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New similarity measures for Pythagorean fuzzy sets with applications
Article in International Journal of Fuzzy Computation and Modelling · January 2020
DOI: 10.1504/IJFCM.2020.10027701
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2. New Similarity Measures for Pythagorean Fuzzy
Sets with Applications
Department of Mathematics/Statistics/Computer Science, University of
Agriculture, P.M.B. 2373, Makurdi, Nigeria
E-mail: ocholohi@gmail.com
Abstract
The concept of Pythagorean fuzzy sets (PFSs) is pertinent in finding reliable
solution to decision-making problems, because of its unique nature of indeter-
minacy. Pythagorean fuzzy set is characterized by 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. The objective of this paper is
to present some new similarity measures for PFSs by incorporating the con-
ventional parameters that describe PFSs, with applications to some real-life
decision-making problems. Furthermore, an illustrative example is used to es-
tablish the applicability and validity of the proposed similarity measures and
compare the results with the existing comparable similarity measures to show
the effectiveness of the proposed similarity measures. While analyzing the relia-
bility of the proposed similarity measures in comparison to analogous similarity
measures for PFSs in literature, we discover that the proposed similarity mea-
sures, especially, s4 yields the most reasonable measure. Finally, we apply s4
to decision-making problems such as career placement, medical diagnosis, and
electioneering process. Additional applications of these new similarity measures
could be exploited in decision-making of real-life problems embedded with un-
certainty such as in multi-criteria decision-making (MCDM) and multi-attribute
decision-making (MADM), respectively.
Keywords Fuzzy set, Intuitionistic fuzzy set, Similarity measure, Pythagorean
fuzzy set
1
1 Introduction
Decision-making is a universal process in the life of human beings, which can
be described as the final outcome of some mental and reasoning processes that
lead to the selection of the best alternative. In many situations, it is difficult
for decision makers to precisely express a preference regarding relevant alter-
natives under several criteria, especially when relying on inaccurate, uncertain,
1This paper has been published. P.A. Ejegwa (2020). New Similarity Measures for
Pythagorean Fuzzy Sets with Applications, International Journal of Fuzzy Computation and
Modelling, 3(1):75-94. https://doi.org/10.1504/IJFCM.2020.106105.
1

3. or incomplete information [1]. Multi-criteria decision-making (MCDM) is de-
fined as the process that involves the analysis of a finite set of alternatives and
ranking them in terms of how credible they are to decision-makers when all the
criteria are considered simultaneously [2]. To this end, the theory of fuzzy sets
was introduced by Zadeh [3] to address multi-criteria decision-making (MCDM)
problems within uncertainty. Fuzzy set is characterized by a membership func-
tion, µ which takes value from a crisp set to a unit interval I = [0, 1]. With the
massive imprecise and vague information in the real world, diverse extensions
of fuzzy set have been developed by some researchers. The notion of intuition-
istic fuzzy sets (IFSs) was proposed by [4–7] as a generalize framework of fuzzy
sets. Sequel to the introduction of IFSs, a lot of attentions have been paid on
developing measures for IFSs, as a way to apply them to solving many decision-
making problems. As a result, some measures were proposed, see [8–11]. Some
applications of IFSs have been carried out in medical diagnosis [12–17], career
determination [18], selection process [19], and other multi-criteria decision mak-
ing problems [20–25], among other, some using measures for IFSs.
The idea of Pythagorean fuzzy sets (PFSs) proposed by [26, 27] is a new
tool to deal with vagueness considering the membership grade, µ and non-
membership grade, ν in such a way that the sum of the square of each of
the membership grade and the non-membership grade is less than or equal
to one, unlike in IFSs. Honestly speaking, the origin of Pythagorean fuzzy
sets emanated from intuitionistic fuzzy sets of second type (IFSST) introduced
in [28] as generalized IFSs. 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. Garg [29] presented an improved
score function for the ranking order of interval-valued Pythagorean fuzzy sets
(IVPFSs). Based on it, a Pythagorean fuzzy technique for order of preference by
similarity to ideal solution (TOPSIS) method by taking the preferences of the
experts in the form of interval-valued Pythagorean fuzzy decision matrices was
discussed. In fact, the theory of PFSs has been extensively studied, as shown
in [30–42].
In connection to the applications of PFSs in real-life situations; Rahman
et al. [43] worked on some geometric aggregation operators on interval-valued
PFSs (IVPFSs) and applied same to group decision making problem. Perez-
Dominguez [44] presented a multiobjective optimization on the basis of ratio
analysis (MOORA) under PFS setting and applied it to MCDM problem. Rah-
man et al. [45] proposed some approaches to multi-attribute group decision
making based on induced interval-valued Pythagorean fuzzy Einstein aggrega-
tion operator. In a nutshell, the idea of Pythagorean fuzzy sets has attracted
great attentions of many scholars, and the concept has been applied to several
application areas viz; multi-criteria decision making (MCDM), multi-attribute
decision making (MADM), aggregation operators, information measures, etc.
For details, see [1,32–36,39,40,46–64].
Similarity measure for PFSs is a function that shows how comparable two or
more PFSs are to each other. In fact, it is a dual concept of distance measure for
PFSs. Similarity measures for PFSs have been studied from different perspec-
tives. In [65], some distance/dissimilarity measures for PFSs and Pythagorean
fuzzy numbers, which take into account four parameters were proposed. It is
observed that, the four parameters are not the conventional features of PFSs.
Also, in [66], a new similarity measure and a new dissimilarity measure for
2

4. Pythagorean fuzzy set were introduced in [66] by incorporating four param-
eters more than the three traditional components of PFSs. The notions of
similarity and dissimilarity of PFSs as extension of the work in [65] were in-
troduced in [67] by incorporating five parameters, and applied to multi-criteria
decision-making (MCDM) problems. Howbeit, the five parameters captured
are not the traditional components of PFSs. In [68], some similarity measures
between PFSs based on the cosine function were proposed by considering the
degree of membership, degree of non-membership and degree of hesitation, and
applied to pattern recognition and medical diagnosis. The notion of dissimi-
larity measure for PFSs studied in [69] incorporated only the three traditional
parameters of PFSs, notwithstanding, the measure failed the metric distance
conditions whenever the level sets of the Pythagorean fuzzy sets are not equal.
A similarity measure for PFSs based on the combination of cosine similarity
measure and Euclidean distance measure featuring only membership and non-
membership degrees were introduced in [70]. Of recent, some dissimilarity and
similarity measures for PFSs which satisfied the metric distance conditions were
introduced in [71] by incorporating the three conventional parameters of PFSs.
As a result of the survey of some similarity measures for PFSs [66–68,70]; es-
pecially, those similarity measures [71] that incorporated the three conventional
parameters of PFSs, the need to propose new similarity measures for PFSs
with more reasonable, reliable, and efficient output, are undeniable. Thus, the
motivation of this work.
This paper explores some new similarity measures for PFSs. By taking into
account the three parameters characterization of PFSs (viz; membership degree,
non-membership degree and indeterminate degree), we propose some new sim-
ilarity measures for PFSs with applications to decision-making problems. The
paper is organized by presenting some basic notions of PFSs in Section 2. In Sec-
tion 3, we reiterate some similarity measures for PFSs studied in [71] with some
numerical examples. Also in Section 3, some new similarity measures for PFSs
are proposed with numerical examples. Section 4 discusses some applications of
the most accurate of the similarity measures for PFSs to decision-making prob-
lems. Finally, Section 5 summarises the resulted outcomes of the paper with
direction for future studies.
2 Basic notions of Pythagorean fuzzy sets
We recall some basic notions of fuzzy sets, IFSs and PFSs to be used in the
sequel.
Definition 2.1 [3] Let X be a nonempty set. A fuzzy set A of X is character-
ized by a membership function
µA : X → [0, 1].
That is,
µA(x) =



1, if x ∈ X
0, if x /
∈ X
(0, 1) if x is partly in X
Alternatively, a fuzzy set A of X is an object having the form
A = {hx, µA(x)i | x ∈ X} or A = {h
µA(x)
x
i | x ∈ X},
3

5. where the function
µA(x) : X → [0, 1]
defines the degree of membership of the element, x ∈ X.
Definition 2.2 [4, 5] Let a nonempty set X be fixed. An IFS A of X is an
object having the form
A = {hx, µA(x), νA(x)i | x ∈ X}
or
A = {h
µA(x), νA(x)
x
i | 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 ∈ X to A, which is a subset of X, and for every x ∈ X,
0 ≤ µA(x) + νA(x) ≤ 1.
For each A in X,
πA(x) = 1 − µA(x) − νA(x)
is the intuitionistic fuzzy set index or hesitation margin of x in X. The hesitation
margin πA(x) is the degree of non-determinacy of x ∈ X, to the set A and
πA(x) ∈ [0, 1]. The hesitation margin is the function that expresses lack of
knowledge of whether x ∈ X or x /
∈ X. Thus,
µA(x) + νA(x) + πA(x) = 1.
Definition 2.3 [26, 27] Let X be a universal set. Then a Pythagorean fuzzy
set A which is a set of ordered pairs over X, is defined by
A = {hx, µA(x), νA(x)i | x ∈ X}
or
A = {h
µA(x), νA(x)
x
i | 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 ∈ X to 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 ∈ X to A defined by πA(x) =
p
1 − [(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 PFSs over X by PFS(X).
4

6. Example 2.1 Let A ∈ PFS(X). Suppose µA(x) = 0.70 and νA(x) = 0.50 for
X = {x}. Clearly, 0.70+0.50 1, but 0.702
+0.502
≤ 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 intu-
itionistic fuzzy sets [71].
Table 1: Intuitionistic fuzzy sets and Pythagorean fuzzy sets
Intuitionistic fuzzy
sets
Pythagorean fuzzy
sets
µ + ν ≤ 1 µ+ν ≤ 1 or µ+ν ≥ 1
0 ≤ µ + ν ≤ 1 0 ≤ µ2
+ ν2
≤ 1
π = 1 − (µ + ν) π =
p
1 − [µ2 + ν2]
µ + ν + π = 1 µ2
+ ν2
+ π2
= 1
Definition 2.4 [26] Let A, B ∈ PFS(X). Then A = B ⇔ µA(x) = µB(x) and
νA(x) = νB(x) ∀x ∈ X.
Definition 2.5 [26, 27] Let A, B ∈ PFS(X). Then the following are defined
thus:
(i) Ac
= {hx, νA(x), µA(x)i|x ∈ X}.
(ii) A ∪ B = {hx, max(µA(x), µB(x)), min(νA(x), νB(x))i|x ∈ X}.
(iii) A ∩ B = {hx, min(µA(x), µB(x)), max(νA(x), νB(x))i|x ∈ X}.
(iv) A ⊕ B = {hx,
p
(µA(x))2 + (µB(x))2 − (µA(x))2(µB(x))2, νA(x)νB(x)i|x ∈ X}.
(v) A ⊗ B = {hx, µA(x)µB(x),
p
(νA(x))2 + (νB(x))2 − (νA(x))2(νB(x))2i|x ∈ X}.
Remark 2.1 [71] Let A, B, C ∈ PFS(X). By Definition 2.5, the following
properties hold:
(Ac
)c
= A
A ∩ A = A
A ∪ A = A
A ⊕ A 6= A
A ⊗ A 6= A
A ∩ B = B ∩ A
A ∪ B = B ∪ A
A ⊕ B = B ⊕ A
A ⊗ B = B ⊗ A
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
5

7. A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C
A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ⊕ (B ∪ C) = (A ⊕ B) ∪ (A ⊕ C)
A ⊕ (B ∩ C) = (A ⊕ B) ∩ (A ⊕ C)
A ⊗ (B ∪ C) = (A ⊗ B) ∪ (A ⊗ C)
A ⊗ (B ∩ C) = (A ⊗ B) ∩ (A ⊗ C)
(A ∩ B)c
= Ac
∪ Bc
(A ∪ B)c
= Ac
∩ Bc
(A ⊕ B)c
= Ac
⊗ Bc
(A ⊗ B)c
= Ac
⊕ Bc
.
Definition 2.6 [17] Let A ∈ PFS(X). Then the level/ground set of A is
defined by
A∗ = {x ∈ X|µA(x) 0, νA(x) 1 ∀x}.
Certainly, A∗ is a subset of X.
3 Similarity measures for Pythagorean fuzzy sets
Similarity measure (SM) for PFSs is a dual concept of distance measure for PFSs
[71]. Firstly, we recall the axiomatic definition of similarity for Pythagorean
fuzzy sets.
Definition 3.1 [71] Let X be nonempty set and A, B, C ∈ PFS(X). The
similarity measure s between A and B is a function s : PFS × PFS → [0, 1]
satisfies
(i) 0 ≤ s(A, B) ≤ 1 (boundedness)
(ii) s(A, B) = 1 iff A = B (separability)
(iii) s(A, B) = s(B, A) (symmetric)
(iv) s(A, C) + s(B, C) ≥ s(A, B) (triangle inequality).
Proposition 3.1 [71] 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).
Now, we state a new result on similarity measure for PFSs.
6

8. Proposition 3.2 If A, B, C ∈ PFS(X), such that A ⊆ B ⊆ C, then
s(A, C) ≤ min[s(A, B), s(B, C)].
Proof. Let A, B, C ∈ PFS(X). Assume that A ⊆ B ⊆ C, then by Proposition
3.1 we have s(A, B) ≥ s(A, C) and s(B, C) ≥ s(A, C). Hence
s(A, C) ≤ min[s(A, B), s(B, C)].
By incorporating the three parameters of PFSs, the following similarity
measures for PFSs were proposed in [71]. Let A, B ∈ PFS(X) such that
X = {x1, ..., xn}, then
s1(A, B) = 1 −
1
2n
n
X
i=1
[|µA(xi) − µB(xi)| + |νA(xi) − νB(xi)|
+ |πA(xi) − πB(xi)|],
s2(A, B) = 1 − (
1
2n
n
X
i=1
[(µA(xi) − µB(xi))2
+ (νA(xi) − νB(xi))2
+ (πA(xi) − πB(xi))2
])
1
2 ,
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
|]
3.1 Some new similarity measures for PFSs
We propose some new similarity measures for Pythagorean fuzzy sets, and ex-
emplify the measures to determine their compliant to Definition 3.1.
Let A, B ∈ PFS(X) such that X = {x1, ..., xn}. By incorporating the
three parameters of PFSs, we propose the following new similarity measures for
PFSs:
s4(A, B) = 1 −
1
4n
n
X
i=1
[|µA(xi) − µB(xi)| + ||µA(xi) − νA(xi)| − |µB(xi) − νB(xi)||
+ ||µA(xi) − πA(xi)| − |µB(xi) − πB(xi)||]
s5(A, B) = 1 −
1
4n
n
X
i=1
[|µA(xi) − µB(xi)| + |νA(xi) − νB(xi)| + |πA(xi) − πB(xi)|
+ 2 max{|µA(xi) − µB(xi)|, |νA(xi) − νB(xi)|, |πA(xi) − πB(xi)|}]
7

9. s6(A, B) = 1 − (
1
4n
n
X
i=1
[(µA(xi) − µB(xi))2
+ (νA(xi) − νB(xi))2
+ (πA(xi) − πB(xi))2
+ 2 max{(µA(xi) − µB(xi))2
, (νA(xi) − νB(xi))2
, (πA(xi) − πB(xi))2
}])
1
2
where
πA(xi) =
p
1 − [(µA(xi))2 + (νA(xi))2]
and
πB(xi) =
p
1 − [(µB(xi))2 + (νB(xi))2].
3.2 Numerical examples
We now verify whether these similarity measures satisfy the conditions in Defi-
nition 3.1 by using the examples in [71].
Example 3.1 Let A, B, C ∈ PFS(X) for X = {x1, x2, x3}. Suppose
A = {h
0.6, 0.2
x1
i, h
0.4, 0.6
x2
i, h
0.5, 0.3
x3
i},
B = {h
0.8, 0.1
x1
i, h
0.7, 0.3
x2
i, h
0.6, 0.1
x3
i}
and
C = {h
0.9, 0.2
x1
i, h
0.8, 0.2
x2
i, h
0.7, 0.3
x3
i}.
Calculating the similarity using the proposed similarity measures above, we
have
s4(A, B) = 1 −
1
12
3
X
i=1
[|0.6 − 0.8| + ||0.6 − 0.2| − |0.8 − 0.1|| + ||0.6 − 0.7746| − |0.8 − 0.5916||
+ |0.4 − 0.7| + ||0.4 − 0.6| − |0.7 − 0.3|| + ||0.4 − 0.6928| − |0.7 − 0.6481||
+ |0.5 − 0.6| + ||0.5 − 0.3| − |0.6 − 0.1|| + ||0.5 − 0.8124| − |0.6 − 0.7937||]
= 0.8505
s5(A, B) = 1 −
1
12
3
X
i=1
[|0.6 − 0.8| + |0.2 − 0.1| + |0.7746 − 0.5916|
+ 2 max{|0.6 − 0.8|, |0.2 − 0.1|, |0.7746 − 0.5916|}
+ |0.4 − 0.7| + |0.6 − 0.3| + |0.6928 − 0.6481|
+ 2 max{|0.4 − 0.7|, |0.6 − 0.3|, |0.6928 − 0.6481|}
+ |0.5 − 0.6| + |0.3 − 0.1| + |0.8124 − 0.7937|
+ 2 max{|0.5 − 0.6|, |0.3 − 0.1|, |0.8124 − 0.7937|}]
= 0.7628
8

10. s6(A, B) = 1 − (
1
12
3
X
i=1
[(0.6 − 0.8)2
+ (0.2 − 0.)2
+ (0.7746 − 0.5916)2
+ 2 max{(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
+ 2 max{(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
+ 2 max{(0.5 − 0.6)2
, (0.3 − 0.1)2
, (0.8124 − 0.7937)2
}])
1
2
= 0.7662
Similarly, we obtain
s4(A, C) = 0.7952, s5(A, C) = 0.6706, s6(A, C) = 0.6654,
s4(B, C) = 0.8976, s5(B, C) = 0.8216, s6(B, C) = 0.8309.
Remarks
1. In all the similarities, it follows that 0 ≤ s(A, B) ≤ 1.
2. s(A, B) = 1 iff A = B.
3. It follows that s(A, B) = s(B, A) because of the use of square and absolute
value.
4. s(A, C) + s(B, C) ≥ s(A, B) holds for all the similarity measures.
Clearly, the Conditions (i)–(iv) of Definition 3.1 hold for all the similarity
measures.
Table ?? contains all the values of similarity measures using Example 3.1,
and explains the properties of the discussed similarity measures.
Table 2: Example 3.1
9

11. SM
s(A, B) s(A, C) s(B, C)
s1 0.7589 0.6702 0.8113
s2 0.7706 0.6726 0.8368
s3 0.7600 0.6100 0.8133
s4 0.8505 0.7952 0.8976
s5 0.7628 0.6706 0.8216
s6 0.7662 0.6654 0.8309
3.2.1 Discussion
The following observations are made from Table ??:
1. s4 is the most accurate of the similarity measures discussed, since it pro-
vides the greatest similarity between A and B, A and C, and B and C,
respectively.
2. s1, s2, s3, s4, s5 and s6 satisfy the conditions of Definition 3.1 and hence,
they are appropriate similarity measures for PFSs.
3. Since s4 is the most accurate of the discussed similarity measures, we
adopt it for the applications to be considered.
Howbeit, we consider a case where the level/ground sets of the PFSs are not
equal, to determine whether the similarities will satisfy the aforesaid conditions.
Example 3.2 Let A, B, C ∈ PFS(X) for X = {x1, x2, x3, x4, x5}. Suppose
A = {h
0.6, 0.4
x1
i, h
0.5, 0.7
x2
i, h
0.8, 0.4
x3
i, h
0.7, 0.2
x5
i},
B = {h
0.7, 0.3
x1
i, h
0.4, 0.7
x3
i, h
0.9, 0.2
x4
i}
and
C = {h
0.6, 0.4
x2
i, h
0.7, 0.3
x3
i, h
0.5, 0.4
x4
i}.
10

12. These PFSs could be rewritten thus:
A = {h
0.6, 0.4
x1
i, h
0.5, 0.7
x2
i, h
0.8, 0.4
x3
i, h
0.0, 1.0
x4
i, h
0.7, 0.2
x5
i},
B = {h
0.7, 0.3
x1
i, h
0.0, 1.0
x2
i, h
0.4, 0.7
x3
i, h
0.9, 0.2
x4
i, h
0.0, 1.0
x5
i}
and
C = {h
0.0, 1.0
x1
i, h
0.6, 0.4
x2
i, h
0.7, 0.3
x3
i, h
0.5, 0.4
x4
i, h
0.0, 1.0
x5
i}.
We rewrite the PFSs for easily calculaions. Using the proposed similarity mea-
sures, we obtain the following similarities;
s4(A, B) = 0.5634, s5(A, B) = 0.4054, s6(A, B) = 0.3855.
Similarly, we get
s4(A, C) = 0.5867, s5(A, C) = 0.3873, s6(A, C) = 0.3846,
and
s4(B, C) = 0.6142, s5(B, C) = 0.4968, s6(B, C) = 0.4625.
Remarks
Although the level/ground sets of the PFSs considered here are not equal, we
get observations that coincide with the observations of Example 3.1. They are:
1. In all the similarities, it is observed that 0 ≤ s(A, B) ≤ 1.
2. s(A, B) = 1 iff A = B.
3. It follows that s(A, B) = s(B, A) because of the use of square and absolute
value.
4. s(A, C) + s(B, C) ≥ s(A, B) holds for all the similarity measures.
Clearly, the Conditions (i)–(iv) of Definition 3.1 hold for all the similarity
measures.
Table 3 contains all the values of similarity measures using Example 3.2, and
explains the properties of the discussed similarity measures.
11

13. Table 3: Example 3.2
SM
s(A, B) s(A, C) s(B, C)
s1 0.3328 0.3070 0.4322
s2 0.3602 0.3524 0.4345
s3 0.4220 0.3620 0.4580
s4 0.5634 0.5867 0.6142
s5 0.4054 0.3873 0.4968
s6 0.3855 0.3846 0.4625
3.2.2 Discussion
The following observations are made from Table 3:
1. s4 is the most accurate of the similarity measures discussed, since it pro-
vides the greatest similarity between A and B, A and C, and B and C,
respectively.
2. s1, s2, s3, s4, s5 and s6 satisfy the conditions of Definition 3.1 and hence,
they are appropriate similarity measures for PFSs.
3. Since s4 is the most accurate of the discussed similarity measures, we
adopt it for the applications to be considered.
4 Applicative examples
In this section, we provide some applications of Pythagoren fuzzy sets using
s4 in the areas of electoral process, career placement, and disease diagnosis,
respectively.
12

14. 4.1 Case 1
Suppose there are five provinces in a certain nation with an electoral qualifica-
tion requirements for a position P represented by PFS
à = {
h0.7, 0.2i
x1
,
h0.8, 0.1i
x2
,
h0.7, 0.1i
x3
,
h0.9, 0.0i
x4
,
h0.8, 0.2i
x5
},
where the five provinces are represented by
X = {x1(A), x2(B), x3(C), x4(D), x5(E)}.
Assume that four candidates C1, C2, C3 and C4 represented by PFSs B̃1, B̃2,
B̃3 and B̃4, respectively are vying for position P. Afetr the voting process,
the four candidates gathered the following votes in Pythagorean fuzzy values as
shown below:
B̃1 = {
h0.5, 0.4i
x1
,
h0.8, 0.2i
x2
,
h0.4, 0.3i
x3
,
h0.6, 0.2i
x4
,
h0.7, 0.2i
x5
},
B̃2 = {
h0.8, 0.2i
x1
,
h0.6, 0.3i
x2
,
h0.6, 0.1i
x3
,
h0.3, 0.5i
x4
,
h0.8, 0.1i
x5
},
B̃3 = {
h0.6, 0.3i
x1
,
h0.7, 0.3i
x2
,
h0.7, 0.2i
x3
,
h0.2, 0.6i
x4
,
h0.8, 0.2i
x5
},
B̃4 = {
h0.7, 0.1i
x1
,
h0.8, 0.1i
x2
,
h0.5, 0.4i
x3
,
h0.8, 0.2i
x4
,
h0.5, 0.3i
x5
}.
The collated result is determined by showing which of the candidates B̃i (i =
1, ..., 4) has the greatest similarity value with respect to Ã.
The Pythagorean fuzzy values are gotten thus: let F(x) be the number of
voters that voted for, A(x) be the number of voters that voted against, and
U(x) be the number of voters that remained undecided or cast invalid votes in
every x ∈ X. From the knowledge of PFS, we have
µ(x) =
F(x)
X
and ν(x) =
A(x)
X
,
implying that
F(x) = µ(x)X and A(x) = ν(x)X.
Then
π(x) =
r
1 − [(
F(x)
X
)2 + (
A(x)
X
)2] =
r
1 − [
(F(x))2
+ (A(x))2
X2
]
=
p
X2 − (F(x))2 − (A(x))2
X
.
Thus,
U(x) =
p
X2 − (F(x))2 − (A(x))2.
That is,
X = F(x) + A(x) + U(x).
Now, we calculate the similarity between à and B̃i for (i = 1, ..., 4) as follows:
13

15. s4(Ã, B̃1) = 1 −
1
20
5
X
i=1
[|µÃ(xi) − µ ˜
B1
(xi)| + ||µÃ(xi) − νÃ(xi)| − |µ ˜
B1
(xi) − ν ˜
B1
(xi)||
+ ||µÃ(xi) − πÃ(xi)| − |µ ˜
B1
(xi) − π ˜
B1
(xi)||]
= 0.8126.
Similarly,
s4(Ã, B̃2) = 0.8526, s4(Ã, B̃3) = 0.9049, s4(Ã, B̃4) = 0.8762.
Table 4: Electoral Results
s4 C1 C2 C3 C4
P 0.8126 0.8526 0.9049 0.8762
From the results in Table 4 above, candidate C3 is will be declared winner of
position P because s4(Ã, B̃3) is the greatest of the similarities.
4.2 Case 2
Suppose three students S1, S2, and S3 represented by PFSs ˜
E1, ˜
E2, and ˜
E3 are
vying to study course C in a certain institution, wherein the subjects require-
ments for admission are represented by the set
X = {x1, x2, x3, x4, x5},
where x1 = English language, x2 = Mathematics, x3 = Physics, x4 = Chemistry,
and x5 = Biology.
The following Pythagorean fuzzy values are the scores of the students or
applicants after they sat for an examination over 100% of multi-choice questions
on the listed subjects, within a stipulated time.
˜
E1 = {
h0.6, 0.3i
x1
,
h0.5, 0.4i
x2
,
h0.6, 0.2i
x3
,
h0.5, 0.3i
x4
,
h0.5, 0.5i
x5
},
˜
E2 = {
h0.5, 0.3i
x1
,
h0.6, 0.2i
x2
,
h0.5, 0.3i
x3
,
h0.4, 0.5i
x4
,
h0.7, 0.2i
x5
},
˜
E3 = {
h0.7, 0.1i
x1
,
h0.6, 0.3i
x2
,
h0.7, 0.1i
x3
,
h0.5, 0.4i
x4
,
h0.4, 0.5i
x5
}.
Assume the institution has a bench-mark for admission into studying C repre-
sented by PFS D̃ as
D̃ = {
h0.8, 0.1i
x1
,
h0.7, 0.2i
x2
,
h0.9, 0.0i
x3
,
h0.6, 0.3i
x4
,
h0.8, 0.1i
x5
}.
14

16. Our task is to find which of the students Ẽi (i = 1, ..., 3) is suitable to study C
from the sense of similarity between Ẽi (i = 1, ..., 3) and D̃. Thus,
s4(D̃, ˜
E1) = 1 −
1
20
5
X
i=1
[|µD̃(xi) − µ ˜
E1
(xi)| + ||µD̃(xi) − νD̃(xi)| − |µ ˜
E1
(xi) − ν ˜
E1
(xi)||
+ ||µD̃(xi) − πD̃(xi)| − |µ ˜
E1
(xi) − π ˜
E1
(xi)||]
= 0.8009.
Similarly,
s4(D̃, ˜
E2) = 0.8182, s4(D̃, ˜
E3) = 0.8314.
Table 5: Qualification Results
s4 S1 S2 S3
C 0.8009 0.8182 0.8314
From the results in Table 5, we can see that s4(D̃, ˜
E3) s4(D̃, ˜
E2) s4(D̃, ˜
E1).
Hence, student S3 is the most suitable to study course C from the career place-
ment exercise.
4.3 Case 3
Let us consider a set of diseases given as D = {D1, D2, D3, D4, D5}, where D1 =
viral fever, D2 = malaria fever, D3 = typhoid fever, D4 = stomach problem,
and D5 = chest problem; and a set of symptoms is thus:
X = {x1, x2, x3, x4, x5},
where x1 = temperature, x2 = headache, x3 = stomach pain, x4 = cough, and
x5 = chest pain.
Suppose a sample of a patient P is collected and analysed. Let us represent
the patient P laboratory result as an PFS F̃ given as
F̃ = {
h0.8, 0.1i
x1
,
h0.6, 0.1i
x2
,
h0.2, 0.8i
x3
,
h0.6, 0.1i
x4
,
h0.1, 0.6i
x5
}.
And then let each of the diseases Di (i = 1, ..., 5) be viewed as PFSs
G̃1 = {
h0.4, 0.0i
x1
,
h0.3, 0.5i
x2
,
h0.1, 0.7i
x3
,
h0.4, 0.3i
x4
,
h0.1, 0.7i
x5
},
G̃2 = {
h0.7, 0.0i
x1
,
h0.2, 0.6i
x2
,
h0.0, 0.9i
x3
,
h0.7, 0.0i
x4
,
h0.1, 0.8i
x5
},
G̃3 = {
h0.3, 0.3i
x1
,
h0.6, 0.1i
x2
,
h0.2, 0.7i
x3
,
h0.2, 0.6i
x4
,
h0.1, 0.9i
x5
},
G̃4 = {
h0.1, 0.7i
x1
,
h0.2, 0.4i
x2
,
h0.8, 0.0i
x3
,
h0.2, 0.7i
x4
,
h0.2, 0.7i
x5
},
15

17. G̃5 = {
h0.1, 0.8i
x1
,
h0.0, 0.8i
x2
,
h0.2, 0.8i
x3
,
h0.2, 0.8i
x4
,
h0.8, 0.1i
x5
}.
Now, we calculate the similarity between F̃ and G̃i for (i = 1, ..., 5) as follows:
s4(F̃, G̃1) = 1 −
1
20
5
X
i=1
[|µF̃ (xi) − µ ˜
G1
(xi)| + ||µF̃ (xi) − νF̃ (xi)| − |µ ˜
G1
(xi) − ν ˜
G1
(xi)||
+ ||µF̃ (xi) − πF̃ (xi)| − |µ ˜
G1
(xi) − π ˜
G1
(xi)||]
= 0.8336.
Similarly,
s4(F̃, G̃2) = 0.8686, s4(F̃, G̃3) = 0.8316, s4(F̃, G̃4) = 0.7817, s4(F̃, G̃5) = 0.7827
Table 6: Medical Diagnostic Results
s4 D1 D2 D3 D4 D5
P 0.8336 0.8686 0.8316 0.7817 0.7827
From the results in Table 6, we can infer that patient P is suffering from malaria
fever since s4(F̃, G̃2) is the greatest of the similarities.
5 Conclusion
We have proposed some new similarity measures for PFSs which satisfied the
properties of similarity measure, by taking into account the traditional param-
eters of PFSs. We verified the authenticity of the proposed similarity measures
with reference to some similarity measures for PFSs that also used the three
parameters characterization of Pythagorean fuzzy sets, as studied in [71], and
found that the proposed similarity measures, especially, s4 yields better out-
put. To test the applicability of the proposed similarity measures, some real-life
problems such as, career placement, electioneering process, and disease diagnosis
were considered via s4, for reliable output. The similarity measures introduced
in this work could be used as viable tools in applying PFSs to decision-making
problems.
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