A Coordinated Allocation Method for Right-Turn Strategy at Signalized Intersections with Optimal Pedestrian and Vehicle Delays

. With strict enforcement of pedestrian right of way at all intersections, the inappropriate right-turn resource allocation from a spatial and temporal perspective will lead to a reduction in the operational efciency of the intersection. In this paper, three spatiotemporal resource allocation schemes for right-turning vehicles are proposed, considering the vehicle and pedestrian trafc efciency in all directions of the intersection. To minimize vehicle and pedestrian delay at the intersection individually, an optimization model is established with the efective green time of each phase and three schemes as decision variables. A right-turn vehicle and pedestrian confict delay model is developed based on the pedestrian-vehicle interaction behavior as the constraints of the optimization model. Te NSGA-II algorithm is used to solve the model, and the quantitative criteria for the exclusive right-turn lane and phase are obtained by sensitivity analysis. Te results of this paper can be used as a guide for trafc design and for planning and controlling the operation of right-turning vehicles at intersections.


Introduction
Intersections have multiple trafc fows in diferent directions; the essence of signal control is to give each trafc fow the right of way in time to minimize conficts and improve trafc safety.In the Chinese trafc scenario, right-turning vehicles are typically permitted to traverse intersections at any stage, potentially leading to interactions with pedestrian fow [1] within crosswalks.According to the latest road trafc safety law, these rightturning motor vehicles are obligated to reduce their speed when navigating crosswalks, with pedestrians holding an absolute right of way.Such trafc regulations are implemented indiscriminately at all intersections, resulting in a serious reduction in the capacity of the right-turn lane and even the intersection.However, without considering such phenomenon, the current right-turn resource allocation strategy cannot satisfy the trafc demand.
Te spatiotemporal resource allocation schemes for right-turning vehicles at intersections include the setting of exclusive right-turn lanes and phases.Compared with shared lanes, exclusive right-turn lanes will eliminate delays to right-turning trafc at the red time [2], which is suitable for intersections with high volumes of right-turning trafc, but will also reduce the space available for straight-ahead trafc.From a system perspective, the fow of straight trafc will also impact the boundary conditions of the exclusive lane.Since the exclusive right-turn phase eliminates the confict between vehicles and pedestrians, vehicle delays are always reduced at four-phase intersections without the exclusive right-turn phase, when the pedestrian right of way is strictly enforced.However, the situation would be diferent if pedestrian crossing efciency is considered.
Many scholars have studied the allocation of spatial and temporal resources at single-point intersections in terms of efciency [3][4][5][6], safety [7,8], and environmental friendliness [3,[9][10][11].Te benefts (e.g., confict severity, delays, emissions, etc.) before and after the implementation of the scheme are used as indicators to obtain the lane function setting and signal timing schemes.Tere are also some studies focusing on the robustness of the signal control system [12].In terms of the study population, most studies focus only on motor vehicles and consider buses when trafc priority signals are available at intersections [9,13].Some studies indirectly consider the efect of pedestrians on intersection trafc operations by adding constraints [9,14,15] but do not include pedestrian trafc efciency in the objective function.Te vast majority of studies [16][17][18] assumed no delays for right-turning vehicles because they could pass through the intersection at any phase stage.However, this model assumption does not apply to urban intersections with a certain amount of pedestrian trafc.
Compared with right-turn movements, left-turn movements [19] tend to receive more attention in countries that drive on the right, and therefore relatively few research results have been conducted for right-turn space-time resource allocation, focusing more on the analysis of the yielding behavior of right-turning vehicles and pedestrians.Schmidt and Farber [20] point out that drivers mainly use parameters of body language, such as leg and head movements or body rotation to predict the intention of the pedestrian and thus decide whether to yield to the pedestrian.Muley et al. [21] investigated the factors infuencing pedestrians crossing right-turning motor vehicle fows in the exclusive right-turn lane and showed that waiting behavior is independent of pedestrian characteristics and depends only on right-turning trafc characteristics.However, under the current road regulations in China, pedestrians have absolute priority in the right of way and right-turning vehicles should give way unconditionally.
Vehicle trajectory can well describe the vehicle motion information in various scenarios [22].Te two-dimensional movement of right-turning vehicles at intersections is a complex trafc issue.How to appropriately describe the driving trajectory of vehicles at intersections is a topic that many academics are dedicated to researching.Taking into account the interaction between various trafc fows, Ma et al. [23] established a two-dimensional simulation model based on a traditional social force model to characterize the turning behaviors of vehicles at crossings.Zhao et al. [24] developed a two-dimensional vehicle motion model for intersections under the presumption of optimal control, with the goal of minimizing terminal costs and operating costs.Te dynamics of the vehicle motion are formulated in distance, which is diferent from traditional approaches.On this basis, Zhao et al. [25] considered the impact of vehicle interactions and added the safety cost of vehicle operation as the objective function to reconstruct the model.Te new model can not only describe vehicle trajectories but also properly predict the sequence of vehicles passing through, making it possible to simulate intersection trafc fow more accurately.Some scholars have also conducted studies on the signal control of right-turning motor vehicles.Wang et al. [26] studied the right-turn signal timing method under mixed trafc and verifed that the conficts between vehicles and bicycles were signifcantly reduced after setting right-turn signals.Wu et al. [27] proposed a signal control model based on the "Degree of Clustered Confict" and formulated the right-turn lane signal control logic considering the confict delay and potential accident risk.
However, previous studies of the spatiotemporal resource allocation scheme for right-turning vehicles only considered right-turning vehicle-related metrics, ignoring the impact on pedestrians and trafc fow in other directions at the intersection, and did not simultaneously optimize lane function and overall intersection signal timing to reduce pedestrian-vehicle delays.Tis study establishes a comprehensive framework to optimize the allocation of intersection time resources and right-turn space resources under different yielding ratios and volume levels by considering both vehicle and pedestrian efciency in the context of new trafc regulations and proposes conditions for setting exclusive right-turn lanes and phases.Te results can be applied to actual road scenarios and are of great signifcance to the improvement of right-turning trafc at intersections.

Spatiotemporal Resource Allocation Scheme for Right-Turning Vehicles
In this paper, the optimization algorithm aims to minimize the vehicle and pedestrian delays, respectively, which are utilized as the measurement of trafc efciency.However, due to the incompatibility of these two delays, instead of a unique solution comprehensively, the current solutions typically make a trade-of of each delay correspondingly to achieve the relative optimality.To ensure sufcient green time for pedestrian phases, motor vehicle travel times are bound to be shortened, leading to increased delays.To properly address the above demand, a new model with a multiobjective framework is developed in this paper to select and optimize the allocation of spatiotemporal resources of right-turning vehicles.Te phase diagrams of the three assignment schemes of the four-phase intersection are presented in Table 1.Among them, scheme 1 is the shared-lane case, and scheme 2 represents the case with an exclusive right-turn lane.In Schemes 1 and 2, the vehicles in each direction are permitted to right-turn on red.Considering that pedestrians at a fourphase intersection are usually released to the straight-ahead trafc in the same direction simultaneously, it is proposed to integrate the exclusive right-turn phase into the left-turnprotected phase to maximize the trafc fow per unit time at the intersection while avoiding conficts between pedestrians and right-turning vehicles.

A Multiobjective Optimization Model of Spatiotemporal Resource Allocation
Scheme for Right-Turning Vehicles 3.1.Model Assumptions.Te assumptions in this paper are as follows: (1) Each arm of the intersection is a two-way six-lane road with an exclusive left-turn lane.
2 Journal of Advanced Transportation (2) Te intersection has unlimited vehicle queuing space, i.e., the short-lane scenario is not considered.(3) As mutually independent events, pedestrians in both directions are assumed to cross without interfering with each other.(4) Pedestrians can completely dissipate during the green time.(5) Vehicles are assumed to merge into diferent lanes without merging delays.(6) Te trafc volumes arriving at each approach at the intersection are similar, so the right-turn resource allocation scheme used is consistent.

Notations. Te layout of the intersections is shown in
Figure 1.To facilitate subsequent illustration, the key symbols used are summarized in Table 2.

Objective Function.
Te optimization objective is to maximize the trafc efciency at the intersection for both motorists and pedestrians.For the given intersection, the frst objective of the model mainly refects the viewpoint of the vehicle driver, i.e., minimizing the average delay of all vehicles at the intersection under each scheme.
Te second objective function refects the pedestrian's perspective and aims to minimize the pedestrian crossing delay under each scheme. (2)

Decision Variables.
Te decision variables of the model include the right-turn lane type (δ 1 ), the right-turn phase type (δ 2 ), and the efective green time (g u ) of each phase.

Constraints.
Te following are common inequality constraints that limit the decision variables.Equations ( 3) and ( 4) restrict the cycle length and the green time of each phase to a certain range.Equation (5) guarantees that there is and only one scheme is selected.
Te model also includes the following equation constraints.
(1) Te length of cycle time: Te cycle length is the sum of the green, yellow, and all-red time for each phase, i.e., (2) Pedestrian clearance time: Te pedestrian clearance time is defned to ensure pedestrians have enough time to cross safely.It is determined based on the length of the crosswalk, pedestrian walking speed, and additional safety time.
(3) Te green time: Te green light durations of parallel crosswalks should be equal.Te sum of the pedestrian green light and fashing light duration should be no greater than the vehicle display green light duration, which is considered here as a binding constraint, taking the case where the equal sign holds.
In addition, the delay model expressions (equations ( 11)-( 15)) developed collectively constitute the constraints of this model.

Vehicle and Pedestrian Delay Models
In this optimization problem, the delays mainly originate from (1) Signal control: Trafc objects at the intersection can only pass during the green time (except for rightturning vehicles under Scheme 1 or Scheme 2), which can cause signal delay.In addition, the Te binary variable represents whether scheme 2 has been adopted, δ 1 � 1, Scheme 2 has been adopted.0, Scheme 2 hasn ′ t been adopted. - Te binary variable represents whether scheme 3 has been adopted, 1, Scheme 3 has been adopted.0, Scheme 3 hasn ′ t been adopted. - Te efective green time of pedestrian signals at the approach i s g u Te green time corresponding to the phase u of the phase sequence is shown in Where vehicle delays are calculated by lane group, the blocking delays of straight-ahead vehicles to right-turning vehicles will be included in this signal delay, as in the following equation: Te average signal delay for pedestrians [28] is calculated according to the following equation: 4.2.Confict Delay Model.Te calculation of confict delay is based on the gap theory.For right-turning vehicles and pedestrians at a four-phase signal intersection, conficts are generated only when the pedestrian signal is green.Two scenarios (concentrated pedestrian arrival and random pedestrian arrival) can be classifed, as shown in Figure 2.
During the pedestrian red time, right-turning vehicles are allowed to cross the intersection without restriction, while pedestrians have to wait.Terefore, at the beginning of the pedestrian green time, there is a concentration of pedestrians arriving at the confict zone.At this point, the right-turning vehicles need to wait for these pedestrians to pass through the confict zone until they can fnd a chance to cross the intersection.When the concentration of pedestrians arrives, it is assumed that they are arranged in a matrix through the confict zone, so the right-turn vehicle delay can be calculated by the following equation: where L ped is the length of the pedestrian matrix, T Pred is the pedestrian red time, and k P is the average area occupied by pedestrians.
When pedestrians arrive randomly, considering the arrival intensity of pedestrians in both directions, the delay [29] of right-turning vehicles can be calculated by the following equations: where T Pgreen is the pedestrian green time, λ mn i is the pedestrian arrival intensity from corner m to corner n at approach i, and G safe is the safe traversable clearance, which can be calculated by the following equation: where t s is the time taken by a right-turning vehicle to pass the confict zone.Te confict delay for pedestrians [30] in this scenario is where μ i is the arrival intensity of right-turning vehicles, including right-turning trafc in both directions (arriving and exiting approach i).In scheme 1 (i.e., the shared lane case), the right-turning vehicles exiting approach i are blocked by the straight-ahead trafc and cannot pass the intersection during the red time, so there is right-turning trafc in only one direction.In summary, all delays have been modeled.Since the vehicle and pedestrian delays vary under diferent scenarios and in diferent directions, it is also necessary to obtain the average delay by weighting the respective fows.

Solution Algorithm and Sensitivity Analysis
For the multiobjective optimization problem (MOP) in this study, there is no absolute optimal solution but only a set of "noninferior solutions."Te "noninferior solution" in this study refers to the fact that among the combinations of spatiotemporal resource allocation scheme and signal duration allocation, the selected combination reduces at least one of the pedestrian delay or motor vehicle delay when it is not possible to reduce both.Te values of such a set of decision variables are taken as a noninferior solution, i.e., a Pareto optimal solution.In this paper, all noninferior solutions are generated by the optimization algorithm and are selected based on the research problem.

Algorithm Comparison and Modeling
Results.From the model structure, the problem in this study is a nonlinearly constrained optimization problem.Due to the complexity and multiplicity of objective functions and constraints, traditional optimization algorithms have difculty in fnding the global optimal solution of this problem smoothly and quickly.
In this study, a heuristic algorithm is employed to tackle the MOP.Various multiobjective evaluation algorithms are available, broadly categorized into three categories: domination-based framework, indicator-based framework, and decomposition-based framework [31].Given the unknown Pareto frontier of the proposed MOP, it is amenable to resolution by either the frst or third type of evolutionary algorithm.
One of the most renowned multiobjective optimization algorithms is the nondominated sorting genetic algorithm II (NSGA-II), introduced by Deb's team in 2001 [32].Tis algorithm optimizes MOPs by simultaneously optimizing all objectives.
Drawing inspiration from decomposition-based concepts, the reference vector guided evolutionary algorithm (RVEA) was proposed [33] to achieve better approximation of frontier surfaces in high-dimensional spaces.RVEA leverages a scalarization approach, known as angle-penalized distance, to balance solution convergence and diversity within high-dimensional objective spaces.
To solve the model in this paper, the Python solver Pymoo [34] is employed.Te termination condition of the algorithm is set as follows: the average variance in the target space is below 0.25% and the diference with the constraints is within 10 −6 ; meanwhile, the maximum number of iterations is set to 1000.
Regarding the primary algorithm parameters, they are confgured as follows: (1) Population Size (PS): To strike a balance between global search capability and computational complexity, two population sizes of 200 and 400 were selected for solving the model, considering its relative complexity.A larger population size enhances global search potential, reducing the risk of local optima.
(2) Crossover Probability (CP): Te crossover operation plays a pivotal role in generating new individuals within evolutionary algorithms.Crossover probability signifcantly infuences search efciency and convergence speed.A value that is too large may introduce excessive randomness into the search process, afecting algorithm performance, while a value that is too small can limit the search range and impede convergence.Hence, two scenarios with crossover probabilities of 0.6 and 0.9 are examined.Te optimal solution of the model is determined according to the multicriteria decision making criterion , i.e., by introducing an augmented scalarization function that assigns certain weights to the two objectives.In this paper, both vehicles and pedestrians are considered two types of trafc bodies, and the weights of the respective objectives are determined by the total trafc fow of each trafc body.
Meanwhile, based on the actual road trafc situation in China, the internal parameters of the optimization model are set as shown in Table 3. Te parameter values linked to intersection geometric design are acquired from actual survey data, while the parameter values related to signal timing are taken from a reference [16].
Assume that all vehicles and pedestrians have no violations; that is, the probability of yielding to right-turning vehicles is taken as 1.Te optimization model is solved for a one-way pedestrian arrival rate of 500 pedestrian/hour (ped/h) with 300 vehicles/hour (veh/h) of left-turning trafc, 300 veh/h of straight-through trafc, and 50 veh/h of rightturning trafc, respectively.
To comprehensively evaluate the algorithms' robustness and efectiveness across diverse parameter confgurations, we conducted 500 runs using diferent random seeds to solve the model.Tese runs incorporated diferent parameter and algorithm settings, and the results of the 500 iterations, encompassing the average execution time (AET), the average delay of vehicles (ADV), the average delay of pedestrians 6 Journal of Advanced Transportation (ADP), the variance of vehicle delay (VDV), and the variance of pedestrian delay (VDP) [35], are presented in Table 4. Table 4 reveals that the model's overall performance is optimized when employing the frst set of parameters with the NSGA-II.Under these optimal parameter settings, we successfully obtained a set of dominant and optimal solutions using NSGA-II, as shown in Figure 3.
According to the optimal solution, scheme 1 is selected with an efective green time of 14.33 s, 12.93 s, 14.33 s, and 13.03 s for each phase, corresponding to the vehicle delay of 72.11 s and the pedestrian delay of 37.31 s for the objective function.
Te focus of this study is to obtain the setting conditions for the exclusive right-turn lane and phase for a single approach, so as to simplify the structure of the optimization model and improve the computational accuracy.Te paper assumes a balanced arrival fow for each approach at the intersection.In addition, the multiobjective optimization framework proposed in this study is well-scalable and can be adapted to actual road scenarios.

Sensitivity Analysis.
To obtain the boundary conditions for the exclusive right-turn lane and phase settings, the trafc demand in each direction on all approaches is assumed to be equal, and the optimal solution is obtained by the previously proposed multiobjective optimization framework.In this paper, we focus on four parameters, namely, vehicle yield probability, straight-ahead vehicle volume, right-turn motor vehicle volume, and pedestrian volume.Assuming that the arrival rate of left-turning vehicles is always 300 veh/h, the following results are obtained.

Vehicle Delay.
In Figure 4, the x, y, and z coordinates denote the arrival rate of right-turning vehicles, the arrival rate of straight-ahead vehicles, and the vehicle delay, respectively.Under the diferent combinations of the yielding probability and one-way pedestrian arrival intensity, the vehicle delays of the optimal spatiotemporal allocation schemes show the same distribution pattern.
Overall, the average motor vehicle delay increases with the increasing arrival rate of right-turning and straightahead vehicles and pedestrians, as well as the probability of yielding.In the case of a low arrival rate of straight-ahead trafc, the delay decreases as the arrival rate of rightturning vehicles increases.Tis is because delays in the lowvolume scenario are primarily generated by signal control while right-turning vehicles are not.Te more rightturning vehicles arrive, the lower the average delay for all vehicles.

Pedestrian Delay.
Assume that the arrival rates of straight-ahead trafc are 100 vehicles/h (veh/h) and 300 veh/ h, and the one-way arrival rates of pedestrians are 50 pedestrians/h (ped/h), 250 ped/h, and 450 ped/h.Figure 5 shows the variation of the average pedestrian delay with respect to the right-turning motor vehicle fow for the optimal scheme with diferent vehicle yield probabilities.
In this paper, multiobjective optimization is performed with the total number of pedestrians and vehicles as their respective weights, so there may be situations when delays at low pedestrian fows will be higher than delays at high pedestrian fows.
Pedestrian delay is relatively low and changes more gently as the probability of vehicles yielding to pedestrians increases.At lower straight-ahead trafc volumes, the average pedestrian delay increases with higher right-turning arrival rates, which is generally caused by the increased confict delays between pedestrians and right-turning vehicles.At higher straight-ahead volumes, the change in average pedestrian delay decreases and then increases.Tis is because, with the increase in right-turn trafc fow, the allocation of the intersection spatiotemporal resource scheme is adjusted, resulting in a decrease in signal delay greater than the increase in conficting delay, which ultimately leads to a decreasing trend in the average pedestrian delay.Especially, when P vtp � 1, the delay of pedestrians is entirely due to signal control.Shared lanes are typically employed when there is low right-turn trafc volume.In this lane situation, the signal timing scheme at the intersection is afected by the volume of right-turning vehicles, resulting in changes in pedestrian delay.When the fow of right-turn vehicles reaches the critical threshold for setting an exclusive right-turn lane, the intersection signal timing scheme is no longer impacted.At this time, the delay of pedestrians essentially remains stable, as shown in the last subgraph of Figure 5.

Spatiotemporal Resource Allocation Scheme.
Figure 6 (horizontal coordinates are the arrival rates of straight-ahead trafc and vertical coordinates are the arrival rates of right-turning vehicles) gives the optimal schemes for diferent vehicles yielding probabilities at one-way pedestrian arrival intensities of 50 ped/h, 250 ped/h, and 450 ped/h.In Figure 6, the horizontal coordinates denote the arrival rate of straight-ahead trafc, and the vertical coordinates denote the rates Journal of Advanced Transportation of right-turning vehicles.From Figure 6, the conditions of exclusive right-turn lane and phase can be summarized as follows: (1) Exclusive right-turn lane: As shown in Figure 6, the setting of the exclusive right-turn lane is mainly infuenced by the fow of straight-ahead trafc and Teoretically, the intersections with strict enforcement of pedestrian right of way do not require an exclusive right-turn phase, which is consistent with the results of the model.
In the case where vehicles and pedestrians have equal right of way (P vtp < 1), the optimal solution is derived considering both pedestrian and vehicle delays, resulting in a decision result that does not present a clear quantitative pattern.Terefore, it is difcult to obtain the setting conditions for the exclusive rightturn phase directly from the model results.
Comparing the distributions under diferent pedestrian volume levels, it can be considered appropriate to set an exclusive right-turn phase when the vehicle yielding rate is lower than 0.7 and the one-way pedestrian arrival rate reaches 450 ped/h based on the setting of an exclusive rightturn lane.

Conclusion
In this paper, three right-turn space-time resource allocation schemes are proposed.A multiobjective optimization framework for the four-phase intersection considering both vehicular and pedestrian trafc efciency is developed, and the delay model considering pedestrian-vehicle interaction behavior and pedestrian crossing form is established.Based on this framework, the optimal right-turn space-time resource allocation scheme and intersection signal timing scheme can be derived for diferent pedestrian and vehicular trafc levels and vehicle yielding probabilities.From the results of the optimization model, it is appropriate to set an exclusive right-turn lane when the arrival rate of right-turn vehicles reaches 310 veh/h and is not higher than the arrival rate of straight-ahead vehicles or the ratio of right-turn vehicles to the arrival fow of straight-ahead vehicles reaches 0.7.Te setting conditions for the exclusive right-turn phase are that the yielding rate of vehicles is less than 0.7 and the arrival rate of one-way pedestrians reaches 450 ped/h based on the exclusive right-turn lane setting conditions.
Te optimization model developed in this paper has good scalability and can be adjusted to better predict vehicle and pedestrian delays according to the actual road conditions.In addition, besides trafc efciency, nonmotorized trafc is also an important factor afecting the right-turn space and time resource allocation at intersections.Tis needs to be studied in depth in the future to further clarify the conditions for setting exclusive right-turn lanes and phases.

( 3 )
Mutation Probability (MP): Te mutation operation is an auxiliary mechanism for generating new individuals, dictating the local search capacity of the evolutionary algorithm.Generally falling within the range of 0.01 to 0.1, this study prioritizes strong search capability and assesses mutation probabilities of 0.05 and 0.1.

Table 1 :
Phase diagram under each scheme.
Confict points between pedestrians and right-turning vehicles.Right-turning vehicles are blocked by queues of straight-through vehicles on red.Pedestrian crossing fow lines.

Table 2 :
Notations and parameters.Index of intersection arms (approaches) or corners, as shown in Figure1, i � 0, 1, 2, 3 -L i Te length of the crosswalk at arm i m δ 1

Table 1 C
Te

Table 3 :
Parameters and their values.

Table 4 :
Comparison of algorithms for diferent parameter values.