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Some Neutrosophic Uncertain Linguistic Number Heronian Mean Operators and

Their Application to Multi-Attribute Group Decision Making

Peide Liu a,b,*, Lanlan Shi a, a,School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong

250014, ChinabSchool of Economics and Management, Civil Aviation University of China, Tianjin 300300, China

*The corresponding author: [email protected] , Tel: +86-531-82222188, Fax: +86-531-88525999

Abstract: Heronian mean (HM) is a useful aggregation operator which is marked by catching the interrelations of the

aggregated arguments and the neutrosophic uncertain linguistic set can be better to express the incomplete, indeterminate

and inconsistent information. In this paper, we combine the Heronian mean and the neutrosophic uncertain linguistic set

and proposed some Heronian mean operators based on neutrosophic uncertain linguistic numbers. Firstly, we introduce

some definition and properties of uncertain linguistic numbers, the single valued neutrosophic set, and some heronianmean (HM) operators including the generalized weighted Heronian mean (GWHM) operator, the improved generalized

weighted Heronian mean (IGWHM) operator, the improved generalized geometric weighted Heronian mean (IGGWHM)

operator. Then, we propose the single valued neutrosophic uncertain linguistic set by combining the uncertain linguistic

numbers and the single valued neutrosophic set. Further, the neutrosophic uncertain linguistic number improved

generalized weighted Heronian mean (NULNIGWHM) operator and the neutrosophic uncertain linguistic number

improved generalized geometric weighted Heronian mean (NLUNIGGWHM) operator are developed and the properties

of them are analyzed. Furthermore, we develop the decision making methods for multi-attribute group decision making

(MAGDM) problems with neutrosophic uncertain linguistic information and give the detail decision steps. At last, an

illustrate example is given to show the process of decision making and the effectiveness of the proposed method.

Keywords: multiple attribute group decision making (MAGDM); Heronian mean; neutrosophic uncertain linguistic set;

geometric Heronian mean

1. Introduction

Multiple attribute decision group making (MAGDM) problems exists extensively in many fields such as politics,

economy, military and culture. The attribute values in the decision-making problems are usually incomplete, indeterminate

and inconsistent due to the complexity and fuzziness of the real world. Zadeh [1] firstly proposed the fuzzy set (FS) theory

which has a membership function. Based on fuzzy set theory, Atanassov [2, 3] proposed the intuitionistic fuzzy set (IFS)which added a non-membership function. The intuitionistic fuzzy set is composed of the membership (or calledtruth-membership) ( ) AT x and non-membership (or called falsity-membership) ( ) A F x , and satisfies the

conditions , [0,1] A AT x F x and 1)()(0 x F xT A A . However, IFSs can merely deal with incomplete information, but

cannot do anything for the indeterminate information and inconsistent information. The indeterminacy (or calledHesitation degree) is x F xT A A 1 which is only given by default and cannot be solely expressed in IFSs. With

respect to this situation, Smarandache[4] developed the neutrosophic set (NS) which consists of thetruth-membership xT A , falsity-membership x F A and indeterminacy-membership x I A and the three variables are

independent completely. NS is a generalization of FS and IFS. Now there are many research achievements about NSs.

Wang et al. [5] further proposed a single valued neutrosophic set (SVNS)which is an special instance of the neutrosophicset by changing the conditions to that , , [0,1] A A AT x I x F x and 3)()()(0 x F x I xT A A A ; Ye [6] proved that the

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cosine similarity degree is a special case of the correlation coefficient in SVNS; Wang et al. [7] proposed the definition of

the interval neutrosophic sets (INSs) in which the truth-membership, indeterminacy-membership, and false-membership

were extended to interval numbers, and discussed some properties of INSs. Ye [8] defined the similarity measures between

INSs on the basis of the Hamming and Euclidean distances, and proposed a method for multi-criteria decision-making

problems.

In real problems, sometimes we can use linguistic terms such as ‘good ’, ‘ bad ’ to describe the state or performance of

a car and cannot use some numbers to express some qualitative information. However, when we use the linguistic

variables to express the qualitative information, it only means the membership degree belonged to a linguistic term is 1,

and the non-membership degree or hesitation degree cannot be expressed. In order to overcome this shortcoming, Wang

and Li [9] proposed the concept of intuitionistic linguistic set by combining intuitionistic fuzzy set and linguistic

variables. For the above-mentioned example, we can give an evaluation value ‘good ’ for the state of the car, however, for

this evaluation, we have the certainty degree of 80 percent and negation degree of 10 percent, and then we can use the

intuitionistic linguistic set to express the evaluation result. Furthermore, Wang and Li [9] proposed intuitionistic

two-semantics and the Hamming distance between two intuitionistic two-semantics, and ranked the alternatives by

calculating the comprehensive membership degree to the ideal solution for each alternative.The information aggregation operators which are widely applied in multiple attribute group decision-making problems

are a meaningful research scopes. Heronian mean (HM) is a useful aggregation operator which is marked by catching the

interrelations of the aggregated arguments. Beliakov [10] had firstly proved that Heronian mean was an aggregation

operator. On the basis of this, Skora [11,12] further extended to the generalized Heronian means and discussed two special

cases of them. Yu and Wu [13] proposed a generalized interval-valued intuitionistic fuzzy Heronian mean (GIIFHM) and a

generalized interval-valued intuitionistic fuzzy weighted Heronian mean (GIIFWHM) which extended Heronian mean

from dealing with crisp numbers to intuitionistic fuzzy numbers, and some desirable properties and special cases of these

operators were discussed. Yu [14] proposed some intuitionistic fuzzy aggregation operators based on HM, including the

intuitionistic fuzzy geometric Heronian mean (IFGHM) operator and the intuitionistic fuzzy geometric weighed Heronian

mean (IFGWHM) operator, and the properties of these operators were studied.

As mentioned above, the interactions among the attribute values are common in the real decision making problems.

Because Heronian mean operator can cope with the interactions among the attribute values and the neutrosophic set can be

better to express the incomplete, indeterminate and inconsistent information. However, there is no research on the HM

operator under neutrosophic uncertain linguistic environment. Hence, in this paper, we will extend the HM operator to the

neutrosophic uncertain linguistic set, and propose some Heronian mean operators based on neutrosophic uncertain

linguistic numbers, including the neutrosophic uncertain linguistic number improved generalized weighted Heronian mean

(NULNIGWHM) operator and the neutrosophic uncertain linguistic number improved generalized geometric weighted

Heronian mean (NLUNIGGWHM) operator, then applied them to multi-attribute group decision-making problems.

To achieve the purpose, the rest of this paper is organized as follows. In Section 2, we introduces the definition ofuncertain linguistic numbers, the single valued neutrosophic set, and some operators based on heronian mean(HM)

operator including the improved generalized weighted Heronian mean (IGWHM) operator and the generalized geometric

Heronian mean (GGHM) operator. In Section 3, on the basis of uncertain linguistic numbers and the single valued

neutrosophic set, we develop the single valued neutrosophic uncertain linguistic set and operational rules of it. Section 4

proposes some Heronian mean operators for the single neutrosophic uncertain linguistic numbers, such as the

neutrosophic uncertain linguistic number improved generalized weighted Heronian mean (NULNIGWHM) operator and a

neutrosophic uncertain linguistic number improved generalized geometric weighted Heronian mean (NLUNIGGWHM)

operator and introduces some properties and special cases of them. In Section 5, we propose the decision-making

methods based on the NULNIGWHM and NLUNIGGWHM operators. Section 6 shows a numerical example according

to our approach. Section 7 summarizes the main conclusion of this paper.

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2. Preliminaries

2.1 The linguistic set and uncertain linguistic numbers The linguistic set is regarded as a good tool to express the qualitative information, we can express the linguistic set

by 0 1 1( , , , )l S s s s ,and 1, 2, , 1( )l s

can be called an linguistic number, l is an odd value which can

be the values of 3,5,7,9,etc. For example, when 9l , 0 1 2 3 4 5 6 7 8( , , , , , , , , )S s s s s s s s s s =(extremely poor, very

poor, poor, slightly poor, fair, slightly good, good, very good, extremely good).

Let i s and j s be any two linguistic numbers in linguistic set S , they have the following characteristics [15,16]:

(i) If ji , then ji s s ;

(ii) There exists negative operator: ji s sneg )( , where il j 1 ;

(iii) If ji s s , max( , )i j i s s s ;

(iv) If i j s s , min( , )i j i s s s .

The continuous linguistic set |S s R , which can overcome the weakness of the loss of information in

the process of calculations, is the extension of original discrete linguistic set ),,,( 110 l s s sS , and

|S s R meets the strictly monotonically increasing condition [21, 22]. Some operational rules are defined as

follows[15,16].

(1) 0i i s s (1)

(2) i j i j s s s ( 2 )

(3) i j i j s s s (3)

(4) 0n

ni i

s s n (4)

Definition 1 [17]. Suppose ],[~ba s s s , ,a b s s S with ba are the lower limit and the upper limit of s~ , respectively,

then s~ is called an uncertain linguistic variable.

Let S be a set of all uncertain linguistic variables, ],[~111 ba s s s and ],[~

222 ba s s s be any two uncertain

linguistic variables, the operational rules are defined as follows [18,19]:

(1) 1 2 1 1 2 2 1 2 1 2[ , ] [ , ] [ , ]a b a b a a b b s s s s s s s s (5)

(2) 1 2 1 1 2 2 1 2 1 2[ , ] [ , ] [ , ]a b a b a a b b s s s s s s s s (6)

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(3) 1 1 1 * 1 * 1[ , ] [ , ], 0a b a b s s s s s (7)

(4) 1 1 1 1 1[ , ] [ , ], 0a b a b

s s s s s

(8)

2.2. The single valued neutrosophic setDefinition 2 [5]. Let X be a universe of discourse, with a generic element in X denoted by x . A single valued neutrosophic

set A in X is

X x x F x I xT x A A A A ))(),(),(( (9)

where )( xT A , )( x I A and )( x F A are the truth-membership, indeterminacy-membership and falsity-membership function,

separately. For each point x in X , we have that ),( xT A ),( x I A )( x F A 0 1 , and 0 ( ) ( ) ( ) 3 A A AT x I x F x .

Definition 3 [20]. Suppose X x x F x I xT x A A A A ))(),(),(( and X x x F x I xT x B B B B ))(),(),(( are two NSs. If and

only if )()( xT xT B A , )()( x I x I B A , )()( x F x F B A for all x in X , then B A .2.3. Some operators based on heronian mean(HM) operatorHeronian mean (HM) is a useful aggregation operator which is marked by catching the interrelations of the aggregated

arguments [21,22] and can be defined as follows.Definition 4 [22]: Let I I H I n :,1,0 , if

n

i

n

i j jin x x

nn x x x H

121 1

2,,, (10)

then n x x x H ,,, 21 is called the Heronian mean (HM) operator.

Definition 5 [21,22]. A GHM operator of dimension n is a mapping, GHM : I I n , so that,

q pn

i

n

i j

q j

pin

q p x xnn

x x xGHM

1

121

,

12

,,, (11)

where 0, q p and 1,0 I . Then q pGHM , is called the generalized Heronian mean (GHM) operator.

It is easy to prove that the GHM operator has the following properties [23].

Theorem 1 (Idempotency)

Let x xi for all ni ,,2,1 , then

x x x xGHM nq p ,,, 21

,. (12)

Theorem 2 (Monotonicity)Let n x x x ,,, 21 and n y y y ,,, 21

be two collections of the nonnegative numbers, if ii y x for all ni ,,2,1 ,

then

nq p

nq p y y yGHM x x xGHM ,,,,,, 21

,21

, . (13)

Theorem 3 (Boundary)GHM operator lies between the max and min operators, min 1 2 max 1 2min , , , , max , , ,n na x x x a x x x , i.e.

nnq p x x xa x x xGHM a ,,,,,, 21max21

,min . (14)

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Definition 6 [13] Let 0, q p , and ni xi ,,2,1 be a collection of nonnegative numbers. T nwwwW ,,, 21 is the

weight vector of ni xi ,,2,1 , and satisfies 0iw ,n

iiw

1

1 .If

q pn

i

n

i j

q j j

piin

q p xw xwnn

x x xGWHM

1

121

,

12

,,, (15)

then q pGWHM , is called the generalized weighted Heronian mean (GWHM) operator.



iiw

1

1 .If

q pn

i

n

i j ji

q pn

i

n

i j

q j

pi ji

nq p

ww

x xww

x x x IGWHM

1

1

1

121

, ,,, (16)

then q p IGWHM , is called the improved generalized weighted Heronian mean (IGWHM) operator.

Similar to Theorems 1-3, it is easy to prove the q p IGWHM , operator has these properties [24].

Theorem 4 (Idempotency)Let x xi for all n j ,,2,1 , then

x x x x IGWHM nq p ,,, 21

, . (17)

Theorem 5 (Monotonicity)

Let n x x x ,,, 21 and n y y y ,,, 21 be two collections of the nonnegative numbers, if j j y x for all n j ,,2,1 ,

then

nq p

nq p y y y IGWHM x x x IGWHM ,,,,,, 21

,21

, . (18)

Theorem 6 (Boundary)

q p IGWHM , operator lies between the max and min operators, nn x x xa x x xa ,,,,,,, 21max21min i.e.

nnq p

n x x xa x x x IGWHM x x xa ,,,,,,,,, 21max21,

21min . (19)

In the following, we can analyze some special cases of the I GWHM operator(1) When 0q , then

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pn

i

n

i j ji

pn

i

n

i j

pi ji

n p

ww

xww

x x x IGWHM 1

1

1

121

0, ,,,

. (20)

Further, when 1 p , there is

n

i

n

i j ji

n

i

n

i ji ji

n

ww

xww

x x x IGWHM

1

121

0,1 ,,, . (21)

(2) When 0 p , then

qn

i

n

i j ji

qn

i

n

i j

q j ji

nq

ww

xww

x x x IGWHM 1

1

1

121

,0 ,,,

. (22)

From here we see that the parameters p and q don’t have the interchangeability. (3) When 1 q p , then

21

1

21

121

1,1,,,

n

i

n

i j ji

n

i

n

i j ji ji

n

ww

x xww

x x x IGWHM . (23)

Based on HM and GHM operators, Yu [14] propose the generalized geometric Heronian mean (GGHM) operator shown as

follows.Definition 8 [14] Let 0, q p , and ni xi ,,2,1 be a collection of nonnegative numbers. If

n

i

n

i j

nn jinq p qx px

q p x x xGGHM

1

12

21, 1

,,, . (24)

then q pGGHM , is called the generalized geometric Heronian mean (GGHM) operator.Similar to GHM operator, the GGHM operator also only takes the correlations of the aggregated arguments into account

and ignores their own weights.



iiw

1

1 .If

n

i

n

i j w

w

nnin

jinq p n

ik k

j

qx pxq p

x x x IGGWHM 1

112

21, 1

,,, . (25)

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then q p IGGWHM , is called the improved generalized geometric weighted Heronian mean (IGGWHM) operator.

Similar to Theorems 1-3, it is easy to prove the q p IGGWHM , operator has these properties [24].

Theorem 7 (Reducibility).

LetT

nnnW

1,,

1,

1,then

nq p

nq p x x xGGHM x x x IGGHM ,,,,,, 21

,21

, (26)

Theorem 8 (Idempotency)Let x xi for all ni ,,2,1 , then

x x x x IGGWHM nq p ,,, 21

, . (27)

Theorem 9 (Monotonicity)Let n x x x ,,, 21 and n y y y ,,, 21

be two collections of the nonnegative numbers, if ii y x for all ni ,,2,1 ,

then

nq p

nq p y y y IGGWHM x x x IGGWHM ,,,,,, 21

,21

, . (28)

Theorem 10 (Boundary)

The q p IGGHM , operator lies between the max and min operators, nn x x xa x x xa ,,,,,,, 21max21min i.e.,

nnq p

n x x xa x x x IGGWHM x x xa ,,,,,,,,, 21max21,

21min (29)

In the following, we can analyze some special cases of the q p IGGWHM , operator

(1) When 0q , then

n

i

n

i jw

w

nnin

in p n

ik k

j

x x x x IGGHM 1

112

210, ,,,

. (30)

From here we see that 0, p IGGHM does not have any relationship with p .

(2) When 0 p , then

n

i

n

i jw

w

nnin

jnq n

ik

k

j

x x x x IGGHM 1

112

21,0 ,,,

. (31)

Similarly, q IGGHM ,0 does not have any relationship with q .

(3) When 1 q p , then

n

i

n

i jw

w

nnin

jin n

ik k

j

x x x x x IGGHM 1

112

211,1

21

,,, . (32)

3. The single valued neutrosophic uncertain linguistic set

Definition 10. Let ( ) ( )[ , ] x x s s S , and X be the given discourse domain, then

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}|)(~

),(~

),(~],,[|{ )()( X x x f xi xt s s x A x x . (33)

is called a single valued neutrosophic uncertain linguistic set (SVNULS) where ,, )()( S s s x x and ( )t x , ( )i x and

( ) f x are three sets of some single value in real unit interval 1,0 which express the truth-membership,

indeterminacy-membership and falsity-membership function of the element x to A separately.

Definition 11. Let { | [ , ], , , | } A x s s t i f x X be a single valued neutrosophic uncertain linguistic set, and

f it s sa ~

,~,~],,[~ is called a single valued neutrosophic uncertain linguistic number (SVNULN).

Suppose 1111~

,~,~],,[~11

f it s sa and 2222~

,~,~],,[~22

f it s sa are any two single valued neutrosophic uncertain

linguistic numbers, the operational laws are defined as follows:

(1) 1 2 1 21 2 1 2 1 2 1 2 1 2, , , ,a a s s t t t t i i f f ; (34)

(2) 212121212121~~~~

,~~~~,~~,,~~2121

f f f f iiiit t s saa ; (35)

(3) ;0,~

,~,~11,,~1111 11

f it s sa (36)

(4)

1111~

11,~

11,~,,~11

f it s sa , 0 (37)

Obviously, these operational results are still the single valued neutrosophic uncertain linguistic numbers.

Theorem 11 . Let 1111~

,~

,~

],,[~ 11 f it s sa and 2222~

,~

,~

],,[~ 22 f it s sa be any two single valued neutrosophic

uncertain linguistic numbers, the operational laws have the following characteristics.

(1) 1 2 2 1a a a a (38)

(2) 1 2 2 1a a a a (39)

(3) 1 2 1 2( ) , 0a a a a (40)

(4) 1 1 2 1 1 2 1 1 2( ) , , 0a a a (41)

(5) 1 2 1 21 1 1 1 2( ) , , 0a a a (42)

(6) 1 1 11 2 1 2 1( ) , 0a a a a (43)

Proof:

(1) Formula (38) is obviously right according to the operational rule (1) expressed by (34).

(2) Formula (39) is obviously right according to the operational rule (2) expressed by (35).

(3) For the left hand of (40), we have

1 2 1 21 2 1 2 1 2 1 2 1 2, , , ,a a s s t t t t i i f f

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then

1 2 1 21 2 1 2 1 2 1 2 1 2, , 1 1 , ,a a s s t t t t i i f f

and for the right hand of (40), we have,

,~

,~,~11,,~ 1111 11

f it s sa 2 22 2 2 2, , 1 1 , ,a s s t i f

then 1 2 1 21 2 1 2 1 2 1 2 1 2, , 1 1 1 1 1 1 1 1 , ,a a s s t t t t i i f f

,i.e.

1 2 1 21 2 1 2 1 2 1 2 1 2, , 1 1 , ,a a s s t t t t i i f f

.

so, we have 1 2 2 1 , 0a a a a . i.e., formula (40) is right.

(4) Similar to the proof of (40), it is easy to prove the formula (41) is right. The proof is omitted here.

(5) For the left hand of (42), we have

111

11

11

1

1111~

11,~11,~,,~

f it s sa ,

222

21

21

2

1111~

11,~11,~,,~

f it s sa

then, ,~11~11~11~11,~,,~~ 21212121

121

1

21

2111111

iiiit s saa

2121

1111~

11~

11~

11~

11

f f f f

i.e.

212121

211

211

21

11111~11,~11,~,,~~

f it s saa .

and for the right hand of (42), we have,

212121

211

211

21

1111~

11,~11,~,,~

f it s sa .

So, we have 0,0,~~~21111

2121

aaa . i.e., formula (42) is right.

(6) Similar to the proof of (42), it is easy to prove the formula (43) is right. The proof is omitted here.

Definition 12. Suppose 1111~,~,~],,[~

11 f it s sa is a single valued neutrosophic uncertain linguistic number, then the

expectation value )~( 1a E of 1~a can be defined as follows.

1 1 1 1 1 11 1 1 1 ( ) / 2 ( ) (2 ) / 6

1(2 )( )

3 t i f t i f E a s s (44)

Definition 13. Suppose 1111~

,~

,~],,[~11

f it s sa is a single valued neutrosophic uncertain linguistic number, then the

accuracy function )~( 1a H of 1~a can be defined as follows.

1 1 1 1 1 1 11 1 1 1 ( ) / 2 ( ) ( ) 2/( )( ) t i f t i f H a s s (45)

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Definition 14. Let 1111~

,~

,~],,[~11

f it s sa and 2222~

,~,~],,[~22

f it s sa be any two single valued neutrosophic

uncertain linguistic numbers, then(1) if )~()~( 21 a E a E , then 21

~~ aa ;

(2) if )~()~( 21 a E a E , then

if )~()~( 21 a H a H , then 21~~ aa ;

If )~()~( 21 a H a H , then 21~~ aa .

4. Some Heronian mean operators based on the single valued neutrosophic

uncertain linguistic variables

In this section, we will extend the IGWHM and IGGWHM operators to aggregate the single neutrosophic uncertain

linguistic variables, and propose a neutrosophic uncertain linguistic number improved generalized weighted Heronian

mean (NULNIGWHM) operator and a neutrosophic uncertain linguistic number improved generalized geometric

weighted Heronian mean (NLUNIGGWHM) operator which can be described as follows.4.1 The NULNIGWHM operator

Definition 15. Let 0, q p , and iiii f it s saii

~,~,~],,[~

ni ,,2,1 be a collection of the single valued neutrosophic

uncertain linguistic numbers. T nwwwW ,,, 21 is the weight vector of nia i ,,2,1~ ,and satisfies 0iw ,

n

iiw

1

1 .

If

q p

n

i

n

i j ji

n

i

n

i j

q j pi ji

nq p

ww

aaww

aaa NULNIGWHM

1

1

121

,

~~~,,~,~

(46)

then q p NULNIGWHM , is called the neutrosophic uncertain linguistic number improved generalized weighted Heronianmean operator.

Theorem 12. Let 0, q p , and iiii f it s saii

~,~,~],,[~


uncertain linguistic numbers. T

nwwwW ,,,

21 is the weight vector of nia

i ,,2,1~ , and

satisfies 0iw ,n

iiw

1

1 , then the result aggregated from Definition 15 is still a NULN, and

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

nq p

ww

aaww

aaa NULNIGWHM

1

1

121

,

~~

~,,~,~

,, 1

1

1

1

1

1

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

ww

ww

ww

ww

s s

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11

,~~11

11

1

1

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji jit t ,~1~1111

11

1

1

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

ii

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

f f

11

1

1~

1~

1111 (47)

Proof

Since

p

i p

i p

ii f it s sa pi

pi

p ~11,~11,~,,~

;

q

jq

jq

j j f it s sa q j

q j

q ~11,~11,~,,~

so,

,~~11,,~~ ji

q j

pi ji

q j

pi ji

q p wwq j

pi

wwww ji ji t t s saaww

ji ji ww

q j

pi

wwq j

pi f ii ~1

~11,~1~11

then

,~~11,,~~

11 111

n

i

n

i j

wwq j

pi

wwww

n

i

n

j ji ji

ji

n

i

n

i j

q j

pi ji

n

i

n

i j

q j

pi ji

q p

t t s saaww

n

i

n

i j

q j

pi

n

i

n

i j

q j

pi

jwiw jwiw

f f ii11

~1

~11,

~1

~11

,,~~1

11

11

111

1

n

i

n

i j

q j

pi jin

i

n

i j ji

n

i

n

i j

q j

pi jin

i

n

i j ji

q p

wwww

wwww

n

i

n

i j ji jin

i

n

i j ji

s saaww

ww

,~~11 1

1

1

n

i

n

i j ji ji wwn

i

n

i j

wwq j

pi t t

n

i

n

i j ji

jin

i

n

i j ji

jiww

n

i

wwn

i j

q j

pi

wwn

i

wwn

i j

q j

pi f f ii 11

1

1

1

1

~1

~11,~1~11 ,

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12

and

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

ww

aaww

1

1

1

~~

,, 1

1

1

1

1

1

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

ww

ww

ww

ww

s s

,~~11

11

1

1

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

t t ,~1~1111

11

1

1

q p

wwn

i

wwn

i j

q j

pi

n

i

n

i j ji

ji

ii

q p

wwn

i

wwn

i j

q j

pi

n

i

n

i j ji

ji

f f

11

1

1

~1

~1111

So,

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

nq p

ww

aaww

aaa NULNIGWHM

1

1

121

,

~~

~,,~,~

,, 1

1

1

1

1

1

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

ww

ww

ww

ww

s s

,~~11

11

1

1

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji jit t

,~1~1111

11

1

1

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

ii

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

f f

11

1

1~

1~

1111

which completes the proof of theorem 12.

The NULNIGWHM operator has the properties, such as idempotency, monotonicity and boundedness.

Theorem 13. (Idempotency) .Let all aa i

~~ for all nii ,,2,1 , then

.~~,,~,~ 21, aaaa NULNIGWHM nq p (48)

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13

Proof

Since all aa i~~ f it s s

~,~,~],,[ for all nii ,,2,1 , then we have

1 1

1 1

1 1

,1 2, , , , ,

n n n n p q p q p q p qi j i j

i j i i j in n n n

i j i ji j i i j i

p qn

w w w w

w w w w

NULNIGWHM a a a s s

,~11

11

1

1

q p

wwn

i

n

i j

wwq pn

i

n

i j ji jit ,

~1111

11

1

1

q p

wwn

i

n

i j

wwq pn

i

n

i j ji ji

i

q p

wwn

i

n

i j

wwq pn

i

n

i j ji ji

f

11

1

1~

1111 f it s s ~

,~,~],,[

Theorem 14.(monotonicity)

Let iiii f it s saii

~,~,~],,[~

and iiii f it s sa ii****** ~

,~

,~],,[~ ni ,,2,1 be two collection of

neutrosophic uncertain linguistic fuzzy numbers, and if *~~ii aa ,i.e. ,, **

iii

i

s s s s ** ~~,~~iiii iit t and *~~

ii f f , for

all nii ,,2,1 , then

)~,,~,~()~,,~,~( *2

*1

*21

,nn

q p aaa NULNIGWHM aaa NULNIGWHM . (49)

Proof

(1) Since ,, **iiii

s s s s then, ,, **qi

qi

pi p

i s s s s

,*qi

pi

qi

pi

s s ,*

qi

pi ji

qi

pi ji wwww s s

,*

11

n

i

n

i j

qi

pi ji

n

i

n

i j

qi

pi ji wwww

s s

.*

1

1

1

1

n

i

n

i j ji

n

i

n

i j

qi

pi ji

n

i

n

i j ji

n

i

n

i j

qi

pi ji

ww

ww

ww

ww s s

(2) Firstly, since *~~ii t t for all i ,and ,0, q p then we can get

,~~~~ ** q j

pi

q j

pi t t t t ,~~1~~1 ** q

j p

iq

j p

i t t t t

,~~1~~11

**

1

n

i

n

i j

wwq j

pi

n

i

n

i j

wwq j

pi

ji ji t t t t and

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14

,~~1~~1 11

1

1

**

1

1

n

i

n

i j ji ji

n

i

n

i j ji ji wwn

i

n

i j

wwq j

pi

wwn

i

n

i j

wwq j

pi t t t t then,

,~~

11~~

11 1

1

1

1

**

1

1

n

i

n

i j ji ji

n

i

n

i j

ji ji wwn

i

n

i j

wwq j

pi

wwn

i

n

i j

wwq j

pi t t t t

so, q p

wwn

i

n

i j

wwq j

pi

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

n

i

n

i j ji ji t t t t

11

1

**

11

1

11

~~11~~11

Secondly, since *~~ii ii for all i and ,0, q p , q

i p

i ii *~1~1 and q j

p j ii *~1~1 ,

q j

qi

p j

pi iiii ** ~

1~

1~

1~

1 ,

n

i

n

i j

wwq j

qi

n

i

n

i j

ww p j

pi

ji ji

iiii

1

**

1

~1~11~1~11 ,

n

i

n

i j ji ji

n

i

n

i j ji ji wwn

i

n

i j

wwq j

pi

wwn

i

n

i j

wwq j

pi iiii 11

1

1

**

1

1

~1~11~1~11 ,

n

i

n

i j ji ji

n

i

n

i j ji ji wwn

i

n

i j

wwq j

pi

wwn

i

n

i j

wwq j

pi iiii 11

1

1

**

1

1

~1

~111

~1

~111 ,

q p

wwn

i

n

i j

wwq j

pi

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

n

i

n

i j ji ji

iiii

11

1

**

11

1

11~1~111~1~111

so,

q p

wwn

i

n

i j

wwq j

pi

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji ji

n

i

n

i j ji ji

iiii

11

1

**

11

1

11~1~1111~1~1111 .

Thirdly, similar with the previous step, we can prove that

q pn

i

n

i j

wwq j

qi

q pn

i

n

i j

ww p j

pi

ji ji

f f f f

1

1

**

1

1

~1

~1111

~1

~1111

According to (1)-(2), we can get

,, 1

1

1

1

1

1

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

ww

ww

ww

ww

s s

,~~11

11

11

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji jit t

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15

,~1~1111

11

11

q p

wwn

i

n

i j

ww p j

pi

n

i

n

i j ji

ji

ii

q p

wwn

i

n

i j

ww p j

pi

n

i

n

i j ji

ji

f f

11

11

~1

~1111

,, **

1

1

1

1

1

1

q p

n

i

n

i j jwiw

n

i

n

i j

q j

pi jwiw

q p

n

i

n

i j ji

n

i

n

i j

q j

pi ji

s s

ww

ww

,~~11

11

1

**1

q p

wwn

i

n

i j

wwq j

pi

n

i

n

i j ji jit t

,~

1~

1111

11

1

**1

q p

wwn

i

n

i j

ww p j

pi

n

i

n

i j ji

ji

ii

q p

wwn

i

n

i j

ww p

j

p

i

n

i

n

j ji

ji

f f

11

1

**1 1

~1

~1111

i.e. )~,,~,~()~,,~,~( *2

*1

*21 nn aaa NULNIGWHM aaa NULNIGWHM , which complete the proof of theorem 14.

Theorem 15. (Boundary).

The q p NULNIGWHM , operator lies between the max and min operators:)~,,~,~max(~),~,,~,~min(~

21max21min nn aaaaaaaa , then

max21min~)~,,~,~(~ aaaa NULNIGWHM a n . (50)

ProofSince aa ~~

min , based on theorems 13 and 14, we can get

)~,,~,~(~ *2

*1

*,min n

q p aaa NULNIGWHM a

and then aa ~~max , max21

~)~,,~,~( aaaa NULNIGWHM n .

so, max21min~)~,,~,~(~ aaaa NULNIGWHM a n , which complete the proof of theorem 15.

In the following, we will discuss some special cases of the NULNIGGWHM operator in regard to the parameters

p and q .(1) when 0 p , then

1 1

1 1

1 1

0,1 2

1 1

, , , , ,n n n nq qq q

i j i j j ji j i i j i

n n n n

i j i ji j i i j i

qn

w w w w

w w w w

NULNIGWHM a a a s s

,~11

11

1

1

q

wwn

i

n

i j

wwq j

n

i

n

i j ji jit

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16

,~1111

11

1

1

q

wwn

i

n

i j

wwq j

n

i

n

i j ji ji

i

q

wwn

i

n

i j

wwq j

n

i

n

i j ji ji

f

11

1

1~

1111 . (51)

(2) when 0q , then

n p aaa NULNIGWHM ~,,~,~

210,

,, 1

1

1

1

1

111

p

n

i

n

i j ji

n

i

n

i j

pi ji

p

n

i

n

i j ji

n

i

n

i j

pi ji

ww

ww

ww

ww

s s

,~11

11

1

1

p

wwn

i

n

i j

ww pi

n

i

n

i j ji ji

t ,~

1111

11

1

1

p

wwn

i

n

i j

ww pi

n

i

n

i j ji ji

i

p

wwn

i

n

i j

ww pi

n

i

n

i j ji ji

f

11

1

1~

1111 (52)

(3) when 1 q p , then

naaa NULNIGWHM ~,,~,~21

1,1

,,

21

1

1

21

1

1

n

i

n

i j ji

n

i

n

i j ji ji

n

i

n

i j ji

n

i

n

i j ji ji

ww

ww

ww

ww

s s

,~~11

21

1

1

1

n

i

n

i j ji ji

wwn

i

n

i j

ww ji t t ,

~1

~1111

21

1

1

1

n

i

n

i j ji ji

wwn

i

n

i j

ww ji ii

21

1

1

1~

1~

1111n

i

n

i j ji ji wwn

i

n

i j

ww ji f f (53)

4.2 The NULNIGGWHM operator

Definition 16. Let 0, q p , and iiii f it s saii

~,~,~],,[~

ni ,,2,1 be a collection of single neutrosophic uncertain

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17

linguistic numbers. T nwwwW ,,, 21 is the weight vector of nia i ,,2,1~ , and satisfies 0iw ,

n

iiw

1

1 .If

n

i

n

i j w

w

nnin

jinq p n

ik k

j

qa paq p

aaa NULNIGGWHM 1

112

21, 1

,,, (54)

then q p, NULNIGGWHM is called the neutrosophic uncertain linguistic number improved generalized geometricweighted Heronian mean (NULNIGGWHM) operator.

Theorem 16 . Let 0, q p , and iiii f it s saii

~,~,~],,[~


uncertain linguistic numbers. T nwwwW ,,, 21 is the weight vector of nia i ,,2,1~ , and

satisfies 0iw ,n

i

iw

1

1 , then the result aggregated from Definition 16 is still a NLUN, and

n

i

n

i jw

w

nnin

jinq p n

ik k

j

qa paq p

aaa NULNIGGWHM 1

112

21, 1

,,,

,,

11

12

11

12 11

n

i

n

i jn

ik k w

jw

nnin

ji

n

i

n

i jn

ik k w

jw

nnin

ji q pq p

q pq p

s s

,~1~1111

1

112

1

q p

w

w

nnin

n

i

n

i j

q j

pi

n

ik k

j

t t (55)

,~~

111

1

112 q pn

i

n

i jw

w

nnin

q j

pi

n

ik k

j

ii

q pn

i

n

i jw

w

nnin

q j

pi

n

ik k

j

f f 1

1

112~~

11

The proof is similar with the theorem 12, and it is omitted here.

Similar to Theorems 13-15, it is easy to prove the NULNIGGWHM operator has the following properties.

Theorem 17. (Idempotency) .Let all aa i

~~ for all nii ,,2,1 , then

.~~,,~,~21

, aaaa NULNIGGWHM nq p (56)

Theorem 18. (Boundary). The q p NULNIGGWHM , operator lies between the max and min operators:

)~,,~,~max(~),~,,~,~min(~21max21min nn aaaaaaaa , then

max21min~)~,,~,~(~ aaaa NULNIGGWHM a n . (57)

Theorem 19.(monotonicity)

Let iiii f it s saii

~,~,~],,[~

and iiii f it s sa ii****** ~

,~,~],,[~ ni ,,2,1 be two collection of

neutrosophic uncertain linguistic fuzzy numbers, and if ,, )(*

)()(*

)( iiiiaaaa s s s s ** ~~

,~~iiii t t and *~~

ii f f , for all

nii ,,2,1 ,then

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18

)~,,~,~()~,,~,~( *2

*1

*21 nn aaa NULNIGGWHM aaa NULNIGGWHM . (58)

In the following, we will discuss some special cases of the NULNIGGWHM operator in regard to the parameters p and q .

(1) when 0 p , then

nq aaa NULNIGGWHM ,,, 21

,0

,,

11

12

11

12 11

n

i

n

i jn

ik k w

jw

nnin

j

n

i

n

i jn

ik k w

jw

nnin

j qq

qq

s s

,~1111

1

112

1

q

w

w

nnin

n

i

n

i j

q j

n

ik k

j

t

,~11

1

1

112 qn

i

n

i jw

w

nnin

q j

n

ik k

j

i

qn

i

n

i jw

w

nnin

q j

n

ik k

j

f

1

1

112~

11 (59)

(2) when 0q , then

n p aaa NULNIGGWHM ,,, 21

0,

,,1

112

11

12 11

n

i

n

i jn

ik k w

jw

nnin

i

n

i

n

i jn

ik k w

jw

nnin

i p p

p p

s s

,~1111

1

112

1

p

w

w

nnin

n

i

n

i j

pi

n

ik k

j

t

,~

11

1

1

112 pn

i

n

i jw

w

nnin

pi

n

ik k

j

i

pn

i

n

i jw

w

nnin

pi

n

ik k

j

f

1

1

112~

11 (60)

(2) when 1 pq , then

naaa NULNIGGWHM ,,, 211,1

,,

11

12

11

12

21

21

n

i

n

i jn

ik k w

jw

nnin

ji

n

i

n

i jn

ik k w

jw

nnin

ji

s s

,~1~1111

21

112

1

n

ik k

j

w

w

nnin

n

i

n

i j ji t t

,~~11

21

1

112

n

i

n

i jw

w

nnin

jin

ik k

j

ii

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19

21

1

112~~

11n

i

n

i jw

w

nnin

jin

ik k

j

f f (61)

5. The decision-making methods based on the NULIGWHM operator and

NULIGGWHM operator

In order to strengthen the efficiency of this decision-making, we can make several experts participate in the

decision-making under neutrosophic uncertain linguistic fuzzy environment.

Considering the multiple attribute group decision making problems with neutrosophic uncertain linguistic fuzzy

information described as follows. Let 1 2{ , , , }m A A A A be a set of alternatives, and 1 2{ , , , }nC C C C be the set

of attributes, and 1 2{ , , , }nW w w w be the weight vector of the attribute ),,2,1( n jC j , where

0 jw , n j ,,2,1 , 11

n

j jw .Let },,,{ 21 t D D D D be the set of decision makers, and T t ,, 21 be the

weight vector of decision makers t e De ,,2,1 , where 0e , 11

t

ee . Suppose

nme

ije h H ][ are the decision

matrices where eij

eij

eij

eeeij f it s sh

ijij

~,~,~],,[

~ takes the form of the single valued neutrosophic uncertain linguistic

variables given by the decision maker e D for alternative i A with respect to attribute jC . Then, the ranking of alternatives is

finally acquired.

The methods involve the following steps:

Step 1. Utilize the NULIGWHM operator

ein

ei

ei

ei hhh NULIGWHM h

~,,

~,

~~21

(62)

or NULIGGWHM operator

ein

ei

ei

ei hhh NULIGGWHM h

~,,

~,

~~21

(63)

to get the comprehensive attribute values of each alternative for decision maker e D .

Step 2. Utilize the NULIGWHM operator

t iiii hhh NULIGWHM h

~,,

~,

~~ 21 (64)

or NULIGGWHM operator

t iiii hhh NULIGGWHM h

~,,

~,

~~ 21 (65)

to aggregate the evaluation values of the single decision maker to the collective comprehensive values for each

alternative.Step 3. Calculate the value )( ih E of ih .

Step 4. Rank mihi ,,2,1 in descending order according to the comparison method of INULNs described in Definition

14.

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Step 5. End.

6. A numerical example

In this section, we will provide an example to illustrate the application of IULFPEWA and IULFPEWG operators.

Suppose that an investment company wants to invest an amount of money to a company. There are four candidate

companies )4,3,2,1( i Ai evaluated by three decision makers },,{ 321 D D D . The weight vector of the decision makers

is T 331.0,355.0,314.0 , and the considered attributes include: 1C (the risk index) , 2C ( the growth index), 3C (the

social-political impact index), and 4C (the environmental impact index). Suppose the attribute weight vector

is T w 40.0,20.0,4.0 .The three decision makers },,{ 321 D D D evaluate the four companies )4,3,2,1( i Ai with respect to

the attributes )3,2,1( jC j by using the neutrosophic uncertain linguistic variables (suppose that the decision makers use

linguistic term set 0 1 2 3 4 5 6( , , , , , , )S s s s s s s s to express their evaluation results) and construct three following decision

matrices 3,2,1][ 44 eh H eij

e as listed in Tables 1-3.

Table 1. Decision matrix 1 H .

C1 C2 C3

A1 <[S 5,S5],(0.265,0.350,0.385)> <[S 2,S3],(0.330,0.390,0.280 )> <[S 5,S6],(0.245,0.275,0.480 )>

A2 <[S 4,S5],(0.345,0.245,0.410 )> <[S 5,S5],(0.430,0.290,0.280 )> <[S 3,S4],(0.245,0.375,0.380 )>

A3 <[S 3,S4],(0.365,0.300,0.335 )> <[S 4,S4],(0.480,0.315,0.205 )> <[S 4,S5],(0.340,0.370,0.290 )>

A4 <[S 6,S6],(0.430,0.300,0.270 )> <[S 2,S3],(0.460,0.245,0.295 )> <[S 3,S4],(0.310,0.520,0.170 )>


C1 C2 C3

A1 <[S 3,S4],(0.125,0.470,0.405 )> <[S 3,S4],(0.220,0.420,0.360 )> <[S 3,S4],(0.345,0.490,0.165 )>

A2 <[S 5,S6],(0.355,0.315,0.330 )> <[S 3,S4],(0.300,0.370,0.330 )> <[S 4,S5],(0.205,0.630,0.165 )>

A3 <[S 4,S5],(0.315,0.380,0.305 )> <[S 4,S4],(0.330,0.565,0.105 )> <[S 2,S3],(0.280,0.520,0.200 )>

A4 <[S 5,S5],(0.365,0.365,0.270 )> <[S 4,S5],(0.355,0.320,0.325 )> <[S 2,S3],(0.425,0.485,0.090 )>


C1 C2 C3

A1 <[S 5,S5],(0.260,0.425,0.315 )> <[S 3,S4],(0.220,0.450,0.330 )> <[S 4,S5],(0.255,0.500,0.245 )>

A2 <[S 4,S5],(0.270,0.370,0.360 )> <[S 5,S5],(0.320,0.215,0.465 )> <[S 2,S3],(0.135,0.575,0.290 )>

A3 <[S 4,S4],(0.245,0.465,0.290 )> <[S 5,S5],(0.250,0.570,0.180 )> <[S 1,S3],(0.175,0.660,0.165 )>

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A4 <[S 3,S4],(0.390,0.340,0.270 )> <[S 3,S4],(0.305,0.475,0.220 )> <[S 4,S5],(0.465,0.485,0.050 )>

6.1 The decision-making method based on NULIGWHM operator Step 1. Get the comprehensive attribute values of each alternative for decision maker e D by the NULIGWHM operator in

Eq.(62).(suppose 1 pq )

11

~h = < 046.3925.2 , s s ,(0.269,0.324,0.401 )>, 1

2~h = <[ 717.2320.2 , s s ],(0.322,0.303,0.373 )>,

13

~h = <[ 146.3810.2 , s s ],(0.376,0.330,0.291 )>, 1

4~h = <[ 274.3953.2 , s s ],(0.388,0.369,0.229 )>,

21

~h = <[ 046.3925.2 , s s ],(0.240,0.469,0.283 )>, 2

2~h = <[ 839.2462.2 , s s ],(0.287,0.438,0.253 )>,

23

~h = <[ 953.2717.2 , s s ],(0.303,0.466,0.218 )>, 2

4~h = <[ 943.2943.2 , s s ],(0.389,0.403,0.188 )>,

31

~h = <[ 690.2139.2 , s s ],(0.251,0.459,0.287 )>, 3

2~h = <[ 156.3049.3 , s s ],(0.227,0.412,0.347 )>,

33

~h = <[ 847.2470.2 , s s ],(0.218,0.558,0.214 )>, 3

4~h = <[ 818.2437.2 , s s ],(0.408,0.420,0.144 )>.

Step 2. Get the collective comprehensive values for each alternative by the NULIGWHM operator in Eq.(64).(s

uppose 1 pq )

1~h <[ 461.2354.2 , s s ],(0.250,0.422,0.323)>, 2

~h <[ 472.2343.2 , s s ],(0.276,0.389,0.323)>,

3~h <[ 385.2245.2 , s s ],(0.298,0.453,0.243)>, 4

~h <[ 434.2322.2 , s s ],(0.391,0.402,0.189)>

Step 3.Calculate the value )( ih E of ih .

208.11)( sh E , 25 5.12 )( sh E , 23 6.13 )( sh E , 42 7.14 )( sh E .

Step 4. Rank mihi ,,2,1 in descending order according to the comparison method of INULNs described in

Definition 16.So, )( 4h E > )( 2h E > )( 3h E > )( 1h E .

i.e., 4 A is the best choice.

Step 5. End.

6.2 The decision-making method based on NULIGWHM operator Step 1 ’. Get the comprehensive attribute values of each alternative for decision maker e D by the NULIGWHM operator

in Eq.(63).(suppose 1 pq )

11

~h = <[

463.4245.4 , s s ],(0.274,0.333,0.394 )>, 12

~h = <[

652.3644.2 , s s ],(0.326,0.308,0.365 )>,

13

~h = <[

94 5.4934.3 , s s ],(0.385,0.331,0.284 )>, 14

~h = <[

329.5326.4 , s s ],(0.390,0.373,0.240 )>,

21

~h = <[

643.4245,4 , s s ],(0.221,0.464,0.307 )>, 22

~h = <[

958.3938.2 , s s ],(0.279,0.459,0.270 )>,

23

~h = <[32 6.4652.3 , s s ],(0.306,0.488,0.217 )>, 2

4~h = <[

314.4314.4 , s s ],(0.384,0.400,0.220 )>,

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31

~h = <[

58 8.3115.2 , s s ],(0.247,0.461,0.293 )>, 32

~h = <[

957.4556.4 , s s ],(0.225,0.413,0.364 )>,

33

~h = <[

961.3942.2 , s s ],(0.218,0.572,0.216 )>, 34

~h = <[

934.3915.2 , s s ],(0.393,0.435,0.178 )>.

Step2 ’. Get the collective comprehensive values for each alternative by the NULIGWHM operator in Eq.(65)(su

ppose 1 pq ).

1~h <[

469.4653.3 , s s ],(0.247,0.422,0.332)>, 2~h <[

601.4747.3 , s s ],(0.275,0.397,0.333)>,

3~h <[

035.4177.3 , s s ],(0.298,0.470,0.239)>, 4~h <[

282.4442.3 , s s ],(0.389,0.403,0.213)>

Step 3 ’. Calculate the value )( ih E of ih .

02 0.21)( sh E , 150.22 )( sh E , 90 9.13 )( sh E , 28 2.24 )( sh E .

Step 4 ’. Rank mihi ,,2,1 in descending order according to the comparison method of INULNs described in

Definition 16.So, )( 4h E > )( 2h E > )( 1h E > )( 3h E .

i.e., 4 A is the best choice.

Step 5 ’. End

6.3 The influences of the parameters q p, on the decision-making problem

Table 4 Ordering of the alternatives by the different parameters p and q in NULIGWHM operatorq p, )( ih E ranking

1,0 q p 13 5.31 )( sh E , 948.22 )( sh E ,

687.23 )( sh E , 56 4.34 )( sh E .3214 A A A A

1.2,0 q p 285.21 )( sh E , 179.22 )( sh E ,

94 0.13 )( sh E , 57 2.24 )( sh E .3214 A A A A

2.2,0 q p 274.21 )( sh E , 173.22 )( sh E ,

93 3.13 )( sh E , 55 9.24 )( sh E .3214 A A A A

10,0 q p 589.21 )( sh E , 675.22 )( sh E ,

429.23 )( sh E , 00 0.34 )( sh E .3124 A A A A

0,1 q p 08 3.31 )( sh E , 548.32 )( sh E ,

24 7.33 )( sh E , 86 3.34 )( sh E .1324 A A A A

0,2 q p 291.21 )( sh E , 64 5.22 )( sh E ,

342.23 )( sh E , 92 6.24 )( sh E .1324 A A A A

0,10 q p 619.21 )( sh E , 889.22 )( sh E ,

548.23 )( sh E , 37 8.34 )( sh E .3124 A A A A

1,2 q p 12 0.11 )( sh E , 202.12 )( sh E ,

16 6.13 )( sh E , 34 8.14 )( sh E .1324 A A A A

1,10 q p 98 8.11 )( sh E , 16 7.22 )( sh E ,

96 2.13 )( sh E , 50 9.24 )( sh E .3124 A A A A

2,1 q p 12 0.11 )( sh E , 137.12 )( sh E ,

10 0.13 )( sh E , 30 5.14 )( sh E .3124 A A A A

10,1 q p 95 9.11 )( sh E , 03 8.22 )( sh E ,88 9.13 )( sh E , 27 4.24 )( sh E .

3124 A A A A

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1,1 q p 20 8.11)( sh E , 25 5.12 )( sh E ,

23 6.13 )( sh E , 427.14 )( sh E .1324 A A A A

Table 5 Ordering of the alternatives by the different parameters p and q in NULIGGWHM operatorq p, )(

ih E

Ranking1,0 q p 03 6.21 )( sh E , 783.12 )( sh E ,

51 0.13 )( sh E , 196.24 )( sh E .3214 A A A A

2,0 q p 01 4.21 )( sh E , 749.12 )( sh E ,

49 4.13 )( sh E , 15 9.24 )( sh E .3214 A A A A

10,0 q p 865.11 )( sh E , 561.12 )( sh E ,

38 0.13 )( sh E , 01 4.24 )( sh E .3214 A A A A

0,01.0 q p 924.11 )( sh E , 445.22 )( sh E ,

15 1.23 )( sh E , 43 5.24 )( sh E .1342 A A A A

0,1 q p 912.11 )( sh E , 423.22 )( sh E ,

127.23 )( sh E , 415.24 )( sh E .1342 A A A A

0,2 q p 899.11 )( sh E , 3940.22 )( sh E ,

099.23 )( sh E , 3941.24 )( sh E .1324 A A A A

1,2 q p 982.11 )( sh E , 219.22 )( sh E ,

97 2.13 )( sh E , 30 6.24 )( sh E .1324 A A A A

1,1 q p 02 0.21 )( sh E , 15 0.22 )( sh E ,

90 9.13 )( sh E , 28 2.24 )( sh E .3124 A A A A

1,10 q p 822.11 )( sh E , 073.22 )( sh E ,

89 2.13 )( sh E , 24 0.24 )( sh E .1324 A A A A

2,1 q p 09 2.21 )( sh E , 085.22 )( sh E ,83 4.13 )( sh E , 24 3.24 )( sh E .

3214 A A A A

10,1 q p 878.11 )( sh E , 649.12 )( sh E ,

47 1.13 )( sh E , 035.24 )( sh E .3214 A A A A

According to the Tables 4 and 5, the ranking results may be different for the different parameter values q p, in

NULIGWHM and NULIGGWHM operators. In general, the best alternative is always 4 A in two tables. We can take the

values of the two parameters as 1 q p , which is not merely intuitive and simple but also takes the correlations of the

aggregated arguments into consideration completely.

7. Conclusions

The MAGDM problems widely exist in real decision making, and the aggregation operators are the important tools

these problems. Especially, Heronian mean (HM) can catch the interrelations of the aggregated arguments. In addition, the

neutrosophic uncertain linguistic set can be better to express the incomplete, indeterminate and inconsistent information. In

this paper, we proposed the neutrosophic uncertain linguistic numbers by combining neutrosophic set and uncertain

linguistic variables, and developed some Heronian mean operators on the basis of neutrosophic uncertain linguistic

numbers, included the neutrosophic uncertain linguistic number improved generalized weighted Heronian mean

(NULNIGWHM) operator and the neutrosophic uncertain linguistic number improved generalized geometric weightedHeronian mean (NLUNIGGWHM) operator and discussed the properties of them in detail. In the meantime, we studied

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the some special cases in consideration of the values of p and q . Moreover, we developed two methods which have the

advantages that they can take the correlations of the attributes into account fully to deal with the multi-attribute group

decision making (MAGDM) problems under neutrosophic uncertain linguistic number environment. We also gave anumerical example to show the steps of the proposed methods and to discuss the influences of different values of p and q

on the ranking results. In the future research, we can extend the application scopes of the proposed operators to other

fields such as option of sponsors, science-technology assessment, the performance evaluation, and so on.

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