# Novel Parameterized Utility Function on Dual Hesitant Fuzzy Rough Sets and Its Application in Pattern Recognition

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Notion of DHFS

**Definition**

**1**

**.**Let U be a given fixed set, then a DHFS $\mathbb{D}$ on U is defined as:

**Definition**

**2**

**.**Let U be a finite and non-empty discourse domain. For any $\mathbb{A},\mathbb{B}\in DHF\left(U\right)$, then the complement of $\mathbb{A}$ (which is denoted by ${\mathbb{A}}^{c}$), the union of $\mathbb{A}$ and $\mathbb{B}$ (which is denoted by $\mathbb{A}\u22d3\mathbb{B}$), and the intersection of $\mathbb{A}$ and $\mathbb{B}$( which is denoted by $\mathbb{A}\u22d2\mathbb{B}$) are defined by

**Definition**

**3**

**.**Let ${d}_{i}=({h}_{{d}_{i}},{g}_{{d}_{i}})(i=1,2)$ be any two given DHF elements. $s\left({d}_{i}\right)=s\left({h}_{{d}_{i}}\right)-s\left({g}_{{d}_{i}}\right)=\frac{\sum \gamma \in {h}_{{d}_{i}}\gamma}{l\left({h}_{{d}_{i}}\right)}-\frac{\sum \gamma \in {g}_{{d}_{i}}\gamma}{l\left({g}_{{d}_{i}}\right)}$ is termed as the score function of ${d}_{i}$, and $p\left({d}_{i}\right)=s\left({h}_{{d}_{i}}\right)+s\left({g}_{{d}_{i}}\right)=\frac{\sum \gamma \in {h}_{{d}_{i}}\gamma}{l\left({h}_{{d}_{i}}\right)}+\frac{\sum \gamma \in {g}_{{d}_{i}}\gamma}{l\left({g}_{{d}_{i}}\right)}$ is termed as the accuracy function of ${d}_{i}$, where $l\left({h}_{{d}_{i}}\right)$ and $l\left({g}_{{d}_{i}}\right)$ are the cardinality numbers of ${h}_{{d}_{i}}$ and ${g}_{{d}_{i}}$, respectively. When ${s}_{{d}_{1}}>{s}_{{d}_{2}}$, it is considered that ${d}_{1}\succ {d}_{2}$; when ${s}_{{d}_{1}}={s}_{{d}_{2}}$, ${p}_{{d}_{1}}={p}_{{d}_{2}}$ it is considered that ${d}_{1}\equiv {d}_{2}$; when ${s}_{{d}_{1}}={s}_{{d}_{2}}$, ${p}_{{d}_{1}}>{p}_{{d}_{2}}$, it is considered that ${d}_{1}\prec {d}_{2}$.

#### 2.2. The Notion of DHFRS

**Definition**

**4**

**.**Let U be a finite and non-empty universe domain. A hesitant fuzzy relation $\mathcal{R}$ on U is represented as a hesitant fuzzy subset where $\mathcal{R}\in HF(U\times U)$ and $\mathcal{R}=\{\langle (x,y),{h}_{\mathcal{R}}(x,y)|(x,y)\in U\times U\}$. For all $(x,y)\in U\times U$, ${h}_{\mathcal{R}}(x,y)$ is a set of the values in $[0,1]$, which is denoted as the possible membership degrees of the relationships between x and y.

**Definition**

**5**

**.**Let U, V be two finite and non-empty universes. A DHF subset $\mathbb{R}$ of the universe $U\times V$ is termed as a DHF relation from U to V , namely, $\mathbb{R}$ is given by $\mathbb{R}=\{\langle (x,y),{h}_{\mathbb{R}}(x,y),{g}_{\mathbb{R}}(x,y)|(x,y)\in U\times V\}$, where ${h}_{\mathbb{R}},{g}_{\mathbb{R}}:U\times V\to [0,1]$ are two sets where their elements are valued in $[0,1]$, expressing the possible membership and non-membership degrees of the relationship between x and y, respectively, in which $0\le \gamma ,\eta \le 1$ and $0\le {\gamma}^{+},{\eta}^{+}\le 1$. Besides, for all $(x,y)\in U\times V$, $\gamma \in {h}_{\mathbb{R}}(x,y),\eta \in {g}_{\mathbb{R}}(x,y)$, it gets that ${\gamma}^{+}\in {h}_{\mathbb{R}}^{+}(x,y)={\bigcup}_{\gamma \in {h}_{\mathbb{R}}(x,y)}max\left\{\gamma \right\}$, ${\eta}^{+}\in {g}_{\mathbb{R}}^{+}(x,y)={\bigcup}_{\eta \in {g}_{\mathbb{R}}(x,y)}max\left\{\eta \right\}$. In particular, if $U=V$, $\mathbb{R}$ is termed as a DHF relation on U.

**Definition**

**6**

**.**Let U, V be two non-empty and finite universes, $\mathbb{R}$ be a DHF relation from U to V. The triple $(U,V,\mathbb{R})$ is termed as a DHF approximation space. For any $\mathbb{A}\in DHF\left(V\right)$, the lower and upper approximations of $\mathbb{A}$ with regard to $(U,V,\mathbb{R})$, suggested as $\underline{\mathbb{R}}\left(\mathbb{A}\right)$ and $\overline{\mathbb{R}}\left(\mathbb{A}\right)$, are two DHF sets of U and are, respectively, specified as $\underline{\mathbb{R}}\left(\mathbb{A}\right)=\left\{\langle x,{h}_{\underline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right),{g}_{\underline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right)\rangle \right|x\in U\}$, $\overline{\mathbb{R}}\left(\mathbb{A}\right)=\left\{\langle x,{h}_{\overline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right),{g}_{\overline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right)\rangle \right|x\in U\}$, where ${h}_{\underline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right)={\displaystyle \bigcap _{y\in V}}\{{g}_{\mathbb{R}}(x,y)\cup {h}_{\mathbb{A}}\left(y\right)\}$, ${g}_{\underline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right)={\displaystyle \bigcup _{y\in V}}\{{h}_{\mathbb{R}}(x,y)\cap {g}_{\mathbb{A}}\left(y\right)\}$, ${h}_{\overline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right)={\displaystyle \bigcup _{y\in V}}\{{h}_{\mathbb{R}}(x,y)\cap {h}_{\mathbb{A}}\left(y\right)\}$, ${g}_{\overline{\mathbb{R}}\left(\mathbb{A}\right)}\left(x\right)={\displaystyle \bigcap _{y\in V}}\{{g}_{\mathbb{R}}(x,y)\cup {g}_{\mathbb{A}}\left(y\right)\}.$ Especially, $\underline{\mathbb{R}}\left(\mathbb{A}\right)$ and $\overline{\mathbb{R}}\left(\mathbb{A}\right)$ are, respectively, termed as the lower and upper approximations of $\mathbb{A}$ with regard to $(U,V,\mathbb{R})$. The pair $(\underline{\mathbb{R}}\left(\mathbb{A}\right),\overline{\mathbb{R}}\left(\mathbb{A}\right))$ is called the DHF rough set of $\mathbb{A}$ with respect to $(U,V,\mathbb{R})$, and $\underline{\mathbb{R}},\overline{\mathbb{R}}:DHF\left(V\right)\to DHF\left(U\right)$ are referred to as lower and upper DHF rough approximation operators, respectively.

## 3. Main Results

#### 3.1. Analysis on DHFRSs

#### 3.2. A Kind of Novel Utility Function on DHFRSs

**Theorem**

**1.**

- (i)
- $0\le {E}_{\mathbb{A}}\left({x}_{i}\right),{E}_{\mathbb{B}}\left({x}_{i}\right),{E}_{\mathbb{C}}\left({x}_{i}\right)\le 1,\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{3.33333pt}{0ex}}i\in \{1,2,\cdots ,n\};$
- (ii)
- ${E}_{\mathbb{A}}\left({x}_{i}\right)=0,\phantom{\rule{3.33333pt}{0ex}}iff\phantom{\rule{3.33333pt}{0ex}}{h}_{\mathbb{A}}\left({x}_{i}\right)=\left\{0\right\},\phantom{\rule{3.33333pt}{0ex}}{g}_{\mathbb{A}}\left({x}_{i}\right)=\left\{1\right\}$;
- (iii)
- ${E}_{\mathbb{A}}\left({x}_{i}\right)=1,\phantom{\rule{3.33333pt}{0ex}}iff\phantom{\rule{3.33333pt}{0ex}}{h}_{\mathbb{A}}\left({x}_{i}\right)=\left\{1\right\},\phantom{\rule{3.33333pt}{0ex}}{g}_{\mathbb{A}}\left({x}_{i}\right)=\left\{0\right\}$;
- (iv)
- ${E}_{\mathbb{A}}\left({x}_{i}\right)\ge {E}_{\mathbb{B}}\left({x}_{i}\right),\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}s\left(\mathbb{A}\left({x}_{i}\right)\right)\ge s\left(\mathbb{B}\left({x}_{i}\right)\right),\phantom{\rule{3.33333pt}{0ex}}p\left(\mathbb{A}\left({x}_{i}\right)\right)\ge p\left(\mathbb{B}\left({x}_{i}\right)\right)$;
- (v)
- ${E}_{\mathbb{A}}\left({x}_{i}\right)\ge {E}_{\mathbb{C}}\left({x}_{i}\right),\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}{E}_{\mathbb{A}}\left({x}_{i}\right)\ge {E}_{\mathbb{B}}\left({x}_{i}\right),{E}_{\mathbb{B}}\left({x}_{i}\right)\ge {E}_{\mathbb{C}}\left({x}_{i}\right)$,

**Definition**

**7.**

#### 3.3. Novel Dual Hesitant Fuzzy Rough Pattern Recognition Method

**Step 1**According to Definition 6, we consider the lower and upper approximations $\underline{\mathbb{R}}\left(\mathbb{A}\right)$ and $\overline{\mathbb{R}}\left(\mathbb{A}\right)$ of DHFSs $\mathbb{A}$ with regard to $(U,V,\mathbb{R})$.

**Step 2**By Equation (5), the utility value of $\mathbb{A}$ with respect to each ${x}_{i}(i=1,2,\cdots ,m)$ is obtained as ${E}_{\mathbb{A}}\left({x}_{i}\right)$. Then, a utility vector is obtained as ${E}_{\mathbb{A}}=({E}_{\mathbb{A},\mathbb{R},{x}_{1}}\left({\lambda}_{1}\right),{E}_{\mathbb{A},\mathbb{R},{x}_{2}}\left({\lambda}_{2}\right),\cdots ,{E}_{\mathbb{A},\mathbb{R},{x}_{m}}\left({\lambda}_{m}\right))$. For convenience, we only consider the situation where all the ${\lambda}_{i}(i=1,2,\cdots ,m)$ are equal.

**Step 3**For any given ${\lambda}^{*}$, an index ${T}_{0}$ is obtained as ${T}_{0}=\left(\right)open="\{"\; close="\}">k|\underset{{x}_{k}\in U}{max}\left\{{E}_{\mathbb{A},R,{\lambda}^{*}}\left({x}_{k}\right)\right\}$, and we choose ${x}_{k}(k\in {T}_{0})$ as the optimal pattern.

## 4. Illustrative Examples

#### 4.1. Example 1

**Step 1**By Definition 6, both the lower and upper approximations of each ${\mathbb{A}}_{k}(i=1,2,3,4)$ expressed in the function of $(U,V,\mathbb{R})$ can be presented as

**Step 2**By Equation (5), the utility values of all the ${\mathbb{A}}_{k}(k=1,2,3,4)$ with respect to each ${x}_{i}(i=1,2,\cdots ,5)$ are obtained as ${E}_{{\mathbb{A}}_{k}}=({E}_{{\mathbb{A}}_{k},{x}_{1}}\left({\lambda}_{1}\right),{E}_{{\mathbb{A}}_{k},{x}_{2}}\left({\lambda}_{2}\right),{E}_{{\mathbb{A}}_{k},{x}_{3}}\left({\lambda}_{3}\right),{E}_{{\mathbb{A}}_{k},{x}_{4}}\left({\lambda}_{4}\right),\cdots ,{E}_{{\mathbb{A}}_{k},{x}_{5}}\left({\lambda}_{5}\right))$. For any given $k=1,2,3,4$, $i=1,2,3,4,5$, take $\lambda =[0,0.01,1]$, a set of values of ${E}_{{\mathbb{A}}_{k},{x}_{i}}\left(\lambda \right)$ can be obtained. The sketch Maps of them are shown in Figure 1.

**Step 3**By Figure 1, one can get the following conclusions. $\left(i\right)$ For any $\lambda \in [0,1]$, ${\mathbb{A}}_{1}$ is sustaining from the disease “malaria $\left({x}_{2}\right)$”, ${\mathbb{A}}_{2}$ is sustaining from the disease “stomach problem $\left({x}_{4}\right)$”, and ${\mathbb{A}}_{4}$ is also sustaining from the disease “chest problem $\left({x}_{5}\right)$”. $\left(ii\right)$ For any $\lambda \in [0.05,1]$, ${\mathbb{A}}_{3}$ is sustaining from the disease “chest problem $\left({x}_{5}\right)$”, and for any $\lambda \in [0,0.05]$, it gets that ${\mathbb{A}}_{3}$ is sustaining from the disease “stomach problem $\left({x}_{4}\right)$”, and therefore, ${\mathbb{A}}_{3}$ needs some more high-technology inspections.

#### 4.2. Example 2

**Step 1**By Definition 5, the lower and upper approximations of each ${\mathbb{A}}_{0}$ with respect to $(U,V,\mathbb{R})$ are obtained as

**Step 2**By Equation (4), one gets the utility values of ${\mathbb{A}}_{0}$ with respect to each ${x}_{i}(i=1,2,3,4)$ as

**Step 3**For any given $\lambda \in [0,1]$, ${E}_{{\mathbb{A}}_{0},{x}_{1}}\left(\lambda \right)$, ${E}_{{\mathbb{A}}_{0},{x}_{2}}\left(\lambda \right)$, ${E}_{{\mathbb{A}}_{0},{x}_{3}}\left(\lambda \right)$ and ${E}_{{\mathbb{A}}_{0},{x}_{4}}\left(\lambda \right)$ are compared, and the following conclusions are obtained: $\left(i\right)$ For any $\lambda \in [0,0.9049]$, the studied residential area ${\mathbb{A}}_{0}$ should be built following the type ${x}_{2}$; $\left(ii\right)$ For any $\lambda \in [0.9049,1]$, the studied residential area ${\mathbb{A}}_{0}$ should be built following the type ${x}_{4}$. For more details on ${E}_{{\mathbb{A}}_{0},{x}_{1}}\left(\lambda \right)$, ${E}_{{\mathbb{A}}_{0},{x}_{2}}\left(\lambda \right)$, ${E}_{{\mathbb{A}}_{0},{x}_{3}}\left(\lambda \right)$ and ${E}_{{\mathbb{A}}_{0},{x}_{4}}\left(\lambda \right)$ please see Figure 2, where Line 1 describes ${E}_{{\mathbb{A}}_{0},{x}_{1}}\left(\lambda \right)$, Line 2 describes ${E}_{{\mathbb{A}}_{0},{x}_{2}}\left(\lambda \right)$, Line 3 is describes ${E}_{{\mathbb{A}}_{0},{x}_{3}}\left(\lambda \right)$, and Line 4 is describes ${E}_{{\mathbb{A}}_{0},{x}_{4}}\left(\lambda \right)$.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Function curves of ${E}_{{\mathbb{A}}_{k},{x}_{i}}\left(\lambda \right)(k=1,2,3,4;i=1,2,\cdots ,5)$.

**Figure 2.**The sketch maps of ${E}_{{\mathbb{A}}_{0},{x}_{1}}\left(\lambda \right)$, ${E}_{{\mathbb{A}}_{0},{x}_{2}}\left(\lambda \right)$, ${E}_{{\mathbb{A}}_{0},{x}_{3}}\left(\lambda \right)$ and ${E}_{{\mathbb{A}}_{0},{x}_{4}}\left(\lambda \right)$.

$\mathit{R}({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}})$ | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ |
---|---|---|---|---|---|

${y}_{1}$ | (0.4,0.0) | (0.7,0.0) | (0.3,0.3) | (0.1,0.7) | (0.1,0.8) |

${y}_{2}$ | (0.3,0.5) | (0.2,0.6) | (0.6,0.1) | (0.2,0.4) | (0.0,0.8) |

${y}_{3}$ | (0.1,0.7) | (0.0,0.9) | (0.2,0.7) | (0.8,0.0) | (0.2,0.8) |

${y}_{4}$ | (0.4,0.3) | (0.7,0.0) | (0.2,0.6) | (0.2,0.7) | (0.2,0.8) |

${y}_{5}$ | (0.1,0.7) | (0.1,0.8) | (0.1,0.9) | (0.2,0.7) | (0.8,0.1) |

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**MDPI and ACS Style**

Wu, Z.; Zhang, F.; Sun, J.; Wang, W.; Tang, X.
Novel Parameterized Utility Function on Dual Hesitant Fuzzy Rough Sets and Its Application in Pattern Recognition. *Information* **2019**, *10*, 71.
https://doi.org/10.3390/info10020071

**AMA Style**

Wu Z, Zhang F, Sun J, Wang W, Tang X.
Novel Parameterized Utility Function on Dual Hesitant Fuzzy Rough Sets and Its Application in Pattern Recognition. *Information*. 2019; 10(2):71.
https://doi.org/10.3390/info10020071

**Chicago/Turabian Style**

Wu, Zhongjun, Fangwei Zhang, Jing Sun, Wenjing Wang, and Xufeng Tang.
2019. "Novel Parameterized Utility Function on Dual Hesitant Fuzzy Rough Sets and Its Application in Pattern Recognition" *Information* 10, no. 2: 71.
https://doi.org/10.3390/info10020071