Generalized q-rung picture linguistic aggregation operators and their application in decision making

Abstract

The q-rung picture linguistic set (q-RPLS) is an effective tool for managing complex and unpredictable information by changing the parameter ‘q’ regarding hesitancy degree. In this article, we devise some generalized operational laws of q-RPLS in terms of the Archimedean t-norm and t-conorm. Based on the proposed generalized operations, we define two types of generalized aggregation operators, namely the q-rung picture linguistic averaging operator and the q-rung picture linguistic geometric operator, and study their relevant characteristics in-depth. With a view toward applications, we discuss certain specific cases of the proposed generalized aggregation operators with a range of parameter values. Furthermore, we explore q-rung picture linguistic distance measure and its required axioms. Then we put forward a technique for q-RPLSs based on the proposed aggregation operators and distance measure to solve multi-attribute decision-making (MADM) challenges with unknown weight information. At last, a practical example is presented to demonstrate the suggested approaches’ viability and to perform the sensitivity and comparison analysis.

Keywords

q-rung-Picture linguistic fuzzy set generalized operations generalized aggregation operators entropy decision-making

1 Introduction

Multi-attribute decision-making (MADM) has been broadly utilized in different areas of science like designing, the board, and financial aspects as of late. It assumes a focal part in the current choice field. The handling of MADM is the characterization of choices as per the detail of models and picking out the finest one from all choices. When we have a target to choose the finest option from the different options that we have, then the decision-making is used to pick out the best option. It is complicated for the decision-maker to choose the best option from all possible options. Fuzzy theory is one of the decision-making subjects that is used to determine the finest option from all possible options. Due to the high capacity of uncertainty and fuzziness [8] of information, we widely used MADM for the purpose of investigating administration and the public economy, as well as agriculture, as the foremost major branches of decision-making. In reality circumstances, decision-makers normally depend on their intellect and previous intelligence to make moral choices. The precondition in the MADM process is to appropriately show vague and fuzzy information due to the complexities of decision-making problems.

Many countless sorts of higher fuzzy sets have been presented since Zadeh’s [29] introduction of fuzzy sets. Atanassov [25] transformed the fuzzy set into an intuitionistic fuzzy set (IFS) including participant degree, non-participant degree, and hesitation functions with condition $0 \leq \hat{R} (t) + \hat{T} (t) \leq 1$ . After that, Pythagorean fuzzy set was proposed by Yager [1] with condition extra that $0 \leq \hat{R} (t)^{2} + \hat{T} (t)^{2} \leq 1$ . But in some cases, the participation degree and non-participation degree can not satisfy the condition of the Pythagorean FS. For instance, if the participation degree and non-participation degree are 0.8 and 0.9, respectively. Then, the condition of Pythagorean fuzzy can’t be fulfilled, such as (0.8) ² + (0.9) ² > 1. To solve these types of problems, Yager [10] proposed q-rung OPFS, which is a more generalized fuzzy set in which qth power of membership and non-membership is less than or equal to one such as $0 \leq \hat{R} (t)^{q} + \hat{T} (t)^{q} \leq 1$ and q < 1. By setting the values q=1 and q=2, the q-rung fuzzy set transforms into intuitionistic (IFS) and Pythagorean fuzzy set. Even though q-ROFS can adapt to incomplete and hazy appraisal data, it can’t manage clashing data well in genuine conditions. For example, in general, in election work, casting ballot outcomes for the town chief’s appointment might be divided into three classes: “vote in favor of”, ”unbiased casting a ballot”, and “vote to opposition”. “Neutral democratic” infers “no casting vote” that the democratic paper will remain white and will mean that he has rejected both candidates. This case happened in actuality; however, the q-ROFS was inadequate in managing these problems. Cuong et al. [5] created a fuzzy set that was more generalized, named picture fuzzy set (PFS), to resolve these problems, that incorporates the data which has three sorts: indeed, unbiased, and negative. Mahmood et al. [18] started spherical fuzzy set (SFS) for managing such difficulties by changing the PFS decide with the end goal that the sum of the squares of membership, neutral, and non-membership is confined to the unit interval.

The subsequent problem is that, in certain circumstances, DMs like to settle on qualitative choices rather than quantitative choices because of time deficiency and an absence of earlier skills. Zadeh’s [30] proposed the linguistic variable that is a very suitable tool to solve these problems. But, Wang and Li [27] called attention to the fact that linguistic variables can communicate leaders’ qualitative inclination yet can’t think about a component’s participation and non-enrollment levels to a specific idea. This way, they presented the idea of ILFS. Du et al. [7] extended the concept of ILFS into PLFS. Liu and Zhang [17] proposed a PLFS.

Wei [26] proposed a different interesting collection of operators for PFS and tended to their pertinence in direction issues. Ashraf et al. [3] identified as in previous functional leads and created an imaginative upgraded aggregation operator to manage vulnerability in muddled genuine world dynamic issues in picture fuzzy set. Khan et al. [14] defined the new augmentation to be specifically summed up Picture fuzzy soft set (PFSS) and tended to its true capacity applications in collective decision-making. However, there were still some difficulties; when an individual gives such numbers, the absolute of which surpasses the unit stretch, the PFS can’t deal with it. Aggregation operators assume a vital part in the decision-making. Various researchers have made notable commitments to presenting aggregation operators for spherical fuzzy conditions. Ashraf and Abdullah [2] introduced a few groups of aggregation operators in view of Archimedean t-norm and t-conorm with spherical fuzzy data. With the passage of time, it was noticed that these operators can’t model DMs judgment when we get data in the type of a trio like (0.6, 0.8,07) where the amount of squares of truth, forbearance, and deception degrees surpasses 1, i.e., (0.6) ² + (0.8) ² + (0.7) ² > 1 To handle such issues, Garg et al. [11] concentrated on some mathematical aggregation operators in view of their created q-rung fuzzy functional rules. Quek and his associates [22] examined some q-rung weighted fuzzy operators of aggregation and applied them to an issue connected with the level of contamination.

Motivations

Our research motivations can be summarized in the following points:

At present, most q-rung picture linguistic fuzzy aggregation operators are based on algebraic operations which ignore the flexibility of the information fusion process. Generalized operations based on Archimedean t-norm and t-conorm possess an adjustable parameter to compensate for the algebraic operations’ deficiency. Therefore it is required to adapt these operations to q-rung picture linguistic fuzzy environment.

The weight information is a vital part of addressing decision-making problems. As far as the authors are concerned, there is no approach in the literature for solving q-rung picture linguistic fuzzy MADM issues in which the weights of criteria are unknown.

Since approaches based on aggregation operators, which are flexible and simple to use, might adapt to a complex and dynamic decision environment. To make full use of q-RPLS, it is necessary to explore this topic to q-rung picture linguistic fuzzy data with modified aggregation operators.

Objectives

The main objectives of our study are delineated as follows:

To define generalized operations based on existing operations and Archimedean t-norm and t-conorm.

To derive q-rung picture linguistic number weighted averaging aggregation (q-RPLNWAA), and q-rung picture linguistic number weighted geometric aggregation (q-RPLNWGA) operators based on defined generalized operations. Moreover, a few basic characteristics like idempotency, monotonicity, bounded, symmetric, and some special cases of proposed operators are also examined.

To propose the entropy measure for q-rung picture linguistic fuzzy data, which will help to find out unknown weights information of the attribute.

To construct an MADM algorithm based on score functions and aggregation operators, to interpret decision-making problems.

The framework of this paper is recorded as follows: Some basic concepts on q-RFNs and the idea of t-norm and t-conorm are briefly discussed in Section 2. Section 3 presents the defined generalized operations and their relevant proofs. Section 4 develops q-RPLNWAA and q-RPLNWGA operators based on the proposed operations along with the related proofs of their necessary properties. The concept of distance measuring for q-RFNs and entropy measure is also added in this part. In section 5, we develop the technique to solve the decision-making problem based on generalized aggregation operators. Section 6 gives an example based on MADM. Section 7 describes the sensitivity analysis in two different ways: changing the values of q-parameters and changing the attribute weights. Section 8 performs the comparison study in detail. The last section gives conclusions and future research lines.

2 Some basic concepts

In this part, we briefly review the relative theories and background knowledge of q-RPLS.

Definition 1. [23] Let F be a universe of discourse, a q-rung orthopair fuzzy set (q-ROFS) on F is described as follows:

$A_{1} = (x, {μ_{A_{1}} (x), ν_{A_{1}} (x)} | x \in F),$ (1) μ_{A
₁} (x)∈ [0, 1] is named as degree of positive membership of A₁, ν_{A
₁} (x)∈ [0, 1] is named as degree of negative membership of A₁ and μ_{A
₁} (x), ν_{A
₁} (x) satisfy the condition: 0 ≤ μ_{A
₁} (x) ^q + ν_{A
₁} (x) ^q ≤ 1 for ∀x ∈ A₁. Then for π_{A
₁} (x) = (μ_{A
₁} (x) ^q + ν_{A
₁} (x) ^q - μ_{A
₁} (x) ^q ν_{A
₁} (x) ^q) ^1/q is named as indeterminacy degree of x in A₁.

Definition 2. [9] Let F be a universe of discourse, a picture fuzzy set (PFS) defined on F is described as follows: $A_{2} = (x, {μ_{A_{2}} (x), η_{A_{2}} (x), ν_{A_{2}} (x)} | x \in F),$ (2)

μ_{A
₂} (x)∈ [0, 1] is named as degree of positive membership of A₂, η_{A
₂} (x)∈ [0, 1] is named as degree of neutral membership of A₂ and ν_{A
₂} (x)∈ [0, 1] is named as degree of negative membership of A₂ and μ_{A
₂} (x),η_{A
₂} (x),ν_{A
₂} (x) satisfy the condition: 0 ≤ μ_{A
₂} (x) + η_{A
₂} (x) + ν_{A
₂} (x) ≤1 for ∀x ∈ A₂. Then for π_{A
₂} (x) =1 - μ_{A
₂} (x) + η_{A
₂} (x) + ν_{A
₂} (x) is named as degree of refusal membership of x in A₂.

Definition 3. [21] Let F be a universe of discourse, a q-rung picture fuzzy set (q-RPFS) on F is described as follows: $A_{3} = (x, {μ_{A_{3}} (x), η_{A_{3}} (x), ν_{A_{3}} (x)} | x \in F),$ (3) μ_{A
₃} (x)∈ [0, 1] is named as positive membership degree of A₃, η_{A
₃} (x) ∈ [0, 1] is named as degree of neutral membership of A₃ and ν_{A
₃} (x) ∈ [0, 1] is named as degree of negative membership of A₃ and μ_{A
₃} (x),η_{A
₃} (x),ν_{A
₃} (x) meet the condition: 0 ≤ μ_{A
₃} (x) ^q + η_{A
₃} (x) ^q + ν_{A
₃} (x) ^q ≤ 1 for ∀x ∈ A₃. Then for π_{A
₃} (x) = (1 - (μ_{A
₃} (x)) ^q + (η_{A
₃} (x)) ^q + (ν_{A
₃} (x)) ^q) ^1/q is named as degree of refusal membership of x in A₃.

Definition 4. [16] Let F be a universe of discourse, and $\vec{S}$ be a continuous linguistic term set of S = {s_j|j = 0, 1, 2, . . . . . , t}, then a q-rung picture linguistic set (q-RPLS) D on F is defined as follows: $A_{4} = (s_{θ} (x), {μ_{A_{4}} (x), η_{A_{4}} (x), ν_{A_{4}} (x)} | x \in F),$ (4) where, $s_{θ} (x) \in \vec{S}$ , μ_{A
₄} (x)∈ [0, 1] is named as positive membership degree of A₄, η_{A
₄} (x)∈ [0, 1] is named as neutral degree membership of A₄ and ν_{A
₄} (x)∈ [0, 1] is named as negative membership degree of A₄ and μ_{A
₄} (x),η_{A
₄} (x),ν_{A
₄} (x) meet the condition: 0 ≤ μ_{A
₄} (x) ^q + η_{A
₄} (x) ^q + ν_{A
₄} (x) ^q ≤ 1 for ∀x ∈ F. Then for π_{A
₄} (x) = (1 - (μ_{A
₄} (x)) ^q + (η_{A
₄} (x)) ^q + (ν_{A
₄} (x)) ^q) ^1/q is named as degree of refusal membership of x in A₄.

For the purpose of simplicity, (s_θ (x),{μ_{A
₄} (x),η_{A
₄} (x),ν_{A
₄} (x)}) is called as q-rung picture linguistic number (q-RPLN) and is written as ζ = (s_θ,{μ,η,ν}).

Definition 5. [12, 16] Suppose that ζ = (s_θ,{μ, η, ν}) be a q-RPLN, then the score function for ζ is defined as: $Sc (ζ) = θ \times (μ^{q} + 1 - ν^{q}) .$ (5)

Definition 6. [12, 16] Suppose that ζ = (s_θ,{μ, η, ν}) be a q-RPLN, then the accuracy function for ζ is defined as: $H (ζ) = θ \times (μ^{q} + η^{q} + ν^{q}) .$ (6)

Definition 7. [12, 16] Suppose that ζ₁ = (s_θ₁,{μ₁,η₁,ν₁}) and ζ₂ = (s_θ₂,{μ₂,η₂,ν₂}) be any two q-RPLNs, Sc(ζ₁) and Sc(ζ₂) be score function of ζ₁ and ζ₂ respectively, H(ζ₁) and H(ζ₂) be accuracy function of ζ₁ and ζ₂ respectively.

If Sc(ζ₁) > Sc (ζ₂), then ζ₁ > ζ₂

If Sc(ζ₁) = Sc (ζ₂), then

If H(ζ₁) > H (ζ₂), then ζ₁ > ζ₂

If H(ζ₁) = H (ζ₂), then ζ₁ = ζ₂.

Definition 8. [6] Let ℧=[0, 1] be a unit interval. A function $\overset{˘}{T} : ℧ \times ℧ \to ℧$ is called t-norm if

$\overset{˘}{T}$ is monotonic, continuous and associative.

$\overset{˘}{T}$ (x,1)= x ∀x∈ [0, 1].

Definition 9. [31] Let ℧=[0, 1] be a unit interval. A function $\overset{˘}{S}$ : ℧× ℧ → ℧ is called t-conorm if

$\overset{˘}{S}$ is monotonic, continuous and associative.

$\overset{˘}{S}$ (x,0)= x ∀x∈ [0, 1].

Next, we define two special types of t-norms and t-conorms.

Definition 10. [13] A t-norm $\overset{˘}{T}$ is called Archimedean triangular norm if

It is continuous.

$\overset{˘}{T} (\tilde{x}$ , $\tilde{x}) < \tilde{x}$ $\forall \tilde{x} \in [0, 1]$ .

Moreover, t-norm is known as strict archimedean if t-norm is strict increasing for every x₀, x₁ ∈ [0, 1].

Definition 11. [4] A t-conorm $\overset{˘}{S}$ is called Archimedean triangular conorm if

It is continuous.

$\overset{˘}{S} (\tilde{x}$ , $\tilde{x}) > \tilde{x}$ $\forall \tilde{x} \in$ [0, 1].

Moreover, t-conorm is known as strict archimedean if t-conorm is strict increasing for every x₀, x₁ ∈ (0, 1).

According to Klement and Mesiar [15], the Archimedean triangular norm can be defined as $\overset{˘}{T}$ (x₀, x₁) = ς^-1 (ς (x₀) + ς (x₁)); ς is an additive function from [0, 1] to [0, ∞] such that ς (1) =0 and ς (0) =1, and similarly, the Archimedean triangular conorm can be expressed as $\overset{˘}{S}$ (x₀, x₁) = φ^-1 (φ (x₀) + φ (x₁)) where φ (x) = ς (1 - x).

3 Generalized operations

Based on Archimedean t-conorm and t-norm, this section extends the existing q-rung picture linguistic operational laws to a more general form.

To do so, the one-to-one mapping $g : [a, b] \subseteq ℝ ⟶ [0, 1]$ defined as $g (t) = \frac{t - a_{1}}{t - a_{2}}, \forall t \in [a_{1}, a_{2}]$ , is utilized [28]. Letting $a_{1} = 0, a_{2} = \ddot{ξ}$ , then $g : [0, \ddot{ξ}] ⟶ [0, 1]$ .

Definition 12. Suppose that ζ = (s_θ,{μ, η, ν}), ζ₁ = (s_θ₁,{μ₁,η₁,ν₁}) and ζ₂ = (s_θ₂,{μ₂,η₂,ν₂}) be any three q-RPLNs and κ be positive real number, then:

Additive operation: $\begin{matrix} ζ_{1} \oplus ζ_{2} & = & (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (θ_{1})) + ς (g (θ_{2})))}, \\ {{(ς^{- 1} (ς (μ_{1}^{q}) + ς (μ_{2}^{q}))}^{1 / q}, (φ^{- 1} (φ (η_{1}) + φ (η_{2})), (φ^{- 1} (φ (ν_{1}) + φ (ν_{2}))} \end{matrix}); \end{matrix}$

Multiplication operation: $\begin{matrix} ζ_{1} \otimes ζ_{2} & = & (\begin{matrix} s_{g^{- 1} (φ^{- 1} (φ (g (θ_{1})) + φ (g (θ_{2}))))}, \\ {{(φ^{- 1} (φ (μ_{1}) + φ (μ_{2})))), (φ^{- 1} (φ (η_{1}) + φ (η_{2}))), (ς^{- 1} (ς (ν_{1}^{q}) + (ς (ν_{2}^{q}))}^{1 / q}} \end{matrix}); \end{matrix}$

Scalar-multiplication operation: $\begin{matrix} κ ⊙ ζ & = & (\begin{matrix} s_{g^{- 1} (ς^{- 1} (κ ς (g (θ))))}, \\ {(ς^{- 1} (κ ς (μ^{q})))^{1 / q}, (φ^{- 1} (κ φ (η))), (φ^{- 1} (κ φ (ν)))} \end{matrix}); \end{matrix}$

Power operation: $\begin{matrix} ζ^{κ} & = & (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ φ (g (θ))))}, \\ {(φ^{- 1} (κ φ (μ)), (φ^{- 1} (κ φ (η))), (ς^{- 1} (κ ς (ν^{q})))^{1 / q}} \end{matrix}) . \end{matrix}$

Theorem 1. Suppose that ζ = (s_θ,{μ, η, ν}), ζ₁ = (s_θ₁,{μ₁,η₁,ν₁}) and ζ₂ = (s_θ₂,{μ₂,η₂,ν₂}) be any three q-RPLNs and κ₁, κ₂, κ₃ ≥ 0 are three scalars, then the following laws are satisfied.

ζ₁⊕ ζ₂ = ζ₂ ⊕ ζ₁ ;

ζ₁⊗ ζ₂ = ζ₂ ⊗ ζ₁ ;

κ⊙ (ζ₁ ⊕ ζ₂) = (κ ⊙ ζ₂) ⊕ (κ ⊙ ζ₁) ;

(ζ₁⊗ ζ₂) ^κ = (ζ₂) ^κ ⊗ (ζ₁) ^κ ;

(κ₁⊙ ζ) ⊕ (κ₂ ⊙ ζ) = (κ₁ + κ₂) ⊙ ζ ;

(ζ) ^κ₁ ⊗ (ζ) ^κ₂ = (ζ) ^{(κ₁+κ₂)} .

Proof. 1. and 2. are obvious, and we prove others.

Now, we will solve the left-hand side of (3) in Theorem 1:

$\begin{matrix} κ ⊙ (ζ_{1} \oplus ζ_{2}) & = & κ ⊙ (s_{θ_{1}}, {μ_{1}, η_{1}, ν_{1}}) \oplus (s_{θ_{2}}, {μ_{2}, η_{2}, ν_{2}}) . \end{matrix}$ Using the concept of g and the generalized additive operation:

$\begin{matrix} = κ ⊙ (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (θ_{1})) + ς (g (θ_{2}))))}, \\ {\begin{matrix} (ς^{- 1} (ς (μ_{1}^{q}) + (ς (μ_{2}^{q}))))^{1 / q}, (φ^{- 1} (φ (η_{1}) + φ (η_{2})))), \\ (φ^{- 1} (φ (ν_{1}) + φ (ν_{2}))) \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (ς (κ ς (g (g^{- 1} (ς^{- 1} (ς (g (θ_{1}) + ς (g (θ_{2})))))))))}, \\ {\begin{matrix} (ς^{- 1} (κ ς (ς^{- 1} (ς (μ_{1})^{q} + ς (μ_{2})^{q}))^{1 / q})), (φ^{- 1} (κ φ (φ^{- 1} (φ (η_{1}) + φ (η_{2}))))), \\ (φ^{- 1} (κ φ (φ^{- 1} (φ (ν_{1}) + φ (ν_{2}))))) \end{matrix}} \end{matrix}) \end{matrix}$ Now solving right-hand side of (3) in Theorem 1: $\begin{matrix} (κ ⊙ ζ_{2}) \oplus (κ ⊙ ζ_{1}) & = & (κ ⊙ (s_{θ_{2}}, {μ_{2}, η_{2}, ν_{2}})) \oplus (κ ⊙ (s_{θ_{1}}, {μ_{1}, η_{1}, ν_{1}})) . \end{matrix}$ Using scalar multiplication operation and additive operation: $\begin{matrix} = (s_{g^{- 1} (ς^{- 1} (κ ς (g (θ_{2}))))}, {(ς^{- 1} (κ ς (μ_{2}^{q})))^{1 / q}, (φ^{- 1} (κ φ (η_{2}))), (φ^{- 1} (κ φ (ν_{2})))}) \oplus \\ ((s_{g^{- 1} (ς^{- 1} (κ ς (g (θ_{1}))))}, {(ς^{- 1} (κ ς (μ_{1}^{q})))^{1 / q}, (φ^{- 1} (κ φ (η_{1}))), (φ^{- 1} (κ φ (ν_{1})))}) \\ = (\begin{matrix} s_{g^{- 1} (ς (κ ς (g (g^{- 1} (ς^{- 1} (ς (g (θ_{1}) + ς (g (θ_{2})))))))))}, \\ {\begin{matrix} (ς^{- 1} (κ ς (ς^{- 1} (ς (μ_{1})^{q} + ς (μ_{2})^{q}))^{1 / q})), (φ^{- 1} (κ φ (φ^{- 1} (φ (η_{1}) + φ (η_{2}))))), \\ (φ^{- 1} (κ φ (φ^{- 1} (φ (ν_{1}) + φ (ν_{2}))))) \end{matrix}} \end{matrix}) \end{matrix}$ Hence, the result of the left and right sides of (3) verifies this property.

Next, we solve left-hand side of (4) in Theorem 1: $\begin{matrix} (ζ_{1} \otimes ζ_{2})^{κ} & = & ((s_{θ_{1}}, {μ_{1}, η_{1}, ν_{1}}) \otimes (s_{θ_{2}}, {μ_{2}, η_{2}, ν_{2}}))^{κ} . \end{matrix}$ Using generalized multiplication and power operations here: $\begin{matrix} = {(\begin{matrix} s_{g^{- 1} (φ^{- 1} (φ (g (θ_{1})) + φ (g (θ_{2}))))}, \\ {\begin{matrix} (φ^{- 1} (φ (μ_{1}) + φ (μ_{2})))), (φ^{- 1} (φ (η_{1}) + φ (η_{2}))), \\ (ς^{- 1} (ς (ν_{1}^{q}) + (ς (ν_{2}^{q}))))^{1 / q} \end{matrix}} \end{matrix})}^{κ} \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ φ (g (g^{- 1} (φ^{- 1} (φ (g (θ_{1})) + φ (g (θ_{2}))))))))}, \\ {\begin{matrix} (φ^{- 1} (κ φ ((φ^{- 1} (φ (μ_{1}) + φ (μ_{2})))))), (φ^{- 1} (κ φ ((φ^{- 1} (φ (η_{1}) + φ (η_{2})))))), \\ (ς^{- 1} (κ ς ((ς^{- 1} (ς (ν_{1}^{q}) + (ς (ν_{2}^{q}))))^{1 / q})^{q}))^{1 / q} \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ (g (θ_{1} + φ (g (θ_{2}))))))}, \\ {\begin{matrix} (κ φ^{- 1} (φ (μ_{1}) + φ (μ_{2}))), (κ φ^{- 1} (φ (η_{1}) + φ ({eta}_{2}))), \\ (κ ς^{- 1} (ς (ν_{1})^{q} + ς (ν_{2})^{q}))^{1 / q} \end{matrix}} \end{matrix}) . \end{matrix}$

Solving right-hand side of (4) in Theorem 1: $\begin{matrix} (ζ_{2})^{κ} \otimes (ζ_{1})^{κ} = (s_{θ_{1}}, {μ_{1}, η_{1}, ν_{1}})^{κ} \otimes (s_{θ_{2}}, {μ_{2}, η_{2}, ν_{2}}))^{κ} \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ φ (g (θ_{1}))))}, \\ {\begin{matrix} (φ^{- 1} (κ φ (μ_{1}))), (φ^{- 1} (κ φ (η_{1}))), \\ (ς^{- 1} (κ ς (ν_{1}^{q})))^{1 / q} \end{matrix}} \end{matrix}) \otimes \\ (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ φ (g (θ_{2}))))}, \\ {\begin{matrix} (φ^{- 1} (κ φ (μ_{2}))), (φ^{- 1} (κ φ (η_{2}))), \\ (ς^{- 1} (κ ς (ν_{2}^{q})))^{1 / q} \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ φ (g (g^{- 1} (φ^{- 1} (φ (g (θ_{1})) + φ (g (θ_{2}))))))))}, \\ {\begin{matrix} (φ^{- 1} (κ φ ((φ^{- 1} (φ (μ_{1}) + φ (μ_{2})))))), \\ (φ^{- 1} (κ φ ((φ^{- 1} (φ (η_{1}) + φ (η_{2})))))), \\ (ς^{- 1} (κ ς ({(ς^{- 1} (ς (ν_{1}^{q}) + (ς (ν_{2}^{q}))}^{1 / q}))))^{q})^{1 / q} \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ (g (θ_{1} + φ (g (θ_{2}))))))}, \\ {\begin{matrix} (κ φ^{- 1} (φ (μ_{1}) + φ (μ_{2}))), \\ (κ φ^{- 1} (φ (η_{1}) + φ ({eta}_{2}))), \\ (κ ς^{- 1} (ς (ν_{1})^{q} + ς (ν_{2})^{q}))^{1 / q} \end{matrix}} \end{matrix}) . \end{matrix}$ Hence, the result of the left and right sides of (4) verifies this property.

Now, we take left-hand side of (5): $\begin{matrix} (κ_{1} ⊙ ζ) \oplus (κ_{2} ⊙ ζ) = \\ (κ_{1} ⊙ (s_{θ}, {μ, η, ν})) \oplus (κ_{2} ⊙ (s_{θ}, {μ, η, ν})) \\ = (s_{g^{- 1} (ς^{- 1} (κ_{1} ς (g (θ))))}, {(ς^{- 1} (κ_{1} ς (μ^{q})))^{1 / q}, (φ^{- 1} (κ_{1} φ (η))), (φ^{- 1} (κ_{1} φ (ν)))}) \oplus \\ (s_{g^{- 1} (ς^{- 1} (κ_{2} ς (g (θ))))}, {(ς^{- 1} (κ_{1} ς (μ^{q})))^{1 / q}, (φ^{- 1} (κ_{1} φ (η))), (φ^{- 1} (κ_{1} φ (ν)))}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (g^{- 1} (ς^{- 1} (κ_{1} ς (g (θ)))))) ς (g (g^{- 1} (ς^{- 1} (κ_{2} ς (g (θ))))))))}, \\ {\begin{matrix} (ς^{- 1} (ς ((ς^{- 1} (κ_{1} ς (μ)^{q})))^{1 / q})^{q}) + (ς ((ς^{- 1} (κ_{2} ς (μ^{q})))^{1 / q}))^{q})^{1 / q}, \\ (φ^{- 1} (φ ((φ^{- 1} (κ_{1} φ (η)))) + φ ((φ^{- 1} (κ_{2} φ (η)))))), \\ (φ^{- 1} (φ ((φ^{- 1} (κ_{1} φ (ν))) + φ ((φ^{- 1} (κ_{2} φ (ν))))))) \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (κ_{1} + κ_{2}) (ς (g (θ))))}, \\ {\begin{matrix} (ς^{- 1} (κ_{1} + κ_{2}) (ς (μ^{q})))^{1 / q}, \\ (φ^{- 1} (κ_{1} + κ_{2}) (φ (η))), \\ (φ^{- 1} (κ_{1} + κ_{2}) (φ (ν))) \end{matrix}} \end{matrix}) \\ = (κ_{1} + κ_{2}) ⊙ (s_{θ}, {μ, η, ν}) = (κ_{1} + κ_{2}) ⊙ ζ . \end{matrix}$ Hence, this satisfies the property.

Now, we take left-hand side of (6): $\begin{matrix} (ζ)^{κ_{1}} \otimes (ζ)^{κ_{2}} = ((s_{θ}, {μ, η, ν}))^{κ_{1}} \otimes ((s_{θ}, {μ, η, ν}))^{κ_{2}} \\ = (s_{g^{- 1} (φ^{- 1} (κ_{1} φ (g (θ))))}, {(φ^{- 1} (κ_{1} φ (μ))), (φ^{- 1} (κ_{1} φ (η))), (ς^{- 1} (κ_{1} ς (ν^{q})))^{1 / q}})) \otimes \\ (s_{g^{- 1} (φ^{- 1} (κ_{2} φ (g (θ))))}, {(φ^{- 1} (κ_{2} φ (μ)))), (φ^{- 1} (κ_{2} φ (η))), (ς^{- 1} (κ_{2} ς (ν^{q})))^{1 / q}}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (φ (g (g^{- 1} (φ^{- 1} (κ_{1}) φ (g (θ))) + φ (g (g^{- 1} (φ^{- 1} (κ_{2}) φ (g (θ)))))))))}, \\ {\begin{matrix} (φ^{- 1} (φ (φ^{- 1} (κ_{1} φ (μ)) + φ (φ^{- 1} (κ_{2} φ (μ) φ^{- 1} (κ_{2} φ (μ))))))), \\ (φ^{- 1} (φ (φ^{- 1} (κ_{1} φ (η))) + φ (φ^{- 1} (κ_{2} φ (η))))), \\ ((ς^{- 1} (ς (ς^{- 1} (κ_{1} ς (ν^{q})))^{1 / q})^{q}) + (ς (ς^{- 1} (κ_{2} ς (ν^{q})))^{1 / q})^{q})^{1 / q} \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (κ_{1} + κ_{2}) φ (g (θ)))}, \\ {\begin{matrix} (φ^{- 1} ((κ_{1} + κ_{2}) φ (μ))), (φ^{- 1} ((κ_{1} + κ_{2}) φ (η))), \\ (ς^{- 1} ((κ_{1} + κ_{2}) ς (ν^{q})))^{1 / q} \end{matrix}} \end{matrix}) \\ = (s_{θ}, {μ, η, ν})^{κ_{1} + κ_{2}} = (ζ)^{(κ_{1} + κ_{2})} . \end{matrix}$ Hence, this satisfies the property. ■

Theorem 2. Suppose that ζ = (s_θ,{μ,η,ν}), ζ₁ = (s_θ₁,{μ₁,η₁,ν₁}) and ζ₂ = (s_θ₂,{μ₂,η₂,ν₂}) be any three q-RPLNs and κ, then the associative laws for the additive and multiplication operations are:

(ζ₀ ⊕ ζ₁) ⊕ ζ₂ = ζ₀ ⊕ (ζ₁ ⊕ ζ₂);

(ζ₀ ⊗ ζ₁) ⊗ ζ₂ = ζ₀ ⊗ (ζ₁ ⊗ ζ₂) .

Proof. Now, we solve the left-hand side of (1) in Theorem 2. By using the concept of g and generalized operations, we have $\begin{matrix} (ζ_{0} \oplus ζ_{1}) \oplus ζ_{2} = ((s_{θ}, {μ, η, ν}) \oplus (s_{θ_{1}}, {μ_{1}, η_{1}, ν_{1}})) \oplus (s_{θ_{2}}, {μ_{2}, η_{2}, ν_{2}}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (θ)) + ς (g (θ_{1}))))}, \\ {\begin{matrix} (ς^{- 1} (ς (μ^{q}) + (ς (μ_{1}^{q}))))^{1 / q}, (φ^{- 1} (φ (η) + φ (η_{1}))) \\ , (φ^{- 1} (φ (ν) + φ (ν_{1}))) \end{matrix}} \end{matrix}) \oplus (s_{θ_{2}}, {μ_{2}, η_{2} ν_{2}}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (g^{- 1} (ς^{- 1} (ς (g (θ)) + ς (g (θ_{1})))) + ς (g (θ_{3}))))))}, \\ {\begin{matrix} (ς^{- 1} (ς ((ς (μ_{1}^{q})^{1 / q})^{q} + (ς (μ_{3}^{q})))))^{1 / q}, (φ^{- 1} (φ (φ^{- 1} (φ (η) + φ (η_{1})) + φ (η_{3})))), \\ (φ^{- 1} (φ (φ^{- 1} (φ (ν) + φ (ν_{1}))) + φ (ν_{3}))) \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (φ (g (θ) + φ (θ_{1} + φ (θ_{2})))))}, \\ {\begin{matrix} (ς^{- 1} (ς (μ^{q}) + ς (μ_{1}^{q} + ς (μ_{2})^{q})))^{1 / q}, (φ^{- 1} (φ (η) + φ (η_{1}) + φ (η_{2}))), \\ (φ^{- 1} (φ (ν) + φ (ν_{1}) + φ ({nu}_{2}))) \end{matrix}} \end{matrix}) \end{matrix}$ Now, we solve right-hand side of (1) in Theorem 2: $\begin{matrix} ζ_{0} \oplus (ζ_{1} \oplus ζ_{2}) = (s_{θ}, {μ, η, ν}) \oplus ((s_{θ_{1}}, {μ_{1}, η_{1}, ν_{1}}) \oplus (s_{θ_{2}}, {μ_{2}, η_{2}, ν_{2}})) \\ = (s_{θ}, {μ, η, ν}) \oplus (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (θ_{1})) + ς (g (θ_{2}))))} \\ {\begin{matrix} (ς^{- 1} (ς (μ_{1}^{q}) + (ς (μ_{2}^{q}))))^{1 / q}, (φ^{- 1} (φ (η_{1}) + φ (η_{2}))), \\ (φ^{- 1} (φ (ν_{1}) + φ (ν_{2}))) \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (φ (g (θ)) + φ (g (g^{- 1} (ς^{- 1} (ς (g (θ_{1}))))))))}, \\ {\begin{matrix} (ς^{- 1} (ς (μ^{q}) + (ς (ς^{- 1} (ς (μ_{1}^{q}) + (ς (μ_{2}^{q}))^{q})))))^{1 / q}, \\ (φ^{- 1} (φ (η) + φ ((φ^{- 1} (φ (η_{1}) + φ (η_{2})))))), \\ (φ^{- 1} (φ (ν_{1}) + φ ((φ^{- 1} (φ (ν_{1}) + φ (ν_{2})))))) \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (φ (g (θ) + φ (θ_{1} + φ (θ_{2})))))}, \\ {\begin{matrix} (ς^{- 1} (ς (μ^{q}) + ς (μ_{1}^{q} + ς (μ_{2})^{q})))^{1 / q}, (φ^{- 1} (φ (η) + φ (η_{1}) + φ (η_{2}))), \\ (φ^{- 1} (φ (ν) + φ (ν_{1}) + φ (ν_{2}))) \end{matrix}} \end{matrix}) . \end{matrix}$ Hence, the result of the left and right sides of (1) verifies this property. Therefore, for additive operation, the associative law is verified. ■

Similarly, we can prove that using the concept of g and generalized operations (2).

4 Generalized q-RPLNs aggregation operator

This section investigates the generalized q-Rung picture linguistic aggregation operators according to defined generalized operations. In addition, q-Rung picture linguistic distance and entropy measures are also expounded in this section.

4.1 q-Rung picture linguistic number weighted averaging aggregation operator

Definition 13. Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, the based on Archimedian t-norm and t-conorm, q-RPLNWAA operator can be defined as:

$q - RPLNWAA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{n}) = \oplus_{l = 1}^{n} (δ_{l} ⊙ ζ_{l}),$ (7)

where δ_l = (δ₁, δ₂, δ₃, . . . , δ_n) ^T is weighting vector, such that δ_l ∈ [0, 1] and $\sum_{1 = 1}^{n}$ δ_l = 1 .

Theorem 3. Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs. Then q-RPLNWAA is defined as

$q - RPLNWAA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{n}) =$ (8) $(\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (ν_{l})))} \end{matrix}) .$

Proof. We prove this by using the method of mathematical induction.

(a) Now, take n=2, we have

$\begin{matrix} q - RPLNWAA (ζ_{1}, ζ_{2}) = δ_{1} ζ_{1} \oplus δ_{2} ζ_{2} = \\ (s_{g^{- 1} (ς^{- 1} (δ_{1} ς (g (θ_{1}))))}, {(ς^{- 1} (δ_{1} ς (μ_{1}^{q})))^{1 / q}, (φ^{- 1} (δ_{1} φ (η_{1}))), (φ^{- 1} (δ_{1} φ (ν_{1})))}) \oplus \\ (s_{g^{- 1} (ς^{- 1} (δ_{2} ς (g (θ_{2}))))}, {(ς^{- 1} (δ_{2} ς (μ_{2}^{q})))^{1 / q}, (φ^{- 1} (δ_{2} φ (η_{2}))), (φ^{- 1} (δ_{2} φ (ν_{2})))}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (ς (g (g^{- 1} (ς^{- 1} (δ_{1} ς (g (θ_{1})))))) + ς (g (g^{- 1} (ς^{- 1} (δ_{2} φ (g (θ_{2}))))))))}, \\ {\begin{matrix} (ς^{- 1} (ς (ς^{- 1} (δ_{1} φ (μ_{1}^{q})))^{1 / q})^{q} + ς ((φ^{- 1} (δ_{2} φ (μ_{2}^{q})))^{1 / q})^{q})^{1 / q}, \\ (φ^{- 1} (φ (φ^{- 1} (δ_{1} φ (η_{1}))) + φ ((φ^{- 1} (δ_{2} φ (η_{1})))))), \\ (φ^{- 1} (φ ((φ^{- 1} (δ_{1} φ (ν_{1}))) + φ ((φ^{- 1} (δ_{2} φ (ν_{2}))))))) \end{matrix}} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{2} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{2} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{2} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{2} δ_{l} φ (ν_{l})))} \end{matrix}) . \end{matrix}$

(b) We suppose our result is true for any n=t is given that,

$\begin{matrix} q - RPLNWAA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{t}) = \\ (\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{t} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{t} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{t} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{t} δ_{l} φ (ν_{l})))} \end{matrix}) . \end{matrix}$ (c) Now we show our result is true for any n=t+1, by using (a) and (b) in this part.

$\begin{matrix} q - RPLNWAA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{t}, ζ_{t + 1}) = \oplus_{l = 1}^{t} (δ_{l} ⊙ ζ_{l}) \oplus (δ_{t + 1} ⊙ ζ_{t + 1}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (ν_{l})))} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{t} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{t} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{t} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{t} δ_{l} φ (ν_{l})))} \end{matrix}) \oplus \\ (\begin{matrix} s_{g^{- 1} (ς^{- 1} (δ_{t + 1} ς (g (θ_{t + 1}))))}, \\ {(ς_{t + 1}^{- 1} (δ_{t + 1} ς (μ_{t + 1}^{q})))^{1 / q}, (φ^{- 1} (δ_{t + 1} φ (η_{t + 1}))), (φ^{- 1} (δ_{t + 1} φ (ν_{t + 1})))} \end{matrix}) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{t + 1} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{t + 1} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{t + 1} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{t + 1} δ_{l} φ (ν_{l})))} \end{matrix}) . \end{matrix}$ Thus the result is fulfilled for n=t+1. Thus the result is fulfilled for the entire n. ■

The suggested aggregation operators meet the characteristics: idempotency, monotonicity, boundedness, and symmetry, which are verified in the following.

Theorem 4. (Idempotency) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, If all ζ_l are equal, i.e., ζ_l = ζ ∀ l, then:

$q - RPLNWAA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) = ζ .$ (9)

Proof. It is given that ζ_l = ζ ∀ l, therefore: $\begin{matrix} q - RPLNWAA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{n}) = q - RPLNWAA (ζ, ζ, ζ, . . ., ζ) \\ = (\begin{matrix} s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (g (θ_{l}))))}, \\ {(ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (ν_{l})))} \end{matrix}) \\ = (s_{θ}, {μ, η, ν}) = ζ . \end{matrix}$

Hence the result is verified. ■

Theorem 5. (Monotonicity) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) and ℘_l= $(s_{θ_{l}^{*}}$ , ${μ_{l}^{*}$ ,η_l, $ν_{l}^{*}}) (l = 1, 2, 3, . . ., n)$ be two collections of q-RPLNs, If ζ_l ≤ ℘ _l for all l, then:

$q - RPLNWAA (ζ_{1}, ζ_{2}, . . ., ζ_{l}) \leq q - RPLNWAA (℘_{1}, ℘_{2}, . . ., ℘_{l}) .$ (10)

Proof. As we know that g is monotonic increasing function and moreover we have ζ_l ≤ ℘ _l. Therefore $\begin{matrix} (s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (g (θ_{l}))))}, {(ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (μ_{l}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (η_{l}))), (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (ν_{l})))}) \leq \\ (s_{g^{- 1} (ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (g (θ_{l}^{*}))))}, {(ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς ({μ_{l}^{*}}^{q})))^{1 / q}, (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (η_{l}^{*}))), (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (ν_{l}^{*})))}) \\ q - RPLNWAA (ζ_{1}, ζ_{2}, . . ., ζ_{l}) \leq q - RPLNWAA (℘_{1}, ℘_{2}, . . ., ℘_{l}) . \end{matrix}$ Hence the proof. ■

Theorem 6. (Boundedness) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l} ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, then

$ζ_{l}^{-} \leq q - RPLNWAA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) \leq ζ_{l}^{+},$ (11) where $ζ_{l}^{-} = min (ζ_{1}, ζ_{2}, . . ., ζ_{n})$ and $ζ_{l}^{+} = max (ζ_{1}, ζ_{2}, . . ., ζ_{n})$ .

Proof. As we know that, ∀l, $\min_{1 \leq l \leq n} s_{θ_{l}} \leq s_{θ_{l}} \leq \max_{1 \leq l \leq n} s_{θ_{l}}$ , $\min_{1 \leq l \leq n} μ_{l} \leq μ_{l} \leq \max_{1 \leq l \leq n} μ_{l}$ , $\min_{1 \leq l \leq n} η_{l} \leq η_{l} \leq \max_{1 \leq l \leq n} η_{l}$ , and $\min_{1 \leq l \leq n} ν_{l} \leq ν_{l} \leq \max_{1 \leq l \leq n} ν_{l}$ , therefore on the basis of Theorem 4.1 and Theorem 4.1. $\Rightarrow \min (ζ_{1}, ζ_{2}, . . ., ζ_{n}) \leq q - RPLNWAA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) \leq \max (ζ_{1}, ζ_{2}, . . ., ζ_{n})$ $\Rightarrow ζ_{l}^{-} \leq q - RPLNWAA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) \leq ζ_{l}^{+} .$ Hence the proof. ■

Theorem 7. (Symmetry) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs. Then, if =, {μ_l′,, be any permutation of ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ), then we have:

Proof. The proof is obvious. Hence, it is omitted. ■

In the following, we investigate different cases of the proposed operator.

Case 1: If ς (t) = - log(1 - t) and φ (t) = - log (t), then q-RPLNWAA operator can be written as: $q - RPLNWAA (ζ_{1}, ζ_{2} . . . . ζ_{n}) = (\begin{matrix} s_{g^{- 1} (1 - \prod_{i = 1}^{n} (1 - g (θ_{l}))^{δ_{l}})}, \\ {(1 - \prod_{l = 1}^{n} (1 - μ_{l}^{q})^{δ_{l}})^{1 / q}, (\prod_{l = 1}^{n} (η_{l})^{δ_{l}}), (\prod_{l = 1}^{n} (ν_{l})^{δ_{l}})} \end{matrix}) .$ (13)

Case 2: If ς (t) = log((2 - (1 - t))/(1 - t)) and φ (t) = log ((2 - t)/t), then q-RPLNWAA operator can be written as:

$E - q - RPLNWAA (ζ_{1}, ζ_{2} . . . . ζ_{n} = (\begin{matrix} s_{g^{- 1} (\frac{\prod_{l = 1}^{n} (1 + g (θ_{l}))^{δ_{l}} - \prod_{l = 1}^{n} (1 - g (θ_{l}))^{δ_{l}}}{\prod_{l = 1}^{n} (1 + g (θ_{l}))^{δ_{l}} + \prod_{l = 1}^{n} (1 - g (θ_{l}))^{δ_{l}}})}, \\ {(\frac{\prod_{l = 1}^{n} (1 + μ_{l}^{q})^{δ_{l}} - \prod_{l = 1}^{n} (1 - μ_{l}^{q})^{δ_{l}}}{\prod_{l = 1}^{n} (1 + μ_{l}^{q})^{δ_{l}} + \prod_{l = 1}^{n} (1 - μ_{l}^{q})^{δ_{l}}})^{1 / q}, (\frac{2 \prod_{l = 1}^{n} (η_{l^{δ_{l}}})}{\prod_{l = 1}^{n} (2 - η_{l})^{δ_{l}} + \prod_{l = 1}^{n} (μ_{l})^{δ_{l}}}), (\frac{2 \prod_{l = 1}^{n} (ν_{l^{δ_{l}}})}{\prod_{l = 1}^{n} (2 - ν_{l})^{δ_{l}} + \prod_{l = 1}^{n} (ν_{l})^{δ_{l}}})} \end{matrix}) .$ (14)

This operator is called Einstein-q-rung picture linguistic fuzzy number weighted averaging aggregation (E-q-RPLNWAA) operator.

Case 3: If ς (t) = log((1 - (1 - α^*) t)/(1 - t)) and φ (t) = log((α^* + (1 - α^*))/t), α^*>0 then q-RPLNWAA operator can be written as:

$H - q - RPLNWAA (ζ_{1}, ζ_{2} . . . . ζ_{n}) = (\begin{matrix} s_{g^{- 1} (\frac{\prod_{l = 1}^{n} (1 - (1 - α^{*}) g (θ_{l}))^{δ_{l}} - \prod_{l = 1}^{n} (1 - g (θ_{l}))^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) g (θ_{l}))^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (1 - g (θ_{l}))^{δ_{l}}})}, \\ {\begin{matrix} (\frac{\prod_{l = 1}^{n} (1 - (1 - α^{*}) μ_{l}^{q})^{δ_{l}} - \prod_{l = 1}^{n} (1 - μ_{l}^{q})^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) μ_{l}^{q})^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (1 - μ_{l}^{q})^{δ_{l}}})^{1 / q}, (\frac{α^{*} \prod_{l = 1}^{n} (η_{l})^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) (1 - η_{l}))^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (η_{l})^{δ_{l}}}), \\ (\frac{α^{*} \prod_{l = 1}^{n} (ν_{l})^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) (1 - ν_{l}))^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (ν_{l})^{δ_{l}}}) \end{matrix}} \end{matrix}) .$ (15) This operator is called Hammer-q-rung picture linguistic fuzzy number weighted averaging aggregation (H-q-RPLNWAA) operator. If α^* = 1, then the H-q-RPLNWAA operator can be written as q-RPLNWAA; If α^* = 2, then the H-q-RPLNWAA operator can be written as E-q-RPLNWAA operator.

Case 4: If $ς (t) = log ((\hat{ζ} - 1) / ((\hat{ζ})^{1 - t} - 1))$ and $φ (t) = \log ((\hat{ζ}) - 1) / ({\hat{ζ}}^{t} - 1))$ , $\hat{ζ}$ >1 then q-RPLNWAA operator can be written as:

$F - q - RPLNWAA (ζ_{1}, ζ_{2} . . . . ζ_{n}) = (\begin{matrix} s_{g^{- 1} (1 - {log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{1 - g (θ_{l})} - 1)^{δ_{l}}}{\hat{ζ} - 1}))}, \\ {\begin{matrix} (1 - {log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{1 - μ_{l}^{q}} - 1)^{δ_{l}}}{\hat{ζ} - 1}))^{1 / q}, (1 - {log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{η_{l}} - 1)^{δ_{l}}}{\hat{ζ} - 1})), \\ (1 - {log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{ν_{l}} - 1)^{δ_{l}}}{\hat{ζ} - 1})) \end{matrix}} \end{matrix}) .$ (16)

This operator is called Frank-q-rung picture linguistic fuzzy number weighted averaging aggregation (F-q-RPLNWAA) operator. If $\hat{ζ} \to 1$ , then F-q-RPLNWAA operator can be written as q-RPLNWAA.

4.2 q-Rung picture linguistic number weighted geometric aggregation operator

Definition 14. Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, the based on Archimedian t-norm and t-conorm, q-RPLNWGA operator can be defined as:

$q - RPLNWGA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{n}) = \otimes_{l = 1}^{n} (ζ_{l})^{δ_{l}} .$ (17)

Theorem 8. Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, we can define the following operator based on Archimedian t-norm and t-conorm and generalized operations:

$q - RPLNWGA (ζ_{1}, ζ_{2}, ζ_{3}, . . ., ζ_{n}) = (\begin{matrix} s_{g^{- 1} (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (g (θ_{l}))))}, \\ {, (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (μ_{l}))), (φ^{- 1} (\sum_{l = 1}^{n} δ_{l} φ (η_{l})), (ς^{- 1} (\sum_{l = 1}^{n} δ_{l} ς (ν_{l}^{q})))^{1 / q})} \end{matrix}) .$ (18)

Proof. The proof of this theorem is the same as Theorem 4.1. We can easily prove it, hence omit it. ■

Likewise, q-RPLNWGA operator also meets some interesting properties, which are asserted (without proof) as below:

Theorem 9. (Idempotency) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, If all ζ_l are equal, i.e., ζ_l = ζ ∀ l, then:

$q - RPLNWGA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) = ζ .$ (19)

Theorem 10. (Monotonicity) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) and ℘_l= $(s_{θ_{l}^{*}}$ , ${μ_{l}^{*}$ , $η_{l}^{*}$ , $ν_{l}^{*}})$ (l=1,2,3,...,n) be two collections of q-RPLNs, If ζ_l ≤ ℘ _l for all l, then:

$q - RPLNWGA (ζ_{1}, ζ_{2}, . . ., ζ_{l}) \leq q - RPLNWGA (℘_{1}, ℘_{2}, . . ., ℘_{l}) .$ (20)

Theorem 11. (Boundedness) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs, then

$ζ_{l}^{-} \leq q - RPLNWGA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) \leq ζ_{l}^{+},$ (21) where $ζ_{l}^{-} = min (ζ_{1}, ζ_{2}, . . ., ζ_{n})$ and $ζ_{l}^{+} = max (ζ_{1}, ζ_{2}, . . ., ζ_{n})$ .

Theorem 12. (Symmetry) Let ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l} ) (l = 1, 2, 3, . . . , n) be the collection of q-RPLNs. Then, if =, {μ_l′,, be any permutation of ζ_l= (s_{θ_l}, {μ_l,η_l,ν_l } ), then we have:

In the followings, we present some particular cases of the q-RPLNWGA operator with respect to a different function.

Case 1: If ς (t) = - log(1 - t) and φ (t) = - log(t), then q-RPLNWGA operator can be written as:

$q - RPLNWGA (ζ_{1}, ζ_{2}, . . ., ζ_{n}) = (s_{g^{- 1} (\prod_{l = 1}^{n} (g (θ_{l}))^{δ_{l}})}, {(\prod_{l = 1}^{n} (μ_{l})^{δ_{l}}), (\prod_{l = 1}^{n} (η_{l})^{δ_{l}}), (1 - \prod_{l = 1}^{n} (1 - ν_{l}^{q})^{δ_{l}})^{1 / q}}) .$ (23)

Case 2: If ς (t) = log((2 - (1 - t))/(1 - t)) and φ (t) = log((2 - t)/t), then q-RPLNWGA operator can be written as:

$E - q - RPLNWGA (ζ_{l}, ζ_{2}, . . ., ζ_{n}) = (\begin{matrix} s_{g^{- 1} (\frac{2 \prod_{l = 1}^{n} (g (θ_{l}))^{δ_{l}}}{\prod_{l = 1}^{n} (2 - g (θ_{l}))^{δ_{l}} + \prod_{l = 1}^{n} (g (θ_{l}))^{δ_{l}}})}, \\ {(\frac{2 \prod_{l = 1}^{n} (μ_{l})^{δ_{l}}}{\prod_{l = 1}^{n} (2 - μ_{l}^{q})^{δ_{l}} + \prod_{l = 1}^{n} (μ_{l})^{δ_{l}}}), (\frac{2 \prod_{l = 1}^{n} (η_{l^{δ_{l}}})}{\prod_{l = 1}^{n} (2 - η_{l})^{δ_{l}} + \prod_{l = 1}^{n} (μ_{l})^{δ_{l}}}), (\frac{\prod_{l = 1}^{n} (1 + ν_{l}^{q})^{δ_{l}} - \prod_{l = 1}^{n} (1 - ν_{l}^{q})^{δ_{l}}}{\prod_{l = 1}^{n} (1 + ν_{l}^{q})^{δ_{l}} + \prod_{l = 1}^{n} (1 - ν_{l}^{q})^{δ_{l}}})^{1 / q}} \end{matrix}) .$ (24) This operator is called Einstein-q-rung picture linguistic fuzzy number weighted geometric aggregation (E-q-RPLNWGA) operator.

Case 3: If ς (t) = log((1 - (1 - α^*) t)/(1 - t)) and φ (t) = log ((α^* + (1 - α^*))/t), α^*>0 then q-RPLNWGA operator can be written as:

$H - q - RPLNWGA (ζ_{l}, ζ_{2}, . . ., ζ_{n}) = (\begin{matrix} s_{g^{- 1} (\frac{\prod_{l = 1}^{n} (1 - (1 - α^{*}) g (θ_{l}))^{δ_{l}} - \prod_{l = 1}^{n} (1 - g (θ_{l}))^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) g (θ_{l}))^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (1 - g (θ_{l}))^{δ_{l}}})}, \\ {\begin{matrix} (\frac{α^{*} \prod_{l = 1}^{n} (μ_{l})^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) (1 - μ_{l}))^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (μ_{l})^{δ_{l}}}), (\frac{α^{*} \prod_{l = 1}^{n} (η_{l})^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) (1 - η_{l}))^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (η_{l})^{δ_{l}}}), \\ (\frac{\prod_{l = 1}^{n} (1 - (1 - α^{*}) ν_{l}^{q})^{δ_{l}} - \prod_{l = 1}^{n} (1 - ν_{l}^{q})^{δ_{l}}}{\prod_{l = 1}^{n} (1 - (1 - α^{*}) ν_{l}^{q})^{δ_{l}} - (1 - α^{*}) \prod_{l = 1}^{n} (1 - ν_{l}^{q})^{δ_{l}}})^{1 / q} \end{matrix}} \end{matrix}) .$ (25) This operator is called Hammer-q-rung picture linguistic fuzzy number weighted geometric aggregation (H-q-RPLNWGA) operator. If α^* = 1, then the H-q-RPLNWGA operator can be written as the q-RPLNWGA operator; if α^* = 2, then the H-q-RPLNWGA operator can be written as E-q-RPLNWGA operator.

Case 4: If $ς (t) = log ((\hat{ζ} - 1) / ((\hat{ζ})^{1 - t} - 1))$ and $φ (t) = log ((\hat{ζ}) - 1) / ({\hat{ζ}}^{t} - 1))$ , $\hat{ζ}$ >1 q-RPLNWGA operator can be written as:

$(\begin{matrix} s_{g^{- 1} ({log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{g (θ_{l})} - 1)^{δ_{l}}}{\hat{ζ} - 1}))}, \\ {\begin{matrix} ({log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{μ_{l}} - 1)^{δ_{l}}}{\hat{ζ} - 1})), ({log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{η_{l}} - 1)^{δ_{l}}}{\hat{ζ} - 1})), \\ (1 - {log}_{\hat{ζ}} (1 + \frac{\prod_{l = 1}^{n} ({\hat{ζ}}^{1 - ν_{l}^{q}} - 1)^{δ_{l}}}{\hat{ζ} - 1}))^{1 / q} \end{matrix}} \end{matrix}) .$ (26)

This operator is called Frank-q-rung picture linguistic fuzzy number weighted geometric aggregation (F-q-RPLNWGA) operator. If $\hat{ζ} \to 1$ , then the F-q-RPLNWGA operator can be written as the q-RPLNWGA operator.

In what follows, distance and entropy measures on q-RPLNs are given along with their required properties.

Definition 15. Suppose that ζ₁ = (s_θ₁,{μ₁,η₁, ν₁}) and ζ₂ = (s_θ₂,{μ₂,η₂,ν₂}) be any two q-RPLNs, then the distance between these two q-RPLNs is defined as: $d^{*} (ζ_{1}, ζ_{2}) = (\frac{| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |}{t + 1}) .$ (27)

Theorem 13. Let d^* (ζ₁, $ζ_{2}) = (\frac{| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |}{t + 1})$ be distance defined for the q-RPLNs, then the distance axioms will be hold for that distance.

Proof.

As we know that $0 \leq μ_{1}^{q} + η_{1}^{q} + ν_{1}^{q} \leq 1$ and $0 \leq μ_{1} 2^{q} + η_{2}^{q} + ν_{2}^{q} \leq 1$

⇒ $0 \leq (\frac{| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |}{t + 1}) \leq 1$

⇒ 0≤d^* (ζ₁,ζ₂)≤1

d^* (ζ₁,ζ₂)=0

⇔ $(\frac{| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |}{t + 1})$ =0

⇔ $| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |$

⇔ ζ₁ = ζ₂ .

d^* (ζ₁, $ζ_{2}) = (\frac{| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |}{t + 1})$

⇒ $(\frac{| \frac{θ_{2}}{t} - \frac{θ_{1}}{t} | + | μ_{2}^{q} - μ_{1}^{q} | + | η_{2}^{q} - η_{1}^{q} | + | ν_{2}^{q} - ν_{1}^{q} |}{t + 1})$ = d^* (ζ₂,ζ₁)

⇒ d^* (ζ₁,ζ₂)=d^* (ζ₂,ζ₁) .

If ζ₁ ⊆ ζ₂ ⊆ ζ₃ then,

⇒ $| \frac{θ_{1}}{t} - \frac{θ_{2}}{t} | + | μ_{1}^{q} - μ_{2}^{q} | + | η_{1}^{q} - η_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} |$ $\subseteq | \frac{θ_{1}}{t} - \frac{θ_{3}}{t} | + | μ_{1}^{q} - μ_{3}^{q} | + | η_{1}^{q} - η_{3}^{q} | + | ν_{1}^{q} - ν_{3}^{q} |$

⇒ d^* (ζ₁,ζ₂) ≤ d^* (ζ₁,ζ₃)

And If ζ₁ ⊆ ζ₂ ⊆ ζ₃ then,

⇒ $| \frac{θ_{2}}{t} - \frac{θ_{3}}{t} | + | μ_{2}^{q} - μ_{3}^{q} | + | η_{2}^{q} - η_{3}^{q} | + | ν_{2}^{q} - ν_{3}^{q} |$ $\subseteq | \frac{θ_{1}}{t} - \frac{θ_{3}}{t} | + | μ_{1}^{q} - μ_{3}^{q} | + | η_{1}^{q} - η_{3}^{q} | + | ν_{1}^{q} - ν_{3}^{q} |$

⇒ d^* (ζ₂,ζ₃) ≤ d^* (ζ₁,ζ₃) .

■

Definition 16. Let ζ = (s_θ,{μ,η, ν}) be a q-RPLN, then entropy measure of ζ is defined as: $E (ζ) = 1 - d^{*} (ζ, ζ^{c}),$ (28) where d^* (ζ,ζ^c) is the distance between the q-RPLN and its complement.

5 Technique for solving decision-making problems based on generalized q-RPLN operators

In this section, we will propose a technique for the solution of MADM problems based on generalized q-RPLN operators. Suppose we have J= ${J_{1}, J_{2}, J_{3}, . . ., J_{n}}$ be any finite collection of n alternatives and we have finite set of m attributes such as H= ${H_{1}, H_{2}, H_{3}, . . ., H_{m}}$ . On the base of q-rung picture linguistic fuzzy set, they will collect the data in the form of ζ = (s_θ,{μ, η, ν}) where, s_θ is taken from linguistic set S = {s₀=Extremely bad,s₁=Abominable, s₂=Unproductive, s₃=Impartial, s₄=Fine, s₅=All right, s₆=Outstanding} and the condition for quantitative part of ζ is 0 ≤ μ^q + η^q + ν^q ≤ 1.

Collection of data:

Collect the DMs’ assessment information in the form of matrix M= $[B_{nm}]$ as $M = (\begin{matrix} B_{11} & B_{12} & \dots & B_{1 m} \\ B_{21} & B_{22} & \dots & B_{2 m} \\ ⋮ & ⋮ & \dots & ⋮ \\ B_{n 1} & B_{n 2} & \dots & B_{nm} \end{matrix}) .$

Normalization:

In this step, the decision matrix $M = [B_{ij}]$ is transformed into the normalized matrix $\bar{M} = [\bar{B_{ij}}]$ by the following formulation:

${\bar{B}}_{ij} = {\begin{matrix} B_{ij}, & if it is from benefit attribute \\ (B_{ij})^{c}, & if it is from cost attribute . \end{matrix}$ (29) where $B_{ij}^{c}$ is called the complement of $B_{ij}$ . Its worthy to note that for any q-RPLN $B = (s_{θ}$ ,{μ,η,ν}) its complement is determined as

$B^{c} = (s_{t - θ}, {ν, η, μ}) .$ (30)

Determine the unknown weight attribute:

Find out the unknown weight vector using the entropy measure concept for q-RPLNs. We use the proposed distance to determine the q-rung picture linguistic entropy measure. Then, we get entropy matrix $E = (\begin{matrix} E_{11} & E_{12} & \dots & E_{1 m} \\ E_{21} & E_{22} & \dots & E_{2 m} \\ ⋮ & ⋮ & \dots & ⋮ \\ E_{n 1} & E_{n 2} & \dots & E_{nm} \end{matrix})$

by using the entropy formulation: $E (B) = 1 - (\frac{| \frac{θ}{t} - \frac{\overset{´}{θ}}{t} | + | μ^{q} - {\overset{´}{μ}}^{q} | + | η^{q} - {\overset{´}{η}}^{q} | + | ν^{q} - {\overset{´}{ν}}^{q} |}{t + 1}) .$ (31)

Now ${\tilde{γ}}_{j} = \frac{E_{ij}}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} E_{ij}} (i = 1, 2, . . ., n, j = 1, 2, . . ., m) .$ (32) Then, find out the unknown weight along each attribute. ${\tilde{ϖ}}_{j} = \frac{1 - {\tilde{γ}}_{j}}{\sum_{j = 1}^{m} 1 - {\tilde{γ}}_{j}}, (j = 1, 2, 3, . . ., m) .$ (33)

Aggregation:

Aggregate the q-RPLNs $B_{ij}$ (j=1,2,3,...,m) for all alternative $J_{i} (i = 1, 2, 3, . . ., n)$ into the general preference value $B$ by applying the proposed q-RPLNWAA or q-RPLNGAA operators.

Mathematically, it can be written as: $B_{i} = q - {RPLNWAA}_{δ} (B_{i 1}, B_{i 2}, B_{i 3}, . . ., B_{im}),$ (34) $B_{i} = q - {RPLNWGA}_{δ} (B_{i 1}, B_{i 2}, B_{i 3}, . . ., B_{im}),$ (35) where δ = (δ₁, δ₂, . . . , δ_n) is the weight vector of attribute that is derived in previous step such that δ_j > 0 and ∑δ_j = 1.

Determine the score values:

Based on Definition 5, find out the score values Sc( $B_{i}) (i = 1, 2, 3, . . ., m)$ of all q-RPLNs $B_{i} (i = 1, 2, 3, . . ., m)$ .

Ranking:

Choose the best one by ranking the all alternatives $J_{i} (i = 1, 2, 3, . ., m)$ using the score values Sc( $B_{i})$ .

For clarity, the proposed decision-making method is depicted in Fig. 1.

Fig. 1

Flow chart of proposed technique.

6 Illustrative example

An illustration of laptop choice is utilized in this subsection to expound on the ramifications and viability of the expressed strategy. It is actually important that the created technique isn’t simply restricted to the laptop determination issue and can be utilized to address various dynamic issues. With the headway of organization innovation, different shopping sites are made by the need. As of now, internet shopping has turned into an irreplaceable way of life for individuals, presenting to us a ton of persuading in our lives. We can buy what we really want without leaving. We never again line to gather cash from the bank for shopping due to the comfort of installment. A customer needs to purchase a laptop and needs to know which laptop sells well on Tmall, Amazon, ebay, and different destinations to make future exchanges more straightforward. He/she calls his/her companions who have mastery in the determination of laptops. The notes that most laptops are evaluated in view of the accompanying standards: execution $(H_{1})$ , pixel $(H_{2})$ , appearance $(H_{3})$ , shading $(H_{4})$ and cost $(H_{5})$ . Then, at that point, he/she chooses the accompanying four top-rated laptops (see Fig. 2) yet wavers which one to buy: Apple $(J_{1})$ , Dell $(J_{2})$ , Hewlett/Packard $(J_{3})$ , Haier $(J_{4})$ . Clearly the determination interaction of laptops is a MCDM issue comprise of five criteria ${H_{1}, H_{2}, H_{3}, H_{4}, H_{5}}$ , four models ${J_{1}, J_{2}, J_{3}, J_{4}}$ and specialist d. Then, at that point, the created approach can be utilized to track down the ideal arrangement.

Fig. 2

Different types of laptops that describe in the above-illustrated example.

Step 1: Collect the data in the form of a matrix as shown in Table 1.

Table 1

q-rung picture linguistic fuzzy decision matrix taken by d

	$H_{1}$	$H_{2}$	$H_{3}$	$H_{4}$	$H_{5}$
$J_{1}$	$‹ s_{2}, {0.2, 0.4, 0.3} ›$	$‹ s_{2}, {0.3, 0.2, 0.5} ›$	$‹ s_{3}, {0.4, 0.1, 0.5} ›$	$‹ s_{5}, {0.4, 0.3, 0.2} ›$	$‹ s_{2}, {0.5, 0.2, 0.1} ›$
$J_{2}$	$‹ s_{3}, {0.1, 0.5, 0.2} ›$	$‹ s_{3}, {0.6, 0.6, 0.2} ›$	$‹ s_{6}, {0.5, 0.4, 0.3} ›$	$‹ s_{4}, {0.6, 0.2, 0.7} ›$	$‹ s_{4}, {0.1, 0.7, 0.3} ›$
$J_{3}$	$‹ s_{3}, {0.5, 0.5, 0.5} ›$	$‹ s_{2}, {0.5, 0.4, 0.2} ›$	$‹ s_{4}, {0.8, 0.2, 0.1} ›$	$‹ s_{5}, {0.6, 0.7, 0.1} ›$	$‹ s_{3}, {0.1, 0.2, 0.8} ›$
$J_{4}$	$‹ s_{2}, {0.3, 0.4, 0.6} ›$	$‹ s_{2}, {0.6, 0.2, 0.7} ›$	$‹ s_{6}, {0.7, 0.6, 0.3} ›$	$‹ s_{1}, {0.5, 0.5, 0.3} ›$	$‹ s_{5}, {0.7, 0.7, 0.1} ›$

Step 2: Normalize the data according to Equation (29).

Table 2

Normalized Matrix

	$H_{1}$	$H_{2}$	$H_{3}$	$H_{4}$	$H_{5}$
$J_{1}$	$‹ s_{2}, {0.2, 0.4, 0.3} ›$	$‹ s_{2}, {0.3, 0.2, 0.5} ›$	$‹ s_{3}, {0.4, 0.1, 0.5} ›$	$‹ s_{5}, {0.4, 0.3, 0.2} ›$	$‹ s_{5}, {0.1, 0.2, 0.5} ›$
$J_{2}$	$‹ s_{3}, {0.1, 0.5, 0.2} ›$	$‹ s_{3}, {0.6, 0.6, 0.2} ›$	$‹ s_{6}, {0.5, 0.4, 0.3} ›$	$‹ s_{4}, {0.6, 0.2, 0.7} ›$	$‹ s_{3}, {0.3, 0.7, 0.1} ›$
$J_{3}$	$‹ s_{3}, {0.5, 0.5, 0.5} ›$	$‹ s_{2}, {0.5, 0.4, 0.2} ›$	$‹ s_{4}, {0.8, 0.2, 0.1} ›$	$‹ s_{5}, {0.6, 0.7, 0.1} ›$	$‹ s_{4}, {0.8, 0.2, 0.1} ›$
$J_{4}$	$‹ s_{2}, {0.3, 0.4, 0.6} ›$	$‹ s_{2}, {0.6, 0.2, 0.7} ›$	$‹ s_{6}, {0.7, 0.6, 0.3} ›$	$‹ s_{1}, {0.5, 0.5, 0.3} ›$	$‹ s_{2}, {0.1, 0.7, 0.7} ›$

Step 3: In this step, we construct the entropy matrix and find out the unknown weight for each attribute as follows:

Now, using proposed technique for (q=2);

${\tilde{γ}}_{1} = 0.2100$ , ${\tilde{γ}}_{2} = 0.2015$ , ${\tilde{γ}}_{3} = 0.1931$ , ${\tilde{γ}}_{4} = 0.2003$ , and ${\tilde{γ}}_{5} = 0.1950$ .

And unknown calculated weights are:

$\begin{matrix} W = ({\tilde{ϖ}}_{1} = 0.1975, {\tilde{ϖ}}_{2} = 0.1996, {\tilde{ϖ}}_{3} = 0.2017, \\ {\tilde{ϖ}}_{4} = 0.1999, {\tilde{ϖ}}_{5} = 0.2013) . \end{matrix}$ Step 4: In this step, we apply aggregation operators (q-RPLNWAA and q-RPLNWGA) with unknown weights that we have found in the previous step.

We obtain the following results:

q-RPLNWAA: $J_{1} = (s_{3.6904}, {0.3069, 0.2163, 0.3764})$ , $J_{2} = (s_{4.1447}, {0.4763, 0.4417, 0.2425})$ ,

$J_{3} = (s_{3.7575}, {0.6778, 0.3536, 0.1578})$ , and $J_{4} = (s_{3.2519}, {0.5113, 0.4424, 0.4832}) .$

q-RPLNWGA:

$J_{1} = (s_{3.1347}, {0.2992, 0.2163, 0.4259})$ , $J_{2} = (s_{3.6544}, {0.3531, 0.4417, 0.3974})$ ,

$J_{3} = (s_{3.4409}, {0.6247, 0.3536, 0.2618})$ , and $J_{4} = (s_{2.1731}, {0.3629, 0.4424, 0.5718}) .$

Step 5: In this step, we calculate the score values of each alternative.

q-RPLNWAA:

$Sc (J_{1}) = 3.2091$ , $Sc (J_{2}) = 4.8412$ , $Sc (J_{3}) = 5.3902$ , and $Sc (J_{4}) = 3.3428$ .

q-RPLNWGA:

$Sc (J_{1}) = 2.8467$ , $Sc (J_{2}) = 3.5329$ , $Sc (J_{3}) = 4.5565$ , and $Sc (J_{4}) = 1.7488$ .

Step 6: Finally, we rank each alternative according to its score values.

q-RPLNWAA:

$J_{3} > J_{2} > J_{4} > J_{1} .$ (36)

q-RPLNWGA:

$J_{3} > J_{2} > J_{1} > J_{4} .$ (37)

7 Sensitivity analysis

In the proposed method, the parameter q and criteria weights are important to ranking outcomes. Thus, a sensitivity analysis is carried out to evaluate our approach’s stability in different scenarios.

7.1 Sensitivity analysis with respect to parameter q

This section is dedicated to examine the influence of the parameter q on the decision-making results. To do this, we solve the same numerical example presented in Section 6 with different values of q.

7.1.1 q-RPLNWAA operator

In this part, we just review the responsiveness examination by utilizing the q-RPLNWAA operator to look at the impact of changed values of the q parameter on the order positioning of all other alternatives; we put various values of q in Table 4. We can find in the table that there is very little change happens as we increment the values of q. Furthermore, we notice the score function values of every alternative become more modest as the value of q increases. As we put q = 2, q = 3, and q = 5, the best option is the same, yet by putting the value of q = 8, q = 10, and q = 10, the best option is changed a little bit. We likewise notice the conduct of all alternatives through the graphical image of its score values displayed in Fig. 3. There is a tiny change appearing in Fig. 3. A parameter q can be seen as the “attitude of DMs.” The q-RPLNWAA aggregation operator is appropriate for pessimistic DMs, while the q-RPLNWAA operator is valuable in reflecting hopeful DMs. Assume we utilize the q-RPLNWAA operator as a collection instrument for the chosen cycle. The higher values of q show that DMs have a more pessimistic mentality and the other way around on account of the q-RPLNWGA operator. In this way, various DMs can pick the value of q based on their mentality.

Table 3
Entropy Matrix

$H_{1}$ $H_{2}$ $H_{3}$ $H_{4}$ $H_{5}$

$J_{1}$ 0.9339 0.9064 0.9596 0.9164 0.8864

$J_{2}$ 0.9746 0.9021 0.8707 0.9496 0.9621

$J_{3}$ 0.9821 0.8939 0.8246 0.8589 0.8246

$J_{4}$ 0.8797 0.9139 0.8107 0.8707 0.8264

	$H_{1}$	$H_{2}$	$H_{3}$	$H_{4}$	$H_{5}$
$J_{1}$	0.9339	0.9064	0.9596	0.9164	0.8864
$J_{2}$	0.9746	0.9021	0.8707	0.9496	0.9621
$J_{3}$	0.9821	0.8939	0.8246	0.8589	0.8246
$J_{4}$	0.8797	0.9139	0.8107	0.8707	0.8264

Table 4

Different ranking by changing the values of parameters

Values of q:	Score function values	Ranking
q=2	$Sc (J_{1}) = 3.3091$ , $Sc (J_{2}) = 4.8412$ , $Sc (J_{3}) = 5.3902$ , $Sc (J_{4}) = 3.3428$	$J_{3} > J_{2} > J_{1} > J_{4}$
q=3	$Sc (J_{1}) = 3.6163$ , $Sc (J_{2}) = 4.7880$ , $Sc (J_{3}) = 4.9584$ , $Sc (J_{4}) = 3.3810$	$J_{3} > J_{2} > J_{1} > J_{4}$
q=5	$Sc (J_{1}) = 3.6798$ , $Sc (J_{2}) = 4.3006$ , $Sc (J_{3}) = 4.4037$ , $Sc (J_{4}) = 3.3560$	$J_{3} > J_{2} > J_{1} > J_{4}$
q=8	$Sc (J_{1}) = 3.6804$ , $Sc (J_{2}) = 4.1759$ , $Sc (J_{3}) = 4.0427$ , $Sc (J_{4}) = 3.2944$	$J_{2} > J_{3} > J_{1} > J_{4}$
q=10	$Sc (J_{1}) = 3.6903$ , $Sc (J_{2}) = 4.1555$ , $Sc (J_{3}) = 3.9313$ , $Sc (J_{4}) = 3.2729$	$J_{2} > J_{3} > J_{1} > J_{4}$
q=12	$Sc (J_{1}) = 3.6904$ , $Sc (J_{2}) = 4.1485$ , $Sc (J_{3}) = 3.8658$ , $Sc (J_{4}) = 3.2621$	$J_{2} > J_{3} > J_{1} > J_{4}$

Fig. 3

Sensitivity analysis with respect to parameter q by using q-RPLNWAA.

7.1.2 q-RPLNWGA operator

In this part, we notice the change in the ranking of the alternative as we change the values of parameter q by utilizing q-RPLNWGA. As with the q-RPLNWAA operator, there are no big changes that occur by utilizing q-RPLNWGA can be seen in Table 5. Some more virilization of score values, mostly everything is equal, can be seen in Fig. 4, where little change occurs as the values of q are changed. This operator is also significant for the DM in choosing the best one.

Table 5
Different ranking by changing the values of parameters

Values of q: Score function values Ranking

q=2 $Sc (J_{1}) = 2.8467$ , $Sc (J_{2}) = 3.5329$ , $Sc (J_{3}) = 4.5565$ , $Sc (J_{4}) = 1.7488$ $J_{3} > J_{2} > J_{1} > J_{4}$

q=3 $Sc (J_{1}) = 2.8153$ , $Sc (J_{2}) = 3.3864$ , $Sc (J_{3}) = 4.4737$ , $Sc (J_{4}) = 1.7077$ $J_{3} > J_{2} > J_{1} > J_{4}$

q=5 $Sc (J_{1}) = 3.0812$ , $Sc (J_{2}) = 3.5403$ , $Sc (J_{3}) = 3.7518$ , $Sc (J_{4}) = 1.9982$ $J_{3} > J_{2} > J_{1} > J_{4}$

q=8 $Sc (J_{1}) = 3.1275$ , $Sc (J_{2}) = 3.6121$ , $Sc (J_{3}) = 3.5201$ , $Sc (J_{4}) = 2.1155$ $J_{2} > J_{3} > J_{1} > J_{4}$

q=10 $Sc (J_{1}) = 3.1329$ , $Sc (J_{2}) = 3.6336$ , $Sc (J_{3}) = 3.4724$ , $Sc (J_{4}) = 2.1464$ $J_{2} > J_{3} > J_{1} > J_{4}$

q=12 $Sc (J_{1}) = 3.13424$ , $Sc (J_{2}) = 3.6443$ , $Sc (J_{3}) = 3.4534$ , $Sc (J_{4}) = 2.1602$ $J_{2} > J_{3} > J_{1} > J_{4}$

Values of q:	Score function values	Ranking
q=2	$Sc (J_{1}) = 2.8467$ , $Sc (J_{2}) = 3.5329$ , $Sc (J_{3}) = 4.5565$ , $Sc (J_{4}) = 1.7488$	$J_{3} > J_{2} > J_{1} > J_{4}$
q=3	$Sc (J_{1}) = 2.8153$ , $Sc (J_{2}) = 3.3864$ , $Sc (J_{3}) = 4.4737$ , $Sc (J_{4}) = 1.7077$	$J_{3} > J_{2} > J_{1} > J_{4}$
q=5	$Sc (J_{1}) = 3.0812$ , $Sc (J_{2}) = 3.5403$ , $Sc (J_{3}) = 3.7518$ , $Sc (J_{4}) = 1.9982$	$J_{3} > J_{2} > J_{1} > J_{4}$
q=8	$Sc (J_{1}) = 3.1275$ , $Sc (J_{2}) = 3.6121$ , $Sc (J_{3}) = 3.5201$ , $Sc (J_{4}) = 2.1155$	$J_{2} > J_{3} > J_{1} > J_{4}$
q=10	$Sc (J_{1}) = 3.1329$ , $Sc (J_{2}) = 3.6336$ , $Sc (J_{3}) = 3.4724$ , $Sc (J_{4}) = 2.1464$	$J_{2} > J_{3} > J_{1} > J_{4}$
q=12	$Sc (J_{1}) = 3.13424$ , $Sc (J_{2}) = 3.6443$ , $Sc (J_{3}) = 3.4534$ , $Sc (J_{4}) = 2.1602$	$J_{2} > J_{3} > J_{1} > J_{4}$

Fig. 4

Sensitivity analysis with respect to parameter q by using q-RPLNWGA.

7.2 Sensitivity analysis with respect to attributes weights

In exemplary strategies of MADM, frequently, it is expected that every utilized data are predetermined. Then, at that point, the final score or alternatives utility is obtained by tackling MADM, though information on choice-making issues is changing as a general rule. We use sensitivity analysis after solving DM’s problems. In the previous section, we did sensitivity analysis by changing the values of parameter q. But in this section, we will change the weights of attributes and see the scores’ results after changing weights. However, a new strategy for responsiveness examination of MADM issues is considered in this paper that works out the changing in the alternative final score of when a change happens in the attribute weight.

According to the considered analysis, we added 0.05 to each attribute and afterward subtracted 0.05 from each attribute weight independently and modified the weight according to the Equation (38) [20]: $ϖ_{a} = \frac{1 - {\tilde{ϖ}}_{a}}{1 - {\tilde{ϖ}}_{0}} ϖ_{0}$ (38) where ${\tilde{ϖ}}_{0}$ is the attribute weight and ${\tilde{ϖ}}_{a}$ is the wieght after adding or subtracting 0.05 from ${\tilde{ϖ}}_{0}$ .

From Table 6, we can clearly see that there is no change in the positioning of alternatives even if we change the weights of attributes. This shows the strength of the generalized aggregation operator. In Fig. 5, we found that there is almost no change in score values as we change the weight of attributes. The order of alternatives using different weight attributes is always $J_{3} > J_{2} > J_{1} > J_{4}$ or some little bit change in it.

Table 6

Sensitivity analysis by changing the attributes weights

New Weights:	Score function values	Ranking
{0.2475, 0.1872, 0.1891, 0.1874, 0.1888}	$Sc (J_{1}) = 3.4355$ , $Sc (J_{2}) = 4.7268$ , $Sc (J_{3}) = 5.2734$ , $Sc (J_{4}) = 3.2217$	$J_{3} > J_{2} > J_{1} > J_{4}$
{0.1852, 0.2496, 0.1891, 0.1874, 0.1887}	$Sc (J_{1}) = 3.4091$ , $Sc (J_{2}) = 4.7050$ , $Sc (J_{3}) = 5.2188$ , $Sc (J_{4}) = 3.2586$	$J_{3} > J_{2} > J_{1} > J_{4}$
{0.1851, 0.1871, 0.2517, 0.1874, 0.1887}	$Sc (J_{1}) = 3.4747$ , $Sc (J_{2}) = 5.0526$ , $Sc (J_{3}) = 5.4761$ , $Sc (J_{4}) = 3.7571$	$J_{3} > J_{2} > J_{4} > J_{1}$
{0.1852, 0.1871, 0.1891, 0.2499, 0.1887}	$Sc (J_{1}) = 3.6699$ , $Sc (J_{2}) = 4.8339$ , $Sc (J_{3}) = 5.5106$ , $Sc (J_{4}) = 3.2682$	$J_{3} > J_{2} > J_{1} > J_{4}$
{0.1851, 0.1871, 0.1891, 0.1874, 0.2513}	$Sc (J_{1}) = 3.5747$ , $Sc (J_{2}) = 4.7631$ , $Sc (J_{3}) = 5.4695$ , $Sc (J_{4}) = 3.1942$	$J_{3} > J_{2} > J_{1} > J_{4}$

Fig. 5

Sensitivity analysis with respect to attribute weight.

8 Comparison analysis

To demonstrate the proposed method’s advantages, we compare with Zhang et al. [16] and Mahnaz et al. [19]. Also, we use the same data as in the illustrative example and the weight vector of attributes $W = (0.1975, 0.1996, 0.2017, 0.1999, 0.2013)$ .

8.1 Comparison of the proposed technique with Zhang et al. [16]

We use Zhang et al. [16] strategy to tackle the above issue, and the results can be seen in Table 7. In Table 7, the score values by using existing q-rung picture linguistic weighted Heronian mean (q-RPLWHM) and q-rung picture linguistic weighted geometric Heronian mean (q-RPLWGHM) operators [16] are calculated and compare the results with the proposed technique. There is no big change in the ranking of each alternative. The best-ranked alternative is the same for both techniques. In Zhang et al. technique [16], the weights attribute should be known, which represents that technique has the least information to find out the ranking of alternatives. But, in our proposed technique, the attributes weight is unknown, and we find it by using the entropy concept, which makes this technique more reasonable and flexible.

Table 7
Comparison of proposed operators with existing operators [16] and [19]

Different operators: Score function values Ranking

q-RPLWHM [16] $Sc (J_{1}) = 5.1918$ , $Sc (J_{2}) = 6.6252$ , $Sc (J_{3}) = 6.8070$ , $Sc (J_{4}) = 3.5026$ $J_{3} > J_{2} > J_{1} > J_{4}$

q-RPLWGHM [16] $Sc (J_{1}) = 2.9013$ , $Sc (J_{2}) = 3.7323$ , $Sc (J_{3}) = 4.6817$ , $Sc (J_{4}) = 1.9923$ $J_{3} > J_{2} > J_{1} > J_{4}$

T-SFFWA [19] $Sc (J_{1}) = - 0.0557$ , $Sc (J_{2}) = - 0.0408$ , $Sc (J_{3}) = 0.3578$ , $Sc (J_{4}) = - 0.1964$ $J_{3} > J_{2} > J_{1} > J_{4}$

T-SFFWG [19] $Sc (J_{1}) = - 0.2181$ , $Sc (J_{2}) = - 0.3292$ , $Sc (J_{3}) = 0.1297$ , $Sc (J_{4}) = - 0.4894$ $J_{3} > J_{1} > J_{2} > J_{4}$

ATS q-RPLNWAA (Proposed operator) $Sc (J_{1}) = 3.2091$ , $Sc (J_{2}) = 4.8412$ , $Sc (J_{3}) = 5.3902$ , $Sc (J_{4}) = 3.3428$ $J_{3} > J_{2} > J_{4} > J_{1}$

ATS q-RPLNWGA (Proposed operator) $Sc (J_{1}) = 2.8467$ , $Sc (J_{2}) = 3.5329$ , $Sc (J_{3}) = 4.5565$ , $Sc (J_{4}) = 1.7488$ $J_{3} > J_{2} > J_{4} > J_{1}$

Different operators:	Score function values	Ranking
q-RPLWHM [16]	$Sc (J_{1}) = 5.1918$ , $Sc (J_{2}) = 6.6252$ , $Sc (J_{3}) = 6.8070$ , $Sc (J_{4}) = 3.5026$	$J_{3} > J_{2} > J_{1} > J_{4}$
q-RPLWGHM [16]	$Sc (J_{1}) = 2.9013$ , $Sc (J_{2}) = 3.7323$ , $Sc (J_{3}) = 4.6817$ , $Sc (J_{4}) = 1.9923$	$J_{3} > J_{2} > J_{1} > J_{4}$
T-SFFWA [19]	$Sc (J_{1}) = - 0.0557$ , $Sc (J_{2}) = - 0.0408$ , $Sc (J_{3}) = 0.3578$ , $Sc (J_{4}) = - 0.1964$	$J_{3} > J_{2} > J_{1} > J_{4}$
T-SFFWG [19]	$Sc (J_{1}) = - 0.2181$ , $Sc (J_{2}) = - 0.3292$ , $Sc (J_{3}) = 0.1297$ , $Sc (J_{4}) = - 0.4894$	$J_{3} > J_{1} > J_{2} > J_{4}$
ATS q-RPLNWAA (Proposed operator)	$Sc (J_{1}) = 3.2091$ , $Sc (J_{2}) = 4.8412$ , $Sc (J_{3}) = 5.3902$ , $Sc (J_{4}) = 3.3428$	$J_{3} > J_{2} > J_{4} > J_{1}$
ATS q-RPLNWGA (Proposed operator)	$Sc (J_{1}) = 2.8467$ , $Sc (J_{2}) = 3.5329$ , $Sc (J_{3}) = 4.5565$ , $Sc (J_{4}) = 1.7488$	$J_{3} > J_{2} > J_{4} > J_{1}$

8.2 Comparison of the proposed technique with Mahnaz et al. [19]

To show the adequacy and legitimacy of the proposed technique in this paper, we compare our proposed technique with Mahnaz et al. [19] technique. In Table 7, the above-illustrated example, with the help of T-spherical fuzzy frank weighted averaging (T-SFFWA) and T-spherical fuzzy weighted geometric (T-SFFWG) operators, are calculated, and compare the results of the proposed technique in this paper with Mahnaz et al. technique [19]. We notice that the ranking of alternatives is almost the same. In Mahnaz et al. technique [19], there is only a quantitative part in the picture fuzzy set, but in our paper, we have more information quantitative part as well as the qualitative part named as linguistic part. We also notice that when we use [19] technique, the score values are almost negative, but in our technique, no score value is negative. This implies that our technique is more suitable and informative.

We can discover from the above examination that our proposed techniques can be effectively applied to decision-making problems. Contrasted and different strategies, our techniques are more adaptable and also reasonable for tending to MADM problems. The benefits and merits of the proposed technique can be finished as follows. First and foremost, the proposed techniques depend on q-RPLSs, and it permits the total and square amount of positive participation degree, unbiased participation degree, and negative enrollment degree to be more noteworthy than one, giving more opportunity for decision-makers to communicate their assessments, and further prompting less data misfortune in the course of MADM. Besides, considering the way that leaders like to make subjective decisions’ quantitative thoughts. Accordingly, q-RPLSs are appropriate and adequate for displaying DMs’ assessments on other options. Thirdly, in most decision-making problems, credits are associated with the goal that the interrelationship between characteristic qualities ought to be considered while intertwining them. Our strategy for MADM is based on q-RPLNWAA or q-RPLNWGA operators, which think about the interrelationship between contentions. In this way, our technique can successfully real MADM issues. In a word, the proposed technique gives DMs another tool to communicate their evaluations and successfully model the course of genuine MADM issues. This way, our strategy is more broad, strong, and more adaptable than different techniques.

8.3 Validation

Spearman’s rank correlation analysis (ρ_r), a nonparametric test used to assess the accuracy of the ranking offered by the MADM techniques (Equation (39)) [24].

$ρ_{r} = 1 - \frac{6 \sum d_{i}^{2}}{m (m^{2} - 1)},$ (39) where m represents the number of alternatives, d_i is the difference between any two MADM methods’ rankings. Table 8 illustrates the meaning of various ρ_r. If the value of ρ_r is more than 0.6, it is obvious from Table 8 that there is a substantial statistical dependence among the considered techniques.

Table 8

Spearman’s rank correlation analysis range

correlation analysis range	Interpretation
0.8 < ρ_r ≤ 1	Very strong
0.6 < ρ_r ≤ 0.8	Strong
0.4 < ρ_r ≤ 0.6	Average
0.2 < ρ_r ≤ 0.4	Weak
ρ_r ≤ 0.2	Very weak

Table 9 displays the rank correlation analysis between the suggested method versus the established selection methods. The correlation coefficient for the majority of techniques is more than 0.7 (except T-SFFWG [19]), as shown in Table 9. We conclude, then, that the ranking supplied by the suggested operators is compatible with the considered?MADM techniques.

Table 9

Spearman’s correlation coefficients

	q-RPLWHM [16]	q-RPLWGHM [16]	T-SFFWA [19]	T-SFFWG [19]	Proposed operators
q-RPLWHM [16]	1	1	1	0.8	0.8
q-RPLWGHM [16]	1	1	1	0.8	0.8
T-SFFWA [19]	1	1	1	0.8	0.8
T-SFFWG [19]	0.8	0.8	0.8	1	0.4
Proposed operators	0.8	0.8	0.8	0.4	1

9 Conclusions

In the present work, various aggregation operators established on the designed ATS operational laws are explored and covered by q-rung picture fuzzy linguistic setting, and it is shown that most of the existing q-rung picture linguistic aggregation operators are special cases. In addition, some required properties of the presented operators are delineated. Following that, an MADM methodology based on the suggested operators is built to address MADM problems with unknown weight information. Lastly, an exemplary case highlights the approach’s flexibility and practical benefits, and the outcome indicates that the technique may give decision-makers more options than ever before. Comparison and sensitive analysis are also accomplished to highlight the supremacy and stability of the established approach.

In the future, we aim to design some advanced aggregation operators for q-RPLS, such as the Maclaurin symmetric mean, weighted prioritized averaging, and weighted prioritized geometric operators, which can be used in MADM applications. The notion of power aggregation operators can also be initiated for q-RPLSs and used in MADM. An exploration of distance and similarity measures is also suggested for future work.

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Generalized q-rung picture linguistic aggregation operators and their application in decision making

Abstract

Keywords

1 Introduction

2 Some basic concepts

4 Generalized q-RPLNs aggregation operator

4.1 q-Rung picture linguistic number weighted averaging aggregation operator

7.1 Sensitivity analysis with respect to parameter q

7.1.1 q-RPLNWAA operator

Table 3 Entropy Matrix H 1 H 2 H 3 H 4 H 5 J 1 0.9339 0.9064 0.9596 0.9164 0.8864 J 2 0.9746 0.9021 0.8707 0.9496 0.9621 J 3 0.9821 0.8939 0.8246 0.8589 0.8246 J 4 0.8797 0.9139 0.8107 0.8707 0.8264

8.1 Comparison of the proposed technique with Zhang et al. [16]

8.3 Validation

References

Table 3
Entropy Matrix

$H_{1}$ $H_{2}$ $H_{3}$ $H_{4}$ $H_{5}$

$J_{1}$ 0.9339 0.9064 0.9596 0.9164 0.8864

$J_{2}$ 0.9746 0.9021 0.8707 0.9496 0.9621

$J_{3}$ 0.9821 0.8939 0.8246 0.8589 0.8246

$J_{4}$ 0.8797 0.9139 0.8107 0.8707 0.8264