1 Introduction

Medical diagnosis or diagnosis is the process of deciding which illness or disease describes a patient’s signs and symptoms. The information necessary for diagnosis is usually collected from a history and frequently, physical examination of the patient seeking medical attention. Over and over again, one or more diagnosis processes, like medical tests, are also conducted during the procedure. Diagnosis is time and again thought-provoking, because many signs and symptoms are uncertain. For example, headache by itself, is a sign of numerous diseases and thus does not show the physician what the patient is suffering from Ejegwa and Onyeke (2020). Consequently, differential diagnosis, in which some possible explanations are juxtaposed, must be performed, which could be best done by Pythagorean fuzzy approach. This involves the correlation of many pieces of information followed by the recognition of patterns via composite relation. In fact, the process of medical diagnosis is more challenging when a patient is showing symptoms of some closely related diseases; this also posed a problem to therapeutic process.

Uncertainties are a huge barrier to reckon with in medical diagnostic processes because of fuzziness. The invention of fuzzy sets technology by Zadeh (1965) brought an amazing sight of relief to medical decision-makers, because of the ability of fuzzy model to curb the embedded uncertainties in medical diagnostic processes. Some medical decision-making problems could not be properly resolved with fuzzy approach because fuzzy set only considered membership grade whereas many medical diagnostic processes have the component of both membership grade and non-membership grade with the possibility of hesitation.

Subsequently, Atanassov (1983, 1986) introduced intuitionistic fuzzy sets (IFSs). This construct captured the non-membership degree (NMD) \(\nu \) together with MD \(\mu \) of fuzzy set with a possibility of hesitation margin (HM) \(\pi \) such that \(\mu +\nu \le 1\) and \(\mu +\nu +\pi =1\). The concept of intuitionistic fuzzy sets and its generalizations have been applied to many decision-making problems (see Atanassov 1999; Chen et al. 2016; Chen and Chang 2016; Chen et al. 2016a; Cheng et al. 2016b; Liu et al. 2020; Zeng et al. 2019; De et al. 2001; Garg and Kumar 2018; Szmidt and Kacprzyk 2001; Jana et al. 2019a, 2020a, b; Jana and Pal 2019a, b). However, in some problems like medical diagnosis, the decision on the medical status of a patient may not be taken just once and so discarding the degrees of membership and non-membership in each of the consultations may lead to a compromised diagnosis. By considering the said degrees in each of the consultations, Shinoj and Sunil (2012) proposed intuitionistic fuzzy multiset (IFMS) which has the same features such as intuitionistic fuzzy set but allowing repetitions of membership grade, non-membership grade and hesitation margin. Some fundamentals of intuitionistic fuzzy multisets have been studied (Ejegwa 2016; Ejegwa and Awolola 2013), and the methods of transforming intuitionistic fuzzy multisets to intuitionistic fuzzy sets and fuzzy sets were explicated (Ejegwa 2015). Myriad of applications of IFMSs have been discussed (Das et al. 2013; Rajarajeswari and Uma 2013, 2014; Shinoj and Sunil 2013; Ulucay et al. 2019).

Assuming a decision-maker has a MD \(\mu =0.7\) and a NMD \(\nu =0.4\) for a particular problem, then the framework of intuitionistic fuzzy sets could not be used since \(\mu +\nu \ge 1\). As a result, Yager (2013) proposed Pythagorean fuzzy set which has MD \(\mu \), NMD \(\nu \) and HM \(\pi \) with the properties that \(\mu +\nu \ge 1\) and \(\pi =\sqrt{1-(\mu ^2 +\nu ^2)}\). Pythagorean fuzzy set is a generalized intuitionistic fuzzy set with a more degree of accuracy. Many studies have applied Pythagorean fuzzy sets to several applicative areas (Ejegwa 2019a, b, c, d, e, 2020a, b; Khan et al. 2019; Rahman and Abdullah 2019; Ejegwa and Awolola 2019; Garg 2018a, b; Zhang 2016; Jana et al. 2019b, c). By allowing the repetitions of MD \(\mu \), NMD \(\nu \) and HM \(\pi \), the idea of Pythagorean fuzzy multisets (PFMSs) was introduced by Ejegwa (2020c) and applied in course placement using a max–min–max composite relation on PFMSs. Max–min–max approach use maximum and minimum values of the parameters of PFMS only without considering the average values of the parameters. To this end, we are motivated to modify max–min–max composite relation on PFMSs by incorporating the average values for accuracy sake and address its applicability in medical diagnostic processes. The objectives of this paper are to

  1. (i)

    show the matrix representation of PFMSs and the transformation of PFMSs to Pythagorean fuzzy sets,

  2. (ii)

    explicate the max–min–max composite relation on PFMSs and modify it for better output,

  3. (iii)

    numerically verify the superiority of the proposed composite relation on PFMSs over the existing one,

  4. (iv)

    apply the proposed composite relation on PFMSs in medical diagnosis to curb embedded fuzziness.

The rest of the paper are outlined as follow; Sect. 2 presents the basic notions of PFMSs, Sect. 3 discusses composite relation on PFMSs, its modified version and numerical verifications to ascertain the advantage of the proposed composite relation on PFMSs, Sect. 4 shows the application of the proposed composite relation in solving the problem of medical diagnosis where symptoms are represented in Pythagorean fuzzy setting with multi-values, and Sect. 5 concludes the findings in the paper.

2 Basic notions of Pythagorean fuzzy multisets

Definition 2.1

(Atanassov 1983) An intuitionistic fuzzy set A of X (where X is a non-empty set) is an object having the form

$$\begin{aligned} A=\Bigg \lbrace \Bigg \langle \dfrac{\mu _A(x), \nu _A(x)}{x} \Bigg \rangle \mid x\in X \Bigg \rbrace , \end{aligned}$$
(1)

where the functions \(\mu _A(x),\; \nu _A(x): X\rightarrow [0,1]\) define MD and NMD of the element \(x\in X\) such that

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

For any intuitionistic fuzzy set A of X, \(\pi _A(x)= 1-\mu _A(x)-\nu _A (x)\) is the intuitionistic fuzzy set index or hesitation margin of A.

Definition 2.2

(Shinoj and Sunil 2012) An intuitionistic fuzzy multiset A of X (where X is a non-empty set) is of the form

$$\begin{aligned} {A}=\Bigg \lbrace \Bigg \langle \dfrac{CM_{{A}}(x)}{x}, \dfrac{CN_{{A}}(x)}{x} \Bigg \rangle | x\in X \Bigg \rbrace \end{aligned}$$
(2)

where

$$\begin{aligned} CM_{{A}}(x)= \mu _{{A}}^1 (x), \ldots ,\mu _{{A}}^m (x) \end{aligned}$$

and

$$\begin{aligned} CN_{{A}}(x)= \nu _{{A}}^1 (x), \ldots ,\nu _{{A}}^m (x), \end{aligned}$$

or simply \(CM_{{A}}(x)= \mu _{{A}}^j (x)\) and \(CN_{{A}}(x)= \nu _{{A}}^j (x)\) for \(j=1,\ldots , m\) are the count MD and count NMD defined by the functions \(CM_{{A}}(x),\; CN_{{A}}(x) :X\rightarrow N^{[0,1]}\) such that \(0\le CM_{{A}}(x) + CN_{{A}}(x)\le 1\), \(N=\mathbb {N}\cup \lbrace 0\rbrace \).

For each intuitionistic fuzzy multiset A of X, \(CH_{{A}}(x)= 1-CM_{{A}}(x)-CN_{{A}}(x)\) is the intuitionistic fuzzy multisets index or count HM of A, where \(CH_{{A}}(x)= \pi _{{A}}^1(x), \ldots , \pi _{{A}}^m(x)\).

Definition 2.3

(Yager 2013) A Pythagorean fuzzy sets \(\mathtt {A}\) of X (where X is a non-empty set) is the set of ordered pairs defined by

$$\begin{aligned} \mathtt {A}= \Bigg \lbrace \Bigg \langle \dfrac{\mu _{\mathtt {A}}(x), \nu _{\mathtt {A}}(x)}{x} \Bigg \rangle \mid x\in X \Bigg \rbrace , \end{aligned}$$
(3)

where the functions \(\mu _{\mathtt {A}}(x),\; \nu _{\mathtt {A}}(x):X\rightarrow [0,1]\) define the MD and NMD of the element \(x\in X\) to A such that \(0\le (\mu _{\mathtt {A}}(x))^2 + (\nu _{\mathtt {A}}(x))^2 \le 1\). Assuming \((\mu _{\mathtt {A}}(x))^2 + (\nu _{\mathtt {A}}(x))^2 \le 1\), then there is a degree of indeterminacy of \(x\in X\) to \(\mathtt {A}\) defined by \(\pi _{\mathtt {A}}(x)=\sqrt{1-[(\mu _{\mathtt {A}}(x))^2 + (\nu _{\mathtt {A}}(x))^2]}\) and \(\pi _{\mathtt {A}}(x)\in [0,1]\).

Definition 2.4

(Ejegwa 2020c) A Pythagorean fuzzy multiset \({\mathsf {A}}\) of X (where X is a non-empty set) is characterized by

$$\begin{aligned} {\mathsf {A}}= \Bigg \lbrace \Bigg \langle \dfrac{CM_{\mathsf {A}}(x)}{x}, \dfrac{CN_{\mathsf {A}}(x)}{x} \Bigg \rangle | x\in X \Bigg \rbrace \end{aligned}$$
(4)

alternatively,

$$\begin{aligned} {\mathsf {A}}=\lbrace \langle x, CM_{\mathsf {A}}(x), CN_{\mathsf {A}}(x)\rangle |x\in X\rbrace \end{aligned}$$
(5)

where

$$\begin{aligned} CM_{\mathsf {A}}(x)= \mu _{\mathsf {A}}^1 (x),\ldots ,\mu _{\mathsf {A}}^m (x) \end{aligned}$$

and

$$\begin{aligned} CN_{\mathsf {A}}(x)= \nu _{\mathsf {A}}^1 (x),\ldots ,\nu _{\mathsf {A}}^m (x), \end{aligned}$$

or simply \(CM_{\mathsf {A}}(x)= \mu _{\mathsf {A}}^j(x)\) and \(CN_{ A }(x)= \nu _{ A }^j (x)\) for \(j=1,\ldots ,m\) are the count MD and count NMD defined by the functions \(CM_{\mathsf {A}}(x),\; CN_{\mathsf {A}}(x) :X\rightarrow N^{[0,1]}\) such that \(0\le [CM_{\mathsf {A}}(x)]^2 + [CN_{\mathsf {A}}(x)]^2\le 1\), \(N=\mathbb {N}\cup \lbrace 0\rbrace \).

For any Pythagorean fuzzy multiset \({\mathsf {A}}\) of X,

$$\begin{aligned} CH_{\mathsf {A}}(x)=\sqrt{1-[CM_{\mathsf {A}}(x)]^2-[CN_{\mathsf {A}}(x)]^2} \end{aligned}$$
(6)

is the count HM of \({\mathsf {A}}\), where

$$\begin{aligned} CH_{\mathsf {A}}(x)= \pi _{\mathsf {A}}^1(x), ..., \pi _{\mathsf {A}}^m(x). \end{aligned}$$

The count HM \(CH_{\mathsf {A}}(x)\) is the degree of non-determinacy of x in \({\mathsf {A}}\) and \(CH_{\mathsf {A}}(x)\in [0,1]\). The count HM is the function that expresses lack of knowledge of whether \(x\in {\mathsf {A}}\) or \(x\notin {\mathsf {A}}\).

Throughout this paper \({\mathrm{PFMS}}(X)\) denotes the set of all PFMS of X.

Definition 2.5

(Ejegwa 2020c) Suppose \({\mathsf {A}}\in {\mathrm{PFMS}}(X)\). Then, the level/ground set of \({\mathsf {A}}\) is \({\mathsf {A}}_*=\lbrace x\in X| CM_{\mathsf {A}}(x)>0,\, CN_{\mathsf {A}}(x)<1\rbrace \). It follows that, \({\mathsf {A}}_*\) is a subset of X.

Definition 2.6

(Ejegwa 2020c) Let \({\mathsf {A}}, \mathsf {B}\in {\mathrm{PFMS}}(X)\). Then \({\mathsf {A}}\) and \(\mathsf {B}\) are said to be equal if and only if \(CM_{\mathsf {A}}(x)=CM_{\mathsf {B}}(x)\) and \(CN_{\mathsf {A}}(x)=CN_{\mathsf {B}}(x)\) \(\forall x\in X\).

Definition 2.7

(Ejegwa 2020c) Suppose \({\mathsf {A}}, \mathsf {B}\in {\mathrm{PFMS}}(X)\), then \({\mathsf {A}}\subseteq \mathsf {B}\) if \(CM_{\mathsf {A}}(x)\le CM_{\mathsf {B}}(x) \) and \(CN_{\mathsf {A}}(x)\ge CN_{\mathsf {B}}(x)\) \(\forall x\in X\). Also \({\mathsf {A}}\subset \mathsf {B}\) if \({\mathsf {A}}\subseteq \mathsf {B}\) and \({\mathsf {A}}\ne \mathsf {B}\).

Definition 2.8

(Ejegwa 2020c) Let \({\mathsf {A}}, \mathsf {B}\in {\mathrm{PFMS}}(X)\). Then the following operations hold.

  1. (i)

    \(\overline{\mathsf {A}}= \Bigg \lbrace \Bigg \langle \dfrac{CN_{\mathsf {A}}(x)}{x},\dfrac{CM_{\mathsf {A}}(x)}{x} \Bigg \rangle \mid x\in X \Bigg \rbrace \)

  2. (ii)

    \({\mathsf {A}}\cup \mathsf {B}= \Bigg \lbrace \Bigg \langle \dfrac{\max (CM_{\mathsf {A}}(x), CM_{\mathsf {B}}(x))}{x},\dfrac{\min (CN_{\mathsf {A}}(x), CN_\mathsf {B}{}(x))}{x} \Bigg \rangle \mid x\in X \Bigg \rbrace \)

  3. (iii)

    \({\mathsf {A}}\cup \mathsf {B}= \Bigg \lbrace \Bigg \langle \dfrac{\min (CM_{\mathsf {A}}(x), CM_{\mathsf {B}}(x))}{x},\dfrac{\max (CN_{\mathsf {A}}(x), CN_{\mathsf {B}}(x))}{x} \Bigg \rangle \mid x\in X \Bigg \rbrace \)

Definition 2.9

Let \(X=\lbrace x_i\rbrace \) for \(i=1,...,n\). Then, the PFMS \({\mathsf {A}}\) of X is a Pythagorean fuzzy set \(\mathtt {A}\) of X by the computations:

$$\begin{aligned} \dfrac{1}{n}\ CM_{\mathsf {A}}(x_i)\ =\ \mu _{\mathtt {A}}(x_i), \, \dfrac{1}{n}\ CN_{\mathsf {A}}(x_i)\ = \ \nu _{\mathtt {A}}(x_i). \end{aligned}$$
(7)

Clearly, every Pythagorean fuzzy set is a PFMS but the converse is not true (in particular, if \(i=1\)).

Example 2.10

If \({\mathsf {A}}\) is an PFMS of \(X=\lbrace x,y\rbrace \) such that

$$\begin{aligned} {\mathsf {A}}&= \Bigg \lbrace \Bigg \langle \dfrac{(0.7,0.5,0.4),(0.3, 0.5, 0.6)}{x}\Bigg \rangle ,\\&\qquad \Bigg \langle \dfrac{(0.8, 0.6, 0.4),(0.4, 0.5, 0.5)}{y}\Bigg \rangle \Bigg \rbrace . \end{aligned}$$

To enhance computation, an PFMS \({\mathsf {A}}\) becomes a Pythagorean fuzzy set

$$\begin{aligned} \mathtt {A}=\Bigg \lbrace \Bigg \langle \dfrac{0.5333,0.4667}{x}\Bigg \rangle , \Bigg \langle \dfrac{0.6, 0.4667}{y}\Bigg \rangle \Bigg \rbrace , \end{aligned}$$

and the \(CH_{\mathsf {A}}(x)\) and \(CH_{\mathsf {A}}(y)\) can be computed using (6).

Definition 2.11

Let \(X=\lbrace x_i\rbrace \) for \(i=1,\ldots ,n\). If

$$\begin{aligned} {\mathsf {A}}= & {} \Bigg \lbrace \Bigg \langle \dfrac{(\mu ^1_{\mathsf {A}}(x_1),..., \mu ^m_{\mathsf {A}}(x_1)),(\nu ^1_{\mathsf {A}}(x_1),..., \nu ^m_{\mathsf {A}}(x_1))}{x_1}\Bigg \rangle ,\\&\ldots , \Bigg \langle \dfrac{(\mu ^1_{\mathsf {A}}(x_n),...,\mu ^m_{\mathsf {A}}(x_n)), (\nu ^1_{\mathsf {A}}(x_n),\ldots ,\nu ^m_{\mathsf {A}}(x_n))}{x_n}\Bigg \rangle \Bigg \rbrace \end{aligned}$$

is a PFMS of X. Then, \({\mathsf {A}}\) can be represented in matrix form as

$$\begin{aligned} {\mathsf {A}} = \begin{array}{c} \begin{pmatrix} \mu ^1_{\mathsf {A}}(x_1) &{} ... &{} \mu ^m_{\mathsf {A}}(x_1) \\ \nu ^1_{\mathsf {A}}(x_1) &{} ... &{} \nu ^m_{\mathsf {A}}(x_1) \end{pmatrix}\\ x_1, \end{array} \ldots \begin{array}{c} \begin{pmatrix} \mu ^1_{\mathsf {A}}(x_n) &{} ... &{} \mu ^m_{\mathsf {A}}(x_n) \\ \nu ^1_{\mathsf {A}}(x_n) &{} ... &{} \nu ^m_{\mathsf {A}}(x_n) \end{pmatrix}\\ x_n \end{array} \end{aligned}$$

Using Example 2.10, we have

$$\begin{aligned} {\mathsf {A}} = \begin{array}{c} \begin{pmatrix} 0.7 &{} 0.5 &{} 0.4 \\ 0.3 &{} 0.5 &{} 0.6 \end{pmatrix}\\ x \end{array} , \begin{array}{c} \begin{pmatrix} 0.8 &{} 0.6 &{} 0.4 \\ 0.4 &{} 0.5 &{} 0.5 \end{pmatrix}\\ y \end{array} \end{aligned}$$

3 Composite relation on Pythagorean fuzzy multisets

In this section, we recall the composite relation on PFMSs (Ejegwa 2020c). Suppose X and Y are non-empty sets. Then, a Pythagorean fuzzy multi-relation (PFMR) \(\mathsf {R}\) from X to Y is a PFMS of \(X\times Y\) characterised by \(CM_{\mathsf {R}}\) and \(CN_{\mathsf {R}}\), denoted by \(\mathsf {R}(X\rightarrow Y)\).

3.1 Max–min–max composite relation on Pythagorean fuzzy multisets

Definition 3.1

Suppose \({\mathsf {A}}\in {\mathrm{PFMS}}(X)\). Then, the max–min–max composition of \(\mathsf {R}(X\rightarrow Y)\) with \({\mathsf {A}}\) is a PFMS \(\mathsf {B}\) of Y defined by \(\mathsf {B}=\mathsf {R}\circ {\mathsf {A}}\) where

$$\begin{aligned} \left. \begin{array}{l} CM_{\mathsf {B}}(y)=\max _{x}(\min [CM_{\mathsf {A}}(x), CM_{\mathsf {R}}(x,y)])\\ CN_{\mathsf {B}}(y)=\min _{x}(\max [CN_{\mathsf {A}}(x), CN_{\mathsf {R}}(x,y)]) \end{array}\right\} , \end{aligned}$$
(8)

\(\forall x\in X\) and \(y\in Y\)

Definition 3.2

If \(\mathsf {Q}(X\rightarrow Y)\) and \(\mathsf {R}(Y\rightarrow Z)\) are PFMRs. Then, the max–min–max composition \(\mathsf {R}\circ \mathsf {Q}\) is a PFMR from X to Z where

$$\begin{aligned} \left. \begin{array}{l} CM_{\mathsf {R}\circ \mathsf {Q}}(x,z)=\max _{y}(\min [CM_{\mathsf {Q}}(x,y), CM_{\mathsf {R}}(y,z)])\\ CN_{\mathsf {R}\circ \mathsf {Q}}(x,z)=\min _{y}(\max [CN_{\mathsf {Q}}(x,y), CN_{\mathsf {R}}(y,z)]) \end{array} \right\} , \end{aligned}$$
(9)

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

Remark 3.3

From Definitions 3.1 and 3.2, the max–min–max composite relation \(\mathsf {B}\) or \(\mathsf {R}\circ \mathsf {Q}\) can be computed by

$$\begin{aligned} \beta =CM_{\mathsf {B}}(y)-CN_{\mathsf {B}}(y)CH_{\mathsf {B}}(y) \end{aligned}$$
(10)

\( \forall y\in Y\). Alternatively,

$$\begin{aligned} \beta =CM_{\mathsf {R}\circ \mathsf {Q}}(x,z)-CN_{\mathsf {R}\circ \mathsf {Q}}(x,z)CH_{\mathsf {R}\circ \mathsf {Q}}(x,z) \end{aligned}$$
(11)

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

3.2 Modified composite relation on Pythagorean fuzzy multisets

Now, we propose the modified composite relation on PFMSs and demonstrate its advantage over the max–min–max composite relation using some numerical examples.

Definition 3.4

Suppose \({\mathsf {A}}\in {\mathrm{PFMS}}(X)\). Then, the modified composite relation of \(\mathsf {R}(X\rightarrow Y)\) with \({\mathsf {A}}\) is a PFMS \(\mathsf {B}^*\) of Y defined by \(\mathsf {B}^*=\mathsf {R}\circ {\mathsf {A}}\) where

$$\begin{aligned} \left. \begin{array}{l} CM_{\mathsf {B}^*}(y)=\max _{x}\Bigg [\dfrac{CM_{\mathsf {A}}(x)+ CM_{\mathsf {R}}(x,y)}{2}\Bigg ]\\ CN_{\mathsf {B}^*}(y)=\min _{x}\Bigg [\dfrac{CN_{\mathsf {A}}(x)+ CN_{\mathsf {R}}(x,y)}{2}\Bigg ] \end{array} \right\} , \end{aligned}$$
(12)

\(\forall x\in X\) and \(y\in Y\).

Definition 3.5

If \(\mathsf {Q}(X\rightarrow Y)\) and \(\mathsf {R}(Y\rightarrow Z)\) be two PFMRs. Then, the modified composite relation \((\mathsf {R}\circ \mathsf {Q})^*\) is a PFMR from X to Z where

$$\begin{aligned} \left. \begin{array}{l} CM_{(\mathsf {R}\circ \mathsf {Q})^*}(x,z)=\max _{y}\Bigg [\dfrac{CM_{\mathsf {Q}}(x,y)+ CM_{\mathsf {R}}(y,z)}{2}\Bigg ] \\ CN_{(\mathsf {R}\circ \mathsf {Q})^*}(x,z)=\min _{y}\Bigg [\dfrac{CN_{\mathsf {Q}}(x,y)+ CN_{\mathsf {R}}(y,z)}{2}\Bigg ] \end{array} \right\} , \end{aligned}$$
(13)

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

Remark 3.6

From Definitions 3.4 and 3.5, the modified composite relation \(\mathsf {B}^*\) or \((\mathsf {R}\circ \mathsf {Q})^*\) can be computed by

$$\begin{aligned} \beta ^*=CM_{\mathsf {B}^*}(y)-CN_{\mathsf {B}^*}(y)CH_{\mathsf {B}^*}(y) \end{aligned}$$
(14)

\(\forall y\in Y\). Alternatively,

$$\begin{aligned} \beta ^*=\,\, & {} CM_{(\mathsf {R}\circ \mathsf {Q})^*}(x,z)-CN_{(\mathsf {R}\circ \mathsf {Q})^*}(x,z)\nonumber \\&\times CH_{(\mathsf {R}\circ \mathsf {Q})^*}(x,z), \end{aligned}$$
(15)

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

Proposition 3.7

Suppose \(\mathsf {R}\) and \(\mathsf {Q}\) are two PFMRs on \(X\times Y\) and \(Y\times Z\), respectively. Then

  1. (i)

    \((\mathsf {R}^{-1})^{-1}=\mathsf {R}\),

  2. (ii)

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

3.3 Numerical examples

This subsection shows the performance index of the modified PFMR in comparison to the performance index of max–min–max PFMR using following examples.

Example 3.8

Suppose \(\mathsf {E},\mathsf {F}\in {\mathrm{PFMS}}(X)\) for \(X=\lbrace x_i\rbrace \) for \(i=1,2,3\). If

$$\begin{aligned} \mathsf {E} = \begin{array}{c} \begin{pmatrix} 0.6 &{} 0.5 &{} 0.7 \\ 0.2 &{} 0.3 &{} 0.1 \end{pmatrix}\\ x_1 \end{array}, \begin{array}{c} \begin{pmatrix} 0.4 &{} 0.7 &{} 0.1 \\ 0.6 &{} 0.4 &{} 0.8 \end{pmatrix}\\ x_2 \end{array}, \begin{array}{c} \begin{pmatrix} 0.5 &{} 0.4 &{} 0.6 \\ 0.3 &{} 0.5 &{} 0.1 \end{pmatrix}\\ x_3 \end{array} \end{aligned}$$
$$\begin{aligned} \mathsf {F} = \begin{array}{c} \begin{pmatrix} 0.6 &{} 0.8 &{} 1.0 \\ 0.2 &{} 0.1 &{} 0.0 \end{pmatrix}\\ x_1 \end{array}, \begin{array}{c} \begin{pmatrix} 0.7 &{} 0.6 &{} 0.8 \\ 0.3 &{} 0.4 &{} 0.2 \end{pmatrix}\\ x_2 \end{array}, \begin{array}{c} \begin{pmatrix} 0.6 &{} 0.5 &{} 0.7 \\ 0.1 &{} 0.2 &{} 0.0 \end{pmatrix}\\ x_3 \end{array} \end{aligned}$$

Using Definitions 3.1 and 3.2, respectively we have

$$\begin{aligned} \min [CM_{\mathsf {R}}(e_i, x_j), CM_{\mathsf {S}}(x_j, f_k)]=0.6,0.4,0.5 \end{aligned}$$

implying that

$$\begin{aligned} CM_{\mathsf {B}}(e_i,f_k)=\max _{x_j\in X} [0.6,0.4,0.5] =0.6. \end{aligned}$$

Applying this to \(\mathsf {E}\) and \(\mathsf {F}\), we see that the minimum of the membership values of the elements in \(\mathsf {E}\) and \(\mathsf {F}\), respectively are 0.6, 0.4 and 0.5.

Also,

$$\begin{aligned} \max [CN_{\mathsf {R}}(e_i, x_j), CN_{\mathsf {S}}(x_j, f_k)]=0.2,0.6,0.3 \end{aligned}$$

implying that

$$\begin{aligned} CN_{\mathsf {B}}(e_i,f_k)=\min _{x_j\in X} [0.2,0.6,0.3] =0.2. \end{aligned}$$

It follows that the maximum of the non-membership values of the elements in \(\mathsf {E}\) and \(\mathsf {F}\), respectively are 0.2, 0.6 and 0.3.

Thus

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

Now, by applying the modified composite relation on \(\mathsf {E}\) and \(\mathsf {F}\) using Definitions 3.4 and 3.5, we have

$$\begin{aligned} \dfrac{CM_{\mathsf {R}}(e_i, x_j)+CM_{\mathsf {S}}(x_j, f_k)}{2}=0.7,0.55,0.55 \end{aligned}$$

that is,

$$\begin{aligned} CM_{\mathsf {B}^*}(e_i,f_k)=\max _{x_j\in X} [0.7,0.55,0.55] =0.7. \end{aligned}$$

Similarly,

$$\begin{aligned} \dfrac{CN_{\mathsf {R}}(e_i, x_j)+CN_{\mathsf {S}}(x_j, f_k)}{2}=0.15,0.45,0.2 \end{aligned}$$

that is,

$$\begin{aligned} CN_{\mathsf {B}^*}(e_i,f_k)=\min _{x_j\in X} [0.15,0.45,0.2] =0.15. \end{aligned}$$

Thus

$$\begin{aligned} \beta ^*=0.7-(0.15\times 0.6982)=0.5953. \end{aligned}$$

From the computations, the modified composite relation gives a better relation between \(\mathsf {E}\) and \(\mathsf {F}\) when compare to max–min–max composite relation.

Example 3.9

Suppose \(\mathsf {G},\mathsf {H}\in {\mathrm{PFMS}}(X)\) for \(X=\lbrace x_i\rbrace \) for \(i=1,\ldots , 5\).

$$\begin{aligned} \mathsf {G} = \begin{array}{c} \begin{pmatrix} 0.6 &{} 0.8 &{} 1.0 \\ 0.4 &{} 0.6 &{} 0.2 \end{pmatrix}\\ x_1 \end{array}, \begin{array}{c} \begin{pmatrix} 0.5 &{} 0.6 &{} 0.4 \\ 0.7 &{} 0.4 &{} 1.0 \end{pmatrix}\\ x_2 \end{array}, \begin{array}{c} \begin{pmatrix} 0.8 &{} 0.7 &{} 0.9 \\ 0.4 &{} 0.6 &{} 0.2 \end{pmatrix}\\ x_3 \end{array}, \begin{array}{c} \begin{pmatrix} 0.7 &{} 0.5 &{} 0.9 \\ 0.2 &{} 0.1 &{} 0.3 \end{pmatrix}\\ x_5 \end{array} \end{aligned}$$
$$\begin{aligned} \mathsf {H} = \begin{array}{c} \begin{pmatrix} 0.7 &{} 0.6 &{} 0.8 \\ 0.3 &{} 0.2 &{} 0.4 \end{pmatrix}\\ x_1 \end{array}, \begin{array}{c} \begin{pmatrix} 0.4 &{} 0.6 &{} 0.2 \\ 0.7 &{} 0.5 &{} 0.9 \end{pmatrix}\\ x_3 \end{array}, \begin{array}{c} \begin{pmatrix} 0.9 &{} 0.8 &{} 1.0 \\ 0.2 &{} 0.4 &{} 0.0 \end{pmatrix}\\ x_4 \end{array} \end{aligned}$$

Certainly, \(\mathsf {G}_*\ne \mathsf {H}_*\).

From Definitions 3.1 and 3.2, respectively, we have

$$\begin{aligned} \min [CM_{\mathsf {R}}(g_i, x_j), CM_{\mathsf {S}}(x_j, h_k)]=0.7,0.0,0.4,0.0,0.0 \end{aligned}$$

and

$$\begin{aligned} CM_{\mathsf {B}}(g_i,h_k)=\max _{x_j\in X} [0.7,0.5,0.4,0.9,0.7] =0.7. \end{aligned}$$

Similarly,

$$\begin{aligned} \max [CN_{\mathsf {R}}(g_i, x_j), CN_{\mathsf {S}}(x_j, h_k)]=0.4,1.0,0.7,1.0,1.0 \end{aligned}$$

and

$$\begin{aligned} CN_{\mathsf {B}}(g_i,h_k)=\min _{x_j\in X} [0.4,1.0,0.7,1.0,1.0] =0.4. \end{aligned}$$

Thus

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

Also, computing \(\beta ^*\) using Definitions 3.4 and 3.5, we get

$$\begin{aligned} \dfrac{CM_{\mathsf {R}}(g_i, x_j)+CM_{\mathsf {S}}(x_j, h_k)}{2}=0.75,0.25,0.6,0.45,0.35 \end{aligned}$$

and

$$\begin{aligned} CM_{\mathsf {B}}(g_i,h_k)=\max _{x_j\in X}[0.75,0.25,0.6,0.45,0.35] =0.75. \end{aligned}$$

Again,

$$\begin{aligned} \dfrac{CN_{R}(g_i, x_j)+CN_{\mathsf {S}}(x_j, h_k)}{2}=0.35,0.85,0.55,0.6,0.6 \end{aligned}$$

and

$$\begin{aligned} CN_{\mathsf {B}}(g_i,h_k)=\min _{x_j\in X} [0.35,0.85,0.55,0.6,0.6] =0.35. \end{aligned}$$

Then

$$\begin{aligned} \beta ^*=0.75-(0.35\times 0.5612)=0.5536. \end{aligned}$$

In this example also, the modified composite relation yields a better relation between \(\mathsf {G}\) and \(\mathsf {H}\). Table 1 provides a quick comparison between the modified composite relation \(\beta ^*\) and the max–min–max composite relation \(\beta \) on PFMSs.

Table 1 Comparison between \(\beta \) and \(\beta ^*\)
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4 Diagnostic processes using composite relations in Pythagorean fuzzy environment

Medical diagnosis/testing is a delicate exercise because failure to make the right decision may lead to the death of the patient. In this section, we present a scenario of mathematical approach of medical diagnosis to ascertain the medical conditions of some patients via a novel composite relation on PFMSs, where the symptoms or clinical manifestations of the diseases are represented in PFMSs framework using a hypothetical approach.

Assuming there are four patients represented by the set \(\mathsf {P}_j\) for \(j=1,2,3,4\), who are billed for medical diagnosis due to the manifestation of some symptoms. After critical analysis on the samples collected from \(\mathsf {C}_j\), the following major symptoms are observed;

$$S=\lbrace s_1, s_2, s_3, s_4, s_5\rbrace ,$$

where \(s_1=\) fever, \(s_2=\) cough, \(s_3=\) shortness of breath or breathing difficulties, \(s_4=\) sore throat, and \(s_5=\) headache.

Suppose \(\mathsf {D}=\lbrace \mathsf {D}_1, \mathsf {D}_2, \mathsf {D}_3, \mathsf {D}_4, \mathsf {D}_5\rbrace \) are set of diseases with relatively common symptoms, where \(\mathsf {D}_1=\) influenza, \(\mathsf {D}_2=\) viral fever, \(\mathsf {D}_3=\) hay fever, \(\mathsf {D}_4=\) pneumonia, and \(\mathsf {D}_5=\) common cold which \(\mathsf {P}_1\), \(\mathsf {P}_2\), \(\mathsf {P}_3\), and \(\mathsf {P}_4\) are likely to be infected with.

The Pythagorean fuzzy multi-relation \(\mathsf {N}(\mathsf {S}\rightarrow \mathsf {D})\) is hypothetically given in Table 2 based on the medical knowledge of the enlisted diseases. The Pythagorean fuzzy multi-relation \(\mathsf {M}(\mathsf {P}\rightarrow \mathsf {S})\) is hypothetically given in Table 3 based on the medical analysis on \(\mathsf {P}_j=\lbrace \mathsf {P}_1, \mathsf {P}_2, \mathsf {P}_3, \mathsf {P}_4\rbrace \). After applying Eq. (7) on Tables 2 and 3 (for the ease of computations), we obtain the values of the membership and non-membership grades of \(\mathsf {B}\) and \(\mathsf {B}^*\) in Tables 4 and 6. After computing the degree of hesitation count via Eq. (6), we calculate \(\beta \) and \(\beta ^*\) as presented in Tables 5 and 7.

Table 2 \(\mathsf {N}(\mathsf {S}\rightarrow \mathsf {D})\)
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The first row represents the count membership degrees while the second row represents the count non-membership degrees, respectively.

Table 3 \(\mathsf {M}(\mathsf {P}\rightarrow \mathsf {S})\)
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The Pythagorean fuzzy multi-values in Table 3 were taken at three different times to curb any chance of error; this is unlike in the instance of Pythagorean fuzzy values. The first column represents the first result in both membership and non-membership grades, the second column and the third column likewise.

Applying max–min–max composite relation, we obtain Tables 4 and 5, respectively.

Table 4 \(CM_{\mathsf {B}}(c,d)\) and \(CN_{\mathsf {B}}(c,d)\)
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Table 5 Result outputs
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The following diagnoses are obtained from Table 5; \(\mathsf {P}_1\) is diagnosed with influenza and viral fever with some symptoms of pneumonia and common cold in that order, \(\mathsf {P}_2\) is diagnosed with influenza (but should be treated for viral fever and pneumonia), \(\mathsf {P}_3\) is diagnosed of influenza and viral fever (but should be treated for pneumonia), \(\mathsf {P}_4\) is diagnosed with pneumonia and common cold (but should be treated for influenza). It is noticed that \(\mathsf {P}_1\) and \(\mathsf {P}_3\) are diagnosed of the same ailments (but \(\mathsf {P}_1\) has a common cold in addition.)

Using the modified approach, we obtain Tables 6 and 7, respectively.

Table 6 \(CM_{\mathsf {B}^*}(c,d)\) and \(CN_{\mathsf {B}^*}(c,d)\)
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Table 7 Result outputs
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From Table 7, we infer that \(\mathsf {P}_1\) is tested positive for viral fever in a severe situation with very prominent symptoms of influenza, pneumonia, common cold and hay fever in that order. \(\mathsf {P}_2\) is tested positive for viral fever in a mild situation with less viral load. \(\mathsf {P}_3\) is tested positive for viral fever in a severe situation with very prominent symptoms of influenza, pneumonia, hay fever and common cold in that order. \(\mathsf {P}_4\) is tested positive for both viral fever and common cold in a mild situation with some symptoms of influenza and pneumonia in that order.

It is observed that \(\mathsf {P}_1\) and \(\mathsf {P}_3\) situations are very severe with the same viral load, follow by \(\mathsf {P}_4\) and \(\mathsf {P}_2\) in that order. In fact, \(\mathsf {P}_2\) has a very less viral load in comparison to the other cases. All the suspected cases test positive for viral fever but with a different viral load to enhance treatment and attention.

Synthesizing the information in Tables 5 and 7, we observe that the diagnoses given by max–min–max composite relation are different from the diagnostic results gotten from the modified approach. Notwithstanding, the diagnostic results of the modified approach are reliable because the average of the membership and non-membership grades of the symptoms were considered unlike the max–min–max approach where the minimum of the membership grades and the maximum of the non-membership grades of the symptoms were considered. Thus, it is meet to say that the results of the modified approach are more precise and reliable.

5 Conclusion

Medical diagnosis is an essential exercise because it determines recovery and so, a wrong diagnosis could lead to death or very critical cases. In this paper, we have addressed the mathematical approach of medical diagnosis using a modified composite relation on PFMSs, where symptoms were captured in Pythagorean fuzzy multi-values to prevent any chance of error-influence on the diagnostic processes. The nexus between Pythagorean fuzzy multi-values and Pythagorean fuzzy values was established with the aid of a proposed formula, and the matrix representation of PFMSs was introduced. Max–min–max composite relation on PFMSs has been studied (Ejegwa 2020c). However, max–min–max approach used maximum and minimum values of the parameters of PFMS only without considering the average values. To remedy this limitation, we have modified max–min–max composite relation on PFMSs to enhance reliable output by incorporating the average values of the PFMSs’ parameters. The modified composite relation on PFMSs was verified to have high performance over the max–min–max composite relation. A hypothetical case of medical diagnosis on some selected patients was considered via the modified composite relation on PFMSs. Medical diagnosis on four patients was considered in the hypothetical case. An algorithmic approach embedded with the modified approach could be employed to address the medical diagnosis of more patients in future research.