A new multi-criteria group decision model based on Single-valued triangular Neutrosophic sets and EDAS method

Abstract

This paper proposes a new method, which combines the Power operator and the evaluation based on distance from average solution (EDAS) method to deal with the fuzzy multi-criteria group decision problem in the Single-valued triangular Neutrosophic sets (SVTNS) environment. Firstly, we define the normalized Euclidean distance between two single-valued triangular Neutrosophic numbers, and propose the Power weighted averaging (PWA) operator, Power weighted geometric (PWG) operator, generalized Power weighted averaging (GPWA) operator and generalized Power weighted geometric (GPWG) operator of SVTNS. Then, those operators and EDAS method are combined to construct a new multi-criteria group decision-making model. Finally, the model is applied to the problem of the investment projects selection, which proves its applicability and feasibility.

Keywords

Single-valued triangular neutrosophic sets power average operator power geometric operator EDAS

1 Introduction

Decision makers need to take sustainability into account in the decision-making process, due to strict government regulations and increased public awareness. With the rapid development of the economy and the high degree of uncertainty in the environment, it is very important for any venture capital company to choose the right investment project. But choosing the right investment project requires consideration of criteria such as pollutants emission, resource and energy consumption, product characteristics, ecological environment and management level. In essence, the process of choosing the best investment project is a multi-criteria decision problem.

Multi-criteria decision-making usually means that the decision makers evaluate and rank alternatives and then select the optimal alternative. However, due to the increasing complexity of the decision-making environment, accurate numbers can no longer meet the needs of decision makers. In order to solve the problem of multi-criteria decision-making in uncertain environments, Zaheh [1] proposed fuzzy sets (FS) theory in 1965. The FS has been widely concerned and popularized by many scholars since its introduction. Correspondingly, the intuitionistic fuzzy sets (IFS) [2], the interval valued fuzzy sets (IVFS) [3], the Vague sets (VS) [4], interval valued intuitionistic fuzzy sets (IVIFS) [5], hesitant fuzzy sets (HFS) [6], dual hesitant fuzzy sets (DHFS) [7] and Pythagorean fuzzy sets (PFS) [8] and so on. Although FS theory has been extensively studied and expanded, FS and its extensions cannot handle discontinuous and inconsistent information. For example, during the voting process, 30% were in favor, 20% were in disapproval, 10% abstained, and 40% remained neutral or failed to vote for other reasons. This situation cannot be handled with FS and its extended sets, so some new theories are needed. The emergence of the Neutrosophic sets (NS) just made up for this deficiency.

Smarandache [9] proposed the concept of the Neutrosophic sets, which uses the truth-membership function, the indeterminacy-membership function and the falsity-membership function to depict the fuzzy information, in which the indeterminacy-membership function, the truth-membership function and the falsity-membership function are independent of each other. Like the example above, it can be expressed as a Neutrosophic number x (0.3, 0.4, 0.2). In order to make the Neutrosophic sets solve the practical problem, different scholars have successively proposed interval-valued Neutrosophic sets (IVNS) [10], single-valued Neutrosophic sets (SVNS) [11], and simplified Neutrosophic sets (SNS) [12]. Considering that the triangular fuzzy number can better represent the fuzzy information in real life, Biswas, Pramanik and Giri [13] used the triangular fuzzy numbers to represent the truth-membership function, the indeterminacy- membership function and the falsity-membership function, and proposed weighted arithmetic averaging operators and weighted geometric averaging operators for Single-valued triangular Neutrosophic sets (SVTNS). Wang et al. [14] considered the conflicting criteria, extended the original VIKOR model to SVTNS, and introduced the specific steps to apply the method.

The aggregation operators present the powerful tool for handling multi-criteria decision-making problem. Among the many existing aggregation operators, the Power average operator proposed by Yager [15] can consider the interrelationship between data through support degree and reduce the negative impact of decision makers’ unreasonable assessments. Xu et al. [16] proposed Power geometric operator, Power ordered geometric operator and Power ordered geometric operator, and applied these operators to the problem of multi-criteria group decision-making in uncertain environments. Zhou et al. [17] expended Power average operator and Power geometry operator to generalized Power average operator and generalized Power geometric operator in order to improve the scope of application. Ju et al. [18] proposed power generalized Heronian mean operator under hesitant fuzzy linguistic environment. Considering the advantages of Power operator and Bonferroni mean operator, Liu and Li [19] proposed IVIFS Power Bonferroni aggregation operators, and Liu et al. [20] proposed linguistic q-rung orthopair fuzzy Power Bonferroni aggregation operators, Wei and Zhang [21] proposed SVNS Power Bonferroni aggregation operators and used to handle multi-criteria decision-making problems. Khan et al. [22] based on this, combined with Dombi operations, proposed INS Dombi Power Bonferroni mean operators. Liu [23] used the Heronian mean operator to deal with the correlation between the criteria and proposed the IVIFS Power Heronian aggregation operators. Yang et al. [24] used the flexibility of Frank operations to propose interval-valued Pythagorean fuzzy Frank power aggregation operators. Wei et al. [25] applied the Power average operator to the Pythagorean fuzzy environment, extending the scope of the Power average operator. Liu et al. [26] proposed a generalized power weighted averaging operator in the Neutrosophic number environment and used to deal with multi-criteria group decision problems. Considering the advantages of Muirhead mean operator, different experts and scholars have proposed Neutrosophic cubic power Muirhead mean operator [27], Pythagorean fuzzy power Muirhead mean operator [28] and T-spherical fuzzy power dual Muirhead mean operator [29] accordingly. All these enrich the study of Power operator. Power average operator has its own unique advantages, which can reduce the impact of unreasonable data on decision results, however, up to now, no one has proposed Power average operator in the SVTNS environment.

Ghorabaee et al. [30] proposed the evaluation based on distance from average solution (EDAS) method for inventory ABC classification. It was presented that the EDAS method is very useful when we have some conflicting criteria. And Ghorabaee et al. proved that EDAS is used for multi-criteria decision-making through comparative analysis, which is more validity and stability than VIKOR and TOPSIS. Ghorabaee et al. [31] used the EDAS method to solve the supplier selection problem in fuzzy multi-criteria decision-making. In recent years, the EDAS method has been widely concerned by different scholars. Ghorabaee et al. [32] extended EDAS method to the interval type-2 fuzzy sets environment to solve the problem of supplier evaluation. Peng et al. [33] proposed the EDAS method of single-valued Neutrosophic soft sets, and proved that the results of EDAS method are more distinguishable than those of similarity measure and level soft set methods. Karaşan et al. [34] proposed interval-valued Neutrosophic EDAS method solve the priority problem of the United Nations national sustainable development goals. Liang et al. [35] integrated the EDAS with elimination and choice translating reality module, and proposed Picture fuzzy EDAS-ELECTRE evaluate cleaner production performance of gold mines. Ecer [36] combined the Fuzzy Analytic Hierarchy Process with the EDAS, put forward a new comprehensive model, and used it to select the suitable third party logistics supplier. Li and Wang et al. [37] extended the EDAS method to Linguistic Neutrosophic environment to solve the problem of fuzzy multi-criteria group decision-making.

With the globalization of economy and the fierce of market competition, project investment selection has become a hot issue of theoretical and practical concern. Making decisions scientifically has important strategic implications for improving investment efficiency. The choice of investment project has become an important part for any venture capital company. Choosing the right investment project can eliminate some environmental impact and improve the competitiveness. Pablo et al. [38] used the Analytic Hierarchy Process (AHP) and the Analytic Network Process (ANP) method to help a solar power investment company to select the best investment project. According to DEMATEL method and patent citation analysis, Altuntas et al. [39] selected the first investment project among the optional projects. Kiliş et al. [40] proposed a selection method for investment projects of type-2 fuzzy sets. Mohagheghi et al. [41] evaluated the research and development project in an interval type-2 fuzzy sets environment.

There will be conflicts in the evaluation criteria of investment projects, which will not guarantee that all the indicators given by the decision makers are completely reasonable and unbiased. Power average operator can reduce the negative impact of decision makers’ unreasonable assessments by support degree. And EDAS method is a useful and an easy-to-calculate method for assessing conflicting criteria. The combination of Power average operator and EDAS method can deal with group decision making problems with conflict criteria. From the existing literature, Power average operator and EDAS method were not combined to deal with SVTNS environment. Therefore, it is great significance propose Power average operator in SVTNS environment. In this paper, we choose SVTNS to represent the evaluation value corresponding to different investment project, which can represent more uncertain information. The selection of the Power average operator can take into account the interrelationship between the criteria, and the selection of the EDAS method can deal with conflicting criteria.

The motivations and goals of this paper are (1) to define the normalized Euclidean distance in the SVTNS environment; (2) to propose the Power weighted averaging (PWA) operator, Power weighted geometric (PWG) operator, generalized Power weighted averaging (GPWA) operator and generalized Power weighted geometric (GPWG) operator of SVTNS; (3) to construct a multi-criteria group decision-making model for the evaluation and selection of venture capital companies investment projects; and (4) to demonstrate feasibility and stability of the proposed method.

The rest of the paper is structured as follows: In the second section, the related concepts of the Single-valued triangular Neutrosophic sets, Power average operator are introduced in detail. The third section proposes the normalized Euclidean distance and the new operators of Single-valued triangular Neutrosophic sets. The forth section establishes a model for selecting suitable investment projects by combining new operators with EDAS methods. The fifth section gives a numerical example to prove the feasibility and stability of the proposed method. And the conclusion is discussed in the last section.

2 Preliminaries

In this section, we recall some basic concepts about the SVTNS and Power average operator, which will be necessary for this paper.

2.1 Single-valued triangular neutrosophic sets

The SVNS has been widely used in multi-criteria decision-making, since the SVNS can effectively express uncertain, incomplete, and inconsistent information. However, due to the limitation of knowledge, emotion and ability, decision makers often use fuzzy numbers to represent the truth-membership function, the indeterminacy-membership function, and the falsity-membership function. Meanwhile, the triangular fuzzy numbers can handle effectively fuzzy information rather than precise numbers. Biswas et al. [13] proposed SVTNS by combining triangular fuzzy numbers and SVNS.

Definition 1 [13]. Let X be a finite set of points (objects), let x denote a generic element in X, and E [0, 1] be the set of all triangular fuzzy numbers on [0, 1]. The SVTNS A in X is characterized by a truth-membership function T_A (x), an indeterminacy-membership function I_A (x) and a falsity-membership function F_A (x). Then, the SVTNS A can be defined as follows: $A = {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 | x \in X}$ where, T_A (x) : X → E [0, 1], I_A (x) : X → E [0, 1] and F_A (x) : X → E [0, 1]. T_A (x) , I_A (x) , F_A (x) can be expressed by the triangular fuzzy numbers as follows. $T_{A} (x) = (T_{A}^{1} (x), T_{A}^{2} (x), T_{A}^{3} (x))$ , $I_{A} (x) = (I_{A}^{1}$ $(x), I_{A}^{2} (x), I_{A}^{3} (x))$ and $F_{A} (x) = (F_{A}^{1} (x), F_{A}^{2} (x),$ $F_{A}^{3} (x))$ . For every x ∈ X, the sum of T_A (x) , I_A (x) , F_A (x) is $0 ⩽ T_{A}^{3} (x) + I_{A}^{3} (x) + F_{A}^{3} (x) ⩽ 3$ .

For convenience, we consider A = 〈 (a, b, c) , (e, f, g) , (r, s, t)〉 as a SVTN number, where $(T_{A}^{1} (x), T_{A}^{2} (x), T_{A}^{3} (x)) = (a, b, c)$ , $I_{A}^{1} (x), I_{A}^{2} (x),$ $I_{A}^{3} (x) = (e, f, g)$ and $(F_{A}^{1} (x), F_{A}^{2} (x), F_{A}^{3} (x)) = (r, s, t)$ .

Definition 2 [13]. Let A₁ = 〈 (a₁, b₁, c₁) , (e₁, f₁, g₁) , (r₁, s₁, t₁)〉,

A₂ =〈 (a₂, b₂, c₂) , (e₂, f₂, g₂) , (r₂, s₂, t₂) 〉 be two SVTN numbers, the rules of operations can be defined as follows:

$A_{1} \oplus A_{2} = 〈 \begin{matrix} (a_{1} + a_{2} - a_{1} a_{2}, b_{1} + b_{2} - b_{1} b_{2}, c_{1} + c_{2} - c_{1} c_{2}), \\ (e_{1} e_{2}, f_{1} f_{2}, g_{1} g_{2}), (r_{1} r_{2}, s_{1} s_{2}, t_{1} t_{2}) \end{matrix} 〉$

$A_{1} \otimes A_{2} = 〈 \begin{matrix} (a_{1} a_{2}, b_{1} b_{2}, c_{1} c_{2}), \\ (e_{1} + e_{2} - e_{1} e_{2}, f_{1} + f_{2} - f_{1} f_{2}, g_{1} + g_{2} - g_{1} g_{2}), \\ (r_{1} + r_{2} - r_{1} r_{2}, s_{1} + s_{2} - s_{1} s_{2}, t_{1} + t_{2} - t_{1} t_{2}) \end{matrix} 〉$

$λ A_{1} = 〈 \begin{matrix} (1 - {(1 - a_{1})}^{λ}, 1 - {(1 - b_{1})}^{λ}, 1 - {(1 - c_{1})}^{λ}), \\ (e_{1}^{λ}, f_{1}^{λ}, g_{1}^{λ}), (r_{1}^{λ}, s_{1}^{λ}, t_{1}^{λ}) \end{matrix} 〉, λ > 0$

$A_{1}^{λ} = 〈 \begin{matrix} (a_{1}^{λ}, b_{1}^{λ}, c_{1}^{λ}), \\ (1 - {(1 - e_{1})}^{λ}, 1 - {(1 - f_{1})}^{λ}, 1 - {(1 - g_{1})}^{λ}), \\ (1 - {(1 - r_{1})}^{λ}, 1 - {(1 - s_{1})}^{λ}, 1 - {(1 - t_{1})}^{λ}) \end{matrix} 〉, λ > 0$

The above operations satisfy the following properties:

A₁ ⊕ A₂ = A₂ ⊕ A₁ ; A₁ ⊗ A₂ = A₂ ⊗ A₁

λ (A₁ ⊕ A₂) = λA₁ ⊕ λA₂ ; (A₁ ⊗ A₂) ^λ = $A_{1}^{λ} \otimes A_{2}^{λ} λ > 0$

$λ_{1} A_{1} \oplus λ_{2} A_{1} = (λ_{1} + λ_{2}) A_{1} A_{1}^{λ_{1}} \otimes A_{1}^{λ_{2}} = A_{1}^{λ_{1} + λ_{2}} λ_{1}, λ_{2} > 0$

Definition 3 [13]. Let A = 〈 (a, b, c) , (e, f, g) , (r, s, t)〉 be a SVTN number, the score function S and accuracy function H can be expressed as follows:

$\begin{matrix} S (A) = & \frac{1}{12} [8 + (a + 2 b + c) - (e + 2 f + g) \\ - (r + 2 s + t)], S (A) \in [0, 1] \end{matrix}$ (1)

$\begin{matrix} H (A) = & \frac{1}{4} [(a + 2 b + c) - (r + 2 s + t)], \\ H (A) \in [- 1, 1] \end{matrix}$ (2)

Definition 4 [13]. Let A₁ and A₂ be two SVTN numbers, according definition 3 the order relations are defined as follows:

if S (A₁) ltS (A₂) , then A₁ ≺ A₂;

if S (A₁) gtS (A₂) , then A₁ ≻ A₂;

if S (A₁) = S (A₂) , H (A₁) ltH (A₂) , then A₁ ≺ A₂;

if S (A₁) = S (A₂) , H (A₁) gtH (A₂) , then A₁ ≻ A₂;

if S (A₁) = S (A₂) , H (A₁) = H (A₂) , then A₁ ∼ A₂;

Definition 5. Let A =〈 (a, b, c) , (e, f, g) , (r, s, t) 〉 be a SVTN number, the complement of A is denoted by A^c and is defined as follows: $A^{c} = 〈 (r, s, t), (1 - g, 1 - f, 1 - e), (a, b, c) 〉$ (3)

2.2 Power average operator

When decision makers deal with real problems, they will be affected in many ways. The decision makers will be biased against an assessment object, so the evaluation data given is not entirely reasonable. Yager [15] proposed a Power average aggregation operator that can handle the situation where the evaluation data given by the decision makers are too large or too small.

Definition 6 [15]. Let a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers. Then, the Power average operator is depicted as follows: $PA (a_{1}, a_{2}, \dots a_{n}) = \sum_{i = 1}^{n} \frac{1 + T (a_{i})}{\sum_{i = 1}^{n} (1 + T (a_{i}))} a_{i}$ (4) where, $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} \sup (a_{i}, a_{j})$ and sup (a_i, a_j) is the support degree for a_i from a_j, which satisfy the following properties:

sup (a_i, a_j) ∈ [0, 1]

sup (a_i, a_j) = sup (a_j, a_i)

if d (a_i, a_j) ltd (a_p, a_q), the sup (a_i, a_j) ⩾ sup (a_p, a_q), where d (a_i, a_j) is the distance between a_i and a_j.

3 Power operators of SVTNS

This section proposes the normalized Euclidean distance between two SVTN numbers, power weighted averaging operator, power weighted geometric operator, generalized power weighted average operator and generalized power weighted geometric operator of SVTNS. Then, we prove several properties of the normalized Euclidean distance and these operators.

3.1 Distance measurement

Definition 7 [14]. Let A₁ = 〈 (a₁, b₁, c₁) , (e₁, f₁, g₁) , (r₁, s₁, t₁)〉 and A₂ = 〈 (a₂, b₂, c₂) , (e₂, f₂, g₂) , (r₂, s₂, t₂)〉 be two SVTN numbers, and the normalized Hamming distance d_h (A₁, A₂) is depicted as follows:

$\begin{matrix} d_{h} (A_{1}, A_{2}) \\ = \frac{1}{9} (\begin{matrix} | a_{1} - a_{2} | + | b_{1} - b_{2} | + | c_{1} - c_{2} | \\ + | e_{1} - e_{2} | + | f_{1} - f_{2} | + | g_{1} - g_{2} | \\ + | r_{1} - r_{2} | + | s_{1} - s_{2} | + | t_{1} - t_{2} | \end{matrix}) \end{matrix}$ (5)

Definition 8. Let A₁ = 〈 (a₁, b₁, c₁) , (e₁, f₁, g₁) , (r₁, s₁, t₁)〉 and A₂ = 〈 (a₂, b₂, c₂) , (e₂, f₂, g₂) , (r₂, s₂, t₂)〉 be two SVTN numbers, and the normalized Euclidean distance d_e (A₁, A₂) is depicted as follows:

$\begin{matrix} d_{e} (A_{1}, A_{2}) \\ = \frac{1}{3} {(\begin{matrix} {| a_{1} - a_{2} |}^{2} + {| b_{1} - b_{2} |}^{2} + {| c_{1} - c_{2} |}^{2} \\ + {| e_{1} - e_{2} |}^{2} + {| f_{1} - f_{2} |}^{2} + {| g_{1} - g_{2} |}^{2} \\ + {| r_{1} - r_{2} |}^{2} + {| s_{1} - s_{2} |}^{2} + {| t_{1} - t_{2} |}^{2} \end{matrix})}^{1 / 2} \end{matrix}$ (6)

3.2 Power weighted averaging operator of SVTNS

Definition 9. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the SVTNPWA operator can be depicted as follows: $\begin{matrix} SVTNPWA (A_{1}, A_{2}, \dots, A_{n}) \\ = \oplus_{i = 1}^{n} \frac{(1 + T (A_{i})) w_{i} A_{i}}{\sum_{i = 1}^{n} ((1 + T (A_{i})) w_{i})} \end{matrix}$ where, $T (A_{i}) = \sum_{j = 1, j \neq i}^{n} w_{j} \sup (A_{i}, A_{j})$ and sup (A_i, A_j) is the support degree for A_i from A_j, which satisfy the following properties

sup (A_i, A_j) ∈ [0, 1]

sup (A_i, A_j) = sup (A_j, A_i)

if d (A_i, A_j) ltd (A_p, A_q), the sup (A_i, A_j) ⩾sup (A_p, A_q), where d (A_i, A_j) is the distance between A_i and A_j.

Theorem 1. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the result aggregated from definition 9 is still a SVTN number.

$\begin{matrix} SVTNPWA (A_{1}, A_{2} \dots, A_{n}) \\ = 〈 \begin{matrix} (1 - \prod_{i = 1}^{n} {(1 - a_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - b_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - c_{i})}^{φ_{i}}), \\ (\prod_{i = 1}^{n} e_{i}^{φ_{i}}, \prod_{i = 1}^{n} f_{i}^{φ_{i}}, \prod_{i = 1}^{n} g_{i}^{φ_{i}}), (\prod_{i = 1}^{n} r_{i}^{φ_{i}}, \prod_{i = 1}^{n} s_{i}^{φ_{i}}, \prod_{i = 1}^{n} t_{i}^{φ_{i}}) \end{matrix} 〉 \end{matrix}$ (7)

Proof. For convenience, let $φ_{i} = \frac{(1 + T (A_{i})) w_{i}}{\sum_{i = 1}^{n} ((1 + T (A_{i})) w_{i})}$ for the purpose of this proof. Then, the SVTNPWA can be proven. $\begin{matrix} φ_{i} A_{i} = & 〈 (1 - {(1 - a_{i})}^{φ_{i}}, 1 - {(1 - b_{i})}^{φ_{i}}, 1 - \\ {(1 - c_{i})}^{φ_{i}}), (e_{i}^{φ_{i}}, f_{i}^{φ_{i}}, g_{i}^{φ_{i}}), (r_{i}^{φ_{i}}, s_{i}^{φ_{i}}, t_{i}^{φ_{i}}) 〉 \end{matrix}$ $\oplus_{i = 1}^{n} φ_{i} A_{i} = 〈 \begin{matrix} (1 - \prod_{i = 1}^{n} {(1 - a_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - b_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - c_{i})}^{φ_{i}}), \\ (\prod_{i = 1}^{n} e_{i}^{φ_{i}}, \prod_{i = 1}^{n} f_{i}^{φ_{i}}, \prod_{i = 1}^{n} g_{i}^{φ_{i}}), (\prod_{i = 1}^{n} r_{i}^{φ_{i}}, \prod_{i = 1}^{n} s_{i}^{φ_{i}}, \prod_{i = 1}^{n} t_{i}^{φ_{i}}) \end{matrix} 〉$

We can prove that theorem 1 is established.

Property 1 (Idempotency). Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and if all SVTN numbers are equal, i.e., A_i = A. Then, $SVTNPWA (A_{1}, A_{2}, \dots, A_{n}) = A$

Proof. For convenience, let $φ_{i} = \frac{(1 + T (A_{i})) w_{i}}{\sum_{i = 1}^{n} ((1 + T (A_{i})) w_{i})}$ for the purpose of this proof. $\begin{matrix} SVTNPWA (A_{1}, A_{2}, \dots, A_{n}) \\ = \oplus_{i = 1}^{n} φ_{i} A_{i} = A \sum_{i = 1}^{n} φ_{i} = A \end{matrix}$

Then, we can prove that property 1 is established.

Property 2 (Boundedness). Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and $A^{+} = 〈 {max}_{i = 1}^{n}$ $(a_{i}, b_{i}, c_{i}), {min}_{i = 1}^{n} (e_{i}, f_{i}, g_{i}), {min}_{i = 1}^{n} (r_{i}, s_{i}, t_{i}) 〉$ , $A^{-} = 〈 {min}_{i = 1}^{n} (a_{i}, b_{i}, c_{i}), {max}_{i = 1}^{n} (e_{i}, f_{i}, g_{i}), {max}_{i = 1}^{n} (r_{i}, s_{i}, t_{i}) 〉$ . Then, $A^{-} ⩽ SVTNPWA (A_{1}, A_{2}, \dots, A_{n}) ⩽ A^{+}$

Proof. According to property 1 we can get $\begin{matrix} SVTNPWA (A_{1}, A_{2}, \dots, A_{n}) \\ = \oplus_{i = 1}^{n} φ_{i} A_{i} ⩽ \oplus_{i = 1}^{n} φ_{i} A^{+} = A^{+} \sum_{i = 1}^{n} φ_{i} = A^{+} \end{matrix}$ $\begin{matrix} SVTNPWA (A_{1}, A_{2}, \dots, A_{n}) \\ = \oplus_{i = 1}^{n} φ_{i} A_{i} ⩾ \oplus_{i = 1}^{n} φ_{i} A^{-} = A^{-} \sum_{i = 1}^{n} φ_{i} = A^{-} \end{matrix}$

Thus, we can prove that property 2 is established.

3.3 Power weighted geometric operator of SVTNS

Definition 10. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the SVTNPWG operator can be depicted as follows: $\begin{matrix} SVTNPWG (A_{1}, A_{2}, \dots, A_{n}) \\ = \otimes_{i = 1}^{n} A_{i}^{(1 + T (A_{i})) w_{i} / \sum_{i = 1}^{n} ((1 + T (A_{i})) w_{i})} \end{matrix}$ where, $T (A_{i}) = \sum_{j = 1, j \neq i}^{n} w_{j} \sup (A_{i}, A_{j})$ and sup (A_i, A_j) is the support degree for A_i from A_j, which satisfy the properties proposed in defined 9.

Theorem 2. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the result aggregated from definition 10 is still a SVTN number.

$\begin{matrix} SVTNPWG (A_{1}, A_{2} \dots, A_{n}) \\ = 〈 \begin{matrix} (\prod_{i = 1}^{n} a_{i}^{φ_{i}}, \prod_{i = 1}^{n} b_{i}^{φ_{i}}, \prod_{i = 1}^{n} c_{i}^{φ_{i}}), \\ (1 - \prod_{i = 1}^{n} {(1 - e_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - f_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - g_{i})}^{φ_{i}}), \\ (1 - \prod_{i = 1}^{n} {(1 - r_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - s_{i})}^{φ_{i}}, 1 - \prod_{i = 1}^{n} {(1 - t_{i})}^{φ_{i}}) \end{matrix} 〉 \end{matrix}$ (8)

The proof process of theorem 2 is similar to theorem 1, and the SVTNPWG operator has the characteristics of idempotency and boundedness.

3.4 Generalized power weighted averaging operator of SVTNS

Definition 11. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the SVTNGPWA operator can be depicted as follows: $\begin{matrix} SVTNGPWA (A_{1}, A_{2}, \dots, A_{n}) \\ = {(\oplus_{i = 1}^{n} \frac{(1 + T (A_{i})) w_{i} A_{i}^{λ}}{\sum_{i = 1}^{n} ((1 + T (A_{i})) w_{i})})}^{1 / λ} \end{matrix}$ where, $T (A_{i}) = \sum_{j = 1, j \neq i}^{n} w_{j} \sup (A_{i}, A_{j})$ and sup (A_i, A_j) is the support degree for A_i from A_j, which satisfy the properties proposed in defined 9.

Theorem 3. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the result aggregated from definition 11 is still a SVTN number.

$\begin{matrix} SVTNGPWA (A_{1}, A_{2} \dots, A_{n}) \\ = 〈 \begin{matrix} ({[1 - \prod_{i = 1}^{n} {(1 - a_{i}^{λ})}^{φ_{i}}]}^{1 / λ}, {[1 - \prod_{i = 1}^{n} {(1 - b_{i}^{λ})}^{φ_{i}}]}^{1 / λ}, {[1 - \prod_{i = 1}^{n} {(1 - c_{i}^{λ})}^{φ_{i}}]}^{1 / λ}), \\ (1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - e_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}, 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - f_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}, 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - g_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}), \\ (1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - r_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}, 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - s_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}, 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - t_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}) \end{matrix} 〉 \end{matrix}$ (9)

The proof process of theorem 3 is similar to theorem 1, and the SVTNGPWA operator has the characteristics of idempotency and boundedness.

3.5 Generalized power weighted geometric operator of SVTNS

Definition 12. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉 (i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the SVTNGPWG operator can be depicted as follows: $\begin{matrix} SVTNGPWG (A_{1}, A_{2}, \dots, A_{n}) \\ = \frac{1}{λ} \otimes_{i = 1}^{n} {(λ A_{i})}^{(1 + T (A_{i})) w_{i} / \sum_{i = 1}^{n} ((1 + T (A_{i})) w_{i})} \end{matrix}$ where, $T (A_{i}) = \sum_{j = 1, j \neq i}^{n} w_{j} \sup (A_{i}, A_{j})$ and sup (A_i, A_j) is the support degree for A_i from A_j, which satisfy the properties proposed in defined 9.

Theorem 4. Let A_i = 〈 (a_i, b_i, c_i) , (e_i, f_i, g_i) , (r_i, s_i, t_i)〉(i = 1, 2, ⋯ , n) be a set of SVTN numbers and w_i be the weight of A_i, where w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the result aggregated from definition 12 is still a SVTN number.

$\begin{matrix} \begin{matrix} SVTNGPWG (A_{1}, A_{2} \dots, A_{n}) \\ = 〈 \begin{matrix} (1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - a_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}, 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - b_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}, 1 - {[1 - \prod_{i = 1}^{n} {(1 - {(1 - c_{i})}^{λ})}^{φ_{i}}]}^{1 / λ}), \\ ({[1 - \prod_{i = 1}^{n} {(1 - e_{i}^{λ})}^{φ_{i}}]}^{1 / λ}, {[1 - \prod_{i = 1}^{n} {(1 - f_{i}^{λ})}^{φ_{i}}]}^{1 / λ}, {[1 - \prod_{i = 1}^{n} {(1 - g_{i}^{λ})}^{φ_{i}}]}^{1 / λ}) \\ ({[1 - \prod_{i = 1}^{n} {(1 - r_{i}^{λ})}^{φ_{i}}]}^{1 / λ}, {[1 - \prod_{i = 1}^{n} {(1 - s_{i}^{λ})}^{φ_{i}}]}^{1 / λ}, {[1 - \prod_{i = 1}^{n} {(1 - t_{i}^{λ})}^{φ_{i}}]}^{1 / λ}) \end{matrix} 〉 \end{matrix} \end{matrix}$ (10)

The proof process of theorem 4 is similar to theorem 1, and the SVTNGPWG operator has the characteristics of idempotency and boundedness.

4 Decision-making method based on EDAS

This section suggests a new investment project model based on EDAS.

Suppose that there are m investment projects X ={ x₁, x₂, ⋯ , x_m }, n criteria C ={ c₁, c₂, ⋯ , c_n } and p experts E ={ e₁, e₂, ⋯ , e_p }. Let w_j (=1, 2, ⋯, n) be the weight of the criteria, where w_j ⩾ 0 and $\sum_{j = 1}^{n} w_{j}$ . Let ω_k (k = 1, 2, ⋯ , n) be the weight of the decision makers, where ω_k ⩾ 0 and $\sum_{k = 1}^{p} ω_{k}$ . Let U = (a_ij) _m×n be a single-valued triangular Neutrosophic decision matrix, where A_ij =〈 T_ij, I_ij, F_ij 〉 is the evaluation value of each investment project under different criteria.

The specific process of the method is depicted as follows.

Step 1: Normalize the decision matrix.

There are two types of criteria for evaluating an investment project, one of which is a benefit criteria B and the other is a cost criteria C. Definition 5 can convert the cost-type criteria into benefit-type criteria. Then, the decision matrix U can be transformed into the normalized matrix R by Equation (11). $A_{ij} = {\begin{matrix} A_{ij}, & c_{j} \in B \\ A_{ij}^{c}, & c_{j} \in C \end{matrix}$ (11)

Step 2: Calculate the supports. $\begin{matrix} \sup (A_{ij}^{k}, A_{ij}^{\tilde{k}}) = 1 - d (A_{ij}^{k}, A_{ij}^{\tilde{k}}) \\ (i = 1, 2, \dots m) (j = 1, 2, \dots n) (k, \tilde{k} = 1, 2, \dots p) \end{matrix}$ (12)

Step 3: Calculate $T (A_{ij}^{k})$ and $φ_{ij}^{k}$ based on Step 2. $φ_{ij}^{k} = \frac{(1 + T (A_{i})) ω_{k}}{\sum_{k = 1}^{p} ((1 + T (A_{i})) ω_{k})}$ (13) $T (A_{ij}^{k}) = \sum_{k = 1, k \neq \tilde{k}}^{p} ω_{\tilde{k}} sup (A_{ij}^{k}, A_{ij}^{\tilde{k}})$ (14) where ω_k is the weight of the decision maker.

Step 4: Calculate the collective matrix.

Aggregate the decision matrixes by using the PWA operator, PWG operator, GPWA operator and GPWG operator of SVTNS.

Step 5: Calculate the average solution AV. $AV = {[{AV}_{j}]}_{1 \times n} = {[\frac{1}{m} \oplus_{i = 1}^{m} A_{ij}]}_{1 \times n}$ (15) $\frac{1}{m} \oplus_{i = 1}^{m} A_{ij} = 〈 \begin{matrix} (1 - \prod_{i = 1}^{m} {(1 - a_{ij})}^{1 / m}, 1 - \prod_{i = 1}^{m} {(1 - b_{ij})}^{1 / m}, 1 - \prod_{i = 1}^{m} {(1 - c_{ij})}^{1 / m}), \\ (\prod_{i = 1}^{m} e_{ij}^{1 / m}, \prod_{i = 1}^{m} f_{ij}^{1 / m}, \prod_{i = 1}^{m} g_{ij}^{1 / m}), (\prod_{i = 1}^{m} r_{ij}^{1 / m}, \prod_{i = 1}^{m} s_{ij}^{1 / m}, \prod_{i = 1}^{m} t_{ij}^{1 / m}) \end{matrix} 〉 (16)$

Step 6: Calculate the positive distance from average (PDA) matrices and the negative distance from average (NDA) matrices.

$\begin{matrix} PDA & = {[{PDA}_{ij}]}_{m \times n} \\ = {[\frac{max (0, S (A_{ij}) - S ({AV}_{j}))}{S ({AV}_{j})}]}_{m \times n} \end{matrix}$ (16)

$\begin{matrix} NDA & = {[{NDA}_{ij}]}_{m \times n} \\ = {[\frac{max (0, S ({AV}_{j}) - S (A_{ij}))}{S ({AV}_{j})}]}_{m \times n} \end{matrix}$ (17) where, S (A_ij) and S (AV_j) are the score function of A_ij and AV_j

Step 7: Calculate the weighted sum of the PDA and the NDA. ${SP}_{i} = \sum_{j = 1}^{n} w_{j} {PDA}_{ij}$ (18) ${SN}_{i} = \sum_{j = 1}^{n} w_{j} {NDA}_{ij}$ (19)

Step 8: Normalize the values of SP_i and SN_i. ${NSP}_{i} = \frac{{SP}_{i}}{max_{i} {SP}_{i}}$ (20) ${NSN}_{i} = 1 - \frac{{SN}_{i}}{max_{i} {SN}_{i}}$ (21)

Step 9: Calculate the appraisal score AS_i. ${AS}_{i} = \frac{1}{2} ({NSP}_{i} + {NSN}_{i})$ (22)

Step 10: Rank the alternatives.

The rule of sorting is that the higher the value of AS_i, the better the alternative.

Step 11: Replace λ with different values and compare analysis.

5 Numerical example

A venture capital company wants to invest in a project from five investment projects X ={ x₁, x₂, ⋯ , x₅ }. The five evaluation criteria C ={ c₁, c₂, ⋯ , c₅ } of investment projects are pollutants emission, resource and energy consumption, product characteristics, ecological environment and management level. The pollutants emission and the resource and energy consumption are cost criteria. The product characteristics, the ecological environment and the management level are benefit criteria. There are three experts E ={ e₁, e₂, e₃ } evaluate these five investment projects under five criteria. Suppose that the criteria weighting vector is (0.17, 0.22, 0.34, 0.18, 0.09) ^T, the decision makers weighting vector is (0.278, 0.395, 0.327) ^T.

The specific process of choose the best investment project is depicted as follows.

Step 1: Normalize the decision matrix.

Transform the decision matrix into the normalized matrix. The results of the normalization of the decision matrix are shown in Tables 1 –3.

Table 1
Decision matrix of e₁

c ₁ c ₂ c ₃ c ₄ c ₅

x ₁ < (0.21,0.34,0.46), (0.38,0.45,0.53), (0.46,0.53,0.71)> < (0.80,0.86,0.92), (0.26,0.28,0.37), (0.05,0.18,0.24)> < (0.42,0.52,0.81), (0.41,0.45,0.58), (0.34,0.42,0.61)> < (0.51,0.55,0.58), (0.25,0.37,0.57), (0.15,0.47,0.62)> < (0.72,0.74,0.85), (0.62,0.68,0.72), (0.42,0.48,0.52)>

x ₂ < (0.61,0.69,0.76), (0.25,0.37,0.42), (0.20,0.29,0.31)> < (0.71,0.84,0.86), (0.51,0.58,0.64), (0.12,0.35,0.42)> < (0.51,0.64,0.68), (0.40,0.51,0.67), (0.34,0.45,0.61)> < (0.80,0.82,0.86), (0.42,0.51,0.67), (0.21,0.25,0.38)> < (0.57,0.61,0.97), (0.53,0.61,0.84), (0.61,0.67,0.75)>

x ₃ < (0.75,0.84,0.91), (0.54,0.62,0.67), (0.47,0.57,0.61)> < (0.64,0.69,0.85), (0.34,0.45,0.52), (0.21,0.29,0.38)> < (0.42,0.57,0.61), (0.45,0.48,0.51), (0.35,0.39,0.42)> < (0.50,0.58,0.61), (0.42,0.51,0.67), (0.31,0.38,0.56)> < (0.71,0.75,0.82), (0.61,0.68,0.72), (0.31,0.58,0.86)>

x ₄ < (0.34.0.42.0.61), (0.65,0.67,0.75), (0.61,0.69,0.76)> < (0.57,0.61,0.97), (0.53,0.61,0.84), (0.28,0.36,0.42)> < (0.87,0.89,0.94), (0.54,0.68,0.71), (0.23,0.35,0.51)> < (0.64,0.84,0.87), (0.51,0.58,0.64), (0.32,0.38,0.42)> < (0.47,0.51,0.67), (0.32,0.45,0.58), (0.12,0.38,0.45)>

x ₅ < (0.42,0.52,0.81), (0.41,0.45,0.58), (0.34,0.42,0.61)> < (0.51,0.58,0.64), (0.34,0.39,0.41), (0.24,0.29,0.34)> < (0.72,0.74,0.85), (0.62,0.68,0.72), (0.42,0.48,0.52)> < (0.71,0.84,0.86), (0.51,0.58,0.64), (0.12,0.35,0.42)> < (0.64,0.69,0.85), (0.34,0.45,0.52), (0.21,0.29,0.38)>

	c ₁	c ₂	c ₃	c ₄	c ₅
x ₁	< (0.21,0.34,0.46), (0.38,0.45,0.53), (0.46,0.53,0.71)>	< (0.80,0.86,0.92), (0.26,0.28,0.37), (0.05,0.18,0.24)>	< (0.42,0.52,0.81), (0.41,0.45,0.58), (0.34,0.42,0.61)>	< (0.51,0.55,0.58), (0.25,0.37,0.57), (0.15,0.47,0.62)>	< (0.72,0.74,0.85), (0.62,0.68,0.72), (0.42,0.48,0.52)>
x ₂	< (0.61,0.69,0.76), (0.25,0.37,0.42), (0.20,0.29,0.31)>	< (0.71,0.84,0.86), (0.51,0.58,0.64), (0.12,0.35,0.42)>	< (0.51,0.64,0.68), (0.40,0.51,0.67), (0.34,0.45,0.61)>	< (0.80,0.82,0.86), (0.42,0.51,0.67), (0.21,0.25,0.38)>	< (0.57,0.61,0.97), (0.53,0.61,0.84), (0.61,0.67,0.75)>
x ₃	< (0.75,0.84,0.91), (0.54,0.62,0.67), (0.47,0.57,0.61)>	< (0.64,0.69,0.85), (0.34,0.45,0.52), (0.21,0.29,0.38)>	< (0.42,0.57,0.61), (0.45,0.48,0.51), (0.35,0.39,0.42)>	< (0.50,0.58,0.61), (0.42,0.51,0.67), (0.31,0.38,0.56)>	< (0.71,0.75,0.82), (0.61,0.68,0.72), (0.31,0.58,0.86)>
x ₄	< (0.34.0.42.0.61), (0.65,0.67,0.75), (0.61,0.69,0.76)>	< (0.57,0.61,0.97), (0.53,0.61,0.84), (0.28,0.36,0.42)>	< (0.87,0.89,0.94), (0.54,0.68,0.71), (0.23,0.35,0.51)>	< (0.64,0.84,0.87), (0.51,0.58,0.64), (0.32,0.38,0.42)>	< (0.47,0.51,0.67), (0.32,0.45,0.58), (0.12,0.38,0.45)>
x ₅	< (0.42,0.52,0.81), (0.41,0.45,0.58), (0.34,0.42,0.61)>	< (0.51,0.58,0.64), (0.34,0.39,0.41), (0.24,0.29,0.34)>	< (0.72,0.74,0.85), (0.62,0.68,0.72), (0.42,0.48,0.52)>	< (0.71,0.84,0.86), (0.51,0.58,0.64), (0.12,0.35,0.42)>	< (0.64,0.69,0.85), (0.34,0.45,0.52), (0.21,0.29,0.38)>

Table 2

Decision matrix of e₂

	c ₁	c ₂	c ₃	c ₄	c ₅
x ₁	< (0.87,0.89,0.93), (0.41,0.48,0.64), (0.23,0.27,0.39)>	< (0.42,0.53,0.69), (0.34,0.37,0.42), (0.42,0.57,0.61)>	< (0.61,0.67,0.81), (0.35,0.42,0.68), (0.12,0.37,0.42)>	< (0.53,0.67,0.82), (0.41,0.48,0.59), (0.71,0.85,0.92)>	< (0.32,0.41,0.59), (0.42,0.58,0.61), (0.54,0.61,0.69)>
x ₂	< (0.42,0.57,0.61), (0.64,0.85,0.93), (0.35,0.41,0.42)>	< (0.64,0.69,0.87), (0.43,0.58,0.64), (0.32,0.39,0.42)>	< (0.82,0.86,0.92), (0.26,0.29,0.37), (0.15,0.18,0.24)>	< (0.61,0.84,0.86), (0.29,0.58,0.64), (0.12,0.37,0.42)>	< (0.51,0.69,0.85), (0.34,0.39,0.52), (0.21,0.31,0.38)>
x ₃	< (0.48,0.61,0.97), (0.53,0.72,0.84), (0.28,0.29,0.42)>	< (0.71,0.76,0.86), (0.36,0.58,0.64), (0.12,0.36,0.42)>	< (0.64,0.69,0.85), (0.34,0.42,0.52), (0.21,0.31,0.38)>	< (0.26.0.42.0.61), (0.65,0.69,0.75), (0.52,0.69,0.76)>	< (0.51,0.52,0.73), (0.41,0.47,0.58), (0.34,0.53,0.61)>
x ₄	< (0.57,0.61,0.97), (0.43,0.59,0.84), (0.56,0.67,0.75)>	< (0.75,0.82,0.86), (0.42,0.49,0.67), (0.21,0.27,0.38)>	< (0.71,0.79,0.82), (0.61,0.64,0.72), (0.31,0.58,0.86)>	< (0.51,0.68,0.85), (0.34,0.48,0.52), (0.21,0.29,0.38)>	< (0.51,0.69,0.76), (0.25,0.37,0.42), (0.23,0.29,0.45)>
x ₅	< (0.57,0.62,0.97), (0.43,0.61,0.84), (0.26,0.37,0.42)>	< (0.62,0.73,0.85), (0.62,0.67,0.72), (0.42,0.45,0.52)>	< (0.47,0.61,0.97), (0.53,0.75,0.84), (0.28,0.32,0.42)>	< (0.37,0.52,0.97), (0.53,0.59,0.84), (0.31,0.36,0.42)>	< (0.55,0.84,0.86), (0.51,0.58,0.65), (0.12,0.32,0.42)>

Table 3

Decision matrix of e₃

	c ₁	c ₂	c ₃	c ₄	c ₅
x ₁	< (0.57,0.84,0.86), (0.18,0.58,0.64), (0.12,0.29,0.42)>	< (0.55,0.66,0.86), (0.51,0.58,0.65), (0.12,0.32,0.61)>	< (0.59,0.76,0.86), (0.36,0.55,0.64), (0.12,0.36,0.45)>	< (0.57,0.61,0.87), (0.33,0.59,0.84), (0.56,0.69,0.75)>	< (0.31,0.52,0.78), (0.41,0.46,0.58), (0.34,0.55,0.61)>
x ₂	< (0.34,0.43,0.59), (0.32,0.34,0.51), (0.44,0.61,0.68)>	< (0.45,0.67,0.97), (0.53,0.57,0.84), (0.51,0.63,0.78)>	< (0.77,0.89,0.93), (0.31,0.47,0.64), (0.23,0.27,0.37)>	< (0.56,0.69,0.85), (0.34,0.35,0.52), (0.21,0.36,0.38)>	< (0.37,0.65,0.95), (0.52,0.56,0.84), (0.28,0.38,0.42)>
x ₃	< (0.42,0.53,0.69), (0.34,0.37,0.42), (0.42,0.57,0.61)>	< (0.31,0.55,0.58), (0.25,0.42,0.57), (0.15,0.47,0.62)>	< (0.44,0.69,0.87), (0.43,0.49,0.54), (0.32,0.39,0.42)>	< (0.80,0.82,0.86), (0.42,0.51,0.67), (0.21,0.25,0.38)>	< (0.51,0.86,0.92), (0.26,0.33,0.37), (0.15,0.38,0.24)>
x ₄	< (0.27,0.52,0.97), (0.33,0.59,0.84), (0.31,0.36,0.56)>	< (0.26.0.32.0.61), (0.45,0.69,0.75), (0.52,0.69,0.76)>	< (0.71,0.84,0.86), (0.41,0.58,0.64), (0.12,0.35,0.42)>	< (0.25,0.72,0.86), (0.42,0.48,0.63), (0.25,0.27,0.38)>	< (0.61,0.84,0.86), (0.51,0.61,0.64), (0.12,0.35,0.42)>
x ₅	< (0.51,0.57,0.85), (0.34,0.39,0.52), (0.21,0.29,0.48)>	< (0.37,0.61,0.97), (0.53,0.71,0.84), (0.18,0.36,0.42)>	< (0.43,0.67,0.82), (0.41,0.78,0.89), (0.51,0.85,0.92)>	< (0.51,0.61,0.97), (0.43,0.61,0.84), (0.61,0.63,0.75)>	< (0.42,0.69,0.85), (0.34,0.39,0.52), (0.21,0.41,0.58)>

Table 4

Comprehensive decision matrix

	c ₁	c ₂	c ₃	c ₄	c ₅
x ₁	< (0.68,0.80,0.84), (0.30,0.50,0.61), (0.22,0.33,0.47)>	< (0.61,0.70,0.84), (0.36,0.40,0.47), (0.15,0.34,0.47)>	< (0.56,0.67,0.83), (0.37,0.47,0.64), (0.16,0.38,0.48)>	< (0.54,0.62,0.79), (0.33,0.48,0.66), (0.42,0.67,0.77)>	< (0.47,0.56,0.75), (0.47,0.56,0.63), (0.43,0.55,0.61)>
x ₂	< (0.46,0.57,0.66), (0.39,0.49,0.61), (0.32,0.42,0.45)>	< (0.61,0.74,0.92), (0.48,0.58,0.70), (0.28,0.44,0.51)>	< (0.74,0.83,0.89), (0.31,0.40,0.53), (0.22,0.27,0.36)>	< (0.67,0.79,0.86), (0.34,0.47,0.61), (0.17,0.33,0.39)>	< (0.49,0.66,0.93), (0.44,0.50,0.70), (0.31,0.41,0.48)>
x ₃	< (0.56,0.68,0.91), (0.46,0.55,0.63), (0.37,0.44,0.53)>	< (0.59,0.68,0.80), (0.31,0.49,0.58), (0.15,0.37,0.46)>	< (0.52,0.66,0.81), (0.40,0.46,0.52), (0.28,0.36,0.40)>	< (0.57,0.64,0.72), (0.50,0.57,0.70), (0.33,0.42,0.55)>	< (0.58,0.73,0.84), (0.40,0.47,0.53), (0.25,0.49,0.50)>
x ₄	< (0.42,0.53,0.94), (0.44,0.61,0.81), (0.47,0.55,0.68)>	< (0.58,0.65,0.88), (0.46,0.58,0.74), (0.31,0.40,0.49)>	< (0.77,0.84,0.88), (0.52,0.63,0.69), (0.21,0.42,0.58)>	< (0.48,0.75,0.86), (0.41,0.51,0.59), (0.25,0.31,0.39)>	< (0.53,0.71,0.78), (0.34,0.46,0.53), (0.15,0.33,0.44)>
x ₅	< (0.51,0.58,0.91), (0.39,0.48,0.64), (0.26,0.35,0.49)>	< (0.52,0.65,0.89), (0.50,0.59,0.65), (0.27,0.37,0.43)>	< (0.55,0.67,0.91), (0.51,0.74,0.82), (0.38,0.49,0.58)>	< (0.54,0.67,0.95), (0.49,0.59,0.78), (0.30,0.43,0.51)>	< (0.54,0.76,0.85), (0.40,0.47,0.57), (0.17,0.34,0.45)>

Step 2: Calculate the supports and the calculation results are depicted as follows: $\begin{matrix} \sup (A_{ij}^{1}, A_{ij}^{2}) = \sup (A_{ij}^{2}, A_{ij}^{1}) \\ = [\begin{matrix} 0.7044 & 0.7456 & 0.8900 & 0.7878 & 0.7978 \\ 0.7533 & 0.9389 & 0.7489 & 0.9233 & 0.7644 \\ 0.8333 & 0.9311 & 0.8867 & 0.8211 & 0.8444 \\ 0.8611 & 0.8778 & 0.8711 & 0.8956 & 0.9089 \\ 0.8700 & 0.7933 & 0.8689 & 0.8667 & 0.9067 \end{matrix}] \end{matrix}$ $\begin{matrix} \sup (A_{ij}^{1}, A_{ij}^{3}) \\ = \sup (A_{ij}^{3}, A_{ij}^{1}) \\ = [\begin{matrix} 0.7144 & 0.7867 & 0.8767 & 0.8067 & 0.8322 \\ 0.7978 & 0.8000 & 0.8389 & 0.9022 & 0.8589 \\ 0.8211 & 0.8456 & 0.9456 & 0.8667 & 0.7289 \\ 0.7944 & 0.7644 & 0.9122 & 0.8956 & 0.8744 \\ 0.9156 & 0.8167 & 0.8078 & 0.7833 & 0.9333 \end{matrix}] \end{matrix}$ $\begin{matrix} \sup (A_{ij}^{2}, A_{ij}^{3}) \\ = \sup (A_{ij}^{3}, A_{ij}^{2}) \\ = [\begin{matrix} 0.8989 & 0.8233 & 0.9578 & 0.8811 & 0.9100 \\ 0.7733 & 0.8433 & 0.9011 & 0.9167 & 0.8744 \\ 0.7789 & 0.8256 & 0.9300 & 0.6878 & 0.8067 \\ 0.8622 & 0.7044 & 0.8567 & 0.9378 & 0.8589 \\ 0.8833 & 0.8700 & 0.8100 & 0.8611 & 0.8756 \end{matrix}] \end{matrix}$

Step 3: Calculate $φ_{ij}^{k}$ based on Step 2 and the calculation results are depicted as follows:

$φ_{ij}^{1} = [\begin{matrix} 0.2771 & 0.2839 & 0.2842 & 0.2830 & 0.2826 \\ 0.2869 & 0.2892 & 0.2817 & 0.2877 & 0.2839 \\ 0.2896 & 0.2909 & 0.2871 & 0.2949 & 0.2863 \\ 0.2858 & 0.2932 & 0.2893 & 0.2858 & 0.2893 \\ 0.2880 & 0.2839 & 0.2889 & 0.2857 & 0.2899 \end{matrix}]$ $φ_{ij}^{2} = [\begin{matrix} 0.3880 & 0.3838 & 0.3847 & 0.3847 & 0.3843 \\ 0.3825 & 0.3868 & 0.3830 & 0.3834 & 0.3819 \\ 0.3829 & 0.3843 & 0.3812 & 0.3787 & 0.3878 \\ 0.3861 & 0.3847 & 0.3809 & 0.3836 & 0.3833 \\ 0.3813 & 0.3842 & 0.3847 & 0.3867 & 0.3810 \end{matrix}]$ $φ_{ij}^{3} = [\begin{matrix} 0.3349 & 0.3323 & 0.3310 & 0.3323 & 0.3331 \\ 0.3307 & 0.3241 & 0.3353 & 0.3289 & 0.3342 \\ 0.3275 & 0.3249 & 0.3317 & 0.3264 & 0.3259 \\ 0.3281 & 0.3220 & 0.3298 & 0.3306 & 0.3274 \\ 0.3307 & 0.3319 & 0.3265 & 0.3276 & 0.3291 \end{matrix}]$

Step 4: Calculate the collective matrix.

Aggregate the decision matrixes by using the PWA operator and the calculation results are shown in Table 4.

Step 5: Calculate the average solution AV and the calculation results are depicted as follows: $AV = [\begin{matrix} < (0.54, 0.65, 0.88), (0.39, 0.53, 0.66), (0.32, 0.41, 0.52) >, \\ < (0.58, 0.69, 0.87), (0.42, 0.52, 0.62), (0.22, 0.38, 0.47) >, \\ < (0.64, 0.75, 0.87), (0.41, 0.53, 0.63), (0.24, 0.38, 0.47) >, \\ < (0.56, 0.70, 0.86), (0.41, 0.52, 0.66), (0.28, 0.41, 0.51) >, \\ < (0.52, 0.69, 0.85), (0.41, 0.49, 0.59), (0.25, 0.42, 0.49) > \end{matrix}]$

Step 6: Calculate the positive distance from average (PDA) matrices and the negative distance from average (NDA) matrices and the calculation results are depicted as follows: $PDA = [\begin{matrix} 0.1293 & 0.0861 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.1461 & 0.1264 & 0.0000 \\ 0.0000 & 0.0328 & 0.0000 & 0.0000 & 0.0123 \\ 0.0000 & 0.0000 & 0.0000 & 0.0669 & 0.0671 \\ 0.0258 & 0.0000 & 0.0000 & 0.0000 & 0.0690 \end{matrix}]$ $NDA = [\begin{matrix} 0.0000 & 0.0000 & 0.0096 & 0.1417 & 0.1701 \\ 0.0414 & 0.0414 & 0.0000 & 0.0000 & 0.0328 \\ 0.0116 & 0.0000 & 0.0053 & 0.0825 & 0.0000 \\ 0.1807 & 0.0668 & 0.0308 & 0.0000 & 0.0000 \\ 0.0000 & 0.0459 & 0.1882 & 0.0538 & 0.0000 \end{matrix}]$ where, S (A_ij) and S (AV_j) are the score function of A_ij and AV_j.

Step 7: Calculate the weighted sum of the PDA and the NDA and the calculation results are depicted as follows: $\begin{matrix} {SP}_{1} = 0.0409, {SP}_{2} = 0.0724, {SP}_{3} = 0.0083, \\ {SP}_{4} = 0.0181, {SP}_{5} = 0.0106 \end{matrix}$ $\begin{matrix} {SN}_{1} = 0.0441, {SN}_{2} = 0.0191, {SN}_{3} = 0.0186, \\ {SN}_{4} = 0.0559, {SN}_{5} = 0.0838 \end{matrix}$

Step 8: Normalize the values of SP_i and SN_i and the calculation results are depicted as follows: $\begin{matrix} {NSP}_{1} = 0.5650, {NSP}_{2} = 1.0000, {NSP}_{3} = 0.1148, \\ {NSP}_{4} = 0.2496, {NSP}_{5} = 0.1465 \end{matrix}$ $\begin{matrix} {NSN}_{1} = 0.4736, {NSN}_{2} = 0.7721, {NSN}_{3} = 0.7778, \\ {NSN}_{4} = 0.3328, {NSN}_{5} = 0.0000 \end{matrix}$

Step 9: Calculate the appraisal score AS_i and the calculation results are depicted as follows: $\begin{matrix} {AS}_{1} = 0.5193, {AS}_{2} = 0.8861, {AS}_{3} = 0.4463, \\ {AS}_{4} = 0.2912, {AS}_{5} = 0.0732 \end{matrix}$

Step 10: Rank the alternatives and the calculation results are depicted as follows: $x_{2} ≻ x_{1} ≻ x_{3} ≻ x_{4} ≻ x_{5}$

Step 11: Replace λ with different values and compare analysis.

In order to consider the influence of parameters and distance on the ranking, this step analyzes the influence of the change of λ and d_h and d_e respectively on the ordering. The effect of the change of λ and d_h and d_e respectively is shown in Table 5.

Table 5

The ranking orders of different λ

λ	Operators	d _h	d _e
1	SVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
2	GSVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
5	GSVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
10	GSVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
20	GSVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
50	GSVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₅ ≻ x₄	x₂ ≻ x₁ ≻ x₃ ≻ x₅ ≻ x₄
1	SVTNPWG	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
2	GSVTNPWG	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅
5	GSVTNPWG	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅
10	GSVTNPWG	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅
20	GSVTNPWG	x₂ ≻ x₃ ≻ x₅ ≻ x₄ ≻ x₁	x₂ ≻ x₃ ≻ x₅ ≻ x₄ ≻ x₁
50	GSVTNPWG	x₃ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₄	x₃ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₄

From the results of Table 5, it can be seen that when λ = 1, the GSVTNPWA and GSVTNPWG operators are degenerated into the SVTNPWA and SVTNPWG operator respectively. As λ gradually increases, the optimal choice for GSVTNPWA is x₂. But when λ = 50, the optimal choice for GSVTNPWG is x₃. Therefore, it can be considered that GSVTNPWA is better than GSVTNPWG when dealing with fuzzy information. The GSVTNPWA operator is superior to the GSVTNPWG operator when dealing with SVTNS fuzzy information. At the same time, the validity and feasibility of the proposed method are proved by the example of investment project selection.

Comparing our proposed method with literature [13, 14], the results are shown in Table 6.

Table 6

Rank of alternatives by different approaches

	order
SVTNPWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
SVTNPWG	x₂ ≻ x₁ ≻ x₃ ≻ x₅ ≻ x₄
[13] TFNNWA	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
[13] TFNNWG	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅
[14] TFNNWA and VIKOR	x₂ ≻ x₁ ≻ x₃ ≻ x₄ ≻ x₅
[14] TFNNWG and VIKOR	x₂ ≻ x₃ ≻ x₁ ≻ x₄ ≻ x₅

According to the results in Table 6, the best choice is x₂, which proves the applicability and feasibility of the method proposed in this paper.

6 Conclusion

In this paper, we use Power operator that consider the interrelationship between data through support and eliminate the influence of biased decision makers on irrational data. And we apply the Single-valued triangular Neutrosophic sets to indicate the evaluation value, which can retain more uncertain information of the object to be evaluated. The main contribution of this paper are proposing SVTNPWA operator, SVTNPWG operator, GSVTNPWA operator and GSVTNPWG operator, and using EDAS method to the selection of investment projects in the Single-valued triangular Neutrosophic sets environment. In this paper, the effectiveness and feasibility of the method are proved by an example of investment project selection.

In the future, we can further observe the scope of application of SVTNPWA operator, SVTNPWG operator, GSVTNPWA operator and GSVTNPWG operator. At the same time, we can conduct different research on EDAS.

Footnotes

Acknowledgments

This work was supported in part by the Fund for Shanxi “1331 Project” Key Innovative Research Team 2017, and in part by “The Discipline Group Construction Plan for Serving Industries Innovation”, Shanxi, China: The Discipline Group Program of Intelligent Logistics Management for Serving Industries Innovation 2018.

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