Multiple attribute group decision making based on Maclaurin symmetric mean operator under single-valued neutrosophic interval 2-tuple linguistic environment

Abstract

Maclaurin symmetric mean (MSM) operator is a powerful tool to integrate multiple input arguments, which has the characteristic of considering the interrelationships among the input arguments. In this paper, we extend the traditional MSM operator to the single-valued neutrosophic interval 2-tuple linguistic environment, propose some novel aggregation operators, and develop a novel method to solve multiple attribute group decision making (MAGDM) problems. Firstly, we put forward the concept of single-valued neutrosophic interval 2-tuple linguistic sets (SVN-ITLSs) by combining the definitions of single-valued neutrosophic sets and interval 2-tuple. Secondly, the Maclaurin symmetric mean is extended to the single-valued neutrosophic interval 2-tuple linguistic environment and three new aggregation operators are proposed, such as the single-valued neutrosophic interval 2-tuple linguistic Maclaurin symmetric mean (SVN-ITLMSM) operator, the single-valued neutrosophic interval 2-tuple linguistic weighted average (SVN-ITLWA) operator and the single-valued neutrosophic interval 2-tuple linguistic weighted Maclaurin symmetric mean (SVN-ITLWMSM) operator. Some desirable properties of the proposed SVN-ITLMSM operator are investigated. Thirdly, an approach to solve MAGDM problem is developed based on the proposed operators. Finally, a numerical example is given to illustrate the application and the effectiveness of the proposed method.

Keywords

Multiple attribute group decision making single-valued neutrosophic interval 2-tuple linguistic sets single-valued neutrosophic interval 2-tuple linguistic Maclaurin symmetric mean (SVN-ITLMSM) operator single-valued neutrosophic interval 2-tuple linguistic weighted Maclaurin symmetric mean (SVN-ITLWMSM) operator

1 Introduction

Multiple attribute group decision making (MAGDM) problem is one of the most important branches of decision making theory, which has received more and more attention during the last several years [1 –6]. In the real world, due to the complexity of the decision problem, sometimes it is impossible for decision makers to adopt precise quantitative evaluation information. Under this situation, decision makers usually use linguistic variables to represent their subjective judgments [7 –11]. In various linguistic models, the 2-tuple linguistic representation model developed by Herrera and Martínez [12] owns the characteristics of effectively avoiding information distortion and loss in the linguistic information aggregation process. Based on the idea of the 2-tuple linguistic representation model, Zhang [13] further proposed the concept of interval 2-tuple linguistic representation model and developed some interval 2-tuple linguistic aggregation operators. The methods based on the 2-tuple linguistic representation model and interval 2-tuple linguistic representation model have been increasing dramatically since their appearance [14 –19].

However, the 2-tuple and interval 2-tuple linguistic model can’t process the indeterminate and inconsistent information. To handle the indeterminate and inconsistent information, Smarandache [20] proposed neutrosophic set theory, which consists of three membership functions, i.e., positive membership function, neutral membership function and negative membership function. Inspired by the idea of neutrosophic set, Wang et al. [21] presented the concept of single-valued neutrosophic sets (SVNSs) by limiting the ranges of positive membership, neutral membership and negative membership to the interval [0, 1], which can be easily used to solve many practical problems [22 –28]. Obviously, such situation can’t be accurately described by the classical fuzzy sets (FSs) [29] or intuitionistic fuzzy sets (IFSs) [30]. For describing the degree of positive membership, degree of neutral membership, degree of negative membership of an element to linguistic label, Ye [31] further proposed the concept of single-valued neutrosophic linguistic sets (SVNLSs). Although the approaches based on single-valued neutrosophic linguistic sets have been successfully applied in solving MADM problem [32 –34], it is not yet used to deal with interval 2-tuple linguistic information, while interval 2-tuple linguistic information is widely used as the evaluation value provided by decision makers or experts. In order to extend the application of the single-valued neutrosophic linguistic approach, we further propose the concept of single-valued neutrosophic interval 2-tuple linguistic sets (SVN-ITLSs) in this paper.

Information aggregation operator is a useful tool for information fusion and plays an important role in solving MAGDM problems. The Maclaurin symmetric mean (MSM) operator, firstly proposed by Maclaurin [35] and then developed by Detemple and Robertson [36], has an ability to capture the interrelationships among multi-input arguments. Recently, a series of research results have been put forward to aggregate uncertain information based on the MSM operator [37, 38]. For example, Zhang et al. [39] proposed the 2-tuple linguistic Maclaurin symmetric mean (2TLMSM) operator to solve the comprehensive evaluation problem with 2-tuple linguistic information. Qin and Liu [40] proposed the dual Maclaurin symmetric mean (DMSM) operator and extended the DMSM operator to accommodate uncertain linguistic environment. Ju et al. [41] extended the MSM operator to intuitionistic fuzzy linguistic environment and proposed a series of operators. However, the existing literature does not deal with single-valued neutrosophic interval 2-tuple linguistic information, while the single-valued neutrosophic interval 2-tuple linguistic information can easily be used to reflect decision makers’ preferences. Therefore, it is very interesting to develop some single-valued neutrosophic interval 2-tuple linguistic information aggregation operators based on the idea of Maclaurin symmetric mean.

In this paper, we focus on our attention on developing some new single-valued neutrosophic interval 2-tuple linguistic aggregation operators based on the idea of Maclaurin symmetric mean, and presenting a novel approach to solve MAGDM problems under single-valued neutrosophic interval 2-tuple linguistic environment. The remainder of this paper is organized as follows. In Section 2, some basic concepts are briefly reviewed, such as the definitions of 2-tuple, interval 2-tuple and single-valued neutrosophic sets (SVNSs). Section 3 proposes the concept of SVN-ITLSs and some operational laws of single-valued neutrosophic interval 2-tuple linguistic numbers (SVN-ITLNs). In Section 4, we propose some new operators to aggregate single-valued neutrosophic interval 2-tuple linguistic information, such as the single-valued neutrosophic interval 2-tuple linguistic Maclaurin symmetric mean (SVN-ITLMSM) operator, the single-valued neutrosophic interval 2-tuple linguistic weighted average (SVN-ITLWA) operator, as well as the single-valued neutrosophic interval 2-tuple linguistic weighted Maclaurin symmetric mean (SVN-ITLWMSM) operator. In Section 5, a novel multi-attribute group decision making method is developed based on the proposed SVN-ITLWA and SVN-ITLWMSM operators. A numerical example is given to illustrate the application and the validity of the proposed method in Section 6. The paper is concluded in Section 7.

2 Preliminaries

In this section, we briefly review some concepts involving this paper, such as 2-tuple, interval 2-tuple and single-valued neutrosophic set (SVNS). The 2-tuple linguistic approach, originally proposed in [12], is based on the concept of symbol translation and denoted by (s_i, α_i), where s_i is a linguistic term of the predefined linguistic term set S = {s₀, s₁, …, s_g} and α_i represents the value of symbolic translation. Obviously, the range of α_i is between 0 and g, which is relevant to the granularity of the linguistic term set S = {s₀, s₁, …, s_g}. To overcome this restriction, we adopt the definition proposed by Tai and Chen [42] in this paper.

Definition 1. [42] Let S = {s₀, s₁, …, s_g} be a linguistic term set and β ∈ [0, 1] be a crisp value representing the aggregation result of linguistic symbolic, then the function Δ used to obtain the 2-tuple linguistic information equivalent to β is defined as follows:

$\begin{matrix} Δ : [0, 1] \to S \times [- \frac{1}{2 g}, \frac{1}{2 g}), \\ Δ (β) = (s_{i}, α) with \\ {\begin{matrix} s_{i} & i = round (β \times g) \\ α = β - \frac{i}{g} & α \in [- \frac{1}{2 g}, \frac{1}{2 g}) \end{matrix}, \end{matrix}$ (1) where round(.) is the usual round operation, s_i has the closest index label to β and α is the value of the symbolic translation.

Conversely, there is always a Δ^-1 function such that a 2-tuple can be converted into a crisp value β ∈ [0, 1] as follows [42]: $Δ^{- 1} [s_{i}, α] = \frac{i}{g} + α = β .$ (2)

Definition 2. [13] Let S = {s₀, s₁, …, s_g} be a linguistic term set. An interval 2-tuple is composed of two linguistic terms and two crisp numbers, denoted by [(s_i, α_i) , (s_j, α_j)], where s_i ≤ s_j (or α_i ≤ α_j if s_i = s_j), s_i (or s_j) and α_i (or α_j) represent the linguistic label of the predefined linguistic term set S and symbolic translation, respectively. An interval 2-tuple linguistic that expresses the equivalent information to an interval value [γ₁, γ₂] (γ₁, γ₂ ∈ [0, 1] , γ₁ ≤ γ₂) is determined by the following expressions: $\begin{matrix} Δ [γ_{1}, γ_{2}] = [(s_{i}, α_{i}), (s_{j}, α_{j})] \\ with {\begin{matrix} s_{i} & i = round (γ_{1} \times g) \\ s_{j} & j = round (γ_{2} \times g) \\ α_{i} = γ_{1} - \frac{i}{g}, & α_{i} \in [- \frac{1}{2 g}, \frac{1}{2 g}) \\ α_{j} = γ_{2} - \frac{j}{g}, & α_{j} \in [- \frac{1}{2 g}, \frac{1}{2 g}) \end{matrix} . \end{matrix}$ (3)

Conversely, there is always a Δ^-1 function such that an interval 2-tuple can be converted into an interval value [γ₁, γ₂] (γ₁, γ₂ ∈ [0, 1] , γ₁ ≤ γ₂) as follows:

$\begin{matrix} Δ^{- 1} [(s_{i}, α_{i}), (s_{j}, α_{j})] \\ = [\frac{i}{g} + α_{i}, \frac{j}{g} + α_{j}] = [γ_{1}, γ_{2}] . \end{matrix}$ (4)

Definition 3. [21] Let X be a universe set, with a generic element in X denoted by x. A single-valued neutrosophic set (SVNS) A in X is characterized by the degree of positive membership T_A (x), degree of neutral membership I_A (x) and degree of negative membership F_A (x), and the ranges of T_A (x), I_A (x) and F_A (x) satisfy the following conditions: 0 ≤ T_A (x) ≤1, 0 ≤ I_A (x) ≤1 and 0 ≤ F_A (x) ≤1.

A SVNS A can be written as follows: $A = {< x, T_{A} (x), I_{A} (x), F_{A} (x) > | x \in X}$ (5)

Further, Ye [31] called the triplet <T_A (x) , I_A (x) , F_A (x)> single-valued neutrosophic number (SVNN), which is denoted by α =< T_A, I_A, F_A >.

3 Single-valued neutrosophic interval 2-tuple linguistic sets

In the following, we will propose the basic concept of single-valued neutrosophic interval 2-tuple linguistic set (SVN-ITLS) and some operational laws among single-valued neutrosophic interval 2-tuple linguistic numbers (SVN-ITLNs).

Definition 4. A single-valued neutrosophic interval 2-tuple linguistic set A in X is defined as follows:

$\begin{matrix} A & = & {< [(s_{(x)}, α_{(x)}), (l_{(x)}, β_{(x)})], \\ (μ_{A} (x), η_{A} (x), v_{A} (x)) > | x \in X}, \end{matrix}$ (6) where [(s_(x), α_(x)) , (l_(x), β_(x))] is an interval 2-tuple, μ_A (x) ∈ [0, 1] , η_A (x) ∈ [0, 1] and v_A (x) ∈ [0, 1] are called the degree of positive membership, degree of neutral membership and degree of negative membership of the element x to the interval 2-tuple [(s_(x), α_(x)) , (l_(x), β_(x))], respectively, and μ_A (x) , η_A (x) , v_A (x) satisfy the following conditions: 0 ≤ μ_A (x) ≤1, 0 ≤ η_A (x) ≤1, 0 ≤ v_A (x) ≤1, and 0 ≤ μ_A (x) + η_A (x) + v_A (x) ≤3, ∀x ∈ X. For x ∈ X, π_A (x) =1 - μ_A (x) - η_A (x) - v_A (x) is called the degree of refusal membership of the element x to the interval 2-tuple [(s_(x), α_(x)) , (l_(x), β_(x))].

For convenience, we call p =< [(s, α) , (l, β)] , (μ, η, v) > a single-valued neutrosophic interval 2-tuple linguistic number (SVN-ITLN).

Definition 5. Let p =< [(s, α) , (l, β)] , (μ, η, v) > be a SVN-ITLN, a score function of the SVN-ITLN p is defined as follows:

$\begin{matrix} S (p) & = & \frac{1}{6} \times [Δ^{- 1} (s, α) + Δ^{- 1} (l, β)] \\ \times (μ + 1 - η + 1 - v) . \end{matrix}$ (7)

Definition 6. Let p =< [(s, α) , (l, β)] , (μ, η, v) > be a SVN-ITLN, an accuracy function of the SVN-ITLN p is defined as follows:

$\begin{matrix} H (p) & = & \frac{1}{6} \times [Δ^{- 1} (s, α) + Δ^{- 1} (l, β)] \\ \times (μ + η + v) . \end{matrix}$ (8)

Definition 7. For any two SVN-ITLNs p₁ =< [(s₁, α₁) , (l₁, β₁)] , (μ₁, η₁, v₁) > and p₂ =< [(s₂, α₂) , (l₂, β₂)] , (μ₂, η₂, v₂) >, then an order relationship for p₁ and p₂ is identified as follows:

if S (p₁) > S (p₂), then p₁ is superior to p₂, denoted by p₁ ≻ p₂;

if S (p₁) = S (p₂), then

if A (p₁) > A (p₂), then p₁ is superior to p₂, denoted by p₁ ≻ p₂;

if A (p₁) = A (p₂), then p₁ is equivalent to p₂, denoted by p₁ ∼ p₂.

Definition 8. Let p₁ =< [(s₁, α₁) , (l₁, β₁)] , (μ₁, η₁, v₁) > and p₂ =< [(s₂, α₂) , (l₂, β₂)] , (μ₂, η₂, v₂) > be two SVN-ITLNs, then some operational laws of SVN-ITLNs are given as follows: $\begin{matrix} (1) p_{1} \oplus p_{2} = < Δ [Δ^{- 1} (s_{1}, α_{1}) + Δ^{- 1} (s_{2}, α_{2}), \\ Δ^{- 1} (l_{1}, β_{1}) + Δ^{- 1} (l_{2}, β_{2})], (μ_{1} + μ_{2} \\ - μ_{1} \times μ_{2}, η_{1} \times η_{2}, v_{1} \times v_{2}) > . \\ (2) p_{1} \otimes p_{2} = < Δ [Δ^{- 1} (s_{1}, α_{1}) \times Δ^{- 1} (s_{2}, α_{2}), \\ Δ^{- 1} (l_{1}, β_{1}) \times Δ^{- 1} (l_{2}, β_{2})], (μ_{1} \times μ_{2}, \\ η_{1} + η_{2} - η_{1} \times η_{2}, v_{1} + v_{2} - v_{1} \times v_{2}) > . \\ (3) λ p_{1} = < Δ [λ Δ^{- 1} (s_{1}, α_{1}), λ Δ^{- 1} (l_{1}, β_{1})], \\ (1 - (1 - μ_{1})^{λ}, (η_{1})^{λ}, (v_{1})^{λ}) > . \\ (4) (p_{1})^{λ} = < Δ [(Δ^{- 1} (s_{1}, α_{1}))^{λ}, (Δ^{- 1} (l_{1}, β_{1}))^{λ}], \\ (μ_{1})^{λ}, 1 - (1 - η_{1})^{λ}, 1 - (1 - v_{1})^{λ}) > . \end{matrix}$

According to Definition 8, we can obtain the following properties of the SVN-ITLNs.

Theorem 1. For any two SVN-ITLNs p₁ =< [(s₁, α₁) , (l₁, β₁)] , (μ₁, η₁, v₁) > and p₂ =< [(s₂, α₂) , (l₂, β₂)] , (μ₂, η₂, v₂) >, 0 ≤ λ, λ₁, λ₂ ≤ 1, then we have

p₁ ⊕ p₂ = p₂ ⊕ p₁ .

p₁ ⊗ p₂ = p₂ ⊗ p₁ .

λ (p₁ ⊕ p₂) = λp₁ ⊕ λp₂ .

λ₁p₁ ⊕ λ₂p₁ = (λ₁ + λ₂) p₁, 0 ≤ λ₁ + λ₂ ≤ 1 .

$p_{1}^{λ_{1}} \otimes p_{1}^{λ_{2}} = p_{1}^{λ_{1} + λ_{2}}, 0 \leq λ_{1} + λ_{2} \leq 1 .$

$p_{1}^{λ_{1}} \otimes p_{2}^{λ_{1}} = (p_{1} \otimes p_{2})^{λ_{1}} .$

$(p_{1}^{λ_{1}})^{λ_{2}} = p_{1}^{λ_{1} λ_{2}} .$

This theorem can be proved according to the operational laws in Definition 8. To save space, we omit the proof.

4 Some single-valued neutrosophic interval 2-tuple linguistic Maclaurin symmetric mean operators

Uncertain linguistic evaluation information can be easily described by single-valued neutrosophic interval 2-tuple linguistic information, while the MSM operator can capture the interrelationships among evaluation information given by decision makers or experts. It is very interesting to develop some novel operators to integrate the SVN-ITLNs based on the MSM operator. The definition of the MSM operator is given as follows.

Definition 9. [35] Let a_i (i = 1, 2, …, n) be a set of nonnegative real numbers, and k = 1, 2, …, n. If

$\begin{matrix} {MSM}^{(k)} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} a_{i_{j}}}{C_{n}^{k}})}^{1 / k}, \end{matrix}$ (9) then MSM^(k) is called Maclaurin symmetric mean (MSM) operator, where (i₁, i₂, …, i_k) traverses all the k-tuple combination of (1, 2, …, n) and $C_{n}^{k} = \frac{n!}{k! (n - k)!}$ is the binomial coefficient.

The traditional MSM operator is usually used on situation where the input arguments are nonnegative real numbers. In what follows, we shall extend the traditional MSM operator to aggregate the single-valued neutrosophic interval 2-tuple linguistic information.

4.1 Single-valued neutrosophic interval 2-tuple linguistic Maclaurin symmetric mean operator

Definition 10. Let p_i = < [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of single-valued neutrosophic interval 2-tuple numbers and k = 1, 2, …, n, then a single-valued neutrosophic interval 2-tuple linguistic Maclaurin symmetric mean (SVN-ITLMSM) operator is a mapping: Pⁿ → P,

$SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) = {(\frac{\underset{1 \leq i_{1} < . . . < i_{k} \leq n}{\oplus} (\otimes_{j = 1}^{k} p_{i_{j}})}{C_{n}^{k}})}^{1 / k},$ (10)

where (i₁, i₂, …, i_k) traverses all the k-tuple combination of (1, 2, …, n) and $C_{n}^{k}$ is the binomial coefficient.

According to the operational laws of SVN-ITLNs described in Definition 8, we can derive the following theorem.

Theorem 2. {Let p_i = < [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs and k = 1, 2, …, n, then the aggregated value obtained from the SVN-ITLMSM operator is also a SVN-ITLN, and $\begin{matrix} SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ = 〈 Δ [{(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}})}{C_{n}^{k}})}^{1 / k}, {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})}{C_{n}^{k}})}^{1 / k}], \\ ({(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, \\ 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}) 〉 . \end{matrix}$ (11) Proof. According to the operational laws in Definition 8, we have $\begin{matrix} \otimes_{j = 1}^{k} p_{i_{j}} & = & 〈 Δ [\prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}}), \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})], (\prod_{j = 1}^{k} μ_{i_{j}}, 1 - \prod_{j = 1}^{k} (1 - η_{i_{j}}), 1 - \prod_{j = 1}^{k} (1 - v_{i_{j}})) 〉 \end{matrix}$ and $\begin{matrix} \underset{1 \leq i_{1} < \dots < i_{k} \leq n}{\oplus} (\otimes_{j = 1}^{k} p_{i_{j}}) = 〈 Δ [\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}}), \sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})], \\ (1 - \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}), \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})), \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}}))) 〉, \end{matrix}$ then we obtain $\begin{matrix} \frac{1}{C_{n}^{k}} (\underset{1 \leq i_{1} < \dots < i_{k} \leq n}{\oplus} (\otimes_{j = 1}^{k} p_{i_{j}})) \\ = 〈 Δ [\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}})}{C_{n}^{k}}, \frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})}{C_{n}^{k}}], \\ (1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}}, {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}})))}^{\frac{1}{C_{n}^{k}}}) 〉 . \end{matrix}$ Therefore, we have $\begin{matrix} SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ = 〈 Δ [{(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}})}{C_{n}^{k}})}^{1 / k}, {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})}{C_{n}^{k}})}^{1 / k}], \\ ({(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, \\ 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}) 〉 . \end{matrix}$ Since $0 \leq 1 - \prod_{j = 1}^{k} μ_{i_{j}} \leq 1$ and $0 \leq \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}) \leq 1$ , then we have $\begin{matrix} 0 & \leq & {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}})}^{1 / k} \\ \leq & 1 . \end{matrix}$ Similarly, we have $\begin{matrix} 0 & \leq & {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k} \\ \leq & 1 \end{matrix}$ and $\begin{matrix} 0 & \leq & {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k} \\ \leq & 1, \end{matrix}$ which completes the proof of Theorem 2.Theorem 3. Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs, k = 1, 2, …, n, then the properties of the SVN-ITLMSM operator are shown as follows:

(Idempotency) If all SVN-ITLNs p_i (i = 1, 2, …, n) are equal, i.e., p_i =< [(s, α) , (l, β)] , (μ, η, v) > for all i = 1, 2, …, n, then $\begin{matrix} SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ = < [(s, α), (l, β)], (μ, η, v) > . \end{matrix}$ (12)

(Monotonicity) Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > and p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be two sets of SVN-ITLNs, if [(s_i, α_i) , (l_i, β_i)], μ_i≤μ_i, η_i≥η_i and υ_i ≥ υ_i for all i = 1, 2, …, n, then $S V N - I T L M S M^{(k)} (p_{1}, p_{2}, ..., p_{n}) \leq S V N - I T L M S M^{(k)} (p_{1}^{'}, p_{2}^{'}, ..., p_{n}^{'}) .$ (13)

(Boundedness) Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs, if $\begin{matrix} [(s_{-}, α_{-}), (l_{-}, β_{-})] = min_{1 \leq i \leq n} {[(s_{i}, α_{i}), (l_{i}, β_{i})]}, \\ [(s_{+}, α_{+}), (l_{+}, β_{+})] = max_{1 \leq i \leq n} {[(s_{i}, α_{i}), (l_{i}, β_{i})]}, \end{matrix}$ $\begin{matrix} μ^{-} = min_{1 \leq i \leq n} {μ_{i}}, μ^{+} = max_{1 \leq i \leq n} {μ_{i}}, η^{-} = min_{1 \leq i \leq n} {η_{i}}, \\ η^{+} = max_{1 \leq i \leq n} {η_{i}}, v^{-} = min_{1 \leq i \leq n} {v_{i}}, v^{+} = max_{1 \leq i \leq n} {v_{i}}, \end{matrix}$ then $\begin{matrix} 〈 [(s_{-}, α_{-}), (l_{-}, β_{-})], (μ^{-}, η^{+}, v^{+}) 〉 \\ \leq SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ \leq 〈 [(s_{+}, α_{+}), (l_{+}, β_{+})], (μ^{+}, η^{-}, v^{-}) 〉 . \end{matrix}$ (14)

(Commutativity) Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > and (i = 1, 2, …, n) be two sets of SVN-ITLNs, and be any permutation of p_i = < [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n), then $S V N - I T L M S M^{(k)} (p_{1}, p_{2}, ..., p_{n}) = S V N - I T L M S M^{(k)} (p_{1}^{'}, p_{2}^{'}, ..., p_{n}^{'}) .$ (15)

Based on Theorems 2 and 3 as well as Definition 7, we can prove this theorem. Here, we omit the proof.Theorem 4. For the given SVN-ITLNs p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) and k = 1, 2, …, n, then the SVN-ITLMSM operator is monotonically decreasing with respect to the parameter k.Proof. For p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n), on the basis of Theorem 2, we have

\begin{matrix} SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ = 〈 Δ [{(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}})}{C_{n}^{k}})}^{1 / k}, {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})}{C_{n}^{k}})}^{1 / k}], \\ ({(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, \\ 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}) 〉 . \end{matrix}

Let

\begin{matrix} F (k) = {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (s_{i_{j}}, α_{i_{j}})}{C_{n}^{k}})}^{1 / k} + {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (l_{i_{j}}, β_{i_{j}})}{C_{n}^{k}})}^{1 / k}, \\ G (k) = {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} μ_{i_{j}}))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}}, \\ M (k) = 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - η_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}} \end{matrix}

and

N (k) = 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - v_{i_{j}})))}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{k}},

then, on the basis of reference [38], we can see that the function F(k) and G(k) are monotonically decreasing with respect to the parameter k, M(k) and N(k) are monotonically increasing with respect to the parameter k.According to the score function in Definition 5, we have

\begin{matrix} S (SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n})) \\ = \frac{1}{6} F (k) (G (k) + 1 - M (k) + 1 - N (k)) \end{matrix}

and

\begin{matrix} S (SVN - {ITLMSM}^{(k + 1)} (p_{1}, p_{2}, \dots, p_{n})) \\ = \frac{1}{6} F (k + 1) (G (k + 1) + 1 - M (k + 1) + 1 - N (k + 1)) . \end{matrix}

For any positive integer k ∈ [1, n - 1], we can obtain the following inequality:

\begin{matrix} S (SVN - {ITLMSM}^{(k + 1)} (p_{1}, p_{2}, \dots, p_{n})) - S (SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n})) \\ = \frac{1}{6} F (k + 1) [G (k + 1) + 1 - M (k + 1) + 1 - N (k + 1)] - \frac{1}{6} F (k) [G (k) + 1 - M (k) + 1 - N (k)] \\ \leq \frac{1}{6} F (k) [G (k + 1) + 1 - M (k + 1) + 1 - N (k + 1)] - \frac{1}{6} F (k) [G (k) + 1 - M (k) + 1 - N (k)] \\ = \frac{1}{6} F (k) [G (k + 1) + 1 - M (k + 1) + 1 - N (k + 1) - G (k) - 1 + M (k) - 1 + N (k)] \\ = \frac{1}{6} F (k) [G (k + 1) - G (k) + M (k) - M (k + 1) + N (k) - N (k + 1)] \leq 0, \end{matrix}

that is, S (SVN - ITLMSM^(k+1) (p₁, p₂, …, p_n)) ≤S (SVN - ITLMSM^(k) (p₁, p₂, …, p_n)) .

If S (SVN - ITLMSM^(k+1) (p₁, p₂, …, p_n)) < S (SVN - ITLMSM^(k) (p₁, p₂, … , p_n)), according to Definition 7, we can obtain SVN - ITLMSM^(k+1) (p₁, p₂, … , p_n)) < SVN - ITLMSM^(k) (p₁, p₂, …, p_n)) .

If S (SVN - ITLMSM^(k+1) (p₁, p₂, …, p_n)) = S (SVN - ITLMSM^(k) (p₁, p₂, … , p_n)), we can infer that the following expressions hold: F(k) = F(k+1), G(k) = G(k+1), M(k) = M(k+1) and N(k) = N(k+1). According to Definition 6, we can obtain the equality: H (SVN - ITLMSM^(k+1) (p₁, p₂, … , p_n)) = H (SVN - ITLMSM^(k) (p₁, p₂, … , p_n)). Based on Definition 7, we can easily prove this theorem.

Theorem 5. Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs and k = 1, 2, …, n, then we have

\begin{matrix} (1) max_{1 \leq k \leq n} {SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, . . ., p_{n})} \\ = 〈 Δ [(\frac{1}{n} \sum_{i = 1}^{n} Δ^{- 1} (s_{i}, a_{i})), (\frac{1}{n} \sum_{i = 1}^{n} Δ^{- 1} (l_{i}, β_{i}))], \\ (1 - {(\prod_{i = 1}^{n} (1 - μ_{i}))}^{\frac{1}{n}}, {(\prod_{i = 1}^{n} η_{i})}^{\frac{1}{n}}, {(\prod_{i = 1}^{n} v_{i})}^{\frac{1}{n}}) 〉 . \\ (2) min_{1 \leq k \leq n} {SVN - {ITLMSM}^{(k)} (p_{1}, p_{2}, . . ., p_{n})} \\ = 〈 Δ [{(\prod_{j = 1}^{n} Δ^{- 1} (s_{i_{j}}, α_{i_{j}}))}^{1 / n}, {(\prod_{j = 1}^{n} Δ^{- 1} (l_{i_{j}}, β_{i_{j}}))}^{1 / n}], \\ ({(\prod_{j = 1}^{n} μ_{j})}^{\frac{1}{n}}, 1 - {(\prod_{j = 1}^{n} (1 - η_{j}))}^{\frac{1}{n}}, 1 - {(\prod_{j = 1}^{n} (1 - v_{j}))}^{\frac{1}{n}}) 〉 . \end{matrix}

Based on Theorem 4, we can easily prove this theorem.

4.2 Single-valued neutrosophic interval 2-tuple linguistic weighted Maclaurin symmetric mean operator

It is noted that the SVN-ITLMSM operator in Section 4.1 is used to deal with the situation without considering the weight information. Nevertheless, in many practical situations, especially in multiple attribute group decision making, the weight information plays an important role in the aggregation process. To overcome the limitation, the single-valued neutrosophic interval 2-tuple linguistic weighted average (SVN-ITLWA) operator and the single-valued neutrosophic interval 2-tuple linguistic weighted Maclaurin symmetric mean (SVN-ITLWMSM) operator are given in this subsection.Definition 11. Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs and k = 1, 2, …, n, w = (w₁, w₂, …, w_n) ^T be the weight vector of p_i (i = 1, 2, …, n), with w_i ∈ [0, 1] , i = 1, 2, . . . , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $SVN - ITLWA (p_{1}, p_{2}, \dots, p_{n}) = \oplus_{i = 1}^{n} (w_{i} \otimes p_{i}),$ (16) then SVN-ITLWA is called the single-valued neutrosophic interval 2-tuple linguistic weighted average operator.Theorem 6. Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs and k = 1, 2, …, n, w = (w₁, w₂, …, w_n) ^T be the weight vector of p_i (i = 1, 2, …, n), with w_i ∈ [0, 1], i = 1, 2, …, n and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregated value obtained from the SVN-ITLWA operator is also a SVN-ITLN, and $\begin{matrix} SVN - ITLWA (p_{1}, p_{2}, \dots, p_{n}) \\ = 〈 Δ [\sum_{i = 1}^{n} (w_{i} \times Δ^{- 1} (s_{i}, α_{i})), \sum_{i = 1}^{n} (w_{i} \times Δ^{- 1} (l_{i}, β_{i}))], \\ (1 - \prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}}, \prod_{i = 1}^{n} (η_{i})^{w_{i}}, \prod_{i = 1}^{n} (v_{i})^{w_{i}}) 〉 . \end{matrix}$ (17) The proof of this theorem is similar to that of Theorem 2.Definition 12. Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs and k = 1, 2, …, n, w = (w₁, w₂, …, w_n) ^T be the weight vector of p_i (i = 1, 2, …, n), with w_i ∈ [0, 1], i = 1, 2, …, n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} SVN - {ITLWMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ = {(\frac{\underset{1 \leq i_{1} < \dots < i_{k} \leq n}{\oplus} (\otimes_{j = 1}^{k} (w_{i_{j}} \otimes p_{i_{j}}))}{C_{n}^{k}})}^{1 / k}, \end{matrix}$ (18) then SVN-ITLWMSM is called the single-valued neutrosophic interval 2-tuple linguistic weighted Maclaurin symmetric mean operator, where (i₁, i₂, …, i_k) traverses all the k-tuple combination of (1,2,…,n) and $C_{n}^{k}$ is the binomial coefficient.According to the operational laws of SVN-ITLNs proposed in Definition 8, we can derive the following theorem.Theorem 7. Let p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) be a collection of SVN-ITLNs and k = 1, 2, …, n, w = (w₁, w₂, . . . , w_n) ^T be the weight vector of p_i (i = 1, 2, …, n), with w_i ∈ [0, 1], i = 1, 2, …, n and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregated value obtained from the SVN-ITLWMSM operator is also a SVN-ITLN, and $\begin{matrix} SVN - {ITLWMSM}^{(k)} (p_{1}, p_{2}, \dots, p_{n}) \\ = 〈 Δ [{(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} (w_{i_{j}} \times Δ^{- 1} (s_{i_{j}}, α_{i_{j}}))}{C_{n}^{k}})}^{1 / k}, {(\frac{\sum_{1 \leq i_{1} < \dots < i_{k} \leq n} \prod_{j = 1}^{k} (w_{i_{j}} \times Δ^{- 1} (l_{i_{j}}, β_{i_{j}}))}{C_{n}^{k}})}^{1 / k}], \\ ({(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - (1 - μ_{i_{j}})^{w_{i_{j}}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k}, \\ 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - (η_{i_{j}})^{w_{i_{j}}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k}, \\ 1 - {(1 - {(\prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - (v_{i_{j}})^{w_{i_{j}}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k}) 〉 . \end{matrix}$ (19) The proof of this theorem is similar to that of Theorem 2.Theorem 8. For the given SVN-ITLNs p_i =< [(s_i, α_i) , (l_i, β_i)] , (μ_i, η_i, v_i) > (i = 1, 2, …, n) and their associated weight vector w = (w₁, w₂, . . . , w_n) ^T, then the SVN-ITLWMSM operator is monotonically decreasing with respect to the parameter k.The proof of this theorem is similar to that of Theorem 4.

5 A novel approach to MAGDM problem based on the SVN-ITLWA and SVN-ITLWMSM operators

In this section, we will propose a novel approach to solve MAGDM problem based on the proposed SVN-ITLWA and SVN-ITLWMSM operators, where the evaluation information given by the decision makers takes the form of the single-valued neutrosophic interval 2-tuple linguistic information. Suppose that A = {A₁, A₂, …, A_m} is a discrete set of alternatives, and C = {C₁, C₂, …, C_n} is an attribute set with the weight vector W = (w₁, w₂, …, w_n) ^T, where w_j ∈ [0, 1], j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ . To choose the most desirable alternative, the individual single-valued neutrosophic interval 2-tuple linguistic decision matrices $R^{t} = (r_{ij}^{t})_{m \times n}, (t = 1, 2, \dots, l)$ are given by the decision makers,

where $r_{ij}^{t} = < [(s_{ij}^{t}, α_{ij}^{t}), (l_{ij}^{t}, β_{ij}^{t})], (μ_{ij}^{t}, η_{ij}^{t}, v_{ij}^{t}) >$ is a single-valued neutrosophic interval 2-tuple linguistic number given by the decision maker D_t (t = 1, 2, …, l) for the alternative A_i (i = 1, 2, …, m) with respect to the attribute C_j (j = 1, 2, …, n). The weight vector of decision makers is λ = (λ₁, λ₂, …, λ_l) ^T, with λ_t ∈ [0, 1] , t = 1, 2, …, l and $\sum_{t = 1}^{l} λ_{t} = 1$ . The proposed approach is summarized as follows.

Step 1. Aggregate all individual single-valued neutrosophic interval 2-tuple linguistic decision matrices $R^{t} = (r_{ij}^{t})_{m \times n}, (t = 1, 2, \dots, l)$ into the collective one R = (r_ij) _m×n by Equation (20), which is shown as follows:

$\begin{matrix} r_{ij} & = & SVN - ITLWA (r_{ij}^{1}, r_{ij}^{2}, \dots, r_{ij}^{l}) \\ = & 〈 [Δ (\sum_{t = 1}^{l} λ_{t} Δ^{- 1} (s_{ij}^{t}, α_{ij}^{t})), Δ (\sum_{t = 1}^{l} λ_{t} Δ^{- 1} (l_{ij}^{t}, β_{ij}^{t}))], \\ (1 - \prod_{t = 1}^{l} (1 - u_{ij}^{t})^{λ_{t}}, \prod_{t = 1}^{l} (η_{ij}^{t})^{λ_{t}}, \prod_{t = 1}^{l} (v_{ij}^{t})^{λ_{t}}) 〉 . \end{matrix}$ (20)

Step 2. Determine the overall assessment value r_i (i = 1, 2, …, m) of all alternatives by Equation (21).

$\begin{matrix} r_{i} & = & SVN - {ITLWMSM}^{(k)} (r_{i 1}, r_{i 2}, \dots, r_{in}) \\ = & 〈 Δ [{(\frac{\sum_{1 \leq j_{1} < \dots < j_{k} \leq n} \prod_{q = 1}^{k} (w_{j_{q}} \times Δ^{- 1} (s_{{ij}_{q}}, α_{{ij}_{q}}))}{C_{n}^{k}})}^{1 / k}, {(\frac{\sum_{1 \leq j_{1} < \dots < j_{k} \leq n} \prod_{q = 1}^{k} (w_{j_{q}} \times Δ^{- 1} (l_{{ij}_{q}}, β_{{ij}_{q}}))}{C_{n}^{k}})}^{1 / k}], \\ ({(1 - {(\prod_{1 \leq j_{1} < \dots < j_{k} \leq n} (1 - \prod_{q = 1}^{k} (1 - (1 - μ_{{ij}_{q}})^{w_{j_{q}}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k}, \\ 1 - {(1 - {(\prod_{1 \leq j_{1} < \dots < j_{k} \leq n} (1 - \prod_{q = 1}^{k} (1 - (η_{{ij}_{q}})^{w_{j_{q}}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k}, \\ 1 - {(1 - {(\prod_{1 \leq {ij}_{1} < \dots < j_{k} \leq n} (1 - \prod_{q = 1}^{k} (1 - (v_{{ij}_{q}})^{w_{j_{q}}})))}^{\frac{1}{C_{n}^{k}}})}^{1 / k}) 〉 . \end{matrix}$ (21)

Step 3. Calculate the score values S (r_i) of alternatives by Equation (22).

$\begin{matrix} S (r_{i}) & = & \frac{1}{6} \times [Δ^{- 1} (s_{i}, α_{i}) + Δ^{- 1} (l_{i}, β_{i})] \\ \times (μ_{i} + 1 - η_{i} + 1 - v_{i}), \end{matrix}$ (22)

If there are equal score values of alternatives, then we can further calculate the accuracy degree H (r_i) by Equation (23).

$\begin{matrix} H (r_{i}) & = & \frac{1}{6} \times [Δ^{- 1} (s_{i}, α_{i}) + Δ^{- 1} (l_{i}, β_{i})] \\ \times (μ_{i} + η_{i} + v_{i}) . \end{matrix}$ (23)

Step 4. Rank all feasible alternatives and select the most desirable one(s) according to Definition 7.

6 Numerical example and comparison analysis

6.1 A numerical example

In this subsection, a numerical example is given to illustrate the application of the proposed method. In addition, comparative analysis is provided to verify the effectiveness of the proposed approach. This example is adapted from Ju et al. [41], which can be described as follows: an emergency management department wants to select the most desirable alternative(s) from four emergency alternatives, which are denoted by A_i (i = 1, 2, 3, 4), according to the following four attributes: emergency process capability (C₁), emergency forecasting capacity (C₂), emergency support capacity (C₃) and after-disaster process capacity (C₄), whose weight vector is W = (0.32, 0.26, 0.18, 0.24) ^T. The feasible alternatives are evaluated by three experts D₁, D₂ and D₃, whose weight vector is denoted by λ = (0.40, 0.32, 0.28) ^T. The single-valued neutrosophic interval 2-tuple linguistic decision matrices of three experts are obtained using the linguistic term set S = {S₀ = extremely poor, S₁ = very poor, S₂ = poor, S₃ = slightly poor, S₄ = fair, S₅ = good, S₆ = very good, S₇ = slightly good, S₈ = extremely good}, which are shown in Tables 1 –3. Then, we utilize the proposed approach to choose the most desirable alternative(s) under the single-valued neutrosophic interval 2-tuple linguistic environment.

Table 1
The SVN-ITL matrix provided by decision maker D₁

Alternative C ₁ C ₂ C ₃ C ₄

A ₁ < [(S₅,0), (S₆,0)], (0.1, 0.5, 0.3)> < [(S₂,0), (S₄,0)], (0.4, 0.4, 0.3)> < [(S₅,0), (S₆,0)], (0.5, 0.3, 0.2)> < [(S₃,0), (S₆,0)], (0.2, 0.5, 0.2)>

A ₂ < [(S₄,0), (S₅,0)], (0.3, 0.4, 0.3)> < [(S₅,0), (S₇,0)], (0.4, 0.3, 0.3)> < [(S₃,0), (S₄,0)], (0.1, 0.7, 0.2)> < [(S₄,0), (S₆,0)], (0.5, 0.3, 0.1)>

A ₃ < [(S₃,0), (S₆,0)], (0.1, 0.5, 0.4)> < [(S₄,0), (S₇,0)], (0.2, 0.5, 0.3)> < [(S₄,0), (S₆,0)], (0.3, 0.6, 0.1)> < [(S₅,0), (S₆,0)], (0.2, 0.5, 0.3)>

A ₄ < [(S₆,0), (S₇,0)], (0.4, 0.2, 0.4)> < [(S₂,0), (S₄,0)], (0.2, 0.6, 0.2)> < [(S₃,0), (S₅,0)], (0.2, 0.5, 0.3)> < [(S₃,0), (S₆,0)], (0.3, 0.4, 0.2)>

Alternative	C ₁	C ₂	C ₃	C ₄
A ₁	< [(S₅,0), (S₆,0)], (0.1, 0.5, 0.3)>	< [(S₂,0), (S₄,0)], (0.4, 0.4, 0.3)>	< [(S₅,0), (S₆,0)], (0.5, 0.3, 0.2)>	< [(S₃,0), (S₆,0)], (0.2, 0.5, 0.2)>
A ₂	< [(S₄,0), (S₅,0)], (0.3, 0.4, 0.3)>	< [(S₅,0), (S₇,0)], (0.4, 0.3, 0.3)>	< [(S₃,0), (S₄,0)], (0.1, 0.7, 0.2)>	< [(S₄,0), (S₆,0)], (0.5, 0.3, 0.1)>
A ₃	< [(S₃,0), (S₆,0)], (0.1, 0.5, 0.4)>	< [(S₄,0), (S₇,0)], (0.2, 0.5, 0.3)>	< [(S₄,0), (S₆,0)], (0.3, 0.6, 0.1)>	< [(S₅,0), (S₆,0)], (0.2, 0.5, 0.3)>
A ₄	< [(S₆,0), (S₇,0)], (0.4, 0.2, 0.4)>	< [(S₂,0), (S₄,0)], (0.2, 0.6, 0.2)>	< [(S₃,0), (S₅,0)], (0.2, 0.5, 0.3)>	< [(S₃,0), (S₆,0)], (0.3, 0.4, 0.2)>

Table 2

The SVN-ITL matrix provided by decision maker D₂

Alternative	C ₁	C ₂	C ₃	C ₄
A ₁	< [(S₄,0), (S₆,0)], (0.1, 0.5, 0.3)>	< [(S₃,0), (S₅,0)], (0.2, 0.5, 0.2)>	< [(S₃,0), (S₄,0)], (0.2, 0.6, 0.2)>	< [(S₆,0), (S₇,0)], (0.4, 0.3, 0.3)>
A ₂	< [(S₅,0), (S₇,0)], (0.4, 0.3, 0.2)>	< [(S₃,0), (S₄,0)], (0.3, 0.4, 0.2)>	< [(S₄,0), (S₅,0)], (0.2, 0.4, 0.3)>	< [(S₃,0), (S₇,0)], (0.2, 0.5, 0.2)>
A ₃	< [(S₄,0), (S₅,0)], (0.2, 0.4, 0.4)>	< [(S₄,0), (S₆,0)], (0.2, 0.5, 0.2)>	< [(S₂,0), (S₅,0)], (0.4, 0.4, 0.2)>	< [(S₃,0), (S₄,0)], (0.3, 0.5, 0.2)>
A ₄	< [(S₅,0), (S₆,0)], (0.3, 0.4, 0.3)>	< [(S₄,0), (S₇,0)], (0.4, 0.3, 0.3)>	< [(S₂,0), (S₅,0)], (0.3, 0.4, 0.3)>	< [(S₄,0), (S₆,0)], (0.2, 0.4, 0.3)>

Table 3

The SVN-ITL matrix provided by decision maker D₃

Alternative	C ₁	C ₂	C ₃	C ₄
A ₁	< [(S₅,0), (S₆,0)], (0.2, 0.4, 0.4)>	< [(S₃,0), (S₄,0)], (0.3, 0.5, 0.2)>	< [(S₄,0), (S₇,0)], (0.4, 0.3, 0.3)>	< [(S₄,0), (S₆,0)], (0.2, 0.5, 0.3)>
A ₂	< [(S₄,0), (S₅,0)], (0.3, 0.5, 0.5)>	< [(S₅,0), (S₇,0)], (0.3, 0.4, 0.2)>	< ([(S₂,0), (S₅,0)], (0.1, 0.6, 0.3)>	< ([(S₃,0), (S₆,0)], (0.4, 0.4, 0.2)>
A ₃	< [(S₄,0), (S₅,0)], (0.2, 0.5, 0.4)>	< [(S₅,0), (S₇,0)], (0.3, 0.4, 0.3)>	< [(S₁,0), (S₃,0)], (0.1, 0.6, 0.2)>	< [(S₄,0), (S₆,0)], (0.2, 0.5, 0.2)>
A ₄	< [(S₃,0), (S₃,0)], (0.2, 0.5, 0.5)>	< [(S₃,0), (S₆,0)], (0.1, 0.5, 0.4)>	< [(S₄,0), (S₅,0)], (0.3, 0.4, 0.2)>	< [(S₅,0), (S₆,0)], (0.4, 0.3, 0.3)>

Step 1. Aggregate the evaluation information of three experts by the SVN-ITLWA operator in Equation (20), and the results are shown in Table 4.

Step 2. Aggregate the evaluation information of four attributes by SVN-ITLWMSM operator (suppose k = 2), and the results are given as follows: $\begin{matrix} r_{1} & = & < [(S_{1}, - 0.005), (S_{1}, 0.047)], \\ (0.07, 0.81, 0.72) >, \\ r_{2} & = & < [(S_{1}, - 0.006), (S_{1}, 0.052)], \\ (0.09, 0.80, 0.70) >, \\ r_{3} & = & < [(S_{1}, - 0.011), (S_{1}, 0.049)], \\ (0.06, 0.84, 0.72) >, \\ r_{4} & = & < [(S_{1}, - 0.010), (S_{1}, 0.047)], \\ (0.08, 0.79, 0.74) > . \end{matrix}$

Step 3. The score values of alternatives are calculated by Equation (22), which are shown as follows:

$\begin{matrix} S (r_{1}) & = & 0.0264, S (r_{2}) = 0.0290, \\ S (r_{3}) & = & 0.0242, S (r_{4}) = 0.0260, \end{matrix}$ which means the rank order of alternatives is A₂ ≻ A₁ ≻ A₄ ≻ A₃. Therefore, the most desirable alternative is A₂.

6.2 Comparison with other method

In order to verify the effectiveness of the proposed method, we compare our method with a method based on the Bonferroni mean [43] operator. Inspired by the idea of [44], we present the expression of the single-valued neutrosophic interval 2-tuple linguistic weighted Bonferroni mean (SVN-ITLWBM) operator:

$\begin{matrix} SVN - {ITLWBM}^{p, q} ({\tilde{p}}_{1}, {\tilde{p}}_{2}, \dots, {\tilde{p}}_{n}) = \frac{1}{p + q} {(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ((w_{i} {\tilde{p}}_{i})^{p} (w_{j} {\tilde{q}}_{j})^{q}))}^{\frac{1}{n (n - 1)}} \\ = 〈 [Δ ({(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(w_{i} Δ^{- 1} (s_{i}, a_{i}))}^{p} {(w_{j} Δ^{- 1} (s_{j}, a_{j}))}^{q})}^{\frac{1}{p + q}}), \\ Δ ({(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(w_{i} Δ^{- 1} (l_{i}, β_{i}))}^{p} {(w_{j} Δ^{- 1} (l_{j}, β_{j}))}^{q})}^{\frac{1}{p + q}})], \\ ({(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - (1 - (1 - u_{i})^{w_{i}})^{p} (1 - (1 - u_{j})^{w_{j}})^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - (1 - (η_{i})^{w_{i}})^{p} (1 - (η_{j})^{w_{j}})^{q})^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - (1 - (v_{i})^{w_{i}})^{p} (1 - (v_{j})^{w_{j}})^{q})^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}) 〉 . \end{matrix}$ (24)

For the collective matrix shown in Table 4, we use the SVN-ITLWBM operator in Equation (24) to determine the overall assessment value r_i (i = 1, 2, …, m) of all alternatives and further calculate the score values S (r_i) by Equation (22). The ranking orders of alternatives are shown in Table 5.

Table 4

The collective single-valued neutrosophic interval 2-tuple linguistic matrix

Alternative	C ₁	C ₂	C ₃	C ₄
A ₁	< [(S₅,–0.04), (S₆,0)], (0.13, 0.47, 0.33)>	< [(S₃,–0.05), (S₄,0.04)], (0.31, 0.46, 0.24)>	< [(S₄,0.01), (S₆,–0.05)], (0.39, 0.37, 0.22)>	< [(S₄,0.03), (S₆,0.04)], (0.27, 0.42, 0.26)>
A ₂	< [(S₄,0.04), (S₆,–0.05)], (0.33, 0.39, 0.3)>	< [(S₄,0.045), (S₆,0.01)], (0.34, 0.36, 0.24)>	< ([(S₃,0.01), S₅,–0.05)], (0.13, 0.56, 0.26)>	< ([(S₃,0.05), (S₆,0.04)], (0.39, 0.38, 0.15)>
A ₃	< [(S₄,–0.05), (S₅,0.05)], (0.16, 0.47, 0.4)>	< [(S₄,0.04), (S₇,–0.04)], (0.23, 0.47, 0.26)>	< [(S₃,–0.06),(S₅,–0.02)], (0.29, 0.53, 0.15)>	< [(S₄,0.01), (S₅,0.05)], (0.23, 0.5, 0.24)>
A ₄	< [(S₅,–0.02), (S₆,–0.06)], (0.32, 0.32, 0.39)>	< [(S₃,–0.01), (S₆,–0.06)], (0.25, 0.46, 0.28)>	< [(S₃,–0.01), (S₅,0)], (0.26, 0.44, 0.27)>	< [(S₄,–0.02), (S₆,0)], (0.3, 0.37, 0.26)>

Table 5

Comparison with SVN-ITLWMSM and SVN-ITLWBM operators

Aggregation operators	Parameter values	Ranking
SVN-ITLWMSM	k = 1	A₂> A₁> A₄> A₃
SVN-ITLWMSM	k = 2	A₂> A₁> A₄> A₃
SVN-ITLWMSM	k = 3	A₂> A₁> A₄> A₃
SVN-ITLWMSM	k = 4	A₂> A₁> A₄> A₃
SVN-ITLWBM	p = 0.1, q = 0.1	A₂> A₁> A₄> A₃
SVN-ITLWBM	p = 0.1, q = 10	A₂> A₄> A₁> A₃
SVN-ITLWBM	p = 10, q = 0.1	A₂> A₄> A₁> A₃
SVN-ITLWBM	p = 10, q = 10	A₂> A₁> A₄> A₃

From Table 5, we can see that the ranking orders of alternatives are unchanged by the SVN-ITLWMSM proposed in this paper, while there are some changes on the ranking of alternatives by the SVN-ITLWBM operator in Equation (24). This verifies the good stability of the proposed method in this paper. Moreover, the advantage of the SVN-ITLWMSM operator proposed in this paper is that the operator only needs one parameter k to capture the interrelationship among arguments, and the parameter k takes on its value from a finite integer set {1, 2, … , n }. The method proposed in this paper reflects the overall interrelationships among multi-input arguments, while the SVN-ITLWBM operator only considers the relationship between two arguments. Therefore, the proposed method in this paper considers much more information among the multi-input arguments, and it is more flexible for solving MAGDM problems under single-valued neutrosophic interval 2-tuple linguistic environment.

7 Conclusions

Single-valued neutrosophic interval 2-tuple linguistic set is a flexible approach of expressing the uncertain linguistic evaluation information provided by decision makers by combining the definitions of single-valued neutrosophic set and interval 2-tuple. Maclaurin symmetric mean operator is a suitable tool for dealing with existing interrelationship among evaluation information. In this paper, we proposed the concept of the single-valued neutrosophic interval 2-tuple linguistic sets (SVN-ITLSs) by combining the definitions of single-valued neutrosophic set and interval 2-tuple, and further proposed three aggregation operators to aggregate single-valued neutrosophic interval 2-tuple linguistic information, i.e., the SVN-ITLMSM, SVN-ITLWA and SVN-ITLWMSM operators. Some desirable properties of the proposed SVN-ITLMSM operator are investigated. Based on the proposed SVN-ITLWA and SVN-ITLWMSM operators, an approach to MAGDM is developed under the single-valued neutrosophic interval 2-tuple linguistic environment. Finally, a practical example of emergency alternative selection is given to illustrate the application of the proposed method. In future research, we will extend the Heronian mean operator to the single-valued neutrosophic interval 2-tuple linguistic environment, and further focus on the applications of the proposed aggregation operators to other domains, such as risk management, services quality assessment, supply chain management, etc.

Footnotes

Acknowledgments

This research is supported by Program for New Century Excellent Talents in University (NCET-13-0037), Humanities and Social Sciences Foundation of Ministry of Education of China (14YJA630019).

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