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Z. Su et al. / Distance and similarity measures for dual hesitant fuzzy sets and their applications in pattern recognition 733
2.2. HFSs and their distance and similarity
measures
Definition 2. [4] Let X be a fixed set, a HFS on X is
in terms of a function that when applied to X returns a
subset of [0,1].
In order to make HFS easily understood, Xia and Xu
[19] defined a mathematical symbol:
A =
{
<x,h
A
(
x
)
>
|
x ∈ X
}
where h
A
(
x
)
is a set of some values in [0, 1], denoting
the possible membership degrees of the element x ∈ X
to the set A, and h = h
A
(
x
)
is a hesitant fuzzy element
(HFE).
In order to calculate the distances between two HFSs
A and B on X =
{
x
1
,x
2
,...,x
n
}
, we should do some
preparing work. First of all, we let l
h
A
(
x
)
be the number
of values in h
A
(
x
)
, l
x
i
= max
l
h
A
(
x
i
)
,l
h
B
(
x
i
)
for each
x
i
∈ X and h
σ
(
j
)
A
(
x
)
be the jth largest value in h
A
(
x
)
.
Secondly, we should make some extensions when
comparing HFEs, for example, assume that h
A
(
x
i
)
=
{
0.2, 0.3, 0.6
}
,h
B
(
x
i
)
=
{
0.5, 0.6
}
, l
h
A
(
x
i
)
= 3 and
l
h
B
(
x
i
)
= 2. It is obvious that l
h
A
(
x
i
)
and l
h
B
(
x
i
)
are not
always equal. With respect to this problem, Xu and
Xia [18] introduced a method that mainly extends the
shorter HFE by adding some values until both of them
have the same length when we compare them. Two
ways extending h
B
(
x
i
)
are available. One is called
the optimist rule which extends h
B
(
x
i
)
as h
B
(
x
i
)
=
{
0.5, 0.6, 0.6
}
while the other is called the pessimist
rule which extends it as h
B
(
x
i
)
=
{
0.5, 0.5, 0.6
}
. The
pessimistic rule has been adopted commonly because
we generally assume that the decision makers are pes-
simistic.
Eventually several distance measures of HFSs [14]
which are based on Hamming distance, Euclidean dis-
tance and Hausdorff distance are introduced:
(1) The generalized hesitant normalized distance:
d
ghn
(A, B)=
⎡
⎣
1
n
n
i=1
⎛
⎝
1
l
x
i
l
x
i
j=1
h
σ
(
j
)
A
(
x
i
)
−h
σ
(
j
)
B
(
x
i
)
λ
⎞
⎠
⎤
⎦
1
/
λ
(4)
(2) The generalized hesitant weighted Hausdorff dis-
tance:
d
ghwh
(A, B) =
⎡
⎣
n
j=1
w
i
max
j
h
σ
(
j
)
A
(
x
i
)
−h
σ
(
j
)
B
(
x
i
)
λ
⎤
⎦
1
/
λ
(5)
(3) The generalized hybrid hesitant weighted dis-
tance:
d
ghhw
(A, B)
=
⎡
⎣
n
i=1
w
i
⎛
⎝
1
l
x
i
l
x
i
j=1
h
σ
(
j
)
A
(
x
i
)
− h
σ
(
j
)
B
(
x
i
)
λ
+ max
j
h
σ
(
j
)
A
(
x
i
)
− h
σ
(
j
)
B
(
x
i
)
λ
1
/
λ
(6)
where w
i
(i = 1, 2,...,n) are the weights of the
elements x
i
∈ X, i = 1, 2,...,n, with w
i
∈
[
0, 1
]
,
n
i=1
w
i
= 1, and λ>0.
There are also many other distances for HFSs which
can be seen as the special cases of the three distances
above and the corresponding similarity measures can
obtained from these distances by defining s
(
A, B
)
=
1 − d
(
A, B
)
, where d
(
A, B
)
can take d
ghn
(A, B),
d
ghwh
(A, B)ord
ghhw
(A, B), respectively.
3. Distance and similarity measures for DHFSs
As a new extension of fuzzy sets, DHFSs have
attracted much attention recently, which can be defined
as follows:
Definition 3. [16] Let X be a fixed set, then a DHFS D
on X is described as:
D =
{
<x,h
(
x
)
,g
(
x
)
>
|
x ∈ X
}
(7)
in which h
(
x
)
and g
(
x
)
are two sets of some values in
[0, 1], denoting the possible membership degrees and
nonmembership degrees of the element x ∈ X to the set
D respectively, with the conditions: 0 ≤ γ, η ≤ 1, 0 ≤
γ
+
+ η
+
≤ 1, where γ ∈ h
(
x
)
=
γ∈h
(
x
)
max
{
γ
}
and
η
+
∈ g
+
(
x
)
=
η∈g
(
x
)
max
{
η
}
for all x ∈ X. For con-
venience, the pair d
(
x
)
=
(
h
(
x
)
,g
(
x
))
is called a
dual hesitant fuzzy element (DHFE) denoted by d =
(
h, g
)
, with the conditions: γ ∈ h, η ∈ g, γ
+
∈ h
+
=
γ∈h
max
{
γ
}
,η
+
∈ g
+
=
η∈g
max
{
η
}
,0≤ γ, η ≥
1, and 0 ≤ γ
+
+ η
+
≤ 1.
Until now, there is no research on the distance and
similarity measures for DHFSs. So we will firstly pro-
pose the axioms for distance and similarity measures
under dual hesitant fuzzy environment:
Definition 4. Let A and B be two DHFSs on X =
{
x
1
,x
2
,...,x
n
}
. d
(
A, B
)
is said to be the distance
between A and B if d
(
A, B
)
satisfies the properties: