# Multiple Criteria Decision Making Approach with Multivalued Neutrosophic Linguistic Normalized Weighted Bonferroni Mean Hamacher Operator

Authors: Bao-lin Li; Juan-ru Wang; Li-hua Yang; Xue-tao Li
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The neutrosophic set and linguistic term set are widely applied in recent years. Motivated by the advantages of them, we combine the multivalued neutrosophic set and linguistic set and define the concept of the multivalued neutrosophic linguistic set (MVNLS). Furthermore, Hamacher operation is an extension of the algebraic and Einstein operation. Additionally, the normalized weighted Bonferroni mean (NWBM) operator can consider the weight of each argument and capture the interrelationship of different arguments. Therefore, the combination of NWBM operator and Hamacher operation is more valuable and agile. Firstly, MVNLS and multivalued neutrosophic linguistic number (MVNLN) are defined, then some new operational rules of MVNLNs on account of Hamacher operations are developed, and the comparison functions for MVNLNs are given. Secondly, multivalued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher operator (MVNLNWBMH) is proposed, and a number of expected characteristics of new operator are investigated. Meanwhile, some special cases of different parametersp,q, and ε are analyzed. Thirdly, the approach utilizing the MVNLNWBMH operator is introduced to manage multiple criteria decision making (MCDM) issue in multivalued neutrosophic linguistic environment. Ultimately, a practical example is presented and a comparative analysis is carried out, which validate the effectiveness and generalization of the novel approach.


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## Body

## 1. Introduction

In real world, due to the complexity of decision information, the fuzzy theory has attracted widespread attention and has been developed in various fields. Zadeh [1] firstly proposed the notion of fuzzy sets (FSs). Then, Atanassov [2] introduced the intuitionistic fuzzy sets (IFSs), which overcome the weakness of nonmembership degrees. Subsequently, in order to address the hesitation degree of decision makers, Torra [3] defined hesitant fuzzy sets (HFSs). Fuzzy set theory has been well promoted, but it still cannot manage the inconsistent and indeterminate information. Under this circumstance, Smarandache [4] proposed neutrosophic sets (NSs), whose indeterminacy degree is independent of both true and false membership. NS is an extension of IFS and makes decision makers express their preference more accurately, so some achievements on NSs and its extensions have been undertaken. Some various concepts of different NSs are defined. For example, Smarandache [5] and Wang et al. [6] introduced single-valued neutrosophic sets (SVNs) to facilitate its application. Ye [7] pointed out the concept of simplified neutrosophic sets (SNSs). Wang et al. [8] developed the concept of interval neutrosophic sets (INSs). However, under certain conditions, the decision makers likely give different evaluation numbers for expressing their hesitancy. Subsequently, the definition of single-valued neutrosophic hesitant fuzzy sets (SVNHFSs) was firstly proposed by Ye [9] in 2014, and then Wang and Li [10] also proposed multivalued neutrosophic sets (MVNSs) in 2015. Actually, the notions of SVNHFSs and MVNSs are equal. For simplicity, we adapt the term of MVNSs in this paper.On the other hand, the aggregation operators, comparison method for neutrosophic numbers, have also been studied. For SVNSs, Liu and Wang [11] employed NWBM operator to solve multiple criteria problem in single-valued neutrosophic environment. Ye [12] gave the definitions of cross-entropy and correlation coefficient. For INSs, Zhang et al. [13] developed some aggregation operators. Liu and Shi [14] not only provided the definition of interval neutrosophic hesitant fuzzy sets (INHFSs), but also discussed the generalized hybrid weighted average operator. Broumi and Smarandache [15–17] studied the correlation coefficients, cosine similarity measure, and some new operations. Ye [18] proposed similarity measures between interval neutrosophic sets. For MVNSs, Ye [9] developed SVNHFWA and SVNHFWG operators for MCDM problem. Peng et al. [19, 20] extended power aggregation operators and defined some outranking relations under MVNS environment. Ji et al. [21] analyzed a novel TODIM method for MVNSs.In real life, owing to the ambiguity of decision makers’ thinking, people prefer to utilize linguistic variables for describing their assessment value rather than the quantization value. Therefore, linguistic variable has attracted widespread attention in the field of MCDM. The linguistic variable was firstly proposed by Zadeh [22] and applied for the fuzzy reasoning. After that, a series of works on it have been made. Wang et al. [23–25] presented a new approach in view of hesitant fuzzy linguistic information. Meng et al. [26] developed linguistic hesitant fuzzy sets and studied hybrid weighted operator. Tian et al. [27] defined gray linguistic weighted Bonferroni mean operator for MCDM.In order to indicate the true, indeterminate and false extents concerning a linguistic term, the NSs and linguistic set (LS) are combined. Several neutrosophic linguistic sets and their corresponding operators are defined, for example, single-valued or simplified neutrosophic linguistic sets and trapezoid linguistic sets [28–31], interval neutrosophic certain or uncertain linguistic sets [32–34]. However, due to the hesitancy of people’s thinking, the trueness of a linguistic term may be given several values, and the case is similar to the false and indeterminate extents. The existing literature does not consider this perspective. Therefore, the multivalued neutrosophic linguistic set (MVNLS) and multivalued neutrosophic linguistic number (MVNLN) in this article are proposed in order to better express the information.Aggregation operator which can fuse multiple arguments into a single comprehensive value is an important tool for MCDM problem. Many researchers have developed some efficient operators [35–42], for instance, the weighted geometric average (WGA) or averaging (WA) operator, prioritized aggregation (PA) operator, Maclaurin symmetric mean operator, and Bonferroni mean (BM) operator. BM operator was originally defined by Bonferroni [43] and has attracted widespread attention because of its characteristics of capturing interrelationship among arguments. Some achievements have been made on it [11, 44–49]. In order to aggregate neutrosophic linguistic information, some researches on aggregation operators under neutrosophic linguistic and neutrosophic uncertain linguistic environments have also been applied [28–34, 50]. Until now, BM and NWBM fail to accommodate aggregation information for multivalued neutrosophic linguistic environment. Motivated by this limitation, we will extend the NWBM operator to MVNLS in this article.T-norms and t-conorms are two functions that satisfy certain conditions, respectively. The Archimedean t-conorms and t-norms are well known, which include algebraic, Einstein, and Hamacher. Hamacher operation is an extension of algebraic and Einstein. Generally, the algebraic operators are common; there are also a few aggregation operations based on Einstein operations. Because Hamacher operator is more general, Liu et al. [51, 52] discussed the Hamacher operational rules. So far, there is no research for MVNLS based on Hamacher operations. Since it is better for MVNLS to depict the actual situation, NWBM operator can capture the interrelationship among arguments, and Hamacher operations are more general, it is of great meaning to study the NWBM Hamacher operators under multivalued neutrosophic linguistic environment for MCDM problems.The main purposes of the paper are presented as follows:(1)
To better express people’s hesitancy, combining the MVNS and LS, we give the notions of MVNLS and MVNLN; besides, the score, accuracy, and certainty functions are also investigated to compare MVNLNs.(2)
Due to the generalization of Hamacher operational rules, we define new operations of MVNLNs based on Hamacher operational rules and discuss their operational relations.(3)
The NWBM considering the interrelationship of different arguments has gained widespread concerns; we extend NWBM operator to MVNLN environment, the MVNLNWBMH operator is defined, and some desirable characteristics are also studied.(4)
In order to verify the effectiveness, an example for MCDM problem utilizing MVNLNWBMH operator is illustrated and a comparative analysis is conducted. We also analyze the influences of different parameter values for the final outcomes, and the results demonstrate that the operator proposed is more general and flexible.The article is arranged in this way. In Section2, we review a number of notions and operations for MVNS, LS, NWBM operator and Hamacher. In Section 3, we propose the definitions of MVNLS and MVNLN and develop the operations of MVNLNs on the basis of Hamacher t-conorms and t-norms. Meanwhile, the algebraic as well as Einstein operations for MVNLNs are also presented, which are special cases of Hamacher operation. Moreover, the comparison method of MVNLNs is also defined. In Section 4, we propose the MVNLNWBMH operator and investigate its properties. Furthermore, when corresponding parameters are assigned different values, the special examples are also discussed. In Section 5, we establish the MCDM procedure on account of the proposed aggregation operators with MVNLS information. Section 6 presents a concrete example, and a comparison analysis is provided to show the practicability of utilizing our method. Finally, in Section 7, some results are presented.


## 2. Preliminaries

Some notions and operation are introduced in this section, which will be useful in the latter analysis.


### 2.1. Linguistic Term Sets

Suppose thatS=s1,s2,…,sι is an ordered and finite linguistic set, in which sj denotes a linguistic variable value and ι is an odd value. When ι is equal to seven, the corresponding linguistic sets are provided as follows:(1)S=s1,s2,s3,s4,s5,s6,s7=extremely  poor,  very  poor,  poor,  medium,  good,  very  good,  extremely  good.In order to avoid the linguistic information loss, the set above is expanded, that is, a contiguous set, S-=sα∣α∈R.Definition 1 (see [53]).
Letsi and sj be any two linguistic variables, the corresponding operations are presented:(2)1λsi=sλ×i,λ≥0;2si⊕sj=si+j;3si⊗sj=si×j;4siλ=siλ.


### 2.2. Multivalued Neutrosophic Sets

Definition 2 (see [9, 10]).
Suppose thatX is a collection of objects; MVNSs A on X is defined by (3)A=x,T~Ax,I~Ax,F~Ax∣x∈X,where T~A(x)=γ∣γ∈T~A(x), I~A(x)=δ∣δ∈I~A(x), F~A(x)=η∣η∈F~A(x), and T~A(x), I~A(x), and F~A(x) are three collections of crisp numbers belonging to 0,1, representing the probable true-membership degree, indeterminacy-membership degree, and falsity-membership degree, where x in X belongs to A, respectively, satisfying these conditions 0≤γ,δ,η≤1, and 0≤sup⁡T~Ax+sup⁡I~Ax+sup⁡F~A(x)≤3. If there is only one element in X, A is indicated by the three-tuple A=T~A(x),I~A(x),F~A(x), that is, known as a multivalued neutrosophic number (MVNN). Generally, MVNSs are considered as the generalizations of the other sets, such as FSs, IFSs, HFSs, DHFs, and SVNSs.


### 2.3. Normalized Weighted Bonferroni Mean

Definition 3 (see [43]).
Letp,q≥0 as well as ai(i=1,2,…,n) be a set of nonnegative values; then the BM is defined as(4)BMp,qa1,a2,…,an=1nn-1∑i,j=1,i≠jnaipajq1/p+q.Definition 4 (see [46]).
Letp,q≥0 and ai(i=1,2,…,n) be a set of nonnegative values, and the corresponding NWBM can be expressed as follows:(5)NWBMp,qa1,a2,…,an=⨁i,j=1,i≠jnwiwj1-wiaip⊗ajq1/p+q,where w=(w1,w2,…,wn) represents the corresponding weighted vector of aii=1,2,…,n, satisfying wi≥0 and ∑i=1nwi=1. The weight vector can be given by decision makers in real problem.
Obviously, the NWBM operator possesses a few characteristics such as commutativity, reducibility, monotonicity, boundedness, and idempotency.


### 2.4. Hamacher Operations

We know aggregation operator is given in accordance with different t-norms and t-conorms; there are some exceptional circumstances listed as follows:(1)
algebraic t-norm and t-conorm(6)a⊗b=ab,a⊕b=a+b-ab;(2)
Einstein t-norm and t-conorm(7)a⊗b=ab1+1-a×1-b,a⊕b=a+b1+ab;(3)
Hamacher t-norm and t-conorm(8)a⊗b=abε+1-εa+b-ab,a⊕b=a+b-ab-1-εab1-1-εab,ε>0.In particular, whenε=1, ε=2, the algebraic and Einstein operations are the simplifications of Hamacher t-norm and t-conorm.


## 2.1. Linguistic Term Sets

Suppose thatS=s1,s2,…,sι is an ordered and finite linguistic set, in which sj denotes a linguistic variable value and ι is an odd value. When ι is equal to seven, the corresponding linguistic sets are provided as follows:(1)S=s1,s2,s3,s4,s5,s6,s7=extremely  poor,  very  poor,  poor,  medium,  good,  very  good,  extremely  good.In order to avoid the linguistic information loss, the set above is expanded, that is, a contiguous set, S-=sα∣α∈R.Definition 1 (see [53]).
Letsi and sj be any two linguistic variables, the corresponding operations are presented:(2)1λsi=sλ×i,λ≥0;2si⊕sj=si+j;3si⊗sj=si×j;4siλ=siλ.


## 2.2. Multivalued Neutrosophic Sets

Definition 2 (see [9, 10]).
Suppose thatX is a collection of objects; MVNSs A on X is defined by (3)A=x,T~Ax,I~Ax,F~Ax∣x∈X,where T~A(x)=γ∣γ∈T~A(x), I~A(x)=δ∣δ∈I~A(x), F~A(x)=η∣η∈F~A(x), and T~A(x), I~A(x), and F~A(x) are three collections of crisp numbers belonging to 0,1, representing the probable true-membership degree, indeterminacy-membership degree, and falsity-membership degree, where x in X belongs to A, respectively, satisfying these conditions 0≤γ,δ,η≤1, and 0≤sup⁡T~Ax+sup⁡I~Ax+sup⁡F~A(x)≤3. If there is only one element in X, A is indicated by the three-tuple A=T~A(x),I~A(x),F~A(x), that is, known as a multivalued neutrosophic number (MVNN). Generally, MVNSs are considered as the generalizations of the other sets, such as FSs, IFSs, HFSs, DHFs, and SVNSs.


## 2.3. Normalized Weighted Bonferroni Mean

Definition 3 (see [43]).
Letp,q≥0 as well as ai(i=1,2,…,n) be a set of nonnegative values; then the BM is defined as(4)BMp,qa1,a2,…,an=1nn-1∑i,j=1,i≠jnaipajq1/p+q.Definition 4 (see [46]).
Letp,q≥0 and ai(i=1,2,…,n) be a set of nonnegative values, and the corresponding NWBM can be expressed as follows:(5)NWBMp,qa1,a2,…,an=⨁i,j=1,i≠jnwiwj1-wiaip⊗ajq1/p+q,where w=(w1,w2,…,wn) represents the corresponding weighted vector of aii=1,2,…,n, satisfying wi≥0 and ∑i=1nwi=1. The weight vector can be given by decision makers in real problem.
Obviously, the NWBM operator possesses a few characteristics such as commutativity, reducibility, monotonicity, boundedness, and idempotency.


## 2.4. Hamacher Operations

We know aggregation operator is given in accordance with different t-norms and t-conorms; there are some exceptional circumstances listed as follows:(1)
algebraic t-norm and t-conorm(6)a⊗b=ab,a⊕b=a+b-ab;(2)
Einstein t-norm and t-conorm(7)a⊗b=ab1+1-a×1-b,a⊕b=a+b1+ab;(3)
Hamacher t-norm and t-conorm(8)a⊗b=abε+1-εa+b-ab,a⊕b=a+b-ab-1-εab1-1-εab,ε>0.In particular, whenε=1, ε=2, the algebraic and Einstein operations are the simplifications of Hamacher t-norm and t-conorm.


## 3. Multivalued Neutrosophic Linguistic Set

### 3.1. MVNLS and Its Hamacher Operations

Definition 5.
LetX be a set of points; an MVNLS A in X is defined as follows:(9)A=x,sθx,T~Ax,I~Ax,F~Ax∣x∈X,where sθ(x)∈S, T~A(x)=γ∣γ∈T~A(x), I~A(x)=δ∣δ∈I~A(x), F~A(x)=η∣η∈F~A(x), and T~A(x), I~A(x), and F~A(x) are three sets of crisp values in 0,1, denoting three degrees of x in X belonging to sθ(x), which are trueness, indeterminacy, and falsity, satisfying these conditions 0≤γ,δ,η≤1, and 0≤sup⁡T~Ax+sup⁡I~Ax+sup⁡F~A(x)≤1.Definition 6.
LetA=x,sθ(x),T~A(x),I~A(x),F~A(x)∣x∈X be an MVNLS; supposing there is only one element in X, then tuple sθ(x),T~A(x),I~A(x),F~A(x) is depicted as a multivalued neutrosophic linguistic number (MVNLN). For simplicity, the MVNLN can also be represented as(10)A=sθx,T~Ax,I~Ax,F~Ax∣x∈X.Definition 7.
Leta1=sθ(a1),T~(a1),I~(a1),F~(a1) and a2=sθ(a2),T~(a2),I~(a2),F~(a2) be two MVNLNs, and λ>0; then the operations of MVNLNs can be defined on the basis of Hamacher operations.(11)1a1⊕a2=sθa1+θa2,⋃γ1∈T~a1,γ2∈T~a2γ1+γ2-γ1γ2-1-εγ1γ21-1-εγ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1δ2ε+1-εδ1+δ2-δ1δ2,⋃η1∈F~a1,η2∈F~a2η1η2ε+1-εη1+η2-η1η2;2a1⊗a2=sθa1×θa2,⋃γ1∈T~a1,γ2∈T~a2γ1γ2ε+1-εγ1+γ2-γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1+δ2-δ1δ2-1-εδ1δ21-1-εδ1δ2,⋃η1∈F~a1,η2∈F~a2η1+η2-η1η2-1-εη1η21-1-εη1η2;3λa1=sλθa1,⋃γ1∈T~a11+ε-1γ1λ-1-γ1λ1+ε-1γ1λ+ε-11-γ1λ,⋃δ1∈I~a1εδ1λ1+ε-11-δ1λ+ε-1δ1λ,⋃η1∈F~a1εη1λ1+ε-11-η1λ+ε-1η1λ;4a1λ=sθλa1,⋃γ1∈T~a1εγ1λ1+ε-11-γ1λ+ε-1γ1λ,⋃δ1∈I~a11+ε-1δ1λ-1-δ1λ1+ε-1δ1λ+ε-11-δ1λ,⋃η1∈F~a11+ε-1η1λ-1-η1λ1+ε-1η1λ+ε-11-η1λ.
Ifε=1, then the operations based on Hamacher operational rules in Definition 7 will be simplified to the Algebraic operational rules as follows:(12)5a1⊕a2=sθa1+θa2,⋃γ1∈T~a1,γ2∈T~a2γ1+γ2-γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1δ2,⋃η1∈F~a1,η2∈F~a2η1η2;6a1⊗a2=sθa1×θa2,⋃γ1∈T~a1,γ2∈T~a2γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1+δ2-δ1δ2,⋃η1∈F~a1,η2∈F~a2η1+η2-η1η2;7λa1=sλθa1,⋃γ1∈T~a11-1-γ1λ,⋃δ1∈I~a1δ1λ,⋃η1∈F~a1η1λ;8a1λ=sθλa1,⋃γ1∈T~a1γ1λ,⋃δ1∈I~a11-1-δ1λ,⋃η1∈F~a11-1-η1λ.
SupposingT~(a1), I~(a1), F~(a1), T~(a2), I~(a2), and F~(a2) contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on algebraic operations proposed by Ye [28].
Ifε=2, then the operations based on Hamacher operational rules in Definition 7 will be simplified to the Einstein operations of MVNLNs presented below:(13)9a1⊕a2=sθa1+θa2,⋃γ1∈T~a1,γ2∈T~a2γ1+γ21+γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1δ22-δ1-δ2+δ1δ2,⋃η1∈F~a1,η2∈F~a2η1η22-η1-η2+η1η2;10a1⊗a2=sθa1×θa2,⋃γ1∈T~a1,γ2∈T~a2γ1γ22-γ1-γ2+γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1+δ21+δ1δ2,⋃η1∈F~a1,η2∈F~a2η1+η21+η1η2;11λa1=sλθa1,⋃γ1∈T~a11+γ1λ-1-γ1λ1+γ1λ+1-γ1λ,⋃δ1∈I~a12δ1λ2-δ1λ+δ1λ,⋃η1∈F~a12η1λ2-η1λ+η1λ;12a1λ=sθλa1,⋃γ1∈T~a12γ1λ2-γ1λ+γ1λ,⋃δ1∈I~a11+δ1λ-1-δ1λ1+δ1λ+1-δ1λ,⋃η1∈F~a11+η1λ-1-η1λ1+η1λ+1-η1λ.
SupposingT~(a1), I~(a1), F~(a1), T~(a2), I~(a2), and F~(a2) contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on Einstein operations.Theorem 8.
Leta1=sθ(a1),T~(a1),I~(a1),F~(a1), a2=sθ(a2),T~(a2),I~(a2),F~(a2), and a3=sθ(a3),T~(a3),I~(a3),F~(a3) be any three MVNLNs, and λ,λ1,λ2>0; then the properties below are correct:(14)1a1⊕a2=a2⊕a1;2a1⊗a2=a2⊗a1;3λa1⊕a2=λa1⊕λa2;4λ1a1⊕λ2a1=λ1+λ2a1;5a1λ1⊗a1λ2=a1λ1+λ2;6a1λ⊗a2λ=a1⊗a2λ;7a1⊕a2⊕a3=a1⊕a2⊕a3;8a1⊗a2⊗a3=a1⊗a2⊗a3.Then, (4) will be proved as follows.Proof of (4).
Sinceλ1,λ2>0, (15)λ1a1⊕λ2a1=sλ1θa1+λ2θa1,⋃γ1∈T~a11+ε-1γ1λ11+ε-1γ1λ1⋃γ1∈T~a11+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1+1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1-ε1+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2·1-1-ε1+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1,⋃δ1∈I~a1εδ1λ11+ε-11-δ1λ1+ε-1δ1λ1·εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2-1·εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2·ε+1-εεδ1λ11+ε-11-δ1λ1+ε-1δ1λ1+εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2-εδ1λ11+ε-11-δ1λ1+ε-1δ1λ1·εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2-1,⋃η1∈F~a1εη1λ11+ε-11-η1λ1+ε-1η1λ1·εη1λ21+ε-11-η1λ2+ε-1η1λ2·εη1λ21+ε-11-η1λ2+ε-1η1λ2·ε+1-εεη1λ11+ε-11-η1λ1+ε-1η1λ1+εη1λ21+ε-11-η1λ2+ε-1η1λ2-εη1λ11+ε-11-η1λ1+ε-1η1λ1·εη1λ21+ε-11-η1λ2+ε-1η1λ2-1=sλ1+λ2θa1,⋃η1∈F~a1εη1λ1+λ21+ε-11-η1λ1+λ2+ε-1η1λ1+λ2⋃γ1∈T~a11+ε-1γ1λ1+λ2-1-γ1λ1+λ21+ε-1γ1λ1+λ2+ε-11-γ1λ1+λ2,⋃δ1∈I~a1εδ1λ1+λ21+ε-11-δ1λ1+λ2+ε-1δ1λ1+λ2,⋃η1∈F~a1εη1λ1+λ21+ε-11-η1λ1+λ2+ε-1η1λ1+λ2=λ1+λ2a1.Therefore, (4)λ1a1⊕λ2a1=(λ1+λ2)a1 can be obtained.Similarly, the other equations in Theorem8 are easily certified in the light of Definition 7.


### 3.2. Comparison Method

The score, accuracy, and certainty functions are important indexes to rank MVNLNs, and their corresponding definition is given below.Definition 9.
Leta=sθ(a),T~(a),I~(a),F~(a) be an MVNLN, and the score, accuracy, and certainty functions are achieved as below.(16)1Ea=1ιT~aιI~aιF~a∑γ∈T~a,δ∈I~a,η∈F~aγ+1-δ+1-η3sθa=s1/ιT~aιI~aιF~a∑γ∈T~a,δ∈I~a,η∈F~aγ+1-δ+1-η/3θa,2Ha=1ιT~aιF~a∑γ∈T~a,η∈F~aγ-ηsθa=s1/ιT~aιF~a∑γ∈T~a,η∈F~aγ-ηθa3Ca=1ιT~a∑γ∈T~aγsθa=s1/ιT~a∑γ∈T~aγθa,where ιT~(a), ιI~(a), and ιF~(a) are the numbers of the values in T~A(x), I~A(x), and F~A(x), respectively.
The linguistic variablesθ(a) is important for an MVNLN. Therefore, the comparison functions defined above in Definition 9 are denoted as the linguistic variable. The bigger the truth degree T~(a) concerning the variable sθ(a) is, the smaller the indeterminacy degree I~(a) and the false degree F~(a) concerning the linguistic variable sθ(a) are, and the higher the MVNLN is. Regarding the function of score, the greater the γ-δ-η corresponding to sθ(a) is, the higher the affirmative statement is. Regarding the function of accuracy, the greater the γ minus η is, the more certain the statement is. Regarding the function of certainty, the bigger the γ is, the more certain the statement is.Based on Definition9, the comparison method between MVNLNs is obtained.Definition 10.
Supposinga1 and a2 are two MVNLNs, the compared approach is achieved as follows:(1)
Supposing thatE(a1)>E(a2), then a1 is greater than a2, represented as a1≻a2.(2)
Supposing thatE(a1)=E(a2), and H(a1)>H(a2), then a1 is greater than a2, represented as a1≻a2.(3)
Supposing thatE(a1)=E(a2), H(a1)=H(a2), and C(a1)>C(a2), then a1 is greater than a2, represented as a1≻a2.(4)
Supposing thatE(a1)=E(a2), H(a1)=H(a2), and C(a1)=C(a2), then a1 equals a2, represented as a1~a2.


## 3.1. MVNLS and Its Hamacher Operations

Definition 5.
LetX be a set of points; an MVNLS A in X is defined as follows:(9)A=x,sθx,T~Ax,I~Ax,F~Ax∣x∈X,where sθ(x)∈S, T~A(x)=γ∣γ∈T~A(x), I~A(x)=δ∣δ∈I~A(x), F~A(x)=η∣η∈F~A(x), and T~A(x), I~A(x), and F~A(x) are three sets of crisp values in 0,1, denoting three degrees of x in X belonging to sθ(x), which are trueness, indeterminacy, and falsity, satisfying these conditions 0≤γ,δ,η≤1, and 0≤sup⁡T~Ax+sup⁡I~Ax+sup⁡F~A(x)≤1.Definition 6.
LetA=x,sθ(x),T~A(x),I~A(x),F~A(x)∣x∈X be an MVNLS; supposing there is only one element in X, then tuple sθ(x),T~A(x),I~A(x),F~A(x) is depicted as a multivalued neutrosophic linguistic number (MVNLN). For simplicity, the MVNLN can also be represented as(10)A=sθx,T~Ax,I~Ax,F~Ax∣x∈X.Definition 7.
Leta1=sθ(a1),T~(a1),I~(a1),F~(a1) and a2=sθ(a2),T~(a2),I~(a2),F~(a2) be two MVNLNs, and λ>0; then the operations of MVNLNs can be defined on the basis of Hamacher operations.(11)1a1⊕a2=sθa1+θa2,⋃γ1∈T~a1,γ2∈T~a2γ1+γ2-γ1γ2-1-εγ1γ21-1-εγ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1δ2ε+1-εδ1+δ2-δ1δ2,⋃η1∈F~a1,η2∈F~a2η1η2ε+1-εη1+η2-η1η2;2a1⊗a2=sθa1×θa2,⋃γ1∈T~a1,γ2∈T~a2γ1γ2ε+1-εγ1+γ2-γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1+δ2-δ1δ2-1-εδ1δ21-1-εδ1δ2,⋃η1∈F~a1,η2∈F~a2η1+η2-η1η2-1-εη1η21-1-εη1η2;3λa1=sλθa1,⋃γ1∈T~a11+ε-1γ1λ-1-γ1λ1+ε-1γ1λ+ε-11-γ1λ,⋃δ1∈I~a1εδ1λ1+ε-11-δ1λ+ε-1δ1λ,⋃η1∈F~a1εη1λ1+ε-11-η1λ+ε-1η1λ;4a1λ=sθλa1,⋃γ1∈T~a1εγ1λ1+ε-11-γ1λ+ε-1γ1λ,⋃δ1∈I~a11+ε-1δ1λ-1-δ1λ1+ε-1δ1λ+ε-11-δ1λ,⋃η1∈F~a11+ε-1η1λ-1-η1λ1+ε-1η1λ+ε-11-η1λ.
Ifε=1, then the operations based on Hamacher operational rules in Definition 7 will be simplified to the Algebraic operational rules as follows:(12)5a1⊕a2=sθa1+θa2,⋃γ1∈T~a1,γ2∈T~a2γ1+γ2-γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1δ2,⋃η1∈F~a1,η2∈F~a2η1η2;6a1⊗a2=sθa1×θa2,⋃γ1∈T~a1,γ2∈T~a2γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1+δ2-δ1δ2,⋃η1∈F~a1,η2∈F~a2η1+η2-η1η2;7λa1=sλθa1,⋃γ1∈T~a11-1-γ1λ,⋃δ1∈I~a1δ1λ,⋃η1∈F~a1η1λ;8a1λ=sθλa1,⋃γ1∈T~a1γ1λ,⋃δ1∈I~a11-1-δ1λ,⋃η1∈F~a11-1-η1λ.
SupposingT~(a1), I~(a1), F~(a1), T~(a2), I~(a2), and F~(a2) contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on algebraic operations proposed by Ye [28].
Ifε=2, then the operations based on Hamacher operational rules in Definition 7 will be simplified to the Einstein operations of MVNLNs presented below:(13)9a1⊕a2=sθa1+θa2,⋃γ1∈T~a1,γ2∈T~a2γ1+γ21+γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1δ22-δ1-δ2+δ1δ2,⋃η1∈F~a1,η2∈F~a2η1η22-η1-η2+η1η2;10a1⊗a2=sθa1×θa2,⋃γ1∈T~a1,γ2∈T~a2γ1γ22-γ1-γ2+γ1γ2,⋃δ1∈I~a1,δ2∈I~a2δ1+δ21+δ1δ2,⋃η1∈F~a1,η2∈F~a2η1+η21+η1η2;11λa1=sλθa1,⋃γ1∈T~a11+γ1λ-1-γ1λ1+γ1λ+1-γ1λ,⋃δ1∈I~a12δ1λ2-δ1λ+δ1λ,⋃η1∈F~a12η1λ2-η1λ+η1λ;12a1λ=sθλa1,⋃γ1∈T~a12γ1λ2-γ1λ+γ1λ,⋃δ1∈I~a11+δ1λ-1-δ1λ1+δ1λ+1-δ1λ,⋃η1∈F~a11+η1λ-1-η1λ1+η1λ+1-η1λ.
SupposingT~(a1), I~(a1), F~(a1), T~(a2), I~(a2), and F~(a2) contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on Einstein operations.Theorem 8.
Leta1=sθ(a1),T~(a1),I~(a1),F~(a1), a2=sθ(a2),T~(a2),I~(a2),F~(a2), and a3=sθ(a3),T~(a3),I~(a3),F~(a3) be any three MVNLNs, and λ,λ1,λ2>0; then the properties below are correct:(14)1a1⊕a2=a2⊕a1;2a1⊗a2=a2⊗a1;3λa1⊕a2=λa1⊕λa2;4λ1a1⊕λ2a1=λ1+λ2a1;5a1λ1⊗a1λ2=a1λ1+λ2;6a1λ⊗a2λ=a1⊗a2λ;7a1⊕a2⊕a3=a1⊕a2⊕a3;8a1⊗a2⊗a3=a1⊗a2⊗a3.Then, (4) will be proved as follows.Proof of (4).
Sinceλ1,λ2>0, (15)λ1a1⊕λ2a1=sλ1θa1+λ2θa1,⋃γ1∈T~a11+ε-1γ1λ11+ε-1γ1λ1⋃γ1∈T~a11+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1+1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1-ε1+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2·1-1-ε1+ε-1γ1λ1-1-γ1λ11+ε-1γ1λ1+ε-11-γ1λ1·1+ε-1γ1λ2-1-γ1λ21+ε-1γ1λ2+ε-11-γ1λ2-1,⋃δ1∈I~a1εδ1λ11+ε-11-δ1λ1+ε-1δ1λ1·εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2-1·εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2·ε+1-εεδ1λ11+ε-11-δ1λ1+ε-1δ1λ1+εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2-εδ1λ11+ε-11-δ1λ1+ε-1δ1λ1·εδ1λ21+ε-11-δ1λ2+ε-1δ1λ2-1,⋃η1∈F~a1εη1λ11+ε-11-η1λ1+ε-1η1λ1·εη1λ21+ε-11-η1λ2+ε-1η1λ2·εη1λ21+ε-11-η1λ2+ε-1η1λ2·ε+1-εεη1λ11+ε-11-η1λ1+ε-1η1λ1+εη1λ21+ε-11-η1λ2+ε-1η1λ2-εη1λ11+ε-11-η1λ1+ε-1η1λ1·εη1λ21+ε-11-η1λ2+ε-1η1λ2-1=sλ1+λ2θa1,⋃η1∈F~a1εη1λ1+λ21+ε-11-η1λ1+λ2+ε-1η1λ1+λ2⋃γ1∈T~a11+ε-1γ1λ1+λ2-1-γ1λ1+λ21+ε-1γ1λ1+λ2+ε-11-γ1λ1+λ2,⋃δ1∈I~a1εδ1λ1+λ21+ε-11-δ1λ1+λ2+ε-1δ1λ1+λ2,⋃η1∈F~a1εη1λ1+λ21+ε-11-η1λ1+λ2+ε-1η1λ1+λ2=λ1+λ2a1.Therefore, (4)λ1a1⊕λ2a1=(λ1+λ2)a1 can be obtained.Similarly, the other equations in Theorem8 are easily certified in the light of Definition 7.


## 3.2. Comparison Method

The score, accuracy, and certainty functions are important indexes to rank MVNLNs, and their corresponding definition is given below.Definition 9.
Leta=sθ(a),T~(a),I~(a),F~(a) be an MVNLN, and the score, accuracy, and certainty functions are achieved as below.(16)1Ea=1ιT~aιI~aιF~a∑γ∈T~a,δ∈I~a,η∈F~aγ+1-δ+1-η3sθa=s1/ιT~aιI~aιF~a∑γ∈T~a,δ∈I~a,η∈F~aγ+1-δ+1-η/3θa,2Ha=1ιT~aιF~a∑γ∈T~a,η∈F~aγ-ηsθa=s1/ιT~aιF~a∑γ∈T~a,η∈F~aγ-ηθa3Ca=1ιT~a∑γ∈T~aγsθa=s1/ιT~a∑γ∈T~aγθa,where ιT~(a), ιI~(a), and ιF~(a) are the numbers of the values in T~A(x), I~A(x), and F~A(x), respectively.
The linguistic variablesθ(a) is important for an MVNLN. Therefore, the comparison functions defined above in Definition 9 are denoted as the linguistic variable. The bigger the truth degree T~(a) concerning the variable sθ(a) is, the smaller the indeterminacy degree I~(a) and the false degree F~(a) concerning the linguistic variable sθ(a) are, and the higher the MVNLN is. Regarding the function of score, the greater the γ-δ-η corresponding to sθ(a) is, the higher the affirmative statement is. Regarding the function of accuracy, the greater the γ minus η is, the more certain the statement is. Regarding the function of certainty, the bigger the γ is, the more certain the statement is.Based on Definition9, the comparison method between MVNLNs is obtained.Definition 10.
Supposinga1 and a2 are two MVNLNs, the compared approach is achieved as follows:(1)
Supposing thatE(a1)>E(a2), then a1 is greater than a2, represented as a1≻a2.(2)
Supposing thatE(a1)=E(a2), and H(a1)>H(a2), then a1 is greater than a2, represented as a1≻a2.(3)
Supposing thatE(a1)=E(a2), H(a1)=H(a2), and C(a1)>C(a2), then a1 is greater than a2, represented as a1≻a2.(4)
Supposing thatE(a1)=E(a2), H(a1)=H(a2), and C(a1)=C(a2), then a1 equals a2, represented as a1~a2.


## 4. The Multivalued Neutrosophic Linguistic Normalized Weighted Bonferroni Mean Hamacher Operator

The NWBM operator not only can take into account the advantages of BM and WBM, but also has the property of reducibility and idempotency. However, the NWBM operator has not been applied to the cases where the input arguments are MVNLNs.Definition 11.
Letai(i=1,2,…,n) be a space of MVNLNs, ai=sθ(ai),T~(ai),I~(ai),F~(ai),p,q≥0, and ω=ω1,ω2,…,ωn be the weighted vector for ai,ωi∈0,1 and ∑i=1nωi=1. Then the operator of MVNLNWBMH is achieved as below, the aggregation result is still an MVNLN. (17)MVNLNWBMHa1,a2,…,an=⨁i,j=1i≠jnωiωj1-ωiaip⊗ajq1/p+q.
According to the operational laws in Definition7, the results are derived below:(18)MVNLNWBMHa1,a2,…,an=s∑i,j=1i≠jnωiωj/1-ωi·θpai·θqaj1/p+q,⋃γi∈T~ai,γj∈T~ajεx-yx+ε-1y1/p+q1+ε-11-x-yx+ε-1y1/p+q+ε-1x-yx+ε-1y1/p+q-1,⋃δi∈I~ai,δj∈I~aj1+ε-1hg+ε-1h1/p+q-1-hg+ε-1h1/p+q1+ε-1hg+ε-1h1/p+q+ε-11-hg+ε-1h1/p+q-1,⋃ηi∈F~ai,ηj∈F~aj1+ε-1uv+ε-1u1/p+q-1-uv+ε-1u1/p+q1+ε-1uv+ε-1u1/p+q+ε-11-uv+ε-1u1/p+q-1=s∑i,j=1i≠jnωiωj/1-ωi·θpai·θqaj1/p+q,⋃γi∈T~ai,γj∈T~ajεx-y1/p+qx+ε2-1y1/p+q+ε-1x-y1/p+q,⋃δi∈I~ai,δj∈I~ajg+ε2-1h1/p+q-g-h1/p+qg+ε2-1h1/p+q+ε-1g-h1/p+q,⋃ηi∈F~ai,ηj∈F~ajv+ε2-1u1/p+q-v-u1/p+qv+ε2-1u1/p+q+ε-1v-u1/p+q,where(19)x=∏i,j=1i≠jnε-ε-1γipε-ε-1γjq+ε2-1γipγjqωiωj/1-ωi,y=∏i,j=1i≠jnε-ε-1γipε-ε-1γjq-γipγjqωiωj/1-ωi,g=∏i,j=1i≠jn1+ε-1δip1+ε-1δjq+ε2-11-δip1-δjqωiωj/1-ωi,h=∏i,j=1i≠jn1+ε-1δip1+ε-1δjq-1-δip1-δjqωiωj/1-ωi,v=∏i,j=1i≠jn1+ε-1ηip1+ε-1ηjq+ε2-11-ηip1-ηjqωiωj/1-ωi,u=∏i,j=1i≠jn1+ε-1ηip1+ε-1ηjq-1-ηip1-ηjqωiωj/1-ωi.Proof.
According to the operational rules for MVNLNs, the results below can be gained(20)aip=sθpai,⋃γi∈T~aiε·γip1+ε-11-γip+ε-1γip,⋃δi∈I~ai1+ε-1δip-1-δip1+ε-1δip+ε-11-δip,⋃ηi∈F~ai1+ε-1ηip-1-ηip1+ε-1ηip+ε-11-ηip,ajq=sθqaj,⋃γj∈T~ajε·γjq1+ε-11-γjq+ε-1γjq,⋃δj∈I~aj1+ε-1δjq-1-δjq1+ε-1δjq+ε-11-δjq,⋃ηj∈F~aj1+ε-1ηjq-1-ηjq1+ε-1ηjq+ε-11-ηjq,aip⊗ajq=sθpaiθqaj,⋃γi∈T~ai,γj∈T~ajεγipγjqε-ε-1γip·ε-ε-1γjq+ε-1γipγjq,⋃δi∈I~ai,δj∈I~aj1+ε-1δip·1+ε-1δjq-1-δip1-δjq1+ε-1δip·1+ε-1δjq+ε-11-δip1-δjq,⋃ηi∈F~ai,ηj∈F~aj1+ε-1ηip·1+ε-1ηjq-1-ηip1-ηjq1+ε-1ηip·1+ε-1ηjq+ε-11-ηip1-ηjq.
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The mathematical induction onn is adopted to prove (21).
( 1 ) Supposing n=2, the equation below is obtained. (22)⨁i,j=1i≠j2ωiωj1-ωiaip⊗ajq=ω1ω21-ω1a1p⊗a2q⊕ω2ω11-ω2a2p⊗a1q=sω1ω2/1-ω1·θpa1·θqa2+ω2ω1/1-ω2·θpa2·θqa1,⋃γi∈T~ai,γj∈T~ajx1x2-y1y2x1x2+ε-1y1y2,⋃δi∈I~ai,δj∈I~ajεm1m2z1z2+ε-1m1m2,⋃ηi∈F~ai,ηj∈F~ajεv1v2u1u2+ε-1v1v2,where (23)x1=1+ε-1·εγ1pγ2qε-ε-1γ1p·ε-ε-1γ2q+ε-1γ1pγ2qω1ω2/1-ω1,x2=1+ε-1·εγ2pγ1qε-ε-1γ2p·ε-ε-1γ1q+ε-1γ2pγ1qω2ω1/1-ω2,y1=1-εγ1pγ2qε-ε-1γ1p·ε-ε-1γ2q+ε-1γ1pγ2qω1ω2/1-ω1,y2=1-εγ2pγ1qε-ε-1γ2p·ε-ε-1γ1q+ε-1γ2pγ1qω2ω1/1-ω2,m1=1+ε-1δ1p·1+ε-1δ2q-1-δ1p1-δ2q1+ε-1δ1p·1+ε-1δ2q+ε-11-δ1p1-δ2qω1ω2/1-ω1,m2=1+ε-1δ2p·1+ε-1δ1q-1-δ2p1-δ1q1+ε-1δ2p·1+ε-1δ1q+ε-11-δ2p1-δ1qω2ω1/1-ω2,z1=1+ε-11-1+ε-1δ1p·1+ε-1δ2q-1-δ1p1-δ2q1+ε-1δ1p·1+ε-1δ2q+ε-11-δ1p1-δ2qω1ω2/1-ω1,z2=1+ε-11-1+ε-1δ2p·1+ε-1δ1q-1-δ2p1-δ1q1+ε-1δ2p·1+ε-1δ1q+ε-11-δ2p1-δ1qω2ω1/1-ω2,v1=1+ε-1η1p·1+ε-1η2q-1-η1p1-η2q1+ε-1η1p·1+ε-1η2q+ε-11-η1p1-η2qω1ω2/1-ω1,v2=1+ε-1η2p·1+ε-1η1q-1-η2p1-η1q1+ε-1η2p·1+ε-1η1q+ε-11-η2p1-η1qω2ω1/1-ω2,u1=1+ε-11-1+ε-1η1p·1+ε-1η2q-1-η1p1-η2q1+ε-1η1p·1+ε-1η2q+ε-11-η1p1-η2qω1ω2/1-ω1,u2=1+ε-11-1+ε-1η2p·1+ε-1η1q-1-η2p1-η1q1+ε-1η2p·1+ε-1η1q+ε-11-η2p1-η1qω2ω1/1-ω2,
and then,(24)
⨁
i
,
j
=
1
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≠
j
2
ω
i
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-
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i
a
i
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⊗
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j
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ω
1
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1
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⊗
a
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⋃
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∈
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~
a
i
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η
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∈
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~
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j
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∏
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,
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=
1
i
≠
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1
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1
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ω
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1
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ω
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·
∏
i
,
j
=
1
i
≠
j
2
1
+
ε
-
1
η
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p
1
+
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-
1
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+
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2
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1
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1
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i
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1
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+
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-
1
∏
i
,
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=
1
i
≠
j
2
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
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q
-
ε
∏
i
,
j
=
1
i
≠
j
2
1
+
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-
1
η
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p
1
+
ε
-
1
η
j
q
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
-
1
.We can make (21) right when n=k.
( 2 ) Supposing n=k, the equation below is right; then(25)
⨁
i
,
j
=
1
i
≠
j
k
ω
i
ω
j
1
-
ω
i
a
i
p
⊗
a
j
q
=
s
∑
i
,
j
=
1
i
≠
j
k
ω
i
ω
j
/
1
-
ω
i
·
θ
p
a
i
·
θ
q
a
j
,
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
⋃
γ
i
∈
T
~
a
i
,
γ
j
∈
T
~
a
j
∏
i
,
j
=
1
i
≠
j
k
ε
-
ε
-
1
γ
i
p
·
ε
-
ε
-
1
γ
j
q
+
ε
2
-
1
γ
i
p
γ
j
q
ω
i
ω
j
/
1
-
ω
i
-
∏
i
,
j
=
1
i
≠
j
k
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
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γ
j
q
-
∏
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1
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≠
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1
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+
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1
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∏
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=
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≠
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≠
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1
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-
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-
∏
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,
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=
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≠
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1
+
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1
+
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1
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p
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ω
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ω
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/
1
-
ω
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-
1
,
⋃
δ
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∈
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~
a
i
,
δ
j
∈
I
~
a
j
ε
∏
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,
j
=
1
i
≠
j
k
1
+
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-
1
δ
i
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·
1
+
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-
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ω
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/
1
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·
∏
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,
j
=
1
i
≠
j
k
1
+
ε
-
1
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i
p
1
+
ε
-
1
δ
j
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+
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2
-
1
1
-
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1
-
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q
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i
ω
j
/
1
-
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i
+
ε
-
1
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
δ
i
p
1
+
ε
-
1
δ
j
q
-
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
1
-
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1
-
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j
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ω
i
ω
j
/
1
-
ω
i
-
1
,
⋃
η
i
∈
F
~
a
i
,
η
j
∈
F
~
a
j
ε
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
η
i
p
·
1
+
ε
-
1
η
j
q
-
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
·
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
+
ε
2
-
1
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
+
ε
-
1
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
-
∏
i
,
j
=
1
i
≠
j
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
-
1
.
Whenn=k+1, we need to calculate the equation below. (26)⨁i,j=1i≠jk+1ωiωj1-ωiaip⊗ajq=⨁i,j=1i≠jkωiωj1-ωiaip⊗ajq⊕⨁i=1kωiωk+11-ωiaip⊗ak+1q⊕⨁j=1kωk+1ωj1-ωk+1ak+1p⊗ajq.The mathematical induction on k is used to testify the equations below.(27)
⨁
i
=
1
k
ω
i
ω
k
+
1
1
-
ω
i
a
i
p
⊗
a
k
+
1
q
=
s
∑
i
=
1
k
ω
i
ω
k
+
1
/
1
-
ω
i
·
θ
p
a
i
·
θ
q
a
k
+
1
,
∏
i
=
1
k
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
k
+
1
q
⋃
γ
i
∈
T
~
a
i
,
γ
k
+
1
∈
T
~
a
k
+
1
∏
i
=
1
k
ε
-
ε
-
1
γ
i
p
·
ε
-
ε
-
1
γ
k
+
1
q
+
ε
2
-
1
γ
i
p
γ
k
+
1
q
ω
i
ω
k
+
1
/
1
-
ω
i
-
∏
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=
1
k
ε
-
ε
-
1
γ
i
p
ε
-
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-
1
γ
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+
1
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∏
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=
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+
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+
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1
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·
∏
i
=
1
k
ε
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-
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+
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+
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+
1
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+
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1
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∏
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=
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+
1
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=
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+
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+
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+
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1
-
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-
1
,
⋃
δ
i
∈
I
~
a
i
,
δ
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+
1
∈
I
~
a
k
+
1
ε
∏
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=
1
k
1
+
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-
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i
p
·
1
+
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-
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+
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+
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i
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+
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1
-
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i
·
∏
i
=
1
k
1
+
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-
1
δ
i
p
1
+
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-
1
δ
k
+
1
q
+
ε
2
-
1
1
-
δ
i
p
1
-
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k
+
1
q
ω
i
ω
k
+
1
/
1
-
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i
+
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-
1
∏
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=
1
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1
+
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-
1
δ
i
p
1
+
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-
1
δ
k
+
1
q
-
∏
i
=
1
k
ε
-
ε
-
1
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i
p
ε
-
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-
1
γ
k
+
1
q
1
-
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i
p
1
-
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k
+
1
q
ω
i
ω
k
+
1
/
1
-
ω
i
-
1
,
⋃
η
i
∈
F
~
a
i
,
η
k
+
1
∈
F
~
a
k
+
1
ε
∏
i
=
1
k
1
+
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-
1
η
i
p
·
1
+
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-
1
η
k
+
1
q
-
1
-
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i
p
1
-
η
k
+
1
q
ω
i
ω
k
+
1
/
1
-
ω
i
·
∏
i
=
1
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
k
+
1
q
+
ε
2
-
1
1
-
η
i
p
1
-
η
k
+
1
q
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i
ω
k
+
1
/
1
-
ω
i
+
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-
1
∏
i
=
1
k
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
k
+
1
q
-
∏
i
=
1
k
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
k
+
1
q
1
-
η
i
p
1
-
η
k
+
1
q
ω
i
ω
k
+
1
/
1
-
ω
i
-
1
,
⨁
j
=
1
k
ω
k
+
1
ω
j
1
-
ω
k
+
1
a
k
+
1
p
⊗
a
j
q
=
s
∑
j
=
1
k
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
·
θ
p
a
k
+
1
·
θ
q
a
j
,
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
⋃
γ
k
+
1
∈
T
~
a
k
+
1
,
γ
j
∈
T
~
a
j
∏
j
=
1
k
ε
-
ε
-
1
γ
k
+
1
p
·
ε
-
ε
-
1
γ
j
q
+
ε
2
-
1
γ
k
+
1
p
γ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
-
∏
j
=
1
k
ε
-
ε
-
1
γ
k
+
1
p
ε
-
ε
-
1
γ
j
q
-
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
γ
k
+
1
p
γ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
·
∏
j
=
1
k
ε
-
ε
-
1
γ
k
+
1
p
ε
-
ε
-
1
γ
j
q
+
ε
2
-
1
γ
k
+
1
p
γ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
+
ε
-
1
∏
j
=
1
k
ε
-
ε
-
1
γ
k
+
1
p
ε
-
ε
-
1
γ
j
q
-
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
γ
k
+
1
p
γ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
-
1
,
⋃
δ
k
+
1
∈
I
~
a
k
+
1
,
δ
j
∈
I
~
a
j
ε
∏
j
=
1
k
1
+
ε
-
1
δ
k
+
1
p
·
1
+
ε
-
1
δ
j
q
-
1
-
δ
k
+
1
p
1
-
δ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
·
∏
j
=
1
k
1
+
ε
-
1
δ
k
+
1
p
1
+
ε
-
1
δ
j
q
+
ε
2
-
1
1
-
δ
k
+
1
p
1
-
δ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
+
ε
-
1
∏
j
=
1
k
1
+
ε
-
1
δ
k
+
1
p
1
+
ε
-
1
δ
j
q
-
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
1
-
δ
k
+
1
p
1
-
δ
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
-
1
,
⋃
η
k
+
1
∈
F
~
a
k
+
1
,
η
j
∈
F
~
a
j
ε
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
·
1
+
ε
-
1
η
j
q
-
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
1
-
η
k
+
1
p
1
-
η
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
·
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
+
ε
2
-
1
1
-
η
k
+
1
p
1
-
η
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
+
ε
-
1
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
-
∏
j
=
1
k
1
+
ε
-
1
η
k
+
1
p
1
+
ε
-
1
η
j
q
1
-
η
k
+
1
p
1
-
η
j
q
ω
k
+
1
ω
j
/
1
-
ω
k
+
1
-
1
.Therefore,(28)
⨁
i
,
j
=
1
i
≠
j
k
+
1
ω
i
ω
j
1
-
ω
i
a
i
p
⊗
a
j
q
=
⨁
i
,
j
=
1
i
≠
j
k
ω
i
ω
j
1
-
ω
i
a
i
p
⊗
a
j
q
⊕
⨁
i
=
1
k
ω
i
ω
k
+
1
1
-
ω
i
a
i
p
⊗
a
k
+
1
q
⊕
⨁
j
=
1
k
ω
k
+
1
ω
j
1
-
ω
k
+
1
a
k
+
1
p
⊗
a
j
q
=
s
∑
i
,
j
=
1
i
≠
j
k
+
1
ω
i
ω
j
/
1
-
ω
i
·
θ
p
a
i
·
θ
q
a
j
,
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
⋃
γ
i
∈
T
~
a
i
,
γ
j
∈
T
~
a
j
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
·
ε
-
ε
-
1
γ
j
q
+
ε
2
-
1
γ
i
p
γ
j
q
ω
i
ω
j
/
1
-
ω
i
-
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
-
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
γ
i
p
γ
j
q
ω
i
ω
j
/
1
-
ω
i
·
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
+
ε
2
-
1
γ
i
p
γ
j
q
ω
i
ω
j
/
1
-
ω
i
+
ε
-
1
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
-
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
γ
i
p
γ
j
q
ω
i
ω
j
/
1
-
ω
i
-
1
,
⋃
δ
i
∈
I
~
a
i
,
δ
j
∈
I
~
a
j
ε
∏
i
,
j
=
1
i
≠
j
k
+
1
1
+
ε
-
1
δ
i
p
·
1
+
ε
-
1
δ
j
q
-
1
-
δ
i
p
1
-
δ
j
q
ω
i
ω
j
/
1
-
ω
i
·
∏
i
,
j
=
1
i
≠
j
k
+
1
1
+
ε
-
1
δ
i
p
1
+
ε
-
1
δ
j
q
+
ε
2
-
1
1
-
δ
i
p
1
-
δ
j
q
ω
i
ω
j
/
1
-
ω
i
+
ε
-
1
∏
i
,
j
=
1
i
≠
j
k
+
1
1
+
ε
-
1
δ
i
p
1
+
ε
-
1
δ
j
q
-
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
1
-
δ
i
p
1
-
δ
j
q
ω
i
ω
j
/
1
-
ω
i
-
1
,
⋃
η
i
∈
F
~
a
i
,
η
j
∈
F
~
a
j
ε
∏
i
,
j
=
1
i
≠
j
k
+
1
1
+
ε
-
1
η
i
p
·
1
+
ε
-
1
η
j
q
-
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
·
∏
i
,
j
=
1
i
≠
j
k
+
1
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
+
ε
2
-
1
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
+
ε
-
1
∏
i
,
j
=
1
i
≠
j
k
+
1
1
+
ε
-
1
η
i
p
1
+
ε
-
1
η
j
q
-
∏
i
,
j
=
1
i
≠
j
k
+
1
ε
-
ε
-
1
γ
i
p
ε
-
ε
-
1
γ
j
q
1
-
η
i
p
1
-
η
j
q
ω
i
ω
j
/
1
-
ω
i
-
1
.
That is, Ifn=k+1, (21) is right. Therefore, for all n, (21) is right.
Then, (18) is right.In the following, the properties of MVNLNWBMH operator will be proved.(  1) Reducibility. Let ω=1/n,1/n,…,1/n. Then MVNLNWBMH(a1,a2,…,an)=MVNLBMH(a1,a2,…,an).Proof.
Sinceω=1/n,1/n,…,1/n, then according to the operations in Definition 11, the result below can be obtained.(29)MVNLNWBMHa1,a2,…,an=⨁i,j=1i≠jnωiωj1-ωiaip⊗ajq1/p+q=⨁i,j=1i≠jn1/n21-1/naip⊗ajq1/p+q=⨁i,j=1i≠jn1nn-1aip⊗ajq1/p+q=1nn-1⨁i,j=1i≠jnaip⊗ajq1/p+q=MVNLBMHa1,a2,…,an.(  2) Idempotency. Let ai=ai=1,2,…,n. Then MVNLNWBMH(a1,a2,…,an)=a.Proof.
For eachi, owing to ai=a, the formula below is obtained on the basis of (5) in Theorem 8. (30)MVNLNWBMHa1,a2,…,an=⨁i,j=1i≠jnωiωj1-ωiaip⊗ajq1/p+q=⨁i,j=1i≠jnωiωj1-ωiap⊗aq1/p+q=⨁i,j=1i≠jnωiωj1-ωiap+q1/p+q=a⨁i,j=1i≠jnωiωj1-ωi1/p+q=a.(  3) Commutativity. Let a~1,a~2,…,a~n be any permutation of a1,a2,…,an. Then(31)MVNLNWBMHa~1,a~2,…,a~n=MVNLNWBMHa1,a2,…,an.Proof.
Asa~1,a~2,…,a~n is permutation of a1,a2,…,an, then the equation below can be obtained. (32)MVNLNWBMHa1,a2,…,an=⨁i,j=1i≠jnωiωj1-ωiaip⊗ajq1/p+q=⨁i,j=1i≠jnωiωj1-ωia~ip⊗a~jq1/p+q.(  4) Monotonicity. Suppose ai=sθ(ai),T~(ai),I~(ai),F~(ai)(i=1,2,…,n) and bi=sθ(bi),T~(bi),I~(bi),F~(bi)(i=1,2,…,n) are two sets of MVNLNs; when sθ(ai)≥sθ(bi), T~(ai)≥T~(bi), I~(ai)≤I~(bi), and F~(ai)≤F~(bi) for each i, then MVNLNWBMH(a1,a2,…,an)≥MVNLNWBMH(b1,b2,…,bn).Proof.
(I) Linguistic Term Part.As p,q≥0, and sθ(ai)≥sθ(bi) for each i, the result below is gained.(33)θpai≥θpbi,θqai≥θqbi⇓θpaiθqai≥θpbiθqbi⇓ωiωj1-ωiθpaiθqai≥ωiωj1-ωiθpbiθqbi⇓∑i,j=1i≠jnωiωj1-ωiθpaiθqai≥∑i,j=1i≠jnωiωj1-ωiθpbiθqbi⇓∑i,j=1i≠jnωiωj1-ωiθpaiθqai1/p+q≥∑i,j=1i≠jnωiωj1-ωiθpbiθqbi1/p+q.
(II) True-, Indeterminacy-, and Falsity-Membership Parts. As T~(ai)≥T~(bi), I~(ai)≤I~(bi), and F~(ai)≤F~(bi) for each i,p,q≥0, then the following results can be proved easily.(34)εxai-yai1/p+qxai+ε2-1yai1/p+q+ε-1xai-yai1/p+q≥εxbi-ybi1/p+qxbi+ε2-1ybi1/p+q+ε-1xbi-ybi1/p+q;gai+ε2-1hai1/p+q-gai-hai1/p+qgai+ε2-1hai1/p+q+ε-1gai-hai1/p+q≤gbi+ε2-1hbi1/p+q-gbi-hbi1/p+qgbi+ε2-1hbi1/p+q+ε-1gbi-hbi1/p+q;vai+ε2-1uai1/p+q-vai-uai1/p+qvai+ε2-1uai1/p+q+ε-1vai-uai1/p+q≤vbi+ε2-1ubi1/p+q-vbi-ubi1/p+qvbi+ε2-1ubi1/p+q+ε-1vbi-ubi1/p+q,where the corresponding x, y, g, h,u, and v are defined in Definition 11.
(III) Comparing M V N L N W B M H ( a 1 , a 2 , … , a n ) with M V N L N W B M H ( b 1 , b 2 , … , b n ). Suppose a=sθ(a),T~(a),I~(a),F~(a)=MVNLNWBMH(a1,a2,…,an) and b=sθ(b),T~(b),I~(b),F~(b)=MVNLNWBMH(b1,b2,…,bn).
Becausesθ(a)≥sθ(b), T~(a)≥T~(b), I~(a)≤I~(b), and F~(a)≤F~(b), then a≥b.
ThenMVNLNWBMH(a1,a2,…,an)≥MVNLNWBMH(b1,b2,…,bn).In the following, a few special examples of MVNLNWBMH operator regarding different valuesε, p, and q will be explored.( 1 ) If q=0, then the MVNLNWBMH operator defined by (18) will be reduced to the generalized multivalued neutrosophic linguistic Hamacher weighted average (GMVNLHWA) operator shown as below.(35)
GMVNLHWA
a
1
,
a
2
,
…
,
a
n
=
⨁
i
,
j
=
1
i
≠
j
n
ω
i
ω
j
1
-
ω
i
a
i
p
⊗
a
j
q
1
/
p
+
q
=
⨁
i
,
j
=
1
i
≠
j
n
ω
i
ω
j
1
-
ω
i
a
i
p
1
/
p
=
⨁
i
=
1
n
ω
i
1
-
ω
i
1
-
ω
i
a
i
p
1
/
p
=
⨁
i
=
1
n
ω
i
a
i
p
1
/
p
=
s
∑
i
=
1
n
ω
i
·
θ
p
a
i
1
/
p
,
v
′
+
ε
2
-
1
u
′
1
/
p
-
v
′
-
u
′
1
/
p
v
′
+
ε
2
-
1
u
′
1
/
p
+
ε
-
1
v
′
-
u
′
1
/
p
⋃
γ
i
∈
T
~
a
i
,
γ
j
∈
T
~
a
j
ε
x
′
-
y
′
1
/
p
x
′
+
ε
2
-
1
y
′
1
/
p
+
ε
-
1
x
′
-
y
′
1
/
p
,
⋃
δ
i
∈
I
~
a
i
,
δ
j
∈
I
~
a
j
g
′
+
ε
2
-
1
h
′
1
/
p
-
g
′
-
h
′
1
/
p
g
′
+
ε
2
-
1
h
′
1
/
p
+
ε
-
1
g
′
-
h
′
1
/
p
,
⋃
η
i
∈
F
~
a
i
,
η
j
∈
F
~
a
j
v
′
+
ε
2
-
1
u
′
1
/
p
-
v
′
-
u
′
1
/
p
v
′
+
ε
2
-
1
u
′
1
/
p
+
ε
-
1
v
′
-
u
′
1
/
p
,where (36)x′=∏i=1n1+ε-11-γip+ε2-1γipωi,y′=∏i=1nε-ε-1γip-γipωi,h′=∏i=1n1+ε-1δip-1-δipωi,g′=∏i=1n1+ε-1δip+ε2-11-δipωi,u′=∏i=1n1+ε-1ηip-1-ηipωi,v′=∏i=1n1+ε-1ηip+ε2-11-ηipωi.( 2 ) When p=1, q=0, then the operator of MVNLNWBMH defined by (18) will be reduced to the multivalued neutrosophic linguistic Hamacher weighted arithmetic average (MVNLHWAA) operator shown as below.(37)MVNLHWAAa1,a2,…,an=⨁i=1nωiai=s∑i=1nωiθai,⋃γi∈T~ai∏i=1n1+ε-1γiωi-∏i=1n1-γiωi∏i=1n1+ε-1γiωi+ε-1∏i=1n1-γiωi,⋃δi∈I~aiε∏i=1nδiωiε-1∏i=1nδiωi+∏i=1nε-ε-1δiωi,⋃ηi∈F~aiε∏i=1nηiωiε-1∏i=1nηiωi+∏i=1nε-ε-1ηiωi.Ifε=1, the MVNLHWAA operator will be reduced to MVNLWAA operator shown as below.(38)MVNLWAAa1,a2,…,an=s∑i=1nωiθai,⋃γi∈T~ai1-∏i=1n1-γiωi,⋃δi∈I~ai∏i=1nδiωi,⋃ηi∈F~ai∏i=1nηiωi.Ifε=2, the MVNLHWAA operator will be reduced to the multivalued neutrosophic linguistic Einstein weighted arithmetic average (MVNLEWAA) operator shown as follows:(39)MVNLEWAAa1,a2,…,an=s∑i=1nωiθai,⋃γi∈T~ai∏i=1n1+γiωi-∏i=1n1-γiωi∏i=1n1+γiωi+∏i=1n1-γiωi,⋃δi∈I~ai2∏i=1nδiωi∏i=1n2-δiωi+∏i=1nδiωi,⋃ηi∈F~ai2∏i=1nηiωi∏i=1n2-ηiωi+∏i=1nηiωi.( 3 ) If p→0, q=0, then the MVNLNWBMH operator defined by (18) will be reduced to the multivalued neutrosophic linguistic Hamacher weighted geometric average (MVNLHWGA) operator shown as below.(40)MVNLHWGAa1,a2,…,an=⨂i=1naiωi=s∏i=1nθωiai,⋃γi∈T~aiε∏i=1nγiωiε-1∏i=1nγiωi+∏i=1nε-ε-1γiωi,⋃δi∈I~ai∏i=1n1+ε-1δiωi-∏i=1n1-δiωi∏i=1n1+ε-1δiωi+ε-1∏i=1n1-δiωi,⋃ηi∈F~ai∏i=1n1+ε-1ηiωi-∏i=1n1-ηiωi∏i=1n1+ε-1ηiωi+ε-1∏i=1n1-ηiωi.Ifε=1, the MVNLHWGA operator will be reduced to MVNLWGA operator shown as below.(41)MVNLWGAa1,a2,…,an=s∏i=1nθωiai,⋃γi∈T~ai∏i=1nγiωi,⋃δi∈I~ai1-∏i=1n1-δiωi,⋃ηi∈F~ai1-∏i=1n1-ηiωi.Ifε=2, the MVNLHWGA operator will be reduced to the multivalued neutrosophic linguistic Einstein weighted geometric average (MVNLEWGA) operator shown as below.(42)MVNLEWGAa1,a2,…,an=s∏i=1nθωiai,⋃γi∈T~ai2∏i=1nγiωi∏i=1n2-γiωi+∏i=1nγiωi,⋃δi∈I~ai∏i=1n1+δiωi-∏i=1n1-δiωi∏i=1n1+δiωi+∏i=1n1-δiωi,⋃ηi∈F~ai∏i=1n1+ηiωi-∏i=1n1-ηiωi∏i=1n1+ηiωi+∏i=1n1-ηiωi.( 4 ) If ε=1, then the MVNLNWBMH operator defined by (18) will be reduced to the MVNLNWBM operator shown as below.(43)MVNLNWBMa1,a2,…,an=s∑i,j=1i≠jnωiωj/1-ωi·θpai·θqaj1/p+q,⋃γi∈T~ai,γj∈T~aj1-∏i,j=1i≠jn1-γipγjqωiωj/1-ωi1/p+q,⋃δi∈I~ai,δj∈I~aj1-1-∏i,j=1i≠jn1-1-δip1-δjqωiωj/1-ωi1/p+q,⋃ηi∈F~ai,ηj∈F~aj1-1-∏i,j=1i≠jn1-1-ηip1-ηjqωiωj/1-ωi1/p+q.( 5 ) If ε=2, then the MVNLNWBMH operator defined by (18) will be reduced to the MVNLNWBME, that is the simplification of multivalued neutrosophic linguistic NWBM Einstein.(44)MVNLNWBMEa1,a2,…,an=s∑i,j=1i≠jnωiωj/1-ωi·θpai·θqaj1/p+q,⋃γi∈T~ai,γj∈T~aj2x-y1/p+qx+3y1/p+q+x-y1/p+q,⋃δi∈I~ai,δj∈I~ajg+3h1/p+q-g-h1/p+qg+3h1/p+q+g-h1/p+q,⋃ηi∈F~ai,ηj∈F~ajv+3u1/p+q-v-u1/p+qv+3u1/p+q+v-u1/p+q.where (45)x=∏i,j=1i≠jn2-γip2-γjq+3γipγjqωiωj/1-ωi,y=∏i,j=1i≠jn2-γip2-γjq-γipγjqωiωj/1-ωi,h=∏i,j=1i≠jn1+δip1+δjq-1-δip1-δjqωiωj/1-ωi,g=∏i,j=1i≠jn1+δip1+δjq+31-δip1-δjqωiωj/1-ωi,u=∏i,j=1i≠jn1+ηip1+ηjq-1-ηip1-ηjqωiωj/1-ωi,v=∏i,j=1i≠jn1+ηip1+ηjq+31-ηip1-ηjqωiωj/1-ωi.From the above analysis, we can obtain that the MVNLNWBMH operator is more generalized.


## 5. The Multiple Criteria Decision Making Approach Based on the MVNLNWBMH Operator

The proposed MVNLNWBMH operator is presented to cope with MCDM problem under multivalued neutrosophic linguistic environment in this subsection.Suppose thatA=A1,A2,…,Am represent m alternatives and C=C1,C2,…,Cn represent n criteria. Let ω=ω1,ω2,…,ωn be the corresponding weights of criteria, where wj≥0(j=1,2,…,n) and ∑j=1nwj=1. The evaluation value of the criteria Cj(j=1,2,…,n) regarding the alternative Ai(i=1,2,…,m) is provided by experts. Each value is represented by MVNLNNs. Suppose that R=aijm×n is the multivalued neutrosophic linguistic decision matrix, where aij=sθ(aij),T~(aij),I~(aij),F~(aij) is the evaluation information which represents the assessment value of alternative Ai(i=1,2,…,m) on criteria Cj(j=1,2,…,n) with respect to the linguistic value sθ(aij), where T~(aij) indicates the satisfaction degree, I~(aij) indicates the indeterminacy degree, and F~(aij) indicates the dissatisfaction degree.Then, the main method for ranking and selecting the best alternative is presented below.Step 1 (the decision matrix is normalized).
Generally, criteria in MCDM problems consist of two types: maximum type and minimum type; the minimum type should be transformed into the maximum type for eliminating the influence of distinguished types. Suppose thatR=aijm×n is the original decision matrix, which can be normalized as follows:(46)bij=aij,for  maximizing  criteriasι-θaij,T~aij,I~aij,F~aij,for  maximizing  criteria.
Thus, the normalized matrixB=bijm×n  is gained.Step 2 (the comprehensive value of each alternative is calculated).
The comprehensive value represented byai  (i=1,2,…,m) can be obtained by utilizing the MVNLNWBMH operator in Definition 11, which can aggregate the overall value for each alternative with respect to all criteria.Step 3 (the compared values of three functions are calculated).
According to the equations given in Definition9, the score value denoted by E(ai), the accuracy value denoted by H(ai), and the certainty value denoted by C(ai) can be obtained.Step 4 (the alternatives are selected).
Based on Definition10, all alternatives Ai(i=1,2,…,m) can be ranked on the basis of E(ai), H(ai), and C(ai), and the best alternative(s) can be selected.


## 6. A Numerical Example

In order to validate the effectiveness and practicality of the novel approach, an investment project is adapted from Ye.An investment company wants to expand its business. Four alternatives will be chosen:A1 represents auto-corporation, A2 represents food corporation, A3 represents computer company corporation, A4 represents weapon corporation. Each alternative is evaluated under three criteria: C1 denotes risk, C2 denotes growth, and C3 denotes the impact of environment, where C3 is the minimizing criteria. The corresponding weighted vector is ω=0.35,0.25,0.4. In real situation, the decision maker may hesitate and give several possible values for the satisfaction, indeterminacy, and dissatisfaction regarding the alternative Ai corresponding to the criteria Cj under the linguistic term set S. Therefore, the assessment value is given in the form of MVNLNs, and the linguistic term set is employed as S=s1,s2,s3,s4,s5,s6,s7={extremelypoor,verypoor,poor,medium,good,very good,extremelygood}.The multivalued neutrosophic linguistic decision matrixR=aij4×3 is shown as follows:(47)R=aij4×3=s5,0.3,0.4,0.5,0.1,0.3,0.4s6,0.5,0.6,0.2,0.3,0.3,0.4s5,0.2,0.3,0.1,0.2,0.5,0.6s6,0.6,0.7,0.1,0.2,0.2,0.3s5,0.6,0.7,0.1,0.3s5,0.6,0.7,0.1,0.2,0.1,0.2s6,0.5,0.6,0.4,0.2,0.3s5,0.6,0.3,0.4s4,0.5,0.6,0.1,0.3s4,0.7,0.8,0.1,0.1,0.2s4,0.6,0.7,0.1,0.2s6,0.3,0.5,0.2,0.1,0.2,0.3.


### 6.1. The Procedure Using the Proposed Aggregation Operator

Step 1 (the decision matrix is normalized).
BecauseC3 is the minimizing criteria, which should be converted to the maximizing criteria, then the normalized decision matrix B=bijm×n  can be obtained as follows: (48)B=bij4×3=s5,0.3,0.4,0.5,0.1,0.3,0.4s6,0.5,0.6,0.2,0.3,0.3,0.4s2,0.2,0.3,0.1,0.2,0.5,0.6s6,0.6,0.7,0.1,0.2,0.2,0.3s5,0.6,0.7,0.1,0.3s2,0.6,0.7,0.1,0.2,0.1,0.2s6,0.5,0.6,0.4,0.2,0.3s5,0.6,0.3,0.4s3,0.5,0.6,0.1,0.3s4,0.7,0.8,0.1,0.1,0.2s4,0.6,0.7,0.1,0.2s1,0.3,0.5,0.2,0.1,0.2,0.3.Step 2 (the comprehensive value of each alternative is calculated).
Derive the comprehensive valueai(i=1,2,…,m) of each alternative Ai(i=1,2,…,m) by using the MVNLNWBMH operator presented in Definition 11. Here let p=q=1, ε=1. The MVNLNWBMH operator is shown as below:(49)MVNLNWBMHa1,a2,…,an=s∑i,j=1i≠jnωiωj/1-ωi·θai·θaj1/2,⋃γi∈T~ai,γj∈T~aj1-∏i,j=1i≠jn1-γiγjωiωj/1-ωi1/2,⋃δi∈I~ai,δj∈I~aj1-1-∏i,j=1i≠jn1-1-δi1-δjωiωj/1-ωi1/2,⋃ηi∈F~ai,ηj∈F~aj1-1-∏i,j=1i≠jn1-1-ηi1-ηjωiωj/1-ωi1/2.
And we have(50)
a
1
=
s
3.9529
,
0.3088,0.3515,0.3309
,
0.3752,0.3429,0.3869,0.3671
,
0.4118,0.3746,0.4199,0.4009,0.4463
,
0.1255,0.1645,0.1482,0.19
,
0.3768,0.4171,0.4049,0.446,0.4109
,
0.4511,0.4382,0.4783
;
a
2
=
s
3.9904
,
0.6,0.6368,0.6275,0.6648
,
0.6345,0.6727,0.6622,0.7
,
0.1,0.1363,0.1335,0.1714
,
0.1882,0.2268,0.2261,0.2629
;
a
3
=
s
4.4850
,
0.5272,0.5644,0.5618,0.6
,
0.263
,
0.2918,0.3273
;
a
4
=
s
2.6487
,
0.5107,0.5951,0.5406
,
0.6232,0.5415,0.6275,0.5743,0.6569
,
0.1363
,
0.1255,0.1645
,
0.2021,0.1623,0.2,0.2373
.Step 3 (the compared values of three functions are calculated).
By using the equations in Definition9, we can obtain E(ai), H(ai), and C(ai)(i=1,2,3,4) as follows:(51)Ea1=s2.3605,Ha1=s-0.2036,Ca1=s1.4879;Ea2=s3.0440,Ha2=s1.6912,Ca2=s2.5930;Ea3=s2.9763,Ha3=s1.1382,Ca3=s2.5267;Ea4=s2.0002,Ha4=s1.0641,Ca4=s1.5461.Step 4 (the alternatives are selected).
By using the compared approach in Definition10, E(a2)≻E(a3)≻E(a1)≻E(a4) can be obtained, so the final raking of alternatives is A2≻A3≻A1≻A4. Apparently, A2 is the best one, and A4is the worst one. We do not need to compare the other functions because the values of score function differ.


### 6.2. Comparison Analysis

We take different value into consideration in Step  2 to select the alternative for discussing the impact of different valuesp, q, and ε. The comparisons are presented in Tables 1 and 2.Table 1
Ranking of alternatives utilizing differentp, q, and ε=1.
ε = 1 ,  p ,  q Score functionE(ai), (i=1,2,3,4) Ranking p → 0, q=0 E ( a 1 ) = s 2.1214 , E ( a 2 ) = s 2.8131 , E ( a 3 ) = s 2.8732 , E ( a 4 ) = s 1.7094 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.01, q=0 E ( a 1 ) = s 2.1907 , E ( a 2 ) = s 2.8460 , E ( a 3 ) = s 2.9575 , E ( a 4 ) = s 1.7629 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.1, q=0 E ( a 1 ) = s 2.2165 , E ( a 2 ) = s 2.8790 , E ( a 3 ) = s 2.9713 , E ( a 4 ) = s 1.8005 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 1, q=0 E ( a 1 ) = s 2.4697 , E ( a 2 ) = s 3.1970 , E ( a 3 ) = s 3.1084 , E ( a 4 ) = s 2.1573 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=0 E ( a 1 ) = s 2.7154 , E ( a 2 ) = s 3.4882 , E ( a 3 ) = s 3.2507 , E ( a 4 ) = s 2.4519 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=0 E ( a 1 ) = s 3.1768 , E ( a 2 ) = s 3.9655 , E ( a 3 ) = s 3.5749 , E ( a 4 ) = s 2.8445 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=0 E ( a 1 ) = s 3.5203 , E ( a 2 ) = s 4.2619 , E ( a 3 ) = s 3.8673 , E ( a 4 ) = s 3.0415 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=1 E ( a 1 ) = s 2.6253 , E ( a 2 ) = s 3.3082 , E ( a 3 ) = s 3.1563 , E ( a 4 ) = s 2.2904 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.01, q=1 E ( a 1 ) = s 2.6151 , E ( a 2 ) = s 3.2980 , E ( a 3 ) = s 3.1492 , E ( a 4 ) = s 2.2778 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=1 E ( a 1 ) = s 2.5377 , E ( a 2 ) = s 3.2202 , E ( a 3 ) = s 3.0964 , E ( a 4 ) = s 2.1839 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=1 E ( a 1 ) = s 2.3605 , E ( a 2 ) = s 3.0440 , E ( a 3 ) = s 2.9763 , E ( a 4 ) = s 2.0002 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=1 E ( a 1 ) = s 2.4824 , E ( a 2 ) = s 3.1918 , E ( a 3 ) = s 3.0464 , E ( a 4 ) = s 2.1664 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=1 E ( a 1 ) = s 2.9208 , E ( a 2 ) = s 3.6644 , E ( a 3 ) = s 3.3532 , E ( a 4 ) = s 2.5870 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=1 E ( a 1 ) = s 3.3239 , E ( a 2 ) = s 4.0427 , E ( a 3 ) = s 3.6946 , E ( a 4 ) = s 2.8680 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=2 E ( a 1 ) = s 2.8597 , E ( a 2 ) = s 3.5712 , E ( a 3 ) = s 3.2871 , E ( a 4 ) = s 2.5589 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.01, q=2 E ( a 1 ) = s 2.8519 , E ( a 2 ) = s 3.5630 , E ( a 3 ) = s 3.2820 , E ( a 4 ) = s 2.5494 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=2 E ( a 1 ) = s 2.7889 , E ( a 2 ) = s 3.4959 , E ( a 3 ) = s 3.2400 , E ( a 4 ) = s 2.4731 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=2 E ( a 1 ) = s 2.5228 , E ( a 2 ) = s 3.2087 , E ( a 3 ) = s 3.0542 , E ( a 4 ) = s 2.1890 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=2 E ( a 1 ) = s 2.5363 , E ( a 2 ) = s 3.2305 , E ( a 3 ) = s 3.0440 , E ( a 4 ) = s 2.2430 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=2 E ( a 1 ) = s 2.8535 , E ( a 2 ) = s 3.5768 , E ( a 3 ) = s 3.2558 , E ( a 4 ) = s 2.5581 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=2 E ( a 1 ) = s 3.2306 , E ( a 2 ) = s 3.9375 , E ( a 3 ) = s 3.5836 , E ( a 4 ) = s 2.8212 A 2 ≻ A 3 ≻ A 1 ≻ A 4Table 2
Ranking of alternatives utilizing differentp, q, and ε=2.
ε = 2 , p , q Score functionE(ai), (i=1,2,3,4) Ranking p → 0, q=0 E ( a 1 ) = s 2.1344 , E ( a 2 ) = s 2.8137 , E ( a 3 ) = s 2.8863 , E ( a 4 ) = s 1.7187 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.01, q=0 E ( a 1 ) = s 2.1944 , E ( a 2 ) = s 2.8478 , E ( a 3 ) = s 2.9631 , E ( a 4 ) = s 1.7661 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.1, q=0 E ( a 1 ) = s 2.2171 , E ( a 2 ) = s 2.8803, E(a3)=s2.9752,E(a4)=s1.8021 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 1, q=0 E ( a 1 ) = s 2.4553 , E ( a 2 ) = s 3.1938 , E ( a 3 ) = s 3.0982 , E ( a 4 ) = s 2.1483 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=0 E ( a 1 ) = s 2.7142 , E ( a 2 ) = s 3.4825 , E ( a 3 ) = s 3.2361 , E ( a 4 ) = s 2.4453 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=0 E ( a 1 ) = s 3.2212 , E ( a 2 ) = s 3.9697 , E ( a 3 ) = s 3.5908 , E ( a 4 ) = s 2.8608 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=0 E ( a 1 ) = s 3.5770 , E ( a 2 ) = s 4.2923 , E ( a 3 ) = s 3.9217 , E ( a 4 ) = s 3.0737 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=1 E ( a 1 ) = s 2.6103 , E ( a 2 ) = s 3.3048 , E ( a 3 ) = s 3.1457 , E ( a 4 ) = s 2.2818 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.01, q=1 E ( a 1 ) = s 2.6011 , E ( a 2 ) = s 3.2948 , E ( a 3 ) = s 3.1394 , E ( a 4 ) = s 2.2700 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=1 E ( a 1 ) = s 2.5312 , E ( a 2 ) = s 3.2188 , E ( a 3 ) = s 3.0925 , E ( a 4 ) = s 2.1809 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=1 E ( a 1 ) = s 2.3702 , E ( a 2 ) = s 3.0464 , E ( a 3 ) = s 2.9849 , E ( a 4 ) = s 2.0055 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=1 E ( a 1 ) = s 2.4994 , E ( a 2 ) = s 3.1940 , E ( a 3 ) = s 3.0556 , E ( a 4 ) = s 2.1751 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=1 E ( a 1 ) = s 2.9637 , E ( a 2 ) = s 3.6734 , E ( a 3 ) = s 3.3833 , E ( a 4 ) = s 2.6091 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=1 E ( a 1 ) = s 3.3787 , E ( a 2 ) = s 4.0718 , E ( a 3 ) = s 3.7516 , E ( a 4 ) = s 2.9010 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=2 E ( a 1 ) = s 2.8577 , E ( a 2 ) = s 3.5652 , E ( a 3 ) = s 3.2720 , E ( a 4 ) = s 2.5525 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.01, q=2 E ( a 1 ) = s 2.8506 , E ( a 2 ) = s 3.5572 , E ( a 3 ) = s 3.2675 , E ( a 4 ) = s 2.5435 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=2 E ( a 1 ) = s 2.7919 , E ( a 2 ) = s 3.4917 , E ( a 3 ) = s 3.2304 , E ( a 4 ) = s 2.4707 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=2 E ( a 1 ) = s 2.5398 , E ( a 2 ) = s 3.2108 , E ( a 3 ) = s 3.0635 , E ( a 4 ) = s 2.1976 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=2 E ( a 1 ) = s 2.5586 , E ( a 2 ) = s 3.2343 , E ( a 3 ) = s 3.0581 , E ( a 4 ) = s 2.2559 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=2 E ( a 1 ) = s 2.8950 , E ( a 2 ) = s 3.5871 , E ( a 3 ) = s 3.2878 , E ( a 4 ) = s 2.5824 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=2 E ( a 1 ) = s 3.2843 , E ( a 2 ) = s 3.9650 , E ( a 3 ) = s 3.6392 , E ( a 4 ) = s 2.8549 A 2 ≻ A 3 ≻ A 1 ≻ A 4In Table1, we take the parameter value ε=1, which is based on algebraic operation, and the MVNLNWBM operator is applied. In Table 2, we take the parameter value ε=2, which is based on Einstein operation, and the MVNLNWBME operator is applied. As we can see from Tables 1 and 2, the ordering of alternatives taking different parameters p, q, and ε may be different, because the different parameters will cause different score function value. However, A2 or A3 is always the best selection, and A4 is always the worst selection. Whether ε=1 or ε=2, the same ranking result is obtained with regard to the same parameter value p and the same parameter value q except for one situation in which p=0, q=1, and ε=1. Specially, if p→0, q=0, MVNLNWBMH will reduce to MVNLHWGA operator. If p→0, q=0, and ε=1, MVNLHWGA will reduce to MVNLWGA operator. When p=1, q=0, MVNLNWBMH will reduce to MVNLHWAA operator. If p=1, q=0, and ε=1, MVNLHWAA will reduce to MVNLWAA operator. If ε=1 in Table 1, the ranking results on the basis of the MVNLWGA and MVNLWAA operators differ, which is due to the two operators emphasis on different major points, and the same situation happens in Table 2. When the parameters q=0 and p→0, p=0.01, p=0.1, respectively, the rankings are identical in two Tables, the ranking order is always A3≻A2≻A1≻A4. When p and q are assigned the other values in two Tables, the ranking order is changed, and the result is A2≻A3≻A1≻A4. That is, the best selection is from A3 to A2 except for one situation where p=0, q=1 in Table 1.For illustrating the effectiveness and flexibility of the novel approach, the method in literature is adopted in multivalued neutrosophic linguistic environment in this paper, and the same ranking orders are obtained in [9] where the SVNHFWA and SVNHFWG operators are adopted to fuse single-valued neutrosophic hesitant fuzzy information. When ε=1, p→0, q=0 and ε=1, p=0, q=0, the two operators are special cases of MVNLNWBMH operator. Therefore, novel operator in this paper has better flexibility and generalization. In actual cases, the decision makers can assign different parameter values ε, p, and q. Generally, for convenience, we can set ε=p=1, which can not only simplify the calculation, but also consider the interrelationship of multiple values.


## 6.1. The Procedure Using the Proposed Aggregation Operator

Step 1 (the decision matrix is normalized).
BecauseC3 is the minimizing criteria, which should be converted to the maximizing criteria, then the normalized decision matrix B=bijm×n  can be obtained as follows: (48)B=bij4×3=s5,0.3,0.4,0.5,0.1,0.3,0.4s6,0.5,0.6,0.2,0.3,0.3,0.4s2,0.2,0.3,0.1,0.2,0.5,0.6s6,0.6,0.7,0.1,0.2,0.2,0.3s5,0.6,0.7,0.1,0.3s2,0.6,0.7,0.1,0.2,0.1,0.2s6,0.5,0.6,0.4,0.2,0.3s5,0.6,0.3,0.4s3,0.5,0.6,0.1,0.3s4,0.7,0.8,0.1,0.1,0.2s4,0.6,0.7,0.1,0.2s1,0.3,0.5,0.2,0.1,0.2,0.3.Step 2 (the comprehensive value of each alternative is calculated).
Derive the comprehensive valueai(i=1,2,…,m) of each alternative Ai(i=1,2,…,m) by using the MVNLNWBMH operator presented in Definition 11. Here let p=q=1, ε=1. The MVNLNWBMH operator is shown as below:(49)MVNLNWBMHa1,a2,…,an=s∑i,j=1i≠jnωiωj/1-ωi·θai·θaj1/2,⋃γi∈T~ai,γj∈T~aj1-∏i,j=1i≠jn1-γiγjωiωj/1-ωi1/2,⋃δi∈I~ai,δj∈I~aj1-1-∏i,j=1i≠jn1-1-δi1-δjωiωj/1-ωi1/2,⋃ηi∈F~ai,ηj∈F~aj1-1-∏i,j=1i≠jn1-1-ηi1-ηjωiωj/1-ωi1/2.
And we have(50)
a
1
=
s
3.9529
,
0.3088,0.3515,0.3309
,
0.3752,0.3429,0.3869,0.3671
,
0.4118,0.3746,0.4199,0.4009,0.4463
,
0.1255,0.1645,0.1482,0.19
,
0.3768,0.4171,0.4049,0.446,0.4109
,
0.4511,0.4382,0.4783
;
a
2
=
s
3.9904
,
0.6,0.6368,0.6275,0.6648
,
0.6345,0.6727,0.6622,0.7
,
0.1,0.1363,0.1335,0.1714
,
0.1882,0.2268,0.2261,0.2629
;
a
3
=
s
4.4850
,
0.5272,0.5644,0.5618,0.6
,
0.263
,
0.2918,0.3273
;
a
4
=
s
2.6487
,
0.5107,0.5951,0.5406
,
0.6232,0.5415,0.6275,0.5743,0.6569
,
0.1363
,
0.1255,0.1645
,
0.2021,0.1623,0.2,0.2373
.Step 3 (the compared values of three functions are calculated).
By using the equations in Definition9, we can obtain E(ai), H(ai), and C(ai)(i=1,2,3,4) as follows:(51)Ea1=s2.3605,Ha1=s-0.2036,Ca1=s1.4879;Ea2=s3.0440,Ha2=s1.6912,Ca2=s2.5930;Ea3=s2.9763,Ha3=s1.1382,Ca3=s2.5267;Ea4=s2.0002,Ha4=s1.0641,Ca4=s1.5461.Step 4 (the alternatives are selected).
By using the compared approach in Definition10, E(a2)≻E(a3)≻E(a1)≻E(a4) can be obtained, so the final raking of alternatives is A2≻A3≻A1≻A4. Apparently, A2 is the best one, and A4is the worst one. We do not need to compare the other functions because the values of score function differ.


## 6.2. Comparison Analysis

We take different value into consideration in Step  2 to select the alternative for discussing the impact of different valuesp, q, and ε. The comparisons are presented in Tables 1 and 2.Table 1
Ranking of alternatives utilizing differentp, q, and ε=1.
ε = 1 ,  p ,  q Score functionE(ai), (i=1,2,3,4) Ranking p → 0, q=0 E ( a 1 ) = s 2.1214 , E ( a 2 ) = s 2.8131 , E ( a 3 ) = s 2.8732 , E ( a 4 ) = s 1.7094 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.01, q=0 E ( a 1 ) = s 2.1907 , E ( a 2 ) = s 2.8460 , E ( a 3 ) = s 2.9575 , E ( a 4 ) = s 1.7629 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.1, q=0 E ( a 1 ) = s 2.2165 , E ( a 2 ) = s 2.8790 , E ( a 3 ) = s 2.9713 , E ( a 4 ) = s 1.8005 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 1, q=0 E ( a 1 ) = s 2.4697 , E ( a 2 ) = s 3.1970 , E ( a 3 ) = s 3.1084 , E ( a 4 ) = s 2.1573 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=0 E ( a 1 ) = s 2.7154 , E ( a 2 ) = s 3.4882 , E ( a 3 ) = s 3.2507 , E ( a 4 ) = s 2.4519 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=0 E ( a 1 ) = s 3.1768 , E ( a 2 ) = s 3.9655 , E ( a 3 ) = s 3.5749 , E ( a 4 ) = s 2.8445 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=0 E ( a 1 ) = s 3.5203 , E ( a 2 ) = s 4.2619 , E ( a 3 ) = s 3.8673 , E ( a 4 ) = s 3.0415 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=1 E ( a 1 ) = s 2.6253 , E ( a 2 ) = s 3.3082 , E ( a 3 ) = s 3.1563 , E ( a 4 ) = s 2.2904 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.01, q=1 E ( a 1 ) = s 2.6151 , E ( a 2 ) = s 3.2980 , E ( a 3 ) = s 3.1492 , E ( a 4 ) = s 2.2778 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=1 E ( a 1 ) = s 2.5377 , E ( a 2 ) = s 3.2202 , E ( a 3 ) = s 3.0964 , E ( a 4 ) = s 2.1839 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=1 E ( a 1 ) = s 2.3605 , E ( a 2 ) = s 3.0440 , E ( a 3 ) = s 2.9763 , E ( a 4 ) = s 2.0002 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=1 E ( a 1 ) = s 2.4824 , E ( a 2 ) = s 3.1918 , E ( a 3 ) = s 3.0464 , E ( a 4 ) = s 2.1664 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=1 E ( a 1 ) = s 2.9208 , E ( a 2 ) = s 3.6644 , E ( a 3 ) = s 3.3532 , E ( a 4 ) = s 2.5870 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=1 E ( a 1 ) = s 3.3239 , E ( a 2 ) = s 4.0427 , E ( a 3 ) = s 3.6946 , E ( a 4 ) = s 2.8680 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=2 E ( a 1 ) = s 2.8597 , E ( a 2 ) = s 3.5712 , E ( a 3 ) = s 3.2871 , E ( a 4 ) = s 2.5589 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.01, q=2 E ( a 1 ) = s 2.8519 , E ( a 2 ) = s 3.5630 , E ( a 3 ) = s 3.2820 , E ( a 4 ) = s 2.5494 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=2 E ( a 1 ) = s 2.7889 , E ( a 2 ) = s 3.4959 , E ( a 3 ) = s 3.2400 , E ( a 4 ) = s 2.4731 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=2 E ( a 1 ) = s 2.5228 , E ( a 2 ) = s 3.2087 , E ( a 3 ) = s 3.0542 , E ( a 4 ) = s 2.1890 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=2 E ( a 1 ) = s 2.5363 , E ( a 2 ) = s 3.2305 , E ( a 3 ) = s 3.0440 , E ( a 4 ) = s 2.2430 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=2 E ( a 1 ) = s 2.8535 , E ( a 2 ) = s 3.5768 , E ( a 3 ) = s 3.2558 , E ( a 4 ) = s 2.5581 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=2 E ( a 1 ) = s 3.2306 , E ( a 2 ) = s 3.9375 , E ( a 3 ) = s 3.5836 , E ( a 4 ) = s 2.8212 A 2 ≻ A 3 ≻ A 1 ≻ A 4Table 2
Ranking of alternatives utilizing differentp, q, and ε=2.
ε = 2 , p , q Score functionE(ai), (i=1,2,3,4) Ranking p → 0, q=0 E ( a 1 ) = s 2.1344 , E ( a 2 ) = s 2.8137 , E ( a 3 ) = s 2.8863 , E ( a 4 ) = s 1.7187 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.01, q=0 E ( a 1 ) = s 2.1944 , E ( a 2 ) = s 2.8478 , E ( a 3 ) = s 2.9631 , E ( a 4 ) = s 1.7661 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 0.1, q=0 E ( a 1 ) = s 2.2171 , E ( a 2 ) = s 2.8803, E(a3)=s2.9752,E(a4)=s1.8021 A 3 ≻ A 2 ≻ A 1 ≻ A 4 p = 1, q=0 E ( a 1 ) = s 2.4553 , E ( a 2 ) = s 3.1938 , E ( a 3 ) = s 3.0982 , E ( a 4 ) = s 2.1483 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=0 E ( a 1 ) = s 2.7142 , E ( a 2 ) = s 3.4825 , E ( a 3 ) = s 3.2361 , E ( a 4 ) = s 2.4453 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=0 E ( a 1 ) = s 3.2212 , E ( a 2 ) = s 3.9697 , E ( a 3 ) = s 3.5908 , E ( a 4 ) = s 2.8608 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=0 E ( a 1 ) = s 3.5770 , E ( a 2 ) = s 4.2923 , E ( a 3 ) = s 3.9217 , E ( a 4 ) = s 3.0737 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=1 E ( a 1 ) = s 2.6103 , E ( a 2 ) = s 3.3048 , E ( a 3 ) = s 3.1457 , E ( a 4 ) = s 2.2818 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.01, q=1 E ( a 1 ) = s 2.6011 , E ( a 2 ) = s 3.2948 , E ( a 3 ) = s 3.1394 , E ( a 4 ) = s 2.2700 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=1 E ( a 1 ) = s 2.5312 , E ( a 2 ) = s 3.2188 , E ( a 3 ) = s 3.0925 , E ( a 4 ) = s 2.1809 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=1 E ( a 1 ) = s 2.3702 , E ( a 2 ) = s 3.0464 , E ( a 3 ) = s 2.9849 , E ( a 4 ) = s 2.0055 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=1 E ( a 1 ) = s 2.4994 , E ( a 2 ) = s 3.1940 , E ( a 3 ) = s 3.0556 , E ( a 4 ) = s 2.1751 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=1 E ( a 1 ) = s 2.9637 , E ( a 2 ) = s 3.6734 , E ( a 3 ) = s 3.3833 , E ( a 4 ) = s 2.6091 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=1 E ( a 1 ) = s 3.3787 , E ( a 2 ) = s 4.0718 , E ( a 3 ) = s 3.7516 , E ( a 4 ) = s 2.9010 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0, q=2 E ( a 1 ) = s 2.8577 , E ( a 2 ) = s 3.5652 , E ( a 3 ) = s 3.2720 , E ( a 4 ) = s 2.5525 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.01, q=2 E ( a 1 ) = s 2.8506 , E ( a 2 ) = s 3.5572 , E ( a 3 ) = s 3.2675 , E ( a 4 ) = s 2.5435 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 0.1, q=2 E ( a 1 ) = s 2.7919 , E ( a 2 ) = s 3.4917 , E ( a 3 ) = s 3.2304 , E ( a 4 ) = s 2.4707 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 1, q=2 E ( a 1 ) = s 2.5398 , E ( a 2 ) = s 3.2108 , E ( a 3 ) = s 3.0635 , E ( a 4 ) = s 2.1976 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 2, q=2 E ( a 1 ) = s 2.5586 , E ( a 2 ) = s 3.2343 , E ( a 3 ) = s 3.0581 , E ( a 4 ) = s 2.2559 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 5, q=2 E ( a 1 ) = s 2.8950 , E ( a 2 ) = s 3.5871 , E ( a 3 ) = s 3.2878 , E ( a 4 ) = s 2.5824 A 2 ≻ A 3 ≻ A 1 ≻ A 4 p = 10, q=2 E ( a 1 ) = s 3.2843 , E ( a 2 ) = s 3.9650 , E ( a 3 ) = s 3.6392 , E ( a 4 ) = s 2.8549 A 2 ≻ A 3 ≻ A 1 ≻ A 4In Table1, we take the parameter value ε=1, which is based on algebraic operation, and the MVNLNWBM operator is applied. In Table 2, we take the parameter value ε=2, which is based on Einstein operation, and the MVNLNWBME operator is applied. As we can see from Tables 1 and 2, the ordering of alternatives taking different parameters p, q, and ε may be different, because the different parameters will cause different score function value. However, A2 or A3 is always the best selection, and A4 is always the worst selection. Whether ε=1 or ε=2, the same ranking result is obtained with regard to the same parameter value p and the same parameter value q except for one situation in which p=0, q=1, and ε=1. Specially, if p→0, q=0, MVNLNWBMH will reduce to MVNLHWGA operator. If p→0, q=0, and ε=1, MVNLHWGA will reduce to MVNLWGA operator. When p=1, q=0, MVNLNWBMH will reduce to MVNLHWAA operator. If p=1, q=0, and ε=1, MVNLHWAA will reduce to MVNLWAA operator. If ε=1 in Table 1, the ranking results on the basis of the MVNLWGA and MVNLWAA operators differ, which is due to the two operators emphasis on different major points, and the same situation happens in Table 2. When the parameters q=0 and p→0, p=0.01, p=0.1, respectively, the rankings are identical in two Tables, the ranking order is always A3≻A2≻A1≻A4. When p and q are assigned the other values in two Tables, the ranking order is changed, and the result is A2≻A3≻A1≻A4. That is, the best selection is from A3 to A2 except for one situation where p=0, q=1 in Table 1.For illustrating the effectiveness and flexibility of the novel approach, the method in literature is adopted in multivalued neutrosophic linguistic environment in this paper, and the same ranking orders are obtained in [9] where the SVNHFWA and SVNHFWG operators are adopted to fuse single-valued neutrosophic hesitant fuzzy information. When ε=1, p→0, q=0 and ε=1, p=0, q=0, the two operators are special cases of MVNLNWBMH operator. Therefore, novel operator in this paper has better flexibility and generalization. In actual cases, the decision makers can assign different parameter values ε, p, and q. Generally, for convenience, we can set ε=p=1, which can not only simplify the calculation, but also consider the interrelationship of multiple values.


## 7. Conclusions

In this paper, the MVNLSs are proposed by combining the MVNS and LS, which not only describe linguistic terms, but also give the quantitative value of three membership degrees concerning the linguistic variables, which has better flexibility in expressing the decision information. Moreover, NWBM is a useful operator which has the trait of taking into account the interrelationship of different arguments and overcomes drawbacks of nonreducibility and nonidempotency. Hamacher operations are the extension of algebraic and Einstein operations, which is more general. Considering these advantages, we have developed Hamacher operational laws for MVNLNs and extended the NWBM to fuse MVNL information. Thus, the MVNLNWBMH operator is proposed, which is appropriate to deal with MVNL information. Some desirable properties of the novel operator are discussed in detail, and some special cases are analyzed. Furthermore, the comparison method for MVNLNs is also studied, and the rankings of alternatives affected by different parametersp, q, and ε are also compared. For verifying the novel approach, we successfully applied the approach to an example. The results show the novel approach has the following advantages: the MVNLNWBMH operator is more flexible and more general and can capture the interrelationship among arguments and express decision information more practically; the decision makers can assign appropriate values according to the real situation. In future, we will explore applying the operator to the different domains, for instance, fault diagnosis, machine learning, and medical diagnosis.As a future possible research, we will extend our research by using the refined neutrosophic set [54]; that is, the truth value T is refined into types of subtruths such as T1, T2; similarly indeterminacy I is refined into types of subindeterminaciesI1,I2, and so forth; and the subfalsehoodF is split intoF1,F2, and so forth.



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*Source: 2432167-2018-03-18.xml*