1 Introduction

Multi-attribute decision-making (MADM) is a process that can give the ranking results for the finite alternatives according to the attribute values of different alternatives, and it is an important aspect of decision-making sciences. The problem of MADM has received a lot of attention recently, since it is connected to how businesses develop and how society makes decisions in many spheres. In the decision process, an important problem is how to express the attribute value more efficiently and accurately. Decision-makers (DMs) use several methodologies to address MADM problems, such as fuzzy sets (FSs) Zadeh (1965), bipolar complex fuzzy sets (BCFSs) Mahmood et al. (2023), cubic intuitionistic complex fuzzy soft sets (CICFSSs) Mahmood et al. (2022), linear Diophantine uncertain linguistic sets (LDULSs) Mahmood et al. (2021), linear Diophantine fuzzy sets (LDFSs) Yousafzai et al. (2023), and T-spherical linear Diophantine fuzzy sets (T-SLDFSs) Quran (2023). As a generalization of the FS and intuitionistic fuzzy set (IFS) Atanassov (1968), Riaz and Hashmi (2019) introduced the idea of a linear Diophantine fuzzy set (LDFS) in which the satisfaction of an attribute is represented in terms of membership, non-membership, and reference parameters between zero and one. In LDFS, it is observed that each object has three membership degree, membership \(\mu _{A}(a)\), non-membership \(\mu _{A}(a)\), and the reference parameters \(\alpha _{A}, \beta _{A}\) which satisfying the condition \(0\le \alpha _{A}\mu _{A}(a) + \beta _{A} \eta _{A}(a) \le 1\) and \(0\le \alpha _{A}+ \beta _{A} \le 1\) for all \(a \in U\). Because linear Diophantine fuzzy numbers (LDFNs) endow the decision-makers with more flexibility, many researchers devoted themselves to studying linear Diophantine fuzzy decision-making. The linear Diophantine fuzzy numbers (LDFNs) has been widely used in many applications Farid et al. (2022) and has been integrated with other set theories such as spherical linear Diophantine fuzzy sets Alshammari et al. (2022). In 2021, Hashmi et al. (2021) proposed the spherical linear Diophantine fuzzy numbers (SLDFNs) and certain properties of SLDFSs and SLDFNs and the spherical linear Diophantine fuzzy soft rough set (SLDFSRS) and spherical linear Diophantine fuzzy soft approximation space to solve multi-criteria decision-making (MCDM) problem. In addition, Parimala et al. (2021) proposed the concepts of the conditions of optimality in Linear Diophantine fuzzy (LDF) networks for the solution algorithm’s design. Furthermore, (Ayub et al. 2021) generalized the notion of LDFS and defined a new concept of f linear Diophantine fuzzy relation (LDF-relation). Also, (Iampan et al. 2021) proposed some aggregation operators for LDFSs, i.e., linear Diophantine fuzzy Einstein weighted averaging (LDFEWA), linear Diophantine fuzzy Einstein ordered weighted averaging (LDFEOWA), linear Diophantine fuzzy Einstein weighted geometric (LDFEWG), and linear Diophantine fuzzy Einstein ordered weighted geometric (LDFEOWG) operators and discussed their applications in multi-criteria decision-making (MCDM). In Kamacı (2021), Kamacı introduced some algebraic properties of finite linear Diophantine fuzzy subsets of group, ring, and field. They also proposed the linear Diophantine fuzzy subgroup and normal subgroup of a group, linear Diophantine fuzzy subring and ideal of a ring, and linear Diophantine fuzzy subfield of a field. In 2022, (Petchimuthu et al. 2022) developed a multi-criteria decision-making (MCDM) process based on interval-valued linear Diophantine fuzzy weighted average (IVLDFWA) and interval-valued linear Diophantine fuzzy weighted geometric aggregation (IVLDFWGA) operators for fusing interval-valued linear Diophantine fuzzy information. Another generalization of LDFSs was introduced in 2022, when Kamacı (2022) introduced complex linear Diophantine fuzzy set (CLDFS), gave examples on them, and investigated their properties. Later, Tahan et al. (2022) introduced linear Diophantine fuzzy subpolygroups of a polygroup, linear Diophantine fuzzy normal subpolygroups of a polygroup, and linear Diophantine anti-fuzzy subpolygroups of a polygroup as a generalization of fuzzy subpolygroups of a polygroup, fuzzy normal subpolygroups of a polygroup, and anti-fuzzy subpolygroups of a polygroup and presented some examples and results on them. Besides, Mohammad et al. (2022) developed a multi-criteria group decision-making (MCGDM) process based on similarity measures of a Linear Diophantine fuzzy (LDF) information. In Prakash et al. (2022), Prakash et al. presented the special forms of linear Diophantine fuzzy bridges, linear Diophantine fuzzy cut-vertices, linear Diophantine fuzzy cycles, linear Diophantine fuzzy trees, and linear Diophantine fuzzy forests. Based on the proposed basic operations and comparison method for q-rung linear Diophantine fuzzy number (q-RLDFN), Almagrabi et al. (2022) strengthened the classical arithmetic average and geometric average to the q-rung linear Diophantine fuzzy environment and developed two q-rung linear Diophantine fuzzy operators. The q-rung linear Diophantine fuzzy set (q-RLDFS) contains the membership degree and non-membership degree which assure that the sum of qth power of membership \(\mu _{A}(a)\), non-membership \(\mu _{A}(a)\) and the reference parameters \(\alpha _{A}, \beta _{A}\) is not greater than one, i.e., \(0\le \alpha ^{q}_{A}\mu _{A}(a) + \beta ^{q}_{A} \eta _{A}(a) \le 1\) and \(0\le \alpha ^{q}_{A}+ \beta ^{q}_{A} \le 1\) for all \(a \in U\). We can conclude that the q-RLDFS is more precise as the required orthopair capacity raises. In addition, the qRLDFS gives membership degrees greater freedom to communicate their linear Diophantine fuzzy information. They also proposed the q-rung linear Diophantine fuzzy weighted geometric aggregation (q-RLDFWGA), q-RLDFOWGA, and q-RLDFHWGA-operators. In 2023, Ashraf et al. (2023) introduced the spherical q-linear Diophantine fuzzy set (Sq-LDFS), an unique generalization of the LDFS, q-LDFS, and spherical linear Diophantine fuzzy set (SLDFS), and also highlighted its significant aspects. Furthermore, aggregation operators contribute significantly to the efficient aggregation of uncertainty in multiple attribute decision-making (MADM) situations. The Sq-LDFS cover the membership \( \alpha _{A}\), neutral \( \gamma _{A}\) and non-membership \(\beta _{A}\) with control factors \( \alpha _{A}, \gamma _{A}, \beta _{A}\) having restrictions \( \alpha _{A}, \gamma _{A}, \beta _{A} \in [0,1]\) are control factors also \(0\le \alpha ^{p}_{A}\mu _{A}(a) + \gamma ^{q}_{A} \nu _{A}(a) + \beta ^{r}_{A} \eta _{A}(a) \le 1\) and \(0\le \alpha ^{p}_{A}+ \gamma ^{q}_{A} + \beta ^{r}_{A} \le 1\) for all \(a \in U\). It gives us an open choice to select the membership, neutral and non-membership values. Subsequently, Iqbal and Yaqoob (2023) proposed an idea of trapezoidal linear Diophantine fuzzy numbers in general (TrapLDFNs). In (Riaz et al. 2023), developed a multi-criteria decision-making (MCDM) process based on linear Diophantine fuzzy weighted average (LDFWA) and linear Diophantine fuzzy weighted geometric (LDFWG) operators for linear Diophantine fuzzy information. The concepts of LDFSs and q-LDFSs have numerous applications in various fields of real life, but these theories have their own limitations related to the membership, non-membership, and reference parameters. To eradicate these restrictions, we introduce the novel concept of (p, q)-rung linear Diophantine fuzzy set ((p, q)RLDFS). The proposed decision-making model of (p, q)-rung linear Diophantine fuzzy number ((p, q)RLDFN) is more efficient and flexible rather than other approaches due to the use of reference parameters \(\alpha ^{p}_{A}+ \beta ^{q}_{A} \in [0,1]\). The (p, q)RLDFS also categorizes the data in multi-attribute decision-making (MADM) problems by changing the physical sense of reference parameters. The inspiration for the proposed work is addressed in every part of this article. This paper has multiple objectives as follows:

  1. 1.

    To define a new linear Diophantine fuzzy set.

  2. 2.

    To define some properties of (p, q)-rung linear Diophantine fuzzy sets.

  3. 3.

    To define a new score function useful for (p, q)-rung linear Diophantine fuzzy multi-attribute decision-making ((p, q)RLDFMADM) applications.

  4. 4.

    To define a new aggregation operators, i.e., the (p, q)-rung linear Diophantine fuzzy weighted averaging ((p, q)RLDFWA) and (p, q)-rung linear Diophantine fuzzy weighted geometric ((p, q)RLDFWG) operators and study some of its properties.

  5. 5.

    To develop optimization (p, q)RLDFMADM models to determine the weights of attributes.

The layout of this paper is systematized as follows: Sect. 2 reviews some preliminaries regarding intuitionistic fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, n, m-rung orthopair fuzzy sets, linguistic intuitionistic fuzzy variables, linguistic Pythagorean fuzzy numbers, and linguistic q-rung orthopair fuzzy numbers. Section 3 proposes the concepts of a linguistic (p, q)-rung orthopair fuzzy variable and defines some basic operations of linguistic (p, q)-rung orthopair fuzzy variables and the score function of a linguistic (p, q)-rung orthopair fuzzy variable. In Sect. 4, we develop linguistic (p, q)-rung orthopair fuzzy weighted averaging operator for linguistic (p, q)-rung orthopair fuzzy variables and investigate their properties. In Sect. 5, we propose linguistic (p, q)-rung orthopair fuzzy weighted geometric operator and discuss some properties and some special cases of these operators. In Sect. 6, an illustrative example of a company selecting the most suitable green supplier is provided to illustrate the application of the developed method. Finally, the concluding remarks are given in Sect. 7.

2 Notation and background

2.1 Notation

We use the following notation throughout the model:

IFS

Intuitionistic fuzzy set

PFS

Pythagorean fuzzy set

q-ROFS

q-rung orthopair fuzzy set

n, m-ROFS

n, m-rung orthopair fuzzy set

n, m-ROFN

n, m-rung orthopair fuzzy number

LDFS

Linear Diophantine fuzzy set

q-RLDFS

q-rung linear Diophantine fuzzy set

(p, q)RLDFS

(p, q)-rung linear Diophantine fuzzy set

(p, q)RLDFN

(p, q)-rung linear Diophantine fuzzy number

\({\mathcal{S}\mathcal{C}}\)

Score function

\({\mathcal{A}\mathcal{C}}\)

Accuracy function

(p, q)RLDFWA

(p, q)-rung linear Diophantine fuzzy

 

   weighted averaging

\(\mathcal{W}\mathcal{A}\)

(p, q)RLDFWA-operator

(p, q)RLDFWG

(p, q)-rung linear Diophantine fuzzy

 

   weighted geometric

\({\mathcal{W}\mathcal{G}}\)

(p, q)RLDFWG-operator

MADM

Multi-attribute decision-making

(p, q)RLDFMADM

(p, q)-rung linear Diophantine fuzzy

 

   multi-attribute decision-making

2.2 Background

In this section, we recall some basic concepts of intuitionistic fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, n, m-rung orthopair fuzzy sets, linear Diophantine fuzzy sets, and q-rung linear Diophantine fuzzy sets. We utilize these basic components for the construction of a hybrid structure called (p, q)-rung linear Diophantine fuzzy sets.

The concept of intuitionistic fuzzy sets was introduced in Atanassov (1968) as a generalization of the notion of fuzzy sets as follows:

Definition 1

Atanassov (1968) An intuitionistic fuzzy set (IFS) \({\mathcal {A}}\) over a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} : a\in U\right\} , \end{aligned}$$
(1)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership and the degree of non-membership, respectively, and \(\mu _{A}(a) + \eta _{A}(a) \in [0,1]\) for all \(a \in U\).

Remark 1

Let \({\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \nu _{A}(a)\right\} : a\in U\right\} \) be an IFS over a fixed set U. For the sake of simplicity, IFS will be denoted by \({\mathcal {A}} = \left( \mu _{A}, \nu _{A}\right) \).

Definition of Pythagorean fuzzy sets is given in Yager (2014) as follows:

Definition 2

Yager (2014) A Pythagorean fuzzy set (PFS) \({\mathcal {A}}\) over a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} : a\in U\right\} , \end{aligned}$$
(2)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership and the degree of non-membership, respectively, and \(\mu ^{2}_{A}(a) + \eta ^{2}_{A}(a) \in [0,1]\) for all \(a \in U\).

In 2017, Yager (2017) presented the concept of q-rung orthopair fuzzy sets which is the generalization of Pythagorean fuzzy sets as follows:

Definition 3

Yager (2017) A q -rung orthopair fuzzy set (q -ROFS) \({\mathcal {A}}\) on a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} : a\in U\right\} , \end{aligned}$$
(3)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership and the degree of non-membership, respectively, and \(\mu ^{q}_{A}(a) + \eta ^{q}_{A}(a) \in [0,1]\) for all \(a \in U\).

In 2022, Ibrahim and Alshammari (2022) redefined the notion of q-rung orthopair fuzzy set (q-ROFS), as a useful extension of q-ROFSs as follows:

Definition 4

Ibrahim and Alshammari (2022) A n ,  m -rung orthopair fuzzy set (n ,  m -ROFS) \({\mathcal {A}}\) on a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} : a\in U\right\} , \end{aligned}$$
(4)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership and the degree of non-membership, respectively, and \(\mu ^{n}_{A}(a) + \eta ^{m}_{A}(a) \in [0,1]\) for all \(a \in U\).

Remark 2

Let \( {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} : a\in U\right\} \) be a n, m-ROFS on a fixed set U. In particular, if U has only one element, \({\mathcal {A}}\) is called a n ,  m -rung orthopair fuzzy number (n ,  m -ROFN).

The theory of linear Diophantine fuzzy sets, which was initially introduced in 2019 by Riaz and Hashmi in Riaz and Hashmi (2019), has been applied to many mathematical branches.

Definition 5

Riaz and Hashmi (2019) A linear Diophantine fuzzy set (LDFS) \({\mathcal {A}}\) on a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} , \left\{ \alpha _{A}, \beta _{A}\right\} : a\in U\right\} , \end{aligned}$$
(5)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership grades and the degree of non-membership grades, respectively, and \( \alpha _{A}, \beta _{A} \in [0,1]\) are reference parameters also \(0\le \alpha _{A}\mu _{A}(a) + \beta _{A} \eta _{A}(a) \le 1\) and \(0\le \alpha _{A}+ \beta _{A} \le 1\) for all \(a \in U\).

In 2022, Almagrabi et al. (2022) defined the q-rung linear Diophantine fuzzy set as an extension of linear Diophantine fuzzy set and q-rung orthopair fuzzy set as follows.

Definition 6

Almagrabi et al. (2022) A q -rung linear Diophantine fuzzy set (q -RLDFS) \({\mathcal {A}}\) on a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} , \left\{ \alpha _{A}, \beta _{A}\right\} : a\in U\right\} , \end{aligned}$$
(6)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership grades and the degree of non-membership grades, respectively, and \( \alpha _{A}, \beta _{A} \in [0,1]\) are reference parameters also \(0\le \alpha ^{q}_{A}\mu _{A}(a) + \beta ^{q}_{A} \eta _{A}(a) \le 1\) and \(0\le \alpha ^{q}_{A}+ \beta ^{q}_{A} \le 1\) for all \(a \in U\).

Now, we introduce the concepts of t-norms and t-conorms which are defined as follows:

Definition 7

Klir and Yuan (1995) A function \(T: [0,1]\times [0,1]\rightarrow [0,1]\) is called a t -norm (TN) if it satisfies the following four conditions:

  1. 1.

    \(T\left( 1, x\right) = x\) for every x of [0, 1].

  2. 2.

    \(T\left( x, y\right) = T\left( y, x\right) \) for every x and y of [0, 1].

  3. 3.

    \(T\left( x, T\left( y, z\right) \right) = T\left( T\left( x, y\right) , z\right) \) for every x, y and z of [0, 1].

  4. 4.

    If \(w\le x\) and \(y\le z\) , then \(T\left( w, y\right) \le T\left( x, z\right) \).

A TN T is called Archimedean t -norm if it is continuous and \(T(x,x) < x\) for every x of [0, 1]

Definition 8

Klir and Yuan (1995) A function \(T^{*}: [0,1]\times [0,1] \rightarrow [0,1]\) is called a t -conorm (TC) if it satisfies the following four conditions:

  1. 1.

    \(T^{*}\left( 0, x\right) = x\) for all x.

  2. 2.

    \(T^{*}\left( x, y\right) = T^{*}\left( y, x\right) \) for all x and y.

  3. 3.

    \(T^{*}\left( x, T^{*}\left( y, z\right) \right) = T^{*}\left( T^{*}\left( x, y\right) , z\right) \) for all x, y and z.

  4. 4.

    If \(w\le x\) and \(y\le z\) , then \(T^{*}\left( w, y\right) \le T^{*}\left( x, z\right) \).

A TC \(T^{*}\) is called Archimedean t -conorm if it is continuous and \(T^{*}(x,x) < x\) for every x of [0, 1]

3 (p, q)-Rung linear Diophantine fuzzy sets

In this section, we describe the concepts of (p, q)-rung linear Diophantine fuzzy set and (p, q)-rung linear Diophantine fuzzy number, and then investigate their operational laws.

The (p, q)-rung linear Diophantine fuzzy sets are generated with the help of n, m-ROFSs and q-RLDFSs. Therefore, we can say that (p, q)-rung linear Diophantine fuzzy set is the generalization of previously defined concepts related to linear Diophantine fuzzy set theory. The (p, q)-rung linear Diophantine fuzzy set gives more opportunities to deal with uncertainty in data.

Definition 9

A (p ,  q)-rung linear Diophantine fuzzy set ((p ,  q)RLDFS) \({\mathcal {A}}\) on a fixed set U is an object having the form

$$\begin{aligned} {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} , \left\{ \alpha _{A}, \beta _{A}\right\} : a\in U\right\} , \end{aligned}$$
(7)

where the functions \(\mu _{A}: U\rightarrow [0,1]\) and \(\eta _{A}: U\rightarrow [0,1]\) denote the degree of membership grades and the degree of non-membership grades, respectively, and \( \alpha _{A}, \beta _{A} \in [0,1]\) are reference parameters also \(\alpha ^{p}_{A}\mu _{A}(a) + \beta ^{q}_{A} \eta _{A}(a) \in [0,1]\) and \(\alpha ^{p}_{A}+ \beta ^{q}_{A} \in [0,1]\) for all \(a \in U\).

Remark 3

By Definition 9, we observe that (Almagrabi et al. 2022) q-RLDFSs are (p, q)RLDFSs with \(p = q\).

Let \( {\mathcal {A}} = \left\{ \left\{ a, \mu _{A}(a), \eta _{A}(a)\right\} , \left\{ \alpha _{A}, \beta _{A}\right\} : a\in U\right\} \) be a (p, q)RLDFS on a fixed set U. In particular, if U has only one element, \({\mathcal {A}}\) is called a (p ,  q)-rung linear Diophantine fuzzy number ((p ,  q)RLDFN). For convenience, a (p, q)RLDFN is denoted by \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) ,\left( \alpha _{A}, \beta _{A}\right) \right) \). Moreover, we denote by \(\mathcal{R}\mathcal{L}(U)\) the collection of (p, q)RLDFNs on a on a fixed set U with \(\mathbb {U} = \left( \left( 1,0\right) ,\left( 1,0 \right) \right) \) and \(\emptyset = \left( \left( 0,1 \right) ,\left( 0,1 \right) \right) \).

Example 1

Suppose that \( \mu _{A} = 0.7, \eta _{A} = 1, \alpha _{A} = 0.9\), \(\beta _{A} = 0.7\) and \(U = \left\{ a \right\} \). Hence, \((0.9)(0.7) + (0.7) (1) = 1.33 > 1\), \(0.9 + 0.7 = 1.6 > 1\), \((0.9)^{2}(0.7 ) + (0.7)^{2}(1) = 1.057 > 1\), \((0.9)^{2} + (0.7)^{2} = 1.3 >1\), \((0.9)^{3}(0.7 ) + (0.7)^{3}(1) = 0.8533 < 1\), \((0.9)^{3} + (0.7)^{3} = 1.072 >1\), but \((0.9)^{3}(0.7 ) + (0.7)^{4}(1) = 0.7504 < 1\), \((0.9)^{3} + (0.7)^{4} = 0.9691 < 1\), and \((0.9)^{4}(0.7 ) + (0.7)^{3}(1) = 0.80227 < 1\), \((0.9)^{4} + (0.7)^{3} = 0.9991 < 1\). Therefore, \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) ,\left( \alpha _{A}, \beta _{A}\right) \right) \) is both (3, 4)RLDFN and (4, 3)RLDFN, but \({\mathcal {A}}\) is neither LDFN nor 2-RLDFN nor 3-RLDFN.

For this relation of less than and equality are defined as follows:

Definition 10

For any two (p, q)RLDFNs \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) and \(\mathcal {B} = \left( \left( \mu _{B}, \eta _{B} \right) ,\left( \alpha _{B}, \beta _{B}\right) \right) \) on a fixed set U, the corresponding operations are defined as follows:

  1. 1.

    \(\left( \mu _{A}, \eta _{A} \right) \preceq \left( \mu _{B}, \eta _{B} \right) \) if and only if \(\mu _{A} \le \mu _{B}\) and \(\eta _{A}\ge \eta _{B}\),

  2. 2.

    \(\left( \alpha _{A}, \beta _{A} \right) \preceq \left( \alpha _{B}, \beta _{B} \right) \) if and only if \(\alpha _{A} \le \alpha _{B}\) and \(\beta _{A}\ge \beta _{B}\),

  3. 3.

    \({\mathcal {A}}\preceq \mathcal {B}\) if and only if \(\left( \mu _{A}, \eta _{A} \right) \preceq \left( \mu _{B}, \eta _{B} \right) \) and \(\left( \alpha _{A}, \beta _{A} \right) \preceq \left( \alpha _{B}, \beta _{B} \right) \),

  4. 4.

    \({\mathcal {A}} = \mathcal {B}\) if and only if \({\mathcal {A}}\preceq \mathcal {B}\) and \(\mathcal {B}\preceq {\mathcal {A}}\).

Motivated by the operations of the LDFNs and q-RLDFNs, in the following, we shall define some operational laws of (p, q)RLDFNs.

Definition 11

Let \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) and \(\mathcal {B} = \left( \left( \mu _{B}, \eta _{B} \right) ,\left( \alpha _{B}, \beta _{B}\right) \right) \) be any (p, q)RLDFNs on a fixed set U. For every element \(\lambda \in \left( 0,\infty \right) \), then their operations are defined as follows:

  1. 1.

    \(\left( \mu _{A}, \eta _{A} \right) \oplus \left( \mu _{B}, \eta _{B} \right) = \left( \root p \of { \left( \mu _{A}\right) ^{p} + \left( \mu _{B}\right) ^{p} - \left( \mu _{A}\right) ^{p}\left( \mu _{B}\right) ^{p}}, \eta _{A}\eta _{B} \right) \),

  2. 2.

    \(\left( \alpha _{A}, \beta _{A}\right) \oplus \left( \alpha _{B}, \beta _{B}\right) = \left( \root p \of { \left( \alpha _{A}\right) ^{p} + \left( \alpha _{B}\right) ^{p} - \left( \alpha _{A}\right) ^{p}\left( \alpha _{B}\right) ^{p}}, \beta _{A}\beta _{B} \right) \),

  3. 3.

    \(\left( \mu _{A}, \eta _{A} \right) \otimes \left( \mu _{B}, \eta _{B} \right) = \left( \mu _{A}\mu _{B}, \root q \of { \left( \eta _{A}\right) ^{q} + \left( \eta _{B}\right) ^{q} - \left( \eta _{A}\right) ^{q}\left( \eta _{B}\right) ^{q}}\right) \),

  4. 4.

    \(\left( \alpha _{A}, \beta _{A}\right) \otimes \left( \alpha _{B}, \beta _{B}\right) = \left( \alpha _{A}\alpha _{B}, \root q \of { \left( \beta _{A}\right) ^{q} + \left( \beta _{B}\right) ^{q} - \left( \beta _{A}\right) ^{q}\left( \beta _{B}\right) ^{q}}\right) \),

  5. 5.

    \(\lambda \left( \mu _{A}, \eta _{A} \right) = \left( \root p \of { 1 - \left( 1 - \left( \mu _{A}\right) ^{p} \right) ^{\lambda }}, \left( \eta _{A}\right) ^{\lambda } \right) \),

  6. 6.

    \(\lambda \left( \alpha _{A}, \beta _{A}\right) = \left( \root p \of { 1 - \left( 1 - \left( \alpha _{A}\right) ^{p} \right) ^{\lambda }}, \left( \beta _{A}\right) ^{\lambda } \right) \),

  7. 7.

    \(\left( \mu _{A}, \eta _{A} \right) ^{\lambda } = \left( \left( \mu _{A}\right) ^{\lambda }, \root q \of { 1 - \left( 1 - \left( \eta _{A}\right) ^{q} \right) ^{\lambda }} \right) \),

  8. 8.

    \(\left( \alpha _{A}, \beta _{A}\right) ^{\lambda } = \left( \left( \alpha _{A}\right) ^{\lambda }, \root q \of { 1 - \left( 1 - \left( \beta _{A}\right) ^{q} \right) ^{\lambda }} \right) \),

  9. 9.

    \({\mathcal {A}} \oplus \mathcal {B} = \left( \left( \mu _{A}, \eta _{A} \right) \oplus \left( \mu _{B}, \eta _{B} \right) , \left( \alpha _{A}, \beta _{A}\right) \oplus \left( \alpha _{B}, \beta _{B}\right) \right) \),

  10. 10.

    \({\mathcal {A}} \otimes \mathcal {B} = \left( \left( \mu _{A}, \eta _{A} \right) \otimes \left( \mu _{B}, \eta _{B} \right) , \left( \alpha _{A}, \beta _{A}\right) \otimes \left( \alpha _{B}, \beta _{B}\right) \right) \),

  11. 11.

    \(\lambda {\mathcal {A}} =\left( \lambda \left( \mu _{A}, \eta _{A} \right) , \lambda \left( \alpha _{A}, \beta _{A}\right) \right) \),

  12. 12.

    \({\mathcal {A}}^{\lambda } = \left( \left( \mu _{A}, \eta _{A} \right) ^{\lambda }, \left( \alpha _{A}, \beta _{A}\right) ^{\lambda } \right) \).

It can be easily proved that the (p, q)RLDFN has the following properties.

Theorem 1

Let \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) , \mathcal {B} = \left( \left( \mu _{B}, \eta _{B} \right) ,\left( \alpha _{B}, \beta _{B}\right) \right) \) and \(\mathcal {C} = \left( \left( \mu _{C}, \eta _{C} \right) ,\left( \alpha _{C}, \beta _{C}\right) \right) \) be any (p, q)RLDFNs on a fixed set U. For every elements \(\lambda , \xi \in \left( 0,\infty \right) \), the following properties hold:

  1. 1.

    \({\mathcal {A}}\oplus \mathcal {B} = \mathcal {B}\oplus {\mathcal {A}}\).

  2. 2.

    \({\mathcal {A}}\otimes \mathcal {B} = \mathcal {B}\otimes {\mathcal {A}}\).

  3. 3.

    \(\left( {\mathcal {A}}\oplus \mathcal {B}\right) \oplus \mathcal {C} = {\mathcal {A}}\oplus \left( \mathcal {B} \oplus \mathcal {C}\right) \).

  4. 4.

    \(\left( {\mathcal {A}}\otimes \mathcal {B}\right) \otimes \mathcal {C} = {\mathcal {A}}\otimes \left( \mathcal {B}\otimes \mathcal {C} \right) \).

  5. 5.

     \(\lambda \left( {\mathcal {A}}\oplus \mathcal {B}\right) = \lambda {\mathcal {A}} \oplus \lambda \mathcal {B}\).

  6. 6.

    \(\left( {\mathcal {A}}\otimes \mathcal {B}\right) ^{\lambda } = {\mathcal {A}}^{\lambda }\otimes \mathcal {B}^{\lambda }\).

  7. 7.

    \(\left( \lambda + \xi \right) {\mathcal {A}} = \lambda {\mathcal {A}}\oplus \xi {\mathcal {A}}\).

  8. 8.

    \({\mathcal {A}}^{\lambda + \xi } = {\mathcal {A}}^{\lambda } \otimes {\mathcal {A}}^{\xi }\).

  9. 9.

    \(\left( {\mathcal {A}}^{\lambda }\right) ^{\xi } = {\mathcal {A}}^{\lambda \xi }\).

For applications of (p, q)RLDFNs in decision-making problems, we need to know how to rank the (p, q)RLDFNs. For ranking a (p, q)RLDFN, we need to calculate the score and accuracy of that (p, q)RLDFN as follows defined in the next definition.

Definition 12

Let \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) be a (p, q)RLDFN on a fixed set U. Then, the score function (accuracy function) of \({\mathcal {A}}\) can be defined as follows:

$$\begin{aligned}{} & {} {\mathcal{S}\mathcal{C}}\left( {\mathcal {A}} \right) = \frac{1 }{2}\left( \mu _{A} - \eta _{A} + \left( \alpha _{A}\right) ^{p} - \left( \beta _{A} \right) ^{q} \right) \end{aligned}$$
(8)
$$\begin{aligned}{} & {} {\mathcal{A}\mathcal{C}}\left( {\mathcal {A}} \right) = \frac{1 }{4}\left( \mu _{A} + \eta _{A}\right) + \frac{1 }{2}\left( \left( \alpha _{A}\right) ^{p} + \left( \beta _{A} \right) ^{q} \right) . \end{aligned}$$
(9)

Remark 4

It is clear that if membership grades \(\mu _{A} (\alpha _{A})\) are bigger, and the non-membership grades \(\eta _{A} (\beta _{A})\) are smaller, then the score value of the (p, q)RLDFN \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) is greater.

It is noted that the score function (accuracy function) \({\mathcal{S}\mathcal{C}}\left( {\mathcal {A}}\right) \left( {\mathcal{A}\mathcal{C}}\left( {\mathcal {A}}\right) \right) \) has some desirable properties as below.

Theorem 2

Let \({\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) be a (p, q)RLDFN on a fixed set U. Then, the following properties hold.

  1. 1.

    \(-1\le {\mathcal{S}\mathcal{C}}\left( {\mathcal {A}} \right) \le 1\).

  2. 2.

    \(0\le {\mathcal{A}\mathcal{C}}\left( {\mathcal {A}} \right) \le 1\).

  3. 3.

    \({\mathcal{S}\mathcal{C}}\left( \mathcal {\mathbb {U}} \right) = 1\) (one property).

  4. 4.

    \({\mathcal{S}\mathcal{C}}\left( \emptyset \right) = -1\) (zero property).

  5. 5.

    \({\mathcal{A}\mathcal{C}}\left( \mathcal {\mathbb {U}} \right) = 1\) (one property).

  6. 6.

    \({\mathcal{A}\mathcal{C}}\left( \emptyset \right) = 1\) (zero property).

Proof

The proof is obvious.

4 (p, q)-Rung linear Diophantine fuzzy weighted averaging operators

In this section, we shall develop some weighted averaging operators with (p, q)RLDFN, such as (p, q)-rung linear Diophantine fuzzy weighted averaging operator.

We now discuss some aggregation operators based on operational rules of (p, q)RLDFNs. Thus, from the above definition, it is clear that (p, q)-rung linear Diophantine fuzzy weighted averaging operators are a generalization of a q-rung linear Diophantine fuzzy weighted averaging operator.

Definition 13

Let \(\Delta = \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U and \(\lambda = \left( \lambda _{1},\lambda _{2},\dots , \lambda _{n}\right) ^{T}\) be the weight vector of \(\Delta \) indicates the importance degree of \(\Delta \) satisfying \(\lambda _{1},\lambda _{2},\dots , \lambda _{n}\in \left[ 0,1\right] \) and \(\sum ^{n}_{i = 1}\lambda _{i} = 1\), and let (p ,  q)-rung linear Diophantine fuzzy weighted averaging ((p ,  q)RLDFWA) \(\mathcal{W}\mathcal{A}: \left( \mathcal{R}\mathcal{L}(U)\right) ^{n}\rightarrow \mathcal{R}\mathcal{L}(U)\) if

$$\begin{aligned} \mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots , {\mathcal {A}}_{n} \right) = \displaystyle \bigoplus ^{n}_{i = 1}\lambda _{i}{\mathcal {A}}_{i}, \end{aligned}$$
(10)

then the function (p, q)RLDFWA is called the (p ,  q)-rung linear Diophantine fuzzy weighted averaging operator ((p ,  q)RLDFWA-operator).

On the basis of the operational rules of the (p, q)RLDFNs on U, we can get the aggregation result described as Theorem 3.

Theorem 3

Let \(\Delta = \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U and \(\lambda = \left( \lambda _{1},\lambda _{2},\dots , \lambda _{n}\right) ^{T}\) be the weight vector of \(\Delta \), where \(\lambda _{j}\) indicates the importance degree of \(\Delta \), satisfying \(0\le \lambda _{1},\lambda _{2},\dots , \lambda _{n}\le 1\) and \(\sum ^{n}_{i = 1}\lambda _{i} = 1\). Then, their accumulated outcome utilizing the (p, q)RLDFWA-operator is again (p, q)RLDFN and

$$\begin{aligned} \mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n} \right) = \left( \begin{array}{ccc} \left( \begin{array}{ccc} \root p \of { 1 - \displaystyle \prod ^{n}_{i = 1}\left( 1- \left( \mu _{A_{i}}\right) ^{p} \right) ^{\lambda _{i}}},&{} \\ \displaystyle \prod ^{n}_{i = 1}\left( \eta _{A_{i}}\right) ^{\lambda _{i}}&{} \end{array}\right) , &{} \\ \left( \begin{array}{ccc} \root p \of { 1 - \displaystyle \prod ^{n}_{i = 1}\left( 1- \left( \alpha _{A_{i}}\right) ^{p}\right) ^{\lambda _{i}}},&{} \\ \displaystyle \prod ^{n}_{i = 1}\left( \beta _{A_{i}} \right) ^{\lambda _{i}}&{} \end{array}\right)&\end{array}\right) . \end{aligned}$$
(11)

From Theorem 3, it is observed that our proposed (p, q)RLDFWA-operator satisfies idempotency, boundedness, and monotonicity properties which are as follows:

Theorem 4

(Idempotency) Let \( \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U. If \({\mathcal {A}}_{i} ={\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) for every \(i=1,2,\ldots , n\), then \(\mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots , {\mathcal {A}}_{n} \right) = {\mathcal {A}}\).

It can be easily proved that the (p, q)RLDFWA-operator has the following properties.

Corollary 1

Let \(\Delta = \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U. Then, the following properties hold.

  1. 1.

    If \({\mathcal {A}}_{i} = \emptyset \) for every \(i = 1,2,3,\ldots , n\), then \(\mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n} \right) = \emptyset \).

  2. 2.

    If \({\mathcal {A}}_{i} = \mathcal {\mathbb {U}} \) for every \(i = 1,2,3,\ldots , n\), then \(\mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots , {\mathcal {A}}_{n} \right) = \mathbb {U}\).

Theorem 5 establishes some properties of (p, q)RLDFWA-operator.

Theorem 5

(Monotonicity) Let \(\left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) and \(\left\{ \mathcal {B}_{i} = \left( \left( \mu _{B_{i}}, \eta _{B_{i}} \right) , \left( \alpha _{B_{i}}, \beta _{B_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be any collections of (p, q)RLDFNs on a fixed set U. If \({\mathcal {A}}_{i} \preceq \mathcal {B}_{i}\) for every \(i=1,2,3,\ldots , n\), then \(\mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1}, \ldots ,{\mathcal {A}}_{n} \right) \preceq \mathcal{W}\mathcal{A}\left( \mathcal {B}_{1}, \ldots , \mathcal {B}_{n} \right) \).

Now, we prove properties of (p, q)RLDFWA-operator.

Theorem 6

(Boundedness) Let \(\left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U. If

$$\begin{aligned} {\mathcal {A}}^{+} = \left( \left( \displaystyle \bigvee ^{n}_{i=1}\mu _{A_{i}}, \displaystyle \bigwedge ^{n}_{i=1}\eta _{A_{i}} \right) , \left( \displaystyle \bigvee ^{n}_{i=1}\alpha _{A_{i}}, \displaystyle \bigwedge ^{n}_{i=1}\beta _{A_{i}} \right) \right) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {A}}^{-} = \left( \left( \displaystyle \bigwedge ^{n}_{i=1}\mu _{A_{i}}, \displaystyle \bigvee ^{n}_{i=1}\eta _{A_{i}} \right) , \left( \displaystyle \bigwedge ^{n}_{i=1}\alpha _{A_{i}}, \displaystyle \bigvee ^{n}_{i=1}\beta _{A_{i}} \right) \right) , \end{aligned}$$

then \({\mathcal {A}}^{-} \preceq \mathcal{W}\mathcal{A}\left( {\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n}\right) \preceq {\mathcal {A}}^{+}\).

5 (p, q)-Rung linear Diophantine fuzzy weighted geometric operators

In the following, we shall propose the concepts and basic operations of the (p, q)-rung linear Diophantine fuzzy weighted geometric operator on the basis of the (p, q)RLDFNs.

The following definition describes the (p, q)-rung linear Diophantine fuzzy weighted geometric operators.

Definition 14

Let \(\Delta = \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U and \(\lambda = \left( \lambda _{1},\lambda _{2},\dots , \lambda _{n}\right) ^{T}\) be the weight vector of \(\Delta \), where \(\lambda _{j}\) indicates the importance degree of \(\Delta \), satisfying \(\lambda _{1},\lambda _{2},\dots , \lambda _{n}\in \left[ 0,1\right] \) and \(\displaystyle \sum \nolimits ^{n}_{i = 1}\lambda _{i} = 1\), and let (p ,  q)-rung linear Diophantine fuzzy weighted geometric ((p ,  q)RLDFWG) \({\mathcal{W}\mathcal{G}}: \left( \mathcal{R}\mathcal{L}(U)\right) ^{n}\rightarrow \mathcal{R}\mathcal{L}(U)\) if

$$\begin{aligned} {\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n} \right) = \displaystyle \bigotimes ^{n}_{i = 1}{\mathcal {A}}^{\lambda _{i}}_{i}, \end{aligned}$$
(12)

then the function (p, q)RLDFWG is called the (p ,  q)-rung linear Diophantine fuzzy weighted geometric operator ((p ,  q)RLDFWG-operator).

The general expression of the (p, q)RLDFWG-operator is constructed in the following theorem:

Theorem 7

Let \(\Delta = \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U and \(\lambda = \left( \lambda _{1},\lambda _{2},\dots , \lambda _{n}\right) ^{T}\) be the weight vector of \(\Delta \), where \(\lambda _{j}\) indicates the importance degree of \(\Delta \), satisfying \(\lambda _{1},\lambda _{2},\dots , \lambda _{n}\in \left[ 0,1\right] \) and \(\displaystyle \sum \nolimits ^{n}_{i = 1}\lambda _{i} = 1\). Then, their accumulated outcome utilizing the (p, q)RLDFWG-operator is again (p, q)RLDFN, and

$$\begin{aligned} {\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n} \right) = \left( \begin{array}{ccc} \left( \begin{array}{ccc} \displaystyle \prod ^{n}_{i = 1}\left( \mu _{A_{i}} \right) ^{\lambda _{i}},&{}\\ \root q \of { 1 - \displaystyle \prod ^{n}_{i = 1}\left( 1- \left( \eta _{A_{i}}\right) ^{q} \right) ^{\lambda _{i}}}&{} \end{array}\right) ,&{}\\ \left( \begin{array}{ccc} \displaystyle \prod ^{n}_{i = 1}\left( \alpha _{A_{i}} \right) ^{\lambda _{i}},&{}\\ \root q \of { 1 - \displaystyle \prod ^{n}_{i = 1}\left( 1- \left( \beta _{A_{i}}\right) ^{q} \right) ^{\lambda _{i}}}&{} \end{array}\right)&\end{array}\right) . \end{aligned}$$
(13)

From Theorem 7, it is observed that our proposed (p, q)RLDFWG-operator satisfies idempotency, boundedness, and monotonicity properties which are as follows:

Theorem 8

(Idempotency) Let \( \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U. If \({\mathcal {A}}_{i} = {\mathcal {A}} = \left( \left( \mu _{A}, \eta _{A} \right) , \left( \alpha _{A}, \beta _{A}\right) \right) \) for every \( i=1,2,\ldots , n\), then \({\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n} \right) = {\mathcal {A}}\).

It can be easily proved that the (p, q)RLDFWG-operator has the following properties.

Corollary 2

Let \(\left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U. Then, the following properties hold.

  1. 1.

    If \({\mathcal {A}}_{i} = \emptyset \) for every \(i = 1,2,3,\ldots , n\), then \({\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n} \right) = \emptyset \).

  2. 2.

    If \({\mathcal {A}}_{i} = \mathcal {\mathbb {U}} \) for every \(i = 1,2,3,\ldots , n\), then \({\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1}, {\mathcal {A}}_{2},\ldots , {\mathcal {A}}_{n} \right) = \mathbb {U}\).

The properties of the (p, q)RLDFWG-operator are stated in the following theorem:

Theorem 9

(Monotonicity) Let \( \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) and \( \left\{ \mathcal {B}_{i} = \left( \left( \mu _{B_{i}}, \eta _{B_{i}} \right) , \left( \alpha _{B_{i}}, \beta _{B_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be any collections of (p, q)RLDFNs on a fixed set U. If \({\mathcal {A}}_{i} \preceq \mathcal {B}_{i}\) for every \(i=1,2,\ldots , n\), then \({\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n} \right) \preceq {\mathcal{W}\mathcal{G}}\left( \mathcal {B}_{1},\ldots ,\mathcal {B}_{n} \right) \).

The following Theorem 10 establishes some properties of (p, q)RLDFWG-operator.

Theorem 10

(Boundedness) Let \( \left\{ {\mathcal {A}}_{i} = \left( \left( \mu _{A_{i}}, \eta _{A_{i}} \right) , \left( \alpha _{A_{i}}, \beta _{A_{i}}\right) \right) : i = 1,2,3,\dots , n \right\} \) be a collection of (p, q)RLDFNs on a fixed set U. If

$$\begin{aligned} {\mathcal {A}}^{+} = \left( \left( \displaystyle \bigvee ^{n}_{i=1}\mu _{A_{i}}, \displaystyle \bigwedge ^{n}_{i=1}\eta _{A_{i}} \right) , \left( \displaystyle \bigvee ^{n}_{i=1}\alpha _{A_{i}}, \displaystyle \bigwedge ^{n}_{i=1}\beta _{A_{i}} \right) \right) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {A}}^{-} = \left( \left( \displaystyle \bigwedge ^{n}_{i=1}\mu _{A_{i}}, \displaystyle \bigvee ^{n}_{i=1}\eta _{A_{i}} \right) , \left( \displaystyle \bigwedge ^{n}_{i=1}\alpha _{A_{i}}, \displaystyle \bigvee ^{n}_{i=1}\beta _{A_{i}} \right) \right) , \end{aligned}$$

then \({\mathcal {A}}^{-} \preceq {\mathcal{W}\mathcal{G}}\left( {\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{n}\right) \preceq {\mathcal {A}}^{+}\).

6 A new multi-attribute decision-making method for (p, q)RLDFNs based on the proposed operators

In this section, we define a multi-attribute decision-making (MADM) method, the so-called (p, q)-rung linear Diophantine fuzzy multi-attribute decision-making ((p, q)RLDFMADM) method. Its adopted from Almagrabi et al. (2022).

In the DM problems, we propose a (p, q)RLDFMADM approach using (p, q)-rung linear Diophantine fuzzy aggregation operators in which weights of the real numbers under (p, q)-rung linear Diophantine fuzzy environment. Here, (p, q)RLDFMADM method is used to find a favorable alternative for a selecting the most suitable supplier. The set \(A = \left\{ A_{1}, A_{2}, A_{3}, \dots , A_{m}\right\} \) contains the numbers of alternatives, while \(G = \left\{ G_{1}, G_{2},\dots , G_{n}\right\} \) represents the numbers of attributes. Let \(\lambda = \left( \lambda _{1},\lambda _{2},\dots , \lambda _{n}\right) ^{T}\) be the weight vector met the criteria as \(0\le \lambda _{1},\lambda _{2},\dots , \lambda _{n}\le 1\) and \(\displaystyle \sum \nolimits ^{n}_{i = 1}\lambda _{i} = 1\). A set of decision-makers (DMs) \(D = \left\{ D_{1}, D_{2}, D_{3}, \dots , D_{k}\right\} \) are invited to appraisal all the alternatives. The expert’s decisions are stated as (p, q)RLDFNs, i.e., \( \Bigg \{ {\mathcal {A}}_{ij} = \left( \left( \mu _{A_{ij}}, \eta _{A_{ij}} \right) , \left( \alpha _{A_{ij}}, \beta _{A_{ij}}\right) \right) : i = 1,2,3,\dots , m, j = 1,2,3,\dots , n \Bigg \} \), such that \(0\le \alpha ^{p}_{A_{ij}}\mu _{A_{ij}} +\beta ^{q}_{A_{ij}} \eta _{A_{ij}} \le 1\) and \(0\le \alpha ^{p}_{A_{ij}}+ \beta ^{q}_{A_{ij}} \le 1\). Thus, the (p, q)-rung linear Diophantine fuzzy decision matrix is given as \(D = \left[ \left( \left( \mu _{A_{ij}}, \eta _{A_{ij}} \right) , \left( \alpha _{A_{ij}}, \beta _{A_{ij}}\right) \right) \right] _{m\times n}\) and presented as follows:

$$\begin{aligned} D = \left[ \begin{array}{ccccc} {\mathcal {A}}_{11} &{} {\mathcal {A}}_{12} &{}\ldots &{} {\mathcal {A}}_{1n} &{} \\ {\mathcal {A}}_{21}&{} {\mathcal {A}}_{22} &{}\ldots &{} {\mathcal {A}}_{2n} &{} \\ \vdots &{} &{} &{} &{} \\ {\mathcal {A}}_{m1} &{} {\mathcal {A}}_{m2} &{}\ldots &{} {\mathcal {A}}_{mn} &{} \\ \end{array} \right] . \end{aligned}$$
(14)

The algorithm follows a (p, q)RLDFMADM method to interpret a (p, q)RLDFMCDM problem under (p, q)-rung linear Diophantine fuzzy information using (p, q)RLDFWA and (p, q)RLDFWG-operators.

Algorithm

Input: (p, q)-rung linear Diophantine fuzzy information

Output: Best alternative

Step 1. Calculate the (p, q)-rung linear Diophantine fuzzy decision matrix.

Step 2. To eliminate the influence of attribute type, we consider the following technique and obtain the standardize (p, q)-rung linear Diophantine fuzzy matrix \(R = \left[ \mathcal {R}_{ij}\right] _{m\times n} \), where \(\mathcal {R}_{ij} = {\mathcal {A}}_{ij} = \left( \left( \mu _{A_{ij}}, \eta _{A_{ij}} \right) , \left( \alpha _{A_{ij}}, \beta _{A_{ij}}\right) \right) \) is (p, q)RLDFN. Then, we have

$$\begin{aligned} \mathcal {R}_{ij} = \left\{ \begin{array}{ll} {\mathcal {A}}_{ij} &{} \text {; for same type input data} \\ {\mathcal {A}}^{c}_{ij} &{} \text {; for different type input data}. \end{array} \right. \end{aligned}$$
(15)

Step 3. Based on the standardize (p, q)-rung linear Diophantine fuzzy matrix, as obtained from step 2, the overall aggregated value of the alternative \(A_{i}\) for all \(i = 1, 2,\ldots , m\), under the different criteria \(G_{j}\) for all \(j = 1, 2,\ldots , n\) is obtained using either

$$\begin{aligned} r_{i} = \mathcal{W}\mathcal{A}\left( \mathcal {R}_{i1},\mathcal {R}_{i2},\ldots ,\mathcal {R}_{in} \right) \end{aligned}$$
(16)

or

$$\begin{aligned} r_{i} = {\mathcal{W}\mathcal{G}}\left( \mathcal {R}_{i1},\mathcal {R}_{i2},\ldots ,\mathcal {R}_{in}\right) \end{aligned}$$
(17)

operator and hence get the collective value \(r_{i}\) for each alternative \(A_{i}\) for all \(i = 1, 2,\ldots , m\).

Step 4. Next, the score values of \({\mathcal{S}\mathcal{C}}\left( r_{i}\right) \) \((i = 1, 2, 3,\ldots , m)\) of the overall (p, q)RLDFNs of \(r_{i} (i = 1, 2, 3,\ldots , m)\) is obtained to rank the alternatives \(A_{i} (i = 1, 2,\ldots ,m)\). If the score values of \({\mathcal{S}\mathcal{C}}\left( r_{i}\right) \) and \({\mathcal{S}\mathcal{C}}\left( r_{j}\right) \) are equal for two alternatives \(A_{i}\) and \(A_{j}\), then it is required to calculate accuracy degrees of \({\mathcal{A}\mathcal{C}}\left( r_{i}\right) \) and \({\mathcal{A}\mathcal{C}}\left( r_{j}\right) \) with respect to the overall collective (p, q)RLDFNs to rank the alternatives \(A_{i}\) and \(A_{j}\), respectively, based on the aforementioned accuracy degrees \({\mathcal{A}\mathcal{C}}\left( r_{i}\right) \) and \({\mathcal{A}\mathcal{C}}\left( r_{j}\right) \).

Step 5. We select the best alternative from the rankings of all alternatives \(A_{i}\) \((i = 1, 2,\ldots , m)\) according to \({\mathcal{S}\mathcal{C}}\left( r_{i}\right) (i = 1, 2,\ldots , m)\).

Step 6.: End.

An illustrative example: We consider a (p, q)RLDFMADM problem given below. To illustrate the proposed (p, q)RLDFMADM strategy, we solve a (p, q)RLDFMADM problem adapted from Almagrabi et al. (2022).

We are suggesting a novel emergency public health solution to determine the best attribute under (p, q)-rung linear Diophantine fuzzy information. We are applying our input data over this algorithm. Three health experts are appointed to evaluate the four alternatives for emergencies \(A = \left\{ A_{1}, A_{2}, A_{3}, A_{4}\right\} \) with respect to five attributes \(G = \left\{ G_{1}, G_{2},G_{3}, G_{4}\right\} \), and the decision matrices \(D = \left\{ D_{1}, D_{2}, D_{3}\right\} \) are constructed, as shown in Table 1. Suppose that the four experts’risk attitudes are \(\lambda _{1} = 0.2, \lambda _{2} = 0.3, \lambda _{3} = 0.4\) and \(\lambda _{4} = 0.1\). It is clear from final weight vectors that \(\displaystyle \sum \nolimits ^{4}_{i = 1}\lambda _{i} = 0.2+0.3+0.4+0.1 = 1\) (Tables 2, 3, 4).

Table 1 Evaluations of decision-makers
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The (p, q)RLDFMADM steps based on (p, q)RLDFWA and (p, q)RLDFWG-operators:

Table 2 Aggregated values of the alternatives using (p, q)RLDFWA-operator
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Table 3 Aggregated values of the alternatives using (p, q)RLDFWG-operator
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Step 1. Calculate the (p, q)-rung linear Diophantine fuzzy decision matrix. These five alternatives \(A_{1}\), \(A_{2}\), \(A_{3}\) and \(A_{4}\) are to be evaluated by an expert under the four aspects \(G_{1}, G_{2}, G_{3}\) and \(G_{4}\) using (p, q)-rung linear Diophantine fuzzy decision matrix \(D = \left[ {\mathcal {A}}_{ij}\right] _{4\times 4}\) and their corresponding rating values are shown in Eq. (18)

$$\begin{aligned} D = \left[ \begin{array}{ccccc} &{} \left( \begin{array}{ccc} (0.9,1),&{} \\ (0.8,0.7) &{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.8,0.9), &{} \\ (0.6,0.9)&{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.8,0.9), &{} \\ (0.7,0.8) &{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.8,0.8), &{} \\ (0.7,0.8)&{} \end{array} \right) \\ &{} \left( \begin{array}{ccc} (1,0.9), &{} \\ (0.7,0.8)&{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.9,0.7),&{} \\ (0.8,0.7)&{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.8,1), &{} \\ (0.7,0.8) &{} \end{array} \right) &{} \left( \begin{array}{ccc} (1,0.8), &{} \\ (0.7,0.8)&{} \end{array} \right) \\ &{} \left( \begin{array}{ccc} (0.8,1), &{} \\ (0.8,0.7) &{} \end{array} \right) &{} \left( \begin{array}{ccc} (1,1), &{} \\ (0.7,0.8)&{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.8,0.9),&{} \\ (0.7,0.8)&{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.8,0.9),&{} \\ (0.8,0.7)&{} \end{array} \right) \\ &{} \left( \begin{array}{ccc} (0.9,0.8),&{} \\ (0.6,0.9)&{} \end{array} \right) &{} \left( \begin{array}{ccc} (1,0.9),&{} \\ (0.6,0.9) &{} \end{array} \right) &{} \left( \begin{array}{ccc} (1,1), &{} \\ (0.8,0.7) &{} \end{array} \right) &{} \left( \begin{array}{ccc} (0.9,0.8), &{} \\ (0.7,0.8) &{} \end{array} \right) \\ \end{array} \right] . \end{aligned}$$
(18)

Step 2. In this case, the input data for all attributes are the same, then we do not require to normalize the data.

Step 3. By following the (p, q)RLDFWA and (p, q)RLDFWG-operators given in Eqs. (16) and (17), we obtain the overall rating values of each alternative \(A_{i}\).

Step 4. Calculate the scores function of \(r_{i}\) for all \(i =1, 2, 3,4\).

Table 4 Score values of alternatives using (p, q)RLDFWA-operator
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Fig. 1
figure 1

Scores of alternative based on the (p, q)RLDFWA-operator

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Fig. 2
figure 2

Scores of alternative based on the (p, q)RLDFWG-operator

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Step 5.

To explain the effect of the parameters p and q on (p, q)RLDFMCDM end results, we have utilized different values of p and q to rank the alternatives (Figs. 1, 2).

The results of ranking order of the alternatives based on (p, q)ROFWA and (p, q)ROFWG-operators are presented in Table  5. When \((p,q) = (3,3), (p,q) = (3,4)\) and \( (p,q) = (4,3)\), we obtained a rank of alternatives as \(A_{2} \ge A_{3} \ge A_{4}\ge A_{1} \); here, \(A_{2}\) is the best choice and \(A_{3}\) is the best second choice, but, when \( (p,q) = (10,15)\), we obtained a rank of alternatives as \(A_{2} \ge A_{4} \ge A_{3}\ge A_{1}\), here, \(A_{2}\) is the best choice, and \(A_{4}\) is the best second choice. Thus, the overall best rank is \(A_{2}\).

This section gives the comparison analysis of the proposed (p, q)RLDFWA and (p, q)RLDFWG-operators under (p, q)RLDFNs with other well-known operator (Table 6).

We compared the results of (p, q)RLDFWA and (p, q)RLDFWG-operators with n, m-ROFWPA-operator Ibrahim and Alshammari (2022), LDFWGA-operator Riaz and Hashmi (2019), LDFEPWA and LDFEPWG-operators Farid et al. (2022), LDFWA and LDFWG-operators Riaz et al. (2023), LDFEWA, LDFEOWA, LDFEWG, and LDFEOWG-operators Iampan et al. (2021), and q-RLDFOWAA, q-RLDFHWAA, q-RLDFOWGA, and q-RLDFHWGA-operators Almagrabi et al. (2022). The results are summed up as follows in Table 7.

According to Table 7, the proposed operators have the following advantages over the current operators:

  1. 1.

    It is clear that our results of (3, 3)RLDFWA, (3, 4)RLDFWA, (4, 3)RLDFWA, (10, 15)RLDFWA, (3, 3)RLDFWG, (3, 4)RLDFWG and (4, 3)RLDFWG-operators are similar to 3-RLDFOWAA, 3-RLDFHWAA, 3-RLDFOWGA and 3-RLDFHWGA-operators, but results of (3, 3)RLDFWA, (3, 4)RLDFWA, (4, 3)RLDFWA, (10, 15)RLDFWA, (3, 3)RLDFWG, (3, 4)RLDFWG and (4, 3)RLDFWG-operators give \(A_{2}\) is the best choice and \(A_{3}\) is the best second choice, but results of (10, 15)RLDFGA operator give \(A_{2}\) is the best choice and \(A_{4}\) is the best second choice. Therefore, our proposed method is more flexible than other existing methods.

    Table 5 Score values of alternatives using (p, q)RLDFWG-operator
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    Table 6 Ranking order of the alternatives
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    Table 7 Comparison between existing work with the proposed work
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  2. 2.

    The L(p, q)ROFMCDM, being the more flexible, accomplished and general model, covers more decision-making problems using different values and befitting the needs of MCDM problems.

  3. 3.

    (p, q)RLDFN is used for the representation of information in (p, q)RLDFWA and (p, q)RLDFWG-operators. The (p, q)RLDFN is the combination of LDFN with n,m-rung orthopair fuzzy set (n,m-ROFS) which gives full details of the evaluation. The (p, q)RLDFS handles the quantitative as well as qualitative information, so the proposed operator is more general.

  4. 4.

    The existing operators given in (Ibrahim and Alshammari 2022; Riaz and Hashmi 2019; Farid et al. 2022; Riaz et al. 2023; Iampan et al. 2021; Almagrabi et al. 2022) do not handle the problems when data are in the (p, q)-rung linear Diophantine fuzzy information form. The results obtained from the proposed approach are more accurate and precise.

  5. 5.

    The LIFPWA, LIFPWG, LPFPWA, LPFPWG, Lq-ROFWA, and Lq-ROFWG-operators are the special cases of our developed operators, but these operators have some limitations as the uncertainty of decision data does not reduce. Therefore, our developed operators are much more efficient.

To compare the new method with previous methods more clearly, please see Table 8.

Table 8 The characteristic comparisons of different methods
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7 Conclusion

The publication provided a brief explanation of how the q-linear Diophantine fuzzy set framework expands all current ideas and offers a solid foundation without constraints. The formal definition of (p, q)-RLDFS was stated which is generalization of q-linear Diophantine fuzzy set by merging it with n, m-rung orthopair fuzzy set to enhance the memberships space. Set theoretical operations were developed and a number of aggregation operators were built under the (p, q)-RLDFN. The suggested aggregating operators’ intriguing characteristics were investigated. A (p, q)RLDFMADM approach was also built using recommended aggregating operators and scoring functions. To illustrate how the proposed technique should be applied, a case study was provided. We only consider four options as a restriction on our analysis to show the validity of the proposed technique. The suggested technique works where the n, m-ROFSs, linear Diophantine fuzzy number did not work. The q-linear Diophantine fuzzy set only works for \(p = q \ge 1\), but (p, q)-RLDFS works for \(p \ge 1\) and \(q\ge 1\). The suggested methodology has the following notable successes when compared to some of the current MAGDM methods: (1) defining a huge information space with flexible membership grades and non-membership grades constraints, (2) covering a range of diverse operations, and (3) decreasing the negative effects of decision-makers’ excessive evaluation values and dynamically adjusting the weights allocated to input arguments. The approach of the provided strategy may be turned into a computer program, which enables us to carry out our investigation for a limited number of characteristics and options while using enormous data and taking into consideration extra elements. Future research aims include extending the recommended operators to the Archimedean norm and examining other aggregating operators, such as Hamacher and Bonferroni.