# Multiattribute Group Decision-Making Method in terms of Linguistic Neutrosophic Z-Number Weighted Aggregation Operators

Authors: Jun Ye; Rui Yong; Shigui Du
Journal: Journal of Mathematics
 (2022)

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## Abstract

To make a fuzzy value more reliable, Zadeh presented the notion of Z-number, which reflects a fuzzy value related to its reliability measure. Since linguistic expression conforms to human thinking habits, linguistic neutrosophic decision-making is one of the key research topics in linguistic indeterminate and inconsistent setting. In order to ensure the reliability of multiattribute group decision-making (MAGDM) problems in the linguistic environment of truth, falsehood, and indeterminacy, we require a new linguistic neutrosophic framework that combines the decision-maker’s linguistic neutrosophic judgment with its reliability measure. Inspired by the linguistic Z-numbers of the truth, falsehood, and indeterminacy, this article first proposes a linguistic neutrosophic Z-number (LNZN) to make the truth, falsehood, and indeterminacy linguistic values more reliable. Then, we define the operational relations, score and accuracy functions, and sorting laws of LNZNs. Next, we establish the LNZN weighted arithmetic mean (LNZNWAM) and LNZN weighted geometric mean (LNZNWGM) operators and indicate their properties. Furthermore, an MAGDM approach is developed based on the two aggregation operators and the score and accuracy functions of LNZNs in the LNZN setting. Lastly, an MAGDM example of industrial robot selection and comparison with existing related methods are provided to verify the applicability and efficiency of the developed MAGDM method in the setting of LNZNs. In general, the developed MAGDM approach not only makes the MAGDM information more reliable but also solves MAGDM problems under the environment of LNZNs.


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## Body

## 1. Introduction

Decision-making is a hotspot of current research problems. It is very significant to establish reasonable information representation and decision-making models. Linguistic representation may be more suitable for human thinking habits, especially reflecting its advantages in qualitative assessment of complex objective things. In this case, linguistic decision-making indicates its importance. Thus, linguistic multiattribute (group) decision-making (MADM/MAGDM) research has attracted the attention of many researchers in the past few decades. Since Zadeh [1] first introduced the concept of linguistic variables, various linguistic MADM/MAGDM methods have been utilized to solve various decision-making problems [2–5]. In terms of a membership/truth linguistic variable and a nonmembership/falsity linguistic variable, Chen et al. [6] proposed linguistic intuitionistic fuzzy numbers (LIFNs) and used them for MAGDM problems. Then, Yager [7] presented the ordinal LIFN aggregation operators, and Zhang et al. [8] used LIFNs to indicate the preferred and nonpreferred qualitative judgments of decision-makers in linguistic MADM problems. Next, some LIFN aggregation operators and their decision-making approaches [9–11] have been proposed and applied in MADM issues with LIFN information. Regarding the truth, falsehood, and indeterminacy linguistic variables, Fang and Ye [12] defined linguistic neutrosophic numbers (LNNs) and their operations; then, they presented the LNN weighted arithmetic and geometric mean operators and their MAGDM approach to solve MAGDM issues with LNN information. After that, various aggregation operators of LNNs and their MAGDM methods [12–16] have been applied in MAGDM issues with LNNs.However, LIFN is a special case of LNN. LNN can describe indeterminacy and inconsistent linguistic information as its highlighting advantage, while LIFN cannot do it. More recently, according to the conceptual generalization of Z-numbers [18], Ding et al. [19] presented the linguistic Z-number QUALIFLEX (Qualitative Flexible Multiple Criteria) method for MAGDM. Next, Du et al. and Ye proposed neutrosophic Z-numbers (NZNs), their weighted arithmetic and geometric mean operators [20], and their similarity measures [21] and then applied them to MADM problems in the environment of NZNs. Yong et al. [22] presented trapezoidal neutrosophic Z-numbers and their weighted arithmetic and geometric mean operators for MADM issues with trapezoidal NZNs. Although NZN and trapezoidal NZN contain the information of the truth, falsehood, and indeterminacy Z-numbers, they cannot represent LNN information. Furthermore, existing LNN lacks the reliability measure of the truth, falsehood, and indeterminacy linguistic values, which shows its flaw. To make up for this flaw, we should introduce the reliability measure to the truth, falsehood, and indeterminacy linguistic values in LNN, propose a linguistic neutrosophic Z-number (LNZN) so as to strengthen the reliability of LNN, and then present some operations and sorting laws of LNZNs to solve MAGDM issues with the information of LNZNs. Therefore, this study aims (a) to propose LNZNs and their operational relations, (b) to define the score and accuracy functions and sorting laws of LNZNs, (c) to establish LNZN weighted arithmetic mean (LNZNWAM) and LNZN weighted geometric mean (LNZNWGM) operators, (d) to develop an MAGDM approach by using the LNZNWAM and LNZNWGM operators and score and accuracy functions of LNZNs, and (e) to apply the developed MAGDM approach to an MAGDM problem of industrial robot selection in the environment of LNZNs.Generally, the critical contributions of this original study are summarized as follows:(a)
The new notion of LNZN proposed in terms of linguistic Z-numbers of the truth, falsehood, and indeterminacy can make the linguistic values more reliable(b)
The defined operational relations, score and accuracy functions, and sorting laws of LNZNs and the proposed LNZNWAM and LNZNWGM operators provide the necessary mathematical tools for modeling of MAGDM issues in the setting of LNZNs(c)
The proposed MAGDM approach can solve MAGDM issues with LNZNs(d)
The proposed MAGDM method can efficiently handle the MAGDM problem of industrial robot selection in the LNZN setting and show its usabilityThe rest of this study is composed of the following structures: in Section2, some basic concepts of LNNs are reviewed as preliminaries of this study. Section 3 proposes LNZNs and their operational relations, score and accuracy functions, and sorting laws. Section 4 develops the LNZNWAM and LNZNWGM operators and indicates their properties. In Section 5, an MAGDM approach is developed using the LNZNWAM and LNZNWGM operators and score and accuracy functions to carry out MAGDM issues in the LNZN environment. Section 6 applies the developed MAGDM approach to an MAGDM example of industrial robot selection for a manufacturing company under the environment of LNZNs and then presents a comparison with existing related MAGDM methods to reflect the efficiency of the developed approach. Lastly, conclusions and future research are indicated in Section 7.


## 2. Preliminaries of LNNs

SetU = {δ0, δ1, …, δv} as a linguistic term set (LTS) with odd cardinality v+1. Fang and Ye [12] first defined the LNN le = <δT, δI, δF> on U such that δT, δI, δF ∈ U and T, I, F ∈ 0,v, where δT, δI, and δF express the truth, indeterminacy, and falsehood linguistic variables, respectively.Regarding two LNNsle1=δT1,δI1,δF1 and le2=δT2,δI2,δF2 on U and any real number p > 0, the following operational relations [12] are defined as follows:(i)
le1⊕le2=δT1,δI1,δF1⊕δT2,δI2,δF2=δT1+T2−T1T2/v,δI1I2/v,δF1F2/v(ii)
le1⊗le2=δT1,δI1,δF1⊗δT2,δI2,δF2=δT1T2/v,δI1+I2−I1I2/v,δF1+F2−F1F2/v(iii)
p⋅le1=p⋅δT1,δI1,δF1=δv−v1−T1/vp,δvI1/vp,δvF1/vp(iv)
le1p=δT1,δI1,δF1p=δvT1/vp,δv−v1−I1/vp,δv−v1−F1/vpRegarding a series of LNNslek= <δT(k), δI(k), δF(k)> with their weights pk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1, the LNN weighted arithmetic mean (LNNWAM) and LNN weighted geometric mean (LNNWGM) operators [12] are proposed as follows:(1)LNNWAMle1,le2,…,len=∑k=1npk⋅lek=δv−v∏k=1n1−Tk/vpk,δv∏k=1nIk/vpk,δv∏k=1nFk/vpk,(2)LNNWGMle1,le2,…,len=∏k=1nlekpk=δv∏k=1nTk/vpk,δv−v∏k=1n1−Ik/vpk,δv−v∏k=1n1−Fk/vpk.Then, Fang and Ye [12] defined the score and accuracy functions of lek = <δT(k), δI(k), δF(k)>:(3)Dlek=2v+Tk−Ik−Fk3lek,forDlek∈0,1,(4)Elek=Tk−Fklek,forElek∈−1,1.Regarding two LNNs,le1=δT1,δI1,δF1 and le2=δT2,δI2,δF2, their sorting laws [12] are defined as follows:(i)
le1≻le2 if D(le1) > D(le2)(ii)
le1≻le2 if D(le1) = D(le2) and E(le1) > E(le2)(iii)
le1 = le2 if D(le1) = D(le2) and E(le1) = E(le2)


## 3. LNZNs

To make LNN more reliable, this section proposes LNZNs in terms of the truth, falsehood, and indeterminacy Z-numbers and then defines the operational relations, score and accuracy functions, and sorting laws of LNZNs.Definition 1.
SetU = {δ0, δ1, …, δv} as LTS with odd cardinality v+1. Then, LNZN on U is defined as lz = <(δRT, δMT), (δRI, δMI), (δRF, δMF)> for δRT, δRI, δRF, δMT, δMI, δMF ∈ U and RT, RI, RF, MT, MI, MF ∈ 0,v, where (δRT, δMT) is the truth linguistic Z-number that combines the truth linguistic term value δRT with the linguistic reliability measure δMT for δRT specified from the LTS U; (δRI, δMI) is the indeterminacy linguistic Z-number that combines the indeterminacy linguistic term value δRI with the linguistic reliability measure δMI for δRI specified from the LTS U; (δRF, δMF) is the falsehood linguistic Z-number that combines the falsity linguistic term value δRF with the linguistic reliability measure δMF for δRF specified from the LTS U.Definition 2.
Let two LNZNs belzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2) on U and any real number p > 0. Then, their operational relations are defined as follows:(i)
lz1⊕lz2=δRT1,δMT1,δRI1,δMI1,δRF1,δMF1⊕δRT2,δMT2,δRI2,δMI2,δRF2,δMF2=δRT1+RT2−RT1RT2/v,δMT1+MT2−MT1MT2/v,δRI1RI2/v,δMI1MI2/v,δRF1RF2/v,δMF1MF2/v(ii)
lz1⊗lz2=δRT1,δMT1,δRI1,δMI1,δRF1,δMF1⊗δRT2,δMT2,δRI2,δMI2,δRF2,δMF2=δRT1RT2/v,δMT1MT2/v,δRI1+RI2−RI1RI2/v,δMI1+MI2−MI1MI2/v,δRF1+RF2−RF1RF2/v,δMF1+MF2−MF1MF2/v(iiii)
p⋅lz1=pδRT1,δMT1,δRI1,δMI1,δRF1,δMF1=δv−v1−RT1/vp,δv−v1−MT1/vp,δvRI1/vp,δvMI1/vp,δvRF1/vp,δvMF1/vp(iv)
lz1p=δRT1,δMT1,δRI1,δMI1,δRF1,δMF1p=δvRT1/vp,δvMT1/vp,δv−v1−RI1/vp,δv−v1−MI1/vp,δv−v1−RF1/vp,δv−v1−MF1/vp
Clearly, the above operational results are still LNZNs.Example 1.
Set two LNZNs aslz1 = <(δ6, δ7), (δ2, δ6), (δ3, δ7)> and lz2 = <(δ7, δ7), (δ2, δ5), (δ3, δ6)> on U = {δ0, δ1, …, δ8} with v = 8 and p = 0.6. Then, based on the above operational relations, one obtains their operational results:(i)
lz1⊕lz2=δRT1+RT2−RT1RT2/v,δMT1+MT2−MT1MT2/v,δRI1RI2/v,δMI1MI2/v,δRF1RF2/v,δMF1MF2/v=δ6+7−6×7/8,δ7+7−7×7/8,δ2×2/8,δ6×5/8,δ3×3/8,δ7×6/8=δ7.75,δ7.875,δ0.5,δ3.75,δ1.125,δ5.25(ii)
lz1⊗lz2=δRT1RT2/v,δMT1MT2/v,δRI1+RI2−RI1RI2/v,δMI1+MI2−MI1MI2/v,δRF1+RF2−RF1RF2/v,δMF1+MF2−MF1MF2/v=δ6×7/8,δ7×7/8,δ2+2−2×2/8,δ6+5−6×5/8,δ3+3−3×3/8,δ7+6−7×6/8=δ5.25,δ6.125,δ3.5,δ7.25,δ4.875,δ7.75(iiii)
p⋅lz1=δv−v1−RT1/vp,δv−v1−MT1/vp,δvRI1/vp,δvMI1/vp,δvRF1/vp,δvMF1/vp=δ8−81−6/80.6,δ8−81−7/80.6,δ82/80.6,δ86/80.6,δ83/80.6,δ87/80.6=δ4.5178,δ5.7026,δ3.4822,δ6.7317,δ4.4413,δ7.3841(iv)
lz1p=δvRT1/vp,δvMT1/vp,δv−v1−RI1/vp,δv−v1−MI1/vp,δv−v1−RF1/vp,δv−v1−MF1/vp=δ86/80.6,δ87/80.6,δ8−81−2/80.6,δ8−81−6/8p,δ8−81−3/80.6,δ8−81−7/80.6=δ6.7317,δ7.3841,δ1.2683,δ4.5178,δ1.9658,δ5.7026
To compare LNZNs, we can present the score and accuracy functions and sorting laws of LNZNs.Definition 3.
Set LNZN aslz = <(δRT, δMT), (δRI, δMI), (δRF, δMF)> on U. Then, the score and accuracy functions of lz are presented as follows:(5)Ylz=2v2+RT×MT−RI×MI−RF×MF3v2,forYlz∈0,1,(6)Zlz=RT×MT−RF×MFv2,forZlz∈−1,1.Definition 4.
Set two LNZNs aslzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> for k = 1, 2 on U. Then, their sorting laws are defined as follows:(i)
lz1≻ lz2 if Y(lz1) > Y(lz2)(ii)
lz1≻ lz2 if Y(lz1) = Y(lz2) and Z(lz1) > Z(lz2)(iii)
lz1 = lz2 If Y(lz1) = Y(lz2) and Z(lz1) = Z(lz2)Example 2.
Set three LNZNs aslz1 = <(δ6, δ6), (δ3, δ7), (δ3, δ6)>, lz2 = <(δ6, δ6), (δ3, δ6), (δ3, δ7)>, and lz3 = <(δ7, δ6), (δ3, δ7), (δ3, δ5)> on U = {δ0, δ1, …, δ8} with v = 8. Then, by equations (5) and (6), the values of their score and accuracy functions are yielded as follows: Y(lz1) = (2 × 82 + 6 × 6 − 3 × 7 − 3 × 6)/192 = 0.651, Y(lz2) = (2 × 82 + 6 × 6 − 3 × 6 − 3 × 7)/192 = 0.651, and Y(lz3) = (2 × 82 + 7 × 6 − 3 × 7 − 3 × 5)/192 = 0.6979; Z(lz1) = (6 × 6 − 3 × 6)/64 = 0.2813 and Z(lz2) = (6 × 6 − 3 × 7)/64 = 0.2344.
According to the sorting laws in Definition4, their sorting order is lz3≻lz1≻lz2.


## 4. LNZNWAM and LNZNWGM Operators

### 4.1. LNZNWAM Operator

Definition 5.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, the LNZNWAM operator can be defined as(7)LNZNWAMlz1,lz2,…,lzn=∑k=1npk⋅lzk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1.
Thus, the following theorem can be presented corresponding to the operational relations in Definition2 and equation (7).Theorem 1.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, the aggregation result yielded by equation (7) is also LNZN, which is calculated by the following equation:(8)LNZNWAMlz1,lz2,…,lzn=∑k=1npk⋅lzk=δv−v∏k=1n1−RTk/vpk,δv−v∏k=1n1−MTk/vpk,δv∏k=1nRIk/vpk,δv∏k=1nMIk/vpk,δv∏k=1nRFk/vpk,δv∏k=1nMFk/vpk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1.In the following, Theorem1 can be verified by means of mathematical induction.Proof.
(1)
Setn = 2. By the operational relations in Definition 2, there is the following result:(9)LNZNWAMlz1,lz2=p1⋅lz1⊕p2⋅lz2=δv−v1−RT1/vp1+v−v1−RT2/vp2−v−v1−RT1/qp1v−v1−RT2/vp2/v,δv−v1−MT1/vp1+v−v1−MT2/vp2−v−v1−MT1/vp1v−v1−MT2/vp2/v,δvRI1/vp1RI2/vp2,δvMI1/vp1MI2/vp2,δvRF1/vp1RF2/vp2,δvMF1/vp1MF2/vp2=δv−v1−RT1/vp1+v−v1−RT2/vp2−v−v1−RT1/vp1−v1−RT2/vp2+v1−RT1/vw11−RT2/vp2,δv−v1−MT1/vp1+v−v1−MT2/vp2−v−v1−MT1/vp1−v1−MT2/vp2+v1−MT1/vw11−MT2/vp2,δvRI1/vp1RI2/vp2,δvMI1/vp1MI2/vp2,δvRF1/vp1RF2/vp2,δvMF1/vp1MF2/vp2=δv−v1−RT1/vp11−RT2/vp2,δv−v1−MT1/vp11−MT2/vp2,δvRI1/vp1RI2/vp2,δvMI1/vp1MI2/vp2,δvRF1/vp1RF2/vp2,δvMF1/vp1MF2/vp2=δv−v∏k=121−RTk/vpk,δv−v∏k=121−MTk/vpk,δv∏k=12RIk/vpk,δv∏k=12MIk/vpk,δv∏k=12RFk/vpk,δv∏k=12MFk/vpk.(2)
Setn = m. Equation (8) can keep the following result:(10)LNZNWAMlz1,lz2,…,lzm=∑k=1mpk⋅lzk=δv−v∏k=1m1−RTk/vpk,δv−v∏k=1m1−MTk/vpk,δv∏k=1mRIk/vpk,δv∏k=1mMIk/vpk,δv∏k=1mRFk/vpk,δv∏k=1mMFk/vpk.(3)
Setn = m + 1. By equations (9) and (10), the operational result is given as follows:(11)LNZNWAMlz1,lz2,…,lzm,lzm+1=∑k=1mpk⋅lzk⊕pm+1⋅lzm+1=δv−v∏k=1m1−RTk/vpk+v−v1−RTm+1/vpm+1−v−v∏k=1m1−RTk/vpkv−v1−RTm+1/vpm+1/v,δv−v∏k=1m1−MTk/vpk+v−v1−MTm+1/vpm+1−v−v∏k=1m1−MTk/vpkv−v1−MTm+1/vpm+1/v,δv∏k=1mRIk/vpkRIm+1/vpm+1,δv∏k=1mMIk/vpkMIm+1/vpm+1,δv∏k=1mRFk/vpkRFm+1/vpm+1,δv∏k=1mMFk/vpkMFm+1/vpm+1=δv−v∏k=1m1−RTk/vpk+v−v1−RTm+1/vpm+1−v−v∏k=1m1−RTk/vpk−v1−RTm+1/vpm+1+v∏k=1m1−RTk/vpk1−RTm+1/vpm+1,δv−v∏k=1m1−MTk/vpk+v−v1−MTm+1/vpm+1−v−v∏k=1m1−MTk/vpk−v1−MTm+1/vpm+1+v∏k=1m1−MTk/vpk1−MTm+1/vpm+1,δv∏k=1mRIk/vpkRIm+1/vpm+1,δv∏k=1mMIk/vpkRIm+1/vpm+1,δv∏k=1mRFk/vpkRFm+1/vpm+1,δv∏k=1mMFk/vpkMFm+1/vpm+1=δq−q∏k=1m1−RTk/qpk1−RTm+1/qpm+1,δq−q∏k=1m1−MTk/qpk1−MTm+1/qpm+1,δv∏k=1mRIk/vpkRIm+1/vpm+1,δv∏k=1mMIk/vpkMIm+1/vpm+1,δv∏k=1mRFk/vpkRFm+1/vpm+1,δv∏k=1mMFk/vpkMFm+1/vpm+1=δv−v∏k=1m+11−RTk/vpk,δv−v∏k=1m+11−MTk/vpk,δv∏k=1m+1RIk/vpk,δv∏k=1m+1MIk/vpk,δv∏k=1m+1RFk/vpk,δv∏k=1m+1MFk/vpk.
Based on the above results, equation (8) holds for any n. Thus, the proof is completed.
Then, the LNZNWAM operator implies the following properties:(i)
Idempotency: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk = lz for k = 1, 2, …, n, then LNZNWAMlz1,lz2,…,lzn=lz.(ii)
Boundedness: setlzk (k = 1, 2, …, n) as a series of LNZNs on U and then set the minimum and maximum LNZNs as(12)lz−=minkδRTk,minkδMTk,maxkδRIk,maxkδMIk,maxkδRFk,maxkδMFk,lz+=maxkδRTk,maxkδMTk,minkδRIk,minkδMIk,minkδRFk,minkδMFk,and then there is lz−≤LNZNWAMlz1,lz2,…,lzn≤lz+.(iii)
Monotonicity: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk ≤ lzk∗ for k = 1, 2, …, n, there is LNZNWAMlz1,lz2,…,lzn≤LNZNWAMlz1∗,lz2∗,…,lzn∗.(iv)
Commutativity: set the LNZN sequencelz1′,lz2′,…,lzn′ as an arbitrary permutation of (lz1, lz2, …, lzn). Then, there is LNZNWAMlz1′,lz2′,…,lzn′=LNZNWAMlz1,lz2,…,lzn.Proof.
(i)
Becauselzk = lz for k = 1, 2, …, n, there is the following operational result:(13)LNZNWAMlz1,lz2,…,lzn=∑k=1npk⋅lzk=δv−v∏k=1n1−RTk/vpk,δv−v∏k=1n1−MTk/vpk,δv∏k=1nRIk/vpk,δv∏k=1nMIk/vpk,δv∏k=1nRFk/vpk,δv∏k=1nMFk/vpk=δv−v1−RT/v∑k=1npk,δv−v1−RT/v∑k=1npk,δvRI/v∑k=1npk,δvRI/v∑k=1npk,δvRF/v∑k=1npk,δvRF/v∑k=1npk=δv−v1−RT/v,δv−v1−MT/v,δvRI/v,δvMI/v,δvRF/v,δvMF/v=δRT,δMT,δRI,δMI,δRF,δMF=lz.(ii)
Since the minimum and maximum LNZNs arelz− and lz+, there is lz− ≤ lzj ≤ lz+. Thus, there exists the inequality ∑k=1npklz−≤∑k=1npklzk≤∑k=1npklz+. Regarding property (i), there is the inequality lz−≤∑k=1npklzk≤lz+, namely, lz−≤LNZNWAMlz1,lz2,…,lzn≤lz+.(iii)
Becauselzk≤lzk∗ for k = 1, 2, …, n, there exists the inequality ∑k=1npklzk≤∑k=1npklzk∗, namely, LNZNWAMlz1,lz2,…,lzn≤LNZNWAMlz1∗,lz2∗,…,lzn∗.(iv)
The commutativity of the LNZNWAM operator is straightforward.
Hence, the proof of these properties is finished.
Especially whenpk = 1/n for k = 1, 2, …, n, the LNZNWAM operator reduces into the LNZN arithmetic mean operator.


### 4.2. LNNWGM Operator

Definition 6.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, we can define the LNZNWGM operator:(14)LNZNWGMlz1,lz2,…,lzn=∏k=1nlzkpk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1.
Based on the operational relations in Definition2 and equation (14), we can give the following theorem.Theorem 2.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, the aggregation result yielded by equation (14) is also LNZN, which is calculated by the aggregation equation:(15)LNZNWGMlz1,lz2,…,lzn=∏k=1nlzkpk=δv∏k=1nRTk/vpk,δv∏k=1nMTk/vpk,δv−v∏k=1n1−RIk/vpk,δv−v∏k=1n1−MIk/vpk,δv−v∏k=1n1−RFk/vpk,δv−v∏k=1n1−MFk/vpk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1. Especially when pk = 1/n for k = 1, 2, …, n, the LNZNWGM operator reduces into the geometric mean operator of LNZNs.By the similar proof way of Theorem1, one can verify Theorem 2, which is not repeated here.Similarly, the LNZNWGM operator also implies the following properties:(i)
Idempotency: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk = lz (k = 1, 2, …, n), there exists LNZNWGMlz1,lz2,…,lzn=lz.(ii)
Boundedness: setlzk (k = 1, 2, …, n) as a series of LNZNs on U and set the minimum and maximum LNZNs as(16)lz−=minkδRTk,minkδMTk,maxkδRIk,maxkδMIk,maxkδRFk,maxkδMFk,lz+=maxkδRTk,maxkδMTk,minkδRIk,minkδMIk,minkδRFk,minkδMFk,and then there exists the inequality lz−≤LNZNWGMlz1,lz2,…,lzn≤lz+.(iii)
Monotonicity: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk ≤ lzk∗ (k = 1, 2, …, n), there exists the inequality LNZNWGMlz1,lz2,…,lzn≤LNZNWGMlz1∗,lz2∗,…,lzn∗.(iv)
Commutativity: set the LNZN sequencelz1′,lz2′,…,lzn′ as an arbitrary permutation of (lz1, lz2, …, lzn). Then, there is LNZNWGMlz1′,lz2′,…,lzn′=LNZNWGMlz1,lz2,…,lzn.By the similar proof of the properties of the LNZNWAM operator, one can verify these properties, which are not repeated here.


## 4.1. LNZNWAM Operator

Definition 5.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, the LNZNWAM operator can be defined as(7)LNZNWAMlz1,lz2,…,lzn=∑k=1npk⋅lzk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1.
Thus, the following theorem can be presented corresponding to the operational relations in Definition2 and equation (7).Theorem 1.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, the aggregation result yielded by equation (7) is also LNZN, which is calculated by the following equation:(8)LNZNWAMlz1,lz2,…,lzn=∑k=1npk⋅lzk=δv−v∏k=1n1−RTk/vpk,δv−v∏k=1n1−MTk/vpk,δv∏k=1nRIk/vpk,δv∏k=1nMIk/vpk,δv∏k=1nRFk/vpk,δv∏k=1nMFk/vpk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1.In the following, Theorem1 can be verified by means of mathematical induction.Proof.
(1)
Setn = 2. By the operational relations in Definition 2, there is the following result:(9)LNZNWAMlz1,lz2=p1⋅lz1⊕p2⋅lz2=δv−v1−RT1/vp1+v−v1−RT2/vp2−v−v1−RT1/qp1v−v1−RT2/vp2/v,δv−v1−MT1/vp1+v−v1−MT2/vp2−v−v1−MT1/vp1v−v1−MT2/vp2/v,δvRI1/vp1RI2/vp2,δvMI1/vp1MI2/vp2,δvRF1/vp1RF2/vp2,δvMF1/vp1MF2/vp2=δv−v1−RT1/vp1+v−v1−RT2/vp2−v−v1−RT1/vp1−v1−RT2/vp2+v1−RT1/vw11−RT2/vp2,δv−v1−MT1/vp1+v−v1−MT2/vp2−v−v1−MT1/vp1−v1−MT2/vp2+v1−MT1/vw11−MT2/vp2,δvRI1/vp1RI2/vp2,δvMI1/vp1MI2/vp2,δvRF1/vp1RF2/vp2,δvMF1/vp1MF2/vp2=δv−v1−RT1/vp11−RT2/vp2,δv−v1−MT1/vp11−MT2/vp2,δvRI1/vp1RI2/vp2,δvMI1/vp1MI2/vp2,δvRF1/vp1RF2/vp2,δvMF1/vp1MF2/vp2=δv−v∏k=121−RTk/vpk,δv−v∏k=121−MTk/vpk,δv∏k=12RIk/vpk,δv∏k=12MIk/vpk,δv∏k=12RFk/vpk,δv∏k=12MFk/vpk.(2)
Setn = m. Equation (8) can keep the following result:(10)LNZNWAMlz1,lz2,…,lzm=∑k=1mpk⋅lzk=δv−v∏k=1m1−RTk/vpk,δv−v∏k=1m1−MTk/vpk,δv∏k=1mRIk/vpk,δv∏k=1mMIk/vpk,δv∏k=1mRFk/vpk,δv∏k=1mMFk/vpk.(3)
Setn = m + 1. By equations (9) and (10), the operational result is given as follows:(11)LNZNWAMlz1,lz2,…,lzm,lzm+1=∑k=1mpk⋅lzk⊕pm+1⋅lzm+1=δv−v∏k=1m1−RTk/vpk+v−v1−RTm+1/vpm+1−v−v∏k=1m1−RTk/vpkv−v1−RTm+1/vpm+1/v,δv−v∏k=1m1−MTk/vpk+v−v1−MTm+1/vpm+1−v−v∏k=1m1−MTk/vpkv−v1−MTm+1/vpm+1/v,δv∏k=1mRIk/vpkRIm+1/vpm+1,δv∏k=1mMIk/vpkMIm+1/vpm+1,δv∏k=1mRFk/vpkRFm+1/vpm+1,δv∏k=1mMFk/vpkMFm+1/vpm+1=δv−v∏k=1m1−RTk/vpk+v−v1−RTm+1/vpm+1−v−v∏k=1m1−RTk/vpk−v1−RTm+1/vpm+1+v∏k=1m1−RTk/vpk1−RTm+1/vpm+1,δv−v∏k=1m1−MTk/vpk+v−v1−MTm+1/vpm+1−v−v∏k=1m1−MTk/vpk−v1−MTm+1/vpm+1+v∏k=1m1−MTk/vpk1−MTm+1/vpm+1,δv∏k=1mRIk/vpkRIm+1/vpm+1,δv∏k=1mMIk/vpkRIm+1/vpm+1,δv∏k=1mRFk/vpkRFm+1/vpm+1,δv∏k=1mMFk/vpkMFm+1/vpm+1=δq−q∏k=1m1−RTk/qpk1−RTm+1/qpm+1,δq−q∏k=1m1−MTk/qpk1−MTm+1/qpm+1,δv∏k=1mRIk/vpkRIm+1/vpm+1,δv∏k=1mMIk/vpkMIm+1/vpm+1,δv∏k=1mRFk/vpkRFm+1/vpm+1,δv∏k=1mMFk/vpkMFm+1/vpm+1=δv−v∏k=1m+11−RTk/vpk,δv−v∏k=1m+11−MTk/vpk,δv∏k=1m+1RIk/vpk,δv∏k=1m+1MIk/vpk,δv∏k=1m+1RFk/vpk,δv∏k=1m+1MFk/vpk.
Based on the above results, equation (8) holds for any n. Thus, the proof is completed.
Then, the LNZNWAM operator implies the following properties:(i)
Idempotency: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk = lz for k = 1, 2, …, n, then LNZNWAMlz1,lz2,…,lzn=lz.(ii)
Boundedness: setlzk (k = 1, 2, …, n) as a series of LNZNs on U and then set the minimum and maximum LNZNs as(12)lz−=minkδRTk,minkδMTk,maxkδRIk,maxkδMIk,maxkδRFk,maxkδMFk,lz+=maxkδRTk,maxkδMTk,minkδRIk,minkδMIk,minkδRFk,minkδMFk,and then there is lz−≤LNZNWAMlz1,lz2,…,lzn≤lz+.(iii)
Monotonicity: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk ≤ lzk∗ for k = 1, 2, …, n, there is LNZNWAMlz1,lz2,…,lzn≤LNZNWAMlz1∗,lz2∗,…,lzn∗.(iv)
Commutativity: set the LNZN sequencelz1′,lz2′,…,lzn′ as an arbitrary permutation of (lz1, lz2, …, lzn). Then, there is LNZNWAMlz1′,lz2′,…,lzn′=LNZNWAMlz1,lz2,…,lzn.Proof.
(i)
Becauselzk = lz for k = 1, 2, …, n, there is the following operational result:(13)LNZNWAMlz1,lz2,…,lzn=∑k=1npk⋅lzk=δv−v∏k=1n1−RTk/vpk,δv−v∏k=1n1−MTk/vpk,δv∏k=1nRIk/vpk,δv∏k=1nMIk/vpk,δv∏k=1nRFk/vpk,δv∏k=1nMFk/vpk=δv−v1−RT/v∑k=1npk,δv−v1−RT/v∑k=1npk,δvRI/v∑k=1npk,δvRI/v∑k=1npk,δvRF/v∑k=1npk,δvRF/v∑k=1npk=δv−v1−RT/v,δv−v1−MT/v,δvRI/v,δvMI/v,δvRF/v,δvMF/v=δRT,δMT,δRI,δMI,δRF,δMF=lz.(ii)
Since the minimum and maximum LNZNs arelz− and lz+, there is lz− ≤ lzj ≤ lz+. Thus, there exists the inequality ∑k=1npklz−≤∑k=1npklzk≤∑k=1npklz+. Regarding property (i), there is the inequality lz−≤∑k=1npklzk≤lz+, namely, lz−≤LNZNWAMlz1,lz2,…,lzn≤lz+.(iii)
Becauselzk≤lzk∗ for k = 1, 2, …, n, there exists the inequality ∑k=1npklzk≤∑k=1npklzk∗, namely, LNZNWAMlz1,lz2,…,lzn≤LNZNWAMlz1∗,lz2∗,…,lzn∗.(iv)
The commutativity of the LNZNWAM operator is straightforward.
Hence, the proof of these properties is finished.
Especially whenpk = 1/n for k = 1, 2, …, n, the LNZNWAM operator reduces into the LNZN arithmetic mean operator.


## 4.2. LNNWGM Operator

Definition 6.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, we can define the LNZNWGM operator:(14)LNZNWGMlz1,lz2,…,lzn=∏k=1nlzkpk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1.
Based on the operational relations in Definition2 and equation (14), we can give the following theorem.Theorem 2.
Setlzk = <(δRT(k), δMT(k)), (δRI(k), δMI(k)), (δRF(k), δMF(k))> (k = 1, 2, …, n) as a series of LNZNs on U. Then, the aggregation result yielded by equation (14) is also LNZN, which is calculated by the aggregation equation:(15)LNZNWGMlz1,lz2,…,lzn=∏k=1nlzkpk=δv∏k=1nRTk/vpk,δv∏k=1nMTk/vpk,δv−v∏k=1n1−RIk/vpk,δv−v∏k=1n1−MIk/vpk,δv−v∏k=1n1−RFk/vpk,δv−v∏k=1n1−MFk/vpk,where pk is the weight of lzk (k = 1, 2, …, n) for pk ∈ [0, 1] and ∑k=1npk=1. Especially when pk = 1/n for k = 1, 2, …, n, the LNZNWGM operator reduces into the geometric mean operator of LNZNs.By the similar proof way of Theorem1, one can verify Theorem 2, which is not repeated here.Similarly, the LNZNWGM operator also implies the following properties:(i)
Idempotency: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk = lz (k = 1, 2, …, n), there exists LNZNWGMlz1,lz2,…,lzn=lz.(ii)
Boundedness: setlzk (k = 1, 2, …, n) as a series of LNZNs on U and set the minimum and maximum LNZNs as(16)lz−=minkδRTk,minkδMTk,maxkδRIk,maxkδMIk,maxkδRFk,maxkδMFk,lz+=maxkδRTk,maxkδMTk,minkδRIk,minkδMIk,minkδRFk,minkδMFk,and then there exists the inequality lz−≤LNZNWGMlz1,lz2,…,lzn≤lz+.(iii)
Monotonicity: setlzk (k = 1, 2, …, n) as a series of LNZNs on U. If lzk ≤ lzk∗ (k = 1, 2, …, n), there exists the inequality LNZNWGMlz1,lz2,…,lzn≤LNZNWGMlz1∗,lz2∗,…,lzn∗.(iv)
Commutativity: set the LNZN sequencelz1′,lz2′,…,lzn′ as an arbitrary permutation of (lz1, lz2, …, lzn). Then, there is LNZNWGMlz1′,lz2′,…,lzn′=LNZNWGMlz1,lz2,…,lzn.By the similar proof of the properties of the LNZNWAM operator, one can verify these properties, which are not repeated here.


## 5. MAGDM Approach in terms of the LNZNWAM and LNZNWGM Operators

This section develops an MAGDM method by utilizing the LNZNWAM and LNZNWGM operators and score and accuracy functions to perform MAGDM issues in the setting of LNZNs.Regarding an MAGDM problem, experts/decision-makers preliminarily propose a set of alternativesN = {N1, N2, …, Nm}, which needs to satisfy the requirements of n attributes in a set of attributes H = {h1, h2, …, hn} with the weight vector of the attributes P = (p1, p2, …, pn). Then, a group of decision-makers/experts G = {G1, G2, …, Gs} is invited along with their corresponding weight vector D = (d1, d2, …, ds) to assess the alternatives over the attributes by LNZNs from the predefined LTS U = {δ0, δ1, …, δv} with odd cardinality v+1. In the assessment process of each alternative Nj (j = 1, 2, …, m) over each attribute hk (k = 1, 2, …, n), each decision-maker can provide the truth, falsehood, and indeterminacy linguistic term values and their corresponding linguistic reliability measure values from U, which are constructed as LNZN. Thus, the LNZNs provided by each decision-maker Gi (i = 1, 2, …, s) can be constructed as each LNZN decision matrix Mi = (lzjki)m∗n, where lzjki=δRTijk,δMTijk,δRIijk,δMIijk,δRFijk,δMFijk (i = 1, 2, …, s; k = 1, 2, …, n; j = 1, 2, …, m) are LNZNs.Thus, we can develop an MAGDM method by using the LNZNWAM and LNZNWGM operators and score and accuracy functions to perform the MAGDM issue with LNZN information and introduce the decision steps below.Step 1: obtain the aggregated matrixM = (lzjk)m×n, where lzjk=δRTjk,δMTjk,δRIjk,δMIjk,δRFjk,δMFjk (k = 1, 2, …, n; j = 1, 2, …, m) is an aggregated LNZN, by using the LNZNWAM operator:(17)lzjk=LNZNWAMlzjk1,lzjk2,…,lzjks=∑i=1sdi⋅lzjki=δv−v∏i=1s1−RTijk/vdi,δv−v∏i=1s1−MTijk/vdiδv∏i=1sRIijk/vdi,δv∏i=1sMIijk/vdiδv∏i=1sRFijk/vdi,δv∏i=1sMFijk/vdiStep 2: obtain the aggregated LNZNlzj for Nj (j = 1, 2, …, m) by using the LNZNWAM or LNZNWGM operator:(18)lzj=LNZNWAMlzj1,lzj2,…,lzjn=∑k=1npk⋅lzjk=δv−v∏k=1n1−RTjk/vpk,δv−v∏k=1n1−MTjk/vpkδv∏k=1nRIjk/vpk,δv∏k=1nMIjk/vpkδv∏k=1nRFjk/vpk,δv∏k=1nMFjk/vpk,(19)lzj=LNZNWGMlzj1,lzj2,…,lzjn=∏k=1nlzjkpk=δv∏k=1nRTjk/vpk,δv∏k=1nMTjk/vpkδv−v∏k=1n1−RIjk/vpk,δv−v∏k=1n1−MIjk/vpkδv−v∏k=1n1−RFjk/vpk,δv−v∏k=1n1−MFjk/vpk,Step 3: calculate the score values ofY(lzj) (the accuracy values of Z(lzj) if necessary) (j = 1, 2, …, m) by equation (5) (equation (6)).Step 4: sort the alternatives based on the score (accuracy) values and sorting laws of LNZNs and then choose the best one.Step 5: end.


## 6. MAGDM Example and Comparison

### 6.1. MAGDM Example of Industrial Robot Selection

Because of the complexity, advanced features, and facilities of industrial robots, selecting an industrial robot for a specific application is a multifaceted task. This requires decision-makers to select the most suitable robot for a specific application in terms of the various features, costs, and benefits, which is an MAGDM issue. To illustrate the applicability and efficiency of the proposed MAGDM approach, this section provides an MAGDM application of industrial robot selection in order to choose the most suitable industrial robot for the flexible manufacturing system of a manufacturing company.A manufacturing company needs to select the most suitable type of industrial robots from robot suppliers. Some experts preliminarily choose four types of industrial robots, which are denoted as a set of alternativesN = {N1, N2, N3, N4} from robot suppliers. Meanwhile, they must satisfy four indices (attributes): the operation dexterity (h1), the payload capacity (h2), the programming versatility (readability, coordination, and intelligent control capacity) (h3), and the man-machine interface (h4). The weight vector of the four attributes hk for k = 1, 2, 3, 4 is given by P = (0.27, 0.23, 0.26, 0.24) to indicate the importance of the attributes. A group of three experts/decision-makers G = {G1, G2, G3} with the weight vector D = (0.37, 0.35, 0.28) is requested to assess the four alternatives over the four attributes by LNZNs from the predefined LTS U = {δ0(extremely low), δ1(very low), δ2(low), δ3(slightly high), δ4(medium), δ5(slightly high), δ6(high), δ7(very high), δ8(extremely high)} with v = 8. Therefore, the LNZNs specified by each decision-maker Gi (i = 1, 2, 3) can form the following LNZN decision matrix Mi (i = 1, 2, 3):(20)M1=δ6,δ7,δ1,δ6,δ2,δ6δ7,δ6,δ1,δ5,δ2,δ5δ5,δ6,δ2,δ6,δ2,δ7δ6,δ5,δ2,δ5,δ2,δ6δ7,δ6,δ1,δ5,δ1,δ6δ7,δ7,δ3,δ6,δ2,δ4δ7,δ6,δ3,δ6,δ2,δ5δ6,δ6,δ2,δ6,δ2,δ5δ6,δ6,δ2,δ6,δ2,δ6δ6,δ5,δ1,δ5,δ1,δ6δ6,δ7,δ2,δ4,δ1,δ5δ6,δ5,δ2,δ6,δ3,δ6δ7,δ7,δ1,δ6,δ2,δ5δ7,δ6,δ1,δ6,δ3,δ7δ7,δ5,δ2,δ5,δ2,δ6δ5,δ6,δ1,δ7,δ1,δ6,M2=δ5,δ6,δ1,δ6,δ2,δ6δ6,δ7,δ3,δ5,δ2,δ5δ6,δ6,δ2,δ6,δ2,δ7δ6,δ5,δ2,δ5,δ2,δ6δ6,δ5,δ3,δ5,δ1,δ7δ6,δ7,δ2,δ6,δ3,δ4δ6,δ6,δ1,δ6,δ3,δ5δ7,δ6,δ1,δ6,δ3,δ5δ5,δ5,δ2,δ6,δ2,δ6δ5,δ5,δ2,δ5,δ1,δ7δ5,δ7,δ2,δ5,δ4,δ5δ7,δ6,δ3,δ6,δ1,δ7δ6,δ6,δ1,δ7,δ2,δ5δ7,δ6,δ1,δ7,δ2,δ6δ6,δ5,δ1,δ7,δ3,δ6δ5,δ5,δ3,δ5,δ1,δ6,M3=δ5,δ5,δ1,δ6,δ1,δ6δ6,δ6,δ1,δ6,δ3,δ5δ5,δ6,δ2,δ5,δ1,δ7δ5,δ4,δ3,δ5,δ1,δ6δ7,δ6,δ2,δ5,δ1,δ7δ5,δ6,δ2,δ7,δ3,δ6δ5,δ5,δ2,δ6,δ1,δ6δ6,δ6,δ2,δ6,δ2,δ5δ4,δ5,δ1,δ6,δ2,δ6δ5,δ6,δ1,δ6,δ2,δ5δ7,δ6,δ1,δ7,δ2,δ5δ7,δ5,δ2,δ5,δ3,δ7δ3,δ6,δ1,δ6,δ3,δ5δ5,δ5,δ2,δ5,δ1,δ5δ6,δ6,δ3,δ6,δ2,δ6δ5,δ5,δ3,δ4,δ1,δ7.Then, the developed MAGDM approach can be used in this MAGDM problem and depicted by the following steps:Step 1: by equation (17), we obtain the aggregated matrix M = (lzjk)4×4:(21)M=δ5.4179,δ6.2664,δ1.0000,δ6.0000,δ1.6472,δ6.0000δ6.4524,δ6.4308,δ1.4689,δ5.2619,δ2.2405,δ5.0000δ6.7254,δ5.6950,δ1.7835,δ5.0000,δ1.0000,δ6.6119δ6.2664,δ6.7858,δ2.3237,δ6.2646,δ2.5821,δ4.4809δ5.2013,δ5.4179,δ1.6472,δ6.0000,δ2.0000,δ6.0000δ5.4179,δ5.3220,δ1.2746,δ5.2619,δ1.2142,δ6.0174δ5.9998,δ6.4524,δ1.0000,δ6.3326,δ2.2405,δ5.0000δ6.6398,δ5.7595,δ1.2142,δ6.0174,δ1.9138,δ6.0360δ5.3969,δ6.0000,δ2.0000,δ5.7014,δ1.6472,δ7.0000δ5.7595,δ4.7483,δ2.2405,δ5.0000,δ1.6472,δ6.0000δ6.2664,δ5.7595,δ1.8232,δ6.0000,δ1.8983,δ5.2619δ6.4308,δ6.0000,δ1.5692,δ6.0000,δ1.5692,δ6.0000δ6.1017,δ6.7858,δ1.6472,δ5.0586,δ1.9725,δ5.0000δ6.7076,δ5.3969,δ2.3050,δ5.7014,δ2.0423,δ6.6119δ6.4524,δ5.3220,δ1.7578,δ5.9195,δ2.3050,δ6.0000δ5.0000,δ5.4179,δ1.9980,δ5.3199,δ1.0000,δ6.2646.Step 2: by equation (18), we obtain the aggregated LNZNs of lzj for Nj (j = 1, 2, …, m): lz1 = <(δ5.7769, s5.9550), (δ1.5877, s5.4988), (δ1.7679, s5.9889)>, lz2 = <(δ6.4423, s6.0917), (δ1.8486, s5.7688), (δ1.6372, s5.5662)>, lz3 = <(δ5.9368, s5.8558), (δ1.6832, s5.5011), (δ1.7857, s5.8611)>, and lz4 = <(δ6.1127, s5.8027), (δ1.4296, s5.8980), (δ1.7936, s5.7792)>, or by equation (19), we obtain the aggregated LNZNs of lzj for Nj (j = 1, 2, …, m): lz1 = <(δ5.7177, s5.8316), (δ1.6841, s5.5431), (δ1.7888, s6.1666)>, lz2 = <(δ6.4270, s6.0216), (δ1.8726, s5.8403), (δ1.7604, s5.7607)>, lz3 = <(δ5.8174, s5.7156), (δ1.7300, s5.5426), (δ1.8309, s5.9682)>, and lz4 = <(δ5.9907, s5.7334), (δ1.4983, s5.9404), (δ1.9052, s5.8523)>.Step 3: by equation (5), we obtain the score values: Y(lz1) = 0.7452, Y(lz2) = 0.7681, Y(lz3) = 0.7450, and Y(lz4) = 0.7535, or Y(lz1) = 0.7343, Y(lz2) = 0.7585, Y(lz3) = 0.7330, and Y(lz4) = 0.7411.Step 4: the sorting order of the four alternatives isN2≻f N4≻N1≻N3, and then the best one is N2.Sorting orders of the four alternatives regarding the developed MAGDM approach using the LNZNWAM and LNNWGM operators are shown in Figure1. It is obvious that the sorting orders of the alternatives and the best one in terms of the LNZNWAM and LNNWGM operators are identical.Figure 1
Sorting orders of the four alternatives regarding the developed MAGDM approach using the LNZNWAM and LNNWGM operators.


### 6.2. Comparison with Related Methods

This part compares the proposed MAGDM approach with the related LNN and NZN decision-making approaches [12,20] to indicate the suitability and efficiency of the proposed MAGDM approach.To conveniently compare the proposed MAGDM approach with the existing related MAGDM approach [12] for the robot selection problem, we only use LNNs in M1, M2, and M3 without considering the linguistic reliability measures in LNZNs since the linguistic neutrosophic MAGDM approach [12] cannot handle such an MAGDM problem with the information of LNZNs. As a special case, the above LNZN decision matrices of the three decision-makers are reduced to the following LNN decision matrices:(22)M′1=δ6,δ1,δ2δ7,δ1,δ2δ5,δ2,δ2δ6,δ2,δ2δ7,δ1,δ1δ7,δ3,δ2δ7,δ3,δ2δ6,δ2,δ2δ6,δ2,δ2δ6,δ1,δ1δ6,δ2,δ1δ6,δ2,δ3δ7,δ1,δ2δ7,δ1,δ3δ7,δ2,δ2δ5,δ1,δ1,M′2=δ5,δ1,δ2δ6,δ3,δ2δ6,δ2,δ2δ6,δ2,δ2δ6,δ3,δ1δ6,δ2,δ3δ6,δ1,δ3δ7,δ1,δ3δ5,δ2,δ2δ5,δ2,δ1δ5,δ2,δ4δ7,δ3,δ1δ6,δ1,δ2δ7,δ1,δ2δ6,δ1,δ3δ5,δ3,δ1,M′3=δ5,δ1,δ1δ6,δ1,δ3δ5,δ2,δ1δ5,δ3,δ1δ7,δ2,δ1δ5,δ2,δ3δ5,δ2,δ1δ6,δ2,δ2δ4,δ1,δ2δ5,δ1,δ2δ7,δ1,δ2δ7,δ2,δ3δ3,δ1,δ3δ5,δ2,δ1δ6,δ3,δ2δ5,δ3,δ1.Thus, we utilize the existing MAGDM approach using the LNNWAM and LNNWGM operators of equations (1) and (2) and the score function of equation (3) [12] for this MAGDM example in the setting of LNNs.First, by equation (1), we obtain the following aggregated matrix:(23)M′=δ5.4179,δ1.0000,δ1.6472δ6.4524,δ1.4689,δ2.2405δ5.3969,δ2.0000,δ1.6472δ5.7595,δ2.2405,δ1.6472δ6.7254,δ1.7835,δ1.0000δ6.2664,δ2.3237,δ2.5821δ6.2664,δ1.8232,δ1.8983δ6.4308,δ1.5692,δ1.5692δ5.2013,δ1.6472,δ2.0000δ5.4179,δ1.2746,δ1.2142δ6.1017,δ1.6472,δ1.9725δ6.7076,δ2.3050,δ2.0423δ5.9998,δ1.0000,δ2.2405δ6.6398,δ1.2142,δ1.9138δ6.4524,δ1.7578,δ2.3050δ5.0000,δ1.9980,δ1.0000.Then, by equations (1)–(4), the decision results are shown in Table 1. For the convenient comparison, the decision results corresponding to the proposed MAGDM approach are also contained in Table 1.Table 1
Decision results of different MAGDM methods.
MAGDM methodScore valueSorting orderThe best oneExisting MAGDM method using equation (1) [12]0.7676, 0.7899, 0.7695, 0.7871N2≻N4≻N3≻N1N2Existing MAGDM method using equation (2) [12]0.7602, 0.7831, 0.7607, 0.7745N2≻N4≻N3≻N1N2Proposed MAGDM method using equation (18)0.7452, 0.7681, 0.7450, 0.7535N2≻N4≻N1≻N3N2Proposed MAGDM method using equation (19)0.7343, 0.7585, 0.7330, 0.7411N2≻N4≻N1≻N3N2In Table1, there is the sorting difference between the existing MAGDM method [12] and the proposed MAGDM method; then, the best one is the same. However, this sorting difference reflects that the new method utilizes the LNZN information constructed by the truth, falsehood, and indeterminacy linguistic Z-numbers, while the existing method [12] only contains the LNN information without containing their reliability measures. Moreover, LNZNs imply much more useful information than LNNs and strengthen the reliability of LNNs. Thus, different MAGDM information and methods may affect the sorting order of alternatives, which illustrates the efficiency and applicability of the new MAGDM method in the LNZN environment. Therefore, the new MAGDM method is superior to the existing one [12].Furthermore, the existing decision-making method in the setting of NZNs [20] cannot carry out such a linguistic decision-making issue with the LNZN information. On the contrary, the new MAGDM approach with the LNZN information especially suits such an MAGDM issue with the LNZN information. Then, in the MAGDM problem with qualitative attributes, the new MAGDM approach shows its main merit since it contains the LNN assessments related to their reliability measures in the LNZN environment.However, the original study reveals the following main advantages:(a)
The proposed LNZN information can express more useful information to strengthen the reliability measure of LNNs and avoid the insufficiency of missing reliability measures in the existing methods(b)
The developed MAGDM approach not only makes the MAGDM process more reliable and reasonable but also provides a new way for linguistic MAGDM problems in the LNZN setting(c)
The proposed MAGDM approach can effectively solve the MAGDM issue of selecting industrial robots with the evaluation information of LNZNs and make the decision result more reliable


## 6.1. MAGDM Example of Industrial Robot Selection

Because of the complexity, advanced features, and facilities of industrial robots, selecting an industrial robot for a specific application is a multifaceted task. This requires decision-makers to select the most suitable robot for a specific application in terms of the various features, costs, and benefits, which is an MAGDM issue. To illustrate the applicability and efficiency of the proposed MAGDM approach, this section provides an MAGDM application of industrial robot selection in order to choose the most suitable industrial robot for the flexible manufacturing system of a manufacturing company.A manufacturing company needs to select the most suitable type of industrial robots from robot suppliers. Some experts preliminarily choose four types of industrial robots, which are denoted as a set of alternativesN = {N1, N2, N3, N4} from robot suppliers. Meanwhile, they must satisfy four indices (attributes): the operation dexterity (h1), the payload capacity (h2), the programming versatility (readability, coordination, and intelligent control capacity) (h3), and the man-machine interface (h4). The weight vector of the four attributes hk for k = 1, 2, 3, 4 is given by P = (0.27, 0.23, 0.26, 0.24) to indicate the importance of the attributes. A group of three experts/decision-makers G = {G1, G2, G3} with the weight vector D = (0.37, 0.35, 0.28) is requested to assess the four alternatives over the four attributes by LNZNs from the predefined LTS U = {δ0(extremely low), δ1(very low), δ2(low), δ3(slightly high), δ4(medium), δ5(slightly high), δ6(high), δ7(very high), δ8(extremely high)} with v = 8. Therefore, the LNZNs specified by each decision-maker Gi (i = 1, 2, 3) can form the following LNZN decision matrix Mi (i = 1, 2, 3):(20)M1=δ6,δ7,δ1,δ6,δ2,δ6δ7,δ6,δ1,δ5,δ2,δ5δ5,δ6,δ2,δ6,δ2,δ7δ6,δ5,δ2,δ5,δ2,δ6δ7,δ6,δ1,δ5,δ1,δ6δ7,δ7,δ3,δ6,δ2,δ4δ7,δ6,δ3,δ6,δ2,δ5δ6,δ6,δ2,δ6,δ2,δ5δ6,δ6,δ2,δ6,δ2,δ6δ6,δ5,δ1,δ5,δ1,δ6δ6,δ7,δ2,δ4,δ1,δ5δ6,δ5,δ2,δ6,δ3,δ6δ7,δ7,δ1,δ6,δ2,δ5δ7,δ6,δ1,δ6,δ3,δ7δ7,δ5,δ2,δ5,δ2,δ6δ5,δ6,δ1,δ7,δ1,δ6,M2=δ5,δ6,δ1,δ6,δ2,δ6δ6,δ7,δ3,δ5,δ2,δ5δ6,δ6,δ2,δ6,δ2,δ7δ6,δ5,δ2,δ5,δ2,δ6δ6,δ5,δ3,δ5,δ1,δ7δ6,δ7,δ2,δ6,δ3,δ4δ6,δ6,δ1,δ6,δ3,δ5δ7,δ6,δ1,δ6,δ3,δ5δ5,δ5,δ2,δ6,δ2,δ6δ5,δ5,δ2,δ5,δ1,δ7δ5,δ7,δ2,δ5,δ4,δ5δ7,δ6,δ3,δ6,δ1,δ7δ6,δ6,δ1,δ7,δ2,δ5δ7,δ6,δ1,δ7,δ2,δ6δ6,δ5,δ1,δ7,δ3,δ6δ5,δ5,δ3,δ5,δ1,δ6,M3=δ5,δ5,δ1,δ6,δ1,δ6δ6,δ6,δ1,δ6,δ3,δ5δ5,δ6,δ2,δ5,δ1,δ7δ5,δ4,δ3,δ5,δ1,δ6δ7,δ6,δ2,δ5,δ1,δ7δ5,δ6,δ2,δ7,δ3,δ6δ5,δ5,δ2,δ6,δ1,δ6δ6,δ6,δ2,δ6,δ2,δ5δ4,δ5,δ1,δ6,δ2,δ6δ5,δ6,δ1,δ6,δ2,δ5δ7,δ6,δ1,δ7,δ2,δ5δ7,δ5,δ2,δ5,δ3,δ7δ3,δ6,δ1,δ6,δ3,δ5δ5,δ5,δ2,δ5,δ1,δ5δ6,δ6,δ3,δ6,δ2,δ6δ5,δ5,δ3,δ4,δ1,δ7.Then, the developed MAGDM approach can be used in this MAGDM problem and depicted by the following steps:Step 1: by equation (17), we obtain the aggregated matrix M = (lzjk)4×4:(21)M=δ5.4179,δ6.2664,δ1.0000,δ6.0000,δ1.6472,δ6.0000δ6.4524,δ6.4308,δ1.4689,δ5.2619,δ2.2405,δ5.0000δ6.7254,δ5.6950,δ1.7835,δ5.0000,δ1.0000,δ6.6119δ6.2664,δ6.7858,δ2.3237,δ6.2646,δ2.5821,δ4.4809δ5.2013,δ5.4179,δ1.6472,δ6.0000,δ2.0000,δ6.0000δ5.4179,δ5.3220,δ1.2746,δ5.2619,δ1.2142,δ6.0174δ5.9998,δ6.4524,δ1.0000,δ6.3326,δ2.2405,δ5.0000δ6.6398,δ5.7595,δ1.2142,δ6.0174,δ1.9138,δ6.0360δ5.3969,δ6.0000,δ2.0000,δ5.7014,δ1.6472,δ7.0000δ5.7595,δ4.7483,δ2.2405,δ5.0000,δ1.6472,δ6.0000δ6.2664,δ5.7595,δ1.8232,δ6.0000,δ1.8983,δ5.2619δ6.4308,δ6.0000,δ1.5692,δ6.0000,δ1.5692,δ6.0000δ6.1017,δ6.7858,δ1.6472,δ5.0586,δ1.9725,δ5.0000δ6.7076,δ5.3969,δ2.3050,δ5.7014,δ2.0423,δ6.6119δ6.4524,δ5.3220,δ1.7578,δ5.9195,δ2.3050,δ6.0000δ5.0000,δ5.4179,δ1.9980,δ5.3199,δ1.0000,δ6.2646.Step 2: by equation (18), we obtain the aggregated LNZNs of lzj for Nj (j = 1, 2, …, m): lz1 = <(δ5.7769, s5.9550), (δ1.5877, s5.4988), (δ1.7679, s5.9889)>, lz2 = <(δ6.4423, s6.0917), (δ1.8486, s5.7688), (δ1.6372, s5.5662)>, lz3 = <(δ5.9368, s5.8558), (δ1.6832, s5.5011), (δ1.7857, s5.8611)>, and lz4 = <(δ6.1127, s5.8027), (δ1.4296, s5.8980), (δ1.7936, s5.7792)>, or by equation (19), we obtain the aggregated LNZNs of lzj for Nj (j = 1, 2, …, m): lz1 = <(δ5.7177, s5.8316), (δ1.6841, s5.5431), (δ1.7888, s6.1666)>, lz2 = <(δ6.4270, s6.0216), (δ1.8726, s5.8403), (δ1.7604, s5.7607)>, lz3 = <(δ5.8174, s5.7156), (δ1.7300, s5.5426), (δ1.8309, s5.9682)>, and lz4 = <(δ5.9907, s5.7334), (δ1.4983, s5.9404), (δ1.9052, s5.8523)>.Step 3: by equation (5), we obtain the score values: Y(lz1) = 0.7452, Y(lz2) = 0.7681, Y(lz3) = 0.7450, and Y(lz4) = 0.7535, or Y(lz1) = 0.7343, Y(lz2) = 0.7585, Y(lz3) = 0.7330, and Y(lz4) = 0.7411.Step 4: the sorting order of the four alternatives isN2≻f N4≻N1≻N3, and then the best one is N2.Sorting orders of the four alternatives regarding the developed MAGDM approach using the LNZNWAM and LNNWGM operators are shown in Figure1. It is obvious that the sorting orders of the alternatives and the best one in terms of the LNZNWAM and LNNWGM operators are identical.Figure 1
Sorting orders of the four alternatives regarding the developed MAGDM approach using the LNZNWAM and LNNWGM operators.


## 6.2. Comparison with Related Methods

This part compares the proposed MAGDM approach with the related LNN and NZN decision-making approaches [12,20] to indicate the suitability and efficiency of the proposed MAGDM approach.To conveniently compare the proposed MAGDM approach with the existing related MAGDM approach [12] for the robot selection problem, we only use LNNs in M1, M2, and M3 without considering the linguistic reliability measures in LNZNs since the linguistic neutrosophic MAGDM approach [12] cannot handle such an MAGDM problem with the information of LNZNs. As a special case, the above LNZN decision matrices of the three decision-makers are reduced to the following LNN decision matrices:(22)M′1=δ6,δ1,δ2δ7,δ1,δ2δ5,δ2,δ2δ6,δ2,δ2δ7,δ1,δ1δ7,δ3,δ2δ7,δ3,δ2δ6,δ2,δ2δ6,δ2,δ2δ6,δ1,δ1δ6,δ2,δ1δ6,δ2,δ3δ7,δ1,δ2δ7,δ1,δ3δ7,δ2,δ2δ5,δ1,δ1,M′2=δ5,δ1,δ2δ6,δ3,δ2δ6,δ2,δ2δ6,δ2,δ2δ6,δ3,δ1δ6,δ2,δ3δ6,δ1,δ3δ7,δ1,δ3δ5,δ2,δ2δ5,δ2,δ1δ5,δ2,δ4δ7,δ3,δ1δ6,δ1,δ2δ7,δ1,δ2δ6,δ1,δ3δ5,δ3,δ1,M′3=δ5,δ1,δ1δ6,δ1,δ3δ5,δ2,δ1δ5,δ3,δ1δ7,δ2,δ1δ5,δ2,δ3δ5,δ2,δ1δ6,δ2,δ2δ4,δ1,δ2δ5,δ1,δ2δ7,δ1,δ2δ7,δ2,δ3δ3,δ1,δ3δ5,δ2,δ1δ6,δ3,δ2δ5,δ3,δ1.Thus, we utilize the existing MAGDM approach using the LNNWAM and LNNWGM operators of equations (1) and (2) and the score function of equation (3) [12] for this MAGDM example in the setting of LNNs.First, by equation (1), we obtain the following aggregated matrix:(23)M′=δ5.4179,δ1.0000,δ1.6472δ6.4524,δ1.4689,δ2.2405δ5.3969,δ2.0000,δ1.6472δ5.7595,δ2.2405,δ1.6472δ6.7254,δ1.7835,δ1.0000δ6.2664,δ2.3237,δ2.5821δ6.2664,δ1.8232,δ1.8983δ6.4308,δ1.5692,δ1.5692δ5.2013,δ1.6472,δ2.0000δ5.4179,δ1.2746,δ1.2142δ6.1017,δ1.6472,δ1.9725δ6.7076,δ2.3050,δ2.0423δ5.9998,δ1.0000,δ2.2405δ6.6398,δ1.2142,δ1.9138δ6.4524,δ1.7578,δ2.3050δ5.0000,δ1.9980,δ1.0000.Then, by equations (1)–(4), the decision results are shown in Table 1. For the convenient comparison, the decision results corresponding to the proposed MAGDM approach are also contained in Table 1.Table 1
Decision results of different MAGDM methods.
MAGDM methodScore valueSorting orderThe best oneExisting MAGDM method using equation (1) [12]0.7676, 0.7899, 0.7695, 0.7871N2≻N4≻N3≻N1N2Existing MAGDM method using equation (2) [12]0.7602, 0.7831, 0.7607, 0.7745N2≻N4≻N3≻N1N2Proposed MAGDM method using equation (18)0.7452, 0.7681, 0.7450, 0.7535N2≻N4≻N1≻N3N2Proposed MAGDM method using equation (19)0.7343, 0.7585, 0.7330, 0.7411N2≻N4≻N1≻N3N2In Table1, there is the sorting difference between the existing MAGDM method [12] and the proposed MAGDM method; then, the best one is the same. However, this sorting difference reflects that the new method utilizes the LNZN information constructed by the truth, falsehood, and indeterminacy linguistic Z-numbers, while the existing method [12] only contains the LNN information without containing their reliability measures. Moreover, LNZNs imply much more useful information than LNNs and strengthen the reliability of LNNs. Thus, different MAGDM information and methods may affect the sorting order of alternatives, which illustrates the efficiency and applicability of the new MAGDM method in the LNZN environment. Therefore, the new MAGDM method is superior to the existing one [12].Furthermore, the existing decision-making method in the setting of NZNs [20] cannot carry out such a linguistic decision-making issue with the LNZN information. On the contrary, the new MAGDM approach with the LNZN information especially suits such an MAGDM issue with the LNZN information. Then, in the MAGDM problem with qualitative attributes, the new MAGDM approach shows its main merit since it contains the LNN assessments related to their reliability measures in the LNZN environment.However, the original study reveals the following main advantages:(a)
The proposed LNZN information can express more useful information to strengthen the reliability measure of LNNs and avoid the insufficiency of missing reliability measures in the existing methods(b)
The developed MAGDM approach not only makes the MAGDM process more reliable and reasonable but also provides a new way for linguistic MAGDM problems in the LNZN setting(c)
The proposed MAGDM approach can effectively solve the MAGDM issue of selecting industrial robots with the evaluation information of LNZNs and make the decision result more reliable


## 7. Conclusion

To make the LNN information more reliable, this study proposed an LNZN notion in terms of the truth, falsehood, and indeterminacy linguistic Z-numbers as a new linguistic neutrosophic framework. Then, the proposed operational relations and score and accuracy functions of LNZNs are to realize reasonable operations and sorting rules of LNZNs in the setting of LNZNs. The proposed LNZNWAM and LNZNWGM operators provided useful information aggregation tools in MAGDM problems with the LNZN information. Then, the established MAGDM approach in terms of the proposed LNZNWAM and LNZNWGM operators and score and accuracy functions can solve MAGDM problems under the environment of LNZNs. Through the application of the proposed MAGDM method in the industrial robot selection problem and the comparison of existing decision-making methods, the decision results demonstrated not only the suitability and efficiency of the proposed MAGDM approach in the LNZN setting but also the superiority of the new MAGDM method over the existing ones.However, the limitations of this study lie in the lack of flexible decision-making methods and quantitative algorithms for reliability measures of LNNs. To overcome the limitations, we shall further study new aggregation operators with a changeable parameter, flexible MAGDM methods, and some quantitative algorithms of the reliability measures. Then, we shall use them in engineering areas such as environmental risk assessment and management, slope stability/risk assessment, and construction engineering management in the LNZN environment.



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*Source: 9509823-2022-03-20.xml*