# Prioritized Aggregation Operators and Correlated Aggregation Operators for Hesitant 2-Tuple Linguistic Variables

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- ${s}_{p}>{s}_{q}\iff p>q.$
- (2)
- For each ${s}_{p}$, there always exist $neg\left({s}_{p}\right)={s}_{q}$, $q=l-p$.
- (3)
- If ${s}_{p}<{s}_{q}$, then $\mathrm{max}({s}_{p},{s}_{q})={s}_{q}$ and $\mathrm{min}({s}_{p},{s}_{q})={s}_{p}$.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

- (1)
- If $S\left({H}_{1}\right)<S\left({H}_{2}\right)$, then ${H}_{1}<{H}_{2}$.
- (2)
- If $S\left({H}_{1}\right)>S\left({H}_{2}\right)$, then ${H}_{1}>{H}_{2}$.
- (3)
- If $S\left({H}_{1}\right)=S\left({H}_{2}\right)$, then
- (a)
- if $V\left({H}_{1}\right)=V\left({H}_{2}\right)$, then ${H}_{1}\sim {H}_{2}$;
- (b)
- if $V\left({H}_{1}\right)<V\left({H}_{2}\right)$, then ${H}_{1}>{H}_{2}$;
- (c)
- if $V\left({H}_{1}\right)>V\left({H}_{2}\right)$, then ${H}_{1}<{H}_{2}$.

**Definition**

**8.**

- 1.
- $\mu (\varnothing )=0,\phantom{\rule{4pt}{0ex}}\mu \left(C\right)=1;$
- 2.
- $E\subseteq F\phantom{\rule{4pt}{0ex}}implies\phantom{\rule{4pt}{0ex}}\mu \left(E\right)\le \mu \left(F\right),for\phantom{\rule{4pt}{0ex}}all\phantom{\rule{4pt}{0ex}}E,F\subseteq C;$
- 3.
- $\mu (E\cup F)=\mu \left(E\right)+\mu \left(F\right)+\alpha \mu \left(E\right)\mu \left(F\right)$ for all $E,F\subseteq C\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}E\cap F=\varnothing ,where\phantom{\rule{4pt}{0ex}}\alpha \in (-1,+\infty ).$

## 3. New Hesitant 2-Tuple Linguistic Aggregation Operators

**Definition**

**9.**

#### 3.1. Hesitant 2-Tuple Linguistic Prioritized Weighted Aggregation Operator

**Definition**

**10.**

**Theorem**

**1.**

- (1)
- (Idempotency): Let $H=\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$ be n H2TLSs; if all ${H}_{j}\phantom{\rule{4pt}{0ex}}(j=1,2,\dots ,n)$ are equal to ${H}_{j}=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$, then$$\begin{array}{c}\hfill H2TLPWA({H}_{1},{H}_{2},\dots ,{H}_{n})=({x}_{1},{x}_{2},\dots ,{x}_{n})\end{array}$$
- (2)
- (Boundedness): Let $H=\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$ be n H2TLSs; then$$\begin{array}{c}\hfill \underset{1\le j\le n}{\mathrm{min}}\left\{{H}_{j}\right\}\le H2TLPWA({H}_{1},{H}_{2},\dots ,{H}_{n})\le \underset{1\le j\le n}{\mathrm{max}}\left\{{H}_{j}\right\}\end{array}$$
- (3)
- (Monotonicity): Let $H=\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$ and ${H}^{\prime}=\{{H}_{{1}^{\prime}},{H}_{{2}^{\prime}},\dots ,{H}_{{n}^{\prime}}\}$ be two collections of H2TLSs; if ${H}_{j}\le {H}_{{j}^{\prime}}\phantom{\rule{4pt}{0ex}}(j=1,2,\dots ,n)$ for all j, then$$\begin{array}{c}\hfill H2TLPWA({H}_{1},{H}_{2},\dots ,{H}_{n})\le H2TLPWA({H}_{{1}^{\prime}},{H}_{{2}^{\prime}},\dots ,{H}_{{n}^{\prime}})\end{array}$$

**Definition**

**11.**

#### 3.2. Hesitant 2-Tuple Linguistic Correlated Aggregation Operator

**Definition**

**12.**

**Definition**

**13.**

- If $\mu \left(A\right)=1$, for any $A\in P\left(C\right)$, then$$\begin{array}{c}\hfill H2TLC{A}_{\mu}({H}_{1},{H}_{2},\dots ,{H}_{n})=\mathrm{max}\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}={H}_{\sigma \left(1\right)}\end{array}$$
- If $\mu \left(A\right)=0$, for any $A\in P\left(C\right)$ and $A\ne C$, then$$\begin{array}{c}\hfill H2TLC{A}_{\mu}({H}_{1},{H}_{2},\dots ,{H}_{n})=\mathrm{min}\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}={H}_{\sigma \left(n\right)}\end{array}$$
- For any $E,F\in P\left(C\right)$ such that $\left|E\right|=\left|F\right|$, where $\left|E\right|$ and $\left|F\right|$ are the number of elements in E and F, respectively, if $\mu \left(E\right)=\mu \left(F\right)$ and $\mu \left({A}_{\sigma \left(j\right)}\right)=\frac{j}{n}$, $1\le j\le n$, then$$\begin{array}{c}\hfill H2TLC{A}_{\mu}({H}_{1},{H}_{2},\dots ,{H}_{n})=\bigcup _{{x}_{j}\in {H}_{j},(j=1,2,\dots ,n)}\u25b5\left(\sum _{j=1}^{n}\frac{1}{n}\ast {\u25b5}^{-1}\left({x}_{j}\right)\right)\end{array}$$
- If $\mu \left(E\right)={\sum}_{{c}_{i}\in E}\mu \left({c}_{i}\right),\phantom{\rule{4pt}{0ex}}for\phantom{\rule{4pt}{0ex}}all\phantom{\rule{4pt}{0ex}}E\subseteq C$ holds, then$$\begin{array}{c}\hfill \mu \left({c}_{\sigma \left(j\right)}\right)=\mu \left({A}_{\sigma \left(j\right)}\right)-\mu \left({A}_{\sigma (j-1)}\right),\phantom{\rule{4pt}{0ex}}j=1,2,\dots ,n\end{array}$$In this situation, the H2TLCA aggregation operator is reduced to the hesitant 2-tuple linguistic weighted averaging (H2TLWA) aggregation operator:$$\begin{array}{c}\hfill H2TLW{A}_{\mu}({H}_{1},{H}_{2},\dots ,{H}_{n})=\bigcup _{{x}_{j}\in {H}_{j},(j=1,2,\dots ,n)}\u25b5\left(\sum _{j=1}^{n}\mu \left({c}_{j}\right)\ast {\u25b5}^{-1}\left({x}_{j}\right)\right)\end{array}$$
- For $A\in P\left(C\right)$, let ${\omega}_{j}=\mu \left({A}_{\sigma \left(j\right)}\right)-\mu \left({A}_{\sigma (j-1)}\right),\phantom{\rule{4pt}{0ex}}j=1,2,\dots ,n.$ ${\sum}_{j=1}^{n}{\omega}_{j}=1$ and $\omega ={({\omega}_{1},{\omega}_{2},\dots ,{\omega}_{n})}^{T}$; if $\mu \left(A\right)={\sum}_{j=1}^{\left|A\right|}{\omega}_{j}$, where $\left|A\right|$ is the number of the elements in A, then the H2TLCA aggregation operator is reduced to the hesitant 2-tuple linguistic ordered weighted averaging (H2TLOWA) aggregation operator:$$\begin{array}{c}\hfill H2TLOW{A}_{\mu}({H}_{1},{H}_{2},\dots ,{H}_{n})=\bigcup _{{x}_{\sigma \left(j\right)}\in {H}_{\sigma \left(j\right)},(j=1,2,\dots ,n)}\u25b5\left(\sum _{j=1}^{n}{\omega}_{j}\ast {\u25b5}^{-1}\left({x}_{\sigma \left(j\right)}\right)\right)\end{array}$$

**Theorem**

**2.**

- (1)
- (Idempotency): Let $H=\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$ be n H2TLSs; if all ${H}_{j}\phantom{\rule{4pt}{0ex}}(j=1,2,\dots ,n)$ are equal with ${H}_{j}=({x}_{1},{x}_{2},\dots ,{x}_{n})$, then$$\begin{array}{c}\hfill H2TLCA({H}_{1},{H}_{2},\dots ,{H}_{n})=({x}_{1},{x}_{2},\dots ,{x}_{n})\end{array}$$
- (2)
- (Boundedness): Let $H=\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$ be n H2TLSs; then$$\begin{array}{c}\hfill \underset{1\le j\le n}{\mathrm{min}}\left\{{H}_{j}\right\}\le H2TLCA({H}_{1},{H}_{2},\dots ,{H}_{n})\le \underset{1\le j\le n}{\mathrm{max}}\left\{{H}_{j}\right\}\end{array}$$
- (3)
- (Monotonicity): Let $H=\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$ and ${H}^{\prime}=\{{H}_{{1}^{\prime}},{H}_{{2}^{\prime}},\dots ,{H}_{{n}^{\prime}}\}$ be two collections of H2TLSs; if ${H}_{j}\le {H}_{{j}^{\prime}}\phantom{\rule{4pt}{0ex}}(j=1,2,\dots ,n)$ for all j, then$$\begin{array}{c}\hfill H2TLCA({H}_{1},{H}_{2},\dots ,{H}_{n})\le H2TLCA({H}_{{1}^{\prime}},{H}_{{2}^{\prime}},\dots ,{H}_{{n}^{\prime}})\end{array}$$
- (4)
- (Commutativity): If $\{{H}_{{1}^{\prime}},{H}_{{2}^{\prime}},\dots ,{H}_{{n}^{\prime}}\}$ is a permutation of $\{{H}_{1},{H}_{2},\dots ,{H}_{n}\}$, then$$\begin{array}{c}\hfill H2TLCA({H}_{1},{H}_{2},\dots ,{H}_{n})=H2TLCA({H}_{{1}^{\prime}},{H}_{{2}^{\prime}},\dots ,{H}_{{n}^{\prime}})\end{array}$$

**Definition**

**14.**

## 4. An Approach to Multi-Criteria Decision-Making with Hesitant 2-Tuple Linguistic Information

**Step 1**: Obtain the decision information matrices and transform the linguistic expression into H2TLSs:

**Step 2**: Apply Equation (7) to calculate the value of ${T}_{ij}^{\left(k\right)}$:

**Step 3**: Utilize the H2TLPWA operator to aggregate individual values to obtain H2TLSs, which are given below:

**R**is defined as $\mathbf{R}={\left({r}_{ij}\right)}_{m\times n}$.

**Step 4**: Utilize the H2TLCA aggregation operator to aggregate all the criteria of the alternative and obtain the overall values of alternatives:

**Step 5**: Sort the overall preference values in descending order by using Equations (5)–(7), and the best value can be identified.

## 5. An Illustrative Example

**Step 1**: Establish the decision matrices ${\mathbf{R}}^{\left(k\right)}={({r}_{ij}^{\left(k\right)})}_{(4\times 3)}\phantom{\rule{4pt}{0ex}}(k=1,2,3)$ and transform the linguistic expression into H2TLSs, which are shown in Table 4, Table 5 and Table 6, respectively.

**Step2**: Apply Equation (7) to calculate the values of ${T}_{ij}^{\left(k\right)}\phantom{\rule{4pt}{0ex}}(k=1,2,3)$, which are given below:

**Step 3**: Utilize the H2TLPWA aggregation operator to aggregate individual values, which are given in Table 7.

**Step 4**: Calculate fuzzy measures of criteria ${c}_{1}$, ${c}_{2}$ and ${c}_{3}$ and their $\alpha $ parameter. We suppose that $\mu \left({c}_{1}\right)=0.3$, $\mu \left({c}_{2}\right)=0.25$ and $\mu \left({c}_{3}\right)=0.37$. Then $\alpha =0.2795$ is determined by using Equation (5). According to the Equation (4) fuzzy measures of criteria sets of $C=\{{c}_{1},{c}_{2},{c}_{3}\}$, we can obtain $\mu ({c}_{1},{c}_{2})=0.5710$, $\mu ({c}_{1},{c}_{3})=0.7010$, $\mu ({c}_{2},{c}_{3})=0.6459$, and $\mu ({c}_{1},{c}_{2},{c}_{3})=1$.

**a.**According to the score function, we can obtain the following:

**b.**For the alternative ${D}_{2}$, the following information can be obtained:

**c.**For the alternative ${D}_{3}$, the following results are obtained:

**d.**For the alternative ${D}_{4}$, the following results are obtained:

**Step 5**: According to Definition 5, we can obtain the mean value of each alternative:

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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

${D}_{1}$ | Between good and very good | Medium | Good |

${D}_{2}$ | bad | Between very good and extremely good | Medium |

${D}_{3}$ | Very bad | Very good | Between medium and good |

${D}_{4}$ | Between very good and extremely good | Good | Medium |

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

${D}_{1}$ | Very good | Bad | Between medium and good |

${D}_{2}$ | Between medium and good | Between very good and very good | Bad |

${D}_{3}$ | Bad | Between medium and good | Between medium and good |

${D}_{4}$ | Very good | Between very good and extremely good | Bad |

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

${D}_{1}$ | Extremely good | Medium | Between medium and good |

${D}_{2}$ | Medium | Extremely good | Medium |

${D}_{3}$ | Medium | Very good | Between medium and good |

${D}_{4}$ | Between very good and extremely good | Very good | Medium |

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

${D}_{1}$ | $\{({s}_{4},0),({s}_{5},0)\}$ | $\left\{({s}_{3},0)\right\}$ | $\left\{({s}_{4},0)\right\}$ |

${D}_{2}$ | $\left\{({s}_{2},0)\right\}$ | $\{({s}_{5},0),({s}_{6},0)\}$ | $\left\{({s}_{3},0)\right\}$ |

${D}_{3}$ | $\left\{({s}_{1},0)\right\}$ | $\left\{({s}_{5},0)\right\}$ | $\{({s}_{3},0),({s}_{4},0)\}$ |

${D}_{4}$ | $\{({s}_{5},0),({s}_{6},0)\}$ | $\left\{({s}_{4},0)\right\}$ | $\left\{({s}_{3},0)\right\}$ |

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

${D}_{1}$ | $\left\{({s}_{5},0)\right\}$ | $\left\{({s}_{2},0)\right\}$ | $\{({s}_{3},0),({s}_{4},0)\}$ |

${D}_{2}$ | $\{({s}_{3},0),({s}_{4},0)\}$ | $\{({s}_{4},0),({s}_{5},0)\}$ | $\left\{({s}_{2},0)\right\}$ |

${D}_{3}$ | $\left\{({s}_{2},0)\right\}$ | $\{({s}_{3},0),({s}_{4},0)\}$ | $\{({s}_{3},0),({s}_{4},0)\}$ |

${D}_{4}$ | $\left\{({s}_{5},0)\right\}$ | $\{({s}_{5},0),({s}_{6},0)\}$ | $\left\{({s}_{2},0)\right\}$ |

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

${D}_{1}$ | $\left\{({s}_{6},0)\right\}$ | $\left\{({s}_{3},0)\right\}$ | $\{({s}_{3},0),({s}_{4},0)\}$ |

${D}_{2}$ | $\left\{({s}_{3},0)\right\}$ | $\left\{({s}_{6},0)\right\}$ | $\left\{({s}_{3},0)\right\}$ |

${D}_{3}$ | $\left\{({s}_{3},0)\right\}$ | $\left\{({s}_{5},0)\right\}$ | $\{({s}_{3},0),({s}_{4},0)\}$ |

${D}_{4}$ | $\{({s}_{5},0),({s}_{6},0)\}$ | $\left\{({s}_{5},0)\right\}$ | $\left\{({s}_{3},0)\right\}$ |

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

${D}_{1}$ | $\{({s}_{6},-0.2321),({s}_{6},-0.1964)\}$ | $\left\{({s}_{3},-0.3)\right\}$ | $\{({s}_{3},0.0523),({s}_{3},0.2628)$, |

$({s}_{4},-0.2109),({s}_{4},0)\}$ | |||

${D}_{2}$ | $\{({s}_{3},-0.1),({s}_{3},0.1)\}$ | $\{({s}_{6},-0.384),({s}_{6},-0.352)$, | $\left\{({s}_{3},-0.3)\right\}$ |

$({s}_{6},-0.208),({s}_{6},-0.176)\}$ | |||

${D}_{3}$ | $\left\{({s}_{2},0.25)\right\}$ | $\{({s}_{5},-0.4251),({s}_{5},-0.2123)\}$ | $\{({s}_{3},0),({s}_{3},0.0597)$, |

$({s}_{3},0.209),({s}_{3},0.2687)$, | |||

$({s}_{3},0.403),({s}_{4},-0.2687)$, | |||

$({s}_{4},-0.209),({s}_{4},0)\}$ | |||

${D}_{4}$ | $\{({s}_{5},0),({s}_{5},0.0294),$ | $\{({s}_{5},-0.0375),({s}_{5},0.1106)\}$ | $\left\{({s}_{3},-0.3)\right\}$ |

$({s}_{6},-0.1912),({s}_{6},-0.1618)\}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, L.; Wang, Y.; Liu, X.
Prioritized Aggregation Operators and Correlated Aggregation Operators for Hesitant 2-Tuple Linguistic Variables. *Symmetry* **2018**, *10*, 39.
https://doi.org/10.3390/sym10020039

**AMA Style**

Wang L, Wang Y, Liu X.
Prioritized Aggregation Operators and Correlated Aggregation Operators for Hesitant 2-Tuple Linguistic Variables. *Symmetry*. 2018; 10(2):39.
https://doi.org/10.3390/sym10020039

**Chicago/Turabian Style**

Wang, Lidong, Yanjun Wang, and Xiaodong Liu.
2018. "Prioritized Aggregation Operators and Correlated Aggregation Operators for Hesitant 2-Tuple Linguistic Variables" *Symmetry* 10, no. 2: 39.
https://doi.org/10.3390/sym10020039