# Multiple Attribute Group Decision-Making Models Using Single-Valued Neutrosophic and Linguistic Neutrosophic Hybrid Element Aggregation Algorithms

Authors: Sumin Zhang; Jun Ye
Journal: Journal of Mathematics
 (2022)

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## Abstract

Multiple attribute group decision-making (MAGDM) issues may involve quantitative and qualitative attributes. In inconsistent and indeterminate decision-making issues, current assessment information of quantitative and qualitative attributes with respect to alternatives only contains either numerical neutrosophic values or linguistic neutrosophic values as the single information expression. However, existing neutrosophic techniques cannot perform the mixed information denotation and aggregation operations of numerical neutrosophic values and linguistic neutrosophic values in neutrosophic decision-making issues. To solve the puzzles, this article presents the information denotation, aggregation operations, and MAGDM models of single-valued neutrosophic and linguistic neutrosophic hybrid sets/elements (SVNLNHSs/SVNLHEs) as new techniques to perform MAGDM issues with quantitative and qualitative attributes in the environment of SVNLNHEs. In this study, we first propose a SVNLNHS/SVNLNHE notion that consists of a single-valued neutrosophic element (SVNE) for the quantitative argument and a linguistic neutrosophic element (LNE) for the qualitative argument. According to a linguistic and neutrosophic conversion function and its inverse conversion function, we present some basic operations of single-valued neutrosophic elements and linguistic neutrosophic elements, the SVNLNHE weighted arithmetic mean (SVNLNHEWAMN) and SVNLNHE weighted geometric mean (SVNLNHEWGMN) operators (forming SVNEs), and the SVNLNHEWAML and SVNLNHEWGML operators (forming LNEs). Next, MAGDM models are established based on the SVNLNHEWAMN and SVNLNHEWGMN operators or the SVNLNHEWAML and SVNLNHEWGML operators to realize MAGDM issues with single-valued neutrosophic and linguistic neutrosophic hybrid information, and then their applicability and availability are indicated through an illustrative example in the SVNLNHE circumstance. By comparison with the existing techniques, our new techniques reveal obvious advantages in the mixed information denotation, aggregation algorithms, and decision-making methods in handling MAGDM issues with the quantitative and qualitative attributes in the setting of SVNLNHSs.


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## Body

## 1. Introduction

In general, there exist both quantitative attributes and qualitative attributes in multiple attribute (group) decision-making (MADM/MAGDM) issues. In the assessment process, the assessment information of quantitative attributes is usually represented by numerical values because numerical values are more suitable to the denotation form of quantitative arguments, while the assessment information of qualitative attributes is usually assigned by linguistic term values because the linguistic value is more suitable to human judgment and thinking/expression habits. Generally speaking, it is difficult to represent qualitative arguments by numeric values, but they are easily represented by linguistic values. In inconsistent and indeterminate situations, a simplified neutrosophic set (SNS) [1], including an interval-valued neutrosophic set/element (IVNS/IVNE) [2] and a single-valued neutrosophic set/element (SVNS/SVNE) [3], is depicted by the truth, falsity, and indeterminacy membership degrees, while a linguistic neutrosophic set/element (LNS/LNE) [4] is depicted by the truth, falsity, and indeterminacy linguistic values. Since the neutrosophic set theories [5], including SNS, SVNS, IVNS, and LNS, are vital mathematical tools to denote and handle indeterminate and inconsistent issues in the real world, they have been widely applied in decision-making issues [6–14]. In the setting of SNSs, some researchers presented various aggregation operators and their MADM/MAGDM models to solve neutrosophic MADM/MAGDM problems [10, 15–20]. Then, other researchers introduced various extended versions of SNSs, including single-valued neutrosophic rough sets [21], normal neutrosophic sets [22], bipolar neutrosophic sets [23], simplified neutrosophic indeterminate sets [24], and neutrosophic Z-numbers [25], and used them in MADM/MAGDM issues. In the setting of LNEs, some researchers proposed several aggregation operators of LNEs and their MAGDM models to carry out linguistic neutrosophic MAGDM problems [26, 27]. Then, some extended linguistic sets, such as linguistic neutrosophic uncertain sets and linguistic neutrosophic cubic sets, were also presented to perform some linguistic neutrosophic MAGDM problems [28]. Unfortunately, the existing neutrosophic theories and MADM models [28, 29] cannot yet resolve the denotation, operations, and MADM issues of the mixed information of SVNEs and LNEs. However, the existing assessment information of the quantitative or qualitative attributes with respect to alternatives only gives either numerical neutrosophic information or linguistic neutrosophic information as a single information expression. In the case of single-valued neutrosophic and linguistic neutrosophic mixed information, existing neutrosophic technologies cannot represent the mixed information of SVNE and LNE nor can they perform mixed operations of the two. Therefore, the mixed information representation and aggregation operations and decision-making problems pose challenges in this study, which motivates our research to address them. To solve these problems, the aims of this article are as follows: (1) to propose a single-valued neutrosophic and linguistic neutrosophic hybrid set/element (SVNLNHS/SVNLNHE) for the mixed information representation of both SVNE and LNE, (2) to present basic operations of SVNEs and LNEs according to a linguistic and neutrosophic conversion function and its inverse conversion function, (3) to propose the single-valued neutrosophic and linguistic neutrosophic hybrid element weighted arithmetic mean (SVNLNHEWAMN) and single-valued neutrosophic and linguistic neutrosophic hybrid element weighted geometric mean (SVNLNHEWGMN) operators for the aggregated SVNEs and the SVNLNHEWAML and SVNLNHEWGML operators for the aggregated LNEs, (4) to establish MAGDM models based on the SVNLNHEWAMN and SVNLNHEWGMN operators or the SVNLNHEWAML and SVNLNHEWGML operators in the setting of SVNLNHSs, and (5) to apply the established MAGDM models to an illustrative example on the selection problem of industrial robots that contain both quantitative and qualitative attributes in a SVNLNHS circumstance.Generally, the main contributions of this article are summarized as follows:(i)
The proposed SVNLNHS/SVNLNHE solves the representation problem of single-valued neutrosophic and linguistic neutrosophic mixed information.(ii)
The proposed weighted aggregation operators of SVNLNHEs based on the linguistic and neutrosophic conversion function and its inverse conversion function provide the effective aggregation algorithms of SVNLNHEs.(iii)
The established MAGDM models can solve MAGDM issues with quantitative and qualitative attributes in a SVNLNHS circumstance.(iv)
The established MAGDM models can solve the selection problem of industrial robots that contain both quantitative and qualitative attributes and show the availability and rationality of the new techniques in a SVNLNHS circumstance.The remaining structure of this article consists of the following sections. Section2 reviews the basic concepts and operations of SVNEs and LNEs as the preliminaries of this study. The notions of SVNLNHS and SVNLNHE and some basic operations of SVNEs and LNEs based on the linguistic and neutrosophic conversion function and its inverse conversion function are proposed in Section 3. In Section 4, the SVNLNHEWAMN, SVNLNHEWGMN, SVNLNHEWAML, and SVNLNHEWGML operators are presented in terms of the basic operations of SVNEs and LNEs. In Section 5, two new MAGDM models are established by the SVNLNHEWAMN and SVNLNHEWGMN operators or the SVNLNHEWAML and SVNLNHEWGML operators. Section 6 presents an illustrative example on the selection problem of industrial robots that contains both quantitative and qualitative attributes and then gives a comparative analysis with the existing techniques to show the availability and rationality of the new techniques. Finally, conclusions and future research are summarized in Section 7.


## 2. Preliminaries of SVNEs and LNEs

This part reviews the basic notions and operations of SVNEs and LNEs.


### 2.1. Basic Notions and Operations of SVNEs

SetU = {u1, u2, …, um} as a universal set. Then, a SVNS ZN in U can be represented as [1, 3](1)ZN=ui,xZNui,yZNui,zZNui|ui∈U,where < ui, xZN(ui), yZN(ui), zZN(ui)> (i = 1, 2, …, m) is SVNE in ZN for uj∈ U and xZN(ui), yZN(ui), zZN(ui)∈ [0, 1], and then it is simply denoted as zni = < xZNi, yZNi, zZNi > .For two SVNEs, zn1 = < xZN1, yZN1, zZN1>, ZN2= < xZN2, yZN2, zZN2>, and β >  0, and their relations are contained as follows [17]:(1)
zn1⊕zn2=xZN1+xZN2−xZN1xZN2,yZN1yZN2,zZN1zZN2(2)
zn1⊗zn2=xZN1xZN2,yZN1+yZN2−yZN1yZN2,zZN1+zZN2−zZN1zZN2(3)
β⋅zn1=1−1−xZN1β,yZN1β,zZN1β(4)
zn1β=xZN1β,1−1−yZN1β,1−1−zZN1βSuppose that there is a group of SVNEs zni = < xZNi, yZNi, zZNi> (i = 1, 2, …, m) with their related weights βi∈ [0, 1] for ∑i=1mβi=1. Then, the SVNE weighted arithmetic mean (SVNEWAM) operator and the SVNE weighted geometric mean (SVNEWGM) operator are introduced, respectively, as follows [17]:(2)SVNEWAMzn1,zn2,…,znm=∑i=1mβi⋅zni=1−∏i=1m1−xZNiβi,∏i=1myZNiβi,∏i=1mzZNiβi,(3)SVNEWGMzn1,zn2,…,znm=∏i=1mzniβi=∏i=1mxZNiβi,1−∏i=1m1−yZNiβi,1−∏i=1m1−zZNiβi.To compare SVNEs, the score and accuracy functions of SVNEs and their ranking laws are introduced below [17].Set zni = < xZNi, yZNi, zZNi >  as any SVNE. The score and accuracy functions of zni are presented, respectively, as follows:(4)Fzni=2+xZNi−yZNi−zZNi3forFzni∈0,1,(5)Gzni=xZNi−zZNiforGzni∈−1,1.Then, the sorting laws based on the score values ofF(zni) and the accuracy values of G(zni) (i = 1, 2) are as follows [17]:(a)
zn1 >  zn2 for F(zn1) > F(zn2)(b)
zn1 >  zn2 for F(zn1) = F(zn2) and G(zn1) > G(zn2)(c)
zn1 = zn2 for F(zn1) = F(zn2) and G(zn1) = G(zn2)


### 2.2. Basic Notions and Operations of LNEs

LetU = {u1, u2, …, um} be a universal set and S=sp|p=0,1,…,r be a linguistic term set (LTS) with an odd cardinality r + 1. Thus, a LNS LH is defined as follows [4]:(6)LH=ui,saui,sbui,scui|ui∈U,where ui,saui,sbui,scui for ui∈U is LNE in LH and saui,sbui,scui∈S are the truth, indeterminacy, and falsity linguistic variables, respectively. For convenience, LNE is simply denoted as lhi=sai,sbi,sci.For two LNEs,lh1=sa1,sb1,sc1, lh2=sa2,sb2,sc2, , and β >  0, and their operational relations are as follows [4]:(1)
lh1⊕lh2=sa1+a2−a1a2/r,sb1b2/r,sc1c2/r(2)
lh1⊗lh2=sa1a2/r,sb1+b2−b1b2/r,sc1+c2−c1c2/r(3)
β⋅lh1=sr−r1−a1/rβ,srb1/rβ,src1/rβ(4)
lh1β=sra1/rβ,sr−r1−b1/rβ,sr−r1−c1/rβSuppose that there is a group of LNEslhi=sai,sbi,sci (i = 1, 2, …, m) with their related weights βi ∈ [0, 1] for ∑i=1mβi=1. Then, the LNE weighted arithmetic mean (LNEWAM) and LNE weighted geometric mean (LNEWGM) operators are introduced as follows [4]:(7)LNEWAMlh1,lh2,…,lhm=∑i=1mβi⋅lhi=sr−r∏i=1m1−ai/rβi,sr∏i=1mbi/rβi,sr∏i=1mci/rβi,(8)LNEWGMlh1,lh2,...,lhm=∏i=1mlhiβi=sr∏i=1mai/rβi,sr−r∏i=1m1−bi/rβi,sr−r∏i=1m1−ci/rβi.Setlhi=sai,sbi,sci as any LNE. The score and accuracy functions of lhi are defined, respectively, as follows [4]:(9)Plhi=2r+ai−bi−ci3rforPlhi∈0,1,(10)Qlhi=ai−cirforQlhi∈−1,1.Then, the sorting laws based on the score values ofP(lhi) and the accuracy values of Q(lhi) (i = 1, 2) are given as follows [4]:(a)
lh1 >  lh2 for P(lh1) > P(lh2)(b)
lh1 >  lh2 for P(lh1) = P(lh2) and Q(lh1) > Q(lh2)(c)
lh1 = lh2 for P(lh1) = P(lh2) and Q(lh1) = Q(lh2)


## 2.1. Basic Notions and Operations of SVNEs

SetU = {u1, u2, …, um} as a universal set. Then, a SVNS ZN in U can be represented as [1, 3](1)ZN=ui,xZNui,yZNui,zZNui|ui∈U,where < ui, xZN(ui), yZN(ui), zZN(ui)> (i = 1, 2, …, m) is SVNE in ZN for uj∈ U and xZN(ui), yZN(ui), zZN(ui)∈ [0, 1], and then it is simply denoted as zni = < xZNi, yZNi, zZNi > .For two SVNEs, zn1 = < xZN1, yZN1, zZN1>, ZN2= < xZN2, yZN2, zZN2>, and β >  0, and their relations are contained as follows [17]:(1)
zn1⊕zn2=xZN1+xZN2−xZN1xZN2,yZN1yZN2,zZN1zZN2(2)
zn1⊗zn2=xZN1xZN2,yZN1+yZN2−yZN1yZN2,zZN1+zZN2−zZN1zZN2(3)
β⋅zn1=1−1−xZN1β,yZN1β,zZN1β(4)
zn1β=xZN1β,1−1−yZN1β,1−1−zZN1βSuppose that there is a group of SVNEs zni = < xZNi, yZNi, zZNi> (i = 1, 2, …, m) with their related weights βi∈ [0, 1] for ∑i=1mβi=1. Then, the SVNE weighted arithmetic mean (SVNEWAM) operator and the SVNE weighted geometric mean (SVNEWGM) operator are introduced, respectively, as follows [17]:(2)SVNEWAMzn1,zn2,…,znm=∑i=1mβi⋅zni=1−∏i=1m1−xZNiβi,∏i=1myZNiβi,∏i=1mzZNiβi,(3)SVNEWGMzn1,zn2,…,znm=∏i=1mzniβi=∏i=1mxZNiβi,1−∏i=1m1−yZNiβi,1−∏i=1m1−zZNiβi.To compare SVNEs, the score and accuracy functions of SVNEs and their ranking laws are introduced below [17].Set zni = < xZNi, yZNi, zZNi >  as any SVNE. The score and accuracy functions of zni are presented, respectively, as follows:(4)Fzni=2+xZNi−yZNi−zZNi3forFzni∈0,1,(5)Gzni=xZNi−zZNiforGzni∈−1,1.Then, the sorting laws based on the score values ofF(zni) and the accuracy values of G(zni) (i = 1, 2) are as follows [17]:(a)
zn1 >  zn2 for F(zn1) > F(zn2)(b)
zn1 >  zn2 for F(zn1) = F(zn2) and G(zn1) > G(zn2)(c)
zn1 = zn2 for F(zn1) = F(zn2) and G(zn1) = G(zn2)


## 2.2. Basic Notions and Operations of LNEs

LetU = {u1, u2, …, um} be a universal set and S=sp|p=0,1,…,r be a linguistic term set (LTS) with an odd cardinality r + 1. Thus, a LNS LH is defined as follows [4]:(6)LH=ui,saui,sbui,scui|ui∈U,where ui,saui,sbui,scui for ui∈U is LNE in LH and saui,sbui,scui∈S are the truth, indeterminacy, and falsity linguistic variables, respectively. For convenience, LNE is simply denoted as lhi=sai,sbi,sci.For two LNEs,lh1=sa1,sb1,sc1, lh2=sa2,sb2,sc2, , and β >  0, and their operational relations are as follows [4]:(1)
lh1⊕lh2=sa1+a2−a1a2/r,sb1b2/r,sc1c2/r(2)
lh1⊗lh2=sa1a2/r,sb1+b2−b1b2/r,sc1+c2−c1c2/r(3)
β⋅lh1=sr−r1−a1/rβ,srb1/rβ,src1/rβ(4)
lh1β=sra1/rβ,sr−r1−b1/rβ,sr−r1−c1/rβSuppose that there is a group of LNEslhi=sai,sbi,sci (i = 1, 2, …, m) with their related weights βi ∈ [0, 1] for ∑i=1mβi=1. Then, the LNE weighted arithmetic mean (LNEWAM) and LNE weighted geometric mean (LNEWGM) operators are introduced as follows [4]:(7)LNEWAMlh1,lh2,…,lhm=∑i=1mβi⋅lhi=sr−r∏i=1m1−ai/rβi,sr∏i=1mbi/rβi,sr∏i=1mci/rβi,(8)LNEWGMlh1,lh2,...,lhm=∏i=1mlhiβi=sr∏i=1mai/rβi,sr−r∏i=1m1−bi/rβi,sr−r∏i=1m1−ci/rβi.Setlhi=sai,sbi,sci as any LNE. The score and accuracy functions of lhi are defined, respectively, as follows [4]:(9)Plhi=2r+ai−bi−ci3rforPlhi∈0,1,(10)Qlhi=ai−cirforQlhi∈−1,1.Then, the sorting laws based on the score values ofP(lhi) and the accuracy values of Q(lhi) (i = 1, 2) are given as follows [4]:(a)
lh1 >  lh2 for P(lh1) > P(lh2)(b)
lh1 >  lh2 for P(lh1) = P(lh2) and Q(lh1) > Q(lh2)(c)
lh1 = lh2 for P(lh1) = P(lh2) and Q(lh1) = Q(lh2)


## 3. SVNLNHSs and SVNLNHEs

This section proposes SVNLNHS/SVNLNHE for the mixed information representation of both SVNE and LNE and then presents some basic operations of SVNEs and LNEs according to a linguistic and neutrosophic conversion function and its inverse conversion function.Definition 1.
LetU = {u1, u2, …, um} be a universe set and S = {sp|p = 0, 1, …, r} be LTS with an odd cardinality r + 1. Then, a SVNLNHS ML is defined by(11)ML=ui,TLMLui,ILMLui,FLMLui|ui∈U,where TLML(ui), ILML(ui), and FLML(ui) are the truth, indeterminacy, and falsity membership functions, and their values are either the fuzzy values for TLML(ui), ILML(ui), FLML(ui) ∈ [0, 1] or the linguistic values for TLML(ui) ILML(ui), FLML(ui) ∈ S and ui ∈ U. Moreover, the SVNLNHS ML is composed of the q SVNEs zni=xZNi,yZNi,zZNi for xZNi, yZNi, zZNi ∈ [0, 1] (i = 1, 2, …, q) and the m − q LNEs lhi=sai,sbi,sci for sai,sbi,sci∈S and ai, bi, ci ∈ [0, r] (i = q + 1, q + 2, …, m).Definition 2.
Suppose thatML1 and ML2 are two SVNLNHSs, which contain q SVNEs zn1i = < xZN1i, yZN1i, zZN1i> (i = 1, 2, …, q) and m − q LNEs lh1i=sa1i,sb1i,sc1i for sa1i,sb1i,sc1i∈S (i = q + 1, q + 2, …, m) and q SVNEs zn2i = < xZN2i, yZN2i, zZN2i> (i = 1, 2, …, q) and m − q LNEs lh2i=sa2i,sb2i,sc2i for sa2i,sb2i,sc2i∈S (i = q + 1, q + 2, …, m). Thus, ML1 and ML2 imply the following relations:(1)
ML1 ⊆ ML2 ⇔ zn1i ⊆ zn2i (i = 1, 2, …, q) and lh1i ⊆ lh2i (i = q + 1, q + 2, …, m), i.e., xZN1i ≤ xZN2i, yZN2i ≤ yZN1i, and zZN2i ≤ zZN1i for i = 1, 2, …, q and sa1i≤sa2i, sb1i≥sb2i, and sc1i≥sc2i for i = q + 1, q + 2, …, m;(2)
ML1 = ML2 ⇔ zn1i ⊆ zn2i, zn1i ⊇ zn2i, lh1 ⊆ lh2, and lh2 ⊆ lh1, i.e., xZN1i = xZN2i, yZN2i = yZN1i, and zZN2i = zZN1i for i = 1, 2, …, q and sa1i=sa2i, sb1i=sb2i, and sc1i=sc2i for i = q + 1, q + 2, …, m.Definition 3.
Set zni = < xZNi, yZNi, zZNi > and lhi=sai,sbi,sci as any SVNE and any LNE, respectively. Then, let a linguistic and neutrosophic conversion function be flhi=ai/r,bi/r,ci/r for ai, bi, ci ∈ [0, r], and then its inverse conversion function is f−1zni=xZNir,yZNir,zZNir for xZNi, yZNi, zZNi ∈ [0, 1]. Thus, some basic operations of SVNEs and LNEs are given as follows:(1)
f−1zni⊕lhi=xZNir+ai−xZNiai,yZNibi,zZNici(2)
zni⊕flhi=xZNi+ai/r−xZNiai/r,yZNibi/r,zZNici/r(3)
f−1zni⊗lhi=xZNiai,yZNir+bi−yZNibi,zZNir+ci−zZNici(4)
βf−1zni=r−r1−xZNiβ,ryZNiβ,rzZNiβ for β >  0(5)
βflhi=1−1−ai/rβ,bi/rβ,ci/rβ for β >  0(6)
f−1zniβ=rxZNiβ,r−r1−yZNiβ,r−r1−zZNiβ for β >  0(7)
fβlhi=ai/rβ,1−1−bi/rβ,1−1−ci/rβ for β >  0
It is obvious that the operational results of (2), (4), (5), and (8) are LNEs and the operational results of (3) and (7), and (9) are SVNEs.


## 4. Weighted Arithmetic and Geomatic Mean Operators of SVNLNHEs

This section proposes some weighted aggregation operators of SVNLNHEs corresponding to the linguistic and neutrosophic conversion function and its inverse conversion function, and then indicates their properties.


### 4.1. Aggregation Operators of SVNLNHEs Corresponding to the Linguistic and Neutrosophic Conversion Function

Let zni = < xZNi, yZNi, zZNi> (i = 1, 2, …, q) and lhi=sai,sbi,sci (i = q + 1, q + 2, …, m) be q SVNEs and m − q LNEs, respectively. Then, based on Definition 3 and the SVNEWAM and SVNEWGM operators of Eqs. (2) and (3) [17], the weighted arithmetic and geomatic mean operators of SVNLNHEs corresponding to the linguistic and neutrosophic conversion function are proposed by the SVNLNHEWAMN and SVNLNHEWGMN operators,(12)SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∑i=1qβi⋅zni+∑i=q+1mβiflhi=1−∏i=1q1−xZNiβi∏i=q+1m1−ai/rβi,∏i=1qyZNiβi∏i=q+1mbi/rβi,∏i=1qzZNiβi∏i=q+1mci/rβi(13)SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∏i=1qzniβi∏i=q+1mfβilhi=∏i=1qxZNiβi∏i=q+1mai/rβi,1−∏i=1q1−yZNiβi∏i=q+1m1−bi/rβi,1−∏i=1q1−zZNiβi∏i=q+1m1−ci/rβi,where βi ∈ [0, 1] is the weight of zni (i = 1, 2, …, q) and lhi (i = q + 1, q + 2, …, m) with ∑i=1mβi=1. Then, the aggregated results of the SVNLNHEWAMN and SVNLNHEWGMN operators are SVNEs.Especially, whenq = m (without LNEs), the SVNLNHEWAMN and SVNLNHEWGMN operators are reduced to the SVNEWAM and SVNEWGM operators [17], i.e., Eq. (2) and Eq. (3).Based on the properties of the SVNEWAM and SVNEWGM operators [17], it is obvious that the SVNLNHEWAMN and SVNLNHEWGMN operators also contain the following properties:(1)
Idempotency: if zni = f(lhi) = zn for i = 1, 2, …, q, q + 1, q + 2, …, m, there are SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=zn and SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=zn.(2)
Boundedness: letzn−=minixZNi,ai/r,maxiyZNi,bi/r,maxizZNi,ci/r and zn+=maxixZNi,ai/r,miniyZNi,bi/r,minizZNi,ci/r be the minimum and maximum SVNEs for i = 1, 2, …, m, and then there are the inequalities zn−≤SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤zn+ and zn−≤SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤zn+.(3)
Monotonicity: if the condition ofzni≤zni∗ (i = 1, 2, …, q) and lhi≤lhi∗ (i = q + 1, q + 2, …, m) exists, SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWAMNzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ and SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWGMNzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ also exist.


### 4.2. Aggregation Operators of SVNLNHEs According to the Inverse Conversion Function

Let zni = < xZNi, yZNi, zZNi> (i = 1, 2, …, q) and lhi=sai,sbi,sci (i = q + 1, q + 2, …, m) be q SVNEs and m − q LNEs, respectively. Then, based on Definition 3 and the LNEWAM and LNEWGM operators of Eqs. (7) and (8) [4], the weighted arithmetic and geomatic mean operators of SVNLNHEs corresponding to the inverse conversion function are proposed by the SVNLNHEWAML and SVNLNHEWGML operators:(14)SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∑i=1qβif−1zni+∑i=q+1mβi⋅lhi=r−r∏i=1q1−xZNiβi∏i=q+1m1−ai/rβi,r∏i=1qyZNiβi∏i=q+1mbi/rβi,r∏i=1qzZNiβi∏i=q+1mci/rβi,(15)SVNLNHEWGMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∏i=1qf−1zniβi∏i=q+1mlhiβi=r∏i=1qxZNiβi∏i=q+1mai/rβi,r−r∏i=1q1−yZNiβi∏i=q+1m1−bi/rβi,r−r∏i=1q1−zZNiβi∏i=q+1m1−ci/rβi,where βi∈ [0, 1] is the weight of zni (i = 1, 2, …, q) and lhi (i = q + 1, q + 2, …, m) with ∑i=1mβi=1. Then, the aggregated results of the SVNLNHEWAML and SVNLNHEWGML operators are LNEs.Especially, whenq = 0 (without SVNEs), the SVNLNHEWAML and SVNLNHEWGML operators are reduced to the LNEWAM and LNEWGM operators [4], i.e., Eq. (7) and Eq. (8).Based on the characteristics of the LNEWAM and LNEWGM operators [4], it is obvious that the SVNLNHEWAML and SVNLNHEWGML operators also contain the following characteristics:(1)
Idempotency: iff− 1(zni) = lhi = lh for i = 1, 2, …, q, q + 1, q + 2, …, m, there are SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=lh and SVNLNHEWGMLzn1,zn2,...,znq,lhq+1,lhq+2,...,lhm=lh.(2)
Boundedness: letlh−=minirxZNi,ai,maxiryZNi,bi,maxirzZNi,ci and lh+=maxirxZNi,ai,miniryZNi,bi,minirzZNi,ci be the minimum and maximum LNEs for i = 1, 2, …, m, and then there are the inequalities lh−≤SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤lh+ and lh−≤SVNLNHEWGMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤lh+.(3)
Monotonicity: if the condition ofzni≤zni∗ for i = 1, 2, …, q and lhi≤lhi∗ for i = q + 1, q + 2, …, m exists, SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWAMLzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ and SVNLNHEWGMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWGMLzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ also exist.


## 4.1. Aggregation Operators of SVNLNHEs Corresponding to the Linguistic and Neutrosophic Conversion Function

Let zni = < xZNi, yZNi, zZNi> (i = 1, 2, …, q) and lhi=sai,sbi,sci (i = q + 1, q + 2, …, m) be q SVNEs and m − q LNEs, respectively. Then, based on Definition 3 and the SVNEWAM and SVNEWGM operators of Eqs. (2) and (3) [17], the weighted arithmetic and geomatic mean operators of SVNLNHEs corresponding to the linguistic and neutrosophic conversion function are proposed by the SVNLNHEWAMN and SVNLNHEWGMN operators,(12)SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∑i=1qβi⋅zni+∑i=q+1mβiflhi=1−∏i=1q1−xZNiβi∏i=q+1m1−ai/rβi,∏i=1qyZNiβi∏i=q+1mbi/rβi,∏i=1qzZNiβi∏i=q+1mci/rβi(13)SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∏i=1qzniβi∏i=q+1mfβilhi=∏i=1qxZNiβi∏i=q+1mai/rβi,1−∏i=1q1−yZNiβi∏i=q+1m1−bi/rβi,1−∏i=1q1−zZNiβi∏i=q+1m1−ci/rβi,where βi ∈ [0, 1] is the weight of zni (i = 1, 2, …, q) and lhi (i = q + 1, q + 2, …, m) with ∑i=1mβi=1. Then, the aggregated results of the SVNLNHEWAMN and SVNLNHEWGMN operators are SVNEs.Especially, whenq = m (without LNEs), the SVNLNHEWAMN and SVNLNHEWGMN operators are reduced to the SVNEWAM and SVNEWGM operators [17], i.e., Eq. (2) and Eq. (3).Based on the properties of the SVNEWAM and SVNEWGM operators [17], it is obvious that the SVNLNHEWAMN and SVNLNHEWGMN operators also contain the following properties:(1)
Idempotency: if zni = f(lhi) = zn for i = 1, 2, …, q, q + 1, q + 2, …, m, there are SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=zn and SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=zn.(2)
Boundedness: letzn−=minixZNi,ai/r,maxiyZNi,bi/r,maxizZNi,ci/r and zn+=maxixZNi,ai/r,miniyZNi,bi/r,minizZNi,ci/r be the minimum and maximum SVNEs for i = 1, 2, …, m, and then there are the inequalities zn−≤SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤zn+ and zn−≤SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤zn+.(3)
Monotonicity: if the condition ofzni≤zni∗ (i = 1, 2, …, q) and lhi≤lhi∗ (i = q + 1, q + 2, …, m) exists, SVNLNHEWAMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWAMNzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ and SVNLNHEWGMNzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWGMNzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ also exist.


## 4.2. Aggregation Operators of SVNLNHEs According to the Inverse Conversion Function

Let zni = < xZNi, yZNi, zZNi> (i = 1, 2, …, q) and lhi=sai,sbi,sci (i = q + 1, q + 2, …, m) be q SVNEs and m − q LNEs, respectively. Then, based on Definition 3 and the LNEWAM and LNEWGM operators of Eqs. (7) and (8) [4], the weighted arithmetic and geomatic mean operators of SVNLNHEs corresponding to the inverse conversion function are proposed by the SVNLNHEWAML and SVNLNHEWGML operators:(14)SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∑i=1qβif−1zni+∑i=q+1mβi⋅lhi=r−r∏i=1q1−xZNiβi∏i=q+1m1−ai/rβi,r∏i=1qyZNiβi∏i=q+1mbi/rβi,r∏i=1qzZNiβi∏i=q+1mci/rβi,(15)SVNLNHEWGMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=∏i=1qf−1zniβi∏i=q+1mlhiβi=r∏i=1qxZNiβi∏i=q+1mai/rβi,r−r∏i=1q1−yZNiβi∏i=q+1m1−bi/rβi,r−r∏i=1q1−zZNiβi∏i=q+1m1−ci/rβi,where βi∈ [0, 1] is the weight of zni (i = 1, 2, …, q) and lhi (i = q + 1, q + 2, …, m) with ∑i=1mβi=1. Then, the aggregated results of the SVNLNHEWAML and SVNLNHEWGML operators are LNEs.Especially, whenq = 0 (without SVNEs), the SVNLNHEWAML and SVNLNHEWGML operators are reduced to the LNEWAM and LNEWGM operators [4], i.e., Eq. (7) and Eq. (8).Based on the characteristics of the LNEWAM and LNEWGM operators [4], it is obvious that the SVNLNHEWAML and SVNLNHEWGML operators also contain the following characteristics:(1)
Idempotency: iff− 1(zni) = lhi = lh for i = 1, 2, …, q, q + 1, q + 2, …, m, there are SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm=lh and SVNLNHEWGMLzn1,zn2,...,znq,lhq+1,lhq+2,...,lhm=lh.(2)
Boundedness: letlh−=minirxZNi,ai,maxiryZNi,bi,maxirzZNi,ci and lh+=maxirxZNi,ai,miniryZNi,bi,minirzZNi,ci be the minimum and maximum LNEs for i = 1, 2, …, m, and then there are the inequalities lh−≤SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤lh+ and lh−≤SVNLNHEWGMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤lh+.(3)
Monotonicity: if the condition ofzni≤zni∗ for i = 1, 2, …, q and lhi≤lhi∗ for i = q + 1, q + 2, …, m exists, SVNLNHEWAMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWAMLzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ and SVNLNHEWGMLzn1,zn2,…,znq,lhq+1,lhq+2,…,lhm≤SVNLNHEWGMLzn1∗,zn2∗,…,znq∗,lhq+1∗,lhq+2∗,…,lhm∗ also exist.


## 5. MAGDM Models in the Environment of SVNLNHSs

In this section, novel MAGDM models are developed in terms of the SVNLNHEWAMN and SVNLNHEWGMN operators and the SVNLNHEWAML and SVNLNHEWGML operators to perform MAGDM issues with quantitative and qualitative attributes in the mixed information environment of SVNEs and LNEs.Regarding a mixed information MAGDM issue in the circumstance of SVNLNHSs, there existt alternatives, denoted by a set of them E=E1,E2,…Et, and then they are satisfactorily assessed over m attributes, denoted by a set of them V = {v1, v2, …, vq, vq + 1, vq + 2, …, vm}, which contains q quantitative attributes and m − q qualitative attributes. Then, there is a group of decision makers G=g1,g2,…ge with their weight vector α=α1,α2,…αe for αk∈ [0, 1] and ∑k=1eαk=1. The assessment values of each alternative over the q quantitative attributes are given by the decision makers gk (k = 1, 2, …, e) and represented by SVNEs znjik=xZNjik,yZNjik,zZNjik for xZNjik,yZNjik,zZNjik∈0,1 (k = 1, 2, …, e; j = 1, 2, …, t; i = 1, 2, …, q), and then the assessment values of each alternative over the m − q qualitative attributes are represented by LNEs lhjik=sajik,sbjik,scjik for sajik,sbjik,scjik∈S (k = 1, 2, …, e; j = 1, 2, …, t; i = q + 1, q + 2, …, m) from the LTS S = {sp|p = 0, 1, 2, …, r}. Thus, all assessed values can be constructed as the e decision matrices of SVNLNHEs Mk = (znjik,lhjik)t× m (k = 1, 2, …, e). Then, a weight vector β=β1,β2,…βm is specified to consider the weights βi of attributes vi (i = 1, 2, …, m) with βi∈ [0, 1] and ∑i=1mβi=1.Thus, two MAGDM models are developed in terms of the SVNLNHEWAMN and SVNLNHEWGMN operators or the SVNLNHEWAML and SVNLNHEWGML operators to perform MAGDM issues with the mixed evaluation information of SVNEs and LNEs.Model 1. A MAGDM model using the SVNLNHEWAMN and SVNLNHEWGMN operators is developed to perform the MAGDM issue with SVNLNHEs. Its detailed steps are presented as follows:Step 1: using the SVNEWAM operator of Eq. (2) and the LNEWAM operator of Eq. (7), the decision matrices Mk=znjik,lhjikt×m (k = 1, 2, …, e) are aggregated into the overall decision matrix M=znji,lhjit×m.Step 2: using the SVNLNHEWAMN operator of Eq. (12) or the SVNLNHEWGMN operator of Eq. (13), the aggregated result for Ej (j = 1, 2, ⋯, t) is obtained by the following equation:(16)znj=SVNLNHEWAMNznj1,znj2,…,znjq,lhjq+1,lhjq+2,…,lhjm=∑i=1qβi⋅znji+∑i=q+1mβiflhji=1−∏i=1q1−xZNjiβi∏i=q+1m1−aji/rβi,∏i=1qyZNjiβi∏i=q+1mbji/rβi,∏i=1qzZNjiβi∏i=q+1mcji/rβi,or(17)znj=SVNLNHEWGMNznj1,znj2,…,znjq,lhjq+1,lhjq+2,…,lhjm=∏i=1qznjiβi∏i=q+1mfβilhji=∏i=1qxZNjiβi∏i=q+1maji/rβi,1−∏i=1q1−yZNjiβi∏i=q+1m1−bji/rβi,1−∏i=1q1−zZNjiβi∏i=q+1m1−cji/rβi.Step 3: the score values ofF(znj) (j = 1, 2, …, t) are given by Eq. (4) and the accuracy values of G(znj) (j = 1, 2, …, t) are given by Eq. (5) if necessary.Step 4: the alternatives are sorted in descending order based on the sorting laws of SVNEs, and the first one is the best choice.Step 5: end.Model 2. A MAGDM model using the SVNLNHEWAML and SVNLNHEWGML operators is developed to perform the MAGDM issue with SVNLNHEs. Its detailed steps are presented as follows:Step 1’: the same as Step 1.Step 2’: using the SVNLNHEWAML operator of Eq. (14) or the SVNLNHEWGML operator of Eq. (15), the aggregated result for Ejj=1,2,…,t is given by the following equation:(18)lhj=SVNLNHEWAMLznj1,znj2,…,znjq,lhjq+1,lhjq+2,…,lhjm=∑i=1qβif−1znji+∑i=q+1mβi⋅lhji=r−r∏i=1q1−xZNjiβi∏i=q+1m1−aji/rβi,r∏i=1qyZNjiβi∏i=q+1mbji/rβi,r∏i=1qzZNjiβi∏i=q+1mcji/rβi,or(19)lhj=SVNLNHEWGMLznj1,znj2,…,znjq,lhjq+1,lhjq+2,…,lhjm=∏i=1qf−1znjiβi∏i=q+1mlhjiβi=r∏i=1qxZNjiβi∏i=q+1maji/rβi,r−r∏i=1q1−yZNjiβi∏i=q+1m1−bji/rβi,r−r∏i=1q1−zZNjiβi∏i=q+1m1−cji/rβi.Step 3’: the score values ofP(lhj) (j = 1, 2, …, t) are given by Eq. (9) and the accuracy values of Q(lhj) (j = 1, 2, …, t) are given by Eq. (10) if necessary.Step 4’: the alternatives are sorted in descending order based on the sorting laws of LNEs, and then the first one is the best choice.Step 5’: end.


## 6. Illustrative Example on the Selection Problem of Industrial Robots Containing Both Quantitative and Qualitative Attributes

This section applies the proposed MAGDM models to an illustrative example on the selection problem of industrial robots that contains both quantitative and qualitative attributes in the circumstance of SVNLNHSs to prove their usefulness, and then gives a comparison with existing techniques to show the availability and rationality of the new techniques.


### 6.1. Illustrative Example

This subsection applies the proposed MAGDM models to the selection problem of industrial robots containing both quantitative and qualitative attributes to illustrate their application and availability in the circumstance of SVNLNHSs.Some industrial company wants to buy a type of industrial robots for a manufacturing system. The technical department preliminarily provides four types of industrial robots/alternatives, denoted as their setE = {E1, E2, E3, E4}. Then, they must satisfy four requirements/attributes: operating accuracy (v1), carrying capacity (v2), control performance (v3), and operating space and dexterity (v4). The weight vector of the four attributes is given by β = (0.25, 0.3, 0.25, 0.2). Thus, three experts/decision makers are invited to satisfactorily assess each alternative over the four attributes by their truth, falsity, and indeterminacy options/judgments, where the assessment values can be specified in the mixed forms of both the SVNEs znjik=xZNjik,yZNjik,zZNjik for xZNjik,yZNjik,zZNjik∈0,1 (k = 1, 2, 3; i = 1, 2; j = 1, 2, 3, 4) regarding the quantitative attributes v1 and v2 and the LNEs lhjik=sajik,sbjik,scjik for sajik,sbjik,scjik∈S (k = 1, 2, 3; i = 3, 4; j = 1, 2, 3, 4) regarding the qualitative attributes v3 and v4 from the LTS S = {very unsatisfactory, unsatisfactory, slight unsatisfactory, medium, slight satisfactory, satisfactory, very satisfactory} = {s0, s1, s2, s3, s4, s5, s6} with r = 6. The weight vector of the three decision makers is given by α = (0.4, 0.35, 0.25). Thus, the three decision matrices are constructed as follows:(20)M1=0.8,0.1,0.20.7,0.1,0.1s5,s2,s2s5,s2,s30.8,0.2,0.10.8,0.1,0.3s5,s1,s2s4,s3,s20.7,0.1,0.10.8,0.2,0.2s5,s3,s2s4,s2,s30.8,0.2,0.20.9,0.2,0.3s4,s1,s2s5,s2,s2,M2=0.7,0.2,0.20.8,0.1,0.2s4,s3,s1s5,s2,s10.8,0.2,0.30.8,0.2,0.3s5,s2,s1s4,s1,s20.8,0.2,0.30.7,0.1,0.1s4,s1,s2s5,s2,s30.9,0.1,0.10.8,0.2,0.1s5,s1,s3s5,s1,s2,M3=0.8,0.3,0.10.8,0.1,0.1s5,s2,s1s4,s1,s10.7,0.1,0.20.9,0.2,0.3s4,s1,s1s5,s1,s10.8,0.1,0.10.8,0.2,0.1s5,s2,s2s5,s2,s30.8,0.1,0.10.8,0.1,0.1s5,s1,s2s5,s3,s2.Thus, the two MAGDM models developed can be utilized in the example to perform the MAGDM issue with SVNLNHEs.Model 1. The MAGDM model using the SVNLNHEWAMN and SVNLNHEWGMN operators can be applied in the example, and then its detailed steps are depicted as follows:Step 1: using the SVNEWAM operator of Eq. (2) and the LNEWAM operator of Eq. (7), the above three decision matrices are aggregated into the following overall decision matrix:(21)M=0.7695,0.1677,0.16820.7648,0.1000,0.1275s4.7254,s2.3050,s1.3195s4.8108,s1.6818,s1.55180.7787,0.1682,0.17470.8318,0.1516,0.3000s4.8108,s1.2746,s1.3195s4.3182,s1.5518,s1.68180.7648,0.1275,0.14690.7695,0.1569,0.1320s4.7254,s1.8455,s2.0000s4.6805,s2.0000,s3.00000.8431,0.1320,0.13200.8484,0.1682,0.1552s4.6805,s1.0000,s2.3050s5.0000,s1.7366,s2.0000.Step 2: by Eq. (16) or Eq. (17), we give the aggregated values:zn1 = < 0.7795, 0.1958, 0.1804>, zn2 = < 0.7921, 0.1884, 0.2392>,zn3 = < 0.7751, 0.2049, 0.2231>, and zn4 = < 0.8290, 0.1761, 0.2178 > .Or zn1 = < 0.7789, 0.2324, 0.1885>, zn2 = < 0.7876, 0.1934, 0.2464>,zn3 = < 0.7749, 0.2276, 0.2754>, and zn4 = < 0.8265, 0.1850, 0.2504 > .Step 3: by Eq. (4), the score values of F(znj) for Ej (j = 1, 2, 3, 4) are given below:F(zn1) = 0.8011, F(zn2) = 0.7882, F(zn3) = 0.7824, and F(zn4) = 0.8117.OrF(zn1) = 0.7860, F(zn2) = 0.7826, F(zn3) = 0.7573, and F(zn4) = 0.7970.Step 4: the sorting order of the four alternatives isE4 >  E1 >  E2 >  E3.Clearly, the sorting orders obtained by the SVNLNHEWAMN operator of Eq. (16) and the SVNLNHEWGMN operator of Eq. (17) are identical in this example.Model 2. The MAGDM model using the SVNLNHEWAML and SVNLNHEWGML operators can also be applied in the example, and then its detailed steps are depicted as follows:Step 1’: the same as Step 1.Step 2’: by Eq. (18) or Eq. (19), we give the aggregated values:lh1 = < s4.6773, s1.1747, s1.0823>, lh2 = <  s4.7528, s1.1302, s1.4353>,lh3 = < s4.6508, s1.2297, s1.3384>, and lh4 = < s4.9739, s1.0564, s1.3070 > ,orlh1 = < s4.6736, s1.3944, s1.1311>, lh2 = < s4.7255, s1.1603, s1.4783>,lh3 = < s4.6494, s1.3657, s1.6526>, and lh4 = < s4.9591, s1.1100, s1.5027 > .Step 3’: by (9), the score values of P(lhj) for Ej (j = 1, 2, 3, 4) are given as follows:P(lh1) = 0.8011, P(lh2) = 0.7882, P(lh3) = 0.7824, and P(lh4) = 0.8117,orP(lh1) = 0.7860, P(lh2) = 0.7826, P(lh3) = 0.7573, and P(lh4) = 0.7970.Step 4’: the sorting order of the four alternatives isE4 >  E1 >  E2 >  E3.Hence, the sorting orders obtained by the SVNLNHEWAML operator of Eq. (18) and the SVNLNHEWGML operator of Eq. (19) are identical in this example.Obviously, the score values and sorting orders between Model 1 and Model 2 reflect the same results. Moreover, we see that whether SVNEs are converted to LNEs or LNEs to SVNEs in the aggregation operations, their final decision results are actually identical. Thus, decision makers can choose Model 1 or Model 2 in MAGDM applications. Therefore, it is obvious that our new techniques are valid and reasonable.


### 6.2. Comparative Analysis with the Existing Neutrosophic MAGDM Models

Since the assessed values of SVNLNHEs are given in this illustrative example, the existing neutrosophic MAGDM models [4, 17] cannot deal with this illustrative example in the situation of SVNLNHEs. Then, our new techniques can handle neutrosophic MAGDM issues with SVNEs and/or LNEs and show the following highlights and advantages:(1)
The proposed SVNLNHEs can conveniently denote the mixed information of SVNEs and LNEs regarding the assessment objects of quantitative and qualitative attributes, which is suitable for human judgment and thinking/expression habits, while existing neutrosophic expressions cannot represent SVNLNHE information.(2)
The proposed SVNLNHEWAMN and SVNLNHEWGMN operators or the proposed SVNLNHEWAML and SVNLNHEWGML operators provide the necessary aggregation tools for handling MAGDM issues in the SVNLNHE circumstance, while the existing SVNEWAM and SVNEWGM operators [17] are only the special cases of the SVNLNHEWAMN and SVNLNHEWGMN operators, and then the existing LNEWAM and LNEWGM operators [4] are only the special cases of the SVNLNHEWAML and SVNLNHEWGML operators. Furthermore, the various existing aggregation operators cannot aggregate SVNLNHEs.(3)
Since the existing MAGDM models with the single evaluation information of SVNEs or LNEs [4, 17] are the special cases of our new MAGDM models, our new MAGDM models are broader and more versatile than the existing MAGDM models [4, 17]. Furthermore, the various existing MAGDM models cannot carry out MAGDM problems with SVNLNHE information.Generally, the new techniques solve the SVNLNHE denotation, aggregation operations, and MAGDM issues in the mixed information situation of SVNEs and LNEs. It is clear that our new techniques are very suitable for such decision-making issues with quantitative and qualitative attributes and overcome the defects of the existing decision-making techniques subject to the single evaluation information of SVNEs or LNEs. Therefore, our new techniques reveal obvious superiorities over the existing techniques in the neutrosophic information denotation, aggregation operations, and decision-making methods.


## 6.1. Illustrative Example

This subsection applies the proposed MAGDM models to the selection problem of industrial robots containing both quantitative and qualitative attributes to illustrate their application and availability in the circumstance of SVNLNHSs.Some industrial company wants to buy a type of industrial robots for a manufacturing system. The technical department preliminarily provides four types of industrial robots/alternatives, denoted as their setE = {E1, E2, E3, E4}. Then, they must satisfy four requirements/attributes: operating accuracy (v1), carrying capacity (v2), control performance (v3), and operating space and dexterity (v4). The weight vector of the four attributes is given by β = (0.25, 0.3, 0.25, 0.2). Thus, three experts/decision makers are invited to satisfactorily assess each alternative over the four attributes by their truth, falsity, and indeterminacy options/judgments, where the assessment values can be specified in the mixed forms of both the SVNEs znjik=xZNjik,yZNjik,zZNjik for xZNjik,yZNjik,zZNjik∈0,1 (k = 1, 2, 3; i = 1, 2; j = 1, 2, 3, 4) regarding the quantitative attributes v1 and v2 and the LNEs lhjik=sajik,sbjik,scjik for sajik,sbjik,scjik∈S (k = 1, 2, 3; i = 3, 4; j = 1, 2, 3, 4) regarding the qualitative attributes v3 and v4 from the LTS S = {very unsatisfactory, unsatisfactory, slight unsatisfactory, medium, slight satisfactory, satisfactory, very satisfactory} = {s0, s1, s2, s3, s4, s5, s6} with r = 6. The weight vector of the three decision makers is given by α = (0.4, 0.35, 0.25). Thus, the three decision matrices are constructed as follows:(20)M1=0.8,0.1,0.20.7,0.1,0.1s5,s2,s2s5,s2,s30.8,0.2,0.10.8,0.1,0.3s5,s1,s2s4,s3,s20.7,0.1,0.10.8,0.2,0.2s5,s3,s2s4,s2,s30.8,0.2,0.20.9,0.2,0.3s4,s1,s2s5,s2,s2,M2=0.7,0.2,0.20.8,0.1,0.2s4,s3,s1s5,s2,s10.8,0.2,0.30.8,0.2,0.3s5,s2,s1s4,s1,s20.8,0.2,0.30.7,0.1,0.1s4,s1,s2s5,s2,s30.9,0.1,0.10.8,0.2,0.1s5,s1,s3s5,s1,s2,M3=0.8,0.3,0.10.8,0.1,0.1s5,s2,s1s4,s1,s10.7,0.1,0.20.9,0.2,0.3s4,s1,s1s5,s1,s10.8,0.1,0.10.8,0.2,0.1s5,s2,s2s5,s2,s30.8,0.1,0.10.8,0.1,0.1s5,s1,s2s5,s3,s2.Thus, the two MAGDM models developed can be utilized in the example to perform the MAGDM issue with SVNLNHEs.Model 1. The MAGDM model using the SVNLNHEWAMN and SVNLNHEWGMN operators can be applied in the example, and then its detailed steps are depicted as follows:Step 1: using the SVNEWAM operator of Eq. (2) and the LNEWAM operator of Eq. (7), the above three decision matrices are aggregated into the following overall decision matrix:(21)M=0.7695,0.1677,0.16820.7648,0.1000,0.1275s4.7254,s2.3050,s1.3195s4.8108,s1.6818,s1.55180.7787,0.1682,0.17470.8318,0.1516,0.3000s4.8108,s1.2746,s1.3195s4.3182,s1.5518,s1.68180.7648,0.1275,0.14690.7695,0.1569,0.1320s4.7254,s1.8455,s2.0000s4.6805,s2.0000,s3.00000.8431,0.1320,0.13200.8484,0.1682,0.1552s4.6805,s1.0000,s2.3050s5.0000,s1.7366,s2.0000.Step 2: by Eq. (16) or Eq. (17), we give the aggregated values:zn1 = < 0.7795, 0.1958, 0.1804>, zn2 = < 0.7921, 0.1884, 0.2392>,zn3 = < 0.7751, 0.2049, 0.2231>, and zn4 = < 0.8290, 0.1761, 0.2178 > .Or zn1 = < 0.7789, 0.2324, 0.1885>, zn2 = < 0.7876, 0.1934, 0.2464>,zn3 = < 0.7749, 0.2276, 0.2754>, and zn4 = < 0.8265, 0.1850, 0.2504 > .Step 3: by Eq. (4), the score values of F(znj) for Ej (j = 1, 2, 3, 4) are given below:F(zn1) = 0.8011, F(zn2) = 0.7882, F(zn3) = 0.7824, and F(zn4) = 0.8117.OrF(zn1) = 0.7860, F(zn2) = 0.7826, F(zn3) = 0.7573, and F(zn4) = 0.7970.Step 4: the sorting order of the four alternatives isE4 >  E1 >  E2 >  E3.Clearly, the sorting orders obtained by the SVNLNHEWAMN operator of Eq. (16) and the SVNLNHEWGMN operator of Eq. (17) are identical in this example.Model 2. The MAGDM model using the SVNLNHEWAML and SVNLNHEWGML operators can also be applied in the example, and then its detailed steps are depicted as follows:Step 1’: the same as Step 1.Step 2’: by Eq. (18) or Eq. (19), we give the aggregated values:lh1 = < s4.6773, s1.1747, s1.0823>, lh2 = <  s4.7528, s1.1302, s1.4353>,lh3 = < s4.6508, s1.2297, s1.3384>, and lh4 = < s4.9739, s1.0564, s1.3070 > ,orlh1 = < s4.6736, s1.3944, s1.1311>, lh2 = < s4.7255, s1.1603, s1.4783>,lh3 = < s4.6494, s1.3657, s1.6526>, and lh4 = < s4.9591, s1.1100, s1.5027 > .Step 3’: by (9), the score values of P(lhj) for Ej (j = 1, 2, 3, 4) are given as follows:P(lh1) = 0.8011, P(lh2) = 0.7882, P(lh3) = 0.7824, and P(lh4) = 0.8117,orP(lh1) = 0.7860, P(lh2) = 0.7826, P(lh3) = 0.7573, and P(lh4) = 0.7970.Step 4’: the sorting order of the four alternatives isE4 >  E1 >  E2 >  E3.Hence, the sorting orders obtained by the SVNLNHEWAML operator of Eq. (18) and the SVNLNHEWGML operator of Eq. (19) are identical in this example.Obviously, the score values and sorting orders between Model 1 and Model 2 reflect the same results. Moreover, we see that whether SVNEs are converted to LNEs or LNEs to SVNEs in the aggregation operations, their final decision results are actually identical. Thus, decision makers can choose Model 1 or Model 2 in MAGDM applications. Therefore, it is obvious that our new techniques are valid and reasonable.


## 6.2. Comparative Analysis with the Existing Neutrosophic MAGDM Models

Since the assessed values of SVNLNHEs are given in this illustrative example, the existing neutrosophic MAGDM models [4, 17] cannot deal with this illustrative example in the situation of SVNLNHEs. Then, our new techniques can handle neutrosophic MAGDM issues with SVNEs and/or LNEs and show the following highlights and advantages:(1)
The proposed SVNLNHEs can conveniently denote the mixed information of SVNEs and LNEs regarding the assessment objects of quantitative and qualitative attributes, which is suitable for human judgment and thinking/expression habits, while existing neutrosophic expressions cannot represent SVNLNHE information.(2)
The proposed SVNLNHEWAMN and SVNLNHEWGMN operators or the proposed SVNLNHEWAML and SVNLNHEWGML operators provide the necessary aggregation tools for handling MAGDM issues in the SVNLNHE circumstance, while the existing SVNEWAM and SVNEWGM operators [17] are only the special cases of the SVNLNHEWAMN and SVNLNHEWGMN operators, and then the existing LNEWAM and LNEWGM operators [4] are only the special cases of the SVNLNHEWAML and SVNLNHEWGML operators. Furthermore, the various existing aggregation operators cannot aggregate SVNLNHEs.(3)
Since the existing MAGDM models with the single evaluation information of SVNEs or LNEs [4, 17] are the special cases of our new MAGDM models, our new MAGDM models are broader and more versatile than the existing MAGDM models [4, 17]. Furthermore, the various existing MAGDM models cannot carry out MAGDM problems with SVNLNHE information.Generally, the new techniques solve the SVNLNHE denotation, aggregation operations, and MAGDM issues in the mixed information situation of SVNEs and LNEs. It is clear that our new techniques are very suitable for such decision-making issues with quantitative and qualitative attributes and overcome the defects of the existing decision-making techniques subject to the single evaluation information of SVNEs or LNEs. Therefore, our new techniques reveal obvious superiorities over the existing techniques in the neutrosophic information denotation, aggregation operations, and decision-making methods.


## 7. Conclusion

Due to the lack of the SVNLNHE denotation, operations, and decision-making models in existing neutrosophic theory and applications, the proposed notion of SVNLNHS/SVNLNHE and the defined linguistic and neutrosophic conversion function solved the hybrid neutrosophic information denotation and operational problems of SVNEs and LNEs. Then, the proposed SVNLNHEWAMN, SVNLNHEWGMN, SVNLNHEWAML, and SVNLNHEWGML operators provided necessary aggregation algorithms for handling MAGDM issues with SVNLNHEs. The established MAGDM models solved such decision-making issues with quantitative and qualitative attributes in the SVNLNHE circumstance. Since the evaluation values of quantitative and qualitative attributes in the decision-making process are easily represented in SVNEs and LNEs that are given in view of decision makers’ preferences/thinking habits, the managerial implications of this original research will be reinforced in neutrosophic decision-making methods and applications. Finally, an illustrative example was given and compared with the existing techniques to show the availability and rationality of the new techniques. Moreover, our new techniques not only overcome the insufficiencies of the existing techniques but also are broader and more versatile than the existing techniques when dealing with MAGDM issues in the setting of SVNLNHEs. However, in this study, the new techniques of the SVNLNHE denotation, aggregation algorithms, and MAGDM models reflected their superiority over existing techniques.Regarding future research, these new techniques will be further extended to other areas, such as medical diagnosis, slope risk/instability evaluation, default diagnosis, and mechanical concept design, in the mixed information situation of SVNEs and LNEs. Then, we shall also develop more aggregation algorithms, such as Hamacher, Dombi, and Bonferroni aggregation operators, and their applications in clustering analysis, information fusion, image processing, and mine risk/safety evaluation in the mixed information situation of both SVNE and LNE or both IVNE and uncertain LNE.



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*Source: 1021280-2022-09-20.xml*