Multiple criteria decision making approach with multi ...vixra.org/pdf/1803.0533v1.pdf · Archimedean t-conorms and t-norms are well-known, which include algebraic, Einstein and Hamacher,

Mathematical Problems in Engineering.

1

Multiple criteria decision making approach with multi-valued neutrosophic

linguistic normalized weighted Bonferroni mean Hamacher operator

Bao-lin Li1，2

, Juan-ru Wang1, Li-hua Yang2, Xue-tao Li2

1.

School of Management, Northwestern Polytechnical University, Xi’an 710072, China; 2.

School of Economics and Management, Hubei University of Automotive Technology,

Shiyan 442002, China

Corresponding author: Juan-ru Wang

E-mail: [email protected]

Abstract The neutrosophic set and linguistic term set are widely applied in recently years. Motivated

by the advantages of them, we combine the mult i-valued neutrosophic set and linguistic set and define

the concept of the mult i-valued neutrosophic linguistic set (MVNLS). Furthermore, Hamacher

operation is an extension of the Algebraic and Einstein operation. Additionally, the normalized

weighted Bonferroni mean (NW BM) operator can consider the weight of each argument as well as

capture the interrelationship of different arguments . Therefore, the combination of NWBM operator

and Hamacher operation is more valuable and agile. Firstly, MVNLS and mult i-valued neutrosophic

linguistic number (MVNLN) are defined, then some new operational ru les of MVNLNs on account of

Hamacher operations are developed, and the comparison functions for MVNLNs are given. Secondly, multi-valued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher o perator

(MVNLNW BMH) is proposed, and a number of expected characteristics of new operator are

investigated. Meanwhile, some special cases of different parameters ,p q and are analyzed. Th ird ly,

the approach utilizing the MVNLNW BMH operator is introduced to manage multip le criteria decision

making issue (MCDM) in mult i-valued 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.

Keywords Multi-valued neutrosophic linguistic, NWBM, Hamacher, Multiple criteria decision making

1 Introduction

In real world, due to the complexity of decision informat ion, the fuzzy theory has attracted

widespread attentions and has been developed in various fields. Zaheh [1] firstly proposed the notion of

fuzzy sets (FSs). Then, Atanassov [2] introduced the intuitionistic fuzzy sets (IFSs), which overcome

the weakness of non-membership degrees. Subsequently, in order to address the

hesitation degree of decision-makers, Torra [3] defined hesitant fuzzy sets (HFSs). Fuzzy set theory has

gained 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 on 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 [54] and

Wang et al. [5] introduced single-valued neutrosophic sets (SVNs) to facilitate its application. Ye [6]

pointed out the concept of simplified neutrosopic sets (SNSs).Wang [7] developed the concept of

interval neutrosopic sets (INSs). However, under certain conditions, the decision makers likely give

different evaluation numbers for expressing their hesitant. Subsequently, the definition of single-valued

neutrosophic hesitant fuzzy sets (SVNHFSs) was firstly proposed by Ye [8] in 2014, then Wang [9]

also proposed mult i-valued 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.


2

On the other hand, the aggregation operators, comparison method for Neutrosophic numbers are also

been studied. For SVNSs, Liu [10] employed NWBM operator to solve mult iple criteria problem in

single-valued neutrosophic environment. Ye [11] gave the definitions of cross-entropy and correlation

coefficient. For INSs, Zhang [12] developed some aggregation operators. Liu [13] not only provided

the definition of interval neutrosophic hesitant fuzzy sets (INHFSs), but also discussed the generalized

hybrid weighted average operator. Broumi and Smarandache [14-16] studied the correlation

coefficients, cosine similarity measure and some new operations. Ye [17] proposed similarity measures

between interval neutrosophic sets. For MVNSs, Ye [8] developed SVNHFWA and SVNHFW G

operators for MCDM problem. Peng [18, 19] extended power aggregation operators and defined some

outranking relat ions under MVNS environment. Ji et al. [20] 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 variab le was firstly

proposed by Zadeh [21] and applied fo r the fuzzy reasoning. After that, a series of works on it have

been made, Wang [22-24] presented a new approach in view of hesitant fuzzy linguistic informat ion.

Meng [25] developed linguistic hesitant fuzzy sets and studied hybrid weighted operator. Tian [26]

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 its corresponding

operator are defined, for example single valued o r simplified neutrosophic linguistic sets and trapezoid

linguistic sets [27-30], interval neutrosophic certain or uncertain linguistic sets [31-33].However, due

to the hesitancy of people’s thinking, the true of a linguistic term may be given several values, and the

case is similar to the false and indeterminate extents. The existing literatures don’t consider this

perspective. Therefore, the mult i-valued neutrosophic linguistic set (MVNLS) and mult i-valued

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 [34-41],

for instance, the weighted geometric average (W GA) or averag ing (WA) operator, prioritized

aggregation (PA) operator, Maclaurin symmetric mean and Bonferroni mean (BM) operator. BM

operator was originally defined by Bonferroni [42], which has attracted widespread attentions because

of its characteristics of capture interrelat ionship among arguments. Some ach ievements have been

made on it [43-49].In order to aggregate neutrosophic linguistic informat ion, some researches on

aggregation operators under neutrosophic linguistic and neutrosophic uncertain linguistic environments

are also been applied [27-33,50].Until to now, BM and NWBM fail to accommodate aggregation

informat ion for mult i-valued neutrosophic linguistic environment. Motivated by this limitat ion, 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

commonly, there are also a few aggregation operations based on Einstein operations. Due to Hamacher

operator is more general, Liu [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

multi-valued neutrosophic linguistic environment for MCDM problems.

The main purposes of the paper are presented in the following:

1. To be better express people’s hesitant, 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.


3

2. Due to the generalizat ion 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 MVNLNW BMH operator is

defined, moreover, some desirable characteristics are also studied.

4. In order to verify the effectiveness, an example for MCDM problem utilizing MVNLNWBMH

operator is illustrated and conduct a comparative analysis. We also analyze the influences of

different parameter values for the final outcomes, the results demonstrate the operator proposed is

more general and flexible.

The article is arranged in this way. In Section 2, we review a number o f notions and operations for

MVNS, LS, NW BM operator and Hamacher. In Sect ion 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 A lgebraic as well as Einstein 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 MVNLNW BMH operator and investigate its properties. Furthe rmore, 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 in formation. Section 6 presents a concrete example, as well as a comparison analysis is

provided to show the practicability 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 that 1 2, , ,S s s s

is an ordered and finite linguistic set, in which

js denotes a

linguistic variable value and is an odd value. When is equal to seven, the corresponding linguistic

set are provided in the following:

extremely poor,very poor,poor,medium,good,very good,extremely good1 2 3 4 5 6 7, , , , , , .S s s s s s s s

In order to avoid the linguistic information loss, the set above is expanded, that is a contiguous set,

.S s R

Definition 1 [53] Let i

s and js be any two linguistic variables, the corresponding operations are

presented:

(1) , 0;

(2) ;

(3) ;

(4) .

i i

i j i j

i j i j

i i

s s

s s s

s s s

s s

2.2 Multi-valued neutrosophic sets

Definition 2 [8,9] Suppose that X is a collection of objects, MVNSs A on X is defined by

, ( ), ( ), ( ) ,A A A

A x T x I x F x x X

Where ( ) ( ) ,A AT x T x ( ) ( ) ,

A AI x I x ( ) ( ) ,

A AF x F x

( ), ( ),A A

T x I x and ( )A

F x are three collections of crisp numbers belonging to 0,1 , representing the

probable true-membership degree, indeterminacy-membership degree and falsity- membership degree,


4

where x in X belonging to A , respectively, satisfying these conditions 0 , , 1, and

0 sup ( ) sup ( ) sup ( ) 3.A A AT x I x F x If there is only one element in X ,A is indicated by the

three tuple ( ), ( ), ( ) ,A A A

A T x I x F x that is known as a multi-valued neutrosophic number

(MVNN).Generally, MVNSs is considered as the generalizations of the other sets, such as FSs, IFSs,

HFSs, DHFs, and SVNSs.

2.3 Normalized weighted Bonferroni mean

Definition 3 [42] Let , 0,p q as well as ( 1,2, )ia i n be a set of nonnegative

values, then the BM is defined as

1

,

1 2, 1,

1( , , , )

( 1)

p q

np q p q

n i ji ji j

BM a a a a an n

Definition 4 [45]Let , 0,p q and ( 1,2, )ia i n be a set of nonnegative values, and

the corresponding NWBM can be expressed as below:

1

,

1 2 1, 1,( , , , ) ( )

p q

i j

i

nw wp q p q

n i jwi ji j

NWBM a a a a a

Where ),,,( 21 nwwww represents the corresponding weighted vector

of 1,2, ,ia i n , satisfying 0

iw and

1

1n

ii

w

. 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 in the following:

(1) Algebraic t-norm and t-conorm

, ;a b ab a b a b ab

(2) Einstein t-norm and t-conorm

, ;

11 1 1

ab a ba b a b

aba b

(3) Hamacher t-norm and t-conorm

(1 )

, , 01 (1 )1

ab a b ab aba b a b

aba b ab


5

In special, when 1, 2, the Algebraic and Einstein operations are the

simplifications of Hamacher t-norm and t-conorm.

3 Multi-valued neutrosophic linguistic set

3.1 MVNLS and its Hamacher operations

Definition 5 Let X be a set of points, an MVNLS A in X is defined as follows:

( ), , ( ), ( ), ( ) ,

x A A AA x s T x I x F x x X

Where ( ),

xs S

( ) ( ) ,A AT x T x ( ) ( ) ,

A AI x I x ( ) ( ) ,

A AF x F x

( ), ( ),A A

T x I x and ( )A

F x are three sets of crisp values in 0,1 , denoting three degrees of

x in X belonging to ( )

,x

s , that are true, indeterminacy and falsity, satisfying these conditions

0 , , 1, and 0 sup ( ) sup ( ) sup ( ) 1.A A AT x I x F x

Definition 6 Let ( )

, , ( ), ( ), ( )x A A A

A x s T x I x F x x X

be an MVNLS, supposing there is

only one element in X , then tuple ( ), ( ), ( ), ( )

x A A As T x I x F x is depicted as a mult i-valued

neutrosophic linguistic number (MVNLN).For simplicity, the MVNLN can also be represented as

( ), ( ), ( ), ( )

x A A AA s T x I x F x x X

Definition 7 Let 11 ( ) 1 1 1, ( ), ( ), ( )

aa s T a I a F a

and

22 ( ) 2 2 2, ( ), ( ), ( )

aa s T a I a F a

be two

MVNLNs, and 0, then the operations of MVNLNs can be defined on the basis of Hamacher

operations.

1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 2

( ) ( )

1 2 1 2 1 2

( ), ( )

1 2

1 2

( ), ( )

1 2 1 2

1 2

( ), ( )

1 2 1 2

(1)

,

(1 ),

1 (1 )

,(1 )( )

(1 )( )

a a

T a T a

I a I a

F a F a

a a

s

;

1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 2

( ) ( )

1 2

( ), ( )

1 2 1 2

1 2 1 2 1 2

( ), ( )

1 2

1 2 1 2 1 2

( ), ( )

1 2

(2)

,

,(1 )( )

(1 ),

1 (1 )

(1 )

1 (1 )

a a

T a T a

I a I a

F a F a

a a

s

;


6

1

1 1

1 1

1 1

1 ( )

1 1

( )

1 1

1

( )

1 1

1

( )

1 1

(3) ,

(1 ( 1) ) (1 ),

(1 ( 1) ) ( 1)(1 )

,(1 ( 1)(1 )) ( 1)

;(1 ( 1)(1 )) ( 1)

a

T a

I a

F a

a s

1

1 1

1 1

1 1

1 ( )

1

( )

1 1

1 1

( )

1 1

1 1

( )

1 1

(4) ,

,(1 ( 1)(1 )) ( 1)

(1 ( 1) ) (1 ),

(1 ( 1) ) ( 1)(1 )

(1 ( 1) ) (1 )

(1 ( 1) ) ( 1)(1 )

a

T a

I a

F a

a s

.

If 1, then the operations based on Hamacher operational ru les in Definit ion 7 will simplified to

the Algebraic operational rules in the following:

1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 2

( ) ( )

1 2 1 2( ), ( )

1 2( ), ( )

1 2( ), ( )

(5)

,

,

,

;

a a

T a T a

I a I a

F a F a

a a

s

1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 2

( ) ( )

1 2( ), ( )

1 2 1 2( ), ( )

1 2 1 2( ), ( )

(6)

,

,

,

;

a a

T a T a

I a I a

F a F a

a a

s

1

1 1

1 1

1 1

1 ( )

1( )

1( )

1( )

(7) ,

1 (1 ) ,

,

;

a

T a

I a

F a

a s


7

1

1 1

1 1

1 1

1 ( )

1( )

1( )

1( )

(8) ,

,

1 (1 ) ,

1 (1 ) .

a

T a

I a

F a

a s

Supposing 1 1 1 2 2( ), ( ), ( ), ( ), ( ),T a I a F a T a I a and

2( )F a contain only one value, then the operations

defined above can be reduced to the operations of SVNLNs based on Algebraic operations proposed by

Ye [27].

If 2, then the operations based on Hamacher operational ru les in Defin ition 7 will simplified to

the Einstein operations of MVNLNs presented below:

1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 2

( ) ( )

1 2

( ), ( )

1 2

1 2

( ), ( )

1 2 1 2

1 2

( ), ( )

1 2 1 2

(9)

,

,1

,2

;2

a a

T a T a

I a I a

F a F a

a a

s

1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 2

( ) ( )

1 2

( ), ( )

1 2 1 2

1 2

( ), ( )

1 2

1 2

( ), ( )

1 2

(10)

,

,2

,1

;1

a a

T a T a

I a I a

F a F a

a a

s

1

1 1

1 1

1 1

1 ( )

1 1

( )

1 1

1

( )

1 1

1

( )

1 1

(11) ,

(1 ) (1 ),

(1 ) (1 )

2,

(2 )

2;

(2 )

a

T a

I a

F a

a s


8

1

1 1

1 1

1 1

1 ( )

1

( )

1 1

1 1

( )

1 1

1 1

( )

1 1

(12) ,

2,

(2 )

(1 ) (1 ),

(1 ) (1 )

(1 ) (1 ).

(1 ) (1 )

a

T a

I a

F a

a s

Supposing 1 1 1 2 2( ), ( ), ( ), ( ), ( ),T a I a F a T a I a and 2

( )F a contain only one value, then the operations

defined above can be reduced to the operations of SVNLNs based on Einstein operations.

Theorem 1 Let 11 ( ) 1 1 1, ( ), ( ), ( ) ,

aa s T a I a F a

22 ( ) 2 2 2, ( ), ( ), ( ) ,

aa s T a I a F a

and

33 ( ) 3 3 3, ( ), ( ), ( )

aa s T a I a F a

be any three MVNLNs, and 1 2

, , 0, then the properties

below are correct:

1 2 2 1(1) ;a a a a

1 2 2 1

(2) ;a a a a

1 2 1 2(3) ;a a a a

1 1 2 1 1 2 1(4) ( ) ;a a a

1 2 1 2

1 1 1(5) ;a a a

1 2 1 2(6) ;a a a a

1 2 3 1 2 3(7) ;a a a a a a

1 2 3 1 2 3(8) .a a a a a a

Then, equation (4) will be proved as follows:

Proof (4) Since 1 2, 0,

1 1 2 1

1 1 2 2 1 1 2

1 1 2 2 1 1

1 1

1 1 2 1 ( ) ( )

1 1 1 1 1 1 1

1 1 1 1 1 1( )

,

(1 ( 1) ) (1 ) (1 ( 1) ) (1 ) (1 ( 1) ) (1 ) (1 ( 1) )

(1 ( 1) ) ( 1)(1 ) (1 ( 1) ) ( 1)(1 ) (1 ( 1) ) ( 1)(1 )

a a

T a

a a s

2 1 1 2 2

2 2 1 1 2 2

1 1

1 1

1 1 1 1 1

1 1 1 1 1 1

1 1

1 1

(1 ) (1 ( 1) ) (1 ) (1 ( 1) ) (1 )(1 )

(1 ( 1) ) ( 1)(1 ) (1 ( 1) ) ( 1)(1 ) (1 ( 1) ) ( 1)(1 )

(1 ( 1) ) (1 ) (11 (1 )

(1 ( 1) ) ( 1)(1 )

2 2

2 2

1 2

1 1 2 2

1 1 1 2

1 1

1 1

1 1

1 1

1 1 1 1( )

1 1

1 1

,( 1) ) (1 )

(1 ( 1) ) ( 1)(1 )

(1 ( 1)(1 )) ( 1) (1 ( 1)(1 )) ( 1)

(1 )((1 ( 1)(1 )) ( 1) (1 ( 1)(1

I a

1 2

2 2 1 1 2 2

1 2

1 1 2 2

1 1 1

1 1

1 1 1 1 1 1

1 1

1 1 1 1( )

1

1

,

))) ( 1) (1 ( 1)(1 )) ( 1) (1 ( 1)(1 )) ( 1)

(1 ( 1)(1 )) ( 1) (1 ( 1)(1 )) ( 1)

(1 )((1 ( 1)(1

F a

2 1 2

1 1 2 2 1 1 2 2

1 2 1

1 2 1 2

1 1

1 1 1

1 1 1 1 1 1 1

( ) ( )

1 1( )

1

))) ( 1) (1 ( 1)(1 )) ( 1) (1 ( 1)(1 )) ( 1) (1 ( 1)(1 )) ( 1)

,

(1 ( 1) ) (1 )

(1 ( 1) )

a

T a

s

1 2 1 2

1 2

1 1 1 2 1 2

1 2

1 1 1 2 1 2

1

1( )

1 1

1( )

1 1

1 2 1

,( 1)(1 )

,(1 ( 1)(1 )) ( 1)

(1 ( 1)(1 )) ( 1)

( )

I a

F a

a

Therefore, equation 1 1 2 1 1 2 1(4) ( )a a a can be obtained.


9

Similarly, the other equations in Theorem1 are easily certified in the light of Definition 7.

3.2 Comparison method

The score, accuracy , and certainty funct ions are important inde xes to rank MVNLNs , and its

corresponding definition is given below:

Definition 8 Let ( ), ( ), ( ), ( )

aa s T a I a F a

be an MVNLN, and the score, accuracy , and

certainty functions are achieved as below.

( ), ( ), ( )( ) ( ) ( )

( )( ), ( ), ( )

( ) ( ) ( )

1 1 1( ) ( )

3

1 1 1(1) E( ) ( )

3

T a I a F aT a I a F a

aT a I a F aT a I a F a

a

a s

s

( ), ( )( ) ( )

( )( ), ( )

( ) ( )

1( ) ( )

1(2) H( ) ( )

T a F aT a F a

aT a F aT a F a

a

a s

s

( )( )

( )( )

( )

1( )

1(3) C( )

T aT a

aT aT a

a

a s

s

Where ( ) ( ), ,

T a I a and ( )F a

are the numbers of the values in ( ), ( ),A A

T x I x and ( )A

F x respectively.

The linguistic variable ( )as is important for an MVNLN. Therefore, the comparison functions defined

above in Definition 8 are denoted as the linguistic variable. The bigger the truth degree ( )T a

concerning the variable ( )as is, meanwhile, the smaller the indeterminacy degree ( )I a as well as the

false degree ( )F a concerning the linguistic variable ( )as are, then the higher the MVNLN is. Relating

to the function of score, the greater corresponding to ( )as is, the higher the affirmat ive

statement is. Relating to the function of accuracy, the greater minus is, the certain the statement is.

Regarding to the function of certainty, the bigger is, the certain the statement is.

Based on Definition 8, the comparison method between MVNLNs are obtained.

Definition 9 Supposing 1a and 2

a are two MVNLNs, the compared approach is achieved in the

following:

(1) Supposing that 1 2( ) ( )E a E a , then 1

a is greater than 2a , represented as

1 2a a ;

(2) Supposing that 1 2( ) ( )E a E a , and 1 2

( ) ( )H a H a ,then 1a is greater than 2

a , represented

as 1 2a a ;

(3) Supposing that 1 2( ) ( )E a E a , 1 2

( ) ( )H a H a , and 1 2( ) ( )C a C a ,then 1

a is greater than 2a ,

represented as 1 2a a ;

(4) Supposing that 1 2

( ) ( )E a E a , 1 2( ) ( )H a H a , and 1 2

( ) ( )C a C a ,then 1a equals 2

a ,

represented as 1 2a a ;


10

4 The multi-valued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher

operator

The NWBM operator can not only take into account the advantages of BM and WBM, but also has

the property of reducibility and idempotency. However, the NW BM operator has not been applied to

the cases where the input arguments are MVNLNs.

Definition 10 Let ( 1,2, , )ia i n be a space of MVNLNs,

( ), ( ), ( ), ( ) ,

ii a i i ia s T a I a F a

, 0,p q and 1 2

, , ,n

be the weighted vector

for , 0,1i ia and

1

1.n

ii

Then the operator of MVNLNWBMH is achieved as below, the

aggregation result is still an MVNLN.

1

1 2, 1

( , , ) ( )1

p qn

i j p qn i j

i j ii j

MVNLNWBMH a a a a a

According to the operational laws in Definition 7, the results are derived below:


11

1

, 1

1 2

( ) ( )1

1

( ), ( ) 1 1

( , , ) ,

1

1 1 1 11 1

p qn

i j p qi j

ii ji j

i i j j

n

a a

p q

T a T a

p q p q

MVNLNWBMH a a a s

x y

x y

x y x y

x y x y

1 1

( ), ( ) 1 1

1

( ), ( )

,

1 1 11 1

,

1 1 1 11 1

1 1 11 1

i i j j

i i j j

p q p q

I a I a

p q p q

p q

F a F a

h h

g h g h

h h

g h g h

u u

v u v u

1

, 1

1

1 1

( ) ( )1

1

( ), ( ) 1 12

1 1 1 11 1

,

1 1

p qn

i j p qi j

ii ji j

i i j j

p q

p q p q

a a

p q

T a T a

p q p q

u u

v u v u

s

x y

x y x y

1 12

( ), ( ) 1 12

1 12

( ), ( ) 1 12

,

1,

1 1

1

1 1

i i j j

i i j j

p q p q

I a I a

p q p q

p q p q

F a F a

p q p q

g h g h

g h g h

v u v u

v u v u

(1)

Where

12

, 1

1 1 1 ,

i j

i

n p qp q

i j i ji ji j

x


12

1, 1

1 1 ,

i j

i

n p qp q

i j i ji ji j

y

12

, 1

1 1 1 1 ( 1) 1 1 ,

i j

i

n p q qp

i j i ji ji j

g

1, 1

1 1 1 1 1 1 ,

i j

i

n p q qp

i j i ji ji j

h

12

, 1

1 1 1 1 ( 1) 1 1 .

i j

i

n p q qp

i j i ji ji j

v

1

, 1

1 1 1 1 1 1 ,

i j

i

n p q qp

i j i ji ji j

u

Proof. According to the operational rules for MVNLNs, the results below can be gained

( ) ( )

( )

( )

, ,1 1 1 1

1 1 1,

1 1 1 1

1 1 1

1 1 1 1

pi i i

i i

i i

pp i

i pa T a p

i i

p p

i i

pI a p

i i

p p

i i

pF a p

i i

a s

( ) ( )

( )

( )

, ,1 1 1 1

1 1 1,

1 1 1 1

1 1 1

1 1 1 1

qj j j

j j

j j

q

jq

j qa T aq

j j

q q

j j

qI a q

j j

q q

j j

qF a q

j j

a s


13

( ) ( ) ( ), ( )

( ), ( )

, ,1 1 1

1 1 1 1 1 1,

1 1 1 1 1 1 1

1 1 1 1

p qi j i i j j

i i j j

p qi jp q

i j p qa a T a T a p q

i j i j

p q qp

i j i j

p qI a I a qp

i j i j

p q

i j

a a s

( ), ( )

1 1

1 1 1 1 1 1 1i i j j

qp

i j

p qF a F a qp

i j i j

Firstly, we need to testify the mathematical formula below.

, 1

, 1

1 12

, 1 , 1

( ) ( )1 12

, 1

( )1

1 1 1 1 1

,

1 1 1 1

i j i j

i i

ni j p q i j

i ji ii j

i j

ni j p q

i ji j ii j

n np q p qp q p q

i j i j i j i ji j i ji j i j

a a n p qp q

i j i ji ji j

a a

s

( ), ( )

1

, 1

1

, 1

12

,

1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1 1 1 1 1 1 1

i ji i j j

i

i j

i

i j

i

T a T an p q

p q

i j i ji ji j

n p q qp

i j i ji ji j

p q p qqp p

i j i j i j i

( ), ( )

1

, 1 , 1

1

, 1

1 12

,

,

1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1 1 1 1 1 1 1 1

i ji i j j

i

i j

i

i j i j

i i

I a I an n

q

ji j i ji j i j

n p q qp

i j i ji ji j

p q p qq qp p

i j i j i j i ji

( ), ( )

, 1 1

.i i j jF a F a

n n

i j ji j i j

(2)

The mathematical induction on n is adopt to prove Eq.(2)

(1) Supposing 2,n the equation below is obtained.

1 2 2 11 2 2 1

1 2

21 2 2 1

1 2 2 1, 1 1 2

1 2 1 2

( ), ( )( ) ( ) ( ) ( )

1 2 1 21 1

1 2

1 2 1 2

( ) ( ) ( )1 1 1

, ,1

,1

p q p q i i j j

i

i j p q p q p qi j

i j ii j

T a T aa a a a

I

a a a a a a

x x y ys

x x y y

m m

z z m m

( ), ( )

1 2

( ), ( )1 2 1 2

1

i j j

i i j j

a I a

F a F a

v v

u u v v

Where

1 21 1

1 21

1 2 1 2

1 1 ,1 1 1

p q

p qp q

x


14

2 11 2

2 12

2 1 2 1

1 1 ,1 1 1

p q

p qp q

x

1 21 1

1 21

1 2 1 2

1 ,1 1 1

p q

p qp q

y

2 11 2

2 12

2 1 2 1

1 ,1 1 1

p q

p qp q

y

1 21 1

1 2 1 2

1

1 2 1 2

1 1 1 1 1 1,

1 1 1 1 1 1 1

p q p q

p q p qm

2 11 2

2 1 2 1

2

2 1 2 1

1 1 1 1 1 1,

1 1 1 1 1 1 1

p q p q

p q p qm

1 21 1

1 2 1 2

1

1 2 1 2

1 1 1 1 1 11 1 1 ,

1 1 1 1 1 1 1

p q p q

p q p qz

2 11 2

2 1 2 1

2

2 1 2 1

1 1 1 1 1 11 1 1 ,

1 1 1 1 1 1 1

p q p q

p q p qz

1 21 1

1 2 1 2

1

1 2 1 2

1 1 1 1 1 1,

1 1 1 1 1 1 1

p q p q

p q p qv

2 11 2

2 1 2 1

2

2 1 2 1

1 1 1 1 1 1,

1 1 1 1 1 1 1

p q p q

p q p qv

1 21 1

1 2 1 2

1

1 2 1 2

1 1 1 1 1 11 1 1 ,

1 1 1 1 1 1 1

p q p q

p q p qu

2 11 2

2 1 2 1

2

2 1 2 1

1 1 1 1 1 11 1 1 ,

1 1 1 1 1 1 1

p q p q

p q p qu


15

Then,

2

, 1

21 2 2 1

1 2 2 1, 1 1 2

2 21 12

, 1 , 1

( ) ( )1

( ) ( ) ( )1 1 1

1 1 1 1 1

,

1

i j i j

i i

i j p qi j

ii ji j

i j p q p q p q

i ji j ii j

p q p qp q p q


a a

i

a a a a a a

s

( ), ( )2 2

1 12

, 1 , 1

21

, 1

2

,

1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 ( 1)

i j i ji i j j

i i

i j

i

T a T ap q p q

p q p q

j i j i j i ji j i ji j i j

p q qp

i j i ji ji j

p q

i j

( ), ( )2 2

1 1

, 1 , 1

21

, 1

12

,

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1

i j i ji i j j

i i

i j

i

i j

I a I ap qq qp p

i j i j i ji j i ji j i j

p q qp

i j i ji ji j

p q qp

i j i j

( ), ( )

2 21

, 1 , 1

.

1 1 1 1 1 1 1

i ji i j j

i i

F a F ap q qp

i j i ji j i ji j i j

We can achieve Eq. (2) is right when n k .

(2) Supposing ,n k the equation below is right, then

, 1

, 1

1 12

, 1 , 1

( ) ( )1 12

, 1

( )1

1 1 1 1 1

,

1 1 1 1

i j i j

i i

ki j p q i j

i ji ii j

i j

ki j p q

i ji j ii j

k kp q p qp q p q


a a k p qp q

i j i ji ji j

a a

s

( ), ( )

1

, 1

1

, 1

12

,

1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1 1 1 1 1 1 1

i ji i j j

i

i j

i

i j

i

T a T ak p q

p q

i j i ji ji j

k p q qp

i j i ji ji j

p q p qqp p

i j i j i j i

( ), ( )

1

, 1 , 1

1

, 1

1 12

,

,

1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1 1 1 1 1 1 1 1

i ji i j j

i

i j

i

i j i j

i i

I a I ak k

q

ji j i ji j i j

k p q qp

i j i ji ji j

p q p qq qp p

i j i j i j i ji

( ), ( )

, 1 1

.i i j jF a F a

k k

i j ji j i j

If 1n k , we need to calculate the equation below.

111

1 1, 1 , 1 1 1 1

( ) ( ) ( ) ( )1 1 1 1

k k k ki j i j k jp q p q p q p qi k

i j i j i k k ji j i j i ji i i ki j i j

a a a a a a a a

The mathematical induction on k are used to testify the equations below.

11

1

1 1

1

11

( ) ( )11

1 12

1 1 1 11 1

12

1 1

( ) ,1

1 1 1 1 1

1 1 1

kp qi k

i kii

i k i k

i i

i k

i

kp qi k

i ka ai i

k kp q p qp q p q

i k i k i k i ki i

p qp q

i k i ki

a a s

11 1

1

1

( ), ( )

1

1 11 1

1

1 11

12

1 1

,

1 1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1 1 1 1

i ki i k k

i

i k

i

i k

i

T a T ak k p q

p qi k i k

i

k p q p q

i k i ki

p q p q

i k i k

11 1

1

1

( ), ( )

1

1 11 1

1

1 11

12

1 1 1

,

1 1 1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1 1 1 1 1 1

i ki i k k

i

i k

i

i k

i

I a I ak k p q p q

i k i ki i

k p q p q

i k i ki

p q p qp q

i k i k i k

11 1( ), ( )

1

11 1

1 1

i ki i k k

i

F a F ak k

p q

i ki i


16

11

11

1 1

1 1

1

1

1( ) ( )1 1 1

1 12

1 1 1 11 1

2

1 1

( ) ,1

1 1 1 1 1

1 1 1

kk j p q

k jkj

k j k j

k k

k

kk j p q

k ja aj k

k kp q p qp q p q

k j k j k j k jj j

p qp q

k j k j

a a s

11 1

1 1

1

1

1

( ), ( )

1 1

1 11 1

1

1 11

2

1 1

,

1 1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1

j k jk k j j

k k

k j

k

k

T a T ak k p q

p q

k j k jj j

k p q qp

k j k jj

p q qp

k j k j

11 1

1 1

1

1

1

( ), ( )

1 1

1 11 1

1

1 11

12

1 1

,

1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 ( 1) 1 1

j k jk k j j

k k

k j

k

k j

I a I ak k p q qp

k j k jj j

k p q qp

k j k jj

p q qp

k j k j

11 1

1 1

( ), ( )

1

1 11 1

1 1 1 1 1 1 1

k jk k j j

k k

F a F ak k p q qp

k j k jj j

Therefore,

1

, 1

111

1 1, 1 , 1 1 1 1

112

, 1

( ) ( )1

( ) ( ) ( ) ( )1 1 1 1

1 1 1

,

i j

i

ki j p q

i jii j

i j

k k k ki j i j k jp q p q p q p qi k

i j i j i k k ji j i j i ji i i ki j i j

k p qp q

i j i ji ji j

a a

a a a a a a a a

s

11

, 1

( ), ( )1 1

1 12

, 1 , 1

1 1

,

1 1 1 1 1 1

1 1 1 1

i j

i

i j i ji i j j

i i

k p qp q

i j i ji ji j

T a T ak kp q p q

p q p q


p q

i j

11

, 1

( ), ( )1 1

1 12

, 1 , 1

1 1

,

1 1 1 1 ( 1) 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

i j

i

i j i ji i j j

i i

kqp

i ji ji j

I a I ak kp q p qq qp p


p q p

i j i

11

, 1

( ), ( )1 1

1 12

, 1 , 1

.

1 1 1 1 ( 1) 1 1 1 1 1 1 1 1 1

i j

i

i j i ji i j j

i i

kq

ji ji j

F a F ak kp q p qq qp p


That is, If 1,n k Eq. (2) is right. Therefore, for all ,n Eq. (2) is right.

Then, Eq. (1) is right.

In the following, the properties of MVNLNWBMH operator will be proved.

(1) Reducibility. Let 1 1 1, , , .

n n n

Then

1 2 1 2( , , ) ( , , ).

n nMVNLNWBMH a a a MVNLBMH a a a

Proof. Since 1 1 1, , , ,

n n n

then according to the operations in Defin ition 10, the

result below can be obtained.


17

1

1 2, 1

1

2

, 1

1

, 1

1

, 1

1

( , , ) ( )1

1

( )1

1

1( )

( 1)

1( )

( 1)

( ,

p qn

i j p q

n i ji j ii j

p q

np q

i ji ji j

p qn

p q

i ji ji j

p qn

p q

i ji ji j

MVNLNWBMH a a a a a

n a a

n

a an n

a an n

MVNLBMH a a

2, ).

na

(2) Idempotency. Let 1,2, , .ia a i n Then 1 2

( , , ) .n

MVNLNWBMH a a a a

Proof. For each ,i owing to ,ia a the formula below is obtained on the basis of Eq. (5) in

Theorem 1.

1

1 2, 1

1

, 1

1

, 1

1

, 1

( , , ) ( )1

( )1

1

1

.

p qn

i j p q

n i ji j ii j

p qn

i j p q

i j ii j

p qn

i j p q

i j ii j

p qn

i j

i j ii j

MVNLNWBMH a a a a a

a a

a

a

a

(3) Commutativity. Let 1 2, ,

na a a be any permutation of 1 2

, , .n

a a a Then

1 2 1 2, , ( , , ).

n nMVNLNWBMH a a a MVNLNWBMH a a a

Proof. Owing to 1 2, ,

na a a is permutation of 1 2

( , , )n

a a a , then the equation below can be

obtained.


18

1

1 2, 1

1

, 1

( , , ) ( )1

( )1

p qn

i j p q

n i ji j ii j

p qn

i j p qi j

i j ii j

MVNLNWBMH a a a a a

a a

(4) Monotonicity. Suppose

( ), ( ), ( ), ( ) ( 1,2, , )

ii a i i ia s T a I a F a i n

and

( ), ( ), ( ), ( ) ( 1,2, , )

ii b i i ib s T b I b F b i n

are two sets of MVNLNs,

when ( ) ( ),

i ia bs s

( ) ( ),i i

T a T b ( ) ( )i i

I a I b and ( ) ( )i i

F a F b for each ,i then

1 2 1 2( , , ) ( , , ).

n nMVNLNWBMH a a a MVNLNWBMH b b b

Proof. (I) Linguistic term part

Owing to , 0,p q and ( ) ( )i ia b

s s

for each ,i the result below is gained.

( ) ( )p pi ia b

and ( ) ( )q q

i ia b

, 1 , 1

1

, 1 ,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )1 1

( ) ( ) ( ) ( )1 1

( ) ( ) ( ) ( )1 1

p q p qi i i i

i j i jp q p qi i i i

i i

n ni j i jp q p q

i i i ii j i ji ii j i j

p qn

i j i jp q p qi i i i

i j i ji ii j

a a b b

a a b b

a a b b

a a b b

1

1

p qn

i j

(II) True, indeterminacy and falsity membership parts

Owing to ( ) ( ),i i

T a T b ( ) ( )i i

I a I b and ( ) ( )i i

F a F b for each ,i , 0,p q then the

following results can be proved easily.

1 1

1 11 1

2 2

;

1 1 1 1

i i i i

i i i i i i i i

p q p qa a b b

p q p qp q p qa a a a b b b b

x y x y

x y x y x y x y

1 11 1

2 2

1 11 1

2 2

1 1;

1 1 1 1

i i i i i i i i

i i i i i i i i



g h g h g h g h

g h g h g h g h

1 11 1

2 2

1 11 1

2 2

1 1.

1 1 1 1

i i i i i i i i

i i i i i i i i



v u v u v u v u

v u v u v u v u

Where the corresponding , , , ,x y g h u and v are defined in Definition 10.

(III) Comparing 1 2( , , )

nMVNLNWBMH a a a with 1 2

( , , )n

MVNLNWBMH b b b

Suppose ( ) 1 2, ( ), ( ), ( ) ( , , )

a na s T a I a F a MVNLNWBMH a a a

and


19

( ) 1 2, ( ), ( ), ( ) ( , , ).

b nb s T b I b F b MVNLNWBMH b b b

Because ( ) ( ), ( ) ( ), ( ) ( )

a bs s T a T b I a I b

and ( ) ( ),F a F b thus .a b

Then 1 2 1 2( , , ) ( , , ).

n nMVNLNWBMH a a a MVNLNWBMH b b b

In the following, a few special examples for MVNLNWBMH operator regarding

different values ,p and q will be explored.

(1) If 0,q then the MVNLNWBMH operator defined by equation (1) will be reduced

to the generalized multi-valued neutrosophic linguistic Hamacher weighted average

(GMVNLHWA) operator shown as below.

1

1

1

1 2, 1

1

, 1

1

1

1

1

( )

1

( ), ( )

( , , ) ( )1

1

1

1

,n p

pi i

i

i i j j

p qn

i j p q

n i ji j ii j

pn

i j p

ii j ii j

pni i p

ii i

n pp

i ii

a

p

T a T a

GMVNLHWA a a a a a

a

a

a

s

x y

x

1 12

1 12

( ), ( ) 1 12

1 12

( ), ( ) 1 12

,

1 1

1,

1 1

1

1 1

i i j j

i i j j

p p

p p

I a I a

p p

p p

F a F a

p p

y x y

g h g h

g h g h

v u v u

v u v u

Where

2

1

1 1 1 1 ,in p

pi i

i

x

1

1 ,in p

pi i

i

y

1

1 1 1 ,in p p

i ii

h


20

2

1

1 1 1 1 ,in p p

i ii

g

1

1 1 1 ,in p p

i ii

u

2

1

1 1 1 1in p p

i ii

v

(2) When 1, 0,p q then the operator of MVNLNWBMH defined by equation (1)

will be reduced to the multi-valued neutrosophic linguistic Hamacher weighted

arithmetic average (MVNLHWAA) operator shown as below.

1

1 21

( )

1 1

( )

1 1

1

( )

1 1

(

( , , )

,

1 1 1

,

1 1 1 1

,

1 1

n

i ii

i i

i ii i

i

i ii

i

i

n

n i ii

a

n n

i ii i

T a n n

i ii i

n

ii

I a n n

i ii i

F a

MVNLHWAA a a a a

s

1

)

1 1

1 1

i

ii

i

n

ii

n n

i ii i

If 1, the MVNLHWAA operator will be reduced to MVNLWAA operator shown

as below.

1

1 2( )

( )1

( )1

( )1

( , , ) ,

1 1 ,

,

n

i ii

i

i i

i

i i

i

i i

na

n

iT ai

n

iI ai

n

iF ai

MVNLWAA a a a s

If 2, the MVNLHWAA operator will be reduced to the multi-valued neutrosophic

linguistic Einstein weighted arithmetic average (MVNLEWAA) operator shown in the

following.


21

1

1 2( )

1 1

( )

1 1

1

( )

1 1

1

( )

1

( , , ) ,

1 1

,

1 1

2

,

2

2

2

n

i ii

i i

i ii i

i

i ii i

i

i ii

na

n n

i ii i

T a n n

i ii i

n

ii

I a n n

i ii i

n

ii

F a n

i ii

MVNLWAA a a a s

1

i

n

i

(3) If 0, 0,p q then the MVNLNWBMH operator defined by equation (1) will be

reduced to the multi-valued neutrosophic linguistic Hamacher weighted geometric

average (MVNLHWGA) operator shown as below.

1

1 21

( )

1

( )

1 1

1 1

( )

1 1

(

( , , )

,

,

1 1

1 1 1

,

1 1 1 1

i

ni

ii

i

i ii

i

i i

i ii i

i

n

n ii

a

n

ii

T a n n

i ii i

n n

i ii i

I a n n

i ii i

F a

MVNLHWGA a a a a

s

1 1

)

1 1

1 1 1

1 1 1 1

i i

ii i

n n

i ii i

n n

i ii i

If 1, the MVNLHWGA operator will be reduced to MVNLWGA operator shown

as below.


22

1

1 2( )

( )1

( )1

( )1

( , , ) ,

,

1 1 ,

1 1

ni

ii

i

i i

i

i i

i

i i

na

n

iT ai

n

iI ai

n

iF ai

MVNLHWGA a a a s

If 2, the MVNLHWGA operator will be reduced to the multi-valued neutrosophic

linguistic Einstein weighted geometr ic average (MVNLEWGA) operator shown as

below.

1

1 2( )

1

( )

1 1

1 1

( )

1 1

1 1

( )

( , , ) ,

2

,

2

1 1

,

1 1

1 1

ni

ii

i

i ii i

i i

i ii i

i i

i i

na

n

ii

T a n n

i ii i

n n

i ii i

I a n n

i ii i

n n

i ii i

F a

MVNLHWGA a a a s

1 1

1 1i in n

i ii i

(4) If 1, then the MVNLNWBMH operator defined by equation (1) will be reduced

to the MVNLNWBM operator shown as below.


23

1

, 1

1 2

( ) ( )1

1

1( ), ( )

, 1

( ), ( )

( , , ) ,

1 1 ,

1 1 1 1 1

p qn

i j p qi j

ii ji j

i j

i

i i j j

i

i i j j

n

a a

p qn

p qi jT a T a

i ji j

qp

i jI a I a

MVNLNWBM a a a s

1

1

, 1

1

1

( ), ( ), 1

,

1 1 1 1 1

j

i

i j

i

i i j j

p qn

i ji j

p qn

qp

i jF a F ai ji j

(5) If 2, then the MVNLNW BMH operato r defined by equat ion (1) will be reduced to the

MVNLNW BME, that is the s implificat ion o f mult i-valued neut rosophic lingu ist ic NW BM

Einstein.

1

, 1

1 2

( ) ( )1

1

( ), ( ) 1 1

1 1

( ), ( ) 1 1

( , , ) ,

2,

3

3

3

p qn

i j p qi j

ii ji j

i i j j

i i j j

n

a a

p q

T a T a

p q p q

p q p q

I a I a

p q p q

MVNLNWBMH a a a s

x y

x y x y

g h g h

g h g h

1 1

( ), ( ) 1 1

,

3

3i i j j

p q p q

F a F a

p q p q

v u v u

v u v u

Where

1

, 1

2 2 3 ,

i j

i

nqp p q

i j i ji ji j

x

1, 1

2 2 ,

i j

i

nqp p q

i j i ji ji j

y

1, 1

1 1 1 1 ,

i j

i

nq qp p

i j i ji ji j

h


24

1

, 1

1 1 3 1 1 ,

i j

i

nq qp p

i j i ji ji j

g

1

, 1

1 1 1 1 ,

i j

i

nq qp p

i j i ji ji j

u

1, 1

1 1 3 1 1 .

i j

i

nq qp p

i j i ji ji j

v

From the above analysis, we can obtain the MVNLNWBMH operator is more generalized.

5 The multiple criteria decision making approach based on the MVNLNWBMH operator

The proposed MVNLNW BMH operator is presented to cope with MCDM problem under multi-

valued neutrosophic linguistic environment in this subsection.

Suppose that 1 2, , ,

mA A A A represent m alternatives, and 1 2

, , ,n

C C C C represent n

criteria. Let 1 2, , ,

n be the corresponding weights of criteria,

where 0( 1,2, , ),j

j n and

1

1n

jj

.The evaluation value of the criteria

jC ( 1,2, , )j n regarding the alternative i

A ( 1,2, , )i m is provided by experts. Each value

is represented by MVNLNNs. Suppose that ij m nR a

is the multi-valued neutrosophic linguistic

decision matrix, ( )

, ( ), ( ), ( ) ijij a ij ij ij

a s T a I a F a

is the evaluation information which

represents the assessment value of alternative ( 1,2, , )iA i m on criteria ( 1,2, , )

jC j n with

respect to the linguistic value ( ),

ijas where ( )

ijT a indicates the satisfaction degree, ( )

ijI a indicates

the indeterminacy degree and ( )ij

F a indicates the dissatisfaction degree.

Then, the main method for ranking and selecting the best alternative is presented in the following.

Step1. The decision matrix is normalized.

Generally, criteria in MCDM problems consist of two types: maximum type and minimum type,

the min imum type should be transformed into the maximum type for eliminat ing the influence of

distinguished types. Suppose that ij m n

R a

is the original decision matrix, which can be

normalized as follows:

for maximizing criteria

for maximizing criteria( )

,

, ( ), ( ), ( ) ,ij

ij

ij

a ij ij ij

ab

s T a I a F a

Thus, the normalized matrix

ij m nB b is gained.

Step2. The comprehensive value of each alternative is calculated.

The comprehensive value represented by ( 1,2, , )ia i m can be obtained by utilizing the

MVNLNWBMH operator in Definit ion 10, which can aggregate the overall value fo r each alternative

with respect to all criteria.

Step3. The compared values of three functions are calculated.

According to the equations given in Defin ition 8, the score value denoted byE( )ia , the accuracy

value denoted by H( )ia and the certainty value denoted by C( )

ia can be obtained.


25

Step4. The alternatives are selected.

Based on Defin ition 9, all alternatives ( 1,2, , )iA i m can be ranked on the basis of

E( ),ia H( )

ia and C( )

ia , and the best alternative(s) can be selected.

6 An numberical example

In order to validate the effect iveness and practical of the novel approach, an investment project is

adapted from Ye.

An investment company wants to expand its business. Four alternatives will be chosen,

1A represents auto corporation, 2

A represents food corporation, 3A represents computer company

corporation), 4A represents weapon corporation. Each alternative is evaluated under three criteria,

1C denotes risk, 2

C denotes growth, 3C denotes the impact of environment, where 3

C is the minimizing

criteria. The corresponding weighted vector is 0.35,0.25,0.4 . In real situation, the decision

maker may hesitant and give several possible value for the satisfaction, indeterminacy and

dissatisfaction regarding the alternative iA corresponding to the criteria j

C 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

extremely poor,very poor,poor,medium,good,very good,extremely good1 2 3 4 5 6 7, , , , , , .S s s s s s s s

The multi-valued neutrosophic linguistic decision matrix

4 3ijR a

is shown as follows.

4 3

5 6 5

6 5 5

6 5 4

, 0.3,0.4,0.5 , 0.1 , 0.3,0.4 , 0.5,0.6 , 0.2,0.3 , 0.3,0.4 , 0.2,0.3 , 0.1,0.2 , 0.5,0.6

, 0.6,0.7 , 0.1,0.2 , 0.2,0.3 , 0.6,0.7 , 0.1 , 0.3 , 0.6,0.7 , 0.1,0.2 , 0.1,0.2

, 0.5,0.6 , 0.4 , 0.2,0.3 , 0.6 , 0.3 , 0.4 ,

ijR a

s s s

s s s

s s s

4 4 6

0.5,0.6 , 0.1 , 0.3

, 0.7,0.8 , 0.1 , 0.1,0.2 , 0.6,0.7 , 0.1 , 0.2 , 0.3,0.5 , 0.2 , 0.1,0.2,0.3s s s

6.1 The procedure using the proposed aggregation operator

Step1. The decision matrix is normalized.

Because 3C is the min imizing criteria, which should be converted to the maximizing criteria, then

the normalized decision matrix .ij m n

B b

can be obtained as follows:

4 3

5 6 2

6 5 2

6 5 3

, 0.3,0.4,0.5 , 0.1 , 0.3,0.4 , 0.5,0.6 , 0.2,0.3 , 0.3,0.4 , 0.2,0.3 , 0.1,0.2 , 0.5,0.6

, 0.6,0.7 , 0.1,0.2 , 0.2,0.3 , 0.6,0.7 , 0.1 , 0.3 , 0.6,0.7 , 0.1,0.2 , 0.1,0.2

, 0.5,0.6 , 0.4 , 0.2,0.3 , 0.6 , 0.3 , 0.4 ,

ijB b

s s s

s s s

s s s

4 4 1

0.5,0.6 , 0.1 , 0.3

, 0.7,0.8 , 0.1 , 0.1,0.2 , 0.6,0.7 , 0.1 , 0.2 , 0.3,0.5 , 0.2 , 0.1,0.2,0.3s s s

Step2. The comprehensive value of each alternative is calculated.

Derive the comprehensive value ( 1,2, , )ia i m of each alternative ( 1,2, , )

iA i m

by using the MVNLNWBMH operator presented in Definition 10. Here let 1, 1.p q The

MVNLNWBMH operator is shown as below:


26

1

2

, 1

1 2

( ) ( )1

1

2

1( ), ( )

, 1

1( ), ( )

, 1

( , , ) ,

1 1 ,

1 1 1 1 1

ni j

i jii j

i j

i j

i

i i j j

i j

i

i i j j

n

a a

n

i jT a T ai ji j

i jI a I ai j

MVNLNWBM a a a s

1

2

1

2

1( ), ( )

, 1

,

1 1 1 1 1i j

i

i i j j

n

i j

n

i jF a F ai ji j

And we have

1 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 s

2 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 s

3 4.4850,

0.5272,0.5644,0.5618,0.6 ,

0.263 ,

0.2918,0.3273 ;

a s

Step3. The compared values of three functions are calculated.

By using equations in Defin ition 8, we can obtain E( )ia , H( )

ia and C( ),( 1,2,3,4)

ia i as

follows:

1 2.3605 1 0.2036 1 1.4879

E( ) , ( ) , ( ) ;a s H a s C a s

2 3.0440 2 1.6912 2 2.5930

E( ) , ( ) , ( ) ;a s H a s C a s

3 2.9763 3 1.1382 3 2.5267E( ) , ( ) , ( ) ;a s H a s C a s

4 2.0002 4 1.0641 4 1.5461E( ) , ( ) , ( ) ;a s H a s C a s

Step4. The alternatives are selected.

By using the compared approach in Definition 9, 2 3 1 4E( ) E( ) E( ) E( )a a a a can


27

be obtained, so the final raking of alternatives is 2 3 1 4.A A A A Apparently, 2

A is the

best one, and 4A is the worst one. We don’t need to compare the other functions because the values of

score function differ.

6.2 Comparison analysis

We take d ifferen t value into cons iderat ion in step 2 to select the alternat ive fo r d iscussing the

impact of different values ,p q and . The comparisons are presented in Table 1 and Table 2.

Table 1 Ranking of alternatives utilizing different ,p q and 1

1, ,p q Score function E( ),( 1,2,3,4)ia i Ranking

0, 0p q 1 2.1214 2 2.8131 3 2.8732 4 1.7094( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

3 2 1 4A A A A

0.01, 0p q 1 2.1907 2 2.8460 3 2.9575 4 1.7629

( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 3 2 1 4A A A A

0.1, 0p q 1 2.2165 2 2.8790 3 2.9713 4 1.8005


1, 0p q 1 2.4697 2 3.1970 3 3.1084 4 2.1573( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

2, 0p q 1 2.7154 2 3.4882 3 3.2507 4 2.4519


5, 0p q 1 3.1768 2 3.9655 3 3.5749 4 2.8445


10, 0p q 1 3.5203 2 4.2619 3 3.8673 4 3.0415( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

0, 1p q 1 2.6253 2 3.3082 3 3.1563 4 2.2904


0.01, 1p q 1 2.6151 2 3.2980 3 3.1492 4 2.2778


0.1, 1p q 1 2.5377 2 3.2202 3 3.0964 4 2.1839( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

1, 1p q 1 2.3605 2 3.0440 3 2.9763 4 2.0002


2, 1p q 1 2.4824 2 3.1918 3 3.0464 4 2.1664


5, 1p q 1 2.9208 2 3.6644 3 3.3532 4 2.5870( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

10, 1p q 1 3.3239 2 4.0427 3 3.6946 4 2.8680


0, 2p q 1 2.8597 2 3.5712 3 3.2871 4 2.5589


0.01, 2p q 1 2.8519 2 3.5630 3 3.2820 4 2.5494( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

0.1, 2p q 1 2.7889 2 3.4959 3 3.2400 4 2.4731


1, 2p q 1 2.5228 2 3.2087 3 3.0542 4 2.1890


2, 2p q 1 2.5363 2 3.2305 3 3.0440 4 2.2430( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

5, 2p q 1 2.8535 2 3.5768 3 3.2558 4 2.5581


10, 2p q 1 3.2306 2 3.9375 3 3.5836 4 2.8212( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 2 3 1 4

A A A A

Table 2 Ranking of alternatives utilizing different ,p q and 2

2, ,p q Score function E( ),( 1,2,3,4)ia i Ranking

0, 0p q 1 2.1344 2 2.8137 3 2.8863 4 1.7187( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 3 2 1 4

A A A A

0.01, 0p q 1 2.1944 2 2.8478 3 2.9631 4 1.7661( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 3 2 1 4

A A A A

0.1, 0p q 1 2.2171 2 2.8803 3 2.9752 4 1.8021( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 3 2 1 4

A A A A

1, 0p q 1 2.4553 2 3.1938 3 3.0982 4 2.1483( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 2 3 1 4

A A A A

2, 0p q 1 2.7142 2 3.4825 3 3.2361 4 2.4453( ) , ( ) , ( ) , ( )E a s E a s E a s E a s 2 3 1 4

A A A A


28

5, 0p q 1 3.2212 2 3.9697 3 3.5908 4 2.8608


10, 0p q 1 3.5770 2 4.2923 3 3.9217 4 3.0737( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

0, 1p q 1 2.6103 2 3.3048 3 3.1457 4 2.2818


0.01, 1p q 1 2.6011 2 3.2948 3 3.1394 4 2.2700


0.1, 1p q 1 2.5312 2 3.2188 3 3.0925 4 2.1809( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

1, 1p q 1 2.3702 2 3.0464 3 2.9849 4 2.0055


2, 1p q 1 2.4994 2 3.1940 3 3.0556 4 2.1751


5, 1p q 1 2.9637 2 3.6734 3 3.3833 4 2.6091( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

10, 1p q 1 3.3787 2 4.0718 3 3.7516 4 2.9010


0, 2p q 1 2.8577 2 3.5652 3 3.2720 4 2.5525


0.01, 2p q 1 2.8506 2 3.5572 3 3.2675 4 2.5435( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

0.1, 2p q 1 2.7919 2 3.4917 3 3.2304 4 2.4707


1, 2p q 1 2.5398 2 3.2108 3 3.0635 4 2.1976


2, 2p q 1 2.5586 2 3.2343 3 3.0581 4 2.2559( ) , ( ) , ( ) , ( )E a s E a s E a s E a s

2 3 1 4A A A A

5, 2p q 1 2.8950 2 3.5871 3 3.2878 4 2.5824


10, 2p q 1 3.2843 2 3.9650 3 3.6392 4 2.8549


In tab le 1，we take the parameter value 1, which is based on Algebraic operation, and the

MVNLNWBM operator is applied. In tab le 2，we take the parameter value 2, which is

based on Einstein operation, and the MVNLNW BME operator is applied. As we can see from

table 1 and table 2, the ordering of alternatives taking different parameters ,p q and may

be different, because the different parameters will cause d ifferent score funct ion value.

However, 2A or 3

A is always the best select ion , and 4A is always the worst select ion. Whether

1 or 2, the same ranking results is obtained with regard to the same parameter

value p and the same parameter value q except fo r one s ituat ion in

which 0, 1p q and 1 .Specially , if 0, 0,p q MVNLNW BMH will reduce to

MVNLHW GA operato r. If 0, 0p q and 1 , MVNLHW GA will reduce to MVNLW GA

operator. When 1, 0p q , MVNLNW BMH will reduce to MVNLHWAA operator. If

1, 0p q and 1 , MVNLHWAA will reduce to MVNLWAA operator. If 1 in Table 1,

the ranking results on the basis of the MVNLW GA and MVNLWAA operators differ, which due to the

two operators emphasis on different major points, and the same situation happens in Table 2.When the

parameters 0q and 0, 0.01, 0.1,p p p respectively, the rankings are identical in two

Tables, the ranking order is always 3 2 1 4A A A A . When p and q are assign the other values in

two Tables, the ranking order is changed, and the result is 2 3 1 4A A A A .That is, the best

selection is from 3A to 2

A except for one situation where 0, 1p q in Table 1.

For illustrating the effective and flexib le of the novel approach, the method in literature is adopted in

multi-valued neutrosophic linguistic environment in this paper, and the same ranking orders are

obtained in ref [8] where the SVNHFWA and SVNHFW G operators are adopted to fuse single-valued

neutrosophic hesitant fuzzy information. When 1, 0, 0p q and 1, 0, 0,p q the

two operators are special cases of MVNLNWBMH operator .Therefore, novel operator in this paper

has better flexib ility and generalizat ion. In actual cases, the decision makers can assign different


29

parameter values ,p and .q . Generally, for convenience, we can set 1,p which can not only

simplify the calculation, but also consider the interrelationship of multiple values.

7 Conclusions

In this paper, the MVNLS 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 to express the decision informat ion. Moreover, NW BM

is a useful operator which has the trait of taking into account the interrelationship of different

arguments, and overcome drawbacks of non-reducib ility and non-idempotency. 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 MVNLNW BMH operator is proposed, which is appropriate to deal

with MVNL information. Some desirable properties of the novel operator are discuss ed in detail, and

some special cases are analyzed. Furthermore, the comparison method for MVNLNs is also studied,

and the rankings of alternatives affecting by d ifferent parameters ,p q and are also compared. For

verify ing 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 flexib le and

more general, and which can capture the interrelationship among arguments and express decision

informat ion more practical, the decision makers can assign appropriate values according to the real

situation. In future, we will exp lore to apply 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 [61],

i.e. the truth value T is refined into types of sub-truths such as T1, T2, etc.,

similarly indeterminacy I is refined into types of sub-indeterminacies I1, I2, etc., and the sub-falsehood

F is split into F1, F2, etc.

Acknowledgement

This paper is supported by the National Natural Science Foundation of China (No. 71371154), the

Humanities and Social Sciences Research Pro ject of Min istry of Education of China (No.

16YJCZH049), Th inking Bank Project of Hubei (No. HBSXK2017055) and the Humanit ies and social

Sciences foundation of Department of Education of Hubei (No. 17Q121).The authors also would like to

express appreciation to the anonymous reviewers and Editor for their helpfu l comments that improved

the paper.

Compliance with ethical standards

Conflict of interest：The authors declare that they have no conflict of interests regarding the

publication of this paper.

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