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

1. Introduction

Providing 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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