Some aggregation operators of credibility trapezoidal fuzzy neutrosophic values and their decision-making application in the selection of slope design schemes

Abstract

Modern decision-making (DM) systems are becoming more and more complex and sophisticated in their demands for information expressions and credibility levels. In the existing literature, a trapezoidal fuzzy neutrosophic value (TFNV) that combines trapezoidal fuzzy numbers with neutrosophic values can be better depicted by truth, indeterminacy, and falsity membership functions. Unfortunately, TFNV implies its defect since it lacks a measure of credibility. To make TFNV more creditable, TFNV should be related to its credibility level. Regarding the motivation for combining TFNV with its credibility level, this paper first proposes the concept of a credibility TFNV (C-TFNV) as a new framework of TFNV associated with the measure of credibility. The advantage of its information expression is that C-TFNV has a more creditable ability to describe indeterminate and inconsistent knowledge and judgments of human beings by the mixed information of a TFNV and a related credibility level (an ordered pair of TFNVs). Next, we propose the operational laws of C-TFNVs and the score function of C-TFNV. Furthermore, we present a C-TFNV weighted arithmetic averaging (C-TFNVWAA) and a C-TFNV weighted geometric averaging (C-TFNVWGA) operators and their properties. Then, a multicriteria DM model based on the C-TFNVWAA and C-TFNVWGA operators and the score function is established in the case of C-TFNVs. Finally, an actual DM example of slope decision schemes is provided to show the applicability and efficiency of the established DM model in the case of C-TFNVs.

Keywords

Credibility trapezoidal fuzzy neutrosophic value credibility trapezoidal fuzzy neutrosophic value weighted arithmetic averaging operator credibility trapezoidal fuzzy neutrosophic value weighted geometric averaging operator decision making

1 Introduction

In the case of inconsistency and indeterminacy, the neutrosophic theory based on the truth, falsity, and indeterminacy membership functions [1] has attracted the interest of many researchers and become a research hotspot. Therefore, it was widely applied in many fields, such as decision making (DM) [2 –7], medical diagnosis/evaluation [8], fault diagnosis [9], image processing [10, 11]. Based on the hybrid form of a trapezoidal fuzzy number (TFN) and a neutrosophic value (NV), Ye [12] first proposed a trapezoidal fuzzy neutrosophic value (TFNV) (a triangular fuzzy neutrosophic value as a special case of TFNV) through the discrete domain of the neutrosophic set was extended to the continuous domain. Since TFNVs are a main research topic in the neutrosophic theory, they have been used in many areas, such as decision making [12 –15] and path optimization [16, 17]. As a special case of TFNV, a triangular fuzzy neutrosophic value was also used for DM [18, 19], linear programming [20], performance evaluation of the medical service [21], and green supplier selection [22], etc. Furthermore, trapezoidal fuzzy neutrosophic numbers (TFNNs) [23] and interval TFNNs [24] were presented in terms of the true, false, and indeterminate TFNs, which are different from the TFNV notion containing TFN and NV, and then applied to DM problems.

As a generalization of fuzzy sets, Ye et al. [25] first presented the notion of a fuzzy credibility number (FCN), which implies a greater ability to depict human knowledge and judgments by the mixed information of a fuzzy number and a credibility level, and then developed weighted aggregation operators of FCNs and their DM model for choosing slope design schemes. Then, Ye et al. [26] also proposed fuzzy credibility cubic numbers (FCCNs), weighted aggregation operators of FCCNs, and their DM strategy for the selection problem of slope design schemes. Ye et al. [27] further put forward intuitionistic fuzzy credibility sets (IFCSs) and their similarity measures for the performance evaluation of industrial robots. Recently, Yong et al. [28] proposed trapezoidal neutrosophic Z-numbers (TNZNs) in terms of truth, falsity and indeterminacy Z-numbers, which are composed of the truth, falsity, and indeterminacy membership functions and their reliability measures, and defined some weighted aggregation operators and the score and accuracy functions of TNZNs for multicriteria DM problems.

Modern decision-making problems are becoming increasingly complex and sophisticated in their demands for the expression and credibility of information. Then, it is known that TFN is a common and typical membership function with continuous fuzzy information. In the representation of fuzzy information, triangular fuzzy numbers and fuzzy values are only special cases of TFNs. Therefore, TFN is obviously superior to the triangular fuzzy number and the fuzzy value. However, in the concept of TFNV [12], the TFNV consisting of TFN and NV cannot reflect the level of credibility due to the lack of a credibility measure of TFNV. Furthermore, TNZN [26] consists of the truth, falsity, and indeterminacy membership functions and their corresponding reliability measures, containing the three trapezoidal fuzzy Z-numbers. Therefore, the TNZN notion is also different from the TFNV notion.

Since TFNV lacks its credibility measure, it is difficult to keep the credibility of the TFNV information in the uncertain and vague environments. Furthermore, the existing literature does not study the mixed information of TFNV and its credibility measure, which shows the gap. To make up for the gap, TFNV should be combined with its credibility level so as to strengthen the credibility level of TFNV. Based on the motivation of the FCN and IFCS notions [12, 27] and strengthening the credibility level of TFNV, we can combine TFNV with its credibility level to describe their mixed information using an ordered pair of TFNVs. To perform such a DM problem with the mixed information of combining TFNVs with their credibility levels, the aim of this study is to propose a multicriteria DM model using aggregation operators of credibility TFNVs (C-TFNVs) in the case of C-TFNVs. To do so, this paper first proposes the concept of C-TFNV based on a novel framework of TFNV associated with its credibility level, which is depicted by an ordered pair of TFNVs. Next, the operational laws of C-TFNVs and a score function for sorting C-TFNVs are defined, and then a C-TFNV weighted arithmetic averaging (C-TFNVWAA) operator and a C-TFNV weighted geometric averaging (C-TFNVWGA) operator and their properties are presented under the environment of C-TFNVs. In terms of the C-TFNVWAA and C-TFNVWGA operators and the score function, a multicriteria DM model is established in the case of C-TFNVs to solve multicriteria DM problems with C-TFNV information. Furthermore, an actual example of the slope design scheme selection problem is provided to demonstrate the applicability and efficiency of the established DM model in the situation of C-TFNVs. However, the main advantage of this study is that the proposed DM model not only makes the assessment information of TFNVs more creditable, but also strengthens the credibility and efficiency of DM problems in the environment of C-TFNVs.

In the original study, the main contributions of the article are summarized below:

The proposed C-TFNV effectively expresses the mixed information of TFNV and a credibility level by an ordered pair of TFNVs in an indeterminate and inconsistent setting.

The presented C-TFNVWAA and C-TFNVWGA operators of C-TFNVs and score function provide the necessary modeling tools for solving multicriteria DM problems in the case of C-TFNVs.

The established DM model in the setting of C-TFNVs strengthens the information expression ability and credibility level of DM problems.

The rest of the study is composed of the following parts. Section 2 briefly depicts the preliminary knowledge of TFNVs for this study. Section 3 proposes the notion of C-TFNV, the operational laws of C-TFNVs, and the score function and sorting rules of C-TFNVs. In Section 4, the C-TFNVWAA and C-TFNVWGA operators and their properties are presented to aggregate C-TFNVs. A multicriteria DM model using the C-TFNVWAA and C-TFNVWGA operators and the score function is established in Section 5. In Section 6, an actual DM example of slope design schemes and a relative comparison are provided to indicate the applicability and efficiency of the established DM model in the setting of C-TFNVs. Section 7 addresses conclusions and further study.

2 Preliminary knowledge of TFNVs

This section briefly depicts the preliminary knowledge of TFNVs for the following study.

Ye [12] first introduced TFNV in a nonempty set U with a generic element u in U denoted by $\tilde{s} = 〈 {\tilde{T}}_{R} (u), {\tilde{I}}_{R} (u), {\tilde{F}}_{R} (u) 〉$ , in which its truth, indeterminacy, and falsity membership functions were defined, respectively, as follows: ${\tilde{T}}_{R} (u) = {\begin{matrix} \frac{u - b_{1}}{b_{2} - b_{1}} {\tilde{T}}_{R}, b_{1} \leq u < b_{2}, \\ {\tilde{T}}_{R}, b_{2} \leq u \leq b_{3}, \\ \frac{b_{4} - u}{b_{4} - b_{3}} {\tilde{T}}_{R}, b_{3} < u \leq b_{4}, \\ 0, otherwise \end{matrix},$ ${\tilde{I}}_{R} (u) = {\begin{matrix} \frac{c_{2} - u + {\tilde{I}}_{R} (u - c_{1})}{c_{2} - c_{1}}, c_{1} \leq u < c_{2}, \\ {\tilde{I}}_{R}, c_{2} \leq u \leq c_{3}, \\ \frac{u - c_{3} + {\tilde{I}}_{R} (c_{4} - u)}{c_{4} - c_{3}}, c_{3} < u \leq c_{4}, \\ 1, otherwise \end{matrix},$ ${\tilde{F}}_{R} (u) = {\begin{matrix} \frac{d_{2} - u + {\tilde{F}}_{R} (u - d_{1})}{d_{2} - d_{1}}, d_{1} \leq u < d_{2}, \\ {\tilde{F}}_{R}, d_{2} \leq u \leq d_{3}, \\ \frac{u - d_{3} + {\tilde{I}}_{R} (d_{4} - u)}{d_{4} - d_{3}}, d_{3} < u \leq d_{4}, \\ 1, otherwise \end{matrix},$

where ${\tilde{T}}_{R}, {\tilde{I}}_{R}, {\tilde{F}}_{R} \in [0, 1]$ are the truth, indeterminacy, and falsity values in the TFNV $\tilde{s}$ with $0 \leq {\tilde{T}}_{R} + {\tilde{I}}_{R} + {\tilde{F}}_{R} \leq 3$ , and $b_{k}, c_{k}, d_{k}, \in R$ (k = 1, 2, 3, 4) are the parameters of TFNs in the TFNV $\tilde{s}$ . If b_k = c_k = d_k (k = 1, 2, 3, 4) in the TFNV $\tilde{s}$ , TFNV can be simply denoted as $\tilde{s} = 〈 (b_{1}, b_{2}, b_{3}, b_{4}); {\tilde{T}}_{R}, {\tilde{I}}_{R}, {\tilde{F}}_{R} 〉$ .

Suppose that there are two TFNVs ${\tilde{s}}_{1} = 〈 (b_{11}, b_{12}, b_{13}, b_{14}); {\tilde{T}}_{R 1}, {\tilde{I}}_{R 1}, {\tilde{F}}_{R 1} 〉$ and ${\tilde{s}}_{2} = 〈 (b_{21}, b_{22}, b_{23}, b_{24}); {\tilde{T}}_{R 2}, {\tilde{I}}_{R 2}, {\tilde{F}}_{R 2} 〉$ and α>0. Then, Ye [12] presented their operational laws:

$\begin{matrix} {\tilde{s}}_{1} \oplus {\tilde{s}}_{2} \\ = 〈 \begin{matrix} (b_{11} + b_{21}, b_{12} + b_{22}, b_{13} + b_{23}, b_{14} + b_{24}); \\ {\tilde{T}}_{R 1} + {\tilde{T}}_{R 2} - {\tilde{T}}_{R 1} {\tilde{T}}_{R 2}, {\tilde{I}}_{R 1} {\tilde{I}}_{R 2}, {\tilde{F}}_{R 1} {\tilde{F}}_{R 2} \end{matrix} 〉 \end{matrix}$ ;

${\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = 〈 \begin{matrix} (b_{11} b_{21}, b_{12} b_{22}, b_{13} b_{23}, b_{14} b_{24}); \\ {\tilde{T}}_{R 1} {\tilde{T}}_{R 2}, {\tilde{I}}_{R 1} + {\tilde{I}}_{R 2} - {\tilde{I}}_{R 1} {\tilde{I}}_{R 2}, \\ {\tilde{F}}_{R 1} + {\tilde{F}}_{R 2} - {\tilde{F}}_{R 1} {\tilde{F}}_{R 2} \end{matrix} 〉$ ;

$α {\tilde{s}}_{1} = 〈 \begin{matrix} (α b_{11}, α b_{12}, α b_{13}, α b_{14}); \\ 1 - (1 - {\tilde{T}}_{R 1})^{α}, {\tilde{I}}_{R 1}^{α}, {\tilde{F}}_{R 1}^{α} \end{matrix} 〉$ ;

${\tilde{s}}_{1}^{α} = 〈 \begin{matrix} (a_{11}^{α}, a_{12}^{α}, a_{13}^{α}, a_{14}^{α}); {\tilde{T}}_{R 1}^{α}, \\ 1 - (1 - {\tilde{I}}_{R 1})^{α}, 1 - (1 - {\tilde{F}}_{R 1})^{α} \end{matrix} 〉$ .

Assume that there is a group of TFNVs ${\tilde{s}}_{i} = 〈 (b_{i 1}, b_{i 2}, b_{i 3}, b_{i 4}); {\tilde{T}}_{Ri}, {\tilde{I}}_{Ri}, {\tilde{F}}_{Ri} 〉$ (i = 1, 2,..., m) with the weight α_i of ${\tilde{s}}_{i}$ for 0 ≤ α_i ≤ 1 and $\sum_{i = 1}^{m} α_{i} = 1$ . Ye [12] proposed the TFNV weighted arithmetic averaging (TFNVWAA) and TFNV weighted geometric averaging (TFNVWGA) operators: $\begin{matrix} TFNVWAA ({\tilde{s}}_{1}, {\tilde{s}}_{2}, . . ., {\tilde{s}}_{m}) = \sum_{i = 1}^{m} α_{i} {\tilde{s}}_{i} \\ = 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} b_{i 1}, \sum_{i = 1}^{m} α_{i} b_{i 2}, \sum_{i = 1}^{m} α_{i} b_{i 3}, \sum_{i = 1}^{m} α_{i} b_{i 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉 \end{matrix},$ (1) $\begin{matrix} TFNVWGA ({\tilde{s}}_{1}, {\tilde{s}}_{2}, . . ., {\tilde{s}}_{m}) = \prod_{i = 1}^{m} {\tilde{s}}_{i}^{α_{i}} \\ = 〈 \begin{matrix} (\prod_{i = 1}^{m} b_{i 1}^{α_{i}}, \prod_{i = 1}^{m} b_{i 2}^{α_{i}}, \prod_{i = 1}^{m} b_{i 3}^{α_{i}}, \prod_{i = 1}^{m} b_{i 4}^{α_{i}}); \\ \prod_{i = 1}^{m} {\tilde{T}}_{Ri}^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{I}}_{Ri})^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{F}}_{Ri})^{α_{i}} \end{matrix} 〉 \end{matrix} .$ (2)

To sort TFNVs, Ye [12] defined a score function of $\tilde{s} = 〈 (b_{1}, b_{2}, b_{3}, b_{4}); {\tilde{T}}_{R}, {\tilde{I}}_{R}, {\tilde{F}}_{R} 〉$ : $S (\tilde{s}) = \frac{1}{12} (\begin{matrix} (b_{1} + b_{2} + b_{3} + b_{4}) \\ \times (2 + {\tilde{T}}_{R} - {\tilde{I}}_{R} - {\tilde{F}}_{R}) \end{matrix}), S (\tilde{s}) \in [0, 1] .$ (3)

Then, sorting rules based on the score functions of two TFNVs ${\tilde{s}}_{1}$ and ${\tilde{s}}_{2}$ are defined below [12]:

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

If $S ({\tilde{s}}_{1}) = S ({\tilde{s}}_{2})$ , then ${\tilde{s}}_{1} = {\tilde{s}}_{2}$ .

Especially when b₁ = b₂ = b₃ = b₄ = 1, TFNVs are degenerated to the neutrosophic values, then Equations (1)–(3) are degenerated to the neutrosophic value weighted arithmetic averaging (NVWAA) and neutrosophic value weighted geometric averaging (NVWGA) operators and the score function of NV, respectively [3]: $\begin{matrix} NVWAA ({\tilde{s}}_{1}, {\tilde{s}}_{2}, . . ., {\tilde{s}}_{m}) = \sum_{i = 1}^{m} α_{i} {\tilde{s}}_{i} \\ = 〈 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Ri}^{α_{i}} 〉 \end{matrix},$ (4) $\begin{matrix} TFNVWGA ({\tilde{s}}_{1}, {\tilde{s}}_{2}, . . ., {\tilde{s}}_{m}) = \prod_{i = 1}^{m} {\tilde{s}}_{i}^{α_{i}} \\ = 〈 \prod_{i = 1}^{m} {\tilde{T}}_{Ri}^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{I}}_{Ri})^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{F}}_{Ri})^{α_{i}} 〉 \end{matrix},$ (5) $S^{'} (\tilde{s}) = (2 + {\tilde{T}}_{R} - {\tilde{I}}_{R} - {\tilde{F}}_{R}) / 3, S^{'} (\tilde{s}) \in [0, 1] .$ (6)

3 C-TFNVs and their operations and score function

Based on a new framework of TFNV related to its credibility level, this section proposes C-TFNV, various operational laws of C-TFNVs, and a score function of C-TFNV for sorting C-TFNVs.

Definition 1. Let U be a nonempty set with a generic element u in U. C-TFNV is defined as $\tilde{z} = (〈 {\tilde{T}}_{R} (u), {\tilde{I}}_{R} (u), {\tilde{F}}_{R} (u) 〉, 〈 {\tilde{T}}_{M} (u), {\tilde{I}}_{M} (u), {\tilde{F}}_{M} (u) 〉)$ for u ∈ U, then its truth, indeterminacy, and falsity membership functions and corresponding credibility measures: ${\tilde{T}}_{R} (u) = {\begin{matrix} \frac{u - b_{1}}{b_{2} - b_{1}} {\tilde{T}}_{R}, b_{1} \leq u < b_{2}, \\ {\tilde{T}}_{R}, b_{2} \leq u \leq b_{3}, \\ \frac{b_{4} - u}{b_{4} - b_{3}} {\tilde{T}}_{R}, b_{3} < u \leq b_{4}, \\ 0, otherwise \end{matrix},$ ${\tilde{I}}_{R} (u) = {\begin{matrix} \frac{c_{2} - u + {\tilde{I}}_{R} (u - c_{1})}{c_{2} - c_{1}}, c_{1} \leq u < c_{2}, \\ {\tilde{I}}_{R}, c_{2} \leq u \leq c_{3}, \\ \frac{u - c_{3} + {\tilde{I}}_{R} (c_{4} - u)}{c_{4} - c_{3}}, c_{3} < u \leq c_{4}, \\ 1, otherwise \end{matrix},$ ${\tilde{F}}_{R} (u) = {\begin{matrix} \frac{d_{2} - u + {\tilde{F}}_{R} (u - d_{1})}{d_{2} - d_{1}}, d_{1} \leq u < d_{2}, \\ {\tilde{F}}_{R}, d_{2} \leq u \leq d_{3}, \\ \frac{u - d_{3} + {\tilde{I}}_{R} (d_{4} - u)}{d_{4} - d_{3}}, d_{3} < u \leq d_{4}, \\ 1, otherwise \end{matrix},$ ${\tilde{T}}_{M} (u) = {\begin{matrix} \frac{u - e_{1}}{e_{2} - e_{1}} {\tilde{T}}_{M}, e_{1} \leq u < e_{2}, \\ {\tilde{T}}_{M}, e_{2} \leq u \leq e_{3}, \\ \frac{e_{4} - u}{e_{4} - e_{3}} {\tilde{T}}_{M}, e_{3} < u \leq e_{4}, \\ 0, otherwise \end{matrix},$ ${\tilde{I}}_{M} (u) = {\begin{matrix} \frac{f_{2} - u + {\tilde{I}}_{M} (u - f_{1})}{f_{2} - f_{1}}, f_{1} \leq u < f_{2}, \\ {\tilde{I}}_{M}, f_{2} \leq u \leq f_{3}, \\ \frac{u - f_{3} + {\tilde{I}}_{M} (f_{4} - u)}{f_{4} - f_{3}}, f_{3} < u \leq f_{4}, \\ 1, otherwise \end{matrix},$ ${\tilde{F}}_{M} (u) = {\begin{matrix} \frac{g_{2} - u + {\tilde{F}}_{M} (u - g_{1})}{g_{2} - g_{1}}, g_{1} \leq u < g_{2}, \\ {\tilde{F}}_{M}, g_{2} \leq u \leq g_{3}, \\ \frac{u - g_{3} + {\tilde{F}}_{M} (g_{4} - u)}{g_{4} - g_{3}}, g_{3} < u \leq g_{4}, \\ 1, otherwise \end{matrix},$

where ${\tilde{T}}_{R}, {\tilde{I}}_{R}, {\tilde{F}}_{R} \in [0, 1]$ and ${\tilde{T}}_{M}, {\tilde{I}}_{M}, {\tilde{F}}_{M} \in [0, 1]$ are the truth, indeterminacy and falsity values in the C-TFNV $\tilde{z}$ for $0 \leq {\tilde{T}}_{R} + {\tilde{I}}_{R} + {\tilde{F}}_{R} \leq 3$ and $0 \leq {\tilde{T}}_{M} + {\tilde{I}}_{M} + {\tilde{F}}_{M} \leq 3$ , and then $b_{k}, c_{k}, d_{k}, e_{k}, f_{k}, g_{k} \in R$ (k = 1, 2, 3, 4) are the parameters of TFNs in the C-TFNV $\tilde{z}$ . If b_k = c_k = d_k and e_k = f_k = g_k (k = 1, 2, 3, 4) in the C-TFNV $\tilde{z}$ , the C-TFNV $\tilde{z}$ can be denoted by $\tilde{z} = (\begin{matrix} 〈 (b_{1}, b_{2}, b_{3}, b_{4}); {\tilde{T}}_{R}, {\tilde{I}}_{R}, {\tilde{F}}_{R} 〉, \\ 〈 (e_{1}, e_{2}, e_{3}, e_{4}); {\tilde{T}}_{M}, {\tilde{I}}_{M}, {\tilde{F}}_{M} 〉 \end{matrix})$ .

In the following, we discuss several cases of C-TFNVs under some conditions:

If b₂ = b₃ and e₂ = e₃ in a C-TFNV $\tilde{z}$ , the C-TFNV $\tilde{z}$ is reduced to the credibility triangular fuzzy neutrosophic value;

If b₁ = b₂ = b₃ = b₄ = 1 and e₁ = e₂ = e₃ = e₄ = 1 in a C-TFNV $\tilde{z}$ , the C-TFNV $\tilde{z}$ is reduced to the credibility neutrosophic value;

If 0 ≤ b₁ ≤ b₂ ≤ b₃ ≤ b₄ ≤ 1 and 0 ≤ e₁ ≤ e₂ ≤ e₃ ≤ e₄ ≤ 1, $\tilde{z}$ is named a normalized C-TFNV, which is used for this paper and simply named C-TFNV.

Definition 2. Set ${\tilde{z}}_{1} = (\begin{matrix} 〈 (b_{11}, b_{12}, b_{13}, b_{14}); \\ 〈 (e_{11}, e_{12}, e_{13}, e_{14}); \end{matrix}$ $\begin{matrix} {\tilde{T}}_{R 1}, {\tilde{I}}_{R 1}, {\tilde{F}}_{R 1} 〉, \\ {\tilde{T}}_{M 1}, {\tilde{I}}_{M 1}, {\tilde{F}}_{M 1} 〉 \end{matrix})$ and ${\tilde{z}}_{2} = (\begin{matrix} 〈 (b_{21}, b_{22}, b_{23}, b_{24}); \\ 〈 (e_{21}, e_{22}, e_{23}, e_{24}); \end{matrix}$ $\begin{matrix} {\tilde{T}}_{R 2}, {\tilde{I}}_{R 2}, {\tilde{F}}_{R 2} 〉, \\ {\tilde{T}}_{M 2}, {\tilde{I}}_{M 2}, {\tilde{F}}_{M 2} 〉 \end{matrix})$ as two C-TFNVs and α>0. Then, their operational laws are defined as follows:

$\begin{matrix} {\tilde{z}}_{1} \oplus {\tilde{z}}_{2} \\ = (\begin{matrix} 〈 \begin{matrix} (b_{11} + b_{21}, b_{12} + b_{22}, b_{13} + b_{23}, b_{14} + b_{24}); \\ {\tilde{T}}_{R 1} + {\tilde{T}}_{R 2} - {\tilde{T}}_{R 1} {\tilde{T}}_{R 2}, {\tilde{I}}_{R 1} {\tilde{I}}_{R 2}, {\tilde{F}}_{R 1} {\tilde{F}}_{R 2} \end{matrix} 〉, \\ 〈 \begin{matrix} (e_{11} + e_{21}, e_{12} + e_{22}, e_{13} + e_{23}, e_{14} + e_{24}); \\ {\tilde{T}}_{M 1} + {\tilde{T}}_{M 2} - {\tilde{T}}_{M 1} {\tilde{T}}_{M 2}, {\tilde{I}}_{M 1} {\tilde{I}}_{M 2}, {\tilde{F}}_{M 1} {\tilde{F}}_{M 2} \end{matrix} 〉 \end{matrix}) \end{matrix}$ ;

$\begin{matrix} {\tilde{z}}_{1} \otimes {\tilde{z}}_{2} \\ = (\begin{matrix} 〈 \begin{matrix} (b_{11} b_{21}, b_{12} b_{22}, b_{13} b_{23}, b_{14} b_{24}); {\tilde{T}}_{R 1} {\tilde{T}}_{R 2}, \\ {\tilde{I}}_{R 1} + {\tilde{I}}_{R 2} - {\tilde{I}}_{R 1} {\tilde{I}}_{R 2}, {\tilde{F}}_{R 1} + {\tilde{F}}_{R 2} - {\tilde{F}}_{R 1} {\tilde{F}}_{R 2} \end{matrix} 〉, \\ 〈 \begin{matrix} (e_{11} e_{21}, e_{12} e_{22}, e_{13} e_{23}, e_{14} e_{24}); {\tilde{T}}_{M 1} {\tilde{T}}_{M 2}, \\ {\tilde{I}}_{M 1} + {\tilde{I}}_{M 2} - {\tilde{I}}_{M 1} {\tilde{I}}_{M 2}, {\tilde{F}}_{M 1} + {\tilde{F}}_{M 2} - {\tilde{F}}_{M 1} {\tilde{F}}_{M 2} \end{matrix} 〉 \end{matrix}) \end{matrix}$ ;

$α {\tilde{z}}_{1} = (\begin{matrix} 〈 \begin{matrix} (α b_{11}, α b_{12}, α b_{13}, α b_{14}); \\ 1 - (1 - {\tilde{T}}_{R 1})^{α}, {\tilde{I}}_{R 1}^{α}, {\tilde{F}}_{R 1}^{α} \end{matrix} 〉, \\ 〈 \begin{matrix} (α e_{11}, α e_{12}, α e_{13}, α e_{14}); \\ 1 - (1 - {\tilde{T}}_{M 1})^{α}, {\tilde{I}}_{M 1}^{α}, {\tilde{F}}_{M 1}^{α} \end{matrix} 〉 \end{matrix})$ ;

${\tilde{z}}_{1}^{α} = (\begin{matrix} 〈 \begin{matrix} (b_{11}^{α}, b_{12}^{α}, b_{13}^{α}, b_{14}^{α}); {\tilde{T}}_{R 1}^{α}, \\ 1 - (1 - {\tilde{I}}_{R 1})^{α}, 1 - (1 - {\tilde{F}}_{R 1})^{α} \end{matrix} 〉, \\ 〈 \begin{matrix} (e_{11}^{α}, e_{12}^{α}, e_{13}^{α}, e_{14}^{α}); {\tilde{T}}_{M 1}^{α}, \\ 1 - (1 - {\tilde{I}}_{M 1})^{α}, 1 - (1 - {\tilde{F}}_{M 1})^{α} \end{matrix} 〉 \end{matrix})$ .

To sort C-TFNVs, the score function and sorting rules of C-TFNVs are defined based on the expected value of TFNV [12] in the following.

Definition 3. Set $\tilde{z} = (\begin{matrix} 〈 (b_{1}, b_{2}, b_{3}, b_{4}); {\tilde{T}}_{R}, \\ 〈 (e_{1}, e_{2}, e_{3}, e_{4}); {\tilde{T}}_{M}, \end{matrix}$ $\begin{matrix} {\tilde{I}}_{R}, {\tilde{F}}_{R} 〉, \\ {\tilde{I}}_{M}, {\tilde{F}}_{M} 〉 \end{matrix})$ as C-TFNV. Based on Equation (3), the score function of $\tilde{z}$ is defined as $\begin{matrix} S (\tilde{z}) = \frac{1}{12} (b_{1} + b_{2} + b_{3} + b_{4}) (2 + {\tilde{T}}_{R} - {\tilde{I}}_{R} - {\tilde{F}}_{R}) \\ \times \frac{1}{12} (e_{1} + e_{2} + e_{3} + e_{4}) (2 + {\tilde{T}}_{M} - {\tilde{I}}_{M} - {\tilde{F}}_{M}), \\ = \frac{1}{144} (\begin{matrix} (b_{1} + b_{2} + b_{3} + b_{4}) (2 + {\tilde{T}}_{R} - {\tilde{I}}_{R} - {\tilde{F}}_{R}) \\ \times (e_{1} + e_{2} + e_{3} + e_{4}) (2 + {\tilde{T}}_{M} - {\tilde{I}}_{M} - {\tilde{F}}_{M}) \end{matrix}) \\ S (\tilde{z}) \in [0, 1] . \end{matrix}$ (7)

Definition 4. Set ${\tilde{z}}_{1} = (\begin{matrix} 〈 (b_{11}, b_{12}, b_{13}, b_{14}); {\tilde{T}}_{R 1}, \\ 〈 (e_{11}, e_{12}, e_{13}, e_{14}); {\tilde{T}}_{M 1}, \end{matrix}$ $\begin{matrix} {\tilde{I}}_{R 1}, {\tilde{F}}_{R 1} 〉, \\ {\tilde{I}}_{M 1}, {\tilde{F}}_{M 1} 〉 \end{matrix})$ and ${\tilde{z}}_{2} = (\begin{matrix} 〈 (b_{21}, b_{22}, b_{23}, b_{24}); \\ 〈 (e_{21}, e_{22}, e_{23}, e_{24}); \end{matrix}$ $\begin{matrix} {\tilde{T}}_{R 2}, {\tilde{I}}_{R 2}, {\tilde{F}}_{R 2} 〉, \\ {\tilde{T}}_{M 2}, {\tilde{I}}_{M 2}, {\tilde{F}}_{M 2} 〉 \end{matrix})$ as two C-TFNVs. If $S ({\tilde{z}}_{1}) > S ({\tilde{z}}_{2})$ , then ${\tilde{z}}_{1}$ > ${\tilde{z}}_{2}$ ; if $S ({\tilde{z}}_{1}) = S ({\tilde{z}}_{2})$ , then ${\tilde{z}}_{1} = {\tilde{z}}_{2}$ .

Example 1. Set two C-TFNVs as ${\tilde{z}}_{1} = (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.4, 0.1, 0.2 〉, \\ 〈 (0.6, 0.7, 0.8, 0.9); 0.8, 0.2, 0.1 〉 \end{matrix})$ and ${\tilde{z}}_{2} =$ $(\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8); 0.7, 0.2, 0.3 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.6, 0.3, 0.4 〉 \end{matrix})$ . Then, both are sorted by their score values of Equation (7) below.

Since $S ({\tilde{z}}_{1}) = {(0.3 + 0.4 + 0.5 + 0.6)$ ×(2+ 0.4−0.1−0.2)× (0.6 + 0.7 + 0.8 + 0.9) ×(2+0.8−0.2−0.1)}/144 = 0.1969 and $S ({\tilde{z}}_{2}) = {(0.5 +$ 0.6 + 0.7 + 0.8) ×(2 + 0.7−0.2−0.3) ×(0.4 + 0.5+ 0.6 + 0.7) ×(2 + 0.6−0.3−0.4)}/144 = 0.166, according to Definition 4, the sorting of both is ${\tilde{z}}_{1} > {\tilde{z}}_{2}$ .

4 Two weighted aggregation operators of C-TFNVs

Since weighted arithmetic and geometric averaging aggregation operators are the most basic aggregation operations in multicriteria DM applications, this section should present the two weighted aggregation operators based on the operational laws of C-TFNVs for the modeling requirements of multicriteria DM problems with C-TFNV information.

4.1 C-TFNVWAA operator

Definition 5. Set ${\tilde{z}}_{i} = (\begin{matrix} 〈 (b_{i 1}, b_{i 2}, b_{i 3}, b_{i 4}); {\tilde{T}}_{Ri}, \\ 〈 (e_{i 1}, e_{i 2}, e_{i 3}, e_{i 4}); {\tilde{T}}_{Mi}, \end{matrix}$ $\begin{matrix} {\tilde{I}}_{Ri}, {\tilde{F}}_{Ri} 〉, \\ {\tilde{I}}_{Mi}, {\tilde{F}}_{Mi} 〉 \end{matrix})$ (i = 1, 2,..., m) as a group of C-TFNVs with the weight α_i of ${\tilde{z}}_{i}$ for 0 ≤ α_i ≤ 1 and $\sum_{i = 1}^{m} α_{i} = 1$ . Then, a C-TFNVWAA operator is defined as $C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \sum_{i = 1}^{m} α_{i} {\tilde{z}}_{i} .$ (8)

Thus, the following theorem is presented corresponding to the operational laws of C-TFNVs and Equation (8).

Theorem 1. Set ${\tilde{z}}_{i} = (\begin{matrix} 〈 (b_{i 1}, b_{i 2}, b_{i 3}, b_{i 4}); {\tilde{T}}_{Ri}, \\ 〈 (e_{i 1}, e_{i 2}, e_{i 3}, e_{i 4}); {\tilde{T}}_{Mi}, \end{matrix}$ $\begin{matrix} {\tilde{I}}_{Ri}, {\tilde{F}}_{Ri} 〉, \\ {\tilde{I}}_{Mi}, {\tilde{F}}_{Mi} 〉 \end{matrix})$ (i = 1, 2, ... , m) as a group of C-TFNVs with the weight α_i of ${\tilde{z}}_{i}$ for 0 ≤ α_i ≤ 1 and $\sum_{i = 1}^{m} α_{i} = 1$ . Based on the operational laws of C-TFNVs and Equation (8), we can obtain the following C-TFNVWAA operator:

$\begin{matrix} C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \sum_{i = 1}^{m} α_{i} {\tilde{z}}_{i} \\ = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} b_{i 1}, \sum_{i = 1}^{m} α_{i} b_{i 2}, \sum_{i = 1}^{m} α_{i} b_{i 3}, \sum_{i = 1}^{m} α_{i} b_{i 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} e_{i 1}, \sum_{i = 1}^{m} α_{i} e_{i 2}, \sum_{i = 1}^{m} α_{i} e_{i 3}, \sum_{i = 1}^{m} α_{i} e_{i 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Mi})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Mi}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Mi}^{α_{i}} \end{matrix} 〉 \end{matrix}) \end{matrix} .$ (9)

Then, the aggregated value of the C-TFNVWAA operator is still C-TFNV. In the following, Theorem 1 can be verified based on mathematical induction.

Proof. (1) If m = 2, there is the following operation:

$\begin{matrix} C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}) = \sum_{i = 1}^{2} α_{i} {\tilde{z}}_{i} = α_{1} {\tilde{z}}_{1} \oplus α_{2} {\tilde{z}}_{2} \\ = (\begin{matrix} 〈 \begin{matrix} (\begin{matrix} α_{1} b_{11} + α_{2} b_{21}, α_{1} b_{12} + α_{2} b_{22}, \\ α_{1} b_{13} + α_{2} b_{23}, α_{1} b_{14} + α_{2} b_{24} \end{matrix}); \\ 1 - (1 - {\tilde{T}}_{R 1})^{α_{1}} + 1 - (1 - {\tilde{T}}_{R 2})^{α_{2}} \\ - (1 - (1 - {\tilde{T}}_{R 1})^{α_{1}}) (1 - (1 - {\tilde{T}}_{R 2})^{α_{2}}), \\ {\tilde{I}}_{R 1}^{α_{1}} {\tilde{I}}_{R 2}^{α_{2}}, {\tilde{F}}_{R 1}^{α_{1}} {\tilde{F}}_{R 2}^{α_{2}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\begin{matrix} α_{1} e_{11} + α_{2} e_{21}, α_{1} e_{12} + α_{2} e_{22}, \\ α_{1} e_{13} + α_{2} e_{23}, α_{1} e_{14} + α_{2} e_{24} \end{matrix}); \\ 1 - (1 - {\tilde{T}}_{M 1})^{α_{1}} + 1 - (1 - {\tilde{T}}_{M 2})^{α_{2}} \\ - (1 - (1 - {\tilde{T}}_{M 1})^{α_{1}}) (1 - (1 - {\tilde{T}}_{M 2})^{α_{2}}), \\ {\tilde{I}}_{M 1}^{α_{1}} {\tilde{I}}_{M 2}^{α_{2}}, {\tilde{F}}_{M 1}^{α_{1}} {\tilde{F}}_{M 2}^{α_{2}} \end{matrix} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{2} α_{i} b_{i 1}, \sum_{i = 1}^{2} α_{i} b_{i 2}, \sum_{i = 1}^{2} α_{i} b_{i 3}, \sum_{i = 1}^{2} α_{i} b_{i 4}); \\ 1 - \sum_{i = 1}^{2} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{2} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{2} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{2} α_{i} e_{i 1}, \sum_{i = 1}^{2} α_{i} e_{i 2}, \sum_{i = 1}^{2} α_{i} e_{i 3}, \sum_{i = 1}^{2} α_{i} e_{i 4}); \\ 1 - \sum_{i = 1}^{2} (1 - {\tilde{T}}_{Mi})^{α_{i}}, \prod_{i = 1}^{2} {\tilde{I}}_{Mi}^{α_{i}}, \prod_{i = 1}^{2} {\tilde{F}}_{Mi}^{α_{i}} \end{matrix} 〉 \end{matrix}) . \end{matrix}$ (10)

(2) If m = n, Equation (9) can keep the following equation:

$\begin{matrix} C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{n}) = \sum_{i = 1}^{n} α_{i} {\tilde{z}}_{i} \\ = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{n} α_{i} b_{i 1}, \sum_{i = 1}^{n} α_{i} b_{i 2}, \sum_{i = 1}^{n} α_{i} b_{i 3}, \sum_{i = 1}^{n} α_{i} b_{i 4}); \\ 1 - \prod_{i = 1}^{n} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{n} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{n} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{n} α_{i} e_{i 1}, \sum_{i = 1}^{n} α_{i} e_{i 2}, \sum_{i = 1}^{n} α_{i} e_{i 3}, \sum_{i = 1}^{n} α_{i} e_{i 4}); \\ 1 - \prod_{i = 1}^{n} (1 - {\tilde{T}}_{Mi})^{α_{i}}, \prod_{i = 1}^{n} {\tilde{I}}_{Mi}^{α_{i}}, \prod_{i = 1}^{n} {\tilde{F}}_{Mi}^{α_{i}} \end{matrix} 〉 \end{matrix}) \end{matrix} .$ (11)

If m = n+1, based on the operational laws of C-TFNVs and Equations (10) and (11), there exists the following result:

$\begin{matrix} C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{n}, {\tilde{z}}_{n + 1}) = \sum_{i = 1}^{n + 1} α_{i} {\tilde{z}}_{i} \\ = \sum_{i = 1}^{n} α_{i} {\tilde{z}}_{i} \oplus α_{n + 1} {\tilde{z}}_{n + 1} \\ = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{n + 1} α_{i} b_{i 1}, \sum_{i = 1}^{n + 1} α_{i} b_{i 2}, \sum_{i = 1}^{n + 1} α_{i} b_{i 3}, \sum_{i = 1}^{n + 1} α_{i} b_{i 4}); \\ 1 - \prod_{i = 1}^{n} (1 - {\tilde{T}}_{Ri})^{α_{i}} + 1 - (1 - {\tilde{T}}_{Rn + 1})^{α_{n + 1}} \\ - (1 - \prod_{i = 1}^{n} (1 - {\tilde{T}}_{Ri})^{α_{i}}) (1 - (1 - {\tilde{T}}_{Rn + 1})^{α_{n + 1}}), \\ \prod_{i = 1}^{n + 1} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{n + 1} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{n + 1} α_{i} e_{i 1}, \sum_{i = 1}^{n + 1} α_{i} e_{i 2}, \sum_{i = 1}^{n + 1} α_{i} e_{i 3}, \sum_{i = 1}^{n + 1} α_{i} e_{i 4}); \\ 1 - \prod_{i = 1}^{n} (1 - {\tilde{T}}_{Mi})^{α_{i}} + 1 - (1 - {\tilde{T}}_{Mn + 1})^{α_{n + 1}} \\ - (1 - \prod_{i = 1}^{n} (1 - {\tilde{T}}_{Mi})^{α_{i}}) (1 - (1 - {\tilde{T}}_{Mn + 1})^{α_{n + 1}}), \\ \prod_{i = 1}^{n + 1} {\tilde{I}}_{Mi}^{α_{i}}, \prod_{i = 1}^{n + 1} {\tilde{F}}_{Mi}^{α_{i}} \end{matrix} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{n + 1} α_{i} b_{i 1}, \sum_{i = 1}^{n + 1} α_{i} b_{i 2}, \sum_{i = 1}^{n + 1} α_{i} b_{i 3}, \sum_{i = 1}^{n + 1} α_{i} b_{i 4}); \\ 1 - \prod_{i = 1}^{n + 1} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{n + 1} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{n + 1} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{n + 1} α_{i} e_{i 1}, \sum_{i = 1}^{n + 1} α_{i} e_{i 2}, \sum_{i = 1}^{n + 1} α_{i} e_{i 3}, \sum_{i = 1}^{n + 1} α_{i} e_{i 4}); \\ 1 - \prod_{i = 1}^{n + 1} (1 - {\tilde{T}}_{Mi})^{α_{i}}, \prod_{i = 1}^{n + 1} {\tilde{I}}_{Mi}^{α_{i}}, \prod_{i = 1}^{n + 1} {\tilde{F}}_{Mi}^{α_{i}} \end{matrix} 〉 \end{matrix}) . \end{matrix}$ (12)

Corresponding to the above results, Equation (9) is true for any m.

It is obvious that the C-TFNVWAA operator implies the following properties:

Idempotency: Set ${\tilde{z}}_{i}$ (i = 1, 2,..., m) as a group of C-TFNVs. If ${\tilde{z}}_{i}$ = $\tilde{z}$ for i = 1, 2,..., m, then there exists $C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \tilde{z}$ .

Boundedness: Set ${\tilde{z}}_{i}$ (i = 1, 2,..., m) as a group of C-TFNVs and let the minimum and maximum C-TFNVs:

${\tilde{z}}^{L} = (\begin{matrix} 〈 \begin{matrix} (min_{i} (b_{i 1}), min_{i} (b_{i 2}), min_{i} (b_{i 3}), min_{i} (b_{i 4})); \\ min_{i} ({\tilde{T}}_{Ri}), max_{i} ({\tilde{I}}_{Ri}), max_{i} ({\tilde{F}}_{Ri}) \end{matrix} 〉, \\ 〈 \begin{matrix} (min_{i} (e_{i 1}), min_{i} (e_{i 2}), min_{i} (e_{i 3}), min_{i} (e_{i 4})); \\ min_{i} ({\tilde{T}}_{Mi}), max_{i} ({\tilde{I}}_{Mi}), max_{i} ({\tilde{F}}_{Mi}) \end{matrix} 〉 \end{matrix}),$ ${\tilde{z}}^{U} = (\begin{matrix} 〈 \begin{matrix} (max_{i} (b_{i 1}), max_{i} (b_{i 2}), max_{i} (b_{i 3}), max_{i} (b_{i 4})); \\ max_{i} ({\tilde{T}}_{Ri}), min_{i} ({\tilde{I}}_{Ri}), min_{i} ({\tilde{F}}_{Ri}) \end{matrix} 〉, \\ 〈 \begin{matrix} (max_{i} (e_{i 1}), max_{i} (e_{i 2}), max_{i} (e_{i 3}), max_{i} (e_{i 4})); \\ max_{i} ({\tilde{T}}_{Mi}), min_{i} ({\tilde{I}}_{Mi}), min_{i} ({\tilde{F}}_{Mi}) \end{matrix} 〉 \end{matrix}) .$

Then ${\tilde{z}}^{L} \leq C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m})$ $\leq {\tilde{z}}^{U}$ exists.

Monotonicity: Set ${\tilde{z}}_{i}$ (i = 1, 2,..., m) as a group of C-TFNVs. If ${\tilde{z}}_{i}$ ≤ ${\tilde{z}}_{i}^{*}$ for i = 1, 2,..., m, then $C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) \leq C - TFNVWAA ({\tilde{z}}_{1}^{*}, {\tilde{z}}_{2}^{*}, . . ., {\tilde{z}}_{m}^{*})$ exists.

Proof. (i) Owing to ${\tilde{z}}_{i} = \tilde{z}$ for i = 1, 2,..., m, there is the following result: $\begin{matrix} C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \sum_{i = 1}^{m} α_{i} {\tilde{z}}_{i} \\ = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} b_{i 1}, \sum_{i = 1}^{m} α_{i} b_{i 2}, \sum_{i = 1}^{m} α_{i} b_{i 3}, \sum_{i = 1}^{m} α_{i} b_{i 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Ri})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Ri}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Ri}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} e_{i 1}, \sum_{i = 1}^{m} α_{i} e_{i 2}, \sum_{i = 1}^{m} α_{i} e_{i 3}, \sum_{i = 1}^{m} α_{i} e_{i 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Mi})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Mi}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Mi}^{α_{i}} \end{matrix} 〉 \end{matrix}) \end{matrix}$

$\begin{matrix} = (\begin{matrix} 〈 \begin{matrix} (b_{1} \sum_{i = 1}^{m} α_{i}, b_{2} \sum_{i = 1}^{m} α_{i}, b_{3} \sum_{i = 1}^{m} α_{i}, b_{4} \sum_{i = 1}^{m} α_{i}); \\ 1 - (1 - {\tilde{T}}_{R})^{\sum_{i = 1}^{m} α_{i}}, {\tilde{I}}_{R}^{\sum_{i = 1}^{m} α_{i}}, {\tilde{F}}_{R}^{\sum_{i = 1}^{m} α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (e_{1} \sum_{i = 1}^{m} α_{i}, e_{2} \sum_{i = 1}^{m} α_{i}, e_{3} \sum_{i = 1}^{m} α_{i}, e_{4} \sum_{i = 1}^{m} α_{i}); \\ 1 - (1 - {\tilde{T}}_{M})^{\sum_{i = 1}^{m} α_{i}}, {\tilde{I}}_{M}^{\sum_{i = 1}^{m} α_{i}}, {\tilde{F}}_{M}^{\sum_{i = 1}^{m} α_{i}} \end{matrix} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 (b_{1}, b_{2}, b_{3}, b_{4}); 1 - (1 - {\tilde{T}}_{R}), {\tilde{I}}_{R}, {\tilde{F}}_{R} 〉, \\ 〈 (e_{1}, e_{2}, e_{3}, e_{4}); 1 - (1 - {\tilde{T}}_{M}), {\tilde{I}}_{M}, {\tilde{F}}_{M} 〉 \end{matrix}) = \tilde{z} . \end{matrix}$

(ii) Since ${\tilde{z}}^{L}$ and ${\tilde{z}}^{U}$ are the minimum C-TFNV and the maximum C-TFNV respectively, there exists ${\tilde{z}}^{L}$ ≤ ${\tilde{z}}_{i}$ ≤ ${\tilde{z}}^{U}$ . Thus, there is $\sum_{i = 1}^{m} α_{i} {\tilde{z}}^{L} \leq \sum_{i = 1}^{m} {α_{i} \tilde{z}}_{i} \leq \sum_{i = 1}^{m} α_{i} {\tilde{z}}^{U}$ . Based on the property (i), there is ${\tilde{z}}^{L}$ ≤ $\sum_{i = 1}^{m} α_{i} {\tilde{z}}_{i}$ ≤ ${\tilde{z}}^{U}$ , namely ${\tilde{z}}^{L} \leq C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) \leq {\tilde{z}}^{U}$ .

(iii) Due to ${\tilde{z}}_{i}$ ≤ ${\tilde{z}}_{i}^{*}$ for i = 1, 2,..., m, there exists $\sum_{i = 1}^{m} α_{i} {\tilde{z}}_{i} \leq \sum_{i = 1}^{m} α_{i} {\tilde{z}}_{i}^{*}$ , i.e., $C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m})$ $\leq C - TFNVWAA ({\tilde{z}}_{1}^{*}, {\tilde{z}}_{2}^{*}, . . ., {\tilde{z}}_{m}^{*})$ .

Thus, we finish the proof of these properties.

Example 2. Set two C-TFNVs as ${\tilde{z}}_{1} = (\begin{matrix} 〈 (0.4, 0.5, 0.6, 0.7); 0.5, 0.2, 0.2 〉, \\ 〈 (0.5, 0.6, 0.7, 0.8); 0.7, 0.1, 0.3 〉 \end{matrix})$ and ${\tilde{z}}_{2} =$ $(\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.6, 0.1, 0.1 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.7, 0.2, 0.3 〉 \end{matrix})$ with their weight values 0.6 and 0.4. Then, the aggregated value of Equation (9) is obtained by the following calculation: $\begin{matrix} C - TFNVWAA ({\tilde{z}}_{1}, {\tilde{z}}_{2}) = \sum_{i = 1}^{2} α_{i} {\tilde{z}}_{i} = α_{1} {\tilde{z}}_{1} \oplus α_{2} {\tilde{z}}_{2} \end{matrix}$ $\begin{matrix} = (\begin{matrix} 〈 \begin{matrix} (\begin{matrix} 0.4 \times 0.6 + 0.3 \times 0.4, 0.5 \times 0.6 + 0.4 \times 0.4, \\ 0.6 \times 0.6 + 0.5 \times 0.4, 0.7 \times 0.6 + 0.6 \times 0.4 \end{matrix}); \\ 1 - (1 - 0.5)^{0.6} (1 - 0.6)^{0.4}, 0 . 2^{0.6} \times 0 . 1^{0.4}, 0 . 2^{0.6} \times 0 . 1^{0.4} \end{matrix} 〉, \\ 〈 \begin{matrix} (\begin{matrix} 0.5 \times 0.6 + 0.4 \times 0.4, 0.6 \times 0.6 + 0.5 \times 0.4, \\ 0.7 \times 0.6 + 0.6 \times 0.4, 0.8 \times 0.6 + 0.7 \times 0.4 \end{matrix}); \\ 1 - (1 - 0.7)^{0.6} (1 - 0.7)^{0.4}, 0 . 1^{0.6} \times 0 . 2^{0.4}, 0 . 3^{0.6} \times 0 . 3^{0.4} \end{matrix} 〉 \end{matrix}) \\ = (\begin{matrix} < (0 . 36, 0 . 46, 0 . 56, 0 . 66); 0 . 5427, 0 . 1516, 0 . 1516 >, \\ < (0 . 46, 0 . 56, 0 . 66, 0 . 76); 0 . 7, 0 . 132, 0 . 3 > \end{matrix}) . \end{matrix}$

4.2 C-TFNVWGA operator

Definition 6. Set ${\tilde{z}}_{i} = (\begin{matrix} 〈 (b_{i 1}, b_{i 2}, b_{i 3}, b_{i 4}); {\tilde{T}}_{Ri}, \\ 〈 (e_{i 1}, e_{i 2}, e_{i 3}, e_{i 4}); {\tilde{T}}_{Mi}, \end{matrix}$ $\begin{matrix} {\tilde{I}}_{Ri}, {\tilde{F}}_{Ri} 〉, \\ {\tilde{I}}_{Mi}, {\tilde{F}}_{Mi} 〉 \end{matrix})$ (i = 1, 2,..., m) as a group of C-TFNVs with the weight α_i of ${\tilde{z}}_{i}$ for 0 ≤ α_i ≤ 1 and $\sum_{i = 1}^{m} α_{i} = 1$ . Then, a C-TFNVWGA operator is defined as $C - TFNVWGA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \prod_{i = 1}^{m} {\tilde{z}}_{i}^{α_{i}} .$ (13)

Similarly, the aggregated value of the C-TFNVWGA operator is also C-TFNV. Thus, the following theorem is presented corresponding to the operational laws of C-TFNVs and Equation (13).

Theorem 2. Set ${\tilde{z}}_{i} = (\begin{matrix} 〈 (b_{i 1}, b_{i 2}, b_{i 3}, b_{i 4}); {\tilde{T}}_{Ri}, \\ 〈 (e_{i 1}, e_{i 2}, e_{i 3}, e_{i 4}); {\tilde{T}}_{Mi}, \end{matrix}$ $\begin{matrix} {\tilde{I}}_{Ri}, {\tilde{F}}_{Ri} 〉, \\ {\tilde{I}}_{Mi}, {\tilde{F}}_{Mi} 〉 \end{matrix})$ (i = 1, 2, ... , m) as a group of C-TFNVs with the weight α_i of ${\tilde{z}}_{i}$ (i = 1, 2,..., m) for 0 ≤ α_i ≤ 1 and $\sum_{i = 1}^{m} α_{i} = 1$ . Based on the operational laws of C-TFNVs and Equation (13), we can obtain the following C-TFNVWGA operator:

$\begin{matrix} C - TFNVWGA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \prod_{i = 1}^{m} {\tilde{z}}_{i}^{α_{i}} \\ = (\begin{matrix} 〈 \begin{matrix} (\prod_{i = 1}^{m} b_{i 1}^{α_{i}}, \prod_{i = 1}^{m} b_{i 2}^{α_{i}}, \prod_{i = 1}^{m} b_{i 3}^{α_{i}}, \prod_{i = 1}^{m} b_{i 4}^{α_{i}}); \\ \prod_{i = 1}^{m} {\tilde{T}}_{Ri}^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{I}}_{Ri})^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{F}}_{Ri})^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\prod_{i = 1}^{m} e_{i 1}^{α_{i}}, \prod_{i = 1}^{m} e_{i 2}^{α_{i}}, \prod_{i = 1}^{m} e_{i 3}^{α_{i}}, \prod_{i = 1}^{m} e_{i 4}^{α_{i}}); \\ \prod_{i = 1}^{m} {\tilde{T}}_{Mi}^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{I}}_{Mi})^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{F}}_{Mi})^{α_{i}} \end{matrix} 〉 \end{matrix}) \end{matrix} .$ (14)

By the similar proof way of Theorem 1, Theorem 2 can be also verified based on mathematical induction, which is omitted here.

It is obvious that the C-TFNVWGA operator also implies the following properties:

Idempotency: Set ${\tilde{z}}_{i}$ (i = 1, 2,..., m) as a group of C-TFNVs. If ${\tilde{z}}_{i}$ = $\tilde{z}$ for i = 1, 2,..., m, then there exists $C - TFNVWGA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) = \tilde{z}$ .

Boundedness: Set ${\tilde{z}}_{i}$ (i = 1, 2,..., m) as a group of C-TFNVs and let the minimum and maximum C-TFNVs: ${\tilde{z}}^{L} = (\begin{matrix} 〈 \begin{matrix} (min_{i} (b_{i 1}), min_{i} (b_{i 2}), min_{i} (b_{i 3}), min_{i} (b_{i 4})); \\ min_{i} ({\tilde{T}}_{Ri}), max_{i} ({\tilde{I}}_{Ri}), max_{i} ({\tilde{F}}_{Ri}) \end{matrix} 〉, \\ 〈 \begin{matrix} (min_{i} (e_{i 1}), min_{i} (e_{i 2}), min_{i} (e_{i 3}), min_{i} (e_{i 4})); \\ min_{i} ({\tilde{T}}_{Mi}), max_{i} ({\tilde{I}}_{Mi}), max_{i} ({\tilde{F}}_{Mi}) \end{matrix} 〉 \end{matrix}),$ ${\tilde{z}}^{U} = (\begin{matrix} 〈 \begin{matrix} (max_{i} (b_{i 1}), max_{i} (b_{i 2}), max_{i} (b_{i 3}), max_{i} (b_{i 4})); \\ max_{i} ({\tilde{T}}_{Ri}), min_{i} ({\tilde{I}}_{Ri}), min_{i} ({\tilde{F}}_{Ri}) \end{matrix} 〉, \\ 〈 \begin{matrix} (max_{i} (e_{i 1}), max_{i} (e_{i 2}), max_{i} (e_{i 3}), max_{i} (e_{i 4})); \\ max_{i} ({\tilde{T}}_{Mi}), min_{i} ({\tilde{I}}_{Mi}), min_{i} ({\tilde{F}}_{Mi}) \end{matrix} 〉 \end{matrix}) .$

Then ${\tilde{z}}^{L} \leq C - TFNVWGA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . .,$ ${\tilde{z}}_{m}) \leq {\tilde{z}}^{U}$ exists.

Monotonicity: Set ${\tilde{z}}_{i}$ (i = 1, 2,..., m) as a group of C-TFNVs. If ${\tilde{z}}_{i}$ ≤ ${\tilde{z}}_{i}^{*}$ for i = 1, 2,..., m, then $C - TFNVWGA ({\tilde{z}}_{1}, {\tilde{z}}_{2}, . . ., {\tilde{z}}_{m}) \leq C - TFNVWGA ({\tilde{z}}_{1}^{*}, {\tilde{z}}_{2}^{*}, . . ., {\tilde{z}}_{m}^{*})$ exists.

By the similar proof way of the properties of the C-TFNVWAA operator, the above properties can be also verified, which is omitted here.

Example 3. Set two C-TFNVs as ${\tilde{z}}_{1} = (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉, \\ 〈 (0.6, 0.7, 0.8, 0.9); 0.8, 0.2, 0.1 〉 \end{matrix})$ and ${\tilde{z}}_{2} =$ $(\begin{matrix} 〈 (0.4, 0.5, 0.6, 0.7); 0.7, 0.2, 0.1 〉, \\ 〈 (0.2, 0.3, 0.4, 0.5); 0.8, 0.1, 0.2 〉 \end{matrix})$ with their weight values 0.3 and 0.7. Then, the aggregated value of Equation (14) is obtained by the following calculation: $\begin{matrix} C - TFNVWGA ({\tilde{z}}_{1}, {\tilde{z}}_{2}) = \prod_{i = 1}^{2} {\tilde{z}}_{i}^{α_{i}} = {\tilde{z}}_{1}^{α_{1}} \otimes {\tilde{z}}_{1}^{α_{2}} \\ = (\begin{matrix} 〈 \begin{matrix} (0 . 3^{0.3} \times 0 . 4^{0.7}, 0 . 4^{0.3} \times 0 . 5^{0.7}, \\ 0 . 5^{0.3} \times 0 . 6^{0.7}, 0 . 6^{0.3} \times 0 . 7^{0.7}); \\ 0 . 7^{0.3} \times 0 . 7^{0.7}, 1 - (1 - 0.1)^{0.3} \times (1 - 0.2)^{0.7}, \\ 1 - (1 - 0.1)^{0.3} \times (1 - 0.1)^{0.7} \end{matrix} 〉, \\ 〈 \begin{matrix} (0 . 6^{0.3} \times 0 . 2^{0.7}, 0 . 7^{0.3} \times 0 . 3^{0.7}, \\ 0 . 8^{0.3} \times 0 . 4^{0.7}, 0 . 9^{0.3} \times 0 . 5^{0.7}); \\ 0 . 8^{0.3} \times 0 . 8^{0.7}, 1 - (1 - 0.2)^{0.3} \times (1 - 0.1)^{0.7}, \\ 1 - (1 - 0.1)^{0.3} \times (1 - 0.2)^{0.7} \end{matrix} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 \begin{matrix} (0 . 3669, 0 . 4676, 0 . 5681, 0 . 6684); \\ 0 . 7, 0 . 1712, 0 . 1 \end{matrix} 〉, \\ 〈 \begin{matrix} (0 . 2718, 0 . 3868, 0 . 4925, 0 . 5964); \\ 0 . 8, 0 . 1312, 0 . 1712 \end{matrix} 〉 \end{matrix}) . \end{matrix}$

5 Multicriteria DM model in the setting of C-TFNVs

In this section, a multicriteria DM model is established based on the C-TFNVWAA and C-TFNVWGA operators and the score function to carry out multicriteria DM problems in the environment of C-TFNVs.

Regarding a multicriteria DM problem, there is normally a set of alternatives Q = {q₁, q₂, … , q_n}, which satisfies the requirements of a set of criteria U = {u₁, u₂, …, u_m}. The weight vector of the criteria is α = (α₁, α₂, . . . , α_m) for specifying the importance of various criteria. Then, decision makers can give their satisfactory assessments of the alternatives over the criteria by the linguistic values of C-TFNVs from the

predefined linguistic term set S = {Very low, Low, Fairly low, Medium, Fairly high, High, Very high}. In the assessment process, the decision makers can assign the linguistic term values of C-TFNVs obtained from the linguistic terms of Table 1 to indicate the satisfactory evaluation values and credibility levels of alternatives over the criteria. Thus, the evaluation information of each alternative q_j (j = 1, 2, ... , n) over the criteria u_i (i = 1, 2, ... , m) is expressed by a form of the specifical C-TFNV: ${\tilde{z}}_{ji} = (\begin{matrix} 〈 (b_{ji 1}, b_{ji 2}, b_{ji 3}, b_{ji 4}); {\tilde{T}}_{Rji}, {\tilde{I}}_{Rji}, {\tilde{F}}_{Rji} 〉, \\ 〈 (e_{ji 1}, e_{ji 2}, e_{ji 3}, e_{ji 4}); {\tilde{T}}_{Mji}, {\tilde{I}}_{Mji}, {\tilde{F}}_{Mji} 〉 \end{matrix}) .$

Table 1
Linguistic term values of TFNVs corresponding to the linguistic terms [12]

Linguistic term Linguistic term value of TFNVs

Very low 〈(0.1, 0.1, 0.1, 0.1) ;0.5, 0.3, 0.3〉

Low 〈(0.2, 0.3, 0.4, 0.5) ;0.6, 0.2, 0.2〉

Fairly low 〈(0.3, 0.4, 0.5, 0.6) ;0.7, 0.1, 0.1〉

Medium 〈(0.4, 0.5, 0.6, 0.7) ;0.8, 0.0, 0.1〉

Fairly high 〈(0.5, 0.6, 0.7, 0.8) ;0.8, 0.1, 0.1〉

High 〈(0.7, 0.8, 0.9, 1.0) ;0.8, 0.2, 0.2〉

Very high 〈(1.0, 1.0, 1.0, 1.0) ;0.9, 0.1, 0.1〉

Linguistic term	Linguistic term value of TFNVs
Very low	〈(0.1, 0.1, 0.1, 0.1) ;0.5, 0.3, 0.3〉
Low	〈(0.2, 0.3, 0.4, 0.5) ;0.6, 0.2, 0.2〉
Fairly low	〈(0.3, 0.4, 0.5, 0.6) ;0.7, 0.1, 0.1〉
Medium	〈(0.4, 0.5, 0.6, 0.7) ;0.8, 0.0, 0.1〉
Fairly high	〈(0.5, 0.6, 0.7, 0.8) ;0.8, 0.1, 0.1〉
High	〈(0.7, 0.8, 0.9, 1.0) ;0.8, 0.2, 0.2〉
Very high	〈(1.0, 1.0, 1.0, 1.0) ;0.9, 0.1, 0.1〉

Thus, we can give the C-TFNV decision matrix $Z = ({\tilde{z}}_{ji})_{n \times m}$ .

Then, we propose a multicriteria DM model using the C-TFNVWAA or C-TFNVWGA operator and the score function to perform multicriteria DM problems with C-TFNV information and give the following decision procedures:

Step 1: The aggregated C-TFNVs ${\tilde{z}}_{j}$ for q_j (j = 1, 2, … , n) are obtained by the C-TFNVWAA operator or the C-TFNVWGA operator: $\begin{matrix} {\tilde{z}}_{j} = C - TFNVWAA ({\tilde{z}}_{j 1}, {\tilde{z}}_{j 2}, . . ., {\tilde{z}}_{jm}) = \sum_{i = 1}^{m} α_{i} {\tilde{z}}_{ji} \end{matrix}$

$\begin{matrix} = (\begin{matrix} 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} b_{ji 1}, \sum_{i = 1}^{m} α_{i} b_{ji 2}, \sum_{i = 1}^{m} α_{i} b_{ji 3}, \sum_{i = 1}^{m} α_{i} b_{ji 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Rji})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Rji}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Rji}^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\sum_{i = 1}^{m} α_{i} e_{ji 1}, \sum_{i = 1}^{m} α_{i} e_{ji 2}, \sum_{i = 1}^{m} α_{i} e_{ji 3}, \sum_{i = 1}^{m} α_{i} e_{ji 4}); \\ 1 - \prod_{i = 1}^{m} (1 - {\tilde{T}}_{Mji})^{α_{i}}, \prod_{i = 1}^{m} {\tilde{I}}_{Mji}^{α_{i}}, \prod_{i = 1}^{m} {\tilde{F}}_{Mji}^{α_{i}} \end{matrix} 〉 \end{matrix}) \end{matrix}$ (15)

$\begin{matrix} {\tilde{z}}_{j} = C - TFNVWGA ({\tilde{z}}_{j 1}, {\tilde{z}}_{j 2}, . . ., {\tilde{z}}_{jm}) = \prod_{i = 1}^{m} {\tilde{z}}_{ji}^{α_{i}} \\ = (\begin{matrix} 〈 \begin{matrix} (\prod_{i = 1}^{m} b_{ji 1}^{α_{i}}, \prod_{i = 1}^{m} b_{ji 2}^{α_{i}}, \prod_{i = 1}^{m} b_{ji 3}^{α_{i}}, \prod_{i = 1}^{m} b_{ji 4}^{α_{i}}); \\ \prod_{i = 1}^{m} {\tilde{T}}_{Rji}^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{I}}_{Rji})^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{F}}_{Rji})^{α_{i}} \end{matrix} 〉, \\ 〈 \begin{matrix} (\prod_{i = 1}^{m} e_{ji 1}^{α_{i}}, \prod_{i = 1}^{m} e_{ji 2}^{α_{i}}, \prod_{i = 1}^{m} e_{ji 3}^{α_{i}}, \prod_{i = 1}^{m} e_{ji 4}^{α_{i}}); \\ \prod_{i = 1}^{m} {\tilde{T}}_{Mji}^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{I}}_{Mji})^{α_{i}}, 1 - \prod_{i = 1}^{m} (1 - {\tilde{F}}_{Mji})^{α_{i}} \end{matrix} 〉 \end{matrix}) \end{matrix} .$ (16)

Step 2: The score values of $S ({\tilde{z}}_{j})$ (j = 1, 2, ... , n) are calculated by Equation (3).

Step 3: The alternatives are sorted in a descending order corresponding to the score values, then the best one is chosen.

Step 4: End.

6 Actual example and related comparison

6.1 Actual example

This section provides an actual example of choosing an optimal slope design scheme for an open pit mine to show the applicability and efficiency of the proposed DM model in the CTFNV setting.

A mining developer/investor wants to choose an optimal slope design scheme for an open pit mine to ensure that miners will not cause accidents due to slope collapse during the mining process. Designers/experts preliminarily provide a set of four potential design schemes (alternatives) Q = {q₁, q₂, q₃, q₄}. Then, the assessment of the alternatives must satisfy the requirements of three criteria:

the economy (cost and construction period) (u₁); the technology (technological capability, safety, and reliability) (u₂), and the environment (climate condition and geographical location) (u₃). The importance of the three criteria is specified by the weigh vector α=(0.35, 0.25, 0.4). Then, the experts indicate that the satisfactory assessments and credibility levels of the four potential alternatives over the three criteria are specified by the linguistic term values of the TFNVs obtained from the linguistic terms of Table 1. Thus, each ordered pair of TFNVs corresponding to the satisfactory assessment and credibility level can form C-TFNV, then all C-TFNVs are constructed as the following C-TFNV decision matrix Z:

$Z = [\begin{matrix} (\begin{matrix} 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉, \\ 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉, \\ 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉 \end{matrix}) \\ (\begin{matrix} 〈 (0.2, 0.3, 0.4, 0.5); 0.6, 0.2, 0.2 〉, \\ 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉, \\ 〈 (0.7, 0.8, 0.9, 1.0); 0.8, 0.2, 0.2 〉 \end{matrix}) & (\begin{matrix} 〈 (0.2, 0.3, 0.4, 0.5); 0.6, 0.2, 0.2 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉 \end{matrix}) \\ (\begin{matrix} 〈 (0.7, 0.8, 0.9, 1.0); 0.8, 0.2, 0.2 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉, \\ 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 \end{matrix}) \\ (\begin{matrix} 〈 (0.2, 0.3, 0.4, 0.5); 0.6, 0.2, 0.2 〉, \\ 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉, \\ 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉 \end{matrix}) & (\begin{matrix} 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉, \\ 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 \end{matrix}) \end{matrix}] .$

Hence, the established DM model can be utilized for this multicriteria DM problem and depicted by the following computational steps:

Step 1: The C-TFNVWAA operator of Equation (15) is utilized to yield the aggregated values ${\tilde{z}}_{j}$ for q_j (j = 1, 2, 3, 4):

= (< (0.385, 0.485, 0.585, 0.685); 0.7648, 0, 0.1>,< (0.46, 0.56, 0.66, 0.76); 0.8, 0, 0.1>),

= (< (0.225, 0.325, 0.425, 0.525); 0.6278, 0.1682, 0.1682>,< (0.51, 0.61, 0.71, 0.81); 0.8, 0, 0.1189>),

= (< (0.49, 0.59, 0.69, 0.79), 0.7648, 0.1275, 0.1275>,< (0.36, 0.46, 0.56, 0.66); 0.7648, 0.0, 0.1>),

= (< (0.265, 0.365, 0.465, 0.565); 0.6682, 0.1275, 0.1275>,< (0.385, 0.485, 0.585, 0.685); 0.7648, 0, 0.1>).

Or the C-TFNVWGA operator of Equation (16) is utilized to yield the aggregated values ${\tilde{z}}_{j}$ for q_j (j = 1, 2, 3, 4):

= (< (0.3770, 0.4786, 0.5797, 0.6805); 0.7584, 0.0662, 0.1>,< (0.4573, 0.5578, 0.6581, 0.7584); 0.8, 0.0613, 0.1>),

= (< (0.2213, 0.3224, 0.4229, 0.5233); 0.6236, 0.1761, 0.1761>,< (0.4974, 0.5994, 0.7008, 0.8019); 0.8, 0.0885, 0.1261>),

= (< (0.4585, 0.5642, 0.6681, 0.771); 0.7584, 0.1363, 0.1363>,< (0.3565, 0.4573, 0.5578, 0.6581); 0.7584, 0.0413, 0.1>),

= (< (0.2603, 0.3617, 0.4624, 0.5629); 0.6632, 0.1363, 0.1363>,< (0.377, 0.4786, 0.5797, 0.6805); 0.7584, 0.0662, 0.1>).

Step 2: The score values of $S ({\tilde{z}}_{j})$ (j = 1, 2, 3, 4) are calculated by Equation (3) below:

$S ({\tilde{z}}_{1}) = 0.2609$ , $S ({\tilde{z}}_{2}) = 0.1689$ , $S ({\tilde{z}}_{3}) = 0.2426$ , and $S ({\tilde{z}}_{4}) = 0.1586$ .

Or $S ({\tilde{z}}_{1}) = 0.2444$ , $S ({\tilde{z}}_{2}) = 0.158$ , $S ({\tilde{z}}_{3}) = 0.2257$ , and $S ({\tilde{z}}_{4}) = 0.15$ .

Step 3: The four alternatives are sorted as q₁ > q₃ > q₂ > q₄ based on the above score values. Thus, the optimal one is q₁. Obviously, the final decision results based on the C-TFNVWGA operator and the C-TFNVWGA operator are identical.

6.2 Comparison of the proposed DM Model with existing related DM models in the situations of TFNVs and NVs

Since this study proposes the C-TFNV notion and the DM model with C-TFNVs for the first time, it is difficult to directly compare the proposed DM model with existing DM models in the environment of C-TFNVs. To conveniently compare the proposed DM model with the relevant DM models with TFNVs [12] and NVs [3], suppose that we only consider TFNVs and NVs, respectively, in the C-TFNV decision matrix Z as the two special cases of C-TFNVs. Then, the C-TFNV decision matrix Z is reduced to the TFNV and NV decision matrices, respectively: $Z^{'} = [\begin{matrix} 〈 (0.4, 0.5, 0.6, 0.7); 0.8, 0.0, 0.1 〉 & 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉 & 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 \\ 〈 (0.2, 0.3, 0.4, 0.5); 0.6, 0.2, 0.2 〉 & 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 & 〈 (0.2, 0.3, 0.4, 0.5); 0.6, 0.2, 0.2 〉 \\ 〈 (0.7, 0.8, 0.9, 1.0); 0.8, 0.2, 0.2 〉 & 〈 (0.5, 0.6, 0.7, 0.8); 0.8, 0.1, 0.1 〉 & 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 \\ 〈 (0.2, 0.3, 0.4, 0.5); 0.6, 0.2, 0.2 〉 & 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 & 〈 (0.3, 0.4, 0.5, 0.6); 0.7, 0.1, 0.1 〉 \end{matrix}],$ $Z^{″} = [\begin{matrix} 〈 0.8, 0.0, 0.1 〉 & 〈 0.8, 0.1, 0.1 〉 & 〈 0.7, 0.1, 0.1 〉 \\ 〈 0.6, 0.2, 0.2 〉 & 〈 0.7, 0.1, 0.1 〉 & 〈 0.6, 0.2, 0.2 〉 \\ 〈 0.8, 0.2, 0.2 〉 & 〈 0.8, 0.1, 0.1 〉 & 〈 0.7, 0.1, 0.1 〉 \\ 〈 0.6, 0.2, 0.2 〉 & 〈 0.7, 0.1, 0.1 〉 & 〈 0.7, 0.1, 0.1 〉 \end{matrix}] .$

In the environment of TFNVs, the decision results can be obtained by the existing DM model using the TFNVWAA and TFNVWGA operators of Equations (1) and (2) and the score function of Equation (3) introduced in [12]. Then in the environment of NVs, the decision results can be obtained by the existing DM model using the SVNNWAA and SVNNWGA operators of Equations (4) and (5) and the score function of Equation (6) introduced in [3]. Thus, all the DM results between the existing DM models [3, 12] and the proposed DM model in the environments of TFNVs, NVs, and C-TFNVs are shown in Table 2 for a clear comparison.

Table 2
DM results between the existing DM models [3, 12] and the proposed DM model in the environments of TFNVs, NVs, and C-TFNVs

Aggregation operator Score value Sorting Optimal one

DM model using TFNVWAA [12] 0.4752, 0.2864, 0.5354, 0.3338 q₃ > q₁ > q₄ > q₂ q ₃

DM model using TFNVWGA [12] 0.4570, 0.2820, 0.5099, 0.3282 q₃ > q₁ > q₄ > q₂ q ₃

DM model using NVWAA [3] 0.8883, 0.7638, 0.8366, 0.8044 q₁ > q₃ > q₄ > q₂ q ₁

DM model using NVWAA [3] 0.8641, 0.7571, 0.8286, 0.7968 q₁ > q₃ > q₄ > q₂ q ₁

DM model using C-TFNVWAA 0.2609, 0.1689, 0.2426, 0.1586 q₁ > q₃ > q₂ > q₄ q ₁

DM model using C-TFNVWGA 0.2444, 0.1580, 0.2257, 0.1500 q₁ > q₃ > q₂ > q₄ q ₁

Aggregation operator	Score value	Sorting	Optimal one
DM model using TFNVWAA [12]	0.4752, 0.2864, 0.5354, 0.3338	q₃ > q₁ > q₄ > q₂	q ₃
DM model using TFNVWGA [12]	0.4570, 0.2820, 0.5099, 0.3282	q₃ > q₁ > q₄ > q₂	q ₃
DM model using NVWAA [3]	0.8883, 0.7638, 0.8366, 0.8044	q₁ > q₃ > q₄ > q₂	q ₁
DM model using NVWAA [3]	0.8641, 0.7571, 0.8286, 0.7968	q₁ > q₃ > q₄ > q₂	q ₁
DM model using C-TFNVWAA	0.2609, 0.1689, 0.2426, 0.1586	q₁ > q₃ > q₂ > q₄	q ₁
DM model using C-TFNVWGA	0.2444, 0.1580, 0.2257, 0.1500	q₁ > q₃ > q₂ > q₄	q ₁

From the ranking results in Table 2, it can be seen that the existing DM models [3, 12] and the proposed DM model have differences in the ranking orders given in the cases of NVs, TFNVs and C-TFNVs. Then, the optimal alternative is q₁ in the cases of NVs and C-TFNVs and the optimal alternative is q₃ in the case of TFNVs. Obviously, the DM models with and without considering credibility levels can affect the sorting orders of alternatives in the DM example, which reveal the importance of the credibility levels in the DM problem. Therefore, the proposed DM model can demonstrate its efficiency and rationality in the environment of C-TFNVs because the presented DM information can strengthen the credibility levels related to the assessment values of TFNVs in the DM process in the environment of C-TFNVs; while the existing DM model with TFNVs or NVs loses the information of credibility levels [12] or the information of both TFNs and credibility levels [3] so as to cause the distortion/irrationality of the decision results. Then, the DM model proposed in the article is a further generalization of the existing DM models with NVs and TFNVs [3, 12]; while the existing DM models with NVs [3] or TFNVs [12] is only a special case of the proposed DM model with C-TFNVs. Clearly, the superiority of the proposed DM model with C-TFNVs compared to the existing model with NVs or TFNVs is unquestionable. In general, the proposed DM model shows its major advantages in terms of DM information representation and methodology in the environment of C-TFNVs and makes the DM results more credible and reasonable.

Furthermore, C-TFNV and TNZN differ in the concept and information expression because C-TFNV contains much more information (the hybrid information of TFNV (containing TFN and NV) and its credibility level) than TNZN (the hybrid information of the true, false, and indeterminate TFNs and their reliability degrees). Therefore, the existing DM model with TNZN information [28] cannot also carry out such a DM problem with C-TFNVs. However, the DM model proposed in the article is a further generalization of the existing DM model with TNZN information [28].

7 Conclusion

To strengthen the credibility level of the TFNV information, this study proposed the C-TFNV notion by an ordered pair of TFNVs to reasonably represent the information of combining a TFNV argument with a credibility level related to the TFNV argument in the case of indeterminacy and inconsistence. Then, we presented some operational laws of C-TFNVs, the score function of C-TFNV, and the C-TFNVWAA and C-TFNVWGA operators so as to provide the necessary modeling tools for performing multicriteria DM problems with C-TFNV information. Next, a multicriteria DM model was presented using the C-TFNVWAA and C-TFNVWGA operators and the score function to perform multicriteria DM problems in the case of C-TFNVs. Furthermore, the proposed DM model was applied in an actual DM example of slope design schemes in the case of C-TFNVs to demonstrate the applicability and efficiency of the proposed DM model. Comparative analysis with the existing DM models with TFNV and NV information indicated that DM models with and without the credibility levels in DM problems can affect the sorting of alternatives, which indicated the efficiency and rationality of the proposed DM model. Meanwhile, the highlighting advantage of this study is that the proposed DM model in the case of C-TFNVs can enhance the credibility levels related to the assessment values of TFNVs and improve the rationality and efficiency in multicriteria DM problems. Since the first study was conducted in the C-TFNV setting, the scope of research and applications needs to be further extended to other fields. Regarding future work, we shall further propose various measure algorithms, such as similarity measures and correlation coefficients, and apply them to clustering analysis, pattern recognition, medical diagnosis, and image processing in the case of C-TFNVs.

Conflicts of interest

The authors declare no conflict of interest.

References

Smarandache

, Neutrosophy: neutrosophic probability, set, and logic, American Research Press, Rehoboth, USA (1998).

Liu

P.D.

and Wang

Y.M.

, Multiple attribute decision making method based on single-valued neutrosophic normalized weighted Bonferroni mean, Neural Computing and Applications 25(7-8) (2014), 2001–2010.

Peng

J.J.

, Wang

J.Q.

, Wang

, Zhang

H.Y.

and Chen

X.H.

, Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems, International Journal of Systems Science 47(10) (2016), 2342–2358.

Zhou

L.P.

, Dong

J.Y.

and Wan

S.P.

, Two new approaches for multi-attribute group decision-making with interval-valued neutrosophic Frank aggregation operators and incomplete weights, IEEE Access 7 (2019), 102727–102750.

Peng

and Dai

, A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017, Artificial Intelligent Review 53 (2020), 199–255.

Yörükoğlu

and Aydın

, Smart container evaluation by neutrosophic MCDM method, Journal of Intelligent & Fuzzy Systems 38(1) (2020), 723–733.

Hezam

I.M.

, Nayeem

M.K.

, Foul

and Alrasheedi

A.F.

, COVID-19 Vaccine: A neutrosophic MCDM approach for determining the priority groups, Results in Physics 20 (2021), 103654. https://doi.org/10.1016/j.rinp.2020.103654

Alia

, Son

L.H.

, Thanhc

N.D.

and Minh

N.V.

, A neutrosophic recommender system for medical diagnosis based on algebraic neutrosophic measures, , Applied Soft Computing 71 (2018), 1054–1071.

, Single valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine, Soft Computing 21(3) (2017), 817–825.

10.

Guo

, Sengur

and Ye

, A novel image thresholding algorithm based on neutrosophic similarity score, Measurement 58 (2014), 175–186.

11.

Jha

, Son

L.H.

, Kumar

, Priyadarshini

, Smarandache

and Long

H.V.

, Neutrosophic image segmentation with Dice coefficients, Measurement 134 (2019), 762–772.

12.

, Some weighted aggregation operators of trapezoidal neutrosophic numbers and their multiple attribute decision making method, Informatica 28(2) (2017), 387–402.

13.

Giri

B.C.

, Molla

M.U.

and Biswas

, TOPSIS method for MADM based on interval trapezoidal neutrosophic number, Neutrosophic Sets and Systems 22 (2018), 151–166.

14.

Saini

R.K.

, Sangal

and Manisha

, Application of single valued trapezoidal neutrosophic numbers in transportation problem, Neutrosophic Sets and Systems 35 (2020), 563–583.

15.

Broumi

, Lathamaheswari

, Tan

, Nagarajan

, Mohamed

, Smarandache

and Bakali

, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids, , Neutrosophic Sets and Systems 35 (2020), 478–500.

16.

Broumi

, Nagarajan

, Bakali

, Mohamed

, Smarandache

and Lathamaheswari

, The shortest path problem in interval valued trapezoidal and triangular neutrosophic environment, Complex & Intelligent Systems 5 (2019), 391–402.

17.

Yang

, Li

and Tan

, Shortest path solution of trapezoidal fuzzy neutrosophic graph based on circle-breaking algorithm, Symmetry 12(8) (2020), 1360. https://doi.org/10.3390/sym12081360

18.

Karamaşa

, Demir

, Memiş

and Korucuk

, Weighting the factors affecting logistics outsourcing, Decision Making: Applications in Management and Engineering 4(1) (2021), 19–32.

19.

Meng

, Wang

and Xu

, Triangular fuzzy neutrosophic preference relations and their application in enterprise resource planning software selection, Cognitive Computing 12 (2020), 261–295.

20.

Edalatpanah

S.A.

, A direct model for triangular neutrosophic linear programming, International Journal of Neutrosophic Science 1(1) (2020), 19–28.

21.

Yang

, Cai

, Edalatpanah

S.A.

and Smarandache

, Triangular single valued neutrosophic data envelopment analysis: application to hospital performance measurement, Symmetry 12(4) (2020), 588. https://doi.org/10.3390/sym12040588

22.

Fan

, Jia

and Wu

, Green supplier selection based on Dombi prioritized Bonferroni mean operator with single-valued triangular neutrosophic sets, International Journal of Computational Intelligence Systems 12(2) (2019), 1091–1101.

23.

Tan

and Zhang

, Multiple attribute decision making method based on DEMATEL and fuzzy distance of trapezoidal fuzzy neutrosophic numbers and its application in typhoon disaster evaluation, Journal of Intelligent & Fuzzy Systems 39(3) (2020), 3413–3439.

24.

Jana

, Pal

, Karaaslan

and Wang

J.Q.

, Trapezoidal neutrosophic aggregation operators and their application to the multi-attribute decision-making process, Scientia Iranica E 27(3) (2020), 1655–1673.

25.

, Song

J.M.

, Du

S.G.

and Yong

, Weighted aggregation operators of fuzzy credibility numbers and their decision-making approach for slope design schemes, Computational and Applied Mathematics 40 (2021), 155. https://doi.org/10.1007/s40314-021-01539-x

26.

, Du

S.G.

, Yong

and Zhang

F.W.

, Weighted aggregation operators of fuzzy credibility cubic numbers and their decision making strategy for slope design schemes, Current Chinese Computer Science 1(1) (2021), 28–34.

27.

, Du

S.G.

and Yong

, Similarity measures between intuitionistic fuzzy credibility sets and their multicriteria decision-making method for the performance evaluation of industrial robots, Mathematical Problems in Engineering 2021 (2021), Article ID 6630898. https://doi.org/10.1155/2021/6630898

28.

Yong

, Ye

and Du

S.G.

, Multicriteria decision making method and application in the setting of trapezoidal neutrosophic Z-numbers, Journal of Mathematics 2021 (2021), Article ID 6664330. https://doi.org/10.1155/2021/6664330.