# New Applications of m-Polar Fuzzy Matroids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- 1.
- $\mathit{If}\phantom{\rule{4.pt}{0ex}}{D}_{1}\in I\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}{D}_{2}\subset {D}_{1}\phantom{\rule{4.pt}{0ex}}\mathit{then}\phantom{\rule{4.pt}{0ex}},{D}_{2}\in I$,
- 2.
- $\mathit{If}\phantom{\rule{4.pt}{0ex}}{D}_{1},{D}_{2}\in I\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}|{D}_{1}|<|{D}_{2}|\phantom{\rule{4.pt}{0ex}}\mathit{then}\phantom{\rule{4.pt}{0ex}}\mathit{there}\phantom{\rule{4.pt}{0ex}}\mathit{exists}\phantom{\rule{4.pt}{0ex}}{D}_{3}\in I\phantom{\rule{4.pt}{0ex}}\mathit{such}\phantom{\rule{4.pt}{0ex}}\mathit{that}\phantom{\rule{4.pt}{0ex}}{D}_{1}\subset {D}_{3}\subseteq {D}_{1}\cup {D}_{2}$.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- 1.
- If ${\mathsf{\tau}}_{1}\in \mathcal{I}$ and ${\mathsf{\tau}}_{2}\subset {\mathsf{\tau}}_{1}$ then, ${\mathsf{\tau}}_{2}\in \mathcal{I}$,where, ${\mathsf{\tau}}_{2}\subset {\mathsf{\tau}}_{1}\Rightarrow {\mathsf{\tau}}_{2}\left(y\right)<{\mathsf{\tau}}_{1}\left(y\right)$, for every $y\in X$.
- 2.
- If ${\mathsf{\tau}}_{1},{\mathsf{\tau}}_{2}\in \mathcal{I}$ and $|supp\left({\mathsf{\tau}}_{1}\right)|<|supp\left({\mathsf{\tau}}_{2}\right)|$ then there exists ${\mathsf{\tau}}_{3}\in \mathcal{I}$ such that
- a.
- ${\mathsf{\tau}}_{1}\subset {\mathsf{\tau}}_{3}\subseteq {\mathsf{\tau}}_{1}\cup {\mathsf{\tau}}_{2}$, for any $y\in X$, ${\mathsf{\tau}}_{1}\cup {\mathsf{\tau}}_{2}\left(y\right)=max\{{\mathsf{\tau}}_{1}\left(y\right),{\mathsf{\tau}}_{2}\left(y\right)\}$,
- b.
- $m\left({\mathsf{\tau}}_{3}\right)\ge min\{m\left({\mathsf{\tau}}_{1}\right),m\left({\mathsf{\tau}}_{2}\right)\}$ where, $m\left(\nu \right)=min\left\{\nu \right(y):y\in supp(\nu \left)\right\}$.

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 3. Matroids Based on $\mathit{m}$F Sets

**Definition**

**9.**

**Example**

**1.**

**Case**

**1.**

**Case**

**2.**

**Definition**

**10.**

- 1.
- ${\left\{{\mathit{x}}_{k}\right\}}_{k=1}^{n}$ is linearly independent,
- 2.
- ${C}_{v}\left({\displaystyle \sum _{k=1}^{n}}{c}_{k}{\mathit{x}}_{k}\right)={\displaystyle \underset{k=1}{\overset{n}{\bigwedge}}}{C}_{v}\left({c}_{k}{\mathit{x}}_{k}\right)$ for all ${\left\{{c}_{k}\right\}}_{k=1}^{n}\subset K$.

**Definition**

**11.**

**Proposition**

**1.**

**Proposition**

**2.**

- ${C}_{v}\left(\mathbf{0}\right)={sup}_{\mathit{y}\in Y}{C}_{v}\left(\mathit{y}\right)$,
- ${C}_{v}\left(a\mathit{y}\right)={C}_{v}\left(\mathit{y}\right)$ for all $a\in K\setminus \left\{0\right\}$ and $\mathit{y}\in Y$,
- If ${C}_{v}\left(\mathit{y}\right)\ne {C}_{v}\left(\mathit{z}\right)$ for some $\mathit{y},\mathit{z}\in Y$ then ${C}_{v}(\mathit{y}+\mathit{z})={C}_{v}\left(\mathit{y}\right)\wedge {C}_{v}\left(\mathit{z}\right)$.

**Remark**

**1.**

**Definition**

**12.**

- 1.
- If ${\eta}_{1}\in \mathcal{C}$, ${\eta}_{2}\in \mathcal{P}\left(Y\right)$ and ${\eta}_{2}\subset {\eta}_{1}$ then, ${\eta}_{2}\in \mathcal{C}$,where, ${\eta}_{2}\subset {\eta}_{1}\Rightarrow {\eta}_{2}\left(y\right)<{\eta}_{1}\left(y\right)$ for every $y\in Y$.
- 2.
- If ${\eta}_{1},{\eta}_{2}\in \mathcal{C}$ and $|supp\left({\eta}_{1}\right)|<|supp\left({\eta}_{2}\right)|$ then there exists ${\eta}_{3}\in \mathcal{C}$ such thata. ${\eta}_{1}\subset {\eta}_{3}\subseteq {\eta}_{1}\cup {\eta}_{2}$,where for any $y\in Y,({\eta}_{1}\cup {\eta}_{2})\left(y\right)=sup\{{\eta}_{1}\left(y\right),{\eta}_{2}\left(y\right)\}$,b. $m\left({\eta}_{3}\right)\ge inf\{m\left({\eta}_{1}\right),m\left({\eta}_{2}\right)\}$,$m\left({\eta}_{i}\right)=inf\left\{{\eta}_{i}\left(x\right)\right|x\in supp\left({\eta}_{i}\right)\}$, $i=1,2,3$.

- $\xd8\notin {C}_{r}\left(\mathcal{M}\right)$,
- If ${\delta}_{1}$ and ${\delta}_{2}$ are distinct and ${\delta}_{1}\subseteq {\delta}_{2}$ then, $supp\left({\delta}_{1}\right)=supp\left({\delta}_{2}\right)$,
- If ${\delta}_{1},{\delta}_{2}\in {C}_{r}\left(G\right)$ and for $A\in \mathcal{P}\left(Y\right)$, $A\left(e\right)=inf\{{\delta}_{1}\left(e\right),{\delta}_{2}\left(e\right)\}$, $e\in supp({\delta}_{1}\cap {\delta}_{2})$ then there exists ${\delta}_{3}$ such that ${\delta}_{3}\subseteq {\delta}_{1}\cup {\delta}_{2}-\{(e,A\left(e\right)\}$.

**Proposition**

**3.**

**Proposition**

**4.**

**Definition**

**13.**

- 1.
- If ${\eta}_{1},{\eta}_{2}\in \mathcal{P}\left(Y\right)$ and ${\eta}_{1}\subseteq {\eta}_{2}$ then ${\mu}_{r}\left({\eta}_{1}\right)\le {\mu}_{r}\left({\eta}_{2}\right)$,
- 2.
- If $\eta \in \mathcal{P}\left(Y\right)$ then, ${\mu}_{r}\left(\eta \right)\le \left|\eta \right|$,
- 3.
- If $\eta \in \mathcal{C}$ then, ${\mu}_{r}\left(\eta \right)=\left|\eta \right|$.

- A trivial example of an mF matroid is known as an mF uniform matroid which is defined as,$$\mathcal{C}=\{\eta \in \mathcal{P}(Y):|supp\left(\eta \right)|\le l\}.$$It is denoted by ${\mathcal{U}}_{l,n}=(Y,\mathcal{C})$ where, l is any positive integer and $\left|Y\right|=n$. The mF circuit of ${\mathcal{U}}_{l,n}$ contains those mF subsets $\delta $ such that $\left|supp\right(\delta \left)\right|=l+1$.Consider the example of a $2$-polar fuzzy uniform matroid $\mathcal{M}=(Y,\mathcal{C})$ where, $Y=\{{e}_{1},{e}_{2},{e}_{3}\}$ and $\mathcal{C}=\{\eta \in \mathcal{P}(Y):|supp\left(\eta \right)|\le 2\}$ such that for any $\eta \in \mathcal{P}\left(Y\right)$, $\eta \left(y\right)=\mathsf{\tau}\left(y\right)$, for all $y\in Y$ where,$$\mathsf{\tau}\left(y\right)=\left\{\begin{array}{cc}(0.2,0.3),\hfill & y={e}_{1}\hfill \\ (0.4,0.5),\hfill & y={e}_{2}\hfill \\ (0.1,0.3),\hfill & y={e}_{3}\hfill \end{array}.\right.$$$$\begin{array}{}\mathcal{C}=& \{\xd8,\left\{({e}_{1},0.2,0.3)\right\},\left\{({e}_{2},0.4,0.5)\right\},\left\{({e}_{3},0.1,0.3)\right\},\{({e}_{1},0.2,0.3),({e}_{2},0.4,0.5)\},\hfill \\ & \{({e}_{2},0.4,0.5),({e}_{3},0.1,0.3)\},\{({e}_{1},0.2,0.3),({e}_{3},0.1,0.3)\}\}.\hfill \end{array}$$The $2$-polar fuzzy circuit of $\mathcal{M}$ is ${C}_{r}\left(\mathcal{M}\right)=\{({e}_{1},0.2,0.3),({e}_{2},0.4,0.5),({e}_{3},0.1,0.3)\}.$ For $\eta =\{({e}_{2},0.4,0.5),({e}_{1},0.2,0.3)\}$, ${\mu}_{r}\left(\eta \right)=(0.6,0.8)$.
- mF linear matroid is derived from an mF matrix. Assume that Y represents the column labels of an mF matrix and ${\eta}_{x}$ denotes an mF submatrix having those columns labelled by Y. It is defined as,$$\mathcal{C}=\{{\eta}_{x}\in \mathcal{P}\left(Y\right):\phantom{\rule{4.pt}{0ex}}\mathrm{columns}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}{\eta}_{x}\phantom{\rule{4.pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}m\text{-polar fuzzy linearly independent}\}.$$For any ${\eta}_{x}\in \mathcal{P}\left(Y\right)$, $|{\eta}_{x}|={\displaystyle \sum _{k=1}^{r}}sup\{{\eta}_{x}\left({a}_{k1}\right),{\eta}_{x}\left({a}_{k2}\right),\dots ,{\eta}_{x}\left({a}_{kc}\right)\}$, ${\eta}_{x}^{*}={\left[{a}_{ij}\right]}_{r\times c}$.Let $A=\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}$ be a set of $3$-polar fuzzy $2\times 1$ vectors over $\mathbb{R}$ such that for any ${\eta}_{x}\in \mathcal{P}\left(Y\right)$, ${\eta}_{x}\left(y\right)=A\left(y\right)$ where,$$A=\left[\begin{array}{cccc}\mathbf{1}& \mathbf{2}& \mathbf{3}& \mathbf{4}\\ (0.1,0.2,0.3)& (0.3,0.4,0.5)& (0.5,0.6,0.7)& (0.7,0.8,0.9)\\ (0.2,0.3,0.4)& (0.4,0.5,0.6)& (0.6,0.7,0.8)& (0.8,0.9,1.0)\end{array}\right]$$Take $\mathcal{C}=\{\xd8,\{\mathbf{1}\},\{\mathbf{2}\},\{\mathbf{4}\},\{\mathbf{1},\mathbf{2}\},\{\mathbf{2},\mathbf{4}\left\}\right\}$ then, $\mathcal{M}\left(A\right)=(A,\mathcal{C})$ is a $3$-polar fuzzy matroid on A. The family of dependent $3$-polar fuzzy subsets of matroid $\mathcal{M}\left(A\right)$ is $\left\{\right\{\mathbf{3}\}$, $\{\mathbf{1},\mathbf{3}\}$, $\{\mathbf{1},\mathbf{4}\}$, $\{\mathbf{2},\mathbf{3}\}$, $\{\mathbf{3},\mathbf{4}\}\}\cup \{\mathit{\eta}:\mathit{\eta}\subseteq A,\left|supp\right(\mathit{\eta}\left)\right|\ge 3\}$. For $\mathit{\eta}=\{\mathbf{2},\mathbf{4}\}$, ${\mu}_{r}\left(\mathit{\eta}\right)=(1.5,1.7,1.9).$
- An mF partition matroid in which the universe Y is partitioned into mF sets ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{r}$ such that$$\mathcal{C}=\{\eta \in \mathcal{P}\left(Y\right):|supp\left(\eta \right)\cap supp\left({\alpha}_{i}\right)|\le {l}_{i},\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}1\le i\le r\}$$
- The very important class of mF matroids are derived from mF graphs. The detail is discussed in Proposition 5. The mF matroid derived using this method is known as m-polar fuzzy cycle matroid, denoted by $\mathcal{M}\left(G\right)$. Clearly $\mathcal{C}$ is an independent set in G if and only if for each $\eta \in \mathcal{C}$, $supp\left(\eta \right)$ is not edge set of any cycle. Equivalently, the members of $\mathcal{M}\left(G\right)$ are mF graphs $\eta $ such that $supp\left(\eta \right)$ is a forest.Consider the example of an mF fuzzy cycle matroid $(Y,\mathcal{C})$ where, $Y=\{{y}_{1},{y}_{2},{y}_{3},{y}_{4},{y}_{5}\}$ and for any, $\eta \in \mathcal{C}$, $\mathit{\beta}\left(y\right)=D\left(y\right)$, $(C,D)$ is an mF multigraph on Y as shown in Figure 1.By Proposition 5, ${C}_{r}\left(G\right)=$ $\{\left\{({y}_{5},0.2,0.3,0,4)\right\}$, $\{({y}_{2},0.1,0.2,0.3)$, $({y}_{3},0.1,0.2,0.3)\}$, $\left\{\right({y}_{1},$ $0.1,$ $0.2,0.3)$, $({y}_{2},0.1,0.2,0.3)$, $({y}_{4},0.5,0.6,0.7)\}$, $\{({y}_{1},0.1,0.2,0.3)$, $({y}_{3},0.1,0.2,0.3)$, $({y}_{4},0.5,0.6,0.7)\left\}\right\}.$$$\begin{array}{cc}\hfill \mathcal{C}=& \{\xd8,\left\{({y}_{1},0.1,0.2,0.3)\right\},\left\{({y}_{2},0.1,0.2,0.3)\right\},\left\{({y}_{3},0.1,0.2,0.3)\right\},\{({y}_{1},0.1,0.2,0.3),\hfill \\ & ({y}_{2},0.1,0.2,0.3)\},\{({y}_{1},0.1,0.2,0.3),({y}_{4},0.5,0.6,0.7)\},\left\{({y}_{4},0.5,0.6,0.7)\right\},\{({y}_{2},0.1,0.2,0.3),\hfill \\ & ({y}_{4},0.5,0.6,0.7)\},\{({y}_{1},0.1,0.2,0.3),({y}_{3},0.1,0.2,0.3)\},\{({y}_{3},0.1,0.2,0.3),({y}_{4},0.5,0.6,0.7)\}\}\hfill \end{array}$$For $\eta =\{({y}_{2},0.1,0.2,0.3),({y}_{4},0.5,0.6,0.7)\}$, ${\mu}_{r}\left(\eta \right)=(0.6,0.8,1.0)$.

**Proposition**

**5.**

**Proof.**

**Example**

**2.**

- ${E}_{\mathit{t}}=\{yz\in supp\left(D\right)|D\left(yz\right)\ge \mathit{t}\}$,
- ${F}_{\mathit{t}}=\left\{H\right|H\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}a\phantom{\rule{4.pt}{0ex}}\mathit{forest}\phantom{\rule{4.pt}{0ex}}\mathit{in}\phantom{\rule{4.pt}{0ex}}\mathit{the}\phantom{\rule{4.pt}{0ex}}\mathit{crisp}\phantom{\rule{4.pt}{0ex}}\mathit{graph}\phantom{\rule{4.pt}{0ex}}(Y,{E}_{\mathit{t}})\}$,
- ${\mathcal{C}}_{\mathit{t}}=\left\{E\left(F\right)\right|F\in {F}_{\mathit{t}}\}$, $E\left(F\right)$ is the edge set of F.
- Clearly $({E}_{\mathit{t}},{\mathcal{C}}_{\mathit{t}})$ is a matroid for each $\mathbf{0}\le \mathit{t}\le \mathbf{1}$. Define $\mathcal{D}=\{\eta \in \mathcal{P}\left(Y\right)|{\eta}_{\mathit{t}}\in {\mathcal{C}}_{\mathit{t}},\mathbf{0}\le \mathit{t}\le \mathbf{1}\}$ then, $(Y,\mathcal{D})$ is an mF cycle matroid.

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

- 1.
- ${\mathit{t}}_{0}=\mathbf{0},\phantom{\rule{1.em}{0ex}}{\mathit{t}}_{n}\le \mathbf{1}$,
- 2.
- ${\mathcal{C}}_{\mathit{w}}\ne \xd8$ if $\mathbf{0}<\mathit{w}\le {\mathit{t}}_{n}$ and ${\mathcal{C}}_{\mathit{w}}=\xd8$ if $\mathit{w}>{\mathit{t}}_{n}$,
- 3.
- If ${\mathit{t}}_{i}<\mathit{w},\mathit{s}<{\mathit{t}}_{i+1}$ then, ${\mathcal{C}}_{\mathit{w}}={\mathcal{C}}_{\mathit{s}}$, $0\le i\le n-1$,
- 4.
- If ${\mathit{t}}_{i}<\mathit{w}<{\mathit{t}}_{i+1}<\mathit{s}<{\mathit{t}}_{i+2}$ then, ${\mathcal{C}}_{\mathit{w}}\supset {\mathcal{C}}_{\mathit{s}}$, $0\le i\le n-2$.

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Definition**

**14.**

**Example**

**3.**

**Theorem**

**4.**

**Definition**

**15.**

**Remark**

**3.**

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**5.**

**Proof.**

**Example**

**4.**

**Theorem**

**6.**

**Proof.**

- (i)
- ${\mathit{\beta}}_{{i}^{*}}\le {\mathit{t}}_{n}$, ${\widehat{\mu}}_{r}\left({\eta}_{1}\right)={\displaystyle \sum _{i=1}^{{i}^{*}}}{\gamma}_{i}\left({\eta}_{1}\right)$.
- (ii)
- For ${\eta}_{2}\in \mathcal{C},{\eta}_{2}\subseteq {\eta}_{1}$ we have, $\mathbf{0}<{\eta}_{2}\left(y\right)\le {\mathit{\beta}}_{{i}^{*}}$ for each $y\in supp\left({\eta}_{2}\right)$.

- (i)
- ${D}_{{\mathit{\beta}}_{i}}$ is maximal in $(Y,{\mathcal{C}}_{{\mathit{\beta}}_{i}}^{{\eta}_{1}})$
- (ii)
- $|{D}_{{\mathit{\beta}}_{i}}|={R}_{j}\left({\eta}_{{\mathit{\beta}}_{i}}\right)$ where, i and j are such that ${\mathit{t}}_{i-1}\le {\mathit{\beta}}_{j-1}<{\mathit{\beta}}_{j}\le {\mathit{t}}_{i}$.

## 4. Applications

#### 4.1. Decision Support Systems

#### 4.2. Ordering of Machines/Workers for Certain Tasks

#### 4.3. Network Analysis

- Input the n number of locations ${L}_{1},{L}_{2},\dots ,{L}_{n}$ of wireless communication network.
- Input the adjacency matrix $\xi ={\left[{L}_{ij}\right]}_{{n}^{2}}$ of membership values of edges among locations.
- From this adjacency matrix, arrange the membership values in increasing order.
- Select an edge having minimum membership value.
- Repeat Step 4 so that the selected edge does not create any circuit with previous selected edges.
- Stop the procedure if the connection between every pair of locations is set up.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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Sarwar, M.; Akram, M.
New Applications of *m*-Polar Fuzzy Matroids. *Symmetry* **2017**, *9*, 319.
https://doi.org/10.3390/sym9120319

**AMA Style**

Sarwar M, Akram M.
New Applications of *m*-Polar Fuzzy Matroids. *Symmetry*. 2017; 9(12):319.
https://doi.org/10.3390/sym9120319

**Chicago/Turabian Style**

Sarwar, Musavarah, and Muhammad Akram.
2017. "New Applications of *m*-Polar Fuzzy Matroids" *Symmetry* 9, no. 12: 319.
https://doi.org/10.3390/sym9120319