a new interval-valued 2-tuple linguistic bonferroni mean...

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

[email protected]

Xi Liu

[email protected]

Zhifu Tao

[email protected]

Ligang Zhou

[email protected]

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

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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.

X. Liu et al.: A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application... 87

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�


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.


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,


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,


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;


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,


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

� �� " #( )#:


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-


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


i; j ¼ 1

/ u�1½puðD�1ðs0i; a0iÞÞ þ quðD�1ðs0j; a0jÞÞ� �

0BBBBBBBB@

1CCCCCCCCA

2666666664

3777777775

0BBBBBBBB@

1CCCCCCCCA

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

2666666666666666666666664

3777777777777777777777775

:

ð16Þ


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


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


i; j ¼ 1

/ u�1 puðD�1ðsi; aiÞÞ� ��

0BBBBBBBB@

1CCCCCCCCA

2666666664

3777777775

0BBBBBBBB@

1CCCCCCCCA

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;;

D u�1 1

pu /�1 1


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


� �;


based interval-valued 2-tuple linguistic mean (ATS-

I2TLM) operator.


123

3. If p = 2, q ? 0, then from the ATS-I2TLBMp,0, it is

obtained that


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:


¼

D u�1 1

2u /�1 1


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


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.


� �� 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:


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


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


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


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:


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:


� �� 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


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


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


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


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


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


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


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


� �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


¼

D /�1 1

2/ u�1 1


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


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

;


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:


¼

D u�1 1

1þ 1u /�1 1


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


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:


� �� 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


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

� �


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�

26664

37775

C1 C2 C3 C4

E2 ¼

X1

X2

X3

X4


26664

37775

C1 C2 C3 C4

E3 ¼

X1

X2

X3

X4


26664

37775

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�

26664

37775

C1 C2 C3 C4

A2 ¼

X1

X2

X3

X4


26664

37775

C1 C2 C3 C4

A3 ¼

X1

X2

X3

X4


26664

37775

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


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�

26664

37775

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


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


I2TLWBM operator


I2TLWBM operator


I2TLWBM operator


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])


I2TLWGBM operatorFig. 6 The values S(A1) for alternative X1 obtained by the ATS-

I2TLWGBM operator


I2TLWGBM operator


I2TLWGBM operator


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


and ATS-I2TLWGBM


and ATS-I2TLWGBM


and ATS-I2TLWGBM


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

� �� 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


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


(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


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


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


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


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

:


123

So, we can obtain that

1

nðn � 1Þ � �n

i 6¼ j

i; j ¼ 1


qÞ

0BBBBBB@

1CCCCCCA

26666664

37777775

1pþq

¼

D /�1 1


i; j ¼ 1


2666666664

3777777775

0BBBBBBBB@

1CCCCCCCCA

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;;

D /�1 1


i; j ¼ 1


2666666664

3777777775

0BBBBBBBB@

1CCCCCCCCA

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

2666666666666666666666664

3777777777777777777777775

1pþq

¼

D u�1 1

pþ qu D�1 D /�1 1


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


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


i; j ¼ 1


0BBBBBBBB@

1CCCCCCCCA

2666666664

3777777775

0BBBBBBBB@

1CCCCCCCCA

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;;

D u�1 1

pþ qu /�1 1


i; j ¼ 1


0BBBBBBBB@

1CCCCCCCCA

2666666664

3777777775

0BBBBBBBB@

1CCCCCCCCA

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

2666666666666666666666664

3777777777777777777777775

:

Thus, the proof of Theorem 4.1 is completed. h


(1) By Theorem 4.1, it has


¼

D u�1 1

pþ qu /�1 1


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


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


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


¼ 1


i 6¼ j

i; j ¼ 1


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,


123


¼ 1


i 6¼ j

i; j ¼ 1


qÞ

0BBBBBB@

1CCCCCCA

26666664

37777775

1pþq

¼ 1


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


¼

D u�1 1

pþ qu /�1 1


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


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


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


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


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.


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


include group decision making, aggregation operators, and combined

forecasting.


123

a new interval-valued 2-tuple linguistic bonferroni mean...

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