An extended ITARA-TOPSIS method for multi-criteria group decision-making problems based on R-number

Abstract

With the continuous improvement and development of various decision-making methods, it has led to the widespread use of fuzzy sets and fuzzy numbers. At the same time, the application of decision-making methods in different fuzzy environments has been very effective in addressing the deficiencies in existing research. At present, triangular fuzzy numbers have been widely used in the evaluation aspects of various decision making methods, and the proposed R-number effectively solve the uncertainty involving problems related to future events, but the existing research based on the TOPSIS method in the R-number environment has not yet been clearly applied to the triangular fuzzy number environment, and the indifference threshold-based attribute ratio analysis (ITARA) method in the fuzzy environment has yet to be extended. Therefore, this paper proposes a fuzzy indifference threshold-based attribute ratio analysis (FITARA) method based on triangular fuzzy numbers for solving the problem of determining attribute weights in the multi-attribute decision-making process. Secondly, the various risks of the decision environment and the impact on future events are considered and R-number are used to solve this puzzle. In addition, the incorporation of risk perception factors in the context of the existing RTOPSIS method considering multiple risk factors and the use of Manhattan distances to optimize the large number of operations in the process of the method resulted in the development of the FITARA-RTOPSIS model. Finally, the proposed FITARA-RTOPSIS method is applied to the problem of siting emergency supplies storage depots, and the effectiveness of the proposed method is verified through comparative analysis.

Keywords

FITARA R-number RTOPSIS Manhattan distance TFN

1 Introduction

Uncertainty about risks can create uncertainty about the impact of existing assessment results on future events. In numerous decision problems, especially about future events, there is a degree of error or mistake in the data analyzed during the decision-making process. The causes of these errors and mistakes are, in turn, unreliable sources of information or effects on future events, and these similar factors are collectively referred to as risk factors [1]. To ensure the comprehensiveness of considerations in the decision-making process, the concept of R-number to model the risks and errors associated with fuzzy numbers was proposed by Seiti [2] et al. in 2019. The various aspects of risk in this study are considered to be the main factors influencing the assessment, such as pessimistic risk, optimistic risk, pessimistic acceptable risk, optimistic acceptable risk, and the perception of risk by different experts. The R-SAW method was also developed for application to the supplier selection problem based on triangular fuzzy numbers. Seiti [3] et al. proposed the R-TOPSIS method for risk-based preventive maintenance planning and used a steel rolling mill as an example to illustrate the effectiveness of the proposed method. Mousavi [4] et al. proposed the risk-based R-VIKOR method, using an explanatory model to eventually propose a simple and practical decision-making tool and to illustrate the effectiveness of the proposed method through its application in new product development projects. Seiti [5] et al. proposed a flexible risk prioritization hierarchy based on the limitations of the traditional FMEA, culminating in a risk-based fuzzy information processing and decision-making approach (R-SECA), illustrating the effectiveness of the proposed model through the case of a steel plant. Zhao [6] et al. proposed a hybrid decision framework incorporating SMAA, ROR, and MABAC methods for models with risk factors and preference relationships, defined new operations for R-number and TFN, and finally illustrated the superiority of the framework through wind energy potential assessment. Cheng [7] et al. proposed an RN-MARICA method for multi-attribute decision-making by considering the flexibility of R-number and ideal comparison analysis method and illustrated the feasibility of the proposed method through the risk analysis of 5G base station project construction. Zhao [8] et al. proposed a hybrid multi-criteria decision-aid framework based on R-number and preference models, firstly proposing new operations on R-number and triangular fuzzy numbers, and later proposing the R-MABAC method for the uncertainty and error problems of triangular fuzzy numbers, and finally using wind energy resource potential assessment as an example for comparative analysis to verify the effectiveness and superiority of the proposed method.

The determination of the attribute weights is the basis for ensuring that the evaluation results are valid. Numerous methods exist for determining attribute weights, which can be divided into two main categories: subjective weights and objective weights. Common methods for subjective weighting include the Analytic Hierarchy Process method (AHP) [9], Best Worst method (BWM) [10], Analytic Network Process method (ANP) [11], etc. This type of approach is heavily influenced by expert preferences, and if the expert lacks knowledge or experience in a particular context, then the results of the weights are unreliable. Common methods of objective weighting include entropy weighting [12], standard deviation [13], CRITIC [14], ITARA [15], etc. Such methods are effective in avoiding bias caused by the lack of knowledge and experience of experts while reducing the discounting of relevant information. Hatefi [15] proposed an attribute ratio analysis method (ITARA) based on an undifferentiated threshold. The method is based on the dispersion of the measured data to determine the attribute weights in a multi-attribute decision problem. Ulutas [16] et al. proposed the CCSD-ITARA-MARCOS multi-criteria decision method and illustrated the reliability and stability of the proposed method with a case study of the selection of the best manual stacker for a small warehouse. Liu [17] et al. proposed a multi-attribute decision probabilistic hesitant fuzzy classification method based on BWM and ITARA, which combines subjective weights with objective weights. The Taxonomy method is also extended to probabilistic hesitant fuzzy sets (PHFS) to illustrate the effectiveness of the method with the selection of research topics. Simic [18] et al. proposed the T2NN-ITARA-EDAS method for determining and ranking semi-objective weights, illustrating the validity of the proposed method through a case study of a transport route.

The various evaluation indicators used to measure the attributes of evaluation objects are all fuzzy. In 1965, Zadeh [19] proposed the concept of fuzzy sets. With the continuous research of scholars, fuzzy sets have developed into intuitionistic fuzzy sets [20], hesitant fuzzy sets [21], Pythagorean fuzzy sets [22], picture fuzzy sets [23], Pythagorean hesitant fuzzy sets [24], and so on. The triangular fuzzy number [25] is a method for transforming fuzzy and uncertain linguistic variables into definite values, and its use in evaluation methods is a good solution to the situation where the performance of the evaluated object cannot be accurately measured. For the time being, triangular fuzzy numbers have been used in a wide range of studies for evaluation sessions.

The TOPSIS method is a method for solving multi-objective decision problems proposed by Hwang [26] et al. in 1981. The method transforms a multi-objective decision-making problem into a bi-objective decision-making problem based on the principle that the decision outcome should be as close as possible to a positive ideal solution and as far as possible from a negative ideal solution. Chen [27] extended TOPSIS to the fuzzy environment and proposed a method to calculate the distance between two triangular fuzzy numbers by calculating the distance between the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS) to determine the ranking of the alternatives. Pei [28] proposed a method to construct an improved TOPSIS approach with linguistic evaluation, which was used to address the inability of fuzzy TOPSIS to different choices in a linguistic context. Yoon [29] et al. proposed the behavioral TOPSIS, which reformulates the traditional TOPSIS and incorporates the behavioral tendencies of decision-makers, and the proposed approach is more in line with behavioral economics. Pei [30] et al. proposed the fuzzy language TOPSIS method for linguistic decision-making, et al. The alternatives are obtained by transforming the linguistic evaluation into a fuzzy linguistic set, aggregating to obtain positive and negative ideal solutions, and using distance to transform into proximity. Silva [31] et al. proposed a new TOPSIS-Sort-C ranking method to prevent rank reversal during the ranking process and illustrated the effectiveness of the method with a numerical application. Chen [32] et al. investigated the effect of entropy weights on the TOPSIS method, finding that entropy weight(EW) enhances the function of attributes with the highest multidimensionality in attribute data and weakens the function of attributes with low DAD in decision making, eventually proposing an entropy TOPSIS method with adjustable weight coefficients. Wang [33] et al. proposed the generalized three-way decision BWM-TOPSIS method, which generalized the classical TOPSIS method by adding a third intermediate point, and eventually proved the feasibility of the proposed model through comparative analysis. Hooshangi [34] et al. proposed a geographical information system (GIS) based TOPSIS method with a more reasonable sensitivity that depends on the degree of uncertainty in the numbers and can change depending on the expert’s knowledge background, ultimately illustrating the validity of the method through an analysis of the assessment of potential addresses for solar power plants in Iran.

With the development of various decision-making methods, fuzzy sets and fuzzy numbers, scholars’ research has been intensified, but there are still areas for improvement and gaps. Firstly, R-number were proposed to solve risk problems such as uncertainty of future events, but the application of R-number in various decision-making methods has not been perfected yet. For example, the existing RTOPSIS method is limited to the environment of trapezoidal fuzzy numbers, and the consideration of risk factors does not incorporate expert risk perception factors, and has not been extended to the field of group decision making. Secondly, the ITARA method is an easy to understand and effective method for determining attribute weights, but the application of the method in a fuzzy environment needs to be extended, especially the application of the ITARA method in a triangular fuzzy number environment is still in a blank state in the existing research. Finally, the proposed R-number method effectively solves the problem of risk in the decision-making process, but its complexity of operations and the large amount of operations in the group decision-making environment still need further improvement.

Faced with the current decision-making environment full of riskiness and the widespread use of various fuzzy numbers, this paper aims to propose the ITARA method in a fuzzy environment and the TOPSIS group decision making method in an R-number environment. The contribution of this paper is reflected in the following:

(1) The ITARA method in a fuzzy environment (FITARA) is proposed to solve the problem of applying the ITARA method in a triangular fuzzy number environment.

(2) The proposed TOPSIS method in a risk environment (RTOPSIS) is extended to group decision making, incorporating expert risk perception factors and using Manhattan distance instead of the traditional Euclidean distance. The improved RTOPSIS method has advantages in terms of application environment and computing.

(3) We have developed a new multi-criteria group decision making method, FITARA-RTOPSIS, which has a wide range of applications and can be applied not only to the case of this paper, but also to other multi-attribute group decision making problems.

(4) Effectively alleviates the computational complexity of the R-number application process without affecting the decision outcome.

The rest of the paper is organized as follows: Section 2 introduces the basic concepts of ITARA, R-number, and TOPSIS. Section 3 carries out the model construction, provides a detailed description of FITARA, and the improved RTOPSIS, and constructs the basic FITARA-RTOPSIS framework. Section 4 ranks the options, using the issue of sitting in an emergency supplies depot as an example. Section 5 conducts a sensitivity analysis and a comparative analysis. Section 6 concludes by explaining the feasibility of the proposed methodology and suggesting future research directions.

2 Preliminaries

This section introduces some basic concepts of ITARA, R-number, and TOPSIS.

2.1 ITARA

Hatefi [15] proposes an ITARA method for determining attribute weights in MADM problems, based on the concept of " Indifference Threshold (IT)". This approach considers the dispersion of the decision maker’s evaluation under each attribute and then uses the information in the decision matrix to determine the attribute weights. The method consists of six steps as follows.

Step 1. Establish a decision matrix. The decision matrix has m alternatives and n attributes, a_ij denotes the evaluation of the j attribute under the i scheme, where i∈ M = { 1, 2, . . . , m } , j ∈ N = { 1, 2, . . . , n }. In addition, determine the undifferentiated threshold for each attribute.

Step 2. Normalize the decision matrix and IT_j. $α_{ij} = \frac{a_{ij}}{\sum_{i = 1}^{m} a_{ij}}, \forall i \in M, \forall j \in N$ (1)

$NI T_{j} = \frac{I T_{j}}{\sum_{i = 1}^{m} a_{ij}}, \forall j \in N$ (2)

Step 3. Sort the normalized ratings in ascending order under each attribute, and redefined as β_ij, where β_ij ≤ β_i+1,j.

Step 4. Calculate the difference between adjacent scores γ_ij by Equation (3). $γ_{ij} = β_{i + 1, j} - β_{ij}, γ_{ij} \geq 0$ (3)

Step 5. Calculate the difference between γ_ij and NIT_j by Equation (4). $δ_{ij} = {\begin{matrix} γ_{ij} - NI T_{j}, γ_{ij} > NI T_{j} \\ 0, γ_{ij} \leq NI T_{j} \end{matrix}$ (4)

Step 6. Determine attribute weights by Equation (5), where p is a model parameter taking values in the range 1 to ∞. When p takes different values, γ_ij can be interpreted as different distance measures. Commonly, when p=1, it represents the Manhattan distance, and when p=2, the Euclidean distance. $W_{j} = \frac{V_{j}}{\sum_{j = 1}^{n} V_{j}}, V_{j} = {(\sum_{i = 1}^{m - 1} {δ_{ij}}^{p})}^{1 / p}, \forall j \in N$ (5)

2.2 R-number

The R-number is based on triangular fuzzy numbers, and the accuracy of the decision result is improved by considering the uncertainty of the fuzzy numbers themselves. Three main factors are considered, fuzzy risk, acceptable risk, and expert risk perception. It also presents the R-number for the beneficial and unbeneficial models, taking into account the pessimistic-optimistic model of risk allocation.

2.2.1 Triangular fuzzy number (TFN)

Definition 1 [35] Triangular fuzzy number in R-number, expressed in terms of the affiliation function as by Equation (6). As shown in Fig. 1. $μ_{\tilde{A}} (x) = {\begin{matrix} (x - a_{1}) / (a_{2} - a_{1}), a_{1} \leq x < a_{2} \\ (a_{3} - x) / (a_{3} - a_{2}), a_{2} \leq x \leq a_{3} \\ 0, otherwise \end{matrix}$ (6)

Fig. 1

Triangular fuzzy number.

where a₁ and a₃ are the lower and upper bounds of the fuzzy number, respectively, and a₂ denotes the modal value.

2.2.2 Definitions and arithmetic rules

Definition 2 [2] The triangular fuzzy numbers with upper and lower bounds are expressed as and in the beneficial-unbeneficial model of R-number in the following Equations (7) – (10).

Beneficial model: $R_{b} (\tilde{A}) = (R_{1 b} (\tilde{A}), R_{2 b} (\tilde{A}), R_{3 b} (\tilde{A})), where$ (7) ${\begin{matrix} R_{1 b} (\tilde{A}) = max (\tilde{A} \otimes (1 ⊖ min (\frac{{\tilde{r}}^{-}}{1 ⊖ \tilde{R P^{-}}}, ɛ) \otimes (1 ⊖ \tilde{A R^{-}})), l_{\tilde{A}}) \\ R_{2 b} (\tilde{A}) = \tilde{A} \\ R_{3 b} (\tilde{A}) = min (\tilde{A} \otimes (1 \oplus \frac{{\tilde{r}}^{+}}{1 ⊖ \tilde{R P^{+}}} \otimes (1 ⊖ \tilde{A R^{+}})), u_{\tilde{A}}) \\ 0 < {\tilde{r}}^{-} < 1, {\tilde{r}}^{+} > 0 \\ 0 \leq \tilde{A R^{-}}, \tilde{A R^{+}} < 1 \end{matrix}$ (8) ∥Non-Beneficial model: $R_{c} (\tilde{A}) = (R_{1 c} (\tilde{A}), R_{2 c} (\tilde{A}), R_{3 c} (\tilde{A})), where$ (9) ${\begin{matrix} R_{1 c} (\tilde{A}) = max (\tilde{A} \otimes (1 ⊖ min (\frac{{\tilde{r}}^{+}}{1 ⊖ \tilde{R P^{+}}}, ɛ) \otimes (1 ⊖ \tilde{A R^{+}})), l_{\tilde{A}}) \\ R_{2 c} (\tilde{A}) = \tilde{A} \\ R_{3 c} (\tilde{A}) = min (\tilde{A} \otimes (1 \oplus \frac{{\tilde{r}}^{-}}{1 ⊖ \tilde{R P^{-}}} \otimes (1 ⊖ \tilde{A R^{-}})), u_{\tilde{A}}) \\ {\tilde{r}}^{-} > 1, 1 < {\tilde{r}}^{+} > 0 \\ 0 \leq \tilde{A R^{-}}, \tilde{A R^{+}} < 1 \end{matrix}$ (10) where ɛ is a number infinitely close to 1, r⁺ and r^- represent fuzzy positive risk and fuzzy negative risk respectively, which can lead to fuzzy numbers getting better or worse in the future. $\tilde{A R^{+}}$ and $\tilde{A R^{-}}$ denote the acceptable positive and negative risks to the decision maker, respectively, and it has a value between 0 and 1. $\tilde{R P^{+}}$ and $\tilde{R P^{-}}$ indicate that the decision maker’s perception of risk is positive or negative, three common scenarios are as follows.∥Active experts: $0 < \tilde{R P^{-}}, \tilde{R P^{+}} < 1$ ∥Neutral experts:0∥Pessimistic experts: $- \infty < \tilde{R P^{-}}, \tilde{R P^{+}} < 0$ ∥Definition 3 [2] For any two R-numbers $R (\tilde{A})$ and $R (\tilde{B})$ , whether in the beneficial or non-beneficial mode, the following operator rules can be established by Equation (11). ${\begin{matrix} R (\tilde{A}) \oplus R (\tilde{B}) = (R_{1} (\tilde{A}) \oplus R_{1} (\tilde{B}), R_{2} (\tilde{A}) \oplus R_{2} (\tilde{B}), R_{3} (\tilde{A}) \oplus R_{3} (\tilde{B})) \\ R (\tilde{A}) ⊖ R (\tilde{B}) = (R_{1} (\tilde{A}) ⊖ R_{3} (\tilde{B}), R_{2} (\tilde{A}) ⊖ R_{2} (\tilde{B}), R_{3} (\tilde{A}) ⊖ R_{1} (\tilde{B})) \\ R (\tilde{A}) \otimes R (\tilde{B}) = (R_{1} (\tilde{A}) \otimes R_{1} (\tilde{B}), R_{2} (\tilde{A}) \otimes R_{2} (\tilde{B}), R_{3} (\tilde{A}) \otimes R_{3} (\tilde{B})) \\ R (\tilde{A}) ø R (\tilde{B}) = (R_{1} (\tilde{A}) ø R_{3} (\tilde{B}), R_{2} (\tilde{A}) ø R_{2} (\tilde{B}), R_{3} (\tilde{A}) ø R_{1} (\tilde{B})) \\ λ \otimes R (\tilde{A}) = (λ \otimes R_{1} (\tilde{A}), λ \otimes R_{2} (\tilde{A}), λ \otimes R_{3} (\tilde{A})) \end{matrix}$ (11) ∥Let two R-numbers $R (\tilde{A})$ and $R (\tilde{B})$ , where $R (\tilde{A}) = ((a_{11}, a_{12}, a_{13}), (a_{21}, a_{22}, a_{23}), (a_{31}, a_{32}, a_{33}))$ , $R (\tilde{B}) = ((b_{11}, b_{12}, b_{13}), (b_{21}, b_{22}, b_{23}), (b_{31}, b_{32}, b_{33}))$ , Then the operation rules are defined in Equations (12) – (16) below.

$\begin{matrix} R (\tilde{A}) \oplus R (\tilde{B}) = & ((a_{11} + b_{11}, a_{12} + b_{12}, a_{13} + b_{13}), (a_{21} + b_{21}, a_{22} + b_{22}, a_{23} + b_{23}), \\ (a_{31} + b_{31}, a_{32} + b_{32}, a_{33} + b_{33})) \end{matrix}$ (12)

$\begin{matrix} R (\tilde{A}) ⊖ R (\tilde{B}) = & ((a_{11} - b_{13}, a_{12} - b_{12}, a_{13} - b_{11}), (a_{21} - b_{23}, a_{22} - b_{22}, a_{23} - b_{21}), \\ (a_{31} - b_{33}, a_{32} - b_{32}, a_{33} - b_{31})) \end{matrix}$ (13)

$\begin{matrix} R (\tilde{A}) \otimes R (\tilde{B}) = ((a_{11} b_{11}, a_{12} b_{12}, a_{13} b_{13}), \\ (a_{21} b_{21}, a_{22} b_{22}, a_{23} b_{23}), (a_{31} b_{31}, a_{32} b_{32}, a_{33} b_{33})) \end{matrix}$ (14)

$\begin{matrix} R (\tilde{A}) ø R (\tilde{B}) = ((\frac{a_{11}}{b_{33}}, \frac{a_{12}}{b_{32}}, \frac{a_{13}}{b_{31}}), \\ (\frac{a_{21}}{b_{23}}, \frac{a_{22}}{b_{22}}, \frac{a_{23}}{b_{21}}), (\frac{a_{31}}{b_{13}}, \frac{a_{32}}{b_{12}}, \frac{a_{33}}{b_{11}})) \end{matrix}$ (15)

$\begin{matrix} λ \otimes R (\tilde{A}) = ((λ a_{11}, λ a_{12}, λ a_{13}), \\ (λ a_{21}, λ a_{22}, λ a_{23}), (λ a_{31}, λ a_{32}, λ a_{33})) \end{matrix}$ (16)

Definition 4 [2] For an R-number $R (\tilde{A}) = ((a_{11}, a_{12}, a_{13}), (a_{21}, a_{22}, a_{23}), (a_{31}, a_{32}, a_{33}))$ , clear values can be obtained by performing a defuzzification operation according to Equation (17).

$\begin{matrix} COA (R (\tilde{A})) = \frac{1}{9} (a_{11} + a_{12} + a_{13} + a_{21} + a_{22} \\ + a_{23} + a_{31} + a_{32} + a_{33}) \end{matrix}$ (17)

2.3 TOPSIS

The steps involved in the TOPSIS method are described below.

Step 1. Forming a decision matrix.

Assuming K decision maker, m options and n criteria. Firstly, the evaluation matrix D_k and the decision maker weighting matrix W are constructed.

$\begin{matrix} D_{k} = {[S_{ij}^{k}]}_{m \times n}, i = 1, 2, . . ., m; j = 1, 2, . . ., n; \\ k = 1, 2, . . ., K; \end{matrix}$ (18)

$W = {[W_{k}]}_{1 \times k}, k = 1, 2, . . ., K;$ (19) where D_k is the evaluation matrix for the K decision maker, $S_{ij}^{k}$ is the assessed value of the K decision maker under the attributes of i program j. W_k is the weighting of K decision makers.

Step 2. Aggregate the decision matrix.

The evaluation matrix of each decision maker is aggregated using Equation (20) and the final aggregation matrix is denoted by D^T.

$S_{ij}^{T} = {\sum_{k = 1}^{K} [S_{ij}^{k} \otimes W_{k}]}_{m \times n}$ (20)

$D^{T} = {[S_{ij}^{T}]}_{m \times n}$ (21)

Step 3. Aggregate matrix normalization.

Using Equation (22) for normalization, the final normalization matrix is denoted by U.

${\begin{matrix} u_{ij} = \frac{S_{ij}^{T}}{C_{j}^{+}}, C_{j}^{+} = {max}_{i = 1}^{n} S_{ij}^{T}, \\ u_{ij} = \frac{C_{j}^{-}}{S_{ij}^{T}}, C_{j}^{-} = {min}_{i = 1}^{n} S_{ij}^{T} \end{matrix}$ (22)

$U = {[u_{ij}]}_{m \times n}, i = 1, 2, . . ., m; j = 1, 2, . . ., n;$ (23)

Step 4. Calculate the weighted normalized decision matrix.

Multiplying the normalization matrix and each criterion weight gives the weighted normalized decision matrix, denoted by V.

$v_{ij} = u_{ij} \times w_{j}$ (24)

$V = {[v_{ij}]}_{m \times n}, i = 1, 2, . . ., m; j = 1, 2, . . ., n;$ (25)

Step 5. Determine the positive ideal solution and the negative ideal solution.

${\begin{matrix} A^{+} = {max v_{ij} | j = 1, 2, . . ., n} \\ A^{-} = {min v_{ij} | j = 1, 2, . . ., n} \end{matrix}$ (26)

Step 6. Calculate the distance between the positive and negative ideal solutions for each alternative.

${\begin{matrix} l_{i}^{+} = \sqrt{\sum_{i = 1}^{m} {(v_{ij} - v_{j}^{+})}^{2}}, wherej = 1, 2, . . ., n \\ l_{i}^{-} = \sqrt{\sum_{i = 1}^{m} {(v_{ij} - v_{j}^{-})}^{2}}, wherej = 1, 2, . . ., n \end{matrix}$ (27)

Step 7. Calculate the relative closeness of the alternative solution to the ideal solution.

${\begin{matrix} C_{i}^{+} = \frac{l_{i}^{+}}{l_{i}^{+} + l_{i}^{-}}, i = 1, 2, . . ., m \\ C_{i}^{-} = \frac{l_{i}^{-}}{l_{i}^{+} + l_{i}^{-}}, i = 1, 2, . . ., m \end{matrix}$ (28)

Step 8. Rank decision alternative.

The requirements are ranked according to their relative closeness and the best solution is then selected.

3 Model

In this section, the fuzzy ITARA(FITARA) method and the improved RTOPSIS method are first proposed, followed by the construction of the FITARA-RTOPSIS model.

3.1 The FITARA method

Step 1. Construct a fuzzy decision matrix E based on triangular fuzzy numbers, as detailed in Equations (29) and (30). $\begin{matrix} E = {[E_{ij}]}_{m \times n} = [\begin{matrix} (a_{11}^{l}, a_{11}^{m}, a_{11}^{u}) (a_{12}^{l}, a_{12}^{m}, a_{12}^{u}) \dots (a_{1 n}^{l}, a_{1 n}^{m}, a_{1 n}^{u}) \\ (a_{21}^{l}, a_{21}^{m}, a_{21}^{u}) (a_{22}^{l}, a_{22}^{m}, a_{22}^{u}) \dots (a_{2 n}^{l}, a_{2 n}^{m}, a_{2 n}^{u}) \\ ⋮ ⋱ ⋮ \\ (a_{m 1}^{l}, a_{m 1}^{m}, a_{m 1}^{u}) (a_{m 2}^{l}, a_{m 2}^{m}, a_{m 2}^{u}) \dots (a_{mn}^{l}, a_{mn}^{m}, a_{mn}^{u}) \end{matrix}], i = 1, 2, . . ., m; j = 1, 2, . . ., n; \end{matrix}$ (29) $where, E_{ij} = (a_{ij}^{l}, a_{ij}^{m}, a_{ij}^{u})$ (30)

E_ij is a TFN, denotes the evaluation value of option i under attribute j.

Step 2. Normalize decision matrix.

Here, the decision matrix is normalized by Equation (31), which after normalization is denoted by Equation (32). ${\begin{matrix} E_{ij}^{N} = \frac{E_{ij}}{E_{j}^{+}}, E_{j}^{+} = \underset{i = 1}{max^{n}} E_{ij}, Beneficialset \\ E_{ij}^{N} = \frac{E_{ij}}{E_{j}^{-}}, E_{j}^{-} = \underset{i = 1}{min^{n}} E_{ij}, Non - Beneficialset \end{matrix}$ (31)

$E^{N} = {[E_{ij}^{N}]}_{m \times n}$ (32)

Step 3. Sort the option evaluations under each attribute in the normalized decision matrix in ascending order.

The ascending order is then recorded as β_ij, where β_ij ≤ β_i+1,j. In this case, the ranking is based on the likelihood of comparing sizes. Set up two TFNs, $\tilde{A} = (a, b, c)$ and $\tilde{B} = (d, e, f)$ , a ≤ d, then there are the following.

(1) If a = d, b ≥ e, c ≥ f, then $p (\tilde{A} \geq \tilde{B}) = 1$

(2) If a ≤ d, b ≤ e, c ≤ f, then $p (\tilde{A} \leq \tilde{B}) = 1$

(3) If a ≤ d, b ≤ e, c > f, then $p (\tilde{A} \geq \tilde{B}) = \frac{{(c - f)}^{2}}{(c + e - b - f) (c + d - a - f)}, p (\tilde{A} \leq \tilde{B}) = 1 - p (\tilde{A} \geq \tilde{B});$

(4) If a ≤ d, b > e, c ≤ f, then $p (\tilde{A} \geq \tilde{B}) = \frac{{(b - e)}^{2}}{(d + b - e - a) (f + b - e - c)}, p (\tilde{A} \leq \tilde{B}) = 1 - p (\tilde{A} \geq \tilde{B});$

(5) If a < d, b > e, c > f, then $p (\tilde{A} \leq \tilde{B}) = \frac{{(d - a)}^{2}}{(c + d - a - f) (b + d - a - e)}, p (\tilde{A} \geq \tilde{B}) = 1 - p (\tilde{A} \leq \tilde{B});$

Step 4. Calculate the ordered distance between neighbors.

Calculate the ordered distance between neighbors using Equation (33), noted as d_ij.

$\begin{matrix} d_{ij} = β_{i + 1, j} - β_{ij}, \\ i = 1, 2, . . ., m - 1; j = 1, 2, . . ., n; \end{matrix}$ (33)

d_ij denotes the ordered distance between adjacent schemes i and i + 1 under the j attribute. The Manhattan distance is used here to calculate the ordered distance between two TFNs. Under the premise that Manhattan distance and Euclidean distance have the same effect, and considering the complexity of R-number calculation, the Manhattan distance is chosen to replace the Euclidean distance to link the problem of high computational effort in the distance calculation link. With two TFNs, $\tilde{A} = (a_{1}, a_{2}, a_{3}), \tilde{B} = (b_{1}, b_{2}, b_{3})$ , then.

$d_{AB} = | a_{1} - b_{1} | + | a_{2} - b_{2} | + | a_{3} - b_{3} |$ (34)

Step 5. Calculate the appreciable ordered distance.

$Q_{ij} = {\begin{matrix} d_{ij} - ξ, d_{ij} > ξ \\ 0, d_{ij} \leq ξ \end{matrix}, ξ > 0; i = 1, 2, . . ., m - 1; j = 1, 2, . . ., n;$ (35)

Where ξ is the undifferentiated threshold parameter. If d_ij > ξ, then the difference is considered greater for attribute j, not within the acceptable range, considered relatively important and therefore given some weight; which are otherwise considered to be within acceptable limits, then Q_ij = 0.

Step 6. Calculate the weights for each attribute.

Calculations were carried out using Equation (36) to determine the final weights for each attribute.

$\begin{matrix} W_{j} = \frac{\sum_{i = 1}^{m - 1} Q_{ij}}{\sum_{j = 1}^{n} (\sum_{i = 1}^{m - 1} Q_{ij})}, i = 1, 2, . . ., m - 1; \\ j = 1, 2, . . ., n; \end{matrix}$ (36)

$W = {(W_{1}, W_{2}, . . ., W_{j})}^{T}, j = 1, 2, . . ., n;$ (37)

3.2 Improved RTOPSIS method

Assuming K decision maker, m options and n criteria. In the RTOPSIS method, the R-number is combined with the traditional TOPSIS method. This paper extends to the field of group decision-making by considering existing research and incorporating expert risk perception factors. And the Manhattan distance is used instead of the Euclidean distance to reduce the arithmetic while ensuring valid results.

Step 1. Construct an evaluation matrix, determine expert weights, and attribute weights.

$\begin{matrix} D^{K} = {[S_{ij}^{K}]}_{m \times n}, K = 1, 2, . . ., K; i = 1, 2, . . ., m; \\ j = 1, 2, . . ., n; \end{matrix}$ (38)

$W = {[W^{K}]}_{1 \times K}$ (39)

$V = {[V_{j}]}_{1 \times n}$ (40) where D^K denotes the evaluation matrix of K experts. $S_{ij}^{K}$ is a TFN, denotes the value of Expert K evaluation of Option i under Attribute j. W^K indicates the weight of expert K, V_j indicates the weight of attribute j.

Step 2. Construct fuzzy positive risk r⁺, fuzzy negative risk r^-, acceptable positive risk AR⁺, and acceptable negative risk AR^- matrices and determine expert risk preferences RP.

$r^{+^{K}} = {[r_{ij}^{+^{K}}]}_{m \times n}$ (41)

$r^{-^{K}} = {[r_{ij}^{-^{K}}]}_{m \times n}$ (42)

$A R^{+^{K}} = {[{AR}_{i}^{+^{K}}]}_{1 \times n}$ (43)

$A R^{-^{K}} = {[{AR}_{i}^{-^{K}}]}_{1 \times n}$ (44)

Step 3. Combine the known r⁺, r^-, AR⁺, AR^- matrix, and RP, and transform the decision matrix into an R-number matrix.

Using Equations (7) to (10), the evaluation matrix of each expert is transformed into an R-number matrix according to each attribute, and the transformed matrix is denoted by R^K.

$R^{K} = {[R_{ij}^{K} (S_{ij})]}_{m \times n}, where$ (45)

$R_{ij}^{K} = {(\begin{matrix} (a_{ij}^{11}, a_{ij}^{12}, a_{ij}^{13}) \\ (a_{ij}^{21}, a_{ij}^{22}, a_{ij}^{23}) \\ (a_{ij}^{31}, a_{ij}^{32}, a_{ij}^{33}) \end{matrix})}^{K}$ (46)

Step 4. Aggregate the R-number matrices of the experts to form a cluster decision matrix.

Using Equation (47), the final group decision matrix is denoted as R^T.

$R_{ij}^{T} = \sum_{K = 1}^{K} (R_{ij}^{K} (S_{ij}) ∙ W^{K})$ (47)

$R^{T} = {[R_{ij}^{T}]}_{m \times n}$ (48)

Step 5. Normalize the cluster decision matrix.

The two attributes, beneficial and non-beneficial, are normalized using Equation (49) and the normalized group decision matrix is denoted by R^N.

${\begin{matrix} R_{ij}^{N} = \frac{R_{ij}^{T}}{C_{j}^{+}}, Benneficialset \\ R_{ij}^{N} = \frac{C_{j}^{-}}{R_{ij}^{T}}, Non - Benneficialset \end{matrix}$ (49)

$\begin{matrix} R_{ij}^{T} = ((a_{ij}^{11}, a_{ij}^{12}, a_{ij}^{13}), \\ (a_{ij}^{21}, a_{ij}^{22}, a_{ij}^{23}), (a_{ij}^{31}, a_{ij}^{32}, a_{ij}^{33})) \end{matrix}$ (50)

${\begin{matrix} C_{j}^{+} = \underset{i = 1}{max^{m}} a_{ij}^{33} \\ C_{j}^{-} = \underset{i = 1}{min^{m}} a_{ij}^{11} \end{matrix}$ (51)

$R^{N} = {[R_{ij}^{N}]}_{m \times n}$ (52)

Step 6. Calculate the weighting matrix.

$R_{ij}^{V} = V_{j} ∙ R_{ij}^{N}$ (53)

$R^{V} = {[R_{ij}^{V}]}_{m \times n}$ (54)

Step 7. Determine fuzzy positive ideal solutions (FPISs) and fuzzy negative ideal solutions (FNISs).

${\begin{matrix} A^{+} = {{\tilde{V}}_{1}^{+}, {\tilde{V}}_{2}^{+}, . . ., {\tilde{V}}_{n}^{+}} \\ A^{-} = {{\tilde{V}}_{1}^{-}, {\tilde{V}}_{2}^{-}, . . ., {\tilde{V}}_{n}^{-}} \end{matrix}$ (55)

${\begin{matrix} {\tilde{V}}_{j}^{+} = \underset{i = 1}{max^{m}} a_{ij}^{33} \\ V_{j}^{-} = \underset{i = 1}{min^{m}} a_{ij}^{11} \end{matrix}$ (56)

Step 8. Calculate the distance between each option to the FPISs and FNISs.

${\begin{matrix} d_{j}^{+} = \sum_{j = 1}^{n} d (R_{ij}^{V}, V_{j}^{+}) \\ d_{j}^{-} = \sum_{j = 1}^{n} d (R_{ij}^{V}, V_{j}^{-}) \end{matrix}$ (57)

Suppose there are two R-numbers $R (\tilde{A})$ and $R (\tilde{B})$ , $R (\tilde{A}) = ((a_{11}, a_{12}, a_{13}), (a_{21}, a_{22}, a_{23}), (a_{31}, a_{32}, a_{33}))$ , $R (\tilde{B}) = ((b_{11}, b_{12}, b_{13}), (b_{21}, b_{22}, b_{23}), (b_{31}, b_{32}, b_{33}))$ , then use the Manhattan distance to calculate the distance between the two R-numbers as in Equation (58).

$\begin{matrix} d (R (\tilde{A}), R (\tilde{B})) = | a_{11} - b_{11} | + | a_{12} - b_{12} | \\ + \dots + | a_{32} - b_{32} | + | a_{33} - b_{33} | \end{matrix}$ (58)

Step 9. Calculate and rank the closeness factor.

${\begin{matrix} C_{i}^{+} = \frac{d_{i}^{+}}{d_{i}^{+} + d_{i}^{-}}, i = 1, 2, . . ., m \\ C_{i}^{-} = \frac{d_{i}^{-}}{d_{i}^{+} + d_{i}^{-}}, i = 1, 2, . . ., m \end{matrix}$ (59)

3.3 FITARA-RTOPSIS model construction

The steps of the specific FITARA-RTOPSIS model are as follows.

Step 1. Construct the evaluation matrix for each expert, determine the r⁺, r^-, AR⁺, AR^- matrix and RP, and determine the expert weights. For details see Equations (38) (39).

Step 2. Transform into an R-number matrix.

Transforming the correlation matrix in step 1 into an R-number matrix using Equations (7) to (10) gives R^K, see Equations (45) (46).

Step 3. Aggregate the expert matrices to form a cluster decision matrix.

Aggregating the expert matrix according to Equation (47), expressed as RT, see Equation (48).

Step 4. Normalize group decision matrix.

Step 5. Determine the attribute weights using the FITARA method.

The attribute weights are calculated using the method in section 3.1 and are denoted as V, see Equation (40).

Step 6. Calculate the weighted cluster decision matrix.

Step 7. Calculate the distance between the options to the FPISs and FNISs, determine the closeness factor and rank them.

Where the distance is calculated using Equation (58) for Manhattan distance.

Fig. 2

FITARA-FTOPSIS model process.

4 Case study

With the development of society and the use of various tools, scholars are committed to using a variety of decision-making methods and tools for analyzing problems, taking into account the development of events and their impact on the future in order to select the appropriate method to solve real-life problems. For example, the modeling study of the evolution of the competitive landscape of inland container ports by integrating the analysis of the overall competitive situation, the evolution of the competitive pattern and the intensity of competition [36], the use of quantitative analysis to analyse the causes of groundings in the context of the melting of Arctic sea ice to promote shipping [37], the study of the changes in the development of logistics in coastal provinces in the context of the economy driven by the Maritime Silk Road [38], and many other studies to promote social development are being further developed.

In this section, the proposed FITARA-RTOPSIS method is applied to the problem of siting emergency supplies stores to illustrate the feasibility of the proposed method. There is uncertainty about the timing of emergencies and the degree of ring-breaking, which will result in unpredictable casualties and economic losses [39, 40]. In recent years, natural disasters have been frequent, especially since the outbreak of Corona Virus Disease 2019 (COVID-19) in 2019, which has had a huge impact on economies around the world, and many people have died from the disease during this period. In February 2023, a 7.8 magnitude earthquake struck Turkey, killing 50,399 people and causing over US $104 billion in damage.

Risk assessment of natural hazards is already an issue for many countries. In April 2022, the United Nations released the “Global Assessment Report on Disaster Risk Reduction to 2022”. The report states that on the one hand, human activities and behavior are causing disaster-prone results worldwide, which will threaten human lives and expose societies and economies to severe shocks; On the other hand, disasters can be prevented, provided that countries invest time and resources to understand and reduce the risks. Based on the severity of past disasters, countries are also aware of the importance of disaster prevention measures, and to date, numerous countries have established stockpiles of emergency supplies or emergency response zones.

It is clear from the above that the resettlement work, the distribution of emergency supplies, and the protection of people’s livelihoods after any emergency are particularly important. Distributing a larger amount of supplies to the affected areas in a shorter period time was the goal of all the preparatory work. This highlights the importance of the location of the emergency stores. The selection of sites for emergency stores needs to be judged based on factors such as the natural environment, probability of disasters, and economic conditions in different areas, as well as an overall assessment taking into account the risk aspects of each factor.

R-number have a large degree of advantage in risk coverage and risk identification, which can effectively improve the accuracy of decision results. Therefore, this paper uses the Delphi method to determine the evaluation criteria, the FITARA method to determine the weight of each criterion, and the RTOPSIS method to rank the alternatives.

In this paper, based on the actual situation of the location, five criteria of C₁ traffic conditions, C₂ economic conditions, C₃ infrastructure development, C₄ cost, and C₅ coverability were finally determined, and four experts were invited to evaluate the five candidate addresses. The expert weights are defined as 0.2, 0.3, 0.3 and 0.2 respectively. The experts evaluated the program based on the linguistic variables in Table 1 and, for risk-related variables, based on Tables 2 and 3. In this case, all ARs are considered to be 0 (AR⁺ = AR^- = 0) and the risk perception of all experts is a pessimistic VL with RP⁺ = RP^- = (-0.3, 0, 0). Table 4 provides a detailed evaluation of the information.

Table 1
Language variables of the evaluation program

Language variables TFN

Very Low (VL) (0.1, 0.1, 0.3)

Low (L) (0.1, 0.3, 0.5)

Medium (M) (0.3, 0.5, 0.7)

High (H) (0.5, 0.7, 0.9)

Very High (VH) (0.7, 0.99, 0.99)

Language variables	TFN
Very Low (VL)	(0.1, 0.1, 0.3)
Low (L)	(0.1, 0.3, 0.5)
Medium (M)	(0.3, 0.5, 0.7)
High (H)	(0.5, 0.7, 0.9)
Very High (VH)	(0.7, 0.99, 0.99)

Table 2

Linguistic variables for assessing r⁺, r^-, AR⁺, AR^-

Language variables	TFN
Very Low (VL)	(0, 0, 0.3)
Low (L)	(0.1, 0.3, 0.5)
Medium (M)	(0.3, 0.5, 0.7)
High (H)	(0.5, 0.7, 0.9)
Very High (VH)	(0.7, 0.99, 0.99)

Table 3

Linguistic Variables for assessing decision makers’ risk perception RP

	Language variables	TFN
Optimism expert	Very Low (VL)	(0, 0, 0.3)
	Low (L)	(0.1, 0.3, 0.5)
	Medium (M)	(0.3, 0.5, 0.7)
	High (H)	(0.5, 0.7, 0.9)
	Very High (VH)	(0.7, 0.99, 0.99)
Pessimism expert	Very Low (VL)	(-0.3,0,0)
	Low (L)	(-0.5,-0.3,-0.1)
	Medium (M)	(-0.7,-0.5,-0.3)
	High (H)	(-0.9,-0.7,-0.5)
	Very High (VH)	(-0.99,-0.99,-0.7)

Table 4

Evaluation matrix given by pessimistic experts

		C ₁			C ₂			C ₃			C ₄			C ₅
		Reviews	r ^-	r ⁺	Reviews	r ^-	r ⁺	Reviews	r ^-	r ⁺	Reviews	r ^-	r ⁺	Reviews	r ^-	r ⁺
DM ₁	A ₁	M	VL	H	L	VL	H	M	L	H	M	L	H	VH	H	M
	A ₂	H	L	H	M	L	M	L	VH	M	VL	H	M	H	VL	VH
	A ₃	M	L	VL	H	M	L	VL	L	L	L	VL	VL	M	M	VH
	A ₄	H	M	M	H	M	L	M	H	M	H	H	M	M	L	L
	A ₅	VH	H	M	M	H	M	H	M	M	M	M	L	L	M	L
DM ₂	A ₁	M	H	L	M	H	VL	M	H	VH	M	H	VH	H	L	VL
	A ₂	H	L	H	H	VL	H	L	M	H	M	M	H	H	VH	M
	A ₃	VH	M	H	M	M	M	VL	L	H	H	M	L	VL	M	H
	A ₄	M	M	M	H	L	L	H	H	M	VH	L	M	M	H	L
	A ₅	VH	H	M	VH	H	M	M	M	M	L	VL	M	L	M	M
DM ₃	A ₁	H	M	M	M	L	M	L	M	H	H	H	M	VH	M	L
	A ₂	VH	M	M	VH	M	H	M	H	M	M	M	H	H	M	L
	A ₃	H	L	M	L	H	L	VL	M	M	L	L	M	M	H	H
	A ₄	M	H	L	L	L	M	H	M	L	M	H	L	L	L	M
	A ₅	M	H	H	M	M	L	M	L	L	M	M	M	M	H	M
DM ₄	A ₁	H	L	M	M	M	L	M	M	H	M	M	H	H	L	M
	A ₂	H	M	L	M	L	M	M	H	L	L	H	M	VH	M	H
	A ₃	M	H	L	H	H	H	L	L	M	M	L	M	L	H	M
	A ₄	L	H	H	H	L	M	M	H	L	H	M	H	M	M	L
	A5	M	M	H	H	M	H	H	M	L	L	M	L	M	H	M

Step 1. Construct the evaluation matrix for each expert, determine the r⁺, r^-, AR⁺, AR^- matrix and RP, determine the expert weights.

Table 4 gives information on the evaluation of the four experts, with expert weights is V = [0.2, 0.3, 0.3, 0.2].

Step 2. Transform into an R-number matrix.

Matching the evaluation information in Table 4 with Tables 1 to 3, the data were transformed into an R-number matrix using Equations (7) to (10) as shown in Tables 5 – 8.

Table 5

R-number matrix in DM₁

	C1	C2	C3	C4	C5
A1	((0.21,0.5,0.7)(0.3,0.5,0.7)(0.415,0.85,1.33))	((0.07,0.3,0.5)(0.1,0.3,0.5)(0.138,0.51,0.95))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.415,0.85,1.33))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.62,1.05))	((0.07,0.297,0.609)(0.7,0.99,0.99)(0.862,1.485,1.683))
A2	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.001,0.003,0.231)(0.1,0.3,0.5)(0.123,0.45,0.85))	((0.03,0.05,0.231)(0.1,0.1,0.3)(0.138,0.17,0.57))	((0.35,0.7,0.9)(0.5,0.7,0.9)(0.769,1.393,1.791))
A3	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.3,0.5,0.91))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.538,0.91,1.35))	((0.05,0.07,0.277)(0.1,0.1,0.3)(0.108,0.13,0.45))	((0.07,0.1,0.3)(0.1,0.1,0.3)(0.1,0.1,0.39))	((0.09,0.25,0.538)(0.5,0.5,0.7)(0.462,0.995,1.393))
A4	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.538,0.91,1.35))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.323,0.65,1.05))
A5	((0.07,0.297,0.609)(0.7,0.99,0.99)(0.862,1.485,1.683))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.15,0.35,0.392)(0.5,0.7,0.9)(0.615,1.05,1.53))	(0.15,0.35,0.646)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.108,0.39,0.75))

Table 6

R-number matrix in DM₂

	C1	C2	C3	C4	C5
A1	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.3,0.5,0.91))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.462,0.995,1.393))	((0.003,0.005,0.323)(0.3,0.5,0.7)(0.415,0.85,1.33))	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.4,0.7,1.17))
A2	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.35,0.7,0.9)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.138,0.51,0.95))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.005,0.007,0.415)(0.5,0.7,0.9)(0.615,1.05,1.53))
A3	((0.21,0.495,0.762)(0.7,0.99,0.99)(0.969,1.683,1.881))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.05,0.07,0.277)(0.1,0.1,0.3)(0.138,0.17,0.57))	((0.25,0.49,0.831)0.5,0.7,0.9)(0.615,1.05,1.53))	((0.03,0.05,0.231)(0.1,0.1,0.3)(0.138,0.17,0.57))
A4	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.538,0.91,1.35))	((0.05,0.21,0.554)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.21,0.495,0.762)(0.7,0.99,0.99)(0.754,1.287,1.485))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.65,1.05))
A5	((0.07,0.297,0.609)(0.7,0.99,0.99)(0.862,1.485,1.683))	((0.07,0.297,0.609)(0.7,0.99,0.99)(0.862,1.485,0.683))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.1,0.3,0.65))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.123,0.45,0.85)

Table 7

R-number matrix in DM₃

	C1	C2	C3	C4	C5
A1	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.138,0.51,0.95))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.21,0.495,0.762)(0.7,0.99,0.99)(0.754,1.287,1.485))
A2	((0.21,0.495,0.762)(0.7,0.99,0.99)(0.862,1.485,1.683))	((0.21,0.495,0.762)(0.7,0.99,0.99)(0.969,1.683,1.881))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.538,0.91,1.35))
A3	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.01,0.09,0.308)(0.1,0.3,0.5)(0.108,0.39,0.75))	((0.03,0.05,0.231)(0.1,0.1,0.3)(0.123,0.15,0.51))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.108,0.39,0.75))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.415,0.85,1.33))
A4	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.05,0.21,0.462)(0.1,0.3,0.5)(0.123,0.45,0.85))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.538,0.91,1.35))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.415,0.85,1.33)	((0.05,0.21,0.462)(0.1,0.3,0.5)(0.123,0.45,0.85))
A5	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.415,0.85,1.33))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.323,0.65,1.05))	(0.09,0.25,0.538)(0.3,0.5,0.7)(0.369,0.75,1.19))	(0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))

Table 8

R-number matrix in DM₄

	C1	C2	C3	C4	C5
A1	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.415,0.85,1.33))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.615,1.05,1.53))
A2	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.538,0.91,1.35))	((0.15,0.35,0.646)(0.3,0.5,0.7)(0.369,0.75,1.19))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.03,0.15,0.385)(0.1,0.3,0.5)(0.138,0.51,0.95))	((0.21,0.495,0.762)(0.7,0.99,0.99)(0.969,1.683,1.881))
A3	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.05,0.21,0.554)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.05,0.21,0.462)(0.1,0.3,0.5)(0.123,0.45,0.85))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.01,0.09,0.308)(0.1,0.3,0.5)(0.123,0.45,0.85))
A4	((0.01,0.09,0.308)(0.1,0.3,0.5)(0.138,0.51,0.95))	((0.25,0.49,0.831)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.323,0.65,1.05))	((0.05,0.21,0.554)(0.5,0.7,0.9)(0.615,1.05,1.53))	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.323,0.65,1.05))
A5	((0.09,0.25,0.538)(0.3,0.5,0.7)(0.415,0.85,1.33))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.692,1.19,1.71))	((0.15,0.35,0.692)(0.5,0.7,0.9)(0.538,0.91,1.35))	((0.05,0.21,0.462)(0.1,0.3,0.5)(0.123,0.45,0.85))	((0.03,0.15,0.431)(0.3,0.5,0.7)(0.369,0.75,1.19))

Step 3. Aggregate the expert matrices to form a cluster decision matrix.

Use Equations (47)(48) to form a cluster decision matrix, see Table for details.

Table 9

Group decision matrix

	C1	C2	C3	C4	C5
A1	((0.146,0.348,0.643)(0.4,0.6,0.8)(0.488,0.89,1.346))	((0.086,0.26,0.531)(0.26,0.46,0.66)(0.293,0.607,1.03))	((0.066,0.21,0.482)(0.24,0.44,0.64)(0.346,0.792,1.235))	((0.058,0.1667,0.477)(0.36,0.56,0.76)(0.471,0.892,1.36))	((0.202,0.453,0.766)(0.6,0.845,0.945)(0.672,1.103,1.439))
A2	((0.218,0.464,0.782)(0.56,0.787,0.927)(0.712,1.222,1.63))	((0.228,0.499,0.757)(0.48,0.707,0.847)(0.646,1.162,1.553))	((0.024,0.121,0.377)(0.2,0.4,0.6)(0.242,0.598,1.022))	((0.03,0.13,0.382)(0.22,0.38,0.58)(0.277,0.586,1.018))	(0.159,0.346,0.665)(0.54,0.758,0.918)(0.694,1.203,1.598))
A3	(0.174,0.396,0.693)(0.48,0.707,0.847)(0.6,1.05,1.415))	((0.07,0.214,0.503)(0.32,0.52,0.72)(0.389,0.762,1.194))	((0.044,0.092,0.3)(0.1,0.14,0.34)(0.125,0.212,0.584))	((0.116,0.262,0.532)(0.26,0.42,0.62)(0.302,0.582,0.972))	((0.038,0.128,0.368)(0.2,0.34,0.54)(0.283,0.595,1.019))
A4	((0.068,0.208,0.491)(0.3,0.5,0.7)(0.358,0.732,1.168))	((0.17,0.378,0.692)(0.38,0.58,0.78)(0.429,0.8,1.236))	((0.072,0.228,0.546)(0.42,0.62,0.82)(0.485,0.868,1.312))	((0.148,0.366,0.672)(0.5,0.727,0.867)(0.612,1.089,1.493))	((0.072,0.228,0.505)(0.24,0.44,0.64)(0.263,0.59,0.99))
A5	((0.062,0.244,0.542)(0.5,0.745,0.845)(0.638,1.168,1.507))	((0.084,0.264,0.568)(0.46,0.687,0.827)(0.568,1.029,1.4))	((0.132,0.32,0.632)(0.38,0.58,0.78)(0.438,0.812,1.248))	((0.076,0.232,0.498)(0.2,0.4,0.6)(0.239,0.555,0.96))	(0.03,0.15,0.408)(0.2,0.4,0.6)(0.243,0.588,1))

Step 4. Normalize group decision matrix.

Normalized using Equation (49), the final normalized matrix is shown in Table 10.

Table 10

Normalized group decision matrix

	C1	C2	C3	C4	C5
A1	((0.09,0.214,0.395)(0.245,0.368,0.491)(0.299,0.546,0.826))	((0.055,0.167,0.342)(0.167,0.296,0.425)(0.189,0.391,0.663))	((0.05,0.16,0.367)(0.183,0.335,0.488)(0.264,0.603,0.941))	((0.022,0.034,0.064)(0.039,0.054,0.083)(0.063,0.18,0.518))	((0.126,0.283,0.479)(0.375,0.529,0.591)(0.42,0.69,0.9))
A2	((0.134,0.284,0.48)(0.344,0.483,0.569)(0.437,0.75,1))	((0.147,0.321,0.487)(0.309,0.455,0.545)(0.416,0.748,1))	((0.018,0.092,0.287)(0.152,0.305,0.457)(0.184,0.456,0.779))	((0.029,0.051,0.108)(0.052,0.079,0.136)(0.079,0.231,1))	((0.099,0.217,0.416)(0.338,0.474,0.574)(0.434,0.753,1))
A3	((0.107,0.243,0.425)(0.294,0.434,0.52)(0.368,0.644,0.868))	((0.045,0.138,0.324)(0.206,0.335,0.464)(0.51,0.491,0.769))	((0.034,0.07,0.229)(0.076,0.107,0.259)(0.095,0.162,0.445))	((0.031,0.052,0.099)(0.048,0.071,0.115)(0.056,0.115,0.259))	((0.024,0.08,0.23)(0.125,0.213,0.338)(0.177,0.372,0.637))
A4	((0.042,0.128,0.301)(0.184,0.307,0.429)(0.22,0.449,0.717))	((0.109,0.243,0.446)(0.245,0.373,0.502)(0.276,0.515,0.796))	((0.055,0.174,0.416)(0.32,0.473,0.625)(0.369,0.662,1))	((0.02,0.028,0.049)(0.035,0.041,0.06)(0.045,0.082,0.203))	((0.045,0.143,0.316)(0.15,0.275,0.4)(0.165,0.369,0.619))
A5	((0.038,0.149,0.332)(0.307,0.457,0.518)(0.392,0.716,0.924))	((0.054,0.17,0.366)(0.296,0.442,0.532)(0.365,0.662,0.901))	((0.101,0.244,0.482)(0.29,0.442,0.595)(0.334,0.619,0.951))	((0.031,0.054,0.125)(0.05,0.075,0.15)(0.06,0.129,0.395))	((0.019,0.094,0.255)(0.125,0.25,0.375)(0.152,0.368,0.626))

Step 5. Determine the attribute weights using the FITARA method.

The normalized matrix is first sorted, followed by the calculation of the ordered distance d_ij between two adjacent solutions under the same attribute, determining attribute thresholds of 0.45, 0.5, 0.62, 0.38, 0.5, respectively., comparing the magnitude relationship between d_ij and ξaccording to Equation (35) and calculating the appreciable ordered distance Q_ij, Finally, the weight V=[0.1163,0.0891,0.1974,0.1509,0.4463] was calculated for each attribute, and the detailed calculation process is shown in Table 11.

Table 11

Determine the attribute weights

	C1	C2	C3	C4	C5
A1	((0.042,0.128,0.301)(0.184,0.307,0.429)(0.22,0.449,0.717))	((0.055,0.167,0.342)(0.167,0.296,0.425)(0.189,0.391,0.663))	((0.034,0.07,0.229)(0.076,0.107,0.259)(0.095,0.162,0.445))	((0.02,0.028,0.049)(0.035,0.041,0.06)(0.045,0.082,0.203))	((0.045,0.143,0.316)(0.15,0.275,0.4)(0.165,0.369,0.619))
A2	((0.09,0.214,0.395)(0.245,0.368,0.491)(0.299,0.546,0.826))	((0.045,0.138,0.324)(0.206,0.335,0.464)(0.51,0.491,0.769))	((0.018,0.092,0.287)(0.152,0.305,0.457)(0.184,0.456,0.779))	((0.022,0.034,0.064)(0.039,0.054,0.083)(0.063,0.18,0.518))	((0.019,0.094,0.255)(0.125,0.25,0.375)(0.152,0.368,0.626))
A3	((0.107,0.243,0.425)(0.294,0.434,0.52)(0.368,0.644,0.868))	((0.109,0.243,0.446)(0.245,0.373,0.502)(0.276,0.515,0.796))	((0.05,0.16,0.367)(0.183,0.335,0.488)(0.264,0.603,0.941))	((0.031,0.052,0.099)(0.048,0.071,0.115)(0.056,0.115,0.259))	((0.024,0.08,0.23)(0.125,0.213,0.338)(0.177,0.372,0.637))
A4	((0.038,0.149,0.332)(0.307,0.457,0.518)(0.392,0.716,0.924))	((0.054,0.17,0.366)(0.296,0.442,0.532)(0.365,0.662,0.901))	((0.101,0.244,0.482)(0.29,0.442,0.595)(0.334,0.619,0.951))	((0.031,0.054,0.125)(0.05,0.075,0.15)(0.06,0.129,0.395))	((0.126,0.283,0.479)(0.375,0.529,0.591)(0.42,0.69,0.9))
A5	((0.134,0.284,0.48)(0.344,0.483,0.569)(0.437,0.75,1))	((0.147,0.321,0.487)(0.309,0.455,0.545)(0.416,0.748,1))	((0.055,0.174,0.416)(0.32,0.473,0.625)(0.369,0.662,1))	((0.029,0.051,0.108)(0.052,0.079,0.136)(0.079,0.231,1))	((0.099,0.217,0.416)(0.338,0.474,0.574)(0.434,0.753,1))
d _ij	0.6967	0.4409	1.2852	0.495	0.2308
d _ij	0.4301	0.4849	0.6607	0.453	0.1599
d _ij	0.4435	0.7005	0.6652	0.2234	2.1984
d _ij	0.6461	0.6384	0.3996	0.7662	0.4424
ξ	0.45	0.5	0.62	0.38	0.5
Q _ij	0.2467	0	0.6652	0.115	0
Q _ij	0	0	0.0407	0.073	0
Q _ij	0	0.2005	0.0452	0	1.6984
Q _ij	0.1961	0.1384	0	0.3862	0
∑	0.4428	0.339	0.7511	0.5743	1.6984
W _j	0.1163	0.0891	0.1974	0.1509	0.4463

Step 6. Calculate the weighted cluster decision matrix.

The weighted cluster decision matrix was calculated using Equations (53)(54) and details are shown in Table 9.

Step 7. Calculate the distance between the options to the FPISs and FNISs, determine the closeness factor, and rank them.

The FPISs and FNISs were determined according to Table 12 and the results were obtained by calculating the distances and closeness factors the results are shown in Table 13.

Table 12

Weighted group decision matrix

	C1	C2	C3	C4	C5
A1	((0.01,0.025,0.046)(0.029,0.043,0.057)(0.035,0.064,0.096))	((0.005,0.015,0.03)(0.015,0.026,0.038)(0.017,0.035,0.059))	((0.01,0.032,0.072)(0.036,0.066,0.096)(0.052,0.119,0.186))	((0.003,0.005,0.01)(0.006,0.008,0.13)(0.009,0.027,0.078))	((0.056,0.26,0.214)(0.168,0.236,0.264)(0.188,0.308,0.402))
A2	((0.016,0.033,0.056)(0.04,0.056,0.066)(0.051,0.087,0.116))	((0.013,0.029,0.043)(0.028,0.041,0.049)(0.037,0.067,0.089))	((0.004,0.018,0.057)(0.03,0.06,0.09)(0.036,0.09,0.154))	((0.004,0.008,0.016)(0.008,0.012,0.021)(0.012,0.035,0.151))	((0.044,0.097,0.186)(0.151,0.212,0.256)(0.194,0.336,0.446))
A3	((0.012,0.028,0.049)(0.034,0.05,0.06)(0.043,0.075,0.101))	((0.004,0.012,0.029)(0.018,0.03,0.041)(0.022,0.044,0.068))	((0.007,0.014,0.045)(0.015,0.021,0.051)(0.019,0.032,0.088))	((0.005,0.008,0.015)(0.007,0.011,0.017)(0.009,0.017,0.039))	((0.011,0.036,0.103)(0.056,0.095,0.151)(0.079,0.166,0.284))
A4	((0.005,0.015,0.035)(0.021,0.036,0.05)(0.026,0.052,0.083))	((0.01,0.022,0.04)(0.022,0.033,0.045)(0.025,0.046,0.071))	((0.011,0.034,0.082)(0.063,0.093,0.123)(0.073,0.131,0.197))	((0.003,0.004,0.007)(0.005,0.006,0.009)(0.007,0.012,0.031))	((0.02,0.064,0.141)(0.067,0.123,0.179)(0.073,0.165,0.276))
A5	((0.004,0.017,0.039)(0.036,0.053,0.06)(0.046,0.083,0.108))	((0.005,0.015,0.033)(0.026,0.039,0.047)(0.033,0.059,0.08))	((0.02,0.048,0.095)(0.057,0.087,0.117)(0.066,0.122,0.188))	((0.005,0.008,0.019)(0.008,0.011,0.023)(0.009,0.02,0.06))	((0.008,0.042,0.114)(0.056,0.112,0.168)(0.068,0.164,0.279))

Table 13

Results of different methods

	ITARA-RTOPSIS		FTOPSIS		RSAW		RVIKOR
	${CC}_{i}^{+}$	rank	${CC}_{i}^{+}$	rank	COA	rank	GG	rank
A ₁	0.6333	2	0.4286	2	0.3816	2	0.4452	2
A ₂	0.6096	1	0.3824	1	0.4047	1	0.4408	1
A ₃	0.7826	5	0.7225	5	0.2358	5	0.4657	5
A ₄	0.7241	4	0.606	4	0.2929	4	0.4584	4
A ₅	0.7104	3	0.5407	3	0.3062	3	0.4551	3

In summary, according to the analysis of the proposed FITARA-RTOPSIS method for the siting problem of the local emergency supplies reserve, it can be seen that more attention should be paid to the size of the coverage area according to the actual local situation, and among the five alternatives, the address of option 2 is the optimal choice for the local emergency supplies reserve.

5 Sensitivity analysis and comparative analysis

5.1 Sensitivity analysis

In this section, we vary the threshold ξ in the FITARA-RTOPSIS method to rank the alternatives. A comparison of the three sets of data with the original threshold results shows that the scheme ranking is still the same, which proves the validity of the proposed FITARA method. The three sets of ξ values were [0.4, 0.45, 0.57, 0.33, 0.45], [0.5, 0.55, 0.67, 0.43, 0.55], [0.3, 0.4, 0.35, 0.2, 0.13], as detailed in Fig. 3.

Fig. 3

Results for different thresholds.

5.2 Comparative analysis

To illustrate the validity of the proposed FITARA-RTOPSIS method, in this section FTOPSIS, RSAW, and RVIKOR are calculated and compared with the proposed FITARA-RTOPSIS method, using the case in Section IV as basic information, and the results are presented in Table 13. As can be seen from Table 13, there is a high degree of consistency in the ranking of the schemes, regardless of the method used, which is more indicative of the stability and validity of the proposed method.

6 Conclusion

In the problem of siting emergency supplies stores, on the one hand, due to the uncertainty of the occurrence of disasters, an effective method of evaluation, in this case, is the use of fuzzy sets, and on the other hand, risk factors in this problem can lead to bias in the assessed values, hence the application of R-number to them. In multi-attribute decision problems, the coverage of evaluation information by fuzzy numbers, the method of obtaining attribute weights, and risk factors are all key aspects of interest to researchers. This paper defines the FITARA method based on the TFN environment and applies it to the obtaining of attribute weights session. In addition, TFN is used as the base evaluation value, so that TOPSIS is combined with R-number, the fuzzy numbers are considered for evaluating the information coverage while incorporating risk factors, and extended to the field of group decision-making. Finally, the feasibility of the proposed FITARA method and the validity of the FITARA-RTOPSIS model are illustrated through the application of the emergency supplies storage site selection problem.

The FITARA-RTOPSIS model proposed in this paper consists of the following advantages. First, the concept of FITARA is defined to consider the evaluation of alternatives by decision-makers in a TFN environment. Secondly, the method extends the existing ROPSIS approach to the field of group decision-making and incorporates risk perception factors that are more relevant to practical applications. Finally, the model replaces the Euclidean distance with the Manhattan distance, optimizing the complexity of the R-number calculation process in process. However, there are shortcomings in this study: On the one hand, there are numerous methods of triangular fuzzy number ranking in the FITARA method, and it is particularly important to choose a ranking method that is suitable for the current decision-making environment; On the other hand, although the R-number operation has been optimized for distance calculation, the application of the FITARA-RTOPSIS model in the TFN environment still suffers from a large computational system and computational complexity.

In the future, we still have much to improve and explore: (1) We can consider extending the ITARA method to other fuzzy number applications, such as picture fuzzy sets, Z-numbers, etc. (2) Extending the proposed risk-based approach to different assumptions, such as prospect theory, consensus building, etc. (3) The operations of various multi-criteria decision methods under the R-number system need to be optimized. (4) Methods of defuzzification and ranking that are consistent with the meaning of R-number are still to be explored.

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An extended ITARA-TOPSIS method for multi-criteria group decision-making problems based on R-number

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 ITARA

2.2.1 Triangular fuzzy number (TFN)

3.1 The FITARA method

Table 1 Language variables of the evaluation program Language variables TFN Very Low (VL) (0.1, 0.1, 0.3) Low (L) (0.1, 0.3, 0.5) Medium (M) (0.3, 0.5, 0.7) High (H) (0.5, 0.7, 0.9) Very High (VH) (0.7, 0.99, 0.99)

5.1 Sensitivity analysis

6 Conclusion

References

Table 1
Language variables of the evaluation program

Language variables TFN

Very Low (VL) (0.1, 0.1, 0.3)

Low (L) (0.1, 0.3, 0.5)

Medium (M) (0.3, 0.5, 0.7)

High (H) (0.5, 0.7, 0.9)

Very High (VH) (0.7, 0.99, 0.99)