# Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

#### 2.1. q-Rung Orthopair Fuzzy Set

**Definition**

**1**

**[33].**Let X be an ordinary fixed set. A q-ROFS A defined on X is given by

**Definition**

**2**

**[34].**Let${\tilde{a}}_{1}=\left({u}_{1},{v}_{1}\right)$,${\tilde{a}}_{2}=\left({u}_{2},{v}_{2}\right)$be two q-ROFNs and$\lambda $be a positive real number. Then,

- 1.
- ${\tilde{a}}_{1}\oplus {\tilde{a}}_{2}=\left({\left({u}_{1}^{q}+{u}_{2}^{q}-{u}_{1}^{q}{u}_{2}^{q}\right)}^{1/q},{v}_{1}{v}_{2}\right)$.
- 2.
- ${\tilde{a}}_{1}\otimes {\tilde{a}}_{2}=\left({u}_{1}{u}_{2},{\left({v}_{1}^{q}+{v}_{2}^{q}-{v}_{1}^{q}{v}_{2}^{q}\right)}^{1/q}\right)$.
- 3.
- $\lambda {\tilde{a}}_{1}=\left({\left(1-{\left(1-{u}_{1}^{q}\right)}^{\lambda}\right)}^{1/q},{v}_{1}^{\lambda}\right)$.
- 4.
- ${\tilde{a}}_{1}^{\lambda}=\left({u}_{1}^{\lambda},{\left(1-{\left(1-{v}_{1}^{q}\right)}^{\lambda}\right)}^{1/q}\right)$.

**Definition**

**3**

**[34].**Let$\tilde{a}=\left({u}_{a},{v}_{a}\right)$be a q-ROFN. Then, the score of$\tilde{a}$is defined as$S\left(\tilde{a}\right)={u}_{a}^{q}-{v}_{a}^{q}$and the accuracy of$\tilde{a}$is defined as$H\left(\tilde{a}\right)={u}_{a}^{q}+{v}_{a}^{q}$. For any two q-ROFNs,${\tilde{a}}_{1}=\left({u}_{1},{v}_{1}\right)$and${\tilde{a}}_{2}=\left({u}_{2},{v}_{2}\right)$. Then,

- 1.
- If $S\left({\tilde{a}}_{1}\right)>S\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}>{\tilde{a}}_{2}$;
- 2.
- If $S\left({\tilde{a}}_{1}\right)=S\left({\tilde{a}}_{2}\right)$, thenif $H\left({\tilde{a}}_{1}\right)>H\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}>{\tilde{a}}_{2}$;if $H\left({\tilde{a}}_{1}\right)=H\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}={\tilde{a}}_{2}$.

#### 2.2. q-Rung Dual Hesitant Fuzzy Set

**Definition**

**4.**

**Definition**

**5.**

- 1.
- If$S\left({d}_{1}\right)>S\left({d}_{2}\right)$, then${d}_{1}$is superior to${d}_{2}$, denoted by${d}_{1}>{d}_{2}$;
- 2.
- If$S\left({d}_{1}\right)>S\left({d}_{2}\right)$, thenif $H\left({d}_{1}\right)=H\left({d}_{2}\right)$, then ${d}_{1}$ is equivalent to ${d}_{2}$, denoted by ${d}_{1}={d}_{2}$;if $H\left({d}_{1}\right)>H\left({d}_{2}\right)$, then ${d}_{1}$ is superior to ${d}_{2}$, denoted by ${d}_{1}>{d}_{2}$.In the following, we define some operations of the q-RDHFEs.

**Definition**

**6.**

- 1.
- ${d}_{1}\oplus {d}_{2}={\cup}_{{\gamma}_{1}\in {h}_{1},{\gamma}_{2}\in {h}_{2},{\eta}_{1}\in {g}_{1},{\eta}_{2}\in {g}_{2}}\left\{\left\{{\left({\gamma}_{1}^{q}+{\gamma}_{2}^{q}-{\gamma}_{1}^{q}{\gamma}_{2}^{q}\right)}^{\frac{1}{q}}\right\},\left\{{\eta}_{1}{\eta}_{2}\right\}\right\}$;
- 2.
- ${d}_{1}\otimes {d}_{2}={\cup}_{{\gamma}_{1}\in {h}_{1},{\gamma}_{2}\in {h}_{2},{\eta}_{1}\in {g}_{1},{\eta}_{2}\in {g}_{2}}\left\{\left\{{\gamma}_{1}{\gamma}_{2}\right\},\left\{{\left({\eta}_{1}^{q}+{\eta}_{2}^{q}-{\eta}_{1}^{q}{\eta}_{2}^{q}\right)}^{\frac{1}{q}}\right\}\right\}$;
- 3.
- $\lambda d={\cup}_{\gamma \in h,\eta \in g}\left\{\left\{{\left(1-{\left(1-{\gamma}^{q}\right)}^{\lambda}\right)}^{\frac{1}{q}}\right\},\left\{{\eta}^{\lambda}\right\}\right\},\lambda >0$;
- 4.
- ${d}^{\lambda}={\cup}_{\gamma \in h,\eta \in g}\left\{\left\{{\gamma}^{\lambda}\right\},\left\{{\left(1-{\left(1-{\eta}^{q}\right)}^{\lambda}\right)}^{\frac{1}{q}}\right\}\right\},\lambda >0$.

#### 2.3. Heronian Mean

**Definition**

**7**

**[45].**Let${x}_{i}\left(i=1,2,...,n\right)$be a group of real numbers, and$s,t>0$. Then, the HM is defined as

**Definition**

**8**

**[46].**Let${x}_{i}\left(i=1,2,...,n\right)$be a group of numbers, and$s,t>0$. Then, the GHM is defined as

## 3. The q-Rung Dual Hesitant Fuzzy Heronian Mean Operators

#### 3.1. The q-Rung Dual Hesitant Fuzzy Heronian Mean Operator

**Definition**

**9.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- 1.
- If $t\to 0$, then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy generalized linear descending weighted mean operator, and we can obtain$$q-RDHFH{M}^{s,0}\left({d}_{1},{d}_{2},...,{d}_{n}\right)=\underset{t\to 0}{\mathrm{lim}}\{{\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\gamma}_{i}^{s}{\gamma}_{j}^{t}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q(s+t)}}\right\},\phantom{\rule{0ex}{0ex}}\left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\eta}_{i}^{q}\right)}^{s}{\left(1-{\eta}_{j}^{q}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{q}}\right\}\}\phantom{\rule{0ex}{0ex}}={\cup}_{{\gamma}_{i}\in {h}_{i},{\eta}_{i}\in {g}_{i}}\left\{\left\{{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left({\gamma}_{i}^{s}\right)}^{q}\right)}^{n+1-i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{qs}}\right\}\uff0c\left\{{\left(1-{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left(1-{\eta}_{i}^{q}\right)}^{s}\right)}^{n+1-i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{s}}\right)}^{\frac{1}{q}}\right\}\right\}\text{}$$

- 2.
- If $s\to 0$, then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy generalized liner ascending weighted mean operator, and we can obtain$$\begin{array}{c}q-RDHFH{M}^{0,t}\left({d}_{1},{d}_{2},...,{d}_{n}\right)=\underset{s\to 0}{\mathrm{lim}}\{{\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\gamma}_{i}^{s}{\gamma}_{j}^{t}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q(s+t)}}\right\},\\ \left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\eta}_{i}^{q}\right)}^{s}{\left(1-{\eta}_{j}^{q}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{q}}\right\}\}\\ ={\cup}_{{\gamma}_{j}\in {h}_{j},{\eta}_{j}\in {g}_{j}}\left\{\left\{{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left({\gamma}_{j}^{t}\right)}^{q}\right)}^{i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{qt}}\right\}\uff0c\left\{{\left(1-{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left(1-{\eta}_{j}^{q}\right)}^{t}\right)}^{i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{t}}\right)}^{\frac{1}{q}}\right\}\right\}\end{array}$$

- 3.
- If $s=t=\frac{1}{2}$, then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy basic Heronian mean operator, and we can obtain$$\begin{array}{l}q-RDHFH{M}^{\frac{1}{2},\frac{1}{2}}\left({d}_{1},{d}_{2},...,{d}_{n}\right)=\text{\hspace{0.17em}}\\ {\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left(\sqrt{{\gamma}_{i}{\gamma}_{j}}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q}}\right\},\left\{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-\sqrt{\left(1-{\eta}_{i}^{q}\right)\left(1-{\eta}_{j}^{q}\right)}\right)}^{\frac{2}{nq(n+1)}}}}\right\}\}\\ \end{array}$$
- 4.
- If $s=t=1$, then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy line Heronian mean operator. It follows that$$\begin{array}{ll}q-RDHFH{M}^{1,1}\left({d}_{1},{d}_{2},...,{d}_{n}\right)={\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}& \{\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\gamma}_{i}{\gamma}_{j}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2q}}\right\},\\ & \left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-\left(1-{\eta}_{i}^{q}\right)\left(1-{\eta}_{j}^{q}\right)\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{q}}\right\}\}\end{array}$$
- 5.
- If $q=2$, then the q-RDHFHM reduces to a dual hesitant Pythagorean fuzzy Heronian mean operator. So, we can obtain$$\begin{array}{ll}q-RDHFH{M}^{s,t}\left({d}_{1},{d}_{2},...,{d}_{n}\right)={\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}& \{\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\gamma}_{i}^{s}{\gamma}_{j}^{t}\right)}^{2}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2(s+t)}}\right\},\\ & \left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\eta}_{i}^{2}\right)}^{s}{\left(1-{\eta}_{j}^{2}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{2}}\right\}\}\end{array}$$
- 6.
- If $q=1$, then the q-RDHFHM reduces to the dual hesitant fuzzy Heronian mean operator proposed by Yu et al. [47]. It follows that$$\begin{array}{ll}q-RDHFH{M}^{s,t}\left({d}_{1},{d}_{2},...,{d}_{n}\right)={\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}& \{\left\{{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-\left({\gamma}_{i}^{s}{\gamma}_{j}^{t}\right)\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right\},\\ & \left\{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\eta}_{i}\right)}^{s}{\left(1-{\eta}_{j}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)\right\}\}\end{array}$$

#### 3.2. The q-Rung Dual Hesitant Fuzzy Weighted Heronian Mean (q-RDHFWHM) Operator

**Definition**

**10.**

**Theorem**

**5.**

**Theorem**

**6.**

**Proof.**

#### 3.3. The q-Rung Dual Hesitant Fuzzy Geometric Heronian Mean Operator

**Definition**

**11.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

- If $t\to 0$, then the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy generalized geometric linear descending weighted mean operator, and we can obtain$$\begin{array}{l}q-RDHFGH{M}^{s,0}\left({d}_{1},{d}_{2},...,{d}_{n}\right)\\ =\underset{t\to 0}{\mathrm{lim}}\{{\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\gamma}_{i}^{q}\right)}^{s}{\left(1-{r}_{j}^{q}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{q}}\right\},\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\mu}_{i}^{s}{\eta}_{j}^{t}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q(s+t)}}\right\}\}\\ ={\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left(1-{\gamma}_{i}^{q}\right)}^{s}\right)}^{n+1-i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{s}}\right)}^{\frac{1}{q}}\right\},\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\left\{{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left({\eta}_{i}^{s}\right)}^{q}\right)}^{n+1-i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{qs}}\right\}\}\end{array}$$
- If $s\to 0$, the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy generalized geometric liner ascending weighted mean operator, and we can obtain$$\begin{array}{l}q-RDHFGH{M}^{0,t}\left({d}_{1},{d}_{2},...,{d}_{n}\right)\\ =\underset{s\to 0}{\mathrm{lim}}\{{\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\gamma}_{i}^{q}\right)}^{s}{\left(1-{\gamma}_{j}^{q}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{q}}\right\},\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\eta}_{i}^{s}{\eta}_{j}^{t}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q(s+t)}}\right\}\}\\ ={\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left(1-{\gamma}_{j}^{q}\right)}^{t}\right)}^{i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{t}}\right)}^{\frac{1}{q}}\right\},\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\left\{{\left(1-{\left({\displaystyle \prod _{i=1}^{n}{\left(1-{\left({\eta}_{j}^{t}\right)}^{q}\right)}^{i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{qt}}\right\}\}\end{array}$$
- If $s=t=\frac{1}{2}$, the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy basic geometric Heronian mean operator, and we can obtain$$\begin{array}{l}q-RDHFGH{M}^{\frac{1}{2},\frac{1}{2}}\left({d}_{1},{d}_{2},...,{d}_{n}\right)\\ ={\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-\sqrt{\left(1-{\gamma}_{i}^{q}\right)\left(1-{\gamma}_{j}^{q}\right)}\right)}^{\frac{2}{nq(n+1)}}}}\right\},\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left(\sqrt{{\eta}_{i}{\eta}_{j}}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q}}\right\}\}\end{array}$$
- If $s=t=1$, the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy line Heronian mean operator, and it follows that$$\begin{array}{l}q-RDHFGH{M}^{1,1}\left({d}_{1},{d}_{2},...,{d}_{n}\right)=\text{}{\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\\ \{\left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-\left(1-{\gamma}_{i}^{q}\right)\left(1-{\gamma}_{j}^{q}\right)\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{q}}\right\},\left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\eta}_{i}{\eta}_{j}\right)}^{q}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2q}}\right\}\}\end{array}$$
- If $q=2$, then the q-RDHFGHM reduces to the dual hesitant Pythagorean fuzzy Heronian mean operator, and can we can obtain$$\begin{array}{l}q-RDHFGH{M}^{s,t}\left({d}_{1},{d}_{2},...,{d}_{n}\right)=\\ {\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\gamma}_{i}^{2}\right)}^{s}{\left(1-{\gamma}_{j}^{2}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{2}}\right\},\\ \left\{{\left(1-{{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}\left(1-{\left({\eta}_{i}^{s}{\eta}_{j}^{t}\right)}^{2}\right)}}}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2(s+t)}}\right\}\}\end{array}$$
- If $q=1$, then the q-RDHFGHM reduces to the dual hesitant fuzzy Heronian mean operator proposed by Yu et al. [47], and it follows that$$\begin{array}{l}q-RDHFH{M}^{s,t}\left({d}_{1},{d}_{2},...,{d}_{n}\right)=\\ {\cup}_{{\gamma}_{i}\in {h}_{i},{\gamma}_{j}\in {h}_{j},{\eta}_{i}\in {g}_{i},{\eta}_{j}\in {g}_{j}}\{\left\{\left(1-{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-{\left(1-{\gamma}_{i}\right)}^{s}{\left(1-{\gamma}_{j}\right)}^{t}\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right)\right\},\\ \left\{{\left(1-{\displaystyle \prod _{i=1}^{n}{\displaystyle \prod _{j=i}^{n}{\left(1-\left({\eta}_{i}^{s}{\eta}_{j}^{t}\right)\right)}^{\frac{2}{n(n+1)}}}}\right)}^{\frac{1}{s+t}}\right\}\}\end{array}$$

#### 3.4. The q-Rung Dual Hesitant Fuzzy Weighted Geometric Heronian Mean Operator

**Definition**

**12.**

**Theorem**

**11.**

**Theorem**

**12.**

## 4. A Novel Approach to MAGDM with q-Rung Dual Hesitant Fuzzy Information

#### 4.1. Description of a Typical MAGDM Problem with q-Rung Dual Hesitant Fuzzy Information

#### 4.2. An Algorithm for q-Rung Dual Hesitant Fuzzy MAGDM Problems

**Step 1.**Standardize the original decision matrix according the following equation:

**Step 2.**For alternative ${A}_{i}\left(i=1,2,...,m\right)$, utilize the q-RDHFWHM operator

**Step 3.**Compute the score functions of all the alternatives and rank them.

**Step 4.**Rank the corresponding alternatives according to the rank of overall values and select the best alternative.

## 5. Numerical Example

_{1}, A

_{2}, A

_{3}, and A

_{4}) remain on the candidates list. To select the best supplier, a set of experts are invited to assess the four suppliers regarding four attributes: (1) relationship closeness (G

_{1}); (2) product quality (G

_{2}); (3) price competitiveness (G

_{3}); and (4) delivery performance (G

_{4}). The weight vector of the attributes is $w={\left(0.17,0.32,0.38,0.13\right)}^{T}$. The DMs are required to utilize DHFEs to express their preference information. The dual hesitant fuzzy decision matrix is shown in Table 1.

#### 5.1. The Decision-Making Process

**Step 1.**As all the attributes are of the benefit type, the original decision matrix does not need to be normalized.

**Step 2.**Utilize the q-RDHFWHM operator to aggregate attributes values, so that the overall assessments are obtained (assume s = t = 1 and q = 3). Due to the relatively large numbers, the overall assessments are omitted.

**Step 3.**Calculate the scores of the overall assessments of alternatives to obtain $s\left({d}_{1}\right)=0.2235$, $s\left({d}_{2}\right)=0.2631$, $\mathrm{s}\left({d}_{3}\right)=0.2097$, and $s\left({d}_{4}\right)=0.0780$.

**Step 4.**Rank the overall assessments so that we can obtain ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$. Therefore, the best alternative is A

_{2}.

_{2}.

#### 5.2. The Influence of the Parameters on the Results

_{2}or A

_{1}. In addition, from Figure 6 and Figure 7, we find that if we let t or s be a fixed value, then when s or t increases, the scores based on the q-RDHFWHM operator become greater and greater. Similarly, from Table 3 and Figure 8, Figure 9, Figure 10 and Figure 11, we can obtain different scores and ranking results when s and t represent different values based on the q-RDHFWGHM operator. No matter what the values of s and t are, the best alternative is always A

_{2}. However, what is opposite to the q-RDHFWHM operator is that if we let s or t be a fixed value, then when s or t increases, the scores based on the q-RDHFWGHM operator become smaller and smaller. The results shown in Table 2 and Table 3 and Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 demonstrate the flexibility of the aggregation processes by utilizing the q-RDHFWHM and q-RDHFWGHM operators. In real decision-making problems, DMs should choose the appropriate s and t according to their preference.

_{2}or A

_{4}based on the q-RDHFWHM, whereas the best alternative is always A

_{2}based on the q-RDHFWGHM operators. In addition, when q increases, both the scores obtained by the q-RDHFWHM and q-RDHFWGHM operators have the tendency to decrease.

#### 5.3. Compared with Exiting MAGDM Methods

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–356. [Google Scholar] [CrossRef] - Son, L.H.; Phong, P.H. On the performance evaluation of intuitionistic vector similarity measures for medical diagnosis 1. J. Intell. Fuzzy Syst.
**2016**, 31, 1597–1608. [Google Scholar] [CrossRef] - Own, C.M. Switching between type-2 fuzzy sets and intuitionistic fuzzy sets: An application in medical diagnosis. Appl. Intell.
**2009**, 31, 283–291. [Google Scholar] [CrossRef] - Chen, S.M.; Cheng, S.H.; Lan, T.C. A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition. Inf. Sci.
**2016**, 343–344, 15–40. [Google Scholar] [CrossRef] - Hwang, C.M.; Yang, M.S.; Hung, W.L.; Lee, M.G. A similarity measure of intuitionistic fuzzy sets based on the Sugeno integral with its application to pattern recognition. Inf. Sci.
**2012**, 189, 93–109. [Google Scholar] [CrossRef] - Wang, Z.; Xu, Z.S.; Liu, S.S.; Tang, J. A netting clustering analysis method under intuitionistic fuzzy environment. Appl. Soft Comput.
**2011**, 11, 5558–5564. [Google Scholar] [CrossRef] - Xu, D.W.; Xu, Z.S.; Liu, S.S.; Zhao, H. A spectral clustering algorithm based on intuitionistic fuzzy information. Knowl. Based Syst.
**2013**, 53, 20–26. [Google Scholar] [CrossRef] - Liu, P.D. Multiple attribute decision-making methods based on normal intuitionistic fuzzy interaction aggregation operators. Symmetry
**2017**, 9, 261. [Google Scholar] [CrossRef] - Wang, S.W.; Liu, J. Extension of the TODIM method to intuitionistic linguistic multiple attribute decision making. Symmetry
**2017**, 9, 95. [Google Scholar] [CrossRef] - Liu, P.D.; Mahmood, T.; Khan, Q. Multi-attribute decision-making based on prioritized aggregation operator under hesitant intuitionistic fuzzy linguistic environment. Symmetry
**2017**, 9, 270. [Google Scholar] [CrossRef] - Zhao, J.S.; You, X.Y.; Liu, H.C.; Wu, S.M. An extended VIKOR method using intuitionistic fuzzy sets and combination weights for supplier selection. Symmetry
**2017**, 9, 169. [Google Scholar] [CrossRef] - Yager, R.R. Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst.
**2014**, 22, 958–965. [Google Scholar] [CrossRef] - Zhang, X.; Xu, Z. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int. J. Intell. Syst.
**2014**, 29, 1061–1078. [Google Scholar] [CrossRef] - Sajjad, A.K.M.; Abdullah, S.; Yousaf, A.M.; Hussain, I.; Farooq, M. Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environment. J. Intell. Fuzzy Syst.
**2018**, 34, 267–282. [Google Scholar] [CrossRef] - Ren, P.; Xu, Z.; Gou, X. Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl. Soft Comput.
**2016**, 42, 246–259. [Google Scholar] [CrossRef] - Ma, Z.M.; Xu, Z.S. Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multi-criteria decision making problems. J. Intell. Fuzzy Syst.
**2016**, 31, 1198–1219. [Google Scholar] - Xing, Y.P.; Zhang, R.T.; Wang, J.; Zhu, X.M. Some new Pythagorean fuzzy Choquet-Frank aggregation operators for multi-attribute decision making. Int. J. Intell. Syst.
**2018**. [Google Scholar] [CrossRef] - Wei, G.W.; Lu, M. Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making. Int. J. Intell. Syst.
**2017**, 33, 1043–1070. [Google Scholar] [CrossRef] - Liang, D.C.; Xu, Z.S.; Darko, A.P. Projection model for fusing the information of Pythagorean fuzzy multicriteria group decision making based on geometric Bonferroni mean. Int. J. Intell. Syst.
**2017**, 32, 966–987. [Google Scholar] [CrossRef] - Zhang, R.T.; Wang, J.; Zhu, X.M.; Xia, M.M.; Yu, M. Some generalized Pythagorean fuzzy Bonferroni mean aggregation operators with their application to multi attribute group decision-making. Complexity
**2017**, 2017, 5937376. [Google Scholar] [CrossRef] - Li, L.; Zhang, R.T.; Wang, J.; Zhu, X.M.; Xing, Y.P. Pythagorean fuzzy power Muirhead mean operators with their application to multi-attribute decision making. J. Intell. Fuzzy Syst.
**2018**, 35, 2035–2050. [Google Scholar] [CrossRef] - Xu, Y.; Shang, X.P.; Wang, J. Pythagorean fuzzy interaction Muirhead means with their application to multi-attribute group decision making. Information
**2018**, 9, 7. [Google Scholar] [CrossRef] - Teng, F.; Liu, Z.; Liu, P. Some power Maclaurin symmetric mean aggregation operators based on Pythagorean fuzzy linguistic numbers and their application to group decision making. Int. J. Intell. Syst.
**2018**, 33, 1949–1985. [Google Scholar] [CrossRef] - Du, Y.; Hou, F.; Zafar, W.; Yu, Q.; Zhai, Y. A novel method for multiattribute decision making with interval-valued Pythagorean fuzzy linguistic information. Int. J. Intell. Syst.
**2017**, 32, 1085–1112. [Google Scholar] [CrossRef] - Xian, S.; Xiao, Y.; Yang, Z.; Li, Y.; Han, Z. A new trapezoidal Pythagorean fuzzy linguistic entropic combined ordered weighted averaging operator and its application for enterprise location. Int. J. Intell. Syst.
**2018**, 33, 1880–1899. [Google Scholar] [CrossRef] - Geng, Y.; Liu, P.; Teng, F.; Liu, Z. Pythagorean fuzzy uncertain linguistic TODIM method and their application to multiple criteria group decision making. J. Intell. Fuzzy Syst.
**2017**, 33, 3383–3395. [Google Scholar] [CrossRef] - Liu, C.; Tang, G.; Liu, P. An approach to multicriteria group decision-making with unknown weight information based on Pythagorean fuzzy uncertain linguistic aggregation operators. Math. Probl. Eng.
**2017**. [Google Scholar] [CrossRef] - Liu, Z.; Liu, P.; Liu, W.; Pang, J. Pythagorean uncertain linguistic partitioned Bonferroni mean operators and their application in multi-attribute decision making. J. Intell. Fuzzy Syst.
**2017**, 32, 2779–2790. [Google Scholar] [CrossRef] - Wei, G.; Lu, M.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Pythagorean 2-tuple linguistic aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst.
**2017**, 33, 1129–1142. [Google Scholar] [CrossRef] - Huang, Y.H.; Wei, G.W. TODIM method for Pythagorean 2-tuple linguistic multiple attribute decision making. J. Intell. Fuzzy Syst.
**2018**, 35, 901–915. [Google Scholar] [CrossRef] - Tang, X.; Wei, G. Models for green supplier selection in green supply chain management with Pythagorean 2-tuple linguistic information. IEEE Access
**2018**, 6, 18042–18060. [Google Scholar] [CrossRef] - Yager, R.R. Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst.
**2017**, 25, 1222–1230. [Google Scholar] [CrossRef] - Liu, P.D.; Wang, P. Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst.
**2018**, 33, 259–280. [Google Scholar] [CrossRef] - Liu, P.D.; Liu, J.L. Some q-rung orthopair fuzzy Bonferroni mean operators and their application to multi-attribute group decision making. Int. J. Intell. Syst.
**2018**, 33, 315–347. [Google Scholar] [CrossRef] - Liu, Z.M.; Liu, P.D.; Liang, X. Multiple attribute decision-making method for dealing with heterogeneous relationship among attributes and unknown attribute weight information under q-rung orthopair fuzzy environment. Int. J. Intell. Syst.
**2018**, 33, 1900–1928. [Google Scholar] [CrossRef] - Bai, K.Y.; Zhu, X.M.; Wang, J.; Zhang, R.T. Some partitioned Maclaurin symmetric mean based on q-rung orthopair fuzzy information for dealing with multi-attribute group decision making. Symmetry
**2018**, 10, 383. [Google Scholar] [CrossRef] - Li, L.; Zhang, R.T.; Wang, J.; Shang, X.P. Some q-rung orthopair linguistic Heronian mean operators with their application to multi-attribute group decision making. Arch. Control Sci. accepted.
- Li, L.; Zhang, R.T.; Wang, J.; Shang, X.P.; Bai, K.Y. A novel approach to multi-attribute group decision-making with q-rung picture linguistic information. Symmetry
**2018**, 10, 172. [Google Scholar] [CrossRef] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Zhu, B.; Xu, Z.S.; Xia, M.M. Dual hesitant fuzzy sets. J. Appl. Math.
**2012**, 2012, 879629. [Google Scholar] [CrossRef] - Wei, G.; Lu, M. Dual hesitant Pythagorean fuzzy Hamacher aggregation operators in multiple attribute decision making. Arch. Control Sci.
**2017**, 27, 365–395. [Google Scholar] [CrossRef] - Khan, M.S.A.; Abdullah, S.; Ali, A.; Siddiqui, N.; Amin, F. Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information. J. Intell. Fuzzy Syst.
**2017**, 33, 3971–3985. [Google Scholar] [CrossRef] - Liang, D.C.; Xu, Z.S. The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets. Appl. Soft Comput.
**2017**, 60, 167–179. [Google Scholar] [CrossRef] - Sykora, S. Mathematical Means and Averages: Generalized Heronian Means; Stan’s Library: Castano Primo, Italy, 2009; Available online: http://www.ebyte.it/library/docs/math09/Means_Heronian.html (accessed on 9 October 2018).
- Yu, D.J. Intuitionistic fuzzy geometric Heronian mean aggregation operators. Appl. Soft Comput.
**2013**, 13, 1235–1246. [Google Scholar] [CrossRef] - Yu, D.J.; Li, D.F.; Merigó, J.M. Dual hesitant fuzzy group decision making method and its application to supplier selection. Int. J. Mach. Learn. Cybern.
**2016**, 7, 819–831. [Google Scholar] [CrossRef] - Wang, H.J.; Zhao, X.F.; Wei, G.W. Dual hesitant fuzzy aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst.
**2014**, 26, 2281–2290. [Google Scholar] - Tu, N.H.; Wang, C.Y.; Zhou, X.Q.; Tao, S.D. Dual hesitant fuzzy aggregation operators based on Bonferroni means and their applications to multiple attribute decision making. Annl. Fuzzy Math. Inform.
**2017**, 14, 265–278. [Google Scholar]

**Figure 2.**Scores of alternative A

_{1}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator (q = 3).

**Figure 3.**Scores of alternative A

_{2}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator (q = 3).

**Figure 4.**Scores of alternative A

_{3}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator (q = 3).

**Figure 5.**Scores of alternative A

_{4}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator (q = 3).

**Figure 6.**Scores of alternatives ${A}_{i}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ when t = 1 and $s\in \left(1,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator (q = 3).

**Figure 7.**Scores of alternative ${A}_{i}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ when s = 1 and $t\in \left(1,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator (q = 3).

**Figure 8.**Scores of alternative A

_{1}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-rung dual hesitant fuzzy weighted geometric Heronian mean (q-RDHFWGHM) operator (q = 3).

**Figure 9.**Scores of alternative A

_{2}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWGHM operator (q = 3).

**Figure 10.**Scores of alternative A

_{3}when $s,\text{\hspace{0.17em}}t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWGHM operator (q = 3).

**Figure 11.**Scores of alternative A

_{4}when $s,t\in \left(0,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWGHM operator (q = 3).

**Figure 12.**Scores of alternative ${A}_{i}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ when t = 1 and $s\in \left(1,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWGHM operator (q = 3).

**Figure 13.**Scores of alternative ${A}_{i}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ when s = 1 and $t\in \left(1,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWGHM operator (q = 3).

**Figure 14.**Scores of alternative ${A}_{i}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ when s = t = 1 and $q\in \left(1,\text{\hspace{0.17em}}10\right)$ based on the q-RDHFWHM operator.

**Figure 15.**Scores of alternative ${A}_{i}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ when s = t = 1 and $q\in \left(1,10\right)$ based on the q-RDHFWGHM operator.

G_{1} | G_{2} | G_{3} | G_{4} | |
---|---|---|---|---|

A_{1} | {{0.3, 0.4}, {0.6}} | {{0.7, 0.9}, {0.1}} | {{0.4}, {0.2,0.3}} | {{0.5, 0.6}, {0.2}} |

A_{2} | {{0.2, 0.3}, {0.5}} | {{0.6, 0.7}, {0.2}} | {{0.7, 0.8}, {0.2}} | {{0.6}, {0.1, 0.2, 0.3}} |

A_{3} | {{0.4}, {0.2,0.3}} | {{0.2,0.3,0.4}, {0.6}} | {{0.7,0.8}, {0.1}} | {{0.7}, {0.2,0.3}} |

A_{4} | {{0.6,0.7}, {0.3}} | {{0.5}, {0.4}} | {{0.3,0.4}, {0.5}} | {{0.4, 0.6}, {0.1,0.2}} |

**Table 2.**Scores and ranking results by using the q-rung dual hesitant fuzzy weighted Heronian mean (q-RDHFWHM) operator (q = 3).

Parameters | Score Function $\mathit{s}\left({\mathit{d}}_{\mathit{i}}\right)\left(\mathit{i}=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)$ | Ranking Results |
---|---|---|

s = t = 1/2 | $s\left({d}_{1}\right)=0.1617$$s\left({d}_{2}\right)=0.2086$$s\left({d}_{3}\right)=0.1532$$s\left({d}_{4}\right)=0.0685$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = t = 1 | $s\left({d}_{1}\right)=0.2235$$s\left({d}_{2}\right)=0.2631$$s\left({d}_{3}\right)=0.2093$$s\left({d}_{4}\right)=0.0780$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = t = 2 | $s\left({d}_{1}\right)=0.3235$$s\left({d}_{2}\right)=0.3375$$s\left({d}_{3}\right)=0.2989$$s\left({d}_{4}\right)=0.0942$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = t = 5 | $s\left({d}_{1}\right)=0.4494$$s\left({d}_{2}\right)=0.4363$$s\left({d}_{3}\right)=0.4174$$s\left({d}_{4}\right)=0.1198$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

s = 1, t = 2 | $s\left({d}_{1}\right)=0.2769$$s\left({d}_{2}\right)=0.3124$$s\left({d}_{3}\right)=0.2626$$s\left({d}_{4}\right)=0.0836$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = 2, t = 1 | $s\left({d}_{1}\right)=0.3001$$s\left({d}_{2}\right)=0.3079$$s\left({d}_{3}\right)=0.2677$$s\left({d}_{4}\right)=0.0930$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = 1, t = 5 | $s\left({d}_{1}\right)=0.4024$$s\left({d}_{2}\right)=0.4118$$s\left({d}_{3}\right)=0.3719$$s\left({d}_{4}\right)=0.1042$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = 5, t = 1 | $s\left({d}_{1}\right)=0.4463$$s\left({d}_{2}\right)=0.3952$$s\left({d}_{3}\right)=0.3825$$s\left({d}_{4}\right)=0.1214$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

Parameters | Score Function $\mathit{s}\left({\mathit{d}}_{\mathit{i}}\right)\left(\mathit{i}=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right)\text{}$ | Ranking Results |
---|---|---|

s = t = 1/2 | $s\left({d}_{1}\right)=0.1516$$s\left({d}_{2}\right)=0.1972$$s\left({d}_{3}\right)=0.1387$$s\left({d}_{4}\right)=0.0861$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = t = 1 | $s\left({d}_{1}\right)=0.1187$$s\left({d}_{2}\right)=0.1819$$s\left({d}_{3}\right)=0.0851$$s\left({d}_{4}\right)=0.0566$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = t = 2 | $s\left({d}_{1}\right)=0.0683$$s\left({d}_{2}\right)=0.1544$$s\left({d}_{3}\right)=0.0055$$s\left({d}_{4}\right)=0.0132$ | ${A}_{2}\succ {A}_{1}\succ {A}_{4}\succ {A}_{3}$ |

s = t = 5 | $s\left({d}_{1}\right)=-0.0029$$s\left({d}_{2}\right)=0.1054$$s\left({d}_{3}\right)=-0.0949$$s\left({d}_{4}\right)=-0.0521$ | ${A}_{2}\succ {A}_{1}\succ {A}_{4}\succ {A}_{3}$ |

s = 1, t = 2 | $s\left({d}_{1}\right)=0.0884$$s\left({d}_{2}\right)=0.1833$$s\left({d}_{3}\right)=0.0423$$s\left({d}_{4}\right)=0.0210$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

s = 2, t = 1 | $s\left({d}_{1}\right)=0.0841$$s\left({d}_{2}\right)=0.1456$$s\left({d}_{3}\right)=0.0279$$s\left({d}_{4}\right)=0.0358$ | ${A}_{2}\succ {A}_{1}\succ {A}_{4}\succ {A}_{3}$ |

s = 1, t = 5 | $s\left({d}_{1}\right)=0.0204$$s\left({d}_{2}\right)=0.1540$$s\left({d}_{3}\right)=-0.0507$$s\left({d}_{4}\right)=-0.0418$ | ${A}_{2}\succ {A}_{1}\succ {A}_{4}\succ {A}_{3}$ |

s = 5, t = 1 | $s\left({d}_{1}\right)=0.0215$$s\left({d}_{2}\right)=0.0917$$s\left({d}_{3}\right)=-0.0653$$s\left({d}_{4}\right)=-0.0206$ | ${A}_{2}\succ {A}_{1}\succ {A}_{4}\succ {A}_{3}$ |

Methods | Score Function $\mathit{s}\left({\mathit{d}}_{\mathit{i}}\right)\left(\mathit{i}=1,2,3,4\right)\text{}$ | Ranking Results |
---|---|---|

Wang et al.’ [48] method based on the DHFWA operator | $s\left({d}_{1}\right)=0.3915$$s\left({d}_{2}\right)=0.4147$ $s\left({d}_{3}\right)=0.3573$$s\left({d}_{4}\right)=0.1198$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

Yu et al.’ s [47] method based on the DHFWHM operator (s = t = 2) | $s\left({d}_{1}\right)=-0.3813$$s\left({d}_{2}\right)=-0.3916$ $s\left({d}_{3}\right)=-0.3960$$s\left({d}_{4}\right)=-0.6147$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

Tu et al.’s [49] method based on the DHFWBM operator | $s\left({d}_{1}\right)=0.3152$$\mathrm{s}\left({d}_{2}\right)=0.3004$ $\mathrm{s}\left({d}_{3}\right)=0.2978$$s\left({d}_{4}\right)=0.0258$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

Wei and Lu’s [42] method based on the DHPFHWA operator | $s\left({d}_{1}\right)=0.2369$$s\left({d}_{2}\right)=0.2196$ $s\left({d}_{3}\right)=0.1284$$s\left({d}_{4}\right)=0.0026$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

The proposed method in this paper | $s\left({d}_{1}\right)=0.2235$$s\left({d}_{2}\right)=0.2631$ $s\left({d}_{3}\right)=0.2097$$\mathrm{s}\left({d}_{4}\right)=0.0780$ | ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ |

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**MDPI and ACS Style**

Xu, Y.; Shang, X.; Wang, J.; Wu, W.; Huang, H.
Some *q*-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making. *Symmetry* **2018**, *10*, 472.
https://doi.org/10.3390/sym10100472

**AMA Style**

Xu Y, Shang X, Wang J, Wu W, Huang H.
Some *q*-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making. *Symmetry*. 2018; 10(10):472.
https://doi.org/10.3390/sym10100472

**Chicago/Turabian Style**

Xu, Yuan, Xiaopu Shang, Jun Wang, Wen Wu, and Huiqun Huang.
2018. "Some *q*-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making" *Symmetry* 10, no. 10: 472.
https://doi.org/10.3390/sym10100472