The new similarity measure and distance measure between hesitant fuzzy linguistic term sets and their application in multi-criteria decision making

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Journal of Intelligent & Fuzzy Systems 37 (2019) 995–1006
DOI:10.3233/JIFS-181886
IOS Press
995
The new similarity measure and distance
measure between hesitant fuzzy linguistic
term sets and their application in
multi-criteria decision making
Liu Donghaia,∗, Liu Yuanyuanaand Chen Xiaohongb,c
aDepartment of Statistics, Hunan University of Science and Technology, Hunan, China
bMobile E-business Collaborative Innovation Center of Hunan Province, Hunan University of Commerce,
Hunan, China
cBusiness School, Central South university, Hunan, China
Abstract. Considering that the existing cosine similarity measure between hesitant fuzzy linguistic term sets(HFLTSs) has an
impediment as it does not satisfy the axiom of similarity measure, we propose a new similarity measure of HFLTSs in the paper,
which is constructed based on the existing cosine similarity measure and Euclidean distance measure of HFLTSs. Then the
corresponding distance measure of HFLTSs is obtained according to the relationship between the similarity measure and the
distance measure. Furthermore, we develop the TOPSIS method to the proposed distance measure in hesitant fuzzy linguistic
decision environment and apply the closeness coefficients to rank the alternatives. The main advantage of the proposed
method is that it not only considers the distance measure from the point view of algebra and geometry but also overcomes
the disadvantage of the existing cosine similarity measure. Finally, an example is provided to illustrate the feasibility of the
proposed method and some comparative analyses are given to show its efficiency.
Keywords: Hesitant fuzzy linguistic term set, similarity measure, distance measure, TOPSIS
1. Introduction
Multi-criteria decision making is a series of proce-
dures in a specific order to help the decision makers
find the optimal alternative. Due to the complex-
ity of the decision making environment, there is
some uncertainty in multi-criteria decision making
problems. In 1965, Zadeh [1] proposed a fuzzy set
A={(xj,μ
A(xj))|xj∈X)}to handle the imprecise
and vague information in decision making problems,
∗Corresponding author. LIU Donghai, Department of Statistics,
Hunan University of Science and Technology, Hunan, China. Tel.:
086-0731-58290977; E-mail: dhliu@hnust.edu.cn.
where μA(xj) is the membership degree of xj∈Xto
the set A.
Since the fuzzy set was proposed, it has received a
lot of attention in many fields, such as pattern recog-
nition, medical diagnosis and so on [2–5]. However,
in some practical decision problems, it is difficult
to describe the membership value of an element
with a single value. For example, two experts are
invited to evaluate the profitability of some airline,
one expert believes that the possibility of its profit
is 0.7, the other expert believes that the possibil-
ity of its profit is 0.6. The airline profit evaluation
cannot be represented by the fuzzy set at this time.
In order to describe the relevant information better,
ISSN 1064-1246/19/$35.00 © 2019 – IOS Press and the authors. All rights reserved

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996 L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets
Torra [6] proposed the concept of hesitant fuzzy set
(HFS), which contains all possible memberships of
an element in [0,1]. Then the airline profit evalu-
ation can be represented by a HFS H={0.6,0.7},
where 0.6 and 0.7 represent its membership degree,
respectively. Since the HFS was proposed, it has
attracted a lot of attention from many researchers
([7–11]). For example, Xu et al. [7] defined the dis-
tance and correlation measures between HFSs under
the assumption that two hesitant fuzzy elements have
the same length and applied these distance measures
to multi-criteria decision making problems. Farha-
dinia [8] investigated the relationship between the
entropy, the similarity measure and the distance mea-
sure for HFSs and interval-valued hesitant fuzzy sets
(IVHFSs), and two clustering algorithms are devel-
oped under a hesitant fuzzy environment in which
indices of similarity measures of HFSs and IVHFSs
are applied in data analysis and classification. Zhang
et al. [10] defined the best additive consistency index,
the worst additive consistency index and the average
additive consistency index to measure the consistency
level of hesitant fuzzy preference relation. Garg et al.
[11] proposed Maclaurin symmetric mean aggrega-
tion operators based on t-norm operations between
dual hesitant fuzzy soft set and applied these opera-
tors to multi-criteria decision making problems.
But in some practical decision making problems,
many criteria should be assessed in a qualitative
form. For example, when the car design is evaluated
online by a customer, he/she thinks that the design
is “very good”, it is suitable to be evaluated in
a linguistic term set(LTS) because the linguistic
evaluation is very close to human’s cognitive
process. Thus, Zadeh ([12–14]) proposed the LTS to
describe the corresponding decision information, the
general seven terms of LTS can be expressed as S=
{s0:very poor, s1:poor, s2:a little poor, s3:
medium, s4:a littlegood, s5good, s6:very good}
So the car design evaluation can be represented
as {s6}. However, if the car design is evaluated
online by many customers, some of them think the
design is good, the others think its design is just
medium, then the car design evaluation cannot be
represented by the LTS with a single value. In order
to express the above information, Rodr´
iguez et al.
[15] proposed the hesitant fuzzy linguistic term
set (HFLTS) based on HFS and LTS. Then the car
design evaluation can be represented as {s2,s
3},
where s3,s
5represent the possible membership
degrees of the car design, they are called hesitant
fuzzy linguistic elements(HFLEs). The HFLTSs are
highly useful in handling situations where people are
hesitant in providing their preferences with regard
to objects in a decision-making process, more and
more multi-attribute decision making theories and
methods under hesitant fuzzy linguistic environment
have been developed ([15–19]). For example,
Dong et al. [16] proposed a novel distance-based
consensus measure and developed an optimization-
based consensus model in the hesitant linguistic
group decision making problems. Yu et al. [17]
developed a new method to deal with multi-criteria
group decision making problems with unbalanced
HFLTSs by considering the psychological behavior
of decision makers. Liao et al. [18] proposed the
hesitant fuzzy linguistic preference utility-TOPSIS
method to select the best fire rescue plan. Yu et
al. [19] introduced some aggregation operators of
HFLTS and proposed a decision method to evaluate
the meteorological disaster that occurred in China.
On the other hand, similarity measure and dis-
tance measure are important topics in multi-criteria
decision making problems, which can describe the
similarity degree and difference between two differ-
ent alternatives. They have been widely studied in
the past ten years ([20–34]). For example, Song et al.
[20] took account into the similarity measure between
intuitionistic fuzzy sets(IFSs) and proposed the cor-
responding distance measure between intuitionistic
fuzzy belief functions. Liao et al. [21] presented
a family of similarity measures and distance mea-
sures between HFLTSs and applied them to rank
alternatives in multi-criteria decision making prob-
lems. Lee et al. [22] presented a similarity measure
between HFLTSs based on likelihood relations. There
still have a lot of related studies consider the sim-
ilarity measure between HFLTSs, we can refer to
[23–26]. Furthermore, the cosine similarity measure
is also a significant similarity measure, which is
defined as the inner product of two vectors divided
by the product of their lengths [27], some schol-
ars studied the cosine similarity measure ([28–31]).
For example, Ye [28] introduced a weighted cosine
similarity measure between intuitionistic fuzzy sets
and applied it to pattern recognition and medical
diagnosis. Furthermore, Ye [29] presented the cosine
similarity measure between interval-valued fuzzy
sets with risk preference and altered its decision
making method depending on decision makers’ pref-
erence. Liao et al. [30] defined the cosine similarity
measure between HFLTSs and extended the TOPSIS
method and VIKOR method to the cosine distance
measure. Considering the interaction between the

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L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets 997
pairs of the membership degrees, Garg [31] proposed
an improved cosine similarity measure between IFSs
and applied it to decision making process. Other
studies on distance measures can refer to [32–34].
Distance measure is often used with TOPSIS method
in multi-criteria decision making problems. Hwang
et al. [35] originally introduced the TOPSIS method
to solve the multi-criteria decision making problem,
the basic notion of the TOPSIS is that the chosen
alternative should have the shortest distance from the
positive ideal solution and the farthest distance from
the negative ideal solution. A number of papers are
devoted to fuzzy TOPSIS method in [36–40].
If two HFLTSs are not equal, the value of the cosine
similarity measure between HFLTSs(Liao et al. [30])
is 1 (the example can be seen in section 3). That is
to say, the cosine similarity measure defined by them
does not satisfy the axiom of similarity measure, it
is not a regular similarity measure. This flaw will
certainly limit its applicability and lead to the initial
information distortion in decision process. Motivated
by this, the objective of the paper is to propose an
innovative method to construct a new similarity mea-
sure of HFLTSs, which not only includes the cosine
similarity measure of HFLTSs in Liao et al. [30] and
the Euclidean distance measure of HFLTSs but also
satisfies the axiom of similarity measure. As a result,
the contributions of the paper are summarized as fol-
lows:
(1) The proposed new similarity measure
improves the cosine similarity measure of
HFLTSs in Liao et al. [30] and overcomes its
disadvantage.
(2) The TOPSIS method is developed to the pro-
posed distance measure, which could improve
the adaptability of HFLTSs in practice and the
effectiveness of processing decision informa-
tion.
The rest of the paper is organized as follows: In
Section 2, the concepts of HFS, LTS, HFLTS and
some related knowledge about HFLTSs are briefly
reviewed. In Section 3, we propose an innovative
method to construct a new similarity measure and dis-
tance measure of HFLTSs, the relevant properties are
also discussed. In Section 4, we develop the TOPSIS
method to the proposed distance measure in hesitant
fuzzy linguistic environment and give a multi-criteria
decision making method. In Section 5, a practical
example is given to illustrate the feasibility of the pro-
posed method and some comparative analyses with
other methods are conducted to show its effective-
ness. Finally, the conclusions and recommendations
for future research are presented in Section 6.
2. Preliminaries
In this section, we review and discuss some related
knowledge, including HFS, LTS, HFLTS, the score
function of HFLTS. Some existing distance measures
and similarity measures between HFLTSs are also
given. Throughout this paper, let X={x1,x
2, ..., xn}
be a discrete and finite discourse set.
2.1. Hesitant fuzzy set
In practical fuzzy decision making problems, the
HFS provides a better representation of reality and
uncertainty to express the preferences of the deci-
sion makers. Torra [6] firstly introduce the concept
of HFS.
Definition 1. [6] Let X={x1,x
2, ..., xn}beafixed
set, a HFS Eon Xis defined as:
E={(xj,e(xj))|xj∈X},
where e(xj) is a set of some values between 0 and 1,
denoting the possible membership degree of xj∈X
to the set E. Furthermore, we call e(xj) a hesitant
fuzzy element.
2.2. Linguistic term set
LTS has been studied in depth and applied to
many fields. Some extension forms of LTSs have
been developed for handling more complicated multi-
criteria decision making problems.
Definition 2. [41, 42] Let S={si|i=
−t, ..., −1,0,1, ..., t}be a finite and totally
ordered discrete linguistic term set, where siis a
linguistic variable, tis a positive integer, it satisfies
the following properties:
(1) The set is ordered: si≤sjif i≤j;
max(si,s
j)=siif si≥sj;min(si,s
j)=siif si≤sj;
The negation operator is defined: neg(si)=sjsat-
isfying with i+j=2t.
2.3. Hesitant fuzzy linguistic term set
HFLTS was used to deal with the situations where
decision makers think of several possible linguistic
values than a single linguistic term for an alternative,

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998 L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets
etc. It is more convenient to reflect the decision-
makers’ preferences.
Definition 3. [15] Given a fixed set X, assume S=
{si|i=−t, ..., −1,0,1, .., t}be a LTS, then a HFLTS
on Xcan be defined as:
Hs={(xj,h
s(xj))|xj∈X},
where hs(xj)⊆srepresents the possible membership
degrees of the element xj∈X. For convenience, the
element of hs(xj) is called hesitant fuzzy linguistic
element (HFLE).
For two HFLTSs H1
sand H2
s, if the numbers of
linguistic terms in H1
sand H2
sare not equal, it is
difficult to calculate the similarity measure between
H1
sand H2
s. In order to deal with this situation, we
assume L=max{L1,L
2}, where L1,L
2represent
the number of elements in H1
sand H2
srespectively.
Zhu et al. [43] proposed the rules of regulation:
the set of fewer number of elements is extended by
adding the linguistic term ¯
Sδ1(xj)=S+
δ(xj)+(1 −
ζ)S−
δ(xj)(0 ≤ζ≤1) until they have the same num-
ber of elements, where S+
δ(xj)=maxLk
l=L1{Sδ1(xj)},
S−
δ(xj)=minLk
l=L1{Sδ1(xj)}j=1,2, ..., n). In prac-
tical decision problems, the optimized parameter is
related to the risk preference of the decision mak-
ers. The optimists want to add the maximum value
S+
δ(xj), while the pessimists want to add the min-
imum value S−
δ(xj). In this paper, we assume that
the maximum element is added to the set with fewer
number of elements.
The score function of HFLTSs is defined as fol-
lows:
Definition 4. [44] Let S={si|i=−t, ..., −1,
0,1, .., t}be a LTS, Hs={Sδk
1(xj)|Sδk
1(xj)∈S}
(L=1,2, ..., Lj,j =1,,2, ..., n) be the HFLTS on
X, then the score function of HSis:
F(Hs)=1
n
n

j=1
¯
δ(xj)
−n
j=11
LjLj
l=1δ1(xj)−¯
δ(xj)2
Var(2t),
where ¯
δ=1
Lj
Lj

l=1
δl(xj), Var(2t)=2t
i=0(i−t)2
2t+1.
Lemma 1. [44] For two HFLTSs H1
sand H2
s,
the comparison rules between them are defined as
follows:
(1) H1
s>H
2
sif and only if F(H1
s)>F(H2
s);
(2) H1
s=H2
sif and only if F(H1
s)=F(H2
s).
2.4. Existing distance measures and similarity
measures between hesitant fuzzy linguistic
term sets
Distance and similarity measures are the two most
important measures for HFLTSs, they are the base
of the TOPSIS method. Here we give the Euclidean
distance measure as follows.
Definition 5. [21] Given a fixed set X, assume that
S={si|i=−t, ..., −1,0,1, .., t}beaLTS.Forany
two HFLTSs H1
s={(xj,h
1
s(xj))|xj∈} and H2
s=
{(xj,h
2
s(xj))|xj∈}(j=1,2, .., n), where hk
s(xj)=
{Sδk
1(xj)|Sδk
1(xj)S,l =1,2, ..., Lj,k =1,2},isthe
maximum number of linguistic terms in h1
s(xj)
and h2
s(xj), assume the weight of xjis ωj(j=
1,2, ..., n), then the weighted Euclidean distance
measure between H1
sand H2
scan be defined as:
DωHFL(H1
s,H2
s)=
⎛
⎝
n

j=1
ωj
Lj
Lj

l=1δ1
l(xj)−δ2
l(xj)
2t+12⎞
⎠
1
2
(1)
Remark 1. For all j=1,2, ..., n,ifωj=1
n,
then the weighted Euclidean distance measure
DωHFL(H1
s,H2
s) is reduced to the normalized
Euclidean distance measure DHFL(H1
s,H2
s):
DHFL(H1
s,H2
s)=
⎛
⎝
1
n⎛
⎝
n

j=1
1
Lj
Lj

l=1δ1
l(xj)−δ2
l(xj)
2t+12⎞
⎠⎞
⎠
1
2
.
Liao et al. [30] defined a cosine similarity measure
between HFLTSs as follows:
Definition 6. [30] Given a fixed set X, assume
S={si|i=−t, ..., −1,0,1, .., t}beaLTS.Forany
two HFLTSs H1
s={(xj,h
1
s(xj))|xj∈} and H2
s=
{(xj,h
2
s(xj))|xj∈}(j=1,2, .., n), where hk
s(xj)=
{Sδk
1(xj)|Sδk
1(xj)S,l =1,2, ..., Lj,k =1,2},isthe
maximum number of linguistic terms in h1
s(xj) and
h2
s(xj), assume that the weight of different element
xjis ωj(j=1,2, ..., n), then the weighted cosine
similarity measure can be defined as:

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L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets 999
CosωHFL(H1
s,H2
s)=n
j=1ωj
LjLj
l=1δ1
l(xj)
2t+1.δ2
l(xj)
2t+1
n
j=1ωj
LjLj
l=1δ1
l(xj)
2t+12.n
j=1ωj
LjLj
l=1δ2
l(xj)
2t+121
2
(2)
Remark 2. For all j=1,2, ..., n,ifωj=1
n,
then the weighted cosine similarity measure
CosωHFL(H1
s,H2
s) is reduced to the normalized
cosine similarity measure CosHFL(H1
s,H2
s):
CosHFL(H1
s,H2
s)=n
j=11
LjLj
l=1δ1
l(xj)
2t+1.δ2
l(xj)
2t+1
n
j=11
LjLj
l=1δ1
l(xj)
2t+12.n
j=11
LjLj
l=1δ2
l(xj)
2t+121
2
(3)
3. The new similarity measure and distance
measure between hesitant fuzzy linguistic
term sets
In this section, we define a new similarity measure
of HFLTSs based on the weighted cosine similar-
ity measure CosωHFL and the weighted Euclidean
distance measure DωHFL.
It is already known that the regular similarity mea-
sure should satisfy the following Lemma 2.
Lemma 2. [21] Let S={si|i=−t, ..., −1,0,1, .., t}
be a LTS, H1
sand H2
sbe any two HFLTSs, if the
similarity measure S(H1
s,H2
s)satisfies the following
properties:
(1) 0 ≤S(H1
s,H2
s)≤1;
(2) S(H1
s,H2
s)=1if and only if H1
s=H2
s;
(3) S(H1
s,H2
s)=S(H1
s,H2
s);
then the similarity measure S(H1
s,H2
s)is a regular
similarity measure.
The cosine similarity measure CosωHFL proposed
by Liao et al. [30] is not a regular similarity measure,
we can see from the following Example 1.
Example 1. Two experts are invited to evaluate
the car design, they provide their evalua-
tions with HFLTSs, for some given LTS S=
{s−3:very poor, s−2:poor, s−1:a little poor, s0:
medium, s1:a little good, s2:good, s3:
very good}, the car design evaluations can be
represented as HFLTSs H1
s={s−1,s
−2}and H2
s=
{s−1,s
−2}, respectively.
According to Lemma 1, it is already known
H1
s/=H2
s. Using formula (3) to calculate the
similarity measure between H1
sand H2
s,wehave
CosHFL(H1
s,H2
s)=1. That is to say, the similarity
measure CosHFL does not satisfy the property (2) in
Lemma 2, it is not a regular similarity measure. If it
is applied in multi-criteria decision making problem,
it may lead to the decision information distortion in
decision process. Motivated by this, we propose an
innovative method to construct a similarity measure
between HFLTSs, which is based on the cosine sim-
ilarity measure CosHFL and the Euclidean distance
measure DHFL. We also know that the proposed new
similarity measure overcomes the disadvantage of the
similarity measure CosHFL defined in Liao et al. [30].
3.1. Definitions and properties
Definition 7. Given a fixed set X, assume S=
{si|i=−t, ..., −1,0,1, .., t}be a LTS. For any
two HFLTSs H1
s={(xj,h
1
s(xj))|xj∈} and H2
s=
{(xj,h
2
s(xj))|xj∈}(j=1,2, .., n), where hk
s(xj)=
{Sδk
1(xj)|Sδk
1(xj)S,l =1,2, ..., Lj,k =1,2},isthe
maximum number of linguistic terms in h1
s(xj) and
h2
s(xj), the new similarity measure S∗
HFL(H1
s,H2
s)
can be defined as:
S∗
HFL(H1
s,H2
s)=
1
2[CosHFL(H1
s,H2
s)+1−DHFL(H1
s,H2
s),
where

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1000 L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets
CosHFL(H1
s,H2
s)=n
j=11
LjLj
l=1δ1
l(xj)
2t+1.δ2
l(xj)
2t+1
n
j=11
LjLj
l=1δ1
l(xj)
2t+12.n
j=11
LjLj
l=1δ2
l(xj)
2t+121
2
,
DHFL(H1
s,H2
s)=⎛
⎝
1
n⎛
⎝
n

j=1
1
Lj
Lj

l=1δ1
l(xj)−δ2
l(xj)
2t+12⎞
⎠⎞
⎠
1
2
.
The similarity measure S∗
HFL(H1
s,H2
s) is defined
based on the cosine similarity measure CosHFL and
Euclidean distance measure DHFL, it deal with the
decision information not only from the point view of
geometry but also from the point view of algebra. It
is more efficient to solve the multi-criteria decision
problems in hesitant fuzzy decision environment than
the existing similarity measures.
Theorem 1. The similarity measure S∗
HFL(H1
s,H2
s)
is a regular similarity measure.
Proof: According to Lemma 2, if the similarity mea-
sure S∗
HFL(H1
s,H2
s) satisfies all properties in Lemma
2, then it is a regular similarity measure. We can prove
it by the following steps.
(1) CosHFL(H1
s,H2
s) can be regarded as the exten-
sion of cosine function (the inner product of
two vectors divided by the product of their
lengths), then 0 ≤CosHFL(H1
s,H2
s)≤1.
Because DωHFL(H1
s,H2
s) is an Euclidean
distance measure, we have 0 ≤1
−DHFL(H1
s,H2
s)≤1 then 0 ≤S∗
HFL(H1
s,
H2
s)≤1.
(2) If H1
s=H2
s,wehaveCos
HFL(H1
s,H2
s)=1,
DHFL(H1
s,H2
s)=0, then S∗
HFL(H1
s,H2
s)=
1. On the other hand, when S∗
HFL(H1
s,H2
s)=
1, we have CosHFL(H1
s,H2
s)+
1−DHFL(H1
s,H2
s)=2, that is
CosHFL(H1
s,H2
s)=1+DHFL(H1
s,H2
s)
. Because 0 ≤CosHFL(H1
s,H2
s)≤1 and
DωHFL(H1
s,H2
s)≥0 should exist at the same
time, so they must have DωHFL(H1
s,H2
s)=0
and CosHFL(H1
s,H2
s)=1. 
When CosHFL(H1
s,H2
s)=1, H1
s=kH2
sis
obtained, where kis a constant; Furthermore,
when,wehaveH1
s=H2
s. That is to say, if
S∗
HFL(H1
s,H2
s)=1, we have H1
s=H2
s.
So S∗
HFL(H1
s,H2
s)=1 if and only if H1
s=H2
s.
(3) Because
CosHFL(H1
s,H2
s)=CosHFL(H1
s,H2
s),
DHFL(H1
s,H2
s)=DHFL(H1
s,H2
s),
then S∗
HFL(H1
s,H2
s)=S∗
HFL(H1
s,H2
s) can be
obtained easily.
Remark 3. If the similarity measure S∗
HFL(H1
s,H2
s)
is a regular similarity measure, we can obtain a new
distance measure D∗
HFL(H1
s,H2
s) based on the rela-
tionship between the distance measure and similarity
measure, which can be given as follows:
D∗
HFL(H1
s,H2
s)=1−S∗
HFL(H1
s,H2
s),
where
S∗
HFL(H1
s,H2
s)=
1
2(CosHFL(H1
s,H2
s)+1−DHFL(H1
s,H2
s))
Theorem 2. The distance measure D∗
HFL(H1
s,H2
s)
satisfies the following properties:
(1) 0 ≤D∗
HFL(H1
s,H2
s)≤1;
(2) D∗
HFL(H1
s,H2
s)=0 if and only if H1
s=H2
s;
(3) D∗
HFL(H1
s,H2
s)=D∗
HFL(H1
s,H2
s)
Proof. Properties (1) and (3) are obvious, we only
give the proof of property (2) here. 
If H1
s=H2
s,wehaveS∗
HFL(H1
s,H2
s)=1, then
D∗
HFL(H1
s,H2
s)=1−S∗
HFL(H1
s,H2
s)=1.
On the other hand, when D∗
HFL(H1
s,H2
s)=0, we
have S∗
HFL(H1
s,H2
s)=1−D∗
HFL(H1
s,H2
s)=1.
Because S∗
HFL(H1
s,H2
s) is a regular similarity mea-
sure, according to Lemma 2, we have H1
s=H2
s.
Thus D∗
HFL(H1
s,H2
s)=0 if and only if H1
s=H2
s.

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L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets 1001
Next we give the definition about the weighted
similarity measure S∗
HFL(H1
s,H2
s) as follows:
Definition 8. Given a fixed set X, assume S=
{si|i=−t, ..., −1,0,1, .., t}beaLTS.Forany
two HFLTSs H1
s={(xj,h
1
s(xj))|xj∈} and H2
s=
{(xj,h
2
s(xj))|xj∈}(j=1,2, .., n), where hk
s(xj)=
{Sδk
1(xj)|Sδk
1(xj)S,l =1,2, ..., Lj,k =1,2}is the
maximum number of linguistic terms in h1
s(xj)
and h2
s(xj), the associated weight vector ω=
(ω1,ω
2, ..., ωn), satisfying with n
j=1ωj=1(0 ≤
ωj≤1), then the weighted similarity measure
S∗
HFL(H1
s,H2
s) can be defined as:
S∗
HFL(H1
s,H2
s)=
1
2(CosHFL(H1
s,H2
s)+1−DHFL(H1
s,H2
s))
Theorem 3. The weighted similarity measure
S∗
HFL(H1
s,H2
s)is a regular similarity measure.
Proof . The proof is similar to Theorem 1, we omit it
here. 
Remark 4. Similar to Remark 3, if we assume that
the weight of element xjis ωj=1(0 ≤ωj≤1) ,
and satisfying with n
j=1ωj=1(0 ≤ωj≤1), then
the weighted distance measure D∗
HFL(H1
s,H2
s) can
be given as follows:
D∗
ωHFL(H1
s,H2
s)=1−S∗
ωHFL(H1
s,H2
s)
Theorem 4. The weighted distance measure
D∗
HFL(H1
s,H2
s)also satisfies the following proper-
ties:
(1) 0 ≤D∗
ωHFL(H1
s,H2
s)≤0;
(2) D∗
ωHFL(H1
s,H2
s)=0 if and only if H1
s=H2
s;
(3) D∗
ωHFL(H1
s,H2
s)=D∗
ωHFL(H1
s,H2
s).
Proof. The proof is similar to Theorem 2, we omit it
here. 
Remark 5. For all j=1,2, ..., n), if ωj=1
n, then
the weighted distance measure D∗
HFL(H1
s,H2
s) and
the weighted similarity measure S∗
HFL(H1
s,H2
s) are
reduced to the distance measure D∗
HFL(H1
s,H2
s) and
the similarity measure S∗
HFL(H1
s,H2
s) respectively.
3.2. Comparison with the existing similarity
measure
In order to illustrate the advantage of the pro-
posed similarity measure S∗
HFL(H1
s,H2
s), a numerical
example is provided to compare with the similarity
measure CosHFL proposed by Liao et al. [30].
Example 2. Assume that there exist three patterns
A1,A
2,A
3, and the unknown pattern B. The fol-
lowing three factors C1: the depth of colour, C2: the
shape,C3: the size are considered. The factors are
expressed with different LTSs, which are given as
follows respectively:
S
1={s−3:very light, s−2:light,
s−1:alittle light, s0:medium, s1:a little
dark, s2:dark, s3:very dark}
S
2={s−3:very round, s−2:round,
s−1:alittle round, s0:medium, s1:a little
square, s2:square, s3:very squre},
S
3={s−3:very small, s−2:small, s−1:
alittle small, s0:medium, s1:a little l arg e, s2:
larg e, s3:very l arg e}.
The concrete evaluations about each pattern are
given in form of HFLTSs, which can be represented
as:
A1={(C1,s
1,s
1),(C2,s
1),(C3,s
2,s
3)};
A2={(C1,s
−1),(C2,s
−1,s
−2),(C3,s
−3)};
A3={(C1,s
2,s
3),(C2,s
1),(C3,s
0)},
the unknown pattern
B={(C1,s
−1,s
−2),(C2,s
−1,),(C3,s
−2,s
−3)},
now we want to make sure which one that the pattern
Bbelongs to, so we calculate the similarity measure
between Band Ai(i=1,2,3), respectively.
At first, we apply the similarity measure CosHFL
proposed by Liao et al. [30] to calculate it, we can get
CosHFL(A2,B)=0.9092,CosHFL(A2,B)=0.5196
Then the pattern Bbelongs to A1. Intuitively,
the pattern Bcannot belong to A1. The reason that
CosHFL(A1,B)=1 is the subscripts of the linguis-
tic terms A1and Bhave the linear relationship. But
if we use the similarity measure S∗
HFL(A1,B)(i=
1,2,3) proposed in this paper to calculate it,
we have S∗
HFL(A1,B)=0.7392 ,S∗
HFL(A2,B)=
0.9041, S∗
HFL(A1,B)=0.5436. It can be observed
that the pattern Bbelongs to A2the result is more

AUTHOR COPY
1002 L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets
reasonable intuitively. So we can say that the similar-
ity measure S∗
HFL shows a better performance than
the similarity measure CosHFL.
4. The TOPSIS method with proposed
distance measure
In this section, we extend the TOPSIS [45]
method to the proposed distance measure D∗
ωHFL
between HFLTSs. For a multi-criteria decision mak-
ing problem, the experts decide to choose the
best one from malternatives according to ncri-
teria. Let S={si|i=−t, ..., −1,0,1, .., t}be a
LTS,H={H1,H
2, ..., Hm}be a set of alternatives
and C={C1,C
2, ..., Cm}be a set of criteria. For
the alternative Hiwith respect to the criterion
Cj, the experts provide their evaluations repre-
sented by HFLTSs Hij
s={Sij
δ1|l=1,2, ..., Lj}(i=
1,2, ..., m;j=1,2, ..., n), ωj(j=1,2, ..., n)isthe
corresponding weight of criteria satisfying with 0 ≤
ωj≤1(j=1,2, ..., n) and
n

j=1
ωj=1, the decision
matrix Hwith hesitant fuzzy linguistic information
can be given as follows:
H=⎛
⎜
⎜
⎜
⎜
⎜
⎝
H11
sH11
s··· H1n
s
H21
sH11
s··· H2n
s
.
.
..
.
.....
.
.
Hm1
sHm2
s··· Hmn
s
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,
where Hij
s(i=1,2, ..., m;j=1,2, ..., n) are
HFLTSs.
In the following, we present the steps of the devel-
oped TOPSIS method in hesitant fuzzy linguistic
environment for multi-criteria decision making prob-
lems, which are given as follows:
Step 1. Normalize the decision matrix H, for the
benefit-type criteria, we need not do anything; for
the cost-type criteria, we should use the operator
neg(si)=si(i+j=2t) to normalize the decision
matrix.
Step 2. Find the hesitant fuzzy linguistic posi-
tive ideal solution H+={H1+
s,H2+
s, ..., H n+
s}
and hesitant fuzzy linguistic negative ideal
solution H−={H1−
s,H2−
s, ..., H n−
s}.For
i=1,2, ..., m;j=1,2, ..., n), the two type of
ideal solutions are given as follows:
Hj+
s=m
max
i=1Hij +
s},Hj−
s=
m
min
i=1Hij −
s}
Under the criteria Cj(j=1,2, ..., n), we can get
the value of F(Hij
s)(i=1,2, ..., m) by the score
function in Definition 3. According to Theorem
1, the order relationship between HFLTSs can be
given as follows: if F(H1
s)>F(H2
s), we have H1
s>
H2
s. So the ideal solution Hj+
sand Hj−
scan be
obtained.
Step 3. Calculate the separation of each alterna-
tive between the hesitant fuzzy linguistic positive
ideal solution H+={H1+
s,H2+
s, ..., H n+
s}and hes-
itant fuzzy linguistic negative ideal solution H−=
{H1−
s,H2−
s, ..., H n−
s}, respectively.
The distance measure between Hi(i=1,2, ..., m)
and H+is: D+
i=n
j=1D∗
ωHFL(Hij
s,H+); the dis-
tance measure between Hi(i=1,2, ..., m) and H−
is: D−
i=n
j=1D∗
ωHFL(Hij
s,H−).
For the given alternative Hi(i=1,2, ..., m),
D+
i=
n

j=1
D∗
ωHFL(Hij
s,H+)=1−S∗
ωHFL(Hij
s,H+
s)
=
n

j=1
1−1
2[CosωHFL(Hij
s,H+
s)+1−DωHFL(Hij
s,H+
s)] =
n

j=1
1
2
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1−n
j=11
LjLj
l=1δij
l(xj)
2t+1.δ+
l(xj)
2t+1
n
j=11
LjLj
l=1δij
l(xj)
2t+12.n
j=11
LjLj
l=1δ+
l(xj)
2t+121
2
+⎛
⎝
n

j=1
ωj
Lj
Lj

l=1δij
l(xj)−δ+
l(xj)
2t+12⎞
⎠
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,

AUTHOR COPY
L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets 1003
D−
i=
n

j=1
D∗
ωHFL(Hij
s,H−)=1−S∗
ωHFL(Hij
s,H−
s)
=
n

j=1
1−1
2[CosωHFL(Hij
s,H−
s)+1−DωHFL(Hij
s,H−
s)] =
n

j=1
1
2
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1−n
j=11
LjLj
l=1δij
l(xj)
2t+1.δ−
l(xj)
2t+1
n
j=11
LjLj
l=1δij
l(xj)
2t+12.n
j=11
LjLj
l=1δ−
l(xj)
2t+121
2
+⎛
⎝
n

j=1
ωj
Lj
Lj

l=1δij
l(xj)−δ−
l(xj)
2t+12⎞
⎠
⎞
⎟
⎟
⎟
⎟
⎟
⎠
Step 4. Calculate the closeness coefficient iof each
alternative Hi(i=1,2, ..., m):
i=D−
i
D−
i+D−
i
Step 5. Rank the alternatives according to the overall
closeness coefficient i, the greater value iis, the
better alternative Hiwill be.
5. Numerical example
In this section, we give a numerical example about
a movie recommender system (adapted from Liao et
al. [21]) to illustrate the feasibility of the TOPSIS
method with proposed distance measure.
5.1. Background
A company intends to give ratings on five movies
(H1,H
2,H
3,H
4,H
5) with respect to the following
criteria: story (C1) , acting (C2), visuals (C3) and
direction (C4). The weighing vector of four criteria
is ω=(0.4,0.2,0.2,0.2). Since these criteria are all
qualitative, it is convenient and feasible for the deci-
sion makers to represent their evaluations in linguistic
term set. The company uses the following LTS to
evaluate the movies with respect to the criteria, which
can be represented as: S={s−3:terrible, s−2:
very bad, s−1:bad, s−2:very bad, s−1:bad, s0:
medium, s1:well, s2:very well, s3:perfect}.For
example, the evaluation about the movie H2under
the criterion C1is between medium and very well,
then the hesitant fuzzy linguistic evaluation can
be represented as {s0,s
1,s
2}. The hesitant fuzzy
linguistic decision matrix H=(Hij
s)5×4can be given
in Table 1
Step 1. Normalize the decision matrix.
As all the criteria are benefit-type criteria, we need
not do anything in this step.
Step 2. Determine the hesitant fuzzy linguistic
positive ideal solution H+and the hesitant fuzzy
linguistic negative ideal solution H−respectively,
which are given as follows:
H+={{s2,s
3},{s1,s
2,s
3},{s1,s
2,s
3},{s2}}
H−={{s−2,s
−1,s
0},{s−1,s
0,s
1},{s0,s
1},{s0,s
1}}.
Step 3. Calculate the distance measure D∗
ωHFL
(Hij
s,H+) and D∗
ωHFL(Hij
s,H−) for each alternative
Hi(i=1,2,3,4,5) , which are obtained in Table 2.
Step 4. Calculate the closeness coefficient i, which
are obtained in Table 3.
Table 1
The hesitant fuzzy linguistic decision matrix provided by experts
C1C2C3C4
H1{s−2,s
−1,s
0}{
s0,s
1}{
s0,s
1,s
2}{
s1,s
2}
H2{s0,s
1,s
2}{
s1,s
2}{
s0,s
1}{
s0,s
1,s
2}
H3{s2,s
3}{
s1,s
2,s
3}{
s1,s
2}{
s2}
H4{s0,s
1,s
2}{
s−1,s
0,s
1}{
s1,s
2,s
3}{
s1,s
2}
H5{s−1,s
0}{
s0,s
1,s
2}{
s0,s
1,s
2}{
s0,s
1}
Table 2
The distance measure of each alternative
D+
iD−
i
H10.2820 0.1090
H20.1730 0.3659
H30.0217 0.3366
H40.1793 0.3426
H50.2821 0.1892

AUTHOR COPY
1004 L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets
Table 3
The closeness coefficient of each alternative
H1H2H3H4H5
i0.2788 0.6790 0.9394 0.6564 0.4014
Table 4
Comparison of different methods
Ranking
Approach from Liao et al. [21] H3H2H4H5H1
Approach from Liao et al. [30] H3H2H4H5H1
Proposed approach based on D∗
ωHFL H3H2H4H5H1
Step 5. Ranking the alternatives Hiaccording to the
value of closeness coefficient i(i=1,2,3,4,5).
We know that H3H2H4H5H1, the
best alternative is H3.
5.2. Comparative analysis
In order to illustrate the feasibility and effective-
ness of the proposed method, different methods are
used to compare with the same numerical example in
section 5.1. The comparison results are displayed in
Table 4.
From Table 4, we can see that the proposed method
produces the same ranking results as the existing
methods, which means the method in this paper is
feasible and effective. Compared with other meth-
ods, it has some advantages in solving multi-criteria
decision making problems.
In Liao et al. [21], they proposed different distance
measures between HFLTSs. Firstly, they obtain the
hesitant fuzzy linguistic positive ideal solution and
hesitant fuzzy linguistic negative ideal solution, then
they calculate the distance measure between each
alternative and the positive ideal solution, the distance
measure between each alternative and the negative
ideal solution, respectively. But the distance mea-
sures in Liao et al. [21] only consider the HFLTSs
from the point view of algebra, which may lead to
the decision information loss in decision process. As
a result, the proposed method in this paper is superior
to the method in Liao et al. [21], because it consider
the distance measure not only from the point view of
algebra but also from the point view of geometry, and
it can avoid the disadvantage of the method in Liao
et al. [21].
In Liao et al. [30], they proposed the hesitant
fuzzy linguistic TOPSIS method based on the cosine
distance measure, which aims at choosing alternative
with the shortest distance from the positive ideal solu-
tion and the furthest distance from the negative ideal
solution. The calculation results indicate that the best
alternative is H3It is already known that the similarity
measure CosHFL proposed by Liao et al. [30] is not
a regular similarity measure, it cannot deal with the
situation that the subscripts of two linguistic terms
have the linear relationship, so the result obtained in
Liao et al. [30] may be unreliable. The similarity mea-
sure proposed in the paper is constructed based on the
cosine similarity measure CosHFL and the Euclidean
distance measure, which can overcome the disadvan-
tage of similarity measure CosHFL and improve the
adaptability of HFLTSs in practical decision prob-
lems; Furthermore, the proposed distance measure
D∗
ωHFL can be applied more widely than the existing
similarity measure CosHFL, because it shows a better
performance in the field of decision making.
6. Conclusions and discussion
The similarity measure is widely used in multi-
criteria decision making problems. Considering that
the similarity measure CosHFL in Liao et al. [30]
is not a regular similarity measure, we propose
an innovative approach to construct a new similar-
ity measure, which combines the cosine similarity
measureCosHFL and the Euclidean distance together.
Then the corresponding distance measure with TOP-
SIS method is developed, and a numerical example
is provided to illustrate the implementation and fea-
sibility of the proposed method.
As a result, the characteristic of the proposed
method can be summarized as follows:
(1) The proposed similarity measure considers the
similarity degree between two HFLTSs from
the point view of algebra and geometry, and it
also satisfies the axiom of similarity measure.
The corresponding distance measure between
HFLTSs is obtained according to the rela-
tionship between the similarity measure and
distance measure, which can deal with the hes-
itant fuzzy decision effectively.
(2) The TOPSIS method was developed to the
proposed distance measure in hesitant fuzzy
linguistic decision environment, which can
improve the effectiveness of handling decision
information and make the fuzzy set theory per-
fect.

AUTHOR COPY
L. Donghai et al. / The new similarity measure and distance measure between hesitant fuzzy linguistic term sets 1005
As future studies, based on the decision results,
they can be extended in the following directions:
•In the proposed method, the subscript of lin-
guistic terms is calculated directly in process of
operations, it does not consider the difference
between linguistic terms in different semantic
situations. We know that the linguistic scale
function [46, 47] can assign different seman-
tic values to the linguistic terms in different
circumstances, which can transform the linguis-
tic information more flexibly. So we will apply
linguistic scale function to deal with linguistic
information under different semantic situations.
•Linguistic large-scale group decision making
problems are more and more common nowa-
days [48, 49], the distance measure is a popular
and effective tool to make the decision, we will
extend our work to deal with large-scale group
decision making problems.
•It would be very interesting to apply the pro-
posed method to solve the real life decision
making problems, such as the site selection for
electrical power plant, the energy policy selec-
tion, the evaluation quality of the movie and the
medical diagnosis, et al.
Conflict of interests
The authors declare no conflict of interests regard-
ing the publication for the paper.
Data availability statement
No additional data are available.
Acknowledgments
This research is fully supported by a grant
by National Natural Science foundation of Hunan
(2017JJ2096), by National Social Science Founda-
tion of China (15BTJ028), by the outstanding youth
project of Hunan Education Department (1713092),
by National Natural Science Foundation of Hunan
(2018JJ3137), by Philosophy and Social Science
Foundation of Hunan(18ZDB012),by Major projects
of the National Social Science Foundation of China
(17ZDA046).
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