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Distance and similarity measures for Pythagorean fuzzy sets
Paul Augustine Ejegwa
Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373, Makurdi, Nigeria
Email: ocholohi@gmail.com; ejegwa.augustine@uam.edu.ng
Abstract
The concept of Pythagorean fuzzy sets is very much applicable in decision science because of its unique nature of
indeterminacy. The main feature of Pythagorean fuzzy sets is that it is characterized by three parameters namely,
membership degree, non-membership degree, and indeterminate degree in such a way that the sum of the square of
each of the parameters is one. In this paper, we present axiomatic definitions of distance and similarity measures for
Pythagorean fuzzy sets, taking into account the three parameters that describe the sets. Some distance and similar-
ity measures in intuitionistic fuzzy sets viz; Hamming, Euclidean, normalized Hamming and normalized Euclidean
distances and similarities are extended to Pythagorean fuzzy set setting. However, it is discovered that Hamming and
Euclidean distances and similarities fail the metric conditions in Pythagorean fuzzy set setting whenever the elements
of the two Pythagorean fuzzy sets, whose distance and similarity are to be measured, are not equal. Finally, numerical
examples are provided to illustrate the validity and applicability of the measures. These measures are suggestible
to be resourceful in multicriteria decision making problems (MCDMP) and multiattribute decision making problems
(MADMP), respectively.
Keywords Distance measure . Fuzzy set . Intuitionistic fuzzy set . Similarity measure . Pythagorean fuzzy set
1
1 Introduction
The theory of fuzzy sets proposed by Zadeh (1965) has achieved a great success in several fields due to its potentiality
in handling uncertainty. Fuzzy set is characterized by a membership function, µwhich takes value from a crisp set to
a unit interval I= [0,1]. Several researches on the application of fuzzy sets have been carried out (see Chen et al.,
2001; Chen and Tanuwijaya, 2011; Chen and Chang, 2011; Chen et al., 2012; Wang and Chen, 2008). With the vast
majority of imprecise and vague information in the real world, different extensions of fuzzy set have been developed
by some researchers. The concept of intuitionistic fuzzy sets (IFSs) was proposed by Atanassov (1983, 1986, 1999,
2012) as a generalize mathematical framework of the traditional fuzzy sets. The main advantage of the IFS is its
property to cope with the hesitancy that may exist due to imprecise information. This is achieved by incorporating
a second function, called a non-membership function, νalong with the membership function, µof the conventional
fuzzy set.
Sequel to the introduction of IFSs and the subsequent works on the fundamentals of IFSs, a lot of attentions have
been paid on developing distance and similarity measures between IFSs, as a way to apply them to solving many
problems of decision making, pattern recognition, among others. As a result, several measures were proposed (see
Hatzimichailidis et al., 2012; Szmidt and Kacprzyk, 2000; Szmidt, 2014; Wang and Xin, 2005), each one presenting
specific properties and behavior in decision making problems. Some applications of IFSs have been carried out using
these measures, (see Davvaz and Sadrabadi, 2016; De et al., 2001; Ejegwa et al., 2014; Ejegwa, 2015; Ejegwa and
Modom, 2015; Hatzimichailidis et al., 2012; Szmidt and Kacprzyk, 2001, 2004, 2000). In addition, (see Chen and
Chang, 2015; Chen et al., 2016; Liu and Chen, 2017) for more works on IFS and its applications.
No matter how robust the notion of IFSs is, there are cases where µ+ν≥1unlike the situation captured in IFSs
where µ+ν≤1only. This limitation in IFS construct is the motivation for the introduction of Pythagorean fuzzy
sets (PFSs). Pythagorean fuzzy set (PFS) proposed in Yager (2013a,b, 2014) is a new tool to deal with vagueness
considering the membership grade, µand non-membership grade, νsatisfying the condition µ+ν≥1. As a generalized
set, PFS has close relationship with IFS. The concept of PFSs can be used to characterize uncertain information more
sufficiently and accurately than IFSs. Since inception, the theory of PFSs has been extensively studied (see Beliakov
and James, 2014; Dick et al., 2016; Gou et al., 2016; He et al., 2016; Peng and Yang, 2015; Peng and Selvachandran,
2017).
1This Article has been Published by Granular Computing (2018). https://doi.org/10.1007/s41066-018-00149-z
1
Pythagorean fuzzy set has attracted great attentions of many scholars, and the concept has been applied to several
areas such as decision making, aggregation operators, and information measures, etc. Rahman et al. (2017) worked
on some geometric aggregation operators on interval-valued PFSs (IVPFSs) and applied same to group decision
making problem. Perez-Dominguez et al. (2018) presented a multiobjective optimization on the basis of ratio analysis
(MOORA) under PFS setting and applied it to MCDMP. Liang and Xu (2017) proposed the idea of PFSs in hesitant
environment and its MCDM by employing the technique for order preference by similarity to ideal solution (TOPSIS)
using energy project selection model. Mohagheghi et al. (2017) offered a novel last aggregation group decision
making process for the weight of decision makers using PFSs. Rahman et al. (2018b) proposed some approaches
to multiattribute group decision making based on induced interval-valued Pythagorean fuzzy Einstein aggregation
operator. For other applications of PFSs and IVPFSs in MCDM and MADM, (see Gao and Wei, 2018; Rahman et al.,
2018a; Rahman and Abdullah, 2018; Khan et al., 2018a,b; Garg, 2018, 2016a,b,c, 2017; Du et al., 2017; Hadi-Venchen
and Mirjaberi, 2014; Yager, 2016; Yager and Abbasov, 2013).
Li and Zeng (2018) stated that PFS is characterized by four parameters and consequently proposed a variety of
distance measures for PFSs and Pythagorean fuzzy numbers, which take into account the four proposed parameters.
It is observed that, the four parameters are not the conventional features of PFSs. Zeng et al. (2018) explored the
notions of distance and similarity of PFSs as an extension of Li and Zeng (2018) by incorporating five parameters, and
applied the notions to MCDM problems. Howbeit, the five parameters captured are not the traditional components of
PFSs. Mohd and Abdullah (2018) studied the similarity measure of PFSs based on the combination of cosine similarity
measure and Euclidean distance measure featuring only membership and non-membership degrees, disregarding the
possibility of indeterminacy. The notion of distance measure for PFSs by Zhang and Xu (2014) is the only one in
literature that incorporated the three parameters of PFSs, notwithstanding, the measure failed the metric distance
conditions whenever the elements of the two Pythagorean fuzzy sets are not equal.
In this paper, we introduce the axiomatic definitions of distance and similarity measures for PFSs, and also
remedy the shortcoming of Zhang and Xu (2014). By taking into account the three parameters characterization of
PFSs (viz; membership degree, non-membership degree and indeterminate degree), and following the basic line of
reasoning on which the definitions of distance and similarity for IFSs were based, we propose some distance and
similarity measures for PFSs and deduce some observations. The paper seeks to establish measures for PFSs which
satisfy metric conditions, by incorporating the three parameters to enhance wider spectrum of applications of PFSs
to MCDMP, MADMP, pattern recognition problems, medical diagnosis/imaging, etc.
The paper is organized by presenting some mathematical preliminaries of fuzzy sets and IFSs in Section 2. In
Section 3, the concept of PFSs is outlined, while Section 4 introduces the axiomatic definition of distance measures
for PFSs and some distance measures with illustrations. Section 5 presents the axiomatic definition of similarity
measures for PFSs and some similarity measures with illustrations. Some observations are made on both distance
and similarity measures for PFSs. Finally, Section 6 summarises the resulted outcomes of the paper and some useful
conclusions are drawn.
2 Preliminaries
We recall some basic notions of fuzzy sets and IFSs as background to PFSs.
2.1 Fuzzy sets
Definition 1 (see Zadeh, 1965) Let Xbe a nonempty set. A fuzzy set Ain Xis characterized by a membership
function
µA:X→[0,1].
That is,
µA(x) =
1,if x∈X
0,if x /∈X
(0,1) if xis partly in X
Alternatively, a fuzzy set Ain Xis an object having the form
A={hx, µA(x)i | x∈X}or A={hµA(x)
xi | x∈X},
where the function
µA(x) : X→[0,1]
defines the degree of membership of the element, x∈X.
2
The closer the membership value µA(x)to 1, the more xbelongs to A, where the grades 1 and 0 represent full
membership and full non-membership. Fuzzy set is a collection of objects with graded membership, that is, having
degree of membership. Fuzzy set is an extension of the classical notion of set. In classical set theory, the membership
of elements in a set is assessed in binary terms according to a bivalent condition; an element either belongs or does not
belong to the set. Classical bivalent sets are in fuzzy set theory called crisp sets. Fuzzy sets are generalised classical
sets, since the indicator function of classical sets are special cases of the membership functions of fuzzy sets, if the
latter only take values 0 or 1. Fuzzy sets theory permits the gradual assessment of the membership of element in a
set; this is described with the aid of a membership function valued in the real unit interval [0,1].
Let us consider two examples;
(i) All employees of XY Z who are over 1.8min height.
(ii) All employees of XY Z who are tall.
The first example is a classical set with a universe (all XY Z employees) and a membership rule that divides
the universe into members (those over 1.8m) and non-members. The second example is a fuzzy set because some
employees are definitely in the set and some are definitely not in the set, but some are borderline.
This distinction between the ins, the outs and the borderline is made more exact by the membership function, µ.
If we return to our second example and let Arepresent the fuzzy set of all tall employees and xrepresent a member
of the universe X(i.e. all employees), then µA(x)would be µA(x)=1if xis definitely tall or µA(x)=0if xis
definitely not tall or 0< µA(x)<1for borderline cases.
2.2 Intuitionistic fuzzy sets
Definition 2 (see Atanassov, 1983, 1986) Let a nonempty set Xbe fixed. An IFS Ain Xis an object having the
form
A={hx, µA(x), νA(x)i | x∈X}
or
A={hµA(x), νA(x)
xi | x∈X},
where the functions
µA(x) : X→[0,1] and νA(x) : X→[0,1]
define the degree of membership and the degree of non-membership, respectively of the element x∈Xto A, which
is a subset of X, and for every x∈X,
0≤µA(x) + νA(x)≤1.
For each Ain X,
πA(x) = 1 −µA(x)−νA(x)
is the intuitionistic fuzzy set index or hesitation margin of xin X. The hesitation margin πA(x)is the degree of
non-determinacy of x∈X, to the set Aand πA(x)∈[0,1]. The hesitation margin is the function that expresses lack
of knowledge of whether x∈Xor x /∈X. Thus,
µA(x) + νA(x) + πA(x)=1.
Example 3 Let X={x, y , z}be a fixed universe of discourse and
A={h0.6,0.1
xi,h0.8,0.1
yi,h0.5,0.3
zi}
be the intuitionistic fuzzy set in X. The hesitation margins of the elements x, y, z to Aare
πA(x)=0.3, πA(y)=0.1and πA(z)=0.2.
3 Concept of Pythagorean fuzzy sets
Definition 4 Yager (2013a,b) Let Xbe a universal set. Then a Pythagorean fuzzy set Awhich is a set of ordered
pairs over X, is defined by
A={hx, µA(x), νA(x)i | x∈X}
or
A={hµA(x), νA(x)
xi | x∈X},
3
where the functions
µA(x) : X→[0,1] and νA(x) : X→[0,1]
define the degree of membership and the degree of non-membership, respectively of the element x∈Xto A, which
is a subset of X, and for every x∈X,
0≤(µA(x))2+ (νA(x))2≤1.
Supposing (µA(x))2+ (νA(x))2≤1, then there is a degree of indeterminacy of x∈Xto Adefined by πA(x) =
p1−[(µA(x))2+ (νA(x))2]and πA(x)∈[0,1]. In what follows, (µA(x))2+ (νA(x))2+ (πA(x))2= 1. Otherwise,
πA(x) = 0 whenever (µA(x))2+ (νA(x))2= 1.
We denote the set of all P F Ss over Xby PFS(X).
Example 5 Let A∈PFS(X). Suppose µA(x)=0.7and νA(x)=0.5for X={x}. Clearly, 0.7 + 0.51, but
0.72+ 0.52≤1. Thus πA(x)=0.5099, and hence (µA(x))2+ (νA(x))2+ (πA(x))2= 1.
Table 1 explains the difference between Pythagorean fuzzy sets and intuitionistic fuzzy sets.
Table 1: Pythagorean fuzzy sets and Intuitionistic fuzzy sets
Intuitionistic fuzzy
sets
Pythagorean fuzzy
sets
µ+ν≤1µ+ν≤1or µ+ν≥1
0≤µ+ν≤1 0 ≤µ2+ν2≤1
π= 1 −(µ+ν)π=p1−[µ2+ν2]
µ+ν+π= 1 µ2+ν2+π2= 1
Definition 6 (Yager, 2013a) Let A∈PFS(X). Then the complement of Adenoted by Acis defined as
Ac={hx, νA(x), µA(x)i|x∈X}.
Remark 7 It is noticed that (Ac)c=A. This shows the validity of complementary law in PFS.
Definition 8 (Yager, 2013a) Let A, B ∈PFS(X). Then the following define union and intersection of Aand B:
(i) A∪B={hx, max(µA(x), µB(x)), min(νA(x), νB(x))i|x∈X}.
(ii) A∩B={hx, min(µA(x), µB(x)), max(νA(x), νB(x))i|x∈X}.
Definition 9 (Yager, 2013a) Let A, B ∈PFS(X). Then the sum of Aand Bis defined as
A⊕B={hx, p(µA(x))2+ (µB(x))2−(µA(x))2(µB(x))2, νA(x)νB(x)i|x∈X},
and the product of Aand Bis defined as
A⊗B={hx, µA(x)µB(x),p(νA(x))2+ (νB(x))2−(νA(x))2(νB(x))2i|x∈X}.
Remark 10 Let A, B , C ∈PFS(X). Then the following properties hold:
(a) idempotent;
(i) A∩A=A
(ii) A∪A=A
(iii) A⊕A6=A
(iii) A⊗A6=A
(b) commutativity;
(i) A∩B=B∩A
(ii) A∪B=B∪A
(iii) A⊕B=B⊕A
(iii) A⊗B=B⊗A
(c) associativity;
(i) A∩(B∩C)=(A∩B)∩C
4
(ii) A∪(B∪C)=(A∪B)∪C
(iii) A⊕(B⊕C)=(A⊕B)⊕C
(iv) A⊗(B⊗C)=(A⊗B)⊗C
(d) distributivity;
(i) A∩(B∪C)=(A∩B)∪(A∩C)
(ii) A∪(B∩C)=(A∪B)∩(A∪C)
(iii) A⊕(B∪C)=(A⊕B)∪(A⊕C)
(iv) A⊕(B∩C)=(A⊕B)∩(A⊕C)
(v) A⊗(B∪C)=(A⊗B)∪(A⊗C)
(vi) A⊗(B∩C)=(A⊗B)∩(A⊗C)
(e) DeMorgan’s laws;
(i) (A∩B)c=Ac∪Bc
(ii) (A∪B)c=Ac∩Bc
(iii) (A⊕B)c=Ac⊗Bc
(iv) (A⊗B)c=Ac⊕Bc.
Definition 11 Let A, B ∈PFS(X). Then A=B⇔µA(x) = µB(x)and νA(x) = νB(x)∀x∈X, and A⊆B⇔
µA(x)≤µB(x)and νA(x)≥νB(x)(or νA(x)≤νB(x))∀x∈X. We say A⊂B⇔A⊆Band A6=B.
Definition 12 Let A, B ∈PFS(X). Then Aand Bare comparable to each other if A⊆Band B⊆A.
4 Distance measures for Pythagorean fuzzy sets
Here, we present some distance measures for PFSs. First, let us recall the definition of metric distance of sets.
Definition 13 A metric distance din a set Xis a real function d:X×X→R, which satisfies the following
conditions ∀x, y, z ∈X;
(i) d(x, y)=0⇔x=y,
(ii) d(x, y) = d(y, x)(symmetric),
(iii) d(x, z) + d(z, y)≥d(x, y)(triangle inequality).
Now we give some known distances for fuzzy sets in IFS and PFS settings, and extend those distances to PFS capturing
all its parameters.
4.1 Distance measures for fuzzy sets
Various metric distances, involving fuzzy sets, have been proposed (see Diamond and Kloeden, 1994; Kacprzyk, 1997).
Some common metrics, which are used for the description of the distance between fuzzy sets, are the following: If
the universe set Xis finite, that is, X={x1, ..., xn}, then for any two fuzzy sets Aand Bof Xwith membership
functions µA(xi)and µB(xi)for i= 1, ..., n, respectively, we have
Hamming distance;
dH(A, B)=Σn
i=1{| µA(xi)−µB(xi)|} (1)
Euclidean distance;
dE(A, B) = qΣn
i=1{(µA(xi)−µB(xi))2}(2)
normalized Hamming distance;
dnH (A, B) = dH(A, B)
n⇒
dnH (A, B) = 1
nΣn
i=1{| µA(xi)−µB(xi)|} (3)
normalized Euclidean distance;
dnE (A, B) = dE(A, B)
√n⇒
5
dnE (A, B) = r1
nΣn
i=1{(µA(xi)−µB(xi))2}(4)
Consider A∈I F S(X), recall that whenever πA(xi) = 0, it becomes a fuzzy set such that µA(xi) + νA(xi)=1. That
is, a fuzzy set in an equivalent IFS setting is
A={hx, µA(xi),1−µA(xi)i|x∈X}.
In (Szmidt and Kacprzyk, 2000), equations 1 to 4 are represented in IFS setting as
dH(A, B)I= 2Σn
i=1{| µA(xi)−µB(xi)|} (5)
dE(A, B)I=q2Σn
i=1{(µA(xi)−µB(xi))2}(6)
dnH (A, B)I=2
nΣn
i=1{| µA(xi)−µB(xi)|} (7)
dnE (A, B)I=r2
nΣn
i=1{(µA(xi)−µB(xi))2}(8)
Again, let us consider A∈PFS(X)for X={x1, ..., xn}. Recall that whenever πA(xi) = 0, we get
(µA(xi))2+ (νA(xi))2= 1 ⇒νA(xi) = p1−(µA(xi))2.
That is, a fuzzy set in an equivalent PFS setting is
A={hx, µA(xi),p1−(µA(xi))2i|x∈X}.
Now, we rewrite equations 1 to 4 in PFS setting. Firstly, by taking into account a Pythagorean-type representation
of a fuzzy set, we can express the very essence of the Hamming distance as
dH(A, B)P= Σn
i=1{| µA(xi)−µB(xi)|+|νA(xi)−νB(xi)|}.
But |νA(xi)−νB(xi)|=|p1−(µA(xi))2−p1−(µB(xi))2|and
|(νA(xi))2−(νB(xi))2|=|1−(µA(xi))2−1+(µB(xi))2|
=|(µA(xi))2−(µB(xi))2|.
Also,
|(νA(xi)−νB(xi))(νA(xi) + νB(xi))|=|(µA(xi)−µB(xi))(µA(xi) + µB(xi))|
implies
|(νA(xi)−νB(xi))||(νA(xi) + νB(xi))|≤|(µA(xi)−µB(xi))||(µA(xi) + µB(xi))|
implies
|(νA(xi)−νB(xi))|=|(µA(xi)−µB(xi))|(µA(xi) + µB(xi))
p1−(µA(xi))2+p1−(µB(xi))2.
Then,
dH(A, B)P=
n
X
i=1
{|µA(xi)−µB(xi)|+|(µA(xi)−µB(xi))|(µA(xi) + µB(xi))
p1−(µA(xi))2+p1−(µB(xi))2}
=
n
X
i=1
{|µA(xi)−µB(xi)|[1 + (µA(xi) + µB(xi))
p1−(µA(xi))2+p1−(µB(xi))2]},
that is,
dH(A, B)P=
n
X
i=1
{|µA(xi)−µB(xi)|(p1−(µA(xi))2+p1−(µB(xi))2) + µA(xi) + µB(xi)
p1−(µA(xi))2+p1−(µB(xi))2}.(9)
We removed the sign of absolute value from µA(xi) + µB(xi)and p1−(µA(xi))2+p1−(µB(xi))2since they are
always positive.
Similarly, the normalized Hamming distance of Pythagorean-type representation of fuzzy sets is
dnH (A, B)P=1
n
n
X
i=1
{|µA(xi)−µB(xi)|(p1−(µA(xi))2+p1−(µB(xi))2) + µA(xi) + µB(xi)
p1−(µA(xi))2+p1−(µB(xi))2}.(10)
6
Also, by the same reason, the Euclidean distance taking into account a Pythagorean-type representation of a fuzzy
set, we obtain
dE(A, B)P=qΣn
i=1{(µA(xi)−µB(xi))2+ (νA(xi)−νB(xi))2}.
However,
(νA(xi)−νB(xi))2= (q1−(µA(xi))2−q1−(µB(xi))2)2
= 2 −(µA(xi))2−(µB(xi))2−2q(1 −(µA(xi))2)(1 −(µB(xi))2).
Then,
dE(A, B)P=rΣn
i=1{(µA(xi)−µB(xi))2+ 2 −(µA(xi))2−(µB(xi))2−2q(1 −(µA(xi))2)(1 −(µB(xi))2)}.(11)
Similarly, the normalized Euclidean distance is
dE(A, B)P=s1
n
Σn
i=1{(µA(xi)−µB(xi))2+ 2 −(µA(xi))2−(µB(xi))2−2q(1 −(µA(xi))2)(1 −(µB(xi))2)}.(12)
4.1.1 Numerical example
We proceed to find the distance between two degenerated fuzzy sets using equations 1 to 4 and 9 to 12, respectively.
Example 14 Let Aand Bbe degenerated fuzzy sets of X={x, y}such that
A={h0.6,0.8
xi,h0.7,0.7141
yi}
and
B={h0.4,0.9165
xi,h0.8,0.6
yi}.
Calculating the distance between Aand Busing equations 1 to 4, we obtain
dH(A, B) = 0.3, dnH (A, B ) = 0.15, dE(A, B)=0.05, dnE(A, B ) = 0.025,
and
dnE (A, B)< dE(A, B )< dnH(A, B)< dH(A, B).
Also, we calculate the distance between Aand Busing equations 9 to 12, and get
dH(A, B)P= 0.5307, dnH (A, B )P= 0.2653, dE(A, B)P= 0.2768, dnE(A, B )P= 0.1957.
Deducibly, the conditions of metric distance enumerated by Definition 13 are satisfied.
4.2 Distances for Pythagorean fuzzy sets
Distance measure is a term that describes the difference between Pythagorean fuzzy sets.
Definition 15 Let Xbe nonempty set and A, B, C ∈PFS(X). The distance measure dbetween Aand Bis a
function d:PFS ×PFS →[0,1] satisfies
(i) 0≤d(A, B)≤1(boundedness)
(ii) d(A, B)=0iff A=B(separability)
(iii) d(A, B) = d(B , A)(symmetric)
(iv) d(A, C) + d(B , C)≥d(A, B)(triangle inequality).
7
We now extend the distances for IFSs (see Szmidt and Kacprzyk, 2000) to the case of PFSs. Following the line of
thought presented in equations 1 to 4, for two PFSs Aand Bof X={x1, ..., xn}, the Hamming distance is equal to
dPFS(A, B)H=1
2
n
X
i=1{|µA(xi)−µB(xi)|+|νA(xi)−νB(xi)|+|πA(xi)−πB(xi)|}.(13)
Taking into account
πA(xi) = p1−[(µA(xi))2+ (νA(xi))2]
and
πB(xi) = p1−[(µB(xi))2+ (νB(xi))2],
we have
|πA(xi)−πB(xi)|=|p1−[(µA(xi))2+ (νA(xi))2]−p1−[(µB(xi))2+ (νB(xi))2]|
and
|(πA(xi))2−(πB(xi))2|=| − (µA(xi))2−(νA(xi))2+ (µB(xi))2+ (νB(xi))2|
≤ |(µA(xi))2−(µB(xi))2|+|(νA(xi))2−(νB(xi))2|.
Then,
|πA(xi)−πB(xi)| ≤ |(µA(xi))2−(µB(xi))2|+|(νA(xi))2−(νB(xi))2|
p1−[(µA(xi))2+ (νA(xi))2] + p1−[(µB(xi))2+ (νB(xi))2](14)
Equation 14 means that the third parameter, πin equation 13 can not be omitted.
The Euclidean distance is equal to
dP F S (A, B)E=v
u
u
t
1
2
n
X
i=1
{(µA(xi)−µB(xi))2+ (νA(xi)−νB(xi))2+ (πA(xi)−πB(xi))2}.(15)
Let us verify the effect of omitting the third parameter, πfrom equation 15, that is
(πA(xi)−πB(xi))2= (q1−[(µA(xi))2+ (νA(xi))2]−q1−[(µB(xi))2+ (νB(xi))2])2
= 2 −[(µA(xi))2+ (νA(xi))2]−[(µB(xi))2+ (νB(xi))2]−γ ,
where γ= 2p(1 −[(µA(xi))2+ (νA(xi))2])(1 −[(µB(xi))2+ (νB(xi))2]). Then we say that, taking into account π
when calculating the Euclidean distance for PFS does have an influence on the final result.
Now, the normalized Hamming distance and normalized Euclidean distance are
dPFS(A, B)nH =1
2n
n
X
i=1{|µA(xi)−µB(xi)|+|νA(xi)−νB(xi)|+|πA(xi)−πB(xi)|}.(16)
and
dP F S (A, B)nE =v
u
u
t
1
2n
n
X
i=1
{(µA(xi)−µB(xi))2+ (νA(xi)−νB(xi))2+ (πA(xi)−πB(xi))2}.(17)
4.2.1 Numerical verifications of the distance measures
We now verify whether these distance measures satisfy the conditions in Definition 15.
Example 16 Let A, B , C ∈PFS(X)for X={x1, x2, x3}. Suppose
A={h0.6,0.2
x1i,h0.4,0.6
x2i,h0.5,0.3
x3i},
B={h0.8,0.1
x1i,h0.7,0.3
x2i,h0.6,0.1
x3i}
and
C={h0.9,0.2
x1i,h0.8,0.2
x2i,h0.7,0.3
x3i}.
8
Calculating the distance using the four proposed distance measures, incorporating the three parameters we get
dPFS(A, B)H=1
2
3
X
i=1{|0.6−0.8|+|0.2−0.1|+|0.7746 −0.5916|
+|0.4−0.7|+|0.6−0.3|+|0.6928 −0.6481|
+|0.5−0.6|+|0.3−0.1|+|0.8124 −0.7937|}
= 0.7232,
dPFS(A, B)E= [ 1
2
3
X
i=1{(0.6−0.8)2+ (0.2−0.1)2+ (0.7746 −0.5916)2
+ (0.4−0.7)2+ (0.6−0.3)2+ (0.6928 −0.6481)2
+ (0.5−0.6)2+ (0.3−0.1)2+ (0.8124 −0.7937)2}]1
2
= 0.3974,
dPFS(A, B)nH =1
6
3
X
i=1{|0.6−0.8|+|0.2−0.1|+|0.7746 −0.5916|
+|0.4−0.7|+|0.6−0.3|+|0.6928 −0.6481|
+|0.5−0.6|+|0.3−0.1|+|0.8124 −0.7937|}
= 0.2411,
dPFS(A, B)nE = [ 1
6
3
X
i=1{(0.6−0.8)2+ (0.2−0.1)2+ (0.7746 −0.5916)2
+ (0.4−0.7)2+ (0.6−0.3)2+ (0.6928 −0.6481)2
+ (0.5−0.6)2+ (0.3−0.1)2+ (0.8124 −0.7937)2}]1
2
= 0.2294.
That is,
dPFS(A, B)H= 0.7232, dPFS(A, B)E= 0.3974,
dPFS(A, B)nH = 0.2411, dPFS(A, B)nE = 0.2294.
Similarly, we obtain
dPFS(A, C)H= 0.9894, dPFS(A, C )E= 0.5671,
dPFS(A, C)nH = 0.3298, dPFS(A, C )nE = 0.3274
and
dPFS(B, C )H= 0.5662, dPFS(B , C)E= 0.2826,
dPFS(B, C )nH = 0.1887, dPFS(B , C)nE = 0.1632.
Discussion
It follows from Definition 15 that,
(a) Condition (i) holds since d(A, B ), d(A, C), d(B, C )∈[0,1].
(b) Condition (ii) is straightforward.
(c) Condition (iii) holds because of the use of square and absolute value, respectively.
(d) clearly, d(A, C) + d(B , C)≥d(A, B)holds for each of the four proposed distances.
9
We observe that
dPFS(A, B)nE < dPFS(A, B)nH < dPFS(A, B)E< dPFS(A, B)H,
which implies dPFS(A, B)nE is the accurate measure. In this case ( i.e. where the elements of the PFSs are equal),
the four proposed distances are distance measures for PFSs.
Albeit, we consider a case where the elements of the PFSs are not equal to ascertain whether the four proposed
distances will satisfy the aforesaid conditions.
Example 17 Let A, B , C ∈PFS(X)for X={x1, x2, x3, x4, x5}. Suppose
A={h0.6,0.4
x1i,h0.5,0.7
x2i,h0.8,0.4
x3i,h0.7,0.2
x5i},
B={h0.7,0.3
x1i,h0.4,0.7
x3i,h0.9,0.2
x4i}
and
C={h0.6,0.4
x2i,h0.7,0.3
x3i,h0.5,0.4
x4i}
We obtain the following distances;
dPFS(A, B)H=1
2
5
X
i=1{|0.6−0.7|+|0.4−0.3|+|0.6928 −0.6481|
+|0.5−0.0|+|0.7−1.0|+|0.5099 −0.0|
+|0.8−0.4|+|0.4−0.7|+|0.4472 −0.5916|
+|0.0−0.9|+|1.0−0.2|+|0.0−0.3873|
+|0.7−0.0|+|0.2−1.0|+|0.6856 −0.0|}
= 3.3360,
dPFS(A, B)E= [ 1
2
5
X
i=1{(0.6−0.7)2+ (0.4−0.3)2+ (0.6928 −0.6481)2
+ (0.5−0.0)2+ (0.7−1.0)2+ (0.5099 −0.0)2
+ (0.8−0.4)2+ (0.4−0.7)2+ (0.4472 −0.5916)2
+ (0.0−0.9)2+ (1.0−0.2)2+ (0.0−0.3873)2
+ (0.7−0.0)2+ (0.2−1.0)2+ (0.6856 −0.0)2}]1
2
= 1.4305,
dPFS(A, B)nH =1
10
5
X
i=1{|0.6−0.7|+|0.4−0.3|+|0.6928 −0.6481|
+|0.5−0.0|+|0.7−1.0|+|0.5099 −0.0|
+|0.8−0.4|+|0.4−0.7|+|0.4472 −0.5916|
+|0.0−0.9|+|1.0−0.2|+|0.0−0.3873|
+|0.7−0.0|+|0.2−1.0|+|0.6856 −0.0|}
= 0.6672,
dPFS(A, B)nE = [ 1
10
5
X
i=1{(0.6−0.7)2+ (0.4−0.3)2+ (0.6928 −0.6481)2
+ (0.5−0.0)2+ (0.7−1.0)2+ (0.5099 −0.0)2
+ (0.8−0.4)2+ (0.4−0.7)2+ (0.4472 −0.5916)2
+ (0.0−0.9)2+ (1.0−0.2)2+ (0.0−0.3873)2
+ (0.7−0.0)2+ (0.2−1.0)2+ (0.6856 −0.0)2}]1
2
= 0.6398.
10
That is,
dPFS(A, B)H= 3.3360, dPFS(A, B)E= 1.4305,
dPFS(A, B)nH = 0.6672, dPFS(A, B)nE = 0.6398.
Similarly, we get
dPFS(A, C)H= 3.4652, dPFS(A, C )E= 1.4481,
dPFS(A, C)nH = 0.6930, dPFS(A, C )nE = 0.6476
and
dPFS(B, C )H= 2.8391, dPFS(B , C)E= 1.2646,
dPFS(B, C )nH = 0.5678, dPFS(B , C)nE = 0.5655.
Discussion
We observe that
dPFS(A, B)nE < dPFS(A, B)nH < dPFS(A, B)E< dPFS(A, B)H,
which implies dPFS(A, B)nE is the accurate measure. This observation coincides with the observation of Example 16.
Notwithstanding, in Example 17, it is observed that Hamming distance and Euclidean distance do not satisfy
all the conditions of distance measure. Therefore, they can not be adopted into finding the distance between PFSs
since they do not satisfy the conditions completely in all cases. Thus normalized Hamming distance and normalized
Euclidean distance are reliable distance measures for PFSs.
4.2.2 Zhang and Xu’s distance for Pythagorean fuzzy sets
Zhang and Xu (2014) proposed a distance measure for PFSs. Let Xbe a finite universe set, that is, X={x1, ..., xn},
then for any two PFSs Aand Bof X, we have as
d(A, B) = 1
2
n
X
i=1
{|(µA(xi))2−(µB(xi))2|+|(νA(xi))2−(νB(xi))2|+|(πA(xi))2−(πB(xi))2|},(18)
where xi∈X.
We apply this distance to calculate the distance for PFSs.
Example 18 Let A, B , C ∈PFS(X)for X={x1, x2, x3}. Suppose
A={h0.6,0.2
x1i,h0.4,0.6
x2i,h0.5,0.3
x3i},
B={h0.8,0.1
x1i,h0.7,0.3
x2i,h0.6,0.1
x3i}
and
C={h0.9,0.2
x1i,h0.8,0.2
x2i,h0.7,0.3
x3i}.
Using Zhang and Xu’s distance, we get
d(A, B) = 1
2
3
X
i=1{|0.62−0.82|+|0.22−0.12|+|0.77462−0.59162|
+|0.42−0.72|+|0.62−0.32|+|0.69282−0.64812|
+|0.52−0.62|+|0.32−0.12|+|0.81242−0.79372|}
= 0.7200,
Similary,
d(A, C)=1.1700, d(B , C)=0.5600.
11
Discussion
In what follows, we see that Zhang and Xu’s distance does not satisfies Conditions (i)–(iv) of distance measure
completely. Thus, it is not a reliable distance measure for PFSs because it is not consistent in satisfying the conditions
of distance measure.
Again, we consider another case where the elements of the PFSs are not equal.
Example 19 Let A, B, C ∈PFS(X)for X={x1, x2, x3, x4, x5}. Suppose
A={h0.6,0.4
x1i,h0.5,0.7
x2i,h0.8,0.4
x3i,h0.7,0.2
x5i},
B={h0.7,0.3
x1i,h0.4,0.7
x3i,h0.9,0.2
x4i}
and
C={h0.6,0.4
x2i,h0.7,0.3
x3i,h0.5,0.4
x4i}
Using Zhang and Xu’s distance, we obtain
d(A, B) = 1
2
5
X
i=1{|0.62−0.72|+|0.42−0.32|+|0.69282−0.64812|
+|0.52−0.02|+|0.72−1.02|+|0.50992−0.02|
+|0.82−0.42|+|0.42−0.72|+|0.44722−0.59162|
+|0.02−0.92|+|1.02−0.22|+|0.02−0.38732|
+|0.72−0.02|+|0.22−1.02|+|0.68562−0.02|}
= 2.8900,
Similary,
d(A, C)=3.1900, d(B , C)=2.7100
Discussion
With this example, Zhang and Xu’s distance does not satisfy the conditions of distance measure. Hence Zhang and
Xu’s distance is not reliable as a distance measure for PFS.
Now we give a modified version of Zhang and Xu’s distance as
d(A, B)m=1
2n
n
X
i=1
{|(µA(xi))2−(µB(xi))2|+|(νA(xi))2−(νB(xi))2|+|(πA(xi))2−(πB(xi))2|},(19)
for A, B ∈PFS(X), where xi∈Xand i= 1,2, ..., n.
We validate this modified Zhang and Xu’s distance using Examples 18 and 19, respectively. For Example 18, we get
d(A, B)m=1
6
3
X
i=1{|0.62−0.82|+|0.22−0.12|+|0.77462−0.59162|
+|0.42−0.72|+|0.62−0.32|+|0.69282−0.64812|
+|0.52−0.62|+|0.32−0.12|+|0.81242−0.79372|}
= 0.2400.
Similary,
d(A, C)m= 0.3900, d(B , C)m= 0.1867
This satisfied Conditions (i)–(iv) of distance measure for PFSs. For Example 19, we obtain
d(A, B)m= 0.5780, d(A, C )m= 0.6380, d(B, C)m= 0.5420.
Also, this satisfied Conditions (i)–(iv) of distance measure for PFSs.
12
5 Similarity measure of Pythagorean fuzzy sets
Firstly, we give the axiomatic definition of similarity measure of Pythagorean fuzzy sets.
Definition 20 Let Xbe nonempty set and A, B, C ∈PFS(X). The similarity measure sbetween Aand Bis a
function s:PFS ×PFS →[0,1] satisfies
(i) 0≤s(A, B)≤1(boundedness)
(ii) s(A, B)=1iff A=B(separability)
(iii) s(A, B) = s(B , A)(symmetric)
(iv) s(A, C) + s(B , C)≥s(A, B)(triangle inequality).
Distance measure for PFSs is a dual concept of similarity measure for PFSs. We state the following propositions.
Proposition 21 Let A, B ∈PFS(X). If d(A, B)is a distance measure between PFSs Aand B, then s(A, B) =
1−d(A, B)is a similarity measure of Aand B.
Proposition 22 Let A, B ∈PFS(X). If s(A, B)is a similarity measure between PFSs Aand B, then d(A, B) =
1−s(A, B)is a distance measure of Aand B.
Proposition 23 Let A, B ∈PFS(X). Then
(ii) d(A, B) = d(Ac, B c)
(iii) s(A, B) = s(Ac, B c).
Proposition 24 Let A, B , C ∈PFS(X). Suppose A⊆B⊆C, then
(i) d(A, C)≥d(A, B )and d(A, C)≥d(B, C ).
(ii) s(A, C)≤s(A, B )and s(A, C)≤s(B, C ).
We omitted their proves because they follow from Definitions 15 and 20, respectively.
Let A, B ∈PFS(X)such that X={x1, ..., xn}. Using the distance measures in Section 4, and Propositions 21
and 22, we have the following similarity measures of PFSs;
(i)
s1(A, B) = 1 −1
2n
n
X
i=1
{|µA(xi)−µB(xi)|+|νA(xi)−νB(xi)|+|πA(xi)−πB(xi)|},(20)
(ii)
s2(A, B) = 1 −v
u
u
t
1
2n
n
X
i=1
{(µA(xi)−µB(xi))2+ (νA(xi)−νB(xi))2+ (πA(xi)−πB(xi))2},(21)
(iii)
s3(A, B) = 1 −1
2n
n
X
i=1
{|(µA(xi))2−(µB(xi))2|+|(νA(xi))2−(νB(xi))2|+|(πA(xi))2−(πB(xi))2|}.(22)
5.1 Verifications of the similarity measures
We now verify whether these similarity measures satisfy the conditions in Definition 20.
Example 25 Let A, B , C ∈PFS(X)for X={x1, x2, x3}. Suppose
A={h0.6,0.2
x1i,h0.4,0.6
x2i,h0.5,0.3
x3i},
B={h0.8,0.1
x1i,h0.7,0.3
x2i,h0.6,0.1
x3i}
and
C={h0.9,0.2
x1i,h0.8,0.2
x2i,h0.7,0.3
x3i}.
13
Calculating the similarity between the PFSs using the three proposed similarities, incorporating the three param-
eters we get
s1(A, B)=1−1
6
3
X
i=1{|0.6−0.8|+|0.2−0.1|+|0.7746 −0.5916|
+|0.4−0.7|+|0.6−0.3|+|0.6928 −0.6481|
+|0.5−0.6|+|0.3−0.1|+|0.8124 −0.7937|}
= 0.7589,
s2(A, B)=1−[1
6
3
X
i=1{(0.6−0.8)2+ (0.2−0.1)2+ (0.7746 −0.5916)2
+ (0.4−0.7)2+ (0.6−0.3)2+ (0.6928 −0.6481)2
+ (0.5−0.6)2+ (0.3−0.1)2+ (0.8124 −0.7937)2}]1
2
= 0.7706,
s3(A, B)=1−1
6
3
X
i=1{|0.62−0.82|+|0.22−0.12|+|0.77462−0.59162|
+|0.42−0.72|+|0.62−0.32|+|0.69282−0.64812|
+|0.52−0.62|+|0.32−0.12|+|0.81242−0.79372|}
= 0.7600.
That is,
s1(A, B) = 0.7589, s2(A, B )=0.7706, s3(A, B )=0.7600.
Similarly, we have
s1(A, C)=0.6702, s2(A, C )=0.6726, s3(A, C) = 0.6100
and
s1(B, C )=0.8113, s2(B, C) = 0.8368, s3(B , C)=0.8133
Discussion
It follows from Definition 20 that, Conditions (i)–(iv) hold. In this case ( i.e. where the elements of the PFSs are
equal), the three proposed similarities are similarity measures for PFSs.
Now, we consider a case where the elements of PFSs are not equal.
Example 26 Let A, B , C ∈PFS(X)for X={x1, x2, x3, x4, x5}. Suppose
A={h0.6,0.4
x1i,h0.5,0.7
x2i,h0.8,0.4
x3i,h0.7,0.2
x5i},
B={h0.7,0.3
x1i,h0.4,0.7
x3i,h0.9,0.2
x4i}
and
C={h0.6,0.4
x2i,h0.7,0.3
x3i,h0.5,0.4
x4i}
Similarly, we obtain the following similarities;
s1(A, B) = 0.3328, s2(A, B )=0.3602, s3(A, B )=0.4220,
s1(A, C)=0.3070, s2(A, C )=0.3524, s3(A, C)=0.3620
and
s1(B, C )=0.4322, s2(B, C) = 0.4345, s3(B , C) = 0.4580.
14
Discussion
Also, the conditions of similarity measure are satisfied completely. Hence, we establish that s1,s2and s3are reliable
similarity measures for PFSs.
6 Conclusion
The concept of PFSs is very much applicable in real-life problems because of its ability to cope with imbedded
uncertainty more effective than IFSs. Some of these applications have been explored (see Mohagheghi et al., 2017;
Perez-Dominguez et al., 2018; Yager, 2013b, 2014, 2016). We have proposed axiomatic definitions of distance and
similarity measures between PFSs. Some distances and similarities were introduced in PFS setting, and it was shown
that both distance and similarity should be measured taking into account the three parameters of PFS. The limitation
of the distance measure for PFSs proposed by Zhang and Xu (2014) was identified and remedied by considering the
number of elements of the PFSs. It was observed that the measures introduced by Li and Zeng (2018); Mohd and
Abdullah (2018); Zeng et al. (2018) are not appropriate for effective result because of the consideration of some
parameters which are not captured in the signification of PFSs. The measures introduced in this work could be
used as viable tools in applying PFSs to MCDMP, MADMP, pattern recognition problems, etc. This paper is not
exhaustible since more distance and similariy measures for IFSs could be investigated for plausible extension to PFS
setting in future studies.
Acknowledgement
The author is thankful to the Editors in-chief: Professor Withold Pedrycz and Professor Shyi-Ming Chen for their
technical comments and to the anonymous reviewers for their suggestions, which have improved the quality of this
paper.
Compliance with ethical standards
The author declares that there is no conflict of interest toward the publication of this manuscript.
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