# Probabilistic Linguistic Power Aggregation Operators for Multi-Criteria Group Decision Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Probabilistic Linguistic Term Sets (PLTSs)

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (1)
- ${L}_{1}(p)\oplus {L}_{2}(p)={\bigcup}_{{L}_{1}^{(k)}\in {L}_{1}(p),{L}_{2}^{(k)}\in {L}_{2}(p)}\left(\right)open="\{"\; close="\}">{p}_{1}^{(k)}{L}_{1}^{(k)}\oplus {p}_{2}^{(k)}{L}_{2}^{(k)}$;
- (2)
- ${L}_{1}(p)\otimes {L}_{2}(p)={\bigcup}_{{L}_{1}^{(k)}\in {L}_{1}(p),{L}_{2}^{(k)}\in {L}_{2}(p)}\left(\right)open="\{"\; close="\}">{({L}_{1}^{(k)})}^{{p}_{1}^{(k)}}\otimes {({L}_{2}^{(k)})}^{{p}_{2}^{(k)}}$;
- (3)
- $\lambda (L(p))={\bigcup}_{{L}^{(k)}\in L(p)}\left(\right)open="\{"\; close="\}">\lambda {p}^{(k)}{L}^{(k)}$ and $\lambda \ge 0$;
- (4)
- ${(L(p))}^{\lambda}={\bigcup}_{{L}^{(k)}\in L(p)}\left\{{({L}^{(k)})}^{\lambda {p}^{(k)}}\right\}$ and $\lambda \ge 0$.

**Definition**

**4.**

**Definition**

**5.**

- (1)
- If $E({L}_{1}(p))>E({L}_{2}(p))$, then ${L}_{1}(p)$ is bigger than ${L}_{2}(p)$, denoted by ${L}_{1}(p)>{L}_{2}(p)$;
- (2)
- If $E({L}_{1}(p))<E({L}_{2}(p))$, then ${L}_{1}(p)$ is smaller than ${L}_{2}(p)$, denoted by ${L}_{1}(p)<{L}_{2}(p)$;
- (3)
- If $E({L}_{1}(p))=E({L}_{2}(p))$, then we need to compare their deviation degrees:
- (a)
- If $\sigma ({L}_{1}(p))=\sigma ({L}_{2}(p))$, then ${L}_{1}(p)$ is equal to ${L}_{2}(p)$, denoted by ${L}_{1}(p)\sim {L}_{2}(p)$;
- (b)
- If $\sigma ({L}_{1}(p))>\sigma ({L}_{2}(p))$, then ${L}_{1}(p)$ is smaller than ${L}_{2}(p)$, denoted by ${L}_{1}(p)<{L}_{2}(p)$;
- (c)
- If $\sigma ({L}_{1}(p))<\sigma ({L}_{2}(p))$, then ${L}_{1}(p)$ is bigger than ${L}_{2}(p)$, denoted by ${L}_{1}(p)>{L}_{2}(p)$.

**Definition**

**6.**

**Definition**

**7.**

#### 2.2. Power Average (PA)

**Definition**

**8.**

- (1)
- $sup({a}_{i},{a}_{j})\in [0,1]$;
- (2)
- $sup({a}_{i},{a}_{j})=sup({a}_{j},{a}_{i})$;
- (3)
- $sup({a}_{i},{a}_{j})\ge sup({a}_{i},{a}_{k})$, if $|{a}_{i}-{a}_{j}|<|{a}_{i}-{a}_{k}|$.

## 3. Probabilistic Linguistic Power Aggregation Operators

#### 3.1. Probabilistic Linguistic Power Average (PLPA) Aggregation Operators

#### 3.1.1. PLPA

**Definition**

**9.**

- (1)
- $sup({L}_{i}(p),{L}_{j}(p))\in [0,1]$;
- (2)
- $sup({L}_{i}(p),{L}_{j}(p))=sup({L}_{j}(p),{L}_{i}(p))$;
- (3)
- $sup({L}_{i}(p),{L}_{j}(p))\ge sup({L}_{i}(p),{L}_{k}(p))$ if $d({L}_{i}(p),{L}_{j}(p))<d({L}_{i}(p),{L}_{k}(p))$.

**Proposition**

**1.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

#### 3.1.2. WPLPA

**Definition**

**10.**

**Proposition**

**2.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Theorem**

**8.**

**Proof.**

#### 3.2. Probabilistic Linguistic Power Geometric (PLPG) Aggregation Operators

#### 3.2.1. PLPG

**Definition**

**11.**

- (1)
- $sup({L}_{i}(p),{L}_{j}(p))\in [0,1]$;
- (2)
- $sup({L}_{i}(p),{L}_{j}(p))=sup({L}_{j}(p),{L}_{i}(p))$;
- (3)
- $sup({L}_{i}(p),{L}_{j}(p))\ge sup({L}_{j}(p),{L}_{i}(p))$ if $d({L}_{i}(p),{L}_{j}(p))<d({L}_{i}(p),{L}_{k}(p))$.

**Proposition**

**3.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

#### 3.2.2. WPLPG

**Definition**

**12.**

**Proposition**

**4.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Theorem**

**14.**

**Proof.**

## 4. Approaches to Multi-Criteria Group Decision Making with Probabilistic Linguistic Power Aggregation Operators

## 5. An Illustrative Example

#### 5.1. Decision Analysis with Our Proposed Approaches

#### 5.2. Comparison Analysis

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Yager, R.R. The power average operator. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2001**, 31, 724–731. [Google Scholar] [CrossRef] - Xu, Z.S.; Yager, R.R. Power-Geometric operators and their use in group decision making. IEEE Trans. Fuzzy Syst.
**2010**, 18, 94–105. [Google Scholar] - Xu, Y.J.; Merigó, J.M.; Wang, H.M. Linguistic power aggregation operators and their application to multiple attribute group decision making. Appl. Math. Model.
**2012**, 36, 5427–5444. [Google Scholar] [CrossRef] - Zhou, L.G.; Chen, H.Y. A generalization of the power aggregation operators for linguistic environment and its application in group decision making. Knowl. Based Syst.
**2012**, 26, 216–224. [Google Scholar] [CrossRef] - Zhu, C.; Zhu, L.; Zhang, X. Linguistic hesitant fuzzy power aggregation operators and their applications in multiple attribute decision-making. Inf. Sci.
**2016**, 367–368, 809–826. [Google Scholar] [CrossRef] - Pang, Q.; Wang, H.; Xu, Z.S. Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci.
**2016**, 369, 128–143. [Google Scholar] [CrossRef] - Bai, C.Z.; Zhang, R.; Qian, L.X.; Wu, Y.N. Comparisons of probabilistic linguistic term sets for multi-criteria decision making. Knowl. Based Syst.
**2017**, 119, 284–291. [Google Scholar] [CrossRef] - Merigó, J.M.; Casanovas, M.; Martínez, L. Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of evidence. Int. J. Uncertain. Fuzziness Knowl. Based Syst.
**2010**, 18, 287–304. [Google Scholar] [CrossRef] - Zhai, Y.L.; Xu, Z.S.; Liao, H.C. Probabilistic linguistic vector-term set and its application in group decision making with multi-granular linguistic information. Appl. Soft Comput.
**2016**, 49, 801–816. [Google Scholar] [CrossRef] - Liao, H.C.; Xu, Z.S.; Zeng, X.J.; Merigó, J.M. Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets. Knowl. Based Syst.
**2015**, 76, 127–138. [Google Scholar] [CrossRef] - Liao, H.C.; Xu, Z.S.; Zeng, X.J. Hesitant fuzzy linguistic vikor method and its application in qualitative multiple criteria decision making. IEEE Trans. Fuzzy Syst.
**2015**, 23, 1343–1355. [Google Scholar] [CrossRef] - Rodriguez, R.M.; Martinez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Liang, D.C.; Liu, D. A novel risk decision making based on decision-theoretic rough sets under hesitant fuzzy information. IEEE Trans. Fuzzy Syst.
**2015**, 23, 237–247. [Google Scholar] [CrossRef] - Gou, X.J.; Xu, Z.S. Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets. Inf. Sci.
**2016**, 372, 407–427. [Google Scholar] [CrossRef] - He, Y.; Xu, Z.S.; Jiang, W.L. Probabilistic interval reference ordering sets in multi-criteria group decision making. Int. J. Uncertain. Fuzziness Knowl. Based Syst.
**2017**, 25, 189–212. [Google Scholar] [CrossRef] - Wu, Z.B.; Xu, J.C. Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information. IEEE Trans. Cybern.
**2016**, 46, 694–705. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.X.; Xu, Z.S.; Wang, H.; Liao, H.C. Consistency-based risk assessment with probabilistic linguistic preference relation. Appl. Soft Comput.
**2016**, 49, 817–833. [Google Scholar] [CrossRef] - Zhou, W.; Xu, Z.S. Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim. Decis. Mak.
**2016**. [Google Scholar] [CrossRef] - Katz, D.A. Faculty salaries, promotions and productivity at a large University. Am. Econ. Rev.
**1973**, 63, 469–477. [Google Scholar] - Bryson, N.; Mobolurin, A. An action learning evaluation procedure for multiple criteria decision making problems. Eur. J. Oper. Res.
**1995**, 96, 379–386. [Google Scholar] [CrossRef] - Merigó, J.M.; Gil-Lafuente, A.M.; Zhou, L.G.; Chen, H.Y. Induced and linguistic generalized aggregation operators and their application in linguistic group decision making. Group Decis. Negot.
**2012**, 21, 531–549. [Google Scholar] [CrossRef] - Zhang, Z.M.; Wu, C. Hesitant fuzzy linguistic aggregation operators and their applications to multiple attribute group decision making. J. Intell. Fuzzy Syst.
**2014**, 26, 2185–2202. [Google Scholar]

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

${\mathit{x}}_{\mathbf{1}}$ | ${s}_{8}$ | ${s}_{6}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{2}}$ | ${s}_{6}$ | ${s}_{7}$ | ${s}_{7}$ |

${\mathit{x}}_{\mathbf{3}}$ | ${s}_{5}$ | ${s}_{8}$ | ${s}_{7}$ |

${\mathit{x}}_{\mathbf{4}}$ | ${s}_{7}$ | ${s}_{4}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{5}}$ | ${s}_{8}$ | ${s}_{6}$ | ${s}_{7}$ |

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

${\mathit{x}}_{\mathbf{1}}$ | ${s}_{6}$ | ${s}_{8}$ | ${s}_{5}$ |

${\mathit{x}}_{\mathbf{2}}$ | ${s}_{5}$ | ${s}_{6}$ | ${s}_{7}$ |

${\mathit{x}}_{\mathbf{3}}$ | ${s}_{7}$ | ${s}_{6}$ | ${s}_{7}$ |

${\mathit{x}}_{\mathbf{4}}$ | ${s}_{8}$ | ${s}_{6}$ | ${s}_{7}$ |

${\mathit{x}}_{\mathbf{5}}$ | ${s}_{8}$ | ${s}_{7}$ | ${s}_{6}$ |

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

${\mathit{x}}_{\mathbf{1}}$ | ${s}_{7}$ | ${s}_{8}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{2}}$ | ${s}_{4}$ | ${s}_{5}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{3}}$ | ${s}_{8}$ | ${s}_{7}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{4}}$ | ${s}_{7}$ | ${s}_{5}$ | ${s}_{8}$ |

${\mathit{x}}_{\mathbf{5}}$ | ${s}_{6}$ | ${s}_{7}$ | ${s}_{6}$ |

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

${\mathit{x}}_{\mathbf{1}}$ | ${s}_{6}$ | ${s}_{7}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{2}}$ | ${s}_{8}$ | ${s}_{7}$ | ${s}_{7}$ |

${\mathit{x}}_{\mathbf{3}}$ | ${s}_{7}$ | ${s}_{6}$ | ${s}_{8}$ |

${\mathit{x}}_{\mathbf{4}}$ | ${s}_{5}$ | ${s}_{7}$ | ${s}_{6}$ |

${\mathit{x}}_{\mathbf{5}}$ | ${s}_{5}$ | ${s}_{6}$ | ${s}_{5}$ |

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

${\mathit{x}}_{\mathbf{1}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{2}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{3}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{4}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{5}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

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

${\mathit{x}}_{\mathbf{1}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{2}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{3}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{4}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

${\mathit{x}}_{\mathbf{5}}$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

Method | Rank |
---|---|

Aggregation-based method of Ref. [6] | ${x}_{3}>{x}_{1}>{x}_{5}>{x}_{2}>{x}_{4}$ |

The method with HFLWA of Ref. [23] | ${x}_{3}>{x}_{1}>{x}_{5}>{x}_{2}={x}_{4}$ |

The method with HFLWG of Ref. [23] | ${x}_{1}>{x}_{2}>{x}_{5}>{x}_{3}>{x}_{4}$ |

Max lower operator of Ref. [12] | ${x}_{3}>{x}_{2}={x}_{5}={x}_{4}>{x}_{1}$ |

ILGCIA with group decision making of Ref. [22] | ${x}_{3}>{x}_{2}>{x}_{1}>{x}_{4}>{x}_{5}$ |

Our proposed method with WPLPA | ${x}_{3}>{x}_{1}>{x}_{5}>{x}_{4}>{x}_{2}$ |

Our proposed method with WPLPG | ${x}_{1}>{x}_{3}>{x}_{5}>{x}_{2}>{x}_{4}$ |

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**MDPI and ACS Style**

Kobina, A.; Liang, D.; He, X.
Probabilistic Linguistic Power Aggregation Operators for Multi-Criteria Group Decision Making. *Symmetry* **2017**, *9*, 320.
https://doi.org/10.3390/sym9120320

**AMA Style**

Kobina A, Liang D, He X.
Probabilistic Linguistic Power Aggregation Operators for Multi-Criteria Group Decision Making. *Symmetry*. 2017; 9(12):320.
https://doi.org/10.3390/sym9120320

**Chicago/Turabian Style**

Kobina, Agbodah, Decui Liang, and Xin He.
2017. "Probabilistic Linguistic Power Aggregation Operators for Multi-Criteria Group Decision Making" *Symmetry* 9, no. 12: 320.
https://doi.org/10.3390/sym9120320