# Heronian Mean Operator of Linguistic Neutrosophic Cubic Numbers and Their Multiple Attribute Decision-Making Methods

Authors: Changxing Fan; Jun Ye
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Many aggregation operators in multiattribute decisions assume that attributes are independent of each other; this leads to an unreasonable situation in information aggregation and decision-making. Heronian mean is the aggregation operator that can embody the interaction between attributes. In this paper, we merge the linguistic neutrosophic cubic number (LNCN) and the Heronian mean operator together to develop a LNCN generalized weighted Heronian mean (LNCNGWHM) operator and a LNCN three-parameter weighted Heronian mean (LNCNTPWHM) operator and then discuss their properties. Further, two multiattribute decision methods based on the proposed LNCNGWHM or LNCNTPWHM operator are introduced under LNCN environment. Finally, an example is used to indicate the effectiveness of the developed methods.


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## Body

## 1. Introduction

In multiple attribute decision-making problems, the general attribute values are usually difficult to be given as the form of real numbers and instead of the form of uncertain variables, such as fuzzy numbers, intuitionistic fuzzy numbers, and linguistic variables (LVs). Fuzzy set (FS) was proposed by Zadeh to measure things that cannot be described with accurate information or accurate probability distribution [1]. The intuitionistic fuzzy set (IFS) proposed by Atanassov consists of the membership and nonmembership information, which is an effective extension of fuzzy set theory [2]. LVs are used to evaluate attributes in multiattribute decision-making while attributes cannot be evaluated by numerical values. Linguistic set (LS)L ={L0,L1,L2,L3,…Lg} (g is an even number) was proposed by Zadeh to deal with the approximate reasoning problems [3, 4]. Since FS, IFS, and LS are effective tools to deal with the approximate reasoning problems, many researches have studied them [5–11]. However, in the literature [7–11], only incomplete information can be effectively expressed, and then indeterminate and inconsistent information cannot be described effectively. In order to make up for the deficiency of literature [7–11], Smarandache put forward the theory of neutrosophic set (NS) consisting of three parts: truthT(x),falsityF(x), and indeterminacy I(x) [12]. Wang and Smarandache also proposed a single value neutrosophic set (SVNS), which satisfiesTx,Ix,Fx∈[0,1], and 0≤T(x)+I(x)+Fx≤3 [13]. IF the three componentsT(x), I(x), and Fx in NS are interval numbers, Wang and Zhang introduced an interval neutrosophic set (INS) [14]. Next, some scholars put NS and LS together to develop their new set of concepts. First, Ye defined the interval neutrosophic uncertain linguistic set (INULS) and its multiple attribute decision-making (MADM) method [15]. Then, Ye proposed single valued neutrosophic linguistic numbers (SVNLNs) for multiple attribute group decision-making (MAGDM) [16]. Further, Fang and Ye proposed the linguistic neutrosophic number (LNN) consisting of the truth, falsity, and indeterminacy linguistic degrees [17]. Recently, Ye also proposed a new concept of a linguistic neutrosophic cubic number (LNCN) to extend neutrosophic cubic sets to linguistic neutrosophic arguments [18]. So far, there is little study on LNCN. However, many aggregation operators in multiattribute decisions assume that attributes are independent of each other; this leads to an unreasonable situation in information aggregation and decision-making. Then, Heronian mean is the aggregation operator that can embody the interaction between attributes. So, in this paper, we propose the LNCN generalized weighted Heronian mean (LNCNGWHM) operator and the LNCN three-parameter weighted Heronian mean (LNCNTPWHM) operator and investigate their properties.The remaining organizations of this paper are listed as follows. Section2 describes some concepts of LNN, LNCN, GWHM, and TPWHM. Section 3 proposes the LNCNGWHM and LNCNTPWHM operators and investigates their properties. Section 4 establishes MADM methods by using the LNCNGWHM and LNCNTPWHM operators. Section 5 provides an illustrative example to demonstrate the application and effectiveness of the proposed methods. Section 6 gives conclusions of this paper.


## 2. Some Basic Concepts

### 2.1. Linguistic Neutrosophic Numbers, Linguistic Neutrosophic Cubic Numbers, and Their Operational Laws

Definition 1 ([17]).
SetL={lj∣l0≤lj≤lg,j∈[0,g]} as a language term set; g is an even number. Then a LNN can be defined as follows:(1)m=lα,lβ,lγ,in which lα,lβ,lγ∈L and α,β,γ∈[0,g],lα,lβ, and lγ represent the truth, indeterminacy, and falsity variables, respectively, in linguistic terms.Definition 2 ([17]).
Setm=lα,lβ,lγ, m1=lα1,lβ1,lγ1, and m2=lα2,lβ2,lγ2 as three LNNs in L and a real numberδ≥0; then the operational laws of LNNs are as follows:(2)m1⊕m2=lα1,lβ1,lγ1⊕lα2,lβ2,lγ2=lα1+α2-α1α2/g,lβ1β2/g,lγ1γ2/g;(3)m1⊗m2=lα1,lβ1,lγ1⊗lα2,lβ2,lγ2=lα1α2/g,lβ1+β2-β1β2/g,lγ1+γ2-γ1γ2/g;(4)δm=δlα,lβ,lγ=lg-g1-α/gδ,lgβ/gδ,lgγ/gδ;(5)mδ=lα,lβ,lγδ=lgα/gδ,lg-g1-β/gδ,lg-g1-γ/gδ.Definition 3 ([18]).
SetL={lj∣l0≤lj≤lg,j∈[0,g]} as a language term set; g is an even number. Then a LNCN can be defined as follows:(6)m=u,c,where u=lαb,lαt,lβb,lβt,lγb,lγt expresses an uncertain LNN and lαb,lαt,lβb,lβt,lγb,lγt represent, respectively, the truth, the indeterminacy, and the falsity uncertain linguistic variables for lαb,lαt,lβb,lβt,lγb,lγt∈L and αb,αt,βb,βt,γb,γt∈[0,g],αb≤αt,βb≤βt,γb≤γt; c=lα,lβ,lγexpresses a LNN, in which lα,lβ,lγ∈L and α,β,γ∈[0,g], andlα,lβandlγ represent the truth, the indeterminacy, and the falsity linguistic variables, respectively, in linguistic terms.Definition 4 ([18]).
Setm=(lαb,lαt,lβb,lβt,lγb,lγt, lα,lβ,lγ) as a LNCN in L, then, one calls m①
an internal LNCN ifαb≤α≤αt,βb≤β≤βt,γb≤γ≤γt;②
an external LNCN ifα∉[αb,αt],β∉βb,βt and γ∉γb,γt.Definition 5 ([18]).
Setm1 =(lαb1,lαt1,lβb1,lβt1,lγb1,lγt1, lα1,lβ1,lγ1), and m2=(lαb2,lαt2,lβb2,lβt2,lγb2,lγt2,lα2,lβ2,lγ2) as two LNCNs in L and a real numberδ≥0; then the operational laws of LNCNs are as follows:(7)m1⊕m2=lαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1=⊕lαb2,lαt2,lβb2,lβt2,lγb2,lγt2,lα2,lβ2,lγ2=lαb1+αb2-αb1αb2/g,lαt1+αt2-αt1αt2/g,lβb1βb2/g,lβt1βt2/g,=lγb1γb2/g,lγt1γt2/g,lα1+α2-α1α2/g,lβ1β2/g,lγ1γ2/g;(8)m1⊗m2=lαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1=⊗lαb2,lαt2,lβb2,lβt2,lγb2,lγt2,lα2,lβ2,lγ2=lαb1αb2/g,lαt1αt2/g,lβb1+βb2-βb1βb2/g,lβt1+βt2-βt1βt2/g,=lγb1+γb2-γb1γb2/g,lγt1+γt2-γt1γt2/g,=lα1α2/g,lβ1+β2-β1β2/g,lγ1+γ2-γ1γ2/glβb1+βb2-βb1βb2/g,lβt1+βt2-βt1βt2/g;(9)δm1=δlαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1=lg-g1-αb1/gδ,lg-g1-αt1/gδ,lgβb1/gδ,lgβt1/gδ,lgγb1/gδ,lgγt1/gδ,=lg-g1-α1/gδ,lgβ1/gδ,lgγ1/gδ;(10)m1δ=lαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1δ=lgαb1/gδ,lgαt1/gδ,lg-g1-βb1/gδ,lg-g1-βt1/gδ,=lg-g1-γb1/gδ,lg-g1-γt1/gδ,=lgα1/gδ,lg-g1-β1/gδ,lg-g1-γ1/gδlg-g1-βb1/gδ,lg-g1-βt1/gδ.Definition 6 ([18]).
Setm=(lαb,lαt,lβb,lβt,lγb,lγt, lα,lβ,lγ) as a LNCN in L; then the score S(m), accuracy A(m), and certain C(m) functions can be defined, respectively, as follows:(11)Sm=19g4g+αb+αt-βb-βt-γb-γt+2g+α-β-γ;(12)Am=13gαb+αt-βb-βt+α-β;(13)Cm=13gαb+αt+α.Definition 7 ([18]).
Setm1=(lαb1,lαt1,lβb1,lβt1,lγb1,lγt1, lα1,lβ1,lγ1), m2=(lαb2,lαt2,lβb2,lβt2,lγb2,lγt2, lα2,lβ2,lγ2) as two LNCNs in L; thenifS(m1) >S(m2), thenm1>m2;ifS(m1) = S(m2) thenifA(m1) >A(m2), thenm1>m2;ifA(m1)>A(m2) andC(m1)>C(m2), thenm1>m2;ifA(m1) = A(m2) andCm1=C(m2), thenm1=m2.


### 2.2. Generalized Weighted Heronian Mean and Three-Parameter Weighted Heronian Mean Operators

Definition 8 ([19]).
Letw=(w1,w2,…,wn) be the relative weight ofmi for mi∈R(i=1,2,…,n), wi∈0,1, and ∑i=1nwi=1 and p,q≥0. Then(14)GWHMp,qm1,m2,…mn=1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q,where λ=∑i=1n∑j=inwipwjq is called a generalized weighted Heronian mean (GWHM) operator.Definition 9 ([19]).
Letw=(w1,w2,…,wn) be the relative weight ofmiformi∈R(i=1,2,…,n),wi∈0,1, and ∑i=1nwi=1 and p,q,r≥0. Then(15)TPWHMp,q,rm1,m2,…mn=1λ⨁i=1n⨁j=in⨁k=jnwipwjqwkrmip⊗mjq⊗mkr1/p+q+r,where λ=∑i=1n∑j=in∑k=jnwipwjqwkr is called a three-parameter weighted Heronian mean (TPWHM) operator.


## 2.1. Linguistic Neutrosophic Numbers, Linguistic Neutrosophic Cubic Numbers, and Their Operational Laws

Definition 1 ([17]).
SetL={lj∣l0≤lj≤lg,j∈[0,g]} as a language term set; g is an even number. Then a LNN can be defined as follows:(1)m=lα,lβ,lγ,in which lα,lβ,lγ∈L and α,β,γ∈[0,g],lα,lβ, and lγ represent the truth, indeterminacy, and falsity variables, respectively, in linguistic terms.Definition 2 ([17]).
Setm=lα,lβ,lγ, m1=lα1,lβ1,lγ1, and m2=lα2,lβ2,lγ2 as three LNNs in L and a real numberδ≥0; then the operational laws of LNNs are as follows:(2)m1⊕m2=lα1,lβ1,lγ1⊕lα2,lβ2,lγ2=lα1+α2-α1α2/g,lβ1β2/g,lγ1γ2/g;(3)m1⊗m2=lα1,lβ1,lγ1⊗lα2,lβ2,lγ2=lα1α2/g,lβ1+β2-β1β2/g,lγ1+γ2-γ1γ2/g;(4)δm=δlα,lβ,lγ=lg-g1-α/gδ,lgβ/gδ,lgγ/gδ;(5)mδ=lα,lβ,lγδ=lgα/gδ,lg-g1-β/gδ,lg-g1-γ/gδ.Definition 3 ([18]).
SetL={lj∣l0≤lj≤lg,j∈[0,g]} as a language term set; g is an even number. Then a LNCN can be defined as follows:(6)m=u,c,where u=lαb,lαt,lβb,lβt,lγb,lγt expresses an uncertain LNN and lαb,lαt,lβb,lβt,lγb,lγt represent, respectively, the truth, the indeterminacy, and the falsity uncertain linguistic variables for lαb,lαt,lβb,lβt,lγb,lγt∈L and αb,αt,βb,βt,γb,γt∈[0,g],αb≤αt,βb≤βt,γb≤γt; c=lα,lβ,lγexpresses a LNN, in which lα,lβ,lγ∈L and α,β,γ∈[0,g], andlα,lβandlγ represent the truth, the indeterminacy, and the falsity linguistic variables, respectively, in linguistic terms.Definition 4 ([18]).
Setm=(lαb,lαt,lβb,lβt,lγb,lγt, lα,lβ,lγ) as a LNCN in L, then, one calls m①
an internal LNCN ifαb≤α≤αt,βb≤β≤βt,γb≤γ≤γt;②
an external LNCN ifα∉[αb,αt],β∉βb,βt and γ∉γb,γt.Definition 5 ([18]).
Setm1 =(lαb1,lαt1,lβb1,lβt1,lγb1,lγt1, lα1,lβ1,lγ1), and m2=(lαb2,lαt2,lβb2,lβt2,lγb2,lγt2,lα2,lβ2,lγ2) as two LNCNs in L and a real numberδ≥0; then the operational laws of LNCNs are as follows:(7)m1⊕m2=lαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1=⊕lαb2,lαt2,lβb2,lβt2,lγb2,lγt2,lα2,lβ2,lγ2=lαb1+αb2-αb1αb2/g,lαt1+αt2-αt1αt2/g,lβb1βb2/g,lβt1βt2/g,=lγb1γb2/g,lγt1γt2/g,lα1+α2-α1α2/g,lβ1β2/g,lγ1γ2/g;(8)m1⊗m2=lαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1=⊗lαb2,lαt2,lβb2,lβt2,lγb2,lγt2,lα2,lβ2,lγ2=lαb1αb2/g,lαt1αt2/g,lβb1+βb2-βb1βb2/g,lβt1+βt2-βt1βt2/g,=lγb1+γb2-γb1γb2/g,lγt1+γt2-γt1γt2/g,=lα1α2/g,lβ1+β2-β1β2/g,lγ1+γ2-γ1γ2/glβb1+βb2-βb1βb2/g,lβt1+βt2-βt1βt2/g;(9)δm1=δlαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1=lg-g1-αb1/gδ,lg-g1-αt1/gδ,lgβb1/gδ,lgβt1/gδ,lgγb1/gδ,lgγt1/gδ,=lg-g1-α1/gδ,lgβ1/gδ,lgγ1/gδ;(10)m1δ=lαb1,lαt1,lβb1,lβt1,lγb1,lγt1,lα1,lβ1,lγ1δ=lgαb1/gδ,lgαt1/gδ,lg-g1-βb1/gδ,lg-g1-βt1/gδ,=lg-g1-γb1/gδ,lg-g1-γt1/gδ,=lgα1/gδ,lg-g1-β1/gδ,lg-g1-γ1/gδlg-g1-βb1/gδ,lg-g1-βt1/gδ.Definition 6 ([18]).
Setm=(lαb,lαt,lβb,lβt,lγb,lγt, lα,lβ,lγ) as a LNCN in L; then the score S(m), accuracy A(m), and certain C(m) functions can be defined, respectively, as follows:(11)Sm=19g4g+αb+αt-βb-βt-γb-γt+2g+α-β-γ;(12)Am=13gαb+αt-βb-βt+α-β;(13)Cm=13gαb+αt+α.Definition 7 ([18]).
Setm1=(lαb1,lαt1,lβb1,lβt1,lγb1,lγt1, lα1,lβ1,lγ1), m2=(lαb2,lαt2,lβb2,lβt2,lγb2,lγt2, lα2,lβ2,lγ2) as two LNCNs in L; thenifS(m1) >S(m2), thenm1>m2;ifS(m1) = S(m2) thenifA(m1) >A(m2), thenm1>m2;ifA(m1)>A(m2) andC(m1)>C(m2), thenm1>m2;ifA(m1) = A(m2) andCm1=C(m2), thenm1=m2.


## 2.2. Generalized Weighted Heronian Mean and Three-Parameter Weighted Heronian Mean Operators

Definition 8 ([19]).
Letw=(w1,w2,…,wn) be the relative weight ofmi for mi∈R(i=1,2,…,n), wi∈0,1, and ∑i=1nwi=1 and p,q≥0. Then(14)GWHMp,qm1,m2,…mn=1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q,where λ=∑i=1n∑j=inwipwjq is called a generalized weighted Heronian mean (GWHM) operator.Definition 9 ([19]).
Letw=(w1,w2,…,wn) be the relative weight ofmiformi∈R(i=1,2,…,n),wi∈0,1, and ∑i=1nwi=1 and p,q,r≥0. Then(15)TPWHMp,q,rm1,m2,…mn=1λ⨁i=1n⨁j=in⨁k=jnwipwjqwkrmip⊗mjq⊗mkr1/p+q+r,where λ=∑i=1n∑j=in∑k=jnwipwjqwkr is called a three-parameter weighted Heronian mean (TPWHM) operator.


## 3. Two Aggregation Operators of LNCNs

### 3.1. GWHM Operator of LNCNs

Definition 10.
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi) (i=1,2,…,n) as a collection of LNCNs in L; then the LNCNGWHM operator can be defined as(16)LNCNGWHMp,qm1,m2,…mn=1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q,whereλ=∑i=1n∑j=inwipwjq, wi∈0,1 and ∑i=1nwi=1.Then we can use Definitions5 and 10 to get the following theorem.Theorem 11.
Setmi=(lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi) (i=1,2,…,n) as a collection of LNCNs in L; then by LNCNGWHM operator, the aggregation result of mi is still a LNCN, which is given by the following form:(17)LNCNGWHMp,qm1,m2,…mn=1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q=lg1-∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ1/p+q,lg1-∏i=1n∏j=in1-αti/gpαtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq1/λ1/p+q,lg-g1-∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq1/λ1/p+q,lg-g1-∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq1/λ1/p+q,=lg1-∏i=1n∏j=in1-αi/gpαj/gqwipwjq1/λ1/p+q,lg-g1-∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq1/λ1/p+q,where λ=∑i=1n∑j=inwipwjq,wi∈0,1, and∑i=1nwi=1.Proof.
(1)(18)mip=lgαbi/gp,lgαti/gp,lg-g1-βbi/gp,lg-g1-βti/gp,=lg-g1-γbi/gp,lg-g1-γti/gp,=lgαi/gp,lg-g1-βi/gp,lg-g1-γi/gp;
(2)(19)mjq=lgαbj/gq,lgαtj/gq,lg-g1-βbj/gq,lg-g1-βtj/gq,lg-g1-γbj/gq,lg-g1-γtj/gq,lgαj/gq,lg-g1-βj/gq,lg-g1-γj/gq;
(3)(20)mip⊗mjq=lgαbi/gpgαbj/gq/g,lgαti/gpgαtj/gq/g,=lg-g1-βbi/gp+g-g1-βbj/gq-g-g1-βbi/gpg-g1-βbj/gq/g,lg-g1-βti/gp+g-g1-βtj/gq-g-g1-βti/gpg-g1-βtj/gq/g,=lg-g1-γbi/gp+g-g1-γbj/gq-g-g1-γbi/gpg-g1-γbj/gq/g,lg-g1-γti/gp+g-g1-γtj/gq-g-g1-γti/gpg-g1-γtj/gq/g,=lgαi/gpgαj/gq/g,lg-g1-βi/gp+g-g1-βj/gq-g-g1-βi/gpg-g1-βj/gq/g,lg-g1-γi/gp+g-g1-γj/gq-g-g1-γi/gpg-g1-γj/gq/g=lgαbi/gpαbj/gq,lgαti/gpαtj/gq,lg-g1-βbi/gp1-βbj/gq,lg-g1-βti/gp1-βtj/gq,lg-g1-γbi/gp1-γbj/gq,lg-g1-γti/gp1-γtj/gq,==lgαi/gpαj/gq,lg-g1-βi/gp1-βj/gq,lg-g1-γi/gp1-γj/gq;
(4)(21)wipwjqmip⊗mjq=lg-g1-gαbi/gpαbj/gq/gwipwjq,lg-g1-gαti/gpαtj/gq/gwipwjq,lgg-g1-βbi/gp1-βbj/gq/gwipwjq,lgg-g1-βti/gp1-βtj/gq/gwipwjq,lgg-g1-γbi/gp1-γbj/gq/gwipwjq,lgg-g1-γti/gp1-γtj/gq/gwipwjq,lg-g1-gαi/gpαj/gq/gwipwjq,lgg-g1-βi/gp1-βj/gq/gwipwjq,lgg-g1-γi/gp1-γj/gq/gwipwjq=lg-g1-αbi/gpαbj/gqwipwjq,lg-g1-αti/gpαtj/gqwipwjq,lg1-1-βbi/gp1-βbj/gqwipwjq,lg1-1-βti/gp1-βtj/gqwipwjq,lg1-1-γbi/gp1-γbj/gqwipwjq,lg1-1-γti/gp1-γtj/gqwipwjq,lg-g1-αi/gpαj/gqwipwjq,lg1-1-βi/gp1-βj/gqwipwjq,lg1-1-γi/gp1-γj/gqwipwjq;
(5)(22)⨁i=1n⨁j=inwipwjqmip⊗mjq=lg-g∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq,lg-g∏i=1n∏j=in1-αti/gpαtj/gqwipwjq,lg∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq,lg∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq,lg∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq,lg∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq,lg-g∏i=1n∏j=in1-αi/gpαj/gqwipwjq,lg∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq,lg∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq;
(6)(23)1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1λ=lg-g∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ,1λlg-g∏i=1n∏j=in1-αti/gpαtj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq1/λ,1λlg-g∏i=1n∏j=in1-αi/gpαj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq1/λ;
(7)(24)1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q==lg1-∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ1/p+q,=lg1-∏i=1n∏j=in1-αti/gpαtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq1/λ1/p+q,=lg1-∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq1/λ1/p+q.
The proof of Theorem11 is completed.Theorem 12 (idempotency).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; if mi = m, then (25)LNCNGWHMp,qm1,m2,…mn=LNCNGWHMp,qm,m,…m=m.Proof.
Sincemi = m for i=1,2,…,n, there is the following result: (26)LNCNGWHMp,qm1,m2,…mn==LNCNGWHMp,qm,m,…m==lg1-1-αb/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg1-1-αt/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,=lg-g1-1-1-βb/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-βt/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,=lg-g1-1-1-γb/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-γt/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,=lg1-1-α/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-β/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-γ/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q==lgαb/gp+q1/p+q,lgαt/gp+q1/p+q,lg-g1-βb/gp+q1/p+q,lg-g1-βt/gp+q1/p+q,=lg-g1-γb/gp+q1/p+q,lg-g1-γt/gp+q1/p+q,lgα/gp+q1/p+q,lg-g1-β/gp+q1/p+q,lg-g1-γ/gp+q1/p+q==lαb,lαt,lβb,lβt,lγb,lγb,lα,lβ,lγ=m.
The proof of Theorem12 is completed.Theorem 13 (monotonicity).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi) and ki = (lαbi′,lαti′,lβbi′,lβti′,lγbi′,lγti′,lαi′,lβi′,lγi′) (i=1,2,…,n) as two collections of LNCNs in L; if αbi≤αbi′, αti≤αti′, αi≤αi′, βbi≥βbi′, βti≥βti′, βi≥βi′andγbi≥γbi′, γti≥γti′, γi≥γi′, then (27)LNCNGWHMp,qm1,m2,…mn≤LNCNGWHMp,qk1,k2,…kn.Proof.
Becauseαbi≤αbi′, αti≤αti′, αi≤αi′, βbi≥βbi′, βti≥βti′, βi≥βi′ and γbi≥γbi′, γti≥γti′, γi≥γi′, we can easily obtain(28)1-αbigpαbjgq≥1-αbi′gpαbj′gq,g1-∏i=1n∏j=in1-αbigpαbjgqwipwjq1/λ1/p+q≤g1-∏i=1n∏j=in1-αbi′gpαbj′gqwipwjq1/λ1/p+q,g1-∏i=1n∏j=in1-αtigpαtjgqwipwjq1/λ1/p+q≤g1-∏i=1n∏j=in1-αti′gpαtj′gqwipwjq1/λ1/p+q,g1-∏i=1n∏j=in1-αigpαjgqwipwjq1/λ1/p+q≤g1-∏i=1n∏j=in1-αi′gpαj′gqwipwjq1/λ1/p+q.
Similarly(29)1-1-βbigp1-βbjgq≥1-1-βbi′gp1-βbj′gq,g-g1-∏i=1n∏j=in1-1-βbigp1-βbjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-βbi′gp1-βbj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-βtigp1-βtjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-βti′gp1-βtj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-βigp1-βjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-βi′gp1-βj′gqwipwjq1/λ1/p+q.
And(30)g-g1-∏i=1n∏j=in1-1-γbigp1-γbjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-γbi′gp1-γbj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-γtigp1-γtjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-γti′gp1-γtj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-γigp1-γjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-γi′gp1-γj′gqwipwjq1/λ1/p+q.
So,LNCNGWHMp,q(m1,m2,…mn)≤LNCNGWHMp,q(k1,k2,…kn). Therefore, the proof of Theorem 13 is completed.Theorem 14 (boundedness).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; letting m-=(min⁡(lαbi),min⁡(lαti),max⁡(lβbi),max⁡(lβti), max⁡(lγbi),max⁡(lγti), min⁡(lαi),max⁡(lβi),max⁡(lγi)) and m+=(max⁡(lαbi),max⁡(lαti),min⁡(lβbi),min⁡(lβti), min⁡(lγbi),min⁡(lγti), min⁡(lαi),min⁡(lβi),min⁡(lγi)), then (31)m-≤LNCNGWHMp,qm1,m2,…,mn≤m+.Proof.
Based on Theorems12 and 13, we can obtain(32)m-=LNCNGWHMp,qm-,m-,…,m-,m+=LNCNGWHMp,qm+,m+,…,m+,LNCNGWHMp,qm-,m-,…,m-≤LNCNGWHMp,qm1,m2,…,mn≤LNCNGWHMp,qm+,m+,…,m+.
Thenm-≤LNCNWHMp,q(m1,m2,…,mn)≤m+.
The proof of Theorem14 is completed.


### 3.2. Three-Parameter Weighted Heronian Mean Operator of LNCNs

Definition 15.
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; then the LNCNTPWHM operator can be defined as (33)LNCNTPWHMp,q,rm1,m2,…mn=1λ⨁i=1n⨁j=in⨁k=jnwipwjqwkrmip⊗mjq⊗mkr1/p+q+r,where λ=∑i=1n∑j=in∑k=jnwipwjqwkr, wi∈0,1, and ∑i=1nwi=1.Then, we can use Definitions5 and 15 to get the following theorem.Theorem 16.
Setmi=(lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; then by LNCNTPWHM operator, the aggregation result of mi is still a LNCN, which is as follows: (34)LNCNTPWHMp,q,rm1,m2,…mn==1λ⨁i=1n⨁j=in⨁k=jnwipwjqwkrmip⊗mjq⊗mkr1/p+q+r==lg1-∏i=1n∏j=in∏k=jn1-αbi/gpαbj/gqαbk/grwipwjqwkr1/λ1/p+q+r,lg1-∏i=1n∏j=in∏k=jn1-αti/gpαtj/gqαtk/grwipwjqwkr1/λ1/p+q+r,=lg-g1-∏i=1n∏j=in∏k=jn1-1-βbi/gp1-βbj/gq1-βbk/grwipwjqwkr1/λ1/p+q+r,lg-g1-∏i=1n∏j=in∏k=jn1-1-βti/gp1-βtj/gq1-βtk/grwipwjqwkr1/λ1/p+q+r,=lg-g1-∏i=1n∏j=in∏k=jn1-1-γbi/gp1-γbj/gq1-γbk/grwipwjqwkr1/λ1/p+q+r,lg-g1-∏i=1n∏j=in∏k=jn1-1-γti/gp1-γtj/gq1-γtk/grwipwjqwkr1/λ1/p+q+r,=lg1-∏i=1n∏j=in∏k=jn1-αi/gpαj/gqαk/grwipwjqwkr1/λ1/p+q+r,lg-g1-∏i=1n∏j=in∏k=jn1-1-βi/gp1-βj/gq1-βk/grwipwjqwkr1/λ1/p+q+r,=lg-g1-∏i=1n∏j=in∏k=jn1-1-γi/gp1-γj/gq1-γk/grwipwjqwkr1/λ1/p+q+r,where λ=∑i=1n∑j=in∑k=jnwipwjqwkr, wi∈0,1, and ∑i=1nwi=1.Theorem 17 (idempotency).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi) (i=1,2,…,n) as a collection of LNCNs in L; if mi=m, then (35)LNCNTPWHMp,q,rm1,m2,…mn=LNCNTPWHMp,q,rm,m,…m=m.Theorem 18 (monotonicity).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi) and ki = (lαbi′,lαti′,lβbi′,lβti′,lγbi′,lγti′,lαi′,lβi′,lγi′) (i=1,2,…,n) as two collections of LNCNs in L; if αbi≤αbi′, αti≤αti′, αi≤αi′, βbi≥βbi′, βti≥βti′, βi≥βi′andγbi≥γbi′, γti≥γti′, γi≥γi′, then(36)LNCNTPWHMp,q,rm1,m2,…mn≤LNCNTPWHMp,q,rk1,k2,…kn.Theorem 19 (boundedness).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; letting m-=(min⁡(lαbi),min⁡(lαti),max⁡(lβbi),max⁡(lβti), max⁡(lγbi),max⁡(lγti), min⁡(lαi),max⁡(lβi),max⁡(lγi)) and m+=(max⁡(lαbi),max⁡(lαti),min⁡(lβbi),min⁡(lβti), min⁡(lγbi),min⁡(lγti), min⁡(lαi),min⁡(lβi),min⁡(lγi)), then (37)m-≤LNCNTPWHMp,q,rm1,m2,…mn≤m+.The proofs of Theorems16–19 are similar to those of Theorems 11–14, so we do not repeat them again.


## 3.1. GWHM Operator of LNCNs

Definition 10.
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi) (i=1,2,…,n) as a collection of LNCNs in L; then the LNCNGWHM operator can be defined as(16)LNCNGWHMp,qm1,m2,…mn=1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q,whereλ=∑i=1n∑j=inwipwjq, wi∈0,1 and ∑i=1nwi=1.Then we can use Definitions5 and 10 to get the following theorem.Theorem 11.
Setmi=(lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi) (i=1,2,…,n) as a collection of LNCNs in L; then by LNCNGWHM operator, the aggregation result of mi is still a LNCN, which is given by the following form:(17)LNCNGWHMp,qm1,m2,…mn=1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q=lg1-∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ1/p+q,lg1-∏i=1n∏j=in1-αti/gpαtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq1/λ1/p+q,lg-g1-∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq1/λ1/p+q,lg-g1-∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq1/λ1/p+q,=lg1-∏i=1n∏j=in1-αi/gpαj/gqwipwjq1/λ1/p+q,lg-g1-∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq1/λ1/p+q,where λ=∑i=1n∑j=inwipwjq,wi∈0,1, and∑i=1nwi=1.Proof.
(1)(18)mip=lgαbi/gp,lgαti/gp,lg-g1-βbi/gp,lg-g1-βti/gp,=lg-g1-γbi/gp,lg-g1-γti/gp,=lgαi/gp,lg-g1-βi/gp,lg-g1-γi/gp;
(2)(19)mjq=lgαbj/gq,lgαtj/gq,lg-g1-βbj/gq,lg-g1-βtj/gq,lg-g1-γbj/gq,lg-g1-γtj/gq,lgαj/gq,lg-g1-βj/gq,lg-g1-γj/gq;
(3)(20)mip⊗mjq=lgαbi/gpgαbj/gq/g,lgαti/gpgαtj/gq/g,=lg-g1-βbi/gp+g-g1-βbj/gq-g-g1-βbi/gpg-g1-βbj/gq/g,lg-g1-βti/gp+g-g1-βtj/gq-g-g1-βti/gpg-g1-βtj/gq/g,=lg-g1-γbi/gp+g-g1-γbj/gq-g-g1-γbi/gpg-g1-γbj/gq/g,lg-g1-γti/gp+g-g1-γtj/gq-g-g1-γti/gpg-g1-γtj/gq/g,=lgαi/gpgαj/gq/g,lg-g1-βi/gp+g-g1-βj/gq-g-g1-βi/gpg-g1-βj/gq/g,lg-g1-γi/gp+g-g1-γj/gq-g-g1-γi/gpg-g1-γj/gq/g=lgαbi/gpαbj/gq,lgαti/gpαtj/gq,lg-g1-βbi/gp1-βbj/gq,lg-g1-βti/gp1-βtj/gq,lg-g1-γbi/gp1-γbj/gq,lg-g1-γti/gp1-γtj/gq,==lgαi/gpαj/gq,lg-g1-βi/gp1-βj/gq,lg-g1-γi/gp1-γj/gq;
(4)(21)wipwjqmip⊗mjq=lg-g1-gαbi/gpαbj/gq/gwipwjq,lg-g1-gαti/gpαtj/gq/gwipwjq,lgg-g1-βbi/gp1-βbj/gq/gwipwjq,lgg-g1-βti/gp1-βtj/gq/gwipwjq,lgg-g1-γbi/gp1-γbj/gq/gwipwjq,lgg-g1-γti/gp1-γtj/gq/gwipwjq,lg-g1-gαi/gpαj/gq/gwipwjq,lgg-g1-βi/gp1-βj/gq/gwipwjq,lgg-g1-γi/gp1-γj/gq/gwipwjq=lg-g1-αbi/gpαbj/gqwipwjq,lg-g1-αti/gpαtj/gqwipwjq,lg1-1-βbi/gp1-βbj/gqwipwjq,lg1-1-βti/gp1-βtj/gqwipwjq,lg1-1-γbi/gp1-γbj/gqwipwjq,lg1-1-γti/gp1-γtj/gqwipwjq,lg-g1-αi/gpαj/gqwipwjq,lg1-1-βi/gp1-βj/gqwipwjq,lg1-1-γi/gp1-γj/gqwipwjq;
(5)(22)⨁i=1n⨁j=inwipwjqmip⊗mjq=lg-g∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq,lg-g∏i=1n∏j=in1-αti/gpαtj/gqwipwjq,lg∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq,lg∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq,lg∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq,lg∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq,lg-g∏i=1n∏j=in1-αi/gpαj/gqwipwjq,lg∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq,lg∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq;
(6)(23)1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1λ=lg-g∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ,1λlg-g∏i=1n∏j=in1-αti/gpαtj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq1/λ,1λlg-g∏i=1n∏j=in1-αi/gpαj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq1/λ,1λlg∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq1/λ;
(7)(24)1λ⨁i=1n⨁j=inwipwjqmip⊗mjq1/p+q==lg1-∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ1/p+q,=lg1-∏i=1n∏j=in1-αti/gpαtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βbi/gp1-βbj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βti/gp1-βtj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γbi/gp1-γbj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γti/gp1-γtj/gqwipwjq1/λ1/p+q,=lg1-∏i=1n∏j=in1-αbi/gpαbj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-βi/gp1-βj/gqwipwjq1/λ1/p+q,=lg-g1-∏i=1n∏j=in1-1-γi/gp1-γj/gqwipwjq1/λ1/p+q.
The proof of Theorem11 is completed.Theorem 12 (idempotency).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; if mi = m, then (25)LNCNGWHMp,qm1,m2,…mn=LNCNGWHMp,qm,m,…m=m.Proof.
Sincemi = m for i=1,2,…,n, there is the following result: (26)LNCNGWHMp,qm1,m2,…mn==LNCNGWHMp,qm,m,…m==lg1-1-αb/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg1-1-αt/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,=lg-g1-1-1-βb/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-βt/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,=lg-g1-1-1-γb/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-γt/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,=lg1-1-α/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-β/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q,lg-g1-1-1-γ/gp+q∑i=1n∑j=inwipwjq1/λ1/p+q==lgαb/gp+q1/p+q,lgαt/gp+q1/p+q,lg-g1-βb/gp+q1/p+q,lg-g1-βt/gp+q1/p+q,=lg-g1-γb/gp+q1/p+q,lg-g1-γt/gp+q1/p+q,lgα/gp+q1/p+q,lg-g1-β/gp+q1/p+q,lg-g1-γ/gp+q1/p+q==lαb,lαt,lβb,lβt,lγb,lγb,lα,lβ,lγ=m.
The proof of Theorem12 is completed.Theorem 13 (monotonicity).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi) and ki = (lαbi′,lαti′,lβbi′,lβti′,lγbi′,lγti′,lαi′,lβi′,lγi′) (i=1,2,…,n) as two collections of LNCNs in L; if αbi≤αbi′, αti≤αti′, αi≤αi′, βbi≥βbi′, βti≥βti′, βi≥βi′andγbi≥γbi′, γti≥γti′, γi≥γi′, then (27)LNCNGWHMp,qm1,m2,…mn≤LNCNGWHMp,qk1,k2,…kn.Proof.
Becauseαbi≤αbi′, αti≤αti′, αi≤αi′, βbi≥βbi′, βti≥βti′, βi≥βi′ and γbi≥γbi′, γti≥γti′, γi≥γi′, we can easily obtain(28)1-αbigpαbjgq≥1-αbi′gpαbj′gq,g1-∏i=1n∏j=in1-αbigpαbjgqwipwjq1/λ1/p+q≤g1-∏i=1n∏j=in1-αbi′gpαbj′gqwipwjq1/λ1/p+q,g1-∏i=1n∏j=in1-αtigpαtjgqwipwjq1/λ1/p+q≤g1-∏i=1n∏j=in1-αti′gpαtj′gqwipwjq1/λ1/p+q,g1-∏i=1n∏j=in1-αigpαjgqwipwjq1/λ1/p+q≤g1-∏i=1n∏j=in1-αi′gpαj′gqwipwjq1/λ1/p+q.
Similarly(29)1-1-βbigp1-βbjgq≥1-1-βbi′gp1-βbj′gq,g-g1-∏i=1n∏j=in1-1-βbigp1-βbjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-βbi′gp1-βbj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-βtigp1-βtjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-βti′gp1-βtj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-βigp1-βjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-βi′gp1-βj′gqwipwjq1/λ1/p+q.
And(30)g-g1-∏i=1n∏j=in1-1-γbigp1-γbjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-γbi′gp1-γbj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-γtigp1-γtjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-γti′gp1-γtj′gqwipwjq1/λ1/p+q,g-g1-∏i=1n∏j=in1-1-γigp1-γjgqwipwjq1/λ1/p+q≥g-g1-∏i=1n∏j=in1-1-γi′gp1-γj′gqwipwjq1/λ1/p+q.
So,LNCNGWHMp,q(m1,m2,…mn)≤LNCNGWHMp,q(k1,k2,…kn). Therefore, the proof of Theorem 13 is completed.Theorem 14 (boundedness).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; letting m-=(min⁡(lαbi),min⁡(lαti),max⁡(lβbi),max⁡(lβti), max⁡(lγbi),max⁡(lγti), min⁡(lαi),max⁡(lβi),max⁡(lγi)) and m+=(max⁡(lαbi),max⁡(lαti),min⁡(lβbi),min⁡(lβti), min⁡(lγbi),min⁡(lγti), min⁡(lαi),min⁡(lβi),min⁡(lγi)), then (31)m-≤LNCNGWHMp,qm1,m2,…,mn≤m+.Proof.
Based on Theorems12 and 13, we can obtain(32)m-=LNCNGWHMp,qm-,m-,…,m-,m+=LNCNGWHMp,qm+,m+,…,m+,LNCNGWHMp,qm-,m-,…,m-≤LNCNGWHMp,qm1,m2,…,mn≤LNCNGWHMp,qm+,m+,…,m+.
Thenm-≤LNCNWHMp,q(m1,m2,…,mn)≤m+.
The proof of Theorem14 is completed.


## 3.2. Three-Parameter Weighted Heronian Mean Operator of LNCNs

Definition 15.
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; then the LNCNTPWHM operator can be defined as (33)LNCNTPWHMp,q,rm1,m2,…mn=1λ⨁i=1n⨁j=in⨁k=jnwipwjqwkrmip⊗mjq⊗mkr1/p+q+r,where λ=∑i=1n∑j=in∑k=jnwipwjqwkr, wi∈0,1, and ∑i=1nwi=1.Then, we can use Definitions5 and 15 to get the following theorem.Theorem 16.
Setmi=(lαbi,lαti,lβbi,lβti,lγbi,lγti, lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; then by LNCNTPWHM operator, the aggregation result of mi is still a LNCN, which is as follows: (34)LNCNTPWHMp,q,rm1,m2,…mn==1λ⨁i=1n⨁j=in⨁k=jnwipwjqwkrmip⊗mjq⊗mkr1/p+q+r==lg1-∏i=1n∏j=in∏k=jn1-αbi/gpαbj/gqαbk/grwipwjqwkr1/λ1/p+q+r,lg1-∏i=1n∏j=in∏k=jn1-αti/gpαtj/gqαtk/grwipwjqwkr1/λ1/p+q+r,=lg-g1-∏i=1n∏j=in∏k=jn1-1-βbi/gp1-βbj/gq1-βbk/grwipwjqwkr1/λ1/p+q+r,lg-g1-∏i=1n∏j=in∏k=jn1-1-βti/gp1-βtj/gq1-βtk/grwipwjqwkr1/λ1/p+q+r,=lg-g1-∏i=1n∏j=in∏k=jn1-1-γbi/gp1-γbj/gq1-γbk/grwipwjqwkr1/λ1/p+q+r,lg-g1-∏i=1n∏j=in∏k=jn1-1-γti/gp1-γtj/gq1-γtk/grwipwjqwkr1/λ1/p+q+r,=lg1-∏i=1n∏j=in∏k=jn1-αi/gpαj/gqαk/grwipwjqwkr1/λ1/p+q+r,lg-g1-∏i=1n∏j=in∏k=jn1-1-βi/gp1-βj/gq1-βk/grwipwjqwkr1/λ1/p+q+r,=lg-g1-∏i=1n∏j=in∏k=jn1-1-γi/gp1-γj/gq1-γk/grwipwjqwkr1/λ1/p+q+r,where λ=∑i=1n∑j=in∑k=jnwipwjqwkr, wi∈0,1, and ∑i=1nwi=1.Theorem 17 (idempotency).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi) (i=1,2,…,n) as a collection of LNCNs in L; if mi=m, then (35)LNCNTPWHMp,q,rm1,m2,…mn=LNCNTPWHMp,q,rm,m,…m=m.Theorem 18 (monotonicity).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi) and ki = (lαbi′,lαti′,lβbi′,lβti′,lγbi′,lγti′,lαi′,lβi′,lγi′) (i=1,2,…,n) as two collections of LNCNs in L; if αbi≤αbi′, αti≤αti′, αi≤αi′, βbi≥βbi′, βti≥βti′, βi≥βi′andγbi≥γbi′, γti≥γti′, γi≥γi′, then(36)LNCNTPWHMp,q,rm1,m2,…mn≤LNCNTPWHMp,q,rk1,k2,…kn.Theorem 19 (boundedness).
Setmi = (lαbi,lαti,lβbi,lβti,lγbi,lγti,lαi,lβi,lγi)  (i=1,2,…,n) as a collection of LNCNs in L; letting m-=(min⁡(lαbi),min⁡(lαti),max⁡(lβbi),max⁡(lβti), max⁡(lγbi),max⁡(lγti), min⁡(lαi),max⁡(lβi),max⁡(lγi)) and m+=(max⁡(lαbi),max⁡(lαti),min⁡(lβbi),min⁡(lβti), min⁡(lγbi),min⁡(lγti), min⁡(lαi),min⁡(lβi),min⁡(lγi)), then (37)m-≤LNCNTPWHMp,q,rm1,m2,…mn≤m+.The proofs of Theorems16–19 are similar to those of Theorems 11–14, so we do not repeat them again.


## 4. MADM Methods Based on the LNCNGWHM or LNCNTPWHM Operator

This section uses the LNCNGWHM or LNCNTPWHM operator to deal with the MADM problems with LNCN information.LetA={A1,A2,…,Am} be a discrete set of alternatives with a set of attributesC={C1,C2,…,Cn}, and the weight vector of Cjj=1,2,…,n is λ=λ1,λ2,…,λn, λj≥0 and ∑j=1nλj=1. A linguistic assessment set L={lj∣j∈[0,g]} is given. Some experts use LNCN to evaluate the alternatives Aii=1,2,…,m under the attributesCjj=1,2,…,n. The assessed values of the experts for Ai with attribute Cj are mij=(lαbij,lαtij,lβbij,lβtij,lγbij,lγtij,lαij,lβij,lγij)i=1,2,…,m;j=1,2,…,n. Then, we can get the neutrosophic linguistic cubic decision evaluation matrix(mij)m×n.Then, the decision-making method based on the LNCNGWHM or LNCNTPWHM operator is described as follows.Step 1.
According to the weight vectorλ=λ1,λ2,…,λnand the LNCNGWHM or LNCNTPWHM operator, we can calculate mi=LNCNGWHM(mi1,mi2,…,mij) ormi=LNCNTPWHM(mi1,mi2,…,mij).Step 2.
Calculate the score value ofS(mi) (accuracy value ofA(mi) and certain value ofC(mi)if necessary) of the LNCN Aii=1,2,…,m according to formula (11) (formula (12) and formula (13) if necessary).Step 3.
According to ranking method of Definition7, we can rank the attributes corresponding to the values ofS(mi) (accuracy valueA(mi) and certain valueC(mi)if necessary).Step 4.
End.


## 5. Illustrative Example

This section considers a decision-making problem adapted from the literature [20]. A mechanical designer wants to design press machine; then he should consider the design of the reducing mechanism and the working mechanism. According to the press machine’s functional requirements, there are four design schemes/alternatives A1,A2,A3andA4 to be proposed by the designers, which are shown in Table 1. The four design schemes must satisfy the requirements of four attributes while being evaluated: the manufacturing cost (C1), the mechanical structure (C2), the transmission effectiveness (C3),and the reliability (C4). The importance of four attributes is given as a weight vector λ=0.3,0.25,0.25,0.2. Then, the experts define the linguistic term setL={lj∣j∈[0,8]}, where L={l0=extremelylow, l1=verylow, l2=low, l3=slightlylow, l4=medium, l5=slightlyhigh, l6=high, l7=veryhigh, l8=extremelyhigh}. Afterwards, they evaluate the four design schemes/alternatives under the four attributes by the form of LNCNs based on L. Thus, the LNCN decision matrix(mij)4×4 can be established, which are shown in Table 2.Table 1
Four design schemes/alternatives for press machine.
Design scheme A 1 A 2 A 3 A 4 Reducing mechanism Gear reducer Gear head motor Gear reducer Gear head motor Working mechanism Flywheel-crank-slider mechanism Flywheel-screw-driving mechanismTable 2
The decision matrix(mij)4×4.
C 1 C 2 C 3 C 4 A 1 l 4 ,  l 6 , l 1 ,  l 2 , l 1 ,  l 3 , l 5  , l 1 , l 2 l 4 ,  l 6 , l 1 ,  l 3 , l 1 ,  l 3 , l 5  , l 2 , l 2 l 4 ,  l 7 , l 1 ,  l 3 , l 2 ,  l 3 , l 6  , l 2 , l 3 l 4 ,  l 7 , l 1 ,  l 2 , l 1 ,  l 2 , l 5  , l 2 , l 3 A 2 l 3 ,  l 5 , l 1 ,  l 2 , l 1 ,  l 2 , l 4  , l 1 , l 1 l 5 ,  l 7 , l 1 ,  l 2 , l 1 ,  l 2 , l 6  , l 1 , l 2 l 4 ,  l 6 , l 2 ,  l 3 , l 1 ,  l 2 , l 5  , l 3 , l 1 l 4 ,  l 5 , l 1 ,  l 3 , l 1 ,  l 2 , l 5  , l 2 , l 1 A 3 l 4 ,  l 7 , l 1 ,  l 2 , l 2 ,  l 3 , l 5  , l 2 , l 3 l 6 ,  l 7 , l 1 ,  l 3 , l 1 ,  l 3 , l 7  , l 2 , l 2 l 5 ,  l 7 , l 1 ,  l 3 , l 2 ,  l 3 , l 5  , l 2 , l 3 l 5 ,  l 7 , l 1 ,  l 2 , l 1 ,  l 3 , l 6  , l 2 , l 2 A 4 l 6 ,  l 7 , l 2 ,  l 3 , l 2 ,  l 3 , l 7  , l 3 , l 3 l 5 ,  l 7 , l 1 ,  l 2 , l 1 ,  l 2 , l 6  , l 1 , l 2 l 4 ,  l 6 , l 1 ,  l 2 , l 1 ,  l 2 , l 5  , l 1 , l 1 l 5 ,  l 6 , l 2 ,  l 3 , l 1 ,  l 2 , l 6  , l 1 , l 2Next, the decision-making methods proposed in Section4 are employed to deal with the decision problem; the description of decision procedures is shown as follows:


### 5.1. The Decision-Making Process Based on LNCNGWHM Operator or LNCNTPWHM Operator

Step 1.
By using (17) (supposep = q = 1) and the weight vector λ=0.3,0.25,0.25,0.2 of attributes, we can obtain the comprehensive evaluation values mi(i=1,2,3,4) of alternative Ai as follows:(38)m1=l4.0000,l6.4628,l1.0000,l2.4379,l1.1967,l2.8039,l5.2546,l1.5999,l2.3849,m2=l3.9271,l5.8066,l1.1967,l2.3849,l1.0000,l2.0000,l4.9402,l1.5297,l1.1967,m3=l4.9402,l7.0000,l1.0000,l2.4379,l1.5372,l3.0000,l5.7560,l2.0000,l2.5578,m4=l5.1800,l6.6362,l1.4814,l2.5028,l1.3041,l2.3243,l6.2161,l1.5499,l2.0274.Step 2.
Calculate the score values ofS(mi) according to (11) for mii=1,2,3,4:(39)Sm1=0.7263;Sm2=0.7412;Sm3=0.7384;Sm4=0.7617.According to the results ofS(mi)i=1,2,3,4, we can rank the alternatives A4≻A2≻A3≻A1, and then the design scheme A4 is the best among all the alternatives.On the other hand, we can use the LNCNTPWHM operator (setp=1,q=1) to deal with this problem.Step  1 ′. By using (34) (supposep = q = 1) and the weight vector λ=0.3,0.25,0.25,0.2of attributes, we can obtain the comprehensive evaluation values mi(i=1,2,3,4) of alternative Ai as follows:(40)m1=l4.0000,l6.4368,l1.0000,l2.4328,l1.1994,l2.8180,l5.2461,l1.5880,l2.3749,m2=l3.9051,l5.7709,l1.1994,l2.3749,l1.0000,l2.0000,l4.9108l1.5319,l1.1994,m3=l4.9108,l7.0000,l1.0000,l2.4328,l1.5632,l3.0000,l5.7142l2.0000,l2.5777,m4=l5.1989,l6.6406,l1.5015,l2.5170,l1.3314,l2.3472,l6.2253l1.6109,l2.0674.Step  2 ′. Calculate the score values ofS(mi) according to (11) for mii=1,2,3,4:(41)Sm1=0.7260;Sm2=0.7400;Sm3=0.7368;Sm4=0.7596.According to the results ofS(mi)  i=1,2,3,4, we can rank the alternatives A4≻A2≻A3≻A1, so the design scheme A4 is the best among all the alternatives.


### 5.2. Analyzing the Effect of the Parameters p, q, and r

Different parametersp, q, and r may have different effects on the decision results. Therefore, this section takes different values of p, q, and r to sort the various alternatives, and then Tables 3-4 present the results.Table 3
The ranking based on the LNCNGWHM operator with different values ofp and q.
p, q LNCNGWHM Ranking P=1, q=0 S( m 1 )=0.7285, S( m 2 )=0.7479, S( m 3 )=0.7377, S( m 4 )=0.7638 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=1, q=0.5 S( m 1 )=0.7263, S( m 2 )=0.7420, S( m 3 )=0.7377, S( m 4 )=0.7609 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=1, q=2 S( m 1 )=0.7277, S( m 2 )=0.7427, S( m 3 )=0.7388, S( m 4 )=0.7645 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=0, q=1 S( m 1 )=0.7269, S( m 2 )=0.7411, S( m 3 )=0.7470, S( m 4 )=0.7688 A 4 ≻ A 3 ≻ A 2 ≻ A 1 P=0.5, q=1 S( m 1 )=0.7257, S( m 2 )=0.7404, S( m 3 )=0.7410, S( m 4 )=0.7635 A 4 ≻ A 3 ≻ A 2 ≻ A 1 P=2, q=1 S( m 1 )=0.7283, S( m 2 )=0.7424, S( m 3 )=0.7350, S( m 4 )=0.7592 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=2, q=2 S( m 1 )=0.7290, S( m 2 )=0.7428, S( m 3 )=0.7352, S( m 4 )=0.7609 A 4 ≻ A 2 ≻ A 3 ≻ A 1Table 4
The ranking based on the LNCNTPWHM operator with different values ofp, q,andr.
p, q, r LNCNTPWHM Ranking p=1, q=1, r=2 S( m 1 )=0.7269, S( m 2 )=0.7406, S( m 3 )=0.7376, S( m 4 )=0.7621 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=1, q=2, r=1 S( m 1 )=0.7166, S( m 2 )=0.7342, S( m 3 )=0.7279, S( m 4 )=0.7772 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=1, q=2, r=2 S( m 1 )=0.7282, S( m 2 )=0.7431, S( m 3 )=0.7362, S( m 4 )=0.7612 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=1, q=2, r=3 S( m 1 )=0.7299, S( m 2 )=0.7444, S( m 3 )=0.7363, S( m 4 )=0.7650 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=2, q=1, r=1 S( m 1 )=0.7278, S( m 2 )=0.7401, S( m 3 )=0.7333, S( m 4 )=0.7556 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=2, q=1, r=2 S( m 1 )=0.7282, S( m 2 )=0.7404, S( m 3 )=0.7342, S( m 4 )=0.7578 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=2, q=2, r=1 S( m 1 )=0.7290, S( m 2 )=0.7419, S( m 3 )=0.7328, S( m 4 )=0.7569 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=3, q=2, r=1 S( m 1 )=0.7302, S( m 2 )=0.7412, S( m 3 )=0.7303, S( m 4 )=0.7531 A 4 ≻ A 2 ≻ A 3 ≻ A 1From Tables3 and 4, we can see that LNCNTPWHM operator can get more stable sorting than the LNCNGWHM operator, and, in addition, when the parameters p, q, and r take different values, the best design scheme/alternative using either the LNCNGWHM or LNCNTPWHM operator is always A4. Therefore, the parameters p, q, and r in the LNCNGWHM or LNCNTPWHM operator have little influence on decision-making.


### 5.3. Comparing with the Related Methods

Firstly, compared with the literature [20], this paper used the decision information under LNCN environment, while the literature [20] used the decision information under intuitionistic fuzzy environment. In the literature [20], only incomplete information can be effectively expressed, and the indeterminate and inconsistent information cannot be described effectively, while LNCN is composed of uncertain linguistic neutrosophic number and linguistic neutrosophic number, where the uncertain linguistic neutrosophic number is represented by the truth, the indeterminacy, and the falsity uncertain linguistic variables, respectively, and the linguistic neutrosophic number is represented by the truth, indeterminacy, and falsity linguistic variables, respectively. So LNCN contains more information than the intuitionistic fuzzy number in [20].Second, compared with the existing related methods based on the LNCNWAA and LNCNWGA operators in literature [18], all the ranking results have been shown in Table 5.Table 5
Decision results based on four aggregation operators.
Aggregation operator Aggregated result Score value Ranking LNCNWAA m 1 = l 4.0000 ,  l 6.5359 , l 1.0000 ,  l 2.4495 , l 1.1892 ,  l 2.7663 , l 5.2892  , l 1.6245 , l 2.4003, S( m 1 )=0.7277, A 4 ≻ A 2 ≻ A 3 ≻ A 1 m 2 = l 4.0199 ,  l 5.9402 , l 1.1892 ,  l 2.4003 , l 1.0000 ,  l 2.0000 , l 5.0448  l 1.5118 , l 1.1892, S( m 2 )=0.7460, m 3 = l 5.0448 ,  l 7.0000 , l 1.0000 ,  l 2.4495 , l 1.4641 ,  l 3.0000 , l 5.8980  l 2.0000 , l 2.4997, S( m 3 )=0.7435, m 4 = l 5.1455 ,  l 6.6340 , l 1.4142 ,  l 2.4495 , l 1.2311 ,  l 2.2587 , l 6.2022  l 1.3904 , l 1.8993 S( m 4 )=0.7686  LNCNWGA m 1 = l 4.0000 ,  l 6.4310 , l 1.0000 ,  l 2.5228 , l 1.2646 ,  l 2.8143 , l 5.2332  , l 1.7160 , l 2.4726, S( m 1 )=0.7205, A 4 ≻ A 3 ≻ A 2 ≻ A 1 m 2 = l 3.8798 ,  l 5.6924 , l 1.2646 ,  l 2.4726 , l 1.0000 ,  l 2.0000 , l 4.8943  l 1.7601 , l 1.2646, S( m 2 )=0.7320, m 3 = l 4.8943 ,  l 7.0000 , l 1.0000 ,  l 2.5228 , l 1.5690 ,  l 3.0000 , l 5.6408  l 2.0000 , l 2.5725, S( m 3 )=0.7343, m 4 = l 4.9946 ,  l 6.5309 , l 1.5193 ,  l 2.5228 , l 1.3163 ,  l 2.3194 , l 6.0040  l 1.6721 , l 2.0962 S( m 4 )=0.7512  LNCNGWHM(p=q=1) m 1 = l 4.0000 ,  l 6.4628 , l 1.0000 ,  l 2.4379 , l 1.1967 ,  l 2.8039 , l 5.2546 , l 1.5999 , l 2.3849, S( m 1 )=0.7263; A 4 ≻ A 2 ≻ A 3 ≻ A 1 m 2 = l 3.9271 ,  l 5.8066 , l 1.1967 ,  l 2.3849 , l 1.0000 ,  l 2.0000 , l 4.9402 , l 1.5297 , l 1.1967, S( m 2 )=0.7412; m 3 = l 4.9402 ,  l 7.0000 , l 1.0000 ,  l 2.4379 , l 1.5372 ,  l 3.0000 , l 5.7560 , l 2.0000 , l 2.5578, S( m 3 )=0.7384; m 4 = l 5.1800 ,  l 6.6362 , l 1.4814 ,  l 2.5028 , l 1.3041 ,  l 2.3243 , l 6.2161 , l 1.5499 , l 2.0274 S( m 4 )=0.7617  LNCNTPWHM(p=q=1) m 1 = l 4.0000 ,  l 6.4368 , l 1.0000 ,  l 2.4328 , l 1.1994 ,  l 2.8180 , l 5.2461  , l 1.5880 , l 2.3749, S( m 1 )=0.7260; A 4 ≻ A 2 ≻ A 3 ≻ A 1 m 2 = l 3.9051 ,  l 5.7709 , l 1.1994 ,  l 2.3749 , l 1.0000 ,  l 2.0000 , l 4.9108  l 1.5319 , l 1.1994, S( m 2 )=0.7400; m 3 = l 4.9108 ,  l 7.0000 , l 1.0000 ,  l 2.4328 , l 1.5632 ,  l 3.0000 , l 5.7142  l 2.0000 , l 2.5777, S( m 3 )=0.7368; m 4 = l 5.1989 ,  l 6.6406 , l 1.5015 ,  l 2.5170 , l 1.3314 ,  l 2.3472 , l 6.2253  l 1.6109 , l 2.0674 S( m 4 )=0.7596The results given in Table5 show that all the aggregated values of the LNCNGWHM and LNCNTPWHM operators are more or less close to moderate values between the aggregated values of the LNCNWAA and LNCNWGA operators. Then, all the ranking orders based on the LNCNWAA LNCNWGA, LNCNGWHM, and LNCNTPWHM operators are identical. However, the LNCNGWHM and LNCNTPWHM operators embody the interaction between attributes and consider the different p, q, and r values to make the decision-making results more persuasive and comprehensive than the LNCNWAA and LNCNWGA operators in literature [18].


## 5.1. The Decision-Making Process Based on LNCNGWHM Operator or LNCNTPWHM Operator

Step 1.
By using (17) (supposep = q = 1) and the weight vector λ=0.3,0.25,0.25,0.2 of attributes, we can obtain the comprehensive evaluation values mi(i=1,2,3,4) of alternative Ai as follows:(38)m1=l4.0000,l6.4628,l1.0000,l2.4379,l1.1967,l2.8039,l5.2546,l1.5999,l2.3849,m2=l3.9271,l5.8066,l1.1967,l2.3849,l1.0000,l2.0000,l4.9402,l1.5297,l1.1967,m3=l4.9402,l7.0000,l1.0000,l2.4379,l1.5372,l3.0000,l5.7560,l2.0000,l2.5578,m4=l5.1800,l6.6362,l1.4814,l2.5028,l1.3041,l2.3243,l6.2161,l1.5499,l2.0274.Step 2.
Calculate the score values ofS(mi) according to (11) for mii=1,2,3,4:(39)Sm1=0.7263;Sm2=0.7412;Sm3=0.7384;Sm4=0.7617.According to the results ofS(mi)i=1,2,3,4, we can rank the alternatives A4≻A2≻A3≻A1, and then the design scheme A4 is the best among all the alternatives.On the other hand, we can use the LNCNTPWHM operator (setp=1,q=1) to deal with this problem.Step  1 ′. By using (34) (supposep = q = 1) and the weight vector λ=0.3,0.25,0.25,0.2of attributes, we can obtain the comprehensive evaluation values mi(i=1,2,3,4) of alternative Ai as follows:(40)m1=l4.0000,l6.4368,l1.0000,l2.4328,l1.1994,l2.8180,l5.2461,l1.5880,l2.3749,m2=l3.9051,l5.7709,l1.1994,l2.3749,l1.0000,l2.0000,l4.9108l1.5319,l1.1994,m3=l4.9108,l7.0000,l1.0000,l2.4328,l1.5632,l3.0000,l5.7142l2.0000,l2.5777,m4=l5.1989,l6.6406,l1.5015,l2.5170,l1.3314,l2.3472,l6.2253l1.6109,l2.0674.Step  2 ′. Calculate the score values ofS(mi) according to (11) for mii=1,2,3,4:(41)Sm1=0.7260;Sm2=0.7400;Sm3=0.7368;Sm4=0.7596.According to the results ofS(mi)  i=1,2,3,4, we can rank the alternatives A4≻A2≻A3≻A1, so the design scheme A4 is the best among all the alternatives.


## 5.2. Analyzing the Effect of the Parameters p, q, and r

Different parametersp, q, and r may have different effects on the decision results. Therefore, this section takes different values of p, q, and r to sort the various alternatives, and then Tables 3-4 present the results.Table 3
The ranking based on the LNCNGWHM operator with different values ofp and q.
p, q LNCNGWHM Ranking P=1, q=0 S( m 1 )=0.7285, S( m 2 )=0.7479, S( m 3 )=0.7377, S( m 4 )=0.7638 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=1, q=0.5 S( m 1 )=0.7263, S( m 2 )=0.7420, S( m 3 )=0.7377, S( m 4 )=0.7609 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=1, q=2 S( m 1 )=0.7277, S( m 2 )=0.7427, S( m 3 )=0.7388, S( m 4 )=0.7645 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=0, q=1 S( m 1 )=0.7269, S( m 2 )=0.7411, S( m 3 )=0.7470, S( m 4 )=0.7688 A 4 ≻ A 3 ≻ A 2 ≻ A 1 P=0.5, q=1 S( m 1 )=0.7257, S( m 2 )=0.7404, S( m 3 )=0.7410, S( m 4 )=0.7635 A 4 ≻ A 3 ≻ A 2 ≻ A 1 P=2, q=1 S( m 1 )=0.7283, S( m 2 )=0.7424, S( m 3 )=0.7350, S( m 4 )=0.7592 A 4 ≻ A 2 ≻ A 3 ≻ A 1 P=2, q=2 S( m 1 )=0.7290, S( m 2 )=0.7428, S( m 3 )=0.7352, S( m 4 )=0.7609 A 4 ≻ A 2 ≻ A 3 ≻ A 1Table 4
The ranking based on the LNCNTPWHM operator with different values ofp, q,andr.
p, q, r LNCNTPWHM Ranking p=1, q=1, r=2 S( m 1 )=0.7269, S( m 2 )=0.7406, S( m 3 )=0.7376, S( m 4 )=0.7621 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=1, q=2, r=1 S( m 1 )=0.7166, S( m 2 )=0.7342, S( m 3 )=0.7279, S( m 4 )=0.7772 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=1, q=2, r=2 S( m 1 )=0.7282, S( m 2 )=0.7431, S( m 3 )=0.7362, S( m 4 )=0.7612 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=1, q=2, r=3 S( m 1 )=0.7299, S( m 2 )=0.7444, S( m 3 )=0.7363, S( m 4 )=0.7650 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=2, q=1, r=1 S( m 1 )=0.7278, S( m 2 )=0.7401, S( m 3 )=0.7333, S( m 4 )=0.7556 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=2, q=1, r=2 S( m 1 )=0.7282, S( m 2 )=0.7404, S( m 3 )=0.7342, S( m 4 )=0.7578 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=2, q=2, r=1 S( m 1 )=0.7290, S( m 2 )=0.7419, S( m 3 )=0.7328, S( m 4 )=0.7569 A 4 ≻ A 2 ≻ A 3 ≻ A 1 p=3, q=2, r=1 S( m 1 )=0.7302, S( m 2 )=0.7412, S( m 3 )=0.7303, S( m 4 )=0.7531 A 4 ≻ A 2 ≻ A 3 ≻ A 1From Tables3 and 4, we can see that LNCNTPWHM operator can get more stable sorting than the LNCNGWHM operator, and, in addition, when the parameters p, q, and r take different values, the best design scheme/alternative using either the LNCNGWHM or LNCNTPWHM operator is always A4. Therefore, the parameters p, q, and r in the LNCNGWHM or LNCNTPWHM operator have little influence on decision-making.


## 5.3. Comparing with the Related Methods

Firstly, compared with the literature [20], this paper used the decision information under LNCN environment, while the literature [20] used the decision information under intuitionistic fuzzy environment. In the literature [20], only incomplete information can be effectively expressed, and the indeterminate and inconsistent information cannot be described effectively, while LNCN is composed of uncertain linguistic neutrosophic number and linguistic neutrosophic number, where the uncertain linguistic neutrosophic number is represented by the truth, the indeterminacy, and the falsity uncertain linguistic variables, respectively, and the linguistic neutrosophic number is represented by the truth, indeterminacy, and falsity linguistic variables, respectively. So LNCN contains more information than the intuitionistic fuzzy number in [20].Second, compared with the existing related methods based on the LNCNWAA and LNCNWGA operators in literature [18], all the ranking results have been shown in Table 5.Table 5
Decision results based on four aggregation operators.
Aggregation operator Aggregated result Score value Ranking LNCNWAA m 1 = l 4.0000 ,  l 6.5359 , l 1.0000 ,  l 2.4495 , l 1.1892 ,  l 2.7663 , l 5.2892  , l 1.6245 , l 2.4003, S( m 1 )=0.7277, A 4 ≻ A 2 ≻ A 3 ≻ A 1 m 2 = l 4.0199 ,  l 5.9402 , l 1.1892 ,  l 2.4003 , l 1.0000 ,  l 2.0000 , l 5.0448  l 1.5118 , l 1.1892, S( m 2 )=0.7460, m 3 = l 5.0448 ,  l 7.0000 , l 1.0000 ,  l 2.4495 , l 1.4641 ,  l 3.0000 , l 5.8980  l 2.0000 , l 2.4997, S( m 3 )=0.7435, m 4 = l 5.1455 ,  l 6.6340 , l 1.4142 ,  l 2.4495 , l 1.2311 ,  l 2.2587 , l 6.2022  l 1.3904 , l 1.8993 S( m 4 )=0.7686  LNCNWGA m 1 = l 4.0000 ,  l 6.4310 , l 1.0000 ,  l 2.5228 , l 1.2646 ,  l 2.8143 , l 5.2332  , l 1.7160 , l 2.4726, S( m 1 )=0.7205, A 4 ≻ A 3 ≻ A 2 ≻ A 1 m 2 = l 3.8798 ,  l 5.6924 , l 1.2646 ,  l 2.4726 , l 1.0000 ,  l 2.0000 , l 4.8943  l 1.7601 , l 1.2646, S( m 2 )=0.7320, m 3 = l 4.8943 ,  l 7.0000 , l 1.0000 ,  l 2.5228 , l 1.5690 ,  l 3.0000 , l 5.6408  l 2.0000 , l 2.5725, S( m 3 )=0.7343, m 4 = l 4.9946 ,  l 6.5309 , l 1.5193 ,  l 2.5228 , l 1.3163 ,  l 2.3194 , l 6.0040  l 1.6721 , l 2.0962 S( m 4 )=0.7512  LNCNGWHM(p=q=1) m 1 = l 4.0000 ,  l 6.4628 , l 1.0000 ,  l 2.4379 , l 1.1967 ,  l 2.8039 , l 5.2546 , l 1.5999 , l 2.3849, S( m 1 )=0.7263; A 4 ≻ A 2 ≻ A 3 ≻ A 1 m 2 = l 3.9271 ,  l 5.8066 , l 1.1967 ,  l 2.3849 , l 1.0000 ,  l 2.0000 , l 4.9402 , l 1.5297 , l 1.1967, S( m 2 )=0.7412; m 3 = l 4.9402 ,  l 7.0000 , l 1.0000 ,  l 2.4379 , l 1.5372 ,  l 3.0000 , l 5.7560 , l 2.0000 , l 2.5578, S( m 3 )=0.7384; m 4 = l 5.1800 ,  l 6.6362 , l 1.4814 ,  l 2.5028 , l 1.3041 ,  l 2.3243 , l 6.2161 , l 1.5499 , l 2.0274 S( m 4 )=0.7617  LNCNTPWHM(p=q=1) m 1 = l 4.0000 ,  l 6.4368 , l 1.0000 ,  l 2.4328 , l 1.1994 ,  l 2.8180 , l 5.2461  , l 1.5880 , l 2.3749, S( m 1 )=0.7260; A 4 ≻ A 2 ≻ A 3 ≻ A 1 m 2 = l 3.9051 ,  l 5.7709 , l 1.1994 ,  l 2.3749 , l 1.0000 ,  l 2.0000 , l 4.9108  l 1.5319 , l 1.1994, S( m 2 )=0.7400; m 3 = l 4.9108 ,  l 7.0000 , l 1.0000 ,  l 2.4328 , l 1.5632 ,  l 3.0000 , l 5.7142  l 2.0000 , l 2.5777, S( m 3 )=0.7368; m 4 = l 5.1989 ,  l 6.6406 , l 1.5015 ,  l 2.5170 , l 1.3314 ,  l 2.3472 , l 6.2253  l 1.6109 , l 2.0674 S( m 4 )=0.7596The results given in Table5 show that all the aggregated values of the LNCNGWHM and LNCNTPWHM operators are more or less close to moderate values between the aggregated values of the LNCNWAA and LNCNWGA operators. Then, all the ranking orders based on the LNCNWAA LNCNWGA, LNCNGWHM, and LNCNTPWHM operators are identical. However, the LNCNGWHM and LNCNTPWHM operators embody the interaction between attributes and consider the different p, q, and r values to make the decision-making results more persuasive and comprehensive than the LNCNWAA and LNCNWGA operators in literature [18].


## 6. Conclusions

This paper proposed MADM methods based on the LNCNGWHM and LNCNTPWHM operators for LNCNs. First, a LNCN generalized weight Heronian mean (LNCNGWHM) operator and a LNCN three-parameter weighted Heronian mean (LNCNTPWHM) operator were proposed and the related properties of these two operators were discussed. Second, the two methods of MADM in a LNCN setting were put forward based on the LNCNGWHM operator and the LNCNTPWHM operator. Finally, these two methods are used to solve a practical problem. In order to make the decision-making result more convincing, the different values of the parametersp,q,andr were taken to observe the sorting results. From the sorting results, we found that the influence of three parameters on the decision results was very small. Furthermore, compared with the relative method, the proposed methods in this paper can get the same selection result as the existing method. Therefore, the proposed methods demonstrate potential applications in handling MADM problems under LNCN environment.



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*Source: 4158264-2018-07-26.xml*