On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis

Abstract

In the present communication, we introduce Pythagorean fuzzy soft matrix and its various possible types. Some binary operations and various properties over the matrices are also being defined with their proof of validity. Further, the Pythagorean fuzzy soft matrices have been taken into account for proposing a new algorithm for decision making by using choice matrix and weighted choice matrix. In addition to this, an algorithm for medical diagnosis problem by making use of score matrix and utility matrix has also been proposed. Numerical examples for each of the applications have been successfully illustrated. A comparative analysis with other existing methods has also been carried out.

1 Introduction

In the literature, there are many theories which are used to deal with vagueness and uncertainties of many problems arising in engineering, economics, social science, etc. But all the theories have their own limitations intuitively because of the parametrization tool involved in it. In order to overcome these difficulties, Molodstov (1999) introduced a novel mathematical tool for dealing with the uncertainties, called soft set, which is free from the inadequacy of parametrization and established various results based on this. In 2001, Maji et al. (2001) studied the theory of soft sets and defined soft binary operations such as AND, OR, union, intersection, equality, complement of the soft sets. Further, Maji et al. (2002, 2003) successfully extended the soft set to fuzzy soft set and intuitionistic fuzzy soft set and studied the application of these soft sets in decision-making problems. Peng et al. (2015) introduced the Pythagorean fuzzy soft set (PFSS) and studied various binary operations over PFSS and also proposed a decision-making algorithm based on the Pythagorean fuzzy soft set. Naim and Serdar (2010) introduced the soft matrices which are representations of the Molodtsov’s soft sets and successfully applied the soft matrices in decision-making problems. Yong and Chenli (2011) and Chetia and Das (2012) extended the matrix representation of soft set to fuzzy soft set and intuitionistic fuzzy soft matrix, respectively, and applied it to decision-making problems. Various other researchers have also worked on the concept of soft set and soft matrices which are available in the literature.

In this paper, Pythagorean fuzzy soft matrices and its types are introduced and defined based on the Pythagorean fuzzy soft set. Also, the various binary operations are analogously proposed for the Pythagorean fuzzy soft matrices and a new decision-making algorithm has been proposed. We have studied some preliminaries and fundamental notions in Sect. 2. In Sect. 3, we have formally introduced the idea of the Pythagorean fuzzy soft matrix and extended various binary operations on them. In Sect. 4, we have framed choice and weighted choice matrix to propose an algorithm to solve decision-making problem by taking Pythagorean fuzzy soft matrix into account and illustrated through an example. In a similar way, an algorithm for medical diagnosis problem by using score and utility matrix has been proposed in view of the Pythagorean fuzzy soft matrices with a numerical example in Sect. 5. A detailed methodology and comparative analysis have also been provided to discuss the reliability of the proposed method. Finally, the paper has been concluded in Sect. 6.

2 Preliminaries

In this section, we recall and present some fundamental concepts in connection with the Pythagorean fuzzy set, which are well known in literature.

Definition 1

(Atanassov 1986) An intuitionistic fuzzy set (IFS) I in X (universe of discourse) is given by

$$\begin{aligned} I=\left\{ <x, \mu _{I}(x), \nu _{I}(x)> \mid x \in X\right\} ; \end{aligned}$$

where $\mu _{I}: X \rightarrow [0,1]$ and $\nu _{I}: X \rightarrow [0,1]$ denote the degree of membership and degree of non-membership, respectively, and for every $x \in X$ satisfy the condition

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

and the degree of indeterminacy for any IFS I and $x\in X$ is given by

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

Definition 2

(Yager 2013) A Pythagorean fuzzy set (PFS) M in X (universe of discourse) is given by

$$\begin{aligned} M=\left\{ <x, \mu _{M}(x), \nu _{M}(x)>\mid x\in X\right\} ; \end{aligned}$$

where $\mu _{M}:X\rightarrow [0,1]$ and $\nu _{M}:X\rightarrow [0,1]$ denote the degree of membership and degree of non-membership, respectively, and for every $x\in X$ satisfy the condition

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

and the degree of indeterminacy for any Pythagorean fuzzy set M and $x\in X$ is given by

$$\begin{aligned} \pi _{M}(x)=\sqrt{1 - \mu _{M}^{2}(x) - \nu _{M}^{2}(x)}. \end{aligned}$$

In case of PFS, the restriction corresponding to the degree of membership $\mu _M(x)$ and the degree of non-membership $\nu _M(x)$ is

$$\begin{aligned} 0\le \mu _{M}^{2}(x) + \nu _{M}^{2}(x)\le 1, \end{aligned}$$

whereas the condition in case of IFS is

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

for $\mu _M(x), \nu _M(x) \in [0,1]$. This difference in constraint conditions gives an additional advantage for a wider coverage of information span which can be geometrically shown in Fig. 1.

Fig. 1

IFS versus PFS

The generalization in terms of development of concepts from Soft Sets to Pythagorean fuzzy soft sets is available with explanatory examples in the literature (Molodstov 1999; Naim and Serdar 2010; Peng et al. 2015).

Let $X={\{x_{1}, x_{2},\ldots , x_{m}}\}$ be the universe of discourse and $E={\{e_{1}, e_{2},\ldots , e_{n}}\}$ be the set of parameters. Consider $A \subseteq E$. The basic notion of soft set, soft matrix and Pythagorean fuzzy soft set is presented below which is well known in the literature:

• The pair $(F_{A}, E)$ is called soft set over X if and only if $F_{A} : A \rightarrow {\mathcal P}(X)$, where ${\mathcal P}(X)$ is the power set of X.

• Let F(X) denotes the set of all fuzzy sets of X. A pair $(F_{A}, E)$ is called a fuzzy soft set over F(X), where F is a mapping given by $F_{A} : A \rightarrow {\mathcal P}(F(X))$.

• The pair $(F_{A}, E)$ is called the Pythagorean fuzzy soft set (PFSS) over X if $F_{A} : A \rightarrow PFS(X)$ and can be represented as

  $$\begin{aligned} (F_{A}, E)=\left\{ (e, F_{A}(e)) : e \in A, F_{A}(e) \in \textit{PFS}(X)\right\} , \end{aligned}$$

  where $\textit{PFS}(X)$ denotes the set of all Pythagorean fuzzy sets of X.

• Let $(F_{A}, E)$ be a soft set over X. Then, the subset $X \times E$ is uniquely defined by $R_{A}=\{(x, e), e \in A, x \in F_{A}(e)\}$.

  The characteristic function of $R_{A}$ is $\chi _{_{R_{A}}}: X \times E \rightarrow [0,1]$ given by

  $$\begin{aligned} \chi _{_{R_{A}}}(x, e)={\left\{ \begin{array}{ll} \begin{array}{c} 1\\ 0 \end{array} &{} \begin{array}{c} \hbox {if} \ (x, e) \in A\\ \hbox {if} \ (x,e)\notin A \end{array}\end{array}\right. }. \end{aligned}$$

  If $a_{ij}= \chi _{_{R_{A}}}(x_{i}, e_{j})$, then a matrix $[a_{ij}]=[\chi _{_{R_{A}}}(x_{i}, e_{j})]$ is called soft matrix of the soft set $(F_{A}, E)$ over X of order $m\times n$.

3 Pythagorean fuzzy soft matrices and various operations

Since matrices play an important role in many computational techniques, handling dimensionality feature of various problems of engineering, medical sciences, social sciences, etc., it motivates to extend the concept of Pythagorean fuzzy soft set to Pythagorean fuzzy soft matrices. In this section, we propose the concept of Pythagorean fuzzy soft matrix with various operations over it.

If $(F_{A}, E)$ be a Pythagorean fuzzy soft set over X, then the subset $X \times E$ is uniquely defined by $R_{A}= \{(x, e), e\in A, x\in F_{A}(e)\}$. The $R_{A}$ can be characterized by its membership function and non-membership function given by $\mu _{R_{A}}: X \times E \rightarrow [0, 1]$   and   $\nu _{R_{A}}: X \times E \rightarrow [0,1]$, respectively.

If $(\mu _{ij}, \nu _{ij})=(\mu _{R_{A}}(x_{i}, e_{j}), \nu _{R_{A}}(x_{i}, e_{j}))$, where $\mu _{R_{A}}$$(x_{i}, e_{j})$ is the membership of $x_i$ in the Pythagorean fuzzy set $F(e_j)$ and $\nu _{R_{A}}(x_{i}, e_{j})$ is the non-membership of $x_i$ in the Pythagorean fuzzy set $F(e_j)$, respectively, then we define a matrix given by

$$\begin{aligned}{}[M]= & {} [m_{ij}]_{m\times n}=\left[ \left( \mu _{ij}^{M}, \nu _{ij}^{M}\right) \right] _{m\times n}\\= & {} \left[ \begin{array}{cccc} (\mu _{11}, \nu _{11)} &{} \quad (\mu _{12}, \nu _{12)} &{}\quad \cdots &{} \quad (\mu _{1n}, \nu _{1n})\\ (\mu _{21}, \nu _{21}) &{} \quad (\mu _{22}, \nu _{22}) &{} \quad \cdots &{} \quad (\mu _{2n}, \nu _{2n})\\ \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots \\ (\mu _{m1}, \nu _{m1}) &{} \quad (\mu _{m2}, \nu _{m2}) &{} \quad \cdots &{} \quad (\mu _{mn}, \nu _{mn}) \end{array}\right] \end{aligned}$$

which is called Pythagorean fuzzy soft matrix of order $m\times n$ over X.

For a better understanding, let us consider $X=\{x_{1}, x_{2}, x_{3}\}$ as a universal set and $E={\{e_{1}, e_{2}, e_{3}, e_{4}}\}$ as a set of parameters. If $A=\{e_{1}, e_{2}, e_{3}\}\subseteq E$ and

$$\begin{aligned} F_{A}(e_{1})=&\{(x_{1},0.6,0.5), (x_{2},0.5,0.8), (x_{3},0.9,0.2)\},\\ F_{A}(e_{2})=&\{(x_{1},0.8,0.5), (x_{2},0.9,0.3), (x_{3},0.6,0.6)\},\\ F_{A}(e_{3})=&\{(x_{1},0.6,0.7), (x_{2},0.5,0.6), (x_{3},0.7,0.5)\}, \end{aligned}$$

then $(F_{A}, E)$ is the parameterized family of $F_{A}(e_{1}),$$F_{A}(e_{2}), F_{A}(e_{3})$ over X.

Hence, the Pythagorean fuzzy soft matrix $[M(F_{A},E)]$ can be written as

$$\begin{aligned}{}[M]= \left[ \left( \mu _{ij}^{M},\nu _{ij}^{M}\right) \right] _{m\times n}=\left[ \begin{array}{ccc} (0.6,0.5) &{} \quad (0.8,0.5) &{} \quad (0.6,0.7)\\ (0.5,0.8) &{}\quad (0.9,0.3) &{} \quad (0.5,0.6)\\ (0.9,0.2) &{} \quad (0.6,0.6) &{}\quad (0.7,0.5) \end{array}\right] . \end{aligned}$$

Suppose $\textit{PFSM}_{m\times n}$ is a collection of all Pythagorean fuzzy soft matrices over X. Subsequently, various kinds of Pythagorean fuzzy soft matrices have been analogously proposed. A Pythagorean fuzzy soft matrix $M=[(\mu _{ij}^{M}, \nu _{ij}^{M})] \in \textit{PFSM}_{m\times n}$ is called:

• Pythagorean fuzzy soft zero matrix if

  $$\begin{aligned} \mu _{ij}^{M}=0 \,\,\text{ and }\,\, \nu _{ij}^{M}=0; \forall i, j \,\,\text{ and } \text{ is } \text{ denoted } \text{ by }\,\, 0 = [0,0]. \end{aligned}$$

• Pythagorean fuzzy soft square matrix if $m=n$.

• Pythagorean fuzzy soft row matrix if $n=1$.

• Pythagorean fuzzy soft column matrix if $m=1$.

• Pythagorean fuzzy soft diagonal matrix if all its non-diagonal elements are zero $\forall$i, j.

• Pythagorean fuzzy soft$\mu$-universal matrix if $\mu _{ij}^{M}=1$ and $\nu _{ij}^{M}=0$$\forall$i and j, denoted by $P_{\mu }$.

• Pythagorean fuzzy soft$\nu$-universal matrix if $\mu _{ij}^{M}=0$ and $\nu _{ij}^{M}=1$$\forall$i and j, denoted by $P_{\nu }$.

• Scalar multiplication of Pythagorean fuzzy soft matrix: For any scalar k, we define $kA=[(k\mu _{ij}^{M},k\nu _{ij}^{M})]$, $\forall$i and j.

Further, we define the following relations over two Pythagorean fuzzy soft matrices $M=[(\mu _{ij}^{M}, \nu _{ij}^{M})]$ and $N=[(\mu _{ij}^{N}, \nu _{ij}^{N})] \in \textit{PFSM}_{m\times n}$:

• Sub matrix:$M\subseteq N$ if $\mu _{ij}^{M}\le \mu _{ij}^{N}$ and $\nu _{ij}^{M}\ge \nu _{ij}^{N}$$\forall$i and j.

• Super matrix:$M\supseteq N$ if $\mu _{ij}^{M}\ge \mu _{ij}^{N}$ and $\nu _{ij}^{M}\le \nu _{ij}^{N}$$\forall$i and j.

• Equal matrix:$M=N$ if $\mu _{ij}^{M}=\mu _{ij}^{N}$ and $\nu _{ij}^{M}=\nu _{ij}^{N}$$\forall$i and j.

• Max–min product of Pythagorean fuzzy soft matrix:

  Let $M=[a_{ij}]=[(\mu _{ij}^{M}, \nu _{ij}^{M})]\in \textit{PFSM}_{m\times n}$ & $N=[b_{jk}]=[(\mu _{jk}^{N}, \nu _{jk}^{N})] \in \textit{PFSM}_{n\times p}$ be two Pythagorean fuzzy soft matrices, then

  $M*N=[c_{ik}]_{m\times p}=\left[ \left\{ \max \left( \min \limits _{j}\left( \mu _{ij}^{M},\mu _{jk}^{N}\right) \right) ,\right. \right.$$\left. \left. \min \left( \max \limits _{j}\left( \nu _{ij}^{M},\nu _{jk}^{N}\right) \right) \right\} \right]$$\forall$i, j and k.

Remark

In the literature, various basic triangular norm (t-norm) and triangular conorm (t-conorm) along with their types and properties have been discussed by Klement et al. (2000, 2004) in detail. In order to define various operations over Pythagorean fuzzy soft matrices, some combinations of them have to be taken into account. However, in the present communication, we have considered the combination of maximum operator (t-conorm) and minimum operator (t-norm). Some other combinations using different types of t-norm and t-conorm given in the above stated literature may also be considered in the future.

Operations over Pythagorean fuzzy soft matrices

Various standard operations over two Pythagorean fuzzy soft matrices $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})]$ and $B=[(\mu _{ij}^{B}, \nu _{ij}^{B})]$$\in \textit{PFSM}_{m\times n}$ can be defined as follows:

• $A^{c}=\left[ \left( \nu _{ij}^{A},\mu _{ij}^{A}\right) \right]$$\forall$i and j.

• $A\cup B=\left[ \max \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) ,\min \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) \right]$$\forall$i and j.

• $A\cap B=\left[ \min \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) ,\max \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) \right]$$\forall$i and j.

• $A\cdot B=\left[ \left( \mu _{ij}^{A}\cdot \mu _{ij}^{B}, \nu _{ij}^{A}+\nu _{ij}^{B}-\nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \right]$$\forall$i and j.

• $A+B=\left[ \left( \mu _{ij}^{A}+\mu _{ij}^{B}-\mu _{ij}^{A}\cdot \mu _{ij}^{B}, \nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \right]$$\forall$i and j.

• $A\,\otimes \, B=\left[ \left( \mu _{ij}^{A}\cdot \mu _{ij}^{B}, \sqrt{(\nu _{ij}^{A})^{2}+(\nu _{ij}^{B})^{2}-(\nu _{ij}^{A})^{2}\cdot (\nu _{ij}^{B})^{2}}\right) \right]$$\forall$i and j.

• $A\,\oplus \, B=\left[ \left( \sqrt{(\mu _{ij}^{A})^{2}+(\mu _{ij}^{B})^{2}-(\mu _{ij}^{A})^{2}\cdot (\mu _{ij}^{B})^{2}}, \nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \right]$$\forall$i and j.

• $A\,@ \,B=\left[ \left( \frac{\mu _{ij}^{A}+\mu _{ij}^{B}}{2}, \frac{\nu _{ij}^{A}+\nu _{ij}^{B}}{2}\right) \right]$$\forall$i and j.

• $A @_{w} B=\left[ \left( \frac{w_{1}\mu _{ij}^{A}+w_{2}\mu _{ij}^{B}}{w_{1}+w_{2}}, \frac{w_{1}\nu _{ij}^{A}+w_{2}\nu _{ij}^{B}}{w_{1}+w_{2}}\right) \right]$$\forall$i and j ; where $w_{1},w_{2}>0$ are the weights.

• $A {\$} B=\left[ \left( \sqrt{\mu _{ij}^{A}\cdot \mu _{ij}^{B}}, \sqrt{\nu _{ij}^{A}\cdot \nu _{ij}^{B}}\right) \right]$$\forall$i and j.

• $A {\$}_{w} B=\left[ \left( (\mu _{ij}^{A})^{w_{1}}\cdot (\mu _{ij}^{B})^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}},\right.$$\left. \left( (\nu _{ij}^{A})^{w_{1}}\cdot (\nu _{ij}^{B})^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}})\right]$$\forall$i and j, where $w_{1},w_{2}>0$ are the weights.

• $A\bowtie B=\left[ \left( 2\cdot \frac{\mu _{ij}^{A}\cdot \mu _{ij}^{B}}{\mu _{ij}^{A}+\mu _{ij}^{B}}, 2\cdot \frac{\nu _{ij}^{A}\cdot \nu _{ij}^{B}}{\nu _{ij}^{A}+\nu _{ij}^{B}}\right) \right]$$\forall$i and j.

• $A \bowtie _{w} B=\left[ \left( \frac{w_{1}+w_{2}}{\frac{w_{1}}{\mu _{ij}^{A}}+\frac{w_{2}}{\mu _{ij}^{B}}}, \frac{w_{1}+w_{2}}{\frac{w_{1}}{\nu _{ij}^{A}}+\frac{w_{2}}{\nu _{ij}^{B}}}\right) \right]$$\forall$i and j ; where $w_{1},w_{2}>0$ are the weights.

Proposition 1

Let A and B$\in \textit{PFSM}_{m\times n}$ be two Pythagorean fuzzy soft matrices, then the following results hold:

• (i) $A\cup B = B\cup A$

• (ii) $A\cap B = B\cap A$

• (iii) $A+B = B+A$

• (iv) $A\cdot B = B\cdot A$

• (v) $(A\cup B)^{c} = A^{c}\cap B^{c}$

• (vi) $(A\cap B)^{c} = A^{c}\cup B^{c}$

• (vii) $\left( A^{c}\cap B^{c}\right) ^{c} = A\cup B$

• (viii) $\left( A^{c}\cup B^{c}\right) ^{c} = A\cap B$

• (ix) $\left( A^{c}+B^{c}\right) ^{c} = A\cdot B$

• (x) $\left( A^{c}\cdot B^{c}\right) ^{c} = A+B$.

Proof

Let $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})],B=[(\mu _{ij}^{B}, \nu _{ij}^{B})]\in \textit{PFSM}_{m\times n}$. For all i and j we have,

• (i)
  $$\begin{aligned} A\cup B= & {} \left[ \max \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) , \min \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) \right] \\= & {} \left[ \max \left( \mu _{ij}^{B}, \mu _{ij}^{A}\right) , \min \left( \nu _{ij}^{B}, \nu _{ij}^{A}\right) \right] = B\cup A. \end{aligned}$$
• (ii)
  $$\begin{aligned} A\cap B= & {} \left[ \min \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) , \max \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) \right] \\= & {} \left[ \min \left( \mu _{ij}^{B}, \mu _{ij}^{A}\right) ,\max \left( \nu _{ij}^{B},\nu _{ij}^{A}\right) \right] = B\cap A. \end{aligned}$$
• (iii)
  $$\begin{aligned} A+B= & {} \left[ \left( \mu _{ij}^{A}+\mu _{ij}^{B}-\mu _{ij}^{A}\cdot \mu _{ij}^{B}, \nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \right] \\= & {} \left[ \left( \mu _{ij}^{B}+\mu _{ij}^{A}-\mu _{ij}^{B}\cdot \mu _{ij}^{A}, \nu _{ij}^{B}\cdot \nu _{ij}^{A}\right) \right] = B+A. \end{aligned}$$
• (iv)
  $$\begin{aligned} A\cdot B= & {} \left[ \left( \mu _{ij}^{A}\cdot \mu _{ij}^{B}, \nu _{ij}^{A}+\nu _{ij}^{B}-\nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \right] \\= & {} \left[ \left( \mu _{ij}^{B}\cdot \mu _{ij}^{A}, \nu _{ij}^{B}+\nu _{ij}^{A}-\nu _{ij}^{B}\cdot \nu _{ij}^{A}\right) \right] = B\cdot A. \end{aligned}$$
• (v)
  $$\begin{aligned} (A\cup B)^{c}= & {} \left( \left[ \left( \mu _{ij}^{A}, \nu _{ij}^{A}\right) \right] \cup \left[ \left( \mu _{ij}^{B}, \nu _{ij}^{B}\right) \right] \right) ^{c}\\= & {} \left[ \max \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) ,\min \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) \right] ^{c}\\= & {} \left[ \min \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) ,\max \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) \right] \\= & {} \left[ \left( \nu _{ij}^{A}, \mu _{ij}^{A}\right) \right] \cap \left[ \left( \nu _{ij}^{B}, \mu _{ij}^{B}\right) \right] = A^{c}\cap B^{c}. \end{aligned}$$

Similarly, (vi), (vii), (viii), (ix) and (x) can be proved easily. $\square$

Proposition 2

Let $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})]$$\in \textit{PFSM}_{m\times n}$ be a Pythagorean fuzzy soft matrix. Then, in view of the definitions, the following results may easily be verified:

• (i) $\left( A^{c}\right) ^{c} = A$

• (ii) $\left( P_{\mu }\right) ^{c} = P_{\nu }$

• (iii) $\left( P_{\nu }\right) ^{c} = P_{\mu }$

• (iv) $A\cup A = A$

• (v) $A\cup P_{\mu } = P_{\mu }$

• (vi) $A\cap P_{\nu } = A$

• (vii) $A\cap A = A$

• (viii) $A\cap P_{\mu } = A$

• (ix) $A\cap P_{\nu } = P_{\nu }$.

Proposition 3

Let A and B$\in \textit{PFSM}_{m\times n}$ be two Pythagorean fuzzy soft matrices, then the following results with respect to the weighed operations hold:

• (i) $(A^{c} @_{w} B^{c})^{c} = A @_{w} B$

• (ii) $(A^{c} {{\$}}_{w} B^{c})^{c} = A {\$}_{w} B$

• (iii) $(A^{c} \bowtie _{w} B^{c})^{c} = A \bowtie _{w} B$

• (iv) $A @_{w} B = B @_{w} A$

• (v) $A {\$}_{w} B = B {\$}_{w} A$

• (vi) $A \bowtie _{w} B = B \bowtie _{w} A$.

Proof

Let $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})],B=[(\mu _{ij}^{B}, \nu _{ij}^{B})]\in \textit{PFSM}_{m\times n}$. For all i, j and $w_{1}, w_{2}>0,$, we have,

• (i)
  $$\begin{aligned} (A^{c} @_{w} B^{c})^{c}= & {} \left( \left[ \left( \nu _{ij}^{A}, \mu _{ij}^{A}\right) @_{w} \left( \nu _{ij}^{B}, \mu _{ij}^{B}\right) \right] \right) ^{c}\\= & {} \left( \left[ \frac{w_{1}\nu _{ij}^{A}+w_{2}\nu _{ij}^{B}}{w_{1}+w_{2}}, \frac{w_{1}\mu _{ij}^{A}{+}w_{2}\mu _{ij}^{B}}{w_{1}{+}w_{2}}\right] \right) ^{c}\\= & {} \left[ \frac{w_{1}\mu _{ij}^{A}+w_{2}\mu _{ij}^{B}}{w_{1}+w_{2}}, \frac{w_{1}\nu _{ij}^{A}+w_{2}\nu _{ij}^{B}}{w_{1}+w_{2}}\right] \\= & {} A @_{w} B. \end{aligned}$$
• (ii)
  $$\begin{aligned} (A^{c} {\$}_{w} B^{c})^{c}&= \left( \left[ \left( \nu _{ij}^{A}, \mu _{ij}^{A}\right) {\$}_{w} \left( \nu _{ij}^{B},\mu _{ij}^{B}\right) \right] \right) ^{c}\\&=\left( \left[ \left( \left( \nu _{ij}^{A}\right) ^{w_{1}}\cdot \left( \nu _{ij}^{B}\right) ^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}},\right. \right. \\&\quad \left. \left. \left( \left( \mu _{ij}^{A}\right) ^{w_{1}}\cdot \left( \mu _{ij}^{B}\right) ^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}}\right] \right) ^{c}\\&=\left[ \left( \left( \mu _{ij}^{A}\right) ^{w_{1}}\cdot \left( \mu _{ij}^{B}\right) ^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}},\right. \\&\quad \left. \left( \left( \nu _{ij}^{A}\right) ^{w_{1}}\cdot \left( \nu _{ij}^{B}\right) ^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}}\right] \\&= A {\$}_{w} B. \end{aligned}$$

  Similar proof for (iii).

• (iv)
  $$\begin{aligned} A @_{w} B&= \left[ \frac{w_{1}\mu _{ij}^{A}+w_{2}\mu _{ij}^{B}}{w_{1}+w_{2}}, \frac{w_{1}\nu _{ij}^{A}+w_{2}\nu _{ij}^{B}}{w_{1}+w_{2}}\right] \\&=\left[ \frac{w_{2}\mu _{ij}^{B}+w_{1}\mu _{ij}^{A}}{w_{2}+w_{1}}, \frac{w_{2}\nu _{ij}^{B}+w_{1}\nu _{ij}^{A}}{w_{2}+w_{1}}\right] \\&= B @_{w} A. \end{aligned}$$
• (v)
  $$\begin{aligned} A {{\$}}_{w} B&= \left[ \left( \left( \mu _{ij}^{A}\right) ^{w_{1}}\cdot \left( \mu _{ij}^{B}\right) ^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}}, \left( \left( \nu _{ij}^{A}\right) ^{w_{1}}\cdot \right. \right. \\&\quad \left. \left. \left( \nu _{ij}^{B}\right) ^{w_{2}}\right) ^{\frac{1}{w_{1}+w_{2}}}\right] \\&=\left[ \left( \left( \mu _{ij}^{B}\right) ^{w_{2}}\cdot \left( \mu _{ij}^{A}\right) ^{w_{1}}\right) ^{\frac{1}{w_{2}+w_{1}}}, \left( \left( \nu _{ij}^{B}\right) ^{w_{2}}\cdot \right. \right. \\&\quad \left. \left. \left( \nu _{ij}^{A}\right) ^{w_{1}}\right) ^{\frac{1}{w_{2}+w_{1}}}\right] = B {\$}_{w} A \end{aligned}$$

Similar proof for (vi). $\square$

Proposition 4

Let A, B and C$\in \textit{PFSM}_{m\times n}$ be three Pythagorean fuzzy soft matrices, then the following results related to associativity of operations hold:

• (i) $(A\cup B)\cup C = A\cup (B\cup C)$

• (ii) $(A\cap B)\cap C = A\cap (B\cap C)$

• (iii) $(A + B) + C = A+ (B + C)$

• (iv) $(A \cdot B)\cdot C = A \cdot (B \cdot C)$

• (v) $(A @ B) @ C = A @ (B @ C)$

• (vi) $(A {\$} B) {\$}C=A {\$}(B {\$}C)$

• (vii) $(A \bowtie B)\bowtie C=A \bowtie (B \bowtie C)$.

Proof

For all i and j, we have

• (i)
  $$\begin{aligned} (A\cup B)\cup C&=\left[ \left( \max \left\{ \mu _{ij}^{A}, \mu _{ij}^{B}\right\} , \min \left\{ \nu _{ij}^{A}, \nu _{ij}^{B}\right\} \right) \right] \\&\quad \cup \left[ \left( \mu _{ij}^{C}, \nu _{ij}^{C}\right) \right] \\&= \left[ \left( \max \left\{ \left( \mu _{ij}^{A},\mu _{ij}^{B}\right) , \mu _{ij}^{C}\right\} ,\right. \right. \\&\quad \left. \left. \min \left\{ \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) ,\nu _{ij}^{C}\right\} \right) \right] \\&= \left[ \left( \max \left\{ \left( \mu _{ij}^{A}, \left( \mu _{ij}^{B},\mu _{ij}^{C}\right) \right) \right\} ,\right. \right. \\&\quad \left. \left. \min \left\{ \nu _{ij}^{A}, \left( \nu _{ij}^{B},\nu _{ij}^{C}\right) \right\} \right) \right] \\&= A\cup (B\cup C). \end{aligned}$$
• (ii)
  $$\begin{aligned} (A\cap B)\cap C&=\left[ \left( \min \left\{ \mu _{ij}^{A}, \mu _{ij}^{B}\right\} , \max \left\{ \nu _{ij}^{A}, \nu _{ij}^{B}\right\} \right] \right. \\&\quad \left. \cup \left( \mu _{ij}^{C}, \nu _{ij}^{C}\right) \right] \\&=\left[ \left( \min \left\{ \left( \mu _{ij}^{A}, \mu _{ij}^{B}\right) , \mu _{ij}^{C}\right\} ,\right. \right. \\&\quad \left. \left. \max \left\{ \left( \nu _{ij}^{A}, \nu _{ij}^{B}\right) , \nu _{ij}^{C}\right\} \right) \right] \\&=\left[ \left( \min \left\{ \left( \mu _{ij}^{A}, \left( \mu _{ij}^{B}, \mu _{ij}^{C}\right) \right) \right\} ,\right. \right. \\&\quad \left. \left. \max \left\{ \nu _{ij}^{A}, \left( \nu _{ij}^{B},\nu _{ij}^{C}\right) \right\} \right) \right] \\&= A \cap (B \cap C). \end{aligned}$$
• (iii)
  $$\begin{aligned} (A + B) + C&=\left[ \left( \mu _{ij}^{A}+\mu _{ij}^{B}-\mu _{ij}^{A}\cdot \mu _{ij}^{B}, \nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \right] \\&\quad +\left[ \left( \mu _{ij}^{C}, \nu _{ij}^{C}\right) \right] \\&=\left[ \left( \mu _{ij}^{A}+\mu _{ij}^{B}\right) +\mu _{ij}^{C}-\left( \mu _{ij}^{A}\cdot \mu _{ij}^{B}\right) \cdot \right. \\&\quad \left. \mu _{ij}^{C}, \left( \nu _{ij}^{A}\cdot \nu _{ij}^{B}\right) \cdot \nu _{ij}^{C}\right] \\&=\left[ \mu _{ij}^{A}+\left( \mu _{ij}^{B}+\mu _{ij}^{C}\right) -\mu _{ij}^{A}\cdot \left( \mu _{ij}^{B}\cdot \mu _{ij}^{C}\right) ,\right. \\&\quad \left. \nu _{ij}^{A}\cdot \left( \nu _{ij}^{B}\cdot \nu _{ij}^{C}\right) \right] \\&=A + (B + C). \end{aligned}$$

  Similar proof for (iv), (v), (vi) and (vii).

$\square$

Proposition 5

Let A, B and C$\in \textit{PFSM}_{m\times n}$ be three Pythagorean fuzzy soft matrices, then the following results related to distributivity of operations hold:

• (i) $A \cap (B \cup C)=(A \cap B) \cup (A \cap C)$

• (ii) $(A \cap B)\cup C=(A\cup C)\cap (B\cup C)$

• (iii) $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

• (iv) $(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$

• (v) $(A\cap B) @ C= (A @ C) \cap ( B @ C)$

• (vi) $(A\cap B)\bowtie C=(A\bowtie C)\cap (B\bowtie C)$

• (vii) $(A\cup B)+C=(A+C)\cup (B+C)$

• (viii) $(A\cup B)\cdot C=(A\cdot C)\cup (B\cdot C)$

• (ix) $A\cup (B @ C)=(A\cup B) @ (A\cup C)$

• (x) $(A\cup B)\bowtie C=(A\bowtie C)\cup (B\bowtie C)$

• (xi) $A @ (B\cup C)=(A @ B)\cup (A @ C)$

• (xii) $A @ (B\cap C)=(A @ B)\cap (B @ C)$

• (xiii) $A {\$} (B\cup C)=(A {\$} B)\cup (A {\$} C)$

• (xiv) $(A\cup B) {\$}C = (A {\$} C)\cup (B {\$} C)$

• (xv) $A\cdot (B @ C)=(A\cdot B) @ (A\cdot C)$

• (xvi) $A\cup (B\bowtie C)=(A\cup B)\bowtie (A\cup C)$

• (xvii) $A\bowtie (B\cup C)=(A\bowtie B)\cup (A\bowtie C)$

• (xviii) $A {\$} (B \cap C)=(A {\$} B) \cap (B {\$} C)$

• (xix) $(A \cap B) {\$} C=(A {\$} C)\cap (B {\$} C)$.

Proof

For all i and j, we have

• (i)
  $$\begin{aligned} A \cap (B \cup C)&=\left[ \left( \mu _{ij}^{A}, \nu _{ij}^{A}\right) \right] \cap \left[ \left( \max \left\{ \mu _{ij}^{B}, \mu _{ij}^{C}\right\} ,\right. \right. \\&\quad \left. \left. \min \left\{ \nu _{ij}^{B}, \nu _{ij}^{C}\right\} \right) \right] \\&=\left[ \left( \min \left\{ \mu _{ij}^{A}, \max \left\{ \mu _{ij}^{B}, \mu _{ij}^{C}\right\} \right\} , \right. \right. \\&\left. \left. \quad \max \left\{ \nu _{ij}^{A}, \min \left\{ \nu _{ij}^{B}, \nu _{ij}^{C}\right\} \right\} \right) \right] . \end{aligned}$$

  Now,

  $$\begin{aligned} (A \cap B) \cup (A \cap C)&=\Big [\big (\min \big \{\mu _{ij}^{A}, \mu _{ij}^{B}\big \}, \max \big \{\nu _{ij}^{A}, \nu _{ij}^{B}\big \}\Big ] \\&\quad \cup \big [\big (\min \big \{\mu _{ij}^{A}, \mu _{ij}^{C}\big \}, \max \big \{\nu _{ij}^{A}, \nu _{ij}^{C}\big \}\big ]\\&=\big [\max \big (\min \big \{\mu _{ij}^{A}, \mu _{ij}^{B}\big \},\\&\quad \min \big \{\mu _{ij}^{A}, \mu _{ij}^{C}\big \}\big ),\\&\quad \min \big (\max \big \{\nu _{ij}^{A}, \nu _{ij}^{B}\big \}, \max \big \{\nu _{ij}^{A}, \nu _{ij}^{C}\big \}\big )\big ]\\&=\big [\max \big (\mu _{ij}^{A}, \min \big \{\mu _{ij}^{B}, \mu _{ij}^{C}\big \}\big ),\\&\quad \min \big (\nu _{ij}^{A}, \max \big \{\nu _{ij}^{B}, \nu _{ij}^{C}\big \}\big )\big ]\\&=\big [\min \big (\mu _{ij}^{A}, \max \big \{\mu _{ij}^{B}, \mu _{ij}^{C}\big \}\big ),\\&\quad \max \big (\nu _{ij}^{A}, \min \big \{\nu _{ij}^{B}, \nu _{ij}^{C}\big \}\big )\big ]\\&= A \cap (B \cup C). \end{aligned}$$

  Hence, $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$ holds.

• (ii)
  $$\begin{aligned} (A\cap B)\cup C&=\Big [\Big (\min \Big \{\mu _{ij}^{A}, \mu _{ij}^{B}\Big \}, \max \Big \{\nu _{ij}^{A}, \nu _{ij}^{B}\Big \}\Big ]\\&\quad \cup \Big [\Big (\mu _{ij}^{C}, \nu _{ij}^{C}\Big )\Big ]\\&=\Big [\max \Big (\min \Big \{\mu _{ij}^{A}, \mu _{ij}^{B}\Big \}, \mu _{ij}^{C}\Big ),\\&\quad \min \Big (\max \Big \{\nu _{ij}^{A}, \nu _{ij}^{B}\Big \}, \nu _{ij}^{C}\Big ]. \end{aligned}$$

  Now,

  $$\begin{aligned} (A\cup C)\cap (B\cup C)&=\Big [\max \Big \{\mu _{ij}^{A}, \mu _{ij}^{C}\Big \}, \min \Big \{\nu _{ij}^{A}, \nu _{ij}^{C}\Big \}\Big ] \\&\quad \cap \Big [\max \Big \{\mu _{ij}^{B}, \mu _{ij}^{C}\Big \}, \min \Big \{\nu _{ij}^{B}, \nu _{ij}^{C}\Big \}\Big ]\\&=\Big [\min \Big (\max \Big \{\mu _{ij}^{A}, \mu _{ij}^{C}\Big \}, \max \Big \{\mu _{ij}^{B}, \mu _{ij}^{C}\Big \}\Big ), \\&\quad \max \Big (\min \Big \{\nu _{ij}^{A}, \nu _{ij}^{C}\Big \}, \min \Big \{\nu _{ij}^{B}, \nu _{ij}^{C}\Big \}\Big )\Big ]\\&=\Big [\min \Big (\max \Big \{\mu _{ij}^{A}, \mu _{ij}^{B}\Big \}, \mu _{ij}^{C}\Big \}\Big ),\\&\qquad \max \Big (\min \Big \{\nu _{ij}^{A}, \nu _{ij}^{B}\Big \}, \nu _{ij}^{C}\Big \}\Big )\Big ]\\&=\Big [\max \Big (\min \Big \{\mu _{ij}^{A}, \mu _{ij}^{B}\Big \}, \mu _{ij}^{C}\Big \}\Big ),\\&\qquad \min \Big (\max \Big \{\nu _{ij}^{A}, \nu _{ij}^{B}\Big \}, \nu _{ij}^{C}\Big \}\Big )\Big ]\\&= (A \cap B)\cup C \end{aligned}$$

  Hence, $(A\cap B)\cup C=(A \cup C) \cap (B \cup C)$.

It may be observed that the results pointed in $(iii)-(xix)$ can be easily proved on similar lines as stated above. $\square$

Fig. 2

Flowchart of the algorithm for decision making

4 Application of Pythagorean fuzzy soft matrix in decision making

In view of a general decision-making problem and taking the idea of Pythagorean fuzzy soft matrix into account, we propose the following revised definitions for choice matrix and weighted choice matrix:

Definition 3

If $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})] \in \textit{PFSM}_{m\times n}$, then the choice matrix of Pythagorean fuzzy soft matrix A is given by

$$\begin{aligned} C(A)= & {} \left[ \left( \frac{\sum \limits _{j=1}^{n}(\mu _{ij}^{A})^{2}}{n}, \frac{\sum \limits _{j=1}^{n} (\nu _{ij}^{A})^{2}}{n}\right) \right] _{m\times 1} \\&\quad \forall \,\, i\,\,\text{ when } \text{ weights } \text{ are } \text{ equal }. \end{aligned}$$

Definition 4

If $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})]\in \textit{PFSM}_{m\times n}$, then the weighted choice matrix of Pythagorean fuzzy soft matrix A is given by

$$\begin{aligned} C_w(A)= & {} \left[ \left( \frac{\sum \limits _{j=1}^{n} w_{j}(\mu _{ij}^{A})^{2}}{\sum w_{j}}, \frac{\sum \limits _{j=1}^{n} w_{j}(\nu _{ij}^{A})^{2}}{\sum w_{j}}\right) \right] _{m\times 1}\\&\quad \forall \,\,i \,\, \text{ where }\,\, w_{j}>0 \,\, \text{ are } \text{ weights }. \end{aligned}$$

Based on the Pythagorean fuzzy soft matrix, choice matrix and weighted choice matrix, we propose an algorithm for solving a standard decision-making problem with the help of the following flowchart (Fig. 2):

Example 1

Suppose an automobile company produces three different types of car $c_1, c_2, c_3$, i.e., $U=\{c_1, c_2, c_3\}$, and let a set of parameters $E=\{e_1, e_2, e_3\}$ represent good mileage, comfort, good power steering, respectively. Suppose that a customer has to decide which car to be purchased? We model the problem by considering the Pythagorean fuzzy soft set (G, E) over U, where G is mapping $G: E \rightarrow {\mathcal P}(U)$ which represents the description of car on the basis of different parameters.

• Step 1 Construct the Pythagorean fuzzy soft matrix on the basis of the parameter as follows:

  $$\begin{aligned} A=\begin{array}{cccc} &{} e_{1} &{}\quad e_{2} &{}\quad e_{3}\\ c_{1} &{}\quad (0.8,0.5) &{} \quad (0.6,0.6) &{} \quad (0.8,0.2)\\ c_{2} &{}\quad (0.6,0.5) &{}\quad (0.7,0.4) &{} \quad (0.8,0.4)\\ c_{3} &{}\quad (0.5,0.7) &{}\quad (0.7,0.6) &{}\quad (0.9,0.3) \end{array} \end{aligned}$$

  It may be noted that the first element (0.8, 0.5) in the above Pythagorean fuzzy soft matrix A gives the degree to which the alternative $c_1$ satisfies the criterion $e_1$ is 0.8 while the degree to which the alternative $c_1$ does not match with the criterion $e_1$ is 0.5. Similar meaning is for the other entries of the above matrix A.

• Step 2

  • Case 1: equal weights

    Compute the choice matrix for the Pythagorean fuzzy soft matrix A as:

    $$\begin{aligned} C(A)=\left[ \begin{array}{c} (0.5467,0.2167)\\ (0.4967,0.19)\\ (0.5167,0.3133) \end{array}\right] \end{aligned}$$

  • Case 2: unequal weights

    If the weights 0.2, 0.6, 0.2 are given for the parameters good mileage, comfort, good power steering, respectively, then the weighted choice matrix for A is as

    $$\begin{aligned} C_{w}(A)=\left[ \begin{array}{c} (0.472,0.274)\\ (0.494,0.178)\\ (0.506,0.332) \end{array}\right] \end{aligned}$$

• Step 3

  • Case 1 (equal weights) It is clear from the choice matrix obtained in Step 2 that if we give equal preference for all the parameters, we have 0.5467 as the highest membership value, i.e., of car $c_1$. Therefore, in this case the most suitable car for the customer would be $c_1$.

  • Case 2 (unequal weights) However, it may also be observed that if the customer gives preference for the parameter “comfort” over the other parameters, then 0.506 is the highest membership value for car $c_3$. Therefore, in this case the most suitable car for the customer would be $c_3$.

5 Application of Pythagorean soft matrix in medical diagnosis

In view of a general medical diagnosis problem and taking the idea of Pythagorean fuzzy soft matrix into account, we propose the following revised definitions for score matrix and utility:

Definition 5

If $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})] \in \textit{PFSM}_{m\times n}$, then the score matrix of Pythagorean fuzzy soft matrix A is given by $S(A)=[s_{ij}]=[((\mu _{ij}^{A})^{2}-(\nu _{ij}^{A})^{2})]$ for all i and j. In the literature, the $(i, j){\hbox {th}}$ entry of the score matrix is considered to be an important index for measuring the optimized magnitude of the belongingness/non-belongingness of $i{\hbox {th}}$ patient having a chance of $j{\hbox {th}}$ disease.

Definition 6

If $A=[(\mu _{ij}^{A}, \nu _{ij}^{A})],B=[(\mu _{ij}^{B}, \nu _{ij}^{B})]\in \textit{PFSM}_{m\times n}$ then the utility matrix of Pythagorean fuzzy soft matrices A and B is given by $U(A, B)=[u_{ij}]_{m\times n}= [S(A)-S(B)]$$\forall$i and j. It may also be noted that the $(i, j)^{\text{ th }}$ entry of the utility matrix represents another important index for measuring the mixed magnitude of the belongingness in connection with its non-belongingness of $i{\hbox {th}}$ patient having a chance of $j{\hbox {th}}$ disease.

Based on the Pythagorean fuzzy soft matrices, score matrix and utility matrix, we propose an algorithm for solving a standard diagnosis problem with the help of the following flowchart (Fig. 3):

Fig. 3

Flowchart of the algorithm for medical diagnosis

In view of Step 6 of the above diagnosis algorithm, in case there are more than one instances where the value of max $U_{i}$ is repeating, then to break the tie, we have to reassess the symptoms. In order to have a proper implementation and understanding of the algorithm, we present a formal methodology followed by a numerical example in the following subsection:

5.1 Methodology

Let us assume that there is a set of m patients $A=\{a_1, a_2, \ldots , a_m\}$ with a set of symptoms $S= \{s_1, s_2, \ldots , s_n\}$ related to as set of k disease $Q=\{q_1, q_2, \ldots , q_k\}$. We apply the concept of Pythagorean fuzzy soft set to diagnose which patient is suffering from which disease. We construct a Pythagorean fuzzy soft set (F, S) over A, where F is a mapping $F: S \rightarrow {\mathcal P}(A)$ (collection of all the Pythagorean fuzzy subsets of A), which gives a collection of an approximate description of patient’s symptoms. This Pythagorean fuzzy soft set gives a relation matrix M called patient symptoms matrix. Then, we construct another Pythagorean fuzzy soft set (G, Q) over S, where G is mapping $G: Q \rightarrow {\mathcal P}(S)$, (collection of all the Pythagorean fuzzy subsets of S), which gives an approximate description of Pythagorean fuzzy soft medical knowledge of diseases and their symptoms. This Pythagorean fuzzy soft set gives another relation matrix N called symptoms disease matrix. We compute the complement of matrices, i.e., $M^c$ and $N^c$ given by $(F,S)^c$ and $(G,Q)^c$, respectively. Further, the max-min product matrix $M*N$, denoted by $R_1$, is computed which gives the maximum membership of occurrence of the symptoms of the disease and consequently, the matrix $M^{c}*N^{c}$, denoted by $R_2$, which gives the maximum membership of non-occurrence of the symptoms of the disease. Using Definition (5), we compute the score matrices $S(R_1)$ and $S(R_2)$ which represent the respective optimized magnitude of the sense of belongingness and non-belongingness of a patient in a certain disease. Next, using Definition (6), the utility matrix is computed based on taking the above two score matrices into account. The elements of the utility matrix correspondingly represent that how well one alternative satisfies the decision maker’s opinion. Since the utilities are real numbers, the preferred alternatives are those which are having higher values.

Example 2

Let $A=\{a_{1}, a_{2}, a_{3}, a_{4}\}$ be the universal set where $a_{1}, a_{2}, a_{3}, a_{4}$ represent the patients in a hospital. Consider a set of symptoms $S=\{s_{1}, s_{2}, s_{3}, s_{4}\}$ which represent temperature, headache, body pain and cough, respectively, and a set of diagnoses $Q=\{q_{1}, q_{2}, q_{3}\}$, which represent malaria, viral fever and typhoid, respectively. We model the problem by considering the Pythagorean fuzzy soft set (F, S) over A, where F is mapping $F: S \rightarrow {\mathcal P}(A)$, which represents the description of patient symptoms in the hospital.

• Step 1

  $$\begin{aligned}&(F, S)=\left\{ \begin{array}{c} F(s_{1})=\left\{ (a_{1},0.8,0.3), (a_{2},0.6,0.5), (a_{3},0.2,0.9), (a_{4},0.3,0.8)\right\} \\ F(s_{2})=\left\{ (a_{1},0.3,0.8), (a_{2},0.4,0.7), (a_{3},0.6,0.5), (a_{4},0.7,0.4)\right\} \\ F(s_{3})=\left\{ (a_{1},0.9,0.3), (a_{2},0.8,0.5), (a_{3},0.7,0.4), (a_{4},0.5,0.6)\right\} \\ F(s_{4})=\left\{ (a_{1},0.8,0.3), (a_{2},0.5,0.6), (a_{3},0.4,0.8), (a_{4},0.4,0.7)\right\} \end{array}\right\} \end{aligned}$$

  By considering the Pythagorean fuzzy soft set, we can convert the problem into the following Pythagorean fuzzy soft matrix as follows:

  $$\begin{aligned}&M=\begin{array}{ccccc} &{} s_{1} &{} \quad s_{2} &{} \quad s_{3} &{} \quad s_{4}\\ a_{1} &{} \quad (0.8,0.3) &{} \quad (0.3,0.8) &{} \quad (0.9,0.3) &{} \quad (0.8,0.3)\\ a_{2} &{} \quad (0.6,0.5) &{} \quad (0.4,0.7) &{} \quad (0.8,0.5) &{} \quad (0.5,0.6)\\ a_{3} &{} \quad (0.2,0.9) &{} \quad (0.6,0.5) &{} \quad (0.7,0.4) &{} \quad (0.4,0.8)\\ a_{4} &{} \quad (0.3,0.8) &{} \quad (0.7,0.4) &{} \quad (0.5,0.6) &{} \quad (0.4,0.7) \end{array} \end{aligned}$$

  It may be noted that the first element (0.8, 0.3) in the above Pythagorean fuzzy soft matrix M gives the degree to which the patient $a_1$ satisfies the symptom $s_1$ is 0.8 while the degree to which the patient $a_1$ does not match with the symptom $s_1$ is 0.3. Similar meaning is for the other entries of the above matrix M.

  Next, consider Pythagorean fuzzy soft set (G, Q) over S, where G is mapping $G: Q \rightarrow {\mathcal P}(S)$ which represents the medical knowledge of the diagnosis and their symptoms.

  $$\begin{aligned}&(G, Q)=\left\{ \begin{array}{c} G(q_{1})=\left\{ (s_{1},0.5,0.7), (s_{2},0.7,0.4), (s_{3},0.4,0.8), (s_{4},0.4,0.9)\right\} \\ G(q_{2})=\left\{ (s_{1},0.8,0.5), (s_{2},0.4,0.7), (s_{3},0.3,0.9), (s_{4},0.82,0.46)\right\} \\ G(q_{3})=\left\{ (s_{1},0.4,0.7), (s_{2},0.5,0.6), (s_{3},0.7,0.4), (s_{4},0.9,0.4)\right\} \end{array}\right\} \end{aligned}$$

  We model this problem into the Pythagorean fuzzy soft matrix as follows:

  $$\begin{aligned}&N=\begin{array}{cccc} &{} q_{1} &{} q_{2} &{} q_{3}\\ s_{1} &{} \quad (0.5,0.7) &{} \quad (0.8,0.5) &{} \quad (0.4,0.7)\\ s_{2} &{} \quad (0.7,0.4) &{} \quad (0.4,0.7) &{} \quad (0.5,0.6)\\ s_{3} &{} \quad (0.4,0.8) &{} \quad (0.3,0.9) &{} \quad (0.7,0.4)\\ s_{4} &{} \quad (0.4,0.9) &{} \quad (0.82,0.46) &{} \quad (0.9,0.4) \end{array} \end{aligned}$$

  Here, the first element (0.5, 0.7) in the above Pythagorean fuzzy soft matrix N gives the degree to which the symptom $s_1$ satisfies the disease $q_1$ is 0.5 while the degree to which the symptom $s_1$ does not have the disease $q_1$ is 0.7. Similar meaning is for the other entries of the above matrix N.

• Step 2 The Pythagorean fuzzy soft complement matrices of the Pythagorean fuzzy soft matrices obtained in Step 1 are computed as follows:

  $$\begin{aligned} M^{c}= & {} \begin{array}{ccccc} &{} s_{1} &{} \quad s_{2} &{} \quad s_{3} &{} \quad s_{4}\\ a_{1} &{} \quad (0.3,0.8) &{} \quad (0.8,0.3) &{} \quad (0.3,0.9) &{} \quad (0.3,0.8)\\ a_{2} &{} \quad (0.5,0.6) &{} \quad (0.7,0.4) &{} \quad (0.5,0.8) &{} \quad (0.6,0.5)\\ a_{3} &{} \quad (0.9,0.2) &{} \quad (0.5,0.6) &{} \quad (0.4,0.7) &{} \quad (0.8,0.4)\\ a_{4} &{} \quad (0.8,0.3) &{} \quad (0.4,0.7) &{} \quad (0.6,0.5) &{} \quad (0.7,0.4) \end{array}\\ N^{c}= & {} \begin{array}{cccc} &{} q_{1} &{} \quad q_{2} &{} \quad q_{3}\\ s_{1} &{} \quad (0.7,0.5) &{} \quad (0.5,0.8) &{} \quad (0.7,0.4)\\ s_{2} &{} \quad (0.4,0.7) &{} \quad (0.7,0.4) &{} \quad (0.6,0.5)\\ s_{3} &{} \quad (0.8,0.4) &{} \quad (0.9,0.3) &{} \quad (0.4,0.7)\\ s_{4} &{} \quad (0.9,0.4) &{} \quad (0.46,0.82) &{} \quad (0.4,0.9) \end{array} \end{aligned}$$

• Step 3 The max-min products of the Pythagorean fuzzy soft matrices obtained in Step 1 and Step 2 have been obtained as below:

  $$\begin{aligned} R_{1}= & {} M*N=\begin{array}{cccc} &{} q_{1} &{} \quad q_{2} &{} \quad q_{3}\\ a_{1} &{} \quad (0.5,0.7) &{} \quad (0.8,0.46) &{} \quad (0.8,0.4)\\ a_{2} &{} \quad (0.5,0.7) &{} \quad (0.6,0.46) &{} \quad (0.7,0.5)\\ a_{3} &{} \quad (0.6,0.5) &{} \quad (0.4,0.7) &{} \quad (0.7,0.4)\\ a_{4} &{} \quad (0.7,0.4) &{} \quad (0.4,0.7) &{} \quad (0.5,0.6) \end{array}\\ R_{2}= & {} M^{c}*N^{c}=\begin{array}{cccc} &{} q_{1} &{} \quad q_{2} &{} \quad q_{3}\\ a_{1} &{} \quad (0.4,0.7) &{} \quad (0.7,0.4) &{} \quad (0.6,0.5)\\ a_{2} &{} \quad (0.6,0.5) &{} \quad (0.7,0.4) &{} \quad (0.6,0.5)\\ a_{3} &{} \quad (0.8,0.4) &{} \quad (0.5,0.6) &{} \quad (0.7,0.4)\\ a_{4} &{} \quad (0.5,0.6) &{} \quad (0.7,0.4) &{} \quad (0.7,0.5) \end{array} \end{aligned}$$

• Step 4 The corresponding score matrices of the Pythagorean fuzzy soft matrices $R_{1}$ and $R_{2}$ obtained in Step 3 are:

  $$\begin{aligned} S(R_{1})= & {} \begin{array}{cccc} &{} q_{1} &{} \quad q_{2} &{} \quad q_{3}\\ a_{1} &{} \quad -0.24 &{} \quad 0.4284 &{}\quad 0.48\\ a_{2} &{} \quad -0.24 &{}\quad 0.1484 &{} \quad 0.24\\ a_{3} &{} \quad 0.11 &{} \quad -0.33 &{} \quad 0.33\\ a_{4} &{} \quad 0.33 &{} \quad -0.33 &{} \quad -0.11 \end{array} \end{aligned}$$
  $$\begin{aligned} S(R_{2})= & {} \begin{array}{cccc} &{} q_{1} &{} \quad q_{2} &{} \quad q_{3}\\ a_{1} &{} \quad -0.33 &{} \quad 0.33 &{} \quad 0.11\\ a_{2} &{} \quad 0.11 \quad &{} 0.33 &{} \quad 0.11\\ a_{3} &{} \quad 0.48 &{}\quad -0.11 &{} \quad 0.33\\ a_{4} &{} \quad -0.11 &{}\quad 0.33 &{} \quad 0.24 \end{array} \end{aligned}$$

• Step 5 The utility matrix of the score matrices $S(R_{1})$ and $S(R_{2})$ obtained in Step 4 is:

  $$\begin{aligned} U=\begin{array}{cccc} &{} q_{1} &{} \quad q_{2} &{} \quad q_{3}\\ a_{1} &{}\quad 0.09 &{}\quad 0.0984 &{}\quad 0.37\\ a_{2} &{}\quad -0.35&{}\quad -0.1816 &{} \quad 0.13\\ a_{3} &{}\quad -0.37&{}\quad -0.22 &{} \quad 0\\ a_{4} &{}\quad 0.44 &{}\quad -0.66 &{} \quad -0.35 \end{array} \end{aligned}$$

• Step 6 Therefore, in view of the values of the elements of utility matrix obtained in Step 5, it reflects that the patients $\{a_{1},a_{2},a_{3}\}$ are suffering from typhoid $\left( q_{3}\right)$ and patient $a_{4}$ is suffering from malaria $\left( q_{1}\right)$.

5.2 Comparative study

In order to have the comparative analysis of the proposed methodology in the decision-making process, we investigate the example taken from the related literature and compare the results with the recent existing methods of medical diagnosis.

Example 3

(Szmidt and Kacprzyk 2004) Suppose a doctor wants to make a proper diagnosis D$=$ {$d_1, d_2, d_3, d_4, d_5$}; where $d_1$ is viral fever, $d_2$ is Malaria, $d_3$ is typhoid, $d_4$ is stomach problem and $d_5$ is chest problem, for a set of patients $P= \left\{ \text{ Ted }, \text{ Al }, \text{ Bob }, \text{ Joe }\right\}$ with the values of symptoms V$=$ {$v_1, v_2, v_3, v_4, v_5$}; where $v_1$ is temperature, $v_2$ is headache, $v_3$ is stomach pain, $v_4$ is cough and $v_5$ is chest pain. We model the problem by considering the Pythagorean fuzzy soft set (F, V) over P, where F is mapping $F: V \rightarrow \mathcal {P}(P)$, which represents the description of patient’s symptoms in the hospital.

• Step 1

  $$\begin{aligned}&(F, V)=\left\{ \begin{array}{c} F(v_{1})=\left\{ (\text{ Al },0.8,0.1), (\text{ Bob },0.0,0.8), (\text{ Joe },0.8,0.1), (\text{ Ted },0.6,0.1)\right\} \\ F(v_{2})=\left\{ (\text{ Al },0.6,0.1), (\text{ Bob },0.4,0.4), (\text{ Joe },0.8,0.1), (\text{ Ted },0.5,0.4)\right\} \\ F(v_{3})=\left\{ (\text{ Al },0.2,0.8), (\text{ Bob },0.6,0.1), (\text{ Joe },0.0,0.6), (\text{ Ted },0.3,0.4)\right\} \\ F(v_{4})=\left\{ (\text{ Al },0.6,0.1), (\text{ Bob },0.1,0.7), (\text{ Joe },0.2,0.7), (\text{ Ted },0.7,0.2)\right\} \\ F(v_{5})=\left\{ (\text{ Al },0.1,0.6), (\text{ Bob },0.1,0.8), (\text{ Joe },0.0,0.5), (\text{ Ted },0.3,0.4)\right\} \end{array}\right\} \end{aligned}$$

  By considering the Pythagorean fuzzy soft set, we can convert the problem into the following Pythagorean fuzzy soft matrix as follows:

  $$\begin{aligned} M=\begin{array}{cccccc} &{} v_{1} &{} \quad v_{2} &{}\quad v_{3} &{} \quad v_{4} &{} \quad v_5 \\ \text{ Al } &{} \quad (0.8,0.1) &{} \quad (0.6,0.1) &{} \quad (0.2,0.8) &{}\quad (0.6,0.1) &{} \quad (0.1,0.6)\\ \text{ Bob } &{} \quad (0.0,0.8) &{} \quad (0.4,0.4) &{} \quad (0.6,0.1) &{} \quad (0.1,0.7) &{} \quad (0.1,0.8)\\ \text{ Joe } &{} \quad (0.8,0.1) &{} \quad (0.8,0.1) &{} \quad (0.0,0.6) &{} \quad (0.2,0.7) &{} \quad (0.0,0.5)\\ \text{ Ted } &{} \quad (0.6,0.1) &{} \quad (0.5,0.4) &{} \quad (0.3,0.4) &{} \quad (0.7,0.2) &{} \quad (0.3,0.4) \end{array} \end{aligned}$$

  Next, consider Pythagorean fuzzy soft set (G, D) over V,where G is mapping $G: D \rightarrow \mathcal P(V)$, which represents the medical knowledge of the diagnosis and their symptoms.

  $$\begin{aligned}&(G, D)=\left\{ \begin{array}{c} G(d_{1})=\left\{ (v_{1},0.4,0.0), (v_{2},0.3,0.5), (v_{3},0.1,0.7), (v_{4},0.4,0.3), (v_{5},0.1,0.7)\right\} \\ G(d_{2})=\left\{ (v_{1},0.7,0.0), (v_{2},0.2,0.6), (v_{3},0.0,0.9), (v_{4},0.7,0.0), (v_{5},0.1,0.8)\right\} \\ G(d_{3})=\left\{ (v_{1},0.3,0.3), (v_{2},0.6,0.1), (v_{3},0.2,0.7), (v_{4},0.2,0.6), (v_{5},0.1,0.9)\right\} \\ G(d_{4})=\left\{ (v_{1},0.1,0.7), (v_{2},0.2,0.4), (v_{3},0.8,0.0), (v_{4},0.2,0.7), (v_{5},0.2,0.7)\right\} \\ G(d_{5})=\left\{ (v_{1},0.1,0.8), (v_{2},0.0,0.8), (v_{3},0.2,0.8), (v_{4},0.2,0.8),(v_{5},0.8,0.1)\right\} \end{array}\right\} \end{aligned}$$

  We model this problem into the Pythagorean fuzzy soft matrix as follows:

  $$\begin{aligned} N=\begin{array}{cccccc} &{} d_{1} &{}\quad d_{2} &{}\quad d_{3} &{} \quad d_{4} &{} \quad d_{5}\\ v_{1} &{} \quad (0.4,0.0) &{} \quad (0.7,0.0) &{} \quad (0.3,0.3) &{} \quad (0.1,0.7) &{} \quad (0.1,0.8)\\ v_{2} &{} \quad (0.3,0.5) &{} \quad (0.2,0.6) &{} \quad (0.6,0.1) &{} \quad (0.2,0.4) &{} \quad (0.0,0.8)\\ v_{3} &{} \quad (0.1,0.7) &{} \quad (0.0,0.9) &{} \quad (0.2,0.7) &{}\quad (0.8,0.0) &{} \quad (0.2,0.8)\\ v_{4} &{} \quad (0.4,0.3) &{} \quad (0.7,0.0) &{} \quad (0.2,0.6) &{} \quad (0.2,0.7) &{} \quad (0.2,0.8)\\ v_{5} &{} \quad (0.1,0.7) &{} \quad (0.1,0.8) &{} \quad (0.1,0.9) &{} \quad (0.2,0.7) &{} \quad (0.8,0.1) \end{array} \end{aligned}$$

Table 1 A comparison of the results of existing methods in medical diagnosis

• Step 2 The Pythagorean fuzzy soft complement matrices of the Pythagorean fuzzy soft matrices obtained in Step 1 are computed as follows:

  $$\begin{aligned} M^{c}= & {} \begin{array}{cccccc} &{} v_{1} &{} \quad v_{2} &{}\quad v_{3} &{} \quad v_{4} &{} \quad v_{5}\\ \text{ Al } &{} \quad (0.1,0.8) &{} \quad (0.1,0.6) &{} \quad (0.8,0.2) &{} \quad (0.1,0.6) &{} \quad (0.6,0.1)\\ \text{ Bob } &{} \quad (0.8,0.0) &{} \quad (0.4,0.4) &{} \quad (0.1,0.6) &{} \quad (0.7,0.1) &{} \quad (0.8,0.1)\\ \text{ Joe } &{} \quad (0.1,0.8) &{} \quad (0.1,0.8) &{} \quad (0.6,0.0) &{} \quad (0.7,0.2) &{} \quad (0.5,0.0)\\ \text{ Ted } &{} \quad (0.1,0.6) &{} \quad (0.4,0.5) &{} \quad (0.4,0.3) &{} \quad (0.2,0.7) &{} \quad (0.4,0.3) \end{array}\\ N^{c}= & {} \begin{array}{cccccc} &{} d_{1} &{} \quad d_{2} &{} \quad d_{3} &{} \quad d_{4} &{} \quad d_{5}\\ v_{1} &{} \quad (0.0,0.4) &{} \quad (0.0,0.7) &{} \quad (0.3,0.3) &{} \quad (0.7,0.1) &{} \quad (0.8,0.1)\\ v_{2} &{} \quad (0.5,0.3) &{} \quad (0.6,0.2) &{} \quad (0.1,0.6) &{} \quad (0.4,0.2) &{} \quad (0.8,0.0)\\ v_{3} &{} \quad (0.7,0.1) &{} \quad (0.9,0.0) &{} \quad (0.7,0.2) &{} \quad (0.0,0.8) &{} \quad (0.8,0.2)\\ v_{4} &{} \quad (0.3,0.4) &{} \quad (0.0,0.7) &{} \quad (0.6,0.2) &{} \quad (0.7,0.2) &{} \quad (0.8,0.2)\\ v_{5} &{}\quad (0.7,0.1) &{} \quad (0.8,0.1) &{} \quad (0.9,0.1) &{} \quad (0.7,0.2) &{} \quad (0.1,0.8) \end{array} \end{aligned}$$

• Step 3 The max-min products of the Pythagorean fuzzy soft matrices obtained in Step 1 and Step 2 have been obtained as below:

  $$\begin{aligned} R_{1}= & {} M*N=\begin{array}{cccccc} &{} d_{1} &{} d_{2} &{}\quad d_{3} &{}\quad d_{4} &{}\quad d_{5}\\ \text{ Al } &{}\quad (0.4,0.1) &{}\quad (0.7,0.1) &{}\quad (0.6,0.1) &{}\quad (0.2,0.4) &{}\quad (0.2,0.6)\\ \text{ Bob } &{}\quad (0.3,0.5) &{}\quad (0.4,0.6) &{}\quad (0.4,0.4) &{}\quad (0.6,0.1) &{}\quad (0.2,0.8)\\ \text{ Joe } &{}\quad (0.4,0.1) &{}\quad (0.6,0.1) &{}\quad (0.7,0.1) &{}\quad (0.2,0.4) &{}\quad (0.2,0.5)\\ \text{ Ted } &{}\quad (0.7,0.1) &{}\quad (0.7,0.1) &{}\quad (0.5,0.3) &{}\quad (0.3,0.4) &{}\quad (0.3,0.4) \end{array}\\ R_{2}= & {} M^{c}*N^{c}=\begin{array}{cccccc} &{} d_{1} &{}\quad d_{2} &{}\quad d_{3} &{}\quad d_{4} &{}\quad d_{5}\\ \text{ Al }&{}\quad (0.7,0.1) &{}\quad (0.7,0.2) &{}\quad (0.7,0.1) &{}\quad (0.6,0.2) &{}\quad (0.8,0.2)\\ \text{ Bob } &{}\quad (0.7,0.1) &{}\quad (0.8,0.1) &{}\quad (0.8,0.1) &{}\quad (0.7,0.1) &{}\quad (0.8,0.1)\\ \text{ Joe } &{}\quad (0.6,0.1) &{}\quad (0.6,0.1) &{}\quad (0.6,0.1) &{}\quad (0.7,0.2) &{}\quad (0.7,0.2)\\ \text{ Ted } &{}\quad (0.4,0.3) &{}\quad (0.6,0.3) &{}\quad (0.4,0.3) &{}\quad (0.4,0.3) &{}\quad (0.4,0.3) \end{array} \end{aligned}$$

• Step 4 The corresponding score matrices of the Pythagorean fuzzy soft matrices $R_{1}$ and $R_{2}$ obtained in Step 3 are:

  $$\begin{aligned} S(R_{1})= & {} \begin{array}{cccccc} &{} d_{1} &{} \quad d_{2} &{} \quad d_{3} &{} \quad d_{4} &{} \quad d_{5}\\ \text{ Al } &{} \quad 0.15 &{} \quad 0.48 &{} \quad 0.35 &{} \quad -0.12 &{} \quad -0.32\\ \text{ Bob } &{}\quad -0.16 &{} \quad -0.20 &{} \quad 0.0&{} \quad 0.35 &{} \quad -0.60\\ \text{ Joe } &{} \quad 0.15 &{} \quad 0.35 &{} \quad 0.48 &{} \quad -0.12 &{} \quad -0.21\\ \text{ Ted } &{} \quad 0.48 &{} \quad 0.48 &{} \quad 0.16 &{}\quad -0.07 &{} \quad -0.07 \end{array}\\ S(R_{2})= & {} \begin{array}{cccccc} &{} d_{1} &{} \quad d_{2} &{} \quad d_{3} &{} \quad d_{4} &{} \quad d_{5}\\ \text{ Al } &{} \quad 0.48 &{} \quad 0.45 &{} \quad 0.48 &{} \quad 0.32 &{} \quad 0.60\\ \text{ Bob } &{} \quad 0.48 &{}\quad 0.63 &{}\quad 0.63 &{}\quad 0.48 &{} \quad 0.63\\ \text{ Joe } &{}\quad 0.35 &{} \quad 0.35 &{} \quad 0.35 &{} \quad 0.45 &{} \quad 0.45\\ \text{ Ted } &{} \quad 0.07 &{} \quad 0.27 &{} \quad 0.07 &{} \quad 0.07 &{} \quad 0.07 \end{array} \end{aligned}$$

• Step 5 The utility matrix of the score matrices $S(R_{1})\,\, \& S(R_{2})$ obtained in Step 4 is:

  $$\begin{aligned} U=\begin{array}{cccccc} &{} d_{1} &{} \quad d_{2} &{} \quad d_{3}&{} \quad d_{4} &{} \quad d_{5}\\ \text{ Al }&{} \quad -0.33 &{} \quad 0.03 &{} \quad -0.13&{}\quad -0.44 &{} \quad -0.92\\ \text{ Bob } &{} \quad -0.64 &{} \quad -0.83 &{} \quad -0.63 &{}\quad -0.13 &{} \quad -1.23\\ \text{ Joe } &{} \quad -0.20&{}\quad 0.0 &{} \quad 0.13 &{} \quad -0.57 &{} \quad -0.66\\ \text{ Ted } &{} \quad 0.41 &{} \quad 0.21 &{} \quad 0.09 &{} \quad -0.14 &{} \quad -0.14 \end{array} \end{aligned}$$

• Step 6 Therefore, in view of the values of the elements of utility matrix obtained in Step 5, it is quite probable that the Al is suffering from malaria $(d_2)$, Bob is suffering stomach problem $(d_4)$, Joe is suffering from Typhoid $(d_3)$ and Ted is suffering form viral fever $(d_1)$.

Observations Comparing the obtained results with the results of various other researchers who investigated the same diagnosis problem, we find that the proposed methodology is completely consistent with the various existing methods. This has been summarized in the following table with their references (Table 1):

Similar comparative analysis can also be carried out by taking example in case of decision-making problem of Sect. 4. In the future, the proposed idea of Pythagorean fuzzy soft matrices may be used in various other aspects, such as group decision making, information retrieval, pattern recognition, dimensionality reduction, data mining.

6 Conclusions

The concept of the Pythagorean fuzzy soft matrix has been well established along with its various types and properties. Valid proofs for the proposed properties over the matrices have also been provided. Further, the proposed algorithms for decision making by using choice matrix and weighted choice matrix and for medical diagnosis problem by using score matrix and utility matrix have been successfully implemented with the help of numerical example for each. Further, the comparative analysis shows that the results of the proposed methodology are equally consistent with the results of various other existing methods available in the literature.