Calibrating a microscopic traffic simulation model for roundabouts using genetic algorithms

Abstract

The paper introduces a methodological approach based on genetic algorithms to calibrate microscopic traffic simulation models. The specific objective is to test an automated procedure utilizing genetic algorithms for assigning the most appropriate values to driver and vehicle parameters in AIMSUN. The genetic algorithm tool in MATLAB^® and AIMSUN micro-simulation software were used. A subroutine in Python implemented the automatic interaction of AIMSUN with MATLAB^®. Focus was made on two roundabouts selected as case studies. Empirical capacity functions based on summary random-effects estimates of critical headway and follow up headway derived from meta-analysis were used as reference for calibration purposes. Objective functions were defined and the difference between the empirical capacity functions and simulated data were minimized. Some model parameters in AIMSUN, which can significantly affect the simulation outputs, were selected. A better match to the empirical capacity functions was reached with the genetic algorithm-based approach compared with that obtained using the default parameters of AIMSUN. Overall, GA performs well and can be recommended for calibrating microscopic simulation models and solving further traffic management applications that practioners usually face using traffic microsimulation in their professional activities.

Keywords

Genetic algorithm traffic microsimulation AIMSUN passenger car equivalent roundabout

1 Introduction

Computer science and transportation engineering have seen and will continue to see the highest application of microscopic traffic simulation models to the transportation systems analysis. New advances on this active field of research are coming thanks to processing power and computational capability of microscopic traffic simulation models, and their great potential for testing road policies, geometric designs, traffic controls and a number of traffic management measures before implementing new solutions in the real world [12]. In recent years, the Highway Capacity Manual [10] has recognized micro-simulation as an alternative approach to highway capacity analysis; it has also embraced this new methodology as an approach complementary to more traditional analytic-based approaches which seek to reduce a system to its elementary elements to understand the types of interaction existing among them [1]. Microscopic frameworks, indeed, seem to be well-suited for capturing the complexity and uncertainty involved in the interaction among vehicles driving along road networks or single road installations. They are based on a combination of car following and lane change models which create a representation of the vehicle movements in a road network, section or whatever road entity under various traffic conditions; they are also based on gap acceptance rules used to model give-way behaviour [29]. Reliability of simulation results depends greatly on these behavioral models which should be estimated using real traffic data or have to be assessed from statistical and behavioral standpoints [34]. Thus, calibration of microscopic traffic simulation models is a very crucial task.

Optimization finds application in every branch of engineering and science. Results of different applications to transportation engineering problems have shown that simulation-based optimization can be usefully applied to improve efficiency of the calibration process of microscopic traffic simulation models [18, 35]. With this in mind, in order to formulate the calibration process of a microsimulation model as an optimization problem, it is necessary to define a criterion that can be used to assess the model performance through an objective function, to identify parameters that are to be optimized, to select the algorithm which is to be used in the calibration process and then minimize the objective function [12, 30]. It is well-known that the objective function expresses the distance between an observable variable and its simulated value generated from the simulation runs [12]. Simulation-based optimization methods and, among these, genetic algorithms combine the strength of simulation and mathematical optimization; this is necessary in order to produce reliable results from the analysis being made and support decision to implement in practice [23]. It should be noted that incorporating the optimization problem within the calibration of a traffic micro-simulation model, the iterative process of manually adjusting the model parameters, that users have to perform, should be automatized. The sensitivity of the model to parameters is also an important step to find most significant parameters. However, engineers often use default values than perform manual calibration or use more expensive automated calibration tools [26]. In order to automate the iterative process of manually adjusting the model parameters, however, microscopic traffic simulation models must be enhanced still further with custom models and/or equipped with various APIs, if available, to perform remote control within simulations. And on the practical side of scientific research in computer science, the question is how to provide integrated software solutions, user-friendly for transportation engineers which need to use microsimulation for real world case studies in their professional activities.

Since results from studies and researches on this topic have been a very positive experience thus far, commercial and/or open-source traffic simulation software which are based on working principles of genetic algorithms should be ready to use. It is noteworthy that many engineers and practioners do not know all the tools available to them or do not have adequate programming skills and often need a computational aid to write own codes and adapt an objective function to the specific problem encountered in practice. It should also be said, however, that software packages with specific optimization tools as for example MathWorks’s^® proposes are ready to use [21]. Note that this paper did not introduce the theory on the genetic algorithms, since it focuses on the use of genetic algorithms to test an automated procedure for calibration purposes; however, for further knowledge of genetic algorithms see e.g. [22, 37] and the online MathWorks’s^® website [21].

1.1 Objectives

In the present paper, a genetic algorithm-based procedure is presented for calibrating a microscopic traffic simulation model. This paper tests an automated procedure which uses genetic algorithms for assigning the most appropriate values to driver and vehicle parameters in AIMSUN [2], that can easily be used by transportation engineers. Focus is made on roundabouts where the capacity is typically estimated for each entry lane. Capacity functions were developed by specifying the well-known Hagring formula [27] for two roundabouts with different layout, using summary random-effects estimates of critical headway and follow up headway derived from meta-analysis [24]. Thus, these capacity functions represented empirical target values to which the simulated output were compared.

As part of the procedure, a sensitivity analysis and a manual calibration were performed for two roundabouts built in AIMSUN to identify the model parameters to be included in the calibration process. Simulation outputs, both obtained with the default values of AIMSUN parameters and derived from manual calibration, were compared to the empirical capacity functions. The genetic algorithm tool in MATLAB^® was then used and the automatic interaction with AIMSUN was implemented. A Python script has been now written to fit the purposes of this paper; see [15]. The genetic algorithm-based calibration procedure gave a better match to the empirical capacity functions than using the default parameters in AIMSUN and simple manual calibration. Since the genetic algorithm approach performs well, the efficiency of the calibration efforts resulted significantly improved. At last, the calibrated model was applied to calculate the Passenger Car Equivalents (PCEs) for heavy vehicles which represent the starting point for the operational analysis of roads and intersections and other traffic management applications.

The rest of this paper is structured as follows. Section 2 presents a review of the literature on genetic algorithms and their applications in highway engineering. Section 3 introduces the data description for the selected case studies and preliminary AIMSUN modeling for the roundabouts built in AIMSUN. Section 4 introduces the structure of the genetic algorithm-based procedure applied to estimate the calibration parameters and presents the results. At last, Section 5 proposes the criterion which used the calibrated model to estimate the PCEs for heavy vehicles driving the roundabouts, while Section 6 concludes the paper.

2 Literature review

In recent years, the use of optimization methods has become widespread to solve real-world design problems even in transportation engineering. Classical optimization methods that are still valid, may sometimes result few effective to enable the solution of problems in complex search spaces. Nevertheless, it is widely recognized that, among evolutionary algorithms, Genetic Algorithms (GAs) are robust search procedures which require little information to solve a problem and find effectively a solution in a search space which can be large or little known [8]. According to the theory of natural selection, this artificial intelligence procedure uses a population of individuals selected at random; evolution to a new generation is implemented using some genetic operators (e.g. selection, crossover, mutation, and so on) [22]. Basically, GAs identify the fittest solutions by giving them greater weight and running further search in the regions where better solutions for the problem under examination can be found [28].

GAs find application in every branch of engineering and science including biology, economics, scheduling, computer-aided design, traffic simulation and so on. Many studies have demonstrated that genetic algorithms are applied in computer-aided design or optimization of a design, mainly to adapt the objective function to the user’s requirements and aid in the analysis. Successful applications to road geometric design were developed by [4]; they proposed a novel optimization technique which used a genetic algorithm to improve design method of highway horizontal alignment. The results of this research encouraged Ahmad et al. [5] which, in turn, proposed a new technique which used genetic algorithms to develop highway alignments in a three-dimensional space. In addition, a heuristic procedure which used a genetic algorithm was proposed for the optimization of speed consistency at single-lane roundabouts [14]; the results highlighted that the model gives design flexibility also for different layouts of roundabouts. Genetic algorithms were also applied to combine the capabilities of the GA and the GIS and then optimize the highway alignments; see e.g. [20, 38]. Genetic Algorithms were also applied to pavement maintenance scheduling [7]; in order to achieve the optimal pavement maintenance and rehabilitation strategy for a highway network, the selection of a set of pavement activities enabled to minimize the maintenance costs and maximize the pavement performances during a certain planning period.

However, GAs have been used to solve a very wide range of practical problems and the results obtained indicate that these models can be considered a valuable toolbox for transportation engineers. Beneficial effects, indeed, have been obtained using the GAs with microscopic traffic simulation [36], especially when GAs have been to some extent integrated into the calibration process. Incorporating optimization into model calibration objectives, the process of manually and iteratively adjusting the model parameters can be automatized; see e.g. [12]. In this regard, some options for using optimization routines based on GAs were proposed to calibrate FRESIM [32] and AIMSUN [12, 33] for freeways. However, automated calibration tools for microscopic traffic simulators have not yet been completed and the microscopic traffic simulation models still need to be equipped with custom models or should be enhanced still further by developing specific APIs in order to remotely control the simulation. Thus, the development of integrated software solutions, user-friendly for engineers which use microsimulation in their professional activities still represents an evolving research area.

3 Data description and AIMSUN modeling

The principles of roundabout design suggest that roundabout treatments for at-grade intersections simplify conflicts, reduce vehicle speeds and provide a clearer indication of the driver’s right of way than other treatments for at-grade intersections. However, geometric design and operations at double-lane roundabouts are more complex than single-lane roundabouts [16]. Single-lane roundabouts have one circulatory lane and a single-lane entry at all legs. In turn, double-lane roundabouts have double-lane entries and exits; two vehicles travelling side-by-side can be accommodated within the circulatory roadway. To maximize the level of efficiency and safety during the early years of traffic operations, a single-lane roundabout may be the interim configuration, initially built to serve the near-term traffic volumes. Expansion from a single-lane roundabout to a double-lane roundabout can be driven by needs of higher capacity and improved traffic performances especially for urban roads and arterials [16]. Since so many conditions can (or cannot) prevent the installation of roundabouts and so many factors have to be considered, it is not easy to specify whether a site could or could not be appropriate for a specified roundabout treatment.

In this study two roundabout layouts were selected: a single-lane roundabout and a double-lane roundabout; hereinafter, they will be named roundabout 1 and roundabout 2, respectively. It is noteworthy that no roundabout treatment with a modern design was installed in our City when this study was carried out. However, different signalized and unsignalized at-grade intersections in operation had been identified in the urban road network. Due to comparable size, some of them were likely to be converted into the roundabout 1 or roundabout 2 chosen for calibration purposes. By way of example, Fig. 1 exhibits an isolated signalized intersection installed at the intersection of S. Lorenzo Street and G. Spadolini Street, along an urban road corridor in Palermo City, Italy, where other roundabouts are currently in operation. Traffic data required for the model were collected from 7:30 to 8:30 am and 5:30 to 6:30 pm on weekdays last winter. On field observations revealed that these periods were the rush hours of the intersection; heterogeneous traffic was observed with a percentage of heavy vehicles never more than 30%. The AIMSUN models of the roundabout 1 and roundabout 2 are presented in Fig. 2, while in Fig. 3 the truck percentage versus time throughout a study period (7:30 to 8:30 am) is given as an example.

Fig.1

Map of the case study from Google.

Fig.2

The roundabout models in AIMSUN.

Fig.3

Truck percentage vs. time throughout a study period given as an example.

As it is well-known, roundabouts produce efficiency through the gap acceptance behavior of drivers. The critical headway and the follow up headway are two key factors in determining the entry capacity; in turn, entry capacity is a function of the circulating flow which follows a specific distribution of arrival headways. Note that, depending on the roundabout layout and the number of lanes on the ring, the circulating flow can be arranged in a single stream (roundabout 1) or two streams of vehicles traveling side-by-side (roundabout 2). Thus, the critical headway and the follow up headway, on which the gap acceptance behavior of drivers is based, were distinguished by entry lane. For the roundabout 1 in Fig. 2, one value of critical headway was considered for the entry lane faced by one antagonist traffic stream on the one-lane ring. In turn, for the roundabout 2 in Fig. 2, two critical headways were attributed to the left entry lane opposed by two circulating streams that move along the two lanes of the ring, while one value of critical headway was considered for the right entry lane faced by one antagonist traffic stream on the outer lane of the ring. The next step consisted of estimating the entry capacity functions for the selected case study. According to [27], estimation of entry lane capacity was performed with reference to a minor stream of vehicles which cross or merge into the major circulating streams, where each of them follows a Cowan’s M3 distribution:

$\begin{matrix} C_{e} = 3600 \cdot \sum_{j} \frac{φ_{j} \cdot Q_{c, j}}{3600 - Δ_{j} \cdot Q_{c, j}} \\ \cdot \prod_{k} (\frac{3600 - Δ_{k} \cdot Q_{c, k}}{3600}) \cdot \\ \cdot \frac{exp (- \sum_{l} \frac{φ_{l} \cdot Q_{c, l}}{3600 - Δ_{l} \cdot Q_{c, l}} \cdot (T_{c, l} - Δ_{l}))}{1 - exp (- \sum_{m} \frac{φ_{m} \cdot Q_{c, m}}{3600 - Δ_{m} \cdot Q_{c, m}} \cdot T_{f, m})} . \end{matrix}$ (1)

where C_e is the entry lane capacity [veh/h], φ_{_} is the proportion of free traffic within each major stream (named the Cowan’s M3 parameter), Q_c is the circulating traffic volume [veh/s], T_{c,_} stands for the critical headway [s], T_{f,_} stands for the follow up headway [s], Δ _ is the minimum headway between circulating vehicles [s] for which a value of 2.10 s was used, while j, k, l, m are the indices for circulating lanes when the above formula is used for multi-lane layouts. For the purpose of this study the above general capacity formula was specified for each entry lane at every roundabout. To calculate the entry capacity functions for the roundabout 1 and roundabout 2 in Fig. 2, summary effects estimates of critical headway and follow up headway derived from meta-analysis were used [24]. The wide application of meta-analytical approaches in lots of research fields (e.g. [31]) encouraged the Authors to perform a meta-analysis through the random effect model.

In a previous study [24], the values of the critical headway and the follow up headway, already estimated for single-lane and double-lane roundabouts in operation around the world, were collected to:

estimate the mean effects or the effect sizes;

evaluate the dispersion in effect sizes;

compute the summary effect for each gap acceptance parameter.

The meta-analytic estimates resulted consistent across all the studies and gave more reliable results for the gap acceptance parameters compared with the values collected from each single site; thus, the meta-analytic estimates were considered representative of driver behavior at the case studies of roundabout. For each entry lane on roundabout 1 and roundabout 2, the capacity functions which used the summary random-effects estimates of critical headway and follow up headway derived from meta-analysis, were used as reference for calibration purposes; they represented target values of empirical capacity to which the simulated capacities were compared. Based on the meta-analytic estimations of the behavioral parameters, the roundabout 1 and roundabout 2 were considered as representative of the class of the compact and conventional roundabouts, respectively; in this regard, see Italian functional and geometric standards for the construction of road intersections [11]. Table 1 shows the geometric design and behavioral parameters of roundabout case studies. Before applying the GA-based procedure to test the model validity, traffic conditions on roundabout 1 and roundabout 2 were reproduced in AIMSUN (version 8.1)

Table 1

Geometric design and behavioral parameters of roundabout case studies

Basic parameter from [11, 16]	Roundabout 1	Roundabout 2
circulatory roadway width or ring width [m]	7.00	9.00
outer diameter [m]	39.00	41.0
entry width [m]	3.75	6.00
exit width [m]	4.50	7.00
leg length [m]	30.00	35.00
	20.00	25.00
T_ci [s]	–	4.16
T_ce [s]	4.27	3.82
T_f [s]	3.10	2.85

using the default parameters. O/D matrices were assigned from all entries with due regard for turning movements. Entry lane capacity was observed at each subject approach lane where a saturated condition was reached. Based on this representation, each capacity was given by the maximum number of vehicles approaching the roundabout 1 and roundabout 2 from the specified entry lane. In each simulation, the comparison between collected and simulated data returned a percentage error always below 10%. Equation (1) was then used to calculate entry lane capacity at roundabout 1 and roundabout 2. This equation was adapted to roundabout 1, in which entry capacity depends on Q_ce, or the flow of circulating traffic on the ring. The same Equation (1) was adapted to roundabout 2, where the right lane capacity is a function of Q_ce (i.e. the flow of circulating traffic in the outer lane on the ring), while the left lane capacity is a function of Q_ci and Q_ce (i.e. the flows of circulating traffic in the inner lane and the outer lane of the ring, respectively). A sensitivity analysis of the AIMSUN parameters was performed with the objectives to better understand their influence on capacity and select the model parameters for calibration purposes. Before applying the optimization procedure, a manual calibration was performed by running AIMSUN many times and adjusting iteratively the model parameters which were identified; the manual calibration was stopped when the simulation model replicated as closely as possible the empirical data (see e.g. [12, 17]). In particular, a one-parameter sensitivity analysis was performed and then two parameters or three parameter manual calibration were carried out: the first activity required that some model parameters were singly tested using different values, while the other activities required that pairs of two parameters or a set of three parameters were tested, together with their combinations of values. The set of the calibration parameters of AIMSUN included:

the driver reaction time R_T [s]: a parameter of the car-following model that is defined at the experiment level (and used with the same value for all vehicles); it measures the period of time for a driver to react to changes in speed of its previous vehicle. Lower reaction time usually means higher capacity;

the minimum gap G_min [s]: the minimum time between the leader vehicle and the follower vehicle; the lane changing model also includes this minimum time as a restriction of the deceleration component (this constraint is applied before updating position and speed of the leader vehicle respect to its follower);

the speed acceptance S_acc: the degree to which the validity of a particular speed limit is accepted by drivers; it is the acceptance level of the maximum legal speed (v_max) on a road section, i.e. the desired speed of the driver will be v_max · S_acc, which will be achieved if the geometric and operating conditions permit.

Note that G_min and S_acc are defined at the vehicle type level assuming a normal truncated distribution of values; thus, the user must provide the mean, the standard variation, the minimum and maximum values for each parameter. The maximum acceleration that a vehicle can achieve under any circumstances was excluded from manual calibration, since significant benefit to this process was not found when the maximum acceleration was combined with other parameters.

According to [12], the GEH statistic was applied as acceptance or rejection criterion for the model; this global indicator is widely used in practice for the validation of traffic micro-simulation models, especially when only aggregated values can be used (e.g. values of entry capacity). The GEH statistic calculates the indicator for each station i by using the following expression: $G E H_{i} = \sqrt{\frac{2 {(x_{i} - y_{i})}^{2}}{(x_{i} + y_{i})}}$ (2)

where x_i is the ith simulated capacity and y_i is the ith empirical capacity. Based on this criterion the model is accepted if a deviation smaller than 5 in at least 85% of the cases is measured between the simulated values and the corresponding empirical measurement [12]. Note that in the three-parameter manual calibration, for some set of parameters (especially for the left lane on roundabout 2), GEH resulted equal to 100%; however, the parameter set which gave the lowest value for each single GEH_i was selected so that the simulated curve of capacity got close enough to the empirical one. Building on these results, in order to improve the calibration process, the parameter calibration problem was then formulated according to the optimization framework described in Section 4.

4 Structure of the GA-based method and calibration procedure

Now the formulation and the solution of the calibration problem is proposed through a GA-based method; the interpretation of the problem will be provided and the solution by applying a genetic algorithm will be described.

The optimization problem minimized the objective function which expressed the distance between the empirical capacity (for roundabouts it is typically estimated for each entry lane) and its simulated value that in turn, is depending on the set of possible values of the model parameters. In the case study, the optimization problem was formulated as follows: $\begin{matrix} \min_{β} J (β) = \min_{β} \frac{1}{N} \sum_{j = 1}^{N} {\sum_{i = 1}^{m} [\hat{C} (β)_{sim, i}^{j} - C_{obs, i}^{j}]}^{2} \end{matrix}$ (3) $subject to : {LL}_{xp} \leq x_{p} \leq {UL}_{xp}, p = 1, 2, \dots, n$

where β is the vector of parameters to be optimized expressed by β = [R_T, G_min, S_acc], N is the number of measures, each one at each location j (i.e. each entry lane), m is the number of time intervals, n is the number of model parameters, J (β) is the objective function (also called the fitness function or the cost function), to be optimized which represents the mean square of the differences between the simulated value of capacity ( ${\hat{C}}_{sim, i}^{j}$ ) and the experimental value of capacity ( $C_{obs, i}^{j}$ ) at location j and time interval i, while LL_xp and UL_xp are the lower bound and upper bound of each parameter x_p, respectively.

It is should be noted that the objective function J (β) varies with the road entity; for a first application to freeways see [33]. Thus, efforts have been made to adapt the proposed procedure to roundabouts. In Section 3 the Hagring model [27] was used to estimate the empirical capacities at each entry lane on roundabout 1 and roundabout 2 based on summary random-effects estimates of critical headway and follow up headway derived from meta-analysis. In turn, simulation output was generated by using AIMSUN which ran with a fixed model based on a suitable set of parameters.

In Equation (3)N was set equal to 32, since the observations were distributed over eight hours. In each hour, 4 simulations ran (one every 15 minutes); each hour corresponded to an antagonist traffic volume so that the capacity function could be plotted by entry lane for every roundabout (see Section 3). The solution of the calibration problem will be given by the vector β^* of parameters that minimized the J (β):

$β^{*} = arg \min_{β} J (β)$ (4)

The problem expressed in Equation (3) can be solved iteratively. For the selected roundabouts in Fig. 2, the optimization framework to search β^* was implemented in MATLAB^® by using the built-in genetic algorithm tool; this tool has been applied in order to minimize the differences between the sets of empirical (entry lane) capacities and simulated data derived from AIMSUN. The automatic interaction with AIMSUN was implemented through an external script in Python that was written.

It is noteworthy that the algorithm starts from a generic initial condition, that is an initial population of 20 individuals randomly generated (each individual corresponded to a set of 3 parameters in each entry lane for every roundabout); the default setting of AIMSUN is used as first individual just to increase the speed of convergence of the algorithm. However, the genetic algorithm is robust with respect to the initial condition choice; thus, whatever the initial condition is, the genetic algorithm will converge towards the global minimum. The genetic algorithm, indeed, generates a set of parameters β (i.e. one β for each individual) and AIMSUN can run with these β; a subroutine written in Python language allowed to transfer the data between AIMSUN and MATLAB^®. The Python script then modified the corresponding parameters in AIMSUN for each individual, simulated the model in console mode and compared the empirical data and simulated capacities to estimate the value of J (β). AIMSUN gives a set of outputs (i.e. one for each β), so that the algorithm can calculate the J (β) associated with each β. Thus, each individual obtains a fitness value; the genetic algorithm then selects the best parameter β and can generate a new set of parameters β. Several genetic operators can be applied to produce a new generation of β; each new generation overlaps the previous generation of β. The cycle is iterative until a predefined stopping criterion is met.

In order to minimize the fitness function using the GA function, a number of variables, as well as lower bound and upper bound for the model parameters were specified; the maximum number of iterations was fixed equal to 50 generations. The initial population was made by 20 individuals; according to [3], the structural bias of GA could increase with a further increase in the population size. Options for the evolutionary parameters included a mutation function: constraint dependent; a crossover function: scattered; a selection function of the type stochastic uniform; an elite count of 2; the crossover fraction of 0.8.

In order to restrict the search domain and avoid generating negative parameters (or parameters without physical meaning), the condition LL_xp ≤ x_p ≤ UL_xp was established for each parameter x_p; LL_xp (i.e. the lower bound) and UL_xp (i.e. the upper bound) were based on the outcome of the manual calibration. For the roundabout 1, we set LL_xp: [R_T = 0.82 ; G_min = 1.50 ; S_acc = 1.00] and UL_xp: [R_T = 0.86 ; G_min = 1.70 ; S_acc = 1.10]. For the roundabout 2, we set the following bounds LL_xp: [R_T = 0.85 ; G_min = 1.00 ; S_acc = 0.90] and UL_xp: [R_T = 0.95 ; G_min = 1.50 ; S_acc = 1.10] for each entry lane. After 50 iterations and 4 hours computing time, the algorithm reached a convergence condition; J (β) then reached a steady-state. Thus, the algorithm provided the optimal solution for β^* in Equation (4). The GA has been shown to give near-global optima when calibrating parameters in AIMSUN.

Figure 4 shows the outline of GA calibration process.

Fig.4

The GA calibration process.

In order to show the sequence of steps needed for the problem under examination, the pseudo code of the Python script is given in the following:

procedure Python script file (pseudocode)

inputFilePath ← path of the.txt file where MATLAB writes the values of reaction time, minimum gap and speed acceptance at each iteration.

networkFilePath ← path of the AIMSUN file where the network is modeled;

outputFilePath ← path of the.txt file where AIMSUN will write the results of the simulation carried out with the parameters’ value given in inputFile;

replicationID ← ID replication number indicated in the AIMSUN file;

detectorID ← ID detector number indicated in the AIMSUN file;

carID ← ID car number indicated in the Aimsun file;

experimentID ← ID experiment number indicated in the AIMSUN file;

inFile ← open(inputFilePath,“reading”) ᐅ open the file inputFilePath in reading mode.

outFileopen (outputFilePath, “writing”) ᐅ open the file outputFilePath in writing mode.

reactionTime ← takes the value from the file in inFile;

minGap ← takes the value from the file in inFile;

speedAcc ← takes the value from the file in inFile; ᐅ In the following is shown the Python code which runs the AIMSUN file of the network in consol mode, using the parameters of reaction time, minimum gap and speed acceptance indicated in inFile. It gives as output the values of capacity.

if console.open(networkFile) then

plugin ← GKSystem.getSystem(). getPlugin(“GGetram”)

model ← GKSystem.getSystem(). getActiveModel()

simulator ← plugin. getCreateSimulator (model)

experiment ← model. getCatalog().find (experimentID)

simTime ← model.getColumn (“GKExperiment::simStepAtt”)

experiment.setDataValue(simTime, reactionTime)

carType ← model.getCatalog(). find(carID)

carType.setDataValueByID (GKVehicle. minimunGapMean, minGap)

carType←model.getCatalog().find (carID)

carType.setDataValueByID (GKVehicle. speedAcceptanceMean,speedAcc)

if simulator.isBusy() == False then

replication ← model. getCatalog(). find(replicationID)

ifreplication! = Noneandreplica tion. isA(“GKReplication”) then

simulator.addSimulationTask (GKSimulationTask(replication, GKReplication.eBatch))

simulator.simulate()

detector ← model.getCatalog (). find(detectorID)

counts←model.getColumn (“DYNAMIC::SRC_GKDetector_ count_0”)

capacities←model.getColumn (“DYNAMIC::SRC_GKDetector_flow0”)

ts1←detector.getDataValueTS (counts)

ts2←detector.getDataValueTS (capacities)

numValues←ts1.size()

for j←0, numValuesdo

outFile.write (j,ts1.getValue (GKTimeSerieIndex(j)), ts2.getValue(GKTimeSerieIndex(j)));

j = j+1;

end for

end if

outFile.close() ᐅ close outFile.

inFile.close() ᐅ close inFile.

end procedure

Table 2 shows the results of the manual calibration and the best combination of the GA-optimized parameters for the roundabout 1 and roundabout 2. The same table reports the J (β) for the GA-based optimization, as well as the J (β) obtained using the default parameters of AIMSUN and the parameters derived from manual calibration. Table 2 also reports the GEH index used to compare the simulated and empirical values of entry capacity, and then support the decision to accept or otherwise reject the model [12]. For every roundabout, since the deviation of each GA-optimized value of entry capacity with respect to the corresponding empirical measurement resulted smaller than 5 in at least 85% of the cases, the model could be considered “calibrated”, i.e. able to reproduce the empirical capacities at entries of the examined roundabouts. Moreover, the root mean squared normalized error resulted less than 0.10, while the mean absolute percent error resulted less than 5%.

Table 2

Results of manual calibration vs GA-based optimization

AIMSUN Parameter	Entry lane (roundabout 1)			Entry lane (roundabout 2)
	Default Values	Manual Calibrated	GA-Optimized	Default values		Manual calibrated		GA-optimized
	Default Values	Manual Calibrated	GA-Optimized	Left	Right	Left	Right	Left	Right
R_t [s]	0.80	0.85	0.86	0.80	0.80	0.95	0.95	0.95	0.94
G_min [s]	0.00	1.60	1.58	0.00	0.00	1.50	1.50	1.33	1.00
S _acc	1.10	1.00	1.00	1.10	1.10	0.90	0.90	0.97	0.95
J (β)	159.04	135.00	126.8	129.16	117.63	61.61	74.20	57.03	60.52
GEH [% ]	56.25	87.50	87.50	78.10	78.10	100.00	96.90	93.80	96.90

By way of example, Fig. 5 shows the values of the objective function J (β) during the optimization period for the left entry lane on roundabout 2. The graph in Fig. 5 plots the mean best fit (the mean value of the objective functions calculated for all individuals of the same generation), and the best fit (the objective function of the best individual within the generation). Since the GA found the minimum of the objective function, the best fitness value for a population is the smallest value for any individual of the population. With reference to the left entry lane (roundabout 2) in Table 2, J (β) resulted equal to 57.03 for the optimized parameters and equal to 61.61 for the parameters derived from manual calibration. Moreover, the parameter tuning would bring significant benefits if the default parameters of AIMSUN were used as the initial condition; indeed, J (β) resulted equal to 129.16 for AIMSUN default parameters. It should be noted that the GA ran for 50 generations in total (about 4 hours of computational time) in order to ensure accuracy in calibration. Indeed, the algorithm has not been stopped when it reached the steady-state (20th generation for roundabout 1 or just under the 20th generation for each entry lane on roundabout 2), but it ran for the other generations up to 50 generations. Thus, after reaching the steady-state, the algorithm continued to generate randomly 20 individuals at each iteration; 600 (or more) sets of potential parameters by entry lane, within the sets bounded from LL_xp and UL_xp, were generated. However, among all these 600 (or more) sets of possible parameters “randomly generated”, none gives a smaller cost function than the set of parameters generated at the 20th generation (for roundabout 1) or just under the 20th generation (for each entry lane on roundabout 2). Indeed, the cost function in Fig. 5 is constant during the remaining generations after the steady-state. At last, for every roundabout, the scattergram analysis of empirical versus simulated capacities of each entry lane was performed to further support the model validation. Only for illustrative intentions, Fig. 6 shows the scattergram analysis of empirical versus simulated capacities for the left entry lane (roundabout 2). The regression line of empirical capacity versus simulated capacity was plotted along with the 95% PI (i.e. the Prediction Interval). High values of R² and most of points were within the confidence band of the regression lines suggested that the model can be accepted since it is significantly close to the reality.

Fig.5

The fitness function J (β) during the optimization period (roundabout 2).

Fig.6

Simulated and empirical relationship with 95% prediction interval (PI) for the left entry lane (roundabout 2).

5 Capacity-based calculation of PCEs at roundabouts

This section presents the criterion which used the calibrated model to estimate the PCEs for heavy vehicles driving the roundabouts. The criterion is based on the equivalence between the proportions of capacity used by passenger cars (i.e. the base vehicle) and heavy vehicles (i.e. trucks). According to [19] which estimated the PCEs under free-flowing, multi-lane conditions, the flow rate of a base stream (including only passenger cars) and the flow rate of a mixed stream (including the proportions of passenger cars and trucks) were calculated. The ratio between the two flow rates having the same level of a measure of impedance (or the density of the two streams) was necessary for calculating PCEs. For each entry lane on every roundabout, the capacity C_car simulated for a traffic demand made with 100% passenger cars and the capacity C_p simulated for a traffic demand with a p percentage of trucks were compared. Thus, the equation C_car = (1 - p) · C_p + p · C_p · E_t was used, where the capacity C_car included the proportion of passenger cars (1 - p) C_p and the proportion of heavy vehicles (p C_p); the last proportion was multiplied by the equivalent factor E_t for questions of homogeneity. Being the passenger car the base vehicle, the capacity function C_car was the base curve, while C_p was the capacity function for a mixed fleet with a p percentage of trucks. Different traffic scenarios were simulated using different percentages of trucks in the traffic demand; specifically, the traffic scenario with 100% passenger cars is corresponding to p = 0%. The dimensional and operational features of the trucks in AIMSUN included: a length between 6.00 m and 10.00 m; a width ranging from 2.00 m to 2.80 m; the maximum desired speed of 85 km/h (ranged from 70 km/h to100 km/h); the maximum acceleration of 1 m/s² (ranged from 0.6 m/s² to 1.80 m/s²), the maximum deceleration of 5 m/s² ranged from (4 m/s² to 6 m/s²). The base curve function was calibrated setting the AIMSUN parameters on the basis of the solution of the GA-based optimization problem. Generation of the C_car and C_p functions required that O/D matrices were assigned to the roundabouts built in AIMSUN. To generate some traffic scenarios, different mixed fleets (i.e. 100% passenger cars, 10%, 20%, 30% up to 100% heavy vehicles) entering the roundabouts were produced. In order to calculate E_t for each entry lane of the selected roundabouts, the plots of the C_car and C_p functions were based upon regressions on simulated data; GENSTAT 17.0 software was used to perform a nonlinear regression analysis on simulated data [9]. Combinations of circulating flows ranged from Q_ci/Q_ce = 0.25 to Q_ci/Q_ce = 1.50 were examined for roundabouts 2, in order to examine possible distributions of the circulating flows on the two lanes of the circulatory roadway. Equation (1), specifically adapted to every roundabout, was the best functional form to associate the regressors with the response variable; in turn, the gap acceptance parameters (i.e. the critical headway and follow-up headway) for each entry lane of every roundabout were the model parameters to be estimated and fitted for different percentages of trucks.

Tables 3 –5 depict the results of the statistical regressions of the model parameters for roundabout 1 and roundabout 2; two critical headways were estimated for the left entry lane of roundabout 2. Estimation of headways in Tables 3–5 correspond to the mixed fleets with more realistic percentages of trucks (i.e. p = 10%, 20%, 30%); the percentage 100% of trucks was also estimated for further insights on the truck behaviour at entries, but not reported here. According to literature on the topic (see e.g. [13 , 26]), the model parameters increase when the percentages of heavy vehicles increase from p = 0% to p = 30%. It should be noted that heavy vehicles reduce the vehicular capacity at roundabouts, especially with a higher entering volume of trucks. Moreover, the space and maneuvering requirements of most trucks entering a roundabout are restricted by size and operational features which, in turn, affect the driver gap acceptance behaviour. As introduced above, the PCEs have been calculated separately for each entry lane of the roundabouts built in AIMSUN (see Fig. 2). Since the capacity functions C_car and C_p are depending on the circulating flow, the equivalent factors E_t are a function of the circulating flow: for roundabout 1 E_t is depending on the circulating flow Q_ce along the one-lane ring; the same trend characterizes the right entry lane on roundabout 2, where the circulating flow Q_ce moves along the outer lane of the ring. For the left entry lane (roundabout 2), the equivalent factors are depending on the inner circulating flow Q_ci and the outer circulating flow Q_ce. Thus, in this case, surface plots have been generated for graphing the PCEs versus Q_ci and Q_ce on the ring.

Table 3
Regressions for critical and follow up headways at roundabout 1

Regressions Fleet

10% hv 20% hv 30% hv 100% pc

Estimation T _ce T _f T _ce T _f T _ce T _f T _ce T _f

est [s] 5.13 2.64 5.43 2.78 5.87 2.87 4.85 2.54

(s.e.) [s] (0.06) (0.02) (0.09) (0.03) (0.09) (0.03) (0.05) (0.02)

t 92.32 151.10 61.71 100.48 68.46 100.28 88.87 150.41

UL [s] 5.24 2.68 5.60- 2.84 6.03 2.92 4.96 2.58

LL [s] 5.02 2.61 5.26 2.73 5.70 2.82 4.75 2.51

R ² 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99

Regressions	Fleet
est [s]	5.13	2.64	5.43	2.78	5.87	2.87	4.85	2.54
(s.e.) [s]	(0.06)	(0.02)	(0.09)	(0.03)	(0.09)	(0.03)	(0.05)	(0.02)
t	92.32	151.10	61.71	100.48	68.46	100.28	88.87	150.41
UL [s]	5.24	2.68	5.60-	2.84	6.03	2.92	4.96	2.58
LL [s]	5.02	2.61	5.26	2.73	5.70	2.82	4.75	2.51
R ²	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99

Note: hv = heavy vehicles; pc = passenger cars; UL = Upper Limit and LL = Lower Limit of the 95% confidence interval (α= 0.05); T_ce = critical headway; and T_f = follow up headway.

Table 4

Regressions for critical and follow up headways for right entry lane at roundabout 2

Regressions	Fleet
	10% hv		20% hv		30% hv		100% pc
Estimation	T _ce	T _f	T _ce	T _f	T _ce	T _f	T _ce	T _f
est [s]	5.19	2.77	5.58	2.90	6.11	2.98	4.54	2.67
(s.e.) [s]	0.04	0.01	0.05	0.01	0.05	0.01	0.03	0.01
t	137.35	269.28	104.89	195.94	123.27	219.25	121.77	256.36
UL [s]	5.27	2.79	5.68	2.93	6.21	3.01	4.62	2.69
LL [s]	5.12	2.75	5.47	2.87	6.02	2.96	4.47	2.65
R ²	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99

Note that: hv = heavy vehicles; pc = passenger cars; UL = Upper Limit and LL = Lower Limit of the 95% confidence interval (α= 0.05); T_ce = critical headway; and T_f = follow up headway.

Table 5

Regressions for critical and follow up headways for left entry lane at roundabout 2

Regressions	Fleet
	10% hv		20% hv		30% hv		100% pc
Estimation	T _ce	T _ci	T _ce	T _ci	T _ce	T _ci	T _ce	T _ci
est [s]	4.74	3.80	5.48	3.98	6.13	4.17	4.01	3.70
(s.e.) [s]	0.09	0.12	0.13	0.17	0.19	0.25	0.09	0.13
t	52.43	32.24	42.26	23.57	31.60	16.74	43.23	29.39
UL [s]	4.92	4.04	5.75	5.24	6.51	4.66	4.19	3.95
LL [s]	4.56	3.57	4.32	3.65	5.75	3.68	3.83	3.46
R ²	0.99	0.99	0.98	0.98	0.97	0.97	0.84	0.84

Note that: hv = heavy vehicles; pc = passenger cars; UL = Upper Limit and LL = Lower Limit of the 95% confidence interval (α= 0.05); T_ce = critical headway (right entry lane); T_ci = critical headway (left entry lane).

Figure 7 shows the estimation of PCEs for the entry lane on roundabout 1 and the right entry lane on roundabout 2; because of the need to compare the equivalent factors corresponding to entry lanes characterized by the same mechanism of capacity, only a mean E_t (corresponding to 10%, 20% and 30% percentages of heavy vehicles simulated with AIMSUN), was plotted. In turn, Fig. 8 depicts the mean surface plot of the PCEs for the left entry lane on roundabout 2; because no significant difference was found among the PCEs for the (more realistic) percentages of heavy vehicles (p = 10%, 20% and 30%), this surface plot can be considered representative for the traffic scenarios with p = 10%, 20% and 30%.

Fig.7

PCE estimations on roundabout 1 and roundabout 2. Note that Q_ce stands for the circulating flow in the one-lane circulatory roadway on roundabout 1 and the circulating flow in the outer lane of the circulatory roadway on roundabout 2, while E_t stands for the equivalent factor.

Fig.8

PCE estimations for left entry lane on roundabout 2. Note that a mean percentage of heavy vehicles among 10%, 20 % and 30% of heavy vehicles is considered; Q_ce stands for circulating flow in the outer lane of the circulatory roadway, while Q_ci stands for circulating flow in the inner lane of the circulatory roadway; E_t stands for equivalent factor.

One can observe from Fig. 7 that E_t increases when Q_ce is increasing from 0 veh/h onwards. For roundabout 1 E_t keeps values around 2, but it does not exceed 2 in the range of variation of the circulating flow when mean traffic conditions are considered. For roundabout 2, the same trend in E_t is shown when Q_ce is ranging from 0 to 600 veh/h (see Fig. 7); E_t exceeds 2 only when Q_ce is increasing from 600 veh/h onwards, or when traffic conditions for the circulating flow reach the saturation level. It is well known that the HCM [10] proposes equivalent factors to adjust the flow rate for each entry movement and takes into account the vehicle stream features. The HCM [10] assumes that a heavy vehicle is equivalent to two passenger cars on roundabouts; however, it does not consider the PCE variation with the circulating flow. By way of example, the effect of heavy vehicles on vehicular operations of the right lane on roundabout 2 (that is a double-lane roundabout) can be misunderstood if one sets equal to 2.0 the PCE for heavy vehicles. The results in Fig. 7, indeed, have highlighted, even for the examined (realistic) mixed fleets including heavy vehicles with a maximum length of 10 m, how possible it is to underestimate the effect of heavy vehicles if E_t = 2 with Q_ce higher than 600 veh/h for roundabout 2 or overestimate the effect of heavy vehicles if E_t = 2 with Q_ce lower than 500 veh/h for roundabout 1 (that is a single-lane roundabout).

Figure 8 is given as an example only to show the impact of heavy vehicles on vehicular operations of the left lane of a double-lane roundabout (roundabout 2) in usual operational conditions. An E_t just under 4 can be reached with 10%, 20% or 30% of heavy vehicles. As a consequence, if one set E_t = 2 to adjust the flow rates for heavy vehicles as the HCM [10] proposes for roundabouts, the effect of heavy vehicles on the quality of traffic flow can be underestimated. Despite the reader is advised that heavy vehicles here considered are only a part of heavy vehicles considered in the simulation model used to estimate the HCM PCE values, these results have been found to be consistent with the conclusions drawn from Lee [6]; indeed, values of PCEs different from what the HCM [10] proposes for roundabouts were recommended, since they are depending on traffic conditions.

6 Conclusions

This paper introduces a procedure based on a genetic algorithm for calibrating a microscopic traffic simulation model. Despite microscopic traffic simulation models are increasingly seen as effective tools to analyze a wide variety of traffic problems associated with complex processes which cannot easily be described in analytical terms, the results of the analysis could be unreliable and misleading without a proper calibration of the model parameters.

The specific objective of this research was to explore the potential advantages of performing model calibration as an optimization problem which searches for an optimum set of model parameters through a GA-based method. Thus, the genetic algorithm-based calibration was applied to traffic simulations with AIMSUN. In order to automate the process of manual calibration of the model parameters, the calibration process was formulated as an optimization problem aimed at minimizing an objective function by using the genetic algorithm tool in MATLAB^®. A subroutine in Python implemented the automatic interaction between MATLAB^® and AIMSUN and allowed the data transfer between the two programs.

Focus was made on two case studies of roundabouts (roundabout 1 and roundabout 2), for which the calibration parameters were preliminarily identified by using sensitivity analysis and manual calibration; for each entry lane on every roundabout, the optimum values for the model parameters were obtained by minimizing the error between the simulated and empirical capacities through a genetic algorithm.

The genetic algorithm-based approach appeared to be effective in the calibration of AIMSUN; indeed, the results showed a better match to the empirical data than simple manual calibration. Despite this research is a first analysis for roundabouts, it should be noted that the comparison between the empirical data and the simulation results obtained with the GA-optimized parameters, gave insights into the performance of the proposed calibration procedure; this is consistent with the sensitivity analysis and manual calibration that had highlighted no further need of a higher number of calibration parameters. Thus, given the number of model parameters here used, the proposed procedure can be applied to other case studies. Based on the results of this research - as summarized by the values of the cost function J (β) in Table 2 – the benefits resulting from using of a GA-based approach can compensate the computational efforts which derive from the application of an optimization technique which automates the iterative process of manually adjusting the simulation parameters.

Another important aspect of this research is the criterion proposed to evaluate the impact of heavy vehicles on traffic conditions at roundabouts as an important component in the estimation of the roundabout capacity. It should be noted that for the examined cases, only a subject type of heavy vehicle was considered; however, this might result more elucidative to better understand the impact of this class of heavy vehicles on operational performance of the roundabouts.

The experience of this initial implementation has also suggested several possible directions for additional research. The proposed criterion to calculate PCEs is general and could be applied to different types of intersection to improve geometric design; however, the same criterion of PCE calculation should be specified to explore more classes of heavy vehicles or specific mixed traffic conditions.

At last, it should be said that the advantages of the proposed procedure are also to improve the efficiency of simulation models to predict different traffic scenarios by varying different initial conditions and to match the safety-affecting progress in vehicles and intelligent infrastructures. Moreover, the application of this procedure to other road facilities should be designed to address further problems and issues on safety analysis through traffic simulation models that transportation engineers are often called to solve. For future developments, more consideration is required to specify the calibration procedure of traffic simulation models through automated processes which implement the latest optimization techniques.

References

Tarko

, Davis

, Saunier

, Sayed

, and Washington

, Surrogate Measures of Safety, White paper, ANB20 (3) Subcommittee on Surrogate Measures of Safety, 2009.

AIMSUN Dynamic Simulator User Manual, Transport Simulation System (TSS) version 8, Barcelona, 2011.

Piotrowski

A.P.

and Napiorkowski

J.J.

, Searching for structural bias in particle swarm optimization and differential evolution algorithms, Swarm Intelligence10 (2016), 307–353.

Ahmad Al-Hadad

, Mawdesley

M.J.

and Stace

, A genetic algorithm approach to highway alignment development, In: International Conference on Computing in Civil and Building Engineering, Nottingham University Press, Nottingham, 2010.

Ahmad Al-Hadad

and Mawdesley

, A genetic algorithm approach to a 3D highway alignment development, In: International Conference on Evolutionary Computation, SciTePress, Valencia, 2010, pp. 129–136.

Lee

, Developing passenger-car equivalents for heavy vehicles in entry flow at roundabouts, Journal of Transportation Engineering141 (8) (2015), 1–1.

Yang

, Remenyte-Prescott

, Andrews

J.D.

, Pavement maintenance scheduling using genetic algorithms, International Journal of Performability Engineering11 (2) (2015), 135–152.

Goldberg

D.E.

, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Longman Publishing Co., Inc, Boston1989.

GenStat for Windows, 17th Edition, V.

10.

Highway Capacity Manual, Transportation Research Board, 5th., Washington DC, Transportation Research Board, 2010.

11.

Italian Minister of Infrastructure and Transport, http://www.mit.gov.it/mit/site.php?p=normativa&o=vd&id=1735&id_cat=&id_dett=0.

12.

Barceló

, Fundamentals of Traffic Simulation, Springer, Heidelberg, 2010.

13.

Dahl

and Lee

, Empirical estimation of capacity for roundabouts using adjusted gap-acceptance parameters for trucks, Transportation Research Record2312 (2012), 34–45.

14.

Rubio-Martín

J.L.

, Jurado-Piña

, and Pardillo-Mayora

J.M.

, Heuristic procedure for the optimization of speed consistency in the geometric design of single-lane roundabouts, Canadian Journal of Civil Engineering42 (2015), 13–21.

15.

Stewart

J.M.

, Python for scientists. Cambridge University Press, Cambridge, 2014.

16.

Rodegerdts

, ., Roundabouts: An Informational Guide, 2nd. NCHRP REPORT 672, Washington DC: Federal Highway Administration, United States Department of Transportation, 2010.

17.

Vasconcelos

, Seco

Á.

, Silva

A.B.

, Hybrid Calibration of Microscopic Simulation Models, In: Sousa

J.F.

, and Rossi

, (eds.) Comuter-based Modelling and Optimization in Transportation, 262, Springer, Cham, 2014, pp. 307–320.

18.

Bierlaire

, Simulation and optimization: A short review, Transportation Research Part C: Emerging Technologies55 (2015), 4–13.

19.

Huber

M.J.

, Estimation of passenger-car equivalents of trucks in traffic stream, Transportation Research Record869 (1982), 60–70.

20.

Jha

M.K.

, Schonfeld

, A highway alignment optimization model using geographic information systems, Transportation Research Part A: Policy and Practice38 (6) (2004), 455–481.

21.

MathWorks^®, Documentation on genetic algorithms [Online]. Available: https://www.mathworks.com/help/gads/geneticalgorithm.html (accessed: 20/07/2017).

22.

Mitchell

, An Introduction to Genetic Algorithms, The MIT Press, Boston, 1998.

23.

Bukharov

and Bogolyubov

D.P.

, Development of a decision support system based on neural networks and a genetic algorithm, Expert Systems with Applications42 (15) (2015), 6177–6183.

24.

Giuffrè

, Granà

, and Tumminello

M.L.

, Gap-acceptance parameters for roundabouts: A systematic review, European Transport Research Review8 (1): 2 (2016), 1–20.

25.

Giuffrè

, Granà

, Marino

, and Galatioto

, Microsimulation-based passenger car equivalents forheavy vehicles driving turbo-roundabouts, Transport31 (2) (2016), 295–303.

26.

Giuffrè

, Granà

, Tumminello

M.L.

, and Sferlazza

, Estimation of Passenger Car Equivalents for single-lane roundabouts using a microsimulation-based procedure, Expert Systems with Applications79 (2017), 333–347.

27.

Hagring

, A further generalization of Tanner’s formula, Transportation Research Part B: Methodological32 (6) (1998), 423–429.

28.

Kramer

, Genetic Algorithm Essentials, Springer, Heidelberg, 2017.

29.

Gipps

P.G.

, A behavioural car-following model for computer simulation, Transportation Research Part B: Methodological15B (1981), 105–111.

30.

Dowling

, Skabardonis

, Halkias

, Mchale

, and Zammit

, Guideline for calibration of microsimulation models framework and applications, Transportation Research Record1876 (2014), 1–9.

31.

Elvik

, Road safety effects of roundabouts: A meta-analysis, Accident Analysis and Prevention99 (2017), 364–371.

32.

Cheu

R.L.

, Calibration of FRESIM for singapore expressway using genetic algorithm, Journal of Transportation Engineering124 (6) (1998), 526–535.

33.

Chiappone

, Giuffrè

, Granà

, Mauro

, and Sferlazza

, Traffic simulation models calibration using speed-density relationship: An automated procedure based on genetic algorithm, Expert Systems with Applications44 (2016), 147–155.

34.

Giuffrè

, Trubia

, Canale

, and Persaud

, Using microsimulation to evaluate safety and operational implications of newer roundabout layouts for European Road networks, Sustainability9 (2084) (2017), 1–13.

35.

Chen

X.M.

, Xiong

, He

, Zhu

, and Zhang

, Time-of-day vehicle mileage fees for congestion mitigation and revenue generation: A simulation-based optimization method and its real-world application, Transportation Research Part C: Emerging Technologies63 (2016), 71–95.

36.

Wang

, Cheu

R.L.

, and Fwa

T.F.

, Highway Maintenance Scheduling Using Genetic Algorithm with Microscopic Traffic Simulation, TRB 2002 Annual Meeting, Washington, 2002.

37.

Hamadi

, Monfroy

, and Saubion

, What Is Autonomous Search? In: van Hentenryck

, Milano

(eds). Hybrid Optimization, Springer Optimization and Its Applications 45, Springer: New York, 2011.

38.

Chan

W.T.

and Tao

, Using GIS and genetic algorithm in highway alignment optimization, Intelligent Transport System2 (2003), 1563–1567.