A New Interval-valued 2-Tuple Linguistic Bonferroni MeanOperator and Its Application to Multiattribute Group DecisionMaking
Xi Liu1,2 • Zhifu Tao4 • Huayou Chen1 • Ligang Zhou1,3
Received: 30 April 2015 / Revised: 15 November 2015 / Accepted: 12 December 2015 / Published online: 21 January 2016
� Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2016
Abstract The purpose of this paper is to introduce some
new Bonferroni mean operators under interval-valued
2-tuple linguistic environment. First, a class of new oper-
ational laws of interval-valued 2-tuple linguistic are pro-
posed. Then, we put forward some new interval-valued
2-tuple linguistic Bonferroni mean (IV2TLBM) operators.
Moreover, properties and special cases of new aggregation
operators are investigated. The main characteristic of the
IV2TLBM is that the interrelationship among the input
arguments and the closed operations are taken into account.
Finally, an approach to multiple attributes group decision
making is presented, and a numerical example is given to
illustrate the proposed method.
Keywords Multiple attributes group decision making �Bonferroni mean operators � Operational laws � Interval-valued 2-tuple linguistic
1 Introduction
Multiple attribute group decision making (MAGDM) is an
important branch of modern decision science, which has
been widely applied in many fields. Its essence is to rank a
set of alternatives and find the best alternative through a
certain way by the existing decision making information.
Therefore, decision makers should give their evaluated
values of each alternative on each attribute.
As the development of modern society and economy,
business scale is continually expanded, the decision system
is becoming more and more complex, and the human
thinking is characterized by ambiguity. Thus, it is more
suitable to provide preference by using linguistic variables
rather than numerical ones. Zadeh [1] introduced the con-
cept of linguistic variable characterized by words or sen-
tences in a natural or artificial language. For instance, when
evaluating a teacher’s service ability, linguistic terms are
usually used, such as, ‘‘good’’, ‘‘fair’’ and ‘‘poor’’. In recent
years, some methods have been proposed for handling real
problems with linguistic information [2–9].
To avoid information loss, Herrera and Martinez [10]
proposed 2-tuple linguistic representation models that
represent the linguistic information by means of 2-tuples,
which is composed of a linguistic term and a real number.
From then on, the 2-tuple linguistic models have been
widely used in MAGDM. Herrera and Martinez [10] pre-
sented the 2-tuple arithmetic mean and the 2-tuple ordered
weighted average (OWA) operator. Jiang and Fan [11]
developed 2-tuple ordered weighted geometric average
(OWGA) operator. Merigo et al. [12] introduced the
induced 2-tuple linguistic generalized aggregation opera-
tors. Wang et al. [13] proposed some 2-tuple linguistic
aggregation operators of multi-hesitant fuzzy linguistic
term element, which extend previous approaches by using
& Huayou Chen
Xi Liu
Zhifu Tao
Ligang Zhou
1 School of Mathematical Science, Anhui University, Hefei,
Anhui, China
2 Department of Foundation, Anhui Occupational College of
City Management, Hefei, Anhui, China
3 Signal and Image Processing Institute, Department of
Electrical Engineering, University of Southern California,
Los Angeles, USA
4 School of Economics, Anhui University, Hefei, Anhui, China
123
Int. J. Fuzzy Syst. (2017) 19(1):86–108
DOI 10.1007/s40815-015-0130-4
generalized means, order-inducing variables by reordering
of the arguments and linguistic information represented
with the 2-tuple linguistic approach. Lin et al. [14] pre-
sented the concept of an interval 2-tuple linguistic variable.
Liu et al. [15] proposed a modified multimoora method
based on interval 2-tuple linguistic variables for evaluating
and selecting HCW treatment technologies. To bring more
convenience to the comparison between two 2-tuple dif-
ferent granularity linguistic term sets, Chen and Tai [16]
gave the definition of generalized 2-tuple linguistic vari-
able. In the above 2-tuple models, the decision linguistic
information is derived from a predefined linguistic term set.
And it is not easy to get only one appropriate linguistic
term set to meet the requirements of all the decision
makers. To overcome such problem, Zhang [17] introduced
a new definition of interval-valued 2-tuple linguistic vari-
able and put forward the interval-valued 2-tuple linguistic
representation model. This model is suitable to deal with
MAGDM problems with multi-granular linguistic contexts.
In recent decades, some interval-valued 2-tuple linguistic
aggregation operators are proposed, such as the interval-
valued 2-tuple weighted averaging operator [17], the
interval-valued 2-tuple OWA operator [17], the interval-
valued 2-tuple ordered weighted harmonic operator and the
interval-valued 2-tuple ordered weighted quadratic opera-
tor [18], the interval-valued 2-tuple prioritized weighted
operator [19]. In many MAGDM problems, the decision
makers are usually unsure of their preferences during the
alternative selection process because of time pressure, lack
of experience and data. By using the interval-valued 2-tu-
ple linguistic representation model, decision makers can
express his information better, and unify the interval-val-
ued 2-tuple linguistic information easily under the multiple
granular linguistic contexts. In this paper, we will introduce
some novel aggregation operators for interval-valued
2-tuple together with their properties.
In the process of aggregating interval-valued 2-tuple
linguistic information, the operational laws defined by Xu
[20] are not closed. Here, we give an example to verify this
problem. Let S ¼ fsiji ¼ 0; 1; . . .; 6g be a linguistic term
set, we have [(s2, 0), (s3, 0)] � [(s4, 0), (s5, 0)] = [(s6, 0),
(s8, 0)], and [(s2, 0), (s3, 0)] � [(s3, 0), (s5, 0)] = [(s6, 0),
(s15, 0)]
Clearly, the results of operation exceed the range of S.
To solve this problem, Lan et al. [21] employed the
extended triangular conorm to deal with linguistic infor-
mation and proposed some aggregation operators. Xia et al.
[22] introduced some new operations on intuitionistic
fuzzy set based on Archimedean t-norm and s-norm and
gave some intuitionistic aggregation operators. Tao et al.
[23] proposed some new operational laws of 2-tuples based
on Archimedean t-norm and s-norm and gave some
aggregation operators by using the proposed operations.
To cope with the situation where the input arguments
have some connections, some aggregation operators are
proposed. Yager [24] proposed the power average operator,
which allows input arguments to support each other in the
aggregation process. Xu et al. [25] proposed some lin-
guistic average operators. Choquet integral [26] is an
effective useful method to model the interdependence or
correlation. Yager [27] introduced the Choquet integral
operator on fuzzy set, and Yang et al. [5] extended the
Choquet integral to 2-tuple linguistic environment. Shapley
[28] proposed Shapley function. In [29], Meng et al.
pointed it out that the Choquet integral only reflects the
interaction between two adjacent coalitions. To solve this
problem, Meng et al. [30, 31] proposed some k-ShapleyChoquet operators and Shapley hybrid operators. Heronian
mean operator was introduced by Beliakov et al. [32],
which can consider the arguments value interrelationships.
Liu et al. [33] proposed some intuitionistic uncertain lin-
guistic Heronian mean operators.
The Bonferroni mean (BM) [34] can provide a tool for
information aggregation lying between max and min
operators. The prominent characteristic of Bonferroni mean
is its capability to capture the interrelationship between
input arguments. In order to enhance its modeling capa-
bility, Yager [35] emphasized the importance of having an
aggregation function to express interrelationship between
the criteria, and proposed some generalizations of the
Bonferroni mean. Yager et al. [36] and Beliakov et al. [37]
introduced another generalized form of Bonferroni mean
operators.
In some group decision making problems, the input
arguments are fuzzy or uncertain. Thus, some new gener-
alizations of the Bonferroni mean [38–43] were developed.
Compared with the Bonferroni mean emphasizing on the
aggregated arguments, Choquet integral or Shapley
aggregation operator emphasizing on changing the weight
vector of aggregation operators. Choquet integral and
Shapley aggregation operator only reflect the correlation of
the aggregated arguments subjectively by decision makers,
while the power averaging operators determined by
weighted vector depend on the values of aggregated
arguments objectively. For a set of attributes
Ci ði ¼ 1; 2; . . .; nÞ, the Heronian mean operator can con-
sider the relationship between each pair of attributes Ci and
Cj (i C j). However, it ignores the relationship between the
Ci and Cj (i\ j). In fact, owing to the existence of
uncertainty of the parameters p and q, the correlation of Ci
over Cj (i = j) is not equal to that of Cj over Ci (i = j).
Furthermore, it is not necessary to consider the correlation
of Ci over itself. Nevertheless, the Heronian mean operator
considers it. Therefore, the Bonferroni mean operator has
some different characteristics that other operators don’t
share, so it can be used to solve these problems effectively.
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123
Motivated by the aforementioned analysis, we investi-
gate some new operational laws of interval-valued lin-
guistic 2-tuples based on the Archimedean t-norm and
s-norm. The most advantage of such operational laws is that
the operations are closed. Then, we extend the BM and
obtain the interval-valued 2-tuple linguistic Bonferroni
mean (IV2TLBM) operator and the interval-valued 2-tuple
linguistic geometric Bonferroni mean (IV2TLGBM) oper-
ator. The proposed operators are taken into account the
interrelationship among the input arguments and the closed
operational laws of 2-tuple linguistic variables. For the
situations where the input arguments have different
importance, the interval-valued 2-tuple linguistic weighted
Bonferroni mean (IV2TLWBM) operator and the interval-
valued 2-tuple linguistic weighted geometric Bonferroni
mean (IV2TLWGBM) operator are defined, based on which
we develop a method for multiple attribute decision making
under interval-valued 2-tuple linguistic environment.
The rest of this paper is structured as follows. In Sect. 2,
we review some concepts and notations. In Sect. 3, some
new operations for interval-valued linguistic 2-tuple based
on Archimedean t-norm and s-norm are proposed; some
properties of these operation and special cases are dis-
cussed in details. Based on the proposed operations, Sect. 4
introduces some interval-valued 2-tuple linguistic Bonfer-
roni mean operators and their properties. Section 5 pro-
vides an approach to multiple attribute group decision
making with interval-valued 2-tuple linguistic information
based on these operators and gives a real-life example to
illustrate the efficiency of the proposed method. Finally,
some remarks are provided in Sect. 6.
2 Preliminaries
In this section, we introduce some basic concept related to
the 2-tuple fuzzy linguistic representation model, Archi-
medean t-norm and s-norm and some existing Bonferroni
mean operators.
2.1 The 2-tuple Fuzzy Linguistic Representation
Model
The linguistic method was first introduced by Zadeh in [1],
which was proposed as an approximate technique that
represents qualitative information by means of linguistic
labels.
Let S ¼ fsiji ¼ 0; 1; . . .; gg be a linguistic term set with
odd cardinality. The term si represents a possible value for
a linguistic variable. For example, S can be defined as
follows:
S ¼ fs0 ¼ neitherðNÞ; s1 ¼ very lowðVLÞ; s2 ¼ lowðLÞ;s3 ¼ mediumðMÞ; s4 ¼ highðHÞ; s5 ¼ very highðVHÞ;s6 ¼ perfectðPÞg
where the mid-linguistic term s3 represents an assessment
of ‘‘approximately 0.5’’ with the rest of the terms being
placed symmetrically around it, and the term set should
satisfy the following characteristics:
(1) The set is ordered: si [ sj , i[ j;
(2) The negation operator: Neg sið Þ ¼ sg�i;
(3) Min operator: minðsi; sjÞ ¼ si , si � sj;
(4) Max operator: maxðsi; sjÞ ¼ si , si � sj.
Based on the concept of symbolic translation, Herrera
and Martinez [10] originally proposed a 2-tuple linguistic
representation model for dealing with linguistic informa-
tion. Continuity is the main advantage of this representa-
tion. A 2-tuple (si, ai) is a 2-tuple linguistic variable, wheresi is a linguistic label of predefined linguistic term set S,
and ai is a numerical value representing the value of
symbolic translation.
Definition 2.1 [10, 44] Let S ¼ fsiji ¼ 0; 1; . . .; gg be a
finite linguistic term set, and b [ [0, g] be a value repre-
senting the result of a symbolic aggregation operation.
Then the function D, which denotes to obtain the 2-tuple
linguistic information equivalents to b, is defined as
follows:
D : ½0; g� ! S ½�0:5; 0:5Þ; ð1Þ
D bð Þ ¼ si; aið Þ with si; i ¼ roundðbÞ;ai ¼ b� i; ai 2 ½�0:5; 0:5Þ:
�ð2Þ
where round(b) is the usual round operation, si has the
closest index label of b and ai is the value of the symbolic
translation.
Definition 2.2 [10, 44] Assume that S ¼ fsiji ¼0; 1; . . .; gg is a collection of linguistic term and (si, ai) is alinguistic 2-tuple. There is always a function D-1, such that
it returns its equivalent numerical value b [ [0, g] from a
linguistic 2-tuple, where
D�1 : S ½�0:5; 0:5Þ ! ½0; g�; ð3Þ
D�1ðsi; aiÞ ¼ iþ ai ¼ b: ð4Þ
The range of b is from 0 to g, which is relevant to the
granularity of the linguistic term set. In the past few dec-
ades, a series of 2-tuple linguistic aggregation operators
have been proposed for aggregating 2-tuple linguistic.
However, these 2-tuple linguistic aggregation operators all
88 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
focus on usual 2-tuples. If the 2-tuples are from the dif-
ferent linguistic term sets with different granularities, they
cannot be aggregated directly. To avoid this problem, Chen
and Tai [16] put forward a generalized 2-tuple linguistic
variable and translation function.
Definition 2.3 [16] Let S ¼ fsiji ¼ 0; 1; . . .; gg be an
ordered linguistic term set, and crisp value b 2 [0, 1] can
be transformed into one 2-tuple linguistic variable by the
following function:
D : ½0; 1� ! S � 1
2g;1
2g
� �; ð5Þ
D bð Þ ¼ si; aið Þ with
si; i ¼ roundðb gÞ;ai ¼ b� i=g; ai 2 � 1
2g;1
2g
� �:
8<:
ð6Þ
Conversely, there the 2-tuple can be converted into a
crisp b [ [0, 1] as follows:
D�1 : S � 1
2g;1
2g
� �! ½0; 1�; ð7Þ
D�1ðsi; aiÞ ¼i
gþ ai ¼ b: ð8Þ
where the value of b ranges between 0 and 1. That is, the
2-tuple linguistic variable is standardized, which makes it
very convenient to compare 2-tuples from different multi-
ple granularity linguistic term sets. In this paper, unless
otherwise mentioned, the 2-tuple linguistic variable is the
generalized 2-tuple variable defined in Definition 2.3.
To avoid linguistic information loss, Lin et al. [14]
presented the definition for the interval 2-tuple linguistic
variable based on Definition 2.1. In view of the advantage
of Definition 2.3, Zhang [17] introduced a new concept of
the interval-valued 2-tuple linguistic variable and proposed
some aggregation operators with interval-valued 2-tuple
linguistic information.
Definition 2.4 [17] Suppose that S ¼ fsiji ¼ 0; 1; . . .; ggis an ordered linguistic term set. An interval-valued 2-tuple
is composed of two linguistic terms and two numbers,
denoted by [(si, ai), (sj, aj)], where i B j, and ai B aj. Ifi = j, si(sj) means the linguistic label of the linguistic term
set S, then ai(aj) is the value of symbolic translation. The
interval-valued 2-tuple that expresses the equivalent
information to an interval-value [b1, b2] (b1, b2 [[0, 1], b1 B b2) as follows:
D b1; b2½ �ð Þ ¼
si; aið Þ; sj; aj� �� �
with
si; i ¼ roundðb1 gÞ;sj; j ¼ roundðb2 gÞ;
ai ¼ b1 �i
g; ai 2 � 1
2g;1
2g
� �;
aj ¼ b2 �i
g; aj 2 � 1
2g;1
2g
� �:
8>>>>>>><>>>>>>>:
ð9Þ
There always exists the inverse function D-1, such that
for each interval-valued 2-tuple, it returns its corresponding
interval value [b1, b2](b1, b2 [ [0, 1], b1 B b2) as follows:
D�1ð½ðsi; aiÞ; ðsj; ajÞ�Þ ¼i
gþ ai;
j
gþ aj
� ¼ b1; b2½ �:
ð10Þ
Zhang [17] proposed the concept of the score and
accuracy function to compare two interval-valued 2-tuples.
Definition 2.5 [17] For an interval-valued 2-tuple
A = [(si, ai), (sj, aj)], its score function is
SðAÞ ¼ iþ j
2gþ ai þ aj
2: ð11Þ
The accuracy function is
HðAÞ ¼ j� i
gþ aj � ai; ð12Þ
where S ¼ fsiji ¼ 0; 1; . . .; gg is an ordered linguistic term
set with g ? 1 linguistic labels. Obviously, 0 B S(A) B 1
and 0 B H(A) B 1.
Definition 2.6 [17] Let A = [(si, ai), (sj, aj)] and
B = [(sk, ak), (sl, al)] be two interval-valued linguistic
2-tuples. It follows that:
(1) If S(A) B S(B), then A B B;
(2) If S(A) = S(B), then
If H(A) B H(B), then A C B;
If H(A) = H(B), then A = B.
2.2 Archimedean t-norm and Archimedean s-norm
Definition 2.7 [46, 47] A triangular norm (briefly called
t- norm) is a mapping
T : ½0; 1� ½0; 1� ! ½0; 1�
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 89
123
such that:
(1) T(1, x) = x, for all x;
(2) T(x, y) = T(y, x), for all x and y;
(3) T(x, y) B T(x, z), if y B z;
(4) T(x, T(y, z)) = T(T(x, y), z), for all x, y, and z.
Definition 2.8 [46, 47] A triangular conorm (briefly
t-conorm or s-norm) is a mapping S: [0, 1] 9
[0, 1] ? [0, 1]
such that:
(1) S(0, x) = x, for all x;
(2) S(x, y) = S(y, x), for all x and y;
(3) S(x, y) B S(x, z), if y B z;
(4) S(x, S(y, z)) = S(S(x, y), z), for all x, y, and z.
A t-norm T(x, y) is called Archimedean t-norm if it is
continuous and T(x, x)\ x for all x [ (0, 1). A s-norm
S(x, y) is called Archimedean s-norm if it is continuous and
S(x, x)[ x for all x [ (0, 1). A strict Archimedean t-norm
is obtained from a continuous additive generator u as
T(x, y) = u-1(u(x) ? u(y)); an additive generator is a
strictly decreasing function: u:[0, 1] ? [0, ?] such that
u(1) = 0. Similarly, applied to its dual s-norm,
S(x, y) = /-1(/(x) ? /(y)) with /(x) = u(1 - x).
Based on the Archimedean t-norm and s-norm [48], Tao
et al. [23] defined the algebra operations for fuzzy numbers
as follows.
Definition 2.9 [48] Let x, y 2 [0, 1], and k[ 0 be a
scalar, then we have
(1) x � y ¼ Sðx; yÞ ¼ /�1ð/ðxÞ þ /ðyÞÞ;(2) x � y ¼ Tðx; yÞ ¼ u�1ðuðxÞ þ uðyÞÞ;(3) k � x ¼ /�1ðk/ðxÞÞ;(4) xk ¼ u�1 ku xð Þð Þ:
If we assign different generator functions, then some
special cases can be obtained:
(1) Let u(x) = -logx and /(x) = u(1 - x) =
-log(1 - x). Then we can get Algebra t-norm and
s-norm as follows:
TAðx; yÞ ¼ xy; SA x; yð Þ ¼ xþ y� xy;
(2) Let uðxÞ ¼ log 2�xx
� �and /ðxÞ ¼ uð1� xÞ ¼
log2�ð1�xÞ
1�x
�. Then Einstein t-norm and s-norm are
obtained following:
TEðx; yÞ ¼ xy
1þ ð1� xÞð1� yÞ ; SEðx; yÞ ¼ xy
1þ xy;
(3) Let uðxÞ ¼ logcþð1�cÞx
x
�; c[ 0 and /ðxÞ ¼
uð1� xÞ ¼ log1�ð1�cÞx
1�x
�. Then we have Hamacher
t-norm and s-norm:
THðx; yÞ ¼ xy
cþ ð1� cÞðxþ y� xyÞ ; c[ 0;
SHðx; yÞ ¼ xþ y� xy� ð1� cÞxy1� ð1� cÞxy ; c[ 0;
(4) Let uðxÞ ¼ log c�1cx�1
�; c[ 0 and /ðxÞ ¼ uð1�
xÞ ¼ log c�1c1�x�1
�: Then we can get Frank t-norm and
s-norm:
TFðx; yÞ ¼ logc 1þ ðcx � 1Þðcy � 1Þc� 1
� �; c[ 1;
SFðx; yÞ ¼ 1� logc 1þ ðc1�x � 1Þðc1�y � 1Þc� 1
� �;
c[ 1:
2.3 Bonferroni Mean and Geometric Bonferroni
Mean
The Bonferroni mean was originally introduced by Bon-
ferroni in [34], which can provide for the aggregation lying
between the max, min operators, the logical ‘‘or’’ and
‘‘and’’ operators, which is defined as follows:
Definition 2.10 [34] Let (a1, a2,…,an) be a collection of
values such that ai [ [0, 1], and p, q C 0. Then the
aggregation function
BMp;qða1; a2; . . .; anÞ ¼1
nðn� 1ÞXni;j¼1i 6¼j
api a
qj
0BB@
1CCA
1pþq
ð13Þ
is called the Bonferroni mean operator.
The geometric Bonferroni mean [49], considering both the
geometric mean and Bonferroni mean, is defined as follows:
Definition 2.11 [49] Let (a1, a2,…,an) be a set of non-
negative real numbers, and p, q C 0. Then we call
GBMp;qða1; a2; . . .; anÞ ¼1
pþ q
Yni;j¼1i6¼j
pai þ qaj� �
0BB@
1CCA
1nðn�1Þ
ð14Þ
the geometric Bonferroni mean operator.
90 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
3 Some New Operations for Interval-valuedLinguistic 2-tuple Based on Archimedeant-norm and s-norm
Based on the Archimedean triangular norms, Xia et al. [22]
proposed some new operations for intuitionistic fuzzy number;
Tao et al. [23] defined the new operational laws of 2-tuple lin-
guistic. Obviously, these operational laws are closed. Enlight-
ened by these ideas, we can get the following closed operational
laws of interval-valued 2-tuple linguistic information.
Definition 3.1 LetA = [(si, ai), (sj, aj)] andB = [(sk, ak),(sl, al)] be two interval-valued 2-tuple linguistic variables.
k[ 0 is a scalar, then we have
(1) A� B ¼ D /�1 / D�1 si; aið Þ� �
þ / D�1 sk; akð Þ� �� � ��
;
D /�1 / D�1 sj; aj� �� �
þ / D�1 sl; alð Þ� �� � ��
;
(2) A� B ¼ D u�1 u D�1 si; aið Þ� �
þ u D�1 sk; akð Þ� �� � �
;�D u�1 u D�1 sj; aj
� �� �þ u D�1 sl; alð Þ
� �� � ��;
(3)k� A ¼ D /�1 k/ D�1 si; aið Þ
� �� � ��;
D /�1 k/ D�1 sj; aj� �� �� � ��
;
(4)Ak ¼ D u�1 ku D�1 si; aið Þ
� �� � ��;
D u�1 ku D�1 sj; aj� �� �� � ��
:
Some special cases can be obtained.
Case 1 If u(x) = - logx, then we have
(1) A�A B¼DðD�1ðsi; aiÞ þ D�1ðsk; akÞ � D�1ðsi; aiÞ � D�1ðsk; akÞÞ;
DðD�1ðsj; ajÞ þ D�1ðsl; alÞ � D�1ðsj; ajÞ � D�1ðsl; alÞÞ
" #;
(2) A�A B ¼ D D�1 si; aið Þ���
D�1 sk; akð ÞÞ; D D�1 sj; aj� �
� D�1 sl; alð Þ� �
�;(3) k�A A ¼ D 1� 1� D�1 si; aið Þ
� �k � �;
h
D 1� 1� D�1 sj; aj� �� �k � �i
;
(4) Ak ¼ D D�1 si; aið Þ� �k �
;D D�1 sj; aj� �� �k �h i
:
The notations �A, �A, �A represent general addition,
multiplicative, and scalar multiplicative operations of
Algebra t-norm and s-norm, respectively.
Case 2 If uðxÞ ¼ log 2�xx
� �, then we have the following
formulas based on Einstein t-norm and s-norm:
(5) A�E B ¼ D D�1ðsi;aiÞþD�1ðsk ;akÞ1þD�1ðsi;aiÞ�D�1ðsk ;akÞ
�;
h
D D�1ðsj;ajÞþD�1ðsl;alÞ1þD�1ðsj;ajÞ�D�1ðsl;alÞ
�i;
(6) A�E B ¼ D D�1ðsi;aiÞ�D�1ðsk ;akÞ1þð1�D�1ðsi;aiÞÞð1�D�1ðsk ;akÞÞ
�;
h
D D�1ðsj;ajÞ�D�1ðsl;alÞ1þð1�D�1ðsj;ajÞÞð1�D�1ðsl;alÞÞ
�i;
(7) k�E A ¼ D ð1þD�1ðsi;aiÞÞk�ð1�D�1ðsi;aiÞÞk
ð1þD�1ðsi;aiÞÞkþð1�D�1ðsi;aiÞÞk
�;
h
D ð1þD�1ðsj;ajÞÞk�ð1�D�1ðsj;ajÞÞk
ð1þD�1ðsj;ajÞÞkþð1�D�1ðsj;ajÞÞk
�i;
(8) Ak ¼ D 2ðD�1ðsi;aiÞÞk
ð2�D�1ðsi;aiÞÞkþðD�1ðsi;aiÞÞk
�;
h
D 2ðD�1ðsj;ajÞÞk
ð2�D�1ðsj;ajÞÞkþðD�1ðsj;ajÞÞk
� i:
The notations �E, � E, � E represent general addition,
multiplicative, and scalar multiplicative operations of
Einstein t-norm and s-norm, respectively.
Case 3 If uðxÞ ¼ logcþð1�cÞx
x
�; c[ 0, then we can get
the formulas based on the Hamacher t-norm and s-norm:
(9) A�H B
¼D
D�1ðsi; aiÞ þ D�1ðsk; akÞ � ð1� cÞ � D�1ðsi; aiÞ � D�1ðsk; akÞ1� ð1� cÞ � D�1ðsi; aiÞ � D�1ðsk; akÞ
� �;
DD�1ðsj; ajÞ þ D�1ðsl; alÞ � ð1� cÞ � D�1ðsj; ajÞ � D�1ðsl; alÞ
1� ð1� cÞ � D�1ðsj; ajÞ � D�1ðsl; alÞ
!266664
377775;
(10) A�H B
¼
DD�1ðsi; aiÞ � D�1ðsk; akÞ
cþ ð1� cÞðD�1ðsi; aiÞ þ D�1ðsk; akÞ � D�1ðsi; aiÞ � D�1ðsk; akÞÞ
� �;
DD�1ðsj; ajÞ � D�1ðsl; alÞ
cþ ð1� cÞðD�1ðsj; ajÞ þ D�1ðsl; alÞ � D�1ðsj; ajÞ � D�1ðsl; alÞÞ
!
2666664
3777775;
(11) k�H A
¼
Dð1þ ðc� 1Þ � D�1ðsi; aiÞÞk � ð1� D�1ðsi; aiÞÞk
ð1þ ðc� 1Þ � D�1ðsi; aiÞÞk þ ðc� 1Þð1� D�1ðsi; aiÞÞk
!;
Dð1þ ðc� 1Þ � D�1ðsj; ajÞÞk � ð1� D�1ðsj; ajÞÞk
ð1þ ðc� 1Þ � D�1ðsj; ajÞÞk þ ðc� 1Þð1� D�1ðsj; ajÞÞk
!
2666664
3777775;
(12) Ak ¼
Dc � ðD�1ðsi; aiÞÞk
ðcþ ð1� cÞ � D�1ðsi; aiÞÞk þ ðc� 1ÞðD�1ðsi; aiÞÞk
!;
Dc � ðD�1ðsj; ajÞÞk
ðcþ ð1� cÞ � D�1ðsj; ajÞÞk þ ðc� 1ÞðD�1ðsj; ajÞÞk
!
2666664
3777775:
The notations �H, �H, �H represent general addition,
multiplicative, and scalar multiplicative operations of
Hamacher t-norm and s-norm, respectively.
Obviously, if c = 1, then (9)–(12) reduce to (1)–(4), and
if c = 2, then (9)–(12) reduce to (5)–(8).
Case 4 If uðxÞ ¼ log c�1cx�1
�; c[ 1, then we have the
following formulas which are based on Frank t-norm and
s-norm:
(13) A�F B ¼ D logcc � ðc� 1Þ
ðcD�1ðsi;aiÞ�1 � 1ÞðcD�1ðsk ;akÞ�1 � 1Þ þ ðc� 1Þ
! !;
"
D logcc�ðc�1Þ
ðcD�1ðsj ;ajÞ�1�1ÞðcD�1ðsl ;alÞ�1�1Þþðc�1Þ
� �� �i;
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 91
123
(14) A�F B ¼
D logc 1þ ðcD�1ðsi;aiÞ � 1ÞðcD�1ðsk ;akÞ � 1Þðc� 1Þ
! !;
D logc 1þ ðcD�1ðsj;ajÞ � 1ÞðcD�1ðsl;alÞ � 1Þðc� 1Þ
! !
2666664
3777775;
(15) k�F A ¼
D 1� logc 1þ ðc1�D�1ðsi;aiÞ � 1Þk
ðc� 1Þk�1
! !;
D 1� logc 1þ ðc1�D�1ðsj;ajÞ � 1Þk
ðc� 1Þk�1
! !
2666664
3777775;
(16) Ak ¼ D logc 1þ ðcD�1ðsi ;aiÞ�1Þk
ðc�1Þk�1
� �;
h
D logc 1þ ðcD�1ðsj ;ajÞ�1Þk
ðc�1Þk�1
� �� ��:
The notations �F, �F, �F represent general addition,
multiplicative, and scalar multiplicative operations of
Frank t-norm and s-norm. Moreover, the following opera-
tional laws can be obtained.
Theorem 3.1 Suppose that S ¼ fsiji ¼ 0; 1; . . .; gg is a
linguistic term set, then X¼ ðsm; amÞ; ðsn; anÞ½ � sm;j sn 2fS; am; an 2 � 1
2g; 12g
h �g is a set of all interval-valued
linguistic 2-tuple based on S. [(si, ai), (sj, aj)], [(sk, ak),(sl, al)] [ X, k C 0 is a scalar, we can get some properties
as follows:
(1) [(si, ai), (sj, aj)] � [(sk, ak), (sl, al)] 2 X;(2) [(si, ai), (sj, aj)] � [(sk, ak), (sl, al)] 2 X;(3) k � [(si, ai), (sj, aj)] 2 X;(4) [(si, ai), (sj, aj)]
k 2 X.
The Proof of Theorem 3.1 can be seen in the
Appendix.
Based on the above definition and theorem, we
introduce the following properties of the operational
laws:
Theorem 3.2 Suppose that A = [(si, ai), (sj, aj)] and
B = [(sk, ak), (sl, al)] are two interval-valued 2-tuple lin-
guistic variables in X, then the relations of these opera-
tional laws are given as
(1) A � B = B � A;
(2) A � B = B � A;
(3) k � (A � B) = (k � A) � (k � B);
(4) (A � B)k = Ak � Bk;
(5) (k1 � A) � (k2 � A) = (k1 ? k2) � A;
(6) Ak1 � Ak2 ¼ Ak1þk2 ;
(7) k1 � (k2 � A) = (k1k2) � A;
(8) Ak1� �k2¼ Ak1�k2 :
where, k, k1, k2, are scalars.
The Proof of Theorem 3.2 can be seen in the Appendix.
Theorem 3.3 Assume that ai; a0i 2 � 12g; 12g
h i; Ai ¼
si; aið Þ; s; ai0ð Þ½ � i ¼ 1; 2; . . .; n; si; s0i 2 S; ai; ai0 2 � 1
2g;
h12g�Þ is a collection of interval-valued 2-tuple linguistic
variables in X. Then:
(1)
�n
i¼1si; aið Þ; s0i; ai0
� �� �¼ D /�1
Xni¼1
/ D�1 si; aið Þ� �" #( )
;
"
D /�1Xni¼1
/ D�1 s0i; ai0� �� �" #( )#
;
(2)
�n
i¼1si; aið Þ; s0i; a
0i
� �� �¼ D u�1
Xni¼1
u D�1 si; aið Þ� �" #( )
;
"
D u�1Xni¼1
u D�1 s0i; a0i
� �� �" #( )#:
The Proof of Theorem 3.3 can be seen in the Appendix.
4 Interval-valued 2-tuple Linguistic BonferroniMean Operators Based on Archimedean t-normand Archimedean s-norm
The operational laws can be used to aggregate the interval-
valued 2-tuple linguistic information.
Definition 4.1 Let s1; a1ð Þ; s01; a01
� �� �; s2; a2ð Þ;½
s02; a
02
� ��;
� � � ; sn; anð Þ; s0n; a0n
� �� �g be a set of interval-valued 2-tuple,
and p, q C 0. If
ATS� I2TLBMp;q ½ðs1; a1Þ; ðs01; a01Þ�; ½ðs2; a2Þ; ðs02; a02Þ�; . . .; ½ðsn; anÞ; ðs0n; a0nÞ�� �
¼ 1nðn�1Þ� �
n
i 6¼ji;j¼1
½ðsi;aiÞ;ðs0i;a0iÞ�p�½ðsj;ajÞ;ðs0j;a0jÞ�
qð Þ
0@
1A
24
35
1pþq
;
ð15Þ
then ATS-I2TLBMp,q is an Archimedean t-norm and s-norm
based interval-valued 2-tuple linguistic Bonferroni mean.
Herein, Ai ¼ si; aið Þ; s0i; a0i
� �� �represents the evaluated
interval 2-tuple of xi 2 X under Ci. And
si; aið Þ; s0i; a0i
� �� �p� sj; aj� �
; s0j; a0j
�h iqis the degree that
xi 2 X satisfies Ci and Cj with given parameters p and q.
Theorem 4.1 Let Ai ¼ si; aið Þ; s0i; a0i
� �� �i ¼ 1; 2; . . .; n;ð
si; s0i 2 S; ai; a0i 2 � 1
2g; 12g
h iÞ be a set of interval-valued 2-
92 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
tuple linguistic variable. Then aggregated value by using
the ATS-I2TLBMp,q operator is also a interval-valued 2-
tuple linguistic variable and
ATS � I2TLBMp;qðA1;A2; . . .;AnÞ
¼ 1
nðn � 1Þ � �n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ�p � ½ðsj; ajÞ; ðs0j; a0jÞ�
qÞ
0BBBBBB@
1CCCCCCA
26666664
37777775
1pþq
¼
D u�1 1
pþ qu /�1 1
nðn � 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ�� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
pþ qu /�1 1
nðn � 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
:
ð16Þ
The Proof of Theorem 4.1 can be seen in the Appendix.
In Theorem 4.1, we obtain the general expression of
ATS-I2TLBM. To analyze ATS-I2TLBM in context multi-
attribute decision making, a formula in detail is needed.
Taking u(x) = -logx for an example, Eq. (16) can be
rewritten as follows:
A� I2TLBMp;qð½ðs1; a1Þ; ðs01; a01Þ�; ½ðs2; a2Þ; ðs02; a02Þ�; . . .; ½ðsn; anÞ; ðs0n; a0nÞ�Þ
¼
D 1�Yni 6¼ j
i; j ¼ 1
1� ðD�1ðsi; aiÞÞpðD�1ðsj; ajÞÞq� � 1
nðn�1Þ
0BBBBBBBB@
1CCCCCCCCA
1pþq
8>>>>>>>>><>>>>>>>>>:
9>>>>>>>>>=>>>>>>>>>;
;
D 1�Yni 6¼ j
i; j ¼ 1
1� ðD�1ðs0i; a0iÞÞpðD�1ðs0j; a0jÞÞ
qh i 1
nðn�1Þ
0BBBBBBBB@
1CCCCCCCCA
1pþq
8>>>>>>>>><>>>>>>>>>:
9>>>>>>>>>=>>>>>>>>>;
266666666666666666666666664
377777777777777777777777775
:
ð17Þ
A-I2TLBM is called Algebra t-norm and s-norm based
interval-valued 2-tuple linguistic Bonferroni mean.
In Eq. (17), let uij ¼ D�1 si; aið Þ� �p
D�1 sj; aj� �� �q
and u0ij ¼ D�1 s0i; a0i
� �� �pD�1 s0j; a
0j
� �q, then
A� I2TLBMp;qð½ðs1; a1Þ; ðs01; a01Þ�; ½ðs2; a2Þ; ðs02; a02Þ�; . . .; ½ðsn; anÞ; ðs0n; a0nÞ�Þ
¼ D ½1�Qni 6¼ji;j¼1
ð1�uijÞ1
nðn�1Þ�1
pþq
8><>:
9>=>;; D ½1�
Qni 6¼ji;j¼1
ð1�u0ijÞ
1nðn�1Þ�
1pþq
8><>:
9>=>;
264
375:
ð18Þ
In Eq. (18), 1 - uij and 1� u0ij indicate the upper and
lower bound of the negative degree of attributes Ci and Cj.
SoQni 6¼ji;j¼1
ð1� uijÞ1
nðn�1Þ andQni 6¼ji;j¼1
ð1� u0ijÞ1
nðn�1Þ can be consid-
ered as the average dissatisfaction bound of attributes Ci
and Cj. Here then we see that A-I2TLBM can capture the
interrelationship between input arguments and assess the
alternatives performance.
Let us investigate some special cases of ATS-I2TLBMp,q
with respect to the parameters p and q:
1. If q ? 0, then by the ATS-I2TLBMp,q, we have
limq!0
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼ limq!0
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 p � uðD�1ðsi; aiÞÞ þ qu D�1ðsj; ajÞ� �� �� �
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 p � uðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞh i �
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼
D u�1 1
pu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðsi; aiÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
pu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðs0i; a0iÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼
D u�1 1
pu /�1 1
n
Xni¼1
/ u�1 puðD�1ðsi; aiÞÞ� �� �" # ! !( )
;
D u�1 1
pu /�1 1
n
Xni¼1
/ðu�1½puðD�1ðs0i; a0iÞÞ�Þ" # ! !( )
2666664
3777775
¼ 1
n� �
n
i¼1½ðsi; aiÞ; ðs0i; a0iÞ�
p� �� �� 1
p
¼ ATS� I2TLBMp;0ðA1;A2; . . .;AnÞ;
which is called the Archimedean t-norm and s-norm
based interval-valued 2-tuple linguistic generalized
mean (ATS-I2TLGM) operator.
2. If p = 1, q ? 0, then from the ATS-I2TLBMp,0, we
have
ATS� I2TLBM1;0ðA1;A2; . . .;AnÞ
¼ D /�1 1
n
Xni¼1
/ðD�1ðsi; aiÞÞ !( )
;
"
D /�1 1
n
Xni¼1
/ðD�1ðs0i; a0iÞÞ !( )#
¼ 1
n� �
n
i¼1½ðsi; aiÞ; ðs0i; a0iÞ�
� �;
which is called the Archimedean t-norm and s-norm
based interval-valued 2-tuple linguistic mean (ATS-
I2TLM) operator.
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 93
123
3. If p = 2, q ? 0, then from the ATS-I2TLBMp,0, it is
obtained that
ATS� I2TLBM2;0ðA1;A2; . . .;AnÞ
¼
D u�1 1
2u /�1ð1
n
Xni¼1
/ðu�1½2uðD�1ðsi; aiÞÞ�ÞÞ" # !( )
;
D u�1 1
2u /�1ð1
n
Xni¼1
/ðu�1½2uðD�1ðs0i; a0iÞÞ�ÞÞ" # !( )
2666664
3777775
¼ 1
n� ð�
n
i¼1ð½ðsi; aiÞ; ðs0i; a0iÞ�
2ÞÞ� 1
2
;
which is called the Archimedean t-norm and
s-norm based interval-valued 2-tuple linguistic square
mean (ATS-I2TLSM) operator.
4. If p = 1, q = 1, then ATS-I2TLBMp,q reduces to the
Archimedean t-norm and s-norm based interval-valued
2-tuple linguistic interrelated square mean (ATS-
I2TLISM) operator:
ATS� I2TLBM1;1ðA1;A2; . . .;AnÞ
¼
D u�1 1
2u /�1 1
nðn � 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½uðD�1ðsi; aiÞÞ þ uðD�1ðsj; ajÞÞ�� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
2u /�1 1
nðn � 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½uðD�1ðs0i; a0iÞÞ þ uðD�1ðs0j; a0jÞÞ� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼ 1
nðn� 1Þ � �n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ� � ½ðsj; ajÞ; ðs0j; a0jÞ�Þ
0BBBBBB@
1CCCCCCA
26666664
37777775
12
:
We now investigate some properties of the Archimedean
t-norm and s-norm based interval-valued 2-tuple linguistic
bonferroni mean.
Theorem 4.2 Let Ai ¼ si; aið Þ; s0i; a0i
� �� �be a collection of
interval-valued 2-tuple linguistic variable, and p, q C 0,
then we get some properties of ATS-I2LBM as follows:
(1) Idempotency: If Ai ¼ si; aið Þ; s0i; a0i
� �� �¼ sk; akð Þ;½
sl; alð Þ� for all i, thenATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼ sk; akð Þ; sl; alð Þ½ �:
(2) Boundedness: If A� ¼ miniAi ¼ mini si; aið Þ;½mini s
0i; a
0i
� ��;Aþ ¼ maxiAi ¼ maxi si; aið Þ;½
maxi s0i; a
0i
� ��, then
A� �ATS� I2TLBMp;qðA1;A2; . . .;AnÞ�Aþ:
(3) Commutativity: Assume that Ai ¼ si; aið Þ; s0i; a0i
� �� �and A0
i ¼ srðiÞ; arðiÞ� �
; s0rðiÞ; a0rðiÞ
�h iði ¼ 1; 2; . . .; nÞ
are two sets of interval-valued 2-tuple linguistic
variables, where A0i ¼ srðiÞ; arðiÞ
� �; s0rðiÞ; a
0rðiÞ
�h iði ¼ 1; 2; . . .; nÞ is any permutation of Ai ¼ si; aið Þ;½s0i; a
0i
� �� ði ¼ 1; 2; . . .; nÞ, then ATS� I2TLBMp;q
ðA1;A2; . . .;AnÞ ¼ ATS� I2TLBMp;qðA01; A
02; . . .A
0nÞ:
(4) Monotoncity: Let Ai ¼ si; aið Þ;½ s0i; a0i
� �� and Ai ¼
si ; ai
� �; s0i ; a
0i
� �� �be two sets of interval-valued 2-
tuples. If si; aið Þ� si; a
i
�and s0
i; a0
i
�� s0
i; a0
i
�for all i, then
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ�ATS
� I2TLBMp;qðA1;A
2; . . .;A
nÞ:
The Proof of Theorem 4.2 can be seen in the Appendix.
Since that the input argument has different importances, we
define the Archimedean t-norm and s-norm based interval-
valued 2-tuple linguisticweightedBonferronimean as follows:
Definition 4.2 Let Ai ¼ si; aið Þ; s0i; a0i
� �� �ði ¼ 1; 2; . . .; n;
si; s0i 2 S; ai; a0i 2 � 1
2g; 12g
h i) be a set of interval-valued
2-tuple linguistic variables, and p, q[ 0. W = (w1,
w2,…,wn)T is the weight vector of Ai (i = 1, 2,…,n), where
wi indicates the importance degree of Ai, satisfying
wi[ 0(i = 1, 2,…n), andPni¼1
wi ¼ 1; if
ATS� I2TLWBMp;qw s1; a1ð Þ; s01; a
01
� �� �; s2; a2ð Þ; s02; a
02
� �� �; . . .; sn; anð Þ; s0n; a
0n
� �� �� �
¼ 1
nðn� 1Þ � �n
i 6¼ j
i; j ¼ 1
ððwi � ½ðsi; aiÞ; ðs0i; a0iÞ�Þp � ðwj � ½ðsj; ajÞ; ðs0j; a0jÞ�
qÞÞ
0BBBBBB@
1CCCCCCA
26666664
37777775
1pþq
;
ð19Þ
then ATS-I2TLWBMwp,qis called Archimedean t-norm and
s-norm based interval-valued 2-tuple linguistic weighted
bonferroni mean operator.
Similar to the Theorem 4.1, we have
ATS� I2TLWBMp;qw s1; a1ð Þ; s01; a
01
� �� �; s2; a2ð Þ; s02; a
02
� �� �; . . .; sn; anð Þ; s0n; a
0n
� �� �� �
¼
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puð/�1ðwi/ðD�1ðsi; aiÞÞÞÞ þ quð/�1ðwj/ðD�1ðsj; ajÞÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puð/�1ðwi/ðD�1ðs0i; a0iÞÞÞÞ þ quð/�1ðwj/ðD�1ðs0j; a0jÞÞÞh i �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
:
Based on the Theorem 3.2, we shall extend the geometric
bonferroni mean operator to accommodate the situations,
where the input arguments are interval-valued 2-tuple lin-
guistic variable. We introduce the Archimedean t-norm-
and s-norm-based on interval-valued 2-tuple linguistic
geometric Bonferroni mean as follows:
94 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
Definition 4.3 Let s1; a1ð Þ; s01; a01
� �� �; s2; a2ð Þ;½
s02;�
a02Þ�; . . .; sn; anð Þ; s0n; a0n
� �� �g be a collection of interval-
valued 2-tuple linguistic variables, and p, q[ 0. Then the
aggregation function
ATS� I2TLGBMp;qð½ðs1; a1Þ; ðs01; a01Þ�; ½ðs2; a2Þ; ðs02; a02Þ�; . . .; ½ðsn; anÞ; ðs0n; a0nÞ�Þ
¼ 1
pþ q� �
n
i 6¼ j
i; j ¼ 1
ððp� ½ðsi; aiÞ; ðs0i; a0iÞ�Þ � ðq� ½ðsj; ajÞ; ðs0j; a0jÞ�ÞÞ
26666664
37777775
8>>>>>><>>>>>>:
9>>>>>>=>>>>>>;
1nðn�1Þ
ð20Þ
is called the Archimedean t-norm and s-norm based inter-
val-valued 2-tuple linguistic geometric Bonferroni mean
(ATS-I2TLGBMp,q) operator.
Similarly, based on the Definition 3.1 and Theorem 3.3,
we can get the following theorem:
Theorem 4.3 Let Ai ¼ si; aið Þ; s0i; a0i
� �� �ði ¼ 1; 2; . . .; nÞ
be a set of interval-valued 2-tuple linguistic variables.
Then the aggregated value by using the ATS-
I2TLGBMp,qoperator is also interval-valued linguistic 2-
tuple, and
ATS � I2TLGBMp;qðA1;A2; . . .;AnÞ
¼
D /�1 1
pþ q/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
uð/�1½p/ðD�1ðsi; aiÞÞ þ q/ðD�1ðsj; ajÞÞ�Þ
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D /�1 1
pþ q/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
uð/�1½p/ðD�1ðs0i; a0iÞÞ þ q/ðD�1ðs0j; a0jÞÞ�Þ
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
:
ð21Þ
Proof Similar to the proof of Theorem 4.1, it is easy to
establish Theorem 4.3. h
In Theorem 4.3, we obtained the general expression of
ATS-I2TLGBM. Similar to Eq. (17), if we let
u(x) = -logx, then (21) becomes
A � I2TLGBMp;q s1; a1ð Þ; s01; a01
� �� �; s2; a2ð Þ; s02; a
02
� �� �; . . .; sn; anð Þ; s0n; a
0n
� �� �� �
¼
D 1� 1�Yni 6¼ j
i; j ¼ 1
1� ½1� D�1ðsi; aiÞ�p½1� D�1ðsj; ajÞ�q� � 1
nðn�1Þ
0BBBBBBBB@
1CCCCCCCCA
1pþq
8>>>>>>>>><>>>>>>>>>:
9>>>>>>>>>=>>>>>>>>>;
;
D 1� 1�Yni 6¼ j
i; j ¼ 1
1� ½1� D�1ðs0i; a0iÞ�p½1� D�1ðs0j; a0jÞ�
q � 1
nðn�1Þ
0BBBBBBBB@
1CCCCCCCCA
1pþq
8>>>>>>>>><>>>>>>>>>:
9>>>>>>>>>=>>>>>>>>>;
266666666666666666666666664
377777777777777777777777775
;
ð22Þ
where A-I2TLGBM is Algebra t-norm- and s-norm-based
interval-valued 2-tuple linguistic geometric Bonferroni
mean.
Let vij ¼ 1� D�1 si; aið Þ� �p
1� D�1 sj; aj� �� �q
; v0ij ¼
1� D�1 s0i; a0i
� �� �p1� D�1 s0j; a
0j
� �q. Then
A� I2TLGBMp;qð s1; a1ð Þ; s01; a01
� �� �; s2; a2ð Þ; s02; a
02
� �� �; . . .; sn; anð Þ; s0n; a
0n
� �� �
¼ D 1� 1�Yni 6¼ j
i; j ¼ 1
ð1� vijÞ1
nðn�1Þ
266666664
377777775
1pþq
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;D 1� 1�
Yni 6¼ j
i; j ¼ 1
ð1� v0ijÞ1
nðn�1Þ
266666664
377777775
1pþq
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666664
3777777775:
ð23Þ
In Eq. (23), 1 - vij and 1� v0ij indicates the lower and
upper bound of the negative degree of x towards Ci and Cj.
SoQni 6¼ji;j¼1
ð1� vijÞ1
nðn�1Þ andQni 6¼ji;j¼1
ð1� v0
ijÞ1
nðn�1Þ can be consid-
ered as the average satisfaction bound of attributes Ci and
Cj. Here then we see that A-I2TLGBM can capture the
interrelationship between input arguments and assess the
alternatives performance.
If we consider the possible values of the parameters
p and q in the ATS-I2TLGBM operator, we can get a group
of particular cases.
1. If q ? 0, then by the ATS-I2TLGBMp,q, we have
ATS� I2TLGBMp;qðA1;A2; . . .;AnÞ
¼ limq!0
D /�1 1
pþ q/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
u /�1 p/ðD�1ðsi; aiÞÞ þ q/ðD�1ðsj; ajÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D /�1 1
pþ q/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
u /�1 p/ðD�1ðs0i; a0iÞÞ þ q/ðD�1ðs0j; a0jÞÞ �h i
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼
D /�1 1
p/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
u /�1 p/ðD�1ðsi; aiÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D /�1 1
p/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
u /�1 p/ðD�1ðs0i; a0iÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼ 1
p� ½�
n
i¼1ðp� ½ðsi; aiÞ; ðs0i; a0iÞ�Þ�
� �1n
¼ ATS� I2TLGBMp;0ðA1;A2; . . .;AnÞ;
which we call the Archimedean t-norm and s-norm
based interval-valued 2-tuple linguistic generalized
geometric mean (ATS-I2TLGGM) operator.
2. If p = 1, q ? 0, then from the ATS-I2TLGBMp,0, it is
obtained that
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 95
123
ATS� I2TLGBM1;0ðA1;A2; . . .;AnÞ
¼ D u�1 1
n
Xni¼1
uðD�1ðsi; aiÞÞ" #( )
;
"
D u�1 1
n
Xni¼1
uðD�1ðs0i; a0iÞÞ" #( )#
¼ �n
i¼1½ðsi; aiÞ; ðs0i; a0iÞ�
� �1n
;
which is called the Archimedean t-norm and
s-norm based interval-valued 2-tuple linguistic geo-
metric mean (ATS-I2TLGM) operator.
3. If p = 2, q ? 0, then from the ATS-I2TLGBMp,0, we
get
ATS� I2TLGBM2;0ðA1;A2; . . .;AnÞ
¼
D /�1 1
2/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
u /�1 2/ D�1ðsi; aiÞ� �� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D /�1 1
2/ u�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
u /�1 2/ D�1ðs0i; a0iÞ� �� �� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼ 1
2� �
n
i¼1ð2� ½ðsi; aiÞ; ðs0i; a0iÞ�Þ
� � �1n
;
which is called the Archimedean t-norm and s-norm
based interval-valued 2-tuple linguistic square geo-
metric mean (ATS-I2TLSGM) operator.
4. If p = 1, q = 1, then ATS-I2TLBMp,q reduces to the
Archimedean t-norm- and s-norm-based interval-val-
ued 2-tuple linguistic interrelated square geometric
mean (ATS-I2TLISGM) operator:
ATS� I2TLGBM1;1ðA1;A2; . . .;AnÞ
¼
D u�1 1
1þ 1u /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½uðD�1ðsi; aiÞÞ þ uðD�1ðsj; ajÞÞ�� �
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
1þ 1u /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½uðD�1ðs0i; a0iÞÞ þ uðD�1ðs0j; a0jÞÞ� �
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼ 1
2� �
n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ� � ½ðsj; ajÞ; ðs0j; a0jÞ�Þ
26666664
37777775
8>>>>>><>>>>>>:
9>>>>>>=>>>>>>;
1nðn�1Þ
:
Similarly, we have some properties of ATS-I2TLGBM as
follows:
Theorem 4.4 Let Ai ¼ si; aið Þ; s0i; a0i
� �� �be a collection of
interval-valued 2-tuple linguistic variable, and p, q[ 0.
Then ATS-I2TLGBM have the following properties:
(1) Idempotency If Ai ¼ si; aið Þ; s0i; a0i
� �� �¼ sk; akð Þ;½
sl; alð Þ� ¼ A for all i, then
ATS� I2TLGBMp;qðA1;A2; . . .;AnÞ ¼ A:
(2) Boundedness Assume that Ai ¼ si; aið Þ; s0i; a0i
� �� �; is a
set of interval-valued 2-tuple linguistic variables, and
A� ¼ mini
Ai ¼ ½miniðsi; aiÞ;min
iðs0i; a0iÞ�;
Aþ ¼ maxiAi ¼ maxi si; aið Þ;maxi s0i; a
0i
� �� �:
Then
A� �ATS� I2TLGBMp;qðA1;A2; . . .;AnÞ�Aþ:
(3) Commutativity If A0i ¼
srðiÞ; arðiÞ� �
; s0rðiÞ; a0rðiÞ
�h iði ¼ 1; 2; . . .; nÞ is any
permutation of Ai ¼ si; aið Þ; s0i; a0i
� �� �, then
ATS� I2TLGBMp;qðA1;A2; . . .;AnÞ¼ ATS� I2TLGBMp;qðA0
1;A; . . .;A0nÞ:
(4) Monotoncity Let Ai ¼ si; aið Þ; s0i; a0i
� �� �and A
i ¼si ; a
i
� �; s0i ; a
0i
� �� �be two sets of interval-valued
2-tuples. If si; aið Þ� si ; ai
� �and s0i; a
0i
� �� s0i ; a
0i
� �for all i, then
ATS� I2TLGBMp;qðA1;A2; . . .;AnÞ�ATS
� I2TLGBMp;qðA1;A
2; . . .;A
nÞ:
Proof The proof of this property is similar to Theo-
rem 4.2, so it is omitted. h
Based on the Definition 4.2, we consider that the input
arguments have different importances; thus we introduce
the following definition:
Definition 4.4 Suppose that Ai ¼ si; aið Þ; s01; a01
� �� �ði ¼
1; 2; . . .; nÞ is a set of interval-valued 2-tuple linguistic
variables, and p, q[ 0, W ¼ w1;w2; . . .;wnð ÞT is the
weight vector of Ai, where wi indicates the importance
degree of Ai, satisfying wi [ 0 i ¼ 1; 2; . . .nð Þ andPni¼1
wi ¼ 1, then the aggregation function:
ATS� I2TLWGBMp;qw s1; a1ð Þ; s01; a
01
� �� �; s2; a2ð Þ; s02; a
02
� �� �; . . .; sn; anð Þ; s0n; a
0n
� �� �� �
¼ 1
pþ q� �
n
i 6¼ j
i; j ¼ 1
ððp� ½ðsi; aiÞ; ðs0i; a0iÞ�wiÞ � ðq� ½ðsj; ajÞ; ðs0j; a0jÞ�
wjÞÞ
2666664
3777775
8>>>>><>>>>>:
9>>>>>=>>>>>;
1nðn�1Þ
ð24Þ
96 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
is called the Archimedean t-norm- and s-norm-based
interval-valued 2-tuple linguistic weighted geometric
Bonferroni mean operator.
Similarly, we can obtain
ATS� I2TLWGBMp;qw s1;a1ð Þ; s01;a
01
� �� �; s2;a2ð Þ; s02;a
02
� �� �; . . .; sn;anð Þ; s0n;a
0n
� �� �� �
¼
D /�1 1
pþq/ u�1 1
nðn�1ÞXni 6¼ j
i; j¼ 1
u /�1 p/ðu�1ðwiuðD�1ðsi;aiÞÞÞÞþq/ðu�1ðwjuD�1ðsj;ajÞÞÞÞ
� �� �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D /�1 1
pþq/ u�1 1
nðn�1ÞXni 6¼ j
i; j¼ 1
u /�1 p/ðu�1ðwiuðD�1ðs0i;a0iÞÞÞÞþq/ðu�1ðwjuD�1ðs0j;a0jÞÞÞÞ
h i �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
5 An Approach to Multiple Attribute DecisionMaking Based on the New Operation
In this section, we shall utilize the interval-valued 2-tuple
linguistic Bonferroni mean operators to multiple attribute
decision making with interval-valued 2-tuple linguistic
information.
5.1 An Approach to Multiple Attribute Decision
Making Based on the New Operation
The Archimedean t-norm- and s-norm-based interval-val-
ued 2-tuple linguistic aggregation operators can be widely
used in solving group decision making, where the decision
information is represented by interval-valued linguistic
2-tuple. for instance, si; aið Þ; s0i; a0i
� �� �represented that the
performance value of one alternative on an attribute is
between 2-tuples (si, ai) and s0i; a0i
� �. In the group decision
making process, for a given alternative and attribute,
decision maker should have a different knowledge (ability
and experience) and use different interval-valued linguistic
2-tuple to express the evaluation. For choosing best alter-
native, we should aggregate that interval-valued linguistic
2-tuples. Obviously, the closer between (si, ai) and s0i; a0i
� �,
the higher accuracy information the decision maker can
give with the original defined linguistic term set. Thus the
difference between D-1(si, ai) and D�1 s0i; a0i
� �can be rep-
resented the degree of precision of decision information.
Generally speaking, we want to assign bigger weights to
the interval-valued 2-tuples with more degree of precision.
Definition 5.1 [45] Let S ¼ siji ¼ 0; 1; . . .; gf g be a lin-
guistic term set. An interval-valued linguistic 2-tuple Ai ¼si; aið Þ; s0i; a
0i
� �� �is composed of two linguistic term from S,
then
DPð½ðsi; aiÞ; ðs0i; a0iÞ�Þ ¼ 1�D�1ðs0i; a0iÞ � D�1ðsi; aiÞ þ 1
g
1þ 1g
ð25Þ
is called the degree of precision of Ai ¼ si; aið Þ; s0i; a0i
� �� �:
Obviously, 0�DPð½ðsi; aiÞ; ðs0i; a0iÞ�Þ �g
gþ1. If
D�1 s0i; a0i
� �¼ D�1 si; aið Þ; i:e: si; aið Þ; s0i; a
0i
� �� �is reduced to
(si, ai), we get DPð½ðsi; aiÞ; ðs0i; a0iÞ�Þ ¼g
gþ1, and the degree
of precision reaches it maximum. Inversely, if D�1 s0i; a0i
� �¼ 0 and D�1 si; aið Þ ¼ 1, we have DP si; aið Þ; s0i; a
0i
� �� �� �¼
0, in this case, we cannot get any useful information from
this evaluation value. Thus we can give a method to
determine the aggregation operator weights in group
decision making as follows.
Definition 5.2 [45] For a given alternative Xi with
respect to attributes Cj j ¼ 1; 2; . . .; nð Þ, the decision maker
Dk k ¼ 1; 2; . . .; tð Þ provides their evaluate information by
sijk ; a
ijk
� �; l
ijk ; b
ijk
� �� �, where s
ijk ; l
ijk 2 S and aijk ; b
ijk 2
� 12g; 12g
h i. Then the weight vector Wij ¼ w
ij1 ;w
ij2 ; . . .;w
ijt
� �of the aggregation operator can be construct as follows:
wijk ¼ DPð½ðsijk ; a
ijk Þ; ðl
ijk ; b
ijk Þ�ÞPt
h¼1
DPð½ðsijh ; aijhÞ; ðl
ijh ; b
ijhÞ�Þ
; ð26Þ
where wijk 2 0; 1½ � and
Ptk¼1
wijk ¼ 1.
Based on these ideas, we develop a method for multiple
attribute decision making under interval-valued 2-tuple
linguistic environment.
We solve a multiple attribute group decision making
problem, where the attribute assessment values are repre-
sented by interval-valued 2-tuple linguistic variable.
Assume that X ¼ X1;X2; . . .;Xmf g is a discrete set of
alternatives, C ¼ C1;C2; . . .;Cnf g is a set of attributes, and
k ¼ k1; k2; . . .; knð Þ is weight vector, where kj 2 0; 1½ � is
associated weight of Cj, andPnj¼1
kj ¼ 1. Let D ¼
D1;D2; . . .;Dtf g be a collection of decision makers. A
decision maker Dk k ¼ 1; 2; . . .; tð Þ gives his/her assess-
ment value of alternative Xi with respect to attribute Cj by
eijk ¼ s
ijk ; l
ijk
� �ðsijk ; l
ijk 2 S; i ¼ 1; 2; . . .;m; j ¼ 1; 2; . . .; nÞ,
and transform eijk ¼ s
ijk ; l
ijk
� �into linguistic interval-valued
2-tuple Aijk ¼ s
ijk ; 0
� �; l
ijk ; 0� �� �
, so that we can get a series
of interval-valued 2-tuple linguistic decision matrices
Ak ¼ Aijk
� �mn
. Besides, each decision maker uses differ-
ent linguistic term set S to express the preference values.
To obtain the best alternative, the following steps are
given.
Step 1 The decision maker Dk gives his/her linguistic
decision matrix Ek ¼ eijk
� �mn
, where eijk ¼ s
jk;
�lijk � s
ijk ; l
ijk 2 S; i ¼ 1; 2; . . .;m; j ¼ 1; 2; . . .; n
� �
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 97
123
represent the interval linguistic value of each
attribute Cj of each alternative Xj.
Step 2 According to the Definition 2.4, we can trans-
form the interval linguistic decision matrix Ek ¼eijk
� �mn
into interval-valued 2-tuple linguistic
decision matrix Ak ¼ Aijk
� �mn
, where Aijk ¼
sijk ; 0
� �; l
ijk ; 0� �� �
is a interval-valued 2-tuple lin-
guistic value.
Step 3 Utilize Definitions 5.1 and 5.2 to calculate the
associated weight vector Wij ¼ wij1 ;w
ij2 ; . . .;w
ijt
� �of the ATS-I2TLWBM operator.
Step 4 Based on ATS-I2TLWBM (ATS-I2TLWGBM)
operator to aggregate all the decision matrix
Ak k ¼ 1; 2; . . .; tð Þ into a collection decision
matrix A ¼ Aijð Þmn, where Aij ¼ ATS� I2TL
WBM Aij1 ;A
ij2 ; . . .;A
ijt
� �ðAij ¼ ATS� I2TLWGBM
ðAij1 ;A
ij2 ; . . .;A
ijt ÞÞ and the weight vector is
Wij ¼ wij1 ;w
ij2 ; . . .;w
ijt
� �.
Step 5 Utilize the ATS-I2TLWBM (ATS-I2TLWGBM)
operator to derive the collective overall prefer-
ence values Ai for the alternative
Xi i ¼ 1; 2; . . .;mð Þ, where the weight vector is
k ¼ k1; k2; . . .; knð Þ.Step 6 According to the Theorem 2.1, rank the
alternatives.
Step 7 End.
5.2 Illustrative Example
In this section, we use practical multiple attribute group
decision making problems to illustrate the efficiency of the
proposed method in dealing with interval-valued 2-tuple
linguistic information. Suppose an investment company
wants to find an optimal investment (adapted from [50]).
There is a panel with four possible alternatives to invest the
money: X1 is a car industry; X2 is a food company; X3 is a
computer company; X4 is an arms industry. The investment
company must take a decision according to the following
four attributes: C1 is the risk analysis; C2 is the growth
analysis; C3 is the social-political impact analysis; C4is the
environment impact analysis. In order to avoid interaction
effect, the decision makers are invited to provide their
preferences for each possible alternative on each attributes
in anonymity and using different linguistic term sets:
decision maker D1 provides his preferences in the set of
nine terms, S1 ¼ s0; s1; s2; . . .; s8f g; decision maker D2
provides his preferences in the set of seven terms,
S2 ¼ s0; s1; s2; . . .; s6f g; decision maker D3 provides his
preferences in the set of five terms, S3 ¼ s0; s1; s2; . . .; s4f g.The weight vector of attributes is k = (0.3, 0.1, 0.2, 0.4).
Then, we utilize the method developed to obtain the best
alternative(s).
Step 1 Each decision maker Dk uses his/her linguistic
term set and give his/her linguistic decision
matrix Ek ¼ eijk
� �44
k ¼ 1; 2; 3ð Þ as follows,
where eijk ¼ s
ijk ; l
ijk
� �sijk ; l
ijk 2 S; i ¼ 1; 2; 3; 4;
�j ¼ 1; 2; 3; 4Þ represent the interval linguistic
value of each attribute Cj of each alternative.
C1 C2 C3 C4
E1 ¼
X1
X2
X3
X4
½s7; s7� ½s1; s1� ½s1; s3� ½s3; s5�½s5; s7� ½s4; s4� ½s3; s4� ½s4; s6�½s4; s7� ½s3; s4� ½s3; s6� ½s1; s1�½s3; s3� ½s0; s2� ½s4; s4� ½s3; s4�
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C1 C2 C3 C4
E2 ¼
X1
X2
X3
X4
½s3; s3� ½s5; s5� ½s1; s3� ½s3; s4�½s5; s5� ½s4; s5� ½s1; s3� ½s3; s5�½s2; s4� ½s4; s5� ½s1; s3� ½s0; s1�½s4; s5� ½s1; s2� ½s3; s4� ½s3; s5�
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C1 C2 C3 C4
E3 ¼
X1
X2
X3
X4
½s2; s3� ½s1; s2� ½s1; s3� ½s3; s3�½s3; s3� ½s2; s3� ½s2; s3� ½s1; s1�½s3; s3� ½s0; s2� ½s2; s3� ½s2; s2�½s1; s1� ½s2; s3� ½s1; s3� ½s3; s3�
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Step 2 According to the Definition 2.4, we can trans-
form the interval linguistic decision matrices
Ek ¼ eijk
� �44
k ¼ 1; 2; 3ð Þ into interval-valued 2-
tuple linguistic decision matrices Ak ¼ Aijk
� �44
as follows:
C1 C2 C3 C4
A1 ¼
X1
X2
X3
X4
½ðs7; 0Þ; ðs7; 0Þ� ½ðs1; 0Þ; ðs1; 0Þ� ½ðs1; 0Þ; ðs3; 0Þ� ½ðs3; 0Þ; ðs5; 0Þ�½ðs5; 0Þ; ðs7; 0Þ� ½ðs4; 0Þ; ðs4; 0Þ� ½ðs3; 0Þ; ðs4; 0Þ� ½ðs4; 0Þ; ðs6; 0Þ�½ðs4; 0Þ; ðs7; 0Þ� ½ðs3; 0Þ; ðs4; 0Þ� ½ðs3; 0Þ; ðs6; 0Þ� ½ðs1; 0Þ; ðs1; 0Þ�½ðs3; 0Þ; ðs3; 0Þ� ½ðs0; 0Þ; ðs2; 0Þ� ½ðs4; 0Þ; ðs4; 0Þ� ½ðs3; 0Þ; ðs4; 0Þ�
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C1 C2 C3 C4
A2 ¼
X1
X2
X3
X4
½ðs3; 0Þ; ðs3; 0Þ� ½ðs5; 0Þ; ðs5; 0Þ� ½ðs1; 0Þ; ðs3; 0Þ� ½ðs3; 0Þ; ðs4; 0Þ�½ðs5; 0Þ; ðs5; 0Þ� ½ðs4; 0Þ; ðs5; 0Þ� ½ðs1; 0Þ; ðs3; 0Þ� ½ðs3; 0Þ; ðs5; 0Þ�½ðs2; 0Þ; ðs4; 0Þ� ½ðs4; 0Þ; ðs5; 0Þ� ½ðs1; 0Þ; ðs3; 0Þ� ½ðs0; 0Þ; ðs1; 0Þ�½ðs4; 0Þ; ðs5; 0Þ� ½ðs1; 0Þ; ðs2; 0Þ� ½ðs3; 0Þ; ðs4; 0Þ� ½ðs3; 0Þ; ðs5; 0Þ�
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C1 C2 C3 C4
A3 ¼
X1
X2
X3
X4
½ðs2; 0Þ; ðs3; 0Þ� ½ðs1; 0Þ; ðs2; 0Þ� ½ðs1; 0Þ; ðs3; 0Þ� ½ðs3; 0Þ; ðs3; 0Þ�½ðs3; 0Þ; ðs3; 0Þ� ½ðs2; 0Þ; ðs3; 0Þ� ½ðs2; 0Þ; ðs3; 0Þ� ½ðs1; 0Þ; ðs1; 0Þ�½ðs3; 0Þ; ðs3; 0Þ� ½ðs0; 0Þ; ðs2; 0Þ� ½ðs2; 0Þ; ðs3; 0Þ� ½ðs2; 0Þ; ðs2; 0Þ�½ðs1; 0Þ; ðs1; 0Þ� ½ðs2; 0Þ; ðs3; 0Þ� ½ðs1; 0Þ; ðs3; 0Þ� ½ðs3; 0Þ; ðs3; 0Þ�
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Step 3 Utilize Definition 5.1 and 5.2 to calculate the
associated weight vector Wij ¼ wij1 ;w
ij2 ;w
ij3
� �of
the ATS-I2TLWBM operator. Then, we have
98 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
W11 ¼ ð0:379; 0:365; 0:256Þ; W12 ¼ 0:379; 0:365; 0:256ð Þ
W13 ¼ ð0:407; 0:349; 0:244Þ; W14 ¼ 0:305; 0:328; 0:367ð Þ
W21 ¼ ð0:287; 0:369; 0:344Þ; W22 ¼ 0:404; 0:324; 0:272ð Þ
W23 ¼ ð0:399; 0:293; 0:308Þ; W24 ¼ 0:327; 0:280; 0:393ð Þ
W31 ¼ ð0:288; 0:297; 0:415Þ; W32 ¼ 0:411; 0:378; 0:211ð Þ
W33 ¼ ð0:322; 0:331; 0:347Þ; W34 ¼ 0:370; 0:297; 0:333ð Þ
W41 ¼ ð0:370; 0:297; 0:333Þ; W42 ¼ 0:336; 0:361; 0:303ð Þ
W43 ¼ ð0:444; 0:356; 0:200Þ; W44 ¼ 0:362; 0:266; 0:372ð Þ
Step 4 Based on ATS-I2TLWBM operator to aggregate
all the decision matrix Ak (k = 1, 2, 3) into a
collection decision matrix A ¼ Aijð Þ44 as fol-
lows, where p = 1, q = 1 and u = -log(x) for
Aij ¼ ATS� I2TLWBM1;1 Aij1 ;A
ij2 ;A
ij3
� �and the
weight vector is Wij ¼ wij1 ;w
ij2 ;w
ij3
� �. In addition,
we can express the result of aggregation used by
linguistic interval-valued 2-tuples derived from
each linguistic term set Sk (k = 1, 2, 3). In this
problem, the final results are expressed by
interval-valued 2-tuples derived from linguistic
term set S1 with nine labels.
C1 C2 C3 C4
A ¼
X1
X2
X3
X4
½ðs2; 0:03804Þ; ðs3;�0:03143Þ� ½ðs1; 0:0182Þ; ðs2;�0:0575Þ� ½ðs0; 0:0606Þ; ðs2;�0:0270Þ� ½ðs2;�0:0175Þ; ðs3;�0:0575Þ�½ðs3;�0:01087Þ; ðs3; 0:0617Þ� ½ðs2;�0:0141Þ; ðs3;�0:0465Þ� ½ðs1; 0:0066Þ; ðs2; 0:0034Þ� ½ðs1; 0:0346Þ; ðs2; 0:0263Þ�½ðs2;�0:0262Þ; ðs3; 0:0108Þ� ½ðs1; 0:0174Þ; ðs2; 0:0238Þ� ½ðs1; 0:0053Þ; ðs2; 0:0615Þ� ½ðs0; 0:0576Þ; ðs1;�0:0368Þ�½ðs1; 0:0430Þ; ðs2;�0:0509Þ� ½ðs1;�0:0615Þ; ðs1; 0:0493Þ� ½ðs1; 0:0440Þ; ðs2; 0:0259Þ� ½ðs2;�0:0207Þ; ðs3;�0:0444Þ�
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Step 5 Utilize the ATS-I2TLWBM operator to derive
the collective overall preference values Ai for the
alternative Xi(i = 1, 2, 3, 4) are shown in
Table 1, where p = 1, q = 1, and u =
-log(x) for Ai = ATS-I2TLWBM1,1(Ai1, Ai2,…,
Ain) and the weight vector is k = (0.3, 0.1,
0.2, 0.4).
Step 6 According to the Definition 2.5, calculate the
score function S(Ai) of Ai (i = 1, 2, 3, 4) as
follows:
SðA1Þ ¼ 0:06423; S A2� �
¼ 0:07361
SðA3Þ ¼ 0:04847; S A4� �
¼ 0:05669:
Then, by Theorem 2.1, S(A2)[ S(A1)[ S(A4)[S(A3), we get A2[A1[A4[A3, which implies that
X2 � X1 � X4 � X3, Thus X2 is the best choice.
It is possible to analyze how the different attitudinal
characters p and q play a role in the aggregation results. As
the values of the parameters p and q change between 0 and
10, different results of a score function betai = S(Ai)
(i = 1, 2, 3, 4) of the collective overall preference values
Ai of the alternative Xi (i = 1, 2, 3, 4) can be obtained.
Figures 1, 2, 3 and 4 illustrate the values bi = S(Ai)
(i = 1, 2, 3, 4) of the four alternatives Xi(i = 1, 2, 3, 4)
obtained by the ATS-I2TLWBM operator in detail.
If we let the parameter p fixed, different values ai =
S(Ai) and rankings of the alternatives can be obtained as
the parameter q changed which was shown in Fig. 5.
From Fig. 5, we can find that
1) when q [ (0, 4.3517], the ranking of the alternatives
is X2 � X1 � X4 � X3; the best alternative is X2.
2) when q [ (4.3517, 10], the ranking of the alterna-
tives is X1 � X2 � X4 � X3; the best alternative is
X1.
By Figs. 1, 2, 3, 4 and 5, We can conclude that as the
values of the parameters p and q change according to the
decision maker’s subjective preferences, we may obtain
different rankings of the alternatives, which can reflect the
decision makers’ risk preference.
If the ATS-I2TLWGBM operator is used in place of the
ATS-I2TLWGBM operator to aggregate the values of the
alternatives in Step3 and Step4,then the different values
and the rankings of the alternatives can be obtained as the
values of the parameters pand qchange. As the values of
the parameters p and q change between 0 and 10, different
results of a score function bi = S(Ai) (i = 1, 2, 3, 4) of the
collective overall preference values Ai of the alternative
Xi(i = 1, 2, 3, 4) can be obtained. Figures 6, 7, 8 and 9
illustrate the value bi = S(Ai) of the four alternatives Xi
obtained by the ATS-I2TLWGBM operator in detail.
If we let the parameter p fixed, different values ai =
S(Ai) and rankings of the alternatives can be obtained as
the parameter q changed which was shown in Fig. 10.
From Fig. 5, we can find that
1) when q [ (0, 3.1515], the ranking of the alternatives
is X2 � X1 � X4 � X3; the best alternative is X2.
2) when q [ (3.1515, 6.1160], the ranking of the alter-
natives is X1 � X2 � X4 � X3; the best alternative
is X1.
3) when q [ (6.1160, 10], the ranking of the alterna-
tives is X1 � X4 � X2 � X3; the best alternative is
X1.
It is worth noted that most of the values obtained by the
ATS-I2TLWBM operator are bigger than the values
obtained by the ATS-I2TLWGBM operator, which indicates
that the ATS-I2TLWBM operator can obtain more unfa-
vorable (or pessimistic) expectations, while the ATS-
I2TLWGBM operator has more favorable (or optimistic)
expectations. Therefore, we can conclude that the ATS-
I2TLWBM operator can be considered as the pessimistic
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 99
123
one, while the ATS-I2TLWGBM operator can be considered
as the optimistic one and the values of the parameters can
be considered as the pessimistic or optimistic levels. So, we
can conclude that the decision makers who take a gloomy
view of the prospects could use the ATS-I2TLWBM oper-
ator and choose the smaller values of the parameter p and
q, while the decision makers who are optimistic could use
the ATS-I2TLWGBM operator and choose the smaller val-
ues of the parameter pand q.
In order to obtain the more neutral results, we can use
the arithmetic averages of the pessimistic and optimistic
results, which is found in Figs. 11, 12, 13, and 14.
Table 1 The result of collect overall preference values
A1 A2 A3 A4
D([0.05072, 0.7774]) D([0.05752, 0.0897]) D([0.03233, 0.0646]) D([0.04492, 0.06845])
[(s0, 0.05072), (s1, -0.04726)] [(s0, 0.05752), (s1, -0.0353)] [(s0, 0.03233), (s1, -0.0604)] [(s0, 0.03449), (s1, -0.0565)]
Fig. 1 The values S(A1) for alternative X1 obtained by the ATS-
I2TLWBM operator
Fig. 2 The values S(A2) for alternative X2 obtained by the ATS-
I2TLWBM operator
Fig. 4 The values S(A4) for alternative X4 obtained by the ATS-
I2TLWBM operator
Fig. 3 The values S(A3) for alternative X3 obtained by the ATS-
I2TLWBM operator
100 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
Fig. 5 Variation of S(Ai) obtained with the ATS-I2TLWBM operator
(p = 1, q [ (0, 10])
Fig. 10 Variation of S(Ai) obtained by the ATS-I2TLWGBM operator
(p = 1, q [ (0, 10])
Fig. 9 The values S(A4) for alternative X4 obtained by the ATS-
I2TLWGBM operatorFig. 6 The values S(A1) for alternative X1 obtained by the ATS-
I2TLWGBM operator
Fig. 7 The values S(A2) for alternative X2 obtained by the ATS-
I2TLWGBM operator
Fig. 8 The values S(A3) for alternative X3 obtained by the ATS-
I2TLWGBM operator
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123
6 Conclusions
In this paper, we investigate the multiple attribute group
decision making (MAGDM) problems with interval-
valued 2-tuple linguistic information. We first study a
further application of Archimedean t-norm and t-con-
orm under linguistic fuzzy environment, and give some
special operational laws for interval-valued linguistic
2-tuples. In particular, we have discussed some prop-
erties of these operations. Based on the new operations,
we have developed several new interval-valued 2-tuple
linguistic aggregation operators, such as ATS-I2TLBM,
ATS-I2TLGBM, and ATS-I2TLWGBM operators. Some
fundamental properties of the developed operators have
been studied. Furthermore, we have used the proposed
operators solve MAGDM problems. Finally, an example
is provided to illustrate that the method is not only
more reasonable but more efficient in practical appli-
cation, because this method captures the interrelation-
ship of the input arguments and considers the degree of
precision of decision information. Operational laws to
other linguistic fuzzy environment will be studied in
our future work.
Acknowledgments The work was supported by National Natural
Science Foundation of China (Nos. 71301001, 71371011, 11426033),
Provincial Natural Science Research Project of Anhui Colleges
(No.KJ2015A379), Higher School Specialized Research Fund for the
Doctoral Program (No.20123401110001), Humanity and Social Sci-
ence Youth Foundation of Ministry of Education (No.
13YJC630092), Anhui Provincial Philosophy and Social Science
Planning Youth Foundation (No. AHSKQ2014D13), The Doctoral
Scientific Research Foundation of Anhui University.
Fig. 11 The value for alternatives X1 obtained by ATS-I2TLWBM
and ATS-I2TLWGBM
Fig. 12 The value for alternatives X2 obtained by ATS-I2TLWBM
and ATS-I2TLWGBM
Fig. 14 The value for alternatives X4 obtained by ATS-I2TLWBM
and ATS-I2TLWGBM
Fig. 13 The value for alternatives X3 obtained by ATS-I2TLWBM
and ATS-I2TLWGBM
102 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
Appendix
The Proof of Theorem 3.1:
(1) According to the Definition 3.1, we know that A�B ¼ D /�1 /ðD�1 ðsi; aiÞÞ þ / ðD�1 ðsk; akÞÞ
� � �;
�D /�1 /ðD�1
� ðsj; ajÞÞ þ /ðD�1 ðsl; alÞÞ�g�. It is
clear that D�1 si; aið Þ�D�1 sj; aj� �
; D�1 sk; akð Þ�D�1 sl; alð Þ; and D�1 : S � 1
2g; 12g
h �! ½0; 1�; / :
0; 1½ � ! ½0;þ1Þ are strictly increasing function,
such that
/ðD�1ðsi; aiÞÞ þ /ðD�1ðsk; akÞÞ�/ðD�1ðsj; ajÞÞþ /ðD�1ðsl; alÞÞ
and
/ D�1 si; aið Þ� �
þ / D�1 sk; akð Þ� �
2 ½0;þ1Þ;/ D�1 sj; aj
� �� �þ / D�1 sl; alð Þ
� �2 ½0;þ1Þ:
Noting that /�1 : ½0;þ1Þ ! 0; 1½ � is also a strictly
increasing function, we have
/�1 / D�1 si; aið Þ� �
þ / D�1 sk; akð Þ� �� �
�/�1
/ D�1 sj; aj� �� �
þ / D�1 sl; alð Þ� �� �
and
/�1 / D�1 si; aið Þ� �
þ / D�1 sk; akð Þ� �� �
;
/�1 / D�1 sj; aj� �� �
þ / D�1 sl; alð Þ� �� �
2 0; 1½ �:
Thus Dð/�1 / D�1 si; aið Þ� �
þ / D�1 sk; akð Þ� �� �
Þ�D /�1 / D�1 sj; aj
� �� �þ / D�1 sl; alð Þ
� �� �� �and
D /�1 /ðD�1ðsi; aiÞÞ þ /ðD�1ðsk; akÞÞ� �� �
;
D /�1 /ðD�1ðsj; ajÞÞ þ /ðD�1ðsl; alÞÞ� �� �
2 S ½� 1
2g;1
2gÞ:
So, we can get
½ðsi; aiÞ; ðsj; ajÞ� � ½ðsk; akÞ; ðsl; alÞ�¼ D /�1 /ðD�1ðsi; aiÞÞ þ /ðD�1ðsk; akÞÞ
� �� �;
�D /�1 /ðD�1ðsj; ajÞÞ þ /ðD�1ðsl; alÞÞ
� �� ��2 X
(2) The proof is similar to that (1), it is omitted here.
(3) From Definition 3.1, we can get
k� A ¼ D /�1 k/ D�1 si; aið Þ� �� � �
;�
D /�1 k/ D�1 sj; aj� �� �� � ��
:
Since D�1 si; aið Þ�D�1 sj; aj� �
;/ : 0; 1½ � ! ½0;þ1Þis a strictly increasing function, then k/ D�1
�si; aið ÞÞ� k/ D�1 sj; aj
� �� �; and k/ D�1 si; aið Þ
� �;
k/ D�1 sj; aj� �� �
2 ½0;þ1Þ where k C 0. Besides,
/�1 : ½0;þ1Þ ! 0; 1½ � is also a strictly increasing
function, we have
/�1 k/ D�1 si; aið Þ� �� �
�/�1 k/ D�1 sj; aj� �� �� �
;
and /�1 k/ D�1 si; aið Þ� �� �
;/�1 k/ D�1 sj; aj� �� �� �
20; 1½ �: Obviously, Dð/�1 k/ D�1 si; aið Þ
� �� ��D
/�1 k/ D�1 sj; aj� �� �� �� �
; and D /�1½k/ðD�1�
ðsi; aiÞÞ�Þ;D /�1½k/ðD�1ðsj; ajÞÞ�� �
2 S � 12g; 12g
h �.
Thus
k� A ¼ D /�1 k/ D�1 si; aið Þ� �� � �
;�
D /�1 k/ D�1 sj; aj� �� �� � ��
2 X:
(4) For any interval-valued 2-tuple linguistic, we get
D�1 si; aið Þ�D�1 sj; aj� �
:. From Definition 3.1, we
have
Ak ¼ D u�1 ku D�1 si; aið Þ� �� �� �
;D u�1 ku D�1 sj; aj� �� �� �� �� �
:
In view of the function u : 0; 1½ � ! 0;þ1½ Þ is strictly
decreasing function, we obtain
kuðD�1ðsi; aiÞÞ� kuðD�1ðsj; ajÞÞ
and ku D�1 si; aið Þ� �
; ku D�1 sj; aj� �� �
2 ½0;þ1ÞCorrespondingly, the inverse function u�1 : 0;þ1½ Þ !
0; 1½ � is also strictly decreasing function, then
u�1 ku D�1 si; aið Þ� �� �
�u�1 ku D�1 sj; aj� �� �� �
;
and u�1 ku D�1 si; aið Þ� �� �
;u�1 ku D�1 sj; aj� �� �� �
2 0; 1½ �:Thus,
D u�1 ku D�1 si; aið Þ� �� �� �
�D u�1 ku D�1 sj; aj� �� �� �� �
andD u�1½kuðD�1ðsi; aiÞÞ�� �
;D u�1½kuðD�1ðsj; ajÞÞ�� �
2 S � 12g; 12g
h �.
Therefore, we have
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 103
123
Ak ¼ D u�1½kuðD�1ðsi; aiÞÞ� �
;D u�1½kuðD�1ðsj; ajÞÞ� �� �
2 X
Combining (1) with (4), we have that such operational
laws are closed and the results of the operation are also
interval-valued 2-tuple linguistic variables in X, which
completes the proof. h
The Proof of Theorem 3.2:
(1) and (2) are easy to be verified, which is omitted;
(3)
k� ðA� BÞ
¼ k�D /�1½/ðD�1ðsi; aiÞÞ þ /ðD�1ðsk; akÞÞ� �
;
D /�1½/ðD�1ðsj; ajÞÞ þ /ðD�1ðsl; alÞÞ� �
" #
¼D /�1 k/ D�1 D /�1 /ðD�1ðsi; aiÞÞ þ /ðD�1ðsk; akÞÞ
� �� �� �� �� � �;
D /�1 k/ D�1 D /�1 /ðD�1ðsj; ajÞÞ þ /ðD�1ðsl; alÞÞ� �� �� �� �� � �
" #
¼D /�1 k½/ðD�1ðsi; aiÞÞ þ /ðD�1ðsk; akÞÞ�
� � �;
D /�1 k½/ðD�1ðsj; ajÞÞ þ /ðD�1ðsl; alÞÞ�� � �
" #
¼D /�1½k/ðD�1ðsi; aiÞÞ þ k/ðD�1ðsk; akÞÞ� �
;
D /�1½k/ðD�1ðsj; ajÞÞ þ k/ðD�1ðsl; alÞÞ� �
" #
¼ ðk� AÞ � ðk� BÞ:
(4)
ðA� BÞk
¼D u�1½uðD�1ðsi; aiÞÞ þ uðD�1ðsk; akÞÞ� �
;
D u�1½uðD�1ðsj; ajÞÞ þ uðD�1ðsl; alÞÞ� �
" #
¼D u�1 ku D�1ðDðu�1½uðD�1ðsi; aiÞÞ þ uðD�1ðsk; akÞÞ�ÞÞ
� �� � �;
D u�1 ku D�1ðDðu�1½uðD�1ðsj; ajÞÞ þ uðD�1ðsl; alÞÞ�ÞÞ� �� � �
" #
¼D u�1 k½uðD�1ðsi; aiÞÞ þ uðD�1ðsk; akÞÞ�
� � �;
D u�1 k½uðD�1ðsj; ajÞÞ þ uðD�1ðsl; alÞÞ�� � �
" #
¼D u�1½kuðD�1ðsi; aiÞÞ þ kuðD�1ðsk; akÞÞ� �
;
D u�1½kuðD�1ðsj; ajÞÞ þ kuðD�1ðsl; alÞÞ� �
" #
¼ Ak � Bk:
Similarly, it is obtained that (5)–(8) hold, which com-
pletes the proof. h
The Proof of Theorem 3.3:
By using mathematical induction on n.
(1) For n = 2, we have
½ðs1; a1Þ; ðs01; a01Þ� � ½ðs2; a2Þ; ðs02; a02Þ�
¼ D /�1½X2i¼1
/ðD�1ðsi; aiÞÞ�( )
;
"
D /�1X2i¼1
/ðD�1ðs0i; a0iÞÞ" #( )#
:
When n = k - 1, k 2 N?, (1) holds, that is
�k�1
i¼1si; aið Þ; ðs0i; a0iÞ
� �¼
D /�1Xk�1
i¼1
/ D�1 si; aið Þ� �" #( )
;
"
D /�1Xk�1
i¼1
/ D�1 s0i; a0i
� �� �" #( )#;
then
�k
i¼1½ðsi; aiÞ; ðs0i; a0iÞ�
¼ D /�1Xk�1
i¼1
/ D�1 si; aið Þ� �" #( )
;D /�1Xk�1
i¼1
/ D�1 s0i; a0i
� �� �" #( );
" #� sk; akð Þ; s0k; a
0k
� �� �
¼
D /�1 / D�1 D /�1Xk�1
i¼1
/ðD�1ðsi; aiÞÞ" # !" # !
þ /ðD�1ðsk; akÞÞ" #( )
;
D /�1 / D�1 D /�1Xk�1
i¼1
/ðD�1ðs0i; a00i ÞÞ" # !" # !
þ /ðD�1ðs0k; a0kÞÞ" #( )
2666664
3777775
¼
D /�1Xk�1
i¼1
/ðD�1ðsi; aiÞÞ !
þ /ðD�1ðsk; akÞÞ" #( )
;
D /�1Xk�1
i¼1
/ðD�1ðs0i; a0iÞÞ !
þ /ðD�1ðs0k; a0kÞÞ" #( )
2666664
3777775
¼ D /�1Xki¼1
/ D�1 si; aið Þ� �" #( )
;D /�1Xki¼1
/ D�1 s0i; a0i
� �� �" #( )" #:
So (1) holds for n = k. Thus (1) holds for all n.
(2) The proof is similar to (1), thus it is omitted. h
The Proof of Theorem 4.1:
Based on the Definition 3.1, we can get
½ðsi; aiÞ; ðs0i; a0iÞ�p � ½ðsj; ajÞ; ðs0j; a0jÞ�
q
¼ D u�1 puðD�1ðsi; aiÞÞ� � �
; u�1 puðD�1ðs0i; a0iÞÞ� � �� �
� D u�1 puðD�1ðsj; ajÞÞ� � �
; u�1 puðD�1ðs0j; a0jÞÞ �n oh i
¼D u�1½puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ� �
;
D u�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ�n o
24
35:
It follows from Theorem 3.3 that
�n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ�p � ½ðsj; ajÞ; ðs0j; a0jÞ�
qÞ
¼
D /�1Xni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D /�1Xni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞh i �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
:
104 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123
So, we can obtain that
1
nðn � 1Þ � �n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ�p � ½ðsj; ajÞ; ðs0j; a0jÞ�
qÞ
0BBBBBB@
1CCCCCCA
26666664
37777775
1pþq
¼
D /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ�� �
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ� �
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
1pþq
¼
D u�1 1
pþ qu D�1 D /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/fu�1½puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ�
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
pþ qu D�1 D /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/fu�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ�
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
¼
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ�� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;;
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ� �
0BBBBBBBB@
1CCCCCCCCA
2666666664
3777777775
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
2666666666666666666666664
3777777777777777777777775
:
Thus, the proof of Theorem 4.1 is completed. h
The Proof of Theorem 4.2:
(1) By Theorem 4.1, it has
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ðu�1½puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ�
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ðu�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ�
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
:
Since Ai ¼ si; aið Þ; s0i; a0i
� �¼� ½ sk; akð Þ; sl; alð Þ
� �; i ¼
1; 2; . . .; n, then
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼D u�1 1
pþ qu /�1 1
nðn� 1Þ nðn� 1Þ/ðu�1½puðD�1ðsk; akÞÞ þ quðD�1ðsk; akÞÞ�Þ� �� �� �� �
;
D u�1 1
pþ qu /�1 1
nðn� 1Þ nðn� 1Þ/ðu�1½puðD�1ðsl; alÞÞ þ quðD�1ðsl; alÞÞ�Þ� �� �� �� �
26664
37775
¼D u�1 1
pþ quðu�1½ðpþ qÞuðD�1ðsk; akÞÞ�Þ
� �� �;
D u�1 1
pþ quðu�1½ðpþ qÞuðD�1ðsl; alÞÞ�Þ
� �� �
26664
37775
¼ sk; akð Þ; sl; alð Þ½ �:
(2) According to the Definition 2.5, we can get
SðAiÞ ¼D�1ðsi; aiÞ þ D�1ðs0i; a0iÞ
2;
SðAþÞ ¼D�1ðmax
iðsi; aiÞÞ þ D�1ðmax
iðs0i; a0iÞÞ
2
SðA�Þ ¼D�1ðmin
iðsi; aiÞÞ þ D�1ðmin
iðs0i; a0iÞÞ
2:
Then, S A�ð Þ� S Aið Þ� S Aþð Þ for all i. Since ATS-
I2TLBM satisfies the idempotency, we have
Aþ ¼ ATS� I2TLBMp;qðAþ;Aþ; . . .;AþÞ andA� ¼ ATS� I2TLBMp;qðA�;A�; . . .;A�Þ:
Besides, u is a strictly decreasing function and / is a
strictly increasing function, we obtain
A� ¼ATS� I2TLBMp;qðA�;A�; . . .;A�Þ
¼D u�1 1
pþqu /�1 1
nðn�1Þnðn�1Þ/ðu�1 puðD�1ðminiðsi;aiÞÞþquðD�1ðmin
iðsi;aiÞÞÞ
� � �� �� �� �;
D u�1 1
pþqu /�1 1
nðn�1Þnðn�1Þ/ðu�1 puðD�1ðminiðs0i;a0ÞÞþquðD�1ðmin
iðs0i;a0ÞÞÞ
� � �� �� �� �
26664
37775
�
D u�1 1
pþqu /�1 1
nðn�1ÞXni 6¼ j
i; j¼ 1
/ðu�1½puðD�1ðsi;aiÞÞþquðD�1ðsj;ajÞÞ�Þ
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D u�1 1
pþqu /�1 1
nðn�1ÞXni 6¼ j
i; j¼ 1
/ðu�1½puðD�1ðs0i;a0iÞÞþquðD�1ðs0j;a0jÞÞ�Þ
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
�D u�1 1
pþqu /�1 1
nðn�1Þnðn�1Þ/ðu�1½puðD�1ðmaxiðsi;aiÞÞÞþquðD�1ðmax
iðsi;aiÞÞÞ�Þ
� �� �� �� �;
D u�1 1
pþqu /�1 1
nðn�1Þnðn�1Þ/ðu�1½puðD�1ðmaxiðs0i;a0iÞÞÞþquðD�1ðmax
iðs0i;a0ÞÞÞ�Þ
� �� �� �� �
26664
37775
¼ATS� I2TLBMp;qðAþ;Aþ; . . .;AþÞ¼Aþ:
Thus, we have A� �ATS� I2TLBMp;qðA1;A2; . . .;
AnÞ�Aþ:(3) According to the Definition 4.1, it has
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼ 1
nðn� 1Þ � �n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ�p � ½ðsj; ajÞ; ðs0j; a0jÞ�
qÞ
0BBBBBB@
1CCCCCCA
26666664
37777775
1pþq
:
If A0i ¼ srðiÞ; arðiÞ
� �; s0rðiÞ; a
0rðiÞ
�h iis any permuta-
tion of Ai ¼ si; aið Þ; s0i; a0i
� �� �ði ¼ 1; 2; . . .; nÞ, then
for any two interval-valued linguistic 2-tuples Ai and
Aj, we have k; l 2 f1; 2; . . .; ng; such that Ai ¼½ðsi; aiÞ; ðs0i; a0iÞ� ¼ ½ðsrðkÞ; arðkÞÞ; ðs0rðkÞ; a0rðkÞÞ� ¼ A0
k
and Aj ¼ ½ðsj; ajÞ; ðs0j; a0jÞ� ¼ ½ðsrðlÞ; arðlÞÞ; ðs0rðlÞ; a0rðlÞÞ�¼ A0
l; then,
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 105
123
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼ 1
nðn� 1Þ � �n
i 6¼ j
i; j ¼ 1
ð½ðsi; aiÞ; ðs0i; a0iÞ�p � ½ðsj; ajÞ; ðs0j; a0jÞ�
qÞ
0BBBBBB@
1CCCCCCA
26666664
37777775
1pþq
¼ 1
nðn� 1Þ � �n
k 6¼ l
k; l ¼ 1
ð½ðsrðkÞ; arðkÞÞ; ðs0rðkÞ; a0rðkÞÞ�p � ½ðsrðlÞ; arðlÞÞ; ðs0rðlÞ; a0rðlÞÞ�
qÞ
0BBBBBB@
1CCCCCCA
26666664
37777775
1pþq
¼ ATS� I2TLBMp;qðA01;A
02; . . .;A
0nÞ:
(4) According to the concept of Archimedean t-norm
and s-norm, we know u is a strictly decreasing
function and / is a strictly increasing function.
Applying Theorem 4.1, we obtain that
ATS� I2TLBMp;qðA1;A2; . . .;AnÞ
¼
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðsi; aiÞÞ þ quðD�1ðsj; ajÞÞ� �� �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞh i �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
�
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðsi ; ai ÞÞ þ quðD�1ðsj ; aj ÞÞh i �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA;
D u�1 1
pþ qu /�1 1
nðn� 1ÞXni 6¼ j
i; j ¼ 1
/ u�1 puðD�1ðs0i ; a0i ÞÞ þ quðD�1ðs0j ; a0j ÞÞh i �
0BBBBBBBB@
1CCCCCCCCA
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
0BBBBBBBB@
1CCCCCCCCA
0BBBBBBBB@
1CCCCCCCCA
2666666666666666666666664
3777777777777777777777775
¼ ATS� I2TLBMp;qðA1;A
2; . . .;A
nÞ:
Thus, the proof of Theorem 4.2 has been finished.h
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Xi Liu is a Ph.D. candidate of
Statistics in the School of
Mathematical Sciences at the
Anhui University. She has con-
tributed several journal articles
to professional journals. Her
current research interests
include decision making theory,
forecasting, information fusion,
fuzzy statistics, and fuzzy
mathematics.
Zhifu Tao is a lecturer of
School of Economics, Anhui
University, China. He received
a PhD degree in School of
Mathematical Sciences from
Anhui University. He has con-
tributed over 10 journal articles
to professional journals such as
Knowledge-based systems and
Applied Soft Computing. His
current research interests
include group decision making,
aggregation operators, and
combined forecasting.
Huayou Chen is a Professor of
School of Mathematical Sci-
ences, Anhui University, China.
He received a Ph.D. degree in
Operational Research from
University of Science Technol-
ogy of China in 2002. He grad-
uated from Nanjing University
for 2 years postdoctoral
research work in 2005. He has
published a book: The Efficient
Theory of Combined Forecast-
ing and Applications (Science
Press, Beijing, 2008) and has
contributed over 120 journal
articles to professional journals, such as Fuzzy Sets and Systems,
Information Sciences, and Group Decision and Negotiation. His
current research interests include information fusion, multi-criteria
decision making, aggregation operators, and combined forecasting.
X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 107
123
Ligang Zhou is an associate
professor of School of Mathe-
matical Sciences, Anhui
University. He received a PhD
degree in operations research
from Anhui University in 2013.
He has contributed over 40
journal articles to professional
journals, such as Fuzzy Sets and
Systems, Applied Mathematical
Modelling, Applied Soft Com-
puting, Group Decision and
Negotiation, and Expert Sys-
tems with Applications. His
current research interests
include group decision making, aggregation operators, and combined
forecasting.
108 International Journal of Fuzzy Systems, Vol. 19, No. 1, February 2017
123