In dealing with the complex, unknown, and uncertain decision-making problems, a group of decision-makers are usually employed to analyze a set of alternatives and to get the optimal result in a certain way. Such a decision-making process is called multiple attribute group decision-making (MAGDM) problem. When making decisions, decision-makers tend to use words such as “excellent”, “good”, and “poor” to express their evaluations for objects. Zadeh proposed a linguistic variable set
S = {
} (
g is an even number) to deal with the approximate reasoning problems [
1,
2]. The linguistic variable is an effective tool, it improves the reliability and flexibility of classical decision models [
3,
4]. In recent years, the linguistic variables have been frequently linked to other theories. Liu proposed the intuitionistic linguistic set (ILS) composed of linguistic variables and IFS, where the first component provides its qualitative evaluation value/linguistic value and the second component gives the credibility of its intuitionistic fuzzy value for the given linguistic value [
5]. Then, Chen et al. proposed the linguistic intuitionistic fuzzy number (LIFN), which is composed of the intuitionistic fuzzy number (the basic element in IFS) and the linguistic variable [
6]. On the other hand, some methods for multiple attribute group decision-making (MAGDM) were proposed based on two-dimension uncertain linguistic variable [
7,
8]. Some improved linguistic intuitionistic fuzzy aggregation operators and several corresponding applications were given in decision-making [
9]. Although the IFS theory considers not only
T(
x), but also
F(
x), IFS is still not perfect enough because it ignores the indeterminate and inconsistent information. Thus, the intuitionistic fuzzy number can only be used for expressing incomplete information, but not for expressing indeterminate and inconsistent information. To make up for the insufficiency of the IFS theory, Smarandache put forward the neutrosophic set (NS) composed of three parts: truth
T(
x), falsity
F(
x), and indeterminacy
I(
x) [
10,
11]. Wang et al. and Smarandache also proposed the concept of a single-valued neutrosophic set (SVNS) satisfying
T(
x),
I(
x),
F(
x)
[0, 1], 0 ≤
T(
x) +
F(
x) +
I(
x) ≤ 3 [
10,
11,
12]. Ye proposed an extended TOPSIS (technique for order preference by similarity to an ideal Solution) method for MAGDM based on single valued neutrosophic linguistic numbers (SVNLNs), which are basic elements in a single-valued neutrosophic linguistic set (SVNLS) [
13]. Liu and Shi presented some neutrosophic uncertain linguistic number Heronian mean operators and their application to MAGDM [
14]. Since the Bonferroni mean (BM) is a useful operator in decision-making [
15], it was extended to hesitant fuzzy sets, IFSs, and interval-valued IFSs to propose their some Bonferroni mean operators for decision making [
16,
17,
18,
19,
20]. Then, Fang and Ye proposed the linguistic neutrosophic numbers (LNN) and their basic operational laws [
21]. LNN consists of the truth, indeterminacy, and falsity linguistic degrees, which can be expressed as the form
a = <
lT,
lI,
lF>, 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.
The rest organizations of this paper are as follows.
Section 2 describes some basic concepts of LNN, the basic operational laws of LNNs, and the basic concepts of BM and the normalized weighted BM.
Section 3 proposes the LNNNWBM and LNNNWGBM operators and investigates their properties.
Section 4 establishes MAGDM methods by using the LNNNWBM operator and LNNNWGBM operator.
Section 5 provides an illustrative example with different values of the parameters
p and
q to demonstrate the application of the proposed methods.
Section 6 gives conclusions.