MRDPGA: a multiple restart dynamic population genetic algorithm for scheduling road traffic

Abstract

A genetic algorithm is a biologically inspired stochastic approach to finding solutions to optimization problems. However, unlike its deterministic counterpart, it cannot guarantee a globally optimal solution since it may be trapped within a local optimum of the search space. Most researchers have focused on proposing new techniques for various parameters of genetic algorithms. That is a mutation, crossover, or selection algorithm. This research proposes a modification to the standard genetic algorithm, which may serve as a framework that can integrate any of these parameters for their contribution to the final solution of the genetic algorithm. The multiple restart dynamic population genetic algorithm (MRDPGA) proposed in this research was used in training the parameters of the adaptive neuro-fuzzy inference system (ANFIS) for scheduling road vehicular traffic flows. The results of training the ANFIS models based on the different clustering methods showed that the MRDPGA-based ANFIS controller performed better with the mean square error (MSE) of 0.299 and root mean square error (RMSE) of 0.547 in the training phase; and MSE of 0.272 and RMSE of 0.521 in the testing phase. Using the controllers for traffic flow scheduling, the results showed that the MRDPGA-trained controllers performed better in terms of average waiting time (AWT) minimization and total arrived vehicles (TAV). The best-performing controller achieved 50.40% AWT minimization and 21.44% TAV improvement. Analyzing the results using a one-tailed t-test for paired two-sample means showed that the MRDPGA algorithm had a significant impact on the controllers. Particularly the FCM controller, where (p = 0.0038) and (p = 0.0003) for AWT and TAV at a 95% confidence level. Thus, MRDPGA algorithms are recommended for further assessment in other optimization problems to ascertain their performance in those problem domains.

Introduction

The concept of optimization, which is concerned with the effectiveness and efficiency of solutions to problems, has become an interesting research topic. It involves determining a mix of problem parameters in appropriate proportions so that the solution is either maximized or minimized as required [1]. Optimization may be deterministic or stochastic. A deterministic optimization approach often fails when the scale of the optimization problem becomes large and complex, or the internal working mechanism of the optimization is not known [2, 3]. Stochastic approaches can deal with these problems associated with deterministic approaches even though they also have the challenge of guaranteeing optimal global solutions [4, 5]. One of the most notable approaches to stochastic optimization is the evolutionary algorithmic technique [6, 7].

Evolutionary algorithms are biologically inspired techniques that essentially perform a search for a global solution within the possible solution space of the optimization [8]. The genetic algorithm (GA) is one of the most notable evolutionary algorithms. It is capable of dealing with large and complex problems, which are either discrete or continuous. It allows for the specification of adequate objective requirements and constraints as well as generates several possible optimal solutions. Therefore, GA has found several applications in diverse optimization problems. Examples include constraint function [6], two-dimensional rectangular packing problem [9], water distribution networks [10], classification problems [11], profit maximization [12], traffic flow control [13,14,15,16,17], and many other areas [18, 19].

Though GAs [20], in their several variants, have several advantages and have been applied in several optimization problems, they are often confronted with some fundamental challenges. Like any stochastic approach, GA has performance bottlenecks, which can be minimized with the application of parallel computing platforms. Also, GA may be trapped in local optima and unable to generate the global optima [4, 9]. Therefore, this research proposes a modified genetic algorithm technique that minimizes the probability of GA being trapped in local optima even with the increase in problem size. That is, the probability of attaining global optima is improved.

To assess the effectiveness of the proposed GA technique, the modified GA was used to tune the adaptive neuro-fuzzy inference system (ANFIS) for adaptive control of road vehicular traffic flows. Even though a classical problem, road traffic control remains a major problem in modern cities of the world [21]. Traffic congestions problems have been approached differently in terms of factors affecting free flows, from queueing [22], driver and driving factors [23], traffic rules violations [24], to signal optimization [25, 26] etc. However, desired results are yet attained in terms of minimization of vehicular waiting time (or delays), especially in modern cities. We have taken the city Kaduna of Nigeria to implement our model. Thus, it's the case study used to assess the performance of the modified GA.

The major contributions of this work include:

• To develop a new genetic algorithm technique based on population modifications and multiple restarts.

• To apply the newly developed genetic algorithm technique for the optimization of ANFIS controller parameters.

• To estimate the vehicular hourly arrival rates at various roundabouts, and the enhancement of road traffic flows within a complex road network.

The remaining part of this paper is organized as follows: the review of the most relevant literature is presented in section “Literature review”. In section “Methodology”, the methodology describes the newly modified genetic algorithm, ANFIS controller, and simulation modeling. In section “Results and discussion”, the results and discussion are presented. Section “Conclusion” presented the conclusion and recommended the future direction of the paper.

Literature review

The standard genetic algorithm (SGA) is a five-step technique that performs population initialization, evaluation, selection, crossover, and mutation [27]. Due to the limitations of SGA, some researchers have attempted to address the challenges by proposing varying modifications or techniques that suit specific problem domains; and a few have considered generic modifications to the standard GA. This section presents a review of the relevant literature survey and presents a summary in Table 1.

Table 1 A summary of related literature

Dao et al. [4] proposed a new GA technique that involved adaptively restarting the GA. The proposed approach was tested on two benchmark functions, Hump with two dimensions and Rastrigin with fifteen dimensions. Results showed that the proposed method achieved better results at estimating the global optima of the functions compared to the other approaches. However, the study only performs population modification when restarting the genetic algorithm. Also, for every restart cycle, a new set of individuals from the population. These imply that the advantages of the best individuals in a given generation do not contribute to the next restart cycle. Also, close observation of the genetic algorithm reveals that though it may appear to have been trapped to local optima, at times, it can break out after several generations. Thus, jumping to restart the next cycle after there has not been an improvement in the best solution for a given number of generations may not necessarily be the best technique.

Potuzak [3] performed road traffic load balancing across multiple road intersections using distributed standard genetic algorithm. To perform the traffic load balancing, the average number of vehicles on each lane is obtained and used for traffic division into various sub-networks of the considered complex road network. Due to the computational bottlenecks of genetic algorithms, a parallel and distributed computing model was used. The results of simulating the proposed approach were reported to have shown significant minimization of computational time requirement and interesting results of load balancing. However, the use of a standard genetic algorithm implies that the proposed approach suffers from the general challenge of local optima associated with standard genetic algorithms. The author of another study combined parallel and distributed computing for the genetic algorithm with graph coarsening to solve the same problem of load balancing [28]. However, the challenges associated with the genetic algorithm and spectral and cut guarantees associated with graph coarsening are the study's limitations.

Luis et al. [13, 14] modeled complex isolated intersection designs and enhance vehicular throughput of the intersections. A model that calculated the intersection inflow’s points of conflict was proposed, and standard genetic algorithm techniques were used to optimize vehicular arrival rates. The implementation of the cellular automata simulator and simulation of various traffic scenarios showed that the proposed approach achieved improvement within the range of 9.21% and 36.98%. However, the utilization of the standard genetic algorithm technique suggests that the proposed approach may suffer from similar challenges to standard genetic algorithms.

Basak [29] considered that the limitations of SGA were due to constant control operators resulting in the proposed adaptive mutation based on rank order. The result reported showed better performance but had the limitation of focusing on a single operator. This may not yield the best result as genetic algorithms require a proper mix of various parameters. Also, in an attempt to address the same problem of constant control operator parameters, Muzid [30] proposed the use of fuzzy logic to determine the probabilities of mutation and crossover operators as well as population size based on fitness value. The results reported showed that the fuzzy logic-based approach to determining crossover and mutation probability showed better results compared to when the standard genetic algorithm was used. However, the challenge of the non-adaptability of fuzzy logic suggests a limitation of the study.

Villalba-morales and Ramírez-Echeverry [31] presented a weight optimization of a three-dimensional steel truss using a meta-heuristic search. The utilization of multi-chromosomal GA with self-adapting variables ensured a good mix of parameters that resulted in the minimization of the weight of three-dimensional steel. The authors reported that 35% minimization was achieved. However, the study may be said to have binary-coded genetic algorithm challenges that include hamming cliffs, uninformed precision, and uneven schema importance are challenges associated with binary-coded genetic algorithms.

Mao et al. [15] presented a single framework containing a genetic algorithm and machine learning for the optimization of traffic flow control. SGA was used for the determination of phase durations, and a machine learning regression technique was used to determine SGA operation parameters. The utilization of the new framework as a controller for traffic control problems with non-recurrent incidences was reported to have yielded promising results. However, the solution suffers from computational complexity and requires that the machine learning regression model be adequately trained.

Al-Madi and Hnaif [32] considered and apply a human community-based genetic algorithm to solve traffic congestion problems. The special-cased genetic algorithm can introduce constraints in the crossover and mutation operations to improve the chances of maintaining diversity in the population. The results of using this algorithm showed that it achieved shorter congestion durations by 13% and minimized congestion by 83% in the considered road traffic congestion scenarios. The challenge associated with the human community-based genetic algorithm is the effectual and adequate modeling of constraints. When constraints are not properly and adequately considered for a given problem, it may not perform as expected.

These studies focused on particular GA operations. Therefore, a generic technique that integrates these proposed techniques is proposed in this research. The new modified genetic algorithm is referred to multiple restart dynamic population algorithm.

Methodology

This section presents the modified genetic algorithm and the traffic control problem used to assess the performance of the genetic algorithm.

Proposed genetic algorithm

The multiple restart dynamic population genetic algorithm (MRDPGA) proposed in this research is presented in Algorithm 1. First, the general and local termination criteria are selected and used to control various iterations of the algorithm. The initial population size, crossover, and mutation percentages are chosen. The population size continually changes during the local or general iterations of the algorithm. To determine when population size is modified during the local termination criteria, a threshold criterion is chosen and used to assume when the algorithm is trapped within local optima. At the restart, the population size modification is performed and a subset of the best individuals from the previous cycle of the MRDPGA are migrated to form part of the newly initialized population of the restart cycle. This is to increase the chance of having the best solution by having the best individuals mixed with an entirely new random population.

Applying the developed algorithm to the particular problem of training the Adaptive Neuro-Fuzzy Inference System, the objective function used for fitness evaluation was Mean Square Error (MSE) and Root Mean Square Error (RMSE). The general and local termination criteria were chosen to be some iterations to be performed. The general termination criteria were either a minimum MSE of 0.0002 or five iterations (C_g = 5); while the local termination criteria were chosen to be 1000 iterations (C_l = 1000). The initial population size, p was fifty (p = 50), and the threshold, T_h of local iterations without improvement in the local solution was ten iterations (T_h = 10).

Since this research focused on proposing a modified genetic algorithm technique, default Roulette-Wheel selection was used for population selection. Forty percent (40%) crossover rate and seventy percent (70%) mutation rate were used for the genetic operations. The trapped threshold value of ten (10) and initial population size of fifty (50) were used. To control population size growth and shrinkage, it was bounded above and below by fifty percent (50%). This was necessary to minimize performance (runtime) bottlenecks of the MRDPGA.

ANFIS controller modeling

To assess the performance of MRDPGA, ANFIS controllers based on the clustering methods were modeled. The grip partition clustering (GPC), subtractive clustering (SC), and fuzzy C-means (FCM) clustering controllers modeled were trained using the default training algorithm, gradient descent, and least squares estimations. They were also trained using a standard genetic algorithm and multiple restart dynamic population genetic algorithm. The objective function for the genetic algorithm was the mean square error (MSE) and root means square error (RMSE). These parameters were used to determine performance variance between the ANFIS default training algorithm, SGA, and MRDPGA. The ANFIS controllers were modeled to have two inputs (waiting time and queue length) and one output (phase duration). The GPC controller was modeled to have a Gaussian membership function and five fuzzy sets for each of the inputs. The SC controller was modeled to have a radius of ten (10), and the FCM was modeled to have ten (10) clusters.

Simulation modeling

Simulation of traffic scenarios was used to assess the performance of the modeled ANFIS controllers trained using the default algorithm, SGA and MRDPGA. Particularly, microscopic simulation was implemented using the simulation of urban mobility (SUMO) open-source platform. To control SUMO objects such as traffic lights and access simulation parameters such as vehicular waiting time and flows queue lengths, a traffic control interface for matrix laboratory (TraCI4Matlab) was used.

The considered road traffic flow network in this research was the road section from Kalapanzi Army barrack and St. Gerrard’s Catholic Hospital in Kakuri to General Hassan Katsina House in Kaduna, Nigeria. To capture a near-reality road network design, OpenStreetMap was used to retrieve the considered road network with little modification to include the new conversion of the Leventis roundabout to an underpass. The considered road network consists of eight roundabouts and one underpass with lanes ranging from one to three.

The traffic flow rates and vehicles per hour (VPH) were based on vehicular sample count performed at the two strategic roundabouts connecting the northern and southern parts of the metropolis. The peak period had an hourly average arrival rate of 350VPH, and the off-peak period had 103VPH. These values were scaled to have different traffic scenarios considered to be representative of various traffic flow possibilities. The vehicles were into categories of light, moderate or heavy vehicles based on the Federal Highway Administration classification. Their speed specification in kilometers per hour (KMPH) was based on these categories, and as provided by Federal Road, Safety Corps recommended speeds for vehicles within cities. Light and moderate vehicles had a speed of 50KMPH, while heavy vehicles had a speed of 45KMPH. The traffic controller programmer was implemented such that a fraction of the assigned green phase duration for each traffic flow is used for the clearing of the roundabouts. This technique enhances the flows and minimizes the probability of deadlocks at the roundabouts. For this research, 40% of the green phase duration was used for the clearing of the roundabouts. That is, the green wave is assigned only to the circulating flows of the roundabouts.

In summarizing the research method, a framework of activities performed is presented in Fig. 1. That is, with traffic training data, the ANFIS models are trained using either a default algorithm (DA), a standard genetic algorithm (SGA), or a multiple restart dynamic population genetic algorithm. The ANFIS controllers resulting from the training of ANFIS models are then used for traffic scheduling. The considered traffic scenarios were based on the extrapolation of traffic count data and estimation performed at selected roundabouts of the considered road network.

Fig. 1

Performed research activities framework

Results and discussion

This section presents the results of training the ANFIS controllers using the default training algorithm, SGA and MRDPGA. The results of the simulation of traffic flow scenarios using the different controllers are also presented.

Controller training performance

In Table 2, the GPC controller training performance results of considered training algorithms are presented. SGA-based GPC-trained (SGAGridPartFIS) had the lowest performance results in terms of MSE and RMSE in the training phase of the controller. The default-based GPC-trained (DefaultGridPartFIS) performed better than the SGA-based GPC-trained but below the MRDPGA-based GPC-trained (MRDPGAGridPartFIS). That is, the GPC controller outperformed the other two controllers to demonstrate its ability at finding the optimal solution to an optimization problem.

Table 2 Grid partition model training performances

In Table 3, the SC controllers' training performance results are presented. The default-based SC-trained controller (DefaultSubCluFIS) underperformed compared to the other two controllers both in the training and testing phases. The MRDPGA-based SC-trained controller (MRDPGASubCluFIS) outperformed the other two controllers having the lowest values of MSE and RMSE both in the training and testing phases. This also shows that the MRDPGA training algorithm was better at searching for the optimal solution for the SC models.

Table 3 Subtractive clustering model training performances

In Table 4, the performance results of using the three considered algorithms for training the FCM model parameters are presented. The results showed that the SGA-based FCM-trained controller (SGASubCluFIS) underperformed compared to the other trained controllers. However, the MRDPGA-based FCM-trained controller (MRDPGASubCluFIS) performed best to outperform both the default-based FCM-trained and the SGA-based FCM-trained controllers. Having lower values of MSE and RMSE both in the training and testing phases demonstrates the ability of the MRDPGA algorithm to optimally tune the parameters of the FCM models.

Table 4 Fuzzy C-means clustering model training performances

The MRDPGA-based controllers showed that irrespective of clustering methods used, the algorithm generally performed better at finding the optimal solution to the optimization of ANFIS-based controllers. Comparing the results of the MRDPGA-based controllers shows that the MRDPGA algorithm performed best at tuning the parameters of FCM model, followed by the GPC model and then the SC model (Figs. 2 and 3). This may be a result of the ANFIS parameters used.

Fig. 2

Summary of controller training performance

Fig. 3

Summary of controller testing performance

Simulation results analysis

The road traffic simulation results were assessed based on the vehicular average waiting for time (AWT) and the number of vehicles that have arrived at their destination [19]—total arrived vehicles (TAV). Five different traffic scenarios are considered in terms of hourly arrival rates.

In Table 5, the road traffic simulation results of the GPC controllers are presented. The results showed that there was no performance variance between the SGA-based and GPC-trained controllers. However, comparing the results of the MRDPGA-based GPC-trained controller to that of the default-based GPC-trained and SGA-based GPC-trained controllers, the results showed that the MRDPGA-based GPC-trained controller performed better in terms of both the AWT and TAV by approximately 33% and 21%, respectively.

Table 5 Grid partitioned model simulation results

In addition, analyzing the impact of MRDPGA algorithm on the performance of the implemented grid partitioned clustered controllers, p = 0.0089 was obtained from the t-Test for paired two sample means in the case of AWT results; and p = 0.0053 in the case of TAV results of GPC-trained and MRDPGA-trained controllers as well as for SGA-trained and MRDPGA-trained controllers. These p values show that at a 95% confidence level, there is a statistically significant difference between sampled means of the results obtained by the different controllers. This demonstrates the ability of MRDPGA at obtaining better global solutions to the problem at hand.

In Table 6, the road traffic simulation results using SC controllers are presented. The results showed that the SGA-based GPC-trained controller slightly underperformed compared to the default-based GPC-trained controller both in terms of the AWT and TAV by approximately 3% and 1%, respectively. However, the MRDPGA-based GPC-trained controller outperformed the default-based GPC-trained controller by approximately 37% in AWT and 20% in TAV. It also outperformed the SGA-based GPC-trained controller by approximately 38% and 21% in AWT and TAV, respectively.

Table 6 Subtractive clustered model simulation results

In addition, analyzing the impact of MRDPGA on the performance of the controllers, p = 0.0263 was obtained AWT and p = 0.0041 for TAV from t-Test analysis performed between Default-trained and MRDGPA-trained controllers. In the case of SGA-trained and MRDPGA-trained controllers, p = 0.0155 for AWT results and p = 0.0050 for TAV results. These also showed that at a 95% confidence level, there is a significant difference between the sampled means of the results obtained using the controllers. A closer look at the p values in these cases showed that the MRDPGA-trained controller had only slight statistical significance variance in the case of AWT results obtained using SGA-trained and MRDPGA-trained controllers. However, it showed higher significance in the case of TAV for the controllers.

In Table 7, the road traffic simulation results of using FCM models as controllers are presented. Comparing the results of the SGA-based FCM-trained controller to that of the default-based FCM-trained controller, the results showed that the default-base FCM-trained controller significantly outperformed the SGA-based FCM-trained controller both in terms of the AWT and TAV by approximately 35% and 2%, respectively. Comparing the results of the MRDPGA-based FCM-trained controller to that of the default-based FCM-trained controller, the results showed that the MRDPGA-based FCM-trained controller outperformed the default-based FCM-trained controller by approximately 33% and 22% in terms of the AWT and TAV, respectively. Also, the results of the MRDPGA-based FCM-trained controller outperformed that of the SGA-based FCM-trained controller by approximately 50% in terms of the AWT and 21% in terms of the TAV.

Table 7 Fuzzy C-means clustered model simulation results

In addition, analyzing the impact of the MRDPGA algorithm on the training of FCM controllers, p = 0.0053 and p = 0.00396 were, respectively, obtained for AWT and TAV in t-Test analysis between Default-trained and MRDPGA-trained controllers. In the case of SGA-trained and MRDPGA-trained controllers, p = 0.0038 and p = 0.0003 were obtained for AWT and TAV, respectively. These show that there are significant differences in the paired two sample means of the results obtained using the Default, SGA, and MRDPGA-trained controllers, which again demonstrate the impact of the MRDPGA algorithm in the training of the FCM-based controller.

These results showed the MRDPGA-based controller always outperformed the other controllers in their respects. This demonstrates the importance of effectively tuning ANFIS parameters, which the MRDPGA algorithm has distinguished itself in this research. That is, it considered the search space and effectively searched for the optimal results.

A comparison of the MRDPGA-based controllers based on the different ANFIS clustering methods showed that the GPC controller outperformed the SC controller by approximately 4% in terms of AWT but underperformed by approximately less than 1% in terms of TAV. FCM controller outperformed the GPC controller by approximately 11% in terms of AWT and by approximately 1% in terms of TAV. Also, the FCM controller outperformed the SC controller by approximately 12% and 1% in terms of AWT and TAV, respectively. That is, the FCM controller trained using the MRDPGA algorithm essentially demonstrated superiority in this research. This follows from the abilities of FCM to flexibly and adaptively assign data points to different clusters of varying shapes and sizes based on the degrees of the membership of each data point. In addition, FCM can determine the number of clusters as well as deal with irregularities in datasets, which may be a result of uncertainties. When trained with MRDPGA, it acquired the added advantage of searching through a possibly large solution space in such a manner as to enhance the chance of obtaining a global solution in terms of cluster shapes, sizes, and numbers.

The summary of the results obtained is presented in Figs. 4 and 5. In Fig. 4, the graphs of the average waiting times recorded when using the different controllers are presented. It shows that the MRDPGA-trained controllers for all ANFIS models recorded the lowest AWT, with the FCM model being the lowest of all controllers. This implies that the MRDPGA-trained controllers performed better in this respect. Also, in Fig. 5, the average total number of vehicles that have arrived at their destinations, called total arrived vehicles (TAV) for all controllers, are presented. The results showed that the MRDPGA-trained controllers recorded the highest TAV numbers. This implies that the MRDPGA-trained controllers performed better.

Fig. 4

Graphical representation of controllers’ average waiting time

Fig. 5

Graphical representation of controllers’ total arrive vehicles

It should be noted that the computational complexity of MRDPGA may be higher compared to SGA, especially in cases where the population size of SGA is equal to the initial population size of MRDPGA. This is a result of the fact that MRDPGA runs multiple iterations based on the global and local termination criteria. However, when the number of iterations and population size differ, it may not be the case.

Conclusion

Optimization problems require techniques or algorithms that effectively mix problem parameters in other to guarantee optimal solutions to the problem. This requires searching through the solution space such that the chance of the algorithm being trapped within a local optimum and failing to obtain the global optima is reduced. This research modified a standard genetic algorithm, which has the above-mentioned challenge. The modified algorithm demonstrated its capability in training ANFIS models based on different clustering methods. It recorded the lowest MSE of 0.299 and RMSE of 0.547 in the training phase; and MSE of 0.272 and RMSE of 0.521 in the testing phase. Using the controllers for traffic flow scheduling, the results showed that the MRDPGA-trained controllers performed better in terms of average waiting time (AWT) minimization and total arrived vehicles (TAV). The best-performing controller achieved 50.40% AWT minimization and 21.44% TAV improvement over other controllers.

The modified algorithm is a framework that can integrate other parameter-based optimization of genetic algorithms. That is, the different adaptive population techniques may be used at the local and global population modification stages of MRDPGA. Also, different selection algorithms may be used instead of the used roulette-wheel selection algorithm. To further assess this modified genetic algorithm, it may be assessed with other known optimization problems used as benchmarks.