# Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Several Types of NS

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. The Distance and Similarity Measures for SVNHFSs

**Definition**

**3.**

- (1)
- $0\le D(A,B)\le 1$;
- (2)
- $D(A,B)=0$ iff $A=B$;
- (3)
- $D(A,B)=D(B,A)$;
- (4)
- If $A\subseteq B\subseteq C$, then $D(A,C)\ge D(A,B)$, $D(A,C)\ge D(B,C)$.

**Definition**

**4.**

- (1)
- $0\le S(A,B)\le 1$;
- (2)
- $S(A,B)=1$ iff $A=B$;
- (3)
- $S(A,B)=S(B,A)$;
- (4)
- If $A\subseteq B\subseteq C$, then $S(A,C)\le S(A,B)$, $S(A,C)\le S(B,C)$.

**Definition**

**5.**

- (1)
- $E(A)=0$ if A is a crisp set;
- (2)
- $E(A)=1$ iff $A=\{0.5,0.5,0.5\}$;
- (3)
- $E(A)\le E(B)$ if A is more crisper than B;
- (4)
- $E(A)\le E({A}^{c})$, where ${A}^{c}$ is the complement of A.

## 3. The Distance and Similarity Measures of PSVNHFS

**Definition**

**6.**

**Definition**

**7.**

**Example**

**1.**

- (1)
- If the probability values are equal for the same type of hesitant membership function, i.e.,$$\begin{array}{c}\hfill {P}_{1}^{T}={P}_{1}^{T}=\cdots ={P}_{L(T)}^{T},{P}_{1}^{I}={P}_{1}^{I}=\cdots ={P}_{L(I)}^{I},{P}_{1}^{F}={P}_{1}^{F}=\cdots ={P}_{L(F)}^{F}.\end{array}$$Then, the normal PNHFS is reduced to the SVNHFS.
- (2)
- If $L(T)=L(I)=L(F)=1$ and ${P}_{1}^{T}={P}_{1}^{I}={P}_{1}^{F}=1$, then the normal PNHFS reduces to the SVNS.
- (3)
- If $I(x)=\varnothing $ (there is also ${P}^{I}(x)=\varnothing $), ${\alpha}^{+}+{\beta}^{+}\ge 1$, then the normal PNHFS reduces to the PDHFS, which can be expressed by $N=\{\langle x,T(x)|{P}^{T}(x),F(x)|{P}^{F}(x)\rangle |x\in X\}$.
- (4)
- If the normal PNHFS satisfies the conditions in (3), and ${P}_{1}^{T}={P}_{1}^{T}=\cdots ={P}_{L(T)}^{T},{P}_{1}^{F}={P}_{1}^{F}=\cdots =\phantom{\rule{3.33333pt}{0ex}}{P}_{L(F)}^{F}$, then the normal PNHFS reduces to the DHFS, denoted by $N=\{\langle x,T(x),F(x)\rangle |x\in X\}$
- (5)
- If $I(x)=F(x)=\varnothing $ (there is also ${P}^{I}(x)={P}^{F}(x)=\varnothing $), then the normal PNHFS reduces to the PHFS, the mathematical symbol is $N=\{\langle x,T(x)|{P}^{T}(x)\rangle |x\in X\}$.
- (6)
- If the normal PNHFS satisfies the conditions in (5), and ${P}_{1}^{T}={P}_{1}^{T}=\cdots ={P}_{L(T)}^{T}$, the normal PNHFS reduces to the HFS, denoted by $N=\{\langle x,T(x)\rangle |x\in X\}$.
- (7)
- If $I(x)=\varnothing $ (there is also ${P}^{I}(x)=\varnothing $), $L(T)=L(F)=1$, ${P}_{1}^{T}={P}_{1}^{F}=1$,${\alpha}_{1}+{\gamma}_{1}\ge 1$, then the normal NHFS reduces to the IFS, denoted by $N=\{\langle x,{\alpha}_{1},{\gamma}_{1}\rangle |x\in X\}$.
- (8)
- If $I(x)=\varnothing $ (there is also ${P}^{I}(x)=\varnothing $), $L(T)=L(F)=1$, ${P}_{1}^{T}={P}_{1}^{F}=1$, and $1-{\alpha}_{1}-{\gamma}_{1}=0$, then the normal NHFS reduces to the FS.

**Definition**

**8.**

**Definition**

**9.**

**Example**

**2.**

#### 3.1. The Method of Comparing PNHFSs

**Definition**

**10.**

**Definition**

**11.**

- (1)
- If ${g}_{1}\le {g}_{2}$, then $I{E}_{1}\le I{E}_{2}$;
- (2)
- If ${g}_{1}\ge {g}_{2}$, then $I{E}_{1}\ge I{E}_{2}$;
- (3)
- If ${g}_{1}={g}_{2}$, then (i) If ${e}_{1}\le {e}_{2}$, then $I{E}_{1}\le I{E}_{2}$; (ii) If ${e}_{1}\ge {e}_{2}$, then $I{E}_{1}\ge I{E}_{2}$;
- (4)
- If ${g}_{1}={g}_{2},{e}_{1}={e}_{2}$, then (i) If ${d}_{1}\le {d}_{2}$, then $I{E}_{1}\le I{E}_{2}$; (ii) If ${d}_{1}\ge {d}_{2}$, then $I{E}_{1}\ge I{E}_{2}$.

**Definition**

**12.**

- (1)
- If $\langle {d}_{A}(x),{e}_{A}(x),{g}_{A}(x)\rangle \ge \langle {d}_{B}(x),{e}_{B}(x),{g}_{B}(x)\rangle $, then $A\ge B$;
- (2)
- If $\langle {d}_{A}(x),{e}_{A}(x),{g}_{A}(x)\rangle \le \langle {d}_{B}(x),{e}_{B}(x),{g}_{B}(x)\rangle $, then $A\le B$;
- (3)
- If $\langle {d}_{A}(x),{e}_{A}(x),{g}_{A}(x)\rangle =\langle {d}_{B}(x),{e}_{B}(x),{g}_{B}(x)\rangle $, then $A=B$.

#### 3.2. Distance and Similarity Measures of PNHFSs

**Definition**

**13.**

- (1)
- ${D}_{IS}(I{S}_{A},I{S}_{B})=0$ iff $I{S}_{A}=I{S}_{B}$;
- (2)
- ${D}_{IS}(I{S}_{A},I{S}_{B})={D}_{IS}(I{S}_{B},I{S}_{A})$;
- (3)
- ${D}_{IS}(I{S}_{A},I{S}_{C})\ge {D}_{IS}(I{S}_{A},I{S}_{B})$, ${D}_{IS}(I{S}_{A},I{S}_{C})\ge {D}_{IS}(I{S}_{B},I{S}_{C})$ when $I{S}_{A}\subseteq I{S}_{B}\subseteq I{S}_{C}$.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Definition**

**14.**

- (1)
- ${D}_{PNHFS}(A,B)=0$ iff $A=B$;
- (2)
- ${D}_{PNHFS}(A,B)={D}_{PNHFS}(B,A)$;
- (3)
- If $A\subseteq B\subseteq C$, then ${D}_{PNHFS}(A,B)\le {D}_{PNHFS}(A,C)$ and ${D}_{PNHFS}(B,C)\le {D}_{PNHFS}(A,C)$.

**Theorem**

**3.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

**Definition**

**15.**

- (1)
- ${S}_{IS}(I{S}_{A},I{S}_{B})=1$ iff $I{S}_{A}=I{S}_{B}$;
- (2)
- ${S}_{IS}(I{S}_{A},I{S}_{B})={S}_{IS}(I{S}_{B},I{S}_{A})$;
- (3)
- If $I{S}_{A}\subseteq I{S}_{B}\subseteq I{S}_{C}$, then ${S}_{IS}(I{S}_{A},I{S}_{B})\ge {S}_{IS}(I{S}_{A},I{S}_{C})$, ${S}_{IS}(I{S}_{B},I{S}_{C})\ge {S}_{IS}(I{S}_{A},I{S}_{C})$.

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Definition**

**16.**

- (1)
- ${S}_{PNHFS}(A,B)=1$ iff $A=B$;
- (2)
- ${S}_{PNHFS}(A,B)={S}_{PNHFS}(B,A)$;
- (3)
- If $A\subseteq B\subseteq C$, then ${S}_{PNHFS}(A,B)\ge {S}_{PNHFS}(A,C)$ and ${S}_{PNHFS}(B,C)\ge {S}_{PNHFS}(A,C)$.

**Theorem**

**6.**

**Proof.**

**Example**

**5.**

#### 3.3. The Interrelations among Distance, Similarity and Entropy Measures

**Theorem**

**7.**

**Proof.**

**Definition**

**17.**

**Definition**

**18.**

- (1)
- $E(A)=0$ if $A=\{\langle x,\{1|1\},\{0|1\},\{0|1\}\rangle |x\in X\}$ or $A=\{\langle x,\{0|1\},\{0|1\},\{1|1\}\rangle |x\in X\}$or $A=\{\langle x,\{0|{P}_{1}\},\{0|{P}_{2}\},\{0|{P}_{3}\}\rangle |x\in X\}$;
- (2)
- $E(A)=1$ if $A=\{\langle x,\{0.5|1\},\{0.5|1\},\{0.5|1\}\rangle |x\in X\}$;
- (3)
- $E(A)=E({A}^{c})$ iff $A=\{\langle x,\{T|{P}^{T}\},\{I|{P}^{I}\},\{F|{P}^{F}\}\rangle |x\in X\}$ holds the requirement that ${\sum}_{b=1}^{L(I)}{i}_{b}{P}_{b}^{I}={\sum}_{c=1}^{L(F)}{f}_{c}{P}_{c}^{F}$, in which ${A}^{c}$ is the complement of A.
- (4)
- $E(B)\le E(C)$ when ${S}_{PNHFS}(A,B)\le {S}_{PNHFS}(A,C)$ or ${D}_{PNHFS}(A,B)\ge {D}_{PNHFS}(A,C)$, in which$A=\{\langle x,\{0.5|{P}_{1}\},\{0.5|{P}_{2}\},\{0.5|{P}_{3}\}\rangle |x\in X\}$.

**Theorem**

**8.**

**Proof.**

- Let $A=\{\langle x,\{1|1\},\{0|1\},\{0|1\}\rangle |x\in X\}$, $A=\{\langle x,\{0|1\},\{0|1\},\{0|1\}\rangle |x\in X\}$ or $A=\{\langle x,\{0|1\},\{0|1\},\{1|1\}\rangle |x\in X\}$, thus the corresponding ISs of A are shown:$$\begin{array}{c}\hfill I{S}_{A}=(1,1,1)\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}I{S}_{A}=(-1,1,1).\end{array}$$Next, the entropy measure of A is calculated as follows:$$\begin{array}{c}\hfill E(A)=MDT(MI{U}_{1}(1),MI{U}_{2}(1),MI{U}_{3}(1))=MDT(1,1,1)=0.\end{array}$$
- $$\begin{array}{c}E(A)=1\hfill \\ \iff MDT(MI{U}_{1}(|{d}_{A}(x)|),MI{U}_{2}(|2{e}_{A}(x)-1|),MI{U}_{3}(|{g}_{A}(x)|)=1\hfill \\ \iff MI{U}_{1}(0)=0,MI{U}_{2}(0)=0,MI{U}_{3}(0)=0\hfill \\ \iff |d(x)|=0,|2e(x)-1|=0,|g(x)|=0,\hfill \\ \Leftarrow {t}_{a}={f}_{c}=0.5,{i}_{b}=0.5.\phantom{\rule{4pt}{0ex}}a,b,c\in \infty .\hfill \end{array}$$
- Let $A=\{\langle x,{T}_{A}|{P}^{T},{I}_{A}|{P}^{I},{F}_{A}|{P}^{F}\rangle |x\in X\}$, then the complementary of A is obtained: ${A}^{c}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{\langle x,{F}_{A}|{P}^{F},{I}_{A}|{P}^{I},{T}_{A}|{P}^{T},\rangle |x\in X\}$. By Definition 9, the following equality is obtained: $I{S}_{A}=I{S}_{{A}^{c}}$. Obviously, $E(A)=E({A}^{c})$.
- Suppose that B and C are two PNHFS of X, $A=\{\langle x,\{0.5|{P}_{a}^{T}\},\{0.5|{P}_{b}^{I}\},\{0.5|{P}_{c}^{F}\}\rangle |x\in X\}$. Thus, the corresponding IS of A is $I{S}_{A}=\{0,0,0\}$. By Theorem 5, the following similarity measures can be obtained:$$\begin{array}{c}\hfill {S}_{PNHFS}(A,B)=MIU(AIO(MIB(MD{U}_{1}(\frac{|{d}_{B}(x)|}{2}),MD{U}_{2}(|{e}_{B}(x)|),MD{U}_{3}(|{g}_{B}(x)|))));\\ \hfill {S}_{PNHFS}(A,C)=MIU(AIO((MIB(MD{U}_{1}(\frac{|{d}_{C}(x)|}{2}),MD{U}_{2}(|{e}_{C}(x)|),MD{U}_{3}(|{g}_{C}(x)|)))).\end{array}$$

**Theorem**

**9.**

**Theorem**

**10.**

**Theorem**

**11.**

**Theorem**

**12.**

**Theorem**

**13.**

## 4. Method Analysis Based on Illustrations and Applications

#### 4.1. Comparative Evaluations

**Note**

**1.**

**Note**

**2.**

#### 4.2. Streamlining the Talent Selection Process

## 5. Conclusions and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

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${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | |

${A}_{1}$ | $\{\{0.3|0.3,0.4|0.3,0.5|0.3\},\{0.1|1\},\{0.3|0.5,0.4|0.5\}\}$ | $\{\{0.5|0.5,0.6|0.5\},\{0.2|0.5,0.3|0.5\},\{0.3|0.5,0.4|0.5\}\}$ |

${A}_{2}$ | $\{\{0.6|0.5,0.7|0.5\},\{0.1|0.5,0.2|0.5\},\{0.2|0.5,0.3|0.5\}\}$ | $\{\{0.6|0.5,0.7|0.5\},\{0.1|1\},\{0.3|1\}\}$ |

${A}_{3}$ | $\{\{0.5|0.5,0.6|0.5\},\{0.4|1\},\{0.2|0.5,0.3|0.5\}\}$ | $\{\{0.6|1\},\{0.3|1\},\{0.4|1\}\}$ |

${A}_{4}$ | $\{\{0.7|0.5,0.8|0.5\},\{0.1|1\},\{0.1|0.5,0.2|0.5\}\}$ | $\{\{0.6|0.5,0.7|0.5\},\{0.1|1\},\{0.2|1\}\}$ |

${\mathit{C}}_{\mathbf{3}}$ | ||

${A}_{1}$ | $\{\{0.2|0.5,0.3|0.5\},\{0.1|0.5,0.2|0.5\},\{0.5|0.5,0.6|0.5\}\}$ | |

${A}_{2}$ | $\{\{0.6|0.5,0.7|0.5\},\{0.1|0.5,0.2|0.5\},\{0.1|0.5,0.2|0.5\}\}$ | |

${A}_{3}$ | $\{\{0.5|0.5,0.6|0.5\},\{0.1|1\},\{0.3|1\}\}$ | |

${A}_{4}$ | $\{\{0.3|0.5,0.5|0.5\},\{0.2|1\},\{0.1|0.3,0.2|0.3,0.3|0.3\}\}$ |

Parameter | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | Ranking |
---|---|---|---|---|---|

${D}^{\varphi =\mu =\nu =\lambda =1}$ | $0.1269$ | $0.0635$ | $0.0989$ | $0.1053$ | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ |

${D}^{\varphi =\mu =\nu =2,\lambda =\frac{1}{2}}$ | $0.1498$ | $0.1063$ | $0.1150$ | $0.1239$ | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ |

${D}^{\varphi =\mu =1,\nu =2,\lambda =1}$ | $0.0956$ | $0.0622$ | $0.0561$ | $0.0792$ | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ |

${D}^{\varphi =\nu =1,\mu =2\lambda =1}$ | $0.1024$ | $0.691$ | $0.0523$ | $0.0733$ | ${A}_{1}>{A}_{4}>{A}_{2}>{A}_{3}$ |

${D}^{\varphi =2,\nu =1=\mu =1\lambda =1}$ | $0.1871$ | $0.1203$ | $0.1071$ | $0.1449$ | ${A}_{1}>{A}_{4}>{A}_{2}>{A}_{3}$ |

Method | Ranking | The Best Result | The Worst Result |
---|---|---|---|

Xu and Xia’s Method | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ | ${A}_{1}$ | ${A}_{2}$ |

Singh’s Method | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ | ${A}_{1}$ | ${A}_{2}$ |

Sahin’s Method | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ | ${A}_{1}$ | ${A}_{2}$ |

$\mathit{RE}$ | $\mathit{DR}$ | |

${A}_{1}$ | $\{\{0.4|0.6,0.6|0.2\},\{0.4|0.6\},\{0.3|0.4,0.4|0.5\}\}$ | $\{\{0.3|0.4,0.6|0.4\},\{0.5|0.5,0.6|0.4\},\{0.3|0.4\}\}$ |

${A}_{2}$ | $\{\{0.5|0.4,0.6|0.3\},\{0.4|0.2,0.6|0.5\},\{0.3|0.4\}\}$ | $\{\{0.6|0.5\},\{0.4|0.3,0.6|0.5\},\{0.4|0.6,0.6|0.3\}\}$ |

${A}_{3}$ | $\{\{0.5|0.7\},\{0.4|0.3,0.5|0.4\},\{0.4|0.3,0.6|0.5\}\}$ | $\{\{0.4|0.5,0.6|0.5\},\{0.5|0.6\},\{0.4|0.4,0.5|0.4\}\}$ |

${A}_{4}$ | $\{\{0.5|0.3\},\{0.2|0.1,0.4|0.5,0.6|0.2\},\{0.5|0.7\}\}$ | $\{\{0.6|0.5\},\{0.4|0.5,0.6|0.5\},\{0.5|0.3,0.6|0.5\}$ |

$\mathit{EV}$ | $\mathit{FL}$ | |

${A}_{1}$ | $\{\{0.7|0.5,0.8|0.5\},\{0.3|0.5,0.4|0.4\},\{0.5|0.6\}\}$ | $\{\{0.5|0.4,0.7|0.6\},\{0.3|0.5,0.5|0.4\},\{0.5|0.4\}\}$ |

${A}_{2}$ | $\{\{0.7|0.3,0.8|0.5\},\{0.6|0.6\},\{0.4|0.5,0.6|0.4\}\}$ | $\{\{0.6|0.4,0.8|0.4\},\{0.4|0.2,0.6|0.5\},\{0.5|0.3\}\}$ |

${A}_{3}$ | $\{\{0.6|0.5\},\{0.4|0.5,0.5|0.3\},\{0.4|0.5,0.6|0.4\}\}$ | $\{\{0.6|0.5\},\{0.5|0.4,0.6|0.4\},\{0.5|0.6,0.6|0.4\}\}$ |

${A}_{4}$ | $\{\{0.6|0.3,0.8|0.5\},\{0.4|0.6\},\{0.5|0.3,0.6|0.5\}\}$ | $\{\{0.6|0.5,0.8|0.4\},\{0.4|0.6\},\{0.4|0.5,0.5|0.4\}$ |

$\mathit{RE}$ | $\mathit{DR}$ | |

${A}_{1}$ | $\{\{0.6|0.5\},\{0.4|0.2,0.6|0.6\},\{0.4|0.6,0.6|0.2\}\}$ | $\{\{0.5|0.4,0.7|0.4\},\{0.6|0.4\},\{0.4|0.6,0.5|0.4\}\}$ |

${A}_{2}$ | $\{\{0.3|0.4\},\{0.5|0.4\},\{0.2|0.2,0.4|0.5,0.6|0.3\}\}$ | $\{\{0.5|0.6\},\{0.6|0.4\},\{0.5|0.3,0.6|0.4\}\}$ |

${A}_{3}$ | $\{\{0.4|0.6,0.6|0.2\},\{0.6|0.3\},\{0.5|0.4,0.6|0.5\}\}$ | $\{\{0.6|0.4,0.8|0.4\},\{0.5|0.3,0.7|0.5\},\{0.5|0.4\}\}$ |

${A}_{4}$ | $\{\{0.5|0.4,0.6|0.4\},\{0.5|0.3\},\{0.3|0.4,0.6|0.5\}\}$ | $\{\{0.7|0.5\},\{0.5|0.6,0.6|0.3\},\{0.5|0.6\}$ |

$\mathit{EV}$ | $\mathit{FL}$ | |

${A}_{1}$ | $\{\{0.5|0.3,0.6|0.5\},\{0.4|0.4,0.6|0.6\},\{0.3|0.6\}\}$ | $\{\{0.6|0.6\},\{0.3|0.5\},\{0.4|0.4,0.5|0.3,0.6|0.3\}\}$ |

${A}_{2}$ | $\{\{0.5|0.4,0.6|0.3\},\{0.5|0.6,0.6|0.3\},\{0.5|0.5\}\}$ | $\{\{0.5|0.6,0.6|0.4\},\{0.4|0.5,0.6|0.3\},\{0.3|0.4\}\}$ |

${A}_{3}$ | $\{\{0.5|0.4,0.6|0.5\},\{0.5|0.4,0.7|0.5\},\{0.5|0.8\}\}$ | $\{\{0.4|0.6,0.7|0.4\},\{0.3|0.4,0.4|0.6\},\{0.5|0.5\}\}$ |

${A}_{4}$ | $\{\{0.5|0.6\},\{0.5|0.5\},\{0.4|0.2,0.6|0.5,0.7|0.3\}\}$ | $\{\{0.5|0.5,0.7|0.5\},\{0.5|0.4\},\{0.4|0.6,0.6|0.3\}$ |

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## Share and Cite

**MDPI and ACS Style**

Shao, S.; Zhang, X.
Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes. *Mathematics* **2019**, *7*, 649.
https://doi.org/10.3390/math7070649

**AMA Style**

Shao S, Zhang X.
Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes. *Mathematics*. 2019; 7(7):649.
https://doi.org/10.3390/math7070649

**Chicago/Turabian Style**

Shao, Songtao, and Xiaohong Zhang.
2019. "Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes" *Mathematics* 7, no. 7: 649.
https://doi.org/10.3390/math7070649