An Integrated Approach of Best-Worst Method (BWM) and Triangular Fuzzy Sets for Evaluating Driver Behavior Factors Related to Road Safety

: Driver behavior plays a major role in road safety because it is considered as a signiﬁcant argument in tra ﬃ c accident avoidance. Drivers mostly face various risky driving factors which lead to fatal accidents or serious injury. This study aims to evaluate and prioritize the signiﬁcant driver behavior factors related to road safety. In this regard, we integrated a decision-making model of the Best-Worst Method (BWM) with the triangular fuzzy sets as a solution for optimizing our complex decision-making problem, which is associated with uncertainty and ambiguity. Driving characteristics are di ﬀ erent in di ﬀ erent driving situations which indicate the ambiguous and complex attitude of individuals, and decision-makers (DMs) need to improve the reliability of the decision. Since the crisp values of factors may be inadequate to model the real-world problem considering the vagueness and the ambiguity, and providing the pairwise comparisons with the requirement of less compared data, the BWM integrated with triangular fuzzy sets is used in the study to evaluate risky driver behavior factors for a designed three-level hierarchical structure. The model results provide the most signiﬁcant driver behavior factors that inﬂuence road safety for each level based on evaluator responses on the Driver Behavior Questionnaire (DBQ). Moreover, the model generates a more consistent decision process by the new consistency ratio of F-BWM. An adaptable application process from the model is also generated for future attempts.


Introduction
According to data from the worldwide road safety status report, annual traffic deaths are reported to reach 1.35 million [1]. According to this report, it was stated that the road safety performance for Hungary is below the EU average. In 2018, the proportion of people died on the roads in Hungary was set at 64 per million, and this statistic increased by 1% compared to the previous year [2]. However, the Road Safety Action Program (2014-2016) was integrated with the Hungarian Transport Strategy in line with the goal of reducing the number of road deaths by 50% between 2010 and 2020. According to the Road Safety Action Program situation analysis, most accidents stemmed from human-induced factors. Therefore, addressing them becomes a dynamic target of road safety actions [3]. According to the estimates of some previous studies, approximately 90% of road traffic accidents have been found to be the sole or major causative factor of human factors [4][5][6].
pairwise comparisons and handles inconsistency in an effective manner versus conventional AHP with triangular fuzzy sets. Therefore, it presents an easy and accurate decision framework.

Literature Review
In the literature, BWM and its fuzzy extended versions are frequently applied to various areas, from manufacturing to supply chain management and transportation [30]. Although plenty of papers have been published in these areas, there are very few contributions applied to the evaluation of driving behaviors for road safety. Most of the recent BWM/F-BWM contributions focus on supply chain design, supplier or green supplier evaluation, and occupational or environmental safety risk analysis [31].
Regarding supply chain performance and supplier assessment, Badi and Ballem [31] studied supplier selection problems in the pharmaceutical industry using an integrated rough BWM method. While the Z-number BWM is proposed by Aboutorab et al. [32] for supplier development problem, intuitionistic F-BWM is applied for the green supplier selection problem by Tian et al. [33]. Wu et al. [34] integrated the interval type-2 fuzzy sets and BWM for green supplier selection problems.
In the risk assessment literature, AHP or F-AHP is mostly used as a weighted factor scoring method for risk parameters. After BWM is introduced, researchers have begun to use it instead of AHP due to its superiorities. BWM/F-BWM is used to assign weights to risk parameters like AHP. In many studies, it is frequently integrated with FMEA [35][36][37][38][39][40]. In some other studies, it is integrated with MCDM such as interval triangular fuzzy Delphi method under 5 × 5 matrix [41], F-TOPSIS [42], and artificial intelligence-based methods such as Bayesian networks [43,44] and business impact analysis [45].
In the light of this brief review of the previous studies related to BWM and F-BWM applied to diverse selection and ranking problems, it is observed that BWM has not integrated yet with fuzzy sets for the addressed problem "driving behavior evaluation". Therefore, in the proposed approach, F-BWM is utilized to determine the importance weight of driver behavior factors related to road safety. As fuzzy extensions, triangular fuzzy numbers are used in the existed study since they reflect uncertainty in the decision-making process well. Additionally, the DBQ is attached to the F-BWM to strengthen the methodology of the study. The benefits of the applied F-BWM methodology are as follows: (1) From the theoretical viewpoint, it has designed a solid decision-making framework with the aid of triangular fuzzy numbers and, modeled uncertainty well. Although the literature covers methods like AHP and BWM in determining the importance weights of factors, the F-BWM methodology fits well with the structure of the problem handled in this study. The full consistency method (FUCOM) is the simplest example. It is proposed by Pamučar et al. [46] and applied by many scholars [47][48][49] to various MCDM problems. (2) From an application viewpoint, the study considers a DBQ and priorities the driving behavior factors relating to road safety. Previous studies regarding driving behavior factor evaluation are mostly based on statistical inference logic either cross-sectional or cross-cultural. However, this study handles the problem in an MCDM manner. Additionally, as a case study, the Budapest city of Hungary is studied to demonstrate applicability. Of course, the approach can be adapted to other cities.

Driver Behavior Questionnaire (DBQ) Survey
The study utilized the driver behavior questionnaire (DBQ) as a tool to collect driver behavior data on perceived road safety issues from Budapest city. To do so, the car drivers having at least fifteen-year driving experience were asked to fill the DBQ by face-to-face method, which increased its reliability. The drivers who participated in this study were the faculty members of the Department of Transport Technology and Economics and the Department of Control for Transportation and Vehicle Systems at Budapest University of Technology and Economics who also have transportation engineering research experience. In addition, the participants were asked to indicate how often they likely to involve in each of the observed driver behaviors in the recent year using Saaty's traditional ratio scale (1)(2)(3)(4)(5)(6)(7)(8)(9). The questionnaire survey was designed in two parts: The first part intended to measure demographic data about the participants and results are tabulated in Table 1. The results stated the mean and standard deviation (SD) values of observed data such as age, gender, and driving experience based on drivers' responses. In addition, we used digits (1, 0) for assessment purposes to explain simply the characteristics of gender. The second part of DBQ designed on Saaty scale (1977) to analyze the significant driver behavior factors related to road safety. For evaluation purposes, the driver behavior factors were designed in a three-level hierarchical structure and symbolized each factor with alphabet 'F' as shown in Figure 1. These driver behavior factors have a significant influence on road safety. Some recent studies considered the specified driver behavior factors for evaluation of road safety performance by different evaluator groups [50][51][52]. engineering research experience. In addition, the participants were asked to indicate how often they likely to involve in each of the observed driver behaviors in the recent year using Saaty's traditional ratio scale (1)(2)(3)(4)(5)(6)(7)(8)(9). The questionnaire survey was designed in two parts: The first part intended to measure demographic data about the participants and results are tabulated in Table 1. The results stated the mean and standard deviation (SD) values of observed data such as age, gender, and driving experience based on drivers' responses. In addition, we used digits (1, 0) for assessment purposes to explain simply the characteristics of gender. The second part of DBQ designed on Saaty scale (1977) to analyze the significant driver behavior factors related to road safety. For evaluation purposes, the driver behavior factors were designed in a three-level hierarchical structure and symbolized each factor with alphabet 'F' as shown in Figure  1. These driver behavior factors have a significant influence on road safety. Some recent studies considered the specified driver behavior factors for evaluation of road safety performance by different evaluator groups [50][51][52].

Overview on Best Worst Method (BWM)
The general BWM method was created by Rezaei (2015) to derive the weights of the criteria with the smaller number of comparisons and more consistent comparisons. The most important factor is the one which has the most vital role in making the decision, while the less important has the opposite role in the decision process. Furthermore, the BMW does not only derive the weights independently but it can also be combined with other multi-criteria-decision-making methods [53][54][55].
The procedure of the BWM can be highlighted as follows:

Overview on Best Worst Method (BWM)
The general BWM method was created by Rezaei (2015) to derive the weights of the criteria with the smaller number of comparisons and more consistent comparisons. The most important factor is the one which has the most vital role in making the decision, while the less important has the opposite role in the decision process. Furthermore, the BMW does not only derive the weights independently but it can also be combined with other multi-criteria-decision-making methods [53][54][55].

•
The procedure of the BWM can be highlighted as follows: We consider a set of elements (e 1 , e 2 , . . . , e n ) and then select the most important element and compare it to others using Saaty's scale (1)(2)(3)(4)(5)(6)(7)(8)(9). Accordingly, this provides the most important element to other vectors would be: E a = (e a1 , e a2 , . . . , e an ), and obviously e aa = 1. However, the least important element to other vectors would be: E b = (e 1b , e 2b , . . . , e nb ) T by using the same scale.
After deriving the optimal weight scores, the consistency has been checked through computing the consistency ratio from the following formula: where Table 2 provides us the consistency index values: To obtain an optimal weight for all elements, the maximum definite differences are w a w j − e aj and w j w b − e jb , and for all j is minimized. If we assumed a positive-sum for the weights, the following problem would be solved: The problem could be transferred into the following problem: By solving this problem, we obtain the optimal weights and ξ * . For further reading on priority criteria, one may refer to [56,57]. While w B presents the importance weights of best criterion, w W shows the e importance weights of the worst criterion. e Bj denotes the evaluation of the best to others, e W j denotes the evaluation of the others to worst.

General Information on Fuzzy Sets
Prior to explaining F-BWM, some fundamental notations regarding fuzzy sets can be useful. Zadeh [58] introduced fuzzy sets for better reflecting of the human judgments and assessment in the decision making process. It is considered as a more robust tool to deal with vagueness, ambiguity, and uncertainty. Many decision-making problems consist of goals, constraints, and possible actions that are not known precisely [58]. The usage of fuzzy sets is better for transforming the linguistic decision of human judgment. Hence, many real-world decision-making problems have used fuzzy sets [59,60]. A triangular fuzzy number consists of lower, medium and upper numbers of the fuzzy as A = (l, m, u) where l, m and u which is crisp and real numbers (x ≤ y ≤ z. The membership function of a triangular fuzzy number can be defined as follows: A triangular fuzzy number is presented in Figure 2. The linguistic terms and triangular fuzzy numbers are also given in Table 3. and uncertainty. Many decision-making problems consist of goals, constraints, and possible actions that are not known precisely [58]. The usage of fuzzy sets is better for transforming the linguistic decision of human judgment. Hence, many real-world decision-making problems have used fuzzy sets [59,60]. A triangular fuzzy number consists of lower, medium and upper numbers of the fuzzy as , , where l, m and u which is crisp and real numbers ( . The membership function of a triangular fuzzy number can be defined as follows: A triangular fuzzy number is presented in Figure 2. The linguistic terms and triangular fuzzy numbers are also given in Table 3.  , , and , , are any two triangular fuzzy numbers, and the mathematical calculation of the two triangular fuzzy numbers is defined as follows: The addition operation: The subtraction operation: The multiplication operation:  A 1 = (l 1 , m 1 , u 1 ) and A 2 = (l 2 , m 2 , u 2 ) are any two triangular fuzzy numbers, and the mathematical calculation of the two triangular fuzzy numbers is defined as follows: The addition operation: The subtraction operation: The multiplication operation: The arithmetic operation: The graded mean integration representation (GMIR) R A i of a triangular fuzzy number for the ranking of triangular fuzzy number is calculate as follows: The BWM was proposed by Rezai (2015) for multi-criteria decision-making problems considering pairwise comparison manner. The best and worst criteria are determined in BWM [33]. Different fuzzy sets-based versions have been proposed as intuitionistic fuzzy sets [32,61], triangular fuzzy numbers [27,62], Z-numbers [34], dominance degree [63], and interval type-2 fuzzy number [64,65]. Mi et al. [66] presented a survey of BWM applications and extensions. The interested readers and researchers may refer to this study in detail.
In BWM, there are n criteria, and the fuzzy pairwise comparisons are applied based on the linguistic terms of decision-makers as presented in Table 3. Then, the linguistic evaluations are transformed into triangular fuzzy numbers. The fuzzy comparison matrix is getting as follows: where a ij denotes the relative fuzzy preference of criterion i to criterion j, which is a triangular fuzzy number; a ij = (1, 1, 1) when i = j.
In this paper, we will present the detailed steps of fuzzy BWM. The detailed steps of fuzzy BWM are used for obtaining the fuzzy weights [62].
Step 1. Construct the criteria system. A set of criteria reflects the performances of different criteria. Suppose there are n decision criteria {c 1 , c 2 , . . . , c n }.
Step 2. Determine the best criterion and the worst criterion. In this step, the best criterion and the worst criterion is determined by experts based on the constructed decision criteria system. The best criterion is denoted as c B , and the worst criterion is also denoted as c W .
Step 3. Perform the fuzzy reference comparisons for the best criterion. According to the pairwise comparison a ij , c B is the best criterion; c W is the worst criterion. The fuzzy preferences of the best criterion over all the criteria can be determined. Then, the fuzzy comparisons are converted to triangular fuzzy numbers. The fuzzy Best-to-Others vector is obtained as follows: where A B denotes the fuzzy best-to-others vector; a Bj denotes the fuzzy comparison of the best criterion c B over criterion j, j = 1, 2, . . . , n. It is known that a BB = (1, 1, 1).
Step 4. Perform the fuzzy reference comparisons for the worst criterion. In this step, the fuzzy preferences of all the criteria over the worst criterion can be determined. They are transformed into triangular fuzzy numbers. The fuzzy others-to-worst vector can be obtained as: where A W denotes the fuzzy others-to-worst vector; a iW denotes the fuzzy comparison of the worst criterion c W , i = 1, 2, . . . , n. It is known that a WW = (1, 1, 1).
Step 5. Determine the optimal fuzzy weights w * 1 , w * 2 , . . . , w * n . In this step, the optimal fuzzy weight for each criterion is determined for each fuzzy pair w B / w j and w j / w W . It should have w B / w j = a Bj and w j / w W = a jW . A solution is obtained that the maximum absolute gaps w B w j − a Bj and w j w W − a jW for all j are minimized to satisfy these conditions for all j. w B , w j and w W in fuzzy BWM are triangular fuzzy numbers. In some cases, we prefer to use w j = l w j , m w j , u w j for optimal criteria selection. The triangular fuzzy weight of the criterion w j = l w j , m w j , u w j is transformed to a crisp value using Equation (11). Consequently, the constrained optimization problem is constructed for obtaining the optimal fuzzy weights w * 1 , w * 2 , . . . , w * n as follows: where Bj and a jW = l w jW , m w jW , u w jW . Equation (12) is transformed to the nonlinearly constrained optimization problem: where ξ = l ξ , m ξ , u ξ . Considering l ξ ≤ m ξ ≤ u ξ , it is supposed that ξ * = (k * , k * , k * ), k * ≤ l ξ then Equation (13) can be transferred as: Step 6. Determine the consistency ratio. The consistency ratio is determined in the same way as BWM. In this step, the consistency index for fuzzy BWM is calculated. The main steps of the proposed F-BWM model are discussed in Figure 3. Step 6. Determine the consistency ratio. The consistency ratio is determined in the same way as BWM. In this step, the consistency index for fuzzy BWM is calculated..
The main steps of the proposed F-BWM model are discussed in Figure 3.

Results
The F-BWM model was applied to evaluate driver behavior factors related to road safety and to compute weight scores. Furthermore, the reliability of the PCs consistency in F-BWM was checked, and it was acceptable for each matrix. In the following, step by step application of F-BWM to the problem is provided. In this application, three main factors, eight sub-factors, and nine sub-subfactors are evaluated. We will first present the F-BWM model for main factors as violations (F1), lapses (F2), and errors (F3). The violations (F1) and lapses (F2) are determined as the most significant and the less significant factor, respectively (Step 2). The fuzzy reference comparisons are applied, and the

Results
The F-BWM model was applied to evaluate driver behavior factors related to road safety and to compute weight scores. Furthermore, the reliability of the PCs consistency in F-BWM was checked, and it was acceptable for each matrix. In the following, step by step application of F-BWM to the problem is provided. In this application, three main factors, eight sub-factors, and nine sub-sub-factors are evaluated. We will first present the F-BWM model for main factors as violations (F 1 ), lapses (F 2 ), and errors (F 3 ). The violations (F 1 ) and lapses (F 2 ) are determined as the most significant and the less significant factor, respectively (Step 2). The fuzzy reference comparisons are applied, and the linguistic terms for fuzzy preferences of the most significant factor and the less significant factor are given in Tables 4 and 5, respectively. Table 4. The linguistic terms for fuzzy preferences of the most important factor.

F1 F2 F3
Best factor (F1) EI FI WI Then, the fuzzy most significant-to-others vector and the fuzzy others-to-less significant can be obtained with respect to Table 3   Then, for obtaining the optimal fuzzy weights of all the main factors, the nonlinearly constrained model is constructed as follows in Equation (14): Then, the following nonlinearly constrained optimization problem is obtained using represented by crisp numbers as in Equation (15). min e s.t.
The optimal fuzzy weights of three factors ('violations', 'lapses', and 'errors') are calculated as follows: Then, the crisp weights of three factors 'violations', 'lapses', and 'errors', are determined as follows: w * F1 = 0.423, w * F2 = 0.251, w * F3 = 0.327. In this process, the consistency ratio is calculated. a Bw = a 12 = (1.5, 2, 2.5) is the largest in the interval, hence, CI is considered as 5.29 using Table 6. The consistency ratio is CR = 0.303/5.29 = 0.0573 which shows a very high consistency because the consistency ratio 0.0573 is very close to zero. According to the results of the F-BWM model, among the main factors at the first level, "violations" (F1) were found to be the most crucial driver behavior factor related to road safety based on the responses given by the assessors in DBQ. One of the previous studies [67] stated that Road Traffic Violations (RTVs) are the most important factor causing certain risks for other road users. Subsequently, "errors" (F3) followed by "lapses" (F2) as shown in Table 7, as the second-ranking factor. Among the second level factors, "aggressive violation" (F12) has emerged as the most crucial driver behavior factor related to road safety. According to the results of a previous study carried for Finland and Iran [68], it is found a significant relationship between aggressive violations and the number of accidents. Additionally, the results demonstrated that "fail to apply brakes in road hazards" (F33) was determined as the second most crucial factor as compared to other related factors. The previous study noticed that more fatalities can occur if the driver does not apply the brakes and has higher impact-speed crashes [69,70]. While "pull away from traffic lights in wrong gear" (F22) is observed as the lowest ranked driver behavior factor related to road safety as shown in Table 8. According to the evaluation results of the third level factors, the most important driver behavior factor related to road safety was identified as "driving with alcohol" (F126). This result is directly proportional to the zero-tolerance policy in practice in drinking and driving according to Hungarian driving laws and can be verified in this context [71]. Subsequently, the model results observed, "failing to yield pedestrian" (F122) as second rank factor followed by "disobey traffic lights" (F123). The previous study revealed that one of the possible causes for the high number of crashes and injuries is due to beating traffic lights [72]. While the results showed "no deterrence of punishing" (F124) as the least rank driver behavior factor as compared to other related factors as shown in Table 9. Due to space limitations, open forms of mathematical models for the remaining two levels (levels 2 and level 3) are provided in the Appendix A. All mathematical models for the F-BWM are solved in GAMS version 23.5.1 as minimization problems by mixed-integer non-linear programming (MINLP).

Comparative Study
In this section, we make a comparative study between the results of the existed approach (F-BWM model) and a recent hybrid study covering AHP and BWM models [51]. Moslem et al. [51] handled evaluation of the driver behavior factors related to road safety using both AHP and BWM. They used AHP in PCMs that have a 4 × 4 or smaller structure. On the other side, they used BWM in 5 × 5 matrices or larger ones. We then observe the variations in factor rankings of both approaches. The results are shown in Table 10. Table 10. Comparative study results of factor ranks. )   F1  1  1  F2  3  3  F3  2  2   F11  7  4  F12  1  1  F21  4  6  F22  6  8  F23  3  3  F31  5  5  F32  8  7  F33  2  2   F111  9  7  F112  6  4  F113  8  5  F121  7  3  F122  3  2  F123  2  3  F124  5  8  F125  4  6  F126  1  1 It is observed from Table 10 that, by both approaches, the ranks of main factors have remained the same. By using the AHP-BWM model of Moslem et al. [51], we notice that the ranks of sub-factors F11, F21, and F22 are changed. Regarding sub-sub-factors, F126 is the most important one by both approaches. When we compare the results obtained by both approaches, we observe that there are very small rank variations between them. The highest difference is observed in sub-sub-factor ranking results. Although we do not observe drastic rank variations between the benchmarking model that have been previously proved in the literature and our current approach, it can be claimed that the application of this approach is new in the application domain. It is also noted that, according to a correlation analysis, which measures the association between the rank of factors, there is a significant and strong positive correlation between both approaches. The Spearman rank correlation coefficient (RHO) values for every three groups are obtained as 1.00, 0.79, and 0.62.

AHP-BWM Model (Moslem et al. [51]) F-BWM Model (Existed Study
From the methodological perspective, there exist some similar contributions in the literature [73][74][75][76]. Gerogiannis et al. [73] studied a group-AHP scoring model. The method used in [73] is differentiated from our current approach considering that the decision is based on the aggregation of both experts' and users' judgements. Similar to our approach, [73] seeks a solution in facing a very large number of PCs. However, in BWM/FBWM-based approaches, decreased PCMs are designed according to the best and the worst criterion. The two studies of [74] and [75] have focused on the improvement of the traditional BWM approach. While the first one proposes a mixed-integer linear programming model approximation, the other deals with a robust solution to BWM. In [76], the same problem which we handle in the current study is aimed to solve by using the analytic network process (ANP). A primitive version of the criteria set which we used in the current study is used to prioritize. It has also taken into account the interrelationships between the decision criteria. In light of the above critics, the integrated BWM approach with triangular fuzzy sets enables decision-makers more freedom in making the final decision and face with a decreased number of PCMs.

Conclusions
The significance of driver behavior factors for road safety is critical and difficult to analyze due to uncertain driver behavior. The novelty of this study is the combined use of the best-worst method (BWM) and triangular fuzzy sets as a supporting tool for ranking and prioritizing the critical driver behavior criteria. For the first level of hierarchical structure, the study evaluation results observed the 'violations' as the most significant factor related to road safety followed by 'errors'. Subsequently, for the second level, the study results observed the 'aggressive violations' as the most significant driver behavior factor related to road safety followed by 'fail to apply brakes in road hazards'. While the study results revealed the 'visual scan wrongly' as the least important driver behavior factor related to road safety. Furthermore, for the third level, the F-BWM model results evaluated the 'drive alcohol use' as the most important factor followed by 'disobey traffic lights' as compared to other specified factors. While 'failing to use personal intelligence' was observed as the least important driver behavior factor related to road safety.
Driver behavior recognition has been noticed as a significant and complex concern to obviate road issues due to the huge amount of driver behavior data and its variation [47][48][49]. In the current study, we explained some AHP drawbacks and then utilized an advanced F-BWM model for estimating the driver behavior factors related to road safety. To collect driver behavior data, the study utilized the driver behavior questionnaire from experienced drivers with fifteen years of driving experience or more. This causes less evaluation time and better understandability for evaluators due to fewer comparisons as compared to conventional methods, like AHP. The acquired model results are more coherent due to more consistent PCs which increase the efficiency of the proposed model.
Considering further research, more applications of the F-BWM model are essential to obtain familiar to analyze different real-world features. The objective advantages are evident: it gives quicker and cheaper survey processes, and undoubtedly the survey pattern can more easily be expanded by this method than employing the classical AHP with complex PC questionnaires. However, this paper only provided one example, but many other applications can ultimately validate the technique. The F-BWM model will help the researchers to enhance their future studies by developing consistency with fewer PCs and save time for analyzing the collected data.
For future directions, BWM can be applied to the same problem under recently released fuzzy extensions such as Pythagorean fuzzy sets [77][78][79], spherical fuzzy sets [80], and hexagonal fuzzy sets [81]. By doing this, a comparative framework may be developed and used to test the solidity of the integration of BWM and fuzzy set extensions.
The consistency of the relative expert's responses regarding the weights of the factors, sub-factors, and sub-sub-factors have been checked. For each level, it is obtained a consistency ratio value lower than 0.1 as given in Table A1 below: