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Soft Comput
DOI 10.1007/s00500-015-1851-x
FOUNDATIONS
Inclusion measure for typical hesitant fuzzy sets, the relative
similarity measure and fuzzy entropy
Hongying Zhang1·Shuyun Yang1
© Springer-Verlag Berlin Heidelberg 2015
Abstract Typical hesitant fuzzy sets (THFSs), possessing
a finite-set-valued fuzzy membership degrees called typical
hesitant fuzzy elements (THFEs), is a special kind of hes-
itant fuzzy sets. Fuzzy inclusion relationship, as the order
structure in fuzzy mathematics, plays an elementary role in
the theoretical research and practical applications of fuzzy
sets. In this paper, a new partial order for THFEs is defined
via the disjunctive semantic meaning of a set, based on
which fuzzy inclusion relationship is defined for THFSs.
Furthermore, inclusion measures are defined to present the
quantitative ranking of every two THFEs and THFSs and
different inclusion measures are constructed. The related
similarity measure, distance and fuzzy entropy of THFSs
are presented and their relationship with inclusion measures
are investigated. Finally, an example is given to show that
the inclusion measure can be applied effectively in hesitant
fuzzy multi-attribute decision making.
Keywords Typical hesitant fuzzy sets ·Disjunctive ·
Inclusion measure ·Similarity measure ·Fuzzy entropy
Communicated by A. Di Nola.
BHongying Zhang
zhyemily@mail.xjtu.edu.cn
Shuyun Yang
yangshuyun2009125@163.com
1School of Mathematics and Statistics, Xi’an Jiaotong
University, Xi’an 710049, Shaanxi,
People’s Republic of China
1 Introductions
Since fuzzy set theory was introduced by Zadeh (1965), many
new approaches and theories treating imprecision and uncer-
tainty have been proposed, such as interval-valued fuzzy sets
(IVFSs) (Zadeh 1975), intuitionistic fuzzy sets (IFSs) theory
and interval valued intuitionistic fuzzy set theory introduced
by Atanassov (1986) and Atanassov and Gargov (1989),
type-2 fuzzy sets Zadeh (1975) and so on. Recently, to deal
with hesitant and incongruous problems, Torra and Narukawa
(2009) and Torra (2010) introduced the concept of HFSs,
which was an extension of the classic fuzzy set and interval-
valued fuzzy set, and arranged the membership degree to a
subset of interval [0,1]. So hesitant fuzzy set is a special case
of type-2 fuzzy set (Zadeh 1975) and many-valued fuzzy sets
originally proposed by Young (1931) and Grattan-Guiness
(1976). HFSs permit the membership degree of an object to
be a set of several possible values between 0 and 1, which
makes HFSs more reasonable and reliable for modeling peo-
ple’s hesitancy and uncertainty in defining the membership
degree of an object. Hesitant fuzzy set theory has obtained
much attention in theoretical and applied research for a few
years. The operators (Bedregal et al. 2014a;Wei 2012;Xu
2014;Yu 2014;Zhao et al. 2014), measurements (Farhadinia
2013;Xu 2014;Zhu and Xu 2014;Zhao et al. 2014) and
applications of hesitant fuzzy sets to group decision making
(Rodriguez et al. 2012;Wei et al. 2013;Yu 2014) and knowl-
edge discovery by rough set theory Yang et al. (2014) and
Zhang et al. (2014) have been studied in the literature. Many
kinds of generalizations of HFS theory were proposed and
their properties and applications are studied in detail, such as
intuitionistic hesitant fuzzy sets (IHFSs) (Xia and Xu 2011),
interval-valued hesitant fuzzy sets (IVHFSs) (Chen et al.
2013;Farhadinia 2013), dual hesitant fuzzy sets (DHFSs)
(Zhu et al. 2012) and its correlation (Farhadinia 2014b), dual
123
H. Zhang, S. Yang
interval-valued hesitant fuzzy sets (DIVHFSs) (Farhadinia
2014b;Ye 2014), high-order hesitant fuzzy sets (HOHFSs)
(Farhadinia 2014a) and linguistic hesitant fuzzy sets (LHFSs)
(Liu and Rodrguez 2014;Rodriguez et al. 2012).
All the research mentioned above have improved the study
and development of hesitant fuzzy sets from theoretical and
practical angles. They frequently consider finite and non-
empty HFEs (Xia and Xu 2011;Xu and Xia 2011) that were
called typical hesitant fuzzy elements (THFEs) (Bedregal
et al. 2014b;Bustince et al. 2006), and the relative HFSs are
called typical hesitant fuzzy sets (THFSs). At the same time,
interval-valued fuzzy sets have been studied in detail. This
paper will pay attention to THFEs and THFSs. Furthermore,
most of them are based on the assumption that the THFEs
for one object in two THFSs have same cardinalities usually
by adding a number of the biggest, the smallest or a para-
meterized one of the THFE with smaller cardinality. But the
mismatching and loss of the transformations are neglected in
spite of its simple framework. In Bedregal et al. (2014a,b),
Bedregal et al. introduced a new partial order between two
THFEs ∨H,leading to a comparison based on selecting the
larger values in the THFE with larger cardinality. The partial
order defined in Bedregal et al. (2014a,b) is a good beginning
to consider the problem of comparing THFEs with different
cardinalities. However, there is still inconsistency in the def-
inition of partial order in Bedregal et al. (2014a,b) because
selection of the larger values in a THFE is too optimistic to
make an objective and reasonable ranking. A THFE is a finite
subset of [0,1]taht represents uncertain and possible values
of a fuzzy membership degree. In this sense, “a set is used as
a disjunction of possible items or of values of some under-
lying quantity, one of which is the right one (Dubois and
Prade 2012)”. So in this paper, we will first construct a par-
tial order ≤∨Hfor all THFEs under the disjunctive semantic
meaning. As we know, ranking of all the THFEs is impor-
tant to decision making in hesitant fuzzy background. Since
the set of all THFEs under the partial order ≤∨His a poset
and inclusion measure (Kitainik 1987;Sinha and Dougherty
1993) on a poset indicates the quantitative degree to which
a given element of a poset is less than another one, it means
that the inclusion measure not only reflects partially ordered
relationships between two elements which possess the par-
tial order, but also shows the exact pairwise less-than degree
between any two elements in a poset. So in this paper, the
inclusion measures of THFEs and THFSs will be studied in
detail, respectively.
The inclusion measures, also known as subsethood mea-
sures, have been studied mainly by constructive approaches
(Fan et al. 1999;Grzegorzewski 2011;Ma et al. 1999;Qiu
et al. 2003;Taká ˘c 2013;Xu et al. 2002;Yao and Deng 2014;
Young 1996;Zeng and Li 2006) and axiomatic approaches
(Cornelis et al. 2003;Fan et al. 1999;Kitainik 1987;Sinha
and Dougherty 1993;Young 1996;Zhang and Leung 1996).
The constructive approach is suitable for practical applica-
tions of inclusion measures. Fan et al. (1999) and Young
(1996) proposed an application of inclusion measure to clus-
tering validity. The assessment criteria of data redundancy in
fuzzy relational database was proposed by the semantic inclu-
sion degree in Ma et al. (1999). Qiu et al. (2003) addressed
an uncertainty analysis method with different inclusion mea-
sures for intelligent systems and other systems such as fuzzy
relational databases. The concept of inclusion measure has
also been introduced into rough set theory and several impor-
tant relationships between inclusion measure and measures
of rough set data analysis have been established in Xu et al.
(2002). The inclusion measure has also been introduced
successfully into fuzzy concept lattice theory in Fan et al.
(2006). On the other hand, the axiomatic approach is appro-
priate for studying the structure of the inclusion measure. In
this approach, a set of axioms are used to characterize the
the inclusion operators. Kitainik (1987) first proposed four
axioms for the inclusion measure according to the crisp inclu-
sion relations properties in 1987. Later, Sinha and Dougherty
(1993) gave a collection of nine axioms, plus three addi-
tional ones. Young (1996) argued that Sinha and Doughertys
axioms are too hard for the applications of the inclusion mea-
sures and proposed four simplified axioms for the inclusion
measure. Zhang and Leung (1996) thought that quantitative
methods were the main approaches in uncertainty infer-
ence which is a key problem for artificial intelligence, so
they presented a generalized definition for the inclusion
measure, called including degree, to represent and measure
the uncertainty information. Fan et al. (1999) proposed a
more generalized the inclusion measure by changing the
monotonicity of Young’s axioms. In Bustince et al. (2006), a
special case of Young’s subsethood measure, DI-subsethood
measure, was presented, which strengthened the monotonic-
ity. Their construction and the conditions under which they
satisfy different axiom operators were also analyzed. Zhang
and Zhang (2009) proposed the hybrid monotonic inclusion
measure which preserves the monotonicity of two variables
and weakens the second condition in D-I subsethood mea-
sure. However, as we know, any research on the inclusion
measures of THFEs and THFSs has not been presented. So in
this paper, we will present a new partial order for THFEs and
THFSs via the disjunctive view. Next, the hybrid monotonic
inclusion measure for THFEs and THFSs will be defined
and constructed under the extended and unextended envi-
ronments and their relationships with the relative similarity
measures and fuzzy entropy will be discussed.
The rest of this paper is organized as follows. In Sect.
2, some preliminaries will be given . In Sect. 3,anew
partial order relationship ≤∨Hon THFEs will be defined,
then some inclusion measures under the new partial order
≤∨Hfor THFEs and THFSs will be defined and constructed
and their properties analyzed in detail. The similarity mea-
123
Inclusion measure for typical hesitant fuzzy sets…
sure and fuzzy entropy derived from the inclusion measure
for THFEs and THFSs will be constructed in Sect. 4.In
Sect. 5, we present an example to show the application
of the inclusion measure of THFEs and THFSs in multi-
attribute decision making. The final section contains the
conclusion.
2 Preliminaries
2.1 Fuzzy logic operators
We first introduce the relevant notions of fuzzy logic opera-
tors (Smets and Magrez 1987). Triangular norm, or t-norm
for short, is an increasing, associative and commutative map-
ping T:[0,1]2→[0,1]that satisfies the boundary
condition: for all α∈I,T(α, 1)=α. The most popular
continuous t-norms include:
•the standard min operator: TM(α, β) =min(α, β);
•the algebraic product: TP(α, β) =α∗β;
•the bold intersection (also called the Łukasiewicz t-norm):
TL(α, β) =max(0,α +β−1).
An implicator is a function I:[0,1]2→[0,1]satisfy-
ing I(1,0)=0 and I(1,1)=I(0,1)=I(0,0)=
1. An implicator Iis called left monotonic (resp. right
monotonic) iff for every α∈[0,1],I(·,α) is decreasing
(resp. I(α, ·)is increasing). If Iis both left monotonic and
right monotonic, then it is called hybrid monotonic. For all
x,y∈[0,1]2,Isatisfies x≤y⇔I(x,y)=1; then it
follows the confinement principle (CP principle).
Several classes of implicators have been studied in the
literature. We recall here the definition of R-implicator (resid-
ual implicator). An implicator is a R-implicator based on
a left-continuous t-norm Tiff for every x,y∈[0,1],
I(x,y)=sup{γ∈[0,1],T(x,γ) ≤y}.
The well-known R-implicators are:
•the Łukasiewicz implicator: IL(x,y)=min(1,1−x+
y), based on TL;
•the G¨odel implicator: IG(x,y)=1forx≤yand
IG(x,y)=yotherwise, based on TM;
•the Gaines implicator: I(x,y)=1forx≤yand
I(x,y)=y
xotherwise, based on TP.
A biimplicator in [0,1]is defined by a↔b=
min{I(a,b), I(b,a)}where Iis an R-implicator. The
biimplicator corresponding to the Łukasiewicz implicator,
the G¨odel implicator and the Gaines implicator are listed
as a↔Lb=1−|a−b|,a↔Gb=1,for a=b;
a↔Gb=min{a,b}otherwise. The main property of the
biimplicator is a↔b=1iffa=b.
2.2 Typical hesitant fuzzy sets
Hesitant fuzzy set is a kind of many-valued quantities hav-
ing subintervals or subsets of [0,1]as membership values
(Grattan-Guiness 1976;Torra and Narukawa 2009;Torra
2010;Young 1931). Most of works on HFSs assume explic-
itly or implicitly that the memberships of the HFSs are finite
and nonempty subsets of [0,1](Xia and Xu 2011), so the typ-
ical hesitant fuzzy sets are introduced in Xia and Xu (2011)
and have been studied in typical fuzzy logic (Bedregal et al.
2014b;Grattan-Guiness 1976;Young 1931).
Definition 2.1 (Typical hesitant fuzzy sets,Xia and Xu 2011)
Let X={x1,x2,...,xn}be a finite and nonempty universe
of discourse and Hbe the set of all finite nonempty subsets
of the unit interval [0,1]; a typical hesitant fuzzy set Aon
Xis a function: hA:X→Hwhich can be expressed as
follows:
A={x,hA(x)|x∈X},
where hA(x)denotes the fuzzy membership degrees of the
element x∈Xto the set A. For convenience, h=hA(x)is
called a typical hesitant fuzzy element (THFE) and l(hA(x))
is the number of values in a THFE hA(x).Thesetofallthe
hesitant fuzzy sets on Xis denoted as THFS(X). Obviously,
typical hesitant fuzzy set defined on the universe Xdegener-
ates to a fuzzy set if ∀x∈X,h(x)contains one element.
Note 2.1 In the following parts, for simplicity, all the values
in each h(x)are arranged in increasing order.
Given a THFE handaTHFS A, some operators are defined
as follows (Torra 2010):
•Lower bound: h−(x)=minh(x),
•Upper bound: h+(x)=maxh(x),
•α-upper bound of h:h+
α={hi∈h|hi≥α},
•α-lower bound of h:h−
α={hi∈h|hi≤α},
•α-upper bound of A:A+
α={h+
α(x)|x∈X},
•α-lower bound of A:A−
α={h−
α(x)|x∈X},
•hc=γ∈h{1−γ},
•|h|=l(h)
i=1hi(x).
Let A=x,hA(x)be a THFS on the reference set X,
then Aαis called the α-cut of THFS Aif it satisfies
Aα={x|hi(x)≥α, i=1,2,...,l(h(x))},α∈[0,1],
where hi(x)is the ith element of THFE h(x).
Let H=x,h(x)be a THFS on the reference set X, then,
empty THFE, empty THFS, full THFE and full THFS, the
set to represent complete ignorance for xand the nonsense
set are defined as follows:
123
H. Zhang, S. Yang
•empty THFE: h(x)={0}for x∈X,h(x)=0 for short;
empty THFS: h(x)={0}for all x∈X,H={
0}for
short,
•full THFE: h(x)={1}for x∈X,h(x)=1 for short;
full THFS: h(x)={1}for all x∈X,H={
1}for short,
•complete ignorance for all x∈X(all is possible): H=
[0,1].
Remark 2.1 (Xu and Xia 2011)FortwoTHFEsh1and
h2,l1≤l2, for simplicity, h1can be extended to have the
same cardinality with h2. The selection of adding values
mainly depends on the decision makers’ risk preferences.
Optimists may add the maximum value of the THFE with
larger cardinality, while pessimists may add the minimum
value. So the adding value h=ηh+
1+(1−η)h−
1, where
0≤η≤1, will be extended to the shorter hesitant fuzzy
element until l1=l2.Ifη=1, h=h+
1(x), the extension of
h1(x)should be considered optimistic; if η=0, h=h−
1(x),
the extension of h1(x)should be considered pessimistic, and
the extension of h1(x)should be viewed as neutral when
η=1
2,h=1
2(h1(x)+h2(x)). But the mismatching and
loss of the transformations are neglected in spite of its simple
framework. There is still a problem that repeating numbers
in a THFE existed by this extension approach.
3 Inclusion measures for THFEs and THFSs
The inclusion measure describes the extent of one element
contained in another one of a partially ordered set. Suppose
≤His a partial order on H;A⊆THF B⇔A(x)≤HB(x)is
a partial order on THFS(X). Then, hybrid monotonic inclu-
sion measure (Zhang and Zhang 2009) for partially ordered
sets (H,≤H)is constructed in Definition 3.1.
Definition 3.1 (HM inclusion measure for THFE )Leth1,h2
∈(H,≤H). A real number Inc(h1,h2)∈[0,1]is called
the inclusion measure between h1and h2,ifInc
≤H(h1,h2)
satisfies the following properties:
(TI1) 0 ≤Inc≤H(h1,h2)≤1,
(TI2) if h1≤Hh2, then Inc≤H(h1,h2)=1,
(TI3) if h=¯
1, then Inc≤H(h,hc)=0,
(TI4) if h1≤Hh2, for any THFE h3,Inc
≤H(h3,h1)≤
Inc≤H(h3,h2)and Inc≤H(h2,h3)≤Inc≤H(h1,h3).
When the partially ordered set (H,≤H)is replaced by
(THFS(X), ⊆THF), then the inclusion measure between any
two typical hesitant fuzzy sets can be defined in the same
way.
Now in the following two parts, the partial orders and HM
inclusion measures for THFEs and THFSs in the extended
environment and unextended environment will be studied in
detail.
3.1 The partial orders and inclusion measures for
THFEs in the extended environment
First, we show the partial orders for THFEs and THFSs in
Definition 3.2 (Xia and Xu 2011) in the extended environ-
ment.
Definition 3.2 (The partial orders for THFEs and THFSs in
the extended environment,Xia and Xu 2011)Givenh1=
{h1
1,h2
1,...,hl
1}and h2={h1
2,h2
2,...,hl
2}are two THFEs
which have been extended to the same number of values l,
and the elements of h1and h2are arranged in increasing
order, hereafter the first partial order relationship under the
extended environment can be defined as follows:
h1≤EH h2iff hi
1≤hi
2,i=1,2,...,l.
Similarly, let A={x,hA(x)|x∈X}and B=
{x,hB(x)|x∈X}be two HFSs defined on the reference
set X, then
A⊆ETHF Biff hA(x)≤EH hB(x), ∀x∈X.
It is apparent that the information in the extended THFEs
and the original ones are different and the difference will cer-
tainly further influence the final decision making, although
the discussion in this framework is simple.
It is obvious that ≤EH and ⊆ETHF are partial orders.
Now, we present the construction approaches to inclu-
sion measures between any two elements in (H,≤EH)and
(THFS(X), ⊆ETHF), respectively. Since any two THFEs h1
and h2∈(H,≤EH)are finite subsets of [0,1]and have
the same cardinalities, the inclusion measures between h1
and h2can be constructed In a similar way to construction
of the inclusion measures between any two fuzzy sets in
Theorem 3.1.
Theorem 3.1 Let Ibe an implicator which satisfies hybrid
monotonicity and CP principle, and hi={h1
i,...,hl
i},i=
1,2,3are three THFEs, then the following functions are
inclusion measures for THFEs under the partial order rela-
tionship ≤EH:
1. Inc≤EH1(h1,h2)=l
i=1I(hi
1,hi
2).
2. Inc≤EH2(h1,h2)=l
i=1λiI(hi
1,hi
2), for all xi∈X,
where λiis positive real numbers satisfying n
i=1λi=1.
3. Inc≤EH3(h1,h2)=1,h1=0,
|h2|/(|h1|∨|h2|), otherwise.
4. Inc≤EH4(h1,h2)=1,h2=1,
(|hc
1|∧|hc
2|)/|hc
2|,otherwise.
123
Inclusion measure for typical hesitant fuzzy sets…
5. Inc≤EH5(h1,h2)=1,h2=1,
|hc
1|/(|hc
1|∨|hc
2|), otherwise.
The inclusion measure between any two THFEs can also
be constructed based on the inclusion measure of α-upper
bound sets Inc(h+
1α,h+
2α).
Theorem 3.2 Let Inc(h+
1α,h+
2α)be an HM inclusion measure
and {αi∈[0,h+]|i=1,2,...,n}, then the HM inclusion
measure for any two THFEs h1and h2∈(H,≤EH )can be
aggregated as follows:
1. Inc≤EH6(h1,h2)=i=1,2,...,nαi·Inc(h+
1αi,h+
2αi)
i=1,2,...,nαi;
2. Inc≤EH7(h1,h2)=i=1,2,...,nf(αi)·Inc(h+
1αi,h+
2αi)
i=1,2,...,nf(αi),
where f (αi)= 0if αi= 0.
They can be proved straightforwardly.
Up to now, we have discussed the inclusion measures of
THFEs in the extended environment. In the next subsection,
we will define a new partial order ≤∨Hbetween any two
THFEs and discuss the HM inclusion measure on the partially
ordered set.
3.2 A novel partial order and HM inclusion measures
for THFEs under the unextended environment
As Dubois and Prade (2012) have interpreted, a THFE pos-
sesses the disjunctive semantic meaning of a set. So in this
part, we define a new partial order for THFEs and THFSs via
disjunctive semantic meaning under the unextended environ-
ment.
Definition 3.3 Let h1={h1
1,h2
1,...,hl1
1},h2={h1
2,h2
2,
...,hl2
2}be two THFEs; an order ≤∨Hbetween h1and h2is
defined as follows:
h1≤∨Hh2iff hi
1≤hi
2,i=1,2,...,l1,if l1≤l2,
hl1−l2+i
1≤hi
2,i=1,2,...,l2,else.
For any two THFSs Aand B,A⊆∨THF Biff
hA(x)≤∨HhB(x), ∀x∈X. The ordered set is denoted
as (THF(X),⊆∨THF).
Theorem 3.3 The order ≤∨His a partial order between
THFEs h1and h2.
Proof 1. It can be proved straightforwardly that ≤∨Hsat-
isfies reflexivity.
2. If h1≤∨Hh2and h2≤∨Hh1, we suppose that l1≤l2,
then hi
1≤hi
2,i=1,2,...,l1and hl2−l1+i
2≤hi
1,i=
1,2,...,l1. Combined with the increasing order, we can
get hi
2≤hl2−l1+i
2≤hi
1≤hi
2and then l2=l1,hi
1=hi
2,
namely, h1=h2. In the same way, we can prove h1=h2
if l1>l2. We conclude that ≤∨Hsatisfies antisymmetry.
3. Now we prove that ≤∨His transitive, namely, if h1≤∨H
h2,h2≤∨Hh3for any three THFEs h1,h2and h3,we
need to testify h1≤∨Hh3.
(i) If l1≤l2,h1≤∨Hh2implies that hi
1≤hi
2,i=
1,2,...,l1based on the definition of ≤∨H.Ifl2≤l3,
then we have hi
2≤hi
3,i=1,2,...,l2. So we can
get hi
1≤hi
3,i=1,2,...,l1, and then we conclude
that h1≤∨Hh3.
If l2>l3, then we have hl2−l3+i
2≤hi
3,i=
1,2,...,l3. We suppose that l1≤l2,l1≤l3, then
hi
1≤hi
2≤hl2−l3+i
2≤hi
3,i=1,2,...,l1that
means h1≤∨Hh3. Otherwise, l1≤l2,l1>l3,
then hl1−l3+i
1≤hl1−l3+i
2≤hl2−l3+i
2≤hi
3,i=
1,2,...,l3and we can conclude that h1≤∨Hh3.
(ii) If l1>l2, we can prove that ≤∨His transitive by the
similar way.
Thus, ≤∨Hsatisfies transitivity. In conclusion, ≤∨His a par-
tial order.
It is obvious that (THF(X),⊆∨THF)is a partially ordered
set by Theorem 3.3.
By analyzing the above two partial order relationships
≤EH and ≤∨H, we find the former is a special case of ≤∨H.
We present an example to show the difference between the
new partial order ≤∨Hand the partial order ≤EH.
Example 3.1 Let h1={0.1,0.4,0.7},h2={0.2,0.5,0.6,
0.9}be two THFEs, obviously h1≤∨Hh2according to Def-
inition 3.3. But under the extended environment, an optimist
will add 0.7 to h1; thus, h
1={0.1,0.4,0.7,0.7},h
1and h2
do not satisfy the partial order ≤∨Hand ≤EH.
In the following part, the inclusion measures for THFEs
via the partial order ≤∨Hunder an unextended environment
will be proposed.
Theorem 3.4 Let h 1and h2be two THFEs, then Inc≤∨H1(h1,
h2)is an HM inclusion measure for THFEs under the partial
order ≤∨H.
1. Inc≤∨H1(h1,h2)
=⎧
⎪
⎨
⎪
⎩
i
I(hi
1,hi
2), i=1,2,...,l1,if l1≤l2,
i
I(hl1−l2+i
1,hi
2), i=1,2,...,l2,else.
2. Inc≤∨H2(h1,h2)
=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1
l1
l1
i=1
I(hi
1,hi
2), i=1,2,...,l1,if l1≤l2,
1
l2
l2
i=1
I(hl1−l2+i
1,hi
2), i=1,2,...,l2,else.
123
H. Zhang, S. Yang
3. Inc≤∨H3(h1,h2)
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
=1,2,...,l1
αi·Inc(hi
1,hi
2)
i=1,2,...,l1
αi,α
i>0,i=1,2,...,l1,if l1≤l2,
i=1,2,...,l2
αi·I(hl1−l2+i
1,hi
2)
i=1,2,...,l2
αi,α
i>0,i=1,2,...,l2,else,
where Iis an implicator which satisfies hybrid monotonic-
ity.
Proof It is obvious that Inc≤∨H1(h1,h2)satisfies (TI1),
(TI2) and (TI3) in Definition 3.1. Now we prove that
Inc≤∨H1(h1,h2)satisfies (TI4), namely the monotonicity.
If h1≤∨Hh2, for all h3∈H,
1. If l1≤l2,forl3≤l1≤l2and l1≤l2≤l3, we can
obtain directly that Inc≤∨H1(h3,h1)≤Inc≤∨H1(h3,h2)and
Inc≤∨H1(h2,h3)≤Inc≤∨H1(h1,h3)by the definition of
Inc≤∨H1(h1,h2)based on the hybrid monotonicity of impli-
cator and the increasing order of THFEs.
If l1≤l3≤l2, based on the hybrid monotonicity of
implicator and the increasing order of THFEs, then we have
Inc≤∨H1(h3,h2)
=
i
I(hi
3,hi
2), i=1,2,...,l3
≥I(hl3−l1+1
3,h1
2)∧I(hl3−l1+2
3,hl3−l1+2
2)
∧...∧I(hl3
3,hl3
2)
≥I(hl3−l1+1
3,h1
2∧I(hl3−l1+2
3,h2
2)∧...∧I(hl3
3,hl1
2)
≥I(hl3−l1+1
3,h1
1)∧I(hl3−l1+2
3,h2
1)∧...∧I(hl3
3,hl1
1)
=
i
I(hl3−l1+i
1,hi
1), i=1,2,...,l1,
=Inc≤∨H1(h3,h1).
Now, we prove Inc≤∨H1(h2,h3)≤Inc≤∨H1(h1,h3).Com-
bining the hybrid monotonicity of implicator with the
increasing order of THFEs, we have
Inc≤∨H1(h2,h3)
=
i
I(hl2−l3+i
2,hi
3), i=1,2,...,l3.
≤I(hl2−l3+1
2,h1
3)∧I(hl2−l3+2
2,h2
3)∧...∧
I(hl2−l3+l1
2,hl1
3)
≤I(h1
2,h1
3)∧I(h2
2,h2
3)∧...∧I(hl1
2,hl1
3)
≤I(h1
1,h1
3)∧I(h2
1,h2
3)∧...∧I(hl1
1,hl1
3)
=Inc≤∨H1(h1,h3).
2. If l1>l2,forl3≥l1>l2and l1>l2≥l3,we
can obtain directly that Inc≤∨H1(h3,h1)≤Inc≤∨H1(h3,h2)
and Inc≤∨H1(h2,h3)≤Inc≤∨H1(h1,h3)by the definition of
Inc≤∨H1(h1,h2)based on the hybrid monotonicity of impli-
cator and the increasing order of THFEs.
If l1≥l3>l2, based on the hybrid monotonicity of
implicator and the increasing order of THFEs, then we have
Inc≤∨H1(h1,h3)
=
i
I(hl1−l3+i
1,hi
3), i=1,2,...,l3
≥I(hl1−l3+1
1,h1
3)∧I(hl1−l3+2
1,h2
3)∧...∧I(hl1
3,hl2
3)
≥I(hl1−l2+1
1,h1
3)∧I(hl1−l2+2
1,h2
3)∧...∧I(hl1
1,hl2
3)
≥I(h1
2,h1
3)∧I(h2
2,h2
3)∧...∧I(hl2
2,hl2
3)
=Inc≤∨H1(h2,h3).
Now, we prove Inc≤∨H1(h3,h1)≤Inc≤∨H1(h3,h2)by the
hybrid monotonicity of implicator and the increasing order
of THFEs.
Inc≤∨H1(h3,h1)
=
i
I(hi
3,hi
1), i=1,2,...,l3
≤I(hl3−l2+1
3,hl3−l2+1
1)∧...∧I(hl3
3,hl3
1)
≤I(hl3−l2+1
3,hl1−l2+1
1)∧...∧I(hl3
3,hl1−l2+l2
1)
≤I(hl3−l2+1
3,h1
2)∧...∧I(hl3
3,hl2
2)
=Inc≤∨H1(h3,h2).
We have proved that Inc≤∨H1(h1,h2)satisfies the conditions
(TI1), (TI2), (TI3) and (TI4) in Definition 3.1, then it is an
HM inclusion measure on (H,≤∨H).
In a similar way, we can prove that Inc≤∨H2(h1,h2)and
Inc≤∨H3(h1,h2)are HM inclusion measures on (H,≤∨H).
When h1≤∨Hh2,wehaveh−
1≤h−
2and h+
1≤h+
2,so
the inclusion measure for THFEs via upper bound and lower
bound can be constructed by the following Theorem 3.5.
Theorem 3.5 Let h1,h2,and h3be three HFEs, then the
following function is a fuzzy inclusion measure for HFEs
under the partial order relationship “≤∨H”.
Inc≤∨H4(h1,h2)=0.5(h+
2−h+
1)+(h−
2−h−
1)
|h+
2−h+
1|+|h−
2−h−
1|+1.
Proof It is straightforward to prove that Inc≤∨H2(h1,h2)sat-
isfies TI1–TI3 in Definition 3.1 and we need to prove TI4. If
h1≤∨Hh2⇒h−
1≤h−
2,h+
1≤h+
2, then for any THFE h3,
when h+
1≤h+
2≤h+
3and h−
1≤h−
2≤h−
3,
1. Inc≤∨H4(h3,h1)=Inc≤∨H4(h3,h2)=0, and Inc≤∨H4
(h2,h3)=Inc≤∨H4(h1,h3)=1 when h−
1≤h−
2≤h−
3from
the construction of Inc≤∨H4.
123
Inclusion measure for typical hesitant fuzzy sets…
2. If h+
1≤h+
2≤h+
3and h−
1≤h−
3≤h−
2, then
Inc≤∨H4(h3,h1)=0 and
Inc≤∨H4(h3,h2)=0.5(h+
2−h+
3)+(h−
2−h−
3)
|h+
2−h+
3|+|h−
2−h−
3|+1
≥0=Inc≤∨H2(h3,h1).
Similarly, Inc≤∨H4(h2,h3)≤Inc≤∨H4(h1,h3)=1.
3. If h+
1≤h+
2≤h+
3and h−
3≤h−
1≤h−
2, suppose h+
2=
h+
1+αand h−
2=h−
1+β, then α, β ≥0,
Inc≤∨H4(h3,h1)=0.5(h+
1−h+
3)+(h−
1−h−
3)
|h+
1−h+
3|+|h−
1−h−
3|+1
=0.5(h+
1−h+
3)+(h−
1−h−
3)
(h+
3−h+
1)+(h−
1−h−
3)+1
=h−
1−h−
3
(h+
3−h+
1)+(h−
1−h−
3)
Inc≤∨H4(h3,h2)=0.5(h+
2−h+
3)+(h−
2−h−
3)
|h+
2−h+
3|+|h−
2−h−
3|+1
=(h−
2−h−
3)
(h+
3−h+
2)+(h−
2−h−
3)
=(h−
1−h−
3)+β
(h+
3−h+
1)+(h−
1−h−
3)+β−α
≥Inc≤∨H4(h3,h1).
Similarly,Inc ≤∨H4(h2,h3)≤Inc≤∨H4(h1,h3)canbe proved.
Similarly, we can prove that Inc≤∨H4(h1,h2)satisfies TI4
if “h+
1≤h+
3≤h+
2,h−
1≤h−
3≤h−
2,h−
3≤h−
1≤h−
2
and h−
3≤h−
1≤h−
2”or“h+
3≤h+
1≤h+
2,h−
3≤h−
1≤h−
2,
h−
3≤h−
1≤h−
2and h−
3≤h−
1≤h−
2”. Then we can conclude
that Inc≤∨H4is an HM inclusion measure.
The HM inclusion measures can also be constructed from
existing HM inclusion measures.
Theorem 3.6 Let Inc(h1,h2)and Inc (h1,h2)be HM
inclusion measures for THFEs h1and h2via ≤∨H, then
the following formulae are HM inclusion measures on
(H,≤∨H):
1. Inc≤∨H5(h1,h2)=T(Inc(h1,h2), Inc(h1,h2)), Tis
a t-norm.
2. Inc≤∨H6(h1,h2)=αInc(h1,h2)+βInc(h1,h2), α +
β=1.
Up to now, we have discussed the HM inclusion mea-
sure on (H,≤EH)and (H,≤∨H). HM inclusion measure
for THFSs on (THFS(X), ⊆ETHF)and (THFS(X), ⊆∨THF )
can be constructed similarly by aggregation of all inclusion
measures of THFEs, respectively. For example, We present
the construction approaches to HM inclusion measures on
(THFS(X), ⊆THF).
Theorem 3.7 Let A,B∈(THFS(X), ⊆THF)be two HFSs
and Ibe an implicator which satisfies the hybrid monotonic-
ity and CP principle, then the inclusion measures Inc(A,B)
can be defined as follows:
1. Inc1(A,B)=1/nn
i=1(Inc(hA(xi), hB(xi))).
2. Inc2(A,B)=n
i=1Inc(hA(xi), hB(xi)).
3. Inc3(A,B)=n
i=1λiInc(hA(xi), hB(xi)), for all xi∈
X, where λiis a positive real number with n
i=1λi=1.
4. Inc4(A,B)=i=1,2,...,nαi·InchA(xi),hB(xi)
i=1,2,...,nαi,α
i>0.
This theorem can be proved directly.
In the next section, we discuss similarity measure, dis-
tance and fuzzy entropy which are related to the inclusion
measure.
4 Similarity measure, fuzzy entropy and distance
between THFSs based on the inclusion measure
In Xu and Xia (2011), the distance and similarity measures
for THFEs and THFSs have been obtained only under the
extended environment (H,≤EH). In this section, we con-
struct the similarity measure, distance and entropy for THFSs
based on HM inclusion measure under the unextend envi-
ronment (H,≤∨H). First, the similarity measure, distance
between THFEs and the fuzzy entropy of THFSs are defined
as given in Definition 4.1 (Farhadinia 2013). The similarity
measure and the distance between THFSs can be defined in
a similar way.
Definition 4.1 Let h1,h2and h3be THFEs. The similarity
measure between h1and h2is defined as S≤∨H(h1,h2), which
satisfies the following properties:
(STHE1) 0 ≤S≤∨H(h1,h2)≤1,
(STHE2) S≤∨H(h1,h2)=S≤∨H(h2,h1),
(STHE3) if h=0or h =1,then S≤∨H(h,hc)=0,
(STHE4) if h1=h2, then S≤∨H(h1,h2)=1,
(STHE5) if h1≤∨Hh2≤∨Hh3, then S≤∨H(h1,h3)≤
min{S≤∨H(h1,h2), S≤∨H(h2,h3)}.
The distance between h1and h2is defined as D≤∨H(h1,h2),
which satisfies the following properties:
(DTHE1) 0 ≤D≤∨H(h1,h2)≤1,
(DTHE2) D≤∨H(h1,h2)=D(h2,h1),
(DTHE3) if h=0orh=1,then D≤∨H(h,hc)=1,
123
H. Zhang, S. Yang
(DTHE4) if h1=h2, then D≤∨H(h1,h2)=0,
(DTHE5) if h1≤∨Hh2≤∨Hh3, then D≤∨H(h1,h3)≥
max{D≤∨H(h1,h2), D≤∨H(h2,h3)}.
Let Aand Bbe two THFSs on X, then Eis called a fuzzy
entropy for THFSs if it possesses the following properties:
(ETHF1) 0 ≤E(A)≤1,
(ETHF2) if A={
1
2},then E(A)=1,
(ETHF3) if B⊆∨THF Aand {1
2}⊆
∨THF B,elseA⊆∨THF
B,then B⊆∨THF {1
2}and E(A)≤E(B),
(ETHF4) E(A)=E(Ac).
Now, we construct the similarity measures for THFEs by
the following formulae, and the similarity measure for THFSs
can be constructed in a similar way.
Theorem 4.1 The similarity measure S≤∨H(h1,h2)between
THFSs h1,h2can be defined as:
1. S≤∨H1(h1,h2)
=⎧
⎪
⎨
⎪
⎩
i
(hi
1↔hi
2), i=1,2,...,l1,if l1≤l2,
i
(hl1−l2+i
1↔hi
2), i=1,2,...,l2,else
where ↔is a bi-implicator.
2. S≤∨H2(h1,h2)=T(Inc≤∨H(h1,h2), Inc≤∨H(h2,h1))
where Inc(A,B)is an inclusion measure for THFEs h1
and h2on X , and Tis a t-norm.
Proof They can be proved straightforwardly by the prop-
erties of bi-implicator and t-norm and definition of HM
inclusion measures.
Since distance is complementary to similarity measure,
we can obtain the following theorem directly.
Theorem 4.2 Let f be a monotonously increasing real func-
tion f : [0,1] →[0, 1] and S≤∨His the similarity measure of
HFSs A and B, then
D(h1,h2)=f(1)−f(S≤∨H1(h1,h2))
f(1)−f(0)
is the distance between the THFSs A and B.
looseness1In a similar way to the construction approach
to the HM inclusion measure of THFSs by that of THFEs, the
similarity measure and distance of THFSs can be obtained
by the aggregation of similarity measure and distance of all
THFEs.
In paper Farhadinia (2013), Farhadinia constructed
entropy based on the distance for THFSs; here, to construct
the entropy derived from the inclusion measure and simi-
larity measure, we define the entropy measure of THFSs as
follows:
looseness1According to the interrelationships among the
inclusion measure, similarity measure, distance and entropy
measure, we construct the entropy measure for HFS Aas
follows:
Theorem 4.3 Let S(A,B)be a similarity measure between
THFSs A and B, then fuzzy entropy for THFSs A is defined
as follows:
E1(A)=S(A,Ac).
Proof It is sufficient to show that E1satisfies the require-
ments (ETHF1–ETHF4):
(ETHF1) It is obvious.
(ETHF2) A={
1
2}⇒Ac=A={
1
2}⇒E1(A)=
S(A,Ac)=1.
(ETHF3) A⊆∨THF Bwhen B⊆∨THF {1
2}⇒A⊆∨THF
B⊆∨THF Bc⊆∨THF Ac⇒E1(A)=S(A,Ac)≤
S(A,Bc)≤S(B,Bc)=E1(B). On the other hand, it
can be proved similarly when 1
2Band A⊆∨THF B.
(ETHF4) It is straightforward according to the property
of similarity measure.
We can conclude that E1(A)is a fuzzy entropy.
5 Illustrative example
In this part, we present an example to show the application
of the inclusion measure in typical hesitant multi-attribute
decision making.
Example 5.1 With the economic development of societies,
energy is an essential factor. Therefore, the correct energy
policy affects economic development and environment
directly. Hence, the most appropriate energy policy selection
is very important. Now, we suppose that there are five energy
projects as alternatives Hi(i=1,2,3,4,5)to be invested
and four attributes (x1: technological; x2: environmental; x3:
socio-political; x4: economic) to be considered. The weight
vector of the attributes is ω=(0.15,0.3,0.2,0.35)T.Sev-
eral decision makers are invited to evaluate the performances
of the five alternatives. For an alternative under an attribute,
though all of the decision makers provide their evaluated val-
ues, some of these values may be repeated. However, here we
only consider all the possible values for an alternative under
an attribute, that is to say these values repeated many times
appear only once. In this case, all possible evaluations for an
alternative under the attributes can be regarded as a THFS .
123
Inclusion measure for typical hesitant fuzzy sets…
Table 1 Typical hesitant fuzzy
decision making matrix (Yang
et al. 2013)
x1x2x3x4
H1{0.3,0.4,0.5} {0.1,0.7,0.8,0.9} {0.2,0.4,0.5} {0.3,0.5,0.6,0.9}
H2{0.3,0.5} {0.2,0.5,0.6,0.7,0.9} {0.1,0.5,0.6,0.8} {0.3,0.4,0.7}
H3{0.6,0.7} {0.6,0.9} {0.3,0.5,0.7} {0.4,0.6}
H4{0.3,0.4,0.7,0.8} {0.2,0.4,0.7} {0.1,0.8} {0.6,0.8,0.9}
H5{0.1,0.3,0.6,0.7,0.9} {0.4,0.6,0.7,0.8} {0.7,0.8,0.9} {0.3,0.6,0.7,0.9}
Table 2 Results obtained by the inclusion measure Inc≤∨H1(Hi,Hj),
i,j=1,2,...,5
H1H2H3H4H5
H11 0.85 0.875 0.8 0.925
H20.775 1 0.9 0.75 0.925
H30.725 0.725 1 0.7 0.775
H40.725 0.7 0.8 1 0.875
H50.675 0.775 0.725 0.675 1
For convenience, we use a hesitant fuzzy decision matrix to
express the results evaluated by the decision makers, which
is given in Table 1.
Here, we use Inc≤∨H1(h1,h2)based on the Lukasiewicz
implicator (IL(x,y)=min(1,1−x+y)) and present the
result in the following inclusion measure table (Table 2):
We aggregate the preference of each Hiby P(Hi)=
5
j=1Inc≤∨H1(Hi,Hj)(i=1,2,...,5). It is obvious that
when P(Hi)is smaller, then Hiwill be more superior.
The preference of each Hi,I=1,2,3,4,5 is listed as
follows:
P(H1)=
5
j=1
Inc1(H1,Hj)=4.45,
P(H2)=
5
j=1
Inc1(H2,Hj)=4.35,
P(H3)=
5
j=1
Inc1(H3,Hj)=3.925,
P(H4)=
5
j=1
Inc1(H4,Hj)=4.1,
P(H5)=
5
j=1
Inc1(H5,Hj)=3.85.
Thus from the results of the values of P(Hi), i=
1,2,3,4,5, we conclude the following relationships: H1
H2H4H3H5.
Table 3 Results obtained by the inclusion measure Inc≤∨H2(Hi,Hj),
i,j=1,2,3,4,5
H1H2H3H4H5
H11 0.89 0.92 0.87 0.96
H20.89 1 0.94 0.85 0.94
H30.78 0.78 1 0.76 0.81
H40.81 0.79 0.85 1 0.91
H50.78 0.7 0.8 0.81 1
If we use Inc≤∨H2(h1,h2)based on the Lukasiewicz impli-
cator (IL(x,y)=min(1,1−x+y)), we obtain the
following inclusion measure table (Table 3):
The preference of each Hi,I=1,2,3,4,5 is listed as
follows:
P(H1)=
5
j=1
Inc1(H1,Hj)=4.63,
P(H2)=
5
j=1
Inc1(H2,Hj)=4.62,
P(H3)=
5
j=1
Inc1(H3,Hj)=4.13,
P(H4)=
5
j=1
Inc1(H4,Hj)=4.36,
P(H5)=
5
j=1
Inc1(H5,Hj)=4.09.
The ranking result of the conclusion computed by
Inc≤∨H2(Hi,Hj)is the same as that by Inc≤∨H1(Hi,Hj).
Remark Compared with the result of the paper Yang et al.
(2013), in which the hybrid inclusion measure Inc1(H1,H2)
was constructed only by the upper bounds and lower bounds
of THFEs, it was a rough inclusion measure. So the inclusion
measure used in this paper is more precise for interpreting
the inclusion degree between any two THFEs and the ranking
result is more accurate.
123
H. Zhang, S. Yang
6 Conclusion
In this paper, we have defined a novel partial order for THFEs
via disjunctive views under the unextended environment.
Then a series of inclusion measures for THFEs and THFSs
have been proposed under both the extended and unextended
environments. Furthermore, similarity measure, distance and
fuzzy entropy have been constructed by the inclusion mea-
sure. Finally, an example is given to show the application of
the inclusion measure in typical hesitant fuzzy multi-attribute
decision making.
Acknowledgments This work was supported by grants from the
National Natural Science Foundation of China (No. 61005042), the
Natural Science Foundation of Shaanxi Province (No. 2014JQ8348)
and the Fundamental Research Funds for the Central Universities.
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of
interest.
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