Generalized Hesitant Fuzzy Sets
Bin Zhu
School of Economics and Management, Southeast University, Nanjing, Jiangsu, China
Keywords: Generalized Hesitant Fuzzy Set (GHFS), Hesitant Fuzzy Set (HFS), Decision Making.
Abstract: The hesitant fuzzy set (HFS) is useful to deal with the situation that decision makers (DMs) assign several
possible values to a fixed set. It is convenient to collect and deal with DMs’ preferences in group decision
making. However, HFSs have the information loss problem and cannot tell DMs from each other in group
decision making. In order to deal with these problems, we develop a generalized hesitant fuzzy set (GHFS)
in this paper, which is an extension of the HFS.
1 INTRODUCTION
Zadeh (1965) introduced the idea of fuzzy sets (FSs)
as a powerful tool to address fuzziness, then several
famous extensions have been developed, such as
intuitionistic fuzzy sets (IFSs; Atanassov, 1986),
type-2 fuzzy sets (T2FSs); (Zadeh, 1975);
(Mizumoto and Tanaka, 1976); (Dubois and Prade,
1980), fuzzy multisets (FMSs); (Yager, 1986),
interval-valued fuzzy sets (IVFSs); (Zadeh, 1975),
interval-valued intuitionistic fuzzy sets (IVIFSs);
(Atanassov and Gargov, 1989), and hesitant fuzzy
sets (HFSs); (Torra, 2010).
Atanassov (1986) introduced the notion of IFSs,
whose basic elements are intuitionistic fuzzy
numbers (IFNs) (Xu and Yager, 2006); (Xu, 2007).
Each IFN is characterized by a membership degree
and a non-membership degree, which satisfies the
condition that their sum is smaller or equal to 1. The
IFN can be used to depict uncertainty and vagueness
of an object, and thus it is a basic tool to express
data information under fuzzy environments (Li et al.,
2009); (Liu, 2009); (Ye, 2010).
FMSs are another generalization of FSs that
permit multiple occurrences of an element, and
correspond to the case where the membership
degrees to the multisets are not Boolean but fuzzy.
FMSs are effective for the application to information
retrieval on the world wide web, where a search
engine retrieves multiple occurrences of the same
subjects with possible different degrees of relevance
(Miyamoto, 2003). However, the basic operations of
FMSs are not applied to FSs and IFSs.
T2FSs, described by membership functions that
are characterized by more parameters, permit the
fuzzy membership as a fuzzy set improving the
modeling capability than the original one. FSs, IFSs
and FMSs all can be considered as particular cases
of T2FSs. Many studies have been conducted on
T2FSs due to their remarkable modeling capability
(Doctor and Hagras, 2005); (Hagras, 2004), at the
same time, T2FSs have some difficulties in
establishing the secondary membership functions,
and difficulties in manipulation (Greenfield et al.,
2009); (Karnik and Mendel, 2001); (Rickard et al.,
2009).
Among existed fuzzy sets, HFSs, originally
introduced by Torra (2010), have close relationships
with IFSs and FMSs, can also be considered as a
particular case of T2FSs. The motivation to propose
the HFSs is that when defining the membership of
an element, the difficulty of establishing the
membership degree is not a margin of error (as in
IFSs), or some possibility distribution (as in T2FSs)
on the possible values, but a set of possible values.
Torra (2010) gave an example to illustrate this
situation: two DMs discuss the membership of
x
into
A , one wants to assign
0.5
and the other
0.6
,
which can be denoted by a hesitant fuzzy element
(HFE),
{0.5,0.6}h  . In such a case, two values
given by two DMs can be collected into a HFE,
which means that HFEs can be used to represent
several preferences provided by different DMs in a
single HFE. This advantage of HFSs contributes to
the preference collection in group decision making.
We also can use FMSs to model this situation, but
the operations of FMSs do not apply correctly to
HFSs (Torra, 2010).
395
Zhu B..
Generalized Hesitant Fuzzy Sets.
DOI: 10.5220/0004137803950401
In Proceedings of the 4th International Joint Conference on Computational Intelligence (FCTA-2012), pages 395-401
ISBN: 978-989-8565-33-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Compared with IFSs, HFSs are a tool to
represent uncertainty by several discrete possible
values, which is convenient to be used to collect
discrete data from the mathematical point of view.
However, as the situation that two DMs discuss the
membership of
x
into A mentioned above, if they
both assign
0.5
to
x
, denoted by a HFE {0.5}h

 ,
we can only save one value, and loss the other one.
In this situation, if the two DMs give their
evaluation values anonymously, we can save one
value reasonably; if the two DMs have different
importance, we have to loss some information. DMs
are of vital importance in group decision making, we
often need to consider their difference in practice, a
leading DM for example. As a significant problem,
it’s common to consider the different importance of
DMs in group decision making, lots of studies
concentrate on the determination of the weighting
vector of DMs (Yager, 1988; 2004); (Yager and Xu,
2006); (Wu et al., 2009); (Zhou and Chen, 2011);
(Chen and Zhou, 2011).
Naturally, the loss of information provided by
important DMs may lend to an ineffective result. To
overcome this limitation, we develop a generalized
hesitant fuzzy set (GHFS) which saves all
information associated with different DMs. And as
an extension of HFSs, GHFEs have close
relationships with existed FSs.
We organize the paper as follows. Section 2
reviews some basic knowledge of IFSs and HFSs.
Sections 3 presents the concept of GHFSs, discusses
some properties of GHFSs, and studies the
relationships among GHFSs, HFSs and IFSs.
Section 4 gives the concluding remarks.
2 MANUSCRIPT PREPARATION
Atanassov (1986) originally introduced the concept
of the intuitionistic fuzzy set (IFS) below.
Definition 1 (Atanassov, 1986). Let X be a fixed
set, an intuitionistic fuzzy set (IFS)
A on X is
represented in terms of two functions

: [0,1]X 
and

: [0,1]X  , with the condition,
0()()1xvx  ,
x
X , where

represents
the degree of membership and

the degree of
nonmembership of
x
into the set A . IFSs are often
represented as
,,
AA
x  , for all
x
X
. For
convenience, Xu and Yager (2006) called
,
AA
   an intuitionistic fuzzy number (IFN).
Atanassov (1986) gave some basic operations on
IFSs, which ensure that the operational results are
also IFSs.
Definition 2 (Atanassov, 1986). Let a set
X
be
fixed, and let
A
,
1
A
and
2
A
be three IFSs,
represented by the functions
A

and
A

,
1
A

and
1
A

,
2
A

and
2
A

, respectively. Then the following
operations are valid:
1) Complement:
{,(),()}
AA
Axxxnm=< >
;
2) Union:
12
12
12
,min{ ( ), ( )},
max{ ( ), ( )}
AA
AA
xxx
AA
xx
mm
nn
ìü
ïï
<
ïï
ïï
=
íý
ïï
>
ïï
ïï
îþ

;
3) Intersection:
12
12
12
,max{ ( ), ( )},
min{ ( ), ( )}
AA
AA
xxx
AA
xx
mm
nn
ìü
ïï
<
ïï
ïï
=
íý
ïï
>
ïï
ïï
îþ

;
4)
Å
-union:
1212
12
12
,() () ()(),
() () |
AAAA
AA
xx x xx
AA
xxxX
mmmm
nn
ìü
ïï
<+-
ïï
ïï
Å=
íý
ïï
>Î
ïï
ïï
îþ
;
5)
Ä
-intersection:
12 1 2
12
12
, () (), () ()
() ()|
AA A A
AA
xxx x x
AA
xxxX
mm n n
nn
ìü
ïï
<+
ïï
ïï
Ä=
íý
ïï
-Î>
ïï
ïï
îþ
.
Torra (2010) defined the hesitant fuzzy set (HFS) in
terms of a function that returns a set of membership
values for each element in the domain as follows:
Definition 3 (Torra, 2010). Let X be a fixed set, a
hesitant fuzzy set (HFS) on
X is in terms of a
function that when applied to X returns a subset of
[0,1] , which can be represented as the following
mathematical symbol:
{,()| }
E
ExhxxX


(1)
where ()
E
hx is a set of some values in
[0,1]
,
denoting the possible membership degrees of the
element
x
X

to the set E . For convenience, we
call
()
E
hx a hesitant fuzzy element (HFE) and H
the set of all the HFEs.
Given three HFEs
h ,
1
h and
2
h , Torra (2010)
defined some operations listed below.
1)
{1 }
c
h
h





;
2)
12 1 2 12
{( | max(,))}hh h h hh hh

 
;
3)
12 1 2 12
{( | min(,))}hh hh hh hh

 
.
Xia and Xu (2010) developed some new operations
as follows:
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
396

1)
{}
h
h





 
,
0


;
2)
{1 (1 ) }
h
h





,
0


;
3)
112 2
12 , 1212
{}
hh
hh



  ;
4)
112 2
12 , 12
{}
hh
hh



 .
Torra (2010) gave a definition below that
corresponds to the envelope of a HFE.
Definition 4 (Torra, 2010). Given a hesitant fuzzy
element (HFE)
h , an intuitionistic fuzzy number
(IFN)
()env h
A
is defined as the envelope of
h . This
number, which will be denoted by
()
env
A
h , is
represented by
(,)

 with

and

defined as
h 


, 1 h 

 , where
max{ | }hh 


and
min{ | }hh 


.
Furthermore, Torra (2010) studied some properties
of
()
env
Ah
:
1)
() ( ())
cc
env env
Ah Ah=
;
2)
12 1 2
()()()
env env env
Ah h Ah Ah=
;
3)
12 1 2
()()()
env env env
Ah h Ah Ah=
.
3 GENERALIZED HESITANT
FUZZY SET AND SOME
PROPERTIES
3.1 Generalized Hesitant Fuzzy Set
Given several HFSs, we propose a Cartesian product
of HFSs to construct a generalized hesitant fuzzy set
(GHFS). The definition is as follows:
Definition 5. Let X be a fixed set,
()
() { }
DD
D
D
hx
hx



 
(1,,)Dm  be hesitant
fuzzy sets (HFSs) on
X . Then, a generalized
hesitant fuzzy set (GHFS), that is
D
h
H
, is defined as
1
() () ()
D
hm
H
xhx hx
(2)
11
1
(), , ()
,( (), , ())
|
mm
m
hx h x
xx x
xX












For convenience, we call
11
1,,1
{( , , )}
Dmm
hmhhm
Hh h





 
a generalized hesitant fuzzy element (GHFE). Let
1
(, , )
m
u

   , we call
u
a membership unit (MU).
Each MU corresponds to the selection of one
membership each in every one of those given HFSs.
Thus, we can save all the information associated the
all the DMs. Based on
u
, a GHFE, H , can also be
indicated by
1
1
,,
{} {( , , )| }
m
uH m
u
H
uuH




 

 
(3)
In group decision making, assume
m
decision
makers (DMs), provided
m
HFSs to a fixed set,
then we can construct a GHFS, which saves all the
information associated with all DMs.
Given a GHFE,
H , we define its “reduced
GHFE” as
1
1
,,
{, , | } {}
m
HmH
u
huH








 
(4)
Obviously, the reduced GHFE is a HFE. For
example, given a GHFE,
{(0.2, 0.3),(0.3,0.3)}H  ,
then the “reduced GHFE” is
{0.2,0.3}
H
h 
, that is a
HFE and it is an unique result. And the
H is
constructed by two HFEs,
1
{0.2, 0.3}h 
,
2
{0.3}h 
. Furthermore, if there is only one MU in
H , then the GHFE is equivalent to a HFE. Assume
a GHFE,
1
{( , , )}
m
H   
1
(,)
m

 
,
consequently,
1
{, , }
m
H

  
, that is a HFE. In
such a case, the HFS is a particular case of the
GHFS, which is stated below.
Proposition 1. HFSs are a particular case of GHFSs.
Consequently, we also have the following
proposition:
Proposition 2. Several HFSs can construct a GHFS.
Consider that IFSs are a particular case of HFSs,
where HFSs are nonempty closed intervals (Torra,
2010). According to Proposition 1, IFSs are also can
be considered as a particular case of GHFSs. We
state this below.
Proposition 3. IFSs are a particular case of GHFSs.
Consequently, we have
Proposition 4. Several IFSs can construct a GHFS.
Given a GHFE, we can get an unique reduced HFE,
we state this as follows:
Proposition 5. Each GHFS has an unique reduced
HFS.
3.2 Basic Operations and Properties
The definition of the complement of a GHFE is
defined as follows:
GeneralizedHesitantFuzzySets
397

Definition 6. Given a generalized hesitant fuzzy
element (GHFE)
H , we define its complement as
11
,, 1
{(1 , ,1 )}
mm
c
hh m
H






(5)
1
1
,,
{(1 , ,1 ) | }
m
m
u
uH






Obviously, the complement of complement of the
GHFE is itself, which can be concluded as below.
Proposition 6. The complement of the GHFE is
involutive, and it can be represented as
()
cc
H
H

.
Given a GHFE H , we define the minimum and
maximum memberships of
H as
1) The minimum membership of H
:
min{ | }
H
H



;
2)
The maximum membership of H :
max{ | }
H
H



.
For example, we assume a GHFE,
{(0.2,0.3),(0.2,0.4)}H  , then
min{0.2,0.3, 0.4} 0.2
H



,
max{0.2, 0.3, 0.4} 0.4
H



.
For two GHFEs,
1
H
and
2
H
, we now define their
union and intersection below.
Definition 7. Assume two generalized hesitant fuzzy
elements (GHFEs)
1
H
and
2
H
, the union of them is
defined as
12
12
12 ( )
{| max( , )}
uH H
HH
HH uu 





(6)
or, equivalently
11 1 2
22
1 2 12 12
,
{ , | , max( , )}
uH H H
uH
H H uu uu 




where
1
H


and
2
H


are the minimum memberships
in
1
H
and
2
H
respectively.
The intersection of GHFE is defined as
12
12
12 ( )
{| min( , )}
uH H
HH
HH uu 





(7)
or, equivalently
11 1 2
22
1 2 12 12
,
{, | , min( , )}
uH H H
uH
H H uu uu 




where
1
H


and
2
H


are the maximum memberships
in
1
H
and
2
H
respectively.
Example 1.
Let
1
{(0.2,0.3),(0.2,0.4)}H 
and
2
{(0.3,0.4)}H 
be two GHFEs, we have
1
0.2
H



,
2
0.3
H



,
1
0.4
H



and
2
0.4
H



. By
Definition 7, we can get
12
{(0.3, 0.4)}HHH


,
12
{(0.2, 0.3),(0.2,0.4), (0.3,0.4)}HHH


.
The operations between GHFEs and HFEs have
close relationship.
Proposition 7. Assume two GHFEs,
1
H
and
2
H
,
let
1
H
h
and
2
H
h
be the two reduced GHFEs of
1
H
and
2
H
, the following are valid:
1)
12 1 2
()
H
HHH
hhh


;
2)
12 1 2
()
H
HHH
hhh


.
Proof. 1) For any two GHFEs,
1
H
and
2
H
, by the
operations of HFEs, and Eq. (4), we can get
12
12 12
12
12 12
,
,|,
max( , )
HH HH
HH
hh

 















(8)
By Eq. (4), it can be shown that
12 12
() ()
{}
HH HH
h






(9)
Since
12
12
12 ( )
{| max( , )}
uH H
HH
HH uu 





(10)
We have
12
12
() 12
{| ( ), max( , )}
HH u
HH
huHHu




 


12
12
12
12 12
12
,
,|, , ( ),
max( , )
HH
HH
uu H H
u

 

















12
12
12
12
12
,
,|,
max( , )
HH
HH

 















(11)
12
H
H
hh 
which completes the proof.
The proof of the intersection of GHFEs is similar
to the proof of union above, which is not listed here.
Example 2 (Example 1 continuation). Since
12
{(0.3, 0.4)}HHH


,
12
{(0.2, 0.3),(0.2,0.4), (0.3,0.4)}HHH


,
we have
12
()
{0.3,0.4}
HH
h 

,
12
()
{0.2, 0.3,0.4}
HH
h 

.
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
398

According to Eq.(4), we have
1
{0.2,0.3,0.4)}
H
h 
,
2
{0.3,0.4)}
H
h 
,
and according to the operations of HFEs, we have
12 12
()
{0.3,0.4}
HH HH
hh h


,
12 12
()
{0.2,0.3,0.4}
HH HH
hh h 

 .
We now develop some operations of GHFEs further.
Definition 8. Given three GHFEs, {}
uH
H
u

  ,
11
11
{}
uH
H
u

  ,
22
22
{}
uH
H
u

  ,
0


, since
membership units
u
,
1
u
and
2
u
can be considered
as three hesitant fuzzy elements (HFEs), we have the
following operations:
1)
{}
uH
H
u



 
;
2)
{}
uH
H
u



 
;
3)
1122
12 12
,
{}
uHuH
H
Huu

 
;
4)
1122
12 12
,
{}
uHuH
H
Huu

 
.
Example 3. Suppose two GHFEs,
1
{(0.1,0, 2),(0.1,0.3)}H =
,
2
{(0.2, 0.3)}H =
,
let
2


, we have
1
22
11
{ } {(0.01,0.04), (0.01, 0.09)}
uH
Hu


,
11
11
2 {2 } {(0.19,0.36),(0.19,0.51)}
uH
Hu


,
12
(0.298, 0.397, 0.396, 0.496),
(0.298,0.397,0.494,0.591)
HH
ìü
ïï
ïï
ïï
Å=
íý
ïï
ïï
ïï
îþ
,
12
(0.02, 0.03, 0.04, 0.06),
(0.02, 0.03, 0.06, 0.09)
HH
ìü
ïï
ïï
ïï
Ä=
íý
ïï
ïï
ïï
îþ
.
On the basis of the relationships between GHFEs
and HFEs, we can further develop the following
proposition:
Proposition 8. For any three GHFEs
H
,
1
H
and
2
H
, and their reduced GHFEs
H
h
,
1
H
h
and
2
H
h
,
0l >
, the following are valid:
1)
()
H
H
hh
l
l
=
;
2)
()
HH
hh
l
l=
;
3)
12
12
()HH
HH
hhh
Å
=Å
;
4)
12
12
()HH
HH
hhh
Ä
=Ä
.
Proof. For any three GHFEs
H
,
1
H
and
2
H
, and
their reduced GHFEs
H
h
,
1
H
h
and
2
H
h
,
0l >
, we
have
1)
{| } {}()
uHH
H
huH h
l
lll
gg
gg
ÎÎ
=Î==
;
2)
{1 (1 ) | }
Hu
huH
l
lg
g
Î
=--Î
{1 (1 ) } ( )
HH
h
l
g
gl
Î
=--=
;
3)
12
11
22
1212
()
,
1122
|,
HH
u
u
h
uHuH
g
g
gggg
Å
Î
Î
ìü
ïï
+-
ïï
ïï
=
íý
ïï
ÎÎ
ïï
ïï
îþ

11
22
1212
,
{}
H
H
g
g
gggg
Î
Î
=+-
12
HH
hh=Å
;
4)
12
1122
() 121122
,
{| , }
HH
uu
huHuH
gg
gg
Ä
ÎÎ
=ÎÎ
1122 1 2
12
,
{}
HH H H
hh
gg
gg
ÎÎ
==Ä
HFEs and IFNs have a close relationship that HFEs
were deemed IFNs when HFEs are nonempty closed
intervals. Given an IFN,
,

 （）, we can get a
corresponding HFE,
h , i.e., [,1 ]h

  if
1



 ; given a HFE, h , the envelope of h is a
IFN, i.e.,
() ,1
env
Ah h h



. The envelope of
GHFEs also has a close connection with IFNs. We
now give a definition of the envelope of a GHFE as
follows:
Definition 9. Given a generalized hesitant fuzzy
element (GHFE)
{}
uH
H
u

 
, the envelope of H
can be defined as
()
env
A
H represented by {( , )}


with

and

defined as follows:
1)
{}u 


;
2)
{1 }u 


.
where
u

and
u

are the minimum and maximum
memberships of
u
, respectively.
It's clear that the envelope of a GHFE is a set of
IFNs. In addition, in the particular case that a GHFE,
H
, is equivalent to a HFE,
h
, proposed in
proposition 1, the envelope of
H
is equivalent to
the envelope of
h
, i.e.,
() ()
env env
A
HAh
. Thus,
()
env
A
h
is also a particular case of
()
env
A
H
, which
is stated below.
Proposition 9. ()
env
A
h
is a particular case of
()
env
A
H .
GeneralizedHesitantFuzzySets
399

Example 4.
Given a GHFE,
{(0.2, 0.3, 0.4),(0.2, 0.3, 0.5)}H =
,
according to Definition 9, we have
( ) { 0.2, 0.6 , 0.2, 0.5 }
env
AH=< >< >
.
Since the reduced HFE of
H
is
{0.2,0.3, 0.4,0.5}
H
h 
,
And according to Definition 4, we can get
( ) 0.2,0.5
env H
Ah 
.
It’s clear that
()
env H
Ah
is an IFN in
()
env
AH
, if
{(0.2, 0.3, 0.5)}H =
, then
( ) { 0.2, 0.5 } ( )
env env H
AH Ah=< >=
.
Thus,
()
env
A
h
is a particular case of
()
env
A
H .
Proposition 10. For any three GHFEs
H
,
1
H
and
2
H
, 0l > , we have
1)
()( ())
env env
AH AH
ll
=
;
2)
() (())
env env
AH AHll=
;
3)
12 1 2
()()()
env env env
AH H AH AHÅ= Å
;
4)
12 1 2
()()()
env env env
AH H AH AHÄ= Ä
.
Proof. For any three GHFEs
H
,
1
H
and
2
H
,
0l >
, we have
1)
() ( {})
env env u H
AH A u
ll
Î
= 
,
{(( ) ,1 ( ) ) | }
uu u
uuuH
ll
-+
-+
Î
=-Î
,
{(( ) , 1 (1 (1 )) ) | }
uu u
uuuH
ll
-+
-+
Î
=---Î
,
{(( ,1 ) ) | } ( ( ))
env
uu u
uuuHAH
ll
-+
-+
Î
=-Î=
;
2)
() ( {))
env env u H
AHA ull
Î
= 
,
(1 (1 ) , 1 (1 (1 ) ))
|
uu u
uu
uH
ll
-+
-+
Î
ìü
ïï
-- ---
ïï
ïï
=
íý
ïï
Î
ïï
ïï
îþ

,
(1 (1 ) , (1 ) ))
|
uu u
uu
uH
ll
-+
-+
Î
ìü
ïï
-- -
ïï
ïï
=
íý
ïï
Î
ïï
ïï
îþ

,
({(,1)|})
uu u
uuuHl
-+
-+
Î
=-Î
(())
env
AHl=
;
3)
1122
12 12
,
()( {)
env env
uHuH
AH H A u u
ÎÎ
Å= Å
11 122 2
1212
1212
,,,
1122
(,
1( ))
|,
uu uuu u
uuuu
uuuu
uHuH
-+ -+
----
++++
ÎÎ
ìü
ïï
+-
ïï
ïï
ïï
ïï
=-+-
íý
ïï
ïï
ïï
ÎÎ
ïï
ïï
îþ

11 122 2
1212 1 2
,,,
1122
(,(1)(1))
|,
uu uuu u
uuuu u u
uHuH
-+ -+
---- + +
ÎÎ
ìü
ïï
+- - -
ïï
ïï
=
íý
ïï
ÎÎ
ïï
ïï
îþ

11 1
22 2
1111
,
2222
,
({(,1)| })
({(,1)| })
uu u
uu u
uuuH
uuuH
-+
-+
-+
Î
-+
Î
=-ÎÅ
-Î


12
() ();
env env
AH AH=Å
4)
12 12
()()
env env
AH H AH HÄ= Ä
11
22
121 12 2
,
({ | , })
env
uH
uH
AuuuHuH
Î
Î
=ÄÎÎ
11 1
22 2
12 12
,,
1122
,
(,1 )
|,
uu u
uu u
uu uu
uHuH
-+
-+
-- ++
Î
Î
ìü
ïï
-
ïï
ïï
=
íý
ïï
ÎÎ
ïï
ïï
îþ

11 1
22 2
12 1 2
12
,,
,
1122
(( ,(1 ) (1 )
(1 )(1 )))
|,
uu u
uu u
uu u u
uu
uHuH
-+
-+
-- + +
++
Î
Î
ìü
ïï
-+-
ïï
ïï
ïï
ïï
=---
íý
ïï
ïï
ïï
ÎÎ
ïï
ïï
îþ

11 1
22 2
1111
,
2222
,
{( ,1 ) | }
{( ,1 ) | }
uu u
uu u
uuuH
uuuH
-+
-+
-+
Î
-+
Î
=-ÎÅ
-Î


12
() ().
env env
AH AH=Ä
For a given GHFS H on X , we have ()
H
x for all
x
in X . Then, we can define the GHFS as the
fuzzy multiset (FMS):
{( , ) | ( )}
HxXu
F
MS x u H x




 
(12)
Thus, we can give the relationship between the
GHFS and the FMS below.
Proposition 11. GHFSs can be represented as
FMSs.
Similar to HFSs, the operations for FMSs do not
apply correctly to the GHFSs.
Given a GHFS
H on X , for all
x
in X , we
can also define the GHFS as the following type-2
fuzzy set (T2FS):
()
1, , ( )
() ( )
0, , ( )
Hx
uu H x
x
X
uu H x









(13)
Similarly, we can derive the following result:
Proposition 12. GHFSs can be represented as
T2FSs.
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
400

4 CONCLUSIONS
We have developed the generalized hesitant fuzzy
set (GHFS) to resolve the information loss problem
of hesitant fuzzy sets (HFSs) in this paper. We have
shown that HFSs and the intuitionistic fuzzy sets
(IFSs) are two particular cases of GHFSs. Given
several hesitant fuzzy elements (HFEs), we can
construct a generalized hesitant fuzzy element
(GHFE) by their Cartesian product. Given a GHFE,
we also can get its reduced HFE. We have also built
the relationship between intuitionistic fuzzy numbers
(IFNs) and the GHFE via the envelope of GHFEs.
As an extension of HFSs, GHFSs can save all the
information associated with different decision
makers (DMs) in group decision making, consider
the difference of DMs, and widen the applications of
HFSs in practice.
REFERENCES
Atanassov, K., (1986). Intuitionistic fuzzy sets. Fuzzy Sets
and Systems, 20, 87-96.
Atanassov, K., Gargov, G., (1989). Interval-valued
intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31,
343-349.
Mizumoto, M., Tanaka, K., (1976). Some properties of
fuzzy sets of type 2. Information and Control, 31, 312-
340.
Torra, V., (2010). Hesitant fuzzy sets. International
Journal of Intelligent Systems, 25, 529-539.
Xu, Z. S., Yager, R. R., (2006). Some geometric
aggregation operators based on intuitionistic fuzzy sets.
International Journal of General Systems, 35, 417-
433.
Wang, Z. J., Li, K. W., Wang, W. Z., (2009). An approach
to multiattribute decision making with interval-valued
intuitionistic fuzzy assessments and incomplete
weights. Information Sciences, 179, 3026-3040.
Yager, R. R., (1986). On the theory of bags. International
Journal of General Systems, 13, 23-37.
Zadeh, L. A., (1975). The concept of a linguistic variable
and its application to approximate reasoning, Part I.
Information Sciences, 8, 199-249.
Zadeh, L. A., (1965). Fuzzy sets. Information and Control,
8, 338-353.
Yager, R. R., (1988). On ordered weighted averaging
aggregation operators in multi-criteria decision
making. IEEE Transactions on Systems, Man, and
Cybernetics, 18, 183-190.
Yager, R. R., (2004). OWA aggregation over a continuous
interval argument with applications to decision
making. IEEE Transaction Systems, Man, and
Cybernetics, 34, 1952-1963.
Yager, R. R., Xu, Z. S., (2006). The continuous ordered
weighted geometric operator and its application to
decision making. Fuzzy Sets and Systems, 157, 1393-
1402.
Wu, J., Li, J. C., Li, H., Duan, W. Q., (2009). The induced
continuous ordered weighted geometric operators and
their application in group decision making. Computers
& Industrial Engineering, 56, 1545-1552.
Zhou, L. G., Chen, H. Y., (2011). Continuous generalized
OWA operator and its application to decision making.
Fuzzy Sets and Systems, 168, 18-34.
Chen, H. Y., Zhou, L. G., (2011). An approach to group
decision making with interval fuzzy preference
relations based on induced generalized continuous
ordered weighted averaging operator. Expert Systems
with Applications, 38, 13432-13440.
Xu, Z. S., (2007). Intuitionistic fuzzy aggregation
operators. IEEE Transactions on Fuzzy Systems,15,
1179-1187.
Ye, J., (2010). Fuzzy decision-making method based on
the weighted correlation coefficient under
intuitionistic fuzzy environment. European Journal
of Operational Research, 205, 202-204.
Li, D. F., Wang, Y. C., Liu, S., Shan, F., (2009).
Fractional programming methodology for multi-
attribute group decision-making using IFS. Applied
Soft Computing ,vol. 9, 219-225.
Liu, P. D., (2009). A novel method for hybrid multiple
attribute decision making. Knowledge-Based Systems,
22, 388-391.
Miyamoto, S., (2003). Information clustering based on
fuzzy multisets. Information Processing and
Management, 39, 195-213.
Doctor, F., Hagras, H., Callaghan, V., (2005). A type-2
fuzzy embedded agent to realise ambient intelligence
in ubiquitous computing environments. Information
Science, 171, 309-334.
Hagras, H. (2004). A hierarchical type-2 fuzzy logic
control architecture for autonomous mobile robots.
IEEE Transactions Fuzzy Systems, 12, 524-539.
Greenfield, S., Chiclana, F., Coupland, S., John, R., (2009).
The collapsing method of defuzzification for
discretised interval type-2 fuzzy sets. Information
Sciences, 179, 2055-2069.
Karnik, N. N., Mendel J. M., (2001). Centroid of a type-2
fuzzy set. Information Science, vol.132, 195-220.
Rickard, J. T., Aisbett, J., Gibbon, G., (2009). Subsethood
of type-n fuzzy sets. IEEE Transactions Fuzzy
Systems, 17, 50-60.
GeneralizedHesitantFuzzySets
401