A bee colony optimization (BCO) and type-2 fuzzy approach to measuring the impact of speed perception on motor vehicle crash involvement

The major challenge of this paper is to examine how various forms of speed perception affect motor vehicle crash (MVC) involvement. To model this relationship, we use a type-2 fuzzy inference system (T2FIS). Another general challenge is to improve the performance of seven created T2FISs in a sense of compliance with the empirical data. This is achieved by a proposal of an algorithm based on the bee colony optimization (BCO) metaheuristic. The main novelty of this algorithm is the way how the testing points are selected in a type-2 fuzzy environment, which influences the execution efficiency. Data collection was carried out in twelve experiments. A total of 178 young drivers assessed the speed level from four positions; three of them relate to the speed perception of other vehicles on the road, while the remaining one represents the assessment of their own speed. At each position, three speed levels were assessed: 30, 50, and 70 km/h. As a result of the implemented methodology, a relationship between the various forms of speed perception and participation in MVCs can be quantified. The BCO-based algorithm achieved an average improvement of 21.17% in the performance of the initial T2FIS structures. The final results indicate that the drivers whose speed perception of the vehicle they are looking at from the rear side, as well as of the own vehicle, is poor have an elevated risk toward participation in MVCs compared to other forms of speed perception. This can be useful in various educational and recruitment procedures.


Introduction
Motor vehicle crashes (MVCs) result in 1.35 million deaths each year, which is around 3700 deaths per day globally (WHO 2018). Further, approximately 30 to 50 million people are either injured or permanently disabled every year. Moreover, MVCs cause great financial havoc of $518 billion every year and thus costing countries from 1 to 2% of their GDP (Ashraf et al. 2019;Lotfi et al. 2019). However, more than 85% of fatalities and injuries on the road occurred in developing countries (Al-Rukaibi et al. 2020). As in other parts of the world, road safety remains a major societal issue within the European Union (Murphy and Morris 2020). This is justified having in mind that more than 25,000 people died on the roads in Europe in 2018 (ETSC 2020).
The fact that MVCs are the leading cause of death for the 15-29 age group (Bassani et al. 2020) motivated us to deeply investigate one possible factor that affects the involvement of young drivers in crashes. It is about the speed perception of young drivers considering their own and other vehicles on the road.
To improve road safety, insight is needed into preventable causes of MVCs. The causes of motor vehicle collisions are complex but broadly depend on the characteristics of drivers (Rolison et al. 2018). Skill level (McGwin & Brown 1999), inexperience (McCartt et al. 2003), risk-taking behaviors (Rolison et al. 2014), and excessive speed (Costa et al. 2020;Gonzales et al. 2005;Lam 2003) have been implicated in the collisions of young drivers compared to drivers in other age ranges (Rolison et al. 2018). Young drivers often tend to risky behavior and also misjudge the speed of the vehicle, which makes them the drivers who are prone to participating in MVCs. In the literature, it is well known that driving speed is one of the most significant factors in road safety.
In-depth MVC analysis provides a comprehensive view of all the factors involved in a crash (Aarts and Schagen 2006). Along with geometric characteristics, road surface conditions, traffic status, and driver behavior (Schlögl 2020), speeding is among the most important variables influencing the occurrence of road crashes. Speed is a key risk factor in MVCs, increasing both the likelihood of a crash and the severity of resulting injuries (PAHO 2018). Speed can be deceiving since many circumstances affect how it is perceived as a risk factor, including the type of vehicle, time of day, weather conditions, and the design and state of the road, etc. (PAHO 2018). Driving speed, for example, is one of the behaviors affected by the driver's perception of the road's safety, and it is not necessarily compatible with the road's design speed (Misaghi and Hassan 2005). If the speed chosen is not appropriate in a given situation, it may result in losing control and runoffroad crashes (Ben-Bassat and Shinar 2011;Janssen et al. 2006). Speed estimation of the own vehicle and other vehicles on the road is an important task for drivers and is also crucial for roadway safety. The results of the study conducted by Fildes and Lee (1993) show that the interaction of driving experience, road type, and drivers' gender affects speed estimation. Perceiving the speed of own vehicle and other vehicles is critical to ensure safe driving and, in particular, to prevent the violations of the traffic law, such as going over the speed limit unintentionally (Wu et al. 2017). Close links exist between real vehicle speed and perceived vehicle speed; namely, based on the research by Zheng et al. 2018, the average driver's perceived speed is approximately twice less than the actual vehicle speed. Inaccurately estimated vehicle speed is the leading contributing factor to MVCs in highway tunnels (Zheng et al. 2018).
There is a question of what can be done to improve traffic safety, save people's lives, and reduce financial losses. A sanctioning system operates to respond to drivers who perform risky and illegal driving behaviors (McDonald et al. 2020); however, how to support the drivers whose perception of vehicle speed is poor? To what extent the drivers who inaccurately estimate vehicle speed are prone to be involved in MVCs and how to act preventively on them? The goal of this paper is exactly to answer these questions. By conducting 12 types of experiments related to vehicle speed perception and combining the results with the participants' driving history, we determine the driver's propensity for MVCs.
To model the driver characteristics, we propose the implementation of fuzzy logic; more concrete the interval type-2 fuzzy inference system (T2FIS). Since speed perception, which is the main topic of this research, involves measurement imprecision and subjectivity, fuzzy logic is a particularly convenient method to apply. There is evidence about the very promising performance of this method compared to the traditional statistic methods (Č ubranić-Dobrodolac et al. 2020a). Additionally, fuzzy logic is a convenient tool to use if the data collection implies descriptive answers; for example, the participant can be asked to assess the vehicle speed as small, medium, or high instead by concrete values. The use of a fuzzy inference system also allows further adjustment of a model aiming to describe certain phenomena as real as possible, following the collected data. This kind of adjustment represents a very complex task of combinatorial optimization and certain metaheuristic algorithm would be an expedient tool to use here. Having the previous advantages in mind, fuzzy logic is widely applied for MVCs prediction. Some of the examples are the papers by Chai et al. (2017), Č ubranić-Dobrodolac et al. (2019, 2020ab, c), Dimitriou and Vlahogianni (2015), Nemet et al. (2019), Thakur (2014), Xiong et al. (2019). In this paper, to test the stability of the initially obtained conclusions and to perform the optimization of T2FISs in the sense of maximizing compliance with the empirical data, we implemented the bee colony optimization (BCO) metaheuristic. We had chosen this approach having in mind the evidence from the literature about the very competitive results of the BCO algorithms, considering both the solution quality and computational efficiency (Marković, 2016(Marković, , 2017Yazid et al. 2019). There are several examples of using BCO for tuning the parameters of a fuzzy inference system (Amador-Angulo and Castillo, 2018;Nikolić et al. 2020;Mijović et al. 2021;Č ubranić-Dobrodolac et al. 2021). However, a special challenge in this procedure is how to choose the domains of search, i.e., how to set the constraints while changing the values of points of the membership functions. A detailed description of this procedure is offered just in the cases of type-1 fuzzy inferences systems (Nikolić et al. 2020;Č ubranić-Dobrodolac et al. 2021); therefore, this was a motive for the authors to consider this phenomenon in the type-2 fuzzy environment as well.
Having the previous explanation in mind, the main contributions of this study can be summarized as follows: (i) For the first time, a relationship between various types of speed perception and MVC involvement is modeled by T2FIS. This leads to useful conclusions in the field of traffic safety about what kind of poor speed perception contributes to a higher likelihood of being involved in MVCs. (ii) An original approach for setting the domains of search within the BCO approach in a type-2 fuzzy environment is proposed. (iii) The proposed methodology is tested and verified in the case study involving 178 young drivers. The obtained findings indicate the effectiveness of the introduced T2FIS and BCO approach, also offering significant recommendations for traffic safety improvement.

Literature review
This section provides an insight into the previous papers published in the field of interest for this research. It is divided into two subsections. The first is related to the human factor in traffic safety and perception issues, while the second considers the fuzzy inference systems and corresponding optimization procedures.

Perception and the human factor in traffic safety
MVCs are the result of the malfunction of a certain component of the driving system which constitutes the human, vehicle, road, and environment. Most of the safety studies conclude that human factors are the main cause of MVCs (Bucsuházy et al. 2020;Dingus et al. 2016;Weber et al. 2018). One of the crucial components concerning human factors is perception. Human perception is a complex phenomenon that affects the cognitive processes of driving. For this reason, it is one of the most important aspects to be considered in road safety research . The perception of the speed of movement is a basic visual function (Wu et al. 2017). Vision science studies show that optic flow is one of the main visual cues used for speed perception (Colombet et al. 2011;Lappe and Grigo 1999;Wu et al. 2017). Speed perception is significant for many aspects of driving, for example, to estimate the (un)safe distance between vehicles depending on the speed. Speed estimation by drivers is affected by a variety of sensory inputs, including visual, auditory, and, less significantly, kinesthetic and vestibular inputs. However, drivers primarily estimate their speed by analyzing visual input through a process called optic flow. As one moves through the environment, the visual field in front expands and passes by (Rudin-Brown 2004). However, the low or high luminance and monotonous road environment make them a weak visual reference system. This weakness leads to the deficiency of the visual reference system for drivers and thus reduces their perception of their actual driving speed, resulting in an extremely high risk of overspeeding and rear-end crashes (Zheng et al. 2018).
The studies that examine driver perception were carried out using different methodologies, by the experiments conducted in driving simulators (Ben-Bassat and Shinar 2011;Martens and Brouwer 2013;Pešić et al. 2019;Rudin-Brown 2004;Wu et al. 2017;Zheng et al. 2018), real traffic situations (Troscianko et al. 1999), tablet PCs and smart boards (Cicevic et al. 2017), by driver self-reports, i.e., questionnaire (Ojsteršek and Topolšek 2019), and also in virtual reality . The sample ranges from 14 to 200 respondents, while statistical analysis was most often used for data processing. Since the beginning of this century, the speed-MVCs relationship was examined in many studies throughout the world, where a basic understanding of the issue can be summarized in the following statements: higher speeds are associated with a higher probability of being involved in an MVC; higher speeds are directly related to higher MVC severity; higher speed variation increases the risk of MVC (Gitelman et al. 2017). Table 1 summarizes the most important characteristics of the mentioned papers, such as the considered problem, research methodology, sample, processing data, and the key research results.
Based on the literature review, previous studies were mainly limited to the investigation of the interactions among different factors that affect speed perception of drivers' vehicles. There is a need to explore more how the speed perception of vehicles influences the occurrence of MVCs.

The optimization of fuzzy inference systems
The optimization of the fuzzy inference system (FIS) represents a tuning of the characteristics of FIS to minimize or maximize the objective function, depending on the type of the considered task. Here, it is mostly the minimization task because the performance of FIS is generally measured as the level of deviation from certain empirical data. There are numerous examples where this procedure is useful. In

Statistical analysis
The speed overestimation of drivers in the middle of tunnels results from the presence of high-frequency visual information, while speed underestimation results from the presence of mediumfrequency and low-frequency visual information Ojsteršek and Topolšek (2019) The research focuses on selected visual and cognitive distractions that the driver is faced with, and on their influence on detecting and perceiving changes in the traffic environment Driver selfreport (213 drivers) Exploratory factor analysis, Confirmatory factor analysis, and Structural Equation Modelling Drivers that visually focus on traffic signals and pedestrians and think about driving speed, and driving rules, tend to notice crucial changes in the traffic environment more often Pešić et al.

T2FIS and BCO metaheuristic
The drivers whose speed perception of own vehicle and vehicle in front is poor, are more prone to MVCs the case of the current research, we implement the optimization of FIS intending to perform a stability analysis of the initial solution in the sense of the relationships between different FISs. Many papers deal with FIS optimization issues. Therefore, here we will offer just a review of the most frequently used techniques in the field in the last three years, from 2019 to 2021, which is shown in Table 2. An interesting fact to notice here is that general principles of FIS optimization set up in the past are valid also nowadays and the changes are in terms of newly applied optimizations methods, which have been proposed in the meanwhile. Guillaume (2001) systemized the procedures for fuzzy rule generations from empirical data and structured the optimization methods as ''shared partitions,'' ''clustering,'' and ''hybrid methods.'' The hybrid methods were based on the implementation of neuro-fuzzy modeling or heuristic algorithms, mentioning genetic algorithms (GA) as the most popular at that time.
One direction in the optimization procedures is related to the implementation of an adaptive neuro-fuzzy inference system-ANFIS (Jang, 1993). Certain authors combine the ANFIS method with other metaheuristics. Nath et al. (2020) combined particle swarm optimization (PSO) with ANFIS to optimize the rainfall-runoff relationship. Chouksey et al. (2020) applied an improved artificial neural network-based particle swarm optimization (IANN-PSO) method to maximize the power from the solar power system.
The development of metaheuristic approaches based on mimicking the behavioral patterns observed in nature has been very popular in recent decades. These techniques were successfully implemented in many cases for solving complex computational tasks, such as optimization of FIS.
As previously mentioned, genetic algorithms (GA) are frequently used. Nagammai et al. (2020) used GA to tune the membership functions of FIS for water level control in a conical tank process. Some authors further improved GA algorithms. For example, Chu et al. (2020) applied a nondominated sorting genetic algorithm-II (NSGA-II), as a multiobjective optimization method derived from GA, to optimize a fuzzy proportional-integral-derivative (PID) controller for automatic train operation. El-Gendy et al.
(2020) proposed a hybrid of GA and PSO to tune the parameters of different adaptive PID controllers. Mahmoodabadi and Nejadkourki (2020) applied FIS to regulate the control parameters of the PID controller for a quarter-car model, where the PSO algorithm is proposed to Our study Driver behavior modeling using speed perception WM-BCO Type-2 A bee colony optimization (BCO) and type-2 fuzzy approach to measuring the impact of speed… 4467 ascertain the optimum gains of the designed controller. The idea of PSO is inspired by the social behavior of bird flocking or fish schooling. The PSO metaheuristic is applied also by Zorić et al. (2019)  The ant colony optimization (ACO) algorithm was applied by Aldair et al. (2019) to tune and find the best parameters of the output membership function of the fuzzy controller for robot moves. Precup et al. (2020) implemented a relatively new metaheuristic called grey wolf optimizer (GWO) inspired by specific leadership styles of grey wolves. Ali et al. (2020) presented a quantum-inspired lightning search algorithm (QLSA) to optimize the performance of the induction motor under different speed and load conditions. Karar et al. (2020) applied the invasive weed optimization (IWO) algorithm inspired by the behavior of weed colonies. Elias and Mat Yahya (2020) applied the bats sonar algorithm (BSA) which is inspired by the echolocation process of a colony of bats to find food or prey.
Mohammadzadeh and Kayacan (2020) proposed the particle swarm optimization and artificial bee colony algorithm (PSO-ABC). The algorithms based on the bees demonstrated very competitive results in optimization procedures. For example, Yazid et al. (2019) demonstrated that the ABC outperforms the GA and PSO approach in optimizing the fuzzy logic controller for trajectory tracking of a quadcopter drone. Nikolić et al. (2020) compared the BCO and simulated annealing (SA) for fine-tuning the MFs of FIS. The BCO algorithm achieved better results in eight cases, while SA was better in seven cases. Mijović et al. (2021) applied the BCO and PSO. The BCO algorithm improved the initial solution by 44.9%, while the use of the PSO algorithm provided the improvement by only 19.9%. The BCO algorithm in the paper by Č ubranić-Dobrodolac et al. (2021) reached a 36% improvement of the objective function compared to the starting FIS. The mentioned BCO algorithms were designed in a type-1 fuzzy environment.
In our study, we combine ''shared partition'' and ''hybrid method'' as segmented by Guillaume (2001). One class of shred partition is ''one rule per pair,'' and the principle proposed by Wand and Mendel-WM (1992) is the most popular here. Therefore, we combine the WM approach with a metaheuristic algorithm based on BCO to perform the optimization of different FISs.

Research methodology
The research methodology is structured into three parts. The first relates to the explanation of performed experiments and the procedure of data collection. Section 3.2 is about forming different T2FIS structures. Finally, the last subsection explains the procedure of T2FIS optimization using the BCO metaheuristic. The concept of research is illustrated in Fig. 1, and a more detailed explanation is offered in the following text.

Driving simulator
Driving simulator experiments have been widely applied in driving behavior studies as well as speed perception studies (Zheng et al. 2018;Pešić et al. 2019). For this study, the experiments took place in a PC-based driving simulator. The driving simulator incorporates three 4200 plasma displays that give the responders a 180°horizontal and 50°v ertical field of view of the simulated environment. Each display has a resolution of 1360 9 768 pixels and a refresh rate of 60 Hz (Bıçaksız et al. 2019;Pešić et al. 2019). This driving simulator can be applied to study issues related to road traffic safety under controlled experimental conditions.
Many studies have classified driving simulators (Hussain et al. 2020Saluäär, 2000;Wynne et al. 2019). Saluäär (2000) in his study classified driving simulators as low, mid-level, or high-level. The low-level simulators are ordinary personal computers equipped with a steering wheel and pedals. High-level simulators usually have huge motion base systems. Simulators between these two categories are called mid-level (Saluäär 2000;Mathur 2010). The simulator used in this study consists of three LCD monitors, three connected computers, and a cockpit for the driver that transposes the vibration from the virtual environment on the driver. According to the above classification, this simulator can be classified as a mid-level one.
Driving simulators have two levels of validity: 1. physical and 2. behavioral. Physical validity measures the degree to which the simulator dynamics and visual system reproduce the vehicle being simulated. The behavioral validity of a driving simulator, according to Blana (1997), is defined as the comparison of driving performance indices from a particular experiment on a real road with indices from an experiment in a driving simulator which is as close as it can be to the real environment. On the other hand, Blaauw (1982) in his study proposed two types of driving behavioral validity: 1. absolute and 2. relative. A driving simulator is absolutely valid if the difference between the magnitudes of critical driver performance variables to example speed, acceleration, etc., observed in the driving simulator and those in the real world is statistically insignificant (Mathur 2010). A driving simulator is relatively valid if the differences with experimental conditions are in the same direction and have a similar magnitude (Yan et al. 2008;Mathur 2010).
The simulator used in the experiment was developed and funded by a team of experts from the Faculty of Transport and Traffic Engineering-University of Belgrade. Before each experiment, the simulator is set up to mimic reality as well as possible. Taking into account all the above, as well as the fact that the simulator used in this study is similar by its characteristics to other simulators used in the published research papers (Boot et al. 2015;Wynne et al. 2019;Pešić et al. 2019;Hussain et al. 2019Hussain et al. , 2020Bıçaksız et al. 2019;Kummetha et al. 2020), the simulator used in this experiment can be considered as valid for this type of research and belongs to the mid-level group of simulators.

Selection of vehicle for experiment and vehicle speed
The vehicle used for the experiment was Peugeot 307, the hatchback version with five doors. Vehicle dimensions are: length 4210 mm, width 1730 mm, and height of 1510 mm. The color of the vehicle is the factory golden yellow (color code: kaw Jaune Persepolis met). The facts from the literature suggest that it is easier to spot the vehicle if there is a higher contrast between the vehicle color and the environment. In general, brighter colored vehicles have a higher contrast with the environment, so it is, therefore, easier to perceive (Pešić et al. 2019;Allen and Clark 1964;Dahlstedt and Rumar 1973). Values of vehicle test speed are permanently determined due to legal restrictions: the limit of 30 km/h in a school zone; limit of 50 km/h in inhabited area and limit of 70 km/h (OGRS 2020; Pešić et al. 2019;Cicevic et al. 2017). The length of the route traveled by the experimental passenger car was 300 m.

Experimental protocol
This paper presents the experiment carried out in the Laboratory of Traffic Psychology, Faculty of Transport and Traffic Engineering in Belgrade (Cicevic et al. 2017). For this testing, twelve driving situations, three speeds from four perspectives, are shown on a driving simulator to the participants. In the driving simulator, the vehicles are shown in different perspectives (with certain speed from ''FRONT view'' (F), as well as from ''REAR view'' (R), from the side of the vehicle ''SIDE view'' (S), and the angle of the ''DRIVER'S SIDE PERSPECTIVE'' (D) (see Fig. 2). The present study focused on the trajectory characteristics of free-flow driving with no roadside interference. The participants viewed the traffic scenarios in a twolane undivided road, during the daytime on a sunny day. The driving environment included usual traffic signalization and vegetation; there were no additional objects added in the traffic scenes to avoid the impact on participants' Fig. 1 A configuration of the research: the parameter S i (i = 1,3) takes the following values: S 1 = 30 km/h, S 2 = 50 km/h, and S 3 = 70 km/h; the parameter P j (j = 1,4) takes the following values: P 1 = front view, P 2 = rear view, P 3 = side view, and P 4 = driver's seat perspective expectations about the movement of the visual targets, and to prevent distraction. At the beginning of the experiment, each participant received his combination of experimental stimulus order, which was chosen by using a random number generator. This procedure was carried out to neutralize the anchoring effect, through counterbalancing. Counterbalancing is accomplished by randomizing the order of presentations of the test stimuli. Each respondent estimated speed for all twelve traffic situations. Speed estimation made by each participant was registered immediately after watching each traffic situation in the driving simulator. No pre-defined answers are given (Pešić et al. 2019). All participants had identical driving environment, identical test speeds, route length, and speed assessment time. At the end of each scenario, respondents gave their answers.
A detailed explanation of each question and requested data from the participants are offered before the start of the experiment. During the experiment, three analysts were present in the Laboratory giving instructions and support to the participants. Particular attention is given to reporting the participation in MVCs. A participant was considered as the one who participated in the MVC only if he/she had acted as a driver and if the MVC had occurred by his/her fault. MVCs that appeared at the parking places or those that can be considered as random events are not taken into consideration. To prove the participation in an MVC, we asked the participants about the detailed description of the MVC, including the court judgments, type and level of penalties, obtained penalty points, etc. A certified traffic safety expert carried out this examination.

Sample
The sample included 178 young drivers aged from 19 to 22. A motive to research young drivers is the fact that MVCs represent a leading cause of deaths in this population (WHO 2018;Jannusch et al. 2020;Wang et al. 2020). Besides the expected valuable conclusions from this research, we believe this kind of study can increase the awareness of young people about the significance of proper behavior in traffic, potentially saving someone's life. We implemented a convenience sampling technique (a nonprobability technique). To collect data on young drivers, 241 students from the Faculty of Transport and Traffic Engineering-University of Belgrade were invited. From this number, 178 fulfilled the conditions of participation and accepted to take part in the experiment. The main conditions were possession of a driver's license and the absence of any previous experience with the driving simulator. These conditions were set to provide the same background for all participants. The response rate was very high, which can be explained by the previous educational connection between the authors and participants. The study adhered to the Code of Ethics and Conduct of the Serbian Psychological Association. Respondents did not receive any compensation for participation.
Because the implemented sampling technique does not include a random selection of participants, it would be welcome to expand the number of participants in future research, considering also other populations of young drivers. Besides, further research covering the drivers of all ages would be also welcome. The essence of the model is the design and testing of type-2 FIS structures. There are seven categories of T2FIS structures that are tested in this research (Fig. 3). They can be classified into two groups. The first (Part a) in Fig. 3 To form the T2FIS, two main tasks are to design the membership functions and the fuzzy rule database. For both purposes, we use the empirical data collected in the experiments previously explained.
In total, there are twelve input variables and one output to be used in different ways in seven categories of T2FIS. The input variables are marked by x i , where i takes values from 1 to 12, and the output variable is marked by y. A description of the used variables and the corresponding descriptive statistics of the sample are shown in Table 3, while the distribution of particular variables in each T2FIS category is presented in Table 4. It can be noticed that certain participants made very huge errors in their assessment. This may be explained by the young age of participants, having in mind that all of them had very limited experience in driving.
The descriptive statistics of the sample is a starting point for the membership functions design. Since the domain of input variables contain both positive and negative values, which represents some sort of a specific situation, we propose an adequate algorithm (Table 5)  values: P 1 = front view, P 2 = rear view, P 3 = side view, and P 4 = driver's seat perspective A bee colony optimization (BCO) and type-2 fuzzy approach to measuring the impact of speed… 4471 The following notation is valid here. The number of membership functions is denoted as 2 N ? 1. A starting point of a variable domain, i.e., the maximum absolute value of the negative error of speed assessment form the sample is E neg max , while the maximum value of the positive error of speed assessment is E pos max . The number of membership functions on the negative side of the variable domain is i neg , and the number on the positive side is i pos . The position of points with the highest degree of membership on the negative side of the variable domain is P j neg l x ð Þ¼1 , while the same parameter for the positive side is P j pos l x ð Þ¼1 . Since the proposed algorithm finds 2 N points with the highest degree of membership at the x-axis, there is one more remaining point to be defined. In each case, it is the point where the error in speed assessment is equal to zero. It should be noted that all the mentioned points are the same for both, upper and lower membership functions. When it comes to the points of upper membership functions where lðxÞ ¼ 0, they are placed at x-axis where adjacent fuzzy sets have the highest degree.
After the position of membership functions is determined, the fuzzy sets formed accordingly should be named. In our case, five membership functions describe each input variable, which means that N = 2. However, the following names of fuzzy sets were used depending on the domain of a variable and position of the points with the highest membership degree: VEU_S i -Assessment is very extremely under S i , EU_S i -Assessment is extremely under S i , SU_S i -Assessment is significantly under S i , MU_S i -Assessment is moderately under S i , A_S i -Assessment is around S i , MO_S i -Assessment is moderately over S i , SO_S i -Assessment is significantly over S i and EO_S i -Assessment is extremely over S i , where the parameter S i (i = 1,3) takes the following values: S 1 = 30 km/h, S 2 = 50 km/h, and S 3 = 70 km/h. An example of how the membership functions for input variables in the case of T2FIS_30 are designed based on the proposed algorithm is shown in Fig. 4. There are four parts (a), (b), (c), and (d) describing the variables x 1 , x 4 , x 7 , and x 10 , respectively.
On the other hand, the output variable is defined just with three membership functions because the interval that should be covered is relatively small. The reason lies in the fact that the participants were young drivers with a small number of MVCs in their driving experience. Therefore, the values near zero belong to the fuzzy set SNA-small number of MVCs; values near 3, which is the maximum number of MVCs reported by the participant, are described x 4 , x 5 , x 6 , y VI T2FIS_S x 7 , x 8 , x 9 , y VII T2FIS_D x 10 , x 11 , x 12 , y by the fuzzy set HNA-high number of MVCs; and in the middle of the considered interval, there is the fuzzy set MNA-middle number of MVCs. After the variables of T2FIS were described by the corresponding fuzzy sets, it was necessary to set the fuzzy rules. Here, we applied the well-known method proposed by Wang and Mendel (1992). This approach is widely used in the literature (Chang et al. 2005;Č ubranić-Dobrodolac et al. 2019Č ubranić-Dobrodolac et al. , 2020aJovčić et al. 2019).
Wang-Mendel's method consists of two general parts. In the first part, fuzzy rules are generated from the empirical data and the missing rules are added based on expert opinion in the second part. The first part is solved in this paper based on the programming code that is offered in the paper by Č ubranić-Dobrodolac et al. 2020a. In this process, depending on the concrete T2FIS category, the specific input-output pairs were taken into consideration as demonstrated in Table 6. The concrete values of selected inputs and output that are used in the calculation process correspond to the total sample of 178 participants. In practice, many rules generated from the empirical data are the same or conflict which results in the fact that in the concrete cases, from 178 input-output pairs, the number of final rules based on the empirical data is from 23 to 38. The total number of fuzzy rules in the considered T2FIS categories is 125 and 625, in the case of T2FIS with 3 inputs and 4 inputs, respectively.
To design a complete fuzzy rule base, the authors' logic was based on the assumption of linear interdependence between input and output variables in the following way: if the errors in speed assessments are higher, then the number of experienced MVCs is higher. In the concrete case, for example in T2FIS_30, one of the fuzzy rules would be: THEN y (the number of MVCs) is HNA (High number of MVCs).
Finally, the T2FIS categories should be tested comparing themselves with each other. For this purpose, we use Eq. (1). Cumulative deviation (CD) is a measure that describes how well the performance of T2FIS corresponds to the empirical data. CD is calculated as an absolute value of the difference between the actual number of MVCs experienced by drivers in the sample y (i) , and corresponding results of T2FIS that is marked as T2FIS(i). A letter i in the mentioned variables denotes each of 178 participants from the sample. This concept of measuring the performance of FIS categories can be found also in other papers (see Cubranic-Dobrodolac et. al., 2019;Jovcic, et al. 2019).

The optimization of T2FIS by a BCO metaheuristic algorithm
BCO metaheuristic was proposed by Teodorović (2001, 2003). The essence is that the artificial bees collectively search for the best solution. They independently search the space of permissible solutions, comparing the obtained solutions in particular moments. In the comparison procedure, the quality of the achieved solution impacts the probability of whether a bee will continue its search on its own path or it will follow some other bee. The probability in decision-making is provided to avoid being trapped in local optimums. When a bee starts its search, this part of the algorithm is called forward pass, while the procedure of returning to the hive and comparison of achieved solutions is called backward pass.
The main attributes of the BCO algorithm are the following (Nikolić, & Teodorović, 2013): B-the number of bees involved in the search procedure, IT-the number of iterations, NP-the number of forward and backward passes in one iteration, NC-the number of solution changes in one forward pass, S-the best-known solution.
The application of the BCO algorithm for T2FIS optimization in our case is based on the following principles.
; y ð178Þ ) 4 , x 5 , x 6 ; y ð1Þ ), (x 4 , x 5 , x 6 ; y ð2Þ ),…, (x ; y ð178Þ ) 8 , x ; y ð178Þ ) 10 , x 11 , x 12 ; y ð1Þ ), (x 10 , x 11 , x 12 ; y ð2Þ ),…, (x 10 , x 11 , x 12 ; y ð178Þ ) Each vertex of upper membership functions (MFs) is considered as a parameter P f ch ð Þ (f ¼ 1; NP; ch ¼ 1; NC) to be changed NC times in one forward pass. In the same time, when upper MF is changed, the lower MF has changed accordingly, where we followed the principle that points with the maximum degree of membership (lðxÞ ¼ 1) for particular MF are the same for upper and lower MF in all changes and the footprint of uncertainty (FOU) is relatively similar before and after the change.
After each change of a parameter, the fuzzy rules should be set. This is performed by applying the WM method. When a T2FIS is completely designed, the effects of each change should be tested on the empirical data by applying Eq. (1).
The concept of the proposed BCO algorithm is shown in Fig. 5. In our research, NC = 5; however, for the simplicity of illustration, here it is assumed that NC = 2 which means that in a forward pass there will be two changes of the parameter. Each parameter P f ch ð Þ is changed by the new value P 0 f ðchÞ according to Eq.
(2) and after each change and generation of new fuzzy rules, the performance of newly created T2FIS is evaluated by Eq. (1).
P fmin is the minimal value of the parameter P f , P fmax is the maximum value of the parameter P f (f ¼ 1; NP), and rand f ;ch is a random number in the interval from 0 to 1 which changes its value NP x NC times in each iteration ðch ¼ 1; NC).
To implement Eq. (2), it is necessary to set the constraints, i.e., the range where P 0 f ch ð Þ can take the values. Accordingly, we need to define P fmin and P fmax . In the following text, the constraints in the case of the first input variable of T2FIS_30 will be presented. The same principle is applied for other variables of this T2FIS and other defined T2FIS structures in this paper. First, the notation used in the constraints should be noticed in Fig. 6. As can be noticed, we use the symbol R u for the parameter of MF that is ''right'' border of this MF, and L u for the parameter that is ''left'' border of considered MF. Letter u in the upper index denotes that it is about Upper MF, while the symbols in the lower index determine the MF that the considered parameter describes. The points at x-axis where MF has the maximum degree (l( SO 30 max , L u EO 30 max represent P fmax . Besides, another factor that appears in the constraints is ODC, representing an overlapping and distance constant. A case when ODC represents the minimum allowed overlapping is presented in part (a) of Fig. 7, while the same value of ODC can be used as the minimum allowed distance between two membership functions for the points with the maximum degree (l(x) = 1) illustrated in the part (b) of Fig. 7. In the proposed algorithm, the value of ODC should be calculated for each variable of T2FIS, by Eq. 3, where LD is the lower border of the domain of the variable, RD is the upper border of the domain of the variable, and n MF is the number of MF that describes the considered variable.
Equation (3) is set according to the authors' opinion; however, the condition about overlapping can be set also in some other way. Some authors even do not set it in the procedure of MF tuning. For example, Nikolić et al. (2020) allowed the cases with minimal overlapping of MFs or even the cases where MFs do not even ''touch'' between themselves, leaving in this way some parts of the variable's domain uncovered by MFs. The authors of the paper Nikolić et al. (2020) accepted and performed testing the FIS structures where some parts of the variable's domain remained uncovered in their algorithm; however, in the further procedure, during the calculation of the objective function of the considered problem, a penalty was added to discourage the algorithm from keeping these solutions. To avoid this kind of procedure, in this paper we introduce ODC to prevent the unwanted FIS structures from the beginning of the algorithm, by that improving the performance of the algorithm execution.
After the explanation of used notation, the constraints in the case of the first input variable of T2FIS_30, with the aim to calculateR It should be noticed that the concrete values in the set conditions, considering this explained variable and also others in the T2FIS structure, are dynamically changing during the execution of the algorithm. It means that each formed T2FIS in the testing procedure has its conditions that characterize the concrete fuzzy system. When a bee finishes its forward pass making two changes (because NC = 2 in Fig. 5), it should decide which of two values will take and bring to the hive for comparison with other bees. This is decided by a certain probability level having in mind the quality of achieved solutions and further by generating a random number and implementing the principle of a roulette wheel.
A principle of bees' solutions comparison is also based on probability. First, a bee should decide to be loyal or not to its obtained solution. If the bee decides not to be loyal to its solution, it chooses which bee of those that are loyal to own solution to follow. In our algorithm, all decisions are based on the principle of the roulette wheel (Marković 2017); however, some other approach is possible, for example, to make decisions based on a fuzzy inference system (Dell'Orco et al. 2017).
In the case of Fig. 5, Bees 1 and 3 are loyal to their previous solutions, while Bees 2 and 4 decide to continue their search following the solutions of Bees 1 and 3, respectively.
The pseudocode of the BCO algorithm applied to optimize T2FISs is shown in Table 7. In the proposed pseudocode, the used symbols are as previously defined (inputs: B, IT, NP, NC; output: S). A case when the proposed algorithm is performed ones, including IT iterations, will be called an experiment (E). It is welcome to repeat the experiment more times and to compare the results. In this paper, we defined the value of IT to be 10 and repeated the experiment 8 times for each of the seven different T2FISs (m). To compare the considered T2FISs, we calculated the mean values of solutions obtained in 8 experiments per each iteration and each T2FIS structure.

Results and discussion
In the first step, we tested a relationship between seven T2FIS structures. Here, each T2FIS structure is designed as explained in Sect. 3.2. The testing of each T2FIS is performed on a sample of 178 young drivers. We selected the young drivers as participants because they are particularly jeopardized road users; as previously stated, for the age of 15-29, MVCs are the leading cause of death globally. The considered sample included the drivers aged 19 to 22 years. There were 99 male and 79 female participants.
The results of the proposed methodology for forming T2FIS structures are shown in Fig. 8. Since the aim is a minimization of the cumulative deviation of T2FIS results from the empirical data, it can be concluded that the best result is achieved by T2FIS_R (CD = 107.60). It is a T2FIS structure that implies the estimation of various speed levels of the vehicle mowing away, i.e., it is about a rear view of the considered vehicle. The second-best result is of T2FIS_D (CD = 110.43). In this case, the speed levels were assessed from the driver's seat perspective. These results lead to the conclusion that the drivers whose speed perception of vehicles in front of them, as well as, the perception of own vehicle speed is poor, are more prone to experience an MVC. To validate this conclusion, the optimization of each T2FIS structure is performed by the proposed BCO based algorithm.
The simulation results based on the implementation of the BCO metaheuristic are shown in Fig. 9. Each curve in Fig. 9 represents the results of a T2FIS structure. By analyzing the total number of 80 simulations per each T2FIS structure, i.e., by considering the mean of the best solutions in 8 experiments with 10 simulations, T2FIS_R achieved the best average result. The mentioned mean value of the best CD results of T2FIS_R from 8 experiments is equal to 96.7863. This leads to the conclusion that T2FIS_R can be A bee colony optimization (BCO) and type-2 fuzzy approach to measuring the impact of speed… 4477 used as a very good predictor of MVCs. If we transfer this experiment to the real-life situation in traffic, it could be said that drivers who misjudge the speed of the vehicle in front of them have an increased risk of MVCs. A propensity for MVCs involvement of young drivers is higher compared to adults, which is confirmed in earlier studies (Chliaoutakis et al. 1999;Ulleberg 2001) and recent as well (Park et al. 2021;Dutta and Vasudevan 2020). The risk is especially elevated while overtaking (Bucsuházy et al. 2020;Figueira and Larocca 2020;Han and Zhao 2020). This is mostly due to the lack of driving experience and poor speed perception that is necessary for safe overtaking Pešić et al. 2019). The importance of speed perception of the vehicle in front is particularly reflected while catching up with the other vehicles on fast roads. In such situations, to conduct a safe maneuver of overtaking or, if there are no conditions for overtaking, to reduce the speed of the vehicle in time, a good assessment of the speed and distance of the vehicle in front is essential. Misjudging the speed of the vehicle in front is also one of the important factors that affect the collision of the vehicle with the bicycle, in circumstances when the vehicle overtakes the cyclist (Moll et al. 2021;Piccinini et al. 2018;Simović et al. 2021).
By considering all the performed 560 simulations, the best-found T2FIS is T2FIS_D where CD = 96.0294 (Table 8). Therefore, the problem of perception of the own vehicle speed is proved to be the most important indicator of experiencing MVC. The most often cause of singlevehicle collisions, for example, runoff-road collisions in a curve or even on a straight lane, overturning the vehicle, etc., lies exactly in the poor speed perception of own speed (Bentaleb et al. 2014). Except for the mentioned types of MVCs, inaccurate perception of own speed increases the risk of collisions with other vehicles. For traffic safety, an accurate assessment of the speed difference between own vehicle and the overtaking vehicle is also important. A study by Dutta and Vasudevan (2020) showed that the estimated speed of the overtaking vehicle affects the distance between the vehicles while overtaking. As the lateral gap increases, so does the increase in relative speed (Dutta and Vasudevan 2020). The traffic situations that include overtaking are very complex for young, inexperienced drivers, which is associated with the risk of participating in MVCs. This is in line with our finding that drivers who misjudge the speed of vehicle speed in front, as well as their own speed, have a higher risk of being involved in MVCs.
Since two T2FIS structures, T2FIS_R and T2FIS_D appeared to be the most convenient tools for describing a relationship between the considered variables in each of them and the number of MVCs, we performed the t-test to determine a relationship between these two T2FIS structures, as well as the relationship between all the considered T2FIS structures. The results of the t-test are shown in Table 9. Considering T2FIS_R and T2FIS_D, as the bestfound T2FIS structures, there is no statistical difference between the two of them. However, if we consider each of these two T2FIS structures individually in relation to other structures, there is a significant difference between each of them and all others (p \ 0.05). This confirms the conclusion that drivers who misjudge the speed of the vehicle in front of them and their own speed, have an increased risk of MVCs. Additionally, these two phenomena of speed  T2FIS_30  T2FIS_50  T2FIS_70  T2FIS_F  T2FIS_R T2FIS_S T2FIS_D Fig. 9 The results of different T2FIS structures optimization-mean values of 8 experiments A bee colony optimization (BCO) and type-2 fuzzy approach to measuring the impact of speed… 4479 T2FIS_30, and T2FIS_70 even more better. By this, it is possible to conclude that the drivers who misjudge the higher speeds are more likely to participate in MVCs compared to drivers who have a problem with the estimation of lower speeds. However, it is interesting to notice that, based on the study by Sourelli et al. (2021), the drivers report a lower level of experienced risk during the overtaking when the speed difference between the involved vehicles is larger. This is explained by the shorter time required to complete the action. The exact best values that each T2FIS structure achieved are presented in Table 8. Here are also the execution times. The T2FIS structures composed of three input variables take 728 min for 80 iterations (8 experiments with 10 iterations). To perform the calculations in the case of T2FIS with four input variables, the computer needs 2440 min. Therefore, the total execution time is 10232 min, which corresponds to 7 days, two hours, and 32 min.
At the level of seven considered T2FIS structures, the average improvement of the values of the objective function is 21.17% (Table 10). This indicates that the proposed optimization algorithm based on the BCO metaheuristic significantly improves the performance of the initial T2FIS structures.
Since the best-found T2FIS can be used also as a decision-making tool, to assess different drivers about their propensity for MVCs based on their abilities related to speed perception, it is useful to demonstrate this T2FIS. T2FIS_D structure that gives the best result is shown in Fig. 10. It can be used for various purposes in the field of traffic safety.
According to the Road Traffic Safety Agency (2020), the data that refer to the Republic of Serbia and the year 2019 indicate that from MVCs with deaths, there are 38.5% of MVCs with at least two vehicles where the cause might be related to a poor speed perception. Some of the representatives of the mentioned MVCs are the cases where a minimum of two vehicles are moving in the same direction, such as the rear-end collision, merging into traffic, and turning left or right in front of other vehicles. The mentioned type of MVC is the most frequently reported type of MVC (46.6%) among the participants in our research who experienced an MVC. Also, Richter et al. (2017) found that severe MVCs on two-lane rural roads in Germany are mostly due to the incorrectly estimated speed of own vehicle, as well as the vehicle in front.
Our results are in line with the findings of the Road Traffic Safety Agency (2020), indicating that drivers who misjudge their own speed and the speed of the vehicle in front of them, have a higher risk of participating in MVCs. These findings could be used for preventive actions and improvement of traffic safety, primarily through the To further examine if there are certain patterns in the relationships between the speed perception and MVC involvement considering various populations from different cultural and residential places, it is useful to make the sample decomposition. Here, we will test if there are differences among participants from the rural and urban areas. In our research, 154 young drivers were living in urban areas, and the remaining 24 were in rural. The implemented methodology in the case of separated groups from the sample is the same as previously described, with just one modification. Because the number of participants in each group is different, the cumulative deviation described by Eq. (1) should be calculated as shown in Eq. (4). This is done to obtain the comparable results of objective function from groups that are not of the same size.
where CD g is a cumulative deviation of the considered group, n is the total number of respondents, and k is the number of respondents in the considered group. The results of seven considered T2FIS structures, improved by the BCO algorithm, are shown in Fig. 11. To further analyze the results from Fig. 11 in sense of differences between participants from an urban and rural area, we implemented the Student's T-Test. Finally, it can be concluded that there are no statistically significant differences between the considered two groups (p = 0.382). This conclusion may be explained by the high mobility of the young rural population in the modern age, bringing to similar capabilities in speed perception between urban and rural inhabitants. On the other hand, when it comes to adults, there are often differences between urban and rural areas considering traffic safety issues Twisk et al, 2021;Watson & Austin, 2021).

Conclusion
In this paper, the main goal was to examine how a speed perception of a driver's vehicle or other vehicles on the road influences the occurrence of MVCs. The data were collected in the driving history questionnaire and by twelve types of experiments in the driving simulator. Based on these data, seven T2FIS structures were tested to conclude what type of speed perception has the highest impact on MVCs occurrence. The aim was to conclude which of the T2FIS structures describe the empirical data in the best way, which is tested by the proposed BCO-based algorithm. Accordingly, the conclusion was reached about the most convenient technique to assess the driver propensity for MVCs based on testing their speed perception capability. The result indicates that drivers who misjudge the speed of the vehicle they are looking at from the rear side, as well as their own vehicle speed, have the highest likelihood of participation in MVCs. This conclusion may be very useful in educational procedures to improve traffic safety.
Since the proposed model is based on the implementation of fuzzy logic, it is also particularly convenient in the case when the researcher would like to offer the participants a possibility to express their speed appraisals by using descriptive statements instead of crisp numbers. Type-2 fuzzy logic systems that we used in the proposed model appear to be more capable of handling problems with high uncertainties, comparing to Type-1 fuzzy systems, and their use in the case of speed estimation is very convenient.
MVCs with the participation of at least two vehicles, where a poor speed assessment can be assigned as its cause is the most common type of MVCs reported in the group of participants in this research (46.6%). These results are in line with the settings of this study so that drivers who misjudge their own vehicle speed and the speed of the vehicle in front of them have a higher risk of participation in an MVC. Such findings should be used for preventive actions and improvement of traffic safety, primarily through the education of attendants in driving schools. For solving the identified problem in traffic safety, a driving simulator can be used. In this way, the future drivers can gain experience in the assessment of their own speed and also the speed of the vehicle in front. This training would be in a safe environment, on the simulator; however, the benefits could be expected in the real world.
A limitation of this study is related to the considered sample. We tested just the drivers aged from 19 to 22 because it is the most vulnerable group on the road. Further research would be welcome to cover the drivers of all ages as well as different vehicle categories and colors, different test speed ranges, different road weather conditions (cloudy, raining, foggy conditions, etc.). Besides, this methodological approach can be implemented when assessing the risk of other categories of riders, such as motorcycles, e-bikes, e-scooters, and also other road users  T2FIS_30 T2FIS_50 T2FIS_70  T2FIS_F  T2FIS_R  T2FIS_S  T2FIS_D urban rural Fig. 11 The results of testing different T2FIS structures considering participants from an urban and rural area-the obtained CD g values of the bestfound T2FISs like children, people over 65, people with disabilities, etc. Also, it would be useful to further analyze the demographic characteristics of the respondents, as well as to compare the results from different countries and cultural areas reaching appropriate conclusions in this way. However, besides the mentioned limitations, the obtained conclusions are very inspiring for the creation of appropriate programs in the field of road traffic safety. The best-found T2FIS can be considered as a decision-making tool that gives information about someone's propensity for MVCs based on an assessment of the speed perception capabilities. The proposed examinations can be used, besides some other testing, in the recruitment procedures for professional drivers to select the drivers with a lower propensity for MVCs. Furthermore, the rehabilitation programs for drivers with traffic law offenses or drivers with an elevated number of MVCs in their driving history can be improved by training of speed perception capabilities. In addition to the above, the results of this paper can serve as an initial idea for designers of driver assistance systems, to develop systems that will help the driver to correct errors when estimating vehicle speed, most often when overtaking (Perumal et al. 2021). These systems would reduce the risk from the occurrence of MVCs, especially in situations that are identified as the riskiest in this paper.