q -Rung orthopair fuzzy uncertain linguistic choquet integral operators and their application to multi-attribute decision making

Abstract

Multiple attribute decision making (MADM) problems widely exist in real decision making, and MADM methods with linguistic information have achieved great success. However, as the complexity of decision making problems is increasing in the real world, it is of great necessity to further develop new expression of evaluation information and aggregation technologies that can reflect the correlation among multi-attributes under uncertain decision-making environment. In response, this paper originally presents q-rung orthopair fuzzy uncertain linguistic set (q-ROULS) by combining q-rung orthopair fuzzy set (q-ROFS) and uncertain linguistic set (ULS). Then operational laws, expected functions and accuracy functions of q-rung orthopair uncertain linguistic variables (q-ROULVs) are also defined. Considering the correlation between q-ROULVs, we propose a family of q-rung orthopair fuzzy uncertain linguistic Choquet integral operators to aggregate q-rung orthopair uncertain linguistic information. Further, a novel MADM technique is presented based on the proposed q-rung orthopair fuzzy uncertain linguistic Choquet integral operators. The developed MADM method with q-rung orthopair fuzzy uncertain linguistic information enriches fuzzy decision-making theory and provides a new way for decision makers (DMs) under q-rung orthopair fuzzy uncertain linguistic environment.

Keywords

multi-attribute decision making

1. Introduction

Multiple attribute decision making (MADM) is the process of ranking the alternatives and selecting an optimal one among a set of alternatives assessed on multiple attributes, which has been widely used in economy, management and other fields in recent years. For example, Roy et al. [1] proposed a rough strength relational decision making and trial evaluation laboratory model to analyze the internal strength and external impacts of key success factors in hospital service quality. Tang et al. [2] proposed an algorithm for MADM and applied it to flood disaster risk evaluation. Chatterjee et al. [3] developed a novel multi-attributive border approximation area comparison methods for non-traditional machining process selection problems in manufacturing domain. Mukhametzyanov and Pamučar [4] provided a model for the sensitivity analysis of MADM and selection of the optimal one. Vasiljević et al. [5] developed a rough group analytic hierarchy process approach to the evaluation supplier criteria in the company for producing metal washers for the automotive industry. Liu et al. [6] presented a multicriteria model for evaluating and selecting a transport service provider based on a single valued neutrosophic number. As MADM is becoming more and more complicated, we have to make decisions in different fuzzy environments. Therefore, one of the biggest difficulties in decision making is how to express decision makers’ assessments over alternatives. In general, the fuzzy information can be expressed by two forms: quantitative description and qualitative description. To appropiroretly describe the quantitative fuzzy information, quite a few tools, such as intuitionistic fuzzy set (IFS) [7], neutrosophic set [8], hesitant interval-valued fuzzy set [9], Pythagorean fuzzy set (PFS) [10 –13] have been proposed. More recently, Yager [14] proposed a new concept of q-ROFS whose prominent feature is that the qth power of the membership degree and the qth power of the degree of non-membership is equal to or less than 1. Evidently, q-ROFS relaxes the constraints of both IFS and PFS, which makes it more powerful to express quantitative fuzzy information. Because of this advantage, q-ROFS has received more and more attention in recent years [15 –19].

In real MADM problems, DMs prefer to make qualitative decisions instead of quantitative decisions due to time shortage and a lack of priori expterise. For example, people can give some linguistic terms like “excellent”, “medium” or “poor” to represent the evaluation values. To effectively address these circumtances, Zhadeh [20] provided a novle tool, called linguistic variables (LVs), to express fuzzy information. Later, Xu [21] proposed the concpt of uncertain linguistic variables (ULVs). After that, the researches on MADM under uncertain linguistic context received wide attention and several different linguistic sets have been developed, such as intuitionistic uncertain linguistic set (IULS) [22], interval intuitionistic uncertain linguistic set [23], hesitant fuzzy uncertain linguistic set [24], Pythagorean uncertain linguistic set (PULS) [25], interval-valued Pythagorean uncertain linguistic set [26], single value neutrosophic uncertain linguistic set (SVNULS) [27], interval neutrosophic uncertain linguistic set (INULS) [28]. An important research topic for these uncertain linguistic set is aggregation operator theory that can aggregate a collection of individual evaluated values into one. To aggregate intuitionistic uncertain linguistic information, Liu P.D. et al. [29], Liu P.D. et al. [30], Liu Z.M. et al. [31] successively extended some existing operators, such as weighted geometric average operator, ordered weighted geometric operator, Heronian mean, Bonferroni OWA and partitioned Bonferroni mean to the IULS. Liu Z.M. et al. [32] proposed intuitionistic uncertain linguistic Hamy mean operators with linguistic scale functions and apply them to health-care waste treatment technology selection. In addition, by combining interval neutrosophic uncertain linguistic variables with the Choquet integral, Liu and Tang [33] presented an interval neutrosophic uncertain linguistic Choquet integral method for MAGDM. Liu and Shi [34] introduced some neutrosophic uncertain linguistic Heronian mean operators for MAGDM. As PULS can describes fuzzy information better than IULS, Liu et al. [35] developed Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation operators and applied them to MADM with completely unknown weight of information. At the same time, they also extended the Maclaurin symmetric mean to PULS and introduced a family of Pythagorean fuzzy uncertain linguistic Maclaurin symmetric mean operator. Considering the partition interrelationship between attributes being aggregated, Z.M. Liu et al. [36] divided attributes into several different groups based on their correlation characteristics, and proposed a Pythagorean uncertain linguistic partitioned Bonferroni mean operator and its weighted form.

From the above analysis, we can find that researches on MADM methods with linguistic information have achieved great success. However, as the complexity of decision making problems is increasing in the real world, we may encounter the following MADM issues simultaneously, and the existing linguistic decision-making methods cannot express and deal with these situations:

There exist some situations where the square sum of the membership degree and the non-membership degree belonging to the uncertain linguistic variable is bigger than 1. For example: performance of a car is perhaps felt to be lower than “good” (s₅) but higher than “fair” (s₃). The membership degree to [s₃, s₅] is 0.9, and nonmembership degree is 0.7. The evaluation result can be denoted as < [s₃, s₅] , (0.9, 0.7)>. Due to the fact that 0.9 + 0.7 > 1 and 0.9² + 0.7² > 1, the evaluation value < [s₃, s₅] , (0.9, 0.7)> cannot be expressed by the existing IULS and PULS. Thus, it is of great necessity to further develop an extension of PFULS and IULS to better express linguistic information.

In some situations, there exist correlations among attributes. For example, to evaluate patients based on the following symptoms of lung diseases: (vital signs, body temperature, cough and hemoptysis), we want to place more emphasis on hemoptysis than on body temperature. However, on the other hand, we also want to pay more attention to patients who have severe hemoptysis and high body temperature, because hemoptysis and hyperthermia are two classical symptoms of pneumonia. However, traditional aggregation operators, such as arithmetic and geometric operators are based on the assumption that the attributes are independent of one another. Therefore, it is meaningful to investigate aggregation technologies that can account the situation.

Motivated by q-ROFS and ULVs, we propose a new concept of uncertain linguistic set named q-ROFULS that can integrate the advantages of q-ROFS and ULVs. Compared to the existing PFULS and IULS, the constraint of new q-ROULS is that the sum of the qth power of membership degree and the qth power of non-membership degree belonging to the uncertain linguistic variable is less than or equal to one. The proposed q-ROULS are more general than PFULS and IULS due to the fact that PFULS and IULS are the special cases of q-ROULS when q = 1 and q = 2, respectively. Obviously, the larger the rung q, the more orthopairs satisfy the bounding constraint and thus the larger fuzzy information space can be expressed by q-ROULS. This feature makes q-ROULS more powerful and useful than PFULS and IULS in aspect of expressing the vagueness and fuzzy information. For the second issue, we note that Choquet integral operator originally developed by Choquet [37] can model correlation among attributes by using a fuzzy measure [38 –42]. The fuzzy measure can be used to define a weight on each combination of criteria. So far, there is no research on MADM methods with q-rung orthopair uncertain linguistic information based on Choquet integral operator. Since it is better for q-ROULS to depict the actual situation, Choquet integral operator can capture the correlation among attributes, it is of great meaning to study the Choquet integral operator under q-rung orthopair uncertain linguistic environment for MCDM problems.

The above analysis describes the motivation behind proposing a comprehensive approach for tackling MADM with q-rung orthopair uncertain fuzzy linguistic information. Based on the above comprehensive analysis, the goal of this paper are: (1) to introduce a new linguistic fuzzy set called q-ROULS; (2) to introduce operational laws, expected functions and accuracy functions of an q-ROFULV; (3) to develop some novel Choquet integral aggregation operators to deal with q-rung orthopair uncertain linguistic information. Compared with existing aggregation operators, the proposed aggregation operators not only consider the correlations among attributes, but also demonstrate high generality than exiting uncertain linguistic aggregation operators; (4) to provide a new train of thought for MADM with q-rung orthopair fuzzy uncertain linguistic information based on the proposed operators.

The remainder of the paper is organized as follows. Section 2 recalls some basic notions, and further proposes a new concept of uncertain linguistic term named q-ROULS. Section 3 proposes a family of q-rung orthopair fuzzy uncertain linguistic Choquet integral operators, such as q-rung orthopair uncertain linguistic Choquet integral averaging (q-ROULCA) operator, q-rung orthopair uncertain linguistic Choquet integral geometric (q-ROULCG) operator, generalized q-rung orthopair uncertain linguistic Choquet integral averaging q-ROULGCA) operator and generalized q-rung orthopair uncertain linguistic Choquet integral geometric (q-ROULGCG) operator. Section 4 presents a novel approach to MADM based on proposed operators and Section 5 provides a numerical instance to show the validity of propose method.

2. Preliminaries

In this section, we briefly review concepts about q-ROFS and Choquet integral, and then we define a new notation of q-ROULS, some operational laws on q-ROULVs are also introduced.

2.1. q-Rung orthopair fuzzy set

Definition 1 [14]. Let X be an ordinary fixed set, a q-ROFS A defined on X is given by $A = {〈 x, u_{A} (x), v_{A} (x) 〉 | x \in X}$ (1) where u_A (x) and v_A (x) represent the membership degree and non-membership degree respectively, satisfying u_A (x) ∈ [0, 1] , v_A (x) ∈ [0, 1] and $0 \leq u_{A} (x)^{q} + v_{A} (x)^{q} \leq 1, (q \geq 1)$

The indeterminacy degree is defined as π_A (x) = (u_A (x) ^q + v_A (x) ^q - u_A (x) ^qv_A (x) ^q) ^1/q. For convenience, (u_A (x) , v_A (x)) is called a q-rung orthopair fuzzy number (q-ROFN) by Liu and Wang [15], which can be denoted by A = (u_A, v_A).

Liu and Wang [15] also proposed operational laws for q-ROFNs.

Definition 2 [15]. Let ${\tilde{a}}_{1} = (u_{1}, v_{1})$ , ${\tilde{a}}_{2} = (u_{2}, v_{2})$ be two q-ROFNs, and λ be a positive real number, then

${\tilde{a}}_{1} \lor {\tilde{a}}_{2} = (max {u_{1}, u_{2}}, min {v_{1}, v_{2}})$ ;

${\tilde{a}}_{1} \land {\tilde{a}}_{2} = (min {u_{1}, u_{2}}, max {v_{1}, v_{2}})$ ;

${\tilde{a}}_{1} \oplus {\tilde{a}}_{2} = ({(u_{1}^{q} + u_{2}^{q} - u_{1}^{q} u_{2}^{q})}^{1 / q}, v_{1} v_{2})$ ;

${\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = (u_{1} u_{2}, {(v_{1}^{q} + v_{2}^{q} - v_{1}^{q} v_{2}^{q})}^{1 / q})$ ;

$λ {\tilde{a}}_{1} = ({(1 - {(1 - u_{1}^{q})}^{λ})}^{1 / q}, v_{1}^{λ})$ ;

${\tilde{a}}_{1}^{λ} = (u_{1}^{λ}, {(1 - {(1 - v_{1}^{q})}^{λ})}^{1 / q})$ .

To compare two q-ROFNs, Liu and Wang [15] proposed a comparison method for q-ROFNs.

Definition 3 [15]. Let $\tilde{a} = (u_{a}, v_{a})$ be a q-ROFN, then the score of $\tilde{a}$ is defined as $S (\tilde{a}) = u_{a}^{q} - v_{a}^{q}$ , the accuracy of $\tilde{a}$ is defined as $H (\tilde{a}) = u_{a}^{q} + v_{a}^{q}$ . For any two q-ROFNs, ${\tilde{a}}_{1} = (u_{1}, v_{1})$ and ${\tilde{a}}_{2} = (u_{2}, v_{2})$ . Then

If $S ({\tilde{a}}_{1}) > S ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

If $S ({\tilde{a}}_{1}) = S ({\tilde{a}}_{2})$ , then

if $H ({\tilde{a}}_{1}) > H ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

if $H ({\tilde{a}}_{1}) = H ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} = {\tilde{a}}_{2}$ .

2.2. The linguistic set and uncertain linguistic set

Let S = {s_i|i = 0, 1, . . . , l - 1} be a discrete linguistic term set with odd cardinality, where the term s_i ∈ S represents a possible value for a linguistic variable. For the linguistic set S, the following conditions should be satisfied:

The set is ordered, namely, the s_i > s_j, if and only if i > j;

Negation operator: Neg (s_i) = s_j, such that j = l - 1 - i;

Maximum and minimum operators: max {s_i, s_j} = s_i, if s_i ≥ s_j; min {s_i, s_j} = s_i, if s_i ≤ s_j.

Take l = 7 for example, a possible set S can be: S= (s₀, s₁, s₂, s₃, s₄, s₅, s₆)= very poor, poor, slightly poor, fair, slightly good, good, very good.

Based on the linguistic set, Xu [21] proposed uncertain linguistic set (ULS).

Definition 4. Suppose $\tilde{s} = [s_{a}, s_{b}]$ , s_a, s_b ∈ S and a ≤ b, s_a and s_b are the lower limit and upper limit of $\tilde{s}$ respectively, then $\tilde{s}$ is called ULS.

In addition, Xu [21] provided some operations for ULVs, which are shown in Definition 5.

Definition 5 [21]. Let ${\tilde{s}}_{1} = [s_{a_{1}}, s_{b_{1}}], {\tilde{s}}_{2} = [s_{a_{2}}, s_{b_{2}}]$ be any two ULVs and λ be a positive real number, then

${\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \oplus [s_{a_{2}}, s_{b_{2}}] = [s_{a_{1} + a_{2}}, s_{b_{1} + b_{2}}]$ ;

${\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \otimes [s_{a_{2}}, s_{b_{2}}] = [s_{a_{1} \times a_{2}}, s_{b_{1} \times b_{2}}]$ ;

$λ {\tilde{s}}_{1} = λ [s_{a_{1}}, s_{b_{1}}] = [s_{λ a_{1}}, s_{λ b_{1}}]$ ;

${({\tilde{s}}_{1})}^{λ} = {([s_{a_{1}}, s_{b_{1}}])}^{λ} = [s_{a_{1}^{λ}}, s_{b_{1}^{λ}}]$ .

2.3. q-Rung orthopair fuzzy uncertain linguistic set

By combing the ULS with q-ROFS, we propose the concept of q-ROULS.

Definition 6. Let $[s_{θ (x)}, s_{τ (x)}] \in \tilde{s}$ and X is a given domain, then a q-ROULS is defined as follows

$A = {〈 x [[s_{θ (x)}, s_{τ (x)}], (u_{A} (x), v_{A} (x))] 〉 | x \in X}$ (2) where u_A : X → [0, 1] and v_A : X → [0, 1], satisfying 0 ≤ u_A (x) ^q + v_A (x) ^q ≤ 1, (q ≥ 1), $s_{θ (x)}, s_{τ (x)} \in \bar{S}$ . u_A (x) and v_A (x) represent the membership and non-membership degrees of the element x to the uncertain linguistic variable[s_θ(x), s_τ(x)], respectively.

For a q-ROULS A, the indeterminacy of x to the uncertain linguistic set [s_θ(x), s_τ(x)] is represented by π_A (x) = (u_A (x) ^q + v_A (x) ^q -u_A (x) ^qv_A (x) ^q) ^1/q. For convenience, we call 〈 [s_θ(x), s_τ(x)] , (u_A (x) , v_A (x))〉 a q-ROULV, which can be denoted by 〈 [s_θ, s_τ] , (u, v)〉.

Based on the operations for ULVs and for q-ROFNs, we provide the following operational laws for q-ROULVs.

Definition 7. Let a₁ = 〈 [s_{θ
₁}, s_{τ
₁}] , (u₁, v₁) 〉 and a₂ = 〈 [s_{θ
₂}, s_{τ
₂}] , (u₂, v₂) 〉 be any two q-ROULVs and λbe a positive real number, then

$a_{1} \oplus a_{2} = 〈 [s_{θ_{1} + θ_{2}}, s_{τ_{1} + τ_{2}}], ((u_{1}^{q} + u_{2}^{q} - u_{1}^{q}$ $u_{2}^{q})^{1 / q}, v_{1} v_{2}) 〉$

$a_{1} \otimes a_{2} = 〈 [s_{θ_{1} \cdot θ_{2}}, s_{τ_{1} \cdot τ_{2}}], (u_{1} u_{2}, (v_{1}^{q} + v_{2}^{q} -$ $v_{1}^{q} v_{2}^{q})^{1 / q}) 〉$

$λ a_{1} = 〈 [s_{λ \cdot θ_{1}}, s_{λ \cdot τ_{1}}], ((1 - (1 - u_{1}^{q})^{λ})^{1 / q}, v_{1}^{λ}) 〉$

$(a_{1})^{λ} = 〈 [s_{θ_{1}^{λ}}, s_{τ_{1}^{λ}}], (u_{1}^{λ}, (1 - (1 - v_{1}^{q})^{λ})^{1 / q}) 〉$ .

Theorem 1. Let a₁ = 〈 [s_{θ
₁}, s_{τ
₁}], (u₁, v₁) 〉 and a₂ = 〈 [s_{θ
₂}, s_{τ
₂}] , (u₂, v₂) 〉 be any two q-ROULVs then the following calculation rules can be obtained

a₁ ⊕ a₂ = a₂ ⊕ a₁;

a₁ ⊗ a₂ = a₂ ⊗ a₁;

λ (a₁ ⊕ a₂) = λa₁ ⊕ λa₂;

λ₁a₁ ⊕ λ₂a₁ = (λ₁ + λ₂) a₁;

$a_{1}^{λ_{1}} \otimes a_{1}^{λ_{2}} = (a_{1})^{λ_{1} + λ_{2}}$ ;

$a_{1}^{λ} \otimes a_{2}^{λ} = (a_{1} \otimes a_{2})^{λ}$ .

To compare two q-ROULVs, we firstly provide the concepts of expected value and accuracy function of a q-ROULV.

Definition 8. Leta =〈 [s_θ, s_τ] , (u, v) 〉 be a q-ROULV, then the expected value E (a) is defined as follows:

$E (a) = \frac{θ + τ}{4} (u^{q} + 1 - v^{q})$ (3)

Definition 9. Let a =〈 [s_θ, s_τ] , (u, v) 〉 be a q-ROULV, then the accuracy function H (A) of a is defined as follows:

$H (a) = \frac{θ + τ}{2} (u^{q} + v^{q})$ (4)

Based on the two above concepts, a comparison law for q-ROULVs can be provided.

Definition 10. Let a₁ =〈 [s_{θ
₁}, s_{τ
₁}] , (u₁, v₁) 〉 and a₂ = 〈 [s_{θ
₂}, s_{τ
₂}] , (u₂, v₂)〉 be any two q-ROULVs, E (a₁) and E (a₂) denote the expected values of a₁ and a₂ respectively, H (a₁) and H (a₁) represent the accuracy function of a₁ and a₂ respectively, then

If E (a₁) > E (a₂), then a₁ > a₂;

If E (a₁) = E (a₂), then

If H (a₁) > H (a₂), then a₁ > a₂;

If H (a₁) = H (a₂), then a₁ = a₂.

2.4. Choquet integral operator

Sugeno [38] initiated the notation of fuzzy measure, which can be used to define a weight on each combination of criteria in the Choquet integral model. In this subsection, we introduce the definitions of fuzzy measure and Choquet integral.

Definition 11 [38]. A fuzzy measure on X is a function ρ : Γ (x) → [0, 1] , with the following conditions:

ρ (φ) = 0, ρ (x) = 1 (boundary conditions)

A, B ∈ X, A ⊆ B, then ρ (A) ≤ ρ (B) (monotonicity)

However, we generally need to determine 2ⁿ - 2 values for n criteria, which make it quite complex, and thus it is not easy to give such fuzzy measure according to Definition 11. Therefore, Sugeno [38] defined the following ρ- fuzzy measure:

$ρ (A \cup B) = ρ (A) + ρ (B) + \partial ρ (A) ρ (B)$ (5)

Where A ∪ B = φ, and the parameter ∂ ∈ [- 1, + ∞) denotes interaction between the attributes. In Equation (5),

If ∂=0, then ∂- fuzzy measure reduces to ρ (A ∪ B) = ρ (A) + ρ (B), which is defined as additive measure.

In this situation, if all the elements in X are independent, we get $ρ (A) = \sum_{x_{i} \in A} ρ (x_{i})$ (6) if all the elements in X are finite, then

$\begin{matrix} ρ (A) = ρ (⋃_{i = 1}^{n} x_{i}) = \\ {\begin{matrix} \frac{1}{\partial} [\prod_{i = 1}^{n} (1 + \partial ρ (x_{i})) - 1], & \partial \neq 0 \\ \sum_{x_{i} \in A} ρ (x_{i}), & \partial = 0 \end{matrix} \end{matrix}$ (7) where x_i ∩ x_j = Φ, for i, j = 1, 2 ⋯ n, and i ≠ j .

If ∂≻0, then ∂- fuzzy measure reduces toρ (A ∪ B) ≻ ρ (A) + ρ (B), which is defined as super-additive measure.

If -1 ≤ ∂≺0, then∂- fuzzy measure reduces toρ (A ∪ B) ≺ ρ (A) + ρ (B), which is defined as sub-additive measure.

When using a fuzzy measure to model the importance of decision criteria set S, a well-known aggregation function is the Choquet integral [37].

Definition 12. Let f be a positive real-valued function on X andρ be a fuzzy measure on X. The discrete Choquet integral of f with respect to ρ is defined as:

$(C) \int fd ρ = \sum_{i = 1}^{n} [ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})] f_{σ (i)}$ (8) where σ (i) denotes a permutation of (1, 2 ⋯ n) such that f_σ(1) ≥ f_σ(2) ≥ ⋯ ≥ f_σ(n), and A_σ(i) = {x_σ(1), ⋯ x_σ(j) } , A_σ(0) = φ .

3. q-Rung orthopair fuzzy uncertain linguistic choquet integral operators

To capture correlations among attributes, in this section, we propose a family of q-rung orthopair uncertain linguistic Choquet integral operators, such as q-ROULCA operator, q-ROULCG, q-ROULGCA operator and q-ROULGCG operator. Moreover, desirable properties of them are discussed.

3.1. The q-ROULCA and q-ROULCG operators

Definition 13. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, then the q-ROULCA operator is defined as: $\begin{matrix} F (C_{1}) \int ad ρ = q - ROULCA (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{i = 1}^{n} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) a_{σ (i)} \end{matrix}$ (9) where σ (i) denotes a permutation of (1, 2 ⋯ n) such that a_σ(1) ≥ a_σ(2) ≥ ⋯ ≥ a_σ(n), and G_σ(i) is the attribute corresponding toa_σ(i), A_σ(i) = { G_σ(1), ⋯ G_σ(i) } , A_σ(0) = φ .

Based on the operational laws for q-ROULVs, the following theorem can be obtained.

Theorem 2. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, andσ (i) denotes a permutation of (1, 2 ⋯ n) such that a_σ(1) ≥ a_σ(2) ≥ ⋯ ≥ a_σ(n), and G_σ(i) is the attribute corresponding to a_σ(i), A_σ(0) = φ, A_σ(i) = { G_σ(1), ⋯ G_σ(i) } , then the aggregated value by the q-ROULCA is still a q-ROULV and

$\begin{matrix} q - ROULCA (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\sum_{i = 1}^{n} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ (a_{σ (i)})}, \\ s_{\sum_{i = 1}^{n} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ (a_{σ (i)})}], \\ ((1 - \prod_{i = 1}^{n} (1 - μ_{σ (i)}^{q})^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})^{1 / q}, \\ \prod_{i = 1}^{n} ν_{σ (i)}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}) 〉 \end{matrix}$ (10)

Proof.

When n = 2, by the operational law (1) and (3) in Definition 6, we have

$\begin{matrix} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) a_{σ (i)} = \\ {[s_{(ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ (a_{σ (i)})}, \\ s_{(ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ (a_{σ (i)})}], \\ ((1 - \prod_{i = 1}^{n} (1 - μ_{σ (i)}^{q})^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})^{1 / q}, \\ \prod_{i = 1}^{n} ν_{σ (i)}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})} \end{matrix}$ for i = 1, 2.

Then $\begin{matrix} q - ROULCA (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{i = 1}^{2} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) a_{σ (i)} \\ = 〈 [s_{\sum_{i = 1}^{2} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ (A_{σ (i)})}, \\ s_{\sum_{i = 1}^{2} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ (A_{σ (i)})}], \\ ((1 - \prod_{i = 1}^{2} (1 - μ_{σ (i)}^{q})^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})^{1 / q}, \\ \prod_{i = 1}^{2} ν_{σ (i)}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}) 〉 \end{matrix}$

Thus, result is true for n = 2.

If Equation (8) holds for n = k, that is

\begin{matrix} q - ROULCA (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\sum_{i = 1}^{k} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ (A_{σ (i)})}, \\ s_{\sum_{i = 1}^{k} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ (A_{σ (i)})}] \\ ((1 - \prod_{i = 1}^{k} {(1 - μ_{σ (i)}^{q})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})^{1 / q}, \\ \prod_{i = 1}^{k} ν_{σ (i)}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}) 〉 \end{matrix}

then, when n = k + 1, by the operational laws (1) and (3) in Definition 7, we have

\begin{matrix} q - ROULCA (a_{1}, a_{2}, . . ., a_{k}) \oplus (ρ (A_{σ (k + 1)}) \\ - ρ (A_{σ (k)})) a_{σ (k + 1)} = \\ 〈 [s_{\sum_{i = 1}^{k} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ (A_{σ (i)})}, \\ s_{\sum_{i = 1}^{k} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ (A_{σ (i)})}], \\ \oplus 〈 [s_{(ρ (A_{σ (k + 1)}) - ρ (A_{σ (k)})) θ (A_{σ (k + 1)})}, \\ s_{(ρ (A_{σ (k + 1)}) - ρ (A_{σ (k)})) τ (A_{σ (k + 1)})}], \\ ((1 - \prod_{i = 1}^{n} {(1 - μ_{σ (k + 1)}^{q})}^{ρ (A_{σ (k + 1)}) - ρ (A_{σ (k)})})^{1 / q}, \\ \prod_{i = 1}^{n} ν_{σ (k + 1)}^{ρ (A_{σ (k + 1)}) - ρ (A_{σ (k)})}) 〉 \\ = 〈 [s_{\sum_{i = 1}^{k + 1} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ (A_{σ (i)})}, \\ s_{\sum_{i = 1}^{k + 1} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ (A_{σ (i)})}], \\ ({(1 - \prod_{i = 1}^{k + 1} {(1 - μ_{σ (i)}^{q})}^{(ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)}))})}^{1 / q}, \\ \prod_{i = 1}^{k + 1} ν_{σ (i)}^{(ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)}))}) 〉 \end{matrix}

Thus, Equation (10) holds for n = k + 1.

Therefore, Equation (10) holds for all n, which completes the proof. □

In the followings, we discuss some desirable properties of the q-ROULCA operator.

Theorem 3 (Idempotency). Let a_i be a collection of q-ROULVs, if all the q-ROULVs are equal, i.e. a_i = a for all i, then

$q - ROULCA (a_{1}, a_{2}, . . ., a_{n}) = a$ (11)

Theorem 4. (Boundedness). Let a_i be a collection of q-ROFULVs, then

$\begin{matrix} \min (a_{1}, a_{2}, . . ., a_{n}) & \leq q - ROULCA (a_{1}, a_{2}, . . ., a_{n}) \\ \leq \max (a_{1}, a_{2}, . . ., a_{n}) . \end{matrix}$ (12)

Theorems 3, 4 can be easily proven, and thus the proofs are omitted.

It is noted that there is a parameter q in the q-ROULCA operator, which makes the aggregation processes flexible and feasible. Thus, we discuss some special cases of the q-ROULCA operator.

Case 1. When q = 1, then the q-ROULCA operator reduces to the intuitionistic uncertain linguistic Choquet integral averaging (IULCA) operator.

Case 2. When q = 2, then the q-ROUCA operator reduces to the Pythagorean uncertain linguistic Choquet integral averaging (PULCA) operator.

Case 3. When m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULCA operator reduces to the following

$\begin{matrix} q - ROULWA (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\sum_{i = 1}^{n} m_{σ (i)} θ (A_{σ (i)})} s_{\sum_{i = 1}^{n} m_{σ (i)} τ (A_{σ (i)})}], \\ ((1 - \prod_{i = 1}^{n} {(1 - μ_{σ (i)}^{q})}^{m_{σ (i)}})^{1 / q}, \\ \prod_{i = 1}^{n} ν_{σ (i)}^{m_{σ (i)}}) 〉 \end{matrix}$ (13) which is called q-rung orthopair uncertain linguistic weighted averaging (q-ROULWA) operator.

$\begin{matrix} q - ROULOWA (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\sum_{i = 1}^{n} m_{i} θ (A_{σ (i)})} s_{\sum_{i = 1}^{n} m_{i} τ (A_{σ (i)})}], \\ ((1 - \prod_{i = 1}^{n} {(1 - μ_{σ (i)}^{q})}^{m_{i}})^{1 / q}, \\ \prod_{i = 1}^{n} ν_{σ (i)}^{m_{i}}) 〉 \end{matrix}$ (14) which is called q-rung orthopair uncertain linguistic ordered weighted averaging (q-ROULOWA) operator.

Case 5. When q = 1, m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULCA operator reduces to the intuitionistic uncertain linguistic weighted averaging (IULWA) operator defined by Liu and Jin [22].

Case 6. When q = 2, m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROUCA operator reduces to Pythagorean uncertain linguistic averaging (PULWA) operator.

Similar to the definition of the q-ROULCA operator, we define the following q-rung orthopair uncertain linguistic Choquet geometric mean (q-ROULCG) operator.

Definition 14. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, then the q-ROULCG operator is defined as

$\begin{matrix} F (C_{2}) \int ad ρ = q - ROULCG (a_{1}, a_{2}, . . ., a_{n}) \\ = \prod_{i = 1}^{n} a_{σ (i)}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})} \end{matrix}$ (15) where σ (i)denotes a permutation of (1, 2 ⋯ n) such that a_σ(1) ≥ a_σ(2) ≥ ⋯ ≥ a_σ(n), and G_σ(i) is the attribute corresponding toa_σ(i), A_σ(i) = { G_σ(1), ⋯ G_σ(i) } , A_σ(0) = φ.

Based on the operational laws for q-ROULVs, the following theorem can be obtained.

Theorem 5. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, and then the aggregated value by the q-ROULCG is still a q-ROULV and $\begin{matrix} q - ROULCG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\prod_{i = 1}^{n} θ {(a_{σ (i)})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}}, \\ s_{\prod_{i = 1}^{n} τ (A_{σ (i)})^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}},] \\ (\prod_{i = 1}^{n} μ_{σ (i)}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}, \\ (1 - \prod_{i = 1}^{n} {(1 - ν_{σ (i)}^{q})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})^{1 / q}) 〉 \end{matrix}$

Parallel to Theorems 3, 4, the q-ROULCG operator have properties such as idempotency and boundedness under same conditions.

By giving different values of the parameters λ, q, we get the following special cases.

Case 1. When q = 1, then the q-ROULCG operator reduces to the intuitionistic uncertain linguistic Choquet integral geometric (IULCG) operator.

Case 2. When q = 2, then the q-ROUCG operator reduces to the Pythagorean uncertain linguistic Choquet integral geometric (PULCG) operator.

Case 3. When m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)), then the q-ROULCG operator reduces to the following $\begin{matrix} q - ROULWG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\sum_{i = 1}^{n} m_{σ (i)} θ (A_{σ (i)})} s_{\sum_{i = 1}^{n} m_{σ (i)} τ (A_{σ (i)})}], \\ (\prod_{i = 1}^{n} μ_{σ (i)}^{m_{σ (i)}}, \\ (1 - \prod_{i = 1}^{n} {(1 - ν_{σ (i)}^{q})}^{m_{σ (i)}})^{1 / q}) 〉 \end{matrix}$ which is called q-rung orthopair uncertain linguistic weighted geometric (q-ROULWG) operator.

Case 4. if $ρ (A) = \sum_{i = 1}^{| A |} m_{i}$ for all A ⊆ X, |A| is the number of elements in set A, m_i = ρ (A_σ(i)) - ρ (A_σ(i-1)), where m = (m₁, m₂, ⋯ m_n) ^T, m_i ∈ [0, 1], $\sum_{i = 1}^{n} m_{i} = 1$ , then q-ROULCG operator reduces to the following $\begin{matrix} q - ROULOWG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\sum_{i = 1}^{n} m_{i} θ (A_{σ (i)})} s_{\sum_{i = 1}^{n} m_{i} τ (A_{σ (i)})}], \\ (\prod_{i = 1}^{n} μ_{σ (i)}^{m_{i}}, (1 - \prod_{i = 1}^{n} (1 - ν_{σ (i)}^{q})^{m_{i}})^{1 / q}) \end{matrix}$ which is q-rung orthopair uncertain linguistic ordered weighted geometric (q-ROULOWG) operator.

Case 5. When q = 1, m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULCG operator reduces to the intuitionistic uncertain linguistic weighted geometric (IULWG) operator defined by Liu and Jin [22].

Case 6. When q = 2, m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULCG operator reduces to Pythagorean uncertain linguistic geometricg (PULWG) operator.

3.2. The q-ROULGCA and q-ROULGCG operators

Definition 15. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, then we define the q-ROULGCA operator: Ωⁿ → Ω, if

$\begin{matrix} F (C_{3}) \int ad ρ = q - ROULGCA (a_{1}, a_{2} \dots a_{n}) \\ = {(\sum_{i = 1}^{m} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) a_{σ (i)}^{λ})}^{1 / λ} \end{matrix}$ (16) where σ (i) denotes a permutation of (1, 2 ⋯ n) such that a_σ(1) ≥ a_σ(2) ≥ ⋯ ≥ a_σ(n), and G_σ(i) is the attribute corresponding to a_σ(i), A_σ(i) = { G_σ(1), ⋯ G_σ(i) } , A_σ(0) = φ.

Based on the operations for q-ROULVs, the following theorem can be obtained.

Theorem 6. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, then the aggregated value by the q-ROULGCA is still a q-ROULV and $\begin{array}{l} q - R O U L G C A (a_{1}, a_{2}, ..., a_{n}) = \\ 〈 [s_{(\sum_{i = 1}^{n} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) θ {(A_{σ (i)})}^{λ}) 1 / λ}, \\ s_{(\sum_{i = 1}^{n} (ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})) τ {(A_{σ (i)})}^{λ}) 1 / λ}], \\ ({(1 - \prod_{i = 1}^{n} {(1 - μ_{σ (i)}^{q λ})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})}^{\frac{1}{q λ}}, \\ {(1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - v_{σ (i)}^{q})}^{λ})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}]}^{\frac{1}{λ}})}^{\frac{1}{q}}) 〉 \end{array}$ (17)

The q-ROULGCA operators have properties similar to q-ROULCA operators such as idempotency, and boundedness under some conditions, which are omitted in order to save space.

By giving different values of the parameters λ, q, we get the following special cases.

Case 1. when λ = 1, then the q-ROULGCA operator reduces to the q-ROULCA operator we proposed.

Case 2. When q = 1, then the q-ROULGCA operator reduces to the generalized intuitionistic uncertain linguistic Choquet integral averaging(GIULCA) operator.

Case 3. When q = 2, then the q-ROULGCA operator reduces to the generalized Pythagorean uncertain linguistic Choquet integral averaging(GPULCA) operator.

Case 4. when m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULGCA operator reduces to the following $\begin{matrix} q - R O U L G W A (a_{1}, a_{2}, ..., a_{n}) = \\ 〈 [s_{{(\sum_{i = 1}^{n} m_{σ (i)} θ {(A_{σ (i)})}^{λ})}^{1 / λ}}, s_{{(\sum_{i = 1}^{n} m_{σ (i)} τ {(A_{σ (i)})}^{λ})}^{1 / λ}}], \\ ({(1 - \prod_{i = 1}^{n} {(1 - μ_{σ (i)}^{q λ})}^{m_{σ (i)}})}^{\frac{1}{q λ}}, \\ {(1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - v_{σ (i)}^{q})}^{λ})}^{m_{σ (i)}}]}^{\frac{1}{λ}})}^{\frac{1}{q}}) 〉 \end{matrix}$ which is the generalized q-rung orthopair uncertain linguistic weighted averaging (q-ROULGWA) operator.

Case 5. if $ρ (A) = \sum_{i = 1}^{| A |} m_{i}$ for all A ⊆ X, |A| is the number of elements in set A, m_i = ρ (A_σ(i)) - ρ (A_σ(i-1)), where m = (m₁, m₂, ⋯ m_n) ^T, m_i ∈ [0, 1], $\sum_{i = 1}^{n} m_{i} = 1$ , then q-ROULGCA operator reduces to the following $\begin{matrix} q - R O U L G O W A (a_{1}, a_{2}, ..., a_{n}) = \\ 〈 [s_{{(\sum_{i = 1}^{n} m_{i} θ {(a_{σ (i)})}^{λ})}^{1 / λ}}, s_{{(\sum_{i = 1}^{n} m_{i} τ {(a_{σ (i)})}^{λ})}^{1 / λ}}], \\ ({(1 - \prod_{i = 1}^{m} {(1 - μ_{σ (i)}^{λ})}^{m_{i}})}^{\frac{1}{q λ}}, \\ {(1 - {[1 - \prod_{i = 1}^{m} {(1 - {(1 - v_{σ (i)}^{q})}^{λ})}^{m_{i}}]}^{\frac{1}{λ}})}^{\frac{1}{q}}) 〉 \end{matrix}$ which is the generalized q-rung orthopair uncertain linguistic ordered weighted averaging (q-ROUGOWA) operator.

Case 6. When q = 1, m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULGCA operator reduces to the following $\begin{matrix} G I U L W A (a_{1}, a_{2}, ..., a_{n}) = \\ 〈 [s_{{(\sum_{i = 1}^{n} m_{σ (i)} θ {(a_{σ (i)})}^{λ})}^{1 / λ}}, s_{{(\sum_{i = 1}^{n} m_{σ (i)} τ {(a_{σ (i)})}^{λ})}^{1 / λ}}], \\ ({(1 - \prod_{i = 1}^{n} {(1 - μ_{σ (i)}^{λ})}^{m_{σ (i)}})}^{\frac{1}{λ}}, \\ 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - v_{σ (i)})}^{λ})}^{m_{σ (i)}}]}^{\frac{1}{λ}}) 〉 \end{matrix}$ which is the generalized intuitionistic uncertain linguistic weighted averaging (GIULWA) operator.

Case 7. Similarly, when m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)), q = 2, then the q-ROULCA operator reduces to generalized Pythagorean uncertain linguistic weighted averaging (GPULWA) operator.

Similar to the definition of q-ROULGCA operator, we define the following q-ROULGCG operator.

Definition 16. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, then we define q-ROULGCG operator: Ωⁿ → Ω, if

$\begin{matrix} F (C_{4}) \int ad ρ = q - ROULGCG (a_{1}, a_{2} \dots a_{n}) \\ = \frac{1}{λ} ({(\prod_{i = 1}^{n} λ a_{σ (i)})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}) \end{matrix}$ (18) where σ (i) denotes a permutation of (1, 2 ⋯ n) such that a_σ(1) ≥ a_σ(2) ≥ ⋯ ≥ a_σ(n), and G_σ(i) is the attribute corresponding to a_σ(i), A_σ(i) = { G_σ(1), ⋯ G_σ(i) } , A_σ(0) = φ.

Based on the operational laws for q-ROULVs, the following theorem can be obtained.

Theorem 7. Let a_i =〈 [s_{θ
_i}, s_{τ
_i}] , (u_i, v_i) 〉 be a collection of q-ROULVs, then the aggregated value by the q-ROULGCG is still a q-ROULV and $\begin{matrix} q - ROULGCG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ θ (A_{σ (i)}))}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})}, \\ s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ τ (A_{σ (i)}))}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})}], \\ ({(1 - {[1 - \prod_{i = 1}^{n} (1 - (1 - μ_{σ (i)}^{q})^{λ})^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})}]}^{\frac{1}{λ}})}^{\frac{1}{q}}, \\ {(1 - \prod_{i = 1}^{n} {(1 - ν_{σ (i)}^{q λ})}^{ρ (A_{σ (i)}) - ρ (A_{σ (i - 1)})})}^{\frac{1}{q λ}}) 〉 \end{matrix}$ (19)

By giving different values of the parameters λ, q, we get the following special cases.

Case 1. When then the q-ROULGCG operator reduces to the q-ROULCG operator we proposed.

Case 2. When q = 1, then the q-ROULGCG operator reduces to generalized intuitionistic uncertain linguistic Choquet integral geometric (GIULCG) operator.

Case 3. When q = 2, then the q-ROULGCG operator reduces to generalized Pythagorean uncertain linguistic Choquet integral geometric (GPULCG) operator.

Case 4. When m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULGCG operator reduces to the following $\begin{matrix} q - ROULGWG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ θ (A_{σ (i)}))}^{m_{σ (i)}})}, \\ s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ τ (a_{σ (i)}))}^{m_{σ (i)}})}], \\ ({(1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{σ (i)}^{q})}^{λ})}^{m_{σ (i)}}]}^{\frac{1}{λ}})}^{\frac{1}{q}}, \\ {(1 - \prod_{i = 1}^{n} {(1 - ν_{σ (i)}^{q λ})}^{m_{σ (i)}})}^{\frac{1}{q λ}}) 〉 \end{matrix}$ which is generalized q-rung orthopair uncertain linguistic weighted geometric (q-ROULGWG) operator.

Case 5. if $ρ (A) = \sum_{i = 1}^{| A |} m_{i}$ for all A ⊆ X, |A| is the number of elements in set A, m_i = ρ (A_σ(i)) - ρ (A_σ(i-1)) , where m = (m₁, m₂, ⋯ m_n) ^T, m_i ∈ [0, 1], $\sum_{i = 1}^{n} m_{i} = 1$ , then q-ROULGCA operator reduces to the following $\begin{matrix} q - ROULGOWG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ θ (A_{σ (i)}))}^{m_{i}})}, s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ τ (A_{σ (i)}))}^{m_{i}})}], \\ ({(1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{σ (i)}^{q})}^{λ})}^{m_{i}}]}^{\frac{1}{λ}})}^{\frac{1}{q}} \\ (1 - \prod_{i = 1}^{n} {(1 - ν_{σ (i)}^{q λ})}^{m_{i}})^{\frac{1}{q λ}}) 〉 \end{matrix}$ which is generalized q-rung orthopair uncertain linguistic generalized ordered weighted geometric (q-ROULGOWG) operator.

Case 6. When q = 1, m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)) , then the q-ROULGCG operator reduces to the following $\begin{matrix} GIULWG (a_{1}, a_{2}, . . ., a_{n}) = \\ 〈 [s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ θ (A_{σ (i)}))}^{m_{σ (i)}})}, s_{\frac{1}{λ} ({(\prod_{i = 1}^{n} λ τ (a_{σ (i)}))}^{m_{σ (i)}})}], \\ (1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{σ (i)})}^{λ})}^{m_{σ (i)}}]}^{\frac{1}{λ}}, \\ {(1 - \prod_{i = 1}^{n} {(1 - ν_{σ (i)}^{λ})}^{m_{σ (i)}})}^{\frac{1}{λ}}) 〉 \end{matrix}$ which is generalized intuitionistic uncertain linguistic weighted geometric (GIULWG) operator.

Case 7. Similarly, when m_σ(i) = ρ (A_σ(i)) - ρ (A_σ(i-1)), q = 2, then the q-ROULGCG operator reduces to generalized Pythagorean uncertain linguistic geometric (GPULWG) operator.

4. A new method to MADM problems with q-Rung orthopair uncertain linguistic information

In this section, utilize the proposed operators to deal with q-rung orthopair fuzzy uncertain linguistic MADM problems.

A typical q-rung orthopair uncertain linguistic MADM problem can be described as: Let X ={ x₁, x₂, . . . , x_n } be a collection of alternatives, G ={ G₁, G₂, . . . , G_t } be a set of attributes. For attribute G_j (j = 1, 2, . . . , t) of alternative x_i (i = 1, 2, . . . , n), decision maker is required to utilize a q-ROULV to express his evaluation information, which can be denoted asa_ij =〈 [s_{θ
_ij}, s_{τ
_ij}] , (u_ij, v_ij) 〉. Finally, q-rung orthopair uncertain linguistic decision matrix can be obtained, which can be denoted as A = (a_ij) _n×t. In the following, based on the proposed q-rung orthopair uncertain linguistic Choquet integral aggregation operators, a novel approach to solve this problem is introduced.

Step 1. Normalize the decision matrix. In real decision making problems, attributes can be generally divided into two type: benefit attributes and cost attributes. Therefore, the decision matrix should be normalized in the following method $\begin{matrix} a_{ij} = 〈 [s_{θ_{ij}}, s_{τ_{ij}}], (u_{ij}, v_{ij}) 〉 \\ {\begin{matrix} 〈 [s_{θ_{ij}}, s_{τ_{ij}}], (u_{ij}, v_{ij}) 〉 G_{j} \in I_{1} \\ 〈 [s_{θ_{ij}}, s_{τ_{ij}}], (v_{ij}, u_{ij}) 〉 G_{j} \in I_{2} \end{matrix} \end{matrix}$ where I₁ and I₂ represent the benefit attribute type and the cost attribute type respectively.

Step 2. Identify the fuzzy measure of the t attributes of G. To avoid the exponential complexity, we take Equation (5) for determining the fuzzy measure.

Step 3. Based on score functions in Definition 8, we calculate E (a_ij)to reorder the a_ijfor each alternative x_i(i = 1, 2, 3, 4) in a descending order. If E (a_ij) and E (a_ik), we compare H (a_ij) and H (a_ik), so the partial evaluation a_ij of the alternative x_i is reordered such that a_i(j) ≤ a_i(j+1).

Step 4. Calculate the overall preference for each alternative. For alternative x_i (i = 1, 2, 3, 4), utilize the q-ROULCA or the q-ROULCA operator or q-ROULGCA or q-ROULGCG to aggregate all the attributes values.

Step 5. According to Definition 8, calculate the expected values of the overall preference value x_i.

Step 6. Rank alternatives x_i (i = 1, 2, 3, 4) according to the rank of corresponding overall values

5. Numerical experiment

To illustrate the validity, advantages and superiorities of the proposed approach, we provide a numerical example as well as some comparative analysis in this section. The example is adopted from Ref [30]. An investment company wants to invest its money to a company and after primary evaluation, there are four possible alternative on the candidates list. They are: (1) x₁ is a car company; (2) x₂ is a computer company; (3) x₃ is a TV company; (4) x₄ is a food company. In order to select the best alternative, the four companies are evaluated from four attributes: G₁ is the risk analysis; G₂ is the growth analysis; G₃ is the social–political impact analysis; G₄ is the environmental impact analysis. Assume that the evaluation results of alternatives x_i (i = 1,2,3,4) with respect to the attributes G_i (j = 1,2,3,4) are expressed by the form of q-ROULVs a_ij =〈 [s_{θ
_ij}, s_{τ
_ij}] , (μ_ij, v_ij) 〉 and the ULS S = { s₀, s₁, s₂, s₃, s₄, s₅, s₆ } = {very poor, poor, slightly poor, fair, slightly good, good, very good}. The decision matrix A = [a_ij] _4×4 is present in Table 1.

Table 1
q-Rung orthopair uncertain linguistic decision matrix A

	G ₁	G ₂	G ₃	G ₄
x ₁	< [s₅, s₅] , (0.7, 0.1)>	[< [s₃, s₃] , (0.7, 0.3) >	< [s₄, s₅] , (0.6, 0.2)>	< [s₄, s₅] , (0.7, 0.2)>
x ₂	< [s₄, s₅] , (0.6, 0.2)>	< [s₅, s₅] , (0.6, 0.3)>	< [s₂, s₃] , (0.8, 0.1)>	< [s₃, s₄] , (0.6, 0.4)>
x ₃	$\begin{matrix} < [s_{3}, s_{4}], (0.7, 0.2) > \end{matrix}$	< [s₄, s₅] , (0.6, 0.2)>	< [s₁, s₂] , (0.7, 0.1)>	< [s₂, s₃] , (0.7, 0.1)>
x ₄	< [s₃, s₃] , (0.5, 0.2)>	< [s₂, s₃] , (0.7, 0.1)>	< [s₃, s₄] , (0.6, 0.3)>	< [s₄, s₄] , (0.5, 0.3)>

5.1. The decision making process

The decision-making steps based on the q-ROULCA operators

Step 1. As all the attributes are benefit attribute, the decision matrix does not need to be normalized.

Step 2. Identify the fuzzy measure of the n attributes of G. Suppose that the fuzzy measures of attributes of G are given as follows: ρ (G₁) = 0.2, ρ (G₂) = 0.3, ρ (G₃) = 0.2, ρ (G₄) = 0.4 The ρ-fuzzy measure is used to calculate the fuzzy measure of attribute sets. Firstly, according to Equation (5), the value of ∂ is obtained:∂ = -0.237,and then the fuzzy measure of attribute sets of G = { G₁, G₂, G₃, G₄ } can be calculated by Equation (7), which are shown as following: $\begin{matrix} ρ (G_{1}, G_{2}) = 0.486, ρ (G_{1}, G_{3}) \\ = 0.400, ρ (G_{1}, G_{4}) = 0.580, ρ (G_{2}, G_{4}) \\ = 0.680, ρ (G_{3}, G_{4}) = 0.581, ρ (G_{2}, G_{3}) \\ = 0.486, ρ (G_{1}, G_{2}, G_{3}) = 0.663, ρ (G_{1}, G_{2}, G_{4}) \\ = 0.840, ρ (G_{1}, G_{3}, G_{4}) = 0.754, ρ (G_{2}, G_{3}, G_{4}) \\ = 0.840, ρ (G_{1}, G_{2}, G_{3}, G_{4}) = 1 \end{matrix}$

Step 3. Calculate expected values E (a_ij) to reorder a_ij. According to expected values of a_ij, we rearrange a_ij in a descending order, which is shown as follows: $\begin{matrix} a_{1 σ (1)} = < [s_{5}, s_{5}], (0.7, 0.1) > a_{1 σ (2)} = < [s_{4}, s_{5}], \\ (0.7, 0.2) >; a_{1 σ (3)} = < [s_{4}, s_{5}], (0.6, 0.2) >; \\ a_{1 σ (4)} = < [s_{3}, s_{3}], (0.7, 0.3) >; a_{2 σ (1)} = < [s_{5}, s_{5}], \\ (0.6, 0.3) >; a_{2 σ (2)} = < [s_{4}, s_{5}], (0.6, 0.2) >; a_{2 σ (2)} \\ = < [s_{4}, s_{5}], (0.6, 0.2) >; a_{2 σ (3)} = < [s_{3}, s_{4}], \\ (0.6, 0.4) >; a_{2 σ (3)} = < [s_{3}, s_{4}], (0.6, 0.4) >; \\ a_{2 σ (4)} = < [s_{2}, s_{3}], (0.8, 0.1) >; a_{3 σ (1)} \\ = < [s_{4}, s_{5}], (0.6, 0.2) >; a_{3 σ (2)} = < [s_{3}, s_{4}], \\ (0.7, 0.2) >; a_{3 σ (3)} = < [s_{2}, s_{3}], (0.7, 0.1) >; \\ a_{3 σ (4)} = < [s_{1}, s_{2}], (0.7, 0.1) >; a_{4 σ (1)} \\ = < [s_{4}, s_{4}], (0.5, 0.3) >; a_{3 σ (2)} = < [s_{3}, s_{4}], \\ (0.6, 0.3) >; a_{3 σ (3)} = < [s_{2}, s_{3}], (0.7, 0.1) >; \\ a_{3 σ (4)} = < [s_{3}, s_{3}], (0.5, 0.2) > \end{matrix}$ where σ (i) denotes a permutation of (1, 2 ⋯ n) such that a_σ(1) ≥ a_σ(2) ≥ ⋯ ≥ a_σ(n) . Since G_σ(i) is the attribute corresponding to a_σ(i), and A_σ(i) = { G_σ(1), ⋯ G_σ(i) } , then we can get $\begin{matrix} A_{1 σ (1)} = {G_{1}}, A_{1 σ (2)} = {G_{1}, G_{4}}, \\ A_{1 σ (3)} = {G_{1}, G_{3}, G_{4}} A_{1 σ (4)} = {G_{1}, G_{2}, G_{3}, G_{4}} \\ A_{2 σ (4)} = {G_{1}, G_{2}, G_{3}, G_{4}} A_{2 σ (1)} = {G_{2}}, \\ A_{2 σ (2)} = {G_{1}, G_{2}}, A_{2 σ (3)} = {G_{1}, G_{2}, G_{4}} \\ A_{3 σ (1)} = {G_{2}}, A_{3 σ (2)} = {G_{1}, G_{2}}, \\ A_{3 σ (3)} = {G_{1}, G_{2}, G_{4}} A_{3 σ (4)} = {G_{1}, G_{2}, G_{3}, G_{4}} \\ A_{4 σ (4)} = {G_{1}, G_{2}, G_{3}, G_{4}} A_{4 σ (1)} = {G_{4}}, \\ A_{4 σ (2)} = {G_{3}, G_{4}}, A_{4 σ (3)} = {G_{2}, G_{3}, G_{4}} \end{matrix}$

Take the alternative x₁ for an example, the weight of the alternative x₁ is calculated as follows: $\begin{matrix} ρ_{1 A_{σ (1)}} = ρ_{G_{1}} = 0.2, ρ_{1 A_{σ (2)}} - ρ_{1 A_{σ (1)}} = ρ_{G_{1} G_{4}} \\ - ρ_{G_{1}} = 0.38, ρ_{1 A_{σ (3)}} - ρ_{1 A_{σ (2)}} = ρ_{G_{1} G_{3} G_{4}} \\ - ρ_{G_{1} G_{4}} = 0.174, ρ_{1 A_{σ (4)}} - ρ_{1 A_{σ (3)}} = ρ_{G_{1} G_{2} G_{3} G_{4}} \\ - ρ_{G_{1} G_{3} G_{4}} = 0.246 \end{matrix}$

Similarly, the weight of other alternatives x_i (i = 1, 2, 3, 4) are obtained. Thus the weight matrix is shown in Table 2.

Table 2
The weight matrix

ρ _{A
_σ(1)} ρ_{A
_σ(2)} - ρ_{A
_σ(1)} ρ_{A
_σ(3)} - ρ_{A
_σ(2)} ρ_{A
_σ(4)} - ρ_{A
_σ(3)}

x ₁ 0.2 0.38 0.174 0.246

x ₂ 0.3 0.186 0.354 0.16

x ₃ 0.3 0.186 0.354 0.16

x ₄ 0.4 0.181 0.259 0.16

	ρ _{A _σ(1)}	ρ_{A _σ(2)} - ρ_{A _σ(1)}	ρ_{A _σ(3)} - ρ_{A _σ(2)}	ρ_{A _σ(4)} - ρ_{A _σ(3)}
x ₁	0.2	0.38	0.174	0.246
x ₂	0.3	0.186	0.354	0.16
x ₃	0.3	0.186	0.354	0.16
x ₄	0.4	0.181	0.259	0.16

Step 4. (Suppose q = 3) For alternative x_i (i = 1, 2, 3, 4), utilize the q-ROULCA operator to aggregate the all the attributes values. Therefore, we can get overall values a_i of alternatives. $\begin{matrix} a_{1} = < [s_{3.820}, s_{4.240}], (0.686, 0.203) > \\ a_{2} = < [s_{3.318}, s_{4.132}], (0.696, 0.189) > \\ a_{3} = < [s_{2.318}, s_{3.318}], (0.685, 0.141) > \\ a_{4} = < [s_{2.979}, s_{3.419}], (0.577, 0.209) > \end{matrix}$

Step 5. Calculate expected values of a_i (i = 1, 2, 3, 4) according to Definition 8, we can get $\begin{matrix} E (a_{1}) = 2.648, E (a_{2}) = 2.478, E (a_{3}) = 1.858, \\ E (a_{4}) = 1.893 \end{matrix}$

Therefore, the rank of the overall values isa₁ ≻ a₂ ≻ a₄ ≻ a₃.

Step 6. Rank alternativex_i (i = 1, 2, 3, 4) according to the rank of a_i. We can get x₁ ≻ x₂ ≻ x₄ ≻ x₃. Therefore, x₁ is the best alternative, which means that investing the car company is the best choice.

The decision-making steps based on the q-ROULGCA operators

Step 1. As all the attributes are benefit attribute, the decision matrix does not need to be normalized.

Step 2. (Suppose q = 3, λ = 2) For alternative x_i (i = 2,3,4), utilize the q-ROULGCA operator to aggregate the all the attributes values. Therefore, we can get overall values a_i of alternatives. $\begin{matrix} a_{1} = < [s_{4.009}, s_{4.589}], (0.688, 0.192) > \\ a_{2} = < [s_{3.782}, s_{4.388}], (0.663, 0.258) > \\ a_{3} = < [s_{2.837}, s_{3.782}], (0.678, 0.140) > \\ a_{4} = < [s_{3.241}, s_{3.615}], (0.604, 0.211) > \end{matrix}$

Step 3. Calculate expected values of a_i (i = 1, 2, 3, 4) according to Definition 8, we can get $\begin{matrix} E (a_{1}) = 2.833, E (a_{2}) = 2.603, E (a_{3}) = 2.165, \\ E (a_{4}) = 2.076 \end{matrix}$

Therefore, the rank of the overall values is a₁ ≻ a₂ ≻ a₃ ≻ a₄.

Step 4. Rank alternative x_i (i = 1, 2, 3, 4) according to the rank of a_i We can get x₁ ≻ x₂ ≻ x₃ ≻ x₄. Therefore, x₁ is the best alternative, which means that investing the car company is the best choice.

5.2. The influence of the parameters q, λ on the final result

The prominent characteristic of proposed operators is that they can model practical decision making problems more flexible with additional parameter. To reflect the influences of parameters λ, q on result, we utilize different values of parameters q, λ to rank the alternatives by the proposed q-ROULGCA operator, and results are shown in Fig. 1 and Table 3.

Fig.1

Expected values when λ = 2 and q ∈ (1, 10) by the q-ROULGCA operator.

Table 3

Rankings with different value of parameter

Parameters	Expected values	Ranking Results
λ = 5	E (a₁) =2.951 E (a₂) =2.916 E (a₃) =2.439 E (a₄) =2.253	x₁ ≻ x₂ ≻ x₃ ≻ x₄
λ = 10	E (a₁) =3.069 E (a₂) =3.227 E (a₃) =2.665 E (a₄) =2.411	x₂ ≻ x₁ ≻ x₃ ≻ x₄
λ = 15	E (a₁) =3.138 E (a₂) =3.377 E (a₃) =2.769 E (a₄) =2.487	x₁ ≻ x₂ ≻ x₃ ≻ x₄
λ = 20	E (a₁) =3.183 E (a₂) =3.462 E (a₃) =2.827 E (a₄) =2.530	x₁ ≻ x₂ ≻ x₃ ≻ x₄
λ = 30	E (a₁) =3.233 E (a₂) =3.554 E (a₃) =2.888 E (a₄) =2.576	x₁ ≻ x₂ ≻ x₃ ≻ x₄
λ = 40	E (a₁) =2.426 E (a₂) =3.603 E (a₃) =2.178 E (a₄) =1.948	x₁ ≻ x₂ ≻ x₃ ≻ x₄
λ = 50	E (a₁) =2.437 E (a₂) =3.632 E (a₃) =2.191 E (a₄) =1.956	x₁ ≻ x₂ ≻ x₃ ≻ x₄

Figure 1 illustrates the expected values of the alternatives obtained by the q-ROULGCA operator if we take λ = 2, q ∈ (1, 10). From Fig. 1, we can easily find the ranking results may be slightly different for different parameters. For example, when we take λ = 2, q ∈ [1, 6] , then the ranking is a₁ ≻ a₂ ≻ a₃ ≻ a₄; when we take λ = 2, q ∈(6, 10] then the ranking isa₁≻ a₂ ≻ a₄ ≻ a₃ ; However, the best choice is always x₁ . Moreover, as the values of q increase from 1 to 10, the score values become smaller and smaller.

Table 3 illustrates the expected values of the alternatives obtained by the q-ROULGCA operator as assigned different values λ. It is easily find that we get different scores in Table 3 when we change the parameter λ, which makes decision making more flexible and can meet the needs of different types of DMs.

6. Comparative analysis

To further illustrate the prominent advantages of our methods and the weaknesses of the existing methods in some situations, we make a minor change to the above illustrative example. We replace a₁₂, a₂₁, a₃₂ and a₄₄ in Table 3 with the value (0.8,0.7), and the changed decision matrix is shown in Table 4.

Table 4

The new uncertain linguistic fuzzy decision matrix

	G ₁	G ₂	G ₃	G ₄
x ₁	< [s₅, s₅] , (0.7, 0.1)>	< [s₃, s₃] , (0 . 8, 0 . 7)>	< [s₄, s₅] , (0.6, 0.2)>	[s₄, s₅] , (0.7, 0.2)>
x ₂	< [s₄, s₅] , (0 . 8, 0 . 7)>	< [s₅, s₅] , (0.6, 0.3)>	< [s₂, s₃] , (0.8, 0.1)>	$\begin{matrix} < [s_{3}, s_{4}], (0.6, 0.4) > \end{matrix}$
x ₃	< [s₃, s₄] , (0.7, 0.2) >)	< [s₄, s₅] , (0.6, 0.2)>	< [s₁, s₂] , (0 . 8, 0 . 7)>	< [s₂, s₃] , (0.7, 0.1)>
x ₄	< [s₃, s₃] , (0.5, 0.2)>	< [s₂, s₃] , (0.7, 0.1)>	< [s₃, s₄] , (0.6, 0.3)>	< [s₄, s₄] , (0 . 8, 0 . 7)>

We solve the illustrative new example by using different MADM methods including the intuitionistic uncertain linguistic weighted geometric average (IULWGA) operator, intuitionistic uncertain linguistic ordered weighted geometric (IULOWG) operator in Ref [22], Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation (PFULPWAA) operator and Pythagorean fuzzy uncertain linguistic prioritized weighted geometric aggregation (PFULPWGA) operator in Ref [35]. The ranking results are shown in Table 5.

Table 5

Comparison of rankings with different aggregation operators

Approaches	Expected values	Ranking results
IULWGA [22]	Cannot be calculated	No
IULOWG [22]	Cannot be calculated	No
PULPWA [35]	Cannot be calculated	No
PULPWGA [35]	Cannot be calculated	No
q-ROULCA	E (a₁) =2.828 E (a₂) =2.605 E (a₃) =2.083 E (a₄) =2.257	x₁ ≻ x₂ ≻ x₄ ≻ x₃
q-ROULCG	E (a₁) =2.579 E (a₂) =2.366 E (a₃) =1.953 E (a₄) =1.941	x₁ ≻ x₂ ≻ x₄ ≻ x₃
q-ROULGCA	E (a₁) =2.939 E (a₂) =2.703 E (a₃) =2.223 E (a₄) =2.341	x₁ ≻ x₂ ≻ x₄ ≻ x₃
q-ROULGCG	E (a₁) =2.418 E (a₂) =2.326 E (a₃) =2.311 E (a₄) =2.163	x₁ ≻ x₂ ≻ x₃ ≻ x₄

From Table 4, we know the elements a_12, a₂₁, a₃₂ and a₄₄ are (0.8, 0.7), respectively. Given 0.8 + 0.7 > 1 and 0.8² + 0.7² > 1, the evaluation attribute value (0.8, 0.7) cannot be expressed by IFNs and PFNs. Thus, as shown in Table 5, IULWGA, IULOWG, PULPWG and PULPWGA operators cannot solve the above problem as the membership degree and non-membership degree do not satisfy the constraint conditions of IFNs and PFNs. However, the proposed approach based on the q-ROULCA, q-ROULCG q-ROULGCA and q-ROULGCA operators can still work as (0.8, 0.7) can be represented by q–ROFNs by adjusting the value of q. Therefore, the applicable range of our approach is wider than methods based on the IULWGA, IULOWG operators, PULPWG and PULPWGA operators. Further, according to Table 5, we find that the ranking results may be slightly different by our MADM method, but the best alternative is always x₁.

By further analysis, we can draw the following conclusions.

The proposed q-ROULS can more precisely ex press the uncertainty than the IFULS and PFULS. In today’s complex decision-making environment, the q-ROULS proposed in this paper are a very powerful tool in decision making.

The approaches in this paper are the optimization of the existing methods. The IULCA and IULCG, PULCA and PULCG operators are the special cases of proposed operators in this paper when q = 1, 2, respectively. Furthermore, the proposed operators can include almost all of the geometric and arithmetic aggregation operators for IULVs, PULVs, and q-ROULVs according to different values of parameter.

Our methods can efficiently take the correlations among the decision data into account. Furthermore, when we change the values of parameters, we get different scores, which makes decision making more flexible and can meet the needs of different types of DMs.

Based on the comparisons and analysis above, the methods proposed in this paper are more superior than other approaches.

Conclusions

In this paper, we have proposed q-ROULS by combining q-ROFS and ULS, which is a further generalization of the concepts of IFULS and PFULS. Then, based on q-rung orthopair fuzzy uncertain linguistic Choquet integral operators, we developed a framework to the MADM problem under q-rung orthopair fuzzy uncertain linguistic environment. The strengths of the proposed method have been discussed via comparative analysis.

The main contributions of this paper are four aspects. First, a new linguistic fuzzy set called q-ROULS are proposed. The proposed q-ROULS more powerful and useful than PFULS and IULS in aspect of dealing with the vagueness and fuzzy information. Second, some novel Choquet integral aggregation operators for q-ROULS are proposed. The proposed operators not only consider the correlations among attributes, but also demonstrate high generality than exiting aggregation operators. Third, a novel approach to MADM with q-rung orthopair uncertain linguistic information is proposed. Compared with the existing MADM methods, the novel method has wider and more flexible applicable scope for MADM problems, and can be further applied to other MADM problems, such as pilot hospitals selection, supplier selection and so on.

In the future study, we can extend the proposed operators to some other fuzzy sets [43 –45], and further explore the generalisation of Choquet integral by combining with other aggregation operators, such as Heronian mean, and Frank operator.

Footnotes

Acknowledgments

This work was partially supported by a Key Project of National Natural Science Foundation of China (NSFC) with grant number 71532002, Fundamental Research Funds for the Central Universities (No.2019YJS055) and Beijing Logistics Informatics Research Base.

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q -Rung orthopair fuzzy uncertain linguistic choquet integral operators and their application to multi-attribute decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. q-Rung orthopair fuzzy set

2.3. q-Rung orthopair fuzzy uncertain linguistic set

3.1. The q-ROULCA and q-ROULCG operators

5. Numerical experiment

Table 1 q-Rung orthopair uncertain linguistic decision matrix A

Table 2 The weight matrix ρ A σ(1) ρ A σ(2) - ρ A σ(1) ρ A σ(3) - ρ A σ(2) ρ A σ(4) - ρ A σ(3) x 1 0.2 0.38 0.174 0.246 x 2 0.3 0.186 0.354 0.16 x 3 0.3 0.186 0.354 0.16 x 4 0.4 0.181 0.259 0.16

Conclusions

Footnotes

Acknowledgments

References

Table 1
q-Rung orthopair uncertain linguistic decision matrix A

Table 2
The weight matrix

ρ _{A
_σ(1)} ρ_{A
_σ(2)} - ρ_{A
_σ(1)} ρ_{A
_σ(3)} - ρ_{A
_σ(2)} ρ_{A
_σ(4)} - ρ_{A
_σ(3)}

x ₁ 0.2 0.38 0.174 0.246

x ₂ 0.3 0.186 0.354 0.16

x ₃ 0.3 0.186 0.354 0.16

x ₄ 0.4 0.181 0.259 0.16