1 Introduction

The theory of fuzzy sets proposed by Zadeh (1965) has several applications because of its ability to handle the imprecision imbedded in real-life situations. A fuzzy set is characterized by a membership function µ which takes value from a crisp set to a unit interval \(I=[0,1]\). Many researchers have worked on the theory of fuzzy sets and its applications (see Chen et al. 2001; Chen and Tanuwijaya 2011; Chen and Chang 2011; Chen et al. 2012; Wang and Chen 2008). The inevitable presence of uncertainty and imprecision in the real world necessitated researchers to develop some mathematical frameworks to cope imprecision more accurately than fuzzy sets. The concept of intuitionistic fuzzy sets (IFSs) proposed in Atanassov (1983, 1986) is one of the generalizations of fuzzy sets with a better applicability.

After the introduction of IFSs theory, many studies that bother on its applications have been carried out, especially in areas like medical diagnosis, electoral system, career determination, appointment procedures, pattern recognition, learning techniques, among others (see Davvaz and Sadrabadi 2016; Chen and Chang 2015; Chen et al. 2016; De et al. 2001; Ejegwa et al. 2014a, b, c; Ejegwa 2015; Ejegwa and Modom 2015; Ejegwa et al. 2016; Hatzimichailidis et al. 2012; Liu and Chen 2017; Szmidt and Kacprzyk 2001, 2004).

Although the notion of IFSs is very resourceful, there are cases where the sum of the membership and nonmembership degrees is greater than one unlike the situation captured in IFSs; where the sum of the membership and nonmembership degrees is less than or equal to one only. This limitation in IFSs led to the introduction of intuitionistic fuzzy sets of second type (IFSST) by Atanassov (1989). The concept was later called Pythagorean fuzzy sets (PFSs) by Yager (2013a). PFS is a new tool to deal with vagueness involved in decision-making. 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 when compare to IFSs. The theory of PFSs has been extensively studied in the literature since inception (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; Yager 2013b, 2014; Ejegwa 2018).

Pythagorean fuzzy set has attracted great attentions of many researchers and subsequently, the concept has been applied to many application areas such as decision-making, aggregation operators, information measures, among others. Perez-Dominguez et al. (2018) presented a multi-objective optimization on the basis of ratio analysis (MOORA) under PFS setting and applied it to multi-criteria decision-making (MCDM) problems. Liang and Xu (2017) proposed the idea of PFSs in hesitant environment and its multi-criteria decision-making (MCDM) by employing the technique for order preference by similarity to ideal solution (TOPSIS) using energy project selection model. Rahman et al. (2017) worked on some geometric aggregation operators on interval-valued PFSs (IVPFSs) and applied the same to group decision-making problem. 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 multi-attribute group decision-making based on induced interval-valued Pythagorean fuzzy Einstein aggregation operator. Garg (2018a) discussed a decision-making problem under Pythagorean fuzzy environment by proposing some generalized aggregation operators. Garg (2018b) proposed an improved score function for solving multi-criteria decision-making (MCDM) problem with partially known weight information, such that the preferences related to the criteria are taken in the form of interval-valued Pythagorean fuzzy sets. Garg (2018d, e) developed a new decision-making model with probabilistic information, using the concept of immediate probabilities to aggregate the information under the Pythagorean fuzzy set environment, and defined two new exponential operational laws about IVPFS and their corresponding aggregation operators with application to multi-criteria decision-making (MCDM). See Gao and Wei (2018); Rahman et al. (2018a); Rahman and Abdullah (2018); Khan et al. (2018a, b); Garg (2016, 2017, 2018c); Du et al. (2017); Hadi-Venchen and Mirjaberi (2014); Yager and Abbasov (2013); Yager (2016) for more applications of PFSs and IVPFSs, respectively.

In this paper, we propose an improved version of max–min–max composite relation for Pythagorean fuzzy sets with application to medical diagnosis. A juxtapositional analysis of the improved composite relation for Pythagorean fuzzy sets and the max–min–max composite relation for Pythagorean fuzzy sets is carried out using some numerical experiments. It follows that the improved version provides a reliable Pythagorean fuzzy relation when compared to max–min–max composite relation for Pythagorean fuzzy sets. Finally, an application of the improved composite relation for Pythagorean fuzzy sets is carried out in medical diagnosis because of its reliable output when compared to max–min–max composite relation. The paper is organized by presenting some mathematical preliminaries of fuzzy sets, IFSs and PFSs, in Sect. 2. In Sect. 3, we present max–min–max composite relation for PFSs, its improved version, and their numerical verifications. An application of the improved composite relation for PFSs to medical diagnosis is explored in Sect. 4. Finally, Sect. 5 concludes the paper and provides direction for future studies.

2 Some basic notions of Pythagorean fuzzy sets

In this section, we recall some mathematical preliminaries of fuzzy sets, IFSs and PFSs.

Definition 1

(See Zadeh 1965) Let X be a nonempty set. A fuzzy set A of X is characterized by a membership function:

$$\begin{aligned} \mu _A:X\rightarrow [0,1]. \end{aligned}$$

That is:

$$\begin{aligned} \mu _A(x) = \left\{ \begin{array}{ll} 1, &{}\quad \text {if}\, x\in X\\ 0, &{}\quad \text {if}\, x\notin X\\ (0,1) &{}\quad \text {if}\, x\, \text {is partly in}\, X \end{array} \right. \end{aligned}$$

Alternatively, a fuzzy set A of X is an object having the form:

$$\begin{aligned} A=\left\{ \left\langle x, \mu _A(x)\right\rangle \mid x\in X\right\} \, \text {or}\, A=\left\{ \left\langle \dfrac{\mu _A(x)}{x} \right\rangle \mid x\in X\right\} , \end{aligned}$$

where the function

$$\begin{aligned} \mu _A(x): X\rightarrow [0,1] \end{aligned}$$

defines the degree of membership of the element \(x\in X\).

Definition 2

(See Atanassov 1983, 1986) Let a nonempty set X be fixed. An IFS A of X is an object having the form:

$$\begin{aligned} A=\lbrace \langle x, \mu _A(x), \nu _A(x) \rangle \mid x\in X\rbrace \end{aligned}$$

or

$$\begin{aligned} A=\left\{ \left\langle \dfrac{\mu _A(x), \nu _A(x)}{x} \right\rangle \mid x\in X\right\} , \end{aligned}$$

where the functions

$$\begin{aligned} \mu _A(x): X\rightarrow [0,1] \, \text {and}\, \nu _A(x): X\rightarrow [0,1] \end{aligned}$$

define the degree of membership and the degree of nonmembership, respectively, of the element \(x\in X\) to A, which is a subset of X, and for every \(x\in X\):

$$\begin{aligned} 0\le \mu _A(x)+ \nu _A(x)\le 1. \end{aligned}$$

For each A in X:

$$\begin{aligned} \pi _A(x)= 1-\mu _A(x)-\nu _A(x) \end{aligned}$$

is the intuitionistic fuzzy set index or hesitation margin of \(x\in X\). The hesitation margin \(\pi _A (x)\) is the degree of nondeterminacy of \(x\in X\), to the set A and \(\pi _A (x)\in [0,1]\). The hesitation margin is the function that expresses the lack of knowledge of whether \(x\in X\) or \(x\notin X\). Thus:

$$\begin{aligned} \mu _A(x)+\nu _A(x)+\pi _A(x)=1. \end{aligned}$$

Example 3

Let \(X=\lbrace x,y,z \rbrace\) be a fixed universe of discourse and

$$\begin{aligned} A=\left\{ \left\langle \dfrac{0.70,0.10}{x}\right\rangle , \left\langle \dfrac{0.85, 0.05}{y}\right\rangle , \left\langle \dfrac{0.50,0.20}{z} \right\rangle \right\} \end{aligned}$$

be the intuitionistic fuzzy set in X. The hesitation margins of the elements x, y, z to A are

$$\begin{aligned} \pi _A(x)=0.20, \, \pi _A(y)=0.10 \, \quad \text {and}\quad \, \pi _A(z)=0.30. \end{aligned}$$

Definition 4

(See Yager 2013a, b) Let X be a universal set. Then, a Pythagorean fuzzy set A which is a set of ordered pairs over X, is defined by the following:

$$\begin{aligned} A=\lbrace \langle x, \mu _A(x), \nu _A(x) \rangle \mid x\in X\rbrace \end{aligned}$$

or

$$\begin{aligned} A=\left\{ \left\langle \dfrac{\mu _A(x), \nu _A(x)}{x} \right\rangle \mid x\in X\right\} , \end{aligned}$$

where the functions

$$\begin{aligned} \mu _A(x): X\rightarrow [0,1] \,\quad \text {and}\quad \, \nu _A(x): X\rightarrow [0,1] \end{aligned}$$

define the degree of membership and the degree of nonmembership, respectively, of the element \(x\in X\) to A, which is a subset of X, and for every \(x\in X\):

$$\begin{aligned} 0\le (\mu _A(x))^2 + (\nu _A(x))^2 \le 1. \end{aligned}$$

Supposing that \((\mu _A(x))^2 + (\nu _A(x))^2 \le 1\); then, there is a degree of indeterminacy of \(x\in X\) to A defined by \(\pi _A(x)=\sqrt{1-[(\mu _A(x))^2 + (\nu _A(x))^2]}\) and \(\pi _A(x)\in [0,1]\). In what follows, \((\mu _A(x))^2 + (\nu _A(x))^2 + (\pi _A(x))^2=1\). Otherwise, \(\pi _A(x)=0\) whenever \((\mu _A(x))^2 + (\nu _A(x))^2 =1\).

We denote the set of all PFSs over X by PFS(X).

Example 5

Let \(A\in PFS(X)\). Suppose that \(\mu _A(x)=0.70\) and \(\nu _A(x)=0.50\) for \(X=\lbrace x\rbrace\). Clearly, \(0.70+0.50\nleq 1\), but \(0.70^2+0.50^2\le 1\). Thus, \(\pi _A(x)=0.5099\), and hence, \((\mu _A(x))^2 + (\nu _A(x))^2 + (\pi _A(x))^2=1\).

Table 1 explains the difference between Pythagorean fuzzy sets and intuitionistic fuzzy sets (Ejegwa 2018).

Table 1 Pythagorean fuzzy sets and Intuitionistic fuzzy sets
Full size table

Definition 6

(See Yager 2013a, b, 2014) Let \(A,B\in PFS(X)\). Then, we have the following:

  1. (i)

    \(A^c=\lbrace \langle x, \nu _A(x), \mu _A(x)\rangle |x\in X \rbrace\).

  2. (ii)

    \(A\cup B=\lbrace \langle x, max(\mu _A(x),\mu _B(x)), min(\nu _A(x),\nu _B(x)) \rangle |x\in X \rbrace\).

  3. (iii)

    \(A\cap B=\lbrace \langle x, min(\mu _A(x),\mu _B(x)), max(\nu _A(x),\nu _B(x)) \rangle |x\in X \rbrace\).

  4. (iv)

    \(A\oplus B=\lbrace \langle x, \sqrt{(\mu _A(x))^2+(\mu _B(x))^2-(\mu _A(x))^2(\mu _B(x))^2}, \nu _A(x)\nu _B(x) \rangle |x\in X \rbrace\).

  5. (v)

    \(A\otimes B=\lbrace \langle x, \mu _A(x)\mu _B(x), \sqrt{(\nu _A(x))^2+(\nu _B(x))^2-(\nu _A(x))^2(\nu _B(x))^2} \rangle |x\in X \rbrace\).

Remark 7

(Ejegwa 2018) Let \(A,B,C\in PFS(X)\). By Definition 6, the following properties hold:

$$\begin{aligned}&(A^c)^c=A\\&A\cap A=A\\&A\cup A=A\\&A\oplus A\ne A\\&A\otimes A\ne A\\&A\cap B=B\cap A\\&A\cup B=B\cup A\\&A\oplus B=B\oplus A\\&A\otimes B=B\otimes A\\&A\cap (B\cap C)=(A\cap B)\cap C\\&A\cup (B\cup C)=(A\cup B)\cup C\\&A\oplus (B\oplus C)=(A\oplus B)\oplus C\\&A\otimes (B\otimes C)=(A\otimes B)\otimes C\\&A\cap (B\cup C)= (A\cap B)\cup (A\cap C)\\&A\cup (B\cap C)= (A\cup B)\cap (A\cup C)\\&A\oplus (B\cup C)= (A\oplus B)\cup (A\oplus C)\\&A\oplus (B\cap C)= (A\oplus B)\cap (A\oplus C)\\&A\otimes (B\cup C)= (A\otimes B)\cup (A\otimes C)\\&A\otimes (B\cap C)= (A\otimes B)\cap (A\otimes C)\\&(A\cap B)^c=A^c\cup B^c\\&(A\cup B)^c=A^c\cap B^c\\&(A\oplus B)^c=A^c\otimes B^c\\&(A\otimes B)^c=A^c\oplus B^c.\\ \end{aligned}$$

3 Pythagorean fuzzy relations

The author has proposed the notion of max–min–max composite relation for PFSs in the other communication. Albeit, in this section, we introduce an improved composite relation for PFSs.

Definition 8

Let X and Y be two nonempty sets. A Pythagorean fuzzy relation (PFR) R from X to Y is a PFS of \(X\times Y\) characterized by the membership function \(\mu _R\) and nonmembership function \(\nu _R\). A PF relation or PFR from X to Y is denoted by \(R(X\rightarrow Y)\).

Definition 9

Let \(A\in PFS(X)\). Then, the max–min–max composite relation of

$$\begin{aligned} R(X\rightarrow Y) \end{aligned}$$

with A is a PFS B of Y denoted by \(B=R\circ A\), such that its membership and nonmembership functions are defined by the following:

$$\begin{aligned} \mu _B(y)=\bigvee _{x}\lbrace \min [\mu _A(x), \mu _R(x,y)]\rbrace \end{aligned}$$

and

$$\begin{aligned} \nu _B(y)=\bigwedge _{x}\lbrace \max [\nu _A(x), \nu _R(x,y)]\rbrace \end{aligned}$$

\(\forall x\in X\) and \(y\in Y\), where \(\bigvee =\) maximum; \(\bigwedge =\) minimum.

Definition 10

Let \(Q(X\rightarrow Y)\) and \(R(Y\rightarrow Z)\) be two PFRs. Then, the max–min–max composite relation \(R\circ Q\) is a PFR from X to Z, such that its membership and nonmembership functions are defined by the following:

$$\begin{aligned} \mu _{R\circ Q}(x,z)=\bigvee _{y}\lbrace \min [\mu _Q(x,y), \mu _R(y,z)]\rbrace \end{aligned}$$

and

$$\begin{aligned} \nu _{R\circ Q}(x,z)=\bigwedge _{y}\lbrace \max [\nu _Q(x,y), \nu _R(y,z)]\rbrace \end{aligned}$$

\(\forall (x,z)\in X\times Z\) and \(\forall y\in Y\).

Definition 11

Let \(A\in PFS(X)\). Then, the improved composite relation of

$$\begin{aligned} R(X\rightarrow Y) \end{aligned}$$

with A is a PFS \(\mathbf {B}\) of Y denoted by \(\textbf{B}=R\circ A\), such that its membership and nonmembership functions are defined by the following:

$$\begin{aligned} \mu _{\mathbf {B}}(y)=\bigvee _{x}\left\{ \dfrac{\mu _A(x)+ \mu _R(x,y)}{2}\right\} \end{aligned}$$

and

$$\begin{aligned} \nu _{\mathbf {B}}(y)=\bigwedge _{x}\left\{ \dfrac{\nu _A(x)+ \nu _R(x,y)}{2}\right\} \end{aligned}$$

\(\forall x\in X\) and \(y\in Y\), where \(\bigvee =\) maximum; \(\bigwedge =\) minimum.

Definition 12

Let \(Q(X\rightarrow Y)\) and \(R(Y\rightarrow Z)\) be two PFRs. Then, the improved composite relation \(\mathbf {R\circ Q}\) is a PFR from X to Z, such that its membership and nonmembership functions are defined by the following:

$$\begin{aligned} \mu _{\mathbf {R\circ Q}}(x,z)=\bigvee _{y}\left\{ \dfrac{\mu _Q(x,y)+ \mu _R(y,z)}{2} \right\} \end{aligned}$$

and

$$\begin{aligned} \nu _{\mathbf {R\circ Q}}(x,z)=\bigwedge _{y}\left\{ \dfrac{\nu _Q(x,y)+ \nu _R(y,z)}{2}\right\} \end{aligned}$$

\(\forall (x,z)\in X\times Z\) and \(\forall y\in Y\).

Remark 13

From Definitions 11 and 12, the improved composite relation \(\mathbf {B}\) or \(\mathbf {R\circ Q}\) is calculated by the following:

$$\begin{aligned} \mathbf {B}=\mu _{\mathbf {B}}(y)-\nu _{\mathbf {B}}(y)\pi _{\mathbf {B}}(y) \end{aligned}$$

\(\forall y\in Y\) or

$$\begin{aligned} \mathbf {R\circ Q}=\mu _{R\circ Q}(x,z)-\nu _{R\circ Q}(x,z)\pi _{R\circ Q}(x,z) \end{aligned}$$

\(\forall (x,z)\in X\times Z\).

Proposition 14

If R andS are two PFRs on \(X\times Y\)and \(Y\times Z\), respectively. Then:

  1. (i)

    \((R^{-1})^{-1}=R\).

  2. (ii)

    \((S\circ R)^{-1}=R^{-1}\circ S^{-1}\).

3.1 Numerical examples

Before applying this relation to medical diagnosis, we discuss the procedures of the approach step-wisely. First, we use max–min–max composite relation and, then, the improved composite relation. A reliability analysis is conducted to ascertain which of the composite relation provides the best relation by comparing the relational values.

Example 15

Let \(E,F\in PFS(X)\) for \(X=\lbrace x_1,x_2,x_3\rbrace\). Suppose

$$\begin{aligned} E=\left\{ \left\langle \dfrac{0.6,0.2}{x_1}\right\rangle , \left\langle \dfrac{0.4,0.6}{x_2}\right\rangle , \left\langle \dfrac{0.5,0.3}{x_3}\right\rangle \right\} \end{aligned}$$

and

$$\begin{aligned} F=\left\{ \left\langle \dfrac{0.8,0.1}{x_1}\right\rangle , \left\langle \dfrac{0.7,0.3}{x_2}\right\rangle , \left\langle \dfrac{0.6,0.1}{x_3}\right\rangle \right\} . \end{aligned}$$

We find the composite relation B using Definitions 9 and 10, respectively:

$$\begin{aligned} \min [\mu _R(e_i, x_j), \mu _S(x_j, f_k)]=0.6,0.4,0.5, \end{aligned}$$

implying that

$$\begin{aligned} \mu _B(e_i,f_k)=\bigvee _{x_j\in X} \lbrace 0.6,0.4,0.5\rbrace =0.6. \end{aligned}$$

Clearly, \(\min [\mu _R(e_i,x_j), \mu _S(x_j,f_k)]\) is gotten by synthesizing Definitions 9 and 10. Applying this to E and F as given in the above example, we observe that the minimum value of the membership values of the elements (i.e., \(x_1,x_2,x_3\)) in E and F, respectively, is 0.6, 0.4, and 0.5.

Again:

$$\begin{aligned} \max [\nu _R(e_i, x_j), \nu _S(x_j, f_k)]=0.2,0.6,0.3, \end{aligned}$$

implying that

$$\begin{aligned} \nu _B(e_i,f_k)=\bigwedge _{x_j\in X} \lbrace 0.2,0.6,0.3\rbrace =0.2. \end{aligned}$$

By explanation, \(\max [\nu _R(e_i,x_j), \nu _S(x_j,f_k)]\) is gotten by synthesizing Definitions 9 and 10. Applying this to E and F as given in the above example, we observe that the maximum value of the nonmembership values of the elements (i.e., \(x_1,x_2,x_3\)) in E and F, respectively, is 0.2, 0.6, and 0.3.

Then:

$$\begin{aligned} B=0.6-(0.2\times 0.7746)=0.4451. \end{aligned}$$

Again, finding \(\mathbf {B}\) using Definitions 11 and 12 with application to E and F, we obtain the following:

$$\begin{aligned} \dfrac{\mu _R(e_i, x_j)+\mu _S(x_j, f_k)}{2}=0.7,0.55,0.55, \end{aligned}$$

implying that

$$\begin{aligned} \mu _B(e_i,f_k)=\bigvee _{x_j\in X} \lbrace 0.7,0.55,0.55\rbrace =0.7. \end{aligned}$$

Again:

$$\begin{aligned} \dfrac{\nu _R(e_i, x_j)+\nu _S(x_j, f_k)}{2}=0.15,0.45,0.2, \end{aligned}$$

implying that

$$\begin{aligned} \nu _B(e_i,f_k)=\bigwedge _{x_j\in X} \lbrace 0.15,0.45,0.2\rbrace =0.15. \end{aligned}$$

Then:

$$\begin{aligned} \mathbf {B}=0.7-(0.15\times 0.6982)=0.5953. \end{aligned}$$

From the aforesaid, the improved composite relation yields a better relation with greater relational value when compared to max–min–max composite relation.

Now, we consider a situation where the elements of PFSs are not equal.

Example 16

Let \(G,H\in PFS(X)\) for \(X=\lbrace x_1,x_2,x_3,x_4,x_5\rbrace\). Suppose that

$$\begin{aligned} G=\left\{ \left\langle \dfrac{0.8,0.4}{x_1}\right\rangle , \left\langle \dfrac{0.5,0.7}{x_2}\right\rangle , \left\langle \dfrac{0.8,0.4}{x_3}\right\rangle , \left\langle \dfrac{0.7,0.2}{x_5}\right\rangle \right\} \end{aligned}$$

and

$$\begin{aligned} H=\left\{ \left\langle \dfrac{0.7,0.3}{x_1}\right\rangle , \left\langle \dfrac{0.4,0.7}{x_3}\right\rangle ,\left\langle \dfrac{0.9,0.2}{x_4}\right\rangle \right\} . \end{aligned}$$

Before calculating the composite relations, we rewrite the PFSs; thus:

$$\begin{aligned} G=\left\{ \left\langle \dfrac{0.8,0.4}{x_1}\right\rangle , \left\langle \dfrac{0.5,0.7}{x_2}\right\rangle , \left\langle \dfrac{0.8,0.4}{x_3}\right\rangle , \left\langle \dfrac{0.0,1.0}{x_4}\right\rangle , \left\langle \dfrac{0.7,0.2}{x_5}\right\rangle \right\} \end{aligned}$$

and

$$\begin{aligned} H=\left\{ \left\langle \dfrac{0.7,0.3}{x_1}\right\rangle , \left\langle \dfrac{0.0,1.0}{x_2}\right\rangle , \left\langle \dfrac{0.4,0.7}{x_3}\right\rangle ,\left\langle \dfrac{0.9,0.2}{x_4}\right\rangle , \left\langle \dfrac{0.0,1.0}{x_5}\right\rangle \right\} . \end{aligned}$$

Now, we find B using Definitions 9 and 10, respectively, as follows:

$$\begin{aligned} \min [\mu _R(g_i, x_j), \mu _S(x_j, h_k)]=0.7,0.0,0.4,0.0,0.0, \end{aligned}$$

implying that

$$\begin{aligned} \mu _B(g_i,h_k)=\bigvee _{x_j\in X} \lbrace 0.7,0.5,0.4,0.9,0.7\rbrace =0.7. \end{aligned}$$

Again,

$$\begin{aligned} \max [\nu _R(g_i, x_j), \nu _S(x_j, h_k)]=0.4,1.0,0.7,1.0,1.0, \end{aligned}$$

implying that

$$\begin{aligned} \nu _B(g_i,h_k)=\bigwedge _{x_j\in X} \lbrace 0.4,1.0,0.7,1.0,1.0\rbrace =0.4. \end{aligned}$$

Thus:

$$\begin{aligned} B=0.7-(0.4\times 0.5916)=0.4634. \end{aligned}$$

In addition, finding \(\mathbf {B}\) using Definitions 11 and 12, we obtain the following:

$$\begin{aligned} \dfrac{\mu _R(g_i, x_j)+\mu _S(x_j, h_k)}{2}=0.75,0.25,0.6,0.45,0.35, \end{aligned}$$

implying that

$$\begin{aligned} \mu _B(g_i,h_k)=\bigvee _{x_j\in X} \lbrace 0.75,0.25,0.6,0.45,0.35\rbrace =0.75. \end{aligned}$$

Again:

$$\begin{aligned} \dfrac{\nu _R(g_i, x_j)+\nu _S(x_j, h_k)}{2}=0.35,0.85,0.55,0.6,0.6, \end{aligned}$$

implying that

$$\begin{aligned} \nu _B(g_i,h_k)=\bigwedge _{x_j\in X} \lbrace 0.35,0.85,0.55,0.6,0.6\rbrace =0.35. \end{aligned}$$

Then:

$$\begin{aligned} \mathbf {B}=0.75-(0.35\times 0.5612)=0.5536. \end{aligned}$$

In this case also, the improved composite relation gives a better relation compared to max–min–max composite relation.

Table 2 gives the comparative analysis of the improved composite relation \(\mathbf {B}\) and max–min–max composite relation B for Pythagorean fuzzy sets. In what follows, the relational value of \(\mathbf {B}\) is greater than that of B. This shows that \(\mathbf {B}\) provides a better Pythagorean fuzzy relation when compared to B.

Table 2 Comparative analysis
Full size table

4 Improved composite relation for Pythagorean fuzzy sets in medical diagnosis

In this section, we present an application of Pythagorean fuzzy set theory to medical diagnosis using the proposed composite relation for PFSs. In a given pathology, suppose that S is a set of symptoms, D is a set of diseases, and P is a set of patients. We define Pythagorean medical knowledge as a Pythagorean fuzzy relation R from the set of symptoms S to the set of diseases D (i.e., on \(S\times D\)) which reveals the degree of association and the degree of nonassociation between symptoms and diseases.

Now, we discuss the notion of Pythagorean fuzzy medical diagnosis via the following methodology:

  1. (i)

    determination of symptoms;

  2. (ii)

    formulation of medical knowledge based on Pythagorean fuzzy relations;

  3. (iii)

    determination of diagnosis on the basis of composition of Pythagorean fuzzy relations.

Let A be a PFS of the set S, and R be a PFR from S to D. Then, the improved composite relation \(\mathbf {B}\) for PFS A with the IFR \(R(S\rightarrow D)\) denoted by

$$\begin{aligned} \mathbf {B}=A\circ R \end{aligned}$$

signifies the state of the patient in terms of diagnosis as a PFS \(\mathbf {B}\) of D with the membership function given by the following:

$$\begin{aligned} \mu _{\mathbf {B}}(d)=\bigvee _{s\in S} \left\{ \dfrac{\mu _A(s)+\mu _R(s,d)}{2}\right\} , \end{aligned}$$

and the nonmembership function is given by the following:

$$\begin{aligned} \nu _{\mathbf {B}}(d)=\bigwedge _{s\in S} \left\{ \dfrac{\nu _A(s)+\nu _R(s,d)}{2}\right\} \end{aligned}$$

\(\forall d\in D\).

If the state of a given patient P is described in terms of a PFS A of S, then P is assumed to be assigned diagnosis in terms of PFS \(\mathbf {B}\) of D, through a PFR R of Pythagorean medical knowledge from S to D which is assumed to be given by a doctor who is able to translate his own observation of the fuzziness involved in degrees of association and nonassociation, respectively, between symptoms and diagnosis.

Now, we extend this concept to a finite number of patients. Let there be n patients \(p_i\) for \(i=1,2,\ldots ,n\) in a given laboratory. Thus, \(p_i\in P\) (or simply, \(p\in P\)). Let R be a PFR (\(S\rightarrow D\)) and construct a PFR Q from the set of patients P to the set of symptoms S. Clearly, the composite relation \(\mathbf {B}\) of PFRs R and \(Q (\mathbf {B} =R\circ Q)\) designates the state of patients p in terms of the diagnosis as a PFR from P to D given by the membership function:

$$\begin{aligned} \mu _{\mathbf {B}}(p,d)=\bigvee _{s\in S} \left\{ \dfrac{\mu _Q(p,s)+\mu _R(s,d)}{2}\right\} \end{aligned}$$

and the nonmembership function is given by the following:

$$\begin{aligned} \nu _{\mathbf {B}}(p,d)=\bigwedge _{s\in S} \left\{ \dfrac{\nu _Q(p,s)+\nu _R(s,d)}{2}\right\} \end{aligned}$$

\(\forall p\in P\) and \(\forall d\in D\).

For a given R and Q, the relation \(\mathbf {B}=R\circ Q\) can be computed. From the knowledge of Q and R, one may find \(\mathbf {B}\) of the PFR for which

$$\begin{aligned} \mathbf {B}=\mu _{\mathbf {B}}(p,d)-\nu _{\mathbf {B}}(p,d) \pi _{\mathbf {B}}(p,d) \end{aligned}$$

is the greatest.

Obviously, R is a significant IFR translating the higher degrees of association and lower degrees of nonassociation of symptoms as well as degrees of hesitation to the diseases, an approach to Pythagorean medical Knowledge. From this approach, one may infer diagnosis from symptoms in the sense of a paired value: one being the degree of association and other the degree of nonassociation.

4.1 Application example

Suppose that four patients, viz, Lil, Jones, Deby, and Inas, visit a given laboratory for medical diagnosis. They are observed to have the following symptoms: temperature, headache, stomach pain, cough, and chest pain. That is, the set of patients P is as follows:

$$\begin{aligned} P =\lbrace \text {Lil, Jones, Deby, and Inas}\rbrace , \end{aligned}$$

and the set of symptoms S is as follows:

$$\begin{aligned} S =\lbrace \text {temperature, headache, stomach pain, cough, and chest pain}\rbrace . \end{aligned}$$

The Pythagorean fuzzy relation \(Q (P\rightarrow S)\) is given hypothetically, in Table 3.

Let the set of diseases D be

$$\begin{aligned} D=\lbrace \text {viral fever, malaria, typhoid, stomach problem, and heart problem}\rbrace . \end{aligned}$$

The Pythagorean fuzzy relation \(R (S\rightarrow D)\) is given hypothetically, in Table 4. The values of the membership and nonmembership functions of the composite relation \(\mathbf {B} =R\circ Q\) are given in Table 5. Note that, the data in Tables 3 and 4 are extracted from De et al. (2001). After finding the degree of hesitation in Pythagorean fuzzy sense (\(\pi =\sqrt{1-[\mu ^2+\nu ^2]}\)), we calculate \(\mathbf {B}\), and is given in Table 6.

Table 3 \(Q(P\rightarrow S)\)
Full size table
Table 4 \(R(S\rightarrow D)\)
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Table 5 \(\mu _{\mathbf {B}}(p,d)\) and \(\nu _{\mathbf {B}}(p,d)\)
Full size table
Table 6 \(\mathbf {B}=\mu _{\mathbf {B}}-\nu _{\mathbf {B}} \pi _{\mathbf {B}}\)
Full size table

4.2 Decisions on the patients medical conditions

With the aid of Table 6, we present the decision-making. Decisions are made based on the greatest value of relation between patients and diseases. In doing this, we present two forms of decision-making approaches.

4.2.1 Horizontal decision

This approach is with respect to the patient against diseases. From the horizontal view of Table 6, we see that:

  1. 1.

    Lil is suffering from malaria fever (0.7170) with some elements of viral fever (0.5601) and typhoid fever (0.5206), respectively.

  2. 2.

    Jones is suffering from stomach problem (0.6644).

  3. 3.

    Deby is suffering from malaria fever (0.7170) with some elements of typhoid fever (0.6293) and viral fever (0.5601), respectively.

  4. 4.

    Inas is suffering from malaria fever (0.6644) with some element of viral fever (0.5083).

4.2.2 Vertical decision

This approach is taken from the vertical view of Table 6, thus:

  1. 1.

    Lil and Deby are suffering from malaria fever with equal severity (0.7170), follows by Inas (0.6644) with less severity compare to Lil and Deby’s cases.

  2. 2.

    Deby is also suffering from typhoid fever (0.6293) and likewise, Lil (0.5206). Clearly, the severity of Deby’s case is more acute than Lil’s case.

  3. 3.

    Lil and Deby are also suffering from viral fever with equal severity (0.5601). Inas is also suffering from viral fever (0.5083) with less severity compare to the cases of Lil and Deby.

  4. 4.

    Jones is suffering from stomach problem with relational value 0.6644.

4.3 Some observations

  1. 1.

    From both horizontal and vertical decision approaches, we notice a similarity among malaria fever, typhoid fever, and viral fever, which conforms to the practice in human medicine.

  2. 2.

    None of the patients is suffering from chest problem, since the relational values are less than 0.5.

  3. 3.

    With the exception to Jones, none of the patients is suffering from stomach problem. It shows the lack of similarity between stomach problem and the set of diseases like malaria fever, typhoid fever, and viral fever.

  4. 4.

    For a better diagnosis, it is expedient to consider and synthesize both horizontal and vertical decision approaches together.

5 Conclusion

The concept of Pythagorean fuzzy sets is a novel mathematical framework in the fuzzy family with higher ability to tackle uncertainty imbedded in decision-making. Some applications of Pythagorean fuzzy sets have been discussed in the literature using different approaches (see Perez-Dominguez et al. 2018; Liang and Xu 2017; Rahman et al. 2017; Mohagheghi et al. 2017; Garg 2018a; Gao and Wei 2018; Khan et al. 2018b; Du et al. 2017; Hadi-Venchen and Mirjaberi 2014; Yager 2016). In this paper, the notion of max–min–max composite relation for Pythagorean fuzzy sets was studied, and the approach was improved and applied to medical diagnosis. A juxtapositional analysis of the improved composite relation for Pythagorean fuzzy sets and the max–min–max composite relation for Pythagorean fuzzy sets was carried out with the aid of numerical experiments. It was shown that the improved version provides a better Pythagorean fuzzy relation when compared to max–min–max composite relation and, hence, the need for its usage to solve medical diagnosis problem. Finally, an application of the improved composite relation for Pythagorean fuzzy sets was carried out in medical diagnosis case using medical database extracted from De et al. (2001), but in Pythagorean fuzzy context. The improved composite relation proposed in this paper could be used as a viable tool in applying Pythagorean fuzzy sets to multi-criteria decision-making (MCDM) problems, multi-attribute decision-making (MADM) problems, pattern recognition problems, etc. Albeit, it is suggestible to consider this approach from object-oriented perspective for quick output in further research.