Multiple attribute decision making method based on DEMATEL and fuzzy distance of trapezoidal fuzzy neutrosophic numbers and its application in typhoon disaster evaluation

Abstract

Trapezoidal fuzzy neutrosophic decision making plays an important role in decision-making processes with uncertain, indeterminate, and inconsistent information. In this paper, we propose a new multi-attribute decision-making method based on decision-making trial and evaluation laboratory (DEMATEL), fuzzy distance, and linear assignment method (LAM), and we express evaluation values as the trapezoidal fuzzy neutrosophic numbers (TrFNNs). First, attribute weights are obtained using the DEMATEL method and the new fuzzy distance of TrFNNs based on graded mean integration representation is defined. Then, alternatives are ranked using the LAM in operations research. In addition, we make two comparative analyses in the end to illustrate the feasibility and rationality of our method. Finally, an illustrative example about typhoon disaster assessment is presented to show the advantages of the proposed method.

Keywords

Multiple attribute decision making trapezoidal fuzzy neutrosophic set DEMATEL fuzzy distance LAM typhoon disaster evaluation

1 Introduction

Multi-attribute decision making (MADM) is an important part of modern decision science, and its theory and methods have been widely used in many fields, such as investment decisions, project evaluations, maintenance services, weapon system performance assessments, factory site selections, and disaster assessments [1, 2]. The principle of MADM is to use existing decision-making information to sort or select a limited set of alternatives in a certain way. However, decision-making problems have become increasingly complicated because of the increasing amount of decision information and alternatives, the inherent uncertainty and complexity of decision problems, and the fuzzy nature of human thinking [3]. In most cases, the decision information is fuzzy, incomplete, indeterminate, and inconsistent. Accordingly, mathematical models, such as fuzzy set (FS) [4], intuitionistic fuzzy set (IFS) [5], hesitant FS [6], picture FS [7], Pythagorean fuzzy set (PFS) [8, 9], have been used to model decision information. These sets use different mathematical representations to represent fuzzy information from different perspectives.

The latest and most general theory is neutrosophic set (NS) pioneered by Smarandache [10]. NSs are characterized by a truth-membership degree (T), an indeterminacy-membership degree (I), and a falsity-membership degree (F) and are a generalization of sets, such as crisp sets, FSs, interval-valued fuzzy sets (IvNSs), and IFSs. They are capable of dealing with incomplete, indeterminate, and inconsistent information. For example [11], in a voting process, 30% voted “yes,” 40% voted “no,” 10% gave up, and 20% were undecided. This situation is beyond the scope of FS and IFS but can be easily expressed by NS as <0.3, 0.2, 0.4>. In recent years, NS has become a research hotspot and has received extensive attention [12 –14]. For instance, Wang et al. [15] proposed single-valued neutrosophic sets (SVNSs). Garg [16] studied the novel neutrality aggregation operators for single-valued neutrosophic numbers. Ye [17] introduced the simplified neutrosophic sets (SNSs) and applied it to MADM. Wang and Li [18] also defined multi-valued neutrosophic sets (MVNSs) and proposed the extended TODIM method under a multi-valued neutrosophic number environment. Yang and Pang [19] defined multi-valued interval neutrosophic sets (MVINSs) and proposed a new MADM method based on decision-making trial and evaluation laboratory (DEMATEL) and technique for order performance by similarity to ideal solution (TOPSIS) for MVINSs. Wang et al. [20] proposed interval neutrosophic sets (INSs) and the use of logic. Broumi et al. proposed various types of neutrosophic graphs by combining the NSs and graph theory, and applied them to solve the shortest path problem and decision problems [21, 22]. Liu et al. [23] studied linguistic neutrosophic sets (LNSs). Garg and Nancy studied the Frank Choquet Heronian mean operator [24] and power aggregation operators [25] for linguistic single-valued neutrosophic sets (LSVNSs). Garg and Nancy [26] also introduced the concept of the possibility linguistic single-valued neutrosophic set (PLSVNS). Abdel-Basset et al. [27] defined the concept of the type-2 neutrosophic number set (T2NNS). Ye and Smarandache [28] proposed a refined single-valued neutrosophic set (RSVNS). Ye [29] proposed trapezoidal fuzzy NSs and applied them to MADM. Biswas et al. [30] presented cosine similarity measure-based MADM with trapezoidal fuzzy neutrosophic numbers (TrFNNs). Tan et al. [31] proposed two multi-attribute group decision-making methods based on TrFNNs, including one based on the trapezoidal fuzzy neutrosophic number hybrid averaging operator and the other one based on the TOPSIS method. However, scant research has been conducted on TrFNNs [29 –31]. In addition, in some complex decision-making environments, TrFNNs can be used to better express information. For example, in emergency decision making, the number of people affected in a certain area should be determined. Due to time constraints, emergency management departments cannot wait for the data from lower administrative departments. Instead, an effective emergency response should be promptly made to protect the safety of people’s lives and eliminate adverse social effects. Although the number of people affected by a disaster can be accurately obtained, emergency departments can use information technology to estimate the maximum possible value of the number of people affected by the disaster and obtain the fluctuation range of the number of people. In this case, TrFNNs can be used to represent decision information, and as such, the decision results will be more reasonable and effective. Thus, in this paper, we focus on the MADM method with TrFNNs.

The DEMATEL method was proposed by Gabus and Fontela [32] in 1972. It is an effective method for factor analysis and identification. This method makes full use of the experience and knowledge of experts to deal with complex social issues, especially for systems whose relationship is uncertain. The DEMATEL method mainly uses graph theory, focusing on the matrix calculus of structure diagrams. Dalalah et al. [33] used the modified fuzzy DEMATEL model to deal with the supplier selection problem. Kabak [34] proposed a fuzzy DEMATEL-analytical network process (ANP)-based multicriteria decision-making approach for personnel selection. Tseng et al. [35] proposed a causal-and-effect decision-making model of service quality expectation using the grey-fuzzy DEMATEL approach. Serdarasan et al. [36] introduced an interval-valued hesitant fuzzy DEMATEL method and applied it to group decision making. Geng et al. [37] presented a two-tuple linguistic DEMATEL technique to adjust the attribute weights obtained by DEA and then derived comprehensive attribute weights. Pamučar et al. [38] proposed a hybrid DEMATEL-ANP-MAIRCA model and applied it to group multicriteria decision making based on interval rough numbers. Govindan et al. [39] applied the intuitionistic fuzzy DEMATEL to green supply chain management. Karaşan and Kahraman [40] proposed a novel intuitionistic fuzzy DEMATEL-ANP-TOPSIS decision-making methodology used for freight village location selection. Keshavarzfard and Makui [41] proposed an integrated IF-DEMATEL-AHP method used for selecting managers in the automobile industry in Iran under a triangular intuitionistic fuzzy number environment. Liang et al. [42] applied the single-valued trapezoidal neutrosophic (SVTN)–DEMATEL module to analyze the causal relationships among criteria and proposed the integration module for information fusion with consideration of interdependencies and different priority levels of criteria under an SVTN environment. Awang et al. [43] studied the Shapley weighting vector-based single-valued neutrosophic aggregation operator in the DEMATEL method. Abdel-Basset et al. [44] proposed a hybrid approach of SVNSs and DEMATEL method for developing supplier selection criteria. Yang and Pang [19] proposed a new MADM method based on DEMATEL and TOPSIS for MVINSs. Based on the literature survey and knowledge of the authors, the study of DEMATEL in the TrFNN environment is yet to be found so far. The more objective determination of attribute weights is a research focus in the decision-making process because many of the existing literature on attribute determination are subjective assumptions. However, DEMATEL can be used to analyze the complex relationship among attributes and their model dependencies to determine attribute weights [45]. In the present study, we determine the attribute weights based on the DEMATEL method in a TrFNN environment.

Distance can represent a far or near relationship between two objects, and its research is a hotspot in decision-making methods. As decision-making environments become increasingly complex, distance measurements based on inexact numbers have received more attention. Liu et al. [46] proposed the distance measure based on the cosine similarity measure and Euclidean distance measure for Fermatean fuzzy linguistic term sets. Liu et al. [47] also proposed the distance measures between hesitant q-rung orthopair FS. In particular, in recent years, various distance measures have been widely studied in decision making based on NSs. For example, Ye [48] defined the single-valued neutrosophic weighted Hamming distance and the single-valued neutrosophic weighted Euclidean distance and then defined the generalized single-value neutrosophic weighted distance measure. Ye [49] examined the normalized Hamming distance and normalized Euclidean distance between two interval valued neutrosophic sets (IvNSs). Liu et al. [50] proposed a comprehensive similarity measure based on Majumdar’s similarity and the Euclidean distance to obtain a new distance measure. Ye [51] also proposed two distance measures of IvNSs based on the normalized Hamming distance measure induced by the Hausdorff metric. Liu et al. [52] proposed the cosine distance measure between neutrosophic hesitant fuzzy linguistic sets. Li et al. [53] defined the distance measurement between two linguistic neutrosophic numbers. Kandasamy and Smarandache [54] proposed the triple refined indeterminate neutrosophic weighted Hamming distance and Euclidean distance and then proposed the generalized triple refined indeterminate neutrosophic weighted distance. Şahin et al. [55] proposed the Hamming distance, Euclidean distance, and their standardized measures between neutrosophic soft sets. In addition, many scholars have proposed distance measures based on various similarity measures in different neutrosophic environments, such as cosine similarity measure [30], tangent similarity measure [56, 57], dice similarity measure [58], and jaccard similarity measure [59]. Looking at the existing research, the study of the distance between TrFNNs is rare, except for the distance measure based on the cosine similarity measure proposed by Tan and Zhang [31] and the normalized Hamming distance proposed by Biswas et al. [60]. However, examining the existing distance measurement studies, most of the measurement results are exact numbers. This method is also a general way to deal with imprecise numbers (fuzzy numbers, interval numbers, intuitionistic fuzzy numbers). The advantage is that the calculation is simple and easy to understand. However, the fuzzy information will be partially lost during the conversion from inexact numbers to exact numbers. Accordingly, in this study, we focuses on the fuzzy distance measurement based on TrFNNs. The purpose of the study is to delay the transition of inexact numbers to exact numbers in the decision-making process. In other words, no conversion is performed in the early and middle stages of the decision-making process, and the conversion is performed in the final stage of the decision-making process. Thus, the premature loss of information is avoided and the objectivity and accuracy of decision results are guaranteed. Therefore, defining a fuzzy distance measure for imprecise numbers considering the existence of inherent vagueness has become an open issue in imprecise frameworks [61]. There have been some constructive studies on fuzzy distance [61 –64]. Among them, the fuzzy distance based on graded mean integration representation (GMIR) is a good method, which is easy to calculate and understand [61]. For instance, Chen and Wang introduced a fuzzy distance of two trapezoidal fuzzy numbers by using the GMIR [65]. Hence, inspired by the fuzzy distance of trapezoidal fuzzy numbers [62, 65], we researched and defined the fuzzy distance between TrFNNs based on GMIR and used it to measure the distance between alternatives and ideal solutions to eventually sort the alternatives.

The linear assignment method (LAM) is one of the less known members of MADM approaches in the literature. LAM mainly depends on the concordance concept and linear programming technique to determine the ranking order of alternatives [66]. The LAM based on crisp numbers was first introduced by Bernardo and Blin [67] for consumer choice. Baykasoğ lu et al. [66] proposed a new fuzzy LAM for MADM with an application to spare parts inventory classification. Zamri and Abdullah [68] developed an LAM to produce the final ranking order within the context of interval type-2 FSs. Chen [69] developed a new LAM to produce an optimal preference ranking of alternatives in accordance with sets of criterion-wise rankings and criterion importance within the context of interval type-2 trapezoidal fuzzy numbers. Chen [70] extended the LAM to multicriteria decision analysis based on interval-valued IFSs. Yang et al. [45] developed a new MADM method based on LAM with linguistic hesitant intuitionistic fuzzy information considering correlation. Yang et al. [71] proposed a new multi-valued interval neutrosophic MADM method based on linear assignment and Choquet integral. Yang et al. [72] extended the LAM to the MADM method based on INSs. Considering the existing literature, the study of extending the linear allocation method to the TrFNNs has not been performed yet. Therefore, motivated by the idea of LAM, we propose an extended LAM in a TrFNN environment.

Typhoons are one of the most serious types of natural disasters, and they primarily impact the eastern coastal regions of China [73], where the population is extremely dense, the economy is highly developed, and social wealth is notably concentrated. As a result, they bring serious economic, personnel, and environmental losses. However, the influencing factors of typhoon disasters are completely hard to describe accurately. For instance, economic loss includes many aspects, such as a building’s collapse, the number and extent of damage to housing, and the affected local economic conditions [2]. The economic loss cannot be precisely described because the estimation is based on incomplete, inconsistent, and indeterminate information. Therefore, FS and IFS have been used for typhoon disaster assessments in recent years. Ma [74] proposed a synthetic evaluation model for typhoon disasters under a fuzzy environment. He [1] proposed a typhoon disaster assessment method based on Dombi hesitant fuzzy information aggregation operators. Li et al. [75] proposed evaluating typhoon disasters method based on the TOPSIS method with IFS. Yu [2] proposed a typhoon disaster evaluation based on the generalized intuitionistic fuzzy aggregation operators. However, these studies indicate that the application of the NS theory in typhoon disaster assessment is yet to be examined. We believe that NS provides powerful theories and methods to enhance typhoon disaster assessments. In this study, we used the extended DEMATEL method to derive the evaluation index weights, using the new fuzzy distance of TrFNNs based on GMIR defined by us to calculate the distance of two neutrosophic numbers and using the extended LAM in operations research to rank the alternatives.

The remainder of this paper is organized as follows: Section 2 briefly introduces some basic definitions of NSs, TrFNNs and so on. Section 3 defines the new fuzzy distance of trapezoidal fuzzy neutrosophic numbers based on graded mean integration representation. A MADM method based on the DEMATEL, fuzzy distance and LAM is given in Section 4. Section 5 uses a typhoon disaster evaluation example to illustrate the applicability of the proposed method and gives the comparative analysis. Finally, Section 6 gives the conclusions.

2 Preliminaries

In this section, we briefly introduce some basic concepts and theories, such as trapezoidal fuzzy neutrosophic sets, trapezoidal fuzzy neutrosophic numbers (TrFNNs), operational rules of TrFNNs and so on.

Definition 1. [29] Let X be a universe of discourse, then a trapezoidal fuzzy neutrosophic set $\tilde{N}$ in X has the following form:

$\tilde{N} = {〈 x, T_{\tilde{N}} (x), I_{\tilde{N}} (x), F_{\tilde{N}} (x) 〉 | x \in X},$ (1) where $T_{\tilde{N}} (x) \subset [0, 1]$ , $I_{\tilde{N}} (x) \subset [0, 1]$ and $F_{\tilde{N}} (x) \subset [0, 1]$ are three trapezoidal fuzzy numbers, $T_{\tilde{N}} (x) = (t_{\tilde{N}}^{1} (x), t_{\tilde{N}}^{2} (x), t_{\tilde{N}}^{3} (x), t_{\tilde{N}}^{4} (x)) : X \to [0, 1]$ , $I_{\tilde{N}} (x) = (i_{\tilde{N}}^{1} (x), i_{\tilde{N}}^{2} (x), i_{\tilde{N}}^{3} (x), i_{\tilde{N}}^{4} (x)) : X \to [0, 1]$ , and $F_{\tilde{N}} (x) =$ $(f_{\tilde{N}}^{1} (x), f_{\tilde{N}}^{2} (x), f_{\tilde{N}}^{3} (x), f_{\tilde{N}}^{4} (x)) : X \to [0, 1]$ with the condition $0 \leq t_{\tilde{N}}^{4} (x) + i_{\tilde{N}}^{4} (x) + f_{\tilde{N}}^{4} (x) \leq 3, x \in X$ .

Definition 2. [30] A TrFNN $\tilde{n}$ is denoted by $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}, b_{4}), (c_{1}, c_{2}, c_{3}, c_{4}) 〉$ in a universe of discourse X. The parameters satisfy the following relations: a₁ ≤ a₂ ≤ a₃ ≤ a₄, b₁ ≤ b₂ ≤ b₃ ≤ b₄ and c₁ ≤ c₂ ≤ c₃ ≤ c₄.

Its truth-membership function is defined as follows:

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

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

$F_{\tilde{N}} (x) = 〈 \begin{matrix} \frac{c_{2} - x}{c_{2} - c_{1}} & c_{1} \leq x < c_{2} \\ 0 & c_{2} \leq x \leq c_{3} \\ \frac{x - c_{3}}{c_{4} - c_{3}} & c_{3} < x \leq c_{4} \\ 1 & otherwise \end{matrix} 〉 .$ (4) When a₂ = a₃, b₂ = b₃, and c₂ = c₃ in a TrFNN $\tilde{n}$ , the number reduces to a triangular fuzzy neutrosophic number, which is considered a special case of the trapezoidal fuzzy neutrosophic number.

Definition 3. [29] Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}$ , b₃, b₄) , (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}),$ (f₁, f₂, f₃, f₄) , (g₁, g₂, g₃, g₄) 〉 be two TrFNNs. Then, the following operational rules apply:

(1) ${\tilde{n}}_{1} \oplus {\tilde{n}}_{2} =$

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

(2) ${\tilde{n}}_{1} \otimes {\tilde{n}}_{2} =$

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

(3) $λ {\tilde{n}}_{1} =$

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

(4) ${\tilde{n}}_{1}^{λ} =$

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

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

Definition 4. [29] Let $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}$ , b₃, b₄) , (c₁, c₂, c₃, c₄) 〉 be a TrFNN. Then, a score function of a trapezoidal neutrosophic number can be defined as:

$\begin{matrix} s (\tilde{n}) & = \frac{1}{3} (2 + \frac{a_{1} + a_{2} + a_{3} + a_{4}}{4} \\ - \frac{b_{1} + b_{2} + b_{3} + b_{4}}{4} - \frac{c_{1} + c_{2} + c_{3} + c_{4}}{4}), \\ s (\tilde{n}) \in [0, 1] . \end{matrix}$ (5) where the larger the value of $s (\tilde{n})$ , the bigger the trapezoidal neutrosophic number $\tilde{n}$ .

Definition 5. [29] Let $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}$ , b₃, b₄) , (c₁, c₂, c₃, c₄) 〉 be a TrFNN. Then, the accuracy function of a trapezoidal neutrosophic number can be defined as:

$\begin{matrix} h (\tilde{n}) & = \frac{a_{1} + a_{2} + a_{3} + a_{4}}{4} - \frac{c_{1} + c_{2} + c_{3} + c_{4}}{4}, \\ h (\tilde{n}) \in [- 1, 1] . \end{matrix}$ (6) where the larger the value of $h (\tilde{n})$ , the bigger trapezoidal neutrosophic number $\tilde{n}$ .

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

Definition 6. [29] Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}$ , b₃, b₄) , (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}),$ (f₁, f₂, f₃, f₄) , (g₁, g₂, g₃, g₄) 〉 be two TrFNNs. Thus, $s ({\tilde{n}}_{1})$ and $s ({\tilde{n}}_{2})$ are the scores of ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ , respectively, and $h ({\tilde{n}}_{1})$ and $h ({\tilde{n}}_{2})$ are the accuracy degree of ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ , respectively. Consequently, the ordered relation between two TrFNNs is defined as follows:

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

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

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

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

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

3 Fuzzy distance of the trapezoidal fuzzy neutrosophic numbers

3.1 Graded mean integration representation (GMIR)

Chen et al. [76, 77] using graded mean integration representation method to defuzzify the generalized fuzzy numbers, L-R type fuzzy numbers, and trapezoidal fuzzy numbers. Now, we briefly review Chen’s GMIR of generalized fuzzy numbers.

Suppose $\tilde{A}$ is a generalized fuzzy number. It is described as any fuzzy subset of the real line R, whose membership function μ_A satisfies the following conditions:

(1) μ_A is a continuous mapping from R to the closed interval [0, 1],

(2) μ_A = 0, - ∞ < x ≤ a₁,

(3) μ_A = L (x) is strictly increasing on [a₁, a₂],

(4) μ_A = ω_A, a₂ < x ≤ a₃,

(5) μ_A = R (x) is strictly decreasing on [a₃, a₄],

(6) μ_A = 0, a₄≤ x < ∞, where 0 < ω_A ≤ 1, and a₁, a₂, a₃, and a₄ are real numbers.

Also this type of generalized fuzzy numbers be denoted as $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; ω_{A})_{LR}$ , when ω_A = 1, it can be simplified as $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})_{LR}$ .

In graded mean integration representation method, L^-1 and R^-1 are the inverse functions of L and R respectively, and the graded mean h-level value of generalized fuzzy number $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; ω_{A})_{LR}$ is h (L^-1 (h) + R^-1 (h))/2. Then the graded mean integration representation of $\tilde{A}$ is $P (\tilde{A})$ with grade ω_A where $\begin{matrix} P (\tilde{A}) = \frac{\int_{0}^{ω_{A}} h (\frac{L^{- 1} (h) + R^{- 1} (h)}{2}) dh}{\int_{0}^{ω_{A}} hdh} \end{matrix}$ with 0 < h ≤ ω_A and 0 < ω_A ≤ 1.

Let $\tilde{B}$ be a trapezoidal fuzzy number, and be denoted as $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ . Then we can get the graded mean integration representation of $\tilde{B}$ by the above formula as $\begin{matrix} P (\tilde{B}) & = \frac{\int_{0}^{1} h (\frac{b_{1} + b_{4} + (b_{2} - b_{1} - b_{4} + b_{3}) h}{2}) dh}{\int_{0}^{1} hdh} \\ = \frac{b_{1} + 2 b_{2} + 2 b_{3} + b_{4}}{6} . \end{matrix}$

3.2 Fuzzy distance of trapezoidal fuzzy number based on GMIR

Chen and Wang [65] introduced a fuzzy distance of two trapezoidal fuzzy numbers by using the GMIR. This new idea has two advantages, this is the fuzzy distance is easy to calculate and easy to understand.

Let A = (a₁, a₂, a₃, a₄) and B = (b₁, b₂, b₃, b₄) be two trapezoidal fuzzy numbers, and their GMIR are P (A), P (B) respectively. Assume $\begin{matrix} P (A) & = \frac{a_{1} + 2 a_{2} + 2 a_{3} + a_{4}}{6}, \\ P (B) & = \frac{b_{1} + 2 b_{2} + 2 b_{3} + b_{4}}{6}, \\ s_{i} & = (a_{i} - P (A) + b_{i} - P (B)) / 2, & i = 1, 2, 3, 4, \\ c_{i} & = | P (A) + P (B) + s_{i} | / i, & i = 1, 2, 3, 4, \end{matrix}$ then the fuzzy distance of A and B is C = (c₁, c₂, c₃, c₄).

3.3 Fuzzy distance of trapezoidal fuzzy neutrosophic numbers based on GMIR proposed by us

3.3.1 Definition of proposed fuzzy distance

In this section, inspired by the GMIR proposed in [62, 65], based on the graphical representation of the trapezoidal fuzzy neutrosophic number as shown in Fig. 1 [31], we define the fuzzy distance of two trapezoidal fuzzy neutrosophic numbers based on the graded mean integration representation. Suppose L_T (x) and R_T (x) respectively represent the right and left straight line of $T_{\tilde{N}} (x)$ , then $L_{T}^{- 1}$ and $R_{T}^{- 1}$ are inverse functions of functions L and R, respectively, and the graded mean h-level value of truth-membership function $T_{\tilde{N}} (x)$ of trapezoidal fuzzy neutrosophic number (TrFNN) is $h (L_{T}^{- 1} (h) + R_{T}^{- 1} (h)) / 2$ as shown in Fig. 2. Then the graded mean integration representation of $T_{\tilde{N}} (x)$ of TrFNN based on the integral value of graded mean h-levels is $P_{T} (\tilde{n})$ . The specific calculation is as follows:

Fig.1

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

Fig.2

The graded mean h-level value of $T_{\tilde{N}} (x)$ of TrFNN.

Suppose $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}, b_{4}), (c_{1}$ , c₂, c₃, c₄) 〉 be a TrFNN,

Since, $L_{T} (x) = T (\frac{x - a_{1}}{a_{2} - a_{1}}), a_{1} \leq x \leq a_{2}$ , $R_{T} (x) = T (\frac{a_{4} - x}{a_{4} - a_{3}}), a_{3} \leq x \leq a_{4}$ ,

Then, $L_{T}^{- 1} (h) = a_{1} + \frac{h (a_{2} - a_{1})}{T}, 0 \leq h \leq T$ , $R_{T}^{- 1} (h) = a_{4} - \frac{h (a_{4} - a_{3})}{T}, 0 \leq h \leq T$ , $\begin{matrix} \frac{L_{T}^{- 1} (h) + R_{T}^{- 1} (h)}{2} = \frac{a_{1} + a_{4} + \frac{h (a_{2} - a_{1} - a_{4} + a_{3})}{T}}{2} . \end{matrix}$ Thus, $\begin{matrix} P_{T} (\tilde{n}) \\ = \int_{0}^{T} h \frac{a_{1} + a_{4} + \frac{h (a_{2} - a_{1} - a_{4} + a_{3})}{T}}{2} dh / \int_{0}^{T} hdh \\ = {\frac{1}{2} [\frac{(a_{1} + a_{4}) h^{2}}{2} + \frac{h^{3} (a_{2} - a_{1} - a_{4} + a_{3})}{3 T}] |}_{0}^{T} \\ / {[\frac{1}{2} h^{2}] |}_{0}^{T} \\ = \frac{a_{1} + 2 a_{2} + 2 a_{3} + a_{4}}{6} . \end{matrix}$ where T represents the value range of the truth-membership function $T_{\tilde{N}} (x)$ , and 0 ≤ T ≤ 1.

Next, we discuss the indeterminacy-membership function of TrFNN. Suppose L_I (x) and R_I (x) respectively represent the right and left straight line of $I_{\tilde{N}} (x)$ , then $L_{I}^{- 1}$ and $R_{I}^{- 1}$ are inverse functions of functions L and R, respectively, and the graded mean h-level value of indeterminacy-membership function of trapezoidal fuzzy neutrosophic number is $h (L_{I}^{- 1} (h) + R_{I}^{- 1} (h)) / 2$ as shown in Fig. 3. Then the graded mean integration representation of $I_{\tilde{N}} (x)$ of TrFNN based on the integral value of graded mean h-levels is $P_{I} (\tilde{n})$ . The specific calculation is as follows:

Fig.3

The graded mean h-level value of $I_{\tilde{N}} (x)$ of TrFNN.

Suppose $\tilde{n} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}, b_{4}), (c_{1}$ , c₂, c₃, c₄) 〉 be a TrFNN,

Since, $L_{I} (x) = I (\frac{b_{2} - x}{b_{2} - b_{1}}), b_{1} \leq x \leq b_{2}$ , $R_{I} (x) = I (\frac{x - b_{3}}{b_{4} - b_{3}}), b_{3} \leq x \leq b_{4}$ ,

Then, $L_{I}^{- 1} (h) = b_{2} - \frac{h (b_{2} - b_{1})}{I}, 0 \leq h \leq I$ , $R_{I}^{- 1} (h) = b_{3} + \frac{h (b_{4} - b_{3})}{I}, 0 \leq h \leq I$ , $\begin{matrix} \frac{L_{I}^{- 1} (h) + R_{I}^{- 1} (h)}{2} = \frac{b_{2} + b_{3} + \frac{h (b_{4} - b_{3} - b_{2} + b_{1})}{I}}{2} . \end{matrix}$ Thus, $\begin{matrix} P_{I} (\tilde{n}) \\ = \int_{0}^{I} h \frac{b_{2} + b_{3} + \frac{h (b_{4} - b_{3} - b_{2} + b_{1})}{I}}{2} dh / \int_{0}^{I} hdh \\ = {\frac{1}{2} [\frac{(b_{2} + b_{3}) h^{2}}{2} + \frac{h^{3} (b_{4} - b_{3} - b_{2} + b_{1})}{3 I}] |}_{0}^{I} \\ / {[\frac{1}{2} h^{2}] |}_{0}^{I} \\ = \frac{2 b_{1} + b_{2} + b_{3} + 2 b_{4}}{6} . \end{matrix}$ where I represents the value range of the indeterminacy-membership function $I_{\tilde{N}} (x)$ , and 0 ≤ I ≤ 1.

Based on similar calculations, we can get the graded mean integration representation of falsity membership function of TrFNN $\tilde{n}$ is: $\begin{matrix} P_{F} (\tilde{n}) \\ = \int_{0}^{F} h \frac{c_{2} + c_{3} + \frac{h (c_{4} - c_{3} - c_{2} + c_{1})}{F}}{2} dh / \int_{0}^{F} hdh \\ = {\frac{1}{2} [\frac{(c_{2} + c_{3}) h^{2}}{2} + \frac{h^{3} (c_{4} - c_{3} - c_{2} + c_{1})}{3 F}] |}_{0}^{F} \\ / {[\frac{1}{2} h^{2}] |}_{0}^{F} \\ = \frac{2 c_{1} + c_{2} + c_{3} + 2 c_{4}}{6} . \end{matrix}$ where F represents the value range of the falsity-membership function $F_{\tilde{N}} (x)$ , and 0 ≤ F ≤ 1.

Definition 7. Let ${\tilde{n}}_{1} = 〈 (a_{1}, a_{2}, a_{3}, a_{4}), (b_{1}, b_{2}, b_{3}$ , b₄) , (c₁, c₂, c₃, c₄) 〉 and ${\tilde{n}}_{2} = 〈 (e_{1}, e_{2}, e_{3}, e_{4}),$ (f₁, f₂, f₃, f₄), (g₁, g₂, g₃, g₄) 〉 be two TrFNNs, and their graded mean integration representation are $P ({\tilde{n}}_{1}) = (P_{T} ({\tilde{n}}_{1}), P_{I} ({\tilde{n}}_{1})$ , $P_{F} ({\tilde{n}}_{1}))$ , $P ({\tilde{n}}_{2}) = (P_{T} ({\tilde{n}}_{2})$ , $P_{I} ({\tilde{n}}_{2}), P_{F} ({\tilde{n}}_{2}))$ respectively. Assume $\begin{matrix} s_{i}^{T} & = (a_{i} - P_{T} ({\tilde{n}}_{1}) + e_{i} - P_{T} ({\tilde{n}}_{2})) / 2, \\ i = 1, 2, 3, 4; \\ s_{i}^{I} & = (b_{i} - P_{I} ({\tilde{n}}_{1}) + f_{i} - P_{I} ({\tilde{n}}_{2})) / 2, \\ i = 1, 2, 3, 4; \\ s_{i}^{F} & = (c_{i} - P_{F} ({\tilde{n}}_{1}) + g_{i} - P_{F} ({\tilde{n}}_{2})) / 2, \\ i = 1, 2, 3, 4; \\ d_{i}^{T, I, F} & = | P_{T, I, F} ({\tilde{n}}_{1}) - P_{T, I, F} ({\tilde{n}}_{2}) | + s_{i}^{T, I, F} . \end{matrix}$ Then the fuzzy distance between ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ is

$\begin{matrix} D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) & = 〈 (d_{1}^{T}, d_{2}^{T}, d_{3}^{T}, d_{4}^{T}), (d_{1}^{I}, d_{2}^{I}, d_{3}^{I}, d_{4}^{I}), \\ (d_{1}^{F}, d_{2}^{F}, d_{3}^{F}, d_{4}^{F}) 〉 . \end{matrix}$ (7)

3.3.2 Metric properties of proposed fuzzy distance

The above defined distance $D ({\tilde{n}}_{1}, {\tilde{n}}_{2})$ satisfies the following properties:

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

(2) If ${\tilde{n}}_{1} \leq {\tilde{n}}_{2} \leq {\tilde{n}}_{3}$ , then $D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) \leq D ({\tilde{n}}_{1}, {\tilde{n}}_{3})$ and $D ({\tilde{n}}_{2}, {\tilde{n}}_{3}) \leq D ({\tilde{n}}_{1}, {\tilde{n}}_{3})$ .

Proof.

(1) Support $\begin{matrix} D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) & = 〈 (d_{1}^{T}, d_{2}^{T}, d_{3}^{T}, d_{4}^{T}), \\ (d_{1}^{I}, d_{2}^{I}, d_{3}^{I}, d_{4}^{I}), (d_{1}^{F}, d_{2}^{F}, d_{3}^{F}, d_{4}^{F}) 〉 ., \\ D ({\tilde{n}}_{2}, {\tilde{n}}_{1}) & = 〈 (d_{1^{'}}^{T}, d_{2^{'}}^{T}, d_{3^{'}}^{T}, d_{4^{'}}^{T}), \\ (d_{1^{'}}^{I}, d_{2^{'}}^{I}, d_{3^{'}}^{I}, d_{4^{'}}^{I}), (d_{1^{'}}^{F}, d_{2^{'}}^{F}, d_{3^{'}}^{F}, d_{4^{'}}^{F}) 〉, \\ d_{i}^{T, I, F} & = | P_{T, I, F} ({\tilde{n}}_{1}) - P_{T, I, F} ({\tilde{n}}_{2}) | + s_{i}^{T, I, F} . \end{matrix}$

because $\begin{matrix} P_{T} ({\tilde{n}}_{1}) - P_{T} ({\tilde{n}}_{2}) & = P_{T} ({\tilde{n}}_{2}) - P_{T} ({\tilde{n}}_{1}), \\ P_{I} ({\tilde{n}}_{1}) - P_{I} ({\tilde{n}}_{2}) & = P_{I} ({\tilde{n}}_{2}) - P_{I} ({\tilde{n}}_{1}), \\ P_{F} ({\tilde{n}}_{1}) - P_{F} ({\tilde{n}}_{2}) & = P_{F} ({\tilde{n}}_{2}) - P_{F} ({\tilde{n}}_{1}), \end{matrix}$

$\begin{matrix} s_{i}^{T} & = (a_{i} - P_{T} ({\tilde{n}}_{1}) + e_{i} - P_{T} ({\tilde{n}}_{2})) / 2 \\ = (e_{i} - P_{T} ({\tilde{n}}_{2}) + a_{i} - P_{T} ({\tilde{n}}_{1})) / 2 = s_{i^{'}}^{T}, \\ s_{i}^{I} & = (b_{i} - P_{I} ({\tilde{n}}_{1}) + f_{i} - P_{I} ({\tilde{n}}_{2})) / 2 \\ = (f_{i} - P_{I} ({\tilde{n}}_{2}) + b_{i} - P_{I} ({\tilde{n}}_{1})) / 2 = s_{i^{'}}^{I}, \\ s_{i}^{F} & = (c_{i} - P_{F} ({\tilde{n}}_{1}) + g_{i} - P_{F} ({\tilde{n}}_{2})) / 2 \\ = (g_{i} - P_{F} ({\tilde{n}}_{2}) + c_{i} - P_{F} ({\tilde{n}}_{1})) / 2 = s_{i^{'}}^{F}, \\ d_{i}^{T} & = | P_{T} ({\tilde{n}}_{1}) - P_{T} ({\tilde{n}}_{2}) | + s_{i}^{T} \\ = | P_{T} ({\tilde{n}}_{2}) - P_{T} ({\tilde{n}}_{1}) | + s_{i^{'}}^{T} = d_{i^{'}}^{T}, \\ d_{i}^{I} & = | P_{I} ({\tilde{n}}_{1}) - P_{I} ({\tilde{n}}_{2}) | + s_{i}^{I} \\ = | P_{I} ({\tilde{n}}_{2}) - P_{I} ({\tilde{n}}_{1}) | + s_{i^{'}}^{I} = d_{i^{'}}^{I}, \\ d_{i}^{F} & = | P_{F} ({\tilde{n}}_{1}) - P_{F} ({\tilde{n}}_{2}) | + s_{i}^{F} \\ = | P_{F} ({\tilde{n}}_{2}) - P_{F} ({\tilde{n}}_{1}) | + s_{i^{'}}^{F} = d_{i^{'}}^{F}, \end{matrix}$ thus $D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = D ({\tilde{n}}_{2}, {\tilde{n}}_{1})$ is established.

(2) Support $\begin{matrix} {\tilde{n}}_{3} & = 〈 (o_{1}, o_{2}, o_{3}, o_{4}), (p_{1}, p_{2}, p_{3}, p_{4}), \\ (q_{1}, q_{2}, q_{3}, q_{4}) 〉, \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) & = 〈 (d_{1}^{T}, d_{2}^{T}, d_{3}^{T}, d_{4}^{T}), \\ (d_{1}^{I}, d_{2}^{I}, d_{3}^{I}, d_{4}^{I}), \\ (d_{1}^{F}, d_{2}^{F}, d_{3}^{F}, d_{4}^{F}) 〉, \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{3}) & = 〈 (d_{1^{″}}^{T}, d_{2^{″}}^{T}, d_{3^{″}}^{T}, d_{4^{″}}^{T}), \\ (d_{1^{″}}^{I}, d_{2^{″}}^{I}, d_{3^{″}}^{I}, d_{4^{″}}^{I}), \\ (d_{1^{″}}^{F}, d_{2^{″}}^{F}, d_{3^{″}}^{F}, d_{4^{″}}^{F}) 〉, \\ D ({\tilde{n}}_{2}, {\tilde{n}}_{3}) & = 〈 (d_{1 ‴}^{T}, d_{2 ‴}^{T}, d_{3 ‴}^{T}, d_{4 ‴}^{T}), \\ (d_{1 ‴}^{I}, d_{2 ‴}^{I}, d_{3 ‴}^{I}, d_{4 ‴}^{I}), \\ (d_{1 ‴}^{F}, d_{2 ‴}^{F}, d_{3 ‴}^{F}, d_{4 ‴}^{F}) 〉, \end{matrix}$ because ${\tilde{n}}_{1} \leq {\tilde{n}}_{2} \leq {\tilde{n}}_{3}$ , a_i ≤ e_i ≤ o_i, b_i ≤ f_i ≤ p_i, c_i ≤ g_i ≤ q_i, $\begin{matrix} P_{T} ({\tilde{n}}_{1}) = \frac{a_{1} + 2 a_{2} + 2 a_{3} + a_{4}}{6} \leq P_{T} ({\tilde{n}}_{2}) \\ = \frac{e_{1} + 2 e_{2} + 2 e_{3} + e_{4}}{6} \leq P_{T} ({\tilde{n}}_{3}) \\ = \frac{o_{1} + 2 o_{2} + 2 o_{3} + o_{4}}{6}, \\ P_{I} ({\tilde{n}}_{1}) = \frac{2 b_{1} + b_{2} + b_{3} + 2 b_{4}}{6} \leq P_{I} ({\tilde{n}}_{2}) \\ = \frac{2 f_{1} + f_{2} + f_{3} + 2 f_{4}}{6} \leq P_{I} ({\tilde{n}}_{3}) \\ = \frac{2 p_{1} + p_{2} + p_{3} + 2 p_{4}}{6}, \\ P_{F} ({\tilde{n}}_{1}) = \frac{2 c_{1} + c_{2} + c_{3} + 2 c_{4}}{6} \leq P_{F} ({\tilde{n}}_{2}) \\ = \frac{2 g_{1} + g_{2} + g_{3} + 2 g_{4}}{6} \leq P_{F} ({\tilde{n}}_{3}) \\ = \frac{2 q_{1} + q_{2} + q_{3} + 2 q_{4}}{6}, \\ P_{T, I, F} ({\tilde{n}}_{2}) - P_{T, I, F} ({\tilde{n}}_{1}) \leq P_{T, I, F} ({\tilde{n}}_{3}) \\ - P_{T, I, F} ({\tilde{n}}_{1}), P_{T, I, F} ({\tilde{n}}_{2}) \\ - P_{T, I, F} ({\tilde{n}}_{3}) \leq P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{1}), \\ s_{i_{1, 2}}^{T, I, F} = (a_{i} | b_{i} | c_{i} - P_{T, I, F} ({\tilde{n}}_{1}) + e_{i} | f_{i} | g_{i} \\ - P_{T, I, F} ({\tilde{n}}_{2})) / 2, (| represents or), \\ s_{i_{1, 3}}^{T, I, F} = (a_{i} | b_{i} | c_{i} - P_{T, I, F} ({\tilde{n}}_{1}) + o_{i} | p_{i} | q_{i} \\ - P_{T, I, F} ({\tilde{n}}_{3})) / 2, (| represents or), \\ s_{i_{2, 3}}^{T, I, F} = (e_{i} | f_{i} | g_{i} - P_{T, I, F} ({\tilde{n}}_{2}) + o_{i} | p_{i} | q_{i} \\ - P_{T, I, F} ({\tilde{n}}_{3})) / 2, (| represents or), \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = | P_{T, I, F} ({\tilde{n}}_{1}) - P_{T, I, F} ({\tilde{n}}_{2}) | \\ + s_{i_{1, 2}}^{T, I, F} = P_{T, I, F} ({\tilde{n}}_{2}) - P_{T, I, F} ({\tilde{n}}_{1}) \\ + (a_{i} | b_{i} | c_{i} - P_{T, I, F} ({\tilde{n}}_{1}) \\ + e_{i} | f_{i} | g_{i} - P_{T, I, F} ({\tilde{n}}_{2})) / 2, \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{3}) = | P_{T, I, F} ({\tilde{n}}_{1}) - P_{T, I, F} ({\tilde{n}}_{3}) | \\ + s_{i_{1, 3}}^{T, I, F} = P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{1}) \\ + (a_{i} | b_{i} | c_{i} - P_{T, I, F} ({\tilde{n}}_{1}) \\ + o_{i} | p_{i} | q_{i} - P_{T, I, F} ({\tilde{n}}_{3})) / 2, \\ P_{T, I, F} ({\tilde{n}}_{2}) - P_{T, I, F} ({\tilde{n}}_{1}) + (a_{i} | b_{i} | c_{i} \\ - P_{T, I, F} ({\tilde{n}}_{1}) + e_{i} | f_{i} | g_{i} - P_{T, I, F} ({\tilde{n}}_{2})) / 2 \\ \leq P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{1}) + (a_{i} | b_{i} | c_{i} \\ - P_{T, I, F} ({\tilde{n}}_{1}) + o_{i} | p_{i} | q_{i} - P_{T, I, F} ({\tilde{n}}_{3})) / 2, \end{matrix}$ thus $D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) \leq D ({\tilde{n}}_{1}, {\tilde{n}}_{3})$ is established.

Simultaneously, $\begin{matrix} D ({\tilde{n}}_{2}, {\tilde{n}}_{3}) = | P_{T, I, F} ({\tilde{n}}_{2}) - P_{T, I, F} ({\tilde{n}}_{3}) | \\ + s_{i_{2, 3}}^{T, I, F} = P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{2}) \\ + (e_{i} | f_{i} | g_{i} - P_{T, I, F} ({\tilde{n}}_{2}) \\ + o_{i} | p_{i} | q_{i} - P_{T, I, F} ({\tilde{n}}_{3})) / 2, \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{3}) = | P_{T, I, F} ({\tilde{n}}_{1}) - P_{T, I, F} ({\tilde{n}}_{3}) | + s_{i_{1, 3}}^{T, I, F} \\ = P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{1}) \\ + (a_{i} | b_{i} | c_{i} - P_{T, I, F} ({\tilde{n}}_{1}) \\ + o_{i} | p_{i} | q_{i} - P_{T, I, F} ({\tilde{n}}_{3})) / 2, \\ P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{2}) + (e_{i} | f_{i} | g_{i} - P_{T, I, F} ({\tilde{n}}_{2}) \\ + o_{i} | p_{i} | q_{i} - P_{T, I, F} ({\tilde{n}}_{3})) / 2 \\ \leq P_{T, I, F} ({\tilde{n}}_{3}) - P_{T, I, F} ({\tilde{n}}_{1}) \\ + (a_{i} | b_{i} | c_{i} - P_{T, I, F} ({\tilde{n}}_{1}) \\ + o_{i} | p_{i} | q_{i} - P_{T, I, F} ({\tilde{n}}_{3})) / 2, \end{matrix}$ thus $D ({\tilde{n}}_{2}, {\tilde{n}}_{3}) \leq D ({\tilde{n}}_{1}, {\tilde{n}}_{3})$ is established.

So, the property (2) is established, this is, if ${\tilde{n}}_{1} \leq {\tilde{n}}_{2} \leq {\tilde{n}}_{3}$ , then $D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) \leq D ({\tilde{n}}_{1}, {\tilde{n}}_{3})$ and $D ({\tilde{n}}_{2}, {\tilde{n}}_{3})$ $\leq D ({\tilde{n}}_{1}, {\tilde{n}}_{3})$ .

Example 1. Let ${\tilde{n}}_{1} = 〈 (0.1, 0.2, 0.3, 0.4), (0.0, 0.2$ , 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉 and ${\tilde{n}}_{1} = 〈 (0.5, 0.6,$ 0.7, 0.8), (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉 be two TrFNNs. Then, according to Definition 7, we obtain $\begin{matrix} P_{T} ({\tilde{n}}_{1}) = \frac{a_{1} + 2 a_{2} + 2 a_{3} + a_{4}}{6} = 0.25, \\ P_{I} ({\tilde{n}}_{1}) = \frac{2 b_{1} + b_{2} + b_{3} + 2 b_{4}}{6} = 0.22, \\ P_{F} ({\tilde{n}}_{1}) = \frac{2 c_{1} + c_{2} + c_{3} + 2 c_{4}}{6} = 0.22; \\ P_{T} ({\tilde{n}}_{2}) = \frac{e_{1} + 2 e_{2} + 2 e_{3} + e_{4}}{6} = 0.65, \\ P_{I} ({\tilde{n}}_{1}) = \frac{2 f_{1} + f_{2} + f_{3} + 2 f_{4}}{6} = 0.65, \\ P_{F} ({\tilde{n}}_{1}) = \frac{2 g_{1} + g_{2} + g_{3} + 2 g_{4}}{6} = 0.65; \\ s_{1}^{T} = (a_{1} - P_{T} ({\tilde{n}}_{1}) + e_{1} - P_{T} ({\tilde{n}}_{2})) / 2 \\ = - 0.15, s_{2}^{T} = - 0.05, s_{3}^{T} = 0.05, s_{4}^{T} = 0.15; \\ s_{1}^{I} = (b_{1} - P_{I} ({\tilde{n}}_{1}) + f_{1} - P_{I} ({\tilde{n}}_{2})) / 2 \\ = - 0.23, s_{2}^{I} = - 0.03, s_{3}^{I} = 0.07, s_{4}^{I} = 0.22; \\ s_{1}^{F} = (c_{1} - P_{F} ({\tilde{n}}_{1}) + g_{1} - P_{F} ({\tilde{n}}_{2})) / 2 \\ = - 0.23, s_{2}^{F} = - 0.03, s_{3}^{F} = 0.07, s_{4}^{F} = 0.22; \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = 〈 (0.25, 0.35, 0.45, 0.55), \\ (0.20, 0.40, 0.50, 0.65), (0.20, 0.40, 0.50, 0.65) 〉 . \end{matrix}$

Example 2. Let ${\tilde{n}}_{1} = 〈 (0.0, 0.0, 0.0, 0.0), (0.0, 0.0$ , 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉 and ${\tilde{n}}_{2} = 〈 (1.0, 1.0,$ 1.0, 1.0), (1.0, 1.0, 1.0, 1.0) , (1.0, 1.0, 1.0, 1.0) 〉 be two TrFNNs. Then, according to Definition 7, we obtain $\begin{matrix} P_{T} ({\tilde{n}}_{1}) & = 0.0, P_{I} ({\tilde{n}}_{1}) = 0.0, P_{F} ({\tilde{n}}_{1}) = 0.0; \\ P_{T} ({\tilde{n}}_{2}) & = 1.0, P_{I} ({\tilde{n}}_{1}) = 1.0, P_{F} ({\tilde{n}}_{1}) = 1.0; \\ s_{1}^{T} & = 0.0, s_{2}^{T} = 0.0, s_{3}^{T} = 0.0, s_{4}^{T} = 0.0; \\ s_{1}^{I} & = 0.0, s_{2}^{I} = 0.0, s_{3}^{I} = 0.0, s_{4}^{I} = 0.0; \\ s_{1}^{F} & = 0.0, s_{2}^{F} = 0.0, s_{3}^{F} = 0.0, s_{4}^{F} = 0.0; \\ D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) & = 〈 (1.0, 1.0, 1.0, 1.0), \\ (1.0, 1.0, 1.0, 1.0), (1.0, 1.0, 1.0, 1.0) 〉 . \end{matrix}$ Here, ${\tilde{n}}_{1}$ is equivalent to 〈0.0, 0.0, 0.0〉 and ${\tilde{n}}_{1}$ is equivalent to 〈1.0, 1.0, 1.0〉, the essence of ${\tilde{n}}_{1}$ and ${\tilde{n}}_{1}$ are single-valued neutrosophic numbers. Their fuzzy distance $D ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = 〈 (1.0, 1.0, 1.0, 1.0), (1.0, 1.0$ , 1.0, 1.0) , (1.0, 1.0, 1.0, 1.0) 〉 is equivalent to < 1.0, 1.0, 1.0 >, this result is consistent with human subjective tendencies.

4 Multiple attribute decision making method based on DEMATEL, fuzzy distance and LAM under the trapezoidal fuzzy neutrosophic numbers environment

At this section, we introduce the DEMATEL method in detail, and give the specific steps of our proposed new MADM methods based on DEMATEL, Fuzzy distance and LAM under the trapezoidal fuzzy neutrosophic environment. To achieve our objectives, the problem description and the proposed method consist of the following several steps that are depicted graphically in Fig. 4.

Fig.4

Framework of the proposed decision making method.

4.1 The DEMATEL Method

The specific calculation steps of DEMATEL method are as follows:

Step 1: Generate the direct influence matrix. Decision makers are asked to give the direct relationship values of the attribute pairs. The relation matrix is established as X^d = (a_ij) _n×n, where X^d is an non-negative matrix, a_ij indicates the direct effect of factor i on factor j. In this paper, a_ij is first given by the expert using linguistic terms that is easy to express, and then converted to a corresponding trapezoidal fuzzy neutrosophic numbers.

Step 2: Generate the normalized influence matrix $\bar{X} = ({\bar{a}}_{ij})_{n \times n} .$

➀ First, calculate the sum of the elements in each row of the matrix $\bar{X}$ , and find the maximum value of the sum Max, where $Max = max_{1 < i < n} \sum_{j}^{n} {\bar{a}}_{ij}$ .

➁ Then divide each element of the direct influence matrix $\bar{X}$ by the maximum value Max to obtain the normalization influence matrix $\bar{X}$ .

Step 3: Generate the comprehensive influence matrix T, where $T = (t_{ij}) = \bar{X} (I - \bar{X})^{- 1}$ .

Step 4: Calculate the degree of influence, the extent of being affected, and the degree of centrality and degree of cause of each element, based on the element t_ij of the comprehensive influence matrix T obtained in step 3, where t_ij indicates the degree of direct impact and indirect impact of factor i on factor j, or the extent to which j is influenced by the factor i.

➀ Degree of influence: The sum of elements in each row of T is called the combined influence value of corresponding elements of the row on all other elements, and is also called degree of influence.

➁ Extent of being affected: The sum of the elements in each column of T is the combined influence of other elements in the column, which is also called the extent of being affected.

➂ Degree of centrality: The sum of the degree of influence and the extent of being affected of each element is called the degree of centrality of the element. It indicates the position of the element in the system and the size of its role.

➃ Degree of cause: The difference between the degree of influence and the degree of influence is called the degree of cause of the element.

a. Cause factor: If the degree of cause is greater than 0, it means that the element has a large influence on other factors and is called a cause factor.

b. Result factor: If the cause degree is less than 0, this element is affected by other factors and it is called the result element.

Here, we use degree of centrality to determine the attribute weights. The degree of influence r_i is expressed as: $\begin{matrix} r_{i} = \sum_{j = 1}^{n} t_{ij}, i = 1, 2, \dots, n . \end{matrix}$ The extent of being affected f_j is expressed as: $\begin{matrix} f_{j} = \sum_{i = 1}^{n} t_{ij}, j = 1, 2, \dots, n . \end{matrix}$ The attribute weights ω_i can be calculated using the degree of centrality as follows: $\begin{matrix} ω_{i} = \frac{r_{i} + f_{i}}{\sum_{i = 1}^{n} (r_{i} + f_{i})}, i = 1, 2, \dots n . \end{matrix}$

4.2 New MADM methods based on DEMATEL, fuzzy distance and LAM with TrFNNs

MADM refers to making decisions with multiple criteria and a limited number of predetermined alternatives. In complex environments, attribute values cannot be represented as exact numbers, often with incompleteness, uncertainty, and inconsistency. Thus, we are faced with a neutrosophic MADM. TrFNNs can not only represent the range of values of attribute values but also represent the most likely range of values for attribute values. In this study, we examine MADM in the TrFNN environment.

DEMATEL is a methodology proposed to solve complex and difficult problems in the real world. It is a systematic analysis using graph theory and matrix tools. It can determine the causal relationship between elements and the status of each element in the system through the logical relationship between the elements in the system and the direct influence matrix. Therefore, DEMATEL is very suitable in dealing with complex data, such as TrFNNs. The determination of attribute weights of existing TrFNN-based decision methods is mostly subjective assumptions, and more objective determination methods are not yet common. Therefore, in this study, we used the DEMATEL method to determine the attribute weights of decision information expressed as TrFNNs.

Let x₁, x₂, ⋯ x_m be a discrete set of m alternatives, and c₁, c₂, ⋯ c_n be the set of n attributes. ω_j is the weight of the attribute c_j (j = 1, 2, ⋯ , n), where ω_j ∈ [0, 1] (j = 1, 2, ⋯ , n), and $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that R = (n_ij) _m×n is the decision matrix, where n_ij takes the form of the TrFNN n_ij = 〈 (a_1ij, a_2ij, a_3ij, a_4ij), (b_1ij, b_2ij, b_3ij, b_4ij) , (c_1ij, c_2ij, c_3ij, c_4ij) 〉 (i = 1, 2, ⋯ m, j = 1, 2, ⋯ n) and (a_1ij, a_2ij, a_3ij, a_4ij) ∈ [0, 1], (b_1ij, b_2ij, b_3ij, b_4ij) ∈ [0, 1], (c_1ij, c_2ij, c_3ij, c_4ij) ∈ [0, 1], 0 ≤ a_4ij + b_4ij + c_4ij ≤ 3 for i = 1, 2, ⋯ m, j = 1, 2, ⋯ n, which describes the evaluation information of the attribute c_j with respect to the alternative x_i given by decision maker. Then, we can rank the order of the alternatives and obtain the best alternatives based on the given information. In the following, we propose a new MADM methods based on DEMATEL, fuzzy distance and LAM for trapezoidal fuzzy neutrosophic numbers. The steps are as follows:

Step 1: Give a direct influence matrix X^d = (a_ij) _n×n of the attribute pairs and the decision matrix R = (n_ij) _m×n provided by decision maker makers in the form of linguistic terms expressed easily by people, and then convert them into trapezoidal fuzzy neutrosophic numbers.

Step 2: Determine the weights of the attributes based on the DEMATEL method. For the specific calculation steps, see section 4.1.

Step 2.1: Obtain the normalized influence matrix $\bar{X} = ({\bar{a}}_{ij})_{n \times n}$ . We need to standardize decision information to ensure consistency of information. In general, attributes can be categorized into two types: benefit attributes and cost attributes. In this paper, the normalization of benefit type attribute values and cost type attribute values is as follows: $\begin{matrix} {\bar{a}}_{ij}^{b} = \\ 〈 \begin{matrix} (\frac{a_{1 ij}}{a_{4 max}}, \frac{a_{2 ij}}{a_{4 max}}, \frac{a_{3 ij}}{a_{4 max}}, \frac{a_{4 ij}}{a_{4 max}}), \\ (\frac{b_{1 ij}}{b_{4 max}}, \frac{b_{2 ij}}{b_{4 max}}, \frac{b_{3 ij}}{b_{4 max}}, \frac{b_{4 ij}}{b_{4 max}}), \\ (\frac{c_{1 ij}}{c_{4 max}}, \frac{c_{2 ij}}{c_{4 max}}, \frac{c_{3 ij}}{c_{4 max}}, \frac{c_{4 ij}}{c_{4 max}}) \end{matrix} 〉, \\ {\bar{a}}_{ij}^{c} = \\ 〈 \begin{matrix} (1 - \frac{a_{4 ij}}{a_{4 max}}, 1 - \frac{a_{3 ij}}{a_{4 max}}, 1 - \frac{a_{2 ij}}{a_{4 max}}, 1 - \frac{a_{1 ij}}{a_{4 max}}), \\ (1 - \frac{b_{4 ij}}{b_{4 max}}, 1 - \frac{b_{3 ij}}{b_{4 max}}, 1 - \frac{b_{2 ij}}{b_{4 max}}, 1 - \frac{b_{1 ij}}{b_{4 max}}), \\ (1 - \frac{c_{4 ij}}{c_{4 max}}, 1 - \frac{c_{3 ij}}{c_{4 max}}, 1 - \frac{c_{2 ij}}{c_{4 max}}, 1 - \frac{c_{1 ij}}{c_{4 max}}) \end{matrix} 〉 . \end{matrix}$ where ${\bar{a}}_{ij}^{b}$ represents the benefit type attribute, ${\bar{a}}_{ij}^{c}$ represents the cost type attribute, and $\begin{matrix} a_{4 max} & = max {max_{0 \leq i \leq m} a_{4 ij}, max_{0 \leq j \leq n} a_{4 ij}}, \\ b_{4 max} & = max {max_{0 \leq i \leq m} b_{4 ij}, max_{0 \leq j \leq n} b_{4 ij}}, \\ c_{4 max} & = max {max_{0 \leq i \leq m} c_{4 ij}, max_{0 \leq j \leq n} c_{4 ij}} . \end{matrix}$

Step 2.2: Get the comprehensive influence matrix T = (t_ij) _n×n = S (I - S) ^-1. For the sake of simplicity, we first convert the normalized trapezoidal fuzzy matrix to its score function value matrix S = (s_ij) _n×n, and then calculate its comprehensive influence matrix.

Step 2.3: Calculate the degree of influence r_i and the extent of being affected f_j to get the attribute weights:

$ω_{i} = \frac{r_{i} + f_{i}}{\sum_{i = 1}^{n} (r_{i} + f_{i})}, i = 1, 2, \dots n .$ (8)

Step 3: Calculate the weighted decision matrix $R^{'} = ({\tilde{n}}_{ij})_{m \times n} = (ω n_{ij})_{m \times n}$ according to the operational rules of Definition 3, where

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

Step 4: Calculate the fuzzy distance matrix R″ = (d_ij) _m×n between each alternative and the ideal solution for each attribute c_j according to Equation (7), and the ideal solution is expressed by I⁺ = 〈 (1, 1, 1, 1) , (0, 0, 0, 0) , (0, 0, 0, 0) 〉, where

$\begin{matrix} d_{ij} = D ({\tilde{n}}_{ij}, I^{+}) = 〈 (d_{1 ij}^{T}, d_{2 ij}^{T}, d_{3 ij}^{T}, d_{4 ij}^{T}), \\ (d_{1 ij}^{I}, d_{2 ij}^{I}, d_{3 ij}^{I}, d_{4 ij}^{I}), (d_{1 ij}^{F}, d_{2 ij}^{F}, d_{3 ij}^{F}, d_{4 ij}^{F}) 〉 . \end{matrix}$ (10)

Step 5: Calculate the score function matrix R^s = (s_ij) _m×n according to Equation (5), where s_ij = s (d_ij).

Step 6: Rank the alternatives according to the score function value s_ij of the fuzzy distances between each alternative value and ideal solution for each attribute c_j, and the s_ij is decreasing order.

Step 7: Determine the rank frequency matrix P = (p_ij) _m×m. Where p_ij is the frequency of alternative x_i ranked as the jth based on the s_ij, and P is as follows $\begin{matrix} P = (p_{ij})_{m \times m} = [\begin{matrix} p_{11} & p_{12} & \dots & p_{1 m} \\ p_{21} & p_{22} & \dots & p_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{m 1} & p_{m 2} & \dots & p_{mm} \end{matrix}] \end{matrix}$

Step 8: Get the linear programming model according to the rank frequency matrix in step 7. $\begin{matrix} max \sum_{i = 1}^{m} \sum_{j = 1}^{m} p_{ij} x_{ij} \\ s . t . {\begin{matrix} \sum_{j = 1}^{m} x_{ij} = 1, & i = 1, 2, \dots, m; \\ \sum_{i = 1}^{m} x_{ij} = 1, & j = 1, 2, \dots, m; \\ x_{ij} = 0 or 1, & i, j = 1, 2 . \dots m . \end{matrix} \end{matrix}$

Step 9: Solve the linear programming model of step 8 using the operations research professional software to obtain the matrix X = (x_ij) _m×m, where x_ij = 0 or 1.

Step 10: Obtaining the alternatives ranking result and the optimal alternative according to the X matrix of step 4. If the x_ij value of the ith row ranked in the jth column is equal 1, then the scheme i is ranked jth.

Step 11: End.

5 Typhoon disaster evaluation based on trapezoidal fuzzy neutrosophic information

In this section, we apply the new algorithm to solve the real problem of typhoon disaster evaluation. The typhoon is a major disastrous weather and may trigger multiple disaster chains such as hydrology, geology, oceans, transportation, environment, agriculture, etc. In China, typhoons primarily impact the eastern coastal regions of China, where the population is extremely dense, the economy is highly developed, and social wealth is notably concentrated. Fujian province is located between 115°50^′∼120°43^′ E, 28°19^′∼23°33^′ N, and is located in the subtropical land-sea transition zone, close to the world’s largest typhoon source. It is one of the most serious typhoons in China. For example [24], in 2017, affected by the successive landings of No.9 “Nasha” and No.10 “Haicang”, many places of Fujian Province were seriously affected. A total of 208,900 people in 59 counties were affected by the disaster, 434 houses were collapsed, and 273,300 people were urgently transferred; 26.73 thousand hectares of crops were affected, 10.19 thousand hectares were destroyed, and 2.19 thousand hectares were not collected. The double typhoon also caused Fujian to cancel 507 flights, 139 trains stopped, 4 highway controls, 6 national highways blocked, 4268 passengers stopped, Pingtan cross-sea bridge closed, the ferry to Taiwan was suspended. In terms of power supply communication, 727 lines were tripped on 10 kV lines, and 716,600 units of power were cut off by low-voltage users; 268 times of communication interruption and 12.9 kilometers of line damage. At the same time, 24 industrial and mining enterprises were discontinued; 11.63 km of damaged dikes were damaged, irrigation facilities and 1426 revetments were damaged, and 91 sluice gates, pond dams, hydrological stations, electromechanical wells and electromechanical pumping stations were damaged. According to incomplete statistics, the total direct economic loss of Fujian Province was 966 million yuan (RNB), of which water resources facilities lost 250 million yuan. With the development of the economy and the frequent activities of human activities, the assessment and treatment of typhoon disasters is particularly important. Therefore, we examine the problem of typhoon disaster evaluation in Fujian Province.

5.1 Illustrative example

We will use some indicators to evaluate the typhoon disaster effectively. The assessment indicators C ={ c₁, c₂, c₃, c₄ } include economic loss c₁, social impact c₂, environmental damage c₃, and other impact c₄ proposed by Yu [2]. Several experts are responsible for this assessment, and the evaluation information is expressed by linguistic terms and trapezoidal fuzzy neutrosophic numbers. For ease of understanding and presentation, the direct influence matrix X^d = (a_ij) _n×n of the attribute pairs and assessment matrix R = (n_ij) _m×n are given in the form of language terminology, and then converted into trapezoidal fuzzy neutrosophic numbers to calculate evaluation result based the correspondence table between linguistic terms and TrFNNs (see Table 1). The a_ij and n_ij are first expressed in linguistic terms. If a_ij is expressed by “Absolutely low” in the linguistic terms, it means that the degree of direct and indirect influence of the index i on the index j is absolutely low. If n_ij is expressed by “Fairly high” in the linguistic terms, it indicates the degree of disaster affected by a typhoon in i city under the j assessment indicator is fairly high. Other language terms have the same meaning and are not explained one by one.

Table 1
Correspondence between linguistic terms and trapezoidal fuzzy neutrosophic numbers

Linguistic terms Trapezoidal fuzzy neutrosophic numbers

Absolutely low 〈 (0.0, 0.0, 0.0, 0.0) , (1.0, 1.0, 1.0, 1.0) , (1.0, 1.0, 1.0, 1.0) 〉

Low 〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉

Fairly low 〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉

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

Fairly high 〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉

High 〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉

Absolutely high 〈 (1.0, 1.0, 1.0, 1.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉

Linguistic terms	Trapezoidal fuzzy neutrosophic numbers
Absolutely low	〈 (0.0, 0.0, 0.0, 0.0) , (1.0, 1.0, 1.0, 1.0) , (1.0, 1.0, 1.0, 1.0) 〉
Low	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉
Fairly low	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Medium	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉
Fairly high	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉
High	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉
Absolutely high	〈 (1.0, 1.0, 1.0, 1.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉

Step 1: First, we invite several experts to give the linguistic values of trapezoidal fuzzy neutrosophic numbers for linguistic terms shown in Table 1, and give the direct influence matrix X^d of the attribute pairs and the assessment matrix R as shown in Tables 2–3.

Table 2

Direct influence matrix of the attribute pairs X^d

Index\Index	c ₁	c ₂
c ₁	〈 (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉
c ₂	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉	〈 (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉
c ₃	〈 (1.0, 1.0, 1.0, 1.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉
c ₄	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉
Index\Index	c ₃	c ₄
c ₁	〈 (1.0, 1.0, 1.0, 1.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉
c ₂	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉
c ₃	〈 (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉
c ₄	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉	〈 (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉

Table 3

Evaluation matrix R

Cities\Index	c ₁	c ₂
Nanping (NP)	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Ningde (ND)	〈 (1.0, 1.0, 1.0, 1.0) , (0.0, 0.0, 0.0, 0.0) , (0.0, 0.0, 0.0, 0.0) 〉	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉
Sanming (SM)	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉
Fuzhou (FZ)	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉
Putian (PT)	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉
Longyan (LY)	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Quanzhou (QZ)	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉
Xiamen (XM)	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Zhangzhou (ZZ)	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉
Cities\Index	c ₃	c ₄
Nanping (NP)	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉
Ningde (ND)	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉
Sanming (SM)	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉
Fuzhou (FZ)	〈 (0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Putian (PT)	〈 (0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Longyan (LY)	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉
Quanzhou (QZ)	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉
Xiamen (XM)	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉	〈 (0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0) 〉
Zhangzhou (ZZ)	〈 (0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3) 〉	〈 (0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9) 〉

Step 2: Calculate the weight of the evaluation indicator.

Step 2.1: Generate the normalized influence matrix $\bar{X}$ as shown in Table 4.

Table 4

Normalized influence matrix $\bar{X}$

Index\Index	c ₁	c ₂
c ₁	〈 (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) 〉	〈 (0.70, 0.80, 0.90, 1.00) , (0.00, 0.25, 0.50, 0.75) , (0.00, 0.25, 0.50, 0.75) 〉
c ₂	〈 (0.70, 0.80, 0.90, 1.00) , (0.00, 0.11, 0.22, 0.33) , (0.00, 0.11, 0.22, 0.33) 〉	〈 (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) 〉
c ₃	〈 (1.00, 1.00, 1.00, 1.00) , (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) 〉	〈 (0.50, 0.60, 0.70, 0.80) , (0.00, 0.50, 0.75, 1.00) , (0.00, 0.50, 0.75, 1.00) 〉
c ₄	〈 (0.10, 0.20, 0.30, 0.40) , (0.44, 0.67, 0.78, 1.00) , (0.44, 0.67, 0.78, 1.00) 〉	〈 (0.70, 0.80, 0.90, 1.00) , (0.00, 0.25, 0.50, 0.75) , (0.00, 0.25, 0.50, 0.75) 〉
Index\Index	c ₃	c ₄
c ₁	〈 (1.00, 1.00, 1.00, 1.00) , (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) 〉	〈 (0.50, 0.60, 0.70, 0.80) , (0.00, 0.50, 0.75, 1.00) , (0.00, 0.50, 0.75, 1.00) 〉
c ₂	〈 (0.30, 0.40, 0.50, 0.60) , (0.20, 0.40, 0.50, 0.70) , (0.20, 0.40, 0.50, 0.70) 〉	〈 (0.70, 0.80, 0.90, 1.00) , (0.00, 0.25, 0.50, 0.75) , (0.00, 0.25, 0.50, 0.75) 〉
c ₃	〈 (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) 〉	〈 (0.70, 0.80, 0.90, 1.00) , (0.00, 0.25, 0.50, 0.75) , (0.00, 0.25, 0.50, 0.75) 〉
c ₄	〈 (0.00, 0.10, 0.20, 0.30) , (0.70, 0.80, 0.90, 1.00) , (0.70, 0.80, 0.90, 1.00) 〉	〈 (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) , (0.00, 0.00, 0.00, 0.00) 〉

Step 2.2: Get the comprehensive influence matrix T = (t_ij) _n×n = S (I - S) ^-1 based on python programming as shown in Tables 5–6.

Table 5

Score function value matrix S

Index\Index	c ₁	c ₂	c ₃	c ₄
c ₁	0.667	0.700	1.000	0.508
c ₂	0.839	0.667	0.517	0.700
c ₃	1.000	0.508	0.667	0.700
c ₄	0.269	0.700	0.150	0.667

Table 6

Comprehensive influence matrix T

Index\Index	c ₁	c ₂	c ₃	c ₄
c ₁	-0.516	-0.715	-0.021	-0.809
c ₂	-0.386	-0.441	-0.482	-0.427
c ₃	-0.290	-0.759	-0.305	-0.576
c ₄	-0.550	0.255	-0.718	0.192

Step 2.3: Using Equation (8) to get the attribute weights as follows: $\begin{matrix} ω_{1} = 0.290, ω_{2} = 0.259, ω_{3} = 0.264, ω_{4} = 0.186 . \end{matrix}$ Step 3: Using Equation (9), we can calculate the the weighted evaluation matrix R′ as shown in Table 7:

Table 7

Weighted evaluation matrix R′

Cities\Index	c ₁	c ₂
Nanping (NP)	$〈 \begin{matrix} (0.000, 0.030, 0.063, 0.098), (0.902, 0.937, 0.970, 1.000), \\ (0.902, 0.937, 0.970, 1.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.027, 0.056, 0.088, 0.124), (0.789, 0.876, 0.912, 0.973), \\ (0.789, 0.876, 0.912, 0.973) \end{matrix} 〉$
Ningde (ND)	$〈 \begin{matrix} (1.000, 1.000, 1.000, 1.000), (0.000, 0.000, 0.000, 0.000), \\ (0.000, 0.000, 0.000, 0.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.268, 0.341, 0.449, 1.000), (0.000, 0.551, 0.659, 0.732), \\ (0.000, 0.551, 0.659, 0.732) \end{matrix} 〉$
Sanming (SM)	$〈 \begin{matrix} (0.000, 0.030, 0.063, 0.098), (0.902, 0.937, 0.970, 1.000), \\ (0.902, 0.937, 0.970, 1.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.000, 0.027, 0.056, 0.088), (0.912, 0.944, 0.973, 1.000), \\ (0.912, 0.944, 0.973, 1.000) \end{matrix} 〉$
Fuzhou (FZ)	$〈 \begin{matrix} (0.295, 0.373, 0.487, 1.000), (0.000, 0.513, 0.627, 0.705), \\ (0.000, 0.513, 0.627, 0.705) \end{matrix} 〉$	$〈 \begin{matrix} (0.088, 0.124, 0.164, 0.211), (0.659, 0.789, 0.836, 0.912), \\ (0.659, 0.789, 0.836, 0.912) \end{matrix} 〉$
Putian (PT)	$〈 \begin{matrix} (0.182, 0.233, 0.295, 0.373), (0.000, 0.627, 0.705, 0.767), \\ (0.000, 0.627, 0.705, 0.767) \end{matrix} 〉$	$〈 \begin{matrix} (0.164, 0.211, 0.268, 0.341), (0.000, 0.659, 0.732, 0.789), \\ (0.000, 0.659, 0.732, 0.789) \end{matrix} 〉$
Longyan (LY)	$〈 \begin{matrix} (0.098, 0.138, 0.182, 0.233), (0.627, 0.767, 0.818, 0.902), \\ (0.627, 0.767, 0.818, 0.902) \end{matrix} 〉$	$〈 \begin{matrix} (0.027, 0.056, 0.088, 0.124), (0.789, 0.876, 0.912, 0.973), \\ (0.789, 0.876, 0.912, 0.973) \end{matrix} 〉$
Quanzhou (QZ)	$〈 \begin{matrix} (0.030, 0.063, 0.098, 0.138), (0.767, 0.862, 0.902, 0.970), \\ (0.767, 0.862, 0.902, 0.970) \end{matrix} 〉$	$〈 \begin{matrix} (0.088, 0.124, 0.164, 0.211), (0.659, 0.789, 0.836, 0.912), \\ (0.659, 0.789, 0.836, 0.912) \end{matrix} 〉$
Xiamen (XM)	$〈 \begin{matrix} (0.030, 0.063, 0.098, 0.138), (0.767, 0.862, 0.902, 0.970), \\ (0.767, 0.862, 0.902, 0.970) \end{matrix} 〉$	$〈 \begin{matrix} (0.027, 0.056, 0.088, 0.124), (0.789, 0.876, 0.912, 0.973), \\ (0.789, 0.876, 0.912, 0.973) \end{matrix} 〉$
Zhangzhou (ZZ)	$〈 \begin{matrix} (0.182, 0.233, 0.295, 0.373), (0.000, 0.627, 0.705, 0.767), \\ (0.000, 0.627, 0.705, 0.767) \end{matrix} 〉$	$〈 \begin{matrix} (0.088, 0.124, 0.164, 0.211), (0.659, 0.789, 0.836, 0.912), \\ (0.659, 0.789, 0.836, 0.912) \end{matrix} 〉$
Cities\Index	c ₃	c ₄
Nanping (NP)	$〈 \begin{matrix} (0.000, 0.027, 0.057, 0.090), (0.910, 0.943, 0.973, 1.000), \\ (0.910, 0.943, 0.973, 1.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.000, 0.019, 0.041, 0.064), (0.936, 0.959, 0.981, 1.000), \\ (0.936, 0.959, 0.981, 1.000) \end{matrix} 〉$
Ningde (ND)	$〈 \begin{matrix} (0.167, 0.215, 0.272, 0.346), (0.000, 0.654, 0.728, 0.785), \\ (0.000, 0.654, 0.728, 0.785) \end{matrix} 〉$	$〈 \begin{matrix} (0.064, 0.091, 0.121, 0.157), (0.741, 0.843, 0.879, 0.936), \\ (0.741, 0.843, 0.879, 0.936) \end{matrix} 〉$
Sanming (SM)	$〈 \begin{matrix} (0.000, 0.027, 0.057, 0.090), (0.910, 0.943, 0.973, 1.000), \\ (0.910, 0.943, 0.973, 1.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.000, 0.019, 0.041, 0.064), (0.936, 0.959, 0.981, 1.000), \\ (0.936, 0.959, 0.981, 1.000) \end{matrix} 〉$
Fuzhou (FZ)	$〈 \begin{matrix} (0.090, 0.126, 0.167, 0.215), (0.654, 0.785, 0.833, 0.910), \\ (0.654, 0.785, 0.833, 0.910) \end{matrix} 〉$	$〈 \begin{matrix} (0.019, 0.041, 0.064, 0.091), (0.843, 0.909, 0.936, 0.981), \\ (0.843, 0.909, 0.936, 0.981) \end{matrix} 〉$
Putian (PT)	$〈 \begin{matrix} (0.167, 0.215, 0.272, 0.346), (0.000, 0.654, 0.728, 0.785), \\ (0.000, 0.654, 0.728, 0.785) \end{matrix} 〉$	$〈 \begin{matrix} (0.019, 0.041, 0.064, 0.091), (0.843, 0.909, 0.936, 0.981), \\ (0.843, 0.909, 0.936, 0.981) \end{matrix} 〉$
Longyan (LY)	$〈 \begin{matrix} (0.027, 0.057, 0.090, 0.126), (0.785, 0.874, 0.910, 0.973), \\ (0.785, 0.874, 0.910, 0.973) \end{matrix} 〉$	$〈 \begin{matrix} (0.000, 0.019, 0.041, 0.064), (0.936, 0.959, 0.981, 1.000), \\ (0.936, 0.959, 0.981, 1.000) \end{matrix} 〉$
Quanzhou (QZ)	$〈 \begin{matrix} (0.000, 0.027, 0.057, 0.090), (0.910, 0.943, 0.973, 1.000), \\ (0.910, 0.943, 0.973, 1.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.019, 0.041, 0.064, 0.091), (0.843, 0.909, 0.936, 0.981), \\ (0.843, 0.909, 0.936, 0.981) \end{matrix} 〉$
Xiamen (XM)	$〈 \begin{matrix} (0.027, 0.057, 0.090, 0.126), (0.785, 0.874, 0.910, 0.973), \\ (0.785, 0.874, 0.910, 0.973) \end{matrix} 〉$	$〈 \begin{matrix} (0.000, 0.019, 0.041, 0.064), (0.936, 0.959, 0.981, 1.000), \\ (0.936, 0.959, 0.981, 1.000) \end{matrix} 〉$
Zhangzhou (ZZ)	$〈 \begin{matrix} (0.272, 0.346, 0.455, 1.000), (0.000, 0.545, 0.654, 0.728), \\ (0.000, 0.545, 0.654, 0.728) \end{matrix} 〉$	$〈 \begin{matrix} (0.019, 0.041, 0.064, 0.091), (0.843, 0.909, 0.936, 0.981), \\ (0.843, 0.909, 0.936, 0.981) \end{matrix} 〉$

Step 4: Using the Equation (7) to get the fuzzy distance matrix R″ between each alternative and the ideal solution I⁺ = 〈(1, 1, 1, 1) , (0, 0, 0, 0) , (0, 0, 0, 0) 〉 for each attribute as shown in Table 8:

Table 8

Fuzzy distance matrix R″ between each alternative and the ideal solution

Cities\Index	c ₁	c ₂
Nanping (NP)	$〈 \begin{matrix} (0.929, 0.944, 0.960, 0.978), (0.927, 0.945, 0.961, 0.976), \\ (0.927, 0.945, 0.961, 0.976) \end{matrix} 〉$	$〈 \begin{matrix} (0.904, 0.918, 0.934, 0.952), (0.837, 0.881, 0.899, 0.929), \\ (0.837, 0.881, 0.899, 0.929) \end{matrix} 〉$
Ningde (ND)	$〈 \begin{matrix} (0.000, 0.000, 0.000, 0.000), (0.000, 0.000, 0.000, 0.000), \\ (0.000, 0.000, 0.000, 0.000) \end{matrix} 〉$	$〈 \begin{matrix} (0.422, 0.458, 0.513, 0.788), (0.223, 0.498, 0.552, 0.589), \\ (0.223, 0.498, 0.552, 0.589) \end{matrix} 〉$
Sanming (SM)	$〈 \begin{matrix} (0.929, 0.944, 0.960, 0.978), (0.927, 0.945, 0.961, 0.976), \\ (0.927, 0.945, 0.961, 0.976) \end{matrix} 〉$	$〈 \begin{matrix} (0.936, 0.950, 0.964, 0.981), (0.934, 0.950, 0.965, 0.978), \\ (0.934, 0.950, 0.965, 0.978) \end{matrix} 〉$
Fuzhou (FZ)	$〈 \begin{matrix} (0.394, 0.433, 0.490, 0.746), (0.213, 0.469, 0.526, 0.565), \\ (0.213, 0.469, 0.526, 0.565) \end{matrix} 〉$	$〈 \begin{matrix} (0.825, 0.843, 0.863, 0.887), (0.727, 0.792, 0.815, 0.853), \\ (0.727, 0.792, 0.815, 0.853) \end{matrix} 〉$
Putian (PT)	$〈 \begin{matrix} (0.688, 0.714, 0.745, 0.784), (0.239, 0.552, 0.591, 0.622), \\ (0.239, 0.552, 0.591, 0.622) \end{matrix} 〉$	$〈 \begin{matrix} (0.716, 0.740, 0.768, 0.805), (0.247, 0.577, 0.613, 0.642), \\ (0.247, 0.577, 0.613, 0.642) \end{matrix} 〉$
Longyan (LY)	$〈 \begin{matrix} (0.806, 0.826, 0.848, 0.874), (0.700, 0.770, 0.796, 0.838), \\ (0.700, 0.770, 0.796, 0.838) \end{matrix} 〉$	$〈 \begin{matrix} (0.904, 0.918, 0.934, 0.952), (0.837, 0.881, 0.899, 0.929), \\ (0.837, 0.881, 0.899, 0.929) \end{matrix} 〉$
Quanzhou (QZ)	$〈 \begin{matrix} (0.893, 0.909, 0.927, 0.946), (0.820, 0.868, 0.887, 0.921), \\ (0.820, 0.868, 0.887, 0.921) \end{matrix} 〉$	$〈 \begin{matrix} (0.825, 0.843, 0.863, 0.887), (0.727, 0.792, 0.815, 0.853), \\ (0.727, 0.792, 0.815, 0.853) \end{matrix} 〉$
Xiamen (XM)	$〈 \begin{matrix} (0.893, 0.909, 0.927, 0.946), (0.820, 0.868, 0.887, 0.921), \\ (0.820, 0.868, 0.887, 0.921) \end{matrix} 〉$	$〈 \begin{matrix} (0.904, 0.918, 0.934, 0.952), (0.837, 0.881, 0.899, 0.929), \\ (0.837, 0.881, 0.899, 0.929) \end{matrix} 〉$
Zhangzhou (ZZ)	$〈 \begin{matrix} (0.688, 0.714, 0.745, 0.784), (0.239, 0.552, 0.591, 0.622), \\ (0.239, 0.552, 0.591, 0.622) \end{matrix} 〉$	$〈 \begin{matrix} (0.825, 0.843, 0.863, 0.887), (0.727, 0.792, 0.815, 0.853), \\ (0.727, 0.792, 0.815, 0.853) \end{matrix} 〉$
Cities\Index	c ₃	c ₄
Nanping (NP)	$〈 \begin{matrix} (0.935, 0.949, 0.964, 0.980), (0.933, 0.949, 0.964, 0.978), \\ (0.933, 0.949, 0.964, 0.978) \end{matrix} 〉$	$〈 \begin{matrix} (0.954, 0.964, 0.974, 0.986), (0.952, 0.964, 0.975, 0.984), \\ (0.952, 0.964, 0.975, 0.984) \end{matrix} 〉$
Ningde (ND)	$〈 \begin{matrix} (0.712, 0.736, 0.764, 0.801), (0.246, 0.573, 0.610, 0.639), \\ (0.246, 0.573, 0.610, 0.639) \end{matrix} 〉$	$〈 \begin{matrix} (0.871, 0.884, 0.899, 0.917), (0.794, 0.845, 0.863, 0.891), \\ (0.794, 0.845, 0.863, 0.891) \end{matrix} 〉$
Sanming (SM)	$〈 \begin{matrix} (0.935, 0.949, 0.964, 0.980), (0.933, 0.949, 0.964, 0.978), \\ (0.933, 0.949, 0.964, 0.978) \end{matrix} 〉$	$〈 \begin{matrix} (0.954, 0.964, 0.974, 0.986), (0.952, 0.964, 0.975, 0.984), \\ (0.952, 0.964, 0.975, 0.984) \end{matrix} 〉$
Fuzhou (FZ)	$〈 \begin{matrix} (0.822, 0.840, 0.861, 0.885), (0.722, 0.788, 0.812, 0.851), \\ (0.722, 0.788, 0.812, 0.851) \end{matrix} 〉$	$〈 \begin{matrix} (0.930, 0.940, 0.952, 0.965), (0.879, 0.912, 0.926, 0.948), \\ (0.879, 0.912, 0.926, 0.948) \end{matrix} 〉$
Putian (PT)	$〈 \begin{matrix} (0.712, 0.736, 0.764, 0.801), (0.246, 0.573, 0.610, 0.639), \\ (0.246, 0.573, 0.610, 0.639) \end{matrix} 〉$	$〈 \begin{matrix} (0.930, 0.940, 0.952, 0.965), (0.879, 0.912, 0.926, 0.948), \\ (0.879, 0.912, 0.926, 0.948) \end{matrix} 〉$
Longyan (LY)	$〈 \begin{matrix} (0.902, 0.917, 0.933, 0.951), (0.834, 0.879, 0.897, 0.928), \\ (0.834, 0.879, 0.897, 0.928) \end{matrix} 〉$	$〈 \begin{matrix} (0.954, 0.964, 0.974, 0.986), (0.952, 0.964, 0.975, 0.984), \\ (0.952, 0.964, 0.975, 0.984) \end{matrix} 〉$
Quanzhou (QZ)	$〈 \begin{matrix} (0.935, 0.949, 0.964, 0.980), (0.933, 0.949, 0.964, 0.978), \\ (0.933, 0.949, 0.964, 0.978) \end{matrix} 〉$	$〈 \begin{matrix} (0.930, 0.940, 0.952, 0.965), (0.879, 0.912, 0.926, 0.948), \\ (0.879, 0.912, 0.926, 0.948) \end{matrix} 〉$
Xiamen (XM)	$〈 \begin{matrix} (0.902, 0.917, 0.933, 0.951), (0.834, 0.879, 0.897, 0.928), \\ (0.834, 0.879, 0.897, 0.928) \end{matrix} 〉$	$〈 \begin{matrix} (0.954, 0.964, 0.974, 0.986), (0.952, 0.964, 0.975, 0.984), \\ (0.952, 0.964, 0.975, 0.984) \end{matrix} 〉$
Zhangzhou (ZZ)	$〈 \begin{matrix} (0.417, 0.454, 0.509, 0.781), (0.221, 0.493, 0.548, 0.585), \\ (0.221, 0.493, 0.548, 0.585) \end{matrix} 〉$	$〈 \begin{matrix} (0.930, 0.940, 0.952, 0.965), (0.879, 0.912, 0.926, 0.948), \\ (0.879, 0.912, 0.926, 0.948) \end{matrix} 〉$

Step 5: Using the Equation (5) to obtain the score function value matrix R^s as shown in Table 9:

Table 9

Score function value matrix R^s

Cities\Index	c ₁	c ₂	c ₃	c ₄
Nanping (NP)	0.350	0.385	0.348	0.344
Ningde (ND)	0.667	0.538	0.573	0.399
Sanming (SM)	0.350	0.348	0.348	0.344
Fuzhou (FZ)	0.543	0.420	0.422	0.371
Putian (PT)	0.577	0.572	0.573	0.371
Longyan (LY)	0.429	0.385	0.386	0.344
Quanzhou (QZ)	0.390	0.420	0.348	0.371
Xiamen (XM)	0.390	0.385	0.386	0.344
Zhangzhou (ZZ)	0.577	0.420	0.539	0.371

Step 6: Rank the alternatives according to the score function value s_ij for each attribute c_j.

For c₁, we can get $\begin{matrix} s (ND) & ≻ s (PT) \sim (ZZ) ≻ s (FZ) ≻ s (LY) \\ ≻ s (QZ) \sim (XM) ≻ s (NP) \sim (SM) . \end{matrix}$ For c₂, we can get $\begin{matrix} s (PT) & ≻ s (ND) ≻ s (FZ) \sim (QZ) \sim (ZZ) \\ ≻ s (LY) \sim (XM) \sim (NP) ≻ s (SM) . \end{matrix}$ For c₃, we can get $\begin{matrix} s (ND) & \sim (PT) ≻ s (ZZ) ≻ s (FZ) ≻ s (LY) \\ \sim (XM) ≻ s (NP) \sim (SM) \sim (QZ) . \end{matrix}$ For c₄, we can get $\begin{matrix} s (ND) & ≻ s (FZ) \sim (PT) \sim (QZ) \sim (ZZ) \\ ≻ s (NP) \sim (SM) \sim (LY) \sim (XM) . \end{matrix}$ Step 7: According to the step 6, we can get the rank frequency matrix P and the ranking results for each attribute of Fig. 5: $P = (\begin{array}{l} 1 2 3 4 5 6 7 8 9 \\ NP 0 0 0 0 0 2 1 1 0 \\ ND 3 1 0 0 0 0 0 0 0 \\ SM 0 0 0 0 0 1 1 1 1 \\ FZ 0 1 1 2 0 0 0 0 0 \\ PT 2 2 0 0 0 0 0 0 0 \\ LY 0 0 0 0 2 2 0 0 0 \\ QZ 0 1 1 0 0 1 1 0 0 \\ XM 0 0 0 0 1 3 0 0 0 \\ ZZ 0 2 2 0 0 0 0 0 0 \end{array})$

Step 8: According to the step 7, we get the linear programming model as shown below: $\begin{matrix} s . t . {\begin{matrix} p_{11} + p_{12} + p_{13} + p_{14} + p_{15} \\ + p_{16} + p_{17} + p_{18} + p_{19} = 1, \\ p_{21} + p_{22} + p_{23} + p_{24} + p_{25} \\ + p_{26} + p_{27} + p_{28} + p_{29} = 1, \\ p_{31} + p_{32} + p_{33} + p_{34} + p_{35} \\ + p_{36} + p_{37} + p_{38} + p_{39} = 1, \\ p_{41} + p_{42} + p_{43} + p_{44} + p_{45} \\ + p_{46} + p_{47} + p_{48} + p_{49} = 1, \\ p_{51} + p_{52} + p_{53} + p_{54} + p_{55} \\ + p_{56} + p_{57} + p_{58} + p_{59} = 1, \\ p_{61} + p_{62} + p_{63} + p_{64} + p_{65} \\ + p_{66} + p_{67} + p_{68} + p_{69} = 1, \\ p_{71} + p_{72} + p_{73} + p_{74} + p_{75} \\ + p_{76} + p_{77} + p_{78} + p_{79} = 1, \\ p_{81} + p_{82} + p_{83} + p_{84} + p_{85} \\ + p_{86} + p_{87} + p_{88} + p_{89} = 1, \\ p_{91} + p_{92} + p_{93} + p_{94} + p_{95} \\ + p_{96} + p_{97} + p_{98} + p_{99} = 1, \\ p_{11} + p_{21} + p_{31} + p_{41} + p_{51} \\ + p_{61} + p_{71} + p_{81} + p_{91} = 1, \\ p_{12} + p_{22} + p_{32} + p_{42} + p_{52} \\ + p_{62} + p_{72} + p_{82} + p_{92} = 1, \\ p_{13} + p_{23} + p_{33} + p_{43} + p_{53} \\ + p_{63} + p_{73} + p_{83} + p_{93} = 1, \\ p_{14} + p_{24} + p_{34} + p_{44} + p_{54} \\ + p_{64} + p_{74} + p_{84} + p_{94} = 1, \\ p_{15} + p_{25} + p_{35} + p_{45} + p_{55} \\ + p_{65} + p_{75} + p_{85} + p_{95} = 1, \\ p_{16} + p_{26} + p_{36} + p_{46} + p_{56} \\ + p_{66} + p_{76} + p_{86} + p_{96} = 1, \\ p_{17} + p_{27} + p_{37} + p_{47} + p_{57} \\ + p_{67} + p_{77} + p_{87} + p_{97} = 1, \\ p_{18} + p_{28} + p_{38} + p_{48} + p_{58} \\ + p_{68} + p_{78} + p_{88} + p_{98} = 1, \\ p_{19} + p_{29} + p_{39} + p_{49} + p_{59} \\ + p_{69} + p_{79} + p_{89} + p_{99} = 1, \\ p_{ij} = 0 or 1, i, j = 1, 2, \dots, 10 . \end{matrix} \end{matrix}$

$\begin{matrix} max z \\ = 2 p_{16} + p_{17} + p_{18} + 3 p_{21} + p_{22} + p_{36} \\ + p_{37} + p_{38} + p_{39} + p_{42} + p_{43} + 2 p_{44} \\ + 2 p_{51} + 2 p_{52} + 2 p_{65} + 2 p_{66} + p_{72} + p_{73} \\ + p_{76} + p_{77} + p_{85} + 3 p_{86} + 2 p_{92} + 2 p_{93} \end{matrix}$

Fig.5

City ranking results for each attribute.

Step 9: Using software to solve the 0-1 integer programming problem of step 8 to obtain the solution matrix S as follows: $S = (\begin{array}{l} 1 2 3 4 5 6 7 8 9 \\ NP 0 0 0 0 0 0 0 1 0 \\ ND 1 0 0 0 0 0 0 0 0 \\ SM 0 0 0 0 0 0 0 0 1 \\ FZ 0 0 0 1 0 0 0 0 0 \\ PT 0 1 0 0 0 0 0 0 0 \\ LY 0 0 0 0 1 0 0 0 0 \\ QZ 0 0 0 0 0 0 1 0 0 \\ XM 0 0 0 0 0 1 0 0 0 \\ ZZ 0 0 1 0 0 0 0 0 0 \end{array})$ Step 10: According to that principle: x_ij value of the ith row ranked in the jth column is equal 1, then the scheme i is ranked jth. Here, in the solution matrix x_ND1 = 1, x_PT2 = 1, x_ZZ3 = 1, x_FZ4 = 1, x_LY5 = 1, x_XM6 = 1, x_QZ7 = 1, x_NP8 = 1, x_SM9 = 1, So the ranking order of the nine cities is ND ≻ PT ≻ ZZ ≻ FZ ≻ LY ≻ XM ≻ QZ ≻ NP ≻ SM.

Step 11: End

5.2 Solutions to linguistic term setting problem

In the case study presented in the previous section, we used linguistic terms to express decision information as linguistic information is more in line with human cognition and preference. However, the limited nature of linguistic terms will make some alternatives indistinguishable for each attribute. Although this problem will not have much impact on decision results, we still propose a solution to the problem of predictive term setting.

In the illustrative example in Section 5.1, there are many equivalent issues when sorting alternatives for each attribute due to the insufficiency of language terminologies. For a precise ordering, in this study, we calculated the comprehensive score value of each scheme and further considered the comprehensive score function value when the alternative score function values are equal, so that more accurate scheme ranking results can be obtained.

Here, the comprehensive score function value of each alternatives is expressed as s (x_i), and s (NP) =0.093, s (ND) =0.142, s (SM) =0.091, s (FZ) =0.115, s (PT) =0.137, s (LY) =0.101, s (QZ) =0.100, s (XM) =0.098, s (ZZ) =0.125.

Therefore, the results of the step 6 to the step 10 of the method of section 5.1 are as follows:

The ranking result of alternatives for each attribute based on comprehensive score function value of each alternatives is as shown below and in Fig. 6:

Fig.6

City ranking results for attribute and comprehensive score function value.

For c₁, we can get $\begin{matrix} s (ND) & ≻ s (PT) ≻ s (ZZ) ≻ s (FZ) ≻ s (LY) \\ ≻ s (QZ) ≻ s (XM) ≻ s (NP) ≻ s (SM) . \end{matrix}$ For c₂, we can get $\begin{matrix} s (PT) & ≻ s (ND) ≻ s (ZZ) ≻ s (FZ) ≻ s (QZ) \\ ≻ s (LY) ≻ s (XM) ≻ s (NP) ≻ s (SM) . \end{matrix}$ For c₃, we can get $\begin{matrix} s (ND) & ≻ s (PT) ≻ s (ZZ) ≻ s (FZ) ≻ s (LY) \\ ≻ s (XM) ≻ s (QZ) ≻ s (NP) ≻ s (SM) . \end{matrix}$

For c₄, we can get $\begin{matrix} s (ND) & ≻ s (PT) ≻ s (ZZ) ≻ s (FZ) ≻ s (QZ) \\ ≻ s (LY) ≻ s (XM) ≻ s (NP) ≻ s (SM) . \end{matrix}$

The rank frequency matrix P² as follows: $\begin{array}{l} P^{2} = (\begin{array}{l} 1 2 3 4 5 6 7 8 9 \\ NP 0 0 0 0 0 0 0 4 0 \\ ND 3 1 0 0 0 0 0 0 0 \\ SM 0 0 0 0 0 0 0 0 4 \\ FZ 0 0 0 4 0 0 0 0 0 \\ PT 1 3 0 0 0 0 0 0 0 \\ LY 0 0 0 0 2 2 0 0 0 \\ QZ 0 0 0 0 2 1 1 0 0 \\ XM 0 0 0 0 0 1 3 0 0 \\ ZZ 0 0 4 0 0 0 0 0 0 \end{array}) \\ S^{2} = (\begin{array}{l} 1 2 3 4 5 6 7 8 9 \\ NP 0 0 0 0 0 0 0 1 0 \\ ND 1 0 0 0 0 0 0 0 0 \\ SM 0 0 0 0 0 0 0 0 1 \\ FZ 0 0 0 1 0 0 0 0 0 \\ PT 0 1 0 0 0 0 0 0 0 \\ LY 0 0 0 0 0 1 0 0 0 \\ QZ 0 0 0 0 1 0 0 0 0 \\ XM 0 0 0 0 0 0 1 0 0 \\ ZZ 0 0 1 0 0 0 0 0 0 \end{array}) \end{array}$ Then, according to the same calculation process, the alternative ranking result can be obtained. In the solution matrix S², x_ND1 = 1, x_PT2 = 1, x_ZZ3 = 1, x_FZ4 = 1, x_QZ5 = 1, x_LY6 = 1, x_XM7 = 1, x_NP8 = 1, x_SM9 = 1, So the ranking order of the nine cities is ND ≻ PT ≻ ZZ ≻ FZ ≻ QZ ≻ LY ≻ XM ≻ NP ≻ SM. Compared with the method of Section 5.1, the two sorting results are basically the same, but there are small differences. The reason is that the case where the score function values of several alternatives are equal is reduced, and the sort result is better.

5.3 Comparative analysis and results discussion

5.3.1 Comparative analysis of evaluation methods based on different distance measures

In this section, in order to illustrate the rationality and effectiveness of our proposed fuzzy distance of TrFNNs, we compared the proposed decision method based on fuzzy distance with some other existing methods based on non-fuzzy distance (because of the study of fuzzy distance for TrFNNs has not been seen in the existing literature). First, the distance measurement proposed in [31, 78] is briefly introduced, and then several methods are compared based on different distances. The results of the comparative analysis are shown in Table 10 and Fig. 7.

Table 10
Ranking results using different distance measures

Cities\Different methods Cosine similarity-based non-fuzzy distance measure proposed by [31, 30] (CSBNFDM) Non-fuzzy Hamming distance measure proposed by Tan [78] (λ = 1) (NFHDM)

Distance from ideal solution Ranking results Distance from ideal solution Ranking results

c ₁ c ₂ c ₃ c ₄ c ₁ c ₂ c ₃ c ₄

Nanping (NP) 0.9646 0.9415 0.9678 0.9773 8 0.9522 0.9003 0.9564 0.9689 8

Ningde (ND) 0.0000 0.4814 0.7293 0.9107 1 0.0000 0.4855 0.6111 0.8639 1

Sanming (SM) 0.9646 0.9684 0.9678 0.9773 9 0.9522 0.9572 0.9564 0.9689 9

Fuzhou (FZ) 0.4451 0.8719 0.8692 0.9587 4 0.4613 0.8169 0.8138 0.9269 4

Putian (PT) 0.7005 0.7348 0.7293 0.9587 2 0.5929 0.6146 0.6111 0.9269 2

Longyan (LY) 0.8550 0.9415 0.9403 0.9773 5 0.7979 0.9003 0.8986 0.9689 5

Quanzhou (QZ) 0.9340 0.8719 0.9678 0.9587 7 0.8894 0.8169 0.9564 0.9269 7

Xiamen (XM) 0.9340 0.9415 0.9403 0.9773 6 0.8894 0.9003 0.8986 0.9689 6

Zhangzhou (ZZ) 0.7005 0.8719 0.4755 0.9587 3 0.5929 0.8169 0.4815 0.9269 3

Cities\Different methods Non-fuzzy Euclidean distance measure proposed by Tan [78] (λ = 2) (NFEDM) Proposed fuzzy distance measurement method

Distance from ideal solution Ranking results Ranking results

c ₁ c ₂ c ₃ c ₄

Nanping (NP) 0.9529 0.9024 0.9570 0.9692 8 8

Ningde (ND) 0.0000 0.5643 0.6717 0.8663 1 1

Sanming (SM) 0.9529 0.9577 0.9570 0.9692 9 9

Fuzhou (FZ) 0.5370 0.8212 0.8182 0.9281 4 4

Putian (PT) 0.6522 0.6755 0.6717 0.9281 2 2

Longyan (LY) 0.8031 0.9024 0.9007 0.9692 5 5

Quanzhou (QZ) 0.8919 0.8212 0.9570 0.9281 7 7

Xiamen (XM) 0.8919 0.9024 0.9007 0.9692 6 6

Zhangzhou (ZZ) 0.6522 0.8212 0.5598 0.9281 3 3

Cities\Different methods	Cosine similarity-based non-fuzzy distance measure proposed by [31, 30] (CSBNFDM)	Non-fuzzy Hamming distance measure proposed by Tan [78] (λ = 1) (NFHDM)
Nanping (NP)	0.9646	0.9415	0.9678	0.9773	8	0.9522	0.9003	0.9564	0.9689	8
Ningde (ND)	0.0000	0.4814	0.7293	0.9107	1	0.0000	0.4855	0.6111	0.8639	1
Sanming (SM)	0.9646	0.9684	0.9678	0.9773	9	0.9522	0.9572	0.9564	0.9689	9
Fuzhou (FZ)	0.4451	0.8719	0.8692	0.9587	4	0.4613	0.8169	0.8138	0.9269	4
Putian (PT)	0.7005	0.7348	0.7293	0.9587	2	0.5929	0.6146	0.6111	0.9269	2
Longyan (LY)	0.8550	0.9415	0.9403	0.9773	5	0.7979	0.9003	0.8986	0.9689	5
Quanzhou (QZ)	0.9340	0.8719	0.9678	0.9587	7	0.8894	0.8169	0.9564	0.9269	7
Xiamen (XM)	0.9340	0.9415	0.9403	0.9773	6	0.8894	0.9003	0.8986	0.9689	6
Zhangzhou (ZZ)	0.7005	0.8719	0.4755	0.9587	3	0.5929	0.8169	0.4815	0.9269	3
Cities\Different methods	Non-fuzzy Euclidean distance measure proposed by Tan [78] (λ = 2) (NFEDM)	Proposed fuzzy distance measurement method
	Distance from ideal solution	Ranking results	Ranking results
	c ₁	c ₂	c ₃	c ₄
Nanping (NP)	0.9529	0.9024	0.9570	0.9692	8	8
Ningde (ND)	0.0000	0.5643	0.6717	0.8663	1	1
Sanming (SM)	0.9529	0.9577	0.9570	0.9692	9	9
Fuzhou (FZ)	0.5370	0.8212	0.8182	0.9281	4	4
Putian (PT)	0.6522	0.6755	0.6717	0.9281	2	2
Longyan (LY)	0.8031	0.9024	0.9007	0.9692	5	5
Quanzhou (QZ)	0.8919	0.8212	0.9570	0.9281	7	7
Xiamen (XM)	0.8919	0.9024	0.9007	0.9692	6	6
Zhangzhou (ZZ)	0.6522	0.8212	0.5598	0.9281	3	3

Fig.7

Ranking results using different distance measures: (a) (b) Cosine similarity-based non-fuzzy distance, (c) (d) Non-fuzzy Hamming distance, (e) (f) Non-fuzzy Euclidean distance measure.

(1) Non-fuzzy distance measure proposed by Tan [31] based on cosine similarity measure [30]. $\begin{matrix} S_{TrFNN} ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = \\ \frac{\sum_{i = 1}^{4} a_{i} e_{i} + \sum_{i = 1}^{4} b_{i} f_{i} + \sum_{i = 1}^{4} c_{i} g_{i}}{\begin{matrix} (\sqrt{\sum_{i = 1}^{4} (a_{i})^{2} + \sum_{i = 1}^{4} (b_{i})^{2} + \sum_{i = 1}^{4} (c_{i})^{2}}) \\ \times (\sqrt{\sum_{i = 1}^{4} (e_{i})^{2} + \sum_{i = 1}^{4} (f_{i})^{2} + \sum_{i = 1}^{4} (g_{i})^{2}}) \end{matrix}}, \\ D_{2} ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = 1 - S_{TrFNN} ({\tilde{n}}_{1}, {\tilde{n}}_{2}) . \end{matrix}$

(2) Non-fuzzy distance measure proposed by Tan [78]. $\begin{matrix} D_{3} ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = {\frac{1}{12} (\sum_{i = 1}^{4} | a_{i} - e_{i} |^{λ} \\ {+ \sum_{i = 1}^{4} | b_{i} - f_{i} |^{λ} + \sum_{i = 1}^{4} | c_{i} - g_{i} |^{λ})}}^{1 / λ}, λ \geq 0 . \end{matrix}$

From the results in Table 10 and Fig. 7, it can be seen that the evaluation results are consistent using four different distance measures. To a certain extent, it illustrates that our proposed evaluation method based on fuzzy distance measure is feasible and reasonable. At the same time, compared with the general non-fuzzy distance measurement, the fuzzy distance can retain the evaluation information to the greatest extent and avoid the loss of information.

5.3.2 Comparative analysis of evaluation methods based on different ranking methods

In this section, we compare our ranking method with the other two most widely used ranking methods (that is, the TOPSIS-based ranking method and the score function and the accuracy function-based ranking method) to illustrate the advantages of the proposed method based on the linear assignment method.

Here, we briefly introduce the evaluation calculation based on the TOPSIS method. First of all, we determine positive ideal solutions P⁺ and negative ideal solutions P^- are respectively: $\begin{matrix} P^{+} & = (\begin{matrix} 〈 \begin{matrix} (1.000, 1.000, 1.000, 1.000), \\ (0.000, 0.000, 0.000, 0.000), \\ (0.000, 0.000, 0.000, 0.000) \end{matrix}, 〉 \\ 〈 \begin{matrix} (1.000, 1.000, 1.000, 1.000), \\ (0.000, 0.000, 0.000, 0.000), \\ (0.000, 0.000, 0.000, 0.000) \end{matrix}, 〉 \\ 〈 \begin{matrix} (1.000, 1.000, 1.000, 1.000), \\ (0.000, 0.000, 0.000, 0.000), \\ (0.000, 0.000, 0.000, 0.000) \end{matrix}, 〉 \\ 〈 \begin{matrix} (1.000, 1.000, 1.000, 1.000), \\ (0.000, 0.000, 0.000, 0.000), \\ (0.000, 0.000, 0.000, 0.000) \end{matrix} 〉 \end{matrix}) \\ P^{-} & = (\begin{matrix} 〈 \begin{matrix} (0.000, 0.000, 0.000, 0.000), \\ (1.000, 1.000, 1.000, 1.000), \\ (1.000, 1.000, 1.000, 1.000) \end{matrix}, 〉 \\ 〈 \begin{matrix} (0.000, 0.000, 0.000, 0.000), \\ (1.000, 1.000, 1.000, 1.000), \\ (1.000, 1.000, 1.000, 1.000) \end{matrix}, 〉 \\ 〈 \begin{matrix} (0.000, 0.000, 0.000, 0.000), \\ (1.000, 1.000, 1.000, 1.000), \\ (1.000, 1.000, 1.000, 1.000) \end{matrix}, 〉 \\ 〈 \begin{matrix} (0.000, 0.000, 0.000, 0.000), \\ (1.000, 1.000, 1.000, 1.000), \\ (1.000, 1.000, 1.000, 1.000) \end{matrix} 〉 \end{matrix}) \end{matrix}$ Then, we use Tan’s distance measure [31] based on cosine similarity measure [30] to calculate the distance $D_{i}^{+} (x_{i}, P^{+})$ between the solution and the positive ideal solution, and the distance $D_{i}^{-} (x_{i}, P^{-})$ between the solution and the negative ideal solution, and calculate the relative closeness coefficients $U_{i} = \frac{D_{i}^{-} (x_{i}, P^{-})}{D_{i}^{+} (x_{i}, P^{-}) + D_{i}^{-} (x_{i}, P^{-})}$ as shown in Table 11 and Fig. 8.

Table 11
Distances and the relative closeness coefficient

$D_{i}^{+} (x_{i}, P^{+})$ $D_{i}^{-} (x_{i}, P^{-})$ U _i

Nanping (NP) 0.901 0.203 0.200

Ningde (ND) 0.239 0.828 0.776

Sanming (SM) 0.901 0.203 0.184

Fuzhou (FZ) 0.441 0.529 0.546

Putian (PT) 0.306 0.660 0.683

Longyan (LY) 0.711 0.293 0.292

Quanzhou (QZ) 0.754 0.274 0.266

Xiamen (XM) 0.795 0.247 0.237

Zhangzhou (ZZ) 0.362 0.617 0.631

	$D_{i}^{+} (x_{i}, P^{+})$	$D_{i}^{-} (x_{i}, P^{-})$	U _i
Nanping (NP)	0.901	0.203	0.200
Ningde (ND)	0.239	0.828	0.776
Sanming (SM)	0.901	0.203	0.184
Fuzhou (FZ)	0.441	0.529	0.546
Putian (PT)	0.306	0.660	0.683
Longyan (LY)	0.711	0.293	0.292
Quanzhou (QZ)	0.754	0.274	0.266
Xiamen (XM)	0.795	0.247	0.237
Zhangzhou (ZZ)	0.362	0.617	0.631

Fig.8

City ranking results for TOPSIS.

As can be seen from Table 11 and Fig. 8, U_ND > U_PT > U_ZZ > U_FZ > U_LY > U_QZ > U_XM >U_NP > U_SM, so ranking order of the nine cities is ND ≻ PT ≻ ZZ ≻ FZ ≻ LY ≻ QZ ≻ XM ≻ NP ≻ SM. Compared with the previous two methods, the ranking results are basically the same, but slightly different. However, the most affected cities and the least affected cities are consistent.

Here, we briefly introduce the application of the TOPSIS ranking method in our examples. The other methods are not described in detail. The comparative analysis results of the four methods are shown in Table 12 and Fig. 9.

Table 12

Ranking results using different ranking methods

Cities\Different methods	Proposed method based on LAM (LAM)			TOPSIS-based method (TOPSIS)
	Frequency values	Ranking results	Support	Closeness coefficients	Ranking results
Nanping (NP)	<0, 0, 0, 0, 0, 2, 1, 1, 0>	8	0.2500	0.2004	8
Ningde (ND)	<3, 1, 0, 0, 0, 0, 0, 0, 0>	1	0.7500	0.7759	1
Sanming (SM)	<0, 0, 0, 0, 0, 1, 1, 1, 1>	9	0.2500	0.1842	9
Fuzhou (FZ)	<0, 1, 1, 2, 0, 0, 0, 0, 0>	4	0.5000	0.5456	4
Putian (PT)	<2, 2, 0, 0, 0, 0, 0, 0, 0>	2	0.5000	0.6833	2
Longyan (LY)	<0, 0, 0, 0, 2, 2, 0, 0, 0>	5	0.5000	0.2917	5
Quanzhou (QZ)	<0, 1, 1, 0, 0, 1, 1, 0, 0>	7	0.2500	0.2663	6
Xiamen (XM)	<0, 0, 0, 0, 1, 3, 0, 0, 0>	6	0.7500	0.2370	7
Zhangzhou (ZZ)	<0, 2, 2, 0, 0, 0, 0, 0, 0>	3	0.5000	0.6306	3
Cities\Different methods	Traditional function-based method (TFBM) [29]		Optimal degree-based method proposed by Garg and Nancy [79] (ODBM)
	Score function values	Ranking results	Optimal degree values	Ranking results
Nanping (NP)	0.0931	8	0.0715	8
Ningde (ND)	0.1420	1	0.1583	1
Sanming (SM)	0.0907	9	0.0618	9
Fuzhou (FZ)	0.1146	4	0.1284	4
Putian (PT)	0.1366	2	0.1523	2
Longyan (LY)	0.1007	5	0.1001	6
Quanzhou (QZ)	0.0998	6	0.1007	5
Xiamen (XM)	0.0982	7	0.0911	7
Zhangzhou (ZZ)	0.1244	3	0.1416	3

Fig.9

Ranking results for four different ranking methods.

Here, the TOPSIS-based method (TOPSIS) calculates the distance $D_{i}^{+}$ between each solution and the positive ideal solution and the distance $D_{i}^{-}$ between each solution and the negative ideal solution to obtain the relative closeness coefficient values for every alternative. The traditional function-based method (TFBM) [29] need to calculate the values of the score function and accuracy function, so as to obtain the ranking results. The optimal degree-based method proposed by Garg and Nancy [79] (ODBM) is used for interval-valued neutral numbers. We simply extend the method to trapezoidal fuzzy neutrosophic numbers. When calculating the likelihood matrix, the trapezoidal fuzzy neutrosophic numbers are first converted to interval numbers. That is, take the average of the first two numbers of the trapezoidal fuzzy number of the three membership degrees as the lower bound of the interval number, take the average of the last two numbers of the trapezoidal fuzzy number as the lower bound of the interval number, and then calculate the optimal degree value to obtain the ranking result. For the proposed LAM method, in order to compare with other methods, we calculated the degree of support for the final ranking result of each evaluation object, that is, the number of votes supported divided by the total number of votes, and the total number of votes was is the number of attributes. As shown in Table 12 and Fig. 9, the evaluation results based on the four methods are basically the same, which shows that our proposed method based on the LAM method is feasible and effective. Moreover, our LAM method can not only see the ranking frequency of each evaluation object for each attribute from the frequency values, but also get its final ranking result.

Compared with other related studies, the advantages of our method mainly include several aspects:

(1) Decision information in complex environments is generally expressed as inexact numbers (e.g., fuzzy numbers, interval numbers, trapezoidal fuzzy numbers, intuitionistic fuzzy numbers, hesitant fuzzy numbers, neutrosophic numbers). Most existing distance measurements convert fuzzy information and uncertain information into accurate numbers. This information processing method is beneficial to humans in recognizing and reducing the amount of computation, but at the cost of losing some information. Moreover, there is not much research on fuzzy distance measurement [64, 65], especially in the neutrosophic number environment. Therefore, our fuzzy distance measure based onTrFNNs research is a supplement and extension to the theory of distance measurement.

(2) DEMATEL can determine the causal relationship between elements and the status of each element in a system through a simple matrix calculation. We use DEMATEL to obtain attribute weights, and the calculation is simple and easy to understand. Compared with the existing subjective assumption methods, our method is more objective and makes the decision results more effective and reasonable. Compared with the existing methods based on information entropy, linear programming, and maximum deviation, among others, our method has a more useful discussion. Moreover, DEMATEL has not been used to determine attribute weights in the TrFNN environment. Therefore, our research is a supplement and extension to the decision-making of NSs.

(3) This study uses the LAM to rank the alternatives in a simple and easy-to-understand method. The LAM method can obtain the frequency value by ranking the evaluation objects from various aspects. Thus, we can not only examine the statistics of each evaluation object at each ranking level by the frequency value but also finally obtain the overall ranking of the evaluation objects.

6 Conclusions

In this paper, a typhoon disaster evaluation approach based on DEMATEL and LAM under the TrFNN environment is proposed. First, we extended the DEMATEL for TrFNN to obtain the attribute weights and defined the new fuzzy distance of TrFNNs based on GMIR. Then, we ranked alternatives using the LAM. Finally, we successfully applied the proposed decision-making method to the evaluation of typhoon disaster assessments. In addition, we made two comparative analyses to illustrate the feasibility and rationality of our method. One is to solve the problem that the solution cannot be compared due to the lack of detailed language terminologies, and the other is to compare our method with the widely used TOPSIS method. The findings of this research will be helpful to deepen the study on typhoon disaster evaluations and improve decision making in disaster reduction and disaster prevention. In future research, we will expand the proposed method and apply it to other natural disaster assessment problems. We will continue to study the related theory of SVNSs, IvNSs, bipolar NSs, neutrosophic hesitant FSs, MVNSs, simplified neutrosophic linguistic sets, LSVNSs, picture FS, PFS, and their applications in typhoon disaster evaluation problems.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Footnotes

Acknowledgments

This work is supported by the National Social Science Foundation of China (No.17CGL058).

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