# Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators for Multi-Attribute Decision Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Single-Valued Neutrosophic Hesitant Fuzzy Sets (SVNHFS)

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- 1.
- $\tilde{{n}_{1}}{{\displaystyle \cup}}^{\text{}}\tilde{{n}_{2}}=\{\tilde{{t}_{1}}{{\displaystyle \cup}}^{\text{}}\tilde{{t}_{2}},\tilde{{i}_{1}}{{\displaystyle \cap}}^{\text{}}\tilde{{i}_{2}},\text{}\tilde{{f}_{1}}{{\displaystyle \cap}}^{\text{}}\tilde{{f}_{2}}\};$
- 2.
- $\tilde{{n}_{1}}{{\displaystyle \cap}}^{\text{}}\tilde{{n}_{2}}=\{\tilde{{t}_{1}}{{\displaystyle \cap}}^{\text{}}\tilde{{t}_{2}},\tilde{{i}_{1}}{{\displaystyle \cup}}^{\text{}}\tilde{{i}_{2}},\text{}\tilde{{f}_{1}}{{\displaystyle \cup}}^{\text{}}\tilde{{f}_{2}}\};$
- 3.
- $\tilde{{n}_{1}}\oplus \tilde{{n}_{2}}={{\displaystyle \cup}}_{{\gamma}_{1}\in \tilde{{t}_{1}},\text{}{\sigma}_{1}\in \tilde{{i}_{1}},{\eta}_{1}\in \tilde{{f}_{1}\text{}}{\gamma}_{2}\in \tilde{{t}_{2}},\text{}{\sigma}_{2}\in \tilde{{i}_{2}},{\eta}_{2}\in \tilde{{f}_{2}\text{}}}\{{\gamma}_{1}+{\gamma}_{2}-{\gamma}_{1}{\gamma}_{2},{\sigma}_{1}{\sigma}_{2},{\eta}_{1}{\eta}_{2}\};$
- 4.
- $\tilde{{n}_{1}}\otimes \tilde{{n}_{2}}={{\displaystyle \cup}}_{{\gamma}_{1}\in \tilde{{t}_{1}},\text{}{\sigma}_{1}\in \tilde{{i}_{1}},{\eta}_{1}\in \tilde{{f}_{1}\text{}}{\gamma}_{2}\in \tilde{{t}_{2}},\text{}{\sigma}_{2}\in \tilde{{i}_{2}},{\eta}_{2}\in \tilde{{f}_{2}\text{}}}\{{\gamma}_{1}{\gamma}_{2},{\sigma}_{1}+{\sigma}_{2}-{\sigma}_{1}{\sigma}_{2},{\eta}_{1}+{\eta}_{2}-{\eta}_{1}{\eta}_{2}\};$
- 5.
- $k\tilde{{n}_{1}}={{\displaystyle \cup}}_{{\gamma}_{1}\in \tilde{{t}_{1}},\text{}{\sigma}_{1}\in \tilde{{i}_{1}},{\eta}_{1}\in \tilde{{f}_{1}\text{}}}\{1-{(1-{\gamma}_{1})}^{k},{\sigma}_{1}{}^{k},{\eta}_{1}{}^{k}\};$
- 6.
- ${\tilde{{n}_{1}}}^{k}={{\displaystyle \cup}}_{{\gamma}_{1}\in \tilde{{t}_{1}},\text{}{\sigma}_{1}\in \tilde{{i}_{1}},{\eta}_{1}\in \tilde{{f}_{1}\text{}}}\{{\gamma}_{1}{}^{k},1-{(1-{\sigma}_{1})}^{k},1-{(1-{\eta}_{1})}^{k}\}.$

**Definition**

**5.**

#### 2.2. The Fuzzy Measure and Choquet Integral

**Definition**

**6.**

- 1.
- $\mu (\varnothing )=0;$
- 2.
- $\mu (A)\le \mu (B)\text{}whenever\text{}A\subset B,A,B\in \mathcal{A};$
- 3.
- $If\text{}{A}_{1}\subset {A}_{2}\subset \dots \subset {A}_{n}\subset \dots ,\text{}{A}_{n}\in \mathcal{A},\text{}then\text{}\mu \text{}({{\displaystyle \cup}}_{n=1}^{\infty}{A}_{n})=\text{}li{m}_{n\to \infty}\mu ({A}_{n})\text{};$
- 4.
- $If\text{}{A}_{1}\supset {A}_{2}\supset \dots \supset {A}_{n}\supset \dots ,\text{}{A}_{n}\in \mathcal{A},\text{}then\text{}\mu \text{}({{\displaystyle \cap}}_{n=1}^{\infty}{A}_{n})=\text{}li{m}_{n\to \infty}\mu ({A}_{n})\text{};$

**Theorem**

**1.**

**Definition**

**7.**

## 3. New Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators

#### 3.1. Single-Valued Neutrosophic Hesitant Fuzzy Choquet Ordered Averaging (SVNHFCOA) Operator

**Definition**

**8.**

**Theorem**

**2.**

**Proof.**

- (a)
- For $m=1$, since$${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1}\}=(\mu ({F}_{\varphi (1)})-\mu ({F}_{\varphi (0)})){\tilde{n}}_{\varphi (1)}={\tilde{n}}_{\varphi (1)},$$
- (b)
- When $m=2,$ since$${\mu}_{\varphi (1)}{\tilde{n}}_{\varphi (1)}=\underset{\text{}{\gamma}_{\varphi (1)}\in \tilde{{t}_{\varphi (1)}},\text{}{\sigma}_{\varphi (1)}\in \tilde{{i}_{\varphi (1)}},{\eta}_{\varphi (1)}\in \tilde{{f}_{\varphi (1)}\text{}}}{{\displaystyle \cup}}\{1-{(1-{\gamma}_{\varphi (1)})}^{{\mu}_{\varphi (1)}},{\sigma}_{\varphi (1)}{}^{{\mu}_{\varphi (1)}},{\eta}_{\varphi (1)}{}^{{\mu}_{\varphi (1)}}\},$$$${\mu}_{\varphi (2)}{\tilde{n}}_{\varphi (2)}=\underset{\text{}{\gamma}_{\varphi (2)}\in \tilde{{t}_{\varphi (2)}},\text{}{\sigma}_{\varphi (2)}\in \tilde{{i}_{\varphi (2)}},{\eta}_{\varphi (2)}\in \tilde{{f}_{\varphi (2)}\text{}}}{{\displaystyle \cup}}\{1-{(1-{\gamma}_{\varphi (2)})}^{{\mu}_{\varphi (2)}},{\sigma}_{\varphi (2)}{}^{{\mu}_{\varphi (2)}},{\eta}_{\varphi (2)}{}^{{\mu}_{\varphi (2)}}\}$$$${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2}\}={\mu}_{\varphi (1)}{\tilde{n}}_{\varphi (1)}\text{}\oplus {\mu}_{\varphi (2)}{\tilde{n}}_{\varphi (2)}\phantom{\rule{0ex}{0ex}}=\underset{\text{}{\gamma}_{\varphi (j)}\in \tilde{{t}_{\varphi (j)}},\text{}{\sigma}_{\varphi (j)}\in \tilde{{i}_{\varphi (j)}},{\eta}_{\varphi (j)}\in \tilde{{f}_{\varphi (j)}\text{}}}{{\displaystyle \cup}}\begin{array}{c}\text{}\\ \text{}\{\{1-{(1-{\gamma}_{\varphi (1)})}^{{\mu}_{\varphi (1)}}{(1-{\gamma}_{\varphi (2)})}^{{\mu}_{\varphi (2)}}\},\{{\sigma}_{\varphi (1)}{}^{{\mu}_{\varphi (1)}}{\sigma}_{\varphi (2)}{}^{{\mu}_{\varphi (2)}}\},\{{\eta}_{\varphi (1)}{}^{{\mu}_{\varphi (1)}}{\eta}_{\varphi (2)}{}^{{\mu}_{\varphi (2)}}\}\}\end{array}.$$
- (c)
- If Equation (9) holds for $m=k$, then$${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{k}\}={\oplus}_{j=1}^{k}{\mu}_{\varphi (j)}{\tilde{n}}_{\varphi (j)}\phantom{\rule{0ex}{0ex}}=\underset{{\gamma}_{\varphi (j)}\in \tilde{{t}_{\varphi (j)}},\text{}{\sigma}_{\varphi (j)}\in \tilde{{i}_{\varphi (j)}},{\eta}_{\varphi (j)}\in \tilde{{f}_{\varphi (j)}\text{}}\text{}}{{\displaystyle \cup}}\text{}\{\{1-{\displaystyle \prod}_{j=1}^{k}{(1-{\gamma}_{\varphi (j)})}^{{\mu}_{\varphi (j)}}\},\{{\displaystyle \prod}_{j=1}^{k}{\sigma}_{\varphi (j)}{}^{{\mu}_{\varphi (j)}}\},\{{\displaystyle \prod}_{j=1}^{k}{\eta}_{\varphi (j)}{}^{{\mu}_{\varphi (j)}}\}\}$$

- If $\mu (F)\equiv 1$, then ${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=max\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\};$
- If $\mu (F)\equiv 0$, then ${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=min\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\};$
- The SVNHFCOA operator reduces to the single-valued neutrosophic hesitant fuzzy weighted averaging (SVNHFWA) operator, if the independent condition $\mu ({x}_{\varphi (j)})=\mu ({F}_{\varphi (j)})-\mu ({F}_{\varphi (j-1)})\text{}\mathrm{holds}.$$$\mathrm{SVNHFWA}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}={\oplus}_{j=1}^{m}(\mu ({x}_{j})\cdot {\tilde{n}}_{j})=\underset{{\gamma}_{j}\in {\tilde{t}}_{J},{\sigma}_{j}\in {\tilde{lZ}}_{J},{\eta}_{j}\in \tilde{{f}_{J}}}{\cup}\left\{\left\{1-{\displaystyle \prod _{j=1}^{m}{(1-{\gamma}_{j})}^{\mu ({x}_{j})}}\right\},\left\{{\displaystyle \prod _{j=1}^{m}{\sigma}_{j}{}^{{}^{\mu ({x}_{j})}}}\right\},\left\{{\displaystyle \prod _{j=1}^{m}{\eta}_{j}{}^{{}^{\mu ({x}_{j})}}}\right\}\right\}.$$
- If $\mathsf{\mu}({\mathrm{x}}_{\mathrm{j}})=1/\mathrm{m},\text{}\mathrm{for}\text{}\mathrm{j}=1,2,\dots ,\mathrm{m}$, then both the SVNHFCOA and SVNHFWA operators reduce to the single-valued neutrosophic hesitant fuzzy averaging (SVNHFA) operator, which is shown as follows:$$\mathrm{SVNHFWA}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\underset{\text{}{\gamma}_{j}\in \tilde{{t}_{j}},\text{}{\sigma}_{j}\in \tilde{{i}_{j}},{\eta}_{j}\in \tilde{{f}_{j}\text{}}}{{\displaystyle \cup}}\{\{1-{\displaystyle {\displaystyle \prod}_{j=1}^{m}}{(1-{\gamma}_{j})}^{\frac{1}{m}}\},\{{\displaystyle \prod}_{j=1}^{m}{({\sigma}_{j})}^{\frac{1}{m}}\},\{{\displaystyle \prod}_{j=1}^{m}{({\eta}_{j})}^{\frac{1}{m}}\}\}.$$
- If $\mu (F)={{\displaystyle \sum}}_{\mathrm{j}=1}^{\left|\mathrm{F}\right|}{\mathsf{\omega}}_{\mathrm{j}}$ for all $F\subseteq X,$ where $\left|\mathrm{F}\right|$ is the number of elements in F, then ${\mathsf{\omega}}_{\mathrm{j}}=\mu ({F}_{\varphi (j)})-\mu ({F}_{\varphi (j-1)}),\text{}j=1,2,\dots ,m,$ where $\mathsf{\omega}={({\mathsf{\omega}}_{1},{\mathsf{\omega}}_{2},\dots ,{\mathsf{\omega}}_{\mathrm{m}})}^{\mathrm{T}}$ such that ${\mathsf{\omega}}_{\mathrm{j}}\ge 0$ and ${{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{m}}{\mathsf{\omega}}_{\mathrm{j}}=1$. In this case, the SVNHFCOA operator reduces to the single-valued neutrosophic hesitant fuzzy ordered weighted averaging (SVNHFOWA) operator as:$$\mathrm{SVNHFOWA}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\underset{{\gamma}_{j}\in \tilde{{t}_{j}},\text{}{\sigma}_{j}\in \tilde{i{z}_{j}},{\eta}_{j}\in \tilde{{f}_{j}\text{}}}{{\displaystyle \cup}}\{\{1-{\displaystyle {\displaystyle \prod}_{j=1}^{m}}{(1-{\gamma}_{{\mathsf{\omega}}_{\mathrm{j}}})}^{{\mathsf{\omega}}_{\mathrm{j}}}\},\{{\displaystyle \prod}_{j=1}^{m}{\sigma}_{j}{}^{{\mathsf{\omega}}_{\mathrm{j}}}\},\{{\displaystyle \prod}_{j=1}^{m}{\eta}_{j}{}^{{\mathsf{\omega}}_{\mathrm{j}}}\}\};$$

**Theorem**

**3.**

- 1.
- (Idempotency) Let ${\tilde{n}}_{j}=\tilde{n}$ for all $j=1,2,\dots ,m,$ and $\tilde{n}=\{\{\gamma \},\{\sigma \},\{\eta \}\}$, then:$$SVNHFCO{A}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\{\{\gamma \},\{\sigma \},\{\eta \}\}.$$
- 2.
- (Boundedness) Let ${\tilde{n}}^{-}=\{min\{{\gamma}_{j}\},\text{}max\{{\sigma}_{j}\},\text{}max\{{\eta}_{j}\}\},{\tilde{n}}^{+}=\{\text{}max\{\text{}{\gamma}_{j}\},min\{{\sigma}_{j}\},min\{{\eta}_{j}\}\},$ so:$${\tilde{n}}^{-}\le \text{}SVNHFCO{A}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}\text{}\le \text{}{\tilde{n}}^{+}.$$
- 3.
- (Commutativity) If $\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}$ is a permutation of $\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}$, then,$$SVNHFCO{A}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=SVNHFCO{A}_{\mu}\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}.$$
- 4.
- (Monotonity) If ${\tilde{n}}_{j}\le {\tilde{n}}_{j}^{\prime}$ for $\forall j\in \{1,\text{}2,\dots ,n\}$, then,$$SVNHFCO{A}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}\le SVNHFCO{A}_{\mu}\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}.$$

**Proof.**

- For $\tilde{n}=\{\{\gamma \},\{\sigma \},\{\eta \}\}$, according to Theorem 1, it follows that$${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\phantom{\rule{0ex}{0ex}}\{\{1-{(1-\gamma )}^{{{\displaystyle \sum}}_{j=1}^{m}(\mu ({F}_{j})-\mu ({F}_{j-1}))}\},\{{\sigma}^{{{\displaystyle \sum}}_{j=1}^{m}(\mu ({F}_{j})-\mu ({F}_{j-1}))}\},\{{\eta}^{{{\displaystyle \sum}}_{j=1}^{m}(\mu ({F}_{j})-\mu ({F}_{j-1}))}\}\}=\{\{\gamma \},\{\sigma \},\{\eta \}\}.$$
- Since $y={x}^{a}(0<a<1)$ is a monotone increasing function when $x>0$, therefore, it holds$$1-{\displaystyle \prod}_{j=1}^{m}{(1-\mathrm{min}\{{\gamma}_{j}\})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le 1-{\displaystyle \prod}_{j=1}^{m}{(1-{\gamma}_{j})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le 1-{\displaystyle \prod}_{j=1}^{m}{(1-\mathrm{max}\{{\gamma}_{j}\})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))},$$$$1-{(1-\mathrm{min}\{{\gamma}_{j}\})}^{{{\displaystyle \sum}}_{j=1}^{m}(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le 1-{\displaystyle \prod}_{j=1}^{m}{(1-{\gamma}_{j})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le 1-{(1-\mathrm{max}\{{\gamma}_{j}\})}^{{{\displaystyle \sum}}_{j=1}^{m}(\mu ({F}_{j})-\mu ({F}_{j-1}))}$$$$\mathrm{min}\{{\gamma}_{j}\}\le 1-{\displaystyle \prod}_{j=1}^{m}{(1-{\gamma}_{j})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}=\gamma \le \mathrm{max}\{{\gamma}_{j}\}$$Analogously, we have$$\mathrm{min}\{{\sigma}_{j}\}\le {\displaystyle \prod}_{j=1}^{m}{\sigma}_{j}{}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le \mathrm{max}\{{\sigma}_{j}\}\text{}\mathrm{and}\text{}\mathrm{min}\{{\eta}_{j}\}\le {\displaystyle \prod}_{j=1}^{m}{\eta}_{j}{}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le \mathrm{max}\{{\eta}_{j}\}.$$
- Suppose $(\varphi (1),\varphi (2),\dots ,\text{}\varphi (m))$ is a permutation of both $\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}$ and $\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\},$ such that ${\tilde{n}}_{\varphi (1)}\ge \text{}{\tilde{n}}_{\varphi (2)},\dots ,\ge {\tilde{n}}_{\varphi (m)}$, ${F}_{\varphi (i)}=\{{x}_{\varphi (1)},\text{}{x}_{\varphi (2)},\dots ,{x}_{\varphi (i)}\},$ then,$${\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}={\mathrm{SVNHFCOA}}_{\mu}\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}={\oplus}_{j=1}^{m}((\mu ({F}_{\varphi (j)})-\mu ({F}_{\varphi (j-1)})){\tilde{n}}_{\varphi (j)}).$$
- Considering ${\tilde{n}}_{j}\le {\tilde{n}}_{j}^{\prime}$ for $\forall j\in \{1,\text{}2,\dots ,n\}$, we have$$\gamma =1-{\displaystyle \prod}_{j=1}^{m}{(1-{\gamma}_{j})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}\le 1-{\displaystyle \prod}_{j=1}^{m}{(1-{\gamma}_{j}^{\prime})}^{(\mu ({F}_{j})-\mu ({F}_{j-1}))}={\gamma}^{\prime},\text{}\phantom{\rule{0ex}{0ex}}\sigma ={\displaystyle \prod}_{j=1}^{m}{\sigma}_{j}{}^{\mu ({F}_{j})-\mu ({F}_{j-1})}\ge {\displaystyle \prod}_{j=1}^{m}{({\sigma}_{j}^{\prime})}^{\mu ({F}_{j})-\mu ({F}_{j-1})}={\sigma}^{\prime},\text{}\phantom{\rule{0ex}{0ex}}\eta ={\displaystyle \prod}_{j=1}^{m}{\eta}_{j}{}^{\mu ({F}_{j})-\mu ({F}_{j-1})}\ge {\displaystyle \prod}_{j=1}^{m}{({\eta}_{j}^{\prime})}^{\mu ({F}_{j})-\mu ({F}_{j-1})}={\eta}^{\prime}.$$

#### 3.2. Single-Valued Neutrosophic Hesitant Fuzzy Choquet Ordered Geometric (SVNHFCOG) Operator

**Definition**

**9.**

**Theorem**

**4.**

- If $\mu (F)\equiv 1$, then ${\mathrm{SVNHFCOG}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=max\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\};$
- If $\mu (F)\equiv 0$, then ${\mathrm{SVNHFCOG}}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=min\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\};$
- The SVNHFCOG operator reduces to the single-valued neutrosophic hesitant fuzzy weighted geometric (SVNHFWG) operator, if the independent condition $\mu ({x}_{\varphi (j)})=\mu ({F}_{\varphi (j)})-\mu ({F}_{\varphi (j-1)})\text{}\mathrm{holds}.$$$\mathrm{SVNHFWG}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}={\otimes}_{j=1}^{m}(\mu ({x}_{j})\otimes {\tilde{n}}_{j})\phantom{\rule{0ex}{0ex}}=\underset{{\gamma}_{j}\in \tilde{{t}_{j}},\text{}{\sigma}_{j}\in \tilde{i{z}_{j}},{\eta}_{j}\in \tilde{{f}_{j}\text{}}}{{\displaystyle \cup}}\left\{\left\{{\displaystyle {\displaystyle \prod}_{j=1}^{m}}{{\gamma}_{j}}^{\mu ({x}_{j})}\right\},\left\{1-{\displaystyle \prod}_{j=1}^{m}{(1-{\sigma}_{j})}^{\mu ({x}_{j})}\right\},\left\{1-{\displaystyle \prod}_{j=1}^{m}{(1-{\eta}_{j})}^{\mu ({x}_{j})}\right\}\right\}.$$
- If $\mu ({x}_{j})=1/m,\text{}\mathrm{for}\text{}j=1,2,\dots ,m$, then both the SVNHFCOG and SVNHFWG operators reduce to the single-valued neutrosophic hesitant fuzzy geometric (SVNHFG) operator, which is shown as follows:$$\mathrm{SVNHFWG}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\underset{\text{}{\gamma}_{j}\in \tilde{{t}_{j}},\text{}{\sigma}_{j}\in \tilde{{i}_{j}},{\eta}_{j}\in \tilde{{f}_{j}\text{}}}{{\displaystyle \cup}}\left\{\left\{{\displaystyle {\displaystyle \prod}_{j=1}^{m}}{({\gamma}_{j})}^{\frac{1}{m}}\right\},\left\{1-{\displaystyle \prod}_{j=1}^{m}{(1-{\sigma}_{j})}^{\frac{1}{m}}\right\},\left\{1-{\displaystyle \prod}_{j=1}^{m}{(1-{\eta}_{j})}^{\frac{1}{m}}\right\}\right\}.$$
- If $\mu (F)={{\displaystyle \sum}}_{\mathrm{j}=1}^{\left|\mathrm{F}\right|}{\mathsf{\omega}}_{\mathrm{j}}$ for all $F\subseteq X,$ where $\left|\mathrm{F}\right|$ is the number of elements in F, then ${\mathsf{\omega}}_{\mathrm{j}}=\mu ({F}_{\varphi (j)})-\mu ({F}_{\varphi (j-1)}),\text{}j=1,2,\dots ,m,$ where $\mathsf{\omega}={({\mathsf{\omega}}_{1},{\mathsf{\omega}}_{2},\dots ,{\mathsf{\omega}}_{\mathrm{m}})}^{\mathrm{T}}$ such that ${\mathsf{\omega}}_{\mathrm{j}}\ge 0$ and ${{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{m}}{\mathsf{\omega}}_{\mathrm{j}}=1$. Then, the SVNHFCOG operator reduces to the single-valued neutrosophic hesitant fuzzy ordered weighted geometric (SVNHFOWG) operator as follows:$$\mathrm{SVNHFOWG}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\underset{{\gamma}_{j}\in \tilde{{t}_{j}},\text{}{\sigma}_{j}\in \tilde{i{z}_{j}},{\eta}_{j}\in \tilde{{f}_{j}\text{}}}{{\displaystyle \cup}}\left\{\left\{{\displaystyle {\displaystyle \prod}_{j=1}^{m}}{({\gamma}_{j})}^{{\mathsf{\omega}}_{\mathrm{j}}}\right\},\left\{1-{\displaystyle \prod}_{j=1}^{m}{(1-{\sigma}_{j})}^{{\mathsf{\omega}}_{\mathrm{j}}}\right\},\left\{1-{\displaystyle \prod}_{j=1}^{m}{(1-{\eta}_{j})}^{{\mathsf{\omega}}_{\mathrm{j}}}\right\}\right\}.$$

**Theorem**

**5.**

- 1.
- (Idempotency) Let ${\tilde{n}}_{j}=\tilde{n}$ for all $j=1,2,\dots ,m,$ and $\tilde{n}=\{\{\gamma \},\{\sigma \},\{\eta \}\}$, then:$$SVNHFCO{G}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=\{\{\gamma \},\{\sigma \},\{\eta \}\}.$$
- 2.
- (Boundedness) Let ${\tilde{n}}^{-}=\{min\{{\gamma}_{j}\},\text{}max\{{\sigma}_{j}\},\text{}max\{{\eta}_{j}\}\},{\tilde{n}}^{+}=\{\text{}max\{\text{}{\gamma}_{j}\},min\{{\sigma}_{j}\},min\{{\eta}_{j}\}\},$ so:$${\tilde{n}}^{-}\le \text{}SVNHFCO{G}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}\text{}\le \text{}{\tilde{n}}^{+}$$
- 3.
- (Commutativity) If $\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}\text{}$ is a permutation of $\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}$, then,$$SVNHFCO{G}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}=SVNHFCO{G}_{\mu}\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}$$
- 4.
- (Monotonity) If ${\tilde{n}}_{j}\le {\tilde{n}}_{j}^{\prime}$ for$\forall j\in \{1,\text{}2,\dots ,n\}$, then,$$SVNHFCO{G}_{\mu}\{{\tilde{n}}_{1},{\tilde{n}}_{2},\dots ,{\tilde{n}}_{m}\}\le SVNHFCO{G}_{\mu}\{{\tilde{n}}_{1}^{\prime},{\tilde{n}}_{2}^{\prime},\dots ,{\tilde{n}}_{m}^{\prime}\}.$$

**Theorem**

**6.**

**Proof.**

## 4. Approaches for MADM with Single-Valued Neutrosophic Hesitant Fuzzy Information

## 5. Numerical Example and Analysis

#### 5.1. Numerical Example

**Step 1.**Get the score matrix of ${\tilde{n}}_{ij}$ calculated by Equation (11), shown in Table 2, and the reordered decision matrix shown in Table 3.

**Step 2.**Suppose that the fuzzy measures of attributes of X are given as follows: $\mu ({x}_{1})=0.362,\mu ({x}_{2})=0.2,\mu ({x}_{3})=0.438$. Firstly, according to Equation (7), the value of $\mathsf{\lambda}$ is obtained: $\mathsf{\lambda}=0.856$. Thus, $\mu ({x}_{1},\text{}{x}_{2})=0.626,\text{}\mu ({x}_{2},\text{}{x}_{3})=0.713,\text{}\mu ({x}_{1},\text{}{x}_{3})=0.936,\text{}\mu (X)=1,$ thus,

**Step 3.**Aggregate ${\tilde{n}}_{ij}(i=1,2,3,4;j=1,2,3)$ by using the SVNHFCOA operator to derive the comprehensive score value ${\tilde{n}}_{i}$ for ${a}_{i}$ $(i=1,2,3,4).$ Take ${a}_{1}$ for an example, the comprehensive score value ${\tilde{n}}_{1}$ of ${a}_{1}$ is calculated as follows:

**Step 4.**Based on the score function of SVNHFEs, we get:

**Step 3′.**Aggregate ${\tilde{n}}_{ij}\text{}(i=1,2,3,4;\text{}j=1,2,3)$ by using the SVNHFCOG operator to derive the comprehensive score value ${\tilde{n}}_{i}^{\prime}$ for ${a}_{i}\text{}(i=1,2,3,4).$

**Step 4′.**Based on the score function of SVNHFEs, we get:

#### 5.2. Comparison Analysis and Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | |
---|---|---|---|

${\mathit{a}}_{\mathbf{1}}$ | $\{\{0.3,0.4,0.5\},\{0.1\},\{0.3,0.4\}\}$ | $\{\{0.5,0.6\},\{0.2,0.3\},\{0.3,0.4\}\}$ | $\{\{0.2,0.3\},\{0.1,0.2\},\{0.5,0.6\}\}$ |

${\mathit{a}}_{\mathbf{2}}$ | $\{\{0.6,0.7\},\{0.1,0.2\},\{0.2,0.3\}\}$ | $\{\{0.6,0.7\},\{0.1\},\{0.3\}\}$ | $\{\{0.6,0.7\},\{0.1,0.2\},\{0.1,0.2\}\}$ |

${\mathit{a}}_{\mathbf{3}}$ | $\{\{0.5,0.6\},\{0.4\},\{0.2,0.3\}\}$ | $\{\{0.6\},\{0.3\},\{0.4\}\}$ | $\{\{0.5,0.6\},\{0.1\},\{0.3\}\}$ |

${\mathit{a}}_{\mathbf{4}}$ | $\{\{0.7,0.8\},\{0.1\},\{0.1,0.2\}\}$ | $\{\{0.6,0.7\},\{0.1\},\{0.2\}\}$ | $\{\{0.3,0.5\},\{0.2\},\{0.1,0.2,0.3\}\}$ |

${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | |
---|---|---|---|

${\mathit{a}}_{\mathbf{1}}$ | $0.65$ | $0.65$ | $0.52$ |

${\mathit{a}}_{\mathbf{2}}$ | $0.75$ | $0.75$ | $0.78$ |

${\mathit{a}}_{\mathbf{3}}$ | $0.63$ | $0.57$ | $0.72$ |

${\mathit{a}}_{\mathbf{4}}$ | $0.83$ | $0.78$ | $0.67$ |

${\mathit{x}}_{\mathit{\sigma}(1)}$ | ${\mathit{x}}_{\mathit{\sigma}(2)}$ | ${\mathit{x}}_{\mathit{\sigma}(3)}$ | |
---|---|---|---|

${\mathit{a}}_{\mathbf{1}}$ | $\{\{0.2,0.3\},\{0.1,0.2\},\{0.5,0.6\}\}$ | $\{\{0.3,0.4,0.5\},\{0.1\},\{0.3,0.4\}\}$ | $\{\{0.5,0.6\},\{0.2,0.3\},\{0.3,0.4\}\}$ |

${\mathit{a}}_{\mathbf{2}}$ | $\{\{0.6,0.7\},\{0.1,0.2\},\{0.2,0.3\}\}$ | $\{\{0.6,0.7\},\{0.1\},\{0.3\}\}$ | $\{\{0.6,0.7\},\{0.1,0.2\},\{0.1,0.2\}\}$ |

${\mathit{a}}_{\mathbf{3}}$ | $\{\{0.6\},\{0.3\},\{0.4\}\}$ | $\{\{0.5,0.6\},\{0.4\},\{0.2,0.3\}\}$ | $\{\{0.5,0.6\},\{0.1\},\{0.3\}\}$ |

${\mathit{a}}_{\mathbf{4}}$ | $\{\{0.3,0.5\},\{0.2\},\{0.1,0.2,0.3\}\}$ | $\{\{0.6,0.7\},\{0.1\},\{0.2\}\}$ | $\{\{0.7,0.8\},\{0.1\},\{0.1,0.2\}\}$ |

**Table 4.**Results obtained by utilizing the different methods based on the same illustrative example.

Methods | Final Ranking | Best Alternative | Worst Alternative |
---|---|---|---|

SVNHFWA operator [24] | ${a}_{4}>{a}_{2}>{a}_{3}>{a}_{1}$ | ${a}_{4}$ | ${a}_{1}$ |

SVNHFWG operator [24] | ${a}_{2}>{a}_{4}>{a}_{3}>{a}_{1}$ | ${a}_{2}$ | ${a}_{1}$ |

Correlation coefficient [25] | ${a}_{2}>{a}_{4}>{a}_{3}>{a}_{1}$ | ${a}_{2}$ | ${a}_{1}$ |

Hamming distance [26] | ${a}_{2}>{a}_{4}>{a}_{3}>{a}_{1}$ | ${a}_{2}$ | ${a}_{1}$ |

SVNHFCOA operator | ${a}_{2}>{a}_{4}>{a}_{3}>{a}_{1}$ | ${a}_{2}$ | ${a}_{1}$ |

SVNHFCOG operator | ${a}_{2}>{a}_{4}>{a}_{3}>{a}_{1}$ | ${a}_{2}$ | ${a}_{1}$ |

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**MDPI and ACS Style**

Li, X.; Zhang, X.
Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators for Multi-Attribute Decision Making. *Symmetry* **2018**, *10*, 50.
https://doi.org/10.3390/sym10020050

**AMA Style**

Li X, Zhang X.
Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators for Multi-Attribute Decision Making. *Symmetry*. 2018; 10(2):50.
https://doi.org/10.3390/sym10020050

**Chicago/Turabian Style**

Li, Xin, and Xiaohong Zhang.
2018. "Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators for Multi-Attribute Decision Making" *Symmetry* 10, no. 2: 50.
https://doi.org/10.3390/sym10020050