Multi-Valued Neutrosophic Number Bonferroni Mean Operators with their Applications in Multiple Attribute Group Decision Making

Abstract

Bonferroni mean (BM) is a very useful aggregation operator, which can consider the correlations between the aggregated arguments and the multi-valued neutrosophic set can be much more convenient to denote the incomplete, indeterminate and inconsistent information, in this paper, we applied the Bonferroni mean to the multi-valued neutrosophic set, and proposed some Bonferroni mean operators of multi-valued neutrosophic numbers (MVNNs). First, we gave some operational laws and a comparison method of MVNNs, then we presented the weighted Bonferroni mean (WBM) operator and weighted geometric Bonferroni mean (WGBM) operator. Further, we proposed the multi-valued neutrosophic weighted Bonferroni mean (MVNWBM) operator and the multi-valued neutrosophic weighted geometric Bonferroni mean (MVNWGBM) operator and some properties of them are also investigated. Finally, the decision making methods are developed based on MVNWGBM operator and MVNWBM operator, and an example about investment selection is given to illustrate the applications of the developed methods and the influence of different parameter values on the decision-making results.

Full text

Multi-Valued Neutrosophic Number Bonferroni Mean
Operators with their Applications in Multiple Attribute
Group Decision Making
Peide Liu
*
, Lili Zhang, Xi Liu and Peng Wang
School of Management Science and Engineering
Shandong University of Finance and Economics
Jinan Shandong 250014, P. R. China
*
peide.liu@gmail.com
Published 13 July 2016
Bonferroni mean (BM) is a very useful aggregation operator, which can consider the correlations
between the aggregated arguments and the multi-valued neutrosophic set can be much more
convenient to denote the incomplete, indeterminate and inconsistent information, in this paper,
we applied the Bonferroni mean to the multi-valued neutrosophic set, and proposed some
Bonferroni mean operators of multi-valued neutrosophic numbers (MVNNs). First, we
gave some operational laws and a comparison method of MVNNs, then we presented the
weighted Bonferroni mean (WBM) operator and weighted geometric Bonferroni mean
(WGBM) operator. Further, we proposed the multi-valued neutrosophic weighted Bonferroni
mean (MVNWBM) operator and the multi-valued neutrosophic weighted geometric Bonferroni
mean (MVNWGBM) operator and some properties of them are also investigated. Finally, the
decision making methods are developed based on MVNWGBM operator and MVNWBM op-
erator, and an example about investment selection is given to illustrate the applications of the
developed methods and the in°uence of di®erent parameter values on the decision-making
results.
Keywords: Multiple attribute decision making (MADM); multi-valued neutrosophic sets;
Bonferroni mean operators.
1. Introduction
In real world, there are many MADM and multiple attribute group decision-making
(MAGDM) problems with incomplete, indeterminate and inconsistent information
which cannot be described by crisp numbers. Under these circumstances, Zadeh
1
proposed the fuzzy set (FS), which is an e®ective method to solve MADM and
MAGDM problems.
2,3
Since FS only has one membership, and it cannot handle some
complicated fuzzy information. Therefore, Atanassov
4,5
proposed intuitionistic fuzzy
set (IFS) by extending Zadeh's FS, IFS is consisted of truth-membership TAðxÞand
falsity-membership FAðxÞ. Further, Atanassov and Gargovit
6
extended IFS to the
*
Corresponding author.
International Journal of Information Technology & Decision Making
Vol. 15 (2016)
°
cWorld Scienti¯c Publishing Company
DOI: 10.1142/S0219622016500346
1
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

interval-valued intuitionistic fuzzy set (IVIFS) and applied it to the MADM and
MAGDM problems.
7–18
Later, Torra and Narukawa
11
and Torra
12
introduced a new
concept of hesitant fuzzy sets (HFSs), then a series of extended researches on HFSs
have been undertaken, for example, operational rules, entropy, distance, correlation
coe±cient, and so on for HFSs.
13–18
Although there is a great deal of research achievements on HFSs, however, they
can only process incomplete information, and cannot handle some complex uncer-
tainties in some practical problems, such as the inconsistent information and in-
determinate information. In IFSs, the indeterminacy degree 1 TAðxÞFAðxÞis
default, and in many complex decision making conditions, IFS also has some
restrictions. For instance, if an expert is invited to make a decision for a decision-
making problem, he (or she) may give the true possibility of the statement is 0.5, the
false possibility of the statement is 0.6 and the uncertain possibility is 0.2.
19
In this
case, this kind of decision information cannot be handled by FS and IFS. Then
Smarandache
20
applied an independent indeterminacy-membership function to IFS
and then proposed the de¯nition of neutrosophic sets (NSs) in 1998. Clearly, NS is a
generalization of the existing main fuzzy sets including FS, paraconsistent set, IFS
and so on. Nowadays, many research achievements have been made, including the
single valued neutrosophic set (SVNS), the entropy measurement and the
correlation coe±cients of SVNSs, simpli¯ed neutrosophic sets (SNSs), interval
neutrosophic sets (INSs).
19,21–28
But sometimes, decision makers cannot give accurate evaluation values for each
membership of the SNSs. For instance, he (or she) may give evaluation values for
true membership 0.6 or 0.7, the evaluation values for false membership are 0.2 or 0.3
and the evaluation values for uncertainty membership are 0.1 or 0.2, this will be out
of the capability of SNSs. In order to process this type of uncertain information,
Wang and Li
29
proposed a new de¯nition called multi-valued neutrosophic sets
(MVNSs) and presented their operations and comparison method.
Obviously, MVNSs can better describe the complex fuzzy information which
exists in real decision making, so it is very necessary to study the MADM or
MAGDM method based on MVNSs. However, there is little research on these de-
cision making methods.
The aggregation operators are an important tool to solve the MADM and
MAGDM problems,
30–40
which can produce the comprehensive values of di®erent
alternatives by aggregating the information from all attributes or from all decision
makers, and then rank the alternatives. Comparing with the only ranking methods,
such as TOPSIS,
41
TODIM,
29,42
and so on, the decision making methods based on
the aggregation operators can rank the alternatives based on the comprehensive
values of di®erent alternatives while the ranking methods can only give the ranking
results not comprehensive values. In addition, in real decision making, there are a
great number of decision making problems with the relationships between the
attributes, such as the cost and quality of production, and Bonferroni mean operator
can consider the relationships among di®erent attributes, and it has a wide
2P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

application in MADM or MAGDM problems. However, the traditional Bonferroni
mean operator can only aggregate the crisp numbers, and cannot aggregate the
MVNSs. So, it is very necessary to extend Bonferroni mean operator to deal with the
information of NVNSs, and propose the MADM or MAGDM methods based on these
aggregation operators. The goal of this paper is to propose some Bonferroni mean
operators for NVNSs, and then to develop the MAGDM method based on these
operators for the decision making problems in which the attributes take the form of
NVNSs and there are the relationships between the attributes.
The structure of paper is organized as follows. In Sec. 2, we review the basic
concepts and operational rules of SVNNs and the de¯nition of MVNSs with its
operations, comparison method and distance, some Bonferroni mean (BM) opera-
tors. In Sec. 3, we propose the multi-valued neutrosophic weighted Bonferroni mean
(MVNWBM) operator and the multi-valued neutrosophic weighted geometric
Bonferroni mean (MVNWGBM) operator. At the same time, we study some prop-
erties about these operators. Section 4develops a MAGDM approach based on these
operators with MVNNs. Section 5gives an example about investment selection to
illustrate the decision-making steps and discusses the in°uence the parameters
changed on the results. Finally, we draw the conclusions in Sec. 6.
2. Preliminaries
2.1. SVNS
De¯nition 1 (Ref. 19). Let Xbe a space of points (objects), and a single valued
neutrosophic set (SVNS) Ain Xis described by
A¼fxðtAðxÞ;iAðxÞ;fAðxÞÞjx2Xg;ð2:1Þ
where tAðxÞ,iAðxÞand fAðxÞdenote the truth-membership function, the
indeterminacy-membership function and the falsity-membership function of the
element x2Xto the set Arespectively. For each point xin X, we have
tAðxÞ;iAðxÞ;fAðxÞ2½0;1, and
0tAðxÞþiAðxÞþfAðxÞ3:
For simplicity, we can use x¼ðtx;ix;fxÞto represent an element xin SVNS, and
the element xis called a single valued neutrosophic number (SVNN).
Ye
23
de¯ned some basic operations of SNNNs shown as follows.
De¯nition 2. Let x¼ðt1;i1;f1Þand y¼ðt2;i2;f2Þbe two SVNNs, then the
operations are de¯ned as follows:
ð1ÞThe complement of xis 
x¼ðf1;1i1;t1Þ;ð2:2Þ
ð2Þxy¼ft1þt2t1t2;i1i2;f1f2g;ð2:3Þ
ð3Þxy¼ft1t2;i1þi2i1i2;f1þf2f1f2g;ð2:4Þ
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 3
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

ð4Þkx ¼f1ð1t1Þk;ik
1;fk
1g;ð2:5Þ
ð5Þxk¼ftk
1;1ð1i1Þk;1ð1f1Þkg:ð2:6Þ
2.2. The multi-valued NS
De¯nition 3 (Ref. 32). Let Xbe a space of points (objects), and a multi-valued
neutrosophic set (MVNS) Ain Xis described by
A¼fxð~
tAðxÞ;~
iAðxÞ;~
fAðxÞÞjx2Xg;ð2:7Þ
where ~
tðxÞ,~
iðxÞand ~
fðxÞare three collections of discrete real values in [0, 1], denoting
the truth-membership, the indeterminacy-membership and the falsity-membership
of the element x2Xto the set Arespectively, satisfying 0  ; ; ’ 1,
0þþþþ’þ3, where 2~
tðxÞ;
2~
iðxÞ;’2~
fðxÞ;
þ¼sup ~
tðxÞ;
þ¼sup~
iðxÞ;’
þ¼sup ~
fðxÞ:
For simplicity, we can use x¼ð
~
tx;~
ix;~
fxÞto represent an element xin MVNS, and
the element xis called a multi-valued neutrosophic number (MVNN).
De¯nition 4 (Ref. 32). Let Abe a MVNN, and its complement ACcan be de¯ned
as follows:
AC¼[
2~
F
f’g;[
2~
I
f1g;[
2~
T
fg
8
<
:9
=
;
:ð2:8Þ
De¯nition 5 (Ref. 32). Let ~
n1¼f
~
t1;~
i1;~
f1gand ~
n2¼f
~
t2;~
i2;~
f2gbe two MVNNs,
then their operational laws can be de¯ned as follows:
ð1Þ~
n1~
n2¼f
~
t1~
t2;~
i1~
i2;~
f1~
f2g
¼[
~12~
t1;~12~
i1;~’12~
f1;
~
22~
t2;~
22~
i2;~
’22~
f2
f1þ212;
12;’
1’2gð2:9Þ
ð2Þ~
n1~
n2¼f
~
t1~
t2;~
i1~
i2;~
f1~
f2g
¼[
~12~
t1;~12~
i1;~’12~
f1;
~
22~
t2;~
22~
i2;~
’22~
f2
f12;1þ212;’1þ’2’1’2gð2:10Þ
ð3Þk~
n1¼[
~
12~
t1;~
12~
i1;~
’12~
f1
f1ð11Þk;k
1;’k
1g;k>0ð2:11Þ
ð4Þ~
nk
1¼[
~
12~
t1;~
12~
i1;~
’12~
f1
fk
1;1ð11Þk;1ð1’1Þkg;k>0:ð2:12Þ
4P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Example 1. Let A¼hf0:6g;f0:1;0:2g;f0:2gi and B¼hf0:5g;f0:3g;f0:2;0:3gi be
two MVNNs, and k¼2, then
(1) 2 A¼hf0:84g;f0:01;0:04g;f0:04gi;
(2) A2¼hf0:36g;f0:19;0:36g;f0:36gi;
(3) AB¼hf0:80g;f0:03;0:06g;f0:04;0:06i;
(4) AB¼hf0:30g;f0:37;0:44g;f0:36;0:44i:
Theorem 2.1. Let A ¼f
~
tA;~
iA;~
fAg,B¼f
~
tB;~
iB;~
fBgand C ¼f
~
tC;~
iC;~
fCgbe three
MVNNs,then we can prove that they can meet the following equations.
(1) kðABÞ¼kA kB;k>0;
(2) ðABÞk¼AkBk;k>0;
(3) k1Ak2A¼ðk1þk2ÞA;k1>0;k2>0;
(4) Ak1Ak2¼Ak1þk2;k1>0;k2>0;
Peng and Wang
47
gave the de¯nition of the score function and accuracy function of a
MVNN as follows.
De¯nition 6 (Ref. 47). Let sðAÞbe the score function of MVNN A¼f
~
tA;~
iA;~
fAg,
aðAÞbe the accuracy function of A¼f
~
tA;~
iA;~
fAg,then
ð1ÞsðAÞ¼ 1
l~
tAl~
iAl~
fAX
A2~
tA;A2~
iA;’A2~
fA
ðAA’AÞ=3;ð2:13Þ
ð2ÞaðAÞ¼ 1
l~
tAl~
iAl~
fAX
A2~
tA;A2~
iA;’A2~
fA
ðAþAþ’AÞ=3:ð2:14Þ
Here A2~
tA,A2~
iAand ’A2~
fA;l~
tA;l~
iAand l~
fAis the number of elements in
~
tA,~
iAand ~
fA, respectively.
De¯nition 7 (Ref. 47). Let Aand Bbe two MVNNs. Peng and Wang
47
gave a
comparison method for MVNNS shown as follows:
(1) If sðAÞ>sðBÞ, then AB;
(2) If sðAÞ¼sðBÞand if aðAÞ>aðBÞ, then AB;
Else if aðAÞ¼aðBÞ, then AB.
2.3. The Bonferroni mean (BM) operator
De¯nition 8 (Ref. 43). Let aiði¼1;2;...;nÞbe a collection of non-negative crisp
numbers, and p;q0, a Bonferroni mean operator of dimension nis a mapping of
BM: Rn!R. Such that,
BM p;qða1;a2;...;anÞ¼ 1
nðn1ÞX
n
i¼1X
n
j¼1;j6¼i
ap
iaq
j
!
1
pþq
:ð2:15Þ
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 5
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Because the BM operator only considers the interrelation of aggregated para-
meters and ignores the self-importance of them. So, Zhou
44
introduced a weighted
Bonferroni mean operator with reducibility to overcome this shortcoming.
De¯nition 9 (Ref. 44). Let aiði¼1;2;...;nÞbe a collection of non-negative crisp
numbers, and p;q0, and the weight vector of xibe W¼ðw1;w2;...;wnÞT,
satisfying wi0 and Pn
i¼1wi¼1. Then a weighted Bonferroni mean (WBM)
operator of dimension nis a mapping WBM: Rn!R. Such that
WBM p;qða1;a2;...;anÞ¼ X
n
i¼1X
n
j¼1;j6¼i
wiwj
1wi
ap
iaq
j
!
1=pþq
:ð2:16Þ
Zhou
44
has proved that the WBM operator has ¯ve properties as follows:
Theorem 2.2 (Reducibility). Let W ¼ð
1
n;1
n;...;1
nÞTbe the weighting vector of
aiði¼1;2;...;nÞ,then
WBMp;qða1;a2;...;anÞ¼ 1
nðn1ÞX
n
i¼1X
n
j¼1;j6¼i
ap
iaq
j
!
1
pþq
¼BM p;qða1;a2;...;anÞ:
Theorem 2.3 (Idempotency). Let aj¼aðj¼1;2;...;nÞ,the n WBM p;qða1;
a2;...;anÞ¼a.
Theorem 2.4 (Permutation). Let ða1;a2;...;anÞbe any permutation of
ða0
1;a0
2;...;a0
nÞ,then WBM p;qða0
1;a0
2;...;a0
nÞ¼WBM p;qða1;a2;...;anÞ.
Theorem 2.5 (Monotonicity). If ajbjðj¼1;2;...;nÞ,then
WBM p;qða1;a2;...;anÞWBMp;qðb1;b2;...;bnÞ:
Theorem 2.6 (Boundedness). The W BM p;qoperator lies in the maximum and
minimum operators,i.e., minða1;a2;...;anÞWBM p;qða1;a2;...;anÞmaxða1;
a2;...;anÞ.
By giving di®erent values to the parameters pand q, some special cases of WBM
can be gotten as follows:
ð1ÞIf q¼0;then WBM p;0ða1;a2;...;anÞ¼ X
n
i¼1
wiap
i
!
1
p
;ð2:17Þ
ð2ÞIf p¼1and q¼0;then WBM 1;0ða1;a2;...;anÞ¼X
n
i¼1
wiai;ð2:18Þ
ð3ÞIf p¼1
2and q¼1
2;then WBM 1
2;1
2ða1;a2;...;anÞ
¼X
n
i¼1X
n
j¼1;j6¼i
wiwj
1wi
ðaiajÞ1
2;ð2:19Þ
6P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

ð4ÞIf p¼1and q¼1;then WBM 1;1ða1;a2;...;anÞ
¼X
n
i¼1X
n
j¼1;j6¼i
wiwj
1wi
aiaj
!
1=2
:ð2:20Þ
2.4. The geometric Bonferroni mean operator
De¯nition 10 (Ref. 45). Let aiði¼1;2;...;nÞbe a collection of nonnegative crisp
numbers, and p;q0, a geometric Bonferroni mean operator of dimension nis a
mapping GBM: Rn!R, such that,
GBM p;qða1;a2;...;anÞ¼ 1
pþqY
n
i¼1Y
n
j¼1;j6¼i
ðpaiþqajÞ1
nðn1Þ:ð2:21Þ
Similar to the BM operator, the GBN operator also ignored the weights of the
aggregated arguments. Sun and Liu
46
further improved the weighted geometric
Bonferroni mean (WGBM) operator.
De¯nition 11 (Ref. 46). Let aiði¼1;2;...;nÞbe a collection of non-negative
crisp numbers, and p;q0, and the weight vector of aiði¼1;2;...;nÞbe W¼
ðw1;w1;...;wnÞT, satisfying wi 0 and Pn
i¼1wi ¼1, then a weighted geometric
Bonferroni mean (WGBM) operator of dimension nis a mapping WGBM: Rn!R.
Such that,
WGBM p;qða1;a2;...;anÞ¼ 1
pþqY
n
i¼1Y
n
j¼1;j6¼i
ðpaiþqajÞ
wiwj
1wi:ð2:22Þ
Sun and Liu
46
have proved that the WGBM operator has ¯ve properties which are
reducibility, idempotency, monotonicity, boundedness and permutation.
Theorem 2.7 (Reducibility). Let W ¼ð
1
n;1
n;...;1
nÞTbe the weighting vector of
aiði¼1;2;...;nÞ,then WGBM p;qða1;a2;...;anÞ¼GBM p;qða1;a2;...;anÞ.
Theorem 2.8 (Idempotency). Let aj¼aðj¼1;2;...;nÞ,then WGBM p;qða1;
a2;...;anÞ¼a.
Theorem 2.9 (Permutation). Let ða1;a2;...;anÞbe any permutation of
ða0
1;a0
2;...;a0
nÞ,then WGBM p;qða0
1;a0
2;...;a0
nÞ¼WGBM p;qða1;a2;...;anÞ.
Theorem 2.10 (Monotonicity). If ajbjðj¼1;2;...;nÞ,then
WGBM p;qða1;a2;...;anÞWGBM p;qðb1;b2;...;bnÞ:
Theorem 2.11 (Boundedness). The WGBM p;qoperator is between the max
and min operators,i.e., minða1;a2;...;anÞWGBM p;qða1;a2;...;anÞmaxða1;
a2;...;anÞ.
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 7
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

By giving di®erent values to the parameters pand q, some speci¯c cases of the
WGBM operators are obtained as follows:
ð1ÞIf q¼0;then WGBM p;qða1;a2;...;anÞ¼1
pY
n
i¼1
ðpaiÞwi;ð2:23Þ
ð2ÞIf q¼0;p¼1then WGBM p;qða1;a2;...;anÞ¼ Y
n
i¼1
awi
i;ð2:24Þ
ð3ÞIf p¼1
2and q¼1
2;then WGBM 1
2;1
2ða1;a2;...;anÞ
¼Y
n
i¼1Y
n
j¼1;j6¼i
1
2aiþ1
2aj

wiwj
1wi;ð2:25Þ
ð4ÞIf p¼1and q¼1;then WGBM 1;1ða1;a2;...;anÞ
¼1
2Y
n
i¼1Y
n
j¼1;j6¼i
ðaiþajÞ
wiwj
1wi:ð2:26Þ
3. Some Bonferroni Mean Operators based on MVNNs
In this section, the WBM and WGBM operators will be extended to aggregate the
MVNNs, and a multi-valued neutrosophic weighted Bonferroni mean (MVNWBM)
operator and a multi-valued neutrosophic weighted geometric Bonferroni mean
(MVNWGBM) operator are proposed as follows.
3.1. The MVNWBM operator
De¯nition 12. Let ~
nj¼ð
~
tj;~
ij;~
fjÞðj¼1;2;...;nÞbe a set of MVNNs with the
weighting vector W¼ðw1;w2;...;wnÞTand wj0;Pn
j¼1wj¼1, then the multi-
valued neutrosophic weighted Bonferroni mean (MVNWBM) operator of dimension
nis the mapping MVNWBM: n!, such that
MVNWBM p;qð~
n1;~
n2;...;~
nnÞ¼ M
n
i¼1;j¼1;i6¼j
wiwj
1wi
~
np
i~
nq
j
0
@1
A
1=pþq
;ð3:1Þ
where is the set of all MVNNs.
Based on the operations (2.9)–(2.12) and De¯nition 13, the following theorems will
be deduced.
Theorem 3.1. Let ~
nj¼ð
~
tj;~
ij;~
fjÞðj¼1;2;...;nÞbe a set of MVNNs with the
weighting vector w ¼ðw1;w2;...;wnÞTand wj0;Pn
j¼1wj¼1, and then the
8P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

aggregated result by the MVNWBM operator in De¯nition 12 can be expressed as:
MVNWBM p;qð~
n1;~
n2;...;~
nnÞ
¼1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1p
iq
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
;
0
B
B
B
@
11Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
;
11Y
n
i;j¼1
j6¼i[
~
’i2~
fi;~
’j2~
fj
ð1ð1’iÞpð1’jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq1
C
C
C
A
:ð3:2Þ
Proof. Since
~
np
i¼[
~
i2~
ti;~
i2~
ii;~
’i2~
fi
ðp
i;1ð1iÞp;1ð1’iÞpÞ;
~
nq
j¼[
~
j2~
tj;~
j2~
ij;~
’j2~
fj
ðq
j;1ð1jÞq;1ð1’jÞqÞ;
~
np
i~
nq
j¼[
~
i2~
ti;~
i2~
ii;~
’i2~
fi;~
j2~
tj;~
j2~
ij;~
’j2~
fj
ðp
iq
j;1ð1iÞpð1jÞq;1ð1’iÞpð1’jÞqÞ;
and
wiwj
1wi
~
np
i~
nq
j¼[
~i2~
ti;~i2~
ii;~’i2~
fi;~j2~
tj;~j2~
ij;~’j2~
fj
ð1ð1p
iq
jÞ
wiwj
1wi;
ð1ð1iÞpð1jÞqÞ
wiwj
1wi;ð1ð1’iÞpð1’jÞqÞ
wiwj
1wiÞ;
then
M
n
i;j¼1
i6¼j
wiwj
1wi
~
np
i~
nq
j

¼1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1p
iq
jÞ
wiwj
1wi;
0
B
B
@Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi;
Y
n
i;j¼1
j6¼i[
~
’i2~
fi;~
’j2~
fj
ð1ð1’iÞpð1’jÞqÞ
wiwj
1wi1
C
C
A
:
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 9
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

So
MVNWBM ¼M
n
i;j¼1
i6¼j
wiwj
1wi
~
np
i~
nq
j
0
B
B
B
@
1
C
C
C
A
1=pþq
¼1Y
n
i;j¼1
i6¼j[
~i2~
ti;~j2~
tj
ð1p
iq
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
;
0
B
B
B
@
11Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
;
11Y
n
i;j¼1
j6¼i[
~
’i2~
fi;~
’j2~
fj
ð1ð1’iÞpð1’jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq1
C
C
C
A
:
In addition, the MVNWBM operator has the some properties as follows:
Theorem 3.2 (Permutation). Let ~
nj¼ð
~
tj;~
ij;~
fjÞðj¼1;2;...;nÞbe a set of
MVNNs.If ~
n0
jðj¼1;2;...;nÞis a permutation of ~
njðj¼1;2;...;nÞ,then
MVNWBM p;qð~
n1;~
n2;...;~
nnÞ¼MVNWBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:ð3:3Þ
Proof. According to De¯nition 12, we can get
M
n
i¼1;j¼1;i6¼j
wiwj
1wi
~
np
i~
nq
j
0
@1
A
1=pþq
¼M
n
i¼1;j¼1;i6¼j
wiwj
1wi
~
n0p
i~
n0q
j
0
@1
A
1=pþq
:
So
MVNWBM p;qð~
n1;~
n2;...;~
nnÞ¼MVNWBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:
Theorem 3.3 (Idempotency). Let ~
nj¼~
n¼f
~
t;~
i;~
fg,where ~
t¼,~
i¼and
~
f¼’,then
MVNWBM p;qð~
n1;~
n2;...;~
nnÞ¼~
n¼f
~
t;~
i;~
fg:ð3:4Þ
Proof. Since ~
nj¼~
n¼f
~
t;~
i;~
fg, for all j, we have
MVNWBM p;qð~
n1;~
n2;...;~
nnÞ¼ M
n
i¼1;j¼1;i6¼j
wiwj
1wi
~
np
i~
nq
j
0
@1
A
1=pþq
¼M
n
i¼1;j¼1;i6¼j
wiwj
1wi
~
np~
nq
0
@1
A
1=pþq
¼~
npþqM
n
i¼1;j¼1;i6¼j
wiwj
1wi
0
@1
A
1=pþq
¼ð
~
npþqÞ1=pþq¼~
n:
10 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Theorem 3.4 (Monotonicity). Let ~
njðj¼1;2;...;nÞand ~
n0
jðj¼1;2;...;nÞbe
two sets of MVNNs.If ~
nj~
n0
jfor all j,we may suppose j0
j,j0
jand ’j’0
j
for all j,then
MVNWBM p;qð~
n1;~
n2;...;~
nnÞMVNWBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:ð3:5Þ
Proof. (1) Since j0
jand p;q>0;wj0 for all j, then
p
iq
j0p
i0q
j;1p
iq
j10p
i0q
j;
Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1p
iq
jÞ
wiwj
1wiY
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð10p
i0q
jÞ
wiwj
1wi;
1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1p
iq
jÞ
wiwj
1wi1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð10p
i0q
jÞ
wiwj
1wi;
1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1p
iq
jÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1=pþq
1Y
n
i;j¼1
i6¼j[
~i2~
ti;~j2~
tj
ð10p
i0q
jÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1=pþq
:
(2) Since j0
jand p;q>0;wj0 for all j,then
ð1jÞpð10
jÞpand ð1jÞqð10
jÞq
then
ð1iÞpð1jÞqð10
iÞpð10
jÞq;
1ð1iÞpð1jÞq1ð10
iÞpð10
jÞq;
Y
n
i;j¼1
j6¼i[
~i2~
ii;~j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi;
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 11
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

1Y
n
i;j¼1
j6¼i[
~i2~
ii;~j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
1Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi;
1Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
1Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
;
11Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
11Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
:
(3) Similar to (2), we have
11Y
n
i;j¼1
j6¼i[
~
’i2~
fi;~
’j2~
fj
ð1ð1’iÞpð1’jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
11Y
n
i;j¼1
j6¼i[
~
’i2~
fi;~
’j2~
fj
ð1ð1’0
iÞpð1’0
jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=pþq
:
According to (1)–(3), we can get
MVNWBM p;qð~
n1;~
n2;...;~
nnÞMVNWBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:
Theorem 3.5 (Boundedness). Let ~
njðj¼1;2;...;nÞbe one set of MVNNs,then
we have
minð~
n1;~
n2;...;~
nnÞMVNWBM ð~
n1;~
n2;...;~
nnÞmaxð~
n1;~
n2;...;~
nnÞ:ð3:6Þ
12 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Proof. Let m¼minð~
n1;~
n2;...~
nnÞ,M¼maxð~
n1;~
n2;...~
nnÞ, since m~
njM,
then the following result can be got by Theorem 3.4.
m¼minð~
n1;~
n2;...;~
nnÞMVNWBM ð~
n1;~
n2;...;~
nnÞ
and
MVNWBM ð~
n1;~
n2;...;~
nnÞmaxð~
n1;~
n2;...;~
nnÞ¼M;
then
m¼minð~
n1;~
n2;...;~
nnÞMVNWBM ð~
n1;~
n2;...;~
nnÞmaxð~
n1;~
n2;...;~
nnÞ¼M:
Next, some special cases of the MVNWBM operator concerning the parameters p
and qwill be demonstrated respectively.
(1) When p¼0, then
MVNWBM 0;qð~
n1;~
n2;...;~
nnÞ
¼1Y
n
j¼1[
~
j2~
tj
ð1q
jÞPn
i¼1;i6¼j
wiwj
1wi
0
@1
A
1=q
;
0
@
11Y
n
j¼1[
~j2~
ij
ð1ð1jÞqÞPn
i¼1;i6¼j
wiwj
1wi
0
@1
A
1=q
;
11Y
n
j¼1[
~
’j2~
fj
ð1ð1’jÞqÞPn
i¼1;i6¼j
wiwj
1wi
0
@1
A
1=q1
A:
When q¼0, then
MVNWBM p;0ð~
n1;~
n2;...;~
nnÞ¼ 1Y
n
i¼1[
~
i2~
ti
ð1p
iÞwi
0
@1
A
1=p
;
0
@
11Y
n
i¼1[
~
i2~
ii
ð1ð1iÞpÞwi
0
@1
A
1=p
;
11Y
n
i¼1[
~
’i2~
fi
ð1ð1’iÞpÞwi
0
@1
A
1=p1
A:
(2) When p¼q¼1, then
MVNWBM 1;1ð~
n1;~
n2;...;~
nnÞ¼ 1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ijÞ
wiwj
1wi
0
B
B
@1
C
C
A
1=2
;
0
B
B
B
@
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 13
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

11Y
n
i;j¼1
j6¼i[
~
i2~
ii;~
j2~
ij
0
B
B
B
@
ðiþjijÞ
wiwj
1wi1
C
C
C
A
1=2
;
11Y
n
i;j¼1
j6¼i[
~
’i2~
fi;~
’j2~
fj
ð’iþ’j’i’jÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1=21
C
C
C
A
:
3.2. The MVNWGBM operator
De¯nition 13. Let ~
nj¼ð
~
tj;~
ij;~
fjÞðj¼1;2;...;nÞbe a set of MVNNs with the
weighting vector w¼ðw1;w2;...;wnÞTand wj0;Pn
j¼1wj¼1, then the multi-
valued neutrosophic weighted geometric Bonferroni mean (MVNWGBM) operator
of dimension nis the mapping MVNWGBM: n!, such that
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ¼ 1
pþqO
n
i;j¼1
j6¼i
ðp~
niq~
njÞ
wiwj
1wi;ð3:7Þ
where is the set of all MVNNs.
Based on the operational rules (2.9)–(2.12) and De¯nition 13, we get the following
theorems.
Theorem 3.6. Let ~
nj¼ð
~
tj;~
ij;~
fjÞðj¼1;2;...;nÞbe a set of MVNNs with the
weighting vector w ¼ðw1;w2;...;wnÞTand wj0;Pn
j¼1wj¼1. Then the aggre-
gated result by the MVNWGBM operator can be expressed as:
MVNWGBMp;qð~
n1;~
n2;...;~
nnÞ
¼11Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
0
B
B
@
0
B
B
@
ð1ð1iÞpð1jÞqÞ
wiwj
1wi1
C
C
C
A
1
pþq
;
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1p
iq
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
;1Y
n
i;j¼1
i6¼j[
~’j2~
tj;~’j2~
fj
ð1’p
i’q
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq1
C
C
C
A
:
ð3:8Þ
Proof.
p~
ni¼[
~
i2~
ti;~
i2~
ii;~
’i2~
fi
ð1ð1iÞp;p
i;’p
iÞ;
14 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

q~
nj¼[
~
j2~
tj;~
j2~
ij;~
’j2~
fj
ð1ð1iÞq;q
i;’q
iÞ;
p~
niq~
nj¼[
~
i2~
ti;~
i2~
ii;~
’i2~
fi;~
j2~
tj;~
j2~
ij;~
’j2~
fj
ð1ð1iÞpð1jÞq;p
iq
j;’p
i’q
jÞ;
and
ðp~
niq~
njÞ
wiwj
1wi¼[
~
i2~
ti;~
i2~
ii;~
’i2~
fi;~
j2~
tj;~
j2~
ij;~
’j2~
fj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
;
1ð1p
iq
jÞ
wiwj
1wi;1ð1’p
i’q
jÞ
wiwj
1wi;
then
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ¼ 1
pþqO
n
i;j¼1
j6¼i
ðp~
niq~
njÞ
wiwj
1wi;
O
n
i;j¼1
i6¼j
ðp~
niq~
njÞ
wiwj
1wi
¼Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@
;
1Y
n
i;j¼1
i6¼j[
~i2~
ii;~j2~
ij
ð1p
iq
jÞ
wiwj
1wi;1Y
n
i;j¼1
i6¼j[
~
’j2~
fj;~
’j2~
fj
ð1’p
i’q
jÞ
wiwj
1wi1
C
C
A
;
further
1
pþqO
n
i;j¼1
i6¼j
ðp~
niq~
njÞ
wiwj
1wi
¼11Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
;
0
B
B
B
@
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1p
iq
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
;
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 15
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

1Y
n
i;j¼1
i6¼j[
~
’j2~
fj;~
’j2~
fj
ð1’p
i’q
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq1
C
C
C
A
:
So
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ
¼11Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
;
0
B
B
B
@
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1p
iq
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
;
1Y
n
i;j¼1
i6¼j[
~
’j2~
fj;~
’j2~
fj
ð1’p
i’q
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq1
C
C
C
A
:
In addition, the MVNWGBM operator has the following properties.
Theorem 3.7 (Permutation). Let ~
nj¼ð
~
tj;~
ij;~
fjÞðj¼1;2;...;nÞbe a set of
MVNNs.If ~
n0
jðj¼1;2;...;nÞis a permutation of ~
njðj¼1;2;...;nÞ,then
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ¼MVNWGBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:ð3:9Þ
Proof. According to De¯nition 13, we can get
1
pþqO
n
i;j¼1
j6¼i
ðp~
niq~
njÞ
wiwj
1wi¼1
pþqO
n
i;j¼1
j6¼i
ðp~
n0
iq~
n0
jÞ
wiwj
1wi:
So
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ¼MVNWGBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:
Theorem 3.8 (Idempotency). Let ~
nj¼~
n¼f
~
t;~
i;~
fg,where ~
t¼,~
i¼and
~
f¼’,then
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ¼~
n¼f
~
t;~
i;~
fg:ð3:10Þ
16 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Proof. Since ~
nj¼~
n¼f
~
t;~
i;~
fg, for all j, we have
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ
¼1
pþqO
n
i;j¼1
j6¼i
ðp~
niq~
njÞ
wiwj
1wi
¼1
pþqO
n
i;j¼1
j6¼i
ðp~
nq~
nÞ
wiwj
1wi¼1
pþqO
n
i;j¼1
j6¼i
ððpþqÞ~
nÞ
wiwj
1wi
¼1
pþqððpþqÞ~
nÞX
n
i;j¼1
j6¼i
wiwj
1wi
¼1
pþqððpþqÞ~
nÞ¼~
n:
Theorem 3.9 (Monotonicity). Let ~
njðj¼1;2;...;nÞand ~
n0
jðj¼1;2;...;nÞbe
two sets of MVNNs.If ~
nj~
n0
jfor all j,we may suppose j0
j,j0
jand j0
j
for all j,then
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞMVNWGBM p;q~
n0
1;~
n0
2;...;~
n0
n

:ð3:11Þ
Proof. (1) Since j0
jand p;q>0;wj0 for all j, then
ð1iÞpð10
iÞp;ð1jÞqð10
jÞq;
and
ð1iÞpð1jÞqð10
iÞpð10
jÞq;
then
1ð1iÞpð1jÞq1ð10
iÞpð10
jÞq;
Y
n
i;j¼1
i6¼j[
~i2~
ti;~j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
Y
n
i;j¼1
i6¼j[
~i2~
ti;~j2~
tj
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi;
1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi;
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 17
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1
pþq
1Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1
pþq
:
So
11Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð1iÞpð1jÞqÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1
pþq
11Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ð1ð10
iÞpð10
jÞqÞ
wiwj
1wi
0
B
B
B
@
1
C
C
C
A
1
pþq
:
(2) Since j0
jand p;q>0;wj0 for all j, then
p
iq
j0p
i0q
j;1p
iq
j10p
i0q
j;
Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1p
iq
jÞ
wiwj
1wiY
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð10p
i0q
jÞ
wiwj
1wi;
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1p
iq
jÞ
wiwj
1wi1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð10p
i0q
jÞ
wiwj
1wi:
So
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1p
iq
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð10p
i0q
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
:
18 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

(3) Similar to the (2), we have
1Y
n
i;j¼1
i6¼j[
~
’j2~
fj;~
’j2~
fj
ð1’p
i’q
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
1Y
n
i;j¼1
i6¼j[
~
’j2~
fj;~
’j2~
fj
ð1’0p
i’0q
jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
pþq
:
According to (1)–(3), we can get
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞMVNWGBM p;qð~
n0
1;~
n0
2;...;~
n0
nÞ:
Theorem 3.10 (Boundedness). The MVNWBM operator lives between the
maximum and minimum operators:
minð~
n1;~
n2;...;~
nnÞMVNWGBM ð~
n1;~
n2;...;~
nnÞmaxð~
n1;~
n2;...;~
nnÞ:
ð3:12Þ
Proof. Let m¼minð~
n1;~
n2;...~
nnÞ,M¼maxð~
n1;~
n2;...~
nnÞ, since m~
njM,
we can get the following result according to Theorem 3.9.
m¼minð~
n1;~
n2;...;~
nnÞMVNWGBM ð~
n1;~
n2;...;~
nnÞ
and
MVNWGBM ð~
n1;~
n2;...;~
nnÞmaxð~
n1;~
n2;...;~
nnÞ¼M;
then
m¼minð~
n1;~
n2;...;~
nnÞMVNWGBM ð~
n1;~
n2;...;~
nnÞ
maxð~
n1;~
n2;...;~
nnÞ¼M:
Next, some special cases of the MVNWGBM operator concerning the parameters
pand qcan be shown as follows.
(1) When p¼0, then
MVNWGBM 0;qð~
n1;~
n2;...;~
nnÞ
¼11Y
n
j¼1[
~
j2~
tj
ð1ð1jÞqÞPn
i¼1;i6¼j
wiwj
1wi
0
@1
A
1
q
;
0
@
1Y
n
j¼1[
~
j2~
ij
ð1q
jÞPn
i¼1;i6¼j
wiwj
1wi
0
@1
A
1
q
;1Y
n
j¼1[
~
’j2~
fj
ð1’q
jÞPn
i¼1;i6¼j
wiwj
1wi
0
@1
A
1
q1
A:
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 19
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

(2) When q¼0, then
MVNWGBM 0;qð~
n1;~
n2;...;~
nnÞ¼ 11Y
n
i¼1[
~
i2~
ti
ð1ð1iÞpÞwi
0
@1
A
1
p
;
0
@
1Y
n
i¼1[
~i2~
ii
ð1p
iÞwi
0
@1
A
1
p
;
1Y
n
i¼1[
~
’i2~
fi
ð1’p
iÞwi
0
@1
A
1
p1
A:
Moreover, the parameters pand qdon't have the property of interchangeability.
(3) When p¼q¼1, then
MVNWGBM p;qð~
n1;~
n2;...;~
nnÞ
¼11Y
n
i;j¼1
i6¼j[
~
i2~
ti;~
j2~
tj
ðiþjijÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
2
;
0
B
B
B
@
1Y
n
i;j¼1
i6¼j[
~
i2~
ii;~
j2~
ij
ð1ijÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
2
;1Y
n
i;j¼1
i6¼j[
~
’j2~
fj;~
’j2~
fj
ð1’i’jÞ
wiwj
1wi
0
B
B
@1
C
C
A
1
21
C
C
C
A
:
4. The MAGDM Approaches Based on the MVNNs
For a MAGDM problem, let A¼fA1;A2;...;Amgbe a collection of Malternatives,
C¼fC1;C2;...;Cngbe a collection of ncriteria, which weight vector is w¼
ðw1;w2;...wnÞTsatisfying wj2½0;1;Pn
j¼1wj¼1. Suppose D¼fD1;D2;...;Dgis
a group of decision-makers, whose weight vector is !¼ð!1;!
2;...;!
Þ, where
!k0, P
k¼1!k¼1. Let Rk¼ðrk
ij Þmnbe the multi-valued neutrosophic decision
matrix, and rk
ij ¼ð
~
trk
ij ;~
irk
ij ;~
frk
ij Þis a MVNN given by the decision-maker Dkfor the
alternative Aiunder criterion Cj. Then the MVNWBM operator and MVNWGBM
operator are applied to solve this MAGDM problem under multi-valued neu-
trosophic environment.
The steps of the proposed method as follows:
Step 1. Normalize the decision matrix.
In MAGDM problems, there are two forms in criteria, that is, bene¯t criteria and
cost criteria. To maintain consistency of the criteria, we usually transform the cost
20 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

criteria into bene¯t criteria as follows:
k
ij ¼rk
ij ;for benefit criteria Cj
ðrk
ij Þc;for cost criteria Cj
(;ði¼1;2;...;m;j¼1;2;...;nÞ;ð4:1Þ
where ðrk
ij Þcis the complement of rk
ij , the decision matrix Rk¼ðrk
ij Þmncan be
transformed into a normalized decision matrix ~
Rk¼ðk
ij Þmn.
The detailed normalized methods are shown as follows.
For the cost criteria, the normalization formula is
k
ij ¼ðrk
ij Þc¼[
2~
frk
ij
f’g;[
2~
irk
ij
f1g;[
2~
trk
ij
fg
8
>
>
<
>
>
:
9
>
>
=
>
>
;
;ð4:2Þ
for the bene¯t criteria, is
k
ij ¼rk
ij ¼[
2~
trk
ij
fg;[
2~
irk
ij
fg;[
2~
frk
ij
f’g
8
>
>
<
>
>
:
9
>
>
=
>
>
;
:ð4:3Þ
Step 2. Utilize the MVNWBM operator
k
i¼ð
~
tk
i;~
ik
i;~
fk
iÞ¼MVNWBM p;qðk
i1;k
i2;...;k
inÞ;ð4:4Þ
or MVNWGBM operator
k
i¼ð
~
tk
i;~
ik
i;~
fk
iÞ¼MVNWGBM p;qðk
i1;k
i2;...;k
inÞð4:5Þ
to derive the comprehensive values k
iði¼1;2;...;m;k¼1;2;...;Þ.
Step 3. Utilize the MVNWBM operator
i¼ð
~
ti;~
ii;~
fiÞ¼MVNWBM p;qð1
i;2
i;...;
iÞ;ð4:6Þ
or MVNWGBM operator
i¼ð
~
ti;~
ii;~
fiÞ¼MVNWGBM p;qð1
i;2
i;...;
iÞð4:7Þ
to derive the collective values iði¼1;2;...;mÞ.
Step 4. Calculate the score function values sðiÞand the accuracy function values
aðiÞ.
Step 5. Select the best alternative(s) based on step 4.
Step 6. End.
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 21
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

5. An Application Example
In this section, we cite the example
23
of an investment appraisal project to illustrate
the e®ectiveness of MVNWBM operator and MVNWGBM operator under multi-
valued neutrosophic environment.
An investment company called ABC wants to select an enterprise to invest from
the following four alternatives ðA1;A2;A3;A4Þ. There are three evaluation index for
each alternative: (1) C1is the risk index; (2) C2is the growth index; (3) C3is
the environmental impact index, where C1and C2are the bene¯t type, and C3is the
cost type. The weighting vector of index is w¼ð0:35;0:25;0:4ÞT. There are
three evaluators ðD1;D2;D3Þto do assessments and their weighting vector is
!¼ð0:3;0:5;0:2ÞT. Moreover, rk
ij ¼ð
~
trk
ij ;~
irk
ij ;~
frk
ij Þis provided by the k-th evaluator
about the alternative Aiði¼1;2;3;4Þunder the criterion Cjðj¼1;2;3Þ. Then the
decision matrix Rk¼ðrk
ij Þ43by D1;D2;D3are shown in Tables 1–3, respectively.
5.1. The evaluation steps by the MVNWBM operator
Step 1. Normalize the decision matrix.
Considering all the criteria should be uniform types, the cost type C3should be
transformed into bene¯t type, and then we obtain the normalized decision matrix
~
Rk¼ðk
ij Þ43as follows (Tables 4–6).
Table 1. The multi-valued neutrosophic decision matrix from D1.
C1C2C3
A1ff0.4g,f0.1g,f0.2gg ff0.5g,f0.2g,f0.1gg ff0.3g,f0.1, 0.2g,f0.4gg
A2ff0.7g,f0.1, 0.2g,f0.2gg ff0.6g,f0.2g,f0.2, 0.3gg ff0.4g,f0.2g,f0.3gg
A3ff0.4, 0.5g,f0.1g,f0.3gg ff0.5g,f0.2g,f0.1gg ff0.4, 0.5g,f0.2g,f0.2gg
A4ff0.6g,f0.3g,f0.1gg ff0.5, 0.6g,f0.3g,f0.2gg ff0.5g,f0.1g,f0.2gg
Table 2. The multi-valued neutrosophic decision matrix from D2.
C1C2C3
A1ff0.6g,f0.1g,f0.1, 0.2gg ff0.5g,f0.2g,f0.2gg ff0.4, 0.5g,f0.1g,f0.3gg
A2ff0.5g,f0.2g,f0.2gg ff0.6g,f0.2g,f0.1, 0.2gg ff0.5g,f0.3g,f0.2gg
A3ff0.4, 0.5g,f0.2g,f0.1gg ff0.5g,f0.1g,f0.3gg ff0.5g,f0.1g,f0.2, 0.3gg
A4ff0.5g,f0.3g,f0.2gg ff0.8g,f0.2, 0.3g,f0.2gg ff0.5g,f0.2g,f0.2gg
Table 3. The multi-valued neutrosophic decision matrix from D3.
C1C2C3
A1ff0.4, 0.5g,f0.2g,f0.3gg ff0.4g,f0.2, 0.3g,f0.3gg ff0.2g,f0.2g,f0.5gg
A2ff0.6g,f0.1, 0.2g,f0.2gg ff0.6g,f0.1g,f0.2gg ff0.5g,f0.2g,f0.1, 0.2gg
A3ff0.3, 0.4g,f0.2g,f0.3gg ff0.5g,f0.2g,f0.3gg ff0.5g,f0.2, 0.3g,f0.2gg
A4ff0.7g,f0.1, 0.2g,f0.1gg ff0.6g,f0.1g,f0.2gg ff0.4g,f0.3g,f0.2gg
22 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Step 2. Use Eq. (4.4) to calculate the comprehensive evaluation values
k
iði¼1;2;3;4;k¼1;2;3Þof each decision maker and each alternative (suppose
p¼q¼1), and have
1
1¼ff0:4270g;f0:3921;0:4334g;f0:2081gg;
1
2¼ff0:5107g;f0:3921;0:4334g;f0:2748;0:3037gg;
1
3¼ff0:3429;0:3746g;f0:3921g;f0:2811;0:3193gg;
1
4¼ff0:4044;0:4332g;f0:5524g;f0:2764gg;
2
1¼ff0:4510g;f0:4334g;f0:2393;0:3128gg;
2
2¼ff0:4009g;f0:3916g;f0:2807;0:3128gg;
2
3¼ff0:3429;0:4199g;f0:4344g;f0:3097gg;
2
4¼ff0:4515g;f0:4724;0:5042g;f0:3127gg;
3
1¼ff0:4360;0:4721g;f0:4334;0:4670g;f0:2629gg;
3
2¼ff0:3828;0:4332g;f0:3426;0:3940g;f0:3128gg;
3
3¼ff0:3088;0:3430g;f0:3916;0:4334g;f0:3768gg;
3
4¼ff0:4644g;f0:3075;0:3552g;f0:2393gg:
Table 4. The normalized multi-valued neutrosophic decision matrix from D1.
C1C2C3
A1ff0.4g,f0.1g,f0.2gg ff0.5g,f0.2g,f0.1gg ff0.4g,f0.8, 0.9g,f0.3gg
A2ff0.7g,f0.1, 0.2g,f0.2gg ff0.6g,f0.2g,f0.2, 0.3gg ff0.3g,f0.8g,f0.4gg
A3ff0.4, 0.5g,f0.1g,f0.3gg ff0.5g,f0.2g,f0.1gg ff0.2g,f0.8g,f0.4, 0.5gg
A4ff0.6g,f0.3g,f0.1gg ff0.5, 0.6g,f0.3g,f0.2gg ff0.2g,f0.9g,f0.5gg
Table 5. The normalized multi-valued neutrosophic decision matrix from D2.
C1C2C3
A1ff0.6g,f0.1g,f0.1, 0.2gg ff0.5g,f0.2g,f0.2gg ff0.3g,f0.9g,f0.4, 0.5gg
A2ff0.5g,f0.2g,f0.2gg ff0.6g,f0.2g,f0.1, 0.2gg ff0.2g,f0.7g,f0.5gg
A3ff0.4, 0.5g,f0.2g,f0.1gg ff0.5g,f0.1g,f0.3gg ff0.2, 0.3g,f0.9g,f0.5gg
A4ff0.5g,f0.3g,f0.2gg ff0.8g,f0.2, 0.3g,f0.2gg ff0.2g,f0.8g,f0.5gg
Table 6. The normalized multi-valued neutrosophic decision matrix from D3.
C1C2C3
A1ff0.4, 0.5g,f0.2g,f0.3gg ff0.4g,f0.2, 0.3g,f0.3gg ff0.5g,f0.8g,f0.2gg
A2ff0.6g,f0.1, 0.2g,f0.2gg ff0.6g,f0.1g,f0.2gg ff0.1, 0.2g,f0.8g,f0.5gg
A3ff0.3, 0.4g,f0.2g,f0.3gg ff0.5g,f0.2g,f0.3gg ff0.2g,f0.7, 0.8g,f0.5gg
A4ff0.7g,f0.1, 0.2g,f0.1gg ff0.6g,f0.1g,f0.2gg ff0.2g,f0.7g,f0.4gg
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 23
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Step 3. Use Eq. (4.6) to calculate the collective overall values iði¼1;2;3;4Þof
each alternative (suppose p¼q¼1), and have
1¼ff0:4392;0:4480g;f0:4195;0:4423g;f0:2343;0:2658gg;
2¼ff0:4326;0:4444g;f0:3799;0:4064g;f0:2864;0:3097gg;
3¼ff0:3346;0:3852g;f0:4099;0:4200g;f0:3160;0:3291gg;
4¼ff0:4385;0:4484g;f0:4592;0:4845g;f0:2827gg:
Step 4. Calculate the score function values sðiÞof the collective overall values
iði¼1;2;3;4Þ.
sð1Þ¼0:0824;sð2Þ¼0:0211;sð3Þ¼0:0315;sð4Þ¼0:0518:
Step 5. Rank all the alternatives Aiði¼1;2;3;4Þbased on the values of sðiÞ,
because sð2Þ>sð3Þ>sð4Þ>sð1Þ, we can get A2A3A4A1and A2is the
best option.
Step 6. End.
5.2. The evaluation steps by the MVNWGBM operator
Step 1. Normalize the decision matrix.
Same as Step 1 in Sec. 5.1, the normalized decision matrix ~
Rk¼ðk
ij Þ43are
shown as Tables 4–6.
Step 2. Use Eq. (4.5) to calculate the comprehensive evaluation values k
iði¼
1;2;3;4;k¼1;2;3Þof each decision maker and each alternative (suppose p¼q¼1),
and have
1
1¼ff0:4369g;f0:3007;0:3185g;f0:2028gg;
1
2¼ff0:5216g;f0:3007;0:3624g;f0:2648;0:2954gg;
1
3¼ff0:3629;0:3955g;f0:3007g;f0:2727;0:3025gg;
1
4¼ff0:4270;0:4603g;f0:4779g;f0:2407gg;
2
1¼ff0:4680g;f0:3185g;f0:2174;0:2920gg;
2
2¼ff0:4252g;f0:3404g;f0:2560;0:2920gg;
2
3¼ff0:3557;0:4289g;f0:3394g;f0:2763gg;
2
4¼ff0:4851g;f0:4170;0:4526g;f0:2920gg;
3
1¼ff0:4382;0:4728g;f0:3624;0:4018g;f0:2601gg;
3
2¼ff0:4390;0:4681g;f0:2508;0:3204g;f0:2920gg;
3
3¼ff0:3164;0:3557g;f0:3404;0:3624g;f0:3678gg;
3
4¼ff0:5128g;f0:2351;0:3003g;f0:2174gg:
24 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Step 3. Use Eq. (4.7) to calculate the collective overall values of each alternative
iði¼1;2;3;4Þ(suppose p¼q¼1), and have
1¼ff0:4504;0:4587g;f0:3230;0:3382g;f0:2226;0:2523gg;
2¼ff0:4618;0:4691g;f0:3043;0:3429g;f0:2675;0:2932gg;
3¼ff0:3485;0:4000g;f0:3264;0:3317g;f0:2966;0:3066gg;
4¼ff0:4724;0:4835g;f0:3922;0:4235g;f0:2554gg:
Step 4. Calculate the score function value sðiÞof the collective overall values
iði¼1;2;3;4Þ.
sð1Þ¼0:0866;sð2Þ¼0:0223;sð3Þ¼0:0324;sð4Þ¼0:0534:
Step 5. Rank all the alternatives Aiði¼1;2;3;4Þbased on the values of sðiÞ,
because sð2Þ>sð3Þ>sð4Þ>sð1Þ, we can get A2A3A4A1and A2is the
best option.
Step 6. End.
5.3. The decision results by changing the parameters p,q
When the parameters p;qadopt di®erent values, we may get di®erent ranking
results, and the corresponding score values with the ranking of alternatives are
shown in Tables 7and 8.
Table 7. The rankings of alternatives by changing values of parameters
p;qin MVNWBM operator.
p;qScore function values sjðj¼1;2;3;4ÞRanking
p¼0s1¼0:0340;s2¼0:0084 A2A3A4A1
q¼0:01 s3¼0:0189;s4¼0:0228
p¼0s1¼0:0149;s2¼0:0034 A2A4A3A1
q¼1s3¼0:0138;s4¼0:0099
p¼0s1¼0:0005;s2¼0:0009 A1A2A4A3
q¼2s3¼0:0095;s4¼0:0012
p¼0s1¼0:0541;s2¼0:0159 A3A2A4A1
q¼10 s3¼0:0065;s4¼0:0425
p¼0:01 s1¼0:0466;s2¼0:0146 A2A3A4A1
q¼0s3¼0:0230;s4¼0:0377
p¼1s1¼0:0221;s2¼0:0091 A2A3A1A4
q¼0s3¼0:0167;s4¼0:0232
p¼2s1¼0:0039;s2¼0:0042 A1A2A4A3
q¼0s3¼0:0117;s4¼0:0104
p¼10 s1¼0:0596;s2¼0:0135 A3A2A4A1
q¼0s3¼0:0062;s4¼0:0372
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 25
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Table 7. (Continued )
p;qScore function values sjðj¼1;2;3;4ÞRanking
p¼1s1¼0:0824;s2¼0:0211 A2A3A4A1
q¼1s3¼0:0315;s4¼0:0518
p¼2s1¼0:0636;s2¼0:0169 A2A3A4A1
q¼1s3¼0:0267;s4¼0:0417
p¼10 s1¼0:0278;s2¼0:0070 A3A2A4A1
q¼1s3¼0:0014;s4¼0:0198
p¼1s1¼0:0623;s2¼0:0157 A2A3A4A1
q¼2s3¼0:0261;s4¼0:0385
p¼1s1¼0:0230;s2¼0:0085 A3A2A1A4
q¼10 s3¼0:0013;s4¼0:0236
Table 8. The rankings of alternatives by changing values of parameters
p;qin MVNWGBM operator.
p;qScore function values sjðj¼1;2;3;4ÞRanking
p¼0s1¼0:0866;s2¼0:0223 A2A3A4A1
q¼0:01 s3¼0:0324;s4¼0:0534
p¼0s1¼0:1253;s2¼0:0312 A3A2A4A1
q¼1s3¼0:0419;s4¼0:0719
p¼0s1¼0:1592;s2¼0:0400 A2A3A4A1
q¼2s3¼0:0503;s4¼0:0899
p¼0s1¼0:2667;s2¼0:0719 A2A3A4A1
q¼10 s3¼0:0771;s4¼0:1545
p¼0:01 s1¼0:1133;s2¼0:0270 A2A3A4A1
q¼0s3¼0:0389;s4¼0:0672
p¼1s1¼0:1536;s2¼0:0355 A2A3A4A1
q¼0s3¼0:0485;s4¼0:0844
p¼2s1¼0:1868;s2¼0:0437 A2A3A4A1
q¼0s3¼0:0564;s4¼0:1009
p¼10 s1¼0:2882;s2¼0:0730 A2A3A4A1
q¼0s3¼0:0803;s4¼0:1584
p¼1s1¼0:0372;s2¼0:0115 A2A3A4A1
q¼1s3¼0:0214;s4¼0:0309
p¼2s1¼0:0636;s2¼0:0178 A2A3A4A1
q¼1s3¼0:0279;s4¼0:0452
p¼10 s1¼0:2236;s2¼0:0589 A2A3A4A1
q¼1s3¼0:0662;s4¼0:1317
p¼1s1¼0:0573;s2¼0:0161 A2A3A4A1
q¼2s3¼0:0265;s4¼0:0410
p¼1s1¼0:2086;s2¼0:0577 A2A3A4A1
q¼10 s3¼0:0640;s4¼0:1281
26 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

From Tables 7and 8, we know di®erent parameters p;qmay in°uence the ranking
results in the MVNWBM and MVNWGBM operators. The MVNWBM operator is
based on the arithmetic operations which emphasized the impact and complemen-
tarity among the overall attributes, and the MVNWGBM operator is based on
geometric operations which highlighted the coordination of all attributes and the role
of the individual attribute. So it is normal that there are the di®erent ranking results
for two aggregation operators.
In general, we can assign the parameter values by p;q¼1, which is not only
simple and straightforward but also taking account of the relationships of the ag-
gregated arguments.
5.4. E®ectiveness of the proposed methods
In order to verify the validity of the proposed methods, we can compare with the
method.
29
Because the method
29
is only suitable for MADM problem, we can use the
MVNWBM operator or MVNWGBM operator to produce the collective decision
matrix, i.e., convert MAGDM problem to MADM problem, and then we can use the
method
29
to rank the alternatives. By the calculation, we get the ranking result
A2A3A4A1. So, there is almost the same ranking result for two methods.
Comparing the proposed methods with method,
29
the proposed methods consider the
relationships of the aggregated arguments while the method
29
considers the bounded
rationality of decision makers. So they have their advantages for these methods.
6. Conclusion
The MVNSs can extend the truth-membership function, the indeterminacy-mem-
bership function and the falsity-membership function of NSs to a set expressed by
several possible values and they can be applied to solve the MADM or MAGDM
problems with incomplete, uncertainty, and inconsistent information. Based on the
IFSs and HFSs, the de¯nition, operations and the comparison method of MVNNs
were presented in this paper. Then, the Bonferroni mean operators were extended for
MVNNS, including MVNWBM and MVNWGBM operators. Furthermore, we
present two new methods for solving the MAGDM problems with multi-valued
neutrosophic information. By an illustrative example, we give the speci¯c decision
steps. Besides, by comparing with the existing method, we give their advantages for
these methods. In the future, the authors will do more researches about the MVNNs
and expand them to more areas of MADM or MAGDM, such as clustering algo-
rithms or classi¯cation algorithms,
48–50
consistency analysis,
51,52
and so on.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (Nos.
71471172 and 71271124), the Special Funds of Taishan Scholars Project, National
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 27
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

Soft Science Project of China (2014GXQ4D192), Shandong Provincial Social Science
Planning Project (No. 15BGLJ06).
References
1. L. A. Zadeh, Fuzzy sets, Inf. Control 8(1965) 338–356.
2. R. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Manage. Sci. 17
(1970) 141–164.
3. R. R. Yager, Multiple objective decision-making using fuzzy sets, Int. J. Man-Mach. Stud.
9(1997) 375–382.
4. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set Syst. 20 (1986) 87–96.
5. K. T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets Syst. 33 (1989) 37–46.
6. K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Set. Syst.
31 (1989) 343–349.
7. X. H. Yu and Z. S. Xu, Prioritized intuitionistic fuzzy aggregation operators, Inf. Fusion
14 (2013) 108–116.
8. Y. T. Chen, A outcome-oriented approach to multicriteria decision analysis with intui-
tionistic fuzzy optimistic/pessimistic operators, Expert Syst. Appl. 37 (2010) 7762–7774.
9. Z. S. Xu and H. Hu, Projection models for intuitionistic fuzzy multiple attribute decision
making, Int. J. Inf. Tech. Decis. 9(2010) 267–280.
10. T. Chaira, Intuitionistic fuzzy set approach for color region extraction, J. Sci. Ind. Res.
69 (2010) 426–432.
11. S. Z. Zeng and W. H. Su, Intuitionistic fuzzy ordered weighted distance operator, Knowl.
Based Syst. 24 (2011) 1224–1232.
12. G. W. Wei, Gray relational analysis method for intuitionistic fuzzy multiple attribute
decision making, Expert Syst. Appl. 38 (2011) 11671–11677.
13. Z. Pei and L. Zheng, A novel approach to multi-attribute decision making based on
intuitionistc fuzzy sets, Expert Syst. Appl. 39 (2012) 2560–2566.
14. J. Q. Wang, R. R. Nie, H. Y. Zhang and X. H. Chen, Intuitionistic fuzzy multi-criteria
decision-making method based on evidential reasoning, Appl. Soft Comput. 13 (2013)
1823–1831.
15. J. Q. Wang and H. Y. Zhang, Multi-criteria decision-making approach based on Ata-
nassov's intuitionistic fuzzy sets with incomplete certain information on weights, IEEE
Trans. Fuzzy Syst. 21(3) (2013) 510–515.
16. J. Q. Wang, R. R. Nie, H. Y. Zhang and X. H. Chen, New operators on triangular
intuitionistic fuzzy numbers and their applications in system fault analysis, Inform. Sci.
251 (2013) 79–95.
17. J. Q. Wang, P. Zhou, K. J. Li, H. Y. Zhang and X. H. Chen, Multi-criteria decision-
making method based on normal intuitionistic fuzzy-induced generalized aggregation
operator, TOP 22 (2014) 1103–1122.
18. J. Q. Wang, Z. Q. Han and H. Y. Zhang, Multi-criteria group decision-making method
based on intuitionistic interval fuzzy information, Group Decis. Negot. 23 (2014) 715–
733.
19. H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued neutrosophic
sets, Multispace Multistruct. 4(2010) 410–413.
20. F. Smarandache, Neutrosophy: Neutrosophic Probability, Set, and Logic (American Re-
search Press, Rehoboth, 1998).
21. P. Majumdar and S. K. Samant, On similarity and entropy of neutrosophic sets, J. Intell.
Fuzzy Syst. 26(3) (2014) 1245–1252.
28 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

22. J. Ye, Multicriteria decision-making method using the correlation coe±cient under single-
value neutrosophic environment, Int. J. Gen. Syst. 42(4) (2013) 386–394.
23. J. Ye, A multicriteria decision-making method using aggregation operators for simpli¯ed
neutrosophic sets, J. Intell. Fuzzy Syst. 26(5) (2014) 2459–2466.
24. H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Interval Neutrosophic Sets
and Logic: Theory and Applications in Computing (Hexis, Phoenix, AZ, 2005).
25. F. G. Lupiañez, Interval neutrosophic sets and topology, Kybernetes 38(3–4) (2009) 621–
624.
26. S. Broumi and F. Smarandache, Correlation coe±cient of interval neutrosophic set, Appl.
Mech. Mater. 436 (2013) 511–517.
27. J. Ye, Single valued neutrosophic cross-entropy for multicriteria decision making pro-
blems, Appl. Math. Model. 38(3) (2014) 1170–1175.
28. J. Ye, Similarity measures between interval neutrosophic sets and their applications in
multicriteria decision-making, J. Intell. Fuzzy Syst. 26(1) (2014) 165–172.
29. J. Q. Wang and X. E. Li, TODIM method with multi-valued neutrosophic sets, Control
Decis. 30(6) (2015) 1139–1142.
30. H. Y. Zhang, J. Q. Wang and X. H. Chen, Interval neutrosophic sets and their application
in multicriteria decision making problems, Sci. World J. 2014 (2014) 1–15.
31. P. D. Liu and Y. M. Wang, Multiple attribute decision-making method based on single-
valued neutrosophic normalized weighted Bonferroni mean, Neural Comput. Appl. 25(7–
8) (2014) 2001–2010.
32. P. D. Liu, Y. C. Chu, Y. W. Li and Y. B. Chen, Some generalized neutrosophic number
Hamacher aggregation operators and their application to group decision making, Int. J.
Fuzzy Syst. 16(2) (2014) 242–255.
33. Z. P. Tian, J. Wang, H. Y. Zhang, X. H. Chen and J. Q. Wang, Simpli¯ed neutrosophic
linguistic normalized weighted Bonferroni mean to multi-criteria decision-making pro-
blems, Filomat (2016), doi: http://dx.doi.org/10.2298/FIL1508576F.
34. Z. P. Tian, J. Wang, J. Q. Wang and X. H. Chen, Multi-criteria decision-making ap-
proach based on gray linguistic weighted Bonferroni mean operator, Int. Trans. Oper.
Res., Doi:10.1111/itor.12220.
35. Z. P. Tian, H. Y. Zhang, J. Wang, J. Q. Wang and X. H. Chen, Multi-criteria decision-
making method based on a cross-entropy with interval neutrosophic sets, Int. J. Syst. Sci.,
doi: 10.1080/00207721.2015.1102359.
36. H. Y. Zhang, P. Ji, J. Q. Wang and X. H. Chen, Improved weighted correlation coe±cient
based on integrated weight for interval neutrosophic sets and its application in multi-
criteria decision making problems, Int. J. Comput. Intell. Syst. 8(6) (2015) 1027–1043.
37. H. Y. Zhang, J. Q. Wang and X. H. Chen, An outranking approach for multi-criteria
decision-making problems with interval-valued neutrosophic sets, Neural Comput. Appl.
27(3) (2016) 615–627.
38. J. M. Merig
o, M. Casanovas and D. Palacios-Marqu
es, Linguistic group decision making
with induced aggregation operators and probabilistic information, Appl. Soft Comput. 24
(2014) 669–678.
39. X. P. Jiang and G. W. Wei, Some Bonferroni mean operators with 2-tuple linguistic
information and their application to multiple attribute decision making, J. Intell. Fuzzy
Syst. 27 (2014) 2153–2162.
40. G. W. Wei, X. F. Zhao, R. Lin and H. J. Wang, Uncertain linguistic Bonferroni mean
operators and their application to multiple attribute decision making, Appl. Math. Model.
37 (2013) 5277–5285.
41. P. D. Liu, Multi-attribute decision-making method research based on interval vague set
and TOPSIS method, Technol. Econ. Develop. Econ. 15(3) (2009) 453–463.
Multi-Valued Neutrosophic Number Bonferroni Mean Operators 29
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.

42. P. D. Liu and F. Teng, An extended TODIM method for multiple attribute group de-
cision-making based on intuitionistic uncertain linguistic variables, J. Intell. Fuzzy Syst.
29 (2015) 701–711.
43. C. Bonferroni, Sulle medie multiple di potenze, Bollettino dell Unione Matematica Itali-
ana 5(3–4) (1950) 267–270.
44. W. Zhou and J. M. He, Intuitionistic fuzzy normalized weighted Bonferroni mean and its
application in multicriteria decision making, J. Appl. Math. 2012 (2012) 1–22.
45. M. M. Xia, Z. S. Xu and B. Zhu, Geometric Bonferroni means with their application in
multi-criteria decision making, Knowledge-Based Syst. 40 (2013) 88–100.
46. M. Sun and J. Liu, Normalized geometric Bonferroni operators of hesitant fuzzy sets and
their application in multiple attribute decision making, J. Inform. Comput. Sci. 10(9)
(2013) 2815–2822.
47. J. J. Peng, J. Q. Wang, X. H. Wu, J. Wang and X. H. Chen, Multi-valued neutrosophic
sets and power aggregation operators with their applications in multi-criteria group de-
cision-making problems, Int. J. Comput. Intell. Syst. 8(2015) 345–363.
48. G. Kou, Y. Peng and G. Wang, Evaluation of clustering algorithms for ¯nancial risk
analysis using MCDM methods, Inf. Sci. 275 (2014) 1–12.
49. G. Kou, Y. Lu, Y. Peng and Y. Shi, Evaluation of classi¯cation algorithms using MCDM
and rank correlation, Int. J. Inf. Technol. Decis. Making 11(1) (2012) 197–225.
50. Y. Peng, G. Kou, G. Wang and Y. Shi, FAMCDM: A fusion approach of MCDM methods
to rank multiclass classi¯cation algorithms, Omega 39(6) (2011) 677–689.
51. G. Kou and C. Lin, A cosine maximization method for the priority vector derivation in
AHP, Eur. J. Oper. Res. 235(1) (2014) 225–232.
52. G. Kou, D. Ergu and J. Shang, Enhancing data consistency in decision matrix: Adapting
Hadamard model to mitigate judgment contradiction, Eur. J. Oper. Res. 236(1) (2014)
261–271.
30 P. Liu et al.
Int. J. Info. Tech. Dec. Mak. Downloaded from www.worldscientific.com
by NANYANG TECHNOLOGICAL UNIVERSITY on 07/17/16. For personal use only.