# 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)
- ${L}_{1}(p)\oplus {L}_{2}(p)={\bigcup}_{{L}_{1}^{(k)}\in {L}_{1}(p),{L}_{2}^{(k)}\in {L}_{2}(p)}\left\{{p}_{1}^{(k)}{L}_{1}^{(k)}\oplus {p}_{2}^{(k)}{L}_{2}^{(k)}\right\}$;
- (2)
- ${L}_{1}(p)\otimes {L}_{2}(p)={\bigcup}_{{L}_{1}^{(k)}\in {L}_{1}(p),{L}_{2}^{(k)}\in {L}_{2}(p)}\left\{{({L}_{1}^{(k)})}^{{p}_{1}^{(k)}}\otimes {({L}_{2}^{(k)})}^{{p}_{2}^{(k)}}\right\}$;
- (3)
- $\lambda (L(p))={\bigcup}_{{L}^{(k)}\in L(p)}\left\{\lambda {p}^{(k)}{L}^{(k)}\right\}$ 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\mathit{x}}_{\mathbf{1}}$ | $\left\{{s}_{8}(0.25),{s}_{6}(0.5),{s}_{7}(0.25)\right\}$ | $\left\{{s}_{6}(0.25),{s}_{8}(0.5),{s}_{7}(0.25)\right\}$ | $\left\{{s}_{6}(0.75),{s}_{5}(0.25)\right\}$ |

${\mathit{x}}_{\mathbf{2}}$ | $\left\{{s}_{6}(0.25),{s}_{5}(0.25),{s}_{4}(0.25),{s}_{8}(0.25)\right\}$ | $\left\{{s}_{7}(0.5),{s}_{6}(0.25),{s}_{5}(0.25)\right\}$ | $\left\{{s}_{7}(0.75),{s}_{6}(0.25)\right\}$ |

${\mathit{x}}_{\mathbf{3}}$ | $\left\{{s}_{5}(0.25),{s}_{7}(0.5),{s}_{8}(0.25)\right\}$ | $\left\{{s}_{8}(0.25),{s}_{6}(0.5),{s}_{7}(0.25)\right\}$ | $\left\{{s}_{7}(0.5),{s}_{6}(0.25),{s}_{8}(0.25)\right\}$ |

${\mathit{x}}_{\mathbf{4}}$ | $\left\{{s}_{7}(0.5),{s}_{8}(0.25),{s}_{5}(0.25)\right\}$ | $\left\{{s}_{4}(0.25),{s}_{6}(0.25),{s}_{5}(0.25),{s}_{7}(0.25)\right\}$ | $\left\{{s}_{6}(0.5),{s}_{7}(0.25),{s}_{8}(0.25)\right\}$ |

${\mathit{x}}_{\mathbf{5}}$ | $\left\{{s}_{8}(0.5),{s}_{6}(0.25),{s}_{5}(0.25)\right\}$ | $\left\{{s}_{6}(0.5),{s}_{7}(0.5)\right\}$ | $\left\{{s}_{7}(0.25),{s}_{6}(0.5),{s}_{5}(0.25)\right\}$ |

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

${\mathit{x}}_{\mathbf{1}}$ | $\left\{{s}_{6}(0.5),{s}_{8}(0.25),{s}_{7}(0.25),{s}_{6}(0)\right\}$ | $\left\{{s}_{8}(0.5),{s}_{7}(0.25),{s}_{6}(0.25),{s}_{6}(0)\right\}$ | $\left\{{s}_{6}(0.75),{s}_{5}(0.25),{s}_{5}(0),{s}_{5}(0)\right\}$ |

${\mathit{x}}_{\mathbf{2}}$ | $\left\{{s}_{8}(0.25),{s}_{6}(0.25),{s}_{5}(0.25),{s}_{4}(0.25)\right\}$ | $\left\{{s}_{7}(0.5),{s}_{6}(0.25),{s}_{5}(0.25),{s}_{5}(0)\right\}$ | $\left\{{s}_{7}(0.75),{s}_{6}(0.25),{s}_{6}(0),{s}_{6}(0)\right\}$ |

${\mathit{x}}_{\mathbf{3}}$ | $\left\{{s}_{7}(0.5),{s}_{8}(0.25),{s}_{5}(0.25),{s}_{5}(0)\right\}$ | $\left\{{s}_{6}(0.5),{s}_{8}(0.25),{s}_{7}(0.25),{s}_{6}(0)\right\}$ | $\left\{{s}_{7}(0.5),{s}_{8}(0.25),{s}_{6}(0.25),{s}_{6}(0)\right\}$ |

${\mathit{x}}_{\mathbf{4}}$ | $\left\{{s}_{7}(0.5),{s}_{8}(0.25),{s}_{5}(0.25),{s}_{5}(0)\right\}$ | $\left\{{s}_{7}(0.25),{s}_{6}(0.25),{s}_{5}(0.25),{s}_{4}(0.25)\right\}$ | $\left\{{s}_{6}(0.5),{s}_{8}(0.25),{s}_{7}(0.25),{s}_{6}(0)\right\}$ |

${\mathit{x}}_{\mathbf{5}}$ | $\left\{{s}_{8}(0.5),{s}_{6}(0.25),{s}_{5}(0.25),{s}_{5}(0)\right\}$ | $\left\{{s}_{7}(0.5),{s}_{6}(0.5),{s}_{6}(0),{s}_{6}(0)\right\}$ | $\left\{{s}_{6}(0.5),{s}_{7}(0.25),{s}_{5}(0.25),{s}_{5}(0)\right\}$ |

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