# Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{T}, l

_{I}, l

_{F}>, but the LIFN and SVNLN cannot express such linguistic evaluation value. In [21], Fang and Ye also presented a LNN-weighted arithmetic averaging (LNNWAA) operator and a LNN-weighted geometric averaging (LNNWGA) operator for MAGDM. However, the Bonferroni mean operator is not extended to LNNs so far. Hence, this paper proposes a LNN normalized weighted Bonferroni mean (LNNNWBM) operator, a LNN normalized weighted geometric Bonferroni mean (LNNNWGBM) operator and their MAGDM methods. Compared with the aggregation operators in [14,21], the LNNNWBM and LNNNWGBM operators can calculate the final weights by the relation between attribute values, which can make the information aggregation more objective and reliable.

## 2. Some Concepts of LNNs and BM

#### 2.1. Linguistic Neutrosophic Numbers and Their Operational Laws

**Definition**

**1**

**.**Set $L=\left\{{l}_{0},{l}_{1},{l}_{2},\dots ,{l}_{g}\right\}$ as a language term set, in which g is an even number and g + 1 is the particle size of L. If a = $\langle {l}_{T},{l}_{I},{l}_{F}\rangle $ is defined for ${l}_{T},{l}_{I},{l}_{F}\in L$ and $T,I,F\in $ [0, g], where ${l}_{T}$ expresses the truth degree, ${l}_{I}$ expresses indeterminacy degree, and ${l}_{F}$ expresses falsity degree by linguistic terms, then a is called an LNN.

**Definition**

**2**

**.**Set a = $\langle {l}_{T},{l}_{I},{l}_{F}\rangle $, ${a}_{1}$ = $\langle {l}_{{T}_{1}},{l}_{{I}_{1}},{l}_{{F}_{1}}\rangle $, and ${a}_{2}$ = $\langle {l}_{{T}_{2}},{l}_{{I}_{2}},{l}_{{F}_{2}}\rangle $ as three LNNs in L, the number $\lambda \ge 0$, they have the follow operational laws:

**Definition**

**3**

**.**Set a = $\langle {l}_{T},{l}_{I},{l}_{F}\rangle $ as an LNN in L, then the expectation E(a) and the accuracy H(a) can be defined as follows:

**Definition**

**4**

**.**Set ${a}_{1}$ = $\langle {l}_{{T}_{1}},{l}_{{I}_{1}},{l}_{{F}_{1}}\rangle $ and ${a}_{2}$ = $\langle {l}_{{T}_{2}},{l}_{{I}_{2}},{l}_{{F}_{2}}\rangle $ as two LNNs, then:

- If E(${a}_{1}$) > E(${a}_{2}$), then ${a}_{1}\succ {a}_{2}$;
- If E(${a}_{1}$) = E(${a}_{2}$) then
- If H(${a}_{1}$) > H(${a}_{2}$), then ${a}_{1}\succ {a}_{2}$;
- If H(${a}_{1}$) = H(${a}_{2}$), then ${a}_{1}\sim {a}_{2}$;
- If H(${a}_{1}$) < H(${a}_{2}$), then ${a}_{1}\prec {a}_{2}$.

#### 2.2. Bonferroni Mean Operators

**Definition**

**5**

**.**Let $({a}_{1},{a}_{2},\dots ,{a}_{n})$ be a set of non-negative numbers, the function BM: R

^{n}→R. If p, q ≥ 0 and BM satisfies:

^{p,q}is called a BM operator.

**Definition**

**6**

**.**Let $({a}_{1},{a}_{2},\dots ,{a}_{n})$ be a set of non-negative numbers, the function NWBM: R

^{n}→R, ${w}_{i}$ (i = 1, 2, …, n) be the relative weight of ${a}_{i}$ (i = 1, 2, …, n), ${w}_{i}\in \left[0,1\right],$ and $\sum}_{i=1}^{n}{w}_{i}=1$. If p, q ≥ 0 and NWBM satisfies:

^{p,q}is called a normalized weighted BM operator.

**Definition**

**7**

**.**Let $({a}_{1},{a}_{2},\dots ,{a}_{n})$ be a set of non-negative numbers, the function GBM: R

^{n}→R. If p, q ≥ 0 and GBM satisfies:

^{p,q}is called a geometric BM operator.

**Definition**

**8**

**.**Let (${a}_{1},{a}_{2},\dots {a}_{n})$ be a set of non-negative numbers, the function NWGBM: R

^{n}→R, ${w}_{i}$ (i = 1, 2, …, n) be the relative weight of ${a}_{i}$ (i = 1, 2, …, n), ${w}_{i}\in \left[0,1\right],$ and $\sum}_{i=1}^{n}{w}_{i}=1$. If p, q ≥ 0 and NWGBM satisfies:

^{p,q}is called a normalized weighted geometric BM (NWGBM) operator.

## 3. Two BM Aggregation Operators of LNNs

#### 3.1. Normalized Weighted BM Operators of LNNs

**Definition**

**9.**

**Theorem**

**1.**

**Proof**

**1.**

- (1)
- ${{a}_{i}}^{p}=\langle {l}_{g{\left(\frac{{T}_{i}}{g}\right)}^{p}},{l}_{g-g{\left(1-\frac{{I}_{i}}{g}\right)}^{p}},{l}_{g-g{\left(1-\frac{{F}_{i}}{g}\right)}^{p}}\rangle $;
- (2)
- ${{a}_{j}}^{q}=\langle {l}_{g{\left(\frac{{T}_{j}}{g}\right)}^{q}},{l}_{g-g{\left(1-\frac{{I}_{j}}{g}\right)}^{q}},{l}_{g-g{\left(1-\frac{{F}_{j}}{g}\right)}^{q}}\rangle $;
- (3)
- $\begin{array}{l}{{a}_{i}}^{p}\otimes {{a}_{j}}^{q}\\ =\langle {l}_{\frac{g{\left(\frac{{T}_{i}}{g}\right)}^{p}g{\left(\frac{{T}_{j}}{g}\right)}^{q}}{g}},{l}_{g-g{(1-\frac{{I}_{i}}{g})}^{p}+g-g{(1-\frac{{I}_{j}}{g})}^{q}-\frac{(g-g{(1-\frac{{I}_{i}}{g})}^{p})(g-g{(1-\frac{{I}_{j}}{g})}^{q})}{g}},{l}_{g-g{(1-\frac{{F}_{i}}{g})}^{p}+g-g{(1-\frac{{F}_{j}}{g})}^{q}-\frac{(g-g{(1-\frac{{F}_{i}}{g})}^{p})(g-g{(1-\frac{{F}_{j}}{g})}^{q})}{g}}\rangle \\ =\langle {l}_{g{\left(\frac{{T}_{i}}{g}\right)}^{p}{\left(\frac{{T}_{j}}{g}\right)}^{q}},{l}_{g-g{\left(1-\frac{{I}_{i}}{g}\right)}^{p}{\left(1-\frac{{I}_{j}}{g}\right)}^{q}},{l}_{g-g{\left(1-\frac{{F}_{i}}{g}\right)}^{p}{\left(1-\frac{{F}_{j}}{g}\right)}^{q}}\rangle \end{array}$
- (4)
- $\begin{array}{l}\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}{{a}_{i}}^{p}\otimes {{a}_{j}}^{q}=\langle {l}_{g-g{(1-\frac{g{\left(\frac{{T}_{i}}{g}\right)}^{p}{\left(\frac{{T}_{j}}{g}\right)}^{q}}{g})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}},{l}_{g{\left(\frac{g-g{(1-\frac{{I}_{i}}{g})}^{p}{(1-\frac{{I}_{j}}{g})}^{q}}{g}\right)}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}},{l}_{g{\left(\frac{g-g{(1-\frac{{F}_{i}}{g})}^{p}{(1-\frac{{F}_{j}}{g})}^{q}}{g}\right)}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}\rangle \\ =\langle {l}_{g-g{\left(1-{\left(\frac{{T}_{i}}{g}\right)}^{p}{\left(\frac{{T}_{j}}{g}\right)}^{q}\right)}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}},{l}_{g{\left(1-{\left(1-\frac{{I}_{i}}{g}\right)}^{p}{\left(1-\frac{{I}_{j}}{g}\right)}^{q}\right)}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}},{l}_{g{\left(1-{\left(1-\frac{{F}_{i}}{g}\right)}^{p}{\left(1-\frac{{F}_{j}}{g}\right)}^{q}\right)}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}\rangle \end{array}$
- (5)
- $\begin{array}{l}{\oplus}_{i=1}^{n}{\oplus}_{j=1,j\ne i}^{n}\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}{{a}_{i}}^{p}\otimes {{a}_{j}}^{q}\\ =\langle {l}_{g-g{\displaystyle {\prod}_{i=1}^{n}{\displaystyle {\prod}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}{(1-{\left(\frac{{T}_{i}}{g}\right)}^{p}{\left(\frac{{T}_{j}}{g}\right)}^{q})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}}},{l}_{g{\displaystyle {\prod}_{i=1}^{n}{\displaystyle {\prod}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}{(1-{(1-\frac{{I}_{i}}{g})}^{p}{(1-\frac{{I}_{j}}{g})}^{q})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}}},{l}_{g{\displaystyle {\prod}_{i=1}^{n}{\displaystyle {\prod}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}{(1-{(1-\frac{{F}_{i}}{g})}^{p}{(1-\frac{{F}_{j}}{g})}^{q})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}}}\rangle \end{array}$
- (6)
- $\begin{array}{l}{({\oplus}_{i=1}^{n}{\oplus}_{j=1,j\ne i}^{n}\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}{{a}_{i}}^{p}\otimes {{a}_{j}}^{q})}^{\frac{1}{p+q}}\\ =\langle {l}_{g{(1-{\displaystyle {\prod}_{i=1}^{n}{\displaystyle {\prod}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}{(1-{\left(\frac{{T}_{i}}{g}\right)}^{p}{\left(\frac{{T}_{j}}{g}\right)}^{q})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}})}^{\frac{1}{p+q}}},{l}_{g-g{(1-{\displaystyle {\prod}_{i=1}^{n}{\displaystyle {\prod}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}{(1-{(1-\frac{{I}_{i}}{g})}^{p}{(1-\frac{{I}_{j}}{g})}^{q})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}})}^{\frac{1}{p+q}}},{l}_{g-g{(1-{\displaystyle {\prod}_{i=1}^{n}{\displaystyle {\prod}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}{(1-{(1-\frac{{F}_{i}}{g})}^{p}{(1-\frac{{F}_{j}}{g})}^{q})}^{\frac{{w}_{i}{w}_{j}}{1-{w}_{i}}}}})}^{{}^{\frac{1}{p+q}}}}\rangle \end{array}$

**Theorem**

**2.**

**Proof**

**2.**

_{i}= T; I

_{i}= I; F

_{i}= F for i = 1, 2, …, n, there are the following result:

**Theorem**

**3.**

_{i}= $\langle {l}_{{{T}_{i}}^{\prime}},{l}_{{{I}_{i}}^{\prime}},{l}_{{{F}_{i}}^{\prime}}\rangle $ (i = 1, 2, …, n) as two collections of LNNs in L, if ${T}_{i}\le {T}_{i}{}^{\prime},{I}_{i}\ge {I}_{i}{}^{\prime},and{F}_{i}\ge {F}_{i}{}^{\prime}$ then $LNNNWB{M}^{p,q}({a}_{1},{a}_{2},\dots ,{a}_{n})\le LNNNWB{M}^{p,q}\left({b}_{1},{b}_{2},\dots ,{b}_{n}\right).$

**Proof**

**3.**

**Theorem**

**4.**

**Proof**

**4.**

#### 3.2. Normalized Weighted Geometric BM Operators of LNNs

**Definition**

**10.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

_{i}= $\langle {l}_{{{T}_{i}}^{\prime}},{l}_{{{I}_{i}}^{\prime}},{l}_{{{F}_{i}}^{\prime}}\rangle $ (i = 1, 2, …, n) as two collections of LNNs in L, if ${T}_{i}\le {{T}_{i}}^{\prime},{I}_{i}\ge {{I}_{i}}^{\prime}and{F}_{i}\ge {{F}_{i}}^{\prime}$ then:

**Theorem**

**8.**

## 4. MAGDM Methods Based on the LNNNWBM or LNNNWGBM Operator

## 5. Illustrative Examples

_{1}), the growth (C

_{2}), and the environmental impact (C

_{3}). The importance of three experts is given as a weight vector $w={\left(0.37,0.33,0.3\right)}^{T}$ and the importance of three attributes is given as a weight vector $\lambda ={\left(0.35,0.25,0.4\right)}^{T}.$ Then, the evaluation criteria are based on the linguistic term set L = $\{{l}_{0}=\mathrm{extremely}\text{}\mathrm{bad},{l}_{1}=\mathrm{very}\text{}\mathrm{bad},{l}_{2}=\mathrm{bad},{l}_{3}=\mathrm{slightly}\text{}\mathrm{bad},\text{}{l}_{4}=\mathrm{medium},\text{}{l}_{5}=\mathrm{slightly}\text{}\mathrm{good},\text{}{l}_{6}=\text{}\mathrm{good},{l}_{7}=\text{}\mathrm{very}\text{}\mathrm{good},{l}_{8}=\text{}\mathrm{extremely}\text{}\mathrm{good}\}$. Thus, we can establish the LNN decision matrix ${R}^{i}$ (i = 1, 2, 3), which is listed in Table 2, Table 3 and Table 4.

#### 5.1. The Decision-Making Process Based on the LNNNWBM Operator or LNNNWGBM Operator

_{1}) = 0.7298, E(a

_{2}) = 0.7508, E(a

_{3}) = 0.7424, and E(a

_{4}) = 0.7864.

_{1}) = 0.7342, E(a

_{2}) = 0.7524, E(a

_{3}) = 0.7449, and E(a

_{4}) = 0.7873.

_{4}is the best choice among all the companies.

#### 5.2. Analysis the Influence of the Parameters p and q on Decision Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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${\mathit{C}}_{1}$ | $\dots $ | ${\mathit{C}}_{\mathit{n}}$ | |
---|---|---|---|

${A}_{1}$ | $\langle {l}_{{T}_{11}}^{y},{l}_{{I}_{11}}^{y},{l}_{{F}_{11}}^{y}\rangle $ | $\dots $ | $\langle {l}_{{T}_{1n}}^{y},{l}_{{I}_{1n}}^{y},{l}_{{F}_{1n}}^{y}\rangle $ |

${A}_{2}$ | $\langle {l}_{{T}_{21}}^{y},{l}_{{I}_{21}}^{y},{l}_{{F}_{21}}^{y}\rangle $ | $\dots $ | $\langle {l}_{{T}_{2n}}^{y},{l}_{{I}_{2n}}^{y},{l}_{{F}_{2n}}^{y}\rangle $ |

$\dots $ | $\dots $ | $\dots $ | $\dots $ |

${A}_{m}$ | $\langle {l}_{{T}_{m1}}^{y},{l}_{{I}_{m1}}^{y},{l}_{{F}_{m1}}^{y}\rangle $ | $\dots $ | $\langle {l}_{{T}_{mn}}^{y},{l}_{{I}_{mn}}^{y},{l}_{{F}_{mn}}^{y}\rangle $ |

C_{1} | C_{2} | C_{3} | |
---|---|---|---|

A_{1} | $\langle {l}_{6}^{1},{l}_{1}^{1},{l}_{2}^{1}\rangle $ | $\langle {l}_{7}^{1},{l}_{2}^{1},{l}_{1}^{1}\rangle $ | $\langle {l}_{6}^{1},{l}_{2}^{1},{l}_{2}^{1}\rangle $ |

A_{2} | $\langle {l}_{7}^{1},{l}_{1}^{1},{l}_{1}^{1}\rangle $ | $\langle {l}_{7}^{1},{l}_{3}^{1},{l}_{2}^{1}\rangle $ | $\langle {l}_{7}^{1},{l}_{2}^{1},{l}_{1}^{1}\rangle $ |

A_{3} | $\langle {l}_{6}^{1},{l}_{2}^{1},{l}_{2}^{1}\rangle $ | $\langle {l}_{7}^{1},{l}_{1}^{1},{l}_{1}^{1}\rangle $ | $\langle {l}_{6}^{1},{l}_{2}^{1},{l}_{2}^{1}\rangle $ |

A_{4} | $\langle {l}_{7}^{1},{l}_{1}^{1},{l}_{2}^{1}\rangle $ | $\langle {l}_{7}^{1},{l}_{2}^{1},{l}_{3}^{1}\rangle $ | $\langle {l}_{7}^{1},{l}_{2}^{1},{l}_{1}^{1}\rangle $ |

C_{1} | C_{2} | C_{3} | |
---|---|---|---|

A_{1} | $\langle {l}_{6}^{2},{l}_{1}^{2},{l}_{2}^{2}\rangle $ | $\langle {l}_{6}^{2},{l}_{1}^{2},{l}_{1}^{2}\rangle $ | $\langle {l}_{4}^{2},{l}_{2}^{2},{l}_{3}^{2}\rangle $ |

A_{2} | $\langle {l}_{7}^{2},{l}_{2}^{2},{l}_{3}^{2}\rangle $ | $\langle {l}_{6}^{2},{l}_{1}^{2},{l}_{1}^{2}\rangle $ | $\langle {l}_{4}^{2},{l}_{2}^{2},{l}_{3}^{2}\rangle $ |

A_{3} | $\langle {l}_{5}^{2},{l}_{1}^{2},{l}_{2}^{2}\rangle $ | $\langle {l}_{5}^{2},{l}_{1}^{2},{l}_{2}^{2}\rangle $ | $\langle {l}_{5}^{2},{l}_{4}^{2},{l}_{2}^{2}\rangle $ |

A_{4} | $\langle {l}_{6}^{2},{l}_{1}^{2},{l}_{1}^{2}\rangle $ | $\langle {l}_{5}^{2},{l}_{1}^{2},{l}_{1}^{2}\rangle $ | $\langle {l}_{5}^{2},{l}_{2}^{2},{l}_{3}^{2}\rangle $ |

C_{1} | C_{2} | C_{3} | |
---|---|---|---|

A_{1} | $\langle {l}_{7}^{3},{l}_{3}^{3},{l}_{4}^{3}\rangle $ | $\langle {l}_{7}^{3},{l}_{3}^{3},{l}_{3}^{3}\rangle $ | $\langle {l}_{5}^{3},{l}_{2}^{3},{l}_{5}^{3}\rangle $ |

A_{2} | $\langle {l}_{6}^{3},{l}_{3}^{3},{l}_{4}^{3}\rangle $ | $\langle {l}_{5}^{3},{l}_{1}^{3},{l}_{2}^{3}\rangle $ | $\langle {l}_{6}^{3},{l}_{2}^{3},{l}_{3}^{3}\rangle $ |

A_{3} | $\langle {l}_{7}^{3},{l}_{2}^{3},{l}_{4}^{3}\rangle $ | $\langle {l}_{6}^{3},{l}_{1}^{3},{l}_{2}^{3}\rangle $ | $\langle {l}_{7}^{3},{l}_{2}^{3},{l}_{4}^{3}\rangle $ |

A_{4} | $\langle {l}_{7}^{3},{l}_{2}^{3},{l}_{3}^{3}\rangle $ | $\langle {l}_{5}^{3},{l}_{2}^{3},{l}_{1}^{3}\rangle $ | $\langle {l}_{6}^{3},{l}_{1}^{3},{l}_{1}^{3}\rangle $ |

C_{1} | C_{2} | C_{3} | |
---|---|---|---|

${A}_{1}$ | $\langle {l}_{6.3176},{l}_{1.5682},{l}_{2.6129}\rangle $ | $\langle {l}_{6.6819},{l}_{1.9641},{l}_{1.5682}\rangle $ | $\langle {l}_{5.0059},{l}_{2.000},{l}_{3.2898}\rangle $ |

${A}_{2}$ | $\langle {l}_{6.7045},{l}_{1.9476},{l}_{2.6308}\rangle $ | $\langle {l}_{6.0524},{l}_{1.6728},{l}_{1.6636}\rangle $ | $\langle {l}_{5.7033},{l}_{2.000},{l}_{2.3074}\rangle $ |

${A}_{3}$ | $\langle {l}_{5.9943},{l}_{1.6636},{l}_{2.6129}\rangle $ | $\langle {l}_{6.0264},{l}_{1.000},{l}_{1.6430}\rangle $ | $\langle {l}_{5.9943},{l}_{2.6613},{l}_{2.6129}\rangle $ |

${A}_{4}$ | $\langle {l}_{6.6819},{l}_{1.2955},{l}_{1.9641}\rangle $ | $\langle {l}_{5.6926},{l}_{1.6636},{l}_{1.6728}\rangle $ | $\langle {l}_{6.0264},{l}_{1.6824},{l}_{1.6170}\rangle $ |

p, q | LNNNWBM Operator | Ranking |
---|---|---|

p = 1, q = 0 | E(a_{1}) = 0.7528, E(a_{2}) = 0.7777, E(a_{3}) = 0.7613, E(a_{4}) = 0.8060 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 1, q = 0.5 | E(a_{1}) = 0.7311, E(a_{2}) = 0.7534, E(a_{3}) = 0.7435, E(a_{4}) = 0.7886 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 1, q = 2 | E(a_{1}) = 0.7329, E(a_{2}) = 0.7545, E(a_{3}) = 0.7453, E(a_{4}) = 0.7897 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 0, q = 0 | E(a_{1}) = 0.7573, E(a_{2}) = 0.7766, E(a_{3}) = 0.7656, E(a_{4}) = 0.8046 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 0.5, q = 1 | E(a_{1}) = 0.7326, E(a_{2}) = 0.7530, E(a_{3}) = 0.7449, E(a_{4}) = 0.7879 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 2, q = 1 | E(a_{1}) = 0.7349, E(a_{2}) = 0.7562, E(a_{3}) = 0.7463, E(a_{4}) = 0.7902 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 2, q = 2 | E(a_{1}) = 0.7343, E(a_{2}) = 0.7537, E(a_{3}) = 0.7458, E(a_{4}) = 0.7884 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p, q | LNNNWGBM Operator | Ranking |
---|---|---|

p = 1, q = 0 | E(a_{1}) = 0.7397, E(a_{2}) = 0.7747, E(a_{3}) = 0.7531, E(a_{4}) = 0.8035 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 1, q = 0.5 | E(a_{1}) = 0.7342, E(a_{2}) = 0.7545, E(a_{3}) = 0.7453, E(a_{4}) = 0.7891 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 1, q = 2 | E(a_{1}) = 0.7343, E(a_{2}) = 0.7548, E(a_{3}) = 0.7457, E(a_{4}) = 0.7889 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 0, q = 1 | E(a_{1}) = 0.7437, E(a_{2}) = 0.7730, E(a_{3}) = 0.7570, E(a_{4}) = 0.8019 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 0.5, q = 1 | E(a_{1}) = 0.7356, E(a_{2}) = 0.7541, E(a_{3}) = 0.7467, E(a_{4}) = 0.7885 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 2, q = 1 | E(a_{1}) = 0.7330, E(a_{2}) = 0.7553, E(a_{3}) = 0.7445, E(a_{4}) = 0.7895 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

p = 2, q = 2 | E(a_{1}) = 0.7334, E(a_{2}) = 0.7530, E(a_{3}) = 0.7441, E(a_{4}) = 0.7877 | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ |

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Fan, C.; Ye, J.; Hu, K.; Fan, E. Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods. *Information* **2017**, *8*, 107.
https://doi.org/10.3390/info8030107

**AMA Style**

Fan C, Ye J, Hu K, Fan E. Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods. *Information*. 2017; 8(3):107.
https://doi.org/10.3390/info8030107

**Chicago/Turabian Style**

Fan, Changxing, Jun Ye, Keli Hu, and En Fan. 2017. "Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods" *Information* 8, no. 3: 107.
https://doi.org/10.3390/info8030107