Some weighted geometric operators with SVTrN-numbers and their application to multi-criteria decision making problems

Abstract

Since the single valued triangular neutrosophic number (SVTrN-number) is a generalization of triangular fuzzy numbers and triangular intuitionistic fuzzy numbers, it may express more abundant and flexible information as compared with the triangular fuzzy numbers and triangular intuitionistic fuzzy numbers. This article introduces an approach to handle multi-criteria decision making (MCDM) problems under the SVTrN-numbers. Therefore, we first proposed some new geometric operators are called SVTrN weighted geometric operator, SVTrN ordered weighted geometric operator and SVTrN ordered hybrid weighted geometric operator. Also we studied some desirable properties of the geometric operators. And then, an approach based on the SVTrN ordered hybrid weighted geometric operator is developed to solve multi-criteria decision making problems with SVTrN-number. Finally, a numerical example is used to demonstrate how to apply the proposed approach.

Keywords

Neutrosophic set single valued neutrosophic numbers triangular neutrosophic numbers geometric operators decision making

1 Introduction

Zadeh [54] proposed the notation of fuzzy set X on a fixed set E characterized by a membership function denoted by μ_nX such that μ_nX : E → [0, 1] which are the powerful tools to deal with imperfect and imprecise information. Then, by adding non-membership function to fuzzy sets, Atanassov [1] presented the notation of intuitionistic fuzzy set K on a fixed set E characterized by a membership function μ_nK : E → [0, 1] and a non-membership function γ_nK : E → [0, 1] such that such that 0 ≤ μ_nK (x) + γ_nK (x) ≤1 for any x ∈ E, which is a generalization of fuzzy set [54]. By Smarandache [29], intuitionistic fuzzy set was extended to develop the notation of neutrosophic set A on a fixed set E characterized by a truth-membership function T_nA, a indeterminacy-membership function I_nA and a falsity-membership function F_nA such that T_nA (x), I_nA (x), F_nA (x) ∈] ^-0, 1 [⁺ which is a generalization of fuzzy set and intuitionistic fuzzy set. The neutrosophic sets may express more abundant and flexible information as compared with the fuzzy sets and intuitionistic fuzzy sets. Recently, neutrosophic sets have been researched by many scholars in different fields. For example; on neutrosophic similarity clustering [5 , 53], on multi-criteria decision making problems [17 , 51] etc. Also the notations such as fuzzy sets, intuitionistic fuzzy sets and neutrosophic sets has been applied to some different fields in [4 , 52].

Aggregation operators, which is an important research topic in decision-making theory, have been researched by many scholars such as; intuitionistic fuzzy sets [15 , 44], intuitionistic fuzzy numbers [19 , 41], neutrosophic sets [2 , 26], neutrosophic number [51], and so on. Especially, Xu and Yager [33], introduced some new geometric aggregation operators, is called intuitionistic fuzzy weighted geometric operator, intuitionistic fuzzy ordered weighted geometric operator, and intuitionistic fuzzy hybrid geometric operator. Also, Wu and Cao [40] presented some geometric aggregation operators with intuitionistic trapezoidal fuzzy numbers. Numerous methods for integrating the OWA with the weighted average, such as the weighted OWA [32], the immediate weights [24, 46] and the OWAWA operator [23] have been developed in order to improve the applicability of operators to decision making problems.

Since neutrosophic numbers [13, 31] are a special case of neutrosophic sets, the neutrosophic numbers are importance for neutrosophic multi criteria decision making (MCDM) problems. As a generalization of fuzzy numbers and intuitionistic fuzzy number, a neutrosophic number seems to suitably describe an ill-known quantity. To the our knowledge, existing approaches are not suitable for dealing with MCDM problems under SVTrN-numbers. Therefore, the remainder of this paper is organized as follows: In Section 2, some basic definitions of fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, single valued neutrosophic sets and single valued triangular neutrosophic number are briefly reviewed. In Section 3, some new geometric operator is called SVTrN weighted geometric operator, SVTrN ordered weighted geometric operator, SVTrN ordered hybrid weighted geometric operator are defined (adapted from [19, 51]. In Section 4, an approach based on the SVTrN ordered hybrid weighted geometric operator is developed to solve multi-criteria decision making problems with SVTrN-number is developed. In Section 5, a numerical example is given to demonstrate how to apply the proposed approach. In Section 7, the study is concluded. The present expository paper is a condensation of part of the dissertation [31].

2 Preliminary

In this section, we recall some basic notions of fuzzy sets [54], intuitionistic fuzzy sets [1], intuitionistic fuzzy numbers [19] and neutrosophic sets [29]. For more details, the reader could refer to [1 , 54]. From now on we use I_nn = {1, 2,. . . , n} and I_nm = {1, 2,. . . , m} as an index set for n ∈ N and m ∈ N, respectively.

Definition 2.1. [54] Let E be a universe. Then a fuzzy set X over E is a function defined as follows: $X = {(μ_{X} (x) / x) : x \in E}$ where μ_nX : E → [0.1].

Here, μ_nX called membership function of X, and the value μ_nX (x) is called the grade of membership of x ∈ E. The value represents the degree of x belonging to the fuzzy set X.

Definition 2.2. [1] Let E be a universe. An intuitionistic fuzzy set K on E can be defined as follows: $K = {< x, μ_{K} (x), γ_{K} (x) > : x \in E}$ where, μ_nK : E → [0, 1] and γ_nK : E → [0, 1] such that 0 ≤ μ_nK (x) + γ_nK (x) ≤1 for any x ∈ E.

Here, μ_nK (x) and γ_nK (x) is the degree of membership and degree of non-membership of the element x, respectively.

Definition 2.3. [29] Let E be a universe. A neutrosophic sets(NS) A in E is characterized by a truth-membership function T_nA, a indeterminacy-membership function I_nA and a falsity-membership function F_nA. T_nA (x); I_nA (x) and F_nA (x) are real standard elements of [0, 1]. It can be written as $\begin{matrix} A & = & {< x, (T_{A} (x), I_{A} (x), F_{A} (x)) > : x \in E, \\ T_{A} (x), I_{A} (x), F_{A} (x) \in]^{-} 0, 1^{+} [} \end{matrix}$ There is no restriction on the sum of T_nA (x), I_nA (x) and F_nA (x), so 0^- ≤ T_nA (x) + I_nA (x) + F_nA (x) ≤3⁺.

Definition 2.4. [35] Let E be a universe. A single valued neutrosophic sets (SVNS) A, which can be used in real scientific and engineering applications, in E is characterized by a truth-membership function T_nA, a indeterminacy-membership function I_nA and a falsity-membership function F_nA. T_nA (x); I_nA (x) and F_nA (x) are real standard elements of [0, 1]. It can be written as $\begin{matrix} A & = & {< x, (T_{A} (x), I_{A} (x), F_{A} (x)) > : x \in E, \\ T_{A} (x), I_{A} (x), F_{A} (x) \in [0, 1]} \end{matrix}$ There is no restriction on the sum of T_nA (x), I_nA (x) and F_nA (x), so 0 ≤ T_nA (x) + I_nA (x) + F_nA (x) ≤3.

Definition 2.5. [31] Let $w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} \in [0, 1]$ and a₁, b₁, c₁ ∈ R such that a₁ ≤ b₁ ≤ c₁. Then, a single valued triangular neutrosophic number (SVTrN-number) $\tilde{a} = 〈 (a_{1}, b_{1}, c_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉$ is a special neutrosophic set on the real number set R, whose truth-membership indeterminacy-membership and falsity-membership functions are defined as follows: $\begin{matrix} μ_{\tilde{a}} (x) & = & {\begin{matrix} \frac{(x - a_{1}) w_{\tilde{a}}}{b_{1} - a_{1}} & (a_{1} \leq x < b_{1}) \\ w_{\tilde{a}} & (x = b_{1}) \\ \frac{(c_{1} - x) w_{\tilde{a}}}{c_{1} - b_{1}} & (b_{1} < x \leq c_{1}) \\ 0 & otherwise \end{matrix} \\ ν_{\tilde{a}} (x) & = & {\begin{matrix} \frac{(b_{1} - x + u_{\tilde{a}} (x - a_{1}))}{b_{1} - a_{1}} & (a_{1} \leq x < b_{1}) \\ u_{\tilde{a}} & (x = b_{1}) \\ \frac{(x - b_{1} + u_{\tilde{a}} (c_{1} - x))}{c_{1} - b_{1}} & (b_{1} < x \leq c_{1}) \\ 1 & otherwise, \end{matrix} \\ λ_{\tilde{a}} (x) & = & {\begin{matrix} \frac{(b_{1} - x + y_{\tilde{a}} (x - a_{1}))}{b_{1} - a_{1}} & (a_{1} \leq x < b_{1}) \\ y_{\tilde{a}} & (x = b_{1}) \\ \frac{(x - b_{1} + y_{\tilde{a}} (c_{1} - x))}{c_{1} - b_{1}} & (b_{1} < x \leq c_{1}) \\ 1 & otherwise, \end{matrix} \end{matrix}$ respectively.

If a₁ ≥ 0 and at least c₁ > 0 then $\tilde{a} = 〈 (a_{1}, b_{1}, c_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉$ is called a positive SVTrN-numbers, denoted by $\tilde{a} > 0$ . Likewise, if c₁ ≤ 0 and at least a₁ < 0, then $\tilde{a} = 〈 (a_{1}, b_{1}, c_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉$ is called a negative SVTrN-numbers, denoted by $\tilde{a} < 0$ . A SVTrN-numbers $\tilde{a} = 〈 (a_{1}, b_{1}, c_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉$ may express an ill-known quantity “about a”, which is approximately equal to a.

Note that the set of all SVTrN-numbers on R will be denoted by Δ.

Definition 2.6. [31] Let $\tilde{a} = 〈 (a_{1}, b_{1}, c_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉$ , $\tilde{b} = 〈 (a_{2}, b_{2}, c_{2}); w_{\tilde{b}},$ $u_{\tilde{b}}, y_{\tilde{b}} 〉 \in Δ$ and γ ≠ 0 be any real number. Then,

$\tilde{a} + \tilde{b} = 〈 (a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2}); w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉$

$\tilde{a} - \tilde{b} = 〈 (a_{1} - c_{2}, b_{1} - b_{2}, c_{1} - a_{2}); w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{b}} \lor y_{\tilde{b}} 〉$

$\tilde{a} \tilde{b} = {\begin{matrix} 〈 (a_{1} a_{2}, b_{1} b_{2}, c_{1} c_{2}); & w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉 \\ (c_{1} > 0, c_{2} > 0) \\ 〈 (a_{1} c_{2}, b_{1} b_{2}, c_{1} a_{2}); & w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉 \\ (c_{1} < 0, c_{2} > 0) \\ 〈 (c_{1} c_{2}, b_{1} b_{2}, a_{1} a_{2}); & w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉 \\ (c_{1} < 0, c_{2} < 0) \end{matrix}$

$\tilde{a} / \tilde{b} = {\begin{matrix} 〈 (a_{1} / c_{2}, b_{1} / b_{2}, c_{1} / a_{2}); & w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉 \\ (c_{1} > 0, c_{2} > 0) \\ 〈 (c_{1} / c_{2}, b_{1} / b_{2}, a_{1} / a_{2}); & w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉 \\ (c_{1} < 0, c_{2} > 0) \\ 〈 (c_{1} / a_{2}, b_{1} / b_{2}, a_{1} / c_{2}); & w_{\tilde{a}} \land w_{\tilde{b}}, u_{\tilde{a}} \lor u_{\tilde{b}}, y_{\tilde{a}} \lor y_{\tilde{b}} 〉 \\ (c_{1} < 0, c_{2} < 0) \end{matrix}$

$γ \tilde{a} = {\begin{matrix} 〈 (γ a_{1}, γ b_{1}, γ c_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉 & (γ > 0) \\ 〈 (γ c_{1}, γ b_{1}, γ a_{1}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉 & (γ < 0) \end{matrix}$

${\tilde{a}}^{γ} = {\begin{matrix} 〈 (a_{1}^{γ}, b_{1}^{γ}, c_{1}^{γ}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉 & (γ > 0) \\ 〈 (c_{1}^{γ}, b_{1}^{γ}, a_{1}^{γ}); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉 & (γ < 0) \end{matrix}$

Likewise, it is easily proven that the results obtained by multiplication and division of two SVTrN-numbers are not always SVTrN-numbers. However, we often use SVTrN-numbers to express these computational results approximately.

Definition 2.7. [31] We defined a method to compare any two SVTrN-numbers which is based on the score function and the accuracy function. Let $\tilde{a} = 〈 (a, b, c); w_{\tilde{a}}, u_{\tilde{a}}, y_{\tilde{a}} 〉 \in Δ$ , then $S (\tilde{a}) = \frac{1}{8} [a + b + c] \times (2 + μ_{\tilde{a}} - ν_{\tilde{a}} - γ_{\tilde{a}})$ (1) and $A (\tilde{a}) = \frac{1}{8} [a + b + c] \times (2 + μ_{\tilde{a}} - ν_{\tilde{a}} + γ_{\tilde{a}})$ is called the score and accuracy degrees of $\tilde{a}$ , respectively.

Definition 2.8. [31] Let ${\tilde{a}}_{1}$ , ${\tilde{a}}_{2} \in Δ$ . Then,

If $S ({\tilde{a}}_{1}) < S ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1}$ is smaller than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} < {\tilde{a}}_{2}$

If $S ({\tilde{a}}_{1}) > S ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1}$ is bigger than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} > {\tilde{a}}_{2}$

If $S ({\tilde{a}}_{1}) = S ({\tilde{a}}_{2})$ ;

If $A ({\tilde{a}}_{1}) < A ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1}$ is smaller than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} < {\tilde{a}}_{2}$

If $A ({\tilde{a}}_{1}) > A ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1}$ is bigger than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} > {\tilde{a}}_{2}$

If $A ({\tilde{a}}_{1}) = A ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ are the same, denoted by ${\tilde{a}}_{1} = {\tilde{a}}_{2}$

3 Geometric operators of the SVTrN-number

In this section, three SVTrN weighted geometric operator of SVTrN-numbers is called SVTrN weighted geometric operator, SVTrN ordered weighted geometric operator, SVTrN ordered hybrid weighted geometric operator is given. Some of it is quoted from [13 , 47].

Definition 3.1. Let ${\tilde{a}}_{j} = 〈 (a_{j}, b_{j}, c_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}}, y_{{\tilde{a}}_{j}} 〉 \in Δ (j \in I_{n})$ . Then SVTrN weighted geometric operator, denoted by G_ngo, is defined as; $G_{go} : Δ^{n} \to Δ, G_{go} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) = \prod_{i = 1}^{n} {\tilde{a}}_{i}^{w_{i}}$ where, w = (w₁, w₂,. . . , w_nn) ^nT is a weight vector associated with the G_ngo operator, for every j ∈ I_nn such that, w_nj ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$

Theorem 3.2. Let ${\tilde{a}}_{j} = 〈 (a_{j}, b_{j}, c_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}}, y_{{\tilde{a}}_{j}} 〉 \in Δ (j \in I_{n})$ , w = (w₁, w₂,. . . , w_nn) ^nT be a weight vector of ${\tilde{a}}_{j}$ , for every j ∈ I_nn such that w_nj ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Then, their aggregated value by using G_ngo operator is also a SVTrN-number and $\begin{matrix} G_{go} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) \\ = 〈 (\prod_{j = 1}^{n} a_{j}^{w_{j}}, \prod_{j = 1}^{n} b_{j}^{w_{j}}, \prod_{j = 1}^{n} c_{j}^{w_{j}}); \\ ⋀_{j = 1}^{n} w_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{n} u_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{n} y_{{\tilde{a}}_{j}} 〉 \end{matrix}$

Proof. The proof can be made by using mathematical induction on n as;

Assume that, ${\tilde{a}}_{1} = 〈 (a_{1}, b_{1}, c_{1}); w_{{\tilde{a}}_{1}}, u_{{\tilde{a}}_{1}}, y_{{\tilde{a}}_{1}} 〉$ and ${\tilde{a}}_{2} = 〈 (a_{2}, b_{2}, c_{2}); w_{{\tilde{a}}_{2}}, u_{{\tilde{a}}_{2}}, y_{{\tilde{a}}_{2}} 〉$ be two SVTrN-numbers then, for n = 2, we have $\begin{matrix} G_{go} ({\tilde{a}}_{1}, {\tilde{a}}_{2}) \\ = 〈 (\prod_{j = 1}^{2} a_{j}^{w_{j}}, \prod_{j = 1}^{2} b_{j}^{w_{j}}, \prod_{j = 1}^{2} c_{j}^{w_{j}}); \\ ⋀_{j = 1}^{2} w_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{2} u_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{2} y_{{\tilde{a}}_{j}} 〉 \end{matrix}$ If holds for n = k, that is $\begin{matrix} G_{go} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{k}) \\ = 〈 (\prod_{j = 1}^{k} a_{j}^{w_{j}}, \prod_{j = 1}^{k} b_{j}^{w_{j}}, \prod_{j = 1}^{k} c_{j}^{w_{j}}); \\ ⋀_{j = 1}^{k} w_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{k} u_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{k} y_{{\tilde{a}}_{j}} 〉 \end{matrix}$ then, when n = k + 1, by the operational laws in Definition 2.6, I have $\begin{matrix} G_{go} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{k}, {\tilde{a}}_{k + 1}) \\ = 〈 (\prod_{j = 1}^{k} a_{j}^{w_{j}}, \prod_{j = 1}^{k} b_{j}^{w_{j}}, \prod_{j = 1}^{k} c_{j}^{w_{j}}); \\ ⋀_{j = 1}^{k} w_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{k} u_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{k} y_{{\tilde{a}}_{j}} 〉 \\ \times 〈 (a_{k + 1}^{w_{k + 1}}, b_{k + 1}^{w_{k + 1}}, c_{k + 1}^{w_{k + 1}}); \\ w_{{\tilde{a}}_{k + 1}}, u_{{\tilde{a}}_{k + 1}}, y_{{\tilde{a}}_{k + 1}} 〉 \\ = 〈 (\prod_{j = 1}^{k + 1} a_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} b_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} c_{j}^{w_{j}}); \\ ⋀_{j = 1}^{k + 1} w_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{k + 1} u_{{\tilde{a}}_{j}}, ⋁_{j = 1}^{k + 1} y_{{\tilde{a}}_{j}} 〉 \end{matrix}$ therefore proof is valid.

Definition 3.3. Let ${\tilde{a}}_{j} = 〈 (a_{j}, b_{j}, c_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}}, y_{{\tilde{a}}_{j}} 〉 \in Δ (j \in I_{n})$ . Then SVTrN ordered weighted geometric operator denoted by G_nogo, is defined as; $G_{ogo} : Δ^{n} \to Δ, G_{ogo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) = \prod_{k = 1}^{n} {\tilde{b}}_{k}^{w_{k}}$ where w = (w₁, w₂,. . . , w_nn) ^nT is a weight vector associated with the mapping G_nogo, which satisfies the normalized conditions: w_nk ∈ [0, 1] and $\sum_{k = 1}^{n} w_{k} = 1;$ ${\tilde{b}}_{k} = 〈 (a_{k}, b_{k}, c_{k}); w_{{\tilde{a}}_{k}}, u_{{\tilde{a}}_{k}}, y_{{\tilde{a}}_{k}} 〉$ is the k-th largest of the n sets ${\tilde{a}}_{j} (j \in I_{n})$ which is determined through using ranking method in Definition 2.7.

It is not difficult to follow from Definition 3.3 that $\begin{matrix} G_{ogo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) \\ = \prod_{k = 1}^{n} {\tilde{b}}_{k}^{w_{k}} \\ = \prod_{k = 1}^{n} (〈 (a_{k}, b_{k}, c_{k}); w_{{\tilde{a}}_{k}}, u_{{\tilde{a}}_{k}}, y_{{\tilde{a}}_{k}} 〉)^{w_{k}} \\ = \prod_{k = 1}^{n} 〈 (a_{k}^{w_{k}}, b_{k}^{w_{k}}, c_{k}^{w_{k}}); w_{{\tilde{a}}_{k}}, u_{{\tilde{a}}_{k}}, y_{{\tilde{a}}_{k}} 〉 \\ = 〈 (\prod_{k = 1}^{n} a_{k}^{w_{k}}, \prod_{k = 1}^{n} b_{k}^{w_{k}}, \prod_{k = 1}^{n} c_{k}^{w_{k}}); \\ \land_{k = 1}^{n} w_{{\tilde{a}}_{k}}, \lor_{k = 1}^{n} u_{{\tilde{a}}_{k}}, \lor_{k = 1}^{n} y_{{\tilde{a}}_{k} 〉} \end{matrix}$ which is summarized as in Theorem 3.4.

Theorem 3.4. Let ${\tilde{a}}_{j} = 〈 (a_{j}, b_{j}, c_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}}, y_{{\tilde{a}}_{j}} 〉 \in Δ (j \in I_{n})$ . Then SVTrN ordered weighted geometric operator denoted by G_nogo, is defined as;

$\begin{matrix} G_{ogo} : Δ^{n} \to Δ, G_{ogo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) \\ = 〈 (\prod_{k = 1}^{n} a_{k}^{w_{k}}, \prod_{k = 1}^{n} b_{k}^{w_{k}}, \prod_{k = 1}^{n} c_{k}^{w_{k}}); \\ ⋀_{k = 1}^{n} w_{{\tilde{a}}_{k}}, ⋁_{k = 1}^{n} u_{{\tilde{a}}_{k}}, ⋁_{k = 1}^{n} y_{{\tilde{a}}_{k}} 〉 \end{matrix}$ (2) where $w_{k} \in [0, 1], \sum_{k = 1}^{n} w_{k} = 1; {\tilde{b}}_{k} = 〈 (a_{k}, b_{k}, c_{k}); w_{{\tilde{a}}_{k}}, u_{{\tilde{a}}_{k}}, y_{{\tilde{a}}_{k}} 〉$ is the k-th largest of the n neutrosophic sets ${\tilde{a}}_{j} (j \in I_{n})$ which is determined through using some ranking method in Definition 2.7.

Example 3.5. (It is adapted from [19]) There are four experts who are invited to evaluate some enterprise. Their evaluations are expressed with the SVTrN-number $\begin{matrix} {\tilde{a}}_{1} & = & 〈 (0.123, 0.234, 0.325); 0.4, 0.5, 0.7 〉, \\ {\tilde{a}}_{2} & = & 〈 (0.234, 0.354, 0.451); 0.3, 0.6, 0.6 〉, \\ {\tilde{a}}_{3} & = & 〈 (0.125, 0.365, 0.465); 0.2, 0.7, 0.5 〉, \\ {\tilde{a}}_{4} & = & 〈 (0.215, 0.345, 0.435); 0.1, 0.8, 0.4 〉, \end{matrix}$ respectively. To eliminate effect of individual bias on comprehensive evaluation, the unduly high evaluation and the unduly low evaluation are punished through giving a smaller weight. Assume that the position weight vector is w = (0.15, 0.35, 0.35, 0.15). Compute the comprehensive evaluation of the four experts on the enterprise though using the neutrosophic ordered weighted averaging operator.

Solving. According to Equation (1), the scores of the SVTrN-numbers ${\tilde{a}}_{j} (j = 1, 2, 3, 4)$ are obtained as follows: $\begin{matrix} S ({\tilde{a}}_{1}) \\ = \frac{[0.123 + 0.234 + 0.325] \times (2 + 0.4 - 0.5 - 0.7)}{8} \\ = 0.102, \\ S ({\tilde{a}}_{2}) \\ = \frac{[0.234 + 0.354 + 0.451] \times (2 + 0.3 - 0.6 - 0.6)}{8} \\ = 0.143, \\ S ({\tilde{a}}_{3}) \\ = \frac{[0.125 + 0.365 + 0.465] \times (2 + 0.2 - 0.7 - 0.5)}{8} \\ = 0.119, \\ S ({\tilde{a}}_{4}) \\ = \frac{[0.215 + 0.345 + 0.435] \times (2 + 0.1 - 0.8 - 0.4)}{8} \\ = 0.112, \end{matrix}$

respectively. It is obvious that $S ({\tilde{a}}_{2}) > S ({\tilde{a}}_{3}) > S ({\tilde{a}}_{4}) > S ({\tilde{a}}_{1})$ . Hence according to the above scoring function ranking method, its follows that ${\tilde{a}}_{2} > {\tilde{a}}_{3} > {\tilde{a}}_{4} > {\tilde{a}}_{1}$ . Hence, we have: $\begin{matrix} {\tilde{b}}_{1} & = & {\tilde{a}}_{2} = 〈 (0.234, 0.354, 0.451); 0.3, 0.6, 0.6 〉, \\ {\tilde{b}}_{2} & = & {\tilde{a}}_{3} = 〈 (0.125, 0.365, 0.465); 0.2, 0.7, 0.5 〉, \\ {\tilde{b}}_{3} & = & {\tilde{a}}_{4} = 〈 (0.215, 0.345, 0.435); 0.1, 0.8, 0.4 〉, \\ {\tilde{b}}_{4} & = & {\tilde{a}}_{1} = 〈 (0.123, 0.234, 0.325); 0.4, 0.5, 0.7 〉 . \end{matrix}$ Using Equation (2), we obtain: $\begin{matrix} G_{ogo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3}, {\tilde{a}}_{4}) \\ = 〈 (0 . 234^{0.15} \times 0 . 125^{0.35} \times 0 . 215^{0.35} \\ \times 0 . 123^{0.15}, 0 . 354^{0.15} \times 0 . 365^{0.35} \times 0 . 345^{0.35} \\ \times 0 . 234^{0.15}, 0 . 451^{0.15} \times 0 . 465^{0.35} \times 0 . 435^{0.35} \\ \times 0 . 325^{0.15}); 0.1, 0.8, 0.7 〉 \\ = 〈 (0.166, 0.333, 0.429); 0.1, 0.8, 0.7 〉 \end{matrix}$

Definition 3.6. Let ${\tilde{a}}_{j} = 〈 (a_{j}, b_{j}, c_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}}, y_{{\tilde{a}}_{j}} 〉 \in Δ (j \in I_{n})$ . Then SVTrN ordered hybrid weighted geometric operator denoted by G_nhgo, is defined as; $G_{hgo} : Δ^{n} \to Δ, G_{hgo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) = \prod_{k = 1}^{n} {\hat{b}}_{k}^{w_{k}}$ where w = (w₁, w₂,. . . , w_nn) ^nT. w_nj ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ is a weight vector associated with the mapping G_nhgo, a_nj ∈ Δ a weight with nω (j ∈ I_nn) is denoted by ${\tilde{A}}_{j}$ i.e., ${\tilde{A}}_{j} = n ω {\tilde{a}}_{j}$ , here n is regarded as a balance factor ω = (ω₁, ω₂,. . . , ω_nn) ^nT is a weight vector of the $a_{j} \in Δ (j \in I_{n}); {\hat{b}}_{k}$ is the k-th largest of the n SVTrN-numbers ${\tilde{A}}_{j} \in Δ (j \in I_{n})$ which are determined through using some ranking method such as the above scoring function ranking method.

Note that if ω = (1/n, 1/n,. . . , 1/n) ^nT, then G_nhgo degenerates to the G_ngo .

Theorem 3.7. Let ${\tilde{a}}_{j} = 〈 (a_{j}, b_{j}, c_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}}, y_{{\tilde{a}}_{j}} 〉 \in Δ (j \in I_{n})$ , w = (w₁, w₂,. . . , w_nn) ^nT be a weight vector of ${\tilde{a}}_{j}$ with w_nj ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Then their aggregated value by using G_nhgo operator is also a SVTrN-number and

$\begin{matrix} G_{hgo} : Δ^{n} \to Δ, G_{hgo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) \\ = 〈 (\prod_{k = 1}^{n} a_{k}^{w_{k}}, \prod_{k = 1}^{n} b_{k}^{w_{k}}, \prod_{k = 1}^{n} c_{k}^{w_{k}}); \\ ⋀_{k = 1}^{n} w_{{\tilde{a}}_{k}}, ⋁_{k = 1}^{n} u_{{\tilde{a}}_{k}}, ⋁_{k = 1}^{n} y_{{\tilde{a}}_{k}} 〉 \end{matrix}$ (3) where ${\hat{b}}_{k} = 〈 (a_{k}, b_{k}, c_{k}); w_{{\tilde{a}}_{k}}, u_{{\tilde{a}}_{k}}, y_{{\tilde{a}}_{k}} 〉$ is the k-th largest of the n SVTrN-numbers ${\hat{A}}_{j} = n ω_{j} {\tilde{a}}_{j} (j \in I_{n})$ which is determined through using some ranking method such as the above scoring function ranking method.

Example 3.8. Let us we consider the Example 3.5.

Assume that the weight vector of the three experts is ω = (0.2, 0.3, 0.3, 0.2) ^nT and the position weight vector is w = (0.4, 0.1, 0.1, 0.4) ^nT. Compute the comprehensive evaluation of the four experts on the decision alternative through using the G_nhgo .

Solving. $\begin{matrix} {\tilde{A}}_{1} & = & 4 \times 0.2 \times {\tilde{a}}_{1} \\ = & 4 \times 0.2 \times 〈 (0.123, 0.234, 0.325); 0.4, 0.5, 0.7 〉 \\ = & 〈 (0.098, 0.187, 0.260); 0.4, 0.5, 0.7 〉 \end{matrix}$ $\begin{matrix} {\tilde{A}}_{2} & = & 4 \times 0.3 \times {\tilde{a}}_{2} \\ = & 4 \times 0.3 \times 〈 (0.234, 0.354, 0.451); 0.3, 0.6, 0.6 〉 \\ = & 〈 (0.281, 0.425, 0.541); 0.3, 0.6, 0.6 〉 \end{matrix}$ $\begin{matrix} {\tilde{A}}_{3} & = & 4 \times 0.3 \times {\tilde{a}}_{3} \\ = & 4 \times 0.3 \times 〈 (0.125, 0.365, 0.465); 0.2, 0.7, 0.5 〉 \\ = & 〈 (0.150, 0.438, 0.558); 0.2, 0.7, 0.5 〉 \end{matrix}$ ∥ $\begin{matrix} {\tilde{A}}_{4} & = & 4 \times 0.2 \times {\tilde{a}}_{4} \\ = & 4 \times 0.2 \times 〈 (0.215, 0.345, 0.435); 0.1, 0.8, 0.4 〉 \\ = & 〈 (0.172, 0.276, 0.348); 0.1, 0.8, 0.4 〉 \end{matrix}$

we obtain the scores of the SVTrN-numbers ${\tilde{A}}_{j} (j = 1, 2, 3)$ as follows: $\begin{matrix} S ({\tilde{A}}_{1}) & = & \frac{[0.098 + 0.187 + 0.260] \times (2 + 0.4 - 0.5 - 0.7)}{8} \\ = & 0.082, \\ S ({\tilde{A}}_{2}) & = & \frac{[0.281 + 0.425 + 0.541] \times (2 + 0.3 - 0.6 - 0.6)}{8} \\ = & 0.171, \\ S ({\tilde{A}}_{3}) & = & \frac{[0.150 + 0.438 + 0.558] \times (2 + 0.2 - 0.7 - 0.5)}{8} \\ = & 0.143, \\ S ({\tilde{A}}_{4}) & = & \frac{[0.172 + 0.276 + 0.348] \times (2 + 0.1 - 0.8 - 0.4)}{8} \\ = & 0.090 . \end{matrix}$

Obviously, $S ({\tilde{A}}_{2}) > S ({\tilde{A}}_{3}) > S ({\tilde{A}}_{1}) > S ({\tilde{A}}_{4}) .$ Thereby, according to the above scoring function ranking method, we have $\begin{matrix} {\hat{b}}_{1} & = & {\tilde{A}}_{2} = 〈 (0.281, 0.425, 0.541); 0.3, 0.6, 0.6 〉 \\ {\hat{b}}_{2} & = & {\tilde{A}}_{3} = 〈 (0.150, 0.438, 0.558); 0.2, 0.7, 0.5 〉 \\ {\hat{b}}_{3} & = & {\tilde{A}}_{4} = 〈 (0.172, 0.276, 0.348); 0.1, 0.8, 0.4 〉 \\ {\hat{b}}_{4} & = & {\tilde{A}}_{1} = 〈 (0.098, 0.187, 0.260); 0.4, 0.5, 0.7 〉 \end{matrix}$ It follows from Equation (3) that $\begin{matrix} G_{hgo} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3}, {\tilde{a}}_{4}) \\ = 〈 (0 . 281^{0.4} \times 0 . 150^{0.1} \times 0 . 172^{0.1} \times 0 . 098^{0.4}, \\ 0 . 425^{0.4} \times 0 . 438^{0.1} \times 0 . 276^{0.1} \times 0 . 187^{0.4}, \\ 0 . 541^{0.4} \times 0 . 558^{0.1} \times 0 . 348^{0.1} \times 0 . 260^{0.4}); \\ 0.1, 0.8, 0.7 〉 \\ = 〈 (0.165, 0.294, 0.387); 0.1, 0.8, 0.7 〉 \end{matrix}$

4 Multi-criteria decision making based on SVTrN-numbers

In this section, we define a multi-criteria decision making method, so called SVTrN-multi-criteria decision-making method, by using the G_nhgo operator. Some of it is quoted from application in [13 , 47].

Definition 4.1. Let X = (x₁, x₂,. . . , x_nm) be a set of alternatives, U = (u₁, u₂,. . . , u_nn) be the set of attributes. If ${\tilde{a}}_{ij} = 〈 (a_{ij}, b_{ij}, c_{ij}); w_{ij}, u_{ij}, y_{ij} 〉 \in Δ$ , then ${[{\tilde{a}}_{i j}]}_{m \times n} = \begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{matrix} \overset{\begin{matrix} u_{1} & u_{2} & \dots & u_{n} \end{matrix}}{(\begin{matrix} {\tilde{a}}_{11} & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & {\tilde{a}}_{22} & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{a}}_{m 1} & {\tilde{a}}_{m 2} & \dots & {\tilde{a}}_{m n} \end{matrix})}$ is called an SVTrN-multi-criteria decision-making matrix of the decision maker.

Now, we can give an algorithm of the SVTrN-multi-criteria decision-making method as follows;

n Algorithm:

Construct the decision-making matrix $[{\tilde{a}}_{ij}]_{m \times n}$ for decision;

Compute the SVTrN-numbers ${\tilde{A}}_{ij} = n ω_{i} {\tilde{a}}_{ij} (i \in I_{m}; j \in I_{n})$ and write the decision-making matrix $[{\tilde{A}}_{ij}]_{m \times n}$ ;

Obtain the scores of the SVTrN-numbers ${\tilde{A}}_{ij}$ (i ∈ I_nm ; j ∈ I_nn);

Rank all SVTrN-numbers ${\tilde{A}}_{ij} (i \in I_{m}; j \in I_{n})$ by using the ranking method of SVTrN-numbers and determine the SVTrN-numbers $[b_{i}]_{1 \times n} = {\tilde{b}}_{ik} (i \in I_{m}; k \in I_{n})$ where ${\tilde{b}}_{ik}$ is k-th largest of ${\tilde{A}}_{ij}$ for j ∈ I_nn;

Give the decision matrix [b_ni] _1×n for i = 1, 2, 3, 4 ;

Compute $G_{hgo} ({\tilde{b}}_{i 1}, {\tilde{b}}_{i 2}, . . ., {\tilde{b}}_{in})$ for i ∈ I_nm;

Rank all alternatives x_ni by using the ranking method of SVTrN-numbers and determine the best alternative.

5 Application

In this section, we give an application for the SVTrN-multi-criteria decision-making method, by using the G_nhgo operator. Some of it is quoted from application in [16 , 47].

Example 5.1. Let us consider the decision-making problem adapted from [16, 51]. There is an investment company, which wants to invest a sum of money in the best option. There is a panel with the set of the four alternatives is denoted by X = {x₁= car company, x₂= food company, x₃= computer company, x₄= television company} to invest the money. The investment company must take a decision according to the set of the four attributes is denoted by U = {u₁= risk analysis, u₂= growth analysis, u₃= environmental impact analysis, u₄= social political impact analysis}. The four possible alternatives are to be evaluated under the above three criteria by corresponding to linguistic values of SVTrN-numbers for linguistic terms (adapted from [47]), as shown in Table 1.

Then, the weight vector of the attributes is ω = (0.1, 0.2, 0.3, 0.4) ^nT and the position weight vector is w = (0.24, 0.26, 0.26, 0.24) ^nT by using the weight determination based on the normal distribution. For the evaluation of an alternative x_ni (i = 1, 2, 3, 4) with respect to a criterion u_nj (j = 1, 2, 3, 4), it is obtained from the questionnaire of a domain expert. Then,

The decision maker construct the decision matrix $\begin{matrix} (\begin{matrix} \begin{matrix} 〈 (0.1, 0.2, 0.3); 0.1, 0.2, 0.3 〉 \\ 〈 (0.3, 0.6, 0.9); 0.3, 0.6, 0.9 〉 \\ 〈 (0.2, 0.3, 0.4); 0.2, 0.3, 0.4 〉 \\ 〈 (0.6, 0.7, 0.8); 0.6, 0.7, 0.8 〉 \end{matrix} \end{matrix} \\ \begin{matrix} 〈 (0.3, 0.4, 0.5); 0.3, 0.4, 0.5 〉 \\ 〈 (0.1, 0.6, 0.9); 0.1, 0.6, 0.9 〉 \\ 〈 (0.5, 0.6, 0.7); 0.5, 0.6, 0.7 〉 \\ 〈 (0.2, 0.3, 0.8); 0.2, 0.3, 0.8 〉 \end{matrix} \\ \begin{matrix} \begin{matrix} 〈 (0.6, 0.7, 0.8); 0.6, 0.7, 0.8 〉 \\ 〈 (0.4, 0.5, 0.6); 0.4, 0.5, 0.6 〉 \\ 〈 (0.7, 0.8, 0.9); 0.7, 0.8, 0.9 〉 \\ 〈 (0.2, 0.7, 0.8); 0.2, 0.7, 0.8 〉 \end{matrix} \end{matrix} \\ \begin{matrix} 〈 (0.1, 0.3, 0.9); 0.1, 0.3, 0.9 〉 \\ 〈 (0.1, 0.6, 0.9); 0.1, 0.6, 0.9 〉 \\ 〈 (0.2, 0.4, 0.8); 0.2, 0.4, 0.8 〉 \\ 〈 (0.2, 0.7, 0.8); 0.2, 0.7, 0.8 〉 \end{matrix}) \end{matrix}$

Compute ${\tilde{A}}_{ij} = n ω_{i} {\tilde{a}}_{ij} (i = 1, 2, 3, 4; j = 1, 2, 3, 4)$ as follows: $\begin{matrix} {\tilde{A}}_{11} & = & 4 \times 0.1 \times {\tilde{a}}_{11} \\ = & 〈 (0 . 1^{0.4}, 0 . 2^{0.4}, 0 . 3^{0.4}); 0.1, 0.2, 0.3 〉 \\ = & 〈 (0.398, 0.525, 0.618); 0.1, 0.2, 0.3 〉 \end{matrix}$ Likewise, we can obtain other SVTrN-numbers ${\tilde{A}}_{ij} = n ω_{i} {\tilde{a}}_{ij} (i = 1, 2, 3, 4; j = 1, 2, 3, 4)$ which are given by the SVTrN-decision matrix $[{\tilde{A}}_{ij}]_{4 \times 4}$ as follows: $\begin{matrix} (\begin{matrix} 〈 (0.398, 0.525, 0.618); 0.1, 0.2, 0.3 〉 \\ 〈 (0.618, 0.815, 0.959); 0.3, 0.6, 0.9 〉 \\ 〈 (0.525, 0.618, 0.693); 0.2, 0.3, 0.4 〉 \\ 〈 (0.815, 0.867, 0.915); 0.6, 0.7, 0.8 〉 \end{matrix} \\ \begin{matrix} 〈 (0.382, 0.480, 0.574); 0.3, 0.4, 0.5 〉 \\ 〈 (0.158, 0.665, 0.919); 0.1, 0.6, 0.9 〉 \\ 〈 (0.574, 0.665, 0.754); 0.5, 0.6, 0.7 〉 \\ 〈 (0.276, 0.382, 0.837); 0.2, 0.3, 0.8 〉 \end{matrix} \\ \begin{matrix} 〈 (0.542, 0.652, 0.765); 0.6, 0.7, 0.8 〉 \\ 〈 (0.333, 0.435, 0.542); 0.4, 0.5, 0.6 〉 \\ 〈 (0.652, 0.765, 0.881); 0.7, 0.8, 0.9 〉 \\ 〈 (0.145, 0.652, 0.765); 0.2, 0.7, 0.8 〉 \end{matrix} \\ \begin{matrix} 〈 (0.025, 0.146, 0.845); 0.1, 0.3, 0.9 〉 \\ 〈 (0.025, 0.442, 0.845); 0.1, 0.6, 0.9 〉 \\ 〈 (0.076, 0.231, 0.700); 0.2, 0.4, 0.8 〉 \\ 〈 (0.076, 0.565, 0.700); 0.2, 0.7, 0.8 〉 \end{matrix}) \end{matrix}$

We can obtain the scores of the SVTrN-numbers ${\tilde{A}}_{ij}$ of the alternatives x_nj (j = 1, 2, 3, 4) on the four attributes u_ni (i = 1, 2, 3, 4) as follows: $\begin{matrix} S ({\tilde{A}}_{11}) & = & 0.308 S ({\tilde{A}}_{12}) = 0.251 \\ S ({\tilde{A}}_{13}) & = & 0.269 S ({\tilde{A}}_{14}) = 0.114 \\ S ({\tilde{A}}_{21}) & = & 0.239 S ({\tilde{A}}_{22}) = 0.131 \\ S ({\tilde{A}}_{23}) & = & 0.213 S ({\tilde{A}}_{24}) = 0.098 \\ S ({\tilde{A}}_{31}) & = & 0.344 S ({\tilde{A}}_{32}) = 0.299 \\ S ({\tilde{A}}_{33}) & = & 0.137 S ({\tilde{A}}_{34}) = 0.126 \\ S ({\tilde{A}}_{41}) & = & 0.357 S ({\tilde{A}}_{42}) = 0.205 \\ S ({\tilde{A}}_{43}) & = & 0.287 S ({\tilde{A}}_{44}) = 0.117 \end{matrix}$

The ranking order of all SVTrN-numbers ${\tilde{A}}_{ij} (i = 1, 2, 3, 4; j = 1, 2, 3, 4)$ as follows; $\begin{matrix} {\tilde{A}}_{11} > {\tilde{A}}_{13} > {\tilde{A}}_{12} > {\tilde{A}}_{14} \\ {\tilde{A}}_{21} > {\tilde{A}}_{23} > {\tilde{A}}_{22} > {\tilde{A}}_{24} \\ {\tilde{A}}_{31} > {\tilde{A}}_{32} > {\tilde{A}}_{33} > {\tilde{A}}_{34} \\ {\tilde{A}}_{41} > {\tilde{A}}_{43} > {\tilde{A}}_{42} > {\tilde{A}}_{44} \end{matrix}$ Thus, we have: $\begin{matrix} {\tilde{b}}_{11} & = & {\tilde{A}}_{11}, {\tilde{b}}_{12} = {\tilde{A}}_{13}, {\tilde{b}}_{13} = {\tilde{A}}_{12}, {\tilde{b}}_{14} = {\tilde{A}}_{14} \\ {\tilde{b}}_{21} & = & {\tilde{A}}_{21}, {\tilde{b}}_{22} = {\tilde{A}}_{23}, {\tilde{b}}_{23} = {\tilde{A}}_{22}, {\tilde{b}}_{24} = {\tilde{A}}_{24} \\ {\tilde{b}}_{31} & = & {\tilde{A}}_{31}, {\tilde{b}}_{32} = {\tilde{A}}_{32}, {\tilde{b}}_{33} = {\tilde{A}}_{33}, {\tilde{b}}_{34} = {\tilde{A}}_{34} \\ {\tilde{b}}_{41} & = & {\tilde{A}}_{41}, {\tilde{b}}_{42} = {\tilde{A}}_{43}, {\tilde{b}}_{43} = {\tilde{A}}_{42}, {\tilde{b}}_{44} = {\tilde{A}}_{44} \end{matrix}$

The decision matrix [b_ni] _1×n for i = 1, 2, 3, 4 are given by; $\begin{matrix} {\tilde{b}}_{1} & = & (〈 (0.398, 0.525, 0.618); 0.1, 0.2, 0.3 〉, \\ 〈 (0.542, 0.652, 0.765); 0.6, 0.7, 0.8 〉, \\ 〈 (0.382, 0.480, 0.574); 0.3, 0.4, 0.5 〉, \\ 〈 (0.025, 0.146, 0.845); 0.1, 0.3, 0.9 〉) \end{matrix}$ $\begin{matrix} {\tilde{b}}_{2} & = & (〈 (0.618, 0.815, 0.959); 0.3, 0.6, 0.9 〉, \\ 〈 (0.333, 0.435, 0.542); 0.4, 0.5, 0.6 〉, \\ 〈 (0.158, 0.665, 0.919); 0.1, 0.6, 0.9 〉, \\ 〈 (0.025, 0.442, 0.845); 0.1, 0.6, 0.9 〉) \end{matrix}$ $\begin{matrix} {\tilde{b}}_{3} & = & (〈 (0.525, 0.618, 0.693); 0.2, 0.3, 0.4 〉, \\ 〈 (0.574, 0.665, 0.754); 0.5, 0.6, 0.7 〉, \\ 〈 (0.652, 0.765, 0.881); 0.7, 0.8, 0.9 〉, \\ 〈 (0.076, 0.231, 0.700); 0.2, 0.4, 0.8 〉) \end{matrix}$ $\begin{matrix} {\tilde{b}}_{4} & = & (〈 (0.815, 0.867, 0.915); 0.6, 0.7, 0.8 〉, \\ 〈 (0.145, 0.652, 0.765); 0.2, 0.7, 0.8 〉, \\ 〈 (0.276, 0.382, 0.837); 0.2, 0.3, 0.8 〉, \\ 〈 (0.076, 0.565, 0.700); 0.2, 0.7, 0.8 〉) \end{matrix}$

We can calculate the SVTrN-numbers $G_{hgo} (b_{i}) = G_{hgo} ({\tilde{b}}_{i 1}, {\tilde{b}}_{i 2}, {\tilde{b}}_{i 3}, {\tilde{b}}_{i 4})$ for i = 1, 2, 3, 4 as follows: $\begin{matrix} G_{hgo} (b_{1}) \\ = G_{hgo} ({\tilde{b}}_{11}, {\tilde{b}}_{12}, {\tilde{b}}_{13}, {\tilde{b}}_{14}) \\ = 〈 (0 . 398^{0.24} \times 0 . 618^{0.26} \times 0 . 525^{0.26} \\ \times 0 . 815^{0.24}, 0 . 525^{0.24} \times 0 . 815^{0.26} \\ \times 0 . 618^{0.26} \times 0 . 867^{0.24}, 0 . 618^{0.24} \\ \times 0 . 959^{0.26} \times 0 . 693^{0.26} \times 0 . 915^{0.24}); \\ 0.1, 0.7, 0.9 〉 \\ = 〈 (0.570, 0.693, 0.784); 0.1, 0.7, 0.9 〉 \end{matrix}$ $\begin{matrix} G_{hgo} (b_{2}) \\ = G_{hgo} ({\tilde{b}}_{11}, {\tilde{b}}_{12}, {\tilde{b}}_{13}, {\tilde{b}}_{14}) \\ = 〈 (0 . 542^{0.24} \times 0 . 333^{0.26} \times 0 . 574^{0.26} \\ \times 0 . 145^{0.24}, 0 . 652^{0.24} \times 0 . 435^{0.26} \\ \times 0 . 665^{0.26} \times 0 . 652^{0.24}, 0 . 765^{0.24} \\ \times 0 . 542^{0.26} \times 0 . 754^{0.26} \times 0 . 765^{0.24}); \\ 0.2, 0.7, 0.8 〉 \\ = 〈 (0.353, 0.590, 0.696); 0.2, 0.7, 0.8 〉 \end{matrix}$ $\begin{matrix} G_{hgo} (b_{3}) \\ = G_{hgo} ({\tilde{b}}_{11}, {\tilde{b}}_{12}, {\tilde{b}}_{13}, {\tilde{b}}_{14}) \\ = 〈 (0 . 382^{0.24} \times 0 . 158^{0.26} \times 0 . 652^{0.26} \\ \times 0 . 276^{0.24}, 0 . 480^{0.24} \times 0 . 665^{0.26} \\ \times 0 . 765^{0.26} \times 0 . 382^{0.24}, 0 . 574^{0.24} \\ \times 0 . 919^{0.26} \times 0 . 881^{0.26} \times 0 . 837^{0.24}); \\ 0.1, 0.8, 0.9 〉 \\ = 〈 (0.323, 0.558, 0.794); 0.1, 0.8, 0.9 〉 \end{matrix}$

$\begin{matrix} G_{hgo} (b_{4}) \\ = G_{hgo} ({\tilde{b}}_{11}, {\tilde{b}}_{12}, {\tilde{b}}_{13}, {\tilde{b}}_{14}) \\ = 〈 (0 . 025^{0.24} \times 0 . 025^{0.26} \times 0 . 076^{0.26} \\ \times 0 . 076^{0.24}, 0 . 146^{0.24} \times 0 . 442^{0.26} \\ \times 0 . 231^{0.26} \times 0 . 565^{0.24}, 0 . 845^{0.24} \\ \times 0 . 845^{0.26} \times 0 . 700^{0.26} \times 0 . 700^{0.24}); \\ 0.1, 0.7, 0.9 〉 \\ = 〈 (0.044, 0.303, 0.769); 0.1, 0.7, 0.9 〉 \end{matrix}$

The scores of $G_{hgo} ({\tilde{b}}_{i})$ for i = 1, 2, 3, 4 can be obtained as follows: $\begin{matrix} S (G_{hgo} (b_{1})) = 0.128 \\ S (G_{hgo} (b_{2})) = 0.143 \\ S (G_{hgo} (b_{3})) = 0.084 \end{matrix}$ and $S (G_{hgo} (b_{4})) = 0.070$

It is obvious that $G_{hgo} (b_{2}) > G_{hgo} (b_{1}) > G_{hgo} (b_{3}) > G_{hgo} (b_{4})$ Therefore, the ranking order of the alternatives x_nj (j = 1, 2, 3, 4) is generated as follows: $x_{2} ≻ x_{1} ≻ x_{3} ≻ x_{4}$ The best supplier for the enterpriseis x₂.

6 Comparison analysis and discussion

In this section we introduce a comparative study with other methods in order to validate the feasibility of the presented method. Therefore, the proposed method is compared the methods that were outlined in Refs. [31] and [51] using SVTrNinformation.

The results from the different methods used to resolve the MCDM problem in Example 5.1 are shown in Table 2.

From the results presented in Table 2, the best alternatives in Proposed Method and Ref. [31] is x₂, whilst the worst one is x₄. In contrast, by using the methods in Ref. [51], the best ones are x₂, whilst the worst one is x₁. There are a number of reasons why differences exist between the final rankings of the [31, 51] and proposed method. For example; different aggregation operators and operations also lead to different rankings. Moreover, different aggregation operators lead to different rankings because the operators emphasize the decision makers’ judgments differently. The [31] is based on an arithmetic average. By comparison, the proposed method in this paper focuses on the geometric aggregation operators, which takes both the subjective and objective weights into consideration. Notwithstanding, the ranking of the proposed method is the same as that of the [31]. Therefore, the proposed method is effective.

7 Conclusion

This paper proposes three geometric operator is called SVTrN weighted geometric operator, SVTrN ordered weighted geometric operator, SVTrN ordered hybrid weighted geometric operator. Then, an approach is developed to solve multi-criteria decision making problems. It is easily seen that the proposed approach can be extended to solve more general multi-criteria decision making problems in a straightforward manner. Due to the fact that a SVTrN-number is a generalization of a triangular fuzzy number and triangular intuitionistic fuzzy number, the other existing approaches of triangular fuzzy number and triangular intuitionistic fuzzy number may be extended to SVTrN-numbers. Therefore, more effective approaches for SVTrN-numbers, how to determine the weight vectors for SVTrN-numbers and an approach of multi-criteria decision-making with weights expressed by single valued neutrosophic sets will be investigated in the near future.

Footnotes

Acknowledgments

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions.

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