A method to multi-attribute decision making problems by using heronian mean operators based on linear diophantine uncertain linguistic settings

Abstract

In the real decision process, an important problem is how to express the attribute value more efficiently and accurately. In the real world, because of the complexity of decision-making problems and the fuzziness of decision-making environments, it is not enough to express attribute values of alternatives by exact values. For this managing with such sorts of issues, the principle of Linear Diophantine uncertain linguistic set is a valuable and capable technique to manage awkward and inconsistent information in everyday life problems. In this manuscript, we propose the original idea of Linear Diophantine uncertain linguistic set and elaborated their essential laws. Additionally, to determine the association among any numbers of attributes, we elaborated the Linear Diophantine uncertain linguistic arithmetic Heronian mean operator, Linear Diophantine uncertain linguistic weighted arithmetic Heronian mean operator, Linear Diophantine uncertain linguistic geometric Heronian mean operator, Linear Diophantine uncertain linguistic weighted geometric Heronian mean operator, and their properties are also discovered. By using these operators, we utilize the multi-attribute decision-making procedure by using elaborated operators. To determine the consistency and validity of the elaborated operators, we illustrate some examples by using explored operators. Finally, the superiority and comparative analysis of the elaborated operators with some existing operators are also determined and justified with the help of a graphical point of view.

Keywords

Linear Diophantine uncertain linguisticsets arithmetic/geometric Heronian mean operators decision-making methods

Abbreviations

LDULS

Linear Diophantine uncertain linguistic sets.

LDUL

Linear Diophantine uncertain linguistic.

LDULAHM

Linear Diophantine uncertain Linear Diophantine uncertain

LDULWAHM

Linear Diophantine uncertain linguistic weighted arithmetic Heronian mean.

LDULGHM

Linear Diophantine uncertain linguistic geometric Heronian mean.

LDULWGHM

Linear Diophantine uncertain linguistic weighted geometric Heronian mean.

MADM

Multi-attribute decision-making.

MAGDM

Multiple attribute group decision-making.

Fuzzy set.

IFS

intuitionistic fuzzy set.

PFS

Pythagorean fuzzy set.

QROFS

q-rung orthopair fuzzy set.

LDFS

Linear Diophantine fuzzy set.

1 Introduction

MAGDM is the basic content of strategic process with their intent is to pick the finest choice by multiple decision-makers from the stock of the available ones under the numerous resources. During accessing these, a set of decision-makers appraises the given choices accepting the different evaluation methods like crisp, interval, linguistic or fuzzy, and so on. However, ambiguity is one of the most utmost concern factors keeping in the mind during any decision-making process and hence it will make some sort of hesitancy to expose the information in a crisp format. To relieve it, DMs usually denote their decisions in phrases of fuzzy variables to examine the ambiguity in the erudition. To address it, Zadeh [1] elaborated the theory of FS, which covers the grade of truth restricted to [0,1]. When the theory of FS was elaborated then numerous intellectuals have exploited it in the natural environment of several areas, for instance, Molodtsov [2] elaborated the soft set (SS). Mahmood [3] explored the novel idea of bipolar SS and its applications. The theory of FS in UP-algebra was utilized by Somjanta et al. [4]. Dokhamdang et al. [5] elaborated the generalized FS in UP-algebra. Tanamoon et al. [6] developed the Q-FS in UP algebra. Kawila et al. [7] discovered the bipolar fuzzy UP-algebra. Some more useful and real life applications of fuzzy sets and their generalizations in decision making problems are discussed in [8, 9].

However, the variety of purposes of FS is thin since decision-makers may challenge the circumstances in which the sum of truth and falsity degrees, then the FS is not capable to accomplish it. Under this occurrence, various assessment materials cannot be precisely conveyed by the FS. To survive with such circumstances, Atanassov [10] elaborated the theory of IFS, which covers the grade of truth and falsity grades with a rule that is the sum of both degrees is restricted to [0,1]. When the theory of IFS was elaborated then numerous scholars have utilized it in the environment of several areas, for instance, Beg and Rashid [11] investigated the intuitionistic hesitant FS. the theory of interval-valued IFS was also elaborated by Atanassov [12]. Kumari and Mishra [13] developed some measures for IFS. Jana and Pal [14] created the bipolar intuitionistic fuzzy soft set and their applications. Joshi and Kumar [15] elaborated the fuzzy time series based on IFS. Fu et al. [16] explored the decision-making procedures based on interval-valued IFS. Meng and He [17] investigated the geometric interaction aggregation operators for IFSs.

But the information expression range of IFSs is limited. They must meet the restriction that the summation of the truth grade and the falsity grade is within [0, 1]. Under this constraint, much complex evaluation information cannot be described because some experts may give some evaluating attribute values that exceed the constraint. For example, if the truth grade and the falsity grade of an evaluating attribute value given by an expert are 0.8 and 0.6, respectively, then the IFNs are not suitable to be used for this kind of problem. With the further development of fuzzy theory, Yager [18] elaborated the theory of PFS, which covers the grade of truth and falsity grades with a rule that is the sum of the square of both degrees is restricted to [0,1]. When the theory of PFS was elaborated then numerous scholars have utilized it in the environment of several areas, for instance, Garg [19] explored the linguistic PFS and their application. Wei and Wei [20] elaborated the similarity measures for PFS. Xiao and Ding [21] created the divergence measures for PFS and their applications. Ullah et al. [22] developed some distance measures for complex PFS and their application in medical diagnosis. Yang et al. [23] discovered the frank power aggregation operators for interval-valued PFS. Garg [24] investigated the improved accuracy function for interval-valued PFSs. Some entropy measures based on PFS were elaborated by Yang and Hussain [25].

Moreover, Yager [26] elaborated the theory of QROFS, which covers the grade of truth and falsity grades with a rule that is the sum of the q-powers of both degrees is restricted to [0,1]. When the theory of QROFS was elaborated then numerous scholars have utilized it in the environment of several areas, for instance, Ali [27] created another view on QROFSs. Liu and Wang [28] elaborated on the aggregation operators for QROFSs. Yang et al. [29] investigated the novel fusion strategies for continuous interval-valued QROFSs. Wang et al. [30] explored the similarity measures for QROFSs. Ali and Mahmood [31] investigated the complex QROFSs and their maclurin symmetric mean operators. Liu et al. [32] cosine similarity and distance measures for QROFSs. Liu and Wang [33] investigated the Archimedean Bonferroni mean operators for QROFSs. Liu et al. [34] explored the linguistic QROFSs. Garg [35] investigated the possibility degree for interval-valued QROFSs. The power aggregation operators for complex QROFSs were developed by Garg et al. [36].

To handle such sorts of concerns, Zadeh [37] investigated the theory of linguistic variable (LV) to describe the preferences of decision-makers. Moreover, the theory of a 2-tuple linguistic set was developed by Herrera and Martinez [38]. Liu and Jin [39] investigated the uncertain LV (ULV). Moreover, HFTLS possibility distribution [40], proportional hesitant fuzzy linguistic term set [41], proportional interval type-2 hesitant fuzzy linguistic term set [42], and K-means clustering for the aggregation of HFLTS possibility distributions: N-two-stage algorithmic paradigm [43]. In approximately actual existence troubles, the sum of truth and falsity grades to which an option filling an ascribe offered by decision-maker could not hold the rule of IFS, PFS, and QROFSs, then the theory of IFS and PFS fail in such situations. To survive with such circumstances, Riaz and Hashmi [44] elaborated the theory of linear Diophantine FS (LDFS), which covers the grade of truth and falsity grades and their reference parameters with a rule that is 0 ⩽ α_Au_A (ϱ) + β_Av_A (ϱ) ⩽ 1. When the theory of LDFS was elaborated then numerous scholars have utilized it in the environment of several areas, for instance, Riaz et al. [45] discovered the theory of linear Diophantine fuzz soft rough sets and their applications. Some algebraic structures based on LDFS were developed by Kamaci [46].

But, up to date no explore the theory of LDULVs and their operational laws. The concepts of IFSs, PFSs, QROFSs, and LDFSs have numerous applications in various fields of real life, but these theories have their limitations related to the membership and non-membership grades. To eradicate these restrictions, we introduce the novel concept of LDULS with the addition of reference parameters and uncertain linguistic terms. The proposed model of LDULS is more efficient and flexible rather than other approaches due to the use of reference parameters and ULVs. LDULS also categorizes the data in MADM problems by changing the physical sense of reference parameters and ULVs. This set covers the spaces of existing structures and enlarges the space for membership and non-membership grades with the help of reference parameters and ULVs. The motivation of the proposed model is given step by step in the whole manuscript. Now we discuss some important objectives of this paper.

The theory of LDULS is more generalized than IFSs, PFSs, QROFSs, LDFSs, and ULVs.

If we choose the information in the form of (0.5, 0.6), then by using the condition of IFSs that is the sum of both terms is limited to the unit interval, but 0.5 + 0.6 = 1.1 > 1, the theory of IFS has been failed for coping with such sorts of issues, the theory of LDULS is very comfortable to resolve the above issues. For this, we choose the reference parameters such as (0.1, 0.2), then by using the condition of LDULS is that 0.1 * 0.5 + 0.2 * 0.6 = 0.05 + 0.12 = 0.17 < 1. We clarify that the IFS is the special case of the proposed LDULS.

If we choose the information in the form of (0.8, 0.9), then by using the condition of PFSs that is the sum of the square of both terms is limited to the unit interval, but 0 . 8² + 0 .9² = 0.64 + 0.81 = 1.45 > 1, the theory of PFS has been failed for coping with such sorts of issues, the theory of LDULS is very comfortable to resolve the above issues. For this, we choose the reference parameters such as (0.2, 0.2), then by using the condition of LDULS is that 0.2 * 0.8 + 0.2 * 0.9 = 0.16 + 0.18 = 0.34 < 1. We clarify that the theory of PFS is the special case of the proposed LDULS.

If we choose the information in the form of (0.1, 0.1), then by using the condition of QROFSs that is the sum of the q-powers of both terms is limited to the unit interval, but 1 + 1 =2 > 1, the theory of QROFS has been failed for coping with such sorts of issues, the theory of LDULS is very comfortable to resolve the above issues. For this, we choose the reference parameters such as (0, 0.1), then by using the condition of LDULS is that 0.0 * 1 +0.1 * 1 =0 + 0.1 = 0.1 < 1. We clarify that the theory of QROFS is the special case of the proposed LDULS.

If we choose the information in the form of ([s₁, s₂] , (0.5, 0.3) , (0.5, 0.4)), then by using the condition of IFSs, PFSs, q-ROFSs, and LDFS have been failed, for coping with such sorts of issues, the theory of LDULS is a very proficient and reliable technique to resolve it. From the above analysis, the theory of IFSs, PFSs, QROFSs, and LDFSs is the special case of the proposed LDULS.

The perceptions of IFSs, PFSs, QROFSs, and LDFSs have frequent applications in innumerable grounds of genuine existence, but these philosophies have their shortcomings associated with the truth and falsity grades. To eliminate these constraints, we announce the narrative hypothesis of LDULS with the supplement of situation parameters. The suggested version of LDULS is additionally inexpensive and accommodating more accurately than other methodologies expected to the usage of suggestion parameters. LDULS also compartmentalizes the information in MADM troubles by modifying the physical meaning of orientation parameters. This set encompasses the areas of accessible assemblies and expands the space for truth and falsity grades with the help of reference parameters. The inspiration of the suggested pattern is offered step by step in the entire script. As shown above the advantages of the operators and keeping the superiority of the elaborated approaches the main points of the elaborated approaches are discussed below:

To elaborate on the LDULS and their fundamental laws. The investigated operational laws are also justifying with the help of some examples.

To determine the association among any number of attributes, we elaborated the LDULAHM operator, LDULWAHM operator, LDULGHM operator, LDULWGHM operator, and their properties are also discovered.

To develop a MADM procedure based on elaborated operators.

To determine the consistency and validity of the elaborated operators, we illustrate some examples by using explored operators.

Finally, the superiority and comparative analysis of the elaborated operators with some existing operators are also determined and justify with the help of a graphical pointof view.

The rest of this manuscript is as follows: In section 2, we briefly recall some definitions such that LDFSs, ULSs, and their operational laws. The theory of HM operators is also reviewed. In section 3, we notified the novel idea of LDULS and elaborated their fundamental laws. In section 4, to determine the association among any number of attributes, we elaborated the LDULAHM operator, LDULWAHM operator, LDULGHM operator, LDULWGHM operator, and their properties are also discovered. In section 5, by using these operators, we acquire a MADM procedure based on elaborated operators. To determine the consistency and validity of the elaborated operators, we illustrate some examples by using explored operators. Finally, the superiority and comparative analysis of the elaborated operators with some existing operators are also determined and justify with the help of a graphical point of view. In section 6, we discussed the conclusion of this manuscript.

2 Preliminaries

In this section, we briefly recall some definitions such that LDFSs, ULSs, and their operational laws. The theory of HM operators is also reviewed. In an overall manuscript, the universal set is denoted by X. The symbols used in this manuscript are discussed in the form of Table 1.

Table 1
Expressed the used variables in these manuscripts

Symbol Name Symbol Name Symbol Name

X Universal set ϱ Element of X u_A (ϱ) Truth Grade

v_A (ϱ) Falsity grade α _A Reference Parameter for Truth grade β _A Reference Parameter for Falsity grade

λ ⩾ 0 Scaler element φ _{A
_{LD
₁}} Accuracy value ς _{A
_{LD
₁}} Score value

Ξ _Γ Linguistic term Γ Last term [Ξ_a, Ξ_b] Uncertain linguistic term

$f ⩾ 0$ Scaler $h ⩾ 0$ Scaler $r$ Nth element

Symbol	Name	Symbol	Name	Symbol	Name
X	Universal set	ϱ	Element of X	u_A (ϱ)	Truth Grade
v_A (ϱ)	Falsity grade	α _A	Reference Parameter for Truth grade	β _A	Reference Parameter for Falsity grade
λ ⩾ 0	Scaler element	φ _{A _{LD ₁}}	Accuracy value	ς _{A _{LD ₁}}	Score value
Ξ _Γ	Linguistic term	Γ	Last term	[Ξ_a, Ξ_b]	Uncertain linguistic term
$f ⩾ 0$	Scaler	$h ⩾ 0$	Scaler	$r$	Nth element

Definition 1. [44] A LDFS A_LD is elaborated by: $A_{LD} = {(ϱ, (u_{A} (ϱ), v_{A} (ϱ)), (α_{A}, β_{A})) / ϱ \in X}$ (1)

With the rules of 0 ⩽ α_Au_A (ϱ) + β_Av_A (ϱ) ⩽ 1, u_A (ϱ) , v_A, α_A, β_A ∈ [0, 1] and 0 ⩽ α_A + β_A ⩽ 1. The symbol ξ_{A
_LD}π (ϱ) _{A
_LD} = 1 - (α_Au_A (ϱ) + β_Av_A (ϱ)) expressed the refusal grade. Simply A_LD = ((u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)) is called a linear Diophantine fuzzy number (LDFN). For any two LDFNs A_LD = ((u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)) and B_LD = ((u_B (ϱ) , v_B (ϱ)) , (α_B, β_B)), then

$A_{LD} \oplus B_{LD} = ((u_{A} + u_{B} - u_{A} u_{B}, v_{A} v_{B}), (α_{A} + α_{B} - α_{A} α_{B}, β_{A} β_{B}))$ (2)

$A_{LD} \otimes B_{LD} = ((u_{A} u_{B}, v_{A} + v_{B} - v_{A} v_{B}), (α_{A} α_{B}, β_{A} + β_{B} - β_{A} β_{B}))$ (3)

$λ A_{LD} = ((1 - {(1 - u_{A} (ϱ))}^{λ}, {(v_{A} (ϱ))}^{λ}), {(1 - {(1 - α_{A})}^{λ}, (β_{A})}^{λ}))$ (4)

$A_{LD}^{λ} = (({(u_{A} (ϱ))}^{λ}, 1 - (1 - {(v_{A} (ϱ))}^{λ})), ({(α_{A})}^{λ}, 1 - (1 - (β_{A}))^{λ}))$ (5)

Definition 2. [44] For any LDFN A_{LD
₁} = ((u_{AG
₁} (ϱ) , v_{AG
₁} (ϱ)) , (α_{AG
₁}, β_{AG
₁})), then the score value (SV) and accuracy value (AV) is elaborated by: $\begin{matrix} ς_{A_{{LD}_{1}}} = ς (A_{{LD}_{1}}) = \frac{1}{2} \\ (u_{{AG}_{1}} (ϱ) - v_{{AG}_{1}} (- ϱ) + (α_{{AG}_{1}} + β_{{AG}_{1}})) \end{matrix}$ (6) $\begin{matrix} φ_{A_{{LD}_{1}}} = φ (A_{{LD}_{1}}) = \frac{1}{2} \\ (\frac{(u_{{AG}_{1}} (ϱ) + v_{{AG}_{1}} (ϱ))}{2} + (α_{{AG}_{1}} + β_{{AG}_{1}})) \end{matrix}$ (7)

To determine the relationship among any number of attributes, we constructed the following rules:

If ς_{A
_LD} < ς_{B
_LD} then A_LD < B_LD,

If ς_{A
_LD} > ς_{B
_LD} then A_LD > B_LD,

If ς_{A
_LD} = ς_{B
_LD} then,

If φ_{A
_LD} < φ_{B
_LD} then A_LD < B_LD,

If φ_{A
_LD} > φ_{B
_LD} then A_LD > B_LD,

If φ_{A
_LD} = φ_{B
_LD} then A_LD ≈ B_LD

Definition 3. [37] A linguistic term sets (LTS) is elaborated by: $S = {Ξ_{0}, Ξ_{1}, Ξ_{2}, \dots, Ξ_{Γ - 1}}$ (8)where Γ must be odd and keep the resulting circumstances:

If Γ > Γ′, then Ξ_Γ > Ξ_Γ′;

The pessimistic operator Neg (Ξ_Γ) = Ξ_Γ′ with a condition Γ + Γ′ = Γ - 1;

If Γ ⩾ Γ′, max(Ξ_Γ, Ξ_Γ′) = Ξ_Γ, and if Γ ⩽ Γ′, min(Ξ_Γ, Ξ_Γ′) = Ξ_Γ.

Let $\tilde{Ξ} = [Ξ_{a}, Ξ_{b}]$ , where Ξ_a, Ξ_b ∈ S, Ξ_a and Ξ_b are the lower and the upper limits, respectively. We call $\tilde{Ξ}$ the uncertain linguistic variable. For any two ULVs ${\tilde{Ξ}}_{1} = [Ξ_{a 1}, Ξ_{b 1}]$ and ${\tilde{Ξ}}_{2} = [Ξ_{a 2}, Ξ_{b 2}]$ , ${\tilde{Ξ}}_{1} \oplus {\tilde{Ξ}}_{2} = [Ξ_{a 1}, Ξ_{b 1}] \oplus [Ξ_{a 2}, Ξ_{b 2}] = [Ξ_{a 1 + a 2}, Ξ_{b 1 + b 2}]$ (9) ${\tilde{Ξ}}_{1} \otimes {\tilde{Ξ}}_{2} = [Ξ_{a 1}, Ξ_{b 1}] \otimes [Ξ_{a 2}, Ξ_{b 2}] = [Ξ_{a 1 \times a 2}, Ξ_{b 1 \times b 2}]$ (10) $λ {\tilde{Ξ}}_{1} = λ [Ξ_{a 1}, Ξ_{b 1}] = [Ξ_{λ * a 1}, Ξ_{λ * b 1}]$ (11) ${({\tilde{Ξ}}_{1})}^{λ} = [Ξ_{{(a 1)}^{λ}}, Ξ_{{(b 1)}^{λ}}]$ (12)

Definition 4. [34] For any $I = [0, 1], f, h ⩾ 0, H^{f, h} : I^{r} \to I$ , then the HM operator is elaborated by: $H^{f, h} (ϱ_{1}, ϱ_{2}, \dots, ϱ_{r}) = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 𝒷}^{r} ϱ_{𝒷}^{f} ϱ_{j}^{h})}^{\frac{1}{f + h}}$ (13)

4 Linear Diophantine Uncertain Linguistic set

As shown above, the existing theories are very useful in the circumstances of investigation of any theories. In this study, by using the existing theories, we elaborated on the novel concept of LDULSs and their operational laws. These investigated theories are also justifying with the help of examples.

Definition 5. A LDULS A_LDULS is elaborated by:

$\begin{matrix} A_{LDULS} = {(ϱ [[Ξ_{θ (ϱ)}, Ξ_{τ (ϱ)}], \\ (u_{A} (ϱ), v_{A} (ϱ)), (α_{A}, β_{A})] / ϱ \in X} \end{matrix}$ (14)

With the rules of 0 ⩽ α_Au_A (ϱ) + β_Av_A (ϱ) ⩽ 1, u_A (ϱ) , v_A, α_A, β_A ∈ [0, 1] and 0 ⩽ α_A + β_A ⩽ 1. The symbol ξ_{A
_LD}π (ϱ) _{A
_LD} = 1 - (α_Au_A (ϱ) + β_Av_A (ϱ)) expressed the refusal grade with $Ξ_{θ (ϱ)}, Ξ_{τ (ϱ)} \in \tilde{S}$ . Simply A_LD = ([Ξ_θ(ϱ), Ξ_τ(ϱ)] , (u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)) expresses the linear Diophantine uncertain linguistic number (LDULN).

Definition 6. For any two LDULNs A_LD = ([Ξ_A-θ(ϱ), Ξ_A-τ(ϱ)] , (u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)) and B_LD = ([Ξ_B-θ(ϱ), Ξ_B-τ(ϱ)] , (u_B (ϱ) , v_B (ϱ)) , (α_B, β_B)), then

$A_{LD} \oplus B_{LD} = (\begin{matrix} [Ξ_{A - θ (ϱ) + B - θ (ϱ)}, Ξ_{A - τ (ϱ) + B - τ (ϱ)}], (u_{A} + u_{B} - u_{A} u_{B}, v_{A} v_{B}), \\ (α_{A} + α_{B} - α_{A} α_{B}, β_{A} β_{B}) \end{matrix})$ (15)

$A_{LD} \otimes B_{LD} = (\begin{matrix} [Ξ_{A - θ (ϱ) * B - θ (ϱ)}, Ξ_{A - τ (ϱ) * B - τ (ϱ)}], (u_{A} u_{B}, v_{A} + v_{B} - v_{A} v_{B}), \\ (α_{A} α_{B}, β_{A} + β_{B} - β_{A} β_{B}) \end{matrix})$ (16)

$λ A_{LD} = ([Ξ_{λ \times A - θ (ϱ)}, Ξ_{λ \times A - τ (ϱ)}], (1 - {(1 - u_{A} (ϱ))}^{λ}, {(v_{A} (ϱ))}^{λ}), {(1 - {(1 - α_{A})}^{λ}, (β_{A})}^{λ}))$ (17)

$A_{LD}^{λ} = ([Ξ_{{(A - θ (ϱ))}^{λ}}, Ξ_{{(A - τ (ϱ))}^{λ}}], ({(u_{A} (ϱ))}^{λ}, 1 - (1 - {(v_{A} (ϱ))}^{λ})), ({(α_{A})}^{λ}, 1 - (1 - (β_{A}))^{λ}))$ (18)

Example 1. For any two LDULNs A_LD = ([Ξ₁, Ξ₂] , (0.5, 0.3) , (0.5, 0.4)) and B_LD = ([Ξ₂, Ξ₃] , (0.7, 0.5) , (0.1, 0.3)), with λ = 2, then by Definition 6, such that $A_{LD} \oplus B_{LD} = ([Ξ_{3}, Ξ_{5}], (0.42, 0.15), (0.55, 0.12))$ $A_{LD} \otimes B_{LD} = ([Ξ_{2}, Ξ_{6}], (0.15, 0.42), (0.12, 0.55))$ $2 A_{LD} = ([Ξ_{2}, Ξ_{4}], (0.75, 0.09), (0.75, 0.16))$ $A_{LD}^{2} = ([Ξ_{1}, Ξ_{4}], (0.09, 0.75), (0.16, 0.75))$

Theorem 1. For any two LDULNs A_LD = ([Ξ_A-θ(ϱ), Ξ_A-τ(ϱ)] , (u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)) and B_LD = ([Ξ_B-θ(ϱ), Ξ_B-τ(ϱ)] , (u_B (ϱ) , v_B (ϱ)) , (α_B, β_B)), then

A_LD ⊕ B_LD = B_LD ⊕ A_LD;

A_LD ⊗ B_LD = B_LD ⊗ A_LD;

λ (A_LD ⊕ B_LD) = λA_LD ⊕ λB_LD, λ > 0;

λ₁A_LD ⊕ λ₂A_LD = (λ₁ + λ₂) A_LD, λ₁, λ₂ > 0;

$A_{LD}^{λ_{1}} \otimes A_{LD}^{λ_{2}} = A_{LD}^{λ_{1} + λ_{2}} λ_{1}, λ_{2} > 0$ ;

$A_{LD}^{λ_{1}} \otimes A_{LD}^{λ_{1}} = {(A_{LD} \otimes B_{LD})}^{λ_{1}} λ_{1} > 0$ ;

Proof.

1. Let A_LD ⊕ B_LD, then $\begin{matrix} A_{LD} \oplus B_{LD} = (\begin{matrix} [Ξ_{A - θ (ϱ) + B - θ (ϱ)}, Ξ_{A - τ (ϱ) + B - τ (ϱ)}], (u_{A} + u_{B} - u_{A} u_{B}, v_{A} v_{B}), \\ (α_{A} + α_{B} - α_{A} α_{B}, β_{A} β_{B}) \end{matrix}) \\ = (\begin{matrix} [Ξ_{B - θ (ϱ) + A - θ (ϱ)}, Ξ_{B - τ (ϱ) + A - τ (ϱ)}], (u_{B} + u_{A} - u_{B} u_{A}, v_{B} v_{A}), \\ (α_{B} + α_{A} - α_{B} α_{A}, β_{B} β_{A}) \end{matrix}) = B_{LD} \oplus A_{LD} \end{matrix}$

Straightforward.

Let λ (A_LD ⊕ B_LD), then $\begin{matrix} λ (A_{LD} \oplus B_{LD}) = λ (\begin{matrix} [Ξ_{A - θ (ϱ) + B - θ (ϱ)}, Ξ_{A - τ (ϱ) + B - τ (ϱ)}], (u_{A} + u_{B} - u_{A} u_{B}, v_{A} v_{B}), \\ (α_{A} + α_{B} - α_{A} α_{B}, β_{A} β_{B}) \end{matrix}) \\ = ([Ξ_{λ \times A - θ (ϱ) + λ \times B - θ (ϱ)}, Ξ_{λ \times A - τ (ϱ) + λ \times B - τ (ϱ)}], {(1 - (1 - u_{A} (ϱ))^{λ} + 1 - (1 - u_{B} (ϱ))}^{λ} \\ - (1 - (1 - u_{A} (ϱ))^{λ}) (1 - (1 - u_{B} (ϱ))^{λ}), {(v_{A} (ϱ))}^{λ} {(v_{B} (ϱ))}^{λ}), (1 - {(1 - α_{A})}^{λ} + 1 \\ - {(1 - α_{B})}^{λ} - (1 - {(1 - α_{A})}^{λ}) (1 - {(1 - α_{B})}^{λ}), {(β_{A} (ϱ))}^{λ} {(β_{B} (ϱ))}^{λ}))) \\ = ([Ξ_{λ \times A - θ (ϱ)}, Ξ_{λ \times A - τ (ϱ)}], (1 - (1 - u_{A} (ϱ))^{λ}, {(v_{A} (ϱ))}^{λ}), (1 - {(1 - α_{A})}^{λ}, {(β_{A})}^{λ})) \\ \oplus ([Ξ_{λ \times B - θ (ϱ)}, Ξ_{λ \times B - τ (ϱ)}], (1 - (1 - u_{B} (ϱ))^{λ}, {(v_{B} (ϱ))}^{λ}), (1 - (1 - α_{B})^{λ}, {(β_{B})}^{λ})) \\ = λ A_{LD} \oplus λ B_{LD} . \end{matrix}$

Straightforward.

Let $A_{LD}^{λ_{1}} \otimes A_{LD}^{λ_{2}}$ , then $\begin{matrix} A_{LD}^{λ_{1}} \otimes A_{LD}^{λ_{2}} = ([Ξ_{{(A - θ (ϱ))}^{λ_{1}}}, Ξ_{{(A - τ (ϱ))}^{λ_{1}}}], ({(u_{A} (ϱ))}^{λ_{1}}, 1 - (1 - {(v_{A} (ϱ))}^{λ_{1}})), ({(α_{A})}^{λ_{1}}, 1 \\ - {(1 - (β_{A}))}^{λ_{1}})) \otimes ([Ξ_{{(A - θ (ϱ))}^{λ_{2}}}, Ξ_{{(A - τ (ϱ))}^{λ 2}}], ({(u_{A} (ϱ))}^{λ_{2}}, 1 - (1 - {(v_{A} (ϱ))}^{λ 2})), ({(α_{A})}^{λ 2}, 1 \\ - {(1 - (β_{A}))}^{λ_{2}})) \end{matrix}$ $= ([Ξ_{{(A - θ (ϱ))}^{λ_{1} + λ_{2}}}, Ξ_{{(A - τ (ϱ))}^{λ_{1} + λ_{2}}}], ({(u_{A} (ϱ))}^{λ_{1} + λ_{2}}, 1 - (1 - {(v_{A} (ϱ))}^{λ_{1} + λ_{2}})),$ $({(α_{A})}^{λ_{1} + λ_{2}}, 1 - (1 - (β_{A}))^{λ_{1} + λ_{2}}))$ $= A_{LD}^{λ_{1} + λ_{2}} .$

Definition 7. For any LDULN A_LD = ([Ξ_A-θ(ϱ), Ξ_A-τ(ϱ)] , (u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)), then the expected value (EV) is elaborated by:

$E (A_{LD}) = Ξ_{\frac{((A - θ (ϱ) + A - τ (ϱ)) \times \frac{1}{2} (u_{A} (ϱ) + 1 - v_{A} (ϱ) + (α_{A} + β_{A})))}{4}}$ (19)

Example 2. For any LDULN A_LD = ([Ξ₁, Ξ₂] , (0.5, 0.3) , (0.5, 0.4)), then the EV is elaborated as below: $E (A_{LD}) = Ξ_{\frac{((1 + 2) \times \frac{1}{2} (0.5 + 1 - 0.3 + 0.5 + 0.4))}{4}} = Ξ_{\frac{((3) \times 1.05)}{4}} = Ξ_{0.7875}$

Definition 8. For any LDULN A_LD = ([Ξ_A-θ(ϱ), Ξ_A-τ(ϱ)] , (u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)), then accuracy value (AV) is elaborated by: $φ_{A_{LD}} = φ (A_{LD}) = Ξ_{\frac{((A - θ (ϱ) + A - τ (ϱ)) \times \frac{1}{2} (u_{A} (ϱ) + v_{A} (ϱ) + (α_{A} + β_{A})))}{4}}$ (20)

Definition 9. For any two LDULNs A_LD = ([Ξ_A-θ(ϱ), Ξ_A-τ(ϱ)] , (u_A (ϱ) , v_A (ϱ)) , (α_A, β_A)) and B_LD = ([Ξ_B-θ(ϱ), Ξ_B-τ(ϱ)] , (u_B (ϱ) , v_B (ϱ)) , (α_B, β_B)), then

If E_{A
_LD} < E_{B
_LD} then A_LD < B_LD,

If E_{A
_LD} > E_{B
_LD} then A_LD > B_LD,

If E_{A
_LD} = E_{B
_LD} then,

If φ_{A
_LD} < φ_{B
_LD} then A_LD < B_LD,

If φ_{A
_LD} > φ_{B
_LD} then A_LD > B_LD,

If φ_{A
_LD} = φ_{B
_LD} then A_LD ≈ B_LD,

4 HM Operators for LDULNs

To determine the association among any number of attributes, we elaborated the LDULAHM operator, LDULWAHM operator, LDULGHM operator, LDULWGHM operator, and their properties are also discovered. By using the investigated operators, the special cases of the elaborated operators are also discussed.

Definition 10. For any LDULNs A_LD-𝒷 = ([Ξ_{A-θ_𝒷}, Ξ_{A-τ_𝒷}] , (u_A-𝒷, v_A-𝒷) , (α_A-𝒷, β_A-𝒷 )) , $𝒷 = 1, 2, \dots, r$ , the LDULAHM operator is elaborated by:

${LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = {(\frac{2}{r (r + 1)} \sum_{= 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h})}^{\frac{1}{f + h}}$ (21)

Theorem 2. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , then by using the Equation (21), we have

$\begin{matrix} {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷}^{f} θ_{A - j}^{h})}^{\frac{1}{f + h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷}^{f} τ_{A - j}^{h})}^{\frac{1}{f + h}}}], \\ (\begin{matrix} ({(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - u_{A - 𝒷}^{f} u_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} ({(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - α_{A - 𝒷}^{f} α_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷})}^{f} {(1 - β_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ (22)

Proof. By using Equation (18), we have ${(A_{LD - 𝒷})}^{f} = ([Ξ_{θ_{A - 𝒷}^{f}}, Ξ_{τ_{A - 𝒷}^{f}}], (u_{A - 𝒷}^{f}, 1 - {(1 - v_{A - 𝒷})}^{f}), (α_{A - 𝒷}^{f}, 1 - {(1 - β_{A - 𝒷})}^{f}))$ ${(A_{LD - j})}^{h} = ([Ξ_{θ_{A - j}^{h}}, Ξ_{τ_{A - j}^{h}}], (u_{A - j}^{h}, 1 - {(1 - v_{A - j})}^{h}), (α_{A - j}^{h}, 1 - {(1 - β_{A - j})}^{h}))$

and $A_{LD - 𝒷}^{f} A_{LD - j}^{h} = (\begin{matrix} [Ξ_{θ_{A - 𝒷}^{f} θ_{A - j}^{h}}, Ξ_{τ_{A - 𝒷}^{f} τ_{A - j}^{h}}], (u_{A - 𝒷}^{f} u_{A - j}^{h}, 1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h}), \\ (α_{A - 𝒷}^{f} α_{A - j}^{h}, 1 - {(1 - β_{A - 𝒷})}^{f} {(1 - β_{A - j})}^{h}) \end{matrix})$

Then $\sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h} = (\begin{matrix} [Ξ_{\sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷}^{f} θ_{A - j}^{h}}, Ξ_{\sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷}^{f} τ_{A - j}^{h}}], \\ (1 - \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - u_{A - 𝒷}^{f} u_{A - j}^{h}), \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h})), \\ (1 - \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - α_{A - 𝒷}^{f} α_{A - j}^{h}), \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷})}^{f} {(1 - β_{A - j})}^{h})) \end{matrix})$ Further $\begin{matrix} \frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h} \\ (\begin{matrix} [Ξ_{\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷}^{f} θ_{A - j}^{h}}, Ξ_{\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷}^{f} τ_{A - j}^{h}}], \\ (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - u_{A - 𝒷}^{f} u_{A - j}^{h}))}^{\frac{2}{r (r + 1)}}, {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - α_{A - 𝒷}^{f} α_{A - j}^{h}))}^{\frac{2}{r (r + 1)}}, {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷})}^{f} {(1 - β_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}}) \end{matrix}) \end{matrix}$ So, $\begin{matrix} {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h})}^{\frac{1}{f + h}} \\ (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷}^{f} θ_{A - j}^{h})}^{\frac{1}{f + h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷}^{f} τ_{A - j}^{h})}^{\frac{1}{f + h}}}], \\ (\begin{matrix} ({(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - u_{A - 𝒷}^{f} u_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} ({(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - α_{A - 𝒷}^{f} α_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷})}^{f} {(1 - β_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ By using Equation (22), we will demonstrate monotonicity, idempotency, and boundedness.

Theorem 3. For any LDULNs $(A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r})$ and $(B_{LD - 1}, B_{LD - 2}, \dots, B_{LD - r})$ , if $A_{LD - r} ⩽ B_{LD - r}$ for all $r = 1, 2, \dots, m$ then ${LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULAHM}^{f, h} (B_{LD - 1}, B_{LD - 2}, \dots, B_{LD - r})$ (23)

Proof. Since A_LD-𝒷 ⩽ B_LD-𝒷 and A_LD-j ⩽ B_LD-j for all 𝔯 = 1, 2, …, m and j = 1, 2, …, m we have $A_{LD - 𝒷} A_{LD - j} ⩽ B_{LD - 𝒷} B_{LD - j}$ then, $A_{LD - 𝒷}^{f} A_{LD - j}^{h} ⩽ B_{LD - 𝒷}^{f} B_{LD - j}^{h}$ thus, $\sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h} ⩽ \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} B_{LD - 𝒷}^{f} B_{LD - j}^{h}$ then $\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h} ⩽ \frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} B_{LD - 𝒷}^{f} B_{LD - j}^{h}$ So, ${(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h})}^{\frac{1}{p + q}} ⩽ {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} B_{LD - 𝒷}^{f} B_{LD - j}^{h})}^{\frac{1}{p + q}}$ i.e. ${LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULAHM}^{f, h} (B_{LD - 1}, B_{LD - 2}, \dots, B_{LD - r})$

Theorem 4. For any LDULNs $A_{LD - j} = A, j = 1, 2, \dots, r$ then ${LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = A$ .

Proof. Since A_LD-j = A, for all j, we have ${LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h})}^{\frac{1}{f + h}}$ $= {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h})}^{\frac{1}{f + h}} = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A^{f + h})}^{\frac{1}{f + h}} = {(A^{f + h})}^{\frac{1}{f + h}} = A$

Theorem 5. For any LDULNs A_LD-𝒷 = ([Ξ_{A-θ_𝒷}, Ξ_{A-τ_𝒷}] , (u_A-𝒷, v_A-𝒷 ) , (α_A -𝒷, β_A-𝒷 )) , $𝒷 = 1, 2, \dots, r$ , if LDULAHM operator lies between the max and min operators, then $\begin{matrix} \min (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ ⩽ {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ ⩽ \max (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \end{matrix}$

Proof. Let $A = \min (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}),$ $B = \max (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r})$ , according to Theorem 4, we have $\begin{matrix} {LDULAHM}^{f, h} (A, A, . ., A) ⩽ {LDULAHM}^{f, h} \\ (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULAHM}^{f, h} (B, B, \dots, B) \end{matrix}$

Further, ${LDULAHM}^{f, h} (A, A, . ., A) = A$ , ${LDULAHM}^{f, h} (B, B, \dots, B) = B$ so, $A = {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = B$ i.e. $\min (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽$

$\max (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r})$ . It is easy to prove that ${LDULAHM}^{f, h}$ the operator does not have the property of commutativity.

As shown above, the reliability and consistency of the elaborated operators, we discussed the special cases of the investigated operators, which are stated below.

For $h \to 0$ , then Equation (22) reduces to an LDUL generalized linear descending weighted mean operator, such that $\begin{matrix} {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = lim_{h \to 0} (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷}^{f} θ_{A - j}^{h})}^{\frac{1}{f + h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷}^{f} τ_{A - j}^{h})}^{\frac{1}{f + h}}}], \\ (\begin{matrix} ({(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - u_{A - 𝒷}^{f} u_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} ({(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - α_{A - 𝒷}^{f} α_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷})}^{f} {(1 - β_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ $= (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} (r + 1 - 𝒷) θ_{A - 𝒷}^{f})}^{\frac{1}{f}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} (r + 1 - 𝒷) τ_{A - 𝒷}^{f})}^{\frac{1}{f}}}], \\ ({(1 - {(\prod_{𝒷 = 1}^{r} {(1 - u_{A - 𝒷}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}, 1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - v_{A - 𝒷})}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}), \\ ({(1 - {(\prod_{𝒷 = 1}^{r} {(1 - α_{A - 𝒷}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}, 1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - β_{A - 𝒷})}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}) \end{matrix})$ (24)

For $f \to 0$ , the Equation (22) reduces to an LDUL generalized linear ascending weighted mean operator, such that $\begin{matrix} {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = lim_{h \to 0} (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷}^{f} θ_{A - j}^{h})}^{\frac{1}{f + h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷}^{f} τ_{A - j}^{h})}^{\frac{1}{f + h}}}], \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} (1 - u_{A - 𝒷}^{f} u_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} (1 - {(1 - v_{A - 𝒷})}^{f} {(1 - v_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} (1 - α_{A - 𝒷}^{f} α_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷})}^{f} (1 - β_{A - j})^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ $= (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} (𝒷 θ_{A - 𝒷}^{f}))}^{\frac{1}{h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} (𝒷 τ_{A - 𝒷}^{f}))}^{\frac{1}{h}}}], \\ ({(1 - {(\prod_{𝒷 = 1}^{r} {(1 - u_{A - 𝒷}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{h}}, 1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - v_{A - 𝒷})}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}), \\ ({(1 - {(\prod_{𝒷 = 1}^{r} {(1 - α_{A - 𝒷}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{h}}, 1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - β_{A - 𝒷})}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}) \end{matrix})$ (25)

If $f = h = 0$ , ${LDULAHM}^{f, h}$ the operator has the linear weighted function for input data.

For $f = h = \frac{1}{2}$ , the Equation (22) reduces to an LDUL basic Heronian mean operator, such that $\begin{matrix} {LDULAHM}^{\frac{1}{2}, \frac{1}{2}} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = \\ (\begin{matrix} [Ξ_{\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} {\sum_{j = 1}^{r} \sqrt{θ_{A - 𝒷} θ_{A - j}}}^{\frac{1}{f + h}}}, Ξ_{\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} {\sum_{j = 1}^{r} \sqrt{τ_{A - 𝒷} τ_{A - j}}}^{\frac{1}{f + h}}}], \\ (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{u_{A - 𝒷} u_{A - j}}))}^{\frac{2}{r (r + 1)}}, {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{(1 - v_{A - 𝒷}) (1 - v_{A - j})}))}^{\frac{2}{r (r + 1)}}), \\ (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{α_{A - 𝒷} α_{A - j}}))}^{\frac{2}{r (r + 1)}}, {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{(1 - β_{A - 𝒷}) (1 - β_{A - j})}))}^{\frac{2}{r (r + 1)}}) \end{matrix}) \end{matrix}$ (26)

For $f = h = 1$ , the Equation (22) reduces to an LDUL line Heronian mean operator, such that $\begin{matrix} {LDULAHM}^{\frac{1}{2}, \frac{1}{2}} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} θ_{A - 𝒷} θ_{A - j})}^{\frac{1}{2}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} τ_{A - 𝒷} τ_{A - j})}^{\frac{1}{2}}}], \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - u_{A - 𝒷} u_{A - j}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{2}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - (1 - v_{A - 𝒷}) (1 - v_{A - j})))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{2}} \end{matrix}), \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - α_{A - 𝒷} α_{A - j}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{2}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - (1 - β_{A - 𝒷}) (1 - β_{A - j})))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{2}} \end{matrix}) \end{matrix}) \end{matrix}$ (27)

Definition 11. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , the LDULWAHM operator is elaborated by:

${LDULWAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} A_{LD - 𝒷})}^{f} {(r ω_{j} A_{LD - j})}^{h})}^{\frac{1}{f + h}}$ (28)where $ω = {(ω_{1}, ω_{2}, \dots, ω_{r})}^{T}$ is the weight vector of $(A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}), ω_{j} \in [0, 1], \sum_{j = 1}^{r} ω_{j} = 1$ . $r$ is a balance parameter.

Theorem 6. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , then by using the Equation (22), we have $\begin{matrix} {LDULWAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} θ_{A - 𝒷})}^{f} {(r ω_{𝒷} θ_{A - j})}^{h})}^{\frac{1}{f + h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} τ_{A - 𝒷})}^{f} {(r ω_{𝒷} τ_{A - j})}^{h})}^{\frac{1}{f + h}}}], \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - u_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - u_{A - j})}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷}^{r ω_{𝒷}})}^{f} {(1 - v_{A - j}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - α_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - α_{A - j})}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷}^{r ω_{𝒷}})}^{f} {(1 - β_{A - j}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ (29)

Proof. ${(r ω_{𝒷} A_{LD - 𝒷})}^{f} = (\begin{matrix} [Ξ_{{(r ω_{𝒷} θ_{A - 𝒷})}^{f}}, Ξ_{{(r ω_{𝒷} τ_{A - 𝒷})}^{f}}], \\ (\begin{matrix} {(1 - {(1 - u_{A - 𝒷})}^{r ω_{𝒷}})}^{f}, \\ {(1 - v_{A - 𝒷}^{r ω 𝒷})}^{f} \end{matrix}), \\ (\begin{matrix} {(1 - {(1 - α_{A - 𝒷})}^{r ω_{𝒷}})}^{f}, \\ {(1 - β_{A - 𝒷}^{r ω 𝒷})}^{f} \end{matrix}) \end{matrix})$ ${(r ω_{j} A_{LD - j})}^{h} = (\begin{matrix} [Ξ_{{(r ω_{j} θ_{A - j})}^{h}}, Ξ_{{(r ω_{j} τ_{A - j})}^{h}}], \\ (\begin{matrix} {(1 - {(1 - u_{A - j})}^{r ω_{𝒷}})}^{h}, \\ {(1 - v_{A - j}^{r ω 𝒷})}^{h} \end{matrix}), \\ (\begin{matrix} {(1 - {(1 - α_{A - j})}^{r ω_{𝒷}})}^{h}, \\ {(1 - β_{A - j}^{r ω 𝒷})}^{h} \end{matrix}) \end{matrix})$ ${(r ω_{𝒷} A_{LD - 𝒷})}^{f} {(r ω_{j} A_{LD - j})}^{h} =$ $(\begin{matrix} [Ξ_{{(r ω_{𝒷} θ_{A - 𝒷})}^{f} {(r ω_{j} θ_{A - j})}^{h}}, Ξ_{{(r ω_{𝒷} τ_{A - 𝒷})}^{f} {(r ω_{j} τ_{A - j})}^{h}}], \\ (\begin{matrix} (1 - {(1 - {(1 - u_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - u_{A - j})}^{r ω_{𝒷}})}^{h}), \\ (1 - {(1 - v_{A - 𝒷}^{r ω 𝒷})}^{f} {(1 - v_{A - j}^{r ω 𝒷})}^{h}) \end{matrix}), \\ (\begin{matrix} (1 - {(1 - {(1 - α_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - α_{A - j})}^{r ω_{𝒷}})}^{h}), \\ (1 - {(1 - β_{A - 𝒷}^{r ω 𝒷})}^{f} {(1 - β_{A - j}^{r ω 𝒷})}^{h}) \end{matrix}) \end{matrix})$ $(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} A_{LD - 𝒷})}^{f} {(r ω_{j} A_{LD - j})}^{h})$ $= (\begin{matrix} [Ξ_{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} θ_{A - 𝒷})}^{f} {(r ω_{j} θ_{A - j})}^{h})}, Ξ_{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} τ_{A - 𝒷})}^{f} {(r ω_{j} τ_{A - j})}^{h})}], \\ (\begin{matrix} (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - u_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - u_{A - j})}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ 1 - (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷}^{r ω_{𝒷}})}^{f} {(1 - v_{A - j}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}}) \end{matrix}), \\ (\begin{matrix} (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - α_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - α_{A - j})}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ 1 - (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷}^{r ω_{𝒷}})}^{f} {(1 - β_{A - j}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}}) \end{matrix}) \end{matrix})$ ${LDULWAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} A_{LD - 𝒷})}^{f} {(r ω_{j} A_{LD - j})}^{h})}^{\frac{1}{f + h}}$ $= (\begin{matrix} [Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} θ_{A - 𝒷})}^{f} {(r ω_{j} θ_{A - j})}^{h})}^{\frac{1}{f + h}}}, Ξ_{{(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} τ_{A - 𝒷})}^{f} {(r ω_{j} τ_{A - j})}^{h})}^{\frac{1}{f + h}}}], \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - u_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - u_{A - j})}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - v_{A - 𝒷}^{r ω_{𝒷}})}^{f} {(1 - v_{A - j}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - α_{A - 𝒷})}^{r ω_{𝒷}})}^{f} {(1 - {(1 - α_{A - j})}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - β_{A - 𝒷}^{r ω_{𝒷}})}^{f} {(1 - β_{A - j}^{r ω_{𝒷}})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix})$

Theorem 7. The LDULAHM Operator is a special of the LDULWAHM operator.

Proof. When $ω = {(\frac{1}{r}, \frac{1}{r}, \dots, \frac{1}{r})}^{T}$ , $\begin{matrix} {LDULWAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r ω_{𝒷} θ_{A - 𝒷})}^{f} {(r ω_{j} θ_{A - j})}^{h})}^{\frac{1}{f + h}} \\ = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} {(r \frac{1}{r} A_{LD - 𝒷})}^{f} {(r \frac{1}{r} A_{LD - j})}^{h})}^{\frac{1}{f + h}} = {(\frac{2}{r (r + 1)} \sum_{𝒷 = 1}^{r} \sum_{j = 1}^{r} A_{LD - 𝒷}^{f} A_{LD - j}^{h})}^{\frac{1}{p + q}} \\ = {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \end{matrix}$

Definition 12. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , the LDULGHM operator is elaborated by: $LDULGH M^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}))}^{\frac{2}{r (r + 1)}}$ (30)

Theorem 8. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , then by using Equation (30), we have $\begin{matrix} {LDULGHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = (\begin{matrix} [Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷} + h θ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷} + h τ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷}^{f} v_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷}^{f} β_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ (31)

Proof. $f A_{LD - 𝒷} = (\begin{matrix} [Ξ_{f θ_{A - 𝒷}}, Ξ_{f τ_{A - 𝒷}}], \\ (\begin{matrix} 1 - {(1 - u_{A - 𝒷})}^{f}, \\ v_{A - 𝒷}^{f} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - α_{A - 𝒷})}^{f}, \\ β_{A - 𝒷}^{f} \end{matrix}) \end{matrix})$ $h A_{LD - j} = (\begin{matrix} [Ξ_{h θ_{A - 𝒷}}, Ξ_{h τ_{A - 𝒷}}], \\ (\begin{matrix} 1 - {(1 - u_{A - 𝒷})}^{h}, \\ v_{A - 𝒷}^{h} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - α_{A - 𝒷})}^{h}, \\ β_{A - 𝒷}^{h} \end{matrix}) \end{matrix})$ then, $f A_{LD - 𝒷} + h A_{LD - j} = (\begin{matrix} [Ξ_{f θ_{A - 𝒷} + h θ_{A - 𝒷}}, Ξ_{f τ_{A - 𝒷} + h τ_{A - 𝒷}}], \\ (\begin{matrix} 1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}, \\ 1 - v_{A - 𝒷}^{f} v_{A - j}^{h} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}, \\ 1 - β_{A - 𝒷}^{f} β_{A - j}^{h} \end{matrix}) \end{matrix})$ $\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}) = (\begin{matrix} [Ξ_{\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷} + h θ_{A - 𝒷})}, Ξ_{\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷} + h τ_{A - 𝒷})}], \\ (\begin{matrix} \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}), \\ \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷}^{f} v_{A - j}^{h}) \end{matrix}), \\ (\begin{matrix} \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}), \\ \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷}^{f} β_{A - j}^{h}) \end{matrix}) \end{matrix})$ ${(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}))}^{\frac{2}{r (r + 1)}} = (\begin{matrix} [Ξ_{{(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷} + h θ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}, Ξ_{{(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷} + h τ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ 1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷}^{f} v_{A - j}^{h}))}^{\frac{1}{r (r + 1)}} \end{matrix}), \\ (\begin{matrix} 1 - (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ 1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷}^{f} β_{A - j}^{h}))}^{\frac{1}{r (r + 1)}} \end{matrix}) \end{matrix})$ $LDULGH M^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}))}^{\frac{2}{r (r + 1)}}$ $= (\begin{matrix} [Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷} + h θ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷} + h τ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷}^{f} v_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷}^{f} β_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix})$

By using Equation (31), we will demonstrate monotonicity, idempotency, and boundedness.

Theorem 9. For any LDULNs $(A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r})$ and $(B_{LD - 1}, B_{LD - 2}, \dots, B_{LD - r})$ , if $A_{LD - r} ⩽ B_{LD - r}$ for all $r = 1, 2, \dots, m$ then ${LDULGHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULGHM}^{f, h} (B_{LD - 1}, B_{LD - 2}, \dots, B_{LD - r})$ (32)

Proof. If $A_{LD - r} ⩽ B_{LD - r}$ for all $r = 1, 2, \dots, m$ , then $f A_{LD - 𝒷} ⩽ f B_{LD - 𝒷}$ then, $f A_{LD - 𝒷} + h A_{LD - j} ⩽ f B_{LD - 𝒷} + h B_{LD - j}$ $\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}) ⩽ \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f B_{LD - 𝒷} + h B_{LD - j})$ thus, ${(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}))}^{\frac{2}{r (r + 1)}} ⩽ {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f B_{LD - 𝒷} + h B_{LD - j}))}^{\frac{2}{r (r + 1)}}$ $\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD - 𝒷} + h A_{LD - j}))}^{\frac{2}{r (r + 1)}} ⩽ \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f B_{LD - 𝒷} + h B_{LD - j}))}^{\frac{2}{r (r + 1)}} .$

Theorem 10. For any LDULNs $A_{LD - j} = A, j = 1, 2, \dots, r$ then ${LDULGHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = A$ .

Proof. $\begin{matrix} LDULGH M^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = \\ \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f A_{LD} + h A_{LD}))}^{\frac{2}{r (r + 1)}} \end{matrix}$ $\begin{matrix} = \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} ((f + h) A_{LD}))}^{\frac{2}{r (r + 1)}} \\ = \frac{1}{f + h} {((f + h) A_{LD})}^{\frac{2}{2}} = A_{LD} . \end{matrix}$

Theorem 11. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , if LDULGHM operator lies between the max and min operators, then $\begin{matrix} m 𝒷 r (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULGHM}^{f, h} \\ (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ ma ϱ (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \end{matrix}$

Proof. Let $A = \min (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}), B = \max (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r})$ , according to Theorem 9, we have $\begin{matrix} {LDULGHM}^{f, h} (A, A, . ., A) ⩽ {LDULGHM}^{f, h} \\ (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULGHM}^{f, h} (B, B, \dots, B) \end{matrix}$ Further, ${LDULGHM}^{f, h} (A, A, . ., A) = A$ , ${LDULGHM}^{f, h} (B, B, \dots, B) = B$ so, $A = {LDULGHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = B$ i.e. $\min (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ {LDULGHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) ⩽ \max (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r})$ . It is easy to prove that ${LDULGHM}^{f, h}$ the operator does not have the property of commutativity.

As shown above, the reliability and consistency of the elaborated operators, we discussed the special cases of the investigated operators, which are stated below.

For $h \to 0$ , then Equation (31) reduces to an LDUL generalized geometric linear descending weighted mean operator, such that $\begin{matrix} {LDULGHM}^{f, 0} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = lim_{h \to 0} (\begin{matrix} [Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷} + h θ_{A - j}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷} + h τ_{A - j}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷}^{f} v_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷}^{f} β_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$

= (\begin{matrix} [Ξ_{\frac{1}{f} {(\prod_{𝒷 = 1}^{r} {(f θ_{A - 𝒷})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f} {(\prod_{𝒷 = 1}^{r} {(f τ_{A - 𝒷})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}}}], \\ (1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - u_{A - 𝒷})}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}, {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - v_{A - 𝒷}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}), \\ (1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - α_{A - 𝒷})}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}, {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - β_{A - 𝒷}^{f})}^{r + 1 - 𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f}}) \end{matrix})

(33)

For $f \to 0$ , the Equation (31) reduces to an LDUL generalized geometric linear ascending weighted mean operator, such that $\begin{matrix} {LDULGHM}^{0, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = lim_{f \to 0} (\begin{matrix} [Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷} + h θ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷} + h τ_{A - 𝒷}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - u_{A - 𝒷})}^{f} {(1 - u_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷}^{f} v_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - α_{A - 𝒷})}^{f} {(1 - α_{A - j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷}^{f} β_{A - j}^{h}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ $= (\begin{matrix} [Ξ_{\frac{1}{h} {(\prod_{𝒷 = 1}^{r} {(h θ_{A - 𝒷})}^{𝒷})}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{h} {(\prod_{𝒷 = 1}^{r} {(h τ_{A - 𝒷})}^{𝒷})}^{\frac{2}{r (r + 1)}}}], \\ (1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - u_{A - 𝒷})}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{h}}, {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - v_{A - 𝒷}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{h}}), \\ (1 - {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - {(1 - α_{A - 𝒷})}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{h}}, {(1 - {(\prod_{𝒷 = 1}^{r} {(1 - β_{A - 𝒷}^{h})}^{𝒷})}^{\frac{2}{r (r + 1)}})}^{\frac{1}{h}}) \end{matrix})$ (34)

For $f = h = \frac{1}{2}$ , the Equation (31) reduces to an LDUL basic geometric Heronian mean operator, such that $\begin{matrix} {LDULGHM}^{\frac{1}{2}, \frac{1}{2}} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = (\begin{matrix} [Ξ_{\frac{1}{2} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (θ_{A - 𝒷} + θ_{A - j}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{2} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (τ_{A - 𝒷} + τ_{A - j}))}^{\frac{2}{r (r + 1)}}}], \\ ({(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{(1 - u_{A - 𝒷}) (1 - u_{A - j})}))}^{\frac{2}{r (r + 1)}}, 1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{v_{A - 𝒷} v_{A - j}}))}^{\frac{1}{r (r + 1)}}), \\ ({(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{(1 - α_{A - 𝒷}) (1 - α_{A - j})}))}^{\frac{2}{r (r + 1)}}, 1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - \sqrt{β_{A - 𝒷} β_{A - j}}))}^{\frac{1}{r (r + 1)}}) \end{matrix}) \end{matrix}$ (35)

For $f = h = 1$ , the Equation (31) reduces to an LDUL geometric line Heronian mean operator, it follows that $\begin{matrix} {LDULGHM}^{1, 1} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = (\begin{matrix} [Ξ_{\frac{1}{2} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (θ_{A - 𝒷} + θ_{A - j}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{2} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (τ_{A - 𝒷} + τ_{A - j}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - (1 - u_{A - 𝒷}) (1 - u_{A - j})))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{2}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - v_{A - 𝒷} v_{A - j}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{2}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - (1 - α_{A - 𝒷}) (1 - α_{A - j})))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{2}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - β_{A - 𝒷} β_{A - j}))}^{\frac{1}{r (r + 1)}})}^{\frac{1}{2}} \end{matrix}) \end{matrix}) \end{matrix}$ (36)

Definition 13. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , the LDULWGHM operator is elaborated by: $LDULGH M^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} f {(A_{LD - 𝒷})}^{r ω 𝒷} + h {(A_{LD - j})}^{r ω j})}^{\frac{2}{r (r + 1)}}$ (37)where $ω = {(ω_{1}, ω_{2}, \dots, ω_{r})}^{T}$ is the weight vector of $(A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}), ω_{j} \in [0, 1], \sum_{j = 1}^{r} ω_{j} = 1$ . $r$ is a balance parameter.

Theorem 12. For any LDULNs $A_{LD - 𝒷} = ([Ξ_{A - θ_{𝒷}}, Ξ_{A - τ_{𝒷}}], (u_{A - 𝒷}, v_{A - 𝒷}), (α_{A - 𝒷}, β_{A - 𝒷})), 𝒷 = 1, 2, \dots, r$ , then by using Equation (31), such that $\begin{matrix} {LDULAHM}^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) \\ = (\begin{matrix} [Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f {(θ_{A - 𝒷})}^{r ω 𝒷} + h {(θ_{A - j})}^{r ω j}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f {(τ_{A - 𝒷})}^{r ω 𝒷} + h {(τ_{A - 𝒷})}^{r ω j}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(u_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - v_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(α_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - β_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix}) \end{matrix}$ (38)

Proof. Let $f {(A_{LD - 𝒷})}^{r ω 𝒷} (\begin{matrix} [Ξ_{f θ_{A - 𝒷}^{r ω j}}, Ξ_{f τ_{A - 𝒷}^{r ω j}}], \\ (\begin{matrix} 1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f}, \\ {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f}, \\ {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} \end{matrix}) \end{matrix}), h {(A_{LD - j})}^{r ω 𝒷} = (\begin{matrix} [Ξ_{h θ_{A - j}^{r ω j}}, Ξ_{h τ_{A - j}^{r ω j}}], \\ (\begin{matrix} 1 - {(1 - {(u_{A - j})}^{r ω 𝒷})}^{h}, \\ {(1 - {(1 - v_{A - j})}^{r ω 𝒷})}^{h} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(α_{A - j})}^{r ω 𝒷})}^{h}, \\ {(1 - {(1 - β_{A - j})}^{r ω 𝒷})}^{h} \end{matrix}) \end{matrix})$ thus, $f {(A_{LD - 𝒷})}^{r ω 𝒷} + h {(A_{LD - j})}^{r ω j} = (\begin{matrix} [Ξ_{f θ_{A - 𝒷}^{r ω j}}, Ξ_{f τ_{A - 𝒷}^{r ω j}}], \\ (\begin{matrix} 1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f}, \\ {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f}, \\ {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} \end{matrix}) \end{matrix}) \oplus (\begin{matrix} [Ξ_{h θ_{A - j}^{r ω j}}, Ξ_{h τ_{A - j}^{r ω j}}], \\ (\begin{matrix} 1 - {(1 - {(u_{A - j})}^{r ω 𝒷})}^{h}, \\ {(1 - {(1 - v_{A - j})}^{r ω 𝒷})}^{h} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(α_{A - j})}^{r ω 𝒷})}^{h}, \\ {(1 - {(1 - β_{A - j})}^{r ω 𝒷})}^{h} \end{matrix}) \end{matrix})$ $= (\begin{matrix} [Ξ_{f θ_{A - 𝒷}^{r ω j} + h θ_{A - j}^{r ω j}}, Ξ_{f τ_{A - 𝒷}^{r ω j} + h τ_{A - j}^{r ω j}}], \\ (\begin{matrix} (1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(u_{A - j})}^{r ω j})}^{h}), \\ (1 - {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - v_{A - j})}^{r ω j})}^{h}) \end{matrix}), \\ (\begin{matrix} (1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(α_{A - j})}^{r ω j})}^{h}), \\ (1 - {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - β_{A - j})}^{r ω j})}^{h}) \end{matrix}) \end{matrix})$ $\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} f {(A_{LD - 𝒷})}^{r ω 𝒷} \oplus h {(A_{LD - j})}^{r ω j} = (\begin{matrix} [Ξ_{\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f θ_{A - 𝒷}^{r ω j} + h θ_{A - j}^{r ω j})}, Ξ_{\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f τ_{A - 𝒷}^{r ω j} + h τ_{A - j}^{r ω j})}], \\ (\begin{matrix} \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(u_{A - j})}^{r ω j})}^{h}), \\ \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - v_{A - j})}^{r ω j})}^{h}) \end{matrix}), \\ (\begin{matrix} \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(α_{A - j})}^{r ω j})}^{h}), \\ \prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - β_{A - j})}^{r ω j})}^{h}) \end{matrix}) \end{matrix})$ $\begin{matrix} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} f {(A_{LD - 𝒷})}^{r ω 𝒷} \oplus h {(A_{LD - j})}^{r ω j})}^{\frac{2}{r (r + 1)}} \\ = (\begin{matrix} [Ξ_{{(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f {(θ_{A - 𝒷})}^{r ω 𝒷} + h {(θ_{A - j})}^{r ω j}))}^{\frac{2}{r (r + 1)}}}, Ξ_{{(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f {(τ_{A - 𝒷})}^{r ω 𝒷} + h {(τ_{A - 𝒷})}^{r ω j}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(u_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ 1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - v_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}} \end{matrix}), \\ (\begin{matrix} 1 - (1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(α_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}}), \\ 1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - β_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}} \end{matrix}) \end{matrix}) \end{matrix}$ $LDULGH M^{f, h} (A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}) = \frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} f {(A_{LD - 𝒷})}^{r ω 𝒷} \oplus h {(A_{LD - j})}^{r ω j})}^{\frac{2}{r (r + 1)}}$ $= (\begin{matrix} [Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f {(θ_{A - 𝒷})}^{r ω 𝒷} + h {(θ_{A - j})}^{r ω j}))}^{\frac{2}{r (r + 1)}}}, Ξ_{\frac{1}{f + h} {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (f {(τ_{A - 𝒷})}^{r ω 𝒷} + h {(τ_{A - 𝒷})}^{r ω j}))}^{\frac{2}{r (r + 1)}}}], \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(u_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(u_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - v_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - v_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(α_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(α_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}}, \\ {(1 - {(\prod_{𝒷 = 1}^{r} \prod_{j = 1}^{r} (1 - {(1 - {(1 - β_{A - 𝒷})}^{r ω 𝒷})}^{f} {(1 - {(1 - β_{A - j})}^{r ω j})}^{h}))}^{\frac{2}{r (r + 1)}})}^{\frac{1}{f + h}} \end{matrix}) \end{matrix})$

5 MADM technique for LDULSs using HM operator

In this part, we promote the examination of HM operators of LDULNs in the MADM problem. We promote weighted HM operators of LDULNs in the MADM problem. The impact of $R$ on the ranking results is investigated thoroughly.

MADM is a way of choosing the best alternative because of some finite attributes using the HM operators of LDULSs. The most favorable thing here is that the information is based on LDULNs which discuss the membership grades and non-membership grades of the information. Let the collection of alternatives be A_k (k is finite) and attributes be G_j (j is finite) which form a decision matrix denoted by D_k×j = (T) _k×j = (Ξ, 𝔯, d) where the terms in triplet denote the membership grades, abstinence, and non-membership grades of the information, where $ω = {(ω_{1}, ω_{2}, \dots, ω_{r})}^{T}$ is the weight vector of $(A_{LD - 1}, A_{LD - 2}, \dots, A_{LD - r}), ω_{j} \in [0, 1], \sum_{j = 1}^{r} ω_{j} = 1$ . A brief algorithm of the MADM process is illustrated in the followingsection.

5.1 Algorithm

The algorithm of MADM based on LDUL information and using LDULAHM, LDULWAHM, LDULGHM, and LDULWGHM operators is proposed as follows:

Step1: In this step, we collect the information about alternatives given by the decision-makers. The decision-makers gave their opinion about alternates in the form of LDULNs which leads to the formation of the decision matrix.

Step 2: If there exist attributes of cost type, we normalize the decision matrix by taking the complement of each triplet in the matrix. If not, then we use the following LDULAHM, LDULWAHM, LDULGHM, and LDULWGHM operators to aggregate the data given in the decisionmatrix.

Step 3: In this step, we compute the scores of the aggregated information using Definition 3.

Step 4: This step is based on the ranking of the alternatives.

An illustrative example to see the viability of the proposed algorithm.

5.2 Illustrated example

Organizing a software company is a very specialized type of management skill, where experienced persons can turn the organizational problem into a unique benefit. For example, having sub-teams spread in different time zones may allow a 24-hour company working day, if the teams, systems, and procedures are well established. A good example is the test team in a time zone 8 hours ahead or behind the development team, who fix software bugs found by the testers.

Software development companies design, develop and maintain applications, frameworks, or other software components for businesses or consumers.

To get a deeper understanding of what this process involves, let’s start by talking about what software development is. Software development is the process of conceiving, specifying, designing, programming, documenting, testing, and bug fixing involved in creating and maintaining applications, frameworks, or other software components.

A software development company puts all of these pieces together. This includes everything from the software’s conception to the final manifestation of the software— research, new development, prototyping, modification, reuse, re-engineering, maintenance, and more.

An example of technology commercialization is adapted from [31] where the selection of the most favorable software enterprise among a list of enterprises is carried out. Let us consider four software enterprises denoted by A_1⩽k⩽4 and the attributes are denoted by G_1⩽j⩽4 where G₁: Advancement of the technology, G₂: Market Potential, G₃: Human resources and financial development, G₄: Creating of Employment and development of technology The weight vector chosen in this case is ω = (0.15, 0.25, 0.41, 0.19) ^T. The selection of weight vector is up to the decision-makers and the information about the alternatives in terms of LDULNs is given in form of Table 2.

Table 2
Original decision matrix

$G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$

$A_{1}$ $(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.5, 0.4), \\ (0.6, 0.3) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.51, 0.41), \\ (0.61, 0.31) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.52, 0.42), \\ (0.62, 0.32) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.53, 0.43), \\ (0.63, 0.33) \end{matrix})$

$A_{2}$ $(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.6, 0.5), \\ (0.4, 0.2) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.61, 0.51), \\ (0.41, 0.21) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.62, 0.52), \\ (0.42, 0.22) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.63, 0.53), \\ (0.43, 0.23) \end{matrix})$

$A_{3}$ $(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.7, 0.1), \\ (0.5, 0.2) \end{matrix})$ $(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.71, 0.11), \\ (0.51, 0.21) \end{matrix})$ $(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.72, 0.12), \\ (0.52, 0.22) \end{matrix})$ $(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.73, 0.13), \\ (0.53, 0.23) \end{matrix})$

$A_{4}$ $(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.8, 0.3), \\ (0.4, 0.3) \end{matrix})$ $(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.81, 0.31), \\ (0.41, 0.31) \end{matrix})$ $(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.82, 0.32), \\ (0.42, 0.32) \end{matrix})$ $(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.83, 0.33), \\ (0.43, 0.33) \end{matrix})$

	$G_{1}$	$G_{2}$	$G_{3}$	$G_{4}$
$A_{1}$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.5, 0.4), \\ (0.6, 0.3) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.51, 0.41), \\ (0.61, 0.31) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.52, 0.42), \\ (0.62, 0.32) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.53, 0.43), \\ (0.63, 0.33) \end{matrix})$
$A_{2}$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.6, 0.5), \\ (0.4, 0.2) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.61, 0.51), \\ (0.41, 0.21) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.62, 0.52), \\ (0.42, 0.22) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.63, 0.53), \\ (0.43, 0.23) \end{matrix})$
$A_{3}$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.7, 0.1), \\ (0.5, 0.2) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.71, 0.11), \\ (0.51, 0.21) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.72, 0.12), \\ (0.52, 0.22) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.73, 0.13), \\ (0.53, 0.23) \end{matrix})$
$A_{4}$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.8, 0.3), \\ (0.4, 0.3) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.81, 0.31), \\ (0.41, 0.31) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.82, 0.32), \\ (0.42, 0.32) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.83, 0.33), \\ (0.43, 0.33) \end{matrix})$

Step 2: In this step, the information given in Table 2 is aggregated using LDULAHM, LDULWAHM, LDULGHM, and LDULWGHM operators. The results of the aggregation are given in Table 3 below.

Table 3

Aggregated values by using different operators

	LDULWAHM operator	LDULWGHM operator	LDULAHM operator	LDULGHM operator
A ₁	$(\begin{matrix} [Ξ_{0}, Ξ_{0.4123}], \\ (0.9495, 0.0508), \\ (0.9681, 0.0703) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{0.4123}], \\ (0.0351, 0.9266), \\ (0.0222, 0.8981) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (0.7597, 0.25), \\ (0.8478, 0.3509) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (0.171, 0.6517), \\ (0.1078, 0.5192) \end{matrix})$ .
A ₂	$(\begin{matrix} [Ξ_{0.4123}, Ξ_{0.8246}], \\ (0.9681, 0.0351), \\ (0.9266, 0.0913) \end{matrix})$	$(\begin{matrix} [Ξ_{4123}, Ξ_{8246}], \\ (0.0222, 0.9495), \\ (0.0508, 0.8677) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{4}], \\ (0.8478, 0.171), \\ (0.6517, 0.4676) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{4}], \\ (0.1078, 0.7597), \\ (0.25, 0.3755) \end{matrix})$
A ₃	$(\begin{matrix} [Ξ_{0.8246}, Ξ_{1.2369}], \\ (0.9825, 0.116), \\ (0.9495, 0.0913) \end{matrix})$	$(\begin{matrix} [Ξ_{0.8246}, Ξ_{1.2369}], \\ (0.0123, 0.8316), \\ (0.0351, 0.8677) \end{matrix})$	$(\begin{matrix} [Ξ_{4}, Ξ_{6}], \\ (0.9158, 0.6242), \\ (0.7597, 0.4676) \end{matrix})$	$(\begin{matrix} [Ξ_{4}, Ξ_{6}], \\ (0.0595, 0.207), \\ (0.171, 0.3755) \end{matrix})$
A ₄	$(\begin{matrix} [Ξ_{1.2369}, Ξ_{1.6492}], \\ (0.9925, 0.0696), \\ (0.9266, 0.0696) \end{matrix})$	$(\begin{matrix} [Ξ_{1.2369}, Ξ_{1.6492}], \\ (0.0053, 0.8994), \\ (0.0508, 0.8994) \end{matrix})$	$(\begin{matrix} [Ξ_{6}, Ξ_{8}], \\ (0.9638, 0.3473), \\ (0.6517, 0.3473) \end{matrix})$	$(\begin{matrix} [Ξ_{6}, Ξ_{8}], \\ (0.0255, 0.5236), \\ (0.25, 0.5236) \end{matrix})$

Step 4: In this step, the aggregated information obtained in Table 4 is targeted and their scores are computed using Definition 3.

Table 4

Score Values of the aggregated operators

Scores	LDULWAHM operator	LDULWGHM operator	LDULAHM operator	LDULGHM operator
A ₁	0.1514	0.053	0.6771	0.2866
A ₂	0.4563	0.1533	2.097	0.7302
A ₃	0.7492	0.2792	3.1486	1.7486
A ₄	1.0531	0.381	4.5771	2.2322

The results portrayed in Table 4 are further described geometrically in the following Fig. 1.

Fig. 1

Graphical expressions of the information of Table 6.

Step 5: Based on the scores obtained in Table 4, the alternatives are ranked, and the ranking results are given in Table 6 as follows:

Table 5

Ranking Values of the information in Table 4

	Ranking analysis
LDULWAHM operator	A₄ > A₃ > A₂ > A₁
LDULWHMG operator	A₄ > A₃ > A₂ > A₁
LDULAHM operator	A₄ > A₃ > A₂ > A₁
LDULHMG operator	A₄ > A₃ > A₂ > A₁

Table 6

Original decision matrix

	$G_{1}$	$G_{2}$	$G_{3}$	$G_{4}$
$A_{1}$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.3, 0.2), \\ (0.1, 0.2) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.31, 0.21), \\ (0.11, 0.21) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.32, 0.22), \\ (0.12, 0.22) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.33, 0.23), \\ (0.13, 0.23) \end{matrix})$
$A_{2}$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.4, 0.3), \\ (0.2, 0.3) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.41, 0.31), \\ (0.21, 0.31) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.42, 0.32), \\ (0.22, 0.32) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.43, 0.33), \\ (0.23, 0.33) \end{matrix})$
$A_{3}$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.7, 0.1), \\ (0.5, 0.2) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.71, 0.11), \\ (0.51, 0.21) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.72, 0.12), \\ (0.52, 0.22) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.73, 0.13), \\ (0.53, 0.23) \end{matrix})$
$A_{4}$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.3, 0.3), \\ (0.2, 0.3) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.31, 0.31), \\ (0.21, 0.31) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.32, 0.32), \\ (0.22, 0.32) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.33, 0.33), \\ (0.23, 0.33) \end{matrix})$
$A_{5}$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.31, 0.21), \\ (0.11, 0.21) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.32, 0.22), \\ (0.12, 0.22) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.34, 0.24), \\ (0.14, 0.24) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], (0.35, 0.25), \\ (0.15, 0.25) \end{matrix})$
$A_{6}$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.41, 0.31), \\ (0.21, 0.31) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.42, 0.32), \\ (0.22, 0.32) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.44, 0.34), \\ (0.24, 0.34) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], (0.45, 0.35), \\ (0.25, 0.35) \end{matrix})$
$A_{7}$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.71, 0.11), \\ (0.51, 0.21) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.72, 0.12), \\ (0.52, 0.22) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.74, 0.14), \\ (0.54, 0.24) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], (0.75, 0.15), \\ (0.55, 0.25) \end{matrix})$
$A_{8}$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.31, 0.31), \\ (0.21, 0.31) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.32, 0.32), \\ (0.22, 0.32) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.34, 0.34), \\ (0.24, 0.34) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], (0.35, 0.35), \\ (0.25, 0.35) \end{matrix})$

As shown above, we obtained the same ranking results, which are discussed in the form of Table 5. The best option is A₄.

Moreover, to investigate the supremacy and consistency of the initiated operators based on the LDULS, we choose any sort of information and resolved it by using elaborated operators. The information which covers the four attributes and eight alternatives are discussed in the shape of Table 6.

By using the information in Table 6, the score and ranking values of the initiated works are discussed in the shape of Table 8.

Table 7

Ranking Values of the information in Table 4

Methods	Score values	Ranking analysis
LDULWAHM operator	0.2324, 0.2430, 0.63745, 0.2635,	A₃ > A₅ > A₇ > A₄
	0.4563, 0.2423, 0.2653, 0.2432	>A₈ > A₂ > A₆ > A₁
LDULWHMG operator	0.3435, 0.3541, 0.7485, 0.3746,	A₃ > A₅ > A₇ > A₄
	0.5674, 0.3534, 0.3764, 0.3543	>A₈ > A₂ > A₆ > A₁
LDULAHM operator	0.4546, 0.4652, 0.8596, 0.4857,	A₃ > A₅ > A₇ > A₄
	0.6785, 0.4645, 0.4875, 0.4654	>A₈ > A₂ > A₆ > A₁
LDULHMG operator	0.1213, 0.1320, 0.5263, 0.1524,	A₃ > A₅ > A₇ > A₄
0.3452, 0.1312, 0.1542, 0.1321	>A₈ > A₂ > A₆ > A₁

Table 8

Comparative analysis of the elaborated operators and existing operators

Method	Ranking analysis
Riaz and Hashmi [44]	A₄ > A₃ > A₂ > A₁
LDULWAHM operator	A₄ > A₃ > A₂ > A₁
LDULWHMG operator	A₄ > A₃ > A₂ > A₁
LDULAHM operator	A₄ > A₃ > A₂ > A₁
LDULHMG operator	A₄ > A₃ > A₂ > A₁

From the above information, we obtained the same ranking result, the best alternative is A₃.

5.3 Comparative analysis

Based on the investigated LDULWAHM, LDULWGHM, LDULAHM, and LDULGHM operators, we determined the reliability and consistency of the developed operators with the help of comparative analysis by using the information of Table 2 shown in 5.2. The information related to existing theories is as followed: Riaz and Hashmi [44] elaborated the aggregation operators for LDFS and their applications. The reasonable analysis of the investigated operators and remaining operators are discussed in the form of Table 8 by using the information of Table 2, by using the values of parameters $f = h = 1, r = 3$ .

The results portrayed in Table 6 are further described geometrically in the following Fig. 2.

Fig. 2

Graphical expressions of the information of Table 7.

As shown above, we chose the linear Diophantine uncertain linguistic types of information’s so the operators, investigated by Riaz and Hashmi [44] are not able to resolve it. However, if we decide the exiting types of information’s then the investigated sort of information can cope with it. therefore, the investigated operators based on LDULSs are more powerful to determine the rationality and consistency of the developed operators.

6 Conclusion

To handle some awkward and imprecise data in this manuscript, we have initiated the novel idea of LDULS and elaborated their preliminary laws. Furthermore, to regulate the association among any number of attributes, we particularized the LDULAHM operator, LDULWAHM operator, LDULGHM operator, LDULWGHM operator, and their properties are also exposed. By using these operators, we develop a MADM procedure based on explained operators. To determine the consistency and validity of the elaborated operators, we illustrate some examples by using explored operators. Finally, the superiority and comparative analysis of the elaborated operators with some existing operators are also determined and justify with the help of a graphical point of view.

In the upcoming time, we will discuss the complex spherical fuzzy sets [47], complex T-spherical fuzzy sets [48], intuitionistic trapezoidal fuzzy number [49], Power-Average-Operator-Based Hybrid Multi-Attribute Online Product Recommendation Model for Consumer Decision-Making [50]; Constructing the geometric Bonferroni mean from the generalized Bonferroni mean with several extensions to linguistic 2-tuples for decision-making [51]; On Generalized Extended Bonferroni Means for Decision Making [52], Managing ignorance elements and personalized individual semantics under incomplete linguistic distribution context in group decision making [53], Revisiting fuzzy and linguistic decision making: Scenarios and challenges for making wiser decisions in a better way [54].

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