# Proposing New Methods to Estimate the Safety Level in Different Parts of Freeway Interchanges.

1. IntroductionProviding an acceptable safety level of traffic facilities is logically vital due to its consequent effects on prevention of fatality and property damage. On the other hand, the low level of safety shall lead to the low performance of these facilities, as well. Since the freeways have always played a significant role in road transportation, determination of the safety level in different segments of the freeways has been one of the main concerns of researchers. Previous investigations are divided into two categories. These two categories include the real-accident data analyzing and the simulation-based safety study. In the real-accident studies, the effect of some factors (such as mainline speed at the beginning of the weaving segments, the speed difference between the beginning and the end of the weaving segment, the logarithm of the volume, the maximum length of the weaving area [1], the heavy vehicle rate, the hourly traffic volume, the speed differential between cars and heavy vehicles, the number of accesses [2], the number of vehicles that enter and exit the freeway at a specific segment, the length of the speed change lane, the speed of off-ramps [3], parallel-type or taper-type exit ramps [4], left-side or right-side merging and diverging areas [5, 6], the number of lanes on freeways, the number of lanes on ramps, and the speeding-related crashes [7]) on the number and/or severity of accidents was separately investigated and some models were developed. When there are no registered data about the accidents in a specific facility or when someone is trying to design that facility or when the facility has not yet been built, simulation-based safety studies/estimations such as conflict analysis by microscopic simulation and surrogate safety measures (SSMs) including time-to-collision (TTC), postencroachment time (PET), proportion of stopping distance (PSD), crash potential index (CPI), unsafe density (UD), max speed (Max S), relative speed ([DELTA]V), kinetic energy (KE), and deceleration rate to avoid collision (DRAC) are often used to estimate the danger or risk of possible collisions. Integrating the rate of TTC variation and the level of hazard associated with TTC as an approximate safety indicator [8] and calculating the risk of sideswipe collisions [9] are some kind of using these measures. An FHWA-sponsored research project has also studied the potential to derive SSMs from existing traffic simulation models. In that research, TTC, PET, and DRAC were used to measure the severity of the conflict, and by use of additional information about the mass of the vehicles, Max S and AV were used to measure the severity of the potential collision [10]. These are some examples of using simulation-based SSMs. These measures are so useful because they occur more frequently than accidents and therefore need shorter periods of the study compared with real-accident investigations [11]. It is important to note that the results of using SSMs mostly showed a good relationship between the proposed SSM and actual accident data [12]. So, using the SSM will help to stay far from long-term real-accident data analysis. But, there are some shortcomings in this way. Different surrogate measures for safety result in different outcomes, and unfortunately, an acceptable method to choose the best outcome especially to use as the basis of collision avoidance systems has not yet been presented. On the other hand, using these measures requires some prerequisites such as providing microsimulation software, simulation experts, and trajectory data. In addition, when it is intended to investigate the safety aspects of interchanges, there is not enough literature which focused well on these traffic facilities to review. Here, the questions are that by focusing on freeway interchanges:

(1) Is it possible to combine the outcomes of different SSMs and have an exclusive outcome which could be a safety indicator of the interchange or not?

(2) Could the level of safety be indicated just by having traffic and geometrical characteristics of the interchange instead of simulation attempts, trajectory data, or accident data?

The first purpose of the paper is to present a method to show that it is possible to give a positive answer to the first question. To do this, the fuzzy inference system (FIS) was used to combine the outcomes of the SSM, and fuzzy logic was applied to define an index called no-collision potential index (NCPI) as a safety indicator of the interchange (i.e., the exclusive outcome). Fuzzy logic could be a suitable technique where qualitative definitions cannot be directly quantified. The next purpose is to develop a model to estimate the safety level in the ramps, weaving, merge, and diverge areas of interchanges based on their traffic and geometrical properties. One of the approaches in developing the model is using a powerful tool dealing with the prediction and classification problems, and that is Artificial Neural Network (ANN) which has been a consistent alternative method to analyze the freeway accident frequency and does not require any predefined underlying relationship between dependent and independent variables [13]. Another approach is using an evolutionary computation algorithm motivated by the behavior of organisms and gathered social psychology principles in sociocognition human agents and evolutionary computations. The algorithm is particle swarm optimization (PSO) which has a simple concept and efficiency in computations and is implemented easily [14-17].

2. Methods

2.1. Description of the Proposed Fuzzy Method

2.1.1. NCPI Definition and Determination. In this paper, a fuzzy-based method was proposed to estimate the safety level in different parts of freeway interchanges. In this method, different outputs of SSMs were combined, and the outcome was NCPI which is defined as a safety indicator. The value of NCPI falls within the range of zero to 100. The higher the value of NCPI, the higher the safety level. Since it was intended to indicate the level of safety despite many other types of research that usually determine the level of risk, NCPI was defined to show the likelihood of no collision in the interchange instead of indicating the probability of collisions to occur. To determine the NCPI, the SSM were categorized into two groups:

Number estimation measures: the measures consist of TTC, PET, PSD, CPI, UD, and DRAC which are usually used to measure the severity of the conflicts. These measures can be used to estimate the number of near-crash events.

Severity estimation measures: the measures include Max S, [DELTA]V, and KE which are usually used to measure the severity of the potential collisions.

Due to the complexity and time-consuming of using all measures, in this research study, the measures of TTC and DRAC were selected to estimate the number of possible collisions, and the severity of these potential collisions was estimated by the measures KE and [DELTA]V.

2.1.2. Definition of the Used SSM. Here, the definitions of the measures TTC, DRAC, KE, and [DELTA]V are briefly described.

TTC: this was first defined by Hayward as the remaining collision occurrence time between two vehicles if the collision course and speed difference were maintained constant [18]. When TTC is low, there is an imminent danger of collision [19]. TTC for rear-end conflicts can be calculated by [20]

TT[C.sub.F](t) = [X.sub.L] (t) - [X.sub.F] (t) - [1.sub.L]/[V.sub.F](t) - [V.sub.L](t) [for all][V.sub.F](t) > [V.sub.L] (t), (1)

where TTC is the time-to-collision, X is the vehicle position (L: leading and F: following), V is the vehicle speed (L: leading and F: following), and l is the vehicle length.

DRAC: deceleration rate is a good measure to detect dangerous maneuvers. DRAC is the rate at which a vehicle must decelerate to avoid a probable collision. For vehicles traveling in the same path, DRAC is defined as [11]

DRA[C.sub.t] = [([[V.sub.F(t)] - [V.sub.L(t)]).sup.2]/2[[X.sub.L(t)] - [X.sub.F(t)] - [l.sub.veh L]], (2)

where [l.sub.veh L] is the length of the leading vehicle and other parameters were described previously. For angled conflicts, the equation changes as

DRA[C.sub.t] = [DELTA][[V.sub.ij(t)].sup.2] / 2[D.sub.i(t)], (3)

where [DELTA][V.sub.ij(t)] is the relative speed of two vehicles engaged in the conflict and [D.sub.i(t)] is the distance between the current position of the vehicle i and point of the intersection ahead of two vehicles.

KE: from Newtonian physics, we know that a moving vehicle has a kinetic energy as [21]

K = 1/2 m[v.sup.2], (4)

where K is the kinetic energy, m is the mass, and v is the speed of the vehicle. The kinetic energy transferred to the target vehicle can be calculated by

K[E.sub.s] = 1/2[m.sub.s][DELTA][v.sub.s.sup.2], (5)

where K[E.sub.s] is the kinetic energy transferred to the target vehicle, [m.sub.s] is the mass of the target vehicle, and [DELTA][v.sub.s] is the change of the target vehicle speed before and after the collision [21].

[DELTA]v: [DELTA]v is the relative speed of vehicles involved in the conflict as a collision severity reflector [22].

2.1.3. Trajectory Data Analysis. Finding the variables' values of (1-5) requires analyzing the data of trajectory. It was assumed that the vehicles had a linear movement with a constant acceleration or deceleration rate between every two consecutive time intervals in the analysis. So, the coordinates of the intersection ahead of two vehicles i and j could be computed. The acceleration or deceleration rate and speed of vehicles could also be determined using these assumptions. It should be noted that any collision in the study areas could either be a rear-end collision or occurs at an angle of [beta]. Thus, the analysis should be done with respect to angled collisions. In a special case in which the angle of collision is zero, it will be a rear-end collision. The collision of two vehicles at an angle of [beta] is depicted in Figure 1.

On the other hand, it is necessary to check whether the coordinates are within the limits of the study area or not. By assuming no changes in the conditions, it can be concluded that two vehicles i and j will never collide with each other if the coordinates of the intersection ahead of the two vehicles are outside of the limits of the study area. The distance between the position of each vehicle to the intersection ahead and the time required to reach this point can be obtained by simple calculations. Assume that [T.sub.i] and [T.sub.j] are the time required to reach the point of intersection of vehicles i and j, respectively. While the absolute value of the difference between [T.sub.i] and [T.sub.j] is less than the TTC threshold, there will be a near-crash event. The value of the TTC threshold varies in several studies. The proposed values of thresholds are 1.5 seconds for urban areas [23], 4 seconds to distinguish between safe and unsafe situations [12], and 1.0 second for critical situations [24]. But, according to the surrogate safety assessment model, which considers a conflict as an event with TTC less than 1.5 seconds [25], a threshold of 1.5 seconds was used in this paper. Among different pairs of vehicles i and j, the number of those that encountered a near-crash event was counted by

[C.sub.TTC] = [Count.sup.j.sub.i] [[absolute value of [T.sub.i] - [T.sub.j]] < TT[C.sub.threshold]], (6)

where [C.sub.TTC] is the number of pairs of vehicles that encountered a near-crash event. Regarding the angled collision of two vehicles i and j, the relative speed of these two vehicles at the time of the collision could be determined by (7) and (8) in each direction of vehicles i and j.

[DELTA][V.sub.ij] = [V.sub.i] - [V.sub.j] cos [beta], (7)

[DELTA][V.sub.ji] = [V.sub.j] - [V.sub.i] cos [beta], (8)

where [beta] is the angle of collision. The minimum required DRAC could be also found by

DRA[C.sub.ij] = 0.5[DELTA][V.sub.ij.sup.2] [L.sub.ij.sup.-1], (9)

DRA[C.sub.ji] = 0.5[DELTA][V.sub.ji.sup.2] [L.sub.ji.sup.-1]. (10)

Maximum deceleration rate for different vehicles was proposed by Maurya and Bokare [26]. If the value of each of the DRACs is more than the maximum deceleration rate, there will be a near-crash event. The DRAC of a couple of vehicles, i and j, is the maximum DRAC of them as written in (11). The number of cases in which a near-crash event takes place will be counted by

DRAC = max{DRA[C.sub.ij] * DRA[C.sub.ji]}, (11)

[C.sub.DRAC] = [Count.sup.j.sub.i]. [DRAC > maximum deceleration rate], (12)

where [C.sub.DRAC] is the number of cases in which a near-crash event occurs. The speed of the vehicle i at the time of collision is calculated by

[V.sub.i-collision] - [a.sub.i][T.sub.i] + [V.sub.i], (13)

where [a.sub.i] is the acceleration/deceleration rate of the vehicle i and [V.sub.i] is the speed of the vehicle i at the time of analysis. Decomposition of the speed vectors at the moment of collision is shown in Figures 2 and 3.

So, the relative speed of two vehicles i and j at the moment of the collision could be determined by

[mathematical expression not reproducible], (14)

[mathematical expression not reproducible], (15)

[mathematical expression not reproducible], (16)

[mathematical expression not reproducible], (17)

[mathematical expression not reproducible], (18)

[mathematical expression not reproducible], (19)

If the difference between the speed of vehicle j before and after the collision is equal to the [DELTA]V obtained above, the amount of kinetic energy transferred in the collision will be defined as

K[E.sub.ij] = 0.5 [m.sub.j][DELTA][V.sup.2], (20)

where [m.sub.j] is the mass of vehicle j.

2.1.4. Probabilistic Framework. Four surrogate measures were used to estimate the number and the severity of possible collisions. But the probability that a near-crash event becomes a real collision should be determined and be considered in the model development. There are two following probabilities:

(1) The probability that after detection of an event as a near-crash event, the event becomes a real collision (it is possible that both or one of the drivers prevents the collision to take place by performing an evasive act).

(2) The probability that the severity of collision does not change (it is possible that both or one of the drivers engaged in a collision reduces the speed to avoid the collision. Even if the collision occurs, the collision severity will become lower).

When both or one of the drivers has a time to react, both mentioned probabilities will appear. In other words, the more the TTC, the less the probability of the collision taking place with a certain severity. A probability density function (PDF) was defined which was sensitive to the perception reaction time of drivers. For this purpose, an exponential PDF was selected among the most appropriate distributions which satisfy the requirements of the problem and was written as

[mathematical expression not reproducible], (21)

where [Pr.sub.i] is the probability that a near-crash event with a certain severity becomes a real collision with the same severity, [t.sub.reaction] is the perception-reaction time of drivers, and [lambda] is a constant value which should be defined based on the problem. Since the probability must be closed to one when TTC approaches zero second, the value of [lambda] should be equal to 1.0.

The American Association of State Highway and Transportation Officials (AASHTO) mandates using a perception-reaction time of 2.7 seconds in most calculations [27]. But, in certain more complex situations, drivers may require more time to react than 2.7 seconds, for example, situations where drivers should detect and react to unexpected events. According to the AASHTO, where a collision avoidance maneuver is needed, the perception-reaction time of 3.0 seconds should be used for rural roads and 9.1 seconds for urban roads. In situations that avoiding a collision needs alterations in the speed, path, and/or direction, the AASHTO recommends a range of perception-reaction time between 10.2 and 11.2 seconds for rural roads, 12.1 and 12.9 seconds for suburban roads, and 14.0 and 14.5 seconds for urban roads [28]. Due to the high speed of vehicles, disorderly movements of vehicles, and a high number of lane changes in a freeway interchange, avoiding a collision certainly requires changing the speed, path, and direction. So, the perception-reaction time of 11.2 seconds was used in this paper. Figure 4 presents the relation between the probability and TTC with [lambda] = 1.0 and [t.sub.reaction] = 11.2.

The number and the severity of collisions in different parts of freeway interchanges could be better estimated by applying the above probability as

[mathematical expression not reproducible], (22)

[mathematical expression not reproducible], (23)

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible], (25)

where [N.sub.TTC] is the number of predicted collisions by the TTC, [N.sub.DRAC] is the number of predicted collisions by the DRAC, [S.sub.KE] is the severity of predicted collisions by the KE, and S[DELTA]V is the severity of predicted collisions by the [DELTA]V.

2.1.5. Fuzzy Logic. The estimated number of collisions using the measures of TTC and DRAC is usually different, and the results of the severity of the predicted collisions using the measure of KE are not the same as that of the measure of [DELTA]V, as well. But, all the outputs from the four measures should be considered all together. So, the NCPI must be a function of the outputs of the four mentioned measures as four variables as shown in

NCPI = F([N.sub.TTC] * [N.sub.DRAC] * [S.sub.KE] * [S.sub.[DELTA]V]). (26)

The determining function of the NCPI was defined using a Mamdani-type FIS with four inputs (the values of the four measures) and one output (NCPI). At first, a membership function with three qualitative classes was defined for each measure. Then, a fuzzification procedure was implemented for the measures. It means that the quantitative value of each measure was automatically placed in one of the three qualitative classes of "low," "medium," and "high" using the membership functions. Between NCPI and its variables, 81 rules were set. These rules determine the qualitative value of NCPI by taking the qualitative values of the four measures into account simultaneously. Table 1 presents the 81 applied rules. A membership function for NCPI was defined with five qualitative classes of "very low," "low," "medium," "high," and "very high." So, a qualitative index was attributed to NCPI based on the rules and the membership function. After all, a defuzzification procedure using the centroid method was accomplished, and the quantitative value of NCPI was obtained by the use of NCPI membership function. The FIS settings for AND/OR methods were set to MIN/MAX and for IMP/AGGR methods were set to MIN/MAX, as well. Figure 5 shows the membership functions of NCPI and [N.sub.TTC], for example, and Figure 6 depicts two samples of 3D profiles of NCPI and its variables.

2.1.6. Field Study. The field study was carried out to reach the following three aims:

(1) To calibrate the simulations

(2) To control the validity of the proposed method in which the safety level is indicated using fuzzy logic and SSM

(3) To check the accuracy of safety estimator models (the safety estimator models will be described in the following sections)

Ten freeway interchanges in Tehran Province in Iran were investigated. All the traffic and geometrical characteristics of the interchanges were gathered and shown in Table 2.

The behavior of the drivers and the trajectory data were extracted by video processing tools. The driving times in different parts of the interchanges, the headways, the real acceleration and deceleration scenarios, the number of lane changes, the route selections of the drivers, the length before the gores which the lane change occurs, and the accepted gap for the lane change were obtained by analyzing the behavior of the drivers. According to the gathered data and analysis on the behavior of the drivers, the simulation was calibrated. The value of the NCPI was also computed for every case by analyzing the real trajectory data and applying the proposed fuzzy method. The interchanges were simulated, and trajectory data were derived. The NCPIs computed using simulation trajectory data were compared with the NCPIs obtained from real trajectory data for controlling the validity of the method. Figure 7 presents the aerial photos of the ten studied interchanges. The results were used to check the accuracy of the future developed safety estimator models, too.

2.2. Development of Safety Estimator Models. In the previous section, it was shown that the safety level could be determined using microsimulation, trajectory data analysis and computations, SSM, and the proposed fuzzy method and computing NCPI. But, providing expensive prerequisites like simulation software, simulation experts, trajectory computations, and the time required is difficult and time-consuming. In this paper, it was intended to develop two models to estimate the level of safety based on geometrical and traffic characteristics of freeway interchanges. Developing the models needs generating a database containing geometrical and traffic characteristics of the ramps, weaving, merge, and diverge areas as inputs and computed NCPIs as output. Then, ANN and PSO were used to develop the models.

2.2.1. Generating Database. Every effective geometrical and traffic variable which may affect the safety level of the interchange was recognized, and a list of these variables comprising the length of the weaving area, the number of lanes in the weaving area, the number of lanes in on-ramp, the number of lanes in off-ramp, the free flow speed in freeway, the free flow speed in on-ramp, the free flow speed in off-ramp, the freeway flow rate, the on-ramp flow rate, the off-ramp flow rate, the length of the acceleration lane, the length of the deceleration lane, the length of the interchange ramp, the average slope of the ramp, and the ramp radius was prepared. Table 3 describes the variables and their range used for the simulations. Combining various values of each variable with those of the other variables resulted in the generation of 13608 different ramps, weaving, merge, and diverge areas with different traffic and geometrical characteristics. Different kinds of interchange ramps, weaving, merge, and diverge areas with different geometry and different traffic characteristics were simulated. The trajectory data were derived from every simulation run. By applying the proposed fuzzy-based method, NCPI was computed for every case. Thus, a database could be generated containing 10368, 2160, 720, and 360 rows of information for weaving areas, merge areas, diverge areas, and ramps, respectively.

As described in Table 4, the database comprised geometrical and traffic characteristics of the ramps, weaving, merge, and diverge areas of the interchange as inputs and computed NCPIs as the output. Using this database, the models could be developed.

2.2.2. ANN Approach. ANNs are massive parallel architectures that can determine answers to demanding problems with the participation of simple mutually dependent processing elements (artificial neurons) [29]. An ANN has powerful aspects of learning and data processing, and thus, it is an effective tool for engineering applications. The multilayer backpropagation network, the most popular ANN paradigm [30], was used for efficient generalization competence. ANNs have the ability to carry out with a good amount of generalization from the patterns on which they are trained [31]. Training incorporated processing neural networks with a set of known specific input-output data using the generated database. The backpropagation ANN included layers with neurons: the input layer, the output layer, and the hidden layers. The learning process was continued in the output layer where the error between the model outputs and desired outputs was calculated and then propagated back to the network with updated weights for the direction in which the performance function decreases very rapidly [32]. The entire training process was repeated for a number of epochs until the desired accuracy in the network output was gained. After training and then, validation of the network, the network was tested by using data that have never experienced before.

Determining the number of hidden neurons is an important task. The number of hidden neurons has a strong effect on the stability of the neural network which is estimated by error (minimal error reflects better stability). The random selection of a number of hidden neurons may cause either overfitting or underfitting of the predicting models [33]. This concern arises when the network corresponds to the data so closely that the ability to generalize over the test data is lost. An excessive number of hidden neurons will cause overfitting where neural networks overestimate the complexity of the target problem. In this sense, determination of the proper number of hidden neurons to prevent overfitting is critical for problem estimation with the capability for steady generalization with the lowest possible deviation in estimation. Accordingly, the number of hidden neurons must be delimited within a reasonable range. To reach the minimum error, the optimized number of hidden neurons was determined. The ANN model had also a number of neurons in the input layer and one neuron in the output layer. In the trial and error attempts, the number of neurons in the input layer was considered equal to or a multiple of the number of variables in every part of the interchange. Subsequently, the neural network results were evaluated to determine the number of neurons that would provide satisfactory results.

To ensure good generalization under ANN processing, dividing the experimental data into three subdivisions is mandatory: training, validation, and testing. For database partitioning, Looney recommended 25% for testing [34], whereas Swingler proposed 20% [35] and Nelson and Illingworth suggested 20 to 30% [36]. Therefore, in the present work, the training dataset consists of 60% of data entries, and the remaining data entries are divided equally between the validation and testing sets. To test the reliability of the neural network model, 20% of the samples were randomly selected as the validation set and 20% of the samples as the test set. The performance of an ANN-based model primarily depends on the network architecture and parameter settings. In this study, the Matlab ANN toolbox [37] was used for ANN applications. All the networks were trained using the Levenberg-Marquardt algorithm with tansig transfer functions between the input and the hidden layers and the purelin transfer function between the hidden layers and the output layer along with a trainlm training function. Performance evaluation was characterized by plots of root-mean-squared error (RMSE) versus epoch for the training, validation, and test performances of the data. In principle, the error is reduced after more epochs of training but may begin to increase on the validation dataset as the network starts overfitting the training data. To reach the best ANN, its performance was deduced from the epoch with the lowest validation error after which there was no more increase in the RMSE error. Consequently, a reliable ANN-based safety estimator model was developed, and therefore, NCPI could be estimated using adequate information about the values of geometrical and traffic characteristics. The structure of the ANN used for estimation of the safety level in the weaving area is shown in Figure 8, for example.

The properties of the designed ANNs are as follows:

Weaving area:

Number of hidden layers: 3

Number of neurons in each hidden layer: 9

Merge area:

Number of hidden layers: 3

Number of neurons in each hidden layer: 7

Diverge area:

Number of hidden layers: 3

Number of neurons in each hidden layer: 7

Ramps:

Number of hidden layers: 3

Number of neurons in each hidden layer: 5

2.2.3. PSO Approach. Recently, Eberhart et al. proposed a PSO algorithm to be used for global optimization on the basis of random exploration methods and models of simple social systems. They stated that the algorithm is very efficient in solving the nonlinear problems. The PSO algorithm is established based on researches on different communities such as birds' community and can be used to optimize both nonlinear-continuous and discrete problems. In addition, the algorithm does not require plenty of time and memory for calculations because of its simple concept [17, 38, 39].

"Any information can be exchanged among the population" and "the behavior of each particle is affected by the behavior of other particles in the community" are the two hypotheses of the PSO algorithm which forms its basic concepts according to the results of a lot of studies on different communities. PSO has been developed in a two-dimensional (x-y) space, and the coordination of each particle is defined in this space. The vectors of the movement of particles are illustrated as [V.sub.x] and [V.sub.y] in each direction as velocity vectors. So, the movements of each particle are depicted by its coordination and vectors. Every particle in the community tries to optimize a specific target function. It knows its own best results and current coordination, and the particle is also informed about the best results of the community. Therefore, the movement vector of each particle can be achieved by [40]

[v.sup.k+1.sub.i] = [w.sub.i] x [v.sup.k.sub.i] + [c.sub.1] x rand x (pbest - [x.sup.k.sub.i]) + [c.sub.2] x rand x(gbest - [x.sup.k.sub.i]), (27)

where [v.sup.k.sub.i] is the movement vector of particle i in iteration k, [v.sup.k+1.sub.i] is the modified movement vector of particle i, rand is a random digit between 0 and 1, [x.sup.k.sub.i] is the current location of particle i in iteration k, pbest is the best result of particle i, gbest is the best result of the community, [w.sub.i] is the weight coefficient of the velocity vector of particle i, and [c.sub.i] is the weight coefficient of each component. The location of each particle can be obtained by [40]:

[x.sup.k+1.sub.i] = [x.sup.k.sub.i] + [v.sup.k+1.sub.i]. (28)

The concept of modifying the search point is depicted in Figure 9.

To ensure the convergence of the PSO, a contraction factor was applied to (27). Thus, the equation of the movement vector is modified as [40]

[v.sup.k+1.sub.i+1] = K x (w x [v.sup.k.sub.i] + [c.sub.1] x rand x (pbest - [x.sup.k.sub.i]) + [c.sub.2] x rand x (gbest - [x.sup.k.sub.i])), (29)

where K is the contraction factor and can be computed by [40]

K = 2/ 2 - [absolute value of [phi] - [square root of [[phi].sup.2] - 4[phi]]], (30)

[phi] = [c.sup.1] + [c.sup.2], [phi] > 4. (31)

All parameters were described previously. Choosing an appropriate w will lead to creating a balance between the general and local search. The coefficient w can be calculated by [40]

w = [w.sub.max] - [w.sub.max] - [w.sub.min]/[iter.sub.max] x iter, (32)

where iter is the number of iterations and [iter.sub.max] is the maximum number of iterations. Based on the above concepts, the following steps were performed to develop the PSO-based safety estimator model in the four parts of freeway interchanges.

Step 1 (defining the general formulation). Equation (33) depicts the general formulation of the PSO algorithm.

[mathematical expression not reproducible], (33)

where NCP[I.sub.i,PSO] is the NCPI calculated by the PSO-based model in iteration i, [t.sub.1] to [t.sub.n] are the traffic variables, [g.sub.1] to [g.sub.n] are the geometry variables, and [a.sub.1,i] to [a.sub.n,i] are the constant parameters and coefficients of the PSO basic equation.

Step 2 (defining the input data). The geometry and traffic variables in the database were considered as the input data of the PSO algorithm.

Step 3 (defining the target output data). The NCPIs (safety level) in the rows of information of the generated database were considered as the target output data.

Step 4 (proposing a basic equation with adequate constant parameters and coefficients). Some equations were proposed, and after a great deal of trial and error attempts, the best equations were chosen. Equations (34-40) are the selected equations. Constant parameters of the basic equations would be defined based on the generated database. Weaving area:

NCP[I.sub.W] = [b.sub.1] sin([b.sub.2][theta] + [b.sub.3]) + [b.sub.4] sin([b.sub.5][theta] + [b.sub.6]) + [b.sup.7], (34)

[mathematical expression not reproducible], (35)

where NCP[I.sub.W] is the NCPI in the weaving segment, [L.sub.W] is the length of the weaving area, [N.sub.W] is the number of lanes in the weaving area, [N.sub.R-ON] is the number of on-ramp lanes, [N.sub.R-OFF] is the number of off-ramp lanes, [V.sub.FW] is the freeway volume, [V.sub.R-ON] is the on-ramp volume, [S.sub.FW] is the freeway free-flow speed, [S.sub.R-ON] is the speed of on-ramp, [S.sub.R-OFF] is the speed of off-ramp, and [a.sub.i] and [b.sub.i] are constant parameters.

Merge area:

[mathematical expression not reproducible], (36)

where NCP[I.sub.M] is the NCPI in the merge area, [L.sub.ACC] is the length of the acceleration lane, N[F.sub.W] is the number of freeway lanes, [N.sub.R-ON] is the number of on-ramp lanes, [V.sub.FW] is the freeway volume, [V.sub.R-ON] is the on-ramp volume, [S.sub.FW] is the freeway free-flow speed, [S.sub.R-ON] is the speed of on-ramp, and [a.sub.i] and [b.sub.i] are constant parameters.

Diverge area:

NCPID = [b.sub.1] tan [absolute value of [b.sub.2] [omega] + [b.sub.3]] + [b.sub.4], (37)

[mathematical expression not reproducible], (38)

where NCP[I.sub.D] is the NCPI in the diverge area, [L.sub.DEC] is the length of the deceleration lane, [N.sub.FW] is the number of freeway lanes, [N.sub.R-OFF] is the number of off-ramp lanes, [V.sub.FW] is the freeway volume, [S.sub.FW] is the freeway free-flow speed, [S.sub.R-OFF] is the speed of off-ramp, and [a.sub.i] and [b.sub.i] are constant parameters.

Ramps:

[mathematical expression not reproducible], (39)

[L.sub.R,mod] = [L.sub.R](1 - [S.sub.LONG]/100), (40)

where NCP[I.sub.R] is the NCPI of the interchange ramp, [L.sub.R] is the length of the ramp, [L.sub.R,mod] is the modified length of the ramp, [N.sub.R] is the number of lanes in the ramp, [S.sub.LONG] is the average slope of the ramp, [F.sub.R] is the ramp flow rate, [R.sub.R] is the radius of the ramp, and [a.sub.i] and [b.sub.i] are constant parameters.

Step 5 (checking the difference between the NCPIs estimated by the PSO-based model and the NCPIs in the database). The results of the NCPI using the basic equations were compared to the NCPI in the database, and the RMSE was calculated by

RMSE = [square root of ([[summation].sup.k.sub.i=1][(NCP[I.sub.i,PSO] - NCP[I.sub.i,DB]).sup.2] /k)], (41)

where NCP[I.sub.i,DB] is the NCPI in the rows of information of the generated database.

Step 6 (obtaining the global best result). After a lot of trial and error attempts and several iterations, the best results of constant parameters (as the global best results) of the basic equations were obtained when the possible minimum RMSE was reached.

2.2.4. Statistical Analysis. The models were evaluated by statistical analysis and field studies. The survey results were compared with the models' outputs. Regarding possible differences between the survey results and the models' outputs, it was necessary to know that these differences were because of either data distribution and their random properties or a significant diversity between the outcomes. Statistical analysis indicated whether there was a significant difference between the surveyed NCPIs and the corresponding NCPIs estimated by the models or not. A pooled t-test was used due to the limited number of samples. Statistical t could be calculated by [41]

t = ([[mu].sub.m] - [[mu].sub.s])[S.sub.p.sup.-1] ([n.sub.m.sup.-1] + [n.sub.s.sup.-1]).sup.-0.5], (42)

[S.sub.p] = [(([n.sub.m] - 1)[[sigma].sub.m.sup.2] + ([n.sub.s] - 1)[q.sub.s.sup.2]).sup.0.5] [([n.sub.m] + [n.sub.s] - 2).sup.-0.5], (43)

where [[mu].sub.m] and [[micro].sub.s] are the mean of the models' population and the mean of the survey population, respectively. [n.sub.m] and [[sigma].sub.m] are the number of samples and standard deviation of the models' results, and [n.sub.s] and [[sigma].sub.s] are the number of samples and standard deviation of survey results, respectively. Computed t should be compared with the tabulated values of the t-distribution table. The tabulated values of the t-distribution table depend on the degree of freedom, f, which represents the number of independent parts. The degree of freedom is defined by (44) in t-distributions [41]:

f = [n.sub.m] + [n.sub.s] - 2. (44)

Once the statistical t is determined, the tabulated values of the t-distribution table yield the probability of a situation in which the t value is greater than the computed value. In order to limit the probability of a "type I" error to 0.05, the difference in the means will be considered significant only if the probability is less than or equal to 0.05. In other words, if the calculated t value falls within the 5% area of the tail, or in other words, if there is less than a five percent chance that such a difference could be found in the same population, the difference in the means will be considered significant. If the probability is greater than 5% (or the computed t value is less than the tabulated values of the t-distribution table), it could be concluded that such a difference in means could be found in the same population and the difference would not be considered significant [41].

3. Results and Discussion

Estimation of the safety level in the ramps, weaving, merge, and diverge segments of freeway interchanges was the main purpose of this research. A huge number of simulated parts of freeway interchanges and the ten existed freeway interchanges in Tehran Province in Iran were investigated, and by using the proposed fuzzy-based method, the NCPI was determined in every case. The models' required database was generated after the NCPI was calculated in all the cases. The database contained, respectively, nine, seven, six, and five traffic and geometric characteristics for weaving areas, merge areas, diverge areas, and ramps as inputs and the NCPIs as output. Therefore, two ANN-based and PSO-based models were developed utilizing the database.

3.1. ANN Results. The ANN-based development process was accomplished, and data training was stopped when the RMSE in the validation set began to increase, which signifies that the ANN generalization stopped increasing. Further analysis was conducted to test the accuracy of estimation by completely new input data to ensure that the reference model of the neural networks provides reliable results. To verify that the given estimation models are suitable to provide reliable results, a complete set of new data (which were not used in the evaluation) was used. The results of the ANN-based model were categorized into three collections of all data, validation data, and test data as indicated in Figure 10 for interchange ramps, for instance. For each collection, ANN outputs were compared with the NCPI in the database (as the target), and the results were presented in the first graph. Error diagram and error distribution of the data of each collection were depicted in the second and third graphs, respectively.

The results of using the ANN to estimate the NCPI in the four parts of the interchange are illustrated in Figure 11, and the correlation between the estimated NCPI by the ANN-based model and the NCPI in the generated database in the interchange ramps was represented in Figure 12, for instance. According to the results, considering the fact that all neural network models showed similar small error means and high coefficients of correlation, the difference between the results of training and validation sets was negligible.

As shown in Figures 11 and 12, the analysis indicated a relative agreement between the NCPIs in the database and estimated data, according to all statistical performance measures. The standard deviation, coefficient of correlation, error mean, and RMSE of the three collections demonstrated a proper development of the ANN-based model. The low values of RMSE and error mean and the high value of coefficient of correlation are commonly recognized as a good estimation of the model.

But, comparing the statistical results of the four parts with each other, the highest coefficient of correlation and the minimum RMSE along with the smallest value of standard deviation were found in the diverge area. The lowest coefficient of correlation was detected in the merge area, and the high values of RMSE and standard deviation were obtained in interchange ramps and merge areas. However, the results of conducted analysis indicated that the proposed ANN-based model could generally be used to estimate the safety level of the four parts of the freeway interchange with sufficient accuracy, and the results clearly depicted that, for all the samples, values estimated with the ANN-based model are strongly consistent with the values obtained by analyzing the trajectory data. It is clear that the modeling results are exceptionally correct; therefore, there is no doubt regarding the accuracy of the estimation performance of the ANN-based model.

3.2. PSO Results. After a lot of trial and error attempts, the maximum number of iterations and the best population size (swarm size) were discovered in the PSO-based model development process. When the algorithm reached the minimum MSE as the best cost, the trial and error attempts were stopped. Therefore, the constant parameters of (34-40) were determined based on the results of iterations. The global best results and their position in the basic equations along with MSE and RMSE of the algorithm for the diverge area are presented in Figure 13, for instance, and a brief overview of applying the optimization process on all parts of interchanges is presented in Table 5.

The values of RMSE indicated a relatively good development of the model. The values are different and vary from 7.66 for the weaving areas to 18.7 for the ramps. Irregular movement of the vehicles in curves may cause a higher value of RMSE in this area.

Although the RMSE values for the PSO-based model are more than those for ANN-based models, it is not enough to decline the application of the PSO-based model in estimating the safety level in freeway interchanges. Constant parameters were determined, and therefore, (34-40) were rewritten as (45-51).

Weaving area:

NCP[I.sub.W] = 16.48 sin (30[theta] - 98.82) + 31.73, (45)

[mathematical expression not reproducible], (46)

Merge area:

[mathematical expression not reproducible]. (47)

Diverge area:

NCP[I.sub.D] = -18.2tan [absolute value of 1.23[omega] + 37.80] + 25.34, (48)

[mathematical expression not reproducible]. (49)

Ramps:

[mathematical expression not reproducible], (50)

[L.sub.R,mod] = [L.sub.R] (l - [S.sub.LONG]/100). (51)

All parameters were described previously.

3.3. Field Studies' Outcomes. As it was mentioned before, field studies were used to evaluate the proposed method and the models. Six weaving areas, five merge areas, five diverge areas, and five ramps were surveyed through the field study on ten freeway interchanges of Tehran Province in Iran. The values of the NCPI were computed using trajectory data analysis and estimated by applying ANN-based and PSO-based models on the interchanges with characteristics brought in Table 2. The results of statistical analysis are illustrated in Table 6. It could be said that no significant differences between the means of the models' population and the real population were found when almost all computed statistical t-values (which are achieved from statistical analysis of the population of the models and studied areas) are less than the tabulated values of the t-distribution table.

4. Conclusion

In this paper, it was intended to propose a new method to have an exclusive safety indicator among different SSMs at the first step and to develop a model to estimate the safety level according to geometrical and traffic characteristics of different parts of freeway interchanges at the second step. Different surrogate measures were combined using fuzzy logic, and an index called NCPI was defined as a safety level indicator. The variables of NCPI including outputs of four surrogate measures of safety were determined by analyzing the trajectory data. The trajectory data could be either achieved from video processing or derived from microsimulation. Then, NCPI was obtained by applying fuzzy rules to the variables. It was done to estimate the level of safety when there are not enough data or information about the number and severity of accidents or the segment is just being designed and has not yet been built. Due to the difficulties of accessing or obtaining the trajectory data, two models were developed by ANN and PSO algorithms to estimate the safety level based on geometrical and traffic characteristics of interchanges. At last, field studies were carried out to calibrate the simulations, controlling the validity of the proposed fuzzy method and checking the accuracy of safety estimator models.

The results indicated an acceptable confidence about the validity of the proposed fuzzy method. The results also showed a good accuracy of the developed models in terms of compliance with the database generated from data analysis and also surveyed data of the field studies. It also became clear that, in most cases, the results of the ANN-based model have more accuracy than the results of the PSO-based model. However, the advantage of using the PSO-based model is that finally, there will be a certain relationship which can be conveniently used, while the ANN-based model has some prerequisites to work, say, database knowledge and MATLAB expertise.

The developed models can also be used for comparing different proposed plans of interchanges from safety aspects and ranking them in the design steps and plan selection process. In these cases, the lower accuracy of some PSO-based models compared with the ANN-based models does not matter.

In general, the proposed models will be valid when the geometric and traffic characteristics of interchange's parts fall within the range of variables' values used for model development. Definitely, the more the difference between the values of these characteristics and the range of mentioned variable values, the more the reduction in the validity of the models. Another conclusion is that the models could be trained to estimate other traffic parameters such as the density, delay, and speed of interchange's parts or other traffic facilities based on their traffic and geometric characteristics.

In this paper, it was proposed to use fuzzy logic and the algorithms of ANN and PSO to estimate the safety level of different parts of freeway interchanges. But, there will be a long way to reach a point that these methods become general and be used in every situation. In this way, we propose that these methods can be applied in a wider range of variables' values and also be used in other segments of the freeway which include traffic conflicts, merging, or diverging.

https://doi.org/10.1155/2018/8702854

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Hamid Behbahani [ID], Sayyed Mohsen Hosseini, Alireza Taherkhani, Hemin Asadi, and Seyed Alireza Samerei

Department of Civil Engineering, Iran University of Science and Technology, P.O. Box 13114-16846, Tehran, Iran

Correspondence should be addressed to Hamid Behbahani; behbahani@iust.ac.ir

Received 18 August 2017; Revised 7 December 2017; Accepted 27 December 2017; Published 29 March 2018

Academic Editor: Alessandro Palmeri

Caption: FIGURE 1: Collision of two vehicles at an angle of [beta].

Caption: FIGURE 2: Decomposition of the speed vectors of vehicle j parallel with and perpendicular to the movement direction of vehicle i.

Caption: FIGURE 3: Decomposition of the speed vectors of vehicle i parallel with and perpendicular to the movement direction of vehicle j.

Caption: FIGURE 4: PDF for a near-crash event becomes a real collision.

Caption: FIGURE 5: The membership functions for NCPI and [N.sub.TTC].

Caption: FIGURE 6: The 3D profiles of NCPI and its variables. (a) Severity estimation variables; (b) number estimation variables.

Caption: FIGURE 7: Case studies: (a) Hemmat FW-Yadegar FW; (b) Imam Ali FW-Khavaran FW; (c) Hemmat FW-Ashrafi; (d) Niayesh FW-Chamran FW; (e) Tehran-Qom FW-Vahnabad; (f) Tehran-Saveh FW-Shahryar; (g) Hakim FW-Sheikh Bahaee; (h) Tehran-Saveh FW-Dehshad; (i) Yadegar FW-Kouhestan; (j) Imam Ali FW-Mahallati.

Caption: FIGURE 8: ANN for estimation of the safety level in the weaving area.

Caption: FIGURE 9: The concept of modifying the search point in PSO.

Caption: FIGURE 10: Ramps: (a) test data comparison of the ANN output and the NCPI in rows of information as target, (b) error diagram, and (c) error distribution.

Caption: FIGURE 12: Correlation between the estimated safety level and the safety level from the database in the ramps.

Caption: FIGURE 13: PSO iteration in the diverge area (best cost = MSE).

TABLE 1: The rules of the FIS. Variables Rule number [S.sup.-1.sub.[DELTA]V] [S.sup.-1.sub.KE] 1 Low Low 2 Low Low 3 Low Low 4 Low Low 5 Low Low 6 Low Low 7 Low Low 8 Low Low 9 Low Low 10 Low Medium 11 Low Medium 12 Low Medium 13 Low Medium 14 Low Medium 15 Low Medium 16 Low Medium 17 Low Medium 18 Low Medium 19 Low High 20 Low High 21 Low High 22 Low High 23 Low High 24 Low High 25 Low High 26 Low High 27 Low High 28 Medium Low 29 Medium Low 30 Medium Low 31 Medium Low 32 Medium Low 33 Medium Low 34 Medium Low 35 Medium Low 36 Medium Low 37 Medium Medium 38 Medium Medium 39 Medium Medium 40 Medium Medium 41 Medium Medium 42 Medium Medium 43 Medium Medium 44 Medium Medium 45 Medium Medium 46 Medium High 47 Medium High 48 Medium High 49 Medium High 50 Medium High 51 Medium High 52 Medium High 53 Medium High 54 Medium High 55 High Low 56 High Low 57 High Low 58 High Low 59 High Low 60 High Low 61 High Low 62 High Low 63 High Low 64 High Medium 65 High Medium 66 High Medium 67 High Medium 68 High Medium 69 High Medium 70 High Medium 71 High Medium 72 High Medium 73 High High 74 High High 75 High High 76 High High 77 High High 78 High High 79 High High 80 High High 81 High High Variables Rule number Function [S.sup.-1.sub.TTC] [S.sup.-1.sub.DRAC] NCPI 1 Low Low Very low 2 Low Medium Very low 3 Low High Low 4 Medium Low Very low 5 Medium Medium Low 6 Medium High Medium 7 High Low Low 8 High Medium Medium 9 High High High 10 Low Low Very low 11 Low Medium Low 12 Low High Medium 13 Medium Low Low 14 Medium Medium Medium 15 Medium High Medium 16 High Low Medium 17 High Medium Medium 18 High High High 19 Low Low Low 20 Low Medium Medium 21 Low High Medium 22 Medium Low Medium 23 Medium Medium Medium 24 Medium High High 25 High Low Medium 26 High Medium High 27 High High Very high 28 Low Low Very low 29 Low Medium Low 30 Low High Low 31 Medium Low Low 32 Medium Medium Low 33 Medium High Medium 34 High Low Low 35 High Medium Medium 36 High High High 37 Low Low Low 38 Low Medium Low 39 Low High Medium 40 Medium Low Low 41 Medium Medium Medium 42 Medium High High 43 High Low Medium 44 High Medium High 45 High High Very high 46 Low Low Low 47 Low Medium Medium 48 Low High High 49 Medium Low Medium 50 Medium Medium High 51 Medium High High 52 High Low High 53 High Medium High 54 High High Very high 55 Low Low Very low 56 Low Medium Low 57 Low High Medium 58 Medium Low Low 59 Medium Medium Medium 60 Medium High High 61 High Low Medium 62 High Medium High 63 High High High 64 Low Low Low 65 Low Medium Medium 66 Low High Medium 67 Medium Low Medium 68 Medium Medium Medium 69 Medium High High 70 High Low Medium 71 High Medium High 72 High High Very high 73 Low Low Medium 74 Low Medium Medium 75 Low High High 76 Medium Low Medium 77 Medium Medium High 78 Medium High Very high 79 High Low High 80 High Medium Very high 81 High High Very high TABLE 2: Characteristics of the interchanges investigated in the field studies. Weaving area Location [L.sub.W] [N.sub.W] [N.sub.R-ON] (m) (-) (-) Hemmat W-E: Yadegar ON and 304.71 5 2 Yadegar OFF Hemmat Freeway W-E: Asharfi 705 5 2 ON and Yadegar OFF Hakim Freeway E-W: Sheikh 296 4 2 Bahaee ON and Chamran OFF Karbala Freeway E-W: Sahne 285 4 2 ON and Hamedan OFF Imam Ali Freeway S-N: N-N 535.04 5 1 U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 256 4 2 Khavaran ON and Khavaran OFF Merge area Location [L.sub.ACC] [V.sub.FW] [N.sub.FW] (m) (veh/h) (-) Hemmat Freeway W-E: merge 145 5023 4 Asharfi N Niayesh Freeway E-W: merge 118 3794 3 Chamran S Tehran-Qom Freeway N-S: 173 2252 3 merge Vahnabad E Tehran-Qom Freeway S-N: 154 1266 3 merge Vahnabad W Tehran-Saveh Freeway W-E: 225 2667 3 merge Shahriar W Diverge area Location [L.sub.DEC] [N.sub.FW] [N.sub.R-OFF] (m) (-) (-) Hakim Freeway W-E: diverge 172 4 1 Sheikh Bahaee S Hemmat Freeway W-E: diverge 202 4 2 Yadegar S Tehran-Saveh Freeway E-W: 215 3 2 diverge Dehshade W Tehran-Saveh Freeway E-W: 180 3 2 diverge Robat Karim E Yadegar Freeway N-S: 152 3 1 diverge Kouhestan E Interchange ramps Location [L.sub.R] [N.sub.R] [S.sub.LONG] (m) (#) (%) Chamran S to Hemmat W 438 2 +3.8 Hemmat W to Chamran S 350 2 +2.3 Imam Ali N to Mahallati W 402 2 +4.2 Mahallati E to Imam Ali N 514 2 +0.5 Niayesh E to Chamran S 470 2 -1.9 Weaving area Location [N.sub.R-OFF] [V.sub.FW] (-) (veh/h) Hemmat W-E: Yadegar ON and 2 6650 Yadegar OFF Hemmat Freeway W-E: Asharfi 2 6850 ON and Yadegar OFF Hakim Freeway E-W: Sheikh 2 2990 Bahaee ON and Chamran OFF Karbala Freeway E-W: Sahne 2 1870 ON and Hamedan OFF Imam Ali Freeway S-N: N-N 2 5860 U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 2 878 Khavaran ON and Khavaran OFF Merge area Location [V.sub.R-ON] [N.sub.R-ON] (veh/h) (-) Hemmat Freeway W-E: merge 1253 2 Asharfi N Niayesh Freeway E-W: merge 909 2 Chamran S Tehran-Qom Freeway N-S: 169 1 merge Vahnabad E Tehran-Qom Freeway S-N: 440 1 merge Vahnabad W Tehran-Saveh Freeway W-E: 361 2 merge Shahriar W Diverge area Location [V.sub.FW] [S.sub.FW] (veh/h) (km/h) Hakim Freeway W-E: diverge 3188 80 Sheikh Bahaee S Hemmat Freeway W-E: diverge 4196 80 Yadegar S Tehran-Saveh Freeway E-W: 4160 120 diverge Dehshade W Tehran-Saveh Freeway E-W: 1895 120 diverge Robat Karim E Yadegar Freeway N-S: 1930 80 diverge Kouhestan E Interchange ramps Location [V.sub.R] [R.sub.R] (veh/h) (m) Chamran S to Hemmat W 1644 60 Hemmat W to Chamran S 943 35 Imam Ali N to Mahallati W 796 60 Mahallati E to Imam Ali N 234 60 Niayesh E to Chamran S 681 60 Weaving area Location [V.sub.R-ON] [S.sub.FW] (veh/h) (km/h) Hemmat W-E: Yadegar ON and 1264 90 Yadegar OFF Hemmat Freeway W-E: Asharfi 839 90 ON and Yadegar OFF Hakim Freeway E-W: Sheikh 895 80 Bahaee ON and Chamran OFF Karbala Freeway E-W: Sahne 185 110 ON and Hamedan OFF Imam Ali Freeway S-N: N-N 157 80 U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 833 80 Khavaran ON and Khavaran OFF Merge area Location [S.sub.FW] [S.sub.R-ON] (km/h) (km/h) Hemmat Freeway W-E: merge 90 50 Asharfi N Niayesh Freeway E-W: merge 80 40 Chamran S Tehran-Qom Freeway N-S: 120 60 merge Vahnabad E Tehran-Qom Freeway S-N: 120 60 merge Vahnabad W Tehran-Saveh Freeway W-E: 120 40 merge Shahriar W Diverge area Location [S.sub.R-OFF] (km/h) Hakim Freeway W-E: diverge 30 Sheikh Bahaee S Hemmat Freeway W-E: diverge 60 Yadegar S Tehran-Saveh Freeway E-W: 60 diverge Dehshade W Tehran-Saveh Freeway E-W: 40 diverge Robat Karim E Yadegar Freeway N-S: 50 diverge Kouhestan E Location Chamran S to Hemmat W Hemmat W to Chamran S Imam Ali N to Mahallati W Mahallati E to Imam Ali N Niayesh E to Chamran S Weaving area Location [S.sub.R-ON] [S.sub.R-OFF] Number of (km/h) (km/h) samples Hemmat W-E: Yadegar ON and 40 40 13 Yadegar OFF Hemmat Freeway W-E: Asharfi 40 40 17 ON and Yadegar OFF Hakim Freeway E-W: Sheikh 40 60 18 Bahaee ON and Chamran OFF Karbala Freeway E-W: Sahne 50 60 21 ON and Hamedan OFF Imam Ali Freeway S-N: N-N 40 60 19 U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 30 40 17 Khavaran ON and Khavaran OFF Merge area Location Number of samples Hemmat Freeway W-E: merge 11 Asharfi N Niayesh Freeway E-W: merge 14 Chamran S Tehran-Qom Freeway N-S: 25 merge Vahnabad E Tehran-Qom Freeway S-N: 19 merge Vahnabad W Tehran-Saveh Freeway W-E: 18 merge Shahriar W Diverge area Location Number of samples Hakim Freeway W-E: diverge 17 Sheikh Bahaee S Hemmat Freeway W-E: diverge 13 Yadegar S Tehran-Saveh Freeway E-W: 22 diverge Dehshade W Tehran-Saveh Freeway E-W: 14 diverge Robat Karim E Yadegar Freeway N-S: 9 diverge Kouhestan E Interchange ramps Location Number of samples Chamran S to Hemmat W 26 Hemmat W to Chamran S 21 Imam Ali N to Mahallati W 16 Mahallati E to Imam Ali N 11 Niayesh E to Chamran S 19 TABLE 3: Variable description and their range used for simulation. Variables (a) Range Weaving area (a) [L.sub.W] (m) 300 to 900 [N.sub.W] (#) 4 to 5 [N.sub.R-ON] (#) 1 to 2 [N.sub.R-OFF] (#) 1 to 2 [V.sub.FW] (veh/h) 750 to 2970 [V.sub.R-ON] (veh/h) 600 to 1600 [S.sub.FW] (km/h) 90 to 120 [S.sub.R-ON] (km/h) 40 to 60 [S.sub.R-OFF] (km/h) 40 to 60 Merge area (b) [L.sub.ACC] (m) 100 to 500 [V.sub.FW] (veh/h) 750 to 2970 [N.sub.FW] (#) 3 to 4 [V.sub.R-ON] (veh/h) 600 to 1600 [N.sub.R-ON] (#) 1 to 2 [S.sub.FW] (km/h) 90 to 120 [S.sub.R-ON] (km/h) 40 to 60 Diverge area [C] [L.sub.DEC] (m) 100 to 500 [N.sub.FW] (#) 3 to 4 [N.sub.R-OFF] (#) 1 to 2 [V.sub.FW] (veh/h) 750 to 2970 [S.sub.FW] (km/h) 90 to 120 [S.sub.R-OFF] (km/h) 40 to 60 Ramps (d) [L.sub.R] (m) 100 to 500 [N.sub.R] (#) 1 to 2 [S.sub.LONG] (%) -3 to 3 [V.sub.R] (veh/h) 600 to 2200 [R.sub.R] (m) 60 to 140 (a) [L.sub.W] is the length of the weaving area, [N.sub.W] is the number of lanes in the weaving area, [N.sub.R-ON] is the number of on-ramp lanes, [N.sub.R-OFF] is the number of off-ramp lanes, [V.sub.FW] is the freeway volume, [V.sub.R-ON] is the on-ramp volume, [S.sub.FW] is the freeway free-flow speed, SR-On is the speed of on-ramp, and SR-Off is the speed of off-ramp. (b) [L.sub.ACC] is the length of the acceleration lane, [V.sub.FW] is the freeway volume, [N.sub.FW] is the number of freeway lanes, [V.sub.R-ON] is the on-ramp volume, [N.sub.R-ON] is the number of on-ramp lanes, [S.sub.FW] is the freeway free-flow speed, and [S.sub.R-ON] is the speed of on-ramp. (c) [L.sub.DEC] is the length of the deceleration lane, [N.sub.FW] is the number of freeway lanes, [N.sub.R-OFF] is the number of off-ramp lanes, [V.sub.FW] is the freeway volume, [S.sub.FW] is the freeway free-flow speed, and [S.sub.R-OFF] is the speed of off-ramp. (d) [L.sub.R] is the length of the interchange ramp, [N.sub.R] is the number of lanes in the interchange ramp, [S.sub.LONG] is the average slope of the ramp, [V.sub.R] is the ramp flow rate, and [R.sub.R] is the radius of the ramp. TABLE 4: The database description. Part Number of rows Number of Function variables (inputs) (output) Weaving areas 10368 9 NCPI Merge areas 2160 7 NCPI Diverge areas 720 6 NCPI Ramps 360 5 NCPI Total 13608 TABLE 5: A brief outlook of applying the optimization process on all parts of interchanges. Part Effective MSE RMSE Number of constant iterations parameters Weaving area 62 58.64 7.66 25 Merge area 21 184.245 13.57 20 Diverge area 121 285.49 16.90 14 Ramps 74 349.69 18.70 8 TABLE 6: Results of statistical analysis between means of the NCPIs estimated by the two safety estimator models versus the NCPIs achieved by field studies. ANN-based model Part Location [[mu].sub.m] [[sigma].sub.m] Waving Hemmat W-E: Yadegar ON 43.34 2.23 area and Yadegar OFF Hemmat Freeway W-E: 28.94 2.89 Asharfi ON and Yadegar OFF Hakim Freeway E-W: 29.75 2.58 Sheikh Bahaee ON and Chamran OFF Karbala Freeway E-W: 21.87 1.07 Sahne ON and Hamedan OFF Imam Ali Freeway S-N: 38.02 1.08 N-N U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 37.38 4.74 Khavaran ON and Khavaran OFF Merge Hemmat Freeway W-E: 16.3381 2.18 area Merge Asharfi N Niayesh Freeway E-W: 14.7116 3.09 Merge Chamran S Tehran-Qom Freeway N-S: 16.1305 4.48 Merge Vahnabad E Tehran-Qom Freeway S-N: 15.2322 2.28 Merge Vahnabad W Tehran-Saveh Freeway 18.1533 3.41 W-E: Merge Shahriar W Diverge Hakim Freeway W-E: 1.6394 0.18 area Diverge Sheikh Bahaee S Hemmat Freeway W-E: 32.335 6.01 Diverge Yadegar S Tehran-Saveh Freeway 49.555 8.72 E-W: Diverge Dehshade W Tehran-Saveh Freeway 5.6069 1.83 E-W: Diverge Robat Karim E Yadegar Freeway N-S: 89.363 14.66 Diverge Kouhestan E Interchange Ramp: Chamran S to 30.45 2.62 ramps Hemmat W Ramp: Hemmat W to 57.93 4.98 Chamran S Ramp: Imam Ali N to 63.02 5.42 Mahallati W Ramp: Mahallati E to 90.06 7.75 Imam Ali N Ramp: Niayesh E to 75.77 6.52 Chamran S PSO-based model Part Location [[mu].sub.m] [[sigma].sub.m] Waving Hemmat W-E: Yadegar ON 47.9 5.90 area and Yadegar OFF Hemmat Freeway W-E: 33.42 6.68 Asharfi ON and Yadegar OFF Hakim Freeway E-W: 27.44 7.35 Sheikh Bahaee ON and Chamran OFF Karbala Freeway E-W: 19.69 2.76 Sahne ON and Hamedan OFF Imam Ali Freeway S-N: 37.31 6.64 N-N U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 38.8 10.67 Khavaran ON and Khavaran OFF Merge Hemmat Freeway W-E: 18.85 4.21 area Merge Asharfi N Niayesh Freeway E-W: 17.06 5.11 Merge Chamran S Tehran-Qom Freeway N-S: 14.41 5.30 Merge Vahnabad E Tehran-Qom Freeway S-N: 13.01 3.12 Merge Vahnabad W Tehran-Saveh Freeway 15.17 4.22 W-E: Merge Shahriar W Diverge Hakim Freeway W-E: 1.61 0.20 area Diverge Sheikh Bahaee S Hemmat Freeway W-E: 36.52 7.30 Diverge Yadegar S Tehran-Saveh Freeway 57.17 10.86 E-W: Diverge Dehshade W Tehran-Saveh Freeway 4.23 1.44 E-W: Diverge Robat Karim E Yadegar Freeway N-S: 99.00 17.63 Diverge Kouhestan E Interchange Ramp: Chamran S to 31.65 3.90 ramps Hemmat W Ramp: Hemmat W to 50.99 10.19 Chamran S Ramp: Imam Ali N to 68.99 13.11 Mahallati W Ramp: Mahallati E to 97.68 13.67 Imam Ali N Ramp: Niayesh E to 69.17 12.32 Chamran S Field study Part Location [[mu].sub.m] [[sigma].sub.m] Waving Hemmat W-E: Yadegar ON 44.88 5.53 area and Yadegar OFF Hemmat Freeway W-E: 30.54 6.10 Asharfi ON and Yadegar OFF Hakim Freeway E-W: 30.43 8.16 Sheikh Bahaee ON and Chamran OFF Karbala Freeway E-W: 21.19 2.97 Sahne ON and Hamedan OFF Imam Ali Freeway S-N: 39.28 6.99 N-N U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 38.03 10.46 Khavaran ON and Khavaran OFF Merge Hemmat Freeway W-E: 17.20 2.12 area Merge Asharfi N Niayesh Freeway E-W: 15.59 3.12 Merge Chamran S Tehran-Qom Freeway N-S: 15.73 4.21 Merge Vahnabad E Tehran-Qom Freeway S-N: 14.53 2.03 Merge Vahnabad W Tehran-Saveh Freeway 17.24 3.07 W-E: Merge Shahriar W Diverge Hakim Freeway W-E: 1.65 0.20 area Diverge Sheikh Bahaee S Hemmat Freeway W-E: 33.83 6.76 Diverge Yadegar S Tehran-Saveh Freeway 52.18 9.91 E-W: Diverge Dehshade W Tehran-Saveh Freeway 5.39 1.83 E-W: Diverge Robat Karim E Yadegar Freeway N-S: 92.65 16.50 Diverge Kouhestan E Interchange Ramp: Chamran S to 30.88 3.80 ramps Hemmat W Ramp: Hemmat W to 55.71 11.13 Chamran S Ramp: Imam Ali N to 65.10 12.37 Mahallati W Ramp: Mahallati E to 92.64 12.97 Imam Ali N Ramp: Niayesh E to 73.65 13.11 Chamran S Statistical pooled t-test Part Location [S.sub.p-ANN] [S.sub.p-PSO] Waving Hemmat W-E: Yadegar ON 4.22 5.72 area and Yadegar OFF Hemmat Freeway W-E: 4.77 6.39 Asharfi ON and Yadegar OFF Hakim Freeway E-W: 6.05 7.77 Sheikh Bahaee ON and Chamran OFF Karbala Freeway E-W: 2.23 2.86 Sahne ON and Hamedan OFF Imam Ali Freeway S-N: 5.00 6.82 N-N U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 8.12 10.57 Khavaran ON and Khavaran OFF Merge Hemmat Freeway W-E: 2.15 3.33 area Merge Asharfi N Niayesh Freeway E-W: 3.10 4.23 Merge Chamran S Tehran-Qom Freeway N-S: 4.35 4.79 Merge Vahnabad E Tehran-Qom Freeway S-N: 2.16 2.64 Merge Vahnabad W Tehran-Saveh Freeway 3.25 3.69 W-E: Merge Shahriar W Diverge Hakim Freeway W-E: 0.19 0.20 area Diverge Sheikh Bahaee S Hemmat Freeway W-E: 6.39 7.03 Diverge Yadegar S Tehran-Saveh Freeway 9.34 10.40 E-W: Diverge Dehshade W Tehran-Saveh Freeway 1.83 1.65 E-W: Diverge Robat Karim E Yadegar Freeway N-S: 15.61 17.07 Diverge Kouhestan E Interchange Ramp: Chamran S to 3.26 3.85 ramps Hemmat W Ramp: Hemmat W to 8.62 10.67 Chamran S Ramp: Imam Ali N to 9.55 12.74 Mahallati W Ramp: Mahallati E to 10.68 13.33 Imam Ali N Ramp: Niayesh E to 10.35 12.72 Chamran S Statistical pooled t-test Part Location [t.sub.ANN] [t.sub.PSO] Waving Hemmat W-E: Yadegar ON 0.93 1.35 area and Yadegar OFF Hemmat Freeway W-E: 0.97 1.31 Asharfi ON and Yadegar OFF Hakim Freeway E-W: 0.34 1.16 Sheikh Bahaee ON and Chamran OFF Karbala Freeway E-W: 1.00 1.69 Sahne ON and Hamedan OFF Imam Ali Freeway S-N: 0.78 0.89 N-N U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 0.23 0.21 Khavaran ON and Khavaran OFF Merge Hemmat Freeway W-E: 0.94 1.16 area Merge Asharfi N Niayesh Freeway E-W: 0.75 0.92 Merge Chamran S Tehran-Qom Freeway N-S: 0.33 0.97 Merge Vahnabad E Tehran-Qom Freeway S-N: 1.00 1.78 Merge Vahnabad W Tehran-Saveh Freeway 0.85 1.68 W-E: Merge Shahriar W Diverge Hakim Freeway W-E: 0.18 0.66 area Diverge Sheikh Bahaee S Hemmat Freeway W-E: 0.60 0.98 Diverge Yadegar S Tehran-Saveh Freeway 0.93 1.59 E-W: Diverge Dehshade W Tehran-Saveh Freeway 0.32 1.86 E-W: Diverge Robat Karim E Yadegar Freeway N-S: 0.45 0.79 Diverge Kouhestan E Interchange Ramp: Chamran S to 0.47 0.73 ramps Hemmat W Ramp: Hemmat W to 0.83 1.43 Chamran S Ramp: Imam Ali N to 0.62 0.86 Mahallati W Ramp: Mahallati E to 0.57 0.89 Imam Ali N Ramp: Niayesh E to 0.63 1.09 Chamran S Statistical pooled t-test Part Location t value (t-dist. table) Waving Hemmat W-E: Yadegar ON 1.711 area and Yadegar OFF Hemmat Freeway W-E: 1.695 Asharfi ON and Yadegar OFF Hakim Freeway E-W: 1.694 Sheikh Bahaee ON and Chamran OFF Karbala Freeway E-W: 1.684 Sahne ON and Hamedan OFF Imam Ali Freeway S-N: 1.692 N-N U-turn ON and Sabalan OFF Imam Ali Freeway S-N: 1.695 Khavaran ON and Khavaran OFF Merge Hemmat Freeway W-E: 1.725 area Merge Asharfi N Niayesh Freeway E-W: 1.706 Merge Chamran S Tehran-Qom Freeway N-S: 1.687 Merge Vahnabad E Tehran-Qom Freeway S-N: 1.692 Merge Vahnabad W Tehran-Saveh Freeway 1.694 W-E: Merge Shahriar W Diverge Hakim Freeway W-E: 1.694 area Diverge Sheikh Bahaee S Hemmat Freeway W-E: 1.711 Diverge Yadegar S Tehran-Saveh Freeway 1.681 E-W: Diverge Dehshade W Tehran-Saveh Freeway 1.706 E-W: Diverge Robat Karim E Yadegar Freeway N-S: 1.746 Diverge Kouhestan E Interchange Ramp: Chamran S to 1.676 ramps Hemmat W Ramp: Hemmat W to 1.684 Chamran S Ramp: Imam Ali N to 1.697 Mahallati W Ramp: Mahallati E to 1.725 Imam Ali N Ramp: Niayesh E to 1.689 Chamran S FIGURE 11: Statistical results of using the ANN-based model. Statistics Weaving Merge Diverge Ramps areas areas areas R (Coefficient of correlation) 0.929 0.882 0.999 0.969 RMSE 3.933 7.897 1.824 8.335 All data Error mean -0.049 -0.192 0.148 -0.038 Standard deviation 3.933 7.897 1.820 8.347 RMSE 3.616 8.590 0.647 12.714 Test data Error mean -0.186 0.436 0.065 1.553 Standard deviation 3.612 8.588 0.646 12.707 RMSE 4.019 7.592 1.215 8.427 Validation Error mean -0.090 -0.113 0.149 -0.454 data Standard deviation 4.018 7.600 1.210 8.474

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Title Annotation: | Research Article |
---|---|

Author: | Behbahani, Hamid; Hosseini, Sayyed Mohsen; Taherkhani, Alireza; Asadi, Hemin; Samerei, Seyed Alireza |

Publication: | Advances in Civil Engineering |

Date: | Jan 1, 2018 |

Words: | 13094 |

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