{j = 1}^{n} = 1$ , and λ is a parameter, with λ ⩾ 0.

Theorem 14. Let a_j (j = 1, 2, . . . , n) be a q-ROTrFLN, the Gq-ROTrFLHWG operator is still a q-ROTrFLN, then $\begin{matrix} Gq - ROTrFLHWG (a_{1}, a_{2}, \dots, a_{n}) = \\ 〈 \begin{matrix} [s_{\frac{1}{λ} \prod_{j = 1}^{n} {(λ \cdot α_{j})}^{w_{j}}}, s_{\frac{1}{λ} \prod_{j = 1}^{n} {(λ \cdot β_{j})}^{w_{j}}}, s_{\frac{1}{λ} \prod_{j = 1}^{n} {(λ \cdot γ_{j})}^{w_{j}}}, s_{\frac{1}{λ} \prod_{j = 1}^{n} {(λ \cdot η_{j})}^{w_{j}}}], \\ (\frac{{(\prod_{j = 1}^{n} A_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ} - {(\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ}}{{(\prod_{j = 1}^{n} A_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ} + (k - 1) {(\prod_{j = 1}^{n} A_{j}^{w_{j}} - \prod_{j = 1}^{n} B_{j}^{w_{j}})}^{1 / λ}}, \frac{\sqrt[q]{k} (\sqrt[\begin{matrix} q \end{matrix}]{\prod_{j = 1}^{n} C_{j}^{w_{j}} - \prod_{j = 1}^{n} D_{j}^{w_{j}}}) 1 / λ}{\sqrt[\begin{matrix} q \end{matrix}]{{(\prod_{j = 1}^{n} C_{j}^{w_{j}} + (k^{2} - 1) \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ} + (k - 1) {(\prod_{j = 1}^{n} C_{j}^{w_{j}} - \prod_{j = 1}^{n} D_{j}^{w_{j}})}^{1 / λ}}}) \end{matrix} 〉 \end{matrix}$ where $A_{j} = {(1 + (k - 1) u_{j}^{q})}^{λ} + (k^{2} - 1) {(1 - u_{j}^{q})}^{λ}$ , $B_{j} = {(1 + (k - 1) v_{j}^{q})}^{λ} - {(1 - v_{j}^{q})}^{λ}$ , $C_{j} = {(1 + (k - 1) (1 - v_{j}^{q}))}^{λ} + (k^{2} - 1) v_{j}^{q λ}$ , $D_{j} = {(1 + (k - 1) (1 - v_{j}^{q}))}^{λ} - v_{j}^{q λ}$ .

While w = (w₁, w₂, . . . , w_n) ^T is the weight vector of them, 0 ⩽ w_j ⩽ 1 (j = 1,2, ... ,n), $\sum_{j = 1}^{n} = 1$ , and λ is a parameter with λ ⩾ 0.

If λ= 1, then the Gq-ROTrFLHWG operator is induced to the q-ROTrFLHWG operator. In addition, the Gq-ROTrFLHWA or Gq-ROTrFLHWG operators have properties similar to Theorem 3 and Theorem 4.

4 Multi-attribute group decision method

In a group decision-making problem, let A = (A₁, A₂, . . . , A_n) ^T be the solution set, C = (C₁, C₂, . . . , C_n) ^T be the property set, S = {s₀, s₁, ... , s_l} be a finite set of linguistic terms, w = (w₁, w₂, . . . , w_n) ^T represents an attribute weight vector, with w_i ⩾ 0, (i = 1, 2, . . . , n), and $\sum_{i = 1}^{n} = 1$ . We assume that the expert’s evaluation of decision scheme A_i under attribute C_j is expressed as a_ij =〈 [s_{α(a_ij)}, s_{β(a_ij)}, s_{γ(a_ij)}, s_{η(a_ij)}, (u (a_ij) , v (a_ij))] 〉 by q-rung orthopair trapezoidal fuzzy linguistic information, u_ij is the membership degree of [s_{α
_ij}, s_{β
_ij}, s_{γ
_ij}, s_{η
_ij}], v_ij is the non-membership degree of [s_{α
_ij}, s_{β
_ij}, s_{γ
_ij}, s_{η
_ij}], u (a_ij) ∈ [0, 1], v (a_ij) ∈ [0, 1], u^q (a_ij) + v^q (a_ij) ⩽1 and R = (a_ij) _m×n is the decision matrix. Here, there are two group decision-making methods based on q-rung orthopair trapezoidal fuzzy information, and the specific steps are as follows:

Step 1: Use the q-ROTrFLHWA and q-ROTrFLHWG integration operators to integrate the ith row of matrix R = (a_ij) _m×n.

Step 2: Calculate the score function E (a_i) (i = 1, 2, . . . , m) for each row according to Definition 5. If the two schemes are E (a_i) = E (a_t) (i, t = 1, 2, . . . , m, i ≠ t), the accuracy functions H (a_i) and H (a_t) (i, t = 1,2, ... , m, i ≠ t) need to be calculated.

Step 3: Sort scheme A_i (i = 1,2, ... ,m) according to Definition 6 to obtain the best scheme.

Step 4: End.

5 The sample analysis

An education department needs to evaluate the teaching quality of several schools. There are 4 schools A = (A₁, A₂, A₃, A₄,) that are evaluated from four aspects, (C₁, C₂, C₃, C₄,) respectively represent the school scale, hardware level, teacher level, and management level, with the weight vectors of attributes are w = (0.15, 0.27, 0.33, 0.25) ^T, its linguistic term evaluation set is S = {s₀ = extremely low, s₁ = very poor, s₂ = poor, s₃ = common, s₄ = well, s₅ = very well, and s₆ = extremely well}, the decision matrix as follow.

The q-ROTrFLHWA or q-ROTrFLHWG operators is used to integrate all the q-rung orthopair trapezoidal fuzzy linguistic information a_ij (i, j = 1, 2, 3, 4) then score function is calculated to sort. When q = 1, it is a intuitionistic fuzzy linguistic set, when q = 2, it is a Pythagorean fuzzy linguistic set. The q-ROTrFLHWA or q-ROTrFLHWG operators is studied when q = 3 or q = 4. Moreover, when k = 1, q-ROTrFLHWA operator is induced to q-rung orthopair trapezoidal fuzzy linguistic weighted average operator, q-ROTrFLHWG operator is induced to q-rung orthopair trapezoidal fuzzy linguistic weighted geometric operator. When k = 2, q-ROTrFLHWA operator is induced to q-rung orthopair trapezoidal fuzzy linguistic Einstein weighted average operator, q-ROTrFLHWG operator is induced to q-rung orthopair trapezoidal fuzzy linguistic Einstein weighted geometric operator. The influences of the operators on the evaluation results are studied when the parameter k is taken to different values, the results are given below.

Two different decision-making methods were used to sort the schemes according to Definition 6. Where the symbol “≻” means “superior to”, when k takes values of 0.00001 and 10000, we consider the influence of k value on the sorting result from the two aspects of small value and large value respectively. It can be seen from Table 2 that the sorting result of q-ROTrFLHWA operator changes with the change of k value, but the optimal scheme does not change, and A₁ is the optimal scheme, thus, it is convenient to choose a favorable k value for calculation in practical problems. In addition, this is also illustrated in Table 3, this example verifies the rationality of the proposed method.

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

Scheme Attribute

C ₁ C ₂ C ₃ C ₄

A ₁ {, (0.7, 0.3)} {[s₂, s₃, s₄, s₅], (0.6, 0.4)} {[s₁, s₃, s₄, s₆], (0.8, 0.4)} {[s₂, s₃, s₅, s₆], (0.8, 0.4}

A ₂ {[s₁, s₂, s₄, s₆], (0.6, 0.5)} {[s₁, s₂, s₃, s₅], (0.8, 0.4)} {[s₁, s₂, s₄, s₅], (0.7, 0.4)} {[s₂, s₃, s₄, s₆], (0.7, 0.3)}

A ₃ {[s₁, s₂, s₄, s₅], (0.6, 0.2)} {[s₁, s₂, s₄, s₅], (0.6, 0.5)} {[s₁, s₂, s₃, s₄], (0.6, 0.3)} {[s₂, s₃, s₄, s₅], (0.7, 0.5)}

A ₄ {[s₁, s₂, s₃, s₆], (0.5, 0.3)} {[s₁, s₃, s₄, s₆], (0.7, 0.4)} {[s₂, s₃, s₄, s₅], (0.7, 0.4)} {[s₁, s₃, s₄, s₅], (0.5, 0.6)}

Scheme	Attribute
A ₁	{, (0.7, 0.3)}	{[s₂, s₃, s₄, s₅], (0.6, 0.4)}	{[s₁, s₃, s₄, s₆], (0.8, 0.4)}	{[s₂, s₃, s₅, s₆], (0.8, 0.4}
A ₂	{[s₁, s₂, s₄, s₆], (0.6, 0.5)}	{[s₁, s₂, s₃, s₅], (0.8, 0.4)}	{[s₁, s₂, s₄, s₅], (0.7, 0.4)}	{[s₂, s₃, s₄, s₆], (0.7, 0.3)}
A ₃	{[s₁, s₂, s₄, s₅], (0.6, 0.2)}	{[s₁, s₂, s₄, s₅], (0.6, 0.5)}	{[s₁, s₂, s₃, s₄], (0.6, 0.3)}	{[s₂, s₃, s₄, s₅], (0.7, 0.5)}
A ₄	{[s₁, s₂, s₃, s₆], (0.5, 0.3)}	{[s₁, s₃, s₄, s₆], (0.7, 0.4)}	{[s₂, s₃, s₄, s₅], (0.7, 0.4)}	{[s₁, s₃, s₄, s₅], (0.5, 0.6)}

Table 2

When q = 3, sorting results as k value changes

k value	q-ROTrFLHWA	q-ROTrFLHWG
k = 0.0001	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 0.01	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 0.1	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 1	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 2	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 10	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 100	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 1000	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 10000	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄

Table 3

When q = 4, sorting results as k value changes

k value	q-ROTrFLHWA	q-ROTrFLHWG
k = 0.0001	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 0.01	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 0.1	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 1	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 2	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 10	A₁ ≻ A₂ ≻ A₄ ≻ A₃	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 100	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 1000	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄
k = 10000	A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₁ ≻ A₂ ≻ A₃ ≻ A₄

Sensitivity analysis is one of the commonly used methods of analyzing uncertainty in evaluating projects. Whether the parameter is a sensitive factor is judged according to whether the small range change of the parameter can lead to a large change of the economic benefit index. However, the disadvantage of this method of analysis is that only one factor is allowed to change at a time and the other factors are assumed to remain constant. In real-world decision making situations, we can choose the appropriate parameter k value for each situation. That is, according to the extent of decision making risk, the adaptive k value is selected. From Table 2 and 3, if the k value is larger, the q-ROTRFLHWA operator is more sensitive to the scheme; if the k value is smaller, the q-ROTRFLHWA operator is less sensitive, whereas q-ROTRFLHWG is the opposite. In addition, as the q value increases, the sensitivity of the q-ROTRFLHWA operator to the scheme also increases.

6 Comparison with Existing Methods

Firstly, the effectiveness is demonstrated by comparing the results of the proposed operators and then its scientificity is illustrated by comparing it with other methods. The Pythagorean fuzzy set has appropriate operation complexity and high precision, so some Pythagorean fuzzy operators are used as the comparison object in this paper. To demonstrate the effectiveness and scientificity of the suggested technique, a systematic comparison analysis was presented with the existing methods from the paper [26 27], and [6]. Then, we shall compare our proposed operators with other information fusing tools such as trapezoidal Pythagorean fuzzy linguistic entropic weighted averaging (TrPFLEWA) operator and trapezoidal Pythagorean fuzzy linguistic weighted averaging (TrPFLWA) operator [26]. We used different fuzzy methods to process the same sample and study the final processing results. If the optimal results are consistent, the proposed method is reasonable; otherwise, it is imperfect. Several fuzzy linguistic information processing methods are as follows.

6.1 TrPFLECOWA operator

Step 1: Let a_j =〈 [s_{α
_j}, s_{β
_j}, s_{γ
_j}, s_{η
_j}] , (u_j, v_j) 〉 be a set of q-ROTRFLNs, according to the Xian et al. [26] proposed the desicion-making method, then by using of the equation $e_{i} = \frac{1}{n} \sum_{i = 1}^{n} \frac{min (u_{i}^{2}, v_{i}^{2}) + π_{i}^{2}}{max (u_{i}^{2}, v_{i}^{2}) + π_{i}^{2}}$ in the paper, we can obtain e₁ = 0.7261, e₂ = 0.615119, e₃ = 0.586538, e₄ = 0.582967.

Step 2: According to the equation $E_{i} = \frac{e_{i}}{\sum_{i = 1}^{n} e_{i}}$ in the paper [26], we can obtain E₁ = 0.289215, $E_{2} = \begin{matrix} 0.244991 \end{matrix}$ , E₃ = 0.232186, E₄ = 0.233608 let ρ = 0.6, according to the ϖ_i = ρw_i + (1 - ρ) E_i, we can obtain $\begin{matrix} ϖ_{1} = 0.205686, ϖ_{2} = 0.259996, \\ ϖ_{3} = 0.290874, ϖ_{4} = 0.243443 . \end{matrix}$

Step 3: According definition 9 in the paper [26], let k₁ = 1/6, k₂ = 2/6, k₃ = 2/6, k₄ = 1/6, and we can obtain V_{u
₁} = 2.535367 ; V_{v
₁} = 1.287903, V_{u
₂} = 2.22501, V_{v
₂} = 1.214295, V_{u
₃} = 1.869178, V_{v
₃} = 1.062337, V_{u
₄} = 2.084197, V_{v
₄} = 1.382513.

Step 4: Let ω_i = 0.5, according definition 10 in the paper [26], we can obtain V (A₁) = 1.911635, V (A₂) = 1.719653, V (A₃) = 1.465758, V (A₄) = 1.733355. and have A₁ ≻ A₂ ≻ A₄ ≻ A₃, therefore, the most desirable alternative is A₁ as well.

6.2 TrPFLEWA operator

According to the Xian et al. [26], we use the TrPFLEWA operator, which is given as follows:

$\begin{matrix} Tr PFLEA (a_{1}, a_{2}, \dots, a_{n}) = \oplus_{j = 1}^{n} \frac{1}{n} a_{j} = \\ 〈 [s_{\oplus_{j = 1}^{n} \frac{1}{n} α_{j}}, s_{\oplus_{j = 1}^{n} \frac{1}{n} β_{j}}, s_{\oplus_{j = 1}^{n} \frac{1}{n} γ_{j}}, s_{\oplus_{j = 1}^{n} \frac{1}{n} η_{j}}], (\sqrt[\begin{matrix} 2 \end{matrix}]{1 - \prod_{j = 1}^{n} {(1 - u_{j}^{2})}^{\frac{1}{n}}}, \prod_{j = 1}^{n} v_{j}^{\frac{1}{n}}) 〉 . \end{matrix}$ (13)

We can obtain as follows: V (A₁) = 1.87564, V (A₂) = 1.725612, V (A₃) = 1.468059, V (A₄) = 1.694612, and have A₁ ≻ A₂ ≻ A₄ ≻ A₃, therefore, the most desirable alternative is A₁ as well.

6.3 TrPFLWA operator

A Comparison Analysis with existing trapezoidal Pythagorean fuzzy linguistic sets, TrPFLWA operator, we can obtain as follows:

S (A₁) = -4.222, S (A₂) = -4.245, S (A₃) = -4.289, S (A₄) = -4.75, and A₁ ≻ A₂ ≻ A₃ ≻ A₄, therefore, the most desirable alternative is A₁ as well.

6.4 PFWA operator

Compared with Pythagorean fuzzy sets, we will turn Table 1 to Table 4, the decision matrix as follow.

Table 4
Pythagorean fuzzy decision matrix

Scheme Attribute

C ₁ C ₂ C ₃ C ₄

A ₁ (0.7, 0.3) (0.6, 0.4) (0.8, 0.4) (0.8, 0.4)

A ₂ (0.6, 0.5) (0.8, 0.4) (0.7, 0.4) (0.7, 0.3)

A ₃ (0.6, 0.2) (0.6, 0.5) (0.6, 0.3) (0.7, 0.5)

A ₄ (0.5, 0.3) (0.7, 0.4) (0.7, 0.4) (0.5, 0.6)

Scheme	Attribute
A ₁	(0.7, 0.3)	(0.6, 0.4)	(0.8, 0.4)	(0.8, 0.4)
A ₂	(0.6, 0.5)	(0.8, 0.4)	(0.7, 0.4)	(0.7, 0.3)
A ₃	(0.6, 0.2)	(0.6, 0.5)	(0.6, 0.3)	(0.7, 0.5)
A ₄	(0.5, 0.3)	(0.7, 0.4)	(0.7, 0.4)	(0.5, 0.6)

We can obtain S (A₁) = 0.410173 S (A₂) = 0.371535 S (A₃) = 0.259755 S (A₄) = 0.225176, and A₁ ≻ A₂ ≻ A₃ ≻ A₄. To sum up, we used various aggregation operators to evaluate the teaching quality of several schools, and the results of each method are shown below.

We illustrated the rationality and validity of the proposed method from two aspects. First, we compared the proposed methods with each other, and then we compared the proposed methods with other methods. As can be seen from Table 5, no matter which aggregation operator is used, the result is that A₁ is the most desirable alternative. In view of the change of k and q values, the results of q-ROTRFLHWA or q-ROTRFLHWG operators are different, but the optimal scheme is same, which shows the rationality of the proposed method. By comparing the proposed method with the TrPFLECOWA, TrPFLEWA, TrPFLWA and PFWA operators, the optimal scheme is also same, which further demonstrates its rationality and validity. We find that the deficiencies of Peng’s [28] based on the PFLWA operator cannot solve some trapezoidal Pythagorean fuzz values, the calculation amount and complexity of Xian’s method [26] are higher than that of our proposed method. All in all, the proposed method can solve the MAGDM problems according to different parameters with have commonality and flexibility.

Table 5

The aggregation operator processes the result

Aggregation operator	Score function	Rank
q-ROTrFLHWA (q = 3, k = 1)	S (A₁) ≻ S (A₂) ≻ S (A₄) ≻ S (A₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
q-ROTrFLHWG (q = 3, k = 1)	S (A₁) ≻ S (A₂) ≻ S (A₃) ≻ S (A₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
q-ROTrFLHWA (q = 4, k = 1)	S (A₁) ≻ S (A₂) ≻ S (A₄) ≻ S (A₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
q-ROTrFLHWG (q = 4, k = 1)	S (A₁) ≻ S (A₂) ≻ S (A₃) ≻ S (A₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
TrPFLECOWA	S (A₁) ≻ S (A₂) ≻ S (A₄) ≻ S (A₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
TrPFLEWA	S (A₁) ≻ S (A₂) ≻ S (A₄) ≻ S (A₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
TrPFLWA	S (A₁) ≻ S (A₂) ≻ S (A₃) ≻ S (A₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
PFWA	S (A₁) ≻ S (A₂) ≻ S (A₃) ≻ S (A₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄

7 Conclusion

The Intuitionistic linguistic fuzzy operators and Pythagorean fuzzy linguistic operators are commonly used fuzzy evaluation methods, they are suitable for aggregating the information taking the form of crisp values, but their application scope is limited and the accuracy is low.

In the paper, we have investigate the MAGDM problems in which the attributes are independent and have different priority level, and the attribute values take the form of q-ROTrFLHs. First, we have defined the q-rung orthopair trapezoidal fuzzy linguistic set and introduced operational laws or the score function based on q-rung orthopair fuzzy sets and trapezoidal fuzzy linguistic numbers. Afterward, we have proposed a series of q-rung orthopair trapezoidal fuzzy linguistic Hamacher aggregation operators. Then, some desirable properties and special cases of these operators have been investigated in detail. Furthermore, according to a different k value, we have investigated the effects of q-ROTrFLHWA and q-ROTrFLHWG operators on MAGDM problems. Moreover, in order to prove the effectiveness of the proposed method, a systematic comparative analysis has made with several methods.

The key contributions of this study are shown as follows: (1) a q-rung orthopair trapezoidal fuzzy linguistic set is proposed. (2) a ranking method for q-ROTrFLHs is given. (3) the standard distance between two q-ROTrFLNs is defined. (4) q-ROTrFLHWA, q-ROTrFLHOWA, q-ROTrFLHHA, and Gq-ROTrFLHWA operators are presented. (5) some desired properties such as commutativity, idempotency, and boundedness are studied. (6) Hamacher operators are degenerate and can be induced on fuzzy aggregation operators, such as the q-ROTrFLHWA operator is induced to a q-rung orthopair trapezoidal fuzzy linguistic weighted average (q-ROTrFLWA) operator. (7) two approaches to MAGDMs with the q-rung orthopair trapezoidal fuzzy linguistic information are presented. (8) This paper makes a systematic comparison and analysis with various methods.

In terms of future research, we shall further generalize these operators by the use of the q-rung orthopair trapezoidal fuzzy linguistic information, or extend the applications of the aggregation operators to other domains, such as investment decisions, project evaluation, fuzzy control and supply chain management. Furthermore, the method proposed in this paper provides a new way of thinking for solving multi-attribute group decision-making problems. Additionally, the attribute weight is determined by selecting a reasonable method, etc. In addition, Although the prediction accuracy of the proposed operators is improved, the complexity of the two methods is also increased. Most possible values of trapezoidal fuzzy sets are interval form, while triangle fuzzy sets are of single value. The study of triangle fuzzy set should be strengthened in the future.

Footnotes

Acknowledgments

This research is supported by National Natural Science Foundation of China (No. 71571019) and (No. 71771025).

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Scheme	Attribute
	C ₁	C ₂	C ₃	C ₄
A ₁	{, (0.7, 0.3)}	{[s₂, s₃, s₄, s₅], (0.6, 0.4)}	{[s₁, s₃, s₄, s₆], (0.8, 0.4)}	{[s₂, s₃, s₅, s₆], (0.8, 0.4}
A ₂	{[s₁, s₂, s₄, s₆], (0.6, 0.5)}	{[s₁, s₂, s₃, s₅], (0.8, 0.4)}	{[s₁, s₂, s₄, s₅], (0.7, 0.4)}	{[s₂, s₃, s₄, s₆], (0.7, 0.3)}
A ₃	{[s₁, s₂, s₄, s₅], (0.6, 0.2)}	{[s₁, s₂, s₄, s₅], (0.6, 0.5)}	{[s₁, s₂, s₃, s₄], (0.6, 0.3)}	{[s₂, s₃, s₄, s₅], (0.7, 0.5)}
A ₄	{[s₁, s₂, s₃, s₆], (0.5, 0.3)}	{[s₁, s₃, s₄, s₆], (0.7, 0.4)}	{[s₂, s₃, s₄, s₅], (0.7, 0.4)}	{[s₁, s₃, s₄, s₅], (0.5, 0.6)}

Some q-rung orthopair trapezoidal fuzzy linguistic hamacher aggregation operators and their applications

Abstract

Keywords

1 Introduction

2 Preliminarie

2.1 q-rung orthopair fuzzy set

2.2 trapezoidal fuzzy linguistic set

2.3 q-rung orthopair trapezoidal fuzzy linguistic set

5 The sample analysis

6.1 TrPFLECOWA operator

6.2 TrPFLEWA operator

6.4 PFWA operator

Table 4 Pythagorean fuzzy decision matrix Scheme Attribute C 1 C 2 C 3 C 4 A 1 (0.7, 0.3) (0.6, 0.4) (0.8, 0.4) (0.8, 0.4) A 2 (0.6, 0.5) (0.8, 0.4) (0.7, 0.4) (0.7, 0.3) A 3 (0.6, 0.2) (0.6, 0.5) (0.6, 0.3) (0.7, 0.5) A 4 (0.5, 0.3) (0.7, 0.4) (0.7, 0.4) (0.5, 0.6)

Footnotes

Acknowledgments

References

Table 4
Pythagorean fuzzy decision matrix

Scheme Attribute

C ₁ C ₂ C ₃ C ₄

A ₁ (0.7, 0.3) (0.6, 0.4) (0.8, 0.4) (0.8, 0.4)

A ₂ (0.6, 0.5) (0.8, 0.4) (0.7, 0.4) (0.7, 0.3)

A ₃ (0.6, 0.2) (0.6, 0.5) (0.6, 0.3) (0.7, 0.5)

A ₄ (0.5, 0.3) (0.7, 0.4) (0.7, 0.4) (0.5, 0.6)