Dimensionality reduction technique under picture fuzzy environment and its application in decision making

Abstract

The notion of soft matrix plays a vital role in many engineering applications and socio-economic and financial problems. A picture fuzzy set has been used to handle uncertainty data in modeling human opinion. In this work, we recall the picture fuzzy soft matrix concept and its different subsequent classes. Also, different kinds of binary operations over the proposed matrices have been provided. The main contribution of this paper is that using the concept of choice matrix and its weighted form and the score matrix, a new algorithm for decision-making has been outlined by considering the picture of fuzzy soft matrices. The current challenge In the decision-making problems is that many qualitative and quantitative criteria are involved. Hence, the dimensionality reduction technique plays an essential role in simplicity and broader applicability in the decision-making processes. We present an algorithm for the reduction process using the proposed definitions of the object and parameter-oriented picture fuzzy soft matrix and the technique to find the threshold value for the provided information. Then, illustrative numerical examples have also been provided for each proposed algorithm. A detailed comparative study of the proposed techniques has also been carried out in contrast with other existing techniques.

Keywords

Picture fuzzy soft set choice matrix score matrix decision-making dimensionality reduction

1. Introduction

In real problems, it is difficult for the decision-maker to achieve the most acceptable decision variant from the set of possible solutions due to the level of the problem complexity. However, it is challenging to conclude which alternative is the best, but still, it is feasible. For example, in decision-making, there are many multi-criteria decision-analysis (MCDA) problems where the criteria can be imprecise, uncertain, ambiguous, and vague [1]. Therefore, this uncertainty and impreciseness in the information are frequently modeled by handling using fuzzy information because the crisp set seems ineffective. Zadeh [2] presented the mathematical notion of a Fuzzy Set (FS) to deal with uncertainty and ambiguity, which has been defined by using the membership function of the alternative. Various researchers have found the applicability of the fuzzy set in different fields like decision-making, medical diagnosis, engineering, socio-economic, finance problems, etc.

As an extension with the fuzzy set by incorporating the idea of hesitancy/indeterminacy, Atanassov [3] introduced the notion of the intuitionistic fuzzy set (IFS) based on the two functions, i.e., membership & non-membership function such that their sum is less or equal to 1. Next, Yager [4] introduced a new extension of a fuzzy set called as Pythagorean fuzzy set (PyFS) based on the membership & non-membership function such that their squared sum is $\leqslant$ 1. It may be observed that the PyFS more effectively enlarged the span of information than IFS. It, too, has many similarities that are evident during the weight assignment process [5, 6]. Further, Cuong [7] shows that the structure of FSs, IFSs, and PyFS are not capable enough to represent the human opinion in a complete sense and introduced the concept of picture fuzzy set (PFS), which has been supported by the example of voting system where the voters are divided into four different groups: ‘one who votes for (yes)’, ‘one who votes against (no)’, ‘one who neither vote for nor against (abstain)’, ‘one who refused for voting (refusal)’. It may be observed that the notion of refusal has not been taken into consideration by any of the earlier sets introduced. It may be noted that Coung’s picture fuzzy sets can handle the uncertainty/ambiguity sufficiently close to human nature/opinion, where all the four parameters, that is, degree of membership, indeterminacy (neutral), non-membership and refusal have been incorporated.

Son [8] developed a generalized picture fuzzy distance measure and utilized it to solve the picture clustering problem. Thong et al. [9, 10] applied a novel picture fuzzy clustering techniques for complex data and particle swarm optimization. Wei [11] studied a technique in light of the picture fuzzy weighted cross-entropy and applied it to the multi-attribute decision-making problem to rank the alternatives under consideration. Also, Son [12] studied the picture association measures and applied them in group decision making. Further, Dutta [13] modified Son’s distance measure and then found the solution of the medical diagnosis problem by utilizing the proposed distance measure. Dinh et al. [14] proposed the distance and dissimilarity measures to measure the difference between two picture fuzzy sets. Wei and Hao [15] introduced the generalized dice similarity measure for picture fuzzy set and solved the computation application of decision making. Next, Wei [16] discussed some similarity measures to quantify the similarity between picture fuzzy sets. Some new aggregation operators for picture fuzzy sets have been studied to solve various problems, viz. Hamacher aggregation approach for assessing the performance of the best enterprise [17], Muirhead mean aggregation approach in evaluating the risk in the financial investment [18] and Dombi aggregation operator in a multi-attribute decision making process [19]. Recently, Khalil et al. [20] developed some fundamental operations for interval-valued picture fuzzy set/picture fuzzy soft set with various properties and applied them to solve the decision-making problems. Zeng et al. [21] studied a novel picture fuzzy discriminant measure based on Jensen’s inequality/exponential entropy in a probabilistic framework with important properties and illustrative examples. Recently, Guleria and Bajaj [22] presented the probabilistic distance measure for picture fuzzy sets where the probability of occurrence/non-occurrence of the picture fuzzy event has been taken into account to solve the classification problems.

The dimensionality reduction approach plays a key role in converting the higher dimensional data set into the lower-dimensional data set. The main aim of the reduction approaches is to enhance the ability to handle the irrelevant/reductant feature of the information. The dimensionality reduction technique also helps visualize data and take care of multicollinearity in the information. Hence, these approaches are a significant study point in various computational application fields with extreme data modality. In statistical science, various researchers have developed the different dimensionality reduction approaches such as “principal component analysis (PCA)”, “linear discriminant analysis (LDA)” [23], “singular value decomposition” & “learning vector quantization approach” [24]. Chen et al. [25] presented the concept of parameterization reduction for the soft set. Then, Xu et al. [26] studied the sequential and simultaneous perspectives technique for reducing informational data. Su et al. [27] provided the two algorithms for the reduction by utilizing the concept “linear sequence discriminant analysis (LSDA)”. Perfilieva [28] provided a dimensionality reduction approach by incorporating the fuzzy transform method. In order to have the approximation of the weights in the MCDA problem, Chaterjee et al. [29] developed a hybrid method by using the pairwise comparison approach. Mukhametzyanov and Pamucar [30] provided a mathematical model to select/choose the best/optimal solution and also carried out the sensitivity of the proposed model by using the different available methods.

There are various mathematical theories that have their limitations/drawbacks in dealing with the vagueness, uncertainty, and impreciseness because of the involvement of parameterizations in the different computational application fields such as engineering applications social-economic problems, and decision-making problems [31]. In order to overcome and develop a new kind of tool that can handle the vagueness, uncertainty, and impreciseness in a better way, Molodtsov [32] introduced a notion of a set called a soft set. Next, in extension with the notion of soft set theory, Maji et al. [33, 34, 35] presented the “fuzzy soft set (FSSs)” and “intuitionistic fuzzy soft set (IFSSs)” along with their various standard binary operations & utilized them to solve the decision-making problems. Further, the concept of Pythagorean fuzzy soft set (PyFSS) and picture fuzzy soft set (PFSS) has been extended by the Peng et al. [36] and Cuong [7] respectively.

Figure 1.

Extensions and generalizations of fuzzy set [37].

In many computational approaches, matrices play a key role to handle the additional feature of dimensionality in various problems of engineering, medical sciences, social sciences, etc. Firstly, Naim and Serdar [38] introduced the notion of soft matrices with their application in the multi-criteria decision-making problems. Further Yong et al. [39] and Chetia et al. [40] extended the matrix representation of soft set to fuzzy and intuitionistic fuzzy setup respectively. Recently, the notion of Pythagorean fuzzy soft matrix has been provided by the Guleria and Bajaj [41]. In addition to this, they also presented two different decision-making algorithm to deal the medical diagnosis problem and MCDM problem. Next, Bajaj and Guleria [42] proposed the notion of Pythagorean fuzzy soft matrix to develop a new dimensionality reduction approach to solve the MCDM problem. In order to better present of the sequential development of various extensions of fuzzy set [37], we present a timeline in Fig. 1.

The concept of matrix plays a vital role in many computational methods and the study of dimensionality features of different engineering problems. As the main contribution of this work, we present a new kind of matrix, called picture fuzzy soft matrix, as an extension of the picture fuzzy soft set. The proposed extension and its format can handle the impreciseness and uncertainty of the incomplete information in a more fitting sense, i.e., picture fuzzy set with four parameters. With the introduction of the proposed notion, the decision-making problems and their dimensionality reduction can be dealt in a better and broader sense of human opinion.

The work in the present manuscript has been structured as follows: The basic definitions and fundamental concepts related to the proposed work have been provided in Section 2. Then, in Section 3, we propose a new kind of matrix termed a picture fuzzy soft matrix along with its various types. Subsequently, different standard binary operations and their operational laws have also been discussed in detail. Next, in Section 4, a new decision-making algorithm has been provided by incorporating the proposed revised choice matrix, its weighted form, and the score matrix for solving a general decision-making problem. Next, Section 5 provides the dimensionality reduction technique along with the definitions of object-oriented and the parameter-oriented picture fuzzy soft matrix. Further, an illustrative application of the decision-making problem with the help of a numerical example has been presented by using the proposed methodology. In addition, the comparative study has been presented in contrast with the existing approaches, which depict the advantages and limitations of the proposed methodology. Finally, we conclude the paper in Section 6.

2. Preliminaries

In this section, we recall some of the basic definitions and fundamental notions related to the picture fuzzy soft sets which are easily available in literature.

Let $U={\{\alpha_{1},\alpha_{2},\ldots,\alpha_{m}}\}$ be the universe of discourse and $\mu:U\rightarrow[0,1]$ , $\eta:U\rightarrow[0,1]$ and $\nu:U\rightarrow[0,1]$ denote the characterizing function for the membership degree, the neutral membership (abstain) degree and the non-membership degree respectively.

•
A picture fuzzy set [7] $A$ in $U$ is given by

$\displaystyle A=\left\{<\alpha,\mu_{A}(\alpha),\eta_{A}(\alpha),\nu_{A}(\alpha% )>\mid\alpha\in U\right\};$ (1)

and for every $\alpha\in U$ satisfy the condition

$\displaystyle\mu_{A}(\alpha)+\eta_{A}(\alpha)+\nu_{A}(\alpha)\leqslant 1.$ (2)

The degree of refusal for any picture fuzzy set $A$ and $\alpha\in U$ is given by

$\displaystyle r_{A}(\alpha)={1-(\mu_{A}(\alpha)+\eta_{A}(\alpha)+\nu_{A}(% \alpha))}.$ (3)

Also, various generalizations and extensions of soft sets are being listed below for ready reference:

Let $U={\{\alpha_{1},\alpha_{2},\ldots,\alpha_{m}}\}$ be the universe of discourse and $P={\{p_{1},p_{2},\ldots,p_{n}}\}$ be the set of parameters. The pair $(\Phi,P)$ is called

•
soft set [32] over $U$ iff $\Phi:P\rightarrow{\mathcal{P}}(U)$ , where ${\mathcal{P}}(U)$ is the power set of $U$ .
•
Pythagorean fuzzy soft set [36] over $U$ if $\Phi:P\rightarrow\textit{PYFS}(U)$ and can be represented as

$\displaystyle(\Phi,P)=\{(p,\Phi(p)):p\in P,\Phi(p)\in\textit{PYFS}(U)\},$ (4)

where $\textit{PYFS}(U)$ represents the set of all PyFSs of $U$ .
•
picture fuzzy soft set [7] over $U$ if $\Phi:P\rightarrow\textit{PFS}(U)$ and can be represented as

$\displaystyle(\Phi,P)=\{(p,\Phi(p)):p\in P,\Phi(p)\in\textit{PFS}(U)\},$ (5)

where $\textit{PFS}(U)$ represents the set of all PFSs of $U$ .

Let $(\Phi,P)$ be a soft set over $U$ . Then the subset $U\times P$ is uniquely defined by $R_{P}=\{(\alpha,p),p\in P,\alpha\in\Phi(p)\}$ . The characteristic function of $R_{P}$ is $\chi_{{}_{R_{P}}}:U\times P\rightarrow[0,1]$ given by

$\displaystyle\chi_{{}_{R_{P}}}(\alpha,p)=\begin{cases}\begin{array}[]{c}1\\ 0\end{array}&\begin{array}[]{c}\textit{if}\,\,\,(\alpha,p)\in P\\ \textit{if}\,\,\,(\alpha,p)\notin P\end{array}\end{cases}.$ (6)

If $a_{ij}=\chi_{{}_{R_{P}}}(\alpha_{i},p_{j})$ , then a matrix $[a_{ij}]=[\chi_{{}_{R_{P}}}(\alpha_{i},p_{j})]$ is called soft matrix of the soft set $(\Phi,P)$ over $U$ .
3. Picture fuzzy soft matrices and operations

In this section, we propose a new kind of soft matrix which is a generalized notion of intuitionistic fuzzy soft matrix and can also be viewed as an extension of picture fuzzy soft set. Next, we introduce various types of binary operations over these matrices.

Let $(\Phi,P)$ be a picture fuzzy soft set over $U$ (universe), then $U\times P$ is defined by $R_{P}=\{(\alpha,p),p\in P,\alpha\in\Phi(p)\}$ . The $R_{P}$ can be defined by its membership, the neutral membership (abstain) and non-membership function given by $\mu_{R_{P}}:U\times P\rightarrow[0,1]$ , $\eta_{R_{P}}:U\times P\rightarrow[0,1]$ and $\nu_{R_{P}}:U\times P\rightarrow[0,1]$ respectively.

If $(\mu_{ij},\eta_{ij},\nu_{ij})=(\mu_{R_{P}}($ $\alpha_{i},p_{j}),\eta_{R_{P}}($ $\alpha_{i},p_{j}),\nu_{R_{P}}(\alpha_{i},p_{j}))$ , where $\mu_{R_{P}}($ $\alpha_{i},p_{j})$ represents membership/belongingness of $\alpha_{i}$ in the picture fuzzy set $F(p_{j})$ , $\eta_{R_{P}}($ $\alpha_{i},p_{j})$ represents the neutral/abstain membership of $\alpha_{i}$ in the picture fuzzy set $F(p_{j})$ , and $\nu_{R_{P}}($ $\alpha_{i},p_{j})$ represents the non-membership/non-belongingness of $\alpha_{i}$ in the picture fuzzy set $F(p_{j})$ respectively, then we propose a matrix, termed as Picture fuzzy soft matrix (PFSM) over $U$ , which is given by

$\displaystyle S=[s_{ij}]_{m\times n}=[(\mu_{ij}^{S},\eta_{ij}^{S},\nu_{ij}^{S}% )]_{m\times n}=\left[\begin{array}[]{cccc}(\mu_{11},\eta_{11},\nu_{11)}&(\mu_{% 12},\eta_{12},\nu_{12)}&\cdots&(\mu_{1n},\eta_{1n},\nu_{1n})\\ (\mu_{21},\eta_{21},\nu_{21})&(\mu_{22},\eta_{22},\nu_{22})&\cdots&(\mu_{2n},% \eta_{2n},\nu_{2n})\\ \vdots&\vdots&\vdots&\vdots\\ (\mu_{m1},\eta_{m1},\nu_{m1})&(\mu_{m2},\eta_{m2},\nu_{m2})&\cdots&(\mu_{mn},% \eta_{mn},\nu_{mn})\end{array}\right].$

For a better understanding, let us consider a hypothetical example where $U={\{\alpha_{1},\alpha_{2},\alpha_{3}}\}$ is a universal set and $P={\{p_{1},p_{2},p_{3}\}}$ is a set of parameters and

$\displaystyle\Phi(p_{1})=\{(\alpha_{1},0.5,0.2,0.2),(\alpha_{2},0.3,0.3,0.4),(% \alpha_{3},0.6,0.1,0.2)\},$ $\displaystyle\Phi(p_{2})=\{(\alpha_{1},0.2,0.5,0.3),(\alpha_{2},0.3,0.3,0.3),(% \alpha_{3},0.2,0.3,0.4)\},$ $\displaystyle\Phi(p_{3})=\{(\alpha_{1},0.5,0.2,0.3),(\alpha_{2},0.4,0.2,0.2),(% \alpha_{3},0.4,0.4,0.1)\},$

then $(\Phi,P)$ is the parameterized family of ${\Phi({p}_{1}),\Phi({p}_{2}),\Phi({p}_{3})}$ over $U$ .

Hence, the picture fuzzy soft matrix $S$ can be written as

$\displaystyle[S]=[(\mu_{ij}^{S},\eta_{ij}^{S},\nu_{ij}^{S})]_{m\times n}=\left% [\begin{array}[]{ccc}(0.5,0.2,0.2)&(0.2,0.5,0.3)&(0.5,0.2,0.3)\\ (0.3,0.3,0.4)&(0.3,0.3,0.3)&(0.4,0.2,0.2)\\ (0.6,0.1,0.2)&(0.2,0.3,0.4)&(0.4,0.4,0.1)\end{array}\right].$

Suppose $\textit{PFSM}_{m\times n}$ is a set of all the picture fuzzy soft matrices over $U$ . Subsequently, different types of picture fuzzy soft matrices have been accordingly provided. A matrix $S=[(\mu_{ij}^{S},\eta_{ij}^{S},\nu_{ij}^{S})]\in\textit{PFSM}_{m\times n}$ is called Picture fuzzy soft:

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zero matrix if $\mu_{ij}^{S}=0,\,\,\eta_{ij}^{S}=0\,\,\mbox{and}\,\,\nu_{ij}^{S}=0;\forall i,j% \,\,\mbox{and is denoted by}\,\,0=[0,0,0].$
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square matrix if $m=n$ .
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row matrix if $n=1$ .
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column matrix if $m=1$ .
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diagonal matrix if all its non-diagonal elements are zero $\forall$ $i, j$ .
•
$\mu$ -universal matrix if $\mu_{ij}^{S}=1$ , $\eta_{ij}^{S}=0$ & $\nu_{ij}^{S}=0$ ; $\forall$ $i$ & $j$ , denoted by $P_{\mu}$ .
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$\eta$ -universal matrix if $\mu_{ij}^{S}=0$ , $\eta_{ij}^{S}=1$ & $\nu_{ij}^{S}=0$ ; $\forall$ $i$ & $j$ , denoted by $P_{\eta}$ .
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$\nu$ -universal matrix if $\mu_{ij}^{S}=0$ , $\eta_{ij}^{S}=0$ & $\nu_{ij}^{S}=1$ ; $\forall$ $i$ & $j$ , denoted by $P_{\nu}$ .
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Scalar multiplication of picture fuzzy soft matrix: for any scalar $k$ , we define $kA=[(k\mu_{ij}^{S},k\eta_{ij}^{S},$ $k\nu_{ij}^{S})]$ , $\forall$ $i$ & $j$ .

Next, we present some set-theoretic relations for given picture fuzzy soft matrices $S=[(\mu_{ij}^{S},\eta_{ij}^{S},\nu_{ij}^{S})]$ and $R=[(\mu_{ij}^{R},\eta_{ij}^{R},\nu_{ij}^{R})]\in\textit{PFSM}_{m\times n}$ .

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Subsethood: $S\subseteq R$ if $\mu_{ij}^{S}\leqslant\mu_{ij}^{R}$ , $\eta_{ij}^{S}\geqslant\eta_{ij}^{R}$ & $\nu_{ij}^{S}\geqslant\nu_{ij}^{R}$ ; $\forall$ $i$ & $j$ .
•
Containment: $S\supseteq R$ if $\mu_{ij}^{S}\geqslant\mu_{ij}^{R}$ , $\eta_{ij}^{S}\leqslant\eta_{ij}^{R}$ & $\nu_{ij}^{S}\leqslant\nu_{ij}^{R}$ ; $\forall$ $i$ & $j$ .
•
Equality: $S=R$ if $\mu_{ij}^{S}=\mu_{ij}^{R}$ , $\eta_{ij}^{S}=\eta_{ij}^{R}$ & $\nu_{ij}^{S}=\nu_{ij}^{R}$ ; $\forall$ $i$ & $j$ .
•
Max Min Product: Let $S=[a_{ij}]=[(\mu_{ij}^{S},\eta_{ij}^{S},\nu_{ij}^{S})]\in\textit{PFSM}_{m% \times n}$ & $R=[b_{jk}]=[(\mu_{jk}^{R},\eta_{jk}^{R},\nu_{jk}^{R})]\in\textit{PFSM}_{n% \times p}$ be two picture fuzzy soft matrices then

$\displaystyle SR=[c_{ik}]_{m\times p}=\left[\left(\max(\min\limits_{j}(\mu_{% ij}^{S},\mu_{jk}^{R})),\min(\min\limits_{j}(\eta_{ij}^{S},\eta_{jk}^{R})),\min% (\max\limits_{j}(\nu_{ij}^{S},\nu_{jk}^{R}))\right)\right];\forall i,j\&k.$
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Average Max Min Product: Let $S=[a_{ij}]=[(\mu_{ij}^{S},\eta_{ij}^{S},\nu_{ij}^{S})]\in\textit{PFSM}_{m% \times n}$ & $R=[b_{jk}]=[(\mu_{jk}^{R},\eta_{jk}^{R},\nu_{jk}^{R})]\in\textit{PFSM}_{n% \times p}$ be two picture fuzzy soft matrices then

$\displaystyle S_{A}R=[c_{ik}]_{m\times p}=\left[\left(\max\limits_{j}\left(% \frac{\mu_{ij}^{S}+\mu_{jk}^{R}}{2}\right),\min\limits_{j}\left(\frac{\eta_{ij% }^{S}+\eta_{jk}^{R}}{2}\right),\min\limits_{j}\left(\frac{\nu_{ij}^{S},\nu_{jk% }^{R}}{2}\right)\right)\right];\forall i,j\&k.$

Remark: It may be noted that Klement et al. [43, 44] have provided different types of triangular norm (t-norm) and triangular conorm (t-conorm) in an elaborated way. Accordingly, different combinations of these norms can also be considered for the proposed picture fuzzy soft matrices. Here, we have only taken the composition of maximum (t-conorm) and minimum (t-norm) operator.

Standard binary operations for picture fuzzy soft matrices:

Suppose that there are two picture fuzzy soft matrices $S_{1}=[(\mu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}})]$ and $S_{2}=[(\mu_{ij}^{S_{2}},\eta_{ij}^{S_{2}},\nu_{ij}^{S_{2}})]$ $\in\textit{PFSM}_{m\times n}$ . Then some of the binary operations may be given as follows:

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$S_{1}^{c}=\left[\left(\nu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\mu_{ij}^{S_{1}}% \right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}\cup S_{2}=\left[\left(\max(\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}),\min(\eta_% {ij}^{S_{1}},\eta_{ij}^{S_{2}}),\min(\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}})\right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}\cap S_{2}=\left[\left(\min(\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}),\min(\eta_% {ij}^{S_{1}},\eta_{ij}^{S_{2}}),\max(\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}})\right)\right]$ $\forall$ ; $i$ and $j$ .
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$S_{1}\otimes S_{2}=\left[\left(\mu_{ij}^{S_{1}}\cdot\mu_{ij}^{S_{2}},\,\eta_{% ij}^{S_{1}}\cdot\eta_{ij}^{S_{2}},\,\nu_{ij}^{S_{1}}+\nu_{ij}^{S_{2}}-\nu_{ij}% ^{S_{1}}\cdot\nu_{ij}^{S_{2}}\right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}\oplus S_{2}=\left[\left(\mu_{ij}^{S_{1}}+\mu_{ij}^{S_{2}}-\mu_{ij}^{S_{1% }}\cdot\mu_{ij}^{S_{2}},\eta_{ij}^{S_{1}}\cdot\eta_{ij}^{S_{2}},\nu_{ij}^{S_{1% }}\cdot\nu_{ij}^{S_{2}}\right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}@S_{2}=\left[\left(\frac{\mu_{ij}^{S_{1}}+\mu_{ij}^{S_{2}}}{2},\frac{\eta% _{ij}^{S_{1}}+\eta_{ij}^{S_{2}}}{2},\frac{\nu_{ij}^{S_{1}}+\nu_{ij}^{S_{2}}}{2% }\right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}@_{w}S_{2}=\left[\left(\frac{w_{1}\mu_{ij}^{S_{1}}+w_{2}\mu_{ij}^{S_{2}}}% {w_{1}+w_{2}},\frac{w_{1}\eta_{ij}^{S_{1}}+w_{2}\eta_{ij}^{S_{2}}}{w_{1}+w_{2}% },\frac{w_{1}\nu_{ij}^{S_{1}}+w_{2}\nu_{ij}^{S_{2}}}{w_{1}+w_{2}}\right)\right]$ ; $\forall$ $i$ and $j$ ; where $w_{1},w_{2}>0$ are the weights.
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$S_{1}\$S_{2}=\left[\left(\mu_{ij}^{S_{1}}\cdot\mu_{ij}^{S_{2}},\eta_{ij}^{S_{1% }}\cdot\eta_{ij}^{S_{2}},\nu_{ij}^{S_{1}}\cdot\nu_{ij}^{S_{2}}\right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}\$_{w}S_{2}=\left[\left(((\mu_{ij}^{S_{1}})^{w_{1}}\cdot(\mu_{ij}^{S_{2}}% )^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\eta_{ij}^{S_{1}})^{w_{1}}\cdot(\eta_{ij}% ^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\nu_{ij}^{S_{1}})^{w_{1}}\cdot(% \nu_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}}\right)\right]$ ; $\forall$ $i$ and $j$ , where $w_{1},w_{2}>0$ are the weights.
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$S_{1}\bowtie S_{2}=\left[\left(2\cdot\frac{\mu_{ij}^{S_{1}}\cdot\mu_{ij}^{S_{2% }}}{\mu_{ij}^{S_{1}}+\mu_{ij}^{S_{2}}},2\cdot\frac{\eta_{ij}^{S_{1}}\cdot\eta_% {ij}^{S_{2}}}{\eta_{ij}^{S_{1}}+\eta_{ij}^{S_{2}}},2\cdot\frac{\nu_{ij}^{S_{1}% }\cdot\nu_{ij}^{S_{2}}}{\nu_{ij}^{S_{1}}+\nu_{ij}^{S_{2}}}\right)\right]$ ; $\forall$ $i$ and $j$ .
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$S_{1}\bowtie_{w}S_{2}=\left[\left(\frac{w_{1}+w_{2}}{\frac{w_{1}}{\mu_{ij}^{S_% {1}}}+\frac{w_{2}}{\mu_{ij}^{S_{2}}}},\frac{w_{1}+w_{2}}{\frac{w_{1}}{\eta_{ij% }^{S_{1}}}+\frac{w_{2}}{\eta_{ij}^{S_{2}}}},\frac{w_{1}+w_{2}}{\frac{w_{1}}{% \nu_{ij}^{S_{1}}}+\frac{w_{2}}{\nu_{ij}^{S_{2}}}}\right)\right]$ ; $\forall$ $i$ and $j$ ; where $w_{1},w_{2}>0$ are the weights.

.

Let $S_{1}$ and $S_{2}$ $\in\textit{PFSM}_{m\times n}$ then the following laws hold:

(i)
$S_{1}\cup S_{2}=S_{2}\cup S_{1}$
(ii)
$S_{1}\cap S_{2}=S_{2}\cap S_{1}$
(iii)
$(S_{1}\cup S_{2})^{c}=S_{1}^{c}\cap S_{2}^{c}$
(iv)
$(S_{1}\cap S_{2})^{c}=S_{1}^{c}\cup S_{2}^{c}$
(v)
$\left(S_{1}^{c}\cap S_{2}^{c}\right)^{c}=S_{1}\cup S_{2}$
(vi)
$\left(S_{1}^{c}\cup S_{2}^{c}\right)^{c}=S_{1}\cap S_{2}$ .

Proof: Let $S_{1}=[(\mu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}})],S_{2}=[(\mu_{ij}% ^{S_{2}},\eta_{ij}^{S_{2}},\nu_{ij}^{S_{2}})]\in\textit{PFSM}_{m\times n}$ . Then $\forall$ $i$ & $j$ we get,

$\displaystyle\text{(i)}\ \ S_{1}\cup S_{2}=\Big{[}\big{(}\max(\mu_{ij}^{S_{1}}% ,\mu_{ij}^{S_{2}}),\min(\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}),\min(\nu_{ij}^{S_% {1}},\nu_{ij}^{S_{2}})\big{)}\Big{]}=\Big{[}\big{(}\max(\mu_{ij}^{S_{2}},\mu_{% ij}^{S_{1}}),\min(\eta_{ij}^{S_{2}},\eta_{ij}^{S_{1}}),\min(\nu_{ij}^{S_{2}},% \nu_{ij}^{S_{1}})\big{)}\Big{]}=S_{2}\cup S_{1}.$ $\displaystyle\text{(ii)}\ \ S_{1}\cup S_{2}=\Big{[}\big{(}\min(\mu_{ij}^{S_{1}% },\mu_{ij}^{S_{2}}),\min(\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}),\max(\nu_{ij}^{S% _{1}},\nu_{ij}^{S_{2}})\big{)}\Big{]}=\Big{[}\big{(}\min(\mu_{ij}^{S_{2}},\mu_% {ij}^{S_{1}}),\min(\eta_{ij}^{S_{2}},\eta_{ij}^{S_{1}}),\max(\nu_{ij}^{S_{2}},% \nu_{ij}^{S_{1}})\big{)}\Big{]}=S_{2}\cup S_{1}.$

$\displaystyle\text{(iii)}\ \ (S_{1}\cup S_{2})^{c}=\Big{(}\big{(}[(\mu_{ij}^{S% _{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}})]\cup[(\mu_{ij}^{S_{2}},\eta_{ij}^{S_% {2}},\nu_{ij}^{S_{2}})]\big{)}\Big{)}^{c}=[\max(\mu_{ij}^{S_{1}},\mu_{ij}^{S_{% 2}}),\min(\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}),\min(\nu_{ij}^{S_{1}},\nu_{ij}^% {S_{2}})]^{c}=\big{[}\big{(}\min(\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}),\min(\eta_% {ij}^{S_{1}},\eta_{ij}^{S_{2}}),\max(\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}})\big{)}% \big{]}=\big{[}\big{(}[(\nu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\mu_{ij}^{S_{1}})]% \cap[(\nu_{ij}^{S_{2}},\eta_{ij}^{S_{2}},\mu_{ij}^{S_{2}})]\big{)}\big{]}=S_{1% }^{c}\cap S_{2}^{c}.$

Similarly, $(iv)$ , $(v)$ and $(vi)$ can be proved easily.

.

Let $S_{1}=[(\mu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}})]$ $\in\textit{PFSM}_{m\times n}$ . Then the following laws hold as per the proposed definitions:

(i)
$\left(S_{1}^{c}\right)^{c}=S_{1}$
(ii)
$\left(P_{\mu}\right)^{c}=P_{\nu}$
(iii)
$\left(P_{\eta}\right)^{c}=P_{\eta}$
(iv)
$\left(P_{\nu}\right)^{c}=P_{\mu}$
(v)
$S_{1}\cup S_{1}=S_{1}$
(vi)
$S_{1}\cup P_{\mu}=P_{\mu}$
(vii)
$S_{1}\cap P_{\nu}=S_{1}$
(viii)
$S_{1}\cap S_{1}=S_{1}$
(ix)
$S_{1}\cap P_{\mu}=S_{1}$
(x)
$S_{1}\cap P_{\nu}=P_{\nu}$ .

.

Let $S_{1}$ and $S_{2}\in\textit{PFSM}_{m\times n}$ . Then the following laws w.r.t. the weighted form hold:

(i)
$(S_{1}^{c}@_{w}S_{2}^{c})^{c}=S_{1}@_{w}S_{2}$
(ii)
$(S_{1}^{c}\$_{w}S_{2}^{c})^{c}=S_{1}\$_{w}S_{2}$
(iii)
$(S_{1}^{c}\bowtie_{w}S_{2}^{c})^{c}=S_{1}\bowtie_{w}S_{2}$
(iv)
$S_{1}@_{w}S_{2}=S_{2}@_{w}S_{1}$
(v)
$S_{1}\$_{w}S_{2}=S_{2}\$_{w}S_{1}$
(vi)
$S_{1}\bowtie_{w}S_{2}=S_{2}\bowtie_{w}S_{1}$ .

Proof: Let $S_{1}=[(\mu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}})],S_{2}=[(\mu_{ij}% ^{S_{2}},\eta_{ij}^{S_{2}},\nu_{ij}^{S_{2}})]\in\textit{PFSM}_{m\times n}$ . Then $\forall$ $i, j$ & $w_{1},w_{2}>0,$ we get,

$\displaystyle\text{(i)}\ \ (S_{1}^{c}@_{w}S_{2}^{c})^{c}=\bigg{(}\left[\bigg{(% }(\nu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\mu_{ij}^{S_{1}})@_{w}(\nu_{ij}^{S_{2}},% \eta_{ij}^{S_{2}},\mu_{ij}^{S_{2}})\bigg{)}\right]\bigg{)}^{c}=\left(\left[% \bigg{(}\frac{w_{1}\nu_{ij}^{S_{1}}+w_{2}\nu_{ij}^{S_{2}}}{w_{1}+w_{2}},\frac{% w_{1}\eta_{ij}^{S_{1}}+w_{2}\eta_{ij}^{S_{2}}}{w_{1}+w_{2}},\frac{w_{1}\mu_{ij% }^{S_{1}}+w_{2}\mu_{ij}^{S_{2}}}{w_{1}+w_{2}}\bigg{)}\right]\right)^{c}=\left[% \bigg{(}\frac{w_{1}\mu_{ij}^{S_{1}}+w_{2}\mu_{ij}^{S_{2}}}{w_{1}+w_{2}},\frac{% w_{1}\eta_{ij}^{S_{1}}+w_{2}\eta_{ij}^{S_{2}}}{w_{1}+w_{2}},\frac{w_{1}\nu_{ij% }^{S_{1}}+w_{2}\nu_{ij}^{S_{2}}}{w_{1}+w_{2}}\bigg{)}\right]=S_{1}@_{w}S_{2}.$

$\displaystyle\text{(ii)}\ \ (S_{1}^{c}\$_{w}S_{2}^{c})^{c}=\left(\left[\bigg{(% }(\nu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\mu_{ij}^{S_{1}})\$_{w}(\nu_{ij}^{S_{2}},% \eta_{ij}^{S_{2}},\mu_{ij}^{S_{2}})\bigg{)}\right]\right)^{c}$ $\displaystyle\qquad=\left(\left[\bigg{(}((\nu_{ij}^{S_{1}})^{w_{1}}\cdot(\nu_{% ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\eta_{ij}^{S_{1}})^{w_{1}}% \cdot(\eta_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\mu_{ij}^{S_{1}})^% {w_{1}}\cdot(\mu_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}}\bigg{)}\right]% \right)^{c}$ $\displaystyle\qquad=\left[\bigg{(}((\mu_{ij}^{S_{1}})^{w_{1}}\cdot(\mu_{ij}^{S% _{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\eta_{ij}^{S_{1}})^{w_{1}}\cdot(\eta% _{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\nu_{ij}^{S_{1}})^{w_{1}}% \cdot(\nu_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}}\bigg{)}\right]$ $\displaystyle\qquad=S_{1}\$_{w}S_{2}.$

Similar proof for (iii).

$\displaystyle\text{(iv)}\ \ S_{1}@_{w}S_{2}=\left[\bigg{(}\frac{w_{1}\mu_{ij}^% {S_{1}}+w_{2}\mu_{ij}^{S_{2}}}{w_{1}+w_{2}},\frac{w_{1}\eta_{ij}^{S_{1}}+w_{2}% \eta_{ij}^{S_{2}}}{w_{1}+w_{2}},\frac{w_{1}\nu_{ij}^{S_{1}}+w_{2}\nu_{ij}^{S_{% 2}}}{w_{1}+w_{2}}\bigg{)}\right]=\left[\bigg{(}\frac{w_{2}\mu_{ij}^{S_{2}}+w_{% 1}\mu_{ij}^{S_{1}}}{w_{2}+w_{1}},\frac{w_{2}\eta_{ij}^{S_{2}}+w_{1}\eta_{ij}^{% S_{1}}}{w_{2}+w_{1}},\frac{w_{2}\nu_{ij}^{S_{2}}+w_{1}\nu_{ij}^{S_{1}}}{w_{2}+% w_{1}}\bigg{)}\right]=S_{2}@_{w}S_{1}.$

$\displaystyle\text{(iv)}\ \ S_{1}\$_{w}S_{2}=\left[\!\bigg{(}((\mu_{ij}^{S_{1}% })^{w_{1}}\cdot(\mu_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((\eta_{ij}% ^{S_{1}})^{w_{1}}\cdot(\eta_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2}}},((% \nu_{ij}^{S_{1}})^{w_{1}}\cdot(\nu_{ij}^{S_{2}})^{w_{2}})^{\frac{1}{w_{1}+w_{2% }}}\bigg{)}\!\right]=\left[\!\bigg{(}((\mu_{ij}^{S_{2}})^{w_{2}}\cdot(\mu_{ij}% ^{S_{1}})^{w_{1}})^{\frac{1}{w_{2}+w_{1}}},((\eta_{ij}^{S_{2}})^{w_{2}}\cdot(% \eta_{ij}^{S_{1}})^{w_{1}})^{\frac{1}{w_{2}+w_{1}}},((\nu_{ij}^{S_{2}})^{w_{2}% }\cdot(\nu_{ij}^{S_{1}})^{w_{1}})^{\frac{1}{w_{2}+w_{1}}}\bigg{)}\!\right]=S_{% 2}\$_{w}S_{1}.$

Similarly, (vi) can easily be verified.

.

Let $S_{1}$ , $S_{2}$ and $S_{3}$ $\in\textit{PFSM}_{m\times n}$ be three matrices then the following laws in connection with the associativity hold:

(i)
$(S_{1}\cup S_{2})\cup S_{3}=S_{1}\cup(S_{2}\cup S_{3})$
(ii)
$(S_{1}\cap S_{2})\cap S_{3}=S_{1}\cap(S_{2}\cap S_{3})$
(iii)
$(S_{1}@S_{2})@S_{3}=S_{1}@(S_{2}@S_{3})$
(iv)
$(S_{1}\$S_{2})\$S_{3}=S_{1}\$(S_{2}\$S_{3})$
(v)
$(S_{1}\bowtie S_{2})\bowtie S_{3}=S_{1}\bowtie(S_{2}\bowtie S_{3})$ .

Proof: For all $i$ & $j$ we write,

$\displaystyle\text{(i)}\ (S_{1}\cup S_{2})\cup S_{3}=\bigg{[}\bigg{(}[(\max\{% \mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}% \}),\min\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}\}]\cup[(\mu_{ij}^{S_{3}},\nu_{ij}^% {S_{3}})]\bigg{)}\bigg{]}=\bigg{[}\bigg{(}\max\{(\mu_{ij}^{S_{1}},\mu_{ij}^{S_% {2}}),\mu_{ij}^{S_{3}}\},\min\{(\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}),\eta_{ij}% ^{S_{3}}\},\min\{(\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}),\nu_{ij}^{S_{3}}\}\bigg{)% }\bigg{]}=\bigg{[}\bigg{(}\max\{(\mu_{ij}^{S_{1}},(\mu_{ij}^{S_{2}},\mu_{ij}^{% S_{3}}))\},\min\{\eta_{ij}^{S_{1}},(\eta_{ij}^{S_{2}},\eta_{ij}^{S_{3}})\},% \min\{\nu_{ij}^{S_{1}},(\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}})\}\bigg{)}\bigg{]}=S% _{1}\cup(S_{2}\cup S_{3}).$ $\displaystyle\text{(ii)}\ (S_{1}\cap S_{2})\cap S_{3}=\bigg{[}\bigg{(}(\min\{% \mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}\},(\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}% \},\max\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}\})\cup(\mu_{ij}^{S_{3}},\nu_{ij}^{S% _{3}})\bigg{)}\bigg{]}=\bigg{[}\bigg{(}\min\{(\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}% }),\mu_{ij}^{S_{3}}\},(\min\{(\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}),\eta_{ij}^{% S_{3}}\},,\max\{(\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}),\nu_{ij}^{S_{3}}\})\bigg{)% }\bigg{]}=\bigg{[}\!\bigg{(}\!\!\min\{(\mu_{ij}^{S_{1}},(\mu_{ij}^{S_{2}},\mu_% {ij}^{S_{3}}))\},(\min\{(\eta_{ij}^{S_{1}},(\eta_{ij}^{S_{2}},\mu_{ij}^{S_{3}}% ))\},\max\{\nu_{ij}^{S_{1}},(\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}})\})\!\bigg{)}\!% \bigg{]}=S_{1}\cap(S_{2}\cap S_{3}).$

Similar proof for (iii), (iv) and (v).

.

Let $S_{1}$ , $S_{2}$ and $S_{3}$ $\in\textit{PFSM}_{m\times n}$ are soft matrices. The following laws in connection with the distributivity hold:

(i)
$S_{1}\cap(S_{2}\cup S_{3})=(S_{1}\cap S_{2})\cup(S_{1}\cap S_{3})$
(ii)
$(S_{1}\cap S_{2})\cup S_{3}=(S_{1}\cup S_{3})\cap(S_{2}\cup S_{3})$
(iii)
$S_{1}\cup(S_{2}\cap S_{3})=(S_{1}\cup S_{2})\cap(S_{1}\cup S_{3})$
(iv)
$(S_{1}\cup S_{2})\cap S_{3}=(S_{1}\cap S_{3})\cup(S_{2}\cap S_{3})$
(v)
$(S_{1}\cap S_{2})@S_{3}=(S_{1}@S_{3})\cap(S_{2}@S_{3})$
(vi)
$(S_{1}\cap S_{2})\bowtie S_{3}=(S_{1}\bowtie S_{3})\cap(S_{2}\bowtie S_{3})$
(vii)
$S_{1}\cup(S_{2}@S_{3})=(S_{1}\cup S_{2})@(S_{1}\cup S_{3})$
(viii)
$(S_{1}\cup S_{2})\bowtie S_{3}=(S_{1}\bowtie S_{3})\cup(S_{2}\bowtie S_{3})$
(ix)
$S_{1}@(S_{2}\cup S_{3})=(S_{1}@S_{2})\cup(S_{1}@S_{3})$
(x)
$S_{1}@(S_{2}\cap S_{3})=(S_{1}@S_{2})\cap(S_{2}@S_{3})$
(xi)
$S_{1}\$(S_{2}\cup S_{3})=(S_{1}\$S_{2})\cup(S_{1}\$S_{3})$
(xii)
$(S_{1}\cup S_{2})\$S_{3}=(S_{1}\$S_{3})\cup(S_{2}\$S_{3})$
(xiii)
$S_{1}\cup(S_{2}\bowtie S_{3})=(S_{1}\cup S_{2})\bowtie(S_{1}\cup S_{3})$
(xiv)
$S_{1}\bowtie(S_{2}\cup S_{3})=(S_{1}\bowtie S_{2})\cup(S_{1}\bowtie S_{3})$
(xv)
$S_{1}\$(S_{2}\cap S_{3})=(S_{1}\$S_{2})\cap(S_{2}\$S_{3})$
(xvi)
$(S_{1}\cap S_{2})\$S_{3}=(S_{1}\$S_{3})\cap(S_{2}\$S_{3})$ .

Proof: For all $i$ & $j$ we write,

$\displaystyle\text{(i)}\ \ S_{1}\cap(S_{2}\cup S_{3})=\bigg{[}\bigg{(}\big{[}% \big{(}\mu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}}\big{)}\big{]}\cap% \big{[}\big{(}\max\{\mu_{ij}^{S_{2}},\mu_{ij}^{S_{3}}\},\min\{\eta_{ij}^{S_{2}% },\eta_{ij}^{S_{3}}\},\min\{\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}}\}\big{)}\big{]}% \bigg{)}\bigg{]}=\bigg{[}\bigg{(}\min\{\mu_{ij}^{S_{1}},\max\{\mu_{ij}^{S_{2}}% ,\mu_{ij}^{S_{3}}\}\},\min\{\eta_{ij}^{S_{1}},\min\{\eta_{ij}^{S_{2}},\mu_{ij}% ^{S_{3}}\}\},\max\{\nu_{ij}^{S_{1}},\min\{\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}}\}% \}\bigg{)}\bigg{]}.$

Now,

$\displaystyle(S_{1}\cap S_{2})\cup(S_{1}\cap S_{3})=\Big{[}\big{(}\min\{\mu_{% ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}\},% \max\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}\}\big{)}\Big{]}\cup\Big{[}\big{(}\min% \{\mu_{ij}^{S_{1}},\mu_{ij}^{S_{3}}\},\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{3}% }\},\max\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{3}}\}\big{)}\Big{]}=\Big{[}\big{(}\max% (\min\{\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\min\{\mu_{ij}^{S_{1}},\mu_{ij}^{S_% {3}}\}),\min(\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}\},\quad\min\{\eta_{ij}^% {S_{1}},\eta_{ij}^{S_{3}}\}),\min(\max\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}\},% \max\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{3}}\})\big{)}\Big{]}=\Big{[}\big{(}\max(% \mu_{ij}^{S_{1}},\min\{\mu_{ij}^{S_{2}},\mu_{ij}^{S_{3}}\}),\min(\eta_{ij}^{S_% {1}},\min\{\eta_{ij}^{S_{2}},\eta_{ij}^{S_{3}}\}),\quad\min(\nu_{ij}^{S_{1}},% \max\{\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}}\})\big{)}\Big{]}=\Big{[}\big{(}\min(% \mu_{ij}^{S_{1}},\max\{\mu_{ij}^{S_{2}},\mu_{ij}^{S_{3}}\}),\min(\mu_{ij}^{S_{% 1}},\min\{\mu_{ij}^{S_{2}},\mu_{ij}^{S_{3}}\}),\quad\max(\nu_{ij}^{S_{1}},\min% \{\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}}\})\big{)}\Big{]}=S_{1}\cap(S_{2}\cup S_{3}).$

Hence, $S_{1}\cap(S_{2}\cup S_{3})=(S_{1}\cap S_{2})\cup(S_{1}\cap S_{3})$ holds.

$\displaystyle\text{(ii)}\ \ (S_{1}\cap S_{2})\cup S_{3}=\big{[}\big{(}\min\{% \mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{2}}% \},\max\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{2}}\}\big{)}\big{]}\cup\big{[}\big{(}% \mu_{ij}^{S_{3}},\nu_{ij}^{S_{3}}\big{)}\big{]}=\Big{[}\big{(}\max(\min\{\mu_{% ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\mu_{ij}^{S_{3}}),\min(\min\{\eta_{ij}^{S_{1}},% \eta_{ij}^{S_{2}}\},\mu_{ij}^{S_{3}}),\quad\min(\max\{\nu_{ij}^{S_{1}},\nu_{ij% }^{S_{2}}\},\nu_{ij}^{S_{3}})\big{)}\Big{]}.$

Now,

$\displaystyle(S_{1}\cup S_{3})\cap(S_{2}\cup S_{3})=\big{[}\big{(}\max\{\mu_{% ij}^{S_{1}},\mu_{ij}^{S_{3}}\},\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{3}}\},% \min\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{3}}\}\big{)}\big{]}\cap\big{[}\big{(}\max% \{\mu_{ij}^{S_{2}},\mu_{ij}^{S_{3}}\},\quad\min\{\eta_{ij}^{S_{2}},\eta_{ij}^{% S_{3}}\},\min\{\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}}\}\big{)}\big{]}=\Big{[}\big{(% }\min(\max\{\mu_{ij}^{S_{1}},\mu_{ij}^{S_{3}}\},\max\{\mu_{ij}^{S_{2}},\mu_{ij% }^{S_{3}}\}),\min(\min\{\eta_{ij}^{S_{1}},\eta_{ij}^{S_{3}}\},\quad\min\{\eta_% {ij}^{S_{2}},\eta_{ij}^{S_{3}}\}),\max(\min\{\nu_{ij}^{S_{1}},\nu_{ij}^{S_{3}}% \},\min\{\nu_{ij}^{S_{2}},\nu_{ij}^{S_{3}}\})\big{)}\Big{]}=\Big{[}\big{(}\min% (\max\{\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\mu_{ij}^{S_{3}}),\min(\min\{\eta_{% ij}^{S_{1}},\eta_{ij}^{S_{2}}\},\eta_{ij}^{S_{3}}),\quad\max(\min\{\nu_{ij}^{S% _{1}},\nu_{ij}^{S_{2}}\},\nu_{ij}^{S_{3}})\big{)}\Big{]}=\Big{[}\big{(}\max(% \min\{\mu_{ij}^{S_{1}},\mu_{ij}^{S_{2}}\},\mu_{ij}^{S_{3}}),\min(\min\{\eta_{% ij}^{S_{1}},\eta_{ij}^{S_{2}}\},\eta_{ij}^{S_{3}}),\quad\min(\max\{\nu_{ij}^{S% _{1}},\nu_{ij}^{S_{2}}\},\nu_{ij}^{S_{3}})\big{)}\Big{]}=(S_{1}\cap S_{2})\cup S% _{3}$

Hence, $(S_{1}\cap S_{2})\cup S_{3}=(S_{1}\cup S_{3})\cap(S_{2}\cup S_{3})$ .

The rest of the laws may easily be obtained on similar lines.
4. Application of picture fuzzy soft matrix in decision making

In this section, we consider a general decision-making problem where the information has been taken in the form of picture fuzzy soft matrix and propose some revised definitions which are utmost essential for solving the problem under consideration.

.

If $S_{i}=[(\mu_{ij}^{S_{i}},\eta_{ij}^{S_{i}},\nu_{ij}^{S_{i}})]\in\textit{PFSM}_% {m\times n}$ , then the choice matrix of PFSM $S_{i}$ is defined as

$\displaystyle\textit{CM}(S_{i})=\left[\left(\frac{\sum\limits_{j=1}^{n}\mu_{ij% }^{S_{i}}}{n},\frac{\sum\limits_{j=1}^{n}\eta_{ij}^{S_{i}}}{n},\frac{\sum% \limits_{j=1}^{n}\nu_{ij}^{S_{i}}}{n}\right)\right]_{m\times 1}\,;\,\,\forall% \,\,i\,\,\text{if weights are same}.$

.

If $S_{i}=[(\mu_{ij}^{S_{i}},\eta_{ij}^{S_{i}},\nu_{ij}^{S_{i}})]\in\textit{PFSM}_% {m\times n}$ , then the weighted choice matrix of PFSM $S_{i}$ is defined by

$\displaystyle\text{CM}_{w}(S_{i})=\left[\left(\frac{\sum\limits_{j=1}^{n}w_{j}% \mu_{ij}^{S_{i}}}{\sum w_{j}},\frac{\sum\limits_{j=1}^{n}w_{j}\eta_{ij}^{S_{i}% }}{\sum w_{j}},\frac{\sum\limits_{j=1}^{n}w_{j}\nu_{ij}^{S_{i}}}{\sum w_{j}}% \right)\right]_{m\times 1}\forall\,\,i\,\,\text{where}\,\,w_{j}>0\,\,\text{are% weights}.$

.

Suppose $S_{1}=[(\mu_{ij}^{S_{1}},\eta_{ij}^{S_{1}},\nu_{ij}^{S_{1}})]\in\textit{PFSM}_% {m\times n}$ . Then its score matrix is defined by $Sr(S_{1})=[s_{ij}]=[(\mu_{ij}^{S_{1}}-\eta_{ij}^{S_{1}}-\nu_{ij}^{S_{1}})]$ $\forall$ $i$ and $j$ . It may be noted that the $(i,j)^{\text{th}}$ element of $Sr(S_{1})$ depicts an index for computing the optimized value of the belongingness (or non-belongingness) of the $i^{\text{th}}$ alternative getting $j^{\text{th}}$ criteria.

Based on the above proposed definitions, we present a new algorithm to deal with the problem of decision-making which is being represent in the flow chart given by Fig. 2.

Figure 2.

Flow chart of the algorithm for decision-making.

The proposed methodology for solving the multi-criteria decision-making problem is being illustrated with a numerical example as follows:

Example 1. Consider there is a financial private limited firm whose objective is to invest a significant amount of money in the best possible sector. Suppose there are four possible investment sectors selected on the basis of an initial survey say:

•

“Mutual Funds $(Z^{1})$ ”

•

“Health Insurance $(Z^{2})$ ”

•

“Share Market $(Z^{3})$ ”

•

“Housing Development Corporations $(Z^{4})$ ”

•

“General Insurance $(Z^{5})$ ”

Consider the investment company is going to take a decision based on the following four important criteria:

•

“Maximum Returns $(C^{1})$ ”

•

“Minimum Risk $(C^{2})$ ”

•

“Easy withdrawal in case of emergency $(C^{3})$ ”

•

“ Transparency $(C^{4})$ ”

Suppose that based on the financial strategies adopted for the welfare of the company, the weight vector is $\omega=(0.2352,0.2638,0.2666,0.2344)^{T}.$

The computational steps for the the above stated problem using the proposed algorithm are below.

•

Step 1. First we write the following picture fuzzy soft decision matrix $R=[(r_{ij})]=[(\mu_{ij},\eta_{ij},\nu_{ij})],$ for the five alternatives $Z^{i}\,\,(i=1,2,3,4,5)$ & the four criteria $\mathcal{C}^{j}\,\,(j=1,2,3,4)$ based on the information provided by the experts:

$\displaystyle R=\bordermatrix{&C^{1}&C^{2}&C^{3}&C^{4}\cr Z^{1}&(0.89,0.08,0.0% 3)&(0.42,0.35,0.18)&(0.08,0.89,0.02)&(0.80,0.11,0.05)\cr Z^{2}&(0.23,0.64,0.11% )&(0.03,0.82,0.13)&(0.73,0.15,0.08)&(0.73,0.10,0.14)\cr Z^{3}&(0.56,0.26,0.05)% &(0.04,0.85,0.10)&(0.68,0.26,0.06)&(0.43,0.13,0.25)\cr Z^{4}&(0.74,0.16,0.10)&% (0.02,0.89,0.05)&(0.08,0.84,0.06)&(0.85,0.09,0.05)\cr Z^{5}&(0.68,0.08,0.21)&(% 0.05,0.87,0.06)&(0.13,0.75,0.09)&(0.65,0.05,0.02)}.$

•

Step 2. Since all the criterion under consideration are of homogeneous type so we donot have to normalize the decision matrix.

•

Step 3:

–

Case 1: Weights are EqualWe find the choice matrix for $R$ as:

$\displaystyle\textit{CM}(R)=\left[\begin{array}[]{c}(0.5475,0.3575,0.0700)\\ (0.4300,0.4275,0.1150)\\ (0.4275,0.3750,0.1150)\\ (0.4225,0.4950,0.0650)\\ (0.3775,0.4375,0.0950)\end{array}\right].$

–

Case 2: Weights are UnequalIf the weights $\omega=(0.2352,0.2638,0.2666,0.2344)^{T}$ are given for the criteria “Maximum Returns $(C^{1})$ ”; “Minimum Risk $(C^{2})$ ”; “Easy withdrawal in case of emergency $(C^{3})$ ” and “Transparency $(C^{4})$ ”, respectively. Then the weighted choice matrix for $R$ is as

$\displaystyle\textit{CM}_{w}(R)=\left[\begin{array}[]{c}(0.5289,0.3742,0.0716)% \\ (0.4277,0.4303,0.1143)\\ (0.4243,0.3852,0.1127)\\ (0.3998,0.5175,0.0644)\\ (0.3601,0.4600,0.0939)\end{array}\right].$

•

Step 4: Next, we compute the score matrices of $C(R)$ and $C_{w}(R)$ evaluated in Step 3 as follows:

$\displaystyle\textit{Sr(CM(R))}=\left[\begin{array}[]{c}\textbf{0.4775}\\ 0.3150\\ 0.3125\\ 0.3575\\ 0.2825\end{array}\right].$

$\displaystyle\textit{Sr}(\textit{CM}_{w}(R))=\left[\begin{array}[]{c}\textbf{0% .4574}\\ 0.3134\\ 0.3116\\ 0.3355\\ 0.2662\end{array}\right].$

•

Step 5:

–

When weights are equal: In view of the Step 4 that if equal preference is assigned to each and every criteria, then we get $0.4775$ as the maximum value of the score matrix, i.e., investment in “Mutual Funds”. Therefore, in this case the most suitable option for investment is “Mutual Funds”.

–

When weights are unequal: Suppose that if a company assumes the importance of the criteria “Easy withdrawal in case of emergency” over the other criteria, then $0.4574$ being the maximum value of the score matrix for option $Z^{1}$ . Therefore, in this case the most suitable option for investment is “Mutual Funds”.

On the other hand, in order to illustrate the effectiveness of proposed method, the same above example was solved by using various existing methods [45, 11, 46, 16, 8] and the obtained results has been tabulated in the Table 1.

Table 1

Comparison with other existing approaches

TODIM method [16]	$Z^{1}>Z^{4}>Z^{5}>Z^{3}>Z^{2}$
Generalized picture fuzzy distance measure [8]	$Z^{1}>Z^{4}>Z^{2}>Z^{3}>Z^{5}$
Picture fuzzy cross-entropy [11]	$Z^{1}>Z^{4}>Z^{2}>Z^{5}>Z^{3}$
Similarity measure picture fuzzy set [46]	$Z^{1}>Z^{4}>Z^{2}>Z^{5}>Z^{3}$
Picture fuzzy projection model [45]	$Z^{1}>Z^{2}>Z^{3}>Z^{4}>Z^{5}$
Proposed method	$Z^{1}>Z^{4}>Z^{2}>Z^{3}>Z^{5}$

Thus from the Table 1 it may be observed that the alternative $Z^{1}$ is the best one. Therefore, the best alternative strategy for the company is to invest in the Mutual Fund.

Comparative remarks:

On the basis of above performed calculation, we can conclude the following important comparative remarks as follows:

•

Various researchers [45, 11, 46, 16, 8] had discussed and dealt with the decision-making problem without utilizing the notion of matrices.

•

The proposed methodology firstly represented the available data in the matrix form and then worked out to determine that the ‘Mutual Fund’ $Z^{1}$ is the best/optimal choice available for the investment among all the five alternatives .

•

The inclusion of the notion of matrix in the proposed methodology gives the enhanced dimensionality feature along with wider span of information.

5. Dimensionality reduction technique

In this section, the definition of “object-oriented Picture fuzzy soft matrix” and “parameter-oriented Picture fuzzy soft matrix” has been presented along with a revised definition of threshold value.

Consider $U={\{\alpha_{1},\alpha_{2},\ldots,\alpha_{m}}\}$ be the discourse of universe and a set of parameter, say, $P={\{p_{1},p_{2},\ldots,p_{n}}\}$ and suppose $M$ to be the PFSM of the PFSS $(\Phi,P)$ .

.

The object-oriented PFSM w.r.t the parameters is given as:

$\displaystyle O_{i}=\left[\sum\limits_{j}\frac{\mu_{ij}}{|P|},\sum\limits_{j}% \frac{\eta_{ij}}{|P|},\sum\limits_{j}\frac{\nu_{ij}}{|P|}\right];$ (7)

and the parameter-oriented PFSM w.r.t the objects is defined as:

$\displaystyle P_{j}=\left[\sum\limits_{j}\frac{\mu_{ij}}{|U|},\sum\limits_{j}% \frac{\eta_{ij}}{|U|},\sum\limits_{j}\frac{\nu_{ij}}{|U|}\right];$ (8)

where, $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$ .

.

The threshold value of PFSM is defined as $Sr(T)=\mu_{ij}^{M}-\nu_{ij}^{M}$ , where

$\displaystyle T=(\mu_{T},\eta_{T},\nu_{T})=\left[\sum\limits_{i,j}\frac{\mu_{% ij}}{|U\times P|},\sum\limits_{i,j}\frac{\eta_{ij}}{|U\times P|},\sum\limits_{% i,j}\frac{\nu_{ij}}{|U\times P|}\right];$ (9)

and $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$ .

Based on the picture fuzzy soft matrix and the above proposed definitions, we proposed an dimensionality reduction technique which may be represented with the help of the following flow chart given by Fig. 3.

Figure 3.

Flow chart of algorithm for dimensionality reduction.

5.1 Application of dimensionality reduction in decision making

In this section, an example for illustrating the step by step implementation of the proposed algorithm has been provided along with the comparative analysis to validate the performance of the proposed algorithm in contrast with the existing approaches.

Consider there is a financial private limited firm whose objective is to invest a significant amount of money in the best possible sector. Suppose there are four possible investment sectors selected on the basis of an initial survey say:

•
“Mutual Funds $(Z^{1})$ ”
•
“Health Insurance $(Z^{2})$ ”
•
“Share Market $(Z^{3})$ ”
•
“Housing Development Corporations $(Z^{4})$ ”
•
“General Insurance $(Z^{5})$ ”

Also, suppose the investment company is going to take a decision based on the laid three folowing important criteria:

•
“Maximum Returns $(C^{1})$ ”
•
“Minimum Risk $(C^{2})$ ”
•
“Easy withdrawal in case of emergency $(C^{3})$ ”
•
“ Transparency $(C^{4})$ ”

For implementing the proposed algorithm taking the above problem into consideration, the computational steps are as follows:

•
Step 1. Construct the PFSM

$\displaystyle M=\bordermatrix{&\mathcal{C}^{1}&\mathcal{C}^{2}&\mathcal{C}^{3}% &\mathcal{C}^{4}\cr Z^{1}&(0.89,0.08,0.03)&(0.42,0.35,0.18)&(0.08,0.89,0.02)&(% 0.80,0.11,0.05)\cr Z^{2}&(0.23,0.64,0.11)&(0.03,0.82,0.13)&(0.73,0.15,0.08)&(0% .73,0.10,0.14)\cr Z^{3}&(0.56,0.26,0.05)&(0.04,0.85,0.10)&(0.68,0.26,0.06)&(0.% 43,0.13,0.25)\cr Z^{4}&(0.74,0.16,0.10)&(0.02,0.89,0.05)&(0.08,0.84,0.06)&(0.8% 5,0.09,0.05)\cr Z^{5}&(0.68,0.08,0.21)&(0.05,0.87,0.06)&(0.13,0.75,0.09)&(0.65% ,0.05,0.02)}.$
•
Step 2. Determine the object-oriented PFSM $O_{i}$ for $i=1,2,\ldots,5$ and the parameter-oriented PFSM $P_{j}$ for $j=1,\ldots,4$ .

$\displaystyle\bordermatrix{&C^{1}&C^{2}&C^{3}&C^{4}&O_{i}\cr Z^{1}&(0.89,0.08,% 0.03)&(0.42,0.35,0.18)&(0.08,0.89,0.02)&(0.80,0.11,0.05)&(0.5475,0.3575,0.07)% \cr Z^{2}&(0.23,0.64,0.11)&(0.03,0.82,0.13)&(0.73,0.15,0.08)&(0.73,0.10,0.14)&% (0.43,0.4275,0.1150)\cr Z^{3}&(0.56,0.26,0.05)&(0.04,0.85,0.10)&(0.68,0.26,0.0% 6)&(0.43,0.13,0.25)&(0.4275,0.375,0.1150)\cr Z^{4}&(0.74,0.16,0.10)&(0.02,0.89% ,0.05)&(0.08,0.84,0.06)&(0.85,0.09,0.05)&(0.4225,0.495,0.0650)\cr Z^{5}&(0.68,% 0.08,0.21)&(0.05,0.87,0.06)&(0.13,0.75,0.09)&(0.65,0.05,0.02)&(0.3775,0.4375,0% .0950)\cr P_{j}&(0.62,0.244,0.1)&(0.112,0.756,0.104)&(0.34,0.578,0.062)&(0.692% ,0.096,0.102)}$

Now, compute the score matrix of the object-oriented PFSM $\textit{Sr}(O_{i})$ and the parameter-oriented PFSM $Sr(P_{j})$ is given as:

$\displaystyle\bordermatrix{&C&C^{2}&C^{3}&C^{4}&O_{i}&Sr(O_{i})\cr Z^{1}&(0.89% ,0.08,0.03)&(0.42,0.35,0.18)&(0.08,0.89,0.02)&(0.80,0.11,0.05)&(0.5475,0.3575,% 0.07)&0.4775\cr Z^{2}&(0.23,0.64,0.11)&(0.03,0.82,0.13)&(0.73,0.15,0.08)&(0.73% ,0.10,0.14)&(0.43,0.4275,0.1150)&0.315\cr Z^{3}&(0.56,0.26,0.05)&(0.04,0.85,0.% 10)&(0.68,0.26,0.06)&(0.43,0.13,0.25)&(0.4275,0.375,0.1150)&0.3125\cr Z^{4}&(0% .74,0.16,0.10)&(0.02,0.89,0.05)&(0.08,0.84,0.06)&(0.85,0.09,0.05)&(0.4225,0.49% 5,0.0650)&0.3575\cr Z^{5}&(0.68,0.08,0.21)&(0.05,0.87,0.06)&(0.13,0.75,0.09)&(% 0.65,0.05,0.02)&(0.3775,0.4375,0.0950)&0.2825\cr P_{j}&(0.62,0.244,0.1)&(0.112% ,0.756,0.104)&(0.34,0.578,0.062)&(0.692,0.096,0.102)&\cr\textit{Sr}(P_{j})&0.5% 2&0.008&0.278&0.59}$
•
Step 3. In this step, we determine the threshold element of PFSM and its corresponding threshold value:

$\displaystyle T=(0.441,0.4185,0.092)\,\text{and}\,\,Sr(T)=0.349.$
•
Step 4. Next, in accordance with the values obtained in step 3 & 4, we remove those alternatives for which condition $\textit{Sr}(O_{i})<Sr(T)$ and those parameters for which condition $\textit{Sr}(P_{j})>\textit{Sr}(T)$ holds. Thus, the desired matrix $M^{{}^{\prime}}$ is given as:

$\displaystyle M^{{}^{\prime}}=\bordermatrix{&C^{2}&C^{3}&O_{i}&Sr(O_{i})\cr Z^% {1}&(0.42,0.35,0.18)&(0.08,0.89,0.02)&(0.5475,0.3575,0.07)&\textbf{0.4775}\cr Z% ^{4}&(0.02,0.89,0.05)&(0.08,0.84,0.06)&(0.4225,0.495,0.0650)&0.3575\cr P_{j}&(% 0.112,0.756,0.104)&(0.34,0.578,0.062)&\cr\textit{Sr}(P_{j})&0.008&0.278}$

Since the score value is highest for the option $Z^{1}$ , therefore, the investor will prefer to invest in the mutual fund $Z^{1}$ .

Comparative remarks and advantages:

In view of the above performed calculation, we can conclude the following important comparative remarks and advantages of the proposed dimensionality technique:

•
Various researchers [45, 11, 46, 16, 8] had discussed and dealt with the decision-making problem without utilizing the reduction technique.
•
The significant amount of available data has been reduced and the proposed methodology worked out to obtain that the optimal alternative as $Z^{1}$ which is best for the investment purpose.
•
The inclusion of the notion of matrix in the proposed methodology gives the enhanced dimensionality feature along with wider span of information. Thus, will be widely applicable in various real world problems.
•
From the obtained results it can be easily observed that the proposed methodology is consistent and better enough to solve the decision making problems.
•
For the higher dimensional data set, the proposed technique may be suitably be applicable using the notion of picture fuzzy soft matrices.

6. Conclusions

This paper introduces the new kind of picture fuzzy soft matrix and various standard binary operations. Also, the properties of the proposed generalization have been established, along with their proof. An algorithm has been presented to solve the decision-making problem using the proposed definitions of choice matrix/weighted choice and score matrix. A numerical example related to the financial investment problem has been solved to illustrate the methodology of the proposed algorithm. We have proposed the technique for dimensionality reduction and established its viability and flexibility by considering the real-world decision-making problem. Finally, the computational analysis depicts the proposed algorithm’s contribution and validity and demonstrates the efficiency of the dimension reduction process. The future research directions can be concentrate on the interaction between membership and non-membership functions in Picture Fuzzy Soft Matrices [47]. The presented generalization can be used for further synthesis of new MCDA methods.

Footnotes

Funding

The work was supported by the National Science Centre, Decision number UMO-2021/ 41/B/HS4/01296 (A.S., B.K. and W.S.).

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