Multiple attribute group decision making methods based on trapezoidal fuzzy neutrosophic numbers

Abstract

For the multiple attribute group decision making (MAGDM) problem, in which the attribute weights are unknown and the attribute value of alternatives is in the form of a trapezoidal fuzzy neutrosophic number, this paper proposes two multiple attribute group decision making methods: one based on the trapezoidal fuzzy neutrosophic number hybrid averaging (TrFNNHA) operator, and the other based on the technique for order performance by similarity to ideal solution (TOPSIS) method. First, the attribute weights are obtained using the truth favorite relative expected value, and the distance measure defined using the cosine similarity measure. Next, a proposed trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging (TrFNNOWAA) operator and a proposed trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging (TrFNNHWAA) operator are used to aggregate the trapezoidal fuzzy neutrosophic information. Then, the score and accuracy functions of a trapezoidal neutrosophic number are used to rank the alternatives and obtain the best alternative in a trapezoidal fuzzy neutrosophic environment. In addition, an extended TOPSIS method is also proposed to deal with trapezoidal fuzzy neutrosophic information. An illustrative example and sensitivity analysis demonstrate the applicability and effectiveness of the proposed group decision making methods.

Keywords

Multiple attribute group decision making trapezoidal fuzzy neutrosophic set trapezoidal fuzzy neutrosophic number (TrFNN)trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging (TrFNNOWAA) operator trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging (TrFNNHWAA) operator TOPSIS method sensitivity analysis

1 Introduction

Multiple attribute decision making problems with quantitative or qualitative attribute values have broad application in the areas of operation research, management science, urban planning, natural science, military affairs, etc. [1]. However, because of the ambiguity of people’s thinking and the complexity of objectives, decision makers have difficulty expressing attribute values using crisp numbers. This valuation information can be expressed in a more reasonable and natural manner with fuzzy information such as fuzzy set (FS) introduced by Zadeh [2], intuitionistic fuzzy set (IFS) studied by Atanassov [3], and neutrosophic set (NS) pioneered by Smarandache [4]. Neutrosophic sets are characterized by a truth-membership degree (T), an indeterminacy-membership degree (I), and a falsity-membership degree (F) and are a generalization of sets such as crisp sets, fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, etc. They are capable of dealing with incomplete, indeterminate, and inconsistent information. For example, a voting scenario, in which thirty percent vote “Yes,” twenty percent vote “No,” ten percent give up, and forty percent are undecided. Such a vote is beyond the scope of IFS to distinguish the information between “giving up” and “undecided.” Hence, further generalizations of fuzzy and intuitionistic fuzzy sets are required.

However, neutrosophic sets are difficult to apply directly in actual scientific and engineering applications. To simplify their practical application, Wang et al. [5] proposed single-valued neutrosophic sets (SVNSs), an instance of NS and Ye [6] introduced and defined the operational laws of simplified neutrosophic sets (SNSs) as well as various aggregation operators. Wang et al. [7] further defined multi-valued neutrosophic sets and multi-valued neutrosophic numbers, and also proposed the TODIM method in a multi-valued neutrosophic number environment. Wang et al. [8] proposed interval neutrosophic sets (INSs) along with their set-theoretic operators and Zhang et al. [9] proposed an improved weighted correlation coefficient measure for INSs and also described its application in multi-criteria decision making. Smarandache [10] proposed the symbolic (or literal) neutrosophic theory, which refers to the use of abstract symbols in neutrosophics. The theory has subsequently been intensively studied and gained popularity owing to the application of the symbols to real-world problems. Broumi et al. [11] combined the concept of single-valued neutrosophic sets and graph theory, and introduced various types of single-valued neutrosophic graphs (SVNG), including strong single-valued neutrosophic graph, constant single-valued neutrosophic graph, and complete single-valued neutrosophic graph. In addition, they investigated various of their properties with proofs and examples. Broumi et al. [12] also introduced the neighborhood degree of a vertex and the closed neighborhood degree of a vertex in single-valued neutrosophic graphs as generalizations of neighborhood degree of a vertex and closed neighborhood degree of a vertex in fuzzy graphs and intuitionistic fuzzy graphs. They also proved the necessary and sufficient condition for a single-valued neutrosophic graph to be an isolated single-valued neutrosophic graph [13]. Further, they defined the concept of bipolar single neutrosophic graphs as the generalization of bipolar fuzzy graphs, N-graphs, intuitionistic fuzzy graphs, single-valued neutrosophic graphs, and bipolar intuitionistic fuzzy graphs [14], and also introduced and investigated the properties of bipolar single-valued neutrosophic graphs, strong bipolar single-valued neutrosophic graphs, complete bipolar single-valued neutrosophic graphs, and regular bipolar single-valued neutrosophic graphs [15]. The application of bipolar neutrosophic sets has also been investigated, with an algorithm also being developed to find the shortest path on a network in which the weights of the edges are represented by bipolar neutrosophic numbers [16]. Broumi et al. [17] also introduced interval-valued neutrosophic graphs, proposed using Dijkstra’s algorithm to solve the shortest path problem in their case [18] and analyzed and introduced various operations for strong interval-valued neutrosophic graphs [19]. Ye [20] recently proposed trapezoidal fuzzy neutrosophic sets and trapezoidal fuzzy neutrosophic numbers, and applied them to multiple attribute decision making. Biswas [21] presented cosine similarity measure based multiple attribute decision making with trapezoidal fuzzy neutrosophic numbers. However, scant research has been conducted on trapezoidal fuzzy neutrosophic numbers [20, 21]. In this paper, we focus on information aggregation operators and ranking methods based on trapezoidal fuzzy neutrosophic numbers.

Operators are a kind of common information aggregation method and have been studied extensively. Ye [22] developed a simplified neutrosophic weighted arithmetic averaging (SNWAA) operator and a simplified neutrosophic weighted geometric averaging (SNWGA) operator for multiple attribute decision making under simplified neutrosophic environment. Ye [23] also introduced interval neutrosophic number ordered weighted aggregation operators. Zhang et al. [24] proposed interval neutrosophic number weighted averaging (INNWA) and interval neutrosophic number weighted geometric (INNWG) operators for multicriteria decision making. Liu et al. [25] proposed a single-valued neutrosophic normalized weighted Bonferroni mean (SVNNWBM) operator and analyzed its properties. Ye [26] proposed interval neutrosophic uncertain linguistic variables, and further proposed the interval neutrosophic uncertain linguistic weighted arithmetic averaging (INULWAA) and the interval neutrosophic uncertain linguistic weighted arithmetic averaging (INULWGA) operators. Peng et al. [27] introduced multi-valued neutrosophic sets (MVNSs) and proposed the multi-valued neutrosophic power weighted average (MVNPWA) operator and the multi-valued neutrosophic power weighted geometric (MVNPWG) operator; they also discussed the desirable properties of two operators. A trapezoidal neutrosophic number weighted arithmetic averaging (TNNWAA) operator, and a trapezoidal neutrosophic number weighted geometric averaging (TNNWGA) operator have also been proposed and applied to multiple attribute decision making with trapezoidal neutrosophic numbers [20]. However, to the best of our knowledge, trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging (TrFNNOWAA) and trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging (TrFNNHWAA) operators for multiple attribute group decision making with trapezoidal fuzzy neutrosophic numbers are absent.

The technique for order performance by similarity to ideal solution (TOPSIS) method [28] is used extensively with multiple attribute decision making problems, with particular focus on choosing the alternative with the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. The traditional TOPSIS is only used to solve decision making problems with crisp numbers, and many extended TOPSIS have been proposed to deal with fuzzy information and intuitionistic fuzzy information [29 –31]. For example, Zhang et al. [32] proposed single-valued neutrosophic/interval neutrosophic TOPSIS method to calculate the relative closeness coefficient of each alternative to the single-valued neutrosophic/interval neutrosophic positive ideal solution. Pramanik et al. [33] proposed the TOPSIS technique for solving single-valued neutrosophic soft set expert-based multi-attribute decision making problems. Broumi et al. [34] extended it to interval neutrosophic uncertain linguistic information, and proposed an extended TOPSIS method to solve multiple attribute decision making problems in which the attribute value takes the form of the interval neutrosophic uncertain linguistic variables. Biswas et al. [35] presented a new TOPSIS-based approach for MAGDM under a simplified neutrosophic environment. In their evaluation process, the ratings of each alternative with respect to each attribute are given as linguistic variables characterized by single-valued neutrosophic numbers. However, to date, the extended TOPSIS method has not been applied to multiple attribute group decision making with trapezoidal fuzzy neutrosophic numbers.

In this paper, we propose two multiple attribute group decision making methods, one based on the trapezoidal fuzzy neutrosophic number hybrid averaging (TrFNNHA) operator and the other based on the TOPSIS method, for the trapezoidal fuzzy neutrosophic environment. First, the attribute weights are obtained using the truth favorite relative expected value, and the distance measure defined using the cosine similarity measure. Next, a proposed trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging (TrFNNOWAA) operator and a proposed trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging (TrFNNHWAA) operator are used to aggregate the trapezoidal fuzzy neutrosophic information. Then, the score and accuracy functions of a trapezoidal neutrosophic number are used to rank the alternatives and obtain the best alternative in a trapezoidal fuzzy neutrosophic environment.

The remainder of this paper is organized as follows: Section 2 briefly introduces various preliminaries. Section 3 introduces the two proposed trapezoidal fuzzy neutrosophic number operators: TrFNNOWAA and TrFNNHWAA. Section 4 proposes two multiple attribute group decision making methods for trapezoidal fuzzy neutrosophic environments based on the TrFNNHWAA operator and the extended TOPSIS method. Section 5 presents an illustrative example and detailed sensitivity analysis to show the effectiveness of the proposed approaches. Finally, Section 6 concludes this paper.

2 Preliminaries

In this section, we briefly outline various essential concepts such as trapezoidal intuitionistic fuzzy numbers (TrIFNs), trapezoidal fuzzy neutrosophic sets, trapezoidal fuzzy neutrosophic numbers (TrFNNs), operational rules of TrFNN, the truth favorite relative expected value of TrFNN, and the score function and accuracy function of a trapezoidal neutrosophic number. We also give the distance measure based on cosine similarity measure.

2.1 Trapezoidal Intuitionistic Fuzzy Number (TrIFN)

Definition 1. A TrIFN $\tilde{A} = < ({\underline{a}}_{1}, {\underline{a}}_{2}, {\underline{a}}_{3}, {\underline{a}}_{4}), ({\bar{a}}_{1}, {\bar{a}}_{2}, {\bar{a}}_{3}, {\bar{a}}_{4}) >$ is an intuitionistic fuzzy set on R, with the membership function $μ_{\tilde{A}}$ and the non-member- ship function $ν_{\tilde{A}}$ defined as $μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - {\underline{a}}_{1}}{{\underline{a}}_{2} - {\underline{a}}_{1}}, & if x \in [{\underline{a}}_{1}, {\underline{a}}_{2}) \\ 1, & if x \in [{\underline{a}}_{2}, {\underline{a}}_{3}] \\ \frac{{\underline{a}}_{4} - x}{{\underline{a}}_{4} - {\underline{a}}_{3}}, & if x \in ({\underline{a}}_{3}, {\underline{a}}_{4}] \\ 0, & otherwise, \end{matrix}$ (1) and

$ν_{\tilde{A}} (x) = {\begin{matrix} \frac{{\bar{a}}_{2} - x}{{\bar{a}}_{2} - {\bar{a}}_{1}}, & if x \in [{\bar{a}}_{1}, {\bar{a}}_{2}) \\ 0, & if x \in [{\bar{a}}_{2}, {\bar{a}}_{3}] \\ \frac{x - {\bar{a}}_{3}}{{\bar{a}}_{4} - {\bar{a}}_{3}}, & if x \in ({\bar{a}}_{3}, {\bar{a}}_{4}] \\ 1, & otherwise . \end{matrix}$ (2) where ${\underline{a}}_{1} \leq {\underline{a}}_{2} \leq {\underline{a}}_{3} \leq {\underline{a}}_{4}, {\bar{a}}_{1} \leq {\bar{a}}_{2} \leq {\bar{a}}_{3} \leq {\bar{a}}_{4}, {\bar{a}}_{1} \leq {\underline{a}}_{1}, {\bar{a}}_{2} \leq {\underline{a}}_{2}, {\underline{a}}_{3} \leq {\bar{a}}_{3}, {\underline{a}}_{4} \leq {\bar{a}}_{4}$ . Other trapezoidal intuitionistic fuzzy number forms have also been proposed [36 –42]. A comparison of these trapezoidal intuitionistic fuzzy numbers (TrIFN) is given in Table 1.

Table 1

Comparison of various trapezoidal intuitionistic fuzzy numbers

TrIFN forms	Graphical representation	Differences
Roseline and Amirtharaj [37]: $\tilde{A}$ is a TrIFN, $\tilde{A} = (b_{1}, a_{1}, b_{2}, a_{2}, a_{3}, b_{3}, a_{4}, b_{4}),$ b₁ ≤ a₁ ≤ b₂ ≤ a₂ ≤ a₃ ≤ b₃ ≤ a₄ ≤ b₄.		This TrIFN representation is a generalized representation similar to that by Ban and Tuse [36].
Ban and Tuse [36]: $\tilde{A}$ is a TrIFN, $\tilde{A} = < ({\underline{a}}_{1}, {\underline{a}}_{2}, {\underline{a}}_{3}, {\underline{a}}_{4}), ({\bar{a}}_{1}, {\bar{a}}_{2}, {\bar{a}}_{3}, {\bar{a}}_{4}) >$ , $\begin{matrix} {\underline{a}}_{1} \leq {\underline{a}}_{2} \leq {\underline{a}}_{3} \leq {\underline{a}}_{4}, \\ {\bar{a}}_{1} \leq {\bar{a}}_{2} \leq {\bar{a}}_{3} \leq {\bar{a}}_{4}, \\ {\bar{a}}_{1} \leq {\underline{a}}_{1}, {\bar{a}}_{2} \leq {\underline{a}}_{2}, {\underline{a}}_{3} \leq {\bar{a}}_{3}, {\underline{a}}_{4} \leq {\bar{a}}_{4} . \end{matrix}$		This TrIFN representation is a generalized representation similar to that by Roseline and Amirtharaj [37].
Rezvani [38]: $\tilde{A}$ is a TrIFN, $\tilde{A} = (a_{1}, b_{1}, c_{1}, d_{1}; a_{1}^{'}, b_{1}, c_{1}, d_{1}^{'}),$ $a_{1}^{'} \leq a_{1} \leq b_{1} \leq c_{1} \leq d_{1} \leq d_{1}^{'} .$		When b₂ = a₂ and a₃ = b₃, this TrIFN representation is the same as that of Roseline and Amirtharaj [37]. Thus, it is a special case of [37].
Jayagowri and Ramani [39]: $\tilde{A}$ is a TrIFN, $\tilde{A} = < ([a, b, c, d]; μ_{\tilde{A}}), ([a_{1}, b, c, d_{1}]; ν_{\tilde{A}})$ >, $\begin{matrix} 0 \leq μ_{\tilde{A}} \leq 1, 0 \leq ν_{\tilde{A}} \leq 1, \\ 0 \leq μ_{\tilde{A}} + ν_{\tilde{A}} \leq 1 . \end{matrix}$		This TrIFN representation is similar to that by Rezvani [38]. Thus, it is also a special case of [37].
De and Das [40]: $\tilde{A}$ is a TrIFN, $\tilde{A} = < (a_{1}, a_{2}, a_{3}, a_{4}); w_{\tilde{A}}, u_{\tilde{A}} >,$ $\begin{matrix} 0 \leq w_{\tilde{A}} \leq 1, 0 \leq u_{\tilde{A}} \leq 1, \\ 0 \leq w_{\tilde{A}} + u_{\tilde{A}} \leq 1 . \end{matrix}$		When a₁ = a and d = d_1, this TrIFN representation is the same as that of Jayagowri and Ramani [39]. Hence, it is a simplified representation of [39].
Das and Guha [41]: $\tilde{A}$ is a TrIFN, $\tilde{A} = < (a, b, c, d); w_{\tilde{A}}, u_{\tilde{A}} >,$ $\begin{matrix} 0 \leq w_{\tilde{A}} \leq 1, 0 \leq u_{\tilde{A}} \leq 1, \\ 0 \leq w_{\tilde{A}} + u_{\tilde{A}} \leq 1 . \end{matrix}$		The representation of this TrIFN is the same as that by De and Das [40].
Parvathi and Malathi [42]: $\tilde{A}$ is a symmetric trapezoidal intuitionistic fuzzy number (STrIFN), $\tilde{A} = [a_{1}, a_{2}, h, h; a_{1}, a_{2}, h^{'}, h^{'}],$ h ≤ h′.		When b - a₁ = d₁ - c, b - a = d - c, and b - a < b - a₁, the representation of this TrIFN is the same as that by Jayagowri and Ramani [39]. Thus, it is a special case of [39].

2.2 Trapezoidal Fuzzy Neutrosophic Number (TrFNN)

Definition 2. [20] Let X be a universe of discourse, then a trapezoidal fuzzy neutrosophic set $\tilde{N}$ in X has the following form: $\tilde{N} = {〈 x, T_{\tilde{N}} (x), I_{\tilde{N}} (x), F_{\tilde{N}} (x) 〉 | x \in X},$ (3) where $T_{\tilde{N}} (x) \subset [0, 1]$ , $I_{\tilde{N}} (x) \subset [0, 1]$ and $F_{\tilde{N}} (x) \subset [0, 1]$ are three TrFNNs, $T_{\tilde{N}} (x) = (t_{\tilde{N}}^{1} (x),$ $t_{\tilde{N}}^{2} (x),$ $t_{\tilde{N}}^{3} (x),$ $t_{\tilde{N}}^{4} (x)) : X \to [0, 1]$ , $I_{\tilde{N}} (x) = (i_{\tilde{N}}^{1} (x),$ $i_{\tilde{N}}^{2} (x),$ $i_{\tilde{N}}^{3} (x),$ $i_{\tilde{N}}^{4} (x)) : X \to [0, 1]$ , and $F_{\tilde{N}} (x) = (f_{\tilde{N}}^{1} (x),$ $f_{\tilde{N}}^{2} (x),$ $f_{\tilde{N}}^{3} (x),$ $f_{\tilde{N}}^{4} (x)) : X \to [0, 1]$ with the condition $0 \leq t_{\tilde{N}}^{4} (x) + i_{\tilde{N}}^{4} (x) + f_{\tilde{N}}^{4} (x) \leq 3, x \in X$ .

Definition 3. [21] A TrFNN $\tilde{n}$ is denoted by $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}, b_{4}), (c_{1}, c_{2}, c_{3}, c_{4}) 〉$ in a universe of discourse X. The parameters satisfy the following relations: a₁ ≤ a₂ ≤ a₃ ≤ a₄, b₁ ≤ b₂ ≤ b₃ ≤ b₄ and c₁ ≤ c₂ ≤ c₃ ≤ c₄. Its truth-membership function is defined as follows:

$T_{\tilde{N}} (x) = 〈 \begin{matrix} \frac{x - a_{1}}{a_{2} - a_{1}} & a_{1} \leq x \leq a_{2} \\ 1 & a_{2} \leq x \leq a_{3} \\ \frac{a_{4} - x}{a_{4} - a_{3}} & a_{3} \leq x \leq a_{4} \\ 0 & otherwise \end{matrix} 〉,$ (4)

Its indeterminacy-membership function is defined as follows: $I_{\tilde{N}} (x) = 〈 \begin{matrix} \frac{b_{2} - x}{b_{2} - b_{1}} & b_{1} \leq x \leq b_{2} \\ 0 & b_{2} \leq x \leq b_{3} \\ \frac{x - b_{3}}{b_{4} - b_{3}} & b_{3} \leq x \leq b_{4} \\ 1 & otherwise \end{matrix} 〉,$ (5)

Further, its falsity-membership function is defined as follows: $F_{\tilde{N}} (x) = 〈 \begin{matrix} \frac{c_{2} - x}{c_{2} - c_{1}} & c_{1} \leq x < c_{2} \\ 0 & c_{2} \leq x \leq c_{3} \\ \frac{x - c_{3}}{c_{4} - c_{3}} & c_{3} < x \leq c_{4} \\ 1 & otherwise \end{matrix} 〉,$ (6)

When a₂ = a₃, b₂ = b₃, and c₂ = c₃ in a TrFNN $\tilde{n}$ , the number reduces to a triangular fuzzy neutrosophic number, which is considered a special case of the trapezoidal fuzzy neutrosophic number. The graphical representation of the trapezoidal fuzzy neutrosophic number is given in Fig. 1.

Fig.1

Truth-membership, indeterminacy-membership, and falsity-membership functions of TrFNN.

Definition 4. [20] Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2},$ b₃, b₄), (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}), (f_{1},$ f₂, f₃, f₄), (g₁, g₂, g₃, g₄) 〉 be two TrFNNs. Then, the following operational rules apply:

$\begin{matrix} {\tilde{n}}_{1} \oplus {\tilde{n}}_{2} \\ = 〈 \begin{matrix} (a_{1} + e_{1} - a_{1} e_{1}, a_{2} + e_{2} - a_{2} e_{2}, \\ a_{3} + e_{3} - a_{3} e_{3}, a_{4} + e_{4} - a_{4} e_{4}), \\ (b_{1} f_{1}, b_{2} f_{2}, b_{3} f_{3}, b_{4} f_{4}), \\ (c_{1} g_{1}, c_{2} g_{2}, c_{3} g_{3}, c_{4} g_{4}) \end{matrix} 〉; \end{matrix}$ (7)

$\begin{matrix} {\tilde{n}}_{1} \otimes {\tilde{n}}_{2} \\ = 〈 \begin{matrix} (a_{1} e_{1}, a_{2} e_{2}, a_{3} e_{3}, a_{4} e_{4}), \\ (b_{1} + f_{1} - b_{1} f_{1}, b_{2} + f_{2} - b_{2} f_{2}, \\ b_{3} + f_{3} - b_{3} f_{3}, b_{4} + f_{4} - b_{4} f_{4}), \\ (c_{1} + g_{1} - c_{1} g_{1}, c_{2} + g_{2} - c_{2} g_{2}, \\ c_{3} + g_{3} - c_{3} g_{3}, c_{4} + g_{4} - c_{4} g_{4}) \end{matrix} 〉; \end{matrix}$ (8)

$\begin{matrix} λ {\tilde{n}}_{1} \\ = 〈 \begin{matrix} (1 - (1 - a_{1})^{λ}, 1 - (1 - a_{2})^{λ}, \\ 1 - (1 - a_{3})^{λ}, 1 - (1 - a_{4})^{λ}), \\ (b_{1}^{λ}, b_{2}^{λ}, b_{3}^{λ}, b_{4}^{λ}), \\ (c_{1}^{λ}, c_{2}^{λ}, c_{3}^{λ}, c_{4}^{λ}) \end{matrix} 〉, λ > 0; \end{matrix}$ (9)

$\begin{matrix} {\tilde{n}}_{1}^{λ} \\ = 〈 \begin{matrix} (a_{1}^{λ}, a_{2}^{λ}, a_{3}^{λ}, a_{4}^{λ}), \\ (1 - (1 - b_{1})^{λ}, 1 - (1 - b_{2})^{λ}, \\ 1 - (1 - b_{3})^{λ}, 1 - (1 - b_{4})^{λ}), \\ (1 - (1 - c_{1})^{λ}, 1 - (1 - c_{2})^{λ}, \\ 1 - (1 - c_{3})^{λ}, 1 - (1 - c_{4})^{λ}) \end{matrix} 〉, \\ λ > 0; \end{matrix}$ (10)

${\tilde{n}}_{1} = {\tilde{n}}_{2}$ if a_i = e_i, b_i = f_i and c_i = g_ihold for i = 1, 2, 3, 4 i.e., (a₁, a₂, a₃, a₄) = (e₁, e₂, e₃, e₄), (b₁, b₂, b₃, b₄) = (f₁, f₂, f₃, f₄), (c₁, c₂, c₃, c₄) = (g₁, g₂, g₃, g₄).

Definition 5. [21] Let $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2},$ b₃, b₄), (c₁, c₂, c₃, c₄) 〉 be a TrFNN. Then, the truth favorite relative expected value of $\tilde{n}$ is defined by ${EV}^{T} (\tilde{n}) = \frac{3 \sum_{i = 1}^{4} a_{i}}{(\sum_{i = 1}^{4} a_{i} + \sum_{i = 1}^{4} b_{i} + \sum_{i = 1}^{4} c_{i})} .$ (11)

Definition 6. [20] Let $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2},$ b₃, b₄), (c₁, c₂, c₃, c₄) 〉 be a TrFNN. Then, a score function of a trapezoidal neutrosophic number can be defined as

$\begin{matrix} S (\tilde{n}) = \frac{1}{3} (2 + \frac{\sum_{i}^{4} a_{i}}{4} - \frac{\sum_{i}^{4} b_{i}}{4} - \frac{\sum_{i}^{4} c_{i}}{4}), \\ S (\tilde{n}) \in [0, 1] . \end{matrix}$ (12) where the larger the value of $S (\tilde{n})$ , the bigger the trapezoidal neutrosophic number $\tilde{n}$ .

Definition 7. [20] Let $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2},$ b₃, b₄), (c₁, c₂, c₃, c₄) 〉 be a TrFNN. Then, the accuracy function of a trapezoidal neutrosophic number can be defined as $H (\tilde{n}) = \frac{\sum_{i}^{4} a_{i}}{4} - \frac{\sum_{i}^{4} c_{i}}{4}, H (\tilde{n}) \in [- 1, 1] .$ (13) where the larger the value of $H (\tilde{n})$ , the bigger the trapezoidal neutrosophic number $\tilde{n}$ .

On the basis of the score function $S (\tilde{n})$ and the accuracy function $H (\tilde{n})$ , we give the following ordered relation between two trapezoidal neutrosophic numbers.

Definition 8. [20] Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2},$ b₃, b₄), (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}), (f_{1}, f_{2}, f_{3}, f_{4}), (g_{1}, g_{2}, g_{3}, g_{4}) 〉$ be two TrFNNs. Thus, $S ({\tilde{n}}_{1})$ and $S ({\tilde{n}}_{2})$ are the scores of ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ , respectively, and $H ({\tilde{n}}_{1})$ and $H ({\tilde{n}}_{2})$ are the accuracy degree of ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ , respectively. Consequently, the ordered relation between two TrFNNs is defined as follows:

If $S ({\tilde{n}}_{1}) > S ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} > {\tilde{n}}_{2}$ ;

If $S ({\tilde{n}}_{1}) < S ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} < {\tilde{n}}_{2}$ ;

If $S ({\tilde{n}}_{1}) = S ({\tilde{n}}_{2})$ , and

$H ({\tilde{n}}_{1}) > H ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} > {\tilde{n}}_{2}$ ;

$H ({\tilde{n}}_{1}) < H ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} < {\tilde{n}}_{2}$ .

Definition 9. [21] Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2},$ b₃, b₄), (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}), (f_{1},$ f₂, f₃, f₄), (g₁, g₂, g₃, g₄) 〉 be two TrFNNs. Then, the cosine similarity measure between ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ is defined as follows:

$\begin{matrix} S_{TrFNN} ({\tilde{n}}_{1}, {\tilde{n}}_{2}) \\ = \frac{\sum_{i = 1}^{4} a_{i} e_{i} + \sum_{i = 1}^{4} b_{i} f_{i} + \sum_{i = 1}^{4} c_{i} g_{i}}{\begin{matrix} (\sqrt{\sum_{i = 1}^{4} (a_{i})^{2} + \sum_{i = 1}^{4} (b_{i})^{2} + \sum_{i = 1}^{4} (c_{i})^{2}}) \times \\ (\sqrt{\sum_{i = 1}^{4} (e_{i})^{2} + \sum_{i = 1}^{4} (f_{i})^{2} + \sum_{i = 1}^{4} (g_{i})^{2}}) \end{matrix}} . \end{matrix}$ (14)

Definition 10. Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3},$ b₄), (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}), (f_{1}, f_{2},$ f₃, f₄), (g₁, g₂, g₃, g₄) 〉 be two TrFNNs. For the relationship between distance and similarity, the distance measure $D ({\tilde{n}}_{1}, {\tilde{n}}_{2})$ between ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ is defined as follows:

$\begin{matrix} D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = 1 - S_{TrFNN} ({\tilde{n}}_{1}, {\tilde{n}}_{2}) \\ = \frac{k - \sum_{i = 1}^{4} a_{i} e_{i} + \sum_{i = 1}^{4} b_{i} f_{i} + \sum_{i = 1}^{4} c_{i} g_{i}}{k} . \\ (k = (\sqrt{\sum_{i = 1}^{4} {(a_{i})}^{2} + \sum_{i = 1}^{4} {(b_{i})}^{2} + \sum_{i = 1}^{4} {(c_{i})}^{2}}) \\ \times (\sqrt{\sum_{i = 1}^{4} {(e_{i})}^{2} + \sum_{i = 1}^{4} {(f_{i})}^{2} + \sum_{i = 1}^{4} {(g_{i})}^{2}})) \end{matrix}$ (15)

Theorem 1. The distance measure $D ({\tilde{n}}_{1}, {\tilde{n}}_{2})$ of ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ satisfies the following properties:

$0 \leq D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) \leq 1$ ;

$D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = 0$ if and only if ${\tilde{n}}_{1} = {\tilde{n}}_{2}$ , i.e., a_i = e_i, b_i = f_i, and c_i = g_i hold for i = 1, 2, 3, 4;

$D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = D ({\tilde{n}}_{2}, {\tilde{n}}_{1})$ .

According to the literature [21], this theorem can be easily proved; therefore, we do not provide the proof here.

3 Aggregation operators based on TrFNNs

The ordered weighted arithmetic averaging operator and hybrid weighted arithmetic averaging operators are usually used for information aggregation in decision making. Based on Definition 4, we propose a trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging operator and a trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging operator.

3.1 Trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging operator

Definition 11. Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}), (b_{1 j}, b_{2 j}, b_{3 j}, b_{4 j}), (c_{1 j}, c_{2 j}, c_{3 j}, c_{4 j}) 〉 (j = 1, 2, \dots, n)$ be a collection of trapezoidal neutrosophic numbers. A trapezoidal fuzzy neutrosophic number ordered weighted arithmetic averaging (TrFNNOWAA) operator of dimension n is defined by

$\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = w_{1} {\tilde{n}}_{σ (1)} \oplus w_{2} {\tilde{n}}_{σ (2)} \oplus \dots \oplus w_{n} {\tilde{n}}_{σ (n)} \\ = \sum_{j = 1}^{n} w_{j} {\tilde{n}}_{σ (j)} . \end{matrix}$ (16) where w = (w₁, w₂, ⋯, w_n) ^T is an associated weight vector with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and (σ (1), σ (2), ⋯, σ (n)) is a permutation of (1, 2. ⋯, n), such that ${\tilde{n}}_{σ (j - 1)} \geq {\tilde{n}}_{σ (j)}$ for j = 1, 2, ⋯, n.

Theorem 2. Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}), (b_{1 j}, b_{2 j},$ b_3j, b_4j), (c_1j, c_2j, c_3j, c_4j) 〉 (j = 1, 2, ⋯, n) be a collection of TrFNNs. Thus, their aggregated value using the TrFNNOWAA operator is also a trapezoidal fuzzy neutrosophic number, and then

$\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = w_{1} {\tilde{n}}_{σ (1)} \oplus w_{2} {\tilde{n}}_{σ (2)} \oplus \dots \oplus w_{n} {\tilde{n}}_{σ (n)} \\ = 〈 (1 - \prod_{j = 1}^{n} (1 - a_{1 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - a_{2 σ (j)})^{w_{j}}, \\ 1 - \prod_{j = 1}^{n} (1 - a_{3 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - a_{4 σ (j)})^{w_{j}}), \\ (\prod_{j = 1}^{n} b_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{n} c_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{4 σ (j)}^{w_{j}}) 〉 . \end{matrix}$ (17) where w = (w₁, w₂, ⋯, w_n) ^T is an associated weight vector with w_j ∈ [0, 1], and $\sum_{j = 1}^{n} w_{j} = 1$ , and (σ (1), σ (2), ⋯, σ (n)) is a permutation of (1, 2, ⋯, n), such that ${\tilde{n}}_{σ (j - 1)} \geq {\tilde{n}}_{σ (j)}$ for j = 1, 2, ⋯, n.

Proof. The proof of Equation (17) can be done by means of mathematical induction.

(1) When n = 2, then

\begin{matrix} w_{1} {\tilde{n}}_{σ (1)} = 〈 (1 - (1 - (a_{1 σ (1)})^{w_{1}}, 1 - (1 - (a_{2 σ (1)})^{w_{1}}, \\ 1 - (1 - (a_{3 σ (1)})^{w_{1}}, 1 - (1 - (a_{4 σ (1)})^{w_{1}}), \\ (b_{1 σ (1)}^{w_{1}}, b_{2 σ (1)}^{w_{1}}, b_{3 σ (1)}^{w_{1}}, b_{4 σ (1)}^{w_{1}}), \\ (c_{1 σ (1)}^{w_{1}}, c_{2 σ (1)}^{w_{1}}, c_{3 σ (1)}^{w_{1}}, c_{4 σ (1)}^{w_{1}}) 〉, \end{matrix}

(18)

\begin{matrix} w_{2} {\tilde{n}}_{σ (2)} = 〈 (1 - (1 - a_{1 σ (2)})^{w_{2}}, 1 - (1 - a_{2 σ (2)})^{w_{2}}, \\ 1 - (1 - a_{3 σ (2)})^{w_{2}}, 1 - (1 - a_{4 σ (2)})^{w_{2}}), \\ (b_{1 σ (2)}^{w_{2}}, b_{2 σ (2)}^{w_{2}}, b_{3 σ (2)}^{w_{2}}, b_{4 σ (2)}^{w_{2}}), \\ (c_{1 σ (2)}^{w_{2}}, c_{2 σ (2)}^{w_{2}}, c_{3 σ (2)}^{w_{2}}, c_{4 σ (2)}^{w_{2}}) 〉, \end{matrix}

(19)

Thus, $\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = w_{1} {\tilde{n}}_{σ (1)} \oplus w_{2} {\tilde{n}}_{σ (2)} \\ = 〈 (1 - (1 - a_{1 σ (1)})^{w_{1}} + 1 - (1 - a_{1 σ (2)})^{w_{2}} \\ - (1 - (1 - a_{1 σ (1)})^{w_{1}}) (1 - (1 - a_{1 σ (2)})^{w_{2}}), \\ 1 - (1 - a_{2 σ (1)})^{w_{1}} + 1 - (1 - a_{2 σ (2)})^{w_{2}} \\ - (1 - (1 - a_{2 σ (1)})^{w_{1}}) (1 - (1 - a_{2 σ (2)})^{w_{2}}), \\ 1 - (1 - a_{3 σ (1)})^{w_{1}} + 1 - (1 - a_{3 σ (2)})^{w_{2}} \\ - (1 - (1 - a_{3 σ (1)})^{w_{1}}) (1 - (1 - a_{3 σ (2)})^{w_{2}}), \\ 1 - (1 - a_{4 σ (1)})^{w_{1}} + 1 - (1 - a_{4 σ (2)})^{w_{2}} \\ - (1 - (1 - a_{4 σ (1)})^{w_{1}}) (1 - (1 - a_{4 σ (2)})^{w_{2}})), \\ (b_{1 σ (1)}^{w_{1}} b_{1 σ (2)}^{w_{2}}, b_{2 σ (1)}^{w_{1}} b_{2 σ (2)}^{w_{2}}, b_{3 σ (1)}^{w_{1}} b_{3 σ (2)}^{w_{2}}, b_{4 σ (1)}^{w_{1}} b_{4 σ (2)}^{w_{2}}), \\ (c_{1 σ (1)}^{w_{1}} c_{1 σ (2)}^{w_{2}}, c_{2 σ (1)}^{w_{1}} c_{2 σ (2)}^{w_{2}}, c_{3 σ (1)}^{w_{1}} c_{3 σ (2)}^{w_{2}}, c_{4 σ (1)}^{w_{1}} c_{4 σ (2)}^{w_{2}}) 〉 \\ = 〈 (1 - \prod_{j = 1}^{2} (1 - a_{1 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{2} (1 - a_{2 σ (j)})^{w_{j}}, \\ 1 - \prod_{j = 1}^{2} (1 - a_{3 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{2} (1 - a_{4 σ (j)})^{w_{j}}), \\ (\prod_{j = 1}^{2} b_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{2} b_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{2} b_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{2} b_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{2} c_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{2} c_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{2} c_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{2} c_{4 σ (j)}^{w_{j}}) 〉 . \end{matrix}$ (20)

(2) If Equation (20) holds for n = k, then

$\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{k}) \\ = w_{1} {\tilde{n}}_{σ (1)} \oplus w_{2} {\tilde{n}}_{σ (2)} \oplus \dots w_{n} {\tilde{n}}_{σ (k)} \\ = 〈 (1 - \prod_{j = 1}^{k} (1 - a_{1 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{k} (1 - a_{2 σ (j)})^{w_{j}}, \\ 1 - \prod_{j = 1}^{k} (1 - a_{3 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{k} (1 - a_{4 σ (j)})^{w_{j}}), \\ (\prod_{j = 1}^{k} b_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{k} b_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{k} b_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{k} b_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{k} c_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{k} c_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{k} c_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{k} c_{4 σ (j)}^{w_{j}}) 〉 . \end{matrix}$ (21)

(3) When n = k + 1, by the operational laws in Definition 4, we get $\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{k}) \\ = w_{1} {\tilde{n}}_{σ (1)} \oplus w_{2} {\tilde{n}}_{σ (2)} \oplus \dots w_{n} {\tilde{n}}_{σ (k)} \\ = 〈 (1 - \prod_{j = 1}^{k} (1 - a_{1 σ (j)})^{w_{j}} + 1 - (1 - (a_{1 σ (k + 1)})^{w_{k + 1}} \\ - (1 - \prod_{j = 1}^{k} (1 - a_{1 σ (j)})^{w_{j}}) (1 - (1 - (a_{1 σ (k + 1)})^{w_{k + 1}}), \\ 1 - \prod_{j = 1}^{k} (1 - a_{2 σ (j)})^{w_{j}} + 1 - (1 - (a_{2 σ (k + 1)})^{w_{k + 1}} \\ - (1 - \prod_{j = 1}^{k} (1 - a_{2 σ (j)})^{w_{j}}) (1 - (1 - (a_{2 σ (k + 1)})^{w_{k + 1}}), \\ 1 - \prod_{j = 1}^{k} {(1 - a_{3 σ (j)})}^{w_{j}} + 1 - (1 - (a_{3 σ (k + 1)})^{w_{k + 1}} \\ - (1 - \prod_{j = 1}^{k} (1 - a_{3 σ (j)})^{w_{j}}) (1 - (1 - (a_{3 σ (k + 1)})^{w_{k + 1}}), \\ 1 - \prod_{j = 1}^{k} (1 - a_{4 σ (j)})^{w_{j}} + 1 - (1 - (a_{4 σ (k + 1)})^{w_{k + 1}} \\ - (1 - \prod_{j = 1}^{k} (1 - a_{4 σ (j)})^{w_{j}}) (1 - (1 - (a_{4 σ (k + 1)})^{w_{k + 1}})), \\ (\prod_{j = 1}^{k + 1} b_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{k + 1} c_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{4 σ (j)}^{w_{j}}) 〉 \\ = 〈 (1 - \prod_{j = 1}^{k + 1} {(1 - a_{1 σ (j)})}^{w_{j}}, 1 - \prod_{j = 1}^{k + 1} (1 - a_{2 σ (j)})^{w_{j}}, \\ 1 - \prod_{j = 1}^{k + 1} (1 - a_{3 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{k + 1} (1 - a_{4 σ (j)})^{w_{j}}), \\ (\prod_{j = 1}^{k + 1} b_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{k + 1} c_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{4 σ (j)}^{w_{j}}) 〉 . \end{matrix}$ (22)

Therefore, considering the above results, we have Equation (17) for any n. This completes the proof. □

Theorem 3.The TrFNNOWAA operator has the following properties:

(1) Idempotency: Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}), (b_{1 j}, b_{2 j}, b_{3 j}, b_{4 j}), (c_{1 j}, c_{2 j}, c_{3 j}, c_{4 j}) 〉 (j = 1, 2, \dots, n)$ be a collection of TrFNNs. If all ${\tilde{n}}_{j}$ are equal, that is, ${\tilde{n}}_{j} = \tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}, b_{4}), (c_{1}, c_{2}, c_{3},$ c₄) 〉 for j = 1, 2, ⋯, n, then $TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) = \tilde{n}$ (23)

(2) Monotonicity: Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}),$ (b_1j, b_2j, b_3j, b_4j), (c_1j, c_2j, c_3j, c_4j) 〉 (j = 1, 2, ⋯, n) be a collection of TrFNNs. If ${\tilde{n}}_{j} \leq {\tilde{n}}_{j}^{*}$ for j = 1, 2, ⋯, n, then

$\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq TrFNNOWAA ({\tilde{n}}_{1}^{*}, {\tilde{n}}_{2}^{*}, \dots, {\tilde{n}}_{n}^{*}) . \end{matrix}$ (24)

(3) Boundedness: Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}),$ (b_1j, b_2j, b_3j, b_4j), (c_1j, c_2j, c_3j, c_4j) 〉 (j = 1, 2, ⋯, n) be a collection of TrFNNs. Assume that ${\tilde{n}}_{min} = 〈 (min_{j} a_{1 j}, min_{j} a_{2 j}, min_{j} a_{3 j}, min_{j} a_{4 j}), (max_{j} b_{1 j},$ $max_{j} b_{2 j}, max_{j} b_{3 j}, max_{j} b_{4 j}), (max_{j} c_{1 j}, max_{j} c_{2 j},$ $max_{j} c_{3 j}, max_{j} c_{4 j}) 〉$ , ${\tilde{n}}_{max} = 〈 (max_{j} a_{1 j}, max_{j} a_{2 j},$ $max_{j} a_{3 j}, max_{j} a_{4 j}), (min_{j} b_{1 j}, min_{j} b_{2 j}, min_{j} b_{3 j},$ $min_{j} b_{4 j}), (min_{j} c_{1 j}, min_{j} c_{2 j}, min_{j} c_{3 j}, min_{j} c_{4 j}) 〉$ for j = 1, 2, ⋯, n, then ${\tilde{n}}_{min} \leq TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \leq {\tilde{n}}_{max} .$ (25)

Proof. (1) Since ${\tilde{n}}_{j} = \tilde{n}$ for j = 1, 2, ⋯, n, then

$\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = w_{1} {\tilde{n}}_{σ (1)} \oplus w_{2} {\tilde{n}}_{σ (2)} \oplus \dots \oplus w_{n} {\tilde{n}}_{σ (n)} \\ = 〈 (1 - \prod_{j = 1}^{n} (1 - a_{1 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - a_{2 σ (j)})^{w_{j}}, \\ 1 - \prod_{j = 1}^{n} (1 - a_{3 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - a_{4 σ (j)})^{w_{j}}), \\ (\prod_{j = 1}^{n} b_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} b_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{n} c_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} c_{4 σ (j)}^{w_{j}}) 〉 \\ = 〈 (1 - (1 - a_{1 σ (j)})^{\sum_{j = 1}^{n} w_{j}}, 1 - (1 - a_{2 σ (j)})^{\sum_{j = 1}^{n} w_{j}}, \\ 1 - (1 - a_{3 σ (j)})^{\sum_{j = 1}^{n} w_{j}}, 1 - (1 - a_{4 σ (j)})^{\sum_{j = 1}^{n} w_{j}}), \\ (b_{1 σ (j)}^{\sum_{j = 1}^{n} w_{j}}, b_{2 σ (j)}^{\sum_{j = 1}^{n} w_{j}}, b_{3 σ (j)}^{\sum_{j = 1}^{n} w_{j}}, b_{4 σ (j)}^{\sum_{j = 1}^{n} w_{j}}), \\ (c_{1 σ (j)}^{\sum_{j = 1}^{n} w_{j}}, c_{2 σ (j)}^{\sum_{j = 1}^{n} w_{j}}, c_{3 σ (j)}^{\sum_{j = 1}^{n} w_{j}}, c_{4 σ (j)}^{\sum_{j = 1}^{n} w_{j}}) 〉 \\ = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}, b_{4}), \\ (c_{1}, c_{2}, c_{3}, c_{4}) 〉 = \tilde{n} . \end{matrix}$ (26)

(2) Since ${\tilde{n}}_{j} \leq {\tilde{n}}_{j}^{*}$ for j = 1, 2, ⋯, n, then $\sum_{j = 1}^{n} w_{j} {\tilde{n}}_{σ (j)} \leq \sum_{j = 1}^{n} w_{j} {\tilde{n}}_{σ (j)}^{*},$ (27) $\begin{matrix} TrFNNOWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq TrFNNOWAA ({\tilde{n}}_{1}^{*}, {\tilde{n}}_{2}^{*}, \dots, {\tilde{n}}_{n}^{*}) . \end{matrix}$ (28)

(3) Since ${\tilde{n}}_{min} \leq {\tilde{n}}_{j} \leq {\tilde{n}}_{max}$ for j = 1, 2, ⋯, n, there exists $\sum_{j = 1}^{n} w_{j} {\tilde{n}}_{min} \leq \sum_{j = 1}^{n} w_{j} {\tilde{n}}_{σ (j)} \leq \sum_{j = 1}^{n}$ $w_{j} {\tilde{n}}_{max}$ . Thus, according to idempotency and monotonicity properties, we have ${\tilde{n}}_{min} \leq TrFNNOWAA$ $({\tilde{n}}_{1},$ ${\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \leq {\tilde{n}}_{max}$ . □

3.2 Trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging operator

Definition 12. Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}), (b_{1 j}, b_{2 j}, b_{3 j}, b_{4 j}), (c_{1 j}, c_{2 j}, c_{3 j}, c_{4 j}) 〉 (j = 1, 2, \dots, n)$ be a collection of TrFNNs. The trapezoidal fuzzy neutrosophic number hybrid weighted arithmetic averaging (TFNNHWAA) operator of dimension n is defined by

$\begin{matrix} TrFNNHWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = w_{1} {\dot{\tilde{n}}}_{σ (1)} \oplus w_{2} {\dot{\tilde{n}}}_{σ (2)} \oplus \dots \oplus w_{n} {\dot{\tilde{n}}}_{σ (n)} \\ = \sum_{j = 1}^{n} (w_{j} {\dot{\tilde{n}}}_{σ (j)}) . \end{matrix}$ (29) where w = (w₁, w₂, ⋯, w_n) ^T is an associated weight vector with w_j ∈ [0, 1], and $\sum_{j = 1}^{n} w_{j} = 1$ ; ${\dot{\tilde{n}}}_{σ (j)}$ is the jth largest of ${\dot{\tilde{n}}}_{j} ({\dot{\tilde{n}}}_{j} = n ω_{j} {\tilde{n}}_{j},$ j = 1, 2, ⋯, n), (σ (1), σ (2), ⋯, σ (n)) is a permutation of (1, 2. ⋯, n), such that ${\tilde{n}}_{σ (j - 1)} \geq {\tilde{n}}_{σ (j)}$ for j = 1, 2, ⋯, n, and ω = (ω₁, ω₂, ⋯ ω_n) ^T is a weight vector, with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient.

Theorem 4. Let ${\tilde{n}}_{j} = 〈 (a_{1 j}, a_{2 j}, a_{3 j}, a_{4 j}), (b_{1 j}, b_{2 j},$ b_3j, b_4j), (c_1j, c_2j, c_3j, c_4j) 〉 (j = 1, 2, ⋯, n) be a collection of TrFNNs. Thus, their aggregated value using the TrFNNHWAA operator is also a trapezoidal fuzzy neutrosophic number, and then

$\begin{matrix} TrFNNHWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = w_{1} {\dot{\tilde{n}}}_{σ (1)} \oplus w_{2} {\dot{\tilde{n}}}_{σ (2)} \oplus \dots w_{n} {\dot{\tilde{n}}}_{σ (n)} \\ = 〈 (1 - \prod_{j = 1}^{n} (1 - {\dot{a}}_{1 σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - {\dot{a}}_{2 σ (j)})^{w_{j}}, \\ 1 - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{3 σ (j)})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{4 σ (j)})}^{w_{j}}), \\ (\prod_{j = 1}^{n} {\dot{b}}_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{b}}_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{b}}_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{b}}_{4 σ (j)}^{w_{j}}), \\ (\prod_{j = 1}^{n} {\dot{c}}_{1 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{c}}_{2 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{c}}_{3 σ (j)}^{w_{j}}, \prod_{j = 1}^{n} {\dot{c}}_{4 σ (j)}^{w_{j}}) 〉 . \end{matrix}$ (30)

This operator also has several properties that are similar to those of the TrFNNOWAA operator. By a similar proof of the properties in Theorem 3, we can prove these properties; those proofs do not need to be repeated here.

4 Multiple attribute group decision making methods based on TrFNNs

Considering the multiple attribute group decision making problems based on TrFNNs, let a₁, a₂, ⋯ a_m be a discrete set of m alternatives, and c₁, c₂, ⋯ c_n be the set of n attributes. ω_j is the weight of the attribute c_j (j = 1, 2, ⋯, n), where ω_j ∈ [0, 1] (j = 1, 2, ⋯, n), and $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that d₁, d₁, ⋯ d_l is the set of l decision makers, and e_k (k = 1, 2, ⋯, l) is the weight of decision maker d_k with e_k ∈ [0, 1] (k = 1, 2, ⋯, l), $\sum_{k = 1}^{l} e_{k} = 1$ . Suppose that $R^{k} = ({\tilde{n}}_{ij}^{k})_{m \times n}$ is the decision matrix, where ${\tilde{n}}_{ij}^{k} = 〈 (a_{1 ij}^{k}, a_{2 ij}^{k}, a_{3 ij}^{k}, a_{4 ij}^{k}), (b_{1 ij}^{k}, b_{2 ij}^{k}, b_{3 ij}^{k}, b_{4 ij}^{k}),$ $(c_{1 ij}^{k}, c_{2 ij}^{k}, c_{3 ij}^{k}, c_{4 ij}^{k}) 〉$ (i = 1, 2, ⋯ m, j = 1, 2, ⋯ n) takes the form of a TrFNN and $(a_{1 ij}^{k}, a_{2 ij}^{k}, a_{3 ij}^{k}, a_{4 ij}^{k}) \in [0, 1]$ , $(b_{1 ij}^{k}, b_{2 ij}^{k}, b_{3 ij}^{k}, b_{4 ij}^{k}) \in [0, 1]$ , $(c_{1 ij}^{k}, c_{2 ij}^{k}, c_{3 ij}^{k},$ $c_{4 ij}^{k}) \in [0, 1]$ , $0 \leq a_{4 ij}^{k} + b_{4 ij}^{k} + c_{4 ij}^{k} \leq 3$ for i = 1, 2, ⋯ m, j = 1, 2, ⋯ n and k = 1, 2, ⋯, l which describes the evaluation information of the attribute c_j with respect to the alternative a_i given by decision maker d_k. Then, we can rank the order of the alternatives and obtain the best alternatives based on the given information.

4.1 Multiple attribute group decision making method based on the TrFNNHWAA operator under the trapezoidal fuzzy neutrosophic environment

In the following, we propose a multiple attribute group decision making method based on the TrFNNH WAA operator to process trapezoidal fuzzy neutrosophic numbers. The steps are as follows:

Step 1: Give the decision matrix and importance of the attributes provided by decision maker makers in the form of TrFNNs, and then normalize the decision matrix. In general, attributes can be categorized into two types: benefit attributes and cost attributes. In order to eliminate the influence of attribute types, we need to convert cost type to benefit type.

Step 2: Determine the weights of the attributes. The information available on attribute importance is often vague or incomplete in the decision making situation. In this case, the importance of the attribute c_j (j = 1, 2, ⋯, n) can be presented by the TrFNNs. Let us assume that $ξ_{j}^{k} = 〈 (a_{1 j}^{k}, a_{2 j}^{k}, a_{3 j}^{k}, a_{4 j}^{k}), (b_{1 j}^{k}, b_{2 j}^{k}, b_{3 j}^{k},$ $b_{4 j}^{k}), (c_{1 j}^{k}, c_{2 j}^{k}, c_{3 j}^{k}, c_{4 j}^{k}) 〉$ is the TrFNN-based weight of attribute c_j for decision maker d_k. Then, the weighted importance of the attribute ξ_j (j = 1, 2, ⋯, n) of the attribute can be determined using the TFNNWAA aggregation operator [20]. Then, the weights of attributes ω_j (j = 1, 2, ⋯, n) are determined as follows: $ω_{j} = \frac{{EV}^{T} (ξ_{j})}{\sum_{j = 1}^{n} {EV}^{T} (ξ_{j})} .$ (31) where $ξ_{j} = e^{1} ξ_{j}^{1} \oplus e^{2} ξ_{j}^{2} \oplus \dots \oplus e^{l} ξ_{j}^{l} = \sum_{k = 1}^{l} e^{k} ξ_{j}^{k},$ (32)

Step 3: Calculate the comprehensive evaluation value of each alternative. According to the TFNNWAA operator [20], we can calculate the comprehensive evaluation value ${\tilde{n}}_{i}^{k}$ of each alternative for each decision maker, where

$\begin{matrix} {\tilde{n}}_{i}^{k} & = & TFNNWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) = \sum_{j = 1}^{n} ω_{j} {\tilde{n}}_{j} \\ = & 〈 (1 - \prod_{j = 1}^{n} (1 - a_{1 j})^{ω_{j}}, 1 - \prod_{j = 1}^{n} (1 - a_{2 j})^{ω_{j}}, \\ 1 - \prod_{j = 1}^{n} {(1 - a_{3 j})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - a_{4 j})}^{ω_{j}}), \\ (\prod_{j = 1}^{n} b_{1 j}^{ω_{j}}, \prod_{j = 1}^{n} b_{2 j}^{ω_{j}}, \prod_{j = 1}^{n} b_{3 j}^{ω_{j}}, \prod_{j = 1}^{n} b_{4 j}^{ω_{j}}), \\ (\prod_{j = 1}^{n} c_{1 j}^{ω_{j}}, \prod_{j = 1}^{n} c_{2 j}^{ω_{j}}, \prod_{j = 1}^{n} c_{3 j}^{ω_{j}}, \prod_{j = 1}^{n} c_{4 j}^{ω_{j}}) 〉 . \end{matrix}$ (33)

Step 4: Aggregate the decision information of each decision maker to the collective information using the TrFNNHWAA operator, and get the group comprehensive evaluation value ${\tilde{n}}_{i}$ of each alternative, where ${\tilde{n}}_{i} = TrFNNHWAA ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) = \sum_{j = 1}^{n} (w_{j} {\dot{\tilde{n}}}_{σ (j)})$ , and ${\dot{\tilde{n}}}_{σ (j)}$ is the jth largest of ${\dot{\tilde{n}}}_{j} ({\dot{\tilde{n}}}_{j} = {le}_{k} ({\tilde{n}}_{i}^{k}))$ .

Step 5: Rank the alternatives:

We can rank alternatives according to the score and accuracy functions and choose the best alternative.

4.2 Extended TOPSIS method for multiple attribute group decision making based on TrFNNs

In the following, we extend TOPSIS to process the TrFNNs. The steps are as follows:

Step 1: Obtain the normalized decision matrix.

Step 2: Determine the weights of attributes according to Equations (31) and (32).

Step 3: Construct an aggregated trapezoidal fuzzy neutrosophic decision matrix according to Equation (33).

Step 4: Identify the positive ideal solution P⁺ and the negative ideal solution P^-, where $P^{+} = (p_{1}^{+}, p_{2}^{+}, \dots p_{n}^{+}),$ (34) $P^{-} = (p_{1}^{-}, p_{2}^{-}, \dots p_{n}^{-}),$ (35) $\begin{matrix} p_{j}^{+} & = & 〈 (max_{j} a_{1 j}, max_{j} a_{2 j}, max_{j} a_{3 j}, max_{j} a_{4 j}), \\ (min_{j} b_{1 j}, min_{j} b_{2 j}, min_{j} b_{3 j}, min_{j} b_{4 j}), \\ (min_{j} c_{1 j}, min_{j} c_{2 j}, min_{j} c_{3 j}, min_{j} c_{4 j}) 〉, \\ (j = 1, 2, \dots, n), \end{matrix}$ (36) $\begin{matrix} p_{j}^{-} & = & 〈 (min_{j} a_{1 j}, min_{j} a_{2 j}, min_{j} a_{3 j}, min_{j} a_{4 j}), \\ (max_{j} b_{1 j}, max_{j} b_{2 j}, max_{j} b_{3 j}, max_{j} b_{4 j}), \\ (max_{j} c_{1 j}, max_{j} c_{2 j}, max_{j} c_{3 j}, max_{j} c_{4 j}) 〉, \\ (j = 1, 2, \dots, n) . \end{matrix}$ (37)

Step 5: Obtain the distance $D_{i}^{+} ({\tilde{n}}_{i}, P^{+})$ between each alternative and the positive ideal solution, and distance $D_{i}^{-} ({\tilde{n}}_{i}, P^{-})$ between each alternative and the negative ideal solution.

Step 6: Obtain the closeness coefficients U_i of each alternative A_i to the ideal solution; after which we get $U_{i} = \frac{D_{i}^{-} ({\tilde{n}}_{i}, P^{+})}{D_{i}^{+} ({\tilde{n}}_{i}, P^{+}) + D_{i}^{-} ({\tilde{n}}_{i}, P^{-})}$ (38)

Step 7: Rank the alternatives:

According to the closeness coefficient above, we can choose a best alternative or rank alternatives according to U_i. In general, the greater the U_i, the better the alternative.

5 Illustrative example

In this section, a multiple attribute group decision making problem adapted from [43] under a trapezoidal fuzzy neutrosophic environment is considered to demonstrate the applicability and the effectiveness of the proposed approach. Let us consider the decision making problem in which a customer intends to buy a tablet from the set of four primarily chosen alternatives A ={ a₁, a₂, a₃, a₄ }. The customer takes into account the following three attributes: (1) features, c₁; (2) hardware, c₂; and (3) customer care, c₃. The four possible alternatives in a_i (i = 1, 2, 3, 4) are to be evaluated using the trapezoidal fuzzy neutrosophic numbers by some decision makers or experts under the three attributes of c_j (j = 1, 2, 3). Assume that the set of four experts is D = {d₁, d₂, d₃, d₄}, and the weight vector of the four experts is e = (0.270, 0.200, 0.280, 0.250) ^T.

5.1 Multiple attribute group decision making method based on TrFNNHWAA operator under the trapezoidal fuzzy neutrosophic environment

According to Section 4.1, software selection using the new MCGDM model contains the following steps:

Step 1. Suppose we invite four experts to consider the possible alternatives, give the linguistic values of trapezoidal fuzzy neutrosophic numbers for linguistic terms shown in Table 2, and each expert gives the reference values and the weights of criteria with linguistic terms. Then, the decision matrix and importance of the criteria provided by the decision maker would be as shown in Tables 3–7.

Table 2
Linguistic values of TrFNNs for linguistic terms

Linguistic terms Linguistic values of TrFNNs

Absolutely low 〈 (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0),

(0.0, 0.0, 0.0, 0.0) 〉

Low 〈 (0.0, 0.1, 0.2, 0.3), (0.0, 0.1, 0.2, 0.3),

(0.0, 0.1, 0.2, 0.3) 〉

Fairly low 〈 (0.1, 0.2, 0.3, 0.4), (0.0, 0.2, 0.3, 0.4),

(0.0, 0.2, 0.3, 0.4) 〉

Medium 〈 (0.3, 0.4, 0.5, 0.6), (0.2, 0.4, 0.5, 0.7),

(0.2, 0.4, 0.5, 0.7) 〉

Fairly high 〈(0.5, 0.6, 0.7, 0.8), (0.4, 0.6, 0.7, 0.9),

(0.4, 0.6, 0.7, 0.9) 〉

High 〈(0.7, 0.8, 0.9, 1.0), (0.7, 0.8, 0.9, 1.0),

(0.7, 0.8, 0.9, 1.0) 〉

Absolutely high 〈(1.0, 1.0, 1.0, 1.0), (1.0, 1.0, 1.0, 1.0),

(1.0, 1.0, 1.0, 1.0) 〉

Linguistic terms	Linguistic values of TrFNNs
Absolutely low	〈 (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0),
	(0.0, 0.0, 0.0, 0.0) 〉
Low	〈 (0.0, 0.1, 0.2, 0.3), (0.0, 0.1, 0.2, 0.3),
	(0.0, 0.1, 0.2, 0.3) 〉
Fairly low	〈 (0.1, 0.2, 0.3, 0.4), (0.0, 0.2, 0.3, 0.4),
	(0.0, 0.2, 0.3, 0.4) 〉
Medium	〈 (0.3, 0.4, 0.5, 0.6), (0.2, 0.4, 0.5, 0.7),
	(0.2, 0.4, 0.5, 0.7) 〉
Fairly high	〈(0.5, 0.6, 0.7, 0.8), (0.4, 0.6, 0.7, 0.9),
	(0.4, 0.6, 0.7, 0.9) 〉
High	〈(0.7, 0.8, 0.9, 1.0), (0.7, 0.8, 0.9, 1.0),
	(0.7, 0.8, 0.9, 1.0) 〉
Absolutely high	〈(1.0, 1.0, 1.0, 1.0), (1.0, 1.0, 1.0, 1.0),
	(1.0, 1.0, 1.0, 1.0) 〉

Table 3

Trapezoidal fuzzy neutrosophic decision matrix R¹

	c ₁	c ₂	c ₃
a ₁	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.0, 0.1, 0.2, 0.3), \\ (0.0, 0.1, 0.2, 0.3), \\ (0.0, 0.1, 0.2, 0.3) 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$

Table 4

Trapezoidal fuzzy neutrosophic decision matrix R²

	c ₁	c ₂	c ₃
a ₁	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.0, 0.1, 0.2, 0.3), \\ (0.0, 0.1, 0.2, 0.3), \\ (0.0, 0.1, 0.2, 0.3) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$

Table 5

Trapezoidal fuzzy neutrosophic decision matrix c₂

	c ₃	c ₃	$\tilde{R}$
a ₁	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$

Table 6

Trapezoidal fuzzy neutrosophic decision matrix R⁴

	c ₁	c ₂	c ₃
a ₁	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0), \\ (0.7, 0.8, 0.9, 1.0) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.0, 0.1, 0.2, 0.3), \\ (0.0, 0.1, 0.2, 0.3), \\ (0.0, 0.1, 0.2, 0.3) 〉 \end{matrix}$

Table 7

Importance of attributes provided by decision makers $(ξ_{j}^{k})$

	c ₁	c ₂	c ₃
d ₁	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
d ₂	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$
d ₃	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.1, 0.2, 0.3, 0.4), \\ (0.0, 0.2, 0.3, 0.5), \\ (0.0, 0.2, 0.3, 0.5) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$
d ₄	$\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8), \\ (0.4, 0.6, 0.7, 0.9), \\ (0.4, 0.6, 0.7, 0.9) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6), \\ (0.2, 0.4, 0.5, 0.7), \\ (0.2, 0.4, 0.5, 0.7) 〉 \end{matrix}$

Step 2. Determine the weights of the attributes using Equations (31) and (32), and Definition 5. The weighted importance of attributes provided by the decision maker is as follows: $\begin{matrix} ξ_{1} & = & 〈 (0.465, 0.566, 0.668, 0.770), (0.348, 0.553, \\ 0.654, 0.865), (0.348, 0.553, 0.654, 0.865) 〉, \\ ξ_{2} & = & 〈 (0.249, 0.350, 0.451, 0.552), (0.000, 0.329, \\ 0.433, 0.637), (0.000, 0.329, 0.433, 0.637) 〉, \\ ξ_{3} & = & 〈 (0.346, 0.447, 0.549, 0.652), (0.230, 0.434, \\ 0.535, 0.736), (0.230, 0.434, 0.535, 0.736) 〉 . \end{matrix}$

Then, the truth favorite relative expected values of ξ_j are as follows: $\begin{matrix} {EV}^{T} (ξ_{1}) = 1.536, {EV}^{T} (ξ_{2}) = 1.716, \\ {EV}^{T} (ξ_{3}) = 1.545 . \end{matrix}$

Thus, the weights of the attributes are as follows: $ω_{1} = 0.320, ω_{2} = 0.358, ω_{3} = 0.322 .$

Step 3. Using Equation (33), we can calculate the comprehensive evaluation value of each alternative for each decision matrix as follows: $\begin{matrix} {\tilde{n}}_{1}^{1} & = & 〈 (0.271, 0.376, 0.483, 0.595), (0.000, 0.296, \\ 0.406, 0.617), (0.000, 0.296, 0.406, 0.617) 〉, \\ {\tilde{n}}_{2}^{1} & = & 〈 (0.443, 0.545, 0.647, 0.750), (0.320, 0.527, \\ 0.629, 0.830), (0.320, 0.527, 0.629, 0.830) 〉, \\ {\tilde{n}}_{3}^{1} & = & 〈 (0.469, 0.584, 0.711, 1.000), (0.000, 0.369, \\ 0.488, 0.654), (0.000, 0.369, 0.488, 0.654) 〉, \\ {\tilde{n}}_{4}^{1} & = & 〈 (0.436, 0.538, 0.640, 0.744), (0.312, 0.519, \\ 0.621, 0.823), (0.312, 0.519, 0.621, 0.823) 〉, \\ {\tilde{n}}_{1}^{2} & = & 〈 (0.327, 0.431, 0.536, 0.644), (0.000, 0.370, \\ 0.478, 0.687), (0.000, 0.370, 0.478, 0.687) 〉, \\ {\tilde{n}}_{2}^{2} & = & 〈 (0.394, 0.512, 0.647, 1.000), (0.000, 0.304, \\ 0.419, 0.579), (0.000, 0.304, 0.419, 0.579) 〉, \\ {\tilde{n}}_{3}^{2} & = & 〈 (0.497, 0.610, 0.735, 1.000), (0.000, 0.468, \\ 0.560, 0.774), (0.000, 0.468, 0.560, 0.774) 〉, \\ {\tilde{n}}_{4}^{2} & = & 〈 (0.606, 0.715, 0.832, 1.000), (0.468, 0.640, \\ 0.688, 0.892), (0.468, 0.640, 0.688, 0.892) 〉, \\ {\tilde{n}}_{1}^{3} & = & 〈 (0.383, 0.487, 0.594, 0.704), (0.000, 0.405, \\ 0.517, 0.729), (0.000, 0.405, 0.517, 0.729) 〉, \\ {\tilde{n}}_{2}^{3} & = & 〈 (0.440, 0.556, 0.687, 1.000), (0.000, 0.411, \\ 0.502, 0.714), (0.000, 0.411, 0.502, 0.714) 〉, \\ {\tilde{n}}_{3}^{3} & = & 〈 (0.483, 0.595, 0.719, 1.000), (0.313, 0.513, \\ 0.592, 0.795), (0.313, 0.513, 0.592, 0.795) 〉, \\ {\tilde{n}}_{4}^{3} & = & 〈 (0.436, 0.538, 0.640, 0.744), (0.312, 0.519, \\ 0.621, 0.823), (0.312, 0.519, 0.621, 0.823) 〉, \\ {\tilde{n}}_{1}^{4} & = & 〈 (0.700, 0.800, 0.900, 1.000), (0.700, 0.800, \\ 0.800, 1.000), (0.700, 0.800, 0.800, 1.000) 〉, \\ {\tilde{n}}_{2}^{4} & = & 〈 (0.379, 0.481, 0.584, 0.688), (0.256, 0.462, \\ 0.564, 0.766), (0.256, 0.462, 0.564, 0.766) 〉, \\ {\tilde{n}}_{3}^{4} & = & 〈 (0.440, 0.556, 0.687, 1.000), (0.000, 0.411, \\ 0.502, 0.714), (0.000, 0.411, 0.502, 0.714) 〉, \\ {\tilde{n}}_{4}^{4} & = & 〈 (0.069, 0.169, 0.269, 0.369), (0.000, 0.160, \\ 0.263, 0.424), (0.000, 0.160, 0.263, 0.424) 〉 . \end{matrix}$

Step 4. Using Equations (16) and (29), we can calculate the group comprehensive evaluation value of each alternative as follows: $\begin{matrix} {\tilde{n}}_{1} & = & 〈 (0.486, 0.600, 0.725, 1.000), (0.000, 0.445, \\ 0.539, 0.755), (0.000, 0.445, 0.539, 0.755) 〉, \\ {\tilde{n}}_{2} & = & 〈 (0.412, 0.520, 0.637, 1.000), (0.000, 0.431, \\ 0.537, 0.726), (0.000, 0.431, 0.537, 0.726) 〉, \\ {\tilde{n}}_{3} & = & 〈 (0.476, 0.591, 0.717, 1.000), (0.000, 0.438, \\ 0.531, 0.735), (0.000, 0.438, 0.531, 0.735) 〉, \\ {\tilde{n}}_{4} & = & 〈 (0.364, 0.473, 0.588, 1.000), (0.000, 0.351, \\ 0.464, 0.659), (0.000, 0.351, 0.464, 0.659) 〉 . \end{matrix}$

Here, the associated weight vector is w = (0.155, 0.345, 0.345, 0.155) ^T, adopted from [44]. The sorting method is based on the truth favorite relative expected value of ${\tilde{n}}_{j}^{k}$ .

Step 5. Using Definition 6, we can calculate the score function value of the group comprehensive evaluation value of each alternative as follows: $\begin{matrix} S ({\tilde{n}}_{1}) = 0.611, S ({\tilde{n}}_{2}) = 0.598, \\ S ({\tilde{n}}_{3}) = 0.615, S ({\tilde{n}}_{4}) = 0.623 . \end{matrix}$

According to Definition 8, the ranking order of the four alternatives is a₄ ≻ a₃ ≻ a₁ ≻ a₂, and the best alternative is a₄. In other words, the customer should buy the fourth alternative to get maximum benefits.

5.2 Extended TOPSIS method for multiple attribute group decision making based on trapezoidal fuzzy neutrosophic numbers

Step 1: Here c₁, c₂, and c₃ are benefit attributes; therefore, we do not need to normalize the decision matrix.

Step 2: The weights of the attributes are determined by the truth favorite relative expected value, and are as follows: $ω_{1} = 0.320, ω_{2} = 0.358, ω_{3} = 0.322 .$

Step 3: Based on the decision matrix obtained in Step 1 in Section 5.1 and the weights of the decision makers in Step 2 in Section 5.1, the group aggregated trapezoidal fuzzy neutrosophic decision matrix based on each decision maker’s opinion can be constructed using Equation (33), as shown in Table 8:

Table 8
Group aggregated trapezoidal fuzzy neutrosophic decision matrix

c ₁ c ₂ c ₃

a ₁ $\begin{matrix} 〈 (0.448, 0.560, \\ 0.683, 1.000), \\ (0.000, 0.442, \\ 0.538, 0.750), \\ (0.000, 0.442, \\ 0.538, 0.750) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.481, 0.592, \\ 0.711, 1.000), \\ (0.000, 0.474, \\ 0.571, 0.784), \\ (0.000, 0.474, \\ 0.571, 0.784) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.420, 0.534, \\ 0.661, 1.000), \\ (0.000, 0.385, \\ 0.486, 0.701), \\ (0.000, 0.385, \\ 0.486, 0.701) 〉 \end{matrix}$

a ₂ $\begin{matrix} 〈 (0.366, 0.478, \\ 0.602, 1.000), \\ (0.000, 0.378, \\ 0.476, 0.684), \\ (0.000, 0.378, \\ 0.476, 0.684) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.502, 0.613, \\ 0.732, 1.000), \\ (0.000, 0.454, \\ 0.566, 0.744), \\ (0.000, 0.454, \\ 0.566, 0.744) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.361, 0.462, \\ 0.564, 0.668), \\ (0.241, 0.446, \\ 0.548, 0.749), \\ (0.241, 0.446, \\ 0.548, 0.749) 〉 \end{matrix}$

a ₃ $\begin{matrix} 〈 (0.376, 0.492, \\ 0.623, 1.000), \\ (0.000, 0.353, \\ 0.451, 0.662), \\ (0.000, 0.353, \\ 0.451, 0.662) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.656, 0.759, \\ 0.865, 1.000), \\ (0.602, 0.740, \\ 0.772, 0.972), \\ (0.602, 0.740, \\ 0.772, 0.972) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.279, 0.383, \\ 0.487, 0.595), \\ (0.000, 0.298, \\ 0.418, 0.586), \\ (0.000, 0.298, \\ 0.418, 0.586) 〉 \end{matrix}$

a ₄ $\begin{matrix} 〈 (0.477, 0.586, \\ 0.702, 1.000), \\ (0.000, 0.483, \\ 0.582, 0.794), \\ (0.000, 0.483, \\ 0.582, 0.794) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.371, 0.482, \\ 0.606, 1.000), \\ (0.000, 0.386, \\ 0.483, 0.691), \\ (0.000, 0.386, \\ 0.483, 0.691) 〉 \end{matrix}$ $\begin{matrix} 〈 (0.364, 0.469, \\ 0.575, 0.686), \\ (0.000, 0.354, \\ 0.478, 0.650), \\ (0.000, 0.354, \\ 0.478, 0.650) 〉 \end{matrix}$

	c ₁	c ₂	c ₃
a ₁	$\begin{matrix} 〈 (0.448, 0.560, \\ 0.683, 1.000), \\ (0.000, 0.442, \\ 0.538, 0.750), \\ (0.000, 0.442, \\ 0.538, 0.750) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.481, 0.592, \\ 0.711, 1.000), \\ (0.000, 0.474, \\ 0.571, 0.784), \\ (0.000, 0.474, \\ 0.571, 0.784) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.420, 0.534, \\ 0.661, 1.000), \\ (0.000, 0.385, \\ 0.486, 0.701), \\ (0.000, 0.385, \\ 0.486, 0.701) 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 (0.366, 0.478, \\ 0.602, 1.000), \\ (0.000, 0.378, \\ 0.476, 0.684), \\ (0.000, 0.378, \\ 0.476, 0.684) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.502, 0.613, \\ 0.732, 1.000), \\ (0.000, 0.454, \\ 0.566, 0.744), \\ (0.000, 0.454, \\ 0.566, 0.744) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.361, 0.462, \\ 0.564, 0.668), \\ (0.241, 0.446, \\ 0.548, 0.749), \\ (0.241, 0.446, \\ 0.548, 0.749) 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 (0.376, 0.492, \\ 0.623, 1.000), \\ (0.000, 0.353, \\ 0.451, 0.662), \\ (0.000, 0.353, \\ 0.451, 0.662) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.656, 0.759, \\ 0.865, 1.000), \\ (0.602, 0.740, \\ 0.772, 0.972), \\ (0.602, 0.740, \\ 0.772, 0.972) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.279, 0.383, \\ 0.487, 0.595), \\ (0.000, 0.298, \\ 0.418, 0.586), \\ (0.000, 0.298, \\ 0.418, 0.586) 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 (0.477, 0.586, \\ 0.702, 1.000), \\ (0.000, 0.483, \\ 0.582, 0.794), \\ (0.000, 0.483, \\ 0.582, 0.794) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.371, 0.482, \\ 0.606, 1.000), \\ (0.000, 0.386, \\ 0.483, 0.691), \\ (0.000, 0.386, \\ 0.483, 0.691) 〉 \end{matrix}$	$\begin{matrix} 〈 (0.364, 0.469, \\ 0.575, 0.686), \\ (0.000, 0.354, \\ 0.478, 0.650), \\ (0.000, 0.354, \\ 0.478, 0.650) 〉 \end{matrix}$

Step 4: Identify the positive ideal solution and the negative ideal solution. Then, P⁺ and P^- are obtained as follows: $\begin{matrix} P^{+} \\ = (\begin{matrix} 〈 (0.477, 0.586, 0.702, 1.000), (0.000, 0.353, \\ 0.451, 0.662), (0.000, 0.353, 0.451, 0.662) 〉, \end{matrix} \\ \begin{matrix} 〈 (0.656, 0.759, 0.865, 1.000), (0.000, 0.386, \\ 0.483, 0.691), (0.000, 0.386, 0.483, 0.691) 〉, \end{matrix} \\ \begin{matrix} 〈 (0.420, 0.534, 0.661, 1.000), (0.000, 0.298, \\ 0.418, 0.586), (0.000, 0.298, 0.418, 0.586) 〉 \end{matrix}) \\ P^{-} \\ = (\begin{matrix} 〈 (0.366, 0.478, 0.602, 1.000), (0.000, 0.483, \\ 0.582, 0.794), (0.000, 0.483, 0.582, 0.794) 〉, \end{matrix} \\ \begin{matrix} 〈 (0.371, 0.482, 0.606, 1.000), (0.602, 0.740, \\ 0.772, 0.972), (0.602, 0.740, 0.772, 0.972) 〉, \end{matrix} \\ \begin{matrix} 〈 (0.279, 0.383, 0.487, 0.595), (0.241, 0.446, \\ 0.548, 0.749), (0.241, 0.446, 0.548, 0.749) 〉 \end{matrix}) \end{matrix}$

Steps 5 and 6: The distance $D_{i}^{+} ({\tilde{n}}_{i}, P^{+})$ , distance $D_{i}^{-} ({\tilde{n}}_{i}, P^{-})$ , and the closeness coefficients U_i are obtained as shown in Table 9.

Table 9

Distances and the relative closeness coefficient

$D_{i}^{-} ({\tilde{n}}_{i}, P^{-})$	$D_{i}^{+} ({\tilde{n}}_{i}, P^{+})$	U _i
a ₁	0.007	0.005	0.592
a ₂	0.006	0.008	0.437
a ₃	0.030	0.024	0.548
a ₄	0.011	0.002	0.848

Step 7: As U₄ > U₁ > U₃ > U₂, the rank of the alternatives is a₄ ≻ a₁ ≻ a₃ ≻ a₂. In addition, the best alternative is a₄. In other words, the customer should buy the fourth alternative to get maximum benefits.

5.3 Comparative analysis of the group decision making methods

As can be seen from the above results, the best alternative generated by the first method (method based on the TrFNNHWAA operator) is in accordance with the one generated by the second method (method based on TOPSIS). However, their ranking results are not exactly the same. The first method uses the TrFNNHWAA operator to aggregate the trapezoidal fuzzy neutrosophic information and score and accuracy functions to rank the alternatives. The TrFNNHWAA operator not only considers the importance of attributes, but also takes the importance of position into account. It also reduces the impact of the highest and lowest scores on the decision result in group decision making. The second method focuses on choosing the alternative with the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution, and uses closeness coefficients to rank the alternatives and get the best alternative. These two methods construct different group decision making methods from different angles under trapezoidal fuzzy neutrosophic environments.

5.4 Sensitivity analysis

Sensitivity analysis can be used to determine the impact of the evaluation criteria weights variation on the decision results, which is key to effective use of the model and the implementation of quantitative decision making. In this section, we perform the above case study while varying the weights to investigate their effect on the assessment results.

Suppose a disturbance is imposed on attribute weights ω_j (j = 1, 2, ⋯, n), and ω_j changes into $ω_{j}^{*}$ ; specifically, $ω_{j}^{*} = γ_{j} ω_{j}$ , where γ_j is the initial variation ratio of ω_j, and 0 ≤ γ_j ≤ 1/ω_j. As the sum of weights should be equal to one, the disturbance to ω_j will make other weights change. Thus, $ω_{j}^{'} = \frac{ω_{j}^{*}}{ω_{1} + ω_{2} + \dots ω_{j}^{*} + \dots + ω_{n}} = \frac{γ_{j} ω_{j}}{1 + (γ_{j} - 1) ω_{j}}, (j = 1, 2, \dots, n)$ , where $ω_{1}^{'}$ is the unitary weight for attribute c_j after ω_j is imposed on a disturbance. The variable β_j is defined as the unitary variation ratio of ω_j after disturbance and is expressed as $β_{j} = \frac{ω_{j}^{'}}{ω_{j}}$ ; thus, we get $γ_{j} = \frac{β_{j} - β_{j} ω_{j}}{1 - β_{j} ω_{j}}$ [45].

5.4.1 Sensitivity analysis of the first method

Firstly, we assumed that ω₁ = 0.320 was the weight being imposed as a disturbance. More than ten schemes were designed with the following respective values: β₁ = 0.01, β₁ = 0.02, β₁ = 0.05, β₁ = 0.1, β₁ = 0.2, β₁ = 0.5, β₁ = 1.0, β₁ = 1.3, β₁ = 1.5, β₁ = 2.0, and β₁ = 2.5. After the unitary variation ratios were designed, the weights under these schemes were recalculated, and the ranking results obtained. The results of the first method based on the TrFNNHWAA operator were shown in Table 10, Fig. 2(a), (b). Similarly, if the disturbance is imposed on other weights, the variations of the results under different unitary variation ratios for ω₂ and ω₃ can be calculated. The corresponding results are illustrated in Fig. 2(c), (d), (e), and (f).

Table 10
Attribute weights and ranking results under different unitary variation ratios for ω₁

β₁ Attribute weights Ranking results

ω ₁ ω ₂ ω ₃

β₁ = 0.01 0.003 0.525 0.472 a₄ ≻ a₁ ≻ a₂ ≻ a₃

β₁ = 0.02 0.006 0.523 0.471 a₄ ≻ a₁ ≻ a₂ ≻ a₃

β₁ = 0.05 0.016 0.518 0.466 a₄ ≻ a₁ ≻ a₃ ≻ a₂

β₁ = 0.1 0.032 0.510 0.458 a₄ ≻ a₁ ≻ a₃ ≻ a₂

β₁ = 0.2 0.064 0.493 0.443 a₄ ≻ a₁ ≻ a₃ ≻ a₂

β₁ = 0.5 0.160 0.442 0.398 a₄ ≻ a₁ ≻ a₃ ≻ a₂

β₁ = 1.0 0.320 0.358 0.322 a₄ ≻ a₃ ≻ a₁ ≻ a₂

β₁ = 1.3 0.416 0.307 0.277 a₄ ≻ a₃ ≻ a₁ ≻ a₂

β₁ = 1.5 0.480 0.274 0.246 a₃ ≻ a₄ ≻ a₁ ≻ a₂

β₁ = 2.0 0.640 0.190 0.170 a₃ ≻ a₁ ≻ a₄ ≻ a₂

β₁ = 2.5 0.800 0.105 0.095 a₃ ≻ a₁ ≻ a₄ ≻ a₂

β₁	Attribute weights	Ranking results
β₁ = 0.01	0.003	0.525	0.472	a₄ ≻ a₁ ≻ a₂ ≻ a₃
β₁ = 0.02	0.006	0.523	0.471	a₄ ≻ a₁ ≻ a₂ ≻ a₃
β₁ = 0.05	0.016	0.518	0.466	a₄ ≻ a₁ ≻ a₃ ≻ a₂
β₁ = 0.1	0.032	0.510	0.458	a₄ ≻ a₁ ≻ a₃ ≻ a₂
β₁ = 0.2	0.064	0.493	0.443	a₄ ≻ a₁ ≻ a₃ ≻ a₂
β₁ = 0.5	0.160	0.442	0.398	a₄ ≻ a₁ ≻ a₃ ≻ a₂
β₁ = 1.0	0.320	0.358	0.322	a₄ ≻ a₃ ≻ a₁ ≻ a₂
β₁ = 1.3	0.416	0.307	0.277	a₄ ≻ a₃ ≻ a₁ ≻ a₂
β₁ = 1.5	0.480	0.274	0.246	a₃ ≻ a₄ ≻ a₁ ≻ a₂
β₁ = 2.0	0.640	0.190	0.170	a₃ ≻ a₁ ≻ a₄ ≻ a₂
β₁ = 2.5	0.800	0.105	0.095	a₃ ≻ a₁ ≻ a₄ ≻ a₂

Fig.2

Ranking results under different unitary variation ratios for ω₁, ω₂, and ω₃: (a) (b) ω₁, (c) (d) ω₂, (e) (f) ω₃.

The results in Table 10 and Fig. 2(a) and (b) suggest that the ranking results did not change until the unitary variation ratio β₁ for ω₁ reached approximately 0.7. For β₁ in the range 0.7 to 1.3, the sorting results are the same. In general, the variation of ω₁ did not have a significant impact on the order of alternatives a₁ and a₂, but significantly influenced alternatives a₃ and a₄. It can also be seen from Fig. 2(a) and (b) that the best alternative is not a₄ after the unitary variation ratio exceeded 1.4. Figure 2(c) and (d) show that the alternative order arranged by score function values began to change substantially at approximately β₂ = 0.70. The score function values of alternatives a₂ and a₄ are kept relatively stable under the more than ten schemes, and others are sensitive to the change in ω₂. However, the best alternative, a₄, remained unchanged. Figure 2(e) and (f) show that all the alternatives are sensitive to changing ω₃, especially alternatives a₂ and a₄, because their score function values begin to sharply change once a small disturbance is imposed (at approximately β₃ = 0.50), and their orders also begin to change. However, the best alternative, a₄, still remains unchanged. A conclusive result that can be given from the above analysis is that different alternatives usually show different sensitivities to weight variations. In the present study, a₁ is sensitive to ω₂ and ω₃, a₂ is sensitive to ω₃, a₃ is sensitive to ω₁, ω₂ and ω₃, and a₄ is sensitive to ω₁.

5.4.2 Sensitivity analysis of the second method

The sensitivity analysis results of the second method based on TOPSIS are shown in Fig. 3.

Fig.3

Ranking results under different unitary variation ratios for ω₁, ω₂ and ω₃: (a) ω₁, (b) ω₂, (c) ω₃.

We also analyze the sensitivity of the second method based on TOPSIS to the variations of ω₁, ω₂, and ω₃ as follows. Figure 3 shows that the alternatives may have different feedback to the variations in attribute weights. Figure 3(a) shows that four alternatives are relatively insensitive to the variation of ω₁, and their orders begin to change at approximately β₁ = 2.0. Whereas, all alternatives are sensitive to variations in ω₂, especially the third alternative a₃, as shown in Fig. 3(b). But the relative order of a₁, a₁ and a₄ remains unchanged. It can also be seen from Fig. 3(c) that the closeness coefficient values of a₄ remain relatively stable, while a₃ shows an increasing trend and a₁ and a₂ suggest a decreasing tendency. This shows that a₁, a₂, and a₃ are sensitive, whereas the best alternative, a₄, is insensitive to changes in ω₃. The conclusive results that can be given from the above analysis is that the different alternatives usually show different sensitivities to weight variations. In the present study, a₁ and a₂ are sensitive to ω₂ and ω₃, a₃ is highly sensitive to ω₂ and ω₃, and a₄ is sensitive to ω₂.

6 Conclusion

This paper proposed two group decision making methods under for trapezoidal fuzzy neutrosophic environments, with attribute weights obtained using the truth favorite relative expected value and distance measure defined using the cosine similarity measure. The first method incorporates two trapezoidal fuzzy neutrosophic number operators, whereas the second extends TOPSIS to deal with trapezoidal fuzzy neutrosophic information. The results of an illustrative example and sensitivity analysis demonstrate the application and feasibility of the two proposed methods. The proposed methods enrich and develop the theory and method of decision making, and provide new ways to solve group decision making problems under trapezoidal fuzzy neutrosophic environments. In future research, the authors will continue to study related distance measures of trapezoidal fuzzy neutrosophic numbers including the distance measure based on centroids, fuzzy distance measure based on graded mean integration representation among others. In addition, they will also apply the trapezoidal fuzzy neutrosophic number aggregation operators for practical applications in other areas such as expert systems, information fusion systems, and medical diagnoses. The authors will also study the related theory of single valued neutrosophic sets, inreval-valued neutrosophic sets, bipolar neutrosophic sets, neutrosophic hesitant fuzzy sets, multi-valued neutrosophic sets, simplified neutrosophic linguistic sets, and also their applications in multi-criteria group decision-making problems.

Footnotes

Acknowledgments

We wish to express our sincere thanks to the editors and the anonymous reviewers for their valuable and insightful comments and suggestions, which have help us to improve the quality of our paper. This research is supported by the Fujian Province Social Science Planning Project of China (No. FJ2016C028), Education and Scientific Research Projects of Young and Middle-aged Teachers of Fujian Province (Nos. JAT160556, JAT160559, JAT160097).

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