Application of multi-attribute group decision-making methods in urban road traffic safety evaluation with interval-valued intuitionistic fuzzy information

Abstract

With the development of the social economy, the level of motorization has been greatly improved, and the traffic safety problem has been paid more and more attention. In recent years, China’s road traffic accident rate showed a trend of decline after rising first, suggests that the Chinese road traffic safety level is on the decline. Road traffic safety evaluation has a positive effect in found risk factors of road traffic safety in time and reduce the traffic accident rate, so the study of traffic safety evaluation method is imperative. And the urban road traffic safety evaluation is frequently viewed as the multi-attribute group decision-making (MAGDM) problem. Depending on the conventional VIKOR method and interval-valued intuitionistic fuzzy sets (IVIFSs), this paper designs a novel IVIF-VIKOR method to assess the urban road traffic safety. In addition, since subjective randomness frequently exists in determining criteria weights, the weights of criteria is [Z1] decided objectively by utilizing CRITIC method. Eventually, an application and some comparative analysis are given. The results show that the designed algorithms are useful for assessing the urban road traffic safety.

Keywords

Multi-attribute group decision-making (MAGDM)interval-valued intuitionistic fuzzy sets (IVIFSs)VIKOR method CRITIC method urban road traffic safety

1 Introduction

Since the process of making decision is filled with uncertainty and ambiguity [1 –3], thus, in order to cope with the accuracy of decision-making, Zadeh [4] defined the fuzzy sets (FSs). Atanassov [5] proposed the intuitionistic fuzzy sets (IFSs). In IFSs, there are three mathematical functions depicting the degrees of membership, non-membership and hesitancy [6 –9]. Gou, Xu and Lei [10] defined the exponential operational law of IFNs. He, He and Huang [11] developed the intuitionistic fuzzy power interaction fused operators. Zhang and He [12] built the intuitionistic fuzzy geometric interaction fused operators. Gupta, Arora and Tiwari [13] defined the fuzzy entropy to IFSs setting. Li and Wu [14] defined the decision method with help of cross entropy distance under IFSs. Liang, He, Wang, Chen and Li [15] gave MABAC method under IFSs with distance measures. Chen, Cheng and Lan [16] developed TOPSIS method with similarity measures under FSs. Jin, Ni, Chen and Li [17]defined two group decision making (GDM) methods which could derive the normalized intuitionistic fuzzy priority weights. Liu, Liu and Chen [18] defined some intuitionistic fuzzy fused operators based on the BM and Dombi operations for MAGDM. Garg [19] presented the GDM with intuitionistic fuzzy multiplicative preference relations and defined several geometric operators. Krishankumar, Ravichandran and Saeid [20] built the intuitionistic fuzzy set based PROMETHEE. Krishankumar, Arvinda, Amrutha, Premaladha, Ravichandran and Ieee [21] integrated analytic hierarchy process (AHP) with IFSs to design a GDM method for effective cloud vendor selection. Bao, Xie, Long and Wei [22] proposed the prospect theory and evidential reasoning with IFNs. Cali and Balaman [23] extended ELECTRE I with VIKOR with IFNs to reflect the decision makers’ preferences. Phochanikorn and Tan [24] incorporated DEMATEL with ANP to determine uncertainties and interdependencies among criteria and modified VIKOR to evaluate the sustainable supplier performance’s desired level under intuitionistic fuzzy context. Gan and Luo [25] employed a hybrid method on the basis of DEMATEL and IFSs to look for cause-effect relationships within factors. Hao, Xu, Zhao and Zhang [26] presented theory of decision field for intuitionistic fuzzy MADM. Khan, Lohani and Ieee [27] built the similarity measure about IFNs. Li, Liu, Liu, Su and Wu [28] proposed the grey target decision making with IFNs. Rouyendegh [29] used the ELECTRE method with IFSs to tackle some MCDM issues. Lu and Wei [30] built the TODIM to deal with performance appraisal in IVIFSs. Wu, Wang and Gao [8] solved tourist destination evaluation with Hamy mean with IVIFSs. Wu, Wei, Wu and Wei [31] defined the Dombi Heronian mean fused operators to evaluate the ecological value in IVIFSs. Wu, Gao and Wei [32] defined the VIKOR to cope with financing risk assessment under IVIFSs.

VIKOR method was initially defined by Opricovic [33] to solve MAGDM issues. Compared with some other MAGDM methods [34 –39], VIKOR method is taking the compromise issues both maximizing the utility of group and minimizing personal regrets into account. This method has been used to various fuzzy environments [40 –43]. For example, Zeng, Chen and Kuo [44] used VIKOR method to design a novel MADM method relying on the developed novel score function of IFNs. Wei, Wang, Lu, Wu, Wei, Alsaadi and Hayat [45] used the VIKOR method to 2-tuple linguistic neutrosophic environment to solve some MAGDM issues. Narayanamoorthy, Geetha, Rakkiyappan and Joo [46] defined the interval-valued intuitionistic hesitant fuzzy entropy and VIKOR techniques which was utilized to select industrial robot. Gao, Ran, Wei, Wei and Wu [47] integrated the conventional VIKOR technique under q-RIVOFS. He, Wei, Lu, Wei and Lin [48] combined P2TLNs with VIKOR method to provide a MAGDM technique. Yang and Pang [49] put forward a hesitant interval-valued Pythagorean fuzzy VIKOR with novel distance measures and aggregation operators.

Unfortunately, we are failure to find the work of VIKOR method with CRITIC method within IVIFSs for the existing literature. Thus, investigating VIKOR with IVIFSs is essential. The fundamental aim of such research is to define an original method which can be more effectively to address some MAGDM issues with VIKOR and IVIFSs. Hence, the highlights of such work are illustrated subsequently. Above all, extend VIKOR method to the IVIFSs. In addition, due to the DMs are restrained within their knowledge, CRITIC method is used to decide each attribute’s weight. Then, an empirical application is offered to show this novel approach and several comparative analysis are given to demonstrate some merits of the novel approach.

The reminder of this paper proceeds. Some necessary knowledge of IVIFSs is introduced in section 2. The improved VIKOR method for MAGDM is extended with IVIFNs and the calculating steps is simply listed in section 3. An empirical application for evaluating the urban road traffic safety is given and some comparative analysis are also given to prove the merits of such method in section 4. At last, we give an conclusion of such work in section 5.

2 Preliminaries

2.1 A. Interval-valued IFSs

Definition 1 [50]. The interval-valued IFSs (IVIFSs) on the X is: $I = {〈 x, {\tilde{μ}}_{I} (x), {\tilde{ν}}_{I} (x) 〉 | x \in X}$ (1) where ${\tilde{μ}}_{I} (x) \subset [0, 1]$ is named as “membership degree of I” and ${\tilde{ν}}_{I} (x) \subset [0, 1]$ is called as “non-membership degree of I”, and ${\tilde{μ}}_{I} (x)$ , ${\tilde{ν}}_{I} (x)$ meet condition: $0 ⩽ sup {\tilde{μ}}_{I} (x) + sup {\tilde{ν}}_{I} (x) ⩽ 1$ , ∀ x ∈ X. For convenience, we call $I = ([μ^{L}, μ^{R}], [ν^{L}, ν^{R}])$ is an IVIFN.

Definition 2 [51]. Let $I_{1} = ([μ_{1}^{L}, μ_{1}^{R}], [ν_{1}^{L}, ν_{1}^{R}])$ and $I_{2} = ([μ_{2}^{L}, μ_{2}^{R}], [ν_{2}^{L}, ν_{2}^{R}])$ be two IVIFNs, the operation formula of them can be defined: $\begin{matrix} I_{1} \oplus I_{2} = \\ ([μ_{1}^{L} + μ_{2}^{L} - μ_{1}^{L} μ_{2}^{L}, μ_{1}^{R} + μ_{2}^{R} - μ_{1}^{R} μ_{2}^{R}], [ν_{1}^{L} ν_{2}^{L}, ν_{1}^{R} ν_{2}^{R}]) \end{matrix}$ (2) $\begin{matrix} I_{1} \otimes I_{2} = \\ ([μ_{1}^{L} μ_{2}^{L}, μ_{1}^{R} μ_{2}^{R}], [ν_{1}^{L} + ν_{2}^{L} - ν_{1}^{L} ν_{2}^{L}, ν_{1}^{R} + ν_{2}^{R} - ν_{1}^{R} ν_{2}^{R}]) \end{matrix}$ (3) $\begin{matrix} λ I_{1} = ([1 - {(1 - μ_{1}^{L})}^{λ}, 1 - {(1 - μ_{1}^{R})}^{λ}], \\ [{(ν_{1}^{L})}^{λ}, {(ν_{1}^{R})}^{λ}]), λ > 0 \end{matrix}$ (4) $\begin{matrix} I_{1}^{λ} = ([{(μ_{1}^{L})}^{λ}, {(μ_{1}^{R})}^{λ}], \\ [1 - {(1 - λ_{1}^{L})}^{λ}, 1 - {(1 - λ_{1}^{R})}^{λ}]), λ > 0 \end{matrix}$ (5)

Definition 3 [52]. Let $I_{1} = ([μ_{1}^{L}, μ_{1}^{R}], [ν_{1}^{L}, ν_{1}^{R}])$ and $I_{2} = ([μ_{2}^{L}, μ_{2}^{R}], [ν_{2}^{L}, ν_{2}^{R}])$ be IVIFNs, the score and accuracy values of I₁ and I₂ can be defined: $\begin{matrix} S (I_{1}) = \frac{μ_{1}^{L} + μ_{1}^{L} (1 - μ_{1}^{L} - ν_{1}^{L}) + μ_{1}^{R} + μ_{1}^{R} (1 - μ_{1}^{R} - ν_{1}^{R})}{2} \\ S (I_{2}) = \frac{μ_{2}^{L} + μ_{2}^{L} (1 - μ_{2}^{L} - ν_{2}^{L}) + μ_{2}^{R} + μ_{2}^{R} (1 - μ_{2}^{R} - ν_{2}^{R})}{2} \end{matrix}$ (6) $\begin{matrix} H (I_{1}) = \frac{μ_{1}^{L} + ν_{1}^{L} + μ_{1}^{R} + ν_{1}^{R}}{2}, \\ H (I_{2}) = \frac{μ_{2}^{L} + ν_{2}^{L} + μ_{2}^{R} + ν_{2}^{R}}{2} \end{matrix}$ (7)

For two IVIFNs I₁ and I₂, according to Definition 3, then $\begin{matrix} (1) if s (I_{1}) < s (I_{2}), then I_{1} < I_{2}; \\ (2) if s (I_{1}) > s (I_{2}), then I_{1} > I_{2}; \\ (3) if s (I_{1}) = s (I_{2}), h (I_{1}) < h (I_{2}), then I_{1} < I_{2}; \\ (4) if s (I_{1}) = s (I_{2}), h (I_{1}) = h (I_{2}), then I_{1} = I_{2} . \end{matrix}$

Definition 4 [53]. Let $I_{1} = ([μ_{1}^{L}, μ_{1}^{R}], [ν_{1}^{L}, ν_{1}^{R}])$ and $I_{2} = ([μ_{2}^{L}, μ_{2}^{R}], [ν_{2}^{L}, ν_{2}^{R}])$ be IVIFNs, the Euclidean distance between two IVIFNs can be given as follows:

$IVIFED (I_{1}, I_{2}) = \sqrt{\frac{1}{4} [{(μ_{1}^{L} - μ_{2}^{L})}^{2} + {(μ_{1}^{R} - μ_{2}^{R})}^{2} + {(ν_{1}^{L} - ν_{2}^{L})}^{2} + {(ν_{1}^{R} - ν_{2}^{R})}^{2}]}$ (8)

2.2 B. Interval-valued intuitionistic fuzzy aggregation operators

Under the context of the IVIFSs, some operators could be reviewed in this section, including interval-valued intuitionistic fuzzy WA (IVIFWA) operator and interval-valued intuitionistic fuzzy WG (IVIFWG) operator.

Definition 5 [54]. Let $I_{j} = ([μ_{I_{j}}^{L}, μ_{I_{j}}^{R}],$ $[ν_{I_{j}}^{L}, ν_{I_{j}}^{R}]) (j = 1, 2, \dots, n)$ be a group of IVIFNs, the IVIFWA operator could be defined: $\begin{matrix} {IVIFWA}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \oplus_{j = 1}^{n} (ω_{j} I_{j}) \\ = ([1 - \prod_{j = 1}^{n} {(1 - μ_{I_{j}}^{L})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - μ_{I_{j}}^{R})}^{ω_{j}}], \\ [\prod_{j = 1}^{n} {(ν_{I_{j}}^{L})}^{ω_{j}}, \prod_{j = 1}^{n} {(ν_{I_{j}}^{R})}^{ω_{j}}]) \end{matrix}$ (9) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$

Definition 6 [54]. Let $I_{j} = ([μ_{I_{j}}^{L}, μ_{I_{j}}^{R}],$ $[ν_{I_{j}}^{L}, ν_{I_{j}}^{R}]) (j = 1, 2, \dots, n)$ be a collection of IVIFNs, the IVIFWG operator could be: $\begin{matrix} {IVIFWG}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \otimes_{j = 1}^{n} {(I_{j})}^{ω_{j}} \\ = ([\prod_{j = 1}^{n} {(μ_{I_{j}}^{L})}^{ω_{j}}, \prod_{j = 1}^{n} {(μ_{I_{j}}^{R})}^{ω_{j}}], \\ [1 - \prod_{j = 1}^{n} {(1 - ν_{I_{j}}^{L})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - ν_{I_{j}}^{R})}^{ω_{j}}]) \end{matrix}$ (10) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1 .$

3 VIKOR method for MAGDM with IVIFNs

In such section, we build the IVIF-VIKOR method for MAGDM with IVIFNs. The calculating steps of designed method can be depicted. Let T ={ T₁, T₂, … T_n } be the group of attributes, t = { t₁, t₂, … t_n } be the attribute weight T_j, where $t_{j} \in [0, 1], j = 1, 2, \dots, n, \sum_{j = 1}^{n} t_{j} = 1$ . Assume H = { H₁, H₂, … H_l } be a set of DMs that have significant degree of h = { h₁, h₂, … h_l }, where h_k ∈ [0, 1], k = 1, 2, …, l. $\sum_{k = 1}^{l} h_{k} = 1$ . Let P = { P₁, P₂, … P_m } be a collection of given alternatives. And $Q = {(q_{ij})}_{m \times n}$ is the overall IVIFN decision matrix, $q_{ij}$ means the value of P_i for T_j. Subsequently, the given calculating steps will be depicted.

Step 1. Build each DM’s IVIFN decision matrix $Q^{(k)} = {(q_{ij}^{k})}_{m \times n}$ and calculate the overall IVIFN matrix Q = (q_ij) _m×n. $Q^{(k)} = {[q_{ij}^{k}]}_{m \times n} = [\begin{matrix} q_{11}^{k} & q_{12}^{k} & \dots & q_{1 n}^{k} \\ q_{21}^{k} & q_{22}^{k} & \dots & q_{2 n}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q_{m 1}^{k} & q_{m 2}^{k} & \dots & q_{mn}^{k} \end{matrix}]$ (11) $Q = {[q_{ij}]}_{m \times n} = [\begin{matrix} q_{11} & q_{12} & \dots & q_{1 n} \\ q_{21} & q_{22} & \dots & q_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q_{m 1} & q_{m 2} & \dots & q_{mn} \end{matrix}]$ (12)

$\begin{matrix} q_{ij} = ([1 - \prod_{k = 1}^{l} {(1 - μ_{q_{ij}^{k}}^{L})}^{d_{k}}, 1 - \prod_{k = 1}^{l} {(1 - μ_{q_{ij}^{k}}^{R})}^{d_{k}}], \\ [\prod_{k = 1}^{l} {(ν_{q_{ij}^{k}}^{L})}^{d_{k}}, \prod_{k = 1}^{l} {(ν_{q_{ij}^{k}}^{R})}^{d_{k}}]) \end{matrix}$ (13) where $q_{ij}^{k}$ is the assessment value of the alternative P_i (i = 1, 2, …, m) for attribute T_j (j = 1, 2, …, n) and the DM H_k (k = 1, 2, …, l).

Step 2. Normalize the matrix Q = (q_ij) _m×n to $Q^{N} = {[q_{ij}^{N}]}_{m \times n}$ with IVIFNs. $q_{ij}^{N} = {\begin{matrix} ([μ_{ij}^{L}, μ_{ij}^{R}], [ν_{ij}^{L}, ν_{ij}^{R}]) Z_{j} is a benefit criterion \\ ([ν_{ij}^{L}, ν_{ij}^{R}], [μ_{ij}^{L}, μ_{ij}^{R}]) Z_{j} is a cost criterion \end{matrix}$ (14)

Step 3. Utilize the CRITIC to derive the attributes’ weight.

CRiteria Importance Through Intercriteria Correlation (CRITIC) method will be proposed in this part which is utilized to decide attributes’ weights. This method was initially defined by Diakoulaki, Mavrotas and Papayannakis [55] which took the correlations relationship between attributes into consideration. Subsequently, the calculating procedures of such method will be designed.

(1) Depending on the normalized decision matrix $Q^{N} = {(q_{ij}^{N})}_{m \times n}$ , the correlation coefficient between attributes could be derived. $\begin{matrix} {IVIFCC}_{jr} = \\ \frac{\sum_{i = 1}^{m} (S (q_{ij}^{N}) - S (q_{j}^{N})) (S (q_{ir}^{N}) - S (q_{r}^{N}))}{\sqrt{\sum_{i = 1}^{m} {(S (q_{ij}^{N}) - S (q_{j}^{N}))}^{2}} \sqrt{\sum_{i = 1}^{m} {(S (q_{ir}^{N}) - S (q_{r}^{N}))}^{2}}}, \\ j, r = 1, 2, \dots, n \end{matrix}$ (15) where $S (q_{j}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{ij}^{N})$ and $S (q_{r}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{ir}^{N})$ .

(2) Calculate attributes’ standard deviation. $\begin{matrix} {IVIFSD}_{j} = \\ \sqrt{\frac{1}{m - 1} \sum_{i = 1}^{m} {(S (q_{ij}^{N}) - S (q_{j}^{N}))}^{2}}, j = 1, 2, \dots, n \end{matrix}$ (16) where $S (q_{j}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{ij}^{N})$ .

(3) Calculate the attributes’ weights. $t_{j} = \frac{{IVIFSD}_{j} \sum_{t = 1}^{n} (1 - {IVIFCC}_{jr})}{\sum_{j = 1}^{n} ({IVIFSD}_{j} \sum_{t = 1}^{n} (1 - {IVIFCC}_{jr}))}, j = 1, 2 \dots, n .$ (17) where t_j ∈ [0, 1] and $\sum_{j = 1}^{n} t_{j} = 1$ .

Step 4. Set up the positive ideal solutions ${IVIFPIS}_{j}$ and negative ideal solutions ${IVIFNIS}_{j}$ by using the following Equations (18-19): ${IVIFPIS}_{j} = ([μ_{j}^{L +}, μ_{j}^{R +}], [ν_{j}^{L +}, ν_{j}^{R +}])$ (18) ${IVIFNIS}_{j} = ([μ_{j}^{L +}, μ_{j}^{R +}], [ν_{j}^{L +}, ν_{j}^{R +}])$ (19) where ${IVIFPIS}_{j} = (\begin{matrix} [max_{j} (μ_{ij}^{L}), max_{j} (μ_{ij}^{R})], \\ [min_{j} (ν_{ij}^{L}), min_{j} (ν_{ij}^{R})] \end{matrix})$ and ${IVIFNIS}_{j} = (\begin{matrix} [min_{j} (μ_{ij}^{L}), min_{j} (μ_{ij}^{R})], \\ [max_{j} (ν_{ij}^{L}), max_{j} (ν_{ij}^{R})] \end{matrix})$ .

Step 5. Obtain IVIFS_i and IVIFR_i index according to Equations (20-21). ${IVIFS}_{i} = \sum_{j = 1}^{n} t_{j} \frac{d ({IVIFPIS}_{j}, q_{ij}^{N})}{d ({IVIFPIS}_{j}, {IVIFNIS}_{j}^{-})}$ (20) ${IVIFR}_{i} = max_{j} {t_{j} \frac{d ({IVIFPIS}_{j}, q_{ij}^{N})}{d ({IVIFPIS}_{j}, {IVIFNIS}_{j}^{-})}}$ (21) where d is the Euclidean distance.

Step 6. Calculate the final value IVIFQ_i on the basis of the Equation (22). $\begin{matrix} {IVIFQ}_{i} = ζ \times \frac{{IVIFS}_{i} - {IVIFS}^{*}}{{IVIFS}^{-} - {IVIFS}^{*}} \\ + (1 - ζ) \times \frac{{IVIFR}_{i} - {IVIFR}^{*}}{{IVIFR}^{-} - {IVIFR}^{*}} \end{matrix}$ (22) where ${IVIFS}^{*} = min_{i} {IVIFS}_{i}$ , ${IVIFS}^{-} = max_{i} {IVIFS}_{i}$ , ${IVIFR}^{*} = min_{i} {IVIFR}_{i}$ , ${IVIFR}^{-} = max_{i} {IVIFR}_{i}$ .

And ζ can be viewed as the decision-making coefficient. If ζ > 0.5, it means “maximum group utility”; if ζ < 0.5, it means “minimum regret”, and if ζ = 0.5, it means “equality”. In this easy, ζ = 0.5.

Step 7. Depending on the calculating results of IVIFQ_i, all the alternatives could be ranked. The alternative with minimum value of IVIFQ_i is optimal alternative.

4 Numerical example and comparative analysis

4.1 Numerical example

With development of modern economy and the improvement of living standard, the traffic[Z2] plays more important role in normal life, and the urban road traffic is especially important in it. The urban road traffic helps the people to move quickly and efficiently, while it could endanger life and possession similar the other traffic style[Z3]. The urban traffic accidents affect and restrict the development of urban transport and threaten the people’s life. To improve the road safety and reduce the traffic accidents, the evaluation systems should be given objectively to the road safety that finds problems and provides urban road traffic safety with corresponding solutions. Then the disposal scheme should be worked out which could improve the urban traffic environment effectively. In this chapter, an empirical application of evaluating the urban road traffic safety could be provided with IVIF-VIKOR method. Since the government wants to select one urban road with the best environmental behaviors and awareness to reward, there are five potential urban roads P_i (i = 1, 2, 3, 4, 5). In order to assess the urban road traffic safety of these urban roads fairly, five experts H = { H₁, H₂, H₃, H₄, H₅ } (expert’s weight h = (0 . 20, 0 . 20, 0 . 20, 0 . 20, 0 . 20) are invited. All experts give their assessment information according to four subsequently attributes: ➀T₁ is road engineering; ➁T₂ is road construction cost; ➂T₃ is traffic operational condition; ➃T₄ is traffic management level. Evidently, T₂ is the cost attribute, while T₁ T₃ and T₄ are the benefit attributes.

Step 1. Build each DM’s IVIF matrix $Q^{(k)} = {(q_{ij}^{k})}_{m \times n}$ (i = 1, 2, …, m, j = 1, 2, …, n) as in Tables 1–5. From these tables and Equations (11–13), the overall IVIF matrix could be derived. The results are recorded in Table 6.

Table 1
IVIF evaluation matrix by H₁

T₁ T₂

P₁ ([0.16,0.22],[0.65,0.78]) ([0.33,0.42],[0.50,0.58])

P₂ ([0.32,0.40],[0.55,0.60]) ([0.17,0.25],[0.70,0.75])

P₃ ([0.43,0.47],[0.50,0.53]) ([0.32,0.40],[0.55,0.60])

P₄ ([0.32,0.39],[0.41,0.61]) ([0.27,0.36],[0.57,0.64])

P₅ ([0.25,0.30],[0.55,0.70]) ([0.44,0.48],[0.50,0.52])

T₃ T₄

P₁ ([0.24,0.30],[0.65,0.70]) ([0.47,0.55],[0.40,0.45])

P₂ ([0.71,0.80],[0.14,0.20]) ([0.60,0.70],[0.25,0.30])

P₃ ([0.57,0.62],[0.30,0.38]) ([0.29,0.36],[0.58,0.64])

P₄ ([0.34,0.40],[0.50,0.60]) ([0.32,0.40],[0.55,0.60])

P₅ ([0.62,0.70],[0.25,0.30]) ([0.60,0.65],[0.30,0.35])

	T₁	T₂
P₁	([0.16,0.22],[0.65,0.78])	([0.33,0.42],[0.50,0.58])
P₂	([0.32,0.40],[0.55,0.60])	([0.17,0.25],[0.70,0.75])
P₃	([0.43,0.47],[0.50,0.53])	([0.32,0.40],[0.55,0.60])
P₄	([0.32,0.39],[0.41,0.61])	([0.27,0.36],[0.57,0.64])
P₅	([0.25,0.30],[0.55,0.70])	([0.44,0.48],[0.50,0.52])
	T₃	T₄
P₁	([0.24,0.30],[0.65,0.70])	([0.47,0.55],[0.40,0.45])
P₂	([0.71,0.80],[0.14,0.20])	([0.60,0.70],[0.25,0.30])
P₃	([0.57,0.62],[0.30,0.38])	([0.29,0.36],[0.58,0.64])
P₄	([0.34,0.40],[0.50,0.60])	([0.32,0.40],[0.55,0.60])
P₅	([0.62,0.70],[0.25,0.30])	([0.60,0.65],[0.30,0.35])

Table 2

IVIF evaluation matrix by H₂

	T₁	T₂
P₁	([0.36,0.41],[0.56,0.59])	([0.41,0.45],[0.50,0.55])
P₂	([0.70,0.80],[0.15,0.20])	([0.36,0.40],[0.57,0.60])
P₃	([0.55,0.62],[0.27,0.38])	([0.29,0.35],[0.60,0.65])
P₄	([0.28,0.46],[0.50,0.54])	([0.53,0.60],[0.35,0.40])
P₅	([0.52,0.60],[0.35,0.40])	([0.46,0.52],[0.40,0.48])
	T₃	T₄
P₁	([0.74,0.80],[0.15,0.20])	([0.52,0.62],[0.32,0.38])
P₂	([0.59,0.65],[0.30,0.35])	([0.66,0.75],[0.20,0.25])
P₃	([0.57,0.62],[0.32,0.38])	([0.60,0.65],[0.30,0.35])
P₄	([0.68,0.75],[0.20,0.25])	([0.35,0.40],[0.55,0.60])
P₅	([0.41,0.52],[0.40,0.48])	([0.58,0.63],[0.30,0.37])

Table 3

IVIF evaluation matrix by H₃

	T₁	T₂
P₁	([0.59,0.62],[0.26,0.38])	([0.63,0.70],[0.25,0.30])
P₂	([0.65,0.75],[0.20,0.25])	([0.35,0.40],[0.55,0.60])
P₃	([0.37,0.40],[0.53,0.60])	([0.42,0.48],[0.50,0.52])
P₄	([0.61,0.65],[0.30,0.35])	([0.38,0.42],[0.52,0.58])
P₅	([0.35,0.45],[0.50,0.55])	([0.61,0.65],[0.30,0.35])
	T₃	T₄
P₁	([0.37,0.45],[0.50,0.55])	([0.55,0.60],[0.32,0.40])
P₂	([0.70,0.80],[0.10,0.20])	([0.52,0.62],[0.30,0.38])
P₃	([0.19,0.25],[0.70,0.75])	([0.59,0.65],[0.30,0.35])
P₄	([0.62,0.70],[0.25,0.30])	([0.37,0.45],[0.55,0.60])
P₅	([0.36,0.40],[0.55,0.60])	([0.55,0.62],[0.28,0.38])

Table 4

IVIF evaluation matrix by H₄

	T₁	T₂
P₁	([0.60,0.66],[0.30,0.34])	([0.35,0.40],[0.50,0.60])
P₂	([0.50,0.50],[0.50,0.50])	([0.28,0.32],[0.60,0.68])
P₃	([0.43,0.46],[0.51,0.54])	([0.49,0.55],[0.40,0.45])
P₄	([0.70,0.78],[0.17,0.22])	([0.33,0.43],[0.51,0.57])
P₅	([0.22,0.34],[0.58,0.66])	([0.41,0.45],[0.50,0.55])
	T₃	T₄
P₁	([0.72,0.80],[0.15,0.20])	([0.28,0.35],[0.58,0.65])
P₂	([0.65,0.72],[0.20,0.28])	([0.41,0.55],[0.37,0.45])
P₃	([0.65,0.70],[0.25,0.30])	([0.62,0.72],[0.20,0.28])
P₄	([0.56,0.62],[0.30,0.38])	([0.19,0.25],[0.19,0.25])
P₅	([0.51,0.60],[0.35,0.40])	([0.33,0.42],[0.50,0.58])

Table 5

IVIF evaluation matrix by H₅

	T₁	T₂
P₁	([0.73,0.80],[0.10,0.20])	([0.46,0.55],[0.30,0.45])
P₂	([0.55,0.60],[0.30,0.40])	([0.34,0.41],[0.53,0.59])
P₃	([0.24,0.35],[0.58,0.65])	([0.27,0.34],[0.60,0.66])
P₄	([0.31,0.42],[0.50,0.58])	([0.59,0.65],[0.27,0.35])
P₅	([0.18,0.30],[0.65,0.70])	([0.40,0.45],[0.50,0.55])
	T₃	T₄
P₁	([0.38,0.45],[0.50,0.55])	([0.57,0.67],[0.26,0.33])
P₂	([0.63,0.75],[0.15,0.25])	([0.69,0.75],[0.18,0.25])
P₃	([0.55,0.60],[0.30,0.40])	([0.39,0.45],[0.48,0.55])
P₄	([0.36,0.41],[0.52,0.59])	([0.26,0.32],[0.60,0.68])
P₅	([0.59,0.65],[0.30,0.35])	([0.34,0.41],[0.52,0.59])

Table 6

Overall IVIF evaluation matrix

	T₁	T₂
P₁	([0.5265,0.5879],[0.2908,0.4079])	([0.4478,0.5187],[0.3848,0.4723])
P₂	([0.5623,0.6406],[0.3011,0.3594])	([0.3034,0.3589],[0.5872,0.6411])
P₃	([0.4125,0.4685],[0.4625,0.5315])	([0.3638,0.4299],[0.5243,0.5701])
P₄	([0.4759,0.5699],[0.3497,0.4301])	([0.4333,0.5053],[0.4275,0.4947])
P₅	([0.3160,0.4106],[0.5152,0.5894])	([0.4702,0.5165],[0.4317,0.4835])
	T₃	T₄
P₁	([0.5356,0.6149],[0.4550,0.3636])	([0.4872,0.5702],[0.4127,0.4153])
P₂	([0.6589,0.7498],[0.1660,0.2502])	([0.5875,0.6829],[0.2511,0.3171])
P₃	([0.5273,0.5805],[0.3471,0.4195])	([0.5144,0.5866],[0.3420,0.4134])
P₄	([0.5314,0.6012],[0.3298,0.3988])	([0.3010,0.3678],[0.5677,0.6322])
P₅	([0.5080,0.5864],[0.3567,0.4136])	([0.4932,0.5582],[0.3658,0.4418])

Step 2. Normalize the IVIF matrix $Q = {[q_{ij}]}_{m \times n}$ to $Q^{N} = {[q_{ij}^{N}]}_{m \times n}$ .

Step 3. Derive the attribute weights t_j (j = 1, 2, …, n) with CRITIC method as listed in Table 8.

Table 7

The normalized IVIF evaluation matrix

	T₁	T₂
P₁	([0.5265,0.5879],[0.2908,0.4079])	([0.3848,0.4723],[0.4478,0.5187])
P₂	([0.5623,0.6406],[0.3011,0.3594])	([0.5872,0.6411],[0.3034,0.3589])
P₃	([0.4125,0.4685],[0.4625,0.5315])	([0.5243,0.5701],[0.3638,0.4299])
P₄	([0.4759,0.5699],[0.3497,0.4301])	([0.4275,0.4947],[0.4333,0.5053])
P₅	([0.3160,0.4106],[0.5152,0.5894])	([0.4317,0.4835],[0.4702,0.5165])
	T₃	T₄
P₁	([0.5356,0.6149],[0.4550,0.3636])	([0.4872,0.5702],[0.4127,0.4153])
P₂	([0.6589,0.7498],[0.1660,0.2502])	([0.5875,0.6829],[0.2511,0.3171])
P₃	([0.5273,0.5805],[0.3471,0.4195])	([0.5144,0.5866],[0.3420,0.4134])
P₄	([0.5314,0.6012],[0.3298,0.3988])	([0.3010,0.3678],[0.5677,0.6322])
P₅	([0.5080,0.5864],[0.3567,0.4136])	([0.4932,0.5582],[0.3658,0.4418])

Table 8

The attributes weights t_j

	T₁	T₂	T₃	T₄
t _j	0.3480	0.1621	0.1358	0.3541

Step 4. Determine the ${IVIFPIS}_{j}$ and ${IVIFNIS}_{j}$ by using the Equation (18-19). $\begin{matrix} {IVIFPIS}_{1} = 〈 (0 . 5623, 0 . 6406), (0 . 2908, 0 . 3594) 〉 \\ {IVIFPIS}_{2} = 〈 (0 . 5872, 0 . 6411), (0 . 3034, 0 . 3589) 〉 \\ {IVIFPIS}_{3} = 〈 (0 . 6589, 0 . 7498), (0 . 1660, 0 . 2502) 〉 \\ {IVIFPIS}_{4} = 〈 (0 . 5875, 0 . 6829), (0 . 2511, 0 . 3171) 〉 \\ {IVIFNIS}_{1} = 〈 (0 . 3160, 0 . 4106), (0 . 5152, 0 . 5894) 〉 \\ {IVIFNIS}_{2} = 〈 (0 . 3848, 0 . 4723), (0 . 4702, 0 . 5187) 〉 \\ {IVIFNIS}_{3} = 〈 (0 . 5080, 0 . 5805), (0 . 4550, 0 . 4195) 〉 \\ {IVIFNIS}_{4} = 〈 (0 . 3010, 0 . 3678), (0 . 5677, 0 . 6322) 〉 \end{matrix}$

Step 5. Obtain IVIFS_i and IVIFR_i index on the basis of the Equations (20-21). $\begin{matrix} {IVIFS}_{1} = 0 . 4770, {IVIFS}_{2} = 0 . 0077, IV {IFS}_{3} = 0 . 5236 \\ {IVIFS}_{4} = 0 . 6964, IV {IFS}_{5} = 0 . 7404 \\ {IVIFR}_{1} = 0 . 1574, {IVIFR}_{2} = 0 . 0077, {IVIFR}_{3} = 0 . 2492 \\ {IVIFR}_{4} = 0 . 3541, {IVIFR}_{5} = 0 . 3480 \end{matrix}$

Step 6. Calculate the final value IVIFQ_i through the Equation (22). $\begin{matrix} {IVIFQ}_{1} = 0 . 5363, {IVIFQ}_{2} = 0 . 0000, \\ {IVIFQ}_{3} = 0 . 7006, {IVIFQ}_{4} = 0 . 9700, \\ {IVIFQ}_{5} = 0 . 9912 \end{matrix}$

Step 7. Relying on the calculating results of IVIFQ_i, all the alternatives can be ordered. The optimal alternative is the one with the lowest value of IVIFQ_i. Evidently, the order is P₂ > P₁ > P₃ > P₄ > P₅ and P₂ is the optimal alternative.

4.2 Compare analysis

In this part, our developed method is made comparison with some other methods to illustrate its superiority.

First of all, our designed method is compared with IVIFWA and IVIFWG fused operators. For the IVIFWA operator, the calculating result is S (P₁) = 0 . 5877, S (P₂) = 0 . 6495, S (P₃) = 0 . 5655, S (P₄) = 0 . 5066, S (P₅) = 0 . 4899. Thus, the ranking order is P₂ > P₁ > P₃ > P₄ > P₅. For the IVIFWG operator, the calculating result is S (P₁) = 0 . 5672, S (P₂) = 0 . 6391, S (P₃) = 0 . 5431, S (P₄) = 0 . 5022, S (P₅) = 0 . 4639. So the ranking order is P₂ > P₁ > P₃ > P₄ > P₅.

What’s more, our presented method is also compared with extended MABAC method under IVIFSs [56]. Then we could obtain the calculating result. The overall value of each alternative is derived as: I₁ = 1 . 7236, I₂ = 3 . 5211, I₃ = 1 . 087, I₄ = 0 . 8833, I₅ = 0 . 2659. Therefore, the order of alternatives is derived: P₂ > P₁ > P₃ > P₄ > P₅.

Besides, our designed method could be compared with GRA under IVIFSs [57]. Then we could obtain the calculating result. The grey relational degrees of each alternative are calculated as: γ₁ = 0 . 8674, γ₂ = 1 . 0000, γ₃ = 0 . 8086, γ₄ = 0 . 7633, γ₅ = 0 . 6597. Therefore, the ranking order of alternatives is P₂ > P₁ > P₃ > P₄ > P₅.

In the end, our presented method is also compared with IVIF MCDM-based CODAS [58]. Then we can obtain the calculating result. The total assessment score (AS) of each alternative is calculated as: AS₁ = 0 . 3256, AS₂ = 1 . 6531, AS₃ = -0 . 0922, AS₄ = -0 . 1262, AS₅ = -1 . 2311. Therefore, the order of alternatives is P₂ > P₁ > P₃ > P₄ > P₅.

Eventually, the results of other methods are depicted in Table 9.

Table 9
Evaluation results of other methods

Methods Order The optimal alternative The worst alternative

IVIFWA operator P₂ > P₁ > P₃ > P₄ > P₅ P ₂ P ₅

IVIFWG operator P₂ > P₁ > P₃ > P₄ > P₅ P ₂ P ₅

The MABAC method P₂ > P₁ > P₃ > P₄ > P₅ P ₂ P ₅

The GRA method P₂ > P₁ > P₃ > P₄ > P₅ P ₂ P ₅

The CODAS method P₂ > P₁ > P₃ > P₄ > P₅ P ₂ P ₅

The developed method P₂ > P₁ > P₃ > P₄ > P₅ P ₂ P ₅

Methods	Order	The optimal alternative	The worst alternative
IVIFWA operator	P₂ > P₁ > P₃ > P₄ > P₅	P ₂	P ₅
IVIFWG operator	P₂ > P₁ > P₃ > P₄ > P₅	P ₂	P ₅
The MABAC method	P₂ > P₁ > P₃ > P₄ > P₅	P ₂	P ₅
The GRA method	P₂ > P₁ > P₃ > P₄ > P₅	P ₂	P ₅
The CODAS method	P₂ > P₁ > P₃ > P₄ > P₅	P ₂	P ₅
The developed method	P₂ > P₁ > P₃ > P₄ > P₅	P ₂	P ₅

Derived from the Table 9, it is evidently that the best alternative is P₂, while the worst choice is P₅. Although the optimal and worst alternatives are same, these methods’ order are slightly different. These methods can effectively deal with MAGDM from different angles, which can be viewed as their merits. The IVIFWA and IVIFWG operators emphasis to fuse evaluation information. The MABAC method with IVIFSs emphasis to consider the risk preferences of DMs and the interactive characteristics between criteria and the positive and negative ideal solutions. The GRA with IVIFSs emphasis similarity degree between two sequences. And the CODAS method emphasis the combination of the Euclidean and Hamming distance. However, compared with these methods, our designed method is more precision, since it not only calculates each alternative’s closest ideal value but also calculates each alternative’s farthest worst value. And the CRITIC method can consider the differences among all alternatives’ performance values, which can minimize subjective randomness while the criteria weights are determined.

5 Conclusion

This paper designs an effective solution for such kind of issue, since it designs a novel IVIF-VIKOR method for evaluating the urban road traffic safety. And then a numerical example for evaluating the urban road traffic safety has been given to confirm that this novel method is reasonable. What’s more, to verify the validity and feasibility of the developed method, some comparative analysis is also designed. However, the main drawback of such paper is that the number of DMs and attributes are small and interdependency of criteria is not considered, which may limit the application scope of the designed method to some extent. Future research can cope with the interdependency of criteria by utilizing some methods including analytic network process (ANP). Furthermore, the designed method can also be applied to many other uncertain environments [59 –61] and the designed method can be utilized to tackle many other MAGDM issues like project selection and site selection [62 –64].

Footnotes

Acknowledgments

This work was supported by Scientific Research Project of Education Department of Shaanxi Provincial Government (Project No. 18JK0450) and Natural Science Basic Research Plan of Shaanxi Province (Project No. 2020JQ-682).

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