# Linear Diophantine Uncertain Linguistic Power Einstein Aggregation Operators and Their Applications to Multiattribute Decision Making

Authors: Tahir Mahmood; Izatmand; Zeeshan Ali; Kifayat Ullah; Qaisar Khan; Ahmed Alsanad; Mogeeb A. A. Mosleh
Journal: Complexity
 (2021)

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## Abstract

Linear Diophantine uncertain linguistic set (LDULS) is a modified variety of the fuzzy set (FS) to manage problematic and inconsistent information in actual life troubles. LDULS covers the grade of truth, grade of falsity, and their reference parameters with the uncertain linguistic term (ULT) with a rule0≤αAMGuAMGx+βANGvAMGx≤1, where 0≤αAMG+βANG≤1. In this study, the principle of LDULS and their useful laws are elaborated. Additionally, the power Einstein (PE) aggregation operator (AO) is a conventional sort of AO utilized in innovative decision-making troubles, which is effective to aggregate the family of numerical elements. To determine the interrelationship between any numbers of arguments, we elaborate the linear Diophantine uncertain linguistic PE averaging (LDULPEA), linear Diophantine uncertain linguistic PE weighted averaging (LDULPEWA), linear Diophantine uncertain linguistic PE geometric (LDULPEG), and linear Diophantine uncertain linguistic PE weighted geometric (LDULPEWG) operators; then, we discuss their useful results. Conclusively, a decision-making methodology is utilized for the multiattribute decision-making (MADM) dilemma with elaborated information. A sensible illustration is specified to demonstrate the accessibility and rewards of the intended technique by comparison with certain prevailing techniques. The intended AOs are additional comprehensive than the prevailing ones to exploit the ambiguous and inaccurate knowledge. Numerous remaining operators are chosen as individual incidents of the suggested one. Ultimately, the supremacy and advantages of the elaborated operators are also discussed with the help of the geometrical form to show the validity and consistency of explored operators.


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## Body

## 1. Introduction

MADM is a technique to discover the ideal option between a family of possibilities and a family of certain opinions. The perception of MADM has extensively been utilized in numerous disciplines. Ambiguity usually happened in genuine life troubles owing to the contribution of some convoluted restrictions, unavailability of evidence/information, and unpredictability of the challenge. To survive with this worry, Zadeh [1] expounded the theory of the fuzzy set (FS), which covers the grade of truth uAMGx 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] initiated the soft sets, Mahmood [2] elaborated the new bipolar soft sets, Somjanta et al. [3] utilized the fuzzy sets in UP-algebra, Noppharat Dokkhamdang et al. [4] utilized the generalized fuzzy sets in UP-algebra, Tanamoon et al. [5] explored the Q-fuzzy sets and utilized in UP-algebra, Kawila et al. [6] explored the bipolar fuzzy UP-algebra, Mahmood and Hayat [7] initiated the characterizations of hemirings by their bipolar-valued fuzzy h-ideals, and Mahmood et al. [8] initiated the lattice ordered soft near ring. However, in certain situations, the principle of FS cannot be working effectively; for instance, when an individual face such sorts of information, which contains the truth and falsity grades, then FS failed. To survive with such circumstances, Atanassov [9] elaborated the theory of intuitionistic FS (IFS), which covers the grade of truth and falsity grades with the rule 0≤uAMGx+vANGx≤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 [10] initiated the intuitionistic hesitant fuzzy sets, Atanassov [11] explored the interval-valued intuitionistic fuzzy sets, Kumari and Mishra [12] investigated the parametric measures for IFSs, Jana and Pal [13] explored bipolar intuitionistic fuzzy soft sets, Joshi and Kumar [14] developed the fuzzy time series model based on intuitionistic fuzzy sets, Fu et al. [15] proposed the correlation measures by using the interval-valued intuitionistic fuzzy sets, and Meng and He [16] elaborated the geometric interaction aggregation operators by using the intuitionistic fuzzy sets.Again, in certain situations, the principle of IFS cannot be working effectively; for instance, when an individual faces such sorts of information which contains the truth and falsity grades with the rule0≤uAMG2x+vANG2x≤1, then IFS failed. To survive with such circumstances, Yager [17] elaborated the theory of Pythagorean FS (PFS), which covers the grade of truth and falsity grades with the rule 0≤uAMG2x+vANG2x≤1. When the theory of PFS was elaborated, then numerous scholars have utilized it in the environment of several areas. For instance, Garg [18] explored the linguistic PFS, Wei and Wei [19] developed the similarity measures for PFS, Xiao and Ding [20] investigated the diverges measures by using the PFS and their application in decision making, Ullah et al. [21] explored some similarity measures based on a novel complex PFS and their application in pattern recognition, Li and Lu [22] proposed some similarity and distance measures for PFSs, Garg [23] proposed the novel improved accuracy function for interval-valued PFS and their application in decision making, and Yang and Hussain [24] investigated some entropy measures for PFS and their application.Again, in certain situations, the principle of PFS cannot be working effectively; for instance, when an individual face such sorts of information which contains the truth and falsity grades with the rule0≤uAMGqx+vANGqx≤1, then PFS failed. To survive with such circumstances, Yager [25] elaborated the theory of q-rung orthopair FS (q-ROFS), which covers the grade of truth and falsity grades with the rule 0≤uAMGqx+vANGqx≤1. When the theory of q-ROFS was elaborated, then numerous scholars have utilized it in the environment of several areas; for instance, Ali [26] created another view on q-ROFSs. Liu and Wang [27] elaborated the aggregation operators for q-ROFSs. Peng and Liu [28] investigated the information measures for q-ROFSs. Wang et al. [29] explored the similarity measures for q-ROFSs. Ali and Mahmood [30] investigated the complex q-ROFSs and their Maclaurin symmetric mean operators. Liu et al. [31] proposed cosine similarity and distance measures for q-ROFSs. Liu and Wang [32] investigated the Archimedean Bonferroni mean operators for q-ROFSs. Liu et al. [33] explored the linguistic q-ROFSs. Garg [34] investigated the possibility degree for interval-valued q-ROFSs. The power aggregation operators for complex q-ROFSs were developed by Garg et al. [35].To handle such sorts of concerns, Zadeh [36] 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 [37]. Liu and Jin [38] investigated the uncertain LV (ULV). In approximate actual existence troubles, the sum of truth and falsity grades to which an option filling an ascribe offered by the decision-maker could not hold the rule of IFS, PFS, and q-ROFSs; then, the theory of IFS and PFS fails in such situations. To survive with such circumstances, Riaz and Hashmi [39] elaborated the theory of linear Diophantine FS (LDFS), which covers the grade of truth and falsity grades and their reference parameters with the rule 0≤αAMGuAMGϱ+βANGvAMGϱ≤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. [40] discovered the theory of linear Diophantine fuzz soft rough sets and their applications. Some algebraic structures based on LDFs were developed by Kamaci [41]. But, up to date, no study explored the theory of LDULVs and their operational laws.The concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs), q-rung orthopair fuzzy sets (q-ROFSs), and linear Diophantine fuzzy sets have numerous applications in various fields of real life, but these theories have their own limitations related to the membership and nonmembership grades. To eradicate these restrictions, we introduce the novel concept of the linear Diophantine uncertain linguistic set (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 enlarge the space for membership and nonmembership 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 study.(1)
The theory of LDULS is more generalized than IFSs, PFSs, q-ROFSs, LDFSs, and ULVs(2)
If we choose the information in the form of0.5,0.6, then by using the condition of IFSs, that is, the sum of both terms is limited to unit interval, but 0.5+0.6=1.1>1, the theory of IFS has been failed for coping with such sorts of issues, and 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, 0.1∗0.5+0.2∗0.6=0.05+0.12=0.17<1. We clear that the IFS is the special case of the proposed LDULS.(3)
If we choose the information in the form of0.8,0.9, then by using the condition of PFSs, that is, the sum of the square of both terms is limited to 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, and 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, 0.2∗0.8+0.2∗0.9=0.16+0.18=0.34<1. We clear that the theory of PFS is the special case of the proposed LDULS.(4)
If we choose the information in the form of0.1,0.1, then by using the condition of q-ROFSs, that is, the sum of the q-powers of both terms is limited to unit interval, but 1+1=2>1, the theory of q-ROFS has been failed for coping with such sorts of issues, and 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, 0.0∗1+0.1∗1=0+0.1=0.1<1. We clear that the theory of q-ROFS is the special case of the proposed LDULS.(5)
If we choose the information in the form ofs1,s2,0.5,0.3,0.5,0.4, then using the condition of IFSs, PFSs, q-ROFSs, and LDFS has been failed; for coping with such sorts of issues, the theory of LDULS is very proficient and reliable technique to resolve with it. From the above analysis, the theories of IFSs, PFSs, q-ROFSs, and LDFSs are the special case of the proposed LDULS.The opinions of IFSs, PFSs, q-ROFSs, and LDFSs have repeated applications in uncountable fields 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.The rest of this manuscript as follows. In Section2, we briefly recall some definitions such as LDFSs, ULSs, and their operational laws. The theory of power aggregation (PA) operator is also reviewed. In Section 3, we notified the novel idea of LDULS and elaborated their fundamental laws. In Section 4, we elaborated the LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators 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 study.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 as shown in Figure1.Figure 1
Geometrical expressions of the explored approaches.


## 2. Preliminaries

Certain scholars have proposed numerous theories such as fuzzy sets, intuitionistic fuzzy sets, and their extensions to cope with awkward and complicated information in real-life issues. But these theories have their own limitations related to the membership and nonmembership grades. To eradicate these restrictions, the theory of the linear Diophantine fuzzy set (LDFS) with the addition of reference parameters was explored by Riaz and Hashmi [39]. The proposed model of LDFS is more efficient and flexible rather than other approaches due to the use of reference parameters. LDFS also categorize the data in MADM problems by changing the physical sense of reference parameters. This set covers the spaces of existing structures and enlarge the space for membership and nonmembership grades with the help of reference parameters. Keeping the advantages of the existing LDFS, in this section, we briefly recall some definitions such as LDFSs, ULSs, and their operational laws. The theory of the PA operator is also reviewed. In an overall study, the universal set is denoted by X.Definition 1 (see [39]).
A LDFSALD is elaborated by(1)ALD=x,uAMGx,vANGx,αAMG,βANGx∈X,with a rule 0≤αAMGuAMGx+βANGvAMGx≤1, with 0≤αAMG+βANG≤1. The symbol ξALDπxALD=1−αAMGuAMGx+βANGvANGx expressed the refusal grade. Simply, ALD=uAMGx,vANGx,αAMG,βANG is called linear Diophantine fuzzy number (LDFN). To determine the interrelationships between any numbers of attributes, we revised some algebraic operation laws which are very important for proposed work in the next study.Definition 2 (see [39]).
For any two LDFNsALD=uAMGx,vANGx,αAMG,βANG and BLD=uBMGx,vBNGx,αBMG,βBNG,(2)ALD⊕BLD=uAMG+uBMG−uAMGuBMG,vAMGvBMG,αAMG+αBMG−αAMGαBMG,βANGβBNG,(3)ALD⊗BLD=uAMGuBMG,vAMG+vBMG−vAMGvBMG,αAMGαBMG,βANG+βBNG−βANGβBNG,(4)λALD=1−1−uAMGxλ,vANGxλ,1−1−αAMGλ,βANGλ,(5)ALDλ=uAMGxλ,1−1−vANGxλ,αAMGλ,1−1−βANGλ.
Furthermore, by using any number of LDFNs, we revised the score values for determining the ordering between any two LDFNs.Definition 3 (see [39]).
For any LDFNALD1=uAMG1x,vANG1x,αAMG1,βANG1, the score value (SV) is elaborated by(6)ςALD=ςALD=12uAMGx−vANGx+α+β.
Furthermore, if equation (6) has been failed by comparing any number of LDFNs, then we used the technique of accuracy function, so, for this, we revised the accuracy values for determining the ordering between any two LDFNs.Definition 4 (see [39]).
For any LDFNALD1=uAMG1x,vANG1x,αAMG1,βANG1, the accuracy value (AV) is elaborated by(7)ϱALD=ϱALD=12uAMGx+vANGx2+α+β.Definition 5 (see [39]).
For any two LDFNsALD=uAMGx,vANGx,αAMG,βANG and BLD=uBMGx,vBNGx,αBMG,βBNG,(1)
IfςALD<ςBLD, then ALD<BLD(2)
IfςALD>ςBLD, then ALD>BLD(3)
IfςALD=ςBLD, then(1)
IfϱALD<ϱBLD, then ALD<BLD(2)
IfϱALD>ϱBLD, then ALD>bLD(3)
IfϱALD=ϱBLD, then ALD≈BLD
When faced with problems that are too complex or ill-defined to be solved by quantitative expressions, linguistic variables can be an effective tool because the use of linguistic information enhances the reliability and flexibility of classical decision models. To handle such sorts of concerns, Zadeh [36] 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 [37]. Liu and Jin [38] investigated the uncertain LV (ULV).Definition 6 (see [36]).
A LTS is elaborated by(8)S=s1,s2,…,sl−1,where l should be odd and hold the following conditions:(1)
Ifl>l′, then sl>sl′(2)
The negative operatornegsl=sl′ with a condition l+l′=l+1(3)
Ifl≥l′,maxsl,sl′=sl, and if l≤l′, maxsl,sl′=sl.
Lets˜=sa,sb where sa,sb∈S, sa and sb are the lower and the upper limits, respectively. We call s˜ the uncertain linguistic variable. To determine the interrelationships between any numbers of attributes, we revised some operation laws for ULVs which are very important for the proposed work in next study.Definition 7 (see [38]).
For any two ULVss˜1=sa1,sb1 and s˜2=sa2,sb2,(9)s˜1⊕s˜2=sa1,sb1⊕sa2,sb2=sa1+a2,sb1+b2,s˜1⊗s˜2=sa1,sb1⊗sa2,sb2=sa1×a2,sb1×b2,λs˜1=λsa1,sb1=sλ∗a1,sλ∗b1,s˜1λ=sa1λ,sb1λ.Definition 8 (see [33]).
The PA operator is elaborated by(10)PAALD−1,ALD−2,…,ALD−n=∑i=1n1+TALD−i∗ALD−i∑i=1n1+TALD−i,where TALD−i=∑j=1nsupALD−i,ALD−j, and supALD−i,ALD−j means the support for ALD−i from ALD−j, which holds the following rules:(1)
supALD−i,ALD−j=supALD−j,ALD−i(2)
supALD−i,ALD−j∈0,1(3)
supALD−i,ALD−j≥supALD−k,ALD−l, if ALD−i,ALD−j≤ALD−k,ALD−l


## 3. 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 9.
A LDULSALDULS is elaborated by(11)ALDULS=xsθX,sτX,uAMGx,vANGx,αAMG,βANGx∈X,with the rule 0≤αAMGuAMGx+βANGvAMGx≤1, with 0≤αAMG+βANG≤1. The symbol ξALDπxALD=1−αAMGuAMGx+βANGvANGx expressed the refusal grade with sθX,sτX∈S˜. Simply, ALD=sθX,sτX,uAMGx,vANGx,αAMG,βANG expresses the linear Diophantine uncertain linguistic number (LDULN).Definition 10.
For any two LDULNsALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG and BLD=sB−θX,sB−τX,uBMGx,vBNGx,αBMG,βBNG, then(12)ALD⊕BLD=sA−θX+B−θX,sA−τX+B−τX,uAMG+uBMG−uAMGuBMG,vAMGvBMG,αAMG+αBMG−αAMGαBMG,βANGβBNG,ALD⊗BLD=sA−θX∗B−θX,sA−τX∗B−τX,uAMGuBMG,vAMG+vBMG−vAMGvBMG,αAMGαBMG,βANG+βBNG−βANGβBNG,λALD=sλ×A−θX,sλ×A−τX,1−1−uAMGxλ,vANGxλ,1−1−αAMGλ,βANGλ,ALDλ=sA−θXλ,sA−τXλ,uAMGxλ,1−1−vANGxλ,αAMGλ,1−1−βANGλ.Example 1.
For any two LDULNsALD=s1,s2,0.5,0.3,0.5,0.4 and BLD=s2,s3,0.7,0.5,0.1,0.3, with λ=2, then by using Definition 10, we get(13)ALD⊕BLD=s3,s5,0.42,0.15,0.55,0.12,ALD⊗BLD=s2,s6,0.15,0.42,0.12,0.55,2ALD=s2,s4,0.75,0.09,0.75,0.16,ALD2=s1,s4,0.09,0.75,0.16,0.75.Theorem 1.
For any two LDULNsALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG and BLD=sB−θX,sB−τX,uBMGx,vBNGx,αBMG,βBNG,(1)
ALD⊕BLD=BLD⊕ALD(2)
ALD⊗BLD=BLD⊗ALD(3)
λALD⊕BLD=λALD⊕λBLD,λ>0(4)
λ1ALD⊕λ2ALD=λ1+λ2ALD,λ1,λ2>0(5)
ALDλ1⊗ALDλ2=ALDλ1+λ2,λ1,λ2>0(6)
ALDλ1⊗ALDλ1=ALD⊗BLDλ1,λ1>0Definition 11.
For any LDULNALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG, the expected value (EV) is elaborated by(14)EALD=sA−θX+A−τX×1/2uAMGx+1−vANGx+α+β/4.Example 2.
For any LDULNALD=s1,s2,0.5,0.3,0.5,0.4, the expected value (EV) is elaborated as(15)EALD=s1+2×1/20.5+1−0.3+0.5+0.4/4=s3×1.05/4=s0.7875.Definition 12.
For any LDULNALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG, the accuracy value (AV) is elaborated by(16)ςALD=ςALD=sua1+va1×1/2uAMGx+vANGx+α+β/4.Definition 13.
For any two LDULNsALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG and BLD=sB−θX,sB−τX,uBMGx,vBNGx,αBMG,βBNG, then(1)
IfςALD1<ςALD2, then ALD1<ALD2(2)
IfςALD1>ςALD2, then ALD1>ALD2(3)
IfςALD1=ςALD2, then(1)
IfϱALD1<ϱALD2, then ALD1<ALD2(2)
IfϱALD1>ϱALD2, then ALD1>ALD2(3)
IfϱALD1=ϱALD2, then ALD1≈ALD2


## 4. The Power Aggregation (PA) Operator

Here, we review the basic Einstein t-norm and t-conorm, which are useful for the elaborated approaches.(1)
a⊗εb=a+b/1+a⋅b,a,b∈0,1(2)
a⊕εb=a⋅b/1+1−a⋅1−b,a,b∈0,1By using any two LDULNsALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG and BLD=sB−θX,sB−τX,uBMGx,vBNGx,αBMG,βBNG,(17)ALD⊗εBLD=sA−θX⋅B−θX,sA−τX⋅B−τX,uAMGx⋅uBMGx1+1−uAMGx⋅1−uBNGx,vANGx+vBNGx1+vANGx⋅vBNGx,αAMG⋅αBMG1+1−αAMG⋅1−αBMG,βANG+βBNG1+βANG⋅βBNG,ALD⊗εBLD=sA−θX+B−θX,sA−τX+B−τX,uAMGx+uBMGx1+uAMGx⋅uBMGx,vANGx⋅vBNGx1+1−vANGx⋅1−vBNGx,αAMG+αBMG1+αAMG⋅αBMG,βANG⋅βBNG1+1−βANG⋅1−βBNG,λALD=sλ⋅A−θX,sλ⋅A−τX,1+uAMGxλ−1−uAMGxλ1+uAMGxλ+1−uAMGxλ,2vANGxλ2−vANGxλ+vANGxλ,1+αAMGλ−1−αAMGλ1+αAMGλ+1−αAMGλ,2βANGλ2−βANGλ+βANGλ,ALDλ=sA−θXλ,sA−τXλ,2uAMGxλ2−uAMGxλ+uAMGxλ,1+vANGxλ−1−vANGxλ1+vANGxλ+1−vANGxλ,2αAMGλ2−αAMGλ+αAMGλ,1+βANGλ−1−βANGλ1+βANGλ+1−βANGλ.Theorem 2.
For any two LDULNsALD=sA−θX,sA−τX,uAMGx,vANGx,αAMG,βANG and BLD=sB−θX,sB−τX,uBMGx,vBNGx,αBMG,βBNG, then(1)
ALD⊗εBLD=BLD⊗εALD(2)
ALD⊕εBLD=BLD⊕εALD(3)
λALD⊗εBLD=λALD⊗ελBLD,λ>0(4)
λ1ALD⊗ελ2ALD=λ1+λ2⊗εALD,λ1,λ2>0(5)
ALDλ1⊗εALDλ2=ALDλ1+λ2,λ1,λ2>0(6)
ALDλ1⊗εALDλ1=ALD⊗εBLDλ1,λ1>0Proof.
The proof of the first two parts is trivial. Additionally, we prove that part 3 is(18)ALD⊗εBLD=sA−θX+B−θX,sA−τX+B−τX,uAMGx+uBMGx1+uAMGx⋅uBMGx,vANGx⋅vBNGx1+1−vANGx⋅1−vBNGxαAMG+αBMG1+αAMG⋅αBMG,βANG⋅βBNG1+1−βANG⋅1−βBNG;then,(19)λALD⊗εBLD=sλA−θX+B−θX,sλA−τX+B−τX,1+uAMGx1+uBMGxλ−1+uAMGx1+uBMGxλ1+uAMGx1+uBMGxλ+1+uAMGx1+uBMGxλ,2vANGx⋅vBNGxλ4−2vANGx−2vBNGx+vANGx⋅vBNGxλ+vANGx⋅vBNGxλ,1+αAMG1+αBMGλ−1+αAMG1+αBMGλ1+αAMG1+αBMGλ+1+αAMG1+αBMGλ,2βANG⋅βBNGλ4−2βANG−2βBNG+βANG⋅βBNGλ+βANG⋅βBNGλ.
For the right hand of the part (3),(20)λALD=sλ⋅A−θX,sλ⋅A−τX,1+uAMGxλ−1−uAMGxλ1+uAMGxλ+1−uAMGxλ,2vANGxλ2−vANGxλ+vANGxλ1+αAMGλ−1−αAMGλ1+αAMGλ+1−αAMGλ,2βANGλ2−βANGλ+βANGλ,λBLD=sλ⋅B−θX,sλ⋅B−τX,1+uBMGxλ−1−uBMGxλ1+uBMGxλ+1−uBMGxλ,2vBNGxλ2−vBNGxλ+vBNGxλ1+αBMGλ−1−αBMGλ1+αBMGλ+1−αBMGλ,2βBNGλ2−βBNGλ+βBNGλ.
Then,(21)λALD⊗εBLD=sλA−θX+B−θX,sλA−τX+B−τX,1+uAMGx1+uBMGxλ−1+uAMGx1+uBMGxλ1+uAMGx1+uBMGxλ+1+uAMGx1+uBMGxλ,2vANGx⋅vBNGxλ4−2vANGx−2vBNGx+vANGx⋅vBNGxλ+vANGx⋅vBNGxλ,1+αAMG1+αBMGλ−1+αAMG1+αBMGλ1+αAMG1+αBMGλ+1+αAMG1+αBMGλ,2βANG⋅βBNGλ4−2βANG−2βBNG+βANG⋅βBNGλ+βANG⋅βBNGλ.
Hence, we getλALD⊗εBLD=λALD⊗ελBLD,λ>0. Similarly, we prove that part 4. Know, for part (5), we have(22)ALDλ1=sA−θXλ1,sA−τXλ1,2uAMGxλ12−uAMGxλ1+uAMGxλ1,1+vANGxλ1−1−vANGxλ11+vANGxλ1+1−vANGxλ12αAMGλ12−αAMGλ1+αAMGλ1,1+βANGλ1−1−βANGλ11+βANGλ1+1−βANGλ1,ALDλ2=sA−θXλ2,sA−τXλ2,2uAMGxλ22−uAMGxλ2+uAMGxλ2,1+vANGxλ2−1−vANGxλ21+vANGxλ2+1−vANGxλ22αAMGλ22−αAMGλ2+αAMGλ2,1+βANGλ2−1−βANGλ21+βANGλ2+1−βANGλ2.
Then,(23)ALDλ1⊗EALDλ2=SA−θXλ1+λ2,SA−τXλ1+λ2,uAMGxλ1+λ22−uAMGxλ2uAMGxλ1+2−uAMGxλ1uAMGxλ2,21+vANGxλ1+λ21+vANGxλ1+1−vANGxλ11+vANGxλ2+1−vANGxλ2,αANGλ1+λ22−αAMGλ2αAMGλ1+2−αAMGλ1αAMGλ2,21+βANGλ1+λ21+βANGλ1+1+βANGλ11+βANGλ2+1+βANGλ2,.
For the right hand of the part (5), we get(24)ALDλ1+λ2=SA−θXλ1+λ2,SA−τXλ1+λ2,uAMGxλ1+λ22−uAMGxλ2uAMGxλ1+2−uAMGxλ1uAMGxλ2,21+vANGxλ1+λ21+vANGxλ1+1−vANGxλ11+vANGxλ2+1−vANGxλ2,αANGλ1+λ22−αAMGλ2αAMGλ1+2−αAMGλ1αAMGλ2,21+βANGλ1+λ21+βANGλ1+1+βANGλ11+βANGλ2+1+βANGλ2,.
Hence, we getALDλ1⊗εALDλ2=ALDλ1+λ2,λ1,λ2>0. Similarly, we prove that part (6).


## 5. Some LDULFPE Operators

In this section, we combine the PA operator and Einstein operations to the LDUL environment, propose the LDULFPEA operator, LDLFPEWA operator, LDULFPEG operator, and LDULFPEWG operator, and discuss the properties of them.Definition 14.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, the LDULFPEA operator is elaborated by(25)LDULFPEAALD−1,ALD−2,…,ALD−n=⊗εi=1n1+TALD−i∗ALD−i∑i=1n1+TALD−i=⊗εi=1n1+TALD−i∗ALD−i∑i=1n1+TALD−i,where TALD−i=∑j=1nsupALD−i,ALD−j, and supALD−i,ALD−j is the support for ALD−i from ALD−j.Theorem 3.
For any LDULFNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, then the result aggregated from Definition 14 is still an LDULFN:(26)LDULFPEAALD−1,ALD−2,…,ALD−D=S∑i=1nA|θi1+TALD−i/∑i=1n1+TALD−i,S∑i=1nA−τiθi1+TALD−i/∑i=1n1+TALD−i,∏i=1n1+uAMG−i1+TALD−i/∑i=1n1+TALD−i−∏i=1n1−uAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n1+uAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1n1−uAMG−i1+TALD−i/∑i=1n1+TALD−i,2∏i=1nvANG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n2−vANG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1nvANG−i1+TALD−i/∑i=1n1+TALD−i,∏i=1n1+αAMG−i1+TALD−i/∑i=1n1+TALD−i−∏i=1n1−αAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n1+αAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1n1−αAMG−i1+TALD−i/∑i=1n1+TALD−i,2∏i=1nβANG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n2−βANG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1nβANG−i1+TALD−i/∑i=1n1+TALD−i,where TALD−i=∑j=1j≠insupALD−i,ALD−j, and supALD−i,ALD−j is the support for ALD−i from ALD−j.Proof.
To simplify equation (26), we suppose ci=1+TALD−i/∑i=1n1+TALD−ii=1,2,…,n; then, equation (26) can be elaborated as(27)LDULFPEAALD−1,ALD−2,…,ALD−D=S∑i=1nA=θici,S∑i=1nA−τici,∏i=1n1+uAMG−ici−∏i=1n1−uAMG−ici∏i=1n1+uAMG−ici+∏i=1n1−uAMG−ici,2∏i=1nvANG−ici∏i=1n2−vANG−ici+∏i=1nvANG−ici,∏i=1n1+αAMG−ici−∏i=1n1−αAMG−ici∏i=1n1+αAMG−ici+∏i=1n1−αAMG−ici,2∏i=1nβANG−ici∏i=1n2−βANG−ici+∏i=1nβANG−ici.
The equation (26) can be proved by mathematical induction on n as(1)
Whenn=1, equation (26) is right obviously(2)
Suppose whenn=k, equation (26) is right, i.e.,(28)LDULFPEAALD−1,ALD−2,…,ALD−k=S∑i=1kA=θici,S∑i=1kA−τici,∏i=1k1+uAMG−ici−∏i=1k1−uAMG−ici∏i=1k1+uAMG−ici+∏i=1k1−uAMG−ici,2∏i=1kvANG−ici∏i=1k2−vANG−ici+∏i=1kvANG−ici,∏i=1k1+αAMG−ici−∏i=1k1−αAMG−ici∏i=1k1+αAMG−ici+∏i=1k1−αAMG−ici,2∏i=1kβANG−ici∏i=1k2−βANG−ici+∏i=1kβANG−ici.
Then, whenn=k+1, we have(29)LDULFPEAALD−1,ALD−2,…,ALD−k+1=LDULFPEAALD−1,ALD−2,…,ALD−k⊗εck+1ALD−i=LDULFPEAALD−1,ALD−2,…,ALD−k⊗εSA−θk+1ck+1,SA−τk+1ck+1,1+uAMG−k+1ck+1−1−uAMG−k+1ck+11+uAMG−k+1ck+1+1−uAMG−k+1ck+1,2vANG−k+1k+12−vANG−k+ik+1+vANG−k+ik+1,1+αAMG−k+1ck+1−1−αAMG−k+1ck+11+αAMG−k+1ck+1+1−αAMG−k+1ck+1,2βANG−k+1K+12−βANG−k+1K+1+βANG−k+1K+1=LDULFPEAALD−1,ALD−2,…,ALD−k+1=SΣi=1k+1A=θici,SΣi=1k+1A−τici,∏i=1k+11+uAMG−ici−∏i=1k+11−uAMG−ici∏i=1k+11+uAMG−ici+∏i=1k+11−uAMG−ici,2∏i=1k+1vANG−ici∏i=1k+12−vANG−ici+∏i=1k+1vANG−ici,∏i=1k+11+αAMG−ici−∏i=1k+11−αAMG−ici∏i=1k+11+αAMG−ici+∏i=1k+11−αAMG−ici,2∏i=1k+1βANG−ici∏i=1k+12−βANG−ici+∏i=1k+1βANG−ici.
So, whenn=k+1, equation (26) is also right. According to steps (1) and (2), we can get that equation (26) is right for n.Theorem 4.
For any LDULNsALD−j=A,j=1,2,…,n and A=sA−θ,sA−τ,uAMG,vANG,αAMG,βANG,(30)LDULFPEAALD−1,ALD−2,…,ALD−n=A.Proof.
SinceALD−i=A, for all i, we have(31)LDULFPEAALD−1,ALD−2,…,ALD−n=S∑i=1nA|θi1+TA/∑i=1n1+TA,S∑i=1nA−τiθi1+TA/∑i=1n1+TA,∏i=1n1+uAMG1+TA/∑i=1n1+TA−∏i=1n1−uAMG1+TA/∑i=1n1+TA∏i=1n1+uAMG1+TA/∑i=1n1+TA+∏i=1n1−uAMG1+TA/∑i=1n1+TA,2∏i=1nvANG1+TA/∑i=1n1+TA∏i=1n2−vANG1+TA/∑i=1n1+TA+∏i=1nvANG1+TA/∑i=1n1+TA,∏i=1n1+αAMG1+TA/∑i=1n1+TA−∏i=1n1−αAMG1+TA/∑i=1n1+TA∏i=1n1+αAMG1+TA/∑i=1n1+TA+∏i=1n1−αAMG1+TA/∑i=1n1+TA,2∏i=1nβANG1+TA/∑i=1n1+TA∏i=1n2−βANG1+TA/∑i=1n1+TA+∏i=1nβANG1+TA/∑i=1n1+TA=SA−θ∑i=1n1+TA/∑i=1n1+TA,SA−τ∑i=1n1+TA/∑i=1n1+TA,1+uAMG1+TA/∑i=1n1+TA−1−uAMG1+TA/∑i=1n1+TA1+uAMG1+TA/∑i=1n1+TA+1−uAMG1+TA/∑i=1n1+TA,2vANG1+TA/∑i=1n1+TA2−vANG1+TA/∑i=1n1+TA+vANG1+TA/∑i=1n1+TA,1+αAMG1+TA/∑i=1n1+TA−1−αAMG1+TA/∑i=1n1+TA1+αAMG1+TA/∑i=1n1+TA+1−αAMG1+TA/∑i=1n1+TA,2βANG1+TA/∑i=1n1+TA2+βANG1+TA/∑i=1n1+TA+βANG1+TA/∑i=1n1+TA=sA−θ,sA−τ,uAMG,vANG,αAMG,βANG=A.Theorem 5.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, if LDULFPEA operator lies between the max and min operators,(32)minALD−1,ALD−2,…,ALD−n=minA,maxALD−1,ALD−2,…,ALD−n=maxA;then,(33)minA≤LDULFPEAALD−1,ALD−2,…,ALD−n≤maxA.Proof.
First, letguAMG−i=1−uAMG−i/1+uAMG−i,uAMG−i∈0,1, we can get g′uAMG−i=−2/1+uAMG−i2<0 by taking derivative, so guAMG−i is a decreasing function. Suppose minuAMG≤uAMG−i≤maxuAMG for all j, we can get 1−maxuAMG/1+maxuAMG≤1−uAMG−i/1+uAMG−i≤1−minuAMG/1+minuAMG; then,(34)1−maxuAMG1+maxuAMG1+TALD−i/∑i=1n1+TALD−i≤1−uAMG−i1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≤1−minuAMG1+minuAMG1+TALD−i/∑i=1n1+TALD−i,∏i=1n1−maxuAMG1+maxuAMG1+TALD−i/∑i=1n1+TALD−i≤∏i=1n1−uAMG−i1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≤∏i=1n1−minuAMG1+minuAMG1+TALD−i/∑i=1n1+TALD−i,1−maxuAMG1+maxuAMG∑i=1n1+TALD−i/∑i=1n1+TALD−i≤∏i=1n1−uAMG−i1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≤1−minuAMG1+minuAMG∑i=1n1+TALD−i/∑i=1n1+TALD−i,i.e.,(35)1−maxuAMG1+maxuAMG≤∏i=1n1−uAMG−i1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≤1−minuAMG1+minuAMG,21+maxuAMG≤1+∏i=1n1−uAMG−i1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≤21+minuAMG,1+maxuAMG2≥11+∏i=1n1−uAMG−i/1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≥1+minuAMG2.
Thus,(36)1+maxuAMG≥21+∏i=1n1−uAMG−i/1+uAMG−i1+TALD−i/∑i=1n1+TALD−i≥1+minuAMG,maxuAMG≥21+∏i=1n1−uAMG−i/1+uAMG−i1+TALD−i/∑i=1n1+TALD−i−1≥minuAMG.
Therefore,(37)maxuAMG≥∏i=1n1+uAMG−i1+TALD−i/∑i=1n1+TALD−i−∏i=1n1−uAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n1+uAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1n1−uAMG−i1+TALD−i/∑i=1n1+TALD−i≥minuAMG.
Second, letfvANG−i=2−vANG−i/vANG−i,vANG−i∈0,1, and we can get fvANG−i=−2/vANG−i2<0 by taking derivative, so fvAMG−i is a decreasing function. Suppose minvAMG≤vAMG−i≤maxvAMG for all i, we can get 2−minvAMG/minvAMG≤2−vAMG−i/vAMG−i≤2−maxvAMG/maxvAMG; then, we have(38)2−minvAMGminvAMG1+TALD−i/∑i=1n1+TALD−i≤2−vAMG−ivAMG−i1+TALD−i/∑i=1n1+TALD−i≤2−maxvAMGmaxvAMG1+TALD−i/∑i=1n1+TALD−i,2−minvAMGminvAMG∑i=1n1+TALD−i/∑i=1n1+TALD−i≤∏i=1n2−vAMG−ivAMG−i1+TALD−i/∑i=1n1+TALD−i≤2−maxvAMGmaxvAMG∑i=1n1+TALD−i/∑i=1n1+TALD−i,2−minvAMGminvAMG≤∏i=1n2−vAMG−ivAMG−i1+TALD−i/∑i=1n1+TALD−i≤2−maxvAMGmaxvAMG,2minvAMG≤∏i=1n2−vAMG−ivAMG−i1+TALD−i/∑i=1n1+TALD−i+1≤2maxvAMG,minvAMG2≥1∏i=1n2−vAMG−i/vAMG−i1+TALD−i/∑i=1n1+TALD−i+1≥maxvAMG2,minvAMG≥2∏i=1n2−vAMG−i/vAMG−i1+TALD−i/∑i=1n1+TALD−i+1≥maxvAMG.
Therefore,(39)minvAMG≥2∏i=1nvAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n2−vAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1nvAMG−i1+TALD−i/∑i=1n1+TALD−i≥maxvAMG.
Forα and β because sA−θmin≤sA−θi≤sA−θmax, sA−τmin≤sA−τi≤sA−τmax for all i, then(40)sA−θmin1+TALD−i/∑i=1n1+TALD−i≤sA−θi1+TALD−i/∑i=1n1+TALD−i≤sA−θmax1+TALD−i/∑i=1n1+TALD−i,sA−τmin1+TALD−i/∑i=1n1+TALD−i≤sA−τi1+TALD−i/∑i=1n1+TALD−i≤sA−τmax1+TALD−i/∑i=1n1+TALD−i,s∑i=1nA−θmin1+TALD−i/∑i=1n1+TALD−i≤s∑i=1nA−θi1+TALD−i/∑i=1n1+TALD−i≤s∑i=1nA−θmax1+TALD−i/∑i=1n1+TALD−i,s∑i=1nA−τmin1+TALD−i/∑i=1n1+TALD−i≤s∑i=1nA−τi1+TALD−i/∑i=1n1+TALD−i≤s∑i=1nA−τmax1+TALD−i/∑i=1n1+TALD−i,i.e., sA−θmin≤s∑i=1nA−θi1+TALD−i/∑i=1n1+TALD−i≤sA−θmax,sA−τmin≤s∑i=1nA−τi1+TALD−i/∑i=1n1+TALD−i≤sA−τmax.
IfLDULFPEAALD−1,ALD−2,…,ALD−n=A=sA−θ,sA−τ,uAMG,vANG,αAMG,βANG, we know that sA−θmin≤sA−θ≤sA−θmax, sA−τmin≤sA−τ≤sA−τmax, minuAMG≤uAMG≤maxuAMG, minvANG≤vANG≤maxvANG, minαAMG≤αAMG≤maxαAMG, and minβANG≤βANG≤maxβANG.(41)sA−θmin+sA−τmin≤sA−θ+sA−τ≤sA−θmax+sA−τmax,minuAMG−minvANG≤uAMG−vANG≤maxuAMG−maxvANG,minαAMG−minβANG≤αAMG−βANG≤maxαAMG−maxβANG.
Therefore,(42)minA≤LDULFPEAALD−1,ALD−2,…,ALD−n≤maxA.Theorem 6.
For any two LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i and BLD−i=sB−θi,sB−τi,uBMG−i,vBNG−i,αBMG−i,βBNG−i and if sA−θi≤sB−θi,sA−τi≤sB−τi,uAMG−i≤uBMG−i,vANG−i≤vBNG−i,αAMG−i≤αBMG−i and βANG−i≤βBNG−i, for all i=1,2,…,n, then(43)LDULFPEAALD−1,ALD−2,…,ALD−n≤LDULFPEABLD−1,BLD−2,…,BLD−n.Proof.
SincesA−θi+sA−τi≤sB−θi+sB−τi, uAMG−i≤uBMG−i,vANG−i≤vBNG−i,αAMG−i≤αBMG−i, and βANG−i≤βBNG−i, for all i, we can get(44)∑i=1n1+TALD−i∑i=1n1+TALD−isA−θi+sA−τi≤∑i=1n1+TALD−i∑i=1n1+TALD−isB−θi+sB−τi,s∑i=1nA−θi1+TALD−i/∑i=1n1+TALD−i+s∑i=1nA−τi1+TALD−i/∑i=1n1+TALD−i≤s∑i=1nB−θi1+TBLD−i/∑i=1n1+TBLD−i+s∑i=1nB−τi1+TBLD−i/∑i=1n1+TBLD−i.
Since1−uBMG−i/1+uBMG−i≤1−uAMG−i/1+uAMG−i,i=1,2,…,n, then(45)∏i=1n1−uBMG−i1+uBMG−i1+TBLD−i/∑i=1n1+TBLD−i≤∏i=1n1−uAMG−i1+uAMG−i1+TALD−i/∑i=1n1+TALD−i21+∏i=1n1−uBMG−i/1+uBMG−i1+TBLD−i/∑i=1n1+TBLD−i≥21+∏i=1n1−uAMG−i/1+uAMG−i1+TALD−i/∑i=1n1+TALD−i.
So, we have(46)∏i=1n1+uAMG−i1+TALD−i/∑i=1n1+TALD−i−∏i=1n1−uAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n1+uAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1n1−uAMG−i1+TALD−i/∑i=1n1+TALD−i≥∏i=1n1+uBMG−i1+TBLD−i/∑i=1n1+TBLD−i−∏i=1n1−uBMG−i1+TBLD−i/∑i=1n1+TBLD−i∏i=1n1+uBMG−i1+TBLD−i/∑i=1n1+TBLD−i+∏i=1n1−uBMG−i1+TBLD−i/∑i=1n1+TBLD−i.
ForαAMG−i,αBMG−i, we can get(47)∏i=1n1+αAMG−i1+TALD−i/∑i=1n1+TALD−i−∏i=1n1−αAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n1+αAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1n1−αAMG−i1+TALD−i/∑i=1n1+TALD−i≥∏i=1n1+αBMG−i1+TBLD−i/∑i=1n1+TBLD−i−∏i=1n1−αBMG−i1+TBLD−i/∑i=1n1+TBLD−i∏i=1n1+αBMG−i1+TBLD−i/∑i=1n1+TBLD−i+∏i=1n1−αBMG−i1+TBLD−i/∑i=1n1+TBLD−i.
Since2−vBNG−i/vBNG−i≥2−vANG−i/vANG−i,i=1,2,…,n, then(48)2−vBMG−ivBMG−i1+TALD−i/∑i=1n1+TALD−i≥2−vAMG−ivAMG−i1+TALD−i/∑i=1n1+TALD−i,∏i=1n2−vBMG−ivBMG−i1+TBLD−i/∑i=1n1+TBLD−i≥∏i=1n2−vAMG−ivAMG−i1+TALD−i/∑i=1n1+TALD−i,1∏i=1n2−vBMG−i/vBMG−i1+TBLD−i/∑i=1n1+TBLD−i+1≤12−vAMG−i/vAMG−i1+TALD−i/∑i=1n1+TALD−i+1,2∏i=1n2−vBMG−i/vBMG−i1+TBLD−i/∑i=1n1+TBLD−i+1≤22−vAMG−i/vAMG−i1+TALD−i/∑i=1n1+TALD−i+1,so(49)2∏i=1nvAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n2−vAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1nvAMG−i1+TALD−i/∑i=1n1+TALD−i≤2∏i=1nvBMG−i1+TBLD−i/∑i=1n1+TBLD−i∏i=1n2−vBMG−i1+TBLD−i/∑i=1n1+TBLD−i+∏i=1nvBMG−i1+TBLD−i/∑i=1n1+TBLD−i.
ForβANG−i,βBNG−i, we can get(50)2∏i=1nβAMG−i1+TALD−i/∑i=1n1+TALD−i∏i=1n2−βAMG−i1+TALD−i/∑i=1n1+TALD−i+∏i=1nβAMG−i1+TALD−i/∑i=1n1+TALD−i≤2∏i=1nβBNG−i1+TBLD−i/∑i=1n1+TBLD−i∏i=1n2−βBNG−i1+TBLD−i/∑i=1n1+TBLD−i+∏i=1nβBNG−i1+TBLD−i/∑i=1n1+TBLD−i.
Thus, we got that(51)LDULFPEAALD−1,ALD−2,…,ALD−n≤LDULFPEAALD−1,ALD−2,…,ALD−n.Definition 15.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, the LDULFPEWA operator is elaborated by(52)LDULFPEWAALD−1,ALD−2,…,ALD−n=⊗εi=1nwi1+TALD−i∗ALD−i∑i=1nwi1+TALD−i=⊗εi=1nwi1+TALD−i∗ALD−i∑i=1nwi1+TALD−i,where TALD−i=∑j=1j≠insupALD−i,ALD−j, supALD−i,ALD−j is the support for ALD−i from ALD−j, and w=w1,w2,…,wnT is the weighting vector of the ALD−1,ALD−2,…,ALD−n, such that wi∈0,1, ∑i=1nwi=1.Theorem 7.
For any LDULFNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, the result aggregated from Definition 15 is still an LDULFN:(53)LDULFPEAALD−1,ALD−2,…,ALD−D=⊗εi=1nwi1+TALD−i∗ALD−i∑i=1nwi1+TALD−i=S∑i=1nA−θiwi1+TALD−i/∑i=1nwi1+TALD−i,S∑i=1nA−τiwiθi1+TALD−i/∑i=1nwi1+TALD−i,∏i=1n1+uAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i−∏i=1n1−uAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i∏i=1n1+uAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i+∏i=1n1−uAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i,2∏i=1nvANG−iwi1+TALD−i/∑i=1nwi1+TALD−i∏i=1n2−vANG−iwi1+TALD−i/∑i=1nwi1+TALD−i+∏i=1nvANG−iwi1+TALD−i/∑i=1nwi1+TALD−i,∏i=1n1+αAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i−∏i=1n1−αAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i∏i=1n1+αAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i+∏i=1n1−αAMG−iwi1+TALD−i/∑i=1nwi1+TALD−i,2∏i=1nβANG−iwi1+TALD−i/∑i=1nwi1+TALD−i∏i=1n2−βANG−iwi1+TALD−i/∑i=1nwi1+TALD−i+∏i=1nβANG−iwi1+TALD−i/∑i=1nwi1+TALD−i,where TALD−i=∑j=1j≠insupALD−i,ALD−j, supALD−i,ALD−j is the support for ALD−i from ALD−j, and w=w1,w2,…,wnT is the weighting vector of the ALD−1,ALD−2,…,ALD−n, such that wi∈0,1, ∑i=1nwi=1.Proof.
Trivial.Theorem 8.
For any LDULNsALD−j=A,j=1,2,…,n and A=sA−θ,sA−τ,uAMG,vANG,αAMG,βANG, then(54)LDULFPEWAALD−1,ALD−2,…,ALD−n=A.Proof.
Trivial.Theorem 9.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, if LDULFPEWA operator lies between the max and min operators,(55)minALD−1,ALD−2,…,ALD−n=minA,maxALD−1,ALD−2,…,ALD−n=maxA;then,(56)minA≤LDULFPEWAALD−1,ALD−2,…,ALD−n≤maxA.Proof.
Trivial.Definition 16.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, the LDULFPEA operator is elaborated by(57)LDULFPEGALD−1,ALD−2,…,ALD−n=⊗εi=1nALD−i1+TALD−i/∑i=1n1+TALD−i,where TALD−i=∑j=1j≠insupALD−i,ALD−j, and supALD−i,ALD−j is the support for ALD−i from ALD−j.Theorem 10.
For any LDULFNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, the result aggregated from Definition 16 is still an LDULFN:(58)LDULFPEAALD−1,ALD−2,…,ALD−n=S∏i=1nA=θici,S∏i=1nA=τici,2∏i=1nuANG−ici∏i=1n2−uANG−ici+∏i=1nuANG−ici,∏i=1n1+vAMG−ici−∏i=1n1−vAMG−ici∏i=1n1+vAMG−ici+∏i=1n1−vAMG−ici,2∏i=1nαAMG−ici∏i=1n2−αAMG−ici+∏i=1nαAMG−ici,∏i=1n1+βAMG−ici−∏i=1n1−βAMG−ici∏i=1n2−βAMG−ici+∏i=1n1−βAMG−ici,where ci=1+TALD−i/∑i=1n1+TALD−ii=1,2,…,n, TALD−i=∑j=1j≠insupALD−i,ALD−j, and supALD−i,ALD−j is the support for ALD−i from ALD−j.Proof.
Trivial.Theorem 11.
For any LDULNsALD−j=A,j=1,2,…,n and A=sA−θ,sA−τ,uAMG,vANG,αAMG,βANG, then(59)LDULFPEGALD−1,ALD−2,…,ALD−n=A.Proof.
Trivial.Theorem 12.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, if LDULFPEG operator lies between the max and min operators,(60)minALD−1,ALD−2,…,ALD−n=minA,maxALD−1,ALD−2,…,ALD−n=maxA;then,(61)minA≤LDULFPEGALD−1,ALD−2,…,ALD−n≤maxA.Proof.
Trivial.Theorem 13.
For any two LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i and BLD−i=sB−θi,sB−τi,uBMG−i,vBNG−i,αBMG−i,βBNG−i and if sA−θi≤sB−θi,sA−τi≤sB−τi,uAMG−i≤uBMG−i,vANG−i≤vBNG−i,αAMG−i≤αBMG−i and βANG−i≤βBNG−i, for all i=1,2,…,n, then(62)LDULFPEGALD−1,ALD−2,…,ALD−n≤LDULFPEGBLD−1,BLD−2,…,BLD−n.Proof.
Trivial.Definition 17.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, the LDULFPEWG operator is elaborated by(63)LDULFPEWGALD−1,ALD−2,…,ALD−n=⊗εi=1nALD−iwi1+TALD−i/∑i=1nwi1+TALD−i,where TALD−i=∑j=1j≠insupALD−i,ALD−j, supALD−i,ALD−j is the support for ALD−i from ALD−j, and w=w1,w2,…,wnT is the weighting vector of the ALD−1,ALD−2,…,ALD−n, such that wi∈0,1, ∑i=1nwi=1.Theorem 14.
For any LDULFNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, then the result aggregated from Definition 17 is still an LDULFN:(64)LDULFPEAALD−1,ALD−2,…,ALD−n=⊗εi=1nALD−iwi1+TALD−i/∑i=1nwi1+TALD−i=S∏i=1nA=θibi,S∏i=1nA=τibi,2∏i=1nuANG−ibi∏i=1n2−uANG−ibi+∏i=1nuANG−ibi,∏i=1n1+vAMG−ibi−∏i=1n1−vAMG−ibi∏i=1n1+vAMG−ibi+∏i=1n1−vAMG−ibi,2∏i=1nαAMG−ibi∏i=1n2−αAMG−ibi+∏i=1nαAMG−ibi,∏i=1n1+βAMG−ibi−∏i=1n1−βAMG−ibi∏i=1n2−βAMG−ibi+∏i=1n1−βAMG−ibi,where bi=wi1+TALD−i/∑i=1nwi1+TALD−ii=1,2,…,n, TALD−i=∑j=1j≠insupALD−i,ALD−j, and supALD−i,ALD−j is the support for ALD−i from ALD−j. w=w1,w2,…,wnT is the weighting vector of ALD−1,ALD−2,…,ALD−n, such that wi∈0,1, ∑i=1nwi=1.Proof.
Trivial.Theorem 15.
For any LDULNsALD−j=A,j=1,2,…,n and A=sA−θ,sA−τ,uAMG,vANG,αAMG,βANG,(65)LDULFPEWGALD−1,ALD−2,…,ALD−n=A.Proof.
Trivial.Theorem 16.
For any LDULNsALD−i=sA−θi,sA−τi,uAMG−i,vANG−i,αAMG−i,βANG−i,i=1,2,…,n, if LDULFPEWG operator lies between the max and min operators,(66)minALD−1,ALD−2,…,ALD−n=minA,maxALD−1,ALD−2,…,ALD−n=maxA;then,(67)minA≤LDULFPEWGALD−1,ALD−2,…,ALD−n≤maxA.Proof.
Trivial.


## 6. MADM Technique for LDULSs Using PEAOs

In this part, we promote the examination of PEAOs of LDULNs in the MADM problem. We promote powered Einstein aggregation operators of LDULNs in the MADM problem. The impact ofℜ on the ranking results is investigated thoroughly.MADM is a way of choosing the best alternative because of some finite attributes using the PE operators of LDULSs. The most favorable thing here is that the information is based on LDULNs which discuss the membership grades and nonmembership grades of the information. Let the collection of alternatives beAk (k is finite) and attributes be Gj (j is finite), which forms a decision matrix denoted by Dk×j=Tk×j=sA−θkj,sA−τkj,uAMG−kj,vANG−kj,αAMG−kj,βANG−kj, where the terms in triplet denote the membership grades, abstinence, and nonmembership grades of the information, where ω=ω1,ω2,…,ωrT is the weight vector of ALD−1,ALD−2,…,ALD−r,ωj∈0,1,∑j=1rωj=1. A brief algorithm of the MADM process is illustrated in the following section.


### 6.1. Algorithm

The algorithm of MADM based on LDUL information and using LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators is proposed as follows:Step 1: in this step, we collect the information about alternatives given by the decision-makers. The decision-makers gave the 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, we use the following LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators to aggregate the data given in the decision matrix.Step 3: in this step, we compute the scores of the aggregated information using Definition3Step 4: this step is based on the ranking of the alternativesAn illustrative example to see the viability of the proposed algorithm.


### 6.2. Illustrated Example

An example of technology commercialization is adapted from [30] 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 A1≤k≤4, and the attributes are denoted by G1≤j≤4, where G1 represents the advancement of the technology, G2 represents the market potential, G3 represents the human resources and financial development, and G4 represents the creating of employment and development of technology. The weight vector chosen in this case is ω=0.4,0.3,0.2,0.1T. The selection of the weight vector is up to the decision-makers, and the information about the alternatives in terms of LDULNs is given in Table 1.Step 1: in this step, the information given in Table1 is aggregated using LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators. The results of the aggregation are given in Table 2.Step 2: in this step, the aggregated information obtained in Table3 is targeted, and their scores are computed using Definition 3.Step 3: based on the scores obtained in Table4, the alternatives are ranked, and the ranking results are given in Table 5.Table 1
Original decision matrix.
G1G2G3G4A1s1,s2,0.4,0.3,0.3,0.2s1,s2,0.41,0.31,0.31,0.21s1,s2,0.42,0.32,0.32,0.22s1,s2,0.43,0.33,0.33,0.23A2s1,s3,0.5,0.4,0.4,0.3s1,s3,0.51,0.41,0.41,0.31s1,s3,0.52,0.42,0.42,0.32s1,s3,0.53,0.43,0.43,0.33A3s2,s3,0.6,0.5,0.2,0.1s2,s3,0.61,0.51,0.21,0.11s2,s3,0.62,0.52,0.22,0.12s2,s3,0.63,0.53,0.23,0.13A4s0,s1,0.7,0.4,0.4,0.5s0,s1,0.71,0.41,0.41,0.51s0,s1,0.72,0.42,0.42,0.52s0,s1,0.73,0.43,0.43,0.53Table 2
Aggregated values by using different operators.
LDULPEA operatorLDULPEWA operatorLDULPEG operatorLDULPEWG operatorA1s1.4546,s2.5757,0.4145,0.3146,0.3147,0.2148s1.5657,s2.6868,0.5256,0.4257,0.4258,0.3259s1.3435,s2.4646,0.3034,0.2035,0.2036,0.1037s1.2324,s2.3535,0.2023,0.1024,0.1025,0.0026A2s1.4546,s3.5757,0.5145,0.4146,0.4147,0.3148s1.5657,s3.6868,0.6256,0.5257,0.5258,0.4259s1.3435,s2.4646,0.4034,0.3035,0.3036,0.2037s1.2324,s2.3535,0.3023,0.2024,0.2025,0.1026A3s2.4546,s3.5757,0.6145,0.5146,0.2147,0.1148s2.5657,s3.6868,0.7256,0.6257,0.3258,0.2259s1.3435,s2.4646,0.5034,0.4035,0.1036,0.0037s1.2324,s2.3535,0.4023,0.3024,0.0025,0.0026A4s0.4546,s1.5757,0.7145,0.4146,0.4147,0.5148s0.5657,s1.6868,0.8256,0.5257,0.5258,0.6259s1.3435,s2.4646,0.6034,0.3035,0.3036,0.4037s1.2324,s2.3535,0.5023,0.2024,0.2025,0.3026Table 3
Score values of the aggregated operators.
ScoresLDULPEA operatorLDULPEWA operatorLDULPEG operatorLDULPEWG operatorA10.82090.98420.66980.5401A20.92161.09050.76510.6298A30.72010.87790.57460.4953A41.12311.30320.95540.8091Table 4
Ranking values of the information in Table3.
Ranking analysisLDULPEWA operatorA4>A2>A1>A3LDULPEWG operatorA4>A2>A1>A3LDULPEA operatorA4>A2>A1>A3LDULPEG operatorA4>A2>A1>A3Table 5
Comparative analysis of the elaborated operators and existing operators.
MethodScore valuesRanking valuesRiaz and Hashmi [39]Cannot be calculatedCannot be calculatedRiaz and Hashmi [40]Cannot be calculatedCannot be calculatedWang et al. [42]Cannot be calculatedCannot be calculatedLDULPEWA operator0.9842,1.0905,0.8779,1.3032A4>A2>A1>A3LDULPEWG operator0.5401,0.6298,0.4953,0.8091A4>A2>A1>A3LDULPEA operator0.8209,0.9216,0.7201,1.1231A4>A2>A1>A3LDULPEG operator0.6698,0.7651,0.5746,0.9554A4>A2>A1>A3As shown above, we obtained the same ranking results, which are discussed in the form of Table4. The best option is A4. The results portrayed in Table 3 are further described geometrically in Figure 2.Figure 2
Geometrical expressions of the values in Table3.


### 6.3. Comparative Analysis

Based on the investigated LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators, we determined the reliability and consistency of the developed operators with the help of comparative analysis by using the information of Table1 shown in 6.2. The information related to existing theories is as follows: Riaz and Hashmi [39] elaborated the aggregation operators for LDFS and their applications, Riaz and Hashmi [40] proposed some operators based on linear Diophantine sets, and Wang et al. [42] proposed interaction Hamacher operators for Pythagorean fuzzy sets. The reasonable analysis of the investigated operators and remaining operators are discussed in Table 5 by using the information of Table 1, by using the values of parameters f=h=1,r=3.The results portrayed in Table5 are further described geometrically in Figure 3.Figure 3
Geometrical expressions of the information in Table5.As shown above, we chose the linear Diophantine uncertain linguistic types of information, so the operators investigated by Riaz and Hashmi [39] are not able to resolve it. However, if we decide the exiting types of information, 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.


## 6.1. Algorithm

The algorithm of MADM based on LDUL information and using LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators is proposed as follows:Step 1: in this step, we collect the information about alternatives given by the decision-makers. The decision-makers gave the 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, we use the following LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators to aggregate the data given in the decision matrix.Step 3: in this step, we compute the scores of the aggregated information using Definition3Step 4: this step is based on the ranking of the alternativesAn illustrative example to see the viability of the proposed algorithm.


## 6.2. Illustrated Example

An example of technology commercialization is adapted from [30] 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 A1≤k≤4, and the attributes are denoted by G1≤j≤4, where G1 represents the advancement of the technology, G2 represents the market potential, G3 represents the human resources and financial development, and G4 represents the creating of employment and development of technology. The weight vector chosen in this case is ω=0.4,0.3,0.2,0.1T. The selection of the weight vector is up to the decision-makers, and the information about the alternatives in terms of LDULNs is given in Table 1.Step 1: in this step, the information given in Table1 is aggregated using LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators. The results of the aggregation are given in Table 2.Step 2: in this step, the aggregated information obtained in Table3 is targeted, and their scores are computed using Definition 3.Step 3: based on the scores obtained in Table4, the alternatives are ranked, and the ranking results are given in Table 5.Table 1
Original decision matrix.
G1G2G3G4A1s1,s2,0.4,0.3,0.3,0.2s1,s2,0.41,0.31,0.31,0.21s1,s2,0.42,0.32,0.32,0.22s1,s2,0.43,0.33,0.33,0.23A2s1,s3,0.5,0.4,0.4,0.3s1,s3,0.51,0.41,0.41,0.31s1,s3,0.52,0.42,0.42,0.32s1,s3,0.53,0.43,0.43,0.33A3s2,s3,0.6,0.5,0.2,0.1s2,s3,0.61,0.51,0.21,0.11s2,s3,0.62,0.52,0.22,0.12s2,s3,0.63,0.53,0.23,0.13A4s0,s1,0.7,0.4,0.4,0.5s0,s1,0.71,0.41,0.41,0.51s0,s1,0.72,0.42,0.42,0.52s0,s1,0.73,0.43,0.43,0.53Table 2
Aggregated values by using different operators.
LDULPEA operatorLDULPEWA operatorLDULPEG operatorLDULPEWG operatorA1s1.4546,s2.5757,0.4145,0.3146,0.3147,0.2148s1.5657,s2.6868,0.5256,0.4257,0.4258,0.3259s1.3435,s2.4646,0.3034,0.2035,0.2036,0.1037s1.2324,s2.3535,0.2023,0.1024,0.1025,0.0026A2s1.4546,s3.5757,0.5145,0.4146,0.4147,0.3148s1.5657,s3.6868,0.6256,0.5257,0.5258,0.4259s1.3435,s2.4646,0.4034,0.3035,0.3036,0.2037s1.2324,s2.3535,0.3023,0.2024,0.2025,0.1026A3s2.4546,s3.5757,0.6145,0.5146,0.2147,0.1148s2.5657,s3.6868,0.7256,0.6257,0.3258,0.2259s1.3435,s2.4646,0.5034,0.4035,0.1036,0.0037s1.2324,s2.3535,0.4023,0.3024,0.0025,0.0026A4s0.4546,s1.5757,0.7145,0.4146,0.4147,0.5148s0.5657,s1.6868,0.8256,0.5257,0.5258,0.6259s1.3435,s2.4646,0.6034,0.3035,0.3036,0.4037s1.2324,s2.3535,0.5023,0.2024,0.2025,0.3026Table 3
Score values of the aggregated operators.
ScoresLDULPEA operatorLDULPEWA operatorLDULPEG operatorLDULPEWG operatorA10.82090.98420.66980.5401A20.92161.09050.76510.6298A30.72010.87790.57460.4953A41.12311.30320.95540.8091Table 4
Ranking values of the information in Table3.
Ranking analysisLDULPEWA operatorA4>A2>A1>A3LDULPEWG operatorA4>A2>A1>A3LDULPEA operatorA4>A2>A1>A3LDULPEG operatorA4>A2>A1>A3Table 5
Comparative analysis of the elaborated operators and existing operators.
MethodScore valuesRanking valuesRiaz and Hashmi [39]Cannot be calculatedCannot be calculatedRiaz and Hashmi [40]Cannot be calculatedCannot be calculatedWang et al. [42]Cannot be calculatedCannot be calculatedLDULPEWA operator0.9842,1.0905,0.8779,1.3032A4>A2>A1>A3LDULPEWG operator0.5401,0.6298,0.4953,0.8091A4>A2>A1>A3LDULPEA operator0.8209,0.9216,0.7201,1.1231A4>A2>A1>A3LDULPEG operator0.6698,0.7651,0.5746,0.9554A4>A2>A1>A3As shown above, we obtained the same ranking results, which are discussed in the form of Table4. The best option is A4. The results portrayed in Table 3 are further described geometrically in Figure 2.Figure 2
Geometrical expressions of the values in Table3.


## 6.3. Comparative Analysis

Based on the investigated LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators, we determined the reliability and consistency of the developed operators with the help of comparative analysis by using the information of Table1 shown in 6.2. The information related to existing theories is as follows: Riaz and Hashmi [39] elaborated the aggregation operators for LDFS and their applications, Riaz and Hashmi [40] proposed some operators based on linear Diophantine sets, and Wang et al. [42] proposed interaction Hamacher operators for Pythagorean fuzzy sets. The reasonable analysis of the investigated operators and remaining operators are discussed in Table 5 by using the information of Table 1, by using the values of parameters f=h=1,r=3.The results portrayed in Table5 are further described geometrically in Figure 3.Figure 3
Geometrical expressions of the information in Table5.As shown above, we chose the linear Diophantine uncertain linguistic types of information, so the operators investigated by Riaz and Hashmi [39] are not able to resolve it. However, if we decide the exiting types of information, 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.


## 7. Conclusion

LDULS is a modified variety of the fuzzy set (FS) to manage problematic and inconsistent information in actual life troubles. LDULS covers the grade of truth, grade of falsity, and their reference parameters with ULT with0≤αAMGuAMGx+βANGvAMGx≤1, where 0≤αAMG+βANG≤1. In this manuscript, the principle of LDULS and their useful laws are elaborated. Additionally, PEAO is a conventional sort of AO utilized in innovative decision-making troubles, which is effective to aggregate the family of numerical elements. To determine the interrelationship between any numbers of arguments, we elaborate on the LDULPEA, LDULPEWA, LDULPEG, and LDULPEWG operators and discussed their useful results. Conclusively, a decision-making methodology is utilized for the MADM dilemma with elaborated information. A sensible illustration is specified to demonstrate the accessibility and rewards of the intended technique by comparison with certain prevailing techniques. The intended AOs are additional comprehensive than the prevailing ones to exploit the ambiguous and inaccurate knowledge. Numerous remaining operators are chosen as individual incidents of the suggested one. Ultimately, the supremacy and advantages of the elaborated operators are also discussed with the help of geometrical form to show the validity and consistency of explored operators.In further research, considering the superiority of new LDULSs, we can extend them to some other spherical fuzzy sets [43], spherical linear Diophantine fuzzy sets [44], complex spherical fuzzy sets [45], aggregation operators [42], such as power mean aggregation operators, Bonferroni mean operators, Heronian mean operators, different methods [46–50], and so on [51].



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